⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡟⠿⣿⠿⣟⢛⡶⠶⠶⠶⠶⢆⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣸⣛⣟⣷⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⢦⠙⠶⣬⠓⢌⠃⠿⣅⠩⢈⡉⠡⠌⣉⠉⡍⠙⡓⢋⠛⡛⢛⠩⢉⠡⡉⠍⡛⢄⣼⣻⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢍⠡⢯⠙⣿⡐⠤⢁⠊⢌⠰⡈⢷⡠⠌⡡⠒⡠⢡⠘⡠⢁⠎⡐⠡⢌⠂⡅⠢⢁⠎⡐⠬⠖⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⣾⠏⢂⠅⡊⠷⣌⠙⡄⢃⠌⠂⢆⠘⣀⠳⢦⣁⠃⡔⢡⣈⣔⣁⣆⣡⣃⣌⡰⠈⢆⠡⠒⡈⠴⣋⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡵⡀⢀⢼⠏⠡⢌⠂⡌⠰⣈⣿⣷⣌⡰⢈⠃⠆⡡⢂⠔⡁⢊⠛⠿⣿⣟⠉⠉⠉⠉⠉⠉⠛⠻⠷⣺⣤⣵⡺⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣱⠿⢉⠢⡑⢠⠊⢄⠡⣼⣿⣻⢿⣿⢿⣽⣦⣥⠨⡐⢌⠄⢃⠌⣈⠙⠫⣗⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⣼⣿⣿⣿⣯⣴⣦⣡⡐⠡⢌⠂⣾⣿⣿⠏⠁⡈⢉⠛⢿⣷⣿⣔⢨⠘⡠⠌⡐⠨⢡⠀⡍⢨⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿⡿⠃⢀⠂⡐⢀⠂⠄⡙⠷⣲⣿⣶⡅⡘⠄⣃⠂⡅⢂⠡⢂⠆⢊⠲⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣻⣿⣿⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠐⢀⢂⠐⠠⡈⠐⡀⢂⠙⢿⢮⠺⣦⢁⢂⠒⣈⠂⡑⢂⠌⢂⠌⡰⢳⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡠⠌⡐⠢⠌⢻⣿⣿⣿⣿⣿⡆⠐⡈⠄⢸⣦⠅⠀⡁⡐⠠⠈⣟⡿⠇⠈⠳⣌⠒⡠⠑⠌⢢⠘⢄⠊⡄⣹⣾⣦⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠤⠑⡨⢐⠡⡘⢸⣿⣿⣿⣿⣿⣄⠡⠐⡈⢙⣛⠛⠃⡀⠄⡁⢄⣿⠁⠀⠀⠀⠘⢮⣡⢉⠰⠁⡌⢂⡑⣤⣿⣿⣧⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠍⠇⡑⠄⢃⠤⢡⣿⣿⢾⣿⣿⣟⣿⣷⣴⣧⣾⣿⢠⣵⢀⣢⣶⣻⠏⠀⠀⠀⠀⣠⠟⡙⠍⠳⣁⠢⢁⠾⣿⣿⣾⠏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢡⡌⢂⡅⡊⣴⢠⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣽⣿⣿⣿⣿⣿⣽⠁⠀⠀⠀⢠⣮⡐⣤⢁⢊⢡⣴⢂⢡⣺⣿⣯⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢱⢋⡘⠎⡸⢁⡘⣿⣿⣿⣿⣿⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⡟⡰⢃⠸⠈⡞⢸⠏⠎⣃⣿⣟⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢐⠡⢂⠔⡨⢐⠂⡜⢿⣿⣿⡿⣟⣾⣯⢿⣾⣳⡿⣽⣯⣿⣿⠟⠁⠀⠀⠀⠀⢀⡞⢀⢃⠢⢌⠡⡐⢡⠊⣰⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠰⡈⠔⡁⠆⢡⢂⠱⢐⡘⣿⣿⡽⠻⣷⣿⣻⣽⣿⣿⣟⣾⣽⣞⡀⠀⣀⣠⣴⣶⣾⣴⣮⢀⠒⡈⢔⣈⢂⣼⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢬⠃⠜⡐⢨⠘⡠⢂⠜⢠⠐⣿⣿⣿⢀⡉⠻⣯⣷⣿⣿⡾⠛⡉⠬⣧⣾⣿⣿⣻⣽⣟⡿⠋⢄⣱⠜⣋⣴⣿⡿⡯⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡚⣈⠒⡈⠔⡨⠄⡡⢘⠠⠚⡟⢡⠘⢢⠈⠅⡄⣉⣯⠁⠤⠑⡨⢐⡈⢿⣷⣯⣿⠾⠛⡠⢑⣞⣥⣾⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠐⡠⠑⡈⠆⢡⠘⢄⠡⠊⠔⡨⠐⣌⢂⠩⡐⠰⢀⠢⢘⠠⠃⡔⠡⡐⠢⢈⠉⡄⠢⢑⢠⢟⣿⡿⣯⣿⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠠⡑⢌⡐⣉⠰⣈⠂⠥⣉⠒⢠⠉⣿⠀⢆⠁⢎⡀⢃⠢⢁⠎⢄⠱⣀⠣⠌⢒⡈⡑⠨⣁⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⠐⡐⢂⠔⡠⠒⢠⠑⢂⠔⡈⠆⠒⣸⡘⠄⡊⢄⡶⢁⠢⢁⣾⢂⠢⢐⠢⠑⡂⠔⡡⢡⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣼⣿⣷⠇⠂⡅⠊⡔⢠⠉⡄⠣⣈⣴⡈⠔⢡⣸⡧⠘⣀⣺⢃⠂⡅⡾⢃⠰⠈⠆⢌⠡⡘⠤⢁⣾⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣯⡟⢀⠣⢐⠡⠌⡄⢣⠐⡡⠄⣿⣿⣾⣿⡏⡐⠡⢤⡏⢄⠊⣼⡇⠌⢂⠩⠌⢂⠂⡅⢢⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⡇⢂⠅⡊⠰⡁⠒⡄⠃⡔⡈⣿⣷⣯⠏⡐⠄⢃⡿⢠⠁⠎⣿⣿⣶⡿⠋⡜⢀⠣⣐⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣷⡿⣷⠈⡔⡈⠅⢢⠑⢨⠡⢐⠰⠸⣿⠏⡐⠌⣘⣼⠇⢂⡉⣆⠿⠿⠋⢄⠱⡀⠎⢰⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣾⣿⣿⡇⢰⢈⡘⡄⣉⢢⢘⡀⢃⡆⣉⠒⡈⣔⣼⡟⢨⠐⡄⢻⣀⠃⡜⣠⠒⣠⣩⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⡰⣿⣿⢿⣇⠃⡎⢱⠘⠹⠆⡋⢇⠋⠞⡉⢆⠳⣾⣿⠃⡎⡘⠱⠃⡏⠸⣀⠃⢞⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡏⣿⣿⣿⣻⣿⠐⡨⠄⣉⢂⡑⢨⠠⢉⠔⡈⢆⢸⣿⣿⠐⠤⡁⠅⣃⠰⢁⡤⣿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⣧⢻⣿⣿⣿⣦⣡⡘⡀⠆⠰⡁⢢⠁⢆⠡⠂⣿⣿⣽⡌⠰⢈⠒⡀⠎⡠⣸⣟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣹⣷⢫⣿⢮⡝⣿⡿⢿⢿⡿⣶⣶⣧⣮⣴⣬⣵⣿⣿⣾⣿⣶⣧⣾⣶⣷⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣤⣿⢧⣻⣿⢳⣚⣿⣏⢯⢮⠵⣿⡟⡼⡭⣏⡟⣿⣟⢿⡻⢿⣿⢿⣿⣿⣿⣿⣿⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡴⣿⣿⣿⣣⢿⣟⢧⣻⡿⢮⡝⣮⢻⡿⣽⣛⣟⡳⣿⣿⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣟⣿⣿⣿⣿⣿⣿⣮⣽⣟⢧⡻⣴⣻⣟⡖⣳⣜⣳⣽⣯⢏⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀