Bake Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⢻⡷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⢛⣻⠋⠈⠛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣿⣿⣯⣉⣙⡹⠛⢲⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠉⠀⠙⠀⢀⣠⣾⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠛⠉⢉⡀⢀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠙⠃⠈⠉⠉⠀⠀⠀⠀⠀⠀⠑⠘⣩⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠀⠠⠀⠀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠲⣖⣟⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢂⣄⢀⣀⡀⡀⠀⡄⠈⢡⣸⣶⣔⣨⣰⢮⣿⣞⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠓⠚⠩⡿⢻⡴⠋⠻⢇⠈⠻⣿⡄⢉⡿⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⡄⠀⠀⠀⣸⢠⡄⠀⣾⡄⢀⣿⡇⢸⣧⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣷⠀⢹⠀⢸⣾⣾⠀⣿⡧⢸⣿⡁⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣿⣄⣿⣆⣿⣿⣯⢸⣿⡇⣼⣿⣼⣿⣿⣿⣠⣴⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢿⡇⣿⡿⠸⢿⣿⢾⣿⡇⣿⣿⣼⣿⣿⣿⣶⣶⢞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠳⠾⠿⠿⢿⣿⣿⣿⣿⣿⡯⠿⠷⠄


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

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⠈⠀⠀⢄⠀⠀⠀⢀⣠⡂⣶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⣀⠀⠀⠀⣁⡀⠀⠀⠀⣄⠀⢀⣤⡶⠋⠀⠀⢔⠃⠀⠈⠀⠘⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⣷⠉⠉⠀⢙⡟⠁⠀⣼⠁⠀⠀⠀⡿⠀⠠⠤⠄⠈⣻⣧⠀⠀⠆⠀⠀⠀⠢⠀⠂
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⠀⠀⣠⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣥⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣇⠀⠀⢰⠀⠀⠀⡄⠀⣠⡗⠀⠀⠀⠀⠀⡀⠀⠂⠀⠃⢀⣀⡀⠀⠀⠀⠈⠰⠰⠄⠠
⠀⠀⠀⠀⢰⠏⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⣠⣿⡀⢀⣰⣇⣴⠟⠁⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠈⠉⡅
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⣴⠀⡀⣠⠀⠀⢀⠠⡂⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠆⠋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠄⠀⠀⠈⠃⡏⠀⠅⠹⠆⠀⠈⣳⠀⠀⢤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡄⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣾⠂⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠄⡐⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠃⠀⡠⠄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠆⠀⠀⠀⣄⣠⣧⠈⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠠⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐
⠀⠀⠀⠀⢀⡀⢀⠀⣀⡌⠀⠀⠀⠀⠨⠟⠉⠊⠋⠉⢙⠈⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠘⠀⠀⠀⠀⢺⠁⠀⠁⠀⠀⠀⠀⠀⢀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠩⠀⢀⡀⠀⡀⠀⡠⢀⣠⠆⠀⠀⢠⠐⠄⢀⡠⠄⠀⡀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠠⡄⠀⠀⠀⠀⡄⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠂⠀⠋⠀⠀⠀⠀⠉⠒⠦⠴⠾⠓⠁
⠀⠀⠀⠀⠀⠀⢀⡄⣂⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠐⠂⠀⠀⠻⠛⠃⠀⠠⠶⠶⠥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠅⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠈⠉⠉⠱⠆⠀⠰⠄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠘⠶⠶⠦⠤⠀⠀⠀⠀⠂⠐⠂⠒⠂
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣒⠠⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡂
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠁⠀⠀⣠⠀⡅
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⢠⡄⠀⠺⠷⠂⠀⠀⠀⠀⠀⠀⠀⠯⢀⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⠀⠀⢈⠴
⠀⠀⠀⢿⣿⣛⡛⡟⣟⣿⢻⢛⣿⢻⣛⣟⣿⣿⠀⠀⠈⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠍⠙⠉⠑⠛⠛⠻⠼⠇
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⠒⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠚⠋⠁

⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠠⠖⠿⢗⠀⠀⡐⠀⠂⠀⠀⠀⠀⠈⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡴⠄⠀⠀⠀⠤⠀⢀⠀⢀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠠⠀⡀⠀⠀⠕⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠁⠈⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⢀⠀⠀⠈⢀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡸⠄
⠀⠀⠀⠣⡀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡀⠤⠂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠟⠃
⠀⠀⠀⢆⠈⠂⠤⣀⠀⠀⠀⠉⠐⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠬⠌⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠆⠊⠁
⠀⠀⠀⠀⠑⠢⣄⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⠀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠈⠐⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⢤⠖⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⣀⠀⠈⠁⠐⠂⠀⠀⠀⠐⠂⠀⠀⠀⢀⣠⡤⠴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠦⠤⣤⣤⠤⢤⣄⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⢠⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠠⠀⠀⠀⠠⠄⠛⠧⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡤⡀⠠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠃⠀⠐⠀⠀⠄⠀⠀⠜⠁⠤⠶⠂⠠⠀⠠⠈⠀⠤⠠⠓⠄⠃⠢
⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠂⠐⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄⠀⠀⠠⠀⠀⠀⠀⠀⡀⢄⠠⠀⠀⠠⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡯⠰⠀⠀⠀⠀⠀⡀⠀⠄⣴⣦⡤⢐⡉⠂⠀⢀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠀⡔⠀⠀⢀⣀⠐⠈⠀⠀⠀⠀⠀⠹⣷⠉⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⢠⠌⢩⠄⠀⠀⠠⠀⠀⡀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠁⠄⠠⠨⠁⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠲⠖⠐⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠈⠉⠉⠀⠀⠒⢌⢀⡠⣖⡁⠀⠀⢀⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣈⣿⡽⠳⡶⣮⠀⠐⢆⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠈⢀⢀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠇⠉⠉⠁⠀⠀⠀⠀⠀⡬⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢽⠀⠀⠄⠀⠀⠈⠀⠀⠀⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠀⠀⢠⠄⠐⠂⢤⢺⡀⠀⠐⠂⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢢⣌⢦⢤⡀⢀⣸⣳⣤⣀⡀⠀⢀⢀⠶⠥⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⢱⡁⠀⠈⢣⡁⡉⣘⢈⡀⣀⠀⣀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠐⠢
⠀⠀⠀⠀⠀⡐⠉⠉⠛⠓⠂⠠⠤⢧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠂⠉⠀⠀⠖⠁⠂⠐⠔⠂⠘⢆⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠐⠀⡎
⠀⠀⠰⠿⠉⠑⡄⠚⠂⡤⠒⢦⠤⠂⣾⠿⠄⣀⡀⠀⠀⠀⠀⠀⢈⡤⣀⠀⠀⢀⣠⠷⠀⠀⢀⠂
⠀⠀⠀⠑⣄⡼⠁⠀⠀⠐⣤⠇⡄⢠⠇⠀⠀⠀⠀⠈⠈⠉⣉⡉⠁⠤⠤⠨⢉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠸⠷⠷⠶⠶⠶⠿⠶⠶⠎⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠐⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⠀⠀⠀⠀⣀⡀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠯⠋⠁⠀⠀⠀⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡇⠀⠀⢀⣀⠀⣀⡀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢻⣠⠎⡅⠀⠀⡀⢀⠱⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠃⠠⠲⢏⠃⠐⠇⢀⠀⣆⣠⠃⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠃⡀⠀⢀⠾⠁⠀⠀⠀⠀⠀⠀⢄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡐⠃⢀⣤⡟⡀⠀⠀⠒⠀⠀⢤⠊⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⠶⠊⠉⢉⠤⠀⢹⡀⢄⠀⠠⠄⣀⠜⠒⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⢁⠀⠞⠁⠐⠂⠀⠀⡑⡌⠛⠒⠛⡂⢃⠏⠀⠀⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡔⠃⠈⠻⣀⠀⠀⠐⠆⠀⠘⠈⠄⢀⣼⠀⢸⠀⢀⠀⣈⣿⢯
⠀⠀⠀⠀⠀⣀⡠⢧⡀⠀⢒⡏⡿⠒⠸⠴⢲⠊⠀⠈⠁⠀⢀⠷⠤⢼⡾⠋⠁⠀⠱⢄⡀
⠀⠀⠰⢖⠉⠀⠀⣀⠍⠋⠩⠳⢧⠀⠀⠀⢸⠀⠀⠀⠀⢀⠎⠀⠀⠀⠙⢦⡀⠈⠀⠀⠙⢦⡀
⠀⠀⠠⠮⠤⡴⠊⠀⠂⠀⡀⠀⠻⠀⠀⠀⠈⠳⡘⠋⠛⢧⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠑⡄
⠀⠀⠀⢠⠎⠀⠀⣀⠀⠀⣀⠀⠀⢳⡀⠀⠀⠀⠙⣄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠘⠆⡀
⠀⠀⢠⡣⠐⠊⢁⡠⠔⠂⢹⠀⢀⣀⣱⣄⠀⠀⠀⠘⢤⣤⠇⠒⠒⣒⣒⣊⡉⠭⠭⠥⠟⠒⢆⠀⡀⠈⣂⡀
⠀⠀⡚⠒⠂⠁⠀⠀⠀⠀⢸⠉⠀⠀⣀⣈⡑⠤⣀⠀⠀⠀⠀⠙⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠊⠁⠈⠉⠂
⠀⠀⢰⠀⠀⠀⠟⢂⠀⠀⠸⠀⠸⣍⡀⠀⠀⠀⠀⢆⣲⠦⠧⠣⣎⣤⠤⠤⠤⠴⠒⠒⠒⠊⠉⠂⡍⠉⠒⠢⠤⢄
⠀⠀⢸⠄⠀⣀⡀⠸⣰⠀⣒⠀⠀⠀⠈⠉⠒⠢⠤⣀⡈⠑⠒⠤⠤⣠⠒⢄⠀⠀⠀⠀⠀⠀⠀⢀⣨⢤⢄⠀⠐⣀⠅
⠀⠀⠈⡃⠈⠓⢤⡉⠂⢄⡺⠀⢀⣀⣀⠀⠀⠀⠀⠀⢈⣑⣒⠤⢄⣈⠣⡀⢫⠁⠂⢒⢰⠛⠂⠐⠚⢳⠘⠒⠉
⠀⠀⠀⠉⠉⠉⠀⠉⠑⠢⣄⡊⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣮⠉⠛⡾⠯⠥⠤⠜⢻⠉⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠛⠒⠓⠤⠤⠤⠤⠤⠤⠔⠒⠒⠊⢉⡇⠀⠀⡇⠲⠴⠶⠆⢸⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⠜⠀⠀⠀⡇⠀⠀⠀⠀⠈⡇⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣤⡄⠀⠀⠀⣀⣀⣤⠖⠁⠀⠀⠀⠀⢇⠀⠀⠀⢀⣠⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣾⠹⠡⢃⠎⡲⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠀⠈⠛⠒⠦⣊⢌⠜⠅
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠤⡆⠀⡠⢡⣆⢠⠀⠈⠳⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠞⣁⣨⠔⠋⠊⠈⠈⡘⣆⠀⢀⣤⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠩⢁⣀⠄⠀⠊⠀⢀⡈⣢⣶⢫⠉⠙
⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⢀⣴⣿⣯⣙⡇⠀⠘⢣⡌
⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢁⢀⣤⣶⡛⣲⡞⠬⠽⠟⠋⠀⠀⠀⡠⣁
⡗⠠⠀⠀⠤⠒⠁⠄⠀⠂⠉⠐⠢⡀⠀⠻⣘⣻⠿⠧⣤⡠⠖⠂⠀⠀⠀⡐⠀⡆⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠰⠂⠒⠡⢼⠐⠁⣂
⡇⠉⢀⢀⠀⠀⠀⠀⡡⢆⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠉⠁⠀⠄⡀⠀⡺⠃⠀⡇⠀⠀⢀⠙⣤⣡⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⠀⠂⠀⠀⠀⠀⠁⢄⠀⡐
⡇⠀⠈⠘⠂⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠒⠈⠀⠈⢉⡉⢀⠘⠀⠈⠀⠀⠀⡋⢀⠯⠥⠆⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⢄⠀⠀⠀⠀⠈⡅⠄⠒⠂
⡦⠤⣀⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠋⠰⢊⡔⠁⠠⡀⠠⡀⠀⡀⠀⠀⠆⠀⠀⢰⠖⠁⠀⠱⠐⢄⠐⠄⠀⠀⢾⣧⠀⠀⢀⡀⠨⠂⠀⠀⠀⡄⠀⠐⠃⡄
⣟⣓⠒⠢⠤⠭⢉⣉⣓⠶⠶⠀⠀⠀⡤⠈⡡⠂⠙⠒⠀⣐⠮⠑⢽⡒⠚⣩⣍⡀⠀⢀⠊⠂⠀⠀⠀⠡⢀⡉⠀⠀⠀⠀⠐⠉⢀⠻⠇⠀⠀⢀⡴⢴⡀⠀⡄⠀⠁
⣧⣬⠭⠽⠓⠊⠉⠀⠀⠀⠀⢀⡔⢨⣄⠀⠹⡇⠐⢒⡦⠩⡄⠈⢠⡔⠂⠌⠘⠻⣶⠁⠀⠀⡀⠀⠀⠀⠃⠀⠀⠀⠈⢀⡀⠈⠁⠀⠀⠀⣀⠴⠤⠦⠤⠤⠴⠶⠖
⡏⣉⣐⣓⣒⡒⠒⠒⠤⣤⣤⠀⢌⠲⡵⠃⠐⢖⠆⢨⣄⡈⠿⣀⠸⠋⡩⠠⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡸⠉⠉⠝⠋⠉⠉⠓⠀⠶⢐⠈⠁⠀⢄
⡗⠒⠲⢤⣤⠭⠝⢛⣉⠅⠠⡀⠠⠑⠄⠙⢆⢠⢦⣠⣦⠀⠹⠋⠰⠠⠄⠔⠈⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡈⠂⢂⠀⡔⠀⠀⠀⠀⠀⠢⡀⠀⡠⠅
⡿⠛⠋⠩⠄⠀⠀⠈⠀⠀⠀⠀⠂⠈⠢⡈⠢⡙⠋⠀⠈⢉⠔⠓⠀⠀⠠⠄⠠⠤⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡠⠀⠀⡀⠈⠁⠑⠈⢡⠈⣠⠠⢀
⡇⠀⠀⠀⠒⠀⠀⠀⠀⠠⠀⠀⠀⠄⠂⠉⠀⠈⠑⠤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⡜⠙⠋⠀⠢⢡
⡏⠢⠀⠀⡀⠔⠀⡀⠀⠀⠈⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⡂⠀⠠⠄⠀⢤⠀⠀⠰⠀⠀⠀⠁⠀⠀⠘⠑⠀⢀⡉⢀⡀⠀⠀⠀⠀⡷⠐⠀⠀⠄⡈
⡏⠢⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⢀⡀⢀⡀⢤⠐⠂⠁
⡇⠄⡁⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡃⠀⠀⠀⡑⠠⠈⠂⠀⠀⠀⠀⠀⠀⢀⡄⣀⠀⠀⡀⢀⡄
⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠈
⡇⠀⠀⠀⠠⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠶⢰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠁⠀⠁⠀⠐⠄⠀⠀⠀⠀⠂⠀⢇
⠀⠀⠀⠀⣰⠢⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠃⠰⠀⠠⠀⠀⠀⠂⢃⠄⠀⠰⣄⢀⠀⣠⠕⠀⠀⢀
⠀⠀⠀⢰⣫⣀⣀⣀⣠⢔⣒⡋⢱⠀⠀⡀⠳⠀⠀⠀⠀⠐⣒⠭⠿⠀⠀⠀⢩⣆⣽⣧⡠⢄⣤⣬
⠀⠀⠀⠑⠚⠤⠠⠤⠄⠊⠍⠀⠀⠀⠀⠈⠐⠤⠤⠄⠠⠤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠈⢯⣽⣭⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⠰⣤⣀⡀⠀⠀⡀⠉⡦⢢⠀⠀⠀⠀⠀⡐⠀⠁⠂⠀⠀⣼⠀⠀⠀⠀⡠⠔⠁⠊⠉⠀⢒
⠀⠀⠀⢤⣄⣀⣀⣀⣀⣀⣀⣤⡔⠀⠀⠀⣤⡂⠀⠀⠀⠀⢄⡴⠁⠀⢀⡤⠈⠀⡄⠀⠀⡀⡠⠁⠄
⠀⠀⢀⢀⠠⠈⠉⠉⠉⢉⡡⠞⠀⣠⠀⠀⠉⠑⢧⢀⣀⠤⠋⠀⠀⠀⠑⠳⠄⠉⠀⣀⣀⠆
⠀⠀⠀⠀⣴⠒⠀⠌⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⡢
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠘⡤⠋⠀⠤⠄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⢋⢥⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⣦⣠⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠞⠋⠁⠀⠀⠠⣱⣤⡀⠀⠀⠀⠀⡇⠀⠠⢙⣛⡀⡇
⠀⠀⠠⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣝⠀⠀⠀⠘⠀⠦⠃⠉⠩⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡏⠉⠀⠀⠈⠉⠉⢉⣭⣭⣍⠉⠉⠉⠉⢀⣠⣤⣀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⢸⣧⠀⣾⡁⠀⣹⡇⠀⢰⡆⣾⡃⠈⣻⡆⠀⠀⠀⣿⡁⢈⣿⠀⣬⡅⠀⠀⠀⠘⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠙⠛⠛⢁⠀⣀⠀⠀⠀⠀⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⠟⢋⡉⢉⠉⢩⠟⠋⠀⠄⠀⠈⠀⠐⠀⠼⠛⠉⠁⠀⠆⡀⠌⠙⢿⣿⣿⣿⣿⣿⡇⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣅⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠉⠉⠉⠉⠉⣩⠟⠉⢉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⡇⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⣠⠋⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⠟⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡯⠙⠿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣟⣁⣀⣈⣉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣿⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣿⡷⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢿⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣥⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠯⠥⠬⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠤⠴⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠓⠶⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃




⣿⣿⡿⠿⣿⠿⢿⠿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠊⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢙⢀⣄⡤⠀⠀⠠⠀⣤⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠀⠀⠀⠀⠀⠀⠛⡟⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡖⠀⠀⠀⠀⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠛⠃⣴⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡀⠠⠤⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠤⠂⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⣿⣷⣼⣦⣴⣾⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠛⠛⠉⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣆⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢾⣿⠇⣿⣿⡀⢀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⡖⠀⢠⡀⠈⣧⠀⠀⠀⠀⠀⠀⣿⣿⠀⢹⣿⠆⣇⡞
⠀⠀⠀⠀⠀⠀⢸⠡⢜⡲⢽⣶⢻⠀⠀⠀⠀⠀⠀⠘⠁⠀⠸⡏⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠙⣺⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡭⢬⡍
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠘⣮
⠀⣀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠔⠛⠁
⠀⠻⠦⠘⣷⠖⠤⠤⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡐⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⠉⠉⠑⠒⠒⠲⢤⣠⣤⣤⠤⢤⣄⣀⣀⣀⣀⠀⠀⠀⢠⠁⠶⢛⣃⡈⠛⣁⡒⢠⡤⠉⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠁⢸⡀⠚⢦⡬⠀⠀⠩⠡⠄⠂⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⢦⣄⡀⠂⠝⠛⠀⠘⠛⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⢤⣄⣀⣀⣉⡛⠓⠶⠶⠦⠤⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⢉⡀⢀⡀⠀⢉⠀⠉⠀⠀⠀⠁⠀⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⠄⠀⢀⡀⠀⠘⣀⢀⠀⣀⣠⡀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣍⠀⠀⠀⠀⠀⠉⣤⠀⣀⠀⠀⠄⡄⠉⠀⠉⠉⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⣀⠀⠀⡀⠀⢠⡄⠀⠀⠁⡀⠀⣰⠣⠀⠀⠀⠤⠀⠀⡠⠔⠁⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠉⠀⠀⠀⣀⠔⠀⣀⠾⠭⠄⠁⠁⠉⠈⠠⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠓⠀⠀⠠⠀⠀⠂⠁⠀⠀⠚⢅⡀⠁⠠⡀⢀⠄⠀⢀⡀⠀⠐⠂⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠂⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠒⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠐⠀⠀⠄⠈⢆⠀⠀⠀⠁⠀⢀⡠⠔⠉⠀⠤⠤⠤⠤⠒⠀⠀⢀⡴⢧⡀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠙⢤⡀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢻⠀⠀⠀⠀⠀⠄⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠈⢀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠑⠄
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠄⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⠀⠈⠢⡀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⣀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡅⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠹⣧⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠙⠒⠂⠉⠉⠁⠀⠈⠀⠀⠀⠑⢄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⣦⠤⠀⠀⠀⠄⠒⠈⠀⠀⠀⠀⠀⠀⠑⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⣰⠁⠀⠀⠑⢦⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⡇⢀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠢
⠀⠀⠠⠔⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠇⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠈⠓⠒⢤⡀⠀⠀⠀⠸⡀⠀⠀⢀⡧⠀⠀⠀⠀⠡⣤⢃⠀⠀⠀⠀⠌⠢⡀⠀⠀⡸⠀⠰⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤
⠀⢀⣀⣀⣀⣈⣳⣄⢀⣀⣉⣁⣀⣈⣿⣄⣀⣀⣀⣀⣀⣈⠀⠀⠀⠀⢀⠠⠀⠀⢀⣁⡀⢇⠀⠀⢀⡀⠠⠔⠋⠁⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠉⠉⠉⣉⢩⠍⠉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠁⠰⠦⢀⠀⠀⠈⠑⠁⠀⠀⠀⠤⠀⠀⠀⢀⣀⣤⣶⣿⠿⠋⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⠀⠠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⢢⡀⠈⠲⣀⣤⣶⣶⣶⣶⡿⠿⠟⠋⠉⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠂⢀⠀⠀⠀⠀⠐⠉⠉⡁⠀⠀⠀⢀⠀⠀⣄⢀⡀⠀⣀⡀⠈⢆⠀⠐⠂⠭⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⡄⠈⡁⠀⠀⠉⠀⠀⠀⣈⠉⠉⠉⢠⣀⠈⠢⣄⠠⠄⠈⠑⠀⢤⠤⠤⠤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠰⠂⠒⠚⠀⠉⠉⠛⠓⠛⠉⠀⠀⠉⠉⢓⣓⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠲⢤⣠⣤⣀⣀⡤⠀⠙⠦⢤⣤⡤⠤⠄⠀⠙⠲⠶⠤⠤⠓⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠤⠴⠴⢶⠶⠆⠀⡀⠀⠀⠶⢶⠶⠆⠡⠈⠛⠛⠛⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⡶⢶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⢄⡠⡀⠀⠀⠀⠉⠀⠀⠀⠐⠚⠒⠢⠀⠠⠤⠆⠤⠄⠀⠀⣴⣿⣿⡆⢀⡟⢹⢹⠀⢴⣶⣶⣶⠄⠀⠀⠠⠀⠁⠑⡀
⠀⠀⠀⠀⠀⠀⣠⣮⡄⢀⣦⣹⡏⠀⠀⠀⠸⠀⡆⠀⠀⠀⠀⠀⠀⣆⣄⣀⣀⣸⡀⠀⠙⢿⡿⠃⠀⢯⣼⡼⠀⢸⣿⣿⣿⠀⠀⠀⠤⠀⠀⠀⠠
⠀⠀⠀⠀⣠⠞⢾⡟⡿⢿⣿⣋⣱⡀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⣿⣷⡗⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠸⣷⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣠⠞⠁⢀⡼⣧⣶⠿⣿⢿⣿⠃⢰⣿⣷⣿⡆⠀⠀⢲⡎⠀⢸⣿⢋⠄⢙⢿⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠘⡟⠀⠀⠀⠁
⠀⢸⣿⣤⢴⣿⡾⢋⡵⠊⠙⠛⣿⣦⡄⠛⠿⠛⠀⠀⠀⢼⡧⠀⢸⣧⡠⠖⠉⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇
⠀⠀⠻⢿⡟⢁⢀⣥⣒⣦⡄⠀⠘⣿⣛⢉⠂⠀⠀⠀⠀⢸⡇⠀⢸⣇⠀⡀⠀⣸⡇⠀⠀⢈⣁⠀⡀⢀⣁⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠉⡇⠆⣿⣿⡇⣴⣿⣿⣿⣿⣿⣿⣿⡦⠀⢸⡇⠀⢸⣿⣦⣤⣴⣿⡇⢀⡐⣿⣿⠀⠃⢸⣿⡷⢀⠀⢸⢹
⠀⠀⠀⠀⠀⠀⡇⠸⣿⣿⣷⣽⣿⡇⠁⡇⢸⣿⣿⡇⠀⣞⣳⠀⠘⠾⢿⣿⡿⠿⠃⢸⣧⣹⣿⣆⢀⣾⣿⣡⣾⠀⠀⠁
⠀⠀⠀⠀⠀⢰⡇⡇⣿⣿⣿⣿⣿⡇⠀⡇⢸⣿⡿⠁⠀⠈⣡⣤⣶⣦⣤⡀⠀⠀⠀⠈⢿⣿⡿⠉⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡠⠞⠁⢃⣿⣿⣿⣿⣿⣿⣦⣧⣼⠛⠂⣄⠀⣾⣿⡽⠛⠲⣜⢿⣆⠀⠀⠀⠀⠈⠛⠐⠀⠘⠉⠁⠀⣴⢤⣤⣤⣤⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠘⠷⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⢸⣿⢸⡁⠀⠀⣸⢺⣿⢀⡠⡀⢠⡄⠠⣀⠄⢠⡤⠀⣰⣿⣿⣿⣿⣯⣆⠀⠀⡀⣿⢸⡇⡷
⠀⠀⠀⠀⠀⣠⢐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣂⡄⠀⠀⠻⣷⣝⣒⣚⣹⡾⠃⢸⣗⡣⡿⢿⠸⠿⡆⢤⣓⠀⣿⣿⣀⣀⣀⣻⣿⠀⠰⡇⢹⢸⢣⡇⣶
⠀⠀⠀⣰⡾⢿⣿⣿⡿⢿⣟⡩⠉⢉⣭⢍⠉⣰⠁⠀⣠⠤⠈⢉⠙⠉⢁⣀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⢀⣷⠿⠿⠿⠿⠿⣼⠂⢀⣇⢸⣸⡾⣇⣿
⠀⠀⠀⠹⣧⣀⢹⣿⣿⣇⣿⠀⠀⣼⠇⠀⡟⠄⠀⠀⠐⠀⣠⣽⣶⣶⣤⣌⣉⣲⢈⣉⣀⣙⠾⣯⣩⠟⠀⢰⣿⠀⠀⠀⠀⠀⢻⡆⢰⡼⣸⢧⣇⡿⣼
⠀⠀⠀⠀⠈⠙⢿⡿⢿⣿⢿⡀⢀⣿⢀⡼⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠁⠉⠀⠀⠀⢸⠿⢿⣿⣶⣾⣿⣿⡇⢻⡇⣿⢸⡅⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠯⠉⠭⠟⠀⠀⠀⠀⠀⠀⢮⡤⢾⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⠶⢬⡵⠀⠸⢤⣴⣿⣿⣿⣿⠿⠇⠘⠇⠿⠸⠇⠇⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠒⠊⢁⡀⡀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢴⠇⢸⠁⡀⠀⠀⠀⢠⡆⢉⣉⡄⠈⣆
⠀⠀⠀⣀⠤⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠄⡐⠀⠀⡱⠀⠀⠀⠀⠀⠀⠐⠂⣩⣤⣄⡤⣾⠀⡎⠀⢠⡞⢿⣻⠙⣗⣛⣻⠇⠀⣼
⠀⠐⢊⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠪⠐⠀⠀⡠⠂⠀⢀⠖⠁⡴⢚⡇⠘⡿⢷⣫⠸⡇⠀⠃⠀⢾⣑⠀⠡⠞⠉⠉⠀⣀⡼⡿
⠀⠀⠀⠑⠒⠤⢀⢀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠈⠀⠀⢀⠆⢀⡼⢧⡸⢗⣜⣻⣶⢓⠜⠁⡈⠀⠀⠐⠽⠛⡛⠛⠒⠚⢋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⠀⠀⠀⢀⡎⢠⣇⣟⣲⢨⡉⣉⠟⠁⠀⣀⠜
⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⡸⠀⢸⣻⠶⠓⠈⠉⣁⡠⠔⠊⠁
⠀⠀⢀⡤⠞⠉⠀⠀⠀⡀⠀⡎⠀⢉⠒⡄⠀⡀⢰⠇⠀⠋⣀⡠⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⡔⠈⠙⣒⠃
⠀⡰⠋⠀⠀⠀⠀⠀⠀⢁⢰⣇⠀⠐⠀⠀⠐⠤⢼⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠰⣤⠎⠀⠀⠀⠳⢀
⢸⡠⠤⠤⣀⠀⠀⠀⠐⣀⠜⢻⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣚⠁⠀⠀⠈⠁⠀⠀⠀⢀⠠⠄⠓⠉
⠙⠀⠀⠀⢈⡷⠤⠒⠉⠀⠀⢪⠀⠐⠁⠀⠀⠀⠀⠀⣠⠄⠒⠲⣺⠀⠀⠀⠀⠉⠀⡀⣀⠀⠀⠂⠁
⠸⣄⠀⠀⠉⠉⠉⠑⠢⠀⠀⠀⠀⠀⢀⣀⡰⠂⠀⠹⡵⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀
⠀⠉⠒⠤⠤⠤⠄⠀⠀⠀⠀⠀⠠⠊⠀⠈⠙⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡀⠠⣦⣤⡶⠉⠀⢹⣋⠞⠋⠉⠉⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢸⡿⠋⠀⠀⣾⠞⠁⠀⠀⢀⡾⢋⡳⠶
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠤⠤⢀⣀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠾⠊⠁
⠀⠀⠀⢀⠤⠒⠉⠲⠿⠥⣬⣍⠲⡀⢀⡄⠈⣠⠤⠖⢒⠒⡶⠤⡤⣄⠙⢢⠀⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⣠⠞⣁⡀⠛⠓⠲⢶⡄⢀⡭⠞⠁⠎⠀⠀⠛⠺⡶⠒⠒⠀⠶⢶⡫⠃⠄⠁⠀⠂⠍⠒⠒⠤⠤⠤⠤⠤⠴⠶⠒⠒⠉⠉⢀⡎
⡼⣃⣰⣉⠭⡙⠻⢆⡤⠞⠉⠀⠀⠀⢀⠀⠐⠀⠌⠉⠘⠓⠚⠫⢁⠰⠀⢠⠀⠀⠸⡄⠠⠀⡀⠀⠀⠐⠀⠐⠀⠀⢀⢀⢡⡾
⠣⠏⠛⠛⠛⢂⠶⠋⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠁⠀⠀⠀⠁⣀⡠⠖⠁⠀⠀⠀⠉⠓⠦⠤⢤⣅⣍⣈⣈⣈⡠⠤⠖⠋

⠀⠀⠀⠀⠀⢀⡠⠔⠒⠀⠀⠐⠒⠤⣀
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⢀⠎⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡄⠀⠀⠦⣀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀
⠀⠀⡘⠀⠀⠐⢤⢀⠆⠀⠀⣴⢀⠀⢤⢀⢀⠘⠀⠀⠀⠀⠀⢼⠃⢀⣀⡀⠻⣷⣄⠀⠀⠀⣠⡀⠘⣯⠀⠀⠀⠀⠀⠀⠀⠈⢀⡏⢮⡁⠄⢰⣣⡀⠰⣀⣠
⠀⠀⢃⠀⠀⢀⡎⢈⡶⠈⠆⠚⠒⠀⠈⢩⠃⢀⠀⠀⠀⠀⠀⢿⡀⠘⠛⠀⢀⠈⠻⣷⣄⠀⠛⠁⠀⡯⠀⠀⠀⠀⠀⠀⠀⠤⠎⠐⠊⠉⠉⠁⠉⠀⠀⠀⠃
⠀⠀⠘⡄⠘⠊⠀⠁⠀⠀⠠⠐⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠹⢇⠀⠀⠔⠃⠀⠀⠈⠙⠦⠀⠀⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠦
⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠑⠦⢄⣀⠀⠀⢀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⣠⣾⠟
⠀⠀⠀⠀⡀⠀⠀⠈⢿⣦⡀⣠⣾⠇⠀⠀⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣛⣸⣷
⠀⠀⠀⢨⡇⠀⠀⠀⢠⣿⣑⣿⡇⠀⠀⠀⠘⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⣷⣾⣿⣧
⠀⠀⠀⢾⡆⠀⠀⠠⢾⣿⣷⣿⣷⠄⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦
⠀⠀⠀⢘⣧⠀⠀⢀⣾⣿⣿⣿⣷⡄⠀⠀⢬⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠿⣷⣄⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⢠⡤⣄⠸⠿⢿⠏⢹⡿⠿⠀⡀⢤
⠀⠀⠀⠀⠘⠁⠀⠘⠿⣿⠛⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠻⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⣻⢳⢳⢻⡟⡟⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⠴⢦⢾⡼⠷⠷⠿⠿⠈⠁


⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢰⠂⢠⡄⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠘⠒⠚⠓⠸⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⡌⠀⠀⢠⣶⣿⣿⣷⣿⣿⣷⣦⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⡀⠀⣀⢀⡄⣀⠀⠀⡀⠀⢀⡀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⢿⡀⠀⠈⠉⠛⠛⠛⠛⠛⠋⡉⠀⠀⢠⠇⠀⠀⠀⠀⣤⣴⣶⣦⣤⣾⡟⣿⡇⠛⢸⡇⡆⣿⠈⢡⣿⠘⢻⣿⠸⢃⣿⡘⣸⣿⢛⣷⣦⣤⣶⣶⣤
⠀⠀⠀⠀⠈⠁⠀⠀⠃⠃⠁⠨⠬⠘⠇⠧⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠙⠿⠿⠇⠛⠸⠇⠶⠸⠸⠧⠻⠜⠋⠿⠰⠌⠿⠇⠿⠿⠿⠏⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠐⢧⣰⠒⣶⣲⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠
⠀⠀⠀⢀⣤⣤⠖⠠⠀⠀⠀⠐⠁⠀⡠⠀⠀⠀⢀⣀⢤⠀⡀
⠀⠀⢠⣿⣿⡿⠦⠀⣤⡖⠞⠁⠀⠒⠀⠀⠀⠀⠈⠁⠀⠀⡀
⠀⠀⢈⣿⣿⡂⠆⠘⠁⠈⠃⢀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠂
⠀⢀⣼⣿⡿⢁⠀⠀⠀⠀⠀⠸⡆⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢀⣼⣿⡟⠀⠘⠀⢀⡐⠂⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡄⠐⠀⢀⠠⠐⠒⠉⠉⣀⢠⡆
⢰⣿⡏⠀⠀⠀⠀⠀⠑⠤⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣻⠃
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠠⠄⠀⠉⠀⠂⠀⠠⡀⠀⠀⠀⣸⣿⣿
⠀⠙⢺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⡄⠒⠆⠁⠀⠀⠀⢉⢽⣿
⠀⠀⠈⠀⢶⠦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠘⠒⠒⠒⠒⠲⠶⠶⢶⠦
⠀⠀⠀⠀⣁⣤⡿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⣼⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⠁⠀⠀⠀⡀
⠀⠀⠀⠸⡻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⢞⢉⣎⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠈⠁⠀⠘⠉⠀⠀⠲⠄⠄⠀⠀⠀⠀⠀⠐⠄⠴⢧
⠀⠀⠀⠀⠀⠁⣧⡠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠀⡄⠀⢀⠀⠀⢀⠀⢠⢀⣦⡀⡁⠀⠀⡔⠁⠀⠀⠀⠈⢑⠄⠠
⠀⠀⠀⠀⠀⠀⡧⢌⡉⠁⠀⠒⠒⣊⣉⣹⠤⡀⠀⠀⠀⠀⢢⣀⠀⠀⠂⠈⠀⠀⠘⠐⢀⣀⣤⣞⠀⠀⠀⢸⡶⣐⢢⣤⣒⠋⠄⠂⠄⠀⢤⡀
⠀⠀⠀⢐⠀⣾⡠⣤⣤⣤⣤⣤⣭⡭⣅⣋⡰⠈⡇⠀⠀⢀⢰⠓⠬⢁⠀⠀⠐⠒⣒⣉⡭⠴⢛⠆⡂⠀⠀⠀⢧⣵⣸⣸⣸⣤⠤⢄⣤⢴⣼⠁
⠀⠀⠀⠀⠑⠦⠭⣖⣀⡀⠀⠀⠲⣒⣋⠩⠀⠈⠀⠀⠀⠈⠓⠭⣄⣀⠀⠀⠀⠀⠀⣀⣀⣠⠥⠚⠁⠀⠀⠀⠀⠀⠉⠁⠀⣻⣹⢹⣹⣫⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠔⢡⡀⢀⠠⠀⠒⠒⠀⠀⠒⠒⠬⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠰⠄⡐⠅⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⡠⠖⠩⠙⢥⠒⣀⣤⣀⢀
⠀⠀⠀⠐⠎⠑⠚⠶⠤⠤⠤⠤⠤⠤⠀⠘⠒⠙⡆⠀⠀⠀⠀⠀⠀⠰⠁⡀⠀⠀⠐⠐⠈⠄⡸⠁⠀⠀⠀⠀⢠⠔⠋⠀⠀⢨⠁⣀⠰⠄⡤⠀⢀
⠀⠀⢠⠀⠶⣤⣤⣀⣀⣀⠀⠀⢀⣈⣀⣠⣤⡾⠀⠀⠀⠀⢠⣔⡊⠀⠡⠀⠀⠀⠡⠀⢐⡔⠁⠀⠀⠀⡠⠚⡀⠀⠐⡀⠉⠁⢎⡀⣀⠆⠀⡠⠁
⠀⠀⠸⠀⠀⠈⣉⠉⠛⠛⠛⠛⠋⠙⠛⢉⡼⠁⠀⠀⠀⠀⠈⢋⠻⢔⡂⠁⠀⠐⣀⠜⠉⠀⠀⠀⠈⠣⢕⡀⠁⠀⠂⠡⠊⠁⠐⡄⠀⠀⠄
⠀⠀⠀⠑⠲⠖⠁⢀⡂⠀⠤⠤⠴⠶⢚⠁⠀⠀⠐⠃⠀⠀⠀⠀⠑⠌⣳⣴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠑⠒⠊⡠⢀⠂
⠀⠀⠀⠀⠀⠀⣟⠉⢁⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠬⠭⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⢀⠵⠀⠀⠀⠀⠀⠉⠀⠱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⢀⣁⠀⠤⠀⠀⠐⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠎⡥⠤⠀⠴⠖⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⡐⠐⠀⠃⢀⣠⡶⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⣧⠀⢀⣼⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡅⣀⣴⠶⠛⠁⠐⠣⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠈⠀⢹⣷⣿⠏⠀⡇
⠀⠀⠀⠀⠀⢐⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⠀⠀⠀⠀⠀⠘⡮⣧⢦⣄⠀⠀⠀⠀⠀⠀⠼⠀⠀⠰⠦⣬⣉⣠⢠⡇
⠀⠀⠀⢴⢈⠜⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⡳⡌⠇⠀⠀⠀⠀⠘⢀⢠⡄⠰⠘⠛⣙⣉⡼
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠔⠒⠊⠉⠉⠀⢀⣀⠀⠀⠀⠑⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠒⠈⠁⠀⠀⣀⣀⣤⣤⠴⣾⣮⠻⣿⡿⣄⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢀⠔⠒⠉⠁⠀⢀⣀⣤⡀⣶⣔⢾⡟⢿⣇⠀⠻⠧⠘⠛⠁⠉⠀⠀⣀⡠⠤⠐⠂⢙
⠀⠀⠀⠀⠀⠈⡷⢄⠀⠀⠀⠈⠻⣿⣿⡾⠛⠛⠋⠀⠁⠀⠀⣀⡠⠤⠐⠒⠉⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡏⡆⠣⡀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⡇⠘⡄⠈⡢⡄⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⡇⣿⣿⣇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠠⠋⠁⠀⠃⠀⠀⠀⠀⢀⡀⢠⡄⣶⠺⣿⠋⢹⡟⠸⠿⠃⠋
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⢾⣷⠟⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠚
⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⡆⠀⠀⣀⠠⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣤⢠⢀⡄⣀⢠⣤⡀⣤⡄⢠⡤⠀⣤⣄⠀⠀⢠⣤⢠⢤⡀⣤⠤⢀⡤⣤⢠⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⡏⢸⢼⡇⣿⢸⣤⠇⣧⡿⢸⠶⠀⡇⢸⠀⠀⣿⠈⢸⣼⠃⣿⠆⢸⣧⣼⢸⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠃⢸⠘⠇⠿⠸⡇⠀⠯⠀⠸⠤⠄⣧⠞⠀⠀⢻⡴⠸⠘⠆⠿⠤⠸⠇⠸⠸⠇⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠆⠀⡆⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⠀⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡇⠀⠘⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⡻⠀⠀⠀⠁⠀⢰⠀⠀⢰⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠥⢀⡔⠁⠀⢀⠌⠀⠀⠀⠀⠀⢸⠀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⠚⠁⠀⢀⠠⠋⠀⠀⠀⠠⠀⢀⠌⢠⠃⠌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢀⠔⠁⠀⠀⠄⡀⠔⠁⡠⠂⡴⠃⢠⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠑⠀⠀⠀⢀⠠⠔⠊⣀⠄⠊⣠⠀⠀⠀⢠⠀⢰⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⢀⠂⠀⠈⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠊⠀⢀⠠⠐⣈⣉⣀⡠⠤⠔⠊⠁⠀⠀⠀⡀⠔⠁⠀⡠⠀⢀⣼⡅⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡥⠚⠉⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⠀⢀⡠⠊⡠⠞⠁⢸⡇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠉⠀⠀⠠⠒⠀⠒⠒⠒⠀⠉⠀⠀⠀⠠⠔⠊⡁⠀⠈⠀⠀⠀⡸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠄⠐⢒⡀⠀⡈⢉⣉⠥⠄⠒⠒⠒⠀⠁⠀⢀⠔⠀⢀⡴⠁⠀⠀⠀⣾⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠘⠃⢀⣠⠔⠚⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣠⠴⠋⠀⠀⠀⢀⡜⠁⢇⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠒⠒⠒⠈⠀⠀⢀⠤⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠖⢹⠀⠀⡸⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠒⠀⡀⠀⢠⠞⠉⠉⠓⠒⠢⠤⠤⠤⠔⠂⠁⠒⠁⠀⠀⢀⠀⠀⠁⠀⠀⠀⣠⠃⠀⣠⠃⢀⠂⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠐⠈⢙⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠁⠀⠀⢀⠤⠊⠀⠀⠀⠀⣀⡤⠚⠁⢀⠔⠁⡐⠁⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠁⢀⣠⠴⠒⠋⠉⠀⠀⣠⠆⠀⠀⠈⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⠀⡠⠒⠉⠉⠉⠉⠁⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣠⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣔⣋⡠⠴⢾⡉⠙⠂⠀⠛⠉⠀⠀⠀⠀⣠⣴⣒⣋⣉⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣄⣀⣀⣀⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⢀⠂⠀⣀⠀⠀⠀⢠⣢⡘⠋⢈⠷⣣⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⢃⡸
⠀⠀⠀⠀⠀⢤⣈⠂⠀⠀⠎⠁⣏⠹⢗⣋⢁⠙⠧⢔⡄⠀⠀⠠⠺⠇⢀⠀⠀⠀⠀⠈⠁⣠⣞⠱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠙⠁⠛⢣⣄⣓⢬⠄⣬⡒⠇⠀⠠⠀⠀⡀⠀⢉⠖⠀⠀⠾⢿⡟⠉⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⢶⢄⣤⣼⢀⠁⠾⠧⠕⡀⡄⠀⠀⡀⣠⢴⠛⢋⠠⢂⠀⠤⣀⡄⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠡⠐⠂⢌⠉⠰⠆⠁⠀⠈⠉⢿⡁⠊⠁⠀⠀⠀⠀⠈⠉⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠂⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⢃⡄⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠀⠀⠈⠑⠂⠤⡀⠀⠀⠁⠈⠼⠀⠀⠀⠈⠙⠷⠤⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠢⣄⠉⠒⠄⣐⠂⠀⣆⡀⠀⢀⠀⣀⠂⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⠀⠄⡠⠀⠀⠤⠀⠀⠀⠠⢠⠐
⠀⠀⠀⠀⠉⠀⣠⣿⡟⠓⠚⢷⣾⡚⠛⠋⠙⠲⠀⠀⠀⠠⡀⣠⡆⠀⠈⢹⣦⡇⣀⠀⠂⠠⠠
⠀⠀⠠⠊⢀⠾⠛⣿⠷⠒⠂⠀⠀⠘⠯⠶⠲⢾⠆⠀⠐⣎⠂⠀⠀⠀⠀⠈⠨⠛⡧⢶⣠⣇⣰
⠀⠀⠀⠈⠁⠜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠁⢀
⠀⠀⠀⠀⠘⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠖⠦⢀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠁⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢡⣶⡀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⣦⠀⠉⠀⣐⠆⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠉⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⢤⣤⣤⣤⣀
⠀⠀⠀⣴⣾⠟⠋⠉⠉⠉⠉⠉⠙⠛⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣄
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣥⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢦⣤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠘⣿⣆⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⣴⣾⠟
⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⢺⣷⡶⠾⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣶⣿⡷⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣷⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠶⠦⠤⢤⣤⣤⣀⣀⣀⣀⣤⣶⠿⠛⠛⠛⠛⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠛⠛⠓⠒⠲⠦⢤⣀
⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠐⠒⠒⠊⠍⠀⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠠⠤⠀⠒⠒⠉⠉⠑⠒⠃⠀⡆
⠀⠀⠀⠀⠀⠀⣄⡀⠀⠰⡀⠀⠉⠒⠂⠸⠀⠀⠀⠀⠀⠀⠑⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡒⠢⢤⡀
⠀⠀⠀⠀⢀⠨⣀⡀⠉⠒⠺⠦⣄⡒⠀⠀⠀⠀⠉⡶⢄⡀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠟⣀⡠⠗
⠀⠀⠀⠀⠑⢦⣈⠛⠷⣶⡤⣴⣀⡀⠑⠖⠦⠤⣼⣦⡀⠈⣳⢤⡔⡇⠀⠀⠀⣀⣀⠤⠴⢒⣊⡭⢴⠒⢛⠟
⠀⠀⠀⠀⠀⠀⠣⣉⠓⠦⣍⣛⠁⠉⣙⠲⢤⣀⣀⣈⢉⣶⣄⠠⠼⠗⣒⣊⢭⠤⠒⠒⠉⣉⡠⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣌⡉⠒⠦⠭⢁⣒⣭⣭⠭⠤⠖⠲⠋⠉⣁⣠⠤⠔⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⢀⣀⣀⣠⠤⠔⠒⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢠⡄⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠈⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⠀⠀⠀⠠⣈⣸⢿⣂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⢀⣄⠀⠒⠀⠐⢰⠈⠀⠁⠀⡅⠄⢀⢠⠀⠀⠀⠀⠀⠅⠀⠘⠙⡂⢸⡈⡃
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠁⢀⠈⠂⡀⠀⠃⠁⠉⠋⠉⠀⠉⠀⠐⢀⡀⠘⠈⠂⠈⠊⠁
⠙⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠒⢛⠃⠁⠀⠀⠀⠐⠂⠀⠀⠀⡈⠀⠀⠀⠀⠀⠠⠴⠈⠀⠀⠀⠀⠀⠀⡀⠀⠆
⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢐⠒⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⠐⣀⠂⢀⡀⠤⠀⢀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡿⠴⠆⠀⠀⠀⠀⢀⡀⠀⠤⣴⣶⣤⠄⣂⠀⠂⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡔⠀⠀⢀⣀⣀⠖⠈⠀⠀⠀⠀⠂⠉⢿⡌⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⡁⠀⠀⢀⡈⠛⣃⢀⠀⠀⢁⠀⢀⢀⢸⠇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠙⢌⡀⠀⢐⠢⢀⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡶⠶⠶⠟⠛⠛⠻⠷⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣛⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠉⠙⠛⠶⣦⣤⣀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⠾⠿⠟⠛⠛⠛⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣴⡾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠉⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⠰⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣀⡀⠀⠀⢿⡇⠀⠀⠀⠀⣀⣤⡶⠟⠋⠉⠁⠈⣿⡄⢠⣤⡾⠋⠀⠀⠐⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⠈⠁⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⣠⣿⠃⠀⢀⣴⣦⣤⡀⠀⠀⠈⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⠈⠛⠉⣿⡇⠀⠀⢀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠆⣀⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠉⠲⢦⣤⣤⣤⣤⣴⠾⠛⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠁⠀⠀⠀⠀⢀⣠⣴⣾⠿⢻⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡈⠉⠀⠀⠀⠀⣀⣠⣴⡾⠟⠛⠉⠀⠀⠀⠈⠛⠿⢿⣶⣶⣶⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠀⢷⣄⠹⣷⡄⢀⣤⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠋⠁⠀⠀⣠⣿⠄⠙⠃
⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠁⠀⠀⢀⣤⣶⠿⠛⠁
⠀⢀⣤⣾⠿⠋⠁⠀⠀⢀⣠⣴⡾⠟⠋
⣴⡿⠋⠀⠀⢀⣤⣴⡾⠟⠋⠁
⢿⣷⣤⣶⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠐⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⢦⣤⣤⣄⣤⣤⠶⠞⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠂⠀⠀⠀⠀⢀⣀⠀⠀⠐⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢷⣤⣀⣀⣀⠀⢀⣀⣀⣠⣠⡴⠗⠋⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠉⠉⠉⠉⡉⠉⠉⠀⢀⡀⠄⠂⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠓⠦⢦⣠⣠⣀⣀⣀⣀⣀⣀⣠⣠⣠⣤⠤⠖⡋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣀⣀⣀⡀⠀⠈⡈⠉⠉⠉⠀⠀⢀⣂⣈⢀⣀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠉⠉⢹⠉⠛⠋⡟⠛⠛⢹⡏⠛⠋⡟⠉⠉⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⠀⠸⡟⠀⠀⣇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⠀⠀⡧⠀⠀⢿⠀⠀⢸⠀⠀⢸⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠁⠀⡇⠀⠀⢸⠀⠀⢸⠐⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢻⠀⠀⢸⠀⠀⢸⠀⠀⡞⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠦⠤⠼⠤⠤⠿⠤⠴⠓⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠉⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠠⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⠉⠀⠙⡁⠐⡖⠀⢠⠎⠉⠽⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠶⠖⠒⠒⠋⠉⠝⠁⠀⠀⠀⠈⠁⠖⠪⢼⣷⡒⣁⠀⠀⠀⠀⠙⠃⠉⠓⠲⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⠂⡄⠀⠀⠀⠀⠀⠉⠑⠂⠐⠦⣤⡤⠴⠊⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⢰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠦⣄⡀⠀⠀⠀⠀⢸⠐⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠖⠒⠛⠋⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⡴⠒⠉⠙⠒⠦⡀⠀⠙⠿⣦⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠑⢤⣀⠀⠀⠀⠀⠀⡧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠙⠷⡄⡅⡆⠰⠀⠂⠔⠀⠀⡠⠀⠀⢀⡠⠤⠤⠬⠿⣷⡢⣄⠀⠀⡇⠠⢡⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠙⠣⢳⣤⢀⢠⠀⢠⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠙⢷⣽⣶⣧⣀⠁⠀⢈⠀⠀⡀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠚⢼⣶⣠⠸⡔⢢⡤⡀⠉⠢⠄⠀⠀⠹⣿⡷⠈⢉⡗⠦⣄⣈⣠⠤⠴⠒⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠽⠏⠀⠁⠀⠀⠀⠀⠙⠲⠽⠗⠛⠛⢲⡄⠀⠉⠓⠦⣌⡀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠹⣳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣄⡀⠀⠉⠛⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⣀⠀⠈⠙⠫⣟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢾⣧⣤⠟
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠃⠀⠄⣢⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⡀⠀⠀⠘⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⢀⣀⢂⣠⢀⡀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠰⠄⠀⠀⠀⢠⠀⡔⠈⠀⠄⠀⠀⠀⠘⡳⡔⠀⠀⢀⢀⡟⠂⣄⠀⠈⠁⠀⢠⡃⠀⠀⡈⠁⠁⠀⠙⡖
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠘⠂⢀⣀⣴⡀⢀⠈⠂⠀⠖⠁⠀⡄⠊⠀⠀⠀⠀⠀⠠⢡⠜⠣⢗⡫⠂⡀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⢸⠁⣀
⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⢀⠈⠀⠈⠁⠀⠀⠐⠦⠀⢀⠎⢠⠞⠆⠀⠀⠀⠀⣔⡄⣴⠨⠌⠠⠔⡧⠀⠒⠀⣀⠀⠀⠀⠀⠀⠀⠆⠀⠁
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢠⠀⠀⢤⡄⠀⢧⠀⠂⠀⠀⠄⠀⠅⠠⣤⠔⠚⢀⡠⠆⠀⠀⠀⠙⠀⠤⠄⠠⠤⠤⠂
⠀⠀⠀⠀⠀⣀⢰⡄⠀⠀⠀⢠⠀⠀⢀⡀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠸⢓⣸⠭⡅⢎⡑⠀⠀⣄⢶⠖⠀⠀⣀⠀⠀⢦⣆⡀⠈
⠀⠀⠀⠀⡶⠈⠁⢁⡀⡀⠀⣀⠀⠀⠘⢁⠾⣸⠀⠀⠀⠀⢀⡀⠀⠀⠀⠦⠀⠀⠈⠉⠛⠃⠉⠀⠀⠴⡉⠀⠀⠀⠀⠁⠀⠀⠀⠈⢱⡄
⠀⠀⠀⠀⠀⠀⡔⠋⢀⡠⠴⠨⢄⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠠⡤⡀
⠀⠀⠀⠀⠀⠀⠹⠶⠃⠁⠀⠀⠀⠈⠆⠀⠀⣥⠀⡆⡔⢰⢰⠀⡄⢀⠇⠀⠀⢰⠠⠰⠄⠂⣦⠀⠀⠀⠀⠀⠀⠘⢸⣁⠄⡓⣧⢆
⠀⠀⠀⠀⠃⠀⠀⠘⠀⠁⠀⠀⠀⢀⠀⠉⠀⠤⠐⠣⠁⠬⠘⠄⠇⠈⠤⠀⠀⠃⠱⠐⠢⠀⠇⠀⠄⠁⠀⠰⡦⡄⢉⠭⠶⠒⠈⠆⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡀⢀⠀⠀⠉⠸⠖⣠⡀⠟⢬⠘⣣
⠀⠀⠀⠀⣠⠀⠀⠢⡆⡀⢀⠀⣠⠄⠀⠀⠙⣣⡆⣀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⢀⡱⠎⠁⠀⠀⣀⠀⠀⠀⠀⠐⠳⠴
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠂⠀⠀⠚⠃⠀⢀⠀⠀⡠⠄⠆⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣠⣄⣀⣄⣀⡀⠀⠀⠠⡄
⠀⠀⠀⢠⠀⠈⠀⠀⠀⢠⡀⠠⠔⠀⠀⠀⢠⠀⠀⠁⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠛⠛⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠁⠲⠀⠀⠀⠀⠘⠀⣴⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠻⡿⠄⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠘⡀⢀⠄⠄⠁⡀⠀⠰⠀⠀⠡⣰⠃⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢢⣟⠁⠀⠀⠀⠀⠜⠀⠀⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠲⠈⠈⢀⢀⠉⠈⠀⠈⠀⠀⠈⠁⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠋⠉⠉⠉⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠴⠏⠀⠀⠘⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠞⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⢀⣠⠼⠃⠀⠀⠀⠀⠀⠸⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣻⠄⠀⠀⠀⡇
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠰⢧⡀
⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡞⠀⢰⠛⢳⡄⠀⠀⠀⠀⠀⠀⢠⠖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢲⠀⠀⠀⠀⠀⢠
⢷⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢀⡞
⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠙⢶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠶⠶⠶⠶⠶⢤⣤⣤⣀⣀⠀⠀⢀⣀⣀⣀⣤⡤⠴⠒⢻⡇
⠀⠀⠀⢸⡄⠀⠀⠉⠉⢻⠉⠁⠀⠀⠀⠀⠀⠉⢻⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠉⠉⠉⠉⢹⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣿⣄⣀⣀⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣀⣀⣀⣼⡥⠴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⣄⣠⠞⠁⠀⠀⠉⢷⣀⣀⣴⡋⠉⠉⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⡼⠁⠀⠀⠀⢀⡴⠞⠉⠁⠈⠙⢦⡀⠀⠀⠙⢦
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣠⡴⠚⠛⠳⣦⣼⠷⠒⠶⠦⣤⡞⠁⠀⠀⠀⠀⠀⠈⢧⠀⣠⡤⠼⢧⡀
⠀⠀⣴⠋⣩⣯⣍⣻⢦⣴⠋⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠙⣆⠀⢀⣀⣀⡀
⠀⠈⣏⠓⣧⠀⢀⣉⣻⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠴⠚⠛⠛⠛⠛⢦⡀⠀⠀⢹⡶⣯⡽⢥⣽⡆
⠀⠀⠈⠳⣍⣻⡟⠛⢻⣦⣀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢸⣿⣷⣤⢾⣴⠃
⠀⠀⠀⠀⠈⠹⡗⠾⣭⣀⣉⠽⠟⠒⠶⢾⣧⣤⣤⣤⣤⣤⣴⣯⣤⣤⣤⣤⣤⣤⡤⢴⡶⠶⠚⡿⠛⢻⠇⢀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⢻⡚⠶⠤⢉⡛⠶⠶⢾⣥⣄⣠⣔⣋⣁⣀⣴⣊⣁⣈⡴⠊⠁⣀⣔⣁⣀⣠⣮⣤⡤⠾⠚⣿
⠀⠀⠀⠀⠀⠀⠈⣧⠐⠶⢬⠍⠛⠛⠀⠲⠤⢭⡍⢉⣭⣍⣉⠉⢩⣉⣉⣙⠉⠉⣭⣭⣭⠤⠀⠰⠶⠚⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠠⣤⣀⣈⠙⠛⠂⠀⠲⠶⠤⠄⠀⠴⠦⠤⠄⠀⠠⠤⠤⠄⠀⠠⠶⠶⠚⢋⣁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⢦⠈⢉⣀⡐⠒⠶⠶⠀⠐⠲⠶⠶⠀⠶⠶⠶⠶⠀⠀⠶⠶⠶⠆⠀⠋⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠲⠤⣬⠉⣛⠁⠘⠓⠒⠒⠀⠐⠒⠒⠒⠀⠐⠒⠒⠒⠀⢐⡒⠚⢋⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣉⣛⣓⠀⠀⠛⠒⠒⠂⠘⠓⠒⠒⠂⠀⣉⣉⣭⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁


⣿⣿⢻⢿⢛⠿⡿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡤⣄⠀⠀⠀⡠⢤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠃⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣤⠀⣀⠈⢧⠀⠐⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⣞⠀⠀⠀⠀⠀⠱⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠍⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣠⡍⠠⠇⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠤⠀⠀⠐⠉⠀⠀⢀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⢘⠂⡢⠀⠀⠀⣇⣸⠁⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⠀⠀⠔⠀⠀⠀⠀⠀⠈⠱⠀⠀⠄⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⠀⠀⢠⣀⠀⠁⠀⡀⠀⠀⠀⢀⣀⠀⠀⠄⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⠀⠀⠀⠚⠋⠀⠀⠄⠁⠀⠀⠀⡀⠀⠀⠠⠀⢠⠀⠀⠈⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠈⠀⠀⠎⢀⡀⢀⠀⠆⠀⣀⣀⠇⠀⠀⠄⠂⠀⠒⠐⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⢀⠔⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠀⠒⠒⠶⠒⠒⠚⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⢠⣿⣧⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣌⣿⣿⡿⣀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢰⣶⠿⣸⣿⣿⣯⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢌⣱⡞⢿⠛⠟⢻⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠐⡚⠛⣋⠩⣴⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠛⢷⣶⣴⣾⠖⢠⣼⢹⡇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠐⢴⠃⢌⣭⡾⠷⢺⣛⣱⡿⠈⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⡕⠊⠉⠀⠀⢘⣠⡞⢁⣀⣤⡾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠉⠉⠙⠲⠖⠒⠉⠙⠛⠿⠛⠉⢶⣶⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠆⠐⠐⠄⢠⠀⢀⡀⠀⡀⠀⡠⣦⡔⢾⢙⣧⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣠⠀⣆⢔⣀⣀⡆⠀⡇⣰⡆⠹⠳⡿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⢇⢻⡀⣄⠙⣿⣀⡿⢁⡗⣾⢦⣷⣼⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢼⡾⡏⣭⠓⢿⣀⣧⣾⡧⣼⣾⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣧⣷⢻⣧⣼⡍⣿⣸⣏⣿⣹⣿⣿⡗⣥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢼⣧⣹⣟⣿⣻⣿⣿⣿⣿⡿⠽⣋⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⡗⠻⢭⠄⠊⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣠⠉⠁⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠉⠀⢠⣀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠰⠃⠀⠀⠀⠀⠉⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢰⡆⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠈⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⡄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠛⠂⢀⠁⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡃⠀⣠⠄⠁⠒⠒⣒⡒⠐⠂⠈⠉⠐⠒⠂⠒⠁⠚⡡⠃⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠄⡁⠀⠀⠀⠀⠈⠁⠀⠀⠐⠋⠀⠀⠀⡀⠄⠊⠠⠴⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣄⠀⠉⠐⢒⠀⠠⠤⠤⠄⠐⠒⠂⠉⣁⠀⠀⣀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠉⠀⢀⡀⠀⠘⠓⠀⠀⠀⠋⠀⠀⠀⢀⣀⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠉⠒⠐⠀⠂⠉⠁⠀⠉⠉⠀⠒⠒⠒⠒⠂⠈⠉⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⣓⣶⣤⣀⠀⠀⠀⠀⣀⣤⣶⣚⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢤⡀⢀⡤⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠙⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠿⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⢋⣴⡄⠀⠀⢀⠀⠀⢸⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⠀⡇⠘⠙⠀⠀⢰⠿⠇⣠⠏⢸⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠋⡞⠀⠀⠘⠤⣀⣀⣀⡠⠴⠊⠁⠀⡼⢀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⡇⠀⢧⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⢀⠜⢱⢿⠃⢠⠃⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠳⡀⠀⠑⠦⣅⣁⡀⢀⣀⣀⠤⠚⠁⠀⠀⢀⡴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢰⣶⠈⠲⠤⣀⣀⠀⠉⠉⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⢸⢹⡦⣄⡈⠀⠀⠀⠀⠈⠉⠉⠛⠒⢚⣻⠉⠉⠀⠀⠀⢀⣤⠞⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠈⡆⡇⢻⠙⡗⢲⢤⡤⣀⣀⣀⣀⣀⣀⣋⣠⠤⡴⢲⠋⡇⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⢱⢸⠀⡇⢸⠈⡇⢸⠀⡇⢸⢸⠀⡇⢸⠀⡇⣸⢸⠁⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡀⡇⢧⠘⡆⡇⢸⠀⡇⢸⢸⠀⡇⢸⢠⡇⡇⢸⢠⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⢇⢸⠀⡇⣇⢸⡄⡇⢸⢸⠀⡇⡏⢸⠀⡇⡏⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢹⢸⢸⠀⡇⢸⠘⡇⡇⢸⢸⢀⡇⡇⢸⢸⢁⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡼⡌⡇⢹⢸⠀⡇⡇⣼⢸⢸⠀⡇⡞⢸⢸⢠⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣇⡇⡇⢸⠸⡀⡇⡇⡏⢸⢸⢰⠇⡇⡞⣼⣸⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⠶⠧⠧⣧⣧⣼⣼⠼⠤⠷⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠛⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠉⠁⠀⠀⠹⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡤⣤⣄⣄⠀⢠⠀⣀⡼⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⣄⣀⡀⡀⢀⢠⢀⣼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠉⠉⠛⠋⠉⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠉⠙⠛⠚⠛⠋⠀⠚⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠛⠓⠶⠶⠴⠤⣤⣦⣦⡤⠼⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⠷⢶⣴⣴⣴⣠⣀⣆⣄⣶⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢸⣸⣠⢀⢀⡃⣀⡰⠈⠉⠉⠩⠅⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠯⠍⠉⠉⠛⠛⠛⠳⠶⡖⣴⣴⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⢸⣯⡝⠛⠛⠛⠿⠿⠿⢿⣷⣶⣶⣶⣶⠏
⠀⠀⠀⠀⠀⠀⠀⢼⣁⠀⢀⡀⠀⠀⠀⠀⠸⣅⣸⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣾⠍⠀⢀⣀⠀⠀⠀⠀⢸⣇⣿⠀⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣉⡽⢿⡚⠳⢿⡵⣦⣤⣄⣒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠛⠛⠻⠿⠿⠿⣶⣶⣯⣁⣀⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢙⣄⡤⠷⡖⠛⠻⣅⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⢀⠤⠄⠀⠢⠈⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⢉⣧⡤⠞⢾⠉⣙⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠉⢀⠀⠐⠔⠀⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⡉⢀⣳⣤⠴⠷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⢀⡀⠀⢤⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠛⢯⠁⣘⣦⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠒⠀⠁⠀⣐⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡖⠚⠻⣅⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠢⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⣦⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠐⢀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣤⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠀⠈⠍⠈⠉⠙⠿⠿⣿⣛⠃⠉⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠒⠊⠡⠤⠤⠍⣤⣥⣄⣠⡤⠤⣤⣤⣀⣀⣀⣈⢀⢀⣠⡧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠂⣉⣁⣐⣀⣀⣈⣀⣀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠃⠀⠀⠏⡁⡀⠀⢀⣽⡿⢋⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⠀⠈⠉⠉⠩⢉⣿⣿⡟⢿⣿⡿⡿⣿⠛⢲⠒⠲⢐⠀⠀⠀⣠⣤⣿⣿⠿⠟⣋⠥⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⠃⠀⠙⠀⣢⣤⣶⣧⣤⣬⡖⠻⠶⠟⣛⠭⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⣀⣀⠠⠀⠰⠆⠈⠻⣿⣿⣿⡿⠟⢁⡠⠄⠒⠉
⠀⠀⠀⠀⢀⠔⠁⠀⠐⣀⣠⣀⣀⣄⣤⣄⣰⣠⢴⠶⠻⠿⢠⣠⢀⢠⣶⣷⠶⠖⢛⣫⠵⠚⠉⠁
⠀⠀⠀⡰⣃⠀⠀⢠⣾⡿⣿⡝⠛⢻⣿⣟⣙⣧⣦⣦⣧⣷⣿⡿⠜⠌⣉⠥⠖⠊⠁
⠀⠀⠀⣷⣿⣿⣿⣠⣽⣾⣿⣿⣶⣾⣿⣿⣷⣿⣿⡿⠿⢛⡯⠗⠊⠉
⠀⠀⠀⠙⠿⣛⡿⠿⣿⣿⣿⣿⣿⣿⠻⡟⢛⡭⠷⠒⠉
⠀⠀⠀⠀⠀⠀⠉⠘⠒⠒⠒⠒⠒⠊⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⡿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠁⠑⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⢟⣴⣜⡿⠿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⣤⡀⠀⠀⠀⠀⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠇⠾⠿⢿⣿⣿⣗⣘⠿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣬⣄⡀⠀⠀⠀⠀⠀⠀⠤⠼⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡏⠺⣿⣿⣿⣿⣿⣿⡿⢟⡂⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣎⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠋⢾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣦⢹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⡐⢁⠉⠛⠛⠛⠋⣉⠑⠒⠒⠶⠶⠒⠊⠁⠙⡀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣵⢷⣤⣤⣤⣤⠶⣮⣭⣭⣉⣩⣭⣴⣷⡹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⣶⣤⣤⡦⣤⡤⢤⣤⣤⢦⣤⠞⣶⡶⢶⡞⠁⠀⠀⠀⠀⠀⣿⣿⣿⣷⡌⠉⠉⢉⡉⢙⣛⠛⢛⡙⠉⡌⠉⣉⠉⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⡇⢿⡇⢸⡇⠀⣿⡇⢰⣿⠀⣿⠃⣿⠃⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⠀⡇⠘⡇⠈⣿⠀⣼⡇⢠⡇⢀⡇⢰⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⠸⣇⢸⡇⠀⣿⠀⢸⡟⢰⡟⢰⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀⢹⠀⣷⠀⣿⠀⣿⡇⢸⠇⣸⠁⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢿⡄⣿⠀⣷⠀⢹⠀⢸⡇⢸⠇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣇⠸⡆⢻⠀⣿⡄⣿⡇⣼⠀⡟⢀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠣⠼⢀⣿⣀⣸⣀⣸⡁⠼⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣥⣼⣥⣿⣥⣿⣥⣭⣬⣵⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠒⠊⠉⡁⠀⣀⡀⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⡉⠀⠀⠀⢙⣷⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢄⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠖⠃
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠁⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣀⡽⠖⠊⠉⠉⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠢⣄
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠘⢦⣀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡿⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣠⣤⣴⣶⣾⡿⠟⠁⠀⠀⠀⠀⠸⡏⠠⠀⠀⠀⠀⠤⢤⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⣀⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣤⣀⣤⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⡀⢀⣰⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⠴⠯⡽⠚⠳⠋⠹⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⢧⢀⣼⡇⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠛⡟⠘⢿⠛⠓⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠁⠀⠀⣥⣾⡿⠁⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣠⣬⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢶⣶⣤⣀⣀⣀⣤⣤⣴⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢧⣀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢰⣿⣿⣿⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⣀⠈⠿⣿⡿⠟⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣠⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠛⠛⠻⠷⠶⠦⠤⠴⠖⠀⣠⡀⠈⠉⠀⢀⡈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⣷⣦⣄⣀⡀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠐⠒⠒⠋⠉⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠦⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⣿⣿⣿⣿⣿⠿⠟⢻⣿⡗⠲⠦⣤⣤⣄⣀⣀⣀⣀⣤⣤⣴⠶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠘⡇⠀⠈⣿⡆⠀⢸⣿⡇⠀⠀⣿⣿⠉⠉⢹⣿⠉⠉⣿⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⠀⠀⣾⡇⠀⢰⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡇⠀⢻⣷⠀⢸⣿⣿⠀⢰⣿⣿⠀⠀⣿⡇⠀⣼⡏⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⣷⠀⢸⣿⠀⢸⣿⣿⠀⢸⣿⣿⠀⢸⣿⠇⢠⣿⠇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣿⡆⢸⣿⡇⢸⣿⣿⡀⢸⣿⣿⠀⢸⣿⠀⣸⣿⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣷⠀⣿⣇⢸⣿⣿⡇⢸⣿⣿⠀⣿⡿⢀⣿⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠚⠛⠛⠓⠚⠛⠛⠒⠛⠛⠋⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣲⣆⠄⠐⣊⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠸⠋⠽⠟⠉⠁⠀⠀⠐⠒⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⠁⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠂⠈⢺⡄⢤⣠⢀
⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣠⡈⡀⠀⠀⠀⠈⠀⠀⠔⡻⠿⢦⣄
⠀⠀⠀⠀⢀⣠⡵⠂⠀⠀⠀⠈⠀⠒⠦⣤⣄⡀⠀⠀⠀⠀⣥⠒⠀⠀⠀⠀⢀⣀⣆⢙⡃⠀⢠⣠⣤⡀⠀⢀⡶⢿⢷⢹
⡠⢷⡀⠀⡳⠘⠿⡎⡂⠀⠀⠠⠀⠀⠀⠀⠉⠑⠢⠤⣀⠀⣧⣤⠤⠄⠒⠂⠉⠀⠀⠁⠒⠒⠋⠉⠫⡙⠓⡊⢤⠾⠀⢺
⢥⡀⠑⠬⡻⠠⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠬⠑⠢⠔⠁
⠀⠀⠀⠀⣀⣀⣁⣀⡶⠶⠒⠒⠒⠒⠒⠲⠤⠤⢄⣀⣀⠀⠄⠀⣀⣀⠀⠀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣓⣛⣿⣭⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⠡⢐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢂⠡⣐⠈⠀⠔⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢂⡡⢔⡪⢶⡿⠃⡠⠂
⠀⠀⠀⠀⢀⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢂⡡⢔⡪⠕⠈⠁⡰⠋⡠⠊
⠀⠀⢀⠔⠁⠀⠀⠀⢰⠀⠀⠀⠀⡀⠄⠂⡁⢄⡢⠕⠊⠁⡀⠄⡒⠋⠁⠉⠑⢄
⠀⢠⠁⠀⠀⠀⠀⠀⡆⢁⡠⠒⡁⠄⡂⠅⠂⠁⣀⠤⠒⢁⠠⠎⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⣀⡀
⠀⠸⠀⠀⠀⠀⠀⠀⡰⠋⡠⠊⠂⠁⣀⠤⡒⠋⠀⠀⠚⢅⡀⠀⠁⠀⠀⠀⠀⠀⠀⡱⠊⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠊⡠⠊⠀⠠⠐⠉⢀⠄⠀⠀⠀⠀⠀⠀⠡⡀⠂⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠄⠀⢀⠈⡠⠊⢀⠀⠒⠀⠐⠎⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠔⠊⠁⠐⠢⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠂⢄
⠀⠀⣇⠔⡡⠊⠀⡠⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢄⠀⠂⠀⠀⠀⠀⠀⡠⠃⠀⡠⡾
⠀⡔⡡⢊⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⣤⠵⠒⠐⠢⢴⠂⠐⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠐⠂⢉⢤⡢⠵⠈
⠀⢸⡋⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢀⠠⡐⠌⠂⠁
⠀⠈⠢⢕⡢⢀⠀⠢⠤⣎⣀⣀⡀⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⢛⠒⠲⠤⠀⠒⠈⡀⢄⠢⠕⠋⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢹⠲⠭⠠⠥⡂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⠠⡒⠕⠊⠁⠰⠀⢀⠤⠤⠤⠾⣀
⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠑⠠⣄⠀⠀⠢⢄⣀⡀⢀⣀⠠⠔⡩⢔⠪⠆⠁⠀⠀⠀⢀⠜⠈⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠯⢔⠤⣀⠀⠀⢀⠤⡢⠕⠈⠀⢸⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⡀⠈⠐⠭⠍⠂⠁⠀⠀⠀⠀⡏⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠐⠤⢀⣀⣀⣀⠠⠳⠄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣀⣀⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢄⠀⠀⠄⠀⠀⢀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⣀⣀⡀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠁⠂⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠢⠤⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢰⣤⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣿⡏⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛⠉⢲⠇⣼⢸⠿⡂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡩⢷⡰⠀⠂⢐⣿⢸⡆⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣼⠁⠀⡹⠁⠀⢾⣿⠈⡆⠀⠠⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢐⢺⡇⠀⠀⡇⠀⠀⠘⣿⠀⢿⣄⠀⠊⠉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠄⠐⣄⠀⠳⣄⠀⠙⠹⣧⡈⢿⣼⣄⣀⡼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⢠⣶⠶⢈⡉⠒⠺⣟⣶⣶⠶⢷⣆⡈⠃⠒⠒⠺⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠘⠿⣼⠁⠀⠀⢀⢀⡀⠈⠉⠛⠁⠠⠤⢀⡀⠀⠀⠒⠊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢘⢧⣄⠈⠢⠤⡔⠋⠀⠙⠲⢄⡀⠀⠀⠀⠀⠉⠲⣏⣿⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⢀⠉⠙⠲⠤⢤⣤⣀⣈⢙⡛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⣀⣀⠙⠓⠾⣦⣠⡆⡀⠈⢉⠑⡒⠂⠢⠤⠈⠉⢉⣡⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠛⣿⡦⠤⣤⣭⣭⣿⣷⣦⣤⣤⣤⡴⠶⠞⣮⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⡇⠀⡇⠀⣿⠀⣾⠉⢹⠇⠀⡿⠀⢸⡇⣼⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⣿⠀⢁⠀⣿⠀⣿⠀⢸⠀⢸⡇⢀⣿⢠⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹⡇⠘⠀⢹⠀⡿⠀⡼⠀⣼⡅⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⢧⠀⠀⢸⠀⣷⢠⣧⣰⣿⢡⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣻⡆⠓⢺⣶⡿⢸⡟⣿⣿⣿⡿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⠋⠓⠚⠚⠛⠋⠉⠁

⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤
⠀⠀⢀⢻⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠎⠉⠁⠀⠀⠀⠀⡀
⠀⠀⡳⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢀⣾⠟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⡿⠻⣶⡀⡀⠀⠀⠀⠀⠘⡇
⠀⢰⣾⢤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠘⢾⣈⣻⣿⡀⠀⠀⠀⠻
⠀⡞⢯⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣅⠀⠀⠀⠈⠻⣙⠁⠀⠀⠆⠇
⠀⣻⣶⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣆⠀⠀⠀⠀⠀⠉⠀⠀⠀⡜
⠀⡶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢸⡟⣺⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⡀
⠀⣛⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠮⠈⠻⡙⢻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⢭⣤⣀⠈⠙⢷⡄
⠀⣫⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠁⠢⣄⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣀⣀⡀⠈⠙⠓⠂⠹⣿
⠀⡷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡻⠀⠀⠀⠀⠀⠀⠈⢷⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠈⠻⣭⣉⠃⠀⣀⠀⠠⣿
⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠉⠛⠛⠛⠃⠀⣸
⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡼⣏⠈⠟⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠰⣯⠛⠳⢤⣀⠀⠀⠀⠀⢹
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠑⠙⠣⠘⠽⠿⣷⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠈⠳⠶⣤⣠⣥⣤⠄⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⣀⣀⠀⠀⠁⠀⠀⠸⣿
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⡅⣀⠀⠀⠀⠀⠀⠀⠈⢫⣻⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢬⣉⣉⣀⣆⣀⠀⢸⣿
⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠉⠉⠉⠉⠉⢠⣾⠏
⠀⠀⠀⠈⠀⠀⠀⠀⢠⡌⣿⡜⢻⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡷⢤⣴⣶⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠘⣧⡜⠣⣽⣧⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠈⠻⣄⠘⣿⣧⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⢀⠻⡀⠀⠀⠀⠀⠈⠳⣏⣿⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠄⠤⢀⡀
⠀⠀⠀⠀⠀⠀⣾⣧⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⣠⣴⡗⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣀
⠀⠀⠀⠀⠀⣼⣿⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⢀⣴⣴⣿⣶⣶⣤⣤⡀⠀⠈⢦⡀
⠀⠀⠀⠀⢀⠱⣿⡔⡽⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠀⢀⣰⡿⠋⠀⠀⠀⠈⠙⣿⣷⡄⠀⢺⣷
⠀⠀⠀⠀⣰⠄⠙⢧⡀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⡁⠀⢘⣿⠁⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⠰⣻⠇
⠀⠀⠀⢰⣧⠀⠀⠀⠈⠲⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢲⣿⠇
⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠄⠀⢿⣧⣄⡀⡀⠀⣀⣠⣾⠏⠀⠠⣿⣿
⠀⠀⠀⣦⠘⢳⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⣄⠀⠘⠛⠛⠟⢿⠟⠭⠉⠀⣠⣬⣿⠃
⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣹⣷⣤⣄⣄⡀⠀⡀⢀⠀⣤⣤⣿⠟⠁
⠀⠀⠀⠘⠫⢷⠆⡤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠳⠿⠷⠷⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣹⣿⣿⣱⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⡟⢿⣿⣿⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢻⣿⣿⣿⣿⣤⡇⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣏⣹⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣉⣿⣿⣻⠟⠁⠀⠀⠀⠀⠀⠙⣯⣯⣽⣭⣯⡝⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⣟⡿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⣿⣤⣼⣿⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣯⣟⣿⢿⣾⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣿⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣟⡿⣿⢷⣷⡇⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣶⣿⣿⣾⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡷⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠻⣿⢉⣿⣿⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠁

⠀⠀⠀⠀⢠⠔⣒⡲⢄⡀
⠀⠀⠀⠀⢻⡘⡀⠈⠐⠝⠦⡀
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠀⠉⠒⠤⣀⡀⠀⠀⣀⡤⠖⠊⠉⠉⠉⠉⠉⠉⠁⠀⠀⡉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⡄⠀⠈⠑⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢎⣁⣳⣄⣀⣀⡀⣀⣉⡒⢤⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⢸⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⠑⠢⢄⣱⠀⠀⠀⠀⡰⠃⠀⠉⠉⠉⠉⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠈⢲⡀⣠⠊⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⢀⣀⠤⠒⠋⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠟⠒⠒⠒⣺⠞⠁⠀⠀⠀⢀⡴⠋
⠀⠲⠮⠭⢤⣄⣀⣀⣀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠺⣄⡠⠴⠚⠒⢒⣒⣋⣀⣀⣠⣤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠴⢥⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋

⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⣤⠖⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣠⠄⠀⠰⠆⠀⠀⠰⠶⠂⠀⠀⡐⠀⠀⢠⠖
⠀⠀⢠⠋⠀⢀⣀⡀⠀⠻⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠠⠆⠀⢀⠀⠀⠀⠀⠀⢟⠊⠀⠀⠔⠀⣴⠶⠀⠀⠀⣀⡀⠙
⠀⠀⡆⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠒⠀⠀⠀⢀⠋⠀⣰⣆⡠⠔⠉⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠆⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠤⠶⠭⠤⠴⠖⠊⠁⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⣠⣄⠀⠀⢀⣀
⠀⠀⢿⡄⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⣀⡀⠀⠘⠁
⠀⠀⠀⠐⠀⠐⠻⠀⠀⠀⠀⡶⠀⡾⠃⠀⠀⣀⣀⠀⠀⢀⣀⣀⠀⠀⢀⡾⢹⠀⠀⠀⠀⣀⠀⠀⠀⠁⢤⣤⠀⠀⠙⠃⢀⡠⠶⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⣼⠇⠸⠟⠛⠛⠛⠿⣿⢟⡋⣘⢂⣤⡼⣧⠃⠀⠀⠀⠀⠈⠁⠐⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⣿⡏⢠⡶⢲⢰⡟⢹⣸⣿⣡⠏⣿⢸⡗⡡⣋⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠈⢁⣿⡧⣼⢻⣏⡞⠻⠏⢷⣄⡭⠤⢥⡈⠣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣀⡴⠋⠈⠓⢡⣤⣤⣴⡶⠞⠋⠙⠓⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠴⠆⠀⠚⠛
⠀⠀⠀⢠⠄⠀⢀⡄⠀⠀⠈⠛⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢠⠤⠶⠒⠒⠈⠛⠢⠀⠀⠀⠀⠀⠀⣠⣏⣶⠶⠂⠉⠑⠂⠀⠁⠀⠶
⠀⠀⢀⡄⠘⠳⠀⠀⠀⠀⠴⠖⠀⠀⠉⠉⠀⠀⠀⠀⡠⢈⡋⠀⠐⠛⢂⠀⠀⣤⠀⢠⡄⠀⠀⠀⠹⠏⢠⡦⢀⣀⣃⣠⣠⠄⠀⠠⡄
⠀⠀⢸⣷⣄⡀⠶⠆⠺⠷⠀⠀⠾⠳⠆⠊⡜⠀⠀⢼⠁⢨⣅⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠖⠋⠀⢠⡄⠀⠀⠉⠂⢴
⠀⠀⢿⡿⠿⠿⠿⠿⣦⣯⠷⠶⠶⢶⣖⠛⣀⠀⠀⢏⠀⢀⡀⢀⠀⠠⠀⠈⠷⠀⠀⡀⠁⠀⠀⠀⠸⠏⠁⠶⠄⠀⠀⠀⠐⠦
⠀⠀⠀⣽⣿⣶⣛⣠⣀⣀⣀⣠⣤⣿⠿⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⠀⣸⢀⡄⠀⠀⠀⢀⣄
⠀⠀⢀⣭⣍⡉⠁⢈⣉⡉⣩⠿⠀⢀⣀⣤⣴⠂⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠁⠀⠀⠀⠊⠙⠀⠀⢸⡇⠀⣤⡀⠀⠀⠉⠀⢠⠈⠇
⠀⠀⢈⣜⡋⠩⠿⠉⠉⠹⠏⠿⠉⢉⢉⣉⡥⡄⠀⠀⠀⠠⠤⡀⠰⠂⠁⠂⠈⠀⠁⠀⣄⠀⠀⠀⠀⠻⡀⠈⠁⠐⠂⠀⠀⠀⠀⢀
⠀⠀⠺⡿⣩⣟⠻⢶⣶⠲⠓⠓⠚⠋⣩⡄⢐⡴⠀⠀⠀⠀⠀⡄⠀⠀⠀⠿⠖⠀⠀⠀⠁⢠⡆⠀⠀⠀⠙⠒⠠⢄⣀⡘⠃⠔⠒⠁
⠀⠀⠀⠙⠛⠛⠲⠮⠉⢤⣤⠤⠶⠤⠶⠚⠉⠀⠀⠀⠀⠀⠤⠀⠴⠄⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⡀
⠀⠀⠀⠀⠀⠈⠰⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠄⢤⣀⠀⠀⠀⠈⠑⠢⢄⣠⡀
⠀⠀⠀⠀⠀⠀⠈⠐⠨⢙⠳⢦⣄⣀⣠⢴⠞⠋⠀⠀⠀⢨⣦⠀⠀⠀⠀⠀⠀⠉⠒⢄⡂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠭⣙⠦⢤⣄⡀⣀⣠⡴⠋⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⢀⣤⠤⠖⠒⠒⠀⠐⠒⠂⠤⣄⠀⠀⠀⠀⠀⢷⡀⠀⣠⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⡐⣧⢻⡁⠀⠀⠀⠠⡤⠞⠂⠀⠤⣿⡗⣀⠀⠀⠀⠀⠹⣄⠒⢵⠲⠤⢾⠛⠁⠈⠢⡀
⠀⠀⠀⠀⠉⠛⣦⣌⠓⠒⠤⠤⠤⠤⠴⠞⢛⣁⡴⠛⢃⡄⠀⠀⠀⠙⣆⠀⠉⠐⠂⢀⣀⣠⣤⣴
⠀⠀⠀⠀⠀⠀⠳⣌⠉⠁⠐⠒⠒⠒⠚⠋⢉⣱⠖⠚⠋⠀⠀⠀⠀⠀⠈⢷⡶⠾⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠤⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗⠹⣇⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢃⠀⠘⠳⢤⣀⠀⠠⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⢦⡀⠀⠀⠈⠻⣶⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⡠⠃⢀⡀⠀⢈⡉⠠⠀⠀⠀⣿⡏⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣠⡤⠒⠹⠥⣎⢁⠴⠚⢛⣫⡴⠏⠻⢄⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠀⢰⡟⠁⠀⠀⠀⠈⣦⣤⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⢀⢀⣰⠏⠀⢸⠀⠀⠀⠀⠀⢠⡿⠹⡛⠋⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢀⣠⠟⠀⠀⠀⣀⡤⠞⠀⠀⢿⡀⢨⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢀⣀⣀⡀⢀⣀⣀⣀⣀⢠⠐⠊⠀⣉⡤⠴⠶⠖⠛⠉⠀⠀⠀⣸⣿⣥⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠻⢏⠡⠈⠉⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⣀⣴⣶⡾⣿⢿⡿⣿⠶⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠙⠲⣤⣤⣤⣴⣾⣿⣤⣄⣀⣤⣴⣶⣾⠿⡟⠛⠱⢞⣁⠀⣔⣱⣮⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡕⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠩⡍⢩⠊⡁⠀⠀⢰⣀⣄⣿⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢹⠹⣟⠢⣶⣄⢠⣦⣠⣤⣰⣦⣴⣶⣤⣽⣦⣶⡾⠿⣿⢛⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⡆⢻⡀⢸⡄⠀⣧⠀⡇⠀⣽⠃⢹⡍⢸⠇⢐⡇⢀⣿⣸⣿⣧⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠘⠘⣇⠘⣧⠀⣿⠀⣧⠀⣿⡀⢺⠀⣻⠀⣼⣇⣸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⣤⣿⡀⣿⠀⢸⡆⢻⠰⣿⡆⣸⠁⣿⣄⣿⢡⣿⣿⣿⣽⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⣅⣿⡧⢸⣿⣿⡗⣿⢷⣿⢹⣿⢻⣿⣾⣿⣿⣿⣾⣿⣿⣿⣿⡿⣷⣶⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⢽⣿⣷⡸⣿⣿⡷⣿⡏⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣶⣦⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⡟⣿⣿⣿⣿⣧⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⠞⢘⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠯⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠳⠛⠛⠛⠛⠙⠛⠛⠁⠉⠉⠉⠁



⣿⣿⣿⣿⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⠿⢿⣿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠱⢃⡴⠖⠶⢤⣀⠀⠀⠀⠀⠀⢠⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡀⠀⠀⢌⠳⢄⡀⠀⢀⣼⣜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠏⠉⠉⣩⠍⣓⠳⢤⣵⡒⣮⠻⠟⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣆⣀⣈⡴⠲⢤⣵⢤⠤⠃⣼⡷⠖⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⡙⠷⣤⡗⣿⠁⠈⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⡄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣛⡷⡟⢸⠀⠀⠀⠀⠀⠀⣴⣋⠭⠖⠒⠂⠈⠁⠀⠉⠀⡨⢿⠿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠟⠀⠛⠀⠀⠀⠀⠀⠀⢸⡷⠛⠛⠓⠶⠶⠶⠶⠶⠶⠶⢦⡀⠀⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣥⣤⣄⣁⣀⣀⣀⣀⣀⣀⣸⣇⠀⠠⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡉⠙⠯⠍⠉⠙⠛⠛⠛⠉⠉⠩⠽⣶⣬⢹⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡓⠆⠀⠀⠀⠀⣀⠤⠤⠶⠤⣄⣀⡀⠀⠸⣧⠿⢽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⣤⣀⣀⣠⠞⠁⢀⣠⣤⡀⠀⠙⣦⠀⠀⢰⢾⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⣴⣿⣿⣿⣿⣷⡄⠘⣿⣿⣿⣛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⡿⢭⣿⣧⠀⢿⣿⣿⣿⣷⣿⠁⢠⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣒⡢⠔⠛⣦⣬⠉⠛⠿⠋⠡⣪⠟⢩⠤⢌⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣗⡀⠀⠀⠙⠳⠶⠶⠖⠛⠁⠀⠠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠐⠩⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣄⣀⣀⣤⣤⣄⣀⣀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠶⠶⣤⣌⡒⠦⣄⡀⠀⠀⠀⠀⠀⠸⣯⣀⣉⣛⠙⠛⠛⠛⠉⢙⣻⣿⠇⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⠬⠙⠓⠶⣭⣓⠦⢄⡀⠀⠀⠀⠈⠉⠙⢛⣛⠛⢛⣛⣋⣉⣁⠀⠀⣀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢙⡻⢶⣮⣀⠤⢤⣤⣄⣀⡀⠀⠀⠀⠀⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠢⡈⠙⠽⠳⢶⣄⡀⠀⠀⠈⠑⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡈⢦⣄⣐⣐⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣄⣭⣤⡁⢂⣡⡼⣁⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⠛⠋⠥⠤⠤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⢚⣛⣛⠒⠲⠦⢤⣄⣀⠀⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠾⠷⠶⠤⣤⣄⣀⠀⠀⠈⠉⠒⠠⢄⡉⠛⠶⣄⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠉⠉⠀⠀⠀⠒⠠⠄⢀⣀⠀⠉⠙⠳⢦⣄⠀⠀⠀⠀⠉⠐⠈⠓⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣠⠴⠟⠛⠛⠛⠛⠛⠛⠲⠦⣤⣀⠀⠀⠀⠈⠀⠀⠠⠄⣈⣙⣦⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⢠⠚⠓⠒⠈⠁⠀⠀⠈⠀⠈⠀⠀⠀⠈⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⡰⠁⣀⣀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠙⠃⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⠁
⠀⠀⠀⡇⠛⠉⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠊⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠤⠖⠊⠉
⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠉⠒⠒⠐⠒⠒⠒⠒⠒⠉⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⡐⣤⡤⠞⡁⠀⣀⡙⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠈⠀⠀⢸⣃⣿⡀⠀⢾⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠘⣆⡄⠙⠛⠛⠻⣷⡄⢀⣽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠈⣾⡖⠨⠶⠦⠴⣿⠶⢸⣿⠟⣿⠿⠿⣗⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠉⡮⠵⠻⡆⢸⡿⣆⡈⣷⢟⣩⠺⣷⣤⣀⢹⡷⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡵⣃⢀⣾⣥⡀⠀⡇⢚⢃⣪⣿⡇⣘⡏⣧⢻⠋⢙⣴⢋⡥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢠⣌⢧⣌⣛⢶⣥⣿⢈⠛⠿⣿⣽⣾⣿⣸⣃⢻⣣⢞⣵⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⢶⣶⠏⣉⣸⡿⡿⠧⠀⢠⡾⠋⣼⡯⣨⣿⡿⣛⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⢑⡚⠿⢛⣿⣿⡓⠒⠓⠐⠿⢷⣶⣿⣿⣿⠿⠟⢋⣩⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⠈⠐⠲⠶⢶⣯⣭⣤⣤⣬⣭⣥⣴⣶⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⣄⠀⡇⠀⠀⢈⠙⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣧⠀⢿⡄⡀⣿⣯⢩⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⢸⡇⢟⣿⣿⢹⣿⣿⣿⣿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢾⣿⣸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⠖⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠄⢣⣶⢶⡄⠀⠀⣰⣿⣷⡀⠀⠀⠀⠀⠀⠻⠆⢣⣾⠻⣆⠀⠀⣰⡿⣷⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣎
⠀⠈⠻⡾⠃⠀⠀⠙⣻⠟⠀⠀⠀⠀⠀⠀⣾⣿⣧⣛⣚⠃⠀⠀⠈⣛⠛⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏
⠀⠀⠟⠉⠀⣀⣀⢾⡁⢰⠄⠀⠀⠀⠀⠀⠙⠛⠃⠛⠁⠂⣤⣄⢺⣁⢰⠆⣀⣀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠈⠺⣉⣹⠆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣽⠃⠉⠁⢺⣏⢹⠂
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⡄⣄⣼⣗⠶⠖⠒⠒⠒⠒⠒⢦⠒⠒⠚⠓⠖
⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣤⣇⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⢀⡆⠈⠁⠀⠀⠈⠉⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⢷⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡈
⠀⠀⠀⠀⠀⢈⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠀⠀⣀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⡠⢄⡀⣤⣶
⠀⠀⠀⠀⠀⢿⠶⡄⠀⢘⡃⠀⠀⠀⠀⠀⠈⠉⠒⠤⣄⣀⠉⠀⠈⠛⣀⡄⠀⠀⠀⠀⠲⣿⡵⠂⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠰⠋⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⢁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠾⡇⠀⢀⠀⠀⠠⠐⠊⠁
⠀⠀⠀⢠⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠈⣄⠀⠀⠸⣿⡂⠀⣀⠀⠀⠀⢰⣾⠚⢗⠀⠀⠀⠀⠀⠀⠀⣴⣂⣽⠒⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⢰⣿⣷⣰
⠀⠀⠀⠀⡅⠀⠀⠀⠘⠻⠃⢻⠀⠀⠈⠉⠿⠶⠞⠁⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠰⣤⢧⠆⠀⠀⠀⡉⢟⣿⣻⢏⠛
⠀⠀⠀⠀⠛⠀⡀⠀⠀⠀⠀⡘⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠛⠞⠀⠀⠀⠀⠀⣼⠈⢑⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠚⠉⠒⡶⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠊⠲⢶⣦⣤⣤⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣠⣾⣷⡿⠂⠀⠀⠀⠉⠋⠛⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⣠⡴⣦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢸⡿⠾⠛⠀⠀⠐⠛⠛⠆⠀⠀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠓⠘⠀⠀⠀⠘⢋⡥⠀⠀⡤⠤⢬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⢀⡓⠂⠀⠀⠀⠀⠀⠀⠀⠇⠛⡄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡰⢊⡴⠋⢋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣕⡀⠙⣿⣷⡀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡇⠀⠀⠀⠀⡠⠋⠄⠚⠡⠀⡀⠀⠀⠘⣰⠋⠀⣀⠤⠛⠖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣴⣄⠈⣽⠆⠰⠋⠰⣄⠀⠀⠀⠀⢾⣿⠀⣼⠀⠀⠀⠀⡀⠀⢀⢈⠂⠀⠉⠜⣁⠠⠳⠁⠐⡛⢁⣠⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣛⣿⡏⢻⣷⣾⡇⠀⠙⠳⢦⣀⠀⠈⢁⣾⠟⣄⠀⠀⡰⠐⢀⠏⢂⠀⣰⣿⣿⣶⣶⣤⣄⣀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⣰⣿⣿⣾⣿⣿⣾⣡⠇⡰⠁⡠⢎⢀⠂⣼⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠛⠻⣁⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣷⣿⠟⡀⡀⡰⠁⢀⡆⣼⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⠴⠋⣷⡄⠀⠈⠲⣄⠀⠀⠀⠀⠅⠀⠀⡘⢿⠉⠁⡘⡀⡐⠁⢀⡼⣽⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢸⡇⠉⠾⠋⢄⠁⡀⠈⢆⠈⢆⠀⠀⠀⠉⠒⣤⣄⣀⡀⢸⡆⢀⢀⡜⠐⠀⡰⠀⢜⣿⣿⣿⣿⣿⡟⠳⣤⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣼⣿⠿⣷⣤⠺⢗⠈⠀⠈⢢⢛⠳⣤⣀⣴⠂⠀⢠⣽⠟⠎⠀⢾⠞⠀⠀⠀⢀⢎⣾⣿⣿⣿⣟⠃⠑⣄⢹⣿⡿⠟
⠀⠀⠀⠀⠀⢹⣤⣟⢾⣿⡄⠀⠱⣿⡄⠀⢡⢢⣌⠉⠑⢀⣤⡀⠉⣤⣶⣄⠸⠙⠓⢶⡀⢮⣿⣿⣿⣿⣿⡏⠰⢂⣇⠜⠉
⠀⠀⠀⠀⠀⣼⢿⣿⠀⠈⢳⣶⠄⢨⠙⣆⠀⢳⡉⣠⠖⡋⠉⠁⠀⠐⠀⢤⣅⠄⣄⡸⠎⢟⡛⠛⣻⠛⡛⠃⠞⢈⠃⠀⠀⠀⠀⠀⣤⣠
⠀⠀⠀⠀⠀⢿⣮⡙⢠⣶⠁⠁⠪⣾⣄⠙⡆⠀⠣⠉⠀⠀⠀⠀⢀⡴⠋⠀⢱⣶⠀⡇⠀⠘⠿⣴⡏⠠⠀⠀⠀⠎⠀⠀⠀⠀⢀⣼⣫⡕⣡⠄
⠀⠀⠀⠀⠀⠀⠻⣿⠆⠉⠂⣠⣬⠟⠣⣠⣧⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠤⠄⠊⡉⢿⣂⣀⣽⠁⠸⠇⠀⡜⠀⠀⠀⠀⠀⠙⠻⢛⠄⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡖⠒⠴⣍⢻⣦⣄⠉⢻⠀⢀⡠⠖⠉⠁⠀⢀⣴⡧⠒⢲⠀⡀⢰⡏⠁⠀⢠⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⢋⡀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣿⣫⠴⠚⠋⠁⠀⠀⣠⠞⠉⠉⠁⠀⠀⠘⣶⡇⢸⣇⢀⡴⠋⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⢂⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢠⠐⡇⢸⠉⢻⣬⣤⣷⠄⢠⡏⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠚⣡⠖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⢠⠃⠀⢣⢬⠔⠁⢃⠀⠁⢠⠞⠀⠀⠀⠀⠀⢿⣤⡼⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢀⣀⣤⠶⣿⠀⠀⣴⠃⠀⠀⠀⠀⠀⠸⡄⠀⠸⣀⡀⠀⣼⣀⣴⠋⠀⠀⠀⠀⠀⠀⡼⠀⢠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠋⣉⠀⠀⢿⡗⠐⠛⠓⠒⠲⠦⠂⠀⠂⠙⡶⠦⢭⡦⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢃⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠆⢸⣷⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡇⠀⣿⡻⡁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠓⠊⠁⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⣤⡴⠛⠁⢀⡼⠃⠀⠀⠀⠀⠉⠉⠉⠁⡀⢀⠐⢀⣇⣗⣱⡀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣶⣆⣀⣉⣀⡈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠸⣧⣄⠀⣤⣤⡄⠤⠤⣤⣄⣀⢤⡤⠤⠤⠤⠤⢀⠆⠀⣿⡇⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠉⠉⠉⠒⠋⠀⠀⠀⠀⠀⠁⠀⠀⠰⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⡆⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠀⠀⣿⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⣿⣄⠈⠻⠆⠀⠀⠀⠀⠀⠀⣠⡎⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠈⢻⡗⣲⣄⠒⠒⠒⠒⠲⣟⠋⠀⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠆

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⢢
⠀⠀⡆⠀⠉⠉⠀⠀⠈⢣⢀⠔⠒⠒⢄⠀⡏⠀⠈⠁⠀⠀⠀⠀⢇⠀⡰⠒⠒⠀⠒⡄⠀⡀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⢀⡼⡎⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⢸⡜⠀⠀⠠⠀⣤⠃⠀⢹⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠫⡅⣸⠀⠀⠀⠁⠀⢳⠉⠁⡆⠀⠀⠀⠘⠋⠁⠱⣀⣀⣀⠀⠈⠣⠀⠸⠀⠀⡞
⠀⠀⡇⠀⠀⠀⠠⠴⢃⠇⠀⠰⠟⠀⠀⠈⡆⠀⢀⠈⠀⠀⢠⠀⠀⡠⠒⠒⠒⢵⡄⠀⣇⠀⠣⢤⠇
⠀⠀⡇⠀⠀⠁⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢱⠀⢸⠀⡀⠀⢸⠀⢠⠇⠀⠠⠂⠐⠁⠀⢸⠀⡞⠉⣢
⠀⠀⠑⢲⢀⠤⠞⠰⣅⣀⡀⡰⢃⠀⠀⢀⠜⠀⢸⡀⠀⣀⡜⠀⠈⠧⡀⠀⠀⠀⠈⢀⠆⠀⠙⠚⠁
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⡘⠃⠈⣧⠒⠋⠀⠀⠀⢿⠉⠉⠀⠀⠀⢀⡙⠙⡄⡰⠂⠉⠀⠀⠰⠄
⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠛⠀⠀⣬⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠈⠁⢠⠈⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠔⠒⠒⠒⠲⠤⢄⣙⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠠⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣄
⠀⠀⠀⣀⡠⠴⠒⡃⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⡔⠋⡍⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤
⠈⠢⢀⠐⠠⢀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠠⠀⠂⠙⡇⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⢀
⠀⠀⠀⠙⠢⢄⡈⠒⠤⣀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡢⣄⠀⣀⠴⡉⠀⠀⠀⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠘⢍⠑⠠⢀⠁⠂⠄⡀⠀⠈⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠆⠒⡒⠇⠈⠉⠀⠀⢀⠧⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡈⠐⠤⣀⠁⠒⠄⠀⠀⠀⠉⠐⢒⠃⠁⠐⡊⠭⠐⠀⠁⠀⠀⠀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡉⠂⠀⡀⠉⠐⠂⠉⡀⠀⠂⠁⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣁⢲⡈⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣉⣲⣦⣤⣴⣎⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣉⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡇⣰⠀⠀⠀⠀⠀⠀⡔⡀⠀⠀⠀⠀⡀⠞⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⡶⢃⠉⠀⠀⠀⠀⢄⡠⠊⠁⠀⠈⠀⣠⠈⠀⠀⠀⠀⠀⠀⠈⣞⡡⠄⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠀⠰⠁⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⣀⡤⠒⠁⠀⠀⠑⢄⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡟⢰⢃⠀⠀⢀⠄⠐⠚⠠⢄⡀⠀⠀⠀⠀⠀⠈⣷⠞⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤⡄⣿⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⠐⢤⡀⢀⣀⡴⣧⡀⠀⢀⡠⠔⠂⠉⠑⠢⣁⠀⠀⡔⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡦⠁⢿⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠹⡤⠾⠒⣚⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣬⣷⣦⡀⠘⡇⣸⢐⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠹⠿⣭⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢛⣛⣿⣧⣭⣁⠹⣼⠀⠅⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⣉⢥⣾⣩⡤⢀⣀⡸⡎⠇⠀⠀⠀⠀⠀⢀⠁⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠸⠀⠀⠤⢼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠩⣐⡭⣥⣀⠹⣄⢀⠀⠀⠀⠀⢠⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠀⠀⢀⢠⠃⡀⠠⠤⠌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣩⣵⢟⡉⠁⠀⢼⣦⣘⠥⢄⢀⠸⡇⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⠎⠀⠀⠒⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⣁⣠⠄⠒⠒⠋⠛⠳⠮⣽⣷⣿⡌⢯⣀⠄⠀⠀⠀⢀⣴⣿⠇⠠⣈⡴⠧⠄⠀⠛⠶⣾⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠠⠈⣉⠁⠰⢖⡶⣋⡀⠀⠀⠀⠀⠀⠹⢦⡙⠋⠛⠟⠋⣠⠾⠿⠶⠚⠉⢀⡀⠀⢦⣴⣶⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠗⠊⠀⠀⣰⡿⠋⡕⠀⣀⣶⠿⢰⣀⠀⠀⠉⠓⠒⠒⠋⢁⡀⠀⢀⠠⡙⠞⠿⣷⡦⣝⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠉⠀⠀⠐⠃⡏⠈⠀⢺⡥⢀⡆⢻⠈⢷⡈⢷⡈⢳⡕⢦⣙⢮⣿⡄⠈⠉⠀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠈⢀⠂⠈⠆⣳⡏⢷⣾⣿⠈⠉⠁⠻⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠐⢸⡇⠀⠈⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠃⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⢾⣄
⠀⠀⠀⠀⠀⠀⠀⣰⢻⠻⡐⠀⠀⠀⢠⡟⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠰⣟⣲⠿⣋⣩⡝
⠀⠀⠀⠀⠀⠀⠀⠉⣥⣀⣤⡀⣤⣤⣠⠁⠀⣠⡮⣭⡁⠀⠘⡆⠀⢸⣿⣿⣫⣭⣽⡿⣻⡆
⠀⠀⡠⠒⠚⠓⠶⣄⠉⣿⣏⠉⣉⣿⠟⢠⠞⠉⠀⣠⠼⠶⠴⢧⡀⠐⣿⣷⣶⣶⣶⣿⣉⠀⠀⣸⢷⣤⡀
⠀⢸⡁⠀⠀⠀⠀⣽⡆⢻⣿⠿⣿⣿⠀⣮⠁⠀⣴⠃⠀⠀⠀⠀⠨⡀⢻⣞⣽⣟⣩⣿⠇⠀⢸⡼⣊⡿⢽⣄
⠀⠀⠇⠀⠀⠀⠖⢾⡀⢸⣿⡟⣿⡟⠀⠀⡏⣶⠻⡄⠀⠀⠀⠀⢸⡇⠀⣿⣯⣭⣿⡏⠀⢲⡟⣓⢻⡆⣽⣷⡆
⠀⠀⢰⣠⢀⣶⣤⡞⠁⠈⣷⡟⣿⠇⠀⠈⠳⣿⣤⢈⣖⠒⢘⣿⡿⠀⠀⢸⣟⠟⣿⡇⠀⠙⣷⣷⣦⣭⣽⣶⠃
⠀⠀⠘⣭⢻⣽⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠋⠀⠀⠀⠘⣯⡿⣿⠀⠀⠀⢻⣿⡿⣷⠺⡟
⠀⠀⠀⢆⣹⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣉⡏⠀⠀⠀⠀⠀⢿⠿⣿⠀⠀⠀⠸⣾⡷⢿⢿⠇
⠀⠀⠀⢸⠷⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣶⢛⠃⠀⠀⠀⠀⠀⢸⣶⡏⠀⠀⠀⠀⢹⣿⢾⡟
⠀⠀⠀⠀⢂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⢀⡴⠛⠉⠉⠙⢦⡀⠀⢀⣤⠖⠛⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⣠⠿⠚⠛⠓⢦⡀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣤⣀⠀⣼⣃⣀⣀⡀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⣧
⠀⠀⠀⣠⡤⢶⡶⢤⣄⠀⢀⡴⠋⠁⠀⠀⢙⡷⠋⠉⠉⠉⠙⠳⢿⣄⠀⠀⠀⠀⠀⠀⠀⢻⣤⡶⠛⠉⠙⠳⣆
⠀⠀⣾⣇⣰⠟⠛⠶⣿⣳⡟⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣀⣀⣸⣏⣀⡀⠀⠀⠀⠸⣇⢀⣤⢶⡶⢦⡀
⠀⠀⢻⣌⣹⣆⣀⣠⣬⣿⠃⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡷⠞⠋⠉⠁⠀⠀⠈⠙⢦⡀⠀⠀⣿⣻⡷⠟⢳⡖⣿
⠀⠀⠀⠙⠷⣌⣿⠉⢉⣻⣦⣄⣀⡀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀⣿⣿⢿⣶⣻⡴⠋
⠀⠀⠀⠀⠀⠈⢻⡛⠿⢤⣤⣨⡽⠛⠛⠛⢲⣿⠶⢤⣤⢤⣤⡤⠴⠯⣤⣤⣤⣤⣤⣤⠤⠶⢶⠾⠻⠛⣻⠋⠉⢹⠀⣠⡟⠉
⠀⠀⠀⠀⠀⠀⠈⣧⠛⠶⠤⢉⡛⠳⠶⠾⣥⣤⣀⣤⣖⣉⣀⣀⣤⣚⣁⣀⣠⠖⠉⠀⣠⣔⣁⣀⣠⣴⣥⣦⠴⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠰⠶⣬⡉⠛⠛⠂⠰⠦⣬⣭⡍⢉⣉⣉⣉⡉⢉⣉⣉⣛⣛⠋⠉⣉⣉⣭⣥⠄⠀⠶⠶⠚⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⣄⡀⠀⠛⠒⠖⠀⠀⠤⣤⣤⡄⠀⢠⣤⣤⣍⠉⠉⣭⣤⣤⡄⠀⠀⣤⡤⠶⠞⠛⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣌⡉⠛⠁⠀⠶⢤⣤⣤⠀⠠⢤⣤⣤⡄⢠⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⠀⠰⠚⠛⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⣁⣈⡛⠓⠶⠶⠀⠴⢤⣤⣤⠀⠠⢤⣤⣤⠄⠀⢤⣤⣤⣤⠀⠠⣤⡤⠶⢂⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣉⡁⠐⠶⢤⡄⠀⠠⢤⣤⣤⡄⠀⣤⣤⣤⣤⠀⠀⠶⠶⠶⢂⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠦⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠒⠒⠂⠀⠐⠒⠠⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢹⠆
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡀⠀⠒⠒⢤⡀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠰⡄⠆⠀⠀⠀⠰⡆⠀⠀⢀⣀⢉⢲⣄⣄⢀⣀⣠⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⢀⠇⠨⠧⠦⠆⠦⠽⠶⠤⠤⠰⠒⢳⣳⣋⡭⠶⠷⢿⠛⠿⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡜
⠀⠀⠀⠀⢰⣅⡰⢀⢠⣀⠴⢬⣀⢀⣠⢀⡴⠖⠉⠈⣃⠀⡀⡄⠈⡆⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣸⠁
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠱⠚⠓⠊⠉⠓⢳⢹⣔⣃⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡼
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢰⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠛⠶⢦⣀⡋
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡴⠛⠂⠉⠒⠀⠤⡀⠈⠑⢆
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣸⡅⠤⣠⣤⡀⠀⠀⠀⠐⠢⣈⡇
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣼⣁⡀⠀⠀⠀⠀⠉⠙⠢⣀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠑⠒⠂⢤⡀⠀⠀⠹⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠶⠴⠶⠒⠐⠋⠁⠀⠀⠀⠀⠀⠀⢸⣀⣀⣤⣤⣀⠀⠀⠀⠈⠒⢴⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠀⠀⠀⠀⠉⠒⠤⡂⠄⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠈⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠠⠄⣀⣤⣈⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠳⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠐⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢤⠂⠁⠠⠀⠐⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢑⠆⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠑⢦⠀⠀⡀⠀⠈⢀⠀⠰⡆⠁⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠤⠚⠃⠀⠀⠁⠀⡃⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀
⠀⠀⠀⠀⠀⠀⡏⡇⠈⠙⣒⠤⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⡇⠑⠖⠋⠉⡷⠀⠀⠉⣉⡒⠲⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠀⠣⣤⠴⠋⠀⠱⣀⡠⠔⠒⢦⠉⠉⠙⣒⣶⠦⠤⠤⣄⣑⣄
⠀⠀⠀⠀⠀⠀⢧⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠑⠒⠉⠀⠀⠙⠒⠶⠤⠖⠛⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⢾⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣴⢯⣉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡜⠛⣦⡼⠟⠛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠳⠶⢿⠁⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠳⠦⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⣴⠎⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣱⣾⡄⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢿⣯⡼⠛⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠘⣃⡀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡿⣶⡛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡾⣿⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠈⢧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢉⣿⣾⠟⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠟⠷⠟⠈⢇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢀⣠⠴⠖⠛⣛⣛⣛⣋⣉⣉⣉⣉⣍⠩⠭⠭⠽⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠤⠤⠤⠴⠶⠶⠶⠶⠶⢶⣟⣛⡳⢶⣤⡀
⢠⡞⢻⡷⠖⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡿⠉⠙⡆
⢻⡄⠈⠳⢦⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣤⣤⡤⠴⠶⠞⠋⠁⠀⠀⠀⢹
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⣄⠀⠀⢀⡤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢠⡇⠀⠀⠀⠀⣠⠛⢳⡀⠀⣸⠀⠀⢧⣀⣠⠟⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢶⡄⠀⠀⠀⢰⠋⠉⢻
⢸⡇⠀⠀⠀⠀⣿⠆⠈⠳⠞⠃⠀⠀⠈⠉⠁⠀⠀⠀⢨⡇⠀⠀⠀⣠⠞⠋⠛⠳⣄⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⣿⠀⠀⠀⢀⡄⢀⡼
⣾⠀⠀⠀⠀⢸⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢠⡇⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢸⠇⠀⠀⣀⣹⣦⣤⡴⠾⠛⢻
⡿⠀⠀⠀⠀⣿⠿⠦⠤⣤⠤⠤⠤⠶⠶⠶⠶⠶⠖⠂⣾⠀⠀⠀⠘⡷⠒⠒⠲⠖⠛⠷⢶⡄⠀⠀⣿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡆
⠱⣄⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⣿⡿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢹⠀⠀⠀⠀⠀⣀⣀⣤⠴⡇
⠀⢹⠶⣮⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⠤⠤⠤⠤⢾⡀⠀⠀⠀⣸⠓⠚⠛⠛⠉⠉⠁⠀⠀⢸
⢀⡟⣁⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠁⠈⠁⣈⡉⠉⠉⡁⠀⢠⡀⢀⣀⡀⢀⡀⠛⣷⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⣿⡠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⡆⠀⡟⠉⢹⠀⠘⡛⠳⣼⣇⢸⡦⠀⠀⠙⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡶⠁
⠀⠈⢳⣾⣥⣀⣀⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠛⠀⠙⠶⠞⠀⠀⠁⠀⠈⠋⠈⢉⣉⣀⡀⣥⣤⣤⣤⣤⠤⠄⠶⠶⠒⠉⠉⠀⠀⡇
⠀⠀⡿⠂⠀⠈⠉⠉⠉⠛⠋⠙⠛⠛⠒⠋⠘⠀⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠰⣟⡡⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠈⠙⣿⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠄⠒⠛⠋⠉⠹⡆
⠀⠀⠀⠸⣧⠀⠈⠉⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠈⠉⠉⠉⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠸⣧⠔⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠃
⠀⠀⠀⠀⠀⠈⢹⡷⢦⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⠶⠶⠶⠖⠊⠉⠉⠸⠁
⠀⠀⠀⠀⠀⠀⠘⣶⡠⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣋⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⠒⠒⠚⠛⠛⠙⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⢂⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⡟⣩⣉⣐⣀⡜⣁⠀⣠⣀⡜⣠⠆⣠⠆⣤⣤⣼⣷⠷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡤⠒⣈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠏⢸⠀⠀⢱⢄⠀⠀⠘⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡂⠀⡟⠀⢸⠀⠀⠈⠚⠝⡲⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣆⣇⣂⡇⠀⠸⠁⠀⠉⠑⠄⠘⡌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⣇⣆⣲⣿⡿⠉⠀⠀⠀⠀⠀⡀⢠⠀⠀⡇⢸⣶⠀⢀⠀⢀⠀⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⡶⠒⡟⣷⣿⡟⠀⠀⣀⠤⠀⠀⠀⡠⠋⠀⠜⠀⡿⣒⣾⠒⢀⣰⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣿⣷⣶⣿⣿⡇⣠⠞⠁⣀⣠⠔⠊⠁⠐⠋⠐⠚⡇⢻⣿⣶⣸⠘⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠛⣿⣧⣄⣿⡉⠉⠉⠉⠁⠀⠀⠀⣠⠀⠀⠀⠀⡰⠁⢀⣿⣿⣿⣺⡛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡧⢄⣿⣟⣀⣀⣠⣤⡴⠖⣋⠠⠀⠀⣠⠞⠁⢠⢿⠹⣾⣾⠻⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠍⠟⢿⡁⠀⠀⠀⠉⠀⠀⠀⠀⢀⣠⠞⠁⢀⡴⠃⣸⠀⣹⠻⢿⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⣥⣿⣃⠀⠀⠀⠀⢀⣀⠤⠖⠋⣀⣤⠖⠋⢀⡴⠋⢀⣿⣦⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⡌⠉⠙⠛⠛⠻⠶⠶⠒⠚⠋⠉⣀⡤⠖⠋⠀⢀⡞⠀⡏⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢽⣇⠀⣀⠀⠀⢀⣀⣀⡤⠖⠋⠉⡁⠀⠀⢀⡠⠋⢀⡼⠁⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣿⠻⢭⣝⣲⠶⠖⠒⠚⠈⣀⣠⣄⣒⡯⠶⠚⠁⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⡄⠀⠀⠉⠉⠉⠉⠙⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣓⠶⠤⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠉⠁⡀⠒⠂⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠂⣀⠐⢀⡀⠐⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠐⠀⣀⠐⢂⣀⠐⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⢁⣀⠒⢂⣀⠉⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣁⠤⠒⢂⠤⡉⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣁⣉⣉⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⢠⠚⠣⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢾⢀⡶⢊⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠢⣄⠘⠀⠀⢃⠀⠀⠀⠀⠀⠀⡔⠒⢚⣻⣙⣞⣋⠐⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠃⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⢉⠜⢁⠛⢄⠙⢆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠁⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠘⠶⠊⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⢠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠒⢄
⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢈⠈⠢⣸⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠴⠂⠉⠉⠙⢢⡀
⠀⠀⠀⠀⠑⣤⣙⣤⡶⠘⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠔⠁⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠛⠤⢤⡴⠿⡤⠞⠳⢆⡉⠀⠀⠀⠀⠀⠀⣀⠤⠀⠠⢤⣀⠞⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⣔⠁⣠⠞⠃⡍⠓⠤⠸⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠈⠉⠁⠀⣀⡇⠀⠀⠀⠀⠀⣀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠁⠁⠀⠙⠒⢤⡀⠀⠀⣀⠴⠊⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠉⠀⠀⠀⣠⠎⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠉⠁⠢⡀⠀⠀⡆⠘⢳⠆
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⠉⠁
⠀⠀⠀⠀⡠⠄⣰⠁⠙⡄⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⢤⠀⠈⠀⢀⠇⠀⠈⠃⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠚⠁
⠀⠀⠀⠀⠀⠁⠒⠠⠊⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⣀⠄⠤⣀⡀⠔⠠⣀⡠⠔⢤⣀⡠⠒⢄⣀⠔⠢⣄⣠⠔⠢⣀⡔⠢⢄⣀⠤⠾⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⠀⠀⠠⠀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⠀⠄⠀⠀⠠⠀⠀⢠⡄⠀⡄⠀⠀⢀⡀⡇⣡⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠸⠀⠀⠀⡇⠀⠀⢸⡆⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡆⠀⠀⡄⠀⠀⠘⠁⠀⠃⠀⠀⠳⡤⣿⡷⢯⡀⠤
⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⢶⠀⡆⠀⠀⠇⠀⠀⠸⠃⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠋⠐⢫⡿⢀⡅
⠀⠀⢀⠄⣨⡀⣵⢉⣁⣀⡀⠀⠀⢠⡄⢡⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠘
⠀⠀⠈⠑⢪⣵⣚⡯⢤⡒⠁⠀⠀⠀⠃⠘⠀⠀⠸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⠀⢀⠀⠀⢠⠀⠀⠀⠃⢀⡆
⠀⠀⠀⢮⠅⢾⠉⡗⠢⠘⠀⠀⠀⠀⠸⠀⡇⠀⠀⡆⠀⠀⣶⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢠⠀⢈⠃
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠀⣀⡁⠀⠀⠛⠀⠀⠀⠂⠀⠀⠃⠀⠀⠚⠀⠀⢘⣀⡀⠠⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁



⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⣿⣶⣶⣼⣶⣾⣶⣶⣦⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣾⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⠿⠿⠿⠿⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿
⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃
⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃
⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⠂⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣀⡀
⠀⠀⠀⠀⣤⠊⠀⠉⠉⠐⢠⠀⠀⠀⠀⠀⡠⢄⡠⠤⣊⡣⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠀⠑⠒⡖⠀⠀⠀⠀⠀⠁⣀⠠⣿⣷⣿⡀
⠀⠀⠀⠈⢀⡀⠀⠚⠁⠘⠂⡄⠀⠀⢰⠀⠡⠒⢀⣚⠀⠀⠈⠉⠹⡀⠀⠀⠀⠠⠠⠃⠀⠀⠠⠀⠉⠿⡆⠀⠀⠔⠈⠀⠀⠀⠉⠀⠈⠢
⠀⠀⠀⠲⠬⢁⣀⣘⡃⠠⠔⠂⠀⠀⢎⠁⠀⢀⣀⣀⣀⣰⣖⣀⣀⡁⠀⠀⠔⠘⡶⠀⠁⠀⠀⠁⠀⢀⡱⠀⣌⡀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠒⡖⡄⠤⢤⢤⠤⢲⡃⠀⠀⠀⠹⡩⠈⠘⠀⠀⠘⠀⡀⡄⠀⠀⠀⠈⣯⣥⣤⣤⣤⣤⣤⣭⠾⠁⠀⠈⢾⠯⣍⣬⣩⣩⣭⢿⠋
⠀⠀⠀⢠⠥⣅⣀⣸⣸⠤⠼⡆⠀⠀⠀⠀⢁⢃⢂⢲⠀⠀⢀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⣄⢠⡄⡄⢸⢸⣺
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢳⡈⣾⣴⢃⣂⡘⠀⠀⠀⠀⠀⠘⣠⠀⡄⢀⢀⢀⢠⣼⠀⠀⠀⠈⡆⡄⡆⣇⣸⣸⡏
⠀⠀⠀⠀⠣⢀⣀⠀⣀⡀⠤⠂⠀⠀⠀⠀⠀⠀⠉⣩⡍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠾⠧⢿⡾⠾⠿⠋⠀⠀⠀⠀⠳⠧⠽⠽⠽⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣷⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⢀⣏⡑⢺⠄⢄⠀⠀⠀⠀⠀⠀⣠⡾⣷⡹⣷⣌⡻⢷⣦⡀⠀⠀⠀⠐⠁⠀⠈⠉⠉⠛⢂⠀⠀⠀⠀⠀⢀⠄⡜⢒⣉⣆
⠀⠀⠀⡔⢣⠼⡼⢍⡽⣉⢆⡀⠀⠀⠀⢰⡙⣷⣝⡻⠦⣽⣿⣾⣟⠣⠀⠀⠀⡇⠀⠄⠀⠠⠀⠀⠈⠀⠀⠀⠀⡀⢎⡹⣉⠧⡼⢤⠓⡄
⠀⢀⠊⠉⠉⠈⠑⡚⠢⠂⠀⠈⠄⠀⠀⣿⣿⣮⣝⣛⠻⠷⣶⣌⣿⣣⠀⠀⠀⡇⠀⠠⣀⠐⠀⠀⠀⠀⠀⢀⠊⠀⠈⢢⠛⡖⠁⠈⠉⠉⢆
⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠀⣿⠻⢿⣿⣿⣿⣷⣬⣿⢟⣻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢨
⠀⠀⠒⠀⠤⠄⠄⠠⠤⠤⠐⢚⠇⠀⡋⢹⣿⣷⠂⡱⠛⠙⠛⠉⠉⠅⠀⠀⠀⣿⡓⠳⠷⡾⠊⢛⣿⠀⠀⠀⣇⠒⠠⠤⠤⠠⠤⠤⠀⠒⠃
⠀⠀⠻⠿⠶⠶⠶⠶⠶⠾⠿⢿⠀⠀⠉⡏⣙⡈⠏⠀⠀⠉⠀⠈⠀⠁⠀⠀⠀⢹⣿⣶⡆⠀⣶⣿⣿⠀⠀⠀⢹⠿⠷⠶⠶⠶⠶⠶⠿⠿
⠀⠀⢰⣶⣦⣤⣤⣤⣤⣶⡶⠟⠀⠀⢰⠀⣿⡇⢘⣤⣤⣤⣤⣤⣾⡄⠀⠀⠀⠘⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⠸⠷⣶⣦⣤⣤⣤⣤⣶⡶
⠀⠀⠈⠒⠠⠄⠉⠁⠠⠄⠐⠁⠀⠀⠀⠙⠉⠛⠺⠿⠿⠽⠿⠿⠋⠁⠀⠀⠀⠀⢻⣿⣷⣴⣿⣿⠇⠀⠀⠀⠀⠒⠠⠤⠀⠉⠁⠤⠐⠂
⠀⠀⠀⠀⡀⠄⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣤⣀
⠀⠀⠀⠀⠀⣀⠀⢿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠴⠞⠀⠁⠈⣇⣀⠀⠀⠀⠀⠀⠀⡀⢀⢀⡔⢂⡀⠀⠀⠀⠀⠀⠀⡾⠃⠀⢀⡀⠀⣑⡄
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢔⠀⠂⠂⠀⠐⠀⠀⠻⡃⠀⠀⢀⠤⠊⡨⠁⠤⠈⠁⠘⠂⢂⠀⠀⠀⢰⠠⡤⠀⢈⣁⠀⠉⢐
⠀⠀⢨⢄⠀⣀⠀⠀⠀⠀⠀⣠⠆⠀⢎⠉⠗⠀⠈⠁⠉⠀⠀⢀⡜⠀⠀⡨⠂⠀⠀⠈⠉⠀⠄⡠⠀⠈⢡⠀⠀⢸⠐⠐⠂⠈⠑⠒⠀⣩
⠀⠀⠀⢱⠳⠤⠥⠤⢥⢤⢿⠃⠀⠀⠀⠷⠶⢤⣤⣤⣤⡴⠶⠾⠀⠀⠀⠈⠲⠰⠑⠛⢹⠙⢉⠹⠙⠙⠁⠀⠀⢨⠉⢹⢹⢙⠙⠋⠉⢀
⠀⠀⠀⢘⡇⠰⠔⡄⢸⢸⣸⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢁⢂⢂⢈⠀⠘⠀⠄⠃⠀⠀⠀⠈⡚⠒⠚⠐⠒⠒⠊⠙
⠀⠀⠀⠀⡟⡄⡆⡇⡎⡇⡏⠀⠀⠀⠀⠈⣤⢀⡄⣤⢠⢠⣠⡔⠀⠀⠀⠀⠀⠀⢢⠈⡞⡄⡆⡴⢠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠙⠳⠶⠿⠷⠓⠋⠀⠀⠀⠀⠀⠛⠛⠷⠿⠾⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠓⠚⠓⠓⠓⠁⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠄⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡤⠤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠒⠚⠛⠒⠒⠒⠒⠲⢄
⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⡏⠀⠐⠤⣀⠀⠀⠀⠑⠢⢀⡀⠀⠀⢀⠀⠉⠢⣄⡀⠐⡄⠀⠀⠁⠂⠀⢑⣀⣀⣠⠤⠔⠋
⠀⠀⠀⠀⠀⠑⠦⠤⠤⠬⣭⣤⣤⣄⣀⣤⣤⣤⣬⡥⠤⠤⠤⠴⠒⠓⠒⠒⠛⠋⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠶⣄⠀⠀⢰⠚⠑⠲⠤⠤⢤⣠⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠚⠁⠀⠀⠀⠈⡆⠀⡜⠀⠄⡀⠀⠀⠀⠀⠀⠈⠉⠉⡉⠝⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡺⠋⠁⠀⠀⠤⠊⠀⠐⠀⢀⡇⠀⢇⣠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⣠⠔⠋⡠⠊⠀⠀⠑⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠉⠉⠉⠑⠒⠒⠒⠢⠤⠤⠤⠤⠄⠐⠒⠁
⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉
⠀⠀⠀⠘⣄⣀⣀⣠⠤⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⢠⡶⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢂⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⣤⣤⠋⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣹⡷⠴⣠⣄⠱⣿⣶⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣚⠋⠶⣦⣝⣿⣷⠘⣿⠇⠸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠞⠉⠐⢿⣦⣈⣿⣷⣻⣧⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠏⡀⠀⠐⠄⠙⡿⣮⣿⠛⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠦⡀⣊⣢⠟⡫⠃⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⠭⠁⠜⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠤⠄⠠⡇⠀⣤⠀⠤⠤⠤⠤⠤⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣂⣉⣀⣲⡤⢶⠤⠃⢀⠰⣶⠦⠴⠄⠀⣀⣀⡀⠀⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠴⢏⣙⣇⣸⡠⡧⠼⣦⠀⠈⠐⢦⣤⠤⠤⠤⢤⣄⠀⠙⠢⡀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣶⢮⣉⣉⠭⠄⣀⠀⠠⠀⣀⣀⠀⠀⠀⣀⡀⠀⠀⠀⢑⣤⠃⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣽⣂⠈⣉⠃⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠀⡄⠐⣁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⣝⡻⣟⠀⣴⡂⠀⠀⠀⢠⣶⡄⢰⡯⢦⡀⢂⡀⣇⢈⣀⡥⡞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢈⢀⡇⠁⠈⠋⢮⡇⠀⠀⠂⢨⠟⡻⢿⠺⠾⠟⠛⠋⠙⠁⠃⠃⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠜⠻⠀⠀⠀⠀⢀⠀⠀⢀⠓⢸⡃⢌⢂⢃⠀⡀⠠⠠⠀⠀⡘⢰⠀⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠃⠀⠀⠀⠀⣠⣆⠀⠤⠅⣼⡷⠶⠶⢄⠀⠀⠁⠆⡀⠀⠃⡈⠀⠁⢈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⣠⡾⠃⠈⠳⠶⠞⣦⣀⠀⡀⢋⣆⠀⠀⠠⠀⡘⢀⠃⠃⠠⢸⢠⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠂⠀⠀⣔⠝⢁⡀⠠⠠⠤⠤⣴⣿⣤⡟⢀⠄⠘⣧⡀⠀⠈⠁⠸⢰⡖⠓⠮⡈⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠉⠠⢄⠀⣠⣶⣯⢁⣿⠃⠀⠀⣀⠘⢟⡄⠀⠀⠃⣼⠀⠀⠀⠡⡆⠀⠐
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠆⠀⠨⡥⣀⡠⢶⣦⣙⡿⣿⠀⢀⡄⠀⠀⡜⡌⠀⠀⢀⠻⠀⣄⠀⠀⠀⠠⢄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⣁⠀⠔⢸⠷⣶⣝⣿⢷⠻⡃⠈⣠⠂⡘⡘⠀⠀⢨⡘⢀⠶⣈⡁⠾⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠴⠐⠀⠀⠀⠀⠰⠘⢆⠶⠻⠷⠼⢋⢾⡈⠤⠁⠐⠁⣲⠌⠀⠀⠀⠀⠃⡈⡀⢁⠁⠆⠀⢠⡸⠃
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠱⠈⠖⢎⢦⡀⠋⠀⠀⠀⠖⢠⡳⠁⠀⠠⢀⠂⠈⠰⢡⠁⠄⡘⠰⠀⠂⡉
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠚⠭⠴⠶⠖⣂⣀⣀⣰⠏⠀⠀⢀⠂⠌⡀⢀⢃⢃⠆⡠⢁⢃⠂⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠐⡐⡀⢀⠎⠎⡰⢁⠎⣀⠎
⠀⠀⠀⠀⠀⠀⠐⠢⠀⠂⠀⠊⠑⣠⣉⠂⠑⢊⠔⡠⠂⠂⠐⠀⠁⡠⢁⠌⠔⠜⠀⠀⠎⠈⠀⣤⠞⣣⠀⡠⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⢂⠠⠀⠄⢉⡰⠛⡰⠦⣤⢀⣀⣀⢀⢀⡀⣀⡀⢀⡤⠀⠄⣐⢞⠕⢁⠈⡡⢊⠐⢁⠔⡡⢂⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠡⠒⢁⠔⢉⠔⢊⠤⠂⠠⠀⠀⠀⡁⠀⡠⠂⠔⠠⠂⡠⢁⠄⡡⠊⡄⠁⠔⢁⠘⠁⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠎⠀⠊⠐⠀⠐⠀⠊⠀⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⣻⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⣴⣶⢾⣿⡄⣿⣲⡝⣛⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⠖⠒⠃⠀⢀⣠⠏⠉⠀⣼⣿⣦⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣀⣴⡖⠉⢀⣠⡴⠾⣟⣛⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣄⣠⣤⡴⠋⠀⡐⣦⣿⣿⣿⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡤⠖⠛⢿⣿⠗⠋⠀⣀⡀⢬⡿⢋⣵⠟⣿⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢠⣶⣉⣿⡯⢾⠋⢁⡴⢫⣶⣿⠷⠚⢋⣹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣼⣿⣿⠟⠁⠀⣰⠟⣴⠛⣻⣿⣿⣿⣿⣿⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠌⢛⠻⣶⣶⡾⠓⢚⡛⣛⠻⣿⢿⢿⣿⣿⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⣿⣿⣴⣆⡙⠈⠣⣀⠠⣾⣿⣛⣻⣷⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠚⣿⣄⣿⣿⣇⡀⢤⣬⣼⣿⣿⢿⣿⣿⣿⡃⣾⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠸⣿⠋⣿⡇⢤⡾⣿⡿⣿⣿⡄⠻⣿⡟⢁⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣧⢸⣿⠀⠀⣿⡇⠀⣿⠇⢰⣿⡇⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣿⡘⣿⣿⢸⣿⠁⢸⣿⠀⣾⡿⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢻⠹⢸⣿⡀⢸⡿⢸⣿⣿⣿⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⢼⣏⣸⣿⣃⣻⣿⣾⣿⣿⣿⣿⣿⣦⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⢿⣿⣿⡿⠍⠉⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⣤⠐⠀⠂⡉⠀⠀⠤⠀⠀⠀⠀⠐⠁⠈⠉⠁⠄⠤⠀⡀⠀⠀⠀⠀⡎⠀⠀⠐⢀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⡀⠧⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⢂⡀⠀⠄⠀⠀⠀⠆⠀⠀⠀⣿⣻⡟⣿⣿⣿⡇⠀⠀⠀⡀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⣫⣵⣽⣿⣿⠇⠀⢠⠀⠀⡠⠀⠃⠀⠰⠀⠈⢀⡀⠀⠀⠠⠄⠀⠀⠀⠀⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠄⡀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠛⠋⢉⠀⠀⠘⠉⠀⠉⠀⠘⠅⠀⢀⠄⠀⢁⡀⠰⡂⠀⢠⠀⢈⡠
⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠁⠀⠀⠀⠀⠰⠞⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⡄⠀⠀⠀⠀⠀⠘⠀⡀⣰⠂⠉⠠⠀⡇
⠀⠀⠀⠄⠀⠀⠀⡾⠃⢈⣢⡤⠄⠂⠨⠁⠠⠐⢧⡠⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⢀⠜⣠⣇⠀⠀⠀⠀⠀⠉⠂⠠⠤⢀⠀⠀⠀⠀⠠⣴⠄
⠀⢆⠀⠀⠀⠀⠀⣄⠁⠈⠁⠀⡋⠈⠀⠀⠐⠀⠁⠀⠀⠀⠎⠀⡀⠀⠀⠀⡀⠰⡆⠀⠈⢲⠓⠑⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⢁⠀⠀⠂⠀⠀⠀⠀⠀⠃⠈⢁⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣀⠀⠀⠀⠀⠠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠱⢀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⡀⠂⢀⠀⠁⠀⡀⠈⠀⠀⠀⠠⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠠⠠⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠁⠀⠀⣀⣀⣤⠠⡀
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⠀⠀⠀⠀⢀⠀⠤⠃⠀⠀⠀⠀⢀⣲⡄⢛⢄⠣⡜⢡⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠒⠃⡤⠀⠀⠀⠀⠀⠀⠀⣼⡿⠏⠀⠀⠀⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠈⢣⠌⢱⡜⢣⠄
⠀⢀⣀⣠⠀⠀⠀⠀⣧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠊⠤⠐⠀⠃⠈⠀⠀⡀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠐⢀⡢⢠⡀⠛⣠⠚⢡⠘⠻
⠀⠀⠈⠀⠀⠀⠀⢹⢸⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠍⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⣄⠰⢆⡀⠦⣀⠶⣀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⢀⢱⡇⠀⠀⠀⠠⠶⠤⠀⠀⠀⠤⠀⡀⢀⠔⠊⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠤⠈⢡⡔⢁⠔⠉⡦
⠀⠀⠀⠇⠀⠀⠐⠺⣙⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠁⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⠹⣄⠘⣀⠜⢃⡐⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⢂⣀⣤⣄⡀⠂⡠⠀⠀⠀⢀⣄⡀⠀⠀⢈⡄⠀⠀⠀⢨⠉⠀⠀⠀⠀⡙⣈⠖⣀⠲⠀⠠⢴
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⠁⠀⠀⠀⢐⡀⠃⠀⠀⠀⠂⠀⠠⠩⠀⠈⠁⣸⡧⠄
⠀⠀⠀⣀⠤⣒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣹⣯⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣠⡗⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋⠉⠀⠀⠀⠀⠀⣇⠐⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⡀
⢰⢎⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⢲⠀⠀⠀⠈⠀⢀⠀⠀⠀⡀⠌⠀⠄
⢸⠀⠀⠰⠄⢀⡀⠠⢆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠂⢀⠠⠖⠁⠘
⢸⠀⠀⠀⠀⠈⡠⠑⠣⠀⠀⠀⠀⢨⠁⠀⣀⣀⣠⣤⣤⣤⣤⣴⡄⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠀⡀⠀⠈⠀⠀⠀⢰
⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⣿⣟⣿⣿⣟⣻⢿⣽⠻⣿⣧⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡄⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠉⢀⠀⠀⠀⠀⠀⣿⣧⣿⣝⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠰⠀⠀⠄⠀⡤⠤⠀⠀⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⡇⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠈
⠀⢇⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⢀⠀⠀⠂⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠑⢄⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇⠈⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣔⣒⣒⣀⣀⢠⣤⣄⣒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠾⣟⣋⣉⠉⠉⠁⠀⠀⠈⣀⠈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠨⣗⡲⠦⢭⣍⡉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣊⢻⣿⣿⣶⡖⠒⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠶⠋⠀⣤⡀⠉⢿⣿⣍⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠀⠰⠶⠶⣾⡋⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⡉⠁⠀⡄⠀⠀⠝⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣍⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠤⢤⣀⣀⠀⢀⣿⣿⣿⣿⣿⣿⣯⡅⠀⠀⢀⡀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣏⡛⠻⠿⠿⠿⠯⠷⠢⢒⣣⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠢⠽⠛⠓⠒⠶⠖⠚⠛⠉⠁⠀⠀⠀⡀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡐⡜⢆⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⢧⢹⣾⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⡿⠿⠿⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠛⠓⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⡴⡚
⠀⠀⠀⡀⠻⠃⣤⠀⠀⠀⠀⠙⡀⢀⠀⠀⠀⠀⠀⠐⠀⢱⣙⡶⠆⣠⠃⠀⠂⠀⠀⠀⢀⠀⠙⠿⡆⡁⡧⡙⠂
⠀⠀⢔⡟⠀⢥⡃⢀⡀⠀⡒⡀⠀⠠⠀⠀⠀⠀⠀⢄⡣⠔⡻⠁⠐⣌⠁⠒⢄⡀⠀⠙⠿⠇⣠⡄⣧⠱⠱⣌⠁⠀⠀⠀⠈
⠀⢰⣎⣃⣀⡂⠀⠀⠉⡀⠀⠀⡒⢲⠀⠄⠀⠀⢀⡞⠀⢾⡁⠀⠀⠈⠑⠦⠀⠈⢆⠀⠀⠀⠿⣃⠘⣦⠱⢄⠙⠂⠀⠀⠀⠐⠂
⠄⢰⡆⠀⠄⠉⢁⠢⠄⠷⠀⠀⠀⠀⠰⠀⠀⢒⢚⡣⠄⠼⠷⠤⣀⠀⠀⠀⠷⠂⠚⠁⠀⢠⣆⡈⢧⡈⠳⢤⣁⠀⠀⠀⢠⠠⠴
⠀⠀⣛⣂⣠⣤⣬⣤⣰⣀⣀⣉⡤⠶⠂⠀⠀⡶⢬⣃⣀⠐⣤⡀⠀⠰⢤⣀⠀⠈⣒⡄⠀⠈⢧⣧⣭⡉⣠⢠⡈⠓⣒⠊⣄⢠
⠀⠀⢹⣽⠘⡇⠸⡇⢸⡇⠸⠁⠃⠀⠀⠀⠀⠈⢲⢰⠉⡍⢩⡉⠙⡋⠉⡍⢡⠀⠂⠀⠀⠀⠀⢩⠘⡇⠈⡇⢹⠁⢸⠁⠘⠸
⠀⠀⠀⠇⠇⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⡇⠀⠃⠀⠃⠈⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠃⠈⠀⠘
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠈⠙⠒⠓⠒⠒⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠚⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠒⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣈⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣠⠆
⠀⠀⠀⠀⠀⢀⠂⠀⡄⢠⡀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠐⢀
⠀⠀⠀⢀⠰⣇⠘⢦⡈⠀⢬⠙⠐⡶⠆⠀⠀⠀⠀⠀⠀⢈⣀⡀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘
⠀⠀⣀⠈⠑⠒⠚⢈⠁⠀⢈⣀⣄⣀⣁⣠⠀⠀⠀⠀⡼⠋⢁⡀⠀⠀⠀⣠⠒⠀⠀⠀⠀⠀⠀⣴⠂⠀⣠⠄⠀⡀
⠀⣄⣿⣙⣒⡂⠉⠀⠈⠉⠙⣦⣀⡀⣀⣀⠇⠀⢠⠸⡃⠀⠈⠒⠢⡀⠀⠙⡄⠀⠀⠀⠀⠹⢄⣹⣄⠀⠻⡀⠀⢱⣄
⠀⡀⠠⠀⠀⠉⠀⠀⠀⠘⠰⣌⣉⣉⣀⣀⠤⠂⠘⠓⠳⠤⣤⡤⠴⠶⠤⠔⠓⠒⠂⠀⣄⠱⠀⠠⢄⠈⠉⢍⡀⠁⠠⠄⠀⠉⡄
⠀⠀⠙⠲⣮⣕⠒⣒⣠⣤⢶⣤⢤⣴⢿⠒⠀⠀⠈⢷⣾⣄⣀⡤⢶⣆⠀⡖⣶⡄⠀⠀⠀⠳⠖⠆⠆⢠⠠⡄⢤⢠⢠⠴⠒⠛
⠀⠀⠀⠀⠇⠹⠈⠙⠀⢹⠀⠸⠀⠈⠀⠀⠀⠀⠀⠈⠙⢸⠀⡇⠈⣿⠀⠁⠙⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠢⢀⡇⠀⢠⣀⠆⠀⠀⠀⠄
⠀⠀⠀⠲⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⣠⠶⠒⠲⢤⡀⡼⠀⠀⠀⠀⠀⠀⠀⠈⢣⡴⠚⠉⢀⣠⣤⣄⡀⢀⡼⠚⠋⠉⠳⣄
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠙⢧⡀⠀⠀⣀⡴⠚⠋⠉⠉⠉⠳⡞⠁⠀⠀⠀⠙⢯⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠳⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢿
⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸
⠀⢀⡎⢠⣏⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⡝⣆
⠀⠸⡇⡀⠉⠓⠶⠤⣤⣀⣀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣠⠿⠃⡿
⠀⠀⠳⣄⠁⠠⠀⠀⡀⠀⠈⠉⠉⠉⠛⠛⠛⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠖⠒⠚⠛⡋⠉⡉⠀⠀⢠⠇
⠀⠀⠀⢹⡓⠦⣤⣀⡀⠈⠀⠐⠀⠀⠂⠀⠀⠀⡄⠀⠄⠀⢄⠀⠰⢀⠐⢀⡀⠂⠀⠁⠄⠀⠂⢀⣠⡴⠋
⠀⠀⠀⠀⢳⡀⠀⠄⠉⠉⠛⠒⠒⠶⠶⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⠤⡤⠤⠤⠴⠶⠖⠚⠋⠉⢰⠃
⠀⠀⠀⠀⠀⠱⣄⠘⠀⢖⠈⡄⠀⣀⠄⢀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⢀⠆⡠⠂⡰⠃
⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡀⠈⠂⠉⠶⠉⠱⠊⠱⠊⠈⠎⠀⠛⠀⠛⠠⠈⠁⠂⠀⢉⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠖⠒⠚⠋⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠀⣿⣦⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣠⣼⣿⡟⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢙⡽⢻⣿⣟⣵⡞⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠐⣺⠿⢗⣼⣿⣿⣿⣾⣹⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠶⣀⣻⡛⠿⢫⢟⢛⣿⣿⢝⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⠀⠀⠒⠚⠛⡋⡙⢡⣾⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠆⠀⠀⠀⠒⢻⠤⠶⣤⣼⡷⠉⣡⡞⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢂⡶⢀⠛⢛⣵⣯⣬⠾⠿⣥⣿⠲⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⢌⡣⠵⠞⠋⠉⢠⠄⣙⠷⠋⠁⢀⣼⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡀⠤⠶⠿⣀⣀⣀⣠⡤⢶⣾⣯⣤⡶⠞⣟⣫⡷⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠉⠀⠁⢻⣿⡿⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠂⠀⠆⠈⡆⠀⢠⠂⠀⡄⠀⡔⢻⣶⠈⡏⢸⢧⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⠀⣦⠴⠔⣆⣬⡀⠀⡇⣀⣷⠀⠏⢻⡟⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡉⢿⠻⣅⢠⡀⠙⡷⠀⡿⠁⡌⢺⡿⣼⣧⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢳⣸⣶⣿⢩⡟⠀⣿⢈⣧⣾⣷⢀⣧⣿⣽⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣧⣧⢸⡾⣿⡀⣿⠛⣿⢃⣿⢿⡟⣿⣿⣿⣯⡯⣄⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠽⣇⣇⣿⡛⢿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣦⣝⣷⣢⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣻⣿⢟⡧⡭⠶⢢⣆⡃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠚⠁⠛⠁⠊⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣠⡶⠒⢶⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣁⠈⠙⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠷⣶⣤⣈⣉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢶⣦⣄⡉⠙⠳⠦⡤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⠀⢀⡞⢉⡼⠛⣫⣷⠆⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣿⣀⡟⢁⡞⢁⡴⠋⣺⣎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣾⡄⣼⡁⣼⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡴⢆⢀⣀⠀⠀⠀⠀⠀⠀⠉⡛⠿⣿⣷⡿⡿⢃⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠢⣼⡈⣻⡇⣸⠃⣸⠤⠄⣀⡴⠖⣋⣩⣥⡷⢤⡉⠧⠆⡼⠀⣉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠚⠢⣼⣿⣿⣿⣿⣿⡇⢠⠞⣡⣶⣟⣭⠵⠶⣒⣚⣗⣛⣒⡇⠀⠀⠀⠀⠀⣄⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣁⠒⣿⣿⣿⣾⣧⣿⢧⡏⣼⣿⣿⣽⢶⣛⣋⣭⡭⣯⠭⢽⠃⠀⠀⠀⢈⡲⢤⡙⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢤⡬⣝⠛⡿⢿⢿⣻⢸⡇⡜⢿⣿⣿⣻⣭⡷⠶⠖⡟⠲⢺⣷⣶⣤⣄⣲⣽⣶⡝⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⣞⣾⠉⢻⣧⡼⣾⡟⣼⣷⣜⠦⣝⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡿⠿⢛⣩⣾⣿⣿⣆
⠀⠀⠀⠀⢠⠖⠦⣄⣇⠀⠘⣇⠀⣸⡏⠀⠸⣼⣿⣿⣿⡷⣤⣉⠀⠀⠀⠉⢉⣉⣉⣩⣭⣥⣴⣶⣾⣿⣿⣿⣿⡿⢿⡆⠀⠀⢀⡀⠀⡀
⠀⠀⠀⣀⣈⣣⣀⠀⢽⣾⣶⣿⣾⣿⣾⣤⢶⣿⣟⠷⣬⡛⠲⢭⣙⠛⠶⠶⠶⠶⠶⠶⠶⠶⢛⣛⣯⣿⣿⣿⣿⡿⣻⣿⠐⢦⡏⠱⡾⢹⣤⢤⡀
⠀⠀⠸⣅⠀⠈⢽⣿⣿⣿⡿⣛⣛⣻⣿⣯⢸⣿⣝⠷⣄⠉⠓⠂⠀⠈⠀⠀⠀⠀⠐⠒⠛⢛⣋⣩⣥⡾⣿⣿⣿⡿⣫⣿⠰⣴⣿⣶⣿⣟⣅⡼⠶⣄
⠀⠀⡴⠚⠛⠓⠺⣿⣿⣯⣾⣿⣿⣿⣿⣿⡞⣿⣟⡳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣩⣥⣶⡾⣿⣿⣿⣿⣫⡿⢬⣿⣿⣽⣯⣿⣿⠦⠶⣅
⠀⠀⢙⡶⠖⠶⠚⢿⣿⣿⣿⣿⣿⣿⣿⣉⠳⢻⣿⣿⢶⣍⡑⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣥⣤⠶⢾⣿⣿⣿⣿⣿⣃⡤⠾⣿⣿⣿⣿⣿⠣⣴⠚
⠀⠀⠘⠲⣦⠶⠒⠚⢛⡿⠋⢹⡏⠙⡄⠈⣳⣿⣿⣿⣷⣦⣍⣙⠒⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⢶⣿⣿⣿⣿⢿⣿⣿⣯⣤⡞⠁⡾⢻⡟⢱⡦⠼⠃
⠀⠀⠀⠀⠿⠶⠴⣖⣉⠀⣠⣿⣇⣠⣿⣶⣿⣿⣿⣿⣿⣿⣦⣭⣙⡓⠒⠶⠤⠤⠤⢤⡶⠶⠶⣿⣿⣿⣿⣿⣿⣟⣛⣛⣻⣷⣾⣧⣸⣓⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣯⣝⣛⠳⠶⠶⠶⠶⠶⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠯⠭⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠚⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⡤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠚⠉⠉⠓⠒⠋⠁⠀⠀⠀⣀⠤⠤⢀⡠⠐⠂⠄⠀⠀⠑⠤⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠼⠁⠀⠀⠀⠑⡤⠀⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠢⠀⢠⠊⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠐⢀⠀⠀⠀⢠⣾⡄⠀⠀⠀⡔⠁⢧⡀⠀⠀⠀⣀⣖
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢱⠀⠀⠈⡏⠉⠛⠒⠚⠒⠒⠒⠛⠛⠛⠉⠉⢀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⡇⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠑⠲⠤⠤⣤⠤⠤⠤⠖⠊⠀⠀⢠⠎⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⣋⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠑⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢜⠉⠀⠀⠀⠈⠀⠀⢉⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣄⡤⠊
⠀⠀⠀⠀⠀⠀⠑⠒⠶⠦⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⠤⠄⠊