Pikmin Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣻⣶⣤⣶⡤⠶⠒⠶⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣋⣀⠐⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣦⣄⠘⢛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⣿⣿⣿⣿⣿⡿⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣅⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠋⠁⠀⠛⢻⣿⠛⠻⠿⠿⠿⠿⠿⣿⡿⢿⣇⣈⢉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣶⣶⣶⣀⣤⠟⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣤⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⠟⠻⠿⠷⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣤⣿⣿⣷⣶⣤⣍⡙⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣻⣿⣟⢿⣿⣿⣿⣿⣿⣿⠛⣯⣽⣿⡿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣿⠁⣿⣿⣿⢻⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣣⣿⠋⢿⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣭⣵⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣾⠋⣥⡀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠉⠈⠛⠛⠋⠉⠁⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠘⣧⡈⠛⣿⣿⢿⣴⠁⠀⠿⠇⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⢡⣽⣿⣿⣿⣿⣿⣿⣿⣷⣟⠛⠀⠈⠻⠟⠂⠀⠀⠀⢠⣶⣿⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢀⣾⠏⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠰⠆⠀⠔⠃⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠙⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠠⢦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢲⡘⢿⣿⡄⣤⡀
⣶⣶⣶⣶⣶⣦⣴⠀⠀⠀⣰⣶⣾⣿⣧⣿⠿⡻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣈⡿⠋⠀⠋⠻⠿⢿⣤⣄
⠿⢿⡿⠿⠟⠙⠃⠀⠠⡗⠹⠿⠛⠋⣭⡅⠀⢻⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠟⠛⡛⢿⣿⣿⣿⣿⠇⠀⠀⠀⠐⠛⠉⡿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠁⠈⠙⣿⡿⠛⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠠⠤⠐⠒⠚⠒⠀⠂⠀⠄⠂⠠⠄⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠒⠦⢀⡀⠀⠀⣠⠏⠉⠀⠈⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠲⣎⠇⠀⠀⠩⠂⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠉⠢⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠐⠂⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠱⡘⠀⡠⠊⠀⠀⠀⢀⣤⣶⠖⣠⠀⠴⠀⠀⣀⣀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⡜⠃⠀⣠⣔⠚⢻⣟⣡⡾⠁⠀⢀⡤⣊⠑⢈⡠⣫⣦⣿
⠀⠀⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⢹⡀⠀⢨⠉⠻⠿⣶⣍⠓⡒⠋⠘⠿⠾⠦⣰⣏⣯⠛⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠈⠇⠀⡆⠀⠀⠒⠒⠶⠤⠄⣠⡀⠀⠈⠉⠑⠞⠀⡼⢿⠉⡆⠀⠀⢠⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠆⢀⢀⢸⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠀⠀⠲⠾⠓⣀⠀⠀⠈⡇⠀⠀⢀⡞⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⢼⣾⣻⠿⢧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣵⠄⣤⠀⠂⡾⠀⠷⠛
⠀⠀⠀⠀⠀⠀⢀⠀⢠⡄⠀⠘⣿⡟⢹⣧⠈⠳⢦⣤⣀⡀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠀⢀⣤⠾⠞⠁⠀⠸⠫⣞⣋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠸⡏⠅⠀⠀⣀⡀⠀⠀⠉⠙⠛⠛⠻⠶⠶⢦⣤⣴⠷⠞⠋⠀⠂⢀⠀⠀⠀⠹⣿⣿⡗⠀⢒
⠀⠀⠀⠀⠀⠀⠚⠀⣶⠙⠈⠳⠁⢾⠂⠘⠒⠦⣄⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡍⠀⠸⢠⣤⣸⠞
⠀⠀⠀⠀⠀⠀⠴⣶⣼⡦⠉⠀⠀⠀⠰⠟⠣⣠⣄⣉⢉⡉⠛⠛⠁⠀⠀⠀⢀⡀⢠⢠⢰⣾⠟⠇⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠓⠓⢻⣻⡶⠀⠒⣀⣀⣀⠘⡛⠿⠟⠈⢾⣿⠸⢠⣾⠂⢾⠸⠣⠿⠮⠂⠀⠀⠐⠁⠀⠀⠁⠋⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢠⠾⠛⠋⠀⢀⡄⣀⣀⣀⠄⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⢤⣤⠀⠀⠀⠀⠀⠀⠐⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⡜⠁⠀⢀⠀⠀⠀⣿⠋⠉⠀⠀⠀⠀⠀⢀⡤⠆⠀⣀⠀⠀⠀⠀⠛⢢⠀⠀⣴⠞⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⢶⣿⢳⣴⣦⣼⣧⣤⠤⠀⠀⣀⠀⠈⠀⠀⠀⡝⠀⢀⣠⣴⡀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠓⠒⠿⠿⠿⢿⡶⢾⠿⣶⢷⣿⠿⢶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠟⠛⠻⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⠈⢿⡄⣠⡶⠟⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠟⠛⠷⢦⣄⡀⠀⠀⠀⠀⠘⣿⠏⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⢻⣦⠀⣸⣦⣤⣿⡀⣠⠔⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠈⠻⣿⠋⠁⠈⠉⠻⣯⠀⢀⣀⣤⣄⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢀⣤⠶⠗⠺⠿⢶⣤⣀⠀⠀⢿⣟⡋⠉⠉⠉⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡇⠀⠀⠀⠀⠀⠙⠛⠻⣦⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⡽⠛⠶⢶⣤⣀⡀⢀⣀⣿⣄⣀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⡾⠋⠀⠀⠀⠀⠈⠉⠛⠛⠋⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠀⠀⠘⠳⣦⣀
⣶⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⢶⡆
⢹⡇⠀⠙⢯⡛⠿⢶⣤⣀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⣠⣴⡶⠟⢛⡽⠋⠀⣼⡇
⠈⣿⡀⠀⠀⠙⠢⣄⠈⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⢁⣠⠞⠋⠀⠀⢠⡿
⠀⠸⣧⠀⠀⠀⠀⠈⠙⢺⡿⠋⠉⠙⠳⣄⠀⠀⠀⠀⠀⣰⠞⠋⠉⠙⢿⠞⠉⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠹⣧⡀⠀⠀⠀⠀⣿⠀⢸⣿⣿⠆⢹⠄⠀⠀⠀⢰⡇⢰⣿⣿⡆⠈⡧⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠘⢿⣦⡄⣀⣀⣹⣦⣈⣉⣁⣤⠟⠀⠀⠀⠀⠈⠳⣄⣉⣉⣀⣼⣃⣀⣀⢤⣶⠟⠁
⠀⠀⠀⠀⠀⠙⠻⢦⣤⣤⣈⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣀⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠀⠀⠀⠀⠀⠀⠀⢰⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣸⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⡆⠸⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢠⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢻⡀⠹⣦
⠀⠀⠀⠀⠀⠀⣠⣿⣄⣄⡎⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢇⣤⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠉⢱⡟⠻⠃⠀⠸⣧⠀⠀⣀⣀⣀⣀⡀⠀⠀⣿⠀⠀⠸⠏⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡏⠀⠉⠉⢻⡀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢾⠀⠀⠀⠀⢘⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢄⣬⣳⡀⠀⢠⣾⣦⡴⢿⣄⠀⠀⠀⠬⠀⠀⢡⠀⠁⢈⠈⣁⣈⢡⡀⢒⣐⣀⣀⢀⣰⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⢁⣀⡀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⡴⠊⠉⠀⠉⠙⢦⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡰⠋⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠜⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠙⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠈⢹⠂⢩⣶⡄
⠀⠀⠀⠀⢀⠞⠁⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⠤⠒⠲⢤⡀⠀⠀⠈⢆⠈⢛⣹
⠀⠀⠀⣰⠋⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⠃⢰⣿⣷⡄⢱⠀⠀⠀⠘⠛⢉⡜
⠀⠀⣰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠸⣄⠘⠛⢋⣠⠎⠀⠀⠀⠀⠞⡏
⠀⢰⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣼⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢻⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠘⡆⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠘⢦⡀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠚⡁⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠉⠒⠦⢄⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡯⡀⡠⠒⢩⠇⠀⠀⠀⠀⠀⢸⠳⣌⠓⢄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠟⠀⣠⠋⠀⠀⠀⠀⠀⠀⣿⠀⠈⠑⠢⠬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⢢⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡇⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠾⠃

⠀⠀⠀⢀⣀⣠⣤⣀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣤⣄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿
⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡍⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣤⣼⣿⣿⡟⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠟⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠋⠙⢿⣿⠁⠀⣠⣿⣿⣿⣿⣦⣄⣀⣴⣶⣶⣄⢀⣴⣶⣶⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⣠⣶⣿⣿⣿⡉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣷⣶⠀⠈⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣿⣿⣿⣿⣿⡿⠟⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⣿⣿⣿⡿⣿⣿⣿⠋
⣿⣿⡟⠀⢸⣿⣿
⣿⣿⣿⣠⣾⡿⠁
⣿⣿⡿⠟⠉
⠉⠁





⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⢀⡴⠋⣁⠈⠛⠋⠉⠳⣦⣤⣤⣀
⠀⢀⣮⣵⠞⠁⠀⠀⢀⣤⡴⠟⠛⠛⢿⣿⣤⡀
⢠⣾⠿⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣃⠈⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡟⠿⠀⢋⡐⠷⣄⠈⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⡿⢏⡱⠶⢈⡰⠆⠀⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⣿⣥⣄⠈⠁⠄⠀⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⠛⣤⡔⠂⠄⠈⣠⡀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣿⣿⠋⢧⣜⣣⣴⡿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣛⣛⣿⣿⠆⠀⣘⡛⠠⠻⠿⠛⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡛⢻⡆⠉⠐⠲⣿⡟⠂⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⡿⠀⣘⣿⣿⣷⣆⡀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣴⢿⣿⣿⣿⣿⠃⢐⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣴⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣠⣤⣤⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⢀⣴⣾⣿⣿⡿⠁⣠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⢀⣴⣾⣿⣿⣿⣿⣿⣷⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢁⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠛⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⣽⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⢀⣠⣾
⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢀⣼⣿⣿⣿⣦⣄⡀⠀⠀⠀⢀⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣄⣀⡀
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠣⠀⡀⠀⢀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠠⡈⠀⢘⡁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⣀⠙⣦⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⣀⠠⠈⠁⡁⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡞⠁⠀⠈⠁⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠈⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢠⡿⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⡀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢀⠀⠀⠔⡀⠀⠍⠀⢈⠑⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠃⢀⡇⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠐⣰⠟⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠁⠈⠁⠐⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠃⠀⣠⢎⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢇⠁⡄⢀⣀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠖⠛⠛⠛⠿⣷⣤⡀⢠⣘⠁⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣢⠧⡼⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠈⠓⠈⠻⣿⣧⣨⣷⣄⡀⢀⣀⣠⣤⣤⣤⣀⣀⡀⠀⠀⠛⠉⠉⠉⠁⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢉⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀⠀⠀⠀⡔⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠻⠷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠚⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠘⠀⠄⠀⠀⠤⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣠⠇⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢀⠀⠇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⡰⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠸⣉⠤⠐⠊⠉⠀⠀⠀⠀⠉⠐⠤⡀⠀⡀⠀⠀⢹⣧⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣠⠤⢄⣤⡶⠒⠈⠲⠀⠀⠀⢸⡇⢴⠋⠀⠀⠀⠀⢀⠔⠊⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠊⠁⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠄⢸⡟⣮⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠰⠀⠐⠛⠉⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠀⢀⡟⠀⠟⠀⠀⠀⠀⠀⠣⣤⣄⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢊⠀⠀⠈⠓⢄⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠆⣠⡾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠄⡀⠀⠀⠀⠀⠹⣎⢦⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠈⠁⠀⠀⠀⢀⣈⡴⠋⢀⡡⠞⠀⠀⠀⠀⠀⢠⠀⣼
⠀⠀⠀⠀⠀⠀⣀⠠⠐⠂⢄⠀⠈⠁⢄⠀⠀⠀⠘⢷⣝⠲⠤⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡴⠟⠉⠂⣠⠄⠓⠀⠀⠀⠀⠙⣆⠤⠟⠁⢄⢀⠤⠆
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢡⠀⠀⠀⠑⢠⣀⠀⠀⣼⠿⠶⢤⣄⣙⡒⢦⣤⣤⣠⣠⡤⠤⠖⢊⠍⠀⡠⠒⠉⠁⠀⠀⠀⠈⠉⡹⠊⠀⠀⠀⠀⠀⠉⢠⠤⠤
⢠⠉⠂⠀⢂⠜⠃⠀⠀⠀⠀⣸⡠⠀⠀⠀⠀⢀⡇⢴⣋⡁⠀⠀⠀⠈⠉⠉⠉⠉⠙⠂⠐⢒⠂⠀⠀⠻⣄⠀⠀⣀⡤⣤⣄⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⡆
⠈⢆⡀⠈⣀⠀⠀⠀⠀⠀⠀⢻⠤⠈⠄⠀⠉⠜⠟⠲⠯⢩⣟⣠⣤⡴⠖⠒⠒⠲⡆⠀⠀⠈⢐⡁⠀⢀⡜⡻⠛⡁⠀⠀⠈⢷⣈⢦⣀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠐⠀⠀⠀⡀⠈⠙⢤⡀⠀⠀⠀⣸⣇⡀⠀⠀⠈⡋⡠⠜⠀⠀⠀⠀⠀⠀⠀⢘⡧⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠱⠆⠠⡅⢀⡟⢀⣁⣀⠀⡉⢑⡀⠀⢸⠉⠀⠉⠒⠒⠉⢉⠀⠀⠀⠀⠇⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⢀⣄⠤⠀⠀⢠⡄⠀⠀⠀⠴⠃⠀⠈⠀⢰⠂⠀⠀⠘⣅⡀⠂⠈⠁⢠⡀⠸⣇⣀⣀⠀⠀⠀⠀⢀⣠⠾⣷⡐⠂⢀⣀⡴⠛⢆⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠠⣖⢀⠀⡤⡆⠃⣀⠀⠀⠠⡄⡄⠀⢀⠀⠀⠀⠀⠀⠈⠀⢠⣦⠖⠄⣤⠀⠠⠹⣯⠀⠀⠀⢸⡏⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠑⣠⠀⠀⠀⢤⡸⠁
⠀⠀⠉⠂⠑⠀⠀⠚⠈⠀⠀⠀⠀⠈⠁⠂⡀⠀⠀⠀⠀⠀⢠⡟⢀⡀⠐⠁⠀⠀⠀⠹⣦⣀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⠟⣷⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠞⢉⣠⡟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠸⣟⠛⢻⣿⣵⠇⠀⠀⠀⣶⠒⡦⡄⣀⣀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡴⡏⢡⣿⣿⣿⠃⠀⠀⠀⣿⡿⠛⠋⠁⠀⠀⠀⢰⡟⢻⣿⡿⣷⣫⣷⡀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠰⡇⣿⣿⡿⠋⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣉⠟⣿⣇⣿⠄⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿
⠀⠀⠀⣹⠟⠁⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠙⠉⠀⠀⠀⠀⠀⡴⢶⣤⢤⡀⠄⢸⣿⣿⣿
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⢾⡙⣿⡟⣿⣿⡀⠘⠛⣳⣿
⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⢿⣻⣿⣥⠟⠈⣱⡿⢼
⠀⢻⡇⠀⠀⠀⠀⢀⠢⡴⠃⠀⠳⣎⢀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠉⠉⠀⠀⠀⣿⣷⡺
⡠⠋⢇⠀⠀⠀⠠⣀⣿⠀⠀⠀⠀⣼⣷⡔⠀⢀⣤⣤⣠⠞⠛⠦⣄⣀⣀⡀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⣰⡏⠨⢻
⣧⠀⣾⢷⡄⠀⠀⢺⣛⠇⠀⠀⠈⠯⣿⠇⠀⠀⠣⣹⣷⢦⠀⣶⣿⡏⣹⠃⠀⠀⣠⡞⠙⢷⣄⠀⠀⠀⢸⣿⣿⠀⢿
⡇⠰⠟⡟⠀⠀⠀⠀⠹⣄⣀⣀⣀⣠⡟⠀⠀⠀⠀⠈⠻⡇⠀⢈⠿⠛⠁⠀⠀⠀⢿⡿⠀⢻⣿⠇⠀⠀⠀⠘⢧⣴⣼
⡜⠀⠈⡀⠀⠀⠀⠀⣰⡍⠉⠉⠉⢿⣿⡄⠀⠀⠀⠀⢀⡇⠀⢻⡀⠀⠀⠀⠀⠀⠈⢳⡄⣼⠃⠀⠀⠀⠀⢀⣾⠀⢹
⠇⠀⠐⣷⡀⠀⠀⢀⡟⣇⠀⠀⢀⣼⡟⢧⠀⠀⠀⢀⡞⡇⠀⣸⠻⡄⠀⠀⠀⠀⠀⣾⡇⢹⣧⡀⠀⠀⠀⣸⢿⠀⢸
⢸⠤⢼⠀⠃⠀⠀⠈⠀⠘⣿⠛⠻⡿⠀⠀⠀⠀⠀⠈⠁⢹⠖⢿⠀⠉⠀⠀⠀⠀⠘⠙⡷⢾⡉⠁⠀⠀⠀⠉⢸⡷⢾
⠘⠀⠘⠂⠀⠀⠀⠀⠀⠀⠛⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠋⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⣬⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄⠀⠀⠀⠸⠗⠛⠿⢾⡂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⣤⠄⠀⠦⠰⠦⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠂⠀⠂⠁⠀⠀⠀⠀⠀⠀⡀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢾⠷⢦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⠀⠀⡄⠀⢰⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠇⠄⠀⠀⠀⠀⠀⠀⠀⢸⣷⠈⠀⡇⠀⡞⡱⠂⠄
⠀⠀⠀⠀⠀⢀⣴⣦⣄⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡿⠆⠘⠀⡀⠇⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣿⣦⣿⠆⠀⢤⣾⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⠀⠉⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡀⠀⠀⠐⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠂⣀⣀⣀⣀⠀⠈⠁⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣀⠴⠞⣿⣻⣿⣿⠷⣄⡀⣰⠃⠀⠀⠀⡠⢜⢻⠏⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢾⠁⠀⢈⠙⠉⠁⢀⡀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠳⣄⣼⣟⣿⣿⣿⡧⢀⣀⠀⠀⠄⠰⡀⠀⡀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⢏⠉⠉⠉⢀⡀⠐⡈⠰⠀⠀⠂⠀⡎⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠴⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⢿⡇⠀⠀⠐⢀⠠⠆⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣧⠀⠀⠀⠸⠄⠰⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢀⡀⠀⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠀⠘⠏⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠒⢻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡚⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣆⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠛⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⢠⡄⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡁⠉⠀⠀⠀⠀⠀⢰⠈⠀⠈⠺⠋⠀⢼⣓⡕
⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⢀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠐⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠚⠂⠉⠀⠀⠀⠀⢀⡀⠀⠀⠄⠠⢀⠀⠐⢘⠐⠀⠀⠐⠀⠄
⠀⠀⡀⠎⠀⠀⠀⠀⠈⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠖⠂⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⣳⠀⠀⠀⢐⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢈⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⣟⠛⢏⡻⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣦⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢀⠈⠀⠠⢄⠀⠙⢻⣿⡿⠂⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠈⠉⠓⠶⢄
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⡈⠤⣀⡀⠀⠀⠀⠂⠐⠀⠀⠙⠃⠀⠀⠐⠂⠈⠄⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠈⠃⠀⠐⠈⠀⠄⣈⠀⠀⣀⠀⠀⠒⠀⠄⠀⠦⠀⠀⠀⠠⠈⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠒⠈⠛⠖
⠠⠀⢠⠀⡂⠘⠀⠀⣰⠃⠀⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡘
⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡠
⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⡌⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠠⠃⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠁⠶⠊⠁⠁⠐⠀⡀⠀⠀⠀⢡⠀⣷⡀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡄⠀⠀⠀⠀⠀⠀⠀⣤⡁⠀⠦⠀⢸⡇⢸⡇⠀⠀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠈⠀⠚⠀⠀⠠⠀⠀⠀⢄⠀⠀⠀⠀⠸⡇⠀⠐⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⡴⢏⠀⠀⢠⡀⠐⢄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁
⠂⠀⣠⠴⠋⠠⠀⠈⣐⣓⡠⠀⠀⠈⠒⡂⠀⠄⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠲⠈⠉⠀⠀⠆⠀⠀⠉⠙⠁⠀⠀⠋
⠴⠋⠁⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⢁⡀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⡀⠀⡰⠤⠄
⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠖⠀⠘⡀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠂⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⢀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠶⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠄⢠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠐⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢸⠑⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡡⠄⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠀⠀⠀⠀⠀⠀⠌⠀⠀⡠⢚⢉⠿
⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠘⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⡄⠈⠢⠀⠀⠃⠀⠀⠐⠠⠡⢀⡆
⢺⠀⠀⠀⠀⠀⠀⠀⣀⠠⣀⠀⠓⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⢀⡀⠀⣤⠄⠻⠑⠀⠓⠀⠀⢣⡀⠀⣀⠀⣰⠇⢀⡞⠁
⡈⠀⠀⠀⠀⠀⣠⠚⠁⠘⠿⠃⠀⠀⣀⠀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣳⠊⠀⠀⠀⠀⡀⠀⡆⠀⠀⡗⠀⠛⢠⠏⠀⠎⠀⠌
⣧⠤⣤⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⣤⠀⠀⠀⡄⠀⢇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⢀⠊⢩⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡕⠒⣹⠃⠀⢀⡌⠀⠀⠸⠇⠀⢠⡀⢠⠀⠀⠀⠀⠰⡠⠂
⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠈⠀⠀⠤⣐⠒⠂⠤⠄⠀⣠⠁⠀⠀⢸⠀⡇⠀⠠⠀⠀⣸⠇⠘⠀⠀⡀⠀⡶⢁⠄
⠀⠀⡈⠘⠂⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢢⡀⠀⠈⠀⠀⠀⢸⠀⡞⠀⠂⠀⢸⡘⠀⠠⠀⠰⠇⣠⢃⠆
⠀⠀⠋⠀⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠈⠁⠠⠤⡀⠀⠀⠀⠉⠓⠠⠀⠀⠀⠀⢾⠃⠀⠀⠀⣷⠁⠀⠆⠀⠐⢠⡏⠆⠀⠐
⠀⠘⠠⠆⢘⠁⠀⠄⠀⡀⢠⣶⡀⠠⠄⠀⠉⠈⠀⠀⠀⠀⠀⢀⠀⠠⠄⠉⠀⠀⠄⡀⠀⠱⠄⠀⠀⠈⢢⡇⠀⠀⠁⠀⠀⠀⠀⠀⡺⠀⠀⢀⠁
⠀⠄⣀⣀⡈⠳⠀⠒⠈⣇⡼⢃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⡀⠀⠀⠑⠂⠀⠂⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⢠⠃⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠁⠀⡅⠀⠀⠠⣤⣤⠀⠅⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠓⠣⣸⠿⣤⣄⣄⣈⣀⠀⠀⠈⠀⠀⠀⢸⡇⡀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠂⠀⠀⠀⠀⠸⠛⠛⢉⣠⣾⠟⠛⢷⣌⡙⠦⠀⢻⣏⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⣾⣿⣿⣷⢠⣈⣹⣷⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠰⣶⣆⣀⣤⣄⢀⣿⣶⣿⣻⣿⣿⣿⣿⣿⣿⣧⠀⣟⣿⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⣠⣿⣃⣰⣇⠀⣿⣿⣾⣴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠋⣷⣶⣆⣾⣿⣿⣯⣿⣇⣿⣿⠿⢿⠟⠿⣻⡛⣿⣿⣿⣟⠿⢟⣿⣿⢿⠀⢸⣿⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⣸⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⠿⡷⠎⠀⡰⠋⠁⣅⣙⡷⢷⣾⣷⣿⣧⠄⢀⢾⣿⣿⣿⣭⡄
⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⠟⠉⣹⣿⣁⡀⠉⠚⠛⠀⠀⠁⢀⠼⡯⠛⠀⠀⠁⠀⠙⠁⠀⢀⣿⠃⠉⠙⠒⠗
⠀⠀⠀⠀⠀⢰⡦⠛⠉⠻⢟⣿⡿⠟⠛⠛⠛⠛⠛⠛⠳⠔⠆⢦⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠊⠴⣂⠘⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣰⠀⡜⢁⠇
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢴⡶⡞⣴⠋⠰⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠈⠑⢿⡄⢀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡆⠸⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⠿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠈⢿⡄⣠⡶⠟⠛⠛⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠛⠷⢶⣄⡀⠀⠀⠀⠀⠘⣿⠏⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⢻⣦⡀⣸⣦⣤⣿⡀⣠⠄⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠈⠻⣿⠋⠁⠈⠉⠻⣧⠀⢀⣠⣤⣄⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⢀⣤⠶⠟⠻⠿⢶⣤⣀⠀⠀⢿⣟⡋⠉⠁⠉⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠀⠀⠀⠀⠀⠙⠛⠻⣦⡿⠋⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⡿⠟⠷⢶⣤⣀⡀⢀⣀⣿⣄⣀⣠⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⡾⠋⠀⠀⠀⠀⠈⠉⠛⠛⠋⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠀⠀⠘⠷⣦⣄
⣶⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢀⣀⣤⡴⢾⡧
⢹⡇⠈⠻⢯⡛⠿⣶⣤⣄⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣀⣠⣴⡶⠟⢛⡽⠋⠀⣼⡇
⠈⣿⡀⠀⠀⠙⠶⣄⠈⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⢁⣠⠞⠋⠀⠀⢠⡿
⠀⠸⣧⠀⠀⠀⠀⠈⠙⢺⡿⠋⠉⠙⠳⣄⠀⠀⠀⠀⠀⣴⠞⠋⠉⠛⢿⡞⠋⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠹⣧⡀⠀⠀⠀⠀⣿⠀⢼⣿⣿⠄⢹⠀⠀⠀⠀⢸⡇⢰⣿⣿⡆⠈⡧⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠘⢿⣦⣄⣀⣀⣹⣦⣈⣉⣁⣤⠟⠀⠀⠀⠀⠈⠳⣤⣉⣉⣠⣼⣃⣀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠙⠻⢦⣤⣤⣌⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣠⣤⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣸⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⡆⠹⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢰⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢹⡄⠹⣦
⠀⠀⠀⠀⠀⠀⣠⣿⣤⣄⡞⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣇⣤⣸⣷⡄
⠀⠀⠀⠀⠀⠀⠉⢹⡟⠻⠃⠀⠸⣧⠀⠀⣀⣀⣀⣀⡀⠀⠀⣿⠀⠀⠸⠋⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡏⠀⠉⠉⢿⡀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢾⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣆⣮⣷⡀⠀⣠⣾⣦⡴⢿⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⣤⣶⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣤⠀⠀⠀⠀⢹⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⡼⠋⠉⠁⠀⠀⠀⠀⠀⠀⣀⠹⣿⣿⣾⣷⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣭⣝⣛⣃⡀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠋⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣷⡆⠀⠀⡾⠁
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠿⠿⠛⠛⠁⠀⢸⠇
⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⠼⣿⣿⡷⠄⠀⠀⠀⠀⢩⢹⣿⣿⣿⣿⡫⡭⠀⠀⠀⢶⣶⣤⣶⣿⣿⣷⣤⣤⣶⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡷⠄
⠀⠀⠀⠀⣧⣿⣿⣧⡇⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠈⢿⣇⢆⣿⣯⡰⣸⣿⡿⠀⠀⠀⠀⣠⣴⣿⣷⣄⠀⠀⠀⠀⠀⣣⣿⣿⣮⡇
⠀⠀⠀⠀⠹⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠙⠉⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠹⣿⣿⡟⠁
⠀⠀⠀⠀⣰⣿⣿⣧⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⡄
⠀⠀⠀⠰⢹⣿⣿⣿⢣⠀⠀⠀⠀⡍⢻⣿⣿⣿⣿⣿⠃⠇⠀⠀⠀⠀⡰⠹⣿⣿⣯⠳⡀⠀⠀⠀⠀⠀⠠⣿⣿⡿⡆⠀⠀⠀⠀⠀⠋⣿⣿⣿⠈⢂
⠀⠀⠀⠃⢸⠿⠿⡏⠈⠁⠀⠀⠀⠀⠈⢿⠟⠛⢿⠏⠀⠀⠀⠀⠀⠀⠁⠀⡿⠛⣯⠀⠁⠀⠀⠀⠀⠀⠃⢻⠿⡇⠁⠀⠀⠀⠀⠈⠀⣿⠿⢿⠀⠈
⠀⠀⠀⠀⠘⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠓⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠿⣿⠁⣶⡎⢻⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣄⡀⣾⠉⢿⡄⠲⢤⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⣺⣕⡦⠴⠛⠻⣿⣶⣦⡝⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠮⢽⠯⢟⣫⣤⣔⠊⠙⠛⢦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣽⡞⠀⢾⣷⠄⣸⠿⣟⡠⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⢩⣿⡥⠌⣿⣇⣀⡤⣿⡻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣖⣠⣤⣖⣮⣟⣳⣽⣳⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢈⡹⠿⡛⠯⢟⠮⠿⡷⠽
⣿⣿⣿⣿⣿⣿⣿⡿⢁⣤⣾⣿⠞⢠⣤⠤⠘⠳⣾⣧⣄⡹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⠉⣭⡿⢛⣻⣿⣍⠉⠙⠻⣧
⣿⣿⣿⣿⣿⣿⣿⠃⠛⢿⣿⣿⣄⠐⠿⠟⠀⣠⣿⣿⠟⠃⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣥⡀⠀⠈⢿⣯⣕⣤⣿⠟⠀⠀⣠⣼⡆
⣿⣿⣿⣿⣿⣿⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢚⣿⣟⣛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣋⣭⠤⢶⡺⠉⢠⣤⠈⠳⣌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣛⡩⠔⠚⠉⠀⠀⠀⣈⣇⠀⠈⣽⡄⠀⢿⠜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⠕⠊⠁⣀⠀⠄⠀⠀⢀⠀⠈⠘⣧⣦⣄⡐⠼⠋⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡥⠔⠉⠀⣀⠀⠀⠉⠀⠢⡄⠁⠰⠀⠄⠀⠀⠉⠁⠀⠀⠀⢃⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⠰⠂⠁⠀⠀⠀⠀⢠⠀⠰⣠⠀⠎⠸⡀⠊⡀⠈⠀⠀⠀⠀⠀⠀⠀⣘⣧⠘⣧
⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠶⠖⠛⡩⠄⠒⠈⠀⠀⠠⠐⠀⢀⠀⠀⠀⡇⠂⠀⢣⠐⡀⢣⠈⠘⡄⢀⠀⠀⠀⠀⠀⣠⠽⢿⡄⠹⣆
⠀⠀⢸⡇⢡⡥⠼⠒⠶⠦⣄⡈⠁⠀⢀⠀⠀⠀⢀⠀⢀⡁⠘⡘⡆⠄⠹⠘⡀⠈⣇⢡⠘⠶⠃⢉⡡⠀⠀⠀⠀⠀⡇⢸⣦⠹⡆⢹⡄
⠀⠀⠀⣷⠀⡇⠀⠲⣶⡆⠘⡇⠠⠀⠠⠀⠀⡄⠰⡀⣋⡀⠀⡇⢡⡌⠄⢃⣉⠕⠘⠋⢠⣀⣠⣌⠀⣀⣀⠤⠔⢒⣙⣲⣴⣞⣁⣈⣧⡀
⠀⠀⠀⠘⣇⠙⢄⠀⠼⠧⣠⣷⠂⠀⠘⡄⠠⠃⠐⣷⠀⠄⢀⡇⠀⠘⠶⠀⠀⠤⣀⠤⠛⢉⡠⢈⣈⣥⣴⠶⠿⣻⣛⣛⠛⠛⣯⠉⠁
⠀⠀⠀⢀⣘⣦⠀⠙⠒⠚⠟⠉⠀⠀⠀⢳⠀⠶⡀⢹⣇⢉⡠⠷⠊⠀⢀⡠⠔⠉⣀⠤⣊⣥⠶⠿⠿⠁⢲⣲⠟⢉⣭⡉⠳⡄⢹
⠀⠀⣠⢺⡇⢹⡎⠠⢱⠄⠀⠀⠀⠀⠀⠀⣇⡀⠱⠖⠁⠀⠀⢀⡠⠒⠁⡀⢔⣪⣷⡟⠫⠀⠀⠀⠀⠀⣈⡁⠀⢘⣻⡀⠀⣿⢸⡀
⢀⡔⠻⢸⡇⢨⢿⡀⠀⢾⡄⣀⣀⡀⠈⠀⠈⠠⠀⠀⢰⠦⠂⠁⠠⢐⣯⡶⠛⡡⠀⠐⠂⡤⠀⠀⠖⠄⠐⢷⣄⡀⡉⣄⡴⠋⠘⡇
⣾⢐⠀⠈⣧⠸⣌⣧⠀⡼⠉⠠⣤⠙⣦⠀⠀⠀⠀⣀⠓⢀⣰⣴⢾⠋⠀⢃⠰⠐⢲⠐⠊⢱⠀⣰⠃⠁⠀⠈⠙⠉⠉⠁⠀⢇⠀⣧
⢻⠘⡇⠀⢹⡀⠈⠛⣶⠀⠳⣄⠙⢂⡾⢠⠀⣂⣭⣶⠞⠋⠁⣼⠘⠂⡄⢸⠀⠀⢸⡆⡀⢸⠀⡴⡄⢰⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿
⢸⠀⡇⠀⠘⣇⠀⢘⡜⠶⠶⠶⠿⠿⠖⡟⠛⣏⠈⣷⠘⣁⣀⡇⠀⠃⠐⢸⠀⠂⡀⡇⠃⠸⠆⢃⣃⠈⠀⢀⣀⠀⠀⠀⢒⣿⡇⢺⡇
⢸⠀⢹⠀⠀⣿⠀⠀⣶⠀⠀⠀⠀⠆⠀⡧⡤⢸⢠⠟⠠⠤⠼⠧⠆⠈⠓⠈⠉⠉⠁⠙⢉⡴⠋⠁⠀⠀⠀⠀⠀⠉⠒⢎⣭⠙⢧⠘⡇
⢸⠀⢸⠀⡀⣿⡀⢀⡝⢉⣉⠑⣄⠀⠀⠀⠀⠀⠀⣄⣠⣀⣄⠀⡀⠠⠤⠤⠤⠤⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡬⠭⠥⣝⣰⠷
⢸⡀⠀⡀⠁⢸⣇⠈⣇⠀⢿⠀⣿⠀⠀⠀⢀⣀⠫⠉⠁⠀⠀⠀⠤⠤⠤⠤⢤⡄⢰⠁⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠸⡇⢣⠃⠸⠀⣷⣀⣈⣱⣰⣾⣥⣯⣤⠤⠴⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠉⠃⡆⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣧⠘⠈⠀⡶⠭⠭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠸⡄⣣⠿⢥⣄⡤⠤⠤⠶⠖⠒⠂⠀⠉⠉⠉⠉⠀⢀⣒⣒⣒⣀⣀⣀⣈⣥⠥⠽⣦⠀⠀⠀⢸⠁⠀⠀⠇⡄⠀⠀⠴⠀⠀⠘⠂⠘⠀⠄⠀⠸
⠀⠀⠱⠏⠠⠤⠤⠴⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⢸⠀⠄⠘⠀⡇⠀⣐⠀⠀⡀⠀⢰⠀⠀⡂⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡆⠀⠁⠈⠈⠁⠙⠀⠉⠈⠉⠈⠀⠁⠙⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣶⣶⣯⣥⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁⠉⣷⢶⡞⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢅⠺⣯⡿⢗⣃⣤⣤⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡟⠻⠯⠉⣝⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⡇⠈⠁⣀⣀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⡆⢰⣿⣿⣿⡗⠀⠛⠛⢛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠃⢘⠿⠿⠿⠑⠠⢄⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢄⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⡄⠑⢀⢄⠀⠀⠀⠀⢀⠂⠐⠘⠃⠃⠄⠀⢈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠞⠀⢠⠄⢠⠟⠑⠊⢹⠁⠁⠸⠇⢀⡀⠛⠀⢻⠉⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡼⠀⠉⠀⣀⠀⠸⠀⠈⢰⡆⠀⠀⠈⠃⢀⠀⡀⠀⢤⠈⢆⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣤⣀⣐⠃⠀⠄⢰⡇⠀⠀⠀⡇⠀⠀⠘⠃⠁⣀⣤⣴⡇⠀⠀⡶⢤⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⢶⣶⣶⡖⠒⢒⡒⠈⠉⣍⠙⣿⣿⣿⣿⣄⣠⣤⡼⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⢠⠄⠀⠀⠀⠸⡄⠘⢿⣿⣿⣯⣿⠋⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⠀⠈⢸⠀⠀⠀⠀⠹⡄⠈⢿⣿⣸⣷⣷⣿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠋⠐⣡⠟⠁⠀⢀⡿⠀⣷⣬⣴⣶⣶⣾⣿⣿⡄⠘⡏⢁⠃⠇⡉⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢰⣏⠀⠀⠀⣸⡇⢠⠋⠙⠛⠛⠛⠻⠏⢻⣿⡄⠀⢸⣿⣿⠈⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⢸⡏⠀⣿⣿⣹⠀⠠⢠⡀⠀⠀⠀⠀⠀⠊⣿⣷⠀⠜⣿⣿⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⢸⡇⢀⣿⣿⡏⢀⠃⢠⠀⠀⠀⠀⢁⠀⠀⢹⣿⡇⢠⡙⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⡀⠈⢄⣼⣿⡟⡃⠃⠀⠀⠀⠀⠀⠀⠧⠀⠀⠈⣿⣿⠸⡅⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⠃⠀⠀⠀⠀⡠⠉⣁⣉⣤⠀⢿⣿⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣼⣤⣤⣤⣤⣁⣤⢤⣤⡤⡤⣿⢾⣿⣧⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⣀⠀⠀⣀⣀⣀⣀⡤⣤⣠⢤⣄⣀⣀⣠⣤⣤⣤⠀⡔⡖⣿⣱⣽⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡼⠿⠘⠁⠀⠓⠈⠐⢛⠛⣫⠏⡇⢟⡕⠙⠟⠛⠉⠉⠉⠉⠉⠁⠉⠩⠤⠀⠉⣼⣯⣿⠅
⠀⠀⠀⠀⠀⠀⢀⢔⡕⠁⠀⢀⢤⡠⡀⠀⠀⡰⣡⣾⡟⡼⢱⢿⡇⠀⡆⠀⠀⢠⣠⣤⣤⡄⠀⠀⠀⠀⡇⢩⣏⣿⡄
⠀⠀⠀⠀⠀⡠⡣⠋⠀⠂⠀⠀⠂⠑⠁⠀⠀⣿⣿⣧⢊⡆⣿⣿⡇⢠⠃⠀⠀⠊⠛⠛⠛⠀⠀⢀⠀⡜⡀⣾⣿⣷⠃
⠀⠀⠀⠀⡔⣵⣅⣤⣤⢠⢂⡀⡴⠴⢂⣀⣀⣰⡿⢃⢪⡇⣿⣫⣄⣁⣀⣀⠃⣀⡀⢃⣐⣛⣫⠭⡜⣸⡞⣼⣿⠃
⠀⠀⠀⢬⣬⣭⣿⠿⣶⠶⣶⣮⣭⣥⣭⣍⣙⣻⣣⣇⣬⣇⣿⣭⣿⣿⣿⣿⣿⣿⣿⣯⡭⠼⠶⠶⡾⣻⣿⣿⠁
⠀⠀⠀⢸⢻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣗⣒⣶⠿⣿⣄⣀⣈⣭⣙⣶⣖⣶⣿⣿⠇
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⢯⣿⠏

⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠛⠛⠿⣿⣯⣿⣿⠀⠀⠀⠣⠀⠀⢉⠉⠉⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠠⡄⠤⠤⡄⣠⠖
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡔⢡⠲⢠⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠃⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠄⠘⠘⠈⣂⠀⠀⠀⠀⠀⠀⠠⠔⠺⠋⠀⠀⠀⠀⢸⡇⠀⠈⠀⡖⠀⡆⠸⠓⢠⠀⠀⠀⠀⢀⢀⡤⣄⠀⢠
⣮⠁⠃⠀⠈⠁⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢢⣾⠃⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠓⠋⠀⠈⠀⠉⢹
⡏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⣀⠀⠀⠀⠌⠁⠀⠀⠀⠐⢰⡿⠁⠀⠀⣠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢒⠀⠀⠀⠀⠀⢸
⠇⠀⠴⠶⠀⢀⣤⣦⡀⢐⠂⡌⢦⠀⠀⢠⣄⡀⠀⠘⠤⠄⠔⠀⠀⠀⠀⠀⠀⢀⣴⣆⣀⣀⣀⡀⠀⣠⠀⠢⠔⠀⡀⠀⠀⠀⢿
⠀⠀⠀⡶⣠⠈⠛⠉⣁⠉⠐⠀⠈⣧⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⠛⠛⠛⠁⠀⠉⠀⠀⠀⠂
⠀⠀⠈⠙⠃⠀⡈⠤⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢠⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⠀⠀⠀⢀⣿⣿⣿⣿⡇⢀⡇⠀⠀⢀⣀⠄⠀⠀⠀⠀⠃⠀⠱⠀⠀⡄⠀⠻⢿⠿
⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⢸⡇⠀⠀⢍⡉⡄⠀⠀⠠⠀⠄⠀⠀⣨⠔⠋
⣀⡀⠀⠀⠀⠀⠊⠀⠀⠐⡄⠀⠀⠀⢩⡿⠃⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇⠀⠀⠈⠁⢰⠀⠀⠀⠓⠊⠡⢈⣸⠄⠀⠀⠀⠀⢀
⠠⢴⠆⡱⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠋⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢸⠇⠀⠀⠀⠀⠊⠓⠄⠀⠀⠀⠀⠁⠀⠀⠀⡴⠾⠉⢁
⠀⢸⠖⠀⠀⠀⢀⣠⣤⣤⣀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠂⠰⠆⣠⠂⠀⠀⠀⠀⠠⡀⠫⠋⠉⠁
⡠⠏⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠈⠛⠛⠛⠓⠒⠚⠋⠀⠀⠀⠀⠀⠔⠒⠋⠀⠀⠀⠀⠀⠀⠀⠈⠐⠣⢄⢀
⠀⠀⠀⠀⠀⠈⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠶⠶⢶⣶⣤⣤⠀⠀⠀⠀⠈
⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣿⣾⡿⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⣴⠇⠀⠀⠀⠀⠀⡴⡒⠄⠀⣿⡇⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢿⣿⣟⣿⣿⣗⣿⣽⣿⠧⠀⠀⠀⠀⠀⠀⠣⠥⠂⢰⣿⠃
⠀⠀⠀⠀⠀⠀⢀⡀⢠⠃⠀⠀⠀⢠⡲⠀⠀⠀⠀⠀⠘⢿⣿⣿⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠇⢠⣀⢀⠀⣼⡿⠀⠀⠀⠀⢀
⠀⠆⠻⠀⠀⠒⠐⠀⢸⠀⠀⠀⠀⠨⠤⠤⠴⠀⠖⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠝⠞⢀⣿⡇⠀⠀⠀⠀⠀⡀
⣠⠔⠒⠒⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠠⠤⠄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣅⣠⣄⢄⣀⣈⣷⠀⣾⡷⠀⠀⠀⠀⠀⠰⠂⢲⠄⠀⠀⠀⠀⠀⠰
⠀⠂⠀⠀⠀⠐⠃⣠⣶⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⢸⡟⠛⢯⣛⣼⠛⢻⡿⠀⠿⠉⠁⠂⠀⠀⠀⠀⠀⢸
⢁⠀⣀⠀⠀⠀⢰⣿⣿⣽⣿⣿⣿⣿⣿⣿⣧⡀⣀⠀⣘⠻⣦⣼⣿⣠⣴⠟⣁⣀⣀⡀⢀⡴⠀⠀⠀⡀⣀⠛⠀⠀⢀⡀⠀⠀⢀
⣿⣿⢛⢻⠛⡟⣻⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⢀⣤⣄⣀⣀⡴⢦⣀⣠⣿⡿⠛⠉⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⣠⠞⠁⠈⠉⢁⣠⣶⡿⠟⠉⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⡾⠁⠀⠀⣠⣴⣿⠟⠉⢀⣠⠶⣤⣀⣀⣠⡄⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⡴⠋⠀⣀⣴⣿⠟⠋⢀⣠⠶⠋⠁⠀⠀⠉⠉⠁⠹⣦⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⣀⣴⣾⠿⠋⠀⢠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣤⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⣤⣾⡿⠛⠁⣀⠀⠀⢻⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣠⣶⡿⠛⠁⢀⣴⠞⠉⠀⠀⣼⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡿⠟⠉⢀⣤⠞⠋⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⢠⣾⣿⣿⠟⠉⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢀⣠⠶⠋⠁⣠⣴⡿⠟⠉
⠀⠀⠀⣰⣿⠋⢸⣿⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣤⣾⡟⠀⠀⠀⠛⠁⣠⣴⣿⠟⠋
⠀⠀⠀⢿⣧⡀⠀⠹⣿⣄⠀⠠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠈⠻⢿⣦⣄⠘⢿⣧⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠀⢀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⠻⣷⣄⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠙⠛⠉⠙⠳⣦⠞⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣆⣀⣠⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢺⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⢋⣩⠼⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡧⢿⣏⡤⠴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢾⢿⣾⣿⣤⡾⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣰⣷⠗⢳⡞⠁⢀⣸⣿⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠁⠀⣼⠴⠚⠉⠀⢸⣿⣷⡆
⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣰⢃⣀⡤⠔⠚⠉⠻⣿⡇
⣶⣯⡽⢋⣉⣿⡧⠿⣯⣿⠷⣰⣦⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡼⠉⠁⠀⠀⠀⠀⣀⣴⡷⠃
⠙⢿⣻⣍⡤⠴⢿⣾⣿⢭⣿⡁⠉⠉⠉⠛⠻⠶⣤⣀⠀⣤⣄⡞⠀⡠⠋⠀⠀⣀⣤⣶⠾⠿⢿⣛⡦⣀
⠀⠈⠛⣿⣷⠶⣿⡁⢀⣀⠤⠽⠲⠤⣄⣀⣀⣀⠀⠉⠙⣿⣿⣧⣊⣤⣴⣶⢿⣛⡉⠤⠤⠒⠚⠛⢻⣿⣧
⠀⠀⠀⠀⠸⣻⣟⣿⠉⠀⠀⣀⠤⠒⠉⠀⠈⠉⠉⣉⣭⡼⢿⣿⡍⡀⠀⠀⠀⠈⠑⠒⠶⡒⠒⠒⠛⢻⣟
⠀⠀⠀⠀⠀⠘⣽⣿⣷⠚⠉⠀⠀⠀⣀⣠⣤⣶⡾⠛⠁⢀⢺⣿⡿⠢⢍⠒⠲⠤⣤⡤⠖⠛⠦⣄⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠘⢽⡿⣤⣤⣤⣶⠿⠛⢋⡁⠾⠁⠀⢠⠎⡄⢸⣿⣧⣄⠙⠢⣀⠈⢻⣶⡀⠀⠈⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢹⣏⣿⡤⠒⢁⣴⠃⠀⣠⠃⢀⠇⢸⠙⣿⡈⠳⣄⠈⠓⢄⠀⠙⣦⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⡏⢀⠔⠁⢧⠔⠊⣿⠀⢸⠀⢸⠀⢸⣷⡀⠈⠳⣤⣀⣿⣺⣷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⠁⠀⠀⡇⠀⠘⡿⠀⡌⠀⣼⠀⠀⢿⡇⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⢷⣦⣼⣄⡀⢀⣇⣀⣧⡠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣯⣿⣾⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠿⣷⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣄⣀⣤⣤⣴⠶⠛⠋⣀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣁⣠⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣾⠿⠟⠛⠻⠿⣶⣤⡐⢶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡉⠙⠛⠻⠿⠿⠿⠏⣰⡿⢋⣤⣶⡿⠿⢿⣶⣄⠙⢿⡆⠹⠿⠿⠟⠛⠋⢁⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣤⡄⢸⡿⢡⣾⣿⡏⢰⣶⣦⣽⣿⣷⡀⣴⡄⢠⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⣿⡇⢸⣿⣿⣷⣬⣈⡙⢿⣿⣿⡇⠘⠃⢸⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣤⣄⢸⣷⡈⢿⣿⣏⠻⠿⠟⣸⣿⡿⢁⣿⠇⣀⣀⣤⣤⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣷⣌⠛⠿⣷⣤⣾⠿⠋⣠⣾⠏⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣧⡌⠛⠿⣶⣦⣤⣴⣶⠿⠛⠡⠾⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡇⡰⡶⣤⣖⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠃⠛⠉⠓⠚⠙⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣍⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠂⠀⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠃
⠀⠀⠀⠰⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⡻⡄⡾⣸⣐⠏⣇⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⣤⣶⣶⣤⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⡈⠀⠁⠀⠁
⠀⠀⠀⡔⣠⢄⣰⠰⠀⠉⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠉⠁⠁⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣷⣾⡏⣿⠏⣷⣼⢹⡿⡿⡏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⣿⣿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⣿⣿⣧⣽⣿⠀⠆
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⡶⣶⡀
⠀⠀⠀⠀⠀⡦⠀⢠⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠁⠉⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠝⢱⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⢰⡰⢼⠐⡤⢳⠂⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠁⠁⠁⠁⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⡄⢠⢀⣠⡠⡄⠰⢂⣀⡄⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠑⠊⠛⠃⠃⠐⠘⠉⠙⠑⠊⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢣⡜⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠤⠶⢤⣤⣀⣤⣤⣦⣼⣾⣿⡟⢹⡏⢻⣿⣷⣦⣴⣄⣤⣀⣤⡤⠶⠤⠶⠤⠤⡄
⠀⠀⠀⠀⠀⠀⣀⡀⠛⣡⠤⠒⠚⠛⠛⠉⠻⣿⡿⠿⠟⣛⣵⣿⠏⢀⣾⣷⡀⠹⣷⣮⣛⠻⠿⢿⣿⠟⠉⠛⠛⠒⠲⠤⣌⠛⢀⣀
⠀⠀⠀⠀⣀⠈⣩⡖⠋⠀⠀⠀⠒⠀⠈⠀⠀⢀⣠⣴⣾⣿⡿⢃⣠⣾⠻⠟⣷⣄⡘⢿⣿⣷⣦⣄⡀⠀⠀⠁⠀⠒⠀⠀⠀⠙⢶⣍⠁⣀
⠀⠀⠀⠀⠋⣴⠋⠀⠀⣀⣤⣴⣶⣶⣶⡶⠖⣯⠀⢸⣿⣿⣾⢿⡿⠃⠰⠆⠘⢿⣿⣷⣿⣿⡇⢀⣽⠲⢶⣶⣶⣶⣦⣤⣀⠀⠀⠙⣦⠙
⠀⠀⢀⡮⢠⠇⠀⣼⣿⡿⠿⠛⣻⡿⠋⣀⠀⠀⣰⡿⠏⠁⠁⢸⡇⠀⠺⠗⠀⢸⡇⠈⠈⠙⢿⣆⠀⠀⣀⠙⢿⣍⠛⢟⢿⣿⣦⠀⢸⡄⢰⡀
⠀⠀⠀⡇⢸⠀⢀⣿⡗⠀⠀⠀⣾⠃⠀⠉⢀⣾⠏⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠻⣧⡀⠉⠀⠘⣧⠀⠀⠀⢻⣿⠀⢀⡇⢸
⠀⠀⢠⡆⢸⡇⢸⡿⠀⠀⠀⢠⡇⢠⡦⠀⢸⡇⠀⠀⠀⠀⠀⠀⣷⠀⢰⡆⠀⣾⠀⠀⠀⠀⠀⠀⢸⡇⠀⢴⡄⢸⡄⠀⠀⠠⢿⡇⢸⠇⢸⡄
⠀⠀⠈⢣⠀⣷⠈⣇⠀⠀⠀⠀⡃⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠸⠀⠀⠀⠀⣸⠁⣼⠀⡜⠁
⠀⠀⠀⢸⡆⢹⡄⠘⡆⠀⠀⠀⣇⠸⠇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠺⠇⣸⠀⠀⠀⢰⠃⢠⡏⢰⡅
⠀⠀⠀⠈⢣⠀⡇⢠⣰⡄⠀⠀⠹⡄⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢨⠃⠘⡅⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢠⠏⠀⠀⢠⣆⡄⣸⠀⡜⠁
⠀⠀⠀⠀⣾⣠⣿⣞⢻⣿⣄⠀⠀⢣⠘⠃⠀⠀⢷⡄⠀⠀⠀⠀⠀⡼⠀⠀⢧⠀⠀⠀⠀⠀⢠⡾⠀⠀⠘⠃⡞⠀⠀⣰⣿⡟⣷⣿⣄⣷
⠀⠀⠀⠀⠈⠉⠙⢏⣈⣹⣿⡿⢷⣌⣷⣀⣄⠀⣞⣿⣦⣤⢀⡀⣴⣷⣶⣶⣾⡦⣀⣀⣠⣴⣿⡳⠀⣠⣀⣾⣡⡾⣿⣿⣏⣀⡹⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠻⢭⣍⣉⠙⠛⠛⠻⠿⠿⠶⠾⠿⠿⠿⠿⠿⠤⠿⠋⠙⠿⠥⠿⠿⠿⠿⠿⠷⠶⠿⠿⠿⠛⠓⠛⣉⣉⡭⠗
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡦⠌⢭⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣭⠡⣤⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢞⠻⠭⣤⣀⣠⠄⠀⠀⠀⣩⡟⠛⢿⠅⠀⠀⢀⠖⠲⡀⠀⠀⠸⡿⠛⢿⡝⠂⠀⠀⢠⣄⣀⣤⠭⠟⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠨⠭⢁⣀⠀⣠⠾⠿⠶⠟⠀⠀⠀⠘⠦⠴⢃⡀⠀⠈⠿⠶⠿⠷⢄⠀⣀⡈⠭⠅⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠛⠀⠓⠀⠀⠃⠘⠐⠈⠃⠀⠈⠘⠛⠘⠘⠁⠘⠀⠐⠚⠚




⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⡶⠶⠶⠶⠶⣶⡶⢶⡶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⠀⣰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇⠀⠀⠀⠀⠘⣇⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡗⠶⢶⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⢸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢄⣠⠱⠀⠀⠀⠀⢻⡿⣿⣦⣀⠀⠀⢸⠀⠘⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡇⠈⠻⢿⣷⣦⣬⣄⣥⣴⣿⣷⣄⣀⣀⣠⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠉⠙⠛⠛⠛⠉⠈⠙⠛⠟⠛⠋⣿⣿⠁⠀⣤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣧⣀⠀⢻⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣆⣠⠞⠉⠀⠀⠀⠀⠉⢻⡿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠋⠀⢀⣿⣇⣀⣀⠀⢠⡶⢶⡀⠀⠀⠀⡏⠀⠘⣿⡅⣴⡾⢛⣻⣷⠶⠿⢶⣿⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣶⣶⡾⠛⠉⠁⠀⠉⠙⢾⣿⡧⠙⢦⡀⢰⡇⠀⠀⠹⣷⣏⠀⢿⠀⠀⠀⠀⢸⣿⣆⣀⣸⡓⠂
⠀⠀⠲⡶⠺⡄⠀⠀⠀⠀⠀⢹⣿⡠⢀⣤⣴⣦⣄⡀⠀⠙⣯⢂⠈⠳⣼⠀⠀⡀⠀⢻⣟⠦⣼⡷⠟⠁⠀⣼⠹⡏⠀⠈⣷⣤⣤⣀⡀
⠀⠀⠸⠓⢴⠆⠀⠀⠀⠀⠀⣾⣿⣏⣾⠋⠀⠀⠉⢷⠀⠀⢸⡷⣄⠀⠈⠀⢠⣧⠀⠀⢿⣄⠀⠀⣀⣤⡾⣣⠾⠷⠶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣷⡀⠀⠀⠀⣸⠃⠀⢸⣯⣿⣷⡄⠀⢸⠙⣧⣠⣬⡿⠻⠛⠛⠉⠰⡏⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⣀⡤⠶⠾⠿⠿⠿⢿⣿⣿⣯⠻⠦⠤⠞⠁⠀⢀⣾⣿⣿⡈⠻⡶⠟⠀⠉⠉⠁⠀⢄⣑⣄⣀⣀⣹⣄⠀⠀⠀⠠⢎⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⡹⢿⣿⣧⣀⡀⢀⣀⣴⣿⣟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⡌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣠⠞⠁⠀⢈⠿⠛⠛⠛⠋⠁⠈⠻⣧⠀⠀⠀⢀⡴⠞⠛⠓⠲⣄⡂⠀⠙⢦⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣰⠋⠀⣠⠴⠶⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠔⢃⡀⠀⠀⠀⠀⠀⣿⣰⠃⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣃⡀⢰⠊⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣥⣴⣶⣶⣶⣶⣤⣌⡛⢿⣷⡀
⠀⠀⠀⢰⣿⣿⡿⠛⠛⠻⢿⣿⣿⡿⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⢀⠿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢟⣣
⠀⠀⢘⠋⡎⠉⠀⠀⠀⣦⠀⠀⠀⠀⢰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠚⡉⠀⠀⡀⡀
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠆⠀⢀⠀⠀⠀⠘⣲⠺⠂⠀⠀⠀⠀⠀⣸⡇⠘⠂⠻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠈⠡⠀⠋⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠰⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠈⠑⢄⠙⠻⢿⣷⠀⢰⣿⠿⠛⢉⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠒⣠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠒⢄⢀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠸⠿⠗⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠮⣹⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠐⠀⠀⠁⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣀⠀⠀⠐⠲⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠴⠈⠄
⠀⠀⠀⠀⣠⠀⡠⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⢁⣠⠼⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢠⠖⢃⣶⡒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠛⠲⣤
⠀⠀⠀⠀⡏⠀⢸⣤⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣤⣋⣸⡿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠁⣴⠦⠀⢀⣁⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⢸⡷⠀⠀⣶
⠀⠀⠀⢀⣷⣤⡔⠉⢢⣠⣼⡄⠀⠀⠀⠀⠀⠀⠘⢛⣋⣝⢷⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠨⡽⠀⣸⡟⠀⠀⠀⠀⠀⠀⠘⠛⣃⠰⠿⠿⠆⠘⠛⠃
⠀⠀⠀⠈⠛⠃⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡬⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠱⠶⠶⠶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠄⠖⠉



⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⢀⢸⣷⣤⣿⣧⡄⠀⠀⠀⠀⠀⠀⠔⠉⠀⠁⣩⣦⠠⢄⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⣻
⠀⠀⠀⠀⣰⣿⣨⡵⢹⣿⠏⠁⠀⠀⠀⠀⠀⠈⠦⢄⠀⢸⣿⡿⠁⠀⣸⠀⠀⠀⠀⠀⠀⢻⡟⢠⠞⠙⠦⠸⣿⠃⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⢛⡇⢀
⠀⠀⠀⠸⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠙⠢⠴⠉⠀⠀⠀⠀⠀⠀⠺⠿⠿⠇⠀⠀⠐⠿⠿⠿⠀⠀⠀⠀⠀⠀⢸⠗⠒⠒⢺⠀⣻⠇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣦⡀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⡤⠤⠤⠤⡄
⠀⠀⠀⢰⣶⠿⣶⡦⣿⣄⣼⡧⠀⠀⠀⠀⠀⢀⣽⣅⠀⠀⠀⠀⢀⣭⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣟⣁⣤⣶⠆⢀⡀⡇
⠀⠀⠀⢻⣧⣀⣼⡟⠈⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣠⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣛⣃⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢩⠘⠚⠻⢳⠇
⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠀⠀⠀⠈



⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣰⢆⡤
⠀⠀⠀⣴⣮⠀⣠⣿⣆⠀⣵⣦⠀⠀⠀⠀⠀⠀⢀⣤⡼⣿⣿⣿⠶⣄⠀⠀⠀⠀⠀⠀⠀⠸⢿⡟⣵⣶⣤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏
⠀⠀⢠⣾⣷⡄⢿⣦⡿⢠⣾⣷⡄⠀⠀⠀⠀⠀⣾⣿⣿⢜⣛⡋⠸⠈⡆⠀⠀⠀⠀⠀⠀⠘⢷⡇⣛⡿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣭⣭
⠀⠀⠀⠀⠐⠠⠬⠛⠥⠤⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠘⠛⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠠⣤⠆⠙⠿⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣿⣿⣶⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⡦⢄⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡀⠀⠀⠀⢀⠔⠈⠀⠠⠈⠀⢀⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠸⠀⢆⠀⠄⠀⡀⠐⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠘⠻⠒⡽⠃⠈⠂
⠀⠀⠀⡠⠂⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠤⠤⠤⢄⣀⠈⠁⠀⠀⠀⠄⠦⠤⠤⠤⠀⠀⡀
⠀⠀⢀⡁⠀⠀⠀⠂⢩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠐⠠⡀⠀⠈⠑⠢⣄
⠀⠀⢼⠀⠀⠀⠂⠐⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⢦
⠀⠀⠘⡀⠀⠀⠀⠈⢼⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⡈⠍⠛⠀⠀⠈⠀⠁⠈⠉⡉⠉⢉⠉⠉⠉⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠂
⠀⠀⠀⠳⠀⠀⠀⣀⠙⠀⠀⠀⠀⠀⠀⠄⠈⠢⠠⠤⠂⠡⠔⠠⠀⠠⠀⠀⢄⡀⠐⠢⠤⠀⡀⡀⢤⣶⠀⠀⠀⣀⠤⠄⠒⠈⢀
⠀⠀⠀⠀⠉⠂⠀⢀⠀⠀⠀⠀⠀⡃⠀⢀⢬⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⠀⠈⠁⠈
⠄⠀⠤⠁⠀⠐⠂⣁⣀⡀⠀⠀⠀⠁⠊⠁⠀⠉⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠠⠄⠈⠍⠉⢀⡀⢀⣐⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⡠⠘⠣⣀⢀⣈⡍⠁⠈⠀⠀⠀⠀⠀⢰⠂⠐⠒⠒⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⡔⠋⡀⢀⠀⡬⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠔⠂⠈⠒⠑⠊⠀⠀⠣⠘⠒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡨⠄
⠀⠀⢀⠂⠤⠀⣀⡀⠀⠘⠀⡠⠔⣊⠤⠊⠀⠀⠀⠀⡀⠀⡀⠀⠐⠉⠉⠉⢨⡂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠈⠑⠒⠦⠬⣉⣹⣁⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⠐⣐⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠜⠀⠰⠤⢀⣉⠐⠒⠠⠤⠤⣸⡄⠊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠢⠤⠆⠂

⢀⣤⣶⡟⠛⣿⠷⢶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⢿⡟⠛⣷⣦⣄
⣾⣿⣟⣿⡟⢻⣶⣾⣿⣶⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣾⣿⣶⣾⠛⣿⣟⣿⣿
⢷⢿⣿⣟⣻⡟⠁⠀⠀⠀⠀⢿⣿⡛⠻⢿⣷⣄⡀⠀⠲⣤⡀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⣀⣴⣿⠿⠛⣻⣿⠇⠀⠀⠀⠀⠙⣿⣛⣿⣿⢷
⠘⣿⢿⣿⣿⣶⣤⣤⣄⣀⣀⠀⢻⣷⡄⠀⠈⠙⠿⣦⡀⠀⠙⢆⠀⠀⠀⢀⠔⠁⠀⣠⡾⠟⠉⠀⠀⣴⣿⠃⢀⣀⣀⣤⣤⣴⣾⣿⣿⢿⡟
⠀⠘⣿⢿⡏⠉⠉⠈⠉⠉⠉⠉⠙⠛⢿⣦⡀⠀⠀⠈⠻⣦⠀⠀⢣⠀⢠⠊⠀⢠⡾⠋⠀⠀⠀⣠⣾⠟⠛⠉⠉⠉⠉⠉⠉⠉⠉⣿⢿⡟
⠀⠀⠸⡿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⢷⡀⢀⣷⣇⠀⣰⠏⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡿⡿
⠀⠀⠀⣷⣿⡿⠿⠿⠿⠟⠛⠛⠒⠒⠶⠶⠶⠾⠿⠶⣄⡀⠀⠻⣴⣿⣷⡼⠃⠀⣀⣴⠾⠿⠶⠶⠶⠖⠒⠚⠛⠻⠿⠿⠿⠿⣿⣷⡇
⠀⠀⠀⢻⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣿⣿⣿⣧⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⠃
⠀⠀⠀⠈⢿⣼⣷⣤⣀⣀⣀⣠⣤⣤⣤⣴⣶⡶⠶⠶⠶⠓⠒⠚⣿⣿⣿⡟⠒⠒⠲⠶⠶⠶⣶⣶⣤⣤⣤⣄⣀⣀⣀⣠⣴⣿⣼⠏
⠀⠀⠀⠀⠀⠉⠻⠾⢿⣿⣿⠿⠿⠿⠯⠥⣤⣤⠴⠶⠒⠛⢋⣩⣿⣿⣿⣯⣉⠛⠓⠲⠶⢤⣤⡤⠭⠿⠿⠿⢿⣿⣿⠿⠾⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣤⣤⠤⠤⠤⣴⡿⠀⠀⠀⣠⡴⠋⣡⢿⣿⣿⢧⡉⠳⣤⡀⠀⠀⠸⣷⡤⠤⠤⣤⣤⣬⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣾⣾⠁⠀⢀⣀⡤⠤⣾⣉⣀⣤⣶⠟⠁⠀⣴⠃⢈⣿⣇⠀⢳⡄⠀⠙⢷⣦⣄⣈⣹⡦⠤⣄⣀⠀⠀⢹⣾⡆
⠀⠀⠀⠀⠀⠀⢿⢿⡶⠟⠉⠀⣀⡴⠊⠉⢁⡼⠁⠀⢠⡾⠁⢀⡜⣿⡟⣆⠀⠹⣦⠀⠀⠹⣄⠉⠉⠲⣄⡀⠉⠙⠷⣾⢿⠇
⠀⠀⠀⠀⠀⠀⠘⣿⢧⣀⣤⠞⠉⠀⠀⢰⡿⠀⠀⣰⠟⠀⢀⠞⠀⣿⡇⠘⢆⠀⠘⢷⡀⠀⠸⣷⠀⠀⠈⠛⢦⡀⣠⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣻⣇⠀⠀⠀⣰⠏⠀⢀⣼⠋⠀⡠⠋⠀⠀⣿⡇⠀⠈⠣⡀⠈⢻⣄⠀⠈⢷⡀⠀⠀⢈⣿⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣴⣏⣀⢀⣾⣃⣤⠞⠁⠀⠀⠀⢹⡇⠀⠀⠀⠘⢦⣄⣻⣆⠀⣈⣷⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣿⣏⣿⣯⡿⠟⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠿⣯⣿⣏⣿⣷⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⡄⠀⠀⠀⠀⠴⢿⣆⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡿⠤⠠⠀⠄⠀⠀⠈⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢈⢁⡤⢏⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠀⠀⠈⠳⢄⠠⠤⠄⠆⠀⢂⡈⠁⢢⡀⠈⢧⡀⠀⡀⠠⣀⠁⡀⠀⠡⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⡀⠁⢈⠀⠀⠀⣂⢀⠂⠀⠀⠀⠒⠦⡀⠀⠀⠀⠀⠑⡌⢀⠊⠀⠀⠈⠀⣀⣀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠠⢀⠂⠐⡰⠇⠄⠀⠀⠈⠱⡀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⢰⠀⠀⠀⡠⠤⠀⠠⣄⠑⢆⠀⠰
⠰⣄⡆⠀⠀⡅⠀⠀⠀⢀⡤⡄⠀⢤⡄⠀⠀⠀⠀⠀⡆⠀⠀⠚⡅⠀⠀⡄⠀⠀⠀⡄⠀⠀⠰⠘⠲⠠⠄⠉⠀⢸⠀⡘
⠚⠻⣿⠋⠀⠁⠀⠀⠀⠀⣤⡄⠀⣄⡌⠀⠀⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⢊⡀⠀⢰⢤⡀⠈⢠⣀⠀⠉⡡⠀⠀⠃⡐
⠀⠀⠈⠳⠦⠴⡀⠀⠀⠈⠘⠀⠐⠒⠀⠄⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠿⠀⢰⣦⣤⡼⠁⡀⢡⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠤⢒⣿⠀⠀⠀⠀⢠⢷⠀⠀⠡⢄⠀⡠⠄⢀⣼⣿⣾⢇⠀⠘⠒⠊⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣴⣵⣤⡄⠀⠈⠋⠑⢀⡤⢄⡠⢴⣋⣿⠀⠀⠀⠀⣀⠀⠀⠀⣠⠋⢀⣨⠦⢀⣀⣀⡠⢒⡏⣀
⠀⠀⠐⢦⣤⣀⣻⠀⠉⠉⠒⠦⢖⠒⠉⡇⠀⠀⠀⠞⠟⠓⠂⠰⡎⡼⠁⠒⠈⠀⠰⠯⠥⠤⣤⡄⠀⢳⠈⠳⠟⠃
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⡟⢧⣰⡇⠀⠀⠀⠀⠀⠀⠀⣠⠿⠴⠶⢲⡆⠀⠀⠀⣀⣴⠟⠁⠀⠈⠷⠤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠁⠀⠀⠀⠀⠀⢠⣾⣥⣤⡀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢂⢀⢤⠊⠀⠩⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⠀⢀⠀⠀⣠⠐⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠹⡨⠄⠉⠁⠀⠀⠀⠄⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡀
⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⠃⠌⠠⠀⠀⣀⣀⣀⢠⡀⠀⠤⠆⢒⠀⠀⠀⠀⠔⡀⠌⠀⠀⠁⠘
⠀⠀⠀⢀⡌⡄⠀⠀⠀⠀⠂⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⡠⠁⠐⠤⠀⠀⠀⠂⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠄⠀⠊⠐⠠⡀
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠘⠀⠀⢂⣠⡀⢠⣀⠂⠀⢸⠀⡐⠀⠀⠄⠀⠀⢀⠈⢂
⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠐⢩⣆⠀⠀⣶⡙⠀⡈⠀⠡⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢃⠀⠀⡀⠀⠀⠀⠰⡦⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠉⠁⢀⡃⠐⢤⠄⣀⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠆⠈⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠑⢤⣀⣀⣀⣀⠴⢋⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠔⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⢣⢸⢇⠎⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⣿⣿⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣵⣿⣯⣽⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧⣟⡿⡾⢾⠶⣟⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠙⡾⢽⣭⣟⣹⡏⢹⣅⡼⡆⠀⠀⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣏⣹⠓⡿⢤⣯⡼⣯⡽⢧⡤⢷⢀⡴⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢧⠀⠀⣸⠦⣯⣹⣿⣾⡟⠚⣟⠶⢻⠒⠛⡏⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣄⣟⢺⠧⢼⡇⠙⣏⢛⣏⠓⣻⡓⢚⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢹⣍⣽⠓⢺⡇⠀⢹⣏⣿⣙⣹⣏⣉⣼⢰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢦⣻⡀⢼⠢⣿⣝⣽⠇⠀⠈⡯⢾⣍⣉⣿⣭⢼⣾⡏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠻⡇⣾⣉⡿⢤⡟⠀⠀⠀⡷⢺⡯⡭⢿⠦⠾⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⡷⢭⣽⣾⠁⠀⠀⢰⣟⣹⡗⠶⣿⡓⣾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡖⢿⡏⠀⠀⠀⢸⣧⣤⣏⣋⣹⣿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⢿⣹⠇⠀⠀⠀⢸⠶⠾⣧⣉⡽⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⢿⠀⠀⠀⠀⢸⣓⢚⣷⠏⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠸⡇⠀⠀⠀⠸⣯⡟⠁⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢻⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢳⣄⠀⠀⠘⡇⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣍⣷⣤⣼⡧⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⡏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣉⢀⣴⠛⠉⠉⠉⠙⠛⠶⣤⣿⣿⣿⣿⣤⠶⠛⠋⠉⠉⠉⠛⣦⡀⣉⠻⣆
⠀⠀⠀⠀⠀⢀⣀⡀⠀⡏⠰⡿⢸⡇⠀⠀⠀⢀⣀⣀⠤⠀⣿⠀⠀⣿⠁⠤⣀⣀⡀⠀⠀⠀⢸⡇⢿⠆⢹⠀⢀⣀⡀
⠀⣴⡿⢛⡿⠿⠛⠛⠛⢿⡀⣤⡘⣷⠛⠛⠛⠋⠉⠀⠀⠀⡿⢸⡇⢿⠀⠀⠀⠉⠙⠛⠛⠛⣾⢃⣤⠀⡿⠛⠛⠛⠿⢿⡛⢷⣦
⣴⡏⢰⡏⠀⠀⠀⠀⠀⠘⣧⠘⠃⠹⣆⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⢸⠀⠀⠀⠀⠀⠀⠀⣰⠇⠘⠃⣼⠃⠀⠀⠀⠀⠀⢹⡆⢹
⠻⡇⢸⡄⠀⠀⠀⠀⠀⠀⠈⢷⡘⣷⡘⣦⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⠀⠀⠀⠀⠀⠀⣴⢃⣾⢃⡾⠁⠀⠀⠀⠀⠀⠀⢠⡇⢸
⠀⣿⡌⢧⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢠⡌⢷⡀⠀⠀⠀⠀⡇⢈⡁⢸⠀⠀⠀⠀⢀⡾⢡⡄⢠⡞⠁⠀⠀⠀⠀⠀⠀⢀⡼⢡⣿
⠀⠙⠿⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠛⠀⢳⡀⠀⠀⠀⡇⢸⡇⢸⠀⠀⠀⢀⡞⠀⠛⣱⠏⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⠿⠋
⠀⠀⠀⢿⣦⠹⡄⠀⠀⠀⠀⠀⠀⠀⢹⡄⢿⡄⢳⠀⠀⠀⠀⣨⣅⠀⠀⠀⠀⡞⢠⡿⢠⡏⠀⠀⠀⠀⠀⠀⠀⢠⠏⣰⡿
⠀⠀⠀⠀⠙⢧⠙⢀⠀⠀⠀⠀⠀⠀⠀⠇⢈⣤⣌⠇⠀⠀⢾⣿⣿⡷⠀⠀⠸⣡⣤⡁⢸⠀⠀⠀⠀⠀⠀⠀⡀⠋⡼⠋
⠀⠀⠀⠠⡶⢾⡟⠿⠳⣦⠀⠀⠀⣴⣶⢀⣘⣻⣫⣤⡀⣶⣶⣾⣷⣶⣶⢀⣤⣝⣟⣃⡀⣶⣦⠀⠀⠀⣴⠞⠿⢻⡷⢶⠄
⠀⠀⠀⠿⠷⣼⡟⠛⠛⠧⣶⣷⠀⣾⡿⠛⢃⣿⣷⣛⠁⣈⣿⡿⢿⣿⣁⠘⣛⣾⣿⡘⠛⢿⣷⠀⣾⣶⠼⠛⠛⢻⣧⠾⠿
⠀⠀⠀⠀⠀⢿⣷⣦⢠⣶⣄⣙⡲⠿⠿⠿⠯⠼⠧⠽⠿⢿⢿⢥⡬⠿⠿⠿⠯⠼⠧⠽⠿⠿⠿⢖⣉⣠⣶⡄⣴⣾⡿
⠀⠀⠀⠀⠀⠀⢯⣿⣷⣶⠒⣾⣿⣋⣿⡿⢿⣏⣹⡿⢿⣏⣹⠟⠻⣏⣹⡿⢿⣏⣹⡿⠻⣿⣙⣿⣶⠒⣶⣾⣿⡽
⠀⠀⠀⠀⠀⠀⢸⣏⠀⢹⣿⣯⣼⡿⢿⣄⣠⡿⢻⣷⣾⡟⢻⣦⣴⡟⢻⣷⣾⡟⢿⣄⣠⡿⢿⣧⣽⣿⡏⠀⣹⡇
⠀⠀⠀⠀⠀⠀⣸⠿⠿⠿⠷⣚⣛⣓⣚⣛⡛⠛⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠿⠛⢛⣛⣓⣚⣛⣓⠾⠿⠿⠿⣇
⣶⣶⡶⣶⣶⣶⣿⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣿⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠉⠁⠒⠢⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⠀⠙⢆⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⢄⡀⡰⠁⡴⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠾⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏⢀⡵⣕⠀⠀⠀⠀⠈⢢⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏
⠀⠁⠢⠀⠀⠒⠄⡀⢀⠔⠀⠈⠊⠐⣈⡻⠃⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⠁
⠀⠈⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠌⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡇⠀⠀⠀⠀⢸
⠀⢰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⢂⡔⠛⠃⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⡇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⡀⢀⡠⠃⠀⠀⠀⢀⠤⣚⡐⢛⣤⠤⣀⡀⠀⠈⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠀⠀⢧⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠞⠁⠄⠀⠣⠄⢠⠤⠊⠉⠓⠢⢤⠴⠃⠀⠀⠀⠀⢀⣀⡤⡞⠁⠀⣸⠀⠀⠈⠳⢤⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠒⢉⣉⡀⠔⠀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠀⡸⠀⠀⡱⠦⠴⡄⠀⠀⢰⡆⠠⣥⠀⢀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⣇⣀⣀⣧⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⢸
⠀⠀⠀⠐⠲⢄⣄⣠⢶⠊⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣄⠀⣰⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠃⠈⠛⣿⣿⣿⡿⠛⠁⠙⠻⣿⣿⣿⠟⠋⠀⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣇⣸⣷⠈⣿⣿⠁⣾⡇⣿⡆⢹⣿⡟⢰⣿⣀⣷⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣈⠛⠛⢃⣼⣿⣿⣧⡙⠛⠛⣡⣿⣿⣿⣄⠛⠛⢋⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣂⠈⣿⣿⢡⣾⣿⣦⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⣩⣴⣶⣿⣿⣡⣾⣿⣿⣆⡛⠟⣋⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⣛⠻⣿⣿⣿⣿⠟⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢼⣿⡷⢸⣿⣿⣇⢸⣿⡿⢸⣿⣿⠟⢛⣛⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣭⣴⣿⣿⣿⣿⣦⣭⣴⣿⣿⣿⡄⣦⡙⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣦⡙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢉⣍⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⢹⣿⣬⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⢿⡿⢿⠿⠿⠿⣿⡿⠿⠿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⡟⠉⢹⡟⠻⠿⠿⠟⠿⡛⠿⠻⠿⠿⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡿⣿⣿⣿⡿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣾⣿⣁⡀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣾⣶⡿⠿⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⡆⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠙⠻⣟⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⠀⢀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⢩⣍⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣰⣿⣏⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⣿⣯⣿⣿⡄⠤⠀⠀⣀⡤⠤⣄⠀⠠⠤⠤⠄⣾⣿⣿⣿⠛⠿⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠛⣿⣿⡅⠀⣶⣦⠀⠀⠀⣠⣴⣾⣷⣦⡀⣿⡟⠙⢯⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⠒⣿⣿⠧⣼⣿⣧⣤⣤⣾⣿⣯⣍⣩⣿⣿⣿⣷⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠇⠈⢿⣿⣿⣿⠿⠁⠉⣉⣽⣿⠏⢽⣿⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣀⣀⣀⣉⣉⣡⣤⣤⣀⠈⣛⣁⣀⣿⣿⡄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠉⠁⠀⠀⠀⢸⣇⠀⠉⠛⢤⣀⠀⣹⢹⣧⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⡀⠀⣰⠞⠶⣀⢿⣄⠀⠀⠀⠈⠉⠻⢀⣽⣇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠴⠶⢾⣿⠇⠀⣿⣦⠀⠈⠃⠙⣷⣦⣀⠀⠀⠀⠀⠙⣿⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⣄⠀⠀⠈⠆⠘⠏⠳⢤⣀⣀⡄⠘⣷⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢷⣤⣤⣤⣿⣶⠀⠀⠀⠐⢶⣿⣾⣿⣻⣿⣦⠈⠇⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠁⠙⢏⡀⠠⣧⡀⠀⠀⠀⠙⢿⣿⣿⡙⢧⣄⠈⠀⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠁⠺⣿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣄⣽⣧⠀⢠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⢀⡀⢀⠈⠉⠛⢿⣷⡄⠐⠀⠀⠀⠀⠋⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢠⣾⣿⣧⣄⠀⠀⠀⠉⠑⠲⣆⡀⠀⠀⠀⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣾⠿⣷⠻⣿⢿⡄⠀⠀⠀⠀⢻⣅⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠀⢠⢜⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢲⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠿⢿⣿⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠠⠴⢬⣭⠷⢶⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠳⢦⣶⣦⣀⠂⠠⠍⢸⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠙⠓⠚⠂⠿⠿⠿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣄⣀⣀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣛⡿⢿⣿⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⢛⠹⠁⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⡿⣿⣙⡶⠿⠿⡶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢶⠿⠿⢶⣋⣿⢿
⠀⢿⡯⣭⡗⠒⢚⣻⡿⢈⣿⣵⣦⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣴⣮⣿⡁⢿⣟⣓⠒⢺⣭⢽⡿
⠀⠈⢷⣏⣹⡏⠉⢙⣿⣮⣉⣻⣷⣮⣑⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣊⣵⣾⣟⣉⣵⣿⡋⠉⢹⣟⣹⡾⠁
⠀⠀⠈⢿⣽⣾⠛⢻⡟⠁⠀⠉⠂⠀⠀⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠐⠉⠁⠈⢻⡟⠛⡷⣯⡿⠁
⠀⠀⠀⠀⢹⣿⡶⢾⣷⣄⡀⠀⠤⠦⣀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⣀⠴⠤⠀⢀⣠⣾⡷⢶⣿⡏
⠀⠀⠀⠀⠘⣿⢿⠛⢿⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⡿⠛⡿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣛⣦⠄⠐⠒⠒⠚⠉⠉⠐⠂⢀⠀⠀⠙⣄⣶⣶⣠⠋⠀⠀⡀⠄⠂⠉⠉⠓⠒⠒⠂⠠⣴⣛⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⡟⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢈⣿⣿⡁⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣶⠿⠿⢿⢿⡿⠿⠶⠶⠶⠶⣿⣿⣿⣿⠶⠶⠶⠶⠾⢿⡿⡿⠿⠿⣶⣶⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣼⠟⣿⡏⠀⠀⠀⠀⣠⠊⠀⠀⠈⠀⢀⠔⢹⢿⡿⡏⢢⡀⠀⠉⠀⠀⠑⣄⠀⠀⠀⠀⢹⣿⠻⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⠟⠉⠉⠉⢁⠁⠀⠀⢀⡤⠒⠁⠀⠃⣾⣷⠘⠀⠈⠓⢤⡀⠀⠀⠈⡀⠉⠉⠉⠛⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⣀⣀⠠⠴⠃⠰⠖⠋⠉⠀⠀⠠⠃⡄⢿⡿⢠⠘⠄⠀⠀⠉⠙⠲⠆⠘⠦⠄⣀⣀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⡏⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⡼⠁⢸⡇⠈⢧⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢹⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣟⣿⣿⣿⣏⠀⠀⢀⡄⠀⢀⠄⠀⡼⠁⠀⠀⠀⠀⠈⢧⠀⠠⡀⠀⢠⡀⠀⠀⣹⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣟⣿⣟⣻⡿⣦⢴⡏⢀⡼⠁⠀⠀⠀⠀⠀⠀⠈⢧⡀⢹⣦⣴⢿⣟⣻⣿⣻⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⠿⠽⢿⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡿⠯⠿⠗⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡕⣃⢹⢰⣿⠷⠲⣦⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⣿⣅⠀⢿⣟⠀⢶⡏⣸⢛⡿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠌⣻⣶⠋⠉⢛⣾⡷⣷⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⢳⣦⡘⠦⢸⣇⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢋⣤⣠⣾⣟⠛⢿⣿⠉⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣴⣶⡤⣤⣞⡃⠀⠀⠀⠙⣿⡧⠾⡛⠛⠛⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣻⣿⣿⣿⣿⣿⡇⢿⣿⡃⠀⠀⢀⣼⠟⠳⣶⣿⣶⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⡇⢿⡋⢡⣄⡀⣀⠏⠀⣧⣉⣰⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠈⠉⣿⡃⢀⣾⢿⣿⡛⢻⠉⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣿⣿⣿⣿⣿⣿⣿⣇⢸⣇⣀⣴⣿⠂⠀⣼⡿⠈⣷⠈⢲⣶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣵⠃⠀⢰⣿⠃⠀⠉⠄⢡⢻⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⢃⠎⠀⡼⠂⠀⠀⠀⠀⠈⢆⢻⡍⢌⢷⣀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⡿⢡⠏⠀⣼⣤⡶⢾⡷⠶⣤⣄⠈⠊⢿⡌⢣⡻⣷⣄⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠛⢰⠋⠀⣴⣿⣿⡀⢸⡇⠀⠈⢻⠀⠀⠈⢻⡄⠀⢹⣾⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢻⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣹⡇⢀⣤⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡌⢿⣕⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣿⠀⣞⠇⡐⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣷⢀⠀⡄⠀⠀⠈⣿⣿⢞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢸⣿⣿⡃⠀⢠⠀⣄⠀⠀⠸⣿⠀⢻⣿⡟⠈⢀⣡⢀⠁⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣾⣿⣿⣧⠀⣞⡀⢿⣷⣤⣠⣿⣤⡿⠋⠀⠀⢿⡟⢸⣼⣈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢠⣿⡇⠀⠀⠀⠉⠛⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⣿⣿⢳⣼⣇⢠⢰⢠⣰⢠⣧⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣧⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠥⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⡂
⠀⠀⠀⠀⢀⣤⣤⣶⡦⠀⠀⢠⣦⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾
⠀⠀⠀⠀⠾⠿⠿⠿⠀⠀⠰⠿⠿⠿⠧⠀⠀⠀⠀⠠⣿⣷⣴⡟⠉⣠⣄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣟⠃⠀⠀⠀⠀⠀⠘⠒⠉⠈⠀⠀⠀⠀⠀⠛⠛⠛⠛⡚⠃⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠻⢿⣿⠁⠀⠀⠀⠀⠀⢀⣀⢤⡤⡀⠀⣤⡄⠰⣛⢖⡶⠶⣳⡦⠀⢰⡟⠁⠉⣷⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠸⠇⠈⠁⠸⠇⠘⠉⠁⠉⠉⠉⠒⠀⠈⢳⡤⣴⠛⠀⠀⠀⢀⠰⣼⠖
⠀⠀⠀⠀⢄⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⣀⠀⠀⠀⠀⣤⠀⣦⢤⡤⠤⠀⠀⠀⢰⣶⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣴⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⣀⣀⣠⣤⣰⣶⢸⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡇⠀⠀⠀⠐⠭⠃⠸⣭⣭⠀⠀⠀⠀⣤⠀⠤⠤⠤⠄⠘⠃⠥⠤⠥⠤⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⡇⠀⠀⠀⠀⠀⢀⡼⠃⡄⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠉⠁⠉⡁⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠀⠂⠀⠀⠀⠀⠀⢠⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢼⠥⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠃⠀⠀⠀⠀⣠⣾⣿⣄⠀⠈⢻⣿⣿⣆
⠀⠀⠀⢠⣾⣶⠌⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠄⣠⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢰⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠚⠃⣰⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠗⢀⣉⣃⡀⠻⠀⢶⠆⢰⡄⣴⡄⠙⠛⣛⠋⠤⣶⢠⣆⢰⡄⠰⠶⠘⠗⠐⠓⠠⠷⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⣴⠟⠛⠛⠓⠦⠈⠀⡀⣶⣶⣶⡇⠘⢠⣭⠄⢷⣶⣤⠀⠀⠀⣀⠀⠀⠠⠽⠿⢷⣦⡀⠉⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠶⠀⣦⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⢡⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠶⠂⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣛⠁⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⢰⣊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢄⠀⠀⠀⠀⢤⡀⠀⠀⠀⣐⢄⠉⢉⣴⡏⡀⠀⠀⠀⠀⡀⠀⠂⠀⠀⣴⠃⠀⠀⡞⣫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⡷⡄⠀⠠⠀⠗⣀⣄⠀⣉⠛⠀⢈⠋⠀⣿⡂⠤⠀⠀⠗⠂⢤⣠⣾⠃⠤⢠⣒⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⣿⠈⢳⡄⢻⣳⣞⢺⣿⡙⠀⣶⡶⠸⠇⢶⣤⣿⡷⠀⠼⢷⡾⣿⣿⣿⠁⠠⠃⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⣴⣶⣿⡦⣳⣥⣤⣽⣯⣭⠭⡶⠶⠶⢬⢭⣭⣭⣥⣦⣤⣻⣿⣿⣿⣿⢆⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣬⢵⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠉⠉⢉⣿⣟⣿⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣛⡍⠤⠀⢀⣀⣤⠀⠐⠀⠘⠒⠒⠚⠀⠉⢀⣁⣍⣉⣉⣏⣉⣹⣻⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⡀⠈⠁⠉⠙⠻⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠒⣒⣁⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⢤⣲⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⢿⡟⣙⣿⣻⣿⣿⣿⣿⣿⣿⣶⣦⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠷⠭⠮⣬⣷⣽⣮⣭⣽⠿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠟⠲⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠈⠀⠁⠁⠀⠀⠰⠆⠀⠀⠀⢁⣀⣪⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃⠀⠒⠂⠀⠀⠐⠁⠀⠀⠀⠀⢀⡆⠀⠀⠀⠈⠉⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠂⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠻⠿⠟⠶⣦⠀⠀⣼⡇⠠⠄⡀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠂⠀⢰⡆⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠘⠀⢸⣿⠇⠀⠰⡇⠀⠀⠀⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠃⠀⠁⠀⣀⠀⡀⠀⠀⢸⡇⢠⣄⡀⠀⢀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠛⠄⠁⠀⠈⢉⠉⠉⠛⠃⠀⠠⣶⠦⡀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⣆⠀⠀⠀⣠⡤⠀⠀⢰⠄⠀⢰⣪⣉⣱⠀⢠⣀⠴⢁⠀⠀⠀⠀⢐⠀⠉⠀⠀⠀⢠⠈⠀⠇⠀⠉⠈
⠀⠀⠀⢠⠀⠀⡀⠀⠘⠘⢝⡱⠀⠀⢈⠁⠄⢀⡀⠀⠄⠀⢉⡉⠀⠄⠀⠀⢤⢄⠠⡄⠀⠀⠀⠉⠀⠠⠄⠀⠙⡀⠀⠠⠀⢀
⠀⠀⠀⡀⠘⠀⢀⠀⡠⠔⠋⠀⠆⠀⠀⠀⣠⠐⡀⠀⠠⠄⠀⠐⠀⠀⠄⡀⠘⠈⠀⠀⢰⣶⡄⠀⣀⣀⡀⠀⠀⠀⠠⠋⠀⠉
⠀⠀⠀⠉⢢⡆⠉⠁⠀⣶⡄⠀⠀⣦⠀⠀⠱⣎⢿⡦⠤⠤⠤⠤⠤⣶⢷⡟⠀⠀⠀⠂⠀⠐⠂⠀⠀⠀⠀⠲⠀⠔⠀⠐⠀⠐⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠁⠀⠀⠀⠈⠉⠰⠀⠀⠀⠀⠂⠈⠉⠀⠀⡗⠀⢂⠘⠂⡀⣄⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠚⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⣰⠀⠠⠄⠀⣀⠀⠀⠀⢀⣇⠀⠂⠀⠀⠀⠁⣀⠀⠀⠈⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠲⡆⠀⠀⠀⠀⢰⠰⣇⠄⠋⠹⠄⠠⠒⠊⠋⠁⡲⡛⠒⣟⠺⠇⠀⠀⠄⠀⠉⠀⠂⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠉⠉⢤⣄⠀⠀⣸⠱⡀⠀⠀⠀⢸⣶⠠⣤⢀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⣤⣟⠀⠀⠀⠀⠀⠢⢤⠔⠀⠀⢠⠤⠂⠁⠀⢿⣸⣷⠿⠴⠀⠀⠈⠀⠀⣠⡄⣠⣾⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠀⣈⠀⠀⠐⠌⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠉⠉⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠰⠀⢀⡀⢰⠀⣀⠀⣀⣀⠀⠀⠀⠀⠈⣱⠄⠀⠀⠀⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡉⠉⠍⠉⠉⠁⠀⠀⠀⠀⠁⠀⠒⠒⠒⠒⣒⡒⠶⢒⠒⠒⠒⠒⠤⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⡶⡶⡶⢤⢠⡤⣄⢀⣀⣀⠀⠀⠀⠀⠀⢂⢠⠕⡑⡁⡠⡪⢂⠖⢂⠔⣰⡐⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⠈⠀⠁⠘⠈⠊⠣⠡⠙⠝⢏⢳⠲⡲⡾⡶⣾⡲⢖⠗⡻⢋⢟⠙⡩⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣷⢿⡐⣶⠲⡶⠶⢴⣦⣤⢤⣤⣄⣀⣈⠈⠊⡷⡡⢊⢔⠑⡣⢊⠔⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢻⣽⡋⠕⡄⠁⠀⠀⠈⠄⠀⠀⢰⣚⡍⠙⡀⢹⡴⡡⠊⠚⠔⣡⠞⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣏⡗⡁⢊⠄⡐⣲⢆⠀⠀⠀⠻⠉⠗⠀⡇⠸⣎⠔⡡⢊⠜⡡⠊⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣇⢘⢬⠀⠋⠈⠟⠀⠀⡄⣄⠐⠀⠀⡇⠀⣧⠎⡴⠡⠊⡠⢊⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠸⠚⠗⠀⠀⠀⠀⠸⠋⠻⠈⠀⠀⡇⠀⡧⢊⠔⡡⢊⠔⣩⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⢀⣔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡷⣡⢊⠔⢣⢖⡵⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡿⢄⡀⠂⢀⣜⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⡥⢁⢜⡵⢊⠜⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣯⣂⠙⠲⠮⠅⣒⡐⠩⠤⣀⡀⠀⠀⠀⢀⠇⠘⣏⢔⠵⡋⠔⡡⢊⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⣨⠒⣬⡈⠑⠒⢤⣀⡈⠉⠒⠒⠠⠬⠭⠵⠞⠀⢘⡗⡫⢊⠕⣪⢞⠕⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢁⢀⠸⠿⣿⣿⠇⠠⡊⢉⢉⢫⠉⠑⠲⠦⠄⠠⢆⣀⢂⣿⢊⠔⣡⢊⠕⣩⢊⣽
⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⣾⣡⣼⢗⠄⠀⠀⠀⠀⠑⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⠗⣡⢜⠕⡡⢎⠕⡡⣺
⠀⠀⠀⠀⠀⣠⢊⢔⡀⠀⠓⠒⠛⠋⠀⠀⠀⠀⠀⣠⡽⣿⣤⣀⡀⠀⠀⠀⠀⠀⡜⢡⢔⠕⡡⣊⠔⡡⡊⠋⣽
⠀⠀⠀⠀⠀⡙⠣⡀⠡⢆⠀⠀⠀⠀⠀⠀⠀⣰⠟⢏⠙⢷⠞⡀⣿⠄⠀⠀⢀⣾⢒⠕⡡⢪⠔⡡⢊⠔⡡⢊⣼
⠀⠀⠀⠀⠀⠇⠀⠈⠑⣤⣌⠢⠀⠀⠀⠀⠀⠫⣣⣤⣈⣲⣿⣾⠟⢀⠀⡡⢛⠕⡡⢊⠔⡡⢊⠔⣡⠎⠠⡡⣻
⠀⠀⠀⠀⠀⠀⠺⠲⢀⡀⣤⠛⠓⠢⠤⣀⡀⠀⠈⠉⠉⠉⠉⠀⠄⢡⣞⠕⡡⣪⢖⡱⢊⠔⣡⠞⡡⡪⣪⢞⢿
⠀⠀⠀⠀⠀⢸⠘⠳⠦⢀⢀⣉⡲⠄⠀⣀⣈⠉⠒⠦⠤⣀⡚⣠⣶⢟⡴⢋⠜⡡⢋⠴⢃⠞⡡⡪⢊⢔⠕⡡⢺⠭⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠈⡉⠃⠀⢠⡱⡄⠀⢠⠁⣍⡋⠓⠦⠤⣤⡈⣗⡕⢋⠔⡡⢊⡴⣋⠔⡡⠪⢊⠔⡱⢃⠔⡡⣿⠭⠅⠀⠂⣀⡀⠀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⡦⠃⡰⢆⠈⠍⠙⠂⢸⠀⣻⡇⢀⣤⢄⣽⡇⣏⠔⣡⠊⡠⢊⠔⡡⢊⢔⡵⡡⢊⡴⠣⢪⣾⣿⡉⠥⠤⠀⢀⣐⢒⡒⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣧⠘⠇⠀⠑⠷⠀⠀⢸⠀⠙⠃⠸⣼⡿⣿⡇⣏⠜⡡⢊⠔⣡⢊⠔⡡⢊⠞⡡⢊⢔⠵⢛⣼⠯⠀⠉⢁⣀⣀⡀⠐⢒⠒
⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠘⣰⣦⣄⣀⢠⡄⢸⡇⣇⢊⡔⢁⠞⡠⡣⢊⠔⡡⢊⡴⢿⢩⣴⡛⠍⠭⠅⠀⣀⣀⡒⠒⠂
⠀⠀⠀⠀⠀⠀⠰⠯⡈⠄⠀⠀⠀⠀⠀⢯⣧⣩⡽⠾⠅⣿⢃⡥⢊⠔⡡⢊⢜⠕⣁⣊⠔⠋⣀⡶⠟⣒⡂⣀⡉⠉⠠⠤
⠀⠀⠀⠀⠀⠀⠀⢳⣌⠒⢄⡀⠀⠀⠀⠀⠈⠑⠛⠷⠲⠏⢠⡔⡡⢊⠔⣡⣢⢾⢋⢠⡳⠿⠉⠀⠀⠒⠒⣒⡒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣀⡌⠑⠂⠤⢄⣀⠀⠀⠀⠀⠀⢸⢚⣵⣣⢞⠍⢅⣤⡾⠉⠉⠉⠭⠭⠭⠭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢀⣀⠀⠉⢉⠓⠒⠲⠞⠻⣩⣖⠯⠾⠿⠽⠷⠬⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠤⠥⠿⠿⠭⠭⠭⠭⠭⠭⠭⠍⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠩⢀⡤⠶⣾⣷⠶⣄⠀⠀⠀⠀⠀⢀⠀⠌⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡾⠁⠀⣰⠋⠀⠀⠈⠻⣷⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠉⣼⠃⠀⠀⠀⠀⠀⠘⣿⡌⢧⠀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⢸⡄⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⢰⣦⡀⠀⠀⣿⢸⡇⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠈⣧⠀⠀⠀⢸⣿⡝⢦⡀⠏⢸⠇⡁⠀⡁⠀⠀⠐⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⠀⣸⠘⣷⡀⢹⡇⢹⠀⠈⡊⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠹⣎⣡⠙⢦⡀⡟⣶⣿⣧⣼⠀⠸⡆⠀⠀⢆⡀⠀⠀⠀⠀⠈⠐⠢⢀
⠀⠀⠀⠐⡆⠀⡀⠀⠀⠀⠀⠀⠈⣻⣷⣤⣉⣓⣟⣉⣵⡇⠀⠀⢧⠀⡓⠒⠒⡄⠀⣀⡀⠀⠀⠀⠀⠙⢀
⠀⠀⣰⣇⠈⡏⠀⠀⠀⡐⠈⠀⣤⣇⠼⣟⣿⣯⣿⡋⡜⠀⠀⠀⠘⣆⠀⠀⠀⡇⠉⣄⠙⠋⠐⠒⠲⠄⠤
⠐⢠⠇⠸⡄⣇⠥⡀⠇⣘⠀⠐⠿⠿⣽⣧⣟⡗⣏⠟⠁⠀⠀⠀⠀⠘⣆⠀⠀⡇⠀⠀⠀⠀⠀⣌⡉
⣠⡏⠀⠀⢹⡇⠃⠐⠀⠄⠾⠂⢼⡆⣼⣙⡮⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡁⠀⠀⠉⠈⠒⠂
⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠃⠈⠋⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢂⡡
⠈⠓⠤⠔⠊⠀⠀⠀⠀⠀⠐⠆⠠⡶⠖⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⢶⣤⡼⢃⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣀⣀⣀⣡⣤⣔⣀⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢧⣼⣿⣿⣿⣿⣧⣼⡿⠿⠋

⠀⠈⠁⣤⣤⣯⠉⠲⣤⡈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠂⠖⠁⠴⠀⠀⠀⠠⠴⠇⠘⢁⡀⠙⡆⠀⠀⠘⢃⠀⠀⠀⠀⡀
⢠⠖⠉⠀⠀⠀⠑⠀⠀⠘⠛⠶⢶⡤⠄⠀⠀⠁⠈⠀⠰⠀⡀⢤⠀⠀⠀⠀⠐⠋⠀⢠⠶⡷⠂⠀⠀⠀⠀⠀⠈⠒⠳⣴⡞⠋⠒⡄⠄⠉⠁
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠃⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⡴⠀⠀⠀⠄⠁⠂
⡀⠀⠀⠀⠁⠀⠀⠀⠐⠤⠀⠀⢠⠀⠀⠀⠀⠀⠱⣄⠠⠐⠢⠌⠀⠀⡄⠀⠄⠃⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠰⠄⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⢀⡀
⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠰⠄⠀⢀
⠀⠈⠓⠄⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⡖⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢨⠀⠈⠀⠒⠀⠀⠀⠃⠀⠀⠉⠀⠈⠀⢠⡴⣀⠀⠀⠀⠀⠘⠀⠠
⠀⣠⠶⠒⠂⠮⡡⠖⠒⠒⠒⠋⠉⠀⡀⠈⣤⠀⣤⢠⡀⠈⠠⠄⠐⠂⠀⢠⡄⠉⠀⡅⠀⠀⠀⢠⠇⠀⢀⠀⠆⠂⠀⠀⠀⠀⠀⠈⠀⠂⠀⣀⡀
⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢋⠀⠀⢠⠀⠀⠆⠐⠳⡄⠀⠁⡀⠀⠗⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⠀⠀⠀⣠⡇⠀⢀⠀⠀⠀⠉⠑⠲⢦
⠀⠀⠀⠀⠀⠐⠀⣤⠜⠀⠄⠀⠀⠈⠁⡄⠃⠬⢀⡎⠈⠓⠒⣀⠀⡰⠃⠘⠉⢁⠠⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠐⠐⠺⣁⠀⠔⠉⠀⠂⢀⠀⠀⢀
⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠄⠀⠀⢒⠄⠀⠙⣀⠀⠀⢩⠈⢹⠄⠀⠀⠀⠀⠀⠀⠀⣀⣑⠖⠶⠀⠀⠀⠐⠀⡀⠀⡈⠒⠂⢄⠀⠀⠈⠁
⠁⢶⣀⣀⣀⠴⣤⡀⢀⠀⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠉⠂⠰⠄⠠⠄⠀⠄⣀⠀⠀⠘⠀⡃⠈⠋⠀⢀⡐⠤⣤⠀⠀⠀⡤⠁⠀⠀⠀⠈⠙⠂⢄⠀⣀⡀
⠀⠀⢠⠟⠀⠀⢀⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠦⡀⠀⠀⡓⢘⡧⠀⠚⠓⠈⡃⠀⢘⠀⠀⠀⠀⠆⢤⣀⡄⠀⠀⢀⡀⠁⠀⢱
⠀⠀⢸⡇⠀⠈⠃⣤⡋⠙⠀⠀⡀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⢁⠰⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠐⠋⠁⣻⠀⠀⠀⠟⠃⠀⡟⢠⠀⠀⠀⠀⠀⠀⢀⡏⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠈⠀⠀⠈⠀⢰⡇⠀⠀⠀⠀⠀⠤⠐⠀⠀⢀⠐
⠀⠀⠀⠹⣆⠀⠀⠟⠈⠀⠻⡆⠀⠀⠀⠀⠀⡀⠈⢃⠧⣄⠒⣶⣦⡄⠀⠀⠀⠀⢀⣄⣶⡖⢠⡤⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠂⠀⡀⠀⠰⣤⣤⠀⠈
⠀⠀⠀⠀⢨⠷⣤⡄⠀⢀⣠⡧⣶⣄⣄⢰⡟⠀⣀⢾⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⣀⠀⠀⠀⠀⠁⠀⢀⡿⠇⠐⠂⠐
⠤⣀⠀⢀⡌⠀⠀⠙⠛⡏⠁⠀⠀⠈⢻⣿⠶⠖⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⣧⢶⡀⢜⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀
⠀⣠⢿⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⢹⡾⠁⠘⣁⠀⠀⠀⠀⠀⠠⠶⠀⠀⠈⠱
⡞⠁⠈⠢⡀⠀⠀⠀⠀⠀⣨⣷⠄⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⢰⠾⠀⠀⠀⣹⠶⠶⠀⠀⠀⠀⠀⡀⢀⡄⣠
⠇⠀⠀⢀⠘⠋⠒⠆⠀⠉⠀⠉⣶⠉⠀⡄⠀⠀⠹⣄⠀⣆⠀⠀⠀⠈⠀⠒⠛⠂⠀⠀⠀⠀⠀⢀⣴⣿⠃⢸⡆⢠⡤⠚⠀⠀⠀⠙⣤⢦⣠⣴⠋⠉
⠀⠀⢀⠀⠀⣿⣷⠀⢠⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠘⠃⣘⡷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⢵⡏⠀⢺⠇⠀⣄⠀⠀⠀⠀⠀⠀⠀⠁⠈⢢
⡄⠀⠈⠄⣀⠄⠀⡰⠂⠀⠀⠀⣸⠄⠀⠀⠈⠀⠀⠀⢠⠿⣿⡸⠙⠂⠀⠀⠀⠀⠀⠄⠒⠁⠀⠀⠋⠀⠀⡈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠳⣀⠀⠀⠀⠀⢛⠀⠀⠀⠀⣠⠏⠀⠀⠀⢠⠀⠀⠀⠀⢀⣼⡧⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠈⠳⣦⣀⡀⣸⠀⣠⡠⠞⠋⢤⡀⠀⠀⠀⠄⠀⠈⠀⠀⣾⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢠⣎
⡀⠶⠒⠓⠦⢈⡁⠉⠀⠀⠀⠀⠀⠹⠖⠀⠀⣠⠀⢀⠐⠂⠀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⣀⣠⠞⠁⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠀⠀⠀⠀⠀⠐⠀⠀⡌⠳⠄⠀⡄⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣤⠆⣠⠖⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣤⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠏
⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⠌⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢠⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⣠⠴⡶⠶⢤⢠⡜⠀⠀⢠⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⠀⠉⠀⠀⠄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣶⣀
⠀⠀⠀⠀⠀⠘⠗⢆⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠑⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠉⢸
⠀⠀⠀⠄⠀⠀⠀⠈⣥⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣋⠀⠀⠀⠀⠆⠀⠀⠀⢠⡆⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠁⠠⠀⠀⠾⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠋⣤⠀⠀⠀⠈⠀⠀⠀⠈⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⣶⣿⣶⡄⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠀⢰⡀⠀⠠⡄⠄⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠤⠀⠀⠀⠀⠀⠀⠀⠸⣦
⠀⠉⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡠⠁⠙⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⡀⠄⠀⠀⠸⠀⠀⠀⠀⠀⡀⠈⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠘⠀⠠⠀⠀⠀⠀⡀⢀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⢶⡤⢦⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠃⠀⠀⠀⠀⠀⡀
⢠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠘⣃⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⡟⣿⢻⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣴⠃⢀⡴⢖⣶
⠀⠀⠀⠀⠀⠀⣿⡀⢸⡅⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤
⠀⠀⠀⠀⢀⣠⣬⣿⣦⣝⣶⣦⣴⣶⣾⣿⣿⣶⣾⣿⣿⣿⣯⣤⣤⣤⣀⣀
⢀⠀⢀⡴⢋⣴⣶⣿⣿⣿⠿⠛⠉⠁⢠⡿⠋⠀⠀⠀⠈⠻⣧⠀⠈⠉⠉⠛⠛⠷⢶⣦⣤⣀⡀
⠙⠶⠋⣰⡿⣻⣿⣿⠟⠁⠀⠀⠀⠀⣿⡇⣾⡆⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄⡀
⣀⣀⣴⠟⣱⢿⣿⠋⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀
⠀⠀⣠⠼⠁⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣶⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡄
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣤⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⢰⣿⠏⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠰⣶⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠃
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠉⠉⠉⢻⡿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠻⣿⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠠⠤⠶⠾⠛⠋⠻⣦⣀⠀⠀⠀⣠⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠉⠉⠉⠛⢿⣦⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠋⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⡄⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⢈⣿⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠉⠻⣷
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡶⠾⠿⠛⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⡠⢫⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⡐⡱⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⢴⣦⣈⠏⡴⣤⣤⣤⣤⣤⡶⠶⠞⠛⠉⠁
⠀⠀⠀⢰⣿⠁⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠹⢿⣦⡀⠀⠀⠀⣠⣿⠋⠊
⠀⠀⠀⣸⡟⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠿⣷⣶⣿⡿⣷⣷⣤⡄
⠀⠀⠀⣿⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣟⣾⣷⣯⣭⡶
⠀⠀⢠⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢠⢊⠟⠿⣾⠏⠉
⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠁
⠀⠀⢸⣿⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⠿
⠀⣀⣨⡍⠉⠉⣿⠋⠉⠉⠉⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠁
⠀⠻⣿⣯⣤⣤⡿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠉⢉⣡⣤⣤⡤⠤⠤⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣧⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢰⡏⠁⠀⡄⠀⢀⣀⡄⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⣤⣀⠀⠀⢰⠀⠈⣻⡆
⣀⣀⣀⣈⣻⣶⣶⣷⣚⣋⣉⣷⣶⣛⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣏⣙⣛⣷⣾⣶⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠦⣀⠉⠉⠙⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⡤⢚⡛⠒⠠⣄⠀⠀⠀⠀⠀⢰⠁⠀⠀⠑⢦⠹⡄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣏⣰⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠘⠛⠁⠀⣀⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢦⣄⠀⠈⠁⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⡀⠍⠳⢦⣤⣀⣀⣤⠞⠉⢀⠀⠈⠐⠢⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢤⡤⠖⠋⠀⠐⠀⢌⠓⠶⠶⡶⠖⠋⣁⠀⠀⠀⠀⠀⠐⠄⠤⠤⠂⡄⠈⠀⡀⠀⠀⠀⠀⠀⠂⠀⠘⡀⠉⠉⠒⠲⠤⣀
⠀⠀⣀⠤⠒⣋⣉⠁⠀⠠⠊⠀⢀⡀⠀⠀⠠⡀⣀⣀⡠⠍⠉⠉⠠⠀⠀⠀⠀⡀⠤⠁⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣴⠒⠛⠂⠀⠙⠲⣄
⣠⠞⠁⠀⠀⠀⢉⡹⠶⠧⠤⠤⠤⠤⢀⠀⠀⠠⠬⠥⠔⠁⠀⠀⠐⠠⠀⠀⠈⠃⠐⠊⠀⠀⣀⡀⠀⠀⠀⠀⠐⠰⠋⠉⠤⠤⣄⠀⠀⠈⠑⠀⠀⠀⠀⠀⠸
⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⠒⠂⠀⠐⠰⠄⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⡆⢠⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⢠
⢳⣰⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⣴⠤⢴⡶⢾⢟⠟⢹⡸⢤⣤⠤⠾⠁⠀⠀⠀⠀⣀⣀⣤⠞⠋
⠀⠉⠓⠶⢤⣤⣀⡀⠀⠀⠈⠓⠒⠒⣋⠜⠛⠻⣿⠛⠿⡛⠛⣷⡖⡞⠋⠛⣿⣯⠉⢉⠇⠹⡷⠔⠉⠀⠘⢦⡠⣀⢉⠖⠒⠶⠶⠶⠶⠒⡟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠋⠉⠉⠉⠣⠤⢀⡴⠃⠀⠀⠈⢛⠟⠀⠳⣄⣴⠃⠸⡉⠁⠀⠀⢸⢄⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⣤⠚⠋⠀⠀⠀⠀⡠⠻⠀⠀⠀⣀⠎⠀⠀⠈⢲⡀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠸⡟⠢⣤⣀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠶⠚⠉⣏⠀⠀⠀⠀⢀⡟⠁⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠉⠂⠤⠤⠤⠿⣄⡀⠀⢀⠜⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣤⡴⠋⠓⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣿⣿⡟⠛⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⢰⣶⣶⠀⠀⡀⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡒⠀⣒⠐⡂⢐⡂⢐⠂⢸⣿⣿⠀⠀⠁⠀⢀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⡀⢈
⠀⠀⠀⠀⠀⠀⠤⠀⠤⠠⠄⠠⠄⠠⠄⢸⣿⣿⠀⠀⠀⠀⡀⠀⡀⠀⠀⣀⡀⠀⢀⣀⣀⣀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠠⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠘⠉⠙⠀⠀⠀⠀⠿⠿⠇⠀⢸⡉⢹⠆⠈⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⢄⠀⠀⢀⠀⠀⠀⡀⠀⡔⠈⠁⠀⠈⠁⠈⠑⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠈⠈⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠰⠄⠀⠠⠤⠀⠀⠦⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄⠤⠠⠄⠤⠰⠀⠁⠄⠠⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣠⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⢠⡞⢀⣴⠀⠠⣶⡇⣠⢂⡴⡾⢄⡔⢋⡉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⠀⢀⡛⠘⢞⣶⡦⢴⣿⣾⣾⣷⢿⡞⣹⣵⣋⣠⣿⠢⣤⢤⢤⣬⡁⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢹⡿⣤⠻⠟⣾⢾⡟⢳⢦⡟⡥⠋⢼⣿⣿⣿⣽⣿⣿⣭⣦⡤⠴⡊⣈⣀⣤⣶⣧⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣤⡀⡆⢿⣿⣧⢞⣿⣟⠹⣱⢿⣿⡟⠋⠁⠈⠉⠉⠀⠀⠀⠀⠈⠙⠛⢛⣿⣿⣿⣿⡟⠁⠀⢈⠁⠂
⠀⠀⠀⠀⢀⠀⠈⢻⣍⡰⢬⣁⣛⣮⣟⣻⣁⣿⣿⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡀⠉⠹⡟⢐⠀⡀⢰⠒⠀⠈
⠀⠀⠀⠀⠀⡀⠒⠝⡉⠳⣸⣿⣻⣿⢹⣿⣯⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⣸⠟⠻⢿⡉⠹⢧⡀⡇⢧⡌⢡⡄
⠀⠀⠀⠀⢀⣗⣋⣴⠃⠈⣽⣿⣿⣿⡿⢟⣻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡁⣸⣿⡷⢶⡀⠀⠙⡀⠈⢹⣷⠘⣆⠘⡀⠂
⠀⠀⠀⠀⢀⡯⠡⠈⣉⠒⣾⠻⣽⡏⣿⣿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣺⣅⣬⣿⣿⣿⣅⡠⠙⢆⢀⡀⠀⠀⢸⡆⢩⣏⠱
⠀⠀⠀⠀⢸⣧⡃⠈⠀⠘⡖⣜⣯⢛⢛⣹⠄⠀⠀⠀⠀⠀⠀⠄⢉⣸⣿⣿⣿⣿⣿⣿⣿⣀⡤⠘⡏⠀⠀⠀⠈⣿⢀⡯⠑
⠀⠀⠀⠀⠈⢽⣇⡀⠀⠀⢿⡿⢿⡿⢿⣿⣇⡀⢀⣀⡤⣂⣵⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣟⠢⠄⠻⠀⠈⠀⢀⡏⣠⡗⠌⠐⠁⠄
⠀⠀⠀⠀⢠⣘⡟⠱⢘⡃⠙⡰⠀⠑⣚⠛⡿⠟⣩⣷⣶⣯⣀⡀⠉⠑⢾⣿⣿⣿⣿⣿⣿⣿⣗⠢⠄⠀⠀⠀⣼⠃⣸⡍⠔⠃⠈⣨⡀
⠀⠀⠀⡀⠈⡝⠓⠣⡈⠛⢦⣄⣉⣉⠃⢠⣶⣿⣿⣅⠀⠀⠀⠀⠀⡀⠄⠿⢻⣿⣿⣿⣿⣿⣿⡤⠀⠀⢀⣼⠟⢰⣿⠁⠌⠀⠁⠰⠂
⠀⠀⠀⠀⢘⠣⣆⡐⡄⠠⠤⠀⠀⣙⡶⠋⠹⣿⡃⠈⠉⠀⡰⠒⠋⡅⠀⠀⠁⡘⣹⣿⣿⣿⡏⠳⠠⣠⡿⠃⠘⣼⠇⣴⠆⠉⢠⣖
⠀⠀⠀⠀⠠⠊⣻⡿⣥⠶⠀⣠⣾⣯⣷⣖⠀⠺⢧⡄⡠⠈⠀⢀⠠⠐⠁⠀⠴⠋⢘⣽⣿⣿⣇⢀⡷⠋⢰⣦⡞⠃⠠⡈⢾⠡⠆⠠⠟⠄
⠀⠀⠀⠀⠀⠀⡿⠛⣁⣴⣻⣹⣿⣿⣻⣿⣿⣦⣀⠈⠃⠤⠖⠉⠀⠀⠀⠂⠀⠀⡼⢈⣯⣿⠾⠋⣶⡾⠏⣈⠿⠙⠰⠂⠤⠘⠁
⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⡿⢿⣟⡿⢿⣯⣿⣿⣶⣀⠐⠀⠀⠐⠳⣦⣴⢤⡤⠶⠖⠛⠉⢀⣿⠾⠉⡯⠂⢉⡀⠊⡘⠃⠰⢈⠀⡅⠲
⠀⠀⠀⠀⠀⣰⣿⠿⡿⢛⠉⢛⠉⠠⠐⠛⢀⡠⠟⢧⠓⢡⣄⣤⠀⡀⣴⣈⣰⣆⠤⣿⠛⡋⡿⠀⠃⠀⠀⠈⠀⠀⠉⠀⠐⠆⠀⠐⠃
⠀⠀⠀⠀⠰⠋⠑⠂⠐⠈⠀⠀⠂⠐⠂⠀⠉⠁⠰⢂⡴⠋⠨⠏⠡⡽⠣⠍⠘⠃⠐⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠂⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣹⣯⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣽⣯⡙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣯⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠦⣤⣀⡀⢸⣿⣿⣿⣿⡏⠀⣀⣤⠴⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡆⢼⣿⡆⠀⠀⢰⡝⢿⣿⣿⡿⢋⡄⠀⠀⢠⣿⣷⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡿⢿⣿⣿⣦⢸⣿⣿⣿⣿⣷⣿⡇⣰⣿⣿⣿⢿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⣼⣇⠈⡟⢸⡿⢛⣿⣿⣏⠻⡇⢹⡇⣠⣷⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣤⣶⣿⣿⣿⣿⣷⣤⣤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠛⠛⠛⠛⠛⠿⣶⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⣰⣿⠏⠀⠀⠀⣴⣾⣿⣦⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⡏⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠸⠿⠿⠛⢀⠶⠄⠘⢿⣉⠀⠀⠀⠀⠀⢠⠃⠀⣿⡟⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀⣼⡿⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠈⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠀⠐⡡⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡟⠋⠀⢀⣤⠟⠀⠹⣿⣿⣿⣶⣶⣶⠆⠀⣰⠀⢀⡷⠀⢸⣶⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣶⠿⠋⠉⣿
⠀⠀⠀⠀⢀⣠⣿⡋⠉⠀⢠⣾⣿⣄⣀⣠⣀⡙⠻⠿⠿⣿⣿⣦⣤⣿⣤⣼⣿⣤⣾⣿⡋⠀⢀⣀⣠⡾⠿⣦⡀⠀⠀⠀⠀⠀⣴⡿⣿⠀⠀⠀⠀⠀⣰⡟
⠀⢀⣴⠾⠛⠉⠉⠉⠻⣆⣸⡟⠀⠉⠉⠙⠛⢿⣦⡀⢀⣼⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⡿⠿⠿⣧⠀⠘⢷⣦⣤⣤⣤⣰⣿⠀⠘⠷⠶⣶⡶⠟⠋
⠀⣾⡇⠀⢀⣤⣤⡀⣀⣿⠟⠁⠀⠀⠀⢠⡄⠀⠙⢷⣿⣟⠛⠛⠛⠻⣿⡟⠹⣿⣛⣉⣀⣀⣠⣤⣶⣿⣶⣦⡄⠉⠁⢨⡿⠟⠃⠀⣠⣶⡾⠟⠀⠴⠂⠄
⠀⠈⠛⠛⠋⠉⠈⠙⡛⠁⡀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⠿⠿⠷⠶⣾⣿⠃⠀⣿⡛⠉⠉⠉⢉⣉⣉⣿⠇⠹⣿⣀⣴⣿⣦⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⢠⠀⠻⢷⣦⣤⣄⣠⣤⣴⣿⠀⢸⣿⣟⡛⠛⠻⠿⣿⣿⡁⢀⠆⠙⠛⠋⠀⠹⣷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠉⠉⣉⣽⣿⡄⢸⣿⡛⠿⣿⣷⣶⣤⣽⣿⢠⡇⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠟⢋⣩⣿⣷⣿⣿⣿⣦⣄⡙⢿⣯⡉⠀⠈⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣯⣴⡾⠿⣿⣿⣁⣠⣿⣿⡿⠿⢿⣷⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠿⢿⣶⣤⣶⠟⠙⠷⣦⡤⡈⠙⣿⠟⢁⣤⣶⠟⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡷⣦⡀⠀⠀⠈⢀⣀⡀⣠⡿⠀⠀⣠⣿⠀⠈⢻⣇⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠁⢸⣿⣤⣤⣾⠿⠛⠛⢿⣤⣴⠾⠋⠙⠷⣦⣤⣿⣦⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠈⠻⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠈⢻⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠈⠙⢻⣦⣤⣄⡀⠀⠀⠀⠘⠟⠀⠀⠀⠀⣿⡷⠾⢿⣷⣾⡏⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠛⣿⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⢈⣩⣿⠆⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⠃⢰⣿⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣾⠟⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⢷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⣤⣄⡀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⢳⡄⠈⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣧⣀⠀⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⣦⣀⣠⠴⠶⠶⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡀⢀⡤⠖⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣠⠟⠹⣶⠋⠀⠀⠀⠻⣄⠀⢀⠀⠀⢠⡴⢰⣾⣇⡼⢧
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢰⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡧⢠⣌⠳⠖⠋⠁⠀⠀⣦
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⠏⢀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⠟⠀⠀⠀⢹⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣧⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠸⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠘⠛⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢷⣿⡄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠙⠿⠁⠀⠀⢀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⢻⣆⠀⠀⠀⠀⠰⣤⡀⢀⣠⡄⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⠀⠀⢳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⢷⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⠀⢠⣀⠀⢉⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣙⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣾⣇⠀⠀⢿⣷⣤⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠉⠉⠛⢳⡶⠤⠶⠚⠛⠉⠀⠀⠘⣏⠀⠴⠤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣤⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⠤⠚⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⠃⠀⠀⣿⡉⣹⠏⠉⠉⠉⠉⢁⣀⠀⣼⠿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠏⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⢰⡏⢉⣿⣿⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣼⢋⡴⠲⡄⠀⠀⣠⠖⠲⡄⠀⠀⠀⠉⠊⢹⡟⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠃⠈⠳⠞⠃⠀⠀⠛⣤⠾⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⣄⠀⠘⢷⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣼⠋⢙⡆⠀⠀⠀⠀⠀⠀⠘⢧⡴⠇⠀⠀⠀⢳⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⡀⠀⠀⠙⠓⠋⠀⠀⠀⢠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠳⣆⡸⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢀⡷⠀⠀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⣤⠶⣄⠀⠉⠉⠀⠀⠀⠺⣅⣿
⠀⠀⠀⠀⠀⠀⢸⡿⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⠿⠀⠀⠀⠀⠀⠀⢀⣀⢼⡇
⠀⠀⠀⠀⠀⢠⡟⠀⠙⣿⡽⣟⠓⣂⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣶⠾⣿⠛⠊⢳
⠀⠀⠀⠀⢠⡎⠀⠀⢰⡟⠀⡏⠀⠀⠀⠀⠈⠘⡿⢻⡄⠈⠉⠉⠉⠛⠛⢻⣿⠹⠉⢻⠀⢹⠀⠀⠘⡆
⠀⠀⠀⠀⠘⢧⣄⡀⠸⡇⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⢃⢸⡇⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠹⠀⠇⠀⠀⠀⠀⠀⠀⠘⠁⠆⠀⠀⠀⣀⣠⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⠛⠿⠶⠶⠶⠶⠾⠿⣶⣶⣶⣶⣶⣶⣶⠶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⡀⣠⡀⢻⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠙⠁⠀⠈⣻⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣺⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⣀⣀⡜⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢀⣨⠟⠛⠷⣤⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣅⣀⣀⢀⣀⣈⣉⣉⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⣿⠿⠿⡿⠿⠿⢏⠹⠿⢿⡿⠿⠿⠿⣿⣿⣿⣿⡟⣻⣛⠛⠛⠛⣛⡛⣛⢻⣿⣿⠛⠛⠛⠿⠿⠟⢻⠛⠛⠻⣿
⣿⣷⣿⣶⣶⣷⣾⣶⣾⣶⣾⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⢠⠔⠒⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠊⠉⠑⢦⢀⡀
⠀⠀⠀⡀⠔⠉⠀⠁⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠁⠀⠂⠤⠤⢄⠀⠀⠀⠀⠀⠀⠘⠂⠁⠢⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠢⠤⠤⡀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠤⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠑⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠁⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⡆⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⣀⣀⣀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⡖⠒⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣒⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣀⣠
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢀⣀⠀⣀⡀⢀⣀⠀⣀⣀⠀⣀⡀⢠⣶⣶⣶⠀⠀⠀⠀⠀⣃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡆
⠀⠀⠀⢈⣉⠀⡉⡁⢈⣉⡁⣉⣉⢀⣉⡁⢸⣿⣿⣿⠀⡇⠀⡇⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠃
⠀⠀⠀⢨⢨⠀⣍⡅⢠⣩⡄⣨⣥⠀⡍⡄⢸⣷⣶⣾⢀⠀⠀⡇⠀⠀⠀⣒⣒⡆⠀⣖⣒⡂⠀⢰⣒⣲⠀⢰⣒⣒⠀⠀⣒⣒⡆⠀⡇⠀⠀⠀⠂
⠀⠀⠀⠤⠠⠀⠄⠄⠠⠠⠄⠤⠤⠠⠄⠄⢸⣗⣒⣺⠸⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡀⢀⡇⠀⠀⠀⢰
⠀⠀⠀⠐⠰⠀⠖⠆⠰⠶⠂⠲⠶⠐⠂⠆⠸⠗⠒⠺⣄⠀⠀⡇⠀⠀⣗⣒⣚⡇⠀⠀⣼⠩⠩⢳⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸
⡔⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣑⣄⣀⣀⣀⣄⣀⣀⣇⣀⠀⠁⠀⠀⠁⠀⠀⠹⡆⠀⠟⠀⠀⡇⠐⡁⢸⢐⡉⢹⢸⡉⢱
⠑⣤⠤⠤⠤⠤⣤⠤⠤⠤⢤⣤⠤⢼⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⣀⣀⣀⣀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠁⠁⠀⠁⠁⠀⠁⠁
⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⢈⠈⡇⣈⢈⡇⣈⢈⡇⢸⠀⠀⠀⢀⡀⠀⠀⠀⡄⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠋⠀⠀⢀⣇⣠⣀⠀⠀⡇
⣴⣿⣷⣶⣶⣶⣿⣶⣶⣶⣾⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣶⣶⣶⣶⣶⣷⣶⣶⣾⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢑⣰⠇
⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠂⡀⢀⠀⠐
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠙⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠠⡀⠈⢧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⡞⠑⠠⣀
⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⣰⡁⠀⡉⡆⠀⢘⠀⠁⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⡀⣀⠀⣢⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⢶⡋⠉⠳⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠲⠶⠃⠀⠀⡀⢀⢄⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠀⠀⠸⡉⠊⠀⠀⠀⠀⠀⠎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⢒⣒⣦⠀⠀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠭⠍⢶⠀⠀⠀⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⣧⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⢀⣼⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⣧⣄⡀⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⣀⣸⡀⣄⣴⡇⠀⠀⠀⣧⢠⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⠋⠁⠀⠙⠶⠶⠚⠉⠉⠙⠓⡧⠥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠷⡿⣄⣀⣀⡼⠀⠀⠀⠀⠀⠀⢀⠀⠄⢒⡥⠶⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⠓⠲⠬⣅⣒⠀⠄⢤⠰⣂⣩⠴⠒⠋⡡⡀⢐⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨⢉⢁⠒⠶⠤⢄⣉⠙⢺⠊⢁⡀⢰⢒⠀⡇⡇⠨⠘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣉⠑⠒⠰⠬⠇⢸⠀⠡⠄⠸⠈⠀⠁⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⠿⠛⠉⠁⠙⠲⢄⡀⠀⢰⠀⠀⠀⠀⢀⠀⠄⠂⢀⢄⢸
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠻⠛⠋⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠈⢒⣼⡀⠠⠐⠀⣀⠀⡆⠂⠐⠀⢸
⠀⠀⠀⠀⠀⡆⠉⠉⢡⡄⣀⡐⠢⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠊⠁⢸⠀⡤⢠⢸⠰⠀⠅⠀⠀⠀⣸
⠀⠀⠀⠀⠀⡇⠀⠀⠸⢵⣒⣸⡇⠀⠑⠦⣀⡠⠴⠒⠉⠁⠀⠀⠀⢠⡦⢸⠀⠋⠙⠀⠀⠀⢀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⡇⢠⠤⢀⡀⠀⠈⠀⠀⠀⠀⡍⠀⠀⠀⢠⡄⠀⠯⠀⠈⠀⢸⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⠀⡇⠀⠂⠠⠄⡉⠐⠀⠠⣄⠀⡗⠰⡬⠀⠈⠁⠀⠀⠀⠀⠀⢸⣠⠴⠚⠁
⠀⠀⠀⠀⠀⢧⣈⠀⠀⠠⢀⠁⠀⠀⢼⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢄⡈⠀⠀⢼⠀⡇⠀⠀⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢬⡀⣃⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣦⣷⣶⣤⣶⣴⣯⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡯⣀⠀⠀⠀⠀⠀⠀⡼⠷⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⢷⣦⡀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⣿⣿⣿⣾⣿⣇⠘⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠴⣻⢿⢿⠿⡟⡏⠁⠉⠛⠛⠛⠛⠻⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⣯⡟⡼⠈⡀⠐⠂⠀⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠠⠤⠐⠈⠀⠀⠀⠀⠀⠀⢘⡟⠠⠁⢸⣧⣀⡰⠼⠟⠓⠓⠓⠛⠃
⠀⠀⠀⣠⣦⣾⣷⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠀⠀⢀⡉⠉⠀⠀⣸⠀⠀⠀⠀⣠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⢀⣤⣿⣷⣤⣥⠶⠛⠛⠛⠛⠛⢣
⠀⠀⠸⠿⠁⠋⠛⡉⣿⢹⠀⢸⠛⠁⠀⠀⢠⣴⣿⠏⠀⠀⠀⢼⣿⢹⡥⡍⠀⠀⣹⣀⣀⣀⣀⣄⡇
⠀⠀⣤⠀⣦⣄⣦⣷⣿⢸⠀⠈⣿⡿⣷⣚⣻⣿⡏⠀⠀⠀⠀⠘⢿⣾⣇⣿⠷⠾⠟⠛⠛⠛⠛⠛⢇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢸⠀⠈⣟⠀⠀⢸⠇⠀⠀⠀⠀⠀⣤⣤⢸⡇⢹⠉⣿⡆⠀⠀⠀⠀⢀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡆⣇⡀⠘⠀⠀⢸⣳⣄⠀⠀⠀⠀⠘⢿⣾⣧⣼⣶⣿⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⣀⣿⣿⣷⣦⡄⠀⠀⠀⠹⠏⠻⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠙⠉⠉⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⣋⣉⣉⣩⣥⣤⣤⡤⠴⠶⠶⡠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⢀⣤⠤⠴⠶⠶⠖⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠘⣿⠀⠀⠀⠀⠄⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⡆⠀⠀⠀⠈⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠤⠶⠶⠖⢚⠛⠻⠹⢩⠴⠚⠓⠒⠮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠷⠟⡛⠛⠉⠉⠁⠀⠀⠐⠀⠈⢁⣀⣀⣠⣤⣤⣎⡀⠀⢌⣠⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠌⠒⠈⢁⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⣿⡿⠿⠿⠟⠛⡗⠹⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡟⢻⣿⣿⣿⢿⡟⣛⣉⣉⣱⠆⢚⣋⣀⣠⣤⣤⢶⣶⠟⣧⢀⡾⣿⣿⣿⣟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⡟⠁⣀⣑⣫⣭⡾⣿⣿⣿⣿⣿⣿⣿⡕⢫⣆⣷⡶⢛⣳⣿⣿⣿⣿⣿⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⡿⠿⠟⠛⣷⠖⢿⣹⢠⣸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⢿⣷⠛⠉⠁⠀⠀⠀⠀⠀⠻⢠⢾⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⢹⡀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠁⠀⠀⠈⢿⣦⣀⡀⣀⢀⣀⡴⠒⢧⠀⠙⣠⣿⣿⡿⣿⣿⣿⣿⣿⣿⡏⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⢹⣇⢤⣮⡳⢭⣽⡿⢻⢤⣹⣿⣿⣿⣿⣿⣷⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⠙⢧⣽⣿⣿⣿⣷⣈⠊⣿⣿⣿⣿⣿⣿⣿⢃⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⡌⢻⣿⣿⣿⣿⣦⣝⣛⣿⣿⣿⣿⣷⣿⣿⣿⣶⣶⣤⣀⣠⣴⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⡿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⣾⠏⣀⣤⠞⠋⠉⠉⠉⠉⠉⠛⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠞⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢿⣄⣀⣀⠀⢻⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣿⣀⣿⣤⡟⠿⠁⢼⡟⠛⠛⣧⣀⣿⣆⠀⠀⠀⠀⢵
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣃⣼⠻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠟⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⡂⢠⠟⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠄⠀⢸⠃
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⡄⢀⡎
⠀⠀⠀⠀⠀⢹⣸⣇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠆⣸⠃
⠀⠀⠀⠀⠀⠀⣿⢸⣇⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠋
⠀⠀⠀⠀⠀⠀⢹⣆⠹⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣠⡾⠃⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠼⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣦⣤⣄⣀⡀⠀⠀⢀⣀⣠⣤⠞⠛⣿⡀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠙⢿⣄⠈⠉⠙⠛⠚⠉⠉⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠛⠿⠶⢶⣤⣀⣀⣀⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣠⡛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⠶⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠶⣦⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⢾⡷⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣧⣄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⡀⢠⣴⠀⢶⣾⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣶⣶⣦⣾⡿⠿⠛⠉⠁⣿⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⢻⡃⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠄⢦⣤⠤⠴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠶⠶⠶⢶⣤⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣠⣤⠚⠛⠛⠛⠛⢳⣄⣼⠋⠁⠀⠀⠀⠀⠈⠙⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢾⣧⣄⠀⠀⠀⠀⠀⠀⣨⣿⣤⣤⣀⣤⣤⣤⡤⠶⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢿⣿⣶⣶⣾⡛⠾⠉⠈⠙⠛⠋⠶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢻⢠⠀⢀⠀⡆⠀⢰⡄⢀⠀⠀⠀⠀⡔⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢶⡿⢁⣤⣈⢃⣠⡀⠁⠀⠀⠻⠃⠀⠀⠀⢸⠇⠀⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢿⡶⢿⣿⣧⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣏⢡⣏⡀⠉⠉⡇⠀⠀⠈⡟⠐⠀⠀⣶⡄⠀⠐⠂⠐⠎⠢⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏⠁⠀⠉⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡸⠇⢪⠅⢤⣍⣥⠀⠀⠚⠙⣶⣄⠀⢿⡄⠐⠐⠢⠀⢠⣦⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⠰⠀⠀⢠⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣄⣠⣴⢲⣟⣽⣥⣤⣤⢤⡤⣬⡇⢀⣠⡄⠀⠉⠈⢦⠈⠁⣿⣄⣙⡀⠚⣧⣴⣶⡀⠀⠀⠀⠘⣿⣿⣗⣶⡶⢚⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣼⠿⢭⣰⣷⣭⡽⠿⡷⠻⢠⡴⠾⡇⠸⠁⠀⠀⠀⠀⠈⢧⠘⢮⣛⣹⠄⣾⡇⠸⣟⠇⠀⠀⠀⣀⣻⣿⣿⣿⣧⣀⣽⡿⠿⠗⠒⠷
⠀⠀⠀⠀⣹⠿⠟⢳⡿⣿⢷⣗⣡⠅⢀⣾⣧⡛⡿⢀⠀⠈⠀⠀⣠⠀⠈⢧⢼⣿⡗⣴⡅⠁⠀⠹⡠⠀⢀⣼⣿⣿⡿⣿⣷⡛⣻⡏⢉⠛⠙⠛⠛
⠀⠀⠀⢠⡟⠀⠀⢳⣀⣿⣾⣿⣟⢁⡸⠙⠫⣷⡇⢨⡀⠐⠀⠚⠇⠄⡀⢸⣎⠹⣽⣿⣥⠛⠀⠰⡇⠀⣼⣿⣫⣵⠞⡩⠺⠌⠁⢁⢌⡃⢦⠴⠋
⠀⠀⣤⣿⡿⠤⢠⣀⣿⣿⣿⣿⣿⠸⣦⣀⣠⠛⠁⠸⠄⠀⣶⡖⠀⢠⣀⠀⠙⢁⡙⠙⣠⡤⠀⢰⢹⣾⣿⣿⡟⣡⠞⠷⣤⠆⠜⠁⠀⣼⡟
⠀⠀⠻⡟⠇⡀⠄⢡⠻⣿⣿⣻⣿⣳⣾⡿⠀⠴⡿⠀⠀⣰⣿⣿⣶⠂⠘⣤⣿⣃⠀⠀⠀⠉⠊⣾⢸⡟⣿⢋⣼⡁⢠⡀⣠⡆⡀⠀⠀⡸
⠀⠀⠀⣾⡟⠗⠒⠒⠀⣿⣿⣿⣿⣿⡿⠀⠀⢶⡄⢸⡜⠻⠿⡿⣿⣴⡾⡏⣥⡔⠀⢀⠁⠑⡘⣟⠘⠋⠁⣿⣿⣿⣿⣷⡿⡟⣿⣦⣸⠃⠀⢠⣾
⠀⠀⢀⣿⣧⡀⠀⠀⠀⣸⣿⣿⣿⢿⠃⣸⣄⡀⠀⢸⡏⠀⠈⠁⠉⠉⠳⢡⠈⢳⣧⣄⣠⣠⠿⠋⢀⡀⢀⢘⣿⣿⣳⣱⣟⢀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⢸⣿⣟⢋⣰⠀⠀⣿⡟⢡⣿⣾⠀⠻⠿⠇⠀⢸⣿⣿⣶⣽⡀⠀⠀⠀⠐⢨⣿⡀⠸⠇⠀⠀⢸⣿⢛⢺⣿⣿⣿⣿⠛⡙⠃⠀⢀⡴⢻⣿⣿
⠀⠀⠈⣿⠿⢿⠇⠀⣰⠋⣰⢏⠹⢿⠀⣴⡶⣶⣦⠘⡽⢿⣿⡟⢿⣇⠀⣀⢀⠚⢻⠃⠀⠀⢀⣠⢸⣿⢨⡜⣿⡏⣿⡿⢚⣄⣠⡄⠛⠀⣸⠻⣿
⠀⠀⢸⣿⠀⢸⣆⣤⣁⠸⣽⣬⣽⣿⡄⢹⣿⣿⣿⢸⣷⡾⠻⢻⣹⠻⣉⡛⠁⠒⠁⢢⣔⣮⠜⠛⢸⣿⣾⠁⠉⠁⠁⣹⣗⡷⣾⡟⢢⣴⠃⠀⢻
⠐⣤⢞⡿⠁⠼⠋⣁⠞⠙⣿⣿⣿⣿⣧⠈⣿⣮⣿⠘⠛⠀⠀⡼⣦⣌⠛⠓⠂⢁⣈⠨⣟⠁⠀⠀⢸⣿⣿⣦⡴⢰⣾⣿⣿⣆⢸⡇⢸⣯⠀⠀⠈
⠀⠉⡿⠀⠀⠀⢴⠇⠀⠀⢻⣿⣿⣿⢻⣧⡘⢿⣿⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠸⠛⠠⠍⡀⠀⠀⢸⣿⡿⢿⣧⢺⡿⣿⡟⠀⠈⡇⢸⢿
⠀⠈⠇⢂⢀⠎⣡⠊⠀⠀⢸⣿⣿⠽⠈⢹⣷⣄⠋⢸⣶⣤⣴⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡄⠀⢸⣿⡇⢸⡿⡈⠀⣾⡇⡆⠀⠃⢸⢸⡄
⠀⠀⠀⠀⠁⢎⡀⠀⡄⠀⠸⣿⣿⡆⠀⠀⣿⣿⣷⠈⠉⠛⢡⠀⠈⠀⠀⠀⠀⠀⠀⠀⢉⣀⠠⠤⢸⣿⡇⠀⢁⣧⠀⢿⡇⠃⠀⠀⢾⠈⡇
⠀⠀⠀⠀⠀⣼⣿⣦⣀⠀⠀⠀⠻⣷⠀⡿⡇⢸⣿⠀⣾⠀⢰⠀⢇⠀⠀⠀⠀⠀⠀⢀⡈⠉⠁⠀⢸⣿⣿⠀⢸⣿⡄⢸⡇⠀⠀⠀⣏⠀⡇
⠀⠀⠀⠀⢰⠿⡏⠉⠻⣷⣄⡀⠀⠀⠰⢿⡀⣾⡿⠀⠙⣿⠦⣁⣘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢸⣿⣿⡀⠀⣿⢱⢸⡇⠀⠀⢰⠛⠀⡇
⠀⠀⠀⠀⡞⠀⡇⠀⠄⡀⠙⣿⣦⡀⠀⠀⠁⠉⠁⠀⠀⡜⠂⠈⠛⠻⠦⢤⣤⣿⣷⢌⠁⠀⠀⠀⢸⣿⣿⡇⠀⢻⠈⡜⡇⠀⠀⠘⣦⢺⠃
⠀⠀⠀⢸⠃⠀⡏⠀⠀⠇⠀⣼⠿⣿⣷⣄⡀⠀⠀⢠⣶⣿⡆⠀⠀⠀⠀⠺⢿⣿⠉⢦⡐⠝⠀⠀⢸⣿⣿⡇⠀⠸⠀⣷⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠖⣫⣶⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⢿⠁⠀⢀⠀⠈⠳⠁⢤⡤⠆⢠⡃⠸⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣧⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⣿⣩⣿⡏⣀⣀⠀⠀⠀⠀⠀⠘⢹⠈⠀⠀⠈⢱⣤⡀⠈⠁⠉⠀⣼⣧⡀⢺⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣧⠀⠀⣤⠴⠀⠶⡄⠸
⠀⠀⣶⠀⢋⣨⣿⠌⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⠏⠀⠁⢠⣶⣏⣛⡿⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⣿⡀
⠀⠀⠀⣰⡘⠃⢉⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⢠⣼⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⣿⣿⢳⠀⣿⡇
⠀⠀⠀⠛⠿⠟⠒⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠻⠖⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠃⠀⠸⠿⠿⠘⠇⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢠⡏⠉⢱⡄⠀⠀⠀⠀⠀⠀⠀⢸⠋⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠙⡇⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⢹⡄
⠀⠀⠀⠳⡖⢾⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡷⢲⠞
⠀⠀⠀⠀⡇⠀⢳⡀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢀⡜⠀⢸
⠀⠀⠀⠀⢻⠀⠀⠳⡀⠀⠀⢀⡎⠀⠀⡤⡄⠀⠀⠹⡄⠀⠀⢠⠏⠀⠀⢀⣤⡀⠀⢳⠀⠀⠀⢀⡞⠀⠀⡏
⠀⠀⠀⠀⡼⢦⠀⠀⠱⡄⠀⡞⠀⠀⠀⠓⠁⠀⠀⠀⠙⡄⢠⠋⠀⠀⠀⠈⠚⠁⠀⠀⢣⠀⢀⠎⠀⠀⡠⣇
⠀⠀⠀⠀⠈⡇⠀⠀⠀⣙⣾⣠⠤⠤⠤⠤⠴⠖⠒⠒⠒⠚⠓⠒⠒⠶⠦⠤⠤⠤⠤⣄⣈⣧⡋⠀⠀⠀⣹⠃
⠀⠀⠀⠀⢀⠷⠒⠋⢉⣀⣀⡤⠤⠤⠤⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠲⠶⠶⠦⠤⠤⠤⢄⣀⣉⡉⠓⠲⢧
⠀⠀⠀⠀⠸⣤⡖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢲⣠⠇
⠀⠀⠀⠀⠀⠀⢧⠀⣀⣀⠤⠤⠤⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠒⠒⠒⠒⠒⠒⠒⠦⠤⠤⢤⣀⣀⠀⡏
⠀⠀⠀⠀⠀⠀⣼⢋⣡⠤⠤⠒⠒⠒⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠲⠤⠤⣌⡙⡇
⠀⠀⠀⠀⠀⠀⠘⠣⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⣠⠄⠀⠀⠁
⠀⠀⠀⡀⣀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠍⣶⠐⠆⠀⠀⠘⡄
⡔⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡈⠁⠀⠀⠀⠀⡅
⠑⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⢿⠆⠂⠀⢴⠀⠃
⠀⠀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⢈⠣⠀⢀⣀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠈⠤⡀⠀⠀⠀⡌⠀⠄
⠀⠀⠀⠄⠀⡀⠘⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠰⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⡄⠀⠀⠀⠢⠤⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⠀⠀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⠑⠒⠒⠀⠋⠭⠀⡁⢀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢩⠇⣾⣿⣿⠀⠀⠀⠀⣤⣀⠴⠋⠻⡟⠀⠀⠀⠀⢀⣀⣀⡀⠤⢤⡤⠀⠀⠀⢸⡟⠀⠀⠀⠀⣻⣦⠀⠀⢀⡔⠒⣶⣴⡶⠒⢦
⠀⠀⣀⣐⣀⣿⣿⣿⠀⠀⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠂⠀⠠⠀⠀⠀⠀⠈⠓⠒⢒⣖⠒⠚⠂⠀⠀⠈⠳⠴⠃⠀⠑⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⣤⡀⠘⡟⠀⣠⡄⠀⠀⠀⠀⢠⠖⠛⣷⣦⡀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⣠⣤⡤⠞⣿⠀⠀⠀⠀⢠⢖⡭⠭⣗⣼⠿⠟⠀⠀⠀⠈⠙⢆⣧⠞⠉⠀⠀⠀⠀⢀⣿⣷⣶⣇⣨⡇⠀⠀⠀⠀⠀⣻⣿⣿⠤
⠀⠀⢚⠡⡄⢰⡆⣿⠀⠀⠀⠀⢾⣏⢾⣿⢻⢾⠀⠀⠀⠀⠀⠀⠀⢼⣿⠄⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣭⣿⣭
⠀⠀⢼⠣⡇⢸⠇⣿⠀⠀⠀⠀⠈⠪⣷⣲⠵⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣷⡀⠀⠀⠀⠀⠛⣿⠛⣿⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡷⠶⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⡶⢖⣶⣰⣶⡶⡆⠀⠀⠀⠀⠈⠉⠉⣧⣤⣤⠀⠀⠀⠀⠀⠰⣿⣿⠛⠁⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣦⣀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⣿⠟⣯⢿⠿⢶⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠀⠈⣻⢿⣿⣷⣿⣾⠟⠀⠀⠀⠀⠐⢻⡿⠂
⠀⠀⣿⣼⣛⣾⣯⣿⡇⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⡆⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠴⠖⠓⠛⠛⢿⣿⣿⣿⣿⣷⠶⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡴⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢠⠀⢠⠀⠀⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣤⢠⣇⢀⣸⣀⣸⣀⣤⣆⣸⣇⣰⣇⡀⢸
⠀⠀⠀⣤⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠁
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⠲⠙⠓⠉⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠉⡻⠛⠲⢤⡀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠀⡎⠀⢀⣀⣀⣀⠤⣤⣤⠤⢂⣁⣀⡀⠀⠹⡄⢠⠦⠀⠀⠠⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⢉⣉⠀⣤⣤⡄⠀⠀⠀⣤⡀⠀⣉⡑⠂⡗⠀⠋⠀⡀⠀⠑⠨⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡚⢡⣿⣿⣿⣷⣶⣤⡄⢠⣤⣶⣾⣿⣿⣿⡆⠙⡄⠀⠞⣡⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠀⠻⣿⣿⣽⣿⡿⠀⠀⢿⣿⣯⣿⣿⠟⠠⢚⣷⠀⠀⠁⠀⠀⠀⠔
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⣚⠁⠈⠛⠛⠛⠑⠛⠚⠊⠛⠛⠋⠁⠀⣐⠃⡼⢂⡠⠀⠀⠀⠀⠀⠮⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⢧⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣠⠏⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⣀⠞⢉⠉⠒⠤⡙⢦⣄⣀⣀⠀⠀⠀⣀⣀⣤⠞⠁⠀⠀⠀⠊⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⡍⠀⡀⠀⠀⠀⠈⣦⠤⠽⢿⣉⣉⣹⠿⠷⠦⢤⠀⡀⠤⢄⠄⡀⠀⠀⠀⢀⠀⠤⠤⠒⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠤⠄⠀⠀⠀⠀⠐⠋⢹⠀⠀⠀⠉⠉⠁⠀⠀⠀⡞⠘⠀⠠⠀⢀⠈⠆⠀⠀⠀⠳⣤⡀⠀⠀⠐⠴⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠠⠀⠀⠀⠐⠁⠠⠒⢄⠀⠀⣸⡀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠋⠀⠀⠠⢹⣦⣌⠛⠿⠶⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⠀⠀⠰⠀⠀⠀⠁⠐⠙⡇⠀⠀⢀⠘⠄⠀⠀⢠⡇⢀⠀⠀⠀⠂⠀⠀⠀⠁⠌⢉⡙⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣄⠤⢤⣿⣄⣀⡴⠛⣄⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⠀⠀⢀⠢⣀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠘⠁⠀⠀⠀⠀⠉⠑⢄⠀⠘⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡞⣿⠃⠉⠁⠀⢬⠀⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠊⣾⠁⠈⠉⠉⠉⠉⠀⠀⠘⣿⣿⣉⠀⠀⠀⢀⡴⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡷⠀⠀⠀⢠⣴⠀⠀⠀⠀⢿⠈⠻⢃⠂⠤⠞⠁⠀⠀⠤⠀⠴
⠀⠀⠀⠀⠀⡴⠊⠉⠉⠙⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢸⢹⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡂⠀⠀⠄⡀⠀⠈
⠀⠀⠀⠀⢸⠀⢀⠀⠈⠐⠄⣸⡆⠀⠀⠀⠀⠀⢀⣠⠊⣿⠀⠀⠀⠀⣾⠘⡆⠀⠀⠀⠀⢷⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢄⣸⣀⣀⣨⠉⠑⠉⠉⠀⠀⠉⡇⠀⠀⠖⠋⠀⢧⣤⡀⠤⠄⡞⠅⠀⠀⠀⠁⠘⠁⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠈⠻⠤⣄⣠⠶⣹⠋⠉⣹⠉⡁⠀⠰⠀⠀⣠⣖⣠⠀⡄⠀⠀⠀⠀⠀⠀⠄⣴⡒⠀⠀⠀⠀⠀⠀⢀⡠⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠈⠠⠴⠃⠠⠁⠀⠀⢀⣴⢋⡽⠁⠀⠀⠀⠀⠀⠀⡲⠘⠻⠶⠻⢦⣄⠀⠀⠃⠉⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣷⡏⠀⠀⠀⠀⡀⠀⡜⠉⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠙⡆⠀⠀⠀⠐⠂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠣⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠡⠀⠀⠂⡠⠇

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡄⢀⣧⠀⠀⢀⡇⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⡄⡄⣤⢠⣤⡄⠀⠀⠀⠀⠀⠀⢸⡄⡇
⠀⣠⣀⡇⢸⣿⣟⡃⠸⣷⣦⣦⣶⠀⢸⣾⣇⣀⣧⠀⠀⢠⣧⣀⣠⣇⠀⡀⠀⣠⣶⣴⣦⣶⣄⣯⠀⣠⣇⣧⣿⣼⣿⣧⠀⣤⠤⢤⣶⣾⣼⡧⣷
⠀⠋⠹⠿⠿⠿⠿⠇⠠⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠷⠀⠿⠶⠷⠶⠶⠿⠿⠀⠘⠿⠿⠿⠸⠿⠇⠐⠶⠰⠰⠿⠿⠻⠗⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣤
⠀⢀⣲⡆⡄⠀⠀⡀⠀⠀⢀⣀⣇⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣤⢸⡇⠀⢀⣤⢤⡀⡇⠀⠀⠀⣾⣿⣷⠤⠤⡤⠤⡇⠀⢸⣿⢤⣤⡄⠀⡂
⠀⣿⣿⣿⣧⣷⣧⡇⢠⣶⣎⠉⣻⣿⠀⣤⣿⣷⣸⣾⣷⠀⢸⣿⣿⣿⣭⡅⢸⣿⣿⣿⣿⣿⣄⠀⢰⣿⣿⣿⡇⣸⣷⢸⣿⠀⢰⣿⣿⣿⣧⣤⣆
⠀⠈⠉⠁⠀⠉⠁⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠁⠈⡁⢈⠉⠀⠈⠉⠉⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢠⠀⢀⢀⢰⠀⠀⠀⢀⣀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡇⢰⣦⡇
⠀⢠⣄⣠⣄⣀⣸⠀⢸⣿⢸⡆⠀⠀⢸⣿⠇⠀⠀⡇⠀⢀⣄⢀⣤⡀⢸⡇⠀⢸⣿⡆⠀⣾⣿⢀⠀⠀⠀⣴⣶⣀⣿⣿⠀⠀⠀⠒⡟⢋⣸⣧⡇
⠀⢿⢿⡿⢿⡇⣿⠀⣼⢿⣾⠷⠶⠤⠨⠿⣷⣿⣿⡇⠀⠸⣿⠿⡯⠽⣿⡷⠀⣿⣿⠿⠿⠿⠿⠿⢷⠄⢾⣿⣿⠩⣿⣿⢶⠆⠸⣿⣿⡿⠉⠉⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢰⢆⡀⠀⢘⠒⠰⠒⠀⢳⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠷⣿⠿⠀⠀⠐⣳⣈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⣀⣠⠤⠶⠛⠲⢾⣀⠁⠀⠀⢠⡿⢮⡷⠦⣤⣀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⣁⡀⠉⢁⣤⠆⠀⠀⡀⠀⠙⢓⣦⡀⠿⠃⠀⠀⢀⠀⠈⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣺⡍⠀⠀⠀⢺⣷⠠⣄⣀⠹⠏⠀⣀⠔⣃⠀⠈⠡⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣙⡥⠀⠀⠀⠀⢈⣀⠖⠀⠀⠈⠐⠛⠁⢠⣿⠀⠀⠀⢌⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠐⠆⠀⡔⠙⠫⢀⡀⠀⠀⠀⠀⢀⣤⠟⠃⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⠀⠀⠇⡀⠀⣹⡇⠀⠀⠀⠀⢸⠄⠉⣤⠀⢀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠈⢿⣧⡀⠀⠀⢀⠀⣸⠶⠛⠁⠘⢿⠦⠳
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⡥⠀⣗⠐⠈⠛⠈⠀⠀⠼⢛⠅⠂⠐⠓⠶⡼⠗⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⢠⠄⡗⠿⠀⢀⠀⠀⠁⠀⠴⠁⠀⠀⠀⣀⢲⣿⢇⠠
⠀⠀⠀⠀⠀⠈⠙⠋⠀⡰⠋⠈⠀⢠⠀⠀⠓⠀⠀⠀⣴⠁⠔⠒⠀⠀⠙⠉⠉⠁⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢛⣵⠋⠁⠐⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡟⠁⠀⠀⠀⠀⠀⠉⠐⠦⣤⣤⣤⡀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢙⠀⠀⠀⠀⠀⠀⢠⠐⠒⠊⡁⠉⠁⠐⠀⣀⡀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠉⠉⠀⠀⠀⠀⠀⠌⠁⣠⡼⡣⠄⠀⠀⣄⠀⠀⠈⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠄⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡄⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠐⠀⢀⠀⢄⠀⠀⠀⠀⠀⢀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠨⢀⠌⠃⠀⠀⠀⠁⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⡀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢄⠁⠀⠠⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠐⠀⢀⠔⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠈⠀⠀⠀⠀⠉⠙⠀⠀⠀⠁⠀⠀⠂⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠢⠋⠀⡠⠊⠈⢐⡄⠀⢄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠰⡄⠀⠀⠀⠀⠀⠀⡀⠀⢸⠄⠈⠜⠀⠀⠀⠀⠀⠀⠉⠐⠠⠄⠀⠀⠠⠄⠀⠀⠀⠄⠿⢀⣆⢠⠎⠀⠀⠀⡜⠀⠀⠀⡅⡀
⠀⠐⠀⠀⢃⠀⠈⢆⠀⠀⢀⠀⠀⡟⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⡀⠀⢀⡌⠀⠀⢀⡜⠀⣸⠂⠐⠐⠒⠂
⠀⠀⠀⠀⠀⠐⠤⠸⠦⣀⠜⠁⠈⠀⠀⠀⠀⠀⠀⢀⠀⠄⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠉⠉⠁⠑⠤⠤⠋⠔⠚⢁⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣷⣶⣶⣾⣿⣼⣿⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠿⣿⣿⠁⠀⠀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⢀⣀⠀⠀⠀⢸⣿⣆⣀⣀⡴⠲⢤⣀⠀⠉⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠉⠁⢀⣀⣀⣤⡤⠤⠶⠖⠚⠛⠋⠉⠙⢦⣀⡀⠀⣿⣟⠉⠁⠀⠀⠀⠈⠙⠓⠦⣄⣀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⡀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢲⠋⠀⠀⠀⠀⠀⢠⡞⠛⢷⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡆⠀⠀⠀⢠⡞⠉⢳⡄⠀⠀⠀⠀⠸⣇⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡆⠀⠀⠀⠀⠀⠘⠧⡤⠞⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣇⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠀⠀⠀⢹⡇⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⠀⢻⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣄⡀⠀⠀⢸⡄⠘⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡞⠀⣸⣿⣧⣀
⠀⠀⠀⢀⣠⣶⡿⠟⠉⢹⣿⡀⠸⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⠀⠈⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡾⠁⣰⣿⠋⠙⠿⣷⣦⣀
⠀⣠⣶⡿⠟⠉⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⣾⣿⡿⠋⣿⣿⠁⢀⣀⠈⢿⣿⣇⠀⢿⠀⢹⣿⠋⣀⡀⠙⣿⣿⡇⠀⠀⣼⠁⢠⣿⠏⠀⠀⠀⠈⠙⠿⣷⣦⡀
⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⠀⢸⣿⡿⠁⠰⢟⡋⠀⣭⣭⠀⢰⣿⣿⠀⢸⡄⠘⣿⣷⣭⡃⠀⣿⣿⡇⠀⣸⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡇⠈⣿⣿⣆⠈⠉⠁⢠⣿⣇⣠⣾⣿⡟⠀⠈⣇⠀⣿⣿⣀⣠⣾⣾⡿⠀⣰⠃⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿
⣿⡏⠙⠻⢿⣶⣄⡀⠀⠀⠈⣿⡇⠀⣧⠀⠘⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⢻⠀⢸⣿⣿⣿⡿⠋⠀⣰⠏⢀⣾⡟⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⣿
⣿⡇⠀⠀⠀⠉⠻⢿⣶⣄⡀⢻⣿⠀⢹⡀⠀⠀⠉⠻⠿⢿⣿⠿⠿⠛⠁⠀⠀⠀⠀⢸⡆⠘⣿⡏⠁⠀⠀⢠⠏⠀⣾⡿⠁⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣷⠀⠀⢠⡏⠀⣼⡿⠁⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⢠⡴⠲⣤⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿⡀⢀⡞⠀⣼⣿⣷⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⡀⠀⠀⠀⠀⠀⠸⣄⣀⡼⠀⠀⠀⠀⠀⠀⣸⡆⠈⣿⣇⡾⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢿⣿⠁⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠛⢦⡀⣀⣀⣠⣤⠤⠴⠶⠒⠚⠛⠉⠁⠀⢀⣀⣾⡿⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠉⠉⢀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⡿⠛⠋⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣶⣿⠿⠿⠿⠛⠛⠛⠉⠙⠿⠟⠉⠙⠻⢿⣷⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠙⢿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢟⠍⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⢀⣀⣠⠤⣔⠚⠁⠁⢠⣞⣶⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⢨⠟⠉⠉⠈⠉⠑⠊⣩⣷⣾⣿⣿⣿⡇⠀⠀⣠⣍⠁⠸⣩⠇⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣀⡀⠀⠀⠀⠸⣿⣿⢿⠿⠟⠃⠀⠀⠻⠟⢠⠖⡆⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⡇⣠⣤⡄⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠈⠛⠁⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⢸⠀⢲⣿⠿⣠⣍⠉⠁⠀⡔⠒⡆⠀⠀⠑⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠟⠀⠀⠀⠙⠚⣁⣠⣤⠴⠖⠛⠉⠉⠢⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡾⣋⡩⠤⠒⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⣾⠇⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢠⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠴⠚⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⣙⣋⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠂⣠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠐⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠉⠙⠻⠟⠃⠀⠀⠈⠻⠛⠉⠀⠙⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⢀⣠⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⡆⠀⣿⣿⣿⣿⣿⣿⡆⠀⢰⣿⣿⣿⣿⡷⠀⠀⣿⣿⣿⣿⣿⣿⠀⢰⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢻⣿⣿⣿⠟⠋⠁⠀⠀⠻⢿⡿⠿⠃⠀⠀⠀⠙⢻⣿⣿⡟⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⠀⠀⣀⣤⡀⠀⠀⠀⣠⣤⣀⠀⢠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣷⣤⣤⣤⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠏⠉⠉⠙⣿⠉⠁

⠀⠀⠀⠀⠀⣠⢶⣶⣖⣲⠔⠴⡂⢐⠖⢒⠖⠒⠂⠛⣿⣻⡿⢿⣿⡄
⠀⠀⠀⠀⠀⡷⡞⢻⡟⠓⡗⠛⣶⠒⠒⢅⣰⠒⠒⡦⡾⠕⠚⢮⣾⡇
⠀⠀⠀⠀⢰⣗⣧⠀⠁⢸⡇⠀⣿⠀⣷⠀⣿⠈⣉⣷⠀⣔⣲⠀⣿⡇
⠀⠀⠀⠀⠀⡇⡯⣄⣠⡯⢃⣀⠿⣄⣉⢠⣻⣈⣁⡿⢦⣀⣀⡴⢹⣇
⠀⠀⠀⠀⠀⡧⠏⣡⠞⠂⠰⡫⢺⠾⢧⡼⡴⢥⣻⠄⢮⠔⠜⣆⢸⣷
⠀⠀⠀⠀⠀⣟⢼⠁⣾⠿⠶⣯⡏⢰⠈⣇⡇⡌⠏⡄⢸⡇⣋⣫⢸⡟
⠀⠀⠀⠀⠀⢟⠸⠦⣈⢉⣠⣿⣁⡖⣆⢸⣇⡿⣀⢇⡸⣇⣉⣻⢘⡇
⠀⠀⠀⠀⠠⠘⠔⠄⠂⠮⠤⣤⣤⣥⣬⣵⣮⣤⣤⣾⣯⣯⣤⣤⣵⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠼⠿⠿⠿⠿⠧⠤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠋⠀⢀⠇
⠀⠀⠀⠠⠄⠤⠠⠠⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠼⠛⠉⠉⠉⠉⠓⣶⠿⠿⠏⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠉⠉⠒⠢⣤⡞⠉⠙⡊⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢅⠂⠞⢷⠈⠹⠢⡀⠀⠀⠸⢯⣭⣩⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢈⡊⠐⠃⠘⠀⢐⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠃⠀⠀⠂⠐⠈⠳⠶⠿⠠

⠀⠀⠠⡀⠀⠀⢤⡀⠀⠂⢦⠀⠀⠚⠀⠀⠀⠀⢹⡆⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠄
⠀⠀⠀⠈⠢⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇⠀⠒⠰⠻⡧⢀⠀⠰⠖⠓⠀⢠⣀⠁⠈⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠒⠄⠀⠈⠁⢈⠴⠚⠁⠀⠀⣀⠁⠀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠰⠁⠀⠀⠀⠉⠠⠐⠊⢀⠄⠈⠩⠵⠂⢀⣄⠄
⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⢸⣶⣆⠀⢆⠀⠀⠈⠀⠉⢁⣀⡤⠀⠒⢉⠜⡠⢀⠃⠐⠂
⠀⠐⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⢸⣿⡏⠀⠀⠁⠀⠀⠀⠀⠠⡶⠿⠷⠗⠚⢊⣴⣧⡀
⠀⠀⠀⠀⡁⠀⠀⡁⢘⠀⠀⠀⠀⣿⠀⠀⠀⠀⢺⣿⡇⠐⠀⠀⠀⢠⡆⠀⠀⠈⠓⠶⠠⠶⠻⠀⠙⠛⣠⠄
⠀⠀⠀⠀⡄⠀⠀⠁⢈⠀⠀⠀⠀⠸⠀⠀⣀⠀⠼⠛⠛⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠹⣿⣄⣀
⠀⠀⠀⠀⡀⠀⣀⠆⠀⢀⠤⠀⠄⠉⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⢸⡇⢈⢀⡤⠐⠂⠀⠀⠁⠀⠈⢘⣿⠟⠋
⠀⠂⠄⠲⠀⠀⠀⠀⠐⠭⠿⠿⡖⢂⡀⠀⠀⠀⢠⣄⡀⠅⠀⠀⢨⣿⡏⢠⡠⠧⠖⠀⠀⠰⠏⡀⠀⠼⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠄⠩⢈⣀⣋⣉⣁⡄⡀⠀⠀⠈⣿⠧⠴⠓⣺⣉⠲⣄⠉⢳⠀⠀⠀⠤⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠟⣻⣥⢬⣴⣿⣽⣷⠂⠀⠀⢘⣿⣿⣅⠀⠀⠀⣨⠟⢀⣀⣤⡴⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠡⠖⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣅⠀⡀⠀⣰⠍⠩⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣦⣶⣶⣴⣦⣴⣿⣴⣷⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⣿⣦⣰⣦⠀⠀⠠⣶⣶⠶⣤⣀⠀⠀⠀⠀⠀⢀⣠⡴⢦⣶⡦
⠀⠀⠀⠀⠀⣰⠿⣿⢿⠏⢻⠄⠀⠀⢸⣷⠈⠳⡉⠳⣄⠀⢀⡴⠋⡱⠋⢰⣿⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⣀⣧⣼⡄⠀⠰⢿⡇⠀⠀⠀⠻⣀⣠⣤⣤⣬⣶⣮⣤⣤⣤⣀⡸⠃⠀⠀⣀⣴⣞⢉⠀⣸⣿⡄⠠⡙⣷⣤⡀
⠀⠀⢀⣾⡟⠈⡈⢹⡆⠃⡼⠀⠀⠀⠀⠀⡾⠁⠈⡥⠈⣿⡏⠢⢌⠀⠹⡆⠀⠀⣴⡧⣌⡉⠀⢸⡇⣿⢹⡄⠀⢉⣁⣿⣆
⠀⠀⠘⣿⡗⠈⠙⠂⣻⣴⠗⠀⠀⠀⠀⠀⠹⣤⣠⣐⡴⠘⠱⡔⣠⣠⡼⠃⠀⠀⠉⠉⠛⢿⣷⣿⡶⠙⢦⣿⣾⠟⠛⠉⠋
⠀⠀⠀⠈⠉⠉⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉
⠀⠀⣶⢶⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣴⣷
⠀⢰⣿⠆⢰⡿⢷⡀⠀⠀⣠⡔⢶⡶⠀⠀⠀⡿⠿⢿⣾⡉⢿⣷⠀⠀⠀⠀⢀⣀⡤⣴⣶⠶⠤⢄⣀⣀⣀⠤⠴⠶⣶⡤⣄⣀
⠀⠸⣅⣀⣀⡑⠈⢧⡤⠋⠙⠡⡿⠀⠀⠀⠀⢳⡷⢴⡟⢃⠀⢈⡇⠀⠀⠀⠛⢷⣵⡈⠉⠐⣀⡴⢿⣿⠷⣄⡐⠈⠉⣱⣽⠿⠃
⠀⢀⡾⠉⢉⡉⢻⣿⠒⠶⣦⠜⠁⠀⠀⠀⠀⠈⢳⣸⡁⢁⣶⠛⠛⢳⡀⠀⠀⠀⠙⠳⣤⠞⠫⠂⠈⣿⠀⠢⠛⢦⡴⠟⠁
⠀⠸⣧⡀⠈⡠⠟⢁⠘⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⡏⠠⠜⠀⣰⡇⠀⠀⠀⠀⠀⠙⣦⣤⣤⡜⠙⣤⣤⣤⡾⠃
⠀⠀⠙⠛⠊⠀⠀⠘⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠆⠥⠵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣴⣦⡤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⣠⡤⠤⠤⠂
⠀⠀⠀⠀⢸⣿⣿⣽⣧⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠞⣡⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠘⢛⣭⣷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣾⠛⡩⠭⠭⣙⠻⣷⡄
⠀⠀⠀⠀⠀⠈⠙⢻⡏⠀⠀⠀⠀⠀⠀⠐⢷⣦⣀⠀⢿⣿⣧⣎⡤⠒⠢⣌⠳⣌⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣀⣤⣤⣤⣶⣶⣶⣾⣿⣿⣾⣿⣿⣷⣬⣁⣉⢎⠣⠈⢦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠉⠉⠀⠀⠀⢠⡾⠋⠁⠀⣈⡙⢿⡉⠙⠛⠿⣷⣵⣄⠙⢦⠙⣿⣄
⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠿⠍⣼⠇⠀⠀⠀⠈⠙⣷⣗⡄⢳⢸⣿
⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠈⢿⣾⣸⢸⣿⡄⢀
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡸⣧⠙⠛
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⣦⣤⣀⠀⠀⠀⠀⢻⣿⣷⣽⣷⡄
⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣷⠉⠀⠀⠀⡾⠟⠻⣿⣿⠿⠊
⠀⠀⠀⠈⠙⠛⠿⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠿⠆⠀⠀⠀⢀⣼⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⣀⣀⣠⣤⠶⠛⠁⠀⢀⣼⡇⠀⠀⠀⢿⣦⣶⡿⠋⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠹⣯⠀⠀⠀⣠⣴⣿⣿⠃⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣿⣿⡿⠛⠁⣰⠟⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢠⣶⣦⣤⣾⣧⣾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠈⠛⠶⠶⠞⠋⢀⣾⠀⠀⠀⢀⣿⠁⠀⠀⠀⠈⠉⢉⣿⣿⣿⡿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠀⠀⣸⡟⠀⠀⠀⠀⣀⣴⡿⠋⠉⠉⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣍⣉⣛⡓⠒⠒⠶⣆⠀⣠⠤⠦⠴⠶⠶⠶⠲⣿⡃⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠒⠒⠒⠶⠶⠶⢾⣿⠛⠋⠉
⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣸⡇⠀⣹⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⣿⠀⢀⡟⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⠿⠆⣼⠁⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠿⠴⠞⠛⠛⠿⢶⣦⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠷⠶⠶⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄⣀⣀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⠈⢿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣀⡠⠄⣨⣤⠖⠀⠀⠀⠀⠀⠀⢾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣎
⣿⣿⣿⣿⣿⣿⣷⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⢻⣿⣿⣿⣯⣴⣤⣾⣿⡿⠃
⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⠉⣿⣯⣼⣿⣿⣿⣿⡿⠟⠁
⣽⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣶⣟⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣹⣿⣿⢏⠈⠿⣿⣿⣿⣿⣿⣿
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢰⣾⠟⠁⢿⣿⠇⢂⣠⣾⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⣀⡀
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⣡⣼⣿⡿⢋⡄⣴⣋⣡⣶⣿⣿⣵⣿⣿⣿⣿⣿⣷⠀⠀⠀⣠⣤⡷⠟⠛⠉⠁
⣿⣧⣤⡶⠂⠀⣀⣠⠶⣿⣿⡿⣯⡴⠋⠀⢀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⣿⠟⠁
⣿⣿⣿⡤⣴⡾⢿⣿⢟⣩⡿⠛⠁⠀⣠⣶⣿⣿⡿⢟⣫⣿⣿⣿⣯⣷⣾⣿⣿⣿⣿⣿⣍⣤⡄
⣿⣿⣏⡀⣤⠒⠊⠀⠘⠉⣀⣤⣾⣾⣿⣿⠿⠋⠁⠈⠁⠀⠀⣀⣼⣿⣿⣿⣿⣿⢿⠋⡉⢷⣤⡀
⣿⣿⠶⠁⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣁⣤⡀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣈⠀⠀⠀⠹⣇
⣿⠀⢀⣴⣷⢄⣠⣭⣼⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠚⠀⠀⠀⠀⢰⣿⣿⣄
⣿⠀⡾⣁⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣿⣿⣿⣿⣿⣿⣿⣟⡿⡄⠀⠀⠀⠀⢙⣿⣿⣿⠃
⣿⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣻⣤⡀⠀⠀⠘⢹⣿⣿⣷⡀
⣿⣷⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⡦⣀⣤⣿⣿⣿⣿⡄
⣿⣿⡿⠿⢿⣿⡿⢿⡿⣻⣿⣿⣿⣿⣿⣦⣴⣦⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⣿⣿⣿⡄
⣿⣷⣦⣠⣀⣤⣀⣾⣴⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⣷
⣿⣿⣯⠁⢹⣿⣿⣿⣿⢿⡿⢿⣿⣿⣾⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣽⣧⣿⣙⡟⣿⡃
⣿⣿⣿⣶⣾⣿⣿⡿⢋⣼⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠚⢿⣿⣿⣿⣿⣿⣏⢻⠀⠻⡟⢿⣷
⣿⣿⣿⣿⣿⡿⢧⣠⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠻⢿⣿⣿⣿⣿⡟⢀⠀⠀⠁⠀⢿⣧
⣿⣿⣿⣿⣿⣇⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠐⠛⢿⣿⣿⣿⣧⣠⠀⠀⠠⠀⠀⢻
⣿⣿⣿⣿⣿⣿⣶⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡿⣿⣿⣿⣿⣿⣦⠄⠶⢿⣿⣿⣿⣿⢻⣦⣴⢳⡄⠀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠉⣰⠛⣿⣿⣿⣿⣿⣤⢴⣶⣿⣿⣿⣿⣾⣾⣏⠻⠟⣄⢿
⣿⣿⣿⣿⣿⣿⠟⠿⠿⠿⠿⢿⡟⠟⠛⣻⣿⠿⠁⠀⠀⡴⠀⠸⣿⣿⣿⣿⡯⠀⠀⣸⣿⣿⣿⣿⣿⡏⣽⣦⡀⣿⣯
⣿⣿⣿⣿⣿⡇⠀⠀⡀⠐⢀⠀⠀⠀⣠⣽⠏⣠⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣾⣶⡀⢀⢦⡳⢌⣰⣿⣷⣾⣿⢿⣾⣦⡀⠀⠀⢸⣿⣿⣿⣿⣤⠴⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣎⣮⣷⣽⣴⣿⣿⣟⣟⣵⣿⣷⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢫⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⡉⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⠀⠀⠀⠀⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣤⣦⣤⣿⣿⣿⣿⣷⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣄⣀⠀⢀⣠⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣾⠀⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠙⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡴⣛⣭⣭⣮⣭⣭⡓⢦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⢫⣾⣿⡿⠛⠈⠻⣿⣿⣦⡹⡎⢻⣿⡿⠿⣿⣿⠿⣿⣟⣯⣵⣿⣿⣿⠁⠙⣿⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⡅⠺⢸⣧⣼⣿⣿⣷⢹⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣧⣀⠈⠀⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡿⣷⢠⡉⢻⣿⣿⣿⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⢷⡹⣿⣿⣧⣈⠈⣁⣼⣿⣿⢣⡟⣴⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠳⣬⡻⠿⣿⣾⡿⠿⣛⡵⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠛⠉⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣄⠀⣠⡄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣘⣦⣾⣷⣾⣿⣿⣿⣿⣿⣿⣧⣾⣃⣼⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⣸⣖⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢬⣲⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣽⣶⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢻⣿⣿⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣾⣿⣿⡿⠁⠀⣀⣤⣤⣤⣄⣀⠀⠀⣀⣤⠖⠛⠉⠉⠉⠙⠻⣶⣄⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠁⣠⣾⣿⠟⠋⠁⠀⠀⠉⠉⠉⠀⠀⣀⣀⣀⣀⠀⠀⢹⣿⣧⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⣿⣿⣿⠁⣼⣿⡿⠃⠀⣠⣴⠾⠛⠳⢶⣄⣠⠞⠉⠁⠀⠉⠻⣆⠀⠙⠛⠀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⡇⠀⠛⠋⠀⢠⡾⠋⠀⢀⣀⠀⠀⢻⡏⠀⠀⢠⣤⡀⠀⢻⡆⠀⠀⠀⠀⠀⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠿⠃⠀⠈⣿⡀⠀⠈⠉⠀⠀⢸⡷⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢀⣼⠛⢿⣦⣄⡀⢀⣠⣾⠃⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣴⣾⣿⠁⠀⠀⠹⣿⣿⣿⣏⣁⡀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠨⢾⡿⠋⣏⠀⠀⠀⠀⠹⣿⡙⠳⠦⠀⠀⠀⠀⠀⠀⠆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⢀⣠⣿⠀⠀⠀⠀⠀⢻⣏⠳⣄⠀⠀⠀⠀⢀⠈⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⡀⠀⠀⠀⠀⣼⡿⠀⣿⠀⠀⢀⠂⠀⠐⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠾⢷⣤⣀⣤⣾⡿⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠠⠀⠀⠨⢙⣟⢻⠁⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⣨⣷⣾⣧⠀⠀⠀⠐⠀⠂⠀⢀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠠⣿⣿⣿⣿⣧⠀⠀⠠⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⠏⠂⠀⠀⠀⠀⠀⢀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⡄⠀⢀⠀⠂⠀⠀⠉⠀⠈⠙⠛⠳⣦⠀⠀⠀⠤⠄⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣥⣀⠁⠄⡀⠀⠀⠀⠀⠄⠀⠘⠄⠸⣷⢀⣠⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠻⠿⠷⠶⠶⠾⠶⠿⠿⠟⠛⠋⠁


⠀⠀⡀
⠀⠁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡟⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⡍⠀⠈⡭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠖⢦⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣉⠀⢸⠀⢈⣁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠁⢹⠁⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣇⣀⡀⠀⢀⣀⣸⣀⣀⠀⠀⣀⣀⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡏⠉⠉⠉⡇⠀⢸⠉⠉⠉⢹⠀⠀⡏⠉⠉⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠀⠀⠉⠉⠉⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠶⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠖⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠴⠚⢲⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠚⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠏⠀⠀⠀⠀⠘⡆⣀⣀⣴⡶⠋⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⡀⣀⠀⢰⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠠⠖⠚⠁⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠛⠶⢠⣿⣿⣿⡀⠀⣿⣿⣿⡇
⠀⠀⢸⡿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⣿⣿⣷⠀⣿⠿⠿⣿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⢸⣿⣿⣿⡇⢰⣿⣿⣿⡇
⠀⠀⢸⣇⣀⣀⣀⣀⣸⡇⢸⣶⣤⣄⣸⢿⣿⣿⠀⣿⣤⣤⣿⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⣸⣿⣿⣿⣇⢸⣿⣿⣿⣷
⠀⠀⢸⡏⠉⠉⠉⠉⢹⡇⢸⡿⠿⠿⡿⠿⠾⣯⠀⣿⣀⣀⣿⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣿⠿⣿⡿⢿⣿⠿⣿⡿⢿⡿⠿⣿⠿⢿⣿
⠀⠀⢸⡟⠛⠛⠛⠛⢻⡇⢸⣇⣀⣀⣇⣀⣀⣿⠀⣿⡀⢀⣿⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣿⣤⣾⣧⣼⣿⣤⣿⣦⣼⣷⣤⣿⣦⣼⣿
⠀⠀⢸⣧⣤⣤⣤⣤⣼⡇⢸⣇⣀⣀⣇⣀⣀⣿⠀⣿⠉⠉⣿⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣿⣉⣹⣏⣹⣿⣉⣿⣏⣹⣏⣉⣿⣉⣹⣿
⠀⠀⢸⡇⢀⣀⣀⡀⢸⡇⢸⣿⡏⠉⠉⠉⣿⣿⠀⣿⠛⠛⣿⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣿⠛⢿⡏⠉⠉⠉⠉⠉⢹⡿⠛⣿⠟⢻⣿
⠀⣤⣾⣿⣿⣿⣿⣿⣿⣧⣾⣿⣧⣤⣤⣤⣿⣿⣦⣿⣶⣶⣿⣶⣶⣿⣤⡄⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣶⣿⣇⣀⣀⣀⣀⣀⣸⣿⣶⣿⣶⣾⣿
⠀⠁
⠀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠾⠿⠿⠿⠿⠿⠿⠿⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⠿⠿⠿⠿⠿⠿⠷⣄
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⢠⡏⠀⠐⠀⢠⣤⡄⠀⠀⠈⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⡉⠉⠉⠉⠉⠁⠀⠀⠁⠀⢀⣀⠀⠀⠀⠂⠀⢹⡄
⠀⠀⠀⠀⡾⠀⢰⣶⡆⠈⠛⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣿⣿⣦⠂⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠻⡆⠀⠀⠀⠀⢷
⠀⠀⠀⢸⡇⠀⠈⠉⠁⣤⣤⠀⠻⠿⠀⠀⠀⠠⠀⠀⠀⠀⠿⣿⣿⠟⠄⠀⠀⠀⡀⠀⠀⣰⣿⣦⣀⣼⣿⡆⠀⠀⠀⢸⡇
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠛⠛⢀⣀⡀⠸⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠹⣿⣿⣿⣿⣿⠇⠀⠀⠀⠸⣧
⠀⠀⠀⣿⠀⠀⠠⡀⠀⠀⠀⠸⠿⠃⠀⢀⠄⢀⠀⣤⣤⣤⣤⠀⠀⣠⣤⣤⣤⡀⡀⠀⡀⠀⠉⠛⠛⠛⠁⠀⢀⠀⢀⠀⣿
⠀⠀⠀⢿⠀⠈⠀⡈⠐⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠀⡈⠀⠠⠀⠀⠀⠀⠀⢁⠔⠁⠀⡟
⠀⠀⠀⢸⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡇
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢳⣄⠀⢀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢄⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠈⠓⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠊⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⠒⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠀⠉⠁⠀⠲⢄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠴⠀⠀⠡⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠈⣀⣀⡀⠼⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠔⢪⡁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⢄⠁⠀⠀⠀⠀⠀⠀⠘⠁⠂⢢⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠈⣤⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡀
⠀⡐⠂⠀⠂⢴⠂⠉⠀⠀⠀⠘⠿⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⡇
⠀⡇⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠙⠒⢳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣰⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠻⣁⡠⠊⠉⠳⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⡐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣧⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⡜⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡴⠋⣠⠀⠀⠀⡜⢀⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⢋⠔⠊⠀⠀⠀⡘⡠⠎⠀⠀⠀⡴⠁
⠀⠀⠀⢀⢀⣀⣀⣀⠠⠄⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⢄⡀⠀⠀⠀⠀⠘⡅⠀⠀⠙⠢⠴⣅⠴⡴⠋⠀⠀⠀⠀⢸⠁
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠚⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⢆⡠⠀⠀⡀⣠⠏
⠀⠀⠸⢅⣀⣤⠤⣤⠤⠤⠠⠤⠔⠒⠒⠚⠛⠉⢉⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⡀⠀⢀⡀⣀⡤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣠⣤⢀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣉⣁⣤⣤⣤⣀⣉⡙⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠄⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣹⣿⣧
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣼⣿⡿⠿⠛⠉⠈⠛⠻⠿⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⢿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠉⠃⠈⠃⢹⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢋⠅⡀
⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠘⠈⠁⠀⢀
⠀⠀⠀⠀⡕⢢⠙⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⢱⠿⠠⠂⠀⢀⠀⠀⠀⠀⢒⡶⢶⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⡗⠚⢾⡔
⠀⠀⠀⠀⠠⢈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠃⠀⠀⠆⠀⠁⢄⢀⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠁⡙⠿⣷⣦⣤⣄⠀⢀⣴⣶⣾⡿⠟⢉⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠁⠂⣀⠉⠙⠋⠀⠜⠙⠋⠀⢠⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⡐⢂⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⢹⣶⡄⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠃⠀⠀⠘⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢤⠀⠥
⠀⠀⠀⠄⡀⠑⠂⠀⠌⠀⠀⠠⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠂⠂
⠀⠀⠈⠁⠜⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿