Pucker Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠶⠶⠶⠶⠾⠿⢿⣷⣿⡟⠓⣾⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣠⣴⣦⡴⠄⠀⠀⠀⠀⠈⠉⠳⢦⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠉⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠖⠳⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠘⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣒⠂⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢤⠀⠀⠠⣼⣅⡀
⢀⡀⠀⠀⠀⠀⠀⠀⠘⢥⣀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡅⢀⡇⢀⣀⠀⠀⠉
⠀⠉⠑⠂⠀⠀⠀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠸⣿⠆⠐⠋⠛⠲⣄
⠀⠙⠂⠀⠀⠀⠀⢣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⠀⠀⢀⣀⠀⡹
⠠⠖⠂⠀⠀⠀⠀⣿⣁⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⣁⡤⠀⠀⠈⠹⠄⠀⠈⠙⢧
⠀⠀⡴⠀⠀⠀⠀⠈⠉⠁⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠛⠓⠚⢿⣍⠀⠀⠀⢨⠆⠀⢠⠨⠖⠊⠐⠂⢀⣴⣯⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣦⠀⠀⠸⣆⡴⢁⡔⢀⡴⢒⣠⠉⢢⡈⠉⠿⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⣼⠃⠘⣧⠀⠀⣿⠀⠚⠶⠋⡴⠋⡀⠀⠈⢧⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠺⠀⣿⠀⠀⠘⢧⡀⠹⡄⠀⠀⠐⣧⢾⠀⠆⠀⠈⣧⠀⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢻⡀⠀⠀⣠⡿⣾⡛⢆⠀⠀⠀⠀⢳⡀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣄⠀⠀⢸⠀⠉⠒⠛⠉⠀⠀⠉⠙⢣⠀⠀⠀⢸⠇⠀⢠⡟⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢠⣸⡆⠀⠠⠄⠀⠐⠃⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠉⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡇⠀⠀⢸⠁⢀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠰⠀⠘⣧⠀⠄⠀⠀⠐⠀⣰⠇⠀⠀⠀⡇⠀⠀⢸⡄⣈⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠘⣧⠀⠀⢀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠖⠋⠘⢷⡀⠛⢀⡿⠀⠀⠀⠀⢀⡇⠀⠀⠀⣧⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⣤⠾⢳⣄⣼⠃⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠟⠁⠀⠀⢹⡏⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⠁⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⢀⡾⢿⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⣴⠟⠀⠘⢧⡀⠀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠠⠲⢤⠀⠀⠀⠘⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠉⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⡠
⠀⢀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠙⠋⠀⠀⠆⠀⠀⠀⠀⢠⢦
⢰⠋⠀⠀⠀⠀⠀⠙⠦⡀⠀⣠⠤⠤⡼⠁⠀⠀⠀⠀⠀⠀⠀⢆⣀⡤⠖⠫⢉⠵⠶⣤⢄⡀⠈⠂⠀⠀⠀⠈⣨
⢸⠀⠀⠀⠀⠀⢀⣠⣀⣐⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⣥⣤⣀⡰⠁⠀⠀⠈⢳⠈⠙⠂
⠈⣇⠀⠀⠀⠀⠈⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰⡇⠀⠀⠀⠉⠲⣤⣴⣒⠟
⠀⠘⢆⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⢿⠁⠀⠀⠀⠀⠀⠀⠀⢸⢄⡀
⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡴⢠⠇
⠀⢀⢀⡼⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⡴⠋
⠀⠫⣍⣀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠧⣀⣀⡤⠴⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⡶⠒⠘⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠂⠤⠆⠀⠀⠀⠀⠀⠀⢀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠁⠒⠀⢢⠤⡤⠤⢄⣀⠜
⠀⠀⠀⠀⠀⠀⣀⢠⣎⠀⠀⠀⠀⣤⣥⠊⠀⡠⠊
⠀⠀⠀⠀⠀⠘⠧⣄⣛⡉⠴⠶⠐⠷⠐⢒⢊⢱⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣴⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢋⣵⢞⣿⣟⡽⣟⣵⡞⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢛⡻⠟⢻⣿⣿⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⣿⠟⣵⣿⣵⣿⣿⣿⣾⡿⠋⣿⣧⡄⠀⠂⣾⣿⣿⡿⢻⣿⣿⡟⣰⠟⣠⣶⣼⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣣⣾⣿⣿⣿⣿⡟⣻⣿⣷⠀⣽⢋⣴⡆⣾⡿⠛⠉⢶⣿⠿⣟⣼⢋⣴⣿⣽⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⢛⣰⣿⣿⣿⣿⣿⣛⣎⣡⣿⣷⣆⣠⣾⠁⠘⠻⠿⠛⠛⢫⣶⣻⣿⣿⣿⣬⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡆⣽⣾⣿⣿⢿⣿⣿⣻⣾⣿⢟⣡⣿⣿⣿⣿⣿⣷⣾⣤⣤⣄⣠⣾⣏⣥⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⢻⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣫⣾⣿⣿⣽⣿⣿⣾⣿⣿⣿⣿⠟⣿⣽⠛⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣷⡅
⠀⠀⠀⠀⢀⣿⡇⠈⢹⣿⣿⣿⣿⣿⣿⣿⣨⣿⣿⣿⣿⣿⠋⢠⣿⣿⣿⣿⣿⢣⣾⠟⠁⠀⣿⣿⣿⣯⣿⣜⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢸⣿⠃⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣟⡿⠁⠀⠘⢿⣿⣿⣿⡿⠛⠁⠀⠀⠀⢹⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⢷⡄
⠀⠀⠀⠀⠈⣧⡀⣻⢸⣿⣿⣿⣿⠋⠉⠘⠛⠃⠉⠀⠀⠀⠀⠀⠸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢻⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣷⣆⢹⣆
⠀⠀⠀⠀⠀⠸⣿⣿⣾⣿⣿⣿⠃⣀⣤⣤⠶⠶⠒⠒⠒⠂⠠⠄⠀⠀⠀⠀⠀⠀⠠⠴⠶⠒⠛⠛⠛⠓⠚⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡆
⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⣿⠟⠈⠉⢀⣀⣠⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣦⣤⣤⣤⣄⡀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⣿⡀
⠀⠀⠀⠀⠀⣠⣿⡿⠉⢿⣿⡄⠀⠒⢻⣿⢿⣽⣿⡿⠉⠹⡇⠀⠀⡆⠀⠀⠈⠀⠀⠿⣿⣿⡿⠃⣠⡽⠇⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⢠⡿⠁
⠀⠀⠀⢀⣾⣿⣻⡅⣠⠸⣿⡇⠀⠀⠀⠉⠛⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠐⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡟⠁
⠀⠀⢀⣾⢏⢦⣿⠀⣿⣧⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⣿⣿⣿⠃⣸⠟
⠀⠀⣾⡟⡜⢿⢏⡜⣸⣿⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⡿⠉⣷⣼⠛⠋
⠀⠀⣿⡇⠀⠀⠈⣠⡟⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣉⣥⢌⣵⣶⠟⠁
⠀⠀⢻⣧⣄⠀⢠⡿⠀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢘⣽⣏⣿⣟⠁
⠀⠀⠀⢹⣿⠁⣾⠇⠀⢸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⡿⣿⡄
⠀⠀⠀⢸⣿⢠⣿⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠤⢤⣶⣿⣿⣿⣿⣯⣿⣿⠤⠖⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣏⢸⡇
⠀⠀⠀⢸⣿⣸⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠻⢿⣭⡿⠿⠷⠿⠿⠃⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⣹⣿⣾⡇
⠀⠀⠀⠀⠉⢻⣯⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣸⣿⣿⣿⣿⣿⣿⡿⠃⣿
⠀⠀⠀⠀⠀⢠⡋⠀⡌⣹⣿⣿⣿⣿⣿⣿⣿⣿⢫⣙⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⣿⣿⣿⣿⣿⣿⣛⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠉⢲⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣷⣏⠻⠷⠤⠤⠤⠤⠴⠶⠶⠿⠋⠀⠀⢰⣿⣿⠷⠟⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⣹⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋⡽⣏⢹⣶⣟⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠛⠳⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣿⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡿⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⠷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⠀⠀⣀⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣄⡀⠀⢀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠠⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⡧⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣤⡤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⢦⣄⠀⠀⢠⣍⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠙⢦⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡰⠃⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣄⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⣀⣤⣾⡁⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⣀⠹⡟⠢⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠒⠒⠲⣤⣤⠞⠋⠁⠀⠀⡿⠀⠀⠀⠀⣺⣷⣤⠖⠦⣀⡟⢦⡿
⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢻⣒⠒⠲⠦⣴⣏⣀⣀⣀⠀⢀⡼⠃⠀⠀⢀⡼⣧⣿⣥⣀⣀⠈⠁⠈⠁
⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠙⣛⡛⣧⡀⢀⣀⣩⠿⠋⠀⠀⠀⢠⠟⠋⢁⣀⣀⣀⠉⢳
⠹⣄⡀⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠃⠈⠛⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣉⣉⣉⣡⠞⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⣴⠓⢦⡀
⠀⠀⠙⠓⠶⠤⠬⠥⠤⠤⢶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡁⣹⠷⢶⣏⣈⣿⣴⠛⢻
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⡘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣈⡗⢻⡶⣿⠥⢿⣤⣞⣿⣿⠶⢤
⠀⠀⠀⠀⠀⠀⠀⣏⢀⡀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠠⡏⢉⡷⡿⠉⢿⣠⡾⣻⣅⣈⣯⡴⠞
⠀⠀⠀⠀⠀⠀⠀⢹⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⣠⡤⠿⣿⢳⠟⢻⣏⣹⢿⣇⣨⡯⣤⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⡖⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠰⠾⢭⣀⡀⠢⣿⣼⣷⣿⠉⢻⣧⡼⢯⡀⣿⢧⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣻⢿⡍⣻⠀⢨⡟⠿⢾⣝⣿⣏⡑⣾⣷⣦⣄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⠷⣳⠀⢸⠃⣀⡴⠚⠉⢸⣿⣦⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⢚⡏⢠⣿⠞⠁⠀⠀⢀⣸⣿⣿⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣃⢸⡟⢿⡁⠠⣴⣾⠛⠉⠉⠉⢹⣇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠈⣋⢳⣦⣀⠀⠀⠀⠀⠀⠀⠀⣴⡳⢿⡦⣭⣸⠃⢀⠈⠻⣅⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣯⡀⠄⣸⡟⠶⣄⠀⠀⠀⣼⡁⣥⡾⠀⢻⠉⠀⠘⢧⣤⣬⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣙⠶⢏⠻⣞⡈⣙⣦⣸⣥⡄⣼⠣⣖⠋⠀⠀⠀⠈⢷⡈⢐⣨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⡏⠀⠸⠦⡼⢷⣬⡍⡿⠟⣶⠏⣠⠞⠁⠀⠀⠀⠀⠀⠻⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢤⣾⠁⠀⠀⠀⠛⠶⠭⣷⣿⣠⢏⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠁⣾⠀⠀⠀⠀⣀⣀⡐⢶⣯⣟⣞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⠀⠀⠀⠀⠀⠀⠀⠉⢹⠃⠋⠀⠀⠀⠀⠀⣀⣀⣤⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⢤⢤⣄⣀⣀⣄⣤⣼⣤⣤⠤⢶⠶⠞⠛⠉⠍⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠰⠀⠀⠀⡇⠀⠀⠇⢀⡇⣠⣯⣼⠶⠞⢰⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠇⢸⠀⠖⢺⣚⣿⠟⣿⠛⠒⠗⠀⣃⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⢿⢻⣛⣻⠾⠦⠦⣼⣤⠶⠴⠷⠧⠴⠶⣾⣝⣽⠋⠉⣓⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡀⠀⠀⠀⠸⢅⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠒⠒⠒⠲⠤⢤⣀⡀⠀⠀⠣⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⡠⢄⠔⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠑⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢄⡀⣠⠖⠁⢀
⠀⣠⠔⠒⠒⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠇⠉⠀⠀⠀⢋⠣⠤⡀⠀⠀⣰
⡼⠁⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⣠⠤⠤⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⡷⡄⣀⠤⠖⠫⢍⡭⠶⢶⡤⣄⡀⠀⠃⠉⠀⠀⠮⡉
⣇⠀⠀⠀⠀⠀⢀⣀⣀⡀⢱⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢴⡧⣉⣀⣀⣀⢰⠃⠀⠀⠀⠈⢦⠈⠑⠦⣄
⢹⡄⠀⠀⠀⠀⠈⡇⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠤⢶⠚⠉⠀⠈⣼⠁⠀⠀⠈⠙⠢⣀⣠⡤⢒⡾⠀⠀⠀⠈⠑⢦
⠀⢳⡀⠀⠀⠀⠀⢱⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⣤⣾⡞⠋⠉⠀⠀⠀⠐⠐⢲⠈⣧
⠀⠀⠳⡄⠀⠀⠀⢸⡀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠹⣄⡠⠒⠁⠀⢉⡇⠀⠀⠀⠀⠀⠀⠒⠉⠀⣩⢹
⠀⠀⠀⢹⠀⠀⠀⢸⠃⠀⠀⠈⠓⢤⣀⠀⣠⠤⠤⠤⣇⠀⠀⠀⠀⠀⠀⠈⠳⠖⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢃⡾
⢀⠤⡤⠋⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⢹⠁⠴⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠛⠉
⠀⠑⠦⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡤⠴⠒⠉⠓⠤⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠊⠁⢀⡀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⣻⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⡰⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠒⠒⠤⠤⢤⢀⣀⣀⣀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡔⡡⠊⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⣎⠉⠹⠳⠤⢄⣀⣀⣽⣉⣁⣠⠴⢅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣤⣶⣶⣴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣾⣿⣿⠿⡿⠿⠻⡿⡜⠻⠿⣿⠿⠟⣿⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣭⣷⠖⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠉⣿⠻⢿⣿⢫⣿⡿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠾⢻⣿⣿⣁⠻⢿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⢾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⣿⡶⣄⡄⠀⠀⢀⣀⡀⣀⣀⣀⡀⠀⠉⠁⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣽⣽⠻⣿⠂⠁⣤⢀⡘⣻⡟⠻⡟⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣿⣣⣶⣇⠀⢀⠁⣿⣿⣿⣿⣶⣿⣷⣴⣿⣷⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡇⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⢿⠿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣼⣿⣿⣿⣦⡀⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣉⠀⢙⣻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠽⢿⣿⣿⣏⣉⡭⣿⣿⡿⢿⠿⠟⢿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣠⣬⣭⣴⣶⣶⣷⣶⣶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠟⢻⠿⠿⡿⢿⡿⠿⠿⠛⠋⣹⣻⣽⣿⣿⣿⣿⣿⣷⣿⣏
⠀⠀⠀⠀⠀⣼⡙⠉⠁⠀⠰⡀⠀⠀⠀⢀⣀⣀⣀⣄⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠠⠾⠃⠐⠀⠀⣸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠿⠛⠛⠉⢿⣿⣿⣿⣿⣷⣾⣷⣷⣮⣟⠛⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⢀⠈⠉⠋⠸⠿⡟⠙⢿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⣠⣬⣿⣏⣷⣦⣾⣿⣷⣾⣿⣿⣿⠿⣿⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡷⡄⠙⣻⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⣀⣈⣛⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⣾⣶⠶⠶⠶⢶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣴⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣁⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣥⡾⠿⠛⠉⠉⠉⠉⠉⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⠟⠛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠘⠀⠀⠀⠀⠀⠀⠲⠦⠤⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⢠⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣿⣥⣿⣯⣤⣶⠖
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣿⣿⡇⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⠀⣠⠞⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢄⡀⠘⣿⣟⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⣠⡞⠁⠀⠀⠀⠀⠀⣿⡄
⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣿⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠋⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⢹⣷
⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣷⠀⠀⠀⠀⠀⠀⣿⣿⠞⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣏⠆⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡧⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⢨⣿⡀⠀⠀⠀⠀⠀⠸⣿⠏⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⡄⠀⢠⣤⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⢻⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠸⣿⣧⠀⠘⣿⠉⢻⣿⣿⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠈⠳⠋⢰⣿⠀⠀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠀⠀⠀⠀⠀⠀⢴⣶⡆
⠀⠈⣿⣿⣿⡄⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⣿⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⡀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⡿⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠙⢦⡀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⣿⡇
⠀⠀⢸⣿⡇⠀⠀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢆⣰⠏⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⢸⣷
⠀⠀⢸⣿⡇⠀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠻⣄⣸⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⢸⣿
⠀⠀⢸⣿⠃⣴⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⢸⣿
⠀⠀⢸⣿⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣼⡟
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣆⠀⠀⠈⠛⠷⢶⣤⣀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠇
⠀⠀⠀⠻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⢈⡑⠢⣄⠀⠀⠉⠛⠿⣶⣤⣀⡀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋
⣤⣤⣤⣤⣼⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣷⣮⣷⣤⣵⣶⣤⣤⣤⣤⣭⣿⣿⣷⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠛⠋⠉⠉⠛⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠛⠛⠛⠛⠿⢿⣿⣦⣀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠛⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠛⣿⠉⣉⣛⣿⣷⣶⣤⣀⠀⠀⠀⣴⣿⡃
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⣵⣶⠶⠿⠛⠻⣿⣏⠁⠀⢉⣿⠀⠀⠀⠘⠻⣿⣦⡀
⠀⣀⣤⣶⣶⣶⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⠾⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢙⣷⣶⡿⠃⠀⠀⠀⠀⠀⠈⢻⣷⡀
⣼⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶⣾⣿⣿⣿⣿⣭⣍⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀
⠹⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠈⠙⠻⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉⠁

⣇⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣛⣾⣛⣷⠿⠘⠃⠙⡑⠟⠡⡶⢛⡛⣹⢇
⡀⢀⠀⢀⡐⠳⣄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣿⡏⠁⠀⠀⠀⠀⠈⢀⢴⣮⣽⣿⡏⣌⣿⣦
⡄⠞⡳⣼⣤⡇⢙⡁⢀⣞⠆⠀⠀⠀⠀⠀⠐⡟⢿⣿⡛⣿⠃⡀⠤⠀⠀⠀⣀⡀⠀⢻⣿⣯⣵⣿⢣⡟
⠐⠂⠛⡶⠈⣐⣟⠡⢫⢋⡀⠀⠀⢀⠀⠀⠀⠈⠡⣹⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢹⣿⣾⣛⠋⠁⠐⠀⠈
⠁⠀⠀⢀⠐⠓⠀⠀⡺⠇⢀⣴⠟⠃⠀⠀⠀⠀⠈⠝⠿⠀⠀⠈⠁⠀⠀⠀⠀⠒⠂⠀⢸⠋⠙⠁⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠴⢀⠬
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠒⠀⠀⠀⠀⠀⠀⠰⠀⠀⢤⠄
⠀⠀⣠⠀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡒⠀⠀⠀⣀⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⠀⠀⠀⠐
⠀⢰⣿⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠉⠀⢦⡈⠀⠀⠞⠝⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⡀
⠀⣌⣿⣧⣬⣨⣦⣤⣀⡀⠀⠀⠀⠈⠃⠈⠀⡀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠛⠂⠀⠀⠀⡀⠀⠁
⣀⠙⠻⣿⣿⣿⣿⠃⠉⠀⠀⠀⠀⠀⠈⠀⢀⣄⡲⡅⡀⢛⠇⡀⢠⠀⠀⠀⠀⢄⠀⠈⠓⠀⠈⠠⣤⠔⠀⠄
⠙⠧⠀⠁⢿⣿⡟⠠⠄⠀⠀⠀⠀⠠⠀⠀⠀⢙⡧⢀⡁⠀⠀⠁⠁⣀⣀⡀⠀⠘⠉⣛⣢⡀⠀⠀⠒
⠛⢻⣾⡓⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠐⠀⡀⠲⠘⠇⠙⠓⠷⣦⠀⠘⣿⣿⡂
⣇⡈⠻⣧⡐⠀⠀⠀⠀⠀⣉⣀⠀⠀⠀⠀⠀⣈⡀⠀⡐⠋⡀⠀⢚⠃⠀⠈⢆⡀⠀⢤⡄⠀⢙⣄
⣷⢄⣠⢿⡷⣾⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⣿⡅⠀⢋⠈⠰⢀⡄⠀⠀⠀⠀⡅⠀⠀⠀⡄⠀⠈⠒⠈⠁
⠁⣉⠧⣦⣤⠏⢳⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣸⣄⣶⢘⡌⢸⣧⡀⠀⠻⠈⠀⠀⢈⣄⠀⢀⡀⢦
⠊⠻⣿⣿⠿⢪⣾⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠨⠿⢟⠆⠀⠀⠭⢡⣶⠆⢠⣄⢰⠆⡝⣶⡆⠐⢠⡀
⠀⠄⠻⣿⣿⣻⡧⠿⠂⡠⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⢸⡏⠀⠀⠀⠀⠀⠀⢆⡋⣭⣺⠿⢇⣾⡥⣰⣯⠇
⣀⠀⠀⣿⠿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠰⢷⣻⡧⠌⠄⠀⠈⠁⣤⡈⠁⠉⠟⢁⣬⠫⠐⣱⡷⠖
⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠤⠎⠀⠀⠀⠀⠁⠈⢦⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠁⠀⠀⠟⠁⠐⠚
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⣀⢀⡠⠐⠉⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡐⠋⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠟⢁⣠⣤⣄⢉⠙⣿⡄⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡠⠀⢀⣏⠈⠉⠛⠀⡈⠿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⢿⣧⣿⣿⡅⠀⣿⣠⡄⠀⠀⠰⠶⠄⠀⠀⠀⣶⠀⠦⠀⠄⠀⠀⠀⠈⢐⠀⠈⡁⠀⠒⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠁⠁⠉⣐⢩⣌⣼⠣⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣄⣤⣋⠐⣦⠀⠸⠆⠀⠀⢁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠙⣿⣿⣋⣾⢻⠇⠀⠀⠀⠀⠀⠀⠘⣷⠀⠿⡛⠀⠈⠉⠰⠖⢠⣤⢰⣈⢶⠆⠀⡄⠀⠀⠀⠀⠀⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⣿⣽⡟⠋⠀⠀⠂⠀⠀⠀⠀⠀⠈⣠⣼⡅⠄⠀⠀⠄⠘⠅⠶⡼⠛⡾⢳⠾⣃⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡍⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⢂⡉⠙⠉⠉⠀⠀⠈⠃⠀⠀⠁⠚⠂⢀⠟⠁⠴⠟⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠁⠰⠤⠤⡆⠀⠁⠀⠀⠀⠀⢀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢃⣦⡉⠀⠀⠀⠀⠀⢠⣿⡀⠁⠀⠀⡀⡀⠀⠀⠂⢢⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠹⠃⠀⠀⠀⠀⠀⢠⣿⣧⣥⣶⣤⠤⠀⠀⠀⠈⡈⠁⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠠⠠⢄⠀⠀⠀⡟⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⣀⠀⢴⡖⢛⠲⣌⠉⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠐⣮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⢦⣄⠀⢳⣾⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡷⠛⠊⠻⠷⢬⣄⢄⠙⡇⠈⠁⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⡄⠀⠤⠀⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣃⣤⣶⣾⣴⣤⡄⠻⣇⠀⠈⠀⠀⢘⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⣤⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⡾⢿⣿⠻⠟⠟⠿⢇⡤⠿⢷⡇⠀⠀⠙⠋⠁⠀⠀⠀⠯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠂⠘⠓⢠⡀⢀⠀⡀⠲⣠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⠋⠀⠀⢀⠀⠀⢤⣴⣷⣾⣡⣷⡀⠀⠀⠀⠀⠀⠀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢘⣓⠶⡧⣩⣄⣴⠅⠀⠀⠀⠀⠀⠟⢿⡝⣿⠀⠤⠄⠀⠀⠀⠀⢽⣿⣴⣿⣼⠁⠀⠀⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠙⠃⣤⠁⢈⣤⠒⢀⣤⠀⠀⠀⠀⢾⣿⠁⠀⠄⠀⠀⠀⢀⡀⠀⣿⠿⠇⠀⠀⠐⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⠒⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⠀⠴⡶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⢀⡄⠘⠂⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⢀⢠⣡⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠃⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣿⡀⠠⣄⡀⠂⠂⠀⠀⠀⠘⠂⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠸⠿⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⢀⡁⠆⠀⠶⠆⢀⠀⠀⠀⠀⠀⠓⠀⠀⢀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡬⢝⡡⣶⣌⠣⠈⠹⣿⡟⠠⠀⠀⠀⠀⠀⠀⠀⢻⢀⡁⠀⠀⠈⣀⣄⠀⠈⠉⣓⢆⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⣀⣤⠈⠉⠹⣷⡑⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⣬⡅⠉⠙⠓⠀⠨⠟⢥
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣼⣿⣿⡀⢈⣷⣠⡀⠀⠀⠶⠶⠀⠀⠀⠀⡶⠀⡧⠐⡀⠀⠀⠀⠈⠂⠀⢙⢀⠀⠃⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠈⠁⠀⠉⡙⢣⣍⣼⢣⣄⠀⠀⠀⠀⠀⠀⠀⢉⠄⣰⣠⡄⡃⢰⣆⠀⠻⠐⠀⠈⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⣿⣿⢫⡿⣻⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢟⡛⠁⠀⠋⠴⠖⢠⡄⣄⠨⢷⠀⢀⡄⠀⠀⠀⠀⢠⢮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠙⣿⣽⡛⠋⠰⠀⠀⠀⠀⠀⠀⠀⠉⣀⣾⠀⠀⠀⠀⠀⠹⠄⢧⡟⢻⡾⢳⣟⡂⠀⠀⠀⠀⠈⠉⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡏⠉⠀⠀⠀⡀⢀⡀⠀⠀⠀⠐⢀⠁⠋⠀⠈⠀⠀⠈⠃⠀⠀⡀⠚⠀⢰⠟⠁⡴⠗⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠁⠀⠀⢀⣀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠛⣀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣷⣦⣄⡀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣩⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⢿⣻⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⡛⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠉⠛⠿⢟⣋⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⠙⡿⠛⠻
⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⢺⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⣿
⠉⠙⠛⠿⣿⣟⣿⣿⣻⣿⣿⣿⠿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣹⣿⣿⣿⣿⣿
⠀⠀⢀⣈⣴⣿⣿⣿⡿⠟⠉⠁⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠱⠾⠛⠀⣤⠀⣘⣂⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠟⢀⣘⠛⠛⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠿⢒⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡀⠘⠋⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠿⡿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣉⣁⣤⣤⣤⣤⣄⢈⠛⢿⣿⠆⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠿⠛⠿⣿⣿⣄⠀⣤⠀⠀⠀⠀⠈⠉⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠻⠔⠄⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡿⠿⢶⡆⠀⠀⠀⢰⣿⠿⣾⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢿⣷⣶⡿⠛⠀⠤⣾⡇⠀⠀⠀⣾⣷⡂⠙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣄⡀⣠⣿⣿⡄⠀⠈⠉⠙⠛⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⢩⣴⠾⣿⢿⣿⣿⣅⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⡄⢻⣿⣿⣿⡿⠻⣿⣿⣿⣿⡆⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣶⣿⣿⣿⣿⣶⣤⣾⣿⣿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣤⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣾⣿⣿⣿
⣿⣷⣶⣄⡀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣾⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡘⢻⣿⣿
⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⡟⠛⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⢻⣿
⠀⠀⠙⠻⠿⠟⠋⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⡶⠉⠉⣹⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠈⢣⠀⠀⠀⠀⠈⠉⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⠟⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠒⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⡀⠀⠀⠀⣀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⣀⠀⠀⠀⢀⣠⣴⡾⠟⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⢻⣧⡀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⢀⣤⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠿⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡆
⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣿⠟⠁⠀⣠⣶⠾⠿⠿⠿⠃⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⣿⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⠿⠿⠿⣶⣤⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡏⠀⠀⠠⣦⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠿⣶⡄⠀⠉⢀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⠟⠀⠀⣿⠃
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠙⠛⠻⣿⡏⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣄⣀
⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠃⠉⠉⠛⠻⢷⣦⣄
⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⣰⡿⠀⣴⡶⢶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⠶⣶⣄⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣠⣾⠟⠁⠸⣿⡀⠀⣽⡇⠀⠀⠀⠀⠀⠀⢿⣇⠀⢈⣿⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠛⠋⠁⠀⠀⠀⠙⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠈⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠈⠛⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠟⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣡⣤⣶⣶⣿⣿⣶⣶⣤⣌⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢠⣿⣿⣯⡈⠻⣿⣿⣿⣿⣿⣿⠟⢁⣼⣿⣿⡄⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢀⣿⣿⣿⣿⣿⣦⣨⣿⣿⣿⣿⣅⣴⣿⣿⣿⣿⣿⡀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠟⢁⣴⣾⣿⣿⣷⣦⡈⠻⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⢿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣿⣿⡿⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡈⢿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⡿⢁⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣈⡙⠛⠛⠿⠿⠛⠛⢋⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣼⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⣀⣠⢀⠀⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠁⠀⠀⠀⠀⠀⠈⡐⠋⢨⠖⢠⡤⠀⠀⠹⣿⠀⠀⠀⠀⣙⢿⣿⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠓⠪⢀⡀⣡⠀⠠⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣒⡀⠀⣼⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢥⣜⠀⠀⠀⠀⠀⣴⠛⠀⠀⠀⠐⠠⢄⠲⣤⣼⡷⠀⠀⠀⢼⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⡀⠠⢿⣦⣶⡦⠀⠀⠀⠈⠁⢊⠀⣄⠀⠀⡀⠸⣜⣿⣽⣿⡇
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⠜⠣⢤⣑⠀⠹⣿⠀⠀⠀⠀⠀⠈⣏⠃⠀⢉⣤⣈⠻⣮⡟⠿⠛⠁
⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⣡⣴⠶⣤⡄⠻⡄⠀⠀⠀⣀⠀⠀⠀⡁⡀⠀⠚⠀⠯⠁⡟⢻⣼⡅
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⣻⣻⠟⠙⠋⠙⠃⢀⣋⠇⣄⠀⠁⠀⠀⠀⢃⢠⣤⡄⣦⡘⠒⠋⣿⡿⣿⡀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⣽⣿⡇⠀⠀⠀⠀⠈⢻⣟⣼⡹⠀⠀⠀⠀⠀⠙⠈⡟⠀⠀⠉⢿⣿⢿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠟⠃⠀⠀⠀⠀⠀⠀⠀⡙⠃⠀⠀⠘⠀⠸⣟⢹⡯⠆
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠤⢀⡁⡀⠀⠀⠀⠀⣰⣁⠀⢀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢩⣝⠀⠀⠀⢸⣯⣷⣤⣾⡄
⠀⠀⠈⠠⠀⠀⠀⠀⠀⠄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⠤⣜⢻⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⠀⠋⠠⠀⠀⠀⠀⢂⡀⠀⠋⠀⠀⠀⠀⠀⠀⣠⣾⠟⣋⡛⠺⣷⡟⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠐⡀⠉⠓⠂⣘⠛⢆⠀⠀⠀⠀⠀⠀⠠⠿⠑⠚⠛⠛⠃⠈⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠎⠐⠀⠀⠀⠈⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠅⠃⠈⠀⠰⠀⠀⠀⠀⠈⠀⠄⠀⠁⠀⠀⠰⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⡼⣻⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⠀⣈⠀⠈⠘⣿⣿⡇⠀⠀⠀⠀⢀⠚⠁⠀⠀⠀⠀⢉⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣢⡌⢈⡉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⢻⣿⣿⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠓⠚⠋⢸⣿
⠀⠀⡠⠖⢒⣛⣛⣛⠒⠲⡄⠀⠀⠀⣾⣿⡿⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⡦⢐⣷⡆
⠀⠀⡓⠶⠤⠭⠭⢭⢤⠔⣻⠀⠀⠈⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⡂⠀⠐⠹⠆⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇
⠀⠀⢏⠒⠦⠤⠤⠤⠀⢚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠗⠷⠚⠛⠒⣄⠀⠀⠀⠀⠀⠀⠈⠲⡕⠄⠀⠀⠀⢸⣿⣶⣿⡿
⠀⠀⠀⠁⠒⠀⠂⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠸⡇⢠⠤⠔⠒⢲⡄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠸⡜⢋⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢹⠀⠀⠀⠀⠀⠀⢻⡈⠉⠁⠀⠈⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠄⠄
⠀⠀⠀⠀⠀⠰⡶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⣀⣰⢶⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠖⠚⠁⠀⠀⠀⠀⠀⠈⠻⣏⣀⡇
⢀⠤⠚⠆⠀⠀⠀⠀⠀⣴⣾⣿⣦⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢹⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⢠⠑⠁⠀⠈⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠰⡒
⠀⠀⠄⠀⢃⠀⠀⠀⠀⠀⠈⠉⠛⢦⣘⣷⠀⡀⣠⣛⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠰⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⢾⣿⣿⣿⣦⠀⠀⠀⠘
⢂⠉⣀⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠛⠛⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠂⠁⠀⠀⠀⠀⠀⠀⠀⢷⣺⣿⣿⣿⡿
⠀⢣⠄⢀⠈⢸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⠛⠛
⠀⠀⠳⡤⢤⠊⡧⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⢀⠀⠰⠷⣶⣶⡶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⡝⢶⡆⠀⠀⠀⠀⠀⠀⠁⠔⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠊⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠤⠒⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡉⠢⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠑⠒⠀⠉
⠀⢰⠀⠀⠀⠀⠀⠤⢂⠔⠁⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⢀⠀⠼⠂⣈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡾⠑⠒⠓⠒⠬⣉⠹
⠀⠀⠓⠀⠀⠀⠀⡀⠙⢄⡠⠀⠀⠀⠀⠀⠀⡔⠁⢠⠀⠈⡀⢀⢠⠀⢑⡬⠂⠀⠀⠀⠀⠀⠀⠁⣀⣠⣶⣒⣓⣓⣶⣀
⠀⠀⠀⠀⢀⡴⠛⠷⡤⠊⠀⠀⠀⠀⠀⢠⠎⠠⠀⠀⠠⠄⠀⢠⡤⠒⠁⠀⠀⠀⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠢⡀⠀⠒⢀⡤⠚⠁⠀⠀⠀⠀⢀⣸⡍⢆⠀⠀⠀⠘⣿⣿⣉⣉⡉⠛⠛⣿⣿⡇⠀⠀⠀⢠⣤⡴⢦
⠀⠀⠘⢄⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡒⢈⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⢧⣠⠏
⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣁⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡇⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣻⣾⣻⡿⠗⠙⠊⢋⡘⠋⠴⢊⡙⢹⣿⣀
⡄⢀⡀⢀⢈⢣⣤⠈⠀⡁⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⠋⠀⠀⠀⣀⠀⢁⡰⢷⣽⣿⣯⢤⣿⣷
⠄⠊⡓⢾⣾⢄⣏⡀⡴⡛⠀⠀⠀⠀⠀⠀⢯⡻⣿⡝⣿⠀⠤⠤⠀⠀⠀⡄⠀⡈⣿⣿⣤⣿⣃⡾
⠘⠈⠹⠗⠀⡽⠉⠚⣍⠤⠀⠀⡀⠀⠀⠀⠀⢀⣽⣿⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣯⠉⠀⠘⠁⠈
⠀⠀⠀⢄⠈⠁⠀⢨⠛⠀⠴⠛⠉⠀⠀⠀⠀⠀⠂⠛⠀⠀⠀⠀⢀⠀⠀⠈⠉⠀⠀⡏⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠘⠡
⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠈⠋⢀⠒
⠀⢀⣞⠈⠓⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠠⠀⠀⣀⢴⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘
⠀⢸⣿⡄⠄⢀⠀⠀⠦⠄⠀⠀⠀⠄⠨⠆⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⡄
⠀⣿⣿⣿⣿⣾⣿⠳⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠀⠀⠂⢀⠂
⢶⡄⠩⠻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡆⠁⠆⠀⠰⠂⠊⠀⠀⠀⠀⠤⢤⢀⠀⠀⠀⣙⠃
⡶⣭⣠⠄⠙⢿⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠐⡮⠐⠂⠀⠀⡀⢴⠖⠿⣢⣄⠀⠐⣮⣷⠄
⡄⠸⣿⡆⢁⠈⠃⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠀⠈⠠⠔⠀⠐⡴⡆⠀⢠⠉⠁⢀⡐⠋⠳⠄
⣯⠀⢀⣿⣆⣆⠀⠀⠀⠺⡿⠆⠀⠀⠀⠀⣶⠃⠀⡗⠐⣀⠀⠂⠀⠀⠀⢙⡀⠈⡉⡀⠀⠙⠦⠀⠂
⠉⠚⣩⣌⣩⠿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⢸⡀⣄⢜⠀⢲⡄⠀⠸⡆⠀⠀⠈⣂⠀⠂⠀⢀
⠔⢿⣾⣿⡿⣲⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡆⢉⣿⢿⠌⠀⠉⠛⢠⣤⠀⣀⠀⣤⢙⢦⣄⠁
⠀⡄⠿⣷⡶⣿⣥⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠐⣰⠏⠀⠀⠀⠀⠀⠰⠔⢹⡳⢷⣾⢨⣧⣤⣞⡇
⡀⠀⠀⣿⠿⡷⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⢶⢿⣇⠉⡄⠀⠈⠀⣀⠙⠈⠻⠃⢠⡟⠁⢋⡮⠄⠂
⠈⠀⠀⡇⠀⠀⠀⠀⠀⠠⠠⠼⠁⠀⠀⠀⠀⠈⢢⣀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠉⠁⠀⠟⠃⠀⠚
⠀⠀⠀⠀⠀⢀⠀⠀⠈⠐⠀⠀⠀⠀⣀⣀⡀⠐⠋⠁⠀⠄⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠰⠍⠉⠉⠋⠙⠭⡆⠀⠀⡠⠤⠤⣤⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⣰⠂⠀⠰⠂⠀⠀⠀⠀⠋⠁
⠀⠀⠀⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢲⡐⣀⣿⠀⠀⠈⢹⠁⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠄⠁⠀⣠⢀⡀⠀⠀⠈⠀⠁⠁⠀⠀⠀⢸
⠀⢀⡴⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂⠈⠉⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠠⠆⠀⠀⠀⢸⠀⠀⠀⠁⠀⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠰⠃⠀⠀⠐⠀⠀⠀⠠⣂⠄⠀⠀⢰⠀⠀⠃⠀⠀⠰⠘⡄⠀⠀⣗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⡠⠄⠈⠁⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠹⠁⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⡀⠀⠀⠀⠀⢀⡠⠒⠉⣠⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⡀⠀⠁⠀⠌⠂⠀⠀⠀⣀⠔⠉⢀⡴⠊⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇⠄⠂
⠀⠀⠀⠄⠈⠀⠠⠀⠀⠀⠀⢀⠤⠊⢀⡠⠞⠁⠀⡠⠔⠋⠀⠀⠀⠀⠀⠀⣀⡠⡠⠁
⠤⠈⠀⡀⠄⠀⠀⠀⠀⡠⠖⠁⣀⠔⠋⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠐⠈⠕⠂⠀⠀⠀⠒⠀⠀⠀⣴⣾⣧
⠘⠁⠀⠀⠀⠀⣀⠔⠊⢀⡤⠚⠁⠀⠀⠂⠀⠀⢀⠔⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠃⠀⠸⣿⣿⡡
⠀⠀⠀⠀⡠⠞⣡⢀⠖⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠰⠋⠁⠀⣠⠺⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢈⠀⣀⠔⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⣀⡀⡄⠀⣀⠈
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣤⣤⠤⠔⠒⠈⠉⠁⠀⠀⠀⠀⠀⠁

⢠⣴⣶⣿⣷⣶⣾⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣷⣦⣶⣶⣆⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄
⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣴⣾⣿⣿⣟⣛⡉⠙⣋⣻⡇⠀⠀⠀⠀⠀⣿⣿⣿⣭⣭⣽⣿⣿
⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⢀⣠⣀⣀⣀⡀
⢀⣴⡿⠛⠛⠛⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣴⣴⣤⣤
⢾⣿⠀⠀⠙⠿⠿⢿⣿⣷⠀⠀⠀⠀⠀⣰⣿⣿⣯⣁⣉⣀⣽⣿⣿⣦⠀⠀⠀⠀⠻⣿⣶⣤⣭⣥⣶⣾⡿
⠘⢿⣷⣤⣤⣤⡤⣾⣿⠃⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣿⡿⠋⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟
⠀⠀⠙⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣠⣄
⢠⣶⠿⠛⠻⠿⠛⠻⢷⣦⡀⠀⠀⠀⠀⠰⣿⣿⣛⣛⣛⣛⣛⣿⣿⡆⠀⠀⠀⠀⣴⣿⣟⡙⠛⠛⣛⣿⣷
⠀⠉⠻⣾⣶⣶⣶⣿⣿⡟⠁⠀⠀⠀⠀⠀⢻⣿⣟⠛⠛⠙⢛⣿⡟⠀⠀⠀⠀⠘⣿⣿⣿⣭⣉⣩⣽⣿⣿
⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣴⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀
⠠⣾⣿⣯⣿⣿⣽⣿⣿⡦⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠠⣶⣿⣿⣛⣿⣟⣿⣿⣷
⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡏⠉⠉⠉⠙⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠘⠉⣲⣤⣀⣀⢀⡀⣀⣀⣤⣖⠉⠁⠀⠀⠀⠙⢿⣿⡿⣿⣿⣷⣤⣠⣤⣼⣿⣿⡿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠸⡄⠙⢿⡿⢟⡸⢧⡻⣿⡿⠋⢠⠏⠀⠀⠀⠀⠀⠉⠀⣿⣿⡗⠀⠈⠀⢸⣿⣿⡇⠈⠁
⠀⠀⠀⠀⠀⠠⠁⢠⣐⣀⠉⠃⠈⠉⣀⣀⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⡀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⡅⠀⣶⠀⠈⠉⠂⠔⠉⠁⠀⣶⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠰⡀⠃⣀⠀⠀⠀⠀⠀⠀⣀⡘⢀⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠢⢷⠄⢤⡀⢀⡤⠠⡾⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠐⠈⠁⠈⠁⠃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⣠⣴⣾⣿⡿⠿⠛⠿⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⠟⠉⣀⣤⣤⣴⣤⣄⡙⠿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣦⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣯⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢹⡿⠟⢻⣿⣿⡿⠟⠉⠉⠉⠙⡅⠀⢀⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠃⡾⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣧⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣧⣤⣤⣾⣿⣿⣶⣶⣴⣶⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠛⠿⣿⣿⠿⠟⠛⠛⠋⢻⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⣼⡄⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡂⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠈⠈⢈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⣟⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⣷⣬⣄⣱⣤⣀⣀
⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠈⠀⣤⡄⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠐⠂⠠⠀⠀⠀⠀⢀⣠⣠⣤⣤⡀⠁⠟⣻⣿⣿⣿⡗⠈⠁
⠃⠈⠀⢀⠀⠀⠐⠂⠙⡓⡆⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣈⡿⢅⣠⣌⠙⢧⡀⠉⢹⣯⣿⠃⣀
⠠⠐⠆⠃⠟⠶⣤⡀⠀⠵⢲⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠁⠐⠀⠛⠟⠶⢲⣝⢆⡀⠟⢷
⠀⠤⢢⠀⠀⢢⠀⠀⢀⣈⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠉⢀⣤⣴⣾⣌⡀⣤⠈⢿⣷⠌⡑⠈
⠀⠐⠀⠀⠀⠀⢑⠀⠀⠈⠀⠀⠉⠉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠂⡡⢴⣞⣻⣿⣯⢣⣿⣿⣿⡐⠀⢸⣷⢀⡆⠀⠀⠀⠰
⠰⠃⠀⣤⠀⠀⢸⡆⠀⠀⠀⠑⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣓⣿⢋⣻⡾⠂⠙⠀⠋⠘⠏⠥⠖⢉⠛⣨⢗⢀
⡀⠃⠀⠻⠃⢀⣀⠀⠁⠀⢀⡘⠳⣀⡀⠢⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠁⠀⢶⣄⣵⣿⡏⢄⣌⢅
⠁⠀⠀⠀⠀⠈⢈⠀⠞⠧⣰⣀⡃⢙⠃⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⡏⢿⣿⡛⣿⠇⢀⣀⣀⠀⠉⠀⡀⠀⠈⢻⣿⣯⣡⣾⠇
⠠⠄⠀⠀⠤⠀⠈⢤⠘⢷⡬⠉⠡⡾⢛⡴⠱⠁⠀⠀⠀⠀⠀⠀⠀⠙⠊⢻⣧⡾⠁⠀⠀⠀⠀⠀⠀⠉⠀⠑⠈⢹⣿⣭⠟⠓⠀⠠⠖
⠂⠐⠀⠀⠀⠐⠖⠀⠀⠀⠁⠠⠾⠀⠀⢠⡇⠉⢀⣤⠦⠉⠀⠑⠄⠀⠰⡏⣿⠇⠀⠀⠒⠀⠀⠀⠀⠀⠤⠀⠀⠸⡟⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠂⠀⠀⢀⡀
⠀⠀⡂⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠄⣄⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠈
⠇⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠉⠂⠀⠀⠀⠀⠀⠀⠈⠀⠤⠂⡁⠂⠀⠀⢀⢰⣴⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⡆⢀⣀⡀⠀⠠⠤⠤⠀⠀⠀⢀⠀⠱⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡔⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠶⣿⣷⣶⣶⣾⡶⠶⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⠀⠡⠄
⠀⠀⠴⡶⠞⠻⠶⣤⠈⠉⡽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡅⠂⣤⠠⠉⢠⠀⠆⠀⠀⠀⠀⢀⢀⣤⠈⠁⠀⠀⠰⠎
⣤⡤⠶⣑⠦⣿⣆⣈⠳⢄⠀⢻⣿⡟⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⡑⠀⡤⠀⠀⠀⠀⢠⢀⢤⠄⠀⠀⠀⢡⡝⢀⠀⠀⠈⠁
⠟⠃⠀⡀⡀⠉⠉⠉⢻⣮⡳⢬⡉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠀⠀⠀⠀⠂⠓⠀⠉⠙⠻⠦⠀⠀⠩⠯⣅
⣠⣶⣿⣿⢻⣶⣴⣃⡈⠛⣯⡐⠀⠀⠀⠀⠀⢈⣁⡀⠀⠀⠀⠀⠀⣀⡄⠀⣐⠊⠀⠀⠐⡉⠀⠀⠈⢂⠀⠀⠤⠄⠀⠈⣱
⣗⣾⠻⡏⢻⡿⣻⣧⠀⣀⡾⡷⣼⡄⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⣿⣅⠀⠛⡀⠐⠄⣀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠑
⠟⠉⠀⠀⠀⠠⠌⠀⡉⢥⣖⣄⡾⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣐⣅⢰⡄⣃⠂⣹⣄⠀⠈⠿⢀⠂⠀⠀⣀⠀⠀⡀⠠⡀
⠀⠀⠀⠠⠄⠀⠠⠘⢻⣶⣿⡿⠔⣶⡆⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡆⠀⣵⡿⠣⠁⠀⠈⠉⠀⠶⡄⢀⣀⠀⠄⢈⢢⣤⠀⠂⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠐⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠘⠂⠇⠀⠀⠈⠈⡃⠀⠀⠀⠃⠀⠀⠀⢀⠄⠼⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⠀⠄⠂⠀⠀⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣤
⠀⠀⠴⠒⠒⠒⠒⠲⠒⠒⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⢱⡆⠀⠀⠀⠀⠀⠄
⠀⠀⡇⢀⠖⠉⠁⠉⠢⡀⢰⠉⠉⡆⡊⠀⠙⠄⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠞⠃⠀⢀⠀⠀⠀⢀
⠀⠀⡇⡌⠀⠀⠀⠀⠀⠱⠈⡲⣒⠁⢙⡶⡊⠀⠀⠀⠀⠀⠀⠀⣀⠀⣃⠀⠀⠠⢀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠛⠄
⠀⠀⡇⠡⡀⠀⠀⠀⠀⡔⢸⠀⠀⡇⣇⠀⢈⠀⠀⠀⠀⠀⠀⠀⡅⠀⠬⠀⠀⢀⣀⣀⡀⠀⠠⠀⠰⠀⠀⠀⠀⠁⠤⠀⠄
⠀⠀⣇⠀⠑⠢⡤⠄⠊⠀⠀⠉⠉⠀⠈⠉⠁⢸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢀⠀⠤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⠌⡆⠀⠀⠀⡇⠘⠀⠀⠀⠀⠅⡀⠀⠀⠀⠀⡈⠂⠀⠰⠰⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠁⠀⠀⠁⠀⠀⢮⡰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡾⢩⠏⢿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢤⢀⠠⢄⡀
⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠄⠀⠀⠀⡅⡆⠀⠹⢍⡘⢹⠀⢀⠔⠛⠋⠲⡄⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠀⠀⢻⡀⢀⢈⡬⢷⡾⠀⢸⠀⠀⠀⠀⡸⠀⠀⠀⠀⡇⠀⢠⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠢⡉⠓⠛⠛⠉⠀⠀⠀⠓⠤⠤⠖⠁⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⢢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⢄⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠒⠀⠘⠀⠀⠀⠀⠸⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠄⠀⠀⠀⠀⡀⠀⠀⠡⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠘⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⣴⣿⣿⠂⠀⠀⠀⣀⣀⠀⢰⣿⣷⣆⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⣿⣿⠏⠀⠀⢀⣾⠏⠉⣷⠀⠹⣿⣿⡆⠀⢠⣿⣿⡿⠃
⠀⠀⠀⣴⣦⠀⢀⡴⠶⣶⣄⠀⠀⠀⠀⠙⠻⠿⣧⠀⢿⣿⠀⠀⠀⣼⡟⠀⢀⡟⠀⠀⢹⣿⡇⢀⣿⠿⠋
⠀⠀⠀⣿⡇⠀⡾⠁⠀⢹⣿⡀⠀⣤⣶⣶⣶⣤⣀⠀⠈⠻⠀⠀⢀⣿⡇⢀⡼⠁⠀⠀⢸⡿⠁⠀⢀⣤⣶⣿⣿⣷⡄
⠀⠀⠀⣿⡇⢸⠃⠀⠀⢸⣿⠃⠀⠛⠿⠿⠿⠿⠛⠓⠀⠀⠀⠀⢸⣿⣠⠞⠀⠀⠀⠀⠀⠀⠀⠴⠛⠛⠻⠿⠿⠛⣀⡀
⠀⠀⠀⣿⡇⡿⠀⠀⢀⣾⡟⢰⣶⠀⠀⢰⣿⡇⠀⠀⣠⡶⠛⣦⢸⣿⠁⣠⠞⢻⡷⠀⣠⣶⠟⢳⠀⣴⡶⣦⠀⣴⣿⡟
⠀⠀⠀⣿⣧⡇⠀⣠⣾⡟⠁⣿⡿⠀⠀⣾⣿⠀⠀⣼⣿⠁⠀⠀⢸⣿⣼⣥⣶⠟⠁⢰⣿⠃⣠⠞⢸⣿⣧⢏⡼⣹⣿⠁
⠀⠀⢸⣿⣿⠱⠿⠛⠉⠀⠀⣿⡇⠀⡼⢻⣿⠀⠀⣿⡇⠀⠀⠀⣼⣿⢿⣯⠀⠀⠀⣿⣿⠋⠁⠀⢈⡿⠟⠋⢠⣿⠇⠀⡴
⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⢿⣷⡼⠁⢸⣿⣤⠞⠻⣷⣤⡤⠞⢹⣿⠄⠻⠷⠤⠔⠻⣿⣤⣤⠖⠋⠀⠀⠀⢸⣿⣀⡼⠃
⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠈⣡⣄⡀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠈⠛⠋
⠀⠀⢀⣀⣀⠀⠀⠀⠀⢉⣿⣏⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⢰⡏⠉⠉⠉⠒⢄⠀⠁⠘⠀⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠁⠀⠀⠀⢀⡠⠒⠉⠉⠉⢹⡆
⠀⠈⠁⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⠀⢀⣤⡄⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣠⠤⠤⢄⠀⠀⠀⠱⣄⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣼⣿⣿⠀⢸⣿⣿⠏⢹⣿⣇⠀⢸⣿⡆⠀⣠⠏⠀⠀⢀⡠⠤⠤⣄
⠀⠀⢸⠁⠀⠀⠀⠉⢦⠀⠀⠘⣆⠀⠀⠀⢸⣿⣿⠀⠀⢰⡏⣿⣿⠀⢸⣿⡿⠀⢸⣿⣟⣰⣿⠟⠀⣴⠃⠀⠀⡔⠁⠀⠀⠀⠈⡇
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⣧⠀⠀⠘⣿⣿⣄⣠⠟⠀⣿⣿⠀⢸⣿⡇⣰⣿⣿⠋⠉⠁⠀⣼⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⢱⠀⠀⢸⣇⠀⠀⠈⠛⠛⠁⠀⠀⠉⠉⠀⢸⣿⡇⠘⠛⠁⠀⠀⠀⣸⠇⠀⢀⡎⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠈⣧⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⣰⣿⠀⠀⣼⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⢸⡆⠀⡟⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⣰⣿⢻⠀⢰⡇⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠈⣿⡜⠁⠘⣟⢦⠀⠀⠐⠉⢈⠉⡲⢖⠉⡁⠉⠂⠀⠀⡼⣻⠃⠈⢣⣿⠁⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣷⡀⠀⠘⢮⡳⣄⠐⢄⣹⣿⠁⠈⣿⣏⡠⠂⣠⢞⡴⠁⠀⢀⣾⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⡀⠀⠳⣌⠳⢤⣈⣷⣀⣀⣿⣁⡤⠞⣡⠎⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠑⠀⠉⠉⠉⠉⠉⠉⠀⠊⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀⠀⡀⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠉⠀⠐⠉⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⣀⡤⠤⢀⡀⠄⢠⠄⣀⡀⠀⠈⢤⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠈⠻⣔⠒⠒⠚⠋⠤⠄⡀⠀⠀⠀⢀⣀⠄⠃⠚⠓⠂⠹⠋⠀⠀⠀⠻⠟⠒⠒⠒⠐⠂⠈⠛⠃⠀⠀⠬⠓⠉⠉⠀⠒⠀⢒⠓⠃
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⡠⠊⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠈⠂⠀⢀⡠⠄⠠⡀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠉⠀⠈⠁⠊⠁⠀⠈⠲⣄
⠀⠀⠀⠀⠀⡀⠐⠁⣀⡄⠐⡐⠂⠒⡴⠀⢩⠔⠈⢰⠑⢄⠀⠄⠀⠀⠀⠀⠀⢀⠄⠊⢀⣀⡄⠰⠒⠒⢰⠆⠒⠰⡆⠄⡀⠑⢤⡀
⠀⠀⠀⠐⠚⠁⠲⢭⣜⢆⣀⡶⡄⠀⠇⠀⣸⠷⠂⠈⠀⠈⠉⡩⠒⠀⠀⠀⠻⣋⠚⠈⠁⠀⣁⣉⣀⠒⠊⡁⣀⣈⣁⣀⠉⠐⠒⠚⡻
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠢⣖⠀⠁⠀⠀⠉⠐⠛⠒⠈⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⠤⣀⡠⠔⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠘⠈⠀⡄⠀⠒⠢⠆⡀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⡰⠃⠀⣀⣀⢠⠂⠁⠀⠀⠀⠀⣓⣦⣀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⣠⠤⠈⡇⠀⢸⠀⠀⠀⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠶⡄⠀⠈⠁⠀⠀⠀⠑⠂⠂⠐⠁⠀⠀⠀⠈⠁⠀⢒⡳⠆⠀⠀⠈⠀⠀⠀⣀⠈⠀⠉⠀⠂⠐⢣⠀⢀⠁⠒⡈⠀⡜⠂
⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠀⠠⠼⠀⢀⡥⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠀⠀⠠⠤⠤⢀⡀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂


⡿⠿⠿⢿⠿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⡬⡄⠨⢍⡙⠒⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⣾⢀⠉⠉⠀⠀⢀⠉⠀⠀⠈⠂⠐⠠⠤⣉⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⢋⡅⠀⠶⢂⢀⠀⡬⠵⣆
⠀⠀⠀⠀⠀⣸⠹⡌⠓⠀⠀⠀⠀⠀⠂⠀⣄⡀⡀⠀⠀⣫⠀⠀⠈⣳⣤⡀⠀⠀⠀⠀⣀⡴⠛⡁⠐⠎⢡⠀⠀⢈⡴⠃⠀⠀⠠⠚⢿⡆
⠀⠀⠀⠀⠀⡏⠀⠹⣆⠠⡀⠀⠀⠀⠀⠤⠴⣿⡞⠳⣞⣋⣀⠀⠀⠘⢣⠙⣆⠀⣠⠞⢋⠠⠸⣤⣀⡀⣸⠦⣶⣏⣀⡀⠀⠀⠈⢁⡞⡧
⠀⠀⠀⠀⠀⡇⠀⠀⠈⠃⢈⠁⠀⠀⠀⠀⠀⠀⠶⠒⢟⠻⢭⡉⠁⠒⠀⠠⡈⢳⡁⠶⠄⠀⣀⣨⣭⣿⣄⠎⠿⡁⠀⠀⠀⠐⢀⠘⢁⡇
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⢀⠀⠀⠀⠈⣇⠀⠈⠲⣄⡀⠠⢨⡄⡇⠐⠊⠉⢉⡴⠋⠰⡏⠁⠀⠑⡈⠀⢀⡴⠋⠀⣸
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣙⡀⠤⠀⠘⢦⡀⠀⢀⠙⠀⠀⣱⠃⡀⢀⡴⠋⠀⠀⠀⠁⣠⠠⣂⣠⠖⠉⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠶⠬⠤⠤⠀⠒⢈⣿⣛⠀⠈⣴⠀⠄⢀⣀⡤⢴⠛⠉⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣼⠫⣝⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⣠⠛
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⣴⡿⠃⠠⠉⠲⣶⣦⣤⣤⣤⣤⠤⠤⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠛⠋⠉⡱⢹⡇⠀⢀⠀⠀⠀⠀⡀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠉⠉⡹⠁⠀⠀⠀⠀⢰⠁⢈⡇⠀⠈⠀⡀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⡼⠀⠘⣇⡀⠀⠀⠙⢄⠀⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠜⠀⠀⢸⠁⠳⡄⠀⠃⠀⠹⡀⠂⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣀⡀⠚⠀⢀⡠⠃⠀⠀⠈⠓⢄⠀⠀⠀⢦⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠴⠁⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⢸⡆⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢰⠃⠘⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⣤⣄⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⠋⠉⠉⠉⠙⠛⠛⠙⠋⢿⠷⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⣀⡴⠋⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢰⡀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⡠⠂⠀⠀⠄⠀⠁⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⡠⠔⠚⠃⠤⣀⠀⠀⠀⠀⡀⠤⠤⠤⢀⡀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠑⢄⠔⠉⠀⠀⠀⠀⠀⠈⠢⡀⠀⣿⠀⠀⡀⠀⢸⡆
⢀⣀⠀⠀⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⠛⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⡿⠀⠀⠀⠀⠘⢧⣤⣤⣶
⢻⡍⠙⠛⠉⠉⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢁⡴⠁⠀⢀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⡆⠀⠀⠀⠻⠃⠀⠀⠀⠀⠃⡷⠚⢉⡇⠀⠀⠀⠀⣰⠏
⠀⠹⢦⣄⣤⣀⡀⠀⠀⠀⠀⢀⡴⠒⠶⢤⣤⠴⠋⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⣡⣤⡞⠁⠀⠀⢀⣾⣁⣄
⠀⠀⣠⣤⠼⠛⠀⠀⠀⠀⠀⢸⡁⠀⠐⢲⡄⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⣠⠞⠁⠈⠲⢤⣀⠀⠀⠀⠄⠊⠀⠈⠻⣇⠀⠀⠀⠀⠀⠈⣡⠟
⠀⠀⠙⠷⣤⣤⣀⣤⣤⠂⠀⠀⠳⣤⣀⣘⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⡌⠉⠁⠀⠀⠀⢀⣱⣮⣥⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠉⣉⣻⣶
⠀⠀⠀⠀⠀⠀⣼⣿⣥⣤⣤⣤⠀⠀⠈⠉⠁⠙⢦⡀⠀⠀⠀⠀⠀⣰⠋⠀⠇⠀⠀⠀⠀⠀⣏⠀⡿⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣍⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠁⢀⣤⣞⣁⣠⣤⡴⢶⣤⡶⠛⠻⢦⣀⠀⠀⣰⠃⠀⠀⡇⠀⠀⠀⠀⠀⢸⣀⡇⠀⠘⣇⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⠿⢤⣄⣠⡇⠀⠀⠀⠀⠀⠀⠈⣧⣤⡤⢿⠀⠀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⡇⠀⠀⣀⣀⣀⣀⣀⡁⠀⠀⠘⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣧⠀⠀⠈⠉⠉⠉⣹⡇⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠙⠳⠶⠤⠤⠶⠞⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⣄⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠛⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠷⢶⡶⠶⠶⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣉⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠟⠉⠉⠉⠙⠛⠿⣿⣶⣄⡀⣠⣴⣿⣿⠟⠛⠉⠉⠉⠉⠛⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⣀⣠⣴⣾⣿⣿⣿⠿⠿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠘⢿⣿⣷⣄
⣠⣾⣿⣿⣯⣤⣤⣤⣶⣶⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣶⣶⣿⣿⣿⣷
⠉⠻⣿⣿⡿⣿⣿⣿⣿⣟⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣿⣿⡿⣿⣿⡿⠋
⠀⠀⠘⢿⣿⣆⠉⠛⠻⢿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⠿⠿⠛⠉⢁⣾⣿⡟⠁
⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠒⠠⠄⠀⠀⣀⣠⡀⠴⠊⠀⠀⠀⠀⠀⠀⣠⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠘⠁⠐⠀⠀⠀⠀⢀⠀⠀⠉⠀⠘⠁⢦⠀⡄⠀⠀⠀⠄⠀⢰⡲⠀⠁⠀⠀⠀⠀⠀⠈⡰⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠃⠀⠀⠠⠄⠀⣀⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠣
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠱⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡈⠈⠀⠠⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⣀⠁⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⢀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠐⠀⠂⠀⠀⠀⢒⠀⠈⠀⠁⠀⡆⠀⢀⡀⠀⠀⠀⠀⠀⠄⠄⠀⠀⡐⠐⠁⠀⠀⠋⠀⢠⠋⠉⡑⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡅⠁⣀⠁⠀⠀⠀⠀⠈⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠘⠂⠆⠀⠐⠠⠀⠀⠆⠀⠀⢘⠀⠀⠀⠤⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢁⠀⠀⠀⠀⠀⠀⠇⠤⠀⠀⣢⠀⠀⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠈⡓⠀⢦⠀⠀⠀⠈⠉⢀⠀⢀⠈⠁⢀⠄
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⢰⠀⠁⣸⢏⠀⠀⠀⠀⠀⠰⠉⠿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣾⣆⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣷⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠾⠇⠙⠃⠙⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠋⠘⠋⠸⠷⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⠀⠈⠳⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠞⠁⠀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠅⠀⢀⣀⣀⠈⠀⢀⣴⠶⠞⠳⠶⣦⡀⠀⠁⣀⣀⡀⠀⠨⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠉⠀⣴⠏⠀⠀⠀⠀⠀⠀⠹⣦⠀⠉⠻⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠆⢸⡇⠀⣀⣀⡀⢀⣀⣀⠀⢸⡇⠰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢻⡾⠋⠉⢻⡟⠉⠙⢷⡟⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣴⠟⠁⠀⢀⣼⣧⡀⠀⠈⠻⣦⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡾⠁⠀⠀⣠⡞⠁⠈⢳⣄⠀⠀⠈⢷⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠼⠋⠀⠀⢀⡾⠋⠀⠀⠀⠀⠙⢷⡀⠀⠀⠙⠧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣧⣤⣤⣤⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣼⣧⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⣄⠀⠀⠀⢀⣠⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠁⠀⠉⠙⢿⣷⣤⣴⣿⠟⠉⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⣿⠉⠛⢿⣦⡀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣿⠀⠀⠀⠙⣿⣶⣶⣾⣿⣇
⢀⣤⣴⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣾⠟⠁⠀⠀⠀⠀⣿⠀⠀⣀⣼⠟⠁⠀⠀⠹⣿⣦
⣿⣿⣿⣿⣶⣶⣶⣶⣶⠶⠶⠿⠿⠿⠿⠟⠛⠛⠉⠙⠛⠿⢷⣶⣤⣄⣀⣠⣿⣶⡾⠟⠁⠀⠀⠀⠀⠀⠘⢿⣧
⠈⠛⠿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠁⠀⠀⠈⠻⣷⣦⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠉⠻⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇⠀⣷⠀⠀⠀⠉⠳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣷⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠻⢿⣿⣿⠿⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⣠⣤⣤⣄⢠⣿⣿⣄⢛⣸⣿⣿⡘⢃⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⣹⣿⡏⣼⣏⣹⣿⣿⢸⣿⣿⡿⢛⣩⣭⣍⣙⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣙⣻⣇⠻⣯⣭⣭⣿⡌⣿⣿⣰⣿⣿⣿⣿⣿⣧⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠈⠙⠋⠉⠀⠈⣩⣤⢹⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣾⠟⠁⠈⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣡⣾⠟⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⣿⣿⣟⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡖⢦⡴⠚⠁⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠛⠀⠀⠀⠀⠀⢣⠘⠚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢆⠀⠀⣠⠤⠤⠤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⡤⠖⠋⠑⢂⡬⠭⠷⣶⠤⣄⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢳⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⣁⣀⣀⠀⣰⠃⠀⠀⠀⠀⠹⡆⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⠙⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠉⠉⠙⠻⢤⡀⢀⣀⡤⠴⡟⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⣹⡯⠿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⣸⠃
⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢉⡴⠋
⠀⠀⣔⠦⠞⠁⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⠢⠛⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠉
⠀⠀⠈⠓⠶⠴⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠤⠶⠒⠉⠙⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠴⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠻⣶⠄⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⢀⠀⣠⠖⠒⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣞⠁⠀⠀⠀⠀⠀⣴⣫⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣁⠀⠯⠑⠒⠠⣤⡤⠤⣏⠬⣤⣤⠞⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠈⠉⠙⠒⠦⣄
⠀⠀⠀⣰⢿⠃⠀⡴⠒⠚⠄⠀⠀⠈⠒⠐⠠⣈⠙⢦⣀
⠀⠀⣸⠇⡇⢀⠼⢥⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢱⡀⠙⢦⡀
⠀⢀⡟⢰⠀⣜⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⢳⡀
⠀⢸⡇⠘⡆⠹⣤⣤⣤⣤⡀⠈⠑⠂⠀⠔⠁⠀⠀⠀⠀⠐⠈⢧
⠀⠘⡇⠀⢧⠀⡆⠀⠀⠀⠀⢀⠔⠁⠦⢆⠀⠀⠀⠀⠀⠈⠀⢸⡄
⠀⠀⢷⠀⠈⢧⠘⢄⠀⢀⡔⠁⠀⠀⡇⠀⠢⡀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⠀⠒⠒⠒⠶⢤⣄⡀
⠀⠀⠈⢆⠀⠈⠳⡀⠉⢧⠀⠀⠀⠀⡇⠀⠀⠈⢄⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠈⢂⠀⠀⠈⠢⢄⠑⠢⠄⣀⡇⠀⠀⠀⢀⣱⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠉⠒⠠⣀⣀⡈⠀⢉⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⣀⠀⠈⠉⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣠⣤⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠶⠚⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣭⣬⣭⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠠⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⢻⠿⠋⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠂⠀⡠⠶⠖⠀⠖⠀⠆⠀⠆⠀⠛⠒⠠⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⣀⣈⠁⠀⢠⠂⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⣀⠀⢐⠀⠀⠁⠀⠀⠄⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠐⠄⣀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢀⡯⠖⠀⠀⢹⠁
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⢸⠉⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⢒⠒⠐⠂⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠓⠋⠉
⠀⠀⠀⠀⠠⡀⢰⠀⠀⡆⡆⠘⠀⠀⡇⠀⠀⡏⠀⠀⡔⢸⣽⢲⡄⢁⣈⠀⠀⢀⡀⠐⠃⠀⢀⠉⠀⠀⠘⡁⠀⠒⠒⢀⡀
⠀⠀⠀⠀⠀⠁⠈⢀⠟⠛⢻⡁⠀⢠⡇⠄⢠⠃⠀⠸⣚⣚⣗⣰⡇⢀⠀⠀⠀⠈⠀⠀⠀⣰⠛⢿⡄⣴⠀⠀⠀⠀⣄⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸⡇⠀⠈⠁⠀⠀⠙⢋⠏⠃⠄⢀⠁⠀⡀⢠⠀⠀⠀⠘⠳⠾⠃⢸⠀⠀⠀⢰⡟⠊⠃⠀⢠
⠀⠀⠀⠀⠄⠀⠀⢸⠀⠀⠀⣿⠀⠘⠓⠀⠀⢸⢠⡖⠀⠸⠀⢰⠀⢸⠀⢀⣇⢀⠀⠀⢰⠛⠉⠀⠀⠁⠀⠀⠀⢸⡇⠀⠀⠈
⠀⠀⠊⠀⠀⠀⠀⠈⠆⠀⢰⠏⣀⠯⠀⠀⠀⠈⢸⡇⠀⠀⠀⢸⠀⠈⠀⠐⠶⢺⡀⢀⡀⠧⠔⠰⠶⠆⠀⠀⠐⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢒⡋⣀⢹⠀⠀⡄⠀⠀⠠⠁⠆⡏⠀⢸⠀⠀⢠⡆⠀⠀⠀⠈⢱⢸⠀⠀⢆⠉⠀⠀⠀⢰⡇⠀⠁
⠀⠀⠛⠒⠋⠁⠀⠀⠀⠸⡝⠹⢸⡀⠀⠳⠀⠀⡄⠀⠀⠁⠴⠞⠀⠤⢹⠀⠘⠌⠀⠀⠈⢘⠀⠀⢸
⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠑⠃⠀⢡⣤⣤⡄⢰⠁⠀⠀⠀⢠⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠇⠀⠀⠀⣀
⠐⠒⠒⠒⠒⠒⠒⠀⠒⠒⠓⠒⢢⣤⣤⣤⡄⢸⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⡄⠀⢀⡃⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠘⠀⠀⠂⠀⠀⠏⠀⠀⠀⠀⠀⠋⠀⠀⠄⠀⠃⠀⠀⠳⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⡥⣄⠀⠀⢄⠀⢀⣠⠀⠀⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠁⠀⠀⢦⠀⠀⠱⠂⠀⢀⣀⣀⣨⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠱⠶⠖⠁⠀⠙⠛⠁⠀⢀⣀⠁⠊⡠⠂⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⣸⡇⠈⠀⠀⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠉⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⣼⡇⠀⠀⠐⠒⠉⡄⠀⠸⣠⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠸⠿⠇⠀⠀⠀⠀⠀⠸⠆⠀⠛⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⠀⠁⠀⠀⠀⠡⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠀⠀⠀⠀⠀⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⠛⠃⠀⠀⠀⠈⠀⠀⣀⡜⠁⣀⠆⠀⠀⠀⠀⠀⠘⣄⡀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⠆⠘⠀⠀⠀⠀⠀⢄⡀⠀⠌⠀⣶⡄⠀⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢠⡄⠀⠀⠀⠀⡀⢄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢌⠙⠓⠐⠀⠀⠠⠌⠈⠋⣵⠋⢺⡧⢀⣀⠀⠘⢻⣿⠆⠀⠄⠀⠐⢲⣶⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⡤⣤⡔⠋⠉⠢⠀⣀⠀⠀⠀⠀⠘⠠⠀⠨⠄⠉⠁⠀⢀⠀⠙⠀⠁⢠⡀⠐⠃⣿⣻⣼⣿⡇
⠀⠀⠀⠀⠀⠀⢼⠁⠁⠉⢻⢖⢻⡄⠈⠀⠀⠀⢀⣬⣄⠠⠀⠀⣀⡀⢀⣠⡀⢢⣤⡀⠚⠓⠀⣼⣯⣿⣿⣿⡇
⠀⠒⠀⠀⠀⠀⠘⠀⠀⠀⠈⢻⣷⠁⠀⠀⠀⠀⢻⣇⢈⣀⠠⠄⠀⠈⡱⡜⢻⠟⠻⡀⠀⠀⠈⠇⣿⣿⢿⣿⡇
⢰⠭⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⣠⡤⠦⢄⠛⣿⣿⣿⠋⠀⠀⠀⠀⢨⡵⣎⠶⣀⠄⠴⣤⣙⢮⣻⣿⣿⡇
⡀⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠚⠑⠻⠦⣥⡤⡛⡏⠀⠀⠀⠀⠀⠀⡷⠒⠀⢠⡶⠶⣌⠻⣷⡉⣋⡈⠁
⡐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⢉⣤⣾⣟⣶⣶⠈⢳⣀⡀⠀⢨⣤⠀⠀⠀⡄⢰⢡⠈⠃⠘⢆⢸⣋⣿⡽⣇
⡨⠅⠀⠀⠀⠀⠀⠀⠀⢀⣼⣽⣯⠟⠉⠁⠁⠉⣒⣭⣽⢣⣄⠀⠀⠀⠀⠀⣱⠴⣷⣾⣺⣦⣙⠛⣘⣿⡯⣿⡄
⢨⠬⠀⠀⠀⠀⠀⠀⢠⡶⣿⣻⡇⠀⡄⠀⢀⠈⠹⣿⣥⣿⡟⡀⢀⠀⠀⠀⠘⠃⣿⠁⠀⠈⠻⣿⣿⣿⣷⣿⡇
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠀⠠⠀⠀⠀⠤⠂⢸⠛⠃⠀⢀⣀⡄⠀⠀⠢⡙⠛⠈⠀⠀⠃⠀⢽⡛⢹⡣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠈⠀⡄⢀⠂⠈⠀⣠⣤⣔⠉⢑⠀⠤⠀⠀⠀⣼⣁⡉⢀⡀
⠺⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠄⢠⠀⠴⠆⠀⠐⠀⠀⠀⠈⢿⣽⡆⠀⠀⠀⣼⡟⣧⣤⠬⡅
⢂⢊⠀⠀⠀⠀⠀⢠⡐⠀⠰⠛⠋⠀⠀⠀⠀⠀⠀⠠⡦⡀⠨⠀⠂⠀⠀⠀⢀⢨⠻⠀⢀⣀⣀⠿⣿⣿⡟⠉⠁
⠊⠁⠀⠀⠀⠀⠀⠀⠐⢊⣤⠠⡶⣀⡄⠀⠀⣀⣈⡁⠀⢰⡆⠀⠀⠈⠀⠀⠀⠀⣤⣶⠿⠷⣮⣷⣽⢿⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠌⡅⠀⢀⣴⠶⢦⣄⠰⣿⣇⠀⠠⠀⠀⠀⠀⠀⠀⣾⣟⣡⣶⣿⣶⠜⠿⠧⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡠⢂⠘⠒⠀⠒⡄⠘⠆⠈⠣⠐⠂⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⡀⢀⠄⠀⠀⠀⠀⢀⠀⣠⠀⢀⠀⡀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⠣⠂⠨⠕⠀⠋⠀⠀⠸⠀⠨⠢⠮⠀⠢⠃⠂⠀⠤⠋⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣠⠤⢤⠄⢰⠒⡩
⠀⠀⢠⣦⠀⠀⠀⠀⠐⠟⠀⢲⠤⠀⠀⢠⠟⠁⣀⡤⠒⠋⡐⠁
⠀⠀⠀⠈⠙⢖⣤⡀⠀⡠⠂⠀⠀⠀⡴⣣⡴⠋⠉⠉⣠⠞
⠀⠀⠀⠀⠀⠈⣇⡉⠙⠳⣦⣤⣤⡴⠿⣿⣓⣈⣉⣁⡀⣤⣖⣚⡉⠛⠳⠶⣄
⠀⠀⠀⠀⠀⠘⣿⠙⢦⡀⠀⠙⢦⠀⠀⠀⠉⠑⢤⣀⢀⣸⡀⠁⠙⠲⢄⡈⠉⠳⣤⣄⡀
⠀⠀⠀⠀⠀⢰⠉⢷⡈⠳⣀⡴⠾⣶⠦⣄⠀⢀⠞⠾⠋⠀⠉⠛⠦⣄⠀⠈⠙⠺⠠⠤⠭⠟⡚
⠀⠀⠀⠀⠀⠀⡄⠀⠹⡄⢹⡇⠀⠀⠁⠫⡹⣞⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠁⠀⣇⠀⠀⠀⠀⠀⠜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⣠⠇⠀⠀⠀⠀⠀⠐⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠋⠀⠀⠀⠀⠀⠀⠀⡿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⠳⢄⣀⣀⣀⣠⣄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠉⠉⠛⠶⢶⣶⣤⣤⣤⠖⠛⠉⠉⠉⠛⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣊⡁⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠀⢀⠤⡈⡓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠨⠖⠛⡁⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣾⢀⣄⣀⣀⣀⠈⢇⢾⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⣡⣶⣶⣿⣆⣄⣦⣤⣷⣦⣶⣶⣆⣠⣀⣀⣠⣴⣾⡟⠛⠉⠛⢿⡋⠉⠁⠈⠀⠘⠛⠻⠿⣷⣦⣄
⠀⠀⠀⠀⢠⣾⣿⣿⠿⠿⠛⠛⠉⠉⠩⠕⠛⠁⠀⠀⡝⣻⠛⠛⢿⠯⡵⠄⣦⠈⠀⠀⠀⠀⠙⠀⠀⠀⠀⢠⢀⡀⢐⣶⣿⠟⠛⠂
⠀⠀⠀⠀⠈⠋⢷⣶⡄⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠁⠘⡏⠀⠀⠀⠀⠀⠀⢦⠀⠠⡆⠃⢉⢴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠻⠀⠀⣄⠀⠀⢠⡇⠐⠀⠈⡄⠀⠀⠀⠃⠙⢁⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠦⡀⠀⠀⠀⠀⡄⣸⠀⠀⠀⠠⠀⠀⠀⠌⠁⠀⠀⢘⣧⠀⣾⡀⣠⣰⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣽⣦⣽⣿⣾⣴⣴⣴⣤⣤⣤⣶⣶⣷⣶⣿⣿⣽⣵⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠉⠉⠛⠛⠛⠓⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣠⣦⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠚⠛⠛⠛⠈⠁⠉⠉⠁⠀⠀⠉⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣠⠖⠋⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⢸⡇⠀⠀⢀⠀⣠⠊⠁⠀⠀⠀⠀⠉⠲⣄⠀⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⢸⡇⠀⠀⠂⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠈⠒⠄⡀⠀⠀⡀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡈⠳⢤⣀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠂⠀⠀⠀⠀⢻
⠀⠀⠀⢸⡇⠀⠀⠀⣼⠀⢀⠔⠈⠀⠀⠀⠈⠢⣀⠀⠈⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⢴⡦⣤⡾⠁⠀⢀⠀⢻⠀⠆⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠲⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀
⠈⢳⡀⠀⠀⠐⣏⠉⢻⠈⡀⠀⠀⠀⠛⠀⠀⠀⠃⠀⠀⠀⠘⠗⠀⠀⠀⠀⡄⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣽
⢰⡶⠿⠆⠀⠀⠈⠲⣶⠆⠡⡀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠙⠒⣚⡉⠙⡆⠀⠀⠀⠀⢶⡶⠶⠛⠁
⣀⡵⠦⠄⠀⠀⠀⡜⠁⠀⠀⠈⠀⠀⠀⠰⣯⠅⠀⠀⠑⠢⠤⠤⠤⠐⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⣠⠇⠀⢀⠀⠀⠀⢉⣻⡖
⠉⣻⣓⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⢴⠚⡳⠒⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠒⠒⠋⠁⢀⣀⣀⣛⣿⡛⠉⠁
⠘⠛⠉⠉⠀⠉⠙⢧⡀⠀⠀⠀⠀⠀⢺⣀⠉⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⢦⣤⠦⣤⣀⣈⣻⣍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣋⣁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠉⠉⠒⠲⠦⠤⢤⣤⣤⡤⠤⠖⠚⠉⠁⠀⠈⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⢠⡾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀
⠀⠀⠀⠀⢠⡟⠀⢘⣦⠶⠶⠻⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠓⠶⢦⣼⠃⠈⣷
⠀⠀⠀⠀⢸⡇⢀⣼⡁⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣇⡀⢹⠄
⠀⠀⠀⠀⠘⢷⣤⡼⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠸⣦⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠲⣾⠏⠉⠁⠀⠀⠀⠀⠀⠀⠹⣶⠶⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣥⣤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣬⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⠾⢷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠉⠻⣷⣄⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣀⣀⣀⣀⣀⣀⠀⠀⠀⢈⣿⡿⠛⠋⠉⠙⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠁⠀⢠⣿⠏⢀⣴⡄⠀⢠⣦⡀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠛⠛⠁⠀⠈⠛⠛⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⣴⡿⠿⢿⣦⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣄⡉⠁⠀⠈⢉⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⣀⣀⣠⣴⣾⠟⠉⠙⠻⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⠛⠛⠛⠋⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣶⣾⣟⠁⠀⠀⠀⠀⠀⠀⢘⣿⣶⣦⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠉⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠈⠉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢷⣶⣶⣾⡿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⣶⡶⠶⠶⢀⠀⠀⠈⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⢲⡄⠀⠀⣰⡟⠀⠐⠷⠙⠃⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠋⣉⣠⠤⠖⠂⠈⣇⠀⠀⣿⠀⠀⢴⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠤⠖⣚⣋⣥⠀⢹⡄⠀⣿⠀⠀⠘⣆⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⢩⢴⠖⡀⠈⣧⠀⢻⣇⠀⠀⠈⠳⣄⠙⠻⣦⣄⠀⠀⠀⢰⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢏⣛⣠⡃⠀⢹⡄⠈⠻⣦⡀⠀⠀⠈⠑⠦⡈⠙⠃⠀⣰⡿⠁⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣀⣀⣭⢼⡿⠛⠋⠁⠀⠀⣹⡿⣶⣤⣀⣀⣀⣀⣠⣴⡾⠋⠀⢰⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠈⣇⠀⠀⠀⠀⣰⢿⡇⠀⠉⠛⠛⢿⣿⢿⡍⠀⠀⠀⣾⢡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⢀⣴⠋⣼⠇⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡾⠃⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣰⣟⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⢿⠷⠶⣶⠶⣦⣤⣤⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣻⡄⢸⡇⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠁⠘⠛⠛⠛⠛⠛⠋

⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⠿⣿⣿⠿⠿⣿⡿⣿⢛⣹⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⡿⢛⣭⣭⡍⣿⣿⣿⣿⣿⡿⣋⢿⣿⡆⣿⡏⣸⣿⠇⢟⣵⣿⣷⠹⢸⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣷⢸⣿⠟⣱⣿⡿⣿⣯⢻⡇⣿⣿⣿⣇⠛⣴⣿⣿⡄⣿⣿⢃⣾⣧⢸⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⡇⢵⣾⣿⣿⣇⢿⣿⠸⣇⢿⣿⢇⣿⢰⣯⣭⣭⣿⣮⣵⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣜⢋⣦⣻⣷⣵⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⣷⡶⠦⠄⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⠀⠀⠠⠴⠾⠿⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣶⣶⢿⣿⣿⣏⢹⡟⣿⢟⣭⣯⢹⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢿⣿⣿⠈⣷⢱⣿⡿⢣⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⢠⡸⣧⠛⣱⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢟⣼⣷⣹⡞⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⡟
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠔⠛⠀⠀⣠⠀⠊⠉⠉⠉⠉⠒⢤⡀⠀⠚⠓⢄⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⢀⡀⢀⡄⢀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠋⠀⠋⠰⠋⠞⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢐⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⡀⠀⠐⠢⠄⠀⠀⠤⠒⠀⠀⣀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠣⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠒⠂⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠤⠀⠠⢔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢂⡠⠮⢥⣀⣀⣀⣀⡭⠤⠄⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠉⠑⠒⠒⠒⠒⠈⠉⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢺⡿⠟⣓⣤⠤⠖⠒⠚⠁⠉⠹⡗⢾⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠚⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢤⣤⠤⠤⠔⠚⠉⠉⠉⠓⠒⠤⠤⠤⠤⠤⠞⠁

⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⡿⣿⣿⡿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣷⣿⣿⣷⣿⣾⣷⣿⣿⣷⣷⣾⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠛⠛⠛⠛⠙⠛⠿⠿⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣷⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⣀⣴⣶⣦⣄⣠⣴⣶⣦⣄⠀⠀⠉⠀⠉⠉⠀⠉⠀⠀⣠⣴⣾⣶⣦⣴⣶⣶⣦⣀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣷⣄
⠀⠀⢀⣴⣿⣿⡟⢿⠙⠏⢿⡿⠹⠋⡿⢻⡿⣿⣧⡀⠀⠀⢠⣾⡿⣿⣿⡟⠿⠘⢿⡿⠃⠿⢻⣿⣿⢿⣷⡄
⠀⠀⠀⢹⣯⣶⣴⣾⣲⣶⣦⣠⣶⣶⣿⣦⣶⣽⡯⠀⠀⠀⠈⣭⣅⣴⢆⣴⣶⡆⣴⣦⣰⣶⣤⡰⣦⣨⡭⠁
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⢿⣾⣿⣿⣧⣿⣿⣿⣿⣿⣷⡿⠋
⠀⠀⠀⠀⠀⠈⠈⠛⠛⠛⠛⠛⠛⠛⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠟⠛⠛⠁

⠀⠀⠀⠀⠀⢀⣠⣴⠞⣿⣷⣾⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣤⣤⣶⣶⣶⣄
⠀⠀⠀⢀⣴⢿⣁⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⣰⣿⣿⣾⣿⡿⠟⠛⠉⠙⠛⠻⢿⣿⣿⣿⣿⣆⠀⠠⣾⣿⣿⣿⣿⣿⠿⠋⠙⠋⠉⠻⠿⣿⣿⣿⣿⣷
⠀⠐⢿⣿⣿⣟⠻⠟⢿⠷⣶⣶⣿⣿⡿⠻⣿⣿⣿⡿⠃⠀⠘⢿⣿⣿⣿⣷⣾⣿⣷⣴⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠙⢿⣿⣷⣴⣯⣶⣧⣴⣧⣼⣷⣾⣿⡿⠋⠁⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠛⠛⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣷⣿⣾⣼⣷⣧⣿⣾⣾⣿⣧⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠆⠀⠀⠦⠄⣀⠀⣀⠄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⡶⠿⠁⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⣀⡴⠚⠉⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢿⡀⠀⠀⡆⢸⣸⣦⣶⣾⣟⣋⠀⠀⠀⠀⠀⠀⢢⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣧⣀⣰⣡⣿⡿⠟⠛⠛⠙⠋⠀⢀⢉⡑⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢇⠀⠀⠀⠀⠁⠀⠉⠃⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠐⠢⢤⠤⢤⣴⠾⠃⠀⢸⣿⣉⣉⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⢾⠟⢋⠟⠀⠀⠀⢘⣹⢿⡅⠀⠀⢀⣈⣑⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠉⠉⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⢸⡇⠐⢳⡄⠀⠀⠀⠈⠉⠙⠦⡀
⠀⠀⠀⠀⠀⡴⠁⠐⠀⠀⣤⡦⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠟⠖⠀⠀⠀⠐⠶⠶⢿⣦
⠀⠀⠀⠀⢰⠁⠀⣤⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠹⠀⠀⠀⠀⠀⠤⠶⢾⣷
⠀⠀⠀⠀⣼⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠊⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣹⣄⡸⠀⠀⢳⣄⠀⠀⠀⠀⠈⠙⠇
⠀⠀⠀⠀⠿⢅⡀⢀⠀⣀⢀⣀⣠⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⢁⡀⡀⠀⠉⠉⠙⠧⣤⣀⠈⠱⡄⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⣇⠀⠀⡴⠻⠿⣆⠀⠘⠳⣄⠀⠀⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠖⠈⠀⠀⡈⠀⠂⠀⠠⠀⢀⠁⣐⡡⠀⠀⠈⢷⣄⣢⡙⣿⣶⣞⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠰⢇⡀⢒⠀⠚⠀⠀⢡⠀⠀⠀⢈⠠⠁⠁⠀⠀⠀⠀⣀⠀⠙⢿⠍⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⠃⠀⠁⠀⠀⠉⠘⠆⠰⡀⢀⣀⣾⣥⠆⢀⠀⠒⠃⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣈⢓⣐⠀⠄⢤⡀⠀⢀⢳⣀⣿⡟⠁⣀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡎⠀⠶⠺⠦⣤⠶⠦⠢⢬⡅⠉⠈⠿⠯⡶⠛⠋⠁⠐⠚⠠⢌⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠎⠉⠃⣻⠡⣄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠄⠁⡀⣸⠆⠀⢀⠈⣰⠆⡇⠀⠀⠄⡀⢔⠢⠀⠀⠀⠀⣴⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠒⢊⡠⠾⠒⠀⠀⠠⡛⠀⠀⠀⠀⠠⠙⢄⠓⠂⠀⠲⡴⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠹⡀⠀⠀⢀⠄⠐⠋⠁⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡈⠐⠄⣀⣀⣠⠎⠀⢀⡘⡀⠀⠀⠀⠀⠀⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣤⣀⠀⠀⠀⠚⠛⠁⢀⣀⣠⡶⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠃⠀⠠⠶⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣄⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣌⡛⠿⣿⣿⠟⢃⣤⣴⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣧⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏⣿⣏⣿⡟⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⣿⣽⣿⣰⣿⣿⢛⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣜⣣⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠋⠉⠉⠛⢿⣧⣴⡿⠛⠉⠉⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⢶⣄⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⣹⡆⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁⠀⢀⣤⡤⠶⢶⣿⣤⣴⣿⣧⣄⡀
⠀⢀⣠⣤⣶⣾⠿⠛⣉⣀⣀⣀⣤⣤⣤⣶⣶⣶⣶⡶⠿⢿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⠉⠻⢿⣦⡀
⠐⣿⣿⣿⡿⠿⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣤⣤⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣶⣤⣄⣀⣀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⠿⠋

⠀⠀⠀⣠⣤⣠⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢻⡿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⣿⣿⡇⠀⢰⡇
⠀⠀⠀⢸⣷⣶⣶⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠅⠀⠀⠀⣿⣿⣷⠀⠸⠇⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⣤⣾⠀⠀⣶⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡏
⠀⠀⠀⠀⣿⣿⣿⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡞⠂⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡷⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡇
⠀⠀⠀⠀⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⣴⣧⠀⣿⡇⠀⠀⠀⠀⣿⣇⡀⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣷⣿⢰⣾⣷⡆⠀⠀⢀⣘⣿⣧⣇⡀
⠀⠀⠀⠀⢿⣿⣿⠃⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⢠⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⢸⣿⢈⡇⠀⠀⠘⣿⠂⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠘⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠈⣿⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣠
⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⣀⣤⣄⣀
⠀⠀⠀⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢹⣶⣾⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢀⡾⠋⠀⠀⠙⢷⡄⠀⠀⠀⠀⣀⣙⣛⣛⣛⣛⣛⣋⣠⠀⠀⠀⠈⠻⠟⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣾⠁⠀⠀⠀⠀⠘⣷
⠀⠀⠀⣿⣿⢶⣶⣶⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠉⠉⠉⠉⠋
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢹⡄⠀⠀⠀⢀⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠻⠲⠒⠓⠓⠚
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠙⢶⣤⣾⡷⠏⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠾⠿⢿⣿⣿⣿⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣿⣿⢚⣛⣾⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣈⡇⠀⠀⠀⠀⠀⣾⠉⠙⠻⣿⣿⠟⠋⠉⣷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⡆⠀⠀⠀⢠⣾⣿⣿⣧⡀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠦⣄⣰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⢷⣤⣀⣀⣀⣀⣴⡾⠃
⠀⠀⠀⠿⠿⠛⠛⠛⠁⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠙⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠲⠶⣤⣀⣀⣀⣤⠶⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠠⠤⣤⣀⣰⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠐⠉⠀⠀⠀⠉⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠂⠿⢿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣷⣦⣄⣀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡤⠙⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠷⠶⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠞⠋⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠈⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠄⠄⠂


⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣛⣿⣿⣛⣛⣻⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣬⢱⣰⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⢆⣷⣫⣼⣿⣯⣷⣀⣠⣄⣀⢀⣠⣆⣠⣄⡰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠠⡤⠂⡶⣼⣶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢿⣈⡿⣿⣫⣿⣿⣿⣯⢷⣀⣄
⠀⠀⠀⠀⢠⢀⣤⠶⣾⣷⣽⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢺⢻⣷⣿⣿⢻⣿⡿⣿⡿⣿⣿⢖⣘⢀
⠀⠀⠀⠀⠀⣿⡻⣆⣏⢉⢛⢦⡙⣉⣜⠟⢿⣿⡿⣿⣿⣇⣾⣳⢶⠌⣼⢻⡿⢻⡿⠷⢿⣿⣯⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠶⠚⠟⠋⣻⠝⠳⠋⠀⠏⠁⠉⠛⠀⠙⠁⠉⠈⠉⠀⠀⠀⠘⠻⠟⣿⠾⣶⢸⣏⡼⢣⣿⣷⣿⣿⣿⣿⣿⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠈⠀⠁⠀⠈⠀⠘⠋⠉⢙⠙⡿⠙⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠈⠁⠈⠀⠁⠈⠉⠀⠀⠁⠚⠂⠓⠀⠀⠀⠄⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠃⣤⠀⠠⣀⠀⠀⠀⠀⠀⠀⣤⣄⣴⣦⣴⣾⣿⣿⣷⡶⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠛⢷⣶⣦⠰⠅⢉⣂⡓⣰⠿⠿⠟⢛⢋⣻⣫⡿⢹⠃⠁⠀⠀⠈⣭⠉⠉⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡁⢠⠆⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠉⠈⠋⠙⠋⠁⠈⢀⢀⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢤⠚⠁⢠⡦⠂⠀⠟⠰⠰⠆⠠⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠴⡆⢴⡷⠈⢦⣉⠁⠀⠀⠤⠨
⠀⠀⠀⠀⠀⠀⠈⠙⠛⣅⣿⠈⡴⠠⣇⠐⡀⠀⠆⢰⣅⢴⡏⢀⣴⠲⠁⡇⠀⢢⡶⠈⠘⡟⢶⢉⢠⠘⡃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠄⡦⢩⠀⢰⠃⢀⠘⠋⢨⠅⠀⢰⢘⠀⡥⠀⢤⣀⡀⠘⡀⢀⠋⡠⢐⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⠰⣺⣶⣽⣠⣤⣵⣀⡷⣟⡙⡺⣷⠎⣏⠉⣡⣴⠇⠈⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠛⠻⠛⠿⠿⠿⠷⠾⠷⠚⠛⠛⠛⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠁⠀⠀⠀⠁⠁⠉⠉⠉⠛⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠤⠋⠀⠀⠙⠢⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠓⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡠⠏⠀⠀⠀⠀⠀⢀⣀⡄⠘⠁⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠀⠀⠀⢷⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠉⠁⠀⠀⠀⠀⣀⣀⠀⡀⠂⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠈⠁⠀⠀⡾⠁⡽⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡭⠟⠁⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠄⠀⠀⠀⠘⢏⠀⠀⢈⣉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣶⣞⣀⡀⠙⠒⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢺⠑⠆⠀⠀⠀⢈⣆⣀⣀⣲⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⢾⠥⠴⠒⠶⠚⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠁⠀⠀⢀⡼⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⠒⠂⠤⣀⣀⣀⣀⣀⣤⠤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢱⡆⠀⠀⠀⣀⣈⡧⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠉⠉⠓⠚⠁⠉⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠠⡄⠀⣧⠀⠀⠀⠀⠀⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢧⠀⠘⡄⠀⠀⠀⠀⣧⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⢗⠉⢣⠀⠀⠀⢐⠇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢧⣨⠀⠀⠀⠸⣆⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⣶⠦⣤⣤⣤⣀⣀⣤⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡏⠀⠀⠸⡇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⣟⣿⣿⣿⣿⣟⡟⠂
⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠞⣟⣿⡇⠻⠻⣿⣿⣏⡿⣯⣿⡿⢿⣿⡏⣸⣿⣿⣿⢏⣿⣿⣿⣿⣿⣧⡤
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⣏⠻⠛⠗⠀⠀⠛⠁⢈⣃⠈⠟⠁⠈⠛⠀⢿⣿⣿⣏⣾⣿⣿⡟⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠸⣿⣿⡿⣿⣯⣽⣿⣿⣿⣿⣾⣿⣿⡄⠈⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣛⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠐⠈⣁⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣬⣥⣴⣶⣾⣿⣷⣿⣿⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⢏⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠹⣿⣿⡿⣿⣿⣿⣿⣻⣯⣿⣯⣿⣿⣿⣿⠛⠻⡟⢻⣿⣿⢻⢻⣿⣿⣿⣿⣿⣻⣷⣴⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣹⣁⡟⢤⣿⡟⣿⡘⣿⣿⣿⣅⣹⢿⣿⣿⣿⣿⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠿⣿⣿⣿⣿⣸⣷⣿⣿⣿⣾⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⣿⣿⣿⣿⠿⠿⠿⠟⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣆⢰⡆⢰⠆⢶⡶⠂⣶⠶⢠⡆⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢻⡾⢷⡟⠀⢸⡇⠀⣿⠛⢈⡃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢆⠀⢀⠤⠤⠤⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣦⡀⠀⠀⠀⢀⣴⣧⠀⠀⠁⠀⢀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣀⣙⠻⢶⣾⠉⠉⣿⡿⠉⢉⣶⡶⠟⣋⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⡬⠀⠀⠙⠀⠀⢹⡇⠀⠘⠃⠀⠀⢤⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠉⠋⠁⠀⠠⣋⡝⠷⠀⠀⠀⠀⠿⣡⢸⠀⠀⠉⠋⠋⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣷⣦⣤⣤⡤⠀⠀⠉⢙⠁⠀⠐⠂⠀⢘⠋⠉⠀⠀⢤⣤⣤⣶⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣀⠤⠒⠒⠒⠢⢄⡘⠀⠀⠀⠀⠀⠀⣇⡠⠔⠒⠒⠒⠤⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⢰⠁⠀⠀⠀⠀⠐⠀⠹⣿⣶⣶⣶⣶⣿⠏⠠⠀⠀⠀⠀⠀⠈⡆⠚⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠸⡀⠀⠀⠀⢂⠠⠀⠐⠈⠙⢻⠛⠋⡁⠀⠀⠀⡔⠀⠀⠀⢀⠇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡑⣄⠀⠀⠈⠳⢤⣐⣀⣠⣼⣧⣄⣀⣀⡬⠚⠀⠀⠀⡠⢊⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠤⣀⠀⢸⠻⣿⣉⣉⣉⣉⣿⠿⡇⠀⣀⠤⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⡀⠀⠀⠀⠀⠀⠀⢀⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣑⡤⠤⢀⠀⠤⢴⣊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⣴⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠋⠙⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⢀⣠⠴⠒⠉⠉⠉⠉⠉⠑⠒⠦⣄⡀
⠀⢀⡔⡟⠁⢀⡤⠔⢆⠉⠀⠉⠐⠄⠤⢀⠙⠳⣄
⢀⡾⢸⠀⢀⣏⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠙⢦⠈⠳⣄
⣼⠁⡆⢀⠇⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠘⣧
⡟⠀⡇⠘⡄⠀⠀⠀⠀⠈⠒⢄⠀⠀⡠⠊⠀⠀⠀⠀⢠⠘⣇
⣇⠀⢻⠀⢈⠉⠉⠉⠉⠁⠀⡠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⢻⠀⠈⢧⠘⣄⠀⠀⠀⡠⠊⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⠀⠈⡇⠀⠀⠀⠀⠀⠤⠤⣤⣀⣀
⠘⣆⠀⠈⢧⡈⠂⠤⠋⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⢀⠂⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢤⣦
⠀⠘⢄⠀⠀⠑⢄⡀⠳⢄⠀⠀⢸⠀⠀⠀⠀⠑⢄⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠈⠢⡀⠀⠀⠉⠢⢄⡉⠀⠚⠀⠀⠠⠤⢰⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠈⠑⠒⠦⠤⠤⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠴⠒⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣷⣶⣶⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⡤⠖⠒⠉⠉⠉⠉⠑⠒⠢⠄⡀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⣰⠏⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⠀⠀⢀⣀⣠⣤⣤⣤⣀⣀⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⢠⠏⣠⣄⣀⠀⣲⡶⠆⠀⠀⠀⠈⢷⠚⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣄
⠀⠀⠘⠶⣤⣤⣤⠶⠞⠋⠁⠀⠀⠀⠀⣾⠋⠁⠹⣧⠞⠉⠙⣆⠀⠀⠀⠀⠈⣷⠒⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢦⣤⣠⣏⣀⣀⣀⣽⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⢀⣤⠶⠶⠶⠒⠒⠲⠶⠶⠤⢤⣤⣼⣀⡘⢿⢿⠃⠉⣿⣧⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⢀⣾⠇
⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣯⡛⠓⠦⣄⣠⠏⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⡴⠟⠃
⢸⣧⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠙⣿⢿⣶⣿⣦⣤⣀⣄⣀⣴⣇⣀⣀⣠⣤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀
⠀⢰⠃⠀⠀⠉⠙⠛⠛⠛⡟⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀
⢀⡏⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⣏⠀⠀⢀⡀⠀⠀⢀⣴⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠻⣄⡀⠀⠉⠛⠛⠋⠁⠀⠙⠻⠶⠶⠶⠶⠶⢦⣤⣤⣤⣀⣀⠀⢀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠈⠙⠛⠛⠛⢳⣄⠀⠀⠀⠀⣀⡶⠶⠶⠦⣤⣤⣄⣀⣀⣉⣀⣋⣠⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠋⠀⠀⠀⠀⠀⣧⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢰⠇⠀⠀⢀⡴⠒⠀⠉⠑⠒⠦⢄⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣸⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡟⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡿⢧⠀⠀⠀⠀⢰⡇⠀⠀⢠⡇⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡇⠈⢧⠀⠀⠀⢸⠃⠀⠀⣸⠃⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⠇⠀⠈⢧⡀⠀⣿⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠀⣠⠴⠦⢷⣤⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⡜⠁⠀⠀⠀⣸⠇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⡄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⡿⠁⠀⠀⣀⣀⣿⠀⠀⠀⡟⠀⠀⠀⣴⠖⠛⠒⠶⠦⣤⣄⣀⡈⢳⡀⠀⠀⢷⠀⠀⠾⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠖⠚⠃⠀⠀⠘⠷⢤⣴⠛⠉⠉⠉⠀⠀⠘⠛⠛⠛⠻⡇⢀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠀⠀⢸⡆⣴⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⣿⢸⠀⠀⢰⡇⠀⠳⡄⠀⠀⠀⠀⠀⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣼⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠀⢠⡟⠀⠀⠀⠀⢸⠀⠀⠀⣿⢿⣄⣀⣸⣇⣀⣠⡟⠛⢓⣶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⣀⣀⣀⣀⡿⠛⠛⠛⠛⠛⠒⢾⡇⠀⠀⠀⠀⣸⣀⣀⣠⡟⠀⠈⠉⠁⠀⠈⣁⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢈⡙⠛⠷⠾⠛⠋⠉⠉⠉⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠛⠲⠶⠶⠤⠤⠴⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉

⠀⠀⠀⠀⠀⠦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠓⢄⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠶⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⢳⣇⠀⢀⡄⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⢻⠀⡜⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⣷⠀⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⡇⠀⢰⠏⠀⠀⠀⠀⠀⣠⠤⠀⠢⣤⠤⢄⡀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠉⠉⢔⡢⢄⠀⢀⣼⠁⠀⠀⠀⠁⠈⠁⠚⠍⡲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢗⣼⣿⠈⠀⠀⠀⠀⢠⠀⠀⠀⠘⠢⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⢆⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⡴
⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⠣⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠸⡀⠀⠀⢘⠨⡀⠀⠀⠀⠀⢀⠰⢢⠲⡶⠶⠘⠂⠠⠤⢈⢶⠄⣀⠀⠠⠀⠀⠴⣾⡝⢁⡂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠿⡄⢣⠀⠀⠀⠂⠁⠀⠆⠀⠁⠀⠈⠀⠀⠀⡜⣸⠀⠀⣉⣐⣶⠶⢏⠁⠀⠀⠀⠀⠀⠬⢔⣦
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠒⠱⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡰⠛⢿⠛⠋⠛⠀⠀⠀⠈⠐⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀
⠀⠀⢀⡠⠔⠒⣉⣙⣒⣊⣉⣉⡒⠂⠄⠠⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠉⠉⠀⠙⢄
⠀⠀⠀⡑⠤⢤⣍⣀⣀⣀⣀⣤⠤⠔⠁⠀⠀⠀⠤⠴⠮⢍⣁⣒⣈⣩⣴⣦⠤⠄⠀⠀⠀⠀⠀⣠⡔⠐⠂⠒⠒⢤⡀
⠀⠀⠀⠘⠢⣄⡀⠀⠉⠁⠀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠢⢄⠀⠀⠈⠓⠒⠁⠀⡄⠀⠀⠀⠐⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠠

⠀⠀⠀⠠⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠄⡀
⠀⠀⠀⠐⠒⠤⢬⣥⣄⣀⣉⣑⣢⣤⠤⠀⠀⠀⠈⠀⣀⣀⣀⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠿⠿⠶⣭⣤⠤⠀⠀⠀⠀⠀⠖⠚⠡⠾⠯⠋⠆⠑⠢⠄
⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⡀⠀⠀⣀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⡠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣀⣀⡤⣄⡀
⠀⠀⠄⢤⣀⣀⣈⠡⠤⠴⠒⠒⠒⢰⠟⠀⠀⠀⠀⡠⠊⠀⣀⣀⣀⣀⠀⠑⡄⠀⠀⠀⢠⣐⣊⣁⣀⣄⣀⣀⣤⣀⣉⣒⣠⡄
⠀⠀⠀⠀⠈⠑⠒⠲⠶⠶⠶⠖⠛⢁⠄⠀⠀⠀⢰⣁⣴⢮⣤⣤⣤⣤⣽⣦⣈⡄⠀⠀⠀⢍⠛⠦⢤⣄⣀⣀⣤⠤⠖⢛⠅
⠀⠀⠀⠀⠀⠀⠂⠄⣀⣀⣀⡀⠄⠊⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠓⢤⣀⠀⠀⠀⢀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠔⠊⠉⠉⠉⠉⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠲⠶⠖⠒⠦⣄
⠀⠀⠀⠀⢀⣴⣋⣉⣉⣉⣉⣭⣦⣄⠀⠀⠀⠀⠢⣤⣄⠠⠥⠤⠒⠒⠒⢀⡾⠃⠀⠀⠀⠀⠁⢀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠙⠻⠿⣾⣿⣿⡿⠟⠡⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀⡀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⢀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠐⠠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢂⠀⠀⠓⢄⢀⡤⠤⠤⣤⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠆⠀⠀⠄⠈⠣⡀⣀⣸⠀⠘⠀⠀⡀⠀⠀⠀⠀⠆⠀⣤⢆⠈⠀⠀⢀⣤⠀⠀⠂
⠀⠀⠀⠠⠀⠀⠀⣀⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠄⠀⠀⠀⠠⠀⠀⠀⠀⣄⠘⢿⣦⣽⣟⣿⣿⡿⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠡⡀⠉⠉⠉⠉⠉
⠀⠀⢈⠀⡀⠀⡀⣦⡀⠀⠀⠀⠈⡄⠀⠀⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠀⠁
⠀⠀⢈⠃⠈⠓⠄⠀⠀⠀⠀⠀⠠⠁⢰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠈⡉⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣
⠀⠀⠀⠈⠢⢀⡀⠀⠀⠀⠀⡄⡀⠀⠠⣈⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠥⠀⠀⠀⠀⠀⠀⠀⠈⠰⡄⠁⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠉⠃
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢣⠀⠀⠀⠠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠠⠀⠀⠠⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⡀⣀⠠
⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⠀⠀⠤⣐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠄⣄⣀⠀⢀⡀⠀⠀⠀⠀⣀⠄⠀⡀⣀⠀⣀⡂⠰⠐⠘⣟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⠉⠀⠁⠈⠡⠀⠂⠘⠹⠅⠆⠚⠁⠘⠀⠉⠁⠒⡈⠀⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⢚⡀⠀⡀⠀⡄⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠠⠀⠉⠀⢰⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠛⠛⣺⣯⠖⠀⣀⠀⠰⠀⠀⠴⠀⠀⣂⣐⠒⠀⠀⠀⠀⠐⡼⡟⠉⠂⡆⠂
⠀⠀⠀⠀⠀⠈⠁⢀⣀⠀⠀⠤⠀⠘⢻⣿⣤⣶⣌⣀⣀⠁⢲⣀⡀⠀⠉⣡⡔⣻⣿⡟⢹⣿⢿⣤⣀⡀⠀⠁
⢀⡠⠤⠀⢀⡞⠈⠀⠀⠀⠀⠒⣆⠴⠛⠉⠑⠙⠋⠋⠙⠛⠳⠙⠋⠉⠞⠉⠉⠉⠉⠃⠀⠀⠀⠀⠀⠉⠙⠛⠒⠄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠁⠉⠈⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠑⠒⠊⠀⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⣤⣤⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠁⣤⣀⣠⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣄⡀⠀⠀⣀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠻⠿⠿⣿⣿⠿⠿⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⣛⣋⣩⣭⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣬⣍⣉⣛⡛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠁⠉⠈⠙⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡄⠐⠒⠒⠒⠒⢀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣷⣶⣤⣴⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢁⣀⣀⣀⡈⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⢻⡄⠀⠀⣀⣠⣤⣤⠴⠶⠶⠶⠶⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠸⡷⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠹⣧⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠙⢷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣷⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯
⠀⠀⠀⠀⠀⢰⡞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠿⠋⢡⣏⣠⠖⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠋⠉⠀⠀⠀⠴⠟⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⢀⡀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢰⡾⠋⠉⠙⣧⠀⠀⠀⠀⠀⠀⣰⠋⠀⠉⢿⡤⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠾
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠈⠛⠁⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠉⠀⠀⠀⠘⠓⠂⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⡀⠀⠀⠀⠀⠀⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣠⡀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠇⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠦⡄⠀⠀⠀⠀⠀⠀⠈⠁⠸⣇⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢿⣀⣒⡀⠀⠀⠀⠀⠀⠀⡞⠁⣠⠶⠚⠲⡄⠘⡄⠀⠀⠀⠀⠀⠀⠀⢠⡘⢷⡀⠀⠀⠀⣠⢦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠉⠉⢳⡀⠀⠀⠀⠀⠀⠙⣦⠙⣆⠀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⢠⣤⣤⡼⠃⠀⠀⣰⠏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠖⣿⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢀⡏⡼⠉⠻⠙⡇⢹⡄⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⣠⡀⠀⠀⠈⠳⣄⡀⠀⢸⡌⢧⣄⣠⠜⠁⣸⠃⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠟⠁⢳⣀⣤⠶⢆⡀⣹⢶⣤⣉⡓⠤⠤⠴⠚⣥⠶⣾⣭⣀⡤⢤⣠⠶⠳⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠛⠁⠀⠈⠉⢻⠀⠀⠀⠀⣿⠀⠀⠀⠉⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉


⣶⣶⣶⣶⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣷⣾⠷⡾⠿⢿⠿⣿⣶⠶⠶⠶⠶⠶⢿

⠀⠀⠀⢸⠉⠉⢲⠀⣠⢄⡀⣠⢄⡀⡁⢀⡤⣀⠀⡀⢤⡀⠀⠀⣎⠉⠙⠄⣀⠤⡀⢀⠤⣀⢸⠀⡠⢄
⠀⠀⠀⢸⠀⠀⣸⢻⡒⠒⡃⡑⠲⡄⡇⣏⠀⢸⢀⡇⠀⡇⠀⠀⡌⠉⠓⡄⡇⠀⡀⡤⠒⣻⢸⢸⡒⠒⡃
⠀⠀⠀⠈⠉⠉⠁⠀⠙⠊⠁⠉⠈⠁⠁⠨⣉⡽⠈⠁⠀⠁⠀⠀⠉⠛⠉⠀⠈⠈⠁⠉⠈⠉⠈⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠐⡆⢂⠆⠀⢠⠠⡠⠠⠰⢤⠀⢀⠄⠀⢰⠐⡱⠀⠀⡤⢄⡀⠄⡆⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠁⠀⠀⠈⠀⠈⠂⠀⠈⠀⠈⠁⠀⠈⠈⠀⠀⠀⠁⠈⠁⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠠⡀⢠⣤⡄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠓⢲⡄⠙⠇⠀⠀⠀⠀⠀⠀⠃⠀⡀⢀⡊⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡀⣹⣤⠀⠀⠰⠆⠀⠀⢰⡂⠆⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣶⡞⣰⡄⠀⠀⠀⠀⠀⣤⢸⣶⠔⠸⠆⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢻⣾⠷⠃⠀⠀⠀⠀⠀⠙⣀⣆⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠠⠀⠀⠀⠠⣉⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠆⢈⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢉⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠘⣤⡀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣤⣤⠽⢥⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠘⢁⣠⣤⣀⡀⠻⣧⠀⠘⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠁⠈⠙⠂⠀⠛⠣
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⠾⠿⣿⠀⢸⡦⡆⠀⠀⠺⠖⠀⠀⠀⢸⡁⠐⠀⠄⠀⠀⠀⠀⡀⠈⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢀⣌⣶⣴⢁⣦⡀⠀⠀⠀⠀⠀⠀⣀⠠⣦⡆⠄⠸⠄⣈⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠸⣿⡥⣿⡱⠁⠀⠀⠀⠀⠀⠀⠻⠀⢩⠆⠀⠀⠀⠉⡄⢓⢬⣤⣹⣀⡴⠂⠀⠀⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠺⠂⠀⠀⠀⠤⠈⠈⠃⢠⠋⢈⡵⠀⣠⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠂⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⣴⠈⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⢠⣿⣧⣤⣦⢤⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠠⠀⠀⠀⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⠀⣲⠚⡒⢤⠉⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⢆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠲⠄⠈⠢⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠉⠛⠲⣤⡠⠙⠇⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⣤⣾⣿⢻⣶⡆⠙⣷⡀⠀⠀⠀⣤⡄⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⡿⠋⠉⠈⠙⠃⢂⣙⣹⢃⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢤⢀⣈⠲⠀⢀⠀⠀⠀⠀⠀⠰⣾⡿⡯⠀⠀⠀⠀⠀⠘⢷⣿⢃⡾⠳⠀⠀⠀⠀⠀⠀⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠐⠦⠉⠞⠱⡋⠀⠀⠀⠀⠀⠀⢈⣷⠇⠀⠀⠀⠀⠀⠀⠈⣿⣿⡛⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠃⠀⠚⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⡌⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⠐⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡀⣀⡀⠒⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣀⠛⢻⣿⣿⠉⠁⠀⠀⠀⠀⠀⣀⠤⡀⠚⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡖⠊⠐⠷⢌⣂⢀⠻⡏⠐⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⢠⠴⣤⣀⠀⢩⣢
⠀⠀⠀⠀⠀⠀⠀⢈⣤⣶⢦⣤⣄⠻⣧⠀⠈⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠒⠂⠀⠍⠀⠤⠈⠓⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠿⠻⠛⠻⠗⠤⠺⢳⡆⠀⠀⠉⠉⠀⠀⠀⠻⠀⠉⠀⢂⠀⣀⠀⢨⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣷⡏⣠⢷⠀⠀⠀⠀⠀⠀⠀⣤⠘⣾⠇⠀⠚⠂⣄⠀⡀⠀⠂⣄⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢿⣶⠯⠞⠀⠀⠀⠀⠀⠀⠀⠛⠀⣰⠀⠀⠀⠀⠰⡌⣭⡲⢦⡭⣠⠾⠀⠀⠀⠀⠘⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠒⠀⠀⠁⠶⠀⢰⠎⢀⡴⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⢀⣶⣶⣶⣶⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣷⣦⣄⠀⠙⠛⠷⠄
⠀⠀⠀⠀⠀⢠⣶⣾⣿⣶⣾⣿⡟⠁⢠⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠿⢿⣁⣿⣿⡿⠋⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣷⣤⣤⣤
⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⠁⣠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⢰⣿⣿⣿⠏⠀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⡄
⠀⠀⠀⢰⣿⣿⡿⠁⢠⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠿⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⣀
⠀⠀⠀⠸⣿⠏⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠿⢿⣿⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠂⠀⠀⠀⠀⠀⠻⣧⠀⢿
⢄⡀⠀⠀⠁⢀⣴⣿⣿⣿⡟⠀⠀⢀⣴⣿⣿⠿⠛⠉⠁⠀⢀⣠⣿⣷⣄⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆⠈
⣼⣄⠀⠀⣠⣾⣿⣿⣿⣿⠇⢀⣴⡿⢟⠁⠀⢀⣀⣀⣠⣴⣿⠋⠉⠻⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡿⠋⢀⣜⣷⣾⣿⣿⣿⣿⣿⣿⣦⣤⣤⣿⣿⣏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⣤⣶⣤⣭⣝⣻⣷⣄⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⡿⣿⣿⣿⣿⣮⣁⡀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⡉⠻⠛⠛⠿⠿⠟⠿⡿⠿⠃⠀⠀⠀⠀⠀⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣴⡿⠿⣿⠿⣿⣿⣿⡿⠟⠋⠹⣿⣿⡇⠀⢀⣴⣶⡄⠀⠀⠀⢰⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠈⠁⠐⠻⡿⠿⠛⢉⠉⠀⠀⠀⠀⢸⣿⠀⠀⣼⣿⣿⣷⡀⠀⠀⠸⣿⣿⡆
⠸⠿⠛⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠚⢀⣤⣶⡀⠀⠀⠀⠀⠀⢸⣿⣆⠀⣻⣿⣿⡿⠗⣤⡀⠀⠉⢻⣇⣀⡀⠀⢀
⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⢀⣤⣶⣿⡟⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⣷⣤⠄⢠⡍⠀⣉⡀⠀⠠⡀
⠀⠀⢈⣷⣿⣿⣿⠇⠀⠀⠀⢸⣿⠿⠉⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁⠀⠀⠐
⠀⠀⠈⠙⣿⣿⣿⡄⠀⠀⠀⢸⠛⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⠻⠟⠉⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠉⠀⠀⣴⣿⠀⣽⠚⡀⢉⠀⠀⠂⣾⠀⣰
⠀⠀⠀⠀⠀⢻⣷⣇⠀⠀⠀⠀⠀⠀⢀⡿⠋⠉⠁⠀⠀⢰⣿⣿⣧⣼⣿⣧⡄⣀⣠⣼⣿⣷⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠸⠁⠻⠂⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⡅⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠿⠿⠟⠚
⠀⠀⠀⢀⣀⠀⠀⠈⢁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⡞⠁⠟⢠⣿⣷⡾⡏⠠⠄⠀⠰⠿
⠀⠀⠀⢸⣿⣇⡀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠚⠁⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⢿⣿⠗⠀⣴⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⣠⣿⠀⡀⣈⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⣿⣿⣿⣿⠀⠙⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⣿⣿⣿⣿⠀⠀⠸⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠿⡏⠸⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠸⠟⠛⠉⠋⠁⣀⣀⣄⢰⣦⠹⣷⡘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡏⢀⡟⢳⣿⣷⣶⠆⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⢹⣯⣷⣾⣿⣿⣿⣧⠀⢢⡙⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⢻⣷⣶⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⡅⠙⣧⣄⣤⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠷⢾⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡿⠛⠛⠻
⣿⣿⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠀⠈⠉⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⢀⣀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⣹⣼⠺⡎⣏⣱⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡺⡶⢤⡐⢾⣾⡟⣼⣈⣵⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡎⢅⠢⡻⣮⡂⢳⣿⡿⠟⣩⣾⣿⣝⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠉⣿⣮⣡⣄⣘⣿⣿⠟⣡⣾⣿⣿⣿⡛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣾⣿⣿⣭⠿⣛⣥⣾⣿⣿⣿⣟⣎⣳⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣻⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣟⢷⣽⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⢿⣿⣿⣿⣿⣿⣶⣿⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣾⣛⣯⠮⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⣠⠾⢁⣴⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⡴⢋⣴⣞⣃⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⣀⣀⣴⣿⣿⣿⣽⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣆⠀⠉⠉⠉⣉⣀⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⣧⣸⠏⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡖⣡⣾⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡤⣶⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⣴⣿⠋⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢰⣯⣴⣿⣱⡿⠟⠘⢴⣀⣀⠤⠐⠀⠂⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠟⠋⠀⠀⢀⠜⠉⠤⠀⠈⠀⠈⠀⠁⠠⠈⠂⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢠⠏⠰⠁⠀⠀⠀⠃⢀⠐⠀⠀⠀⠀⠐⡀⠀⠠⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠡⠀⠀⠀⠀⠄⠠⢀⠀⠈⠀⠀⠂⢸⠶⢶⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠄⠀⠤⠀⠐⠣⠠⢄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠠⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⡠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⣀⣈⣡⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢀⣀⠀⠀⠀⠀⢀⡀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠉⠁⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⢀⣠⣦⣀⣤⣏⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠈⢻⠛⠉⠉⠉⠻⠿⠿⠋⠻⡛⠙⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠓⢤⡀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠓⠶⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈
⣿⡿⠿⠿⠻⠿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠲⡦⢄⣀⣀⡠⠔⠒⠀⠀⠂⠀⠀⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠉⠑⠛⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣠⣔⡒⢄⠀⠀⠀⠀⠀⡀⣤⡤⠤⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡼⣿⣿⣿⠆⠃⠀⠀⠀⢰⣿⣿⣿⡆⠘⠄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢴⠀⠱⣙⠛⠋⡠⠃⠀⠀⠀⠨⡙⠿⠟⠁⡠⠂⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢒⠤⠄⢀⣀⣀⠀⠀⠤⡖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡤⠚⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠗⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠔⠚⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣆⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣴⣦⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⡿⢿⣿⡿⣿⡿⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⠈⠁⠉⠉⢁⣀⡀⢀⣤⣤⣤⣤⣶⣷
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠟⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢈⠻⣿⣿⣿⣿⣿⣿⣯⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⢀⣤⣄⢺⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢀⣶⣿⣿⣿⣍⠁⠀⠀⢀⣀⣀⢠⣾⣿⣦⡸⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠉
⠀⠀⣀⣿⣿⣿⠟⠃⣠⣶⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⡇
⠀⠀⣿⠿⠋⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠻⣟⠿⣿⣿⣿⣿⠙⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣶⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠉⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠰⣿⣯⣭⣥⣤⣄⣀⣉⣁⣠⣤⣬⣭⣿⣿⠂⠀⠀⠀⢸⣿⣿⣿⣯⠤⠒⠒⠛⠛⠿⠿⠿⠿⠟⠛⠓⠒⠒⠤⣽⣿⣿⣿
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠈⠑⡄⠀⠀⢀⠖⠁⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠸⠀⠀⠇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣯⠤⠐⠒⠛⠒⠒⠚⠒⠠⠤⣱⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⡄⠀⡎⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣀⣠⠞⠀⠀⠙⢦⣄⡀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⣠⠊⢠⠔⡏⢡⡌⠄⠀⠀⠀⠆⣦⢨⠓⠤⠱⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⡤⠚⠁⠀⠀⠀⠚⠉⠉⠀⠀⠀⠀⠉⠉⠉⠂⠀⠀⠈⠢⢄⣀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠤⢤⡤⡒⠉⠁⠐⠦⣀⣀⡠⠔⠀⠈⠑⡲⢤⠤⠰⠤⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⣀⡠⠼⢤⡄⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠒⠒⠷⢤⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣙⠤⠒⠒⠒⠤⣄⠚⠉⠁⠀⠉⠑⠋⣡⠤⠭⠴⠤⢤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⣀⢔⠀⠀⢀⣄⠀⣀⠜⠀⠀⠘⢄⡀⠀⣀⠀⠀⠀⣄⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠒⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡿⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢾⠆⠂⠀⠀⢀⠐⠘⠳⢉⡙⠙⣛⡥⠀⣀⠀⠀⠀⠁⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠠⢈⠀⠀⠋⠀⠚⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⡈⠇⠀⠀⠀⠀⠀⣠⡆⠣⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣷⠆⠀⠀⠀⠀⠀⣻⣇⣀⣠⣀⠂⠂⠀⠀⠀⠘
⠀⠀⡄⠄⠀⠀⠀⣀⡈⠀⠀⠠⠄⠀⠀⠀⠐⠀⠀⠀⠒⠀⠀⠀⢀⣠⡤⠤⣀⠛⢻⣿⣿⡏⠁⠀⠀⠀⠀⠀⢠⡰⠄
⠀⠀⠀⠤⠢⢄⣀⠀⢔⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡶⠺⠲⣷⣌⣃⣁⠻⣟⠐⠂⠀⠀⠀⠂⠀⠀⣵
⠀⠀⠀⠒⠀⠀⠈⠀⣀⠉⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⢋⣠⣴⢦⣄⡄⠹⣧⠁⠈⠀⠀⠐
⠀⠀⠠⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡶⢿⣿⠿⠸⠿⠿⣀⠼⠧⣆⠀⠀⠙⠛⠀⠀⠀⠸⠇
⠀⠸⠐⠀⠓⢀⣈⢀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⡀⣺⣿⣿⠋⠀⠀⠀⠀⠠⣤⣾⣾⢁⣶⡄⠀⠀⠀⠀⠀⢀⣬⡀
⠀⠀⠀⠀⠀⠀⠠⡈⣃⠶⢆⣭⣠⠞⠀⠀⠀⠀⠀⠍⢿⣍⡯⠀⠀⠀⠀⠠⠀⠘⢿⣧⣿⡰⠁⡀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠐⠄⠁⠈⠁⠌⠀⢡⡔⢀⣤⠄⠀⠀⠀⠘⢿⠁⠀⠀⠀⠀⠀⠄⠀⢸⠛⠗⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠈⠒⠄
⠀⠈⠀⠀⠀⠀⠀⢠⣦⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⠴⠶⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣤⣤⣦⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠀⠃
⠀⠀⡀⢴⡖⢛⠲⣄⢉⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢦⠁⠄⠉⠀⠐⠀⠀⠀⠀⠤⡀⠀⠀⠒⠀⠀⠀⠀⠁⠀⠀⠈
⠀⢸⡷⠉⠈⠛⠓⢮⣄⠌⠻⡅⠁⠀⠀⠀⠀⠀⠀⠰⠆⠂⠀⠀⠰⠚⠷⢤⡀⠐⣮⡆
⠀⢀⣤⣾⣟⣶⣴⠄⠻⣆⢀⠀⠀⢀⣅⡀⠀⠀⠀⣠⠀⣄⠁⠀⠚⠁⠀⠢⠀⠰⠄⠈⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢘⠿⠙⠋⠋⠛⠣⢜⡛⡻⣀⠀⠀⠉⠀⠀⠀⠀⠛⠀⢡⢀⡀⠂⢠⠀⠠⡆⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠉⠉⠉⠀⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⠔⠂⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠈⠁⠉⠉⠈⠓⠥⣬⠠⠅⠭⠨⠠⠅⢭⡨⠄⠃⠉⠈⠁⠁⠉⠐⠠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡛⠘⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠸⣦⡀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢀
⠀⠛⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣲⣶⡟
⠀⠘⡀⠈⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠋⡙
⠀⠀⠡⠀⠀⠀⠈⠉⠛⠿⠿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡿⠿⠟⠋⠁⠀⠰⠁
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⠿⣷⣶⣶⣾⣿⣶⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢈
⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠄⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠙⠙⠃⠚⠙⠈⠃⠛⠙⠁⠂
⣀⣀⣀⣄⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢸⣿⡆⠀⠀⠈⣹⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⡄⠀⢸⡇⠀⡀⠀⢿⣿⠂⠀⣴⣶⣶⣶⣦⣴⣶⣶⣶⡄
⠀⠀⠀⣼⣿⣧⡀⠀⣾⣿⣿⣆⠀⠘⣿⣿⣿⣿⡇⠀⢠⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣶⠀⠀⡇⠀⢸⡇⢰⡇⠀⢸⡟⠀⠀⡿⠉⠙⡟⠉⠙⡟⠉⠙⡇
⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⢘⣿⡀⠀⠀⠀⠀⠀⢾⣿⠀⠀⣿⡇⠀⣿⠀⠀⡇⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠻⠦⠶⠿⠦⠾⠷⠦⠾⠃
⠀⠀⠘⣿⣿⣿⠃⠀⣿⣿⣿⣿⠀⠀⢿⣿⣿⡿⠀⠀⢸⣿⡇⠀⢸⣿⡆⠀⢺⣿⠀⠀⣿⡇⠀⣿⠀⢠⣷⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠐⠀⠐⠒⠒⠒⠒
⠀⠀⠀⢻⣿⡿⠀⠀⢿⣿⣿⡟⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⠇⠀⢸⣿⠇⠀⢸⣿⠀⠀⣿⠇⠀⣿⠀⠐⣿⠀⢸⡇⠸⡇⠀⠀⡇⠀⠀⠚⠂⠚⠛⠛⠛⠒⠒⠒⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣾⣯⡀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⠀⠀⢀⣿⣿⡟⠃⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣿⡟⡇⠀⠀⢠⣤⡹⢿⠃⠀⢰⣿⣦⣤⣤⣴⣾⣷⣶⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡄⠀⠀⠀⠀⣬⣬⡉⠀⠀⠀⣿⣿⣿⣿⠿⢿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⡇⠀⠀⠼⠿⠇⠀⠃⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣷⠀⠘⢠⣶⣿⣿⣷⣤⠀⠀⣿⣿⣿⣿⣶⣾⣿⣿⣿⡏
⠀⠀⢸⣿⣿⡗⠀⢁⠰⠶⠀⡇⠈⠀⠀⠈⠻⢿⡿⣿⣿⣿⣿⣿⠀⠀⣼⣇⠀⠀⣼⣷⠀⠈⡏⠀⢠⣿⡈⠄⠀⢸⣿⣿⣿⣿⣿⠀⠀⣀⣀⣀⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠐⠿⠿⠇⠀⠸⠦⠤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠁⠀⠸⠿⠿⠃⠘⠿⠿⠃⠀⠁⠀⠻⠿⠗⠀⠀⠘⠿⠿⠿⠿⠟⠀⠀⢿⠿⠿⠿⠷⡿⠿⠿⢿⡇
⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⢀⣄⣄⣄⣤⣠⣀⣠⡀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⢹⣟⢸⠠⠀⠆⠀⠀⠀⣿⣿⣧⣤⣼⣿⢿⡃⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣴⡿⢻⡟⡝⠙⢻⣷⡀⠀⠀⠀⠀⣼⣿⣄⠀⠀⠀⠀⠿⠶⠶⠶⠶⠷⠶⠶⠾⠇
⠀⠀⢸⡇⢸⢐⠀⡇⠀⠀⢸⡏⠈⡉⠀⠉⠁⢹⡇⠀⠀⠀⣿⣿⣄⠀⠀⠀⢸⣿⢠⡿⡱⠀⢀⡿⢻⡷⠀⠀⠀⢀⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣄⡀
⠀⠀⢸⡇⢸⢈⠀⡇⠀⠀⠘⡅⠀⠁⠀⠀⠀⢨⡇⠀⠀⡆⢡⣠⡌⢰⠀⠀⠘⣿⣿⡱⠁⢀⣾⣣⣿⡏⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣞⣩⣶⠟⠋⡲
⠀⠀⢸⡇⢸⠘⠀⠇⠀⠀⠀⡟⠈⠉⣿⣿⣿⣿⠁⠀⠀⣇⠈⠉⠀⣾⠀⠀⢀⠈⢻⣿⣶⣾⣿⣿⠋⣀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡿⠛⠁⢀⣤⡇
⠀⠀⠈⠃⠘⠈⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠒⠛⠋⠀⠀⠈⠛⠛⠓⠒⠒⠚⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⠀⠀⣀⠴⠚⠳⠦⣴⡋⣸⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢷⣦⣄⠀⣀⡴⠛⠁
⠀⠀⢈⣩⠿⠛⠻⢯⣉⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⢠⡟⠁⠀⠀⠀⠀⠈⢧⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢰⣿⢿⢏⣤⡄⠀⠀⠀⢸⣿⣿⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣷⣶⡆
⠀⢾⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣰⠋⢋⣼⣿⣧⡀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠘⣧⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣿⣿⣦⣤⣄⣀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠔⠁⠀⠘⣿⣿⣿⠆⠀⠀⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠙⠲⠦⠶⠚⠋⠀⠀⠀⠀⠻⠿⠛⠤⠬⠿⠀⠀⠀⠀⠘⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠿⠿⠿⠃⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⡍⣿⠛⠿⠿⠿⠻⠟⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⣫⣶⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢋⣽⣿⣿⣿⣿⣿⣿⡿⠟⠋⠛⠿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢸⣿⣿⣿⣿⣿⣿⣋⣤⣀⠀⠴⠛⠛⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠙⢩⣤⠉⠀⢠⣤⠀⣸⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⣿⠻⣿⠀⠀⠈⠉⠀⠠⣄⠁⠀⠈⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⣿⠀⢻⡀⠀⠀⠀⠀⠠⠼⠂⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡼⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⣿⠀⠀⠀⢀⣤⣾⣇⡴⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠖⠋⠉⠀⠉⠀⠀⠀⢠⠏⠀⢠⠴⠋⠘⠛⠁⠀⢀⣺⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣸⠄⢀⣠⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣄⣀⣀⣀⣤⠴⠞⠋⠀⠀⠀⡴⠃⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡞⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⡇⠀⠀⠀⠀⠀⢠⠞⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⢠⠏⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠀⠀⠀⢠⠏⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼


⣟⣿⣿⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣻⣿⣟⣟⣻⣟⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⢷⣴⢿⣤⡤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⢰⠇⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⡇⠀⠀⠀⡀⢀⡴⠋⠀⠀⠀⠀⠀⠙⠦⣄⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠰⠀⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠈⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⡀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⡇⠀⢀⠀⢰⡇⠀⠀⠄⠀⠁⠈⠉⠐⢄⠀⠀⠂⠁⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⢠⣄⣀⣸⠇⠀⠀⠀⢸⡇⢰⠃⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠘⢧⠈⠀⠀⠀⣶⠒⠲⡇⠀⠀⠀⠀⠰⠶⠀⠀⠀⡄⠀⠀⠀⠠⡶⠀⠀⠀⠀⠈⠀⠈⢂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢋
⢠⣄⣷⣄⠀⠀⠘⢦⣤⣡⠈⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠤⠒⠢⡄⠀⠀⠀⠀⢀⣠⣤⡴⠟
⠈⢧⣀⠀⠀⠀⠀⢀⡟⠁⠀⠈⠂⠀⠀⠀⣠⠶⠊⠀⠑⠄⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢶⠂⠀⢀⡟⠀⠀⠀⠀⠈⠛⠶⣤
⠾⢿⣭⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣠⣤⣏⣥⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡤⠴⠛⠀⠀⠐⢶⣶⠶⠶⠞⠋
⠰⠿⠶⠚⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⡇⠰⠆⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣎⡀⢀⣀⠀⠀⠰⣾⠛⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠙⠦⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁⠈⠙⠛⠉⠛⠓⠶⠾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⢰⣶⠶⠶⠶⢦⣤⣤⣈⣉⣹⡟⠛⠛⠛⠋⠀⠀⠈⢻⡶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⢸⡇⠈⠉⠉⠻⣤⣀⠀⠀⠀⢀⣤⠟⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣼⡇⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢠⠏⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⣦⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠈⠻⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠓⠺⣧⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢶⡶⠶⠶⠶⠶⠶⠶⠶⣦⣤⡤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡾⠃
⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠿⠛⠛⠛⠛⠛⠛⠿⠶⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡻⣦⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣦⣤⣤⣀⡀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⢶⣬⣭⣍⣉⣉⣩⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣈⡉⠛⠻⢶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢠⡿⠷⣦⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣉⣉⡛⠻⢶⣄⡙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣶⣍⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⡟⠁⠀⠈⠻⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡟⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⣀⣤⡶⠀⠀⠀⠀⢻⡆⣿
⠀⠀⠀⠀⠀⣠⣤⣴⣿⢸⡇⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠰⠾⣋⣁⡀⠀⠀⠀⠀⢸⡇⣿⣦⣤⣄
⠀⠀⠀⢠⡿⠋⠀⠀⣿⢸⡇⠀⠀⠀⢠⡾⠋⠉⠛⣷⡀⠀⠀⠀⢠⡾⠛⠉⠛⢷⡄⠀⠀⢸⡇⣿⠀⠀⠙⢷⡄
⠀⠀⠀⣿⠁⠀⠀⠀⣿⣿⡇⠀⠀⠀⢿⡅⠀⠗⠀⣸⡇⠀⠀⠀⢿⡇⠀⠗⠀⢸⡇⠀⠀⢸⡇⣿⠀⠀⠀⠈⣿
⠀⠀⠀⣿⠀⠀⠀⠀⣿⠿⠃⠀⠀⠀⠈⠻⣶⣴⡾⠟⠀⠀⠀⠀⠈⠻⢶⣤⡶⠟⠀⠀⠀⠘⠿⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠹⣧⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠈⠻⠶⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣴⣄⣤⣤⣶⣶⣶⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠹⣯⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣶⣤⣤⣤⣤⣶⠶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣦⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⢀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣠⠀⠀⠈⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⣸⠛⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⣯⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣾⣿⣭⣷⡙⠋⠙⣉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠓⠛⠓⠚⠛⠛⠛⠋⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⣿⣯⡟⠚⠃⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣸⡇⢠⠀⠀⠀⡀⠀⢩⡄⢦⣭⠛⢳⡿⢂⡞⡜⠀⠠⠀⠀⠀⠀⠈⠒
⠀⠒⠀⠀⠀⠀⠰⠄⠀⠀⡏⠉⠋⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠲⠈⠋⠜⠛⠂⠂⠀⠀⠀⠲⠂⠀⠀⠁⠠⠎⠀⢀⣼⠚⠁⣠⡶⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣪⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡠⠀⠐⠁⠀⠈⠁
⠀⠀⠀⠂⠐⠆⠀⠀⠀⠀⠀⠠⠐⠀⠠⠀⠀⠀⠀⠀⠀⠔⢢⠄⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠲⠆⠺⢯⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣥⠀⡃⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠱⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢽⡏⠀⠀⠀⠀⠀⠀⠀⣈⣿⣧⣡⣘⣤⣄⣉⠉⠀⠀⠀⠐⠀⠊⢀⠠
⢀⠀⡄⠀⠀⠀⠠⠀⢀⠈⠂⠀⠀⢤⡄⠀⠀⠁⠀⠐⠂⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠛⠿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠈⠀⠠⠂⣀⠀⠀⠈⢉⡰⢄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⣴⣦⣈⡽⢯⣰⣦⠙⢦⠐⠙⣿⣿⠏⢀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢫
⠀⠢⢃⠀⠋⠻⠷⠄⠀⢓⡿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠃⠈⠁⡈⠛⠛⢳⣶⡲⠄⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠘⡉⠀⠀⠁⢄⠀⠠⠄⠀⠈⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡻⠃⢉⣤⣴⣾⠿⣷⣤⣆⡈⠻⣷⡢⠀⠀⠀⠀⢀⣁⡀⠀⠀⠀⠀⢀⣀
⠀⡠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣦⣾⣿⣟⢿⡻⡿⢻⣿⢌⡀⢿⡇⣾⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⢸⣧
⡆⠡⠀⣽⡀⠀⠘⠀⠀⠀⠀⣤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣾⣷⡶⠋⠁⠀⠀⠀⠢⠁⣉⡡⣦⣴⠏⢢⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⡁⠀⠀⠀⠀⠊⠛⠀⢶⡀⡄⠠⢱⡧⠈⢀⡄⠀⠀⠀⠀⠀⠀⡦⣴⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠊⠙⣿⣿⠟⣠⡿⠻⠇⠀⠀⠀⠀⠀⠀⠀⠐⠳
⠀⡀⠀⠀⢀⠀⠛⡄⢰⣌⡛⠛⣤⠶⣰⠫⠂⠀⠀⠀⠀⠀⠀⠐⠎⢿⣇⡿⠀⠉⠉⠀⠀⠀⠒⠂⠄⠹⣿⣟⣿⠷⠟⠀⡤⠀⠄
⠂⠐⠀⠀⠀⠰⠆⠀⠀⠈⠀⣤⠀⠀⢠⡆⠉⢀⣤⡔⠂⠀⠀⠀⠸⡚⣿⠁⠀⠠⠀⠀⠀⠀⠀⣄⠀⠀⢿⠛⠗⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠊⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⢀⡐⠀⠔⠃⠀⠀⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠀⠁⠀⠀⠀⡀⢤⣀⠐
⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠁⠀⠀⠤⠀⠀⠀⠈⠀⠂
⠙⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢄⠐⠀⠀⠤⡼⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠂⠀⠀⠀⠀⠀⠀⠀⣻⣿⣀⡂⣤⣀⠈⠉⠁⠀⠀⠀⠂⠐⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⡥⠂⠀⠀⠀⠀⠐
⠀⠀⢀⣀⣠⣤⡀⠘⠿⠿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⢰⡂⠀⠶⠇⠀⢀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠁⢀⣀⠈⠀⠀⠀⠠⡄⠀⠀⠀⠠
⣤⣀⣩⠿⣁⣤⡈⠳⡀⠘⢹⣿⣿⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠃⠀⠀⠁⠁⠀⡀⠀⠀⠒⠒⠢⡄⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⠟⠁⠁⠈⠛⠓⠳⣤⣒⠠⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⢉⠅⠀⠀⢄⠲⠋⠻⠵⣦⣄⠀⠵⣬⣖
⢁⣤⣶⡾⢧⣦⣰⡀⠹⢿⡂⠄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⠁⠀⠖⠂⠀⠰⣆⠀⠀⣀⠉⠙⢓
⣿⣿⡿⣗⢾⠿⣿⡇⢀⣸⣿⢰⡆⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⣿⡁⠀⠏⠀⠠⠀⡀⠀⠀⠀⢈⡄⠀⠀⢀⠀⠈⠑⠄
⠞⠋⠀⠀⠀⠀⠀⢈⠋⢴⣀⣼⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠀⢼⡄⣦⢘⡀⢸⣆⠀⠘⠇⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀