Barking Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢤⡀⠀⠀⠀⠔⠀⠀⠀⠠⡀⠀⠀⠀⠰⢿⣾⣿⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣁⣤⣤⣤⣀⠉⢦⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠘⠁⠀⠈⢳⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠙⢷⡄⠃⠀⠀⠘⠒⠂⠀⠀⠀⠀⠂⠀⠀⣠⣾⣦⡀⠀⣷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠦⠤⠤⣤⣀⠀⠹⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣷⣀⡍
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡇⣿⣿⣿⣿⣿⡇⡋
⠀⠀⠀⠀⠀⠐⠀⠰⣄⣀⡠⠖⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠉⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠐⢒⢬⣍⣙⢄⠽⠆⣡⣒⡉⣑⠒⠄⠀⢀⣀⢻⡆⠀⠀⠀⠀⠀⣾⠿⠿⢿⠛⠻⢿⡿⠋⠀⠀⠀⠉⠁⠒⠒⠒⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠢⣕⡒⢛⡲⠦⠬⡇⢜⡓⠼⠎⣤⡤⠀⠈⠂⠀⠀⠀⠀⢇⠀⠀⠀⠑⠀⠀⢳⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡳⡄⠀⠀⠀⠈⠢⠄⢭⠭⢟⠲⠆⠀⠀⠀⠀⠀⠀⠠⡀⠈⠚⠻⠟⠳⡄⠀⣸⣯⠜⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣤⠌⠤⠀⠀⠀⠀⠑⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣰⠃⠈⠀⠀⣠⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⣄⡂⠀⠀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⢧⣀⡀⣠⠞⢻⡁⢀⣠⠞⠋⠀⠀⠀⠀⠈⠒⠤⠀⠠⠔⠋⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠘⠀⠀⠀⢀⡟⠒⠛⣻⠿⠿⠿⠛⠛⠉⠀⠀⠉⠉⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠴⢲⠞⠋⢠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠒⠈⠉⠻⠀⠀⡗⠂⢠⠒⣦
⠀⠀⠀⠀⠰⠶⠿⠞⠀⠳⠖⠀⠰⠶⠀⠐⠶⠀⠘⠶⠃⠆⠀⠰⠆⠀⠶⠶⠖⠀⠲⠶⠀⠰⠶⠶⠶⠖⠀⠲⠶⠶⠂⠀⠐⠶⠶⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠔⠒⠛⠻⠿⣶⡂⠀⣀⣤⣀⣀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣻⣿⠄⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⠀⣀⣴⣇⡀⢀⣤⣦⠐⣦⣄⡀⠀⠀⠀⠐⣻⣿⣿⣆⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠫⡀⠀⠉⠂⠈⠉⠉⠉⠉⠉⠙⠁⠀⠀⠀⠀⠲⠿⣿⣿⣆⠀⠀⠀⠘⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⣰⣴⠇⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣀⠀⠉⢻⣷⣤⣀⣀⡸⣿⣯⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡷⢤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⡹⣿⣧⡀⠘⠛⠛⣿⠛
⠀⠀⠀⠀⠀⠀⠀⢉⢀⡴⠛⡒⡒⡖⣉⡖⣀⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣿⣿⣿⡄⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⣸⣛⣿⣿⣿⣷⣹⣷⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⢿⣿⠿⣿⣿⠿⣿⣿⡏⣈⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡦⠷⠁⠏⠀⣿⣿⡑⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⣴⠛⢀⠞⠉⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡼⢀⡴⠋⣀⠔⠁⠀⠀⡀⠂⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏⣀⠞⠁⣠⣶⣂⣀⣠⠴⠃⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢠⠄⠚⠻⣿⣿⣯⣤⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠉⠁⠀⠀⠀⠉⠐⠖⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⣠⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡏⠀⠰⠋⠉⠁⠀⠀⠉⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⡇⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⠁⡇⠰⠁⠀⠈⠀⠀⠎⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⣀⠤⠤⠤⢤⡁⠀⠀⢀⣴⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡸⢰⠁⣷⣶⡆⢰⠀⢰⣤⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣇⣼⣒⣉⡛⢦⣌⡀⣸⣿⠃⣀⣜⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⡏⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠚⢉⡤⠠⠬⡁⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠛⠷⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠏⢠⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣠⠃⣰⠁⠀⠈⠢⢄⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢠⠃
⠀⠀⠀⠀⠓⢤⣈⣙⠒⠒⠒⠒⢚⡇⠀⠐⠛⠂⠤⠔⠊⠀⣠⡴⢮⡉⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⢀⡜⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠳⣄⠀⣠⠖⠒⠒⠒⠉⡞⠁⠀⠁⠀⠀⡞⠀⠀⣀⣠⠔⠊⠁⠀⠀⠀⢢⠀⠀⡠⠊⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⡀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠞⠡⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠀⠉⠓⠈⠀⠉⠢⣀⡜⢠⠄⠒⠁⠀⠀⠊⠀⠀⠙⣦⠐⠀⠀⠀⠀⠀⠀⢰⠇⠀⠰⠀⠀⠀⠛⠖⠀⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⢀⠀⠀⡆⠀⠁⢰⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⡴⠃⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠸⣀⠠⠀⡀⠀⣄⠀⠀⠀⠃⠀⠀⠁⠀⠄⡰⠁⠀⢀⡠⠴⠚⠁⠀⠀⠀⠈⠓⠢⠤⠤⣀⡀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⣀⡀⠀⡄⠀⣸⢄⠀⠐⠀⠀⠀⠀⢀⠀⡇⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⡔⠉⠀⠀⠈⡄⢸⡜⠁⠀⠑⡆⠀⠀⠀⠀⠀⠀⢸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠑⢄⣀⣀⠴⢀⡜⠀⠀⠀⠀⠱⡀⠣⢀⡀⠔⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠶⠋⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠔⠈⣽⠀⠀⢸⠋⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢀⣾⠿⣆⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠋⠠⣾⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⣴⠟⠁⠀⠘⣧⠘⡆⠀⠃⠀⠀⠀⠀⢀⠀⠀⣠⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣿⠀⠹⡄⢠⠀⠀⠀⠀⡄⢀⡴⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠙⣶⠀⠀⠀⢠⢠⠎⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠱⠀⠀⠀⠀⣼⠀⠀⠀⠙⡆⠀⢀⡼⠃⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢰⡄⠀⠀⠀⡿⠀⠀⠀⠀⠈⠓⠛⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡅⢸⣇⡀⢀⢌⣴⡞⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠈⠙⠢⢄⡀⠀⠀⠀⠀⢀⠘⠋⣹⠏⠈⡽⠁⠃⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠈⠙⠲⠦⣤⣄⣁⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠈⠲⢤⣀⣀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠶⠶⠾⠋⠉⠛⠶⠤⣄⣀⡀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠈⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⢻⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡏
⠀⠀⠀⠀⠀⠸⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⢹⡀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡎⢠⠆
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣀⠴⠖⠛⠲⢦⡀⠀⠀⠇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠱⡄⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⠀⢸⡂⣸⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢦⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡠⢿⠁⢹⠀⣿⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠙⠦⣄⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡥⢾⠀⢦⡀⠉⠁⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢸⡄⠀⠉⠀⠒⣼⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⣀⣠⠞⠀⠀⠛⠀⠒⠒⠒⠚⠛⠀⠂⠛⠘⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣯⣤⣤⣤⠴⠶⠶⠒⠚⠓⢰⡆⢀⣀⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠎⠉⠉⠇⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡠⠀⠒⠚⠒⠃⠒⠚⠃⠃⠘⠤⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢣⡀⠙⠂⠀⠀⠀⠀⠐⠃⠀⣠⠧⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠀⠀⠙⠦⣀⡀⠀⣀⣠⠴⠖⠋⠀⠀⢰⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⡠⠊⠀⠱⡀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠀⣼⣤⣤⣤⣤⣷⡀⠄⠀⢰⠀⠀⢸⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡇⠀⠀⢸⣿⡿⠛⠛⠛⢿⣧⠘⠀⢸⠀⠀⢸⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⠸⡿⠶⠶⠶⠶⠼⡏⠀⠀⠀⡀⠀⢸⡇⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠃⠙⢤⡀⠀⢀⠜⣁⠀⠀⠀⡇⠀⢸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠉⠙⠛⠉⢀⠀⠀⢀⠇⠀⡼⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢱⠀⠀⠀⠀⠈⢓⡶⠒⠛⣅⠀⠀⠀⠀⣰⠃⠀⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⢦⡀⠀⠀⢀⠎⠀⠀⠀⠈⠳⣤⣤⠾⠁⠀⠀⠀⡇⠀⠀⠡⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠙⠲⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡄⢀⡴⠁⢸⠇
⠀⠀⢀⠀⣀⠀⠀⠀⡟⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⢷⠉⢀⣠⡎⠀⡀
⡠⢈⠄⠀⡠⠄⠀⢨⠇⠀⠀⠀⠈⣆⠄⠀⠸⡄⠀⠀⠉⡆⠀⠀⠱⡀⠀⡇⠀⠀⠀⠀⠈⢯⡉⠀⡀⠲⡌
⠣⢼⣀⡜⠀⢀⡠⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠱⣄⡇⠀⠀⠀⠀⠀⠀⠉⠒⠳⠤⣟
⠀⠀⣠⠜⠋⠉⠀⠀⠀⠀⣠⠤⠴⠃⠀⠀⠀⢹⢤⣄⣠⣧⠄⠀⠀⠀⠘⠓⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠳⣀⡀⠀⢀⣀⡴⠋⡀⠀⠀⡀⠀⠀⢀⠈⢢⡀⡼⠁⠀⡔⠀⠀⠠⡀⠀⢠⠈⣧⣀⡀⣀⣀⣀⣤⠶
⠀⠀⠀⠈⠉⠉⠉⠁⠑⠚⣧⣄⣤⣧⣄⣤⣼⢤⠴⠃⠙⠓⠚⠳⢤⣤⣤⠗⠤⠖⠉⠁⠈⠉⠉⠉⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠲⡆⠀⠀⣀⣀⣀⣠⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣄⡀⠀⠉⠑⠛⠻⠷⣶⣶⣶⣶⣝⢿⣏⠉⠉⠉⠉⠒⢦⣤⠀⠀⠀⣀⣀⣤⣤⣤⣨⣵⣷⣿
⠀⠀⠀⠀⠀⠹⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣀⣨⣿⣿⣿⣶⣦⣄⣠⣽⣿⣿⣿⣿⣿⣿⠿⠿⠿⣶⣿⣿⣿⠉⠈⠙⣿⣿⣿⣿⣿⡿⣿⠃
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠐⠙⠒⠛⣡⠟⣀⣤⣾⣿⣿⣿⢏⠉⠆⠂
⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⢁⣨⣶⣾⡟⠛⠛⠋⠋⠈⠈⠇
⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠐⠂⠀⠀⡀⠀⠀⡀
⠀⠀⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣶⣦⣤⠤⣤⣭⣥⡼⠃
⠀⠀⠙⣿⣿⣿⣿⡿⠿⠟⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠙⠋⠁⢠⢖⣭⣶⠗⣀⣤⣀⣀⣀⣐⢦⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠘⠙⠁⠙⠋⡩⠿⣷⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠃⠀⠀⠀⠀⣀⡈⠀⠒⠈⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡀⠈⠀⠀⠈⡇⠉⠁⠀⠀⠀⠀⠈⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢀⢸⣷⣾⣿⠀⢧⠀⠀⠃⠀⠀⠀⠀⠘⢶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢣⠀⠨⡘⠿⠛⠛⠰⠸⡆⠂⡆⠀⠀⠀⠀⠀⠀⠅⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⡇⠀⠂⠒⠒⠒⠈⢸⢣⢐⢇⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⣸⠀⠈⠆
⠀⠀⠀⠀⠀⡀⠀⣠⠋⠀⠀⡰⠁⠈⠑⠒⠒⠒⠚⠡⢀⠉⢀⠃⠀⠀⢸⠁⠀⠀⠀⠀⢸⠁⠀⢀⠇
⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠃⠀⠀⠀⠀⢣⠀⠀⠀⠀⠘⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠃⠀⠀⠀⠀⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡤⢼⡇⠘⠀⠀⠀⠀⠀⠀⠀⠘⡄⠄⠠⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⢸⠀⢀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡄⠘
⠀⠠⡀⠀⠀⣀⣘⡄⠀⠀⣸⠀⠀⠀⠸⣀⠼⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡜⠀⠀⡔⢀⡘
⠀⠀⠈⠉⠉⠀⠀⠈⠓⢆⣠⠦⣀⣀⠦⠜⠒⠓⠛⣄⠀⠠⠀⠀⡆⢀⠀⠀⠀⠀⢀⠼⠠⠔⠈⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣠⠀⠈⠀⢰⡅⠀⡄⠂⡆⠀⠉⡦⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠐⠊⠑⠂⠒⠖⠊⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢆⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣄⠱⡀⠀⠀⠀⠀⠀⢰⠁⡧⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢦⠘⣤⠔⠒⠒⠒⠉⠘⠳⣆⢸⢇⡠⠤⠰⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠈⠀⢀⣀⣀⡱⠐⡰⢮⡽⠋⠀⠘⢫⣹⡌⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⠇⠀⠀⠡⠺⠋⠠⠊⠰⢹⠃⠀⠀⠀⠈⠁⢀⠇⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⣀⢀⣀⣠⣴⠋⠀⡜
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⠀⠀⠀⠀⡜⠉⠑⣄⠀⠀⠀⠀⠉⣡⠀⡜⠀⢸⣁⡠⠔⠃
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣻⡿⡀⠀⠀⢸⠀⡰⢳⢈⡓⢤⣀⣀⠴⢧⡞⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡎⢱⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⡇⡇⠁⠀⠀⡜⡄⡇⠚⣹⣣⢦⣹⢦⡏⠒⠋
⠀⠀⠀⠀⠀⡁⠀⠡⣀⣀⣀⠤⠒⣲⠏⢠⣄⣙⡜⡀⠀⠀⢃⠘⢴⡀⡟⠇⠘⠃⠀⣀⣀⣀⡀⠀⠀⠸⣁⠈⠁⠒⠢
⠀⠀⠀⠀⠀⢣⠀⠈⠋⠀⠀⠀⠀⠉⠀⠀⠀⠹⣆⠱⡀⠀⢸⡀⣴⠀⠳⣼⣩⠖⠉⣁⡀⠀⢙⡆⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣹⠱⡄⠀⢳⡹⣀⢶⣷⠁⠈⡽⠉⠉⠉⠉⠀⠀⡤⢄
⠀⠀⠀⠀⠀⠀⠸⡄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉⢦⢌⠢⣀⠙⠓⠐⠁⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⣀⡠⠒⢰⡀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠼⠚⠷⢤⡝⣖⣒⠖⢉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡰⢺⢹⠠⠀⠸⡤⠋⠀⢀⡠⠶⣯⠤⡴⠊⠑⡤⡼⠀⠀⡠⢶⡀⠳⢯⡀⡞⣏⠔⠋⠉⢲
⣇⠈⢀⡤⣄⠀⠁⢀⠴⠋⠀⢰⠻⡀⢣⡀⠀⠑⢁⡴⠋⢳⡄⠉⠢⡀⠙⢦⡇⠀⠀⠀⡼
⠘⠦⣘⣄⣨⠆⠀⡏⠀⠀⠀⠘⣆⠀⣠⠀⠄⢠⡼⠀⣠⠞⠀⠀⠀⠈⠓⠤⣄⣠⠤⠊
⠀⠀⠈⠑⢤⣠⠜⠁⠀⠀⠀⠀⠈⠳⢌⣒⣃⡸⠋⠉⠁

⠀⠀⠀⠀⢀⣴⣾⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣆
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠿⠿⠿⣿⣿⣿⣿⡄⠤⠤⠤⠤⠤⠤⢴⣿⣿⣿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⢀⠀⡠⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⢤⣿⣄⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢰
⠀⠀⠀⢠⣶⣿⣿⣶⣦⣄⠀⠀⠀⠒⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⠎⠀⢸
⠀⣠⠔⠉⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠒⠊⠉⠁⠀⠀⠀⠀⠀⠈⠉⠒⠴⠃⠀⡠⠃
⡜⠁⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉
⢇⠀⠀⣀⠠⠔⠓⢄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠙⡄⠉⠢⠂⠀⠈⠜⢻⢠⠋⣆⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠘⡄⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗⠀⠈⠙⢿⣿⣿⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡀⠀⠈⠀⠀⠘⠃⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢣
⠀⠀⠀⠀⣖⣒⠒⠢⠤⠄⣀⣈⣹⠃⠀⠀⠀⠠⠀⠀⠀⠀⡰⢿⠟⢱⢄⡀⠀⠀⠀⠀⠀⠀⢀⠀⢐⠃⠈⢆⠀⢣
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⠇⠀⠎⠀⡃⠀⠖⠒⠒⠒⠉⡍⠀⡌⠀⠀⠈⢆⠈⠧⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⡜⠀⡜⠀⠀⢠⠀⢰⠀⠀⢀⣠⠁⠰⠀⠀⠀⠀⡨⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢀⠤⠴⠁⢰⡀⠀⠀⠈⡄⢸⠀⢸⡁⠁⠀⠈⢳⠀⠀⠀⠣⠤⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢈⠀⠀⠀⡆⠀⠦⢤⡝⢀⠦⣀⡜⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡰⠋⠢⠊⠀⠀⠎⠁⠀⠀⠀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠀⡋⠉

⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡰⠚⠳⡄
⠀⢀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⢠⣾⣧⣴⣶⣿⡄
⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⣠⠋⠉⠉⠉⠉⠀⠱⣀⣀⣠⡤⠖⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠬⢝⠒⠉⠀⠀⠉⣡⠴⠆⠀⡇⠀⠈⡇
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣴⡶⠲⣶⣄⡀⠀⠀⠀⠁⠀⠀⠀⠳⢤⡴⠃
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡙⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠙
⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡼⢁⡴⠒⠢⢤⣀⠀⠀⠀⠀⠀⠉⠓⠦⠤⠤⣤⣤⣀⣀⣠⡼⠁⢀
⠀⠀⠀⠀⣀⣀⣀⡤⠀⠀⠀⠀⢶⣿⠿⢋⡴⠋⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠹⡝⡏⠉⠉⠉⠉⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠛⠁⠀⠀⠀⠈⠒⢶⣇⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⣆⠔⠁⠠⠁
⠀⠀⢠⡠⠀⠀⢠⠁
⠀⠀⡆⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠱
⠀⢸⠃⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠐⠪⡙⠢⣄⠀⢀⣄⠀⠠⠔⠁⣀⠀⡇⠀⣀
⠀⠈⡄⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠘⡆⠈⠛⠁⠀⢉⣉⡀⠀⢀⣷⠶⣊⠀⢈⡄
⠀⠀⢨⣄⠀⠀⠀⠘⠣⠀⣀⠤⠄⠒⠒⠒⠒⠒⠢⠤⢄⣀⡀⢀⠀⣄⣀⠀⡇⠨⠓⠹⠀⠀⠀⠰⠓⠛⠓⠀⠁⠀⠀⢀⣈⣇⣸⡀
⠀⠀⠀⠢⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠑⠉⠈⠻⠤⠄⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡴⢎⠿⢚⠶⠌⠠⠄⠂
⠀⠀⠀⠀⠈⠑⠠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⣜⡡⡈⢀⠊⠀⠀⠀⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠐⣄⠁⠈⣽⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠⠚⠁⠀⠀⠀⠀⠀⠀⠉⠂⠄⠀⠀⠀⠀⠀⢀⡀⠈⠳⢬⣤⠯⡇⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠑⠦⠤⠤⠞⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠌⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡴⣅⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢀⡤⠊⠀⠎⠑⢭⡆⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⢀⣴⠁
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢰⠏⠀⠀⠈⢄⠀⠀⢇⠀⠀⠀⢠⣔⠊⠁⠀⢀⣀⣠⡀⠤⠒⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢸⠀⠀⠀⠀⠈⢆⠀⠸⡀⠀⠀⢸⠀⠈⠉⠉⠁⠀⠀⠱⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠘⠇⠀⠀⠀⠀⠀⠳⠄⣇⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⢈⡧⠀⠀⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠙⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠓⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠠⠈⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢠⣄⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠇⡤
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠁⠀⠀⢀⣠⠤⠐⠒⠲⣄
⠀⠀⠀⠀⠀⢀⠔⣡⠴⠤⠀⠀⠀⣍⢢⠀⠀⢀⣀⡁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⠀⠙⠀⠀⡠⢖⣩⠀⠀⠀⢀⡴⠈⠉⠛⠲⣄
⠀⠀⠀⠀⢰⡧⠊⠀⠀⢰⠀⠀⡌⢨⣷⠉⠉⠉⠁⣋⠀⢀⠇⠀⠀⠀⠀⠀⠀⠂⠤⠀⣁⣀⣤⢮⡖⠉⡇⠀⠀⢀⡞⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠘⣆⠀⠈⠒⠊⠀⠀⣀⣀⣈⠓⢺⠀⠀⢰⣺⣶⣶⣶⠒⠊⠉⠀⠀⠀⠐⠭⠴⠁⠀⠀⢸⠁⠀⠀⠀⠀⠀⡏
⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠘⣆⠀⢠⡖⠋⡉⠀⠥⠉⠙⠉⠐⠋⠀⠙⡿⠛⠁⠀⠀⢀⣀⣤⣀⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡼
⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠽⣷⡄⠀⠀⠀⡀⠀⠀⠐⠂⠀⠀⢧⣤⢤⢴⠊⠉⠈⠉⢀⠁⠉⢳⣄⠀⢰⠇⠀⠀⠀⠀⢰⣧⡀
⠀⠀⠀⠈⠢⣄⠀⠀⠀⣠⠖⠁⠀⠈⠻⣿⠀⠀⣀⡔⠋⡆⠈⠂⠀⡁⠈⠀⠀⢁⡦⠄⠚⠉⠉⠉⠙⠛⢋⡄⢸⠀⠀⠀⠀⣠⠟⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠸⡿⠒⠋⠀⠀⠀⠀⠀⠀⢻⣷⣦⠌⠀⡞⠀⠀⠀⠈⢀⠆⠰⢍⣀⡠⠴⠒⠉⠉⠉⠓⠲⣄⡀⢸⡀⠀⣠⠞⠁⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⢉⣩⠽⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠈⠉⠐⠁⠀⠀⠀⠀⠀⠀⠒⠒⠮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠊⠀⠀⠀⠀⠀⠀⣀⡤⠀⣤⠀⡄⢸⠢⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⢠⠀⠀⠀⠌⠁⠈⠁⢉⡹⠒⠙⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠒⠉⡴⠀⠀⡰⠋⠉⢡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣒⢴⠇⠀⠀⢇⡀⢴⡁⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠄⠀⠀⠀⠀⠀⠀⠙⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⡠⠦⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⡠⠊⠀⠀⠀⠈⠐⠂⠠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢽⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠳⣍⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠀⠠⣠⡥⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡧⠒⠉⠁⠁⠀⠀⠇⠀⠘⠧⡀⠀⢀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠌⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠱⠙⠘⠦⣎⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠴⠴⠁⠈⡑⠀⠀⠀⣸⠁⠀⠀⢣⡀⢀⠐⠀⠧⠌⠍⠠⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⢒⠚⡀⢠⠉⠐⡤⢰⣀⣀⣀⡀⠀⢼⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠥⠄⡈⠉⠃⠁⠀⢰⠃⣆⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣤⣄⣿⣄⢢⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡏⠀⠹⠿⠏⠉⣶⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠜⠀⠀⠀⢀⠠⢪⢋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠥⠄⠀⠵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⢋⣉⡤⠤⠤⠐⠒⠂⠠⢤⣄⣀⡄⠈⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢛⡡⠔⠊⢉⡤⠴⢒⣉⣉⠉⠀⢀⣉⡄⣿⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠔⠉⢀⣤⠾⠕⠒⠉⠁⠀⢀⣀⡤⢶⡏⣸⢿⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢼⣅⣀⣔⠉⠀⠤⢤⣤⣴⡖⠚⠋⠉⠀⣾⢠⢛⢨⠳⣄⠀⠀⢀⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠉⠙⠛⡛⠓⠒⠛⠛⠛⠛⠃⢠⣇⢨⠘⡈⢦⡀⠉⠛⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠐⠋⠛⠀⠐⠛⠳⠀⠀⠀⢸⡟⣎⠣⣜⠢⣍⣲⣶⡶⢟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⣦⢰⠀⣠⡄⠀⠀⠀⠈⢷⣌⠓⢦⣭⢷⣆⠐⠒⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢘⡷⠃⠘⠟⠁⠀⠀⠀⠀⠀⠙⠻⢋⡴⠂⢸⣆⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡈⠛⠂⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣎⢿⣉⣉⣉⣉⡓⠦⡄⠘⣧⠀⠀⠀⠹⢤⣄⣤⠶⢻⠃⠀⠀⠀⠀⠀⠐⣦⣀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣴⣿⣮⣿⣿⣿⣿⣿⣷⣿⠀⠈⠳⣄⠀⠀⠀⢸⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠿⠿⣿⣿⣿⡿⠿⠋⡇⠀⠀⠀⠈⠳⢦⡀⠈⠉⠁⠀⠀⠀⢀⣀⣤⠄⠀⢹⡓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠙⠳⣤⣤⡴⠶⢚⣻⣿⠀⠀⠀⣸⠇⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢋⡁⢸⡆⣴⠚⠁⠀⢠⠏⠉⠛⠛⠛⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⠤⣄⡴⢤⣀⣤⡀⣰⠏⠁⢹⡀⣹⣼⣧⡆⠱⡄⢀⡏⠀⠀⢀⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢼⣄⡀⠀⠀⠀⠀⠀⢸⡯⣭⠗⠚⣉⣭⡿⢊⣽⣿⠠⡀⠘⠿⠓⠺⠿⠒⠚⣹⡾⠀⠀⣄⢸⠇
⠀⠀⠀⠀⠀⠀⡾⡏⠀⠀⠀⠈⠙⢦⡀⠀⢀⣠⠼⠿⣥⣶⠏⠩⠿⣏⠷⢶⡶⠾⢿⣖⢦⡈⠉⠒⠊⠉⠀⠀⠀⠀⠹⣾⡀
⠀⠀⠀⠀⠀⠀⣧⢳⡀⠀⠀⠀⠀⠀⠙⢦⣯⣾⣿⣿⣿⣮⣿⣗⡶⠦⣍⣻⡖⢮⡝⣻⣿⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⡞⢿⢧⡙⣆⠀⠀⠀⠀⠀⠈⢻⡍⠉⠉⠉⣹⣋⣧⣴⣿⡭⠷⠛⠋⠉⠉⠉⢳⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠠⡇⠈⢳⡳⣌⠳⣄⠀⠀⠀⠀⠀⡷⣤⠞⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢛⡆⢸⡇⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢳⡀⠀⠹⣌⠳⣌⠳⢤⣤⣤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⢚⡩⢔⣿⠔⠛⢛⡉⠓⣄⡀⢀⣀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⠳⢬⣳⣄⡀⢆⡽⢩⠷⢤⣤⣤⣤⡤⠶⣚⣫⡭⠶⢒⣫⡥⠾⠧⢴⡾⠛⠉⠀⠀⠈⢿⠛⢤⠙⠛⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠈⠉⠉⠉⠀⠈⠲⣤⡤⠖⣒⣚⣩⡭⠴⠒⠛⠉⠁⢀⣠⠶⢋⣠⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⣀⣀⣤⠴⠾⣟⣛⣒⡯⠯⣁⣠⣤⠴⣶⣟⣁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠚⠋⠉⠉⠉⠁⠀⠀⠰⣏⣭⣭⣤⠞⠉⠀⠀⠀⠀⠻⢬⣥⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠐⠀⣠⠀⠙⠀⡀⠀⠄⠀⠘⠀⡀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠐⠀⡍⠢⡉⠀⠁⢀⠈⠢⠘⠀⡀⠀⠁⢃⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⢄⠂⠐⢦⡐⢄⡰⢸⢢⡀⠆⠃⠤⣀⠡⡀⠀⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠔⠢⡉⠀⠒⡈⠢⠀⣌⢀⠱⡠⢮⠞⠉⠳⡘⠈⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠒⠀⠁⣈⣲⡄⢰⡄⢹⡄⠝⢦⡹⠁⠀⠉⢠⠃⠀⠂⡠⠂⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⢁⣻⡈⠉⠻⡌⠋⡹⠀⠈⠇⠈⢀⣴⠁⠁⠠⠦⠶⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣬⡝⠓⠀⠀⢀⠀⠀⠀⠀⢀⠀⠁⠘⡶⠨⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⡀⠀⢰⡕⠀⠀⠀⣠⡂⠀⢰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⠀⠀⠀⠉⠀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢀⠀⠀⠀⠀⠐⠂⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢆⠢⠀⠤⠄⢌⡤⡁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠉⠂⠄⠀⠈⡰⠑⠀⠀⠈⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠕⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠨⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⢊⠄⠀⠈⠆
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠂⠄⡰⢑⠀⢀⠀⢠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠤⠀⢸⠁⡠⢀⠀⠘
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢤⢜⡉⠀⠈⠄⠀⣀⠀⠀⠤⠷⠃⣴⣄⣠⠀⠀⠀⣾⡵⣶⢗⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠒⢔⠀⡠⣿⠶⠶⠢⠂⡄⠾⠀⠀⠀⠀⠒⠘⠿⢓⡽⡂⢸⠁
⠀⠀⠀⠀⠀⢄⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠵⠚⠁⠀⠳⣀⠓⠄⠀⠀⠀⠀⠐⠘⡗⢊⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠡⡀⠀⡀⠁⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠀⠀⠀⠀⠀⠀⢩⡄⡠⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠄⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⡀⠀⠀⠀⠀⠈⡇⠒⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠢⠤⣀⣀⣀⠀⠀⠀⠁⠀⠀⠂⠂⠀⠀⡔⠁⢂⠄⠔⠀⠀⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢂⠀⡠⠀⡀⠀⡠⠈⠀⠐⠂⠀⠐⢊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⠈⠔⠁⠠⠀⠠⢊⠔⠀⠔⢁⠔⠠⠁⠌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠐⠁⠠⠀⠄⠀⠊⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠉⢆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⠐⠀⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠋⠀⠀⠀⠀⡾⢈⡄⠔⠊⠙⠒⠺⠒⠐⠲⢤⡊⠀⣿⠀⠀⠀⠀⠙⠒⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠗⠁⠀⠀⡊⠀⠀⠀⠀⣈⡂⠀⠉⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠱⠀⢀⠊⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠀⢄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⣸⡷⡄
⠀⠀⠐⣆⠈⠇⠀⢀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠴⠿⣾⡇⢻⠷⠛⠀⠄⠀⡀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆
⠀⠀⠀⠃⠀⡀⢠⠈⠆⡄⠀⠑⢢⣄⣀⣀⢀⡠⢊⣀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠄⣑⣄⡀⠀⢀⠔⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠋⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠆⠀⠀⠀⠀⢠⠃⠀⠀⠀⠠⣿⣿⡆⠀⢀⣾⣿⣦⠀⠀⠀⢸⠀⢆⠀⠀⠀⢠⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠠⠀⠀⠀⠀⡐⠀⢦⠀⠀⠀⠀⠙⠋⠁⠀⠀⠛⠛⠃⠀⠀⢀⠜⠀⡨⠓⠊⠀⠀⠇⢀⠃⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⠀⠈⠒⠈⠳⣄⠀⠉⠒⠀⠀⠂⠀⠀⠐⠀⠀⠤⠤⠐⠒⠁⡠⠊⠀⠀⠀⠀⠀⣧⣾⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⡀⠇⠩⣅⡠⠚⠉⢁⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⠀⠀⠀⠀⠀⠀⣐⡘⠛⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙⢂⠀⠑⠄⡉⠁⠔⠊⣷⣤⡀⠀⠀⠀⣀⡀⢀⣴⠟⠀⠀⠀⠀⠀⠀⡖⠉⠉⠉⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠺⠿⠛⠀⢀⣦⠀⠀⠘⣷⣦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡴⠒⢩⠛⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⡜
⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠀⠑⡄
⠀⠀⠀⠀⠀⠀⡿⢍⡐⠢⡀⠀⠈⠂⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⠉⠢⡍⠐⡒⠂⠀⠀⠀⠈⠢⠄⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⠀⠀⠤⠄⠀⠉⠉⠁⠀⠀⢀⣤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⡆⠆⠀⠀⠀⠈⠢⣄⣲⡀⠐⢄⡀⠀⠀⢀⣀⡀⠈⠑⠈⠁⠀⠀⠈⠀⠀⠀⣀⠤⠤⢀⣀⡠⢒⡬⢽⠋⡖
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣀⠀⠀⠀⠈⠂⠈⠑⠊⠉⠿⠒⡶⠨⢇⣀⣒⠭⠉⠩⠅⣂⣒⠮⠀⣶⠒⠦⠶⠂⠁⠀⢘⣠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣶⣤⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣂⣨⣠⣤⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠷⠶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⡿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠶⠋⠟⢻⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠛⠛⣻⣿⡛⠁⢠⣤⣤⣉⡛⠿⣷⣤⣄⠈⠉⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⡵⠚⠉⠀⠀⠞⠋⠁⠐⢛⡛⠛⢿⣯⣉⠁⠠⠿⣿⣦⡀⠐⠶⣮⡙⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢃⡼⠁⢀⡄⠤⠶⠶⠚⠴⣾⣿⣯⣁⠲⠿⠿⠛⠃⣢⡀⠻⣿⣿⣆⠀⠈⢿⠂⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠈⠛⢢⣿⣾⣶⣶⠶⠖⠀⠀⠠⠾⠟⠛⠃⠀⠀⠈⢻⣷⠦⠀⠙⣿⣷⣦⡀⠀⠀⠀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠒⢦⣤⢈⣫⣿⣿⠟⣇⠀⠀⠀⠈⠉⠛⠳⢦⣀⠀⠀⠀⠀⣿⣦⣀⠚⠻⠿⠟⠃⠀⠀⠀⠀⠠⡘⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠋⠉⠁⠾⠷⠦⢍⠒⠾⠃⣠⠀⠀⠀⣀⣀⣀⣀⣙⣦⠀⠀⠀⣿⣿⣿⣧⡀⠘⣷⠞⠁⠀⠀⠀⠀⢷⠸⡆
⠀⠀⠀⠀⠀⢀⡼⣷⣋⣤⠴⠒⣁⣤⣤⣤⡗⢠⣌⣙⠛⠶⢤⣿⡞⠛⠿⠀⢨⣭⡤⢀⣴⠟⢸⣿⡏⠃⠀⢹⠀⠀⢀⡀⠈⢶⣼⡇⠃
⠀⠀⠀⠀⢀⡾⠁⠈⠉⢀⡴⠛⠉⠀⠒⠈⠁⢤⡌⠙⠿⣤⣀⡈⠉⣀⣶⡾⢛⣿⠁⣾⠃⠀⢸⢻⠁⠀⠀⣾⠀⠀⠀⢷⣀⠈⣿⠓
⠀⠀⠀⠀⢸⠻⣶⣶⡶⠛⣠⣴⣾⣿⣿⣿⡆⠈⣁⣀⣀⡀⠉⠛⣿⣿⠁⢀⣸⣿⡄⣿⡀⠀⠀⠸⠀⠀⠀⢸⢰⠀⠀⠈⢿⣷⣽⡆
⠀⠀⠀⠀⠸⣦⣨⣿⣦⣴⣶⣿⣿⣿⣿⣯⣤⣤⣬⣍⣀⣡⡴⠂⠸⣿⠀⢸⣿⣿⡇⢹⣧⠀⠀⠀⣷⢠⠀⠀⠸⣇⠀⠀⠀⠙⣿⠁
⠀⠀⠀⠀⠀⠈⠻⡖⢒⣋⣭⣴⣎⠉⢻⣿⣿⣿⡿⣿⡏⢥⡤⠀⠀⡿⠀⠘⠁⣿⡇⢾⡏⢀⠀⣀⡏⢸⠀⠀⠀⢿⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢄⢫⠉⠻⣿⣆⠘⣿⡇⢻⡇⣿⣷⡄⠁⠀⣰⡇⠀⠀⠀⡟⢀⡾⠁⣼⠀⢹⠇⣾⠀⠀⢰⡜⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠧⠉⢿⠹⣧⣽⣿⠘⡇⢸⣿⣿⡀⠀⢹⠇⠀⠀⠀⢀⡾⠁⣴⠏⠀⠀⢰⡏⠀⠀⣼⠇⣿⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢱⣿⡝⠿⠀⠁⣼⣿⣿⡇⠀⠀⠀⠀⠀⣠⠞⢀⡼⠋⠀⣠⢀⣿⠁⠀⠚⡟⠀⣿⠇⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡙⣛⣿⣤⣤⣾⣿⣿⣿⢃⡜⠁⠀⠀⠀⠀⠀⡜⠁⠀⣠⣿⡾⠃⠀⠀⠀⠀⣼⣿⡾⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠉⠉⠀⠀⠘⠻⢫⣾⣀⡾⢸⣠⣾⡇⠀⣀⣴⣾⣿⠟⠁⠀⠀⠀⠀⣼⣿⠟⠁⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣤⣀⡀⠀⣀⣴⠏⠈⡿⠁⠸⡿⢿⣷⣾⣿⣿⠟⠁⡀⠀⠀⡀⢀⡾⠋⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠛⠛⢿⡟⠀⠀⠀⠀⠀⠁⠈⠻⣿⠟⠁⣠⣾⣴⡶⠞⠁⡜⠁⠀⢠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⡴⠂⠀⠀⠀⣠⠀⠀⠉⠀⡼⠋⠀⠁⠀⢠⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣁⣀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⣽⣿⡞⣠⡾⠀⣠⣶⣿⣾⠟⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠿⣿⣿⣿⣥⣾⢻⣿⡿⠋⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⡿⠁⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣻⣿⠿⠿⢿⣶⣶⣤⣀⠀⠀⠀⠀⠀⢠⢤⣄⡀
⠀⠀⠀⢰⣷⣄⠀⠀⠀⢠⣴⣶⣶⣿⣟⡿⢾⣝⠻⣷⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣷⣄⠀⠀⠀⡏⡞⡞⡟
⠀⠀⠀⠀⢿⠘⣧⠀⢠⣿⢫⣿⣿⠉⠙⠻⣶⣝⡻⣶⣭⣝⣛⣛⣟⣛⣛⣿⡿⣟⣽⣿⣿⡀⠀⢸⠃⠛⢳⠃
⠀⠀⠀⠀⠘⣧⠘⣇⢸⣿⣾⢹⣿⠀⠀⠀⠀⠙⠿⣶⣮⣭⣽⣛⣻⣯⣭⣵⣾⠿⠛⠉⢿⣿⡄⢸⠀⣠⠏
⠀⠀⠀⠀⢀⣿⡄⢸⣿⣧⣿⣿⠏⢠⣤⣤⣄⠀⠀⠀⠉⠉⠛⠛⠛⢉⣉⣉⡀⠀⠀⠀⠸⣿⡷⣾⣼⡁
⠀⠀⠀⠀⢸⠀⣷⠀⢿⡟⠛⠁⠀⢀⡞⣉⡉⠻⡟⠛⠀⠀⠈⠛⣿⣋⡉⠙⣏⠁⠀⠀⠀⢿⢠⣧⡏⡷
⠀⠀⠀⠀⠸⣇⢸⡇⢸⡇⠀⠀⠀⢸⡟⣻⣿⡆⣿⠀⠀⠀⠀⢰⡟⣻⣿⡆⢹⠀⠀⠀⠀⠸⣿⣼⣰⠇
⠀⠀⠀⠀⠀⠘⢷⣿⠀⣧⠀⠀⠀⠘⣧⣽⣿⣇⣿⠀⢀⣀⡀⢈⣧⣼⣿⣃⣾⠀⠀⠀⠀⢠⣇⡿⠋
⠀⠀⠀⠀⠀⠀⢨⡿⡆⢿⠀⣠⠶⣀⣭⣉⡁⠀⠀⠀⠋⠐⠙⣀⣭⣍⣉⣩⣭⣘⢷⡄⠀⣸⣹⡄
⠀⠀⠀⠀⠀⠀⢸⡄⣷⢸⠀⠃⠈⠁⠙⢧⣹⠽⢿⢷⣶⣲⠞⠁⡠⢈⡧⠋⠀⠀⠀⠃⢀⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣷⣿⠿⠶⠶⢾⣹⡆⠀⠈⠛⠶⠶⠾⠥⠴⠾⠛⠉⣦⣀⠀⣾⣹⠶⠾⠿⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣯⢠⠤⠶⠖⢻⣯⣄⣀⣀⣀⡀⠉⠛⠛⠋⠀⠀⣀⡙⠟⠀⡬⣿⠓⠶⠦⢤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠒⠚⢋⣉⡷⠀⠈⠉⠻⢥⣀⣀⣀⣀⣠⡼⠟⠁⠀⠀⢘⣏⣙⡓⠲⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠛⢋⣉⣩⠟⠀⠀⠀⠀⢀⡈⠉⠉⢉⣁⡤⠖⢁⠀⠀⠸⢯⣉⣉⠛⠃⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣉⣽⢿⣄⠀⠀⠀⠀⠀⠀⠉⠉⢭⠭⠤⠖⠚⠁⠀⠀⠀⢀⣸⣿⣍⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⢤⣄⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠋⠀⠀⠀⠀⠀⢀⣠⣴⢶⣿⣿⣯⣷⣶⣶⣶⣾⣿⣿⠿⠿⠿⠿⠿⠿⠛⠿⢿⣷⣤⠀⠀⠙⢶⣤⡀
⠀⠀⠀⢀⣤⡾⢛⣩⡶⠛⠛⠉⣩⡽⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣀⠀⠀⣿⢿⣷⣄
⢀⣤⠶⠛⣡⡴⠋⠉⠀⣀⡼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠰⣿⢿⣢⡀
⠟⣁⣴⡞⠁⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣄⠀⠀⠀⠘⢿⣧⣿⡛⣄
⣾⡟⠁⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⣻⣿⣿⣿⣿⣷
⠏⠀⠀⢀⡴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠉⠉⠉⠙⠳⣦⣆⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠀⡰⣷⡟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠀⠀⠀⠀⠀⠀⣌⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⣿⣿⣿⣿
⣼⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣫⣛⠁⣀⣠⡤⠀⠁⢀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢙⣷⣿⣿⣇⣠⣴⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⣿⣿⣿⣿
⣿⡿⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠛⠘⠲⠿⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⣠⢀⠔⣄⣠⠀⠀⢈⣿⣿⣿⣿⡟
⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢼⣮⣅⣾⣿⡟⠉⢻⡆⠀⣠⠄⢀⡼⣱⠏⠎⠀⠀⠀⣀⠈⣼⣿⣿⣿⠇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠹⣿⠉⣣⣔⣧⣨⣧⣼⣿⣴⣿⡿⡃⠂⠀⠀⠀⠈⠛⠆⣿⣿⣿⡟⠘
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡰⠃⣴⣿⣿⣿⣤⣿⣿⣿⣿⣿⡿⠁⡀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣶⣿⡿⠁⢀⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⣯⠞⠁⠀⡀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⡿⠋⡀⢠⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⢂⠀⠀⢀⡼⢃⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣷⡿⠛⡟⠀⠈⢙⣦⡙⣿⠃⠙⠻⣿⣿⣿⣿⣿⣿⣷⣄⠀⢠⡔⣴⡟⢱⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⢸⠃⠀⠀⠈⠻⣿⣿⠀⠀⠀⣿⣿⠀⠉⠛⠻⣿⣿⣷⣶⣶⠏⠘⣾⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⡍⠨⠃⠀⠀⠀⠈⠛⠀⠀⠀⣙⠘⢄⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠾⠀⠀⣸⣿⠛⣻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣶⣾⡿⢃⣰⣿⣿⣿⡟⢻⣷⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣴⣿⣿⠏⣹⠏⠀⢨⣿⣾⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠐⢀⠀⣴⣦⠄⠀⠀⢀⣈⣤⣾⣿⣿⣿⣿⣿⡿⣿⣴⠊⠁⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣽⣿⣿⣿⣷⣀⣠⣴⣿⣿⣿⢿⣻⣿⣿⡿⠋⢐⡁⠀⠶⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣼
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣭⣴⣿⣿⠟⣿⣦⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣯⣿⣿⣿⣿⣟⣏⣥⣾⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠐
⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⡿⢟⣩⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢾
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠝⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣩⣾
⣄⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠟⠀⠀⠀⠀⣀⣤⢸⣿⣿
⣿⣷⣤⡀⣀⠀⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣤⣶⣿⣿⡿⠛⠁⢀⣤⣶⣾⣿⠿⠃⠻⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣟⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣭⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡓⠂⠀⠀⠀⠚⡉⠂⠀⠀⠉⠉⢉⢩⡅⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠿⠆⠀⠀⠸⠿⠃⠀⠀⠀⠀⢱⡆⠃⣀⠔⢸⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣈⢴⠊⠀⠐⠋⠉⢉⠍⣠⠞
⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⠀⢧⠀⢤⠤⠤⢀⡠⠄⠀⠀⠀⡀⠡⡼⢶⠶⠿⠭⠄⠒⠈⠁
⠀⠇⠀⠄⠋⠓⣦⢤⡀⠀⣠⣤⣷⡄⠻⠉⠹⠁⠀⠀⢀⡔⣀⣴⣿⣷⣤⣤⣀⡀
⠀⡄⠀⠀⠓⠀⡌⠀⢰⣾⣿⣿⡏⢹⡄⠒⠃⠀⣠⣴⡫⢕⡫⡾⠃⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠈⢦⡀⠈⠠⢁⣴⣿⣿⣿⣿⣿⣿⣿⣶⢖⣋⠉⡀⠀⠁⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠄⠊⠙⠻⣿⣿⣿⣿⣿⣿⣿⣍⠽⠳⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⣿⣿⣿⣿⣿⣦⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⣿⣿⣿⣿⣷⡄
⡀⠀⠀⠀⠀⡇⠀⠀⣠⠶⠟⠋⠨⠟⠛⠛⠛⠛⠛⠛⠛⠳⣬⣙⠿⠿⢿⡿⠟⠛⢷⣽⣿⣿⠟⠋⠉⠙⢄
⡇⠀⠀⠀⢠⢂⠴⠋⠀⢀⠤⠤⣀⣈⣀⣤⡀⡠⣤⣠⣤⣾⣿⣿⣷⣿⠛⠀⠀⠀⠀⠈⠉⠉⠀⠂⠀⠀⠀⠙⠢⢄
⠱⠤⠤⠄⠊⠁⠀⠀⠀⢸⣿⣿⣿⠿⣿⠿⡿⢿⣿⣟⣿⣟⡁⠉⠉⠙⢼⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣿⢿⣿⣽⣿⣿⣿⣿⣟⣳⣆⠀⠀⢀⠊⠐⠤⠐⠀⠂⠉⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⢄⠀⠀⢼⠛⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⠿⠁⢈⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠉⠐⠒⠦⠤⠭⠭⠉⠭⠭⠽⠯⠥⠜⠂⠲⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⣦⣀⣀⣤⣄⣀⣀⣠⠞⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣤⠶⠛⠉⠉⠀⠀⠀⠈⠉⠙⠳⢦⣄⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⣸⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠢⠀⠀⠀⠀⠀⢺⡗⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⡒⠀⠀⠀⠀⠀⣀⣀⣐⡷⠤⠿⠤⣤⡀⠀⢀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠸⣅⡀⠀⠁⠀⠀⠀⢀⡿⣠⡿⣅⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣽⠦⣄⠐⠒⠈⠉⠛⠒⠶⢖⣛⡿⠟⠉⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢀⠀⠉⠀⠀⠀⠒⠒⠛⠉⣁⠤⠤⢄⣀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⢠⠀⠀⡈⠳⡄⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⡠⠷⠖⠶⢤⡀⠀⠀⠀⠀⠈⠉⠣⣀⣠⣇⣀⡟⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠉⠀⠀⠀⣀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢈⡇
⣴⠚⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣤⠼⠦⡤⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠈⢢⠱⡀
⣧⠀⠀⠘⢇⠀⠀⣀⡤⠶⢄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⢠⠞⡠⠃
⠸⣦⡀⠀⢀⠏⡋⠀⣀⣠⠀⢧⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⠜⡆
⠀⠈⠻⠶⢥⡤⠷⠞⠋⠹⣆⠈⠳⠼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠰⠉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⠼⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⢤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⠀⠀⠉⠻⢶⣤⣄⣀⣀⣀⣀⣀⣀⣤⣴⠞⠉⠀⠀⢀⢹⡀⠀⠊⢈⡀⢀⠀⠀⠀⢀⡱⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠀⡀⠀⢠⡞⠁⠈⠉⠉⠉⠉⠉⠉⠀⠈⢷⡀⠀⢀⠈⡟⠀⢠⡀⠀⠉⠐⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⠟⠚⠁⠀⠀⠙⠢⢤⣀⣀⣀⣀⠤⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠞⠩⠖⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠓⠒⠒⠒⠒⠧⢄⡀
⠀⠀⠔⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡶⠆⠀⠀⠀⢠⣦⡀⠈⠳⣄
⠀⢘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡞⢰
⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡈⠀⠀⡀⠀⠰⠀⡀⠀⠀⠂⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡄⡄
⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢃⠀⠠⡷⠀⢀⠀⡿⠀⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⢠⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⠀⠀⠢⠀⠀⠠⠈⠀⠀⠀⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠙⠤⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢠
⡴⠒⠶⡏⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⡿⠖⠲
⢹⠤⣠⡇⠀⢀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⡄⠀⠀⣧⡠⢼
⠈⢦⣀⣑⣶⠋⢀⡤⢈⣉⠛⠒⠢⠤⢄⣀⣀⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣀⠤⠤⠒⠚⢋⣉⠠⣄⠈⢲⣖⢁⣠⠊
⠀⠸⢄⣀⣨⡟⠉⠀⠀⠀⠉⠉⠒⠒⠒⠤⠤⠤⠭⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠥⠤⠤⠔⠒⠒⠊⠉⠁⠀⠀⠈⠙⣧⣀⣁⠼
⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⡇
⠀⠀⠀⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢹⢹⡏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣤⣀⣀⡇⢸⢸⡇⣇⣀⣠⡤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠉⠁⢸⢸⡃⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣯⣿⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣾⣿⣷⣶⣶⣷⣶

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠈⠓⠎⠻⠁⠔⡥⠉⣁
⠀⠀⠀⢀⠔⣉⢀⣀⠄⠀⡀⣀⠤⣀⡀⠀⠀⠚⠫
⠀⠀⠰⠟⠉⣤⡊⠤⢒⠁⠀⠀⠀⠀⠠⡄⠀⠀⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⢢⡀⠈⠑⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠲⡆⢠⠤⡀⠀⠙⣄⢼⠀⠀⠀⣀⣀⠀⠀⡠⠉⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡇⠀⠘⠄⠀⠈⢎⢧⢀⠊⠤⠈⠣⠘⡀⠀⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠸⡄⠀⢸⠀⠀⠈⠿⠋⠘⠁⢀⡄⠀⡇⠀⠙⠃⢐
⠀⠀⠀⠀⠀⢀⡠⠐⠀⠉⠢⠤⣴⠋⠓⠓⠀⠀⠀⢀⡀⠘⢇⠉⠀⠀⠈⠑⠤⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⢤
⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠈⠄⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣀⣀⡌⠀⠀⠀⠀⠚⢢
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾⠁⠘⠁⡀⣀⣶⣿⡄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠀⠄⠀⠀⢤⠀⠀⠀⠐⠂⡇
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢠⡡⠖⠂⠈⠁⠈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣁⢎⠀⠀⠀⠀⠀⠀⠦⣀⠤⢧⡨
⢇⠀⠀⠀⠀⢀⣀⠤⠔⠂⠁⠀⠀⠈⢁⠐⠈⠉⠉⠈⠉⠉⠛⠇⠀⡆⠀⠀⡠⢄⡀⠀⣐⠩⠃⠀⠈⠢⠕⢀
⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⣀⠠⠐⠚⢦⡀⠘⠀⢰⠀⢠⠉⣳⠉⠛⢄⠈⢄⠀⠀⡴⠎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⢰⢦⡸⠷⠤⡇⠀⠒⠈⠁⢀⢧⠀⣀⣁⡦⠑⠢⢄⣠⠀⠃⠒⠈⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠑⢄⣤⠆⠀⢀⣠⢊⣠⡘⢮⡁⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠒⠉⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⡨⠷⠦⠤⡤⠔⢉⡸⠋⠁⡟⠈⣳⠀⠀⠈⠢⣀⣀⣠⠴⠒⠒⠂⡏⠀⠀⠀⠀⠀⢹⡀⠀⢀⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠘⢊⠝⠃⠌⠀⢸⠺⡇⢀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵⡎⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠘⠃⠐⡄⠀⠐⡄⠀⢐⠂⠒⢸⡀⢹⣍⠀⠀⠀⠀⢀⡀⠤⠐⠒⡖⠀⠀⠀⣀⣤⡞⠥⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠠⠰⡅⢀⡼⠛⠛⠛⠀⠀⡌⢣⡆⢹⡯⡑⠶⠈⠀⠀⠀⠀⣸⠀⡠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡄⢀⡀⢀⢒⠟⠁⠀⠀⠀⠀⠀⢈⠷⡆⠐⠀⠓⠈⠆⠡⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢧⠀⠀⠀⠀⠀⠀⠘⠓⠢⢌⠁⠀⠀⠀⠑⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡆⠀⠀⠀⣀⣀⠠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⢉⣭⠡⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣀⣄⣀⣄⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠤⠤⠐⠲⠈⠙⠛⠛⠉⠃⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠢⢤⣈⠙⠫⢍⡉⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⠤⣀⡀⠀⠀⠀⠀⠙⠻⢔⠛⠲⠄⠀⠑⢌⣛⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠚⠛⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠒⠒⢒⡒⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣀⣀⣤⣤⣤⠄⠀⠀⡄⠀⠉⠉⠑⠲⠒⠀⠒⠄⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠉⠁⠐⢲⡆⠙⣄⠀⠠⡀⠀⢀⠀⣠⢃⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡣⠼⠀⠤⣀⢀⡀⢴⢅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠐⣿⡳⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠉⠀⠨⡋⢻⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠃⠀⠀⢇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠓⠢⡌⠘⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠚⠉⢀⡀⡇⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉⡇⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢤⠄⠈⠲⠤⢀⡀⠀⠠⠔⠁⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠘⡤⠊⠉⢢⠀⡁⠀⠀⠀⠀⡠⢴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢹⠀⠀⠀⠀⠘⠀⠀⠹⣀⠔⠃⠀⢇⠀⢀⠔⣻⣶⡞⠛⣶⠞⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡌⠀⠀⣀⣀⣨⡇⠀⠀⠙⠢⠀⠀⢹⣤⣿⣯⣮⠡⠟⠋⢁⢺⡟⡽⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢸⠃⠀⠀⠙⢢⣿⣧⣤⣤⠄⣀⣠⣤⣼⣋⡽⠁⠀⠀⠀⠀⣆⠀⠘⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠀⠀⠀⠀⡸⠢⣀⠿⠿⣀⡸⠿⣇⣀⣿⠛⠒⢄⠀⠀⠀⠀⠑⢆⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣧⣀⠀⢀⠞⠀⠀⠈⢢⣘⠿⢇⣀⡽⡿⠃⠀⠀⠀⠑⣄⠀⢀⠄⢤⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⢳⠃⠀⠡⠉⣲⣦⠃⠀⠀⠙⢤⡜⠛⡷⠁⠀⠰⣀⣤⠚⠁⠀⠈⢀⠀⢧⠠⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⠒⠺⣤⠇⠑⠒⠉⡏⠀⠀⠀⠀⠈⢳⡜⠀⠀⠀⠀⢻⡛⠒⠒⠒⠂⠀⢤⠎⡀⠠⠧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠈⢓⠒⢂⣷⣰⠉⣉⠉⣁⠉⠀⠁⠀⢀⢀⡁⣈⣀⣀⣀⣈⣉⣉⣉⣉⠉⣇⡼⢰⠗⠂⠤⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠈⠙⠧⠏⡠⠊⣀⠤⠤⠀⢐⡀⠐⠒⠒⠒⠒⠂⠠⠤⠤⠄⣀⠀⠀⠑⡄⠉⠉⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠸⣦⣄⣛⣓⠒⠒⠒⠒⠒⠐⠂⠀⠀⣈⣉⣠⣼⠂⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠘⠂⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣠⣭⣛⣛⡛⢛⣯⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⡍⣽⣾⣿⣿⣿⣿⣿⣷⣭⢩⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡿⠃⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠿⣿⠿⣿⣿⣿⠀⠀⣷⠀⣹⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣵⣶⣭⣿⡿⠃⠀⣴⠋⣾⠁⢰⠋⠀⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣛⠋⠀⠀⠀⣘⣁⠘⠃⠘⢃⣴⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⢤⣾⣿⣿⣿⠷⣀⣀⣾⡿⣫⣶⣿⣟⣮⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣝⡻⡿⢟⣺⣿⣿⣟⣱⣿⣾⣿⣿⣟⣸⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⠴⠤⠐⠀⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⣀⡀⣀⡀
⠀⠀⠀⠀⡴⡋⠁⠀⠐⠋⠉⠀⠀⠀⠀⠀⣉⠁⢦⠀⠀⠀⠀⡀⠒⠊⣀⡩⢀⣁⡶⠊⠉⠉⠉⠳⠀⣍⠃⠬⣵⢦⡀
⠀⠀⢀⠞⠸⠁⠀⠀⠀⠀⠀⠀⠀⠄⣀⣦⡴⠖⣿⠀⠀⠀⢸⠀⢀⣀⠦⡄⠣⡿⠧⣀⠀⠘⣦⠀⠀⠀⢳⡀⠀⠀⠹⣆
⠀⠀⡖⢈⠁⡀⠀⠀⢀⣀⠴⠓⣻⡯⢟⠝⠀⠀⢻⠀⠀⠀⡸⠁⢀⣰⠚⠀⡘⠀⠀⠈⢷⣦⣘⢀⡀⠀⠘⢧⠀⠀⠀⡎⣇
⠀⣘⡔⠁⠀⠁⠀⡠⠁⣠⢶⣰⣿⠓⠀⠀⢹⣷⢀⠀⠀⠀⠀⠠⢴⡷⠈⠀⠱⣿⠍⠀⠀⠸⣟⣦⣽⣆⠈⠊⡀⠀⠀⠆⢸
⢠⣢⠀⠀⠀⠄⠠⠂⢾⣬⢮⣿⣿⠀⠀⠀⠀⠀⠈⠒⠒⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢫⠸⣭⡋⣦⠀⢷⡀⠀⠇⢸
⢀⢋⠐⣤⠀⠀⡞⠃⠀⢹⣿⡏⡿⠀⠀⡀⠀⢀⢣⠤⠀⠀⠀⠠⠒⠀⠑⢲⣔⡀⠀⠀⠀⠀⠀⠰⣧⡾⣿⠀⠘⢱⠀⢀⡾
⠸⠇⢀⠈⠉⠺⡇⢀⣴⢉⣿⣿⣄⡀⠀⠘⢰⠿⠟⠁⠀⠀⠀⠀⠀⠠⠀⠈⠘⢶⣌⠂⠀⠀⠀⠀⣿⡟⣿⡻⠀⠀⣸⡶⡇
⠀⢁⠘⠀⠷⣄⡀⠂⡹⡦⣽⡿⣿⣷⣷⣶⣉⡂⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⠀⠀⠀⠀⣴⣿⡟⣿⣷⢀⣴⣷⣷⡇
⠀⠠⠤⠂⠒⠓⢩⠴⠾⠁⠀⢹⠿⠛⠛⠛⠛⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⢘⣟⣤⣿⣿⣇⣾⣥⢄⣇⣞⣇⡾⠉
⠀⠁⠀⠀⠀⠈⠈⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠲⠀⠒⠀⠀⢀⠀⠀⠳⠿⠛⠉⠉⠙⢿⠛⣩⢿⡿⣿⣉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠻⣦⡴⣶⠂⢲⢦⣶⠾⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠙⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⠤⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⡁⠀⠀⠀⣷⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣟⡛⠷⣤⣴⠏⠉⠛⠛⠁⣀⣀⡠⠤⢤⠤⠤⠤⡤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⢏⡋⠀⢴⠆⠀⢰⣺⡆⠀⡠⢒⡩⠷⣒⠰⠟⣹⣿⣉⣝⣽⣶⣖⠮⢽⡓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡿⠋⠀⠀⠀⠀⢒⡎⢼⠃⠀⠹⢬⣓⡊⠭⠉⣿⣧⣍⣹⣿⣦⡤⠿⠟⣚⡧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠓⠶⠦⠦⢸⡇⠀⠀⠀⠈⠉⠙⠛⣿⠒⠒⣒⣲⣶⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡷⠶⣒⡀⠀⢒⢒⠿⢥⣀⣀⠀⠀⣀⣠⠖⠋⢉⡟⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠛⠁⠀⠈⠓⢛⢄⡙⠊⣥⣄⠈⠻⣏⠉⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢠⡟⠲⣄⣴⠋⠀⠀⣠⠆⣀⣀⣬⠭⠥⠴⠾⠷⠶⠦⢼⣷⣀⣠⠞⠁
⠀⠀⠀⠀⠀⢸⣧⣤⣼⡇⠀⠀⣸⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⢶⡛⠉⣠⣿⠀⠀⢠⡟⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠳⠞⠁⢸⡄⠀⢸⡇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⡇⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠹⣦⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣼⡏⠀⠀⠸⡷⠶⣤⣤⣤⣤⠶⣶⠶⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⣰⡏⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣰⣿⣇⣠⣴⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠂

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⢠⡎⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣇⣀⡿⠈⠳⢤⠴⢃⡤⠶⠦⣤⡴⠋⠉⠙⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆⠈⠁⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣠⢶⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡌⠉⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⢀⡴⣡⠏⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣙⡁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢀⡴⠚⠛⠻⣶⣡⢞⡥⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠉⢳⡀⠀⠀⠀⣴⠋⠀⠀⢀⡖⠋⠉⠓⣶⠋⠉⠙⢯⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢠⠶⣦⣤⣷⡀⠀⠀⠻⣄⡀⢰⡞⠓⠒⠒⠒⠚⠒⠒⠒⠚⢳⠀⠀⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⣤⠧⠤⠼⣧⡀⠀⠀⣸⠃⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⠞⠉⠀⠀⠀⠀⠀⠉⠳⣄⢹⡀⠀⠈⠳⢤⣀⠀⠀⢀⣠⠴⠋⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠙⢲⠆⠀⠀⠈⡟⠉⣿⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⠀⠀⠀⠾⠭⠿⠽⠷⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣀⠀⠀⡞⣦⠀⠀⢀⡏⠀⠈⠳⣤⣠⣤⡄⠀⠀⠀⠀⢸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⠛⠒⠛⠉⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⣀⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠤⠤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠀⠠⠀⠁⢠⢧⠰⢔⠀⠀⡦⠀⠐⢀⢸⠃⡧⡂⡀⢰⠰⠆⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⠀⠀⠀⣀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠈⠁⠁⠀⠀⠂⠀⠀⠀⠂⠈⠀⠀⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⢬⣽⣿⣿⣿⣍⣉⣉⣛⡳⣾⣷⠶⠶⠶⠢⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠖⣚⣻⣦
⠀⠀⠀⠀⠀⠰⣲⣤⣀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⣿⣎⣿⣤⣤⣤⣀⣄⣀⣉⣿⣦⣠⣤⣄⣶⣶⣿⣿⣿⣷⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⡀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⣀⣠⣤⣼⣿⣿⣿⣿⣿⣶⣬⣷⣾⣿⣿⣿⣿⣿⠿⠿⠛⠛⠿⠿⣿⣿⢿⡇⠀⠈⣈⣿⣿⣿⣿⣿⡿⠿⠿
⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠂⠈⠉⠉⠉⣁⠾⢁⣠⣴⣿⣿⣿⣿⢿⡅⢹⠘
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠈⠁⣠⣵⣶⣿⠟⡛⠻⠉⠛⠈⠀⢹
⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡧⠤
⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣯⣴⣀⢀⣀⡀⠑⢒⣈⡶⠂
⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠙⠛⠋
⠀⠀⠀⠘⣻⣿⡿⠟⠋⢉⡥⢒⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣤⣤⣤⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⢡⢾⠿⣫⣵⠴⣶⣤⣶⠦⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠔⠋⠻⠿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠿⠿⠿⠀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡀⢠⣴⡶⣦⡄
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣸⣿⣿⣿⣷⣶⣶⣶⢂⣼⣿⠟⣿⣿⡇⢸⣿⣿⣯⣭⣍⠁⠛⣿⣿⡟⠛⠁⣼⣿⢿⣿⣤⡄
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠿⠿⠿⠁⠀⠀⠀⠠⠿⠿⠛⠛⠿⠿⠇⠶⣾⢿⣿⡿⠟⠀⠰⠿⠿⠀⠀⠀⠙⠛⠛⠙⠛⠃
⠀⠀⣸⣿⣿⣿⣿⣤⣤⣤⣤⣤⡄⣾⣿⠇⢰⣿⡿⢠⣿⣿⠿⢿⣿⡷⢀⣾⣿⢂⣴⣾⡿⠿⣿⣷⡄⣸⣿⡗⢠⣿⣿⢂⣶⣿⣿⠿⡿⠟
⠀⢠⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⢸⣿⣿⣠⣿⣿⠃⣾⣿⠿⣿⣿⣍⠁⣼⣿⡏⢸⣿⣿⣀⣴⣿⡿⢠⣿⣿⣁⣼⣿⠏⣨⣛⣛⣻⣿⣿⠆
⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠀⠈⠉⠉⠀⠉⠉⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉










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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢸⣿⠆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠁⠀⠀⠀⢠⣾⣿⣶⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣶⣾⢟⡵⣾⣻⣷⣯⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⣿⣿⡭⣉⠭⣉⠥⡩⢥⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣆⠀⠀⠀⠻⣿⣿⣽⣯⣯⣽⣯⣯⣽⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⣐⣒⡀⠀⠀⠰⣿⣿⢯⣷⣭⣿⣭⣮⢽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣵⣿⣿⣿⣿⣷⣄⠀⠉⣻⣮⣭⣭⣭⣭⣭⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠻⣿⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⡏⠝⣿⡿⠁⠀⠀⠀⠀⠘⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⡋⣵⣶⢰⣿⣿⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢫⣾⣿⡮⣡⣿⡟⡵⠘⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⣿⡿⢟⡣⡿⢟⣫⣴⣿⡏⣼⣇⠺⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠁⠀⠹⣤⡍⡟⣼⣿⣿⡏⣶⣶⡎⣿⣷⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠙⠳⣾⣿⣿⣿⣷⣭⣋⣵⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣦⣤⣴⣤⡀⣀⡀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⢛⠿⣋⣝⣛⣡⣶⣮⡛⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⣩⣴⠿⣿⣿⣿⣿⣿⡿⠿⣇⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡎⣿⣟⠛⠲⣿⣿⣿⡷⢚⢛⣿⢱⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣌⢿⣿⣷⣾⣿⣿⣿⣿⣷⣿⣿⢇⣽⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣜⢿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠜⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⠘⢿⣿⣷⣶⣶⣿⣿⡿⠣⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣬⣛⠿⠟⣛⣥⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⠀⠀⣿⣿⡄⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠨⣿⣿⡿⠿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⣾⣿⠀⠀⠈⠀⠀⠀⣿⣿⡁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢶
⣿⣿⠁⠈⣿⣷⠀⠀⠀⠀⠀⠀⢿⣿⠁⠈⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡅⠠⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⡂⡀⣿⣿⣿⡷⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣾⡷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿
⣿⣿⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⣿⣿⠁⠈⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿⠀⠂⠀⠀⢾⡷⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⣾⣷⠀⠀⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⡁⠈⣿⣿⣿⣿⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⢾⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⢈⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠈⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿
⣿⡧⠀⠀⣿⣿⣿⣿⠄⠠⣿⣿⡁⠈⠀⠀⠀⠀⣿⣿⠁⠀⠀⢀⣿⣿⣿⣿⠆⠀⠀⠀⣿⣿⠀⠀⣾⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠈⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⠀⢈⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⡁⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣾⣿⣿⣷⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠈⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⠀⠀⣿⣿⣿⣿⣷⣿⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⣾⣷⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠉⠸⣿⣿⠉⠁⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿
⣿⣿⠀⠠⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡀⢀⣿⣿⠀⠀⣿⣿⣿⣿⣷⡷⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡿⠀⠀⣿⣿⣿
⣿⣿⠀⠀⠀⠀⣿⣿⠆⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠘⣿⣿⠂⠐⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⠉⠉⠉⠉⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠈⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠩⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⣶⣶⠀⠀⠈⠁⠀⠀⠉⠉⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⢐⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⡉⠉⣿⣿⠀⠀⣿⣿⡀⠀⠀⠀⣿⣿⣿⣿⡁⢈⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠉⢈⣿⣿⣿⣿⡀
⣿⣿⣿⣿⠀⠀⠉⠉⣿⣿⣿⣿⠀⠀⠉⠈⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠉⠉⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠈⢈⣿⣿⠀⠀⠈⠉⣿
⣿⣿⠉⠉⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⠈
⠀⠀⣿⣿⣷⣷⠀⠀⣾⣷⠀⠀⠀⠀⠈⠁⣿⣿⣿⣿⠀⠀⣾⣷⠀⠀⠀⠀⣾⣷⠀⠀⣿⣿⠀⠀⠀⠀⠀⠨⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠉⠉⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠉⠉⠀⠀⣾⣿⣿⣿⠀⠀⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠈⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠉⠉⠀⠀⣶⣶⠀⢈⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿
⠀⠀⠀⠀⣶⣾⣿⣿⡉⠉⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣿⣷⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿
⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣶⣾⣿⣿⠀⠀⠀⠀⠀⠠⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣷⣶⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡖⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⢿⣿⣿⠿⠿⣿⡿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⡿⠿⢷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠿⠛⣉⣤⣴⠆⣸⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠋⣡⡀⢿⣿⣼⡟⢠⣿⣿⣿⣿⣿⠟⠛⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠋⣰⣿⣿⣷⣄⠙⠛⢀⣾⣿⣿⡿⢋⣠⠀⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿⣿⣿⠏⣠⣿⣿⣿⠏⣠⣾⠃⠀⣈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⣾⣿⣿⣿⡿⠛⣡⣾⣿⣿⣿⠏⣰⣿⣇⢸⣦⣿⡆⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣷⣶⣤⠀⠀⠀⢸⣿⡿⠟⠋⣉⣤⡈⠻⠛⢋⣡⣴⣾⣿⣿⠿⠿⠏⢰⣿⣿⣿⣄⠙⠿⠇⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠂⠀⠀⠀⢿⣄⡈⠛⠛⣿⡟⢠⣾⣿⣿⣿⣿⣿⠃⣴⣶⣤⣤⣤⠹⣿⣿⣿⣦⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠿⢁⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠙⢿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⡈⠛⠟⠛⣡⣾⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠸⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣀⣤⣤⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠹⣿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣙⣉⠠⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⠿⠛⢁⣌⡙⠛⠋⣡⣾⣿⣿⣦⣈⠛⢿⣿⣿⣿⣿⣿⣿⣷
⢰⣿⣿⣿⣿⣿⣿⣿⣟⠻⠿⠿⠋⣠⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡿
⠈⢿⣿⣿⣿⣿⣿⣿⡿⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⠿⣿⣿⣿⣿⣿⣷⣦
⢀⣼⣿⣿⣿⣿⣿⡉⢠⣾⣿⣿⣯⣤⣤⣍⡙⢻⣿⣿⣿⣿⣿⡟⢋⣡⣤⣤⣼⣿⡷⠀⣼⣿⣿⣿⣿⣿⣿⠂
⣾⣿⣿⣿⣿⣿⣿⣿⠄⣿⣿⣿⠟⠉⢉⡉⠛⢿⣿⣿⣿⣿⣿⡿⠛⢉⡉⠉⠛⣿⡇⢸⣿⣿⣿⣿⣿⣿⡁
⠹⣿⣿⣿⣿⣿⣿⠟⢠⣿⣿⣿⣾⣷⣌⣡⣶⣼⣿⣿⣿⣿⣿⣧⣶⣌⣡⣾⣷⣿⣷⣄⠛⢿⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣶⣦⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣴⣾⣿⣿⣿⠿⠃
⠀⠀⣩⣿⣿⣿⣿⣿⣿⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠛⣉⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣷⡀
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢻⣿⣿⣿⣿⣿⠃
⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣷⣤⣉⣉⣉⣤⣾⣿⣿⣿⣿⣿⠏⣠⣾⣿⣿⣿⡉⠉⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠃⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⢀⡈⠻⢿⣿⣿⣿⣿⣿⣿⠿⢋⡀⠀⠀⠀⠈⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣦⣌⣉⣉⣉⣉⣤⣶⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠚⠛⠛⠋⠉⠉⠛⠛⠛⠓⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠶⠄⣀⠀⠸⠀⠀⠒⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠹⡇⠀⠀⠀⣠⡴⠯⠚⠉⠀⠉⠙⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡤⢤
⠀⠀⠀⣀⡀⡀⢀⣀⣰⠲⢾⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣷⣰⠶⢤⠴⠒⠲⢤⣀
⢀⣴⣭⣍⡑⠀⠉⠀⢡⠘⣾⠃⠀⠀⠀⢠⣀⣀⡼⢤⣤⠚⠀⠀⠀⠘⣿⠀⠈⠁⢀⠞⠉⠉⠙⡀
⠎⡠⠤⠤⣙⠀⠀⡄⠀⢁⣿⠀⠀⠀⠀⠀⡤⠈⠀⠀⠁⠐⠄⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠄⠠⢵
⠀⡆⠀⠀⠀⣁⣇⡇⠀⠀⠻⣤⣀⡀⠀⢀⡑⠚⠤⠤⠜⢁⡠⠀⢂⣿⠀⠀⠀⢀⣌⠐⠂⠉⠙⡞
⠀⢿⠀⠀⠀⡏⠁⠉⠀⠀⠀⠘⣇⠠⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡞⡿⠀⢠⠔⠉⠈⣧⠀⠀⢠⡇
⠀⢸⡄⠀⠀⣇⠀⠀⠀⠰⡀⠀⠹⣆⠫⠀⡀⠀⠀⣀⠨⠁⠀⠀⣹⠃⠀⣼⠀⠀⢀⡏⠀⠀⢸⠃
⠀⢸⠇⠀⠀⠈⠦⣀⠀⠀⠹⣄⠀⠘⢧⡉⠀⠀⠀⠀⠀⠀⠀⣶⠏⠀⢀⡇⢀⡤⠚⠀⠀⠀⢻⡀
⢠⠇⠀⠀⠀⠀⢀⠈⢣⠀⠀⠙⣦⠀⠀⠙⠚⢷⣤⣤⣤⡴⠞⠁⠀⢀⣾⠁⢀⠀⢄⠀⠀⠀⠀⢱
⠇⠂⠀⠀⢉⡀⠙⣤⠂⠀⠀⠀⠈⢷⣄⠀⠀⠈⠀⠀⠀⠈⠂⠀⢀⡾⠃⠀⠈⡧⠋⠀⠀⠀⠀⠀⡵
⠈⠲⠦⠑⠦⠜⠒⠋⠀⠀⠀⠀⠀⠀⣹⠷⢤⣤⣀⠀⠀⣤⡤⠴⣏⠁⠀⠀⠀⠈⠒⠅⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⢀⡤⠴⠀⠀⢠⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠁⠀⠀⠀⠀⠁⢄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢠⠀⠀⠀⠀⠀⠈⣧⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠉⠀⠀⠀⠀⠀⡀⠃⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣯⣽⣵⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣷⣶⣷

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

⠀⠀⠀⠀⠑⠦⣄⠀⢠⣾⣿⠃⠒⠤⣀⡀⠀⠀⣠⡀⠀⢀⡀
⠀⠦⢤⣀⣀⡀⠀⠀⠙⣿⣥⣀⠀⠀⠀⠉⣑⠚⠭⣧⠔⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠘⠟⠒⢤⠀⠈⠐⠲⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⢄
⠀⠒⠒⠒⠒⠒⠒⠀⣀⠀⢠⣄⣀⣀⣸⠃⠀⠀⠀⠀⠀⣨⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⢠⠖
⠀⠀⠀⠀⠀⠀⠀⣼⠛⠓⠋⠉⠁⠀⠀⠀⣀⣀⣀⡤⣚⠵⠋⠀⠈⠉⠑⠒⠒⠢⠤⠤⠤⠤⠤⠤⣄⣀⡴⢁⣰⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠻⠤⠔⠒⠒⠒⠉⠉⠉⠁⠀⢸⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡄⠀⡸
⠀⢀⡞⢫⠟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡇
⢀⠎⠉⠉⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠃
⠓⠶⠶⠖⠒⠲⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⡀⠐⠀⠀⠀⠀⠈⠁⠔⠲⠊⠀⠠⠀⠀⢠⠄⠀⠀⠀⠀⡐⠦⠀⠀⠀⠄⢶⠀⠠⠀⠀⠀⠀⠁⠀⠶⠀⡈⢠⡄⠀⠀⢀⠠
⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠓⠀⠢⠄⢠⠐⠂⠀⠈⠐⠚⠃⠀⠀⠀⠀⠂⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉
⠀⠀⠸⡄⠀⣄⠀⣌⠆⠀⠈⠉⠠⠀⠈⠀⠀⠀⡀⠀⢠⡄⠀⠀⠀⠂⢀⠀⠀⠋⠀⠋⠑⠂⠀⠀⡀⡖⠢⠠⠄⠄⠁⠤⠸
⠀⠀⠀⠀⠁⢀⠠⠦⠀⠄⢀⠤⠐⠀⡠⢂⠘⠂⠠⢠⢁⠤⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⡉⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀
⢤⡳⠄⠀⠀⠡⠁⠀⠀⠄⠀⠐⠁⠀⠀⠤⠈⠁⠤⠃⠘⠀⠀⠀⠀⠀⠀⠡⣀⠠⠀⠠⠀⠀⠈⠠⠄⠈⠃⠁⠀⠐⠀⡀⠠
⠀⠅⠊⠈⠄⠁⠁⠀⠒⠀⠀⠐⠀⠀⠈⠂⠐⢄⠀⠉⠂⢀⣘⠁⠀⠀⢤⡁⠉⢀⣀⠈⠰⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄⣀
⠁⠀⠀⠘⠁⠀⣠⠂⠀⠀⠀⠂⠁⠄⠐⠤⠀⠠⠙⠂⠚⠀⠁⠐⠀⡿⠀⠀⠀⠀⠀⠀⣀⡰⠀⠀⣀⠠⠄⠀⠀⠁⠀⠀⠑
⡀⠀⠀⠀⠀⡤⠀⠲⠰⠀⠈⠐⠄⠀⠤⠄⠀⠉⠀⢀⢀⡀⠈⠁⠄⠈⠀⠀⠀⡀⠀⢀⠁⠰⠀⠂⠉⠀⡑⡀⢀⠐⢀⡀
⠀⢀⣀⠁⠀⠀⣄⠀⠈⠀⠄⠀⠀⡤⠀⠀⠠⠀⢀⢌⠓⠀⠈⠀⡀⠀⠀⣀⠀⣉⠐⢷⠀⠨⠒⠀⢀⠀⠁⠁⠂⠲⠉⠁
⠀⠀⠉⠀⠀⠲⠄⠁⠔⠐⠀⠀⠂⢒⢾⡀⠨⡀⠀⢠⠌⢁⡄⡼⠀⠀⢁⠁⢤⠀⢀⠠⣄⠀⠈⢈⠉⠀⠀⠀⠀⠄⠂⠀⠂
⠀⠀⠀⠀⠋⠀⠀⢀⠀⡠⠀⠀⠀⢘⠀⠉⠀⠀⠀⠀⠀⠀⠀⢪⠀⣔⠚⠀⠀⠀⠁⠀⠁⠀⠈⠁⠀⠀⡈⡁⠰⠊⢠⡖
⠀⠆⠀⠀⠀⠀⠤⠀⠀⠈⡢⠉⢀⡡⠀⡀⠚⠐⠂⠀⠁⡀⠄⠐⠐⠀⠰⠠⠆⡤⠈⠀⢊⠀⣀⠬⠀⠀⠀⠒⠢⠀⠀⠀⠰
⠀⠀⠼⠃⡀⠆⠈⡈⠀⠀⢀⠀⠀⢀⠀⢠⠀⠀⠠⡤⠝⠀⡣⠀⠀⠀⠀⠀⠀⠀⡀⠀⡁⡀⠀⠀⠈⠁⠃⠀⠀⣠
⠀⠠⠄⠠⠐⠂⠀⠀⠀⣀⠀⠀⠀⠙⠀⠁⠀⠃⠐⢀⠀⠠⡚⠜⢁⠐⠃⠀⠘⡆⠈⡷⠅⠠⠀⠀⠄⠄⡀⠐⠤⠀⠀⣀
⠉⠀⢀⣀⠦⠀⠀⢀⠀⠁⠁⠤⠀⠀⢷⠆⣈⠐⠁⠀⠀⠈⠀⠀⠀⠥⡁⠀⠀⠆⠀⠀⠒⠀⠀⠐⠈⡄⠀⣉⠀⠄⢀⠈⠁
⠀⠀⠈⠀⠄⠁⠀⣨⢡⢀⠤⠀⠆⢠⠁⠀⠈⠀⠀⠀⡐⠐⢀⡀⠈⠈⠀⠀⣀⠀⠐⠒⠂⠐⠀⠀⢀⠆⠦⠁⠒⠈
⠠⢄⠀⠀⢀⠀⠀⠐⠖⠉⠀⠠⠎⠉⠀⠀⠀⠀⠀⠀⣾⠁⠄⢀⠀⡄⠀⣌⠂⠀⠒⠄⠀⡀⠀⠀⠁⠀⠠⠋⠀⢈⠄
⠠⡉⠐⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠄⠀⠉⠂⣀⠘⢃⠀⠀⢀⡙⠁⠠⣄⠀⠸⠇⠢⡄⠀⠀⠀⠀⠀⠠⠀⢀⣁⠐⡄⡥⣘
⢰⡇⠀⠀⠄⠀⠀⢠⡄⠐⠂⠀⠀⠀⠂⠈⠂⠉⠆⠀⠀⠀⠀⠀⠀⡄⢠⠐⠀⠀⠒⠀⠀⢐⠄⠀⠐⠀⠀⠀⣂⡀⠀⠑⢮
⠙⡃⠀⠀⠀⠀⠀⠀⠀⠒⠀⡈⠁⠀⠀⠀⠀⠠⠀⠒⠀⠀⠀⠀⠂⣁⠃⠂⠠⡄⠣⠊⠀⠄⠐⠁⠀⠀⠲⠂⠁⢀
⢹⡇⠒⠀⠐⠀⠂⠢⠽⠈⠀⠀⠠⢄⣠⠀⢠⠀⠀⠁⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣀⠀⠄⠀⡀⠀⠰⡚⠁⠛⠀⠐⡆⠐
⢀⣇⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⢠⡦⠄⡀⠀⠰⠔⠉⠐⠢⠁⡄⠀⠀⠀⠀⠀⠀⠈⡉⠂
⠐⠆⠁⠀⠀⠐⠃⡈⠁⠶⡂⣐⠀⢀⠀⠀⠀⠀⡷⠐⠀⠀⠀⠣⠌⠀⠀⠀⠀⢐⠂⠀⡐⠂⠀⠠⠶⠔⢀⡀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠚⠛⠛⠛⠛⠛⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⣀⣤⠶⠚⣻⠃⠀⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡴⠚⢯⡀⠘⢀⠏⠀⠀⠀⣀⣤⠶⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⠛⣶⠎⠀⠀⣠⠔⢩⣧⡀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣀⡴⠋⠀⠀⣰⠋⢀⡟⠯⢹⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⣄⠀⠀⣿⠀⠀⣀⣤⠶⢯⣀⣠⡴⠞⠉⠀⢀⣀⣰⠇⠀⠀⢳⣄⢠⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣼⠃⠀⠹⡄⠛⠧⢤⡄⢠⡏⠀⠀⠀⠀⢰⡏⠉⠛⠛⢳⠀⠀⠈⠛⠁⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣀⣤⣄⠀⠹⣆⠀⠘⣧⡟⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣤⡼⠃⠀⠈⠳⣤⡈⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠋⠁⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠦⣤⡤⠶⠶⢤⣀⣀⣀⣀⠀⠀⠀⠉⠓⠦⣤⣄⣀⣀⡀⠀⢀⣀⣀⣀⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠈⠉⢳⣤⣄⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⡀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠟⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⢀⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⢠⣤⣀⣠⡴⠳⢤⣤⠾⠃⠀⠈⠳⣤⠀⠀⠀⠀⠀⠈⢹⡄
⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⣤⣤⠴⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢼⡃
⠀⠀⠀⠈⣷⠀⠀⠀⠀⢀⣠⠏⠀⠀⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠈⠳⣤⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⢳⠀⠀⢀⣠⣤⣌⡉⠀⠀⠀⠀⠀⢉⣠⣤⣀⠀⢰⡇⠀⠀⠀⠀⢠⡞
⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⣾⠀⠀⠈⠉⠷⠟⠛⠀⠀⠀⠀⠀⠛⠷⠟⠈⠁⠘⢧⣀⠀⠀⠀⠀⢻
⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣴⠋
⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⠿⣄⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠦⣤⣤⡤⠆⠀⠀⠀⠀⢀⡴⠛⠀⠀⣤⣤⠾⠃
⠀⠀⠀⠀⠀⠙⠒⢦⡀⠀⠀⠀⢀⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠖⠋⠀⠀⡟⠶⣄⡀⠀⠀⠀⢀⣠⠶⣿⠁⠈⠷⣤⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠉⠛⠛⠛⠉⠁⠀⠻


⡖⠒⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠒
⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⢠⠤⢀⠀⡀⠤⠤⠀⣠⠀⢠⠤⣄⢠⠀⠀⡠⠄⠀⢀⠤⠄⠠⡤⠄⡠⢤⡀⡠⠤⣀⢠⠀⡀⢠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⢸⠀⢈⠀⡇⠀⡂⢰⠄⢆⢸⠒⢣⢸⠀⠀⠗⠀⠀⠀⠑⢆⠀⡇⠀⡓⢶⠰⡇⠀⣸⢸⠺⡀⢸⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣷⣶⣶⣶⣶⣶⣾⣷⣶⣾⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣾⣶⣶⣾⣶⣾⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣀⣀⣀⣀⡠⠄⠒⠒⠛⠓⠢⠤⠄⣀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠙⠳⠶⣶⣶⠊⢀⣠⣶⠆⢀⣠⣀⡱⡄⠀⡀⡀⠀⡀⣀⠀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡵⡄⠀⢀⡄⠀⠚⠁⠀⡄⠉⢸⡅⢿⠗⠘⠶⠇⠟⠲⣣⣍⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡸⠆⠀⣀⡼⠀⠸⣇⠱⣌⠁⠀⣉⣉⠤⠴⠖⣙⢆⣩⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣸⠁⠀⠀⣨⠗⠀⢸⠃⠠⠃⠀⠜⡽⠉⠁⠀⠀⠉⢿⣿⣿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⡇⠀⠠⠟⢻⠃⠀⢰⣅⠀⠈⠆⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⢻⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⡇⠰⢲⠀⠈⠳⣄⠀⠀⣠⠏⣰⣿⣷⣶⡖⣶⣶⣶⡾⣤⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⡎⢠⡄⠀⠀⠀⠀⢿⣄⠈⣆⡄⠀⠘⠄⠀⣵⢒⢻⣿⢻⢻⣷⣿⡿⠁⣸⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣤⡼⠧⠈⠃⠀⣠⠀⠀⠀⠈⠃⠹⠅⠀⠀⡄⢠⡏⠈⣾⣿⠇⢸⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠴⠶⢶⡄⠀⢀⣿⣀⡀⠀⢀⠀⢀⠀⠀⣼⡁⠸⣧⣦⢹⣿⠸⢿⡉⣿⠇
⠀⠀⠀⠀⠀⠀⣠⠖⠠⠥⠤⠤⠖⠒⠳⡄⠻⢦⣌⠙⠻⣿⠀⠸⡆⠸⣠⡆⠀⢻⡀⠻⡇⢼⣽⣷⣿⣿⣥⣆
⠀⠀⠀⠀⠀⠸⣡⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠳⡄⢹⡇⠀⢿⣄⠁⠸⣄⡄⠁⢀⠉⢸⣷⣤⠤⠤⠤⡔⡴⠒⠐⠒⠲⠖⠀⠶⠚⣉⣑⣢⡴⠶⠶⣄⡀
⠀⠀⠀⠀⠀⣰⠇⣼⠃⢠⡇⢠⣤⠀⠀⢹⣆⡀⠀⠀⡀⠈⡇⠀⣾⠋⣀⣀⡙⠃⠀⠘⠂⠀⠻⠷⠤⠤⠔⠁⠀⠀⠀⠀⡄⢄⠀⠀⠬⣄⠀⠀⠈⢳⣤⣀⡬
⠀⣠⠶⠦⣾⣿⢀⣿⠀⣾⡀⠸⣿⡆⠀⢿⠉⠉⠙⠲⣣⠀⠃⠀⠁⡴⠋⠉⠉⠁⢀⣤⠖⢀⠼⠀⠀⣀⠀⢴⠀⢀⣷⢠⡗⢸⣷⠷⠶⠙⠿⢶⣄⢠⠦⣹⠟
⡀⡁⣠⠞⠛⢃⣼⣿⢰⣿⡇⠀⡏⡄⠀⠈⡇⠀⠀⠀⠸⣄⣀⣀⣀⣁⠀⣀⡤⠶⠿⠏⠀⠀⠒⠒⠚⠉⠁⠈⢶⣾⣷⢂⢄⠸⠟⣤⡀⢠⡖⢤⠿⠿⣦⠟
⠷⣷⢃⡀⡔⠉⠁⢰⠟⠋⠀⠀⠈⢮⠚⣨⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣸⣥⠾⠛⠿⠿⣇⠎⠀⠘⠉
⠀⠸⠺⢶⠇⣄⠀⢸⣀⠀⠀⣠⣼⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠘⠃
⠀⠀⠀⠀⣶⠿⣶⣾⣏⢿⣻⣛⡵⠋
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢠⡄⠀⠀⠀⡠⠄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠁⠀⠀⠀⠀⢠⣿⠀⠀⣙⠒⠖⠀⠐⠤⢖⡌⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠉⠈⠐⠀⠀⠀⠈⠁⠀⠘⠛⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠈⠀⢈⠀⠀⠀⠀⠀⣐⠦⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⣀⠤⢒⡀⣴⠖⠷⠤⠖⠒⠛⠺⣾⠀⠘⢦⡀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⣷⠘⠳⠖⠖⠲⠲⠆⠚⠁⠀⣾⡀⠙⣦⢴⢪⣡⣾⣷⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢸⡝⣆⠀⠀⠀⠀⠀⠀⠀⢀⡼⣹⠁⣴⣧⢾⡹⣟⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠰⣆⠀⢸⡇⢻⡷⠄⡀⠀⢀⣤⠐⡿⢁⡏⣸⣯⢧⢃⡵⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢹⠀⠀⡇⠀⣧⠞⠻⣤⣼⠉⣻⠅⢸⢃⣿⣧⣿⡞⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⠀⠀⢻⡀⢹⡄⠀⣿⣟⣷⣃⢀⣾⣾⣿⣷⠿⢸⠇⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⣌⡇⠀⡾⣿⣻⣼⢋⢀⣽⣯⢀⢠⣏⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⢸⡧⠖⠦⢤⡀⡄⢸⠀⠀⠀⢸⠏⣷⠀⡿⢸⣿⠻⡟⠚⠉⠀⣼⢹⣸⣒⠉⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⣰⠋⢧⡞⣦⠀⠀⢸⡀⣿⠀⡇⢸⡇⢰⠇⠀⢀⡖⣏⣸⠈⢦⢱⡄⠀⢸⡇
⠀⠀⠀⢀⣀⣀⣸⣇⠀⠀⣿⠀⠒⢧⣼⠀⠀⣘⣧⣿⣀⣇⣈⣧⣿⠤⢤⣼⣼⠉⠀⠀⢸⣇⡇⠀⣼⣀⣀⣀⣀⣀
⠀⠀⠀⣶⡯⠩⠷⠭⣿⣖⣿⣦⡉⢀⣼⣛⡟⣡⠖⠂⠀⠉⠁⠀⠀⠸⡆⠙⣿⣄⣀⣀⣼⣯⣴⣾⢏⡁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢤⡉⣠⠎⠀⠉⠛⠛⢩⡟⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⣹⠏⠉⠉⠉⠉⠉⠙⣦⠐⠀⠀⡀
⠀⠀⠀⠀⠐⠩⢭⡞⠑⠃⠀⠀⠀⠰⠾⠿⢿⡶⣤⠭⢍⣉⣀⠀⠠⠖⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠼⣧⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⢲⣤⣬⣭⣍⡉⠉⠉⠉⠉⣉⣭⣭⣤⣤⠒⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠼⣇⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣀⠔⢤⠲⡚⠉⠹⢻⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠘⢧⡀⠀⣤⣀⠀⠀⢸⡆⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠉⠙⡖⢢⠔⢄⣀⡀
⡔⠃⠀⠈⠆⠈⡧⠀⠞⠂⠸⡇⠀⣠⡶⠋⠁⠀⠀⠀⠀⢀⣀⠀⠀⠙⠶⠏⠉⠛⠲⠾⢃⣀⠀⠀⠈⠙⠓⠶⠦⣤⠤⠀⡿⠀⠚⠆⠀⡋⠀⠎
⠧⠤⠄⠤⠈⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣉⠻⠂⠀⠀⠀⠀⠀⠀⠺⢋⣁⣀⡀⠀⠀⠀⠀⠀⣿⠀⢸⠃⠀⠀⠀⠀⠀⠈⠠⠀⢤
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⡇⠀⠀⠀⠀⠘⠋⠙⠋⠙⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠉⠙⠀⠀⠀⠀⠀⣿⣀⡏⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⢹⠀⠉⠓⠲⠤⠤⢤⢀⠀⡞⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⡆⢀⡀⠤⠤⠴⠒⠛⠁⢸⠂
⠀⠀⠈⡷⢄⠀⠀⠀⠀⠀⠈⠑⡙⠦⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠞⡑⠁⠀⠀⠀⠀⠀⠀⠴⡏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠰⡀⠈⣧⠀⠀⠀⠀⠀⠀⡤⠰⢤⣤⡤⠤⠤⣤⣤⠴⠢⡄⠀⠀⠀⠀⠀⢠⡏⠀⡴⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠙⢄⠘⢧⠀⠀⠀⠀⠀⠹⣠⠎⠀⠀⠀⠀⠀⠈⢦⡸⠁⠀⠀⠀⠀⢠⠟⢀⠜⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠲⣆⣀⣀⣀⡀⢀⣙⡾⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣞⡁⠀⣀⣀⣀⣀⡴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⠝⠋⠀⠀⠀⡾⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⠋⡀⠀⠀⠈⠛⢅⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠈⠉⠀⢤⠒⠘⢺⠃⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠒⠚⠛⠉⠁⠀⠘⣦⣗⠐⠋⢢⠄⠈⠁⠘⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢂⠟⠂⢠⠜⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡄⠀⠘⠄⠀⠚⠑⢢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠰⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣄⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠉⠛⠉⠙⠋⠉⠉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠢⠇⠀⠀⠂⠀⠀⠇⠀⠀⠦⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣁⣠⡀⣀⣀⣀⡄⠀⠀⣅⣀⣀⣀⣠⣀⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⣀⣀⣀⣀⡽⠀⠘⣅⣀⣀⣀⣀⣀⣀⡁
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⢧
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⣠⣾⣷⣶⣾⣿⡿⣧
⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠉⠉⠉⠁⠀⠘⣆⠀⠀⣀⣠⠤⠶⠒⠲⢦⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣈⡛⠭⠭⠐⠂⠀⠈⠉⠉⣡⣄⠀⢰⠃⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣀⣀⣀⣈⣁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠸⡄⠀⠀⣸
⠒⠦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠭⠟⠁⢹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡗⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡉⠙⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢷
⠀⢤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⡼⠁⣠⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⣀⣀⡀⠀⠀⠀⣀⡼⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠟⢀⡞⠁⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⢠⡞⠁
⠀⠀⠀⠀⠤⠴⠒⠒⠛⠉⠀⠀⠀⠀⠀⠙⠿⢭⡥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠈⢻⡙⣏⠛⠛⠛⠋⠉⣿⠳⡄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣄⣀⡀⢀⣀⡷⠟⠀⠀⠀⠀⠰⢧⣀⣹⠀⠀⠀⢀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣶⣶⣾⣯
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠿⠿⠿⠿⠿⡿⠋⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⠘⠋⠁⠀⠀⢀⠀⠉⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠁⠀⠀⢀⣠⡄⠀⠈⡔⢤⡄⠀⠀⢷⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⡙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣻⡌⠋⠁⠀⠀⢰⠀⠈⠓⠶⠇⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡀⠀⠀⠚⠛⢦⡀⠀⠀⠀⣠⠔⠀⠀⡟⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠯⣉⠉⠙⢺⣷⡄⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⣼⠃⠀⠀⠀⢘⣿⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠉⠙⣶⡞⢹⡛⢶⣄⠀⠀⠀⠄⠴⠰⠀⣠⣾⠃⠀⠀⠀⠰⢹⣿⢷⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⣹⠟⠇⡀⢸⠇⠀⠈⢹⡗⠶⠶⢶⠛⠻⡍⣿⢷⡄⠀⠀⠠⡞⢷⣂⠨⢞⡥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣻⠴⠷⣶⣬⠀⠀⢰⡟⠀⡟⠀⠈⡇⠀⠀⠈⣧⠙⣦⡀⠀⠹⡎⢁⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡾⡗⢹⢰⠀⡇⠘⠧⣴⠟⠀⠀⢀⡀⠀⢰⢹⡆⢳⡈⢦⣧⡿⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⢀⢾⡇⡇⢸⢸⠀⣿⢸⠀⣿⠀⠀⠀⠹⡜⣤⠆⠘⠑⣺⢃⡈⠻⡀⠀⣸⠃⠸⡆
⠀⠀⠀⠀⠀⠀⣸⢸⡇⣷⣸⠸⠀⣿⢘⣀⣷⠇⠀⢠⡄⣷⠀⠀⠀⣀⡭⠞⠀⠀⣟⢠⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⢸⡇⢻⣿⠀⣀⣿⠞⠋⢡⠀⠀⣸⣇⡝⢀⡴⠋⢁⣴⣦⣤⡼⠙⢾⠁⠀⢀⡇
⠀⠀⠀⠀⠀⢸⣿⢸⡇⡆⢸⠿⣿⣅⠀⠀⠘⡟⡟⢫⡽⠁⡞⡟⡾⢫⡿⠈⠳⣄⠀⢀⣤⠠⠞⠦⣄⡀
⠀⠀⠀⠀⠀⢸⣿⠀⡇⣼⢸⠀⡇⢹⣆⠀⠀⠸⣷⠋⠀⣴⠁⡇⡟⣼⠇⠀⠀⢈⠟⠡⠤⠤⠄⠀⢄⣀⠝⣢⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠈⣿⣦⣧⣜⣸⢀⠃⢠⠹⣦⠀⠀⠀⠀⢠⠟⠀⣇⣵⣿⢀⣀⣤⣼⡏⠒⠒⠖⠈⠉⠉⠀⠀⡟⢿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠙⠛⠛⠛⠿⠾⠿⢷⡄⠀⢠⡿⢾⣿⣿⣿⣿⣿⣿⣿⠀⠹⢦⣀⠀⠀⣀⣀⡴⠚⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣈⣉⣉⣁⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣽⡇
⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠛⢿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠻⣦⠈⠙⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣤⣤⣤⣤⣼⣿⣄⡀⠀⠀⠀⠈⢷⡀⢸⡇⣈⣽⣿⡟⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠙⢷⣄⠀⠀⠈⢳⣾⣿⣿⣿⣿⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⣷⡀⣠⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠛⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⣫⣾⣿⣶⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠾⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣠⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⡏⠋⣽⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⡿⢦⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠈⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣷⣾⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⡄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⣄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣯⣹⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣶⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⣿⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣹⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⡄⠁⠀⠀⠀⠀⠢⡅⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣅⡌⠀⠀⠀⢐⠋⠀⠐⢰⠐⢦⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠂⠀⠴⠋⠑⠒⠂⠈⠑⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠘⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠘⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⣄⢠⠈⠂⠀⢀⠄⠐⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⢈⢁⣼⡇⠈⡀⠈⢀⠀⠈⠁⠀⠀⠐
⠀⠀⠀⠀⠐⠁⢀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠉⠪⢙⡿⠀⠀⡀⠀⠀⠁⠀⠀⠈⡢⠀⠈
⠀⠀⠀⠜⠀⠐⠋⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠂⠀⠀⠀⡠⠂⠀⠀⠑
⠀⡀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⢀⢂⡀⡉⠲⡢⠤⡤⠄⠀⢀⠜⢀⡀⡘⠀⠀⠀⠀⠀⠀⡆⠀⠀⠙⢀⠀⡤
⡈⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠸⡀⠙⠄⢠⠋⠳⡅⠀⠔⠁⡀⠄
⠁⠀⠀⠀⠀⡰⠂⠰⠀⠀⠀⠀⠀⠀⠁⠢⢕⣄⠴⠋⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠒⠄⠤⠠⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠠⡄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⢏⣁⢀⡀⣀⡀⠀⠀⠀⠂⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠈⠁⢀⣧⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣹⣦⣀⣴⠀⠀⠈⠀⠀⢰⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢙⡛⠛⠴⣶⣶⣵⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠉⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢡⠀⡀⠀⡄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⢦⣀⠀⢣⣍⡙⠙⡄⠀⠲⠀⢀⠄⣀⢤⠀⠀⠀⠀⠀⠁⠀⠳⢄⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⢀⠀⠀⠀⠀⠀⠀⢈⡇⠛⠢⣈⠖⠒⢧⠒⠀⢆⡠⠊⠀⢠⠀⠀⠀⠈⠋⢉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠛⠐⣀⡀⠀⠛⠀⠉⡐⠢⣀⠀⠰⢇⠀⢼⣿⠇⠁⢺⠈⡱⣞⢱⡆⢀⠚⠁⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⡀⠉⠀⠀⠀⠀⠀⠄⠀⠀⠐⠂⠐⠌⢂⣀⠈⣂⢷⣜⡧⡤⣼⡤⢕⣦⣶⢭⣛⡄⠋⠭⠁⣰⠆⢁⠀⠀⠄⠐⣆
⠀⠀⠀⠀⠀⠀⢀⡀⠁⠠⠀⠄⠈⠀⠂⠄⠀⠄⠀⠀⢌⠁⡚⢈⠻⣯⣭⠝⢣⠬⢈⣧⣁⠀⣈⣶⡧⢿⠊⠑⠈⡋⠤⠉⠓⢑⠄⡀⠈⠊⡀⠀⠃⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠬⠄⠢⢀⠀⢁⠠⠤⠧⠤⠤⢄⢡⠂⠀⠀⠄⠙⣧⠀⠴⠴⠲⢮⣭⣻⣿⢖⣳⠶⢧⠀⣀⠚⠓⠚⠢⠉⠀⠀⠻⣄⡁⢀⠈⠆⢻
⠀⠀⠀⠀⠀⠲⠠⠄⠄⠀⠀⠄⢈⠀⠀⠀⢀⠀⠀⠀⠈⠳⣤⠀⡀⠠⠜⠛⡆⠄⢨⣜⣍⠛⠋⣹⣀⠀⠤⠔⠈⠂⠀⠀⠀⠹⣆⠐⠀⠈⠀⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠾⠀⠀⡀⠉⠀⠀⠈⠳⡄⠐⠈⠣⣄⠀⠀⠀⠒⡮⢴⡡⠄⠰⠛⠛⠂⣹⠇⠀⠀⢺⡀⠒⠂⠀⠀⠀⠀⠀⠀⠀⣷⠀⠁⢀⠁⠈⢦⡈⢸⡇
⠀⠀⠀⠀⣸⠀⠂⣁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⡙⠀⠀⠒⠀⠡⠄⠀⢈⠿⠁⠈⢻⡇⠀⠀⠀⠀⠀⡠⠀⠀⠀⠤⡈⠀⠀⢄⣀⡠⡄⢺⠁
⠀⠀⠀⢠⡇⠀⠁⠀⠀⠀⠉⠐⡟⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠤⠀⠀⠀⡰⢀⢀⠆⣟⠦⣤⡤⠬⣝⠐⣄⠀⠀⠀⠀⠀⡔⠣⢆⠐⠄⠠⠀⠀⠀⢩⣼
⠀⠀⠀⢸⠠⠂⠀⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠰⠤⠤⠘⣄⢀⠀⠀⢦⠠⡦⠌⠒⠂⡌⠉⠒⢈⡀⠃⠠⡍⠲⠀⠑⠧⠌⠀⠀⡀⠰⣄⣀⡀⢨⡟
⠀⠀⠀⢸⠀⠆⠀⠉⠉⠉⠉⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢦⠰⢦⡆⣄⠀⠀⠴⡇⠁⠐⠆⡀⠚⢄⡆⠀⢈⠀⠀⠐⠀⠀⠀⢀⣾⠁
⠀⠀⠀⢸⡁⡆⠀⠀⠀⠀⠀⡈⢣⡀⠰⢦⣀⠀⠀⢸⡳⣄⡀⠐⢧⡢⠄⠀⠀⠙⢂⠅⠋⢶⠘⢷⣹⠙⠷⣥⠡⠄⣐⡦⢤⠠⠃⠔⠚⢢⡤⣰⠩⡍⢆⢀
⠀⠀⠀⠈⡇⡔⠤⠄⠀⠤⢆⠐⠈⡈⢦⠉⠀⠉⠆⠀⠀⠈⠂⠀⠀⠱⡆⠂⠀⠘⠀⠀⡐⠀⠘⠷⠸⣄⢀⢘⣋⣋⣭⡴⢂⣙⣆⠀⠀⢚⣰⣨⣷⡞⣛⢡⣹⡄
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⢈⡀⡦⢬⠤⢕⣤⡀⣡⣹⠂⡲⣄⠀⠀⠀⠈⠙⠲⢤⡀⢂⡈⢀⣄⣀⣏⠙⠉⠋⠉⢁⡤⣼⣧⡘⣿⠳⣤⣨⣿⣿⠟⢿⡿⠞⠷
⠀⠀⠀⠀⠀⢻⣆⡀⣀⡀⠰⢄⠍⣃⠀⠀⠉⠻⠈⠙⠺⠀⡄⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣁⣺⡉⠁⣹⢈⣃⡐⣉⣛⣁⢰⣋⣉⣙⣉⣁⣁⢀
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢢⠀⠙⠉⡩⠉⠒⣄⠺⣄⠀⢤⠀⠈⠑⠒⠐⠆⢠⡀⣀⣀⡴⠿⢶⠚⢿⠀⠉⠉⠡⠙⢩⣭⠏⠂⢯⡓⠈⠩⠉⣿⣉
⠀⠀⠀⠀⠀⠀⠀⣳⡄⠀⠀⠀⠈⠁⠲⣄⠒⠒⠴⠚⡄⠉⠑⣇⠲⢀⠁⠸⠄⠒⣀⢠⠀⠀⠀⠐⠀⠁⠀⣈⣂⣀⡤⢬⡦⣬⠙⢂⠈⠣⣢⣸⠓⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢪⡀⠀⠀⠀⢀⠈⠓⠒⠶⣥⠱⣤⣤⠷⠈⠉⠩⠁⠈⢠⣀⠄⣀⠁⠀⢀⡠⠴⠒⠉⠁⠀⠀⣌⢆⠈⠻⢅⣇⡀⢌⡀
⠀⠀⠀⢀⣠⣤⣄⠀⠃⠙⢫⡷⢦⣀⠁⠀⠀⠀⠀⠘⠀⠓⠸⡇⠰⢤⣄⣠⣬⡁⠀⣀⣤⠴⠛⠉⠀⠀⠀⠀⣄⠀⠀⠈⠀⠠⠀⠘⡌⠇⢀⡈⠁
⠀⠀⠐⠓⡿⡕⠚⢷⠀⢀⡴⢉⠓⠠⢛⠿⡒⠶⢅⣀⣀⣀⣀⣀⣀⣀⣡⡤⠤⠗⠊⠉⠀⠀⢀⣠⠄⢀⡀⠀⠐⠀⠁⠀⠀⠀⠀⢀⠇⢀⠈⠉
⠀⠀⠀⠀⠈⠻⢶⣦⣷⣭⣀⣠⠤⠚⢅⠒⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠔⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢽⣃⠒⠺⡭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠫⢠⣲⣶⣶⣖⣒⡒⠢⣤⣤⡤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢦⡄
⠀⠀⠀⠀⢰⣦⣄⠀⠀⠈⠉⠙⠛⠛⢻⣿⣿⣤⣤⣤⣄⣨⣿⣦⣤⣤⣶⣾⣿⣿⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠙⣿⣷⣦⣄⡀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡟⠻⠿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠚⠿⠛⡽⠁⣀⣶⣿⣿⣿⡛⠏⠍
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠐⢀⣽⣶⡿⠛⠛⠙⠋⠁⠷
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠒⠀⠀⢀⡀⠀⡀
⠀⠈⣽⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡷⠖⠲⠦⠶⠞⠁
⠀⠀⠘⣻⣿⠿⠛⣋⠭⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣷⣀⣤⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠔⠟⡯⠵⠲⡷⠶⣲⣺⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⢿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣼⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡟⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢳⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⢀⡇⢰⡏⡆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⣄⠈⢧⣤⠖⠒⠛⠒⠛⠉⠉⠛⠳⣧⠀⢸⠇⢀⣠⠤⠴⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣇⠈⠏⠀⠀⠀⠀⠀⠓⡔⠤⢠⠼⠓⣾⠖⠉⠀⠰⣞⠓⢦⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣿⠀⠀⠀⠴⠒⣶⡴⠖⠁⡠⠚⣩⢿⠃⠀⠀⠀⠀⠈⣧⠞⠀⢸⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢴⠃⠀⠀⠀⠐⠋⠋⠀⢠⠎⠀⠸⠁⡏⠀⠀⠀⠀⠀⡀⠀⡆⣠⡟⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠁⣠⡵⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢀⡾⠓⢤⡀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠋⠉⡇⠀⠀⡟⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡼⡗⠀⠀⠀⠀⠀⡼⠀⠀⣀⠙⢦⠀⠀⠀⠀⠀⠀⢀⡟⠀⡼⠁⠀⠀⠧⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⣉⣿⢳⡀⠀⠀⠀⢰⡇⠀⡜⠙⡇⣠⡙⠢⣄⣀⣀⡤⠾⢴⡞⠁⠀⠀⠀⠀⢠⡔⠒
⠀⠀⠀⠀⠀⠀⠀⡞⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⣯⢸⡀⠳⠀⠀⠀⢸⢧⠀⡇⢰⠟⣽⢂⣶⡈⣿⣅⣸⠓⠦⠽
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⣻⡄⢇⠀⠀⠀⠀⣿⠘⣆⢣⠀⢸⢿⣾⠀⣿⠇⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⡤⠖⠠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢙⣦⣤⢤⡤⠒⠉⠉⣝⠹⠀⠾⠧⢤⡼⠘⡆⠀⠀⠀⢸⠀⠈⠻⡄⢿⠈⢃⣀⠀⠀⣀⣠⡤⠤⠤⣤⡀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢷⡀⠹⡄⠀⠀⠸⡇⠀⣴⠆⠘⢷⣽⣍⡽⠛⠉⠀⠀⠀⠀⠀⢙⡆⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠳⡙⣄⠀⠀⢻⡜⢹⠀⢠⡜⣀⡟⠀⠖⢻⠋⠉⠙⠒⠒⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠴⢶⡇⠘⢦⡀⠀⠻⢌⡲⠋⢛⡿⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⡞⢦⡙⠦⣀⠀⠉⠙⠉⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⢀⢀⡴⠊⠳⡄⠈⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣅⠀⢰⠥⠚⠧⣤⣀⡙⣓⠒⢒⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡠⣴⢫⠏⡄⠀⠀⡇⢠⠎⠀⠀⠀⢀⣠⣴⣚⣀⡀⡠⠔⠦⠀⠀⣸⠀⠀⠀⠀⣀⡀⠉⣇⣿⠚⠉⡠⢾⠛⣩⠽⠒⠢⡄
⡼⠁⡇⠸⡄⠃⠀⠈⣷⠋⠀⠀⢀⡴⠋⠈⢻⡿⠉⣽⠁⠀⠀⣟⣩⠇⠀⢀⣴⡚⠙⠦⣀⠙⠉⢷⣐⠃⢣⠞⠁⠀⠀⠀⢹
⢷⠠⠀⢀⡴⢦⡀⠀⠀⠀⣠⠞⠋⠀⠀⢰⡏⢧⡀⠸⣄⠀⠀⠈⠁⢠⠞⠉⠀⢹⡆⠀⠈⠳⣄⠀⠙⠦⡏⠀⠀⠀⠀⢀⡟
⠘⢦⣀⠸⣇⠀⢹⡄⠀⢸⠇⠀⠀⠀⠀⠘⣧⠀⠁⢀⡠⠄⡀⠀⣄⡾⠀⠀⢠⡞⠁⠀⠀⠀⠈⠓⢦⣀⣀⢀⣀⣠⡴⠋
⠀⠀⠈⠓⢮⡛⠋⠀⢀⡾⠀⠀⠀⠀⠀⠀⠈⢷⡀⠸⣤⣠⡇⠀⣿⠤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠉⠓⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⡿⠿⠿⠿⢿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋⠉⠀⠀⢀⡀⠀⠀⠀⠀⠈⠙⢻⣷⣄
⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⢺⣹⠆⣔⢄⠀⠀⠀⠀⢹⣿⡄
⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⣀⣀⣔⢻⣅⢤⡥⠦⠃⠀⠀⠀⠀⣿⡇
⢠⣿⠟⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠑⣰⠇⠀⠀⠀⠀⠀⢰⣿⠁
⣾⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⢀⣿⠃
⣿⣇⠀⠀⠀⠀⠀⠀⠙⢤⣤⣤⣤⠶⠋⠀⠀⠀⠀⠀⢠⡾⠃
⠸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠈⠻⢿⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⡶⠟⠋
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠋⢉⡴⢒⡶⠀⠀⠀⠀⠀⢠⡤⠖⠒⠒⠢⢄⠀⣀⣤⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢩⠒⡄⠀⠀⠀⢀⡴⠛⠓⠀⠀⠀⠀⠉⠁⠐⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⢀⣴⣣⠀⣀⣤⣤⣞⠳⢤⣀⣤⣀⠀⠀⠀⢨⣽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣁⢃⣾⣿⠫⢹⠙⣧⣰⣟⢻⡛⣷⣦⡀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠛⡟⠘⢦⡾⠀⣿⠻⣄⣼⠀⢸⡎⣧⡜⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡇⢻⣄⠀⢀⣰⣿⣄⠀⠀⢀⣼⠁⢸⡇⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠈⠉⠙⠁⠙⠙⠳⠶⠛⠁⠀⢀⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⣄⣀⣀⣠⣤⠴⣶⢦⣄⠀⠀⣼⢃⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠈⠉⠉⠁⠈⠳⢬⡼⠉⢀⣼⠟⢁⣴⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⡀⠀⠀⠀⠀⠀⢀⣤⢿⣿⣟⣛⣦⠀⠀⠀⠀⢀⡼⢁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢹⣦⡀⠀⠀⠀⠀⠀⣴⢻⡇⠘⠟⠛⠛⣲⡿⠟⢻⠁⢸⡏⠉⢻⡅⠀⠀⠀⠀⣞⠳⠞⣯⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠣⡙⣆⠀⠀⠀⢠⡇⠀⢷⠀⠀⠈⠉⠀⢀⡴⠃⠀⢸⡇⠀⠀⢿⠀⠀⠀⠀⠈⡟⣠⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢨⢸⡄⠀⠀⣼⠁⠀⢸⡇⠀⠀⠀⠐⠉⠀⠀⠀⠀⡇⣤⠀⠘⣇⠀⠀⠀⡼⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⢦⣧⠀⢀⣿⠀⠀⣼⡧⠶⠶⠤⢤⣄⡀⠀⠀⠀⣷⣿⡆⠀⣹⣦⣤⢶⡛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠾⣿⢤⡟⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠉⠓⠒⠛⠉⢸⣿⡄⠀⠀⣏⣾⣟⡛⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠴⣿⣷⣿⣠⡶⠋⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠷⣄⣸⣿⡏⠈⢉⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠉⣹⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣟⣿⠛⠛⠁

⠉⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠁⠐⠉⠊⠀⠁⠙⠁⠚⠃⠈⠉⠀⠂⠂⠀⠀⠉⠂⡐⠂⠀⠂⠀⠀⠐⠆⠤⠄⠀⠐⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⠀⠈⠐⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠃
⠀⢀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠁⠀⡀⠀⠄⠈⠂⠀⠀⠠⠀⠀⠄⠀⠁
⣠⢌⡀⠀⠀⠐⠀⠈⠀⠀⠀⠂⠀⠁⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠀⠒⠀⠀⠠⠀⡀⠁⡀⠄⠄⠑⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠠⠔⠄⠀⢀⠀⠢⠠⠀⠀⠀⠐⠄⠅⠀⠀⠀⠰⠀⠁⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠰
⠠⠄⠑⠀⠈⠀⢀⠀⡌⠀⠀⠀⡤⠀⡀⠈⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢀⠠⠀⠄⠀⠐⠂⠄⢀⠄⠢⠐⠀⠀⠀⡐⠀⠐⠠⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡀⡠⡀⠐⣀⢄⠀⢀⠀⠢⠂⠜⠠⣴⡐⠃⠀⡒⠠⠄⠘⠀⠀⠀⠀⠀⠒
⠶⡲⢄⡐⠀⠼⠀⠠⢉⠀⠠⠄⣠⠦⡄⠀⡈⠥⠀⢀⠀⠀⠰⠰⠀⢤⠀⠠⠀⠀⠒⠀⠀⠀⠄
⣨⡖⢠⢰⠄⠔⢳⠒⠤⡿⠤⠆⠉⡀⢄⠆⡴⠠⠆⣠⠼⠮⠐⡁⠐⠅⠐⠀⠀⠀⠠⠄⠀⠠
⡯⢠⢶⠚⣆⣩⡟⣷⣸⠛⣑⣼⠠⣦⠙⠃⢐⡄⠐⠀⠀⡠⣪⡁⠀⠀⠐⠈⠀⠀⠀⠈⠂
⢲⣾⣷⣗⣁⡺⡾⣿⣾⣆⣏⢙⡿⣷⣶⡆⢖⠙⠀⡾⣆⢐⠄⠖⠀⠴⡆⠐⠐⠐⡰⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂
⣿⣽⠷⣟⡿⢛⡼⣭⢖⣿⢶⣷⣿⠙⢛⣃⠘⡡⢷⠗⣠⠁⢖⣀⠀⠐⡓⢲⠀⠠⠀⢂⠀⠐⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⡀
⢿⣻⣟⣝⡷⣿⣟⣷⣿⣿⡏⣽⡴⢏⣾⣿⣽⣷⣶⠇⡝⣉⣵⠄⠸⡡⠉⠌⡁⠀⠂⠀⠡
⣿⣿⡿⣿⣭⣛⣛⣿⣻⣏⣷⣛⣿⣿⣿⣽⣿⣙⠛⡷⢋⣩⣇⢞⠹⣿⠀⠝⠁⡈⠐⠀⡀⠀⢀⠊⠐⠀⠀⠀⠁⠀⠂
⣿⣯⣿⣿⣛⣫⣣⠈⣿⣏⣺⣯⡟⠿⣧⣿⣾⣷⣽⢦⣺⣹⠟⣊⠰⣓⣏⠐⠁⣄⠂⢒⣄⢰⠠⠁⠐⢀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⡿⠿⠻⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡿⠛⣉⡤⣶⡖⢰⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⣡⣴⡈⢿⣷⡿⢁⣿⣿⣿⣿⠟⢋⣩⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿⣦⡌⢁⣾⣿⣿⠟⣠⡖⠘⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿⡿⠋⣰⣿⣿⡿⢃⣼⡟⢠⠐⣦⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⢸⣿⣿⣿⠿⠛⡁⢾⣿⠿⠛⣉⣴⣿⣿⣿⣿⠃⣾⣿⣷⡘⢿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢿⣁⠰⠶⠿⣿⠆⣰⣶⣿⣿⣿⣿⢋⣤⣄⣀⣉⠻⣿⣿⣦⣅⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠈⢿⣿⣶⠸⠏⣰⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣦⠀⠀⠀⠙⢿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣄⡙⠻⠛⣁⣼⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣀⣠⣴⣶⣤⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢿⣿⣿⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣴⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣾⣶⡄
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⠋⣰⣿⣿⣿⣿⣿⣿⣿⠇
⠀⣀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡿⠋⡙⠻⠿⠛⣡⣾⣿⣦⡙⠻⣿⣿⣿⣿⣿⣿⣦⡀
⢰⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠋⣰⣶⣶⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠘⢿⣿⣿⣿⣿⣿⣿⢀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⢿⣿⣿⣿⣿⣿⣦
⢠⣾⣿⣿⣿⣿⣏⠠⣾⣿⣿⣧⣤⣤⣍⣹⣿⣿⣿⣿⣿⣏⣩⣤⣤⣼⣿⡿⢀⣼⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣟⣉⡀⣤⠉⠻⣿⣿⣿⣿⣿⠟⠉⡄⢈⣉⣿⡇⢸⣿⣿⣿⣿⣿⣄
⠘⣿⣿⣿⣿⣿⣏⠀⢿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⡆⢉⣿⣿⣿⣿⣿
⠀⠻⢿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣏⠁
⠀⢠⣾⣿⣿⣿⣿⣿⣄⡙⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣧⠸⣿⣿⣿⣿⣿⡇
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣙⠛⠛⠛⣉⣿⣿⣿⣿⣿⡿⢉⣴⣾⣿⡟⠛⠛⠁
⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⠃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⣈⠛⢿⣿⣿⣿⣿⣿⣿⠟⢋⠀⠀⠀⠙⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣬⣉⣉⣉⣩⣤⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⢃⣽⢿⡏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠻⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣖⣠⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠘⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⠘⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣯⡛⣿⣷⣶⣶⣿⣿⢟⡋⣽⣿⠟⠛⠛⠛⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⠟⣹⣿⣷⣦⣴⣶⣿⣿⣿⡿⢫⣽⣿⣿⣿⣿⠿⢛⣿⣿⢋⡔⠛⣿⣀⣶⣶⣶⣤⡄⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⢘⣵⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠁⢀⡼⠃⠀⣿⣿⣿⣿⡀⢠⣿⣿⣿⣿⣿⡟⠁⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣀⣠⡾⣫⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣠⣾⡗⢀⡾⣿⣿⣿⣿⣠⣿⣻⣿⣿⡟⣧⣿⡄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⡿⠋⠏⠀⠀⠀⠀⣠⣾⣿⡿⠀⠞⣵⣿⣾⣿⣿⣿⣶⣿⣿⣏⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠀⢸⣿⠟⠋⠀⠀⢀⣴⣿⠀⣼⣿⣿⣿⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⢸⠏⠀⠀⠀⣰⣿⣿⣿⣾⣿⣿⣿⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣁⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠛⣿⣿⣿⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⠀⢠⣿⣿⡏⢀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣿⣿⣿⡇⣿⡀⢹⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⢸⣷⣄⡙⠢⢹⣿⢿⣿⣿⣆⠛⠉⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣦⣌⡙⠾⡇⠀⠀⠀⠈⠶⣄⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠃⠈⠻⣤⡈⠉⠳⠿⠷⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣦⣤⣠⣽⣿⣿⣷⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠮⣍⣛⣛⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡴⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣹⠆⠣⢤⠜⣠⠤⠤⣠⠖⡒⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠈⠀⠀⠀⢀⡼⠀⠀⠀⢀⠾⢍⠓⢄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣳⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⡰⠋⠯⢆⠑⡌⢧⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠒⢦⡀⠀⠀⢀⠜⠂⠀⢀⡜⠓⠰⠷⠐⣠⠼⣺⠄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⢤⣄⣵⠀⠀⠸⣄⡀⢠⡮⠔⣒⡫⢦⡀⡟⠉⠁⠀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡼⠼⠖⠾⠦⣀⠀⢸⠁⢀⣭⠭⠤⠤⠤⠝⣥⢄⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⡰⠋⠀⠀⠀⠀⠀⠈⠳⡘⢦⣈⠳⢕⢤⡛⠒⠷⢸⢸⠀⢰⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠥⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡏⠀⠀⠙⢮⡲⣤⣃⠎⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡀⠀⡖⣆⠀⠀⡞⠀⠑⢤⣀⣠⡀⠈⠛⠁⡸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠛⠋⠁⣠⠞⠁⠀⠀⠀⠀⠀⠓⠤⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣙⣒⣚⣉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠂⠀⠀⡂⡰⠐⠄⠀⠰⠀⠐⠀⠂⠀⠠⠀⠠⠄⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠒⠲⣄⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠩⡷⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢀⣀⠤⢶⣒⣖⣲⣖⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⡀⢀⣄⣤⣴⣂⡀⢐⡏⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⣠⠞⣁⠤⢍⠺⢷⡿⢶⣑⣴⠇⠀⠀⠀⣀⣤⣶⣒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠔⠋⢉⣹⣿⡏⢈⣻⡛⢦⡗⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣧⣬⠷⠾⡻⢯⣜⣫⣶⠟⠉⣀⣤⣤⣴⠗⠋⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠂⠀⠘⠳⠞⣱⣏⡿⠃⡾⠀⠲⠆⢀⡀⠀⠀⠀⠸⣇⠀⠀⠀⢀⣀⣈⣳⣿⡛⠈⠀⠀⠰⣿⡏⠁⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡠⠶⠒⢮⣶⣄⡾⠁⠀⠀⠀⠘⠛⠀⠯⠆⢠⡟⠁⠘⡖⠝⢃⡜⢋⣤⡇⠀⠀⠀⣀⡠⠽⠶⡚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢳⣅⡀⢀⣨⠽⠫⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⠚⠳⠋⠁⠘⡇⠀⠀⡞⠁⠚⢂⡠⢾⡀
⠀⠀⠀⠀⠀⣠⠶⠋⠉⠉⠙⢳⣾⠁⢿⣿⡶⠒⠒⠶⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠟⢚⡧⢃⡠⢤⠔
⠀⠀⠀⠀⣴⠿⣶⣤⡀⠀⠀⠈⢿⣶⡾⠟⠁⠀⠀⢀⣠⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠈⢓⡉⢉⡁⣼⢿⣇⡬⠃
⠀⠀⢀⡾⠁⢀⡾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⢻⡟⠛⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⣤⠀⠀⠀⠀⠀⠀⠸⡍⢻⡟⢡⡗⠛
⠀⠀⢸⠃⠀⣾⢸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⢠⡇⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣤⠀⠀⠀⢠⡗⠤⣈⣹⡄
⠀⠀⠘⣇⠀⠘⢾⡀⠀⣀⡀⠀⠀⠀⣀⣿⡤⣄⣴⠋⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⢧⣈⣁⣀⣀
⠀⠀⠀⠈⠳⣄⠀⠙⠛⠑⢧⣄⠀⠰⣿⠈⠀⢈⣀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠛⠦⠤⠶⠤
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⣀⣺⠃⠀⢻⣃⣤⠟⠉⡶⠳⠞⠛⢃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢧⠙⠚⠋⠁⠀⠀⠀⠉⢧⡀⠉⠙⠛⠛⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⢀⣠⡜⢁⡞⠁
⠀⠀⠀⠀⠀⠀⡟⠁⠀⡟
⠀⠀⠀⠀⠀⢸⠗⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢻⡀⠀⠈⠛⠛⠓⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⣀⡴⠿⣿⣶⣿⣿⠆
⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⠀⠀⠉⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣯⣴⣶⣶⣦⡙⢿⡋
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠶⠀⠀⠀⢸⣿⣿⣿⣷⣄⡀⠀⠀⣰⣶⣾⣿⣿⣿⡿⣿⣿⣟⡷⣿⢳⠀⠓⢤⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡁⠀⠀⢀⡙⠋⠻⠿⢿⣿⡟⠛⠉⢛⣿⣿⣿⠟⠀⠈⠛⠿⠷⠶⠋⠀⠀⠠⣿
⠀⠀⠀⣠⠏⠀⠀⠀⠀⣸⠇⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢹⣻⡄⠀⠀⠀⠀⠀⣾⠋⡷⠖⠒⠋
⠀⠀⠀⢷⡀⠀⢠⠶⠚⢹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣦⣰⣄⠀⠀⠻⣄⢧⡀
⠀⣤⣀⠀⢳⡀⠈⠧⣄⠘⣦⡈⠳⣄⡀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠉⠉⠙⠛⠛⠿⠟
⠰⢬⠏⢣⠟⠓⠰⡔⠮⠻⠘⠿⠀⠀⠙⠒⠦⠿⣄⠀⠀⣤⠄⠀⠀⠀⣠⠾⠁⠀⣠⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣌⠳⡀⠈⠳⠯⣉⡉⠳⠦⠤⠤⣻⠴⠻⠡⢏⠾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠃⠈⠳⢤⣤⡠⡾⡇⢀⠀⡐⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⢀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⡀⠀⠐⠀⢐
⡀⠘⠀⢠⠂⡁⠀⢀⡀⢐⢀⠀⠁⢀⡤⠩⠀⡀⠀⢠⡀⠀⢠⠆⢆⠀⢈⣡⡀⡃⠒⣤⠢⣄⠀⠀⠀⠀⡀⡈⠜⠠⠀⠁⠠⠔⢀
⢁⠉⢆⡠⣀⡀⠠⠌⠷⢠⣌⠀⠖⠺⡅⢰⣈⡇⠠⠊⠐⢂⢙⡀⢉⣻⣠⠉⠆⢘⠠⢀⠒⠙⢰⢇⣈⡄⠻⣿⣄⣆⡀⡝⡀⡒
⠾⢶⣴⣦⡭⣝⣳⡴⠓⣟⣱⣞⣷⡶⣾⢦⣽⣯⣾⢡⣲⠇⢲⣋⢠⣍⡛⠾⡟⢘⢸⡽⣿⡾⣯⣯⣟⣤⢓⡿⡙⣖⣿⡖⢧⣇⡀
⣞⣿⣧⣶⣷⣿⣿⣿⣟⣽⣺⣟⣯⣿⣽⡿⣾⣿⣷⣿⣿⣿⣾⣿⣿⣿⣧⣷⣿⣿⣾⣿⣾⣷⣿⣿⣯⣿⣿⣦⣿⣯⣯⣷⣾⣿⢳
⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣟⣽⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣟⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠓⠀⠀⠐⠈⠹⠃⢀⠈⠅⠀⢀⡌⢁⠂⡄⡱⠀⣘⠃⠈⠀⠀⠀⠑⠀⡙⠠⢂⠒⠀⠀⠀⠀⠈⠀⠀⣁⠀⠈⡀⠀⠀⠉⠈⠄⢉⢤⠠⠕⠀⠀⠀⠅⢀⡐⠡⠄
⠀⢐⠟⡸⠂⠒⠡⠂⡁⠡⠰⢅⠈⡠⢀⠒⢀⡀⡃⡀⠀⠒⠀⡴⢀⠂⢠⠀⠀⡀⠄⠠⠈⠀⠀⠈⠀⠑⠀⠰⡁⠀⠂⠠⢀⠀⠂⡡⠀⣈⠀⡀⠤⠐⠚⠰⢄⡠⢠
⠀⢔⢀⡘⠁⣀⠀⢀⣀⢰⠀⠀⡂⢠⣀⡀⠐⠡⢄⡁⠀⠌⠢⠱⠀⠂⠈⢀⠐⠁⠀⠁⠴⠱⠈⠃⠁⠀⠀⠀⠀⢀⠔⠃⠀⠐⢡⠀⠄⡄⠈⠨⠤⠄⢂⠈⡔⠀⠄
⢁⠌⠐⠐⠂⡀⠀⢈⡈⡀⠐⠅⠋⠥⠀⡀⠀⡅⢄⠁⠃⠠⢈⣀⠄⢠⢐⡀⢀⠀⣄⡋⠄⠀⠠⠐⠁⠂⠒⠀⠀⠐⠂⢀⠘⠒⡈⠁⣠⡈⠐⡀⠀⠀⡀⣅⠁⡠⠂
⢡⠀⠉⠁⢠⠄⠀⠁⠂⠉⠄⡉⠎⡈⠈⠰⢒⡀⠀⠀⢀⠄⠀⠔⠂⡀⠀⠁⠐⠂⠨⠀⠀⠴⠐⢈⠀⠀⠀⠀⠀⢒⠰⠰⠄⠃⠁⠈⠠⠁⢂⡁⠀⠀⢁⣠⡊⠦⠄
⡀⠀⠐⠀⡀⣄⡃⣀⠀⠐⠀⠀⠂⠡⢈⠄⢂⠰⠉⠀⠊⢄⠀⠐⢉⠀⢑⠄⠐⠀⡑⠌⡐⠛⠠⡂⠈⠀⡀⠉⠣⠂⠀⠀⢠⢀⡰⠂⠅⠀⠂⠀⠐⢈⠐⠤⢠⠰⠈
⠀⠘⠀⠍⠀⢈⠀⠀⠀⢀⠀⠠⣀⠘⠁⡉⠡⠁⠠⡀⠈⡀⠍⠀⠈⠒⢀⠄⠈⠀⢀⠀⡠⠐⣒⠀⠀⢣⠁⡀⠁⠁⡄⠈⠀⠅⠀⢀⢡⡀⢆⠐⠨⠒⠘⢁⠀⠂
⠴⠀⢀⠁⢀⠁⠀⠂⣠⠄⠑⠈⠀⠈⠀⡀⠒⡀⢀⡀⡒⠖⣀⢀⡄⢁⠈⠀⠠⠀⢀⠄⠀⢰⠆⡀⠀⠀⠂⢀⠠⢀⠂⠀⣈⠁⠈⢀⠣⠹⢄⢀⠀⠈⠉⡀⠀⠀⣀
⠡⠀⠊⢌⠃⣄⠀⠂⠍⠀⢀⠄⢂⠉⢁⠀⠁⠁⠢⠀⠄⢙⠤⠁⠀⢈⡠⠀⠀⠐⠈⠀⡀⠀⡀⠄⡀⠠⠐⠔⠂⠁⢔⠋⠘⢀⠢⠂⠥⠀⢀⠀⠐⡀⡈⠄⠢⠄
⠁⠠⡁⠠⠀⢀⠐⠦⠀⠐⠀⡉⠀⢁⠆⠰⠀⡄⠐⠁⠀⠀⡀⢄⢩⡅⠀⠀⠀⠈⠌⠐⠀⠀⠑⠈⠰⣆⢐⠂⠀⢨⡀⠀⠀⠘⠀⠀⠂⣠⢠⠫⠀⢆⡅⠠⢠⡀⠊
⠠⠃⢉⡀⣉⠰⠀⠐⠈⠀⢼⠀⠀⠘⠂⠘⢠⠁⠄⠀⠐⠐⢂⠊⣁⠀⠌⠀⠠⠀⠆⡀⠀⠀⠀⠀⠀⠴⠌⠀⣠⡆⠀⠀⣢⡀⠁⠘⠡⠀⢂⣰⠄⠘⠂⠀⠐⠈⠈
⢀⡄⢀⠀⠀⠘⠈⡄⠀⠁⠄⡐⠀⠀⢈⠁⢀⡉⢁⢈⡀⠀⢊⠐⠘⠂⢀⠐⠁⠈⠀⡁⢀⠖⡂⠀⠤⡀⠂⠀⠈⠁⠀⠁⠄⣱⠂⠰⠀⡀⢹⠀⠂⠘⠀⠄⠠⠀⠅
⣄⠀⠀⢁⢀⠀⠄⠈⢀⢁⡀⠀⠀⠂⢠⢂⠄⠠⢊⢂⢸⠋⠑⠀⠁⠈⣤⢀⣄⢢⡒⢒⠃⠠⠀⠀⠲⠐⠈⠀⠀⠔⢇⠢⠈⠀⣌⠨⡅⠀⠇⠠⠈⠄⠀⠣⠀⢠⡀
⢈⡠⠄⠄⠤⠨⠖⠈⣨⠀⠑⠤⠙⢐⠁⠈⠀⢀⠴⠀⢃⠈⡀⠀⡀⠠⠈⠀⣠⢨⡀⠘⠀⠈⠂⠅⡀⠇⡴⠈⠀⡀⠐⠀⢤⠀⢇⠀⣂⢠⠍⠁⠐⠅⡁⠔⡀⠀⢀
⡍⠄⠀⠂⢁⠔⠄⠁⠁⠠⠀⠀⠄⠀⠎⠀⠀⢂⠀⠀⠠⣀⠅⠂⡅⠁⡠⠀⢀⡀⠀⠢⡠⠀⠀⠦⣘⠂⡐⠐⢤⠈⣐⠤⡈⢀⠀⠀⠀⠀⠜⠁⠀⠂⠄⢀⠁⠠⢈
⠈⠀⠀⡑⡈⡂⠌⡀⡬⢙⠒⠢⠐⠃⠊⠁⠀⢂⢤⠀⢁⠀⠀⠀⠃⡀⠈⠀⠁⠆⢁⠈⡂⠘⠔⠄⡀⠀⠀⠞⠅⠐⠀⢅⠀⠈⠋⠐⠀⠒⠠⡀⠉⠄⠈⠥⠴⢀⠄
⠔⠂⠄⠐⣄⢀⢠⠀⠀⢰⠀⠀⠀⢅⠀⢢⢉⠂⡂⠀⠆⡴⠀⠠⠈⠁⠨⠀⠲⠠⠈⡀⠁⠀⢠⠀⠡⠈⠆⡠⠄⠀⠀⠒⡃⠲⠀⠆⠀⠈⠈⣀⠀⢲⠀⢦⡤⠒
⠁⠀⠀⢈⡆⡠⠁⠈⠁⠀⠈⢁⠓⠀⠄⠀⡁⠈⠅⠃⠤⠀⠢⡈⡠⠀⠊⠀⣀⢰⣦⠃⠒⢘⠂⠀⠀⠈⠐⡰⢤⠉⠐⠀⠁⡰⣀⠀⠂⢬⣰⠀⠂⠐⠂⠀⠠⡄⠁
⠤⢀⠈⡌⢌⠠⢀⠁⣀⠐⡂⢑⢠⠈⠀⠈⠂⠐⠠⠠⠞⠀⠀⢁⠀⣈⠆⠡⠀⠆⠁⠀⠀⢀⡁⡀⠂⠀⣈⡋⢰⠌⢀⠀⠲⠄⠀⠆⡨⠉⠀⠁⣠⢀⠁⠄⠁⠀⠈
⠁⡈⢰⡉⢠⠆⠀⠐⠀⡄⠉⠀⡀⠀⢂⢀⠀⠄⠠⠀⠂⠀⠲⠀⠆⡔⢂⠰⠆⠀⠀⠀⣈⠀⢠⠐⡀⠘⠠⡔⠂⣄⠐⢀⡂⢉⠢⠁⢀⠀⠀⠨⠑⠄⠀⠌⠀⡀⣁
⡤⢏⠀⠈⡀⠀⠠⠀⢀⠐⡋⠡⠈⠠⠁⡒⠂⠀⠀⠀⣡⠘⡓⡂⠴⠢⠄⠀⠘⠀⠊⠄⠄⡔⠲⠂⠀⡐⠁⠄⠐⢃⠁⠀⣀⠀⠂⠸⡄⠁⢠⢂⠌⠀⠐⠠⠔⢇⠜
⡤⠀⠀⠢⠀⡁⠀⢀⠈⡀⢡⠆⠃⠀⢰⡐⠀⠀⠁⠀⠁⠀⠐⠜⠀⢒⠈⢀⡐⢀⠄⠀⠐⠰⢡⠋⠰⠀⡂⠀⠁⠊⡁⠠⠀⠠⡒⠁⡂⠀⠘⠩⠐⡁⠡⠀⠄⠠⢘
⠎⠀⠀⡁⠄⠐⢀⡠⠐⠀⠀⠁⡅⠀⠀⠊⠐⠐⠈⢂⡀⠳⣠⠀⠁⠄⡀⠀⠄⡀⠤⠀⠠⡀⢀⡖⠴⠃⠀⣢⡄⠐⠄⠌⢀⠉⠀⢰⠀⢂⠈⡃⢀⠀⠔⠂⠀⠂⡁
⠡⣟⠀⡄⢪⡅⡈⠀⠐⠉⠀⠰⠆⠰⠂⡄⣁⠐⠲⠦⠌⠜⠃⢀⠀⠢⠀⠆⠀⠀⢤⠠⠁⠀⠔⠲⠰⠐⡁⠀⡈⠂⠼⠀⠁⣤⢀⠀⠈⢁⠈⠱⠤⠦⠐⠈⠀⠀⠑
⠐⠗⠄⠂⠃⠀⢀⠀⢀⠁⠀⠠⠀⠀⠀⠘⡀⢂⠀⠈⠀⠀⢁⠁⠈⡀⠠⠂⠁⠌⠈⠀⠅⡠⠁⢀⠳⣜⠃⠁⠠⠄⠀⠘⢘⠀⠀⢮⠠⣀⠹⠁⢆⠠⠀⡑⠐⠈⡑
⠦⣟⢀⣱⢁⣀⣀⠀⡀⠹⠀⡀⢁⠀⣡⠂⠁⠊⠤⡐⣃⠀⠀⠈⣈⠀⢈⡈⣁⠀⠀⢀⠀⠀⡄⠀⠀⢀⡀⠀⠃⠐⠃⡈⠀⠂⢀⣀⡀⠠⠀⠈⣀⠐⡀⠃⠀⠀⠣
⠐⡟⢡⠑⠐⠀⠀⢠⠀⠀⠀⢀⠀⠉⠄⢀⢀⠐⠂⣒⠠⡁⠌⠨⡀⠀⠆⠈⢠⡕⠈⠊⠀⡄⢀⠀⢀⡀⠠⠔⠇⠩⠀⠀⠠⠀⢁⠀⣁⠆⠘⠰⠌⠀⢀⡀⠰⠀⠒
⠑⡿⡄⠀⠀⠁⠈⠂⠠⠀⠐⠌⠀⡤⠂⠀⡈⣁⠡⠌⠁⠉⠀⠂⢀⠠⡐⠀⡈⠀⠠⠁⠄⠁⠈⠀⢀⠈⠤⡠⢁⡀⠀⠄⠂⠀⠀⠂⠦⡀⠀⠲⠀⠆⠈⠀⠀⠂⡀
⠐⣟⠸⠂⠐⠠⠠⠀⠘⡁⠄⡊⠂⠠⠀⡄⠐⠀⠀⠨⠠⠀⠠⠀⠁⢰⡠⡨⠀⠁⠋⠀⡦⠠⠁⠀⠀⢀⠠⠠⢀⠀⣤⠚⣄⠎⠁⠄⠀⡐⠑⠸⠀⢉⡄
⡀⠻⢀⠀⡄⠠⠁⡄⢄⠀⢔⠁⠑⠀⠀⡁⠐⠉⠀⠀⠀⣂⡀⠀⠆⢡⠀⠐⡀⢠⢀⠰⣉⡀⠁⡀⠀⠀⠠⠈⠀⠐⢰⠀⠂⢀⠀⠀⠒⠀⢈⡄⠀⢀⠒⠈⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⢰⡇⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣾⣿⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡜⣡⣾⣿⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣴⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⣄⢠⢣⣿⣿⣿⣿⠟⠋
⠀⣸⣦⡄⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⡤⢊⣡⣴⣶⣿⣿⣿⣿⣶⣝⣿⣿⣿⠟⠁
⠀⢿⣿⣿⣿⣿⣷⣦⣭⣑⣒⣲⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢎⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⣤⣥⣵⣿⣿⣿⣿⣿⣿⣿⡇⠀⡰⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠉⠛⠙⠛⠉⠉⠀⣹⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢔⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⣿⣿⠏⠀⣿⣿⣿⣝⣿⣿⡟⣿⣿⣬⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣇⠀⠐⣿⣿⡿⢿⣿⣿⣿⣝⠏⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡞⣿⣿⣿⣿⣿⣷⢠⡀
⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣯⠝⠋⡀⠀⠹⣎⣛⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⢸⠁⣽⣿⣿⣿⠹⡟⢰⡇
⠀⠀⠀⠀⠀⠀⠸⠛⣿⡿⠇⠀⠐⠀⠀⠀⠹⠿⠟⠻⣿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠃⡟⠀⢱⣿⠻⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄⠀⠀⡌⢠⣿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⡸⠁⠀⠀⢳⡀⠙⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠻⠿⠿⠃⢀⠜⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⡟⠁⠀⠀⠀⠀⠙⢦⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠲⡟⠲⡞⢇⢀⣼⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡄⠀⡘⣦⡙⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⡿⣉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⡼⣿⣿⣿⣿⣿⣧⢻⣿⡿⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢤⣴⣾⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣎⠭⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠐⠀⠘⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠺⣻⣿⠿⢿⣛⣻⣯⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠾⣄⣀⡰⠤⠜
⠀⠀⢪⡀⠀⠀⠀⣀⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⠛⠉⢩⢤⡍⠘⠩⠞⠁
⠀⠀⠀⠈⠉⠓⠚⠁⠰⡀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠼⠤⠂
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⡶⢿⣿⣿⠿⢿⣟⣛⣛⡁⠠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢟⣵⣾⠟⣱⡿⡩⢖⣛⣛⣛⣻⣇⣤⢄⡸⣦⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡵⠿⠿⣗⣠⣿⢠⢞⣭⣽⣷⣶⣶⣿⠿⢟⢡⣿⡇⢹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⣛⣒⣚⣻⣿⣿⣿⣧⣵⣦⣄⣛⡛⠿⢿⣿⢿⣛⡍⣼⡿⠀⣟⣛⣛⠇
⠀⠀⠀⠀⠀⣛⣛⣿⣶⣿⣿⣿⣛⣉⣙⣻⣿⡟⣩⡛⠿⠿⠿⢛⣠⢹⣿⡟⣱⡿⢡⢸⡌⣿⣿⡇⣆
⠀⠀⠀⠀⠲⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣛⣻⣿⣿⣿⣣⣿⢏⣼⠏⣠⡟⣸⠇⣿⣿⢡⣿⡆
⠀⠀⠀⠀⢼⠂⡶⢙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⢋⣾⣿⠟⠡⠚⠵⣾⣿⢣⡿⢸⣿⠇⣾⣿⡇
⠀⠀⠀⠀⠈⠀⠃⠈⣿⡮⢻⣿⣿⣿⣿⣿⣿⣿⡇⣸⡿⠿⢛⡅⠠⢞⡻⡿⣎⠋⣾⢃⣿⡟⣼⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⡏⣿⣿⠻⠏⢿⣿⣿⣇⣿⣷⣾⠏⣴⣾⣿⠿⡇⢈⣄⣿⢸⣿⢸⡟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇⣿⣿⣿⣿⣿⣿⠿⠛⣿⣟⠇⣼⣿⣿⣿⠇⢀⣾⣷⠻⣌⠇⠋⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠘⢿⡿⢏⣼⣿⠘⠉⠉⢵⣒⡶⣾⣿⣿⠀⣿⡟⣼⡿⣰⡿⢟⠋⣁⠬⣠⣾⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⢿⡟⠀⠀⠀⠀⠈⢻⣶⡝⣿⣇⢻⡇⣯⠇⢉⡴⢣⡞⣡⣾⡿⢋⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡷⠈⢿⣎⢿⢹⣧⡾⣱⡏⠼⣛⣩⡶⣣⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢿⡌⢻⣎⢇⢿⢳⣿⣿⠿⣛⣭⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡙⢧⡈⠛⠵⡾⠿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣍⡓⠀⠉⢻⣶⣍⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣼⣷⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠤⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠡⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⡠⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡀⠘⡤⢤⠀⡀⠠⠐⠀⠒⢢⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠉⢇⠐⠈⠁⢉⢹⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡟⠛⠖⠒⠾⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠁⡠⠠⢀⢠⠂⠀⠒⠀⠀⠄⠀⠐⣄⠀⠀⠂⠳⠶⠖⠒⠳⠵⠰⡈
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠣⡀⠀⢽⡃⠔⠀⠠⠄⠒⣰⠀⠨⡄⠀⠀⠀⠀⣀⠄⣀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠐⠒⠋⠀⠂⠀⠔⠊⠁⠀⠀⠈⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠊⠀⠈⠈⠁⠀⢀⣀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⣁⠀⠄⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⠠⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣒⣔⠀⣨⠡⡝
⠀⠀⠀⠀⠀⠀⢡⠃⠀⠁⠀⠰⣘⠆
⠀⠀⠂⠠⠈⠐⠄⢠⢒⠁⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢄⠀⠀⠀⠀⠀⠀⠀⠔⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠘⠠⠌⠒⠒⠠⣄⠀⢀⠎⠀⢹
⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢸⠁
⢠⠖⡂⠐⠃⠀⠀⠀⠀⠠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠸⠆⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣗⠢⢀
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⢃⠒⠆⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠛⠐⢄⡀⠀⡀⠀⠀⠀⠠⠪⠁
⠀⠀⠈⠀⢸⠋⠀⠈⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠐⡀⢀⠄⠆
⠀⠀⠀⣤⠂⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⢄
⠀⠀⠸⠁⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠴⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠁⠠⣀⡠⠤⠄⠤⠒⠉⠈⢹⠋⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⣀⠀⠀⠀⠀⠀⢸⠂⠄⠀⠀⡠⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡔⠁⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢇⠀⠀⠀⠀⠸⡀⠀⠀⠑⢄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠈⠑⠒⠤⡤⠤⠤⠤⠖⠊⠀⠀⠀⠀⠓⢄⠀⠀⠀⢣⠀⠀⠀⠀⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⡰⢉⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⢇⠀⠀⠀⠘⠶⡀⠀⠀⠈⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠈⠑⠂⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠈⢆⠀⠀⠀⠀⠣⢀⠶⠤⡰⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⢆
⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⢠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢧⡀
⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⡑
⠀⠀⠀⠀⠀⠃⠔⠒⠓⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠦⡀⢤⣀⠜⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣽⣭⣯⣬⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢇⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠀⠀⠀⠀⣀⣀⡀⠂⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⡀⡀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠃⠀⠀⠐⠀⠀⠀⡀⠀⠀⢰⣇⣷⣄⢀⠀⠀⠁⠀⠀⠀⠀⣄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠀⢀⢺⣿⣿⠏⠘⡆⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠈⠂⠀⠄⠠⠤⠄⠂⢣⠀⡄⠀⠀⠀⠀⡠⠀⠀⠀⠀⢢⠀⠠
⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡃⠀⢀⠀⠀⠂⠠⠤⠐⠒⢁⠙⢈⠆⠀⠀⠎⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠂⠀⠈⠉⠁⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⠊⠀⠀⠀⠀⡄⠀⠀⠀⠘⠤⠔⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⢠⠖⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⣀⠀⠄⠄
⠀⠀⠀⠁⠀⣰⣿⣿⣿⠇⠀⠦⠀⠈⠪⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢀⣨⣤⣴⣶⣶⣾⣿⣦⠀⠐
⠀⠀⠀⠃⠘⠛⠛⠿⠋⠀⠀⠀⠀⢰⡀⠀⠀⠃⠀⠀⠀⠀⠀⡔⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⢠⡀⠀⢠⠠⠂⠂⡅⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠒⠂⠘⠢⢤⠂⣁⠀⠀⣌⡲⠤⠼⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠀⠀⠀⢀⡈⡀⠀⠀⢀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠰⠀⠊⠃⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠆⠀⠀⡅⠀⠀⢀⠀⠀⠀⣀⡼⠀⠀⠀⢸⠄⠸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠄⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⣢⡀⢠⠀⢠⠀⢅⠀⣾⠀⠁⠀⠀⠀⠘⠀⡅⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠐⠒⠓⠚⠈⠁⠉⠀⠚⠤⠄⠚⠛⠒⠉

⠀⠀⠀⠀⠺⣟⡛⠛⢲⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠙⢾⡓⠿⠉⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣤⢤⣤⢬⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠦⠉⠃⠉⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢈⣛⣶⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⡏⠉⠛⠿⣍⡉⠉⠛⠛⢩⣭⡭⠿⣿⣿⠳⢶⡶⠶⠶⠒⣶⣤⠴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⡀⠀⠀⢸⡇⠀⠀⣤⣈⡙⢄⠀⠀⠀⠬⣽⣿⡗⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠜⠉⠀⠀⠀⠀⢷⠀⠀⠞⠀⠀⠀⠀⠀⠀⠈⣿⡋⠉⠀⠀⠀⠘⠁⢀⣠⡤⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢻⣻⣿⡹⠀⠀⠀⠀⠀⢸⣿⣿⣳⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠉⠁⠀⡀⠀⢀⠀⠠⠉⠉⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠘⠀⣀⣉⣶⣏⣀⠐⠀⠂⢀⣠⠴⣿⠋
⠀⠀⠀⢀⣾⠇⠀⠀⠀⣸⣿⣦⡀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠉⠓⢦⣄⣀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠘⢧⡀
⠀⢀⡤⠞⠁⢀⡀⠀⢰⡟⠀⠈⠻⣦⡀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⣴⢋⡤⠖⢠⡏⠀⢀⡿⠀⠀⠀⠀⢸⣿⡋⣀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠈⠙⠷⠤⠴⠛⠛⠛⠁⠀⠀⣀⡴⠋⠁⠘⢛⣷⡄⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⢀⣰⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⣢⡀⠀⣨⣿⠿⠒⠛⠋⠛⠓⠶⠶⠶⠶⠚⠛⠋⠉⠛⠻⣦⡀⠀⢀⡐⢿⣿⠑⠀⠈⠛⢦⣄⡀
⠀⢀⡤⠶⠖⠚⠛⠉⠁⠀⠀⠀⢀⣾⠿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⡟⠃⠀⠀⠀⣶⣄⣈⠙⠳⠶⣤⣀⡀
⠀⠿⠧⣤⠖⠀⠀⣠⡆⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣼⡋⠀⠁⢀⠀⢠⣄⡀⠉⠙⢶
⠀⠀⠀⠻⠤⠴⠚⠁⠹⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡅⠈⣳⣄⢈⡿⠶⠖⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣯⣥⣬⣽⣧⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠆⣸⠗⠐⠻⠏⠀⠀⣈⠻⠛⠙⠷⠈⠰⢶⡁⠀⠀⠈⠀⠉⠀⠐⠀⠀⠀⠉⠊⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠘⠁⠀⠀⠀⠀⠀⠉
⠀⡉⠷⠆⠀⠀⠀⠀⠉⠁⠀⠠⢦⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠄⠀⠀⠀⠂⠀⢀⡀⠀⠀⠠⣤⡀
⠀⠀⠀⣀⠠⠀⠀⣤⣤⠀⠈⡃⠀⠀⠀⢠⡞⠀⠀⠠⣤⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠆⠀⠀⠀⠀⠰⠆⠀⠀⠀⡀⢰⡜⢇
⠀⠀⡖⠀⠀⠀⠀⠀⣠⠴⣣⡀⠀⠑⢃⣤⡀⠀⠀⣀⣅⠀⠀⠀⠀⠀⠀⠀⠐⠀⣶⠊⠁⠀⠀⠀⠀⣐⠀⠀⠀⢉⠀⠈⠹
⠸⠟⠐⠀⠀⠀⠀⠆⠐⠶⠟⠁⢀⣴⠂⠈⠁⢀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢈⣀⣴⡆
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢁⠀⠀⠀⠀⠀⢀⡀⣀⠠⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⢄⣿⠃⠀⠀⠉⠉⢸⡦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠟⠉⠀⠀⠀⠀⠈⠀⠙⠃⠀⠀⠀⠀⠀⣶⡾⠀⠀⠀⠀⠘⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠻⡦⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢉⡤
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠤⠀⠤⠀⠀⠀⠀⠐⠠⠀⠀⢰⣤⡀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢺⡀
⠀⠀⢠⡖⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣆⠀⠘⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑
⠀⠀⠈⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠘⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⡀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⣋⠅
⣿⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⢀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠀⢶⡯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⢨⡅⠈⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣯
⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⢠⠎⠀⠀⠂⠀⠐⢂⠀⢀⠄⠀⠉⠉⠀⣒⠋⠀⠀⠀⣈⣿⢁⢀⠀⢀⠀⡁⠲⠏
⣠⠀⠈⠀⠀⡂⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⢀⠀⠠⠄⡀⣤⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠉⠉⠐⠀⠀⠀⠒
⡌⠁⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠲⢶⠄⠈⠀⢀⡸⠧⠀⠀⠀⠀⠀⠀⠀⠽⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠂⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣉⡀⢀⠀⠀⢠⠀⠁⠀⠀⠈⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣤⠀⠀⠀⣄⢶⠆⠀⠀⠂⠈⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⢀⠀⠀⠀⠠⠀⠀⠀⢠⡄⠋⠀⠀⠀⠁⠀⣀⠀⠀⠀⣀⡄⠀⠈⠁⠀⠀⠀⠀⠈
⠁⠀⠀⠀⠀⠈⠃⠀⠈⠀⠀⠀⠀⠀⠀⢀⡰⢾⡠⠊⠁⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣸⣀⠀⠀⠀⠀⠀⡰⣻⡳⣀⠳⣦⣄⠀⢀⣠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⠊⠁⠈⠉⠓⠒⠒⠚⠣⢿⢫⣴⣵⠘⠀⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⡗⠪⠒⠛⠋⠀⢀⠀⠀⢠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢄⣀⣀⡀⢤⣾⠇⠀⠀⠀⠀⠀⠈⡗⠒⠁⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠒⠒⣒⠮⢡⠎⠀⠀⢀⡰⠃⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠀⠛⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⡐⠑⣂⡥⠚⠒⢚⡿⠀⠀⠀⠀⣿
⠀⠀⠀⡄⠰⢆⠀⢸⣀⡀⠀⢠⠃⡤⠤⢀⡀⠀⠈⠒⠒⢊⠠⠒⠉⠉⡧⠖⣋⠀⠀⠀⠀⠀⡿
⠀⠀⡜⠻⠁⠘⠋⠉⠈⠉⠑⡇⢘⠄⠀⠀⢳⠀⢀⠔⠊⠁⣠⠔⢊⠼⠚⠉⡀⢱⠀⠀⠀⠀⡇
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠑⣭⢞⣀⡠⠁⣸⠐⣕⡀⢰⡟⡡⠚⠁⠀⣠⠜⣱⠃⠀⠀⠀⢰⠃
⣤⠇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡤⠞⠁⠀⠀⠉⢹⣞⢁⠄⢠⠚⠁⠔⠁⠀⠀⠀⠀⡞
⡍⠇⠀⠀⠀⠀⠀⡴⢋⠀⢀⣴⠋⢳⣀⡀⠀⠀⢀⡤⠋⠈⠘⢲⠋⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠇⠀⠀⠀⠀⠀⠀⠈⠛⡤⠄⠈⠆⢸⠀⠙⣶⠊⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠸⡀⠀⠀⠀⢀⠀⡸⠳⠇⠀⠀⢸⠂⠀⠀⡽⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠱⣄⠀⠀⠘⡗⠁⠀⠀⠀⠀⠀⠀⢀⡴⠃⣀⡠⠔⠊⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠛⠛⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⢀⣠⣤⡶⠞⠛⠋⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⢤⣹⠀⠀⣠⣴⣾⣿⣿⣷⣦⣤⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣀⠻⡆⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡠⠴⠶⠶⠞⠀⢀⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣆⡃⣿⣿⣿⣿⣿⣿⣿⣿⡟⢒⣛⡂⠀⡀⠀⡀⣄⠒⠁⠄⠀⣸⡵⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠘⠁⣽⣿⣿⣿⣿⣿⣿⠟⠁⠀⠁⠀⠀⠀⠁⠡⠶⠚⠀⠀⢠⢃⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣼⠏⠈⠛⠛⠛⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡻⢊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠙⣴⣏⠀⠀⠀⠀⠀⢀⡤⠛⠀⠀⠀⢀⠷⡄⠀⠀⠀⠀⠀⢠⣇⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠠⣫⣿⠀⠀⠀⠀⣠⠚⣅⡆⠀⠀⠀⢀⣀⠽⡀⠀⠀⠀⠀⣾⠟⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣿⠀⠀⠀⠰⠃⠀⠀⠙⢦⣤⡴⠟⠁⠀⠃⠀⡄⠀⢠⢿⣒⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣴⣈⣹⣶⠠⢀⠀⠀⢀⠀⠀⠀⠇⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢻⣿⣿⣧⢀⣴⣤⠴⠞⠇⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⣀⠀⠲⠄⡀⠄⠀⠀⠤⠔⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⡟⡇⠈⠙⡶⡀⠀⠀⠀⠉⠉⠀⠀⠀⣰⣶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡺⠛⠁⠀⡇⠁⠀⠀⢷⠟⠳⠶⠦⠤⠤⠶⣶⠋⠁⡟⣿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠐⠉⠀⠀⠀⠀⢧⠀⠀⠀⠈⠜⡄⠀⠀⠀⠀⢠⠃⠀⢠⠃⠛⠃⠀⠙⠲⢤⡀
⠀⠀⠀⢠⡶⠾⣝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢠⠟⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⢤⣀
⠀⣰⠏⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠢⠄⠀⣀⡀⠀⣀⣀⡤⢞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠦⣄
⢠⡏⠀⠀⠀⠀⠀⠀⢸⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠙⢦
⣼⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡇⠀⠀⠀⠀⠈
⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣛⣋⣛⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡠⠐⠈⠉⠉⠁⠀⠒⠠⡀
⠀⠀⠈⠀⠀⠀⡔⠐⡄⠀⠀⠀⠈⡄
⢀⠔⠂⠀⢀⠊⠁⠈⣉⠢⠀⠀⠀⠑⢄
⠇⠀⠀⡴⢏⣿⣷⣉⣧⣠⢗⠆⠀⠀⢸⠀⢀⣀⣉⡛⢭⣑⠲⠠⠄⣀
⠀⣡⠆⠘⢯⣭⠉⠉⢩⡼⠃⠀⠀⡠⠊⠀⠀⠀⠀⠉⡱⠦⠀⠤⡄⠐⠡⡄
⠀⡁⠀⠀⠀⢀⠕⠘⠃⡀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠣⡀⠀⢨⠀⢀⡚⡀
⠀⠈⠂⠄⠀⠀⠀⠁⠀⣀⡠⠒⠢⡔⠁⠀⠀⠀⠀⠀⠀⣸⣀⠘⣢⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠒⢢⠀⠀⠀⠀⠄⠙⣄⠈⡣⠹⣇⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢆⡠⠔⠊⠈⠀⠉⠀⠀⠙⠘⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠤⠤⠤⠤⠤⠄⡄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠐⠐⡄⢣⡠⠢⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⡀⠉⠉⠉⠀⢣⠜⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⢃⢱⡀⠀⢀⠤⠂⡰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣖⠳⣖⠣⢄⡐⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣹⠘⣏⠀⠀⠀⠀⠀⢈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠏⡼⣏⡭⡁⢀⢈⡹⢧⠤⠚⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡗⢬⡕⠰⠒⠂⠀⠁⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡙⠋⠁⢡⣡⡀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠱⠀⠀⠀⠀⢣⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⣇
⠀⠀⠀⢀⣀⡠⠠⢀⡀⠀⠀⠀⠀⢀⠠⠔⠒⢉⡡⣤⠈⢣
⠞⠉⠉⠀⡀⠀⠀⠀⠈⠉⠀⠈⠈⢱⠴⠒⠉⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⡂⠀⠀⠂⠀⠀⠂⢢⠀⠀⡴⠁
⠀⠀⠈⠀⠄⣀⡀⠀⠀⣨⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⠀⠀⠀⠀⢀⣠⣄⢦⠤⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⠏⠉⠑⠒⠛⢻⣶⢊⠀⢠⠄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠁⠀⠀⠀⣰⠆⠙⠒⠁⠀⠸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠒⢋⡥⠀⠀⠀⠀⠀⡇⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡄⠀⢀⡞⠀⠀⢀⠗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡼⠀⠀⠀⡜⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⢹⡄⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡏⠀⠀⡼⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⣇⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡰⠁⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⢹⠀⠈⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢹⢰⠃⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⡟⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣼⢸⡀⠀⠀⠀⢠⠇⠀⠀⣀⠴⠒⠚⠛⠂⠀⠀⠈⢧⠀⢠⠇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢳⣄⠀⢠⠎⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡜⠀⠀⠚⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠈⠓⡟⠲⣤⣄⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢴⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⣉⠀⣿⠀⢠⠧⢄⢘⣦⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⣿⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⡉⠁⠀⢸⣄⡔⠋⠑⠒⠒⠠⠤⠭⠦⠀⠀⠀⠐⢤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣇⣀⡜⠀⠲⠅⢬⡤⠦⠤⠤⢤⣤⣤⠤⠤⠴⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢳⠀⠀⠀⠀⠀⠀⠀⢀⠻⣏⠜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣇⠸⠀⣄⣀⠀⠀⠀⠀⢸⠁⠀⠘⢻⡀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠁⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⣾⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢧⠀⡆⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡾⠉⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡖⠋⠀⠀⠀⣀⠈⡇⠀⠀⠀⠀⢤⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⣧⠀⠀⠁⠊⣀⡜⢁⡀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⡵
⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠉⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⢀⠤⠒⢉⣀⠠⠄⠚⠁
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⢸⠇⠈⠁⠀⠀⠀⢀⠊⠁
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⡀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠁⠋⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⣎⠁⠐⢠⠇⠀⠀⠀⠀⠀⠀⣠⠤⠀⡇⡾⠀⡆⠀⠨⢤⡀⠀⠀⠀⠀⠀⠀⢸⠀⡜⡦⢀⠀⠀⠀⠀⠀⠠⡀⢸
⠀⠀⣀⣀⣠⠞⠀⠀⢀⡀⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣇⡇⢸⢱⡇⠀⢀⡟⠀⠀⠀⠀⠀⠀⣸⠸⢱⢃⡞⠀⠀⠒⠻⡰⡀⠹⡜⡀
⣴⠋⠁⠀⢀⠀⠀⠀⠀⠈⠆⠈⠀⢢⠀⠳⠀⠀⠀⠀⠀⢀⠈⠃⢀⢻⡇⡾⢸⠀⣴⠋⠀⠀⠀⠀⠀⠀⢠⠃⣠⢋⡞⠀⠀⠀⠀⠀⣑⡅⠀⠙⠄
⢣⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡎⠀⡇⢷⠘⡄⠚⠁⠀⠀⠀⠀⢠⣎⢹⣚⡡⠊⠀⠀⠀⠀⠀⠀⠟⠀⠀⣰⢢
⣤⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⢔⠒⠒⠒⠛⠛⠂⠀⠀⠀⠒⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⡑⡆⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⠔⢣⣆⡔⠁⠀⠀⠸⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠣⡎⠀⣴⣁⡠⠤⡄⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠉⠑⠒⠦⠴⡀⠀⣅⠀⠀⠀⣸⠀⢰⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠉⠉⠁⠀⠈⠄⠀⠀⡏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠀⠀⠀⠤⠐⠂⡙⣀⠔⠦⡀⠀⣀⣸⠀⠀⠁⡠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡫⠿⠥⠦⣶⣶⠀⡇⠀⠀⠀⠐⢀⣊⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠄⠂⠉⠱⠾⢿⡇⠧⠤⠞⢄⡀⠠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠒⠀⢰⣦⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠢⣄⣸⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠙⢄⡈⢙⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠠⢖⡪⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣀⡐⠠⡑⠬⡑⠊⠀⠐⢄⡠⡒⡄⠀⠀⠀⠀⠀⠀⠁⠆⢣⢄⣀⠅⠴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢇⠠⡈⠇⠦⠀⠀⠁⠒⢠⣄⣁⡇⠘⡖⠂⠀⣖⡰⠂⠠⢌⠳⡀⡽⠤⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⢈⣟⢀⠭⠭⠤⠭⠅⢿⠁⠓⠘⠓⠦⠇⠀⢀⠇⠈⠀⠀⠠⠁⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠄⠀⠀⠀⠈⠂⠀⠸⡀⢆⠀⠀⠀⠙⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡀⠀⠀⡔⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠃⠀⡰⡙⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⡤⢤⣷⢎⠎⡵⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡄⠀⠀⠖⠁⠀⡇⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⢤⠤⠼⠞⣄⠀⠀⠀⠀⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠤⠠⠤⠐⠚⠻⠤⠬⠇⠀⠀⠀⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠉⠉⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣁⡤⠤⠤⠤⢤⣀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠞⣆⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣸
⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⠀⠘⠀⡜⠀⢻⡀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠀⠀⠒⠲⢤⡞⠀⣠⠏
⠀⡀⢀⡠⠴⠒⠚⠉⠉⠉⠉⠐⠀⠀⠀⠀⠀⠀⡀⠀⢀⠃⠀⠈⢧⡀⠀⠀⠘⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠁
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⢊⣀⠀⢀⠤⡈⠀⠈⢀⠀⠀⠀⠛⢦⣤⡴⠃⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡰⠁⠀⡇⡇⠀⠘⣆⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠸⢧⠁⡷⡷⠀⢸⠀⠀⠘⢄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⠦⠤⠋⠀⠹⣄⣀⡜⠀⠀⠑⡌⠢⣄⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣠⠊⠉⠉⠉⠈⠉⠉⠙⢢⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣇⠀⠲⠀⠀⢀⡾⠆⠀⡼⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠘⠦⣜⣀⣆⣨⠧⡤⠞⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⣇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡇⠹⡄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⡤⠤⡞⠉⠳⣶⢶⡖⠛⡉⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢸⠁⠀⠹⣄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠛⠛⠉⠀⢹⠀⠈⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⣸⠃⠀⠀⡼⠂⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣠⡴⠯⠚⢀⡴⠁⠀⠀⣰⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡄⠀⠀⠀⠠⡜⠲⠖⠁⠀⠀⡀⠀⣀⡴⢚⠁⢀⠄⠀⣼⠁⠀⠀⢰⠃⠀⢠⠏
⠀⢀⡠⢤⣀⣀⡀⠀⠀⣀⣀⣠⠴⠞⠁⠀⠀⡟⠙⠓⠒⠒⠘⠲⣄⠀⠀⠀⠹⣎⠀⠈⠛⠖⠾⠴⠖⡵⠋⠉⠉⠉⠀⢠⠏
⣰⠋⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⢀⣀⣤⠼⠗⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠘⠧⡜⠀⠀⡆⠀⢹
⣇⣠⡇⢠⠀⠀⢀⠀⢀⣀⣀⣤⠴⠞⢋⡥⠖⠒⠊⠉⠉⠉⠀⠀⠀⢀⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠒⠚
⠀⠉⠛⠾⢦⣤⡼⠏⠉⠉⠀⠀⠀⠀⡟⢠⠂⠀⡖⠀⠀⣴⠶⠚⠋⠉⠉
⡀⢀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠓⣻⣤⣴⣧⣤⡾⡁⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⡀⣀⢀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠲⡤⠒⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠠⠄⢀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡠⠂⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢡⠒⣶⢂⣒⠒⢒⡒⠒⠓⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠙⡄⠐⠀⠉⠁⡀⡏⡎⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢀⡱⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⡖⡶⠶⠶⢶⠑⠁⠀⠰⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠠⠊⢀⡀⡀⠀⢠⠖⠁⡆⠀⠀⢄⠀⠑⢤⠀⠀⠠⠄⡧⣤⣤⣤⣼⣚⡄⢤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠑⢼⡀⠀⠑⣷⠂⡠⠂⢀⣁⠤⠤⡀⠈⢳⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⣆⠑⠦⢀⣹⠇⢂⣈⠁⣀⡠⠞⠁⠀⠘⠦⢄⣀⡤⠤⠚⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠄⡠⠬⠥⠀⠀⠀⠀⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠘⠚⠐⢀⠠⠀⠀⠆⠀⢔⠚⠛⠚⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠈⠀⢤⠐⡔⠰⢤⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⣀⠔⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠄⣲⣻⠤⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⡖⢄⣠⡵⢬⠃
⠀⠀⠀⠀⠄⠠⠊⣸⠣⡈⠁⠀⠡⣘⣯⠐
⠀⠀⠂⣁⡀⠃⠠⡁⢀⠼⠁⠐⠖⠘⠃
⠀⠀⠀⠀⠀⠀⢀⢀⠅⠑⠤

⠤⠤⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠳⣄⠀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣓⣒⣲⡤⣤⡘⣆⠨⡄⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢉⣩⣤⡤⠬⠽⣷⣯⣿⣄⡵⠁⢀⣀⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⣠⠞⠋⢁⣠⣤⣶⣶⡶⢾⣿⣿⠿⣿⣿⣿⣿⣭⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠞⠁⢠⣶⠿⠛⠉⠀⠀⢀⡾⠉⢳⣄⡀⠈⢻⡆⠉⠉⠙⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣴⡿⠁⠀⠀⠀⠀⠀⠘⣇⠀⢺⣼⠟⠦⣼⡏⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⡟⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⣦⣤⣴⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡵⠟⢀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣦⣤⣤⣾⡿⠟⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣶⡆⠀⠀⣠⣾⠋⠁⠸⡍⠙⢶⡀⠀⠰⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠁⠀⠀⣼⠋⠘⡄⠀⠀⢧⠀⠀⢻⣄⠀⠀⠈⢹⣿⠛⠛⠛⠛⠿⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣸⠏⠀⠀⢣⠀⠀⢸⣀⣤⠾⠛⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡿⠀⡇⠀⠈⣤⠴⠛⠉⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⡇⠀⣠⡴⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠸⣷⠞⢉⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢰⣄⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠶⠟⣿⡿⠿⠿⠿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⡷⠶⣦⣤⣀⡀
⠀⣀⣠⣀⣠⣴⡿⠛⠉⠀⠀⠀⢠⣿⠀⠀⠀⠀⠈⢳⣄⠀⡠⠞⠉⢋⠉⠲⣄⠀⣠⠟⠹⣿⠀⠀⠈⠙⠻⣶⣄⡀
⣼⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⢸⠀⠀⢈⡿⠋⠀⠀⢻⣇⠀⠀⠀⠀⠀⠙⢿⣶⣶⣶⣄
⠸⢿⣿⣿⡿⠇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⢸⠀⣠⠟⠁⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠂
⠀⠀⠉⠘⠃⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠁⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢛⣿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⣿⡿⠿⠿⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⠤⠼⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡤⠴⠶⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⢀⣆⣀⣤⣆⠙⣦⠀⠀⠀⠀⠀⠀⠀⣰⠋⢡⣤⣄⣀⣅⣀⣴⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⣭⣥⣤⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣤⣬⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿