Egypt Emojis & Symbols

⠀⠁⠴⠯⠭⠽⢷⠀⠀⠁⠀⠀⠀⠀⠁⠀⢀⠾⠭⠭⠽⠆⠉⠀⠀⠁⡴⠯⠭⠭⢷⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡾⠭⠭⠽⠆⠉
⢀⣤⣛⣛⣛⣛⡻⠁⠀⠀⠠⠶⠦⠀⠀⠀⠘⢟⣻⣛⣛⣛⣤⡀⢀⣤⣛⣛⣛⣛⡻⠁⠀⠀⠠⠶⠦⠀⠀⠀⠘⣟⣛⣛⣛⣓⣤
⢸⡷⠶⠶⣾⣿⠀⠀⠀⠀⢘⣛⣛⠀⠀⠀⠀⠈⡿⡷⠶⠶⢿⡇⢸⡷⠶⠶⣾⢿⠀⠀⠀⠀⢘⣛⣛⠀⠀⠀⠀⠈⡿⡷⠶⠶⢿
⡌⢙⣛⣛⡇⢸⡀⠀⠀⠀⢨⣭⣭⠀⠀⠀⠀⢰⠆⢿⣛⣛⡀⢡⡎⢙⣛⣛⡇⢸⡀⠀⠀⠀⢨⣭⣭⠀⠀⠀⠀⢰⠆⢿⣛⣛⡀
⠁⠸⠶⢎⠀⢨⠇⠀⠀⠀⢘⣛⣛⠀⠀⠀⠀⢸⡄⠀⡷⠶⠆⠈⠁⠸⠶⢎⠀⢠⠇⠀⠀⠀⢘⣛⣛⠀⠀⠀⠀⢸⡄⠀⡷⠶⠇
⠀⠀⠀⢁⣠⠏⠀⠀⠀⠀⣸⢿⢿⣀⠀⠀⠀⠀⠳⣄⠈⠀⠀⠀⠀⠀⠂⢀⣠⠏⠀⠀⠀⠀⣸⢿⢿⣀⠀⠀⠀⠀⠳⣄⠈
⠶⠖⠚⠋⠀⠀⠀⠀⢘⣿⣷⣿⣷⣯⣏⠀⠀⠀⠀⠈⠙⠒⠶⠶⠶⠶⠚⠋⠁⠀⠀⠀⢘⣿⡷⣿⣷⣿⣏⠀⠀⠀⠀⠈⠙⠒⠶
⠲⢲⡶⢄⠀⠀⠀⢀⣺⠏⣧⡿⣷⠏⢿⣂⠀⠀⠀⢀⡤⢶⡲⣖⠲⢲⡶⢄⠀⠀⠀⢈⣺⠏⢧⡿⣷⠏⢿⣂⠀⠀⠀⢀⡤⢶⡲
⠀⣾⣱⣏⣷⠀⠀⠀⠀⠄⠀⠰⠀⠀⠄⠀⠀⠀⢀⣮⣻⡜⣇⣬⠀⣾⣱⣯⣷⡀⠀⠀⠀⠄⠀⠰⠀⠀⠀⠀⠀⠀⢀⣮⣻⡌⣇
⢠⣄⠹⣗⣒⣣⡀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠀⢀⣞⣒⣺⠇⣠⡄⢠⣄⠹⣗⣚⣳⡀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣒⣺⠇⣠
⠀⠁⠴⠯⠭⠽⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠭⠭⠽⠆⠉⠀⠀⠁⠴⠯⠭⠽⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠭⠭⠽⠆⠉
⢀⣤⣛⣛⣻⣟⡻⠁⠀⠀⠰⠶⠦⠀⠀⠀⠘⣟⣻⣛⣛⣛⣤⡀⢀⣤⣛⣛⣻⣝⡻⠁⠀⠀⠰⠶⠦⠀⠀⠀⠘⣟⣹⣛⣛⣛⣤
⢸⡷⠶⠶⣾⢿⠀⠀⠀⠀⢘⣛⣛⠀⠀⠀⠀⠈⡿⡷⠶⠶⢿⡇⢸⡷⠶⠶⣾⢿⠀⠀⠀⠀⢘⣛⣛⠀⠀⠀⠀⠈⡿⡷⠶⠶⢿
⡌⢙⣛⣛⡇⢸⡀⠀⠀⠀⢨⣭⣭⠀⠀⠀⠀⢸⠆⢿⣛⣛⡀⢢⡎⢙⣛⣛⡇⢸⡀⠀⠀⠀⢨⣭⣭⠀⠀⠀⠀⢸⠆⢿⣛⣛⡀
⠁⠸⠶⢎⠀⢸⠇⠀⠀⠀⢘⣛⣛⠀⠀⠀⠀⢸⡄⠀⡷⠶⠇⠈⠁⠸⠶⢞⠀⢨⠃⠀⠀⠀⢘⣋⣛⠀⠀⠀⠀⢸⡄⠀⡷⠶⠇
⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⣸⢿⢿⣀⠀⠀⠀⠀⠳⣄⠈⠀⠀⠀⠀⠀⠀⢁⣠⠏⠀⠀⠀⢀⣸⢿⢿⣀⠀⠀⠀⠀⠳⣄
⠶⠶⠚⠋⠀⠀⠀⠀⢘⣿⢷⣿⡶⣿⣏⠀⠀⠀⠀⠈⠙⠒⠶⠶⠶⠶⠚⠋⠀⠀⠀⠀⢈⣿⣷⣿⣷⣷⣏⠀⠀⠀⠀⠈⠙⠒⠶
⠲⢲⡶⢄⠀⠀⠀⢀⣻⠋⣧⡿⣷⠏⢿⣂⠀⠀⠀⢀⡤⢶⡲⣖⠲⢲⡶⢄⠀⠀⠀⢈⣺⠋⣧⡿⣷⠏⢿⣊⠀⠀⠀⢀⡤⢶⡲
⠀⣼⣱⣯⣵⡀⠀⠀⠀⠀⠀⠰⠈⠀⠄⠀⠀⠀⢀⣮⣻⡌⣇⣨⠀⣾⣱⣯⣵⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣮⣻⡌⣇
⢠⣄⠹⣗⣚⣣⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⢀⣞⣒⢺⠇⣠⠄⢠⣄⠹⡗⣚⣣⠀⠀⠠⠄⠀⠀⠀⢀⠀⠀⠀⢀⣞⣒⣺⠇⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢾⣴⣾⣷⣦⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠄⣾⣿⣿⣦⣿⣿⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⣿⡏⢻⡿⠉⠛⣿⣿⡇
⠀⠀⢀⠔⠋⡝⠙⠃⣼⢿⣿⣿⡿⢯⠌⠛⠻⠙⠖⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣧⣴⣿⣿⣿⣄
⠀⢠⠻⣄⣴⡆⠀⣞⠉⠀⠚⠓⠀⠉⢳⡄⠠⣷⣀⠞⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡛⠟⣿⣿⣿⡏
⠀⢸⠶⢾⣿⡇⠘⡇⠀⣼⣶⢴⣿⠀⢠⠇⢀⣿⣿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠈⠖⣛⣿⣻⣄⠘⣤⣾⣿⣿⣿⣤⠎⢠⡞⣿⣟⢒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣻⣿⣿⣿⣧
⠀⠀⠈⢧⠜⣻⣟⣷⣼⣿⣿⣿⣿⣧⣴⣻⣿⠻⣤⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠙⢥⠟⡿⣿⣷⣿⣿⣷⣻⢿⡛⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠙⠲⢿⣿⣿⣿⣿⣿⠆⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡾⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠻⢯⣯⠿⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠘⢀⣴⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣆⣴⣶⣷⣶⣠⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⣤⣶⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣇⣉⣻⣽⣿⣿⣿⣟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠙⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⣿⣿⣿⣿⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢿⣿⣿⣽⠆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⣼⣿⣿⣿⡆⢸⣿
⠀⠀⠀⠀⢀⣤⠀⠀⠀⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣿⣿⣿⣿⡇⠘⠟
⠀⠀⠀⠀⠘⢿⣷⣦⢠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢀⡾⡆⠀⣦
⠀⠀⠀⠀⠀⠀⠙⠿⣤⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠉⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⢋⠇⢸⠃⢿⠀⣏⠢⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢔⣻⣛⢿⣗⣿⢟⣻⣃⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣛⠟⠛⡣⣽⡥⢛⡛⢟⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠃⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣻⣿⣿⣾⡏⣿⣾⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢉⣿⣿⡧⣿⣿⣿⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⡗⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣩⣿⣿⣾⣯⣽⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⣿⣯⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉

⠀⠀⠀⠀⠀⡔⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⣀⠼⠘⢀⠀⠀⠀⠀⠀⠀⠀⣰⠸⢿⣇⢴⡄⠁⠀⠀⠀⠀⠀⠄⣀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠠⢒⣿⣿⡶⠂⠀⠰
⠀⠀⠀⣰⠇⠐⠁⣀⠀⠀⠀⠀⠀⠀⡰⠉⢀⡀⠁⢀⡀⠀⠀⠀⠀⠀⢠⡾⢿⣥⡀⠀⠀⠈⠆⠀⠀⠀⠀⠃⢈⠳⠟⢃⠄
⠀⠀⠀⠀⡍⠌⠖⢋⠴⠀⠀⠀⠀⠀⠁⠀⠈⠹⢤⣤⠁⠀⠀⠀⠀⠀⠻⠦⠈⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣄⣀⠈
⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠒⠲⠀⠀⠀⠰⠀⠀⠀⠀⠀⠸⠆⠉⠹⠀⠀⠀⠰
⠀⠀⠀⠀⠐⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠊⡄
⠀⠀⠀⠀⠀⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⢴⡄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⡜⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣹⣄⡼⣿⠀⠀⠀⠘⣄⡓⠐⠗⠀⢠⠿⣿⡠⢤⡀⠐⠗⠀⢃⡰⠀⠀⠀⢰⡿⣕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡭⣿⡿⠘⠂⠀⠀⠀⠙⠷⣤⣀⡀⠂⢤⠈⠀⡄⠀⢀⣀⡴⠟⠁⠀⠀⠀⠉⠀⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠭⡴⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠗⢰⡋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⠆⠗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣆⣶⠖⢂⣶⠶⠖⠲⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢮⡀⣸⡇⢨⣧⣴⠟⢁⣤⡾⠟⠋⣉⣤⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡿⣿⣿⣿⣿⣿⠏⣠⡿⠋⣠⣶⠿⣛⣩⣭⣴⡾⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⡿⡗⢸⡿⠹⠏⢰⡟⢡⡾⢋⣵⢿⣿⣿⣿⣿⣶⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⣿⣾⣇⣀⠀⠀⠀⠈⠑⢿⣵⣿⣿⣿⣧⣤⣴⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡿⠒⠙⠛⠋⣉⣉⣭⡳⣦⣴⣿⣿⣿⣿⣏⣩⣤⣤⣤⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⣟⣛⣿⡟⡿⣟⣭⣭⣹⢿⣿⣿⣿⡿⣿⣍⣭⣥⣤⣤⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣛⡟⠛⠟⢰⠃⠈⠉⠉⠁⣀⣬⣽⡿⢿⣍⣯⣭⣤⣤⣤⣤⣴⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⣛⣉⣷⡀⢷⣷⠾⠇⢀⢀⣸⣿⣿⣿⣷⠻⣿⣭⣭⣤⣤⣤⣤⣤⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣛⣉⣉⣩⣿⣼⣿⠶⢶⣴⣿⣿⣿⣿⣿⣿⣷⣯⣤⣤⣤⣤⣤⣤⣴⣶⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⣉⣉⣭⣭⣭⣷⢶⡶⢷⣾⣿⣿⣿⣿⣿⢻⣿⣧⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣋⣉⣭⣭⣭⣭⣤⣷⣾⣷⣶⣾⣿⣿⣿⡿⠁⢸⣿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠭⠤⢤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣟⣿⡗⣄⣸⣿⣷⣶⣾⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣾⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠫⣹⡟⠨⣽⣿⣿⡿⠋⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⣿⡇⣿⣧⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⢕⡵⣧⣍⣷⢿⣿⠟⣡⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢹⣿⡽⣷⢞⣿⢿⣿⣶⣙⣯⣴⣮⡵⢿⣟⣧⣟⣵⣟⣵⣾⣿⣿⣿⠿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢻⣿⣿⣿⣥⣾⣁⡟⣹⡍⢹⣇⣽⣤⡿⢟⣫⣷⣿⡿⣫⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢿⠻⠿⣿⣯⣿⣿⡿⢿⠿⢋⣡⠾⢛⡽⠟⡩⠞⠁⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠀⠸⣿⣿⣶⣄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠆⡀⣠⣄⣀⣤⣤⣴⠛⠀⠀⠀⠀⣨⣿⣿⣿⣿⣷⣄
⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣥⣼⣷⣶⣤⣶⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⢿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠛⠛⠻⠛⠿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣸⠇
⣦⣴⣤⣤⡀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⠛⠋⠀⠛⠁
⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁⠀⢀⣀⣠⡀⣀⣤⣦
⣿⣿⣿⣿⣿⣷⣿⣿⣿⣆⣀⢀⣀⡀⠀⠀⣀⣀⢀⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⡰⠄⠿⠀⠋⠖⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠬⢆⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⣒⣡⡿⠆⠴⠁⢤⢘⣇⠀⠀⠀⠀⠀⠄⣠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⣭⣾⡀⠀⠀⠀⠸⡶⠥⡆⠀⠀⠀⠠⡄⠘⠨⠉⠀⠀⠀⡀⠀⠀⠀⠀⣠⠀⠀⡦
⠀⠀⠀⠀⠀⠶⢶⣾⣷⢶⡖⢴⣿⣶⣶⣿⠀⠀⠀⠀⠣⣔⣆⣴⢾⠀⠀⣯⣀⣀⣠⣴⠇⠀⢰⡿
⠀⠀⠀⠀⣨⣿⣛⣛⢹⣿⣧⡿⠩⠭⣽⣿⠆⠀⠀⠀⠀⠀⠉⠁⠘⠧⣺⡟⠉⠉⠉⠁⠀⢀⣾⡃⠁
⠀⠀⠀⠀⣿⣿⣇⣒⣿⣷⣾⢇⢨⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠛⠛⠀⠛⠛⠘⠋⠂⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠁⡼⠀⠀⠀⠑⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠤⠖⠏⠽⣆
⠀⠀⠀⠀⠀⠇⠀⢀⣠⣄⣷⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣽⣴⠦⠀⠀⠂⢸⣇
⠀⠀⠀⠀⠀⡆⠀⠸⠛⡿⠟⣫⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡾⠿⣧⣶⢶⣶⣾⣿⡄
⠀⠀⠀⠀⠀⡇⠀⠵⣼⣦⠞⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠈⢉⠃⠀⠾⢿⡏⠙⠛
⠀⠀⠀⠀⠀⡇⠈⠁⢏⣡⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠈⠈⠁⠀⠁
⠀⠀⠀⠀⠀⡇⠃⠀⢺⣇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⡇⠀⢈⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠋⠀⠀⠀⠀⠀⠀⠩⠉⠙⠛⠉⠀⠈
⠀⠀⠀⠀⠀⡇⠀⠀⠙⢹⢄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡵⡤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⢡⡩⡠⠿⠲⢠⣀⡤
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇⡔⠊⠉⠉⠉
⠀⠀⠀⠀⠀⠐⠀⢿⣦⣀⠀⠀⠀⠀⠀⠀⢹⣿⡦⣄⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣏⣷⡄⣤⡄
⠀⠀⡀⠀⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠀⠀⠀⠀⠹⣿⣼⣿⣿⡟⢻⣽⡿⠃
⠀⠈⠁⠀⠁⠈⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠛⣿⣿⡿⠛⠉

⡇⠀⣠⠤⠤⠤⠤⡀⠀⡇⠀⡀⠀⠀⠀⠀⢀⡀⢸⢀⡀⡀⣀⢀⢀⡀⡀⢸⠀⣀⡀⠤⠤⠄⣀⡀⢸⠀⢀⣀⢀⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⠀⠀⣤
⡇⡎⠀⠠⣿⡷⠀⠈⡆⡇⠀⡃⠀⢠⡄⠀⠘⠁⢸⠀⠈⠈⠀⠁⠉⠈⠈⢸⠀⠉⠒⠒⠒⠒⠂⠉⢸⠀⢸⡇⠀⠀⣤⢠⣄⠀⡇⢸⣿⣄⠀⠀⠀⣿
⡇⡇⠀⠀⢰⣦⠀⠀⡇⡇⠀⡅⢠⣭⣯⡄⠀⠀⢸⠀⣿⠀⠀⢀⢳⡦⠀⢸⠀⣖⣒⣒⣒⣒⣒⡆⢸⠀⢸⡇⠀⠘⠿⠸⠟⠀⡇⠀⢿⡿⣧⡀⠀⣿
⡇⡇⠀⠀⢹⡏⠀⠀⡇⡇⠀⠇⠀⠀⠀⠀⠠⠀⢘⠀⣿⠀⢠⣸⠟⠂⠀⢸⠀⠁⠀⠁⠈⠈⠀⠀⢸⠀⠸⠷⠄⠀⠀⠀⠀⠀⡇⠀⠈⠀⠈⢛⣀⣿
⡇⡇⠀⠀⠘⠋⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠈⢋⣥⣀⠀⢸⠀⢠⡆⣠⣤⡄⣶⡀⢸⠀⠀⢀⣤⣤⠀⠀⠀⠀⡇⢀⣀⠤⠤⠤⢄⣀
⡇⡇⠀⠿⠿⠿⠇⠀⡇⡇⠈⠭⠭⠭⡭⡽⠦⠀⢸⠀⠛⠶⠆⠀⠙⠛⠃⢸⠀⠘⠇⠙⠛⠃⠿⠁⢸⠀⢠⠟⠁⠘⣇⠀⠀⠀⡇⠉⠑⠒⠀⠐⠒⠉
⡇⡇⢠⠦⡀⣠⢶⠀⡇⡇⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⣀⣀⣀⣀⣀⣀⡀⢸⠀⣄⠀⠀⠀⠀⠐⡀⢸⠀⠙⠛⠀⠀⠘⠛⠃⠀⡇⣶⠀⢀⡀⢀⣴⣶
⡇⡇⠘⠠⠾⠦⠘⠂⡇⡇⢀⣀⣤⠼⠿⠛⠛⠁⢸⠈⠁⠙⠛⠛⠛⠋⠁⢸⠀⠛⠓⠒⠒⠒⠛⠇⢸⠀⠀⢴⣆⣤⣀⣠⠀⠀⡇⣿⠀⢸⣧⠀⠉
⡇⠈⠒⠤⠤⠤⠴⠊⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⢀⣄⠀⠀⠤⠤⠀⢸⠀⠀⢰⡆⠀⠀⠀⠀⢸⠀⠀⠈⠀⠙⠙⠏⠀⠀⡇⣿⠀⢸⣿⣀⠀⢠
⡇⢠⡄⠒⠒⠂⠀⢤⡀⡇⢀⣴⣶⠀⠀⢶⡶⠀⢸⠀⠉⡏⠁⢰⡒⢂⡄⢸⠀⠀⣿⡇⠰⠒⡒⡆⢸⠀⠲⣶⣶⣶⣶⡶⢦⠀⡇⠿⠀⠈⠉⠉⠀⠸⠁
⡇⠀⠈⠉⠁⠈⠁⠀⠀⡇⠈⠉⠁⠀⠀⢺⡗⠀⢸⠀⢀⣇⠀⠀⠉⠉⠀⢸⠀⠛⠛⠃⠐⠐⠃⠃⢸⠀⠀⠀⣉⠉⢀⠀⠀⠀⡇⠀⠀⠠⠶⠦⠀⢠⡄
⡇⠒⠖⠲⠲⠒⠒⠖⠂⡇⢠⣶⠀⠀⠀⠻⠟⠀⢸⠀⠘⠟⠀⠀⠀⣶⡄⢸⠀⠀⠲⡄⠀⡰⠆⠀⢸⠀⠀⣼⡏⠀⢸⣧⠀⠀⡇⠀⠀⠘⠿⠟⠀⠸⠁
⡇⢀⣤⣤⡀⣠⣤⣄⠀⡇⠀⠿⠀⢀⣄⡀⠀⠀⢸⠀⢀⣀⣰⣶⣶⣿⡇⢸⠀⠀⠀⡾⡿⣶⠀⠀⢸⠀⢰⡿⠰⣿⠆⢿⡇⠀⡇⣠⠖⠂⣀⡀⠒⢦⡀
⡇⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⢀⡿⠉⠻⣄⠀⢸⠀⠈⠉⢉⣉⠉⠉⠁⢸⠀⠀⠊⠘⡿⠈⠃⠀⢸⠀⠏⠀⠀⣿⠀⠀⠹⠀⡇⡇⠀⠈⠻⠋⠀⠀⡇
⡇⠀⠀⠀⠀⢸⣧⠀⠀⡇⠠⠶⠞⠀⠀⠴⠿⠀⢸⠀⠀⠀⣯⣟⡇⠀⠀⢸⠀⠀⣀⣤⣄⠀⠀⠀⢸⠀⢰⡇⠠⣿⠄⢰⡆⠀⡇⡇⠀⢰⣷⡀⠀⠀⡇
⡇⢸⠁⡉⢹⢰⣿⣀⠀⡇⠀⠀⠀⢀⣀⠀⢠⡄⢸⠀⠀⣀⡉⠋⣁⡀⠀⢸⠀⠉⠉⠉⠉⠉⠉⠉⢸⠀⠈⠁⠀⠉⠀⠈⠁⠀⡇⡇⠀⢀⣿⠁⠀⠀⡇
⡇⠈⠀⠉⠁⠀⠉⠉⠀⡇⠀⠀⣠⣾⡿⠀⠸⠃⢨⠀⠈⠉⠁⠈⠉⠉⠀⢸⠀⢴⡦⢠⣶⠀⣶⡆⢸⠠⠲⠔⠔⠦⠢⠢⠔⠄⡇⡇⢀⢨⡉⣀⠀⠀⡇
⡇⢀⣇⠀⠀⠀⠀⣰⠀⡇⠀⣴⠿⣿⠃⠀⢰⡄⢸⢠⡞⠀⡿⠛⠃⢸⡆⢸⠀⠘⠃⠘⠛⠀⠻⠀⢸⠀⣀⣀⣀⠀⣀⣀⣀⠀⡇⡇⠙⠒⠒⠒⠚⠀⡇
⡇⠈⠋⠉⠉⠉⠉⠉⠀⡇⠈⠃⠀⠀⠀⠀⠘⠃⢸⢸⣷⠀⡇⠀⠀⢸⣿⢸⠀⢀⣤⡀⠀⢀⣤⠀⢸⠀⠈⠙⠉⠀⠈⠉⠁⠀⡇⡇⠀⠀⣶⡆⠀⠀⡇
⡇⢠⠴⠤⠦⠦⠤⢤⠀⡇⢠⡤⠒⠒⠒⠂⢤⡀⢸⢸⠉⠐⡇⠀⠀⠈⢻⢸⠀⢸⠇⢙⣶⡁⢸⡆⢸⠀⠠⣤⠤⠤⠤⠤⠄⠀⡇⢧⡀⠀⠛⠀⠀⣠⠃
⠇⠈⠉⠉⠉⠉⠉⠉⠀⠇⠀⠀⠉⠉⠉⠉⠀⠀⠸⠀⠀⠀⠁⠀⠀⠀⠀⠸⠀⠀⠀⠉⠉⠉⠀⠀⠸⠀⠀⠈⠉⠉⠉⠉⠉⠀⠇⠀⠉⠉⠉⠈⠉⠁
⠀⡃⠀⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢡⣟⢻⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠀⢁⢀⡗⢸⡀⡀⠀⠑⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⣀⠐⣆⠀⣀⠘⡿⡇⠈⠷⠂⣀⠀⡰⠂⠈⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣎⢧⡘⡆⢻⣄⣷⡗⢶⡜⡀⡈⣰⢁⣞⡴⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⢷⠸⡌⣷⣽⡯⢴⣯⠔⢠⢇⣾⣞⣽⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠽⢿⣿⣿⣿⣗⠋⠁⢺⣷⡜⡿⠈⠙⢾⣿⣿⣃⡂⣐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠩⢽⣿⣿⣿⣿⠀⠀⠘⢻⡞⠁⠀⠀⣶⣿⠷⠄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⢒⣛⣻⣿⣿⠏⠖⠲⠦⠀⠀⠴⠒⠒⠸⣯⣍⡉⣉⣁⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠤⠤⠶⣿⣿⣿⠐⠿⠟⠚⠃⠈⠑⠻⠛⠂⠻⠒⠒⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣂⡀⢉⣩⣽⣿⡿⣆⠀⠀⠀⣀⣀⠀⠀⠀⣠⢪⠭⠭⠍⠉⢉⣉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠀⠀⠐⠒⣺⣿⣿⣟⢳⡀⢀⡈⢥⣄⠀⠀⣹⣿⠂⠀⠒⢒⡒⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢩⠭⠥⠤⠤⠿⣿⢏⢻⣆⠑⠈⠓⠂⠀⠀⣰⡿⡙⠅⠀⠀⠤⠤⠭⠭⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣒⡀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣶⣦⣴⣶⠏⣽⣿⣽⠀⠀⠀⠀⠀⢀⣀⣚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⢲⡄⠨⣿⣿⣿⣿⣿⠛⠉⠀⠀⣿⣿⣿⠀⠀⢠⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣀⢹⣿⣧⡻⠁⠀⠀⠀⠀⣹⣿⣯⣀⣠⠟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⢿⣿⣿⣿⡟⠻⠷⠤⠤⠴⠚⠋⢋⡿⢻⠏⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢿⡄⠙⠳⢤⣀⣀⣀⣀⣤⠴⠋⢀⠜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠈⠂⠤⣀⡀⠉⠉⢉⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠁⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠐⠂⠀⠀⢳⡀⠀⠀⠀⠎⠨⠅⠑⠆⠙⡉⢢⡀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⡷⡶⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣉⠀⠀⠈⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣆⠀⣷⠀⢥⠀⡇⠸⡇⠀⠀⠀⠀⠀⣎⣉⡙⠥⠀⠀⠤⣙⣉⣙⡄⠀⠀⠀⠀⠀⠀⠰⠁⣵⣶⠖⠠⠀⠀⠀⠸⡆
⠀⠀⠠⡀⠀⠀⢀⠤⠴⠆⢿⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⡼⠁⢽⡁⠒⠂⠐⠒⠈⣹⢈⢱⠀⠀⠀⠀⠀⠀⡇⠘⣿⠇⠀⠶⠀⠀⢠⠆⣇
⠀⠀⠀⠱⠀⢀⡧⠂⠁⠀⠈⡄⠀⠀⢰⠀⠘⡅⠨⠅⠘⠀⢺⡄⠀⠀⢰⡧⠤⡬⢦⡀⠀⠀⢀⡤⡧⣤⠴⣧⠀⠀⠀⠀⠀⡇⠀⢿⡀⠀⠄⠰⢆⡾⢀⣿
⠀⠀⠀⠀⠘⡄⠀⠀⠈⠀⠀⠁⠀⠀⠀⡀⠀⡇⠀⠀⠀⡇⠸⡇⠀⠀⢧⡒⠠⠠⠆⣿⠂⠀⣿⠆⠄⠐⢀⡼⠂⠀⠀⠀⢀⣇⠀⠈⣿⡀⠀⠀⡀⣰⠉⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠀⠀⠀⠀⢃⠀⢹⠀⠩⠀⢲⠀⣷⠀⠀⢀⣈⣳⠛⠛⡏⠀⠀⠈⡗⠛⢻⢋⣀⠀⠀⠀⢰⠛⣾⡄⢠⠘⠿⢶⣾⡿⢻⡆⠀⢿⣇
⠀⠀⠀⠀⢀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠸⠀⠈⡀⠀⡀⢀⡀⢹⠀⠀⠀⠁⢈⣏⣉⣷⡆⢰⣼⣏⣋⡏⠈⠉⠀⠀⠀⡜⠀⣿⠀⠘⠀⡇⠐⠒⠂⠈⡧⠀⠸⣿
⠀⠀⠀⠀⠀⠑⠄⠀⣿⣿⡄⠀⠀⠀⠀⠀⡇⠀⠇⠀⠇⠀⠃⢸⡆⠀⢂⢰⡀⣍⣉⣯⣼⣿⢸⣏⣩⠅⡐⢠⠀⠀⠀⠀⠀⢻⡀⠀⠀⣇⠀⠀⠀⠀⣇⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠗⠀⠀⠀⠀⠀⠠⢀⣸⠤⠴⠦⠶⢬⡇⠀⠘⢧⣝⣺⢷⣥⣬⣭⣬⡵⢷⣛⡵⠃⠀⠀⠀⠳⣄⣈⣇⣒⣚⣋⣈⣉⣉⣀⣠⣤⣤⠞
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠠⠤⠤⣈⡇⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢰⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠆⠀⠼⡇⠀⠀⠀⠀⠀⠀⢀⣠⠖⡛⠟⠛⡷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣄⡀
⠀⠀⠀⢸⡗⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⢋⡾⠙⢮⢂⣀⡇⢤⣙⡆⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠈⢿⣆
⠀⠀⠀⣸⣷⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⠀⠐⡆⢰⣧⣀⡀⠀⠀⠀⠀⢇⣒⣚⡖⠛⠉⠡⠄⠀⢼⣿⡀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⠀⠈⠙⢣⡄
⠀⠀⢰⡟⠈⠁⠀⠉⠀⣀⣀⠐⠀⠀⠀⠀⠡⡀⢀⠀⡀⢠⢸⣿⣿⡟⠀⠀⠀⢸⠸⠻⠿⠧⡄⠀⠀⠀⠀⢸⣋⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡅⠀⠀⠀⡏⠁⠀⠀⠀⡰⠀⠀⠀⠀⠈⠂⠁⠊⠉⠶⠋⠉⠁⠀⠀⠀⢆⠀⠤⠨⠭⢷⠀⠰⡄⢠⡮⠤⣼⠀⠀⠀⠀⠀⠀⡄⠀⠈⠉⠉⠀⠀⢸⡇
⠀⠀⢨⠛⠉⠀⠀⢶⡀⠀⠀⠀⡇⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢸⠋⢙⣏⠱⠶⣇⠀⠑⢸⡏⢹⢇⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠒⠢⢾⡄
⠀⠀⢇⠀⢀⡀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠑⡈⢢⡀⠀⠀⠀⠠⢂⡞⠀⠀⠀⠘⡓⠉⡉⢗⠀⢸⣦⠰⡈⣟⠙⣿⡆⠀⠀⠀⢀⠎⠀⠀⠀⢢⡀⠀⠀⠀⣷
⠀⠀⠘⠀⢸⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠀⠹⡀⠀⠀⢁⣾⠀⠀⠀⠀⠀⢳⡄⠀⣘⡝⢂⣿⠠⠕⣟⣻⢹⡇⠀⠀⠀⣌⡀⠀⢸⠓⣤⠗⢒⡒⠒⢿
⠀⠀⢰⠀⣶⠀⠀⢀⠂⠀⠐⢠⠁⠀⠀⠀⠀⠀⠃⠀⡏⠁⠙⠺⣿⠀⠀⠀⠀⠀⠀⠻⣦⣝⣿⣷⣾⣷⢶⣟⣧⡿⠃⠀⠀⠸⡘⠀⠀⠘⣅⠀⠀⠀⠀⠀⣹⡄
⠀⠀⠘⠀⠘⠛⠆⢼⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⢉⡁⠀⠀⠀⠀⡇⠀⠀⠀⢠⣿⠀⠀⠀⠀⢀⣿⡟⡀
⠀⠀⠀⡇⠀⡀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢦⣀⣁⣊⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠤⠀⣵⡄⠀⠱⢄⡀⠀⠈⢣⠀⢹⡦⡀⣼⣿⣇⠃
⠀⠀⠀⣁⠀⣇⡴⢺⠄⠀⠀⠀⡇⠀⠀⠀⠀⣠⣞⣛⣻⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⠀⠀⠀⢸⣿⡄⠀⠀⠹⡀⠀⠈⡆⠀⢷⠀⢼⣿⣿
⠀⠀⠀⡿⠁⠐⠖⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⣿⣿⡿⠛⢋⠁⢉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠺⣿⣿⣿⠃⠀⠀⠀⢳⠀⠀⠸⡀⠸⣇⠸⣿⣿
⠀⠀⢀⡇⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣧⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠉⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢧⠀⠀⣧⠀⠙⠺⠿⣿
⠀⠀⡎⠀⠀⡏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⢀⣈⣼⣿⠀⠀⠀⠀⠀⠂⠀⠤⠤⠤⣾⡀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠘⢄⣀⢸⣧⡀⢀⣆⡿
⠀⠀⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡠⠺⣿⣿⣿⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⢠⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⡇⠀⠀⠰⠀⠀⠿⣿⣿⡆⠀⠹⣿⣿⣿⠀⠀⢔⠀⠀⠀⠀⠀⢩⠟⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠁⠐⠒⠂⠤⡀
⠀⠸⡄⠀⠀⢸⠧⠀⣁⠐⠄⠀⠀⠀⠀⠐⡀⠀⠀⠛⢛⣋⣀⣀⣙⣋⣹⠀⠀⠀⠀⠀⠀⠈⠁⠒⠲⠤⢤⣤⣀⡀⠀⣶⢠⣀⣄⣿⣿⡆⣤⣰⣴⡷
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⡄⢿⡾⢡⣤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⣾⣇⢈⢧⡷⣿⣿⣿⣿⢾⣼⡁⣸⣷⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀⠹⣿⡎⢿⣿⣾⡋⢙⣷⣿⡿⢱⣿⠏⢀⣴⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣤⣈⠻⣿⣄⢻⣿⡌⣇⢿⣿⣿⡿⣸⢡⣿⡟⢠⣿⠟⣁⣤⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠉⢿⣿⣷⡘⣿⣦⠿⠗⠛⣾⣾⣿⣶⡛⠺⠿⣴⣿⢃⣾⣿⡿⠉⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡋⠀⣠⡤⠴⢿⣷⣾⡿⠤⢤⣀⡀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣀⣀⣸⣿⣿⣷⡿⣁⣀⣀⠈⠛⠛⠁⢀⣀⣈⢿⣾⣿⣿⣇⣀⣀⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⣿⣽⣿⣿⣿⠍⣉⢿⣿⡍⢩⣿⡟⣉⠨⣿⣿⣿⣯⣿⠿⠿⠿⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⣤⣤⣤⣤⣼⣽⣿⣿⡟⠓⠛⠛⠻⡇⢸⠟⠓⠛⠚⠻⣿⣿⣯⣧⣤⣤⣤⣤⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠟⠛⠛⠛⠛⢿⢨⣿⣧⡀⠀⠀⡐⣁⣈⢂⠀⠀⠀⣼⣿⡅⡿⠛⠛⠛⠛⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣷⠀⠀⣉⣿⣿⣉⠀⠀⣼⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠛⠛⠛⠛⠉⠉⠉⠹⣿⣿⣿⡔⠛⣍⣉⣉⣩⠛⢣⣿⣿⣿⡏⠉⠉⠉⠛⠛⠛⠛⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄⡴⣿⣿⢦⣠⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣿⣿⣿⣿⢾⣿⣿⡷⣿⣿⣿⣿⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⣉⣉⣉⣀⣤⣤⣤⣤⣿⡿⣿⣿⣽⣿⣿⣯⣿⣿⢿⣿⣤⣤⣤⣤⣀⣈⣉⣉⡩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⠿⠛⠛⠛⠛⢛⣿⣷⡘⢿⣻⣿⣿⣗⡿⢋⣼⣿⡛⠛⠛⠛⠛⠿⠿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣿⣿⣿⡿⢿⣿⡿⢳⣼⣿⣿⣿⣿⣧⡾⢹⣿⡿⢿⣿⣿⣿⣷⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣤⣤⣶⣾⣿⠐⡄⠸⣿⣿⣿⣿⠇⢠⠃⣿⣷⣶⣤⣤⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣉⣉⣩⣿⠀⠈⢆⣿⣿⣿⣿⡰⠁⠀⣿⣍⣉⣉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠹⠀⠀⠀⠻⣿⣿⠟⠀⠀⠀⠟⠻⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⡛⢛⣉⣩⣍⢩⡉⣭⣍⣛⢛⣛⠿⢿⣿⡇
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⠉⣄⠰⣶⣾⣿⡆⢿⠸⣿⣿⣿⠈⡇⢻⣿⠸⡟⣿⣖⡇⣾⡁
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⡁⢤⣶⣾⣿⣇⢹⡄⠻⠟⣛⣉⣬⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣭⣉⡋⠻⡇
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⢉⣤⣾⡌⣧⠘⣿⠿⢛⣩⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠈⢿⣿⣿⣿⢏⣤⢶⠘⣿⣿⠗⣈⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢻⡿⢁⡌⢿⡎⠗⢉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠠⣟⣿⠊⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⠉⠉⠉⠛⠛⠻⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠿⠛⠛⠛⠉⠉⠉⠉⠉⠛⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⡾⠋⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⢀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡼⠋⠀⠀⠀⠀⠞⠛⠉⣉⣄⣈⣍⠉⠻⠷⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠂⢉⣀⣤⣤⣤⡀⠀⠀⠀⠀⢸⣟⣭⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠫⠥⢾⡿⠇⠴⠄⠀⠀⠀⠸⣷⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣤⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣉⣉⣁⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠉⠻⢷⣶⣤⣄⣀⣀⠀⠀⠀⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡐⢊⠰⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⢯⠐⢡⠄⢂⡘⠗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢛⣤⠚⡈⠤⠈⠛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠰⢆⡰⠆⡁⠌⠻⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⠈⢡⡌⢁⡌⠀⡌⠻⢆⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠰⠇⣠⠖⣀⠁⠀⠙⠁⠷⠆⣠⣀⢀⣀⢀⣤⠻⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠉⢴⡅⡘⠃⡄⠀⠀⠀⠈⠉⡈⢉⠈⠡⢀⠈⠉⠹⠋⢉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡈⠋⠴⢀⡁⢐⠀⠀⠀⠀⢐⠀⢁⡄⠘⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⠀⠉⠈⠉⠛⠀⠉⠁⠀⠀⣀⡴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣏⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠉⠻⠦⣵⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠠⣘⣿⠏⠂⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠈⠻⢦⢄⣤⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⡀⠀⠀⠀⢈⣾⣿⣿⣿⣿⣇⣾⣿⣿⣟⣿⣤⣀⣀⣤⣤
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿
⣿⡛⠛⠛⠛⠻⠿⠛⠛⢛⠛⠟⠛⠻⠿⢿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿
⣶⣦⣴⠖⠒⠒⠠⣤⡀⢙⣛⠿⢛⡓⠂⣄⣤⡍⢀⣉⣈⣙⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠉⠉⢹⣿⣿⣿⣿⣿⣿
⣿⣯⣭⣅⣀⣀⣀⣦⣼⣾⣿⣿⢿⣗⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣀⠀⠉⠙⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣤⢤⣤⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⡆⣸⣇⣸⣿⣿⣷⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⢶⡌⢿⠘⣿⢿⣿⢿⣿⣿⣿⣇⣸⠇⣴⠆⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣉⣙⢦⢻⡸⣇⢿⣼⣿⣠⣿⣿⣿⢻⡟⣼⢡⣞⣉⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⣉⣉⣙⠦⠃⠉⠀⠘⠛⠈⠛⠋⠀⠈⠑⠣⣛⣉⣉⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣉⣍⢁⡼⠋⠉⠙⠓⠀⠐⠛⠉⠉⠳⡀⢉⣉⣉⣉⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⣩⡏⣀⠤⠮⠿⠏⠆⠀⠀⠎⠿⠯⠦⣅⠉⣍⣉⣉⣉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⣉⣩⣥⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣡⣍⣉⣉⣉⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⣉⣭⣥⣤⡄⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⣄⣬⣭⣉⣉⣉⣉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣩⣭⣥⣤⣤⣤⡄⠀⠀⠁⠀⠀⠁⠀⠀⣤⣤⣤⣬⣭⣭⣉⣉⣉⣡
⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣥⣤⣤⡤⠤⠤⠶⠶⡄⠀⣤⣤⣤⡄⠀⡴⠶⠤⠤⢤⣤⣤⣤⣭⣭⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠲⣖⣒⣒⣺⣿⣿⡿⠯⠅⠀⠀⢸⣯⣿⡇⠀⠠⠽⢽⣿⣿⣗⣒⣒⣒⠶⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠱⣖⣒⣒⣛⣛⣻⡅⠀⠀⢸⣿⣿⡇⠀⠀⣟⣛⣛⣒⣒⣒⡞⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣭⠽⠶⠶⢶⣒⡂⠀⠀⢸⣿⣿⡇⠀⠀⣒⡶⠶⠶⠯⢭⠅⣀
⠀⠀⠀⠀⠀⠀⠀⣸⠁⣿⠀⣿⣛⣭⣭⣭⡭⠅⠁⠀⢸⣿⣿⡇⠰⠀⢭⣭⣭⣭⣙⣻⡇⢹⡄⢿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠸⣿⠶⠶⢶⣒⣚⡂⠛⠛⢺⣟⣿⠿⠛⠀⣓⣒⣶⠶⠶⢽⡇⢸⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⣯⣭⣭⣭⠭⠄⠛⠷⠾⣿⣿⠷⠾⠇⠭⣭⣭⣭⣽⣿⠇⢸⠇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡆⢹⡆⠸⣷⣒⣚⣛⣿⡁⠷⢶⣤⣤⣤⡴⠶⠇⣽⣟⣓⣒⣺⡟⢀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠻⣦⡈⠻⢶⣒⣒⣲⣦⣤⣤⣤⣤⣤⣤⣆⣒⣒⣲⠾⠋⣠⡞⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⡈⠛⠷⢮⣭⣭⣁⣀⣈⣉⣉⣉⣉⣀⣈⣉⣭⣥⠶⠟⠉⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣤⣀⣈⣉⣉⠉⠉⠉⠉⠉⢉⣉⣉⣀⣠⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡆⢸⡏⣄⢸⠀⢾⡇⢈⣰⡆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢻⣆⠈⣿⣅⠀⢈⣾⡁⣿⡇⢸⡽⠀⣸⠏⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⡈⢷⡀⢻⡄⢸⣧⣆⣶⣿⣇⢻⡇⣰⡆⢰⡟⢀⡞⣡⠶⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣮⣷⡈⣿⠈⠿⡤⡾⠚⠛⣯⡇⠛⠃⠾⢁⣾⣿⣧⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣀⣀⣙⣿⡿⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⣿⣥⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⣉⣉⡄⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣴⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠛⠛⠛⠛⢹⠁⠀⢰⣶⡄⠄⠀⠀⠀⡄⢶⣶⠀⠀⢸⡌⢉⣀⣀⣉⣉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠟⠛⠛⠃⠈⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣉⣉⣉⣉⣉⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠾⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠘⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⠶⠶⠶⠶⠶⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⠤⠶⠶⠶⠶⠶⠖⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⢛⠛⡃
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠶⠶⠶⠶⢶⣒⣚⣛⣛⣛⡛⠀⠀⠀⠀⠀⠀⢀⠀⠀⠩⠭⠭⠭⠭⢭⣭⣭⣭⣭⣭⡍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣋⣭⣭⠭⠭⠭⠭⠶⣖⡂⠀⠀⢈⠒⠶⠚⠉⠀⠀⢀⣛⣛⣛⣒⣒⣒⣒⣒⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⣛⣛⣛⣉⣉⣭⡭⠅⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠶⠶⠶⠤⠤⠤⠤⠅⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⣩⠭⠤⠶⠶⠖⠒⠒⠀⠀⠀⡆⠀⠀⡇⠀⠀⢈⣉⣉⣛⣛⣛⣛⣛⣀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠒⠚⢛⣋⣉⣉⣉⢀⣀⣀⡁⢚⠠⡃⣀⣀⠀⣤⣤⣤⣤⣬⣭⣽⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣏⣉⣉⣩⣭⣤⡤⠐⠿⣭⣧⢨⠀⣯⣭⢿⠠⠶⠶⠦⠤⠤⠤⢾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣸⣿⣯⠤⠤⠤⠴⠶⠖⠀⣟⣛⣿⢸⣀⣗⣛⣻⠀⣛⣛⣛⣒⠒⠒⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡼⣏⣿⣷⠒⠒⠒⠛⣛⣋⠀⠯⢭⣽⣿⣿⣭⡭⠿⠀⣉⣉⣉⣉⣉⣉⣿⡟⣼⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢻⡜⣿⣏⣭⣭⣭⠭⠭⠀⣟⣗⠾⠿⠿⠶⣞⣻⠀⠶⠶⠶⠶⠶⢶⡿⣱⢏⣿⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⢻⣜⢷⣶⠶⠶⠒⠒⠀⣭⣭⣿⣿⣿⣿⣭⣭⡄⣙⣉⣋⣉⣿⡿⣱⢏⣼⠃
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡙⣮⡻⣿⣿⣯⣙⠷⠷⠦⠤⣤⣤⠤⠶⠶⠟⣛⣿⣿⡿⣋⡾⣫⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣙⢶⣝⡻⢿⣿⣶⣶⣶⠶⢶⣶⣶⣾⣿⡿⢟⣭⠾⣫⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣮⣝⣳⠾⢯⣽⣿⣿⣿⣿⣭⡽⠶⣛⣭⡵⠟⠋
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠶⠶⠶⠶⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣏⠛⠿⠿⠛⠛⠛⠛⠻⠟⠻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠌⣻⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⢻⣷⣎⠁⣿⣿⣿⣿⣿⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠹⣿⣼⡿⡿⢻⠛⣋⣠⣤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⠿⠛⠛⠛⠛⣛⣙⣫⠛⠛⠃⠀⣅⣤⣤⠶⠞⠛⣫⣩⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣒⠀⠀⠐⢋⣡⣤⣤⣤⡤⠀⢰⣿⠓⣒⣢⢔⣮⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠃⠀⠀⠈⠁⠙⠁⠉⠀⢀⣀⠳⣏⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⣾⣿⣿⣿⣿⣿⡿⠿⠛⣛⣭⠟⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡀⣛⢛⣋⣭⣥⡦⠶⠚⠋⡩⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠺⠛⠛⢉⣁⣡⠤⠴⣒⣴⣯⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢀⠀⠀⠀⣠⡴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠤⢷⣞⣿⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃

⠈⠉⢁⣀⡀⠀⠀⠀⠀⢀⣴⣒⣾⡆⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⡜⠂⢀⠀⠐⢥⠀⠀⠀⠋⠉
⠀⠀⢩⠵⠯⣄⠀⠀⢹⣾⠟⠉⠉⠷⡤⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠤⠔⠀⠈⠀⠀⠂⠄
⠀⠀⠀⠙⠁⡀⠀⢀⣿⣁⣀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠒⠀⠀⠀⠀⠀⡀
⠀⢀⠀⠀⠈⢀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠐⠂
⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡉⠉⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⢀⠻⣿⣿⡇⠀⠀⠀⠀⠀⠠⠄⠢
⢻⡿⡝⠷⠀⠀⠀⢠⢶⠏⠛⠀⠛⠁⠀⠀⠀⢤⣀⡆⢆⣠⠄⠀⢘⠀⠀⠙⠃
⠀⠀⠀⡀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠐⠀⠂⠀⠀⢩⣷⣮⠁⠀⠀⠈⠀⢀⣠⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢻⠏⣀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⡀⢀⠀⠀⠈⢿⣿⣿⣄
⢐⣿⡂⠀⠳⠀⠀⠀⣶⣾⢟⣒⡄⠀⠀⠀⠀⠈⠁⡐⠀⠉⠀⠀⢠⣿⣿⠋⠀⠀⠀⠐⣠⣄⠂⠀⠀⠀⠹⣿⣿
⠰⠶⠆⠀⠀⠀⠀⠀⠐⠘⢙⣒⣻⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠈⠿⣿⠀⠀⠀⢰⡀⠀⠀⢀⠇⠀⠀⠀⡿⠿
⠀⠀⠶⢖⡦⠀⠀⠀⣄⣸⣿⠾⠿⣇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⠃⠈⡙⡀
⠀⠀⠚⢭⠍⠃⠀⠀⣼⠟⠁⠀⠀⡩⠋⠒⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠈⠅⣀⠀⠀⡀⡀⠁
⠀⠀⠀⠀⠠⠆⠀⠐⠛⠒⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆
⠀⠐⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠤⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⠀⠁⠀⡀
⣶⣿⣶⣆⠀⠀⠀⠀⣀⡤⣶⠂⣤⠄⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢰⠈⠛⢿⡇⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠘⠋⠃⠉⠀⠀⠀⣼⡝⠁⠀⠀⠀⢀⣀⡀⠀⠙⠷⣇⣱⠜⠁⠀⠰
⠀⠀⠠⠂⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⢀⣴⣾⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤
⢀⣤⡀⠀⣆⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠰⠆⢙⠁⠶⠀⠀⠀⣾⣿⡿⠋⠀⠀⢀⠀⠐⡂⠀⠀⠛⢿⣿⣷
⢐⣛⡂⠀⠀⠀⠀⠀⢛⢻⢵⣒⣧⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠸⣿⣿⠀⠀⠀⢀⠀⠚⠓⠀⡄⠀⠀⠈⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡐⠶⣶⡀⢦⡐⢶⣄⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢿⣬⣉⠻⣦⡈⢻⣆⢻⡌⢿⡣⠙⠘⣿⣄⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣚⠛⠶⣬⡙⢷⣌⢻⣆⠹⡆⢿⠻⣿⣿⡄⢿⡷⡋⠒⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣉⠛⢶⣌⡻⣦⡹⣦⢹⡆⠿⠸⣧⣬⣭⣿⣨⢶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⣬⣍⠻⢶⣌⠻⣎⡳⠜⠁⠀⠀⠀⢀⣤⣠⡿⠘⢹⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠶⣦⣍⡛⠶⣬⣻⢞⡁⠀⠀⣠⠔⠒⠚⠉⠙⢧⣠⡬⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠶⣦⣭⣙⡻⢶⣿⠳⣬⣢⠎⠁⠀⢉⣩⣿⣶⡤⠉⣠⡬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⠿⠶⢶⣤⣭⣝⣿⢍⠻⣿⣿⡀⠀⠛⠻⠭⠟⠍⠀⢰⡿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣛⠛⠳⠶⠶⢤⣭⣷⠸⡇⠉⠉⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣉⣉⣛⣛⣛⠻⠶⠶⢿⣷⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣭⣭⣭⣭⣭⣉⣛⣛⣻⣿⢏⡿⢧⡀⠀⠀⠀⠀⠐⠫⣍⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡒⠒⠒⠶⠶⠶⠶⠬⠭⢽⣏⣿⡇⡄⠙⠲⣤⣄⣀⡀⠀⠈⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣑⣏⡩⡭⢍⣿⣩⡭⠽⣟⣒⣿⡦⠞⠁⠀⠀⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣧⣒⢝⡫⠭⣒⡒⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠠⡓⢬⣙⣒⢲⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣛⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣁⠳⣍⡳⠦⡭⣽⢃⠈⠀⠀⠀⠀⠀⠙⠿⣿⣯⣭⣭⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠉⢢⣍⠛⡶⠋⢦⠳⣀⠂⠄⡀⡀⡠⠐⢹⠻⣶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢤⡙⠒⠦⠤⠤⠒⠁⡐⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠖⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣬⣤⡚⠹⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣟⣁⠞⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⢿⣷⢀⣿⣿⣿⣿⣿⠿⠟⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⡿⠿⠛⠛⢉⣁⣤⡴⠾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⢛⡉⠉⠉⠁⢀⣤⡤⠤⠖⠀⠈⠁⢀⣶⡾⠶⠟⠛⢉⡤⣶⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⢭⡀⠀⠀⠉⣤⢶⣶⢶⣶⠖⠀⢸⣷⠴⣶⡶⢖⣯⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠈⠀⠀⢠⣤⣙⣛⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⠿⠿⠛⣛⢩⣅⡶⠞⢋⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣅⣤⣠⣥⣴⠴⠶⠺⠛⢛⠈⣁⢔⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠋⠈⠀⠀⠶⠖⠚⢛⣭⣾⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣀⠀⢐⣢⡴⣞⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⢖⣛⣭⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟




⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣯⣉⣭⣭⣥⣭⣉⣯⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣥⣭⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⠚⠓⣤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⡍⠹⡄⢳⣤⣤⡞⢠⠏⢩⠷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠳⢄⠙⣦⡽⠾⣿⣿⠷⢯⣴⠋⣠⠞⠉⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡉⠙⠲⣬⠋⠁⠀⠀⠈⠁⠀⠀⠈⠙⣤⠖⠋⢉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠛⠲⡏⠀⣙⡳⠦⠀⠀⠴⢞⣋⠀⢹⠖⠋⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠛⡞⣶⣇⠺⠿⠟⠃⠀⠀⠘⠿⠿⠗⣸⢖⢳⠚⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠙⣇⠸⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠇⣸⠋⠉⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠉⠉⠛⠦⣬⡇⠀⠀⠙⠦⠴⠋⠀⠀⢸⣥⠴⠛⠉⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠙⠛⠒⠒⠦⢷⠀⢠⠤⠒⠒⠤⡀⠀⡾⠴⠒⠚⠛⠋⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠓⠒⠒⠶⠤⢤⣈⣧⡀⠑⠀⠀⠈⢀⣼⣁⡤⠤⠶⠒⠒⠚⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠒⠶⠦⠤⢤⣀⣀⣿⡙⡶⢤⣤⣶⢋⣿⣀⣀⡤⠤⠴⠶⠒⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⣄⣀⣀⡀⠈⣿⢿⡟⠲⣥⣼⡿⣿⠁⢀⣀⣀⣤⣤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⠉⠙⣯⠀⣿⠛⡥⣾⠀⣽⠋⠉⠉⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⠙⠓⠒⡯⠀⢻⣙⠖⡟⠀⢹⠒⠛⠋⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠓⠒⠲⡿⠦⣼⢬⡛⣧⠴⢿⠖⠒⠚⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠒⠒⡇⠀⠸⡶⣯⠇⠀⢸⠖⠒⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠓⠚⡇⠀⠀⠙⠋⠀⠀⢸⠒⠚⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⣉⡇⠀⠀⠀⠀⠀⠀⢸⣉⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠇⠀⠀⠀⠀⠀⠀⠸⠽⠛


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡆⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢠
⠀⠀⠀⢷⠀⠀⠸⡍⠛⡗⠢⣄⡀⠀⠀⠀⣠⣾⠿⣿⡿⠻⠀⠀⠀⠀⠀⢀⣀⠔⠚⠟⢩⡇⠀⠀⡾
⠀⠀⠀⠸⡆⢀⡀⠿⠄⠀⠹⡟⠀⠉⠁⢲⠿⠾⣤⣿⡆⣤⡥⠴⡖⠀⢉⠠⢻⠟⠀⢠⠿⢀⡀⢰⠇
⠀⠀⠀⠀⠻⡿⢿⣀⠈⠠⡀⠈⠢⣀⠁⠛⠚⠀⠀⣿⡿⠀⠀⠃⠛⠀⣀⠔⠁⢀⠤⠁⣀⡿⢿⠟
⠀⠀⠀⠀⠀⠙⠫⣞⣱⢢⡄⡀⠀⢀⠑⠢⠄⣀⡂⠉⠁⠀⣠⡠⠔⠊⡁⠀⠀⢀⡐⢎⣳⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⠵⣧⣆⣆⣬⢡⣬⣗⠀⠀⠀⣇⡌⣥⣰⣰⡼⠮⠗⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠷⠔⢲⣯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠈⠁

⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡶⢷⣧⣶⣴⣿⣿⣶⡶⠦⡔⠲⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣷⣾⣷⣿⣿⣿⢍⡉⠁⠀⠁⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠷⡤⠙⠿⣽⣿⣽⣙⣿⣇
⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡗⠄⠀⠀⠢⢝⢿⠉⠙⢿⣏⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣦⣝⡃⠀⡄⠐⠦⢀⡐⢀⡀⢸⠺⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣧⣤⣴⣼⣿⠀⠈⠛⠿⣶⠤⣀⡐⢹⢹⣿⣇⢸⢠⡄⠠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣷⣶⡦⣷⣿⣿⠛⠃⡶⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢹⣿⡷⣤⡿⣿⣿⣿⣿⣿⣿⠀⠀⠉⣿⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⢸⣿⣄⢸⣷⣾⣿⣿⣿⣿⣿⢦⣦⣼⣿⣇⢩⣤⣤⣤⣤⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⣴⣿⣿⣴⣿⣿⣿⣿⡏⢹⣯⣸⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⡆⣿⣾⣿⣿⣿⣿⣧⣿⣽⣴⣿⣿⣿⣷⣀⣀⣀⣀⣿⣿
⠆⠀⢉⢻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣯⣿⣿⣏⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡆⣿⣿⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣉⢁⡤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠈⠉⢁⣀⡀⠀⠀⠀⠀⢠⣴⣒⣾⡆⠀⠐⡂⠀⠀⠀⠀⠀⠀⠁⠀⠉⠙⠀⠀⠀⡜⠂⢠⠄⠐⢥⠀⠀⠀⠋⠉
⠀⠀⣩⠔⠮⡄⠀⠀⢹⣾⠟⠉⠉⠷⡤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠙⠁⡀⠀⢀⣿⣁⣀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⢀⠀⠀⠈⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠐⠂
⣀⣠⣀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠉⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⢀⠻⣿⣿⡇⠀⠀⠀⠀⠀⠠⠄⠢
⢻⠿⡝⠶⠀⠀⠀⢠⢶⠏⠛⠀⠛⠀⠀⠀⠀⢤⣀⡆⢆⣠⠄⠀⢘⠀⠀⠙⠃
⠀⠀⠀⡀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠐⠂⠂⠀⠀⢩⣷⣮⠁⠀⠀⠈⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢻⠏⣀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⡀⠀⠀⠀⠈⢿⣿⣿⣆
⢐⣿⡆⠀⠓⠀⠀⠀⣶⣿⢟⣒⡀⠀⠀⠀⠀⠈⠁⠐⠀⠉⠀⠀⢐⣿⣿⠋⠀⠀⠀⠐⡠⣄⠂⠀⠀⠀⠹⣿⣿
⠰⠶⠆⠀⠀⠀⠀⠀⠀⠘⢙⣒⣳⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠿⣿⠀⠀⠀⢰⡀⠀⠀⢀⠆⠀⠀⠀⡿⠿
⠀⠀⠶⢖⡦⡀⠀⠀⣤⣰⣿⠾⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠁⠀⡙
⠀⠀⠚⢭⠍⠃⠀⠀⣼⠟⠁⠀⠀⡩⠋⠂⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠂⠀⠐⠛⠒⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠤⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⡀
⣶⣿⣶⣀⠀⠀⠀⠀⣀⡤⣶⠂⣤⠄⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢰⠈⠛⢿⡃⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠘⠉⠃⠉⠀⠀⠀⣼⡝⠁⠀⠀⠀⢀⡀⡀⠀⠉⠶⣇⡱⠜⠁⠀⠠
⠀⠀⠀⠂⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⢀⣴⣾⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤
⢀⣤⡀⠀⣂⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠰⠆⠙⠁⠶⠀⠀⠀⣾⣿⡿⠋⠀⠀⢀⠂⠀⡂⠀⠀⠛⢿⣿⣷
⢐⣛⡃⠀⠈⠀⠀⠀⢛⢻⢵⣚⣧⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⢀⠀⠚⠓⠀⡀⠀⠀⠈⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠒⠈⠉⠁⠀⠀⠀⣀⣀⣀⣀⣀⡀⣀⣀⣀⠀⠀⠀⠈⠁⠀⠐⠂⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢣⢀⣀⡤⠄⠒⠒⠈⠉⠁⠁⠀⠀⠀⠀⡃⠀⠀⠆⠀⠈⠈⠈⠉⠙⠐⠒⠠⠤⢄⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠂⠀⣀⣠⣤⠤⠶⠶⢲⡷⡃⠓⠓⣇⠀⠃⠢⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡇⡤⠂⢀⣤⢡⠀⠀⢹⠀⠀⠀⢹⣿⣿⣿⣿⣶⣄⠀⠀⣀⣴⣿⢿⡄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡰⠖⠂⠠⠆⢆⠀⠈⡇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷
⠀⠀⠘⠀⠰⡄⡀⠰⢂⠞⠿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠶⠒⠀⡸⡜⡀⠀⢻⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾
⠀⠀⠀⠀⠀⠀⠃⠀⠜⠀⠓⢹⣿⣿⣿⣿⣿⣿⣿⣿⡔⠖⠂⠀⣤⢣⠀⠸⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡒⠙⠻⠋⡝⢉⡍⠫⠝⡛⢧⠡⠶⠥⢠⡌⡆⠀⢿⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠗⢈⣀⣠⣉⣀⡀⠀⠀⠈⠀⠀⠆⢹⣤⠆⣁⣱⢁⠀⢸⡇⠀⠀⣻⣿⣿⠟⠛⢿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠀⠀⠠⠉⠙⠒⠂⠀⠀⠁⠀⠗⡠⢌⠻⡼⠃⠘⣿⠀⠀⢙⠉⠀⠀⠀⠈⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣿⠟⣿⣗⣶⠦⠤⠤⠴⠦⠀⠛⠂⡀⠘⢠⡇⣇⠀⢿⠂⠀⢘⣧⡀⠀⣠⣤⣄⣻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⣀⡀⠀⠀⠀⠃⡐⢃⡇⡇⠀⢹⡇⠀⠀⣿⣷⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡁⢀⣀⠆⣸⠉⢡⢸⠇⠀⢸⡇⠀⠀⢹⣿⣿⡆⠈⢿⠿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣿⣿⢀⡤⢣⠏⠀⠀⣸⡇⠀⠀⢸⣿⣿⠟⣀⠄⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣈⣵⠏⠀⠀⠀⣿⠁⠀⠀⠰⣿⣿⣧⠁⣾⣿⣦⢸⡿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠂⠀⠀⢸⡟⠀⠀⠀⠐⢿⣿⣿⣷⣬⠉⢁⣬⣶⣿⣿⣿⠿⣏⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣴⡟⠀⠀⠀⠀⠈⢿⣿⣿⣭⣴⣿⣿⣿⣿⢿⡏⢹⡄⢻⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢠⣰⡾⢏⠀⠀⠠⠀⠀⠀⣿⣿⣿⣿⣿⢿⡟⣏⢹⡆⣿⠈⣧⠘⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⢉⠀⠀⠂⠀⠄⠀⠀⠀⡞⡏⢹⡅⣷⠸⡇⢻⠀⣷⢸⡇⢻⡀⢻⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠄⠠⠀⠀⠀⠂⠀⣶⡅⣿⢸⡇⣿⠀⣿⠸⡇⢸⡄⣿⠘⣇⠘⣧⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡴⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠄⠐⠀⢁⠀⠄⠀⣦⣿⡇⣿⢼⡧⢼⡆⢿⡄⣷⠘⣷⢺⡇⢻⡉⣿⣀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡚⣏⢿⡎⣧⡀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠂⠀⡀⢸⣄⣿⣿⣹⠏⣹⠏⢛⡿⣹⢻⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣠⠴⣞⢻⠄⠈⢧⠹⣎⢿⡤⢷⡀⠀⠀⠀⠀⠀⢸⣿⠀⣰⡄⣼⣀⣿⣾⣿⣿⣧⡟⣶⠏⢀⣼⢣⣯⣙⢸⣧⡸⡌⠙⢿⣿⣿⣿⣿⡇
⠀⣠⠖⢛⣇⢳⢹⣆⢻⡄⢈⢧⡙⣎⠻⣎⣹⠦⣄⠀⠀⠀⢸⣿⣴⣿⣷⣿⣿⣿⣿⣿⣿⣯⣼⠋⠀⣶⢏⡾⢠⣏⣿⣉⢫⠃⠀⠈⢿⣿⣿⣿⡇
⠀⠘⡔⠛⢛⣧⢣⡼⣦⠻⣄⠂⠱⣌⠳⣌⠻⣶⣏⡟⢲⠤⣿⢿⢿⡿⡿⢿⠿⡿⢿⢿⡿⣿⠁⣀⣹⢏⡾⣩⠏⣾⣍⣍⡏⠀⠀⠀⠈⣿⣿⣿⣷
⠀⠀⠘⡔⠛⢛⣦⡱⣞⣷⣜⢣⣄⠈⠳⢦⣙⠲⢭⣝⣛⠶⣧⢸⠀⡇⢕⢸⠀⡇⢸⠠⡇⣿⢦⡾⣣⡟⣳⢏⣾⣇⡽⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠈⢌⢋⢉⣷⣜⢣⡿⢷⣍⠳⣤⣀⠉⠛⠶⢤⣬⣹⣷⢸⠀⡇⢸⢸⠀⡇⢸⠀⡇⢿⣫⡾⢧⠜⣡⠿⣧⡬⠅⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠡⣍⢀⣼⠷⣜⠻⣼⡛⣦⣭⣛⠲⢤⣄⣀⣨⣿⣸⣀⣇⣸⣸⣄⣇⣸⣀⣇⣼⣥⢛⣡⣞⠻⡦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠂⢅⠠⡿⠛⣦⣝⡓⠾⢭⡟⢳⠶⣦⢽⣿⣹⣏⣏⣹⣩⣏⣿⣹⣭⣏⣽⠶⣿⣈⠙⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢔⠋⢀⡿⠟⣶⣤⢭⣙⣓⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠮⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⣚⡡⠀⠿⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡖⢲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢎⠙⡄⠈⡇⢹⣏⢨⠃⢠⠏⣱⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠚⠙⢆⣠⣼⣷⣿⢄⢰⡜⢹⡎⢀⢾⣶⠣⣄⡴⠉⠓⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣧⣀⠑⡄⠈⢏⠩⢿⡽⣿⣾⠃⠠⣷⣿⡯⢻⡏⡜⠀⡠⠊⣠⣼⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠤⣄⣙⢷⡈⢆⠘⡆⠑⢿⢿⣿⠀⠀⢺⡿⣿⠚⢸⠁⡜⢠⡞⣋⣠⠤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢤⣌⡑⢷⡈⡆⢱⠀⠸⠄⢿⢀⠀⡸⠁⠇⠠⠇⡸⢀⡿⢚⣁⡤⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠧⠤⠤⠤⣄⣉⣢⠗⠉⠀⠀⠀⠀⠸⣦⣦⠇⠀⠀⠀⠀⠀⠛⣴⣉⣤⠤⠤⠤⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠤⠤⠤⠤⠤⣌⣹⡶⢋⠀⠀⠀⠀⢀⣈⣉⣀⠀⠀⠀⠀⡀⢲⣟⣡⠤⠤⠤⠤⠤⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠐⠂⠀⠀⠤⠤⣤⣬⣿⣵⡂⠈⢀⠀⢄⢄⡱⢀⠀⢈⠉⠐⡮⣿⡭⣤⠤⠤⠄⠀⠐⠒⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠂⠀⠒⠲⠲⠤⣿⢋⣿⣟⣡⣤⣤⣉⠋⠀⠁⠘⣉⣠⣤⣄⣹⣟⡟⢿⠶⠶⠖⠒⠐⠒⠐⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠒⠒⠒⠀⠀⠴⠶⠾⡜⡟⣋⠉⠘⠛⠀⠂⡀⠀⠰⠄⠛⠃⠉⢹⢻⣏⠾⠶⠶⠄⠀⠀⠒⠒⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⡿⠒⠒⠒⠒⠒⠒⠒⠒⠒⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠒⠒⠒⠛⠜⠻⣇⠀⠀⠀⠑⠒⠒⠒⠀⠀⠀⢠⡟⠣⠛⠒⠒⠒⠒⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠠⠤⠬⠭⠿⠛⠛⠻⠿⠿⠿⠿⢟⣿⡀⠀⠐⠀⠀⠀⠀⠒⠀⢀⣾⠻⠿⠿⠿⠿⠟⠛⠛⠛⠭⠭⠥⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⣦⣌⣉⢉⢉⢉⣉⣀⣀⣉⡛⠒⢺⣿⢦⣄⠈⠉⠉⠁⢠⡴⢾⣗⠒⣚⣃⣀⣀⣉⣉⠉⡉⣁⣠⣴⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣶⠤⠦⠤⠀⠀⠀⠀⠤⢽⣿⠀⠉⠓⠒⠶⢾⡏⠀⢸⣯⠥⠤⠀⠀⠤⠶⠤⠤⣴⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣗⡛⣷⠀⠀⠒⠒⠒⠒⠒⢸⣿⠀⠀⡄⠀⠀⢐⡇⠀⢸⡧⠐⠒⠒⠒⠒⠂⠀⣸⢿⣚⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠡⡇⣻⣅⣀⣀⡈⡉⣉⣉⣹⣿⣆⠀⠃⠀⠀⠸⠁⣠⣿⣏⣉⣉⣉⣉⣁⣀⣠⣟⢸⠽⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⢘⣯⢻⣧⠀⠀⠀⠀⠀⣈⣹⡿⠿⣷⣤⣤⣤⣤⣾⡿⢻⣏⣈⠀⠀⠀⠀⠀⣼⡻⣈⡟⢼⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣎⣷⢽⢻⡦⠤⠤⠤⠤⠤⢼⣿⣔⡼⣝⣟⣹⣹⢵⣰⣽⡧⠤⠤⠤⠤⠤⢠⡿⣷⢝⢣⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣞⡜⢏⢞⣷⠀⠀⠀⠀⠀⢸⡿⠾⣵⣿⣟⣻⣻⣧⡵⢾⡧⠀⠀⠀⠀⠀⣾⣻⢯⢣⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⢊⡳⣽⡆⠀⠀⠀⠀⢸⡇⡢⣔⣲⣬⣥⣒⡪⢐⡺⡷⠀⠀⠀⠀⢰⣿⠾⡁⣳⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣏⡪⣛⣶⣶⠶⢒⡾⠯⣵⢢⣩⣩⣩⣍⣖⡦⡭⢳⣒⡶⣶⣶⣟⣵⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠽⡷⣭⣛⢳⠶⠿⣤⣬⣥⣖⣖⣲⣒⣤⣥⣤⡽⠷⡞⣛⣭⣿⠯⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠭⠷⢒⣶⡾⡿⠯⠼⠿⣼⣶⣖⠺⠯⠛⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠻⣿⣿⣿⣿⣻⣿⣶⣶⣦⡄⠀⠀⠀⠀⠠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢸⠟⣿⣿⢿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⠴⠾⣷⡟⠀⡿⠿⢨⣿⣶⠦⣄⠀⠀⠀⠀⠀⠀⢸⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢰⠃⠀⠀⢹⡇⠀⣿⡆⠘⣿⠃⠀⠘⡆⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡀⣀⡤⠀⢧⢤⡟⢣⣠⠁⢀⣒⡂⢷⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⠐⠀⣷⠀⠀⠀⠀⠀⠀⠀⣼⡖⠂⢻⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⠉⢃⠀⠀⠀⠀⠀⢰⠁⣇⠀⠸⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⠀⣽⣿⣹⡗⣧⣶⡇⠀⠙⢤⣀⠈⠑⢦⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣼⢠⠛⠁⣉⠉⣹⣟⠷⡀⠀⠀⠈⠓⠦⣄⡙⠢⠾⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢰⡏⢸⠀⠀⠀⠀⠁⠻⣿⠁⠀⠀⠀⠀⠀⠈⠙⠦⣼⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢡⡌⠀⠀⠀⠀⠀⣰⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⣀⣻⠃⠀⠀⠀⠀⢠⢦⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡏⢹⠀⠀⠀⠀⠠⢊⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣟⡀⠀⡠⠈⠀⠈⢣⣤⣻⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡝⠉⢧⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⠁⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⣇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡃⠀⠀⠀⠘⣆⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⣯⡇⠀⠀⠀⠀⢻⡏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠓⠤⡀⠀⠀⠀⣻⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠖⠲⠦⠬⠿⠀⠀⠳⠴⠒⠦⠤⠽⠟

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⣤⠒⢹⢸⣁⠠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠯⠗⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠢⠂⠀⢀⣀⣠⡟⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢒⣞⢓⣾⣯⡭⢷⣖⣒⣋⣭⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⠀⠀⢀⠘⠎⡏⠈⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠲⠁⣀⠭⠋⣹⣛⡛⢻⢛⣲⡲⠖⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠁⠀⠀⠋⠀⢸⠁⠀⣀⠎⠀⠀⠀⠀⣀⠀⠠⠴⠒⠋⠁⠀⣀⣴⢞⠫⠅⠐⢠⡟⠬⠁⣐⡒⠭⠭⣭⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣀⣀⠀⢸⠀⣸⣤⣶⡖⡎⠉⠀⠀⠀⠀⢀⡠⣤⣲⠿⣽⡫⠉⠀⢀⡴⢿⣄⠅⡀⠤⣀⣩⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠎⢻⣝⣶⠀⠘⣗⣫⣿⠏⣻⣴⡤⡴⣲⢢⣏⠾⣝⣱⣼⠛⠓⣀⣤⢶⡋⠂⠒⠕⢝⡢⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⢆⡀⠈⢻⠀⠀⠿⣟⠁⠸⡯⣟⣾⣱⣥⠿⠺⡋⣉⣿⡿⢶⢛⠉⡀⠀⣑⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣻⡲⣼⠤⠤⠀⠀⠉⠚⢿⣯⣿⣷⡵⠵⠟⠛⢩⢚⠖⠈⢂⣨⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣻⣙⣗⡦⣄⡀⠀⠈⠙⠛⠧⣀⠈⢱⣀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣟⣽⡙⣷⣮⣽⣙⡷⡦⣄⡀⠀⠀⠙⠫⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⡛⡾⣭⣞⢏⢛⢿⣭⣔⡘⣟⣒⢦⢤⣆⡸⠉⠀⠀⠐⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢠⢳⠁⠈⠛⠻⠦⣤⣶⡋⡹⡻⢼⣾⣷⣿⣿⣖⢦⣭⣭⣮⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⠃⠤⣙⣀⠀⠰⠈⠋⡳⠷⣅⣎⡇⠈⠈⠈⠈⠉⢈⣻⠿⡉⠀⠁⡂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠉⠒⠦⢴⣀⡅⠸⠀⠉⠙⠓⡶⠶⡶⠶⡚⣧⡪⡢⡐⢅⠢⢕⠢⣍⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀⠀⢱⠀⢀⠀⠌⠌⢔⢌⢎⠢⡙⠦⡙⠢⣍⠲⢤⣉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢌⡀⠘⡌⠌⢪⠣⡑⢌⠢⡈⠳⣌⠛⢦⣈⠛⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠓⠸⣄⢳⡙⢎⠳⡌⠲⡌⠙⢦⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠉⠙⠓⢞⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠉⠙⡗⠒⢲⠚⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠐⠲⢤⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣀⠤⣠⣇⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣒⣉⣀⣤⠔⠊⠀⠀⠀⠀⠉⠉⠓⠂⠤⠀⢀⣀⣀⣀⣀⠀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⢟⡿⣻⢿⣿⣿⣿⣿⠟⢋⣩⣭⡉⠉
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣹⢟⣷⣿⣯⣿⣽⠏⢀⣁⡠⣤⡉
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇⣿⢻⣿⣿⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡒⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⢤⡝⢾⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠙⡧⠹⡿⣖⠤⢜⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠋⡯⢙⡛⡛⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠜⣆⢄⠀⠉⠛⢛⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡊⠤⠖⣂⢒⢫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠃⠚⠈⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠚⢹⠍⢽⠛⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣋⢚⣤⣶⣿⣿⣿⣷⣦⣜⢊⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣩⣿⣿⣿⣿⢿⡻⠻⣿⣿⣯⡁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⢦⣿⣿⣽⣯⣿⣿⣀⠀⡿⣿⣿⡇⠶⣧
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡘⡷⠞⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⡗⠖⡟⣀⣀⣀⣀⣀
⠀⠀⠀⣠⠶⡾⠋⢩⡟⠛⣻⠟⢳⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⣰⠛⢿⡛⠛⣯⠉⠻⡶⢦⡀
⠀⢀⡾⠃⠀⡇⠀⡞⠀⡴⠁⠀⣀⣱⣍⣨⠛⠿⣿⣿⣿⡿⠟⢫⢌⣽⣁⡀⠀⠱⡄⠸⡆⠀⡇⠀⢻⣆
⢠⡿⣿⡄⠀⣷⣰⣇⡼⠁⠀⣤⣿⡭⠈⠑⠦⢤⣘⣢⣺⣠⠤⠖⠉⠬⣿⣦⡆⠀⠹⣄⣷⣠⡇⠀⣼⡿⣧
⣼⠁⠈⢿⣶⣿⣿⣿⠁⠀⠀⣿⡷⠂⠀⠀⢀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠶⣿⡆⠀⠀⢹⣿⣿⣷⣾⠏⠀⢹
⣿⣀⣀⣸⣿⣿⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⣾⢣⡾⣄⡼⣦⢹⡇⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣀⣀⣸
⢿⠛⠋⠹⣿⣽⠿⣽⡄⠀⠀⢻⡆⠀⠀⠀⣿⣸⣷⡏⣷⣿⣸⡇⠀⠀⢀⣿⠁⠀⠀⣼⡿⢿⣽⡿⠉⠛⢻
⠸⣦⣴⠾⢿⣿⣿⢿⣷⡄⠀⠈⢿⣄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣾⠃⠀⠀⣼⣿⣿⣿⣿⠿⢶⣦⡾
⠀⢻⡀⣀⣼⣿⣿⣄⣹⣿⣆⠀⠀⠻⣦⣿⣿⣿⢿⣿⣿⣿⡿⣿⣧⡿⠁⠀⢀⣾⣟⢀⣾⣿⣿⣄⡀⣸⠃
⠀⠀⢻⡏⠁⢈⣿⣿⣿⣿⢿⣷⣄⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣴⣿⢿⣿⣿⣿⣏⠀⠉⣿⠃
⠀⠀⠀⠹⣶⠟⠁⢙⣿⣿⣤⣨⣿⣿⣾⣿⣿⣿⣝⢿⣿⣿⣿⢿⣿⣾⣿⣯⣠⣾⣿⣟⠁⠙⢷⡾⠁
⠀⠀⠀⠀⠈⠳⣶⠟⠁⢹⣿⣿⣿⣿⡿⣿⣣⣾⡿⣷⡿⣿⣣⣿⡿⣿⣿⣿⣿⣿⠁⠙⢷⡶⠋
⠀⠀⠀⠀⠀⠀⠈⠳⣶⡟⠀⣨⡿⢿⣷⣬⡻⣫⣵⣿⣷⣭⡻⣫⣵⣿⠿⣯⡀⠘⣷⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣟⠀⣠⣿⣿⡿⠻⣿⣿⣿⡿⠿⣿⣿⣧⡀⢘⣷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣍⣿⣿⢛⣿⣿⢻⣿⣧⣽⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⣽⠟⣿⡟⢻⣧⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠈⠻⢟⣤⣿⣷⣌⠿⠋⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⡤⢤⣤⣤⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣂⣴⣫⣼⣿⣿⣿⣶⣿⠳⣄⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣱⣿⣿⣿⠿⣿⣿⣿⣷⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣼⣿⣶⡆⣶⣾⣿⣴⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⢻⣧⣿⡇⣿⣤⡿⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⣽⣟⣫⡿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⢂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⢀⠀⣺⡀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⢧⠁⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡊⣶⣶⣸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠴⠶⠋⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⡝⠉⠃⡠⠂⠀⠠⠀⠀⠀⠀⠀⠀⢀⠴⠚⠛⢉⣅⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢠⠊⠀⠀⡈⢠⠒⠀⠀⡰⠀⠀⠀⠀⣰⠀⢠⡤⡤⠴⡖
⠀⠀⠀⠀⠀⠀⡜⠀⠂⢈⣀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣰⠀⠀⠀⠨⡏⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⢀⢸
⠀⠀⠀⠀⠀⠇⠀⠀⠀⣸⣧⡼⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⠀⠀⠀⣰⠀⠀⠀⣰⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⡏⠀⢀⢾⡏⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠃⠀⠘⢸⡇⢸⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠰⠀⠀⠀⠛⠃⠸⣿⣿⠀⠃⠀⠀⠀⠀⠀⠀⡒⠀⢸
⠀⠀⡶⢶⡆⠀⠀⡀⠀⠀⠹⢻⠀⠀⠀⠀⠀⠀⠀⢰⠉⠳⢸⠠⠆
⠀⠀⠀⠋⠀⠀⠀⡯⣤⠀⠀⢤⠀⠹⠖⠀⠆⠰⠂⢸⠂⢧⢸⠀⠋⠀⢀⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⠀⢈⠀⠀⠀⠀⠂⠀⠀⠘⠀⠈⢸⠄⠴⠚⠉⠀⠀⢀⠀⠀⠘⠁⠀⠀⠀⠆
⠀⣶⡆⡀⢀⡌⠀⠀⠀⠀⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠄⢲⢀⣠⠴⡢⠘⡠⢤⠆
⠀⠄⠀⠑⠘⠃⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠂⠀⡀⠑⠈⣄⠰⠉⠛⣁⣯⣴⡿⠁⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠂⠀⠀⡀⠀⡀⢀⠀⡀⢀⠀⠠⠮⠬⠽⠐⠋⠀⠀⡐⠀⠀⠀⡠
⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢀⡾⠄
⠀⠀⠀⠀⠈⠘⠇⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⡴⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠⣫⢞⣵⣿⠏⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⠟⠋⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠠⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⢀⣀⣀⡐⠀⠀⠒⠀⠀⠀⢀⣶⠟⠁
⠀⢀⣀⣀⠀⠀⠀⣀⠠⣔⠢⣠⡄⠀⠀⣀⠀⡄⠀⣨⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋
⠀⠀⣩⡅⠀⠀⡜⡡⢐⠲⠿⠛⠀⠀⠀⠀⠀⠓⠀⣿⣵⠀⠀⠀⠐⠒⠒⠃⠀⠈⠉⠉⠁⠀⢠⠞⠡⠆⢀
⠀⠀⠄⠀⠀⢰⣷⢡⡏⡇⠠⣄⠀⢠⣦⠀⠀⠀⠀⠈⡁⡀⡀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤
⠀⠈⠀⠈⠀⠈⠍⢾⡇⡷⠚⢿⡴⢸⡿⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠠⡔⠂⠀⠐⣿⢀⡂⠀⠀⠀⠀⠛
⠀⠈⢀⠀⠀⣴⡾⠾⢧⣇⡀⠔⠁⢸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⣿⡄⠀⠀⣿⣿⡇⠀⠀⠀⠀⢰
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠿⠇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⣤⣤⣤
⠀⠀⣉⠁⠀⠀⢠⠀⠀⠀⠀⢀⣀⠸⠄⠀⠀⠀⢠⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠀⡔⣷⣼⣿⣷⣦⣤⣾⣿⣤⣤⣤⣤⠀⠈⠉⢀⣀⣀⠀⠀⠀⠀⠀⠀⠤⠘⠃⠀⠀⠈⢹⣧⣀
⠀⠀⢠⢀⡇⣿⢇⠐⠿⠋⠛⠛⠻⠿⣿⣿⣿⠀⢒⣊⢥⣤⢬⠁⠠⠀⠂⠀⠀⠈⠀⠘⠃⠀⠀⠀⠙⠛⠷
⠀⠀⠛⠘⡇⣿⣼⣦⣤⣤⣤⣤⣤⡤⠀⠉⠉⠀⠈⠉⡸⠋⠀⠀⢀⣀⡀⢀⣀⣀
⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⢰⣶⣶⡿⢱⡟⢀⣾⣟⣁⣀⣀⣰⣟⣩⠟⠀⠀⠂
⠀⠀⠀⠀⠷⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠟⠉⢁⣿⠁⠌⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠠⠤
⠀⠀⠀⢀⠂⣿⣤⢤⣤⠀⠀⠀⠀⣿⠀⢶⠶⡆⠀⠀⠀⠀⠀⣺⣿⡄⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⣀
⠀⠀⢀⠈⠀⣿⠀⠀⣽⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡇⠀⢆⠀⠊⠀⢠⡿⠀⠀⠀⠀⠀⠁
⠀⢀⡄⣶⣶⣿⣄⣄⣾⣄⣀⣀⣀⣿⣿⣷⣶⣿⢶⣶⢶⡇⠀⠀⠘⠃⠀⢠⠀⢀⣠⣿⠇⠀⣀⣀⡀
⠀⠀⠀⣿⣿⣀⣷⣯⣾⣴⣇⣟⣮⣀⣠⣦⣦⣼⣤⣤⡼⠇⡀⠀⠀⠀⢰⣾⣶⣿⣿⣿⠀⢉⠤⣤⡬⢁⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠁⡿⡀⢀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠃⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣴⣦⣴⣦⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⣠⡀⢿⣾⢷⡾⣷⡿⢀⣄⠀⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⢈⠀⠀⠀⠀⡐⠸⣷⢿⡾⣫⣥⣭⣝⢷⡻⣬⠇⢂⠀⠀⠀⠀⡁⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⠇⠀⠀⣠⣀⡉⠑⢾⣾⣰⣿⢽⡯⣿⣆⣿⡿⠎⢈⣀⡀⡀⠀⠸⣴⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢀⣠⠎⡀⠙⣿⣇⣬⡦⢻⣿⡿⣿⣿⡟⢴⣥⡺⣿⠋⢀⠹⣄⡀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⢠⣴⣿⠀⠀⣧⢞⠩⣇⠀⠏⠛⠘⠁⠻⠑⠀⣸⠍⣳⣼⠀⠀⣿⣦⡀⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⣧⠉⠛⠆⠺⣛⢍⠑⡇⡤⢄⡄⠜⠣⢠⡠⢤⢸⠚⡹⡛⠟⠰⠛⢋⣼⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣯⡄⠀⠉⠻⢿⠤⣤⣍⣆⣠⠁⠬⠥⠈⣄⣰⣙⣤⣤⡿⠟⠈⠀⢀⣽⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣧⣄⢀⠀⠀⠉⠉⠉⠧⠄⡂⢰⡆⠀⠀⠽⠉⠉⠉⠀⠂⡀⣠⣼⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⡽⣷⣬⣶⡶⠿⢦⡀⠢⡈⣻⣟⢁⠄⢀⡴⠿⣶⣶⣧⣾⢯⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⢃⢼⡇⠋⣬⠉⢛⠿⣾⣿⡿⢿⣾⣷⡿⢿⣿⡷⠿⡛⠉⣥⠑⢸⡧⡘⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⡣⢼⣳⡏⠁⢙⠊⠉⠋⢸⣧⠙⡷⢮⡵⣾⠋⣾⡆⠉⠙⠑⡋⠈⢹⣞⡯⢞⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡿⠙⣼⡥⢄⠀⣶⡠⠞⠋⠘⣷⣌⠺⠗⣡⣾⡃⠙⠳⠄⣶⠀⡠⠬⣧⠏⢿⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡏⠀⢰⠒⠹⣘⡆⠂⠩⠅⢀⠀⡚⠃⢹⢿⡿⡏⠸⢳⡀⡀⠨⠍⠔⢰⣂⠋⠒⡆⠀⢹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⢸⡗⣟⡋⢇⠀⠰⠀⠻⡖⠀⠐⢙⢸⡇⡋⠀⠀⢶⠟⠀⠆⠀⡸⠙⣻⢺⡇⠀⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣰⣾⡟⣾⢳⣿⣦⠀⠁⠢⠀⠃⠴⣶⢼⡇⣶⡦⠘⠀⠔⠈⠀⣴⣿⡞⣷⢻⣷⣆⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠻⢿⡉⠀⢴⡄⠀⡁⠀⠀⢺⡗⠀⠀⢈⠀⢠⡆⠀⢉⡿⡏⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⢟⣾⣳⣿⢻⠆⠀⠃⠁⡼⢹⡏⢧⠈⠘⠐⠰⡟⣷⣟⣷⡽⠃⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄⠀⠈⠉⠱⢼⢹⠾⡰⢆⠞⡉⢸⡇⢉⠳⡐⠄⠴⡟⣧⠎⠉⠁⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠁⠩⠷⣟⣶⣶⣴⣺⣇⣦⣶⣶⣷⠾⠏⠉⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠈⠙⠻⢭⣭⣭⡭⠟⠋⠁⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠿⣿⣯⣿⠿⠿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠘⣿⣿⣤⡜⠯⠏⣤⣤⣀⣿⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣇⣈⣀⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⢿⣿⢿⣿⣿⣿⡿⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣾⣿⣷⣾⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⣛⣛⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣟⣿⣹⣿⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⣿⣾⣿⣿⣿⣾⡗⢰⡎⢻⣿⣿⣿⣿⣿⣿⣿⡿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⢯⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⢻⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣭⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠾⠶⠷⠼⠷⠏⠛⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠁⢀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢦⠀⠄⠘⢄
⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⡺⠂⠀⠀⣤⠀⠄⠈⠒⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣥⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢱⣄⠀⠀⠀⠐⠂⢠⡮⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⢐⠇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢸⠀⠀⠢⣀⠀⠀⠀⠑⠀⠠⠠⠀⠁⡀⠂⠀⠀⠀⠀⢂⠀⠀⠤⠤⠖⠊⠁⠃⠀⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⣒⣤⠀⠁⡀⡛⠀⠀⠀⠀⠀⠀⠸⢀⡀⠀⡀⠀⠔⣠⠄⠀⠀⠁⠀⠀⠀⡗
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠄⠐⠐⢤⣆⡀⢀⡰⠆⠀⠀⠀⠁⠈⠔⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⡲⠤⣆⣴⡟⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠀⢀⠀⠈⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢸⠀⠀⠀⠉⠁⠀⠴⠁⠀⠀⢀⠗⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣴⣾⡾⡿⣿⣿⢷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣧⠀⠀⠀⠀⢠⣿⡿⠻⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀
⠀⠀⠀⡼⡠⣈⠔⡂⠀⠒⢀⡁⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⠀⠀⠀⢀⣜⣿⡀⠂⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀
⠀⠀⠰⠓⠸⡇⠈⠁⠀⠉⠀⡾⠒⠻⡀⠀⠀⠀⠄⠀⠀⠨⠄⠀⠿⢻⣆⠀⠀⢮⣿⠿⠃⠀⠻⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀
⠀⢠⠛⠓⠓⠛⢆⡀⠂⣠⡞⠛⠛⠚⢣⠀⢀⡀⠉⠂⠒⠲⡄⠀⠀⠀⣿⠀⠀⢸⣇⠀⠀⠀⠒⢒⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠿⠿⠛⠛⠂
⠀⠈⠻⢌⢃⡛⣿⠃⠀⠻⡇⡈⢉⡱⠟⠀⠀⡇⠀⠀⠀⢀⡡⠀⠄⠤⡟⠀⠀⢊⢿⠀⠀⠀⠐⢺⣿⡏
⠀⠠⠶⠼⣀⣀⣭⠀⠰⢀⣇⣀⡼⠠⠶⠀⠀⡇⠀⠀⠀⠚⠁⠀⠀⠀⢸⠀⠀⠈⣸⡇⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠐⠀⢀⡀⠀⣙⡄⠐⠂
⠀⠀⠀⡀⣇⣀⣽⢰⣨⣿⣅⣀⡇⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢲⠀⠸⠀⠀⠀⡿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⡇⠀⢀
⠀⠀⣄⠳⣜⣤⠿⠼⠿⠼⢧⣴⣡⢇⡆⠀⢠⢃⠀⠀⠀⠄⠀⠀⢸⡀⢠⠀⠀⠀⢸⣿⠀⠀⠰⣿⡏⠀⠀⠀⢠⠏⢀⡔⠀⣀⣤⡀⠈⢦⡈⢳
⠀⠀⠈⠛⠾⠭⣭⣿⣿⣿⣽⠽⠖⠋⠀⠀⠈⠞⠒⠒⠒⢺⣀⡴⠾⠁⠘⠀⠀⠀⢸⣿⠀⠀⢰⣿⡇⠀⠀⠀⠸⡄⢸⣿⠎⠁⠀⠙⣾⣿⠂⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠞⠹⡠⠤⣤⠤⣼⠛⠗⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣤⣀⡆⠀⡃⠀⠀⠀⣼⣿⣄⣀⣸⣿⣧⠀⠀⠀⠀⠀⠸⡶⠷⣤⣤⡴⠿⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣷⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠢⠭⠉⢸⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠐⢢⣥⠖⠊⠁
⠀⠀⠀⠀⡠⠀⠀⠀⣭⣿⣿⣿⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⢆⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠁⣁⠀⠀⣺⣿⣿⣿⣷⡄⠀⠀⠀⡇⠀⠀⠀⢀⠀⠈⡇⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡂⠄⠀⠤⣷⡄
⠀⠀⡰⠁⠀⠙⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⠀⡇⠀⠀⠀⠈⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠡⠴⠀⠀⠀⠀⠀⣼⣯⡀⠀⠀⢰⣿⡆
⠀⠀⠀⠀⣼⣇⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⠀⠀⢀⠀⣌⠀⠀⠀⡄⠀⠀⠀⠀⠀⣋⡀⠀⠄⢀⣘⡇⠀⠘⠛⠿⠗⠺⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠤⢀⠀⠤⢿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠛⣿⣿⣇
⠀⠀⠀⠀⠛⠃⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠒⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠈⣁⠈⣦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⠀⠀⢸⣷⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠄⢀⠀⠀⢀⠠⠽⠿⠿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠐⠶⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠐⠂⢳⡄⠀⠖⠒⠋⢿⡀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⣿⣦⡄⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠜⠛⠋⠀⠀⠈⠙⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⢳⠈⠳⠀⠀⠀⠁⠂⠐⠃⠐⠂⠚⠉⠉⠁⠀⠀⠀⠃⠈⠃⠃
⠀⠀⠀⠐⠀⠠⠤⢤⣼⡀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠰⡄⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠁⢠⣦⡀⠀⠀⠀⠀⢀⣾⣿⣷⣄⠀⠀⠀⠀⢀⣄
⠀⢀⡁⠀⠀⠀⠀⢈⠟⠃⠀⠀⠀⠀⠀⠰⠀⠈⠉⠁⠀⠀⠠⠤⢀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣄⠀⢀⣾⣿⣿⣿⣿⣷⣄⠀⠀⣼⣿⣷⣄
⠀⠀⠀⠀⠁⠀⠒⠒⠶⠤⢤⣀⣀⠀⠀⡆⢠⣀⣄⣿⣷⡀⣄⣤⣴⡆⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣴⣾⡎⣿⢚⣓⣿⢱⣿⣦⢄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡛⢿⣿⣎⢿⣿⢸⣷⣮⡇⣿⡿⣡⣿⡿⢛⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣯⣟⣧⡹⡙⡎⣿⡟⢻⣟⢷⣿⢣⢋⣏⣴⣿⣭⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣮⢯⣷⢹⡿⢹⣫⣿⣿⣝⡏⢿⡟⣾⡽⣵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣏⡷⢮⣽⣿⣿⣯⡵⢾⢹⣻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣒⣻⣯⣶⡠⠜⠻⣿⣿⠟⠣⢄⣦⣽⣟⣒⣲⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⢭⣽⣿⠃⣋⣛⣦⡌⢡⣴⣛⣙⠚⣿⣯⡭⣭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣛⣛⣛⣷⣽⠈⠛⠛⠛⠇⠸⠛⠛⠛⠁⣯⣾⣛⣛⣛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⠿⠿⠿⠿⣿⣿⣦⠀⠠⣤⣤⠄⠀⣴⣾⣽⠿⠿⠿⠿⠽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣛⣛⣛⣛⣛⣿⣿⢹⡄⡤⠾⠷⢤⢠⡏⣿⣿⣛⣛⣛⣛⣛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣭⣭⣭⣽⣿⣿⣿⣿⣿⡇⠀⠛⠛⠀⣸⣿⣿⣿⣿⣿⣯⣭⣭⣭⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⣿⢻⣿⣿⣿⣶⣾⣷⣶⣿⣿⣿⡟⣿⠋⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠙⢻⣿⣿⡟⠋⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠈⢿⡿⠁⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⠸⠇⠀⠀⢸⡿⠃

⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣤⣤⣀⣀
⠀⠀⣠⣴⣶⣿⣿⡿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⣿⣿⣿⣷⣶⣤⣀⡀
⠀⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣷⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣷⣦⣤⡀
⠀⠀⠀⢀⣴⣿⡿⠋⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉⠛⠿⣿⣷⣶⣤⣄⣀⣀⡀
⠀⠀⣰⣿⡿⠋⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⡆
⢀⣾⣿⣏⣀⣀⣀⣀⣀⣀⡻⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⣾⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠁
⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠙⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠙⠿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢶⣦⡀
⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢠⠶⣆⠹⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⠃⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣶⣦⣤⣤⣄⣀⣀⣀⣤⣤⣶⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠂⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢾⣿
⠀⠀⠀⠀⠀⠀⠸⣆⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⢈⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢩⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢏⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠘⠃⠤⠄⠀⣀⠀⠀⠠⠄⠘⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⢧⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⢀⣄⡀⠀⠀⣀⢠⣄⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⢸⠻⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⢉⠟⠛⢉⡰⣊⢵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠸⣆⡒⠘⠁⣉⢶⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⢓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠑⠤⢙⡃⠘⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠐⠒⢶⡎⢛⠻⣯⠀⠀⠀⣾⢎⡵⢋⠏⣸⠋⠐⠢⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢱⠁⠀⠀⠀⡟⣤⠓⢦⠀⠀⠀⢻⡂⠴⢊⡴⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠋⠐⢻⠀⠀⠀⢼⠥⠒⢻⠘⠆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠰⠾⣦⠤⠴⣶⠌⡸⠃⠘⠦⠄⠤⠚⠀⠠⡆⢣⠘⣏⣀⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⣀⣷⠘⣧⠾⠂⠀⠀⠀⠀⠠⠶⠾⣾⠀⠱⠄⠀⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡋⠈⠉⠉⠉⠁⠀⠀⢀⠇⠀⢰⠀⠀⠘⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠀⠀⠀⠤⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⡷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡀⠀⠀⢡⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡁⠀⠀⠉⠀⠶⢠⣜⠀⠀⠀⠀⠙⢄⠀⠀⢳⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⣰⠁⠈⠉⠁⠀⠒⠒⣿⠀⠀⠀⠀⠀⠈⠲⣄⢀⡽⣄⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡤⠁⡆⠀⠀⠀⠀⠀⠀⠈⢳⡬⠊⠑⣿⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠲⠇⠀⠸⠃⡆⠀⠀⠀⠀⡜⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣾⢋⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠧⡀⠀⢳⠀⠀⠀⢀⡞⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠁⣧⡀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⡽⢿⢂⢸⡇⠀⡤⠃⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠰⠛⠛⢿⣠⠊⣀⠀⠀⣀⣀⢀⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠀⢷⡜⠉⠀⠈⠀⠏⠈⠀⠋⡀⠈⠉⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠇⠀⠀⠀⠘⠀⠀⠀⠈⢡⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢳⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢱⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢹⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⠀⡆⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠀⠀⢴⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢆⠀⠀⠀⠀⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⡇⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠃⠤⣀⢸⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⡇⠀⠀⠀⠀⠘⠃⠤⣀⣀⡀⠀⠀⠀⠀⠸⠀⢀⣀⣀⣀⣀⠘⠃⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠉⠉⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠿⠿⠿⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⡞⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⣇⢻⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⠁⡼⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠷⢙⣚⣫⠭⣶⣿⡄⠀⠀⠀⢀⠃⡼⢁⡞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠈⠲⠶⠂⡸⣯⣴⣾⣿⣿⣿⠀⠀⠀⠆⡜⢡⠏⡼⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡇⠀⢌⠞⡴⡡⢊⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣪⠊⢀⠔⣫⢞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⠇⢕⠡⣊⠴⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿⠂⠐⠨⢐⣫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣾⣿⣿⣿⣿⣿⢿⢿⢡⣿⣿⡥⠤⣒⣨⣭⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⡿⠭⢝⢒⣫⣅⣿⣾⣿⣿⣿⣇⣒⣒⣒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣀⠤⣒⣭⣾⣿⣿⣿⡿⢗⠒⢀⣐⡲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡖⣫⡿⢛⣽⣿⡿⣟⡫⠭⢀⡲⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢾⣿⢿⠟⠫⠕⣂⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⣀⡠⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⡿⠿⠿⢶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠉⠀⣰⡿⠁⢘⢿⣦⡀⠀⠀⠈⠉⠿⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⣄⠀⠀⢀⣼⡟⣵⠀⣸⣷⡝⢿⣦⡀⠀⠀⠀⢀⣀⣹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢠⣾⢏⣼⡇⠀⣿⣿⣿⣦⡹⣿⣄⠀⠀⠀⠀⠘⠛⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⣰⡿⢣⣾⣿⠃⡇⣿⣿⣿⣿⣿⣮⡻⣷⣤⣤⣤⣴⡾⢷⣮⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣤⠀⠀⢀⣼⡟⣡⣟⠋⠉⣼⣧⡍⢿⣿⣿⡿⠿⣡⡿⠉⠋⠘⠙⡆⠀⡏⠙⠙⠋⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠓⠂⠀⢠⣾⠏⣴⣿⣿⣿⡇⣼⢀⣝⣒⣒⣒⡀⣰⡟⠀⠀⢠⠖⠛⠁⠈⠛⠒⠄⠀⣄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠃⠀⣰⡿⢃⣾⡛⠛⠿⠟⠁⣿⠸⠿⣿⣿⠟⣼⡟⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠱⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣼⡟⢡⣿⣿⣿⣶⣤⡍⢻⡟⢛⣛⠲⠶⣾⡟⠀⢰⠏⠀⢀⣄⡀⠀⠀⠀⣠⣤⠀⠸⢄⠁⢻⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⠏⣰⣿⣿⣿⣿⣿⣿⠇⣿⡇⣾⣿⡟⣸⣿⠁⠀⢸⡄⠀⠈⠉⠁⢀⣶⠀⠈⠉⠀⠀⣸⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⢋⣼⣿⣿⣿⣿⣿⣿⡿⢰⣿⠀⣿⣿⣤⡻⣿⣆⠀⠀⠹⡆⠀⠀⠀⠈⠉⠀⠀⠀⠀⡞⠁⢠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⣼⡿⠁⣾⣿⣿⣿⣿⣿⣿⣿⠃⣼⡏⢠⣿⣿⣿⢃⣾⠟⠀⠀⠀⣻⣀⣀⠀⠀⠀⠀⠀⣀⣸⠁⠀⢺⣯
⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⡙⠻⢿⣿⣿⣿⡟⢠⣿⡇⢸⣿⣿⢃⣾⠏⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡈⠛⠿⠃⣸⣿⠀⣾⣿⢃⣾⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⣀⣿⡟⠀⢛⣋⣾⣿⣷⣶⣤⣤⣴⣾⣷⡶⠶⠶⠶⠶⠶⠶⠦⠄⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⡿⠿⠿⢿⣿⡏⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⣤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣧⣀⣸⣧⣼⣬⣭⣥⣽⣦⣭⣬⣭⣧⣭⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣭⣿⣧⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀
⠀⠀⢀⠀⠈⢀⣠⢀⠀⠀⢀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⡑⠀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠁⠄⠀⠀⠈⠀⠀⠀⠀⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄⣀⠀⠈⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⢃⡈⠀⠂⠒⠠⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢀⠀⠸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠠⠌⠀⠃⠸⠆⢸⠀⡧⠀⠀⠄⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠁⠈⠂⠇⠰⠛⠀⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠄⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡠⡀⠀⢀⠠⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⠔⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠐⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⠃⢀⠄⠐⠄⠀⠀⠀⠈⠃⠠⢶⡄⢰⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑⠀⠀⠒⠀⠀⠠⠀⡂⠀⠀⠀⠀⠣⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠊⠀⠀⠀⠀⠀⢡⢄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠒⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠐⠁⠂⡁⠀⠀⠐⠆⠀⢠⡘⡀⠀⠀⠀⠘⠈
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠁⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣬⡁⡀⡀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠊⠁⠀⣀⠀⠆⢀⣰⠞⠁⠈⠀⠀⠀⠀⠂⢀
⢀⠀⠠⠚⠀⠀⠀⠀⠀⠀⠀⢥⣈⠀⠸⠇⠀⠀⠄⠀⠀⠀⢠⠀⣄⡀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠂⡀⢰⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⢀⡀⠉⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠐⠀⠀⠂⠀⠀⠈⠛⡁⠀⠠⠄⠴
⠀⠀⠀⠀⠀⠀⠀⠠⡖⠒⠂⠐⠂⠀⠀⠈⠀⠀⠀⠀⠠⠀⠦⠀⠐⠠⠔⠀⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠁⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠁⠀⠀⠀⠀⠒⢀⣴⠀⠐⠃
⠀⠀⠀⠀⠀⠈⠀⠘⠀⠂⠀⠀⠈⢁⡀⠀⠀⡀⠚⣀⡀⠉⠀⢀⠀⢀⠠⢦⠀⠀⠄⠄⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⢧⠢⣤⠀⠁⠀⠠
⠀⠀⢀⡀⠄⠠⠀⠀⠀⠀⢀⠡⠈⡉⠀⠐⠀⠀⠀⠈⠛⠀⠒⠀⠀⠐⠦⢄⠀⢂⠰⡀⠰⢀⠀⠈⠁⠁⠀⠀⠀⠈⠚⠀⠀⢀⡸⠿⡖⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⢣⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂⠀⠀⠀⠸⣀⠒⠂⠀⠠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠄⡀⠀⠀⠀⠁⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠚⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠲⠄⠎⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⡄⠁⠀⠠⠀⠀⠈⠓⢀⣀⢀⠀⠀⠀⠀⠀⠠⡀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠁⠀⠀⠀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠄⠇⠀⠀⠀⠀⢠⠀⠀⠑⠀⢀⠀⠪⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠇⠀⠀⠀⡠⠀⠀⠁⠄⢐⠀⣂⠀⠈⠀⠁⣤⠀⠐⠀⡀⣮⠈⠀⠐⠆⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠤⠊⣀⠀⢠⠠⠀⠃⠀⢠⣬⠀⠀⠁⠐⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡁⠀⣸⠀⠃⠀⠀⠈⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⢀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡘⠂⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠉⠀⠐⠀⠠⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠌⠀⠂⠀⠀⠄⠀⣀⠐⠂
⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠑⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠮⠭⣢⠕⠪⣔⠭⠵⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⣠⠜⣫⣿⣅⣨⣽⣝⠳⣄⠩⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢗⣫⡝⣱⣿⣵⣸⣇⣸⣧⣮⣿⣎⢫⣝⡺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠥⣺⢱⣇⣀⣉⡙⠦⠴⢋⣉⣀⣸⡌⣗⠬⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡡⢊⣽⢸⠩⠿⠿⢭⡉⢉⡭⠿⠿⠍⡇⣮⡑⢌⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡵⣣⢾⢸⠘⠻⠿⠳⠄⠀⠞⠿⠟⠃⡇⡷⡜⢮⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⡶⢡⢻⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡟⡌⢶⡿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡈⣷⢃⡎⡏⣷⠀⠀⣀⣀⣀⣀⠀⠀⣾⢩⢱⡘⣾⢡⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢠⢙⢻⣠⡇⣿⣧⡀⠈⠒⠒⠀⢀⣼⣿⢸⣄⡟⡋⡀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⡸⢸⠹⣧⣿⣿⠉⠓⠶⠶⠚⠉⣿⣿⣼⠏⡇⢃⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡏⡄⡇⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⢸⢠⢱⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡇⢸⣇⠀⠀⠀⠀⠀⠀⣸⡇⢸⣼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠂⠹⣿⣿⣿⣿⣿⠓⠦⠤⡤⠴⠚⣿⣿⣿⣿⣿⠏⠘⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠄⣧⣤⣤⣬⣿⣈⠉⠐⠂⠁⣁⣿⣧⣤⣤⣼⠠⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⢛⠻⠿⠿⡌⣋⡛⢛⣛⢡⠿⠿⠟⡛⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣄⣀⣀⣡⣌⣀⣀⣠⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠋⠉⠉⠀⠀⠀⠀⠈⠉⠉⠛⠳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⡛⠛⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣠⣤⡶⠶⡿⢿⡀⡷⢺⠅⣸⠿⠶⠶⣤⣄⣀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢴⡞⠉⠉⣡⣤⣴⠶⠬⢿⣧⣸⡾⠧⠿⢶⣤⣤⡉⠉⠙⣶⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢀⣦⠶⠛⠉⠀⣾⡟⠛⢾⣇⢈⣿⠚⠛⣷⠄⠉⠙⠳⢶⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⣘⣛⣛⣋⣉⣉⣉⣛⣛⣋⡀⠀⠀⠀⠀⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⡶⠛⠋⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠙⠛⢲⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⡇⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢰⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠴⣣⡴⠞⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠓⠶⣌⠳⣄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠃⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⢷⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⠀⠛⠉⣉⡙⠛⠷⠀⠀⠴⠟⠛⣉⠉⠙⠃⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⣿⠈⠻⢟⣻⣿⣻⣆⠀⠀⢠⡞⣻⣿⣹⠿⠃⣿⡏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡆⠀⠀⠈⠁⠀⠈⠀⠀⠉⠀⠈⠁⠀⠀⢀⡿⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣧⠀⠀⠀⢤⠦⠥⠬⠵⠤⠄⠀⠀⢰⣷⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⡽⣟⢧⡀⠀⠀⠑⠒⠒⠚⠁⠀⠀⣰⢫⡟⣶⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⣭⣥⣿⣼⡟⠦⣄⡀⠀⠀⠀⣀⡴⠛⣧⣾⣭⣭⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣸⣆⣧⣸⣸⡇⠀⠈⠉⠛⠛⠋⠁⠀⠀⣇⣼⣰⣆⣦⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⣶⠒⡛⢋⡙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣉⢛⡓⣶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⢛⡘⣩⠤⠐⡈⠰⢌⠛⢦⣄⡀⠀⠀⠀⠀⢀⣀⣴⠞⣁⠶⢁⡐⠠⣭⡉⣘⢃⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⠉⣡⡔⢛⣥⢌⡠⠈⠱⠆⣈⢙⠛⠛⠛⠛⡋⢩⡀⠦⠉⠰⣀⢦⡙⠂⣬⠋⣲⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡡⠲⢎⡐⠋⡴⢃⡄⡀⠁⠘⢁⠓⠘⢃⠋⠈⢑⠈⣄⢳⡝⠆⡁⠶⠠⣱⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠊⠡⡶⣀⠙⠰⠃⡟⣰⢠⣦⡄⣦⢶⠸⡎⠧⠙⣀⠰⡎⠉⢢⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣬⡀⠉⠑⠗⢤⡄⣉⢈⡈⢃⡉⣨⡀⣤⠰⠆⠉⢐⣨⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣄⣄⡉⡈⢉⢈⢁⢈⣡⣐⣬⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁
⣶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣂⣻⣥⣭⣭⣭⣬⣥⣭⣥⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣽⣿⣿⢷⣿⣿⣷⣿⢾⡷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⡿⣾⣿⣟⣿⣿⣿⣻⣾⣿⣿⣷⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⡎⢙⠀⠀⠀⠒⢡⠤⠔⠒⠒⠀⠀⢄⢠⡄⠀⠘⠭⠧⡀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢐⣆⠀⠀⠀⣶
⠀⠀⠴⢶⡶⠦⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⢀⠀⢀⡇⠀⣤⢀⡀⠀⠀⣰⢨⡇⠀⠀⠀⢻
⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⢿⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠊⠑⠺⣆⠀⢈⡇⠀⠉⠀⠉⠉⠉⠉⢸⣧⣀⣀⣠⣼⠄
⠀⠀⠀⣈⠁⠀⠀⠀⠀⠈⠁⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⢁⡉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⣿⣄⠀⡀⠀⢸⣿⣆⠀⠓⠂⠀⡠⠚⢿⣿⠇⠲⣤⠁⠀⠀⢹⡟⠀⠀⠐⣷⣤⣄⣀⣀⠀⠀⠀⢼⡄⠀⠀⠀⠀⠠⣄
⠀⠀⠀⡿⣿⣯⡀⠀⠈⣿⣯⣷⣀⠀⠀⠀⠀⢸⢟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣿⠿⣿⣿⡀⠀⠀⢸⠇⠀⠀⠀⣤⣰⣿
⠀⠀⠴⠡⠼⠝⠁⠠⠤⠿⠅⠘⠧⠀⠀⠀⠀⠈⠀⠑⠀⠑⠀⠀⠀⠸⠇⠀⠀⠀⠂⠃⠀⠃⠘⠁⠀⠀⠀⠐⠀⠀⠀⠸⠿⠷
⠀⠀⠀⣄⢀⡆⠀⢀⣀⣀⣤⣄⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢴⠀⠀⠀⢀⣶⡄⠀⠀⠀⠀⢂⣄⠂
⠀⠀⠜⠿⠁⠀⠀⠀⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⣀⡸⣿⣧⠀⢠⡾⢿⣾⢷⡀⠀⣿⠀⢠⣤⡀⢘⣿⣿⣿⠀⣠⣤⠄
⠀⠀⡰⠠⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣉⣩⣭⣽⠿⠁⠀⢰⣶⣿⡄⠀⠀⡳⠘⠋⠸⠀⠀⢻⠀⠀⠉⠉⠙⠻⠿⠛⠉⠉⠁
⠀⠀⠁⠈⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠉⢉⠀⠀⡀⠈⠉⠉⢉⠁⠀⠀⠉⢠⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡰⠀⠀⣤⣶⣟⠛⠁⠀⠀⡀⠂⠀⢀⣠⣿⡆⠀⠀⢀⣼⣷⠴⠇⠀⠀⠀⠉⡉⠐⠀⠀⠰⣷⣥⣄⠀⠀⠀⣿⣶⣄
⠀⠀⠁⣿⠄⠀⣦⣿⡟⠀⠀⢠⡀⣷⠀⠀⢠⠁⣹⡇⠀⠀⣼⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠋⠉⠃⠀⢠⡵⠁⡀⡀
⠀⠀⠴⠿⠀⠀⠸⠿⠇⠀⠀⠈⠿⠿⠀⠀⠨⠥⠤⠄⠀⠼⠷⠶⠲⠀⠀⠀⠀⠴⠳⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠨⠅⠸⠁⠸
⠀⠀⣶⠀⣦⢰⣆⠀⠀⡀⢰⣶⠀⠀⠀⠀⢰⢶⡀⠀⠀⠀⠀⠀⢀⣶⡆⠀⠀⢠⣄⠀⠀⠀⢰⣶⡄⢰⡄⠀⠀⠀⠀⢠⣄
⠀⢠⣿⣸⣿⣸⣇⠀⠀⠓⣾⣿⣄⠀⠀⠀⠈⣈⣿⣆⣸⡇⠀⠠⠈⣿⡅⠀⠠⠼⠛⢧⠀⠀⠸⣿⢿⣮⠁⠀⠀⠘⠦⢾⣿⣳⡄
⠀⠘⠻⠛⠻⠙⣷⠀⠠⠀⠿⠃⠸⠀⠀⠈⠉⢉⡉⠛⣏⠁⠀⠀⠀⠏⡇⠀⠀⠀⣴⢲⡀⠀⠀⣿⢷⠻⣷⢄⠀⠀⢿⠘⠛
⠀⠈⠈⠉⠉⠁⠉⠀⠘⠒⠚⠉⠀⠀⠀⠀⠘⠛⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠐⠁⠘⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠉⠉⠉
⠀⠀⡄⠀⠀⢀⡀⠀⠢⣤⣿⣦⠀⠀⠙⣶⡆⣶⡞⠀⠀⡀⠀⣰⣄⣶⣀⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⣧⣤⡇⠀⠀⣀
⠀⢴⣷⣦⣤⣤⣁⠀⠀⢀⠘⠉⠀⢤⠀⣿⣧⣿⡇⢠⠄⢴⡿⠿⢻⣟⣿⠏⠓⢦⣴⣤⣦⠀⢀⡼⠁⠀⢠⣿⠀⣿⠀⠀⠙⣄
⠀⠤⠾⠟⠿⠿⠿⠂⠀⠸⠀⠱⠀⠘⠄⠸⠛⠋⠀⠊⠀⠀⠉⠐⠒⠛⠛⠂⠉⠀⠀⠀⠀⠀⠈⠛⠲⠦⠾⣿⣿⡿⠶⠖⠚⠉
⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢀⢆⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠰⠦⡄⠀⠀⠀⠀⣀
⠀⠑⣄⣠⣾⣿⣦⠀⠀⠀⠀⠿⠿⠟⢀⠀⠀⠀⢸⣸⠀⠀⠈⢹⣿⣶⣶⣶⣄⠀⠀⢸⣶⠀⠀⠀⢠⣤⣶⣶⣦⣄⡀⠉⠉⢲⡄
⠀⠀⠈⢳⠘⠟⠉⠀⣎⢀⣀⣀⣁⣀⣨⣤⠂⠀⠀⡏⠀⠀⠀⢀⣿⠛⢛⠟⡇⠀⠀⣾⣿⠀⠀⠀⠀⡉⡟⠛⠋⠛⠛⠦⠴⠞⠁
⠀⢀⠐⠛⠚⠚⠃⠀⠀⠈⠉⠉⠉⢉⠁⠈⠀⠀⠀⠛⠀⠀⠀⠘⠈⠀⠘⠘⠃⠀⠘⠛⠛⠂⠀⠀⠘⠋
⠀⢨⢿⠆⣰⢶⡄⢀⡖⢦⠀⠀⢨⠁⠐⠀⠀⠠⣼⣦⠀⢠⣦⡀⠀⠀⠀⠀⠀⢀⣷⣄⠀⠀⠀⣌⣴⣎⣂⠀⠀⠀⠘⢇⡀
⠀⣝⡖⠀⣿⢸⡇⢸⡇⡞⠀⠀⠀⢈⠀⠀⠀⠀⢈⠋⠀⠈⢿⣻⣆⠀⠀⠀⣠⡏⣿⡟⡆⠀⠀⣿⣿⣿⡇⠀⠀⢠⣄⣿⠃
⠀⠻⣧⣠⡿⠸⣇⣸⠇⢧⣀⠀⠀⢐⠀⠀⠀⢀⠋⠃⠀⠀⢸⣿⠙⡷⠀⢀⣸⣧⠤⠄⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣳⠍⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⢀⣼⣟⣷⣾⣿⣿⣶⣟⣷⣆⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡁⠒⢳⡮⢹⡗⡖⣰⢄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣧⡐⣖⢿⠿⣶⠟⠒⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣙⣃⣰⣿⣿⣿⣿⣿⣿⣽⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡂⣘⢋⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⡱⣞⣋⣏⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣛⣷⢊⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⣩⡿⢻⣿⣿⣯⣿⣛⣿⣿⣿⣿⣿⡫⢻⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣍⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠛⣴⠋⢻⣿⣿⢿⣿⣿⣿⣿⠿⣾⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣄⠛⢻⣦
⠀⠀⠀⠀⠀⠀⢠⡾⠃⣨⡟⠳⢤⣿⣿⣾⢟⣿⣿⢟⣁⠀⠀⠈⠁⠀⢀⣈⣻⣿⣿⣿⣿⣿⡏⢸⠌⢷⡅⡘⢷⡄
⠀⠀⠀⠀⠀⢰⡟⠴⣱⠏⠀⠀⢠⣿⣟⣵⢿⣿⣿⣤⣼⣿⡗⠀⠀⣿⣿⣬⣬⣿⣿⣿⣿⣿⣿⠋⠀⠀⢻⡜⡤⢻⣄
⠀⠀⠀⠀⢠⡿⣠⢰⡏⠀⠀⠸⣤⣿⣿⣿⣿⣿⠉⠙⠈⠁⠀⠀⠀⠉⠈⠉⠉⢙⣿⣿⣿⣿⣿⣤⣰⠀⠀⢻⡄⣀⢿⡄
⠀⠀⠀⠀⣾⠃⠌⣾⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡿⠁⢀⠀⠈⣧⢡⠘⣧
⠀⠀⠀⠀⡿⢠⢶⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣠⣤⣤⡀⠀⠀⠀⣿⣼⣿⣿⣿⡟⠷⠒⠁⠀⠀⢻⣰⣄⣿
⠀⠀⠀⢸⡷⣀⢸⠇⡶⢻⣿⣿⣿⣻⣿⣿⣿⡿⣿⣧⡀⠀⠀⠶⠆⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⢦⠀⠀⠀⢸⡇⡀⢹⡇
⠀⠀⠀⢸⡇⢬⣸⡌⢧⣼⣿⡿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣶⣤⣤⣤⣤⠾⢿⣿⣿⣾⣿⣿⣿⣿⣷⠏⠄⠀⠀⢸⣇⡵⢾⡇
⠀⠀⠀⠘⣇⠀⡸⡇⠀⠈⣿⣥⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⠿⠿⠛⠁⠀⠘⣿⣿⣻⣟⣿⣿⣿⣿⣷⣀⠁⠀⣸⢋⡀⣾⠁
⠀⠀⠀⠀⣿⡀⣣⢿⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣟⣿⣿⣿⣿⣿⡀⢀⡟⣌⢀⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⡟⣽⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣼⣿⣿⢻⣿⣿⣷⣿⣿⣿⣿⣿⣇⠀⢀⠔⢀⣀⡤⢖⠉⠈⡀⠀⠀⠙⣿⢿⣿⣿⣼⣿⡿⣿⠃
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣿⣿⣼⣿⢏⣵⣿⣿⣿⡟⠋⠉⣠⡿⠞⠋⣁⠔⠈⠂⣁⠄⣠⠴⠚⣻⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠶⠚⣋⠥⢀⡠⠙⠀⣀⡴⢟⡿⠛⠁⠀⢠⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠲⢦⣌⣿⣿⡿⣿⠿⣿⣿⣿⢟⣁⠤⠒⣋⡵⠞⢁⣠⠴⠞⠉⣰⠋⠠⠀⣀⣴⡿⠿⣿⣿⣿⣿⣠⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣻⣾⣿⣿⣿⣿⡋⣠⠶⠋⠁⣀⡴⣛⡿⣦⡀⠀⠁⢀⠻⣄⣿⣿⣶⣿⣿⣟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣯⣻⣿⣿⡿⣿⣧⣄⣤⡾⣫⣾⢫⠟⣮⣿⣢⣤⣨⣶⣿⣿⣿⣿⢋⣽⣿⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣿⣿⣿⣿⣿⣿⣿⣷⡿⣽⣿⣿⣿⣧⣼⣿⣿⣷⣟⢿⣿⣿⣿⣿⣾⣿⣿⣯⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣻⣿⣿⣯⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣽⣿⣟⣿⡋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⡿⠿⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠷⢿⣯⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣠⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⢀⣤⠴⢾⣇⣠⠤⢶⣶⣿⣿⠉⢴⣹⣿⣿⢿⣿⣿⡿⣿⣽⣏⡵⠉⣿⣿⣶⡦⠤⣀⣸⡷⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⢈⡃⠾⠿⣶⡦⠼⠿⠛⠛⠀⠀⠁⠻⢿⣿⣿⣽⣟⣿⠻⠁⠀⠀⠛⠛⠿⠧⣴⣾⠿⠟⢐⡁
⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⡉⠉⠛⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣹⣿⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠝⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢠⣶⣦⠀⠀⣤⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣶⡤⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣶⣿⠟⠁⣍⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠞⠛⠛⠻⠷⣤⡀⠀⠀⠀⠀⠀⠀⢈⡇⣤⣦⠆⠀⢴⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠈⡇⣿⡟⠃⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⡇⢸⣷⡶⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣾⡛⠂⠀⠀⡇⣈⣉⣁⣀⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢄⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⢸⣿⡇⠀⠀⠀⡇⣯⠉⢽⣿⠉⣉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣙⣠⣤⠤⠤⠤⠔⠃⠈⠷⠼⠏⠀⠀⠀⢠⡇⠉⠀⣈⡉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣉⡈⣠⣴⡴⢶⡶⠁⢀⠀⠀⠀⠀⠀⠀⠠⡇⠙⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣠⡾⠋⠀⣿⠀⠀⠈⠁⡞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢡⡌⢀⡐⢿⡁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣚⣦⣯⣤⣿⡇⠁⣀⣀⡾⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠻⣝⡦⢹⡏⢀⣀⣙⡃⢿⣷⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣀⣀⠈⠑⠿⠋⢺⠤⠭⠹⠀⡻⠀⣀⢠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⣍⣙⠤⠠⣤⠤⢄⡀⠀⠀⡀⠀⠉⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢃⠙⠒⠚⠛⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣄⣀⣈⣸⠀⠀⠀⠐⢄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠈⠻⠿⢿⣧⣎⡀⠀⠀⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠈⠾⢛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠾⣷⠂⣠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠁⠈⢡⠈⢀⡛⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠇⠀⠈⢆⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡘⠋⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠁⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠷⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠐⠀⠀⠀⠀⠁⠐⠂⠀⠐

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣠⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠘⢻⢸⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠠⠠⠤⠤⠰⠀⠀⠱
⡄⠀⣄⠀⠈⠿⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣤⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡈
⣿⠿⠿⠃⠀⠰⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⠀⢀⠔⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⠘⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⢄⣀⠀⠠
⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡀⠀⠂⠀⠀⢀⠤⡋⢠⠀⠀⢀⣒⢬
⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠠⢊⢃⡎⠀⠀⠀⠀⠴⠊⠘
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡌⠼⣭⡻
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣠⡀⢸⠀⠀⠀⠀⠐⠀⠀⠰⠀⠴⠁⠀⠀⠀⠀⠀⠉
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠁⠁⠀⠱⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⣴⡂⠐⣚⠂⢐⡦⠀⠀⠀⡠⢤⠤⠤⠤⢲⠀⢸⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣁⣼⣿⣧⣾⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠣⡄
⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠉⠉⠉⡛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠖⢀⡄⠀⠀⠀⠀⡤⢘⡠⡄⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⣀⡠⠀⠀⢀⠀⠀⠀⠀⠠⠂⠘⠀⠢⠀⠀⠐⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠂⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠸⡤⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⢀
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⣰⡈⡌⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠂⠀⠀⠀⠀⠀⡠⠒⡖⠒⠒⠒⠷⠀⢧⠀⠀⣚⡥⠗⠐⠀⡄⠀⠀⠀⠀⠰⣄⠀⠀⠀⠈⢿⣤⡀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⡀⠈⢿⣦⡀⠈
⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢬⠗⡆⠫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠁
⠀⠀⠀⠀⣶⠣⠿⠟⢴⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠈⠀⠀⢰⣿⡀⠀⠀⣾⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⡆
⢀⣂⠀⣘⣿⣝⣀⣘⣋⣝⡁⠀⠀⢀⢀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⡀⠈⠀⣖⠀⠉⠉⢙⡀⡃⢀⡀⢀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠴⠴⠶⣶⠆⣷⠀⠀⠀⣶⠀⡇⠀⠀⢰⠆⠀⡇⠀⠠⣦⠀⠀⡇⠀⢀⣄⠀⢘⡇⠀⢠⡄⠀⡇⠤⠀⠀⠒⠆⢸⠀⡤⡄⠀⡄⠀⡇⢦⣀⡴
⢀⣂⠀⢊⠁⣱⠐⠖⢟⡏⠀⣧⠤⠤⠿⠤⠄⡇⠠⣿⡏⡋⠀⡇⠀⡏⢏⡇⠀⡇⣀⣐⣀⡀⣇⠀⠬⠭⠄⠀⢸⠀⡇⣀⠇⡀⠀⡇⢴⠉⠆
⠀⠙⠛⠋⠀⢽⢀⠀⠀⡀⡀⣷⢰⣷⣾⣷⠀⡇⠀⠸⠿⠇⠀⡇⣊⣛⠚⠛⠀⡇⠈⠁⠀⠀⢻⠀⠠⣤⠶⠀⢸⠠⠤⡤⢀⠄⠀⡇⢀⣿⣀
⢰⣶⠶⠿⠉⢸⠈⠀⢄⠀⠁⣿⠀⣿⢸⡇⠀⡇⠀⠐⣶⠀⠀⡇⣸⣏⢳⡀⠀⡟⠿⡛⣻⠇⢸⠀⣉⠀⠀⠀⢸⢀⣀⠉⠉⠉⠁⡷⢿⣇⡩
⠈⠀⠁⣀⠁⢸⠀⣀⠉⠀⠀⢸⠀⠉⠈⣁⠀⡇⢐⡂⠐⠖⠀⡇⠇⠧⢬⡁⠀⡟⢠⠟⠼⠀⢸⠘⣿⠣⣶⠄⢸⢠⣿⠱⠀⢶⠀⡇⠐⠛⠃
⠀⣀⡴⡏⠀⢸⠀⣤⠄⢠⠀⢸⠀⠀⣠⡏⠀⡇⠈⠙⠛⠉⠀⡇⠀⠀⢀⢙⠂⡇⢠⠀⠀⡀⢸⠀⠿⠀⠉⠀⢸⠈⠉⠑⠢⠈⠂⣿⣤⡆⠤
⠀⠉⠀⠉⡀⢸⠀⠸⠄⠈⠂⢸⠀⠈⠛⠅⠀⡇⢺⡗⡇⡟⠛⡗⡉⡉⠉⢿⠀⡇⣈⣉⡉⡍⢸⢠⠠⠠⠀⡀⢸⠀⢀⣾⠃⠀⠀⡟⠁⢃⡄
⠀⢺⢴⠦⡇⢸⠠⡶⠶⠒⠂⢼⠀⡴⠉⠷⡄⡇⠐⠂⡥⡧⠀⡇⠠⣤⣤⡤⠁⡇⢰⢦⢰⠦⣸⠀⠹⠻⠏⠀⢸⠀⠀⠘⢛⠀⠀⡇⣀⣼⠃
⠀⢈⠘⡀⠁⢸⠀⢻⢺⠇⠀⢺⠀⠘⢦⠖⠁⡇⠀⠀⠀⠀⠀⡇⠀⠀⢠⡄⠀⡇⢸⠰⠝⠀⢻⡤⠤⠴⣴⡄⢸⠀⠸⣼⡼⠀⠀⡏⠉⠅
⠀⣀⣀⣉⠀⢸⠀⢘⠘⠀⠀⣾⠀⠀⣈⠁⠀⡇⠀⣉⠉⠽⠀⡇⢠⡴⢟⡀⠀⡇⠀⠐⡂⠀⢸⠀⠆⠀⢝⠂⢸⠀⠀⠘⠃⠀⠀⡇⣠⡿⠄
⠀⢠⣭⣄⠀⢸⠀⣌⡟⣫⡆⣿⠀⣴⡿⠤⠀⡇⠀⠄⠉⠦⠀⡇⠀⠀⠀⠠⠀⡇⠀⠈⠀⠀⣾⠈⠛⠟⠛⠀⢸⠀⠴⠍⠽⠆⠀⡇⠀⢿⠃
⠐⠉⠍⠍⠇⣿⠈⠉⠉⠉⠁⡟⠈⢹⣧⡆⠀⣇⣀⠀⠀⢀⠀⡇⢴⣶⠾⠟⠛⡇⠤⠤⠤⠀⡿⠀⢀⠻⠛⠁⢸⠀⠘⠣⡜⠁⠀⡇⠀⠉
⠀⡀⠀⠀⠂⡟⠤⠀⠀⠤⠤⡇⢀⣀⡉⢁⠀⡏⡋⠡⠤⠄⠀⡇⠀⡈⠂⡀⢰⡇⢶⠂⡶⠀⡇⠀⡠⣄⣠⠀⣼⠀⣀⣀⣁⣀⢐⡇⢠⣭⡄
⠀⠀⢠⠈⠀⡇⠀⠀⠘⠃⠀⡇⢸⡍⢹⣿⠂⡇⠁⠈⠉⠁⢰⡇⠀⣤⣤⡄⢸⡇⠘⠀⠃⠀⡇⠀⣧⡎⠳⡇⡇⠀⠀⢀⣀⣥⢼⡇⣠⣤⠁
⠀⠀⠈⢀⠀⡇⠈⢿⠁⠘⢀⡇⠘⠇⠀⣧⠀⡇⠀⢀⣀⣯⢸⠇⠀⢠⡄⠀⢸⡃⠀⠀⣀⠀⡇⠈⠉⠉⢉⠉⡇⠀⣲⣛⣓⣂⢸⡇⢿⣆⡀
⠈⠉⠉⠉⠁⡇⠀⠿⠏⠀⠀⡇⠀⠂⠀⠁⠀⡇⠘⠉⠻⠆⢸⢀⣀⣰⣇⣀⣸⠃⠀⠲⠀⠀⡇⠀⢀⣠⣾⠁⡇⠀⠁⢡⣄⠀⢸⡇⢈⡛⠃
⠀⣴⣆⠀⠀⡇⠀⠀⠀⠀⠀⡇⢀⠰⠾⠦⠀⡇⠀⣾⠓⠀⢸⠀⢠⠀⡄⢠⢸⠀⠐⠤⠌⠃⡇⢀⣈⠁⣈⠀⡇⠈⣇⠨⠇⠀⢸⠄⢈⡅
⠀⢸⡇⠰⠀⡇⠀⠐⣄⠀⠀⡇⠀⠀⠁⠁⢱⡇⠈⢹⣷⠆⢸⠀⠸⠉⡏⠣⢸⠀⠀⠀⠘⠁⡇⠸⠯⠥⠮⠄⡇⠀⣤⣤⡄⠀⢸⠀⢨⡇
⠀⢤⡥⠄⠀⡇⠀⢀⣤⡀⠀⡇⠀⡉⠩⠽⢸⡇⠰⡆⣴⣦⢸⠀⠀⠀⠀⠀⢸⠀⣁⣈⡁⠀⡇⠀⢤⡬⠆⠀⡇⠐⠃⣀⠒⠀⢸⠔⠶⠶⠆
⠈⠉⣶⡄⠀⡇⢠⣿⡏⣀⢠⡇⠀⠄⠉⠿⢸⡇⠰⡇⠈⡇⢸⠀⠘⠁⠀⡆⢸⡤⠬⠿⣤⠀⡇⢈⣉⣐⣀⠀⡇⠀⠀⢛⠀⠀⢸⠀⢀⡀
⠀⠀⡈⡀⡀⡇⠀⠸⠿⣏⠀⡇⠀⣀⣀⣀⣸⡇⠘⠃⠀⠁⢸⠐⠆⠀⠠⠤⢸⠀⢀⡀⠀⠀⡇⠈⠈⠁⠄⠀⡇⠀⠺⡿⠆⠀⢸⡈⠉⠋⡃
⠨⣉⠹⠏⡇⡇⠀⠀⣤⠀⠀⡇⢢⠀⣠⡄⠈⡇⠉⡍⠉⠉⢸⡆⢰⣶⣶⣄⢸⡇⢨⡅⠸⠆⡇⣶⡶⢾⠟⠃⡇⠀⠀⡇⠀⠀⢸⡷⣿⠆⡇
⠠⠈⠱⠀⠗⡇⠀⢸⠹⡷⠀⡇⠈⠂⠸⠀⠀⡇⣠⣦⣤⣄⢸⡟⠉⠥⠸⠈⢾⡇⢨⡇⠀⠇⡇⠁⠑⠀⠈⠀⡇⠀⠀⡍⠀⠀⢸⡇⢿⠀⡇
⠐⢶⡶⡄⠀⡇⠘⠙⢛⡙⠓⡇⠀⠀⢀⣾⠀⡇⠀⣤⣄⠀⠸⡇⠠⠄⢠⠀⢸⡇⢠⣶⠦⠀⡇⠀⠖⠲⠒⠂⠇⠊⠀⡷⠒⠂⢸⡇⠤⠤⠤
⢰⢻⡇⠘⠂⣇⠀⣠⠚⢣⡀⡇⠐⠚⠛⡍⠀⡇⠀⢻⠛⡀⠀⡇⠰⡖⠸⠀⠀⡇⠾⣷⣠⡄⡇⠀⡘⠂⢸⠀⢸⢀⣤⣥⣴⠶⠀⡇⠢⠤⠉
⠈⠀⠉⠙⢦⣿⠀⠲⠶⠾⠀⡗⠦⠤⠤⠤⠄⡇⠀⠸⠄⠁⠀⡇⠀⠃⠐⠀⢀⡇⠀⠛⠻⠄⡇⠀⠧⠷⠾⠇⢸⠘⠋⠃⠀⠠⠀⡇⠁⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⣮⡙⢧⡙⢷⡘⣿⢹⣷⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠻⢶⣌⠻⣦⠻⣆⢻⡟⢿⣿⣿⣿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣙⠛⢶⣤⡙⢷⡌⢷⣼⣷⣿⣿⣿⣿⣿⣟⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣍⣙⠻⠶⣬⡙⢶⣽⣿⣷⣏⢥⣀⡘⣿⠛⠟⣡⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣬⣉⣙⡛⠶⢮⣝⣿⣿⣿⠋⣠⣤⣤⠉⠀⣞⣉⣥⡴⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣬⣭⣉⣛⡛⠶⣾⣿⡿⣧⠀⠀⠀⠀⠀⠀⠉⢋⣡⣤⠶⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣤⣤⣤⣭⣭⣍⣛⣿⡷⡠⠉⠑⠀⠀⠀⠀⣀⡀⣬⣤⡴⠶⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠘⣓⣶⣶⣶⣾⣿⣿⡧⢠⢶⣀⠀⠀⠀⠙⠿⣳⣶⡾⠯⠑⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⠹⣿⠏⡟⠻⢶⢤⣤⢤⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣤⣉⠟⠀⠀⢀⣿⣿⡼⢣⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣏⡰⠀⡀⢜⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠟⢧⣯⣟⡱⣜⣽⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⣹⣿⣿⣿⣿⣿⡷⣄⡈⠀⠈⠉⠀⡀⣸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⣿⣿⣿⣿⣿⣿⣮⣛⠻⠿⣿⣷⡾⢟⡽⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⡿⠨⠙⠛⠷⠶⠶⠾⠛⡁⢾⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠿⠿⠛⠻⠛⠙⠢⠬⣁⡑⠀⠔⢐⣁⠵⠋⠛⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⣤⠀⣤⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠾⢷⣦⡈⠻⣷⡌⢷⡌⢻⣦⡙⣯⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣙⠻⢷⣦⡙⢿⣦⠈⢿⡄⢿⣀⣿⣭⣄⠀⢽⡟⣤⢎⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣌⡙⠻⣦⣈⠻⣦⠙⣷⡈⢿⡘⣯⠙⢿⣿⣧⢻⡿⣫⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢶⣤⣉⠛⢷⣤⡙⢧⡘⢷⡘⡷⠘⢃⠻⣶⡶⢶⣿⣤⣇⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠷⢦⣬⡙⠻⣦⣌⠻⣮⡻⠊⠁⠀⠀⠀⠀⢠⣦⣴⡿⠈⠩⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢶⣦⣌⡙⠳⣦⣍⣻⢮⡀⠀⠀⣀⡤⠖⠚⠛⠉⠙⢧⣠⣨⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢶⣦⣤⣍⣛⠻⢶⣽⡿⢧⣝⣦⡞⠉⠀⠂⢉⣛⣶⣦⣄⠉⢁⣀⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢶⣶⣦⣬⣍⣛⡛⢷⣟⡻⣶⣿⣿⠀⠀⠶⠾⣟⣻⣿⡛⠀⢠⣿⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⢶⣦⣤⣬⣍⣙⡛⢻⡇⢿⡌⠛⠿⢧⣄⠀⠀⠀⠁⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⣶⣤⣤⣭⣭⣙⣛⣧⣈⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠶⠶⠶⠶⠶⠤⣬⣭⣍⣙⣿⣿⡾⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛⠛⠲⠶⠶⠶⠶⠦⢬⣿⡟⣷⡟⠓⢤⡀⠀⠀⠀⠀⠀⠙⠶⢶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣉⣉⣉⣛⣛⣛⣛⣓⡒⠶⢿⣇⢿⢇⡇⠀⠙⢷⣶⣤⣤⣀⣀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣾⣩⣻⡒⣖⣾⣓⣚⠯⠭⢭⣽⣿⡶⠚⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣛⢽⡲⠭⣝⣒⢺⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢨⡳⢬⣙⡒⠶⢽⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠷⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠁⢧⡙⠶⣭⣙⣛⣿⠀⠐⠀⠀⠀⠀⠀⠙⢿⣿⣿⣟⣛⣛⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠲⣍⡓⠶⢌⡼⢷⡑⣌⠂⡄⡀⠀⠀⠀⠈⡭⢿⣿⣭⣍⣥⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠈⠛⠃⠎⠀⠀⠹⣦⢑⠦⣅⣙⣘⣘⣡⡠⠏⢸⠛⢦⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣌⣉⣉⣁⣀⡴⠃⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠏⣿⢿⣿⣿⣿⣿⣿⣻⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡸⢰⡇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⢸⣿⣿⢸⢸⣻⣿⡿⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⡵⢭⣾⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⢸⣼⣿⠰⣿⣿⣿⣇⢸⢹⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⢀⠀⢤⡤⣞⢠⣌⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⢈⣻⣿⣤⣿⣿⣿⡏⠈⢻⣿⣿⡷⣾⢋⣿⢿⡿⣿⣦⣤⡘⢱⡶⣦⣯⡜⠛⠾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠐⠻⢿⣿⣿⣿⣿⡃⠀⠸⣿⣿⣇⣿⢘⣿⢻⡇⣷⣿⣿⡯⠾⢿⡋⣸⣿⠴⣄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣽⣿⣿⡇⠀⡁⢸⣿⡏⣿⠘⡿⣿⢿⣿⣿⣿⣷⣷⣼⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣾⡇⠀⢀⣬⠛⣿⣿⣿⣇⠀⢙⢿⣿⡷⠿⢦⡷⠹⠿⠿⠛⣻⣿⡿⠿⣿⣿⠿⠏⠉
⠀⠀⡀⣠⣄⠀⢤⠤⠀⠀⠀⢸⣿⣿⣿⣿⣧⡀⠀⢿⣷⣽⣿⣿⣿⡀⠘⠞⠿⣧⣤⣤⣤⡤⠤⡤⡤⢤⣶⣽⣦⡶⠾⡟⠓
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣋⣤⣀⠀⢝⠻⣽⣿⣿⣿⣿⣾⣶⣤⣬⣷⣻⣿⣶⣿⣦⣤⢀⣍⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣬⣶⣏⣽⣻⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡙⣿⡌⠛⢻⣥⡮⢤⣄
⠀⠀⢠⡶⣾⣿⣿⣦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠏⢿⠿⢿⡿⡿⡿⠿⣿⣍⣉⣿⣩⣄⣉⣄⣀⠈⢉⢁⠈⢀⡀
⠀⠀⠋⠙⠛⠛⠛⠛⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠐⠒⠲⠷⠒⠂⠰⠆⠾⠟⠛⠛⠛⠿⠛⠛⠟⠛⠓⠂⠉⠁⠉⠈⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⡠⣿⠉⣀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⠀⢀
⠀⠀⢀⠆⠐⢈⠂⠀⠀⠀⠀⡘⠁⡙⠁⡁⠀⠀⠀⠀⣠⣾⣖⠉⠀⠀⠄⠀⠀⠀⠀⠻⡿⠂
⠀⠀⠀⠍⠐⠊⠄⠀⠀⠀⠀⠀⠀⠑⢤⡅⠀⠀⠀⠀⠻⠌⠉⠒⠀⠀⠀⠀⠀⠀⠁⢡⣤⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠦⠀⠆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠏⢹⡇⠀⠀⢠⠰⠂⡤⠀⣰⣿⢂⡀⠀⡄⠲⢠⠀⠀⠀⣴⠅
⠀⠀⠀⠀⠀⠀⠀⠞⢽⡿⠱⠀⠀⠀⠻⢤⣀⣀⣡⠘⢠⣅⣀⣠⠴⠋⠀⠀⠘⠁⣛⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⡗⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⣍
⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡇⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⠁⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠘⠿⣿⣿⣿⣿⣿⣿⠟⠁⢸⣄⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣽⣿⣿⠭⣤⡀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⣿⣶⣿⣣⣶⣬⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣹⣿⣿⣿⣿⣼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠿⣻⣿⣟⣿⢻⣿⡟⣻⣿⣷⣿⣦⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢄⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣟⣿⣟⣿⣿⣿⣿⣿⣿⡟⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠈⢻⣿⣿⣿⣿⣿⣿⣿⡟⠁⠘⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣷⣤⣀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⢰⣷⣴⣯⣌⣙⠛⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣻⣿⣿⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣿⣦⣠⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢽⣿⢽⡿⣽⡿⣱⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢏⣽⡖⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⣺⣿⣿⣟⠿⣥⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡠⣟⣾⡝⢫⣽⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠟⠾⣋⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣾⣿⣿⡂⠀⠀⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⠙⠛⠛⠒⠒⠀⠀⠀⠘⠛⠛⠉⠉⠙⠛⠓⠒⠒
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⣿⡟⢿⣏⠙⠛⠛⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⢀⣾⢏⠀⣶⡝⢷⣄⠀⠀⠈⠉⠙⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠉⠉⠀⢠⡾⢣⡿⠀⣿⣿⣮⡻⣷⡀⠀⠀⠉⠙⠿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⣰⡟⣱⣿⡇⡆⣿⣿⣿⣿⣮⡻⣦⣠⣤⣤⣴⢷⣼⣷⣤⣄
⠀⠀⠀⠀⠀⠀⢰⣿⠥⠄⠀⢀⣼⢏⣼⣋⠩⢴⡧⡝⢿⣿⠿⠿⣱⡟⠉⠉⠉⡇⢀⡏⠉⠙⠙⢿⣄
⠀⠀⠀⠀⠀⠀⣾⣏⡉⠀⢠⣾⢃⣾⣿⣿⡟⣸⢡⣯⣯⣭⡍⣴⠏⠀⠀⠞⠉⠁⠀⠉⠑⠄⢲⠌⢿⡆
⠀⠀⠀⠀⠀⠀⣿⠋⠀⣰⡿⣡⣿⣍⡉⠩⣅⣿⣈⣛⣛⣛⣼⠏⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠌⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣇⣼⠟⣴⣿⣿⣿⣿⣷⢸⡏⢰⣶⡍⣽⡟⠀⢸⠁⠀⠺⠖⠀⣀⠀⠺⠃⠀⢹⠀⢸⣷
⠀⠀⠀⠀⠀⠀⣹⣿⢋⣼⣿⣿⣿⣿⣿⡇⣾⠇⣿⣿⡰⣿⣇⠀⠘⢷⡀⠀⠀⠀⠿⠀⠀⠀⢠⠞⢀⣤⣿⠆
⠀⠀⠀⠀⠀⣰⡿⢡⣾⣿⣿⣿⣿⣿⡿⢠⣿⢀⣿⣿⡿⣸⡿⠀⠀⠈⣇⣀⠀⠀⠀⠀⢀⣀⠇⠀⢸⣏
⠀⠀⠀⠀⠀⠙⠿⣦⣌⡛⠿⣿⣿⣿⠇⣼⡏⢸⣿⡿⣱⡿⠁⠀⠀⢠⡏⠉⠀⠀⠀⠀⠀⠈⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⡙⠛⢀⣿⠇⣸⡿⣱⣿⠁⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣼⣿⣠⣬⣴⣿⡿⠛⠛⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⣹⣿⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠖⠒⠒⠂⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣖⠚⠻⢯⣉⣽⠟⠓⣲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠧⠤⣤⣍⣙⣶⠖⠛⠳⢶⣋⣩⣤⡤⠼⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠶⠶⠶⢶⣤⠬⣿⢓⢤⠀⣤⠞⢿⠥⣤⡶⠶⠶⠶⠾⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠷⠶⠒⠒⡶⢞⠙⣈⣥⣿⣮⣦⠀⣴⣷⣿⣬⣁⠊⡹⢶⡒⠒⠲⠾⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣋⣤⡴⠶⣶⠿⢍⣴⡾⡿⣸⢌⢶⡇⠉⢸⡷⡅⣯⢿⢿⣦⡁⠻⣶⠶⢦⣤⣙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢋⣠⣤⣶⢏⢡⣾⡓⡗⡖⢼⢢⣿⣇⠀⢸⣿⡵⡧⢲⢺⣚⣿⡜⡹⣶⣤⣄⡉⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣿⠞⠋⣁⣴⡏⢢⡿⡭⢩⠉⡛⢛⠺⣇⡙⠛⠋⣹⠟⡛⢛⠉⡉⢩⢿⡔⣹⣦⣈⠙⠳⣿⣞⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣹⣡⡴⠞⣉⣽⠡⢸⣯⣀⣀⣐⡁⠈⠀⠉⠙⠻⠋⠉⠠⠁⢈⣀⣀⣀⣽⡇⠅⣧⣍⠻⢦⣌⣏⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣴⡿⠋⣠⡾⠋⣽⣈⣼⠟⠛⣛⣛⣛⣛⠛⠻⠿⠿⠿⠟⠛⣛⣛⣛⣛⡛⠻⣿⠀⣿⡙⠳⣄⠙⢿⣦⣣
⠀⠀⠀⠀⠀⠀⢰⡿⣹⣧⡾⠉⣠⠞⣻⠰⣿⠀⠋⠉⠉⠉⠛⠻⠷⠄⠀⠰⠾⠟⡛⠉⠉⠉⠙⠂⣿⠀⣿⠳⣄⠈⢳⣼⣏⢻⡆
⠀⠀⠀⠀⠀⠀⣘⣱⢋⡟⢀⡾⠃⣰⣿⢉⢿⠐⠲⣾⠛⣻⣿⠷⣄⠀⠀⠀⣠⠾⢿⣿⢛⣷⠶⠂⡟⠉⣽⣆⠈⢷⡄⢹⡜⣏⣧
⠀⠀⠀⠀⠀⠀⣿⠋⣸⣿⠏⢀⡼⠃⣿⠐⢸⡇⠀⠀⠉⠛⠛⠋⠙⠂⠀⠐⠋⠙⠛⠛⠉⠁⠀⢠⡇⠂⣿⠈⢷⡀⠹⣿⣇⠘⣿
⠀⠀⠀⠀⠀⢰⡏⢰⠟⣿⣠⠟⢀⡼⢻⡄⠼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢅⡟⢧⡀⠹⣆⣽⠻⡆⢹⡆
⠀⠀⠀⠀⠀⢸⣁⡟⢠⢿⠇⢀⡾⠁⣼⢷⣆⣿⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣠⡾⣧⠈⢷⡀⠘⣿⡄⢹⣀⡇
⠀⠀⠀⠀⠀⢸⣿⠁⡾⢸⣷⡿⠁⣰⠇⣸⠉⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⠉⣇⠸⣆⠈⢿⣶⡇⢱⠈⣿⡇
⠀⠀⠀⠀⠀⠈⡏⣸⠃⢰⢿⣇⣴⡿⠀⣿⠀⢹⣿⡀⠀⠀⠐⠶⠞⠭⠭⠭⠳⡶⠂⠀⠀⢀⣿⡏⠀⣿⠀⢻⣆⣸⡿⡆⠈⡇⢸
⠀⠀⠀⠀⠀⠀⢻⡟⠀⡾⠈⡟⢹⡇⢠⣿⠀⢸⣿⣿⣄⠀⠀⠀⠙⠒⠒⠒⠋⠁⠀⠀⣠⣾⣿⡇⠀⢻⡆⠸⡏⢻⠁⢻⠀⢹⡾
⠀⠀⠀⠀⠀⠀⠘⣇⢰⠇⢰⡇⢸⣷⣿⡇⠀⢸⣿⣿⣿⠷⣤⡀⠀⠀⠀⠀⠀⢀⣠⠞⣿⣿⣿⡇⠀⢸⣿⣶⡟⠸⡇⠘⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣼⠀⣼⠃⣽⢿⣤⣾⣿⣿⣿⠀⠀⠙⠳⠶⠶⠶⠞⠋⠁⠀⣿⣿⣿⣷⣤⡾⣿⠈⣷⠀⣷⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⡏⠀⣿⠀⡇⢸⡏⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢹⡇⢸⡀⣿⠀⢹⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣰⡇⢸⡇⢸⡇⠸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃⢸⡇⢸⡇⢸⣇⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣼⡃⢸⡇⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠈⡇⢸⣇⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⣼⡇⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⣧⣼⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢸⣟⢿⠋⠐⠿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣧⣼⣿⣿⣿⣿⣿⡿⠂⡘⡟⣻⡇
⠀⠀⠀⠀⠀⠀⢿⡜⣏⠄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠙⠻⠷⠶⠶⠶⠶⠟⠋⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠠⣹⢡⡿
⠀⠀⠀⠀⠀⠀⠘⣷⡘⡌⠄⢻⡇⡇⢸⢸⠉⡇⡏⢹⡏⡘⠃⠰⠆⣴⠄⣦⠰⠶⠙⢃⢸⡏⢹⠹⠉⡇⢿⢸⢸⡿⠤⢡⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡘⢆⢻⣿⣿⣿⣾⣷⣷⣿⣾⣷⠆⣥⢂⡐⠠⠀⠄⠂⡐⣬⡰⣾⣷⣾⣾⣶⣷⣿⣿⣿⡟⣰⢃⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡳⣜⠻⢿⣿⣿⣿⣿⣿⡏⠸⠃⣾⢱⡇⣿⢸⡇⢿⠘⠇⢹⣿⣿⣿⣿⣿⡿⠿⣣⠞⣱⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣮⡳⢬⡐⠩⢙⠛⠛⠃⣶⠀⣤⢀⣤⠉⣤⡁⣤⡀⣶⠈⠛⠛⡋⠍⢂⡩⢞⣥⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣭⣓⠢⢤⣁⣂⠆⢠⢀⠀⡉⣀⢉⠀⡈⡄⠀⢐⣈⣤⠴⣚⣭⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⣭⣭⣬⣉⣀⣱⣟⣮⣅⣉⣭⣭⣭⡶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⡶⣶⣦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣉⠛⠻⢿⣶⣭⡻⢿⣶⡌⠳⡙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣶⣄⠙⢿⣿⣦⠻⣿⡀⠀⠀⠉⠀⣠⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⣿⣷⣼⣿⣶⡄⢛⣿⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣽⣿⠋⢘⣈⣱⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡉⠙⠛⠛⠛⢿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⢻⡄⢸⠛⢿⣧⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⠛⠿⠿⠿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠼⣧⢸⡆⠀⣏⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠛⠿⠿⠿⠿⠶⢿⣿⣥⡶⠚⠁⠐⣉⣛⡓⢶⠘⠏⢴⣛⣏⣉⣙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠿⠿⠷⠶⠶⠶⣶⣾⣿⣿⡇⠀⠙⠻⢟⡿⠻⠀⠀⢸⢿⣿⠛⠛⢋⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⢶⣶⣶⣶⣶⣶⣶⣴⡟⠛⢹⣿⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠿⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣶⠶⠿⠿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣤⣤⣤⣭⣍⣉⣉⣩⣿⡏⠈⣻⣧⡀⠀⠀⠀⠀⠠⠔⠒⢰⣾⣧⣤⣶⣶⣶⣶⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⢍⣉⣉⠉⠉⣉⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠈⢸⣿⣿⣤⣤⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡝⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠉⠉⠙⣿⣷⣶⣶⣿⣿⣿⣟⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠻⢿⠏⠀⠀⠀⠀⠀⠀⠀⠙⡟⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠛⠋⠻⡀⠈⢀⡀⠀⠀⠀⠀⠀⠠⢶⠁⠀⠋⠙⠛⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⢐⡀⠀⠁⠄⠀⠀⠀⣀⠔⠁⠀⢀⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢛⣛⣛⣉⣩⣭⡭⠭⠭⠭⠽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣉⣥⠴⠖⠚⠋⠉⢉⣁⣀⣀⡤⠤⠤⠤⠤⠤⢼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡿⠟⣋⡥⠶⠚⠉⣁⣠⠤⠖⠒⣋⣉⣭⣤⣤⣶⣶⣶⣶⣶⣶⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠟⣋⡥⠖⠋⢁⣠⠴⠞⣋⣭⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠛⢁⣠⠴⢛⣩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣋⡴⠚⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣦⣄⣀⡀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⠛⠉⠁⠀⢀⣀⣀⣀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⡀⠀⠀⣿⣿⣿⣏⣠⣴⠾⠛⠋⠉⠉⠉⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠛⠉⠀⢀⣤⡴⠶⠶⣤⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⠛⠙⣧⡻⣄⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⣀⣀⡈⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠋⠀⣠⠾⠋⢹⣾⡇⠀⠀⠀⠀⠀⠀⠙⠻⣥⣸⣿⣿⠆⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢀⣴⠟⠁⠀⠀⠸⣇⣳⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠈⠉⠉⠉⠛⠓⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣭⣭⣛⠻⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠙⠛⠿⣷⣬⡻⢿⣶⣄⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡙⢿⣷⣄⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡙⠿⣷⣤⣤⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣬⣉⣙⣛⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉

⠀⡤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿
⠀⠹⡿⣝⡷⢤⡀⠀⠠⣤⣄⠀⢠⣤⡄⠀⢀⣠⠾⢏⢷⠟⠀⣿⣿⣿⣏⠿⢿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠿⣟⡽⣿⣿⣿
⠀⠀⠙⣿⣮⡢⡙⢶⣾⣿⣿⣷⣿⣿⣷⣴⢛⡥⣨⣽⠟⠀⠀⣿⣿⣿⣿⣧⡤⣝⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⡡⣮⣾⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣶⣿⣿⣿⣿⠍⣿⣿⣿⣿⣷⣿⣿⠟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣯⣭⣙⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠈⠀⠀⢀⡀⡄⠰⣄⠀⠀⠀⠀⠀⡀⠘⠋⠀⠈⡁⠚⠒⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠃⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠂⠄⠀⢄⠀⣤⣀
⢠⠊⠈⢃⠀⠀⢰⣼⣿⠀⢠⠀⠀⠀⠀⠉⠀⡄⠀⣇⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣴⠃⠂⠀⠃⠀⠄
⠀⠄⢄⠈⠀⠀⠈⣒⡟⠀⢸⡇⠀⠀⢀⠀⠀⠃⠀⠇⠁⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠠⣪⢰⣀⡀⠀⣤⣥⡀⡈⡇⠀⠠⠄⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⠃
⠐⠉⠉⠉⢁⣔⣀⢀⣀⣶⡰⡃⠀⠀⠀⠀⠐⠤⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠦⠦⠀⠰
⢸⡀⠀⠢⡀⠈⠹⡹⠿⢋⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⣌⡶⠀⠀⠀⠀⠀⠈⠈⠉
⠈⠧⡄⠀⠈⠀⠂⠀⡉⠀⢾⠀⠀⠀⢠⠃⠀⠀⠀⡰
⠀⠠⡁⠀⡀⠀⣎⣰⡝⠠⠚⠀⠀⠀⠈⠀⢀⡠⠈⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢀⠀⠀⠀⠁⠁
⠀⢰⡀⠀⠰⠄⠙⠉⢁⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠠⡄
⠀⠈⠓⠸⠠⠤⠤⠤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⡀
⠀⠀⠉⠐⠖⠀⠶⠴⡂⠚⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢆
⠀⠀⠈⠀⠇⠄⡄⣼⠃⠨⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠄⠄⠀⠂⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⡀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠃⡇⠀⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠈⠈
⠀⠀⠀⢀⡃⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⢈⠱⠠⠲⠂⠉⠽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠩⢤⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣬⣤⠁
⠀⠀⠀⠣⢇⡄⠀⠀⠁⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠰⠰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣬⡤⠂⠀⠀⠀⠠
⠀⠀⠀⠀⠈⡃⠠⠤⠤⠆⠊⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⠠⠀⠀⢌⡑⢤⡄⠀⠒
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠒⠂⠀⠀⠀⠑⠆⣷⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠛⠃⠀⡇⠛⠁⠀⡁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠄⠀⠀⠀⠀⠀⠈⣯⡇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⡀⠀⠀⠒⠚⠀⠀⢀⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢄⠡⠀⠈⠀⠀⠀⠘⠁⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠙⠁⢀⠈⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⡠⠋⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠂⠸⠀⠀⠀⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠐⠠⠤⠤⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢭⢈⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⣿⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡄⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠀⠀⠀⠀⠠⠀⠈⠙⠂⠀⠀⠁⢀⡀⠄⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢶⡞⠛⢻⡟⢻⡟⠛⢳⡶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡟⠉⢹⡄⢷⣠⣞⠓⠚⣳⣄⡾⢠⡏⠉⢻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⣆⠹⡄⠀⢷⢸⡇⠙⡦⢴⠃⢸⠇⡞⠀⢠⠏⣰⠛⢷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⢻⡙⢦⠀⠘⢆⠹⡄⠘⡆⢻⣤⣗⣺⣤⡟⢸⠃⢠⠏⡴⠁⠀⡴⢋⡟⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠹⣎⢧⠀⠘⢆⢻⣤⣿⣼⣿⡧⢼⣿⣧⣿⣤⡟⡸⠁⢀⡼⣱⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣙⢻⡆⣘⣦⠷⠖⠛⢿⣯⣉⣉⣿⣆⣸⣿⣉⣉⣽⡿⠛⠲⠾⣴⣃⣰⠟⣋⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠙⢿⠉⠀⠀⢀⣀⣨⣭⣯⣭⣤⣤⣤⣬⣭⣽⣭⣅⣀⡀⠀⠀⠉⡿⠋⠁⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣉⡙⠒⢾⣧⡴⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢦⣼⡗⠚⣋⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⡉⠉⠙⠛⢿⡇⠠⠖⠛⠛⠻⠶⣶⣄⠀⠀⣠⣶⠶⠟⠛⠛⠲⠄⢸⡿⠛⠋⠉⢉⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣧⣬⣉⣉⡛⣶⠾⣷⠠⣤⡶⠶⣶⣶⣄⠀⠀⠀⠀⣠⣶⣶⠶⢦⣤⠄⣾⠷⣶⢛⣉⣉⣥⣼⣧
⠀⠀⠀⠀⠀⠀⣸⣏⣀⠀⠀⠉⠉⢳⠚⢿⠀⠀⠙⠺⠿⠷⠚⠧⠀⠀⠼⠓⠾⠿⠗⠋⠀⠀⡿⠃⡟⠉⠉⠀⠀⣀⣹⣇
⠀⠀⠀⠀⠀⢠⣿⣭⣉⣉⣙⠓⠒⠾⣧⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣼⠷⠒⠚⣋⣉⣉⣭⣿⡄
⠀⠀⠀⠀⢀⣿⣇⣀⣀⠈⠉⠉⠛⠛⠻⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠟⠛⠛⠉⠉⠁⣀⣀⣸⣷⡀
⠀⠀⠀⠀⣼⣏⣀⣈⠉⠉⠛⠒⠶⠦⢤⣬⣹⡄⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⢰⣏⣥⡤⠴⠶⠒⠛⠉⠉⣁⣀⣹⣧
⠀⠀⠀⢰⣿⠀⠉⠉⠉⠛⠛⠲⠶⠦⣴⠿⣏⣷⠀⠀⠀⠰⠶⠮⠭⠭⠵⠶⠆⠀⠀⠀⣾⣹⠿⣦⠶⠶⠖⠛⠛⠉⠉⠉⠈⣿⡆
⠀⠀⠀⣿⡏⠛⠛⠲⠶⠶⢤⣤⣄⣴⢏⣆⢿⡉⢷⡄⠀⠀⠈⠛⠒⠒⠛⠁⠀⠀⣠⡾⢉⡟⣴⡹⣦⣠⣤⡤⠶⠶⠖⠛⠛⢹⣷
⠀⠀⢰⣿⠛⠛⠒⠶⠶⠦⣤⣤⣼⣏⣛⡿⢘⣿⢻⡟⠳⣄⡀⠀⠀⠀⠀⢀⣠⠞⢻⡟⣿⡃⢿⣓⣹⣧⣤⡤⠴⠶⠶⠒⠛⠛⣿⡆
⠀⠀⢼⣿⣶⣤⣤⣤⣄⣀⣀⣀⣏⢩⡉⣛⠛⡟⢻⡇⠀⠈⠙⠓⠲⠖⠚⠋⠁⠀⢸⡟⢻⠛⣟⢙⡍⣹⣀⣀⣀⣠⣤⣤⡤⣴⣿⡧
⠀⠀⠀⠉⠻⣷⣤⣤⣀⣈⣉⣙⠛⠻⠛⠿⠿⠷⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠾⠿⠿⠛⠛⠛⣋⣉⣁⣀⣤⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠻⣧⡉⠉⠉⠉⠉⠛⠛⠛⠒⠒⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠒⠛⠛⠛⠉⠉⠉⠉⢉⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠙⣿⡓⠒⠒⠶⠶⠦⠤⠤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠤⠤⠴⠶⠶⠒⠒⢚⣿⠋
⠀⠀⠀⠀⠀⠀⢠⣶⡾⣿⠓⠒⠒⠶⠶⠶⠶⠶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠶⠶⠶⠶⠒⠒⠚⣿⢷⣶⡄
⠀⠀⠀⠀⠀⠀⠘⣿⡤⢹⣷⠲⠶⠶⠶⠦⠤⢴⡟⠻⢶⣤⣀⣀⣀⣀⣀⣠⣤⡶⠟⣿⡦⠤⠶⠶⠶⠶⠶⣾⡏⢤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡥⢿⡟⠓⠒⠒⠒⠒⢾⣿⠗⣤⡄⣉⠉⠙⠋⠉⣉⢠⣄⠺⣿⡶⠲⠒⠒⠒⠒⢻⡿⢬⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡹⢿⣶⠶⠶⠶⠶⢾⡏⣐⠠⢀⡉⡘⡃⢘⢃⢉⡀⠄⣂⢻⡷⠶⠶⠶⠶⣶⡿⢏⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣎⠻⣿⣟⡛⠛⢻⣿⠟⣼⢃⡖⣤⢡⡌⣦⢶⡘⣧⠻⣿⡟⠛⢛⣻⣿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣛⠿⣷⣾⡷⢠⣄⢈⡁⠛⠘⠃⠛⢈⡁⣠⡄⢾⣷⣾⠿⣋⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣭⣐⡨⢁⠘⡃⡚⠃⠘⢛⢘⠃⠈⢅⣂⣭⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠶⣶⣷⣾⣶⠶⠶⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣞⢻⠉⢙⣏⣹⡋⠉⡟⣷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⢳⡀⠸⡌⣷⠻⡄⢠⠟⣾⢰⠇⢀⡞⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⢻⡁⠘⢆⢳⡀⢳⢹⣄⡯⢽⣠⡇⡞⢀⡞⡰⠃⢈⡟⡶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠈⢳⡱⡄⠘⢆⢣⣨⣆⣿⣯⣽⣿⣰⣅⡞⡰⠁⢠⢎⠞⠁⠀⣻
⠀⠀⠀⠀⠀⠀⠀⢸⡯⣝⣻⣄⣱⠽⠖⠛⢿⣯⣭⣽⣧⣼⣯⣭⣹⡿⠛⠲⠧⣎⣠⣟⣩⢽⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣥⣀⡉⣿⠀⣀⣠⣤⠴⠿⠷⠶⠶⠶⠶⠾⠿⠦⣤⣄⡀⠀⣿⢉⣀⣼⣇
⠀⠀⠀⠀⠀⠀⢀⡿⠶⠤⢭⣹⣾⠋⣁⣤⣤⣄⣀⡀⠀⠀⢀⣀⣠⣤⣤⣈⠙⣷⣏⡭⠤⠶⢿⡀
⠀⠀⠀⠀⠀⠀⣼⣓⠒⠶⠤⣤⣿⡀⣁⣠⣤⣬⣍⠛⠀⠀⠛⣩⣥⣤⣄⣈⢠⣿⣤⠤⠶⠒⣚⣧
⠀⠀⠀⠀⠀⣰⡏⠉⠉⠙⠓⣇⢾⡇⠈⠛⠾⠿⠭⠧⠀⠀⠾⠭⠿⠧⠛⠁⢸⠷⣼⠚⠋⠉⠉⢹⡆
⠀⠀⠀⠀⢠⡿⣭⣉⣙⣓⠒⠾⡄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠷⢒⣚⣋⣩⣭⢿⡄
⠀⠀⠀⢀⣾⠧⣤⣄⣀⡉⠉⠙⠳⣼⡇⠀⠀⠀⠀⡀⡀⢀⠀⠀⠀⠀⠀⢸⣧⠞⠋⠉⢉⣁⣠⣤⠼⣷⡀
⠀⠀⠀⣼⠷⠦⠤⣤⣈⣉⣉⠛⣲⣾⣧⠀⠀⠀⣀⣠⠄⠠⣄⣀⠀⠀⠀⣼⣶⡖⢛⣉⣉⣁⣤⠤⠴⠾⣇
⠀⠀⢠⣿⠤⣤⣤⣀⣀⡈⠉⣹⢏⣹⡿⣆⠀⠀⠙⠭⣭⣭⠭⠉⠀⠀⣰⢿⣏⡹⣏⠉⢁⣀⣀⣤⣤⠴⣿⡄
⠀⠀⣼⡷⠶⠤⢤⣤⣍⣉⣹⡏⠿⠏⣷⢾⡷⣄⡀⠀⠀⠀⠀⢀⣠⢾⡷⣾⢻⠷⢹⣏⣉⣩⣤⠤⠤⠶⢾⣧
⠀⠀⣿⣦⡤⣤⣤⣀⣀⣈⣟⣻⢛⡛⡿⢾⡇⠀⠙⠓⠶⠶⠚⠋⠀⢸⡗⢿⢛⡛⣟⣻⣀⣀⣀⣤⣤⢤⣴⣿
⠀⠀⠀⠙⢷⣦⣤⣤⣬⣉⣉⣛⣛⠛⠛⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⢛⣛⣛⣉⣉⣥⣤⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠙⣷⣀⣀⣀⡈⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠉⠉⠉⠉⣀⣀⣀⣠⣾⠋
⠀⠀⠀⠀⠀⢀⣈⣿⣤⣭⣉⣉⣉⣉⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣉⣉⣉⣉⣭⣤⣿⣁⡀
⠀⠀⠀⠀⠀⠸⣯⡙⣧⢤⣤⣤⣀⣀⣉⡿⢶⣄⣀⠀⠀⠀⠀⣀⣠⡶⣿⣉⣀⣠⣤⣤⡤⣼⢋⣹⠇
⠀⠀⠀⠀⠀⠀⢻⣎⢽⡶⠶⠶⠶⠶⠶⣿⠆⣬⢉⠛⠛⠛⠛⡉⣥⠰⣿⠶⠶⠶⠶⠶⣶⡯⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣮⢿⣖⠒⠒⠒⢲⣏⣐⢀⠌⠀⡃⢙⠀⠅⡀⣂⣻⡖⠒⠒⡒⣲⡿⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠿⣭⣉⢹⣿⠃⠟⣼⢰⢲⡖⡆⣧⠻⠘⣿⣏⣉⣭⠿⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣮⣙⡻⢛⠰⠆⡦⢠⡌⢡⡅⢴⠰⠆⡛⢟⣫⣵⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣦⣤⣤⣤⣤⣤⣤⣴⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⡀⣤⠔⡶⠒⢐⠲⣦⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢽⡏⢰⠃⠀⣶⣽⠃⠀⢄⡀⢀⠄⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣮⡿⠀⡞⠀⣸⣻⡧⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣠⠇⠸⠁⠸⠉⢍⡀⣀⣊⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡛⢹⡟⠤⣾⠠⢼⣧⡌⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣽⣿⡃⢠⣿⡚⢿⣿⡇⠀⠀⠀⠤⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⣼⣿⣿⣿⣿⡂⢙⣫⡷⣏⣽⡇⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡱⣈⡅⡅⢸⣿⡇⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡻⡙⢏⢻⣜⡀⡀⣼⣿⣧⡞⡡⠎⡔⠰⢲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⢂⢳⡱⠌⠣⣿⣿⣿⣿⣿⡷⢚⣡⢞⡰⡠⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠨⡢⡙⠪⡑⢾⣿⣿⣿⣿⣗⡉⠔⣋⠚⡅⠠⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡇⠐⠄⠀⠁⠨⡓⠌⣸⣿⣿⡿⡿⠇⢲⡈⠖⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠠⠬⠁⠉⠅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢂⣬⣄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠜⢣⠓⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣋⣿⠟⣽⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠻⠿⠃⣿⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⡀⢼⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠙⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣿⣿⣿⣿⣷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⣿⣿⣿⣿⣿⣿⠟⣬⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⣾⣿⣿⢿⣇⠹⣦⣿⣇⢛⣿⣦⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣾⡟⣿⣿⣧⠫⠼⠉⠛⠉⠙⠏⢙⠛⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⣿⡟⠀⠰⡍⠛⠀⡄⢨⣈⢩⡄⠒⠈⠁⡙⣧⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣉⣛⡄⠤⠤⠾⠄⢲⢄⣰⣿⣿⣷⡆⣘⠂⠈⠉⠻⠷⣶⣀⡀⠀⠐⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⡇⡴⠦⠶⠖⠻⡟⠋⣅⠀⠀⠌⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠂⠙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠈⣀⠉⡀⠿⠠⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠿⠈⠓⠀⣦⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⣍⢤⡳⢟⠈⠃⠐⣤⣆⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢰⠘⠟⠀⠀⠀⢨⢈⠿⢢⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠰⢧⡜⢃⡈⣿⣿⣾⠇⠲⣀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⡾⠀⠈⢠⠈⠀⠁⠀⠹⢀⣠⢈⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⢗⠀⢃⡀⢰⢆⣤⣤⣤⢉⠉⢻⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡆⠺⠃⠘⠂⢀⢾⠿⠟⠿⠿⢰⡿⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠉⣀⠇⠸⠀⡀⠉⢴⠀⠀⢰⣷⡾⣁⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢀⡃⢀⣶⠀⣁⠰⣾⣶⣶⣾⡟⢑⡁⢠⡄⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢶⠈⣷⠀⢁⡀⢁⡀⠿⠛⠟⢿⣯⠢⡛⢠⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣾⠀⠁⡦⠈⠃⣈⣱⡀⠀⡀⢸⡅⣄⢿⣎⢡⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⡘⠃⣿⣿⣿⣿⣿⣿⡟⠛⡁⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⡆⠘⢣⣾⠻⢟⠛⣿⣿⣿⣁⣶⠁⣄⣷⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⢀⠛⠄⠀⠀⠀⠀⠀⡄⠀⠀⣉⣀⣠⡀⠤⠀⠀⠀⠀⣄⡎⢋⡈⣴⡄⡀⠀⠀⠀⠀⠀⠔⢈⣤⣤⠤⣄⠢⡀
⠀⠀⠀⠀⢠⠃⠀⠈⢆⠀⠀⠀⠈⠁⠁⠀⠀⢓⡚⠛⠀⠀⢀⣜⣗⠈⣀⠀⠀⣀⠐⠿⠄⠀⠀⠀⣄⣼⡟⠉⠀⢉⠪⣷⡀
⠀⠀⠀⣠⠃⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⣖⣲⣶⠀⠁⠀⠀⠄⢠⣯⣬⡅⠀⠀⡆⣿⡀⠄⣛⡃⢃⣽⠇
⠀⠀⣰⣇⠂⠀⠀⠀⠀⠈⢧⠀⣐⢒⠒⠐⠀⡀⠀⠂⠀⠀⣒⣒⣲⣧⡀⠠⠀⡠⠭⡭⢭⠃⠀⠀⠀⡙⢿⣦⣤⣔⡽⠋⠄
⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠀⠓⠒⠲⡖⠒⠀⠀⠀⠀⠀⠀⠚⢻⣥⡇⠀⠀⣿⣭⠝⠋⠀⠀⠀⠀⠈⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠉⠐⠊⠁⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⣶⠆⠀⢸⠠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣻⣆
⠀⠀⣼⣿⢦⠀⠘⣦⠏⠀⢠⣠⠀⣄⣠⣀⣄⡄⠀⢀⡖⢛⡓⢦⠀⠠⠤⠤⣔⣯⣭⣭⣽⣀⠄⠀⠀⠀⣰⠿⠶⠶⠿⣆
⠀⠀⡷⣿⢸⠀⠈⢹⠁⠀⠘⣉⠛⣙⣃⣙⣘⠃⠀⠘⣆⠙⢋⡼⠁⠀⠈⠛⠿⢼⡿⠿⠶⣛⡳⠀⠀⡰⠯⠭⠭⣭⡭⠭⣆
⠀⠀⣷⢇⢸⠀⠀⢸⠀⠀⢀⣲⣦⡭⢭⣭⣭⠀⠀⠀⠀⠉⠉⠀⠀⠀⠰⣗⣠⠞⠈⠉⠉⠀⠀⠀⣰⣛⣛⣛⣒⣒⣒⣒⣚⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣖⣲⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠻⣦⣠⠟⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠿⣿⣿⣿⣷⡄⠧⢸⢠⣾⣿⣿⣿⠿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣼⣅⣂⣲⢸⢹⣏⢹⡏⣇⣲⣸⣢⣧⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣼⠁⠚⣛⡿⠿⠟⠛⠿⢟⣛⠓⠀⣇⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢽⡍⠻⠽⠿⠷⠀⠀⠾⠿⠯⠟⢁⡿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢳⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣱⡜⣇⠀⠀⢚⣛⣛⡓⠀⠀⣰⢣⢎⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣓⣓⣛⣹⡷⢄⡀⠀⠀⢀⡠⢾⣏⣛⣚⣊⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣽⣽⣽⡇⠀⠉⠙⠛⠉⠀⢸⣧⣭⣭⣥⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡛⣿⣿⣿⣿⣿⣿⣿⣿⣗⠦⠤⠤⠤⠤⠴⣚⣿⣿⣿⣿⣿⣿⣿⣿⢏⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡜⢾⡭⠻⢿⣿⣿⣿⠃⠏⠭⢭⣭⢭⠹⠘⣿⣿⣿⡿⠟⢭⡷⢣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡛⢵⡢⢈⠚⠴⡴⣤⣀⣀⣀⣀⣤⢦⠦⠓⡁⢔⡮⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢭⣓⠮⣶⣄⣅⡅⡍⢨⢨⣨⣰⣴⠽⣚⡭⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠯⠭⠭⠭⠭⠭⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⠀⠀⣀⡀⣠⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣯⣵⣄⠊⠉⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⡠⠀⠉⢷⡤⢆⣤⠀⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡯⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣷⣷⣶⢟⣁⣤⣴⡄⠀⠀⠨⣿⡧⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣥⠆⠀⣼⣿⣏⣸⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⣿⣿⣿⣤⠴⠄⠀⠀⠶⣷⡏⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡼⠗⠀⢸⣿⡏⠉⠁⠀⠈⠣⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣷⡄⠀⠀⠐⠷⣦⡙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⣶⠶⠀⢀⢾⣿⣅⠀⠀⣤⣶⣆⡁⠚⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣏⣿⣿⣷⣌⠢⡀⠈⠷⢶⡃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣙⠏⠀⠀⢡⣾⣿⣿⡆⠀⠐⢿⡟⠹⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠻⣌⠂⠀⠙⣛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠍⢀⠊⣰⠟⢉⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣦⠘⢷⡀⠄⠩⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⠏⣰⡿⠁⣅⣠⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡉⢿⣿⣿⣿⣿⣿⣿⡿⠃⠹⣷⡀⢻⡌⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣸⠋⣴⡟⠀⠀⢋⣤⣄⡀⠀⠀⠀⠀⠁⡸⢿⣿⣷⣿⣿⣾⣿⣿⢿⣿⣿⣿⣇⠀⠀⠘⣿⡄⢳⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢠⠇⣰⡿⠀⠀⠀⠸⣯⡍⠉⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠳⠀⠀⠸⣿⡀⠇⠐⢠⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣷⡇⢸⠀⣿⠇⠀⠀⠀⠀⢫⡀⠀⢀⣠⣤⣶⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢿⣇⢸⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣼⣿⣿⠏⠉⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠷⣄⠀⢸⣿⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⡇⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⡀⢸⣿⠀⢀⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣷⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⡀⡇⢸⣿⠀⣸⣿⣿⡶⠃
⠀⠀⠀⠀⠀⠀⠱⣾⣿⣿⡇⡀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⡟⠇⠀⠀⣀⡴⣯⠿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣿⡇⣼⡏⡄⣿⣿⣯⡶⠁
⠀⠀⠀⠀⠀⠀⠐⢮⣿⣿⣷⢁⠸⣿⡀⠀⠀⠀⠀⢠⣶⣿⣿⣿⠟⠀⣀⡤⠞⢽⣥⡴⠾⠛⠉⠉⠙⠛⢻⣿⣿⣿⣿⢻⣿⠡⢀⣿⣿⣯⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⠂⠹⣷⡀⠀⠀⣀⣾⣿⡿⢛⠳⠾⣟⣉⣶⠷⠞⠉⠀⡀⠠⠐⠀⢀⣉⣻⣿⣿⣿⣧⣿⠁⢂⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣻⣿⣿⣶⣝⣷⣴⣾⡿⢋⣀⣠⣼⡷⠶⠛⠉⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣾⣿⣿⣿⣿⣶⣿⣿⣋⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⣿⡟⠿⡶⠿⠛⠉⠉⠀⣀⣤⠔⣂⡅⠀⠀⠀⠐⣀⡭⠲⠊⢉⣉⣶⣶⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢾⣏⢙⣂⠁⠀⣀⡠⠴⠚⡩⠔⠊⣁⢤⣖⠾⠝⣋⣡⣴⡶⠿⠛⠛⠋⢹⢻⣿⣿⣷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣾⡿⣾⣭⡶⣋⣀⣀⠀⣀⣤⣴⠶⠍⢃⣠⣴⢾⣿⣿⡏⠃⠀⠀⢀⡤⣦⣼⣿⣿⣶⡞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣤⠞⣥⢞⣭⣶⣯⣬⣻⢿⡟⣦⡉⢀⣴⠞⠿⠉⡇⠀⢿⣿⣧⡴⢞⣛⣯⣼⣯⣶⣬⣛⢮⡛⣦⣀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡛⢡⢾⣵⣿⡷⢶⣮⡝⣿⢷⣯⣔⢽⣿⡁⢐⠀⠀⠀⠀⢈⣽⢫⣶⣿⠟⣻⢟⣽⣲⠶⣿⣷⣹⡆⢹⣇⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣿⣿⢿⣾⡿⠿⢃⣼⢃⣽⣟⣦⡙⢿⣷⣄⣀⣤⣾⢟⣱⠟⢱⣿⣄⣿⡘⠿⢟⣻⣮⣿⣿⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣴⠾⣟⢛⣻⣷⣻⣿⣿⣵⣿⣹⣿⣭⣿⠷⠶⠾⠿⠷⠾⠿⠷⢶⣬⣿⣿⣾⣿⣿⣿⣱⢿⣟⣛⣧⣶⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢯⡻⣵⣿⣿⡿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠷⣿⣿⣧⣟⣫⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠖⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⡐⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⣏⠀⢀⣠⣴⣶⣾⣿⡿⠿⠟⠛⠋⢉⣉⣉⣉⣀⣀⣀⣀⣉⣉⣉⡉⠙⠛⠻⠿⣿⣿⣶⣶⣤⡀⠀⢈⣿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⣿⣿⣿⣿⣿⡧⠶⠶⠛⠛⠓⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠘⠛⠛⠒⠶⠦⣿⣿⣿⣿⣿⣷⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠷⠶⢻⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡿⠿⠶⠾⢿⣿⣷⣀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣟⣭⣶⡄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⢣⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠀⢸⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣹⠿⣿⢿⣿⣿⣿⢿⣶⢲⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⢿⣯⢣⣯⣼⣿⣧⣾⡴⣿⡿⠟⠋⢻⠈⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣯⡼⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣽⣶⡿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣿⣿⣿⣁⡀⠀⢀⣀⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⠿⠿⠇⠿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣏⣹⣇⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⣽⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢈⣿⣿⡿⠿⠿⠿⢿⠿⠿⠿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣼⣿⣿⡟⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⣿⣿⡟⠁⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣸⡟⠀⣾⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢰⣿⠃⢸⣿⣿⡟⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠠⢷
⠀⠀⠀⠀⠀⣿⡏⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⡗⠒⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠐⠉⠈⠙
⠀⠀⠀⠀⣿⠇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀
⠀⠀⠀⠀⠛⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠿⢷⣦⣄⣀⣀⡀⠀⠀⢿⠿⠿⠿⠿⣦⣄⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡄⠰⠀⢦⠀⠀⠀⠀⠀⡸⠁⠀⢉⡁⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠃
⠀⡠⢠⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢠⡃⢠⠀⡈⡧⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⠄⠸⠦⠀⠀⠀⠀⠀⠀⠄
⠘⠀⠀⠇⠀⠈⣛⣿⠀⢰⠀⠀⠀⠀⠀⢀⠀⣳⡆⠀⠐⠀⠀⠀⠀⠀⠀⠈⠄⠀⠐⠏⠈⠄⠀⠀⠃
⠀⢉⡶⡀⠀⠀⡒⠛⠀⢸⡄⠀⠀⠂⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⡄⠀⠀⠀⠀⠀⠀⠃
⠠⠔⠓⠚⢂⠤⠙⠋⣂⢨⠀⠀⠈⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣠⡀⢀
⢰⠀⠀⡀⠈⢻⡖⣿⠟⣡⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡐⡄⠀⠀⠀⠒⠀⠒⠒
⠘⢦⠀⠈⠑⠠⠀⠈⠈⣽⠀⠀⠀⡐⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠬⠁⢀⠀⢰⣡⣼⠀⠚⠀⠀⠀⠁⠀⣀⠔⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⢰⠀⠀⠇⠈⠉⠁⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠄⠠
⠀⠈⡁⡓⠒⠒⠒⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠁⢉⠁⠘⢊⡇⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⢀⣨⠐⢻⢸⠁⠘⠀⠀⠀⠀⠂⠀⠒⠂⠀⠈⠁⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠂⠀⠐⠀⠐⠀⠀⠀⠀⠠⠈⠀⠀⠠⠡
⠀⠀⠀⢸⠀⠘⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠶⡄⣠⡬⠳⢰⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⠶⢒⡉⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⣁⠄
⠀⢀⠰⣷⢂⠀⠀⠂⠁⠁⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⣅⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣹⣉⠄
⠀⠀⠀⠀⠙⠀⠠⠤⠤⠔⠀⠐⠀⠀⠀⠀⠂⠀⠀⠈⡉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⢤⠉⠡⢐⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⠒⠀⠀⠀⠈⢢⣴⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠚⠓⠀⠖⠛⠁⢈⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠱⢄⠀⠀⡀⠀⠀⢿⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢁⠀⠀⠓⢚⡀⠀⣸⠇⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠈⠁⣠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⢉⠁⠠⠁⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡄⠀⠀⠀⠐⢅⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠋⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣻⠑⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢘⠄⡀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠉⡀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠔⠐⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢰⣦⣄⠀⠀⠀⠰⠀⠀⠀⠀⣤⠀⡆⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢰⠀⠀⠀⢰⣿⡄⡇⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⡶⠶⠶⠶⠶⢶⠶⠲⠒⠲⠾⣿⣿⠒⣶⠀⠈⠀⠀⠐⣿⣿⡿⠿⢿⣿⠿⠿⠻⠿⠛⠛⠻⢻
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⣿⣿⠀⣹⡀⢀⣀⣀⣠⣉⣿⡏⠛⠙⣿⠁⠀⠘⠛⠁⠉⠀⠈⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣿⣿⠀⢉⢉⣻⣿⣿⠹⣯⣿⡇⢠⠀⣿⠀⠀⣀⡄⠀⠀⠀⠀⢁
⡄⣤⣴⣶⡆⠀⠀⠀⠀⢄⣸⣿⣄⠄⢠⢀⣿⣿⢤⣿⣾⣨⣿⡿⣾⣿⣿⣇⢼⢠⣿⡆⣦⣯⣷⠂⡂⠀⠀⠘⣶⡶⢶
⡀⣀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⡟⠀⠈⠸⣿⣿⡟⣿⣿⣿⣿⣿⣿⡿⣿⡿⣾⠚⣿⡇⢹⣿⣿⡃⠃⠀⠀⠀⣿⣿⢴
⣿⠿⠿⠿⠃⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⣻⣻⡆⣿⣿⣿⣿⣿⢿⣿⣿⣧⣿⣸⣿⣇⣸⣿⣿⣇⠀⠀⠀⢀⣻⢿⣄
⣠⣤⣷⣶⣿⠂⠂⠀⠀⠸⠛⡿⠟⠛⠛⢻⣿⣿⣧⣿⡿⠿⠟⠋⠿⢵⣿⣿⣿⣿⠿⠛⠛⠫⡿⠟⠉⠛⠉⢉⡛⣻⣿
⣿⣿⣿⣿⣿⣇⣠⣤⣤⢤⠀⢄⡀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠈⢛⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠃⢄⠈⣢⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⣙⡲⣀⡀⠀⠀⠀⠈
⣿⡿⣿⠟⠡⡅⠙⢠⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠍⠻⢤⡤⢀⠉
⣉⣁⠙⠀⣀⠀⣷⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠖⠒⠐⢉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢹⡏⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠈⡇⢠⠿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠸⡇⠀⡇⢸⠀⣽⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣯⢹⡄⣷⢠⡇⣾⢠⡇⣸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢿⡘⡆⢧⢹⢸⡇⡇⡼⢰⢇⡼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⢮⢳⣽⣼⣾⣾⣿⣧⣷⣯⣞⡴⢋⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣽⣶⠿⢛⣛⣉⣭⣍⣛⣛⠻⢿⣶⣯⢞⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⣩⣴⡾⣿⡿⢛⣛⣛⠻⢿⣿⣶⣬⡙⠻⣷⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⢋⣵⣾⠟⠉⢸⡏⣰⣿⣯⣉⣷⡌⢿⡄⠙⠻⣷⣌⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⣷⣌⠙⢿⣦⡀⢸⣇⠻⣿⣿⣿⣿⠇⣾⠃⣀⣴⡿⢋⣴⣟⣻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⣋⣩⠽⢷⣦⡙⠻⣶⣿⣦⣈⣛⣛⣥⣾⣿⣾⠟⣋⣴⡿⠯⣭⣍⣙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣋⣥⠴⢚⣩⠿⣻⣷⣦⣭⣙⣛⣛⣛⣛⣋⣭⣶⡿⣿⡳⣭⣛⠲⢤⣍⡙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠋⣉⡤⠞⢋⡴⢋⡵⢫⢟⣿⣿⣿⣿⣿⣿⣿⡿⡽⣟⢦⡙⠦⣍⠓⢦⣌⡉⠛⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣣⠴⠚⢁⣠⠞⢉⡴⠋⡴⢋⡞⡼⢹⢿⣿⡏⡏⣯⢷⡹⣌⢳⡙⢦⡈⠓⢦⡈⠙⠲⣤⣻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⠁⣀⡴⠋⢀⡴⠋⢠⠞⢡⠟⣰⠃⡟⢸⢸⡇⢿⢸⡎⢷⠘⣆⠙⣆⠙⢦⡀⠉⠳⣄⡀⠉⠻⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠈⠈⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠤⠖⢾⡉⢉⡗⠶⢤⣤⣀⡀⠀⠀⠀⢤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⡉⢣⠀⠣⠎⠉⠽⠅⠉⠁⠈⡰⠃⣽⠒⣄⣀⣀⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡛⢿⣄⠻⡄⢃⠀⢷⣦⠘⠱⣄⡤⢠⠃⡼⢁⣼⠟⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣬⡙⢦⡙⣆⢳⡈⣦⣬⢿⡂⢲⠀⠀⡜⢸⢃⡞⣡⢞⡡⠾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣤⣬⣦⡱⡘⣆⣧⣡⠭⢼⠀⠘⡤⠠⣇⣇⡞⡰⣡⣮⣤⣤⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣤⣤⣤⣵⣟⣭⡴⢖⣒⣛⠆⢸⣓⣒⠶⣬⣍⣳⣥⣤⣤⣤⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⣤⣤⣿⠒⠉⠀⠀⠘⠮⠴⠁⠀⠈⠑⢪⣧⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣤⣄⣀⣏⡽⢻⣔⣩⣭⣍⠓⠀⠐⠛⣩⣭⣙⣺⠗⠮⣻⣀⣠⣤⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⣉⣉⣉⣃⢠⡏⠑⠛⠋⠃⠀⠀⠚⠘⠓⠉⣩⡄⣍⣉⣉⣉⣉⣉⣷
⠀⠀⠀⠀⠀⠀⠀⣼⣉⣉⣉⣉⣉⣉⣹⡀⢽⠀⠀⠀⢀⠀⣀⠀⠀⠀⠐⡧⢘⣉⣉⣉⣉⣉⣉⣙⣇
⠀⠀⠀⠀⠀⠀⠀⠛⠻⢯⡍⠍⣉⣉⣉⣿⢼⡆⠀⠀⠀⠈⠀⢀⠀⠀⣸⡧⣟⣉⡉⣉⠭⣩⠽⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⠿⢽⠫⢽⣄⠀⠐⠢⠒⠀⠀⣰⠭⢿⠭⠗⠋⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⣝⣺⠘⣺⡟⠑⠤⠄⠤⠴⠊⣿⣲⢸⣂⡍⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢺⣉⠙⠲⠮⢼⣇⠀⠀⠀⠀⠀⠀⣿⠬⠶⠖⠉⣀⡗⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢸⠒⠒⠒⠒⢘⡇⡁⠒⠠⠄⠒⢁⣼⣃⣒⡒⠒⢒⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⡯⠭⠭⠭⠬⣟⢿⡶⡶⡶⣶⡿⣿⠤⠤⠤⠬⢹⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠱⡖⠒⠒⠒⣏⢳⣯⣭⣭⡵⡞⣹⣒⣒⠒⣒⠎⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠬⢭⣉⣉⡟⠻⠿⠼⠿⠶⠟⢻⣀⣄⡬⠥⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢐⣷⣿⡿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣦
⠀⠀⢀⡤⠤⢤⠜⣿⣿⣿⣿⡟⢦⢤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣿⣧⡙⣣⣤⣿⡛⠁
⠀⠰⢇⣠⡄⢠⡞⠋⠹⠯⠉⠻⡄⠀⣤⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⢿⣿⣿⡟
⠀⢰⢼⣿⡇⠸⡁⢰⣷⣴⡧⠀⡝⢀⣿⡷⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⣿⣿⠁
⠀⠘⣋⢿⣻⣄⠱⣼⣿⣿⣷⡜⢁⣞⣿⢏⡟⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿
⠀⠀⠐⢎⡽⣯⣷⣿⣿⣿⣿⣷⣯⡿⣍⠊⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⢿⣿⠃
⠀⠀⠀⠀⠙⠴⣹⣿⣿⣿⣿⣏⡣⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢨⠿⣟⣟⡿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠸⢉⣹⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢳⣤⣶⣶⣦⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣏⣝⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢻⣿⣿⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢸⣿⣿⣿⠀⣿
⠀⠀⠀⠀⣀⠀⠀⠘⢿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣼⣿⣿⣿⠀⠿
⠀⠀⠀⠀⠻⣷⡄⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢰⢶⠀⣶
⠀⠀⠀⠀⠀⠈⠳⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢨⣇⣾⢘⣆⣣⡑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣯⡿⣿⢻⣷⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⣽⢻⣥⣶⣶⠾
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠃⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣝⣿⣿⢸⣿⣿⢻⡏
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⢻⣿⣿⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⡿⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠋

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠈⠁⠀⠈⣧⠀⠀⠀⡀⢠⡀⢉⠀⡅⢱⠀⠀⠀⠀⠀⣠⣲⡾⠿⠿⠿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠠⢪⠅⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢹⡀⠀⠀⠇⠀⡇⠘⠀⠁⠈⡇⠀⠀⠀⢠⠧⢤⢔⣂⠀⣐⠢⡤⢴⡀⠀⠀⠀⠀⢠⠁⣾⡿⢉⡀⠀⠀⡈⡇
⠀⠀⠐⡀⠀⣔⠩⠍⠉⠇⠀⠀⢀⠀⢥⠀⡄⢰⡀⣧⠀⠀⠀⣞⢀⣻⡀⠀⠀⠀⢀⣟⡀⣃⠀⠀⠀⠀⢸⠀⢿⠃⠀⠁⣀⢠⡁⣿
⠀⠀⠀⠀⡄⠁⠀⠐⠀⠊⠀⠀⠘⠀⢨⠀⠀⠀⡀⢸⠀⠀⢸⡉⢉⠩⢱⡤⠀⣴⡏⠉⠍⢉⡆⠀⠀⠀⢸⠀⠈⣷⠀⠀⠀⢉⡔⢻⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠈⡆⠸⠄⠃⠸⡇⠀⢀⡙⠶⠖⠾⠁⠀⠙⠶⠶⡖⢉⠀⠀⠀⠰⢫⡇⠀⠘⠷⣤⣾⡿⡇⠀⢿⡆
⠀⠀⠀⠀⡀⠀⠈⣿⠀⠀⠀⠀⠀⠡⠀⢂⠀⡀⢸⡀⡇⠀⠈⠉⢸⣀⣸⣶⠀⣶⣃⣠⠃⠉⠀⠀⠀⡆⢸⡇⠀⠀⡇⠐⡀⠀⢿⠀⢸⣧
⠀⠀⠀⠀⠈⠄⢰⣿⣷⠀⠀⠀⠀⢸⠀⠸⠀⠒⠀⠃⣿⠀⠠⡰⣄⣄⢼⣾⣿⣹⡅⡬⣠⢢⠂⠀⠀⡅⠈⡏⠀⢀⡇⠀⢀⡀⠩⠠⠴⢿
⠀⠀⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡄⠴⠒⠚⠛⠓⢾⠀⠀⠙⠮⣽⣳⣶⣶⣶⣞⣯⠵⠋⠀⠀⠀⠑⠤⠼⣭⣭⣥⣤⣤⡤⠶⠶⠖⠋
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠐⣶⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡠⢚⡧⣒⠁⠢⣚⠹⡄⠀⠀⠀⠀⠀⠀⠀⡔⠉⠙⢶⣄
⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡄⢠⣁⠀⠀⠀⠀⢀⣧⠽⡦⠞⠚⢫⠁⢉⣿⡀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠛⢦⡀
⠀⠀⣼⠋⠉⠀⠐⢀⣀⡀⠀⠀⠀⠈⢄⠀⠀⡀⢃⢸⣿⣿⠀⠀⠀⡘⡞⠞⠿⡄⠀⠀⠀⠀⡏⣇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠏⠁⠀⠀⢰⠀⠀⠀⠀⠑⠈⠀⣙⡚⠉⠉⠀⡀⠀⢅⡀⢂⠸⠽⠀⢲⡄⣶⣴⣾⠀⠀⠀⠀⠀⡆⠀⠈⠉⠀⠀⢸⡇
⠀⠀⡌⠉⠀⠈⠷⡀⠀⠀⢸⠀⠀⠢⠀⡀⠀⠀⠀⠀⠀⠠⡞⠀⠀⢈⣁⣵⣍⡒⢧⠀⠁⣿⣉⣧⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠐⠺⡆
⠀⠀⡇⠀⡤⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠈⢦⠀⠀⢀⢂⡞⠀⠀⠀⠸⡀⠀⠰⣀⠸⡇⠠⣹⣤⣿⡇⠀⠀⢀⠎⠀⢠⡀⢳⣀⣀⡀⣿
⠀⠀⡄⢀⠀⠀⢠⠀⠀⠀⠼⠀⠀⠀⠀⢀⠀⠧⠀⠤⣼⡇⠀⠀⠀⠀⠻⣄⡲⣧⣔⣿⣀⣸⣶⣾⠇⠀⠀⣈⠆⠀⢸⠘⠊⠑⠂⠀⢽
⠀⠀⠃⠘⠶⡄⡰⠒⠒⠒⠷⠁⠀⠀⠀⠐⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠛⠷⠿⠿⠿⠾⠛⠁⠀⠀⢠⠉⠀⠀⢠⡗⠀⠀⠀⠀⣾⡇
⠀⠀⢰⠀⡀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣒⣒⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢁⠈⣦⡀⠘⢄⣀⠀⠘⠃⠐⣦⢀⣸⣿⣟
⠀⠀⢈⡀⣥⣴⣾⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣻⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⣸⣷⠀⠀⠙⡄⠀⠸⡀⢸⡆⢸⣿⡏
⠀⠀⢰⠃⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠩⠀⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠙⠿⣿⡟⠀⠀⠀⠹⡀⠀⢳⠀⢷⢘⣿⡇
⠀⠀⡼⠀⢠⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣿⣿⣦⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠠⠀⢹⡄⠀⠀⠀⢃⠀⠘⣆⠈⠉⡛⡇
⠀⢰⠁⠀⢻⠀⠀⠀⠀⠀⢸⠀⠀⠀⢘⣿⣿⣿⠀⢰⣶⣿⠁⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠈⠐⠂⠻⠦⠤⠗⠃
⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠀⠀⠀⠁⢻⣿⣿⠀⠈⣿⣿⣷⠀⠀⡀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⢀⠀⠛⠀⠀⠠⢀⢀
⠀⢸⠀⠀⢠⣇⠀⠂⢀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠇⠀⠘⠟⣻⡇⠀⠁⠀⠀⠐⠂⠤⠤⣄⣀⣀⠀⠀⡄⢀⢀⣲⣶⡄⣀⣄⣴
⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠀⠒⠲⢀⡲⠶⡶⠶⣤⣤⣤⠶⣶⠶⢖⠂⠖⠒⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢰⠂⠠⠀⠀⣀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠑⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣤⡄⢤⠐⣠⣤⠀⠀⠐⡗⠀⠀⠃
⠀⠀⠀⠀⢀⡐⠁⠈⢂⠀⠀⢀⣶⡆⠀⠀⠀⠙⠁⠀⠠⢤⡄⠀⡠⠤⡀⣄⠀⠀⠀⢃⡠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠈⠀⠉⠉⠀⠐⠀⠋⣙⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢼⣿⣿⢿⢻⢟⠿⣦
⠀⠀⠀⠀⢠⣴⡆⠀⡰⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡀⠀⠀⠀⢣⣸⡿⠺⡿⡏⠀⡠⠉⠃⣜⣧
⠀⠀⠀⣠⣿⣿⠀⢨⠀⠀⠁⠀⠀⠀⠠⠆⢀⠀⠠⠠⠀⢀⠀⠆⠀⠀⠀⠀⠀⠉⠒⠠⠠⠒⠊⠀⢀⠂
⠀⠀⢠⣿⣿⣿⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡤⠄⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛
⠀⠀⣼⣿⡿⠻⣿⣿⡀⠤⢀⠀⠀⠀⠀⠀⢄⢿⣧⢰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⢹⠃⠀⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⡔⠦⣀⡀⠀⠀⠀⠀⠀⠉⠉⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠾⠇⠞⢿⣿⡄⠀⠀⠀⠀⢀⡀⠘⠋⠁
⠀⠀⠀⠀⠂⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠴⡆⠶⠀⠰⠰⡦⢭⡀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠆⠀⢠⢀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣚⣋⢃⠀⠤⠀⣘⣛⣓⡂
⠀⠀⠀⠀⣠⠃⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡖⣚⣀⡀⣀⣓⡒⣶⡄⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⡗⠀⠀⠀⠀⢾⡅⠶⣯⣇⣿⡶⢆⡿⠃⠀⠀⠐⠚⠉⣠⠆
⠀⠀⠀⠀⠈⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠓⠒⠒⠿⠟⠁⠀⠀⠀⠓⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠘⢩⢦⡈⠉⠀⣴⠎⠘⡖⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⠔⠀⣰⠀⡏⠀⡳⠶⢾⠁⠀⡇⡀⢀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⣄⣁⣐⣀⣸⡆⢰⠀⣿⡁⣹⠀⡜⣰⣧⣤⣄⣠⡀⡌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢛⡍⢉⡍⡅⢐⢂⢻⡌⢣⣿⠀⣽⡞⣴⠏⢖⣰⢠⡚⠉⠉⡚⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣈⣾⠤⠽⠵⠾⠿⠶⣿⢠⣧⣀⣸⡗⣿⠒⠒⠒⠶⠿⢤⣇⣈⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⡇⠀⠂⠀⠀⠀⠀⣸⣸⡧⠤⢼⣧⣸⣀⣄⣠⠀⠀⡄⢠⡏⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢘⡞⠉⠉⠉⠉⠉⠉⠉⠙⠚⠉⠁⠁⠀⠉⠉⠉⠹⡟⠋⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡀⠀⠀⢸⠁⠐⠚⠛⠲⢦⡀⠀⠀⠀⠀⠀⣤⠶⠒⠒⠦⠀⢷⠀⠀⢠⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⢰⠃⢰⠃⡿⠸⢶⡶⠶⡶⣦⡀⠀⠀⠀⠀⠀⣰⣿⠷⠶⣴⠖⢸⡄⣆⠀⢣⢠⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡆⠁⡄⡎⠀⡇⠀⠀⠐⠳⠴⠗⠃⠀⠀⠀⠄⠘⠓⠦⠚⠒⠀⠀⠸⡇⠸⡰⡘⡄⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡽⠙⡳⡶⡧⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣷⡴⡷⣿⠻⠹⡽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠇⠆⣇⣣⠇⢸⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡏⡄⡇⡌⡆⠇⣇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠰⢠⠸⡼⠀⣾⣀⡂⠀⠀⠀⠀⠀⠀⠙⠲⠴⠊⠁⠀⠀⠀⠀⠀⢰⣡⡇⡇⢸⢸⣱⠰⢸⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⣆⡜⡇⡃⠀⢿⣏⣹⡄⠀⠀⠀⠀⣠⣄⣀⣁⣀⣤⡀⠀⠀⠀⢀⣏⣽⡇⣧⢸⠘⠉⣔⣏⣧⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠙⠛⠻⠛⢷⠋⡀⠈⢿⣄⠀⠀⠀⠉⠦⢤⣤⠤⠎⠁⠀⠀⣠⡞⠁⡀⠙⣿⠞⠻⠛⢛⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠘⠠⡆⠀⢺⠨⠭⠀⣸⡏⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⡇⠸⡽⠀⡼⠀⠀⡇⠘⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡄⠀⠀⢸⠀⠀⠘⢢⢤⡴⠛⡃⠀⠀⠙⠶⣤⣀⣀⣠⠴⠛⠁⠀⠘⠷⣤⡤⡴⠁⠀⠀⢹⠀⠀⠀⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⢇⣠⡀⠀⠀⠸⠁⣻⣶⡗⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠼⣒⣛⢻⠃⠀⠈⠀⣸⣀⣵⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⢰⠁⠋⡟⢲⠶⣄⠑⠒⢃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠨⠭⠋⣴⡴⠶⠚⡏⣣⢸⢸⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣸⢸⡇⡇⢸⢠⢉⡷⢶⠛⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⡾⡇⡇⢹⠀⡇⣿⢸⢸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣼⣿⢸⠇⡇⣾⢸⢸⠑⣾⢣⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣷⣇⡇⢣⢸⠀⡇⢻⢈⣾⣷⡇
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠓⢬⡙⠛⢶⣇⡇⣈⠘⣰⣿⣄⠳⣄⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢁⡴⢿⢋⢡⢸⢸⣃⣷⡼⠛⡩⠞⠛⠒⠤⣄
⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠙⢆⠀⠌⠙⠿⣤⣇⡿⠈⢶⣌⠙⠦⣄⣉⠙⠛⠒⠚⠛⠋⣁⣤⠖⢋⣴⢿⣸⣬⣼⠋⢉⡜⣡⠞⠁⠀⠀⠀⠀⠈⠳⣄
⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠢⡀⠈⢯⠁⠀⠈⢯⠻⡶⢤⣈⣉⠙⠛⠛⠛⣉⣡⣴⢞⡿⠁⠈⢉⡴⠁⣠⢎⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⢆⠀⠑⢦⡀⠀⠑⣝⢆⠈⣟⠽⢿⢻⡉⣩⢗⡵⠋⠀⣀⡴⠋⢀⡴⣣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⠳⣄⠑⢦⡀⠉⠓⠦⢌⣻⡿⣧⣓⣊⣼⢿⣵⣯⡤⠶⠛⠁⢀⡴⢋⣼⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠂⠈⠛⢦⣍⡛⠶⢤⣤⣼⢸⢡⡉⢙⡈⡇⣃⣀⣀⣤⠴⢚⣩⠶⡏⢸⠀⢳⡀
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⡜⠀⢰⠀⠀⠀⠈⠙⠓⠶⠦⠼⣎⠻⢛⡹⠕⣡⠯⢤⡤⠶⠚⠋⠁⠠⠁⠀⢀⠈⢧
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠘⣧⣿⣛⣛⣿⣭⣉⣏⣙⣏⣹⣟⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣺⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣠⣶⣶⣿⣿⣿⣿⣿⡿⣿⣿⣷⠿⠗⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣠⣤⣄⣀⣠⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡹⠛⠛⢛⣿⣿⣿⣿⣦⠉⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠚⣿⠏⠘⡇⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣼⣿⣀⣤⡴⠆⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠙⠋⢉⣿⢿⡟⠙⢿⣿⡙⠻⣏
⠀⠀⠀⠀⣀⣤⣶⣷⣶⣿⣿⣿⣿⣿⣿⡿⢶⣄⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡸⠃⣸⠃⠀⠀⢻⡇⠀⠈
⠀⠀⠀⠐⠁⠈⣉⣹⣿⣿⣿⣿⣿⣿⣍⠀⠀⠈⠁⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢠⡾⠿⠋⣹⣿⣿⡿⣷⠙⠿⣷⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠟⠀⠀⣼⡿⠉⣿⡇⠸⣧⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠛⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣀⣠⣤⡞⠉⠓⠦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣀⣀⡴⠞⠋⠉⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠓⠤⣤⣄⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣰⢿⣷⣄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠻⡄⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢰⡏⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠻⣿⠀⠹⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢀⡿⠀⢸⣿⣿⣿⣿⣠⣤⠴⠦⠤⠤⠤⠤⠄⢰⣿⠇⢈⠇⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣸⠃⠀⠀⢿⣿⣮⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠈⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡟⠀⠀⠀⢸⣏⡿⣫⣭⣽⣳⣄⠀⠀⣴⣿⡿⣭⡐⢷⡀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢠⡖⢈⣿⡀⠙⠀⠀⠉⠛⠃⠀⠛⠋⠀⠀⠁⣴⣁⡀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠈⣿⠀⢻⡷⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠸⠋⠋⢀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠘⢦⣸⡇⠀⠀⠀⠀⠚⠛⠛⠛⠀⠀⠀⠀⢀⣠⠊⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠉⣧⢰⡀⠀⠴⠶⠶⠶⠶⠄⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⠛⢦⠀⠀⠚⠂⠀⠀⠠⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣦⠀⠀⠀⠀⠀⣷⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣦⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡀⠀⠀⠀⠀⠀⠀⣿⠻⣟⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⣠⠞⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⢻⣷⡀⠀⠀⠀⠀⠀⢹⡄⣿⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠸⣇⠘⠻⢿⣿⢿⣿⡿⡿⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣀⠀⠀⠀⢀⣀⣀⠸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠻⠿⠿⠃⠀⢰⣿⣿⡿⢰⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⡾⢶⡾⢃⣴⣿⣿⣿⠃⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⣿⣿⣿⣿⠟⠀⠸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⣾⣿⠟⢙⡟⢿⠀⠀⠀⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠙⠛⠛⡁⠀⠘⠛⠉⠛⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠋⠉⠉⠙⠛⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⣿⡇⠀⠀⣿⠿⠿⠟⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣿⡏⠛⠀⠀⡇⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣧⣶⠆⢀⡇⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣤⣤⣄⠀⠀⠀⠇⠀⠻⠿⠿⠀⠈⠁⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢸⣿⣦⠀⠀⠀⠀⠠⠴⣶⣶⢶⣶⣶⠶⠶⠶⡷⠄⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⠀⠸⣷⡿⠀⠀⠀⠀⢸⢸⣧⣀⣸⣿⣿⣀⣀⣀⡀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠒⠠⣤⡀⠀⠤⢤⣤⠒⠋⠀⢻⣤⣴⠟⠀⠀⠀⠀⠀⣸⠈⠉⠉⠉⣉⠉⠉⠉⠉⠋⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠆⠀⠎⠁⣀⣀⣀⣀⣀⣤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣹⠐⠶⠶⠿⠿⠿⠶⠶⠶⠆⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⣴⣴⠇⢰⡿⠻⠏⠙⣿⠃⣠⣹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠤⡛⠁⠚⣀⠻⡧⠀⠀⠀⠈⠐⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡙⠲⢅⡀⢶⣍⠳⡤⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡈⢕⣠⣈⣷⡬⢿⡆⠀⠀⠀⣠⡜⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠱⢾⡿⢏⠹⣷⠀⠐⠋⠙⠛⣤⡌⣿⡙⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣎⡱⢿⡄⠐⣍⣉⣻⠾⠆⢿⡟⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣀⣀⡀⠈⠙⠻⠋⠉⡧⠤⠀⠚⠳⠀⠻⠀⣠⣤⠤⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡋⢉⠉⠳⠀⠤⢀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠁⡾⠴⠇⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠛⠒⠁⠰⠦⠾⠶⠄⠛⠛⠁⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢇⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣤⣄⣀⠀⠀⢀⠆⠀⠀⠀⠀⠣⣀⠀⠀⠁⠢⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⡌⠙⠿⣿⣿⣿⣀⣷⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⢻⠀⠀⠆⢠⣽⢿⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⣀⣰⣦⣄⠀⠒⠀⢀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⣶⠀⡼⣼⣷⣼⠏⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⣠⣶⣴⡟⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣄⠀⠀⣼⢧⠈⠃⠀⠉⠘⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡟⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠧⣭⠀⢀⠿⣆⣀⠀⣤⣀⡄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡸⠃⠁⠘⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⡶⠞⢱⣿⡾⠃⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⠉⠀⠀⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠸⡗⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡴⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠟⠛⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⡶⢶⠶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⡾⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣮⣿⣿⣿⣿⣛⣿⣿⣦⡅⣴⣇⠻⢷⣶⡃⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣠⣟⠁⣱⣶⣿⣟⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣡⣍⣹⣿⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⣸⣿⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⡯⢠⣿⣿⣿⣿⣿⠃⠉⠙⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣻⠿⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⣀⣰⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣆⣄
⠀⠀⠀⠀⢤⣽⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣯⡤
⠀⠀⠀⣬⣷⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠺⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣥
⠀⠀⢐⣳⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⢠⣴⣶⣦⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣶⡂
⠀⠀⠰⢿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣶⣤⣄⣀⡀⠉⠿⢇⣀⣠⣤⣶⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⡿⠆
⠀⠀⠀⢻⣿⣿⣿⣿⣽⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡿⣯⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⡝⢿⣿⣿⣿⡟⠈⣿⣿⣿⣿⠁⢻⣿⣿⣿⡿⣫⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠉⠀⠀⢸⣿⣿⡇⠀⠀⠁⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⣤⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠈⠀⠀⣿⣿⣿⣿⠀⠀⠁⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣿⣶⣬⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⣀⠀⠐⣿⣿⣿⣼⣿⡖⡁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⣿⣏⢻⡿⣉⢹⣿⠿
⠀⠀⣠⠚⠉⠋⠉⣡⡿⠿⣿⡿⢟⣄⠉⠉⠍⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⣶⣿⣿⣿⡄
⠀⠰⠛⣤⣾⠆⢰⠏⠈⡐⠁⢁⡁⠘⣇⠀⣾⣦⠞⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣙⠋⣿⣿⣿⠃
⠀⢸⠒⣿⣿⡄⠸⡄⠀⣿⣟⣿⡇⢀⡏⢀⣿⣿⡒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢋⡽⣿⣹⣄⠙⣾⣿⣿⣿⣷⠎⣠⣾⣽⠯⣍⠃⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠈⠲⣉⠿⣭⣷⣿⢿⣿⣿⣿⣾⣯⡿⣍⡳⠁⠀⠀⠀⠀⠀⠀⢄⢼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠿⣿⡟⠁
⠀⠀⠀⠀⠈⠲⣩⢛⣿⣿⣿⣾⣷⡟⢧⡱⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢿⢿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣿⠛⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣞⠀⣙⣛⣊⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⢿⣿⣟⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⡿⣿⡷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⣿⣿⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠟⢻⣿⢿⣿⠿⠟⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠘⣿⣿⡿⣿⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣷⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢀⣿⣿⣿⣿⡆⣼⡇
⠀⠀⠀⠀⢸⣿⣤⡀⢀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣤⡀⢠⡄
⠀⠀⠀⠀⠀⠙⢿⣗⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠞⡇⢸⠇⡇⠸⡳⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⢾⠷⣾⣤⣿⡶⣷⢌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⢏⣻⣉⠾⢿⠷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⣾⣴⡛⣧⣽⡷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠋⠀⠐⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⣿⣿⣯⣿⣿⡛⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⣿⡷⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⢫⣿⣶⣿⡏⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣿⣭⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣿⢿⠿⡿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡠⡀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⠁⢠⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⢂⠨⠀⠀⠀⠀⠀⢀⣀⣠⢄⡁⣀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠑⠐⠊
⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠩⠀⠀⠀⠀⠀⠈⠐⢺⣾⠂⠁⠀⠀⠊⢀⠀⠀⠀⡀⠀⡀⠀⠁
⠀⠀⢀⠄⠀⠀⣠⠄⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⡆
⠀⠀⠊⠀⠐⠐⠃⠀⠀⠚⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠮⣟⣂⠀⠀⠀⠀⠀⢀⡀⠀⠢⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⢠
⠀⠀⠀⠀⠀⣥⣛⡃⠀⠀⠀⠀⠠⡤⢶⠤⠤⠡⡤⢤⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠀⠖⠂⠐⢠⠂⠀⠉
⠀⠀⠀⠀⠀⢹⡯⠀⠀⠀⠀⠀⠠⠠⡏⠎⠰⠱⠈⡆⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⡎⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠁⠉⠀⠀⠀⠀⠀⠛⠒⠓⠓⠚⠃⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠓⠀⠀⠐
⠀⠀⠀⠀⠠⠰⠒⠠⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⢠⢀⠤⠡⠆⢠⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⡘⠲⠀⠀⠀⠀⠀⠀⠁⢤⠀⠈⣡
⠀⠀⠀⢠⠀⢐⡖⠀⢠⠀⠀⠀⠐⠀⠀⢆⠀⡰⠊⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⢂⠈⠀⠀⠀⠀⠀⠀⠁⠐⠈⠄
⠀⠀⠀⠈⢇⡹⠿⣀⠝⠀⠀⠀⠈⠀⠀⡠⣽⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⢰⠖⡜⠈⠀⠀⠀⠀⠀⠀⠠⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⡠⠤⡄⢀⡀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⡠⠱⠄⠈⠀⠀⠀⠀⠠⣣⣠⡈⠁⠀⠀⠀⠀⠜⠭⡄⠁⢐⣲⡄⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠉⠂⠀⡀⠀⡀⠀⠈⠁
⠀⠀⠀⠀⡠⠊⠀⢠⠀⠀⠀⠀⠀⠠⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⢪⠔⢠⡐⢾⠀⠀⠀⠀⢀⣀⠈⣛⡤⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣳
⠀⠀⠀⠀⠠⠄⢈⠂⠀⠀⠀⠀⠀⠸⠋⣷⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠉
⠀⠀⠐⠂⠀⠀⠠⠤⠀⠀⠀⠀⠀⠠⠰⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠅⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠙⡝⣤⡟⢺⣦⢯⠋⡳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⡈⢎⢇⢱⢣⣼⣯⡼⡎⡰⡱⠁⣽⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢤⡙⣵⣬⢾⡾⢿⣿⣿⡿⢷⡷⣥⣮⢊⡤⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⣉⣿⢁⣠⡤⠿⠿⠿⠿⠿⠿⠤⣄⡈⣹⣋⠬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣑⣒⠻⡏⠴⠒⠶⢦⠄⠀⡤⠶⠖⠦⢹⡟⢒⣊⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡓⠒⠒⢯⢿⠙⠻⠿⠷⠄⠀⠾⠿⠿⠃⣾⡽⠖⠒⠚⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠿⠭⠭⣻⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣟⠭⠭⠽⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣭⣙⣒⡒⠢⣽⣇⠀⠀⢀⣀⣀⡀⠀⠀⣸⣯⠴⢒⣒⣋⣭⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠦⠤⠤⣍⣽⣽⣟⣆⠀⠙⠥⠤⠋⠀⣠⢻⣯⣯⣩⡤⠤⠴⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣙⡒⢺⠿⠿⠿⣿⠑⠦⢤⡤⠴⠊⣿⢿⠿⠿⣗⢒⣋⣉⣉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠭⠭⢭⣟⣛⣛⣿⠀⠀⠀⠀⠀⠀⢿⣛⣛⣻⡭⠭⠭⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡒⠢⠤⠤⠤⡇⠀⠀⠀⠀⠀⠀⢸⡤⠤⠤⠔⢒⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⣯⠭⠭⠭⣩⡷⣤⣀⡀⢀⣀⣤⢾⣏⠉⠭⠭⣽⢋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣹⣏⣉⣛⣻⡟⠒⠬⠩⠭⠥⠒⢹⣟⣋⣉⣹⣟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣶⣖⣺⡟⠶⡶⣤⣤⢴⠶⢻⣗⣲⣶⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣽⣁⣃⠂⠆⠰⠐⢚⣈⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣸⣥⡤⠄⣠⡤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣎⣸⣇⣼⣧⡞⢁⣴⠞⢋⣩⣬⣷⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣻⢿⣿⢿⡟⣰⠟⣡⡶⢟⣭⣶⣶⡾⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⣼⡏⠈⠁⠛⠸⢫⣾⣿⣿⣿⢛⣛⣋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡷⠘⠿⠟⢒⣒⣲⣤⣴⣿⣿⣿⣟⣛⣉⣭⣭⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣟⣿⡞⡿⣿⣭⣽⠿⣿⣿⡿⣿⣭⣭⣭⣤⡶⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣭⡏⢙⣱⢇⡈⠁⢀⣤⣶⣿⣻⣯⡷⠶⠶⠶⠶⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠶⠶⢿⣜⣿⣛⣁⣢⣾⣿⣿⣿⣯⡿⠒⠛⢛⣛⣛⣛⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢚⣛⣛⣛⣻⣭⣭⣯⣿⣿⣿⣿⡿⣿⣟⣉⣉⣉⣭⣭⣭⣭⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣛⣩⣭⣭⣭⣽⣴⣷⣾⣿⣿⣿⡟⠁⣿⡯⠤⠶⠶⠶⠶⠶⠶⢾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠲⢶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⢿⣿⣿⣿⣿⡛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⢹⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣽⠧⣸⣿⣿⠏⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣿⣿⣿⣿⣿⣛⣿⣻⣅⣦⣾⣟⣷⣿⣟⣵⣾⣿⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢻⣿⣿⣯⣼⣹⢏⣏⢹⣏⣧⣽⢿⣻⣿⡿⣿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠟⠻⠿⠿⠿⠿⠿⠛⠩⠖⠫⠿⠋⠞⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣌⣤⣤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⠁⠈⠈⢋⣤⠞⠋⣁⣬⠵⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣙⠳⡁⡀⠀⠦⠈⣡⠶⣋⡥⠴⠒⠒⠚⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡯⢭⣿⣿⡅⠀⡥⢾⣷⣿⣵⣲⡯⠭⢭⣉⡙⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⢺⡟⢽⣯⡀⠀⣠⣤⣠⣤⣠⡏⢰⢢⡐⡄⠙⡆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⣿⣿⠛⠉⢁⢀⠉⠉⢉⣉⣛⠻⣿⣿⠃⢠⠀⢡⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠀⢹⣯⣽⣓⣈⠊⠐⢋⢭⡴⣤⡄⢸⡟⠀⠘⡀⢸⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠈⡟⠒⠉⠙⠀⠀⠉⠀⠛⠉⠀⣼⡇⠀⠀⡇⠈⡇⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⢿⣷⣴⠀⢧⠀⢳⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠇⡶⢿⣿⣷⡀⠈⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡷⣼⡀⠸⡄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠿⡇⣻⢻⣿⣷⡀⠠⠀⠈⠀⠀⣀⣴⣿⣿⣿⣇⢸⣿⢻⡷⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡥⠘⣷⣻⣿⣷⣄⣀⣠⡴⠞⢁⣿⣧⣿⣿⣿⣆⢻⣎⢷⡀⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⠟⠁⠀⣿⡇⣿⣿⣿⣟⠉⠀⠀⠈⣿⣿⣿⡿⢿⣿⣷⣝⡷⣝⢷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣁⠤⢊⣼⢟⣽⣿⣿⡿⠋⠀⠀⠀⠀⠘⣿⡷⡅⠈⠻⢿⣿⣿⣮⣷⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⣟⣵⣾⣿⣿⣿⣿⡟⣯⣃⠀⠀⠀⠀⠀⠀⢹⣿⡱⢄⠀⠀⢤⡬⣛⣿⣿⡿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⡿⠉⣿⠃⣿⣿⡟⠐⢼⡞⢄⡀⠀⠀⠐⡈⠅⣽⢻⣦⣅⠢⠀⢿⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⢮⣿⡿⢷⡀⠀⠱⡆⠈⠒⢒⠈⠠⣰⠋⠀⣽⣿⡴⢠⠃⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢆⠀⠱⢄⠀⠙⢦⢆⣲⣤⠞⠁⣠⠞⠁⣻⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠈⡳⣄⠀⠉⠒⢦⠻⡎⢴⠖⠊⠀⣠⣶⡡⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢒⡤⢸⡔⡋⡾⠠⢐⠼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠊⠁⠁