Logger Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⢣⣄⡀⠐⣄⡀⣠⠀⠀⢾⣴⡿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢳⣶⡇⡟⠿⣳⣸⡿⠤⣀⠀⢸⣿⣯⡏⣿⣾⡆
⠀⠀⠀⠀⠀⠀⢀⣠⡤⢒⣿⣏⣻⣿⠿⠟⢟⠉⠛⠛⢷⢈⣤⣤⣼⣿⣅⣿⣿⣿⣨⣴⣦⠄
⠈⢁⣠⣬⣶⣾⣿⣿⡷⣾⣿⡟⣋⣁⣜⢛⣼⢯⡞⠛⣀⣜⣻⣿⣿⡉⢉⣉⣠⣤⣤⣤⣤⢤⠠⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠂⠉⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⢭⣤⠴⠦⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣚⡭⠖⠚⠛⠓⠶⣭⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡵⠞⠉⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣵⠏⠀⠀⠀⣀⣀⠤⠤⠄⠀⠀⠤⠤⢤⣀⠀⠀⠙⢷⡙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣼⠃⢀⠀⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠻⣜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢱⡧⠊⠁⠀⠀⣀⡤⠤⠒⠒⠒⠒⠒⠓⠒⠒⠤⢤⣀⠀⠀⠑⢿⡎⣇⠀⠀⣀⣠⡤⠤⣶⣒⣒⣒⣒⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣱⠟⠀⢀⡴⠚⠉⣀⣤⡆⠀⠀⠀⠀⠀⠻⣿⣷⣶⣶⡄⠉⣲⣤⡀⠻⣮⣓⣯⠵⠖⠛⠉⠉⠁⠀⠀⢸⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣣⡟⣠⣴⡇⢀⡴⠟⠛⠋⠈⠆⠀⠀⠀⠀⠀⠴⠂⠶⡀⠀⠀⢻⣿⣿⣦⣼⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣾⢩⡌⢻⣿⠇⠀⢠⠋⣤⣶⠀⠀⡀⠀⠀⢀⠀⢾⡿⠀⡠⠀⠀⠈⢻⡿⡉⠏⠂⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡏⠃⠰⣼⡟⠀⠀⠈⢦⣈⡍⠀⢀⠃⠀⠀⠀⢧⠀⠴⠞⠁⠀⠀⠀⠀⣷⠁⠀⣰⢏⡹⣆⠀⠀⠀⠀⠀⠀⢿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡻⣦⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣼⣶⣟⡁⠾⠿⣹⣆⡀⠀⠀⠀⠀⢸⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣂⣽⣿⣄⠀⠀⠀⠀⠀⣀⣤⣧⠀⠀⠀⠀⠀⣼⣦⣀⠀⠀⠀⣀⣴⣿⣟⣽⠿⠟⠛⠉⣀⡔⠀⠀⠀⠀⢸⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢉⣿⣯⣿⣿⣷⣶⣾⣿⣟⣛⣿⣿⣶⢶⣶⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠴⣞⣫⣭⡻⠶⠶⣖⣚⣫⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⢿⣿⡏⣿⣿⣿⣿⡟⠉⠉⠉⠙⠡⠴⠋⠉⠉⠉⠈⢻⣿⣿⡟⣿⣿⣿⣿⢣⠞⠉⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡼⢿⣷⣿⡟⣿⣿⣇⠀⠀⠀⠀⣷⣶⣏⡀⠀⢀⣠⣾⣿⣿⣷⣿⣿⠿⠉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⠦⣄⢀⣠⡷⢆⣻⣿⣿⣿⣿⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣏⢶⣚⠁
⠀⠀⠀⠀⠀⣠⠴⣚⣵⠾⢏⠻⣮⡽⠖⠛⠉⢁⣈⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡿⠟⠻⠁⠉⠳⢬⡳⢄⡀
⠀⠀⣀⡤⢞⣱⠟⠫⠄⠰⡀⠱⡈⢳⣴⣶⣿⠿⠋⠁⠀⠆⠙⠛⠿⠿⣿⣿⣿⣿⣿⠿⡿⠋⠉⠀⡐⠀⡇⠀⠀⠀⠙⠶⣝⡢⣄
⠀⢰⡯⡾⠋⠁⡀⠤⠀⣠⠧⡴⠳⡛⠉⠁⠀⠀⠀⠀⠸⠀⠧⡀⠀⠀⠠⠍⠻⠭⠥⠂⠀⢀⢄⠜⠁⠀⢀⠀⠀⠀⠀⠀⠈⠙⠾⣝⡢⣄
⠀⠈⠳⣝⡛⣿⠁⠤⠄⡳⠖⠋⠈⠀⠐⢄⠀⠀⠀⢠⠃⠐⠀⠰⢆⠈⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡻⡄
⠀⠀⠀⠀⢹⢹⡀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠂⠀⣀⡜⠀⡇⠀⠃⠀⢢⠀⠀⠀⣀⠄⠂⠁⠀⠀⠀⠀⠁⢰⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⢸⡧⡇
⠀⠀⠀⠀⠈⢎⢳⣄⠀⠀⠀⠀⢀⡇⠀⣀⡼⣋⢹⡇⢰⠀⠰⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠈⡇⠀⢾⠀⣿⢼⣭⢹⣁⠒⠒⠀⠀⠈⣩⢗⡿
⠀⠀⠀⠀⠀⠈⠳⢭⣟⣒⣒⣛⣛⣶⣚⡭⠵⡏⣿⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡀⣿⢸⣸⢮⡏⠀⠈⠉⢉⡿⣩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠁⠀⢸⢳⡇⠀⠇⠀⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡏⠀⠀⡇⢸⡞⡟⣼⠁⠀⠀⢀⡞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⣾⠓⠠⢀⢠⡟⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣀⡀⠦⢠⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⡀⢠⡗⠈⠉⠁⠀⠀⠀⣨⡟⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡯⣹⢧⡾⠶⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⠤⠶⣦⣼⣇⠀⠀⠀⠀⠀⢠⡏⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⣛⣣⠏⠉⠉⠙⠛⠓⠒⠒⠒⠒⠒⠛⠙⣇⢯⣽⢟⣳⠶⣤⣤⣀⣹⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠋⠉⠙⠛⠲⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠀⢀⣀⣀⠀⣀⣠⠄⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣾⣶⡷⢄⣼⠉⣁⣀⡼⠷⣄⠸⢁⡸⣁⣙⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣷⢸⣭⣭⣿⡄⣬⡁⣸⣅⡼⠧⠴⠿⠤⠾⠤⠾⠤⠧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⠘⣯⣭⣿⡤⢼⣅⣼⡏⠰⠆⠠⠀⣆⠲⢶⣦⡐⢆⢦⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⣿⣿⣿⣿⣿⣿⣿⡷⢶⣶⣿⡄⣿⣭⣽⡇⢠⣧⣠⠀⠺⠶⠿⠀⣿⣷⣦⡹⠿⢦⣀⣬⡄⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⢸⣶⣾⣦⢤⣧⢼⠀⠛⠂⢀⠀⠀⠒⠦⠀⠀⠀⠀⠀⠈⠉⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⢽⣿⣿⣿⣿⣿⣿⣿⣭⣿⣟⣳⣾⠷⠶⣧⣤⣦⣴⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⠂⠁⠈⠁⠀⢱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠁⠀⠈⠛⠛⠛⣿⢏⣿⣿⠛⣿⠓⠺⣿⣿⡀⠀⣠⠖⠃⠀⠀⠀⢠⣤⠄⠶⠀⣿⢸⠸⢸⠿⠁⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣞⣿⣇⠿⣿⡀⠀⣿⣿⢙⣟⡏⢠⣶⣶⣤⠀⠁⢰⣦⡀⠐⠊⠀⠉⠙⠓⠀⠯⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠶⠿⠿⠿⠿⠻⢾⣿⡇⣿⡏⠉⣿⡇⢀⣈⣉⣀⣐⣂⠭⠭⠥⠥⠐⠒⣚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡶⠇⢾⣿⣯⣭⣦⢤⣤⣤⠶⣶⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣿⣿⡿⠁⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉








⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⠠⡤⢤⡤⢤⠤⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠶⠶⠶⢗⠒⠒⠒⠒⠶⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠈⠱⣀⠔⠀⠁⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠄⠀⡇⠀⠄⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣩⠏⠀⠣⣀⠀⠀⠀⢀⡼⠳⢄⡀⢀⣠⠞⠀⠀⠙⣏⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⡄⢀⡖⠉⠉⢏⡁⢈⡖⠁⠀⠀⠀⠈⠙⣒⠾⠛⠒⠒⠒⠨⣍⠀⠀⠀⠀⠀⠈⠳⣁⠀⡹⠉⠉⢑⡄⢢⠄⢠⡀
⠀⠀⠀⠀⡔⠊⠁⠈⠳⠀⠸⡆⠉⡸⠣⠀⡼⠋⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠙⢧⠀⠤⠧⠉⢐⠏⠀⡜⠁⠀⠉⠢
⠀⠀⠀⠀⡀⠀⠀⠀⢀⠟⠐⠁⠁⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⠁⠚⢀⡔⠳⣄⢀⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⡎⠀⠀⠀⢤⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⡤⠀⠀⠀⠱⡀⠀⠀⠀⠀⣸⠁⠀⠿⠸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⣦⣄⡀⠀⠀⠀⠉⡋⢑⡊⢙⠉⠀⠀⠀⠀⣀⣴⠋⠀⠀⠀⠀⠀⢣⠀⠀⢰⡋⠉⡈⡟⠀⡰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣤⣤⣥⣤⣤⣤⣧⣤⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢺⡄⠀⠀⠁⡀⠀⠀⠉⢡⠔
⠀⠀⠀⠀⠀⠀⠀⣠⠤⢲⠤⢤⣸⠈⡆⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢠⠃⣵⣤⠒⠚⢧⡀⡴⠊⠁
⠀⠀⠀⠀⠀⣠⠞⢹⡄⠀⡇⠀⠸⣷⡉⢦⡀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠿⢿⡿⠿⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⣴⢋⣼⡏⠀⠳⣄⠀⡸⠃
⢀⡤⠖⠒⠘⠁⠀⡤⠓⠐⢳⠆⠀⠳⣝⠦⣙⠶⢄⣀⠀⠀⠉⠛⠿⠿⠀⠀⠀⠀⠀⠀⠿⠿⠟⠋⠀⠀⣀⣠⠶⣋⡴⣋⠞⠁⠐⡖⠛⠋
⢏⣀⡤⢶⢆⠀⢀⡽⠀⠀⡜⠁⠛⠐⡜⠷⣬⡛⠶⢭⣩⣛⠒⠖⠶⣶⡶⠤⠤⠤⠤⢶⣶⡶⠲⡒⣛⣉⡭⠶⢚⣩⠶⠿⠀⠛⠂⢻⡀
⠀⠙⠖⠁⢲⡀⢹⠄⠀⢠⡅⣄⢀⣀⠀⠀⠄⠉⠻⠶⣤⣉⣉⣉⠟⠁⠙⢦⠀⠀⡰⠋⠈⠻⣌⣉⣉⣤⠴⠚⠉⠀⠀⠀⠀⢀⣀⠨⣇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⡜⠀⠀⠀⠉⠈⠙⠁⠐⠖⠒⠖⠀⠼⠁⢴⠤⠤⠤⠱⡮⠤⡤⠤⠤⠈⠯⠀⠰⠒⠒⠂⠀⠋⠉⠉⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠠⠤⠄⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣀⣀⠠⡤⠤⠖⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠁⠁⠛⠀⠒⠒⠒⠒⠒⡗⠒⠒⠒⠒⠐⠘⠃⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⡉⠐⠚⠂⠤⠦⠄⣤⣀⣀⣀⢀⡀⠀⡀⠀⢀⠲⢀⡇⠀⠀⠀⢀⠀⢀⡀⣀⣄⡀⢠⠀⠤⠤⠀⠒⠂⠈⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠉⢁⠈⠐⠒⠃⠀⠤⠤⠴⠀⠠⠄⣀⣍⢉⡙⠁⠈⠋⠉⠉⡁⣩⣁⡨⠄⠀⠦⠤⠴⠀⠀⠀⠀⠈⢈⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢘⠀⠀⠀⠘⠂⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠒⠀⡀⠈⠀⡇⠀⠀⠀⠀⠀⠦⠀⠀⠀⠓⠀⠀⠀⠉⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠉⠁⠀⠀⠐⠒⠂⠀⠠⠄⠤⠤⡄⠒⠒⠀⠐⢀⡦⠤⠠⠤⠀⠐⠒⠒⠀⠀⠀⠉⠁⠀⠁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡖⠤⠤⣀⣀⣀⡀⠀⠀⢀⠀⠀⡀⠀⠀⢀⣀⣀⣀⠠⠤⠰⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⢼⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣿⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠁⠀⠀⢀⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⣀⠀⠀⠀⠙⢦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣶⣶⣶⣶⣭⣵⣦⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣶⣾⣭⣶⣶⣶⣶⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣱⠀⠀⢸⡇⠀⠀⠀⢸⣿⡴⠀⠀⠀⣿⣄⠀⢸⠁⠀⠀⠀⠀⠀⠘⠓⠀⠀⢸⣿⡇⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⣷⣆⡆⠀⠀⣿⡆
⠀⠀⡇⠀⠀⢸⣷⠀⠀⠀⠘⣿⢷⢀⠀⠀⣿⡏⠀⢸⡇⠀⠀⠀⠀⠀⢹⣯⠀⠀⢸⣿⡇⠀⠀⢸⡇⠠⣿⠀⠀⠀⠀⢸⣿⡟⡇⠀⠀⣯⡀
⠀⢰⣿⠀⠀⢸⣏⠀⠀⠀⢸⣿⣽⠎⠀⠀⣿⣧⠀⢠⡅⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣷⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⣟⢸⡇⠀⠀⣷⡀
⠀⢸⣯⠀⠀⣾⡟⠀⠀⠀⢸⣿⢿⠀⠀⠀⣿⠞⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇⠀⠀⢸⡇⠘⡿⠀⠀⠀⠀⢸⣿⢻⡇⠀⢠⣿⡇
⠀⢸⡿⠀⠀⢻⣷⠀⠀⠀⢸⣿⢾⠀⠀⠀⣿⣦⠀⢸⠇⠀⠀⠀⠀⠀⢸⣷⠀⠀⢸⣯⡇⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⣿⣷⡇⠀⢘⣿⡂
⠀⠈⠧⠀⠀⠘⠟⠀⠀⠀⠘⠿⢘⠀⠀⠀⣿⡯⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣶⡇⠀⠀⢸⠗⠀⠀⠀⠀⠀⠀⣸⣷⣻⠇⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢺⠆⠀⢀⣿⣾⠄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣷⠆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣈⠀⠀⠈⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⣿⣟⡀⠀⠀⠀⠀⠀⠀⠠⣄⠤⠀⢸⡟⣿⠃
⣭⣽⣟⣿⣭⣽⡋⠥⢤⠤⢊⡙⢉⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣎⡁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⣸⡅
⠉⣙⣛⣛⣛⡻⠿⠿⡿⠿⠿⠿⠿⠧⢀⡀⠀⠀⣀⣀⢠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠿⠬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇
⠿⢿⣶⣷⣶⣶⣦⣴⣭⣷⣦⣠⡄⠀⠀⣷⠀⠀⠉⢹⣇⠀⡀⠰⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⡇
⣬⣤⣶⣤⣩⡉⠹⠿⠻⠛⠛⠛⠓⠀⠠⠀⠀⣿⣤⠘⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⣾⣿⣤⡇⠀⠀⠠⣤⣤⠆
⣤⣀⣉⣉⣉⡛⡛⢛⣓⡻⠶⣶⡇⠀⠀⣦⣄⠉⠙⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠰⠶⣿⣿⣿⣿⠛⠓⠂⢀⣤⣠
⠨⠙⡉⠹⣛⠻⠿⠿⠿⠀⠀⠻⠷⠄⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠠⣴⣿⣿⡿⠿⠶⠄⠂
⣀⣀⠉⠀⠈⠁⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠿⢿⣿⣿⣷⣶⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⢀
⠀⠈⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡟⣟⣉⢙⣓⡀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠴⠉⠁⠭⠹⡟⠉⠀⡄⠉⣏⡁⢐⢖⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⡬⠶⠲⠒⠶⢽⠏⠃⢠⠀⠀⠀⠀⠀⠠⡆⠷⠀⠠⢠⣿⡶⠓⠀⠂⣽⣶⣦⣼⣿⠇
⠀⢠⣄⠀⠀⠀⠀⠀⠈⢿⢫⠟⣻⡟⠂⢚⡿⢻⠁⠓⣀⠀⠀⠀⢀⡔⠚⠟⠒⣶⡔⢚⠉⠓⢶⡶⠋⠉⠛⢫⠛⠋⢦⡀
⠀⢠⡟⠇⠀⡀⠀⠀⠀⠸⣝⡀⡀⠀⢄⣤⠄⡼⠀⠃⠈⠀⠀⠀⢸⢢⣶⠉⠋⣸⡀⣤⡌⠃⢺⡧⠀⠀⠀⢀⣄⣀⣰⡷
⠀⠶⡄⢠⣾⠁⠀⠀⠀⠀⠘⢧⡁⠐⠲⠶⣄⠄⠌⢰⠟⠀⠀⣠⠜⠶⣭⣁⡴⠿⠦⢥⣤⡴⠿⠳⢄⣀⠴⠿⠻⢿⡛⠓⡒⣆
⠀⢳⡄⢸⡟⢢⡀⠀⠀⣠⢿⡄⠛⣶⣤⣤⣿⣦⣆⡈⠀⠀⢠⠀⠀⠃⠀⣿⠠⠀⠀⠀⡿⢀⠀⠀⠈⣧⠀⠀⠀⠀⡿⠖⣻⣸
⠀⠀⠁⠽⠿⣦⡝⠶⣛⠽⣯⡿⣴⢀⣠⣺⣿⣿⣿⣇⠀⠀⠈⢷⣈⣀⣠⣿⣄⣀⣀⣠⣿⣨⣀⣀⣴⣧⣤⣀⣠⣴⣿⣿⣿⠟
⠀⠀⠀⠀⢰⣿⠿⣷⣮⡿⣛⠿⣦⣿⣿⣷⣏⣽⣿⡠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣤⢶
⠀⠀⠀⠀⣿⠋⣤⢙⡏⢀⠑⠤⣙⣿⣿⡇⡇⣿⣼⣧⠛⣧⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⢸⡇
⠀⠀⠀⠀⣧⡜⢠⣿⡇⡿⠄⣾⣿⣯⣹⢏⡏⣹⣿⡹⣆⣹⠀⠀⠀⠀⠀⢀⣞⣁⣠⣀⣭⣹⣇⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠿⢷⣾⣸⡇⣛⠠⡗⡇⡿⣿⣷⣍⠻⣿⢷⠙⡖⠀⢀⣀⣤⣤⣿⣿⣿⣿⣽⣉⣹⣏⣉⣉⣉⣉⣉⣉⣁⡀⠀⣀⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠛⠛⠛⠋⠛⠛⠻⣾⡇⠉⠀⠀⠘⠒⠒⠖⠛⠉⠀⠈⠀⠀⣾⠃⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢰⡄⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣹⡜⡇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⡷⡆⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⢘⡖⠋⣿⣇⠿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡿⠛⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⡏⢿⠘⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠾⠏⢸⣧⣀⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⢸⣿⡸⠀⠀⠀⠀⠐⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⡇⠀⢈⢉⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⣿⡇⣶⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⣶⠀⠀⢠⡇⠀⣸⣅⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⡟⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡘⢘⡁⠀⠀⡀⠹⣿⣔⡀⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⢀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠿⣷⣄⠄⠐⠚⠃⠀⠀⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣷⠀⠈⣿⣦⡀⠀⠀⡀⠀⠀⢳⡀⠠⠀⠀⠀⠀⠀⠰⣿⣷⣿⡛⣿⡆⠈⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡆⣃⠂⢠⡞⠁⢙⡛⢶⣄⡀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⡿⣰⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⣯⡹⣄⠉⠉⠓⢀⣀⣀⡀⠀⠰⠆⠀⠀⠀⠹⠿⠗⠉⠛⢿⣿⣇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡟⠛⠀⠀⢠⣄⣙⣦⠀⠀⠀⠀⠀⠈⠀⠐⠚⠃⠀⠀⠀⡀⢨⣿⣿⣶⡽⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⠀⠀⣠⣿⣿⣧⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠳⠿⠟⢁⡼⡿⢆⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡗⠒⢀⢿⣿⣿⣿⣿⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣦⣠⣤⣶⣿⣿⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠛⢠⡆⠀⠙⢿⣿⣿⣿⣟⣻⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠉⢰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠂⠜⠀⠀⠀⠀⠈⢿⣿⣿⣧⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⢸⠀⠊⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣽⠇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⡄⠀⠀⢀⡀⠀⠀⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡟⡄⠁⠀⠀⠸⢧⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡈⢠⣿⠿⡄⣿⠀⡄⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣠⠾⣾⣿⡟⢀⠁⠈⠐⡇⠀⠀⣾⣧⣄⠀⠈⠙⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣤⠎⢐⣴⣿⡟⢀⣻⡿⢴⡄⠀⠀⢸⣿⠿⣧⣅⣄⠀⠀⠀⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇⣶⣾⣿⣿⠟⢋⢁⣴⣿⠋⠀⠀⠈⠀⠀⠀⢿⡀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢷⡄⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣾⣯⠦⠀⠑⠋⠉⠁
⠀⠰⠂⠈⠀⠸⠗⠒⠿⢷⣾⣿⣿⣿⠷⠃⠀⠀⠘⠚⠙⠛⠛⠙⠛⠻⠿⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡍⣽⠟⠻⠿⠟⠛⠟⠻⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡕⣧⡀⠀⠀⠐⡼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣻⢜⢶⣿⣌⢿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢀⣤⣄⣤⠁⢂⡜⢻⡿⡏⠘⣿⣿⣛⢸⣽⡿⠿⠿⠷⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⢊⣨⠗⢋⣁⣀⣘⢂⣿⡍⠻⣿⣧⠀⠈⠉⡀⢸⣿⣧⣤⣄⣀⡒⠒⠀⢸⣿⢻⡄
⠀⠀⠀⠀⠄⣀⠤⢒⣈⠤⠗⠋⡉⠈⢹⡍⢀⡏⢰⣿⣿⠆⠆⠀⠋⠈⢋⣥⣤⣀⠉⠉⣿⣧⡈⡏⡎⣧
⠀⠀⠀⢀⡯⠵⢚⣉⣠⣄⢷⣿⠿⢿⣿⣿⣿⡭⢽⣿⣿⠶⢾⣟⠏⣼⠟⠛⠛⠻⣶⣶⡶⣶⡿⣷⣾⣿
⠀⠀⠀⠀⣛⣿⣿⢻⣿⡿⣟⣿⣷⣶⣾⣿⣿⣛⣟⡿⠩⠽⠛⠷⣶⠇⠰⢫⠃⠀⢸⣿⣿⡟⠛⣿⣌⡞
⠀⠀⢀⢀⣿⣏⣱⣯⣿⣡⣟⣙⣿⣿⣿⣷⣽⣏⣛⣿⠛⠿⠿⠑⠛⠷⣄⡈⢁⣀⣾⣿⣿⣿⣿⣿⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡀⠈⠢⡀⢠⠔⠆⠠⣄⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣇⠀⠀⠀⠀⠀⠷⡇⠀⠀⠀⠈⡣⡀⢠⠃⠀⢄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣰⡀⢀⣠⠼⠦⣤⣤⠄⠚⠚⢶⣄⠀⠀⠘⠓⢬⣿⠀⠂⠘⡂⢀⠗⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡈⣖⠉⢀⠙⣏⠀⠀⠀⠈⣿⠀⠀⠀⠀⠹⢿⡇⠀⠀⠀⣹⢷⡆⢠⣥⠊⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠝⠟⠷⢮⡳⢤⣀⡠⠿⣆⣀⢀⣠⡜⠋⢝⡲⣖⡬⠀⠀⡇⢸⠎⣷⣀⣿⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢧⠀⠀⠀⢈⣿⣿⣮⣝⡲⢄⣛⣧⣽⡶⣊⠕⢊⡽⠓⣄⠀⠇⢸⢠⠳⢸⣿⡀⠈⣆⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⡶⠦⣖⡮⠀⣄⠚⠛⠛⠚⠓⠒⠒⠛⣛⠛⢻⣽⡒⠛⠓⡇⢸⠸⡃⢸⣟⠳⣀⠈⢂⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡴⢳⣋⣃⣾⣀⣀⣦⣠⣬⣧⣉⣈⣶⣆⣀⣀⣩⢷⠀⣀⠈⠳⠑⢾⣷⡀⠘⣇⠀⣇⡇
⠀⠀⠀⠀⠀⠀⢀⡠⢤⡧⠿⣦⣿⠶⠶⠶⠶⠶⠶⠶⣶⠲⠶⠶⠶⠶⠶⠶⢾⣟⠒⢻⠀⢀⣶⡀⡿⡟⢆⡀⠈⢻⢸⠁
⠀⠀⠀⠀⠀⠀⢎⠈⠀⡇⢀⡟⢹⠀⠀⠀⠀⠀⠀⢀⣿⣆⠀⠀⠀⠀⠀⠀⢸⡇⠳⣼⠀⣸⡆⠛⣏⣇⠀⠙⢦⣼⢸⢟
⠀⠀⠀⠀⠀⠀⡽⠀⠆⣧⠋⡇⢸⡄⠀⠀⠀⠀⣠⢚⢸⡛⠠⡀⠀⠀⠀⠀⣸⣇⣀⣿⣧⣼⣷⡀⣿⡏⠑⢴⡀⡏⢸⠎
⠀⠀⠀⠀⠀⠀⠈⠋⢉⣯⣭⣯⢻⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⢻⡄⠀⠆⠀⠀⢸⣻⠟⡧⣀⡈⢈⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⢲⠾⢿⠁⠀⣀⡀⠀⠤⠤⠤⠤⠀⣀⡀⠀⡰⢃⠜⠛⢻⠃⢺⠀⢸⡏⠀⡇⠀⠙⢺⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣶⣤⠋⣥⣤⠶⠤⠬⠭⠭⠽⠷⠾⣶⡍⢇⠈⣲⢲⣊⣀⣸⠀⢸⡇⠀⡟⢺⢤⣼⢠⡯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⣿⢀⢡⡤⠴⠶⢶⠶⡶⡶⡶⣶⡌⡇⢰⢀⣿⣸⡴⠖⠚⠀⢸⠷⠶⢗⣼⡶⠟⢸⣦⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠎⢉⣉⣉⣁⣸⢸⡆⢸⠀⣿⠀⣿⡇⡇⡿⣿⡗⢸⣚⣤⣤⣦⣤⣻⣖⣺⣶⣷⣿⣯⠭⣤⡼
⠀⠀⠀⠀⠀⠀⠀⣷⡾⠿⡽⠷⡄⢸⣿⡇⢸⠀⣿⠀⡿⡇⣇⡇⣿⡇⢰⣷⠛⣟⣛⣿⣿⢩⡍⢩⣿⡷⢰⢲⡅⣄
⠀⠀⠀⠀⠀⠀⠀⡟⣛⣒⣓⣒⣓⢺⣿⣧⣿⣠⣽⣤⣧⣧⣭⣿⣿⡇⠚⠛⠋⢉⣉⢹⡟⣶⠒⢺⣿⣇⣄⡧⣿⣼
⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛⠛⠛⠋⢹⣿⡋⠉⠉⠉⠉⠉⠉⠉⠉⣿⡏⠉⠉⠉⠉⠉⠉⣿⣻⠀⢸⣿⣏⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣇⣀⣠⣤⣤⣤⣬⣿⣧⣤⣤⣤⣤⠤⠤⠤⠤⡟⡧⠤⠤⠤⠤⠤⣤⢿⡿⠶⠘⣿⣿⣿⡗⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⢻⠛⢻⡟⠓⠒⠒⢻⠛⠛⢻⣟⢙⢉⣏⡽⠉⠉⡏⠉⢹⢘⢻⠀⠀⣿⡟⣧⣷⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⠼⢇⣠⣤⣤⠼⠒⠒⠚⠀⠉⠉⠀⠀⠀⠀⣿⠀⢸⣸⠘⠦⠤⠿⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣿⢷⣶⣦⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣯⣽⣷⡀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠉⠉⠙⢿⣿⡀⠀⠐⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠈⢿⣿⣿⣏⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⠀⠺⡷⢶⡀⣴⣶⡆⠀⡿⡆⠀⠀⠀⠈⣿⣿⣿⣿⠿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠸⠄⢈⡁⠀⣴⣇⠁⠀⠀⠀⣀⡼⢿⡋⣥⣴⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠷⠦⠾⣿⣿⣿⣇⡤⠴⠒⠉⠁⠀⡀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣿⣿⣿⣷⣶⣶⣿⣿⣿⣤⣀⣀⣠⣴⣶⣾⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠖⠚⢉⣹⣿⣿⡿⣹⡿⣿⣿⣿⣿⣿⡿⠉⣿⠙⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⣁⣠⣤⣴⣾⡷⢿⣿⡿⠁⣿⣷⠉⠛⠛⠛⠋⠀⣰⠇⠀⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠲⡒⣏⣡⣶⣿⣿⡿⠟⠛⢉⣡⠖⠋⠀⢻⡌⠀⠉⢓⢦⡄⠀⣰⠞⠁⠀⠀⠀⠀⠈⠑⠢⢤⣀
⠀⠀⠀⣔⡀⠲⠄⠐⢹⠛⣿⠟⠋⠁⢠⡖⠁⠀⠀⠀⠀⠀⠀⠛⢦⠀⡼⠛⢿⣸⡇⠀⠲⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣄
⠀⠀⢠⣿⣹⣇⠈⣠⠖⠀⠁⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠈⣿⣿⣷⢞⣁⠄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣷
⠀⠀⠀⠹⣿⠃⣺⠁⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠃⠀⠀⠀⣀⡠⠃⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⢲⡀⠀⠀⠀⠀⣿⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⣠⠎⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢳⡀⣴⠇⡠⠛⣿⣄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⡟⠁⠀⠀⠀⠀⢸⡀
⠀⠀⢰⠀⠀⠀⠀⠀⠀⢹⡄⢀⠘⣿⡿⠊⠀⠀⢯⡹⡌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⡶⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠇⢸⢀⡯⠀⠀⠀⠀⠘⡧⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡛⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠢⣄⣀⠀⠀⣻⣠⣾⡟⠁⠀⠀⠀⠀⠀⣿⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠘⣃⣤⣶⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠛⠻⠶⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡟⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣯⡿⠟⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠰⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠸⡅⣀⠤⠖⠛⠛⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠙⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⠀⠀⠀⠀⢸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⡆⠀⠀⠀⠈⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢰⣎⢠⡆⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡤⣾⣉⡏⣉⡀⠀⠟⠀⠀⠀⠀⢸⣿⢈⠁⠀⠀⠈⡀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠚⢃⣠⣶⢻⣿⣿⢿⣿⡟⣿⣹⣦⣬⣉⣛⡀⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⡇⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢦⡍⢿⣿⣿⣿⣿⣿⣿⣽⣿⠋⣿⣿⣿⣿⡿⣄⠀⠀⠀⠌⢿⣆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⢀⣥⠈⠿⢻⣿⣿⣿⣿⡿⣽⡧⣴⢟⣿⣯⣵⣿⣷⠀⠀⢖⡀⡏⠀⠀⠀⠀⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⢸⣿⣿⣿⣿⣿⣏⣼⣳⣿⣿⣯⣿⣼⡟⠀⠀⣾⠃⡇⠁⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⣿⣿⣿⡿⣾⣯⣿⣿⣿⣿⣗⣿⠀⠀⠀⣿⣶⠃⠀⠀⠀⠀⠀⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣐⠟⡹⠿⠿⠿⠛⠉⠉⠁⢻⣿⡇⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⢻⣿⣆⣱⣀⣀⣀⠀⢀⠀⢀⣼⠿⠁⠀⠀⠀⣿⣇⡇⠀⠀⠀⠀⣦⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣾⣯⠈⠛⠛⠿⠿⠿⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⡀⠇⠀⠀⠀⠙⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢿⣿⡆⠀⣀⡴⠖⠊⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡄⠀⠀⠀⠀⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠐⣷⣼⡿⠂⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⣠⠏⢹⣿⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⠁⡀⠀⠀⠀⠀⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣼⠃⠀⢸⣿⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢨⡜⠀⢁⠀⠀⠀⠠⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⡠⠄⡰⠇⠀⠀⠀⣿⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣼⠘⡇⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⣏⣩⡞⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢿⠀⠀⡄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⣿⣤⣥⠔⠀⠀⠀⠀⠀⠀⠸⣿⣦⡄⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣬⣄⠀⠇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⠤⠶⠉⠃⠀⠀⣀⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣿⡀⠀⠀⠃⠀⠸
⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⣴⣾⣻⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⠹⠇⠀⠀⠘⢀⢰⠂
⠀⠀⠀⠠⠀⣀⡀⠀⠀⠘⠛⡛⠛⠁⠀⠀⢀⡀⠠⠴⢂⠠⠄⠀⠀⢸⡷⣄⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⢏⢲⠂⠀⠀⠀⠈⢸⡇
⠀⠀⠀⠨⠛⠃⠀⠀⢀⣀⠀⠀⠒⠉⠁⠈⠀⠀⠛⠚⠶⠂⠀⠀⠀⠈⢻⣀⣰⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢷⡄
⠀⠀⠀⠀⣈⣉⣋⠀⠄⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⣤⣶⠟⠋⠀⣰⡃⠀⣤⡄⠀⢠⣦⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⣷⣄⠀⠀⠠⠀⠺⠟⠻⠿⠛⢉⠄⣼⢿⠃⠀⠸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⡀
⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣿⡆⠀⠀⠀⢀⣠⣀⣴⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣧
⠀⠀⢀⡆⠛⢲⡀⠀⠀⣴⠋⢲⣤⣄⠀⠀⠀⠀⠀⠚⠒⠻⢲⡄⠀⠀⠈⠉⠛⠗⠟⣦⠉⠈⠀⠀⠀⠠⣶⡆⠀⠀⠀⢨⣿⣿⣿⣿
⠀⠀⢸⣅⣀⣨⡇⠀⢰⠛⣿⠑⢹⣿⣷⡀⠀⠀⠀⠀⢀⣀⣈⡇⠀⠀⠀⠀⠀⣀⣀⣸⡇⠀⠀⠀⢠⡖⠛⣄⠀⡀⠄⢸⣿⣿⣿⣧
⠀⠀⠈⢸⣿⡇⢠⠖⠙⢶⡟⠋⠛⣿⣿⣇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⢹⡇⠀⠀⠀⠀⠈⣷⣦⣽⣦⠤⠠⢼⣿⣿⣿⣿
⠀⠀⠀⢸⠿⡇⠘⠄⠀⠼⠧⡀⡠⠿⠟⠛⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠀⠀⡿⠸⡇⠀⠀⠀⠀⠀⢸⣿⠙⣿⠀⢀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠘⠳⢿⡿⠿⠿⠿⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣶⡦⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡾⢿⡟⠀⠀⢿⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣴⡾⣷⢶⡞⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠁⠀⣿⣇⠀⠀⢿⠀⠀⠀⠀⠀⣤⢰⣶⡈⣇⠀⠀⢻⣿⣿⣿⣦⠀⠀⠀⠀⢠⣿⣧⣿⣾⣀⣀⣸
⠀⠀⠀⠀⠀⠀⣀⡠⣿⡇⠀⠀⠀⣤⠀⣿⣿⢠⡀⣻⡄⠀⠀⠀⠀⠙⠺⠙⠋⠁⣠⣄⠈⣿⣿⣿⠟⠃⠀⠀⠀⠀⢹⣿⣿⠈⣿⣿⡏
⠀⠀⠀⠀⢰⣶⣿⣷⣿⡇⠀⠀⢀⣿⣾⣿⣿⣾⣧⣿⡇⠀⠀⠀⠀⠀⣀⣤⣴⠆⠈⠛⢄⢨⣭⣤⣤⣶⡄⠂⠀⠀⠸⡿⣿⠀⣿⢿⠇
⠀⠀⠀⠀⠘⣿⣿⣿⡿⠿⢂⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠙⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠠⠤⠽⠿⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⣿⠀⠀⠀⠐⠻⠿⠛⠟⠛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠦⠀⢀
⠀⠀⠀⠀⠀⠀⠠⣤⡰⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠀⢀⣤⣤⣤⣼⡯⣭⣿⣿⣽⣯⡭⣽⣿⣿⣿⣿⣯⣤⣴⣦⡄⠀⠀⠀⠀⠀⠀⢈⣯⣧⣄⠀⢸⣼⡇
⠀⠀⠀⢾⡿⡿⡿⣿⣿⣿⢟⢿⢿⢟⢿⣿⣿⢿⢿⢿⢿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠻⣏⠀⢷⡾⢻⡆⢀⣸⢠⢀⡀⣶⡄⢀
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠈⠋⠈⠋⠀⠀⠀⠈⠉⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣶⣿⣶⣾⣷⣶⣷⣾⣿⣿⣾⣴⣿⣷⣾⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢀⣠⠤⠀⠀⠁⡑⡱⠤⢥⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣭⣽⣟⢶⡄⠀⠀⢀⡴⠚⠛⠛⠋⠉⠉⠛⠛⠚⡇
⠀⠀⠡⠂⠀⠛⢂⠄⡴⢤⠤⠴⠒⠂⠀⠀⠁⠀⠀⠀⠀⠀⠘⡃⠀⠀⢠⣿⣿⣿⣿⣿⣿⣮⣷⠀⢰⣋⠤⠶⠒⠄⢀⣀⣀⠠⠤⠤⣇
⠀⠀⠉⠉⠉⠁⠒⠒⠂⠀⠁⠘⠓⠐⠐⠂⠢⠂⠆⠄⠄⠀⠊⠃⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⡈⠛⠷⠾⠤⠤⠖⠒⠒⠒⠒⠋
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠰⠲⠀⠀⠙⢿⣿⣿⣟⣿⠿⠁⢠⠞⣥⣄⢄⣠⣄⣀⣀⣠⢠⠀⠀⠀⠔⡀
⠀⢠⡇⢀⡆⣰⡤⣠⣠⣰⢷⣥⢤⠆⢠⣤⢤⣄⡄⣼⡟⠀⠀⠀⢀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠉⠉⡬⢟⣹⠋⠉⢀⣡⠏⠀⠀⠀⠀⡇
⠀⠀⠁⠈⠀⠉⠁⠁⠁⠉⠁⠈⠈⣴⠏⠈⠀⠀⠀⠀⠀⠀⠀⢰⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠤⠤⠬⠡⠀⠀⠀⠐⠶
⠀⠀⡰⠒⠐⠲⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠖
⠀⠀⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⠒⠒⠑⠀⠀⠀⠀⠀⠀⠉⠁⠐⠐⠒⠂⠀⠤⠤⠤⠠⠔⠂⠒⠀⠉⠉⠁⠀⣠⣆⡀
⠀⠀⡇⠀⠀⠀⡟⠒⠒⠀⠀⠀⠈⠛⠚⠁⠀⠀⠈⠬⣉⠭⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠤⠤⢤⠄⢒⠆⢰⠀⠀⠸⡛⢻⠁
⠀⠀⢡⡄⠶⢰⠇⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢰⣢⣉⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣢⠀⠜⣠⠀⠀⠀⠙
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠠⠉⠀⠀⠀⠀⠁⡀⠀⢰⡀⠀⡸⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣦⡆
⠀⠀⢰⠒⠒⠲⣦⠀⠀⠃⠀⠀⠺⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⣠⡖⠒⠒⠒⠲⣤⡀⠀⠀⢀⡚⣷⣀⠀⢀⣤⣤⡀⠀⠀⠠⣀⠀⠀⣟⡃
⠀⠀⢘⠀⠀⠀⡼⠀⠀⠐⢄⡀⠀⠀⣀⡐⠁⠀⠀⢀⣠⣶⣿⣽⡇⠀⠀⠀⢀⣿⣏⣷⣤⡈⣗⢹⡹⡆⢈⣀⣠⡄⢤⠀⠀⠃⠀⠀⣿⡇
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⣁⡀⠁⠀⠀⠀⢠⣮⣧⣼⣿⣾⣷⢄⠀⢀⣮⣿⣧⣼⣦⣵⡄⠓⠵⠃⠀⡄⠀⠃⠈⠂⠤⠇⠀⠀⢺⡗
⠀⠀⡰⠊⠁⠀⠈⠐⠄⠀⠀⣿⡯⢭⠆⢀⡀⢠⣿⣉⣉⣹⣿⣹⣏⣫⣲⣍⣉⣿⣏⣽⣍⣉⣿⡄⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⢼⡃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠟⢢⡸⣤⡾⢁⣎⣹⣉⣉⣸⣿⣹⣏⣉⣿⣉⣀⣿⣇⣯⣀⣀⣿⣿⡀⠀⠀⡆⠀⠃⠀⣦⡤⠀⣄⠀⡾⠇
⠀⠘⠛⠉⠉⠙⠉⠛⠛⠃⠀⣿⠈⢹⡿⠁⣼⡟⣿⠻⢿⣿⣿⠟⣻⣿⣿⢿⣅⣹⣿⣿⣿⡛⣿⡟⣧⠀⢀⠀⢠⠀⠀⠀⠰⠆⠀⠀⣿⡇
⠀⠀⠀⠀⣀⣠⢄⠀⠀⠀⠀⣿⠠⠀⡇⠀⣿⠟⣿⠃⢻⣷⠞⢯⣁⡻⣿⢹⠿⣏⣡⠟⠁⠉⣿⡏⣿⠀⢸⠀⠀⠀⠀⠀⠘⠀⠀⠀⡷⡇
⠀⠀⣌⠁⠘⠙⣮⡀⠀⠀⠀⡏⢀⢀⡇⠀⢿⣷⣿⣧⣾⣿⣷⡾⢿⣿⣿⣿⣖⠛⢷⣲⡆⣶⣿⣿⡟⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⣟⡇
⠀⠀⠓⠒⢄⠀⠘⠧⠄⠀⢀⠙⠛⠚⠁⠀⠘⢇⣯⣜⣿⡏⢹⣷⣾⣬⣧⣜⣿⣿⡿⢷⣴⣤⣿⣷⠁⠀⢸⠀⠈⡄⠀⠀⠀⠀⠀⠀⡧⡃
⠀⠀⠆⠀⠈⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠳⣶⣟⣿⣥⣹⣟⣸⣇⣸⣏⣽⣿⣿⣬⣀⣿⠇⠀⠀⠀⠓⠒⠁⠀⠀⠀⠁⠀⠀⣷⡃
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠈⠉⠀⠀⢀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠸⣿⠃
⠀⠀⡄⠀⣤⡄⠀⠀⠀⠀⢀⡄⠀⠀⢠⣠⠀⠀⠀⠀⣠⠀⠀⠀⠀⣴⣰⠀⠀⠀⠀⠀⠀⠀⢈⣤⢾⠁⠀⠀⢰⢿⣿⢿⠀⠀⢠⡞⠁⣾⠂
⠀⠸⡅⣇⠜⣟⡷⡞⢷⡆⢾⠀⠀⢠⡎⡳⠇⠀⠀⠀⠏⠖⠶⠻⠾⡏⡀⣿⠀⠀⠀⠀⠀⢻⠛⠇⠈⠀⠀⠀⠸⣿⣮⠟⠀⠀⢢⣓⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⣶⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡀⢀⣠⣤⠄⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠻⠓⠓⠈⣿⣿⣿⣿⢟⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡁⢀⣠⡀⠀⠰⣿⣿⣿⣿⡿⣿⣿⣿⠋⠘⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠀⠀⢨⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇⡀⠔⣠⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢘⣿⢋⡼⠛⣙⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣴⢡⡞⢀⣴⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠿⣿⣿⡿⢿⣿⣿⣿⠟⠋⠙⠁⠀⠁⠘⣠⣿⣿⣿⣿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⣾⣯⡔⠻⢿⡉⢷⣜⠟⣩⣿⣇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣻⡿⠿⢯⣽⢯⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣥⣶⠟⠉⣷⣏⣀⣠⣿⣶⣏⣴⡿⢃⣽⣆⠀⢰⡣⢠⣿⣿⡏⠉⠛⣿⢿⣷⣶⣴⣶⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣾⠛⣧⣦⣤⣻⣿⣿⡏⠉⢿⣉⣽⣿⣿⢛⠿⠖⠋⠙⢿⣿⣟⠻⡷⣶⣇⣠⢀⣽⡿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⡀⢿⣿⡿⡋⣐⣯⣷⣴⣾⡿⢻⣿⢡⠂⠀⢈⡀⠀⢀⣹⣿⠇⢀⠋⠉⠻⣿⣿⡏⠚⢻⣿⣝⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠚⢶⣽⣟⠉⢿⣿⡏⢿⡿⠁⢲⣿⣼⣿⣿⡶⢰⣦⡄⣳⣿⣿⡷⢺⢾⣿⣷⣴⣿⣿⣿⣿⠛⠛⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢧⣾⡟⠠⢳⣾⠻⣅⣼⣷⠶⢿⣿⡿⠏⠉⠉⢽⢿⡇⠈⢻⣿⣄⢀⣨⣿⣿⢿⣿⣅⠛⠽⣦⡈⢨⣿⣿⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣸⠿⠷⣦⣼⣿⣄⡟⣻⠷⠀⢸⡿⠉⠁⢍⠱⡀⢿⡆⠐⢿⠟⢻⡘⢿⣿⣿⡿⣿⣿⠠⣿⡿⣷⠙⠛⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⣿⣿⢦⡀⣾⣿⡁⣿⣷⣿⣿⣼⡏⠀⠤⠠⡈⠇⢰⣿⣿⣦⣄⠀⠀⠀⠙⠿⣟⣿⣿⣿⡆⢀⣙⣿⣥⣤⣾⣿⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣧⣤⣿⣁⠈⣿⣯⡛⣿⣿⣿⢿⣿⣈⣷⠀⠘⣯⣧⣾⣿⣿⣿⣿⢿⣦⠀⠀⠉⠀⢈⠻⢿⣿⣿⣬⣽⣿⣿⣿⣝⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⢻⣿⣟⣻⣿⣯⣷⣿⣿⣿⡿⠛⠉⣀⡄⠈⣿⣿⣿⣿⠋⠽⠀⢸⣻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢨⣾⣩⠶⠛⠛⠛⠿⠛⠋⠁⠀⠀⠰⣼⣿⣶⡿⣋⣿⣟⣀⣨⣅⣬⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣿⠟⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⢤⣽⣿⣿⣿⣿⣟⣿⣿⠍⠙⠏⠉⠻⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⢀⡋⠀⠀⠀⠀⠀⠀⢠⣠⣴⡶⣤⣾⣿⡿⣿⣿⠟⣰⢿⣿⠀⠀⡀⠀⣈⣤⢸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠨⣤⣀⣀⣀⣄⣤⣷⣾⡿⠿⢥⣾⣿⣿⣿⣿⣿⣿⣿⣓⢾⣿⠿⠛⠛⠚⠛⠿⠉⠙⢿⣿⣿⣿⣦⣀⣀⠀⠀⠀⣠⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢭⡿⣏⠿⠹⠟⠫⣁⠐⡵⣿⣿⣿⣿⡿⢫⠝⢙⡋⠀⠀⣿⠀⠀⠀⠀⢰⣀⠀⠨⣿⣬⣿⣿⣿⣿⣿⣿⣶⢾⣿⣿⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢬⢿⡷⠶⣴⣢⠟⠋⠀⢻⣿⣿⣿⣿⠛⠪⠿⠿⠯⠿⢿⠁⠀⠀⠀⠐⠀⠀⠀⠉⠀⠁⠙⠛⠉⣿⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠈⠁⠀⠀⠀⢸⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⡄⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⠀⠀⠐⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣶⣿⣗⣤⣈⡉⠛⣷⣶⣶⣦⣤⣀⣀⠀⠀⠀⠀⡀⠄⠒⠋⠉⠉⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⡟⠈⠉⠛⠉⠉⠉⠙⣿⣿⠉⠉⠙⠛⠿⢿⣿⣶⣶⡛⠁⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠟⠉⠁⠀⠇⢀⣀⠀⢀⣠⡄⠀⢻⠿⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⡄⠀⠀⣠⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⡆⠀⠀⠀⣼⢤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣲⣶⡛⠀⣠⣾⣿⣏⠀⢀⣦⠀⠀⠀⠀⠀⠀⣠⡴⠚⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣼⣇⣰⣟⠿⠿⠾⢿⠿⠿⣿⣿⣿⡄⣼⣿⣧⠀⣀⡶⠖⠋⢁⣤⣶⣿⣿⠿⠛⠛⣿⣿⣆
⠀⠀⠀⠀⠀⣠⣿⣿⠋⠀⠀⠀⠀⠀⣠⡀⠀⢰⣿⣿⣾⣿⣄⢀⢀⣀⣠⡼⢿⣿⣿⣿⣿⣿⠷⠚⠉⣀⣴⠞⠋⣱⡄⠁⠀⠀⠀⠀⠈⢻⣿⣧⡀
⠀⠀⠀⠀⢠⢟⡟⠀⠀⠀⠀⠀⠀⢀⣿⣥⠘⣻⡿⠻⢛⣿⣿⣯⣿⣿⣏⠀⣸⣿⡿⠟⢋⣡⠴⣾⡿⠿⢿⡛⢠⣿⣗⠀⠀⢀⠀⠀⠀⠀⠹⣿⢧⡀
⠀⠀⠀⢠⢯⡿⠁⠀⠀⠀⣴⡀⠀⣸⣿⣿⣿⠋⠀⠀⠈⠀⠘⠋⠀⣀⡥⠖⢉⡁⠴⠚⠉⡑⠈⠀⠀⠀⠈⢹⣿⣿⣿⣦⠀⣾⣇⠀⠀⠀⠀⠘⣎⣧
⠀⠀⢠⡎⡼⠁⠀⢰⡀⢰⣿⡇⠀⣿⣿⣿⡿⣠⠀⠠⠤⣠⠔⠒⢋⡁⠔⠚⠉⠀⠀⠈⢀⠠⠆⠁⠀⠀⠀⠈⣿⣿⣿⣿⣸⣿⣿⠦⠀⣄⠀⠀⢹⣿⣧
⠀⠀⣼⢳⡇⠀⠀⣿⣇⣸⣿⣿⣼⣿⣿⠿⠛⠁⠀⠀⠀⠈⠠⠖⠋⠀⠀⠃⠀⠀⠈⣩⡏⠀⠀⠀⠀⠀⠀⠐⢼⣿⣿⡿⢿⣿⣿⠆⣰⣿⡆⠀⠀⣿⢻⡄
⠀⢠⡟⣿⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⡿⠳⣦⠀⠀⠀⠀⡠⠷⢤⣄⡤⠠⠆⠀⠀⠀⣿⣷⢀⡴⠓⠀⠀⠐⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⡟⣇
⠀⢸⣇⡿⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⢷⡀⢀⠎⠀⠀⠀⡸⠛⠻⢶⢤⡤⣶⣽⣿⣾⣥⣶⠠⠄⢀⠀⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠘⣇⣿
⠀⢸⣿⡇⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⡄⠛⢿⣀⠀⠀⡰⠄⠀⠀⠀⠀⠀⠀⠀⠨⠍⠍⠏⠙⠲⣄⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⣿⣿
⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠼⡇⣧⣀⠀⠉⡿⠿⣤⣀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿
⠀⢸⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⠈⠁⠐⠇⠀⠀⠙⠿⣷⡄⠀⠀⠀⠀⠐⠂⠀⠀⠆⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿
⠀⠀⣧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢾⠀⠀⠀⣦⠀⠀⢀⡀⢀⡉⠓⠦⢤⣤⣀⣀⣠⣄⣄⣀⣠⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇
⠀⠀⢹⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠈⣿⣿⣿⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⠁
⠀⠀⠀⢷⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠛⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢠⠇
⠀⠀⠀⠈⣧⠹⣄⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⢹⠀⠀⠀⢺⠂⠀⠀⠒⠀⠆⠀⠀⠀⠃⠀⠉⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢡⡏
⠀⠀⠀⠀⠈⢧⢙⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣾⣾⣷⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣞⢡⠏
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠐⠀⠀⠀⠒⠲⠀⠀⠀⠀⠀⠤⠀⠀⠀⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢜⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠀⠀⣀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠒⠒⠀⠒⠒⠒⠒⠒⠄⠀⠀⠦⠬⠤⠤⠤⠤⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⡤⠤⠄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢐⢁⡍⡠⠀⠀⡀⠀⠀⠀⠀⠁⠠⣠⠔⡩⠒⠀⠀⠀⠀⠤⢀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⡜⣄⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠄⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢛⣌⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠮⠎⠤⠴⠶⠶⠶⠤⠤⠤⠤⠤⠠⠶⠶⠤⠤⠤⠤⠤⠤⠯⠽⠗⠒⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⡀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⣤⡤⠄⣾⣿⣿⣿⣿⣿⣿⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠈⠉⢁⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣶⣿⣿⣿⣿⣿⡇⠀⢹⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⢀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⢘⢫⢏⣔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣟⣿⣿⢻⣿⣿⠋⠙⠀⠁⢀⣾⣿⣿⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⡼⠟⣿⠉⣑⣬⣎⣴⠟⣇⠀⢀⢀⣾⡿⠿⢿⣿⣭⣉⣳⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⠛⣧⣤⣿⣿⡏⢹⣉⣳⡊⠛⠐⠙⢻⣿⡛⢶⣾⣄⣩⣟⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢏⣤⡿⣯⣤⣷⡿⢛⣯⣿⣧⣁⣀⢀⣤⣾⣣⣤⣄⣹⣿⣧⣬⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣼⡟⠳⣼⢟⣸⡧⢾⣿⠟⠛⠓⣿⠀⣻⣯⣀⣹⣿⣿⣟⠛⣧⡀⣹⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⡟⠻⣾⡿⣾⣽⣇⣬⠃⠀⢠⠠⣹⣀⠙⠋⠊⢿⣿⣿⣿⡞⠻⣿⣉⢹⣿⢻⡆
⠀⠀⠀⠀⠀⠀⢸⣧⣀⣿⠉⢷⡿⣷⣿⣿⣏⣿⠈⢣⣆⣴⣿⣿⣷⣄⠀⠀⠙⠿⣿⣷⣘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⡋⢹⠿⣹⣿⣿⣿⡿⠟⠋⢠⣄⣸⣿⣿⠋⠃⠸⣿⣆⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⢥⡿⠊⠁⠀⠈⠁⠀⠀⢀⣽⣿⣿⣾⣿⡖⠒⠚⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⢰⠟⠸⠁⠀⠀⢀⣀⣤⢶⣠⣾⣿⣿⣟⡵⣿⣀⣀⣀⣠⡘⣿⣿⣦⡀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠘⣦⣷⣶⣶⣶⡿⠛⣉⣿⣿⣿⣿⡿⠿⠛⢻⠉⠉⠉⠉⠁⢈⣟⣿⣿⣶⣦⣄⣤⣾⢟⠟⠁
⠀⠀⠀⠀⠀⠀⠘⠯⠽⡦⢤⡮⠓⠁⣿⣿⡿⠾⠂⢿⠆⠾⠃⠀⠀⠂⠀⠈⠋⠉⠛⠻⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣖⣒⣒⣿⣿⠉⢑⠶⠟⠉⣲⣎⠁⣲⠒⡶⠲⠤⡄
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣟⣛⣻⡏⢙⠛⢢⠚⢻⣇⣸⣯⣀⣿⢉⡎⣰⣙⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡇⣟⣛⣻⣉⣹⡇⢻⡟⢩⠤⣈⢉⠭⣭⣭⡁⢄⢩⡱⡀
⢹⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⡇⢹⣟⣛⣏⢈⣏⢉⠃⢴⣬⣾⠸⣷⣦⣙⣿⣦⡑⣅⣃⠀⣤
⠨⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⠸⣭⣭⣧⣠⣏⣹⠀⠶⢄⠀⠀⠙⠥⢄⠀⠀⠀⠈⠉⠐⠢⣄
⠸⣹⣿⢩⣿⣿⣿⣿⣿⣯⣿⣿⣻⣿⡤⢿⣯⣯⣠⣅⣨⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⡴⠒⠀⠛⠀⢱⡀
⠀⠉⠉⠻⠿⠟⠉⠉⠙⠿⠟⠻⣿⣻⣿⡿⢿⡶⠿⣿⣿⡀⠀⣠⠖⠒⠀⠀⠀⣤⣤⠠⠇⢸⣷⡷⣶⡿⠏⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣻⣿⡟⢿⡇⠀⣿⣿⢻⣛⡏⣠⣶⣦⣄⠈⠁⣴⣄⠀⣒⡒⠒⠒⠓⠀⠮⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠷⠿⠿⠿⠿⠻⢾⣿⣁⣿⠏⠩⣿⣄⣀⣉⣁⣠⡶⠭⠭⠭⠭⠒⢒⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⠺⠀⣿⣿⣍⠥⠤⣶⡶⠖⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⣿⣿⠏⠀⠀⠀⠀⠀⠈⠙⠛⠋
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⡀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣆⡇⠀⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠏⣁⠄⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠷⣶⣿⠀⣾⣤⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠻⣿⡇⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⢶⡑⠤⣀⡀⡏⠛⠳⣆⡀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣤⠍⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⡙⢶⠃⠈⠃⠀⠀⠈⠉⢑⣒⣋⡘⠿⠛⠀⠀⠀⣿⣿⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣷⣶⡶⠶⠒⠚⠉⠉⠉⠉⠀⠀⠀⠀⣄⣀⣤⣾⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⠋⡄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢩⣟⣼⣯⡋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⠃⠀⠠⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⢀⡼⠉⢿⣿⣷⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢁⡜⠀⠀⠈⢻⣿⣷⢧⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢭⣇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠹⣿⣷⣇⠀⠀⠀⠀⠀⠀⠸⣏⣿⡋⢣⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⢠⣟⠿⢿⡃⢸⡆⠀⢰⡀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢠⣾⠋⠀⢾⡇⠈⠃⡂⠀⠁⠂⡀⠙
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⠰⠎⠃⢀⣠⣶⣯⠁⡀⠀⠧⠀⠒⠂⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠋⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿⢓⣴⠶⠖⠛⠁⠉⠁
⠀⠀⠀⣀⣀⣠⣴⣾⣻⠟⠉⠛⠃⠀⠀⠉⠉⠉⠉⠉⠛⠿⠛⠁
⠀⠀⠈⠉⠉⠙⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⢄⡀⡠⠊⠀⠀⢀⣉⣒⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡰⠋⠣⡜⣠⡞⠤⣝⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢰⠁⠀⠀⢷⢹⠆⠈⣀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣾⣟⠉⠒⠺⢿⣷⣾⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣩⡝⠛⢿⣿⣿⣿⡿⣤⡉⠍⡀⠐⣲⠖⡉⠓⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⠋⠉⠹⣿⠤⡀⠙⠿⣟⡀⢸⣿⠲⡄⠀⠉⢣⠉⠉⢻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⡅⠀⠀⢻⣇⠉⠉⠀⠩⠂⠀⣿⡗⡙⣄⠀⡰⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠡⠀⠀⠈⣿⠀⠀⠀⠐⢣⠀⣿⣇⡀⠛⣻⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣗⠀⠀⠀⠹⠂⠀⠀⢹⡇⠀⠀⠀⣀⠀⣺⡗⠄⠀⡅⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠁⡐⠀⢾⡀⠀⠀⢸⣿⠀⠀⠀⠘⠀⢸⡧⠷⣬⣂⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠂⠀⠈⡇⢳⠀⠀⠀⣿⠀⠀⠀⠀⠀⣺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠙⢣⠈⣇⠀⠀⣿⡇⠀⠀⠐⠂⢺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢳⡗⢐⢐⠹⠇⡀⠀⢈⠥⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠹⢶⣶⣶⣶⣶⣶⣾⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠚⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠠⡔⢂⠀⠀⠂⡄⢰⣶⢲⠂⠠⠄⠀⠀⠀⠀⠀⠀⡠⠰⠛⠣⠠⠀⡟⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠈⠉⠉⠢⣄
⠀⠀⠀⠀⠻⠸⠀⠀⠀⠁⠌⠘⠀⠛⠀⠀⠀⠀⠀⠀⠀⡄⠈⠲⣀⢶⣶⡾⠄⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⡠⠔⠒⣂⠀⠀⠀⢳⠒⠔⠋⠀⠀⠀⠀⠀⢸⡇⢠⠀⡇⠈⡆⠀⡼⠁⠐⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡷⢦⡀⣀⣴⠤⢀⣰⡇
⠀⠀⠀⠀⢺⠀⠀⢈⣁⡈⠁⡽⠔⢊⡖⠉⠀⠀⠀⠀⠸⡇⡀⠠⠶⠀⠂⠀⢻⠀⢠⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠘⣆⣠⠿⣷⣿⣾⠧⡄⡼⠁⠀⠀⠀⠀⠀⠀⠻⣷⠼⣦⣴⣎⣁⣈⣀⢏⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡼⢛⠻⣦⣀⣸⣧
⠀⠀⠀⠀⠀⠘⣏⠀⢾⡩⠆⠀⡴⠁⠀⠀⠀⠀⠀⠀⠄⣖⢋⠑⠉⠉⢉⠉⡯⡇⡟⡃⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠈⠢⣸⠁⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠘⣀⠀⠀⠘⠀⠃⢀⠘⠃⠀⠀⠀⠀⠀⠀⠠⠐⠢⡀⠀⠀⠉⠻⠿⠛⠁⠀⠀⢀⡔⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠦⠼⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠗⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂⡀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⢀⣠⠔⠊⠁
⠀⠀⠀⠐⡆⢅⠁⠆⡂⡄⢸⠺⠙⣀⠐⠀⠀⠀⠀⠀⠀⢀⡴⠋⣄⠀⣀⠀⢹⠐⣇⠀⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠙⠒⠤⣀
⠀⠀⠀⠀⠛⠸⠀⠀⠇⠁⠈⠄⠀⠃⡀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡀⣁⡉⠀⡽⠘⠉⠋⠁⠀⠀⠀⠀⠀⠠⣀⡠⠊⠁⠀⠀⠀⢀⡀⠀⠀⠀⠑⢤⣀⡤
⠀⠀⠀⠀⢀⠀⠀⡠⠀⠈⠀⡄⠀⣀⠀⠀⠀⠀⠀⠀⣼⠀⢀⣸⣿⣽⣿⣶⡷⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠤⠤⣰⢄⠀⣀⠀⣹⣀⠤⠄⠀⠀⠀⠀⠀⢸⣞⡟⠉⠸⣏⠩⡅⠀⢠⡞⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⢀⡀⠈⣃
⠀⠀⠀⠀⠀⠀⢈⣀⣤⡄⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠤⡟⡃⠈⡀⠉⠁⢸⣎⠟⢆⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢉⠈⠉⠁⢸⡇
⠀⠀⠀⠀⣄⡀⠚⢿⣶⣿⠾⠋⠢⣀⡀⠀⠀⠀⠀⠀⠀⡟⢰⢀⠀⠀⠈⡘⢱⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣞⡛⣧⣀⣿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⢙⢿⣿⣿⠿⠛⢘⠇⣠⡀
⠀⠀⠀⠀⠀⡀⡀⡀⡀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣼⡇⠀⠠⢿⡄⠰⢦⡀⠈⣴⠟⠀⠈⠲⡄
⠀⠀⠀⠙⡇⢂⠀⡂⡀⡆⢸⠸⠘⣤⠈⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⠀⠈⠙⢦⡀⠀⠀⠀⢰⣧⠀⠀⣿⡇⠀⠀⠀⡟⠀⠀⠀⣾⡏⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠃⠸⣄⢀⠀⠈⠀⣄⠤⠁⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⣤⣤⡼⣀⠌⠀⠀⢳⡀⠀⠀⣸⠀⠀⠀⢘⣇⠀⠀⠀⡇⠀⠀⠀⣿⣧⠟⠀⠀⠀⣍⠁
⠀⠀⠀⠀⡏⠉⠀⠦⡄⣠⠄⣸⠈⠉⡇⠀⠀⠀⠀⢸⡇⠀⣬⠄⠼⠀⠏⢠⡋⠀⣆⡀⠈⡇⠀⠀⣿⠀⠴⠲⣾⠛⠓⠒⠂⠉⠀⠀⠈⠉⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⢧⠀⠀⣠⠦⣄⢀⡞⠀⢀⡇⠀⠀⠀⠀⠘⣷⣧⡷⡄⡚⠀⡀⠨⠆⣠⠘⣾⢰⡇⠀⠀⢿⠆⠀⢠⡇⠀⠀⠀⣀⣀⣤⠶⠒⠚⢿⠋⠁⠈⣿⠟
⠀⠀⠀⢒⠘⣦⡼⠻⣿⣿⠿⠛⠇⡞⠐⠂⠄⠀⠀⠀⡏⣿⠉⣩⡍⠉⢱⠒⠒⠁⣘⢩⠙⠀⠀⠀⠸⣧⣧⣤⣿⣶⣾⣿⣿⣋⠀⠀⠀⠀⣠⠀⠀⣾⠋
⠀⠀⠉⠈⠁⠛⢆⠒⢳⠘⠃⢀⠞⠃⠉⠊⠁⠀⠀⠀⠇⡿⠀⣸⣇⣰⣘⣀⣂⡂⠾⢀⠰⠀⠀⠀⠀⠀⠀⠈⠉⣿⡿⡿⢻⠛⠿⠿⢿⣿⣽⡿⠛⠁
⠀⠀⠀⣀⠀⠠⠘⠳⣜⢀⡴⠛⠢⢀⢀⡀⠀⠀⠀⠀⠧⠑⠁⣩⢹⠁⠏⣾⠉⠛⠉⠐⠤⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⢀⠀⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠚⠲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠴⠚⠋⠁⠀⠀⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠓⠉⠀⠀⠀⠀⢀⣠⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣼⣿⣤⡀⠀⠀⠀⠀⣀⡤⠔⠊⠁⠀⠀⠀⠀⢀⡠⠔⠚⠉
⠀⠀⠀⠀⢀⣠⣴⣾⡏⢀⣿⣿⣿⣿⣿⣷⣦⠴⠖⠋⠁⠀⠀⠀⠀⣀⣤⠶⠚⠁
⠀⠀⠀⢰⢿⣿⣵⣽⡇⢸⣯⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⢸⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣟⣿⣾⣷⠴⠒⠉
⠀⠀⢀⣮⣿⠿⢛⣻⠀⢸⣿⣿⣿⣿⣿⡿⠟⠋
⠀⣰⣿⠿⠏⠙⣿⣇⠀⣾⣿⠿⠟⠋⠁
⣰⠟⠁⠀⣠⣾⣿⡿⠆⠁
⠻⣦⣴⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣷⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏⢀⣀⣤⣀⠀⢀⣤⣤⣤⣤⣤⡈⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠉⠉⠁⠀⠀⠉⠁⠀⠀⠀⠁⠘⣿⣿⣿⣿⣷⠀⠀⠀⠀⠘⢠⣤⣤⣤⣤⠀⠀⠀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡃⠀⠰⡶⠐⠂⠀⠀⠀⠻⠆⠀⠀⠀⢻⡿⠁⠀⠻⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⣤⣀⣀⣀⣴⣶⣶⣤⡀⣠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠤⠀⢸⡿⠀⠀⠀⣸⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡿⠀⠀⠀⣀⣀⣤⣤⣶⡶⢸⠇⠀⠀⢀⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣀⣰⣿⣿⣦⣀⣠⣿⣿⣿⣿⣿⣷⣾⠛⠛⠉⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠀⠁⠀⠒⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⡄⠀⣶⠾⢿⣿⣿⣿⣿⣿⠈⠿⣿⣿⣿⡿⠋⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣦⡼⠂⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡀⠀⠈⠻⠏⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠢⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⣠⣶⠾⠛⠋⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⢿⣿⣧⠀⠀⢀⣼⣿⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠐⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⢸⣿⣿⡆⢠⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⢀⣾⣿⡏⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣴⣿⡿⠋⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⡀⠀⠀⢨⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣷⣾⢿⣿⣿⣿⣿⣿⣿⡶⠿⣿⡿⠿⠛⠃⠀⠁⠚⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣇⡀⠠⠀⠀⠀⠶⠀⠀⠀⣻⣇⣀⣀⣤⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣾⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡤⠶⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⡀⠀⠈⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⣴⢤⡉⠑⠋⠀⢹⣿⣦⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢳⠈⣆⣰⣶⣷⣿⣿⣥⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢤⣄⣀⠘⢿⣿⣿⣿⠿⠍⠤⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣻⠟⠉⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⡈⠁⠀⠀⠀⠀⠀⠠⠀⠐⡾⠃⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⠀⠀⢿
⠀⠀⢀⠀⠒⢦⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡠⠂⠀⠀⠀⢸
⠀⠀⠉⠛⠦⡼⢿⣀⠄⡀⠈⠙⠒⠒⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⢀⠈⠠⢾
⠀⠀⠀⠀⠀⠀⠀⠈⢰⣖⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣴⣿⣿⣧⠄⠴⠇⠀⠀⠁⠀⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣁⠈⣷⣀⢀⡠⢂⣀⣠⣤⣶⣿⣿⣿⣟⡿⠃⠁⢀⣠⡤⠂⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠓⠹⠟⠝⡿⠉⠀⠙⢛⠋⢉⢉⠉⠉⠀⠈⠈⠈⠉⠂⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢺⡁⠀⠀⠀⢵⣶⡎⢱⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢸⣦⣬⡝⠓⠂⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⢀⡄⣢⣼⠇⡿⡇⢸⡂⠀⠀⠀⢀⡀⠔⠊⢂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⠿⠋⠁⠀⣤⣿⢸⣷⢪⡀⡇⠉⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⣿⣿⣷⣿⠟⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⣿⣿⡿⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣰⡇⢀⠠⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣬⠈⣱⣄⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢱⠱⠋⢰⡄⣀⡀⠀⠀⢠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⠀⠰⠀⠁⠈⠁⠈⠉⠉⠇

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠩⠉⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡟⠒⠒⠒⡖⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⠀⠀⠀⠐⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⠁⠀⠀⠀⡜⠀⠀⢸⣅⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣴⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉⠉⠉⠉⡉⠉⠉⠉⠉⠉⠉⢸⠃⠀⡜⠀⢰⠇⠀⢀⡿⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠲⣄
⠀⠀⠀⠀⣠⠞⠉⢀⣀⣀⡈⠙⠦⣤⠤⠤⠒⠒⠒⠂⠀⠀⠀⠀⢀⠈⠴⠄⠀⠀⠀⠀⣁⠤⠚⠁⠀⡜⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠐⠒⠋⠀⢀⣀⣀⣉⠷⡄
⠀⠀⢀⠞⠁⡠⠊⠉⠀⠀⠈⠑⢄⠈⢷⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠙⣆
⠀⢠⠏⠀⠜⢀⠐⠈⠉⠉⠒⢄⠀⢳⠈⣧⠀⠀⠀⠀⠀⠀⠬⠕⠒⠒⠒⠒⠒⠒⣀⣉⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣒⣐⡰⠶⠶⠖⠒⠒⠂⠀⠀⢸
⢠⡏⠀⠀⠠⢁⠔⠂⠉⠀⠀⠀⢃⠈⡄⢸⡄⠀⠤⠀⠀⠀⠤⢄⣀⣀⣤⠤⠖⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈
⣼⠀⠐⢀⠃⠀⡠⢊⠅⢀⢠⠰⡄⠀⠀⠀⡗⠒⠒⠀⠀⠀⠐⠊⠉⠀⠀⠤⠖⠊⢉⣉⣑⠒⠒⠒⠒⠒⠒⠒⠒⣀⣀⡀⡀
⡏⠀⠀⢸⠀⠀⠇⠰⠄⠜⢸⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⠈⠥⢒⠒⠈⠁⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠈⠃⠉⠉⠀⢀
⡇⠀⠀⠀⠀⢢⡀⠀⠤⠒⣁⠀⢠⠀⡀⢀⡇⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠨⠭⢉⣀⣉⡭⠀⠀⠀⠒⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⠷⠒⠒⠒⠀⠀⠀⢀
⢷⠀⠸⠀⠠⡀⠉⠒⠒⠊⠁⠀⠂⢀⠃⣸⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠤⠤⠤⠤⠴⠶⠒⠓⠒⠒⠈⠈⠉⣁⣀⣀⣀⣀⣀⣐⠶⠖⠚⢈⡉⠤⠤⠒⡾
⠘⣆⠀⢧⠀⠈⠒⠤⠤⠤⠒⠁⢀⠌⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣩⡍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠐⠂⠀⢀⣈⣁⣀⣀⡀⠀⠀⠀⠀⠀⢠⠞
⠀⠘⢦⡈⠳⢄⣀⠀⢀⣀⠤⠐⠁⢠⠞⠀⠀⠀⠀⠀⠠⠤⠀⠤⠤⠒⠈⠀⠀⠀⠐⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠈⠀⠀⠀⠀⠰⠶⠖⠀⢀⣠⡴⠋
⠀⠀⠀⠙⠲⢤⣀⣀⠀⢀⣀⣠⠖⣃⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠤⠤⠬⠭⠭⠭⠭⠭⠭⠤⠤⠤⠤⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠹⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣛⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⡗⠖⠚⠉⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⣠⠄⢀⠀⣿⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⣥⡶⣄⠉⠈⠉⠀⢈⣿⣿⡶⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡂⠩⣀⣠⣶⣶⣾⣿⣿⡀⠀⠀⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠀⠀⣿⣮⣿⣿⣿⣿⣟⢋⡉⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⣶⣮⣿⡿⠟⠋⠉⠁⠀⠉⠷⣄⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⢠⣴⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⣁⣽⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠈⠉⢂⠀⡀⠀⠐⢧⡀⡀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠊⠀⠀⢀⠀⢸⠁
⠀⠀⠀⠉⠳⢦⠼⢷⣀⡠⢀⠀⠈⠉⠛⠒⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠠⠁⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⣤⣾⣿⣿⣄⢀⡴⡆⠀⠀⠘⠁⠀⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣴⠀⡀⣶⡀⠀⠀⢀⠄⠄⠀⢐⣀⣴⡾⠿⢛⢻⢿⣫⠁⠁⠀⠀⣀⡠⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⢄⣻⡷⠶⣾⣭⠖⠚⢻⣿⣽⠿⠿⠿⠃⠺⠋⠁⠠⠤⠴⢟⡉⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⡇⠀⠀⠑⠄⣰⠇⠀⠀⠈⠠⡄⢈⡠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢃⠀⠀⠀⠀⢿⠀⠀⠀⡄⠐⡿⣿⣂⢸⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⠀⠀⢀⢰⣧⣮⡆⡿⢙⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⢀⣀⡧⣸⠷⣿⡟⠀⢸⣦⢸⠇⡀⠀⣇⠀⠔⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣴⣿⡽⠛⠁⢸⠀⣷⢁⢼⣿⢸⡄⢇⠄⡉⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢠⣸⢹⣿⣾⡀⣿⣬⠟⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣜⣿⣿⣇⠟⠁⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⢛⢎⠀⠀⠀⠀⠀⠄⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡇⣬⠇⠀⠘⠀⠀⠀⠀⠈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡼⡏⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⡇⠉⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠡⠄⡿⠉⢦⣄⡀⠀⠀⠀⠀⠀⣠⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⢠⠈⢿⠉⡓⠒⠀⠐⠚⠅⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠛⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣠⣅⡀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠘⠛⣻⠇⠲⠾⠿⠷⠀⢹⡿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⢀⡍⠀⠠⠀⠀⠀⣰⣟⡄⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣯⠾⢃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣷⣶⣶⣷⣶⣾⣿⡯
⠀⠀⠀⠀⠀⠀⣠⣾⣷⣌⢿⡟⠁⠀⠈⢆⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣶⣶⣤⣿⣿⣿⡿⡁
⠀⣀⣠⣤⣴⣾⣿⣿⣿⠋⠀⠘⢆⠀⠀⠈⠣⡀⡠⠔⢺⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡟⢆
⠀⠈⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⢳⣀⡀⠀⠙⣄⠀⠀⠀⣻⠈⢻⣿⣿⣿⣿⠿⠋⢀⣾⠀⠈⠳⣀
⠀⠀⠀⠛⠿⢿⡿⠃⠀⠀⠀⢀⡴⠋⠈⠉⠓⡄⠈⠳⠄⠀⠀⠀⠀⠈⣽⣯⣉⣣⣤⣾⣿⡀⠀⠀⠈⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⣤⠇⠙⢿⣿⣿⣿⡟⠉⡆⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⡁⠀⠀⠀⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⠈⠲⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠈⠙⠢⠀⠀⠀⠀⠁⢦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠅⢀⠀⠀⠀⠀⠀⣖⠀⠀⠀⠐⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠰⠚⠅⠀⠀⠀⠀⠀⢛⣷⡀⠀⠀⠀⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠗⣀⣴⣒⠒⠶⣤⡀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡄⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣾⣩⠞⢁⣠⣤⣤⣤⣙⣦⣴⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡜⠛⠁⣾⢦⠀⠀⠀⠈⠂⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠿⣡⠞⣩⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠈⢣⡀⠀⠀⣀⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⣴⠋⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠃⠀⠀⠑⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠰⣌⣷⡀⠳⠀⠔⠴⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣤⣤⣤⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠿⠿⠛⠛⠋⢉⣁⣀⡀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⢠⣶⡾⠿⠟⠛⢻⣿⡆⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣶⠄⠀⠀⢿⣧⡀⠀⢀⣼⡿⠃⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡋⠀⢾⠿⢿⠆⠀⠀⠀⠙⢿⣶⠿⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⣀⣿⣏⠙⢿⣦⡀⠀⠀⢀⣶⣿⣦⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⣧⠀⠙⢿⣦⡀⠀⠛⠟⠃⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⡀⠀⠀⠙⢿⣦⡀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⠟⠁⠙⢿⣦⣄⠀⠀⠙⣿⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⠟⠁⣠⣶⠆⠀⣨⣿⢷⣦⣾⠟⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⠟⠁⣠⣾⠟⠁⣠⣾⣿⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⠟⠁⣠⣾⠟⠁⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠁⠀⠈⠛⠁⣠⣾⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠁⠀⠀⠀⠀⣠⣾⠿⠅
⠀⠀⠀⠀⠀⠀⣦⣾⠟⠁⠀⠀⠀⠀⣠⣾⠿⠅
⠀⠀⠀⠀⢴⣾⠟⠁⢀⡄⠀⠀⣠⣾⠿⠇
⠀⠀⠀⢀⣿⡇⠀⢤⠀⠀⣠⣾⠿⠇
⠀⠀⠀⠈⠙⣿⣦⣀⣠⣾⠿⠇
⠀⠀⠀⠀⠀⠉⠉⠙⠻⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡁⠙⣟⠿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⢂⡀⠉⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⣾⠏⢦⠀⠀⢣
⠀⠀⠀⢀⠀⠀⠀⠀⠇⢂⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢸⡆⠱⡄⠀⢆
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⣹⡇⠀⠙⡄⠀⢣
⠀⠀⠀⣧⢠⠀⠀⢀⠀⢠⣸⣂⣴⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣧⣀⣀⠑⣀⣸⡀
⠀⠀⠀⡇⠀⠀⠀⢸⢰⣬⣍⠻⣿⠋⠢⠤⢀⠀⠀⣠⡤⠀⢀⣤⠾⠏⢹⣷⣬⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⢦⣀⡠⠊⠀⠀⠀⡴⠙⠛⠛⢛⡹⣿⣿⡇
⠀⠀⠀⡀⡀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡰⠁⢀⣠⣶⣿⣿⣿⣿⣷⡀
⠀⠀⠀⡇⠃⠀⠀⠀⠀⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⣰⣾⣿⡟⠙⡏⢻⣿⣿⣧
⠀⠀⠀⣷⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢰⣾⣿⣿⠟⠀⠀⠈⠄⠻⣿⣿⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⣿⡟⠀⠀⠀⠀⠀⠈⢆⠻⣿⣿⣦⡀
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡸⣿⣿⡄
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢈⣆⠀⠀⠀⠀⠀⠀⠀⣸⢰⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⣿⣿⡆
⠀⠀⠀⡟⠀⠀⠀⠀⠀⡀⠀⠉⢈⠀⠀⠀⠀⠀⠀⣠⣵⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄
⠀⠀⢠⡿⠀⠀⠀⠀⠀⠃⠈⣦⡌⠀⠀⢠⡟⠀⢠⣷⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣄⣤⢴⣶⣿⣿⣿⡇
⠀⠀⠿⠁⡀⢧⡀⠀⠀⠠⣀⠀⠠⡄⠀⠈⢿⣶⣾⡿⣿⣿⠾⠷⠾⠷⠆⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⠀⠀⠈⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣝⣭⣍⣽⣫⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣶⣶⣶⠤⣤⣠⣶⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⣯⣥⣾⣿⣿⠿⠟⠿⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣡⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⣿⡟⠁⠀⢘⡿⠿⠿⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣀⠀⠀⠈⠁⠀⠀⠀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡈⠢⠀⠀⠀⠀⠀⢻⣯⣝⢿⣿⣿⣦⡀⠀⠀⢠⣤⣶⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡶⠿⠀⠀⠀⠀⠈⢉⡛⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⣤⣀⣾⡟⠀⡀⠀⠄⠀⠀⠀⠀⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣯⣿⡛⠋⠂⠀⠀⠄⠀⠀⠀⢻⡿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡟⣹⢿⣿⠟⠁⠙⣷⣤⠀⠀⠲⠀⠄⠀⢀⣼⡇⠈⠉⠛⠎⠙⠛⢿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠐⠩⣿⣫⢫⡿⠋⠏⣼⢏⡿⠃⠀⠀⠀⠈⢿⣧⠀⠸⠆⢀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣅⠂⠜⣡⠟⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⣄⠘⣾⣿⣗⡳⣤⣀⣀⡀⠀⠀⠀⠀⣷⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈⠛⠖⣰⠋⠀⠀⠀⠀⠀⠀⣀⣠⣾⣧⠀⣿⠀⢿⣿⣯⣿⣿⡷⢠⣞⡀⠀⠀⠀⢻⣗⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⣀⣤⣶⠿⣿⡉⢻⣿⣆⠘⡧⠸⡿⣿⣽⣿⣦⣸⣿⣿⣦⠀⢀⣾⣿⢁⣀⢐⣤⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠋⣷⣶⡟⢿⣿⠿⣿⡀⢳⡀⢠⣼⣿⣿⠿⣻⡉⢿⡍⠁⢾⡿⠻⣷⠻⡎⠱⡆⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢲⣦⣼⡛⠿⡧⠀⢹⣧⣽⣷⠀⠁⠀⢃⠘⣿⡄⢹⣅⣀⣻⣄⢀⣿⣀⣸⣄⢹⣄⢠⣤⣴⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⢸⡿⠻⣆⣠⣿⣶⣿⣿⡿⢿⣇⠀⠀⠀⠦⠼⣿⠟⣿⡟⠻⣿⠈⢻⣿⠛⢿⠛⢿⠉⢻⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠷⣿⣧⣀⣸⡿⠙⡏⠀⢸⣿⣤⣿⠀⠀⠴⠶⠒⠛⡀⢸⣧⣄⣿⣦⡄⣿⣆⣼⣷⣾⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣴⡏⣨⡿⠏⣀⠀⣾⣷⣿⣿⣿⣿⠀⠀⡴⠐⠛⠁⠹⠿⣿⠋⢻⣿⠙⢻⡟⢛⣿⠙⣇⢀⣷⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣹⠋⢻⣿⣧⣀⣿⣆⣉⣹⣇⣼⣿⡟⠀⠀⠀⠈⠉⣉⢈⢡⣿⣤⣤⣿⣦⣸⣧⢠⣿⣖⣿⣹⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡿⠿⣶⣿⣤⣽⣿⣷⣿⣿⣿⣿⡿⠛⡡⠀⠀⠀⠀⣸⣿⣆⠹⣿⡟⠉⣿⡏⠛⣿⠉⣿⡇⣻⠉⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣾⠷⢶⣿⢿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⣠⣿⡟⠿⢃⡈⠣⣄⣿⡇⡤⣿⣴⣿⡷⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠷⣶⣿⣤⣞⣉⣈⠉⠛⠛⠋⠁⠀⠀⠀⣠⣾⠟⢠⣶⣾⣿⣿⣿⣦⡀⠉⠓⢌⠛⢀⡀⣿⢀⣿⣀⣥⣼⣿⠿⡆
⠀⠀⠀⠀⠀⢀⣼⡿⣶⡿⠗⢀⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣻⡿⠁⣴⣿⣿⠛⢛⣿⠉⢹⣷⣄⠀⠀⠑⠈⠙⠅⢺⡿⠻⢿⣿⣿⣤⠈⠂
⠀⠐⠀⠀⠐⣿⡟⣠⣿⢇⣤⡾⠁⠀⠀⠀⠀⠀⠀⢐⣦⣀⡼⠋⣠⣾⣿⣿⣿⣥⣼⣿⣶⣿⣿⠟⢳⡄⠀⠀⠀⠀⢸⣧⣦⣼⣷⢿⣿⡆
⠀⠀⠀⠀⠸⣿⣧⣾⢿⣶⣿⡇⣄⣃⣠⡄⣰⣆⣿⣿⠙⠃⣠⣾⣿⣿⠿⢿⣿⠛⢹⣿⢀⣠⣿⣤⣤⣿⣦⠀⠀⣰⣿⣇⣸⡏⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠛⣿⣯⢀⡙⣿⡇⣻⣿⣿⣷⣽⣿⠗⠉⠄⢾⣿⣯⣿⣷⣶⣾⣿⣿⣿⣿⠛⠻⣿⠁⠘⡟⠙⣷⢰⣿⡿⣿⣿⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⣿⣿⣯⡛⠿⠟⠉⡠⠆⠁⠀⢸⣿⠙⣿⡟⠛⢻⡏⠁⢸⣯⣀⣰⣿⡤⣴⣿⡄⢠⣿⣟⣁⣼⡟
⠀⠠⠀⠀⠀⠀⠀⠈⠙⣤⣵⣬⣿⢻⡖⠛⠃⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠹⣿⠁⠈⠇⠀⠘⠛⠉⠛⢻⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢿⡼⣟⠙⣿⡁⢹⣇⠀⠀⢿⠀⠀⠜⠀⠀⠀⠀⠉⠉⠀⠿⢺⡟
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠂⠻⠻⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⡀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣟⠤⣲⡀⢠⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⠛⠻⠿⠃⢾⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⢣⡉⡢⠦⡆⠈⠈⠷⢤⣀⣀⡠⣄⣠⣄⠀⠀⢿⣷⡝⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣭⢃⣀⢀⢀⣀⣀⡤⠤⠭⠶⠈⠉⠀⠀⠀⠘⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⡿⠻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣾⡿⡿⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⠿⣻⢦⡇⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣧⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⢐⡿⣿⣿⣦⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢰⠋⠀⠘⢿⣿⣧⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣣⠈⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡏⠀⠀⠀⠈⢻⣿⣯⡆⠀⠀⠀⠀⠀⠀⣸⢿⣿⠋⡙⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⢰⢺⣿⡿⠀⡇⠀⢀⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢠⡯⠃⢠⡿⠀⠃⠀⠀⠓⠄⠈⠳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⠀⡠⠶⠁⣀⣠⣴⡓⠀⠀⢣⠀⠠⠤⠕⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢼⣿⡟⠁⠀⠀⠀⠀⠀⢀⣀⣸⣿⣟⢢⣴⠶⠐⠛⠉⠉
⠀⠀⠀⣀⣀⣠⣴⣞⡿⠟⠛⠓⠀⠀⠉⠉⠙⠉⠉⠛⠟⠛⠁
⠀⠀⠉⠉⠉⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠠⠃⠂⠰⠂⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣟⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠖⣲⢶⠀⣶⡶⢺
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡿⣿⡀⠀⣈⣀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣄⣀⣤⡀⣀⣤⣾⣇⠉⢻⣿⣿⣟⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⢻⡟⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢿⠛⠛⠛⠻⣿⡟⢸⣿⡿⠿⠿⠿⠟⠋⡇⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢁⣼⡆⠀⢰⣾⣿⣦⣿⣟⠣⡀⠀⠀⠀⠀⢀⣌⣻⡿⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⢫⠃⠀⣨⣿⣿⣻⣿⣿⣿⣿⡁⡀⣠⣴⡿⢿⣿⣷⣤⡆⠈⠛⢦
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠃⠀⣿⣿⡿⢿⢛⠋⢀⣯⣏⣸⣴⣿⣄⣸⣿⣏⠸⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣷⢾⣿⡏⠹⣿⡟⠙⠛⣿⡓⠻⣿⣿⣿⠀⠀⠀⠀⠸⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠛⠀⠀⢠⣆⣀⣻⣿⣿⣈⣭⣧⣤⣽⣷⣤⣤⣼⣧⣤⣿⣿⣿⣦⣀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡀⠀⠙⠭⢹⣿⣿⡏⢛⣻⡉⢙⣉⡉⠉⢻⠋⢹⣿⣿⢻⠙⣧⡀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢳⣈⠠⡉⢸⣿⣿⡷⢾⣿⡏⠻⣿⡿⣿⣿⣷⣾⣿⠿⠿⠀⠀⠙⠷⠀⢀⡀⠀⠘⡂
⠀⠀⠀⠀⠀⡼⠃⠀⠀⠰⣾⣿⣿⠉⣽⣿⣿⣷⢸⣿⣦⣴⣾⣦⣤⣾⡀⠀⡀⠂⠀⠀⠀⠀⠀⠄⢰⡇⠀⢠⠇
⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⣈⣿⢙⠀⢾⣿⣿⣿⢹⣿⣛⢙⣿⠋⠋⣻⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠘⡟⢀⡏
⠀⠀⠀⠀⠀⠻⡄⢀⣤⣶⣿⠇⢸⠀⠘⣿⣿⣿⠻⣿⡿⢿⣿⡷⠿⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋
⠀⠀⠀⠀⠀⠀⠙⠚⠛⠛⠁⠀⣸⠀⠰⢸⠛⠻⣯⣽⣷⣼⣿⣦⣶⣾⡇⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣈⣧⢰⣶⣾⣿⣿⠿⣤⣭⣽⣦⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣣⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠩⢝⣳⣿⣿⣿⣿⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡻⠷⠘⠲⠦⣤⣄⣘⠛⠛⠋⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⡛⠆⠀⠈⠙⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⣀⣄⡀⠈⠋⣴⠷⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡿⠀⣀⣴⡾⠿⢾⣿⣽⡿⢻⣷⠀⠶⠿⠂⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⡟⣡⡶⠷⣶⡿⣫⣶⠟⣻⡇⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠘⣿⠀⠙⣿⣟⣡⣾⣿⣅⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣿⠀⢀⣿⠀⠀⢸⣿⣋⣤⢈⣿⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣠⡾⠋⣀⠿⠃⠀⠀⠀⠉⢉⣥⡾⠋⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢁⣾⠟⠀⠀⣤⣴⠶⠟⠛⠉⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠁⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠈⠉⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠘⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣦⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⡀⠄⠈⠙⠙⠀⠀⢀⡐
⠀⠀⠀⠀⠀⠰⠖⠶⠦⣄⡀⠀⠀⠀⢀⡤⠀⠀⣠⠄⢰⣶⡂⠂⠀⠀⠀⢁⠀⣁⠤⠤⠒⠂
⠀⠀⠀⡔⠘⢦⠀⠀⠀⠀⠈⠂⠀⠀⠀⠨⠄⠀⠴⠆⠠⠈⠉⠀⢀⣀⠩⠝⠉⠁
⠀⠀⠀⣇⠠⡄⢀⡹⠂⠀⢀⢳⡤⢐⠢⠍⠀⠠⠀⣀⠀⠠⠒⠀⠈
⠀⠀⠀⠡⢆⠤⠌⠅⠀⠀⠈⠠⠀⠀⢀⠔⠀⠀⠉
⠀⠀⠀⠀⠘⢦⣀⠀⡀⠤⣢⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠁⠁⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⢦⠦⢰⡤⠄⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⡞⠷⡂⢀⡄⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠄⠘⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣇⡇⠀⢠⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡟⠛⠷⣄⠀⠀⠀⠀⠔⠊⠷⠀⠚⡼⠁⠀⠈⠐⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡁⠀⣼⣿⣧⠀⠀⠀⠀⠀⠱⣤⣾⠁⠔⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡆⠂⠀⠀⢠⣆⢿⡏⣤⡆⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⣠⣾⢽⣿⣿⠏⡼⠠⣲⣦⣫⣤⣻⣼⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⣶⣾⣤⣶⠿⠦⣿⣿⣿⣿⣿⣣⠴⠯⠶⢙⠟⣧⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⠀⡀⣀⠀⡰⠊⣭⣽⣵⣿⣿⣿⣿⡿⠿⠟⢋⠵⠛⠗⠒⠚⠛⠻⣿⣯⠩⢷⢆⣿⣷⡶
⠀⠀⠀⠀⠀⠔⠒⠷⠛⠛⠉⢉⡉⢛⣿⣑⣾⣿⣿⣡⣾⣿⣿⣷⠆⠀⢠⡟⠀⠀⢀⣀⣠⣾⣿⣿⣿⣷⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠷⣽⣶⠶⣿⣞⣻⣿⣿⡿⠿⠿⢣⣾⣾⣷⣾⣤⣄⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠮⠽⠟⠛⠉⡉⠀⠠⣤⣾⣧⣧⠞⠛⢉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠿⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠶⠾⠚⠯⠽⢿⣟⣛⣩⠉⠩⠝⠉⠙⠓⠸⠟⢡⠠
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣶⣶⣿⣶⣿⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠂⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣠⣾⣿⣿⡿⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠟⠁⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⠶⣽⡢⢾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⣍⠊⢟⡁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⡿⢁⣴⣿⣿⣾⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⢀⡿⠋⠀⠀⠑⣆⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣿⣷⣿⣯⢟⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠈⣠⢖⡠⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣟⣿⣿⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⢠⠈⠑⠋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣴⣿⣿⣿⣿⢻⣷⡆⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⡿⣷⣿⣧⣭⡛⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣺⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⠢⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣴⠉⠉⠲⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡟⠛⠋⠁⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢩⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂⠈⢹⣿⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠑⢢⢄⡀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠀⠁⠈⠁⠀⠸⣯⡀⣼⠇
⠀⠀⠀⠀⠺⣿⣿⣷⣦⡶⡄⠀⠈⠹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⣤⣤⣤⣾⣿⣿⣹⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃⠙⠆⠀⠀⠀⠈⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠍⠉⠉⠁⠀⠀⠉⠙⠢⠀⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠳⣄⡀⡈⢢⣀⣀⣠⠤⠶⠒⠚⠉⠋⠙⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠛⠛⠋⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡝⠉⠙⠻⠿⠿⠿⢦⣄⣀⣀⣀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠑⠉⠉⢿⣏⡡⢂⣄⣰⣦⣴⣦⣤⣤⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⣀⣀⣤⣄⣿⣿⡟⡿⠿⠛⠿⣿⣿⠏⣛⡌⠀⠀⠙⠎⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠋⠉⠈⠉⠛⠛⠃⠀⠀⠀⠀⠈⢻⣀⣾⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢃⣴⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⣠⣶⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣵⣟⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⠟⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠠⠟⠁⣀⣤⣶⡶⠶⠶⢶⣦⣤⡀⠀⢻⣄⠈⢳⠄⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⡀⠀⠀⢀⣤⣦⠄⠀⠀⠀⣀⣠⣿⡿⠟
⠀⠀⠠⠴⠖⠋⠉⢁⣀⡀⠀⠀⠀⠈⢻⡁⠈⠋⠉⠐⠁⠙⠀⠀⠀⠀⠀⠀⠂⠐⠀⠔⠃⡀⠘⠁⣠⠔⠻⠋⠁⢀⣠⡴⠾⠛⢋⣁⣀⣀
⠀⠀⠠⠐⠒⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠤⠒⠉⠀⠀⠄⠐⠈⢁⠤⠴⠖⣛⣛⣭⠭⠤
⠀⠀⠀⠉⠉⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢻⣿⣷⣾⡭⠅
⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣠⣄⣠⣤⣀⣀⣤⣀⣤⣤⣀⣠⣄⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣽⣿⣿⣿⣟⣃
⠀⠀⠘⠀⠛⠃⠙⠟⠟⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤
⠀⠀⢸⣷⣦⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⣁⣀⣀⣀⣒⣀⣒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠂⣀⣀⣀⠀⢀⣤⠀⠀⠒⠒⠛⠛⠆
⠀⠀⢸⣯⠑⠢⠀⠀⢨⠁⠀⠐⣦⠀⠀⠉⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣀⡀⠀⠀⠀⠉⠀⠐⠒⠀⠠⡀
⠀⠀⢸⣷⣒⣛⣁⠀⠚⠀⠠⠞⠛⠀⠀⠤⠀⠋⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠀⠀⠰⣿⣉⣁⡀⠀⠀⠀⣴⠚⠛⠙⠛⢻⣶⠀⠈⠂
⠀⠀⢸⣿⣯⣷⣒⣀⣀⣀⠀⠀⠄⠠⠠⠤⠐⡒⠒⠒⠒⠒⠀⠀⠀⠐⠲⠒⠒⠒⠤⠤⠤⠄⣀⣀⠈⠑⠢⢀⣀⡈⡛⡛⠻⠛⣁⡠⠄
⠀⠀⢸⣿⣿⠿⣿⠷⣶⡶⠂⠙⠁⠀⢈⣤⣬⡭⠥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠍⠐⠠⠤⢄⣀⣤⡍⠉⠁
⠀⠀⢸⠟⢋⣿⣿⠻⠖⠀⠀⠀⠀⠀⠐⣒⣛⠓⠊⠐⠂⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢺⣿⣿⣿⣿⣶⣶⣶⣤⣶⣶⣶⣾⣿⣿⣾⣷⣶⣶⣦⣶⣤⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠀⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠬⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣖⠂⠀⠀⣰⣶⡆
⠀⠀⠀⠀⠀⠀⣙⣛⣓⣐⠒⠒⠒⠒⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠁⠀⠀⣈⣻⣿⠃
⠀⠀⢀⣀⡀⠒⠒⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠀⠀⠉⠒⠶⣦⠀⠀⠀⠀⠀⠀⢉⣉⣙⣻⡿⠗
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡿⣿⣿⣿⣟⡳⣤⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣳⣿⣨⣿⢿⣿⣿⣿⣄⠀⠀⠀⠀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠉⠉⠉⠙⠛⠷⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⡘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠒⢢⡀⠘⠛⣉⡉⠙⡀⠘⣿⣿⠿⣿⣿⡇⢀⣿⣿⣿⢛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠙⠃⠀⠉⠈⠉⠁⠀⠀⣿⠡⠂⣼⡏⠀⢸⣿⣿⠃⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠄⠀⣀⢢⣀⠀⠀⣠⣾⣿⠙⣰⡟⠁⠘⠛⢿⣏⢸⣿⣋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⠾⡟⠀⠀⠀⢀⠊⢙⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣤⣬⣭⣿⣿⣿⣿⣿⣿⠀⢷⡄⠀⠀⠀⠀⣸⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢸⡟⢢⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠈⡃⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⣿⣿⡏⢿⡿⣿⣿⣟⡁⠀⠀⠀⠀⠀⡗⠀⠀⠋⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠚⠁⠀⠀⠘⢿⣧⠈⠀⢀⠙⣟⣤⡢⠀⠀⢀⡼⠀⠀⠊⠀⠀⠀⠈⠉⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⣠⠛⠂⠛⠉⠛⢁⠔⠙⠇⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⣀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠢⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠶⠛⠓⠦⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣈⣁⣙⠳⣤⣼⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢐⣄⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⣤⡀⠉⢀⡇⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣽⣿⡇⠚⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠁⠀⢠⡿⡇⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠒⠄⡀⢌⠑⠀⠰⣄⠀⠀⠀⠀⠀⢈⣤⣶⡂⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠲⣤⣄⣿⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣈⠐⠂⡀⠄⠁⣀⣀⠀⠉⠲⢀⠀⠠⣿⣟⣥⣤⣀⡀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⣼⠁⠂⠁⠀⣀⣀⣙⣺⣾⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⡟⠈⢻⡍⠁⠀⠀⠀⠀⠈⠐⠛⠛⠛⠒⠦⠤⣄⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠁⢀⣤⡴⠋⠉⠀⠀⠀⠈⠉⠁⠒⠤⣿⠀⠀⠀⠀⠀⢀⠀⠀⠈⢉⢿⣿⡇⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙
⠀⠀⠀⠀⢀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢰⣆⣸⠃⠀⠀⠀⠀⠙⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⡆
⠀⠀⠀⠀⠠⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣹⡶⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡈⠀⣀⣈⣀⣀⣀⣤⣤⡴⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⣠⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠆⡆⠀⠀⠀⠀⠀⠆⡎⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣭⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣣
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣴⣾⣿⣿⡿⠓⠀⣤⣻⡟⢓⣀⠈⠀⠀⠀⠀⠀⢀⡈⢿⡿⣠⠀⠴⣶⣶⣶⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣧⣾⣿⣿⣿⣷⣏⠀⠀⠀⢹⣿⠀⣋⠿⣷⣦⡆⢰⣤⣾⣿⡏⠀⣿⡟⠀⠀⠈⢳⣿⣿⣿⣿⣮⣱
⠀⠀⠀⠀⠀⠀⣴⣓⣿⣿⣿⣿⡟⢟⠀⠀⠀⠈⣿⣄⠉⠉⠀⠘⠃⠘⠃⠉⠉⠉⣠⣿⠁⠀⠀⠀⢿⢿⣿⣿⣿⣿⠿⣄
⠀⠀⠀⢀⣠⣼⣿⣿⣿⣭⣖⢯⡑⠄⠆⠀⠀⠀⣹⣿⣆⣀⢄⣗⠦⢴⣺⣠⣀⣰⣿⡏⠀⠀⠀⠠⠃⠌⡹⢯⣝⣿⣿⣿⣆⣀
⠀⠀⠐⡟⠛⢿⣿⣿⣿⠋⠀⠑⠙⢦⡘⢦⡀⠘⢷⣧⢿⣷⣿⢿⣻⣟⡿⣿⣾⢷⣽⡾⠃⠀⣠⠂⡠⢊⠔⠁⠹⣿⣿⣿⣿⠟⢷⠄
⠀⠀⠀⠈⠣⣈⠹⢿⠃⠀⠀⠀⠈⠢⡉⠳⣿⣤⣾⣋⠉⢿⣿⣶⣶⣶⣶⣿⡿⢉⢻⣷⣤⣺⡵⠋⠐⠁⠀⠀⠀⠹⣿⠟⠃⠐⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣠⠘⠁⢹⠙⣿⣿⡄⠀⠘⠙⠹⠛⠛⠁⠀⣼⣿⢻⡟⡏⠁⠉⠠⢀⠀⠀⠀⠀⠑⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣈⡤⠖⠀⠹⣆⠀⢸⠀⠀⣿⣷⣠⣀⢀⠀⠀⢠⢠⣸⣿⡟⠈⠁⠁⠀⠀⢀⡾⠓⠀⣀⠀⢀
⠀⠂⠀⠆⠄⠠⠴⢦⣾⠟⠤⠐⠄⠲⠼⠦⠀⠆⣶⣾⣿⣿⣾⣾⣴⣶⣶⣾⣿⣿⣶⣶⠀⠐⠠⠤⠾⠀⠄⠆⠀⠛⢶⡶⠄⠐⠀⠲
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣷⠀⠀⠙⢸⣿⣿⡌⠛⢿⣿⣿⡿⠟⢹⣿⣿⠃⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⡿⠟⠋⠀⠀⠀⠀⡿⠀⠀⠀⠀⣿⣿⣇⠀⠘⣧⢀⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠤⣾⣿
⠀⠀⠀⠀⠀⠀⡴⣢⠔⠉⠀⠀⠀⠙⣧⠀⠀⠀⠠⣻⣿⣟⡀⠁⣿⡌⠀⢀⣿⣿⣷⡀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠐⢤⡨
⠀⠀⠀⠀⠀⠈⣴⠃⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⢻⣿⣿⡅⠀⢸⡇⠀⠈⣿⣿⣿⠃⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⣼⠇⣀⠄⠀⠀⠀⠀⠀⣼⣿⣀⠀⠀⢸⣿⣿⠀⠐⢸⡇⠀⠀⣿⣿⣿⠀⠀⢰⣼⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢠⣿⠞⠁⠀⠀⠀⢀⣠⣼⣿⠋⠀⠀⠀⠀⠀⠈⠁⠀⠒⠧⠁⠀⣿⣿⣿⣀⣤⢾⣿⣟⣽⣖⢤⣀⠈⠙⠲⣼⣿
⠀⠀⠀⠀⢸⡏⠀⠀⠒⠚⣻⡛⣿⣿⡿⢠⣄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣟⡾⠉⠃⠀⠀⣀⠀⣾⣿
⠀⠀⠀⠀⢸⣇⡠⠀⠀⠸⠉⠓⣿⠋⠻⣿⠟⠋⠻⣮⡙⠶⣦⣄⡀⠄⢀⠀⠀⠀⠀⠀⠀⣿⡄⠘⠀⠀⠀⠀⢀⠈⢻⣿⣿
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠙⢷⣬⣙⠿⣿⣶⣦⣤⣤⣀⣀⣠⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⠁
⠀⠀⠀⠀⢈⣹⡷⠀⠀⡀⠀⠀⠀⢀⢠⣀⣀⠀⢀⣈⣉⣩⣭⣽⣿⣶⣭⣛⠻⢿⣿⣿⣿⣿⢳⠀⢀⠀⢠⣠⣤⣾⡏⡄
⠀⠀⠀⠀⠘⠸⣷⣤⡆⣇⠀⢰⠀⢸⢸⣿⣿⡿⠿⠿⠛⢛⣛⣋⣩⣥⣽⣿⣿⣿⣿⣿⣿⣟⣾⣶⣷⣶⢿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠣⠻⠟⠛⠿⢿⣥⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠓⠐⠒⠒⠒⠒⠒⠲⠦⣄⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡤⠴⠶⠶⠶⣤⠶⠶⠦⣤⣀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⠖⣫⣄⠄⣠⣬⠓⠀⠀⠀⠉⢳⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡞⠈⣄⣻⠿⣄⣻⡟⠀⠀⠀⠀⠠⣪⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠁⠈⠉⠁⠈⣍⠀⢀⣀⣀⠀⠀⣓⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⣀⠀⠀⠀⠀⠀⢈⡇⠀⣈⠉⠁⠀⠹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠉⠙⠓⠒⠛⠉⢀⡴⠃⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠓⠤⠤⠤⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢀⣠⠤⠀⠂⢴⠇⠀⠒⠲⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠄⢀⣀
⠀⠀⠀⠀⣰⠏⠀⠀⠀⢰⠏⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠺⡏⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⡏⠀⠂⠂⠀⡸⠳⠶⢤⣤⡤⠴⠶⠚⠛⠉⢹⠀⠀⠀⢻⠀⠀⠄⠀⠀⠀⠹⣄
⠀⠀⠀⢸⠇⠀⠀⡾⠰⠖⠒⠲⢶⣦⠀⡀⠀⠀⠀⠀⣿⠀⠤⣄⢸⡀⠀⠀⠀⢠⠀⠀⠈⠳⣄
⠀⠀⠀⢼⠀⠀⢠⡇⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠙⠇⠀⠀⠀⡸⠀⠀⠰⠀⢹⡄
⠀⠀⠀⢸⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣇⠀⢀⠀⠀⠀⢷
⠀⠀⠀⢸⠀⠀⠄⢹⡆⠀⠀⠀⠀⠀⠀⢳⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⡶⠷⠆⠀⠀⠘⣇
⠀⠀⠀⢰⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⠏⠀⠀⡀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠰⡀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⣰⣀⠀⠀⡠⢿⡀⠀⠀⠀⠀⠠⠀⡸⠀⢀⠀⠀⠀⠀⠠⢴⢚⠋⠢⡀⠀⣈⠇⠉
⠀⠈⠉⠉⠉⠉⠹⡇⠀⡟⠚⠶⠴⠄⣀⣀⣇⣀⣀⡀⠤⠶⢚⣉⣠⡏⠀⠀⢱⠋
⠀⠀⡇⣤⣤⣤⣄⣷⠀⢸⠓⠛⠶⠤⠴⢦⠴⠶⠾⠶⠐⠚⠋⠉⠀⢱⠀⠀⠀⢳⣚⡰⠆
⠀⡀⡙⠁⠀⠀⢨⡅⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⣾⡀⡀⠀⢈⣩⠴⠃
⡇⡙⠁⠀⠀⠀⠀⠱⠀⠈⣄⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠛⢋⠈⢄⠀⠀⠀⠀⠀⣠⡀
⠧⢀⠀⣀⠀⣀⣠⢆⡦⠀⢱⠀⠀⠀⠀⠀⡎⢳⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢣⠈⢆⠀⠀⠀⠀⣶⠈⠑⣄
⠀⠀⠉⠀⠉⠀⠈⠊⠀⠀⠀⡆⠀⠀⠀⢀⠁⠘⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢣⠀⢂⠀⠀⢀⡁⠀⠠⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢐⠂⠀⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢣⠀⠂⢀⡌⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣧⣀⣀⣸⠀⠀⠐⣇⡀⠀⠀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⡗⠛⠎⠀⠀⠀⠀⠀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠈⢇⠀⠀⠘⠈⠉⢉⡉⡑⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⡴⠶⠤⠶⢤⣤⡃⠀⠀⠀⠘⡀⠀⡄⠀⡠⢸⣃⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢹⡦⠖⠚⠉⠉⠉
⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡇⠀⡇⢀⠔⠋⠉⠉⠉⠙⠳⣤⠀⠀⠀⠀⠀⠉⠚⠋
⠀⠀⠀⠀⠀⠘⢦⣄⣀⡀⠀⠀⢀⣠⠟⠒⠒⠃⠀⠓⢯⡀⠀⠀⠀⠀⠀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⢉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣈⣙⢶⣤⣤⣤⣤⡔⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡿⢿⡿⠻⡿⢿⢿⠿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣾⣷⣾⣷⣾⣷⣾⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠿⣋⠉⠉⠉⠉⠉⠉⠙⠛⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⣽⣿⣀⡀⡎⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⢀⡿⠋⠀⢈⣸⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠉⠁⠹⣄⣛⣦⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣹⣀⣀⠀⣦⠀⠀⠀⠀⠀⢰⢀⠄⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⡇⠀⠠⢼⡇⠐⠒⠁⢀⢴⡿⢧⣄⠀⠈⠱⣄⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢆⡀⠀⠀⠈⡒⣤⡀⢀⡞⠁⠈⠙⠲⢼⡵⢃⣤⠂⣸⣶⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣽⣷⣶⣶⡏⠀⠚⠉⠒⣶⣶⣶⣶⡖⣰⣍⡉⢖⡽⢃⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢰⣿⣿⣿⣿⣿⣿⢆⣀⣠⣼⣿⣿⢏⣴⣿⣷⣾⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣷⣿⣷⣷⣇⡟⠀⠀⠉⠉⠓⠒⠶⠦⣤⣄⣀⣀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⠻⠿⣿⡿⠶⠦⢄⣀⠁⠀⠀⠀⠀⠀⠈⠙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠀⠀⠀⠉⠻⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢄⣀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⡀⠘⣿⣿⣿⣿⡆⠀⠀⢀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⣿⡿⠾⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠱⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠶⠦⢠⣄⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⣀⡤⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠉⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠉⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡤⠶⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⢁⣀⣀⣠⣀⣀⣀⡈⠉⠑⠦⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠶⢄
⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣠⠴⠎⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠱⠦⣄⠉⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡴⠊⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣰⠃⢠⠎⠀⠀⠀⠀⠰⠋⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⡄⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⣰⠇⢰⠇⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠸⡆⠀⠀⠀⠀⠠⣎
⠀⢠⠴⠋⠀⠀⠀⠀⠀⠀⠇⢀⠇⠀⠀⠀⠀⠀⠀⠈⠓⠖⠰⠊⠁⠀⠀⠀⠀⠀⠀⣠⠶⠶⠦⡀⠀⠸⠀⢺⠀⠀⠀⠀⠀⠀⠙⠦⡀
⡔⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠘⡄⠀⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢀⡴⠀⠀⡇⠀⡇
⠑⢦⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⣇⠀⢇⠀⠀⠀⠀⠀⠀⠰⠉⠁⠈⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡺⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠘⢇⠘⢧⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡸⠃⠀⠀⠀⠀⠈⢣
⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢳⡄⠀⠀⠀⠈⠲⠆⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡄⠉⠲⠦⣤⣀⡀⠀⠀⢀⣠⣤⠴⠖⠉⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠈⠑⠒⠂⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠉⠲⠂⣤⣤⣀⡈⠀⣀⣤⣤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⣠⠄⠀⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣠⠔⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢢⣄⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠒⠠⠄⠒⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣤⣤⣤⣤⣤⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣶⣶⣶⣶⣶⣦⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢰⣶⡖⣄⢶⡦⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⢰⣖⠖⠀⠀⠀⠀⠀⠀⠠⣶⢶⣶
⢸⣿⣿⡙⣼⣧⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢘⣿⣿⣾⢦⠆⠀⠀⠀⢀⣰⣿⢸⣿
⢸⣿⣿⣿⢿⣿⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠻⣿⢿⣧⣶⠀⠀⠢⠘⣿⣿⢸⡟
⢸⣿⣿⣿⣾⣿⢴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣻⣿⣤⣤⣀⡷⠗⣿⡇⣼⡇
⢸⣿⠟⣿⣿⣿⡬⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠿⢃⠜⠁⡽⣃⡂⠙⠇⣼⡇
⢸⣯⠍⣿⣿⣷⢂⣠⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⣿⣿⡟⠀⢸⣿⣶⢧⠟⠟⠃⠀⠀⣿⡇
⢰⡇⣣⣿⣿⣿⢸⣿⠀⢀⣼⣿⣿⡇⠀⡄⠀⡀⠀⠀⢠⠀⣿⣿⣿⡇⠿⢸⣿⣿⠠⢀⠀⣠⣄⠀⣻⠇
⢈⡁⠻⠋⠉⢩⡈⠛⠰⣿⣿⣿⣿⣦⣴⣷⡀⡅⠀⠆⣿⢺⣿⣿⣿⢡⣶⣀⣿⣟⣰⣬⣤⣬⣿⣷⣮⡃
⠘⢁⠀⢀⣠⣿⣿⣶⣦⣿⣿⣿⣿⣧⣴⣤⣀⣀⣀⣀⠀⠀⢈⠉⢉⣩⣍⣭⣉⠉⢉⡀⢩⣷⡿⣿⣿⠉⡁
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣶⠊⣷⣟⣿⣿⣿⣿⣾⣷
⢸⣿⣿⣿⣿⣿⣿⣿⠟⠀⠟⢉⣿⡿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠙⠭⣿⣿
⠀⢩⢿⣿⣿⣿⣿⠇⢀⣴⣿⣼⣿⣷⣿⡿⠿⣿⠟⠛⠛⠉⠉⣿⣿⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀
⠘⠈⠀⣬⡁⣽⡟⠀⠈⢹⢹⢛⡉⠁⠛⠛⣠⣀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⢀⣀⠀⣀⣉
⠘⠡⣾⣿⣧⣤⠀⠀⣠⣿⡬⣾⣿⠁⠀⠛⠛⠿⠷⠿⠷⠛⠂⠀⠐⠶⣾⣿⠿⢋⣡⣴⣶⣾⣿⡝⠉⣉⡘
⢰⣾⣿⣿⣿⡏⢀⣼⡿⠛⢲⣿⣿⣷⣷⣖⣢⣷⣾⡆⢀⣴⣶⣆⣀⣀⡀⣀⠀⠘⠛⠻⠿⠿⠿⠿⠿⢿⣿
⠸⠿⠿⠿⠿⠠⠾⠿⠥⠾⠟⠑⠿⠹⠿⠿⠏⠻⠟⠵⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠦⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠛⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠈⡆⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠘⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠘⢆⠀⠀⠈⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠢⡀⠀⠀⠘⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⢒⡀⠀⠀⠙⢤⡀⠀⠀⠀⠸⡀⢀⣀⡀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⣄⡀⠀⠀⠀⠀⢀⣿⡀⠀⡳⢶⡄⣤⠀⠉⠢⣀⠀⠀⠉⢉⠆⢇⣀⡤⣄⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢇⡊⡁⠈⠻⣦⠀⠀⠀⢸⡇⠻⣄⠀⠈⠳⠶⢶⣀⡄⠈⠙⠦⣄⡀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠳⣄⣀⡎⠀⠀⠉⠑⠋⠒⠮⢷⣄⣰⢏⠘⢄⡈⠷⣤⡀⠀⠙⠒⢲⡆⣶⣀⠀⠉⢁⣰⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠨⠁⠀⠀⠀⠀⠀⢀⠄⠀⠈⠑⠁⣀⠀⣉⡿⠮⠿⠛⠲⠶⠦⠭⠥⠾⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢤⡴⠖⠊⠁⠀⠃⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠖⠒⠛⠛⠛⢛⡿⠛⣄⠀⠀⠀⠀⡠⠂⠘⢝⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣼⡟⡂⠘⠁⠀⠀⠀⠘⣖⡝⣧
⠀⠀⠀⢀⣠⣤⡤⠤⣤⣤⣀⠀⠀⢀⡴⠋⠀⢀⣴⠟⠁⡗⡇⠀⠀⠀⠀⠀⠀⠹⡐⣸
⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠉⠙⠲⠋⠀⠀⣰⠞⠁⠀⢰⡇⣇⠀⠀⠀⠀⢀⣀⠤⠞⠃
⣰⠋⠀⣠⠞⠋⠉⠉⠉⠛⠲⣄⠀⠀⢠⡞⠁⠀⠀⠀⢸⡇⠈⠉⠉⠉⢻⡉
⡟⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠻⣆⠀⠀⠀⠘⣇⠀⢀⡶⠤⠤⠬⠗
⢧⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢿⡄⠀⠀⠀⠈⠳⣬⡇
⠘⣆⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇
⠀⠘⢦⡀⠈⠳⣦⣀⠀⠀⠀⠀⣠⡞⠀⢀⡿
⠀⠀⠀⠙⠦⣄⠀⠉⠛⠛⠛⠛⠁⠀⣠⠟⠁
⣀⣀⣀⣀⣀⣈⣙⣲⣦⣤⣤⣤⣖⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⠀⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠐⠒⠲⠤⣀⢠⠊⠀⠀⠀⢀⣠⣤⣤⣱⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠆⣠⠞⠙⣄⠘⢁⣴⠃⣤⣉⠛⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⠁⠀⠀⠘⣧⠺⡇⠀⠐⠈⠀⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣷⣤⢤⣀⡀⠫⣤⣇⢀⣠⣦⣤⣦⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢀⣸⢿⣿⣀⣀⣉⡉⣻⣿⣿⣿⣿⣿⣿⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠑⠻⣿⣿⣿⣿⣿⣿⣤⡤⠌⣉⡉⠙⢒⣶⠞⡀⠉⠑⢲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡛⠍⢿⣷⣄⠀⠉⢻⣿⣻⠏⢠⠸⣿⡶⣄⠈⠉⠉⠻⡀⠉⠒⠲⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠁⠚⡤⠐⠀⠈⣿⣆⠱⠦⠤⠁⠿⠞⠃⠁⣿⣷⡌⣣⡀⠀⠀⢈⠂⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡌⠈⠀⠓⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⡀⡀⣹⣿⠲⠐⢿⣄⣀⠎⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠠⢀⠁⠈⢠⠀⠈⠀⠀⣿⡇⠀⠂⠐⠀⠛⡇⠀⢸⣿⣂⠐⢎⠙⠏⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠄⡌⢀⠁⢻⠄⠀⠀⡀⢸⣷⠀⠀⠀⠀⠈⢱⠀⢸⣿⠄⡕⠠⣉⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠤⢀⡈⢀⣩⠴⠀⠃⢁⠈⣿⡆⠐⠀⠀⠀⢹⠀⢸⣿⠖⡄⠢⢾⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠩⠐⡀⠨⠀⢈⡻⡆⠂⢁⠀⠀⣿⡇⠀⡀⢁⠀⠨⠀⢸⣿⣹⠋⠓⠧⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡂⠀⠁⠀⠀⠀⡇⠘⣃⠀⡀⠄⢺⣿⠀⠀⢀⠀⠁⡀⢸⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⡐⠒⠚⠛⢇⠀⢻⠄⠀⢂⢸⣿⡀⠠⠠⠀⠤⠄⢸⣿⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢂⠀⠀⠀⢧⡨⠉⢨⠨⠼⣿⡆⠰⠐⠀⠲⡆⢸⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢻⣅⣐⠀⠒⡉⠃⠐⠘⠀⣾⣍⣝⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠹⡛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⡆⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣷⠀⠀⠀⠀⠀⢀⣀⣤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣄⠀⠀⡠⠚⠉⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠿⣿⣿⣿⣿⣞⠁⠀⠀⠀⣠⣀⣼⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⣴⡿⠷⠈⢻⣿⡏⣿⡿⣦⣄⣴⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⢯⣴⠛⢋⣈⣁⣀⠸⣿⣿⣿⣇⢈⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣴⣶⣿⣿⣷⣴⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⡟⠛⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⡭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠉⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⡴⠋
⠀⠀⠀⠀⠀⣀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡉
⠀⠀⣠⠾⢏⢫⣱⣞⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣯⠙⢌⣶⣏⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⠀⠀⠈⠙⢿
⣴⣛⣭⣿⣿⡿⠿⢀⣼⡛⠉⠁⠈⠛⢿⣿⡿⠿⠿⠛⠛⠁⠀⠀⣀⣠⣤⠗⢸⡆
⠘⢿⡿⠛⠻⠷⣆⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣆⣠⣼⠃⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣀⣀⣀⣀⣀⣐⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡟⣠⣾⣿⣿⣯⡀⢀⠔⠒⠉⠉⡆
⠀⠀⠀⠀⠀⠀⢀⢀⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠙⠛⠛⠛⠛⠉⠁⠀⣴⣿⠿⣿⡻⣿⡿⣧⣄⠀⣴⣴⣷⣦⡀
⠀⠀⠀⠀⡠⠋⠀⣈⠢⡠⠊⠀⠀⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣵⠚⣉⣀⢸⣿⣿⣙⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠘⣄⠡⠟⠈⣿⣿⢲⡀⢀⣠⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⣿⣾⣿⣿⣿⣿⣿⠿⠟⠉⠑
⠀⠀⠀⠀⠨⣶⠿⡷⠟⠉⠛⢿⡉⠉⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⠔
⠀⠀⠀⠀⠀⣯⣿⡷⠂⡀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢦⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠔⢊⡐⠴⠙⠉⠀⢠⣿⣧⡤⠀⠠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣅⣠⣾⠍⣿⡿⢿⣿⣿⣿⣿⣿⠿⠅⠀⢉⣻
⣤⣵⣾⠃⡄⠀⠀⠲⠟⠛⠋⠁⠀⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠿⠿⣅⢸⡇⠀⠀⠈⠉⠉⠁⠀⠀⣀⣈⣋⣸⠇
⠀⠀⠀⠑⠷⠄⡤⠤⠴⠶⣶⣶⣾⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠤⠤⠤⠴⠶⠶⠶⣿⣿⡿⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠂⠉⠉⠁⠔⡒⣔⣀⡠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠕⢁⠔⢈⠉⠬⠌⢐⠀⡙⠦⢌⣀⣀⣀⣀⠀⠀⠀⠐⠂⠀⠤⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂⠔⢁⠄⡡⡂⠉⠁⢀⠑⡀⠌⢀⣆⣀⡀⠉⠉⠉⠁⠒⠒⡂⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡈⢰⢡⢊⠎⢄⠂⢸⠀⡄⠰⢛⠉⢩⠈⠉⠉⠉⠉⠒⠒⠒⠂⠀⠉⠉⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⢀⠘⠨⡀⠐⠆⢘⠀⠀⠀⠸⠀⠸⠀⠀⠀⠒⠒⠀⠀⠀⠀⠤⠤⠤⢄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⡀⢆⠢⠊⠀⢀⠁⢂⠔⢀⠆⡀⠃⠐⠂⠀⠀⠠⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⡑⢄⠁⠢⠄⠀⠂⢁⡴⢃⢜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠦⣄⠈⣗⠒⢒⣈⣡⣴⠵⢣⡄⠀⣲⠆⣠⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⣀⠎⡀⠄⠀⠀⠨⢈⠻⠫⠁⠘⡛⡫⢃⡠⠐⡗⠀⠀⠌⡘⠲⡲⠖⠒⠛⠒⠒⠒⠛⠃⠀⠀⠤⠤⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⣤⢁⠈⠠⢂⠤⠀⢄⠀⠁⠀⢀⢮⠋⡐⠁⡠⢊⠭⠤⢄⡀⠨⠠⣀⣬⠦⠶⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠐⠀⡉⠲⠠⣁⢁⠀⢦⠁⠈⠀⣆⠃⠐⢠⠊⠔⢕⣸⠉⢠⡐⡀⢳⠉⠀⡘⡀⠀⠠⠤⠀⠀⠀⠐⠒⠊⠁⠈⠁⠀⠀⢸
⠀⠀⠀⠀⠄⠀⠀⡃⠀⠃⠁⠀⡀⡄⠀⠀⠉⠀⠀⡆⡘⡸⠑⢐⡊⠨⠁⡇⢀⠀⡇⢰⠆⠠⠤⠄⢀⣤⠤⠤⠒⠂⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠈⠀⠀⠡⠀⡑⠈⠌⢈⠄⡀⢀⡇⠀⡀⢁⠰⠨⡪⠂⠈⡢⡣⠁⡌⢀⠃⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣒⢉⣠⠞
⠀⠀⠀⠀⠀⠈⣌⠢⠑⠀⠄⡨⠄⠊⡠⠂⡸⣥⡐⡀⠓⢄⠈⠀⠁⢊⡠⠊⢀⠎⡐⡙⠁⠀⠀⣀⣀⠤⠀⢒⡊⠩⠁⠚⠉
⠀⠀⠀⠀⠀⠀⠈⠓⢄⠐⠀⢰⠂⢈⣤⡞⠀⠔⠛⢮⡂⠄⠈⣈⢁⢁⠠⠒⣁⡜⠊⠀⠒⠋⠩⠄⠒⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠑⠈⠁⠀⠀⠀⠀⠀⠚⠴⢀⣺⢤⣀⠰⠼⠞⠐⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⢤⡀⣴⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⠀⠙⠉⠀⠠⠀⠀⠤⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠒⠐⢀⠀⠄⠀⠀⠀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⢠⠀⠀⠈⠑⠦⡀⠀⡄⠇⠀⢸⡀⠄⠀⠈⠂
⢀⡀⠈⠀⠀⠀⠀⠉⠁⠀⠒⠒⠂⠃⠀⠀⠂⠀⠤⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⡇⠀⢀⡇⠢⠀⡠⠀⢄⡀
⢨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠒⠣⡥⠉⠄⠀⠀⠀⠂⠀⠀⠀⠀⡀⣤⠀⠈⡇⠀⠄⡬⢀⠀⠁
⠀⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠃⠟⠀⢰⠁⠀⠀⢀⡌⠁
⠀⠀⠀⠀⠉⠀⠉⠋⠁⠒⠒⠐⠠⠤⠀⠤⠤⢀⠀⠀⣀⣀⣀⢀⡀⢀⢀⣀⣜⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠆⠂⠀⢸⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡜⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡏⡀⠀⠠⠏⠺⠂⠐⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⢭⡇⠀⠀⠀⠀⢤⣹⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡀⡟⠢⡀⣰⠆⠼⣥⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠂⢀⡔⣿⣨⡝⢲⣉⣿⣿⡆⣿⣿⣿⣿⣽⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⣠⣼⠦⣴⠃⠀⣡⣏⣈⣿⢺⡇⠈⣿⣿⣯⠱⢱⣶⡾⠛⠛⠛⠿⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⣀⡿⠟⢉⣀⡀⢀⡗⢈⣯⣀⠫⣿⢻⡧⠀⠈⠁⠀⠀⢸⡟⣷⣤⣦⡀⠀⠛⠛⠂⠀⢸⡯⠟⢷
⠀⠀⠀⠀⠀⡀⠀⢀⣠⠄⠊⣁⣷⠶⠛⠋⠉⠉⣟⠉⠈⡟⢢⣼⢾⣿⡀⣄⠀⠸⠇⡈⠟⢋⣉⡉⠉⠉⠙⢳⢾⡂⠈⡇⡆⢺⡀
⠀⠀⠀⠀⠘⡦⠚⣩⡤⠖⠋⠁⣦⣠⣴⣷⣤⣼⣿⣶⣾⠦⠖⣿⣻⡿⣄⣤⣤⣤⡄⣠⣾⣿⢿⣿⣷⣀⣀⣈⣉⣹⣦⣧⣙⣼⡇
⠀⠀⠀⠀⠐⣟⣩⣤⣤⣶⣾⣯⣚⣿⣯⡙⠙⢻⣿⣾⣿⣋⣙⣻⣿⢿⠭⡭⠥⠆⢱⡟⠉⣂⣄⣀⠙⢿⣿⣿⣸⣿⣯⣯⠛⡟⣯
⠀⠀⠀⠀⠀⣬⣿⣿⢿⣹⣯⣯⢿⡵⣮⣿⣿⣿⣟⢿⡯⣥⠿⠴⢇⣚⣛⣉⣉⢻⣿⠃⠘⠑⣎⠀⠀⢸⣿⣿⣿⣉⣉⣿⣷⣴⠋
⠀⠀⠀⣀⣠⣿⣏⣉⣺⣵⣲⣭⣾⣁⣸⣿⣿⢿⣿⣶⣽⣧⣻⣿⡿⠛⠿⠯⠍⠚⠻⠻⣄⡀⠉⣀⣤⣾⣿⣿⣿⣿⣿⣿⣇⣈⣀
⣤⣶⣤⣴⣦⣤⣤⣬⣼⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣴⣭⣴⣤⣽⣥⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠾⠟⠛⠛⢻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢰⡏⢡⠀⠸⠁⠟⠉⢸⡇⠀⠀⠀⠀⢀⣠⡀⠀⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣀⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢛⠆⠂⠚⠀⡛⣺⡇⠀⡦⠄⠐⠻⠁⠀⠀⠘⢿⣿⡟⠋⢐⡄⠀⣤⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠒⠓⠚⠒⢿⠛⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⣼⠃⠀⠻⣿⣿⣿⣿⡿⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⣶⠿⠿⢿⣿⣿⣿⣿⣿⡇⢰⡏⠓⠶⢤⠝⠟⠋⣡⣴⣾⣥⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣋⣥⣶⣎⣇⣀⣠⣾⣿⣿⣿⣿⣿⠁⠠⢿⣿⠿⠀⠘⢷⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⣴⣾⣿⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⡿⢋⠀⠚⣆⣉⣀⣤⣾⣻⣿⣿⡿⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣴⣿⣿⣿⣿⡿⠟⡻⢿⣿⣿⠿⠟⠛⠁⠀⠀⠀⠀⣩⡷⠉⠉⠉⠉⠉⠛⢿⣦⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⡴⢿⣿⠟⠛⢭⠐⠋⠀⣀⡀⠨⠄⣀⠠⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠰⠒⣤⣷⣶⢫⠃⠀⢸⠞⠁⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⡿⣧⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣧⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢟⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣿⡏⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⢹⡄
⠀⠀⠀⠀⣤⠀⠐⠀⠀⠀⠀⢾⡟⡡⠊⢸⡇⠀⠀⡌⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠈⣧
⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣧⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣿
⠀⠀⠀⠀⣯⠀⠀⠁⠀⠁⠀⠀⠀⢀⠀⢸⣿⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⠀⠀⠀⠀⢻⣿⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⣿⠀⣿
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⡀⣿⡀⠀⢠⠂⠀⠀⠀⠀⠀⢸⣿⡇⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠇⢹⡇⠀⠸⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠘⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⣸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢀⣾⣷⣦⣄⡀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡷⠤⠤⠗⠒⠒⣒⠒⠒⠒⠛⠓⠒⢦⠤⠤⢄⣾⣿⣿⣿⣿⣿⣿⣿⠏⣹⠏
⠀⠀⠀⠀⠀⠈⠢⠄⣀⣤⣶⡿⠛⢻⣯⡅⠀⣀⢀⣀⣀⡄⣼⢫⣄⣤⠄⣴⡆⢠⡌⣀⣠⠀⢠⣭⣩⡿⠻⢿⣿⣃⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠆⢠⣿⠎⠰⣏⣿⣿⢿⠿⠷⠻⠞⠙⠷⠛⠹⢷⡿⠾⠿⣿⣴⣛⢢⣯⣤⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢴⠿⠶⠶⠉⣀⣠⣤⠤⠤⠦⠤⠀⠀⣤⣤⣿⣥⡄⠀⣀⣀⠉⠙⠛⠗⠂⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠚⠉⠑⢻⣿⣇⣀⣤⠒⢿⣿⣷⡘⣐⣐⣂⣤⣄⣤⣀⣛⣛⣻⣻⣆⣀⣀⣼⣿⣿⠓⢦⣄⣸⣿⡟⠂⠉⠳
⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⣿⣯⣅⣩⡄⠈⠉⢙⠃⡇⢻⢸⠇⡿⢹⠘⠏⡇⢻⠘⡇⡾⢸⠋⠉⠁⢠⣼⣤⣯⣿⡧⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠿⠛⠣⢒⠿⢿⣿⣿⣿⠀⠐⠛⠒⠳⣾⣾⡶⠓⠚⠒⠒⠓⢾⣶⡷⠓⢚⣚⠂⠀⣾⣿⣟⠿⣿⣶⠾⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⣿⣿⣤⣴⣿⣦⣴⣿⣿⣄⣀⣶⣶⣦⣀⣬⣿⣿⠷⣿⣿⡷⠾⣿⣿⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠘⠛⠃⢸⣿⣿⠋⠁⢿⣿⡿⠉⠛⣿⣿⣆⠈⠉⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣶⣿⠿⣿⣦⣀⠀⠀⠀⣀⣼⣿⠟⠻⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠻⢿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠴⢤⣀⡀⠀⢀⠀⠀⠀⢀⡀⠀⡀⠀⣸⠃⠀⠹⣆⠀⢀⠀⢀⡀⠀⠀⠀⡀⠀⢀⣀⣤⠦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⠿⣦⡟⢣⡼⠛⣶⠃⠀⠀⠀⠹⡴⠛⣧⡞⢳⡴⠿⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣶⣄⠂⡠⠀⢰⣿⣿⣯⣿⣿⣿⣿⣿⡿⠋⠛⠻⠿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣹⠻⣿⣿⣿⠟⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠈⢹⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢸⣿⣏⠐⠈⣿⡟⠀⠀⠀⠀⠤⣀⠀⠄⣼
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⣿⣿⠀⢂⣿⣧⠀⠀⠀⠐⠒⠸⠷⣦⡷
⠀⠀⠀⠀⠀⠀⢰⣤⣍⣛⣻⡿⠁⠀⠀⠀⠀⠀⠀⠛⠟⠒⣿⣿⣿⣆⠀⢀⣀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡿⢿⣿⣿⠟⣤⣍⣀⠀⠀⠀⠀⡴⠆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠂
⠀⠀⠀⠀⠀⠸⣿⣶⣶⡿⠀⠂⢐⣉⠙⢂⣤⣄⡙⠣⢀⠀⠈⠻⣿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠂⠈⠉⠁⠺⢏⣽⠻⣷⣤⣀⠂⠰⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⣙⠛⠛⠶⣦⣭⡝⠓⢄⣌⠃⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢯⣁⡀⢀⠖⠀⡦⢉⠻⢦⡀⠉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⡟⣁⡞⣀⣾⠓⢤⣕⢠⣻⣿⣶⡼⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣀⣿⠉⢺⡏⠛⣶⡏⠳⣥⠹⣿⣿⣦⡙⠙⢷⣄⡀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⢧⣴⣏⣉⣹⡆⣌⣿⠟⣷⡟⢿⡿⣿⣿⣿⠶⣤⠠⢛⣴⠃⣴⣦⣴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣾⣯⡿⢨⣿⠣⢼⣟⠢⣬⡿⢢⣿⣷⣼⣷⣿⢿⣿⡆⢻⡃⠛⠁⢀⣽⣰⢻⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢱⣿⣿⡷⠘⣿⡿⣾⣷⣼⣿⣽⣶⣟⣽⣷⣯⣽⡿⣿⣿⠀⡧⠀⡀⢁⣾⡁⢿⢟⣾⣧⠄⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠧⣾⣿⣵⣿⡏⠉⣿⣾⣿⣻⣿⡇⣿⣼⣷⣿⣿⡶⠁⠀⣤⣿⠿⣋⡴⢿⣿⣙⣥⣄⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣼⣿⠆⢿⡛⢻⣿⣯⣿⣟⣿⣿⣽⣿⣿⣿⠟⠁⢀⣨⠿⢅⡞⠁⠀⠀⠈⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣸⣿⡁⣾⣇⣈⣿⡧⣿⣻⣿⣿⣿⡿⠋⠀⠀⢠⢼⠟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⣿⡉⣿⠏⣹⣿⣨⣿⣗⣿⠿⠛⠋⠁⠀⠀⢠⡠⡷⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣻⠉⣿⣶⠻⣿⣦⡿⠋⠁⠀⠀⠀⠀⢠⣼⠜⢃⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡤⠉⢿⣆⡽⠋⠀⠀⠀⠀⠀⠀⡀⢠⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡟⠀⡄⠀⠀⠀⠀⠀⢀⡰⠟⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⣿⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣘⣦⡀⠀⠀⠀⠀⣀⠤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠀⠀⠀⠄⠠⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠌⠉⠉⠓⢲⡀⠆⠀⠀⠀⠠⠤⠤⡤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠤⡀⠀⠐⠂⠀⠀⠀⠀⠠⠤⢀⡤
⠀⠀⠀⠀⣀⣠⠶⠐⠾⢞⠊⠁⠀⡘⠀⠀⠀⠀⠀⡇⠀⡎⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢀⠈⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢠⣶⣿⠻⣛⢀⠀⡰⠉⠁⡀⢀⠁⠠⢄⠀⢀⠀⠀⠀⠃⡀⠤⠄⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⢀
⠀⠈⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠂⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢠⣄⢂⠢⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⢸⡒⢊⡅⣤⣬⠀⢀⠉⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢈⠨⢤⡦⣤⣼⣭⣽⣶⡴⠗⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡬⠾⡥⢮⠵⢌⠀⠠⠤⠤⡵⠼⠧⠀⠁
⠀⠀⠁⠀⢀⢂⡑⡄⢀⣊⠂⠠⠤⠤⠼⠤⢔⠳⠚⠉⣛⠖⡀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠊⠑⠃⠑⠌⠈⠀⠀⠀⠀⠀⠈⠎⠙⠀⠣⠔⠃⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠰⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⠂⣀⡀⠀⢈⡀
⠀⠀⠀⠀⠀⠊⠀⠀⣤⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⡊⠀⠠⠀⣠⣤⡀⣠⣤⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠈⠷⠌⡑⣤⠄⡀⠘⠁⠂⠁⢐⡀⠀⠀⠀⠀⠀⠀⠘⠷⠧⠃⠀⠀⠀⠀⠀⠷⠧⠃⠻⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⢇⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣀⡀⠁⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠳⠤⠖⢳⡄⠀⠀⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠄⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠠⠼⠄⠂⠀⠴⠆⠀⠐⣫⣏⠉⣩⣭⠙⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠀⠈⠀⠀⠀⠘⠀⠀⠀⠻⠟⠀⠛⠟⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠖⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠚⢿⠤⠴⠶⠀⠀⠀⠀⠀⠀⠘⢇⢠⣿⡦⢄⠀⠀⠀⠀⠀⠀⠠⠤⠠⠀⠐⠠⡀⠈⠃⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠦⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠛⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣄⠈⠓⢄⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢈⣳⣖⠒⠒⢒⢂⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⢀⠀⠀⣀⠀⠀⡀⠈⡀⠀⠀⠀⠀⢀⡀⠘⢆⡀
⠀⠀⠀⠀⠀⠀⠰⠈⠿⠿⠃⠁⠈⠘⠯⠟⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⢠⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⡆⠀⠀⠀⢸⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠁⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣶⣆⠀⠀⠀⠀⠀⣸⠃⡀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡿⠟⠛⠻⡆⠀⠀⠀⠀⣿⡄⠏⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣶⠄⢀⣤⣿⣿⣿⣿⣿⣿⢾⣇⠛⠦⠖⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⡆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠲⣿⣷⣿⣿⣷⣿⣷⣿⡿⣿⣿⣿⣿⣦⠀⠀⠀⢠⢻⣇⠀⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠈⡂⢻⡿⣿⣿⣿⣿⣿⣿⣇⣹⢿⣿⣛⣻⣷⡄⠀⢸⠈⠻⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠞⠛⠀⠀⣿⣿⣿⣿⣿⣟⣱⣯⣿⣿⣿⣏⣿⠃⠀⢸⡇⠠⠂⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⣿⣿⣿⣻⣿⡏⠀⠀⢸⣷⠆⠀⠀⠀⠀⠠⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⡺⠛⠿⠷⠿⠟⠋⠉⠙⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣙⣿⣧⣨⣆⣀⣀⣀⣀⠀⢠⡿⠟⠀⠀⠀⢸⣿⢼⠀⠀⠀⠀⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢰⣿⡅⠙⠛⠛⠛⠛⠛⡏⠀⠀⠀⠀⠀⠀⠀⢸⡿⡄⠃⠀⠀⠀⠉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⠿⣷⠀⣠⡶⠛⠉⠀⢳⠀⠀⠀⠀⠀⠀⠀⢸⠇⣧⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢻⣶⣿⠁⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⢸⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣠⠏⠀⣿⡆⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢸⢟⡇⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⢰⠇⠀⠀⣿⡅⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⣦⢠⠀⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷⢍⡰⠋⠀⠀⠀⠸⣏⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⣬⢸⠀⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡙⠖⡛⠁⠀⠀⠀⠀⠀⣿⣆⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸⠀⠀⡇⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠶⠠⠀⠀⣨⣻⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠄⠀⠀⠆⠀⡆
⠀⠀⠀⠠⠖⠋⠁⠀⠂⣀⣤⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡖⡀⢻⡇⠀⠀⠢⠀⢡
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠋⠀⠀⠀⠀⢀⡤⠀⣀⠀⠀⢸⣷⣀⡀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣇⡿⢸⡄⠀⠀⠀⠁⢸
⠀⡈⠀⡰⠞⠉⠀⠀⠀⣀⠁⠰⠐⠂⠒⠈⠐⠶⠶⠥⠀⠀⠀⠘⢿⣉⣡⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠆⠁⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⣚⣒⡂⠤⠙⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⣤⡾⠟⠉⢸⡆⠀⣠⡀⠀⣠⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣟⣧⡀⠀⠀⠄⠀⠿⠳⠿⠟⠋⠅⢼⢿⠃⠀⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠇
⠀⠀⠀⢴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢿⣧⣱⡄
⠐⣦⣀⠙⢿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠀⠀⠀⠙⢿⣿⣦⡀
⢙⣿⣋⣴⣿⣶⡇⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠂⡀⠀⠀⢀⣴⣶⣼⣿⣶⠂⠀⠀⠀⠹⣿⣿⣆
⠾⣿⣿⣄⣿⣿⣿⠶⣄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠟⠻⠒⠀⠀⠀⠀⢸⢻⣿⡄
⠀⣿⡏⠋⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣲⣖⣶⣾⣶⠆⣤⠀⠀⠀⢸⡄⣿⣷⠶⡴⡆
⠀⣿⡇⢀⣿⣇⡀⠀⠀⢀⠖⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠛⢛⣿⢸⣿⡏⣿⢧⢿⡀⠀⠈⣇⣼⡇⢸⢠⣿⣶
⠈⣿⡇⢠⣿⣭⣷⣤⣤⣼⣠⣼⣷⡆⠀⢠⣤⠀⠀⢀⡟⠛⣿⣿⣿⣿⣿⠈⠹⣿⣿⣾⣼⣽⠀⠀⠛⠛⢻⣿⣿⠛⠛
⢸⣿⡇⢠⣿⣷⡇⣼⢻⡿⢿⣟⣿⠃⠐⡞⢿⣤⠀⢩⣧⢤⣯⣿⣿⣿⣿⣤⣶⣶⠀⣠⣭⡭⠀⢸⣿⣿⠾⠿⠿⢾⣿
⣿⣿⣷⣾⣿⣿⣷⣾⣿⣶⣶⣿⣷⣶⣾⣿⣿⣿⣶⣶⣿⣷⣶⣾⣷⣶⣶⣶⣿⣿⣶⣾⣿⣿⣶⣾⣿⣿⣶⣶⣶⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣻⣻⣿⣻⢷⡄⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⠿⢿⣷⣿⠀⠀⢻⣿⠛⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢉⠁⠀⠀⣀⡀⢿⡿⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠿⡇⠸⠛⠛⢸⡇⠃⠀⢸⡷⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⣀⣦⣠⣤⣤⣾⡏⠁⠀⠀⣇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⣿⣍⣽⣿⣿⣷⠒⠂⢴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣿⣿⣿⠻⠿⠟⠋⠀⡿⠇⠟⠛⠲⠦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⣮⣤⠁⢹⣯⣿⣄⣀⣀⡀⣠⣴⣤⣤⠀⢀⣠⣿⣍⠁⠩⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⣾⡿⣲⣾⣿⠤⠀⠐⣿⣿⣾⣧⡉⡀⢿⢿⠿⠿⠀⠨⢽⣿⡿⠀⠀⠐⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠄⠠⠀⠀⢸⣯⣄⣸⡛⠁⠘⠂⠀⠀⠀⣃⡉⠋⠀⣈⣘⡀⠀⠀⠀⢠⠀⣠⣾⡷⠋⠉⡇
⠀⠀⢀⡀⠀⠀⠈⠙⡾⠗⠂⠀⠀⢀⣾⡯⠀⣿⣷⣤⣴⡇⠀⢀⣿⣿⡯⠇⠀⢹⣿⣿⡆⠀⠤⣼⣿⣿⡟⠁⣀⢰⡇
⠀⠀⠈⠙⠒⠋⠉⠫⡀⠀⠘⠀⠀⣸⣷⣤⣀⣿⣿⣿⡿⠟⠀⠈⢉⠁⠀⠀⠀⠀⡉⠩⠀⠀⠀⣹⣿⣹⣷⣾⣿⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣴⣟⠋⠹⣿⣿⢿⣿⣷⣆⡀⠀⣼⣶⡦⢀⡆⠀⣿⣿⠂⢰⣾⣿⠍⢻⣿⢿⣧⣤⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⠻⣿⣶⣿⣿⡿⢿⣿⣿⡇⠀⢹⠿⠗⠀⠀⠐⠿⠛⠀⠈⢻⡿⠀⠀⢿⣾⠿⣿⡇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢹⣿⡟⠁⠈⣿⣿⠀⠀⢰⣤⣄⢀⣀⣴⣦⡄⠀⡄⣼⡇⠀⠀⠘⠿⣻⣷⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣵⠃⠀⠀⢻⣿⣡⡀⠸⣿⡿⠹⠏⠹⣿⡇⠠⢶⣿⡏⠀⠀⢠⣼⠋⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⢸⣿⢸⠀⠀⢀⠀⠀⠀⢀⣂⠀⠀⠀⠄⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢠⠀⢸⡿⠉⠁⢻⣏⠀⠇⣶⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠃⠀⠉⠉⠹⠄⠁⠀⠀⠐⠒⠿⣯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣷⠖⠒⠶⠦⠤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⢀⡠⠐⣩⡿⣓⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠰⣾⣿⣷⣾⣿⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡽⠯⠛⠛⠆⠘⠟⠋⠀⢉⠉⠍⠭⠀⠀⠀⠀⠀⡆⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠄⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠬⠅⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣁⡀⣰⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢛⣠⣼⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠻⣿⣿⣁⣀⣨⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⣿⠛⠋⠀⠀⠀⢈⣿⣿⣯⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⢻⣷⠻⡄⢢⣄⠀⠁⢸⣿⢻⡿⣧⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⣽⣦⡀⠸⠿⠛⢷⣿⣷⣾⡿⢿⠀⢿⡟⣷⣮⠟⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣸⡟⢻⣾⣟⠀⠰⢶⣿⠻⣿⣶⣷⠼⡇⢸⣷⡟⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣾⡿⢿⡿⢰⣿⢅⣽⡇⠀⣤⣿⣶⠻⣏⢻⣤⡇⢸⣿⢷⣀⢀⣀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣧⣾⣷⢴⣿⣷⠀⢻⣅⣿⣾⣿⠛⣇⡀⠼⣿⠀⠉⠻⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣹⣧⣼⣷⠾⣿⣷⠀⢸⣿⢹⣧⣼⣾⡟⠁⣴⣿⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠋⠁⠀⠀⣾⣯⠄⢺⣿⠿⡟⢹⣦⣿⢀⣿⡟⠀⠀⠀⣿⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢠⣿⣿⠀⢀⣴⣾⡿⢿⣏⣧⣸⣿⡅⠀⠀⠀⣿⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠘⠿⣿⠀⣸⣿⣿⣿⣿⣿⣿⠌⣿⠇⠀⠀⡰⠃⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠠⣄⠀⠀⠀⠈⠢⣿⣿⣿⣿⣿⣿⣿⠸⠇⠀⠀⠘⠃⣀⡄⢼⡛⠋⢓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡔⣀⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣾⢢⡄⠀⠀⠀⠙⢿⠿⡿⠛⠉⢁⣤⣁⠀⠀⠀⠈⠁⠚⣾⡷⠔⢺⠇⠀⠀⠀⢀⣀⣠⣤⠴⠚⠋⠉⠁⠀⠀⠄⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢷⣌⡓⠄⠀⠀⠀⠐⢀⣴⣿⣿⣿⣿⣷⡀⢰⣶⣾⡿⠯⠀⢀⣭⡤⠶⠚⠋⠉⠀⠀⢰⡄⠤⠄⠀⠀⣠⣤⣤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⣷⣽⣿⣷⣦⣅⡀⠀⠀⠉⠉⢩⡏⢻⠏⢓⣿⠟⠉⠁⠀⠀⠙⡄⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⡠⢋⣀⠰⣾⡏⢰⠂⠀⠀⠀⠀⣄⡤⠤⠖⠚⣻⣿⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣬⣴⡿⠃⠲⣿⠀⢉⣀⣤⠄⠒⠀⠀⠀⠀⢀⠴⡻⠇⠀⠀⠀⣠⣾⣽⡇⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢧⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣄⣈⣰⣤⣤⣶⣿⣿⣿⣷⡀⠀⠀⠂⠀⠀⠀⠈⡿⠁⢣⠀⠀⠘⣿⡿⢿⠷⠀⠀⠀⢸⣿⣼⣰
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣤⣝⣛⣻⣿⣿⣿⣿⠃⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠙⠀⠀⠨⡄⠀⠀⠀⠙⠚⠀⠀⠀⢀⣼⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠃⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡈⢿⣿⣿⣿⣿⣿⡿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⣫⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢴⠾⠿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣀⠁⡠⠀⢸⣿⣿⣷⣶⡆⢳⡀
⠀⠀⠀⠀⠀⠀⣴⠛⠺⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣤⢹⡄
⠀⠀⠀⠀⠀⡰⠋⣤⡰⢿⣿⣿⡿⠀⠀⠀⠴⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣶⣄⢳⡀
⠀⠀⠀⢀⠞⠀⠘⢿⣿⡿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣛⠭⠽⠂⠀⠇
⠀⠀⠀⡞⠀⠀⠀⠀⠉⢱⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠉⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣋⣻⣧⣬⣭⣭⣤⣥⣬⣥⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡰⢦⡀⠀⠀⠀⠀⠠⠓⢦⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣠⣤⣤⠤⡄⣤⠠⢤⣤⣀⡀
⠀⠀⢰⡀⠹⣶⣐⣀⣄⡃⠀⠆⢱⣦⣤⣤⡀⢤⣠⣿⣿⣿⣿⣦⣄⠀⠂⣻⣅⢻⣯⣿⣻⣿⣷⣿⢪⣙⠿⣾⣳⣮⣿⣷⣄
⠀⠀⢸⡇⠚⢿⣿⣿⡉⢹⣤⣷⡀⢿⣷⣟⣵⡊⢈⠁⠃⠙⣿⣯⠹⣆⠀⢸⣿⣿⣿⣯⣶⣬⣭⣺⣿⣿⣿⣿⣷⣼⢿⣿⣿⠲⣤⡀
⠀⣼⣋⣐⣷⣽⣿⣿⣾⣷⣿⡽⣿⣾⣿⢿⣟⠻⣤⠀⠀⢀⣿⣿⠃⢼⣆⣼⣿⡛⠟⡫⣭⣿⣿⣿⡿⢿⣿⣍⣿⠈⠳⡌⠃⠀⠺⣿⣶⣄
⢰⣿⣿⣟⣠⣿⣿⠏⣬⠿⣿⣿⡿⣿⣿⡆⠙⡀⣨⠀⣷⣾⢻⡿⠀⣿⣿⣿⡿⠉⠞⠓⠛⠹⣿⣏⠧⣀⠙⢿⠿⢇⣽⣿⣆⠀⠉⢩⣭⠯⢦⡀
⠘⡿⠻⢌⡿⠋⠙⠷⠁⠀⠀⠀⠀⢹⠈⢷⠀⣹⣟⡗⠹⡏⠸⣿⡀⢽⡟⠈⠁⠀⠀⠀⠐⠀⠁⠀⠂⠙⠂⠀⢨⣿⣻⣿⡿⠁⠀⠀⠀⢰⣿⣿⣆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⢿⡆⠀⣼⡿⢧⠀⠀⡏⢸⠏⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠉⠙⠀⠀⠀⠀⠀⠀⠁⡭⢯⣷⡀
⠀⠀⠀⠀⣴⣶⡄⠀⢠⡿⠃⠀⠀⠀⢠⡜⠛⠀⠌⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢼⣿⣿⣿
⠀⠀⠀⠀⠙⣿⣤⣴⠏⠀⠀⠀⣴⣿⣇⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣿
⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⢸⣿⣿⣿⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⢀⡀⣀⠈⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠
⠀⠀⠀⠀⠁⠀⠀⠀⠒⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠆⠀⠉⠀⠀⠀⣀⡀⠂⠲⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠉⠋⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠒⠛⠛⠉⠉⠛⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣥⣄⠈⣧⡤⠶⠚⠋⠉⢀⣀⣤⣤⠶⠒⠛⠋⠉⢳⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⡴⠋⠁⢈⣿⡅⠀⠀⠀⢠⠞⣉⡵⠖⠛⠀⠀⠀⠀⢀⣼⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣥⠞⠉⠀⣠⡶⢋⡼⠁⠀⠀⠀⠘⣾⠋⠀⠀⠀⠀⠀⢀⣴⠟⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡅⠀⣠⠞⣡⠔⣫⠴⣚⣿⡲⣄⠀⠈⠳⢤⣀⣀⣤⡶⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣷⣾⠵⢛⡥⢞⠕⡏⡇⢸⠈⡦⡙⢦⠀⠀⠈⠀⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠲⠒⠒⠲⠶⠶⠶⠒⢺⡿⣄⠀⣴⢋⣾⠙⡀⡇⢰⠈⡄⢡⢸⡎⣧⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⢛⡛⢻⠷⣌⠳⣌⠳⣝⢄⡇⠧⢸⠀⡇⢸⢜⡵⠃⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣩⠟⣋⠝⢳⡈⢳⡌⢮⡣⣸⠈⡤⢛⡵⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣸⣧⠟⣫⠔⢋⡤⣾⢿⢙⣦⡙⢦⠹⣌⣫⠞⠉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢦⡾⢞⡩⠔⢉⡤⢞⡭⡎⣡⣪⠟⡩⢻⣎⠳⣌⢁⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣲⠟⡫⠖⠁⣠⢖⡫⡖⢹⣠⣿⠞⣡⢊⣵⠯⠼⠳⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣏⡿⣋⠕⠋⣀⡴⣫⠕⠋⠀⣧⣾⠟⣡⢞⣵⠧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣸⣡⢟⡡⠊⠀⣠⢞⠵⠊⠀⠄⣰⣾⠟⣡⣼⣵⢯⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣿⡴⢋⠕⠋⢀⣴⡾⠕⡇⠀⣇⢴⡿⠋⣡⣼⣿⠧⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡦⣼⡴⡫⠔⠁⢀⣴⡿⠛⡇⠀⣧⢔⡵⠋⡠⣾⢿⠷⠟
⠀⠀⠀⠀⠀⢀⡤⢼⡷⣫⠚⠁⣠⣶⡟⠉⡇⠀⡧⣾⡵⠋⡠⣪⢾⡷⠾
⠀⠀⠀⠀⡶⢶⡷⢫⠞⠁⣠⣾⡟⠁⡇⢀⠷⣪⠞⠁⡴⣪⢾⡷⠾⠃
⠀⠀⢠⡦⣿⢎⠔⠁⣠⣾⠿⠁⢠⣠⢟⡵⠋⢁⡤⣪⢾⡗⠾⠃
⠀⣤⢬⡿⡡⠃⢀⣼⢿⠁⢸⡠⣪⠞⠉⢀⠔⣫⢾⡟⠛⠃
⣀⣼⡞⡘⠁⢰⡿⡏⢈⣔⡿⠛⠁⢀⢔⣥⢾⠓⠚⠃
⠙⣶⡇⠃⠀⢿⣑⣷⠵⠋⠀⣠⢖⡵⢻⠙⠛
⠾⣽⣏⢆⠀⠀⠀⠀⣀⠔⣪⣖⣻⠛⠛
⠀⢸⠽⣷⢭⣭⣬⣭⣶⡿⢼⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠉⠁⢈⡹⠿⢷⣤⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡶⠒⠋⠁⠀⠀⢿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠏⠑⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠾⣇⡀
⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣤⣀⣠⣀⣤⣀⣠⣀⣠⣀⣀⣀⣸⠁⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⣿⣿⣿⣶⣶⣶⣦
⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠁⠈⠉⠉⢩⠀⠀⠀⣿⣿⠀⠀⠠⢀⠀⠀⠀⠀⠀⠠⠘⢿⡎⠙⢻⣿⠛⠿⢿⣿⣦⣄
⠀⠀⠀⠻⡄⣀⣀⣀⣀⣀⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣰⠒⣆⣿⣿⡄⠀⠀⠀⢸⠀⠀⠀⠊⠐⠀⠚⡇⠀⠀⠉⠀⠀⠀⠈⠙⣿⡇
⠀⠀⠀⠀⠉⠉⠛⠉⠋⠙⠋⠉⠋⠙⠉⠉⠙⠛⠛⠛⠛⠛⠚⠛⠻⠽⢶⣿⣿⡿⣀⠀⠀⠀⢠⣀⡔⠀⠀⢠⣖⡆⢸⣤⣤⣤⣤⣤⣤⣤⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠛⠒⠒⠓⠚⠒⠿⠧⠤⠬⠝⠥⠚⠉⠉⠉⠉⠉⠉⠉⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡇⠘⠟⢿⣿⡿⢛⡛⣿⡿⠿⣿⣿⣿⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠛⢿⣿⡇⠘⣿⢈⡟⠀⣯⣼⠟⢀⣶⣦⡿⠛⣲⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠛⣁⠀⣷⡀⠿⢡⣀⣋⣼⣷⣤⣭⣷⣄⣾⣿⠟⢀⣾⠟⠋⣙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡄⠹⣆⠘⣷⣶⣿⣿⣿⣿⣿⡿⣿⣿⣿⡐⢀⣴⣾⣆⠰⠿⠋⣠⣼⣿⣿⣿⣦
⠀⠀⠀⢠⣾⣿⣿⣇⡈⠻⡟⢀⣿⣿⣿⣿⣿⣿⠟⣩⣄⣀⣀⣩⣩⣿⣿⣿⣿⣿⣿⣄⣚⠛⠛⢛⠻⣿⣿⣧
⠀⠀⢀⡾⣿⣿⣧⡈⠻⡆⣠⠿⣿⣿⣿⣿⣿⡇⣾⠋⠙⠿⠿⠟⠛⢿⢻⣿⣿⣿⣿⣿⣧⡰⣿⣿⣥⣾⣿⣿⣧
⠀⠀⣸⡇⣄⡉⠛⠿⣦⡤⠋⣠⣿⣿⣿⣿⣿⡇⡿⠀⠀⠀⠀⠀⠀⢸⡄⣿⣿⣿⣿⣿⣿⣿⣀⣉⡉⢉⣁⣈⣿
⠀⠀⣿⣿⣬⣹⣷⣦⣀⣠⠿⣯⣿⣿⣿⣿⣿⢿⡁⣀⢀⠀⠀⠀⣀⠈⡷⣿⣿⣿⣿⣿⣿⣿⣿⠋⢉⣰⣭⣿⣿
⠀⢸⣿⣿⣿⡟⠁⠀⠉⢁⣴⣿⣻⣿⣿⣿⣿⣿⠏⠉⠛⠁⠀⠛⠋⠑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⢸⣿⠿⠋⠀⠀⠀⣴⣿⣿⣗⠈⠓⢝⢿⣿⣧⡀⠀⠀⣀⣀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢂⡻⣧⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣌⠐⠽⣿⠀⣂⣀⣈⣁⣀⣐⣂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠑⢍⠻⢴⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⡿⠛⠙⠛⠙⠻⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣶⣍⣛⣻⣿⣿⣿⡿⣿⣿⣿⣷⠁⠘⣷⣄⣀⠀⠀⠀⡇⣸⣇⣤⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣶⡟⠛⠛⡏⠘⣿⢿⣏⣻⣇⢲⠈⣃⡟⣷⣎⣼⠛⠛⢻⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⡿⣧⣄⣀⢿⠀⢻⣀⣹⣇⣸⣾⠿⠿⣶⡿⢻⣿⣷⣶⣾⣿⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⢤⣄⣿⣿⣿⠘⣇⢘⣿⣿⡇⠈⣁⢲⠶⠶⣖⠾⠀⠿⢿⣿⣄⡈⢽⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠈⠛⣿⡝⣶⣷⡈⠋⢉⢹⣶⣿⡟⣿⢛⡿⣿⣷⣿⠛⠛⠛⠛⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⢀⡀⠸⣷⠟⢿⡇⢹⣯⣽⠻⠿⢧⣿⣽⣷⡟⠿⡍⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠠⣗⠤⣙⠢⡍⠂⠈⡇⢸⣿⣿⠀⠀⣸⣿⣿⢡⡁⣠⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠫⡙⠮⢓⡀⣠⣿⣿⣾⣤⢾⣿⡿⠷⠞⠛⠛⠋⠈⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⡄⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⣠⠴⠛⠛⠛⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠙⢷⣄⣼⠅⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣤⣤⣀⣤⣀⠀⠀⠀⠀⠀⠉⠳⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣳⠀⠀⠀⠀⠀⠀⠈⠛⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣻⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣝⢦⡀
⠀⠀⠀⠀⢀⡴⠛⠉⠉⠁⠉⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣦⡀
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⢀⣤⣶⣶⠞⠓⣛⠿⣝⢦⣀
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⢀⣴⡾⠟⠿⠿⠿⣿⣦⡀⠀⣰⣿⠛⠁⠀⠀⠀⠀⢱⢹⡄⠉⠳⢤⡀
⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠈⠻⣿⡄⣿⡇⠀⠀⠀⠀⠀⠀⣸⣾⣣⠴⣻⡟⢿⣷⣦⡄
⠀⢀⣤⠽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣽⢻⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⢀⣼⣿⣿⠗⠉⠀⠀⠀⠀⢹⡽⡆
⢰⠏⠀⠀⠈⠳⣄⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡜⣆⠀⠀⠀⠀⠀⠀⢀⣿⠟⢾⣿⣦⣤⣤⣴⡿⢿⢳⠃⠀⠀⠀⠀⠀⠀⠀⡇⡇
⣿⠀⠀⠀⠀⠀⠘⢧⢸⡀⠈⠹⡄⠀⠀⠀⠀⠀⢀⣀⣤⣤⣽⣮⣶⣄⣀⣄⣤⣴⣿⣿⠿⠿⠿⣿⣟⢯⣁⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⢿⠀⠀⠀⠀⠀⠀⠈⠳⣷⠦⡴⠇⠀⠀⠀⣰⣾⠟⢛⡒⠒⠛⠻⢝⢿⡛⠛⢻⣿⠋⠀⠀⠀⠀⠈⣿⡆⠈⠳⣼⣏⢧⡀⠀⠀⠀⠀⣠⣿⠿⣄
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢧⣷⠀⣿⣇⠀⠀⠀⠀⠀⢀⣿⠇⣀⣦⣠⣽⣷⣿⣶⣶⣶⣾⣿⣷⣾⡿⢿⣶⣄
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡐⡏⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠲⣼⣿⣄⠀⠀⠀⣠⣾⣿⡯⠽⠚⠓⠒⠒⢮⣻⡄⣠⣿⠋⠁⠀⠀⠀⠙⢿
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣜⢆⠀⠀⠀⠀⠀⢀⡴⣳⣧⢶⣾⣛⡿⢿⣿⠿⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⢷⢿⣿⡃⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⣿⣽⣶⣶⣶⣿⠽⡾⣻⠞⠉⠀⠀⠈⠑⢼⢳⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣸⣷⣳⣄⡀⠀⢀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⢀⣤⣾⠿⠯⠭⢝⡿⣷⣄⢸⡷⡇⠀⠀⠀⠀⠀⠀⠈⡇⣿⣜⢦⣀⠀⠀⠀⠀⣀⣴⣻⠏⠈⠙⠻⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠙⣟⣿⢿⣧⢧⡀⠀⠀⠀⠀⠀⣠⣷⠏⠈⠙⠿⢿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⠀⠙⠿⣝⣶⠤⠤⣶⣾⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣿⣷⣀⠀⠀⠀⢀⡴⣻⠇⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠚⠛⠛⠛⠛⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢱⠀⠀⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⢇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠥⠀⠀⠈⢣⠀⠀⠀⠑⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠄⡀⠀⠀⠱⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡅⠀⠀⠀⠘⢆⠀⠀⠀⠑⠒⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⢹⣀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠡⣀⡤⠲⡄⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡛⠳⢦⣀⠀⠀⠀⠀⠀⡾⢷⡀⠠⠛⢹⡆⠶⠀⠀⠉⠢⣄⡀⠀⠀⠘⠁⠣⠤⠞⢢⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡞⠑⢌⣀⠄⠀⠙⢷⡀⠀⠀⢠⠷⠈⠳⣄⠀⠀⠙⠛⢻⡤⠴⠀⠀⠙⠓⢤⣀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠓⠤⣄⡾⠀⠀⠀⠀⠉⠉⠐⠒⢿⣦⣰⠏⠀⠓⣄⠈⠳⣦⣀⠀⠀⠛⠋⣇⢹⡧⡀⠀⢀⣈⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡇⠀⠀⠀⣠⠄⢹⣷⠄⠀⠀⠁⠂⣀⡀⢄⣩⠿⠶⠟⠛⠛⠲⠶⠶⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠖⠋⠁⠀⠐⠀⣄⣠⣥⣠⡾⡇⠀⠀⠀⠀⠀⠀⠹⣯⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠶⠶⠒⠛⠛⠛⢛⡿⠳⡀⠀⠀⠀⠀⢀⡔⠈⠩⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣸⡗⠒⠒⠊⠀⠀⠀⠀⢱⠀⠹⣆
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⠉⡧⠀⠀⠀⠀⠀⠀⠀⠀⢇⠁⢹
⠀⠀⣠⡶⠛⠉⠉⠀⠈⠉⠉⠛⠲⢤⡴⠋⠀⠀⣠⠞⠁⠀⢠⡇⡄⠀⠀⠀⠀⠀⠀⠀⣈⡶⠏
⢀⡾⠋⠀⣀⡤⠶⠶⠶⠶⢤⣄⠀⠀⠀⠀⣨⡾⠃⠀⠀⠀⢸⠃⠙⠂⠀⠀⢰⡖⠚⠉⠁
⣾⠁⠀⣼⠋⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠻⣄⠀⠀⠀⠀⠸⡇⠀⠀⣀⡀⠀⠙⠦⣀
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⡆⠀⠀⠀⠀⠳⣄⠀⡏⠉⠉⠉⠉⠁
⢹⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿⠀⠀⠀⠀⠀⠈⠓⠛
⠀⢳⡀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣿
⠀⠀⠙⢦⡀⠈⠛⢶⣤⣀⣀⣀⣀⣴⠟⠁⢀⣼⠃
⠀⠀⠀⠀⠙⠦⣄⡀⠈⠉⠉⠉⠉⠀⠀⣠⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⡙⣿⢿⣿⣿⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣿⡷⡈⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⡿⠁⡟⢄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⡿⠙⢦⣧⠞⢣⠙⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⣿⣥⡴⠶⠟⠚⢻⠳⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⡿⠁⢼⣧⣶⣶⡶⣿⠀⠘⢄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⣿⡿⠃⠀⢸⣿⣿⡷⣷⢿⠀⠀⠈⣢⠹⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⢣⣿⣿⡷⣄⠀⠀⣿⣆⠔⣣⢾⠀⠀⡴⠃⣷⡘⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣰⢃⣿⣿⠃⡇⠘⢦⢐⣛⡭⠬⠥⢽⢁⡞⠁⠀⣗⠘⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣰⢣⣾⣿⠃⢀⣇⣀⣈⣻⣇⣀⣀⣀⣸⣿⣤⣤⣤⣿⣤⠼⢦⠻⣿⣿⣿⣿⣷⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆
⠀⠀⠀⠀⠀⢳⣾⣿⡟⠉⠉⠏⠉⠉⠹⣍⠉⠀⠀⠸⠀⠀⠀⠀⢿⠀⠀⠀⣷⠧⠤⠭⠬⠥⡭⠭⠭⠭⠭⢉⣍⣉⣉⣍⣋⣛⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⣿⣽⡟⣹⠃⠀⠀⠀⢘⣿⣽⠟⡹⡯⠀⠀⠀⣿⡀⠀⠀⠀⠀⡏⢖⠰⣦⢠⡔⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠬⡭⢼⠀⠀⠀⠀⢈⣿⣭⡩⢤⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⣱⢉⣻⡇⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠿⠼⠷⠿⠀⠀⠀⠀⠸⣿⢠⣵⣋⡧⠀⠀⠀⣿⡀⠀⠀⠀⠀⡇⡲⣤⣦⣸⡇⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡇⠀⠀⢸⠀⠀⠀⠀⢰⠂⠀⠀⠐⣦⠀⠀⠀⣿⣥⡤⠤⠦⣶⡷⠶⠀⠦⠀⠄⡷⢚⣒⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⢶⡷⢶⣶⡼⣧⣤⣤⣤⣼⣤⣤⣀⣀⣿⣀⣀⣀⣿⣂⣄⣛⣓⣤⣧⣤⣤⣦⣤⣤⢷⠶⠶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠓⠒⠛⠺⠶⠶⠼⠧⠿⢽⣿⣽⣭⣏⣿⣻⣯⣭⡯⠽⠼⠶⠶⠚⠛⠚⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⠋⠉⣉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠉⠘⣉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣛⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣷⣤⡀⠉⣸⣿⣷⣦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠢⢞⣏⣩⡤⢞⠳⣻⢠⢏⢿⠯⢿⢭⣷⣇⠄
⠀⠀⠀⠀⠀⠀⣬⢁⠹⣤⠽⣖⣻⣛⣏⣹⡼⡒⣺⢓⡗⠲⠾⣭⡁
⠀⠀⠀⠀⠀⠀⠃⣌⣻⣟⣮⣭⣭⣭⢽⡤⢷⡯⡭⣽⢭⣯⣟⣾⡧⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣰⣲⠧⣽⣿⣿⢛⣶⣿⠚⣟⣻⣗⠲⠶⡾⢶⣾⣯⣭⠀⠀⠀⠐⠺⣿⠟⢠
⠀⠀⠀⠀⢰⢥⣭⣽⣿⠋⢻⣩⣿⣹⣉⣭⣭⣧⣍⣛⣛⣻⣿⣺⣿⠂⠀⠀⠀⣀⠽⣴⣿⠆
⠀⠀⠀⢀⡟⣿⡻⠿⠃⠀⠀⢣⣿⢼⠤⠴⠖⡷⠶⢮⣯⣽⣿⠟⠋⠁⠀⠠⠃⠁⠀⠘⢷⡄⢠⣿⡶
⠀⠀⠀⠘⠿⠛⠉⢻⡆⠀⠀⣼⣿⣛⣓⣛⣃⣏⣭⣿⣗⡚⠁⠀⠀⠀⢁⠄⠀⠀⠀⠀⠈⠻⠔⠁
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢾⢿⡯⠿⠆⠷⠤⡦⠇⠁⠀⠰⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣌⣈⠉⡉⡉⠉⠀⠀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠛⢁⣠⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⢁⣠⣶⣿⣿⣿⡿⣿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⣰⡿⠿⠛⠉⠃⠉⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡴⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣭⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⢹⣧⣤⠤⠶⠒⠛⠋⠉⠉⠀⠀⠠⠀⠐⠚⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠶⠛⠁⠐⢀⡀⠀⠀⠀⠀⠀⠂⠈⣉⠀⠤⠄⠀⠀⠀⠈⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢶⣟⠋⠉⠩⠄⠐⠒⠀⠁⠀⠀⠀⠠⠄⠐⠂⠀⠁⠀⠀⠀⠠⠀⠒⠀⠀⣀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⢰⢏⠔⠀⠂⠘⣧⠀⠀⠐⠂⠈⢉⡀⠠⠄⠀⠀⠐⠂⠈⢁⣀⠠⠄⠀⠀⠂⠈⢉⣀⠠⠄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⡟⠀⡀⣀⠑⠐⠸⡆⠈⠁⡀⠀⠀⠠⠄⠐⠂⠀⠀⡀⠀⠀⠠⠀⠐⠂⠈⡀⠀⠀⠄⠐⢂⣰⠏
⠀⠀⠀⠀⠀⠀⡇⠀⡅⠰⠀⠀⢀⡇⠀⢀⡀⠀⠀⠀⠀⠒⠀⠉⢀⠀⠤⠀⠀⢀⣀⣩⣤⠤⠶⠖⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠐⡐⣸⠃⠀⠀⠀⠤⠀⠒⣀⣁⣤⡤⠴⠶⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣎⠀⢐⣵⣏⣤⠤⠶⠖⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢻⠛⡟⣛⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⡠⢊⠔⡠⢀⣴⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠄⠔⡀⠀⠀⠀⠀⠀⠀⠈⠀⠙⡸⠋⠀⠀⠀⢀⠜⠄⢁⠀⣴⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢠⡤⠊⠀⠂⠉⠀⠀⠀⡀⠀⠀⣠⡐⠋⠀⠀⠀⠀⠀⡀⠀⠔⣡⣾⠟⢏⠀⡇⡆
⠀⠀⠀⡄⣀⡀⠀⠘⠛⠛⠿⠗⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⢰⠜⠁⠀⠀⣀⣀⣀⠀⠀⢀⣾⠋⢿⠀⣮⣀⡴⠃
⠀⠀⠀⡭⣽⣆⣿⣿⡇⣾⣶⡆⡀⠁⠀⠀⢠⠞⠛⠺⣤⢪⡾⠃⠀⠀⠀⠀⠙⠋⠛⠛⠻⠿⠁⠀⢸⣶⣿⢾⠿⢾⡿⠇
⠀⠀⠀⠛⠋⠃⠙⠛⠋⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣰⠟⠀⢀⣀⣀⡤⢐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠬⠥⠐⣶⠒⠂
⠀⠀⠀⠀⢠⣴⡿⠀⠀⢾⣥⡷⠿⠛⠁⢀⣤⣏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣥⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⣿⢰
⠀⠀⠀⢠⢴⡟⠁⠀⢀⡸⡟⠀⠀⣀⣴⣷⣟⡿⣃⠀⠀⠀⠀⠀⡆⠀⠀⢸⠩⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⡄⢨⣿⣩
⠀⠀⠀⢀⣿⠁⠀⢠⣸⢓⣦⣸⣶⣿⢾⡽⠃⣠⠏⠀⠀⠐⢄⠀⠃⠀⠀⠀⣗⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⠅⢠⡿⢮⡄
⠀⠀⠀⣼⡏⠀⠀⠈⠇⠈⠁⠹⣭⣋⠑⠞⠋⠀⠀⠀⠀⠀⠈⢲⡀⠀⠀⣀⡖⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⣸⡞⠁
⠀⠀⠘⠥⢷⣄⠀⠀⠀⠀⠀⠀⣿⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠈⣹⣿⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠖⠀⠀⠀⠀⠀⠈⠀⠀⠈⢹⠓⠀⠀⠀⠀⠈⠀⠀⡆⠀⠀⠀⢚⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣇⠀⠀⠉⠀⠀⠀⠘⠀⠀⠃⢿⡀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠂⣯⣿
⠀⠀⠰⠀⠲⡆⢀⡀⠀⠀⠈⣿⡄⠀⠀⠀⡟⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣥⣨⠀⠀⠀⠈⣿⡇⠀⠀⠀⡇⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠒⢒⠒⠒⠒⠆⠠⢄⣀
⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⢽⣷⠀⠀⠀⡟⠀⠀⠉⠀⠀⠀⠀⠀⣠⠞⠉⡤⠶⠶⣾⠛⢛⣃⣞⡣⠴⠶⠀⠀⠑⢢⡀
⠀⠀⠈⠀⢠⣿⡗⠀⠀⠀⠀⢸⣿⠀⠀⢀⣥⠀⠀⠆⠀⠀⠀⠀⢰⣾⣄⡀⠤⢄⠴⠒⠊⡽⠋⢏⣉⣉⣩⡚⠂⠀⢀⡇
⠀⠀⠃⠀⠈⠋⠃⠀⠀⢰⠖⠋⠘⠀⠀⢸⣤⣄⣀⡀⠀⠀⠀⠀⠘⣿⣾⣿⣶⣦⣤⣀⣈⣀⣀⣉⣉⣁⢤⡤⣶⣾⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠋⠁⠀⠀⠀⠀⠀⠈⠙⠻⢿⡏⣷⣿⣿⣾⣶⣿⠯⡟⣾⡿⡭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡎⣿⠀⣾⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⠿⠿⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣧⣿⠀⠻⢷⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠲⠂⠲⠂⣿⠇⠀⣸⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣤⣄⣀⣿⠀⠀⡿⠄⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⡿⣷⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣼⣿⡟⠀⢀⡇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⡿⠟⣽⢿⡆
⠀⠀⠀⠀⠀⡴⠠⠦⡈⣱⢲⣒⠀⢀⣐⣀⡴⢿⠛⡟⠻⠿⠟⢻⠟⣷⠶⢶⡷⠦⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠛⠛⠃
⠀⠀⠀⠀⠀⠙⢛⠉⠉⡁⠈⠉⣩⠿⡏⠀⠀⣿⠲⡀⠢⠀⠐⠁⡖⢿⠀⠄⢹⡄⡌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡇⠀⠰⣯⠀⠀⠁⣯⠂⠉⠀⠘⡆⠀⢸⠇⠈⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⡟⣿⣿⢻⣧⣿
⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠿⠄⠀⠀⣿⠀⢸⡇⠀⠀⠀⠏⡀⠀⠀⠁⡇⠀⢼⣧⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⠇⢻⡿⠸⣷⡿
⠀⠀⠀⠀⠀⢰⣧⠀⠀⠘⢧⠈⢀⣤⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢐⡇⢀⡞⠘⠻⣦⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠈⣿⣆⠀⣀⣠⣠⡼⠋⢸⡗⠈⠃⠀⠀⠀⠰⠂⠀⠀⢸⡆⢸⡇⠀⢸⠷⠒⠒⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣟⡋
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠋⠀⠀⠸⡇⢨⢷⠀⠀⠀⠀⠀⠀⠀⢺⠀⣾⠀⢠⠇⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢻⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠸⠀⠀⠀⠀⠀⠀⠂⢹⡀⣿⡠⠊⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣦⠀⠀⠀⠀⠀⠀⠀⢸⠄⠁⠀⠐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢶⣶⣶⣶⣶⡶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠠⠽⡄⠀⠀⠀⠀⠀⠀⡿⢋⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⢿⣿⣿⣿⡷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠒⠲⢯⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⠟⠻⠛⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⠿⠿⢯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣁⠠⢤⣶⢾⢷⣶⣄⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣉⣿⣯⣭⣴⡎⠉⠉⢻⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡙⠋⢹⣿⡿⣿⣇⣀⣀⢼⣷⡇
⠀⠀⠀⣀⣶⡜⠻⠗⠿⠿⠶⠾⠷⣴⣶⣦⣤⣴⣄⣤⣤⣀⣀⣄⢀⣀⠀⢀⠀⠀⠀⠀⣀⣿⡷⠶⠾⡿⢏⡙⠙⢻⣥⣼⣿⣰⣷⣄⡀
⠀⠀⢠⡞⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠛⠟⠿⠲⠷⣶⣿⣼⣠⣔⣰⣴⣾⣿⣟⠘⠀⡞⣫⠿⣿⣍⠃
⠀⠀⠘⣷⣷⣧⣿⣶⠔⣠⣶⣶⣶⣵⣤⣦⣤⣤⣤⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣟⠻⠿⠿⢯⣴⣴⣿⣧⣤⣿⣿⣀⡀⢸⠓⡦⠄
⠀⠀⠀⠹⠿⢿⣭⣥⣿⣿⣛⣛⣻⣿⣿⣿⣟⡋⠀⣤⠞⠋⠩⢶⣽⣿⣿⢿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⢯⣹⣿⣿⣿⣿⣿⣶⣷⣿⣦⠛⠷⣀⣢
⠀⠀⠀⠀⠀⠈⠈⠀⠙⠉⠉⠋⠉⠙⠛⠉⠛⠛⠛⠛⠛⠻⠿⠟⠻⠿⠒⠾⠷⠾⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡏⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣟⣍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠊⠩⠭⠗⠚⠋⠉⠁

⣶⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣦⣾⣤⣿⣥⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⢉⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢛⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢩⠉⢰⠎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣨⣵⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡆⢿⣧⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⡀⡃⠔⠀⣊⡂⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠻⣇⠰⠈⠙⣿⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⡄
⠀⠐⠀⠰⢄⡀⠀⠀⠀⠀⡐⠒⡾⠁⠥⠀⢆⠷⢡⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠰⠏⣵⣧⣄⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠓⡀
⠀⠀⠈⠛⠀⠀⠁⠀⠀⠆⠠⠞⢁⠀⠀⠀⠓⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⡃⣰⢥⡾⠋⠀⠂⠠⠤⣌⡁⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⠓⠶⠟⠛⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠉⣽⣿⣿⡁⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⡀⠤⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠁⣺⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⠈⢏⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠂⣿⡿⢎⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⠂⠀⠀⠀⢁⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠃⢀⠼⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠛⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⣠⡼⠁⠀⠘⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⡁⠘⣷⠀⠀⠀⠀⠀⠀⠀⠆⣠⣶⠋⠀⠀⠀⠀⠱⣄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣾⠁⠀⠀⠀⠀⠀⠱⠜⡎⠀⠀⠀⠀⠀⠀⡇⣐⡿⠃⠀⠀⠀⠀⠀⠀⠨⣀⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⡟⠀⠀⠀⠀⠀⠀⠀⡄⢧⠀⠀⠀⠀⠀⠘⠦⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠋⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠉⠉⠉⠉⠀⣿⣷⠀⠀⢀⡀⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠙⠟⠃⠛⠟⠁⣿⡿⠀⠀⢸⡧⠐⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠀⠀⣀⡀⠀⠀⣸⡗⠀⠀⢸⣿⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣴⡿⣟⡿⣷⣼⣿⠇⠀⠀⣼⠀⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣿⣷⣿⣿⣟⡀⣀⣀⣿⢀⠁⠀⢀⣿
⢀⠤⣤⣶⣪⢻⣛⡲⡤⠤⠤⠔⠒⠒⠒⢂⣸⣿⣿⢻⡿⠻⢿⣿⣿⠿⢿⡟⢿⡶⣧⣤⣾⠶⣤⣸⣿
⢯⣤⣤⡤⠭⠾⠟⢁⠾⠶⠶⢒⣶⣶⠛⠛⠁⣼⣿⣀⠳⣄⠀⠀⠀⢀⠞⣁⣸⣧⠀⠀⠙⣦⠀⠙⠣⡄
⠀⠀⢸⠀⠀⠠⠔⠁⠀⠀⣠⠞⢹⡇⠀⠀⠀⣿⡏⠹⢅⣈⠳⣀⠔⢁⣀⡏⢹⢿⠀⠐⠂⢸⡇⢰⣄⠘⢢⡀
⠀⠀⡇⠀⠀⠀⠀⠀⣠⠞⠀⡀⠀⡇⠠⠀⢸⢿⠀⠀⠀⠀⠉⣧⡞⠉⠀⠀⢸⢸⡇⠀⠀⢸⡇⠀⠈⠀⠀⠙⠢⡀
⠀⢰⠀⠀⠀⠀⠀⣰⣏⠀⠀⠁⠀⣿⡀⢀⢸⢸⠀⠀⠀⠀⠀⡟⠀⠀⠠⠀⢸⡻⡇⠈⠀⣸⣷⠀⠐⠆⠈⠀⠰⠘⡄
⠀⢸⡆⠀⠀⠀⢸⣍⠊⢧⠀⠀⠀⣼⡇⠀⢸⣿⠀⠀⠀⠀⠀⡇⠒⠀⠀⠀⠀⡇⡇⠀⠀⣿⢿⣷⣀⠀⠀⠀⠀⠀⢹
⠀⣿⣷⠀⠀⠀⠀⠿⣆⠈⢠⣠⣤⢿⡗⠀⢸⢿⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣿⡇⠀⣠⡏⠀⠙⠻⣾⡤⠤⠀⠀⠁
⠀⢻⣿⡆⠀⠀⠀⠀⢨⢀⢸⡿⠋⢸⣿⠆⢸⢸⠀⠀⠀⠀⠀⢸⢠⠀⠰⠂⠀⣿⡇⠈⢹⡇⠀⠀⢸⣯⡭⠭⣤⢤⢶
⠀⠈⢿⣿⣤⣴⣖⣿⣿⣾⠏⠀⠀⠀⣿⠀⢸⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣿⡇⠀⣾⠀⠀⢰⡿⠁⠀⠀⣠⣿⠟
⠀⠀⠈⠙⠛⠋⠉⠁⠈⠀⠀⠀⠀⠀⣿⠀⢸⢸⡆⠀⠀⡀⠀⠈⠀⠀⠀⠀⢈⣿⠀⢸⡟⠀⢀⡿⠁⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠸⡎⡇⠀⠀⠀⠀⢃⠚⠀⠀⠖⠈⣿⠂⢸⡇⢀⡾⠁⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⡇⡇⠀⠀⠀⠀⢠⠀⠐⠀⠀⠀⣿⠉⠜⠋⠁⠀⣰⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢷⡇⠀⠄⠀⠀⢸⠀⠀⠀⠀⠀⡿⢃⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣮⣱⣄⣀⣀⣀⣸⣀⣀⣀⣀⣸⣷⢋⡀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⡀⢴⣤⠞⣠⡔⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣷⣮⣿⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣽⣿⣿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣿⣿⡇⠀⢀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⡿⠻⣿⡏⠀⠛⠿⠃⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡀⣻⣿⣶⣄⣤⣦⣾⣿⣻⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢻⣿⣿⣿⣿⣿⣷⣮⢿⡧⢹⣯⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣻⣧⡟⠁⠈⢻⣿⣿⣿⣿⣿⣟⣻⣏⣰⣿⠙⠞⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣶⣭⣄⡀⠂⠀⠀⠙⠿⠿⠿⣿⠇⢿⠇⣼⣯⡿⣖⣋⠛⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡻⢿⣿⣎⡽⣿⢡⡀⠀⠀⠀⠀⠀⢠⣿⢀⣟⡃⢾⣷⢿⣸⠏⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣵⣎⣤⠎⡙⠻⡶⣿⢿⣉⣦⡄⠀⠀⢠⣿⡅⢸⣿⠄⣿⠿⣿⠁⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⠋⡿⢻⣾⣷⣾⡁⢻⡽⢚⣿⢑⣮⡑⢎⣿⡧⠸⣿⠒⡿⠀⠘⣆⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣇⣴⡍⣸⣟⣯⣾⣿⣺⣿⢉⣿⣤⣴⡿⠀⣾⠧⣼⠃⠀⠀⠈⠓⠶⠒⠁⠀⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠛⠙⠟⠛⢾⣿⣿⡷⢬⣭⢨⣯⢨⣯⡀⢼⣶⢴⣿⠂⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⣿⣿⡷⢾⣧⢠⣧⢀⣿⠆⣾⡾⢸⣿⠁⡗⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠂⠀⠀⠀⢀⣼⣿⣿⡃⣿⡇⢺⣟⣸⣟⣸⣿⠅⠶⡶⢼⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠠⢾⣷⡶⠃⢸⡧⣯⢨⣿⡅⣿⡏⣽⡍⣽⣷⢀⣿⡇⣿⠀⠀⡀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⢿⣧⡀⣼⢁⡿⢼⣯⠠⣿⠇⣾⠖⣾⣏⢨⣿⠄⠿⠀⠈⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⢿⡏⣸⡗⢾⣷⠠⣾⠴⢿⣏⣯⡷⠴⠿⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣿⠑⢿⠏⣻⡇⠀⣿⠀⣻⣧⡴⠀⠀⢀⡀⢠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢹⡀⢸⣿⣾⣧⣤⣽⣿⠿⠋⠀⣠⣶⣿⣿⣿⣿⣿⠟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠈⢧⣾⣿⣿⡿⠟⠋⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⡉⠸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠟⠋⠁⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣭⡤⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡏⢠⠐⣠⣶⣿⣿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⣶⣒⣀⡐⠲⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠄⡁⠀⢀⣀⣶⣿⣿⣿⡿⠿⣿⣿⡏⠀⠀⠈⠻⣿⣿⣿⣿⣷⣶⣤⣤⠽⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣎⠻⡏⠀⠀⢩⣄⢹⣿⣿⣿⣿⣿⡀⠀⢹⡿⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣾⣿⣾⣿⣿⠿⣿⣿⣿⠟⣡⠀⣾⠃⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠟⢁⣼⣿⣿⣯⣞⠁⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠋⠉⠀⣾⣿⣿⣿⣿⣿⠋⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣯⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⣶⠃⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡃⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⣿
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣯⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣄⡿
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⢹⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠷⠀⠧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣷⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⢸⠀⠀⣿⠀⠀⠀⣿⡆⠀⠀⣿⠀⢸⠀⠀⠀⠀⢈⡁⠀⢸⣿⠀⠀⢸⠀⣷⠀⠀⠀⣿⣦⡆⠀⢿
⠀⢸⡀⠀⣿⠀⠀⠀⣿⣧⠄⠀⣿⠀⢘⠀⠀⠀⠀⢸⡇⠀⢸⣿⡄⠀⢸⠀⡿⠀⠀⠀⣿⠳⡇⠀⣻
⠀⣾⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⠆⢸⠀⠀⠀⠀⢸⡇⠀⢸⣿⠁⠀⣿⠀⡷⠀⠀⠀⣿⣖⡇⠀⣿⠁
⠀⣿⠀⠀⣿⡀⠀⠀⣿⠇⠀⠀⣿⠀⠸⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⣿⠀⠓⠀⠀⠀⣿⣿⠇⠀⢿
⠀⠈⠁⠀⠈⠀⠀⠀⢹⣤⠀⠐⣿⡄⠀⠀⠀⠀⠀⠈⣧⠀⢸⣿⠀⠀⠉⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠃
⠀⢀⠀⠀⠀⠀⠀⠠⣷⡁⠀⠸⠿⠄⠀⠀⠀⠀⠀⠈⠓⠂⢸⢟⠀⠀⠀⠀⠀⠠⠤⠀⣿⢿
⣿⣿⣿⣷⣯⣐⣒⣢⣽⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠮⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡆
⣤⣶⣶⣬⣩⣉⣫⣉⣉⠉⠀⣆⠀⠒⢦⡁⠀⢠⠠⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⡇
⣦⣬⣭⣉⣙⣟⠛⠛⠛⠃⠠⠁⢰⣦⠘⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⡆⠀⠀⠤⠄
⠶⣶⣮⣭⣭⣩⡭⠛⣿⡄⠀⠲⡆⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠇
⠀⠐⠀⠒⠈⠉⠁⢀⡀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁
⠻⠿⣾⣶⡆⠀⠀⠈⠃
⠀⠀⠀⠀⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣿⣿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀⡀⠀⢘⣿⡿⠁⣴⠶⠀⡶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠂⠀⠐⠂⠠⡽⠀⠸⡘⣧⠀⠉⢀⣀⡉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢼⡁⠀⠀⠙⣿⣧⡚⠛⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢰⡯⢍⡉⢩⡝⣿⢿⣿⣿⣿⡿⢗⡒⠲⠤⠤⣀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣄⣀⢬⣉⣿⡿⣟⡐⠘⢦⡈⠉⠉⢹⡮⣿⣧⣦⣤⡀⠈⠉⢻⣀⠁⠏⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⣾⣋⠴⣙⣶⠿⣽⣄⣞⣇⠀⠀⢨⣟⣻⣽⣿⡜⣯⠯⡙⠖⠿⣾⡒⢒⠉⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠧⣝⢳⡾⣽⢻⠃⣾⠩⡯⣦⠀⡸⢿⠼⡖⣿⠾⣩⣧⢹⡄⠀⢉⣩⠟⠫⢌⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣤⣀⣹⣛⡏⣽⢻⡷⢶⠶⡿⢿⣷⢾⡷⣿⣿⡽⣿⣽⡞⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠺⠬⣿⢹⢥⣏⣸⡏⢹⠒⡗⠚⣯⠟⣏⢹⣿⡿⢩⠋⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢓⣛⣻⠷⣾⡂⣯⢽⣟⢻⡓⣿⠒⣿⡟⣿⣹⣿⡇⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠉⠀⠀⠍⣯⡷⣿⣸⣇⣸⣉⣏⣉⣏⣏⣿⣸⣿⣇⠘⠄⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠂⠉⠈⠙⢿⡃⡗⢼⡯⢽⠭⣯⢩⣯⡯⣿⢼⣿⠈⠓⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢻⡷⣿⣧⣼⣤⣧⣸⣴⣧⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡂⠀⠀⠻⣿⣦⣼⢼⡧⢼⢾⠦⡟⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡄⠀⠀⢠⣼⠞⣃⣘⢺⣾⣷⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠘⢿⡘⠿⠿⣰⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠒⠋


⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣷⠀⠀⢸⡇⠁⠀⠀⠀⣿⣧⠄⠀⠀⢸⣧⡀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⣿⡇⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⣷⣆⣧⠀⠀⢹⡇
⠀⠀⣇⠀⠀⢸⣷⠀⠀⠀⠀⣿⢾⡄⡀⠀⢸⣿⠁⠀⣿⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⢸⣿⡇⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⣿⡟⠛⠀⠀⢼⡁
⠀⢐⣿⠀⠀⢸⣟⠃⠀⠀⠀⣿⣞⡶⠃⠀⢸⣿⡆⠀⣩⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡷⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⣿⠛⡇⠀⠀⣼⡅
⠀⢸⣿⠀⠀⢸⡟⠃⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⠇⠀⣿⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡿⡇⠀⠀⢸⡇⠘⣿⠀⠀⠀⠀⢸⣿⣷⡇⠀⠀⣿⡏
⠀⢠⡿⠀⠀⢸⣿⡄⠀⠀⠀⣿⢷⡇⠀⠀⢸⣧⡀⠀⣿⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⣸⣟⡇⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⡗
⠀⠘⣧⠀⠀⠰⣿⠇⠀⠀⠀⣿⠈⠁⠀⠀⢸⡿⠁⠀⠁⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣿⢾⡇⠀⠀⢸⠗⠀⠈⠀⠀⠀⠀⢸⣿⡟⡇⠀⠀⠹⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⡇⠀⠀⢹⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⢺⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠇⠀⠀⠀⠈
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣿⡸⠁⠀⠀⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠐⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⣁⠀⢸⣧⢶⡄
⣒⣲⡶⣖⣢⣴⠆⣀⣀⣀⡀⠛⠓⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣯⢸⡇⠀⠀⠀⠁
⠻⠻⠿⠿⠿⢿⣿⣶⣮⣶⣼⣿⣶⣯⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⣿⢘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⣏
⣴⣶⣽⣿⣭⣭⣤⣀⣐⣔⣂⠀⠀⠈⠀⢹⡀⠀⠘⠻⣭⠉⠀⠀⣀⣠⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂⡿⠀⠀⠀⢀
⡬⢍⣉⡛⢛⠛⢛⣛⡿⡿⠿⣿⣷⣀⠀⡈⠁⢠⣄⠀⠿⠦⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⡀⠀⠀⠀⣀⢀⣠
⠙⠛⠛⠿⠷⠶⠶⣦⠄⣴⣤⣆⣐⠀⠀⢀⠀⠘⠛⠇⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⡏⢣⠀⠀⠀⠈⠉⠁
⠛⠻⠿⠷⣶⣶⣦⣴⣶⠂⠀⢹⣿⡀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠘
⠈⠀⠂⠈⠐⠂⡁⠉⠉⠁⣄⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠈
⣀⣀⣀⣀⣠⣀⠀⠀⠀⠀⠸⢧
⠉⠙⠛⠻⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢠⠠⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⠛⢛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⣤⡽⠿⣛⠿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡶⢿⣿⣯⣿⠀⠀⢀⣧⡆
⠀⠀⠀⢠⣠⣶⣦⣴⣤⣤⣤⣄⣠⣤⣀⣀⡀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⣼⣿⣿⠿⠿⣟⣸⣿⢧⣤⡀
⠀⠀⣨⡟⠉⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠟⠻⠷⠿⠷⢾⡶⠴⢦⣼⣿⣧⡤⢠⣭⣿⣷⡶⡏⠽⢟⣿⣿⡛⠂
⠀⠀⠺⣷⣾⣼⣶⡤⣠⣤⣤⣼⣦⣤⣀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⠿⠿⠿⣏⣭⣽⣷⣌⣿⣿⣀⠉⠽⢶⣤
⠀⠀⠈⠹⢿⣭⣭⣾⣿⣻⣿⣿⣿⣿⣿⣋⢁⣤⠞⠋⡵⣾⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣿⣿⣿⣿⣿⣷⣷⣷⡞⠻⢆⣀
⠀⠀⠀⠀⠀⠀⠁⠉⠁⠈⠉⠉⠉⠉⠙⠛⠉⠛⠋⠛⠛⠛⠻⠛⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡇⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⣮⣙⢿⣿⣿⢿⣿⣿⠿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉

⠀⠀⠀⠀⠀⢀⣠⣠⣀⣀⣠⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⢐⢣⠠⠐⢸⣏⣀⡐⡄⢯⣁⠽⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⢸⡀
⠀⠀⠀⣀⣼⣮⣁⣈⡼⠿⣏⣁⡤⠾⢿⡿⠿⣃⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⣉⣍⣭⣹⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⢸⣤⣴⠶⢹⡇⣀⠰⠬⣯⠀⠀⠀⢱⡀⣊⣧⠀⠀⠀⢀⡤⣤⢴⣿⣿⣿⣿⣭⣿⡯⣭⣭⣭⣭⣥⣀⣀⣀⣢⠗
⠀⢀⡴⣾⣉⣠⠾⠧⣏⣀⠴⠿⢌⣀⠴⠿⢿⡿⠷⠦⡄⠀⠈⠁⠉⠋⠉⠁⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠉
⠀⡏⣤⠂⢸⣇⠠⠀⠈⡇⡀⠀⠈⣯⠀⠀⠀⣿⣒⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠐⣀
⠀⠛⠬⠤⠴⠷⠤⠤⠴⠷⠽⠤⠾⠷⠤⠤⠶⠿⠿⠟⠃⠀⠀⢠⣀⣠⣤⣭⣭⣄⣀⣼⠿⠤⠿⠿⠽⠿⠯⠛⠩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡟⣷⣿⣘⣂⣦⡶⡅⠀⠀⠀⠀⠀⠛⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⠀⢀⡀⠀⠀⠀⠀⠈⣿⠟⠿⢯⡦⠠⣼⣿⠆⠆⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⣿⡿⠋
⠀⢘⣿⠄⠀⠀⠀⠀⢸⡿⠀⠉⢁⣀⡉⢸⠠⠂⠃⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠶⡁⢠⡞⠀⠀⠀⠀⠹⣏⠃⠲⠿⣆⠄⠐⢨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢳⡄⣿⡓⣄⠀⠀⣠⢿⡈⠳⣦⣤⣼⣧⣆⡈⠀⠀⠀⠀⠀⢠⢀⠴⠦⠄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠁⠟⢻⣮⡓⢿⢿⣯⣿⢖⣀⣾⢿⣿⣿⢅⠀⠀⠀⠀⠀⡜⠌⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⣾⡟⢿⣷⠟⢍⠻⣾⣿⡷⣯⢼⣿⡞⣷⡀⠀⠀⠀⠦⠂⠉⠉
⠀⠀⠀⢸⡏⡜⢠⡇⣤⠀⣢⣶⣿⣧⣤⢼⣯⢿⡹⡇⠀⠀⠀⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣀⣠⣄⣀⣀
⠀⠀⠀⢸⣿⣠⠟⡗⢿⢠⣟⣻⣷⣯⡳⢾⣿⣎⢷⡇⠀⠀⠀⢿⣿⡛⣻⣭⣉⣉⡉⣋⣁⣴⣾⣏⣡⠾⣛⣁⣈⣉⡀
⠀⠀⠀⠀⠉⠉⠙⠇⠚⠸⠷⠾⠿⠿⠿⢦⣽⡟⠚⠁⠀⠀⠀⣿⣿⡿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⡗⣠⣤⣶⣶⣬⡭⣍⣋⣀⣠⣄⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣾⣿⣻⣷⡖⠒⠶⢿⣿⣿⣿⣿⢿⣿⣷⣬⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⠯⢏⣹⣯⣷⣾⣿⣟⣻⢿⣶⣬⢛⢝⡿⣿⣷⣽⣿⣿⡿⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠭⠻⢷⣼⠿⣛⣫⢭⠬⠭⠿⣿⣿⣿⣷⣝⢿⣿⢿⣿⣿⣿⣿⣮⣷⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡋⠔⠀⠀⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⡻⡎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⢯⣿⡜⣿⣿⣿⣿⣿⣿⣿⡇⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⣷⣿⣿⣿⣿⣿⣿⣿⣿⢸⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢸⡇⣠⡤⣤⠀⠀⠀⠀⡤⠤⣤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⣿⣿⣿⣷⡙⣿⡆⠇⢻⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⡇⢸⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⣽⣿⣧⢿⣧⡄⠸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠀⣧⢸⠖⣤⡀⠒⠀⠀⠀⠀⠀⠐⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⢿⣿⠟⡇⡍⣹⡏⣿⣷⠈⠀⢀
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣿⣼⢻⣾⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣼⠟⠁⢀⠿⢀⣿⡇⣿⣽⡇⠀⢸
⠀⠀⡄⠙⢄⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⠏⠀⢀⣾⠀⡜⢹⢀⡏⢻⣻⡄⢸
⠀⠀⠘⢄⠀⠑⠄⠀⢦⠀⠀⡿⣿⢸⣿⠀⠀⠀⠈⠒⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢋⠏⠀⠀⣼⠃⡼⠀⣾⣼⠀⠈⣿⡇⠘
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠃⠀⠃⡇⣾⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠈⠀⠀⢰⠃⢰⡇⣰⣿⡇⠣⠀⠸⣷⡏
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢵⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⡠⢁⡟⢠⣿⣿⣇⠀⠈⢦⡿⢁
⠀⠀⠀⠀⠑⢄⠈⠃⠀⠀⠀⠀⠘⠇⢸⡄⠀⠆⠀⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠃⣼⢡⠿⠋⠙⠿⠷⣄⠘⡧⢸⡄
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⣷⡀⠘⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢶⡀⠀⠙⢷⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢸⣷⡀⠘⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠈⠐⢄⠀⠹⣆⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⣿⣷⡀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣼⡄⠀⠀⠠⡀⣀⠇⠀⢹⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠹⣿⣧⠓⠆⠀⠀⠊⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣰⣿⠓⠀⠀⠀⢻⣧⣤⣶⢟⣼⠃
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣠⠋⣿⠀⢠⠀⠀⠸⠀⠀⠒⠛⠁
⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡟⠠⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⡰⢏⡴⠋⠀⢠⠃
⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠁⠀⠀⠀⠀⠀⠀⢠⣕⠪⠃⠀⠀⣠⠴⠃
⠀⡞⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇⡤⠤⠒⠋⠁⠀⢀⡄
⠀⡇⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠈⠑⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⠃
⠘⢇⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠉⠺⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣲⠄⠀⠀⠴⠢⡤⠀⠀⠀⠁⢀⠀⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠧⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⣎⡃⢡⠤⣄
⠀⠀⡠⠂⠂⠀⠀⠀⠀⠀⠀⠠⡐⠧⠟⠶⠀⠀⠈⣁⡀⠀⡸⠁⠠⠀⠁⠀⠀⠀⢠⡤⠐⠂⠀⠀⠀⡸⠛⠲⣄⠀⠀⣄
⠀⠀⠂⠀⠀⠀⠀⡀⢠⠤⠂⠁⠀⠀⢀⠔⠂⠉⠈⠁⠈⠶⠋⠒⠺⠄⢤⡈⠀⣐⠁⠐⠀⡠⠀⠀⠰⠁⠀⠀⢌⠀⠀⠐
⠀⠀⢄⠤⠂⠂⠁⠀⠈⠀⠀⠀⠀⠈⠉⡀⠀⠙⡲⡄⠀⠀⠀⠀⠀⢀⠂⠀⡶⠉⠙⠒⠮⠤⣤⣠⣇⣠⠀⠀⠀⠑⠀⠀⠈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢠⡄⠀⠈⡔⠢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⡄⠀⠀⠀⠀⣀⡤⠚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠁⠀⠀⠈⢙⠓⠒⡎⠠⠲⠦⣄⣀⠀⠀⠀⠀⠀⢱⠒⠒⠒⠉⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠀⠑⠦⣄⡀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠈⠉⢛⠛⠛⠃⠉⢇⠀⠀⠀⠀⡄⢀⡰⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠉⠉⠛⠒⠚⢦⣄⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⡏⠙⠛⠉⠋⠁⣀⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁⠉⠓⠒⠲⠶⡦⠤⠤⠤⠶⠒⠁⠀⠀⠈⣉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⠦⣄⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠀⠀⠀⠀⠀⠀⠄⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡅⠀⠉⠙⠛⢻⠀⠠⠤⢀⣀⣀⣀⣀⣀⣀⣀⣤⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠇⢀⢀⣠⠀⠸⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠒⠘⠂⠐⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣿⣀⠄⠰⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⣀⠀⠀⠠⠤⠀⠤⠄⠤⠔⠤⠀⠁⠀⠈⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠈⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣀⣭⠔⠀⠊⠁⠀⠄⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠠⠠⠀⠀⠀⠀⣀⠀⠙⠻⠟⠧⠄⠊⠀⠈⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⢀⣀⠘⠁⠂⠀⠀⠀⠄⠦⠈⠠⠄⠀⠀⠀⣀⣀⠒⠒⠃
⠀⠀⠀⠀⠀⢀⡠⠘⠉⠉⠉⠛⠳⢦⣄⠀⠀⠀⠀⠠⠴⡊⣁⠀⠠⣾⠁⠀⢽⡿⢭⡉⠠⢀⠠⢀⢀⣠⣔⣂⡴⠒⠒⠃
⠀⠀⠀⠀⡰⠀⠙⣦⢀⠁⠀⢀⠀⠀⠀⢣⠀⠀⠀⣀⠀⠠⠤⠀⠀⠴⠖⠀⠤⠈⠨⠅⠀⠀⢀⡁⠡⠼⠊⠁
⠀⠀⠀⠀⡜⠀⢠⡈⠀⠐⠳⠄⠀⠈⠈⡦⣀⣀⡠⠤⠖⣊⠄⠈⢀⠀⠁⠊⠀⣁⡠⠄⠠⠋⠉⠁
⠀⠀⠀⠀⠷⠃⠸⠃⠀⢶⠈⠐⠀⠈⣾⠐⠛⠀⠖⠩⠋⠁⠀⣈⣡⠤⠾⠂⠉
⠀⠀⠀⠀⠈⢛⡠⠒⠃⠐⠂⡀⠂⠈⠀⢐⠃⢀⠀⢀⡰⠋⠁
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠠⠄⠁⢄⡔⣫⢀⡠⠐⠊⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂⣂⡤⠞⠋⠁⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⡴⠶⠶⠶⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡖⡫⣍⢰⠍⡀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣾⣟⣳⣤⣋⣀⣉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⡴⢏⣼⡷⡟⠖⠀⠀⠀⠀⠀⠤⡉⢆⠈⣷⡀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠙⡋⠉⠁⠀⠀⠀⠀⠈⠉⠉⠻⡄
⠀⠀⠀⠀⠀⣸⠃⣤⣿⣋⣄⠀⠀⠀⠀⠀⠀⠀⠰⢤⡛⠀⢳⠀⠀⠀⠀⢀⠏⠀⣁⠉⢀⣀⣀⣀⣤⣀⣄⣀⣀⣀⣀⡀⢹⡄
⠀⠀⠀⠀⠀⣯⠐⣿⣜⣏⠀⠧⣀⣠⠂⠀⠀⠀⠀⠄⠐⠂⠘⡆⠀⠀⠀⣾⣴⣿⣿⣿⣿⣽⡯⠷⠊⠀⣀⣠⠂⠀⡜⠉⣏⣧
⠀⠀⠀⠀⢸⠃⣾⡾⢳⠎⢡⠠⢮⡀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⢠⡟⢿⣿⣿⠉⠁⠙⢷⠳⣆⠚⠁⠀⠙⣦⢶⣶⠟⢺
⠀⠀⠀⠀⢸⠤⣮⣡⡞⣧⠛⢄⡸⠁⠀⠀⡠⠄⠀⠀⠀⠀⠈⣽⠀⠀⢸⡇⣼⣿⣿⠀⠀⠀⢸⡳⢍⣆⠀⠀⠀⣷⢬⡏⠀⢸
⠀⠀⠀⠀⢸⡶⠛⠟⠣⠔⠤⠀⠀⠀⠀⠀⣀⠐⢃⡤⠀⠀⠀⡟⠀⠀⠘⣧⣿⣿⣿⣧⣤⣴⣿⡶⢒⡽⣦⡤⢴⠋⠸⣇⡀⢸
⠀⠀⠀⠀⠘⣗⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⢨⠀⢦⡠⠈⠀⢠⡇⠀⠀⠀⢿⡿⠿⠿⠿⠿⠿⠿⠛⠚⠓⠚⠲⠾⠶⠶⢼⣽⣿
⠀⠀⠀⠀⠀⢻⠉⣿⣵⡶⠋⡴⠀⣄⠀⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠹⡄⠸⡙⢂⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠈⣧⠸⣿⡟⡞⠱⠲⣅⠀⠀⠀⢀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢳⣖⣿⣿⣿⣿⣿⣿⣿⣯⠏⢻⢷⠲⢦⣤⡞
⠀⠀⠀⠀⠀⠀⠈⢧⣹⣿⣇⡰⢦⠛⠀⠀⠀⢄⠐⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢻⡉⠉⢻⣿⣿⠋⠀⠈⢧⠀⢠⠟⢹⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡼⣷⣶⡖⠀⠀⠀⠐⠂⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣸⣿⣿⣦⣀⣠⣟⠲⡇⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⢏⡶⣦⡛⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⣿⡇⠾⠟⠻⠧⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⢷⡼⠓⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠤⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣟⡧⠦⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣶⠖⠒⠲⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠈⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡄⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠠⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠒⠀⠀⢀⠀⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠠⠄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣛⣛⣙⣛⣿⣛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣏⣩⣭⣽⣿⣿⣛⣛⣻⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢰⠀⠀⠛⠛⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⠀⣀⠀⠀⠀⠀⣠⡀⢀⠀⠠⠀⠀⠐⠂⠤⠄
⠀⢀⠤⠐⠒⠒⠒⠂⠀⠀⠉⠉⠁⠈⠁⠁⠀⠀⠀⠒⠒⠒⡞⠹⢿⡀⠀⠀⠀⠀⠄⠈⣹⡗⠦⠆⠀⠠⣶⡦⠀⠄⠀⠀⡀
⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⣼⣧⠀⠀⠀⠀⠉⠉⠈⣤⣀⠐⠃⠀⠈
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⢉⣿⣦⣠⣤⣾⣤⣤⣾⣿⠛⣵⡿⠥⡤⠤⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⢟⠿⢛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠓⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⡄⣶⠄⡖⡔⢢⡴⢢⡴⣤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣉⣉⣁⠐⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠁⠁⠁⠈⠁⠈⠉⠁⠉
⠀⠀⠀⠀⠀⢰⠀⠀⠀⡆⠀⡁⠀⠉⠘⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⠠⠿⠼⡅⠼⠇
⠀⠀⠀⠀⠀⠀⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⢠⢀⠀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⣄⠀⡀⠀⠀⣠⡔⡀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠓⠚⠓⠚⠘⠘⠂⠃⠋⠃⠃
⠀⠀⠀⡀⠀⠀⠀⠀⠉⠉⣈⠟⠁⠀⡇⠀⠈⠑⠒⡒⡂
⠀⠀⠀⠑⠠⣀⠀⠀⠀⠠⠂⠀⠀⠀⢌⢀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠃⠐⠐⠀⢀⠀⠀⠀⠁⠀⠠⠂
⠀⠀⠀⠢⠀⠀⠈⠀⠀⠈⢆⠀⠀⠀⢁⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠂⠐⠁⠀⠑⠀⢀⣀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢀⣾⣽⣿⣿⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⣴⣿⣿⣧⢹⣿⣶⣶⣦⣄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣶⣶⣶⣄
⠀⠀⠀⠘⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⢻⣿⣾⣿⣹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢠⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⠀⢀⣤⣤⣝⢯⣤⣽⣭⢻⣿⣿⣿⣶⣆⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢸⣿⣹⣿⢻⣿⣹⣿⡗⣿⣿⣿⣿⡟⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠈⠻⠿⠞⠛⠯⠿⠽⠟⠛⠋⠉⠁⠀⠀⠀⠙⠛⠋⠈⠻⠿⠟⠓⠛⠋⠉⠀⠀⠀⠈⠻⠿⠞⠛⠿⡿⠽⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠍⠁⠀⠈⠉⠙⠻⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⡷⣶⠀⣤⣤⣤⡄⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⡏⠀⡀⠀⠀⠀⣼⢯⢺⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡴⠂⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣧⣴⣶⣶⣤⣾⣿⠁
⠀⠀⠀⠀⠀⢀⣴⣿⣌⢻⡿⠁⠀⠐⡄⠀⠀⠀⠀⠀⣠⣿⣿⣶⣦⣤⣽⣿⣿⣿⠋
⠀⣀⣤⣴⣾⣿⣿⣿⠟⠁⠘⢆⠀⠀⠘⢆⢀⠤⠒⡛⣿⢿⣿⣿⣿⣿⣿⣿⡿⠃⡞⢢
⠀⠈⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢱⣄⣀⠀⠳⣄⠀⠀⠹⠌⠻⣿⣿⣿⣿⠋⠁⣰⡇⠀⠑⢄⡀
⠀⠀⠀⠙⠛⢿⠃⠀⠀⠀⣠⠞⠁⠀⠈⠑⣀⠈⠃⠀⠀⠀⢠⡀⡿⣦⣬⣴⣾⣿⣧⡀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠈⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠘⠷⠃⠈⢿⣿⣿⡏⠁⠛⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⡀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠛⢦⣀⠀⠀⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠀⠀⠀⠀⠀⡀⠀⠀⠓⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠤⠂⠀⠀⠀⠀⠀⡷⡄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⣷⠀⣠⡤⠆⢤⣀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⣄⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⢠⣟⡽⠋⣀⣤⣀⣈⣳⣄⣴⡂⠀⠀⠀⠀⠀⠀⠀⠀⢘⡽⠟⢠⡧⡀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⢋⡴⢋⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠱⡄⠀⠀⣠⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⡟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣶⠀⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠳⣽⣆⠉⠉⠋⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠶⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣦⣀⠀⠀⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣼⣿⣿⣿⣿⣿⣷⠞⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣿⣿⡟⢍⢻⣿⡟⠉⠁⠀⢀⣀⠀⠠⣄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⡇⢰⢀⣿⡆⠀⠀⠀⢨⣍⣻⣦⣬⣿
⠀⠀⠀⠀⠀⠀⢠⡘⠛⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠻⣿⣧⣤⣾⣿⣿⡀⠀⠀⠀⠀⡉⠉⠙⠁
⠀⠀⠀⠀⠀⢠⣼⣿⣿⣷⣾⣋⠐⠄⠀⠀⠀⠀⠀⣠⠂⠀⠸⣿⣿⣿⣷⣶⣶⣶⣾⣤⣤⡤
⠀⠀⠀⠀⠀⢲⣿⣽⣿⣿⡏⢙⣛⡲⢦⣤⡀⠀⠸⣧⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⠼⣧⡈⢷⣯⣛⣿⣿⣶⣤⣉⠒⢀⡀⣽⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⣶⣬⣿⣿⣏⣝⠿⢿⣶⣌⠛⢿⡟⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡧⠲⠂⠉⣉⠙⠻⣿⣍⠫⣻⣶⣌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⢶⡴⢠⡞⢡⣼⡓⢦⡙⢷⣟⣁⠙⠿⢄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣴⣾⢦⣾⣯⡉⢻⣮⡲⣹⣿⣾⣅⢀⣺⣦⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠷⣾⣇⣉⣟⠢⣽⣿⣌⣻⣿⣻⣿⣿⣷⣭⡙⣿⠶⣌⠰⡎⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣶⣿⢶⣼⣿⠲⣼⣟⢻⣿⡏⢿⢻⣿⣿⣯⠻⣿⣷⣹⡿⠁⣿⣿⣶⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣻⣿⣾⠇⣿⣟⣚⣿⣏⣲⣾⡏⢻⣿⣿⢾⣿⣿⣿⣿⡆⢻⡉⠉⠀⣨⣿⣿⣻⣿⣿⡧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⠉⣿⣿⣻⣿⣧⣿⡿⣾⣿⣯⣿⢿⣷⣿⠿⣿⣷⢀⡇⠀⣴⣩⣿⣧⣿⢫⣿⣿⡆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡯⢽⣿⣷⣿⣿⣈⣹⣿⣿⣿⣿⣿⡦⣿⢾⣿⣿⣿⡞⠀⢀⣼⡿⢿⣿⠿⠿⣿⣯⣷⣦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡶⣿⡷⠚⣿⠻⣿⣿⣾⣿⣹⣿⣷⣿⣿⣿⡿⠋⠀⢀⣼⡿⢃⡟⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣿⣟⣸⣿⡦⣸⣿⢼⣿⣿⣿⣿⣿⠟⠋⠀⡀⣠⣽⢟⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢿⣏⣹⡿⢸⣿⣧⣿⣿⣻⡿⠟⠛⠉⠀⠀⢀⢨⢵⡿⡣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢸⡏⢹⣷⡟⣿⣷⣸⠟⠋⠀⠀⠀⠀⢠⣠⣼⡟⢃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠞⢻⣷⣨⠟⠁⠀⠀⠀⠂⠀⣀⡺⣬⡑⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢉⣼⠋⣠⡆⠀⠀⠀⠀⠀⣈⢷⡟⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⢸⡇⠀⠀⠁⠀⠀⠀⠀⡰⠍⢃⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢳⣄⠀⠀⠀⠀⠀⣈⣡⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣬⣠⠤⠴⠋⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⡭⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⢀⣠⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⢀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⠞⠋⠁⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣿⣿⡄⢀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⡿⢚⣫⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⢋⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⠛⠫⣽⣿⣿⣿⣟⢩⣤⢚⠛⣙⣲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣳⣾⠶⡗⣿⠛⣿⢻⡏⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣟⢻⣿⣾⣿⠿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢻⣿⡷⣿⢿⣽⣿⣿⠀⠀⡀⠉⢉⠉⢀⡹⣬⣵⣤⣷⣭⣽⣿⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⠀⠀⢸⣿⣇⣿⣏⣿⣿⣿⢛⠛⠛⠛⠛⢻⡟⣛⠉⠁⠒⠒⠀⠀⠀⠀⠈⠰⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣶⣾⠜⢆⠀⠀⠀⠀⢸⣿⣿⣿⣏⣿⣿⣿⣷⣿⣿⣷⣾⣾⣿⣿⠂⠐⠛⠓⠠⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⣿⣿⠅⢠⡵⣄⠀⣀⣼⣿⣿⡏⣷⣿⡿⣿⣿⡗⣿⡈⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⠓⠒⠿⢻⡅⣀⢵⣿⣿⠿⢿⣿⣿⡇⣿⢹⡇⡞⣿⠀⢾⣇⣻⣏⣿⡿⠶⠶⠶⠶⣶⣦⣤⣴⣧⣿⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠛⠋⠉⠀⠀⠘⠿⣿⣇⣃⣺⣾⣿⣻⢀⣽⢸⣂⣸⣿⠇⠀⠀⣄⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠐⢒⠀⢠⠀⠀⠀⠀⢀⣈⡉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠴⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠘⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠩⡍⢑⢶⡦⠄
⠀⠰⢀⠉⣁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢀⡀⠀⣀⠀⠀⠸⠀⠀⠂⠀⠀⣼⠁⡖⠓⣁⠼⠂
⠀⠀⠀⠡⡀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⠀⠀⠀⢠⠎⠁⠀⠀⠀⠑⠤⣀⣀⣀⣀⡠⠖⠁⢐⣋⡥⠵⢶⡖⢉⡇⠦
⠀⠀⠀⠀⢡⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢐⢠⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡏⢠⣀⠤⣥⣤⢾⡄
⠀⠀⠀⠀⠁⠀⠐⠠⠋⠀⠀⢰⡀⠀⠀⠀⠈⠊⢁⢀⠙⢆⠈⠐⢤⠀⠀⠀⡰⠁⡴⠀⠁⢏⣂⡤⠔⠳⣆⠚⠀⠀⠀⠀⠀⡾⠿⠟
⠀⠀⠀⠀⠀⠐⢄⠄⠈⠠⠂⠠⠂⣠⡤⢄⠀⠀⢠⡆⠀⠀⢄⠀⠀⠁⠤⠊⠀⠀⠀⠀⠀⠀⠉⠲⠋⠑⠤⠁
⠀⠀⠀⠀⠀⠀⠈⡆⠳⠀⠀⠀⢠⠀⠀⢠⠓⠃⠀⠂⠀⠀⠀⠉⠉⠉⢙⡿⢶⣤⣍⠲⣔⠤⣈⡁⠀⠀⠀⠀⠀⡤⢢⢄
⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⡄⠀⠀⠀⢏⡀⠤⡠⠇⣀⣤⣴⣶⡆⠀⠈⠲⣨⣟⣿⣷⣼⠀⢬⣿⡇⠀⠀⠈⠣⠤⣤⣬⣑⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠤⠀⠄⠡⠀⢇⣾⢩⠁⡄⣸⠇⠀⠀⠀⠈⠝⣭⡛⣿⢦⣶⣿⠖⠂⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠛⠃⠀⠀⠐⠈⠂⢷⠘⠣⠿⠋⠀⠀⠀⠀⠈⣆⣿⣧⠯⣈⠿⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠄⠈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢅⠤⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⣧⠘
⠀⠀⠀⡀⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠀⠀⠀⠀⢀⣶⠉⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠀⠀⠀⠀⣷⠀⠀⠀⠀⡀⠠⠀⢰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠚⠃⣦⡀⠀⢠⠆⠉⠁⠀⠀⠀⠙⠲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁
⠀⠀⠀⠀⠀⢠⠷⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⠃⢀⠋⠀⠀⠀⣴⣆⠀⠀⠀⠈⡃
⠀⠀⠀⠀⢀⡎⠀⠀⠀⢀⣀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢂⠀⠀⢀⠀⢁⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠐⠀⠀⢺⡄⠀⠀⠀⡀⠄⠀⠈⠭⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⣀⣀⠀⠀⠂⠈⠀⠃
⠀⠀⠀⠀⢸⠁⠊⠉⠁⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡬⠀⢸⡆
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣠⠶⠒⠀⢀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠺⡁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣫⣿⠄⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⣀⠄⠤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⣶⠟⠿⣦⡉⢠⡄⢤⠀⠀⠀⠐⢀⠀⢀⣤⣈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⣀⣤⣴⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠘⣷⠀⢻⡌⠃⠀⠀⠀⠀⠀⣿⣿⣿⠇⡸⠀⠀⠀⡀⠀⠠⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡇⠀⠀⢀⣀⡀⠸⡀⢶⣷⠠⠀⠀⠀⠀⠀⣤⣽⣿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣀⡾⣿⣿⢿⣓⡇⠈⢁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢿⣧⣥⣥⡷⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⢠⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⢸⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣹⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠁⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣦⠀⠀⠀⠀⢠⡏⢠⡀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣠⣤⣤⢠⣼⣻⡟⣋⠈⠹⠆⠀⠀⠀⢸⣷⠘⠀⠀⠀⢀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⢇⣠⣶⣿⣿⡿⣿⡟⣿⣹⣦⣍⣙⣃⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⢻⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠳⡍⢿⣿⣿⣿⣿⣿⣿⣿⡋⢿⣿⣿⡿⢷⡀⠀⠀⢾⢿⡆⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣴⡈⠛⢻⣿⣿⣿⣿⣿⢷⡾⣽⣿⣮⡽⣿⠀⠀⣯⡀⡁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣷⡏⠀⠀⣿⢀⠇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢛⣿⣿⣟⣿⣿⠿⠿⢿⣿⣿⠁⠀⠀⢸⡟⠰⠀⠀⠀⠀⠫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣎⠰⡉⠉⠉⠀⠀⠀⠀⣿⡿⠀⠀⠀⣸⡇⡆⠀⠀⠀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣴⡟⠻⠿⢿⣾⣶⣶⡖⠀⠙⠉⠀⠀⠀⠀⢿⡏⠁⠀⠀⠀⠸⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣿⣷⠀⠀⣠⡤⠐⠀⢧⠀⠀⠀⠀⠀⠀⠀⢸⢹⡀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⣤⣾⠇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⠀⠀⠀⠀⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡴⠉⣿⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⢻⠁⠀⠀⠀⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⡾⠀⠀⣿⡃⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢹⡈⢀⢀⠀⠀⠀⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣴⢂⠜⠁⠀⠀⠸⡗⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⢈⡁⡟⠸⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠷⢶⠋⠀⠀⠀⠀⠀⣿⣄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠨⡇⠟⠀⠰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⡴⡆⠀⠀⠀⣻⣿⡿⠁⠀⠀⠀⠀⠀⠀⢻⡟⠓⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣆⠀⠀⠠⠀⢠
⠀⠀⠀⠠⠖⠋⠁⠐⠂⣀⡤⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢰⢂⠘⣧⠀⠀⠰⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠉⠀⠀⠀⠀⡠⠀⣀⡀⠀⠸⣿⣀⢀⡃⠀⠀⠀⠀⠀⠀⠀⣸⣹⡇⢮⠀⠀⠀⠘⢸⡄
⠀⠀⠀⡰⠚⠉⠀⠀⣀⠈⠀⠆⠒⠐⠂⠁⠒⠲⠬⠄⠀⠀⠈⢻⣍⣡⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⡶⠈⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⣛⣛⠀⠤⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡄⠀⠀⠀⠀⢠⣴⠟⠋⣁⣵⠀⢀⣄⠀⢠⣤⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⣷⣄⠀⠀⠀⠘⠛⠻⠟⠋⠡⠠⠿⡏⠀⠸⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⡿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡯⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⡄⠀⠀⠄⠀⠀⠀⠲⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢄⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⠄⠁⠀⠀⠀⢀⢀⣧⠘⢀⠀⠀⠀⣀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠂⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⣀⠈⠀⠀⡸⠀⠀⣰⠛⠀⠀⠃⠀⠀⠀⠀⡟⠀⠀⠀⢐⡏
⠀⠀⠀⡀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠋⢠⠀⠀⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢫⡁⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡼⠀⠀⢀⠆⠀⣰⠃⢀⡄⢠⠠⠀⠀⠀⠀⠀⢸⡇⠀⠀⢦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠟⢠⠀⢸⠇⠀⠀⣻⠀⢰⡇⠀⢉⡴⠋⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣾⠀⢐⢠⡇⠀⢠⠁⠃⡾⢠⣾⣥⡦⢂⠄⠀⠀⣸⠁⠀⠀⠀⠈
⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠿⣿⢁⡀⣼⣼⣣⢁⣾⣿⠡⠀⣡⡆⠀⣰⣧
⠃⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠖⠧⠀⣰⡞⢠⡿⡴⣶⣽⡏⠘⣡⢷⣟⣧⣾⣿⣏⣧⣬⠍⣠⣼⣿⡿
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡔⠀⠀⣠⢁⣾⡷⣽⢃⣮⠀⢠⣿⣿⣽⣿⣿⣿⣿⣾⡏⣰⢿⣿⠟⠁⢠⠄
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠋⠀⠈⠀⠠⠁⡸⠛⠩⠇⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⡂⠀⡀
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠀⠀⠀⠈⠂⡀⠀⠀⢀⠀⠀⠀⡃⠀⠀⣵⡾⢁⠔⠀⡠⢰⣾⣿⣿⣿⢟⡿⠃⡻⣿⣿⠿⠁⠂⡿⠟⣴⠍⠀⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⢦⣀⣠⠀⠐⠀⠀⢸⠀⠀⠀⠀⢸⠀⡏⠁⠰⠄⣴⠁⢀⣿⢡⡿⠛⢄⠀⣾⣿⡿⡼⠐⠀⢸⢃⢸⡇⠀⠖⠠⠆
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⣼⢿⡟⠿⠡⠀⠀⡙⠀⠀⠀⢀⠃⠀⠋⡐⠁⠀⠁⢰⣿⠿⣿⢹⠷⣎⣾⣿⣿⣣⣢⢃⢠⣏⢎⣠⠃⠈⠀⠁
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡀⠀⠸⡭⣻⠸⠀⠀⣦⣿⠀⠀⠀⢸⠀⠀⠘⠀⠀⠸⠀⢀⣿⢰⡇⡄⣿⣾⣿⣿⣿⣿⣿⣷⠏⢀⡼⠁⠀⣠⢷⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⣅⠠⠱⡉⣄⠀⠐⠉⢹⡀⠀⠀⠸⠀⠀⣍⠀⠀⢦⡀⢸⣟⢿⣷⢳⣿⣿⣿⣿⣿⣿⣿⡟⠀⡟⢁⠀⠛⠉⣸⡟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠠⡀⢙⣿⣟⠢⡀⠀⣇⠀⠀⠀⡀⠀⢹⣧⡆⣼⡇⣸⣿⢀⣿⣫⣿⣿⣿⣿⣿⣿⣿⠁⢸⣁⣇⠂⠈⠀⠛⠄⠀⠀⠀⠀⢀
⠄⠀⠀⠙⠀⠀⠀⠀⠀⠘⢦⣄⢀⡈⣯⣿⣍⣦⡓⣤⣸⡄⠀⠀⣇⢈⠻⣿⣿⣿⣧⢺⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠚
⠰⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠈⢻⣍⢌⠻⢿⡿⠀⠯⣿⢹⡀⠀⢸⣔⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣹⡂⠈⠀⠀⡄⠀⠀⢀⠇⠀⠀⢀⡶
⠀⢣⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠠⣀⠐⠂⠀⠀⠁⠀⣆⣿⢉⣹⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠸⣇⠀⠀⠀⠀⠀⠰⠿⠈⠀⠀⠈⡷
⣤⣤⣧⣤⣤⣴⣦⣤⣴⣦⣦⣤⣤⣦⣤⣮⣷⣾⣷⣿⣦⣴⣦⣴⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣦⣿⣤⣤⣤⣦⣤⣤⣦⣷⣤⣾⣦⣴⣴
⣿⣧⣒⣯⣭⣭⣭⣭⣩⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠿⠋⠉⠉⠙⠻⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠛⠉⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣦⣄⡀⠀⠘⠻⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⢀⣴⣿⣿⣿⠿⠿⠿⣿⣿⣿⣦⡀⠀⠀⠀⠙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡗⠀⠀⣾⣿⣿⣿⣿⣷⣄⠀⠀⠙⣿⣿⣷⡀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠀⠀⢸⣿⣿⡇⠈⢻⣿⣿⣷⣄⠀⢸⣿⣿⡇⠀⠀⠀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠘⣿⣿⣧⠀⠀⠙⠻⣿⣿⣷⣼⣿⣿⠇⠀⢠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⠀⠹⣿⣿⣷⣤⣀⣀⣈⣿⣿⣿⣿⠏⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣤⣀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⡿⣿⣿⣦⣀⠀⠀⠀⢀⣠⣾⣿⣿⣶⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⣤⣠⣾⣿⣿⡿⠋⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠙⣿⣿⣷⣦⣀⣀⣀⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⣠⣤⣤⣤⣄⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⣤⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀⢀⣠⣤⣀⣤⡀⡀⠀⠀⢀⣀⣀⣠⣤
⠀⣟⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣭⣾⠿⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⠀⢋⣿⣿⡏⣿⣿⣿⣷⣿⠧⡅⣼⡄⣤⣿⣿⠋⣿
⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⢀⣘⣿⣷⣿⣿⡇⣏⣿⣿⣿⣿⣟⠀⢀⣿⣷⣤⣿⣯⣔⣟
⠀⣿⣿⣿⣿⣿⡿⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣄⠀⠖⠾⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣯
⠀⣿⣿⣿⣿⣿⣷⣽⣿⣟⣿⣿⣿⣿⣿⡟⡟⠿⢻⣿⠋⣹⠁⠀⢰⣿⣿⣦⣐⣾⣿⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢭⢰⡇⠓⠐⠿⢿⡇⢹⠀⣠⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿
⠀⣿⣿⣿⣯⡅⢹⣁⠵⢿⡿⠟⠛⣨⠙⡇⠁⠀⠀⠀⠀⢹⠈⢹⣿⣿⣿⣿⣁⠈⠿⢯⣽⣿⣛⡻⠿⠿⣿⣿⣿⣻⣿⣿⠿⠛⣹⣿⣿⣿⢻
⠀⢿⣿⣿⣿⠁⣸⣯⠀⠈⠀⠀⠈⠁⢠⣧⠀⠀⠀⠀⠀⢸⣠⣾⣿⡟⠉⣽⣿⣶⠀⢀⣄⣠⣩⣉⡛⠷⢶⣦⣭⣭⣥⠑⣷⠂⠿⠿⠿⣿⣮
⠀⠟⣿⣏⠉⣻⣿⣿⣷⣶⣤⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⢸⣷⡌⣿⣧⣾⣿⣿⣿⣿⣿⣿⣛⠛⠻⠿⠻⣾⣷⣶⣾⣿⣶⣿⡷⠀⠀⠒⠀⠘
⠀⠀⠹⣿⣿⣿⣿⡽⣿⣿⣿⣋⣵⣿⣷⣳⣾⣿⣿⣷⣄⣸⣿⣿⣿⣿⣿⢾⡟⢿⣋⣿⣿⣯⣥⣿⣧⣤⣿⢿⣴⣴⣴⣿⣿⡧⠔⠚⠉⠀⢤
⠀⠀⠀⠉⢿⠏⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠯⣹⣿⣽⣿⣮⡽⣿⠟⠂⣠⣬⣿⣿⣿⣿⣿⣷⣼⣿⣿⣷⣂⡀⠀⣴⣦
⠀⣀⣀⣀⣀⣀⣼⠿⢿⣿⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢈⡛⣿⣿⣿⣿⣿⣿⣿⣧⣀⣴⣿⠿⠿⣿⣿⣿⣿⡿⣿⣿⣟⣿⡿⠷⠶⠟⠿
⠀⠀⠀⠉⢩⣿⣭⡉⢸⣿⣤⡾⣿⣿⣧⣿⢻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣽⣯⣿⣿⣿⢿⡿⠿⡿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣬⣭⣭⣭⣛
⠀⡀⢁⣄⠂⢙⣤⢲⡄⠿⣷⠴⣿⡇⠷⢷⣶⡿⣶⣯⠶⠿⢾⡿⢟⡻⢿⣿⠿⢿⡿⡗⠀⠀⠡⠀⠉⣩⣦⣴⣿⡗⠀⠀⠀⠀⠀⠉⠉⠛⠛
⠀⠉⠁⠁⠘⠋⠛⠉⠉⠁⠀⠈⠉⠈⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠞⠉⠉⠉⠈⠁⢀⡀⣿⣯⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣶⣦⣤⣀⣀⡀⠀⠢⣄⣠⡿⣿⠁⢾⡧⡔⢶⠢
⠀⠀⠔⠀⠀⠀⠀⠙⠛⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣮⣼⣉⣙⡻⠶⠿⣿⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠻⣿⣿⣿⣿⣿⣷⣶⣶⣯⡍⠛⠛⠿⠀⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠂⠀⠀⠃⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⠶⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⣉⠡⠤⠔⢶⡌⢳
⠀⠀⠀⠀⠀⢠⣇⠈⠉⠛⠿⣿⣶⣦⣀⠀⠀⠀⠀⠀⡇⢸⡀⠀⣀⣀⣀⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠿⣷⣄⠀⠀⠀⠈⠙⠻⢿⣦⣤⣀⠀⣷⠈⢉⠥⠤⠔⠒⢲⣘⣧⠴⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠹⢷⣦⡀⠀⠀⠀⠀⠈⠙⠻⣷⣿⡄⠙⠂⢀⡽⠞⠋⠁⠀⣀⣠⠤⠤⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢷⣤⡀⠀⠀⠀⠀⠀⠙⠻⣄⠈⡿⠀⢀⡴⠾⠍⣈⡃⠐⠒⣛⣒⡚⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡷⣦⣀⠀⠀⠀⠀⠈⠙⡇⠀⢸⣧⡀⠀⠈⠉⠛⠻⢶⣬⡹⡄⠀⣈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣤⣠⣀⠀⢀⡇⠀⢸⣧⡝⢦⠀⣠⠤⣀⡀⠀⠉⠛⠤⣀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣾⠁⠀⢸⠙⠒⢊⣀⣈⣷⠦⠟⠀⠀⠀⢀⠈⠀⠒⠚⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢀⡶⢤⣀⡛⠯⠳⢦⣄⠀⢐⠀⠀⠀⣠⣀⠐⢌⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣬⣸⡏⢿⣶⠍⢻⢉⡭⢿⠀⢸⠀⠀⠀⡇⢨⠷⢤⡈⠒⢄⡙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⣳⣇⡀⠉⠃⢸⢐⣖⣸⣀⢤⣭⡀⠀⡇⢸⠀⠀⠉⠓⠤⡈⢢⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠉⠲⢤⣼⢀⣠⣛⣇⠳⣹⡗⠈⠓⠚⠶⣤⣀⡀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⠦⠤⠛⠙⠒⠢⠬⠭⣉⣋⡘⣠