Honeybee Emojis & Symbols

⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠂⠀⠒⡂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⠒⠀⠀⠀⠘⠐⠛⢶⣦⡴⣲⣶⣿⣶⣶⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣹⣿⣿⡆⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣧⣤⡤⣤⠽⠶⢶⣿⣿⣿⣿⠿⠋⢻⣿⣿⣿⡿⢿⣏⣿⡏⠱⡀⢣⠀⠀⡆⠀⠀⠀⠀⠀⢀⡠⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣿⣿⣶⣶⣶⡶⢣⣹⠗⢻⣾⣿⣷⣶⣿⣿⠿⣿⣷⣬⣾⣇⠀⠁⠀⠀⢀⠁⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠀⢻⡼⡟⢛⣩⣽⣿⣿⣿⣬⣷⣵⣽⣿⣿⣿⢿⡇⣀⣬⡉⠛⠛⠛⣷⠀⠀⠀⢸⡆⠀⢠⠞
⠀⠀⠀⠀⠀⠀⢸⣿⣆⠈⢓⣾⣿⣿⣯⠉⢻⣿⣿⣿⢻⣿⡿⠛⠻⠿⢻⣿⣽⣧⢺⡤⣠⣎⣥⣴⣠⡟⠀⠠
⠀⠀⠀⠀⠀⢠⠀⠻⡿⣷⣦⣉⢝⣿⣛⣴⢟⣩⡟⣱⢻⡟⠀⠀⠀⠀⢸⡟⢠⣼⣾⣻⡿⠿⣿⣿⡟⠀⡐⠁
⠀⠀⠀⠀⠀⠸⢶⣤⣽⣮⣟⣛⣿⣿⣟⣵⠟⠉⠀⢹⣼⣄⠀⠠⠤⠤⣼⣄⣰⣿⠟⠁⠀⠀⢨⣿⡶⠋⢀⣰⣖⣉
⠀⠀⠀⠀⠀⣄⠀⣈⣙⣻⣿⣷⡷⠛⢭⣬⣷⣤⣄⠀⠹⣿⣷⣄⠀⠀⢇⣽⡿⠻⠀⠀⠀⢀⣽⣿⡷⠒⠛⠛⠉⠀⠠⢄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠞⠀⠀⠀⠈⠉⠉⢿⣿⣦⡉⠁⠸⠧⢾⣿⣿⣿⠀⢀⣀⣽⣿⡯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣠⠝⠛⡝⡄⣀⣠⠴⢿⣿⣿⡷⠟⠋⡞⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢥⠤⠒⠙⠉⠁⢀⣠⣼⢟⡟⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⣀⠠⠤⢴⠾⣿⡟⠁⢈⡧⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠔⠉⠀⠀⠐⠁⡰⠛⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠙⠲⢄⠀⠀⢀⡠⠖⠋⠀⢀⡹
⠀⠀⠠⠤⠠⠀⠀⡀⠀⠀⠀⠀⠙⠲⡄⠀⠀⠀⠀⠀⢈⣳⡄⡎⠀⠀⠀⠀⠀⠀⢠⡴⠊
⠀⢀⠀⠀⠐⠀⠢⠈⠙⠲⢤⣀⠀⠀⠈⠛⠦⢄⡀⢰⣏⣨⣥⣅⣼⡀⠀⣀⠴⠊⠁⠀⠀⢀⡠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡙⠒⠤⣀⠀⠘⢷⢀⣽⣿⣿⡿⢯⣀⡼⠁⠀⣀⡤⠔⠞⠁⠂⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⠈⠁⠀⠈⢙⣶⡜⢿⣿⣿⣿⡁⠀⡿⠧⣒⠉⠀⠀⠨⠀⠀⠀⠈⠂⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⠀⠀⠉⠁⠀⠠⠬⠿⠓⣞⣿⣿⣿⣿⣚⣻⠶⠥⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠡⠂⣀⠤⣤⣾⣿⣿⡿⢿⣿⣿⣤⣤⢦⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢀⣩⣿⣿⡇⣨⣾⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⣠⠋⢽⣿⣿⣿⣿⣷⠄⠙⢆⠀⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⡇⠀⣤⣤⣼⣿⡇⣤⣴⡀⢹⠀⠀⠈⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠀⠀⠀⢸⡇⠀⣉⣙⣛⣛⣛⣋⣩⡁⣸⡇⠀⠀⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣭⣭⣭⣭⣝⣉⣥⠄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢉⣉⣉⣭⣙⣉⣥⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠈⣉⣛⣛⣛⣋⠁⠀⠀⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁

⠀⠀⠀⠀⠀⠀⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⣴⣿⣿⣿⣷⣄⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠐⠤⠀⣀⣠⣴⣾⣿⡟⣿⡛⢿⡛⣿⣯⣿⣿⣿⣿⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⠛⠯⣉⣚⢿⢾⣿⣿⣿⣟⣯⣻⣿⣷⠦⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⡀⠀⠀⠀⠀⣠⣴⣿⢻⡄⠁⠀⠀⠄⢀⠜⣫⠞⢋⣭⣿⣯⣿⣿⣿⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣫⣬⡅⠉⢿⣷⠀⠀⢠⠏⠰⠁⡄⠠⢻⢫⣟⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⡀⠂⠀⠀⣀⣀⣀⣠⡴⠖⠀⠀⢻⣿⣷⢇⡀⠀⢿⠀⢡⡇⠀⡇⣾⣟⣶⣴⣿⣿⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⡸⠟⠛⠛⠛⠋⠁⢠⣽⣿⣗⠀⠀⠀⠈⠻⣧⡾⣷⡤⣈⣷⣤⣦⣴⣶⣿⢿⣻⣽⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠁⠻⠀⢠⠀⠀⡀⠀⠁⠀⡀⡹⣏⣵⣆⠀⠀⠀⠀⢈⡷⢿⡤⠟⠻⢿⡷⠾⣷⣿⣫⡿⠾⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢈⡇⠀⠚⣇⠀⠄⠈⢀⣀⠤⠀⢀⠀⢿⣭⡶⢣⡀⠀⠀⠴⠛⡡⠂⠁⠀⢸⣛⣿⠿⠛⠉⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡄⠈⣇⠁⠸⢿⡄⠀⠀⠉⠀⣀⠔⠀⠀⡀⠴⢶⠟⠋⠁⠤⠐⠀⠀⠀⠀⢸⣯⡁⠀⠀⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡜⣿⣯⢳⠀⠹⡀⠀⠀⠻⡀⠁⠀⠉⠀⠀⠀⠐⣧⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠹⣿⣷⠀⠀⠀⢀⡞⠋
⠀⠀⠀⠀⠀⠀⠀⣾⡖⣿⡿⣅⢇⠀⠙⡀⠀⠀⠙⠦⠀⠐⢀⣀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡸⠠⠾⣿⣮⢱⣄⠀⢄⠈⠏⠀⠙⠦⣤⠛⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⡎⠍⠡⠹⣄⠾⢛⣳⣜⢦⡀⠢⡅⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⢀⣿⣆⠀⠀⢌⠢⡈⢳⣾⣿⣔⡦⣀⡙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠸⣿⣿⣶⣆⣠⣵⣼⣣⣉⣻⣿⡿⠿⠿⠾
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠰⠦⢀
⠀⠀⠀⣰⣇⣘⢆⡆⡀⠀⠀⠀⠀⠈⢢⠶⠘⠀⠀⠀⠀⠀⠠⢿⣥⣜⣹⢢⡀⠀⠀⠀⠀⠈⡹⢧⠉
⠀⠀⠀⠀⠛⣽⣿⢞⣱⠀⠀⠀⠀⠀⣞⣚⣧⠀⠀⠀⠀⠀⠀⠈⢡⣻⠷⣫⠞⠀⠀⠀⠀⢠⣟⣻⡇
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⢀⡢⠔⣦⣀⣀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⣠⡳⢸⣦⣤⡄
⠀⠀⠀⠀⠀⡟⠯⠀⠠⠀⠀⠀⠀⠈⢻⢙⠙⢻⠯⠆⠀⠀⠀⠀⠰⠛⠒⠘⠢⠀⠀⠀⠀⠈⡍⢹⠉⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠶⠶⣤⠠⠤⠴⠄⠀⠀⠀⠀⠀⠰⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣷⣶⡆⠁⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣺⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡿⠄⠠⠀⠀⠀⠀⠀⣠⠖⢡⠛⠀⠀⢸⣿⣾⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠟⠛⠻⣿⣧⣤⡀⠰⣬⢀⡴⠋⠀⠀⣏⡀⠀⢀⣰⣿⣿⡿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⠿⠉⠐⠁⠀⣼⡻⣯⣆⣾⢿⡟⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⡇⠀⠀⠀⠀⣼⡿⠃⢻⡟⠉⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠀⢀⣠⣤⣤⣄⣠⠴⠋⠀⠠⠃⢠⠅⢀⡴⠛⠁⠀⠘⣇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⢿⣆⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠘⠀⠀⢀⣿⠇⢸⡇⠀⠀⠀⠀⠀⠿⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⡅⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⣾⠇⠀⢸⣿⠀⠀⠀⠀⡰⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣿⣾⣶⣶⣴⣼⣷⣶⣶⣾⣿⣿⡏⠀⠀⣸⣿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠁⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡃⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣝⣿⡏⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠾⢦⣲⢢⣶⣶⣶⣾⣿⣿⡿⠁⠀⠀⠀⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠛⠛⠻⡿⣿⡿⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣠⣤⣄⣴⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⣟⡛⢻⣻⣻⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠀⠀⠀⠈⠠⠀⠠⠈⠀⠀⠀⠀⣀⡤⠄
⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀⢰⣽⣶⣶⣶⣾⣷⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡀
⠀⢔⠁⠀⠑⢒⠖⢒⣀⡩⣒⢂⠤⢀⡀⠀⠘⠿⣆⡽⠉⠉⢛⣿⣅⡾⠟⠀⠀⣀⠠⢤⢒⡪⣁⣐⠒⢖⠒⠁⠀⢑⡄
⠀⠈⠁⢄⠈⠈⢁⠄⠤⠄⣔⠂⠓⠂⠠⠽⣷⣶⣾⡁⠀⠀⢰⣿⣿⣶⣶⡿⠥⠀⠒⠃⠐⡎⠠⠤⠜⠉⠈⢀⠄⠉
⠀⠀⠀⠀⠉⠩⢁⡠⠄⢂⢠⣎⣁⡀⢤⡾⢟⣿⣿⣷⣀⠀⣸⣿⣿⣿⣟⠷⣦⠄⣀⣉⣢⢀⠂⠤⣀⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠙⠤⢤⣊⣠⣔⠉⢄⣒⣭⣭⣿⣿⣾⣿⣿⣯⣭⣕⣂⠌⢑⡠⣈⣢⠤⠜⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⣩⣿⣿⣿⣏⣉⣻⣿⣿⣟⣯⡉⠻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣠⣿⠟⡟⠙⠛⠛⣿⣿⣯⡟⢿⣧⡀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⢰⣧⣤⣀⣀⣼⣿⣿⢰⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠘⣧⣉⣉⣉⣹⣿⣿⡘⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢯⣉⠛⠛⢻⣿⣏⠇⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⢯⣭⣽⣿⣿⠞⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠲⡄⢀⡠⠊⠁⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⢠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⡴⠛⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⣠⠴⠒⢲⡄⠀⣀⣠⣀⠀⣸⠇⣴⡏⡤⠂⠀⠁
⠀⠀⠀⠀⠴⢋⡠⠒⠶⣄⣻⣾⣿⠟⣿⡆⠋⢸⢻⠗⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⣿⣗⣿⣇⣿⣅⢄⠀⣀⣤⣾⣧⣯⡻⡄⠙⣷⡄⠈⣳⣦⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢿⣿⣧⣼⣿⣿⣼⣿⣿⣿⣿⣿⣿⣷⣀⢴⣿⡇⠀⣺⣿⡗⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⣾⣿⡿⢃⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⢻⣿⣿⣻⣿⡿⣿⢿⣿⣿⣤⣿⣿⡿⣻⢆⣿⣿⣟⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⡿⠛⠀⢸⢾⣿⣿⡽⠛⣠⣾⣯⣾⡿⣫⣾⡿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣛⣥⣾⠟⢿⣿⣾⣿⣿⣯⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⣿⡖⠄⠈⠉⠹⢿⣯⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠘⢧⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠗⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⢀⠔⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡛⣻⠛⣿⡟⣋⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⣾⢻⣷⡿⠼⠔⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⢿⣶⣿⡿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣿⣼⣿⡷⢤
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣱⠮⠬⠽⣷⡜⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠘⡯⢬⢽⡟⠄⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣚⣿⣿⣿⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢷⣚⣿⠃⠘⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢈⣷⠀⢤⣞⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⣾⣇⠂⠀⠀⢸⣿⣆⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢆⠸⣿⣿⣀⢀⠀⣼⡿⣿⠀⠀⢠⢎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⣄⣈⣿⣿⡿⡿⣿⣶⣤⣄⣠⡳⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠁⠀⠀⠈⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣭⠇⠀⠀⠀⠀⠀⠀⠈⣿⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢝⣍⠄⠀⠀⠀⠀⠀⠀⠀⣼⣆⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⣤⠖⣁⠂⣺⣷⣷⣇⣠⣆⢠⣠⣼⣿⣿⡭⢂⠀⠑⢤⣀⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠃⡈⠉⠀⠰⢃⣉⣉⣿⠟⠿⠏⣿⡿⠟⣏⣗⣋⠉⢆⠀⠀⠛⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⡄⣮⠷⣿⣀⠀⠁⠀⠀⣠⣤⣾⠛⠲⠭⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠰⡎⣧⠀⠉⠉⠁⠉⠉⠉⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠃⠀⡀⠱⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡆⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡰⣿⠧⠄⠀⡀⠀⢀⡀⠀⠄⠂⠺⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡼⠡⡀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣧⠀⠈⢯⡂⠂⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠿⠻⠄⠀⠀⡀⢀⣠⠬⠘⠇⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣄⡴⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠐⠒⠒⠚⢋⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠳⢦⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠈⠻⣄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠉⠛⠻⠶⣤⣀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⣠⡾⠋⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡟⠀⢀⡤⠶⠖⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⢀⡼⠓⢛⡭⠿⣿⡲⢯⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢄⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡴⠋⠀⡰⠁⠀⢀⠀⠹⡄⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠁⠀⠀⡇⠸⣶⣿⠃⢠⠋⢀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡙⠢⣄⡈⠛⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠘⠦⢤⠤⠔⠛⡘⠿⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠐⠀⠀⠈⢉⣻⣶⡶⠤⠤⢤⣤⣀⣤⣤⢼⡀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠒⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠻⣿⣿⣦⡀⠀⠀⠙⣿⣿⣆⠹⣄⠀⠀⠹⢦⣤⣤⡤⢖⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⠈⣿⣿⡆⠈⠙⡶⠦⠤⠤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⢻⣿⣷⡀⠀⠀⠸⣿⣷⠀⣸⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣦⠀⠀⠈⣿⣿⣇⠀⠀⠀⣿⣿⡴⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⠀⠀⢀⡿⠋⢰⡟⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠉⠛⠿⣿⣿⣇⠀⠀⣸⣿⣿⠴⢶⡏⠀⣴⣿⠃⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⣼⠏⠀⣾⠃⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠋⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠈⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢚⠃⠀⣀⡠
⠀⠀⠀⡤⠶⠶⣄⠀⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡤⠤⣜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣁⣉⢔⠚⠉⠁
⠀⠀⠀⣧⠀⠀⠈⢷⡸⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡘⣖⣒⣾⣂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣫⣏⠄⠤⠊⠁
⠀⠀⠀⠈⠳⣤⣀⠀⠻⣜⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣩⠃⢠⣯⣉⣉⡽⣧⠀⢫⣉⠲⢤⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋⠝⠟⢩⣱⠤⡀
⠀⠀⠀⠀⠀⠀⢉⣹⠒⣿⡷⠉⢲⣶⡖⠀⠂⠀⠀⢠⣞⡡⡶⠟⣩⠖⠁⠺⢭⣭⣷⠏⠙⢮⡙⠷⡦⣝⣦⠀⠀⠀⠀⠀⠘⡾⡿⠢⣴⠓⢯⣼⣪⡿⣦
⠀⠀⠀⠀⠀⣔⣳⢿⣙⡦⠗⠤⠛⠟⠀⠀⠀⠀⠀⠀⠀⠘⠗⠋⠁⠀⠀⠘⣻⠿⠇⠀⠀⠀⠈⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣹⠀⠀⣇⠙⢏⣻
⠀⠀⠀⠀⠰⣉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡏⠉⠲⢄⡀⠀⠀⡢⢄⢠⠀⢀⡠⠔⠉⢹⠀⠀⢀⣀⣀⡀⠀⠀⠀⢂⣠⢶⡄⡄⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⡀⠀⠀⢠⢀⣶⡂⡠⠀⠀⠀⣀⣀⡀
⠀⠀⠘⢦⣤⣤⣬⣱⣾⡟⢻⣧⣶⣭⣤⣤⡴⠃⠀⠀⢅⠀⠈⠩⠙⠛⠒⢸⣯⣉⣽⠗⠒⢟⠋⠉⠀⠀⡜⠀⠐⢏⡀⠀⣉⣉⠩⢤⣿⠿⣿⡶⠬⣉⣉⠀⠀⣹⠂
⠀⠀⠀⠈⠒⠀⠬⢙⢛⠿⠟⡻⡋⠅⠦⠐⠃⠀⠀⠀⠀⠁⠒⠄⠄⠤⠀⣾⣟⣒⣿⣷⠀⠆⠤⠀⠂⠁⠀⠀⠀⠀⠉⠙⠤⠤⡶⠚⣟⠋⣛⢟⠶⠤⠤⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠠⣭⣭⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣷⣶⡞⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⢯⠭⣽⠘⠆⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠏

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠌⠉⠉⠳⣦⡀⠁⣀⡀⠀⢀⡴⠛⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⡔⠉
⠀⠀⠐⢤⣀⣀⠀⠙⠦⣟⣫⠶⠋⠀⣀⣀⡤⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠴⠧⢼⣥⠶⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⡖⠋⠓⠲⠦⣄⣀⡇⢹⣀⡤⠴⠒⠛⠓⡆
⠀⠀⠀⠀⠈⠉⠉⠉⢠⣓⣂⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⢀⣠⠶⠋⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠙⠲⣤⠤⢀⣀⢬⠛⠫⣥⣀⠠⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣟⣒⣒⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠹⣛⡻⢾⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢺⣭⣭⡄⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣭⣵⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠠⢤⣐⠲⢄⠀⢀⡤⠖⣠⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⣄⠀⢀⡤⠖⠂
⠀⠀⠀⠠⡀⠀⠈⠳⣄⣷⣿⡤⠞⠁⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠸⣷⠏⠀⣀⣀⣀
⠀⠀⠀⠠⠉⠙⢛⡶⣾⣿⣿⡷⣖⠛⠋⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣟⣋⣉⣹⡶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⣠⣴⣴⣶⣶⣴⣤⡀⠈⠉⠢
⠀⠀⠀⠐⠤⠖⠋⠀⢭⣶⣾⡅⠈⠛⠦⠔⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⣯⣉⣉⣹⡇⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠐⢤⣖⣯⠟⠋⣶⣶⣶⡌⠛⢯⣗⣦⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣠⠴⠊⠈⢧⣤⠏⠘⠲⢤⣀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⣶⠆⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣀⣀⠀⠀⣼⠃
⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠶⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠛⠉⠉⠛⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⠶⠶⢦⣄⡀
⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢶⣄⡀⠀⠀⠀⠀⣠⠟⠁⠈⠀⠀⠀⠀⠁⠈⠻⣄⠀⠀⠀⠀⢀⣤⡶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⣿⠟⠉⠉⠉⠉⠻⣿⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⠾⠟⠛⠉⠀⣴⣯⣤⣤⣤⣤⣤⣤⣽⣆⠀⠉⠛⠻⠷⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣩⠿⠛⠉⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⠉⠛⠿⣌⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣼⣷⡄⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣀⡴⠛⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠛⢦⣀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠉⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠚⠉⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⣠⠏⠀⠀⢠⠖⠒⠒⠢⣄⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡏⠁⠀⠀⠀⢿⡀⠀⠀⠀⢸⠆⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⠿⠳⠶⠖⠋⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⢀⡠⢔⣒⡒⠿⠙⠛⠒⠒⠤⣴⡃⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⣨⣴⣥⠤⠤⠤⢄⣀⡀
⠀⠀⣠⠚⢁⠎⣡⣤⣍⡆⠀⣀⣀⠈⢷⠀⠙⢆⠀⠀⠀⢀⡟⠀⠀⢀⣤⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⠍⠲
⠀⡼⠁⠀⠘⢆⣿⣷⠼⡗⢉⣠⣌⠱⡈⠀⠀⠈⣇⠀⢀⡞⢀⡤⠚⠉
⢸⠃⠀⠀⠀⠀⠀⠀⠀⣇⢿⣿⣥⣥⠇⠀⠀⠀⢹⢠⣾⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢸⡄⠀⠀⢰⠢⢄⣀⡀⠈⣓⠒⠚⠁⠀⠀⠀⠀⡿⠛⠷⢶⣤⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠖⠋⠁
⠀⢳⡀⠀⠸⣶⣦⡤⠍⠤⡼⠁⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠙⣧⣠⠤⠤⣤⣬⣭⣉⡉⠉⠉⠁
⠀⠀⠙⠦⣄⡉⠓⠤⠔⠚⠁⠀⠀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⣿⣿⣿⠉⠓⢦⣄⡀
⠀⠀⠀⠀⢠⠏⣹⠛⠒⠒⢲⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢸⣿⣿⣿⠀⠀⠘⣿⣿⣦⣄
⣎⠉⠒⠚⣁⠞⠁⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡾⠁⠀⠀⢀⣾⣿⣿⡏⠀⠀⢸⣿⣿⣿⠈⠳⣄
⠘⠳⠶⠚⠁⠀⠀⠀⠀⡠⠞⣠⠟⠷⢤⣄⣀⣀⣤⣤⠞⠋⠀⠀⠀⣠⣾⣿⣿⠏⠀⠀⢀⣿⣿⣿⡏⠀⠀⢹⣷
⠀⠀⠀⣠⠤⠤⠤⠐⠋⣠⡾⠁⠀⠀⢀⡠⠝⣋⡭⣧⠀⠀⣀⣤⣾⣿⣿⡿⠁⠀⠀⢠⣾⣿⣿⣿⠁⠀⠀⣾⣿
⠀⠀⠀⠹⣦⣤⣤⡴⠞⠋⠀⠀⢠⠞⠁⣴⠚⠁⣀⠜⣷⣿⣿⣿⡿⠟⠁⠀⠀⢀⣴⣿⣿⣿⡿⠃⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣰⠇⢠⠊⢠⠞⠁⠉⠻⢧⣤⣀⣀⣤⣶⣿⣿⣿⡿⠟⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣇⠀⠋⠛⢦⠀⠀⠀⠈⠉⠛⡿⢿⡿⣿⣯⣤⣤⣤⣴⠾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠛⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⢸⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⡇⠀⠀⠀⡇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⡇⠀⠀⠀⣇⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡼⠃⠀⠀⠀⠸⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⠃
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠠⢐⣩⣄⣉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣶⣷⣄⣰⣶⣤⡀⢀⢀⢶⣶⣦⣦⣤⢂⢀⠈⣵⣾⣻⣿⣿⡿⠋⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣟⣿⡟⢛⡃⢸⣧⡔⢉⣉⣉⠑⢠⣾⠀⢋⡛⣿⣿⣿⡽⣶⣶⣧⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⢹⣿⣿⣿⣿⣌⠳⣌⠃⢸⡧⢽⣿⣿⠀⢋⡴⣋⣤⣿⣿⣿⣿⣿⠿⠏⠃⢸
⠀⠀⠀⠀⠀⢀⠀⠐⢁⠈⠉⠁⠀⠀⠀⠈⠉⠉⠀⢨⣤⡞⠓⠀⣿⣿⣧⣬⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⣁⠂⠄⡀
⠀⠀⠀⡔⠈⠰⣾⡡⡁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⣧⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡱⠸⠋⠀⠀⡑
⠀⠀⢠⠀⠀⣠⣸⣿⣶⣕⠀⠠⠄⠀⠂⠀⢉⡉⠀⠀⠚⣿⣿⣿⡿⣿⡟⠂⠀⠈⣉⠁⠀⠂⠀⠤⠀⠐⣥⣾⡽⣶⣶⣦⡇⠇
⠀⠀⠀⠀⠀⣿⣿⣼⣿⣿⣿⣿⡦⠀⠀⡾⢫⣍⠺⡶⠒⡻⠃⠙⠻⣿⡓⠢⡾⢊⣍⢻⡆⢤⠀⡦⣿⣿⣿⣿⣿⣿⠿⠋⠂
⠀⠀⠣⣀⠐⠿⣿⣹⣿⣿⠿⣛⣥⣤⣉⠲⣛⡿⢁⠞⢺⡶⠾⠿⠿⣿⣶⠚⢂⠸⣿⠶⢈⣢⣴⣬⣐⠙⠭⣿⣿⣿⠖⢀⡠⠂
⠀⠀⠀⠈⠓⠀⠈⠛⢩⣴⣿⠿⠋⠁⠈⠑⠢⢁⣸⠀⣮⣤⣰⣶⣶⣾⣧⡄⢸⣀⠀⠀⠀⠀⠉⠛⢿⣿⣦⡈⠉⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠀⠀⠀⠀⠀⠀⠀⢨⠏⣶⢹⣥⣤⣴⣶⣿⣯⢣⡎⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡼⢫⠀⣿⡀⢧⣀⣩⣽⣧⠍⢼⡇⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⡿⣞⡌⠻⣿⡿⢏⣼⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡈⠙⠿⣶⣴⡶⠿⠃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⡈⠀⢀⡠⠆⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡾⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄⠀⠀⠀⠀⣠⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⣉⡉⢉⣉⣓⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⢠⣾⣿⠿⠉⠁⠈⠉⢿⣿⣷⡄⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠢⣀⣹⣭⣿⣷⣿⣶⣶⣿⣬⣷⣬⣀⡀⡔⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠇⠐⣿⣿⣿⡿⠈⢻⣿⣶⡆⠈⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠠⣶⣾⣿⣿⣿⣶⠀⢠⣿⣿⢧⣶⠌⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢶⠂⠀⠀⠀⢐⣠⠚⠛⣿⣿⣿⣇⠀⠀⣰⣿⣿⣿⠛⠓⣤⡂⠀⠀⢀⠀⢦⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⡀⢦⡀⠜⠁⠀⠀⢿⠻⣿⣿⡏⢈⣿⣿⠛⡋⠀⠀⡌⠳⡀⡴⢉⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣫⠶⠁⢠⠘⠘⣄⠈⢻⠛⠋⡉⠁⢠⠁⠀⡆⠘⠦⠙⢎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠋⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣾⣷⣶⠇⠀⠀⠈⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠢⠀⠀⠀⠀⠀⡀⠀⢰⣿⣿⣿⠟⢿⣿⡆⠀⢀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠜⠉⠉⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⣷⣦⣶⣄⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠿⠿⠿⠿⠟⠻⠿⠛⠛⠛⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⣿⣶⣶⣦⣦⣤⣤⣠⣀⣤⣤⣤⣶⣿⠀⢰⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⠁⠸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⡉⠀⠘⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡀⠀⠀⠀⣿⣶⣶⣶⣶⣤⣤⡄⢠⣤⣴⣶⣿⡏⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠉⠀⠀⠀⠀⠈⢿⣟⡛⠛⠛⠛⠁⠈⠉⢈⣛⡟⠀⠀⠀⠀⠀⠑⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣶⣶⣶⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠿⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠄


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣯⣽⣽⣮⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣷⣷⣾⣿⣷⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣄⠀⢀⣤⣤⣄⠀⠀⠀⢀⡴⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣤⣜⣧⣄⣶⣯⣴⣦⣄⠀⠀⠀⠀⢠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⣀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢰⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠹⣿⣿⣿⡿⠾⠻⠿⢿⣿⣿⠏⠀⣠⠟⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣹⣶⡆⠂⠈⠀⠀⠀⠀⠠⠀⠤⣴⣾⣿⠿⠋⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠀⢴⡿⣅⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣖⢲⣤⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⣿⣿⣿⣶⠿⠟⠛⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢋⢿⣿⡟⣧⡀⠀⣠⠆⠠⠤⣁⡀⢛⣼⣷⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣫⣾⣿⠟⣅⣟⣹⢷⣤⣤⣤⢾⣥⢏⠛⣿⡝⢦⣝⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁⠾⡻⢹⣿⠐⣋⣭⣽⣷⣶⣷⣾⣿⣅⡛⣢⢸⣿⠧⠈⠳⢬⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡯⠊⠝⡜⠁⣂⢣⠞⠿⣿⠿⡿⣶⢶⢾⢟⣿⣿⠓⡄⢿⠻⣦⡐⠄⠑⢍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢊⣾⠁⡸⣿⣎⡾⣤⣏⣇⡇⢸⣼⢸⢸⣿⣸⡄⣷⠈⢧⣙⢷⣄⠀⠈⢳⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣴⠕⣱⠃⡜⡽⠁⣿⣷⣦⣍⡇⠀⠈⠉⠀⣈⣫⣵⣾⢿⡄⠈⢿⣄⢣⡑⢤⣦⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣠⣫⢎⡼⠀⠀⣿⡀⢿⡿⣿⣷⣶⣾⣿⣿⡿⠟⢡⣸⠇⠀⠈⠻⣦⡑⡄⠉⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⢜⣴⣹⠟⠀⠀⠀⢻⣿⣷⣦⣀⠀⠉⠉⣨⣵⣶⣾⣿⡿⠀⠀⠀⠀⠙⢿⣟⣆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣴⠷⢋⡼⠃⠀⠀⠀⠀⠈⣿⣭⣿⣿⣿⣿⣿⣿⡿⡿⣻⣿⠇⠀⠀⠀⠀⠀⠀⠙⢷⣑⠤⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⢁⡤⠊⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣦⣤⣤⣼⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣍⣉⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓
⡤⠶⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⣾⣆⠒⠉⠉⣒⣦⣀⣄⣠⡒⠒⠒⠂⣠⣾⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢄
⠣⡀⣀⢀⣀⠂⣈⣉⠓⠦⣄⡀⠘⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣀⣿⠏⠀⢀⣠⠔⢒⠙⠃⠀⠀⠉⢀
⠀⠙⢄⠀⠀⠀⠁⢄⣈⣃⡀⠙⠳⢮⣿⣷⣖⠉⠀⠀⠀⠀⠉⠳⣾⣿⣥⠖⠋⠁⣌⠉⠠⠊⠀⠀⠀⡠⠃
⠀⠀⠀⠑⠤⢤⡤⠽⠤⣀⡇⠤⢬⣥⣤⣾⢿⡆⠀⠀⠀⠀⢀⣟⣻⣓⡚⠂⠚⢙⣀⡠⠦⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠻⣛⠉⠍⠉⡽⢿⡁⠀⠀⠀⠀⢀⡽⢯⡉⠉⢉⣻⠛⠓⠂⢄
⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣊⣁⣤⠠⠎⠀⢠⣿⣬⣿⣿⣶⣿⡄⠀⠙⢎⡀⠨⠉⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠀⣀⣴⣿⣃⠀⠉⠉⠉⡀⣀⣼⣿⣦⣀⠈⠿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠁⠀⣼⣿⡇⢸⣿⡟⠹⣿⣿⣿⣿⣿⡃⢼⣿⣆⠀⢸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣟⠁⠸⣿⣦⠑⣿⣿⣿⣿⣿⠇⠈⣻⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⢿⣶⡉⣿⣶⣶⣾⠿⠀⠀⢿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁⠀⠀⠀⠙⢧⣰⣿⣿⠟⠀⠀⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠓⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠰⢧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡄⢠⠤⠤⢄⡀⠀⠀⠀⣀⡠⠤⡀⠀⣠⡎
⠀⠀⡏⠉⠀⢉⠙⡒⠲⠤⣄⡀⠀⠀⢹⣷⠀⠀⢠⣶⣿⣷⣷⣾⣷⣤⠀⠀⢰⣿⠃⠀⠀⠀⣀⡤⢴⡒⠊⠩⠍⢱
⠀⠀⠘⢆⠂⠐⠐⠐⠖⠒⠠⣍⠓⢦⣄⣹⣷⣤⣘⡿⠟⠛⠛⠛⠿⢯⣀⣴⡿⣉⣠⠴⠒⠏⠄⢃⠤⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠱⣄⠀⠀⠀⢵⠒⠒⡆⠀⠈⠽⠛⣻⣷⡄⠀⠀⠀⠀⠀⣦⣿⣿⣿⠉⢠⣤⡞⠒⢰⠁⠈⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⢉⣒⣭⣥⠭⠿⠿⢿⣫⠏⠀⠀⠀⠀⠀⢿⣾⠷⠶⠤⢤⣤⣶⣒⠉⠑⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⠁⠀⡨⠖⠂⠁⢠⠟⠙⣶⡈⣄⣠⣄⣠⡞⠉⠫⡀⠄⠲⠇⠄⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⣶⣶⠖⠊⢁⣀⡴⠻⠿⠿⠿⠟⠛⠣⣠⣄⠉⠲⢤⣥⡤⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠉⢀⣴⣿⢻⣷⣤⣤⣤⣤⣤⣴⣾⣿⢿⣶⡄⠈⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢸⣿⡇⢸⣿⣟⠄⣿⣿⣿⣿⣿⣯⠸⢿⣿⠆⠀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠈⣻⣛⡤⣿⣿⣿⣛⣻⡏⠀⢰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡟⠀⠀⠘⢿⣇⢻⣿⣿⣿⡟⠁⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠈⠙⠾⡿⠟⠋⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⢀⠂⠀⣀⠀⠀⠀⢠⣢⡘⠋⢈⠷⣣⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⢃⡸
⠀⠀⠀⠀⠀⢤⣈⠂⠀⠀⠎⠁⣏⠹⢗⣋⢁⠙⠧⢔⡄⠀⠀⠠⠺⠇⢀⠀⠀⠀⠀⠈⠁⣠⣞⠱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠙⠁⠛⢣⣄⣓⢬⠄⣬⡒⠇⠀⠠⠀⠀⡀⠀⢉⠖⠀⠀⠾⢿⡟⠉⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⢶⢄⣤⣼⢀⠁⠾⠧⠕⡀⡄⠀⠀⡀⣠⢴⠛⢋⠠⢂⠀⠤⣀⡄⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠡⠐⠂⢌⠉⠰⠆⠁⠀⠈⠉⢿⡁⠊⠁⠀⠀⠀⠀⠈⠉⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠂⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⢃⡄⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠀⠀⠈⠑⠂⠤⡀⠀⠀⠁⠈⠼⠀⠀⠀⠈⠙⠷⠤⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠢⣄⠉⠒⠄⣐⠂⠀⣆⡀⠀⢀⠀⣀⠂⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⠀⠄⡠⠀⠀⠤⠀⠀⠀⠠⢠⠐
⠀⠀⠀⠀⠉⠀⣠⣿⡟⠓⠚⢷⣾⡚⠛⠋⠙⠲⠀⠀⠀⠠⡀⣠⡆⠀⠈⢹⣦⡇⣀⠀⠂⠠⠠
⠀⠀⠠⠊⢀⠾⠛⣿⠷⠒⠂⠀⠀⠘⠯⠶⠲⢾⠆⠀⠐⣎⠂⠀⠀⠀⠀⠈⠨⠛⡧⢶⣠⣇⣰
⠀⠀⠀⠈⠁⠜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠁⢀
⠀⠀⠀⠀⠘⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠖⠦⢀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠁⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢡⣶⡀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⣦⠀⠉⠀⣐⠆⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠉⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⢀⣠⠤⠶⠶⠦⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠚⠛⠛⢲⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠶⠶⠦⢤⣄
⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀⠀⠀⡴⠋⠈⠀⠀⠀⠀⠈⠳⡄⠀⠀⣀⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣾⣿⣿⣿⣿⣶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢾⡿⠉⠉⠉⠹⣿⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣤⡤⣤⣴⡶⠾⠛⠋⠁⢀⣾⣶⣶⣶⣶⣶⣾⣆⠀⠉⠛⠻⠶⣶⣤⡤⢤⣤⣤⠤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⠛⣦⠀⠀⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⣠⣏⣀⣠⣤⣤⣤⣤⣤⣤⣀⣈⣧⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣠⡼⢻⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠻⢤⡀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠘⣇⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡟⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠛⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⠈⠀⡀⢀⠀⠁⠀⠀⠀⠀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⢠⣻⣦⣶⣶⣴⣿⡆⢀⣤⠞⠉
⠀⣰⠊⠉⠓⠀⢤⠬⠭⠁⡲⠠⠄⣀⠀⠀⠀⠙⣿⣄⣻⠛⠛⠻⣿⣟⣠⣿⠏⠀⠀⠀⣀⠠⠄⢒⠈⠭⠥⡤⠄⠚⠉⠑⣆
⠀⠈⠒⡀⠐⠂⠳⢈⡉⠉⢘⠥⠵⠦⠌⢁⣒⣦⣤⣽⡏⠀⠀⠠⣽⣯⣯⣤⣴⣖⡈⠡⠴⠮⠬⡃⠉⢉⡁⠞⠐⠂⢀⠒⠁
⠀⠀⠀⠈⠑⢲⡒⠁⣀⠠⠄⣓⠖⠀⠀⣀⣸⡿⣿⣿⣧⡀⠀⢐⣿⣿⣿⣿⢿⣇⣀⠀⠐⠲⣊⠡⠄⣀⠈⠒⡒⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠐⢒⣁⡉⠝⢉⠭⠚⣑⠪⠟⣛⣿⣿⣭⣽⣿⣿⣛⠻⠕⣊⠓⠬⡉⠫⢉⣈⡒⠂⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⣨⣷⡶⠛⣻⣿⣿⡿⠿⢿⣿⣿⡿⣟⠛⢶⣾⣅⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠒⠛⠋⠁⢀⣾⣵⣿⠿⢿⣶⣾⣿⣿⢿⣮⣷⡀⠈⠙⠛⠒⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⢀⣇⠀⠀⠀⠘⣿⣿⡞⡀⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⢸⡟⠿⠿⠿⢿⣿⣿⡇⡇⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⡿⠶⣶⣶⣶⣿⣿⢶⠀⠀⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⢹⡶⢤⣤⣼⣿⣿⡎⠀⠀⠀⠀⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⡄⠀⠀⠀⠀⠀⢳⠀⠀⠄⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠙⢦⡀⠀⠀⠀⢸⡀⠀⡆⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠉⠢⡀⠀⠈⠃⣰⠃⢠⣤⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢐⣯⣤⣝⢿⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⠷⢦⣄⠀⠀⠀⣼⣿⣿⣿⣷⡙⠉⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⣬⡀⠘⢷⡘⠿⣿⣿⠿⠷⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠉⠀⢿⣿⣷⣄⠈⠻⢷⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⣿⣆⠀⠹⢿⣿⣷⣦⡀⡄⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠙⠻⣫⠆⢸⣆⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡉⠻⢿⠿⠂⣰⠏⠀⢸⠈⢦⡀⠀⠀⠀⠙⠦⣄⡀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠋⠀⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠲⡄⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠒⠀⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠄⡔⠒⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠒⠒⠒⠒⡖⠒⠒⠒⡖⠲⠒⠒⠂⠐⠂⠒⠀⠐⢺⠁⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⡀⠀⢰⠀⠃⠀⠀⠀⠘⠀⡇⠀⠀⠀⡇⠀⡇⠀⠀⢰⠀⠃⠀⠀⠀⢰⠀⡄⠀⠀⡈⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡇⠀⠈⡀⢰⠀⠀⠀⠀⠀⢁⠀⠀⠀⡇⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⡄⠀⠃⠀⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡇⠀⠀⡇⢸⠀⠀⠀⠀⡇⠸⠀⠀⠀⠁⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠧⠤⠔⠃⠚⠒⠒⠒⠒⠓⠚⠒⠒⠒⠒⠒⠚⠒⠒⠚⠚⠒⠒⠒⠒⠓⠚⠒⠒⠒⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠐⠒⠂⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠒⠠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢀⠎⣀⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⢤⡈⢇
⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠐⠒⠒⠒⠖⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡒⠒⠒⠒⢲⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠚

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣠⣭⣧⣤⣄⠀⠀⠀⠐⢂⡂⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢠⠴⡂⠄
⠀⠐⢄⡘⢷⣍⣿⣿⣤⡀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⡟⠋⠀⠀⡀
⠀⠀⠀⠈⠉⠋⠹⣿⣼⣿⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢷⣾⣗⡥⠴⠒⠦⢗⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠇⠈⠀⠀⠀⠀⠀⢀⢶⣶⣦⡈⠈⢱⢉⣴⣧⣤⢠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⣤⣀⠀⠀⢀⣿⣴⣿⡿⢛⣿⣿⣽⣧⣤⡚⣿⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠟⢿⣶⣤⡷⢿⢇⢴⣾⣋⣻⣿⣷⠔⢿⠟⣳⣾⣶⣶⢿⠍⠾⡿⠇⠀⢀⡀
⠀⠀⠀⠀⣾⢿⣿⣿⣧⣈⡿⡣⣷⣶⢥⣾⣿⠟⣩⣭⡱⣿⣿⣿⡟⣿⡯⢟⣾⣮⡅⢸⡇⣶⡞⠁
⠀⠀⠀⠀⠈⢿⣶⣴⣿⠻⣷⡔⠭⠹⢒⣨⣷⡐⢿⣿⡿⣨⣥⡁⣾⣿⣷⡼⣿⣿⠿⠘⠺⢟⡃⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠙⠿⢯⣽⣯⣶⣿⣿⢛⣛⡛⠁⣶⣦⡔⣿⠿⠋⣭⣛⢋⣴⣶⣄⣾⣿⣧⠸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⣿⣌⠀⡉⠸⠿⣿⠘⠛⠛⣁⣶⣦⠺⠿⠻⣈⣒⠚⣫⣬⣝⣵⣶⣶⢰⣿⣷⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⢿⠇⡋⢰⡆⠰⠈⢙⡇⠙⠛⠩⡄⢘⣈⠉⢻⣇⡉⠛⣋⣙⠚⣛⡈⣻⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡤⠀⣀⠊⣴⡆⠀⠞⠀⣠⠈⠐⣦⡀⠀⠿⠀⠶⠊⠁⠻⢿⡿⢠⣍⠰⣟⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⣷⣚⠁⠨⡄⠠⠗⠀⠡⡄⠀⣭⣠⠶⣷⡄⠈⠹⠇⢀⡒⠀⠸⠿⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣿⡶⣾⣄⣨⠇⣀⡐⣂⠀⣏⡈⣀⣼⣃⠸⣶⠔⠀⡟⣈⣻⠖⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣯⣹⣿⡧⠀⠿⠟⠓⠛⠋⠁

⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠺⣿⣿⣿⣿⣶⡖⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣬⡻⢿⠟⢋⡀⢰⣮⣶⡄⢠⣶⡶⠟⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣾⡿⠆⢩⠟⣿⠷⠆⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢻⣿⣿⣿⣿⣿⣿⣿⣷⡶⢾⣷⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⠀⢠⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠈⠸⣏⣹⡏⢉⣿⣧⣀⠀⠈⠠⠘⠀⣺⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠛⠿⠛⢿⡶⢦⡿⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣀⣀⡄⠀⠀⡘⠷⠾⠧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡿⠛⠒⠩⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠀⢧⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡤⣄⠀⠀⠀⡠⢤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠃⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣤⠀⣀⠈⢧⠀⠐⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⣞⠀⠀⠀⠀⠀⠱⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠍⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣠⡍⠠⠇⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠤⠀⠀⠐⠉⠀⠀⢀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⢘⠂⡢⠀⠀⠀⣇⣸⠁⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⠀⠀⠔⠀⠀⠀⠀⠀⠈⠱⠀⠀⠄⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⠀⠀⢠⣀⠀⠁⠀⡀⠀⠀⠀⢀⣀⠀⠀⠄⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⠀⠀⠀⠚⠋⠀⠀⠄⠁⠀⠀⠀⡀⠀⠀⠠⠀⢠⠀⠀⠈⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠈⠀⠀⠎⢀⡀⢀⠀⠆⠀⣀⣀⠇⠀⠀⠄⠂⠀⠒⠐⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⢀⠔⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠀⠒⠒⠶⠒⠒⠚⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⣅⣀⣀⠀⣤⣶⣦⣀⣴⣄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠛⠉⠉⠙⠉⠋⢻⣄⢬⣿⡈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠤⠤⠤⠤⠤⠐⢒⣿⢉⣶⡶⣶⣤⠶⠶⢦⣸⣿⣿⣿⣿⡆⢸⠶⠶⠤⡤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⠤⣀⠶⢾⡍⠙⠁⡈⠍⠒⠒⠒⣶⠒⠉⢀⣸⠗⣸⣿⣷⣿⣇⣠⣤⣾⣧⡙⢿⣷⣾⡟⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⡴⠋⠀⠘⠁⠀⠈⠀⠂⠙⢆⢀⠀⠀⠀⣀⡤⠖⡹⢣⡆⠸⣿⡇⢻⣿⣿⣿⣿⡿⠀⠈⠻⣿⠃
⠀⠀⠀⠘⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⣀⣤⣶⣿⠏⠁⣼⠁⣿⣧⣀⠿⣿⣶⣿⣻⣝⡿⠆⠀⢀⣤⡾
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠲⠶⠶⣶⣟⡅⠈⢻⠁⣠⣾⣿⡀⢻⣿⣿⡧⠀⠉⢻⣿⣨⡷⠀⢀⣼⣿⡿⢷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⣿⣿⣆⠀⠉⠡⠈⣿⣷⣈⣿⣿⣷⣴⣶⢾⣿⣇⣠⠴⠋⠀⠀⠈⠻⣿⣷
⠀⠀⠀⠀⠀⠀⣠⡴⠶⣶⠟⣵⣷⣤⡈⠻⣿⣄⠀⠀⢲⡿⠛⠉⣭⠟⣱⡷⣁⠋⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⣠⠶⠛⠓⠐⣽⣦⡘⢿⣿⣷⡀⠘⣿⣷⣀⣸⡿⢶⠞⠀⣼⠟⣿⡿⣶⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⢿⣷⡈⠻⣿⣷⣄⠘⣿⣿⠏⢀⣼⣿⡿⠋⢀⢛⣠⡿⠉⠻⣭⣑⣷⣤⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠻⣿⣄⠙⣿⣿⣦⣽⠓⠈⠀⠀⢉⡽⣦⠊⠘⣽⠃⠀⠀⠈⠻⣿⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣷⣜⢿⣷⣼⡿⢿⡇⠀⠀⢀⡴⠊⢰⡁⠀⢰⠇⠀⠀⠀⠀⠀⠸⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⣾⠃⠀⠐⡋⢀⡠⢂⠀⢛⠏⠀⠀⠀⠀⠀⠀⠀⢿⣁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠓⠉⠀⠁⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣯⠀⠀⢀⠤⢋⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠟⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠞⠀⠉⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⠀⠀⣴⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⢀⣾⣿⣿⡿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠘⠟⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⡈
⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡷⠆⠀⠤⠀⠠⠂⠀⠀⠀⠀⡰⠒⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⡇⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠠⠃⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠰⢻⠃⠀⠘
⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠈⠙⠧⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢸⠀⠀⢀⠄
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡠⠃
⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠄⢠⠑⠤⢀⠌
⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠁⠂⠀⠀⠄⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋
⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠁
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡟
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⢑⣶⣶⣶⡶⠊⠀⠀⠀⣠⡿⠁
⠀⠀⠐⠈⠉⠁⠉⠉⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠂⠋⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠉⠑⠢⢀⠀⠀⠀⠀⠹⣿⣤⠿⠛⠋⠉⠉⠙⠛⠿⣤⣿⠟⠀⠀⠀⠀⡀⠔⠊⠉⠁⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⣀⡀⣸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⢀⣀⠄⠒⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠠⢉⣹⣿⣿⣶⣶⣧⣶⣶⣼⣶⣶⣿⣿⣏⣉⠄⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⢀⣴⡿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⢿⣦⡀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⢰⣿⠟⠃⢉⣉⣉⣉⣥⣶⣶⣬⣉⣉⣉⡁⠙⠻⣿⡆⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢠⣿⠀⠀⠁⠀⠁⠉⠉⠛⠃⠘⠛⠉⠉⠈⠀⠈⠀⠀⣿⡄⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⣼⡟⠀⠀⠆⣀⣀⣀⡀⣀⣠⣄⣀⢀⣀⣀⣀⠶⠀⠀⢻⣧⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠰⣿⠇⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠸⣿⠆⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⡀⠀⠀⠘⣦⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣴⠃⠀⠀⢀⣿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠤⢄⠦⣔⣀⠂⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⣿⡀⠀⠀⠀⠘⠁⠀⢠⠖⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢒⠙⣿⡄⠈⣿⣷⣄⠀⠈⣆⠀⠀⢸⠀⣠⠞⣠⣔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠁⣾⠀⢹⣿⡦⣍⣙⢉⡑⠶⠉⢄⣀⡼⢄⣿⠀⣹⡿⡧⠀⣕⣆⠀⠐⠒⠀⢀
⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⣄⠘⢷⣮⣍⣙⣬⡯⠞⢣⡴⠆⡰⠊⠂⢚⣉⠀⠹⣿⠃⠀⢂⢈⡆
⠀⠀⠀⠀⡐⠁⠀⠀⠀⢀⡹⢦⣤⣈⣹⣿⡝⠞⠏⠀⠀⣎⠀⠀⠀⠘⠿⣿⣃⠀⢱⠊⠀⠀⣱
⠀⠀⠀⡐⠀⠀⠀⠀⠀⣀⡳⢤⡩⠛⠃⠀⠀⠀⡇⠀⢀⡏⠀⠀⠀⠀⢀⠛⣿⣷⡅⠀⠀⠀⢸
⠀⠀⠠⠃⠀⠀⠀⣰⡶⠛⠳⠈⠀⠀⠀⠀⠀⠀⣈⣠⠞⢻⡆⠀⠀⠀⠂⠁⠀⠟⣷⡆⠀⠀⠀⡀
⠀⠀⠄⠀⣠⠼⠋⠁⠀⠀⠀⠀⠀⡀⠀⠠⡤⠖⠋⠀⠀⠀⠳⢔⠂⠈⠀⠀⠀⠀⠈⠹⣿⡀⠀⡇
⠀⣠⡴⠚⠥⠂⠐⠂⠒⠂⠈⠁⠀⠁⠀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠉⠒⠤⠤⣀⣀⠀⠀⠀⠙⣄⢠
⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠢⢽⣧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣿⣿⣿⡄⠀⠀⠀⠆⠀⠈⡀⠀⠀⢼⣷⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⠀⠀⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⣠⠿⠃⠿⠿⠿⠿⣆⠀⠀⠰⡀⠀⠁⠀⠀⣾⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⠈⠀⢀⣼⣿⣿⡇⣿⣿⣿⣿⣿⣧⡀⠀⠱⡀⠠⠂⢀⣭⣭⣏⣉⣉⣙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠄⠁⠀⣰⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⣦⡤⠊⠀⣴⣿⡉⠀⠙⢿⣿⣿⡿⠀⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⢀⣠⠚⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⡟⠠⠂⢸⡿⠃⢤⡄⠀⠀⠹⢿⣷⡄⠀⠀⠀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢰⣏⣉⠀⣩⣭⣭⣭⣭⡁⣩⣭⣭⣭⣭⣭⡸⠀⠀⡈⠀⠀⠠⠄⠀⠀⢰⡀⡘⠉⠀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⠀⠀⠀⡆⠀⣀⣀⣀⣀⣿⣿⡇⣿⣿⠿⠿⠿⠃⠛⣻⠿⠿⣿⣿⡇⠀⢀⠀⠀⠤⠀⠀⠀⠐⠀⢐⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠃⠉⠁⠀⠀⠀⠉⠉⠙⠓⠢⢄⡀⠀⠀⠀⠉⢬⠀⠘⢁⡿⠙⠁⢆⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠉⠳⡄⠀⠀⠀⠚⠀⠘⢇⣠⣤⣈⣦⣤⠀⠊⢀⠞⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢻⠵⠶⠷⣖⣛⠃⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠙⢃⠀⣀⣉⡙⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠉⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣋⠄⠀⠀⠀⠀⠀⠑⡅⠐⠴⠀⠀⠉⠙⠝⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠛⢥⢀⣀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢀⣀⡀⠠⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⢀⣴⡾⠿⣷⣤⣄⡀
⢿⣯⣴⣾⠟⠉⠛⠁⠠⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠙⢿⣦⡀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣶⣤⣀⣀⡀⠀⠀⠀⠈⣛⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢹⣿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⢻⣿⠀⠀⠀⠀⠛⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣀⡀⠀⠀⢀⠀⠀⢸⣿⠀⠀⢰⣷⠀⠀⣼⡇⠀⠀⣙
⠀⠀⠀⠀⠘⠿⠿⠟⢛⣛⡀⠘⠛⠛⠛⠛⠛⠛⢛⡛⢛⡛⠛⣿⡇⠀⠀⣿⡇⠀⢸⣿⠀⠀⣾⡇⠀⢀⣿⠃⠀⣸⡟
⠀⠀⠀⠀⢠⣄⠀⢰⣿⠛⢿⣦⠀⠀⣀⣀⠀⢠⣿⠋⠙⣿⠀⢸⣷⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⠇⠀⣸⡿⠀⢠⣿⠃
⠀⠀⠀⠀⢸⡟⠀⠘⢿⣶⣾⠟⢠⣿⠟⢻⣷⠀⠻⢷⡾⠟⠀⢸⣿⣤⣤⣿⡇⠀⢸⡇⠀⢸⣿⠀⠀⣿⠇⠀⣼⡏
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⢿⣶⣾⠟⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⣿⡇⠀⣿⡇⠀⢸⣿⠀⢰⣿⠀⢰⣿⠁
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠸⣿⠀⠀⣿⡇⠀⠈⠁⠀⣾⡇⠀⣾⡇⠀⣿⡇
⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⣿⡄⠀⣿⡇⠀⣠⡀⠀⣿⠇⢀⡀⠀⣸⡿
⠀⠀⠀⠀⢸⣿⠀⠀⣾⡟⠛⠗⠀⠀⢀⣀⡀⠀⣼⡟⠉⢻⣧⠀⠛⠁⠀⢸⣇⠀⣿⡇⢸⣿⠀⣼⡟⢀⣿⠇
⠀⠀⠀⠀⠀⣿⠀⠀⠻⣷⣶⡦⠀⣼⡟⠛⣿⡆⠘⠿⣶⠿⠃⠀⣴⡄⠀⢸⣿⠀⢿⠇⢸⡿⠀⣿⠃⣸⡟
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠹⣷⣶⡿⠃⠀⠀⠀⠀⠀⠀⣿⣧⣤⣼⣿⠀⠀⠀⣾⡇⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⢹⣿⠀⠀⠀⣿⠃⠀⠀⣾⡏
⠀⠀⢸⣷⠀⣿⡇⠀⠀⢀⣀⡀⠉⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⣿⡇⠀⢸⣿⠀⠀⢸⣿⠀⠀⢰⣿
⠀⠀⢸⣿⠀⢻⣇⠀⢰⣿⠛⢻⣧⠀⠀⣀⡀⠀⢠⣿⠋⠙⣿⠄⣿⡇⠀⢸⣿⠀⠀⠸⠟⠀⢀⣿⠇
⠀⠀⠀⢿⣇⠘⣿⡄⠘⢿⣶⣾⠏⠠⣿⠟⠻⣷⡀⠻⢷⠾⠟⢰⣿⠁⠀⢸⣿⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠘⣿⣆⠘⢿⣦⡀⠀⠀⠀⠀⠠⣶⣾⠟⠀⠀⠀⠀⣠⣿⠏⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠈⢻⣧⡀⠙⠿⣶⣤⣀⣀⠀⠀⠀⠀⣀⣠⣤⣾⠟⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠉⠀⣶⣦⣄⡉⠛⠛⠻⠿⠿⠛⠛⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⣀⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠲⣤⣀⣤⠴⠂⠐⠦⣤⣠⣤⠖⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣧⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⣼⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣩⡄⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠛⢻⠀⠉⠀⠀⠉⠀⡟⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⡄⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣸⡟⠁
⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀⠀⠈⠻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⡿⠋⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⢠⡾⠋⠉⠙⠓⠶⢦⣤⣄⣀⣀⣙⣦⣍⠉⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠛⠛⠉⠉⢉⣥⣞⣃⣀⣀⣤⣤⠶⠖⠛⠉⠉⠻⣦
⠀⠀⠀⠈⢧⣤⣤⣄⣀⣠⠞⠀⠀⠉⠻⣍⣉⣉⣙⣿⣿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⢿⣿⣟⣉⣉⢉⡿⠉⠁⠀⠘⢦⣀⣀⣤⣤⣤⠏
⠀⠀⠀⠀⠀⠙⠓⠦⣬⣉⣻⡶⠞⠛⢋⣝⣋⣭⠖⠃⠀⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⡀⠀⠐⠯⣍⣙⣟⠛⠛⠶⣾⣛⣩⡴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣀⣤⣒⠉⢁⣸⣁⣤⢶⣿⣿⠟⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⢿⣿⣷⢦⣄⣹⣄⠉⢑⣢⣄⣨⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉⠉⢀⣴⣿⠟⢁⣴⣿⣿⣿⣋⣉⣉⣉⣉⣻⣿⣿⣷⣄⠙⢿⣷⣄⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠿⠋⢀⣴⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣷⣄⠈⠻⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠋⠉⠀⠀⠀⣸⡏⠀⢸⣿⣤⣤⣄⣀⣤⣤⣀⣠⣤⣼⣿⠀⠈⣿⡀⠀⠀⠈⠉⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠈⣿⣤⣤⣀⣈⣉⣉⣀⣀⣤⣼⡏⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⢻⡿⠿⠿⠿⢿⠿⠿⠿⠿⣿⠃⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠈⣿⣶⣶⣦⣤⣤⣶⣶⣾⡏⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣉⣉⣉⣩⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠱⢃⡴⠖⠶⢤⣀⠀⠀⠀⠀⠀⢠⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡀⠀⠀⢌⠳⢄⡀⠀⢀⣼⣜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠏⠉⠉⣩⠍⣓⠳⢤⣵⡒⣮⠻⠟⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣆⣀⣈⡴⠲⢤⣵⢤⠤⠃⣼⡷⠖⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⡙⠷⣤⡗⣿⠁⠈⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⡄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣛⡷⡟⢸⠀⠀⠀⠀⠀⠀⣴⣋⠭⠖⠒⠂⠈⠁⠀⠉⠀⡨⢿⠿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠟⠀⠛⠀⠀⠀⠀⠀⠀⢸⡷⠛⠛⠓⠶⠶⠶⠶⠶⠶⠶⢦⡀⠀⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣥⣤⣄⣁⣀⣀⣀⣀⣀⣀⣸⣇⠀⠠⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡉⠙⠯⠍⠉⠙⠛⠛⠛⠉⠉⠩⠽⣶⣬⢹⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡓⠆⠀⠀⠀⠀⣀⠤⠤⠶⠤⣄⣀⡀⠀⠸⣧⠿⢽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⣤⣀⣀⣠⠞⠁⢀⣠⣤⡀⠀⠙⣦⠀⠀⢰⢾⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⣴⣿⣿⣿⣿⣷⡄⠘⣿⣿⣿⣛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⡿⢭⣿⣧⠀⢿⣿⣿⣿⣷⣿⠁⢠⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣒⡢⠔⠛⣦⣬⠉⠛⠿⠋⠡⣪⠟⢩⠤⢌⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣗⡀⠀⠀⠙⠳⠶⠶⠖⠛⠁⠀⠠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠐⠩⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣄⣀⣀⣤⣤⣄⣀⣀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠶⠶⣤⣌⡒⠦⣄⡀⠀⠀⠀⠀⠀⠸⣯⣀⣉⣛⠙⠛⠛⠛⠉⢙⣻⣿⠇⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⠬⠙⠓⠶⣭⣓⠦⢄⡀⠀⠀⠀⠈⠉⠙⢛⣛⠛⢛⣛⣋⣉⣁⠀⠀⣀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢙⡻⢶⣮⣀⠤⢤⣤⣄⣀⡀⠀⠀⠀⠀⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠢⡈⠙⠽⠳⢶⣄⡀⠀⠀⠈⠑⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡈⢦⣄⣐⣐⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣄⣭⣤⡁⢂⣡⡼⣁⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⠛⠋⠥⠤⠤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠆

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⢹⡆⢀⣠⠞⠁⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⢀⣶⠞⠁⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡴⠋⡀⠁⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣴⠟⠃⠀⠀⠀⠀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⣦⡀⠀⠀⠀⢀⠀⠀⠀⢸⡿⠀⣴⡟⢉⣥⠄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⣁⣤⡀⠘⣿⣄⣶⠿⣿⣿⣆⠀⡿⢀⣾⣿⡖⠉
⠀⠀⠀⠀⠀⠛⢉⣤⠞⠁⠈⠛⢷⣬⣿⢻⣿⡅⢻⣿⡆⠁⠘⠃⠟⠕⠀⣀⢀⣠⣀⡠⢤⣤⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⣿⣿⣸⣿⣧⣼⣯⡀⢆⡀⣂⣠⣤⣿⣯⣼⣯⡽⠤⠀⢙⣿⣷⠀⠉⣿⣷⡄
⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣧⣽⣿⣹⣿⣨⣿⣿⣿⣿⣿⣿⣿⣿⣗⡒⠀⣴⣿⣿⠀⠀⢼⣿⣿⠆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠃⢀⣴⣿⣿⡿⠁⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⡟⠁⠀⢹⣿⣿⣿⣿⣾⣿⡿⣿⣿⣿⣿⣿⡟⢻⣀⣶⣾⣿⣿⠟⣁⣾⣿⣿⡧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠀⠀⠈⣽⣿⡿⣍⣿⣿⣧⠿⠿⢷⣾⣿⣿⣾⣿⣿⣿⢟⣭⡇⣾⣿⣿⡟⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⡿⠻⠃⠀⣿⣾⣿⣿⣏⠽⠻⠀⣰⣿⣯⣴⣿⠟⢫⣾⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⣼⡿⢻⠟⣁⣴⣿⣿⣛⣻⣍⣴⣿⣿⠿⣫⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠛⢇⣶⣾⣿⣿⡉⡈⠛⠿⢿⣿⣭⣿⣿⠿⢟⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣷⢰⠀⠀⠀⠀⠙⠻⢿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠈⢿⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡤⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣧⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠄⢣⣶⢶⡄⠀⠀⣰⣿⣷⡀⠀⠀⠀⠀⠀⠻⠆⢣⣾⠻⣆⠀⠀⣰⡿⣷⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣎
⠀⠈⠻⡾⠃⠀⠀⠙⣻⠟⠀⠀⠀⠀⠀⠀⣾⣿⣧⣛⣚⠃⠀⠀⠈⣛⠛⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏
⠀⠀⠟⠉⠀⣀⣀⢾⡁⢰⠄⠀⠀⠀⠀⠀⠙⠛⠃⠛⠁⠂⣤⣄⢺⣁⢰⠆⣀⣀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠈⠺⣉⣹⠆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣽⠃⠉⠁⢺⣏⢹⠂
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⡄⣄⣼⣗⠶⠖⠒⠒⠒⠒⠒⢦⠒⠒⠚⠓⠖
⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣤⣇⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⢀⡆⠈⠁⠀⠀⠈⠉⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⢷⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡈
⠀⠀⠀⠀⠀⢈⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠀⠀⣀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⡠⢄⡀⣤⣶
⠀⠀⠀⠀⠀⢿⠶⡄⠀⢘⡃⠀⠀⠀⠀⠀⠈⠉⠒⠤⣄⣀⠉⠀⠈⠛⣀⡄⠀⠀⠀⠀⠲⣿⡵⠂⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠰⠋⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⢁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠾⡇⠀⢀⠀⠀⠠⠐⠊⠁
⠀⠀⠀⢠⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠈⣄⠀⠀⠸⣿⡂⠀⣀⠀⠀⠀⢰⣾⠚⢗⠀⠀⠀⠀⠀⠀⠀⣴⣂⣽⠒⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⢰⣿⣷⣰
⠀⠀⠀⠀⡅⠀⠀⠀⠘⠻⠃⢻⠀⠀⠈⠉⠿⠶⠞⠁⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠰⣤⢧⠆⠀⠀⠀⡉⢟⣿⣻⢏⠛
⠀⠀⠀⠀⠛⠀⡀⠀⠀⠀⠀⡘⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠛⠞⠀⠀⠀⠀⠀⣼⠈⢑⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠘⣷⡄⠀⠙⢿⣿⣿⣿⣿⣿⡟⡄⠀⠀⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣿⣾⣿⣿⣧⣴⣿⣿⣿⡿⠛⣿⠇⠁⠀⠀⢸⣯⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⢸⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⣀⣤⣴⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⠛⠻⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠛⠿⢶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠿⠿⣿⣿⡄⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣀⣠⣤⠶⠿⠛⠋⣩⠵⠶⢦⡀⠀⠀⠉⠻⠷⣦⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⠛⠛⠉⠁
⣿⣍⠀⠀⠀⠀⢸⣧⣤⣤⣼⣧⣄⠀⠀⠀⠀⠀⠈⠻⢿⣼⣿⣿⣿⣿⡍⢻⣿⣾⣿⣿⣼⣷⣷⣿⣿⡿⠀⣼⣿⣿⣿
⣿⣿⣷⣄⠀⠀⠀⠉⢹⣏⣀⣤⡿⠀⣀⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣷⠀⠙⢿⣿⣿⣿⣿⣿⣿⡟⣀⣼⣿⣿⣿⡏
⠈⠻⣿⣿⣷⣀⠀⠀⢀⣉⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠈⠛⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠙⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⡿⢛⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠀⠀⠉⠛⠶⣤⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠹⠿⠟⢉⣠⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⣷⡀⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⣴⡎⠎⠁⠉⠙⢹⣶⣄⠀⠀⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢷⡀⠀⣾⣿⣧⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⡄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢳⠀⠻⣿⢿⡄⡄⢆⣠⣾⠟⠛⠆⠀⠀⢠⠎⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢞⣇⣀⢤⣿⣿⣿⢻⠻⣿⣶⣷⣤⣄⡴⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡋⠀⠁⠀⠀⠀⠀⠀⠁⠫⣷⣽⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢢⠼⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢨⣷⠒⠀⡀⠀⠀⠀⠀⡀⢀⣀⣰⣿⣤⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠤⠖⠋⡐⡐⣲⢿⣿⣾⣈⡀⣶⠀⠀⢱⣬⣾⣿⣩⡭⠦⠀⢀⠙⢤⠄⣀⣀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠛⠲⠖⠛⠒⠛⠋⠀⠀⡼⠁⠉⠉⢸⣷⡟⡿⡗⢄⢾⢹⠟⣟⢻⡒⠚⠃⢣⠀⠀⠀⠻⠓⠒⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠆⡐⢀⡶⣛⣿⠏⠉⠀⠂⠀⠈⠋⠁⠀⣸⠿⣿⣙⡲⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠋⠈⠻⡍⠻⠳⠤⢤⠀⣀⠠⠔⠾⠓⠛⠃⠰⠈⠉⡁⠀⠀⠀⠀⠀⠀⠘⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⡠⠀⠀⢸⡄⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⢰⠃⠀⡀⠀⢳⣌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡅⠀⣰⢿⡧⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⢾⠀⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⢟⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠀⢀⡄⠀⠈⢧⡢⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡼⠃⠈⣧⣀⣄⠀⠀⠀⠀⠀⠀⠀⠠⠠⢿⡿⠀⠀⠀⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠋⠋⠙⠂⠀⠤⠤⠄⠤⠐⡶⠂⠈⠇⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡞⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠒⠂⠤⠠⠤⠖⠖⠛⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⣀⠀⣀⣀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁


⣶⣶⣶⠶⡶⢶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠒⠒⠂⠀⠐⠒⠠⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢹⠆
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡀⠀⠒⠒⢤⡀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠰⡄⠆⠀⠀⠀⠰⡆⠀⠀⢀⣀⢉⢲⣄⣄⢀⣀⣠⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⢀⠇⠨⠧⠦⠆⠦⠽⠶⠤⠤⠰⠒⢳⣳⣋⡭⠶⠷⢿⠛⠿⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡜
⠀⠀⠀⠀⢰⣅⡰⢀⢠⣀⠴⢬⣀⢀⣠⢀⡴⠖⠉⠈⣃⠀⡀⡄⠈⡆⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣸⠁
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠱⠚⠓⠊⠉⠓⢳⢹⣔⣃⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡼
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢰⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠛⠶⢦⣀⡋
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡴⠛⠂⠉⠒⠀⠤⡀⠈⠑⢆
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣸⡅⠤⣠⣤⡀⠀⠀⠀⠐⠢⣈⡇
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣼⣁⡀⠀⠀⠀⠀⠉⠙⠢⣀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠑⠒⠂⢤⡀⠀⠀⠹⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠶⠴⠶⠒⠐⠋⠁⠀⠀⠀⠀⠀⠀⢸⣀⣀⣤⣤⣀⠀⠀⠀⠈⠒⢴⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠀⠀⠀⠀⠉⠒⠤⡂⠄⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠈⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠠⠄⣀⣤⣈⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠳⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠐⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢤⠂⠁⠠⠀⠐⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢑⠆⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠑⢦⠀⠀⡀⠀⠈⢀⠀⠰⡆⠁⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠤⠚⠃⠀⠀⠁⠀⡃⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢀⡀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠉⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⣀⠀⠄⠐⠀⠀⠀⡀⠀⠀⢀⠛⠁⢂⠀⠀⠀⠤⠜⠉⠀⠀⠀⠈⠀⠀⠂⠀⠉⠄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⠞
⡔⠈⠁⢱⠀⠀⢲⠂⠀⠀⠁⡄⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⣌⡀⠀⠀⣄⠀⠠
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠐⠄⣀⣀⣠⢴⠢⢄⠀⠀⠃⠀⠀⠀⠀⠀⢨⠁⠈⠉⠃⠀⠆⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⠀⠘⠁⠐⠀⠀⠈⠛⠁⠠⠖⠀⢀⠀⠀⠠⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠃⠈⠀⠆⠀⠀⠀⠀⠀⠁⠀⠒⠢⡀⠀⢄⠀⢀⠠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⠈⠁
⠀⠀⠀⡤⠯⠤⠤⠤⠄⣀⣀⣀⣀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡓⠀⠂⠀⠂⠀⠀⠆⠀⠀⠃⠐⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠘
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠰⠀⠀⢀⡀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣄⡤⠾⣴⣦⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⡇⣤⣿⣿⢇⡎⢀⣀⡠⠖⠉⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢳⣤⠞⠛⠋⡔⣽⠾⠋⠀⠀⠀⠄⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢭⠀⠀⢀⢱⣤⣾⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠒⢋⣁⠣⣀⢀⣾⣿⣿⣃⣤⠄
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠀⠀⢀⠔⠙⠁⠳⣿⡟⣛⠩⣧⠬⠅⠀⠰⢤⣀
⠀⠀⠀⣀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠆⢻⠟⠁⠀⠀⣠⠶⠦⠀⠀⠘⠁⢀⣄
⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠸⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢧⡀⢀⣀⠴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⢿⣧⣠⡽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠙⠛⠽⠟

⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⠀⠀⠀⠀⠀⠀⢀⠀⠐⢢⣀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤
⠀⠤⠀⠠⡄⠲⣄⣤⣷⣦⢀⡔⢀⠄⠀⠠⠀⠙⢦⣷⣿⣶⠴⠁⠀⠀⠀⠠⠘⠦⣰⢿⣿⣂⠴⠡⠄
⠀⠃⢄⠀⠀⠀⢘⣿⣦⣿⡟⠁⠀⠀⢀⠔⠀⠀⠔⣿⣿⡿⠢⠀⠀⠐⢄⠀⠀⢀⣿⣿⣿⣟⡁⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠴⢿⣿⠛⣿⣿⠦⠐⠒⠁⠀⠴⠃⠀⣿⠶⡇⠀⠑⠦⠀⠀⠉⡠⢻⣿⠭⢽⡿⡳⣄⠉
⠀⠀⠀⠀⠀⠀⢸⢻⣉⡿⠻⠀⠀⠀⠀⠀⠀⢀⡴⣟⣛⣷⣤⠀⠀⠀⠀⠈⠁⢸⠸⡭⡽⢳⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠙⠙⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠉⠈⠛⠁⠉⠋⢹⠙⠋⠈⠙⠉⠉⠁⠀⠀⢸⣿⣿⣿⣷⣾⣷⣴⣿⣶⣴⣶⣦⣾⣷⣦⣾⣷⣿⣿⣿
⠀⠀⠀⠀⢠⣦⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠙⢿⣧⣿⢿⣦⣄⣬⡴⠶⢦⣄⣀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡉⠉⢠⣈⠉⠻⠟⣋⣉⣛⠿⢿⣿⣿⣿⣿
⠀⠀⢀⣀⣞⣀⣉⣛⣾⣭⣵⣞⣛⣛⣾⣯⡀⠀⠀⠀⢸⣿⣿⣿⠿⢱⠿⡶⣬⠑⠒⠋⣡⢶⣬⡉⠚⢿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣴⢯⣉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡜⠛⣦⡼⠟⠛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠳⠶⢿⠁⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠳⠦⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⣴⠎⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣱⣾⡄⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢿⣯⡼⠛⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠘⣃⡀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡿⣶⡛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡾⣿⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠈⢧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢉⣿⣾⠟⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠟⠷⠟⠈⢇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢀⣠⠴⠖⠛⣛⣛⣛⣋⣉⣉⣉⣉⣍⠩⠭⠭⠽⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠤⠤⠤⠴⠶⠶⠶⠶⠶⢶⣟⣛⡳⢶⣤⡀
⢠⡞⢻⡷⠖⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡿⠉⠙⡆
⢻⡄⠈⠳⢦⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣤⣤⡤⠴⠶⠞⠋⠁⠀⠀⠀⢹
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⣄⠀⠀⢀⡤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢠⡇⠀⠀⠀⠀⣠⠛⢳⡀⠀⣸⠀⠀⢧⣀⣠⠟⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢶⡄⠀⠀⠀⢰⠋⠉⢻
⢸⡇⠀⠀⠀⠀⣿⠆⠈⠳⠞⠃⠀⠀⠈⠉⠁⠀⠀⠀⢨⡇⠀⠀⠀⣠⠞⠋⠛⠳⣄⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⣿⠀⠀⠀⢀⡄⢀⡼
⣾⠀⠀⠀⠀⢸⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢠⡇⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢸⠇⠀⠀⣀⣹⣦⣤⡴⠾⠛⢻
⡿⠀⠀⠀⠀⣿⠿⠦⠤⣤⠤⠤⠤⠶⠶⠶⠶⠶⠖⠂⣾⠀⠀⠀⠘⡷⠒⠒⠲⠖⠛⠷⢶⡄⠀⠀⣿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡆
⠱⣄⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⣿⡿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢹⠀⠀⠀⠀⠀⣀⣀⣤⠴⡇
⠀⢹⠶⣮⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⠤⠤⠤⠤⢾⡀⠀⠀⠀⣸⠓⠚⠛⠛⠉⠉⠁⠀⠀⢸
⢀⡟⣁⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠁⠈⠁⣈⡉⠉⠉⡁⠀⢠⡀⢀⣀⡀⢀⡀⠛⣷⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⣿⡠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⡆⠀⡟⠉⢹⠀⠘⡛⠳⣼⣇⢸⡦⠀⠀⠙⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡶⠁
⠀⠈⢳⣾⣥⣀⣀⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠛⠀⠙⠶⠞⠀⠀⠁⠀⠈⠋⠈⢉⣉⣀⡀⣥⣤⣤⣤⣤⠤⠄⠶⠶⠒⠉⠉⠀⠀⡇
⠀⠀⡿⠂⠀⠈⠉⠉⠉⠛⠋⠙⠛⠛⠒⠋⠘⠀⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠰⣟⡡⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠈⠙⣿⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠄⠒⠛⠋⠉⠹⡆
⠀⠀⠀⠸⣧⠀⠈⠉⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠈⠉⠉⠉⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠸⣧⠔⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠃
⠀⠀⠀⠀⠀⠈⢹⡷⢦⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⠶⠶⠶⠖⠊⠉⠉⠸⠁
⠀⠀⠀⠀⠀⠀⠘⣶⡠⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣋⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⠒⠒⠚⠛⠛⠙⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⢂⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⡟⣩⣉⣐⣀⡜⣁⠀⣠⣀⡜⣠⠆⣠⠆⣤⣤⣼⣷⠷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡰⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⠤⠖⠉⠀⠠⠜⢤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠸⣅⠈⠁⢠⠀⠈⠁⣰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣄⠀⠠⠄⠀⣠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠋⠀⢗⠂⠐⡿⠀⠙⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠄⣡⣌⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⡴⣹⡠⠊⠉⠉⠙⠉⠉⠉⠑⠄⠏⢆⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠂⢄⠀⢀⡼⠥⠛⠂⠄⡀⠀⠀⠀⠀⢀⠠⠒⠚⠬⢧⡀⠀⠠⠐⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⢢⡠⠒⠁⠀⠀⠀⠀⠀⠀⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠒⢄⡔⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢰⡑⡐⣠⢠⠁⠈⠒⠒⠒⠒⠒⠒⠒⠂⢤⠄⠒⠒⠂⠒⠒⠒⠒⠒⠁⠈⡄⣄⢂⢪⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠖⠂⠁⠢⢄⠀⠀⠀⠀⠘⠁⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣈⣐⣤⣀⣀⠤⠔⠁



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⢠⡶⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢂⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⣤⣤⠋⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣹⡷⠴⣠⣄⠱⣿⣶⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣚⠋⠶⣦⣝⣿⣷⠘⣿⠇⠸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠞⠉⠐⢿⣦⣈⣿⣷⣻⣧⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠏⡀⠀⠐⠄⠙⡿⣮⣿⠛⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠦⡀⣊⣢⠟⡫⠃⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⠭⠁⠜⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠤⠄⠠⡇⠀⣤⠀⠤⠤⠤⠤⠤⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣂⣉⣀⣲⡤⢶⠤⠃⢀⠰⣶⠦⠴⠄⠀⣀⣀⡀⠀⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠴⢏⣙⣇⣸⡠⡧⠼⣦⠀⠈⠐⢦⣤⠤⠤⠤⢤⣄⠀⠙⠢⡀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣶⢮⣉⣉⠭⠄⣀⠀⠠⠀⣀⣀⠀⠀⠀⣀⡀⠀⠀⠀⢑⣤⠃⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣽⣂⠈⣉⠃⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠀⡄⠐⣁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⣝⡻⣟⠀⣴⡂⠀⠀⠀⢠⣶⡄⢰⡯⢦⡀⢂⡀⣇⢈⣀⡥⡞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢈⢀⡇⠁⠈⠋⢮⡇⠀⠀⠂⢨⠟⡻⢿⠺⠾⠟⠛⠋⠙⠁⠃⠃⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠜⠻⠀⠀⠀⠀⢀⠀⠀⢀⠓⢸⡃⢌⢂⢃⠀⡀⠠⠠⠀⠀⡘⢰⠀⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠃⠀⠀⠀⠀⣠⣆⠀⠤⠅⣼⡷⠶⠶⢄⠀⠀⠁⠆⡀⠀⠃⡈⠀⠁⢈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⣠⡾⠃⠈⠳⠶⠞⣦⣀⠀⡀⢋⣆⠀⠀⠠⠀⡘⢀⠃⠃⠠⢸⢠⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠂⠀⠀⣔⠝⢁⡀⠠⠠⠤⠤⣴⣿⣤⡟⢀⠄⠘⣧⡀⠀⠈⠁⠸⢰⡖⠓⠮⡈⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠉⠠⢄⠀⣠⣶⣯⢁⣿⠃⠀⠀⣀⠘⢟⡄⠀⠀⠃⣼⠀⠀⠀⠡⡆⠀⠐
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠆⠀⠨⡥⣀⡠⢶⣦⣙⡿⣿⠀⢀⡄⠀⠀⡜⡌⠀⠀⢀⠻⠀⣄⠀⠀⠀⠠⢄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⣁⠀⠔⢸⠷⣶⣝⣿⢷⠻⡃⠈⣠⠂⡘⡘⠀⠀⢨⡘⢀⠶⣈⡁⠾⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠴⠐⠀⠀⠀⠀⠰⠘⢆⠶⠻⠷⠼⢋⢾⡈⠤⠁⠐⠁⣲⠌⠀⠀⠀⠀⠃⡈⡀⢁⠁⠆⠀⢠⡸⠃
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠱⠈⠖⢎⢦⡀⠋⠀⠀⠀⠖⢠⡳⠁⠀⠠⢀⠂⠈⠰⢡⠁⠄⡘⠰⠀⠂⡉
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠚⠭⠴⠶⠖⣂⣀⣀⣰⠏⠀⠀⢀⠂⠌⡀⢀⢃⢃⠆⡠⢁⢃⠂⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠐⡐⡀⢀⠎⠎⡰⢁⠎⣀⠎
⠀⠀⠀⠀⠀⠀⠐⠢⠀⠂⠀⠊⠑⣠⣉⠂⠑⢊⠔⡠⠂⠂⠐⠀⠁⡠⢁⠌⠔⠜⠀⠀⠎⠈⠀⣤⠞⣣⠀⡠⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⢂⠠⠀⠄⢉⡰⠛⡰⠦⣤⢀⣀⣀⢀⢀⡀⣀⡀⢀⡤⠀⠄⣐⢞⠕⢁⠈⡡⢊⠐⢁⠔⡡⢂⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠡⠒⢁⠔⢉⠔⢊⠤⠂⠠⠀⠀⠀⡁⠀⡠⠂⠔⠠⠂⡠⢁⠄⡡⠊⡄⠁⠔⢁⠘⠁⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠎⠀⠊⠐⠀⠐⠀⠊⠀⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠲⢄⣀⣀⡠⠔⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣅⠀⠀⠀⣾⣉⣹⣿⣿⣷⠀⠀⠀⣨⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠳⠂⢀⣻⡄⡽⢻⢿⣷⡄⠐⠞⠁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⡠⠒⠈⠉⠅⠀⠀⠀⠀⠀⠀⠀⠐⠂⠼⢦⣿⠁⠉⢀⣾⣗⣿⣿⡷⠧⠐⠂⠀⠀⠀⠀⠀⠀⠀⠨⠉⠁⠂⢄⡀
⠀⠀⢯⡀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⢀⣾⣅⢰⢾⡯⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡵
⠀⠀⠀⠈⠓⢄⣀⣀⡀⠀⠤⠀⠀⠒⠊⡉⠉⢀⠔⠿⣿⣿⣿⣿⣿⣿⡿⠦⡀⠉⢉⠑⠒⠀⠀⠤⠀⢀⣀⣀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⣤⡾⠓⢤⡲⡄⣠⡼⣿⠝⢹⣿⣾⡿⢯⣄⢠⢖⡤⠚⢷⣤⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⠀⠉⣻⡁⡰⠁⣀⣀⣀⣙⣿⠂⢈⡟⠋⠀⠀⠀⠻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠋⢹⣿⠿⠟⠛⠻⢿⣿⣷⡏⠙⡆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣻⠀⣸⣁⡀⣐⣶⣶⣶⣾⣧⣁⠀⣟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡏⠀⢹⣟⣉⣁⣈⣉⣙⣿⡟⡏⠀⢹⡟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣅⠏⠀⠀⠘⣿⠿⠶⠿⠿⣿⠿⢿⠃⠀⠀⠹⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠃⠀⠀⠀⠀⠸⣦⣤⣠⣶⣿⣷⠇⠀⠀⠀⠀⠘⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠈⣙⢿⣾⣿⣿⣿⣶
⣿⣟⣴⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣿⣿⣿⣿⣿⣿⢿
⣿⣿⣿⡷⣿⡿⢶⣿⣷⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠀⠁⡀⠰⣿⣍⢛⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡿⢿⣿⣿⣿⣿⣿⣿
⣿⣯⠀⠀⠀⠈⠆⠙⠿⠿⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡠⠿⠛⠛⢻⡿⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⡄⢠⣄⠀⡴⢠⡄⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠃⠀⠀⠒⠂⠚⠉⠁⠂⢀⣿⣿⡆⠀⡏⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠠⠤⠶⠒⢂⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⣀⣠⠤⠤⠤⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⢀⠀⡠⠤⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣀⠈⣭⣿⣥⣤⠀⣀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠍⠉⠛⠂⠙⠉⠉⠀⠠⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⢤⣤⣰⣾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⣿⠿⢻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡏⠚⠀⠀⠀⠙⣿⣿⣿⣿⣆⠀⠀⡻⠿⠿⠿⣿⣿⣿⣿⣦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⢿⣿⣿⢿⣿⡟⠁⠀⠀⠀⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⡘⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄⠀⢀⠄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠚⠂⠑⠛⠘⠀⠃⠀⠃⠂⠃⠋⠞⠁⠐⠃⠀⠋⠘⠁⠐⠑⠃⠀⠂⠘

⠀⠀⠀⢠⡤⣶⣀⣄⡀⠀⠀⠀⠀⠀⠠⣤⣤⣶⣠⣤⣀
⠀⢀⠀⢸⡟⣿⣟⣿⡇⠀⠀⠀⠠⣴⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠻⠟⠛⠁⠀⠀⠀⠀⠈⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣦⣭⣽⣿⣿⣿⣿⣾⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠈⠱⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠛⠛⢦⣤⣤⣄⠀⠙⠿⠿⠛⠋⠉⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣾⣦⣾⣶⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣨⡿⠛⠻⣧⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⢀⣿⢧⠀⢀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣸⣿⣷⣿⣿⣴⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠛⠉⠈⣍⠉⠙⠛⢉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠿⠿⠿⠿⠟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⣷⣶⣤⣤⣶⣿⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣦⣭⣭⣭⣥⡶⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⠈⢯⣭⣭⡽⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣕⣯⣭⣭⣭⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣔⣒⣒⣀⣀⢠⣤⣄⣒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠾⣟⣋⣉⠉⠉⠁⠀⠀⠈⣀⠈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠨⣗⡲⠦⢭⣍⡉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣊⢻⣿⣿⣶⡖⠒⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠶⠋⠀⣤⡀⠉⢿⣿⣍⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠀⠰⠶⠶⣾⡋⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⡉⠁⠀⡄⠀⠀⠝⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣍⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠤⢤⣀⣀⠀⢀⣿⣿⣿⣿⣿⣿⣯⡅⠀⠀⢀⡀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣏⡛⠻⠿⠿⠿⠯⠷⠢⢒⣣⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠢⠽⠛⠓⠒⠶⠖⠚⠛⠉⠁⠀⠀⠀⡀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡐⡜⢆⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⢧⢹⣾⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⡿⠿⠿⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⠛⠛⣿⡛⠛⠛⢿⣿⣿⣿⣿⣿⣿⡟⡛⠛⠛⣿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢳⣤⣼⣿⣶⠀⣠⣿⣿⣿⣿⣿⣿⢫⠂⠀⠀⢻⣟⣷⣦⣤⣴⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⢸⣟⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣟⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⣀⣿⣿⣃⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠙⠻⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣯⣅⡀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⡏⠀⠈⠙⠛⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣴⣶⣾⣿
⠀⠀⠀⢀⣀⣤⣴⡶⠾⠟⡛⠉⠛⠛⠿⣷⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋
⣤⡶⠟⠛⠋⠉⢀⡴⠛⠛⠙⡇⠀⠀⠀⠀⠉⠛⠷⣦⣘⡿⣻⡟⣿⣿⣿⢿⣿⣿⣻⡟⣿⢻⡟⡟⣿⣿⣿⠛⢸⣿⣿⣿⠉
⣿⣦⡀⠀⠀⠀⠐⠷⠶⣶⡾⠛⢶⡄⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣾⣿⣿⣿
⢿⣿⣿⣦⡀⠀⠀⠀⠘⢯⣤⠶⢟⣁⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠻⢿⣿⣿⣿⣿⣿⢏⣴⣾⣿⣿⣿⣿
⠀⠙⢿⣿⣿⣦⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠙⢿⣿⣟⣿⣿⣿⡿⠿⣿⣿⡏⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠘⢛⣋⣠⣴⣿⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⡶⠒⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣁⠈⠙⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠷⣶⣤⣈⣉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢶⣦⣄⡉⠙⠳⠦⡤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⠀⢀⡞⢉⡼⠛⣫⣷⠆⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣿⣀⡟⢁⡞⢁⡴⠋⣺⣎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣾⡄⣼⡁⣼⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡴⢆⢀⣀⠀⠀⠀⠀⠀⠀⠉⡛⠿⣿⣷⡿⡿⢃⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠢⣼⡈⣻⡇⣸⠃⣸⠤⠄⣀⡴⠖⣋⣩⣥⡷⢤⡉⠧⠆⡼⠀⣉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠚⠢⣼⣿⣿⣿⣿⣿⡇⢠⠞⣡⣶⣟⣭⠵⠶⣒⣚⣗⣛⣒⡇⠀⠀⠀⠀⠀⣄⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣁⠒⣿⣿⣿⣾⣧⣿⢧⡏⣼⣿⣿⣽⢶⣛⣋⣭⡭⣯⠭⢽⠃⠀⠀⠀⢈⡲⢤⡙⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢤⡬⣝⠛⡿⢿⢿⣻⢸⡇⡜⢿⣿⣿⣻⣭⡷⠶⠖⡟⠲⢺⣷⣶⣤⣄⣲⣽⣶⡝⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⣞⣾⠉⢻⣧⡼⣾⡟⣼⣷⣜⠦⣝⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡿⠿⢛⣩⣾⣿⣿⣆
⠀⠀⠀⠀⢠⠖⠦⣄⣇⠀⠘⣇⠀⣸⡏⠀⠸⣼⣿⣿⣿⡷⣤⣉⠀⠀⠀⠉⢉⣉⣉⣩⣭⣥⣴⣶⣾⣿⣿⣿⣿⡿⢿⡆⠀⠀⢀⡀⠀⡀
⠀⠀⠀⣀⣈⣣⣀⠀⢽⣾⣶⣿⣾⣿⣾⣤⢶⣿⣟⠷⣬⡛⠲⢭⣙⠛⠶⠶⠶⠶⠶⠶⠶⠶⢛⣛⣯⣿⣿⣿⣿⡿⣻⣿⠐⢦⡏⠱⡾⢹⣤⢤⡀
⠀⠀⠸⣅⠀⠈⢽⣿⣿⣿⡿⣛⣛⣻⣿⣯⢸⣿⣝⠷⣄⠉⠓⠂⠀⠈⠀⠀⠀⠀⠐⠒⠛⢛⣋⣩⣥⡾⣿⣿⣿⡿⣫⣿⠰⣴⣿⣶⣿⣟⣅⡼⠶⣄
⠀⠀⡴⠚⠛⠓⠺⣿⣿⣯⣾⣿⣿⣿⣿⣿⡞⣿⣟⡳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣩⣥⣶⡾⣿⣿⣿⣿⣫⡿⢬⣿⣿⣽⣯⣿⣿⠦⠶⣅
⠀⠀⢙⡶⠖⠶⠚⢿⣿⣿⣿⣿⣿⣿⣿⣉⠳⢻⣿⣿⢶⣍⡑⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣥⣤⠶⢾⣿⣿⣿⣿⣿⣃⡤⠾⣿⣿⣿⣿⣿⠣⣴⠚
⠀⠀⠘⠲⣦⠶⠒⠚⢛⡿⠋⢹⡏⠙⡄⠈⣳⣿⣿⣿⣷⣦⣍⣙⠒⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⢶⣿⣿⣿⣿⢿⣿⣿⣯⣤⡞⠁⡾⢻⡟⢱⡦⠼⠃
⠀⠀⠀⠀⠿⠶⠴⣖⣉⠀⣠⣿⣇⣠⣿⣶⣿⣿⣿⣿⣿⣿⣦⣭⣙⡓⠒⠶⠤⠤⠤⢤⡶⠶⠶⣿⣿⣿⣿⣿⣿⣟⣛⣛⣻⣷⣾⣧⣸⣓⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣯⣝⣛⠳⠶⠶⠶⠶⠶⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠯⠭⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠚⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⡤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⡆⣾⣿⣴⣿⢀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡀⣿⣦⣿⣷⢸⡷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣷⡹⠟⣼⣫⠟⠁⠀⠀⠈⠻⣿⣯⣽⣿⠟⠀⠀⠀⠈⢿⣝⣧⠻⢏⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⢀⣠⣤⣤⣤⣭⣭⣤⣤⣤⣄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⣠⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣄⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣾⣿⣿⠗⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠸⣿⣿⣶⣾⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠻⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⣦⣌⣛⠿⢿⣿⣿⡿⠿⣛⣡⣴⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣥⣚⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⢻⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣝⡻⠿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠒⠉⠙⠛⠿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⠴⠂⣀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⠃⠀⠀⣠⠤⠚⠁⠀⠀⣔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡷⡄⠀⠀⠀⠈⢸⠀⠀⠀⢀⣹⣦⣤⣤⣴⣁⠀⠀⢠⢞⠀⠀⠀⠀⢠⢺⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢦⡀⠀⠀⠸⣙⢦⡀⣬⣠⣿⣇⣿⣿⣠⡇⣐⡶⠋⠀⠀⢀⠶⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠷⢦⠀⠀⠈⠑⠟⢃⢉⢻⣯⣿⣯⣛⡾⠏⠀⠀⢀⣼⣶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠂⠠⣤⡤⠀⠐⠟⢮⡟⣾⡟⢵⠷⢒⢁⢤⣦⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠀⠠⠀⠀⠀⠐⠂⠀⠉⠹⣶⣌⠉⣺⡿⢿⣶⣮⣿⢘⡘⠋⠁⠁⠀⠀⠀⠀⡀⢈⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⢀⠀⠤⠀⠀⠚⠁⠀⠉⠀⣻⣹⣼⣦⣿⠁⠀⠈⠑⠂⠀⠀⠠⠀⠀⠀⠀⡀⠑⡄⠒⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢀⢀⠤⠖⠈⠁⠀⡀⠀⠀⠀⢠⠂⠐⡂⢀⡠⢹⢋⢀⣨⡒⢰⡔⠀⠀⠄⠀⢀⠀⠐⢄⡀⠁⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠐⠀⠀⡀⠀⠀⠀⢰⡿⠠⠛⠋⣭⢿⣿⣷⣿⠿⠛⠆⠑⢦⠀⠀⠈⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡴⢣⠃⢴⣶⣶⣧⣴⠾⢶⠶⣶⣤⣶⠀⢿⣇⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠃⠈⣠⣤⣍⡉⠛⠁⠙⢋⣉⣉⡠⠀⠈⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢉⠛⠿⢿⣿⣿⣿⠿⠿⠟⡛⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡟⠀⠀⠀⠘⢧⣿⣶⣲⣶⣶⣶⣿⣟⠅⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠁⠀⠀⠀⠀⠀⠲⠾⢤⣭⣭⡿⢷⠈⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠘⠳⣤⣤⣤⠶⠂⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡀⠀⢠⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⣰⣞⣋⣙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⡄⠀⠀⠀⠀⣠⡤⠶⠴⠢⢤⣤⠤⠤⠬⢧⣤⡉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠶⠶⠖⠒⠒⠛⠻⠀⠀⠠⠄⠀⠀⠀⠄⠀⠀⠀⠢⣹⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠖⠒⠒⠒⠀⠀⠀⠘⠒⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡍⠳⣿⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠁⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣐⠃⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡤⠤⠀⠹⣇
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡄⠀⡤⠀⢠⠲⡀⠀⠁⠈⠁⠘⠒⠄⠀⠀⢿⡀
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠃⡅⡌⠉⢠⠋⠀⢱⠸⠙⢦⡃⢰⠖⠑⡞⢀⡱⠞⡔⠢⣀⠔⡆⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠐⠃⠀⠈⠈⠙⢸⡗⢠⣄⠑⢪⡄⠀⡇⠡⢄⠀⠘⢆⢠⠊⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⡀⠔⡆⢠⢰⠈⠂⠘⠃⠀⡀⠑⡄⠀⠀⠇⠏⠁⡀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠳⠤⠡⢠⠀⠢⠤⠌⠘⠒⠀⠀⠘⠢⠃⠣⠤⠤⠇⠀⢧⠃⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⣤⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠀⠀⠀⠈⠠⠀⠠⠈⠀⠀⠀⠀⣀⡤⠄
⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠙⣦⣄⢰⣽⣶⣶⣶⣾⣿⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⢴⠁⠀⠑⢒⡖⠒⣈⣩⣒⣢⠤⢀⡀⠀⠘⢿⣆⡽⠋⠉⢻⣿⣅⣾⠟⠀⠀⣀⠠⢤⣖⡪⣉⣐⠒⢖⠒⠁⠀⢑⡆
⠀⠈⠁⢆⠈⠈⢙⡤⠤⠈⣖⠒⠓⠒⠤⢽⣷⣶⣾⡇⠀⠀⢰⣿⣿⣶⣶⣿⠥⠔⠒⠓⢒⡎⠤⠤⢌⠉⠈⢀⠆⠉
⠀⠀⠀⠀⠉⠫⣁⡠⠔⢒⢠⣏⣁⡠⢤⡿⢟⣿⣿⣷⣀⠀⣽⣿⣿⣿⣟⠿⣧⠤⣀⣉⣧⢑⠒⠤⣀⡩⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠤⢤⣊⣠⣔⡩⢄⣚⣭⣭⣿⣿⣿⣿⣿⣯⣭⣝⣂⠬⣑⣤⣈⣢⠤⠜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⣩⣿⣿⣿⣏⣉⣿⣿⣿⣿⣯⡉⠻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣠⣿⠟⡟⠛⠛⠻⣿⣿⣿⡟⢿⣧⡀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⢰⣷⣤⣄⣀⣽⣿⣿⢰⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠘⣯⣉⣉⣉⣻⣿⣿⡸⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢯⣉⠛⠛⣻⣿⣏⠇⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⢯⣭⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠰⡀⠀⠀⠁⠈⠁⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣄⣠⣶⣶⣷⣶⣶⣦⡀⠀⠀⠙⠒⠀⠂⠁⠤⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣴⡿⠃⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠋⠀⠀⢿⠆⣤⢸⣿⣿⡿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣰⣿⣤⡄⠀⠀⠀⠀⠘⣿⣿⣷⣶⡀
⢠⡶⠂⠀⠀⠀⠀⠀⠻⠃⡠⠀⢀⢀⣤⣤⣀⠀⢠⣿⣿⣵⣶⣾⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣣⠀⠀⠀⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡁⠉⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢛⠉⠉⢰⣶⣶⣴⣶⣤⠀⣉⠋⠀⠀⠀⠀⠀⣤⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣴⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠛⢿⣿⣿⣿⡦⡤⢤⣀⡀⠀⠀⠀⢀⣸⣿⣶⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠒⠒⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠏⠉⠉⠉⠉⠉⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠻⣿⣿⣇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣸⠿⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠢⢀⠀⡁⠀⡀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣻⣿⢷⡤⣀⠀⠀⠀⠀⠀⠐⢄⡀⢠⣹⣥⣔⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⡶⣦⢳⣱⣿⠿⠿⢯⡀⠀⠀⠀⠀⠀⠁⠀⠀⠏⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡳⣿⣮⡷⠉⠁⠐⢆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⡃⣿⢿⣿⡿⣃⡈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢀⡜⠀⡄⠐⢇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠁⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣄⢠⣤⡄⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣠⣄⠀⣭⡈⠀⣰⡈⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠚⠛⠃
⠀⠀⠀⢀⣀⠀⢈⣀⠁⢉⡀⠉⣁⡉⢉⣩⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢄⡀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠶⠚⠉⠁
⠀⠀⢠⡋⠉⠁⠁⢉⡀⠉⣹⠚⠉⢓⠋⢉⡋⠁⠀⠀⠀⣠⣼⠾⣏⠙⣧⠈⣧⢈⣠⠤⠔⠚⠉⠉
⠀⠀⠈⠇⠙⠈⡃⠉⠛⠐⠊⠑⠘⠛⠃⢘⠋⠀⠀⠀⠀⡏⢹⡆⢹⣆⢽⣶⣾⡿⠃
⠀⠀⠀⠘⠃⡈⠛⠆⠛⠛⠀⠞⠛⢆⠘⠀⠶⠀⠀⠀⠀⠸⣿⣿⣷⣿⣾⣿⠛⠁
⠀⠀⠀⠈⠳⠶⡄⠜⠶⠤⠋⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⠨⡝⠃⠀⣚⠠⠌⢹⡻⠦⣥⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⡀⠈⢘⠷⢤⣀⣈⣵⡒⠜⢃⡴⠋⣛⣻⣿⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡉⠀⠀⠈⠐⠯⢁⠵⠔⣩⡿⠙⠈⠉⠒⢊
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⢶⣴⠀⢳⢉⣶⢻⡇⢠⡴⠖⠚⠩
⠀⠀⠀⠀⠀⠀⠀⠶⣤⣀⣀⠀⠀⠐⢀⣺⣖⢂⣸⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⣦⡙⠛⠟⠀⢸⡏⠀⠀⣠⢤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢉⠛⠳⠛⠛⢻⡯⡹⠁⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡤⡼⠧⠤⠞⠛⠉⡇
⠀⠀⠀⠀⠀⠤⠀⠐⣷⠀⠁⠀⠒⢄⢚⡄⣿⡝⠯⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣤⣄⠠⢀⡀⠀⠀⠀⠀⠄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠠⠝⠻⣦⡵⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡛⠊⠿⢶⣥⠴⡧⠤⠆⠂⠈⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠓⠀⠀⡠⠀⠳⡟⠃⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠅⠀⠀⠈⣀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣦⡇⢀⢓⣶⣾⠿⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⡤⠀⡶⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉⢁⡾⠁⣀⣒⣲⠆⠀⡱⠯⠀⠀⠐⠀⠈⠀⣈⡀
⠀⠀⠀⠀⠀⡄⠒⢤⠀⠀⠀⣀⣀⡤⠴⠒⣲⠟⠉⠉⢡⣿⣥⠄⠀⠈⠉⠉⠙⠒⠒⠠⠎⠀⢨⠆
⠀⠀⠀⠀⠀⠣⡀⠀⠓⠒⠉⠁⠀⠀⣠⣞⣡⣤⠤⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠌⢢⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⡜⠀⠀⢈⠔⢀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢧⠀⠀⢀⠠⠀⢀⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⡀⠀⠈⠄⠀⠁⠠⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⢀⠔⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠢⣀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡠⠋⠈⠉⠓⠒⠒⠶⠶⠖⠒⠒⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⡀⠀⠀⠀⠀⠈⠁⢰⠀⠀⡀⠀⠐⠼⠁⠀⠀⠀⠀⢀⠂
⠀⠈⠄⠀⠰⡄⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⢸⡄⡞⠀⠀⠀⠀⢆⢤⠀⠐⠀⠆⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣦⠘⢄⠒⠲⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡇⡄⠀⠀⠀⠊⠁⠔⠂⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⡀⠸⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⡀⠠⠀⠀⡀⠀⢀⠖⠀⠊⢀⠁
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠄⠀⠀⠀⠀⡦⠃⠀⢠⠀⡇⢀⠀⢰⡁⠀⢸⠀⡰⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠱⠠⠀⠀⠘⠁
⠀⠀⠤⠀⡤⠤⠒⡠⠔⠀⠀⠀⠀⠂⠀⡀⢀⡒⢦⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠀⠤⠠⠠⣄
⠐⢀⣠⡀⠄⠀⠀⡀⠀⠀⣀⠐⠁⢀⠔⠁⠠⠉⠋⠁⠀⠀⠀⠀⠀⢀⣻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠰⠲
⠀⠘⠛⠉⠀⠈⠀⠀⠉⠁⠙⠶⠛⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠢⣄
⠀⠀⠀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠀⠀⠀⠁
⠀⡄⢠⠀⠀⠀⢐⡀⣠⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠋
⠀⠻⠶⠘⡆⠀⠀⡼⠘⠛⠾⠧⠢⠀⠠⠄⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠄⠀⠘⠂⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠁⠉⠓⠶⣄⡀⠀⠀⠀⠀⠀⢀⣠⠞⠃⠀⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⣀⠀⠀⠒⠀⠀⠀⠀⠀⠙⢦⡀⠀⣠⠞⠉⠀⠀⠀⠀⠒⠋⠀⠀⠀⡄
⠀⠐⠀⠐⡒⠐⠂⠂⠠⣀⡀⠀⠀⠀⠀⠀⠈⠻⢇⣠⠀⠀⠀⠀⠀⠀⣤⣿⡆⡿⠀⣀⠀⠀⠀⠀⠀⣀⣐⠞⠋
⠀⠀⣀⣀⡈⠀⡆⠉⣇⣀⠈⠑⠲⢤⣀⠀⠀⠀⠀⠙⠶⢤⢀⠀⢰⣯⣀⣼⣤⣤⣰⣥⡀⠀⢀⡰⠖⠋⠀⠀⠀⠀⢀⣠⠤⠖⠋⠁⠀⠀⠀⢀⡠
⠀⠀⠐⢀⠈⠁⠈⠡⠀⠀⠀⠁⠀⠀⠸⡛⠲⢤⣀⠀⠀⠀⠸⣧⠈⣩⣿⣿⣿⣯⣟⡉⠁⣴⠋⠀⠀⠀⣀⣠⠴⠞⠉⠤⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠈⢀⢙⠢⠄⢈⠛⠦⣤⡀⢻⣶⣿⣿⣿⣿⠏⠁⠙⣶⡋⣀⡤⠖⠛⢁⠠⠆⠀⠀⠀⠀⢄⣀⣀⣀⡄⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠄⠠⠀⠀⢀⡁⠀⠐⠀⠠⠿⢿⡉⢹⣿⣿⣿⣿⣧⣀⣰⣏⠋⡧⠤⣄⠀⣂⡈⣉⡀⠀⠀⠂⠀⠀⠀⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⢄⠀⠀⠀⠀⠀⢀⡠⢖⠞⠛⠉⣯⣻⣿⣿⣿⣿⣁⣩⣿⠛⢓⠢⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠁⠀⣡⠊⢄⡠⠤⣤⣼⣿⣿⣿⣟⡿⡿⣿⣿⣤⣤⡴⡄⠁⠀⠁⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠉⢉⣛⣿⡿⣇⠸⣩⣷⡋⠉⠉⠈⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢀⡴⢻⣿⣿⣿⣧⣿⣟⡋⠙⢧⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠀⢠⡏⠐⠙⠿⠿⣿⣿⣷⠿⠛⠂⠀⢹⡄⠀⠈⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠀⠀⠀⠀⣿⠀⢠⣶⣶⣶⣿⣿⡟⣒⣶⣾⠀⢸⡇⠀⠀⠀⠙⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠈⣿⠀⠈⠛⠛⠻⠿⣛⠿⠎⢚⣉⡁⢸⢷⠀⠀⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢨⡿⠀⠀⠛⠛⠿⠿⠟⠿⠽⠛⠋⠁⠘⣿⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠅⠀⠀⠿⠿⣿⣿⣿⣿⣵⠿⠿⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠏⠀⠀⠀⢰⣤⣤⣴⣶⣶⣤⣤⡖⠀⠀⠀⠘⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠈⣉⣉⣛⣛⣛⣋⣩⠀⠀⠀⠀⠀⠀⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣻⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⡔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢃⠀⢠⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⢀⣠⠖⠂⢉⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠦⠄⠀⣴
⠀⠀⠈⡄⠀⠃⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⡀⡎⠀⠀⠠⢴⠋⠀⣀⠴⠟⠀⠀⠀⢀⡀⠱⠶⠄⠀⣒⡲⠔⠒⠀⠀⠀⠈⠁
⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠴⠀⡀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⡇⠀⠶⢅⡀⠀⣱⠛⠁⠀⢰⠃⠀⠈⠠⠀⠈⠀⢀⠂⠁⠀⠈⠀⠀⢀⠂
⠀⠀⠀⠀⢱⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠄⠐⠀⠐⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠀⠀⠀⠀⠀⠈⠀⠀⠡
⠀⠀⠀⠀⢀⠷⠊⠉⠭⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠄⠀⢠⡶⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢦⠂⢀⠤⠒⠒⠚⢆⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠸⠎⢀⠄⠐⠀⠉⠙⡄⠀⠀⠀⢀⠀⠀⠘⠀⡄⠀⠀⢠⠀⠀⢸⠀⠀⠆⠀⢀⡇⠀⡤⠀⠀⠀⡓⠀⢼⣷⡆⠀⠀⠀⠀⠀⠄⠋
⠀⠀⠀⠀⠀⠻⠁⡠⠒⠊⠉⠁⣇⠀⠀⠀⢸⠀⠀⡇⠀⣇⠀⠀⢸⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠂
⠀⠀⠀⠀⠀⠀⠙⢤⢀⠀⡠⠀⠁⠀⠀⠀⠆⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⢀⡀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠠⢄⡀⠐⠋⠁
⠀⠀⢀⠀⣀⠙⠀⠀⡇⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠤⡀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡀
⠀⠀⢈⢺⡇⠀⠀⠀⠄⠂⠀⠀⡀⠀⠠⣌⠀⢉⢢⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣠⠜⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢋
⠀⠁⠀⠜⠀⠀⠀⠀⠀⠀⠀⣄⣉⣉⠉⠉⠉⣩⡝⢛⢛⠉⠋⢱⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⡐⠐⠉⠀⠈⠀⠘⠀⢄⠉⠀⢌⠀⠈⠁⠀⠀⣁
⠀⠀⠀⢀⠀⠀⠐⠒⠀⠀⠀⠜⠀⠀⠀⠠⠜⠉⠉⠀⠉⢷⠓⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢁⠈⠠⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠘⠆⠀⠀⡀
⠀⣁⠰⠀⣅⠀⠀⡁⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠙⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⡁⠂⠠⠌⠙⠁⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⢤⠀⢄⠀⠀⡀⢠⠄⠁⠀⠀⠲⠀⢠⠀⠀⠀⠀⢦⣈⠀⠀⠀⠀⠀⠀⠀⡨
⠀⠂⠀⠀⠐⠔⠀⠈⠈⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠑⠈⠒⠘⠂⠀⠀⠠⠀⠠⠈⠀⠈⠀⠀⠀⠀⠀⠛⠥⣴⣀⣀⣀⡩⠔⠁