Baby Emojis & Symbols

⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⢀
⠀⠀⠀⠀⠀⠴⡯⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠐⠐⠦⡤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣁⢌⠀⠀⢀⠆⠀⠀⠀⠣⠁⢠⠀⠀⠀⠒⠦⠨⠒⠁
⠀⠀⠠⠆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⢀⣨⢲⣿⠾⢢⣶⡁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⠀⣤⣠⡇⠀⠀⠀⠀⠀⠘⠀⠍⠀⠂⠉⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠉⠉⠉
⠀⠀⠀⠊⠒⠀⢀⣦⠀⠼⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠠⠀⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠂⠘⠆⠠⠁⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣠⠆⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠰⡘⡤⠒⠈⠁⠀⠀⡖⢈⡆⠀⠀⠀⠀⠀⢠⢆⣄⡀
⠀⠀⠀⠙⠀⠍⠀⠀⡀⡀⠃⣤⢀⡄⠀⠀⠚⠁⠀⠀⠀⣑⠄⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⡈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⢠⣽⠀⠀⠀⠀⢀
⠀⢤⡠⡀⠀⠃⠀⠀⣀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠄
⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠂⠂⠀⢠⠆⠀⢈⠁⢀⠤⣴⠃⠀⠘⠀⢴⣄⠀⠀⠀⠈⠀⠘⠉⠉⠁⡀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠀⠀⠀⠀⠘⠆⠀⠈⠀⠘⠛⠋⠀⠁⠀⠀⠈⠉⠀⠀⠀⠈⢀⡁⠀⠀⠀⠈⠀⠀⠀⠀⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠓⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⡀
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠰⠀⠀⠐⡀⠀⠀⠀⢈⠀⠀⠀⡀⠀⠠⠀⠢⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠃⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠘⠆⣠⣞⣧⢀⡀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡆⢀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠐⠀⡀⢀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠰⠂⡾⠋⣉⠞⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠫⠛⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠦
⠀⠀⠀⣄⠀⠀⢀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠂⠁⠘⠒⠛⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠃⢰⡖⠂⠀⠀⠀⠐
⠀⠀⠀⠎⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣿⠇⠀⠀⠀⠠
⠀⠀⠀⠒⠀⠠⠄⠀⠈⡀⠀⢢⠶⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠈⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠓⠀⠇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠆⠠⠦⠿⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣾⣏⣰⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣅⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⠉⠙⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⡿⠁⡀⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⠀⠀⣤⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠆⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⣏⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⢀⣴⣶⣾⣷⣶⣶⣤⣶⡆⠉⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣤⣄
⠀⠀⠀⠀⡾⣥⠀⠀⢘⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⡿⣌⠳⢦⣤⠤⠶⠶⠚⠛⢛⣛⣿⣿⣿⣿⣿⠂⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢰⣷⡌⠙⠒⠦⠀⠀⠠⠖⠿⠛⣉⠝⠛⠛⠿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⡈⢻⣦⡄⠀⠀⠀⠀⠐⠃⢸⠃⠀⠀⠀⠀⠀⠁⠀⣸⠉⠉⢉⠛⢿⣿⣿⣿⣯⠉⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⣳⣄⡉⠛⠊⠀⠀⠀⠀⠀⠀⣦⣷⣦⣄⣀⢀⣠⣴⡃⠀⠀⠠⠀⠀⠓⠄⠀⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⡇⠈⠻⣟⣃⡀⢠⠤⢤⣤⡀⠀⠉⠛⠿⢿⣿⣿⣿⠃⠀⠈⠓⠦⠂⠀⠀⡠⠀⠁⡀⣿⠟⠻⢿⣿⣿⡇
⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠰⠟⠁⠠⠀⠀⠿⣋⠀⠀⢠⠟⠛⠃
⠀⠀⠀⠀⠀⠉⠛⠦⣤⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⣈⣽⠖⠀⠀⢠⡿⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⢿⣍⡉⠙⠛⠻⠿⡷⣶⡤⣤⠾⡿⠟⠠⠂⠀⠈⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣄⡀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢟⡔⢡⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠈⠀⢛⢞⠞⢿⠿⠃⠞⢻⣿⠻⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠟⠘⣿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣷⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠹⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡤⠤⠤⠀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡤⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠘⡆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠋⠉⠁⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡀⠈⢯⡆⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠙⢦⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⡇⢀
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢧⣠⠞⠀⢠⠀⡤⠂⠔
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⡄⠀⠀⠀⠀⠸⠀⠀⣀⣈⠀⠘⠀⠀⣡⡂
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⡞⠉⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠒⢉⡽⠀⠀⠀⢠⢃⡠⠞⠀⢃⠀⠀⢀⡠⠒
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⣇⠠⠁⢍⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠉⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠈⠦⣀⠀⢀⣠⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠈⠉⠉⠀⠈⠑⠦⣄⣀⣀⣀⣀⣀⣀⣀⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⡠⠂⠁⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⡠⠔⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣧⣀⠉⢄⠔⣰⣀⠄⠂⠉⠀⠀⢄⡀⠀⠀⠀⠀⠚⠉⡩⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⠀⠀⡰⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⡅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡤⠀⢱⠀⠁⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⣀⣱⠤⠒⠉⠀⠀⠀⣄⠀⠀⠀⠀⠀⡎⠀⠀⢸⠂⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠒⠘⠣⣄⠀⠀⠸⠀⠀⠀⠸⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣀⣠⠴⠋⠓⢤⣀⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠈⡇⡠⠧⣰⠀⠀⠒⡤⣠⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⠒⠀⠤⠄⣀⢠⢾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠐⠄⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣄⢀⠀⠀⠀⠀⠀⠉⠢⡀⠑⢤⠔⢄⡈⠀⢀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠰⡀⠒⠤⡤⠁⠀⠁⠒⠒⠒⠀⠀⠀⠁⠀⣨⣀⠸⢀⡏⠀⠷⣒⠊⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠑⢄⠀⠣⠀⠂⠤⣀⡀⠀⠀⢀⠠⠓⠀⠘⠒⠀⠠⠘⠚⠃⠁⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠐⠂⠀⠈⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠂⠀⠴⠒⠚⠛⠛⠛⠛⠓⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠦⢤⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠘⠲⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣰⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠘⢦⡠⠏⠴⠂⣀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⣨⠽⠛⢶⡀⠀⠀⠹⡄⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⡞⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣰⠋
⡀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠣⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡼⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡃⠈⠉⠳⣤⣤⣴⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣸⠁
⢳⡀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠛⢧⠀⠀⠀⠀⢀⠤⣄⡀⠀⢠⡇⠀⠀⠀⢀⣠⡴⠖⠂
⠀⠳⣄⠀⠈⠢⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠐⠦⠄⠤⠒⠁⠀⠀⠹⡆⢸⠀⠀⠀⣰⠟⠉
⠀⠀⠈⠑⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠠⢤⡀⠀⠀⠀⠀⢀⣰⢒⡤⢦⠀⠀⠀⢹⢘⡀⠀⣰⠃
⠀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⡴⠁⠁⠈⣠⣃⡤⡀⠀⠀⠄⠀⠀⡏
⠀⠀⣰⠃⠈⠀⠉⢧⠀⠀⠀⠀⠲⢤⡀⠀⢀⠀⠀⠀⠀⠈⡹⠃⠀⠀⠀⢉⣥⢾⡁⠀⠀⠀⠀⠸⡇
⠀⡞⠙⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⣀⡤⠊⠀⠀⠀⣀⣠⠔⠒⠋⠀⠀⠀⠀⠀⠀⠃
⠀⡇⠀⠀⠀⠀⠀⠈⡟⢦⡀⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠀⣠⠞⠁⠘⠆
⠀⡇⠀⠀⠀⠀⠁⠀⠸⣄⠉⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⡤⠖⠋⠁
⠀⠀⠳⣄
⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⠿⠿⠿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⠀⠀⠀⢸⠿⢿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣤⣴⣾⣷⠀⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠠⣀⢸⣿⣿⣿⡿⠟⠀⠀⣠⣶⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠉⠁⠀⠀⠀⢰⣿⣯⣤⣅⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⡀⠀⣶⣶⣄⣸⣿⣿⣿⣿⣿⡆⠀⠤⠀⠐⠂
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠙⠻⠿⢿⣿⡿⠿⠿⣿⣿⡿⡿⠗
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠤⠄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠟⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⡀⠀⠀⠐⡆⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀⠄
⠀⠀⠀⠀⢀⡻⠿⠛⠋⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣾⣶⣿⣶⣾⣿⣿⣿⣿⣷⡿⠿⠿⠶⡁⠰⠶⠖
⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⣄⣀⢂⡀⣉⣁⣽⡿⣟⣽⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⣃⣀⣤⣾⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢔⣿⣿⣿⣿⣿⠟⠿⠿⠁⠀⠀⠀⠀⠀⠀⢻⣿⣟⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠋⠉⡠⠛⢦⠀⠀⢰⡛⠻⠥⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠉⠉⠈⠉⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣀⡶⠂⠀⠀⠀⠙⠳⣄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⣶⠀⠀⣸⡥⠒⠚⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠋⠀⠀⠙⡦⠺⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠊⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃⠀⢀⠀⠀⡟⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠙⢦⡀⠀⠀⠀⠘⠓⢦⣤⣤⣤⣤⡴⠛⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠲⣄⣀⡀⠀⠀⠉⠓⠚⠚⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⢀⡠⠈⠉⠉⢛⡖⠲⣖⠒⠋⠁⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠙⠦⣄⡀⢸⠁⢀⣨⠗⠚⠙⣦⠀⢠⡏⢺⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠠⠤⡖⠛⡶⢧⡘⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠘⠓⠢⢕⡠⣄⣀⣀⣀⣀⣀⣀⣀⣤⡀⣀⠀⢲⠞⠁⠀⠉⠖⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠂⠀⠀⠀⠈⠓⠦⣬⣉⣉⡉⠁⠀⠘⣌⣙⡶⠻⡦⠤⠤⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠉⠉⠉⠉⠉⠉⠀⠀⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⠖⠉⠉⠉⠉⠳⢄⠀⡠⣀⡀⠀⠀⠉⠀⠀⣘⣶⠒⠐⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠛⠐⠁⢋⡗⠀⢀⡴⠋⠀⠘⠆⠈⠀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⠦⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⣸⠤⠖⠛⠦⠤⠤⣀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⢀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⢀⣰⣷⡶⠟⠛⠉⠀⠀⡀⢠⠈⠳⣤⡘⢷⡄⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢶⡀⣿⠀⢸⣿⣿⣿⡿⠿⢶⣒⣠⠴⣫⢞⡟⢠⠐⡌⢻⣾⣿⣦⣄⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢀⣷⠀⠀⠙⢶⣿⣿⣶⣶⣶⣾⣿⡿⠿⣛⣫⣵⠞⣡⡞⣰⢸⡇⠹⢄⠹⣿⣿⣿⣷⣿⣧⠀⠘⡆⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⣾⣦⣉⣛⣛⣋⣭⣭⣭⣴⣶⣾⡿⠟⢋⣴⡾⠋⢠⠇⣾⣿⠀⠘⣆⠹⣿⣿⣜⠿⣿⣦⠀⢿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢁⣤⣾⡿⠋⠀⣴⠏⣼⣿⣿⡄⠀⣿⡀⢹⣿⣝⢆⠹⣿⣇⢸⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⡿⢛⣵⣿⡿⠛⣁⣴⣾⠿⠋⢁⠄⣠⡾⢋⣾⣿⣏⣿⡇⠀⢸⡇⠈⣧⢹⡄⠀⢸⡿⠘⢸⠇
⠀⠀⠀⠀⣀⠀⢀⣀⣤⣾⣿⣿⠿⢋⣴⣿⠟⢉⣴⡾⠛⠉⢀⡠⠚⣡⣾⢏⣴⣿⣿⡟⢸⣿⠀⠀⣿⡇⠀⣿⠀⣿⠀⠘⠃⢺⠟⠀⠀⠀⠀⢴⢤⣀⡄
⠀⠀⠀⠀⠈⠉⠉⢛⣿⢫⡾⢋⡴⠫⠛⣡⠖⠋⠁⠀⠐⠊⣡⣴⡿⢋⣵⣾⣿⣿⡟⢀⣿⡏⠀⢁⣿⣱⢀⡟⠀⣿⡇⠀⣆⢠⡈⠁⠀⢷⡄⣼⣿⣿⣃
⠀⠀⠀⠀⠀⠀⢀⣾⢣⡟⠀⠊⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⠛⣡⣴⣿⢟⣽⣿⠏⢀⣾⡟⠀⠊⣼⣿⠇⣸⠃⠀⣿⡏⠀⠨⣳⣝⡻⣶⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⡾⠁⡞⠀⠀⠀⠀⠀⠀⠒⠋⠛⢋⣩⣤⣶⣿⠿⢋⣴⣿⠟⠁⣠⣿⡟⠀⠀⣸⣿⠏⡰⠃⠀⢰⢟⡇⠂⢀⠈⠛⢿⣿⣿⣿⣛⣋⡤
⠀⠀⠀⠀⠀⢸⠇⡀⠀⠀⠀⠀⢽⣖⣒⣒⣿⣿⡿⠿⠛⣉⣤⣾⡿⠟⠁⣠⣾⡿⠋⠀⢀⣴⣿⠃⠀⠀⠀⣠⠏⠈⡼⠀⠸⡄⠀⡀⣿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⢸⠀⣱⠀⡌⡀⠀⢸⠁⠈⣁⣀⣤⣴⣶⠿⠟⠋⠁⣠⣴⣾⡿⠋⠀⠀⣠⣾⠟⠁⠀⣀⣤⠞⠁⢀⡼⠁⠀⠀⣇⠀⣷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⠃⠀⠚⠦⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⡿⠛⠁⠀⣀⣴⣿⠟⠁⠀⠀⠈⠁⠀⣀⡴⠟⠀⠀⠀⠀⣿⠀⣿⢸⡷
⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠀⠒⡲⠿⠿⠿⠟⠛⠋⠙⠋⠴⠶⠾⠛⠛⠉⠀⠀⠀⠀⠠⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⣿⢸⣿⢸⠇
⠀⠀⠀⠀⠀⠀⣀⣘⢸⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣼⡟⣚⣀
⠀⠀⠀⠀⢠⣿⣿⠿⡷⢿⡀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⣄⠀⠀⠀⠀⠀⡟⢠⡟⠾⢿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⠀⢀⡈⠣⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠋⢀⠀⢦⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣤⡿⠿⣦⡀⠀⠀⢀⢠⣶⣤⣾⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢿⣿⣷⣤⣦⢀⠀⠀⠀⢠⡾⠿⣷⣼⣿⡏
⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⢹⡇⠀⠀⠨⠿⢿⡿⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⠿⠯⠀⠀⠀⣿⠁⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣈⣡⡄⠀⠀⠰⠾⠧⠀⢽⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⠟⠿⠶⠂⠀⠀⢠⣉⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡗⠀⠀⣳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⣒⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⠦⠤⠀⠀⠀⠀⠀⠀⠒⠾⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣧⠀⠀⠀⠀⠀⠀⠇⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣼⢸⣿⣿⣧⡀⠀⠈⠄⠀⠀⠀⠀⣹⣿⢀⣤⣶⣾⣿⣿⣷⣶⣤⣀⢸⣿
⠀⠀⠀⠀⠘⠀⠀⠀⡌⠀⠀⠀⠀⠀⣠⣾⣧⢠⣶⣶⣶⣷⣄⠀⠐⡀⠀⠀⠀⢿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢀⣴⣿⣿⣿⢸⣿⣿⣿⣿⣿⢰⣄⡀⠂⢠⣾⡛⠙⠻⣿⣿⣶⡄⠈⠙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⡔⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⠏⠠⠠⣿⠋⣡⡀⠀⠙⢿⣿⡄⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡀⢼⣩⡅⣩⣭⣭⣭⣭⢈⣭⣭⣭⣭⣭⡈⠀⢀⠀⠀⠀⠄⠀⠀⢆⢠⠋⠀⠀⠀⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠸⢀⣠⢠⠤⣬⣿⣇⡿⠿⠟⠛⠛⠚⢛⡟⠛⢿⡿⣁⣀⠄⠀⠐⠀⢀⠀⢀⠀⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⠀⠀⠀⠐⢀⠈⢰⡏⠁⠐⡄⠀⠀⢀⠔⡁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡡⠄⠀⠀⠀⠙⠄⠀⠀⢉⡀⠈⡷⢖⣶⣞⣻⣄⣁⠔⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠘⠉⠭⣽⠖⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⠀⠀⠀⠀⠀⢀⠓⢒⡄⠀⠒⠦⡤⠤⠤⠴⠒⠋⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠒⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠐⢃⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣀⠀⠤⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠴⠀⠀⠀⠀⠀⠀⣿⣿⢿⠇

⢀⣤
⢸⢛⠀⠀⠀⠲⣀⠀⢀⠀⠀⠀⠀⠀⣐⠀⠀⢰⠆⠤⢦⣤⣤⣄⣀⣀
⢸⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⢀⣀⡆⠸⠃⠀⢀⠙⠚⡻⢯⡙⠲⠨⣧⠈⣠⡂⠀⠀⠀⠀⠀⢠⡆
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⣿⡇⠸⠇⠀⠀⡤⢤⠁⢸⡟⠙⠙⠿⣷⠊⠁⠀⠀⠀⠀⠀⣼⡁⢀⡇
⢸⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠁⡹⡇⢰⠀⠀⢀⠀⠀⠀⢿⢿⠀⠒⢰⡘⢷⠀⠀⠀⠀⣶⣷⡿⠀⣩⡇
⢸⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⣠⣤⣿⡇⢸⡇⠀⠷⠄⠀⠀⣾⣿⠷⡄⠠⣉⣻⣆⣤⡰⣆⣿⣿⠁⠀⣿⣇
⢸⣿⣦⣄⠀⠀⠀⠀⢨⡄⠀⠀⠈⠙⢛⣾⡇⢸⡁⠀⢐⣶⡄⠀⢹⣿⠠⠆⠀⠉⢹⣿⣿⠷⣿⡿⡟⡇⠀⢼⣤
⢸⣿⡿⣿⠨⠀⠄⠀⣼⢇⠀⠀⠀⠀⢠⢿⡧⢸⠆⢠⠤⠹⠇⠀⣾⣿⠃⣤⡀⠀⢸⣯⣟⡿⣛⠀⠷⠁⣴⣿⣿
⢸⣿⠀⢈⡁⠀⠀⢠⣃⡘⡀⡀⠀⠀⠀⠈⣇⢸⣷⣟⣾⣶⢀⠀⢿⣿⣶⣿⣿⣿⣿⣏⣿⡇⠉⠀⢐⣻⣿⣽⣿
⢸⣿⣴⣬⣿⡀⠀⠾⠟⠽⢷⡇⠀⠀⠀⠀⣿⢸⡿⣿⣿⡁⠀⣲⢾⣯⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⡽⣿⣿⣿⣿
⢸⣿⣿⣿⡃⢀⣦⣀⣀⣀⣼⡇⡄⣤⣴⡴⡿⢸⡇⠐⣇⣟⠀⠉⣿⡏⠙⠛⠋⠛⢹⡟⠙⣸⠟⠛⠿⡿⣿⣿⣿
⢸⣿⡿⣛⣿⣿⣇⠀⠄⠀⢻⡇⠰⣮⠔⣯⡇⢸⡇⢀⣿⣿⠦⠀⢿⠓⠂⠀⠀⠀⢸⡆⠀⢸⠄⠀⠀⠀⠼⢿⣿
⢸⣿⣿⣥⣾⣿⠛⠀⠀⠈⢻⣿⠷⣴⣾⣿⡿⢸⣿⣗⣯⣉⣁⡤⠯⠥⠤⠤⠤⠄⠘⠗⠒⠒⠒⠛⠓⠀⠈⠉⠁
⠘⠿⠿⠿⠿⠛⠂⠒⠒⠒⠛⠛⠋⠉⠉⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⣀⢠⣤⣴⣶⣶⣶⣦⣄⣀⣀⣀⠠⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡴⠊⣇⠑⢄
⠀⠀⢀⠤⠒⢉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢆⠀⠀⠀⢠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢣⢂⣀⣁
⠀⠎⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⡄⣀⢈⣄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⠀⠀⠀⠠⠤⠤⠀⠀⢀
⢰⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡆⠀⠩⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠖⡀⠀⠠
⢸⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠁⠀⢀⠔⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠕⢈⡴⢨
⠀⡄⠀⠀⢹⣿⣿⣿⣿⣿⡈⠁⠀⠀⠀⠀⠀⠀⠀⢨⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣠⣶⠟⠃⠂
⠀⠘⡄⠀⠀⠙⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢊⣴⣾⠟⠁
⠀⠀⠈⠢⡄⠀⠀⠹⣿⣿⣿⣆⠀⠀⠈⠀⠀⠀⠀⡠⠃⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⣾⡿⠊
⠀⠀⠀⠀⠸⠒⠠⠀⠀⠉⢩⠍⠋⠖⡀⣶⡒⠒⡘⠁⠐⠀⠶⣓⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁
⠀⠀⠀⠀⠀⠀⢰⠀⢠⠄⡄⠀⠀⠀⡜⠀⠀⠩⠄⠀⢀⠀⠀⠀⠀⠉⠉⢀⠒⠄⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢰⠀⢸⣴⡾⡄⠀⠀⠹⣶⣦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠆⢁⡠⡆⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠻⡀⠀⠔⠛⢿⣿⣿⣿⣿⣿⣿⣿⡖⠒⢶⣻⣿⠿⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣈⠩⠀⠉⠉⠛⠛⠿⠿⡇⠀⡟⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠠⠤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⠀⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⢿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠀⢤⣄⡈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠛⢿⢧⠖⠢⣤⠿⠾⣷⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠹⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣴⣶⣶⡞⣠⣾⣅⣀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣿⣿⣓⣻⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡿⣿⣿⠡⣽⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠀⠨⠀⠍⠀⣽⠉⡇⠈⡉⠨⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣂⡇⣄⠙⠀⠀⠀⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⡖⣋⢹⠛⢀⣥⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣭⣿⡛⠛⠛⠻⠟⢃⣻⠞⠛⠻⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣉⣉⣉⣙⣷⢶⣾⣒⣛⠿⠛⠶⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣁⡀⠈⠙⣒⣦⣤⡶⠒⠒⠲⠶⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣝⡛⠛⡟⠛⠃⢐⣒⡂⢀⡠⠔⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣧⣀⠈⠙⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣠⣾⡿⠟
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⣿⣿⣿⣏⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠻⡿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⡇⠀⢰⡄⠀⠹⣶⣾⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠉⢁⣾⣿⠏⣿⣿⢸⣿⣿⠁
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⠀⢸⣿⣶⣀⣤⡟⠉⠁⠀⠀⠀⠀⠹⡿⠿⣿⣿⠀⣀⣀⣠⣾⣿⣿⣿⣿⣶⣶⡘⠿⠟⠁⠀⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠘⠿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢀⣠⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠈⠙⠿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣤⣾⣿⣿⣿⡿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣖⣠⣶⣶⣾⣿⠀⠉⠉⠙⠯⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣧
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡀⡀⡀⠈⠉⠉⠉⢉⠉⠉⠉⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⢀⣤⣶⣆⣤⠶⠂⠀⠐⠂⡒⠀⢀⣠⠤⠆⠤⠤
⠀⠀⠀⠀⢸⣿⣿⣿⣦⣤⣪⡷⢾⢿⣥⣿⣦⠹⣿⣦⢀⠀⠃⠀⠰⢿⡀⡀⠄⠀⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣤⣤⣤⣀⣠
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣻⠿⣻⣧⣶⣿⣿⣿⣿⣷⠒⠀⠌⡉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠉⢹⣋⣙
⠀⠀⠀⠀⢸⣿⣿⣟⣣⡿⢿⣿⡏⠉⠐⠘⡟⠛⠟⠁⠉⠻⠷⡶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣿⣿⣻⢟⡿⣶⡆⣀⣉⡉⠉⠋⠠⠀⠀⠊⠒⡖⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣿⣿⣿⣯⣼⣦⣽⣿⣿⣢⠌⠀⢀⣄⠀⠀⢘⡁⢀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⢸⣿⣿⠀⢝⣯⣶⣬⣿⡿⣿⣶⣶⣿⣿⣿⣷⣌⣟⠻⢿⡀⠀⢠⣄⣄⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡀⢀⡅⠀⠻⠤⠨
⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⣿⣄⡸⠿⣿⠟⢻⡌⠛⠁⡈⢘⢿⠏⣄⠀⠈⠉⠩⠭⠉⠁⠀⠀⠀⠀⠀⠀⢀⡷⣆⢈⡉
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣻⣟⣽⣷⡤⠀⣎⣁⡀⣀⣄⠀⠀⠈⠊⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⣛⡘⡇
⠀⠀⠀⠀⢸⣿⣿⡛⣿⣿⣿⣿⣿⣼⠉⡡⣇⡤⢁⠀⠛⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢰⣠⣿⣿⣿⡆⢘⠿⠟⠾⢿⣧⣶⣶⣰⣖
⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⠟⢷⠓⡶⠏⠀⠃⢠⠄⠀⠀⠂⢀⣶⣦⣽⣷⣶⣶⣤⣤⣽⣿⣉⣀⠀⠁⣿⣀⠀⢴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠻⣿⡿⠿⠽⢶⣄⠈⠤⠀⣀⣶⣤⣯⣿⣦⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣦⣬⣿⣾⣼⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣠⣿⣿⣴⡖⠀⣠⠁⢀⣠⡉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡛⣿⣿⡛⠛⠛⢻⡟⠛⢛⣛⣛
⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣯⣄⣷⣷⣶⣿⣿⣿⣿⣿⣶⣾⣿⡿⠿⡿⢿⣟⣿⠟⠛⠻⣿⢿⣽⡿⠛⠛⠛⠛⠛⠛⢹⡟⢉⣉⣉⢉
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣉⣉⣙⣛⡛⣋⣀⣀⣀⣉⡉⢁⣀⢀⡇⠀⠉⠁⠀⠀⠀⣾⠠⣿⢣⡀⠀⠀⠐⠂⠀⢸⡧⠽⠛⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡯⠿⠋⠉⣻⡟⢻⢿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣠⡶⣦⣤⣤⣼⣧⣤⣤⣤⣤
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⣡⣴⡶⠟⠋⣉⣵⡦⠻⠿⠿⠛⠛⠉⠙⠻⢻⣿⣿⡿⣟⡛⢻⠛⣻⣿⣾⡛⣟⠛⠛⣿⣿⣿⣷⣿⣿⣭⣿
⢀⣀⣀⣀⣸⣿⣿⠿⢿⣿⣯⣥⣤⣴⣾⣿⣿⣿⣿⣯⣆⣀⣀⣠⣤⣴⣿⣿⣋⡙⠫⣝⣳⣿⡻⢿⠿⠁⢧⠈⠹⠿⠿⠏⠛⠛⡻⠿⠟
⠀⠉⠻⢿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠏⠒⠉⠓⠀⠁⠉⠉⠫⠖⡿⠿⠾⠟⠋⠈⠀⠐⠋⠉⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠈⠻⢿⣿⠋⠉⠛⠛⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⠢⠠⡁⠀⠀⠀⡠⠀⠒⠲⠶⢦⡄⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠄⠀⡈⠀⠀⠀⠀⠀⠈⠙⠱⡄⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠁⢈⠀⠄⣄⠠⠀⡐⠀⠀⡀⠈⡀⠃⠐⠠⡀⠀⢐⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠁⠴⢡⡄⣀⢠⡄⠲⠀⠀⠀⠀⠀⠄⡀⣀⣴⠇⢀⠀⠀⠉⠁⠐⠓⠄⡀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠃⠀⠘⢺⠟⠋⠙⠙⠿⡗⠃⠀⠀⠀⠀⠀⠀⠉⠁⢆⡭⠌⠀⠂⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠀⠂
⠀⠀⠂⠀⠀⠀⣎⠀⠀⠴⠀⠀⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⢎⡀⣀⠀⣀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠿⠓⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⣀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⣦⣄⢨⣦⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⣲⠀⠀⠀⠀⠀⠀⢀⣀⣐⣒⣦⠀⠀⠀⠀⠀⠀⠀⡆⠤⡤⢻⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢻
⠀⠀⠀⢿⣤⠿⡇⠀⠀⠀⠀⠀⠀⠀⢩⣤⡤⠜⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⡞⠀⠀⠀⠀⠀⠀⠀⣿⣉⣁⣜⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠟
⠀⠀⠀⢈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⢇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⠀⠀⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀
⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠐⠀⠒⣦⡀⠀⠀⠀⢰⣦⣄⣀⣀⣀⣀⣠⣾⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⣼⠋⠙⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⣄⡀⠀⠝⠀⢡⠀⠀⠀⠳⠒⠾⠟⠛⠲⠲⠋⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄
⠀⡟⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠒⠠⠬⠭⢤⠴⠖⠛⡆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠃⠀⠀⠀⠐⠀⢀
⢀⣇⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠻⠟⠿⠛⠃⠀⢘⠓⠒⠶⠶⠶⠖⠒⠒⡃
⠈⡏⠉⠛⠉⠛⠋⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠄⣀⣀⣀⠠⠤⢴⡇⠀⠀⠁⠀⠀⣀⣀⠀⠀⠺
⠀⢻⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣦⣤⣤⣄⣤⣤⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠀⠙⡄⠀⣾⡇⠀⠀⠀⠀⡂⠀⠈⠆
⠀⠘⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠄⠀⣿⡇⠀⠀⠀⠀⠁⠐⠂⠀⠀⣰
⠀⢀⡇⠀⠀⠀⠀⠘⣆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠖⠀⠐⠒⠛⠃⠀⠀⠀⣸⠈⠓⠆⠀⠀⠀⠀⢻⡇⠀⠀⡀⠀⠀⠀⠀⢀⣄⣿
⢠⣾⠉⠉⠘⠁⠉⠀⣹⠀⠀⠐⠓⠀⠠⠦⠤⠄⠀⠛⠀⠀⠀⠈⢀⠀⠀⠀⠀⢀⣡⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠁⠀⠀⠉⠉⠉⠉⢠
⠀⠻⢟⣳⣒⡲⠆⠞⠃⠀⠀⠀⠠⢄⣀⣀⣀⣠⠤⠒⠀⠀⠀⠀⢀⣈⣙⣿⣿⠿⠃⠀⠀⠙⠶⣤⣀⣀⣀⣀⡤⠴⠃⠀⠀⢆⠘⢦⣀⢀⠼⠀⡨






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⡓⠙⠀⡴⣫⠹⡆⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣄⠀⠀⠈⠁⠀⡘⡇⠙⠚⠁⠘⡇⢘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢞⠉⠑⠈⢹⠃⠄⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠀⠀⠑⢄⡠⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⡉⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⡿⣿⣿⣿⡟⠙⢿⣿⣿⡿⠿⠟⠀⠀⣤⣤⣶⡶⡞⢩⣬⠳⣄⡀
⠀⢺⡿⠛⢙⣿⣷⣿⣿⣿⠟⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠄⠛⠀⠙⢷⢌⠳⣄
⠀⠀⠀⢠⣿⡟⠛⠛⠉⠁⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⠐⠦⣌⠳⣄⠰⣷⡘⡆
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡌⠂⠈⣿⢘⠆⣹⢄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⡿⣡⠁⡞⢠⠇⡰⢠⣅⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢁⠊⡰⢉⠞⠡⠎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠛⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⠋⡐⢡⢎⡄⠃⡰⠎⣸⠁⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⣿⢿⣐⠁⡌⠃⠀⠀⠫⡀⢁⠘⠁⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣷⣉⠃⢘⠀⢠⡦⠆⢶⡘⡌⠳⠦⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣿⣿⣿⣿⠴⣨⣣⣌⣀⡀⠰⠃⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⠉⠉⠻⣆⠀⣸⣿⣿⣿⣿⣷⣽⣿⣿⣶⣷⡇⠀⠀⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⢻⣷⣾⠀⣿⢠⣿⢟⣛⣛⣛⢿⣿⣿⣿⣿⣿⡇⠀⢀⣀⣈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠛⢁⣼⠃⣾⣽⣟⣿⢿⣿⣧⢹⣿⠛⢿⣿⡇⢠⠏⣡⠤⣀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠈⠀⣸⣿⣝⢯⠼⣻⣿⡿⣸⣿⣷⡌⢻⡇⣊⢸⣠⣦⠀⢻⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣫⡾⢿⣷⣦⣬⣾⣧⠸⡀⠉⠁⢀⣼⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣶⣿⣿⡿⣫⣾⠟⠀⠾⢿⣿⣿⣿⣿⣄⠙⠒⠒⠋⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠂⠂⠙⠻⠿⣛⣿⢋⣾⣽⣿⠶⠤⣤⣤⣿⣿⣿⣿⣿⠟⠢⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠒⠋⡝⣻⠟⢡⣿⡿⣹⣾⣿⣷⣶⣝⢿⠿⠿⠿⠟⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠂⠀⠘⠋⠆⢸⣿⡇⣿⣿⡟⠛⣿⣿⡘⠂⠘⠿⠶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠈⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣯⣿⣿⣿⣿⣿⠇⠈⠂⠀⠀⠀⠀⠀⢔⡢⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⣀⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠲⣤⣀⣤⠴⠂⠐⠦⣤⣠⣤⠖⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣧⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⣼⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣩⡄⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠛⢻⠀⠉⠀⠀⠉⠀⡟⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠁⠀⠉⠒⠤⡀⠀⣐⡤⢦⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡤⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡟⠨⢡⢁⢛⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⡡⠊⠁⡷⡜⢄⠈⢦⣀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣥⢃⡎⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡜⠀⠀⠀⠹⣌⡢⣙⠲⠬⣝⣢⡕⢌⢢⠀⠀⠀⠀⢸⡂⢃⡒⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠐⠀⠀⠀⠀⠀⠀⠩⠽⠝⠲⠠⣌⡙⢦⡳⣋⡃⠀⠀⠠⠇⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣇⠠⠀⣀⠀⠀⠀⠀⠀⠄⠐⠂⠀⠈⠹⣕⢝⡷⠈⡀⠀⠸⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⠢⣾⢁⣀⡀⠀⠀⠈⣴⣷⡶⠀⠀⠀⠈⣿⣵⠊⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢥⣤⡈⠀⢰⠀⠈⠁⠀⠂⠀⠀⠀⠹⣻⣷⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠉⠀⡜⠀⠀⠀⣲⡖⠀⠀⠀⠀⡹⢝⣼⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠉⠁⠐⠮⠥⡁⠀⠀⠀⡔⠀⠈⢏⠙⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠷⡇⠂⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⠉⠆⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⠓⠁⠀⠀⠠⠤⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⢄⡌⠀⠀⠀⠀⠀⢀⠜⢩⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⡠⠀⠀⠀⠀⠀⡰⣶⣶⠘⣄⠀⠀⠀⣠⠴⢻⣙⡀⢠⡚⠤⢀⣀⡀⠤⠐⢁⠔⢁⡀⠤⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠰⣶⣄⠀⠠⢥⠈⠉⠀⠙⠄⣠⠊⢀⡄⠀⠝⡴⠁⠈⠒⠂⠀⠀⠒⠊⠁⡰⠁⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠙⠛⢀⣀⡠⠐⡄⠀⠀⠿⣈⠖⠁⣉⣲⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠠⠀⠀⠂⠀⠀⢠⢺⣤⠀⠀⠀⠀⠑⠤⠥⠘⠀⠀⡰⠓⠁⠀⢀⡜⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠰⢗⠉⠀⠡⠔⣬⣤⣭⣖⣤⣀⠀⠀⠀⠤⢀⡠⢖⣷⣚⣓⣒⠳⢬⣿⠀⠀⠈⢉⣓⠒⠒⠒⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠈⠰⢆⠀⠀⠀⠀⠀⠈⠫⠋⠙⢯⡥⠚⣋⠁⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠉⠁⠀⠀⠀⠀⢇⣀⣦⣘⣆⠤⠴⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⣀⣀⠀⠀⡠⠞⡸⢛⠕⣠⠄⠀⠀⣀⡠⡀⠀⠀⠑⠢⢄⠀⠀⠀⠀⠀⠀⠘⡤⠠⠄⠀⠀⠒⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠈⠃⠐⠶⠑⢘⠥⠊⠀⠀⠀⡀⠈⢦⠀⠀⠀⠀⠈⠒⠠⢀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠱⣀⡀⡀⣀⠀⠀⢠⡞⢀⠄⠀⠑⢄⡀⠀⠀⠀⠀⠀⠈⠐⠢⢄⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠦⠀⠀⠀⠠⡄⠚⠀⠀⡟⢠⠏⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⠠⡈⠢⡀⢸⡇⣼⠀⠀⠀⠀⠀⠀⠀⠍⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣌⠳⣌⣓⢊⡿⠁⠃⠀⣠⠔⠂⠁⠀⡰⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡊⠑⠂⠙⠧⣀⠀⠀⢸⠁⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠁⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⣍⣉⣶⢰⡂⠀⣢⣀
⠀⠀⠀⠀⠀⠀⢀⠴⡅⣁⣫⡉⡕⠒⢻⡛⠛⣽⡇
⠀⠀⠀⠀⠀⡠⠿⣈⣻⡛⢩⠐⠳⠿⠛⢃⡞⠹⠁
⠀⠀⠀⠀⣴⠂⠙⢻⣛⠅⠺⢴⣶⡿⣠⡟⠀⢲⣇⣀
⠀⠀⠀⣎⡈⠛⠛⠿⠙⣗⣦⣶⡇⣰⣗⣀⢀⡟⡶⣭⣿⣭⠑⠲⡂⠠⣤⣄⣀⡀
⠀⠀⠀⡆⠂⠈⠉⠁⠒⠒⠂⠠⡾⢹⡣⣀⡾⣿⣛⣻⣋⡹⢵⡶⠏⢿⣤⣨⣦⡜⠛⣓⠒⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⠇⣧⣟⢮⡷⢿⣛⠚⠵⣶⣿⢴⢉⣉⣯⣉⠿⢿⠟⣼⠟⠀⠀⠀⠀⠀⠀⣤⢰⢀⣰⣶⣦⣬⣯⣟⣤⣿⠿⣾⣷⢄
⠀⠀⠀⠗⠠⠄⣀⠀⠀⠀⠀⠀⢷⣿⠣⡿⢑⠀⠀⠀⠈⠁⠐⠀⠂⠭⠟⣋⠙⠶⠾⣿⡇⣷⣀⣀⠀⠀⠀⠀⡏⢆⢹⢥⣭⣾⣧⣤⣷⢷⣬⣷⣿⣤⣿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⣫⢇⢃⣿⡂⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢙⡝⣤⠿⣶⣶⡷⣶⡀⣀⣃⠀⣳⠛⠛⠛⢟⢍⢝⠝⠍⡡⠉⠈⠀⢨⣗⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡶⢿⠫⡫⠑⠒⠢⠤⢄⣀⡀⠀⠀⠀⠀⠈⠸⣡⡟⢀⣿⣷⠶⠯⢿⣤⣈⣷⢷⠔⢕⠔⠀⠑⡡⠠⢊⢀⢔⣀⣴⣸⣏⣀⠍
⠀⠀⠀⢶⠂⠒⣄⣀⡀⠀⠀⠀⣏⡾⠀⢸⠕⢌⠂⠀⠀⠀⠀⡆⠀⠉⠉⠒⠒⠤⠼⣯⡐⢟⣫⣽⡆⡔⣈⢛⣏⢡⢿⡛⠋⠛⠛⠘⠙⠉⠉⠉⠉⠉⠉⠁
⠀⡠⠁⠼⢽⠿⠏⢹⣢⣤⣾⣶⠛⡁⡈⢿⣤⣔⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠦⠵⢿⣿⣳⡿⢹⡍⡁
⣔⠀⢷⣦⣴⡇⢰⣅⣉⣳⣴⠃⡸⠀⣱⠟⠅⡩⠉⠉⠉⠒⠒⠧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⢂⢀⣀⡀
⡏⠉⠐⠒⠂⠠⠬⠽⢟⣿⠃⢰⢡⡷⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠤⠀⣀⡀⠀⠀⠀⠀⡆⣹⡼⠃
⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣱⠎⠉⠁⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠉⠣⠿⠁⠀⠩⠠⠄
⠙⠢⠄⣀⣀⠀⠀⠠⠀⣆⡾⠃⠐⠒⠠⠀⠀⡠⠤⣁⣐⣾⣣⣤⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⢀⡠⠤⣤⣆⣐⣾⣯⣭⡿⢆
⠀⠀⠀⠀⠀⠈⠉⠉⠒⠛⠂⠀⠉⠉⠉⠀⠀⠈⠻⡛⣯⣉⣱⡿⣶⡼⠧⣄⡀⠀⠀⠀⠀⠀⠀⢮⠹⣣⣤⣼⠿⣶⡶⣟⠚⡿⠓⢌⣽⣉⣺⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠫⡉⣏⠉⠐⣗⣤⣼⣧⣌⡀⠀⠀⠀⠀⠀⢣⠠⡯⣉⣹⠾⣢⣴⡾⢧⣿⣿⢽⠿⠿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠘⢎⠻⢿⡟⠁⠽⣄⣀⣹⣆⠀⠀⠀⢆⠀⢧⠬⠿⢷⠖⢞⠟⢉⢍⠕⠁⠀⠀⠀⣽⣒⣒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⣌⠧⣶⠾⠛⣛⢫⢞⡇⠀⠀⠌⢆⣸⢀⠔⠁⠀⠅⠔⡁⢠⣪⣨⠦⠾⠛⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠣⠤⠖⢛⡬⢢⣾⡏⠅⠠⠄⠈⠷⣥⠦⠶⠶⠺⠮⢙⣉⣁⠀⠐⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢐⡇⢄⢐⣐⣭⠿⠃⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⡷⠛⠁⠀⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣖⠾⠟⠛⠙⠓⠲⠶⣤⣀
⠀⢠⠤⠤⠤⢤⠤⠤⠤⢤⠤⠤⠤⠤⣤⣤⣤⣤⣤⠤⠤⢤⣤⣴⣿⡟⠋⠩⣖⣤⡔⢾⠳⣦⢶⢶⣍⠉⠁⢤⣤⠤⢤
⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣿⣿⣫⠞⠉⠉⠀⡀⠈⠛⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠿⣭⣭
⠀⢺⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠀⠀⠀⠀⠙⠲⢄⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢺⣿
⠀⢸⣿⣿⣿⣿⡿⠿⠛⢛⡉⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣀⣀⠀⠀⠀⢃⠰⠄⣀⡐⠀⣀⠀⠀⢠⠿⣼⣿
⠀⢸⣿⣿⡿⠃⠀⠒⠀⠀⠁⡀⠈⠦⡼⢿⣿⣿⣿⣿⣿⣿⠏⠀⢀⡄⠈⠳⡀⠀⠐⠃⠑⠘⠇⠁⠁⠀⠀⠈⢀⣻⣿
⠀⢸⣿⣿⢄⠀⠄⢠⣖⠒⢪⠏⢑⡌⠀⠈⣿⣿⣿⣿⠛⠁⠀⠘⠛⠗⠖⠀⠉⠀⣨⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⢼⣿
⠀⢸⣿⡏⠀⠁⢨⡁⠀⠀⠀⠀⠉⠀⠀⣀⣸⣿⣿⣿⣦⠲⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⢮⠀⠀⠀⣻⡆⠠⢃⣾⣿
⠀⢸⣿⣿⡐⠀⠌⣂⡀⠀⠀⣠⡄⠀⠐⠋⠉⢿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⣯⣵⡼⣀⠠⡄⡤⣠⡀⡫⠿⠞⢿⣿⣿
⠀⢸⣿⣿⡵⢰⠳⠟⠀⠀⠘⠈⠀⠀⠆⣘⡀⢸⢻⠿⣿⡏⠁⠀⠀⠀⠀⠀⠈⠛⢭⣚⠿⠾⣒⡈⢸⠁⡀⣶⢾⣿⣿
⠀⢸⣿⣿⣿⣿⣦⡔⢂⡄⠀⠀⠀⠀⠈⠉⢀⠎⢘⡠⠄⣉⣋⡷⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠰⣜⡴⣿⣶⣿⣿⣿
⠀⣾⣿⣯⡿⠟⠉⠀⠀⠹⡢⠄⠤⠤⢐⡲⠧⠤⣥⠔⡋⠉⠀⠀⠀⠈⠻⡉⠉⠁⠀⠀⠀⠀⠀⠀⢿⣼⣾⣿⣿⣿⣿
⠀⣩⡾⠁⠀⠀⠀⠠⠀⠀⠀⠄⣀⡉⠰⡅⠀⠀⠀⢱⡀⠀⢀⣀⣀⣀⣠⠧⠶⠃⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿
⠀⡿⠀⠀⠀⠀⠀⡆⠀⢀⣀⡠⠀⠀⠀⠉⠚⠜⣒⠯⠔⠚⠛⠫⣝⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⢰⠁⠉⣩⠅⠒⠒⠲⡶⠦⠤⣐⠒⠒⣒⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠙⠪⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣮
⢿⡀⠘⠀⠀⠀⠀⡎⠀⠂⠤⣲⠦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠐⠫⢓⣲⠴⠴⠶⣵⡶⠲⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠋⢁⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣶⢀⡁⠀⠀⣨⡄⠀⣼⣿⢷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣦⣿⠟⠉⣹⢻⠟⢰⠿⡯⠊⠀⠀⢀⣠⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠈⠛⠋⢠⣷⠃⠀⠀⠀⢀⠴⠛⠛⠝⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠸⠋⡀⠀⢠⡶⢋⡀⠀⠀⢸⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢠⣼⣧⡀⣾⠁⠀⢀⣀⠜⢻⡟⠀⠝⠓⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢀⠀⠉⠉⢉⡳⠦⢶⣿⡀⠄⣈⠓⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢺⡋⠈⠐⠂⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⢦⡄⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣿⣟⠁⠀⠀⢧⡀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣸⠁⠈⠀⠀⠈⠿⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⢐⠃⠀⠀⠀⠀⠀⠀⠘⢢⡀⠀⠀⠠⠀⠐⠈⠠⡄⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⣿⠀⠀⠀⠀⠐⢘⠉⠢⢄⢳⠀⠀⢀⢀⣀⠀⢠⠄⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⠁⢀⡇⠀⠀⠀⢀⡀⠀⠒⠀⠢⠘⢀⣠⣼⡏⠀⠁⠐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡋⠈⣿⡇⠀⠀⠀⠈⠀⠁⠀⢸⣶⣯⡍⢿⢺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠀⠀⣾⠇⠀⠀⠀⣀⠀⠀⠀⢸⣿⡞⢁⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠀⣸⡿⠀⠀⠀⠀⠉⠀⠀⠀⢸⣿⣧⢨⡄⠀⣀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡯⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⡛⣁⣿⣾⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠯⡍⣛⣿⠟⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⢿⣟⣿⣿⣗⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡴⣦⡄
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠂⠀⠤⠶⠖⠛⠛⠛⠛⠛⠳⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⣄⣀⣀⣠⣤⠶⠟⠛⠉⠉⠀⠀⠀⠈⠉⠛⢦⡀⠀⠀⠐⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡼⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⣴⢋⡄⠄⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠤⠒⠒⠛⠉⠉⠙⠛⢓⣶⢤⣀⠀⠀⠈⢳⡄⠩⢤⡶⠃⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣴⠋⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠀⠙⣆⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣸⠃⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠋
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⢸⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣰⠃
⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⠀⠀⠈⠱⢶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢰⠃
⢻⡄⠀⠀⠸⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠛⢦⠀⠀⠀⠀⠀⡠⠦⣤⡀⠀⠀⡏⠀⠀⠀⠀⢀⣤⡶⠞⠋
⠀⠻⣆⠀⠀⠙⢦⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⢦⣤⡤⠔⠋⠀⠀⠀⠹⣆⢰⡇⠀⠀⢀⣴⠟⠁
⠀⠀⠈⠑⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠠⠤⣄⡀⠀⠀⠀⠀⠀⠀⢀⡖⣦⡤⡄⠀⠀⠀⢹⡘⡇⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⣠⠎⠛⠠⠋⣰⠃⣀⠀⠀⠀⡇⠇⠀⢸⡇
⠀⠀⠀⡼⠉⠋⠙⠦⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢴⠃⠀⠀⠀⠰⠓⢋⡽⠃⠀⠀⠀⠈⠀⢸
⠀⢠⠴⠇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠙⢦⡀⠐⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⢘⣋⡽⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠙⢆⣀⣠⠤⠖⠉⠀⠀⠀⢀⡴⠶⡏⠁
⠀⡏⠀⠀⠀⠀⠀⡀⠀⢯⠛⢦⣀⠀⠀⠀⠀⠀⠌⠉⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠙
⠀⣧⠀⡆⠀⠀⠀⠁⠀⠈⢦⡀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⡤⠔⠊⠁
⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠙⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠄⠀⠀⠀⠀⠀⢀⣤⠈⢕⠆⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠆⡀⢠⣈⡷⠊⠑⠦⢄⣀⡘⠂⠈⠓⠁⢀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠧⠛⠊⠉⠀⠀⠀⠀⠀⠠⡛⢍⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⣀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢣⡀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⡀⠀⠀⠀⠹⡀⠀⡇
⠀⠀⠀⠀⢀⡀⠀⢀⠎⠀⠀⠀⡠⠒⠉⠉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠉⢱⠀⠀⠀⠀⡇⢠⠃
⠀⠀⢐⡮⠳⣄⢱⠃⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⢣⠃
⠀⠀⠆⠀⠀⠀⠘⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟
⠀⠀⠱⡀⠀⠀⠀⠀⢠⠃⠀⣀⣠⠄⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠸⡄⠀⠀⢀⠔
⠀⠀⠀⠙⣄⠀⠀⣰⠁⢀⠺⣅⢸⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⠌⠙⠢⠒⠁
⠀⠀⠀⠀⠈⠙⠒⠉⠙⠉⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⢀⡠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣓⡦⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⡴⠖⠚⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⠁⣠⡶⠉⠀⠙⠳⣝⡆⠀⠀⣠⡴⠋⠉⠙⠋
⠀⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⡟⠀⠀⣦⠀⠀⠀⠉⠲⣴⠋⠀⠀⠀⡼⠁
⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠈⠁⠀⠚⠁⠀⠀⠀⠀⠀⠘⣾⡄⠀⢸⠁⠀⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣟⠛⠃⢀⣴⠀⠲⠦⣀⠀⠀⠀⠀⣸⠀⠀⢹⣷⡀⠘⣇⠀⠳⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣿⠃⣀⡼⠀⠈⠓⠀⠀⣠⠏⠞⠀⢀⡇⠱⡄⠈⠳⢦⡙⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⡀⢀⣀⣀⡀⠔⠁⠀⠀⢀⡼⣧⠀⠙⣆⠀⠀⠙⠲⣝⣦⡙⠦⣄⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢿⣝⣒⣾⠋⠉⠛⠲⢾⣉⡁⠈⠳⢄⡈⠳⢤⡄⠀⢀⣹⣿⣷⣄⣙⣦⣤⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣥⣒⣒⣒⣒⣀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠷⠶⠶⢄⠁⠉⢙⡛⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⠀⠀⢀⣀⣀⣀⠹⣎⢻⡄⠈⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠉⠉⠉⠙⠇⠹⠆

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

⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⢀⢀
⠀⠀⠀⠀⠀⠈⠀⡀⠀⢀⠀⡤⠐⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠰⠶⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⡎⠀⢀⠀⠄⠀⠢⠀⡀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢀⠀⠀⡀⠔⠢⠄⠀⢴⠁
⠀⠀⠀⠀⡄⠀⠀⠀⠈⢂⣡⡀⠀⠚⠃⠀⢆⡐⠈⠀⠀⠀⠀⠘⡀⣪⣄⢀⣍⢠⠐⠁⠀⠀⠀⠀⠀⢣⠀⠂⢀⠐⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠚⢄⠀⠬⠑⢀⣾⠖⠉⡁⠀⠀⠀⠀⠉⢲⠀⠄⠀⠂⠐⡀⠙⠫⡘⠛⠀⡃⠀⠀⠀⠀⠀⠀⢀⠀⠙⢚⣢⣄⣢⣄⣔⡀
⠀⠀⠀⠀⣈⠑⠐⠘⠣⠜⣄⡀⠀⠀⡀⠀⠀⠀⠀⠆⢀⠀⠀⠈⠐⠠⠈⠉⠤⠊⠑⠄⢀⠀⠀⠀⠀⠀⠟⠃⠀⠀⠀⠀⠈⠉⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡱⠀⠀⠀⠧⠀⡇⠁⠀⠀⠈⡤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠰⠄⣼⠀⠀⠀⠀⠀⠈⡵⠲⢦⣀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠒⠁⠀⠀⠀⢀⡰⡀⠀⠀⠀⡠⠠⢦
⠀⠀⠀⢸⠀⠂⠀⠀⠀⡀⠇⠄⠐⠉⠁⠀⠀⠀⡀⠀⠀⠀⠠⠀⠄⢅⠀⠀⠖⠁⠃⠀⠘⠁⠀⠀⡄⠀⠙⠒⠤⠀⠀⢀⡀⠠⠄⠊
⠀⠀⠮⣀⣀⠖⠒⠒⠉⠂⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⡀⠁⠀⠀⠀⠸⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠙⡲⠦⣔⠊⠀⠀⡎⠀⠀⠂⡄
⠀⠀⠀⠀⠀⠀⡲⠀⠀⠀⠀⠀⠠⠄⠸⠀⣒⡀⠁⠁⠁⠀⠀⠀⠐⢠⠀⠄⠒⠐⠐⠂⠤⢄⠀⢠⠈⢀⣤⠉⢁⠀⠀⠘⠋⠉⠉⠑
⢀⠀⠀⢀⠔⠀⠀⠂⠀⣀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠂⠈⠁⠤⠀⠱⠄⠀⠄⠔⠂⠠⠃⠀⠼⡇⠉⡁⠀⠿⡃⠀⠀⠀⠀⠂
⠨⠀⠀⠁⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⠃⠉⠉⠉⠙⢊⠁⠀⢀⡀⠀⠁
⠀⠳⡀⠀⠀⠀⡠⠀⠌⠀⠀⠀⠀⠐⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠐⠀⣄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠⠀⠀⣄⠀⠐⠀⠀⠉
⠀⠀⠈⠒⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠶⣄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⠄⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢰⠇⠀⢼⢊⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠄⠀⠀⢸⡯⠥⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠢⡀⠀⠀⠐⠢⣤⣤⣠⣤⠜⠂⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠲⠤⣄⣀⣀⣙⠛⣉⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠰⣄⠀⠀⡘⠉⢫⣥⠤⡀⠀⠠⢤⠲
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⠎⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠋⠠⡤⠟⡴⡙⠈⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠙⠒⠤⣤⠤⣤⣄⣀⣠⠤⣄⣀⠀⡶⠁⠀⠱⠂⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠒⠦⠤⠤⠤⠵⠤⠛⠙⡇⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠔⠒⠒⠲⢄⠀⣀⠀⠀⠀⠒⠉⠲⡤⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢠⠃⠀⠀⠀⠀⠀⠙⠈⠙⡵⠀⡠⠂⠉⠙⠀⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠢⠤⠤⠤⠤⠤⠤⠤⠓⠊⠁⠐⠒⠠⠤⠤⠃


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⠋⠉⢹⠋⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣼⣿⣿⣿⣿⠃⠈⠙⠛⠛⠉⠀⠀⠀⢀⠅⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⣼⠟⠀⢀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⢀⣾⠋⠀⢀⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⣼⡏⠀⣰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢠⠾⠛⢓⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⢀⣿⢀⣼⣿⣿⣿⣿⣿⣿⣿⠏⠠⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⡇⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⣈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠀⠀⠀⠑⢄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠹⡿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣼⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠙⢮⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣄⡀⢀⣴⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠎⠀⠀⠀⠉⠉⠉⠉⠙⠒⠲⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⣠⣶⣿⣿⣷⣈⣿⣝⢿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣀⣀⠀⠀⣠⣾⣿⣿⣟⠷⠞⠉⠉⠉⠀⠟⠿⢿⣿⣷⡄⠒⠚⢉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢸⣿⣯⣹⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠐⢶⣷⣦⣿⣿⠀⠀⠉⠉⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⡜⠁⢀⢀⣿⣿⣿⣿⣿⣏⣻⠀⠀⠀⣀⠈⠑⡄⠀⠀⠀⠀⢙⣿⣻⣧⡤⠖⠚⠉⠁
⠀⠀⠀⣠⠴⠚⠛⢛⣷⣶⣾⣅⡀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⡿⢿⠛⠉⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⣠⠟⠋⠁
⠀⠀⠀⢀⣐⠔⣀⠄⠉⠛⣿⣟⡿⣶⣄⠀⠀⠀⢸⠙⢿⡿⠻⠷⢋⠀⠀⠀⠀⠀⠀⠀⢀⠞⡀⠉⠣⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠉⢻⣷⣶⣤⣠⠄⠘⠿⣾⣿⣷⣄⢀⡼⡆⠠⢐⠀⠀⠸⡄⠀⠀⢸⣶⣄⠀⠆⠀⠙⢢⠖⡠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣾⣁⠄⠈⠿⣭⡻⣿⣼⠃⠄⠨⠀⠀⠀⠙⣆⠀⠀⠙⠛⣀⠤⣢⠖⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠘⢶⣤⡤⢀⡀⠙⠿⣿⣦⡀⢀⠈⠹⣯⣿⣶⣂⠎⠀⠀⣀⣤⣾⡷⠒⠒⠉⠁⣘⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠙⢿⣿⣔⢀⡀⠀⠘⠿⣷⡴⠂⠀⠈⠉⠛⠦⣀⠹⠿⠛⠁⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⡤⠀⢀⠀⠀⠉⠑⠲⢶⣄⣀⠄⠈⠙⠲⢤⣤⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠥⣖⠁⠀⠀⠀⠀⠉⢻⣿⣦⣴⠄⣀⠈⢙⣦⣀⠈⠑⠶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣴⣭⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠮⣁⠤⠀⠀⠹⢿⣿⣯⡁⠀⣀⠘⢿⣷⣄⠀⠀⠉⠓⢦⡀⠀⠀⢴⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁⠀⠀⠙⠻⠿⣌⠄⠀⠀⠹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠓⠦⠤⠖⠋⠙⢦⣀⠆⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣄⠈⠀⠀⠀⠀⠀⠀⠀⢀⡔⠩⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠙⠲⣦⡀⠀⠀⠀⣰⠏⠀⠀⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⣠⠞⢳⡝⠲⣤⣼⠃⠀⠀⠀⠠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠚⠒⠂⠘⠁⠀⠀⠙⠛⠛⢻⡀⠀⠀⠀⠀⠈⠛⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⢠⡀⠀⢀⡴⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡤⠤⠤⣤⣤⣼⠳⣖⠋⠀⠀⠀⢠⠞⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⠀⠀⠀⠈⢹⡟⠛⠳⣤⣀⠀⠘⣷⠶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠈⠛⣏⠁⠰⣞⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠃⠀⠀⠀⢰⣏⠀⠀⠀⠀⣀⠀⣾⣄⠀⠈⠙⠚⠦
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣤⡞⢻⣄⠀⡼⢻⣉⠛⠀⠸⣄⣠⡄
⠀⠀⠀⠀⣀⠀⣰⠏⠉⣿⢣⠾⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣰⠀⡜⣼⠛⠡⠚⣻⡗⣾⡓⠀⠀⠑
⠀⠀⠀⢀⠋⠉⠁⠀⢠⣿⡆⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⣼⡉⠀⠀⠀⡙⠀⠈⠙⢆⠀⠤⣄⣀
⠀⠀⣀⣈⣣⡀⠀⠀⣺⣿⣥⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠝⠀⠀⠙⣇⣀⢄⠁⠀⠀⠀⠀⢻⡤⠞⠁
⢀⡴⠋⠀⠉⠻⢦⡾⢻⣯⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠒⢲⡶⠚⠑⡟
⠈⠻⣄⠀⠀⢠⡟⢁⡾⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡤⠀⠀⠀⠐
⠀⣠⠟⠀⣴⢋⡴⠛⡁⢺⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡑⢾⣷⠂⢈⡁⠐
⠰⡏⠀⠀⣻⣸⡇⠈⠀⢈⡇⠀⣠⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⡀⣀⡹⡄⠈⠂⠈⠃
⠀⠙⢦⠄⠹⠏⠀⣀⣀⠀⣇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⠼⠀⠉⠻⣏⠛⢻⣄⡀
⠀⠀⠀⢀⡠⠖⠚⠁⠀⣬⠁⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⣀⠇⠀⠀⠛⠛⢧
⠀⠀⠐⢋⠀⠀⠀⠀⢀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⠀⠀⠀⢀⣻⠆⠀⠀⢀⠠⠬⣄
⠀⣟⠉⠁⠐⣀⠀⢀⣼⠀⠀⣠⣎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡁⠀⠀⠈⠉⠛⡞⢻⣶⡶⠋⠀⠀⣠⡿
⢀⡞⠁⠀⠀⠈⣳⣮⡟⠀⠘⡇⠈⠀⢀⡴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄⠀⠀⠀⣸⠋⠈⢷⣄⠀⠘⣯
⠈⠳⠳⣄⡀⠛⠙⠏⠀⠀⣠⠇⠀⢀⣜⣳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⠜⡌⠀⠀⠀⠛⢦⠄⠚⠃
⠀⠀⠐⠈⣷⡐⠠⣦⣤⣞⢙⡆⠀⠀⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢦⣤⠴⢦⣄⣼⠃
⠀⠀⠀⠀⠀⣀⡀⠀⡼⠀⠈⠑⠀⠀⠚⠉⢻⣄⠀⢀⣀⡀
⠀⠶⡤⠀⢀⡟⠈⠉⠁⠀⠀⠀⢀⡴⠒⠢⠼⠋⢱⠎⠀⠘⢧
⠀⠈⠐⠐⡺⣁⣀⡀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⣨⠁⠀⠀⠀⠉⢱⠆
⠀⠀⠀⣼⠁⠀⠀⠉⠋⠰⢾⣅⠀⠀⠈⠑⢦⣠⣿⣀⣀⣀⣠⣤⣞⣀⢀⣠⡀⣦⠢
⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⣇⡞⠀⠀⠀⠀⠰⠿⣾⡋⠉⠉⠁⠀⠈⠙⡆
⠀⢀⠀⠀⠀⡟⠀⠀⠀⠛⠻⠤⣄⠀⠀⠈⠀⠀⠸⡇⠀⠀⠀⠀⢀⣼⠁
⠀⠈⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⢳⡀⢀⣴⠟⠉⠁
⠀⠘⠀⠀⠀⠀⠀⠀⠘⠦⣤⠖⠢⢬⡷⠛⠓⢦⡴⠋⠉⠛⠁
⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⡄⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣿⣿⡇⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣐⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣀⠀⣸⡿⡱⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠋⠀⠈⣉⣭⣿⣿⣿⣿⣿⣿⣿⡏⠟⠉⠉⠉⠁⠀⠀⢀⠀⠈⣷⡿⠐
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⢟⣽⣿⣿⣿⣿⣿⠏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢩⡿⢡⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⣻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠏⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣷⣤⣀⣀⡾⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠠⢊⣵⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠘⠻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠛⢿⣿⣿⣇⡀⠀⠀⠈⢿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆⠀⠀⠀⠉⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠻⡟⡀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡷⠁⠀⠀⠀⠀⠀⠀⢨⣿⡇⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠛⠲⢤⣄⣀⠀⢀⣠⣿⠟⣿⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⣹⡄⠀⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⣿⣿⣿⣿⣷⣦⣤⣶⣟⣀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⣶⡤⢤⣄⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠛⢿⡕⢦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠠⢴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⡶⠄
⠀⠀⠀⠀⠀⠀⠰⢾⡏⠀⢀⠂⠈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⢀⠤⠒⠈⠁⠀⠀⠒⢀⠤⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠠⠖⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠰⣶⣄⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠒⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠈⢻⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠁⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⢿⣻⡃⠀⠀⠀⠀⡄⠐⠒⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣢⣀⢀⠀⠠⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠠⡔⠢⢄⡀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠙⠲⠀⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⢘⡣⠀⠀⢣⣈⣉⠉⣉⣀⠔⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠃⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠄⠀⠀⠠⡠⠳⠈⠉⡐⠀⠀⠀⠀⠀⢠⠂⠈⢢⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣀
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠠⠊⠀⠈⠢⠠⠜⠀⠀⠀⠀⠀⠘⢄⣀⠜⠀⠈⠑⢄⠀⠀⠀⠀⠈⠀⠈⠓⣄
⠀⠀⠀⠀⣀⢀⠀⣠⠞⠒⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠶⢤⠀⢀⠀⠀⢀⢺⡄
⠀⠀⠴⠛⠁⠨⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢬⡉⠁⠉⠒⠶⠋
⠀⢸⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⣄⡀
⠀⠸⡄⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠠⠀⣀⣤⠞⠛⠓⠦⣤⣀⣀⡀⣀⣀⣠⠶⣄⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠙⠦⣄⡀⠀⣨⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⣇⠀⠇⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⢀⣀⠠⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⣩⣷⣶⣶⣶⣶⣶⣯⡈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠻⠿⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⣐⣒⣠⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡉⠙⠛⠉⢹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡿⣿⣿⣿⠿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣿⣿⡿⢻⣿⡟⠉⠈⠹⣿⣿⠻⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣀⣺⣿⣇⠀⠀⢀⣿⣿⣇⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⢀⡴⠋⠳⢦⡀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡷⠶⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢡⣦⢹⡏⠀⠀⠀⠀⢹⠡⣶⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠴⠊⠀⠀⠀⠀⠀⠈⠳⠴⠞⠁
⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢤⣤⣀⣀⡀⠀⠀⠀⣀⣀
⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢾⣿⣿⡧
⠀⠀⠀⢸⣿⠈⠁⠉⠈⡇⢸⠀⠉⠈⠅⢹⠀⠍⠈⠁⢹⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠁
⠀⠀⠀⢸⣿⢀⠀⣀⠀⡇⢸⠀⣀⠀⡃⢸⠀⡃⢀⡀⢸⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣄⡀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠀⠀⠀⠀⠀⠁⠀⢩⠉⠁⠀⠀⠀⣿
⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢀⠀⢠⣤⡄⠀⠀⢀⣠⣄⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⠿⠃⠐⣾⡆⠈⠋⠀⠀⣦⠸⢿⠿⠀⠀⠀⠀⠉
⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠰⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣛⡿⠟⠻⢿⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣤⣆⠉⣷⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠉⠙⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⡠⠄⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡏⠀⠀⣅⣠⠀⠀⢹⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠤⠄⠒⠒⠒⠛⠛⠓⠒⠒⠤⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠤⠴⠶⠶⠶⠶⠶⠤⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠾⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣄⡀
⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄
⠀⠀⣰⣿⠟⠉⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠀⠀⠀⠈⠛⠷⢶⣶⠶⠾⠋⠁⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠉⠻⣿⣆
⢀⣾⡿⠃⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⣠⣴⡾⠟⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠀⠀⣠⣤⠀⠀⠀⠀⠀⠘⢿⣷
⣼⣿⠃⠀⠀⠀⣠⣶⠿⠋⢁⣤⣶⠿⠋⠁⠀⢀⣠⣶⠿⠛⠁⠀⢀⣠⣴⡿⠛⠁⠀⢀⣤⣶⠟⠋⠁⠀⢀⠀⠀⠀⠀⠘⣿
⣿⣿⠀⠀⠀⠀⠉⣀⣴⡾⠟⠉⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⣤⣶⠟⠋⠁⠀⣠⣴⡾⠛⠉⠀⠀⣀⣤⡾⠟⠁⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠰⠾⠛⠉⠀⠀⣀⣴⡾⠛⠉⠀⠀⣀⣴⡾⠟⠉⠀⢀⣠⣴⠿⠛⠁⠀⠀⣠⣴⡾⠛⠉⠀⠀⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠁⠀⢀⣠⣴⠿⠛⠁⠀⣀⣤⡾⠟⠋⠀⠀⢀⣠⣶⠿⠋⠁⠀⢀⣠⣴⠿⠋⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠠⣾⠟⠋⠀⠀⣀⣤⡾⠟⠋⠀⠀⣠⣴⡿⠛⠉⠀⠀⣀⣴⡾⠟⠉⠀⠀⣀⣤⣾⠟⠋⠀⢀⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠉⠀⢀⣤⣶⠟⠋⠁⠀⢀⣠⣴⠿⠛⠁⠀⢀⣠⣴⡾⠛⠉⠀⣀⣴⡾⠟⠁⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣶⠟⠋⠁⠀⣀⣴⡾⠟⠉⠀⠀⣀⣤⣶⠟⠋⠁⠀⢀⣠⣶⠿⠋⠁⢀⣠⣴⠿⠋⠁⢀⣠⡄⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠁⠀⠀⣀⣴⡾⠟⠉⠀⠀⣀⣤⡾⠟⠋⠀⣀⣤⡾⠟⠋⠁⢀⣤⣶⠟⠋⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣾⠟⠋⠀⠀⢀⣠⣶⠿⠋⠁⠀⢀⣠⣴⡿⠛⠁⠀⣠⣴⡿⠛⠉⠀⣀⣴⡾⠟⠋⠀⣀⣴⡆⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠀⠀⢀⣤⣶⠟⠋⠁⢀⣠⣶⠿⠋⠁⢀⣠⣶⠿⠛⠁⢀⣠⣴⠿⠛⠁⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣴⡿⠛⠉⠀⠀⣠⣴⡾⠛⠉⠀⣀⣤⡾⠟⠉⠀⣀⣤⣶⠟⠋⠁⣀⣤⣶⠟⠋⠁⣀⣤⣶⠇⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠁⠀⢀⣠⣴⠿⠛⠁⢀⣠⣴⠾⠛⠉⢀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠟⠉⠀⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣤⡾⠟⠋⠁⢀⣤⣾⠟⠋⠁⢀⣠⣶⠿⠛⠁⢀⣠⣶⠿⠛⠁⢀⣠⣶⠿⠛⠁⢀⣠⣶⠿⠃⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠉⠀⣀⣴⡾⠟⠋⠀⣤⣴⣾⠿⠋⠀⣠⣤⣾⠟⠋⠀⣠⣴⣾⠿⠋⢀⣠⣴⣾⠿⠋⠀⣠⣤⠀⠀⠀⠠⣿
⢹⣿⠀⠀⠀⢀⣶⠿⠋⠁⢀⣤⣰⠘⠉⠉⢀⣀⡀⠿⠟⠉⢀⣠⣶⡿⠟⠉⢀⣤⣶⡿⠟⠋⢀⣠⣴⡿⠟⠁⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⣀⣤⣶⠟⠛⠉⢀⣀⡿⠛⠛⠁⠀⣀⠾⠟⠋⠁⠀⣠⡾⠟⠋⠉⣀⣴⣾⠿⠛⠁⣀⣤⡶⠀⠀⠀⢀⣿
⢸⣿⠀⠀⠀⢠⡾⠟⠋⠀⣠⣴⡾⠟⠋⠀⣠⣴⡿⠟⠋⠀⣠⣴⡿⠟⠋⠁⣠⣴⡾⠟⠋⠁⣠⣴⡾⠟⠋⠁⠀⠀⠀⠈⣿
⢸⣿⠀⠀⠀⠀⢀⣠⣶⠿⠛⠁⢀⣠⣴⠿⠛⠁⢀⣠⣴⠿⠛⠁⢀⣠⣶⠿⠛⠁⢀⣠⣴⠿⠛⠁⢀⣠⣶⠇⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠰⠟⠋⠀⣀⣤⣾⠟⠋⠀⣀⣤⣶⠟⠋⠁⢀⣤⣶⠟⠋⠀⢀⣤⣾⠟⠋⠀⣀⣴⡾⠟⠋⢀⡀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⢀⣠⣴⡾⠛⠁⢀⣤⣶⠟⠃⠀⠀⠀⠀⣿
⢸⣿⡄⠀⠀⠘⠋⠁⢀⣠⣶⠿⠋⠁⢀⣠⣶⠿⠋⠁⢀⣠⣶⠿⠛⠁⣀⣤⣶⠟⠋⠁⣀⣴⡾⠛⠉⣀⣤⡆⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⣤⡾⠟⠋⠀⣀⣤⡾⠟⠋⠀⣀⣤⡾⠟⠋⠁⣀⣴⡾⠟⠉⢀⣠⣶⠿⠛⠁⣀⣴⡾⠟⠉⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠁⠀⣠⣴⠿⠛⠁⠀⣠⣴⡿⠛⠉⢀⣠⣶⠿⠛⠁⣀⣴⡾⠟⠋⢀⣠⣶⠿⠛⠁⣀⣴⡾⠃⠀⠀⠀⢸⣿
⠸⣿⣇⠀⠀⠀⠐⠿⠋⠁⢀⣤⣶⠟⠋⠁⣀⣤⡾⠟⠋⠀⣠⣴⡾⠛⠉⣀⣤⣶⠟⠋⢀⣠⣴⠿⠛⠁⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⡄⠀⠀⠀⠀⠀⠾⠛⠉⢀⣠⣴⡾⠛⠉⢀⣤⣶⠟⠋⢁⣠⣴⡾⠛⠉⣀⣤⡶⠟⠋⢁⣤⠖⠃⠀⠀⠀⠀⢠⣿⣿
⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠐⠿⠋⠁⣀⣴⡾⠟⠋⢀⣠⣶⠿⠋⠁⣠⣴⡾⠛⠉⠀⠤⠞⠋⠁⠀⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠈⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠛⠉⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠈⠻⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⣶⣾⣿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣴⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣤⡆⠀⠰⣿⣿⣿⣿⣿⣶⣶⣶⣶⠂⠠⣤⣤⣄⣀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄⡀⣥⠄⢿⣃⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡈⢿⣿⣉⣁⣸⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⡋⠃⣥⡀⢈⣭⠀⠀⡀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣟⠳⣤⠜⠀⠸⣿⠀⠠⣤⣴⣿⣿⡟⠋⠁⠈⣨⣿⣿⡿⣾⣿⡆⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣵⠀⠀⠀⠀⢀⣿⠀⢠⠜⠋⠉⠻⡟⠀⣠⣴⣿⠛⣿⡟⠻⢿⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⣸⣟⡳⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣶⣟⠻⠟⠁⠀⠁⢀⡈⣏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠻⠇⠀⠀⣤⠀⠀⠀⢨⣿⠀⠀⣀⣀⣠⣤⣿⡟⠿⣏⡗⠀⣀⠀⠀⠛⢹⡀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣤⣤⡀⠀⠀⠀⢺⣿⠀⠈⣉⣽⣿⠋⠉⣡⣾⣿⣧⡅⠉⡀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠋⠛⠋⠂⠀⠀⣿⣿⠇⠈⣋⣘⡉⠠⠀⠙⣻⠉⠈⠀⠚⠅⢤⠇⠀⣷⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣥⡜⠂⠀⠀⠀⠀⠀⣿⠘
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠐⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣴⣾⣿⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣉⣹⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠉⠉⢩⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠘⠿⠿⢏⡙⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢻⣿⢄⠀⢠⣾⣿⣿⣿⠋⢁⣤⣤⣾⣿⣿⣿⣿⣷⣶⣔⠄⣤⣦⣶⣿⣮⡑⠦⣼⢇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣌⣿⡇⠃⠘⠛⠛⠛⠛⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣀⣿⣼
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⣿⣷⣄⣤⡄⢠⣶⣶⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣹
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡸⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣀⣵
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠊⠱⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⡿⠟⡋⠁⠀⢀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠟⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⠶⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣠⠤⣄⣀⣠⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⢠⣄⣀⣀⣀⣴⡀⠀⢐⠶⠭⠶⣒⠲⠭⠶⣚⠲⠅⠀⠐⢀⠀⠀⠀⢀⠐⢀⠀⠀⣠⡜⠻⢿⡿⠛⢢⣀⠀⠀⠀⡔⢢⣀⣀⡠⠲⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⢨⣝⣒⣫⣭⣝⣒⣫⣭⣝⡂⠀⠈⠁⡀⠀⠀⠀⠉⡀⠀⢸⣿⣧⣀⡔⢢⢠⣾⣿⡆⠀⠀⣿⣾⣿⣿⣿⣾⡄
⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⠐⠮⣭⠵⠒⠮⣭⠵⠒⠦⡅⠀⠀⡀⠀⠀⠀⠀⠠⠁⠀⠘⢿⡏⠀⣸⡁⠀⢹⡿⠃⠀⠀⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⢨⣛⠒⣛⣭⣛⠒⣚⣭⣝⠂⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠁⠀⠈⢱⣶⣶⠀⠀⢠⣶⣶⡄
⠀⠀⠀⠀⠀⢠⢳⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⠀⠸⠿⠿⣤⣤⣼⠿⠿⣧⣤⣄⠀⢀⠤⢄⡀⢀⡠⠤⡀⠀⠀⠀⣤⢾⡛⠟⡳⣦⡀
⠀⠀⠠⡄⣤⣪⣿⣑⡄⡤⠄⠀⠀⣻⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⠀⠠⠀⠀⠀⠀⠀⠠⠇⠀⠀⣸⢣⠱⣪⣬⡪⠊⣷
⠀⠀⠀⠈⣋⣿⢿⣯⡎⠀⠀⠀⠀⢿⡿⠉⠭⠉⢿⡿⠀⢸⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⢠⡀⠁⠀⠀⠀⢀⠄⠀⠀⢹⣞⣔⢝⡛⢝⢄⡿
⠀⠀⠀⢠⠧⠋⠉⠫⢧⠀⠀⠀⠀⠈⠛⠦⠠⠤⠛⠁⠀⠈⠉⠉⣷⣶⣾⠉⠉⣷⣶⣶⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠿⣯⣬⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠛⠛⠛
⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡴⢖⣿⡲⣦⡀⠀⠀⠀⠀⢼⣿⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠚⠛⣃⡀⠀⠀⠙⠛⠃⠀⢠⡟⠞⢕⣤⡪⡹⣻⡄⠀⠀⠀⢸⣯⣤⣬⣿⠀⠀⠀⠀⣤⢤⣴⣻⣟⣤⠤⣤⠀⠀⠘⠟⣉⠀⠀⣈⡻⠇
⠀⠀⠀⠀⠻⠋⡤⠀⠀⠀⠀⠀⠸⣇⢌⢜⣿⡣⡢⣹⠇⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠙⣽⣿⣿⣻⠊⠀⠀⠀⠀⠘⠿⠣⡄⠻⠟
⠀⠀⠀⠈⠀⠠⠀⠄⠀⠀⠀⠀⠀⠙⢷⣵⣶⣮⡾⠋⠀⠀⠀⠸⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⠽⠓⠚⢭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈
⠀⣠⣤⣄⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣉⡉⠉⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢿⣿⣿⣀⣀⣸⣿⣿⣀⣀⡀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠠⣭⣛⣫⠥⢭⣛⣫⡭⢭⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⢨⠀⠃⠛⠚⠁⡔⠀⢐⡲⠭⠴⣚⡲⠬⠵⣚⡲⠤⠀⣀⣴⠞⠿⠿⠳⣦⣀⠀⢀⣾⣧⠀⢈⡏⠀⣸⣷⡄
⠀⣶⣶⣾⠉⠉⢱⣶⣾⠉⠉⠁⠀⡏⠀⠀⢀⠀⠂⢹⠀⠨⣝⣓⣛⣭⣝⣓⣛⣭⣽⡓⠸⣿⣧⠀⠀⠀⢀⣼⡿⠀⠸⣿⡿⠃⠃⠜⠁⢻⣿⡟
⠀⠙⠛⠛⣤⣤⡜⠛⠛⣤⣤⡄⠀⠙⠀⠀⠀⠀⠀⠋⠀⠐⠶⣭⡵⢖⠶⣭⡵⠖⠲⣭⠀⠀⠙⠦⢤⠤⠾⠋⠀⠀⠀⠈⠣⣤⣾⣷⣦⡜⠋
⠀⠀⠀⠀⠿⠿⠇⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠚⠉⠙⠲⠚⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⣻⣟⡹⢟⡿⠛⡷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠈⠛⠉⠈⠉⠴⠿⠛⠁⠸⣽⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⡏⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⢓⠄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠈⠀⡃⠀⠀⠀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠀⢰⡧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠄⠀⠀⢀⡔⢢⡀⠀⠀⠀⠀⠀⠀⠀⡾⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⢠⡰⠆⠀⣀⠀⠐⠚⠁⠀⡀⠀⠀⠀⣀⣜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡤⠤⢤⣀⠀⠃⠀⠀⠈⠁⠂⠠⠒⣿⠀⠀⠀⠈⠡⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠠⣙⣂⣲⣄⣀⠀⠱⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡀⢳⡀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠉⠳⡀⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠁⡦⡅⠈⠑⠲⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⠀⠀⠀⠀⠑⢈⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠇⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠣⡀⠀⠀⠀⠀⠁⠀⠀⠀⣦⢀⣠⠴⠒⠉⠀⢀⣜⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠑⢄⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠑⠢⣄⣀⡠⠎⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⠖⠢⣄⣀⣀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⠀⠒⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠒⣒⠒⠂⣤⢞⡓⢠⡀⠀⠀⠀⠐⠂⠄⡀⠉⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠈⢀⡠⠔⠓⠈⠉⠁⠀⠀⠈⠙⠢⣙⡀⠀⠀⠀⠀⠀⠈⠓⢄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠁⠀⢠⠀⠈⠳⣄⠀⠈⠢⡀⠀⠀⠀⠱⡌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⡗⡄⠀⠘⣄⠀⠀⠀⠘⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠈⠋⠀⠀⠙⡄⡜⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⡀⠀⠘⡄⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠤⢀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠤⡀⠇⠉⠀⠀⢠⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⣀⣀⠔⠛⣀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⡜⠁⠀⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢨⡇⣹⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠤⠤⠞⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⡆⠈⠁⠒⠒⠒⠀⠐⠒⠖⠈⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅⠇⠗⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⡄⠒⠖⠀⢲⠀⠀⠀⠀⠀⢀⠀⡄⠘⢰⠃⠀⠀⠀⠀⠀⠀⠀⡀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠤⠤⠠⠤⠔⠋
⠀⠀⠀⢣⡀⠀⢀⡜⠀⠀⠀⠀⠀⠈⣃⠱⡀⠈⢢⡀⡄⠀⠀⠀⢡⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠉⠚⠃⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⠀⠀⠙⠷⡄⣤⡄⣺⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⠀⠘⣧⠀⠙⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⠀⠀⢙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠒⠄⠀⠀⠀⠈⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢁⡜⠁⠀⠀⠀⠀⠀⡰⠁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠬⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣨⠙⠋⠉⠉⠉⠉⣓⠀⠀⠀⡠⠊⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⡠⠚⠉⠐⠶⡊⠀⠀⠀⠀⠀⣀⣀⡘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢤⠒⠲⠆⠉⠀⠀⠀⠀⡼⠖⠜⠢⡀⠀⡰⠋⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠤⠠⠠⠤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠱⡄⠀⠀⠶⠦⠄⠀⠀⠓⠂⠀⠈⢲⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⠆⠀⠑⠀⠂⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣉⠀⠀⠀⠀⣀⡐⠂⠀⠀⠀⠀⠀⠐⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⠀⠀⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⡀⢀⣀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠣⡄⠤⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠈⠀⠈⠉⠁⠀⠚⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⡄⠒⣦⠀⠠⠾⢄⠀⠀⠠⠀⠀⠀⠀⠀⠸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠠⡇⣰⠀⠀⠀⠀⠈⠑⢦⣴⠀⠀⠀⠀⢰⠟⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠉⠙⠀⠠⢤⡄⣀⣰⠀⠀⠀⠀⣦⠃⠀⠀⠀⢸⣤⠖⠋⠀⠀⠀⠀⢠⣯⣄⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⠀⢀⣀⣽⣀⣀⡤⠞⠉⠀⠀⠀⠀⣀⡤⠞⠋⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⣣⠶⠀⠀⠀⠀⠀⠀⠀⠤⠴⠊⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⡜⣡⢀⠀⠀⣀⠄⠲⣮⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠈⢙⣡⣟⢼⣘⠁⠀⠀⠀⢀⣾⠃⣌⡉⠀⠒⠒⠒⠒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⣀⠀⢹⣏⠀⠉⠲⣂⣤⣤⣬⢻⡗⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠟⠃⠀⠉⢐⠖⢿⢉⠁⣠⠜⠀⠀⠀⠀⠀⠀⣠⣊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠴⡄⠀⠐⠘⠃⠀⠀⡰⢏⠀⢢⡔⠚⠉⠀⠱⡀⠀⣇⠀⠀⠀⠀⠀⣀⠞⠋⠁⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢸⡠⠋⠀⠈⠑⠠⠃⠀⠀⠀⠀⠈⠁⠉⠠⠤⠒⠚⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⠒⠈⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠉⡌⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣉⡀⠈⠓⠀⠀⠀⠐⣧⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢰⠃⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠐⠂⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⠝⢠⠀⠀⡐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⠂⠀⠀⠀⠀⠀⠀⠈⠀⡀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡄⠄⠐⡆⠀⠀⠀⠀⠀⣠⡤⠷⠶⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⡬⠯⣴⠉⠀⠈⢶⣄⣀⡀⠀⠈⠁⠈⢁⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠁⠀⠀⠀⣈⢠⠀⠀⠀⠉⠙⠛⠳⠶⠾⠿⠛⢉⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⣨⣟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢿⣧⣰⣀⣸⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠉⠀⠀⠀⠀⣀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠉⠀⠁⠠⡀⠀⠀⠀⠀⠀⢸⡃⠶⣴⡶⣴⠄⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⣦⡀⠀⠀⠀⢀⠐⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡌⠻⣶⣵⣉⣈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡆⠀⠀⠀⠀⠀⠀⠸⢿⣶⣄⣀⡀⠀⠀⠀⢾⣽⠁⠀⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢀⡀⠀⠀⠀⠀⠀⠉⠿⠋⠁⠀⠀⢀⣿⡇⠀⠀⠀⠀⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⡝⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⢀⣾⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣝⡿⠁⠀⠙⢿⣶⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠈⠙⠢⣄⡘⠿⠋⠀⠀⠀⠀⢀⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⡀⠠⠀⠐⠂⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⣠⠞⠛⠛⠛⠛⠷⠶⣦⣄⣀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣄⠀⠀⠈⢻⣆
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⠀⢿⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣼⡇⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢻⡆⠀⢀⣠⣶⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⣿⡿
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⢠⣆⠀⠀⣾⣿⡆⠀⠘⠁⠀⠀⠀⢀⣤⡶⠚⠂⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠈⠷⠀⠈⢿⣷⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠂⠐⢿⣷⣾⣿⣿⣿⣿⣀⣤⣶⡷⠀⠀⢠⣄⡀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣬⣭⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠉⠻⣦⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣶⣤⣤⣤⣤⡤⢀⣠⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠉⣡⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣨⡆⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠻⣷⡀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⢻⣷⡀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢋⣿⡏⠀⠀⢻⣷⡄⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⣸⣿⠁⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⢠⣿⡏⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣀⣴⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⣸⣿⣧⣴⣶⣶⣿⠆⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠛⠉⠀⠀⠀⢠⣶⣦⠀⠀⠀⣰⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⢸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⡿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠻⠟⠻⠿⠿⠿⠿⠿⢿⡿⠿⢓⣠⡽⠿⠿⠟⠿⠿⠿⠟⠻⠏⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⢀⡠⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠄⠂⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠋⠁⠙⠀⠀⠀⠀⠐⢇⢀⣀⡀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡆
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠟⠃⠙⠓⠀⠛⠀⢳⢲⢀⡀⠀⠀⡇⠀⠀⢀
⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢢⠀⡄⠀⠀⡄⠀⠀⠀⢸⢸⠀⠀⠈⠉⠁⢠⡐⠁⠀⠀⠀⠀⠁⠀⠰
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣀⢠⣀⡀⠧⠀⠤⠤⠼⠴⣤⣤⡀⣀⣸⢸⡆⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠑⠀⡀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠸⡏⠀⠀⠀⢠⠆⡈⠈⢢⢠⡎⠀⠀⠱⡀⠿⠘⠃⠀⠀⠀⠀⠀⠤⢀⡇
⠀⠀⠀⠀⠀⠱⠛⠂⠉⠀⠀⡀⡠⣂⣀⡇⠀⠀⠀⠸⣄⡁⢀⡴⠂⢦⣀⣁⡼⠁⠀⡀⠀⠄⠀⠀⠀⠀⢀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⡳⠀⠀⠀⢀⠑⠀⠀⠀⠀⠀⢀⣀⠤⠤⠶⠒⠋⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⡄⠀⠉⠢⣄⠀⠀⠀⠘⡄⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠴⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠈⠳⡀⠀⢡⢸⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠢⡀⠀⠘⢄⠀⢠⠀⠸⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡤⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠢⡀⠀⠀⠈⠢⠀⠘⣴⠃⠀⠀⣇⠀⠈⠀⢀⡴⠦⡄⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠢⢄⠀⢀⣡⠔⠁⠀⠀⠀⢹⠀⠀⠀⢸⡀⠀⣿⠀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⡀⠉⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠸⡆⢀⣀⠬⠇⠀⠉⠉⢉⡉⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢘⡸⠁⢀⣀⠉⠓⡤
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠱⣀⠙⠉⠀⠀⡂⠀⠐
⠀⠀⠀⠀⢨⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⢠⠂⠉⠁⠀⠀⠻⡀⠌
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠱⠄
⢠⠂⢁⣀⠈⠁⠀⠀⠤⡤⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡠⠠⠀⢀⡇⠀⠀⠀⠀⢰⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠠⠊⠓⢤⣀⡀⠀⠀⢀⣠⠶⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⢈⠧⠠⢤⣄⠊⠀⡀⠀⠀⠀⠀⠀⠀⠑⠠⠒⠀⠈⠉⠘⢅⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠌⠀⠀⠀⠀⠀⠁⣷⣠⠀⠄⢰⠀⡄⠠⠀⠰⠀⠄⠀⠀⠆⢠⢀⣆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⣧⣴⣴⣦⣶⣤⣦⣴⣶⣧⣴⣴⣷⣴⣦⣼⣿⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿
⣿⣭⣭⣯⣽⣥⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣾⣿⣿⣿⣿⣾⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠶⢲⡆⠀⠘⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣠⣤⣀⣤⠴⠦⣄⡀⠀⠀⠀⠀⠀⢁⣴⠋⡀⠀⣴⣏⣤⡄⠀⣴⣋⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠙⠏⠁⠚⠉⠀⠀⠀⢸⡇⣀⡀⠀⠀⠀⠛⠛⠋⠁⠀⠉⠉⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣠⠶⠶⣄⠀⠀⠀⠀⣴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠘⣧⣀⡀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠙⢳⡄⠀⠀⣠⡤⠴⠶⠶⢦
⠀⠀⠀⠀⠀⣽⠆⠀⠀⠈⢮⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⡶⢿⣤⣤⣤⡄⠀⠀⠀⠘⢿⣦⣼⠃⠀⠀⠀⠀⢘⡷⠶⠤⣤⣀
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⢸⡷⠶⠶⠞⠋⠁⢀⠀⠤⣄⡀⠈⠙⢦⡤⠀⠀⠀⣼⠃⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠈⢷⣶⠒⠒⠦⢤⣀
⠀⢀⣠⣾⣅⠀⠀⠀⠀⠀⠀⠀⢀⣙⣷⣄⣾⠁⢠⠶⢦⣄⠀⢀⣀⣠⡾⠁⠀⠀⠀⢻⡄⢀⣴⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠸⡏⠳⡄⠀⠀⠈⠉⠓⠦⣄
⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡇⠀⣿⠀⠀⣹⡦⠈⠉⠁⠀⠀⢢⡶⠃⠘⣷⡾⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⡄⠀⠀⠀⠀⠀⠈
⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠛⠶⠚⢿⠃⠀⠀⣰⠀⠀⢸⡇⠀⢰⡿⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢷
⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠘⠦⠤⠟⠁⠀⠀⣸⠇⢀⡾⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠸⡆
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣷⠀⠀⠀⢀⣀⡠⠋⠀⣾⠃⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⡉⠂⠛⠋⠉⠀⠀⣸⠏⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠋⡳⣶⣶⣤⣤⣴⡿⠀⠀⠀⠀⠀⢀⡍
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⣴⠾⠛⣃⠀⠉⠙⠛⣿⠃⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⣠⡶⠟⣉⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠛⢷⣿⣋⣴⠿⣫⣾⡿⠛⢾⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠦⣤⣄⣀⣀⣀⣈⣿⣿⡿⠿⠛⠁⠀⠀⢸⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⣝⡛⠛⠁⠀⢻⣿⣿⠇⣴⣾⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣸⣿⡏⣸⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡇⣿⣿⣟⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠇⣿⣿⣿⣿⡝⢿⣿⣿⣦⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣤⠈⣿⣿⣿⣿⣿⣆⠙⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⠀⠘⠿⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⠿⠛⠁




⣏⠉⣻⣛⣛⣛⣛⣻⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣟⣻⣿⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⢰⣾⣷⣦⡖⡚⢿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⢋⠉⣷⣦⣤⣤
⠀⠀⢠⣶⣿⣿⣿⣿⢸⣷⢸⣿⣿⣿⣿⣿⠁⣿⡿⠋⡬⢸⡏⢋⣸⡆⣿⡁⢟⡀⢿⣿⣿⣏
⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠀⢏⡆⢻⣿⣿⢰⣿⢧⣈⣡⣘⣳⡾⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣧⡀
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣧⣼⣿⣿⣄⢛⣾⣿⣿⣿⡟⢰⣿⣿⣿⠿⣿⠟⡛⣿⡟⢫⣬⣷⡀
⠀⠈⠉⠉⠉⠛⠛⠛⠛⠿⣧⣾⣿⡟⠛⠁⣿⠛⠻⣿⠿⢿⡇⣻⠿⣿⡇⢰⣏⠸⡁⠿⡁⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣶⣤⣿⡄⠀⠀⢀⣾⣇⢹⡇⢸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⣿⣿⣄⣠⣿⣿⣿⣤⣷⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠋⠀⢿⣿⣿⣿⣿⠇⢰⣿⣿⣿⣿⣿⡟⠁⠀⠈⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠘⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⠋⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⡏⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⡿⠁⣀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣼⣿⣿⣿⣟⣁⣈⣛⣛⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣋
⣿⣯⣭⣯⣭⣽⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⠀⠐⠿⢛⠻⠓⣛⣠⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⡿⠃⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠁⠒⠆⢀⡀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣠⣾⣿⣿⣿⣿⣿⣷⣆⠀⠀⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⢫⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠉⣡⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠲⠂⠀⠛⣿⠟⠋⠍⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣄⢀⡿⣷⣄⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠓⠈⠻⡿⠀⣠⡌⠉⣼⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠀⢀⠀⡈⠁⠀⠀⠀⠞⠻⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⣛⠛⠹⢄⠀⠀⠀⠆⢹⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠊⠁⣤⣶⣶⣦⣤⣴⣟⠀⢹⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⠀⠀⣛⢷⡄⠀⠀⠀⠐⢶⡜⠷
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠿⣿⠀⠀⢀⠙⣿⣿⣿⣯⢽⠿⠁⠛⠟⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣦⡀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡯⠁⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⣿⡿⠁⠈⠓⠆⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠉⠉⠻⣀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠖⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠈⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣴⠿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠉⠉⠛⠓⣦
⠀⠀⠀⣠⣴⡾⠋⠀⠀⠛⢟⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢰⣿⡀
⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢹⡟⠁
⠀⠀⠀⠙⣿⣧⠀⣀⣀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣿⠛⠓⠊
⠀⠀⠀⠀⠙⠻⠾⣿⡟⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣂⠄⠀⠀⠀⣼⠛⠃⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⠀⠀⠀⠀⠀⣴⠒⠺⠿⠿⠟⠓⡄
⠀⠀⠀⠀⠀⡔⣻⠆⢹⣿⣿⣿⣦⣤⠀⣿⠀⠀⠀⠀⠀⢀⣾⡗⠀⠀⠀⠀⠀⣀⣈⣄⣿⣿⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡀⠛⠋⢠⠀⠉⢻⣿⣿⣿⡇⠙⢧⣄⣀⣀⣀⣼⡿⠃⠀⠀⠀⠀⠈⠻⣧⣬⣿⣿⣿⣀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣠⡄⢀⡶⣦⡀⢀⣴⣿⣿⣥⡏⠁⠀⠀⠙⣿⡟⠉⠁⠀⠀⠀⠀⢀⠄⣷⣄⣠⣤⣿⢿⣿⠿⣀⠀⠘⢿⣤⣤⣤⣤⣤⣶⡧
⢰⡿⣧⠟⠘⢷⡾⢃⣿⣿⣛⡷⠎⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢀⠈⡾⠛⢿⣯⣉⣿⣤⣿⠷⣇⣤⡀⠀⠀⠀⣿⡇⠀⠈⠁
⠈⢧⣄⠘⠛⠀⣠⣾⠿⣿⡝⠁⠀⠀⠀⠀⣀⠀⣿⣷⣄⡀⠀⠀⠀⣿⠀⣠⣦⠈⠋⣁⣿⣿⣿⡇⣸⣬⣷⠀⠀⠀⣿⡃
⠀⠀⠙⢦⣀⣾⣿⣿⡾⠃⠀⠀⠀⠀⠀⠚⠁⣞⣿⣿⡟⢿⣶⣄⡀⠈⠀⢿⣹⣷⣿⠟⣿⣿⣿⣼⣿⠟⠋⠀⡀⠀⠿⠃
⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⢀⡄⢠⠀⣠⡀⣿⣿⣿⡒⠈⠀⠻⡆⠀⠀⠈⠙⠛⠿⠿⠟⠛⠉⠀⣠⠖⠁⠀⠀⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢣⢀⡀⠉⣀⣿⣿⣿⣿⡇⠠⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠠⣶⡧⠀⠈⠋⣫⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⠣⣾⣿⣿⣿⣿⣿⣿⣾⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣭⣅⣀⣀⣀⡈⣫⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⣿⣿⣿⣿⣿⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⡀⣀⣀⣀⣀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡏⢉⢿⣿⡯⠁⠛⣿⣻⣟⠈⢹⣿⡿⠙⣻⣿⣿⡿⠉⢹⡇⠀⣴⣤⡀⠀⢴⣦⣶⠀⠀⢠⢦
⠀⠀⠀⠀⠀⠀⠀⠿⠴⠦⠾⠻⠷⠼⠿⠿⠻⠴⠾⠿⠝⠷⠾⠷⠿⠻⠿⠶⠿⠇⠈⠙⠛⠃⠀⠘⠉⠋⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⣽⣿⣿⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢣⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢠⠏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⢿⡟⣰⡟⢰⣟⣾⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠛⢡⡟⣰⡿⢡⣿⣿⡿⢟⣻⣭⣶⠖⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⡞⣰⡿⢣⡾⢛⣵⡿⠟⠙⠛⠁⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⠖⣋⣾⡟⣡⠏⢨⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⢿⣿⠏⡼⠃⣰⣿⡿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣾⢏⡞⠀⣰⣿⢻⡇⠀⠀⡴⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⣾⣿⡞⠀⢰⣿⠇⠘⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢨⠀⣿⡿⠁⠀⣿⡏⠀⠀⠹⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⡄⢻⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⢿⡆⠀⢿⡀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠑⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣤⣀⣀⣠⠄⠀⢀⠠⡀⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⣝⠛⠛⠋⠁⠀⠀⠈⠑⠮⣄⢠⣀⡀⠀⠀⠀⠀⠀⠃⠀⣨⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣭⣙⠓⠲⠶⠦⠤⠴⠶⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⣠⣠⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡁⣛⣀⣈⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠁⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣄⣀⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⡟⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⡀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣶⣶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⣿⣿⡟⠛⠻⠿⣿⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⣿⠃⢰⣿⡿⠀⠀⠀⠀⠀⠻⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⡇⠀⣿⣿⠁⠀⠀⠀⠀⠀⠈⡤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢸⣿⣅⡀⠀⠀⠀⠀⠀⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⢸⣿⣟⠀⠀⠀⠀⠈⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠠⡇⠀⢸⣿⣇⡀⠀⠀⠀⠀⣈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇⠀⢸⣿⣿⣷⠯⠭⠀⠾⠛⣱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⢸⣿⣿⡆⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇⠀⢸⣿⣿⡀⠀⠀⠀⠀⠐⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠁⠀⢸⣿⣿⠀⠀⠀⢀⣀⣀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡆⠀⢸⣿⣧⠀⠀⠀⠈⣽⠋⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡦⠀⠸⣿⡏⠀⠀⠀⢠⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⠀⠀⣿⣿⡿⠒⠐⠋⣻⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠀⠀⣿⣿⣷⡀⠀⠤⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⣿⡆⠀⣿⣿⣿⡇⢤⣖⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢰⡿⣻⡟⣿⡿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣮⣶⣾⢧⡽⡿⢿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⣼⡏⠉⣻⡿⢿⡆
⠀⠀⠀⣠⣤⣠⣴⣶⣶⣤⣶⣶⡶⣦⣄⠘⠛⠁⠀⠚⠃⠘⠋⠀⠈⣁⢀⣠⣤⣀⣀⣤⣤⣄⡀
⠀⠀⣰⣿⣿⡇⣟⢟⡯⡹⣳⣸⣷⣸⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⡟⡛⣽⢿⣠⣾⣾⣱⣼⠜⢿⣆
⠀⠀⠛⠛⠛⢺⣾⣿⡇⡿⣸⣟⣻⣯⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⡸⣿⡏⣟⢽⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⠋⢳⣿⠉⠉⣸⡾⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⢷⡏⠉⢩⣿⠿⣿⡆
⠀⠀⠀⠀⠛⠛⠁⠀⠐⠛⠀⠘⠛⠁⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⠀⠚⠃⠐⠛⠃⠀⠙⠁
⠀⠀⠀⠀⠀⢀⠀⣀⣤⣤⣄⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣰⡶⣶⡖⣾⢿⣿⡟⡛⡟⡿⢛⠿⢶⣄
⠀⠀⠀⢀⡼⡿⣟⣿⠘⢟⣰⣽⠟⣅⡷⢫⢿⡆⠀⠀⣰⣿⣧⣣⢻⢾⣌⣼⣤⣧⣇⡧⣩⣆⣯⢿⣇
⠀⠀⠀⢾⣿⣾⣾⣜⣿⢹⢿⢫⠟⣽⣏⢷⣳⣿⠀⠈⠛⠿⠿⠿⣿⣼⣇⢟⠛⢸⡃⣟⣿⣼⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢦⣷⣾⠷⠿⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⡟⢮⣾⣾⠛⠛⢻⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣀⣼⡿⠁⠀⣹⣿⠀⣠⣾⠟⠈⣿⠟⠀⠀⠀⣠⣼⣿⠏⠀⢀⣿⡿⠀⣠⣼⡿⠁⠈⣿⠇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠁
⣶⡶⠶⠶⠶⠶⢶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠙⠛⠿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠛⠀⣀⣴⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣀⠀⠐⠾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠙⣿⡆⠀⠀⠀⠀⠐⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⡿⠋⠀⣠⣴⣆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⣿⣿⣿⣿⡉⠉⠀⠀⠀⠉⠛⠛⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣿⣿⣷⣄⠈⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀
⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣾⣿⣯⡀⠀⠙⢿⣿⣿⡆⠀⠈⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦
⣾⡿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠻⣿⣿⣦⡀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⢀⣴⣿⣿⣿⣧
⠙⣿⣦⠀⠐⢶⣾⠂⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣦⣤⣴⡾⠟⠀⠀⠘⠛⠉⣀⣼⣿⡆
⠀⠈⠻⣷⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣧⣤⣤⣀⡀
⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠻⣿⣶⣤⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠘⡷⣤⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⣿⣿⠿⣿⣦
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣶⣤⣤⣶⠟⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠋⠀⠀⠸⠟⠁⢀⣾⣿⣷⡀
⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡁⠀⠀⠀⠀⠀⠀⠠⡿⠿⠛⣿⣷
⠀⠀⠀⠀⠀⠹⣿⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠋⠁⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⡀⠀⠈⠻⠄⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣏⠀⠈⠙⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⢻⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⠿⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠈⠁⠀⠀⠀⠀⠉⠉⠁⡀⠀⣿⣿⠃⠀⣠⡀⠀⠀⣠⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⣾⣿⣿⣿⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⣷⣤⣶⣿⣤⣾⣷⣴⣿⣿⣷⣾⣿⣿⣶⣾⣿⣷⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⠿⠿⠿⢿⣿⣿⡟⠋⠀⢈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⢿⣧⣤⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠙⠛⠛⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⢀⡀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣹⡆⠊⠤⠶⠂⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣈⠀⠛⠋⣱⠉⢶⡄⠀⢀⣠⡿⣀⡿⢛⣷⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⣌⠷⢺⣷⠀⠄⠀⠀⠀⠀⠀⠈⠻⢿⠳⠀⠀⠋⠀⣀⡤⠖⠚⠷⢶⢿⣏⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣝⢐⡛⠀⠀⡠⠔⠈⠉⠉⠒⣄⠀⠉⠀⠀⣠⠞⠁⠀⠀⠀⠀⠺⡾⠛⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣄⠀⠿⠀⠼⠁⠀⠀⠀⠀⠀⠸⡆⢠⣠⡞⣁⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠤⠭⣶⣤⠞⠀⠀⠀⠃⢀⠀⢸⡇⣼⠏⣿⣿⠏⣀⣀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠤⠤⣤⡤⠤⣀⡀⢠⠟⠛⠒⠁⠼⠋⠀⠀⠀⠀⠀⠁⠀⡾⣰⢃⣤⠾⠛⠉⠉⠉⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⡴⠋⠀⠀⠀⠀⠹⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⡼⣱⡿⠋⠀⠀⠀⠘⡀⠀⠀⠀⠀⠈⢳⠄
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢸⠁⠀⠀⠀⢠⠀⠀⠀⠈⢳⡀⠀⠀⢀⣤⠀⠀⠈⣠⣯⡾⢋⠀⡀⠀⠀⠀⠐⡤⢤⡶⣦⣀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⣄⠀⣸⣄⠀⠀⠀⠀⠀⠀⡠⠀⠀⢳⠀⠀⢸⣿⠇⣠⣾⡿⠋⠸⣿⡿⠃⠀⠀⠀⢠⣶⠾⢧⣽⠋⣹⣿⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣈⡃⠉⠾⠶⠆⠀⠀⠤⠚⠁⠀⠀⢸⠀⠀⠘⣡⣾⠿⠻⡉⠻⠁⠀⢄⠀⠀⠀⠀⢀⢻⣿⣋⣏⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣼⠁⠻⡆⠀⠀⠀⠀⢀⣄⠀⠀⡸⢀⣴⡾⠋⠁⠀⠰⢻⠤⣍⠁⠀⢡⡄⠀⠀⠀⠜⠿⠋⠙⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⠔⠂⠐⠢⡀⢸⣿⡗⣰⣷⡿⣋⣤⠴⠶⠒⠒⠾⢄⠀⠀⠀⢐⡷⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⢹⠘⢛⡿⣫⡾⠛⠉⠀⠀⢀⢀⠀⠀⠀⠘⣂⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡞⣴⢋⡀⠀⠀⠀⠀⠀⠻⠛⠿⠲⠗⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣼⢃⡀⢀⣀⢀⡀⠀⠀⠀⠀⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠲⠦⣌⣷⡿⣦⣄⠀⠀⠀⠀⠀⣾⣿⠙⠿⠃⠛⠓⠿⣥⣖⡀⠁⠀⣀⠀⠔⠁
⠀⠀⠀⠀⣀⣛⣻⢿⠏⠋⠀⠀⠈⠻⣄⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠃⣿⣭⡄
⠀⠀⠀⠀⢼⣿⡙⠃⠀⠀⠀⠀⠀⠀⠹⡼⣷⣿⣿⠁⢸⣞⡤⠀⠀⠀⠀⠈⣿⠶
⠀⠀⠀⠀⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣿⡟⠀⠀⠁⠁⠀⠀⠀⠀⢀⡾⠷
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠒⠀⠀⠀⠀⠀⡇⢸⣿⠃⣀⣠⣤⠀⠀⠀⠀⠀⢠⡿⠆
⠀⠀⠀⠀⠀⠈⠒⠤⠀⠀⠠⠀⠀⠀⢰⠃⣿⡏⠾⣿⡿⠃⠀⠀⠀⠠⠀⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡏⣰⣿⠔⠒⠓⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠⠂⢀⡞⣰⠟⡁⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠞⣰⠋⠀⠁⠰⠊⣁⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⠜⠁⠀⠀⠀⠀⠀⠉⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⡷⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣦⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠈⢻⠾⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠟⠁⣼⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣷⣾⣿⡧⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⡿⠛⠋⠉⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢹⣦⣦⣀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣟⣛⢫⡉⠂⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⣾⣿⣿⠘⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⣰⣿⣿⢿⡄⠀⠘⢷⣄⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠃⣴⣿⣿⡏⢸⣧⠀⠀⠘⣿⣆⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⡇⢸⣿⠀⠀⠀⢹⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⠇⠸⢿⠄⠀⢀⡤⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⣶⣿⠉⢹⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠀⠘⣿
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠆⠀⠀⢻⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣿⣿⠀⠀⠀⠈⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠀⠀⠀⠀⠻⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡠⡄⠀⠀⠀⠀⠀⠀⢰⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⣿⠻⡏⠀⠀⠀⠀⠀⠀⠸⣄⣿⣷⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⣸⣿⠐⣂⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⢻⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢿⣿⣷⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡍⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠁⢻⣿⠞⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠑⣿⣿⡏⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣖⣡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠚⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣲⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣯⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠈⠈⣿⡇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠐⠛⢿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⢹⠉⠉⠻⡿⠀⠀⠀⠀⠀⠀⠔⠀⠀⠐⠂⠄⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠂⠀⡀⣴⡀
⠀⠀⠀⠴⢶⣦⣤⢶⣿⣿⣶⣶⡶⣶⣦⠀⠀⠀⠀⠈⠀⠤⠌⠀⠀⡔⠀⠀⠀⠀⠀⠀⣾⠆⠴⡤⠐⠀⣐⠀⠀⣏⠴⠚⠉⠁⠀⢰⡆⠘⡍⠠⠙
⠀⠀⠀⠀⠀⠉⠛⠂⠿⢿⣟⣛⣳⣿⠋⠀⠀⠀⠀⠀⢠⢄⣠⠾⡐⠁⠀⠀⠀⠀⠀⠀⠉⣀⠀⠐⠐⠃⠍⠀⠀⠘⢖⢉⡤⠶⠀⠀⠁⠰⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠋⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠞⠌⠝⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠂⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢟⡿⠛⢿⣿⣿⠀⠀⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢸⣿⣿⣿⠟⠛⠋⣹⡿⠊⠉⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⡎⠐⣆⣾⣿⣿⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢸⣿⣿⣧⣦⡀⠀⣿⣇⠋⠀⢆⣼⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠷⡀⠀⢻⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣁⣛⣿⣿⣯⣿⣿⣿⠀⠸⣿⣿⡟⣿⣿⠀⣿⣿⣷⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣧⠄⠙⢆⣬⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣇⣿⣿⣟⣿⣿⡿⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⠀⢈⣾⣿⣿⣿⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣿⣷⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡿⢿⣷⠞⢻⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠀⠀⣿⣿⣿⠀⠘⠟⠀⢸⣿⣿
⠀⠀⣿⣿⡿⣿⣿⣿⣾⣿⢿⠿⣿⣿⣿⣿⠀⠀⣿⣿⡏⢀⠊⠁⠀⣹⣿⣿⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣦⣤⡻⣿⠻⣿⣿⣿⣿⣟⢿⠯⣿⠀⠀⣿⣿⣧⣀⢀⣴⢰⣿⣿⣿⠀⠀⣿⣯⣼⣿⣯⣿⠞⣅⣽⣿
⠀⠀⣿⣿⣿⣿⣿⣶⣽⣟⣫⣥⣀⡤⣾⣿⠀⠀⣿⣿⣿⣟⣹⣿⣏⣿⣿⣿⠀⠀⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣿⣿⣿⣯⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢸⣿⣿⣿⣿⣷⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⣿⣿⣿⣿⣿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠿⠿⠿⠿⠿⠟⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠀⢀⠐⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠁⣝⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠰⢨⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠙⠉⡁
⠆⠀⠈⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⡠⠁⠀⠰⡍⠀⠀⠀⡀⢤⠀⢀⡏⠉⠀⠄⠐⠃
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠈⠓⡘⠀⠤⠀⠀⠀⢠⡀⠀⠉⠁⠀⠀⢀⡀⢀⡠
⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⣠⣴⡾⡿⢋⣿
⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⢻⢛⠉⡩⡯⠋⠈⠑⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠄⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠣⠈⠀⠓⠂⠀⠗⠀⠀⠓⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠐⠂⠀⠐⠂⠐⠠⢄⡀
⠀⠀⠀⠠⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠎⠀⠀⠌⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠓⠀⠰⠄⠀⠈⡂⠀⡼⠀⠀⢠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠤⠔⠱⡀⢠⢥⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠆⠀⠐⣐⡐⠀⠘⠄⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣄⣀⣆⣾⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠛⠛⠋⠙⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠖⢧⠀⠀⠀⠀⠀⠀⠀⠈⣑⠦⡀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠺⢄⠀⠀⠀⠀⡴⠃⠠⣤⡇⠀⠀⠀⠀⠀⢠⠋⠀⣤⡈⢦⠀⠀⢀⡠⢼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠠⡞⠀⠀⠻⡿⡇⠀⠀⠀⠀⠀⡏⠀⠸⣿⠇⠘⢇⠀⠁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⣀⠷⠒⠒⠒⠒⠃⠀⢀⣀⠀⢀⣧⠤⠤⠦⠤⣴⡀⠀⠀⠀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣈⣀⣈⡉⠁⢀⡀⠀⠀⠀⠀⠙⠄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⢁⣀⣠⠟⠉⢁⡈⠉⢲⡀⠀⠀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⠀⠉⠀⡜⠁⢹⢳⠀⡇⠠⠀⠀⠀⠀⠀⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢧⡀⠀⠀⠀⠀⠋⠳⢄⡀⠀⠈⠓⢲⡀⠱⠤⠷⢊⡼⠁⠀⠀⠀⠀⠀⡰⠁⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠳⢤⡀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠙⢲⡴⠒⠋⢀⡤⢤⣤⣤⠴⠊⠀⠀⠀⠀⠙⣏⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠉⠓⠒⠶⠖⠒⠛⠛⠿⣖⣚⣛⣉⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠢⡄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⣀⣀⠀⠀⢀⣀⣠⣄⠀⠀⠀⠀⣀⡠⠔⣻⠋⠀⢸⡏⠙⢦⠀⠀⠀⠀⠀⠀⠸⣄⡼
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣾⠙⠮⣭⣉⡉⠁⠀⠈⠉⠛⠋⢉⣡⡴⠚⠁⠀⠀⠘⠀⢀⡜⣇⠀⠀⠀⠀⠀⣠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⡇⠀⠀⠀⠈⠉⠉⠛⠓⢻⠟⠛⡏⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣸⠶⠤⠴⠖⠊⠁⠀⠉⠢⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⢀⡔⠋⠳⠤⣀⣀⣀⣠⠴⢾⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢻⡀⠀⠀⠀⠀⠀⡼⠁⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠼
⠀⠀⢀⡠⠖⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣄⠀⠳⣄⣀⠀⠀⠀⣀⣠⠞⠀⠀⠀⠈⠛⠒⠒⠒⠒⠋⢡⠞⠉⠀⠀⣤⣀⣀⠀⠀⠀⠀⣀⠈⠉⠛⠒⠦⡀
⠠⣞⣁⣀⡠⠔⢫⠇⠀⢀⢴⠀⠀⢠⡶⠤⣄⣀⣀⡵⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠴⠒⠋⠁⠀⠈⠙⠦⣄⣀⠈⢹⠖⠒⠒⠒⠁
⠀⠈⠉⠁⠀⢠⡏⢀⡰⠋⢸⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠈⠧⠤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣛⠿⠛⠁⠸⣿⣿⡆⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢠⣿⣟⢸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡻⠿⣿⣿⠇⣿⣿⣿⣿⣝⢿⣿⣿⣦⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⣿⣿⣿⣿⣿⡆⠹⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡗⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠒⠒⠒⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠈⠉⠙⠉⠙⠉⡛⠉⠁⡅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⣠⣀⣀⠀⣀⣀⣇⣸⣤⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠮⢭⣭⣭⣭⣭⡭⠥⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠡⠤⠔⠒⠒⠒⠒⠒⠒⠠⠼⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠑⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠉⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠐⠫⢶⣢⣤⣤⣍⣉⣁⣀⣀⣀⣀⣀⣩⠭⠕⣂⡠⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⣈⡉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢻⣾⣶⠃⣼⣷⠚⣷⠖⢰⣿⠃⢸⣭⣟⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⢀⡀⠀⠀⠀⠀⠉⠉⠀⠛⠀⠘⠛⠂⠈⠀⠀⠀⢀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠘⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠤⢄⣉⣉⡐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠤⢀⣀⡀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠶⠶⠶⠒⠒⠋⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⣑⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠉⠓⠊⠉⠛⠓⠒⠒⠒⣒⣶⣖⣒⣚⣯⠝⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡂⢤⣀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⣀⣤⠖⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢮⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣋⣥⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠏⠛⠟⠉⡛⠛⣟⠛⣛⡿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠈⠉⠉⠉⠉⠉⠐⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢻⣏⠁⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠴⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⢨⡇⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠔⢢⠀⠀⠀⠘⢦⡀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣠⠞⠀⠀⡔⠒⡄
⠀⡔⠒⢤⣆⠀⣇⢀⠖⠉⡆⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠂⠐⢄⢀⠃⢀⡧⠊⠉⡆
⣀⣑⣄⡀⠉⠀⠈⠉⠀⣠⠃⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡀⠀⠀⠐⠁⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢣⡀⠈⠁⠀⠈⠀⣠⠮⠤
⠣⢄⡈⠙⠀⠀⠀⠀⠀⠣⠤⣀⡀⠘⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣈⣭⡷⠾⣯⡁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢀⡜⣀⠤⠔⠃⠀⠀⠀⠀⠈⣀⣠⡔
⠀⠯⢁⣉⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢄⠀⠀⠀⠀⠀⢸⡠⢊⠕⣿⠃⠀⠀⠀⢻⡄⠀⢀⠃⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠈⠁⠰⣄⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢸⠁⠎⠀⢻⣆⡀⠀⣀⣾⢣⠀⡜⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⡰⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⢘⠈⠲⣄⡀⢨⠘⡗⠤⣀⠉⠛⠛⠋⣸⢸⠋⠀⠀⠀⣀⠴⠋⡅⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⡀⠘⠀⠀⠀⠉⠻⣄⠙⢤⠀⠀⠀⡈⠀⢁⣎⡀⠤⠔⠊⠁⠀⠀⠆⢀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⡆⠀⠀⠀⠀⠈⠳⢤⡀⣀⠀⣠⠔⠈⠈⠀⠀⠀⠀⠀⠀⠀⡛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠊⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠐⠒⠂⠠⠤⠤⠤⠄⠀⠀⠀⠀⠤⠤⠤⠤⠀⠒⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⣀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠟⠘⠃⠙⠉⠛⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠛⠉⠛⠙⠋⠸⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⡰⠋⠙⠃⠠⠤⣀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠔⠋⠉⢦⠀⠁⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⣠⠞
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣶⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣱⢾⣿⠟⢽⣿⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⢉⡁⢾⠇⣀⠀⢁⢨⡷⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡝⠀⠉⠈⠈⠉⠀⠀⠀⠠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠠⠄⠀⠀⠀⣤⡄⠀⠀⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡞⠛⠂⠀⠈⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠠⠁⠀⠄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠁⢀⢴⠏⠀⠀⠀⠀⠘⠓⢍⢹⣄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡘⠀⠀⠀⠙⠆⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠦⡀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠁⡅⣀⠀⠀⢛⠀⡆⠐⣯⣤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣈⠁⠀⠀⠀⢠⠃⠀⣀⢔⠜⠁⠀⠠⡀⠀⢀⠁⡄⢉⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⢻⡿⠋⠀⠀⠀⠀⢿⣿⣿⣥⡘⠀⠀⠀⠀⢰⡀⠃⣼⣽⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣇⠀⠀⠀⢠⠘⣡⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⠿⠛⠛⠛⠛⠋⠀⠑⠀⠀⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠐⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⣛⡛⢻⡻⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡤⠤⣄⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀
⠀⠀⠀⡀⡼⢡⠖⠓⢎⢳⠀⠀⢞⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠘⡈⡏⠒⢄
⠀⢀⣺⡍⠃⠳⣀⡤⢃⡞⠀⠀⠸⡄⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⢣⠸⠀⠀⠑⢄
⣴⢋⠴⠒⢄⢀⡰⣞⠉⠀⠀⠀⠀⣡⠀⢀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣴⣿⡿⣈⠳⣄⠈⡆⢣⠀⠀⠀⠣⡀
⣇⢺⣀⣠⠞⡼⠙⢦⣱⠖⠈⠉⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⠤⣀⡀
⠈⠓⠦⠶⠚⠁⠀⣴⣛⣙⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣦
⠀⠀⠀⠀⠀⠀⢰⠇⣺⡉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣞⣚⣚⣇
⠀⠀⠀⠀⠀⠀⡿⠶⠶⠃⣠⣴⣶⣿⣿⣷⣶⣦⣈⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢀⣤⣶⣾⣿⣿⣷⣶⣄⡀⠀⠀⠠⡹⢖⣒⣒⣿
⠀⠀⠀⢀⠤⠤⠧⡀⠀⣼⣿⠟⡫⠅⠀⠨⣙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢋⠍⠀⠀⢍⠻⣿⣿⡄⠀⠀⠉⠭⠭⠭⠽
⠀⠀⠀⠸⡀⠀⠀⣀⣼⣿⡏⠌⠠⠊⠉⠐⡈⠆⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣿⣿⢡⠁⠔⠉⠁⠆⠑⢸⣿⣧⠀⠀⠠⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢃⠐⢄⣀⡠⢀⠆⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡐⡀⠢⣀⡀⠆⡰⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣑⡂⠀⣐⣡⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣌⣒⠄⠀⣊⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⠿⠟⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⣿⡿⣙⠋⡉⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⠟⠋⢻⣿⣿⣿⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣆
⠀⢸⣿⠇⠛⠆⠛⠘⣿⡿⣉⠉⣉⢹⣿⡇⢸⣿⣧⡀⠀⣤⣬⣛⣿⣿⣿⣿⣿⣿⠀⠀⢠⡖⠦⣄⣤⣤⣤⣄⣠⠤⣤⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⠀⠀⠀⠀⠀⣿⠇⠙⠲⠙⠀⣿⡇⢸⡿⠩⣧⡀⠈⣿⣿⣿⣾⣳⢹⣿⣿⠀⠀⠀⢧⣾⣿⣿⣿⣿⣿⣿⣤⡏⠀⠀⣿⣿⣿⣿⠟⢻⡟⠛⠿⢿⣿⣿⣿
⠀⢸⣿⣿⣧⣤⣼⣷⣿⠀⠀⠀⠀⠀⣼⡇⣼⣿⣤⠝⡇⢀⣾⣿⣿⠾⠿⣿⣿⣿⠀⠀⣀⣿⡟⢉⣻⣿⣿⣋⠛⣿⡆⠀⠀⣿⣿⠏⠀⣤⡄⢠⣶⡄⣾⣿⣿⣿
⠀⢸⣿⣿⣻⣿⣿⣿⡿⠟⣃⣀⣘⡛⠿⠇⣿⣿⣿⠈⣵⣿⣿⣿⣱⡿⢿⣮⣿⣿⠀⠀⠠⠼⢿⣶⣾⣟⣿⣧⣴⣿⠁⠀⠀⣿⡏⢸⡷⠈⢰⣦⠉⠀⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⣿⣿⣿⣦⣙⢿⣿⣿⣜⡿⠿⣫⣿⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣾⣿
⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⢈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠃⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⢀⠀⠀⣤⣶⣤⠀⠀⢀⣴⣶⣄⠀⠀⠀⣾⣿⣿⣿⣿⡻⠽⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⡿⠻⢿⣿⣿⠟⢿⣿⣿⣷
⠀⢸⠀⠀⠘⣛⠃⠠⡴⠟⢻⣿⣿⣧⠀⠀⣿⣿⣿⡃⠈⠁⠞⠿⠻⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡿⣷⣻⣧⢿⣿⢿⣿⣿⠀⣿⡿⣋⣀⠀⠀⠛⠁⠀⣀⣙⠿⣿
⠀⠈⠀⠰⣞⠙⡶⠂⠀⢠⣾⣿⢏⣿⡄⠀⣿⣿⡿⣍⣤⠀⣀⣷⣶⣿⣿⣿⣿⣿⠀⣿⡿⣾⣭⣽⡿⢿⢿⣿⡾⠿⣿⣿⠀⣿⣿⣿⠋⠀⣸⠃⢧⠀⠙⣿⣿⣿
⠀⢠⠀⠀⠛⣛⠓⠀⠀⠸⣿⡏⣼⢿⠇⠀⢿⣿⣿⣿⣿⠂⣻⣿⣿⡿⠛⠻⣿⣿⠀⣿⣿⣿⣞⣿⣿⣿⣿⣿⣟⣛⣿⣿⠀⣿⣿⣿⡀⢸⠇⣀⡸⡇⠀⣽⣿⣿
⠀⢰⠠⣤⣼⣿⣦⣤⠀⠀⠙⢷⣿⡄⠀⠀⣼⣿⣿⣿⣿⣾⣿⡇⠀⢱⠄⠀⢹⣿⠀⣿⣿⣿⣻⣟⣿⣿⢿⣟⣾⣭⣿⣿⠀⣿⣿⣿⣷⣌⠀⣿⠁⣡⣾⣿⣿⣿
⠀⢸⠀⢸⣿⣿⣿⠁⠀⣄⢀⣼⠟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠒⢀⣿⣿⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡿⢋⣴⣿⣦⡙⢿⢻⣿⡿
⠀⠀⢀⣈⣁⣀⣈⣀⣀⣉⣉⣁⣀⣀⣀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠠⢤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣄⣀⣀⣀⣠⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢇⣹⣿⡇⠀⠀⠀⠠⣿⣻⣿⣻⡇⠀⠀⠀⠀⠀⠀⣿⣿⣏⠞⢻⢏⣿⡙⡿⠻⣸⣿⣿⠀⠀⠀⠀⣸⢟⠻⣿⠟⠻⣧
⠀⠀⣿⣿⣿⡿⠿⠛⢛⡟⢀⣠⣾⣿⣿⣇⠀⠀⠀⠀⢈⣿⣿⡍⠀⠀⣀⡀⠀⠀⠀⣿⢯⣭⣰⣿⣿⠿⣿⣾⡆⣭⡽⣿⠀⠀⠀⢠⣿⣼⣧⠿⣼⣧⣿⡆
⠀⠀⣿⣿⣏⣀⣤⣶⣿⠟⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⣾⣿⣿⣷⠀⢸⠿⠇⠀⠀⢀⣿⣾⢿⡿⣩⣾⣿⣷⣍⢿⡿⢷⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢸⣿⣿⣿⣿⣿⡿⠃⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⣿⣾⡇⣿⣿⣿⡿⣿⢸⣷⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣾⣿⣇⣤⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢼⣿⣿⣿⣿⡿⠎⠀⠀⠀⠀⢸⣿⣿⣿⣿⣮⣛⣿⣛⣵⣿⣿⣿⣿⠀⠀⠀⠘⠋⢻⣿⣿⣿⡟⠙⠃
⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠻⠛⠛⠋⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋