Normal Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢀⠄⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⢠⣿⠁⣠⡾⣹⣿⠃⠀⠀⠀⢠⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⢠⣿⢇⣼⠋⣰⣿⠃⠰⣾⣧⠆⠈⠧⠀⢠⡟⠀⠀⠀⣀⠀⠀⡀⢀⣀⠀⢰⠆⡀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⣰⣿⣯⡿⠁⣰⣿⠃⠀⠀⢋⣙⣀⣀⠀⠀⣾⠃⠀⠀⠀⣟⣸⣿⢧⣏⣾⣠⢺⠏⠉⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣰⣿⣿⠏⠀⣰⣿⠃⠀⣀⣀⠀⣹⠆⣼⣷⢀⣋⢀⣀⠀⢀⣉⡈⠁⠈⣁⣈⠉⠋⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⣰⣿⠃⣠⣾⣯⡉⢠⡟⣰⣿⢣⣿⣿⢻⣿⣧⢞⣿⡇⢠⣾⠏⢸⣿⡗⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⢠⣿⠃⢠⣿⠇⣹⡗⠋⣼⡿⢣⣿⡿⢡⣿⡿⢣⣿⡟⢠⣿⠃⣠⣿⡟⣰⣿⠃⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠀⠀⠀⣼⡏⠀⠘⢿⣶⠟⠀⣼⡿⠡⣿⠟⠀⣿⡟⠁⢺⣿⡴⠻⣯⡴⢿⣿⠼⢿⣧⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢓⠀⠀⠀⠀⠀⢸⣿⠁⠀⢀⠀⠀⠀⣤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡟⣿⢛⡍⠀⠀⠀⠀⠻⠤⢀⡏⠃⠆⠀⣹⢲⡞⠁⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡏⣏⢻⣿⣟⣼⣛⠄⠀⢸⣷⠹⣇⡯⣏⣸⡏⠁⠀⡞⣧⣾⣷⣞⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⠏⠒⠒⠶⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠛⠛⠛⠉⠉⠛⠿⠳⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠚⠛⠛⠓⠲⠶⢤⣄⡀⠀⠐⢦⠀⠀⠀⠔⠀⠀⠉⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠳⣄⠀⠆⠀⡘⢀⠔⠊⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠳⣄⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⡘⢷⡀
⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢀⡀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⣄⢹⣿⡔⠒⠀⠀⠀⠀⠀⡀⠀⠈⠳⣿⣆
⠀⠀⠀⠀⢀⣼⢁⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣬⣿⠋⠀⠀⠀⠒⢀⠀⠀⠀⠰⠀⠀⠈⢻⣄
⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉⠉⠉⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢀⠀⠀⢻⡄
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⢠⠀⠀⠈⡄⠀⠈⣷
⠀⠀⢳⡀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠆⠀⠀⠠⠀⠀⢸
⠀⠀⠈⠳⠦⣠⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⣤⡄⠀⠀⠀⠹⣦⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢸⣯⡄⡤⠀⠉⠙⠛⠛⠓⠂⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡟⡇⢀⣠⣤⡶⠶⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠤⢤⣤⣀⠀⠀⠀⠀⠀⢹⠀⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢸⡇⡇⠻⣿⠏⠀⣾⣿⣿⢃⠀⠀⠀⠀⠀⣠⠎⢡⣶⣿⣦⠘⣿⣷⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⢸⣧⠏⠀⠀⠀⠀⠀⠿⠟⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⢿⠟⠀⠇⠀⠀⠀⠀⠀⢸⠇⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠇⠀⢠⠀⠀⠀⢸
⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⠀⠀⡈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡰⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠠⠴⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⣼⠃⠄⠀⠀⠁⠀⠀⠀⠀⠀⣼
⣠⣴⡿⢚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡠⢴⠟⠀⠀⠀⠀⢀⡼⠁⠄⠀⠀⠀⠀⠀⠀⡄⢀⡴⠃
⠀⠀⣰⠟⠀⢷⡀⠀⠀⠀⠀⠀⢤⣶⠁⠀⠉⠀⠀⠠⠄⠀⠉⢀⡀⠂⠀⠀⠀⣀⣴⣋⣤⡞⠀⠀⡌⠀⠀⠀⢠⣴⠟⠁
⠀⠈⢷⡀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠙⠂⢤⣤⠤⣤⡤⠀⠒⠊⠁⠀⠀⠀⠰⠟⠛⣩⣿⠋⠀⠀⠰⠁⠀⣠⡴⠋
⠀⠀⠀⠙⠷⣤⣀⡙⢷⣄⠀⠀⠀⠀⠐⠠⠤⠤⠤⠀⠂⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⣠⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠶⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠞⠛⠿⠧⠴⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⣿⠛⠛⢉⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠊⠉⠀⠠⠊⠘⠟⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣀⢰⣿⣾⡀⠀⡄⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠡⠚⣁⣾⣿⡀⠛⠞⡁⢠⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡁⠀⠀⢿⣿⡿⠀⠀⠀⠀⣾⡇⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠈⠳⣄⠀⠘⣿⠁⠀⢀⢠⣾⠉⢇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⣧⠀⢹⠃⠀⠀⣰⠃⠀⠘⣷⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡴⠛⡇⠀⠀⠀⠐⠿⣦⡀⢰⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠃⠀⣤⠀⠀⠀⠀⢙⡾⡇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠶⠾⠉⠷⠶⠶⠛⠉⠳⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠛⢿⠟⢦⠀⣴⠟⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠸⣆⠙⢧⣄⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢸⡇⠀⢻⡀⠀⠀⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣧⠀⠸⣇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣏⠀⠀⣉⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⣿⠀⠀⢹⡄⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠀⢘⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡆⠀⠸⡇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣤⣼⠇⠀⠀⢣⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠉⠀⠀⣿⠀⠀⣠⠞⠁⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠀⠴⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠢⡄⠘⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠀⠀⠀⡀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⢉⣉⣂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢠⠀⠈⠀⢠⣶⣿⠀⠀⠀⠀⠈⣰⡘⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠈⠠⠄⠀⠈⠛⢛⡀⠀⠀⠀⠛⠻⠿⠟⠟⠋⠁⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⠀⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⣸⣿⣿⣛⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⣶⣆⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣿⣿⡀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠉⠉⠉⢠⣄⣀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣛⣅⠀⠀⠘⠿⢿⣄⡀⠀⣂⠀⣶⣿⣿⣿⣿⣿⡆⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⢸⣿⣿⣿⠀⢀⣀⣀⣄⠉⣿⣼⣿⠀⣿⣿⣿⣽⣿⣿⡇⢸⠻⠇
⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠀⢻⣿⣿⣿⠿⠟⠃⣀⡀⣠⣄
⠀⠀⠀⠀⣠⠀⠠⠾⣿⣿⣿⠿⣿⣿⢿⠿⢿⣿⣿⣿⣿⠋⠈⠉⠉⠁⠀⠀⣀⣀⣤⣼⣶⡄⠰⣾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠈⠉⠛⠋⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⠀⠀⠀⠀⠀⠠⠀⠀⠠⣤⡀⢀⡀⡀
⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠛⢿⣷⡀⠀⠘⣂⡁⠀⠞⢾⣿⡧⣛⣿⠃⠂⢀
⠀⠀⠀⠀⢸⠿⡇⠀⠒⠋⠻⠀⠀⠀⣀⣠⣀⡄⠀⠀⠀⢸⣿⡡⠊⠉⠁⠙⢷⡄⠀⠙⢄⠐⡀⠀⠛⠃⢩⡿⠂⠀⢀⡀⠀⠀⢠
⠀⠀⠀⠀⠨⠝⠀⢀⡄⢀⣠⠀⣴⣿⣿⡿⠋⠙⣦⠀⠀⠘⣿⣧⢠⡆⠀⠛⠈⣿⣄⠀⠀⢀⣀⠀⠰⠀⠀⠁⡒⠂⠉⠁⢁⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡯⠒⠀⠈⠙⠀⠀⠀⢿⣷⣄⠀⢠⣴⠀⠻⣿⣧⣄⠘⡋⠀⠀⠑⠀⠀⠐⠀⠾⠂⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⣿⣿⣿⠃⠀⠴⠶⢀⣇⠀⠀⢸⣿⡟⣿⡄⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⣸⣿⡿⢻⣷⠀⠀⢠⣾⠃⠀⠀⠈⡿⡿⠛⠀⠀⠀⠀⠀⣼⣿⠁⠀⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⢀⠀⡀⠀⠄⠀⠀⠀⣾⢻⣤⣿⠃⠀⠀⡈⠛⠒⢀⣀⣀⣡⣤⣤⣴⣶⣶⣶⡆⢻⡟⠖⠲⢦⡤⡬⣷⣶⣶⣶⣶⡶⡴⠛⢻⣿⣿
⣴⣾⣿⣿⣿⣿⣷⡀⠀⠀⠙⠾⣿⣷⣷⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠑⢐⣿⣿⣿⣿⣿⣿⠇⠄⣀⢈⠛
⣿⣿⣿⣿⣿⣯⡿⣷⠀⠰⣶⣿⣿⣿⣿⣿⡿⠋⠉⠁⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠒⠀⣼⣿⣿⣿⣿⣿⣿⠁
⣿⣿⠃⠘⠯⠉⠴⠋⠀⠀⣹⣿⣿⣿⣿⣿⠇⣄⠀⡀⠀⠀⠀⣼⣿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠐⠐⢾⣿⣿⣿⣿⣿⣿⣷⡄
⠠⣽⠀⠀⠀⢀⠀⠀⡀⢠⣼⣿⣿⣿⣿⣿⠀⠉⠀⠉⠀⢀⣴⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠉⠙⠛⢿⣿⠿⠿⠆⠀⠀⠘
⠀⠘⠳⣄⠉⠉⠁⠘⣶⣿⣿⣿⣿⣿⣿⣿⠠⡄⠄⠀⠀⠀⢁⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⡀⠀⠔⠄⡆⠃
⠀⠀⠀⢠⣽⣤⡆⠀⠉⠙⠛⠛⠿⢿⣿⣿⣦⠀⠀⠀⠀⠐⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⢧⡀⡀⠀⠀⢀
⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⢉⣩⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣴⡄⣿⣿⡦⠶⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠃⠀⠈⠉⠡⢀⣿⣿⣿⣿⣆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡇⠀⠁⠀⠀⠐⢀⣃⣽⠿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠒⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠋⠁⠀⠀⠀⢸⣽⠛⠣⠎⠀⠀⠈⠉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⣶⡖⡉⠀⠙⠂⢴⣄⣠⣦⠀⠐⢶⣤⣦⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢤⠀⠠⠀⠓⡸⣿⣷⣿⡀⠃⠼⢡⠄⢀⠀⠻⠷⠤
⠀⠀⠀⠀⢀⣠⣶⡿⠃⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣮⣍⠉⠀⠀⠀⠀⠀⠁⠀⡀⡀
⠀⠀⠀⢾⡿⠟⠉⠀⠀⠀⠀⠄⠤⢄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠠⠀⠀⠀⠀⠀⠈⠁⠉⠈⠁
⠀⠀⠀⠄⠀⠀⠀⡅⠐⠀⣀⠈⠑⠚⠳⠄⠀⠀⢀⠔⠀⠀⠀⠤⣄⢠⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢴⣶⣤⣄⣙⠇⠀⠈⠉⠁⠀⠀⠀⠂⠀⣬⣀⠁⠠⠀⠈⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠂⠆⠀⠈
⠀⠀⠀⠙⠿⠋⢀⣤⠑⣶⠀⢀⠀⠓⡠⠀⡺⡄⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠐⠐⠐⠤⠀⠀⠂⢀⣾⠆
⠀⠀⠀⠀⠀⠂⢻⣬⡴⢟⠀⠈⠳⠈⡀⠴⠶⠺⠂⠒⠓⠁⠀⠀⠀⠀⠀⠐⡀⢉⣤⢤⣀⠀⠢⢟⣿⡇
⠀⠀⠀⠀⠀⠠⠒⠀⢔⣉⠊⠀⠀⠀⠀⢲⠀⠁⠀⠀⠀⠀⠀⢀⠀⢀⠀⡄⡁⠈⠁⠀⡇⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⣀⢠⣶⢰⡦⠀⠀⠀⠀⠀⣐⣿⠀⠀⠀⠀⠀⠀⢉⢈⠀⠐⠃⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠠⠑⢿⡷⣤⠄⠀⠀⢀⣀⣤⠰⠾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠐⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢚⠁⠀⠀⠀⠀⠹⠁⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⣾⣶⣮⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠰⢷⡀⠹⠶⠈⠐⠄⠀⢄
⠀⠀⠈⠀⠀⠙⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⣨⡅
⠀⠀⠀⠀⠀⠁⢀⠀⠀⢶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀⠀⢞⠀⠋⣀⠼⡄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⢠⣿⣦⡷⣄⣀⠈⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠛⠛⠘⠛⡉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⣢⡤⠄⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢌⣻⣉⣠⠎⢹⡯⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⣾⣆⠘⣏⣿⡏⠀⣎⠀⠀⣡⠙⠳⣤⡁⠐⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠉⠙⠣⣿⡿⠁⠀⠙⠷⠾⠃⠀⢸⣿⡇⣠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢀⣾⡧⠀⠀⠀⠀⢠⣀⡼⢿⢏⣾⠋⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⠀⢀⣿⡏⠉⠁⠀⠀⠀⠀⠀⢀⣯⣿⣇⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠀⠀⠀⠀⠐⣿⣿⣷⡄⠀⠀⠀⠂⢠⢡⣾⠏⣿⣿⡎⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠲⣿⣿⣾⣷⠀⣴⣿⣿⣿⡿⠶⠘⠓⠛⢲⣾⡟⠐⣿⡯⠷⠒⠚⠚⡋⡽
⠀⠀⠀⠀⠀⠈⠀⢀⢀⣊⠉⠛⠛⠛⠛⠿⣿⣿⣿⣶⣶⣖⣰⣶⣿⡷⠞⠋⠀⠀⠠⡀⢕⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡄⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⣿⣿⣤⣀⣠⡀⣨⢤⣿⣿⠿⠋⠀⣸⣿⣤⣶⣶⣼⣿⣏
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⢧⣙⢿⣿⣿⣿⠃⠀⢹⢃⢀⠀⣰⣿⣿⣿⡿⡛⠉⠉⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⣆⢲⡿⠿⢿⣷⣦⣇⣎⣎⣾⣿⣿⣿⣿⡿⠁⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠈⢡⣽⠶⠦⠔⠚⢿⣏⣛⢳⢽⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡃⢰⡖⠁
⠀⠀⠀⠀⠀⠀⠈⠹⠶⣶⣶⡶⣾⡟⣡⣿⢬⠙⣻⣽⠛⡫⢛⡻⣻⣿⣿⠇⠀⠀⢀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠹⠻⣷⣾⣿⡿⠓⣿⢛⠎⠻⠟⢓⣟⠳⢛⢻⣿⠀⠀⠀⢸⡿⠊⢓⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣓⠀⢉⣉⠄⠟⣳⣿⣿⣼⡦⠿⠿⠿⠤⠦⢾⡇⠀⠀⢀⡿⠽⠳⠊⠑
⠀⠀⠀⠀⠀⢀⠀⠘⠂⠠⠀⣊⣵⠭⠟⠉⠀⠀⠁⠚⠿⣤⣄⡀⠀⣸⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠐⠦⠀⠄⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⢯⡀⢠⣯⠤⠠⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣿⣿⡏⠁⠨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣶⣴⣿⣿⣿⣿⣷⣾⣿⣿⣿⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⢹⣿⡟⠛⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠁⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡌⠀⢡⣿⠟⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⣶⡏⠀⠉⠉⠉⠉⠉⠉⣿⠃⢷⠉⣀⣈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡍⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣾⣿⣿⣿⣆
⠀⠀⠀⠀⣿⡆⠀⠐⠀⠀⠀⠀⢸⣿⠉⠉⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠻⣿⣿⡉⠉⠀⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠶⠶⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠉⠁⢠⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣿⢗
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⠀⠀⡈⣹⣆⠀⠙⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢈⣣⣐⣻⡿⠷⣶⣆⠀⢀⣀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣷⡄⢀
⠀⠀⠀⠀⣿⡇⠀⠀⠼⣿⣿⣿⣿⣿⣿⣾⣵⠞⠋⠙⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠨⣿⡗⠀⢸⣿⡇⠘⠘⣆
⠀⠀⠀⠀⣿⡇⠀⠀⢠⣿⡏⣿⣿⣿⣿⡿⠁⣴⣿⣿⣦⡈⢷⣄⠁⠈⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⢿⣧
⠀⠀⠀⠀⠿⠇⠀⠀⠀⠉⠀⠙⠉⢫⠈⠀⣾⣿⣿⣿⣿⣷⡀⠤⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠈⣡⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣼⣿⣿⣿⣿⣿⣿⣷⠐⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠰⠯
⠀⣤⣤⣤⣤⠀⠀⠀⠀⢰⣷⣤⣾⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶
⠀⠿⠿⢿⣿⠀⠀⠀⢺⣿⣿⣿⠇⢀⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⡀⣿⣀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠸⠿⠇⠀⠸⠿⠿⠿⠿⠀⠼⠇⠸⠿⠿⠿⠿⠿⠿⠿⠰⠇⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠁

⠀⠀⣀⠰⡿⠚⠀⠀⠀⠠⠀⠉⠁⠈⠉⠉⠨⢶⡿⠀⠁⠀⠈⠀⠀⠀⠁
⢠⣾⠋⠉⠁⠀⡀⠀⠠⠔⢂⠀⠀⠀⠼⠃⠰⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢺⡧⠀⣀⡄
⠈⠀⠀⠀⠀⠺⣏⠔⠀⠀⠀⡀⠀⣄⠀⠤⠀⠀⠀⠀⠀⠀⠙⠀⡁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⡱⠂⢠⣴⠀⣭⠅
⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠈⠉⠒⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢀⡷⠀⣩
⠀⠀⠀⠀⠚⢶⠈⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⢠⣀⣤⣶⠀⠉
⠀⠀⠀⠀⠀⠉⠀⠆⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠐⠊⠁⢈⣠⠀⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣵⣦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠈⠀⡄⠂⠈⢀⠀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡿⠟⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠈⢈⠀⠄⠆
⠄⠀⠤⢀⠠⠍⠀⠂⠀⠀⠀⠀⠀⠀⠙⢾⣧⡀⣰⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠘⢈⠀⡄⠄
⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠉⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡙⠁⠀⠀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠐⠛⠀⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣧⢰⠆⣖⠀⣿⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⡀⠀⠀⠀⠀⣤⡄⠰⠚⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠘⢿⣿⣿⣿⣿⡇⡀⢳⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣠⣤⠀⠀⠀⠀⠀⡤⠄⢰⣲⣀⠧⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⢘⡟⠛⡿⠿⠇⢀⢸⠁⢸⢿⣇⡀⠀⠀⠀⠀⠠⠸⠄⠀⠀⠀⠀⠛⠀⠘⢻⣿⠃⠀⡠⠄⠒⣿⡍⢩⠥⠀⠖⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠞⠷⡟⢺⠀⣈⡁⣠⣼⡤⠀⠀⠀⠀⠀⠀⢀⠥⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡟⣋⣤⣤⣤⡿⠗⠐⢲⠒⢶⣴⠒⠓
⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⣿⡁⢿⡇⣥⠀⠀⣠⡀⠀⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⣠⡘⣿⣿⣯⣅⢺⡁⣿⡇⢸⣿⠀⣿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⣿⣿⠃⠛⣶⠘⣧⠙⢀⠀⠃⠀⠀⠀⠘⠂⠐⠀⢻⠟⠛⡟⠋⠛⠛⠘⠁⠛⠂⠀⠟⠀⠋⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⡀⡁⠱⡀⠈⠋⠀⠠⡿⢋⣿⡗⠉⠀⠈⠁⠀⠀⠁⠈⠁⠀⠀⠀⠀⡴⠶⢆⠉⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢋⢎⢟⢄⣌⡄⠙⡄⠀⠀⠈⠁⠚⠀⠛⠠⡶⣶⢶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠂⢴⣬⣤⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣧⢊⢾⡎⠀⠑⢜⢌⡪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢃⡏⢬⣾⣀⡴⠀⠊⠹⠅⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⡾⣨⢊⢷⣧⡄⠀⠀⢡⡂⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣤⢿⡣⠁⡾⠁⠀⠀⠀⠀⠈⠿⣹⡵⣤⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡴⠉⡆⣱⡇⠀⠀⠀⠀⠀⠀⠀⢀⡉⢳⡹⡝⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠖⣶⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢣⡕⠿⠃⢸⡥
⠀⠀⠀⠀⠀⠀⠀⠠⠋⢸⡇⣶⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⢶⡘⣗⢕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠘⠏⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠾⠃⠀⠀⠐⠆⣀⣀⠶⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡎⠀⠀⠀⡄⢀⡴⠀⠀⢀⠀⢠⡲⢦⣌⣶⠛⢖⠀⡖⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠹⣿⡁⠀⠀⣥⣟⣶⣿⣆⠀⠀⠸⡧⣀⣿⣛⡿⣟⠴⠛⣥⠧⡽⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢹⣿⣧⠀⢡⡜⣿⢿⣿⡏⠀⠀⠀⣷⡼⠎⠉⠣⢰⣂⡈⠀⠘⠧⡽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠋⠰⠚⣽⣷⣄⠙⠈⠙⠛⠛⠀⣀⠴⠟⠟⠻⡛⠷⣌⢈⠀⡀⠘⡧⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠞⡡⠟⠏⠑⡶⠦⠘⠃⠉⠀⠀⠀⠀⠀⠘⠛⠻⣌⢣⠩⠒⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠟⠿⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠻⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠃⠠⠀⠴⠀⠀⠀⠀⢀⠀⢠⠀⠀⢿⡀⠀⠀⣀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠣⠀⠐⠋⠀⠀⠀⢞⠛⠬⠽⡎⠁⠀⠀⠀⠀⢺⠅⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠠⠆⠀⠈⠁⢀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢤⣤⣤⢤⣤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠁⠀⠀⡰⠛⡘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣼⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⣁⠀⠃⢸⢦⡀⠀⠀⠀⠀⠀⢦⣄⣀⠀⠀⠀⣀⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢶⠹⡟⣭⣿⠀⠉⠀⠈⠲⣤⡀⠀⡀⠀⠛⠟⠉⠉⠉⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣧⣀⣆⢻⣮⠁⠘⣷⠄⠠⠈⠇⠘⡿⡿⣶⣦⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⢹⠈⠙⠗⣛⣑⠎⢰⠶⠀⠀⠀⠀⠀⠉⠉⠓⠂⠰⢶⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠐⠄⡁⠈⢱⡍⢀⡀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠘⠁⡞⣤⣇⣁⣅⣀⠁⣀⠈⠡⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢐⣈⡅⢱⠅⣨⠀⠓⠄⠉⣐⠆⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣁⡻⠀⣤⠠⠖⢁⠽⣏⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⢡⠿⠚⠉⠉⠉⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢃⣴⠞⠁⢠⣄⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢇⠀⠓⣆⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⡀⠀⠀⠀⢰⡟⠀⠀⠀⠈⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠖⠃⢠⠜⠻⠳⣄⣾⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡴⠁⣄⢳⠃⢱⣽⠊⡴⠉⢹⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣾⡂⠀⠀⠀⠀⡀⠀⠀⣦⠊⣀⣠⠟⠋⠀⠈⡿⠀⣧⡶⠘⠀⣌⠀⠀⢀⡤⠒⣄⡴⠊⣫⠀⠈⣇
⠀⠀⠀⠀⠀⠀⡰⠃⠀⣰⢯⣒⣶⣏⠉⠁⠠⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣿⠀⠉⠷⡆⣠⠋⠀⡴⠋⠀⠰⡭⢖⠫⡉⠀⠀⠸⡄⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣰⠁⠀⢠⣿⠋⠉⠀⠈⣧⢀⠀⠀⠀⠀⠀⠀⢰⠒⠶⠶⠾⠿⠶⠶⠶⠿⡇⠀⠰⣇⣀⣀⣠⠽⠊⠁⢱⠀⠀⠀⢷⠀⢀⡴⠋⣥⣽⣉⡖⠲⠤⡄
⠀⠀⠀⠀⢠⠇⠀⠀⢸⢹⠃⠚⠁⠉⢹⠟⡆⠀⠀⠀⠀⢰⡿⡀⠀⠀⠀⠀⠀⠀⠀⢟⠻⢄⠀⠈⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⠀⠛⢦⣬⣉⣡⠋⠐⢀⠞⠁
⠀⠀⠀⠀⡞⠀⠀⠀⣿⠘⠦⣠⣤⠤⠞⠀⢷⠀⠀⠀⠀⢈⡇⠑⢒⠆⠀⠀⠀⠀⠰⠚⠁⠈⢆⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⡇⠀⠘⠈⠀⠉⠑⢲⠁
⠀⠀⠀⣸⠁⠀⠀⠀⡏⠀⠀⠀⠘⣆⠀⠀⢸⠀⠀⠀⢀⣾⣀⣀⣧⣀⠀⠀⠀⢀⡰⢇⡀⠀⣼⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣧⠀⠀⠀⡕⠀⠀⠘⡇
⠀⠀⠀⡟⠀⠀⠀⢀⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣼⠃⠀⢽⠁⠀⠉⠉⠉⠉⠀⢽⣿⠂⠈⢇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢳
⠀⠀⢠⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡏⠀⠀⠀⡇⠀⢠⣏⣈⡉⡿⠀⠀⠀⠀⠀⠀⠀⠞⠉⠷⣿⣾⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⢸⠀⠀⠀⠀⢸⡆
⠀⠀⢸⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⣸⠤⠤⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠹⠻⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠁⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⠀⠀⠀⣷⠀⡿⠤⢄⣸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠶⠒⠒⠚⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛

⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⡇⢠⡀⠀⠀⠀⠀⠀⠠⠤⠄⢀⡄⡄⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⡇
⠀⠀⢹⡎⣠⣤⣿⣒⡀⢠⣶⣿⣷⣆⣣⠇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⣷⢀⣤⣽⣦⣀⠀⣠⣀⣤⣤⡘⣼⠁
⠀⠀⢸⠋⠻⠿⢿⠟⣡⡌⠻⢿⣿⡏⠙⡆⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡟⠙⢿⠿⡿⢋⣤⡙⠿⣿⣿⠉⢻
⠀⠀⠈⢳⣦⡄⠀⠰⠿⠿⠀⠀⣭⢤⠊⠁⢰⡇⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⡇⠀⠙⢶⣤⡀⠀⠸⠿⠇⠀⢈⡥⣤⠊
⠀⠀⠀⢸⠈⢿⣦⡀⠀⠠⢠⣴⡿⢸⠀⠀⢸⡇⠀⢧⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⠀⡇⠀⠀⢰⢹⣧⣤⣖⣂⢰⢠⣼⡇⡅
⠀⠀⠀⠈⢦⠈⠈⠘⠉⠋⠉⠉⢀⡼⠀⠀⠀⣧⡀⢸⠟⠀⠈⣉⣉⣉⡛⠒⢄⠀⠀⢔⡚⢋⣉⣉⡉⠀⠘⣧⡏⢸⡇⠀⠀⠸⣄⠉⠉⠉⠁⠈⠉⠙⢁⠇
⠀⠀⠀⠀⠀⠑⢄⣀⠀⢀⣀⡴⠋⠀⠀⠀⠀⢻⣿⡇⢠⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣦⠈⣇⣾⠁⠀⠀⠀⠈⠢⣀⡀⠀⢀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠇⢀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⠁⠈⠷⣭⣿⣿⣿⠟⠁⣠⣿⣧⠈⠛⠿⣿⣿⣿⡵⢋⠀⠉⢻⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠙⢾⡀⢠⠤⠀⠀⠀⠁⠀⢠⣿⣿⣿⣇⠀⠀⠒⠒⠒⠚⢁⠀⣠⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢹⡟⣶⣶⡄⠀⠀⠀⠚⠿⠏⠿⠏⠀⠀⠀⠀⣴⡖⢻⡞⠁⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⣇⢠⠄⠀⠤⠀⠀⠀⠀⠀⠄⢠⡄⡆⠀⠀⠀⠘⡇⠈⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡇⢩⡇⠀⠀⠀⠸⡀⣄⢀⡤⠄⠀⠀⠀⠤⠄⠀⣴⠰⡇
⠀⠀⢹⡇⣾⣿⣿⣷⠄⢰⣿⣿⣿⣦⣵⠃⠀⠀⠀⠀⡇⠀⠹⣿⠾⣭⣍⠉⠉⠉⠙⢚⣉⣩⡵⣿⡿⠇⢸⡇⠀⠀⠀⠀⢿⣁⣴⣾⣿⣦⠀⣼⣿⣿⣶⣌⣾
⠀⠀⢸⡁⠙⠿⠿⠋⣴⣆⠙⠻⠿⠃⠉⠇⠀⠀⠀⠀⣧⢠⠀⠘⠙⠲⠬⣹⣙⣙⢙⣉⡩⠷⠋⠹⠀⠀⢸⡇⠀⠀⠀⠀⣟⠘⠻⠿⠟⢃⣶⡌⠻⠿⡿⠃⢹
⠀⠀⠀⢱⢷⣆⠀⠐⠛⠛⠀⠀⣴⢲⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠶⣴⡄⠀⠺⠛⠇⠀⠀⡆⡶⠁
⠀⠀⠀⢸⠀⣿⣶⣄⣀⣐⣠⣾⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠅⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⢻⣧⣦⣬⣬⣴⣀⣴⡇
⠀⠀⠀⠘⣄⠘⠛⠻⠛⠟⠿⠛⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣞⣀⣀⣀⣠⣏⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠻⢻⠿⠿⢿⠿⠟⠃⡇
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⣤⠊
⠀⠀⠀⠀⠀⠀⠙⠚⠒⠒⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣦⣍⣤⣮⣩⣯⣥⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⣤⡄⠀⠀⠸⣩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠏
⠀⠀⠀⠀⠘⠞⠁⠀⠀⠀⠀⣀⣴⣶⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⣠⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢾⠇⠀⣾⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⣀⣙⠛⠿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⣀⣈⠙⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⢈⠉⡉⢀⠀⢈⣉⡉⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⢴⡦⠀⡀⠠⣶⠬⠙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⠈⠉⠁⢸⠀⠈⠉⠁⢠⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣄⠀⠀⠤⠤⠄⠀⢀⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠀⠀⠤⠤⠄⠀⢀⣼⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢹⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⠐⠀⠀⠀⠀⠀⡀⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠃⠀⡀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠄⠈⠀⡀⠈⠀⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀⠀⠐⢁⣤⣄⣤⠈⠓⠁⢀⠠⠀⠠⠀⠀⠀⢀⢂⠁⡀⠐⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⢀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⢊⠀⠀⠀⠀⢺⠛⠙⠛⣫⠆⠐⠂⢤⠤⠤⢤⣤⠤⠈⠀⠠⠄⠉⢠⠀⡁⢀⠌⠂⢀⣀⠀⠁⠀⠀⣀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠊⠙⠀⠀⡤⠢⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠈⠀⡀⠒⠀⢀⠦⡈⢀⡔⢠⡶⠄⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠁⠀⠀⠀⠀⠀⠌⠁⠀⠠⠂⠁⢀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠠⠀⠄⢨⠈⣡⡄⠴⠛⠉⠀⠀⢀⣴⣆⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠄⠂⣠⠤⣾⣿⣷⠀⠀⠀⠀⠂⠁⠐⢀⠀⠁⠀⣈⠀⣤⣴⣄⣀⡠⠋⡶⠳⣆⠑⢀⠶⠐⠉⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⠀⠄⠉⣀
⠀⠀⠀⠀⠐⠁⠀⡻⢿⣶⣿⡿⠁⠈⠐⠊⠄⡀⠲⣤⡀⠀⠈⢂⡨⠀⠚⠛⠉⠄⡀⢀⡤⠒⠋⢁⠀⠠⠀⠀⡀⠀⠀⠠⠚⠟⠛⠟⠿⠁⠈⠱⣤⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠿⠋⠄⠂⠐⣤⣀⠀⠁⢀⣹⡟⠂⠈⠜⠀⢀⠀⠀⠠⠀⠈⠣⡠⠐⠀⢀⠂⣐⠤⠔⠀⠒⠋⢀⠀⠌⠠⠀⡂⠀⢠⣼⣿⣿⡗⠈⢁
⠀⠀⠀⠀⠐⠄⠀⣤⠛⣀⠒⠐⣀⠂⠀⠀⠐⢀⠠⠿⠿⠛⠟⠀⡎⠁⠀⠄⠀⠈⠁⠀⢠⠀⠀⠈⡄⠁⠤⠂⠀⠀⠀⢀⡰⠁⠀⢀⠁⠀⠀⢀⢿⣯⡇⠀⠀⡱
⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠟⢚⠋⠁⠀⠀⢀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢀⠀⠀⠀⠀⣜⠀⠐⢀⡄⠀⠀⠐⠈⠀⠡⣔⠉⠕⠂⠀⠀⠀⠖⢦⡘⣿⣿⠀⢀⠣
⠀⠀⢀⠂⠀⠀⣠⣀⣀⣉⣀⣀⡀⠀⠠⠊⢀⣠⠈⠂⠀⠀⠉⠶⢉⡀⠀⠀⠀⠀⢀⣜⠁⠀⢀⡀⠧⠀⠤⣀⠐⠤⠐⠃⠤⠄⠂⠀⡴⠁⠀⠀⠳⡟⠁⢀⡎
⠀⠀⠀⠀⠀⢰⣿⣿⡟⠉⢁⠀⠄⠙⢧⠀⡃⣀⠇⢄⠓⣶⣾⡏⠁⣽⠁⣶⣶⣾⣿⠟⠂⠀⠁⠀⠀⠄⢠⣃⠁⢀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠈⡢⣞⠀⢊
⠀⠀⠀⠄⢰⣿⣿⣟⠀⠀⡈⠐⣈⢤⠘⣷⡄⡄⠔⠊⢡⠈⠉⠁⢀⢀⣀⠀⣆⣹⠩⠉⠀⠒⠖⠀⠀⡆⠀⠈⢳⣼⡀⠀⢀⠤⠃⠀⠀⠔⠀⠀⠀⠀⠀⠈⢀⠜
⠀⠀⠂⢠⠊⠁⠉⢻⣄⠠⠃⠉⢐⠀⠴⠌⢉⠓⢲⠒⠋⢀⡄⡪⠑⠉⢀⢀⣀⣉⣱⣒⡂⠠⠀⠀⠀⠿⡀⠠⠈⠉⠉⡚⠛⠆⠀⠐⠡⢀⠂⠀⠀⠀⠄⠀⠀⢀
⠀⠘⠀⢸⠀⠘⢷⣤⠁⠀⠀⢴⣮⠍⢑⠤⢀⣀⣈⣀⣴⣮⠞⠀⠲⠊⠀⠀⠐⠀⠠⠠⠁⠀⠓⣌⠀⠀⠀⣁⡎⠒⠂⠠⠀⠀⠀⣠⠆⠀⢌⢀⠀⠌⠀⠀⠀⢀
⠀⠀⠁⠀⢳⡀⠀⠹⠖⣦⡀⠈⠱⠄⡠⠴⣏⢀⣉⣰⣀⣀⢀⣀⠀⠀⠀⢠⣄⢠⠀⠀⠀⡀⢀⡿⠛⠁⠉⠀⠀⠀⠀⠀⠠⠐⢁⠘⠀⠀⠸⠋⠑⠀⠤⡄⠀⠀⠁
⠀⠀⠀⠀⣾⣿⣷⣶⢁⣧⣶⠀⠀⠀⠀⠀⠀⢿⣁⢠⣀⡈⠡⠘⣿⡆⠀⢺⡯⣿⡆⠀⠀⢁⠙⣦⣁⠀⠀⠀⠠⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⡆⢀⡀⡰⠀⠁
⠀⠀⠆⡼⠉⠛⠻⢏⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠹⣄⠁⠻⢀⠃⣿⡇⠀⠁⢀⠒⢱⠎⠈⠉⠚⠍⡙⢷⣄⣠⠤⠂⠘⢂⠐⡤⠀⠀⠀⠀⠀⠀⢆⡀⣄⠀⠀⠂
⠀⠀⢰⢁⠄⠀⡠⢁⠘⠻⠿⡿⣄⠀⡀⠀⠀⠀⠀⠘⣧⠀⠱⢾⠟⠁⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⡰⠢⠯⢀⡀⠀⢠⠅⠚⡇⠀⠀⠒⠂⢠⢠⣾⣿⣾
⠀⠀⠈⠘⠆⢠⠅⣁⠰⠀⠀⠂⠍⠃⠀⠤⠂⠀⣀⣔⣀⢀⡤⠀⢄⣥⠀⠀⠀⡌⠰⡼⣖⡠⡂⢠⡋⣊⣍⠀⠀⠁⠀⢀⠀⠇⠀⠀⠀⠀⠸⠸⠟⠛⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⢀⣀⠀⠀⠈⠀⠀⠙⠣⢉⣉⣀⣵⠀⠎⠈⠉⠀⡘⠀⠑⠵⠿⠇⠀⣾⠀⣖⣘⡧⠀⠂⠀⠤⠄⠁⣀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠙⠀⡇⠀⠀⠀⢸⣿⣿⠃⠀⢀⡁⠀⠀⢨⡤⡁⡈⠉⢹⡿⠛⢛⠃⠀⠀⠂⠀⠀⣸⠛⢰⠶⠬⣘⢧⡀⠀⠀⠀⠄⢠⣤⣤⣄⣡⠄
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠙⠷⠈⢡⢀⠊⠙⡘⠁⠊⠁⢿⠀⢸⢧⠐⡗⠄⣀⠈⠀⠀⢰⣇⣭⣤⣦⠺⠿⢊⡛⢷⡁⠀⠀⢀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠤⣻⡘⡀⠐⠀⠁⡺⢏⠱⠀⠀⠛⠛⠀⢤⣌⣀⣢⣿⣇⢿⣿⣭⡟⠐⠀⠤⢀⡿⠒⠀⠿⠟⠁⠠⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⢄⢡⠠⠾⠄⠈⠒⢄⣉⣁⠀⡠⠾⣿⡿⠿⣿⣿⠷⢭⣭⣷⡀⠀⠛⠋⠀⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠂⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⢂⢠⠀⠈⢚⡁⠈⠉⠀⠄⠈⠙⠛⠳⠂⠀⠂⢀⠠⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢀⠔⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣾⣧⠀⣴⠂⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⢿⣷⠉⣻⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⣿⣧⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠷⠀⠀⠐⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠸⠛⠛⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠙⠛⠋⠁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠛⠛⠛⡉⠉⠁⠀⠀⣀⣀⣤⣴⣶⣿⣡⠀⠀⠀⣰⠀⡀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⢠⡄⠀⣠⠀⠀⠀⠀⣠⠀⣠⣤⣤⣷⣴⣶⣾⣿⣿⣿⣿⣿⡟⠡⣿⡄⡂⠀⣿⡂⠠⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⢀⣸⡄⡠⠲⢾⡇⡀⠀⢀⡷⢤⡴⠒⠛⡦⠌⣿⣿⡿⠻⠙⠟⠛⡿⢏⢁⠀⠘⠿⠇⣈⢰⣿⣇⠲⠇⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⡿⠛⠋⠀⣠⠚⠉⠒⠣⠀⠂⠀⢠⣆⢁⠔⠀⠀⠀⠀⢄⠈⢰⡆⠀⠁⠋⠳⠴⢄⣆⡀⠁⠉⠑⣄⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠾⠋⢀⠤⠀⠀⢊⠉⠁⣷⢠⠀⢢⠤⠀⠀⠀⠀⡀⠠⠒⠉⠀⢠⡆⣼⣇⠀⠐⠂⠀⠠⠐⠩⠙⡄⠀⠀⡇
⠀⠀⡀⠀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠡⠿⠜⠂⢀⡀⠀⠀⠀⢰⡇⣀⣐⡀⠀⣼⣧⡿⠿⠖⠒⢂⠀⠀⠀⠠⠀⡇⠀⠀⠇
⠀⠀⠇⠠⠄⠀⠠⠄⠁⠒⠂⠀⠶⣿⣿⣽⣷⠀⠀⠉⠁⠀⠚⠃⠼⣷⡐⢉⡁⠨⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠈⠀⠈⠠⠆⠀⠆⢴⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠂⠀⠠⠤⢄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠣⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡀⢀⠤⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠠⡀⠀⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠐⠀⢀⠀⠀⡀⠀⠂⠁⠠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠁⠀⠁⠀⠀⠀⠀⠀⠈⠐⢄⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⡘⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⣂⠐⠀⡀⠂⢀⡀⠀⢀⠀⡑⠀⢀⡠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠀⢀⡀⠀⠀⣌⢻⣿⣷⡀⠀⡀⢠⠀⠀⠀⠀⢀⠀⡀⢀⣀⠔⠀⠀⠄⠁⠀⠄⠀⠠⢄⠁⠠⠀⠀⠀⠀⠁⠀⠊⠁
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢢⣀⣀⡆⠬⠈⠋⠘⢣⣌⠀⠀⠀⡎⠀⠄⠀⠀⠀⠀⠁⠀⠐⠂⠀⠐⠀⠂⠐⡠⣀⣼⣷⠄⠀⠀⠀⠀⢀⠔
⠀⠀⠀⢀⠆⠀⠀⠀⠀⢚⠣⠀⠐⡀⣀⠁⠂⠠⢌⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠠⠀⣈⠴⠟⠋⠀⠀⠀⠀⠀⠔⢁
⠀⠐⢀⠂⠀⠀⡠⠀⠈⠀⠀⢈⣁⣁⣠⣀⣀⡀⠰⢀⠀⣤⡀⡐⠀⠀⠇⠀⠈⠀⠀⣀⡀⠀⢀⠓⠤⠺⣷⣦⡴⠔⠉⠁⠀⠀⠀⠀⣠⠈⠀⡠⠆⠁⠀⡀
⠀⠀⠈⠀⠐⠰⠁⠀⠀⡤⠞⠋⠉⠉⠙⣿⣿⣿⣷⣦⡉⠀⠹⡄⠱⣦⡰⠶⣾⢀⠹⠿⠋⢁⣀⠀⡠⠔⠉⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⡆⠀⠀⠀⠐⡄
⠀⠀⠅⠀⡇⠀⢀⠀⠌⡠⠀⠛⠄⢀⠴⠿⡿⠃⠀⠙⢿⣆⠀⠘⣄⠹⠿⢂⠊⠁⠀⠀⢀⡀⠈⠣⡀⠐⠈⠀⠀⢠⠀⠂⡄⠉⠀⠸⠿⠃⠐⠀⠈
⠀⠀⠀⠀⠀⢰⣾⠄⢨⣰⠀⠄⠁⠰⡤⠀⠂⠉⠁⠀⠺⣿⡆⠒⢡⠀⣤⡆⢀⠀⠀⡀⠀⠀⠋⠀⠀⠄⠀⠀⠠⠀⠁⠈⢀⠀⠀⠀⣾⣷⡄⠀⠉
⠀⠀⢀⡀⠀⠀⢻⣄⠀⠈⠀⠀⠐⠲⠃⠤⠀⠀⣈⠁⠦⠈⠃⠀⠈⠁⠠⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠂⢠⠀⠀⠀⠁⡀⠀⠀⣸⣿⠿
⠀⠀⠎⠀⠐⠉⠈⠉⠙⠲⣾⡇⠘⠀⠐⠛⠋⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⡤⠐⠂⠐⠂⠤⠉⠀⢠⣿⣟⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠒⠀⠈⠀⠀⣤⠐⠐⠀⠀⣀⡀⠀⡀⠀⠤⠈⠄⢠⡤⠐⠖⣷⡄⠀⢴⣶⡿⠟⠀⠀⠀⠀⠐⠀⠲⠄⠀⢠⣿⣿⡁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢰⣿⣿⢿⣦⣄⠀⠀⡀⠀⠀⠀⡀⠄⠀⢳⡀⠁⠒⠀⣸⣦⠀⠁⠀⠀⠈⠀⠀⢈⢸⠈⠉⠀⠁⠆⠀⠀⡀⠀⠀⢰⠿⢿⡇⣀⠀⠁⠁⢀⠄⠰
⠀⠀⠸⠀⢸⣿⣅⣀⠈⠿⠷⠀⠆⠀⡠⠲⠄⠀⠀⣰⣏⠐⠐⠛⠻⣿⠇⠀⠠⡢⠌⠀⠀⠀⢉⣁⣖⠀⠰⡄⠀⠀⠙⠀⡀⢀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⢠⠀⣄⠙⠻⠟⠀⠀⠀⠀⠘⠀⠈⠁⢀⡀⣸⠟⠁⠀⠐⠀⠃⠀⢠⢢⡞⢀⡀⠂⠁⠈⠂⠀⠀⠀⠀⠘⢢⣀⡀⣀⠀⠻⢿⣿⣦⡀⢤⠀⠀⠄
⠀⠀⠸⡀⠸⣷⠂⠀⠀⠀⠈⠀⠀⠠⣐⠀⢪⡅⠁⠀⠀⡴⢀⠁⠤⢠⠔⠉⠁⠀⠉⠀⠀⢠⡀⣀⡀⠀⠀⠀⠘⣠⣶⠶⣦⡀⠈⢻⣿⣿⠈⠂⠆
⠀⠀⠀⠜⢀⣿⠀⠀⠀⠀⠳⠴⢂⡀⠀⠀⠛⠐⠚⠠⣶⣴⠋⢀⡤⠆⠀⠐⠔⠲⣤⡀⠀⢸⣱⣶⡀⠀⠰⠀⢠⣿⠁⢐⠄⡗⠀⢸⠿⠋
⠀⠀⠘⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠉⠃⡰⠈⢤⠀⠀⢢⡀⠀⢻⡇⠀⠊⠀⠀⠀⠀⠀⠄⠸⠿⣄⠀⠀⠀⣠⣿⣀⠂⠈⠠⠀⢀⠃
⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠈⢞⠰⠀⠀⠈⠠⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠈⠉⠀⠀⢉⣿⠄⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠹⠋⠿⢿⣶⡶⠶⣶⣶⣿⣷⣄⠀⠀⢀⠀⠀⢀⠬⣽⡇⠀⠀⣀⠨⠋⢸⠀⠀⠀⠀⡀⢀⣠⠌⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⠀⠀⠑⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⡄⠈⠀⠀⠀⠀⠀⠘⠈⠈⠁⠀⠀⠳⠒⠃⢀⠤⠋⠀⠈⠉⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠁⠀⠀⡀⠃⠰⡘⡀⠀⡀⢡⠈⣀⣠⡀⠀⢠⣄⣠⡄⠀⢠⣴⣶⣶⠞⠓⢦⣤⣤⡄⠒⠇⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠉⠉⠀⢀⠈⠁⠉⠃⠘⠀⠸⠿⢿⡇⠀⣴⡀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠠⠀⠀⠉⠀⣀⠐
⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠰⠀⠀⠀⠀⡄⠀⠁⠀⠀⠄⠀⠀⠀⠀⠐⠘⠄⠤⠀⠀⠈⠀⠙⠓⠀⠀⢂⠀⠀⡀⠀⠈⠐⠀⠀⢀⣀⣀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠊⠀⠀⠀⠀⠔⠠⠂⠂⠀⠀⠀⢀⠰⠀⠀⠀⠀⠂⢀⠦⠀⠄⠀⠀⠀⠀⠀⠀⠦⣀⠌⡀⠀⠀⠀⠀⠀⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠂⠠⠄⠊⠀⡀⠂⡀⠀⠈⣀⠆⠀⠂⠈⠀⠀⡠⢄⣀⠔⠁⠀⠀⠀⠐⠈⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣀⣀⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢴⡻⣏⢹⡿⣿⢿⠉⢉⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣶⢲⣿⣿⣦⣄⠙⠋⢀⣼⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣾⣿⣿⣿⠛⠂⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠈⠻⢿⣿⣿⣿⡿⣿⣳⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⠙⣿⣿⣦⣤⣴⡀⣩⣽⡇⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁⠡⣾⣻⣿⣿⣿⣿⣿⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⢨⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⢀⣘⡀⠴⠟⠁⠑⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡈⣸⣿⡶⢶⣿⣷⠠⣄⡀⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠐⠃⠐⠂⠀⠀⠀⠀⠀⠀⠁⢻⣿⣿⣶⣶⣿⣿⣿⣿⢷⣄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⢻⠛⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡛⢻⡟⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣀⡈⡀⠀⠀⠀⠀⣀⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣾⣿⣟⣉⠉⠀⠉⢙⡖⠁⠀⢩⣿⣾⣿⡄⠢⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠰⣾⣦⣀⠀⠀⠀⣀⠛⠛⠛⠿⠻⠷⠦⠀⠀⠼⠦⠿⠿⠿⢿⣿⣆⢦⡀⠀⠀⠀⢘⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠦⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⠀⢠⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⣉⡭⠓⠢⡤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⡊⠐⠛⣉⡔⠒⠂⠀⠠⣬⣀⠸⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⣀⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⣴⠗⢈⡩⢀⣴⠀⠀⢙⣿⡏⠳⠿⠀⠆⠀⠀⠀⠀⠀⠀⣾⠟⠛⠹⢿⡝⠿⢉⣍⡆⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣃⣴⣯⣾⠿⠿⠶⢶⣶⢾⡇⠀⠐⠈⠀⠀⠀⠀⠀⠀⣾⢿⠁⣰⠶⠶⣶⣪⣋⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣻⡽⠃⠋⠀⣀⡠⣼⣿⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⣯⡏⢰⣿⠀⠀⠈⠉⠉⡏⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣃⣤⣴⣾⢖⣲⣼⡟⠁⢺⡟⠀⠀⠀⠀⠀⠀⢀⠻⡿⡿⠉⢀⣀⡄⠀⢠⡇⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠟⢿⡈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠉⠀⠀⠚⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠻⣿⣿⡿⢿⣿⠟⠀⠈⢣⠀⠀⠀⣠⠆⠒⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠔⢉⡠⠄⢀⠤
⣀⣠⠤⠤⠐⠒⠉⡄⠁⠀⠄⠈⠀⠀⠀⠉⠀⠀⠀⠃⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠖⠰⠀⠚⣄⠠⠒⠁⠀⠠⡀
⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠂⠀⢆⠀⡑⢀⠀⠉⠁⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⣀⠔⠒⠦⠤⠐⠁⠀⡀⠤⠒⠈⠁⠀⠘⠀⠈⢆⠀⠀⢩⠋⠉⠙⠲⣶⡄⠐⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⣄⠐⠀⠘⡄⠀⠀⡠⠚⠁⠀⠀⢸⠀⢀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢂⡐⠁⠁⠄⢀⡄⢹⣿⣄⠁⠀⠀⠀⠀⡄
⠶⣲⠇⠀⣼⠀⠀⠀⠀⠘⠄⢥⠀⠲⡄⠃⠀⠇⠀⢶⡀⢸⠀⣿⣤⣄⡀⠑⢄⠀⠀⢠⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⢐⠈⣿⣿⣆⠀⠀⠀⠀⢧
⣾⣟⣰⢠⡏⠀⠀⠀⠀⠐⠄⠀⠀⠀⣇⠀⠀⠠⠀⣼⢡⡎⢠⡿⣷⡻⣛⢦⡈⢣⡀⣼⠇⢠⠂⠑⣄⠀⠁⠀⢰⠀⠀⠀⢻⢸⣿⣄⠀⠀⠀⠈⣠
⠾⣯⠟⡿⠀⠄⠀⠀⠀⠀⠈⠆⠀⠀⠸⠀⠀⠀⠴⠋⠼⠃⠸⢿⠻⢿⠌⠳⢦⠀⠃⠿⠠⠏⠀⠀⠈⢧⠀⠀⠸⠀⠀⠀⠈⠿⢿⠿⣦

⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢲⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿
⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢺⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸
⠀⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣿⣿⣶⣾⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⡀
⣼⡟⠢⡆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣮⠶⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡿⠀⣼⣳⠀⠀⠀⠀⠛⠿⠿⣿⣿⣿⣇⠀⣻⣿⣿⣄⣠⣄⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠁⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠿⠁⠀⠙⢸⣿⣶⣦⣤⣦⣤⠄
⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⠋⠁⠀⠀⠈⠙⣿⣿⣿⣶⣤⣶⣶⣧⣴⣿⣶⣿⣿⣿⡟⠉⠁
⢀⣼⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠃⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣻⣷⢷⣤⡄⠀⠀⠀⠀⠀⠀⠀⣿⠝⣿⣿⠟⠋⣿⣷⠛⠻⣿⣿⣿⣿⣇⠀⢀⣴⣶⣆⢀⣼⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢯⣻⣿⣿⠀⠀⠀⠀⠀⠀⣀⣦
⣿⠁⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⣿⡾⣟⣗⣦⣂⠘⠻⡄⠀⣸⣻⣿⡿⣿⣷⣏⣨⣿⣉⣿⡟⣹⣯⣿⣿⣿⣉⠀⡈⣈⡹⢦⣛⣿⡟⠀⠀⠀⠀⠀⢠⣽⣇
⠣⠄⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⠻⢿⣿⣿⣯⢟⣿⣷⣼⣟⣴⡿⣹⠋⠻⢿⣿⣮⢥⣿⣿⣿⡶⠿⣿⣷⣽⣿⣿⣾⣉⣷⠁⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⣥⠄⢀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣷⣿⣿⡟⣿⣾⡚⠿⠀⠉⠹⣷⣻⡿⣦⣜⣳⣾⣿⣿⠿⢋⢉⣨⣷⡮⡿⠋⢡⣿⢛⢛⣻⠄⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡄
⠏⠀⠀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠼⡿⣿⢻⠳⢻⣧⠙⣷⣴⣤⣤⣭⣿⣿⣿⣿⣿⡟⢻⣶⣯⣼⠿⣿⠛⢂⢿⣶⡛⣿⡿⣘⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⣟⡉⢁⣇⣸⣶⣾⣶⣿⡿⠿⠟⢻⣿⣿⣿⣭⣬⣍⡛⠛⢻⣿⣿⣿⣟⣻⣏⣍⣽⣋⡭⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⣸⣿⣿⠀⢀⣀⠀⠀⠀⠀⠶⣿⣶⣿⠷⠾⠛⣭⣽⠏⠁⣠⣴⡶⠂⠘⠿⢿⣿⣿⣿⣿⣤⣄⠀⠈⠉⠉⢿⣿⣟⣿⣿⣿⣗⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢠⣿⣿⡯⢠⣾⣿⠀⠀⠀⠀⠀⠾⣿⣷⣶⣶⣶⣾⢚⣀⣀⡐⠞⣠⣤⣴⣶⣾⣿⣿⣿⣿⣻⣿⣷⠘⢶⢶⠾⠯⠧⠻⣏⣹⣿⣿⣿⠯
⢸⣿⣿⣇⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣙⣿⣿⢻⣻⣟⣠⣽⣿⣿⣿⣿⣿⡿⠟⠛⠛⠉⠁⠀⠘⢂⣀⣤⣤⣤⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄
⣿⣿⣿⣽⣿⣿⠇⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠈⢻⣿⣿⢎⣿⣿⣿⣿⣿⣿⣿⢛⠁⠀⠀⣀⣶⠶⢿⣾⡷⡟⣿⣿⡯⠈⢰⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠻⡿⠁⣀⠀⠀⢸⡟⢻⣇⣿⣿⣿⣿⣿⠟⢿⣿⣿⡿⢿⣿⠏⣈⢹⣏⢰⣿⣿⡞⠁⣀⣈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣿⠂
⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠛⠀⠀⢸⣿⠐⠉⢿⣿⣡⣿⣏⢠⡌⣿⢻⠁⢾⢷⣾⣿⣯⣀⣻⣿⣿⡇⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠻⠂⠀⠀⠆⢸⡿⢀⣴⣾⣿⣿⣿⣿⣿⣇⣸⠈⠀⣼⣾⣿⣿⣿⡏⠉⢸⣿⠄⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠶⠃⢈⡛⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⠓⠀⢸⣿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢸⣿⣿⡄⠀⢸⣿⣿⣿⡟⢿⣿⣿⠀⢻⣿⣿⠏⠀⠉⠀⠀⢸⣿⠀⠈⠁⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⣿
⣿⣿⣿⣿⡇⢀⣼⣷⣦⠀⠀⠀⠀⠀⠈⠀⠀⠈⠻⠏⠀⠀⠸⣿⣿⣿⡇⠈⣿⣿⣿⣿⡿⠁⠀⠀⣀⠀⠀⢸⣿⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣏
⣿⣿⣿⣿⣷⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⣿⣿⣿⡇⠀⣿⠻⡿⠋⣸⣿⣿⣿⠁⠀⠀⠸⠃⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠸⣿⣿⣿⡇⠀⢿⡇⠀⠀⠸⠟⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠉⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⣿⣿⣷⣄⠀⠀⠙⣆⢷⡜⠓⠒⣏⣴⠨⢙⡪⠿⣿⣵⣿⣿⣽⡽⣿⢉⣿⣯⣭⡝⠃⠀⠈⡿⠁⣸⠀⠹⣷⠈⠻⠇⠹⠋⢿⡴⣆⣼⠟⠈⠴⡿⠁⠈⡛
⣿⣿⣿⣿⣿⣿⣯⣄⡀⠿⠛⠀⠀⠷⢦⠀⢹⡦⣺⡿⣥⣿⣿⣾⡿⡿⣟⣋⡯⣥⣰⠐⠒⢴⠆⠀⠤⢠⡁⠀⠀⠀⠀⠀⠘⣿⠏⠀⠰⡄⠈⠆⢀⡰⢯
⣿⣿⣿⣿⣿⣟⠻⢿⣍⡉⣶⣄⢀⠀⠈⠀⡘⠀⣡⢐⣦⣶⣿⣿⢧⣼⣬⣛⡹⠫⠍⠀⣀⣾⣷⡳⣴⡆⠁⠀⡀⠀⠀⠀⠀⠻⠀⢀⡤⠀⠹⡞⠉⣠⣮
⣿⣿⠟⠛⠛⣛⡷⢻⣿⠃⣛⢋⠟⠉⠀⠀⠀⠀⣿⠛⠳⠄⠈⠉⠀⠀⠈⠉⠉⡀⠀⠘⠯⠉⢹⡿⡹⢣⠄⣀⠋⠤⢀⠈⠓⠂⠄⠀⠀⣠⣶⡿⠞⠛⠁
⣿⣿⣿⣿⣿⣿⣀⣼⣿⡾⠿⣛⣤⣔⣎⠀⠴⠷⠖⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠟⠀⠀⣴⠂⠘⠃⠃⠀⣴⡄⢀⡀⠈⠢⢤⣦⡀⢠⠀⢀⡟⠀⠃⠤⢴
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣞⣛⡱⡄⡿⠛⠉⠁⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠞⢓⠦⡆⢈⣏⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⡏⢁⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⣶⢾⢚⣿⣼⡇⠀⠀⠀⢴⠆⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣾⣧⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⡿⠙⡇⣶⡾⣻⠿⣞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣼⡗⠀⠀⠀⠀⠘⢉⠁⠀⠁⠀⠀⠀⣀⠀⠉⠀⣀⠲⠚⠛⠘⣷⠐⠉⣹⠛⠀⢀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠵⠂⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⣤⠀⠘⠃⠄⠀⠀⠡⠴⠃⡟⠀⠀⠁⠀⠀⠀⠘⠻⣆⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡶⣢⡶⠖⡋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠲⠶⠙⢦⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠙⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣴⢏⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠚⠰⠮⠁⠀⠀⣀⣀⠀⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣵⣛⠁⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠿⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣏⣾⡁⠀⠀⠀⠀⠨⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⣿⠆⠀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⢰⡹⠳⣷⡶⠚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠔⠏⢠⡶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⢘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⣐⡛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⢸⡶⣂⣒⠀⠀⠐⡯⡷⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠝⣰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠾⠿⠓⠂⣼⢊⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡟⢉⣽
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠂⣄⢀⡀⠀⢿⠄⠙⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⠾⢸⡇⣾⢿⣻⣅⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠳⠶⣼⣿⣻⣤⣽⢿⠓⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢝⡢⠿⢟⠓⣡⣯⣟⡉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⠀⠙⡃⢿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠲⣏⡝⠁⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠲⣄⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⣠⠏⠈⠙⠲⡄⠀⠀⠻⠏
⠀⠀⠀⠀⢠⠇⣀⡤⠖⠋⠁⠀⠀⣀⣀⣼⡀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠴⠟⢤⡀
⠀⠀⠀⠀⢈⠟⠁⢀⣀⣤⡴⠞⠋⠉⠀⡀⣧⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠻⠻⠟⠛⠛⠛⠛⠛⢿⣦⠀⣫⡀⠀⣠⢷⡀
⠀⠀⠀⠀⢸⡖⠚⠉⠈⠁⠀⠀⠀⠀⣀⣡⡟⠸⣿⡄⠀⠀⠀⠀⠀⠀⣤⠀⠀⢴⠀⠀⠀⠀⠀⡠⠛⠉⠀⠉⠉⢁⣠⠧⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠳⣄⢀⣀⡠⠤⠖⠚⢩⡁⠐⢳⠀⠈⢿⡆⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⢀⡤⠓⠒⠛⠉⠉⠉⠉⠀⠀⠈⢇⠀⠈⡿⠃
⠀⠀⠀⠀⠀⠀⢈⡽⢇⠀⠛⠀⠀⠈⣥⠷⠞⠓⠛⠛⠓⠒⠲⠤⣄⠀⠣⠈⡈⠠⠁⠀⢀⣏⠤⠤⠤⠤⠤⣀⣀⣀⣀⣀⣠⠾⣦⠴⠤⢤⣀
⠀⠀⠀⠀⠀⠀⡎⢠⢟⣧⡀⢠⣹⠟⠁⢠⠒⠒⠒⠚⠉⠉⠦⠤⣄⠙⢦⡀⠀⠀⠀⠀⡎⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣦⣄⡈⠳⣄
⠀⠀⠀⠀⠀⢸⠀⣾⣼⠀⣟⡞⢁⡤⠖⠃⠀⠰⠆⠀⠀⠗⠀⢀⠸⣆⡀⠙⣆⠀⠀⠀⡷⠂⠚⠉⠙⠋⠉⠑⠋⠉⠲⠉⠳⠲⣞⠛⠿⣿⣿⡄⠸⡄
⠀⠀⠀⠀⠀⠘⡄⢹⡽⣄⡿⠀⣸⠀⠀⠁⠀⠀⢤⣀⣠⠆⠀⠈⠀⠈⡹⠀⢻⡀⠀⠀⡇⠀⠀⠀⠀⡲⠀⠀⠀⣴⠀⠀⣠⡶⡍⠙⡆⢹⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠱⡄⠙⢼⡇⢺⡁⠀⠒⠀⠀⢀⡴⠦⣄⠀⠈⠀⠀⡀⠙⡆⠈⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢯⡉⢡⠟⠋⣩⠉⠛⢯⣼⠁
⠀⠀⠀⠀⠀⠀⣀⠈⠢⣤⡇⢀⡇⠀⠁⠀⠐⠸⣄⠀⣨⠂⢂⣀⣀⣀⡸⠥⠼⠷⠶⠶⢿⣄⣤⣀⣀⡘⠲⠤⠖⠀⠀⠹⢤⣾⠀⢀⠉⠁⠀⡀⠹⡄
⠀⠀⠀⠀⠀⠺⣿⡇⠀⠀⢳⠈⠣⡄⠐⠀⠄⠠⢌⡽⠛⢋⡩⠤⠤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣍⡉⠓⠶⡄⣠⠤⠶⠾⣘⣆⠈⠦⠤⠘⠋⠀⡷⢒⡦
⠀⠀⠀⠀⠀⠀⢀⣀⢀⢤⣈⣧⣀⡣⣄⠀⠀⡄⢸⣦⣀⠳⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢀⡠⣟⡡⠖⢲⠦⡌⢮⠙⢦⣄⣀⣠⣞⠁⢈⡹
⠀⠀⠀⠀⠀⠀⣸⠉⡍⢀⣉⠆⠀⠙⢾⡦⠴⢄⣈⡇⠈⠉⠓⠒⠒⠬⠭⠭⠭⠭⠭⠭⠭⠥⠔⠒⠒⠉⠉⠀⣏⢀⣠⡟⠀⢸⢸⡆⠀⣠⣁⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠸⣅⢄⣠⠏⠀⠀⣄⠀⠀⢳⡦⠤⠭⢿⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢰⠋⠉⠁⠀⣠⠏⡸⠁⠀⠿⠟
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⡄⠀⣧⠀⠀⠈⢇⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⢀⣮⣤⡤⠴⠚⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠙⠀⠠⠜⠁⠀⣸⠀⠀⠀⠈⢣⡀⠀⠀⠀⠈⠢⣄⣀⣀⡠⠞⠀⠀⠀⠀⣠⠾⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⢤⡤⠤⠤⠖⠁⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⠛⠉⠉⠉⠉⠛⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⢀⣠⠴⠒⠋⠉⠉⠁⠉⠉⠙⠒⠦⣄⠀⠀⠀⠀⡷⢦⡀⠀⠀⠀⠐⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡤⠞⠉⠀⠀⢀⣀⣤⠤⠤⠤⠴⠤⠤⠤⢬⣙⣦⡀⣰⡃⠀⠹⡄⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢰⡷⠋⠀⢀⡤⠖⠊⠉⠀⣀⣠⡤⠤⠶⠒⠒⠛⠛⠛⠛⠛⢧⠈⠳⡄⢹⡄⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⢠⠟⠀⡠⠚⠁⣀⠠⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠘⡄⡇⠀⠀⣦
⠀⠀⠀⠀⠀⢀⠏⡠⠋⣠⠖⠋⠁⣀⣀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠣⣀⡀⠀⠀⠀⠀⠹⡄⢰⣿⠀⠀⠁
⠀⠀⠀⠀⠀⣾⠞⡵⠋⠀⠀⣀⣬⠶⠒⠒⠒⠦⣄⠀⠀⠀⠀⣀⠤⠤⠬⢭⣕⡀⠀⠀⢷⢸⡇
⠀⠀⠀⠀⣸⣿⠞⡇⠀⠀⠀⠻⡏⠀⢠⣿⣿⣤⠘⡆⠀⠀⢰⠁⣶⣿⣄⡀⠘⣿⠂⠀⢸⣿⠁
⠀⠀⠀⢀⡟⢹⡀⡇⠀⠀⠀⠀⠃⣀⣈⡻⠟⠋⢰⠁⠀⠀⢸⠀⠻⠿⠟⠁⣠⠁⠀⠀⢸⢉⡇
⠀⠀⠀⠈⠁⠀⠙⢷⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠓⠋⠉⠉⠉⠁⠀⠀⠀⣼⠞
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢾⡙⠦⣄⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⣰⡏⣀⠴⢋⡷
⢀⣀⡠⠤⠤⠝⠂⡀⠙⢦⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⣠⣼⡿⠋⠀⠐⠯⠤⠤⢤⣄
⣞⡭⠭⠭⠀⠀⠈⠉⢳⢸⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢺⠟⠙⡇⡞⠉⠁⠀⠀⠭⠭⣝
⢣⡤⠤⠤⠴⠶⠖⣶⠋⠘⣇⠀⠀⠀⡰⢫⡿⡓⠒⢒⣶⡒⠒⣿⠙⣄⠘⠮⠷⣾⠁⠑⣶⠶⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠘⠶⠶⠊⠁⣸⠁⠈⠒⠻⣤⠋⠒⠘⡇⠈⠓⠶⠟⠃⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⡤⠞⢹⠀⠀⠀⠀⠀⠀⠀⠀⢽⠢⢤⣀⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣄⣀⣀⣀⣀⣤⡤⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣞⠦⣤⣤⣤⡤⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣠⠞⠘⠧⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡘⠳⣄⠀⣠⠜⢃⣹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣦⡟⢠⡼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⡏⢁⡾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⡶⢿⡿⠀⠰⠃⠀⣠⠀⠀⠀⢰⡄⠀⣀⠀⠢⠀⠀⠀⠐⣿⡷⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠝⠇⠃⠀⠀⠀⠀⢂⠀⠟⠀⠀⠠⠈⠀⠀⠟⠀⠀⠀⠀⠛⠋⢤⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣠⢸⡇⠀⠀⠀⠀⡂⠀⠃⠀⣤⣤⡀⠀⠀⠛⠛⠙⠛⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣌⣳⠀⠀⠀⣤⠶⠋⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⢨⣍⡇⠙⢷⡏⠀⠐⠀⣴⠀⠀⣀⣀⠀⠀⠐⢀⣠⡀⠈⢶⣶⣾⡷⠆⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠅⠀⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⣿⣟⢾⣿⣿⣇⠀⢀⣄⣠⣤⣤⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⣀⣠⣤⡾⣻⣿⣿⣿⣿⠏⠉⠙⠻⣶⣾⣏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣿⠃⠀⠉⠉⠉⠉⠰⣶⣤⠄⠀⠀⠙⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣴⣿⡇⠀⠀⣠⣾⠇⠀⠀⣠⣴⣖⣷⠀⠀⣀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣀⣀⠀⠀⢶⠷⠆⠉⣻⠃⠄⠀⣴⢶⣶⣧⠀⠙⠿⠿⠟⠀⠰⠟⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠹⠉⠃⠀⠁⠀⠀⠰⣿⠅⠀⠀⠙⠺⠟⠁⠀⡀⠠⠀⣀⣀⠀⠀⠀⠠⠰⠟⣡⣦
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⠠⠈⢻⣿⠀⠀⠄⠀⠀⠀⠙⢡⣾⠄
⠀⠀⠀⠀⢴⣿⡾⠀⠀⠀⠀⡀⠠⠀⠠⠀⠀⠀⠘⠛⠁⢠⠄⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣭⢳⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡀⠀⣤⡦⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢀⠀⠀⠿⣠⣌⠸⡷⠃⠀⠀⠀⡋⠁⢸⠮⠁⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠿⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠟⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠨⠏⠀⠘⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠀⣦⠈⠀⠀⠀⣀⡶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏⠀⠈⠛⠓⠛⠋⠁⢰⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡌⠁⠈⠀⠁⠈⠀⠀⠀⠀⢹⣷⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠐⠀⠀⠀⠠⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⢿⡏⠛⠿⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣀⠤⠒⠈⠁⠀⠀⠀⠀⠉⠉⠒⠤⣀⠀⣀⡾⠇⠒⠂⠀⠀⠀⠒⠢⠌⣹⣆⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠰⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠖⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠁⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠁⣠⡤⠶⠖⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠁⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣀⣤⣤⠀⠙⣦⠀⠀⠀⠀⠀⣠⠞⢉⣀⣀⠀⠈⠳⣆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⣽⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⣿⣿⣿⣤⣄⢸⡇⠀⠀⠀⢠⡏⢰⣿⣿⣯⣀⡀⠀⢈⡆⠀⠀⠀⠀⠀⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⢿⣿⣿⡿⠃⢸⠇⠀⠀⠀⠘⣇⠸⣿⣿⣿⣿⠇⠀⢨⡇⠀⠀⠀⠀⠀⣽⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣼⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠙⠆⢈⠉⣭⠡⠤⠤⠟⠀⠀⠀⠀⠀⠀⡿⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣾⠋⠻⢦⣀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣀⡴⠞⠙⣷
⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠉⠳⢦⣄⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⡴⠞⠋⢀⣤⠞⠁
⠀⣠⠶⠖⢒⡛⠛⠛⠋⢉⣀⣀⠀⠘⣧⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⣴⠟⣽⠋⠀⢀⣀⡉⠙⠛⠛⢛⣒⣶⠶⣤
⣼⢛⣉⣉⣬⣅⠀⠀⠀⠈⠉⠉⢻⡀⣞⠀⠀⠀⠉⠠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠁⠀⣿⠀⡖⠁⠉⠀⠀⠀⠀⢨⣭⣭⣁⡉
⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠃⢻⡀⠀⠀⠀⠀⠀⢙⡷⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡞⠋⠁⠀⠀⠀⢀⡿⠈⢧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠘⠷⠶⠚⠛⠛⠛⠛⠛⠋⠹⡏⠀⠀⠘⣧⠀⠀⠀⠀⣠⡞⠁⠀⣿⣌⠉⠙⢛⣻⢿⣿⣟⠋⠉⢿⡄⠈⢳⣄⠀⠀⠀⠀⣸⠃⠀⠀⢩⡏⠙⠛⠛⠛⠛⠛⠓⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠘⠷⣤⡴⠞⠋⠀⠀⢠⣿⠈⠦⠒⠉⢻⡿⡿⠉⠑⠎⠸⣇⠀⠀⠙⠳⠦⠤⠾⠃⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣿⠀⠀⠀⠀⢠⣴⣷⠀⠀⠀⠀⣿⢦⣀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⡤⠤⠤⠶⠚⠋⠀⠀⢛⠀⠀⠀⠀⠘⠙⠻⡇⠀⠀⠀⢸⡀⠉⠛⠶⠤⢤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣶⣶⣿⣧⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣿⣿⢿⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⢿⣿⠟⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠓⠒⢲⡆⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢨⡄⠀⣸⠁⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⣸⠇⠀⢿⡀⠀⠀⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢀⣴⠟⠁⠀⠀⠀⠻⣦⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⢀⡀⣀⣀⣀⠀⠀⣸⣧⣤⣤⣤⣬⣷⡄⣀⣀⣠⣾⣥⣤⣤⣤⣤⣿⡀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⠋⠍⠁⠀⢀⡄⠀⠈⠘⠿⠛⣲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣃⠤⠒⠂⠀⠀⠈⠉⠀⠀⠀⠐⠨⠾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⢫⣿⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡆⣦⣦⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢇⡾⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣄⣷⣿⣷⣿⣿⣟⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⠊⢀⡀⠈⠀⠀⠀⠀⠀⠈⠀⠀⣧⢸⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣇⣼⣄⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠏⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡵⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⡤⢶⣿⡿⠿⠛⡉⣩⣯⣤⣾⣿⣿⣿⡍⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡆⡀⠀⢀⣼⡃⣽⢋⠃⠀⢸⢃⣿⡿⢻⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣸⢿⣿⡏⠈⠀⠀⠀⣸⣿⣷⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⠟⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⡟⣿⣿⡧⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠈⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⣿⣿⣿⣟⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠝⣿⣿⣿⣷⣹⢿⣿⣿⣭⣽⣷⣮⣿⡿⠿⠃⠀⠀⠀⣀⡛⣳⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠋⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⡷⠡⡤⠈⡙⢃⡖⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠄⠨⠭⠀⠓⣆⡉⢙⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣦⣤⡀⠀⠰⣿⣿⣿⣿⣿⢿⣿⣿⢻⣿⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠿⠻⠛⠛⠿⢿⣿⣿⣭⠭⠉⠉⠉⠁⣈⣈⣄⣿⣿⣿⣿⣽⣿⣷⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⣿⣿⣿⣿⣷⠄⠀⠁⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡿⣿⣿⣿⡿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣟⢱⣿⣿⣿⣧⡀⠉⠉⠀⠀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⣿⡿⠀⠈⣿⡿⢀⣀⣀⡀⣿⣿⣿⣰⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠰⣿⠇⢸⣿⣿⣇⠛⠋⠁⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⣾⣿⣷⡸⠛⠉⢸⠁⢀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠿⠋⢹⡇⠀⣀⡘⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠸⠟⠉⣿⠀⢀⣘⢳⣾⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⢹⡇⠀⢀⠛⣶⣿⣧⢸⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢳⣾⣿⠀⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠃⠘⠟⠿⠀⠻⠿⠗⠘⠿⠛⠃⠻⠿⠟





⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⣀⣤⡴⠶⠿⠛⠛⠿⠶⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣁⢀⣈⣉⡿⣶⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠩⡓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠃⠀⠁⠀⠀⠉⠀⠙⣅⠀⢀⡤⠄⠒⠒⠒⠒⠒⠠⠄⡀⠀⠈⠢⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⢀⡀⢀⣀⡀⠀⠉⢠⠋⠀⡀⠠⠶⠄⠀⠤⠄⣀⠀⠑⢄⠀⠙⢆⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠁⠀⠀⣠⣤⣌⠑⠄⠀⣴⣀⡐⠂⠈⠉⠐⠲⢤⡀⠑⢄⠈⢆⠀⠈⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢠⠃⠀⣠⠊⢁⡠⠤⠤⠽⣶⠴⠟⠉⠁⠀⠈⠉⠑⠢⣄⠉⢂⠀⠡⠈⣁⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠠⠆⠀⡠⢁⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠄⠀⠈⠣⡀⠢⠀⠆⠙⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡷⠀⠀⡄⠂⣤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⢀⡀⠀⠀⠡⠀⠃⠀⡀⠀⠀⠘⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⡇⠈⣁⡄⠀⠀⣄⠀⠀⠀⠀⠀⡜⠀⢀⠀⠀⠈⠦⠀⠀⠀⣼⠀⣧⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡇⠀⣾⠀⠸⠁⠀⣠⣤⠀⢱⠀⠀⠀⠀⡇⠐⠿⠃⠀⠀⠐⠂⠀⠀⢧⡞⠁⠀⠀⡘⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠰⡆⢸⡄⢠⠀⠀⠈⠁⠀⡸⠀⠀⠀⠀⠐⢄⣀⠀⣀⡠⠃⠀⠀⢀⣀⠇⠀⠀⣰⠃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠙⣇⠀⠀⠓⢄⣀⡀⠖⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢰⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠇⠀⠀⢹⣶⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠙⡀⠀⠀⠀⠀⢀⡴⠂⠖⠂⠲⡄⠀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⠀⠀⠀⢀⡄⠀⢰⡏
⠀⠀⠀⠀⠀⢀⡴⠶⠶⠦⣾⡇⠀⠀⠀⠱⣄⠀⠀⠀⡟⠀⠀⠻⠂⠀⡀⠀⣀⣤⡶⠟⠋⢻⡄⠀⠐⠀⠀⠃⠀⠜⢡⣴⡟
⠀⠀⠀⠀⣰⠏⡠⢦⠀⠀⢸⠃⠀⠀⠁⠀⣾⠳⠶⣶⣧⣀⢀⣀⣠⠶⢻⣿⠋⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⢿⣆⠀⠊⢠⠄⠁⠀⠀⠀⣠⡾⠃⠀⠀⠀⠈⢉⡽⣿⡀⠀⣠⠟⢷⣤⣤⣀⡀⠀⠀⠈⠙⠛⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀⣀⣤⡴⠞⠋⠀⠀⢀⡴⠚⠛⠻⣗⣈⣉⢉⣁⡠⠏⠀⠀⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⣰⡏⠀⠀⠀⠀⠉⠀⠉⠈⠁⠀⠀⠀⠤⡀⠀⣹⠿⣦⠀⠀⠀⠀⠀⣠⡶⠟⠛⠻⠷⠶⣤
⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠹⢄⡰⠃⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⢸⠈⢁⢠⣿⡀⣠⡴⠞⠛⠉⠀⠛⠃⣀⠰⢎⣻
⠀⣠⣤⣶⠿⠈⠳⢦⣤⣄⣀⣀⣀⠀⣴⠟⢣⣄⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⢶⣏⠀⠈⡿⠋⠀⠀⢀⣤⠞⠛⠿⠿⠷⠾⠋
⣸⠟⢀⠍⢠⢈⣓⣀⠀⠈⠉⠉⠉⠛⠋⠀⣸⡿⠉⢻⡗⠀⠀⠀⠴⡆⠀⠀⠀⣸⡇⠀⠀⠹⣦⡀⠁⢀⣤⠾⠋⠁
⠛⢿⡷⠶⣶⠟⠉⠉⠛⠛⠛⠷⠶⠶⣦⣴⠿⠁⠀⠈⣿⠀⠀⠀⢀⡀⠀⠀⠀⣿⠇⠀⠀⠀⠘⠷⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠻⣤⡀⠀⠈⠁⠀⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⡰⠃⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⢐⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠰⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⡆⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⣰⡟⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢰⠀⠀⠀⢹⡇
⠀⠀⠀⠀⢰⣟⠀⠆⠻⠷⠏⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠈⠻⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠉⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠉⠁⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣆⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠾⠛⢿⡉⠂⣀⡀⣠⣴⣄⠀⠀⠀⠁⠀⠀⢀⣤⣤⣀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠺⣿⠉⠈⣿⠟⠳⣶⠶⠾⠛⠋⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠀⣿⠀⡀⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠛⠛⠁⠀⣿⡀⠀⢻⡄⠀⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⣭⣭⣴⢦⣭⣿⠃⠀⢸⣧⣶⣉⢉⣉⣿⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣷⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠲⠊⠐⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⡠⠁⠀⠀⠀⠈⠒⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⠀⡡⠀⠀⠀⠈⢀⣄⠠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⡀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠰⠉⠲⢄⢀⣞⡀⠀⠀⢀⠐⠀⠀⠀⠀⠀⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠠⠄⠙⢀⠀⢀⠀⠀⠀⠀⠰⠀⣀⢀⣉⢍⠻⣆⢄⠂⠀⠀⠀⠀⢀⠞⡕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠀⠐⠀⠀⠂⠈⠀⠀⠀⠀⠀⠐⣏⢯⣴⣦⡡⠈⠂⢐⢄⠀⠀⣴⢋⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠜⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠺⡹⣿⣆⠀⠀⠀⠈⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡤⡘⢿⣧⣂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⢀⡑⢦⡀
⠀⠀⠀⢀⣴⣦⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠀⠀⠘⠝⢈⠻⢷⡰⠄⠀⠀⠀⠀⠀⠀⠀⡠⢞⠢⣤⡏⢠⣁⠀⢈⠂
⠀⠀⢠⠏⠈⠚⠃⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠑⢄⠀⠈⠡⠅⠃⠀⠀⠁⠀⠀⠀⠀⠀⠮⣤⣷⣇⡃⠻⢅⠁⠔
⠀⢀⡟⠧⠀⣦⣿⡋⢉⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢽⡧⢄⡾⠂
⠀⠘⢙⣿⠆⠸⣿⣿⣿⣿⠟⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠂⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎
⠀⠀⢀⡁⠀⠀⢻⣿⣿⣿⣎⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⡇⠁
⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⡈⠄⠀⠂⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢷⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠚⡄⡀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠴⢊⠁⣈⠦⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⢀⠀⠀⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⡸⠀⠑⠠⢈⡓⠒⠀⠀⠚⠂⣀⡀⠀⠀⠀⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠠⠒⠈⠁⠀⠀⠀⠀⠈⠀⢙⡿⠙⠆⠊⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⢤⣮⣝⣛⢷⠀⠀⠀⢀⠀⠀⠰⢿⡬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠹⣿⣿⠋⠁⠈⠁⠀⠀⠀⠒⢩⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠤⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠙⠁⠀⠒⠒⠂⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠀⠀⠆⠀⠀⠀⠀⢐⣾⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⠻⢍⠀⠀⠤⠶⠀⠀⠐⠒⠈⠠⠆⠀⠀⠀⠥⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣒⠒⠉⠙⠉⠉⠉⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠁⠀⠀⠈⠉⠓⠦⡀⠀⠀⡌⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢢⣼⠆⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠉⢢⡿⠋⠀⠀⠀⠁⠀⠀⠈⢷⡀⠀⠀⠻⠛⠛⠛⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⣀⠤⠔⠒⠛⠉⠉⠉⠉⠉⢧⡀⠀⠀⠀⢀⠀⠐⠄⠈⢧⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠈⠀⠘⡆⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣠⠎⠁⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣀⡤⠔⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⣀⣀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⡆⣶⡟⢋⣥⣆⠀⠀⠀⠀⠠⢖⣾⣟⢶⣦⠀⠀⠀⡇⠁⠀⠄⠀⠀⠀⢸⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠃⠈⠀⠈⠽⠋⠀⠀⠀⠐⠀⠈⠹⠟⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠂⠀⠀⠀⠀⠀⡼⠀⢺⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠧⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠛⡽⢳⠀⠀⠀⠀⠀⣠⠠⠦⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⢎⡀⠀⢠⠀⠀⢠⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠺⡄⠈⢣⡀⠀⠀⢰⠛⠀⠀⠀⠀⠛⣆⠀⠀⠀⠰⠛⣻⠟⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⢀⠤⡀⠀⠈⠓⠦⢬⣦⣄⣀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣠⣴⣟⣁⣀⡠⠖⠊⠁⡠⠒⠄
⠀⠀⢀⣠⢤⠬⠧⠌⠢⡄⠀⠀⠀⠀⠀⠉⠙⠓⠒⣶⠒⠒⡞⠋⠉⠀⠈⠉⠉⠀⠀⠀⢠⠊⠀⠖⠒⠶⠶⢦
⠀⠀⠋⠉⠉⠀⠀⠀⢲⢼⡀⠀⠀⠀⠀⠀⣀⡴⠖⣩⡀⠈⠘⢉⡓⢀⡀⠀⠀⠀⠀⢀⡥⠐⠉⠀⠀⢈⣉⣉⠃
⠀⠀⠉⠈⠉⠀⠐⠂⠛⢶⠉⠒⢦⣀⡠⢞⡝⠴⣫⣿⡛⠤⠆⢸⣄⠀⠙⠦⣀⡴⠚⠁⣰⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠹⠀⣸⠀⠀⢻⡏⠀⠀⢠⠟⠈⠳⢄⡀⠘⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠊⠁⠑⣄⠘⡇⠀⢂⠏⠀⠀⠀⠀⠉⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠚⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⡶⣀⣀⡠⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠀⠈⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⣧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠰⡆⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⢹⡵⠋⠉⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠜⣃⢃⣇⣋⠸⢹⡘⣆⢳⡘⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣛⣒⡒⠶⢭⣖⡡⢝⠆⢳⡘⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣍⣉⡉⠛⠳⣦⣝⠲⣝⢆⢷⢹⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣶⡄⠙⣧⡘⣆⢫⡃⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠘⣷⡘⢧⡳⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣿⣿⡇⠀⠙⣿⠈⠇⣅⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⣿⣿⡇⠀⡴⣋⠞⣱⠙⡆⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⠀⣿⣿⣿⡇⠸⣇⢿⢄⠙⢷⣍⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢌⣼⣿⣿⣿⣿⡄⠹⡎⢎⠳⣄⠙⢷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⣿⡿⠿⠿⠟⠁⠀⣿⠸⡆⣿⠂⣠⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⠏⠀⠀⠀⠀⢀⣼⡿⢠⡇⢱⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠛⠁⠀⠀⠀⠀⢠⣾⡟⢁⡾⠈⣼⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡵⠁⣀⣀⠀⠀⠀⣠⣿⡟⢀⣾⢃⡇⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡺⣹⢡⣾⠿⠛⠛⠿⠿⠛⢋⣠⡾⠃⣼⠃⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⢇⡇⣸⡏⣰⢋⡭⢟⡭⣋⢭⢡⠒⣛⣡⣴⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡇⣿⠃⣟⢸⡇⢿⢸⠁⡏⣼⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⢰⢿⣦⠀⠀⠀⠀⠀⠀⣴⢿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠛⠀⠀⢸⡗⠁⠀⠛⣷⠀⠀⠀⠀⠘⠃⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠟⠉⠁⠀⠀⠀⠀⠀⠈⠛⢷⣤⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣴⠆⠰⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠘⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠠⡿⠻⠿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣿⡉⠉⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢷⣶⠟⢉⣽⣿⣍⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠙⠀⠙⠀⠙⠏
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⣀⣴⣿⣶⣾⡆⢀⣴⣶⣶⣴⣶
⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠼⠃⢘⣙⣿⣛⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠐⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⢠⡾⣿⡆⠀⠀⠀⠀⠀⠀⣼⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⠉⠹⡆
⠀⠀⠀⠀⠈⠀⠀⣠⣴⣦⣄⠀⠈⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⠃⠀⠘⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣾⠿⣧⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣤⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠿⠿⠆


⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⡆⢀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠘⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⠋⣀⠿⣿⠟⠿⠻⣖⣶⣇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⢸⣿⠃⠀⠀⠀⠀⠈⠹⣿⣿⣿⠀⣄⠀⠀⠛
⠀⠀⠀⠀⣤⣤⣀⣀⣤⣤⡄⠀⠀⠈⣿⣿⡟⢸⣿⣦⠀⠀⠀⠀⠀⠀⠙⠿⣿⠆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣦⣴⣿⣿⣿⡏⢸⣿⣿⣷⣦⡀⢀⣀⣀⣠⣼⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠛⠋⠋⠛⠛⠋⠋⠋⠛⠙⠁⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⡿⠿⠿⠿⠿⠻⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⢛⣿⢛⡛⢻⣿⣿⠟⢿⠿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⡤⠖⠚⠉⢩⣟⣥⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣀⡤⢶⡦⣄⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠇⠀⢠⠀⠀⢀⠀⠀⠀⠤⠤⠒⢀⡾⠋⠀⠀⠘⢦⠀⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠠⡀⢌⣆⣱⢄⣉⠙⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠈⡇⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣬⠽⢛⣥⣶⣯⠉⠉⠉⠁⠀⠀⠀⣠⣤⣀⣀⠀⣷⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢸⣴⡿⠛⣉⣁⡀⠀⠀⠀⠀⠀⠀⢙⣛⡿⢿⡇⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠈⡇⣠⠊⠉⠉⠉⠳⡄⠀⠀⠀⡼⠉⠉⠙⢣⡀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⠻⣤⣷⠇⠀⠀⠀⠰⠆⢻⠀⠀⢸⡇⠀⠘⠃⠀⣧⡿⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠈⢢⡀⠈⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡅⠀⠀⠀⠀⣾⣧⠞⢉⡉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢷⢰⡇⠇⠀⠘⣆⠀⠀⠀⠀⢀⡼⢀⣀⡀⠳⢄⣀⣀⡴⠃⠀⡼⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⡂⣠⠀⠈⠛⠲⠶⠒⠋⠀⠉⠐⠛⠀⠀⠀⠀⠀⠀⠘⢀⠇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢾⡿⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⢤⣤⣤⣤⣤⡤⠴⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢹⣆⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡾⠀⠀⠀⠛⠛⠛⠋⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠟⢷⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣠⡞⠁⠀⠀⠙⢦⣌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⠀⠀⠀⠀⠙⠋⡀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢿⡻⣆⢻⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⢷⣌⣷⡘⢿⣷⠘⣧⡾⢻
⠀⠀⠀⠀⠀⢰⣦⡀⣿⣇⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣦⠉⠛⠁⠈⠉⠀⠛⢃⡟
⠀⠀⠀⢠⣦⣌⣿⡻⣿⢿⡿⣿⣴⢻⡆⠀⠀⠀⠀⠀⠈⠻⢦⣤⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠹⢮⡛⠃⠀⠀⠀⠙⢣⡞⠀⢀⣀⠀⣤⡀⢠⣄⠀⠈⢿⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢠⡏⠀⠀⢸⡝⣿⡇⣿⣿⢹⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢀⣤⡀⣼⢧⢀⡶⣦⢸⡇⠀⣶⠾⣷⠈⠃⠘⠋⢾⡾⢻⠆⣿⠀⠀⢠⣦⡀⣿⡆⢀⣤⡄
⠀⠀⡶⣼⡏⣷⣿⢸⣿⠃⡿⢸⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⣰⠏⠀⣿⢀⠀⢸⡌⣷⡟⣿⣸⢃⣇⣴⡦
⠀⠀⢷⡘⣿⠘⠟⠈⠛⢸⣧⡿⣦⠀⠀⠀⠹⡆⠀⠀⠀⣴⠃⠀⠀⣿⣏⠻⣾⡇⠻⠃⣿⡟⢸⡟⣹⠇
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢁⣴⡃⠀⠀⠀⢠⡇⠀⠀⠀⣿⠀⠀⠀⣿⠙⢷⡀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⢀⡇⠀⠀⠀⢠⡴⠋⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢿⡀⠈⠷⣄⣀⠀⠀⠀⢰⡏
⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢹⡄⠀⠀⠈⣷
⠀⠀⢀⣿⠀⠀⠀⠀⢸⡇⠀⣼⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠸⣇⠀⠀⠀⠘⣧⠀⠀⠀⢸⣇
⠀⠀⣸⠃⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢻⡆⠀⠀⠀⢻
⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡴⣂⣄⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡰⣿⣿⢠⡿⠁⠀⠀⠀⠀⠀⢸⠑⢿⣿⢇⡀⠀⠻⣿⡿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣶⢻
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⢻⣾⣶⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠃⣾⣿⣿⣶⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣤⡖⣧⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣿⢀⣀⣀⣀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⣿⣿⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣿⣽⣿⣿⣿⣥⣼⣿⣿⣦⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⣿⠉⠉⠉⠉⠉⢻⣿⣿⡏⠉⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠋⠛⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⣿⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣤⣽⣤⣥⣭⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠀⠀⢸⣧⣤⣧⣼⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣿⣿⣿⣿⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠉⣋⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣄⢀⢸⣧⣤⣧⣼⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⢻⣿⣿⣿⠛⣿⠃⠀⠀⠀⠀⠀⠀⡠⠔⠒⣿⡟⣿⣿⠅⣺⠽⡆⠤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⢉⣹⣯⣨⣧⣸⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⣿⣿⣿⠋⣿⠀⠀⠀⠀⠀⠀⣼⠀⢀⡴⠗⠻⠿⣿⣶⠁⠀⣣⠀⠘⡄⠀⠀⠀⠀⠈⢀⣿⣿⡆⠐⠉⣎⣙⣏⣸⣏⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⢺⣿⠀⣿⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⠜⠀⠉⠙⢄⠀⠀⢀⡴⠁⠀⠀⠀⠀⢄⣁⡻⣿⠁⠠⣀⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠸⢿⣗⣒⠒⠒⠖⠒⢒⣒⢻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡺⣶⢧⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⣤⣤⠜⣆⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⣷⡶⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡿⠟⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠦⢤⢦⣴⣿⣿⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠞⠁⠉⠈⢢⣹⣿⣧⣜⣧⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢺⠀⠐⠃⠀⢀⠿⠏⠙⡀⠉⠛⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⢠⣿⠄⠀⠀⠀⠀⠀⠀⠓⠤⠤⠴⠋⠀⠀⠀⠑⠦⠄⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠶⠶⠖⠒⠒⠶⠶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡴⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢶⡄⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣴⠖⢳⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢷⣶⣄⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢸⡇⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢼⣿⣿⡀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢠⣾⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣸⣇⣀⣸⣿⣿⣇⠀⠀⣴⠛⠉⠀⠀⢀⣿⣀⣀⣼⣿⣿⣀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠻⢦⣤⡄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⢸⡟⠋⠁⠀⠀⠀⠈⠉⢻⡇⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣦⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⢠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠐⠃⠐⠂⠘⠃⠀⠀⠀⠀⠤⠲⠄⠀⠀⠀⠀⠀⠴⢆⣀⠀⠀⠀⠀⠤⢤⣀⡀
⠀⠀⢈⠀⠅⠀⠥⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠦⠶⠴⠦⠀⠀⠀⢀⢆⣠⣀⢄⠀⠀⠀⢀⠀⡄⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠀⠦⠴⠀⠀⠀⠀⠀⠐⠂⠨⠇⠀⠀⠀⣄⢀⢤⠤⢴⠆⠀⠀⠀⠀⠨⡤
⠀⠀⢠⡀⠀⢄⣀⡀⠀⠀⠀⠤⠐⣖⠰⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠆⠈⠐⠓⠀⠀⠀⠀⣀⣀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉

⠀⠀⠀⠀⠄⡀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠀⢀⠦
⠀⠀⢐⠂⠀⠈⠐⠂⠀⠀⠀⠂⠒⠁⠀⠀⠀⠀⠀⢰⠌⡂⢠⠀⠀⠀⡈⠉⠦⠂⢢⠀⠀⠀⠀⠘⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⢀⡆⠒⡄⠂⠀⠀⠔⠂⠀⠀⠀⠂⠀⠀⠠⠒⠀⠉⠀⠀⠀⠀⠠⠄⠗⠀⡀


⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⢀⡄⠃⠀⠀⠀⠀⠈⠓⠅⠀⠀⠀⠀⠀⠙⢭⡗⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠐⣋⢈⡆⠠⣀⠀⠀⠀⠤⠤⢁⣆⠀⠀⠀⢀⣠⣄⠈⠂⠀⠀⠀⠀⠙⠐⠘⠁⠀⠀⠐⠐⠒⠒⠒⠒
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠁⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢺⡐⠭⣧⠇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣸⣥⣤⠖⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢀⣀⡀⠀⠀⠀⠒⡂
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⢶⣄⠀⠀⡀⠀⢖⡲⠀⡊⠁⡆⠈⠉⠁⠀⣼⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⡝⠛⣏⣹⠄⠀⣠⡄⠉⠈⠁⠀⠀⠀⠀⠀⡟⠻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⢡⡀⠀⢽⠆⠀⢀⢠⠀⠀⠀⡀⡔⠃⠀⠁
⠀⠀⠀⠀⠀⡀⠖⣀⣀⠀⢠⠖⠒⢄⠀⢿⡀⠀⠀⠀⠀⠄⠂⠀⠠⢇⣹⠃⠀⠄⠒⢌⠂⠀⢀⣼⠃
⠀⠀⠀⠀⣼⣿⠉⠉⠁⠀⠹⣀⣀⠜⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠸⡀⠀⡠⢀⢠⣾⡿
⠀⠀⢀⣴⠟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠉⠀⠀⢨⡿⣿⡀
⠀⠰⠟⠁⠠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡗⣭⢛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣰⡏⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⢰⣟⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣷⣾⣷⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠛⠀⠀⠀⢀⣀⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⣶⣶⠞⠀⠀⠀⢤⣤⡴⠀⠀⣤⡄⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠹⣆⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡟⠀⢀⣤⡀⠘⣿⠁⢠⡄⢸⡇⣠⣼⠇⣄⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇⠀⢸⣿⣷⣾⣿⠀⣿⣿⣿⡇⠙⢻⡇⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⠀⣿⡿⢿⡇⣶⣾⠻⡆⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇⠀⠸⣿⡏⠈⣿⠀⠙⠃⣸⡇⠉⠸⠆⠐⠃
⠀⠀⣠⣴⣶⣶⣿⡇⠀⠀⠀⣤⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠸⣿⡿⠃⠀⠠⣿⣿⡀⠀⠀⠀⠈⠛⠃⠀⠀⠉⠁
⠀⠾⠿⠛⠛⠉⠉⣿⣷⣶⣾⣿⡇⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛



⠀⡁⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠤
⠀⠴⠂⣀⠠⠄⢤⣈⢡⠀⠀⢸⡿⢿⡿⡟⡟⣿⠀⠀⠀⣔⠠⠀⠈⠄⠀⠀⠈⠀⠠⡄⣤⣤⠠⠀⠐⠀⠀⠉⣒⢀⠀⠀⠘⡆⠐⠂⠀⠈⠄
⠀⠐⠜⠁⠀⠀⠀⠈⠁⠀⠀⢸⣧⣼⣧⣥⣧⣿⠀⠠⠎⠀⠀⠁⠀⠀⠂⠀⠈⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠁⢀⣠⡤⠰⠄
⠀⠀⠀⠀⠀⠒⡀⠀⠸⠀⠛⢻⣿⢸⣰⣆⣾⣿⠀⠀⠀⠀⠈⠀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣴⣆⠀⠀⠘⠀⡀
⠀⠀⠀⠢⣀⠀⠀⠔⡂⠀⠀⠈⠙⠋⠛⠉⠋⠉⠀⠀⠀⠀⠀⠉⠉⠁⠄⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⢈⠀⠀⠀⠀⠀⡀⢀⡀⠀⣐⠨⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠚⠀⢰⠩⢙⠫⠳⢒⣶⠶⡤⡤⡄⠀⠀⠀⠁⠃⠀⠀⠀⠈
⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠹⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠺⣬⢶⣤⣯⣐⣃⣆⣁⡰⡅
⠀⠀⠘⢄⠀⠀⠀⠀⠀⡠⠋⠀⠠⠌⢀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠘⢃⠀⠀⠀⠀⠀⠀⡇⣽⢃⡘⣏⡸⡯⣩⠏⠀⠰⠢⠶⠶⠶⠶⠔⡆⠠
⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⢀⠀⠸⠖⢾⠒⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠓⠲⠞⠷⠟⠀⠀⠀⡀⠀⠀⠀⠀⡬
⠀⠀⠀⠁⠀⠀⠀⠁⢢⠀⠀⡀⠀⠀⠀⠀⠀⠀⣼⡅⠀⠀⣠⠶⠶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⢲⣦⣄⠀⠀⠄⠀⠢⠈⠂⠤⠤⢈⣩⠄
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⡀⠀⢨⡽⠀⠀⢸⠇⠀⢤⠦⢤⢴⠴⡇⢰⠶⠶⠤⡶⠀⠻⢿⣧⠀⢀⣀⡀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⢂⠀⠀⠀⠀⠀⡠⠃⠀⠀⡀⢀⡀⠀⢀⠉⠀⠀⠀⢸⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠿⠃⠀⠀⠀⠀⠐⠠⣆
⠀⠀⠀⠈⠐⠀⠒⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠰⠤⠴⠾⠶⠶⠧⠀⠶⠶⠾⠶⠶⠤⠐⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⡠⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡞⠂⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⣄⠀⠀⠠⢁⠀⢀⠀⠀⠀⠀⠀⠀⠃⠐⠄⠀⠀⠨⠴⠀⠀⣐⠀⠀⠀⡀⠀⡀⠀⡐⠁⠀⢀⠀⠀⢰⢂⠀⠀⠐⢀⠀⠉⠆
⠀⠀⠠⠂⠘⠀⠔⠈⠀⠀⠀⠀⠀⠀⠂⠒⠒⠀⠀⠃⠀⠀⠀⠀⠀⠠⠁⠀⠈⠀⠃⢄⠀⠀⠀⠀⠀⠌⠀⠀⠉⠉⠀⡃⠈⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⠄⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠠⡀⠂⠀⠔⠀⠀⢀⣙⠂⠀⢠
⠀⠀⠘⠷⣤⣀⣀⣤⡶⠟⠉⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠟⠃⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠂⠁
⠀⢀⣀⡀⠈⠙⠋⠅⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⢀⠀⣠⠊⠁⠀⠀⠀⢐⠋⢐⡂
⠀⠀⢀⠠⠤⠴⠀⢠⠆⠀⠀⠐⠁⠀⠀⢑⠀⠀⠀⠑⠀⠀⢰⠀⠘⠤⠤⠄⠀⠀⠀⠈⠉⠀⠰⠀⢀⡠⠈⡀⠀⡄⠈⠂⠀⠀⠀⠀⠈⡀
⠀⢀⠈⠙⠁⠈⢀⣀⠄⠀⡀⠀⡀⠒⠌⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⠀⠀⠆⠀⠀⠀⠀⡈⠀⠀⠀⠈⡀
⠀⠀⣀⢀⡄⠤⠤⡄⠀⠀⡇⡎⠀⠀⠀⠐⡎⡀⠀⡜⠀⠁⠀⠀⠈⠀⠰⡄⠀⢠⣰⠆⠀⠀⠀⢀⠀⠀⣀⡂⠈⠣⠀⠀⠀⠆⠀⠀⠀⠀⠇
⠀⠀⠅⠀⠀⠀⠀⢳⠀⠀⠀⠓⡀⠀⠄⠠⠋⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣀⢴⠀⠀⠈⠁⠀⠄⢱⠀⠀⡀⠀⠀⠀⠐⠀⠴⠂⠒⡄
⠀⠀⠀⢀⣤⣤⠀⠀⠄⠀⠀⠀⢈⠂⠒⡀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⢈⢈⠀⠀⠛⠉⠁⠀⠀⠀⠈⡉⣈⣀⣈⠅⠀⠀⠀⠄⠀⠀⠀⠠⠁
⠀⠀⠀⠀⣤⣹⠆⠀⢈⠀⠀⠔⡁⠀⠀⢀⠀⡀⠠⣁⠀⠀⠀⠀⠀⠀⢀⠝⠀⠀⣿⢂⣽⡃⡀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠒⠂⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠠⠤⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠉⠉⡉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢠⣴⠿⠛⠛⠷⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡂⠀⠀⠐⠒⢀
⠀⠀⠀⠀⠈⠉⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠟⠁⠀⠀⢀⠄⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠷⣦⡀⠰⣄
⠀⡄⠀⣴⡿⠿⣦⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠈⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢈⡿⠀⣿
⡀⠁⣼⡏⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠉⠀⠀⠉⣀⣴⠋
⣇⡀⣹⡇⠀⢀⡘⠀⠺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣠⡿⠛⠿⠿⠟⠋⠁
⠘⠟⠛⠁⠀⠈⠛⢷⣄⡈⢡⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡇⢀⣴⠆⠀⠀⠀⠀⠀⠈⠳⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣤⣤⣤⣀⣤⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣾⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⢀⣤⣴⡌⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣿⣿⣦
⣾⣿⣿⣇⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⠻⠿⠛
⠀⠉⠉⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⢀⡄⠻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⢛⣀
⣴⣿⣧⡀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏⢣⣤⣾⣿⣷
⠿⣿⠿⠋⠁⢸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡄⠀⠀⠀⠸⣿⣿⡇⠀⠈⠛⠿⠛
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠁⠙⡆⠀⠀⠀⠀⠀⠀⡐⠋⠉⢿⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠉⣠⠇⠀⠀⠀⠀⠀⠀⢃⡘⠁⣼⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⢀⠀⠀⠙⠊⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠙⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⢀⣄⣀
⠀⡞⠉⠉⠉⠙⢡⠀⠀⠀⠀⠀⠀⠀⣀⠿⠿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⣃⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠉⠳
⡆⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⣀⠚⠉⠀⠀⠈⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉⢿⣆⠈⠓⣤⠀⠀⠀⠀⣠⠃
⠀⠉⠉⠁⠰⢦⡀⠈⢷⡀⠀⢀⣿⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⣼⡅⠀⢀⡴⠉⢀⡰⠆⠈⠉⠉
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠹⣦⠛⠙⢳⡔⣤⡾⠁⠀⠀⠀⠀⢀⡤⠤⡤⢀⣠⠄⠐⢤⡄⠀⠀⢹⣿⡼⠋⠙⣴⠎⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠛⣄⠀⠀⢀⡴⠎⠁⣿⠇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠉⠁⠀⠀⠀⡇⠀⠀⠀⣿⡿⢦⡀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠔⠛⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⢹⣇⠀⠙⠢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⢀⡠⠞⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣄⣤⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣴⣷⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⣿⠀⠀⠀⣿⡇⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⢹⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿⠀⠀⠀⢸⡇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣿⠀⠀⠀⢸⣇⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⢘⣿⠀⠀⠀⢸⣟⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣷⣄
⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⣀
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠍⠀⠈⠁⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠳⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣶⣶⣶⣜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠏⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡟⢦⣸⣦⡘⠀⠀⠀⢀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⣿⡸⡏⠛⡇⠀⣴⠺⣆⠀⠈⢀⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⠙⣯⡻⣿⣶⡙⠄⠀⠈⣟⡻⠃⠀⠀⣛⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣀⣠⡴⠾⠛⠚⠙⠛⠐⠺⢟⣛⠛⠀⠀⠀⠀⠈⠠⠔⠊⠁⣡⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⢀⠔⠁⠀⠀⠀⠈⠉⠓⠒⠠⢤⡐⢂⣤⠄⠀⠀⠉⠓⣤⢤⡀⠀⠀⠀⢀⡄⠀⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⣿⠀⠀⠀⢸⣘⡷⠋⡆⠀⢠⣾⠧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣰⠁⠀⠀⠀⣾⡅⠀⠀⠀⠀⣧⠁⢀⡄⠀⠐⠀⣀⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠸⠀⠀⢾⡁⠸⠄⠀⣼⠆⠀⠁⠄⠸⢁⠛⠂⠉⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠘⠀⠀⠀⠠⣸⡇⠐⡆⠀⢿⣦⠈⠳⡆⢠⣆⣀⣀
⠀⠀⠀⠀⠀⠘⡀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢰⡀⢧⢡⠀⢱⠀⢻⠊⠃⠀⢀⠀⡉⠉⠛⠇⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡀⠀⠀⠀⢀⠌⠀⠀⠀⠐⡀⠀⠀⡀⣷⢸⡄⠀⢤⡆⠀⠤⡀⠀⠀⠉⠑⠀⠢⠶⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢷⠸⣄⢱⡙⣆⢻⡄⠘⣷⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠢⡘⢶⢇⠙⡎⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠂⠀⠓⠈⠈⢇⠃⠀⠀⣄⣀⣀⣀⠀⠀⠀⠀⢠⣤⣶⠶⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡭⠴⠾⠟⠀⠀⡛⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣦⣴⣿⡷⠶⠆⢰⡶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠦⠼⣭⣏⣁⣁⣤⠠⠀⣤⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠤⢄⡀⠀⠀⠀⠀⠹⡉⠙⠉⠛⢠⢰⣧⣀⣴⠃⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠴⢦⣀⣷⣄⠀⣤⣼⠦⡙⠯⣁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠰⣦⡀⢠⣶⣦⠀⣠⠶⠀⠀⠀⠀⠀⠀⠀⣤⠼⣿⡷⠀⠀⠀⠀⠀⢿⡿⢹⡆⠀⠀⠀⠀⠀⠀⢾⣿⠆⢺⡿⠃
⠀⣴⣶⠀⠀⠀⠙⣮⣽⣥⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⣾⣶⣦⡀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⣴⣶⣷⣶⠛⠁
⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣏⢽⠟⠁⠀⠀⠀⢺⣿⡄⠀⠀⠀⠀⠀⠀⢿⣯⢻⣿
⠀⣿⣿⣶⣶⣆⣀⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⢹⣿⠿⠿⠿⠿⣿⡿⣿⡿⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⢸⣟⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠻⠟⠀⠀⠀⠀⠘⠃⠙⠁⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠹⠟⠿⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠿⠻⠟

⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡄
⠀⣘⣿⣋⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⣼⣿⠗⣿⢻⡄⠀⠀⠀⣸⠏⠉⠉⠁⢩⣽⡍⢿⡀⠀⠀⠀⠀⢀⣤⣽⣭⡄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣉⣩⣤⠀⢀
⠀⣿⣦⣿⠸⡇⠀⠀⣠⣿⣤⣤⣤⣤⣬⣿⣥⣼⣧⡀⠀⠀⠀⠘⢿⣇⣿⡟⣿⣿⣿⠀⠀⠀⢤⣤⣤⣶⣾⠾⠟⢻⣿⡴⣿
⠀⣿⣿⣿⢸⣷⠀⠀⣿⣨⣿⣿⣿⣿⣿⣿⣿⣏⣸⡇⠀⠀⠙⡟⣿⣿⣿⣿⠛⠛⢻⠃⠀⠀⠀⠀⢻⡏⠀⣠⡶⠿⣿⣁⣿
⠀⣿⣿⣿⠀⠀⠀⠀⢻⣿⡟⠛⠛⠛⠛⠛⠛⣿⣿⠃⠀⠀⢠⡇⢿⣯⢹⡿⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⢰⡟⣴⣿⣿⣿⡿
⠀⠙⠉⠛⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠈⠃⠈⠛⠛⠁⠀⠀⠘⠁⠀⠀⠀⠀⠚⠓⠈⠁⠛⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡺⠁
⣶⣿⣆⠺⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⣿⠀⣀⡀⠖⠉
⠘⣿⣿⡆⣿⣶⡶⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⡿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠄⠀⠀⢸⣿⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣤⣤⣿⣿⣭⣭⣽⣿⣧⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣾⣭⣭⣽⣧⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⠠⢖⢀⡒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢐⡲⡄⠀⠀⠀⢜⣳⠀⠀⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠄⠡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢄⠀⠀⠀⢀⠐⣤⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⡄⠀⢀⡼⣀⣁⣼⣀⠀⠀⠣⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠁⣠⣇⠠⢈⡁⠤⢈⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⢙⠎⡴⢃⣀⣠⣅⣀⡜⢢⡻⣈⡽⠟⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠤⠴⠾⢤⡗⠉⠀⠀⠀⠀⠉⠲⡧⠿⠀⠶⣄⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠓⢳⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣉⣉⣑⠢⣄⡼⠁⠈⠉⠓⠒⠒⠒⠒⠒⠒⠒⠉⢁⠀⠀⠀⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠋⠀⡀⠀⠱⣼⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⢇⣀⠠⠏⠀⠀⣸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠄⠒⠄⠐⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢬⣓⠆⠀⠀⣠⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠰⣄⣄⣿⡎⠁⠉⠀⠀⠀⠀⠀⠀⠈⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣉⡀⠹⣷⡀⠀⠀⢸⠉⠉⣹⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⣍⠇⠠⢁⠉⠀⠀⠀⠃⠠⠃⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠘⣄⡰⠙⣄⣤⢀⡿⠀⣠⠴⡇
⠀⠀⠀⡀⠀⠀⢠⡄⣽⠑⠾⠀⠈⠃⠀⠙⠈⠃⠰⠞⠁⠐⠇⢀⣠
⠀⠀⠀⠙⢟⠃⠶⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⢩⠃
⠀⠀⠠⣤⠦⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⣠⠶
⠀⣀⣀⠨⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠏⣀⣀
⠈⠙⠒⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠒⠊⠉
⠀⣠⣔⣋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠦⠄
⠀⠀⠀⠀⣼⣁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣘⣆
⠀⠀⠀⠈⠁⠀⣯⠴⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠹⡄⠁⠀⠀⠀⠀⠀⡴⠈⠀⣠⠛⢩⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣇⠞⠻⢀⡜⢧⡞⣆⡞⠳⣄⣯⠙⠶⣄⡘⢆⠀⠀⠀⢀⣼⡗⠀⠀⢿⠀⠛⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠟⠀⠀⠀⠈⠀⠀⠀⠙⠀⠀⠀⠙⠺⣦⠀⠀⣼⠏⠀⠈⠁⠀⠀⠠⡀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⢸⡇⢀⡖⠂⠀⠀⠐⣒⣚⡇⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠚⠁⠀⠀⠀⣾⢡⡀⠀⠀⠀⠀⠀⠀⠈⣡⠀⢡⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣾⣇⠀⣠⠀⠀⣀⠀⠀⣿⡀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⠹⣤⣤⠋⠀⣴⢟⣧⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⠖⠟⠛⠛⣡⣠⠆⣀⣀⣴⡿⠋⠘⣿⣷⣄⠀⠙⣿⡳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⣾⠛⠄⠀⠀⠈⠃⠉⠀⠉⠛⣯⠠⠶⣄⠈⠉⠻⠷⣦⡈⠹⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠼⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠱⢦⡀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⣤⣹⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡶⡄⢀⠀⠀⠀⣿⣿⠟⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠃⠀⡈⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⣀⡀⠀⠀⣦⣀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣠⡤⠀⢀⣴⣿⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄⠀⠀⢿⠇⠀⠀⠀⠉⣿⠨⠲⠕⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠀⠘⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡗⠀⠀⠀⢧⠀⠀⠀⠀⣿⡄⠀⡢⠀⡀⢅⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⢣⠀⠀⠀⠸⡇⠂⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⡏⠀⢦⡄⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠫⠄⠀⠀⠀⠇⢠⠈⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⠆⡀⢀⠀⠀⠀⢠⡄⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠣⣈⣀⣴⠆⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠉⠑⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⢀⣿⣿⡄⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⣶⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠤⠠⠀⣀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡀⠀⠈⢲⣷⠋⠀⠀⠀⠀⠀⠀⠇⠈⠔⠀⠁⠌⠀⡇⠀⠀⠀⠀⠀⣠⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⣿⣿⣷⣾⣿⣶⣶⣶⡄⠀⠀⠀⡇⠀⢄⡀⢀⠄⠄⠀⠀⠀⠀⠀⠀⣿⣿⣶⣟⠛⠒⡆⠀⠀⠀⠀⠀⢠⡑⢡⡄⢤⣿⡄
⠀⠀⠀⠁⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢈⣏⠁⠘⣦⣀⡇⠀⠀⠀⠀⠠⣿⡷⢿⡿⢿⣿



⠀⠀⠀⢠⡄⠖⠒⠒⡒⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⢀⣾⡗⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⡖⠁⠀⠀⠀⠀⠀⠀⠐⠿⠗⠒⠿⠟⠀⠀⠀⠀⠀⢠⢶⣟⠶⠶⣶⣶⣦⣀
⠀⠀⠈⢹⡇⠀⡠⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠤⣾⡆⠀⠀⠀⠀⠀⠋⠈⠁⠉⠀⠉⠉⠉⠙
⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣶⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢉⣉⠉
⠀⠀⠀⠚⠃⠀⠀⠚⠳⠀⠀⠀⠀⠀⢠⣮⣆⣶⣿⣶⡤⣽⣆⠀⠀⠀⠀⠘⠙⠀⠀⠀⠘⠋⠃⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⣴⣷⣾⡆⠀⣠⣀⠀⠀⠀⠀⠀⠘⠛⠙⠿⠿⠿⠇⠙⠋⠀⠀⠀⢠⣿⣼⣷⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢤⣴⣷⣤
⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢰⣿⠿⠒⡒⠺⣿⡧⠀⠀⠀⠀⠀⢠⠒⠁⡁⠘⢢⡀⠀⠀⠀⠀⠀⠀⢳⠐⠈⠁⠂⡞⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠐⢀
⠀⠀⠀⡏⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠣⠀⠀⡇⠀⠀⠀⠀⠀⠀⡅⠀⢠⡶⠂⠨⠀⠀⠀⠀⠀⠰⠀⠀⠠⠄
⠀⠀⠀⢱⣄⠈⠀⢀⡜⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⢀⡰⠂⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠚⠛⠛⠉⠉⠉⠉⠉⠙⠛⠒⠒⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠤⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠴⠶⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠇⠀⠀⠀⠀⠈⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠛⡋⡙⠁⠀⠀⠐⠶⣤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡋⠀⠀⢀⡄⢸⠀⠀⢠⡄⠀⢸⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦
⠀⠀⠀⠀⠀⢀⡡⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢣⠀⠀⠘⠇⢸⠀⠀⠹⠇⠀⠸⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⡀⠀⡂
⡤⠖⠉⠀⠉⠈⠀⠀⠈⢢⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠓⠀⠀⠀⠊⠡⢀⡀⢀⠔⠁⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠔⠁⠀⠉⠉⠁⠈⠰⠆
⠑⢄⠀⠀⠀⠀⠀⠀⠀⣨⡀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠈⠁⠀⠀⠉⠁⠀⠈⢻⣦⠀⢀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⢤⠀⠀⠀⠀⠀⣿⣄⡀⠀⣠⡾⠋⠐⠀⠐⠒⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠙⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⡀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⣧⡄⠀⠀⠀⠀⠀⠻⣶⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⡁⠠⠽⠟⠃⣄⡐⠰⠶⠀⠉⢠⡀⠀⢀⣴⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡁⢋⡳⠀⢀⡤⠀⠈⠠⢦⣤⣲⡖⠀⠀⠘⢿⠯⢚⡽⠿⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠂⠘⣷⣤⣜⠁⠘⠋⠀⠩⣳⣹⡿⡶⠓⡤⡄⢈⢻⣧⡶
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢾⡆⠹⢏⣿⣷⠂⠀⢀⣀⢸⡿⣿⠏⠘⠛⠻⢻⡟⣳⣄⠢⠐⣀
⠀⠀⠀⠀⠀⠀⢀⣿⡗⠦⣷⠬⣅⡶⠎⠻⢯⡄⠂⠻⠿⠸⢿⠿⠀⣀⠀⠀⠀⢰⣿⣅⣤⠖⡭⠅
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⢢⣴⣿⠶⠼⠾⠀⢀⣀⠠⢀⣰⣤⣾⣿⣧⣤⣀⠙⣏⢀⣀⡰⢶⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣈⠉⠀⠀⠐⠂⠀⠈⠁⣠⠀⠊⠋⡉⠉⠙⣄⠈⠀⢀⡀⠈⠃⠄⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠠⠞⠉⠙⠻⣿⣶⡷⢤⣤⣌⣀⠀⠀⠀⠵⠀⠘⠉⠉⢘⣿⡟⠂⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠐⠀⠀⠀⠀⠙⠴⠛⠧⣾⣻⣾⠃⠐⢒⡀⠀⢐⡛⠘⣿⣁⣀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⢝⡷⢸⡀⠈⢀⠺⢻⡿⠁⣒⠠⣴⣮⠤⠀⣉⣡⠀⠈⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⣷⣾⣧⣼⣠⣴⢫⠀⠸⠟⢳⠎⢡⣾⣿⠉⠉⠀⣰⠚⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣿⣟⢊⢸⣿⡒⠀⡀⢀⠍⢶⣦⡉⠹⠷⢘⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣼⣀⡐⠻⠞⠃⠀⠀⠀⢀⣀⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⢛⣿⣿⣿⣿⣿⣿⣯⠻⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣷⣮⣭⣭⣭⣭⣭⣴⣾⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠭⠯⠭⠯⠏⠛⠛⠉

⠀⡠⠀⣶⣶⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣤⠶⢚⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣖⡲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⡞⢁⣾⣿⡟⠛⠻⣇⠀⠀⠀⠀⠀⢀⡿⠋⠙⢿⣿⣆⠹⣆⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣇⢼⣿⣿⣦⣀⣠⡿⠀⠀⠀⠀⠀⠸⣷⣤⣤⣿⣿⣿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⢀⣤⣶⣶⣦⣄⠀⢀⣴⣶⣶⣦⡀⠀⣀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⠛⠙⠻⣿⣿⣿⡿⠋⠙⠿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠶⠶⠶⠶⢶⣦⣤⣄⣀⡀⠀⢀⣠⣼⠏⠀⠀⠈⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠛⠿⣶⣄⠀⠀⠀⠘⣿
⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠶⠶⠶⣶⣦⣤⣄⡀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠹⣧
⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⢿⣷⡀⠈⢿⣤⡿⠀⠀⢀⣴⣿⠿⣦⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⣿⣧⠀⠘⣿⡇⠀⢠⣿⣿⣿⣤⣼⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠃⠀⠀⣿⡇⠀⠈⢿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣼⠟⣷⡄⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⢾⡄⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢃⣾
⠀⠀⠀⠀⠀⢠⡈⢿⣆⡀⠈⠻⢷⣤⣀⣀⡀⠀⣀⣀⣠⣴⡾⠟⠉⠀⠀⠀⠀⠀⠈⠙⠿⢶⣶⣦⣶⡶⠟⠋⢀⣼⠏⣤
⠀⠀⠀⠀⠀⠈⢿⣄⠙⠻⣶⣤⠀⠈⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢫⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠻⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⠋⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠾⠃⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⢿⡄⠀⠀⡄⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡇⢿⠠⠛⠀⡜⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⣤⢰⢔⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣈⣉⣡⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣨⢅⣸⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣵⣻⠜⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣫⢟⣿⣿⡌⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣫⣾⣹⢇⢩⣵⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣼⣷⣫⢖⡯⢸⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡽⠛⢫⣭⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠛⠋⠭⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣞⣲⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢣⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣾⠇⠀⠀⠨⠿⠿⣟⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣸⡟⠀⠀⠀⣰⣦⠦⣠⣤⣀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢱⡿⠁⠀⠀⠀⣿⣿⣿⣿⣷⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢇⡿⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⣾⠃⠀⠀⠀⠀⢨⣿⣿⣛⣿⣿⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣼⠏⠀⠀⠀⠀⠀⣠⣤⣈⣉⣉⡙⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣱⠏⠀⠀⠀⠀⠀⢀⣿⣾⣿⣿⣭⣿⣿⣟⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣉⣈⠙⠛⠋⠶⠶⠿⢭⣭⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠾⠿⠿⣭⣿⣿⣛⣛⣿⣿⡿⠿⢿⣿⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣷⣶⣶⣦⣤⣤⣤⣉⣉⡁⠈⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢠⣿⡟⠙⠿⣿⣷⣶⣶⣶⣾⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣷⠀⠀⢀⣤⣾⣿⠿⠟⠛⠛⠛⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠋⠀⠀⢀⣤⣤⣄⡀⠀⠙⢿⣿⣄
⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⢰⣿⠟⠛⢻⣿⡄⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠁⢀⣀⣼⣿⠇⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⣿⡿⠛⠉⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣷⣦⡀⠀⢿⣿⡀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⣾⡿⠋⠁⠀⠈⠙⠻⢿⣷⣦⣤⣤⣶⣿⡿⠛⠉⠀⠀⠈⠻⣿⣦⠈⢿⣷⣄⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠙⣿⣷⣤⣀⡀⠀⠀⢀⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢰⣿⠏⢛⣿⣿⠃⣠⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣸⣿⠇⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⣠⣤⡀⠀⢸⣿⣀⣿⡿⠀⣾⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⠀⠀⠘⠛⠋⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⢸⣿⣿⣿⣷⣦⡈⠉⠁
⠀⠀⠀⠀⠀⠀⣼⣿⠋⠉⢿⣿⣿⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⡏⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣠⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⡄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠋⠁⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠉⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⡀⢀⣠⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀
⠀⠀⠀⠀⠀⠐⣆⣶⣆⡴⠀⠀⠀⠀⠀⠀⠀⡴⣤⡔⡆⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡄
⠀⠀⣴⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡟⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣼⡏
⠀⠀⣿⣿⣿⣿⣤⣿⣷⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⡇⠀⠀⠈⡿⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠇

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢠⠻⣿⢆⠀⢾⡷⡀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤⠤⢤⡀
⠀⠀⠘⣿⣿⡟⢸⣿⣿⠻⠄⠀⠀⠀⠀⠀⠀⣿⠇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⢤⣇⣀⣼⣿⣤⣀⣣⠤
⠀⠀⠀⠈⠉⠀⠺⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣙⣟⣛⣟⣉⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣶⣷⣷⡆⠀⠀⠀⠀⠀⠸⠿⠋⠉⠉⠉⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢈⣉⣉⡑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡾⠋⠉⠈⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⣠⠟
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠠⢀
⠠⠊⣠⠶⢦⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠶⣄⠲⡀
⡇⢸⠇⠀⢸⠇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠠⡄⠋⣠⠇
⢧⡼⠇⠀⢶⣄⠙⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠳⣄⠀⠀⠀⠀⠀⠀⣴⠃⡴⠶⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠈⠛⠾⠃⢀⡄⠀⣼⠛⠳⣦⡀⠀⠀⢠⡏⠀⠀⠿⣄⠀⠀⠀⢠⡀⣛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⣻⡄⠀⠀⠙⣦⠴⣾⡄⠀⢀⣀⣽⠂⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣤⠿⠚⠛⠛⠻⠟⠛⠛⠛⠳⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡶⢦⣤⡀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⡟⢀⣀⣤⠾⠋⠀⠀⠈⠙⢷⣄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣤⣴⠾⠛⠁⠸⣿⠛⠋⠉⠀⠀⠀⠀⢠⡄⠀⠀⠹⣷⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⢉⣿⠉⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠖⠤⠀⠀⠀⠀⠀⢀⠤⢦⠀⠀⠀⣼⠿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢨⠃⢀⠃⠀⠀⠀⠀⢿⠒⢸⠀⠀⠀⣿⠀⣷
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠰⠀⠀⠙⠁⠀⠀⢠⡿⠀⢹⡆
⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠻⣦⣀
⠀⠀⠀⠀⣾⡋⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⣻⠆
⠀⠀⠀⠀⠈⠻⢦⣄⣀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⣈⣉⠛⠛⠶⠿⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠟⠛⠋⠉⣠⣀⡀
⠀⠀⠀⠀⣠⠉⠉⠁⠀⡄⠀⠀⠀⠀⣠⠝⠻⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⡿⣄⠀⠀⠀⠀⢠⠈⠀⠉⠈
⠀⠀⠀⠀⠑⠀⠀⢠⡀⠘⣄⠀⠀⡾⠁⠀⠀⠀⣭⠙⠛⢷⠶⠶⠶⠞⠛⡿⠀⢸⡇⠈⢳⠀⠀⣠⠇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠣⠔⠙⣦⣠⡔⢁⠞⠀⠀⠈⠓⠤⠀⠤⠞⠀⠀⠸⣧⣤⠋⠳⠔⠁⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⣀⠞⠁⠀⡟⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢸⡏⠈⠳⣄⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⡇⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⠆⠀⢸⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠣⢄⣀⠤⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠲⠲⡶⠒⢶⠖⠶⢶⡶⠲⣶⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣼⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⢸⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣁⣿⠀⠀⢸⣏⣉⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣶⡀