Mossy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡷⠀⣸⢒⢂⠀⢖⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⠀⡄⠀⢀⣀⣀⢁⡼⢢⣹⣍⠰⠸⣷⣹⣼⣿⡀⠀⢠⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣿⣾⢾⡁⠈⢰⠿⢿⢻⣿⣧⣴⡿⡿⣏⡀⠿⣭⣟⣻⣒⡈⣿⠣⣶⣟⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠹⣟⣏⣳⢀⠘⠂⡀⣴⣯⣧⡂⢸⣟⡂⢛⣿⠻⣯⣿⣿⣯⣾⣾⡀⠸⠟⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣼⣿⣍⣻⣤⢄⠞⡋⢻⡹⣿⡷⡽⡿⠟⠸⡷⡙⣯⡽⢶⠍⢘⠻⠯⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⠊⢉⢋⡽⣾⡾⠫⠁⡾⣇⡡⣰⣕⣃⣾⢄⢣⣶⢷⣼⢷⡠⣾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣼⢷⣦⣼⡌⢼⣯⣤⢸⣧⣙⠀⢼⠉⠷⣯⠘⡿⠎⠉⠛⠁⠿⠝⣧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡛⠙⡩⠰⠒⠀⣡⠞⠓⠈⠷⣶⣶⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⢆⠀⠉⠀⠀⠀⠀⠛⠉⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣜⣾⡷⢠⡀⣴⣰⣠⣀⣸⣤⣼⣠⣼⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⣲⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣛⣿⡏⢹⣿⣿⣯⣜⡍⠗⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⣿⣿⣿⢿⡿⢩⣿⣽⣿⣿⣯⣿⣟⣤⣿⣿⣵⣿⣯⣿⣿⣷⣯⣥⣎⡽⡈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣾⣿⣿⣿⣧⣾⣷⣾⣿⡿⣟⣿⣿⣿⣿⣿⣼⣿⣭⣿⣯⢿⡿⢿⣿⣿⣷⣿⣾⣿⣷⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⣿⣿⣿⣿⣟⣾⣿⣿⣟⡙⡿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⡿⢣⡝⢿⣯⣿⣟⣿⣟⣽⠯
⠀⠀⠀⠀⠀⠠⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣻⣽⢟⣿⣿⢟⡻⣟⣻⡽⡓⣿⣛⣛⣿⣿⣽⣿⡿⣗⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣯⣾⣿⣿⣿⣿⢿⣿⣿⣿⡟⣷⣿⡟⣻⣿⣷⣿⣷⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼⠏⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠹⠟⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣟⣾⣿⣿⣽⡾⣿⢿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡖⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⢿⡿⣿⣷⢟⣿⣿⣮⣿⣿⣿⣿⢿⡟⣞⡿⣿⣿⣿⣿⣼⣿⣿⣯⠍⠳⠝⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠿⠻⠟⠁⠈⠉⠻⠿⢿⣿⣼⠛⠓⣻⣿⣿⣿⣿⡿⠿⠏⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠉⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣷⠟⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣇⠀⠀⠀⢠
⠀⠀⠀⢀⡀⠀⣼⣯⠅⠀⠀⠀⠀⠀⢠⣷⡄⠀⠀⢰⣿⣿⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣆⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣧
⠀⠀⠀⢺⣷⢀⢚⣷⣆⢠⣾⠿⠂⠀⠈⣿⣿⠀⠀⢀⣿⣿⡆⠀⣴⣿⡇⠀⠀⠀⠀⠁⠚⠇⡠⢀⠀⠃⠀⠀⠀⠀⣤⣠⡄⠀⠰⣶⣿⣧⡦
⠀⠀⠀⠨⣿⣧⠀⣬⣿⣿⠏⠀⠀⠀⠀⣿⣿⣷⠀⢿⣿⣿⠇⠈⢿⣿⣷⠀⠀⠀⠀⠠⡀⠸⣀⠌⠀⠀⠆⠀⠀⠰⣿⣿⡶⠆⢶⣾⣿⣧⡄⢠⣰⠆
⠀⠀⠀⠀⠀⣟⣷⣟⠃⠀⠀⠀⠀⠀⠀⢘⣿⣿⡄⣸⣿⣿⣆⢠⣿⣿⡃⠀⠀⠀⠀⠀⣸⠃⡝⠀⢲⡀⠀⠀⠀⠘⣻⣿⣟⠃⣴⣾⣿⠏⠠⣾⠿⠍
⠀⠀⠀⠀⠀⠺⣷⣄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠻⣿⣿⡿⠘⣿⠿⠃⠀⠀⠀⠀⠀⠘⡀⢃⡄⠞⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠈⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠘⠻⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠟⡀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠰⡄⠀⢸⡄⠀⢰⠀⠀⠀⠀⠀⢦⣞⠀⠠⣥⠄⠀⠠⣗⡀⠀⠀⠀⠀⠀⠀⣼⡟⣠⡇⠀⠀⠀⠀⠀⠀⠈⣙⢋⡁⠀⠰⢦⣨⡅
⠀⠀⠀⠸⡀⠀⢹⡀⣼⠁⠀⣾⠀⠀⠀⠀⢨⡿⠇⠀⣽⡿⠀⠀⢈⣛⡃⠀⠀⠀⠀⠀⢸⣿⢰⡿⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣥⣛⣤⢤⣴⠄⡤
⠀⠀⠀⢀⠳⡄⣼⣧⣿⢀⡾⢃⡞⠀⠀⠀⠐⡿⢿⡀⢲⡯⢇⠨⢿⡓⠀⠀⠀⠀⠀⢰⡌⣿⡆⣿⠀⡖⠀⠀⠀⠀⠀⠰⣈⣁⣶⠖⡶⠦⢾⣗⡛
⠀⠀⠀⠀⣧⢹⣿⣿⣿⢏⡴⠋⠀⠀⠀⠀⠀⠨⠭⠄⣰⠯⠭⠈⣷⣿⠀⠀⠀⠀⢀⡾⣱⣿⢃⡿⢸⡿⠀⠀⠀⠀⢈⣝⣿⣤⣿⣟⠴⠦⢶⡿⠃
⠀⠀⠀⠀⢸⡎⣿⣿⡇⣸⠁⠀⠀⠀⠀⠀⠀⠸⠿⢗⡮⣭⠁⣰⢿⡛⠀⠀⠀⠀⠘⣇⣿⡇⣾⢱⡿⠁⠀⠀⠀⠀⠀⠒⣊⢶⡶⢾⣛⡳⠶⣟⠛
⠀⠀⠀⠀⠀⠻⣾⣿⣧⠟⠀⠀⠀⠀⠀⠀⠀⢩⡽⣫⣭⣗⣐⣻⡟⠀⠀⠀⠀⠀⠀⠹⣿⣇⣿⣸⠇⠀⠀⠀⠀⠀⠀⠀⠱⢚⡿⠚⠋⠙⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⡀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣾⣶⣶⣾⡻⢻⣿⣷⣖⠀⢠⣄⡀⡀⠀⠀⠀⣤⣀⣤⠀⠛⣿⣸⣿⠸⠇⢸⠉⠁⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⢶⡄⠀⠀⠀⠀⣷⡄⢢⡐⣒⣺⡿⣿⣷⢸⣷⣿⣿⣿⡥⣼⣿⣿⣿⣿⣶⡘⣾⣿⣿⣿⣀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⡉⢻⣿⣿⣿⠄⠻⣷⣰⣿⣿⣧⣼⣿⣿⣾⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⣰⣆⣠⡄
⠀⠀⠀⠀⠀⠀⣠⣶⣤⣄⣠⣿⣿⣷⣿⣿⣿⣿⣦⣔⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣻⣿⣿⣿⠀⣿⡿⠛⠉⠁
⠀⠰⢾⣾⣷⠈⢥⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣧⣿⣿⡷⠘⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠏⣽⣴⣷⣿⣿⣿⣏⠀⣤⣿⣇
⠀⠀⠦⢻⣿⣧⡿⠷⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣆⣹⣿⣏⠀⠀⠀⢹⢿⣿⣿⣿⠿⣿⣿⣿⣿⣦⣿⣿⣿⡟⠋⣿⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢻⣧⣦⡉⠛⣻⣽⣿⣿⣿⣿⣿⣴⣤⠀⠀⠀⣿⣿⢛⣿⣿⣿⡿⣿⠛⠹⠋⠉⠉⠉⠉⠛⠉⠉⠀⠀⠀⠀⢀⣀⣤⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢥⣘⣿⣿⣿⣾⠠⣾⣿⣟⣿⣿⢽⣿⣿⣿⣶⣤⣾⡟⣸⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣸⡦⠙⠻⠻⡟
⠀⠀⠀⠀⠀⠀⠀⠦⣾⣿⣿⠋⢻⣯⣿⡼⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣧⣶⣄⣠⣶⣶⣴⣦⣶⣦⣼⣷⣶⣎⣷⣿⡶⠏⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠖⠘⠛⢿⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⣿⣿⡟⠛⠻⣹⣿⣿⣯⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠾⢻⡏⠀⠈⢉⠁⣿⠻⣷⣿⡿⣿⣿⣿⣿⣿⡿⠿⠿⡿⢿⣿⣿⣿⢇⣀⣀⣿⡝⣿⣿⣿⣿⣿⣿⡿⣿⠄
⢀⣶⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡜⢻⣷⣟⢿⣿⣿⣏⣀⣀⣠⡷⠚⠛⠛⢟⣿⢿⣿⣿⡟⠛⣯⠋⠉⠉⠟⠀⠉⠀⠀⠀⢀⡀
⠀⣿⣿⣿⣧⣶⣄⣠⣴⣆⣤⣄⡀⢀⣠⡀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠙⢳⣿⣿⣿⠟⠉⠉⠁⠀⠀⠀⠀⠈⠙⠀⠉⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣆⣀⣿⣦⣤⣤⣀⡅⣀⣠⢀⣿⣿⣇⣄⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣦⣀⠀⣠⡀⣀⣴⣶⣠⡄⠀⠀⣤⡄
⠀⣿⣿⣿⣿⢿⠯⣽⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣯⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠠⠿⠷⠿⠷⠦⠦⠽⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠼⠧⢿⣾⠿⠿⠷⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣶⡆⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡀⠀⠀⠠⠘⣦⠀⣅⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⡈⠉⣀⣖⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠛⢩⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢴⢾⣤⣖⢓⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠂⡼⣾⣿⣿⣿⣿⣄⡀⠠⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠘⠁⡀⠀⣢⣴⣿⡟⠩⢉⡿⠋⡝⠁⠀⠓⠒⠂⠀⠀⠀⠀⡠⢄⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠆⡱⠁⣠⣾⡧⠃⢙⠇⠂⢸⠗⢸⠓⡀⣦⠀⠀⠀⢀⡆⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⣱⠋⠀⠀⣠⡴⠀⠀⠀⠀⠇⠀⠉⠉⠈⠉⠁⣛⡆⠀⠀⠹⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢹⣷⣦⢼⠗⡀⡀⠀⠀⠀⠀⠀⠙⠣⠔⠂⢊⡥⣠⡖⣠⡞⠛⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠋⠀⠀⠇⠃⠀⠈⠈⠀⠀⢀⣠⣀⠀⢲⢰⠉⠁⣿⠟⠶⠇⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⠁⣈⣈⠃⠷⢆⡙⠦⣀⡤⠉
⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⢰⡦⢿⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⣴⠟⠩⣿⠂⠰⣰⡏⡂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⠏⠀⠀⠀⠀⠀⠠⣸⣀⠀⠠⣿⣬⣾⡇⡤⢠⣤⣠⡔⠤⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠈⡧⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⣀⣻⡶⣾⠉⡏⢨⣿⠖⠁⠤⠀⣾⣿⣁⡀
⠀⠀⠀⠀⠠⢿⣦⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡹⠃⠈⡩⡀⡇⡜⠏⠀⣴⣖⢄⠘⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠛⢇⣤⣤⣼⣟⠛⠉⠀⠀⠀⠀⠀⠐⠒⠚⡇⠉⠒⢒⡆⣷⡌⠀⠀⠀⣤⣎⡇⠀⠋
⠀⠀⠀⠀⠐⡀⠎⠀⠉⠉⢿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠤⢈⡁⢻⣿⢠⡀⡁⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠸⣿⠒⠒⠀⠀⠀⠀⠀⠰⠦⠀⠁⣴⠒⠆⠀⢸⣇⡄⠀⠁⠀⣿⡏⠀⡀
⠀⠀⠀⠀⠀⠀⡄⢿⣶⠀⣠⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣭⠟⠀⢸⣿⡇⢸⡄⠄⠉⠁⣰⠇
⠀⠀⠀⠀⠀⠀⠹⢌⠿⣰⣿⢿⣄⠀⠄⠢⢀⣑⠃⢀⣤⠄⠛⠣⡄⠀⣄⣠⡿⠻⢌⠿⡤⢆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠈⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢂⣂⡄
⠀⡀⠀⠀⠀⠀⠀⠀⢄⢶⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆⣸⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣾⣿⣿⠃
⠀⢠⣆⡀⠀⠀⠐⢌⣿⣯⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣴⠈⠸⣆⠐⣄⡟⢐⣥⣿⣠⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡖⠀⠙⣿⡟
⠀⠈⣿⣿⢆⠀⢬⡻⣿⣿⠯⠭⠀⠀⠀⠀⠀⠀⠀⠀⢤⡘⢆⢹⡆⠀⠻⣤⡿⠛⣫⡥⠄⠀⠈⠁⠀⠀⠀⠀⠀⠈⣿⡟⠀⢠⡄⣿⠀⠀⢀⣤⣤
⠀⠀⠽⣿⣯⡼⣲⣿⣷⣿⣟⠂⠀⠠⠀⠀⠀⠀⠀⢲⣌⣳⣌⣛⢛⣿⣾⣯⠦⠞⣅⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠀⠀⠉⣇⣿⠀⠀⢨⡿⠋
⠀⠀⠈⢿⣿⣿⣷⣿⣿⣿⠷⡀⠐⠶⠄⠂⠀⠀⠀⠀⠀⡽⣿⠛⢻⡿⢹⠓⣞⢻⡟⠻⣖⠲⠀⠀⠀⠀⠀⣺⣶⣿⣹⣆⣶⣤⣿⣹⢲⣴⣾⣴
⠀⠀⠀⠀⠹⣿⣟⣿⣿⠛⠠⠅⣼⣿⣦⠀⠀⠀⠀⠀⠁⢰⠁⠠⠞⢳⡈⠡⠋⠇⠩⠀⠈⠓⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣷⠤
⠀⠐⣫⠠⢙⣏⠽⠛⠋⣿⣤⣶⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠛⠿⠛⠋⠉⠁⠀⠀⠀⠀⢸⡈⠑⠈
⠀⠀⠿⠁⠻⠿⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣤⡀⠀⠀⠀⠀⠀⡀
⠀⢀⡀⠀⠐⠋⣀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣦⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣇⣸⣀⣼⢁⣤
⠀⣦⣉⠀⣤⣄⢉⣤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣯⣿⣿⣮⠄⠀⠠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣿⠿⣿⣿⣯⣝⠃⠈
⠀⠀⠋⣉⠹⠿⣟⣻⣞⠰⠄⠃⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⣭⣿⣵⡯⠀⣀⣰⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⣁⠚⣇⢻⣿⣿⣇
⠀⢠⠾⠺⠻⡎⢛⣧⠍⠰⢾⣶⡀⠀⠀⠀⠀⠀⠀⢰⣀⣼⣦⣴⣿⣿⣿⣯⣷⣿⣿⣯⣅⣤⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠙⢿⣿⣷⣤⠠⡄⡤⢀
⠀⠀⠀⣴⣿⣧⣪⣿⣄⣴⣶⣶⠆⠀⠀⠀⠀⢀⢤⣴⣿⣿⣿⣷⣿⣿⣿⣻⣻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣿⢇
⠀⠀⠞⢋⣾⣿⣿⠹⡏⠛⡏⠛⠀⠀⠀⠀⠀⠃⠒⠁⡯⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠿⠻⣿⣿⡆
⠀⠀⢀⣾⣟⣻⣷⡶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⠿⠆
⠀⠀⠀⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣯⣤⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀
⠀⠈⣿⣿⣴⡄⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣿⣏⣽⣷⣿⣿⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣄⠀⠀⠀⢀⣴⣿⣡⣤⣤⣿⡃
⠀⠀⠘⡿⠿⣽⣿⡵⣂⠐⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢠⡿⠞⠛⠉⣥⣶⣦⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣾⣿⣦⣿⣿⣿⣿⣿⣯⣤
⠀⠀⠀⣚⣿⣿⣿⢟⡥⠀⣀⣠⣶⣾⡿⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⣿⡗⠀⠀⠀⣠⣠⡄⠀⠀⠀⠀⠀⠀⣘⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡷
⠀⠀⢀⠀⢶⣿⣟⣿⣾⣵⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣷⠀⠚⠛⣻⡿⢷⡆⠀⠀⠀⢀⣘⣟⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠐⢶⣤⣤⣥⣾⣟⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣦⢀⣼⡿⠁⠀⠉⠀⠀⠀⠈⢰⠋⠨⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⡧⡄
⠀⠀⠙⠻⢿⣿⣷⣿⠟⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠛⣿⣿⠟⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠋⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠈⢋⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⢤⣤⡄⣀⢠⠤⡄⠁⠀⠀⠹⠎⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⠃
⠀⠀⠀⠀⠀⠐⠋⠃⣀⠊⠘⠛⠀⠀⣆⠀⠀⢠⡤⣶⢠⣤⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡈⠀⠀⠀⠀⠀⠀⠀⠑⠐⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠙⡠⠊⠀⠀⢀⠇⠀⠐⠤⢀⠠⠈⡐⠀⠈⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠁⠀⠲⠌⠀⠋⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⣈⡁⢐⠀⠉⠉⠐⠠⡀⠀⠲⠀⢰⡤⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠄⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠋⠀⠀⠀⠀⠀⠐⣶⠀⢈⣤⠿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢂⠀⠀⠈⠁⠀⠄⠠⠀⠀⠀⠂⡀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠀⠦⠀⠀⠀⠀⠀⠀⡀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⠂⠤⠼⠶⠤⠖⠒⠢⠄⠀⠅⠀⠀⠀⠀⡄⠘⠀⣂
⠀⠀⠀⠀⠀⠊⢀⠈⠁⠂⠰⠋⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⠈⣡⠀⢀⣀⠀⠀⠀⠀⠁⠀⠄⣀⠐⡄⠀⠨⠇⠀⢐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢲⠌⣭⠀⠀⠡⡀⠀⠀⢉⢀⣤⠀⢈⣁⠠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⡀⠙⡆⢀⣿⡆⢰⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡣⠭⠀⠀⠀⠈⢁⠀⠉⠁⢀⡴⠁⢁⠀⠰⡞⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠐⢄⠀⠛⠁⢀⡀⠈⡘⠃⠀⠈⠉⠉⠀⠐⢸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⠀⠁⠀⠀⠈⠀⠀⠐⠰⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⡻⢘⣿⠂⣀⠄⠀⠀⠀⠙⢢⡈⢢⠀⠘⠂⢛⡴⠈⠸⠷⡆⠀⠀⠀⠀⠀⢶⡂⠚⠁⢰⠀⠚⠘⠈⠁⠈⠀⠠⡀⠀⠀⢀⠲⠆
⠀⠀⠀⠀⠀⠀⡭⠀⡁⠀⡉⠲⡄⠀⢀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠉⢠⠴⠀⠀⠠⠠⣦⠄⠀⠀⠈⠁⠀⠀⢃⣀⠀⠀⠰⣍⠀⠀⢢⠀⠠⡄⠀⠁
⠀⠀⠀⠀⢴⡈⠅⠀⠀⢠⠀⠱⣤⡀⠨⠓⠄⢀⠠⡤⢄⠀⣄⠒⠀⡁⢀⢠⠀⢀⣀⣈⣁⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠓⡜⠢⠀⠀⣐⡀
⠀⠀⠀⠀⠈⠉⠢⣀⠉⠉⠀⠀⡈⠀⣀⠀⢀⠈⠐⠋⢸⠀⠈⠠⠆⢀⠸⣏⠁⠘⠏⠁⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣄⣀⢒⣂⠐⠈⠙⠂⢠⣤⠍⠑⢻⣆⢈⣦⠀⠀⡞⠰⠚⣀⠤⢧⠀⠉⠁⠫⠄⠀⣉⣉⡄
⠀⠀⣤⡌⠁⠄⠀⠠⠗⠀⠋⠙⠉⠙⠆⠀⠀⠀⠀⠀⠙⠀⠘⢶⠒⠠⠙⠂⠀⠀⢆⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⠀⢀⡀
⠀⠀⠉⠈⠀⢀⡀⠁⡐⢤⠠⣀⡀⢹⠾⠇⠀⠀⠀⠀⠀⢀⠤⢄⡛⠒⢶⣾⠃⠀⠀⠉⡀⠲⠖⠤⠀⠀⠀⠀⠀⠿⣁⠀⠁⠂⠈⡟⠋
⠀⠀⠀⡤⠈⠀⠐⠂⠘⠉⠀⢀⡀⠀⠴⢒⢤⠀⠚⠞⠡⠂⣈⠓⡉⠉⠻⠏⠀⢲⠤⠬⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈
⠀⠀⠀⣀⣀⢹⠟⠀⠀⠀⠀⡿⠗⠀⠸⠁⢸⠖⣆⠀⠀⠀⠀⠑⠦⠉⠐⡠⠈⢯⠀⢀⣀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡜⠁⠀⢀⡄⠰
⠀⠀⠰⡷⠀⢀⠀⠀⠰⢙⡮⠀⢢⠄⠀⠈⠁⠑⠊⠀⠀⠀⠠⣤⠆⠀⡄⠀⡀⠀⠑⡌⠉⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠁⣀⣰⡒
⠀⠀⠀⠀⠀⡆⡔⠀⡀⠀⠀⠀⠘⠳⠄⠀⠀⠀⣀⠀⠀⣴⣂⣀⠀⠈⠀⠀⠙⠒⠂⠩⣀⠙⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡔⠋⠀⣀⠤⠤⠄⣠
⠀⠀⠀⠀⠐⢣⡇⢈⠀⢀⡀⠀⠈⠀⠈⠒⠋⠉⠡⢹⡯⡄⢧⠀⠁⢢⠤⠰⡦⠀⡀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣀⡤⠀⢉⡀⠤⣀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⣎⣀⠁⡖⠀⠀⡆⢀⢄⡀⠀⠀⠀⠀⡔⠌⠃⠀⠀⠀⠀⠂⠀⣬⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡡⠖⠉⠁⠀⠀⠀⠀⢀⡐⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠄⠀⠁⣆⠀⠑⠀⠀⢠⠀⠁⠀⣠⣄⠐⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠃⠴⠺⠊⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⡿⠿⠛⢉⢠⣤⣀⡀⠠⢠⣂⣜⠋⠈⠐⠒⢊⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣯⣤⣤⣬⣍⣉⣹⣿⠀⠀⣾⣽⣯⣵⣶⣾⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⡟⣟⣯⣽⠿⡟⠛⢻⠛⠙⠉⠛⢛⠛⠻⠿⢿⣿⣏⠁⠀⢢⣈⢷⣦⣔⡀⠀⠉⠉⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣛⣵⠞⠻⠍⠑⠢⠀⠙⣴⡚⠀⠀⠀⠀⠀⠀⠀⢈⠀⣹⣿⣴⣾⣿⠁⠀⣻⣿⣿⣇⣤⠄⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡾⠋⠀⠀⠒⣒⢻⣿⣾⠶⠛⢿⣾⣿⣷⣶⡿⠒⢫⣗⣵⠿⣷⣿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣥⡟⠀⠤⠠⠄⢰⢾⣛⣻⣿⠅⡀⢸⣿⣟⣿⠟⣶⣶⣷⣾⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠿⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⢚⡉⣻⣿⣧⠀⠀⣿⣷⡇⠆⣾⣿⠿⠟⠁⠀⠀⠐⣶⡄⠀⢠⡄⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⢀⣠⣤⣤⣴⣾⣿⣿⣿⣿⣶⣶⣶⣄⠒⠓⠀⠀⠟⠁⠀⠀⣿⠁⠀⠀⠀⠀⠁⠀⣿⣷⠀⠈⣿⣦⣤⣄⣀⣀⡼⠋
⠀⢠⣶⡿⠛⠋⠉⠉⡀⠀⠀⠀⠀⠈⠉⠉⠛⢷⣤⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⠄⢤⠤⠠⠿⠻⠀⠀⠸⠿⠛⠋⠉⠁
⠀⠻⡏⠀⠀⠠⠶⠾⣿⡉⠁⣰⡦⠄⠀⠀⠀⣠⠟⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⣤⣴⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠉⠳⠶⣤⣤⣴⣽⠅⠀⠙⣵⢦⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⢛⢙⣫⣭⠭⣭⣉⠛⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠳⢦⣭⣿⣿⡇⠀⢸⣿⣷⣦⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠐⢿⡄⠀⠘⣿⠃⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⣷⣤⣀⢀⣈⢶⣾⣴⣶⣿⡿⢏⣶⣧⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣄⠀⠈⠹⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣟⣿⠏⠀⠀⣠⣾⢡⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⡟⠙⠛⠋⠉⠀⣀⣴⣠⣿⣏⣵⣟⠘⠚⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠀⠀⠐⠚⠻⠿⣿⣿⣿⣿⣿⣿⢷⡆⣰⣆⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡄⣞⣿⣟⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⡾⣿⣿⡼⣣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡶⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠚⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠙⣿⣧⣷⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠄⠉⠈⠉⠋⠉⠉⠁


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁⠉⠎⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠉⠛⠶⠀⡦⠀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠈⢩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⢀⡘⣏⡼⢏⠀⣁⠀⠀⣚⠒⠀⠂⠴⣤⡄⠀⢀⡀⢀⣼⣦⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢴⠰⠆⠶⠀⠀⡴⠄⠠⢀⣈⠀⢻⣷⣾⣦⣸⠀⠀⠀⠀⠉⠂⠄⠀⣰⣽⣶⠀⣴⠿⠺⠟⠋⠀⠀⠀⠀⠿⡄⠀⢸⠆
⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⢦⠀⡀⢠⣯⠀⠀⣹⣼⣿⠉⠠⠁⠀⠀⠀⠀⠀⠘⢋⣈⠷⢟⡧⢴⣦⠜⠓⢶⡎
⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠠⠜⠶⠆⠘⣿⡏⠀⠀⡜⢋⡟⠁⠈⠘⣁⣀⠀⠀⠀⠀⢴⡿⣿⣆⠀⠀⢠⡶⠋⠀⠀⠙⠘⠃
⠀⠀⠀⠀⠀⢀⣀⠖⠀⠐⢂⣸⣤⣽⣧⣼⡇⣄⢸⣷⣿⣿⣿⣿⡘⡁⠘⠀⠀⠀⠀⢹⣶⠿⡿⠀⡴⠋
⠀⠀⠀⠀⠀⠀⢻⣄⡠⡀⠤⣭⣿⣿⣿⢿⡷⣼⠟⠉⣏⣿⣿⡿⠘⡆⠀⠀⠀⠀⠠⢶⡧⠏⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢮⡀⣠⡀⢀⢀⣼⠁⠈⠈⢺⠄⣀⠀⠐⢢⡴⠛⠁⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣶⣦⡄⠙⣦⠘⠀⢻⣿⠀⠀⠰⠊⠁⠈⠀⠀⣻⠟⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢃⡀⢀⡀⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢉⠿⣿⡿⡦⠘⢦⠀⠀⢸⠀⣷⢀⠘⠻⢯⡄⢠⡏⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⣥⣾⠉⣀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢏⣟⠛⠁⠁⠆⠈⣧⠀⠘⣧⣀⠘⠁⠀⠈⠟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⠀⠈⢿⣿⣦⡜⠀⢀⡹⣏⡀
⠀⠀⠀⠀⠀⠘⣁⣤⡀⠀⠀⠀⠙⢦⠀⠀⢈⡄⠘⣇⠀⢸⡿⠀⠀⡀⠀⢀⡈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠠⢀⣿⡹⡿⣧⣾⠟⠉⠠⠈
⠀⠀⠀⠀⣶⣿⣿⣿⣵⠆⢀⠀⠀⠈⢇⠀⠀⠀⠀⠘⡄⠀⠀⣴⣾⣧⠀⠈⣣⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⢌⡀⢀⡶⢤⡀⠄⠀⣾⢿⣷⣾⣿⠧⠄
⠀⠀⠀⠀⣿⣬⡿⠋⠹⠷⠈⣧⣤⠀⠀⠀⢤⠀⠠⠀⢯⡀⢠⣿⣿⡏⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣄⠀⠹⠆⠠⠘⠛⠁⠌
⠀⠀⠀⠛⢻⣾⣹⣽⠇⢀⣼⣿⣷⡆⠀⠀⠀⠀⠀⠀⣎⠀⢸⣿⣿⡃⡀⠀⠘⠁⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠘⢷⡄⢠⡀⠀⣠⠳⠘⠲⠀⠈
⠀⠀⠠⣶⡿⣌⡋⠁⠰⢀⡈⠁⠙⠋⠀⠀⠀⠀⠘⢧⡿⡦⢈⣯⠻⠧⠄⠐⠀⠀⠀⠀⠀⠠⣬⠃⠠⠄⠀⠀⠀⠀⠀⠈⣿⠈⣷⠀⣡⠄
⠀⠀⢰⣮⠀⠈⠁⠂⠀⠀⠈⠣⠴⢄⠀⠀⠀⠃⡄⠀⠙⣧⠈⠡⡶⠄⠀⠀⠀⠠⡴⠀⠀⠠⠀⠀⠀⠶⠈⠀⠀⠀⠀⠀⢸⠀⠏⠀⣷⠀⠀⠰⠂
⠀⠀⠘⢿⣾⣷⡀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣤⣤⡤⠂⡀⠀⡉⠀⡀⠃⢀⣀⢀⡔⢆⡀⠀⡀⠀⠀⢀⠁⢠⠒⠂⠀⠀⠀⣾⡄⠀⢰⠏⠘⠲⠁⠀⠀⠀⡀
⠀⠀⠀⠘⠻⢦⣁⠀⠀⢺⣿⡶⠻⠃⠀⠀⠻⢋⠀⠀⠁⣷⣽⣠⠇⠀⢾⡟⢿⣿⣾⢿⣠⣁⣠⠖⠈⢳⠀⠀⠸⡄⠀⠀⣿⡀⠀⣸⡇⠒⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠉⠛⣠⣞⠁⢀⡀⠀⠀⣀⠀⠹⣶⣷⣄⠠⣿⣿⣄⢀⡀⠉⠂⠙⢿⣧⠛⣿⣄⠀⣰⣾⡧⣼⠟⠃⠀⢠⣿⣿⠀⠉⣠⣤⠞⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠋⠑⠿⠿⢦⣭⣄⣉⡀⠐⢿⣿⣿⣄⠀⠀⠀⠌⢿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⢠⡟⠘⢁⡴⠊⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠄⠀⠐⠀⠀⠀⠀⠀⡈⠙⠛⠿⣷⣦⠈⣙⠻⣧⡀⠀⠀⣈⣿⣿⣟⡁⡀⣠⡄⣀⣀⣤⣿⠀⢰⡛
⠀⠀⠀⠀⠀⠀⠀⠐⣦⠉⠛⡙⠆⣹⡆⠀⣠⣴⠃⣤⠀⠀⠈⠛⢿⣌⣳⠌⢻⣦⣀⠈⣿⣙⠻⢧⣭⣭⡽⠿⠛⣾⣿⠀⠀⡄⠀⢀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡐⠶⣉⠀⢻⣶⣾⣾⡇⡀⣨⢇⠐⢈⡷⠀⠀⠂⠀⠻⣿⡆⢠⣽⣿⣷⡄⠹⢶⣴⣀⣀⣤⢶⣿⣿⡟⠀⠀⠀⠀⠉⠁⣾⣿⣿⡏⠓⢁⡀
⠀⠀⠀⠀⠀⣹⠀⠀⠉⠀⠈⠁⠹⣿⡽⣿⡉⢈⢀⠈⠀⠈⠡⠀⢪⣤⣬⣇⠘⣋⢿⣿⢳⣶⣄⠉⢻⡟⠙⠁⢀⣹⠇⠀⢀⣤⣴⠟⢺⣯⠳⡇⠀⠞⠎
⠀⠀⠀⠀⠀⢠⡂⣠⣄⠀⠀⠀⠀⠈⢻⣟⣿⣦⢧⡐⣂⠀⢼⢇⣸⣿⣿⣇⡘⣧⡘⣿⡀⢿⣿⢷⡌⠁⠀⢀⣴⠏⠀⡼⠟⠉⠀⠀⠀⠙⠃⠉⠁
⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣧⣄⠀⠀⠀⠙⠂⠙⣟⣿⣿⢶⣶⣾⢿⣏⢀⠸⣷⡼⠁⣼⡇⠼⠃⠀⢹⡄⠠⠊⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠛⡋⠄⠀⠀⠀⠸⣦⡌⢐⣧⣽⣿⣿⡇⠹⣿⣶⣿⣿⠀⢿⡏⠀⠀⠀⠘⣧⠠⢀⣾⣿⠀⠀⣰⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠰⣧⣤⣴⣦⣄⠀⠈⠻⣿⣛⣿⣷⡾⠶⣦⡽⣯⣿⣿⣀⣿⣿⠀⢹⠀⠀⠈⡀⢀⣹⣿⢀⡴⠃⠉⢻⡁⠐⠓⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠛⠘⠻⣄⡙⠳⣦⣤⣽⣿⣿⡿⠑⠂⠈⠻⣎⡹⠋⠉⠑⢿⣆⠀⢣⠀⠀⢡⠀⢻⡿⣯⡀⠀⠀⠂⠀⠀⡀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢄⠀⡈⠉⠉⠋⠁⠀⡄⠀⠀⠹⣧⡘⢄⣠⢀⠀⢁⠈⠁⠀⠈⠀⣛⣶⡜⣷⣏⠀⠀⠐⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣻⣿⣷⣾⣿⣣⡄⠁⠀⠀⠀⢻⣇⢠⣴⣿⣿⠉⠀⢣⡀⠀⠀⣹⡯⠞⠃⠘⡄⣀⢀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠋⠀⠳⠉⠉⠀⠀⠀⠀⠀⠸⣿⢺⣏⣿⡛⠇⠀⠈⢁⠄⠀⠀⠀⠀⠀⢸⠧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣯⡀⠽⠲⠀⠀⠀⠄⡀⠀⠀⠠

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠂⠀⣴⠆⠀⣀⡀⠀⡬⠀⠀⠃⠀⠘⡅⠀⠘⡇⠀⠁⠀⡈⠃⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠁⠀⠀⡁⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⠀⠀⠰⠦
⠀⠀⠀⠀⢤⡶⠀⣄⣙⣷⠀⣼⠄⠃⠤⠀⠻⣄⠉⠢⡇⠀⣀⠰⡟⠃⠀⠆⠀⠀⠀⠀⣴⢲⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠓⠃⠀⠉⠁
⠀⠀⣠⣀⠀⠀⠠⢀⣴⠀⠒⠀⠰⠂⠲⣴⠆⠀⠀⠄⠀⠁⢈⠃⠦⠅
⠀⠀⡀⠁⣀⣁⣀⠀⢀⡀⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠛⠺⡝⠱⠾⣦⢜⠀⣈⠶⣅⠀⠀⢀⢀⡀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠂⠴⠄⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⠐⠉⠁⠀⠚⠛⠛⢛⠛⠓⠒⠂⠀⠀⠀⠀⢀⠀⠀⢡⢀⠀⠀⠘⡏⠀⠀⠀⠰⠟
⠀⠀⣠⣇⠐⠀⠀⠀⢠⡀⡀⠀⠀⠶⢶⠀⢈⣨⣍⢰⠀⠀⠀⢄⠀⠄⠀⠈⠂⢃⣈⠀⠀⠀⠀⢧⠀⠀⠀⡄⠀⠀⠤
⠀⠀⡈⠋⠀⠙⠁⠀⠾⠋⠁⠀⠀⠀⠀⠀⣤⠀⣀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠
⠀⠈⢍⡲⠻⡆⢠⣐⣶⢇⢀⣈⠁⢀⠀⣄⢻⣇⣿⣿⠇⠀⠀⣶⡀⢸⡇⠀⠀⣦⠀⡀⠀⡀⢠⠀⣠⢈⠀⡀
⠀⠑⢚⡗⠛⠆⡈⠁⠉⢠⠠⡀⠀⣾⡇⠘⣧⣿⣟⡟⠀⠀⢐⡿⣇⢠⡥⢠⢂⣿⠀⡆⢀⡇⠈⠃⡀⢸⣿⣧⡀⠀⠰
⠀⢠⡀⡅⠁⠀⠩⡐⢸⡜⠃⠑⠸⣿⡇⠀⡎⠙⡟⠀⠀⠠⣶⣿⡵⡀⠀⠈⢸⣿⠀⡇⢸⠇⠀⠀⠀⠂⠽⠞⠁⠀⢈
⠀⢸⠀⢣⠀⡤⠢⠀⠀⠁⠀⢂⠸⣿⡇⠀⢤⡀⡄⢀⡀⢘⡺⢷⣿⣖⡀⠑⢸⡿⠀⠃⢸⢀⡆⠀⠠⠀⠀⢂⠀⡆
⠀⠘⢧⠘⠀⢀⠃⠀⠀⠀⠘⠌⠈⠿⠅⢀⣈⣿⠁⠈⠁⢨⢭⣿⣿⠭⠄⡀⣼⡇⠀⠂⠀⢸⠀⠀⠀⠀⠀⢼⠅
⠀⠈⣴⡇⠀⠉⠴⠷⠆⢰⠶⡦⠀⠠⠒⠃⡾⠃⠸⠿⠇⠰⣴⣤⡄⠒⠀⠀⠉⠀⠀⠀⠠⠿⠆⠀⣀⠀⠀⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣔⣀⠠⠤⣀⡀⣀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⡿⣴⣾⣿⣿⣿⣿⣏⣩⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢿⣻⠿⢿⣾⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣉⠀⢘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠟⢶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠏⢹⣿⣿⣿⣿⣷⣈⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣸⣿⠿⢿⣿⣿⣿⣆⣠⣶⣿⣿⣿⣿⢟⠋⢉⠻⣿⠶⠉⠀⢀⡨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣴⣿⣿⣿⣿⣿⣿⣏⠁⠘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣴⣴⣾⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣰⣿⣿⣿⣿⡟⢿⠿⢟⠥⢀⣠⠀⣖⣤⣤⣶⣦⡀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⢿⣿⣿⣿⣿⣷⣴⡞⠛⠂⣉⣵⣾⣿⣿⣿⣿⣿⣿⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⡟
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⡀
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖
⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⠛⠛⠛⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛⣛⣉⣉⣉⣉⣉⣉⣙⣿⣻⣿⣋⣉⣉⣁⣀⣈⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣹⣛⣉⣛⣟⣽⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⢀⡷⣄⠀⠠⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⣄⣄⢠⡜⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠘⣿⣃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠏⠀⠀⠰⠀⠀⠀⠛⠿⣥⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀⠢⠀⢀⡄⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠈⢡⣤⠔⢀⣁⠄⠐⠀⠀⢘⣷⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⡴⠗⠀⠈⠉⠗⢦⣤⠛⠿⣶⣏⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⢀⠐⢀⠀⠀⣸⠃⠀⠀⠛⡟⠓⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢢⠀⢬⡆⠀⣿⠀⣴⡀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣀⡸⠻⣠⠀⢰⡿⢷⠄⣰⡾⣶⠾⡋⢃⣧⡀⢾⠛⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⣶⠶⠤⢦⡴⠃⡀⠈⠁⠶⢸⣶⡀⠌⠿⢿⣌⢀⣻⣿⢳⣾⣄⣷⣔⡀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢺⡆⠢⢰⣿⣀⣠⠋⠀⠰⠋⠀⠛⢿⠀⣹⡞⢻⡿⣿⣿⣅⣿⣿⡿⢿⡙⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠟⠀⣼⣾⡁⣹⡿⣠⠀⢀⣤⣤⡀⣀⢠⣿⣷⠾⡿⢿⣿⠻⣽⡗⡼⠋⠳⣠⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⢀⣶⡿⢻⡏⠀⠀⢠⡄⡾⠗⠀⢻⣿⢿⣿⡷⡒⠀⠸⠿⠞⡛⢈⠓⢛⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢰⣿⡿⠀⣞⢡⣦⣀⠘⠁⠉⢤⣠⣻⣟⣏⣿⡦⠀⠀⣤⠀⠸⣛⠾⠃⣻⡆⠸⡀⠀⠈⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢷⣾⣟⠁⣤⣌⠁⠀⡘⢿⠶⡀⢀⡉⢀⣿⣿⡿⢧⡄⠙⠋⠀⠑⢹⠀⠀⠘⠀⠀⢼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠓⠛⣸⣣⣴⣿⠿⡜⢸⣇⠉⠀⢰⣾⣧⠸⣿⣿⣧⢘⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠁⣿⡿⠃⠻⣸⡟⠀⡖⣃⢨⣿⠾⣿⢛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢀⠀⣺⣧⣾⣼⡀⢈⡤⢌⣶⡞⠁⢤⠀⠟⢻⠃⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⠼⣿⠛⠁⡼⢳⣿⡄⢀⡿⠃⠁⢈⣽⣦⡼⡀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠋⠀⠀⠀⠦⠀⠲⡟⢠⣄⠈⢋⢦⠐⡮⠏⣿⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠀⠹⡿⠇⠄⠀⠀⠁⠀⡿⠀⠈⠙⠀⢠⣳⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠈⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁





⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠉⢉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢤⣤⣤⡀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡄
⠀⠀⢀⡒⢈⣿⡆⠀⠀⠀⡂⡠⠻⣳⣾⣾⣯⠄⠀⠀⠀⠀⠀⣾⣷⠋⠀⠀⠀⠀⣰⣎⣀⠀⠀⣼⠶⠞⣶
⠀⠀⢠⣸⡩⣿⡇⠀⠀⠀⠀⠂⡀⢽⣿⣿⡷⠆⢄⠀⠀⠀⣸⣿⠗⠀⠀⡇⠀⠀⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⢀
⠀⠀⢀⡼⠷⢻⡇⠀⠀⠀⠀⠠⠁⣾⢜⡝⠁⠘⠀⠀⠀⣰⣿⡏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⡀⠀⢀⣼⣀⣀⢀⢀⡀⣤⡄
⠀⠀⣾⣢⡄⠻⣧⠀⠀⠀⢀⢌⣤⠗⠛⠁⠀⠀⠀⠀⢰⣯⣿⡅⠀⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠰⢧⣾⣿⣿⣿⣿⣿⣿⣿⠎
⠀⠀⣿⢉⠻⡟⣇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⡞⠀⠀⠀⠐⢦⠀⠀⢀⠀⠀⠐⣿⣿⣿⣿⠻⠿⠞⡿⢷⣷⡆
⠀⠀⡻⣦⡰⡟⣩⠁⠀⠀⠰⣤⡀⣀⢴⠄⠀⠀⠀⢨⣿⣿⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡎⡇⠀⠀⠀⠉⠈⠉
⠀⠀⣦⣽⣿⡸⠗⠀⠀⡀⠠⡟⢌⠑⠁⣴⡆⠀⢀⣿⣿⣿⠀⠀⠀⢸⣁⡾⠦⠀⠀⠀⠀⠀⡿⢻⣠⡤⠂⠀⣷⠀⠀⠀⢤⣀⣴
⠀⠀⠀⠉⠈⠁⠀⠀⠀⠁⠀⠙⠀⠀⡀⢙⠂⠀⢀⣯⡿⠟⠀⠀⠀⠐⡁⠀⠀⣆⡀⠀⠀⢀⣇⣀⣣⡀⠀⢸⣻⣄⣀⡀⠐⠿⢧⡤
⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⢰⣶⠾⠛⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠁
⠀⠐⠒⡆⠀⠀⢀⣼⠇⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠂⠀⠀⠀⠀⢠⣞⣶⡀⢄⣠⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠸⡆⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⣴⣧⢾⣦⢀⠀⠀⢸⠀⠀⢴⣷⣮⡄⠀⠀⠀⠈⠋⢀⠃⠀⠃
⠀⠀⠀⠀⠐⠌⠀⠀⠀⠀⢀⠀⢀⢀⠀⠈⢶⡠⣽⣡⠔⠚⠡⠀⠀⠸⠀⠀⠌⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢠⣄⡦⡴
⠀⠀⠀⠀⠁⠀⣰⠾⠂⠈⣰⢱⠗⠀⠀⠀⡚⠑⠩⣥⣶⡟⢀⠀⠀⠀⠀⠐⣶⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⢸⣇⣿⠃
⠀⠀⠀⠀⠀⢨⣿⠃⠀⠀⠀⣾⣬⠀⠀⠀⠠⢌⣅⡁⣒⣡⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣤⠀⠀⢀⠀⣄⠀⣼⣯⣴⠀⡠⠟⠉
⠀⠀⡤⣤⠤⠘⠀⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠐⠈⠋⠁⠀⠀⠀⠀⠀⠈⠹⢿⡿⣿⣿⠅⠀⢀⡃⠀⠀⠀⠀⠘⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠙⣫⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠲⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣶⠴⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡜⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⡊⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠑⠀⡄⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⣠⠀⠀⠀⢸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠀⣀⠈⠘⣆⠀⢰⡄⠘⡄⠀⠘⠆⢀⠀⢤⠁⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢀⠀⠀⠉⠣⣀⠈⠛⢿⡄⠀⠁⡀⠃⠘⠦⡐⠀⡆⢠⡀⠀⢀⣠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⢄⣀⠈⠀⢄⠀⠙⠰⣆⠸⡤⢄⠀⠀⠘⣇⠈⡁⠀⢨⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠨⠄⠀⠷⠈⠑⠛⠀⣹⠒⠂⠄⠢⠵⠀⣈⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠠⣄⠀⢄⣀⠁⣢⣀⣁⣀⡀⣠⣬⠅⠳⠖⠁⢈⠛⠠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠡⠖⠉⠀⠈⠀⠌⠉⠈⠉⠉⠁⠀⠀⠄⡀⠀⠀⠀⠊⠀⠠⠆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣴⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⢢⠐⠀⠀⠀⠀⠀⡀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣿⠄
⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠁⠁⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠰⡆⣄⠐⠀⠀⠀⠀⠀⡀⡀⣁⠀⠀⠀⠀⠀⠾⢟⡁
⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⠀⠀⠀⠂⢠⡈⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠋⣫⣧⣤⠀⠀⠀⠀⠀⣥⢾⣿⣱⠀⠀⠀⠀⠀⢺⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢀⣬⠿⣗⣦⣀⠀⠀⢀⣞⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⡄⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠸⣯⢶⡿⠿⠃⠀⠀⠀⢻⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⡭⠷⠋⠀⠀⠀⢲⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⣼⢻⣯⢧⡄⠀⠀⠀⠀⠞⢽⣿⡤⠀⠀⠀⠀⠀⣀⣀⣾⣧⠆⠀⠀⠀⠀⠰⡦⣦⣏⣑⡀⠀⠀⠀⢀⣾⣷⣾⢶⡀⠀⠀⢠⠏⣻⡆
⠀⠀⠀⠀⢘⣾⣽⣶⠀⠀⠀⠀⢰⣛⢾⡿⡷⠀⠀⠀⠀⠀⠈⡿⣿⣯⠇⠀⠀⠀⠀⠸⠭⠿⠉⠹⠃⠀⠀⠀⠀⠉⢿⣿⠉⠀⠀⠀⠈⠆⠐
⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠂⣠⠀⠀⠀⠀⢠⣆⢶⣶⣤⠀⠀⠀⠀⣠⣾⣷⣤⡄⠀⠀⠀⢀⣶⡟⠛⣟⣷⡀⠀⠀⠀⠈⣦⣥⠀⠀⠀⠀⣴⣲⣴⢶⢠
⠀⠀⠀⠀⠰⢴⡶⠿⠃⠀⠀⠈⢻⣿⠿⠿⠿⠀⠀⠀⠀⠻⠿⢿⠿⠿⠇⠀⠀⠸⠋⡀⠃⠐⢽⠃⠀⠀⠀⠑⠷⠗⠀⠀⠀⠀⠚⠛⠓⠟⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣼⠀⢌⣃⡀⠀⠀⠀⠀⣾⢷⣾⡬⡧⠀⠀⠀⠰⡟⠆⠀⠀⠀⠀⠀⠈⡐⢈⡂⠀⠀⠀⠀⠥⢾⣼⢻⡵⠀⠀⠀⣠⣶⣿⣶⣄⡀
⠀⠀⠀⠀⠈⠩⠷⠀⠀⠛⠀⠀⠀⠀⠉⠶⠿⠞⠁⠀⠀⠀⠀⠩⠁⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠋⠙⠾⠳⠿⠀⠀⠀⠙⠉⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⣿⣿⣿⣶⡄⠀⠀⠀⢸⣠⣾⠀⠀⠀⠀⠀⠉⢀⢀⡀⠀⠀⠀⠀⡀⢠⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⣧
⠀⠀⠀⠀⠘⠛⠟⠛⠛⠃⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠠⠶⠄⠀⠀⠀⠀⠀⠐⠺⠗⠂⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠢⣴⠉⠑⠦⡀⢀⣀⠀⣀⡼⠀⠈⠚⠉⠹⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡢⡈⠀⠀⠀⢈⡃⣀⠉⠉⠀⠀⠀⠀⠀⡀⠔⢻⠀⡀⢀⢠⠈⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠐⠪⡀⠌⠈⢇⠀⠀⠀⠀⠀⠁⠀⠀⠘⣇⣲⢎⡀⠛⢶⠚⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠘⠃⠀⠀⠁⠠⠔⡄⢠⢄⡄⠀⠀⠘⡟⠉⠠⠄⠄⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠑⠀⠀⠀⠁⠐⠘⠒⢦⡞⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⢡⡁⠀⠀⠀⠊⠹⢶⣐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⣀⠞⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⢙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠚⢉⠙⠢⡜⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠠⢂⡁⠀⠀⠙⢆
⠀⠀⠀⡠⠴⠲⣏⠔⠒⣦⠖⠀⠀⠀⠐⢄⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠻⣉⠉⠳⡀⢀⣈⠉⠲⣄
⠀⢀⠎⠀⠀⠀⡇⠀⣰⢃⠤⠖⠦⣄⠀⠀⠀⠀⠀⣠⠄⠂⠈⠁⠒⠄⠄⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠣⡞⢁⣉⡆⢈⠑⣆
⠀⡎⠀⠀⠀⠐⠙⠒⠃⠣⣄⠀⠀⠈⠑⢤⣀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡨⠄⠀⠈⠀⠹⠄
⠀⠣⣀⠀⠀⠀⠀⠀⢀⣀⠜⠀⠀⠀⢀⢀⢀⠀⠑⠢⠤⣄⣀⡀⠠⠀⠀⠀⡀⠀⠆⠠⠀⠀⠀⠁⠠⠀⠀⢀⠀⠀⡀⢀⠀⠀⠀⠄⠒⠠⠄
⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠤⡀⠀⠉⢡⡄⠀⠀⡨⠀⠀⠀⠀⠋⠀⠀⠀⠘⠁⠰⠾⠶⠒⠋⠉⠁⠁⠁⠉⢨⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⠠⠄⢠⠑⠒⠂⠊⠁⠈⠆⠔⠑⠆⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠠⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠂⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⣤⡶⣷⢸⢋⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠠⢠⡠⢳⡵⢦⣅⡏⡤⠀⡲⢙⣗⣏⣻⢟⢮⢰⡄
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⡀⠆⣿⣾⣧⠌⢺⡥⢀⠀⠈⢻⠞⣿⡿⣊⠰⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠿⢭⣵⣄⣿⠀⠘⠟⠀⠀⠀⢠⢀⡟⠐⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⢴⠞⡹⡟⠛⢆⠀⢸⠀⠀⠀⢸⡼⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠁⠀⠀⠑⢌⠂⠀⢀⡼⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠀⠐⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠁⠀⠀⡄⢰
⠀⠀⠀⠀⠀⢠⣄⢼⣿⣻⢸⡄⠀⠀⣄⠀⢀⠀⠀⠀⠀⢀⠀⢈⡂⠀⠀⠀⠘⢄⠀⠀⠘⠀⠀⢀⠀⢀
⠀⠀⠀⢠⣿⣙⠻⣾⠁⢻⢈⣥⡠⠊⢉⡫⣍⠀⠀⠀⠀⠀⠀⠂⠈⠑⠀⠀⠀⠀⠱
⠀⠀⠀⣾⡾⣯⣴⣌⣦⣞⣾⣭⠤⠰⡎⠀⣸⣷⠀⠀⠀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀
⠀⠀⠀⣿⣿⣾⠿⡿⣮⠴⡟⢻⣜⣛⣳⣆⣙⣽⠆⠀⠀⢢⡀⠁⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢘⣿⣻⣿⣾⣷⣯⣾⣿⡿⣬⣿⣭⣽⡋⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢂⠀⠄
⠀⠀⠀⠉⡽⢿⢿⡿⡿⢩⣿⣇⣿⣾⡜⣷⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡆⢀⠆
⠀⠀⠀⠀⠀⠀⠋⠛⠁⢸⡇⠟⠵⡆⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶
⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠛⢙⣷⣶⣶⣶⠞⠿⠶⣤⣤⣤⣴⣶⠾⠿⣿⣿⣿⣭⣴⣤⣢⣄⣠⣤⣤⡤⣖⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠛⠋⠹⠘⠁⠛⠶⠶⡾⠉⠉⢀⣀⣤⣼⠿⠛⠉⠁⠀⠉⠙⠛⠿⣟⢻⡹⠟⠉⠁
⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣀⣀⣤⣼⣷⣶⣿⢿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣘⣁⣀⣀⣀⣀⡀
⠀⠀⠘⣿⣶⣒⣒⣻⡿⠷⣛⣛⠋⣀⣩⣿⠟⠛⠙⠀⢸⢀⣸⣤⣄⣀⣀⣤⣤⣤⣶⠿⠛⠉⠉⠛⢛⣿⣿⡿⢿⣿⠿
⢀⣀⣤⣄⣀⣍⣍⣤⣴⣿⢻⡟⢻⠋⠉⣀⣀⣀⣠⣴⠿⠋⠁⠈⣩⠟⠉⠉⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠸⠃
⠉⠁⠀⣿⢻⢿⡏⠀⢀⣈⣀⡀⣈⡴⠛⢿⣻⣿⡏⠀⣀⠴⠒⠚⠁⠀⠀⠀⠀⢀⣤⣶⢿⣿⣋⡟
⠀⠀⠀⠉⠀⠘⠁⠉⣉⡽⢻⣿⠉⠀⠀⢘⡇⠸⠀⠋⠀⠀⠀⠠⠔⠒⠒⣲⡿⢟⣿⠅⣹⡗⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⢻⡇⠀⠙⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣖⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⢾⣿⣿⣟⣻⣿⣿⣿⣿⣿⡳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣭⣧⠿⡼⣿⣾⣯⣿⣽⣟⣻⣟⡧⡭⢀⡂⠲⣄
⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣿⢿⣿⣾⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷⣧⣭⣀⢥⡀⠀⠀⠀⠀⠀⢀⠄
⠐⠒⠀⠈⢠⣤⣤⢐⠀⠀⠀⠀⠉⠓⢷⣾⣯⣿⠿⢻⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡄⠀⠀⢀⡜
⠀⠀⠀⠨⢩⠿⡉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣯⠀⠸⠟⠿⠿⢿⣯⣛⣿⢽⠿⠿⠋⠀⠀⢀⠊
⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣭⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⠲⠿⣳⠤⢄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⢦⣎⣩⢉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠄⠀⠀⠀⠀⢸⣿⠿⡞⠙⣠⡤⣴⢶⠶⡶⢶⡤⣀⣀⡀
⠠⠂⢐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⠁⠈⡐⠃⢹⣤⣼⣾⡞⢻⣶⣿⣵⣯⣫⠳⣄⢀⣀⣀⣀⡀
⠀⠀⢀⡠⠤⠤⠤⡀⣀⣰⡀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⠀⣷⣶⣿⣟⠿⣄⠄⢀⣀⣈⣽⣟⣛⣿⣿⣿⣷⣿⣿⣟⢷⡀
⠀⠀⢘⠀⢤⣀⡠⠭⣭⣥⠏⢳⠀⠀⠀⠐⠀⠀⣿⢿⡀⢀⣿⣿⣿⠿⠿⠛⢿⣾⣯⣼⣷⣿⣽⣏⣿⣷⣶⣾⣿⣿⣖⣿
⠀⠀⠈⢳⣤⣤⣄⠀⣁⣤⢤⡾⠀⠀⠁⠈⢲⢳⣍⢎⡇⢸⣿⣯⣳⣾⣵⣜⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣿⣯⣾⠟
⠀⠀⠀⠀⠁⢿⡏⣷⣿⣠⠂⠀⠀⣄⡀⣰⢌⣿⣿⢹⠁⠀⠈⠙⠛⠿⢿⣿⣻⣿⣿⣿⣿⣿⣽⡿⠟⠋⠉⠉⠉⠉
⠀⠀⠀⠀⢀⣤⣙⣌⠓⠿⣀⣿⣇⢸⡙⣄⣿⣿⣿⢸⠂⠀⣟⢲⡀⠀⠀⠈⠻⣿⣧⠁⢿⡿⠋
⠀⠀⣠⠖⠋⣉⣁⣉⠙⡆⠨⣿⣿⣺⡇⣿⠧⣼⣿⣿⠄⠀⣷⣾⣧⣠⠒⣀⣀⣿⡿⠀⠘⣶⣖⣘⣂⣠⣄⣀⡀
⠀⢰⠃⣴⣿⡟⠋⡍⢡⣷⡀⠱⣿⣿⠁⣿⡆⣿⠿⡿⠀⠀⣿⣿⢾⣟⣃⣿⣿⣿⡧⠀⣴⣯⣭⣩⣭⣽⣿⣍⣿⣷⣦⡀
⠀⠈⠣⠻⠛⢷⠀⢷⢿⢻⢳⡀⠈⠻⡿⡉⠙⣻⣷⢲⠁⢀⣻⣾⣼⣘⣻⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⣷
⠀⠀⠀⠀⢰⢮⡇⠸⣮⠁⠀⢟⡢⢤⣸⣅⢠⣿⣤⠿⣰⣿⣿⣻⠃⠙⣿⣿⣟⡟⠀⠀⢸⣿⣿⣿⡿⠏⠫⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠂⠈⠈⡇⠀⢻⡀⢀⣾⡟⠓⣬⢡⣾⡟⠙⠠⡝⢿⣿⡿⢿⣿⣿⣽⣿⣷⣀⢀⣾⣥⣷⣿⣿⠂⠀⣟⠛⣋⢉⠉⠐
⠀⠀⠀⠀⣀⣶⢿⡀⢀⢃⢈⣿⣧⡀⠰⣿⠟⠗⣚⣃⣓⣞⡏⠀⠘⢿⣿⣿⣿⣾⣏⣽⣿⣿⣟⠿⣇⡠⣼⠟⠿⠠⠮
⠀⠀⠀⠀⣈⠁⠐⠉⠉⣤⣤⣼⣿⡿⠋⠙⠻⠛⠁⣮⣙⣿⣇⡀⢀⣸⡿⣿⣧⣉⣭⣍⠉⠀⠀⠀⠐⠀⠀⣀⢀⡈⠒⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠾⣾⣿⡇⠀⠀⠀⣠⣶⣿⣿⣿⣿⣯⣿⣿⡿⠟⠶⠬⠄⠤⠤⠄⠀⠀⠀⠈⣀⢀
⠀⠀⠀⠀⠐⠀⠀⠚⠁⠀⢱⣿⣧⡣⣄⣠⢶⣟⣿⣿⣿⠟⠛⠛⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⡻⠯⠿⠛⠑⠏⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠛⠛⠛⢈⣉⣉⠙⠻⢻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⣋⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣌⣛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⢿⡏⢭⡘⠇⡵⢘⡟⢿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⢿⣿⣢⣿⣧⣦⢠⣶⣽⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠿⠿⠿⠿⣿⣿⣿⣿⢱⢈⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⣠⣪⣥⣬⣿⣷⣶⣦⣄⡀⠀⠀⠈⠙⠛⢻⠀⠸⡏⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢀
⠀⠀⣀⣤⣾⡿⠛⢛⣉⣙⣛⣉⣉⣛⡻⠶⡄⠀⠀⠀⠸⡃⠀⡇⠀⠀⠀⠀⡀⠀⣠⣤⣴⣶⣿⣿⣿⣿⣶⣌⣦⡀
⠀⣬⡾⣻⣭⣴⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⠆⠀⠀⠀⢠⠁⢰⣿⠀⠀⠠⣿⣥⣶⣶⣶⣶⣽⣦⣒⣶⣲⣶⠮⢥⣙⣄
⠀⢋⣴⣿⣿⣿⣿⣶⣷⣿⣿⠏⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣇
⠀⠈⠛⠛⠋⠉⠀⠀⠈⢛⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⡄⠀⠀⠸⠿⠿⠿⠛⠛⠿⣝⡿⢙⣭⣉⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡇⢸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⡟⡁⠸⠛⠛⠻⠿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣮⣿⣝⣛⣿⣿⣶⣄⠀⠀⢸⠃⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠸⡄⠀⠀⠀⠀⣸⠄⢀⣿⣿⣯⣽⣯⡩⣿⣿⣶⣦⠀⠀⣿⠀⠀⠀⠀⠀⠀⡾⠅
⠀⠀⡆⠄⡀⠀⠀⠀⠅⠂⠀⢷⣦⢸⡆⠀⠀⠀⢺⣋⢸⡟⣿⣿⡇⢸⡇⠉⠉⠉⠉⠀⢀⣿⠐⡆⠀⠀⠘⡷⠇
⠀⠀⠴⠂⠁⣤⠀⠀⠉⠀⣀⠈⠻⣦⠣⡀⠀⠈⣉⣇⣼⡇⠉⠉⡇⢸⢁⠀⢼⡀⠀⠀⣾⡇⢰⠀⠀⠸⡷⠷
⠀⠀⠀⠶⡶⠇⠀⡀⠑⢒⠀⡄⠀⠹⡇⠑⣥⠀⠤⡇⢸⡇⠀⠀⡇⣿⡘⢈⣻⠂⠀⣾⠏⢀⡆⠀⠲⣦⣧⠀⡀⠐
⠀⠀⠀⠀⠠⠤⠉⠃⡀⢈⡀⣴⠀⠀⣹⣤⣌⠳⣄⣏⣹⡇⣸⣠⣧⡟⡇⠐⠷⢦⣾⠏⠈⠈⠀⡄⣠⡅⠀⠶⠀⠵
⠀⠀⠀⠆⠀⢾⡅⢀⣞⡀⠓⢀⣤⣈⢿⡶⢾⡦⡿⣿⣿⡥⢸⣿⣻⢃⡍⢃⣤⡾⢡⠆⣴⢠⠰⢿⡏⠑⡇⡠⡀⡀
⠀⠀⠛⢀⡖⣀⢩⣽⡞⠗⠈⢴⡌⠀⣾⠁⣸⡧⠿⣿⣿⣿⣼⡿⢇⡾⣻⣿⡿⣱⡇⢱⣏⢨⣀⡈⠯⢧⠨⡤⠈⠉
⠀⠀⠀⠀⠈⢉⠆⢊⠸⠘⣤⠘⣋⣷⣿⠀⣿⡀⠀⣻⠟⢻⣿⣴⣾⣿⣿⣿⠁⠉⢀⣿⣿⣿⣾⣵⣄⣼⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠉⣀⣾⣦⣥⣾⣿⣿⣿⣛⣿⡋⠻⣿⣠⡾⠁⠈⣹⣿⣿⡇⠀⢀⣾⣿⣿⡿⣿⣿⣿⣿⣿⠌⠐
⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢧⣿⣧⠶⠷⡿⣿⣿⠷⣾⣿⠻⢿⣿⣿⠟⠛⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⠋⠛⠛⠋⠑⠒⠿⠃⠙⠲⠦⠂⠈⠁⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁⠀⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡄⡤⣘⣁⡀⠀⢄⡤⡞⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠠⡈⢀⠋⠐⠋⢣⡭⣤⡾⠁⢀⣀⠨⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠲⠀⢶⡰⣤⠅⠃⣦⡗⢀⢂⡤⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⢉⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠙⠄⠃⠸⣧⠘⠢⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣤⢻⡶⠓⠆⠀⠋⠀⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⢀⠺⠃⠁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⡄⠀⡀⠀⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠈⠂⠀⠀⠀⠀⣼⣅⠂⠀⠼⣄⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠁⠁⠰⠀⠀⠀⠀⣀⠄⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠴⠚⡤⠀⠄⠀⠀⡀⣠⠄⢸⡧⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡇⢀⠳⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠠⠞⠠⢶⣦⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠁⠀⠀⠀⣈⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠒⠀⢠⠀⠀⠀⡀⠊⠀⠀⠐⠂⠈⠀⠀⠀⠀⠑⠀⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢒⡩⠝⠛⠛⠛⠲⣄
⠀⠀⠀⠀⠀⣀⣠⣤⣄⣠⣴⣴⣟⣿⣿⣿⣿⡟⣿⢿⣿⣆⠀⠀⠀⠀⠀⣀⠤⠄⠂⣀⡄⡒⠙⡡⠒⠁⠀⠀⠄⠤⠤⡀⠈⠃
⠀⠀⢠⣶⣿⣿⣾⣯⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣹⡟⠀⠀⠀⢠⠎⠁⠀⠈⠁⠛⠁⡠⠊⠀⠶⠀⠀⠀⠀⠀⡀⠘⠂
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡶⠀⣀⡀⡀⠀⠀⠂
⠀⠀⠸⣿⣿⢿⣿⣿⣿⣿⣻⣿⣷⣬⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠈⢅⠀⢤⠀⠀⠀⠀⠔⢆⠀⠙⠏⠀⠄⠀⠀⠂
⠀⠀⠀⠈⠻⠿⣿⣿⢿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣤⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠉⠉⠙⢳⣦⣤⠀⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⣽⣟⣻⢿⣿⣾⣿⣿⣷⣶⠖⢫⣵⣤⡀⠀⠀⠀⠀⠀⠀⠼⡚⠑⠃⡖⢤⡀⠉⠀⠀⠀⠝⣭⠿⠟⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⠠⠀⡀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣷⣿⣽⣭⡿⠟⠋⠀⠀⠀⠀⠀⠀⣦⠀⠀⠁⠢⢀⠀⠀⠀⠀⠀⠁⢉⡙⠉⠉⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⡄⠅⠈⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠆⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠃⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⡀⠀⠠⢀⠀⠀⠰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⣿⢿⣿⣿⣿⣿⣷⡧⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣸⡇⢰⡀⢠⠀⠀⠁⣝⣄⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠃⠿⢿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠨⣛⣾⣷⣿⣶⡀⣰⠟⠉⠀⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣠⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣶⣤⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⡿⠿⡿⠛⢿⠛⢿⢻⠻⠿⠿⢿⠿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⢛⡁⠀⠤⠠⣬⣔⣵⣄⣦⣄⣨⢞⠁⠂⡀⠉⠙⠛⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⣤⣀⣈⣅⢀⣙⣿⢉⢬⠠⢉⣿⣿⣶⣟⣻⣶⣶⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⣛⠂⢀⠀⢹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡴⣶⡴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⡀⠀⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡞⠀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⡼⠁⠠⢀⣾⠀⠀⢸⡄⣠⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣷⠃⣼⠀⡞⡏⠀⢰⠿⡷⢁⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡈⢿⣿⣄⡿⣾⢡⣵⣧⣷⣾⠗⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠈⠟⠃⠛⠸⠏⠟⠁⠉⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠂⠀⣴⠆⠀⣀⡀⠠⡬⠀⠀⠂⠀⠘⡅⠀⠘⡇⠀⠁⠀⠈⠃⠀⢀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠁⠀⠀⡁⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⣀⠀⠀⠰⠦
⠀⠀⠀⠀⢤⡶⠀⣄⣙⣷⠀⣼⠄⠃⠠⠀⠹⣄⠉⠢⠇⠀⣁⠲⡟⠃⠀⠆⠀⠀⠀⠀⣴⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠓⠃⠀⠉⠁
⠀⠀⢠⣠⠀⡀⠠⢀⣴⠀⠒⠐⠰⠂⠲⣴⠆⠀⠀⠄⠀⠁⢈⠃⠦⠅
⠀⠀⡀⠁⣀⣁⣀⠀⢀⡀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠈⠛⠲⡝⠳⠾⣦⢜⠀⣈⠶⣅⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠂⠴⠄⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠋⠐⠉⠁⠀⠘⠛⠛⢛⠛⠓⠒⠂⠀⠀⠀⠁⢀⠀⠀⢡⢀⠀⠀⠈⡏⠀⠀⠀⠰⠟
⠀⠀⣠⣇⠐⠀⠀⠀⢠⡀⠀⠀⠀⠶⢶⠀⢈⣨⣍⢰⠀⠀⠀⢄⠀⠄⠀⠈⠃⢃⣈⠀⠀⠀⠀⢧⠀⠀⠀⡀⠀⠀⠤
⠀⢀⡈⠋⠀⠙⠁⠐⠾⠋⠁⠀⠀⠀⠀⠀⣤⠀⣀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠠⠀⠀⢀⡀⢠⣤⡄⢡⣠⣀⡄
⠀⠈⠍⡶⠻⡆⢠⣐⣶⠃⢀⣈⠁⢀⠀⣤⢻⣇⣿⣿⠃⠀⠀⣶⡀⢸⣧⠀⢀⣄⠀⣄⠀⣠⢠⡄⣬⢙⠁⣴⠃⢀⣀
⠀⠙⢚⡗⠛⠆⡈⠁⠉⢠⠠⡀⠀⣾⡇⠸⣧⣿⣟⡟⠀⠀⢐⡿⣇⢠⡯⢠⢦⣶⠀⡦⢠⡆⠀⢢⡚⢼⣿⣿⡄⠸⠹
⠀⢠⡀⡅⠁⠀⠩⡐⢸⡘⠃⠑⢸⣿⡇⠀⡮⠙⠟⠀⠀⠠⣶⣿⡵⡀⠀⠈⢸⣿⠀⡇⢸⡇⠀⠀⠀⠂⠽⠞⠁⠀⢈
⠀⢸⠀⢣⠀⡠⢢⠄⠀⠁⠀⢂⠸⣿⡇⠀⣤⡀⡄⢀⡀⢘⡺⢿⣿⣖⡀⠑⢸⣿⠀⠃⢸⢀⡆⠀⠠⠀⠀⢂⠁⡆
⠀⠘⢧⠘⠀⢀⠃⠀⠀⠀⠘⠌⠁⠿⠅⢀⣈⣿⠁⠈⠁⢨⢽⣿⡿⠭⠄⡀⣼⡇⠀⠂⠀⢸⠀⠀⠀⠀⠀⢼⠅
⠀⠈⣴⡇⠀⠉⠴⠷⠆⢰⠶⡦⠀⠠⠒⠃⡾⠃⠸⠿⠇⠰⣴⣤⡀⠒⠀⠀⠉⠀⠀⠀⠠⠿⠆⠀⣀⠀⠀⠋⠁

⠀⠀⢀⠀⣀⡀⣀⡄⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠁⠈⠛⠡⠯⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢆⡀⠀⠀⠀⢀⠤
⠀⠤⠠⠈⠈⠲⠇⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⢈⠁⠀⠀⢀⡤⠒⢉⠖⠰⠄
⠄⣁⠀⠩⠅⣀⡄⠣⠑⡀⠀⢀⢁⠀⠀⠐⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣴⡤⠤⣤⠀⣄⠀⣀⠀⣀⡰⢋⠀⣴⠁⠀⠰⠁⡀⠠⠘
⠐⣤⢠⠄⣰⠞⣷⣄⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠃⠹⠿⠟⠿⠶⠒⠀⠀⠃⠓⠉⠐⠠
⠀⠁⠀⠀⠠⠞⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⡐⠢⠀⠀⠀⠀⠀⢠⡀⢀⠀⠰
⠀⡀⠘⠀⡥⠘⠆⠀⠀⠀⠀⠀⠀⠒⠑⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠁⠸⠁⣤⠀⠀⠂⠀⠀⡁⠀⠂
⡎⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠡⠀⠠⠃⠀⡼⠀⠀⠀⢀⠆
⠀⠀⠀⢠⠙⢄⡈⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⢴⣏⡅⠠⠀⣀
⠚⠉⠁⠘⠂⠈⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡾⠺⡾⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠃⠀⠁⠀⢀⠐⠊⠠
⠀⠀⠀⠀⡀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠳⡄⠀⠈⣀⠀⠀⢶
⠀⡠⠂⠀⠈⡰⣾⠫⠀⠀⠀⠬⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣄⠀⠉⠂⠠⣄
⠒⠁⠀⠒⢁⠔⢉⡗⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⡀⠀⠀⠀⠀⠀⠈⠙
⠀⠠⠴⠛⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡦⠀⠂
⠆⠁⠀⢰⠋⠘⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠠⣀⡠⠔
⠐⠀⠀⢸⠖⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠈
⠀⠀⠀⠞⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡤⢀⡕⠛⠃⠀⠀⠉⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⠈⠁⠈
⢠⣾⠾⡏⠉⢀⠀⠀⠈⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣿⡀
⢠⡿⠈⠋⠉⠀⠀⠀⠃⠄⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⢀⠀⠀⢤⡆
⠏⠐⠒⠂⠀⠀⠀⠀⠀⠃⠉⠠⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠶⡶⡴⣥⠓⢠⡄⠀⠙⢲⠄
⠀⢄⠀⠠⢀⣀⡘⣶⠛⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤
⠀⠀⠠⠂⡼⠉⠀⠇⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⡀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡄⠀⠀⠀⡄⢀⡀⠀⠀⠀⠈⠉⢀⠀⡀⠀⠀⠈⠠⢤
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⢀⠀⠚⠁⣀⠀⠀⠀⠒⠀⠤⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠳⠀⠀⡤⠀⠀⠘⡂⢤⣀⣄⠀⠚⠀⠀⠐⣀⢀⠀⠈
⠚⠳⡶⠀⠀⠀⠐⠄⢀⠀⠀⠀⠀⠀⠤⡎⠒⠁⠀⠀⠀⠉⠑⢯⣄⣀⡀⣀⡀⡠⠀⠞⣁⠀⠀⠀⠀⠀⠣⠔⠀⠀⠀⡐⠁⡀⠈⡬⢿⢦⣄⡀⠘⠋⠀⠣⡀
⠀⠘⠓⢤⣀⠀⠀⠄⠀⠃⠀⠀⠛⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⣻⡅⠛⢇⠉⠀⠤⡼⣤⠄⠀⠀⠀⠀⠀⠰⠀⠀⣥⡄⠃⠀⡖⠺⢶⡒⠚⠙⠛⠃⠒⠥
⠀⠀⠀⠁⠐⠀⢘⡞⠀⠀⠀⠀⢴⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣻⠮⠁⠀⢀⠜⠙⢷⠤⢀⡐⢰⠃⠀⠀⠀⡀⠀⠁⠚⠩⠇⠀⠁⠨⠀⣤⠁⡄⠀⠀⠀⠀⠈
⠤⠀⠀⠀⠀⠄⠁⢦⠔⢀⣀⠀⠈⢤⣑⠀⠀⠀⠀⠀⠀⠀⣠⠏⢃⡀⠀⢠⠐⠁⠀⠀⠀⢉⠙⠀⠀⠀⠀⠀⢄⠰⠆⠐⠀⠀⠀⠀⠀⠀⠀⠉⠃⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⠐⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⢐⣘⡆⠐⢀⠀⠀⡀⢠⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⢊⠒⠆⡩⣉⠇⠀⠁⠔⠒⠙⠁
⠀⠀⠀⠀⠀⠠⠀⡠⠀⠀⡒⠀⠥⠜⠛⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠢⢠⠀⣤⡄⠰⠚⠂⠀⠀⢀⣤⠤⠔⠊
⠀⠀⠀⠀⠁⢀⡄⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⢀⣤⣈⡦⠄⢀⠁⠄⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢦⡀⠁⠀⢤⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠒⠠⠤⡷⠂⠠⠌⣠⠂⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣋⣝⢡⠒⣡⠬⠦⣀⡀⡽⠦⢀⠨⡁⠠⠴⠂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃⠈⠀⠀⠀⠿⠒⠓⠒⠾⠄⠽⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠊⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⣎⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠛⠿⠁⢙⠆⢠⣴⣥⠶⢀⣠⣤⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⠀⠁⠀⠀⠀⠘⠋⡫⣁⡄⠘⠃⠀⠀⠀⠀⠘⠃⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣭⡀⢉⣅⣈⣉⢀⡴⠈⡿⢿⡆⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡀⠿⣥⠬⣉⣀⣠⡟⠛⢏⡫⠋⠀⠀⡀⠀⢀⡀⠀⠀⠠⠈⠆⠲⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠲⢢⠀⡉⢈⠡⠌⠀⠀⡰⠈⢀⣄⡀⠀⠀⠀⣀⣀⡤⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⠀⣄⡀⠀⠀⠀⠀⠋⠀⠀⢀⠐⠛⣦⡒⢚⡛⠈⠛⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠀⣼⡿⠲⣤⣄⣠⣴⢛⡀⠹⠂⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢇⡻⠩⠀⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠫⣷⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠤⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠐⣸⠟⠁⠀⠀⠀⠉⠳⣤⠰⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠆⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠋⠁⠀⠀⠀⠀⠀⠀⢀⣷⠀⢀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⢤⡀⠀⠘⣦⠀⢠⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣄⠀⠀⠀⠀⠀⠀⡴⠚⠒⢺⣭⡀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣍⡼⠃⠀⠀⢣⠀⢠⡟⠀⢸⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡐⢸⠇⠀⠈⢧⠀⠀⠀⠀⡼⡇⠀⠀⢸⢘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⡇⠈⠁⠀⢸⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠘⣇⠀⠀⢸⣴⡇⢀⡤⣿⢸⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⢸⡄⠀⠀⠉⢠⡏⠀⡿⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠢⡀⠀⠀⠀⢇⠀⢠⠇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣧⠀⠀⠀⣸⠀⢠⡇⡏⠲⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⠃⠀⢳⠀⠀⠀⠘⡀⡞⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢰⠀⣠⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⢸⢠⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⢸⡀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⢢⣗⠻⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⢸⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠉⡄⠀⠀⠨⠁⢀⠁⠀⢸⡄⠀⠈⢪⣿⡏⢱⣤⣧⠀⠀⢸⡇⠀⠀⡇⠀⢸⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡇⠀⡇⠀⠀⠓⠄⢌⣳⡀⠈⡇⠀⠀⣸⣿⡤⣹⣿⣿⡄⠀⠘⡇⠀⢠⡇⠀⢸⢸⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⢃⠀⠀⠀⢉⣋⢯⡄⠀⢷⠀⠀⢸⣏⠀⢹⣿⡋⢀⣀⠀⡇⢀⢘⡇⠀⢸⢸⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⢠⢸⠀⠀⡀⠀⠑⡸⢿⡆⠸⡔⢀⣺⡏⢀⣿⡿⠖⢸⣿⣦⣇⣾⣿⣧⠀⣿⣸⢀⣸⣅
⣴⠾⢶⠀⢲⢰⣇⢴⣦⣄⠀⠀⠀⠀⢸⠀⠀⢾⠠⣽⣾⡀⣸⣧⢤⡄⠈⠘⠳⣌⣷⣿⣾⠃⣼⣏⡀⠀⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣫⠄
⠻⣤⣾⠇⣻⣷⣧⢾⣿⡟⠁⡘⢄⣄⣺⢴⣆⣾⣦⠻⣿⣷⣾⣍⣿⠇⠀⠀⠉⠉⠛⣿⢩⡾⠟⠋⣁⡐⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⠥⢀⣄
⠰⣮⣿⠂⣙⣿⡇⣾⣿⠁⠙⠲⢼⣷⢿⣿⣿⣿⣿⡾⣿⣿⠏⣾⡏⠀⠀⢀⣀⢀⣶⣿⣿⠆⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢂⣼⢿
⠀⢹⣿⣇⠈⣿⣵⣿⠃⠀⡄⡐⠒⠻⣿⣿⣿⡌⣻⣄⣾⡏⢨⡿⠁⢠⡀⠁⠀⢿⣿⣿⣿⣷⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣋⣵⣾⣿⠏
⠀⠈⣿⣿⡟⢿⣿⣿⢋⣿⣿⡏⠀⠈⠉⠹⣟⣿⣿⣿⡿⣇⣼⡁⢨⣾⣷⣾⣴⣿⣿⣿⣿⣿⠿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠿⠛⠋
⠀⠀⠀⠉⠻⢾⣿⣿⡾⣫⣡⣤⣖⡦⠠⢤⣼⣿⣿⣻⣿⣿⣿⣷⣶⣾⣿⣾⣿⣿⣿⣷⣿⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⠂
⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣂⠈⠉⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢆
⠀⠀⠀⢀⣤⣖⣾⣿⣿⣿⣿⣿⣿⣿⣿⡺⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡍
⠀⠀⠀⠀⢁⡔⠹⡟⢡⣽⣿⣿⣿⣯⣿⢇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠇⠈⠀⠀⢈⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡓⢑
⠀⠀⠀⠀⠈⠀⠋⣀⢋⣵⣿⣿⣿⣿⢃⡠⢾⠟⣤⠟⢻⣻⣿⣿⣿⣿⣿⣿⠿⡻⠀⠀⠀⠀⠁⠑⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢤
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠃⠉⠀⠀⠀⠀⠘⠜⠟⢿⡿⡿⣿⣍⠛⠓⠂⠀⠀⠀⠀⠀⢸⣿⠼⣿⡿⣿⣿⣿⣿⣿⣿⣯⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡏⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣾⡟⠄⡨⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠈⡏⠲⢀⠇⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢂⢸⠀⠀⠀⠀⠃⠀⠀⠀⠆⣏⠀⢾⠠⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⢘⡀⠀⠀⠀⠃⠀⠀⡄⠈⣏⠁⠾⡽⢿⢆⣀⠊⠀⠘⠇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⢀⣈⡀⢻⡀⠀⠂⠀⠀⠇⣠⣿⣼⠷⡅⠃⡺⢸⡀⠀⢈⣺⡄⠀⠀⠀⠀⢠⡄⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⡀⠒⣾⠁⠀⡅⠀⣸⣦⣿⣿⠏⠀⠀⠀⠸⣛⠇⠀⠀⢉⡇⠀⢀⠀⠀⠠⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠛⠻⠆⠀⠀⢀⡟⣫⣿⣿⠃⠁⢠⣀⡔⢻⡄⠀⠀⢈⡇⠀⠈⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⢥⠀⢿⡇⠀⢀⣾⣶⠛⠻⣿⡆⠀⠀⠀⢀⠉⡅⠀⠀⠄⠄⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⢀⣼⠀⣸⡁⠀⣾⣷⠀⠀⠀⣿⣧⠀⠀⠀⣾⢸⡇⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠦⣿⣿⢻⠃⠀⣿⡏⠀⠀⡐⣿⣿⠀⡀⢀⡟⢸⡇⠀⠀⣴⠇⠀⡀⠀⠀⠀⠸⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣄⠀⠀⢸⠇⠘⣿⡇⠀⠀⣿⢁⡄⠀⠀⣿⣿⠀⠠⢤⠀⡞⠀⠀⠀⣼⠀⠀⠀⠀⠀⢠⡄⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡌⢠⣄⣹⣇⠀⠀⡿⠀⠀⠀⢀⣿⡟⢀⠂⢰⣾⡇⠀⠀⢰⡏⠀⠹⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠠⠀⡤⠀⠇⠀⢰⡗⠀⠀⢹⣿⠀⠐⢿⡀⠀⠀⢸⣿⠇⠀⣼⣡⡟⠀⢀⠀⡞⠀⠀⡘⠀⠀⠀⠁⠀⢰⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠆⢀⠈⣿⠀⠀⢸⡿⠀⠜⠈⣧⠀⠀⣾⡿⢀⣽⣯⡽⠀⠀⠘⡾⠀⠀⢲⠃⠀⠀⠛⠀⠀⢸⠀⠀⠀⠀⠠⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠨⠐⠂⠠⠆⠘⡄⠀⣿⠃⠀⠀⠀⠉⠀⢸⠏⢃⣻⣻⠟⢣⡀⣀⡼⠁⠀⠒⣾⠀⠀⠉⡃⠀⠚⠏⠀⠀⠀⢠⠃⢀⡁
⠀⠀⠀⠀⠈⠲⢇⡼⡄⠸⠀⢿⣿⠑⢸⣏⣠⠀⠃⠀⡰⠃⣾⠀⣦⣿⣿⠀⣶⣿⣿⠃⠀⡄⢠⡅⠀⢰⠦⠆⢠⣤⡇⠀⠀⠀⡈⠀⡜
⠀⠀⠀⠀⠀⣵⣾⡇⢹⣶⢿⣦⣽⣷⣿⣿⣿⣿⣧⢯⣁⣀⡀⣌⣹⡛⣿⣾⣯⢽⣿⢠⣞⣤⣬⠀⠀⠣⣼⣄⣠⣿⠀⠀⠀⢀⠁⠀⠇
⠀⠀⠀⠀⣆⣿⣿⣿⣌⡟⢹⠿⠛⡿⣿⠛⠿⢾⠿⣼⢷⣿⠷⣾⡷⢿⣬⣿⣻⣛⣿⠻⡟⢷⣿⠀⣼⣼⠉⣘⢉⠏⠀⠀⠀⠈
⠀⠀⠶⠀⣟⣿⣿⢿⣟⣱⣿⢳⡞⢣⠾⡆⠲⠚⢴⢿⣶⣟⣿⣿⡷⣻⣿⢶⡟⢹⣯⡟⣻⣷⡞⣻⣯⣼⠛⡿⢻⡀⠀⠀⠀⠆⠀⢸
⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣧⠸⠇⢻⣀⠁⠀⠀⣄⠞⣴⣿⣿⣹⡾⣷⣿⢻⡏⢸⡏⣽⣛⡋⣹⣍⣷⡖⢣⣿⣸⡇⠀⠀⠸⢀⣀⡍
⠀⠀⠀⠀⠈⣹⠟⡟⢸⠁⠷⢸⡆⠈⠀⠉⠀⠀⠀⢠⡙⠋⡽⣟⢷⠛⣦⣿⠱⢺⡇⡉⠙⠙⠃⢉⡿⢧⣜⣹⠇⣿⠀⣠⠇⠸⢱⡇
⠀⠓⠀⠀⢀⠀⠀⡰⠂⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠨⠀⢨⣿⣿⣼⡅⢡⠠⠀⠀⠈⢁⣾⣯⡝⠲⣿⣶⣏⣠⢣⡟
⠀⠈⢂⠐⡈⢆⠀⠸⣇⡙⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣄⠀⢇⠘⠃⢸⠈⠁⠀⠀⠀⠀⠀⢈⡿⣷⣯⣾⠯⢀⢛⣿⡟⠁⠀⠀⢠
⠀⠀⠈⡄⠐⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡁⠀⠀⠀⠀⠈⠅⠟⢛⠟⠻⣻⠳⢇⢈⠋⠀⢀⠶⢴⠏
⠀⠄⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠐⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠠⡟⠀⢴⠏⠳⡿⢪⠀⢀⡄⣀⡄
⠄⠐⠀⠀⠡⣀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣶⣤⢶⠍⡏⢀⠈⠘⠙
⢰⠄⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢈⠀⠂⢸⡗⠦⠚⠷⢪⡼⠃⠁
⠀⣷⠀⠀⠀⠰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢱⡶⢶⠁⢠⡞⣀
⠀⠀⠄⢸⡄⠠⠄⠀⠀⠐⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡾⠁⢀⠗⠒⠰⠁
⠀⠀⢰⠀⠸⢶⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⣼⠀⠆⠃
⠀⠀⠈⠆⠠⢿⡁⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⢀⣸
⠀⠀⠀⠘⡄⠀⠁⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠈⠹
⠀⠀⠀⠀⢡⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢨⡇⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠀⠀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠙⢶⣺⡶⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠟⠚⠀⠈⠙⣧⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠀⠲⣷⡦⠂⠀⠀⠀⠀⠀⠀⠀⢠⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⢈⡅⠒⠢⡀⠀⠀⠀⢩⣟⠐⠀⠀⠀⠀⡀⠀⠀⠀⢀⣴⡁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⠉⢉⠀⣶⣤⣀⣀⠀⢘⢿⣦⠀⠀⠀⠘⣏⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠀⠉⠙⠴⠿⢴⣿⠷⠤⠀⠀⢼⣧⣄⠀⢀⣠⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡄⠒⠻⢃⠀⠀⠀⠂⠀⠀⠘⠻⣤⠄⡀⠀⢠⡈⢷⣶⣿⠁⢀⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡿⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠙⠧⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠏⠀⠀⠑⠈⠉⠻⢛⡛⠛⠛⠛⠲⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣄⣀⡀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋⠋⠁⠌⠛⢷⣤⡄⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡿⠁⠀⠀⠀⠀⠀⣀⡠⠄⠋⢉⠀⠀⠀⠀⠀⠀⠐⠛⠛⠻⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⠀⢀⣀⣰⡄⠀⠢⡈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⠾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⢀⡤⢖⣋⣵⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠿⣆⡀⠀⣰⣿⣾⡿⢋⢉⡉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠠⠴⠲⠶⠿⠿⢿⣶⣿⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⢀⣠⠬⠿⠿⣿⠏⡇⠀⠀⣿⣿⣩⠥⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠒⠠⠃⡇⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠁⠀⠹⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠆⠑⢧⠘⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡇⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠃⠀⠛⠿⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡏⠁⣀⠀⠀⠘⠢⢀⠀⠀⠁⠀⠀⠀⠀⡄⠀⣰⡟⠁⠀⠀⠀⠀⠈⠂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠻⠓⣿⠙⢣⣀⣱⣶⣄⣲⣀⠀⠀⢠⢸⡿⢀⠰⠄⠂⠀⠀⠄⣤⣷⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠀⠈⡟⠛⠁⠈⠉⠁⠀⠀⠸⡿⢟⠁⡀⣀⣤⡦⢄⡀⡀⡽⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠝⣿⣿⠅⠀⢸⣿⣫⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡇⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣧⠀⠀⠀⠀⢰⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢸⡄⠀⠀⠀⢸⡷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠐⠒⠚

⣿⣿⡿⠿⣿⡿⢿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡿⠿⠀⠀⠀⠀⢉⡛⣿⣿⣿⣿⣿⡿⠋⠀⠀⣩⣭⢹⣷⡄⠉⠀⠀⠸⣿⣿⣿⡿⣽⣿⣁⣀⠀⠀⠀⠀⢶⣿⣿⣿⣿
⠁⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⡏⢉⠁⠀⠀⠀⢿⡏⠀⠙⠓⠀⠀⠀⠀⣤⣶⣾⣿⢿⡏⣿⣟⠃⠀⠀⠀⣾⣿⣿⣿⣿
⢀⣤⢀⣴⠀⠈⠙⠉⣹⠿⠋⠁⢀⠀⠀⠀⠀⠈⢀⠀⢀⣤⣶⡇⠀⠀⠻⢿⠟⣱⣿⣿⠻⣿⠇⠀⠀⠀⠛⠛⠻⠟⢠
⣸⣿⣼⣿⣶⡾⠂⠀⠀⠀⠀⠀⢹⣿⣷⣦⠀⠠⣾⣷⣾⣿⣿⣷⣶⣴⣦⡄⠀⠻⠿⠟⠀⠀⣠⣤⡄⠀⠀⠀⢶⡄⣿
⠸⣧⣙⢿⣶⡄⠀⢠⣶⣶⣦⠄⠘⣿⣿⣿⠃⢠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⢠⣄⣀⠀⠀⠀⠿⠿⠁⠀⠰⣤⡘⣿⠸
⣷⣝⠻⢧⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⡠⠘⣿⣿⣿⣿⣿⣿⣤⡀⠀⢾⣿⣿⣷⡄⠀⠀⠀⠀⠙⠷⣝⢷⡘⣷
⠈⠙⠀⠀⣠⣶⠀⠀⠀⣾⣿⣶⢠⣶⣶⡄⠈⠀⠠⠟⢿⣿⣿⠿⡿⠛⠁⠀⠸⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠿⠈⢿⠘
⠀⢀⡀⠈⠛⠁⢠⣶⣦⡹⣿⣿⢹⣿⣟⣁⠀⠀⠀⠀⠘⠻⠏⠀⠀⠀⢠⣤⣀⠀⢀⠠⣄⠀⣾⡇⣴⣟⡀⠀⣤⣄⡀
⣿⣿⣷⡄⠀⠀⠘⠿⢿⡿⣮⣿⣾⣽⣿⡿⠃⠀⠀⣀⡄⢀⡄⠀⣀⠀⠈⠙⠋⠀⠘⠿⠟⡀⣨⣤⣌⠉⠁⢀⣿⣿⣿
⣿⣿⣿⡶⠃⠀⠀⢴⣿⡿⢿⡿⠙⠿⠷⠀⠀⠀⣰⠟⣴⡿⣡⣾⣿⣴⡖⢀⣀⠀⠀⢠⠋⠀⠙⠻⠄⢠⣴⣤⣙⣿⣿
⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⡀⠀⠀⢸⢇⣾⢅⣼⠛⣹⠿⣻⣾⣿⣥⠄⠀⠀⠀⠀⠀⠀⢀⣄⠈⠙⢻⣿⣿
⢀⣀⣀⠀⠀⡀⠀⣰⣄⠀⠀⣴⣼⣿⡇⠀⠀⠀⢸⠟⣼⣿⢸⣷⢠⣿⣵⣿⣷⣶⣶⡤⠀⠀⢶⣿⣾⣿⣤⠀⠈⠉⠉
⠿⣿⣿⣧⠀⠁⠀⣿⣿⣿⡄⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⠏⢸⣿⠀⢿⡇⠙⢿⡀⠀⠀⠀⣠⣶⣿⣿⣿⣷⡈
⠀⠀⠉⠉⠀⠀⢀⣈⣿⣿⣿⣿⣿⣯⣴⣶⣶⡤⠀⠀⠀⠀⠀⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⠿⡿
⣆⢠⣷⠀⢠⡄⠀⠙⢿⣿⠿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣤⣾⣀⣤⣀⡀⠀⠀⢠⣾⣿⣦⠀⠉⠈⠀⠀⠀⣀⣀⡀
⡿⢸⣿⢠⣿⡇⣠⣶⠀⠀⠆⠙⠻⢿⣦⠀⠀⠀⠀⠠⠀⢿⣿⣿⣿⣿⡿⠋⠀⠀⣾⠿⠿⠋⠀⡀⠀⠀⠀⠀⠈⠻⢿
⣷⡾⣁⣺⠟⣼⣿⢃⣤⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣥⣤⡀⠀⠀⠀⠀⠀⢀⣾⣿⣦⠀⠀⠀⠀⠘⣻
⣿⣿⣿⣥⡴⠛⣱⡿⢃⠀⠀⠀⢻⣿⣷⡀⠀⠀⢰⠿⢻⣿⣿⡻⣿⣿⣿⣷⠀⠀⠀⣿⣷⣼⣿⣿⣿⣠⣴⣶⠀⠀⠉
⠰⠿⠛⣫⣶⠿⣉⡴⠏⠀⠀⠸⢿⢿⡿⠇⠀⣀⣀⠀⠘⢿⣿⠇⠈⠋⠉⠉⠁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡿⠄
⠀⢀⠈⠉⠁⠞⠋⠀⠀⠀⠀⣀⠀⠋⠀⠀⠀⠙⠻⠃⠀⠈⠛⠀⠀⣰⣾⣶⣦⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⠀⠀⠀⢠⣤⣄⣄⠀⣰⣿⣆⠀⠀⠀⡀⠀⣤⣤⡀⠀⠀⠀⢸⡿⠟⠛⢁⣀⠀⠉⠉⠿⠛⠙⢻⡟⠙⠋⠀⠀⣤
⣿⡟⠀⠀⠀⢘⣿⣿⣿⣧⣹⣿⣿⢁⣀⠀⢻⣿⣮⡛⡛⢿⣆⠀⠈⠀⣠⣄⣸⣿⣧⣴⡄⠀⠀⠇⠀⠀⠀⠲⣿⣿⣿
⠀⠁⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡶⠙⢻⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠒⠻⠟⠀⠀⠀⠀⠀⣠⣤⣤⣴⡆⠀⠀⠀⠀⠐⠊⠁⠉⠁⣀⠐⠂⢀⠀⠀⠁⠋⠙⠀⠀⠀⠁⠀⣀⣀⣀⣀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢠⡎⢴⡉⢹⡿⠓⠐⠢⡀⠐⠖⠾⠿⠟⢻⠿⠯⠛⠛⠃⠀⠀⠀⠀⠊⠃⠀⠐⠉⠉⠉⠛⢹⠋⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠀⠙⠋⠂⠀⠀⠀⠀⠀⠂⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣄⢀
⠀⠀⠀⠀⣤⡀⣠⡄⡀⠀⠀⠀⠀⠀⢷⣴⠖⣀⣈⠀⠂⠀⠀⠀⠀⡀⣸⡄⠀⠀⠀⠀⠀⠀⠀⣤⢠⡆⠀⠀⠀⣄⡀⠀⣰⠨⠜⡇
⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋⠉⠉⠀⠀⠀⠀⠀⠹⠿⠏⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠈⡝⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⢠⣤⣄⡀
⠀⠀⠀⠀⣠⠴⢒⣶⣦⣄⠀⠀⣤⣤⣤⡤⣶⠶⠶⠶⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠋⠉⠉⠀⢀⠀⠀⠐⠶⠄⠂⠀⠀⢀⡆
⠀⠀⠀⠀⣉⠒⢚⡿⠝⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠙⠲⢶⣤⣶⣤⣤⠜⠉
⠀⣤⡚⠽⠝⠋⢉⡃⣠⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠟⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠈⢉⣻⠃
⠀⠈⠁⠀⢣⠀⢰⡆⢤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⠇⢀⣀
⠀⠀⠀⠀⠈⣦⣾⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠁⠸⠁⠻⠿
⠀⠀⠀⠀⠈⠛⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣤⣤⣿⠄⠀⠀⠀⢀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠑⢶⠁⢐⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰⣤⣾⡟⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠺⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣎⡇⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢻⣧⠀⠀⠀⠀⠀⢀⡀⠀⠶⠆⠀⠛⠁⠈⠛⠀⠘⠻⠀⢠⣄⠀⠀⠀⠀⠀⠀⠠⣿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡷⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠀⣿⠂⣴⡄⢀⣶⢀⣤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⣄⠈⢿⡄⢸⡇⠀⢻⣶⠟⠀⠘⣿⣼⠏⠀⣼⠏⢀⣄⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣄⠻⢷⣾⡇⢸⣇⠀⣼⠇⢠⣶⠀⣿⠃⠀⠀⣿⣶⠿⢃⣤⠀⠀⠀⠀⠰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⣠⣀⣀⣿⣇⠀⠈⣿⠀⢿⡿⠋⠀⣸⣷⡾⠋⠀⠀⢸⡏⠀⢀⣿⣇⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣌⠉⠋⠉⠛⠻⠿⣿⡄⠈⢿⡄⣰⡟⠀⣼⠇⣠⡀⣼⡿⠿⠛⠋⠉⠉⢉⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠈⣿⡄⠈⣿⡿⠀⠀⣿⣴⠟⣸⡏⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣦⣄⣿⣇⣀⣀⠀⠀⠀⠸⣧⠀⣿⡇⠀⢠⣿⠀⢠⣿⠀⠀⠀⢀⣀⣀⣿⣇⣤⡶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⠋⠉⠉⠉⠛⠻⣦⣄⠀⣿⠀⣿⡇⠀⢸⡇⠀⢸⡇⢀⣤⡾⠛⠋⠉⠉⠉⠻⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣿⢰⡿⣷⠀⠘⣿⠀⢸⣧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣦⠀⢤⣦⣶⣦⠀⢤⣶⣦⣿⣿⣿⣧⣾⣷⣴⣿⣦⣾⣿⣧⣴⣦⠄⢠⣶⡤⠀⣴⣦⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⢀⣤⠀⢠⣤⣤⣤⣤⣤⠀⣤⠀⢀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⠀⣤⠀⠀⠀⣤⣤⣤⣤⣤⡀⣤⡀⠀⣤⣤⣤⣤⣤⠀⠀⢠⡄⢠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠈⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉




⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣻⣿⣿⣿⣿⣿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢧⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⢀⠔⠚⠳⠦⣤⣤⣤⠴⠋⠉⠲⣄⠀⠀⠀⠜⠉⠁⠀⣀⠑⣤⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠈⠁⠀⠉⠘⠰⢦⡄
⠀⠀⠀⠀⠀⠀⡏⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡆⠀⠘⡄⢐⢀⡈⡦⠙⣊⠵⡀⠐⠀⠀⠂⠀⡇⠀⠰⡙⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠉
⠀⠀⠀⠀⠀⢸⡆⣞⢀⠀⢀⠀⢀⠀⢠⠀⢀⢐⡇⣻⠀⠀⣏⣈⣩⡥⠔⠚⣡⠂⠁⠀⠨⣳⠈⡇⢀⣀⠀⠀⠀⠀⠄⢀⠁⠸⠉⠓⡃⣀⣀
⠀⠀⠀⠀⠀⣸⠀⡗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠘⡆⠀⠃⢄⡀⠀⠀⠐⠁⠀⠀⠀⠀⡀⠀⢀⢀⠀⢨⠀⠀⠀⠀⠀⠀⠜⠂⣶⠹⠓⠲⡇
⠀⠀⠀⠿⠀⡇⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣧⠀⠀⠀⠈⢢⠀⣴⠂⠀⠀⠀⠀⢃⠀⠈⠈⠀⠈⠀⠀⠀⣄⠈⠀⠀⠸⢝⡥⢠⡐
⠀⠀⠀⠀⢰⠁⠀⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⠒⡇⠀⢸⠀⠀⠀⠀⠀⠙⣻⠀⠀⠿⠃⠀⠀⠃⠈⠀⠀⢀⣀⣀⠀⠁⠓⠀⡠⠀⠁⠀⢸⠇
⠀⠀⠀⠀⡎⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⠠⠂⠐⠄⢀⠇⠀⠀⠀⠀⢀⠊⠁⠀⠈⠉⠀⡟⢹⠢⡀⠀⠈⠱⠤⠀⠈⠁
⠀⠀⠀⠀⠃⠀⢀⡇⠀⠃⠀⠀⠠⠀⠀⠀⠀⠀⣇⠀⠀⡆⠀⠒⠐⣂⠈⡌⢶⡀⠀⢠⠾⠉⡕⢒⡤⢄⡼⠁⠀⠀⠘⢄⠀⠀⢠⢤⡄
⠀⠀⠀⠀⣦⣄⡼⢳⠶⠖⠒⠒⠒⠒⠒⠒⠒⢲⠻⣀⣀⡇⠀⠉⠀⠈⠀⠈⣶⠛⢠⣾⣶⣦⣤⣌⣀⣾⢓⠤⠆⠀⠀⠈⢑⡄⠀⢠⡇
⠀⠀⠀⠀⣇⣀⡇⠈⠳⠶⠶⠒⠒⠒⠤⠶⠴⠋⠀⢻⡀⣧⠀⠀⠀⠀⠀⠀⠸⠆⠚⠿⠷⣿⣿⣿⣿⣀⣈⣰⡀⢀⡠⠒⠁⠀⢀⡞⢀⣄
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠶⠀⣀⡀⡀⢀⢀⡀⠀⢀⠉⡁⣠⠠⢤⠒⡴⠖⠀⠀⠀⠀⠀⣾⣿⣟⢉⡿⡿⡏⠋⠁⠀⠀⠀⠀⠾⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⠑⠶⣤⣄⠀⠀⠀⠀⣶⡒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⢻⠛⣷⡄⠀⠀⠈⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣀⣀⡀⠀⠈⣦⢻⣷⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⣇⠐⠋⠀⠀⠙⡽⣦⣌⡛⠋⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣞⣶⡿⠃⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢠⣧⣼⠿⣷⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢰⡏⢧⠀⠀⠀⠀⠀⠙⠻⣦⣭⠀⠀⡀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⡄⠀⠀⣼⠁⢿⠀⠀⠀⠀⠀⠀⢀⡴⠿⡏⠉⠱⣜⢷⣾⠁
⠀⠀⠀⣠⣖⣈⣥⣴⣾⣷⠤⣤⣤⣄⠀⠀⠀⠀⢹⠆⢠⠀⢠⡏⠀⢸⡄⠀⠀⠀⠀⣠⠋⠀⠀⠙⠀⠀⢹⣾⣿⠆
⠀⠀⣸⠇⠈⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⠁⢿⠇⢸⠀⢸⠁⠀⠘⡇⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣟⢉⣀
⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠰⣤⡯⠀⢸⠀⢸⠀⠀⠀⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⠉⠛⠛⢦⡀
⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⡇⠀⠀⢠⣷⠶⠒⠶⣄⣠⠞⠁⢸⠀⠀⠀⣀⡔⠀⢻⣆
⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡆⠀⠀⣧⠀⣰⠛⡇⠀⠀⠀⡿⣃⠀⠀⢸⡇⢀⣾⣿⠀⠀⢸⣿
⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣫⢠⠃⢸⡇⠀⢀⡞⠁⢹⡆⠀⠀⡇⠀⢸⣿⡀⠀⣼⠋
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣇⠀⠀⢸⢸⠀⢸⠇⠀⡾⠀⢿⡾⠋⠀⠀⣿⠈⠛⠿⠿⠛⠁
⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⣿⠀⠀⢸⡾⠀⣼⠀⢠⢃⣤⢤⡀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⠀⠀⢸⡇⠀⣿⠀⢸⠏⣹⠀⣷⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⠀⠀⣸⡇⢀⡁⠀⠿⣬⠃⢀⡏⠀⠀⢠⡟
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⢀⣿⠀⢸⠃⠀⡇⣿⠀⣼⠁⠀⠀⣼⠃
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣸⠃⠀⢰⠇⢀⡞⢸⠀⣿⠀⠀⡇⠇⢰⠇⠀⠀⢠⡏
⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⡼⢀⡾⠀⢸⢀⡇⠀⢀⠇⢠⠟⠀⠀⠀⡜⢠⡄
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢰⡇⠀⠀⡇⣸⠃⠀⢼⢸⠁⠀⢾⠀⡜⠀⠀⠀⢰⠇⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠁⠀⢰⢁⡟⠀⠀⣿⣬⣀⠠⡇⢰⠃⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠀⠀⢸⢸⠁⣀⢲⣿⢣⣜⡧⢀⡏⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⣀⠸⣿⠃⠀⠀⠀⣿⠸⣇⠀⠀⠀⡏⠀⠀⣿⣾⠀⣬⠛⢻⣿⣿⡇⣼⠁⠀⠀⠀⢸⠇⠀⣼⡟
⠀⠀⠀⠀⠀⠘⣷⢸⣶⡇⠀⠐⣿⠺⣿⡇⠀⣤⣥⣅⢢⡈⣿⣰⢿⣷⣦⣿⡟⣷⣏⡤⠀⠀⠀⣼⠀⠀⡿
⠀⠀⠀⠀⠀⢰⡽⣿⣻⣇⣴⠀⢾⣱⡿⣧⡔⢃⢻⣟⡟⢯⣿⣿⣾⣿⣿⢹⣿⣿⡏⠀⠀⢰⣄⢧⡄⢸⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⠃⠀⢈⣿⣿⡿⠀⢹⣿⣿⣷⣾⣿⣿⣿⣿⣿⣾⣿⣯⣿⡆⡀⢮⣿⣾⣁⣼⡖⠀⣠⣶⣿⡿⢀⡠⠆
⠀⠀⠀⣀⠀⢀⠘⢿⣿⠁⠀⢤⣄⣹⡟⠀⠀⠀⣴⣿⣿⣿⣿⣿⣼⣿⢹⣿⣿⣿⣿⣷⣿⣾⣿⣣⣽⣿⢯⣶⣿⠟⠉⢀⣸⢇⣶
⠀⣰⣄⣻⣆⠈⣳⣾⣿⣾⣇⡀⢹⣿⣤⡾⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣟⣤⣼⣿⡿⠟⢃
⠘⢿⣿⣿⣿⣦⣝⢻⣿⣿⣿⣿⢾⣿⣿⣀⣸⡿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣉⣠⣾⣿
⠀⠀⠚⠛⠛⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠿⠿⣿⡿⠛⣻
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢷⣶⣾⣷⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠿⣬⠻⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠉⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⢀⣾⡿⣄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠽⠰⠀⠎⣹⢷⡿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢇⡙⢦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠶⠟⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢜⣦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠲⠛⠩⣽⠛⣿⣿⣇⣯⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡺⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠉⠳⠿⠟⠲⠂⢻⣥⡽⠩⣏⠏⣿⣯⣙⣶⣉⣳⡮

⠀⠀⢄
⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠴⠒⠉
⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢹⣿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠈⠢⡀⠀⠀⢀⠀⠀⡌⠀⢠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⣷⠄⠈⠁⠀⠀⠀⠀⢠⣤⣤⠐⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠁⠀⢿⣄⡛⠉
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠄⠀⠀⠀⠀⢈⢿⣶⢷⣼⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀⠀⠣⢤⣀⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡈⢧⡈⠓⠶⠶⠄⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠹⡀⡔⣠⠀⡟⢂⠂⡄⢰⠴⣠⢦⢰⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠈⠀⠉⠀⠉⠁⠈⠀⠁⠈⠈⡄⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⣸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⢄⠀⠀⡇⠀⢀⡠⠊⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢠⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⣀⠱⣴⣡⠔⠉⠀⠀⠀⠑⠤⣀⠀⠀⢳⠀⠀⢠⠏⠀⠀⠀⢀⡤⠂⠁
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⠈⠻⡿⢉⠉⠁⠉⠀⠀⠀⠈⠙⢦⣘⡇⢠⠏⠀⢀⡤⠞⠁⠀⠀⢀
⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⣿⣿⣴⣞⡉
⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠁⠉⠉⡉⠑⠒⠤⡀
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠴⢶⣦⣖⣒⡦⣤⣶⣒⡢⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⣴⠾⠛⠛⠇⣹⣿⡾⢳⡾⣧⡍⢁⣀⠈⠳⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠋⠁⠀⣾⣷⣖⠀⢠⡀⢦⡄⠰⣶⠀⡏⡀⣞⢛⠴⢇⢈⡉⢲⡀
⠀⠀⠀⠀⠀⢀⠞⡁⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⣡⣿⠦⡴⠿⠿⡿⣐⠋⢠⡀⣶⣯⠇⠈⠻⢷⣦⡀
⠀⠀⠀⠀⡜⠁⠀⡤⢊⡰⠶⠄⠀⠀⠐⠻⠛⠛⠿⢣⠾⢦⣴⠀⠰⠟⠫⣦⣙⠙⣭⠇⠀⠀⢀⢈⣷⡄
⠀⠀⠀⠀⡇⠾⢞⣠⡿⣛⣰⡶⠀⣠⡄⠀⠀⠀⢤⣀⡔⣆⠿⠉⠀⢀⠀⠉⠛⠛⠻⢇⣷⢼⠘⠇⠈⢻
⠀⠀⢀⡞⢡⣼⢻⣥⡾⢽⣏⡀⠀⢘⢻⠄⠀⠀⠀⠉⠋⠉⠀⠀⠀⠈⢧⢰⡗⢆⣠⣘⢳⢠⣶⡶⣖⢾⡄
⠀⠀⣨⠇⠈⠉⠋⠉⠱⣆⢙⡇⠀⠈⣏⠇⠀⠀⠀⣀⡀⠀⠰⣺⠆⠀⠈⠋⠁⠀⢙⠉⠀⠀⠁⣈⠀⠀⠳⣄
⠀⢀⣿⠰⣒⣶⣄⣀⡄⠙⠻⢁⣠⠧⡘⠃⠀⠀⠀⠙⣷⣞⡿⠿⠂⠀⠀⠀⠀⠀⠀⠑⢄⡆⠀⠘⢧⣠⡀⢸
⠀⢸⡟⣿⠿⢿⣿⣽⣿⣇⠀⣠⠟⠆⡠⠤⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠐⠪⢀⢤⣌⡁⢦⣰⡐⠀⠘⡆
⠀⠘⣿⢻⣷⣦⡀⣿⣟⠁⣨⡐⠀⠘⣥⣄⠷⠄⠀⠀⠈⡷⠞⣱⠀⠀⣀⡀⣿⠀⡄⠀⠀⠁⢉⠈⠀⢸⡆⣠⠇
⠀⠀⢻⡌⣧⠙⢿⣿⣯⡰⣯⣵⣇⡀⠀⠈⠀⠀⠀⠀⠀⠑⢚⠛⠀⠰⠤⣿⢻⡞⠀⠀⠀⡀⢸⢣⡆⠀⢹⡁
⠀⠀⠀⢻⡇⢰⣿⣿⣫⣯⢉⣽⣝⣿⣷⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣦⢿⢛⡒⣽⠀⠳⣄⢺⡠⢀⡼⠁
⠀⠀⠀⢰⠃⠈⠙⢿⡇⠹⣾⣻⡜⢿⡉⠁⠀⢤⠀⢰⡄⡄⢙⡧⣲⠂⠯⢚⡼⠲⢨⣅⠴⣤⢈⠋⠁⣾⠆
⠀⠀⠀⠈⠣⡀⠀⠘⢴⠻⡋⠻⣶⣾⣧⠀⠙⠓⢷⣱⣽⣉⡘⣦⣩⣟⠂⠩⢕⢩⡌⣿⣂⡘⢹⠇⣼⠟
⠀⠀⠀⠀⠀⠻⣄⢀⡟⢰⢛⠀⠈⠉⠉⠀⠐⠒⢲⣿⢯⣉⠀⢻⢟⠞⣉⡸⢋⠃⢾⡿⠇⠐⢟⡼⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠓⣾⡘⠛⠫⢭⣻⠚⠈⢚⣛⣹⡾⣍⠧⢞⡋⠉⠈⠇⠀⠿⢻⣅⡴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣞⠋⣛⣟⢾⣿⡽⢷⣿⣄⠘⢧⣷⠀⠈⠀⠸⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣤⣾⣿⣿⣾⣦⣤⣉⣀⡴⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣼⣿⣿⣶⣿⣿⣎⢿⣿⠀⠀⠀⢀⠀⢸⡇⠀⠉⠁⠀⠀⠀⠀⠀⠀⣈⠻⣿⣿⣿⣿⣯⣽⣿⡘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣴⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣈⣍⣀⠀⠀⠀⠑⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡻⣿⣷⣿⣿⣿⣿⡟⣼⣹⣿⢃⡜
⠀⠀⠀⠀⠀⢀⣭⣷⣿⣿⣿⠻⣿⣿⣿⣿⣿⡜⣿⣿⣿⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣝⠿⣿⣿⣿⢟⣵⢟⣿⠿⠃
⡀⠀⠀⠀⠀⢿⣿⣿⣟⣛⣿⣿⣚⣻⣽⣿⣿⢇⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣛⠿⢶⣷⣶⠿⢫⡾⠁
⣷⠀⠀⠀⠀⢈⠛⠿⠿⠿⠛⣭⣭⣭⣭⣭⢷⣻⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴
⣿⣧⡀⠀⠀⠸⣿⣿⣿⣿⣷⣾⣯⣭⣽⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣷⡀⠀⠀⠉⠙⠻⢿⣿⣿⡿⠃⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢶⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢻
⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⠀⠘⡆⠀⠀⠀⠰⠀⢀⣤⣶⣿⣿⣿⣿⡗
⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣷⣜⣷⣿⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠚⠛⡿⠿⡁⠀⠀⠾⠿⠿⠿⠛⠋
⠀⠀⠀⣴⣾⣶⣶⣤⣀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿
⣀⡀⢸⣿⣿⣿⣿⡿⣛⠂⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠀⠀⢀⣤⣶⣶⣦⣄⠀⣠⣴⣶⣶⣶⣶⣦
⣿⣿⣞⣿⣿⣿⡏⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣤⡀⠀⠀⣰⣿⣿⣿⣿⡿⠃⠀⣰⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴
⣿⣿⣿⣿⣱⣶⣿⣾⣿⣿⣿⣿⣟⣿⣿⣛⢿⣾⣿⣷⠀⢰⣿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣟⣭⣅⣀⠀⠀⠀⠀⠸⣿
⣿⣿⣿⣿⡘⢿⡿⠏⠋⠉⠙⢻⣾⣿⣿⣿⣷⣿⣿⣿⠀⠸⠿⠟⠋⠀⠀⠀⠀⠀⠻⣿⣟⣻⣿⣿⣿⣿⣟⣵⣿⣿⡆⣿⣿⣿⣿⣆⠀⠀⠀⠀⢙
⣿⣿⣿⣿⣿⡗⠀⠀⣶⡆⠀⠈⣿⣿⣿⣿⣿⣿⠟⣁⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣿⣿⣿⣿⠿⠉⠉⠈⠻⢿⣿⣵⣿⣿⣿⣿⣿⣦⠀⢀⣴⣷
⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⢀⣾⣿⣿⣿⢟⣵⣿⣿⣿⣷⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠰⡶⠀⠀⢘⣿⣿⣿⣿⣟⣛⡛⠋⠀⢸⣿⣿
⠿⣻⣿⣿⣿⣿⣿⣇⣶⣷⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠻⣿⣿⣿⡿⣿⣿⣿⣿⣧⣀⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠙⠿
⣨⣛⣿⣿⣿⣿⣿⣧⠻⣿⣿⣿⡿⠟⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣿⣿⣷⣯⣛⣛⡛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⠙⠛⠋⠁
⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡝⣛⠿⠿⣻⣵⣿⣿⣿⣿⠃⠀⠀⠀⢰⣦
⠀⠙⠻⠟⠛⣿⣿⣿⣿⣿⣿⣷⠙⠛⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢸⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⢠⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠈⠋⠽⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣾⠃⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣄⢀⣀⣠⣈⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢋⡁⠀⣀⠸⣿⣿⣆⠀⠀⠀⠀⠀⢰⡀⠀⣩⣤⠄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣮⡿⣿⣿⣦⡉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡄
⠀⠀⠀⠀⢰⣖⣴⢄⡀⠀⠈⠓⠶⠚⠁⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⣙⣻⣷⣎⡼⠆⠀⢀⡄⢠⣄⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⠃
⠀⠀⠀⠀⢩⣭⠿⣿⡴⢎⠾⣌⣷⣾⡿⢭⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠙⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠷⣿⡿⣯⣙⠓⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⠰⡷⡄
⠀⠀⠀⣀⢶⣘⣷⢰⡀⠀⠀⠘⠳⠆⠉⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠒⠒⠀⠛⠛⠃
⠀⠀⠈⠫⡟⣻⡿⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠁⠉⠀⠀⠀⢺⣷⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡮⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠘⠳⠶⠒⣶⠀⠀⠀⠀⠀⠀⠆⡀⣶⣂⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢳⠆⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⢀⡜
⠀⠀⣔⢁⢀⣼⣷⠀⠀⠀⠀⢨⢀⠡⡙⢹⣿⣿⣯⡤⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⢀⠀⠀⠀⢀⣾⣁⣀⠀⠀⠀⡿⠿⠛⠱⠧⠀⠀⠀⠀⠀⢀
⠀⠀⣀⢙⣋⣾⣷⠀⠀⠀⠀⠐⠀⠀⠀⢠⣿⣿⣿⣶⠆⡀⠀⠀⠀⠀⢀⣾⣿⠿⠀⠀⠘⡇⠀⠀⢸⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠄⠀⠀⠀⣀
⠀⠀⠀⣫⣧⡼⣿⠀⠀⠀⠀⠀⠀⠀⠁⣸⣈⡷⠻⠃⢠⠆⠁⠀⠀⠠⣾⣿⡟⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢐⡄⠀⠀⠀⠀⠁⠂⢉
⠀⢀⣞⡁⠘⣴⣻⠀⠀⠀⠀⠀⠀⠠⠀⣽⢡⡲⠃⠀⠀⠀⠀⠀⢀⣼⢿⣿⠃⠀⣀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢄⣼⠀⣄⣴⣾⣕⣶⣞⣰⣦⣾⣔⡹
⠀⣘⡏⡿⣍⣘⣏⠆⠀⠀⠀⠀⠀⠱⡛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠸⡻⣾⣿⠃⠀⠀⠉⡁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠢⠠⠜⢋⣿⢿⣷⣾⣿⣿⣿⢿⣯⣿⢏⡁
⠀⣿⡏⠤⠙⣛⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⢿⣿⠇⠀⠀⠀⣸⠁⠀⠀⠀⠀⠉⣧⠀⠀⠀⡀⠀⠀⠀⠻⢿⣿⢿⣿⡞⠛⠟⣛⠿⠻⢯⡽
⠀⠙⣻⣤⢱⣽⢈⠏⠀⠀⠀⠀⢦⣠⡀⢀⡠⣴⠠⠀⠀⠀⠀⣨⣿⣿⡏⠀⠀⠀⠀⣿⠀⠀⠐⠀⠀⠀⠨⠀⠀⢰⠸⡄⠀⠀⠀⠀⠀⠀⠙⠁
⠈⢨⣄⣻⠵⠄⣾⠅⠀⠀⠀⠀⢘⢿⡀⠫⠔⠁⣤⡄⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⡟⢠⢶⣀⠀⠀⠀⠀⠀⠀⢸⣧⣷⠂⣀⣠⠀⠀⢦⠄⠀⠀⠀⢀⠀⢀⣄
⠀⠐⠺⠿⠛⠇⠀⠀⠀⠀⠒⠀⠸⡇⠀⠀⠀⠘⢟⣃⠀⠀⢸⢿⣿⣿⡇⠀⠀⠀⢀⢉⠘⠃⠀⣀⠀⠀⠀⠠⠀⢸⠃⠘⡟⠉⠀⠀⢰⣿⠀⠀⠀⠀⠈⣧⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠠⠀⢠⣀⠀⠀⠸⠟⠛⠉⠀⠀⠀⠀⠀⢠⡄⠀⠀⠓⠤⠀⠀⠀⠑⠾⠶⠦⠼⠦⣤⣤⣼⣮⣧⡴⠶⠀⠀⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⢸⣿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡃⠀⠴
⠐⠊⢳⠄⠀⠀⠀⢠⣾⠅⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠚⡏⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢢⡞⣴⢂⠠⣀⣠⠀⠀⠀⠀⠀⡠⠀⣤
⠀⠀⠀⢳⡄⠀⢀⡿⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⢀⣴⢋⣴⣦⠀⠀⠀⠀⠀⡇⠀⠀⣠⣶⣤⣇⠀⠀⠀⠀⠀⠘⠟⠉⠐⡄⠌⠌
⠀⠀⠀⠈⢣⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠁⠻⢷⠏⢉⣙⡧⠁⠀⠀⢠⡇⠀⠀⠙⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⣀⠁⠀⠀⣀⠀⠀⢐⢦⣀⡔⠢⠄⠀⠀⣙⣄⠹⣶⠖⠉⢀⡀⠒⠀⠀⠀⠁⠀⢀⠁⢸⣿⣿⣷⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⢸⠆⡀⠀⢰⡷⢺⢱⠋
⠀⠀⠀⠀⠀⠀⣠⡽⠻⠁⠀⠠⠷⠸⠃⠂⠀⠀⠀⡉⢐⡀⢲⣶⡿⣏⢁⡄⠀⠀⠀⠀⠀⠈⣶⡿⣯⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠁⠀⠀⣇⣾⡿
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢐⣿⡬⠄⠀⠀⠀⢀⠦⣀⣀⠄⣒⣀⡄⠀⠀⠀⠀⠀⠀⢸⣽⣿⣽⣿⣿⣿⣄⡄⠀⠀⠀⡀⣀⣀⠀⣸⣿⣁⡶⠀⣀⠟⠉⠁
⠀⢀⣠⡀⣀⢠⠇⠀⠀⠀⠀⠀⠀⠹⣭⡀⠀⠀⠀⠀⠈⠀⠉⠐⠟⠉⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⡏⠀⠀⠀⡏⠀⠀⠈⠀⠀⠈⠛⠣⠞

⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣠⣤⠀⠀⠈⠲⠦⢤⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠀⠀⠀⠀⣶⠂
⠀⠀⠀⠀⠄⠛⣧⢀⣀⣀⠀⡀⢀⡀⠀⠀⠻⡙⢶⡀⠤⣀⡀⢀⣴⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠠⠀⠀⠀⠀⠀⣀⠈⠁⠠
⠀⠀⠀⠀⠄⠀⠸⠏⠡⠀⠉⠁⠉⠀⠀⠀⠀⠀⠂⠑⠄⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⣀⠀⠀⠀⠀⠀⠐⢀⠄
⠀⠀⠀⠀⡄⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠀⠰⠀⠐⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠁⡀⠀⡀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠉⣠⡄⠀⡀⠀⠀⠀⠀⢀⡦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⣶⣽⡆⠀⠀⠀⠀⢀⠈⣴⠞⠈⠉⣷⡶⠷⠀⠀⠀⣠⠾⠆⠀⠀⢀⠀⠀⠀⠀⠤
⠀⠀⠀⢠⡀⢡⣄⡀⠤⠙⢆⠀⢐⠄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠛⠁⠀⠀⠀⠀⠈⠀⣿⠿⠛⠚⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣏⣀⠈⠦
⠀⠀⠀⠈⠁⠤⡤⠉⠀⠀⠀⠁⠈⠀⠀⠀⠀⠐⠃⠐⠆⠀⡠⠄⠀⠀⢀⡀⢀⡴⠀⠀⠀⠸⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡼⣿⠇⠀⠀⠀⠀⣄⣶⡄⠀⠀⢀⠀⠈⠓⠂⠀⠀⠀⠀⠀⢀⡶⡴⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠿⢿⣭⣷⠋⠀⠀⠀⠀⠀⠉⠛⠓⠀⠈⠉⠀⠀⠀⢤⠀⠀⠀⠀⠀⠈⠉⠳⠀⠀⠀⠀⠀⠀⣩⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⢀⣂⠤⠴⢶⣤⣼⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠑⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⡄⠀⠀⠀⠁⠀⠠⠆⠀⠀⠀⠸⡏⠀⢀⠀⠙⠿⠟⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⡴⠠⠄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣋⠁⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠌⠔⠁⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠀⠖⠀⠈⠑
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠲⠀⠀⠀⠀⠱⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣆⠀⠀⠰⠀⠀⢦⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡎⣼⠋⠀⠀⡀⠀⠀⠞⠁⠀⡀⠀⠀⣄⡰⠋⠀⠀⠃⡀⠀⠀⠀⠀⣾⣄⠀⠀⠀⠸⠟⠊⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⣇⠀⠀⢄
⠀⠀⠐⢶⡀⢹⠄⠂⠔⠉⠉⠀⠀⠀⠀⠢⣽⠆⠜⠓⠒⠀⠀⣠⠀⢱⡄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⠤⠀⣿⠀⠃⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠂⠁⠋⣠⠡⠐⠀⠀⠀⠤⣠⠂⣤⡜⠋⠊⠀⠀⠀⠀⠨⠀⠊⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠠⠈⠁⠰⠀⢸⠀⢀⠴⠀⠀⠀⠂⠙
⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠻⣤⢯⣤⡂⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠞
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠑⠀⠙⠀⠀⠀⠀⠋⠠⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡠⢤⣽⢿⣿⣿⣷⡤⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠐⠿⡆⣠⠰⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠄⢼⡇⠀⠀⠀⠀⠀⠀⠀⡀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢡⡀⣠⠀⠑⠂⣀⣌⣡⡜⡒⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠁⠀⣸⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⢘⣤⡷⠷⡒⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⣶⣷⣶⣿⣿⣶⣄⣤⣀⣀⠀⠀⠀⠰⠊⠉⠀⠀⠀⣀⠀⢆⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠈⠋⠉⠉⠂⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠿⠷⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠶
⠠⠀⠠⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣤⣴⣦⣤⣤⡄⠠⠤⠤⠤⠠⠄⠀⠀⠄⠀⠀⠀⡀⡄⠀⠀⠀⠀⠰⣶⠀⣾⠆
⢠⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⢩⢍⣹⣿⣯⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡎⠉⠁⢣⡇⠀⠀⣴⡦
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⢠⣦
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⣤⡄
⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⣠⣄⠀⣀⠀⠉⠁
⠨⠤⢤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣠⣤⠤⢤⣠⣤⡁⠙⠟⠛⣛⣛⣛⣛⣛⣛⡛⠛⢛⠛⠣⠤⠤⡜⡇⠀⠀⠙⠋⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡀⠀⣀⠀⣀
⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⡇⠀⠀⠘⠃⠈⠛⠀⠛⠁
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⡇
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⠃
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠐⠿
⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠄⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⢀⣀⠀⠀⠀⠠⠌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⠏⠁⠀⠘⠛⠀⠀⠀⠀⠀⣠⡌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠁⠀⠀⠀⠙⠒⠦⠤⣄⡀⣀
⠀⠀⠀⠀⠀⢀⣤⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡾⠇⠀⠀⠀⠀⠚⠚⠀⠀⠀⠀⠚⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠐⠦⣄
⠀⠀⠀⠀⣴⣿⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠⡦⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣀⢀⣸⡧
⠀⠀⠀⠀⠙⠻⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠞⣉⡭⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣛⣿⠶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⢤⡴⠶⠒⠒⡨⠛⣉⡠⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣼⣿⣮⣬⣉⣙⣭⡉⠉⠉⠉⣹⡙⠄⣀⣴⣤⣶⡮⠶⠶⠛⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⢻⣿⠟⠛⠃⠐⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠄⠀⢸⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣼⠻⠳⠦⠆⢶⡊⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘⣿⣷⣾⣤⠶⠚⠛⢧⠀⠀⠀⢠⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⡀⡇⠀⣿⡿⠃⠀⠀⠀⠀⠘⡆⠀⠀⡟⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠹⢄⢳⡇⠀⣿⡏⠀⠀⠀⠀⠀⠀⠹⠀⢸⡇⣠⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡎⣾⡙⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⢠⡇⢰⠁⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣧⡿⣷⣸⣿⣦⠀⠀⠀⠀⡇⠀⠀⠀⣿⢸⣀⡾⠊⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⠑⣯⣿⡏⣿⠀⣀⠀⢠⠀⠀⠐⠀⢻⡞⢠⠇⢰⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣸⢧⣿⡇⣿⠂⡇⠀⡆⠀⠀⠃⠀⣿⡇⡾⢠⡟⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣾⣿⡗⣿⣰⠀⢸⠀⡄⠀⢸⠀⣿⣿⣵⡾⢋⣉⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣿⣿⡄⢸⡀⣷⢰⣸⣾⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠷⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⢩⠇⠱⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠠⠂⠈⢼⠀⠠⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⢄⡀⠀⣸⡰⠁⠀⠙⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡀⠈⠑⢿⠣⣠⡹⢑⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡂⠂⠁⡿⢀⠤⣌⡡⢻⢦⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠒⡒⣒⠊⡉⠐⠒⢉⠢⡀⠀⠀⠀⠀⠀⡃⠄⢀⡠⠀⢾⢚⢰⠙⡡⣳⠁⠀⠀⠀⣸⠟⠢⢂⠄⢐⣒⠰⢒⠋⠁
⠀⠀⠀⠀⠀⣌⡷⡦⣄⠀⠈⢆⠈⠉⠧⡐⣌⠠⠇⠄⡎⠀⠀⣯⠚⡡⠆⡴⣩⣀⡔⣃⠂⡃⠘⢀⠜⠀⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠩⡀⠁⠀⠁⠒⠀⠀⣀⠈⠉⢷⠀⢠⢬⣨⡦⡾⡟⠔⣡⠈⠉⠓⢷⢀⡇⣀⡂⠷⠀⠒⠒⠂⠀⠐⠁
⠀⠀⠀⠀⠀⢀⣗⠦⠠⠀⠄⠠⠐⠉⠓⢢⣈⣇⠸⠃⡎⢼⠁⣟⠴⠁⡆⠀⠀⣸⠝⠁⣭⡄⠉⠀⡀⠀⠀⢇
⠀⠀⠀⠀⠀⠈⠀⠐⢤⠀⠀⠔⢀⡀⠀⣻⡬⡽⠳⢄⠱⣘⣆⢳⠂⠀⠀⢠⢚⢧⡂⢀⠐⠂⢀⠖⡾⢢⡤⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣈⠁⡌⠀⠘⢷⢒⠆⠳⢤⣈⠟⠀⣠⡞⠉⠁⠈⢿⡸⠆⠯⠷⠞⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⣱⣞⡠⣤⠄⡿⠷⡖⠷⡲⢿⡶⠶⡍⠂⡤⠆⢀⣄⠻⢆⣀⣤⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣠⢖⡡⠖⢺⡟⢸⠀⠆⠀⠈⢈⢠⢆⣡⠞⠀⠀⠈⠲⣀⣠⠊⣉⣀⢨⡟⢭⣱⡚⠵⠦⠄
⠀⠀⠀⠀⠀⠀⠰⠗⠚⠀⠢⠬⠤⢌⣑⠚⠂⠀⣝⡠⠛⠁⠀⢀⠀⠀⠀⠈⢂⣩⠵⠎⢄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢎⡟⣄⡀⠀⠀⠀⠀⠀⢀⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡄⠀⠀⠀⠀⠀⠀⠀⣸⠛⣄⢨⢽⣻⠃⠀⠀⠀⢀⡴⠋⢠⠷⣤⣄⣤⡴⣋⣱⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢇⢳⣄⡀⠀⠀⠀⠀⢠⡟⠀⠨⢹⡼⡽⣇⠀⠀⡠⠫⠒⡫⠨⢍⠢⣢⣴⠾⢙⡼⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡴⠃⢸⡀⢻⡇⠀⠀⠀⠀⣾⢷⠀⠉⠀⣧⡎⠙⡷⡞⡡⠻⢷⣅⢎⣦⠿⣯⡤⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢢⠸⡇⢠⠹⡄⠀⠀⢸⢭⠂⠑⣄⠀⡇⣧⠅⢭⡚⡐⠉⣦⣾⠛⠁⠣⢈⡿⢯⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⢂⣿⡔⠀⢙⢦⣠⡏⠀⢆⠀⠈⢦⢻⡇⠉⠙⢇⣨⡶⠛⠍⠈⣷⠴⢿⣽⣢⡧⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡜⣀⣯⠀⠀⠀⢙⢣⠀⠀⢂⠀⠈⣶⠃⢀⣤⠟⡉⢊⢶⡊⢴⡟⣱⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠫⠎⠕⣻⡄⠀⡀⡞⠀⠇⠠⢤⡂⠀⢹⡧⠟⡁⠀⠈⡆⣸⠃⠤⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠀⣿⠚⠘⠀⢈⢧⡙⠁⠀⠀⣎⠀⠀⡸⠞⠛⠑⠿⠶⣥⣔⣲⠇⡠⣀⢛⠷⠒⢦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡗⡌⠓⠤⣀⡿⠀⠀⢠⡀⢸⣧⠀⢀⠼⣠⣥⣞⠓⡄⢢⡀⢀⠤⠚⠉⢩⠝⣃⠶⠟⠧⠶⣶⣶⣑⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⣻⡄⠀⢀⠁⡀⠀⢀⡇⠸⣯⣄⠈⠂⠌⠁⠈⠑⢾⡟⠊⢁⣀⣀⣱⡯⠎⠀⠀⠀⠀⠀⣬⡥⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢣⠀⠀⠀⢰⠀⣸⢁⠌⠸⣫⣄⣀⡀⠀⠀⠀⢠⠞⠉⠀⠀⢩⡯⠶⠒⠛⠋⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡽⠀⢀⠆⠀⠀⢰⣤⣯⡿⣀⠾⠿⣻⠞⢛⡓⠶⠶⣁⠀⠄⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠀⠈⠐⠦⢢⣾⡿⣩⠞⠋⠓⢀⣀⠀⠀⠈⡑⢦⡈⢑⠖⢦⣤⣄⣀⠀⠉⠛⠓⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠂⠀⠈⡿⣫⢶⣑⣂⠀⠀⠀⠀⢈⠑⠆⠀⠀⡠⠊⢠⠔⠻⡌⠉⠋⡐⠂⣀⣬⣷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢆⢂⠔⠓⠒⠒⠉⢭⣐⠀⠀⠀⠀⢠⡶⠶⠿⠒⠛⠛⠒⠚⠓⠶⢅⣜⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⣰⠃⠀⡅⠀⠐⠂⠁⠨⠉⢓⢦⣄⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠋⠉⠙⠓⠦⢄⣄⣀⠸⡷⢄⣛⣁⡌⠻⣦⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠙⠻⠞⠛⠷⠄
⠀⠀⠀⠀⠀⢀⠠⢐⠨⠒⠁
⠀⠀⠀⠀⢀⠕⠂⠁
⠀⠀⠀⠴⠁

⠀⠀⢐⠀⠂⠂⠂⠒⠐⠐⠐⠀⠂⠂⠒⠐⠐⠐⠐⠂⠂⠒⠐⠐⠐⠐⠂⠂⠂⠒⠐⠐⠀⠂⠂⠂⠒⠐⠐⠐⠀⡂
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⡀⢀⡀⡀⡀⢀⡀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⡄⡀⢀⣠⣀⡁⢠⠀⣀⣅⡀⢠⣄⠠⣠⠈⡀⣠⣀⡄⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⢐⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠄⠠⠤⠄⠤⠤⠤⠄⠄⠀⠤⠤⠀⠦⠄⠠⠴⠆⠠⠠⠤⠤⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠨⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠨⠀⠀⠀⠀⠀⠈⢢⡀⢐⣀⣸⣀⣠⠆⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢸⡈⡄⠀⠀⠀⠀⠀⠅
⠀⠀⢐⠀⠀⠀⣀⡀⠀⢢⡝⠉⠀⠀⠈⠙⢶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢠⣧⢱⠀⠀⠀⠀⠀⡂
⠀⠀⠐⠀⠀⠀⠀⢈⣉⡾⠒⠲⡀⠀⠀⠀⢸⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠲⠤⢄⣀⠀⡟⠀⠀⠀⠀⠀⠆
⠀⠀⠨⠀⠀⠀⣀⠤⢴⡃⠀⣀⣹⣤⣤⡄⣸⠉⠙⠒⠂⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⠙⢻⡄⠀⠀⠀⠀⠅
⠀⠀⢈⠀⠀⠀⠀⣶⣿⣿⡿⠉⢿⣿⣿⠿⡁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣀⣴⠃⠀⠀⠀⠀⡁
⠀⠀⢐⠀⠀⠀⠀⢈⣿⡻⠥⠤⠤⠚⠛⣳⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⡶⠶⢖⡺⠋⡇⠈⢧⠀⠀⠀⠀⡂
⠀⠀⠠⠀⠀⠀⠀⠈⡧⠴⢒⣒⣊⠉⣭⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡞⠀⢀⠞⠁⢠⠇⠀⢸⠀⠀⠀⠀⠄
⠀⠀⠨⠁⠀⠀⠀⠀⢻⣛⣓⠚⠩⠽⠓⠚⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣼⠀⡰⠃⠀⠀⡜⠀⠀⢸⠀⠀⠀⠀⠅
⠀⠀⢐⠀⠀⠀⠀⠀⢸⡀⠀⢠⠊⠑⢆⢳⢸⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⠇⡰⠁⠀⠀⣰⠁⠀⠀⢸⠀⠀⠀⠀⡂
⠀⠀⠐⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⢰⠄⠆⠈⡇⠀⠀⠀⠀⠀⠀⡸⠀⡞⢰⡃⠀⠀⢀⡇⠀⠀⠀⡾⠀⠀⠀⠀⠂
⠀⠀⠨⠀⠀⠀⠀⠀⠀⢷⠀⠃⠀⠘⠁⠸⠀⢣⠀⠀⠀⠀⠀⢠⠇⢰⠃⣸⣹⣷⣦⣾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠅
⠀⠀⢈⠀⠀⠀⠀⠀⠀⢸⡄⢰⠀⠀⠀⠀⠇⢸⡀⠀⠀⠀⠀⡜⢦⣸⠀⠁⠉⢻⠿⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⡁
⠀⠀⢐⠀⠀⠀⠀⠀⠀⠈⡇⠀⡄⠀⠀⠀⠲⠀⡇⠀⠀⠀⢸⠁⠀⣿⣀⠀⡠⠋⠀⣧⣤⣶⢶⡞⣿⣇⠀⠀⠀⡂
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢳⠀⠂⠀⠀⠀⢸⠀⢻⠀⠀⠀⡎⠐⠂⢸⣍⣉⠀⠀⡸⣽⣿⠷⠸⠇⠃⠋⠀⠀⠀⠄
⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⡄⠀⠀⢸⠀⢸⡄⠀⢰⠁⡀⣀⣘⠷⣈⡛⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠘⠦⠤⠃⠀⢀⡇⠀⢸⠈⠁⢹⡉⠒⠲⠮⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⣶⠶⢒⠿⠛⣃⣿⠀⠸⠒⣠⡄⣬⣉⣒⢲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣓⣒⣩⡭⠥⠴⠺⡄⠀⠈⠁⠟⠉⠉⣅⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⣀⡤⠤⠴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⢐⠀⡤⡤⡤⣤⢠⢤⢠⡤⡤⡤⣤⢀⢠⢤⢤⡤⡤⡤⣤⢤⢠⡤⡤⡤⡤⣤⢤⢤⠄⡤⡄⡤⣤⢤⢤⢤⠀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⡄⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠀⠀⠀⠀⣆⠒⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡰⠄⢄⡈⠐⠁⠆⠀⠀⠉⠋⠐⠐⢦⣀⠄⠀⠀⠀⡘⠶⣀⡔⠉⠀⠀⢩⡀⠄⢠⠔⠲⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⣀⢀⠑⢤⠀⢈⡀⠉⠖⠃⠀⠀⠀⠀⠆⠀⢀⠀⡉⠘⠁⠀⠐⡀⠁⢸⠁⠀⠀⠀⠀⡿⣯⠀⢀⡀⠈⡄⣀⠄
⠀⠀⠀⠀⠀⠀⢄⠈⠊⠄⠀⠂⠈⡃⠀⠀⠀⠀⠈⠢⡨⠄⠀⠹⠀⠀⠀⠀⠐⠁⣹⣠⣼⠀⢀⣠⡤⠞⣅⡇⠀⢨⠀⢸⠙⠁⠠
⠀⠀⠄⠄⠲⠤⠀⠙⠊⠁⡄⠀⠀⠳⠂⠀⠀⠀⠀⠀⠘⠂⠀⠐⠂⠑⠒⠐⠀⠀⠛⠟⠛⠻⠁⠀⠀⠀⠀⠑⠰⠂⠠⢆⠐⠚⠀⠀⠂⠴
⠀⠀⠀⠀⠐⣀⡀⠀⠀⠑⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠈⠀⠀⠈
⠀⠀⠀⢉⠁⠀⠈⢲⡋⡖⡁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⣆⢁⠠⠀⠄⠤
⠀⠀⠀⠈⢄⡀⠀⠀⢿⣺⡇⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠂⡠⢀⠄⡰⢠⢀⢆⡄⠔⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠘⣺⠞⠀⠀⠀⡆
⠀⠀⡜⠁⠀⠉⠛⠚⢾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⠌⠀⠀⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣏⣀⣀⡤⠆⠤⡀
⠀⠐⠀⠀⠀⠀⢀⣴⡋⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠜⠁⠁⢀⠈⡤⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⣿⣟⠁⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⢩⣃⠒⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⡁⠄⠁⠇⠐⠁⠏⠸⠀⠃⠺⠰⠇⠊⠇⠼⠀⠁⠀⠀⠀⠀⠀⠀⣽⢫⠀⠱⡄⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠐⠉⡈⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠊⠉⠇⠃⡈
⠀⠀⢀⣀⣁⠀⡀⢀⡀⢥⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡀⠈⢠⡌⠀⠠
⠀⠀⠀⠀⠀⠀⡠⠋⠁⡂⠀⠉⠀⠙⠦⣠⡤⠴⠞⠋⣿⠿⢽⠈⠁⢁⠉⡉⠀⡀⠀⠨⠏⢄⠀⠀⠀⠀⠀⠙⡀⢀⠀⠐⢁⠙⠠
⠀⠀⠀⠀⠀⠀⢀⠔⠱⡃⠀⠃⠀⠷⡼⠁⠀⣀⠀⠀⡇⠀⠹⠒⠁⢀⠐⠀⠀⠁⠀⢐⠂⠈⠃⠀⠀⢀⡀⠘⠆⠀⡦⡀⠈⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⣐⡠⠄⠪⡟⡄⠀⠀⠀⣸⠷⣼⡐⠀⠀⠈⢁⠳⣤⣌⡀⠀⠀⠀⠀⢀⠘⡈⠒⠢⢐⡔⠀⠈⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠈⢁⡤⡈⠁⠀⠀⠀⠀⠀⠌⠀⠀⠈⠘⠀⠀⠀⠐⠙⠢⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠀⠤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⣰⡦⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠁⠀⠏⢀⣴⠞⠋⠉⠁⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠲⠶⠦⣄⡉⠀⠀⠀⢛⠃⠠⠀⠀⠈⠀⠀⠀⠀⠀⣤⣶⣾⢿⣦⡀
⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢿⡄⠀⢵⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢿⣇
⠀⠀⣾⡏⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡆⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⢀⣄⣠⣀⣤⣶⣶⣿⣿⣿⣿⣿⡿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⡟
⠀⠸⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣾⠟⠁
⠀⠀⢻⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠙⠿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⠏

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠉⠻⣟⠓⢛⡟⠉⠀⠀⠀⠀⠀⢸⠿⣶⢒⡛⢻⡛⢙⣧⢰⡘⣇⠀⠀⠀⠄⠤⠀⢠⠶⢚⡋⡉⠀⠀⣠⠀⠛⠀⢃⢈⠆⢠⡈⠻⣆
⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⣻⢸⣷⡞⡇⡎⠘⠜⣧⠸⣆⠀⠐⣦⣴⠀⠈⠒⢬⡙⢡⣄⡆⢨⡄⠱⡀⠘⠋⠀⡈⢁⡄⠈⠛⢲⡄
⠀⠀⠀⠀⠀⢰⠟⢻⠀⠀⠀⠀⠀⠀⢀⡟⢀⠸⡎⠟⣷⢀⡇⢶⢠⡙⣧⠹⣆⠀⠈⠁⠀⠀⠀⠀⠛⠶⢦⣄⡀⠹⠶⠟⠀⠆⠈⠃⢀⠙⠦⠴⠀⠻⢄
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⣼⡷⣼⠀⢿⡀⣿⠘⣿⠸⣧⢷⣘⢧⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣶⣶⣾⠋⠛⠓⠒⠒⠉⠉⠑⠋⠁
⠀⠀⠀⠀⠀⢸⠀⠸⡆⠀⠀⠀⠀⠠⣿⣧⡏⢸⢸⣇⡟⣠⢟⣼⣧⣠⣽⡷⠟⠁⡠⠶⠒⠶⠒⠦⣄⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⢸⡀⠀⢷⠀⠀⠀⠀⠀⠈⠿⠷⠷⢾⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢸⣿⡇⠀⠀⠐⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠘⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⢻⠀⠀⠀⠀⢸⡇⠂⠠⣤⡄⠀⠀⠀⣷⠀⠀⠀⠀⣼⣿⡇⢠⣀⠀⡀⠀⣱⠀⣇⡀⡄
⠀⠀⠀⠀⠀⠀⠹⣆⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡸⣿⡇⠀⠀⠀⠈⠳⠶⢤⣿⣀⠀⢠⡞⠁⠀⠀⠀⢠⣿⣿⠁⠘⠛⠛⠛⠛⢉⠋⠉⠉
⠀⠀⠄⠠⠀⠀⠀⠈⠲⠾⠃⠀⣰⣿⣷⣆⠀⠀⠀⠀⠀⣿⡧⠿⢿⡀⠀⠀⠀⠀⣰⡿⠁⠈⠉⠉⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠉⣁
⠀⢔⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⢧⡉⢉⡼⣣⠀⠀⠀⠀⡏⢠⣦⣤⣷⡀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠠⡀⢠⠤⠔⠀⠀⠀⠀⠀⣿⠀⠋⠍⠀⠸⡆⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⢀⣰⡟⢴⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠈⠁⢔⡀⠀⠀⠀⠀⢠⣇⣩⠖⠦⠦⠴⢹⡀⠀⠀⣿⣿⠿⢉⣿⠙⡇⠀⣿⡇⠀⠀⠀⢀⣠⣶⣿⠏⢀⣾⠃⠀⠀⠀⠀⡀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠦⣤⠀⠀⢀⠎⠀⠁⠀⠀⠀⠀⠀⢱⡀⠀⢹⡇⡄⣿⣿⣦⣷⠀⠙⠃⠀⢀⣴⣿⣿⠟⠁⢀⡾⠃⠀⠀⠀⣤⣴⣷⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠦⢴⣦⣤⠖⠛⠈⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⡿⣁⢂⣴⠟⠁⠀⠀⡠⢿⣿⠟⠿⠿⢛⣷⡆
⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⢸⠉⠈⡀⠀⠀⠀⠀⠀⠘⣿⣇⣉⣉⡿⠀⠀⠀⠸⣿⣿⣷⡴⠟⠁⠀⠀⠀⠈⠢⠤⡤⣤⢤⢄⡈⠐⠛⡄
⠀⠀⠀⠀⠘⣾⠱⢧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢀⣰⠉⠁⠁⠁
⠀⠀⠀⠀⡄⢊⠈⠀⢄⠀⠀⠀⠀⠀⡇⠀⠰⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢠⠶⣶⠲⣤⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿
⠀⠀⠀⣘⠊⠉⠉⢈⣠⠇⠀⠀⠀⠀⢡⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⠷⢏⣦⠸⠟⢡⡍⠛⠓⢄⠀⠀⠀⠀⠀⠀⣸⡧⡟⠀⠀⠀⢀⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠒⠂⠈⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⣴⠃⠀⠀⠐⠂⠀⠀⠀⠀⢀⡤⠒⠒⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⠄⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⣤⡀⠀⢀⠀⠀⠀⢠⢀⠀⠁⠁⠀⠀⠀⠀⠠⠤⣄⠢⡀⠒⠀⠀⠀⠘⠀⠆
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⢁⠀⢐⠀⠐⣧⣤⢦⣠⣖⠶⣄⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠘⣷⣦⠀⠀⢀⣀⠂
⠀⠀⠀⢀⢀⠀⠁⠀⢸⡗⠊⠀⠀⠠⠠⠀⣬⠽⠋⠈⠙⢭⣤⣆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⡿⢀⡴⠉
⠀⠀⠠⠁⠈⠀⠀⠀⢠⠄⢠⡄⠐⢀⠀⡀⠁⠀⠒⠢⡀⢸⣿⠉⡄⠀⠠⠐⠐⠒⠀⠀⠀⠂⠄⢸⠄⡞
⠀⠀⠀⠄⠀⢠⠂⠐⢿⣶⡶⢳⣀⣀⡀⡀⠀⠀⠀⠀⠙⠘⣿⢰⠅⠀⠐⠀⠀⡀⠀⠀⠀⠀⢰⣷⣄⠇
⠀⠀⠀⠂⠀⠚⠀⠀⠀⠈⢷⣤⣈⣄⠘⠉⠑⢄⡀⣀⠆⠀⢻⣎⠣⠄⠁⠐⠒⣿⡀⠀⠤⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠔⠊⠈⠉⠉⣁⠐⣷⣦⣤⣦⡄⢠⠊⢙⣷⣀⡀⠀⠀⢈⣷⣄⠀⠀⠀⢨⡟⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣸⠟⠛⠀⠀⡈⠑⠋⠁⢘⣿⠋⠉⠀⠀⠀⠉⠹⣧⣤⣶⡇⠀⡘
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀⢀⡠⠤⠿⠆⣀⠴⠒⠛⠂⢀⠀⠈⠉⢳⡃⡴⠀⠈⠙⠢⠀⣿⡛⢷⣤⣇
⠀⠀⠀⠀⠀⠂⠀⠀⣀⠐⢳⣾⣥⠀⠂⠉⠀⠀⠀⠀⠀⠀⠠⢀⣀⡠⠂⠀⠀⠀⠀⠀⢠⢹⠁⠀⠀⢛⣳
⠀⠀⠀⠀⠀⠀⠈⠀⡈⠀⢼⡟⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠣⡀⢰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠚⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠷⢮⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠉

⠛⣁⣀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠀⢴⣁⠀⠀⡀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡈⠉⡃⠀⠉⠑⠓⠄⠀⠀⡀⠀⠰⠶
⠈⠛⠊⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⣤⠂⢀⠀⠀⠀⠐⠇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⢈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢠⠀⠀⠈⠊⠂⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠈⢤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⣀⣀⠀⠠⠄⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡀⠀⠈⠁⠤⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣈⣀⠈⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠆⠀⠀⠀⠤⠀⠀⠀⣀⣉⣉⣂⣄⠀⠠⠴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⣁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡄
⠀⢀⠤⠀⠀⠀⡰⠖⠋⠀⠀⠀⠈⠛⢉⡀⢀⠀⠀⠀⠀⠀⡀⢌⢀⡀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠀⠀⠀⠀⢸⠀⠀⠀⠈⢻⣿⠋
⠀⠀⠀⠀⠀⠁⠀⠠⠔⠂⠀⠰⡅⠀⠲⠇⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠚⠶⠤⣄⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠙⠐⠒⠤⢈⣢⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠇⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣀⠀⠀⡀⠀⠀⣀⠠⠀⠀⠀⣀⣀⡀⠀⢠⠏⢠⡀⡜
⠀⠀⠠⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠃⠘⠁⠀⠀⠀⠀⡃⠤⣀⡉⠢⣠⢸⠘⠿⠁⡠⠔⢒⡂
⠀⠴⢀⡄⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠐⣀⠀⠀⣠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣨⣙⣓⣼⣷⣴⣇⠠⠔⢀⡰⠁
⠀⠀⠀⠀⡀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⡘⠟⠈⣠⡾⢁⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⢠⠆⠘⠉⢉⠉⢛⣿⣿⣿⠟⠋⡉⠉⠉⠑
⠁⠀⠀⠀⠹⠒⢤⡄⠀⡠⠀⠀⠀⢀⠈⢁⣷⠟⣁⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⢠⣧⡴⠺⠋⠈⡴⠫⠟⠻⢍⢠⣄⠀⠈⠁⢀⠈⠂
⠀⠁⠀⠀⢠⠐⠢⠀⠀⠀⠀⠀⠀⢀⡶⠏⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠠⠖⠈⠀⠘⠀⡴⠘⣤⠌⢆⠉⠁
⢄⠀⡄⠒⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⢀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀
⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠁⠀⠐⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠘⡀⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⡠⢔⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢠⠂⢰⠀⠀⠇⡗⠀⡦⠊⠉⡁⠀⠀⠀⠀⠀⠠⠌⣡⡟⠀⠀⠀⠀⠠⠀⠄⢐⣾⣧⡀
⢀⠀⠀⠀⠀⡀⢀⠀⠀⠰⠅⠀⠹⠂⣁⠴⠋⠀⠀⠀⠀⣶⡞⠁⠁⠀⠀⠀⠀⠀⠀⢀⠁⠁⠈⠁
⡆⢸⠀⠀⢘⠀⠜⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠴⠄⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠇⣸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠴⠤⠆⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠀⠂
⠁⠠⠔⠁⣀⣄⣈⠉⠀⠂⠑⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⠃⠀⠈
⠀⠀⠀⠤⠲⢄⣀⣈⠉⠥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠀⣥⣿⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⡄
⠐⠲⠂⠒⠀⢀⠈⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠁⠀⠐⠀⢁⡀⠀⠀⠇⠀⠀⠀⠉⠒⢦⣀⠁⠀⢱⠆⣀⣧⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠁⠉⠀⠀⠀⠀⢠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⠩⢷⣤⡀⠠⠈⡉⠦⡃⡀
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠷⠆⠈⠙⠻⢧⣶⣀⠁⠡⡄
⣥⣷⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣐⣂⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣈⣐⣚⣁⣈⣉⣙⣂⣀⣀⣀⣀⣀⣀⣀⣀⣐
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠈⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⠂⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢁⣀⣤⣤⡴⢴⣶⠶⢤⣤⣄⡀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⣞⣋⣯⢦⢺⡇⠀⣿⡴⣌⢿⣿⣿⠄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣈⠻⢿⣷⣯⣼⡇⠀⢸⣿⠽⠿⣛⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢦⡀⠀⠀⠀⠈⠙⠒⠲⠶⢾⠃⠀⢸⠗⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠏⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣾⠀⠀⠀⠀⣠⣴⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⠉⠉⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⢰⠇⠀⢀⡏⠀⣤⡶⢛⠉⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢾⣿⣯⣿⠉⣿⣻⣳⣦⠀⣷⠀⠀⠀⢀⡟⠀⠀⣼⠁⢸⡇⠁⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠬⡏⠀⣯⣭⡭⠥⠞⠁⠀⠀⢀⡞⠀⠀⢠⠇⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡟⠀⠀⠀⠀⠀⢨⡟⢳⡶⠶⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡄⠀⠀⠀⠀⠀⢰⠇⠀⠀⣼⠁⠀⠀⠀⠀⣠⠏⠀⡼⠀⠀⢀⣀⡴⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⡀⠀⠀⠀⠀⡾⠀⠀⢀⡇⠀⠀⠀⠀⣴⠃⠀⣼⠁⣴⠚⠋⠉⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⡀⠀⢳⡀⠀⢳⡄⠀⠀⢰⡇⠀⠀⢸⠁⠀⠀⠀⡼⠁⠀⣼⠃⠀⠻⣜⢯⣹⡗⢢⣄⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠛⠶⣄⢳⡀⠀⠙⣆⠀⢸⡇⠀⠀⣼⠀⠀⠀⣸⠃⠀⣸⠃⠀⠀⠀⠈⢱⠟⢀⣟⣺⢟⣠⠟
⠀⠀⠀⠀⠀⠀⣼⠟⠅⠀⠀⠀⠀⠀⣺⠇⢻⡄⠀⠈⢧⣾⢀⠀⡄⣿⠀⠀⢠⡟⠀⢠⡏⠀⠀⠀⣀⡴⠋⢠⠞⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⡤⢴⡖⠋⠁⠀⠀⢻⡄⠀⠘⣿⡸⢠⢲⣻⡆⠀⣸⢷⢰⢸⡇⢀⣤⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠻⣄⠀⠀⠀⠀⢷⢁⠄⢻⣏⡿⡘⣿⣷⠀⣿⣾⣾⢸⣷⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠻⣶⣶⣶⡜⣟⢄⡠⢿⡹⡱⡣⠹⡇⣿⠿⢡⣾⣿⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣤⣶⣶⣶⠏⢻⣃⡆⠊⣿⣴⣥⡼⠃⢹⡇⡴⠰⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⢧⣥⡼⠏⠀⠀⠀⠀⠀⠙⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡴⣀⣤⣤⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠑⣀⡁⠶⡉⣣⡜⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⣟⡄⡸⣾⠀⠀⠀⠐⠠⠀⡐⠂⡂⢢⡀⠀⠀⢀⠰⣄⠩⢤⠘⢣⣄⠹⣆⡙⣥⠚⡾⠋⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠚⠛⠘⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡈⢤⡈⢤⡀⠻⡤⠑⢢⡈⢵⠆⠓⠀⠈⠈⠀⠸⢡
⠀⠀⣤⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠱⠦⠉⠠⠚⠓⠂⠉⠀⠀⠀⠀⠀⠀⠉⠀⠖⠃
⠀⠀⠀⠀⠛⠃⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢠⡤⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⣀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣠⡤⠧⠀⢀⡤⠀⠀⠀⠀⠀⠀⡤⢀⣀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣀⡀⠛⠀⣠⡶⠀⠀⠀⠠⠇⠉⠉⠉⠉⠛⠉⠉⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁⢀⡤⠖⠀⠀⣀⣠⢠⣤⣤⠶⠶⠴⠁⠀⠄⠀⠀⠀⡄⠀⠀⠀⠀⣀⡀⠰⠁⠀⡀
⠀⠀⠀⠀⠀⢈⠛⠃⠀⠀⠀⠀⢲⣶⣾⡦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠘⠲⠆⠒⠊⠀⠀⠀⠀⠈⠉⠠⠈⠀⠁⠉⡗⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠓⠁⣀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠄⢤⠀⠀⠧⠎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠹⠁⠠⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠈⡦⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠰⠀⠘⠀⠀⠃⠀⠅⠀⠐⠀⠀⠀⠀⡶⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣐⢁⠺⠁
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠁⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠈⠀⠐⠀⠀⠀⢄⠀⠀⠀⠀⠀⠰⠀⠀⠀⢠⠤⠊
⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠠⢤⠄⠈⠀⠀⠀⠈⡀
⠀⡀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⢆⠴⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠉⠁⠉⠀⠀⠀⠂⠀⠠⠤⠄⠤⠔⠈
⠀⠀⠀⢀⡀⡧⠀⠴⠆⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⠀⠀⠀⢂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠙⠓⡼⠆⠀⣀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠐⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⠈⠁
⠀⠀⠀⠐⠒⢤⡄⠀⠈⠁⠀⠀⡀⠀⠛⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠈
⠀⠀⠀⢀⡀⠀⠀⠟⠀⠀⠴⠀⢁⢄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣚⠁
⠀⠀⠀⠀⠁⠀⠷⠀⢰⡀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠐⠆⠀⠀⠌⠀⠀⠀⠀⠀⡰⡀⠀⠀⠀⠀⠀⠀⠀⠒⠹⠀⠀⠀⠀⠀⡶⢠⡀⢒⠝⣇
⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠠⠈⠃⠀⠀⠀⣀⠀⠀⠀⠈⠇⡈⠕⠉⠀⠉⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⢁⠀⠃⠠⠄⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠰⠆⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠆⡠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⠈⠇⠈⠀⢠⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣰⠐⠂⠀⠀⠀⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠔⠀⠀⠀⠀⠀⢴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠃⡄⠀⠀⠀⠀⡠⠒⠊⠀⠘⠀⠄⠀⠀⠀⠁⢀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⢀⠀⠀⠠⢀⡰⡞⠀⠀⠂⠀⠐⠒⠂⠀⠐⠄⠀⠀⠀⢀⣠⠔⠲⠒⠀⠀⠀⠀⠀⠉⡇⠀⢀⣀⠀⠀⠀⠀⠁⠀⠀⣡⠔⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠠⠀⠀⠉⠹⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠋⠅⠒⠒⠐⠀⠀⠀⠠⠄⠀⠒⠀⠀⠠⢀⠀⠀⢤⡄⠀⢈⣀⠄⢰⠆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⡀⡄⠀⠤⠀⠀⠀⡞⡤⠀⠀⠀⡂⠀⠂⠀⠀⢠⠉⡼⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⡉⢹⢦⠀⢑⡄⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠈⠀⠈⠀⠂⠐⡂⠀⠒⢿⣤⣴⢶⣆⣂⣯⡹⢧⣤⠀⣧⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠁⠀⠀⠀⠊⠁⠈⡿⣷⣄⠠⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠦⣧⠼⠋⠀⠈⠁⢰⠀⠀⢉⡙⠿⣿⠋⠙⠷⠮⢐⣃⣘⡄⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠃⠀⠀⡀⠑⠀⠙⣿⣿⠁⠀⣠⠟⠉
⠀⠀⡴⠒⠤⠂⠀⠀⠀⠀⠦⠁⠀⢠⠀⢰⠈⠋⠈⠁⠈⡿⠛⠋⠀⠀⠀⡆⢹⣻⣿⠷⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣼⠉⢰⡼⠋
⠐⡴⢃⡀⠀⠁⠀⠀⠀⢀⡳⠀⠈⣶⠀⠘⠣⣀⠄⣀⡄⡠⠂⠈⠉⠙⣄⡀⠀⣷⡏⠀⡇⠀⠀⠀⡔⠉⠉⠑⠉⠠⠀⠀⠂⠁⡀⠄⢠⡷⠀⡾
⠀⠢⠀⠀⠀⢀⡤⠀⣠⣼⣷⣴⣤⡽⣆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠏⠰⠿⣧⠀⡇⠀⠀⣠⠕⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⣠⣾⣄⠀⡇
⠀⢠⠉⠀⠀⢹⠄⠀⠠⡈⠛⢿⣇⠀⠘⠲⠶⠒⢶⣶⢤⣀⠀⠀⠀⠀⠀⠆⠀⣰⣇⠀⡇⠀⠀⣀⠀⠀⣀⣹⡦⠀⠀⡀⠀⢄⢃⠘⠝⣿⣧⠱
⠀⠀⠑⠀⠀⠉⠀⠀⢠⠐⠀⠐⠿⣿⣦⡀⣼⣆⠈⠁⠀⠈⠳⣄⣀⣠⠼⠁⠀⠉⣻⢷⣘⠲⠇⠀⠀⠉⠉⢻⣿⡀⠦⠔⠒⠠⡀⠀⠀⢙⣿⠀⢃
⠀⠀⠀⠀⠂⠠⠃⠀⠐⢀⢀⡤⠚⠁⠈⠛⠉⠉⠁⠀⠺⣿⢦⣀⣤⣤⣤⡄⢀⡠⠉⠈⣿⣧⠀⠀⠀⠀⠀⠀⢙⣿⣆⠀⠀⠀⠰⠀⠁⣿⡿⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢛⣦⣼⣿⠊⠍⡛⢫⡿⢤⡾⠣⠀⣤⣿⣿⠟⠷⠒⠀⠀⠀⠘⠿⢿⣏⠆⠄⢒⢰⡔⠛⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠐⠂⠈⣼⠋⠁⠈⠀⠀⠁⢠⠀⠀⠀⡀⠀⢠⣽⣇⠀⠀⠀⠀⢀⡠⢄⣀⠀⠻⢷⣟⣿⣿⡇⠀⠀⡌
⠀⠀⠀⠀⠀⣀⠠⠂⠀⠄⠀⣄⣀⠈⠈⠀⠤⣴⣿⡿⠆⠀⢰⠶⠂⠚⠓⠀⡐⠀⠀⠈⠁⠈⢿⡏⢀⡴⠀⠀⠀⠉⠓⠄⠀⣿⣿⠹⠿⣦⣄⣧
⠀⠀⠀⠀⠰⡥⠀⠀⠀⠀⠀⣀⣭⣤⢷⠏⠀⠀⡠⠴⠛⠉⠁⠀⠀⠀⠀⠀⠉⠀⡇⠀⠐⡈⢈⣃⠌⠀⠀⠀⠀⠀⠀⠀⢠⢸⡋⠁⠀⠙⠩⢿⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡠⠓⠢⣆⠀⠀⣿⣿⡿⠗⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢷⠀⠀⠈⠀⢠⠤⠝
⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠭⠂⠄⢿⡏⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠳⡄⠀⣻⠨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠡⢒⡿⠃⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣝⡤⡘⠒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡠⠄⠄⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠋⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⡴⠋⠻⠆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡿⢩⡏⠉⠁
⠀⠀⠰⡇⠀⣀⢠⡞⢲⢸⢡⣆⣦⣶⢰⢰⡏⠳⠀⠀⣰⠃⣸⢓⣷⡴⢠⣆⣴⠛⠷⣾⢲⠆
⠀⠀⠀⠙⠛⠋⠈⠛⠉⠘⠈⢉⠁⠈⠉⠁⠁⠀⠀⠀⠋⠀⢣⣏⠸⡶⠙⠋⠈⢁⣀⠀⢽
⠀⠀⠀⠈⢄⣀⠀⢞⣠⠚⡄⠈⠠⠀⠀⢠⠀⠀⡄⣀⢀⣀⢀⠀⣀⠀⠀⠀⠀⠘⢯⣠⠟
⠀⠀⠀⠘⢀⠏⠱⢆⠋⡄⠀⠈⠀⠀⠀⠘⡞⢳⠡⠻⡼⠎⠸⠓⠙⡆
⠀⠀⠀⠀⠀⠘⢀⠎⢀⠑⠀⠠⢀⠀⢀⠖⠂⢠⡀⡆⣀⣀⠀⠀⡤⢤⡄
⠀⠀⠀⠀⠀⠐⠁⠐⠀⠀⡏⠠⠁⠀⠈⠒⠂⠓⠊⠃⠓⠉⠒⠃⠁⠀⠚
⠀⢀⡄⠊⠣⢤⣠⣤⢐⣮⠥⠊⠢⠀⢰⠖⠂⠈⢹⢰⠀⠀⢲⢸⠉⠉⠉⠙
⠀⠀⠁⠂⣄⢀⠈⠈⠃⠄⡚⢛⠂⠐⢸⣄⣠⣤⡼⢸⡀⠀⢸⣸⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠃⠚⠀⠁⠀⢀⡩⡄⠀⣀⠀⠀⡀⢀⡀⠀⠀⠀⠉⠀⠀⠀⣠⣀⡴⡆⠀⡤⠀⣜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣀⡠⢮⣨⠗⣏⡩⠺⠧⠤⠄⠀⠀⠀⢀⠔⠜⠇⠁⢙⠞⡄⠻⣄⢸⣠
⠀⠀⠀⠀⠀⢠⠒⢠⣄⠆⡤⡄⣄⣄⡴⠂⠐⣦⠀⠀⠀⠀⢀⡔⡡⣄⠤⣤⢣⣈⡤⢀⡄⠈⠻⢙⡀
⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⢤⠠⠆⠒⠲⡄⢠⠀⠲⠶⣤⠀⠬⠼⢅⣿⣤⢏⣸⣿⢰⣟⣘⣧⣜⣶⠇
⠀⠀⠀⠀⠀⠀⢸⣁⠀⠀⢸⠀⣀⣀⣀⣇⢸⡀⠀⠀⢻⠀⠀⠀⡸⠀⠀⠐⢠⠀⠈⠃⠡⠌⠉⠋⠉⠐
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠀⠉⠁⠀⠁⠀⠉⠉⠉⠉⠀⠀⠀⠐⠂⠁⠰⠀⠁⠀⠈⠀⢄⠀⠆
⠀⠀⠀⠀⡠⠒⠄⣀⢀⡀⡀⣀⣀⡀⠀⠀⣀⣀⢀⠀⡀⣀⡀⠀⠀⢀⣀⠀⡀⢀⠠⡄⠀⠀⡀
⠀⠀⠀⠀⠧⡤⠼⡤⠟⠳⠧⠸⠒⠊⣇⡀⣗⡀⠘⠛⠁⣴⢀⢧⡅⢸⠀⡾⢳⡈⡶⠅⠈⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⢠⢤⣐⣠⣄⡀⡀⢠⠺⢤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⠦⡴⠓⢄⣈⠥⢼⠁⠀⠈⠀⣈⣍⡀⢀⣈⣭⡀⠀⠀⠀⣀⠄⠤⠂⠉⢱⣆⡀
⠀⠀⠀⠀⢤⡔⠚⠀⠀⠀⡌⠀⢠⡇⠀⢀⡴⠋⠁⠀⠉⠀⡐⠈⠳⣄⠀⢘⡅⠀⠀⠀⢀⠈⠀⢸⡇
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠄⠀⠀⣠⠞⠁⠈⢧⠀⠀⠀⠀⠔⠀⠀⢀⡼⠂⡞⠀⠀⠀⢀⠂⠀⠀⠀⢙
⠀⠀⠀⠰⠣⡄⠀⠀⡜⠀⠀⠀⠈⢓⣄⢠⠏⠀⠀⢀⠈⠀⠀⠀⠹⣄⠀⣹⠆⠀⡠⠁⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⣇⡠⠴⢄⠀⢠⠒⠤⢽⡄⢸⡀⠀⠀⠂⠀⠀⠀⠀⠀⣸⠠⡇⠀⡐⠀⢀⠀⣠⠶⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⠛⠶⠥⠖⠒⠦⣤⠉⠉⠁⠀⠉⠉⠓⠖⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠏⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⠉⠁⠀⠀⠉⠁⠂⠓⢤⡁⡘⠆⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢅⠀⠄⣀⠒⠾⠰⠀⣠⡀⠀⡀⡀⠀⢠⢆⠠⢠⠀⠀⣀⣤⡜⢯⣳⣷
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠈⠐⡶⡤⣤⣄⣀⣠⣁⡤⠀⠤⠤⠀⠠⠄⠁⠉⠠⠤⠀⠋
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⡐⠶⠐⠆⠂⠀⠀⡐⠀⠂⠀⠀⠀⠀⠀⢱⠀⠀⢠⣅⠀⠈⠉⠻⢇⣿
⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠠⠿⣦⠗⠛⠉⠛⢾⣿⣤⡄⠈⠓⠀⠀⠀⠀⢃⠀⢸⠇⠉⠐⣄⢀⣠⣽⡇
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⣸⡃⠀⠀⠀⠀⠘⣿⣆⠐⠂⠀⢯⠀⠀⠀⢸⡄⢈⠀⠀⠀⠀⠈⠀⣿
⠀⠀⡞⠀⠀⠀⠀⠀⠀⣾⡥⣿⡇⠀⠀⠀⠀⠀⠈⢻⣄⡈⠁⠭⠄⠀⠀⢸⠇⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢳⠀⠀⠀⠀⠀⠀⠛⣷⢹⡇⠀⠐⠤⠠⢄⠀⣸⣷⡥⢌⠍⠁⠀⠀⢸⠁⢠⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢊⡚⢻⡆⠀⠀⠀⠀⠛⢻⡇⠝⠃⠀⠀⠀⡀⠃⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠘⡅⠀⣀⡀⠠⠊⠀⠀⠀⢸⠄⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠈⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠢⣤⠀⠸⣼⠀⠀⠀⠀⠀⣼⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠲⢷⣦⣿⠇⣰⠀⠀⡀⢹⢧⣮⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⡜⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⡤⣀⣡⣾⡜⢷⢣⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠰⣀⠏⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠁⠯⡟⡁⢣⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠠⡀⠀⠀⠀⠀⠀⠀⡘⠱⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢦⡆⠟⡧⠹
⠀⠀⠀⠀⠀⠀⡀⠸⢸⢋⠀⢘⡆⠘⡂⣨⠀⣶⢟⣸⡀⢀⠀⠘⠀⠀⡀⠀⡀⣠⢏⡛⣆⠇⢀⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠾⠖⠿⣧⡇⠿⡈⡇⡟⠈⡏⠃⠎⠐⡀⢠⣰⡾⢶⡟⠉⠀⢳⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠰⠈⠃⠘⠠⠇⢾⠚⠷⠔⠋⠀⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠻⣄⣀⠟⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⣠⠴⠖⠚⠛⠋⠙⠛⠓⠲⠦⣄⠀⠈⠓⠒⠋
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠻⠭⣉⠙⠒⠒⠶⠤⣤⣄⡀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢀⣠⣤⠤⠴⠒⠒⠋⣉⠭⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄⡀⠀⠀⣹⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⣏⠀⠀⢀⣠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠁⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⡀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⡸⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢇⠀⠀⠈⠙⠲⠦⣄⡀
⠀⠀⠀⠀⢀⣠⣴⣚⣉⠤⠤⠴⠒⠚⠋⠉⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠉⠙⠒⠒⠢⠤⠤⣉⣓⣦⣄⡀
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⡀⠀⠉⠓⠲⠤⢤⡤⢤⡤⠤⠖⠚⠉⠀⢀⡴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡴⠃⠀⠀⠀⠀⠀⠀⣼⠉⠉⣧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⣀⣴⣞⣡⡄
⠀⠀⠀⠀⠀⠀⡔⣿⣿⣮⢞⡵⣫⠝⠁
⠀⠀⠀⠀⢠⢖⡣⣞⡿⠁⢀⣜⣡
⠀⠀⠀⠀⣎⣿⣿⣿⣷⣠⠜⠻⡀⠀⠀⢀⡀⠀⠀⠀⠈⠁⢀⣀⡀
⠀⠀⠀⠀⠙⠿⣿⣿⠿⣇⠀⠀⠱⣄⠀⢉⡤⢤⣀⠀⢀⠞⠁⠀⢸⠂
⠀⠀⠀⠀⠙⠓⣼⣅⠀⠀⠑⢢⡄⠀⠙⠺⣆⠀⠈⠳⣼⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⡎⢻⣷⣄⠀⢀⣭⡓⢤⡀⠈⢣⡀⠀⠈⢳⡭⠥⠤⠤⠤⠤⠔⠒⠚⠉⠁⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠳⢿⡈⢮⣑⢮⡀⠈⠙⢿⣦⡀⠙⠲⢤⠤⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⢀⡤⢔⡛⠿⠆⠀⠀⠀⠃⠀⠀⢹⡉⠁⠐⠒⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠘⠋⣉⠴⠚⢇⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠻⠻⡖⠚⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢣⠀⠀⠙⠛⠓⠿⣶⡄⢀⣤⣶⠿⠛⣁⣸⠾⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠹⡭⠞⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠧⣄⠀⠀⣾⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠷⢖⠤⣈⣋⢿⠢⣄⠀⠀⠀⠀⠀⠀⠀⠸⡁⢠⡼⠂⠀⠙⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠔⣊⡥⠄⠒⠂⠀⠀⢲⡀⢱⠀⠁⠀⠱⢬⡑⢤⡀⠀⠀⠀⠀⠀⣈⡦⡇⠀⠀⠈⠳⡞⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣈⠳⢄⡠⢊⡵⠊⠁⠀⠀⠀⠀⠀⢰⡣⠷⠞⠀⠀⠀⠀⠈⢯⠓⢬⡒⠒⣒⡩⠥⠒⠊⠹⣤⣀⡀⠠⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠑⣒⣻⡀⠀⣀⠉⣉⣉⡹⠂⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡩⠕⠊⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣤⣀⣀⣀⣀⣀⣀⢤⠤⣴⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⠧⡔⠈⢋⠫⠏⠉⢖⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣍⠋⢀⢿⣾⣿⡛⠺⣧⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷⣶⠷⠞⠿⢳⣾⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⡴⢩⠃⠀⠀⣠⠎⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⡀⠈⣛⣦⣀⡀⣀⣨⠇⣴⡉⠀⠀⢉⣑⣾⣖⠒⣒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣏⡫⢟⣽⡩⠊⠀⠀⠉⠒⠭⣓⠿⣏⡩⣟⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠽⠷⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠠⠀⠀⣠⣤⣄⡀⢠⠔⠆⠀⣀⣀⠰⣇⠆⢰⠀⣦⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢰⣛⣯⣍⢹⣿⠷⠤⠞⠿⠷⠥⠤⠇⠀⠘⠶⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠈⠀⠀⠀⣰⠛⠂⢀⣄⣜⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠎⠛⠛⠛⠙⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⠀⠀⠀⢀⡀⡀⠀⣀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡄⠀⠀⠀⣧⣉⣭⣿⣟⣿⣉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⠉⣉⡉⣙⣻⠀⠀⡁⠀⠀⠀⠀⣁⠀⠀⠀⢰⣶⣶⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠈⠀⢰⠶⠓⠀⢰⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡖⣴⠾⣾⠏⣽⠐⣮⡀⣞⠚⠛⣧⣟⣶⡄⠀⠎⠉⠉⠉⠉⠉⠉⡆⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣇⣴⣖⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠚⣿⣇⣼⣄⠁⠈⠅⢈⡀⣁⠀⢄⡀⢀⣤⣤⣤⣤⣤⣤⣴⣧⣿⠷⣶⠀⠀⠀⣤⣀⣼⣯⣈⣉⣿⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣿⣿⡘⣿⣿⣿⣿⣿⣾⣿⣿⣤⣼⣿⣿⣿⣽⣽⣿⣿⣷⣤⣤⣄⣀⣠⣾⣥⣤⣿⣿⣭⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠀⠂⠀⠀⠀⠀⠀⡄⠀⢂⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠛⠀⢈⠉⠁⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⡇⠀⣀⣠⠀⠑⠀⠂⠀⠰⠂⠀⡀⣉⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈⠀⠈⠉⠉⠁⠈⠀⠉⠉⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠠⠤⠤⠀⠀⠀⠠⠀⠀⠀⠀⠤⠀⠠⠤⠀⠀⢠⣄⣀⣀⣀⠀⠀⠀⣀⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⣠⠊⢸⡈⠢⡀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠈⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠐⡀⠀⠀⠀⠈⢢⠀⠀⢀⠎⠙⢦⣸⣃⡴⠙⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠈⢄⠀⠀⠀⠀⠱⡀⡼⣄⠀⠀⢸⠁⠀⠀⠸⠀⢀⠄⠀⠀⠀⠀⠀⡠⠻⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⢸⠗⢹⠙⠔⣧⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠈⢢⠀⠀⠀⠀⠙⣷⡈⠳⢄⣸⡀⣀⡤⣾⠠⠂⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⡟⡆⢸⠀⢀⢿⠀⠀⠀⠀⢀⠎
⠠⡀⠀⠀⠀⠑⡀⢀⡀⠀⠈⢷⣄⠀⢸⠁⠀⡴⠑⠀⠀⠀⠀⠀⡠⠃⠀⠀⢀⠀⠀⠀⠀⢠⢿⢦⡀⠀⠀⠀⡇⠱⡘⡄⡸⢸⠀⠀⠀⣠⡿⡷⡀
⠀⠐⢄⠀⠀⠀⣸⠋⠉⠷⡢⠄⠿⠀⢸⠀⢼⣃⣴⠛⠉⠉⠢⠜⠀⠀⠀⠀⢸⠀⠀⠀⠀⡎⠘⡆⠑⣦⣀⠀⡇⠀⠱⡷⠁⢸⢀⣠⡔⠋⢠⠅⢣
⠀⠀⠀⠢⡀⠀⢸⣆⠀⠀⠈⠓⠦⠤⣼⠤⠔⠋⠁⠀⠀⠀⣰⠀⠀⠀⠀⡠⢾⠀⠀⢠⣤⠇⠀⠘⣄⢰⠀⡏⢣⠀⠀⡇⠀⡜⡏⠀⡇⠀⡼⠀⠸⣿
⠀⠀⠀⠀⠀⢄⠀⠫⡳⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⢾⠃⠀⠀⡠⠊⠀⢰⠀⠀⠀⡏⠒⠦⠤⣈⣾⠀⢻⠸⡄⠀⢱⠀⡇⢧⠀⢳⡰⢁⡠⠎⣹
⠀⠀⠀⠀⠀⠀⠢⡀⠈⠢⣑⠢⢤⣀⢸⢀⣀⡤⠔⡫⠖⠁⠀⣀⠊⠀⠀⠀⢸⠀⠀⠠⣾⣀⠀⠀⠀⠀⠳⣼⠀⡇⠀⢸⠀⡇⠸⣀⠜⠉⠁⠀⣀⢽⣃
⠀⠀⠀⢠⠋⢫⡉⠻⢄⡀⠈⠳⡄⠈⢹⠁⠀⠠⣎⣀⡤⠒⠉⡠⠋⠳⡄⠀⠸⠀⠀⢠⣧⣀⠉⠉⠉⠈⠉⣉⡳⢇⡀⢸⠀⢱⡴⠛⠒⠒⠚⠉⢀⡨⡿
⠀⠀⠀⠸⣆⠀⠈⠓⠤⣈⡙⠚⠁⠀⢸⠀⠀⠀⠈⠀⢀⡤⠊⠀⠀⢀⡇⠀⠸⡶⣔⠒⠃⠀⠉⠉⠉⠉⠉⠀⠀⠀⠉⢺⠖⠉⠉⠑⠒⠒⠒⠚⠉⠀⠉⣩⡇
⠀⠀⠀⠀⠹⣷⢄⠀⠀⠀⠉⠑⠒⠤⢼⠤⠤⠴⠒⠊⠁⠀⠀⢀⣴⠏⠀⠀⠀⠘⢌⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⡽
⠀⠀⠀⠀⠀⠈⠓⢷⣤⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠈⠳⣄⢀⡽⠛⠒⠒⠲⢤⠤⠒⠒⢺⠒⠒⠒⢶⡖⠒⠚⠉⠓⢤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡕⠲⠤⠤⣸⣀⠤⠤⢒⡩⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⣀⣀⣴⠋⠀⠀⠀⢸⠀⠀⠀⣀⣉⣢⡤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠄⠀⠀⢸⠀⠀⠀⢟⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠉⠉⠉⡟⠉⣹⠋⠉⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢍⣁⡀⠀⠀⢸⠀⣀⡠⠔⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⣇⢀⠇⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⢀⠔⠙⢦⡀⠉⠉⠉⢹⠉⠀⠀⠀⡠⠋⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⢢⡀⠀⢸⠀⣀⣠⠞⠁⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢹⡉⢹⠉⢡⠎⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣸⡿⢿⡟⡿⠿⣋⣛⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⡴⠖⠚⠛⢋⠙⢻⡟⣉⣑⡒⢂⣶⠀⣠⡀⠀⠂⠀⡸⠏⠃⠉⠟⢛⡒⠻⠿⠧⠃
⠀⠀⢀⣠⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣭⣾⠗⠛⠛⠛⠙⠟⠻⢿⡿⣿⡿⠃⢯⡖⠭⣀⣂⣉⣥⣤⣷⣬⣶⣿⠯⠋
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣯⣭⣿⣿⣿⣟⢾⣢⠼⠫⠛⠉⠁
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⠿⢿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠶⠾⠧⢤⣀⡀⠈⣙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢷⣾⣿⣿⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠢⠠⠄⠘⠂⠤⠄⠘⠿⠿⠿⠿⠷⢶⠖⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠉⠉⠉⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠈⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⢀⣀⣛⣛⡛⠻⠿⢯⣛⠿⢦⠀⢦⣼
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⢁⣀⠀⠀⠚⠋⣉⣩⣭⣟⠛⠶⠀⠈⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⣿⣡⠴⠀⠀⠸⠟⣦⣀⣀⣠⣴⣶⡄⠀⠀⠀⠀⣠⡶⠶⣤⠀⣿
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢠⡟⣡⠾⠋⠀⠀⠀⠀⢉⣩⢴⣿⡽⠂⠀⠀⠠⡾⠋⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢸⠃⣠⠆⠀⠀⣀⣾⣯⣶⣯⡵⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢻⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣴⠶⣠⣾⠃⠀⠀⠀⣉⡵⢞⣽⠏⠀⠀⠀⠀⠀⠀⠀⠿⠶⡶⠿⠦⠀⠙⢷⣄⠀⠀⣠⣄
⠀⠀⠀⣀⡴⠿⠀⠀⣠⢴⡟⠀⢸⠇⡿⠀⠀⠀⠀⠉⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⡟⠉⠻⢶⣤⡶⢶⣄⣀
⠀⠀⣾⠋⣰⠟⢡⣾⡟⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠀⠀⠀⠀⠙⠷⢾⣧⣽⡿⣦⠀⠀⣠⠶⡆
⠀⠸⣏⠀⠿⠀⡿⠘⠇⠀⠀⠰⣄⡀⠀⠀⠀⠲⣦⣀⣀⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢿⣶⣞⣡⠞
⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣦⣤⣴⢿⣯⣩⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣙⠶⢤⣤⣤⠶⠋⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢧⣤⣤⣤⡤⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡖⢀⡟
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠙⠻⠶⠖⠂⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠿⣤⡌⠻⠓⢶⣦⣤⣀⡀⠀⠀⠀⢠⡏⢀⡾
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣴⠏⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡅⠀⠀⠈⢻⡄⠀⠀⠓⠀⠈⣹⠟⠲⠦⣼⣁⡾⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⣶⣶⠶⠛⠁⠀⣀⣼⣄⣠⣄⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⣿⠆⠀⠀⢀⡾⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠉⠛⠉⠈⠙⢷⣶⣶⣤⣴⣟⠉⠙⠛⠓⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠄⠶⢦⣤⣄⣀⠀⠀⠙⠿⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠉⠙⠻⠶⣤⡀⠀⠸⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣴⠶⠛⠂⠀⠀⣀⠀⠀⠈⠙⢦⠀⣤⣿⣧⡀⠀⣀⡠⣄⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠴⠋⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⢶⣶⡉⣿⠟⢋⣡⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠗⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠀⠀⠈⢻⣽⡏⠀⡈⢻⡿⠳⠶⢦⣤⣤⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣿⠛⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠛⣅⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⣼⠂⠀⠀⠀⠀⢿⣿⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣗⠀⠈⠙⠷⣦⡀⠀⠀⢀⣴⣿⠋⠀⠚⠁⠀⠀⠀⠀⠀⠈⢿⡉⠉⠛⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⠛⣷⠀⠈⠙⠆⣴⡟⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠙⠳⢶⡶⣦⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⣰⠿⡅⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⠀⠀⠀⠀⣼⠃⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢿⣀⡞⠁⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠒⠒⠀⠲⣖⡶⠒⢲⠖⠒⠒⠒⠒⠒⠒⢒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⠤⡄⠀⠀⠎⠂⠉⢦⠈⠀⢠⠛⣄⠠⠒⠦⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⢸⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠦⡀⠀⠀⠀⢸
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡀⠐⠀⠀⠘⡄⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢰⠀⠀⣠⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠒⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠣⣀⠤⠆⠒⠊⠉⠀⠈⠉⠉⠉⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠢⣄⡀⠀⠀⢀⣀⡇⠀⠀⣇⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⡀⢸
⠀⣂⣀⣀⠀⠌⠀⡄⢨⣶⣀⡉⠦⢀⠀⣀⣀⡀⠀⠀⠉⠒⠉⠀⠀⡇⠀⠀⡏⠉⠁⠀⠈⠉⠁⠐⠒⠁⠀⠈⢣⣀⣀⠔⠒⠢⠖⠊⠁⠀⠀⠈⢾
⠀⠀⠀⠀⢈⠀⡘⢝⢸⣇⡜⢧⡤⠀⢩⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⡇⠀⠀⡏⠉⠉⠁⠉⠉⠀⠀⠐⠒⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠒⢸
⠀⠀⠀⠀⠈⡶⢣⡀⡱⡊⠈⢉⠇⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠙⠠⣈⣚⣒⣚⠩⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡀⢀⢆⠀⣾⠅⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⣾⣷⣶⠶⠶⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣀⡐⣌⣾⣈⠀⡯⠀⣜⣰⠕⢲⠆⣀⣤⣴⣾⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⡋⢰⡥⣟⡹⢒⡷⢻⣹⣏⣿⣝⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠱⣦⣄⡀⠀⢠⣄⣿⣿⣏⣠⠄⠀⢀⣠⣶⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢻⣿⣧⣴⡄⢻⣿⣿⣿⠏⣦⣰⣿⣿⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⣿⣿⣽⣿⣿⣏⣀⡀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠰⠾⠿⠟⠻⠛⠉⠈⠙⠟⠻⠿⠿⠦⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⢻⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡏⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⢸⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠒⠒⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⢀⠀⠀⠰⣯⣷⠀⠀⠀⠀⢀⠀⢀⢠⣴⣤⣀⣀⠀⣀⠀⠀⠀⠀⠀⠂⠀⠣⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠋⢰⣿⡿⣶⡜⠛⠀⠀⠀⢀⢄⠀⠘⠹⢿⣿⣭⣿⡆⠀⡥⠀⠀⠀⠀⠀⠀⠎⠴⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⠙⣆⣀⡹⢆⠀⠀⠀⠈⠈⠀⠰⣤⣶⣰⡾⠿⠅⠀⠀⠀⠀⠀⢀⣠⣠⣦⣤⠄⢰⠺⢍
⠀⠀⠀⠔⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠂⠒⢠⢴⣦⢷⣀⠌⠁⠀⠀⢀⣨⣽⣯⣃⡀⡰⠲⣤⠀⠀⢆⣹⡿⠛⠋⠛⠿⠘⠀⠘⠣
⠀⣀⡩⠀⡞⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢭⡷⠀⠀⠴⠀⠀⣿⣿⢿⡉⠀⡜⠀⢉⠇⢠⡀⠋⠀⠀⠘⠁⠁⠀⢀⠀⢠
⢘⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⣀⠤⡷⡄⠀⠀⠀⠀⠀⠜⠀⠈⠑⠀⠀⠊⠁⠀⠀⠈⠣⠶
⢀⠀⠀⠀⠀⠀⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⠉⠐⢉⠁⡿⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣆⣤
⠡⠀⠀⠀⠛⣶⡀⠀⠀⠀⠄⠐⢀⠀⠀⠀⠀⠀⣠⣴⣆⡊⠀⠀⠀⢨⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⡿⠛⠽⠒⠶⡠⡀⠤
⠀⠠⠀⠈⠦⢬⠟⠀⠀⠀⡆⠉⣉⣷⡄⠀⠀⠀⠯⣉⡃⠘⢄⠃⠀⠘⣶⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⢻⡅⠀⠀⠀⠀⠀⠹⠤⠰⠀⠀⠀⡍⠐⠊⠓⠢⠴⠟⠁⢠⠐⠶⠤⠴⠂⠂⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⣴⣾⣦⣄⡠⢀⠠⢄⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⢻⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡋⠁⡟⠁⣀⠳⠙⠷⣄⠸⡄⠀⠀⠀⢰⠿⠾⣿⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⡦⢐⠐⣄⠀⠉⢿⣿⣦⣐⢶⢼⣦⣷⠟⠂⠡⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠓⢦⣀⠻⣿⡿⢻⣿⣿⣧⣆⠀⠀⠄⠀⢀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠐⣲⡀⠀⠀⠙⠓⣶⣾⣷⣿⡟⠻⠀⠉⠐⣿⣧⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠤⠈⡄⠀⠠⠀⢠⠀⡀⠀⠉⠈⠀⠀⠀⡄⠖⠋⠡⠞⠋⠉⠉⠋⠛⠿⣦⣝⠂⠻⢧⡙⠅⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡅⢁⣠⣠⣴⠷⠖⠠⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢾⣷⡀⠀⢀⡿⠾⣷⣮⡛⠶⣶⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡏⠁⠀⠀⠨⡷⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⡟⠀⠙⢿⡷⠶⣤⣭⠿⣿⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣤⡴⠶⠟⠛⣋⣉⠁⠄⠀⠀⠀⠀⠀⡎⠉⠀⠀⠻⢿⡀⠈⠳⠏⢬⣿⠀⢀⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣍⣠⠤⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠈⠁⠉
⠀⠐⠐⠂⠶⠴⠦⡴⣄⣠⣼⣯⠿⠋⠂⠉
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⢷⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠁⠀⠀⠈⠻⣦⣉⡉⠛⠛⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠻⡄⠀⠀⠀⠀⢈⠻⢯⣄⣀⣠⣤⣀⠀⣀⡀⠀⠀⠀⢀⣀⣀⣤⡶⠋⠁⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠈⣷⡴⠛⠉⠀⢀⡼⠛⢉⠉⣳⣄⣀⣤⠾⠛⠁⠀⠀⢀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠐⠲⠶⡶⠋⠀⠀⠀⡰⠋⠀⠀⢩⡟⢉⣉⣉⠀⠀⠀⠀⠀⠐⠛⠉⠁⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣤⣤⣀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⡙⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡄⠀⠀⠀⢠⡿⠹⣟⠛⠿⣷⠶⠶⠶⠾⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠈⣧⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⠀⠀⠀⠀⢸⣇⠀⠀⠘⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⠀⠀⠀⠀⢻⡄⠀⠀⢹⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡆⠀⠀⠀⠸⣧⠀⠀⢸⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣧⠀⠀⠀⠀⢻⡄⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣿⠀⠀⠀⠀⠘⣧⠀⠀⠇⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢠⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⣼⠿⢶⣤⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠶⣤⡀⣰⡏⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⠿⠿⢟⣛⣛⣭⣭⣭⣭⣝⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢗⡲⣶⣿⣿⣿⣿⠫⣵⡤⣶⠝⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⣵⣿⣿⡿⠿⢿⣿⣮⡃⠫⠾⠇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣛⣫⢍⢠⣚⠿⢸⣿⣿⣿⣮⡻⢏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣝⡋⠾⠇⢟⣣⣿⣿⣿⠿⠿⣿⣿⣿⣀⠀⣤⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⢟⣼⣿⣿⣿⡤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⢰⣭⡝⠻⠿⣿⣿⣿⣿⠛⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣮⣣⣿⣿⣌⢵⡞⣴⡹⣶⢎⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣤⣭⣥⣶⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣾⣶⣶⣶⣶⣶⣖⣲⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣶⣒⣶⣶⣶⣶⣶⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⣟⡿⣿⣿⠿⠿⠛⠛⠻⠟⠛⠛⠛⠛⠻⠿⠻⢿⢟⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠊⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠏⠁⠈⡄⠉⢳⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⢐⡀⠀⠀⣤⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡏⠀⠀⢀⣠⣀⡈⢷⡄⠀⠐⠀⠀⣰⠃⣀⣀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡃⢉⣿⠀⠙⠷⠶⠶⠚⣡⣏⡹⠈⠛⣦⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⢸⣙⣓⣿⠀⠀⠀⠀⠀⠐⣯⠢⡅⣢⠜⣸⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⢺⡇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢷⡤⡤⠴⠋⠀⠠⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣇⡀⢸⣒⣔⣶⣘⣆⡠⡤⠀⠀⠀⠀⣰⣶⣶⣼⡆⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⣀⣀⣒⣻⣿⣿⣻⣶⣒⣦⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠉⠉⠉⠹⠿⣛⢛⡛⠛⢛⣛⡛⣛⣛⣛⣛⣛⠛⠛⢻⠋⠋⠋⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡄⡇⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣉⣉⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⡉⠉⠉⢁⣀⣀⣀⣈⣉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠇
⠀⠀⠀⠀⠐⣶⡀⠀⣹⣦⠀⣴⠁⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡅
⠀⠀⠀⠀⠀⢻⣿⡷⣾⠗⣿⡿⡄⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠸⣿⣷⢸⢹⣿⠃⣇⣀⣀⣀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠀⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣮⣼⣿⠀⠠⠀⠀⡆⢰⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡀⡆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢻⣠⣤⣦⣤⣥⡼⠿⠻⠛⠻⢳⣶⣶⣶⣶⣶⣶⠶⠿⠾⠿⠿⢦⣷⣶⣦⣬⡄
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢻⣯⣤⣤⣤⣤⣤⠜⠛⠛⠛⠛⠻⠛⢤⣤⣤⣤⣤⣤⡿⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢟⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠀⢀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢄⠀⢀⠌⠄⠀⢀⡴⠋⠀⠀⠒⠀⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠢⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⡞⠀⣴⠒⠀⠀⠠⠀⠈⠂⠀⠀⡌⠀⠀⠔⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢷⣾⠅⢰⠇⠀⠀⢀⠀⠁⠀⠀⠀⢰⠀⢀⠤⠒⠂⠠⠤⡀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⡀⠀⠀⡐⢰⡆⠀⣿⠀⣾⠀⢀⡴⠃⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠙⠀⢹⣇⣿⣴⠏⡄⠀⠀⠀⠠⠑⡠⠀⠀⠀⠀⠀⠀⠠⠄⠐
⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠀⠑⣀⠀⠀⠀⠀⣼⣿⣿⢋⣰⡶⠋⠁⠀⡠⠊⠀⡤⠔⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣿⣿⡿⠛⠃⠀⠀⡠⠊⠀⠜⠁⠀⠀⠀⠀⠀⡔⠀⢀⠟⡬⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀⣀⣽⣦⠀⠀⣿⡿⠁⠘⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢰⣋⠒⢁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠻⣿⣷⣀⡘⠃⠸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⠤⠴⠒⠲⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⢈⣿⠏⠀⠀⠜⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡁⠉⠲⢅⡄⠇⠀⠀⠀⠈⠃⣀
⠀⠀⠀⠀⠀⠁⠀⠂⣀⢁⡀⠀⠀⠀⠀⠀⠀⢀⡪⠔⢲⠀⠀⠀⠀⠀⠀⢀⠔⠑⠉⠉⠉⠙⠇⣀⠀⡀⠀⠙⢦⡠⠀⠉
⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠉⠙⠓⠢⡀⠐⠋⠀⠀⣸⠁⡆⠀⠀⠀⡠⠁⡀⣀⠀⠀⠀⠀⠈⢹⡄⢀⠖⠄⠀⠉⠒⠒⠂
⠀⠀⠀⠀⠈⠙⠛⣷⠀⠀⠀⠀⢤⡀⢀⣧⠀⠀⠀⠀⡏⠀⠀⠂⠐⣁⡀⠼⣠⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⣶⡶⠾⠛⠛⢻⡄⠀⠀⠀⣧⠀⢠⠔⠁⠀⢸⠀⠀⣷⢠⠠⠂⠀⠈⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⣹⠀⠃⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠦⢋⣀⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠴⠀⠀⠀⠀⠀⠀⠑⠠⠄⠨⠀⠀⠀⠀⢀⡀⠤⠒⠒⠉⠀⠉⠉⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠀⠁⡜⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣅⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀⣀⣀⣀⣀⣀⠀⣀⣀⡀⣀⣀⡀⣀⣀⢀⣀⢀⣀⣀⡀⣀⣀⡀
⠀⢠⣞⠈⠀⢲⠆⠛⠋⢡⠍⢉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄
⠀⠀⡄⠀⠀⠸⡄⠰⠄⠘⠂⠸⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡀
⠀⠐⠂⢤⠒⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡉⠀⡌
⠐⡁⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢄⡀⠄⠈⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠁
⠀⠚⢠⠀⠷⠂⠀⢎⠠⠢⠄⠐⠆⠀⢀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠙⠀⠀⠉
⠀⠂⠀⢀⣅⠫⠔⠇⢐⠀⡤⠀⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣀⢠⡤⠀⠈⡦
⢈⣀⡐⠨⠍⠁⢈⠇⣾⠀⠑⠀⠀⠢⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢠⣠⡀⠀⠄⠀⡇⠀⠤⠀⠁
⢀⠈⠀⣴⢞⠿⡹⠆⢘⣰⠆⠀⢄⢶⠀⠠⠀⠘⠀⠀⠀⠀⠀⢀⣀⣀⣲⢸⣺⡇⠃⢰⡇⠀⠄⠓⠂⠄⢈⠀⠙
⠀⣤⠄⡂⠙⣫⣤⠀⠌⠙⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠴⠀⠁⣺⡇⠀⠓⢀⡈⠂⠀⠀⢶
⠀⢃⢀⡡⠶⠋⣨⠀⠛⠮⠀⠀⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣇⠀⢹⣦⣇⠀⠧⠊⠉⠐
⢀⡅⠀⠠⠬⢤⣄⠈⠀⠺⠂⠀⠂⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠋⠀⢴⡞⠀⠀⠒⠊⣄
⠈⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠸⠋⠉⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⢀⣦⣤⣤⡀⠀⠀⣤⣾⣿⣿⠃⠀⠀⠀⠂
⠀⠀⠀⡀⠀⠀⠀⠈⠂⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠹⢿⣽⣆⣸⡿⣓⣾⠉⠀⠀⢤⣄⢀⣈⣀⠦
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠒⠒⠒⠒⢊⣠⡤⢀⠄⢀⡀⢿⣿⣿⡶⣻⣥⣤⣶⣶⡶⣶⣖
⠀⠀⠈⠀⠀⠀⠐⠒⠒⠂⠀⡁⠋⣁⣀⠀⢰⡖⣪⡴⠟⠓⠛⠉⢉⣙⠻⢿⣿⣯⣩⣡⣴⣰⣮⣷⣿⠋
⠀⠀⠀⠀⠀⠐⢄⠀⢀⠔⠉⠀⠀⢈⠻⢷⣿⣾⡿⠶⢶⡄⢖⡭⢜⡩⠂⣸⣿⣏⠻⠿⣿⣿⣧⡠
⠀⠀⠒⠀⠈⠉⠀⠀⠀⠀⠀⠀⣠⡤⢶⡾⠿⠿⣛⢚⣧⠘⢉⠔⠙⠁⠀⠿⠿⣿⣧⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⠀⠈⠉⠰⠟⠀⠀⠀⡘⡃⠀⢰⠏⠘⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣀⣀⡄⠀⣀⡀
⠤⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣀⠀⠀⠈⠏⠰⠀⠀⠘⢸⡏⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⢫⠍⠃⠠⠄⠀⠀⢀⠐⠒⠚⠓⠀⠀⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⡍⠁⠀⠐⣦⠸⠅⠸⡄⠳⠜⠇⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠧⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡐⠀⠀⠀⡠⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⡀⡀⠃
⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⢠⡄⡀⠀⠀⠀⠀⡠⠀⠀⠀⣈⠁⡄⠀⠀⠀⠘⠠⠈⠀⠀⠐⠿⢩⣝⡢
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⡀⠁⠀⠠⠄⠏⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠏⠮⣶⠀⢹⠅
⠀⠀⠀⠀⠀⠠⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⠢⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡅⡆⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡜⠠⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⠀⠀⠀⠀⡀⠀⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠁⠀⠰⠆⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣶⣈⣃⡀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⠀⠀⠀⠀⠀⢀⠛⠻⠟⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐
⠀⠀⠀⢡⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⡀
⠀⠀⠉⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⠀⠀⠔⠀⠀⠀⠀⠀⣠⢐⠐⠀⠄⠀⠀⡠⠤⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠁⠀⢀⠂
⠀⠀⠂⡙⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢐⠄⠒⣾⠪⠀⠀⠀⠀⠀⡀⠇⠀⠙⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡂⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠓⡄⠀⠀⢀⠤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠈⠀⢁⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠙⢀⠔⠁⠀⠘⢢⠀⠈⣉⣒⡶⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣄⠀⠀⢀⡼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠥⠴⠃⠀⠉⠉⡁⠀⠰⣌⠣⡀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠦⣤⠴⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢸⡆⠀⢧⡆⠀⠈⠉⠉⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⠀⠀⠀⠀⢀⡤⠀⠉⢷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⡰⠉⠀⠈⠉⠉⠙⠉⠉⠀⠀⣠⠞
⠀⠀⠀⢀⠔⠋⠙⠲⡴⠋⠉⠉⠙⢦⡤⠶⠶⠶⠤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢦⠀⠀⠀⠀⠀⢰⡇⠀⠀⡀⠀⠀⠀⢀⡟⠛⠉
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠉⠛⠋⠛⠦⢤⠴⠚
⠀⠙⠒⠒⠻⣄⡀⢀⣀⡼⠲⠶⠶⠚⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢧⣀⡀⢀⣤⣀⠀⢀⣠⠏⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⡖⠁⠀⢄⠀⠀⠀⢈⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠺⠓⠂⠐⠂⠢⡀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢣⠀⠀⢨⡜⠀⠀⠀⠀⠂⠀⠀⠀⢀⠇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⢸⠁⠀⠀⡄⠀⠸⡀⠀⢨⠅⠀⠀⠀⠀⠁⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠄⢸⠊⠀⠀⠁⠀⠀⡇⢠⠃⠀⠀⠀⠀⠀⡀⢀⡰⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠤⣀⢢⡀⠀⠀⠀⠀⠱⡀⠠⡇⠀⠀⠘⣾⠀⠀⠀⣀⠀⠀⣄⠃⠀⠐⠁⠀⠀⢀⡑⠃⠀⠀⢀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠢⡄⠀⠀⠈⠂⢠⠀⠀⡀⢸⡀⠀⠀⡇⠀⢠⠍⠀⠀⢀⠂⠀⢀⠎⠀⣀⠀⠄⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⢆⠀⠀⠀⠀⣆⠀⠐⠀⡇⠀⠀⣇⡴⠁⠀⠀⠀⠈⢀⡔⠚⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠺⡆⠀⠐⠀⣸⡀⠀⠀⡇⠀⢠⠊⠀⠀⠀⠠⣄⠔⠋⠀⠀⣀⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣀⠀⠈⢧⠀⠀⢳⡰⠃⠀⣀⣤⠊⠁⢀⣀⢀⠔⠊⣀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠛⠻⠶⠿⣧⠖⠋⠉⠁⠈⠀⠉⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢸⢆⢠⢣⢀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⠒⠦⢈⣃⣀⣁⣈⡋⠸⠔⡹⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣏⠉⡠⠊⠁⠀⠀⠀⠀⠈⠑⠢⡉⢡⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢠⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠉⠉⠛⢶⡄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣔⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠱⠓⠒⠉⠉⠉⠑⠒⠢⢄⡀⠀⠀⠀⢸⠠⣔⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⢀⡌⢠⡐⠓
⠀⠀⠀⠀⢠⠔⠒⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠⠊⠰⡒⠳
⡠⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⣓
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳
⣬⣵⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣯
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⢀⠞⡇⠀⠀⢀⠠⠊
⡀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠴⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⡴⡁⠀⢰⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡇⠈⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⢆⠑⡖⠉⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣸⠀⡇⠀⢀⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⠶⠉⠣⡈⢢⡑⡗⢊⡹
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢠⢀⠎⠀⠘⣳⠀⠀⠀⠀⣠⡴⠀⠀⢸⠀⠀⠀⢀⠔⢣⡀⠑⣄⠈⠢⡉⡇⣁⠔⠓⡀
⠀⡦⢀⡀⠠⢄⠀⠘⡐⡄⠀⡇⠀⠀⡐⠹⠀⢀⡶⢌⠂⢒⣄⠀⢸⠀⠀⣰⠉⠳⡀⠑⢄⠈⠑⠤⡈⡏⢀⡠⠒⠉⣆
⠀⡑⢄⢫⠙⠈⠏⠢⡐⡌⢂⢰⢀⠔⠀⢸⢀⢾⠀⣴⠒⠁⠘⠀⢸⠀⣰⠁⠳⡀⠈⠢⡀⠉⠢⢄⠈⠉⠁⡠⠔⠉⡠⢦
⠠⡜⠎⠲⢇⡀⠈⡄⠈⠣⡀⠘⡇⢀⠔⠉⢀⠜⠫⢀⠈⡱⠃⠀⢸⠀⡏⠣⣄⠈⠒⢄⠈⠑⠤⣀⠉⡟⢉⡠⠔⠊⢀⡤⢣
⠀⠘⠲⡖⠀⠬⠱⠶⢄⡀⠈⢄⡷⠃⢀⠔⢁⡤⠒⠉⠟⠀⠀⠀⢸⠀⡗⢄⠈⠑⢤⡀⠈⠒⠤⣀⠉⡏⢁⡠⠔⠊⠁⣠⠔⡆
⠀⠀⠀⠉⠛⠂⠂⡍⠈⠉⠢⡀⡇⣠⠦⠒⠫⠤⠤⠤⣄⣀⠀⠀⢸⠀⠱⡀⠑⠦⡀⠈⠑⠠⢀⡀⠉⠃⢀⣀⠤⠒⠉⠀⢀⠃
⠀⠀⠀⢐⡍⠛⠤⣀⣀⣀⣀⠐⣶⠥⠒⠂⠐⢲⠲⠍⣀⣉⠰⡤⢸⠀⠀⠑⢄⠀⠈⠑⠢⠄⣀⠀⢹⠉⠀⠀⣀⡠⠔⠊⡝
⠀⠀⡠⠛⡩⢷⠊⠉⢀⡤⠖⠙⡗⠔⠁⠉⢃⠒⠧⡀⠀⠉⠒⣱⠜⠀⠀⠀⠈⠁⠢⢀⡀⠀⠀⠉⡛⠈⠉⠉⠀⢀⠠⠊⠁⠀⠈⠉⠉
⠀⠺⠒⠉⢠⣬⠴⠟⠑⠁⠀⠀⢠⢂⠀⠀⠀⠡⡀⠈⠑⠜⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⡱⠒⡤⣧⠐⠐⠂⠁
⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⢸⣈⡆⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⢠⠃⡜⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡎⢠⠁⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⢰⠀⠀⠠⠂⠀⠀⠀⠀⡜⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡠⠂⠀⠀⠀⠀⠀⣎⣀⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⣀⠀⢀⣤⠤⠴⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⡘⠛⠫⣉⠀⠀⠈⠉⠙⠛⠷⢶⣤⣄⠀⠀⠀⠈⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⠁⠀⠉⠙⠻⢦⣄⡑⠀⠀⠀⠀⠀⠀⠀⠉⠙⣇⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀⢁⠤⡀⠀⠀⠀⠨⢿⠃⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠠⠴⠒⢠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⠛⠛⠛⠛⠛⠶⢦⣜⡆⠀⠀⠀⠐⠋⠁⠠⠤⠀⠞⠐⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠷⠶⢦⡀⠀⠀⠀⠀⢀⡠⠭⣿⠀⠀⣴⠿⠛⠛⠻⠿⠿⠷⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠽⡄⠀⠀⢰⣏⣤⡴⠶⠶⠶⠶⢧⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢷⣦⡀⠐⢤⠀⠀⢸⣿⣅⡀⠀⠀⠀⠀⠐⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡛⢷⣾⣷⠤⠬⠷⠭⠿⢄⡀⢀⡏⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢟⣫⠭⠖⠒⠊⠉⠉⢙⡿⣿⣿⣧⣤⡤⣤⣤⣀⣀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⡊⠁⠀⠀⠀⠀⠀⠀⢀⣜⣾⠟⠉⠀⠀⠀⠀⠀⠉⠛⢿⣵⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⠧⠠⡀⠀⠀⠀⠀⠀⠀⣠⠤⠺⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⢸⡆⠀⠀⠀⢠⡿⢧⣀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡀⢀⣀⣀⠎⠀⠀⠀⠀⡜⠀⢀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⣀⠀⠔⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈

⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⠀⠂
⣀⡀⠀⠈⠀⣀⡰⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⡀
⠀⠠⢀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⠤⠖⠀⢤⠀⠀⠀⠀⠀⡄⠤⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⢀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪
⠄⠄⠂⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢿⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⡐⠀⠀⠀⠀⠀⢄⢰⢠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢄⡋⠠
⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠰⠂⢀⣀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⣷⠈⠻⠀⠀⠀⠤⠀⣁⠤⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠴⠀⠀⠱⠀⠀⠹⠆⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠁⠀⠀⠊
⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠁⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠸⡄⢸⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⡀⠁⠀⢁⣀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠴⠋⠁⠀⠉⠲⡀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⡄⠀⠀⠀⠀⠀⡇⢀⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠢⡀⢀⠡⣄⡈⠋⣀⡴⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠁⠀⠀⠀⠀⠁⢀⡉⠉⢁⡠⠤⢄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣀⣀⠀⠀⠐⠋⠀⠀⠁⠐⠀⠀⠙⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⢠⠔⠀⠀⣀⣠⣤⣄⣀⠀⢀⡤⠶⠒⠒⠦⢤⡀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠋⢀⡴⠊⠁⠀⠀⠀⠀⡴⠋⢀⠀⠀⠀⠀⠀⠙⠀⠉⠁⠉⠙⠢⠤⠔⠀⠈⠉⠁
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠚⠀⢀⣀⡤⠤⠤⠤⢤⣀⡀⠀⠀⠀⢀⡠⠴⠒⠒⠲⢦⣄
⠀⠀⠀⣰⠛⠁⠀⠀⠀⣠⠞⢩⠂⡀⢐⠆⠀⠀⠀⡭⠗⣦⠰⠋⠀⠀⠀⢀⣀⠀⠈⠳⣄
⠀⠀⢰⠃⠀⠈⠀⠀⡰⠁⠀⠀⠀⢀⢨⠆⠀⠀⠀⠦⣀⠉⢳⠀⠀⠠⣆⢠⠚⣢⠀⠀⠘⣇
⠀⠀⢸⠀⠀⠀⡀⢠⠃⠀⠀⠀⠉⠁⠀⢀⠠⠀⠀⠀⠀⠀⠸⡆⠀⠀⠈⠢⠜⠁⠀⠀⠀⣿
⠀⠀⠘⣧⠀⠀⠀⢸⡀⠀⠠⠄⢠⠀⠀⢙⠀⠆⢐⠄⠀⠀⢰⡇⠀⠄⠀⠀⡠⠀⠀⠀⣰⠃
⠀⠀⠀⠈⠓⠦⢤⠀⢷⡀⢸⢀⡠⠃⠀⠈⠑⠁⠀⠀⠀⢀⡞⠀⠰⠦⢤⣤⣥⡥⠾⠛⠁
⠀⠀⠀⠀⠀⠀⢲⡀⢠⠙⢦⣉⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⣄⠸⡄⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⣷⠈⢰⠂⠸⡍⠛⠓⡚⠛⢁⠄⠀⠀⠀⠀⠸⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⡟⠀⣾⠀⠀⠙⡄⢠⠇⣰⠋⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⢠⠇⠀⡇⠀⠀⠀⢸⠀⣸⠁⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⣠⠶⠒⠒⠲⠦⡀⠀⠀⡏⠀⣿⢠⣀⣀⠀⢀⡤⠤⣸⠁⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢫⡇⠀⣿⠀⠀⠈⠙⠲⠄⠀⠀⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠘⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠀⠀⠒⠂⠠⠌⠋⠙⠫⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠋⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢶⣶⣤⣴⣄⣄
⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⠀⠀⠐⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣿⣿⣿⣿⣇⠀⠤⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⣿⣿⣿⣿⡀⠀⠀⠿⡿⠇⢠⣤⣶⠀⠀⠀⠀⠀⠀⢀⣀⡺⣿⣿⣿⠯⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⠉⠙⠙⠿⠃⢀⣀⢤⣤⣤⣾⣿⠿⠁⠀⠀⠀⣠⣰⠋⢙⣿⣽⣿⠟⣀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣾⠛⠉⣽⣿⣿⠁⠀⠀⠀⠀⣠⣌⣻⡟⠲⠆⠉⠈⠻⠟⢯⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⣹⣿⡇⠀⠀⠀⠀⣉⣙⠋⠳⠀⠀⢠⠀⠀⠀⡉⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠀⠀⠀⠀⠙⠛⡠⠦⠀⠀⠈⢳⡄⢾⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣤⣄⢁⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⢾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢋⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡆⠀⠀⠀⠀⠀⠈⠐⣶⡄⠀⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣿⣿⣷⠀⠀⢀⡀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠉⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡧⣿⣿⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⢻⣿⣧⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢹⣿⣿⡆⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⣿⣿⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠶⡰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⢺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡟⡑⢿⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢧⣼⠟⣻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣆⢹⠆⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡡⢚⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠐⡆⠀⠀⠀⠀⢀⣰⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢡⠎⠀⠀⠀⠀⠀⣾⡏⢸⠁⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠗⠁⠀⠀⠀⠀⠀⢸⣿⢀⠇⢠⣾⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠀⣠⣤⡄⠀⠀⠀⣾⣿⡞⠀⣾⢇⠎⠀⠀⢀⣀⠀⡠⠚⠉⢀⡤⠖⠂⣩⠟⡿
⢀⣤⣄⡠⡀⠀⠀⠁⢹⣿⠞⣴⣿⠁⣀⡀⠀⢻⣿⠀⢸⣿⠏⠀⠀⡆⠙⣿⡿⠀⠀⡴⠃⠀⣠⠞⠁⢠⡇
⢠⠿⠣⣅⠓⢰⣀⣀⠈⠉⢰⣿⠇⣼⡏⡼⢀⡞⠻⡤⢾⣿⠀⠀⡼⢧⣠⡿⠃⠀⠀⣳⢶⠊⠀⠀⢀⡜
⠈⢹⠓⠈⠳⢌⡁⢠⡇⠀⢸⣿⠀⣿⣏⣠⠊⠀⠀⠀⢸⣿⣠⠞⠁⠀⠀⠀⠀⠠⠊⠁⠀⠑⠒⠛⣉
⠀⠀⢡⡏⠀⣨⣹⠲⢍⡀⠀⠉⠉⠈⠉⠁⢠⣤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒⡏⠁⠀⠳⣤⡀
⠀⠀⠀⠉⠛⠁⠁⠀⠀⢉⣀⣀⣀⣀⣀⣀⣿⡏⠀⠀⣀⣀⡤⠤⠈⠁⢠⣤⠊⢹⣿⠁⣿⡇⠀⠈⠒⠒⠋⠺⠃
⠀⢠⣶⠛⢢⡀⠀⠀⠐⠉⠀⠀⠈⠉⠉⣻⣿⠛⠋⠉⠉⠁⠀⠀⠀⠀⣾⡏⠀⣼⡇⢰⣿⠃⠀⣀⡀
⢠⣿⢁⣴⣿⠁⠀⣰⣦⠀⠀⣴⡆⠀⠀⣿⡏⠀⢀⣾⠂⠀⣠⡀⠀⠀⡿⠃⠀⠈⠀⢸⣿⠀⠀⣿⡟⠉⢉⣷
⠘⠟⠋⣽⡟⠀⣰⣿⠇⢀⣼⣿⠇⠀⡰⢻⣷⣠⢾⣿⠀⢰⣿⡇⠀⡼⠁⠀⠀⠀⠀⢸⣿⠀⢸⣿⠇⠀⣰⣿
⠀⠀⠀⢿⣇⡼⠉⢿⠴⠋⣼⣿⠀⡼⠁⠀⠉⠀⠘⢿⡴⠚⣿⠀⡼⠃⠀⠀⠀⠀⠀⠐⢿⣀⠼⠟⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠴

⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣾⠍⠀⠀⠀⠀⣠⠀⠀⠀⣿⡧⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⢸⣧⠈⣿⣄⢠⡶⠀⠀⢻⣷⠀⢀⣿⣧⠀⣨⣿⠀⠀⠀⠀⠰⠄⡀⠀⠋⠀⠀⠀⣄⡀⠀⢤⣿⣯⡄
⠀⠀⠈⢿⣇⣨⡿⠟⠁⠀⠀⢺⣿⡇⠸⣿⣿⠀⣻⣿⠆⠀⠀⠀⢄⠪⡄⠀⠰⠀⠀⢐⣿⣷⡂⠾⣿⣷⠄⣤⡆
⠀⠀⠀⠈⣿⡛⠀⠀⠀⠀⠀⠀⣿⣷⢴⣿⣿⡶⣿⣿⠀⠀⠀⠀⠸⠀⠁⠸⠀⠀⠀⠰⠟⠟⠀⠛⠟⠃⠘⠏
⠀⠀⠀⠀⠚⠿⡄⠀⠀⠀⠀⠀⠉⠋⠀⠙⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡄⣀⠀⠀⠀⠀⠀⣀⣋⡀⠀⠀⢠
⠀⠀⠀⠀⠱⡀⣸⠀⢰⠀⠀⠀⠠⣿⠂⢨⡟⠀⠰⣷⠀⠀⠀⠀⢠⡿⣱⠏⠀⠀⠀⠀⠀⠴⠴⠠⢄⡑⣚⢀
⠀⠀⠀⢣⣠⣇⡿⣠⢟⡄⠀⠀⠘⡿⡄⢹⣷⠀⣾⡏⠀⠀⠀⠀⡜⣷⢹⡀⡄⠀⠀⠀⠠⣎⣩⡽⣯⠭⣽⡜
⠀⠀⠀⣧⢿⣿⡿⣱⠋⠀⠀⠀⠀⣹⣗⢰⣿⠀⣿⡧⠀⠀⠀⣴⣱⡟⣼⣳⡇⠀⠀⠀⠩⢽⠶⢿⣕⡒⢺⣋
⠀⠀⠀⠸⣜⣿⣇⡇⠀⠀⠀⠀⠀⠯⣭⣽⡇⢴⡷⠂⠀⠀⠀⢸⣿⡼⣇⡟⠀⠀⠀⠀⠈⢒⡻⣭⡷⢯⣽⠆
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠊⠙⠉⠉⠁⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣇⠀⡇⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠘⡆⠀⠀⠀⣿⠀⡇⠀⠀⠀⢀⡏⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢳⡀⠀⢀⠀⢻⠀⠀⡀⢻⠀⡇⠀⠘⠀⣼⠀⢀⠀⠀⡼⠁⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢳⡀⠈⢆⠘⣇⠀⣇⢸⠀⡇⠀⡇⢠⡇⠀⡌⢀⡾⠁⢀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠣⡀⠻⡄⠘⡆⢹⡀⢸⢸⡀⡇⢰⠇⡾⠀⡼⢀⡞⠁⣠⠋⢀⡼⠁⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠢⣀⠙⢦⡀⠙⣆⠹⡄⠹⡌⣧⠘⡼⡅⡇⢸⢠⠇⣸⢁⡞⢀⡼⢁⡴⠋⢀⡤⠂⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠈⠑⢄⡙⢦⡈⢣⡙⣄⠱⡸⡀⢇⠁⠀⡇⡜⢰⢃⠎⣠⠏⡴⠋⣀⠔⢃⣠⠶⠋
⠀⠀⠀⠀⠤⣄⣀⠀⠀⠀⠒⢤⣈⠙⠦⣄⠙⢦⡙⢆⠑⡌⠆⠁⠁⠈⠀⠀⠀⠁⠂⠊⡐⡡⠊⣠⠞⣡⠶⠋⢁⡠⠔⠈⠀⣀⣠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢤⣄⡈⠙⠲⢬⣑⠦⣈⠢⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠔⢋⡠⠴⢚⣁⡤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⣉⡛⠲⠤⢌⡑⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠥⠖⠚⣉⣀⣤⠤⠖⠒⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠐⠒⠒⠲⠶⠤⠤⢤⣤⣄⣀⣉⣉⠓⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡈⣭⠥⠤⠶⠶⠒⠒⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠭⠭⠭⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠶⠶⠖⠒⠂
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠤⠤⠴⠖⢒⣒⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢌⣉⣉⠙⠛⠓⠒⠒⠶⠦⠤⠤⠤⣀⣀
⠀⠀⠈⠉⠉⠉⠁⠀⢀⡠⠤⠔⠒⠚⢉⣉⡤⢔⣂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣉⠒⠦⢬⣉⡛⠒⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⢛⣩⠴⠒⣉⠴⣊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢤⡑⠮⣝⠲⢤⣀⠉⠛⠲⠤⣄⣀
⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⣀⠤⠚⠉⣀⡴⢚⡡⠞⣡⢎⡔⠠⠀⠀⠀⠀⠀⠀⠀⢀⠠⡑⣌⠲⣝⠦⣈⠙⠦⣌⠙⠒⠄⠀⠀⠈⠙⠒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢁⡴⠋⣠⠞⣡⠎⣰⢃⠇⡜⢰⠀⢀⢡⠈⡌⣆⠹⡌⢦⡈⠣⡈⠳⠄⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⢀⠔⠁⢀⡴⢃⡴⠁⣰⢃⡞⢠⠇⡎⢰⢈⡼⡄⢳⠘⡆⠹⣄⠳⡄⠘⢦⡀⠑⠄⠀⠈⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⢀⡴⠋⢠⠞⠀⡼⠁⡞⠀⡾⢠⠇⢸⠈⡇⡇⠘⣇⠹⡄⠙⣆⠘⢆⠀⠙⢦⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠁⠀⡼⠁⠘⠁⢰⠇⢸⠀⢸⠀⡇⢹⠀⢹⡀⠱⠀⠘⣆⠀⠁⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⢀⡞⠁⠀⠁⠀⡞⠀⡌⠀⢸⠀⣧⠘⠀⠈⣇⠀⠁⠀⠈⢧⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⠀⣿⠀⠀⠀⢸⡄⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⢹⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠘

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢸⣇⠀⣿⠀⢸⡇⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣄⠀⢻⡦⠘⣋⣤⠥⠴⠶⠦⠬⣥⣘⠃⢰⡟⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠷⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠾⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡈⠻⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡰⠟⠃⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢃⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡰⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣃⣸⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠲⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡴⠚⠉⠉⠉⠙⢦⡙⠦⠤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠈⠙⠁⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠛⠮⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣤⣤⡀
⠀⠀⠀⠀⢀⣤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⠀⠀⠀⠀⠀⠀⠀⣼⢡⣄⡉
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡆⠀⠀⠀⠀⢀⡼⢡⣄⠉⠛⠷⠂
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠀⠀⠀⣀⡴⠋⣴⣄⠙⠃
⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡕⠧⠴⠖⣋⡁⢲⣆⠈⠻⣷
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢻⡆⠸⣧⠀⠉
⠙⢦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠏⠀⠈⠀⠀⠹⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⢳⡇⠀⡰⠀⠀⠐⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⠀⡁⠀⠀⠀⢸⠹⢀⠁⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠁⠀⠀⡁⠀⠀⠀⢸⠀⢚⣰⣇⠀⡀⠀⣠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣄⠀⡀⠀⢀⠀⢸⡆⣬⢋⠊⣢⠄⠀⠉⡄⠀⢰⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⢈⠦⣼⠀⠇⠀⢸⣬⣻⡟⠁⠀⠊⢰⡇⠀⠈⢿⠀⢸⠀⠀⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠿⠀⠀⢀⡟⣿⣿⡇⠀⠀⠀⠞⡇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⠀⢻⠀⠀⣼⠿⠉⢻⣇⠀⠀⠀⡈⡁⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠘⠀⠀⡄⠀⠀⣾⣤⣾⠀⢸⡟⠀⠀⢸⣿⠀⠀⠸⣗⡇⠀⠰⣄⠀⠀⠀⠀⢀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⢰⠀⢀⠀⠀⣴⢙⣿⡏⠀⢼⡷⡄⠀⢸⣿⠀⢀⡐⣹⠃⠀⢀⡏⠀⠀⠀⠀⠈⠀⢀⠃
⠀⠀⠀⠀⠀⠈⠀⠘⠀⢀⡃⣤⣹⡇⠀⣸⠃⠀⠀⢸⣿⠀⠈⣱⡿⠀⠀⣼⠁⠘⠂⠀⠀⠀⠀⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢐⠀⢸⡄⠁⢸⣷⠀⠹⣦⠀⠀⣿⠏⢀⣶⣽⠀⠀⣀⠇⠀⡀⠀⠀⡈⠀⢰⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠃⠈⢀⠘⠇⠀⢸⡇⠈⠀⠈⠀⢸⡿⢡⠾⣷⠃⠀⢨⠏⠀⢨⠇⠀⠠⠁⠀⣾⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⢠⡀⠀⠀⠀⣤⡾⢀⡿⠀⠠⠀⠀⠄⡈⠀⢹⡻⠁⣧⣶⡏⠀⡀⡘⠀⠀⣄⠧⣀⡇⠀⠀⠀⠀⠰
⠀⠀⠀⠀⣠⣿⠘⣶⡶⣋⣷⣶⣿⣿⣾⣱⣈⠀⢀⣒⣶⣿⣶⣬⣿⣀⣴⣁⡅⠀⢀⣜⢀⣽⠁⠀⠀⠀⠀⡆
⠀⠀⠀⢸⣿⣿⣦⣟⢻⡟⢛⢿⡗⠋⠿⢳⢿⣾⣛⣿⣾⡿⢿⣿⣿⣟⡟⡿⣇⠠⢤⡁⣉⡌
⠀⠀⡄⢸⣿⣟⣿⣹⡏⣹⢫⡟⠃⠁⠀⢚⡿⣿⣿⡟⣿⣿⣿⡇⡿⢿⡿⡟⠫⢷⢛⡙⣎⡇⠀⠀⣠⠀⢴
⠀⠀⠀⠈⡸⢿⢿⡟⢿⠉⠀⠵⠤⠀⠀⠉⢹⠿⣿⣻⡽⣫⣾⡇⣷⠻⠷⢧⠐⢾⣁⡜⣷⢻⠀⢀⠁⢶⡜
⠀⠄⠀⠀⠀⠉⠈⠁⠀⢈⠃⠀⠀⠀⠀⠀⠀⠈⠉⠡⠁⠹⣿⡇⡘⢀⠀⠀⠐⠋⣽⣿⠧⣾⣤⠎⣀⣼⠃
⠀⠀⠀⠀⠀⡀⢺⡈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⠀⠃⡃⠁⠈⠀⠀⢀⢈⡷⣾⣧⠟⠀⠛⡿⠃⠀⠀⡀
⠀⠀⠁⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⣡⠋⢡⣀⣀⡀⠀⠀⠐⠳⠁
⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⡏⢀⡛⢊⡃⡄⠀⠖⠤⠂
⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆⠒⢹⣧⢉⡧⢀⠄
⠀⡄⠀⠀⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠴⡂⢠⢏
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⠁⡎
⠀⠀⠆⠀⠸⠐⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢼⡅⢠
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠘
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡚⠄⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠂⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠂⠀⠀⡀⢉⠉⡡⠆⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢰⠃⣀⠈⠢⡄
⠀⠀⠀⠀⣄⠢⢀⢸⣏⢛⠂⠤⠀⠀⠌⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠉⠀⠀⠀⠠⠀⠀⠀⠄⠀⠤⠆⠀⠀⠀⠘⠀⠀⠀⢀⡄⠀⠀⠀⠂⠀⠈⡀⠀⠁
⠀⠀⠀⠀⣈⠁⠈⠁⠀⠀⠓⡄⠀⠀⡀⠀⢀⠀⠀⠘⠃⠀⠀⢀⠀⠄⠁⠁⠠⠀⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⠀⠀⠀⠑⠢⠀⠀⢀⠀⠀⢁⠀⢰
⠀⠀⠄⠀⠀⠘⠀⠠⢀⠓⠀⠀⠁⡄⡄⠀⢀⡀⠀⠐⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠁⠀⢠⡄⠃
⡲⠍⢇⢲⡀⠈⢢⠄⠊⠀⠈⠁⠉⠙⠃⠀⠀⣀⠀⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡘⠁⠀⠐⠉⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠁
⠁⠁⠘⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠒⠀⠀⠀⠉⠀⠀⠠⠖⢰⠀⠀⠙⢄⣀⠀⠀⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠰⠂⠁⠀⠀⠀⠀⠀⠦⠀⡰⠀⠀⠄⡈⠀⠀⠀⠈⢉⠀⠀⠁
⠈⠁⠀⠀⡄⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠆⠀⠜⠀⠀⠀⠀⠀⠀⠀⠠⠆⠃⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠁⠠⠀⠺⠁⣀⠀⠤
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣀⡀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⢀⠄⠀⢦⡀⠀⢠⡄⠀⠀⠀⠀⠀⡀⠂⠀⠀⠈⠁⠈⠉⠀⡀⠀⠔⠀⠀⢉
⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⡀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡄⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠠⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠄⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠄⠀⢠⠀⠀⠰⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠈
⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠸⠀⠀⠸⠀⠀⠐⠈⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣀⡀⠀⠑⠂⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠠⠄⠀⡄
⠠⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠀⠀⠀⠄⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠴⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠂⠐⠆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡀⠀⢀⠆
⠀⢠⠀⠀⣀⠀⠀⠈⠀⠀⠺⠀⠀⠀⠂⠀⠀⠀⡄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠩⠿
⠀⠀⠺⡄⠁⠑⢤⠀⠘⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠃
⠀⠀⠀⡀⠀⠀⠀⠁⠒⠠⢅⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⢀⠀⠠⠀⠄⠀⠠⠀⡠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣐⣤⣀⠀⡠⠄⠀⡀⠀⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠆⠀⢠⢲⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢁⠏⢹⠍⠀⠁⠀⠒⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠈⠑⠂
⠀⠀⠀⣀⣀⡘⢦⢾⡙⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠩⠀⠁⠀⡌⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠉⠈⠁⠀⠀⠀⠀⠓
⠀⠀⠘⠳⠜⣷⢸⣪⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠔⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠄
⡀⢀⣀⢀⣀⣉⣀⣉⣀⣀⢀⢀⣀⣀⢀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⡀⣀⣌⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠀⢀⠜⠁
⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⢀⣬⠗⠚⠋⠉⠉⠛⠒⠾⣄⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠉⠙⠒⠦⢄⣀⠀⠀⠀⠀⢀⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⠀⠀⠀⠀⠀⣀⡠⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠤⠤⡤⠤⠴⠶⠒⠛⠛⠉⠉⠉⠉⠉⠛⠓⠒⠦⠤⣤⡤⠤⠴⠶⠒⠛⠉⣉⣉⣭⣦⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠲⠤⢤⣤⣤⣀⣀⣀⣀⣤⣤⣤⠴⠶⠒⠚⠋⠉⠉⠒⠯⣒⣒⣒⣒⣻⣭⣭⣥⣤⣀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉
⣤⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠷⠶⣦⣤⣄
⠀⠉⠛⠳⠶⠦⠤⠴⠶⠶⠾⠿⠛⠛⠛⠉⠉⠙⠓⠲⠶⢶⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠖⠛⠛⠋⠉⠉⠙⠒⢬
⠀⠀⣀⣀⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⠦⢤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣿⣷⣶⣾⣿⣿⠀⠀⠉⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣛⣿⣽⣿⣿⣿⣿⣿⠿⠿⢿⣷⣾⣶⣿⣟⣿⣿⣿⣿⣿⣿⣳⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣇⣈⠛⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠘⠁⠸⣿⣿⣿⣿⣿⠋⠉⠉⢙⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣟⣤⣯⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣿⣿⣿⣿⣿⣿⣷⣤⣴⣿⣿⡿⠛⠿⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣷⣀⣴⣶⣿⣿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣻⣏⠉⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠈⢹⣿⣿⠿⣿⣿⣿⣿⡿⠁⣴⣿⣉⣿⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣽⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣁⣸⣿⣿⣁⣨⣿⣿⣿⣿⣠⣤⣤⣴⣿⣿⣤⣿⣿⣿⡟⠀⣀⣉⣿⣿⡿⣿⡄
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣧⣾⣿⣿⣿⣿⣿⣉⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⣹⣿⡉⣿⣿⡾⠿⢿⣿⡟⠀⢻⣿⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣶⣿⣿⣿⡿⢿⣿⡏⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⡿⠛⢿⣷⡀
⠀⢀⣾⣿⣿⣿⣿⣿⡿⣿⠏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣼⣿⣿⣾⣿⡟⠙⠛⠿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣶⣿⣿⣷
⠀⣼⣿⣿⣿⣿⣏⢙⣥⣿⣤⣤⣾⠟⠻⣿⣶⣿⡿⠿⣿⣿⣿⣿⣿⡟⠙⢿⣿⣿⣋⣹⣿⣿⣿⣿⣾⣿⣿⣿⣴⣿⣿⣿⣶⣶⣾⣿⣿⣿⡟⠛⣿⣇⣠⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣦⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣾⣟⠛⣿⣿⣿⣿⣟⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⡟⢛⣿⣟⣹⣿⣿⣿
⠀⢻⣿⠛⠉⢿⣿⢿⣿⣛⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣯⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⡏
⠀⠈⠻⡆⠀⠀⠁⠀⠛⣿⢿⡇⠀⠀⡀⠉⠉⠉⠉⠉⠛⠛⠛⢻⣿⠟⠻⠿⠛⠿⢿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⠁
⠀⠀⠀⠈⠁⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣷⣄⠀⠠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠻⣷⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣶⣷⣦⣀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠾⠛⠉⠉⠁⢹⣿⣿⣿⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⣿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠻⠃⠀⣠⣴⣾⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣹⣿⣿⣷⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⢠⣿⣿⣿⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠙⢿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣽⣿⣿⣿⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢤⣶⣾⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁