⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢿⣮⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣗⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⠛⠋⠛⣿⢻⣿⣿⣿⣿⢹⡟⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣧⠆⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⠃⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢲⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠹⠿⡇⠄⡐⠀⡴⠞⠛⠗⠀⡁⠀⠀⠀⠀⠘⠟⠳⢦⡄⠀⠀⢻⡿⠝⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡿⠀⠑⠢⠔⠒⢶⡶⢆⠀⠀⠀⠄⣴⢦⣶⣆⣀⡄⠀⠀⠀⡆⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⢦⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠚⠀⠀⠀⠰⠀⠀⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⡄⢈⢦⠑⡀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣶⣤⡠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠒⠥⠮⠐⠊⣧⠊⠌⣄⠀⠀⠀⡔⢹⣿⣿⣿⣿⣿⣿⠰⣄⠀⠀⠀⡀⠌⠓⡀⢀⣀⡠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣫⠐⠌⢄⡐⠜⠀⠀⠉⠙⠛⠏⠉⠀⠀⠈⠆⠀⠆⠁⢂⠄⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣧⢘⡂⠈⠐⠓⢶⡶⠴⠦⠴⠶⡶⠒⠀⠀⢠⠊⢐⣌⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣶⡈⡀⠀⠀⠉⢑⡐⠚⠉⠀⠀⠈⣠⠗⣡⡾⢾⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⠷⣅⡀⠀⠀⠈⠁⠀⠀⠀⠀⡤⢛⠔⡫⡡⢘⣿⢿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢫⢷⡙⡶⣤⡀⡀⠀⠀⣀⢠⠞⡐⡡⢊⠔⠀⣼⢯⡿⣽⣳⣦⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⡧⠀⠑⢹⠔⣩⠙⡞⠳⠖⠉⠂⡀⠨⠈⠀⠀⣰⣟⣯⣟⣷⣻⢾⡽⣿⣽⣳⡴⣄⡀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⢿⣾⣯⣿⠁⠀⠀⠀⠘⠤⣛⣬⠕⠈⠂⠈⠀⠀⠀⠀⢠⡿⣞⣷⣻⢾⡽⣯⢿⣽⣿⣽⢿⣻⣽⣻⢶⣧⣤
⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣾⣿⣿⣽⣻⠀⠀⠀⠀⢠⡆⠘⢁⡂⠀⠀⠀⠀⠀⠀⢀⣾⣻⡽⣾⡽⣯⢿⡿⣿⣻⣟⣾⣻⢯⣷⢯⣟⣾⣳
⣀⣴⣿⣿⡿⣿⣟⣿⣿⢿⣿⣿⣻⣽⣛⠀⠀⠀⠀⠻⣷⠥⡤⣻⡧⠀⠀⠀⠀⠀⣼⣳⢯⣟⣷⢻⢟⣏⢽⣳⣟⣾⣳⢯⣟⡾⣯⣟⡾⣽
⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⡿⣿⣿⢯⡇⠀⠀⠀⠀⠀⣸⢲⢻⠀⠀⠀⠀⠀⠀⣨⡷⣯⣟⡾⣽⢦⣤⣤⣾⣳⣿⣳⢯⣟⡾⣽⣳⡽⢯⣷
⣿⣿⣿⣿⣻⣿⣿⣽⣟⣯⣿⣿⣿⣟⡧⠀⠀⠀⠀⢰⠇⣍⡣⡀⠀⠀⠀⠀⢀⡿⣽⣳⢯⡿⣽⢯⡿⣽⣾⣟⡷⣯⣟⡾⣽⣳⢯⣞⣿⣾
⣿⣿⣿⢿⣟⣿⢿⡷⣯⣟⣷⣿⣿⢾⡳⠀⠀⠀⢠⣟⣃⡃⢈⠠⠀⠀⠀⠀⣿⣽⣳⢯⡿⣽⢯⡿⣽⣷⣻⠾⣽⣳⢯⣽⡳⢯⣛⡾⣽⡾
⣿⣿⡿⣿⣻⡿⣿⣿⣳⢯⣿⣿⣿⢿⡳⠀⠀⠀⣻⡧⠦⢌⠐⠍⠀⠀⠀⣼⣳⢯⣟⣯⢿⡽⣳⢿⣽⠾⣽⣻⣳⢯⣻⢶⣻⢯⢿⣽⢯⣿
⣿⣿⣿⣟⣿⣟⣿⣷⣻⢯⣷⣿⣿⣯⢷⠀⠀⢜⡇⢲⢘⠀⠀⠍⠀⠀⣸⢷⣯⣟⡾⣭⣷⣻⣽⡿⣽⣻⣳⣏⡷⣯⠷⣯⣳⢯⣿⣾⣟⢾
⣿⣿⣟⣿⡿⣼⡿⣼⣿⢿⣼⣿⣧⣟⣿⠀⠀⡸⣇⠘⠀⠀⠀⡣⠀⢠⣟⡿⣼⣛⡿⣧⡟⣧⣟⣿⣣⢧⣧⢟⣻⡼⣿⢧⣻⢿⣼⣿⡼⣟
⣿⣿⣿⢯⣿⣿⣽⣟⣯⡿⣞⣿⣿⣻⢾⠀⠸⢖⠩⠀⠀⠄⠀⠁⢀⡾⡽⣯⢷⡯⢿⣵⣻⣟⡾⢧⣟⣻⡼⣯⢷⣛⣧⡿⣽⢯⣿⣿⡽⣫
⣿⣿⡿⣿⣻⣾⢷⣯⢿⣽⢯⣿⣟⣯⣟⠀⣄⠽⠂⡆⢃⠀⢀⠀⣰⡿⣽⢯⣷⢻⣟⣾⣳⢯⣽⡻⣞⣳⣽⣳⢯⣟⡾⣿⣭⢿⣿⣿⣳⣟
⣿⣿⣿⣿⣽⣻⣟⡾⣯⢿⣯⣿⣿⣟⡾⡴⢒⡚⡇⠇⢰⢠⠐⢨⡿⣽⣳⡟⣞⣿⣞⡷⣯⣟⡶⡿⣝⣯⠷⣯⣟⡾⣽⣳⢯⣿⣿⣿⡷⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢶⣶⣷⣖⣶⣢⣤⣲⣷⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠛⢿⣧⠶⠛⠉⠉⠉⠛⠻⢾⣽⣿⣿⢧⡒⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣺⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠳⠛⢶⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⣾⣻⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⣿⣿⣿⣷⡷⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣴⣿⣿⣿⡿⡽⣿⣿⣾⣾⠋⢀⣠⣄⣤⣄⣄⠀⠰⠤⠤⠤⣄⣀⠀⣹⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣄⣺⣟⣿⣿⣿⣿⣽⡿⣿⣾⣾⡦⠀⠈⢩⣼⣾⠿⣿⣿⣶⣄⣉⠉⣳⠋⠁⣀⣧⣄⡀⠀
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠐⠒⢦⡀⠸⠽⣷⣿⠟⠻⢿⣿⣿⣿⡓⠀⠄⣀⡟⠃⠀⣰⡶⡾⢿⣟⡿⣿⠻⣿⣾⣭⡌⡏⠀⠀
⠀⠀⠀⠀⠀⣰⠃⢠⠆⠀⠀⠀⢳⠀⠀⡼⢡⠞⠲⣦⠋⣿⠟⢀⣈⠀⠮⡭⣷⣿⣯⠼⠿⠧⢿⠀⡦⠀⠘⢵⣧⢹⠇⠀⠀
⠀⠀⠀⠀⣰⠃⠀⡞⠀⠀⠀⠀⠈⡇⠀⣧⠸⡞⢹⡏⣇⣿⠀⠃⠱⠈⠋⠁⠁⠐⠛⠐⢲⠖⠋⠀⠀⠀⠀⠀⠙⢾⠀⠀⠀
⠀⠀⠀⢠⠇⠀⢠⠃⠀⠀⠀⠀⠀⢹⡀⣿⡀⠉⠘⠳⢰⣿⣂⠜⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀
⠀⠀⠀⢼⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢧⢽⠹⡆⠾⠃⣼⣿⣿⡟⢿⣇⣾⡴⠶⢀⢆⡀⡀⢀⣴⠖⠒⠒⣄⠀⡀⠀⠀⢳⠀
⠀⠀⢠⠇⠓⠂⢻⠀⠀⠀⠀⠀⠀⠀⢸⣸⡆⢣⣀⢸⣾⣿⣽⣿⣿⣿⣝⣿⡴⢋⣩⣥⣾⣝⡇⣤⣤⣀⠈⠃⠚⠀⠀⢸⡄
⠀⠀⡜⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⡅⠘⣯⣿⣿⣿⣿⣿⣽⣾⣷⣿⣿⣿⣿⣿⣿⣿⣽⣦⡾⣬⣵⣆⣀⠐⠄⠈⡇
⠀⠀⣇⠀⠀⠈⡟⡇⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢹⢻⣽⣻⣻⣿⣿⢽⡿⣏⢻⡭⢭⣋⠛⠋⠉⠙⠃⠙⣿⢖⠉⠻⠤⠴⠃
⠀⠀⣇⠀⠀⠀⢃⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⣟⣯⣷⣿⣷⣿⣿⣿⣟⢆⠳⣾⣩⣏⡷⠺⣾⣚⡞⠋⠁⠀⠀⠀⠀⠀
⠀⢠⠋⠣⠀⠀⢸⢇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢠⣾⡽⣿⡿⣿⣿⣿⣾⣿⣫⣧⡈⠳⠷⢷⣖⣿⢻⣥⠀⠀⠀⠀⠀⠀⠀
⠀⣼⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡟⡨⣿⣿⣷⣷⣟⡿⣾⣾⣿⣿⣺⣦⣄⣀⣀⣤⣾⣿⣇⣤⣤⠀⠀⠀⠀
⢸⠙⡄⠀⠀⠀⠀⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣳⣹⣿⣷⣼⣯⣿⣟⣿⣿⣿⣿⣾⣿⣿⣻⣿⠿⠻⠉⢉⡜⠀⠀⠀⠀
⣈⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇⠻⣹⣿⣿⣏⣿⣿⣿⣿⣿⣿⡟⣿⡿⠃⠀⢠⡞⠙⢳⠖⠢⡄⠀
⠇⠀⠀⠀⠀⠀⠀⠈⡯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⡽⡵⢻⢽⣿⣷⢿⣾⣿⣿⣾⢟⠅⠀⠀⠉⠹⣏⠏⣀⠞⠓⡄
⠘⠀⠀⠀⠀⠀⠀⠀⠸⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡯⡁⠹⣿⢈⣫⣯⠿⠛⠙⢿⡑⠀⠀⠀⠀⠀⠀⣠⠀⢁⣀⠖⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⣰⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠚⠲⠒⢹⡃⢀⣁⠠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡇⠀⣼⠿⡾⡽⡿⢿⡿⠚⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢠⡷⠋⠵⠷⠦⣇⣨⣥⣼⠳⠔⠃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠇⠀⠀⣀⠔⠊⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣀⣠⠤⠖⣫⢖⣁⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠳⠶⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀