⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠂⠁⠀⠈⠉⠀⡉⠉⠦⠠⣤⡀⠀⠀⠀⠀⠀⡠⠤⠒⡁⡭⠦
⠀⠀⠀⠀⢀⠠⠀⠒⠤⠄⠂⠁⠀⠀⠁⢆⠀⡀⡀⠠⠀⠂⠠⡀⠂⠀⠀⠈⠢⣄⠌⠈⡑⢀⡀⠈⠀⡀⠠⠠⠀⠤⢤⠀⠈⠋⠁⠀⠀⠀
⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠉⠑⢆⠀⠀⠀⠀
⠀⠀⠀⠑⢄⠀⠀⠀⠀⢀⣀⢀⠀⣀⣀⣀⢠⢀⣀⣀⠤⢀⠄⠁⠀⠀⠀⠀⠈⢠⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠋⠁⠉⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠨⠁⠗⠩⠙⠒⡛⠃⠒⠰⠐⠲⠢⠆⠗⠈⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡘⣄⡀⣖⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣠⣳⣄⢐⡉⣱⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡟⠃⠀⢫⢌⠃⢠⡄⠀⠀⠀⠀⢣⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡌⣩⠏⡀⠀⠂⠌⢆⣱⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠈⠭⠃⠀⠀⠀⠀⠀⢄⢽⡨⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠄⠑⠅⠀⠀⠀⠀⡀⠀⣀⡅⢦⣵⡰⡌⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣤⠏⠉⠁⠛⠉⠀⠀⠀⢹⠀⠀⠀⠀⠀⣦⢳⡼⠀⠂⠀⡀⡄⠘⣘⣄⡻⣇⡇⡸⣛⡏⠍⡏⡡⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢰⣓⠼⡾⣯⣷⠱⡳⡧⡥⣗⡗⡎⡧⢱⣒⣩⣇⣏⣬⣯⣷⡬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡄⠂⠒⡖⠉⠁⠀⠀⠈⠙⢙⣟⣝⣭⡿⢋⠂⠀⠈⠈⠀⠀⠀⠀⠀⠀⡤⠤⠤⡄⢫⣿⣾⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠅⠀⠀⠀⠀⠀⠈⢁⢋⡯⠯⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⡇⠀⠈⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠨⠀⠀⠀⠀⠀⠀⢪⠾⡏⡉⠂⠀⠀⠀⠀⣴⣷⣶⢶⣶⣦⡠⠄⠀⠀⠀⢀⣐⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡁⠀⠀⠀⠀⠀⠀⠨⣿⢝⠪⡄⠀⠀⠀⠀⣿⡿⠂⠈⣏⣿⠩⠀⠀⠀⠀⠊⡄⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡁⠀⠀⠀⠀⠀⠀⠐⣭⡧⠀⢘⠀⠀⠀⠰⣹⣻⠙⡟⣷⡿⢒⠀⠀⠀⠀⢁⣇⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⡻⣞⢘⡖⣀⢀⢠⢀⣿⣿⣦⢳⣨⣿⣔⡥⢤⡤⣖⣦⣯⣿⡠⠄⠤⠄⠒⠦⠤⡀⣀⠀
⠔⠒⠉⠉⠒⠒⠐⠉⠏⠉⠁⠉⠈⠁⠉⠁⠈⠈⣷⢿⣿⠏⡽⠯⡏⣳⣿⣷⣿⣿⣿⣿⣿⣿⡛⠏⣫⠻⣿⣻⣇⠀⠀⠐⠒⠂⠒⠠⠄⠀
⠀⠀⠖⠒⠒⠂⢀⡀⠔⠈⠉⠉⠉⠒⠂⠂⠀⠀⠸⣷⣏⡗⠕⣌⣓⣷⡾⡿⣿⣷⣻⣿⡯⣿⣟⠝⡻⢏⡻⣹⢖⣶⠀⠀⠉⠐⠢⣄⡀⠀
⠀⠀⠀⠀⣀⣁⣀⡀⢐⣚⣡⣅⡁⠑⠢⠤⠀⠀⠀⠕⠭⡿⣿⣪⣭⣻⣽⡧⣧⣞⢿⡯⣢⣴⣦⣮⡿⡷⠟⣾⣾⣏⡇⠀⠠⠄⠄⠢⠁⠀
⠀⠀⠈⠁⠉⠉⠒⣀⠀⠀⠀⠀⠈⠙⠒⠉⠀⠀⠀⠀⠅⣐⡺⣇⠪⢁⣓⣭⣿⡵⢫⢳⡦⣶⣢⣲⡫⣹⣽⠯⣛⣴⣼⠂⠲⠈⠉⠁⠁⠃
⠀⠤⠔⠒⢊⣉⠤⠄⠬⠭⠵⠦⣀⣀⡀⡤⠤⠄⠀⠀⠀⢋⢿⢎⠷⣚⠷⢯⣓⡲⠲⠣⣍⣧⣽⣭⣮⣆⡔⣟⡿⢓⣭⣉⣄⠀⠒⠢⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠁⣝⣷⣷⡫⠎⣑⢖⢛⢁⢌⢄⣁⢉⣙⡋⠿⠵⡶⣿⠗⣳⡁⢠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣦⣦⢦⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣷⣭⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⠛⠛⠻⠛⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣶⣶⡄⢠⣶⣒⢦⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠉⠉⢈⡅⠀⠁⠉⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡦⣴⠿⠛⢻⢫⣶⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⣫⣄⣈⣍⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣿⣿⠿⠟⠉⢸⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣀⣤⣶⡺⢿⣳⣿⡰⠈⠄⠁⠀⢀⣚⣿⢌⣓⣦⣤⡀⡀⠀⠀⠀⠀⠀
⠀⠀⢀⣴⣾⣷⣭⣟⣿⣽⣿⣿⣿⣧⠐⠀⠀⢠⣟⡿⣟⡾⢭⢻⣞⣿⣶⡶⣄⠀⠀⠀
⠀⢠⣿⢿⢀⡅⢿⣻⡍⡇⢩⣯⡟⡫⠄⠀⠀⠱⠊⣿⣿⠈⠉⢽⣹⡇⢀⢩⣿⢇⠀⠀
⠀⡯⡇⣿⡏⣧⢾⣳⣇⣇⡌⣿⡽⢳⠄⠀⠀⢠⡜⣿⣿⠀⢤⢬⣧⣳⣨⣿⣏⠄⡆⠀
⢰⣯⣧⣿⣷⣟⣿⣿⣽⣾⣻⣿⣷⣻⣷⠀⠀⢸⣡⣿⣿⡽⣚⣍⣿⣿⣿⣧⣿⡼⣿⡀
⣾⢯⣿⣿⣿⠌⣏⣯⡔⠒⣭⡏⣿⡡⢾⠀⠀⠸⠙⣿⢹⣇⠉⢉⣻⣻⢍⣿⣿⡷⢿⡇
⣿⡆⢿⣿⣿⢚⡇⣷⣎⣱⢻⣧⡿⢧⣾⠀⠀⠰⡿⣼⣟⡯⢦⢥⣿⠿⢜⣿⣾⢣⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣍⠢⢖⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡰⠤⠬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⠈⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⡀⠀⠀⠀⠀⠀⣠⠤⡴⠒⡉⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠃⠁⡠⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⢀⣬⠔⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠙⢖⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠈⢧⡈⠭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢖⠢⢀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⢍⣲⠀⠀⣸⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠠⠐⠂⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⢀⡠⢴⡿⠾⡓⠦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⡉⠉⠉⢁⣦⣾⢧⡇⡇⠀⠀⠧⡆⢽⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⢠⣿⡝⡏⠀⠀⡇⠀⠀⠀⣳⣡⣸⡄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠻⣄⠀⣧⠀⣇⢰⣿⣿⡀⣷⠀⠀⠀⠀⠈⠀⠐⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣦⠀⠈⢣⡽⡗⡻⠿⣙⣾⣴⣻⡐⠒⠤⣀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⢸⡧⢤⣟⠁⣠⣿⣿⣄⠈⣿⡟⡿⡣⡀⠀⢀⣱⠒⠋⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠸⢸⣇⠀⠋⠙⡏⡟⢧⠹⢏⡉⠀⡷⡷⠌⠋⠀⣀⠜⠢⢄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠹⣏⡇⠀⠀⡇⣧⡸⡷⢄⡈⠑⠲⢧⣀⡠⠞⠁⠀⠀⠀⠁⡲⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣯⡇⠀⠀⡇⠙⣆⠙⢆⠈⠉⠓⢢⡑⡀⠀⠀⠀⠀⡠⠊⠀⠀⡌⠀⠀⣀⠬⠓⠊⠉⠉⢉⡕⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣏⠩⡉⠙⣞⠀⠀⠸⣄⠈⢦⠈⠳⡀⠀⠀⠳⠌⠓⠢⢤⣜⣀⣀⣀⣪⠤⠶⠭⠥⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢸⠙⣆⠐⡄⠘⢦⡀⠀⠈⠳⡄⠳⡀⠀⠀⠐⠒⠦⣄⣀⠀⠉⠀⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⠈⢑⢆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡜⠀⢫⣣⡈⢆⠀⠹⣄⠀⠀⣇⣄⢳⠀⢀⣀⠀⠀⠀⠀⠉⠙⠓⠒⢤⣤⠬⠿⢏⡉⠀⠀⠀⠉⠳⣧⠣⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡁⠀⠀⠳⣍⠚⠂⠀⠸⣧⠀⠈⢦⡉⠀⠀⠀⠉⠉⠑⠲⠶⠶⠶⠾⠷⠶⠤⣄⡀⠈⠀⠂⠀⠀⠀⠹⡄⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠇⠀⡟⡇⠀⢰⠀⠈⠓⠲⡤⣅⡛⣧⡀⠀⠙⣄⠀⠢⡀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⣅⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣎⠁⠀⠘⠀⢡⠀⠘⡀⠀⠀⠀⠀⠀⠀⠊⢾⣄⠀⠈⠑⢦⡈⢢⡀⠀⠘⢆⠀⢹⡷⣦⡀⠀⠀⠀⠀⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆⡤⢄⣸⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⡦⣄⡀⠙⠳⣿⣦⠀⢬⢳⡈⣷⠈⠙⠒⢄⡀⠀⠉⢿⡿⡷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠏⠉⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⣷⠒⠤⠽⣻⣿⣻⠿⠮⣧⠀⠀⠀⠉⠢⣀⠈⣷⠹⣿⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠀⠀⠐⢦⡀⠤⣄⣀⡠⠤⠤⠤⠠⠤⠖⠓⡞⠉⠒⠢⠄⢀⠋⠁⠀⠀⢸⡳⠀⠀⠀⠀⠘⢦⡌⡆⢻⣷⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⡆⠀⢀⠃⠀⠀⠀⠀⣷⠤⢤⡤⠤⠤⠤⠤⠤⠴⣦⠞⠉⠀⠀⠀⠤⠇⠀⠀⢀⣴⣿⢷⡑⠀⠀⠀⠀⠈⢳⡇⠠⢹⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⣻⠀⠀⠀⢠⢺⠏⠀⢨⡇⠀⠀⠀⠀⠀⢠⠏⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠙⡏⠀⠙⠅⠀⠀⠀⠀⡸⠀⠖⠀⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢨⠃⠀⠀⢀⢦⡇⠀⠀⢸⠃⠀⠀⡆⠀⢰⠈⠒⢲⢹⠁⠀⠀⠒⠠⠘⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠑⠅⠀⠰⡸⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡜⠀⠀⢀⡞⢸⠁⠀⠀⣸⠀⠀⠀⡇⠀⠀⡆⠀⠈⡎⢄⣀⣀⡠⣤⣀⡤⡄⠴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⢠⣿⢀⠇⠀⠀⠀⡇⠀⠀⠀⣧⠀⠀⢹⠀⠀⢱⢰⡝⠁⠀⣽⠏⢿⠀⠐⠈⠣⡀⠈⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢠⠇⢀⣔⠉⢸⠉⠰⣄⣀⡴⡇⠀⠀⢀⣿⡀⠀⢸⡄⠀⢸⣶⡇⠀⠐⠈⠀⠀⢀⠀⢀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠃⠀⡎⠀⡴⠃⠤⠼⠀⠀⠀⠀⠀⠛⠶⠦⠾⠱⠑⠶⣫⠥⠞⠉⣿⡁⢀⠀⠀⠀⠀⡈⡀⠴⠯⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠒⠒⠒⠒⠒⠒⢶⠒⠂⣀⠌⠀⠀⠀⠀⠀⢘⣃⠞⠑⠢⠤⠀⠤⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠀⠈⠂⠀⣀⠀⠀⠀⠀⣰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠉⠙⠋⠫⠔⠒⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄⢀⠴⠁⠀⠑⠤⠴⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠒⠂⠀⠀⠰⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⢀⠤⠰⢒⡶⠒⠶⠖⡒⡖⠲⠒⠒⠒⠒⠒⠒⠒⠒⠳⠶⠶⢶⣶⡶⠶⡶⠶⣖⠖⠶⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣻⣧⣄⡛⠺⡿⡿⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣽⣤⠸⠿⣼⡃⣠⡿⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣬⣰⣾⣿⡟⢷⣏⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢳⣄⠀⢠⡄⢀⡠⣀⣀⢀⠤⣄⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢡⣾⣿⡂⠀⠀⠀⠀⠀⠀
⠐⠄⢢⡀⠀⣄⢀⣯⡟⠑⢆⠉⢻⣛⣧⢯⣀⡬⣯⣿⣢⡈⡿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⡋⠉⣗⣺⣤⣬⡏⢠⣷⠄⠀⠀⠀⠀⠀
⣶⣾⠆⠃⣠⣻⣟⡛⢱⠗⢺⣤⡼⣿⢻⣬⣳⣝⣿⣧⣀⡑⣤⣾⣄⠀⠀⠀⠀⣤⠤⢤⣀⠒⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⡿⢟⣠⣿⡏⠛⣯⣀⣬⢏⠀⡤⠠⣄⢀⣀
⣿⠻⣤⣄⢳⣿⠛⢿⢠⠔⠞⠉⠛⠋⠈⠈⠉⠁⠀⠀⠘⣳⡉⣠⠛⣀⠀⠀⢀⣸⢀⢬⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣙⡿⠛⠋⠀⠀⠀⠀⠸⣟⠛⠿⠞⢦⠀⢊⣤⡴
⠟⢵⣾⣲⡿⠯⡱⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⢀⣙⠋⣸⠂⠀⠐⠦⡌⣁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠶⡇⠈⢡⠀
⠈⠀⢀⣈⣑⡄⠰⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⡋⢀⠱⠒⠢⠚⡉⠑⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢡⠶⢠⠀
⠀⠀⠀⠀⢁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⠡⢠⢿⠛⢲⠂⢨⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⠶⠊⠀