Vagus nerve Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠤⣄⣀⡤⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⠀⡰⢋⢜⡃⡙⣤⠀⠤⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⢊⣠⡹⣈⠎⢵⣡⡏⣤⡰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣿⣦⢜⣷⠤⠆⣼⢣⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡴⢷⢰⡆⡾⢦⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⡾⠁⠀⠀⠰⠂⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⡀⠀⢰⡆⠀⠀⠀⣸⣿⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡛⠼⣿⣟⢻⣿⣷⣨⣄⣤⣾⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢶⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⠛⣿⣟⡻⣷⣿⣿⣾⢿⣻⣿⠛⠛⢿⣾⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢊⣤⣺⣿⠻⣿⣿⣿⣿⣿⣿⠻⣿⣦⣄⠑⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢛⡌⣻⣷⡼⢿⣿⡟⢹⣟⡿⢧⣾⣿⣫⡗⣤⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡟⣼⣿⠗⢻⣷⣰⣿⡿⢿⣿⣆⣾⡿⠻⣿⣥⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣯⠆⣀⣨⣿⣿⣹⣿⣿⡛⢿⣿⣅⡀⢰⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣯⣿⡟⢩⣷⣼⣍⢻⣿⢿⢿⡿⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⡉⠁⠀⢀⣿⣿⣿⣼⣏⡿⣿⣿⡀⠢⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢹⣿⢸⣿⣿⠇⣿⡟⣷⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣮⣿⣿⣯⣿⣾⣿⣠⠿⡇⡀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣼⣿⣍⢻⠿⣿⣿⢿⢟⣫⣿⣷⣴⣶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠈⠉⠙⢿⣶⣿⣿⣷⠿⠋⠉⠉⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⢀⣴⠟⠉⠉⠻⣦⡀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⢻⣿⣷⣶⣯⣽⣶⣶⣿⣿⣯⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⠘⢾⣯⣿⣿⣿⣿⣿⣿⢽⣿⣗⢸⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣷⢨⢿⡓⣾⣿⣿⣿⣿⣟⢺⣿⡟⣺⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣏⡷⣿⣿⡟⠛⠛⢿⣿⣿⡿⣵⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢬⣿⣟⣦⣴⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⡟⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡀⡀⡀⠠⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣪⣥⣶⠷⠶⠿⡟⠳⠟⠻⠿⢵⣆⡪⢑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠫⢠⠀⠀⠀⢀⠰⡀⡅⠀⠀⠀⢈⠟⠳⣟⢟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢟⢰⡁⢃⠀⢇⡀⠓⣄⠁⣄⣡⣄⠴⢜⣁⠈⢢⠘⣯⣳⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⡿⡅⠨⡄⡇⠜⠄⠰⠓⢐⣤⡴⠴⢻⠤⠭⡑⢤⠉⠀⠀⢹⣎⣷⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⣽⠃⠈⠂⢄⠑⠄⠀⡰⡻⠓⠐⠲⠯⡉⣹⠀⣈⠆⠀⠀⠀⣺⠋⠹⠎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⠄⢈⣤⣴⣾⣷⡅⡴⠂⡴⠂⣘⠄⣈⡀⠀⢀⣠⡴⡋⠀⠈⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣷⠀⠀⢠⡟⠀⡀⠀⠻⣾⠃⠚⠙⣆⢹⣀⣏⠉⠉⠉⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠋⢧⣠⣿⠄⠀⡀⠁⢴⡇⢀⣄⣴⠣⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⠢⡙⠿⣦⡄⢃⢈⢸⡇⢀⡟⠞⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠉⡑⠛⠻⢿⣿⡇⢸⣇⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠈⣶⣠⠺⣿⣷⢸⡟⠙⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢱⢨⣾⡽⣿⢸⣿⣿⢧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢸⣷⣿⡿⣿⢸⣿⣶⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣷⣺⣿⡟⣾⣿⣿⡾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣬⣿⣿⣾⠃⡿⣿⣛⣿⠃⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⢗⠳⠏⠰⠿⡘⠙⠈⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣆⣶⢰⣶⡆⣴⣶⣦⣶⣶⣶⣴⡷⡦⠀⣶⣦⣶⢴⡶⢶⣶⣶⣴⣦⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⣿⣷⣿⣷⣷⣿⣿⣿⣙⢿⣦⠀⣿⣿⣿⣿⡟⢺⣿⣿⢸⣿⣿⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠛⠛⠛⠙⠛⠛⠙⠛⠋⠙⠛⠋⠀⠛⠙⠋⠛⠛⠚⠛⠛⠈⠛⠛⠘⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣠⣶⢂⢠⡀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣿⠰⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⢶⣾⣦⣤⣿⣿⣿⣟⣿⣿⢹⣿⣴⢳⡙⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣿⠛⠀⢠⣿⣿⣏⠋⠙⢹⣿⣿⢸⣿⣿⡈⢷⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣷⣟⣿⡹⣿⣄⠀⣾⢿⣿⢹⣿⣿⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢋⡈⣠⡦⠺⢻⡇⣰⣼⣿⣟⣿⡖⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣶⣿⣿⣿⣿⣄⢺⡅⢿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠀⠐⠂⠀⢹⣿⣿⣿⣧⠀⣹⡇⣼⣿⣿⣿⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠾⣿⡇⣹⣿⣿⣿⣿⣝⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡋⠋⠁⠀⣿⣿⢿⣿⣿⢹⣯⣿⢎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⢠⣾⣿⢿⣼⣿⣿⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠙⢿⣿⣿⣏⣼⣿⣿⣿⣾⣿⣿⣿⢧⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⡏⣿⣿⣿⣿⣯⣻⣏⡻⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠈⠈⠀⠀⠀⣿⣿⢻⣿⣿⣿⣿⣿⣉⡿⠇⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠤⠤⠤⠄⠀⢽⡟⣾⣏⣿⣏⠉⠉⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⣿⣿⣿⣿⡁⠀⠀⠀⠀⠈⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠁⢙⣿⣍⣓⣷⣶⠄⠁⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣶⣿⡷⣿⣯⡿⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣻⡿⠛⣠⣠⣌⢁⣘⣀⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠷⠟⠉⢰⣾⡿⠿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⠀⠀⠀⢼⣿⡀⢰⣿⠁⣌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠀⣶⣿⣿⣷⣾⣿⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⢲⣿⣿⡟⣿⣿⡿⣿⠂⠀⠈⠛⠿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣏⣴⣿⡿⠿⣽⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠟⠋⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⣶⣶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠀⠀⠀⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢡⣄⠀⠀⠀⠀⠀⠀⣀⠄⠚⠁⠀⠀⠀⠀⠀⢩⡷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠛⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠢⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣀⡀⠻⢷⠀⠀⠀⠀⠀⠀⢠⣤⠀⢀⡀⠙⠣⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠻⣿⣁⠀⠈⠀⠀⡀⠀⣴⣶⡾⠟⠛⠿⣿⣤⡀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⠀⠀⠐⠀⣼⣿⡀⠛⢀⣿⠋⡀⢀⡠⠠⠀⠀⢨⣿⠀⢠⡄⠀⢸⣯⣤⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠠⠀⠀⠀⢿⠃⠀⠀⠁⢀⣀⠀⠉⠺⠟⣿⡟⢙⡃⠀⠀⠰⠷⠀⠸⠿⣿⣄⠀⠀⣠⣾⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠐⠀⡀⠀⠻⠋⠀⠔⠁⠀⠁⠂⠀⠋⣿⡟⠿⠛⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢡⠀⢀⠠⢄⠀⠐⠁⠠⠠⠀⠀⠀⠰⠋⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠖⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⡀⠀⠀⢹⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠁⠀⠙⠋⠁⠀⠉⠉⠑⠒⠦⠠⢄⡀⠀⠀⠀⠀⠀⠀⠩⠈⠉⠏
⠀⠀⠀⡁⠀⠀⢈⡀⠄⠀⠀⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠢⢀⠀⠀⠀⠀⣀⡀⠃
⠀⠀⠀⠉⠄⠀⢀⡀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠙⠁
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠱⣄
⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠁⠀⠀⠈⢷⡀
⠀⠀⠀⢀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠱
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⢧⠀⠀⣰⠀⠰⠀⠀⠀⠀⠙⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰
⠀⠀⠀⠸⠀⠀⠘⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠑⠢⣅⠂⠤⣤⠀⠀⠀⠀⠸⠁⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⢄⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠈⠢⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡦⡮⠙⣿⠉⣯⡵⣦⡀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠴⢧⡀⢸⡇⡏⢠⢿⠝⢦⡀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⡬⡼⡧⠶⡷⢿⣒⠲⣝⢶⡀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠀⠀⢀⣀⣀⠁⠀⠀⠀⠀⢠⡞⣇⣠⠌⢠⠇⢶⠱⠐⣌⣛⡿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢷⣀⠘⠤⠾⠤⠟⠒⠓⢋⣁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠛⠶⠶⠶⠶⠞⠛⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⡱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣤⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣡⣿⡟⢿⣿⣿⣯⣿⣅⣈⠹⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣾⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣷⣭⣉⣿⣿⣿⣽⣿⣿⣿⡿⢟⣿⡍⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠿⠿⢿⣿⣿⣿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢏⠈⠛⢿⣦
⠀⠀⠀⠀⠀⢸⣿⣷⣼⣿⣿⣿⡿⠛⣣⣶⣶⣶⣒⣶⣤⣤⠤⠄⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⡟⢁⣸⣿⣿⣻⣿⣟⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⡏⠀⣸⣿⣿⣿⡿⠋⠉⡀⣠⣶⣾⣷⣶⣦⣤⣄⡀⠀⠀⠀⠢⢈⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠸⣿⣿⡿⣿⣿⣿⣦⣬⣩⣍⣉⡀⠀⠀⠀⠈⠉⠉⠉⢉⣛⣛⣻⣿⣷⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢩⣴⣄⣿⡿⣿⣿⣿⣿⣿⣿⡌⠀⠠⢀⣠⣦⣭⣤⣦⣾⣧⡬⠛⢦⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣽⣿⣷
⠀⠀⠀⠀⠰⢀⡀⠀⠭⢟⣙⣛⣛⣛⣛⣛⣿⣿⣿⣶⡶⠛⠛⠙⠋⣽⣿⣿⣭⠬⠅⠈⢳⡌⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢀⠠⠀⠀⢀⣀⣄⡀⠀⡀⠀⠹⠁⣠⠴⢋⡀⠄⠀⠘⡀⠛⣻⣿⣿⣧⣥⣀⠀⢿⡿⣿⣹⡿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠈⢠⣾⣿⣿⣿⣿⣆⣂⣤⡴⠞⠙⠀⢀⡈⢉⣨⣿⡾⠋⢿⣿⣿⣿⡿⠉⠁⣾⣿⣛⣙⣙⣫⣟⣙⣻⣿⣋⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⡟⠉⠁⠀⠀⠤⣤⣤⡴⢚⡿⠋⠀⠀⠂⠻⠿⣿⣿⣧⠅⠹⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠠⣄⠀⠀⠙⠻⠿⠿⢛⡁⠤⠶⢲⠖⢻⡏⠅⢰⣿⡋⠴⠁⠠⠄⠀⢸⣯⣿⣟⣲⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠜⠉⣱⠋⠀⠁⠀⠀⠀⠄⠰⢡⠃⠈⠁⢠⢒⣿⣿⠀⠀⠀⠠⠀⠚⠉⣿⣿⣿⣷⡞⠁⢿⣿⣿⣿⣿⣿⠟⠉⠉⠁
⠀⠀⠀⠀⢁⠿⣄⠀⠀⠀⠀⠐⢁⠠⠞⠀⠀⠀⠀⣻⡿⠀⠀⠄⠀⠠⠀⠄⠰⢊⣿⣿⣿⣷⢇⢯⢿⠇⠉⠡⡏⠁⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠉⠀⠉⠉⠓⠀⠀⠉⠀⠀⠀⠔⠀⠀⡴⢻⠃⠀⠀⠀⠀⢀⠌⠠⡏⠀⠀⢉⣿⣿⣼⣧⣸⡆⠀⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄⣠⠴⠋⣰⠃⠀⠛⢁⠄⠆⢀⠀⠀⡇⠀⡴⠛⠻⣿⣿⣿⣽⡇⠀⠂
⠀⠀⠀⠀⠨⠶⠄⠀⠀⠲⠬⠓⠒⠒⠋⠠⡶⠊⠁⢀⠊⡀⠁⠐⠁⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠠⠄⠐⢂⢀⠐⠈⠀⠈⠀⠂⠀⠀⠃⠀⠈⠀⣷⠀⠀⠀⠀⢹⣿⣻⣿⣿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣌⠙⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⠉⠀⠉⠳⠀⠆

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠂⠐⠸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣾⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢠⣄⡘⠉⠁⣰⡄⢸⡟⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠠⡄⠋⢡⣾⣿⠁⠀⠹⠿⣇⠀⠀⠘⢻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⡾⠀⠀⠀⢠⣼⠻⣿⡏⢀⠀⣀⡀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠻⠞⠛⠓⠻⠄⠠⣠⡽⣧⣧⡈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⣐⣇⣿⠉⠬⠻⠀⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢤⣷⠀⣿⡜⠀⢧⡴⣧⡁⠈⢿⣷⡽⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣾⣿⠉⣿⡇⢀⠌⠑⣿⣷⣆⠘⠟⢻⠀⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⠀⣴⣿⠂⠈⠻⠀⠀⠃⠀⢐⡀⠂⠀⠀⠀⠙⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣽⡇⠀⠀⠀⠀⢀⠀⠙⠀⠙⠳⠤⢀⣴⣤⡀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⠀⢨⡧⠀⠀⠀⠀⠀⠠⡄⠈⢷⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⢤⡀⠹⣿⡀⠀⠀⠀⢠⠀⢃⠀⠀⠀⠀⠀⠀⣄⡈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⣀⣀⣨⠓⠲⢿⣇⡄⠀⠀⢀⠀⠀⠀⠂⡀⠐⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢂⡀⣸⡟⠙⠉⠷⡄⠀⠉⠀⠀⠁⢸⣿⣷⡄⠀⠉⡀⠀⠀⠀⠘⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠰⣯⣴⠛⠁⠀⠀⠛⠁⠰⠋⠀⠀⣌⠻⣿⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠚⠁⠀⢀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠹⢃⣹⠀⠀⠙⠀⠀⠀⠀⠀⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠖⠁⠀⠀⢀⡼⠁⠴⠆⠀⠀⠢⡶⢻⡇⠀⠸⣟⠀⠀⢘⠛⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠐⣫⣴⠆⣴⡷⡖⠀⠊⠃⠀⠀⢻⠂⠀⠸⠆⠀⠀⠀⠀⠈⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣰⣆⡸⠋⣁⠐⡋⠻⠋⠠⠀⠀⠀⠀⣈⠀⠀⣀⠀⠀⠀⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠙⣼⡃⢸⠣⠶⠂⠀⠠⡅⠀⠀⠀⠀⡇⠀⠀⡿⡤⠀⣀⠁⠄⠉⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢰⠃⠀⠀⢠⡿⢦⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠴⠤⢠⣦⣤⣀⡀⢘⠀⠀⡼⠁⠘⠂⠀⡄⠣⠉⢦⡐
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢀⣠⠴⠛⠉⠁⠀⡀⠐⢂⣀⣤⡆⠉⠉⠃⣦⣾⡁⠀⡄⠀⠈⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣠⣶⠖⣀⣒⣉⣁⣀⢀⣀⣀⣤⣴⣾⣿⣿⣿⠀⠀⠀⠀⠈⠛⠛⠢⣞⡀⣼⡁⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣮⣽⣿⡯⠓⠏⢨⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⣨⡑⢦
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣿⡆⠈⠀⣶⣅⡘⠛⠉⠁⣨⣍⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠀⢠⣾⢟⡛⠁⠀⠺⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⡁
⠀⠀⠀⠀⠀⠀⣠⣄⠶⠾⠛⣋⣿⡛⠃⠀⡉⠉⠲⢷⡤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣬⡇
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡿⠇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡐⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣤⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⡆⠀⠀⣠⠖⠊⠉⠒⠤⣄⠀⠀⠀⡝⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⡤⠚⣁⠀⠀⠀⠀⠀⢸⡗⢤⣸⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⣤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢸⡇⠾⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠰⠠⠄⡀⢀⠀⣸⣿⠀⠀⠀⠀⠀⢸⡇⣤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢿⣿⠘⠀⠁⠘⠈⢹⣭
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢀⣼⠙⠄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⣠⠞⢸⠀⡄⠀⠀⠀⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⠀⠜⠀⡄⢸⢈⡁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠀⠀⣿⣾⢸⠇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡟⠀⢀⠘⣿⠘⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⣿⠀⢸⠀⣿⠀⠀⠀⠀⠀⠈⠡⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣪⠀⡿⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⢿⠀⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢠⠠⠄⠆⠰⠰⡇⡇⢸⣶⠓⡿⣿⠀⠂⠰⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏⠀⣇⣿⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢈⡇⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⠁⣀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣰⠀⢸⣋⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⠈⡄⠸⣏⠄⠸⣦⠤⠤⠤⠤⠤⠴⠶⠶⠦⠤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⣀⡤⠴⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠐⡀⢉⢺⡄⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠲⠦⢤⣄⣀
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠻⡄⠹⡄⠀⠈⠙⠛⠶⠖⠖⠚⠛⢋⠉⠉⠉⠙⠛⠳⠶⣤⣄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠘⡄⡝⢢⡄⠀⠤⣀⣠⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢴⡀⠉⠢⢄⠀⠀⠈⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⣿⢄⠀⠀⠛⢄⠁⠲⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⢄⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠞⠛⠛⠲⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠀⠈⠢⣄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⢀⠠⠄⠒⠒⠒⠄⢀⠉⠛⠛⠶⠶⢶⣶⡶⠞⠋⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠉⠉⠉⡁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣠⡴⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠦⣤⣤⣤⣤⣤⣤⠤⠤⣤⠤⠶⠞⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡷⣄⣰⡇⠙⠁⢠⣍⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢳⡍⡄⢨⣤⠀⡘⠃⡶⠶⠀⠜⢠⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡷⢴⠟⣤⠹⡤⠿⢀⠀⣠⠀⠠⠾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠷⣦⢶⠀⠈⠡⢤⣼⠀⣈⣙⣀⣙⣄⡀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣿⣦⣿⣿⣷⣦⣹⡂⠛⠛⢿⣟⡉⣝⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠛⣻⣿⣿⣿⣿⣿⣿⣿⡗⠛⠿⣿⡞⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣿⡿⣿⣍⢻⣿⣿⣄⡉⠙⠇⠀⡸⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⢀⡟⣿⡧⣿⣄⠀⣭⡥⢴⣦⢀⠿⢿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣧⣿⣿⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠋⠙⢹⣿⡿⠿⣿⣿⣿⣿⣿⡷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢸⢿⣿⢳⣾⣦⣀⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⢰⠀⣽⡿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⡁⢸⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣼⣿⠇⣿⠆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⡟⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠉⢻⠀⠀⣿⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠒⠀⠠⣿⣦⠼⣿⣄⠠⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⢿⠃⠀⢿⡇⡀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠆⠺⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⣀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⡦⢾⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⢾⣄⠀⢂⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⡀⢀⢀⣰⡀⠀⠀⠺⡇⢙⠶⡀⠀⣡⡀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⣷⠘⠀⠑⠿⠃⢹⡄⠘⠄
⠀⠀⠀⠀⠀⣴⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠰⢷⡤⠤⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠋⠙⢀⣄⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠘⣆⠀⠀⡀⠀⠛⠃⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠂⢉⣦⣙⣁⠀⢰⠚⡁⠈⣇⠀⠀⠀⠘⡇⠀⢰⡀⢄⡰⢾⡁⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠈⠛⠶⠎⠙⢛⠛⢻⡿⣃⣸⡧⠤⠀⠀⠘⣿⣦⣼⣻⣌⡱⢦⠤⠾
⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⣸⣁⡦⠴⢻⠁⠛⠀⠄⠀⠀⠉⠇⠉⣿⠁⡴⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⢡⠉⣰⠀⠀⠀⠘⠂⠀⠀⠀⢸⡆⠀⢻⠜⣣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠐⣿⣤⢏⡀⠀⠀⢘⠁⠠⡀⠀⣸⠃⢴⠀⠀⢈⠃⠒⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⠂⢸⡟⠈⠀⠈⢀⢸⠁⢀⡞⠃⠀⠀⢘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠰⠀⠈⠙⢀⠈⠃⠀⢱⡄⢻⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣰⣿⠀⢀⡀⠀⠀⠀⠈⡆⠀⠀⠈⡅⠀⠀⠛⠀⡄⠰⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢸⣇⠀⠈⠁⠀⠀⠀⠘⠁⠀⠄⠀⠀⠀⠀⠀⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢨⠃⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⠀⢀⠇⡭⢱⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣤⠐⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⢛⢤⠂⠌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣑⢤⢐⣌⣅⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⡿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣻⣧⣿⣷⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⢼⠄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠄⣦⠀⣧⢺⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠁⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣶⡀⣄⡂⠀⣐⣰⡈⢵⡦⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⢁⠀⠘⢹⡏⠙⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⢶⠀⡇⡠⠈⢚⠘⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡜⠀⠀⢹⠇⠜⣧⡇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⠀⠀⠀⠀⠀⠀⠗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣗⠻⠏⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣹⡿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣍⣛⣩⣶⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⡇

⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠋⢉⣿⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣋⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⣀⠀⠀⣀⣀⠀⣀⠀⣀⢀⣀⡀⠀⢀⣀⢀⡀⢀⣀⡀⢀⣀⡀⢀⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣬⡏⣸⣿⣇⣾⡋⣤⡀⣿⠀⣿⠸⢿⣥⠀⢸⡿⣾⡇⣿⡷⠄⢸⣧⡿⠸⣧⣿⠸⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠛⠉⠛⠈⠛⠛⠁⠙⠛⠃⠘⠛⠋⠀⠘⠃⠙⠃⠙⠛⠃⠘⠁⠛⠀⠛⠃⠈⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣤⣶⣶⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢀⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⡬⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⢠⣶⣤⠀⠀⠁⢰⣷⠀⢿⡿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠛⢟⣠⣿⣿⡏⠀⠂⠹⠿⣿⡆⠘⠀⠙⢿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⢤⣀⠀⢀⣴⡟⠁⠀⠈⢿⣆⣤⣿⣿⣿⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠃⠘⢿⣧⣀⣀⣤⣤⣿⣿⣭⣿⠏⠀⠊⠀⢺⡇⢢⡈⠻⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⣿⣿⡿⠿⠛⠉⠉⠙⠟⠀⣿⠄⣿⣿⡚⢿⡄⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠙⠓⠀⠀⣠⡼⠋⠀⠀⣾⣿⣦⠿⣇⠹⣆⠻⠂⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠉⠠⣤⣿⠀⢸⣿⡿⠧⢿⣧⣼⣶⠑⠀⠹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣶⣿⣿⣇⣾⣿⡏⢦⠘⢿⢾⣿⣷⣀⠈⢿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⡿⠈⣿⣿⡇⠸⣇⠀⠈⠿⣿⠟⠎⣀⠀⠙⠃⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡷⠀⣻⣿⡏⠀⠀⢹⣦⠀⠀⠉⢀⠀⠹⡀⠈⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣼⣿⣧⠄⠀⠈⢱⡄⣄⡀⠈⠋⠀⠛⢻⡶⣄⢀⣽⣦⡄⢦⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡷⠆⠠⠀⢸⡆⣽⣇⠀⠰⠿⠂⣄⠀⣄⣈⡉⠙⢻⣾⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⡄⠀⠀⡘⠁⣿⠃⢷⠀⠀⠀⠈⠠⠈⠉⠉⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⢈⣽⣦⣤⢿⣿⡵⠀⠃⢷⠀⢿⠤⠰⠧⠀⠀⠀⠀⠂⠁⡀⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣀⣀⣠⣼⣿⣿⣿⣿⣇⡆⠀⠈⡀⠀⡀⠀⠀⠀⠂⣀⡀⠻⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡀⢀⣿⣟⡿⠏⣿⣿⣿⣿⡟⠁⠄⠛⠃⠰⣿⣿⣷⠀⠀⠙⣷⠘⢀⠀⢰⡉⠻⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⢡⣾⣯⣽⣷⣶⡉⠛⢻⣦⠀⢀⣶⠁⠀⢀⣻⢻⣿⣷⡄⠀⢹⡄⣀⠀⠈⠅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠀⢈⣿⠷⠛⠋⠉⣹⡖⠀⠙⠦⠚⠁⠀⠀⠀⢹⣿⢻⣿⠀⠀⢸⡷⠀⢀⡄⢀⡀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢁⣾⠟⠉⠀⠀⣰⣾⡟⠋⠁⠀⠀⠀⢀⡀⠂⠀⠈⠻⣶⣿⠀⠀⠀⣧⢀⡈⠀⠊⠁⠀⠈⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢠⡞⠁⠀⠀⠀⢠⡿⠋⣤⣾⠃⠀⢀⣼⣿⣿⣿⣇⠀⠀⣿⣿⠀⠀⠀⣿⠈⠁⠀⠀⢠⣶⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⡜⠃⠀⠀⠀⢀⣴⣟⡱⠞⢉⣴⠀⣤⣿⣿⣿⣿⣿⠟⠀⠀⠛⣿⠀⠀⠀⣿⠃⠀⠀⠀⣶⠈⡛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⠀⠀⣼⠋⣴⣾⣿⠋⣰⡿⣿⣿⡇⣽⣧⡄⠀⠀⠃⣻⠀⠀⠀⡁⠀⠀⠀⠀⢈⡜⠁⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠤⠀⠀⠀⢠⣿⣷⡿⠋⣤⢶⣿⣷⠽⠁⢸⣿⣋⡇⠀⠀⢸⣿⠀⠀⢴⡦⠀⠀⠀⣰⣞⣻⣧⣄
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠀⠀⠀⠀⠈⢻⡿⣇⢸⡏⣾⠿⠿⠃⢠⣾⣟⣛⣇⠀⠀⢸⡿⠀⠀⢸⠃⡀⠀⠄⢈⡉⠛⠯⠙⣷⡄
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠿⠋⠈⠀⠀⠐⠐⠂⠲⠿⣯⡉⠇⠀⠀⢸⡃⠀⠀⡿⠀⠁⠀⠀⣾⣿⣦⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⣀⣰⣇⡹⠀⠀⠀⠈⠀⠀⢠⡷⠀⠰⣦⠀⠙⣿⣟⠻⣶⡀⠄
⠀⠀⠀⠀⠀⠀⢸⡾⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠛⠛⠉⠍⠙⠧⠉⣿⠷⣶⣤⣆⠀⢀⣾⠀⠀⡇⠀⢠⣿⣤⠉⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠁⠀⠀⠀⠀⠒⣀⣤⣴⣾⠇⠀⠀⠈⠻⣿⣿⣇⡀⠀⠃⠀⢨⣇⠈⠱⣄
⠀⠀⠀⠀⠀⢂⣡⣶⡶⣎⣑⣚⠉⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠉⠉⠛⢻⣶⣽⠀⢰⣏⠙⠁⠙⠇
⠀⠀⠀⠀⠀⣨⡝⢳⣾⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⠈⠋⠰⠛⢲⡟⢳⣄
⠀⠀⠀⠀⠀⣱⣿⣿⣿⡿⣋⠉⠛⢿⡟⠛⠿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⢻⣿⣷⡌⠳
⠀⠀⠀⠀⢴⠿⠿⢿⠿⠿⠿⣷⠦⠀⠙⣿⣶⣯⣉⠓⠀⣠⣶⣿⣼⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣎⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠠⣾⡿⢛⣋⠸⠄⠀⠻⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢘⣿⣿⣧
⠀⠀⠀⠀⣀⢤⣤⣾⡿⢿⣿⣿⡶⠿⠀⠁⠙⠛⢳⣶⣶⣤⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣾⣿
⠀⠀⠀⣾⣿⣷⣶⣶⣶⣾⣿⡿⠃⠶⠒⠛⠁⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣯⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⠈⣿⣿⠟⠋
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠿⣽⣷⡿⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣉⣤⣷⡶⠟⠋
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣠⡤⠤⠀⠒⠒⠛⠛⠋⠉⠉
⡀⠀⠀⣀⠀⢙⣿⣿⣿⣿⣛⡉⠀⣀⠀⠀⡀⠀⠀⠀⣉⡉⣈⣁⡀⠀⣀⡀⠀⠀⣀⣀⠀⠀⢀⣀⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⣀
⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⠀⠀⢀⡀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢃⣤⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡴⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⢀⢸⠅⠘⡻⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠂⠀⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢣⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣠⠞⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢠⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠢⠜⠋
⠀⠀⠀⠀⣠⣾⡫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⡛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⠄⠊⣀⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠘⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⢿⣷⡴
⠀⠀⠀⠈⣆⡀⠸⢰⠄⢸⠂⣃⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡀⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠿⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠑⠀⠍⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⣟⡄⠀⠀⠀⠀⠀⠀⠰⡇⠠⠈⠓⢬⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢤⠀⠒⢀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠁⢢⣠⠇⠐⣾⠿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⢠⠀⠀⠀⠐⠄
⠀⠀⠀⠠⡄⠄⠽⠞⠛⠒⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢉⣹⣿⣄⣌⢠⣤⡇⠀⠀⠀⠀⠀⣴⠁⠀⠄⠂⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠁⢠⡖⣤⣀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣟⠃⠀⢽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠆⠈⠈⢙⣏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠐⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢤⡄⣁⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡧⠤⢤⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤
⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⣂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣾⡅
⠀⠀⠈⠀⠀⠸⡇⠄⢀⠰⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣆⢰⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⠿⢿⣤
⠀⠀⠀⠀⠀⢠⡇⠀⠨⠄⢻⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡸⠛⢠⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⠁⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⢥⣝⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⣶⣛⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⡤⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⡠⠈⠁⠁⣀⣠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡄⠀⠀⠐⢢
⠀⠀⠀⠀⣇⣈⡤⠞⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣦⣤⣦⣼⣤⣶⢴⡆
⠀⠀⠀⠀⣿⠃⠀⠐⠀⠠⣶⡏⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠲⣦⣀⡟⠒⠒⠿⣶⣦⠉
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠆⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡴⠋⠀⢸⢸⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⡇⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⠁⠀⠀⢸⠘⢚⡥⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢸⠀⡇⡇⢸⠀⠀⠀⢀⢇⡀⠀⠀⣿⡀⢠⡀⠀⢣⠀⠀⠀⢹⠀⠀⢹⢀⣀⠀⠸⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⡇⢸⢸⠀⡇⡇⢸⠀⠀⠀⢸⢸⠁⠀⢠⡧⡵⣌⡏⢶⢬⣇⠀⢸⠈⡇⠀⠀⡏⠉⡇⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⣀⣧⣸⣇⣇⣸⣸⣀⣇⣇⣸⣀⠀⠀⢸⢸⠀⠀⡾⠰⠀⠈⠉⠀⠆⢹⠀⢸⠀⡇⠀⣠⠕⠸⢵⡒⠒⠚⠉⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡆⢸⡇⠀⠀⠠⠄⠀⠀⢸⡆⢸⢀⡇⠀⠛⠚⠒⠚⠃⠀⢀⣀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠳⡙⣾⠙⢤⡀⠻⠟⢀⡴⢋⡹⣼⡞⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⢤⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠸⠬⢷⠀⠀⠀⠀⠀⠈⠳⠞⢦⡏⠑⠊⢹⡤⠚⠓⠙⠀⠀⠀⠀⠀⠀⢠⢴⡏⠀⠀⠀⠹⡽
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢐⣒⣒⡂⢸⠀⠀⠀⠀⠀⣀⣠⡴⢻⠁⠀⠀⣈⣟⠦⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡞⠉⠀⠈⠙⡇⠈⠭⠭⠅⢸⠀⠀⡰⠋⠉⠁⢼⠀⡏⠀⠀⠈⠀⢸⠀⣹⠈⠉⠓⣄⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣤⡤⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠷⠶⢶⠶⠖⠚⠀⠀⡇⠀⠀⢀⡜⠁⡗⠒⠒⠒⠒⢻⠈⢣⡀⠀⠀⢸⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠛⠓⠲⡇
⠀⠀⠀⠀⢠⠜⠃⠀⣄⢸⢀⡄⠀⠸⢤⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡏⠀⡘⠀⠀⠀⠘⡃⠀⠀⡇⠀⠀⠠⡏⠉⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢏⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⢸⡀⢹⠀⠀⠀⢠⠃⣰⠃⠀⡆⠀⠀⡇⠀⠀⡇⠀⣀⣀⣉⣉⣓⣀⡀⠀⡇
⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⢧⠈⡇⠀⠀⡼⢠⠏⠀⠀⡗⠀⠀⢻⠀⠀⡇⠀⠭⠭⠭⠭⠭⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠘⡆⢻⠀⢰⠇⡞⠀⠀⢰⡇⠀⠀⢸⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⣸⡀⣀⣀⣸⣸⣆⣜⣼⢀⣀⣀⣸⡀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠉⠈⠁⠉⠉⠈⠁⠉⠁⠁⠉⠁⠉⠉⠉⠉⢹⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠯⠥⢜⣦⣠⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣿⡹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣶⠟⠁⠀⠀⢀⡧⣙⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠤⠤⠴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⢲⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡐⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣭⡛⠫⠭⠙⠉⠒⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣆⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠛⠢⢄⠠⣴⣵⡄
⠀⠀⠀⠀⠀⢸⡷⠲⠶⢶⠒⣿⠀⠀⠀⣀⣹⣄⠀⠀⠘⡿⠿⠿⠿⠶⣶⡶⠶⣤⣧⣀⡏⠉⠁⠒⢲⠀⠈⡆⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣧⣀⣠⣸⠀⢋⡄⠀⣾⡟⠙⠻⣇⣀⣴⠗⣤⠀⠀⢀⣀⣈⣲⡀⠈⣿⣿⠀⠀⠀⢸⠀⠀⡇⣿⣿⡇
⠀⠀⠀⢠⡞⠉⠛⠾⠭⠭⠷⣿⠙⠓⠛⢻⡷⢾⠿⠿⣥⣾⣷⣤⡤⠴⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⢸⠀⠀⣟⣿⣿⡇
⠀⠀⠀⢸⠉⠓⠒⠒⠀⠐⢺⣿⣿⠛⠉⢹⡇⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇⢀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⠀⡖⠲⠤⠤⠤⠞⣿⣻⡶⠶⣾⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣤⣿⡛⣉⣃⡨⡳⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡉⠉⠀⢀⣲⡄⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⣽⣿⣷⣿⣿⣯⠨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠻⠿⠿⠿⠿⠿⠂⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⡓⣒⣒⣛⣩⡵⡀⠀⠀⢸⢿⣿⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠰⢤⡰⡶⠶⠀⢸⣷⢄⣀⠀⠀⠀⠀⠀⢀⣷⢤⣄⣈⣉⣉⣁⣤⢴⡟⠋⠉⠙⠛⠻⠷⢤⣄⣀⡀
⠀⠀⠀⢸⠀⠧⢄⣀⣀⡇⠀⣴⠋⠀⠈⢲⠄⠀⣿⣿⠟⠻⠿⠿⠿⠿⠓⡏⠙⠓⠒⠶⠒⠒⠊⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓
⠀⠀⠀⠘⠦⣄⣀⣀⣀⡇⠀⣈⡳⣤⡔⣁⠀⠀⣿⡟⠀⠠⡤⠒⠚⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⢀⣠⡟⠧⢤⣀⣷⣷⡶⡟⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣤⡴⠊⠁⠀⢀⡤
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠿⠿⣿⣿⣿⠇⢿⠇⡀⠀⣠⣯⣧⠤⠂⡼⣿⠢⢤⣄⣀⣀⣠⠤⠖⢻⠿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠉
⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠊⠁⠘⠿⠾⠉⠉⡟⠒⠚⠋⢻⡀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠉⠉⠛⠻⠖⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠼⣧⣤⠴⠠⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠁
⠀⠀⠀⣿⠤⠤⠤⠀⠐⠒⠒⠒⠒⠂⠉⢻⠀⠘⠶⠒⠊⠉⠁
⠀⠀⠀⠹⠴⠶⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠒⠤⢤⡀⠀⣀⣀⣰⡿⢯⡙⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠞⠀⠀⠀⠁⠀⡀⠀⠈⠜⢇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢀⠃⠀⢠⡇⡟⠁⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡴⠫⣤⣤⡀⠀⠀⠀⡎⠀⠀⣸⠁⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠩⠀⠀⠈⠁⠂⢀⣀⣼⡄⠀⠒⠋⠀⠀⠈⠁⡄⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠑⠀⠀⠀⠁⠀⢀⡰⣄⠀⣀⡔⣇⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠈⠁⢏⠲⢶⡶⠸⢇⠀⣀⡀⠀⠸⠋⠙⠉⠀⡞⠓⠀⠣⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡁⠀⢀⣨⡷⡾⠀⠀⠀⢢⡿⠿⠶⣾⣦⣶⠖⠛⠛⠂⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠧⣤⠼⠷⣶⣿⡏⠀⠈⢤⠞
⠀⠀⠀⡀⣠⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣶⣿⢶⡄⠀⠈⠓⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⢻⠀⠒⠀⠄⠤⠄⠑⠒⠊⠘⠉⡉⠹⣭⣠⡦⠴⠉⠒⠈⠓⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡨⠀⣠⡆⠀⠀⠂⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖
⠀⠀⠀⡠⠄⠀⠀⠐⢾⣆⠀⠀⠀⠀⠀⠀⡘⠁⠀⠃⠁⠀⠀⠈⣀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠊⠀⢄⠀⠈
⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠜⠋⠔⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣸⠶⠞⠋⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢧⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠤⣤⣤⡤⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⣱⣿⣗⣁⣾⣷⣝⣛⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠯⣾⣿⣼⣿⣿⢻⣿⣿⣿⣿⡷⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣾⣿⣾⣿⣿⠟⣵⣶⡎⠿⣿⣯⣾⣿⡎⣛⡃⠤⣶⣶⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣝⢻⣿⣿⣿⣿⣬⢛⣵⣿⣿⣿⣿⣿⡇⠿⠃⣀⣈⣡⠞⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⣛⣛⣫⣛⠿⣟⡻⣿⡯⠛⣻⣿⡇⡿⢀⣉⣉⣥⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣎⡻⠿⣳⠆⠈⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⡇⠀⠀⠘⠛⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠙⠃




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⡿⠿⣿⡿⠿⠿⠿⠿⠿⣷⣶⣦⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣟⡋⠉⠻⣿⣆⠀⣤⡈⣠⣶⣄⣀⣴⣆⠉⠁⠈⠹⠟⢛⡛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⠋⠁⠈⠻⠷⠀⣤⣾⠏⠀⠙⠿⠟⠋⠛⠛⠛⠿⣶⣿⠂⠀⢰⣿⠋⠀⠾⠿⠿⣶⣄
⠀⠀⠀⠀⠀⣸⡿⢁⣀⣴⣾⠿⢿⣦⠉⣁⣀⣀⣘⠿⠿⠿⠿⣶⣦⣄⣠⣿⡇⠀⠈⠻⠿⢷⣶⣶⣶⠈⣿⡇
⠀⠀⠀⠀⢀⣿⠇⢘⣿⡟⠁⣠⣾⡿⠿⠛⠛⠛⠛⠿⠿⣶⣤⡀⠉⠙⢛⣩⣶⡿⢿⣿⠿⣿⣿⡆⠀⢰⣿⣧
⠀⠀⠀⠀⣼⣿⡄⢸⣿⠀⣼⡿⠁⣠⣴⣶⣶⣶⣶⣦⣤⡀⠙⣿⣆⠰⡿⠛⠁⠀⢻⣷⡀⠀⠙⠿⠀⠀⢸⣿
⠀⠀⠀⢀⣿⠏⣡⣼⣿⢠⣿⡃⢰⣿⡉⠀⠀⢠⣾⠿⠛⢿⣷⡿⣃⣀⣴⣶⣶⣦⣄⠉⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⢸⣿⠀⠉⠙⠿⠀⢿⣧⡀⠻⣷⡄⢀⣼⣿⣶⣶⣾⣟⡘⠛⢻⣿⠇⠙⣯⠿⣷⡀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠸⣿⣿⠿⣷⣦⡀⠈⠻⢿⣶⣿⣷⡿⠋⢸⣿⠈⠉⠛⢿⣦⣼⣿⡾⢇⡈⢰⣿⣧⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⢿⣧⡀⠈⠻⢿⣶⣤⣤⡄⠉⠁⢀⣶⡿⠋⠀⠀⠀⠈⢿⣟⣻⣷⣘⠋⣀⣼⡿⢠⣤⣾⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠻⣿⣦⣄⣀⣈⣉⣉⣁⣀⣀⣼⡟⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣼⣿⠟⠁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⢿⣇⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠉⠉⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⣼⣿⣷⣦⣤⣴⣿⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⣼⡟⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢹⡶⠶⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⣠⡤⠤⣤⡀⠀⠀⢀⣤⠤⢤⣄⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣼⠃⠀⢀⠀⢹⡄⢠⡏⠀⡀⠀⠘⣧⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⠴⠒⠛⠶⣶⡋⠀⢸⡇⢸⡇⠀⢙⣶⠶⠛⠒⠦⣄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⡼⠁⣠⠶⠆⠀⠀⠙⠀⢸⡇⢸⡇⠀⠋⠀⠀⠰⠶⣄⠈⢧⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣇⠀⢿⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⡿⠀⣸
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⣤⠿⢦⡀⠀⢲⣄⣀⣀⣀⢸⡇⢸⣇⣀⣀⣀⣠⡖⠀⢀⡴⠿⣤⡀⠀⠀⣰⠛⢳⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡾⠁⣠⠆⠀⠀⠀⠈⠉⠁⠉⢹⡇⢸⡏⠉⠈⠉⠁⠀⠀⠀⠰⡄⠈⢷⠀⠀⠘⢶⡞⠁
⠀⠀⠀⠀⠀⢀⡼⠷⡄⠀⠀⣧⠀⠻⣄⡀⠀⠀⢀⣀⡀⠀⢸⡇⢸⡇⠀⢀⣀⡀⠀⠀⢀⣠⠟⠀⣼⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⢧⣤⠏⠀⠀⠘⢷⣄⣀⣁⣀⡴⣯⡉⠀⢠⣸⡇⢸⣇⡄⠀⢉⣽⢦⣀⣈⣁⣠⡾⠃⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠁⣀⠀⠛⠀⠀⢹⡇⢸⡏⠀⠀⠛⠀⣀⠈⠉⠉⠉⡇⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠘⣦⠀⢀⡞⠉⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠉⢳⡀⢀⣰⠃⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⠙⢻⠁⠀⡀⠀⣸⠇⢸⡇⢸⡇⠸⣇⠀⢀⠀⠈⡟⠋⠁⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠘⢧⡀⠉⠋⠁⢀⡾⠁⠈⢷⡀⠈⠙⠉⢀⡼⠃⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠙⠓⠒⠛⠉⠀⠀⠀⠀⠉⠛⠒⠚⠋⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠲⠶⠶⠾⣇⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠉⠉⠛⢶⣄⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢻⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡀⠀⠀⠀⠀⠀⠉⠳⢌⠀⣠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠹⠃
⡼⠁⠀⠈⢆⠀⠀⣀⡀⠀⠀⠀⢶⠘⠻⣆⠀⠀⢀⡤⠂⠀⠀⠙⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠂⠀⠀⠀⠁
⢷⡀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⣤⣀⣄⣦⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒
⠈⢷⡀⢦⠀⣈⣆⠀⠀⠀⠀⠀⢰⣲⡀⠀⠀⢠⣤⣄⠀⠀⢄⡀⠀⠀⢘⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⡶⠏⠀⠀⠀⠀⠀⣄
⠀⠈⣷⣾⣇⠹⣿⣧⠀⠀⠀⢀⣿⢿⠆⠔⠛⠃⠊⠚⠻⣶⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠛⢉⣡⣴⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠘⣿⣿⣆⠹⣿⣆⠀⠒⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠫⠶⡀⠀⠀⠀⠀⠀⢷⣠⣶⣾⣿⣿⣿⠶⠚⠋⠁
⠀⠀⠀⠘⣿⣿⣦⡟⡿⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠛⠛⠃⠀⠸⣍⠀⠀⠀⠠⣙⣦⣄⠀⠀⠀⢀⣀⡀⢠⠤⠖⠁⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠳⢦⣄⣠⣜⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡀⠀⠀⠀⠀⢀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠙⠻⣿⣶⣌⠙⠿⣿⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢣⠀⠈⠻⢿⣷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠹⢦⣄⣠⣀⡤⡤⢒⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠾⣿⣷⣴⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠒⠦⡴⠲⠶⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣛⣛⠃⠉⠄⢀⡀⢠⠴⠀⡿⢑⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⣷⣶⢟⠾⠷⠿⠤⠧⣴⣘⠀⠈⠁⠀⠀⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣴⠿⠉⢇⠀⠀⠠⠿⠀⠀⠀⠉⠙⠒⠶⣄⡀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣻⠋⢭⠙⢦⣀⠀⠡⡄⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠉⠢⢄⠸⣟⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠟⡛⠂⣄⠂⠈⠑⢄⠑⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠓⠜⢿⣷⣄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠈⠁⠀⠀⠑⠦⣄⠀⠉⠣⡀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⠾⣿⢦
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢦⣴⠂⠀⠀⢀⠨⠑⠂⡤⢼⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣾⣷⣹⣸⢧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⡞⢆⠑⠂⠠⡀⠑⠄⡀⡀⠀⠛⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠖⣿⢃⡇⡞⠸⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⡟⣦⣴⠉⠢⣀⠈⠑⠤⠀⣈⣦⣀⢠⠠⠀⠀⢀⡇⠀⠀⣐⣯⠞⢀⢯⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠩⠛⣻⣿⡖⠢⣽⣷⣶⣦⣀⣸⡑⠃⣦⣠⠀⣸⣷⡖⠞⠥⠁⡄⣰⣵⣿⡟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡷⣤⣶⣿⣿⣿⣻⣿⣿⣾⣾⣿⣷⠟⣋⣠⠖⣐⠋⣛⣉⣹⣏
⠀⠀⠀⠀⢿⣿⣿⣿⣽⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣛⣿⣭⡿⢶⣻⣽⡟⠛⣩⣿⣻⣽⣿⣿⣷⣄
⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⣿⠻⣶⣶⣿⡿⢯⣿⣶⣿⣧⣶⣶⣶⣿⣿⣿⣿⣿⢿⠾⢿⣿⣿⣿⣿⣿⡿⣿⣷⣅⢁⣈⣙⡆
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣻⣷⣿⣿⣿⣿⢿⣿⣿⣿⡿⣋⣭⣍⣿⣶⡟⢿⣿⣝⠀⠐⠿⠿⣿⠛⠋⡴⡉⢻⣟⠀⠀⠛⠧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⢸⣷⣿⣿⣷⣶⠛⠓⠄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣄⣿⣿⣿⣹⣿⣿⣿⣿⣿⣶⡄⠀⢀⣼⢏⣾⡯⠝⡇
⠀⠀⠀⠀⠀⠈⠙⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢧⡽⣿⣿⣿⣿⣿⣿⣿⣯⣿⠿⠿⠿⡿⠯⢻⠷⢍⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠛⠻⠛⠻⠛⠛⠻⠿⠿⠿⠻⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⣩⢥⠖⠻⠀⠚⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠁⠀⠀⠿⠟

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⢿⡟⠉⠀⠀⠀⣰⠏⠀⠀⠀⢀⣤⡿⠛⣹⣿⣿⣿⠿⠛⠉⠉
⣸⣿⣿⡇⠀⠀⠀⣰⡏⠀⠀⢀⣴⡿⠋⣠⣾⡿⠟⠉
⣿⣿⡿⠁⠀⠀⣰⡟⠀⣀⣤⣿⣿⣥⣼⣿⣯⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⣀
⣿⣿⠃⠀⠀⢠⡟⠀⣠⣿⣿⣯⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇
⣿⣿⣤⠶⣶⣿⣥⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡿⠃⢠⣿⣿⣿⣿⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠃⠀⣾⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣟⢀⣾⣿⠟⠁⠀⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡴⠢⠐⠲⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⢀⠰⠋⡇⢀⡄⢀⠈⣀⣯⢂⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡊⠘⠘⡄⣄⠈⡧⠂⠀⣠⢰⠈⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠑⠀⢀⡗⠀⡉⠋⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⠀⢢⠔⠀⢠⠄⠱⠀⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠓⠁⠀⠀⠋⠔⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉

⡿⠋⠙⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢙
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⡇
⡇
⡇
⡷
⣧⠀⠀⠀⠁⠈⠁⠘⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠘⠀⠁⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⡏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠐
⡇
⡇
⡇
⡇
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠈⠉⠉⠐⣒⡂⠄⠁⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠒⠊⠁⠠⣄⢠⣤⢤⠀⢤⠈⠄⠋⠁⠠⠤⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣴⣿⡿⣼⣿⣫⣷⢿⣷⣤⡄⠒⠀⠃⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠩⡿⢿⣿⣿⣿⣿⣽⣿⡿⠋⠉⠈⠀⠀⠀⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⠛⠱⢤⡈⣿⡿⢹⣷⣿⣿⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣤⡄⠬⠐⠂⠀⠀⡩⠁⠿⢻⣿⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡘⠛⠣⠈⠀⡒⡐⠚⣤⠀⣾⣷⣀⣁⣹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢁⠤⡛⠀⣻⣷⣿⣿⢡⣶⣤⣾⣿⣿⣿⣿⣷⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠀⢀⣂⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠁⠀⠉⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⠇⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⣀⠐⠀⠀⠉⣀⠀⠀⠀⠀⢀⣠⡔⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣿⡿⠛⠁⠀⢈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⠿⠿⣿⡿⠟⣋⠈⠉⠀⢠⠄⣄⠠⠐⣠⡖⠉⠲⡤⠀⠀⠈⠿⣿⣿⣿⣿⣾⡷⠆⣠⣴⣿⠿⠛⠉⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣤⡀⠀⠖⣠⢔⡈⣿⣿⣾⣧⣠⣶⣴⣦⣰⣦⣍⡉⠩⣄⠀⠙⠉⠉⠀⠀⠸⠿⠃⠠⣤⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡄⠀⡀⢀⣄⢘⠶⠴⠄⠴⠁⢶⣤⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣶⣤⣬⣴⠶⠂⠀⠶⣶⡂⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢠⡉⠼⠋⠀⠹⡄⠀⢈⠃⠈⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠰⣶⣶⣦⡍⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠀⢀⣤⣴⣤⣭⣭⣅⠀⢠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣭⡛⢿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⣉⣛⣛⣻⣿⠏⠀⢐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⡿⠟⠛⠃⠀⢈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁
⡇
⣷⣀⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣄⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⣤⡤⣄⣂⣀⠈⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠦⣄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣀⢻⢠⡤⣄⡄⡄⣄⢠⠀⠈⠳⡄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠃⠀⠀⠀⢀⠠⡔⣏⠹⠼⠛⠋⠓⠃⠿⣅⠹⠂⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠠⠀⣾⣈⠳⠛⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠠⠝⢟⠛⠾⠁⠀⠀⠡⠀⠀⠐⠐⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⡏⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣀⣤⢴⢾⣷⡄⣠⠖⢠⠖⢠⡐⠄⣀⠀⠀⠀⢀⣠⠞⠁⠀⠁⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⣴⠋⠀⠀⠀⠻⣦⠇⠀⠛⢲⠀⣷⢀⡏⠉⠛⠉⠉⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢧⠀⣰⡏⠀⠀⠂⠀⢐⡟⠀⣀⢀⡼⠀⠘⠉⠁⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⠻⢿⣇⠀⠀⠂⠀⢨⡇⠀⢸⢋⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠁⠒⠭⢛⢦⣦⣴⣾⡇⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢙⣿⡇⢸⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⠽⡇⢸⡇⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⡧⢨⣇⠈⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠅⢠⡇⢸⠝⠲⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠤⣸⠁⡿⡣⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣂⣱⡏⢰⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡂⠐⡼⠀⡼⠀⠉⠋⠀⠀⠀⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠈⠁⠈⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠐⠲⠒⠲⢒⣒⣒⠀⠦⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠉⠢⣴⠂⣤⠁⠀⠉⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠠
⠀⠀⠀⠀⠀⠀⠀⠘⢵⠀⠀⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠁⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠈⢆⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣃⡾⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠐⣱⣀⠀⢀⠸⡇⡇⠀⢀⣤⠾⠋⠉⡁⠒⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠄⡙⣧⣀⠀⠀⠥⢃⣴⡟⠁⠀⠀⠀⠀⠀⠈⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠩⢥⠀⡈⣄⢸⠋
⠀⠀⠀⠀⢀⡀⠴⠤⠤⠄⣀⠀⠀⠇⢀⠲⣎⠀⡉⢸
⠀⠀⠀⠠⠀⠀⣀⣡⡶⢶⣽⣦⣻⡎⣦⠉⣻⡆⠐⢺
⠀⠀⠰⡠⣆⠞⠉⠠⠀⠀⠘⡀⠉⠳⢬⡷⡙⢷⠘⢫⡀
⠀⢀⣏⠰⠏⠀⢀⡴⠚⡳⠷⠿⣷⣶⣤⣙⠟⢸⣆⢰⡇⠀⠀⢀⣤⠴⠶⠶⢄
⠀⠘⢃⡼⠀⣠⠊⠀⡸⠀⠀⠀⠀⠠⡉⢪⡻⣎⣿⢸⣇⡤⠚⠉⠀⠀⠀⠀⠈⣇
⠀⢈⢠⡗⠰⠃⠀⠀⡇⠀⠀⠀⠀⠀⠈⠢⡙⢾⣿⢸⡏⠀⠀⠀⠀⠀⠀⠀⠄⡿
⠀⠈⣹⡵⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⣻⣼⡇⠀⠀⠀⠀⠀⠀⣸⢀⡇
⠀⢸⡿⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡍⡿⠖⢢⣄⠀⠀⢠⡟⡸⠁
⢀⡿⠁⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⣧⡟⢦⣄⣹⣷⡶⠿⣴⡃
⠈⠁⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢀⣇⣼⡃⢻⣴⠿⣭⡏⣿⡒⠀⠌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢦⣀⣀⣨⠿⣍⣻⣼⠃⢸⠃⠀⡐⢀⢻⣿⡄⡀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⣧⡘⠓⡞⠀⠀⠃⠀⢈⣿⣧⠀⠈⠉⠂⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⡇⠀⠀⠰⢀⠊⡸⢹⡆⠤⠀⣀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⣄⠀⠀⠈⡿⠀⠀⠀⠈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡧⠀⠀⠀⢹⠀⠀⠠⢽⣧⠐⠒⠂⠄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠀⠀⠀⠀⢃⠀⠀⠈⣸⣇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡂⣿⠀⠀⠀⠀⠀⢣⠀⠀⠀⣿⣄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠀⠀⠓⡄⠀⠀⠿⣤⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⢻⣦⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠙⢇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣈⠷⡀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡈⠈⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⢉⣭⣭⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣤⣤⣴⣶⣶⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡿⠶⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠿⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠁⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣨⠟⠁⠀⠀⠀⡰⣺⠇⣴⡄⠠⣄⣄⣠⡴⢻⠖⠒⠂⠀⠀⠈⠹⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠉⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠠⣆⣤⡠⣤⢤⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣴⢧⡰⢦⢤⡠⢆⡴⡄⠀⠊⣧⠤⠴⡄⢘⡷⠲⢤⡟⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠠⢶⢀⣀⢀⣄⠀⣀⠀⠀⢀⡼⢃⣤⢀⣴⢴⠟⣠⠄⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠉⠈⠚⠃⠓⠉⠃⠀⠀⠀⠈⠊⠘⠋⠈⠊⠐⢄⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠒⠊⠀⢠⡄⠀⠄⣴⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠏⡼⠻⠜⠏⠘⠁⠹⠤⢫⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⠀⠀⢀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣦⣄⣀⣀⣀⣀⠀⠀⠀⠈⢀⣀⣠⣴⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠂⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⡀⣶⣶⣶⠀⣶⡶⣶⠆⣴⣖⣶⣶⣆⠀⠰⣖⣶⣶⣶⣶⣶⠶⢶⣶⡶⠂⡶⡀⠀⠠⠀⠒⣲⠆⡄
⠀⠀⡗⠀⢼⣯⡘⣿⣷⣿⣀⠛⢋⣾⣿⣿⠄⠀⢿⣿⣿⣿⣿⡁⠀⠀⠛⢁⣤⣤⠀⠀⠀⡀⠀⠁⢀⡆
⠀⠀⣧⡘⠎⣿⠇⠿⢯⡙⢿⣿⣺⣿⣫⣾⣿⣯⢳⣎⢿⣿⡟⣡⡤⣾⠇⢘⠛⠋⠀⠈⠀⠁⠀⠀⣼⠁
⠀⠀⣻⣿⠂⣴⣿⣷⣆⠹⡇⣿⣿⡁⣿⣿⣿⣿⣾⣿⠘⡿⠃⠋⠀⣉⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣿⣧⢳⣝⢿⣿⣿⢀⡇⠛⠋⠁⠈⠈⠉⠙⣿⠃⣴⣶⢿⣿⡟⣩⣶⣶⣶⠄
⠀⠀⢿⠛⣦⣭⣈⡿⢏⠈⠀⣀⢰⣿⣷⡄⣤⡄⠀⢸⣿⡟⠀⢿⢰⣿⣱⣿⣿⡇⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡆⢸⣿⣿⠹⣷⠈⠀⠀⢻⣿⠿⢛⣼⣿⣷⠀⠀⢶⢶⠄⠸⠏⣋⠿⠿⢋⡀⠉⠀⠀⠀⠀⠀⠈
⠀⠀⣷⠸⢿⣿⣿⡏⠀⠀⣸⣷⣅⣶⡜⣿⣿⣿⠆⠀⣠⢀⣌⣙⠿⣉⣴⣞⣭⣤⡀⠀⢀⣀⣀⡤⣮⡄
⠀⠀⡽⠿⢷⣜⣶⣔⣋⡀⠈⣩⣾⣿⡇⣿⡿⠁⠀⣰⣿⣍⣿⣿⣷⠘⠿⠇⢀⣎⣤⣄⣿⡿⣫⣾⣿⡇
⠀⠀⡹⡐⠦⣻⣮⠹⡟⢷⢀⣙⠁⠿⠃⠀⠀⡀⢰⣿⣿⣿⣼⣾⣷⡇⢰⡄⣻⣿⣿⠃⣭⣾⣿⣿⣿⡇
⠀⠀⢿⠶⠄⠿⢩⡆⣿⡚⠋⣿⣷⣄⣿⣷⠀⠛⢸⣿⣿⡝⣿⣿⣿⠇⣿⣇⠈⢿⡿⠈⣿⣿⣿⣿⡟⠁
⠀⠀⣿⣿⡇⣄⣬⣿⣿⣿⡇⣿⣿⣿⣎⢡⣤⣶⣌⡿⢿⠏⠿⠟⣉⠸⠿⣿⣷⣄⠁⠰⠿⠿⠟⢉⣠⡄
⠀⠀⣿⣿⡇⢿⣿⣿⣿⣿⣏⠻⣿⣿⡅⢸⣿⣿⡿⠛⣡⡰⣶⠯⢃⣶⣶⣦⣤⣼⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣷⡼⣿⣿⣿⣿⣿⣿⡇⡝⠀⠘⣿⢃⣶⣄⠻⢣⣾⣿⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣮⣻⠿⣿⡿⠟⠈⠀⠒⣀⣤⡀⠉⣤⢠⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣾⣿⣿⣵⣧⠙⠈⢸⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢄⠀⠹⣿⣿⣝⠟⢸⣷⡨⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠛⢋⣴⣌⢿⡆⡘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠀⢠⡃⠀⢿⣿⡟⢸⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⡿⣿⣿⣭⠑⠂⢸⡟⠀⣿⠇⣄⢈⡛⣧⢿⡿⣿⠇⠀⠀⠉⡍⢉⣭⣿⣭⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⡋⢠⣿⢃⡀⢄⠹⣿⣦⢳⣄⡀⢴⣶⣾⣿⣷⣾⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠤⢻⢸⡼⣿⠸⠿⠆⠂⡻⠿⠋⠛⠿⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣛⣟⡛⠛⠛⠉⠥⢀⣴⣶⣶⣀⡈⠁⠀⠀⠀⠀⠀⣴⣾⣿⣿⣦⠘⣿⣿⣮⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣽⣿⣄⠀⠀⠀⠀⢸⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⣲⣆⢻⣿⣿⣿⣧⣸⣿⣿⠃⠉⡉⢉⣻⣿⣿⡇
⠀⠀⠉⠉⠈⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠻⡄⠀⣠⠴⠂⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠴⣾⠁⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⡿⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣼⢃⡴⠚⠙⠲⣤⡀⣾⡟⠀⠀⠀⠀⠀⠹⣆⠀⠚⠉⡉⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠈⠙⢯⡿⢦⡀⠀⠀⠀⠀⠙⢧⡀⡿⠙⡟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣿⡄⠀⠀⠀⢀⣀⣭⡇⢀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠦⢤⣽⣷⣿⠤⠖⠚⠉⠉⠀⠙⠛⠓⠞⠷⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠴⠶⠶⠶⢦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢦⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠄⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠻⠶⢾⣷⣤⣬⣵⣤⣤⣵⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⣙⢦⡀⠀⣠⡟⠀⠀⠶⠖⣻⣿⠟⠓⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠁⣿⠂⢙⡾⠋⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡶⠟⠁⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾⠀⠀⠸⣧⠶⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣄⣤⠶⡿⢲⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣽⣷⣭⡙⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⣈⡝⢿⡀⠙⢿⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⣇⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣠⣶⣶⠾⠷⠿⠿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡯⠀⣿⠷⢼⡇⢰⠜⠃⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣄⢹⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠠⣾⣥⣤⣤⣤⡴⠛⣻⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢳⠀⠸⡇⠈⡇⢸⠀⢸⡿⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢷⣜⣧⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⢷⠀⣿⢸⠇⢸⣇⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⣄⡈⠻⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠸⡇⢸⡿⠀⠈⣿⢹⣆⣿⣤⢴⣶⡤⠶⣶⣆⢹⣷⠤⢀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠛⠋⠛⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢿⡀⠀⠸⡇⠀⣿⣸⣧⣤⠶⠛⠉⠉⠀⠀⢰⣿⠀⠀⣿⡽⠦⠿⠄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠛⠶⠖⠲⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠃⠀⠀⢿⡤⠼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣀⢹⡗⣶⠶⢆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣒⣚⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠈⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⡚⣣⡇⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠈⠙⠓⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠸⣿⡃⠀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⡟⠉⠉⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⡏⠙⣿⠉⢻⣏⣉⣠⡤⠶⠚⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⢿⣆⣠⣿⠀⠘⣏⠉⠀⠀⠀⠀⣿⣤⠤⠤⠴⠶⠶⠶⠖⠚⠛⠛⠛
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠁⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠂⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣧⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣤⠤⣤⠼⠟
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠛⢛⣻⣿⣿⣿⠿⠧⠤⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠒⠒⠂⣉⡿⠋⠀⣮⠈⠛⠓⠶⠶⢠⡿
⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠁
⢀⣀⣀⣤⣤⣤⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⣤⠴⠶⢿⡷⠻⠛⠒⠒⢠⡄
⠻⣤⣀⣀⣈⠹⣦⣹⣯⣭⡿⠿⠛⠛⠋⠉⠁⠀⠀⠉⠉⠐⠶⠾⠶⠧⠶⠶⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠻⠯⠤⠤⠤⠤⠭⠭⠭⠽⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣭⣙⣒⣒⣒⣒⣒⣒⣒⣒⣚⣩⣬⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠁⡏⢹⠉⡏⢹⠉⡏⢹⠉⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠾⣦⣧⣼⣤⣧⣼⣤⣧⣼⣤⠷⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠶⠶⢶⡶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠻⠶⠶⠶⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠀⠀⠀⠀⢀⡴⡖⢶⡄
⠀⠀⠀⠀⠀⢀⡴⠶⣤⣤⣄⡀⠀⠀⣿⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⣿⡀⠀⠀⣴⠟⢮⣁⣼⡇
⠀⠀⠀⠀⠀⢸⣄⣀⣸⠐⠚⣻⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⠆⠘⣧⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠋⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣉⣸⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣼⠧⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡾⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡞⠁
⠀⢀⣤⠶⠶⠶⠶⠶⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⢠⡟⠉⠢⡀⠈⠠⡀⠀⠈⠙⠶⣄⣿⠛⠒⠒⠒⠓⠛⠛⠋⠉⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣿⢧⠀⠀⠈⠳⣄⠈⢦⡀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠘⣆⠀⠀⠀⠈⠳⣄⠙⢦⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠸⣆⠈⢢⡀⠀⠄⣀⠈⠳⢄⠙⢦⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠁⠀⢀⡾⠁
⠀⠘⢦⡀⠙⢦⡀⠀⠀⠀⠀⠑⢄⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠙⢦⣀⠈⠓⠶⢤⣤⣀⣤⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣤⣤⠶⠟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠉⢉⣽⠟⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⠾⠋⠁⠠⠄⠀⢀⡤⠞⠉⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⠀⢀⣀⣠⠴⠛⠁⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⡠⠚⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⠦⠤⢤⣄⣀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⣠⠖⠁⠀⣠⠖⠉⠀⠐⠀⠀⣠⠎
⠀⠀⠀⠀⣠⣴⡛⠛⠲⢤⣀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢴⣟⡁⢀⣠⠞⠁⠀⣀⣀⡤⠖⠋⢁⡼
⠀⠀⠀⣰⠏⠈⠙⠲⢦⣀⡈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠈⠉⠉⠉⠉⠉⢉⣀⣠⡤⠶⠋
⠀⠀⠀⣿⣄⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢤⣀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⢹⣏⠑⠦⣄⡀⠀⠂⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢦⣤⢿⡆
⠀⠀⠀⠀⠻⣆⠀⠀⠉⠓⠒⠤⠤⣀⣀⣀⣀⣬⠽⠶⠒⠉⠀⢀⡿
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⠀⠀⣀⣀⣀⣤⡴⠞⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⢠⣄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢀⣀⣤⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣾⣧⣴⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣦⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠿⣿⣿⣧⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣹⣿⣿⣿⣿⣿⣿⠿⠻⠟⠛⠻⠇⠈⣹⣾⣿⣤⣿⣿⣿⢿⣿⡿⠷⣾⣿⣷⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣯⣿⣯⣿⣿⣛⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣷⣿⣻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣴⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠸⣇⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣧⣽⣫⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣤⣠⠶⠀⠀⠀⠀⠀⢠⢠⠒⡏⢘⠻⣯⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⢀⣼⣿⣿⣴⠆⠀⠀⣀⢺⡅⠸⢀⣴⣻⢶⣿⠿⣿⣿⡇
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⡄⢠⠾⣥⣎⡙⣶⣷⣾⠃⢢⡁⢾⣿⣿⡇
⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣏⢖⣠⣿⣽⢿⣿⣄⡑⠎⣳⣾⠿⣿⡇
⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⡏⢁⣭⣾⣷⣿⣿⣾⣳⣯⡉⠾⠟⠀⢴⣿⡇
⣿⣿⣟⣿⣿⣷⣤⣄⠀⠀⠀⠀⢀⣀⣄⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⡿⠋⢰⠞⠁⠈⢻⣿⣿⢿⡟⣿⡗⠀⢠⢀⣼⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣻⣿⣿⣡⣿⣿⣿⣿⣿⣿⢿⣿⣻⣿⣿⣷⣿⣧⡔⠁⠀⠀⠀⠈⠱⠍⠆⣹⠟⠀⠐⣡⠻⣿⣿⡇
⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⠀⠀⣼⠋⣷⣿⣿⡇
⣿⣿⣿⣿⣯⣿⣿⣿⣿⠟⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠀⢄⠀⣈⣷⣿⠈⣿⡇
⠉⠛⠿⣿⣿⣿⣿⣿⠗⢠⢿⡿⠛⠋⣟⠛⣉⠉⣟⣹⣻⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠐⠀⣀⣹⣿⣅⣠⣿⡇
⠀⠀⠀⠈⠙⠻⢿⠏⣰⠇⠀⣩⣴⣾⣿⢿⡟⣠⣿⡿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠠⢀⡸⠛⣽⣏⣾⣁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡰⣿⠛⠛⠁⢺⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢰⣮⢹⣾⡹⠏⢴⣉⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠇⣹⣆⠀⠀⠠⢮⣈⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢠⠛⣩⣶⠻⡿⠣⣘⣦⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠙⣽⠀⠰⡖⠀⠀⠀⢽⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⣼⣿⣿⠟⠹⠂⡐⢫⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⢾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣯⣛⡿⠃⠀⠀⠂⣽⠆⣘⣿⣿⡇
⠀⠀⠀⠀⠀⢠⡆⠀⢀⣤⣆⣀⠘⠀⠀⠀⠀⣼⣾⣯⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠻⣿⠏⣰⠄⠀⢴⣮⢅⣾⡯⠰⣿⡇
⠀⠀⠀⠀⢀⣼⢃⣴⣿⣿⣿⣿⡆⠴⣤⣴⣾⣿⣿⣿⣿⣿⣻⣷⣦⡀⠀⠀⠀⠀⠀⠀⡀⢚⣚⠉⠺⠁⣶⣠⡞⢀⣰⡏⢀⣴⣿⡇
⠀⠀⠀⠀⢈⣴⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⢽⡭⠁⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠰⢏⠉⠁⡰⢦⡄⢀⣤⣼⠃⠀⣈⣿⣿⡇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⣽⣿⡭⢿⣿⣿⣿⡏⠁⠐⢁⣾⣿⡿⣵⣿⣿⣦⠀⠀⠀⠀⠢⡈⠐⢋⠙⡆⠸⠏⠛⠁⢀⡸⣿⣽⣿⡇
⠀⠀⠀⠈⢻⣿⣿⣿⡟⢻⣿⡛⢥⣺⠻⣾⠇⢈⡴⣾⣿⣁⣟⣳⣴⠿⢿⣷⣆⠀⢀⡤⠄⠠⠐⠀⢇⣸⡷⠀⡀⠀⣨⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠛⠙⠛⠛⠊⠁⠚⠛⠚⠣⠘⠋⠸⠿⠿⠯⠼⠿⠿⢿⢿⣿⣿⣷⡎⢳⠸⣷⠖⣴⡟⢉⣄⣺⣽⣷⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠁⣀⠀⠂⠂⠠⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤⠀⠶⠀⡣⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠻⣷⣿⣯⣿⣵⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⡿⣾⣿⡇⣿⣿⡾⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⢿⠟⣟⣻⠿⠃⠀⠀⠀⠀⠸⣟⣿⣭⣿⡿⣿⣯⣿⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⣿⠇⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣾⠇⣸⡣⠀⢨⠆⠀⠀⠀⢰⣤⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢛⢻⣿⠾⢹⣏⣄⣸⣇⢤⠀⠀⠀⢙⣿⣿⣿⣧⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠴⢿⣿⣿⣿⣷⣾⣯⣤⠤⠤⡍⠀⠤⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀⡄⣶⣿⣿⣧⣨⠟⠛⠉⠏⠋⠒⠲⠒⠡⠞⣁⢾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢛⡙⠺⣫⣬⣿⣯⡀⠤⠾⣿⢷⣶⡦⠖⠒⠚⣿⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⢲⣖⣚⡯⡀⣼⠛⠁⡄⠀⠈⣎⣹⢃⣀⣠⡔⠋⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣷⢘⣿⠍⡉⢀⡰⢿⡇⢶⣿⠀⠀⡿⠛⠉⠁⠀⠱⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⣼⣇⢹⡟⠀⠀⣇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣿⠞⡀⠰⠖⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡴⠶⠶⢦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠛⠈⠛⠛⠛⠛⠀⠛⠒⠒⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣤⠴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⡏⠀⠀⠀⣬⢻⡄⠠⣤⣤⣤⣤⣤⣤⣤⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣇⠻⣦⡾⠁⣸⠇⠠⡤⠤⠤⠤⠤⠤⣤⠄⢀⣿⠴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠶⢤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠐⠒⠶⢦⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠞⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣀⣀⣀⣀⢸⡇⠐⠶⠶⠶⠶⠶⠶⠶⠞⢧⣀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢧⡀⠀⠀⢀⡼⠁⠐⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠁⠒⠒⠛⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠁⠀⠀⠈⢳⡄⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠛⠛⠛⠛⢸⠇⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠦⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠧⣶⡄⣶⣠⣶⢄⣶⢰⠆⡖⠂⡗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠀⠉⠀⠉⠈⠉⠈⠀⠉⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣴⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡾⠿⠛⠛⠉⠉⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦
⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⣠⡾⠀⠀⠀⠀⣤⣀⢀⣄⢠⣤⢀⡄⣴⠀⠀⢀⡄⣠⣀⣄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⢰⣿⠀⠀⠀⠀⠘⢹⠙⠼⠧⠻⠛⠌⠒⠛⠄⠀⠉⠓⠁⠋⠈⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⣼⠇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⠀⣀⠀⣾⢠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣴⡖⣦⠴⣄⣖⡼⣴⠶⢏⡴⢇⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⣿⠀⠀⠀⢀⣦⣀⢀⡄⢀⣄⠀⣤⢀⡄⠀⠀⢀⠀⡀⣀⠀⠀⠉⣩⡆⣠⠖⠀⢀⡀⠀⠀⠀⠀⢸⡇
⠀⠘⣿⡀⠀⠀⠘⢹⡙⠺⠇⠸⠛⠆⠓⠚⠃⠀⠀⠙⠂⠑⠉⢇⠀⠀⠀⠃⠈⠊⠳⠾⡁⠀⠀⠀⠐⣿⠁
⠀⠀⢹⣧⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟
⠀⠀⠀⢻⣆⠀⠀⠀⠃⠀⠀⠀⠆⠀⣄⡄⣤⠤⡄⣸⡃⣠⢇⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠈⠂⠈⠀⠁⠀⠉⠁⠙⠋⠀⠓⠒⠒⠁⠀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠹⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣢⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⡀⠀⠀⠀⠀⠀⢾⣿⡧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣷⣶⣦⠼⠋⠁⠀⠠⣴⡶⡷⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡈⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡀⡀⠀⠀⠀⠀⢀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡀⢀⠀⠀⠀⠀⠀⠀⠐⢚⣍⣟⣻⣾⡿⠟⠉⢀⣴⡿⢛⡛⢛⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠀⠀⢁⠀⠀⠀⠀⠀⠀⠁⠂⠍⠉⠁⠀⠀⠀⠀⠙⠙⢶⢾⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢿⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠶⢦⣤⣤⣾⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⡧⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠉⣷⠀⠀⠀⠀⠘⠀⠀⠠⠄⠐⢤⠀⠠⠤⢤⣶⣶⡞⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣿⣿⣿⣧⡂⠸⢷⣶⣄⠀⠘⠀⠀⠀⠀⠀⠀⡄⢀⣀⣠⣘⣿⢄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡿⣕⡀⠈⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⠋⣽⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠺⢿⠿⠂⡴⢿⣤⠀⠘⣿⣿⣿⣿⣿⣦⣤⣜⣧⣤⣤⣤⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠿⣇⢄⣀⣭⣈⡀⠋⠈⢝⣷⢄⠀⠉⠉⠛⢿⣿⣟⡿⢛⣴⣿⣯⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢰⡾⠅⡏⢻⡓⢸⣤⣥⣾⡱⣬⡀⢟⣠⠈⠄⢢⠷⡄⠀⠀⠘⠃⡀⡔⠣⠝⠋⣒⣹⣽⣟⣲⣦⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠰⣑⣂⠶⡨⣳⣟⢲⡸⠁⠀⠴⠲⠠⢝⣷⣦⣀⠽⠞⡳⢄⠀⡐⣹⡏⢴⣾⡿⠋⠇⠀⠈⠀⠈⠷⢾⠹⠶⡶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠐⠳⢏⠎⢲⠔⠃⡈⠀⢘⡂⠀⠄⠀⠀⡨⢻⣍⡶⡀⡀⠄⢛⡦⡙⠽⣟⣏⣴⠁⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠙⠻⠶⠆⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠰⠃⠀⢁⠠⢤⢂⢉⡵⠈⡤⡠⠈⠷⠄⠄⡀⠨⠛⢿⣿⣾⣿⣾⣷⣏⣡⡿⣿⠁⢀⠀⠠⠀⠀⠀⠀⣿⣶⣶⣶⣤⣤⣤⣤⠄⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠘⠤⠀⠤⠦⢼⠥⠷⠞⠷⠟⠾⠶⠶⢿⢷⠶⢤⡦⠽⠿⠿⠿⠿⠟⠫⡭⠇⠀⠉⠀⠀⠀⠀⠀⠀⠹⠿⠍⠫⠿⠟⠛⠿⠆⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⢛⣛⣛⣛⣛⡛⠛⢛⣛⣛⣛⣛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣏⠀⢠⣿⣿⣿⠠⡏⠀⢠⣿⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠈⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡔⠚⢛⣒⣳⣦⢀⡖⠛⢛⣚⣲⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠳⠦⠼⠿⠿⠛⠀⠳⠤⠾⠿⠿⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣠⠤⠤⠤⢤⣄⠀⣠⠤⠤⠤⢤⣄⢸
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣼⣤⣡⣀⣀⣛⣛⣛⠘⢇⣀⣸⣿⣿⡿⢸
⠀⠀⠀⠀⠀⢰⣿⡿⠿⢿⣿⣿⡿⠿⢿⣿⣿⡿⠿⢿⣿⣿⣿⠿⢿⣿⣇⣀⣀⣀⣀⣀⠀⢸
⠀⠀⠀⠀⠀⢸⣿⣠⠞⢁⣿⣿⣠⠔⠉⣿⣿⣁⠔⠋⣿⣿⣇⠴⠋⣿⣿⣿⣿⣭⣭⣽⣷⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣶⣾⣿⣿⣿⣶⣾⣿⣿⣿⣶⣾⣿⢸⣿⣿⡟⠛⠛⠁⢸
⠀⠀⠀⠀⠀⢸⣿⠟⢉⡿⣿⣿⠟⢉⡿⣿⣿⠟⢉⡻⣿⣿⡟⠋⣹⡟⠘⠛⣛⣷⣶⣖⣦⣸⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣖⣉⣠⣿⣿⣶⣋⣠⣿⣿⣶⣋⣀⣿⣿⣷⣊⣠⡇⠀⡞⠉⠀⠀⢸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠈⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣷⣀⣀⣀⣈⣭⣭⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣾⣿⡀⠐⠤⠾⢆⠀⡠⠊⢹⣿⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⣿⣿⢸⣿⣿⣶⣂⣴⣾⣠⠃⠠⠞⠉⠉⠙⢦⣽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⡆⠀⠀⠀⡯⢻⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠠⣴⣾⣿⣟⣇⣿⣿⣿⣿⣿⣿⣿⠃⢀⣸⠃⠀⠀⠰⢧⣸⠁⠀⢻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠉⢠⣿⣿⣿⡏⠁⠀⣿⣿⣷⣿⣿⡇⠀⢀⣻⡄⠀⠀⠀⠀⣧⠀⠒⠿⣌⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⢳⣿⣿⣿⡟⠋⠉⠉⠉⠀⣀⣀⣠⣿⡀⠀⠀⢻⡏⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠙⠛⠲⢤⣿⣿⣿⣿⡇⠀⠀⣏⣾⣿⠏⢘⡇⠀⠀⠀⠐⠋⠉⠁⠀⠀⠈⠳⠄⠀⣷⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡦⠀⠰⠶⣬⣉⣿⡃⠀⠀⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠘⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡿⠃⠀⠉⠀⠳⠎⡀⠙⢅⠀⠀⣿⣿⣿⠃⢸⣧⠤⠴⠶⠶⢾⠀⠀⠀⠀⠀⣤⣤⠾⠁⢰⡾⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣛⠃⡠⠂⠀⣄⠀⠀⠈⠀⠈⢧⣰⡿⢽⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠓⠀⠀⠀⣯⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢟⠴⢅⡄⠁⢀⢡⠀⠀⠀⠀⢾⣇⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢀⠀⣀⠞⠀⠀⠀⢀⡀⠀⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠂⠀⠰⠎⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⢸⠀⢸⡇⠀⠀⠀⠀⣸⠟⠁⠀⠀⠈⠉⠉⠉⠑⣆⠀⠚⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠐⡤⠔⠚⠆⠀⠀⠀⠙⡇⠀⠈⠀⢸⣿⠗⠒⠛⠻⣧⠀⠀⠀⠀⠀⢀⡄⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣓⠀⠀⠀⠀⢀⣀⣀⣴⠀⠀⢀⠈⠙⠀⠀⠀⢸⡏⠀⠀⠀⣀⠀⠙⠓⠲⠶⠒⠊⠀⢰⣾⠃⣸⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⢘⣩⣧⣿⠀⠱⠴⠀⠀⠀⠀⠃⢸⡇⠀⠀⠀⣹⣦⣀⣀⣠⣦⣀⣀⣀⣰⣿⣄⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⢸⡛⠿⣿⠀⠀⠀⢀⢠⡄⣠⡄⢸⡇⠀⠀⢠⡟⠀⠈⡉⠉⠀⣈⣩⣉⠀⠀⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢨⡙⠲⡾⠀⢀⠂⠀⢸⣷⣿⣿⣾⡇⠀⣤⠟⠁⠀⣰⠿⢦⡞⠉⠀⠈⢷⣀⣸⠏
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣳⡖⣧⣈⡹⠃⢀⡞⠀⢀⡆⠈⣿⣿⡟⣷⠚⠛⠶⠶⠚⠁⠀⣀⡷⠀⠀⢀⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣌⠙⣧⠄⠟⣡⣦⣾⣿⣿⣟⣿⣿⡇⠀⠀⠀⣦⣄⣴⡷⠟⠁⢠⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣿⣬⣿⢿⣿⣿⣿⣿⣿⡏⣧⣤⠾⠞⠛⠉⠁⢠⠀⣀⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⣽⣾⣿⣿⠟⠀⠱⣀⣀⣰⣇⣠⣴⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠦⡄⢀⡴⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠧⣴⠀⣿⢸⠀⣳⡤⠷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣴⠛⠀⠈⠁⣿⢸⠀⠁⠀⠛⣦⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⡌⠰⣄⣀⣀⣿⢸⣀⣀⣠⠆⢠⠼⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣞⠀⠀⠀⠀⠀⣿⢸⠁⠀⠀⠀⠀⣳⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣬⣁⡴⢿⡉⢠⣿⢸⡤⢈⡽⢦⣌⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡴⠂⠀⠀⣿⢸⠀⠀⠀⢦⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⣄⣰⠇⡿⢸⠐⢧⣠⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠞⠁⠀⠓⠦⠴⠚⠁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠙⠓⠲⠤⠤⠤⠤⢤⣄⡀⠀⠀⢀⡤⠞⠉⠈⡽
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠚⠉⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⣞⠉⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠉⠉⠉⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠈⢧⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢦⠈⣿⠉⠉⠀⠉⠉⠙⠒⠒⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣨⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢀⠀⠀⢀⡠⠅⡀⡐⡀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⢿⣿⣿⣿⢿⣿⣿⣿⢷⡿⣶⢣⠀⢤⠀⠐⠽⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡟⡛⠀⠀⠀⠁⠐⢊⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠻⣿⣤⣿⣿⡇⠙⢿⣿⣿⣿⡈⡀⠀⠀⠀⠈⡉⠸⣿⣿⣿⣿⣿⠟⠙⣿⣤⣿⣿⡟⠙⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠑⠀⠱⢿⣿⣿⣭⡤⣤⣿⣿⣿⠏⢅⠄⠀⠀⠀⠀⡜⣐⣿⣿⣿⣟⡛⢒⣲⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠥⢛⠉⡧⢩⠑⠀⠀⠀⠀⠐⠳⢧⣿⣿⣿⡿⠦⡉⡉⠛⠟⠟⢻⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠀⠣⠀⠀⠀⠀⠀⠀⣷⣽⣿⣿⣿⣶⡆⠀⠐⠀⠀⠉⠀⠠⠙⣷⣻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⢙⠀⠀⠀⠀⠀⡙⢻⣿⣿⣿⣿⣯⣷⣌⠀⠀⠀⠀⠀⠀⠂⣞⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡶⠔⠀⠀⠀⠀⠀⠵⣿⣿⣿⣙⠛⢻⡭⠏⠦⡀⠀⠀⠀⠀⢀⢻⡵⣿⣿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⠂⠀⠀⠀⠀⠀⣺⣿⣿⣿⣯⢠⠄⠅⠀⠀⠀⠀⠀⠀⠀⠉⡘⢦⣿⣾⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠤⠉⠚⢏⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⠉⡿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠒⢟⠆⠘⡺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣦⣀⣠⣬⣦⣀⡠⣴⣶⡿⣽⣿⣟⣤⡀⢠⣿⣿⡧⡀⠀⠀⠀⠀⠀⠀⠲⡄⠺⠿⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠋⠉⢉⠳⣶⣿⣿⣿⣿⣿⠛⠟⠿⠿⠟⠁⠐⢄⠀⠀⠀⠀⠀⠀⠈⢡⡗⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡼⠶⠀⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠾⠶⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡤⠖⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠠⢾⢻⡗⠀⡀⢴⠀⢀⡀⢶⠀⠀⣠⣹⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⡙⠋⠞⠓⣀⠂⠞⢾⣸⠀⠜⠻⠃⠸⡀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠰⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣄⣘⣿⣷⠀⢀⣀⣤⣦⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠈⠉⠉⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠛⠉⠙⠓⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢀⣀⣀⠤⠤⠖⠒⠲⠤⣀⣀⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠉⠉⠙⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⡆⢠⡄⠀⢀⡶⠚⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡼⠉⠛⠉⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃


⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⡤⠶⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠑⠢⢍⣛⠲⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠑⠦⣌⣉⠒⠤⡉⠛⠷⢶⣤⡀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⣑⠢⢭⣑⠢⢍⡙⠒⠤⣀⠀⠀⠈⠙⢷⣄⠐⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠓⠮⣍⣒⠦⢍⣓⠢⠄⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⣭⣷⣦⣌⣁⠐⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣯⣭⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣤⣄⣠⣤⣴⣟⣁⣤⣤⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢻⣿⢿⣿⠻⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⢛⣛⣛⣻⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⡜⠘⣯⠉⣿⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠖⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠁⠀⠘⢧⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⢶⡟⣀⣠⣤⣴⣦⣬⣿⣿⣿⣿⣿⣿⣏⣉⣉⣉⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⢀⣤⡶⠟⠛⠉⠁⠀⠀⣼⣿⠏⠉⠀⠀⠀⠀⠂⠠⠤⠤⠤⠤⢤⣭⣉⣉⣉⣉⣉⣉⣛⣛⣛⠛⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⢶⡾⠿
⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠒⠒⠒⠒⠤⠬⠭⠭⠭⠭⠭⠍⠉⠁⠀⠀⠀⢠⡟⠳⢤⣀
⣼⠃⠀⠀⠀⠉⠑⠢⠄⠀⠀⢿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣒⡒⠒⠒⠢⠤⠤⠤⠤⢬⣭⣉⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠛⠶⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠄⢀⣀⣀⣀⣀⣈⣉⣉⣉⠉⠉⠉⠉⠉⠉⣛⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣀⣀⣤⣽⡷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣦⣤⣭⣭⣭⣭⣽⣿⣿⣿⣿⣷⣶⣶⣶⣶⡦⠄⠤⠄⠀⠀⠀⠉⠁⣀⣀⠀⠀⠙⣟⠛⠋⠉⠉⡟
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢤⣀⠀⠀⠀⠀⠉⠉⠉⠉⣉⣉⣩⣭⡿⠿⠟⠛⠛⠛⠛⠛⠒⠶⠶⠾⠿⠛⠀⠀⠀⢸⠃
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⠞⠛⠉⠁⢀⡀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⡀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠴⠶⠶⠒⣛⣉⣉⣤⡤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣈⡳⣭⣑⠒⠀⠀⠀⠀⠒⠚⢉⣉⣉⣩⣭⣤⡶⠶⠖⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⣿⣷⣿⣿⡿⠿⠿⣿⣿⣿⣿⡆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⡿⠿⠇⣿⠀⠿⠿⣿⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣇⠛⠛⣿⠟⠛⢂⣿⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣇⠛⢀⣿⣿⣿⣿⡇⠀⠀⣿⣧⡴⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢃⣠⣶⠟⠉⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⣀⣠⣤⣤⣄⣀⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠈⠉⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⣄⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠚⠛⠛⠋⠙⠛⠛⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢼⡗⠶⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⠾⡇
⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⢧⡀⠘⢧⣀⡀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠙⢦⠀⡏⠉⠙⠛⠒⠒⠒⠒⠒⠚⠛⠉⢹⣠⠤⠶⠶⠶⠦⣄
⠀⠀⠀⣀⣧⣤⡤⢤⡀⠀⠀⠀⠈⢳⡏⠁⠒⠲⠤⠤⠤⠤⡄⠀⠀⠀⢸⡟⠒⠒⠀⠒⠒⢻⡇
⠀⢠⢞⡏⠀⠀⠈⢦⠱⡄⠀⠀⠀⠀⣧⠀⠀⠀⢐⢒⠆⠀⡇⠀⠀⠀⣸⠟⠓⠒⠒⠒⠋⠻⣤⣄⣀
⠀⡏⢸⠀⠀⣠⠤⠬⣇⢹⡀⠀⠀⠀⢹⠀⠀⠀⠘⣳⠆⠀⣷⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠙⠒⠤⣀
⠀⣧⠸⡄⢀⡧⢤⡞⣿⢸⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡏⠒⠦⠤⠤⠤⠤⠖⢻⠀⠀⠀⠀⢠⠞⠙⡆
⠀⠸⣆⢓⠁⣀⠞⢱⢇⡞⠀⠀⠀⠀⢸⠲⢤⣀⣀⠀⢀⣀⣼⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡔⠀⢀⠇⡠⠒⠳⣤
⠀⠀⠈⠳⠭⠿⠶⢿⡋⠀⠀⠀⠀⠀⢸⣄⠀⠀⠉⠉⠉⠁⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⢤⣇⡇⢸⠀⢧⣀⣰⡾⣤⣀
⠀⠀⠀⠀⠀⢀⣠⡤⠭⣗⠒⠶⠒⠛⠉⠉⠙⠲⠤⠤⠤⠤⠤⠴⠒⠉⢷⡢⢤⣀⣀⣀⣠⢴⠏⠀⠈⠉⠛⠛⠉⠈⠉⠀⠀⠉⠛⠲⢤⣀
⣠⠤⠤⡤⢴⣅⡀⣠⡤⠏⠀⠀⠀⠀⠀⠀⢠⣒⠲⠶⢦⡀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠚⠁⠀⠀⢠⠤⠶⠒⢤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠁
⠳⢤⣀⣁⣀⣀⡿⠀⠀⠀⠀⢠⠔⢒⠒⠲⣮⣑⣒⣒⣻⣡⣴⠚⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⢨⠀⢀⣸⠆
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢟⠢⠼⠤⢖⡿⠀⠉⣡⠋⠀⠈⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠳⣄⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠦⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠒⠦⡀⠀⠀⠀⠀⠀⣠⠔⢲⢰⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⡆⡇⠀⠀⠀⠀⠀⣼⠕⠂⢀⠀⢠⠂⠀⠈⢆⠀⠀⠀⡜⠁⠀⢸⢸⠴⣚⠥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⢸⠀⡇⡇⠀⠀⠀⠀⡸⠁⠀⢠⢇⡴⠃⠲⡀⠀⠘⡄⠀⠀⡇⠀⠀⢸⠔⠊⠁⢀⡇
⠀⠀⠀⠀⠀⠀⡞⢳⢸⠀⢸⠀⡇⡇⠀⠀⠀⢀⠇⢀⡴⣷⢍⠀⠀⡠⣈⢦⠀⠱⡄⠀⢹⠉⡣⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⢀⣧⣸⣼⣠⣼⣄⣧⣇⣤⡀⠀⢺⠊⣏⠀⠐⠆⠀⠀⠰⠀⠀⣳⢄⣹⢀⡼⠃⠯⡖⠒⠊⠁⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⣷⡀⠀⣐⣂⠀⢀⠟⡏⢀⡞⠈⠉⠉⠉⠁⠀⡤⡄⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⢹⢦⡀⠀⠀⠀⠹⣄⣇⠙⡦⢍⡡⡴⠋⢠⢃⡟⠀⠀⠀⠀⠀⢀⡴⠋⠙⢢⡴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⠭⠬⠉⡇⠀⠀⠀⠀⠈⢁⣹⡇⠀⠀⣸⣾⡓⠋⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡴⠚⠉⠓⡇⠀⣛⣛⠀⡇⠀⢀⠔⢺⠉⠉⢣⡈⢲⡞⢁⠇⠈⠙⡶⠦⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠰⡁⠀⠀⡀⠳⠤⢤⠤⠄⠇⠀⠆⠀⣸⠀⠀⢘⠑⢻⡟⠋⠃⠀⠀⡇⠀⢹⠀⠀⢸⠖⠖⠖⠒⠲⠚⠳⠲⡇
⠀⠀⠀⢀⠔⠁⠀⢄⣇⡠⠀⠘⢦⠀⠀⢰⠀⠀⢻⠀⠀⠘⠖⠑⣞⠒⠂⠀⠀⡷⠀⠈⡄⠀⢸⠀⠀⠰⣉⣱⠀⠀⠀⡇
⠀⠀⠀⠸⣄⠀⠀⠀⠉⠀⠀⠀⢠⠇⠀⢸⠀⠀⣸⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⣧⠀⠀⡇⠀⢸⠀⠤⠤⠤⠬⠤⠄⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣇⣀⡀⠏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡸⠀⠀⢳⠀⢸⣀⣉⣉⣉⣉⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠒⠬⣷⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⣟⡩⠭⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡞⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⢳⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣇⠀⣠⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢼⣄
⠀⠀⠀⠀⠀⠀⠤⢤⡴⣫⡿⠓⠒⢾⢶⡥⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣳⣯⣻⣦⡤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠺⠛⡏⠀⠀⣀⠜⠓⠚⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⠀⣼⠉⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡄⢸⣿⠋⡀⣿⢴⡀⡀⠐⠅⠀⠀⠀⠀⠀⠀⢻⢶⣾⣦⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢉⠻⣼⣿⣸⡇⢸⣧⢱⠁⢀⣓⠀⠀⠀⠀⠀⠀⠀⠱⡴⢹⣽⣿⣿⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡧⣹⡏⣴⣶⣬⣥⡥⠞⠉⣄⣁⣈⠐⠂⠀⠀⠀⠹⠷⠿⢿⢾⣿⣿⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠛⢷⢻⢸⡿⠛⢻⢦⣖⣿⣿⣹⣉⣁⠨⠄⠠⠙⢛⠿⠛⠿⠟⢻⡼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⣰⠰⣾⣟⠸⢬⢳⣄⣤⡾⠋⠉⠀⠹⠄⠋⠉⠉⠱⠒⠃⢲⠚⢁⡠⢛⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⠾⠛⠮⣭⡅⠌⢘⣷⣿⡙⠀⠀⢀⣴⣾⣷⣦⣤⣤⣀⣨⡤⠤⠴⢿⣇⠀⢡⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡖⢻⡇⠐⠒⠀⢘⣵⠟⠛⠉⣹⣭⡔⠈⠉⠈⠙⢿⠹⣿⢻⡁⠀⠀⠀⠀⣨⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⢸⡷⠘⣿⢙⣝⢉⣉⡿⢄⠀⢸⣟⠀⠀⢰⡆⠀⠀⠘⣇⠠⠋⣀⣀⣄⣤⡞⠁⠈⠓⠅
⠀⠀⠀⠀⠀⠀⠀⠈⣶⠀⢹⢿⡋⠉⠉⠀⠀⢰⡿⣿⡄⣶⣼⡇⠀⠀⠀⣿⠶⠚⠋⠉⠁⠉⠹⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠹⠧⠾⠋⠢⠉⠁⠀⠒⠉⠀⣿⠇⢠⠿⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⣠⠨⢰⠇⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⣃⠿⡀⠈⢶⠶⢀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⣼⠇⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠉

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