Planting Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠁⠒⢉⠁⠲⡄⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢞⣡⡕⢦⡑⢄⢨⡀⡄⠀⠀⡆⠀⢀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡠⠉⠙⠏⠀⠀⠙⢦⠙⠗⢢⠈⠀⡀⠐⠀⠀⠤⢠⡄⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠈⠁⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⢩⡻⠀⠈⠏⣷⣀⠀⠀⠄⢀⡠⠎⠀⠄⠘
⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠔⠊⠈⡈⠃⠈⠀⠰⢿⠀⠉⣠⠟⠉⠐⠀⠠⠀⢁⠂
⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠖⠀⠀⡀⠐⣀⠀⠀⠀⠉⣶⣄⣴⠥⠔⠒⠠⣄⠀⠀⠂⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⠁⠡⠄⠀⠂⠈⠡⠀⣂⠀⣠⢈⡅⠄⠒⠀⣉⡉⠉⠉⠖
⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠠⠁⢋⠐⠄⠰⠀⠀⡈⠄⠀⠀⠀⠉⢀⠔⢎⠋⠁⠀⠙⣖⣠
⠀⠀⠀⠀⠈⠘⠂⠀⠀⠄⠀⢀⡀⡐⠁⠀⠀⠀⠀⠃⠀⠠⠀⠀⠉⠷⠼⡅⠀⠀⠀⠐⢤⢰⠖⠀⠉⠀⣲⣴⣧⡆⠄
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢳⡀⢃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⢈⠑⣓⡀⣐⣶⠆⠁⠑⠆⠀⠤⣈⠄⡁⠈⠉⠢⡀
⠀⠀⠀⠀⠈⠀⠈⠂⠀⠀⠀⠀⠀⠘⣎⢺⡉⠁⢈⠀⠠⠀⠀⠀⠀⠀⡄⠀⢸⣴⡀⠙⠐⠢⡄⠀⠀⠴⡀⠊⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠠⠑⡧⠀⠸⠀⠀⠀⠀⠂⠀⠈⠁⠀⢀⡿⢷⡓⠂⠒⠀⣀⠀⠔⠁⠀⠀⢀⠎⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠠⠀⠠⡀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣁⠀⠀⡀⣆⡠⠖⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠂⠀⠈⠚⢜⢄⠀⠀⠀⠀⠀⠀⠀⢸⠹⠋⠀⠈⠉⠈⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣁⠀⠀⢀⡐⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠳⡤⡀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠐⡀⠀⠀⡌⠀⠂⠀⠀⠀⠀⠤⠤⠴
⠀⠀⠀⠀⡁⠀⠀⠘⠀⠀⠀⠸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠘⢟⣄⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⢁⠀⠀⣧⠧⢀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠇⠀⠀⡄⠀⠀⠀⠀⡆⠀⠀⡅⠀⠀⠀⠀⠀⠀⠜⠀⠁⠑⡀⠀⡄⢠⠀⠀⠀⠀⠀⠜⡠⠚⠀⠀⠀⠑⡀⣀⡄⠊⠄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠒⠄⠀⠀⡀⠀⠀⠣⠄⢀⠀⠀⠀⠀⠈⠢⡀⠂⠙⣀⠃⠰⠆⢀⣀⣀⣀⡜⠀⠀⠀⠀⠈⢹⠉⠀⠀⢀⠀⠈⡇
⠀⠀⠀⠤⠓⠒⠒⠒⠴⠊⠀⠉⢡⣤⣤⣤⣤⣼⣇⢤⡠⠄⠐⠰⠍⠶⡞⠉⠑⠒⠃⠤⢨⡁⠀⢐⣄⣀⣀⠤⠄⠒⠒⠂⣀⡘⠀⣐⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⠀⡈⠁⠴⠆⠀⠀⠲⠀⠀⠉⣋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⣿⡏
⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⢻⣷
⠀⠀⢠⣶⣿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣷⣤⡀⢠⣿⠋⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣇⣠⣤⣶⣶⠿⠿⠿⠿⢿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⣠⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⢶⣦⣄⠀⠀⠈⣿⡇⠀⠀⢠⣾⠟⠉⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⢸⣿⠀⢠⣿⠏⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣸⣿⠀⣾⡟⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⣀⣀⢀⣀⣀⣹⣿⣿⣷⣿⣇⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠟⠛⠛⣿⣆⣻⣿⣟⠁⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀⠀⠀⠀⠘⣿⣿⠛⠻⠿⣷⣶⣶⣶⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡉⠉⠛⢿⣷⣄⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠈⢿⣧⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢠⣤⣈⣿⡆⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀⠙⢻⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣠⣶⡿⠿⢿⣷⣄⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⡿⠛⠉⠉⠙⠛⠁⠀⠀⠀⠘⠿⠛⠉⠉⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⡿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣉⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠛⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⣠⣶⠋⠁⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣠⠴⠾⠉⠀⢸⢷⡀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠀⠶⠀⠀⠸⡇⣷⡄⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢆⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣀⡀⠀⢸⠇⡏⢻⡀⣠⠶⠛⠓⣆
⠀⠀⠀⠀⢀⣤⠀⠀⣀⠀⠸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⢯⣾⠶⠚⠛⠁⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⢾⠈⡇⠀⣿⠑⣆⠙⣾⣇⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⣤⣀⠘⢾⡇⠀⠹⣄⢸⣤⣿⠿⠤⠤⠴⠛⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢧⠈⢳⣼⢀⠴⢲⠏⣹⢣⣧⠤⠤⢤⣀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⢀⣀⣙⡚⣿⡿⠴⠋⡴⣳⠋⠓⠀⠀⠀⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠛⠤⡤⠽⢿⣧⠀⡼⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⢸⣸⡼⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⣴⠋⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠈⣇⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣼⠃⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢹⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠸⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣼⢿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⡇⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣼⠃⠸⡇⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠙⠳⠶⠦⣤⡤⠴⠶⠛⠉
⠀⠀⠀⠀⣀⣤⠶⠛⠛⠶⣤⡀⠀⢀⣀⣼⣇⡀⠀⠀⠀⣼⠃⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⢀⣴⠟⠁⠀⠀⠀⠀⣠⡴⠟⢛⣋⠉⠀⠀⠉⠛⣶⠾⠃⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⣶⣋⣀⣀⣀⣀⣀⣀⣿⣁⣰⣟⣉⣁⣀⣀⣀⣀⣰⠏⠀⠀⠀⣀⣤⠞⠋⠛⠲⠶⠶⠶⠶⠶⣿⠀⠙⠷⢤⣄⣀⡀⠀⢀⣀⡴⠋
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠛⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠒⠒⠒⠒⠒⠒⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡄⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⣆⠀⠀⣰⠃⠀⠀⣠⣴⡾⠋⠁⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠸⡆⠀⣿⠀⡤⣾⠟⠉⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠹⣝⢦⡀⠀⡇⠀⣸⢟⣵⣃⠀⠀⣀⣀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠈⢧⡻⣤⠇⣰⢋⡞⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⢤⣤⠤⢷⡙⣦⠏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠉⠉⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠰⠷⠀⠀⠰⠄⠀⠀⠀⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠴⠀⠀⠀⢀⡤⠤⢷⣄⡀⠀⠀⠶⠀⠀⢀⡈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⡀⠀⠀⣼⠀⠀⢀⣄⠀⠸⣄⠀⠀⠀⠉⠛⠲⠤⣤⣀⣘⣋⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠙⠲⣤⡇⠀⠀⠀⠁⠀⠀⠙⢲⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠙⠓⠒⠲⠒⠒⠛⠛⠋⠉⠛⠛⠲⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠉⠙⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀⣀⣀⣀⣀⣀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⡀⠀⠄⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠇⠀⠈⠀⠀⠀⠀⠀⡄⣐⠋⠀⠄⠀⠰⠏⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠤⠰⠆⡤⠤⠤⠤⠠⡄⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⠚⢄⠀⠋⠑⢤⣀⡀⢰⠀⠀⢹⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⢠⠃⠖⠂⠉⠺⠊⠉⠁⠈⠙⢦⡐⡉⢣⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⠃⠀⢻⠀⠀⢆⠀⠀⠀⠳⠀⠀⠀⠀⢁⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠜⠁⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢄⡀⠀⠀⠐⠚⠁⠀⠀⢈⡗⠈⠁⠢⠀⠙⡅⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠋⠑⡄⢡⠀⠀⠀⠀⠈⠉⠉⠉⠁⢩⡗⠀⡞⠀⠀⠀⠀⠣⠀⠘⠀⡇⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⣦⢇⠀⠀⠀⠀⡀⠀⢀⠀⠸⠓⢪⠇⠀⠀⠀⢸⡀⠑⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⡡⠖⠊⢇⠀⠀⡜⠀⠀⡀⠀⠀⠉⠑⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠞⡅⠀⠀⠈⢦⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡸⠀⠙⡄⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢰⠃⠀⠀⠘⡄⠀⠀⡈⠀⠀⠐⠀⠀⠀⠀⠀⠈⢡⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⢠⠀⠟⠒⠂⠤⣠⣙⡄⢀⠁⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢂⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⡠⢚⠋⠀⠉⠒⠢⣀⠌⡆⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⡥⠆⠥⠠⢀⡠⢖⠒⡳⣤⣄⣤⡤⠤⠔⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡅⢀⣌⣷⣤⣤⡦⠆⠀⠐⡄
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠀⠀⠀⡠⠒⠄⠀⠀⠀⡎⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠖⠁⢰⠁⠀⡰⠀⠀⠀⢀⡀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠂⢲⠀⠁⡀⠀⠀⢀⢀⠔⠊⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢉⠀⠀⣸⠀⠀⢣⠀⠐⣼⠄⢀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠱⡈⠂⠀⢹⡖⠐⠠⡀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣈⢣⠀⠀⣺⠱⡀⠀⠘⣤⠇⡰⠊⠁⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠙⢧⠀⣿⠀⠈⠀⡄⣹⠈⡀⠀⠠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠒⠀⠀⠈⢧⣿⠄⠀⢠⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠁⢰⢃⡠⠔⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡏⠁⠀⠀⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠢⡇⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⣿⠀⢀⡞⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣧⢀⣀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⡆⢀⣤⠀⠀⠀⠀⢹⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠠⠆⠀⠀⠀⠀⠈⠠⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢄⠀⠁⠀⠠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⡄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⡐⠀⠠⠀⠀⣀⠈⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠰⠁⠀⠐⠒⠀⠀⠀⠈⢠⠃⠤⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡃⢀⠀⠀⠀⠀⠠⣄⠀⠈⠢⡀⠀⢀⠀⠒⠀⠉⡌⠘⢸⡇
⠀⠀⢐⣤⡍⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⢠⣷
⠀⣾⡿⠋⢀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠐⠉⠀⢣⠀⠁⠀⠀⠀⠀⠀⢸⣿⣆
⣾⡏⡀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⢿⣿⡿⠷⠂
⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢀⠸⡀⠀⠸⠘⡀⣀⢤⠀⠀⠀⠀⠀⠀⠀⢠⣝⡾⢷⣫⣤⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠄⠀⣀⡀⠀⠀⠈⠁⠀⠀⣴⣑⣁⠈⠀⠈⢣⣄⠙⡄⠀⢘⡿⣷⡾⢿⡛⠯⠝⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⡀⠀⢠⠀⡈⡄⢰⢀⣀⣄⢺⣷⢷⡆⠁⠠⠐⡀⡂⡏⠀⠉⠉⠉⠉⠋⣿⡟⢻⠀⠀⠈⢹⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠰⢃⢀⡠⠐⠉⠀⠀⠀⢀⠤⠿⠚⠂⠑⡃⣚⣿⡠⢄⡅⠀⠀⠀⠀⠀⠀⣿⡧⢸⠀⠀⠀⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⡶⠤⠀⠀⠀⠠⠄⠀⠀⣀⣀⢀⡤⠖⠉⠀⠀⠉⠑⠫⢧⣤⡀⠀⠀⠠⣻⡅⢸⠀⠀⠀⠚⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⢐⠀⠀⡎⠠⠂⢀⠀⠀⠀⠀⠀⠀⢀⣰⣷⣶⠀⠀⠀⠀⠀⠀⠀⣤⣽⡷⡤⠀⠀⢹⡇⢸⠀⠨⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠂⡋⠠⠁⠀⠐⠀⠀⠀⠀⠐⠂⡰⠁⠛⠻⠀⣾⢒⠶⡲⢦⠀⠿⠿⠙⣽⡀⠀⣸⣇⢸⠀⠀⢠⠒⠟⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢠⣆⣀⣈⠁⠊⣁⣀⣀⣀⡀⣰⠃⠀⠀⠈⠀⢸⣯⣶⣡⣾⠀⠐⠖⠂⠙⣿⣿⣿⣿⣿⣷⣃⣾⣷⠾⣷⣷⢦⡀
⠀⠀⠀⠀⠐⠀⠀⢀⡠⢒⠘⠊⠉⠉⠭⢙⣳⣧⣍⠀⠀⠈⠉⠻⠀⢰⠖⠒⢿⢳⣶⣴⠖⣾⡗⠒⡂⠑⠒⡛⠋⣛⡛⣛⠛⠓⢈⣕⡺⣿⣿⣞⢧
⠀⠀⠄⠀⠀⠀⠀⠀⣛⠙⠙⠚⠓⠺⢶⠋⠁⠀⢹⢀⠀⠀⠀⠀⠀⢸⠁⠆⢸⣿⢹⡿⣛⢽⠉⡞⢹⠀⠀⡏⠉⣿⡏⣹⠉⠉⠉⠉⠁⠻⢯⡁⠣⡀
⠀⠀⠀⠴⠤⣤⣤⣴⣟⣰⠀⡀⠀⠀⠰⠀⠉⠁⢸⣸⣤⣄⠴⠴⠀⣸⣦⣀⠾⠿⠾⠧⢥⣼⣶⣀⣲⣀⣺⣷⣴⣿⡷⠿⠿⠶⠶⠶⠗⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠀⠁⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⡶⢖⢲⠛⠛⠋⢉⡍⢍⡝⡻⠛⡳⠒⣶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡚⠫⣍⠳⢌⠢⡙⢌⠀⠇⠀⡇⠸⠀⠜⠔⣡⡞⠡⢊⡡⠖⣋⠤⠞⢉⠹⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠲⢌⡑⡂⠍⡒⠅⠢⣈⣤⣬⣄⣡⠞⠛⠲⠞⠉⠉⠳⠴⠶⣭⣴⠶⡶⣄⠒⣩⡷⣤⠉⡓⣦⣀
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠳⡄⠁⠀⣴⣾⣿⣿⣿⣿⣿⡴⢶⢶⠶⣶⠖⣶⡶⠒⢿⠾⠶⢷⣿⣰⡿⠧⣿⡟⠉⠁⠈⠳⢦⡀
⠀⠀⠀⣠⠞⠹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠶⣾⣿⣿⣿⣿⣿⣿⡿⠶⠶⡶⣶⢶⡾⠁⠀⠀⠀⢱⡄⢸⣿⢹⡇⠈⣿⡇⠀⠀⠀⢀⣠⠿⢦⡀
⠀⠀⡼⠃⠀⠀⠀⠁⠀⠀⢄⣠⣤⠞⠁⠀⣀⣷⣿⣿⣿⣿⣿⣿⣿⣈⣓⣙⣙⣈⣻⠇⠀⠀⠀⠀⠀⣧⢸⣿⢸⡇⢰⣿⡇⠀⠀⠀⠈⢡⠀⢈⣷⡄
⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣟⣏⣙⣁⣛⣘⣃⣸⣤⣤⣤⣀⣀⣠⣯⣸⣿⣼⣅⣈⣿⣇⣀⣀⣀⣀⠀⠀⠈⠀⠻⡄
⣸⣷⣶⣶⣶⣦⣤⣤⣀⣀⣀⣠⣤⠤⠤⠶⠶⣟⣛⣿⡿⠿⡟⢛⣛⣯⠿⠟⠛⣋⣭⡤⠶⢚⣫⡭⠖⠋⠁⣠⡴⠛⠉⢀⣩⠟⠉⡽⠉⣟⠛⣟⠲⣶⢿
⣿⠢⠬⢭⣅⣉⢉⣙⠛⠻⠻⡿⣿⣶⣦⡶⠟⢉⣁⣀⣤⠾⠛⠉⠁⣀⣤⠶⠛⠋⢁⣤⠞⠋⠁⠄⣠⡴⢛⠁⠀⢀⡴⠋⠀⠀⡾⢁⢰⣗⢀⣹⣄⣈⣻
⣏⠑⠮⣄⡠⡉⠙⠓⢦⣴⣠⣠⣴⢛⡩⢀⣤⣞⡋⠉⠉⣀⣤⣶⣻⣥⣤⣤⣤⣞⠋⠀⠀⠀⣠⡾⠋⠀⢁⡀⢠⡟⣁⣤⠶⠟⣩⠉⠉⠉⢉⣉⣁⡤⢾
⢻⢳⢤⣤⣉⣳⣷⣿⣛⣫⣯⣥⣬⣩⣿⣿⣟⣉⣹⣻⣿⣽⣽⣭⣭⡁⠹⠛⠟⢻⢧⣀⣠⡾⠋⠀⠀⠀⡚⣵⠿⠭⠕⠒⠉⠀⠀⠀⠀⠠⢝⣛⣓⣟⣿
⠈⢿⣭⣼⣻⣿⣿⡿⣿⠷⣿⣷⣾⣷⣿⣿⣿⣿⡻⣿⣿⣿⠿⣿⣯⣟⡿⣷⢶⣤⣈⣻⣿⡟⣽⢳⣠⢄⣼⣟⣿⣟⣷⣖⣶⣦⣀⣤⣤⣀⣀⡈⠉⣨⠏
⠀⠈⢷⠦⣠⡠⠍⢿⣟⣷⣾⣅⣇⢉⢻⣷⡿⡻⣿⢤⢿⡿⣶⣿⣻⣿⣿⣿⣯⣽⡟⣛⣮⣏⡙⠾⢯⣿⣿⣽⢧⡿⣿⢻⣭⡟⣽⣛⢷⢶⡭⠽⣽⠏
⠀⠀⠀⠳⣄⠹⣯⠭⠇⡹⢤⢿⠿⣿⣦⣻⣶⣻⣫⢿⣷⣟⡻⣯⣿⣟⣯⣟⠿⡽⣷⢿⣆⡿⣓⣾⢧⣸⡟⣛⣧⠹⡷⣟⣲⡧⠯⣝⡟⣼⣭⡿⠁
⠀⠀⠀⠀⠈⠳⢿⡅⣢⢍⡛⣿⡁⣹⣿⡇⣿⣿⣟⣏⢿⣾⣷⣛⣟⣳⣝⣯⣾⡟⣫⣻⣞⢿⣭⢿⣦⣯⣿⣧⣞⣷⢛⡟⢾⣓⢦⣈⣳⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣴⣐⢢⣟⡙⣿⣿⣿⣿⣼⣿⣭⡑⢿⣚⣿⣴⣿⡷⢿⢯⣿⣫⣿⣿⡛⣾⡷⣾⣩⣿⣝⣿⣟⣿⣟⣽⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣥⣼⣿⣿⣿⡟⡷⢮⣉⠟⣾⡿⡻⣼⣗⡿⢳⣿⣿⣮⣽⣛⣷⣿⣿⣯⣴⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠿⢶⣟⣚⣇⣿⣯⣍⣽⣷⣿⣿⣿⣿⣭⣭⠾⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠃⠀⠚⠛⠓⠀⠀⣶⣦⣄⠀⢼⠇⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣋⠁⠀⠀⠀⠀⠀⠘⠢⣀⡙⠛⠛⠓⢨⣄⠋⠉⢴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢷⣄⠾⠏⢠⣀⠀⠀⠀⠀⠀⠀⢠⣬⣛⠓⠒⠀⣙⡀⠀⠀⠠⣌⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⢲⣯⣀⢠⣴⣶⠤⣶⣬⡉⢉⣅⢠⣾⠿⣃⡄⣠⣄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⢠⡄⠲⣶⡍⠉⠈⢻⠀⠀⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠰⠞⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠈⠠⠃⠔⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡔⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⠁⠠⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠝⣯⢻⠝⣯⠿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⡟⣲⢻⡞⣳⠛⣶⠻⡦⡤⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣉⢏⣹⣏⡹⣉⣹⣏⠑⡞⢱⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡿⠶⠷⠾⠷⠾⠶⠿⠴⣯⠯⡨⢏⡹⡇⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⢸⡇⢰⣲⡂⠀⠀⣖⣖⠀⣿⠛⠛⠚⠛⢻⣐⣲⣲⣒⣇⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣈⣀⡀⣠⢀⡀⣀
⠀⠀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠚⠂⠀⠀⠛⠒⠀⣿⢸⣿⢽⠀⢼⡐⢲⢣⠃⡜⠉⠉⠉⠉⠀⠉⠀⢄⡀⠀⢤⡀⠀⠠⣀⠀⠢
⣀⡜⠆⠰⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣛⡋⢹⡋⣛⣳⠀⣯⠀⢀⠀⠀⢹⠬⠾⠷⠧⢼⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠙⢦⣀⠈⠳⣄
⠈⠁⢀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠚⢻⡇⠀⣉⣏⣹⣉⣙⣹⠀⡿⠀⣖⣾⠄⣿⠀⣀⠀⠺⠠⠀⠀⠀⠀⠰⠒⢀⡀⠀⠀⠘⢦⠀⠙⣦⡀⠘
⠀⠈⠉⠉⠉⠉⠛⠦⣤⡴⠒⠛⠉⠁⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠁⠀⡇⠀⠈⠀⠀⣿⠘⠙⠃⠀⢀⠀⡀⠉⠄⠀⠀⠀⠂⠀⠀⠀⠈⢳⡀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⢀⣀⠀⠀⠼⠷⠲⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠛⠚⠒⠒⠒⠚⠶⠶⠤⢤⣄⣀⣀⣀⣀⣀⠀⠀⠀⢷⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⠴⠓⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠈⢍⠜⢧⣄⣀⠘⣷⠀⢹
⠀⠀⠀⠀⢀⣠⡴⠁⠀⠀⠈⢳⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠦⠮⠦⠾⠄⠀⠉⠙⠣⢈
⠀⣀⠴⠚⠉⠀⡇⠀⢀⡀⠀⢸⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈
⠊⠁⠀⡞⠉⠙⠧⠀⠫⠠⠇⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣛⣲⡆⠠⠀⠄
⠀⠀⣀⣷⡀⠀⠀⢀⡤⠤⡄⠀⠀⠠⠾⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⢄
⠀⡼⠁⢀⣤⣤⡀⠈⠓⠚⠁⠀⡤⠴⣦⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⢷⠀⠈⠢⠜⠃⠀⠀⢀⠤⢄⠈⠀⠁⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠑⢤⡀⠀⠀⡀⣀⠘⠤⠜⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠸⡄⠀⠘⢍⠹⠃⠀⣀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⣀⣀⣀⣄⣀⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠤⠄⠂⠁
⠀⠀⢀⣄⠙⢦⠤⠤⣤⠶⠋⠀⢉⡉⠥⡤⠀⣠⡀⠀⠀⠀⡀⢀⠄⠀⠀⡽⠠⣦⠄
⠀⠛⠐⠗⠈⠁⠀⠀⠀⠀⡆⠀⠘⡃⠚⡟⠈⠀⠀⠏⠈⠠⠇⢨⠀⠙⠰⠇⠘⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡾⠟⠁⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⡀⠀⣸⠇⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⣸⡏
⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣶⡿⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⣀⣼⠟
⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠶⠖⠶⠶⢦⣤⣄⡙⠃⣴⠋⠀⠀⠀⠀⣀⣠⣴⡾⠛⠁
⢿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⣴⣶⡶⠟⠛⠉⠁
⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⣦⣄⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⣶⣶⠶⠶⠾⠛⠛⠉⠀⠀⠀⠀⠙⠿⣦⡀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠛⣉⠛⢿⡇⠀⢸⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⠙⠻⠿⣦⣤⡟⠉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣶⣤⣄⣀⣀⣀⣀⣀⣀⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢘⣿⣟⣿⠋⠉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠈⣿⣿⠟⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⠀⠀⠈⠻⠾⣿⣷⣶⣶⠶⠶⠶⠶⠶⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢩⣷⣤⣀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠈⠙⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⣿⠋⣿⡏⣿⡏⠉⠁⠀⠀⠀⠈⠉⠙⠻⢶⣄⡀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢸⡏⠀⢻⣧⠘⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢿⡇⠀⠀⠹⢷⣄⡉⠛⠛⠿⣦⡀⠀⠀⠀⠀⠀⠀⢿⡆⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠘⣿⡀⠀⠀⠀⠙⠛⠿⣶⣶⠿⠃⠀⠀⠀⠀⠀⠀⢸⡇⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠈⠻⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠈⠉⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠷⠶⢶⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠀⠀⠁⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⣴⣿⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠁⠈⠉⢁⣘⣿⣿⣿⡟⠛⠀⠀⠀⠀⢀⡀⢀⣾⡿⠛⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣼⣿⣿⣦⠀⠀⢻⣿⣿⣿⣿⡇⢸⡀⠀⣠⣤⠘⠓⢸⣿⣶⣿⡶⠆⠁⡀
⠀⠀⠀⠀⡠⠐⠀⠈⠀⠀⠀⠀⢀⡀⠀⣀⡀⢀⣿⣿⣿⣿⣄⣀⣀⣿⣿⣿⣿⣧⣾⣄⠀⠙⣿⣷⣤⣤⣿⣿⣷⣶⣶⣗⣀⠄
⠀⠀⢰⣏⠀⣾⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣈⣻⣿⣿⣿⡟⣿⣥⣶⣿⣿⣿⣿⣿⣥⣤⣀⣿⣿⢰⠀⣿⣿⣯⣯⠰⢀⣴⣲⠖⡃
⠀⠸⠿⠯⣭⠄⠶⢷⡨⠉⠀⠀⠀⠀⠉⠉⠀⠉⢙⡛⠛⠛⠋⡛⡛⠋⠉⠙⠋⠉⠙⠋⠛⢛⠉⡉⠀⠀⠉⠉⠙⢃⠀⠁⠀⠀⠀⠠⠀⣀
⠀⠀⠀⢀⣂⡀⢁⣀⠀⠠⣀⢀⢠⣠⠄⡁⣄⡀⡡⣀⣄⠤⠀⠀⠀⠀⢠⠀⢀⠀⠀⡈⡀⠄⠄⢀⠀⠀⠀⠠⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⡈
⠀⠀⠘⡧⣥⠙⢠⠶⠁⢹⣇⡛⡒⣍⣳⡇⢠⣰⡗⢁⡺⣔⡧⠀⢰⠉⠋⢃⠛⣀⠇⢀⣴⢏⠉⡻⠰⢏⡔⢩⣬⣧⣘⡀⠁⢐⡄⢷⡃⢂⣀
⠀⠀⠠⣿⢸⢡⣞⠞⢠⡈⢻⡄⣰⣍⡏⢸⠊⢹⡇⠪⡽⢙⡿⣶⡆⠙⢰⢢⠟⠟⢠⠟⢙⣶⣋⡁⠜⣿⣌⡹⠄⢰⡴⣄⢠⢼⠁⢻⠿⠉⡉
⠀⠀⠐⣿⢸⡷⣻⠨⣄⣽⢿⡀⣷⣌⡷⠲⡗⢸⣧⣸⡵⠌⣃⢤⣹⡣⡆⣼⢖⠃⢰⡖⠁⠹⢘⢹⣇⣉⣮⠁⢈⡑⣚⣉⠙⣀⢠⡼⠳⢒
⠀⠀⠤⣯⣸⡇⣻⢰⡇⠀⣿⠂⠻⢮⣧⣈⣟⢼⡅⣽⣟⠈⡗⣘⢩⡅⠜⣩⣻⡇⢸⡇⠾⢧⣄⢨⣿⣡⢯⡿⠸⡷⣩⣯⡵⠈⠟⠶⠃⡤
⠀⠀⠠⠿⢤⣀⣹⡈⣁⠃⣿⠐⠶⢓⡷⠼⠇⢠⡇⣺⣁⣘⣇⠐⢱⣍⡎⠉⢹⡃⢜⡃⠶⡫⠅⢼⣯⠘⢡⣿⣸⡟⣩⣟⡛⢘⡟⠸⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠚⠓⠦⠤⢤⣧⣤⣄⣘⡇⠈⠁⢸⣟⠀⠉⠛⢘⡱⠈⠥⠀⢸⣟⠀⣀⣷⣸⡏⣰⠏⠋⢸⡟⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠓⠶⠦⠤⢼⣅⡀⣀⠀⠰⡿⠀⠀⠁⢸⡏⠁⠑⠒⢺⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠚⠓⠲⠰⠦⠬⠧⣀⡀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⢀⣀⣀⣀⣀⠉⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠀⢠⡾⠛⣿⡏⣿⣽⡿⣷⣦⣄⡀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠉⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⡇⠀⠻⠀⣿⠟⡄⠉⣸⠿⠿⢷⣤⣀⠉⠓⢦⡀⠀⣠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⢰⣿⣿⡀⠸⣧⠀⠸⠀⣿⠀⠁⠀⠁⠀⠀⣴⡿⣻⢷⣤⡀⢩⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⢸⡿⠈⢷⡀⠹⣧⡀⠀⠘⠀⠀⠀⠀⠀⢞⡔⠁⠁⣴⠿⣿⡿⠁⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢸⣏⠀⠈⢷⡀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡵⣾⡿⠁⣰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⢿⡶⢤⡀
⠀⠀⠀⠀⠀⠈⣿⡀⠀⣨⣷⣄⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⣾⠃⠀⣿⡇⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣼⡇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠸⣷⣀⠟⠁⠙⢷⣄⠀⠈⠓⢦⡀⠀⠀⢸⣿⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣸⡟⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠉⠳⢦⡀⠀⠉⠓⠶⣬⣿⡷⣄⡀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⣀⣴⠞⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠉⠳⢤⣀⡀⠀⢹⣷⠀⠙⠳⢤⣀⠀⠉⠙⠓⠶⠶⠶⠶⢶⣶⠶⠶⠖⠀⠀⠈⢀⣠⠴⠚⠉⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣷⡀⢄⠀⠈⠙⠓⠲⠦⠤⣤⣄⣀⣠⣤⣤⡤⠤⠶⠒⠋⠉⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢃⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣤⣴⣿⣷⣜⣆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠈⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣤⣄⣀⣀⣀⣀⣤⣴⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠒⢠
⠀⠀⠀⠘⠢⠐⣄⠀⠀⠀⠴⠢⡄⠀⢀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣀⡀⢠⣤⠀⠀⣀⠀⠀⣀⣤⣄
⢰⣤⣤⠴⠶⠤⠾⠶⠾⠀⠀⠀⠒⠂⠛⠓⠰⠿⠧⠴⠷⠾⠦⠤⠤⠴⠆⠀⣤⠤⢤⡄⠁⠀⠀⢰⣋⣁⡀⠉⡿⠔⠢⡄⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠋⠙⠁⠛⠋⠁⠠⠾⠿⠿⠿⠶⠤⠤⠤⠧⠄⠘⠛
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠉⠀⠀⠠⣴⣞⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⡀
⠄⡀⠀⡀⢠⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉⡉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠁⠁⠁⠈⠉⠉⠁⠉
⡆⣄⡷⠀⡼⣄⠀⢀⡀⠀⣸⠡⢛⣀⣧⣻⣽⣇⣀⣀⣀⣀⣀⠀⣀⣀⣴⡷⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡆⠏⡟⢡⣟⣾⣾⢀⣇⣆⣟⠰⠮⡉⠁⣀⠀⠀⠀⠂⠀⠀⠀⢠⡇⠀⠘⣻⢿⣯⡆⠆⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⠘⠀⡄
⠟⢰⣰⡧⣩⡎⠙⠞⠒⠈⠀⢀⡠⠔⠈⠀⠀⠀⢀⠐⢀⡠⣤⡚⠀⠀⠈⠻⢿⣷⢣⢸⡌⠆⢠⠀⢹⠀⡅⢀⠀⢀⠀⠰⢀
⣟⣸⡿⠛⠊⠁⠀⢀⣠⡴⠚⢉⠀⠄⠀⠀⠀⠄⠁⢀⠞⡤⠺⠁⠀⠀⠀⢀⣤⣶⢟⡋⢿⢀⢈⡄⡄⣀⢣⡈⠀⣸⠃⠀⣾⠈⡀
⠁⠀⠀⠀⢀⡤⠴⠋⢁⡀⠢⠀⠀⡠⣂⠄⠀⠀⡰⠋⠘⡽⠀⠈⢠⠀⠀⠀⢈⡿⣸⡅⣻⣈⡌⢰⡟⢋⠀⡇⠁⠇⠀⠂⢸⢸⣷
⢀⣠⠴⠚⠁⠀⢀⡸⠁⠀⠀⣐⣄⠣⠀⠀⠀⡜⢡⢠⣼⠃⠀⠀⣾⠀⠀⠀⢈⣸⣛⣛⣋⢿⣃⡘⣧⠄⠀⣷⢠⠀⡼⠀⡖⢨⡾
⠉⠀⠀⣠⣶⠙⠋⠀⢀⢠⣳⠹⠌⠀⠀⢠⠌⠄⣠⡻⠇⠀⠀⠐⠇⠀⠀⠀⠩⠑⠛⠋⢉⢴⡇⣧⢸⣧⠀⠁⡿⢡⣻⠀⡿⡘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⠿⠭⠙⢋⠭⠿⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⠤⠕⠀⠀⠀⠀⠀⢀⠀⠀⢠⠜⡖⠂⢱⢾⡷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠛⠋⠀⠀⠃⠋⢃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠁⣀⠀⢀⡉⠀⠈⠣⠢⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠫⢊⠃⠹⠉⠞⡑⢗⠠⡀⠀⠀⠀⠀⢀⢄⡻⢈⠏⠡⠉⠏⠘⣗⠄⠀⠀⠀⠀⠀⠲⢈⠣⠤⣄⠛⣂⡁⣉⡉⣅⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠂⠆⠄⠠⠴⠐⠐⠈⠈⠀⠀⠀⠀⠉⠑⠀⠀⠤⠀⠦⠌⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠂⠄⠀⠐⠁⠈⠂⠀⠀⠀⠀⠀⠖⠂⠴⢴⠦⢶⠴⠆⠒⠂⠀⠀⠀⠀⠀⠀⠀⣠⠀⠁⠀⠈⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣀⠀⣉⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⢰⣆⣶⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠿⡥⢼
⠀⠀⠀⠀⠀⠀⠀⢨⠅⠰⠤⠂⠄⡲⠤⠀⠭⠭⠅⠀⠀⠀⠀⠠⢤⢤⣀⣈⣁⣍⡀⣠⢤⠤⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣸⣇⣸
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⢒⠐⠖⠘⠒⠘⠐⣒⡀⠀⠀⠀⠠⣾⢟⡽⠁⡏⢸⠈⡇⠹⣝⢿⡦⠀⠀⠀⠀⠀⠐⠛⡐⠖⣂⠻⢎⡺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠳⠐⠘⠖⠠⠙⠆⠀⠒⠀⠀⠀⠀⠀⠐⠁⠁⠈⠠⠈⠂⠈⠀⠈⠐⠀⠀⠀⠀⠀⣀⣴⢀⢱⣗⣉⣰⡌⢉⣦⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⡇⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡋⠀⠀⢙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠄⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢺⠒⡄⠐⣿⠇⠀⠁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⢶⡇⢵⠇⠸⣹⠱⡀⣆⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠖⠀⠈⠘⠞⠀⠀⠀⠀⠀⠀⠀⠀⠛⡶⠀⠀⣎⡆⠀⠀⢳⠆⠀⠀⠀⠀⠀⠀⣆⠙⡋⡘⠋⢈⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠈⠋⠃⠈⠁⠂⠘⠁⠀⠒



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣦⡀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⣤⣤⣴⡄⢀⡀⠀⠀⣀⣀⠀⠀⢠⣤⣿⣧⣤⣤⠀⡀⠀⣤⣤⣴⣾⣷⣄⣤⣤⣴⠦⣠⣾⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣶⣤⢖⣤⣤⣡⣸⣾
⠛⢻⢻⢛⠛⠃⡛⠛⠓⢛⠛⠻⠛⠛⠛⠛⠛⠃⠉⠁⠀⠈⠀⠈⠉⠙⠉⠉⠛⡿⠟⠛⠛⠉⠉⠀⠀⢰⠆⠀⠀⠀⠸⡭⠿⣿⣿⠾⣿⣿⣿⠿⣿
⢠⠁⠀⡀⠀⠢⠈⠀⢈⠀⠀⠐⠂⠀⠀⠀⠄⠀⠀⠀⠒⠂⢀⡀⠈⠁⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠲⢤⡀⠉⠒⠤⣀⠒⠤
⡘⢰⠀⡧⠘⡀⠀⢄⠀⠑⠄⡀⠀⠢⠈⠁⠢⢀⠀⠢⢄⠀⠀⢀⠀⠂⠀⠤⠄⠀⣀⠤⠂⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠐⠦⢀⠙⠲⠤⣀⠙⠲
⠁⢼⠀⢰⠀⢠⠀⠀⢂⠀⠀⠙⢦⡀⠀⠁⠀⡀⠙⠢⡀⠙⠢⣀⠁⠦⣀⠀⠈⠉⠀⠀⠚⠀⠀⠐⠂⠀⠠⠔⠒⠈⣀⡠⠤⠂⢀⠁⠲⣄⠈⠑⢤
⠀⣼⠀⢸⡄⠀⠇⠀⠀⠳⡄⠀⠀⠹⠄⠀⠀⠈⠢⠄⠈⠲⠄⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⠐⠖⠆⠀⠠⢠⠒⠈⠁⢄⠈⠙⢦⡀
⠀⠸⠀⠀⠧⠀⠘⠀⠀⠀⠡⠬⠍⠀⠀⠀⡀⠤⠈⠀⠀⢀⡐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠶⠆⠀⠀⡤⠤⠀⠳⠀⡀⠃
⠠⠀⠐⠀⠀⠂⠀⣀⠀⠠⡄⠀⠢⠀⠒⠀⠁⠀⠀⡀⠀⠀⠉⢀⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠉⠀⠈⠛⠒⠀⠀⠠⠤⠀⠀⠲⠒⢀⠉⢀⠤
⢶⠀⠘⠀⠉⠁⠀⠀⡀⠀⠠⠤⠄⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⡐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠄⠀⠐⢀⣤⠄⠀⠈⠁⣀⡤
⠀⠀⢀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⣀⡄⠀⠀⢀⣠⠄⠀⠀⠈⠀⠰
⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⡀⠀⠈⠀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠒⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⣀⣀⠀⠀⠀⠀⣠⡴⠆
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⢀⡠
⠀⠀⠀⠀⠉⠙⠒⠤⠀⠀⠀⠀⠀⠀⠘⠓⠂⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠊⠁⠀⠀⣀⠀⠀⠈
⢾⣶⡄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠰⠓⢤⠄⡀⠀⠀⠀⠀⠀⠄⠠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠉⠉⠉
⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠉⠁
⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠚⠂⠀⠀⠀⠀⠀⠰⠗⠂⠂⠒⠀⠀⠀⠀⠚⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⢤⡤⠤⠖⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⢀⣤⠀⠀⠀⠈⣿⡟⠁
⠀⠀⠀⠈⠉⠁⠄⠀⠈⠙⠛⠓⠀⠀⠀⠀⠀⢀⡴⠾⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠤⠀⠀⠄⠒⠂⠀⠀⠀⠘⠉⠉⠉


⠀⡀⠄⣠⠄⣀⡀⠀⡀⠀⣀⠀⡀
⢦⡈⠲⢄⡉⠒⢭⡑⠂⠤⢀⠋⠥⣂⠢⠤⣐⡂⠤⢀⣀
⠀⠙⢦⡀⠙⠢⣀⠈⠓⢦⣀⠉⠲⢤⣌⡁⠂⠬⣉⠓⠲⠭⣝⡲⠤⠤⣄⡀
⢳⡀⠀⠙⣦⠀⠈⠳⣄⠀⠈⠳⢶⣀⠈⠙⠳⢤⣀⡉⠑⠶⢤⣍⠙⠒⠦⣭⣙⠒⠦⠤⣄⣀
⠈⢻⣤⠀⠈⢳⡄⠀⠈⠻⣤⠀⠀⠉⠳⣤⡀⠀⠈⠑⠶⣤⣀⠈⠙⠲⢦⣀⠉⠙⠲⢦⣄⡉⠛⠲⢦⣄⡀
⠀⠀⢷⡀⠀⠀⠙⣤⠀⠀⠈⢣⣄⠀⠀⠈⠻⣦⣄⠀⠀⠈⠙⠷⣤⡀⠀⠉⠛⠶⣄⡀⠈⠙⠲⣦⣀⠈⠉⠛⠶⣄⣀
⡇⠀⠈⢿⡆⠀⠀⠙⣷⡀⠀⠀⠙⠷⡀⠀⠀⠀⠙⢷⣦⠀⠀⠀⠈⠛⢷⣤⡀⠀⠈⠛⠷⣤⣀⠀⠉⠳⢦⣤⣄⡈⠙⠳⢦
⣿⣶⠶⠾⠿⣶⣶⣶⣿⣿⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣾⣿⣿⣿⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠁⠐⠀⠠⢄⣀⠀⠒⠀⢄
⣀⣀⣀⣀⡀⢀⡀⠤⠤⠤⠤⠄⠀⠀⠁⠒⠒⠒⠒⠒⠒⠐⠒⠒⠂⠙⠛⠲⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠤⠂⠈⠙⠉⠉
⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠈⠉⠉⠁⠉⠁⠂⠀⠁⠀⠀⠢⠄
⠀⠀⠀⣴⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠄
⠤⠶⠚⠛⠃⠑⠂⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠀⠀⠰⠶⣤⠠⠴⠚⠈⠉⠁⠀⠈⠁⠈⠉⠂⠀⠀⠀⠀⡈⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠤⠤⠒⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠙⠛⣛⡋⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⠄⠤⠒⠒⠂⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠰⣾⣿⠟⠀⣿⣿⣿⡄⠀⠀⠤⠀⠐⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠁⠠⠀⠛⠛⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢠⣾⡃⢀⣀
⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣧⡤⣌⠀⠀⢀
⠀⠀⠀⠀⠿⣿⣿⣿⣿⠛⣆⠃⠸⠷⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠂
⠀⠀⠀⠀⠀⠀⢀⡠⡟⣿⠓⢸⡗⢦⡀⠀⢀⡴⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡐⠒⠂
⠀⠀⠀⠀⣠⠖⠁⠀⠿⢤⣻⣹⡥⠶⠊⢦⣘⣠⠎⠙⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠹⣿⣿⣧⣷⣶⣄
⠀⠀⠀⢰⡇⠀⢠⡀⠀⠀⠘⡯⠠⡄⠀⠀⣨⣇⠀⠀⠀⠀⠁⠀⠀⠊⠉⠀⠀⠐⠁⠘⠉⠃⠈⠻⠿⠿⢿⣿⣦⡀⠀⠈⠑⠒
⠀⠀⠀⠈⣷⠀⠀⢻⣦⢻⣧⣈⣱⢴⣪⣄⠉⠉⣩⣧⣀⣤⡴⠀⠀⠀⠀⠀⠠⠀⢸⣿⠀⣿⡇⢸⣇⣸⣿⣶⣦⣄
⠀⠀⠀⠀⢻⣷⠞⣉⣿⡓⠿⠻⠟⠻⠅⠢⠤⣾⣾⣿⣿⠿⢇⠀⠀⢀⠠⠄⠤⠤⠤⠤⠤⠄⠀⠘⠛⠻⠿⠿⠿⠿⠷⠦⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⣿⣿⡟⠀⠙⠳⣄⠀⠀⢀⡀⢀⢘⠉⠁⣏⢀⠌
⠀⠀⠀⠀⠀⠻⠿⣿⣳⢦⣤⣈⡙⠓⠉⠙⠋⣿⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠀⠒⠈⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⢿⣶⣶⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⡿⠿⢟⡦⠀⠀⢠⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣟⠃⠀⠀⣠⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⣿⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⢰⠟⠀⠀⠁

⠀⠀⠀⠀⠈⠉⠃⠀⠒⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠂⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⣰⠚⠉⠙⢢⡀⠀⣰⠄
⠀⠀⠀⠀⠀⣄⣤⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠛⣶⠒⢲⡞⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢠⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⢀⠀⠀⠀⢀⣀⣿⣀⣸⣇⣀⠀⠀⠀⢀⠀⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⢸⠉⠉⠉⠉⠉⢹⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠉⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⣿⠓⠶⣄⠀⠀⣠⠴⠚⣿⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⡟⠓⢦⡀⠀⠀⣠⠖⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠈⢧⡴⠃⠀⣠⠇⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠹⡄⠀⠙⣆⡼⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣸⣇⡤⠞⠁⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠙⠲⢤⣸⣧⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⣦⠀⢰⡖⠶⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠸⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂



⡏⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡉
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⣀⡀⢀⠀⢀⣀⠀⡀⠀⣀⡀⠀⣀⠀⣀⡀⢀⡀⠀⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⡇⡇⢸⠀⡌⢧⢸⠾⡁⡇⠀⠇⠀⠀⠣⡄⢸⠁⡷⢼⢸⠀⢸⢸⢮⠀⡇
⣶⣶⡶⣶⠶⣶⣶⣶⣶⣿⣾⣶⣷⣾⣶⣶⣾⣾⣿⣶⣿⣶⣷⣶⣶⣾⣷⣾⣶⣷⣾⣶⣿⣷⣾⣶⣷⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⡞⢦⡤⠞⠉⠀⢒⣻⡷⠖⠀⠁⠀⢀⡀⠀⢑⡄⠈⢐⣤⣷⣤⣀⡠⠴⠞⠓⢻⣿⣟⣳⣷⣜⣏⠁
⠉⡵⠋⠀⠀⠀⠄⠈⠈⠃⢀⣀⣠⣞⡀⠀⠀⠈⠂⠉⠀⠀⡠⠊⠁⠀⠀⠀⡰⠉⠀⠀⠀⠈⣟⠁⠀⠀⠀⡠
⠎⠀⣀⡀⠀⠀⢀⡦⠞⠋⠁⠀⠀⠀⢽⡇⠠⡲⡶⠙⠷⠴⣄⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡟⠁⠐⠶⠁
⠿⠈⠀⠀⠉⠲⣅⠀⠀⠀⠀⠀⢀⣰⣾⡯⠀⠜⠀⠀⠀⠀⠈⢿⠀⠀⠀⠈⠃⢄⣠⣄⣤⠞⠀⠀⠀⠀⠙⡭⠛
⢆⠀⠀⠀⠀⠀⠈⢳⡤⠐⠀⠈⠈⠈⠙⠄⠠⠴⠀⠀⠀⠀⠀⢸⠂⠀⠀⠈⠂⠑⢷⡔⣩⠟⢳⡤⡀⠀⠀⡄
⡌⠣⣄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠀⠀⠀⢀⣤⣹⣄⡀⠀⠈⠀⢤⢼⡇⠁⠀⣸⠃⠀⠑⣤⠔⠒
⢹⣠⠞⠋⠀⠈⠤⠴⠦⠤⣤⣤⣶⠯⠝⠒⠶⣆⠀⠀⡔⠁⠀⠀⠀⠘⡦⢬⣥⠿⠯⠀⠀⠴⠧⡀
⡟⠈⢦⡀⠀⠀⢀⡤⠒⠉⠁⠀⣀⠀⠀⠀⣰⠧⣤⣸⠀⠀⠀⠀⠀⢠⡏⠁⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀
⡶⣤⣄⠑⢤⠀⢯⣀⠀⠀⠀⢠⠀⠉⠉⠘⠳⣄⠀⠈⠣⠄⢠⣤⣴⡟⠀⠀⠀⠀⠠⠤⠴⡶⠞⠋⠀⠈⢷⡴⠒
⠜⢉⡼⠇⠰⡄⠀⠈⠓⠶⠶⢆⠀⠀⠀⠀⠀⠈⠗⠀⠀⠀⠀⠾⣍⣻⣄⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⣿
⢰⠟⠀⠀⠀⠈⠳⣤⡀⡀⠀⠈⠢⣀⡀⠀⣠⠀⠀⠀⠀⠀⠟⠀⠀⠀⠈⠙⢶⡓⠦⣤⠾⠿⠛⣻⣦⡾⠃
⢏⠀⠀⠀⠀⠂⠐⣏⠀⠀⠀⠀⠀⠠⢭⣍⡄⠀⠀⠀⣠⡶⠁⠀⠀⠀⠀⠀⠀⣷⠖⠁⠀⠀⢀⡞
⠈⢣⡀⠒⠀⠀⠀⠈⢦⡀⠀⠀⠀⢰⠀⠈⠡⣤⣤⣴⠋⡀⠀⠀⠀⠀⠀⠶⠋⠁⠀⠀⢀⡴⠋⢠⠓⠶⡶⢆⡀
⣄⠀⠈⠓⢦⣄⣢⣙⣦⣽⢦⣄⠀⠀⣄⠀⠀⠀⠀⢸⡄⠀⢀⠀⠉⠆⠀⢀⣦⠄⠠⠶⠋⢀⠠⠜⠀⠀⠀⠀⠈
⣹⠊⠉⢉⡽⠛⠉⠁⠀⠀⠈⠈⠹⠓⠶⢢⡀⠀⢀⡟⠙⠢⣌⠁⠀⠀⢀⡈⢿⠀⠀⠀⠀⠀⠀⢲⡀
⠁⡉⠳⠶⠶⢤⡀⠀⠀⠀⠀⠐⠁⠀⠀⠸⠈⠀⣿⡇⠀⠀⠈⠙⠲⢭⣶⣽⣾⠶⠈⠒⡀⢀⣢⣼⣟⠲⣤⡤⠖
⠤⠁⠀⠀⠀⠀⠙⢦⡀⣠⡴⡇⠀⠀⠀⠉⠉⠑⣷⠟⢲⠠⠤⠒⠁⢀⣴⠟⠁⠀⠀⢀⣂⣾⠿⠙⢛⠻⠿⣃⣀
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠀⠀⠀⡀⠀⠀⠀⠸⣅⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠔⠉⠉⠀⠀⠀⠀⢱⠔⢾⣏
⠀⣀⣠⠶⠀⠠⡟⠀⠀⠀⠀⠀⡀⠘⣆⡀⠀⠀⠈⠙⠶⣤⠶⣶⣻⡀⠀⣀⡴⠂⠀⠀⠀⠀⢀⣠⡶⠋⢄⣀⣿
⠉⠉⠀⠀⠀⠀⢧⡀⠀⠀⢀⠔⠡⡟⠋⠙⠲⠦⠴⠦⠤⠿⠀⠀⠀⠙⡏⠀⠉⠉⠻⠛⠛⠛⡟⠢⣄⣐⣒⡿⠻
⠒⢶⠀⠀⠀⢀⣠⣽⡉⠉⠀⠀⢀⡱⠤⠀⢀⠀⠀⠈⠀⢁⠀⠀⠀⠀⠱⣄⠠⠦⠀⠀⠀⠈⣿⡀⢀⣀⡠⠤⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡷⢀⡀⣩⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠙⠛⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⣽⠓⢀⠀⠀⢠⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡿⠘⣷⠘⠂⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⣾⠇⠀⠁⠀⣼⣿⠁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⣰⣿⠇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣿⣿⣿⣿⣿⣿⢀⡴⠛⢷⡄⡀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣤⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⢫⠗⣺⡤⣎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠁⠀⠤⠀⠠⠄⠀⢀⡠⠆⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣹⣿⣿⣿⣿⣿⣛⣯⣷⣤⣀⣀⣸⠁⠀⢀⣄⣠⠤⠀⣤⡄
⠀⠀⠀⢠⢁⡀⠀⣴⠉⠁⠀⠀⠀⠀⠀⠀⠤⠴⣄⣠⣤⠶⠾⠛⠋⠛⠃⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣏⣉⡉⠉⠉⡇⠀⠉⠉⠳⡤⢶⠋
⠀⠀⠀⠀⠈⠙⠢⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢹⣿⣿⡆⠸⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠘⣿⣿⣿⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⠀⢿⣿⣿⣆⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⠆⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣧⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠸⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⣿⣿⠀⠀⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣶⣿⣿⣿⣿⠀⠀⣿⣿⣿⡛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣤⣠⣀⣤⣤⣤⣤⣀⣠⣤⣿⣿⣿⡿⠛⠉⠉⠁⢀⣴⣿⣶⣬⣿⣿⠋⢀⣴⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠟⣹⣿⣿⣿⣿⣿⣿⣿⠟⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠶⣿⣿⣿⡟⠛⠻⠛⠁⠘⠃⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡾⣿⠿⠿⠿⠙⠋⠀⠀⣀⣴⣶⡆⣴⣶⣶⣶⠶⠾⠛⠋⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠀⠀⢀⠀⠀⣠⠞⣠⡾⣫⠿⣫⠟⠋⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠁⠘⠁⠞⠉⠚⠁⠀⠁

⣦⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠐⡦⠄⣠⣀⢀⣤⣦⣀⣀⣠⣂⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠚⠻⠿⠿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡐⢴⣤⣤⣭⣉⣛⢛⠩⣁⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣁⣮⣿⣿⣿⣿⣟⣛⣷⣉⡛⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠿⢿⣿⣿⣿⡿⠿⠿⢿⣿⣿⡟⣾⣿⠋⠈⡀⠃⠀⠀⢂⡀⢂⠰⣶⠀⣾⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⠉⠛⠀⢴⣶⢾⡿⡿⠛⠁⠀⠁⠀⠀⢌⠙⢻⣷⣮⣝⡺⡷⠙⢤⣿⡗⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣖⣾⣿⣶⣿⣛⠻⠿⠦⣦⡀⠈⣿⣅⡀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣉⣛⣻⣋⣀⣴⣾⣿⣷⠆⠁⣼⣾⣶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠾⠚⠡⠚⠚⠿⠿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⡆⣾⣿⣿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠛⠛⠋⠉⠉⠀⠀⠀⠈⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣊⣁⣀⣀⣙⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⠢⠭⢗⠀⠀⠀⠀⠀⠀⠀⢸⠨⠭⣒⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡊⠐⡉⠀⠀⠒⠓⠒⢶⢶⠒⠒⢶⠒⠚⠉⠒⠂⠙⠢⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠄⡴⡪⠀⢀⠖⠁⠀⠀⢸⢸⠀⠀⡈⠀⠒⠦⡀⠀⠀⡀⢄⠳⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⢣⠎⡜⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⡇⠀⠀⠀⢱⠐⠀⠈⢢⠣⡱⠘⡄
⠀⠀⠀⠀⠀⠀⢀⢣⢂⠏⠐⠀⡸⠠⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠣⢳⠀⢱
⠀⠀⠀⠀⠀⠀⢸⢈⡜⢠⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠇⠀⢃⠀⠀⠂⢇⠀⡆
⠀⠀⠀⠀⠀⠀⡌⠀⠇⢸⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⢰⠀⠀⢀⠀⠀⠀⠀⠘⠀⠀⢰⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠸⠀⠀⢸⠀⠀⠸⠀⠈⢀⡄⠸⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢇⣀⠔⠒⡇⠀⡜⠉⡉⠑⠦⠀⠤⠤⣀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠢⠤⢤⣤⠒⠒⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⡹⠶⠒⠪⢍⡑⢦
⠀⠀⠀⠀⡰⢁⠔⠁⠀⠂⠀⠀⠀⠀⠙⠂⡀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡢⡁
⠀⠀⠀⠐⡠⠃⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠑⠥⠤⠄⠤⠤⢀⣀⣁⡀⠤⠤⠬⠄⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⡀⡀⠀⢀⠀⢀⣀⠀⠀⠀⠀⣣




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠙⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢀⣠⠴⠞⠓⠚⠓⠲⡄⠀⠀⣠⠖⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡤⠞⢋⣁⣀⣀⡀⠀⠀⢀⡏⠀⢸⠷⣄⣰⢯⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣧⠀⠀⢀⣴⠇⠀⠀⠀⠀⢀⡤⠞⠋⣉⣤⣶⠋⠁⢤⣄⠀⣹⣀⡴⠋⠀⣰⠋⣠⠟⠁⠀⠈⣆
⠀⠀⠀⢲⠒⣶⣦⣌⡣⣑⣻⡏⠀⣰⣿⠧⢰⠀⠀⠀⢠⡏⠀⠀⢸⡇⠌⠋⠀⠀⢈⡁⠀⢻⠁⠀⠀⡴⠁⣰⣯⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠈⢧⡀⠙⢿⣿⡄⢹⡇⢰⣿⣿⠀⡌⠀⠀⠀⠘⢧⣀⣀⣼⠛⠀⠀⠀⠀⠁⠬⣿⠋⠀⢀⡜⠁⣼⠃⢹⣄⣀⠀⠀⠀⣘⣧
⠀⠀⢤⣄⣀⠛⠶⠤⠬⠷⣾⠃⢸⣟⣩⠔⡁⠀⠀⠀⠀⢀⣭⣽⠏⠀⠀⢳⣤⣀⣀⣈⡏⠀⢠⠞⢀⡼⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠸⡌⢹⣿⣷⣦⠀⠀⣟⠟⢩⣾⡿⢫⠃⠀⠀⠀⣰⠏⠀⣿⡀⠀⠀⢸⣿⠟⠉⣏⠉⣻⠋⢀⣾⣄⡀
⠀⠀⠀⢳⡈⠩⣿⣿⡇⢸⣹⢠⣿⢟⣵⣧⡶⠾⢿⠛⣿⣀⣀⣈⠛⡶⠶⡿⢣⣤⡤⣽⡰⠃⢠⡞⠁⠈⠛⢦⡀
⠀⠀⠀⠀⠉⠓⠒⠚⢳⣇⡟⠉⢈⣴⠟⠁⠀⠀⢸⡎⠛⢻⡉⠙⣧⢻⡸⣱⠋⣸⣛⠛⠳⢤⡟⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⢠⡟⠁⠀⠀⠀⠀⠘⡇⠀⠘⡇⠀⠈⠊⠃⠃⣴⢯⣉⠙⠂⠀⠹⣆⠀⠀⢀⣀⣀⣬⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣧⡿⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠀⠀⠀⠀⣴⠿⢤⣈⠁⠀⠀⠀⢻⣤⣞⠋⠉⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⡇⠀⠀⠀⠀⠀⢠⢀⡿⠀⠀⣿⣶⡶⠶⢶⡿⠶⣤⡀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠳⠶⠶⢤⣤⣤⣿⡆
⠀⠀⠀⣀⣤⠞⠛⣻⠿⠿⠾⠟⠓⢦⡀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⣠⡿⠁⢀⡾⢿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡀
⣴⠞⠋⣽⢶⣤⣾⣥⣄⢀⣤⡀⠀⠀⢻⣄⡀⣿⡟⠀⠀⠀⠀⠀⣰⠟⠀⢀⡾⠃⠀⠉⠙⠛⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠻⣤⣤⣷⠀⠘⣿⡀⠈⣏⠈⢹⡟⠻⣼⠏⢻⣿⠁⠀⠀⠀⢀⣼⠏⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⢻⡆⠀⠙⡇⠀⢻⠀⠸⡇⠀⠹⠀⣼⡟⣀⣤⣤⣶⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣷⣤⡀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠘⠳⢦⣤⣿⣄⣼⣄⣀⣧⣀⣠⣾⣿⣿⣿⡿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣿⣶⣾⣿⣅
⠀⠀⠀⠀⠀⠀⠀⢈⣭⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣦
⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⡟⢰⡇⢰⠋⠀⠀⠀⠀⣠⢀⡀
⠀⠀⠀⠀⠀⠀⣰⣏⣿⢿⡶⣴⠖⢨⡙⢾⡇⢸⠀⣀⣀⠀⠀⢿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀⠀⣀⡀⣠⣶⣦⣄
⠀⠀⠀⠀⢠⣤⣿⣷⣿⠈⠛⠛⠀⠈⢡⠈⠇⠸⡆⠿⠿⢧⣤⣼⣵⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠴⢶⠦⠴⠿⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠗
⠀⠀⠀⠀⠸⠿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣫⣻⠿⣿⣿⠉⡴⣋⣩⡆⠀⠀⡀⠀⡀⠀⠀⠀⠀⠸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢢⣄⠀⠀⢹⣷⣶⣶⣶⠿⠚⠋⠘⠓⠾⢿⣷⣶⣄⡳⠄⠀⠀⢸⣏⡄
⠀⠀⠀⠀⠀⠀⠁⠒⠀⠠⠄⠀⠀⠚⠀⠋⢓⠦⠃⡞⢿⡔⢎⠻⢿⣿⣿⣶⢒⡄⠀⠀⠀⠀⠀⠉⠙⠓⠳⠀⠀⠘⣷⠁
⠀⠐⢢⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡃⠉⢙⠢⡀⢠⢿⣤⣤⢻⣿⣯⣤⣸⣿⣷⣦⣤⣐⠂⠀⠀⠀⠀⠀⠀⣀⣉⣠⣄
⠀⠀⠀⠀⠈⠉⠑⠬⠷⣦⡤⣀⠀⣿⠻⠀⠀⢈⣙⢦⢁⡞⣿⣿⣾⣿⢰⣮⣅⡉⢿⣿⣿⣿⣿⣷⣶⣤⣰⠞⠉⠁⠀⣠⣴
⠀⠀⠀⠀⠀⠈⠐⢠⣴⣾⣿⣷⡀⣿⢸⡇⠀⢠⣮⡳⠤⣿⣿⣿⣸⣿⣿⣿⢿⣇⡤⢭⢿⢿⣿⣯⠛⠿⣿⣿⣿⡿⠉⡿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⢻⡞⡅⠀⢸⣷⣝⣖⣿⣿⣬⣄⢳⡸⣕⠸⢿⣧⠀⢨⠹⣿⣿⡆⠀⢹⣿⣿⣟⠀⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡈⢿⣼⠀⣿⣿⣿⣿⣿⣿⣿⣨⣮⣻⣮⠻⡶⣍⠙⠸⡄⣿⣿⡇⠀⢸⣿⣿⣿⠀⣷⣾⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡶⣦⡀⠁⢶⣎⣉⠿⡶⠘⣿⢿⣯⠀⠿⢛⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠟⠻⢿⣿⡄⠈⢷⣿⣶⣷⣽⣯⣬⣿⣶⣿⣿⣿⡿⣻⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣀⠈⣿⣇⣠⣿⣿⣿⠿⠟⣿⣿⣿⣽⣿⣟⣟⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⡯⠁⠀⠉⠁⠀⠀⠀⠀⠙⠿⣿⣿⣿⣾⣷⠀⠘⠈⣿⣿⣿⣿⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣆⣀⣴⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⢽⣿⡿⠿⠛


⣿⡿⡛⣿⠿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣥⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣚⣛⣉⣛⣛⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠖⠋⠉⠀⠀⠀⠉⠙⠳⢬⡛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠈⢷⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⡏⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢿⣶⣀⠀⠀⠀⠻⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣸⠀⠀⠀⠀⠀⠀⢀⡴⠛⣴⢞⡵⢋⡼⠻⣦⡀⠀⠀⢻⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⠀⠀⠀⢀⡴⢏⣱⢟⡵⢋⡴⠋⠀⠀⠈⠻⣆⠀⠀⣷⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢿⠀⠀⢀⣴⢏⣠⢟⡵⢋⡴⠋⠀⠀⠀⠀⠀⠀⠈⠳⣤⢸⡎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣆⣴⠟⣡⢟⡵⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠟⣡⢞⡵⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣾⡵⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠛⡿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⢋⡴⢛⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢦⣤⣤⣀⣀⣀⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⢋⣴⢋⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⡴⠟⠳⠶⣤⣍⣉⠉⠙⠛⠛⠒⠾⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⢋⣴⢟⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⣀⠀⠈⠳⣄⠀⠀⠀⠀⢀⡴⢋⡴⢋⣴⢟⣥⠟⠁
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢮⣷⣄⢀⡴⢋⡴⢋⣴⠟⣡⠞⠁
⠀⠀⢸⣿⣍⢹⡶⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⢋⣴⠟⣡⠟⠁
⠀⠀⠈⢧⡈⠻⣇⡿⢀⠀⣤⡞⢋⡽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⣡⠞⠁
⠀⠀⠀⠈⠙⠒⠛⠳⣿⢸⣿⡷⠋⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠛⠶⠞⠋
⢀⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠻⢤⣤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠞
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⣶⡶⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⢀⠀⠈⢢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠴⣫⠶⠛⢻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣵⣿⠁⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⣷⠀⠀⠀⠀⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢷⣤⣠⡴⠃⢸⠙⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠙⢦⠀⠁⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⢆⠀⠀⠀⠀⣇⠀⠃⢀⡄⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣰⠀⢸⣆⡄⠶⠶⣿⣥⣄⡟⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⣰⢻⡧⠊⠙⠒⠋⠀⠀⠀⠘⣏⠁⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠃⢀⡀⠰⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡠⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠹⣇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠑⠢⣼⡤⠤⢤⡀⠀⠀⠀⠀⡃⠀⠈⢳⡀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢠⠚⠁⠀⢐⡶⣬⠆⠀⠀⠀⡇⠀⠀⠀⢷⡀⣠⣴⠻⠤⠖⠒⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⡷⢤⣴⠚⠲⠿⠘⠃⠀⠀⠀⡇⠀⠀⠀⠈⠿⠃⠹⣆⣀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡄⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⢹⡀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡏⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡇⠸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢈⠁⠀⡇⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⡿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢸⠁⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⢧⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡛⠀⢠⡏⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠇⠀⠘⢻⠦⠤⠄⠒⠘⠒⠤⣀⠀⠀⠀⠀⠀⠸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠀⠀⠀⢰⠀⠀⠀⣿⣤⣤⠶⠶⣤⣤⡤⠞⠀⠀⣀⡀⢀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠋⠙⠋⠙⠉⠙⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⡠⠄⠂
⣀⣀⣀⣀⣀⣀⣀⣈⡳⢶⣖⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢻⡿⠿⢿⢿⡙⠿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣟⣿⣟⣛⣿⣛⣻⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠲⣄⠀⠀⠀⠀⠀⠀⣠⢴⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢉⣗⢓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣇⠀⢘⣦⠴⠶⣦⠄⠀⢹⣸⣛⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠤⢼⣁⠜⣆⣀⠀⠀⠀⠀⠀⠀⣀⣭⣷⣤⡈⢳⡞⢀⣴⠯⢴⣤⡀⢀⡏⠈
⠀⠀⠀⠀⠀⠀⢰⣶⣖⡚⠗⠶⣼⣥⠞⢋⡯⣧⠀⢀⣀⣀⣾⠗⠲⠦⣌⣳⣸⣣⣎⣠⠞⠉⠉⠁⡾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣜⡙⠾⣔⣹⣗⣀⡾⠁⣿⣴⠟⠁⠀⠈⢻⡄⡼⠉⠉⠉⠉⠉⠉⣧⠀⠀⢸⣧⡇⡇⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠾⠽⠶⠶⠿⢿⠋⣹⡏⠀⠀⠀⠀⢰⡇⢧⠀⠀⠀⠀⠀⠀⡾⠀⠀⢸⢸⠀⡇⢰⡀⠀⠀⠀⢠⣿⢁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠉⠉⠉⠉⡏⠀⣿⠀⠀⠀⠀⣠⠟⠀⠈⠳⠴⠴⠶⠴⠚⠁⠀⠀⢸⠈⠀⡇⠈⠙⠓⠒⠚⣻⡏⠉⠉⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠼⠁⠀⣿⠀⢀⣠⠞⠁⠀⠀⣠⠞⠲⡄⣼⣿⠀⠀⠀⠀⠘⠦⢴⣃⡀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⡴⠛⠹⣦⢀⣠⠶⣿⠚⠉⠀⠀⠀⠀⣴⡿⠀⢀⡾⢹⡏⠀⠀⢠⣾⡇⢰⠏⢸⣷⠀⠀⠀⣰⣿⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠁⢀⣤⣿⠋⠀⠀⢻⡇⠀⠀⠀⣠⠞⠹⣧⣠⠞⠀⢸⡇⠀⣰⠋⢹⣇⡟⠀⢸⣿⠀⠀⡼⢻⡇⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⣾⣧⡏⣰⠋⣸⡟⠀⠀⠀⠈⢷⡀⣀⡼⠋⠀⠀⠀⠀⠀⠀⠸⣧⠞⠁⠀⢸⣿⠇⠀⠸⣇⣠⠞⠁⠈⣧⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠻⠶⠋⠀⠀⠀⠀⢀⣈⠉⠉⠀⠀⡴⢻⡇⠀⢠⡴⠒⠒⢦⡀⠀⠀⣿⡀⠀⠀⠉⣡⢾⣿⠀⠙⠓⠋⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⢀⠀⠀⠀⢀⡏⢹⣧⠀⠀⡼⠁⢸⡇⢠⣿⠁⠀⠀⠀⢷⠀⣼⠉⣿⡀⠀⣰⠃⢘⣿⠀⠀⠀⢻⣟⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡶⣇⡷⣦⠀⢸⠃⠈⣿⡄⣼⠁⠀⢸⡇⢸⣿⠀⠀⠀⠀⣸⢀⡇⠀⢻⣇⢠⠏⠀⢸⣿⠀⠀⢀⣼⡅
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⡿⠩⠉⡩⠽⠇⣾⠀⠀⢻⣷⠃⠀⠀⢸⡇⢸⣿⠀⠀⠀⢀⡟⢸⡇⠀⢸⣿⡟⠀⠀⠈⣿⠀⢀⡞⡃⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡏⠈⠻⠋⠰⠀⢿⠀⠀⠘⠋⠀⠀⠀⢸⡇⠀⢻⣆⠀⢀⡜⠁⢸⡇⠀⠀⠛⠁⠀⠀⠀⣿⠀⢸⠃⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⢻⡞⢾⣧⠀⡇⢸⣿⡿⢾⣿⠀⣿⠀⡿⢻⣷⡷⣿⡇⢸⡁⢸⣿⢾⢏⣿

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠋⠂⠀⠀⠀⠀⠀⡵⠃⠀⠀⠀⠺⠉⠉⠉⠀⢠⣄⠀⠀⣠⠏⠀⠘⣏⢂⠀⠀⠚⠊⠁⡔⠀⠀⢰⠏⠀⠀⠈⠓⠦⠕⠀⢩⡁⠀⠀⢯⡄⠀⠉⠋
⠀⣀⠀⠘⣇⠀⠀⠀⢀⣄⣠⠾⠀⠀⢰⣀⢠⡈⢇⠀⠀⠁⠀⠀⠀⠈⠳⠇⠀⠀⠀⠀⠹⣤⠀⠸⠄⠀⠀⠀⠰⡄⠀⠐⣸⠃⠀⠀⣄⠠⣆⡀
⣨⠟⠀⠐⡼⠀⣤⠀⠀⠉⠀⠀⢀⡀⠀⠈⠉⠁⠀⠀⠀⢤⣄⣤⣄⠀⠀⠀⡴⠒⠗⠀⠀⠀⠀⠀⢀⣠⠴⠀⠀⣷⠀⠙⠃⠠⣠⠞⠁⠀⠀⠙⠛
⠀⠀⣀⡀⠀⠀⠈⠳⠶⠀⢀⣀⡼⠃⠀⠀⠀⠛⠶⠶⠒⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠋⡳⣄⠀⠛⠂⢀⡤⣀⠉⠀⣀⠀⠀⢠⡔⠚⠃⢀⡶
⠀⠀⠑⠭⡷⠀⠀⢠⣤⣄⡀⠁⠀⠀⠈⠳⠤⠤⠀⠀⠠⠖⠒⠶⠀⠀⠘⢧⡂⠀⠀⣀⠀⠈⠁⢀⣤⡀⠀⠙⠲⠖⠀⠉⠳⡄⠈⠀⠀⠚⠋⠀⣀
⡳⣄⠀⠀⠀⢠⡀⠀⠁⠒⠋⠀⢠⣄⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⣀⡀⠀⠙⠂⠀⠉⢳⣄⢰⡋⠈⠀⣠⡀⠀⣀⠀⠀⠈⠃⠀⢤⣀⣀⣠⠄⠈
⠈⠉⠀⢀⡀⠘⢦⣀⠀⢰⠀⠀⠀⠈⢳⡀⠚⠒⠚⠉⠀⠀⡽⠀⠈⠋⠍⠳⠀⠸⡲⢤⡀⠉⠀⠀⣴⠆⠈⠉⠉⠉⢠⡀⠀⠀⣀⣠⡀⠀⣄
⠶⠀⠀⢼⠁⠀⠀⠉⢀⣸⢀⡀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠚⠃⢠⢄⠀⠀⠀⣀⣀⠀⠉⣅⠀⠀⢰⡇⠀⠀⠀⢀⡀⠀⢣⠀⠘⠓⠂⠁⢸⡋⢀⣠
⡀⠀⠀⠘⠆⠀⢠⠄⠈⠁⠀⠉⠓⠲⠃⠀⠀⠀⠊⠃⠀⠀⠀⠈⠳⢬⡄⠈⠃⠉⠝⠀⠈⣇⠀⠀⠀⠘⠓⠚⠉⠀⠀⠙⠀⠀⢀⡤⠦⠘⠃⠈⠁
⠍⠀⠀⠀⠰⠵⠋⠀⠀⣰⠆⠀⢀⣀⠀⣠⠖⠋⢀⡀⠀⠚⠛⢒⠄⠀⠀⢹⠀⣠⣄⡀⠀⠋⠀⠶⠚⠻⠆⠀⠐⠒⢤⡀⠀⠰⠍⠀⠀⠀⠀⠀⢀
⠀⠘⢳⡀⠀⠀⢀⠀⢸⠁⠀⢠⢊⡏⠀⠁⢀⠀⠈⠙⣢⠀⠀⢀⡀⠀⠀⠛⠀⠀⠐⠹⠆⠀⠀⠀⠀⠀⠴⢶⣄⠤⠀⠃⢀⢐⡆⠰⢦⣤⣤⠤⠸
⡤⣄⡀⠃⠀⢠⠏⠃⠀⠀⠀⠈⠋⠀⠘⠋⠋⠛⠀⠀⠀⠀⠀⣎⠁⠠⣆⣠⣤⠄⢀⡄⣀⡴⠀⠘⢝⢢⡄⠀⢀⠤⣠⠄⠗⠋⢀⡀⠀⠀⣀
⠈⠈⠉⠠⣄⠈⠁⠀⢀⡴⠚⠃⠀⠀⠀⡶⠀⠀⠸⣆⢀⠀⠀⠿⠀⢀⡀⠀⠀⠀⠀⠉⠁⣠⡄⠀⠀⠈⠀⠀⠛⠋⣁⠀⠀⠀⢠⠇⣠⣠⠞
⠀⢶⠀⠀⢈⢳⡀⠀⠛⢀⡀⠀⠀⠴⠊⠀⠀⣰⠀⢀⣉⠁⣤⠀⢠⡏⠁⠀⠉⢢⡀⠀⠎⠀⠀⠰⠥⠤⠟⠀⢀⡀⠉⢳⣄⠸⠟⢀⡈⠀⠀⡟
⣄⠏⠀⠀⠀⠉⠁⠀⢠⠏⠃⢠⣄⠀⠀⠶⠞⠁⢠⣋⠝⠀⣇⠀⠸⠇⠀⣤⠀⠈⠁⠀⠀⠀⣠⠄⠀⠀⠀⠀⡸⠁⠀⠀⣉⠀⠀⠈⢉⢦⠀⢧
⠀⢀⠀⣤⡶⠵⠃⠀⠘⠂⠀⠀⢈⣧⠀⠀⠤⠤⣬⡀⠀⠀⠉⠀⠀⠀⠀⢻⡀⠰⢦⡀⠉⠉⢁⣄⡦⠝⠀⠛⢁⡀⠀⣀⡟⠀⠀⠀⣀⠉⠁⠀⣤
⡰⢋⠃⠀⠀⠀⠀⠀⣀⠤⠀⢀⡠⢤⠄⠀⠀⠉⠀⠉⢀⡀⠀⢰⠀⢠⢄⠀⠀⠀⠐⠹⠆⠀⠀⠀⢠⠀⠀⢠⠏⠁⠈⠋⠀⠄⡸⠃⠹⡕⠀⢠⢸
⠧⠁⠀⢴⡤⠟⠃⠴⠑⠁⠀⠛⠐⢁⡀⠀⡦⡀⠮⠖⠚⠁⠀⣸⠀⠈⢧⡀⠀⠈⠙⣆⠀⠙⠛⢉⣉⠀⠀⠸⠀⠀⣠⡀⠘⠚⠁⣀⣤⠈⠋⠈⠋
⠀⠀⠀⢠⣀⡀⠀⢀⡀⠀⠀⠀⠀⢸⠁⠀⠙⠯⡄⠀⣦⠀⣀⠁⠀⠀⠈⠁⠀⣶⠈⠺⠀⢀⣠⡄⠉⠙⡆⠀⢄⡴⠋⣀⠀⠀⠴⠅⠀⢠⡄⠄⡀
⣀⣤⠄⠀⠀⠙⣦⠀⠉⠒⠚⠀⠀⠈⠃⠀⠀⠀⠀⡴⠃⠀⠙⠝⣶⠀⠀⠀⣠⠏⠀⠀⠴⠉⠀⢀⣀⠀⠀⠀⠀⠀⠰⡝⢰⡟⡄⠀⠀⠀⠈⠓⠶
⠁⠀⠀⢀⡀⠀⡤⠀⠀⠀⠀⡷⠀⠀⣀⠀⢿⠀⠀⠀⢠⠔⠦⠀⠀⠀⣀⠀⢀⣄⠀⠀⡀⠀⣴⡵⠋⠘⠛⠙⡳⠀⠛⠁⠀⠷⠁⢲⡄⢺⡢⡀
⠀⢠⡞⠙⠃⠀⠙⢦⡄⠀⠴⠃⡠⣠⠏⠀⠘⡄⠀⠀⠟⠈⠀⠀⣤⠴⠁⠀⢠⡌⠉⠉⠁⢀⡀⠀⢀⢴⠄⠀⠀⢀⡤⠲⠂⠀⠀⢠⡇⠀⠳⣴
⣀⠀⠀⠰⢤⠀⠀⢀⡀⣀⡀⠀⠛⠁⠀⠀⢀⡀⠀⣀⣠⠾⠃⠀⠀⠀⠀⠐⣸⠃⢀⠀⠀⢠⡇⠼⠁⠈⠀⣰⠆⠘⠀⠀⠘⢦⡀⠀⢠⡤⠶⡶
⠪⢳⠀⠀⠁⢷⠀⠈⢙⠛⠋⠀⢀⡴⠒⠂⢸⠁⠀⠉⠀⠀⠀⠛⠓⠲⡄⠈⠁⠀⠉⢳⣄⠉⠀⠀⠠⡄⠀⣧⠁⠀⠀⡆⠀⠐⠽⠀⣠⠀⠀⣀
⡄⠀⢴⠒⠲⡦⠀⠀⠈⡇⠀⠀⠋⠀⠀⠀⠈⠧⠀⠉⠓⠴⠆⠀⠀⠀⣠⠚⠦⣤⡄⠀⢀⡀⠀⣠⠞⠀⠀⠀⠀⡤⠎⠀⠀⠀⢠⡴⠋⠀⡼⠁
⢜⡆⠀⠀⣀⣀⡀⠀⠀⠿⠀⣠⢴⠂⢠⣤⠀⣠⡀⢀⣠⢤⠄⠀⢄⠴⠋⢀⡔⠂⠀⠀⠀⡙⣆⠉⠐⠦⣄⣤⠀⢀⡤⡖⠖⠀⠀⠀⠀⠘⠃
⢀⡤⠶⠀⠉⠀⠉⠃⣤⠀⠘⠁⠁⠀⢀⣈⠉⠉⠀⠈⠀⠁⣦⠀⠀⠀⠀⠸⠂⠀⡠⣰⠆⠈⠛⠀⣀⠀⠀⡄⠀⡈⠁⠀⡀⣠⠟⠀⣶⠐⣶⡢⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠠⠤⢯⣑⠢⠀⢀⣢⠟⠢⡸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⢀⡤⠝⡢⠴⣇⣔⢪⢇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠨⣰⢦⡈⠑⢦⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⢀⠀⠢⠀⣯⠾⢣⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⡔⠈⠙⠲⣤⣄⣀⣠⣦⢸⣶⡤⠑⠦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⡺⠦⠀⢈⣶⣿⣿⣿⣿⣿⣿⣟⣸⡇⠀⠀⢀⣀⣑⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣱⠀⠀⠙⢿⣿⡿⢿⠛⠛⠛⠿⠛⠛⠊⠉⠁⠀⠀⠀⠀⠀⠁⠂⢄⣀⣀⡀
⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⢜⠫⠹⠢⡀⠀⠘⢃⠀⢀⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠒⠒⠤⠄⣀
⠀⠀⠀⠀⠈⠁⠂⠤⢀⡀⠀⠑⡄⠱⡘⢄⠀⠀⠉⠷⠤⠄⠀⡀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠚⠲⠾⠚⠂⠀⠀⠒⠒⠒⠒⠦⢌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠉⠁⠀⠀⠂⠐⠒⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⣀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠉⠁⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠀⠀⠀⠀⠄⣀⣀⣀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⠤⠀⠁⠐⠊⠁

⠀⠄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠃⠓⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠂⠀⣒⣒⠲⣂⠀⠓⣀⣒⡂⠐⣖⣚⡀⠛⣂⣙⣂⡛⣆⠐⢂⣘⣳⠀⣲⣘⠃⠠⠓⠈⢻⠂⠛⢂⡼⠏⢻⡧⠛
⠂⢀⡒⠀⠀⢹⡆⠀⠘⢳⠀⠀⠉⣷⠆⠀⠈⠳⡀⠀⠒⣃⡀⠚⠃⡀⠘⠁⢠⡘⠃⠀⣬⡋⠀⢀⣍⠀⠀⣍
⣀⠘⠉⠀⠀⡈⠉⠀⠀⢘⣳⣀⠀⠉⢉⣥⡀⠈⠙⢀⣄⣉⠉⠀⢈⣹⡁⠀⠈⠍⠀⠀⠉⣁⣀⠀⠈⢀⣴⣤
⡛⢓⠄⠦⠀⢹⡓⠤⠒⠀⠍⢉⡁⠀⠈⣁⡈⠀⣀⠭⠉⢉⠁⡀⠈⢙⠋⡄⠈⠙⠣⠀⠀⢙⣙⡂⠄⠠⠉⠉⡄
⣿⣿⣾⣷⣶⣾⣿⣷⣾⣾⣾⣿⣿⣿⣿⣿⣷⣶⣿⣷⣿⣾⣿⣿⣷⣾⣿⣷⣿⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣾⣿
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡟⢲⡄⡾⢁⡞⢹⡖⢰⡆⡶⢠⡇⢀⠀⠀⢸⠖⢶⡆⣰⠋⣷⢚⡗⡶⡶
⠀⠀⠀⠀⠀⠀⠀⢱⠗⠛⠀⠓⠈⠓⠚⠃⠙⠈⠓⠛⠒⠉⠀⠀⠛⠈⠈⠋⠙⠐⠁⠚⠁⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠚⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⢰⣾⣿⣹⠒⡄⠀⣠⣾⣾⣷⣦⠀⢀⣦⣷⣧⣤⠀⠀⣷⢾⣿⣆⡀⠀⣠⣴⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢿⣟⣿⣿⠁⠀⢻⣿⣿⣿⡿⠀⢰⣷⣿⣿⣿⠂⠰⢿⣼⣾⠹⠀⠀⣷⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠈⠉⠙⠀⠀⠀⠘⠁⠈⠁⠀⠀⠀⠉⠋⠀⠀⠀⠈⠉⠻⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠐⢂⣀⡀
⠀⠀⠀⠀⠀⠠⣄⠉⠁⡗⠀⠀⠀⣀⣀⡀⠀⠀⢀⡀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀
⠀⠀⠀⠀⠀⠘⠁⢸⡲⣤⠀⠀⠀⢉⣀⡁⠀⠀⠀⠂⢀⣴⢿⣝⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⠀⠀⠀⠀⠯⠤⠐⠆
⣋⣠⣄⣶⣀⣀⣀⢸⡇⠰⠄⠀⢀⣌⣡⢁⡀⠀⠐⡀⠙⠁⠈⠉⠀⠻⠆⠀⠀⠀⠀⠀⠀⢠⡄⠐⠀⠀⠈
⠞⠋⠉⠉⠉⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠉⠉⣀⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠀⡤⠀⣤⣤⣤⣬⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠬⠿⠿⠷⣶⣶⣦⣤⣀⣠⡤⠶⠇⠀⠀⠀⠀⢰⠄⠉⠉⠉⢦⡀
⠀⠀⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⢀⡤⠴⠟⠉⠀⠀⠀⠄⠀⢿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⣀⣀⠀⠄⣐⠀⠀⣤⠠⠞⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⢀⠀⠠⠂⢘⠈⠉⠰⢶⠿⠉⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁
⠀⠀⠀⠀⠀⡆⠤⠀⠀⠋⡉⠤⠄⡒⠒⠛⡀⠈⠈⠉
⠀⠀⠒⠈⢀⡁⠤⠐⠀⠉⠗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠤⢒⠫⠴⠓⠂⢁⣀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣿⣄
⠄⠀⠀⠠⠀⠔⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣴⣿⣿⣧⡀
⠤⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣺⣿⣿⣷⣿⣾⣤⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡀⠀⠀⠀⣀⣠⡠⢀⠤⡤⢤⠄⡠⢀⢀
⠀⠀⠀⠀⠀⠀⠀⣤⢴⢾⣷⡒⠢⢶⣖⣿⣿⣲⣢⣄⡄⠀⠋⠉⢈⢎⣎⠳⢣⠚⠋⠏⠞⠱⠁⠏⠸⠅⠷⠹⠜⠣⠱⠛⠜
⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⡀
⠀⠀⠀⠀⠀⠷⠀⠺⠿⠻⠷⠀⠀⠀⠀⠶⢄⣀⠀⠀⡀⠀⢀⣀⠀⢀⣤⡀⠀⢀⡀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉⠀⠈⠛⠁⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⢾⣿⠖⠒⠒⠚⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠛⠛⠃⠀⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⣠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠐⠛⠛⠛⠛⠀⠀⢰⠏
⠚⠛⠃⠀⠛⠓⠒⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠀⠒⠛⠛⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢦⣄⠀⠀⠀⣀⣀⣛⣛⣀⣀⠀⠀⠀⣠⡴⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠿⠿⠿⠿⠿⠟⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷
⠀⠀⠀⢰⣶⣄⡀⠀⠀⠀⣠⣴⣶⠀⠀⠀⠀⠀⣹⣿⣿⣇⠀⠀⠀⠀⢰⣶⣄⠀⠀⠀⠀⣠⣴⡶
⠀⠀⠀⠀⠻⣿⣿⡄⠀⣼⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠈⢿⣿⣷⡀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠛⠗⡘⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠻⢇⡸⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣀⣀⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⠙⣛⣛⠋⣀⣀⣀⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣱⣿⣿⣮⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⡤⢄⡀
⠀⠀⠀⠀⡰⣿⠀⠀⠹⣆
⠀⠀⠀⡸⢠⢹⠀⠀⠀⠹⡳⣄
⠀⠀⢰⠁⢸⢸⠀⠀⠀⠀⠱⡹⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠖⠒⠒⠒⠒⠀⠉⠉⠉⠁⠒⠒⠒⠦⢄⡀
⠀⠀⡎⠀⠈⠈⡄⠀⠀⠀⠀⠱⡱⡈⠢⡀⠀⢀⡠⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⣈⡉⠭⠍⠛⠛⠛⢳
⠀⢀⠃⠀⡇⠀⡇⠀⠀⠀⠀⠀⢣⠱⡀⠘⣮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⡠⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⣠⠎
⠀⢸⠀⠀⠁⠀⠇⠀⠀⠀⠀⠀⠀⢇⠀⠀⠘⡗⢄⠀⠀⠔⠊⠁⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡿⠃
⠀⡈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠘⡌⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡤⣪⢵⠏
⠀⡇⠀⠀⠰⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⠘⡄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡧⢒⠥⠊⡴⠁
⠀⡇⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢱⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⣤⠇⢀⣗⠁⢀⠞
⠀⢁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢇⠀⢱⠀⠀⣀⠤⠖⣫⠔⠁⠀⣸⠈⠳⡅
⠀⢸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⡗⠉⠀⢀⠞⠁⠀⠀⠀⠋⠀⠀⠹⡄
⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⡀⠀⡇⠀⢁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠀⢱
⠀⠀⠸⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢠⠀⡄⡇⠀⠇⠀⣨⠤⡞⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢸
⠀⠀⠀⠱⡀⠀⠀⠀⢳⠈⢣⢢⠀⠰⡀⠀⠸⠀⡇⠀⡸⠀⢠⠃⣠⡇⠀⠀⠀⠀⠀⠀⢠⠿⠁⠀⢸
⠀⠀⠀⠀⠱⡄⠀⠀⢄⠳⡀⠣⣣⡀⢣⠘⡄⢀⠃⣰⣡⠔⠉⠁⢸⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠢⣀⠈⠳⣝⢤⡙⢕⣦⣧⡿⠛⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠚⠛⠛⠓⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢰⠃⠀⠀⠀⡀⣰⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⡘⡆⠀⠀⢀⠇⡇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡃⠀⠀⢀⠜⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣶⣶⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣶⣶⣶⣿⣶⣷⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣰⣿⡀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡏⠉⠋⠙⠈⠉⠙⠿⠿⠿⠙
⣯⣭⠿⣯⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⠀⠀⣈⠋⡀⢹⣿⠯⣛⠳⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶
⠟⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠱⣶⣿⠗⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣟⠀⣠⣶⣻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⡀⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡃⣀⣹⣿⣿⣿⣿⣿⣿⡭⠉⠿⣿⣿⡏⠀⠀⠀⠀⣷⣦⣶⣳⡄
⠀⠀⠀⠀⠀⠀⢠⢀⣻⡗⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡹⣷⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿
⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣾⣿⠃
⣮⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⠃
⠻⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⡇⠀⠀⠀⠀⠀⠀⠰⠇⠲⣤⣶⡔
⠈⠞⢠⢴⣤⢃⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠠⣤⣦⣦⢑⢶⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠄⠐⠤⠀⠴⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⡌⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢰⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⡊⠀⠈⡆⠀⠀⠀⠀⠀⢀⠀⢤⠈⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡏⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣀⣄⣀⢤⠔⠄
⠀⠀⠀⠀⠇⠀⠀⠐⢠⡤⢐⠂⠉⠀⠀⠀⠒⠀⠀⠁⠂⠄⠀⠀⣀⡀⣠⠔⠈⠔⠠⠠⡓⠄⠈⢄⡠⠄⠀⣀⢄⣠⠊⠒⠦⠠⠀⠀⠓⠊⠀⠀⠉
⠀⠀⠀⠀⢠⠀⠐⠀⢨⠑⠢⠌⡁⠒⠀⠄⠀⠀⠀⠀⠀⠄⠂⠈⠤⠿⠃⠀⠁⠸⠀⠀⠈⠢⠀⠀⠑⠀⠲⣁⣀⣠⣤⣤⣤⡤⠖⣶⢒⡲⣖⣲⣒⠒⠐⠄
⠀⠀⠀⠀⠀⠒⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠚⠀⠖⠈⠀⠀⠀⠀⠀⠈⠢⣀⠂⠀⠀⣀⡠⠔⣂⡥⠙⡊⠕⠋⡁⠔⠊⢁⠠⠐⡠⠒⡁⠔⡡⠔⠉⠩⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠤⠀⠘⠁⠀⠐⠳⠆⢉⡅⠊⢁⠠⢰⣍⣠⠔⢁⡠⠐⠈⡠⠐⢁⠄⠊⡠⠊⠀⢀⠔⠁⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠬⢑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢺⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢰⠃⠀⢰⠆⣁⡠⠒⢁⠠⠂⠁⠔⠁⠀⡠⠂⠁⡠⠈⢀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠑⡀⠁⠂⠄⡀⠀⠀⡀⠀⠈⠉⣨⠄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣠⣰⠀⠈⢉⡘⡧⢤⡊⠁⡀⠔⠁⠀⡠⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠂⠒⠒⡈⠁⢀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠑⠋⠈⡁⠠⠀⠂⣉⣙⠈⠱⠒⡰⠣⡄⣠⠂⡠⠔
⠀⠀⠀⠀⠀⠈⠛⠓⠒⠒⠒⠃⠀⠀⢀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠈⢉⡁⠠⠂⡰⠀⠀⠀⠀⠀⢀⠠⠒⠉⢄⡒⠢⣌⣉⠒⠩⢴⡒⢬⠉⡑⢄⡈⣑⠒⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⡧⠀⠒⡿⠀⠀⠀⠀⠀⠀⡀⠄⠒⠈⠀⠠⠒⠈⠀⠀⠐⠀⣠⢊⡀⠎⡉⠒⠠⢍⣉⠒⠒⠅⣓⠢⠬⠢⣒⠆⢅⠙⠠⡁
⠀⠀⠀⠀⠀⠘⡗⠀⠁⠀⠀⡇⢀⣠⣿⠀⢀⠤⠐⠀⠁⠀⠄⠂⠁⠀⠠⠀⠀⠀⡠⢒⢊⠣⣈⠦⡑⡂⠚⢷⡤⢀⡈⠓⠢⢤⡁⠒⢄⠈⠣⢄⠈⠓
⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠰⡏⢁⡠⠚⠀⠁⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⢀⣤⣊⣀⠛⠩⣉⠲⠤⡀⡉⠂⠤⣍⠢⢉⡐⠢⠄⢈⡃⠂⢌⡐⠤⡁⠒⢀
⠀⠀⠀⠀⢀⡠⡇⠀⡁⣀⠤⠊⠁⠀⢀⠠⠂⠀⠀⠀⢀⣀⡄⠂⠀⠰⠐⢈⣘⠆⠁⡑⠦⠄⢉⡒⢄⢉⠒⠤⢈⠐⠢⠬⡉⠒⢢⡈⠒⠠⢈⠑⠢⢕⡂
⠀⠀⠀⠀⠀⢀⣡⠆⠙⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤⠌⠀⠀⣠⠟⢁⡘⠉⡀⠱⡄⣈⠑⠂⣤⠜⣠⢄⠉⠲⢄⡙⠢⠀⡈⠢⢄⡉⠢⢀⡀⠉⠠⢀⡉
⠀⠀⠀⠀⠴⠛⠁⠀⠀⡀⠈⠐⠀⠀⠀⠀⠀⠀⠘⠀⢰⣤⠞⠃⢾⡁⠈⠢⣌⠒⢄⢀⠙⠂⢄⠀⠘⢄⠑⠢⠤⡈⠢⣄⠉⠢⡀⠈⠉⠂⢌⠠⢄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡞⠀⡎⠉⠦⡀⠑⠠⡀⠑⠆⣀⠙⠢⡀⠙⢤⡀⠉⠒⠢⣈⠒⠤⠉⠒⣌⡒⠤⡀⠀⠈⠢⣉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠒⠀⠉⠒⠀⠘⠂⠀⠀⠑⠀⠈⠐⠀⠈⠂⠀⠀⠈⠀⠀⠂⠀⠀⠁⠀⠉⠂⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⡌⢫⢁⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⢷⡴⡾⠁⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⢾⡜⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡁⢳⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⣼⡧⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⣛⣟⣛⣛⣛⢓⣚⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⡞⣿⠻⢗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣡⢦⣄⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢰⠀⠀⣿⠀⢰⠇⠀⣰⠂⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠙⢦⠀⠹⡀⠀⠘⡇⠀⡏⠀⡼⠀⢠⠃⠀⢀⡞⠁⠀⠀⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣄⠀⠙⢦⡀⠳⡀⠘⠀⠀⠃⠀⠃⠀⠃⢀⠃⠀⡠⠋⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⡀⠉⠢⢄⠙⠦⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠔⠁⢀⠴⠊⠀⠀⠀⠀⣀⣤⠴⠾⣭⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡽⠖⠒⠲⣤⣀⡈⠁⠀⢠⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠤⠔⠚⠩⠋⢀⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠹⠞⠙⢆⢸⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠖⠂⠀⢼⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣦⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣐⣶⣲⣆⣾⣾⣿⣦⣼⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡀⢼⣆⠀⠀⠀⠀⠐⢰⠑⢺⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣇⣀⣀⣀⣼⣿⣫⣿⣿⣿⣿⣿⣿⣷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣶⣶⣤⣄⣠⣯⡴⣯⠷⡶⢶⠗⡶⠲⡶⢦⡤⣤⣤⣦⣼⡀⢸⣯⣿⡽⠷⠿⣟⣛⣛⣿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⡟⣿⢩⠇⣿⢁⡏⢠⡏⣼⢁⡇⣸⠁⡾⢠⠏⠴⠡⠞⣱⠯⢛⣫⠥⠖⣂⣩⡭⠤⠔⠒⣒⣒⣒⣒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⣸⢰⡇⣸⠁⠾⠀⣋⣨⡤⢤⠲⡖⢦⠒⡺⠖⣉⠔⢊⡩⠔⣚⣩⠥⠒⢒⣉⣉⡭⠭⠭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⣉⣴⢲⡙⣏⠳⡜⢆⠳⠈⡠⢊⡠⢊⡠⠚⣩⠔⢋⣡⠴⠚⣋⡩⠤⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠷⠘⡆⢱⡘⢠⠎⡠⢋⡴⢋⡴⢋⡤⠚⢉⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠉⠀⠉






⡿⢿⠿⠿⠿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠚⠟⢉⡄⠀⠀⠀⠀⠀⠀⠀⢠⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣤⡄⠀⣀⣀⡀⠀⠀⠐⠋⠍⣾⣟⣡⡞⠂⣶⣀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⣭⢬⡄⢠⣿⡿⢿⣿⢷⡲⢀⣺⣿⣿⡿⣷⢀⣭⠂⠠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⠀⡶⠀⠴⣧⢬⡼⣏⢸⣿⡧⣴⣄⢸⣿⢿⠁⣽⣷⢿⢘⣭⠄⡀⣈⢻⣿⡆
⠀⠀⣠⣠⣤⡄⠸⠀⠇⠀⠀⠉⠈⡇⢑⠚⠛⠓⠛⠛⠾⡿⠏⠛⠋⡙⠖⢋⡤⢸⠁⠘⡆⣿⣇⢴⢦⠄
⠀⠀⠈⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢳⢀⣀⣀⣀⣔⢴⣶⣶⣿⣦⡀⠛⠶⢋⣤⢸⡀⢀⠇⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣺⣷⠀⢠⣤⢄⣚⣋⡤⢌⡲⣿⡟⢉⢿⣿⣇⣛⡒⢏⡵⡀⠳⣊⡴⠀⣰⣶⡶⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡺⢿⣶⣿⣿⣶⣶⣶⣶⣖⡻⣿⣇⣸⣿⣿⣿⣿⣿⣯⣽⢿⣦⣀⣠⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠋⠁⠀⠀⠉⠼⣿⣿⣾⠿⣿⣿⡿⠛⠋⠉⠉⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⣤⠀⢠⡆⢠⠀⠀⠀⠀⠀⢠⣤⠀⢻⣇⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣰⠄⡀
⠀⠀⠀⠀⢀⡻⣼⡸⢡⡞⡀⡆⢀⣤⡀⢘⡿⠀⢸⡿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣾⢇⡄⣠⠂
⠀⠀⠀⢀⣿⢳⡟⣧⣿⣹⣷⡇⠀⠛⠛⢿⣿⡀⣿⡿⠿⠗⠀⢀⣠⣀⣀⠀⠀⠀⠲⣼⣿⣿⣿⣿⣿⣿⠞⡁
⠀⠀⠀⠀⢿⣸⡿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⣟⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⠿⣿⣯⣿⡟⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠅⠀⠀⠀⠹⢿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠸⣿⣷⣾⣶⣾⡿⠀⠀⠀⢰⣤⣤⣿⣤⣤⡆⠀⠰⣯⣿⣿⣿⣧⡆⠀⠀⠀⣷⣶⣿⣿⣶⣾⠇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⠀⠀⠸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠀⠀⠀⠀⠀⠘⠿⠿⠿⠃⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⠀⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣶⠒⣦⣰⡆⠀⢰⡖⣦⢠⣦⡀⣿⠐⢶⡖⠀⠀⢠⡶⣦⢶⣄⡶⢲⡄⢠⣶⣤⣶
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⢿⡇⠀⣾⣅⣻⢸⡏⣧⣿⠁⣿⡇⠀⠀⢸⡇⠉⣿⢱⣷⠈⣷⢿⡁⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠺⠧⠶⢿⠉⠿⢸⡇⠙⠿⠀⠿⠇⠀⠀⠘⢧⠀⠿⠈⠷⠼⠏⠸⠇⠀⡿
⢀⠀⢆⠀⢣⠀⠀⡄⠘⢠⡀⠀⢂⢄⠀⠀⢄⣀⡀⠀⠐⠒⠀⠐⠂⣀⡀⡄⠄⠊⣀⠤⠈⢀⠄⠀⠠⠊⢀⠀⠀⡄
⠈⢣⡀⠳⣄⠙⢦⡈⠓⣤⡉⠓⣤⣈⠘⠶⢤⣄⣉⡁⠛⠋⠁⠉⣁⣉⣠⡤⠔⠋⣁⣤⠞⢁⣠⠚⢀⡴⠊⣠⠜⢀⡔
⠳⡀⠙⢄⠈⠷⣄⠉⢆⡈⠙⠧⢤⡉⠙⠲⠦⣤⣉⣉⢉⡉⢉⣉⣉⣩⣤⡴⠶⠋⣉⣤⠖⠋⣡⠄⠉⣠⠞⠁⡠⠋⢀
⣆⠘⢷⡄⠱⣦⡀⠳⣦⡈⠛⠦⣄⣈⠛⠒⠦⣤⣄⣀⣉⣉⣉⣉⣉⣥⣤⠶⠖⠚⣉⣤⠴⠛⢁⡴⠟⢁⣴⠊⢠⡾⠃
⣿⣷⣄⠘⠷⣄⡙⠶⣤⡉⠻⢶⣤⣉⠛⠳⠶⢦⣤⣬⣭⣭⣭⣭⣤⣤⣤⠶⠞⠛⣉⣤⡶⠟⢁⣤⠶⢋⣠⠖⠁⣠⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣛⣉⣥⣴⣶⣶⣦⢹⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠿⠛⠛⠀⠙⠻⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡄⠀⠀⠀⠀⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣄⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢹⣿⣿⣿⣦⡀⠀⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣌⣿⣿⣿⣿⣷⡀⠀⢹⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠿⠿⢶⣤⣄
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣶⣿⠟⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣄⡙⠿⢿⣿⣿⣿⣿⣿⠿⢋⣡⣾⣿⣿⠟⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣶⣤⣬⣭⣥⣴⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⠿⠿⠟⠛⠁

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠂⠄⣀⠐⠒⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⡠⠄⢀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠈⠀⠀⠀⠀⠀⡴⠀⠀⠠⢀⠀⠀⠔⢢⣄⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠲⡆
⠀⠀⠀⠀⠀⠌⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢀⡀⢴⣦⣤⣀⠀⠀⠀⠀⣀⡀⢼⣦⣤⣄⡀⠀⠀⠀⢀⢀⣀⡤⢴⣧⣴⣤⣄⠀⠀⠀⠀⢀⣠⠤⢔⣧⢴⣦⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡏⢢⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠢⢀⠈⠻⣾⡔⢹⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⣤⣇⢱⠢⢸⣻⡏⢭⣑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠩⠯⢻⣷⡶⣸⠀⠑⡢⡶⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠯⠐⠉⢺⠀⢰⣰⡷⣛⢒⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⡙⠛⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡤⠤⢄⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⣤⣤⣤⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⣀⣤⠖⡚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠉⠑⠂⠀⠠⠤⠤⣀⣴⡖⠊⠋⠉⢀⠤⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⠤⠼⠃⢀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡀⠀⠀⠤⠄⠀⠠⠤⠠⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠐
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⢄⠀⠀⠀⠐⠈⡀⠀⠊⠀⠈⠀⠀⠨⠃⠀⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠾⠀⠰⠆⠀⢀⠐⠢⠀⠘⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠸⠀⡼⠀⣴⠐⠀⠠⠀⠢⠉⠀⠄⠀⠀⢀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⡀⠐⠀⠀⠀⠀⢁⡀⠀⠃⠐⠉⠛⢘⡂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠁⡄⠀⠐⠀⠀⠂⠄⠀⠲⠤⠀⢂⠐⣸⠁⠀⠠⠀⢠⠤⠰⠋⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠀⣠⠀⠢⣀⠀⠀⣰⡦⠊⡁⠉⢁⠀⠀⠠⢂⠈⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⡑⣦⠀⠀⢠⡘⠃⡐⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠉⢧⠀⠈⣱⡠⠑⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠔⠲⢤⣄⣠⡀⠶⡀⢳⡄⢠⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⣵⢷⣞⣭⣯⣝⣟⡻⣷⣄⡛⣿⢸⡀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠛⠋⠉⠉⠉⠉⠙⣟⣿⣿⣿⢿⡜⡣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠈⣩⠻⠿⣿⠣⡓⢠⡄⣀⢠⠤⠒⡀⢀⠄⠠⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠈⣿⣿⣷⣦⡀⠀⠻⣶⣾⣷⣷⡵⡞⠉⢦⣾⡀⠘⢁⠁⠀⠐⢲⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣆⠀⢹⣿⣿⡿⣷⠃⢀⠞⣽⣟⣈⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠈⠛⠿⡿⠿⣀⣾⣿⣿⣿⣟⡴⣡⣾⡿⠉⠉⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⣿⣿⡟⡠⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⢸⣿⡆⣿⢿⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣷⣴⣶⣶⣾⢿⣿⣿⣿⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⡟⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣟⢽⣿⣿⣿⣽⣿⢻⡇⢹⠈⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣟⣟⠷⢫⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣟⣪⣿⣿⣿⡿⣿⣿⣿⢻⣯⡌⢧⡀⣤⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡛⢿⠛⣿⣹⢷⣁⡅⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⠿⡿⢿⠿⡿⠛⠛⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢶⣠⡆⠀⠀⢰⣦⠘⣷⡀⣦⠵⣶⠶⠜⠃⣶⡀⣴⡆⣰⠶⣆⢰⣆⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠞⢸⡇⠀⠀⣿⢹⡆⣿⢧⣿⠒⣿⠐⠒⠂⣿⣧⣿⡇⡟⠀⣿⢸⢿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣧⣤⣴⡟⠙⡇⣿⠘⣿⠀⣿⠀⠀⠀⣿⠘⢸⡇⢿⣤⡿⢸⡌⠃⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡠⠊⢁⣼⠝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⣯⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢀
⠀⠀⠀⠀⢀⡜⠁⠠⣿⡏⡸⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⣿⢃⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋
⠀⠀⠀⢀⣞⠆⠀⣊⣿⡿⠁⢸⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⣨⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⡞⠈⣀⣠⣾⣿⣅⠀⡞⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢠⣾⡿⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⢸⠐⣰⢃⣻⣿⣻⠃⡸⠀⠀⠀⠀⠀⠀⠀⡈⠀⡴⢀⣿⣿⡑⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⣾⣿⡟⠂
⠀⠀⡇⡈⣟⣽⣿⣟⠇⡰⠁⠀⠀⠀⠀⠀⠀⠀⢃⠀⣋⣿⡿⡑⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣿⣿⣟⠂
⠀⠀⣯⣿⣷⣯⣝⢎⠜⠁⠀⠀⢀⣤⣄⠀⠀⠀⢸⣾⣾⣿⣓⠂⠈⠀⠀⠀⣀⣤⣀⠀⠀⠀⣿⣿⣿⣟⠓
⠀⠀⠱⣽⣿⣿⠞⠁⢀⡔⠛⢺⣻⣿⡎⢷⠀⠀⠣⣿⣿⡷⠋⠁⣠⠒⠛⢿⣻⣷⠙⡆⠀⠀⢹⣿⣿⠟⠋⠀⠀⠀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⢱⣸⣿⠈⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣇⣿⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡅⠀⢸⡧⢷⡄⣅⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⢻⡼⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢷⣀⠐⣿⡟⠁⡏⠀⠀⠀⠀⠀⠀⠀⢹⡆⡆⡀⢸⣿⣿⢸⠃⠀⠀⠀⠀⠀⠀⠀⢖⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⠀⣿⣫⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷⣳⡇⠘⣿⡟⣼⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⠀⣿⣿⠃⠄
⠀⠀⠀⠀⣠⣔⠲⢄⠀⠀⢿⣧⢁⡟⣟⣾⠁⠀⠀⣠⠴⠲⢄⠀⠀⠘⣿⡌⣰⣿⢣⡟⠀⠀⠀⡀⣤⣤⡀⠀⠀⢻⣿⣟⣿⣿⣠
⠀⠀⠀⡼⣵⡻⣹⢿⢇⠀⠀⠻⣧⣿⣿⠏⠀⠀⢰⢧⠈⣇⠁⢧⠀⠀⠈⢷⣸⣿⡿⠁⠀⠀⠘⠀⠻⣿⣿⡄⠀⠀⠹⢿⣿⣿⠃
⠀⠀⢀⣗⡯⡪⠇⠈⢸⠀⠀⠀⠀⠉⠁⠀⠀⠀⡟⡪⠀⠁⠀⠸⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠈⠁
⠀⠀⢸⣼⣏⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠆⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⡇
⠀⠀⢸⢵⣡⡎⡀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⣃⠶⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢸⣾⠿⣷⡗⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠟⡳⠶⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⣆⡀
⠀⠀⠸⣅⠻⢿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠙⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠟
⠀⠀⠀⠙⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡃⠂
⢺⣷⣶⣦⣄⢀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠃⠀⠀⡑⠀⠀⠀⢁
⠚⢯⠏⠉⠉⡟⣀⠀⠀⠃⢀⣓⠂⠒⠐⠐⠂⠐⠒⡓⠒⢒⡒⣿⣶⣶⣄
⠀⠀⠀⠀⠡⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠃⠀⠈⠁⣟⡹⠉⠁⠀⠢⡀
⠀⠀⠀⠀⠀⠈⠡⠒⡲⠂⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠄⠀⠀⣀⡄⣀⠀⠀⡀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⢠⠟⢻⣯⡇⡀⠁⠀⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠄⠀⠈⠶⠠⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠃⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⢾⢉⡧⢼⢁⡞⢉⡏⣰⢇⡇⣸⠃⡀⠀⢀⡟⡑⣾⣰⠏⣹⢊⡟⡗⡗
⠀⠀⠀⠀⠀⠀⠀⡞⠉⢁⣈⣁⢉⣉⣁⣈⣈⣉⣉⣁⣀⣀⣈⡁⠀⠉⠀⠉⠁⠈⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣈⣁⡀⠀⠀⠀⢀⣀⡀⠀⠀⢀⢀⣀⡀⠀⠀⢈⡉⣉⣉⠉⠉⠉⢁⣀⣠⣄
⠀⠀⠀⠀⠀⢹⣿⣷⣯⡵⠀⢰⣿⣿⣿⣷⠀⠰⣿⣽⣷⣖⠀⢸⡟⢿⣿⠆⠀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠿⣿⣿⠇⠀⠘⠻⡿⢿⠛⠀⠙⢻⠟⠿⠛⠀⠉⠻⢿⠏⠁⠀⠻⡿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⣀⠀⡀⠀⠀⡀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⣀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣤⣤⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳⣸⣠⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠷⠶⣿⣿⣿⣵⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣫⣶⣍⡑⢦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⣿⡿⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣣⣿⠋⠇⢸⣿⠌⠻⣝⣝⠉⠒⡄⣀⡀⢀⣠⣤⣬⣵⠀⠀⠀⠀⠀⣻⣻⠃⠁
⠀⠰⠖⠂⠀⠀⠒⠲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣋⡙⠿⢠⣿⣯⣿⠶⠆⣀⣴⣼⣴⣄⣀⠠⠤⠤⠶⣶⣶⡆
⠀⠀⠶⠶⠶⠶⢦⣠⣤⣄⡍⠿⠿⢝⣻⣟⣻⣿⣿⢸⣿⣃⢸⣿⣿⣿⣏⣽⣦⣶⣼⣿⡏⠉⠉⢹⣿⣄⣀⣤⣤⣿⣿⣛⣭⠁⠐⠒⠂⠐⠒⠒
⠀⢠⣤⣤⣤⣤⣤⣤⣬⣤⣖⣲⣿⣿⣿⠿⠿⠟⠛⠘⠛⠋⣻⣟⣛⠛⠋⠙⠛⠛⠘⠛⠃⠀⠀⠚⠿⠿⠿⠛⢿⣭⣷⣴⣾⣴⣶⣶⣤⣤⣠⣤⡀
⠀⠸⠿⠿⠟⠛⠛⠉⢉⡩⠁⢀⣠⣴⣶⡄⠀⠀⠀⠀⠀⠀⢙⣿⣏⣤⣤⣰⡤⣴⣤⠴⣶⣤⣤⠦⢀⣀⣀⡀⣀⣛⣻⠿⠿⠿⡿⠿⠟⠻⠿⠿⠇
⠀⠀⣀⣿⣇⣤⣜⣅⣀⣦⣤⣏⠙⣿⣟⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣟⣱⡇⠢⣦⣿⣿⠃⣠⠈⢻⣿⣤⠈⠛⢛⣦⣤⣤⠂⠀⠀⠀⡀⢀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⣿⣶⣀⡀⠀⠀⡀⠀⠀⠈⠙⠿⣿⣿⣿⣷⢀⡟⡿⠿⣢⣿⡀⢸⣿⠿⣆⠀⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⡀
⠀⠠⣶⣾⣿⣽⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⡷⠗⠖⠺⡿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣇⣹⢇⣿⣿⠗⣾⣶⣾⣿⡶⣶⣾⣿⣾⣶⣶⣶⣶⣿⣿⡄
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣨⣯⣗⣀⣐⣶⣧⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣻⣻⣛⣿⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣸⣾⣧⣴
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢇⣽⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣦⣤⡀⠀⢿⣿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⢸⠀⠰⠟⠛⠛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⢀⣤⣤⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣼⣿⡿⠃⠀⠀⠀⢠⣿⣿⣿⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⠙⢻⠋⠀⠀⠀⠀⠀⢸⣿⠟⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠙⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⢠⣿⡄⠀⠀⠀⢀⣤⣄⡄⠀⠀⠀⠀⢀⣤⣴⠂⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠘⠛⠁⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣆
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⢀⡟⠀⡄⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣴⠃⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⡄⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠈⣧⡀⠻⣤⠀⠀⣠⠇⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠷⣄⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⢰⠇⢀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠸⣆⡀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣬⣥⡤⠞⠁⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⠤⠴⠶⠶
⠀⠀⠀⠀⠀⠛⠛⠓⠶⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠦⣄⡀⠀⣀⣠⡴⠖⠛⠉⠁⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠶⠒⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠘⠲⠶⢤⣤⣀⠀⠀⠀⠀⠀⣀⣤⠿⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢆⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠋⠉⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁
⠀⠀⠀⠀⠈⠉⢓⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠫⠉⠒⠂⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠈⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⢀⠄⠀⠈⠙⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⠁⠀⠀⠐⡀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢑⠔⠛⠢⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠃⢠⣃⣠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡰⠀⠀⠀⣀⠴⢟⠉⡀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠠⠁⡉⠍⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠐⠁⡀⢸⠄⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠘⠁⢤⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⡘⡆
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠂⠀⠀⡀⢵
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⡀⠀⢠⣅⢀⠀⠄⠂⠁⢀⠘⡆
⠀⡟⠁⠉⠒⠴⢀⣀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠴⠶⡆⠀⠀⠀⠐⠲⣄⠀⠀⠊⠉⠙⣇⠀⠉⠉⡇⠀⠀⠀⠂⠀⠀⡀⠠⡀⢠⡇
⠀⣇⠀⠀⠀⠀⠀⠉⠀⠄⠀⠈⠀⠀⠻⣄⠀⠀⠀⣧⠀⠀⠀⡴⢂⠈⡆⠀⠀⠀⠀⠰⡆⠀⠀⢱⠀⠀⠀⠤⠄⠲⡀⠲⠀⢴⡇
⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⣀⠀⢸⠀⠀⠀⡇⢀⠀⣿⠀⠀⠀⢤⠀⠀⠰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡀⠀⠀⠸⡆⠀⠀⠛⠂⢀⠇⠀⠀⠀⢸⠀⠐⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠸⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⡀⡈⠇⠀⠀⠂⠘⠀⠀⠐⠠⠔⠋⠀⠀⠐⠀⠘⠀⠀⠀⠂⠀⠁⠀⡄⠀⠀⠀⠀⠀⢐⠃
⠀⠀⠀⢣⡀⠈⠢⠐⠋⠀⢀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⡄⠀⢐⠀⠀⢀⠆
⠀⠀⠀⠀⢣⡀⠀⣠⠐⠘⠀⠀⠐⠀⡂⠀⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠢⡈⠶⠐⠃⠀⡌⠀⢠⠋
⠀⠀⠀⠀⠀⠱⡎⠀⠀⠀⢰⠂⠊⠐⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣤⠐⡏⠀⠀⢠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢳⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢂⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠌⢀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠤⠀⣀⣀⡀⠀⠀⣀⠀⢀⢀⡠⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠀⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠰⢆
⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒
⠀⠀⠀⠀⠀⠀⠘⢀⠠⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⠆⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⡀⠈
⠀⠀⠀⠀⠀⠖⠒⡺⢆⠀⠀⠀⠀⢨⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⣀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡆⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠭⠭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠈⠂⠀⠀⡄⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢠⠄⠉⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠈⢇⠉⠀⠂⠒⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⡉⠐⡶⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠐⠉⠐⡧⠲⠎⠀⠒⠄⠀⠀⠀⠀⠐⠈⠁⠉⢦⠒⠁⠀⠀⡀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠖⠊⠓⢄⠀⠤⣀⡔⠀⠀⠀⠀⢠⠀⠀⠀⣸⠂⠄⠀⠐⠁⠀⠀⠀⠀⠀⠐⠦⠤⠼⠃⠠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠤⠀⠀⠀⠀⠀⠄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣰⠆⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠁⠀⠀⠁⠀⢀⡀⠀⠠⠤⠤⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠁⠂⠄⠀⠀⠀⠀⠀⠓⠒⠁⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠰⣦⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⢠⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠚⠥⡁⠘⠄⡀⠢⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠀⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠑⢄⠀⠈⠄⠀⠠⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠁⠀⠀⠠⠤⠤⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠀⢣⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢆⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠱⣄⠈⠢⡀⠀⠀⠀⠀⠀⠀⢰
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀
⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠑⡀⠀⠀⠀⠃⠀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠘⠄⠐⢀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠘⠃⢸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠉⡀⠀⠀⠀⢀⡀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⠘⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠈⠁⢐⠉⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⡀⠀⡂⠠⣈⠡⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢴⠔⠀⠀⠀⠇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⡠⠈⠀⡐⠂⠑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢂⢀⡶⠀⢀⠀⠀⢴⡀⠈⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠚⠁⢠⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦
⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣭⣽⣮⣭⣽⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣾⣷⣶⣤⣼⣿⣿⣭⣭⣭⣽

⡿⡍⠉⠉⢹⠉⠉⠉⡏⠉⠉⡿⠉⠉⢉⡟⠉⠉⠉⡹⠋⠉⠉⠉⠉⡩⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠹⡄⠀⠘⡇⠀⠀⡇⠀⢠⡇⠀⢀⡞⠀⠀⢀⠞⠁⠀⠀⢀⡴⢊⡂⠤⠀⠀⠀⣀⠤⠀⠀⠋
⡇⠀⢹⡄⠀⢻⠀⢀⡇⠀⣸⠀⠀⡾⠀⠀⣰⠋⠀⠀⣀⠔⠉⠀⢀⡀⠠⠀⠀⠀⠛⠁⠀⠀⠀⠀⣀⠠
⡷⣄⠀⢳⡀⠸⡇⢸⡇⢀⡟⠀⡼⠁⢀⡞⠁⠀⣠⠞⠁⠀⣄⢐⣵⡄⠀⠀⠀⠀⠀⣀⡤⠔⠂⠁
⡇⠘⢧⡀⢷⡀⣷⢸⡇⣸⠃⣼⠣⠴⠏⢀⡴⠞⠁⠀⠀⠀⢀⠀⠘⠃⣀⡤⠴⠚⠉
⡟⢷⣌⠻⣌⢷⠹⠞⠇⠿⣼⣧⡴⠂⣿⣿⣷⣶⣶⣶⣤⣤⣽⡵⠶⠌⡏⠀⣀⣀
⡟⢶⣍⡳⠉⠀⠀⠀⠀⠀⠀⣿⡿⠁⢿⣇⣝⣙⡿⠻⢿⣿⢻⠃⣰⣧⠤⢶⣽⣿⣷⡄
⣟⡳⠮⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠀⢸⣿⠿⣿⣹⡙⠿⢿⠸⣦⡟⡀⠈⣿⣿⣿⣿⣻⠉⠉⠉⠀⠀⠀⠀⠀⡀
⣯⣭⡅⠀⠀⠀⠀⠀⠀⠒⣆⠀⢠⣤⠾⠏⢼⡿⡛⣾⣿⣿⣿⣿⡟⢇⣠⣿⡟⠀⣿⢋⣀⣀⠀⠀⠀⢀⠞⠁
⡷⠶⣖⡀⠀⠀⠀⠀⠠⡷⢾⠤⠄⠸⠦⠀⠸⣆⠢⣸⢿⣿⡏⠉⢁⣆⠙⢀⡋⣰⠏⠀⠀⡩⠋⠀⠀⠀⠰⠈
⡷⠞⠫⠶⠀⠀⠀⠀⠤⡭⠠⣤⠤⠤⢤⣤⣘⣹⣼⣾⣬⣟⣀⣀⣠⣩⣇⣸⡁⢛⣂⣐⣀⣀⣀⣠⣤⣤⣤⣤⣄

⠲⣾⣴⡶⠀⠀⠀⠀⠀⠀⠀⢀⣤⠒
⢺⡟⡿⠿⡷⠀⠀⠀⠀⠀⠠⠞⠛⠒⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⣾⣿⣟⣤⣰⣾⣦⠛⣶⣿⣿⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣚⣳⣶⣶⣦⣾⣿⣿⣷⣶⡶⠀⣠⠀⢶⣶⣾⣾⣿⣶⣆⣀⣀⣤⣠⣄⡀⣀⣤⡀⢰⣷⣶⣶⣂⡀
⠻⣿⠿⠿⣿⠿⠿⠳⠾⠿⠿⠤⠁⠀⠀⠀⠀⠸⠆⠀⠀⠉⠉⠉⠉⠉⠿⠏⠁⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠏⠋⡉⠁⠉⠉⠉⠉⡉⡍⡛⢛⠋
⡋⠤⠚⣁⠤⠖⠊⠉⣀⠴⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠠⠉⠈⢉⡉⠀⠒⢀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠡⠀⢈⠀⠈⠀⠂⠠⢀⠀⡀⢆
⣠⠔⠋⢁⡠⠴⠒⢉⡠⠔⠂⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠠⢄⡀⠀⠠⣄⠀⠐⠀⢀⠀⠀⢀⠄⠂⠀⠠⠄⠈⠀⠀⠀⠀⠀⠄⠁⠀⠐⠀⢀⠀⢸⠀⠇⠸
⢀⣠⠔⠉⠀⡠⠀⠉⠀⠠⠄⣀⡀⠑⠒⠤⠄⡀⠁⠐⠀⠀⠀⠐⠂⠀⠉⠋⠀⠀⢀⡤⠀⢀⠠⠊⠁⡀⠔⠁⠀⠀⠊⠀⠀⣠⠞⠀⠀⠀⠌⠀⢀⠄⠀⡀⠀⢷
⠋⠁⢀⠔⠋⠀⡀⠚⠀⢤⠀⡀⠉⠠⠶⡶⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠖⠉⢀⡠⠊⠁⣀⠄⠁⠀⠀⢀⠞⠀⠀⠀⣠⠋⠀⠀⠐⠀⢀⡇⠀⣸
⠀⠐⠍⠀⣠⠎⠀⣀⣀⠀⠁⢀⣄⡄⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢉⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂⡀⠂⠀⠀⠀⡄⠀⣼⠀⠀⡿
⠀⣀⠐⠢⠀⠠⣄⡈⠉⢀⡀⠀⠐⠀⠠⣤⡖⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠃⠀⠀⠀⠁⠀⠚⠀⠤⠀⢀⠀⠀⡐⠀⠀⠄⠀⠠⠀⠀⣀⠀⢀
⠣⠈⠁⠀⠐⠦⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⡀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⣀⠀⠁⠀⠘⠃⠀⠰⠖⠀⡤⠀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢶⡦⠠⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣴⣦⣼⣷⣿⣶⣾⣧⣿⣿⣧⡀⠀⠀⠀⠀⠀⣴⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⡀⢀⣤⣿⣿⣁⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿
⠀⠠⠶⢶⢶⡴⣶⣦⠴⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣤⣤⣤⣤⢈⡙⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣷⣶⣴⣆⡷⣀⣠⣴⣠⣤⣤⡤⣶⣯⣠⢙⣛⡟⡿⣿⢿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣨⣸⣇⣷⣽⣞⢧⣿⣿⣿⣿⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⢿⣯⣻⣷⣟⢿⣿⡿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠸⣿⣧⠙⣿⣦⠙⢿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢺⣿⣿⣧⣿⣿⣎⢿⣿⣿⣆⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣇⡙⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣎⣽⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⡇⠆⠋⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡿⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣾⣿⣿⣿⣿⠻⡾⠯⢹⠿⢿⣿⣿⣿⣿⢿⢦⠙⠟⠀⠠⢿⣿⣿⢿⡿⣿⡿⠄⠀⠂⠉⢻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠾⣀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢼⠖⣞⢁⣦⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢻⡏⠛⢋⠏⢁⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⢠⣴⡆⠀⠀⠀⠀⢀⡜⢷⠷⡌⣤⣞⠯⢺⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡊⠀⣄⣀⡄⠀⢀⠙⣯⡘⢳⣠⡄⢹⣧⠜⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⢠⠊⠈⠀⠀⠀⠀⠀⠄⠉⢧⢸⠤⠋⢋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⠀⠀⠀⠀⠀⠀⢌⣹⢢⠀⡦⠀⣄⠀⠀⠀⢠⣈⡳⢑⠀⡰⢅⣰⣨⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠠⠦⠀⠀⠀⠀⠀⠞⠁⠙⠒⠶⠚⠃⠀⠀⠀⠰⠏⠹⠓⠲⡰⠛⠫⠍⠁
⠒⠀⠉⠁⠀⠒⠒⡄⠀⠀⠀⠐⠀⠁⠙⠂⠒⠒⡀⠀⠀⠀⠐⠁⠁⠙⠓⠒⠒⠂⠀⠀⠀⠐⠘⠋⠚⠛⣛⣶⣆⠀⠀⠀⢐⠻⠿⠿⠿⣿⣶⣦
⠠⠤⠤⠄⠄⠤⠰⠁⠀⠀⠀⠀⠤⠄⠤⠀⠤⠠⠁⠀⠀⠀⠀⠤⠄⠤⠄⠤⢰⠃⠀⠀⠀⠀⠤⠤⠤⠀⠡⠀⠃⠀⠀⠀⠀⠠⠤⠤⠄⠡⠔⠎
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣀⣄⣀⣴⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⡀⠀⣀⠀⠀⠀⢀⠀⡀⢀⣰⡀⠀⢀⠀⠀⠀⢀⠀⡀⢀⣸⡄⠀⢀⠀⠀⠀⢀⡀⡀⢀⣠⡆⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣈⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⢃⡄⢐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣀⠄
⠀⠀⠀⠓⢜⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⡏
⠀⠀⠀⣀⡐⣶⣷⡂⠀⠀⣴⠾⢤⣀⣀⠈⠀⠒⠀⢀⣠⠄⠊⢸⣿⠋⡹⢣⡀
⠀⠀⠀⠈⠁⠀⠠⢤⡖⢀⣻⢲⡶⢻⣿⣟⣦⣤⣞⣉⠤⠀⠘⣼⠃⣠⠇⣸⡏⢉⣳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⡜⠉⢲⣼⣾⣥⣬⣇⣈⣚⣉⢣⠀⠀⠉⢡⢇⣼⠃⠀⢻⣇⢈⡟⠻⡗⠛
⠀⠀⠀⠀⠀⠀⠀⢷⠐⡼⣻⡟⡏⡟⠙⠟⡿⢻⠿⡿⣿⣷⣾⣿⣾⣿⣼⣾⣿⣾⣷⣤⣿⠰⠦
⠀⠀⠀⠴⣶⣶⣦⣬⣯⣀⣐⣧⣶⣿⣿⣷⣷⣮⣤⣄⠧⣇⡅⡥⡁⡻⡿⠟⢟⣟⣷⠛⢿⣶⠄⠤
⠀⠀⠀⠀⠀⠀⠨⣭⣍⣀⣀⣀⣈⡉⠉⠙⠛⠻⠷⠿⢾⣿⣛⣽⢓⠛⢧⠀⢘⠁⠈⡄⠈⠉⠁
⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣶⣶⣶⣶⣴⠦⠶⠶⠦⠤⠤⠤⢤⡭⣭⣝⣛⣳⡆⢀⣀⠀⠄
⠀⠀⠀⠈⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⣤⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣈⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣿⠀⢀⠤⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡆⠀⠀⠳⣀⠈⣷⠀⠀⣇⠴⢯⡁⢀⣠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠲⡄⠀⣏⢄⣠⠃⠀⠀⠀⠈⠉⠁⠱⡼⠃⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣳⣎⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⢀⡴⠚⠉⠉⠉⠙⢦⡀⠀⠀⠀⠀⣠⠖⠋⠙⢏⡉⠳⣄⠀⠀⠀⠀⢀⡴⠚⠉⠙⢏⠙⠦⡀
⠀⠀⠀⡞⠀⠀⠀⠀⡄⠀⠀⢹⡄⠀⠀⢰⠃⠀⠀⠀⣀⣩⠆⠈⣧⠀⠀⠀⡞⠀⠀⢠⣴⡞⠀⠀⢹⡄
⠀⠀⠘⡇⠀⠀⢰⡿⠃⠀⠀⢀⡇⠀⠀⢿⠀⠀⠀⢺⠟⠁⠀⠀⣼⠀⠀⠘⡇⠀⠀⠨⡾⡱⠀⠀⢀⡇
⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠘⢦⡀⠀⠈⠁⠀⠀⣰⠃⠀⠀⠀⠻⣄⠀⠀⠃⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠈⠓⠶⠤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⠖⠋⠁⠀⠀⠀⠀⠀⠈⠓⠶⠤⠴⠚⠋


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

⠚⢀⡈⢥⢚⠀⢛⡠⠚⠁⢙⡇⢸⠂⠀⠀⠘⠡⣿⡇⠀⢀⠀⠀⢀
⠓⠆⢤⠌⠭⡴⠂⠀⠱⠤⠀⠀⢀⠁⠀⢠⠀⠠⢉⠴⢶⣜⣁⡤⣴
⠸⢏⡴⢀⡔⠄⣰⠯⡴⠂⢠⡉⡘⢁⠄⠸⠆⠀⡈⠂⠞⠁⠉⣤⡾⠛⠂
⠀⠃⠀⠤⠐⠠⠀⠀⠜⣓⢚⠂⠀⠈⠈⢤⠀⠀⣀⠡⡦⠀⠀⣈⣳⡿⡁
⡀⢀⠀⠈⠠⠂⢀⡠⠤⠜⣷⠀⡄⣹⠀⠀⠀⠠⠄⠈⠈⠁⠈⠈⠑⠶
⠑⠈⠀⠀⢀⠀⠃⠀⠈⠀⠀⠾⠂⠨⠇⡀⠀⠀⠀⠀⢀⠔⡠⠠⠀⠄
⠀⢠⠀⠈⠌⠀⢐⡄⠀⣂⡰⣄⢠⠌⢾⢉⠀⠎⠀⠀⠀⠀⠁
⠀⠈⠀⠀⠀⠀⠀⢨⡦⠀⡅⠁⠈⠒
⠀⠀⠘⠀⠀⠀⠀⠀⠈⠳⠧⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢠⡇⢀
⠈⠁⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣐⠒⢀⠉⠂⣀⠀⡙⠄⠙⠂⢡⣨⠔⠈⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠠⠂⢂⡀⡌⢡⠰⠀⠂⠀⠀⠀⠁⠀⠀⠀⢰⠢⢄⡠⠈⠗
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠣⡈⠉⠀⠀⠒⢀⠆⠊⠏⢀⠤⠀⢀⡄⢠⠄⠠⠠⠒⠈⠀⠉⠀⠈⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢠⠁⠄⠀⠀⠀⢠⠠⡄⠀⠀⠈⢠⠀⠠⠀⠀⠒⢔⠇⠞⠮⠴⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠘⠀⢲⡀⠐⠀⡀⠀⠀⡑⠀⠀⠚⢠⠇⡅⠀⠀⠘⠀⢁⠄⠀⡀⡀⢠⠀⠠⣀⣽⡌⡉
⠀⠀⠀⠀⠈⠘⢤⠀⢂⠀⢀⢀⠀⠀⠀⠀⠀⠆⠀⠂⠊⠀⠀⠀⠀⠀⠀⠠⠁⡏⠰⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠂⡄⠀⠠⡏⠠⠉⠀⠀⠀⠀⠀⠰⠃⠀⠙⠀⠐⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠂
⠤⠁⠀⠀⠀⠀⠘⢶⠃⠈⢲⠁⠀⢄⢀⠀⠀⠠⠁
⢀⠀⠀⠀⠀⠀⠀⠀⠑⠂⡀⡀⠀⠌⡐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⣀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢶⡀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠤⣼⠁⠀⣿⠋⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡄⢀⡜⠓⠚⠳⣄⣠⠼⠓⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡞⠉⠁⠉⡏⠀⠂⠀⠀⠈⠇⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣄⣤⡅⠀⠀⠀⠀⠀⡜⠉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠱⣌⠀⠀⢀⡔⠣⣀⣀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣤⣤⡾⠁⠉⢹⡁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣠⡎⢹⡟⠲⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢹⠇⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣠⠟⣰⣥⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠒⠤⣀⢽⠃⢰⢏⡤⠒⠒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⠀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠋⠉⠉⠉⠙⠉⠉⠉⠓⢦⡤⠖⠶⢤⣀⣤⣀⣤⠴⠶⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠋⠀⠀⠀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠹⠴⢏⠀⢀⠔⠉⢀⡠⠂⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⣀⣀⣀⣀⣀⣷⣃⣀⡴⠋⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠤⠛⠲⢄⠀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣨⠶⢦⡴⠚⠁
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠦⠤⠤⠶⡞⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⠛⠛⠛⠷⠶⠶⣤⣤⣄⣀⣀⣀⣠⣤⡾⠛⠁
⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠉⠲⠦⢀⣀⣀⣀⡴⠟
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⣀⣀⣀⠀⠀⠀⠀⢤⣀⡀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⣬⡤⠠
⣶⠶⠶⠆⠀⠀⠐⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠉⠭⠅⠈⠉⠩⠍⠁⠠⠈⠀⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⠤⠼⠇⠐⠀⠐⠐⠓⠒⠒⠀⠀⠀⠀⠒⠒⠀⠄⠀⠠⡄⣀⠀⠠⠄⠒⠂⠀⠚⠛⠒⠂⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⠛⣛⡋⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠒⠒⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠺⣿⠋⠀⣿⣿⣧⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⡀⠀⠀⠀⠀⠉⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠃⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣠⠞⣀⣤⢶⣾⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢟⣵⠾⠋⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣵⣿⡏⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠻⣷⠀⠀⠀⠀⢀⡧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢷⣄⣀⣠⠞⠁⣼⠉⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢻⡄⠀⠩⠉⠁⠀⠀⡏⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢻⡀⠀⢃⠀⠀⠀⡁⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⡀⠀⢷⠀⠀⠀⠀⢠⣧⠄⠀⢠⡟⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⣠⣿⠀⣨⣧⠧⢈⡉⠙⠷⠶⠶⣼⣀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⢰⠟⢿⡖⠁⠘⠓⠉⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⢀⠀⠀⠋⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⢀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠈⢿⡇⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⣆⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠙⠢⣄⣀⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣧⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⢀⡼⠟⠋⠉⣉⠓⢤⠀⠀⠀⠀⢸⠀⠀⠀⠹⣧⠀⠀⠀⣀⣼⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⣠⠋⠀⠀⠀⠠⣭⣛⣌⠁⠀⠀⠀⢸⠀⠀⠀⠀⠹⣇⣠⡾⣏⠉⠛⠋⢁⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡿⠶⢶⡾⠋⠛⠚⠃⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠋⠀⠹⣧⣀⡀⢻⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⡼⡏⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢸⠁⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⣸⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⡏⠀⢿⡆⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⡟⠀⠀⣼⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢰⡇⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⡆⢸⡄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣸⠀⠀⢸⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢹⡎⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡟⠀⠀⣾⠁⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢠⠇⠀⠀⠻⣷⠤⠤⠤⠤⠀⠓⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡏⠀⢀⣠⣀⣀⠀⠀⢀⡹⠀⠀⠀⢠⠀⠀⠐⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣴⡆⠀⢀⡀⠀⠀⠀⠘⠛⠛⠋⠉⠉⠉⠛⠛⠉⠀⠀⣀⣾⠛⣤⠶⠶⡶⠶⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⡠⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢹⡀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣍⡉⠉⠓⠦⣄⣿⠀⢸⡇⠀⢸⡄⠀⠀⢀⣠⡤⠴⠶⠖⢒⡚⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠈⣿⠀⢸⡇⠀⢸⠃⣠⠞⠋⠁⠀⣀⡤⠞⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡹⣆⢹⡄⠸⡇⢀⣏⡞⠁⢀⡤⠖⠋⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⠷⣄⠀⢷⢠⣷⢿⢏⣠⠞⠉⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠾⠿⢿⣿⣷⣾⣿⢁⣾⣏⣤⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢒⣺⣷⣶⣯⣿⣻⠟⣁⣀⣤⣶⣚⣛⠉⠉⠉⠓⢦⡀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠡⠖⠛⢉⡴⢞⡉⠀⣸⣽⠟⠛⠿⣭⣉⠉⠉⠉⠛⠒⠶⠀⢹⡄⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⣶⣶⠾⠿⠚⠉⣙⠋⢻⣿⠉⠙⠛⠒⣯⣭⣿⣛⠒⠲⠶⢤⣄⣳⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣠⠟⣉⢠⡖⠀⠀⠘⣉⠁⢸⢸⡀⠚⠳⠀⣤⠀⠀⠉⢳⡄⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡾⠹⡆⠀⠀⠀⠀⠀⣄⡟⣠⡞⠁⠀⠃⢀⠀⣴⠒⠀⠁⠐⠺⡎⠳⠆⠀⠀⠈⠀⠘⠳⠀⠻⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⠠⡶⠁⠀⠀⠈⢻⡇⠀⡶⠀⠀⠉⢠⠄⠀⠀⠖⠀⠤⢀⣀⠀⠒⠂⠀⠙⠂⠀⣤⡹⠀⡀⠈⠹⣧⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣸⠁⠀⢹⡄⠈⠀⠀⣀⣀⠀⢳⠀⠀⠀⠆⢠⠈⠀⠛⠀⢠⠄⠐⣯⡀⠀⢰⡄⠀⠀⠶⠀⠘⠃⠈⡇⠀⠀⢸⡶⠔⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢻⡄⢰⠋⠀⠀⢳⠘⡇⠀⠀⠀⣈⣀⣀⣐⣁⠈⠀⣄⠁⠀⠀⢀⣡⠴⠶⠶⢤⣄⡀⠀⡆⢀⡴⠋⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠻⣼⡀⠀⠀⢸⢦⡧⠤⠶⠛⠉⠉⠉⠉⢉⡙⢦⣀⣠⣤⣤⣯⠤⠤⢤⡀⠀⠀⠉⠳⣤⡾⠁⠀⣀⡀⠀⠰⣼⢿⡇
⠀⠀⠀⠀⠀⠠⣧⡖⠀⠀⠀⠀⠻⣷⣤⣖⣡⠏⠀⠀⠀⠀⠀⠀⣴⠋⠉⣩⢿⠋⠀⠀⠀⠉⠓⢦⣤⠏⠀⠀⠀⠀⠸⡇⠀⡞⠁⠈⡆⣠⠏⣼
⠀⠀⠀⠀⠀⠀⠙⠻⢤⣄⡀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠠⣄⣠⡬⠟⠋⠁⠘⠷⣶⡉⠉⣱⠀⠀⠙⢶⡀⠀⠀⠀⠀⢻⡀⢱⠀⣰⠟⠁⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⡓⢦⣀⡈⢷⠆⠀⣀⣠⠴⠚⠋⠁⠀⠀⢠⠒⠒⣠⣤⠽⢿⡁⠀⠀⠀⠀⠙⢦⡀⠴⠢⡌⢧⣀⣷⠃⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣌⡙⠛⠓⠛⠉⠀⠀⠀⠠⣄⠀⠀⠘⣲⡞⠉⠀⣠⣄⠙⢧⡄⠀⠤⣤⠈⠙⠒⠶⢶⡤⣬⣀⣀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣻⣧⠀⠀⢀⣀⣤⠴⠖⢻⠟⠻⣧⣉⣠⠼⠀⠀⠉⢳⡶⠶⠶⠶⠶⠖⠋⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣁⣀⡤⠖⠻⣄⣀⣀⡽⠶⠦⣤⡴⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁

⠀⠀⢠⣶⠀⣠⣤⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⣠⡀⣀⠤⣄⠀⢠⣤⡶⠶⣶⡦⣤⡤⠄
⠀⠀⢸⣿⡞⠁⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⢿⣿⠀⠀⠀⠀⣿⣿⠃⠀⣿⡆⠀⠀⠀⢰⣿⡇
⠀⠀⢸⣿⠁⠀⣸⠏⠀⠀⣿⡇⠀⠠⠴⠒⢲⡿⢾⣿⡦⠤⠀⢀⣿⡇⠀⠀⣿⠃⠀⠀⠀⣼⣿⠀⠀⢀
⠀⠀⢾⣿⠴⠚⠁⠀⠀⢰⣿⡇⠀⠀⠀⠀⡾⠀⠈⣿⡇⠀⢀⣾⣿⠁⠀⠀⣿⠀⠀⠀⡄⣿⣿⠀⠀⡼
⠀⠀⠸⣿⠀⠀⠀⢠⠔⣾⣿⣅⣀⣀⡀⢸⠃⠀⠀⢹⣷⠀⣼⠹⣿⠀⠀⠀⣿⡀⠀⡼⠀⢻⣿⠀⣰⠃
⠀⠀⠀⠛⠀⠀⠀⠙⠶⠛⠁⠀⠈⠉⠀⠉⠀⠀⠀⠀⠙⠛⠁⠀⠉⠀⠀⠀⠈⠓⠉⠀⠀⠈⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠘⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠢⣄⠀⠌⠀⠀⠀⠇⠀⢀⡠⠊⢹⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠆⠀⠀⠈⠺⡀⠀⠀⠀⡧⠔⠁⠀⠀⡌⠀⠀⠀⠁⠀⠀⠠⠀⠄⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠙⡀⢀⠎⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⡀⠘⠂
⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢱⠆⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠂⠘⠐⠀⠐⠊
⠀⠀⠀⠀⢤⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠
⠀⠀⠄⠀⠀⢂⡀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⡠⢁
⠀⠀⠀⠀⠀⠤⠤⠥⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⠤⠬⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢀⠀⠃
⠂⢠⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠉⠀⠄⢀⠀⠀⠀⠉⠠⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡠⠈⠁⠀⠀⢀⠀⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠲⠀⣀⠀⠀⠒⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠐⠂⠀⠀⡀⠰⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢠⠈⠀⠀⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠀⠈⢠⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣾⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣬⣿⣿⣿⣯⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣶⡄⢀⠊⢩⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⣤⡶⣛⣋⣝⣿⣿⣿⠛⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⡋⣉⣻⣿⡏⢞⢃⣿⣿⠃⠠⠐⣃⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡠⢂⠥⠊⠉⢩⣿⠀⠀⢸⣿⠱⡎⣦⢅⡐⠀⠠⠬⢿⣿⡾⠂
⠀⠀⠀⠀⠀⠀⠀⣷⢦⡀⢠⣾⣿⣟⡷⣒⠋⠁⠛⢰⣸⡘⣇⣿⢃⣯⣤⣄⣀⣤⠶⢟⡏
⠀⠀⠀⠀⠀⠀⠀⠫⠀⠘⠋⠛⠛⢹⡿⣢⣶⣶⠿⢫⡿⠟⢿⣎⠾⠳⣦⠐⠦⣐⠊⠁⣴⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢤⣄⣠⣼⠚⡛⠿⠶⠶⠷⢚⡋⠘⠠⠤⢀⣤⣌⣢⣌⠳⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣀⣍⠀⠀⢀⣟⡋⠉⠙⠛⢦⣀⠀⠀⠉⠉⠉⠋⠉
⠀⠀⠀⠀⣀⣄⣤⠤⠄⠀⠀⢚⢩⡿⠃⠀⠈⠳⠀⣀⢀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠓



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠠⠄⢯⣑⠢⠀⢀⣢⠏⠠⡘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⢀⡠⠝⡢⠴⣇⣔⢪⢇⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠨⢰⢦⡈⠙⢮⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠢⠀⣾⠷⢣⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⡔⠉⠙⠒⣤⣄⣀⣠⣶⢸⣦⡤⠙⠦⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡺⠦⠀⢈⣶⣿⣿⣿⣿⣿⣿⣟⣰⡆⠀⠀⢀⣀⣙⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣱⠀⠀⠉⢿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⢄⠀⢀⡀
⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⢘⠣⠹⡢⡀⠀⠈⢃⠀⢀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠁⠒⠠⠀⡀⠀⠙⢄⠱⡜⢄⠀⠀⠉⠳⠤⠤⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠲⠺⠊⠃⠀⠒⠒⠒⠒⠒⠖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡀⢀⠀⠱⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠈⠁⠁⠀⠁⠀⠒⠂⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⣠⠤⠔⠒⠉⠉⠀⠀⠀⠀⠈⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠠⣀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⣀⠤⣀⣈⠐⠊
⣿⣻⣿⣻⣛⣻⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣻⣟⣻⣿⣻⣻⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⣠⣤⢠⣄⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⣄⠀⢸⡄⣿⠀⣧⣤⣴⣦⣤⣤⡀⡀
⠀⠀⠀⠻⠿⠿⠿⢿⣇⣼⣼⣿⣾⣶⣦⣤⣄⡀⠀⠀⠀⢀⡀⠀⢠⣴⣾⣦⣤⣀⠀⠀⠀⠀⠀⢰⠁⣶⢲⡈⠇⠈⣧⣿⠆⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠿⢹⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⡁⠉⠉⠁⠁⠀⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣽⣇⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⢄⠀⢀⣶⣿⠿⠿⠿⡁⢦⠒⣾⣿⠯⢍⢻⣆⢠⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦⠱⡀⢻⣷⣾⣿⡟⡿⠛⠁⠉⠛⠛⠿⠷⢿⣿⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⣿⣿⣖⢣⡀⢻⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠢⢸⣿⣯⡋⣿⣿⣿⣿⣿⣉⣵⣤⣬⣉⡙⠛⠖⢲⣦⣤⡄⠀⠲⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡇⠆⠀⣿⣷⣏⢾⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣶⣄⠀⠁⢠⣴⣶⣾⣴⣶⣤⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⢠⣿⣿⣶⣿⣿⢻⣿⣿⣿⣿⣶⣯⣿⣛⠸⡿⡿⣧⠀⢸⣿⣿⣿⣿⠫⣙⣿⣾⡄
⠀⠀⠈⠁⠐⠐⠀⢀⡀⠀⠈⢻⣿⣿⣷⣾⣿⣿⣷⣿⡇⠀⠉⠉⠉⢐⣿⣿⣿⣦⡉⢻⣶⣿⡇⢸⡿⠞⣻⣟⠂⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠲⢬⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣼⣿⡿⠛⣿⣿⡇⢻⣶⣦⣴⣶⣞⠙⢿⣷⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠆⢹⣿⣿⣿⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣄⣀⠀⢿⣿⣁⣠⣿⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣷⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣤⣄⣀⠀⠀⢠⠞⠋⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠀⠀⠀⠀⠈⠙⠷⣤⠏⡠⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠈⠉⠃⣈⡈⠀⢴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⢉⠇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠒⠶⠛⠉⠉⠉⠙⢷⣄⣀⣤⣀⣀⣀⣀⣤⠤⠤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠻⡴⠋⠀⢀⡠⠘⠉⢁⢨⠽⠶⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠖⠒⠒⠧⠤⠤⠶⠚⠉⠉⠙⠒⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⡏⠀⣠⠖⠁⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⠤⠶⠒⠒⠒⠚⣛⣺⣥⡀⠀⠀⠀⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠙⣆⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠉⠉⠑⠒⠠⠤⠀⠩⢷⠖⠋⠁
⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠤⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊
⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠁
⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠈⠓⠦⣄⣀⠀⢀⣠⠞
⣶⣶⠶⠶⠶⠶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿