Earthworm Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠖⡩⡁⠀⠀⢀⣉⡓⢤⡀
⠀⠀⠀⠀⢀⡞⢁⠤⢤⡀⠀⠀⠀⡠⠤⣀⠱⡄
⠀⠀⠀⠀⡞⠀⡇⣾⣿⣷⠀⠀⢸⣿⣿⠌⠀⢹⡀
⠀⠀⠀⠀⡇⠀⠙⠸⠟⢃⣀⣀⡈⠛⠯⠊⠀⠀⡇⣀⡤⠖⠒⠛⠻⣟⠒⠒⠤⣄⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠏⠹⠃⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠙⠢⣄⠀⢀⡠⠔⢒⡻⠷⠶⠦⣄
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢸⠳⡉⠀⡰⠁⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⢹⢦⡀⠀⠀⠀⠀⠀⠀⢀⡠⡾⠀⠙⢦⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⠀⠈⠲⣅⡀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⣸⠀⠈⠓⠲⠤⠤⠖⠚⠉⠀⢧⠀⠀⠈⣇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠉⠙⠟⠀⠀⠀⠀⠀⡗⠒⠒⠛⢲
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣄⠀⣽⠀⢀⣀⡤⠴⠧⣤⣀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠇⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⣠⠷⠚⠉⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣠⣯⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⣀⣀⣤⣮⠥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⢤⣠⣎⣀⡤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⡤⢤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⣤⣄⡀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⣾⡿⠿⠻⢦⣄⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡁⠀⠀⢠⣶⣴⠋⠃⠁⣶⣾⣄⠙⣷⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⣄⣳⡿⠏⠀⣀⣢⣼⣿⠯⠀⠘⣷⠔⠀⢐⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣰⣼⠿⠉⢀⣶⡬⠟⢡⠋⠁⠀⠀⣺⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠠⠤⠖⠒⠖⠶⣤⣄⡀⠀⠀⠀⢸⠀⠀⠰⣿⣥⣤⣨⣿⠿⠴⠚⠉⠀⠀⠀⢀⣿⡄⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠴⠋⠀⢀⠀⠀⠀⠀⠀⠀⠈⢿⡓⠶⢤⣘⣆⠀⠀⠈⠊⠈⠙⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠐⣸⠁
⠀⠀⠀⢰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠶⣤⣬⣤⣄⣀⣀⠀⠄⠀⢀⣰⠞⠋⠂⠀⠀⠠⣹⠋
⠀⠀⠀⡏⠀⠀⠀⣺⠷⠿⠿⠷⣷⣣⢀⡄⡀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠉⠉⣿⠟⠛⠉⠀⠀⠀⠀⠀⢠⡴⠃
⠀⠀⠀⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⡠⢀⡀⢀⡴⠀⠀⠀⠀⠀⡛⠀⠀⠀⢠⠀⠀⠀⡰⠊
⠀⠀⠀⢸⡀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠾⢧⡀⠀⠀⠀⠀⣿⣠⠀⣀⢀⡬⠖⠉
⠀⠀⠀⠀⠹⣄⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠙⠊⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠓⢆⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠉⠉⠉⠓⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡔⢁⡀⠑⡄⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠰⠀⢿⡿⠂⡘⠀⡰⢀⣈⠑⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠑⠤⠠⠔⠁⠀⡁⠿⡿⠃⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠐⢒⡢⠀⠈⠒⠀⠂⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡦⠀⡀⠀⠀⠀⢀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠀⠀⠒⡇⠀⠀⠀⣀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠊⠁⠀⠀⠊⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⡧⠄⠤⠐⠂⠀⠀⠀⠈⡞⠁⠀⠀⢀⠜⠀⠀⢸⡤⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠶⠾⠁⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠔⠈⠀⠀⠀⠘⠀⠀⠀⠀⠆⠹⡄
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢈⢦⠀⠀⠀⠀⠀⠀⢠⠀⣠⠞⢢⡠⠄⠤⠐⢄⠀⠀⡠⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠲⠤⠤⠤⠤⠼⠋⠙⠁⡾⠀⠀⠀⠀⠈⣢⠾⡄⠀⠀⠀⡇
⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⣀⡤⠋⠀⢸⠒⠒⠈⠣⠤⠶
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠈⠀⠀⠀⠀⠈⠉⠀⠀⢀⠀⢸⡀⠀⠀⠀⠀⢀
⠀⢀⠴⠺⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠓⠒⣲⠚⣧
⣴⠃⠀⠀⠈⢇⠀⠀⣀⡀⠀⠀⡠⠚⠉⠑⢆⠀⢀⣀⡤⣄⡀⠀⠀⠀⠀⣀⠤⢄⠀⡠⠴⢤⠀⡠⠊⠉⠙⡶⠞
⠙⠒⠒⠒⠚⠛⠓⢯⣀⣼⢛⣲⣇⣀⣀⣀⣸⣴⠋⠀⠀⠀⠈⠑⠴⢦⡜⠁⣀⣨⡎⢀⣀⣀⣷⠗⠒⠒⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣀⣉⣳⣤⣤⣤⣴⣶⣶⣞⣋⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡞⠛⢻⡟⠛⣿⠾⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠁⠸⡇⠀⢸⡇⠀⡿⠀⠀⡼⠛⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣟⠈⢿⡄⠀⣷⠀⢸⡇⢀⡇⠀⣼⠁⠀⢀⡿⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢶⣟⠀⢻⣆⠈⣷⠀⢻⡀⢸⣇⢸⡇⢠⡟⠀⢠⡞⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠟⠉⠀⠀⢻⣆⠀⠻⣦⣹⡷⠾⠛⠟⠙⠛⠻⠾⢷⣤⣿⠀⣠⡾⠁⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣶⠶⣿⠀⠀⠀⠀⠀⠀⠹⢷⣤⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢿⣄⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠛⣿⠉⢹⡇⠀⣧⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⣠⣼⡏⠀⢿⠀⢸⡀⢸⡇⠀⢹⠀⠀⣷⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⢀⣾⠃⠈⢷⡀⠘⣇⠈⣇⠀⣧⠀⣸⡇⢠⣿⣆⣀⣤⣾⠟⠁
⠀⠀⠀⢰⡟⠙⢳⡄⠈⢷⡀⢹⣄⣿⡶⠟⠿⠛⠛⠛⠉⠙⠛⠉
⠀⠀⢀⡾⢷⣄⠀⠙⢦⣈⣿⠛⠉
⠀⠀⣸⣧⡀⠙⠳⣤⣌⡿⠁
⠀⠀⣿⠁⠙⠓⠶⢤⡿
⠀⢠⡿⠛⠛⠲⠶⠤⣟
⠀⠸⣷⣤⣤⡤⠤⠤⣿⡀
⠀⠀⣿⠁⠀⣀⣤⠴⠾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⣷⡀
⠀⠀⠻⣿⠟⠉⢀⣠⠶⠻⣦⡀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠻⣇⠀⢸⣷
⠀⠀⠈⣿⣀⣰⠟⠁⢀⡴⠛⣿⠿⣷⠿⠷⡾⠋⢹⡟⠉⠻⡏⠉⠙⡷⠟⠷⣶⠶⣶⣤⣶⣦⣴⠶⣶⠟⠛⣧⠀⠙⣆⠠⣿
⠀⠀⠀⠈⢻⣿⠀⣴⠏⠀⣰⠇⠀⡟⠀⢰⡇⠀⠸⡇⠀⠀⣧⠀⢰⡇⠀⠀⡿⠀⠀⡏⠀⢸⡇⠀⠘⡆⠀⠈⣇⠀⠈⣿⡟
⠀⠀⠀⠀⠀⠻⣷⣟⠠⢠⡟⠀⠀⡇⠀⠈⣇⠀⠀⣷⠀⠀⣿⠀⣸⠀⠀⢀⡇⠀⠀⡇⠀⠸⡇⠀⠀⣷⠀⠀⢸⣄⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣾⣇⠀⢀⣧⠀⠀⣻⣦⣤⣿⣷⣶⣿⣤⣿⣅⣀⣾⠃⠀⢰⡇⠀⠀⣇⠀⠀⣼⠀⠀⣸⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠿⠟⠋⠉⠉⠁⠀⠀⠀⠈⠁⠉⠙⠙⠻⠶⠾⠷⣦⣶⠿⢶⡶⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢸⣁⡤⠴⢦⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠫⢠⣶⢨⣴⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⡈⠁⠙⠉⠀⠀⠀⢸⠃
⠀⠀⠀⠀⢀⡴⠋⣠⠇⠀⠀⣠⠀⠀⣸
⠀⣀⡤⠞⢉⣠⠞⠁⣀⣠⠞⣽⠀⠀⡿
⣿⣁⣤⡶⠛⣿⠀⠈⣍⣠⣴⡏⠀⢠⡇
⠈⠉⠁⠀⠀⡏⠀⢀⠸⠛⣿⠁⠀⢸⠃
⠀⠀⠀⠀⠀⡇⠀⢣⡉⠋⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠛⠛⠳⢦⣤⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⢸⠃⠉⠂⠉⠑⠒⠋⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣧
⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠐⢄⠀⠈⣧⠀⠀⣰⠂⡇⠀⠀⠀⡼⠁⣿⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢳⣀⡿⠟⠛⠛⢾⣅⠀⠀⠊⠠⠊⠀⢿⡀
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠛⠶⣄
⠀⠀⠀⠀⣿⠉⠒⠤⠤⠤⠤⠴⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠹⣷⠐⠂⠁⠀⠀⣨⠿⢦⣄⣀⣀⣤⠾⠫⣄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⡾⠋⠲⣄⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠚⠁⠀⢀⠈⠉⠉⠀⠀⠀⠘⣆⠀⠀⠀⢠
⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⣀⡠⠛⣷⣤⣀⣠⣤⡾⠿⡁⠀⠀⠈⢷⡀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⡏⠀⠀⠀⢠⠀⠀⠀⢸⡆⠀⣠⣾
⠀⠀⠀⠀⠀⢿⡍⠙⠛⠉⠁⠀⠀⠀⠉⠉⠉⠉⠑⠀⠀⠀⠀⠀⠈⣷⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⣀⡼⠁⠀⠀⠀⢸⡄⠀⠀⢸⣧⣾⠟⠁
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣤⣄⣀⣀⣀⣼⣥⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⡼⢀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣾⣥⣾⣴⠶⠶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⣿⢻⡟⡿⢿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⢻⣇⣼⣿⣷⣷⣾⣿⣷⣾⣀⡞⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⡶⠟⣿⣁⣿⣤⣾⣿⣿⠟⠟⠛⠉⠉⠛⢿⣿⣴⢋⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣹⡷⢦⣤⣤⡤⣤⣤⣶⡾⣿⣁⣼⣷⣾⣿⡿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⡡⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣿⣧⣼⣅⣿⣀⣿⣤⣿⣫⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣤⣖⣫⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠧⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠿⠴⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⠁⠀⠀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡖⢒⣶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢽⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⢿⡳⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⣿⢿⣷⣽⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⢿⣿⣶⣶⢶⡶⣿⣿⣿⢿⡿⡿⡟⢻⢹⡜⣯⣷⣷⣼⣷⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⣿⣏⢻⢹⣿⢸⣀⣿⣿⣾⣷⣿⣷⣿⣼⣶⣷⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⢿⣿⣶⣾⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢻⣿⣿⣷⣟⣽⣿⣦
⠀⠀⠀⠀⠀⣰⣿⢿⣛⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣼⣿⣟⣲⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣧⣿⣷⣼⠿⠋⠉⠉⠻⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣥⣭⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠻⣧⣼⣿⡟⠁⠀⠀⠀⠀⠀⠨⣿⣷⣤⣿⡀
⠀⠀⠀⣾⣿⡷⣺⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣙⣶⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢶⣿⡇
⠀⠀⠀⣿⣿⣭⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢿⣌⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣛⢻⣿⡇
⠀⠀⠀⣿⣿⠖⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡻⣿⣿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠻⣿⣿⡿
⠀⠀⠀⣿⣷⢟⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢻⣿⣿⣙⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣉⣿⣿⡟
⠀⠀⠀⢻⣿⢋⣽⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⢹⣿⡿⣯⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢮⣽⣿⠟
⠀⠀⠀⠘⣿⣟⣠⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢁⣿⣿⣟⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣛⣾⣿⠏
⠀⠀⠀⠀⠘⣿⣯⡼⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡞⣻⣼⣿⣿⣿⣹⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣷⣿⣭⡿⢿⣿⣶⣤⣀⣀⣀⣀⣀⣀⡤⢶⣟⣹⣧⣼⣿⡿⢿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣯⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣿⣯⣹⣟⣻⣿⠻⣿⡿⢿⣿⣿⣿⠿⣿⣯⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠷⠾⠿⠾⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⢶⡏⠉⠙⢿⠛⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠙⢦⠀⠀⢻⠀⠀⢸⠀⠀⠈⡏⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠉⠳⣄⠀⢈⠇⠀⠘⠀⠀⠘⠀⠀⡼⠁⢀⡟⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⢦⡀⠀⢸⣤⠴⠚⠋⠉⠉⠉⠉⠛⠲⣄⠠⠊⠀⠀⣸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⣄⡀⠙⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⡠⠖⠁⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠾⣅⠀⠀⢹⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⣀⠞⠙⣧
⠀⠀⠀⢀⣀⡤⠾⢧⣄⠀⠑⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠁⠀⢀⡞⢧⡀
⣴⠚⠉⠉⠙⢳⡀⠀⠘⣆⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠖⠋⠀⢰⠟⠳⣤⣤⡀
⢻⡀⠉⠀⠀⠀⡷⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠴⠋⠀⢠⠏⠀⠙
⠀⠉⠓⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣔⣁⣠⡤⠞

⠀⠀⠀⠀⢀⣀⣠⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡤⣤⣀⣄⣄⡄⠀⣀
⠀⢀⣤⡾⣟⣻⣿⣭⣿⣿⣻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣾⣼⣿⣷⣦⣤⣤⣤⣤⣶⣿⣵⡾⡿⣻⢛⣛⢛⡛⠿⢿⣿⣤⣥⣔⡄
⠀⣼⣯⣿⣿⠟⠋⠉⠛⠛⠿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡿⠿⠯⠦⠂⠬⠽⣛⡻⠿⣿⢿⣛⠽⠝⠓⠉⠉⠀⠀⠀⠉⠉⠐⠲⠬⠭⠿⠿⠂
⠀⣿⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠙⢯⣩⣸⡿⣿⣦⣀⣀⣀⣤⣶⣽⣿⢇⡄⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠹⣿⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠦⣞⡿⣿⣿⣿⣿⣟⣫⠗⠉
⠀⠀⠈⠛⠿⣿⣿⣿⣶⣤⡄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠛⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠖⠛⠛⠛⠛⠛⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠙⠆⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠶⠛⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣠⡾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠖⠒⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠙⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⠛⠁⠀⠀⠀⠈⢙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⢰⠏⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢉⡀⠀⠀⠈⢿⡈⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡴⠶⠦⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⠀⠈⢛⣦⣤⣀⣈⠁⠀⠀⣤⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠃⣰⠏⠁⠀⠈⠉⠛⠶⣤⣿⠀⠀⠀⠀⣽⠉⠛⠲⢦⣄⣀⡀⠀⠀⣀⣴⠏⠀⠙⠛⢦⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⢀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣐⠋⠀⠀⠀⢠⡄⠉⠉⠛⠛⣿⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠛⠁⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣤⣀⣸⡇⠀⠀⠀⠀⠙⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠤⠤⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠖⠂⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠸⠃⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⣴⠛⠻⣶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠙⣶⡏
⠻⢦⣌⡁⠀⠀⠉⣉⠛⠛⠓⠒⠒⠛⡿⠁⠀⢀⣴⠏
⠀⠀⠈⠉⠛⠶⠾⢯⣤⣄⣀⣀⣀⣈⣣⣤⠶⠛⠁
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣀⡠⠤⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢤⡱
⠀⠀⠀⢀⡴⠊⢀⢤⡀⠥⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠠⠞⠁
⠀⠀⢠⠚⢠⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠘⡄
⠀⠀⠏⠀⠴⠁⠀⠀⠀⠀⠀⣀⠤⢖⠊⢉⠀⢨⠉⠢⡄⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⡲⢄⡀
⠀⠀⡆⠀⢢⣀⣀⣀⡠⠴⡊⠁⢀⡠⠖⠚⠉⠙⢖⡁⢘⡄⠀⠀⠀⠀⠀⠀⠀⠙⠻⢄⡠⠈⠑⡦⢄⡀
⠀⠀⠑⢤⡆⢀⡅⠀⣰⡀⠔⠊⠁⠀⠀⠀⠀⠀⡸⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⣀⡀⠉⢢⡀
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⡀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠂⠉⣃⠀⢠⡠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⢉⠃
⠀⠀⠀⠀⠀⠀⡠⠊⠁⣀⡸⠆⠘⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠔⠉⠣⡠⠋
⠀⠀⠀⠀⢀⠜⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢈⠉⠩⠉⠈⠁⠐⣀⡈⠖⠊
⠀⠀⠀⢠⠊⢠⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠀⠈⠁⠉
⠀⠀⠀⠈⠲⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⠠⢠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⡒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠈⠉⠀⢘⡀⢠⣀⣢⠀⡄⠉⡱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣀⣰⠶⠷⣤⠂⠑⣄⠀⠀⠀⢀⠔⠊⠀⢄⡠⠖⠚⠉⠁⠀⠀⠀⠀⠀⠉⠺⣠⠄⠓⡄
⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠜⠁⠀⠀⠈⢲⠁⠈⡄⠀⠀⠈⠢⠼⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢱
⠀⠀⠀⠀⠀⠀⣜⠁⢠⠎⠀⠀⠀⠀⠀⠈⡇⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠂⠼
⠀⢘⠁⠓⠒⠋⠀⣡⠊⠀⠀⠀⠀⠀⠀⠀⠹⣀⠘⢢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠒⠲⠒⢠⠀⠴⠲⠏⠁⡴⠁
⠀⠈⠓⠆⠷⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣆⢠⠅⢹⠀⠀⠀⠀⠀⠀⠑⠚⠶⠶⠀⠤⠀⠤⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⠒⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠁⠀⣠⣤⣤⣄⡀⠀⡈⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢠⣾⡟⠉⠉⠉⠉⠛⠻⠆⡀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠛⣄
⠀⠀⠀⠀⣀⣤⡺⠿⠛⠛⠛⢫⢻⣖⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠈⢃
⠀⠀⢀⡾⠃⢀⠀⢀⣰⣀⣤⣀⡇⣎⠉⠝⡿⠛⠳⠤⡀⠀⠀⠀⠀⢀⣿⣿⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢧
⠀⠀⡼⠉⠀⢀⣻⣿⠿⠿⠿⢿⣿⣿⣿⠏⠀⠀⠀⠀⡽⣤⣀⣀⣠⣾⣿⠍⠄⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢻
⠀⠀⡯⠀⢀⣿⠏⠁⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⢠⡎⠀⠀⠈⠉⠉⠁⠁⠀⡠⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠱⡆⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠺⢿⣷⣦⣠⢀⣴⣖⣒⡾⠛⠁
⠀⠀⠀⠸⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠉⠠⠀⠃⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠒⠁⠀⠀⠂⠠⠊⠀⡀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠱⢀⠀⠀⠀⠁⡜⠉⢉⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⠀⠀⠆⠀⠀⠀⠄⠂⠒⠐⠐⠒⠢⠄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢻⠀⡦⠀⢣⠀⡀⠀⠐⠀⡀
⠀⠀⠀⠀⠁⠀⢀⠀⠀⠤⠂⠀⠊⠀⠁⠀⠀⢆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⠁⠀⢀⡂⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠐⠢⠄⠆⠃⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⢀⠀⠀⢢⠀⠀⠉⠀⠀⠄⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⠀⢀⠜⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠐⠂⠤⡀⠀⠀⠀⠀⡀⠘⠀⠀⠈⠁⠀⠀⠀⠈⠀⠡⡀⠀⠂⠃⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠙⠢⠶⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠄⠀⠐⠀⡀⠂⠀⠀⠀⠈⠀⢀⠇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢁⠀⠀⠀⠱⡄⠠⡀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠀⠀⠠⠂⠈⠁⠢⠄⠀⠀⠖⠒⠉⠀⠀⠀⠀⠀⢀⣀⡎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠂⠈⠀⡀⠀⠀⠀⠂⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⢴
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋⠁⠀⠀⠀⠀⢀⠀⠒⠊⠁⠀⠀⠈⠉⢐⠂⢄⡀⠠⡀⠀⠀⠀⠄⠀⢀⡄
⠀⠀⠀⡀⠀⠀⠀⠀⡥⠤⢅⠀⠀⠀⠀⠀⢸⠀⠀⠀⣄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⢩⠀⠀⠀⠠⠀⠀⢀⠶⠤⢄⠐⡄⠑⠦⣀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠈⡄⠀⠀⠀⠀⡄⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠊⢀⠐⠂⠀⠀⠀⡀⠀⢘⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠠⣈⠀⠚⠊⢀⡀⠔⠠⠀⠀⠀⠀⠂⠀⠒⠘⠒⠓⠀⠀⠂⠄⢀⠀⠀⠤⠒⠁⠀⠀⠀⠀⡀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠫⣁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠂⠉⠃⠐⠃⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠈⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⡤⠤⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⢻⠀⠀⢼⠀⠀⡜⠁⠀⢈⣹⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠹⡀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⣰⠛⢦⡀⠀⠉⢀⣤⠴⠖⠒⠒⠒⠲⢤⡀⠀⠀⠴⠛⠉⠹⡄
⠀⠀⠀⠀⠀⢀⡴⡿⠃⠀⠀⣸⠳⣄⠀⠀⢠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⡤⠴⣧
⠀⠀⠀⢀⡴⠋⡞⠁⠀⠀⢠⣧⣀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡾⠑⢾⠁⠀⠀⠀⢸⡇⠈⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠉⠉⠉⡿
⠀⠀⣼⠁⠒⡏⠀⠀⠀⠀⢸⡟⠓⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠛⠲⢶⠃
⠀⢀⡇⠠⠶⢧⠀⠀⠀⠀⢸⡧⣤⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢦⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠸⡇⢀⡤⠼⣄⠀⠀⠀⣼⠆⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢤⡀⠀⣨⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⡉⠉⢳⣆
⠀⠀⢷⠀⠀⠀⣨⠗⢶⠞⢧⡀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣰⠋⢀⠀⠲⣤⣀⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣀⣀⣠⣸⠆
⠀⠀⠈⢧⡀⠀⠁⠀⠰⠀⠀⠁⠀⣠⠞⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠓⢦⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠈⣩⠾⠋
⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣠⡤⠞⠋⠀⠀⠀⠀⠀⠀⢸⠇⠀⠙⠓⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠆⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠘⢋⡽⠛⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠶⠦⡄⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠴⠒⠋⣠⠜⢁⡟⢳⣄⡀⠀⠀⠀⠀⢠⡞⠓⠦⡀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠴⠃⠀⡼⠁⡟⢳⣄⣤⠶⣏⠉⢢⡀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠀⠀⠀⢸⠁⠃⠀⢸⡄⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⣸⠀⠋⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠤⠴⠷⠶⠶⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⣀⠀⠀⡔⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠁
⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡡⠀⠠⠀⢄⡀⢤⡂⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⡖⠠⠖⠀⠈⠁
⠀⠐⠁⡀⠀⠀⠀⠐⠀⠀⠙⣴⠀⢃⣀⡈⠁⠀⣷⡀⡏⡀⠀⠀⠀⠀⠀⠀⢀⠀⡞
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⠿⠿⠇⠿⣻⣷⣤⠀⠉⠁⠛⡁⠀⡀⠀⠀⠀⠈⠀⡉⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠈⡿⠒⠦⠀⠀⡀⠀⠀⠀⠀⠈⠂⠀⠎⠡⠄⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣰⣤⠀⠁⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⠀⠀⠀⠓⠦⢀⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢠⣾⠚⣡⣶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠑⠛⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢨⠇⢰⣿⡉⣿⣋⣿⣿⠶⠀⠂⠀⡀⠉⠀⠀⠀⠀⣠⡶⠀⢄⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢹⢸⣿⣟⣋⣻⣿⣿⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⡴⠋⠛⠾⠲⠀⠳⠤⠂⠠⢄⢀⠎
⠀⠀⠀⠀⠀⠰⣆⣼⣿⣿⣿⣏⣿⠀⠀⠀⢰⡞⠀⠐⠂⢄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠉⠿⡥⣝⢿⢿⢻⠄⠀⡀⠀⠃⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⢀⡀⠀⠨⠛⠀⠀⠁⠂⠸⣤⡄⠁⠀⠰⣄⡠
⠀⠀⠀⠀⠀⠱⠄⠶⠂
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢝⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⢌⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⠯⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠋⠈⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⣾⣅⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠂⠀⠀⠀⠀⢀⣤⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠈⠛⠛⢿⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠀⠀⠀⣠⣏⡜⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⢀⡼⣛⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡄⢠⣀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠀⢀⣠⡎⠀⠀⠀⢠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠶⠿⢷⣤⣄⡀⠀⠀⠈⣻⣿⣿⣿⣿⣾⣄⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣟⣟⣟⣻⣿⣿⣿⣿⣿⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣽⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⣛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣯⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⢿⣿⣿⢻⣿⣿⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⢿⣿⡟⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⢸⣿⡄⣸⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠃⠞⠁⠟⠹⡆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⠀⠀⠂⠀⠀⠀⠀⠃⠉⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣶⢦⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠶⡡⢿⡞⡏⠀⠀⠀⠀⢰⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡂⢀⠟⡉⠀⠁⣘⢳⢿⡄⢸⠀⠀⢨⠻⠻⣤⣇⢀⣖⠦
⠀⠀⠀⠀⠀⠀⣴⣥⠄⡀⠀⣹⡗⣡⣾⣿⠃⣰⡇⡸⠀⣿⠿⣄⣠⣿⢣⡀⠈⠹⣾⠁⠀⠈
⠀⠀⠀⠀⠀⠠⠋⠉⠛⠳⢾⣯⣾⣻⢳⠃⢠⣿⠇⡇⠀⢿⣾⡘⡈⢻⣇⢻⣤⠬⢶⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⠉⠭⠙⠒⠋⠛⢻⣿⡄⠚⠛⠓⠓⠘⠻⡏⢁⡤⠔⠈⣷⣀⠄
⠀⠀⠀⠀⠀⠢⡀⠀⠀⡀⠘⠇⠀⠀⠀⠀⠀⠀⢸⢸⡇⠖⠂⠀⠀⠀⠀⣹⣀⠀⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠈⠑⠤⠷⠞⢀⣀⠀⠀⠀⠀⠀⡄⢸⠃⠀⠀⠀⠀⢀⠌⠀⢠⠋⠱⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠈⠉⠀⠉⠉⢉⡉⠉⢩⡀⠴⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⠀⠀⠀⢀⠀⢐⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⣝⡀⠀⢀⠁⡊⠀⠀⠈⢻⢦⠀⠴⢂⡿⡄⠀⠀⠹⡦⡐⠐
⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠈⠓⢍⠓⢤⣁⠀⠀⠀⠀⡟⢀⠀⢀⠞⠀⡟⠀⠀⢠⠀⠹⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⢧⠀⠈⠳⡀⠐⣦⡙⢮⠢⢸⠁⢠⠃⠀⡠⠋⠀⠈⡇⠀⢀
⠀⠀⠀⠀⠀⠀⢖⠈⣀⣀⡀⠀⠀⢸⡤⠀⠀⢧⢠⣇⠹⡌⢳⠘⣆⡿⢀⡞⠁⠀⠀⢀⠆⢀⠎⢣⠀⡀
⠀⠀⠀⠀⠀⠈⠀⣈⠉⢳⡎⢢⡀⢸⠀⠶⡀⠈⠻⠂⠀⢱⠀⠀⡿⢽⣼⠀⢠⠀⢀⡌⢀⠠⡀⡏⠐⠁
⠀⠀⠀⠀⠀⠀⠈⢹⠙⠸⡇⢀⠱⣾⡄⢁⠈⠢⡀⢠⡄⡞⠀⢀⠇⡄⠋⡠⢁⢣⠏⠀⢸⢣⢰⠇⢀⢦
⠀⠀⠀⠀⠀⢀⣆⢺⡊⠀⠹⣇⠰⠠⢄⠀⠂⠀⠰⠂⢠⠃⢀⡞⡐⢸⠐⠀⢠⠃⠀⡰⠃⣸⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠘⡏⢦⣀⠀⠀⠈⠳⣝⢤⡈⠲⠀⠀⡄⢸⢠⠟⠐⠀⢾⠀⠂⡇⣠⠏⠀⣀⡎⠀⣠⢀⡸
⠀⠀⠀⠀⠀⠀⠱⣄⠈⠑⠢⢤⡀⠘⡄⠱⠀⠀⠀⠆⠈⠿⠀⠀⠀⠀⠀⡆⢹⠇⡄⢠⠎⢀⡰⣻
⠀⠀⠀⠀⠈⢻⠆⠈⠲⢄⠀⣄⡙⠶⣗⠀⠁⠀⠀⠘⢄⠈⠂⠰⠀⠀⠀⠐⡤⠊⢄⣮⠖⢋⡜⠁⣦⡀
⠀⠀⠀⠀⠀⠘⠒⠠⠱⠀⠱⡀⠂⠑⠄⠀⡄⠀⠀⠀⠀⠉⠂⣤⠀⠀⠀⠀⠄⠐⠎⠥⢈⠏⡀⠀⠁⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠢⡀⡇⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢐⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠙⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠈⠨⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠓⢤⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠂⢀⠁⠀⠀⡀⠁⠀⠔⠀⡀⠀⡀⠀⠀⢐⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠐⠀⠀⠈⠈⠀⠀⠁⠀⠈⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣦⣤⣤⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⢿⣻⠛⠿⡿⢿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣏⠛⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⢿⣿⠃⠀⢀⣸⣿⣷⡖⠋⡀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠋⠹⣷⣄⣿⠀⢀⣼⣿⣿⣿⣟⠉⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⣀⣤⣿⣿⣿⢯⠀⢹⠀⠠⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠦⠤⣊⣉⢱⣿⣛⣿⣿⣧⣈⠇⢀⣼⣿⣿⣶⣶⣶⡶⢶⣶⣶⣴⣄⡠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⣛⣭⣿⣿⣿⣿⠟⠟⠀⣸⡿⠟⠛⠁⠀⠛⠉⠉⠉⠉⠀⠉⠉⠒⠦⠤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠾⢿⣯⠄⣀⣴⣾⣿⠿⠿⠛⣩⠄⠈⢠⠞⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦
⠀⠀⠀⠀⣠⡾⠛⠛⠠⠄⢀⣿⡿⢻⣋⠴⠀⡠⠊⢀⣾⣷⣯⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠄⠀⡄⢀⡀⣄⣤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⣠⠟⠉⠀⠀⠀⣠⣿⣿⡿⠗⡽⡡⢂⠈⣰⣎⣾⠿⠛⠛⠣⢦⣴⡾⠿⡿⣿⣿⣿⠿⠶⢾⣤⣶⣶⣿⣏⠀⠀⠀⠀⠀⠀⠀⣈⣤⣆⣤⠖
⠀⠀⣿⣾⡿⠓⠀⠀⠼⡟⣷⠹⠞⣼⢡⠟⣰⣿⡟⠣⠀⠀⠀⠀⠀⡁⢉⠰⣾⣿⡯⢿⣶⣤⣤⣿⣿⣿⣿⣿⣀⣀⣤⣄⣠⡤⠈⠛⠉⠉
⠀⠀⠉⠁⠀⠀⠀⠀⠀⣱⣿⡆⠀⢺⢻⢸⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠉⠉⠁⠀⠀⠄⠀⠀⡀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡁⠀⡇⣆⣿⣿⣏⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⡄⠀⠤⠒⠂⠀⣀⣠⣴⣾⠋⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⣿⡉⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣡⣤⣤⣶⣶⣿⣿⣿⡿⠉⠁⢻⡆⠀⣠⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⢿⣿⣛⣚⣁⣹⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⡿⠃⠀⠀⠀⣿⡾⠟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣿⣿⣿⣿⣿⣿⡆⠀⠀⠠⣸⡇⠬⣿⢫⣿⡿⠟⠋⠁⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⡏⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⣿⣿⣿⡿⣿⣧⠀⡀⠀⣽⣷⡾⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠟⣴⠉⠈⡆⢸⣿⣷⡄⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⠀⠀⢀⣾⣿⣯⣀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠓⠂⠀⠀⠉⠀⠁⢻⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠙⠿⣿⣿⣿⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠙⠀⠀⠀⠀⠈⠙⢳⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠖⠂⠀⠀⠀⠀⠀⠀⠙⣏⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣳⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣦⣀⣀⡀⢀⣰⠛⠀⢀⡀⠀⠀⠀⠀⠀⣀⣴⣋⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣥⣉⠿⡿⢿⠋⠀⠉⠀⠉⠶⠂⢠⣼⣿⠋⠉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠉⠁⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⣀⡀⣀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠋⠟⠻⠉⠇⠯⠄⠵⠼⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⣀⠄⠀⠠⣤⠀⠀⢀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⣤⡄⢤⡄⢤⠄⡠⠤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⡗⢁⠞⣉⣀⣀⣀⣀⣀⠠⠤⠕⠺⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠋⠓⠋⠓⠘⠚⠓⠒⠛⠘⠂⠀⠀⠀⠀⠀⢀⡴⢁⡴⠋⠀⣸⠟⠉⠁⠀⠀⠈⠉⠐⠶⣴⣇⡠⠞⠛⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⢋⡠⠛⢀⣴⣿⣛⣋⣀⠀⠀⠀⠀⠡⠀⢘⣯⣌⣁⠒⠒⠒⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢐⠛⠶⠋⠀⣰⡿⠿⠭⢭⣉⡑⣴⠏⣤⠄⣀⡴⣟⠒⠂⠭⣉⠛⠦⢤⣀⣀⠶⠒
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠋⢸⠶⠁⠀⣀⡤⠾⠥⢤⣐⡶⠞⠁⠲⣥⣴⡒⢛⡉⡑⠢⣌⠤⡀⡀⠉⠑⠒⠬⠵⢶⡶⠶⠚⠋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡀⠀⠐⠀⠀⠀⠀⠒⠾⢋⣠⠔⢂⠐⠒⢀⠑⠠⡄⢀⣀⣿⣷⣶⠂⠈⠑⠂⠑⠀⢩⣩⡤⠈⠐⠂⠤⣾⡷⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣷⣾⡀⢰⣾⡷⠒⠊⠀⠀⢀⠙⠓⡀⠈⢠⡴⠓⢄⠂⢀⠀⠀⠀⠀⠈⠑⠢⠤⠤⢤⠐⡞⠩⢉⠉⠙⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠉⠀⠀⠀⠀⠉⠀⡀⠀⠁⠐⠈⠉⣚⠛⠚⠋⠀⠀⠀⢢⠂⢀⠀⠀⠀⠀⠀⢡⠀⠂⢈⠢⡐⠢⣀⠀⠀⡀⢅
⠀⠀⠀⠀⠀⠀⠀⡀⡈⢳⠄⠀⠀⢠⣔⠃⠉⢩⣼⡏⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⢸⠐⢀⡀⠈⠀⣀⠀⠁⡴⣷⠀⠀⠘⠓⠠⠩⣲⡀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠂⠀⠐⠀⠀⡞⠈⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠃⠀⠀⠀⢠⠀⠀⠀⠙⢷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢁⣬⠗⠂⠀⡀⠀⠀⢠⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠉⠀⠀⠀⠂⠀⠸⠀⠀⠳⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠀⡰⠋⠀⠀⠀⠚⢱⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢀⡤⠄⢠⠄⠀⠀⣰⠃⠀⠀⠀⠀⠙⠛⠓
⠀⠀⠀⠀⠠⠂⡁⠀⠀⠤⠊⠁⠐⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣰⠋⢀⠀⠀⠀⠀⣸⠋⠀⣦
⠀⠀⠀⠀⠒⠁⠀⠀⠀⣠⠄⡥⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠠⠀⡇⠀⠀⠀⠀⠠⠀⡧⡰⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠔⠀⠀⠀⢷⠀⠀⠀⠀⠀⡂⢰⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠋⠀⠠⠀⠀⠀⠀⠀⠀⣈⡷⣆⡀⠀⠀⠰⢀⠇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠩⠀⠀⠀⠀⠀⠀⠠⠀⢀⡤⠖⣩⠀⠀⣑⠢⢄⡀⠈⠒⠤⣤
⠀⠀⠀⠠⢠⣈⡒⠀⠀⠀⠒⠀⠠⠀⠀⠀⢀⣀⡴⢋⠀⠀⠀⠀⢂⠀⢂⣠⠴⠊⠉⠀⠐⠃⢠⠐⠁⢠⠀⠉⠓⠦⣄⠄⠂⠀⠀⢤⡀
⠀⠀⠀⠀⢀⣈⣁⠄⠀⣀⡀⠀⠀⠀⠉⠙⢉⡽⠁⠀⠠⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡄⠀⠀⠹⠁⠀⠿⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠙⢢
⠀⠀⠀⠀⠀⠞⢳⡴⠈⠁⠈⠉⣀⡘⠃⠶⣾⠁⠁⠀⠀⠀⡀⢰⡟⠀⠙⠐⠂⠰⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠉⠀⠀⡏⠄⠀⠀⠄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⡠⠀⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⡀⠀⠀⢿⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠒⠀⡀⠀⣀⠀⠔⠁⠀⠀⠲⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠂⠀⠠⠈⠛⠾⢥⣄⡀⢀⣊⢀⣀⠀⢠⠀⠀⠀⠘⠁⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠂⠐⠀⠂⠀⡀⠀⠈⠉⠓⠾⢤⣦⡤⠀⠀⠀⠠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣀⠀⠁⠀⢀⠀⠀⠀⠘⠀⠀⡈⠙⠳⠦⣤⠘⠃⠁⠠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⢀⡠⠔⠂⠈⠉⠓⢶⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠐⢣⠔⠒⠈⠁⠒⢤⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢼⣿⣷⠁⣾⣿⠆⠀⠀⠀⠱⠄⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠉⠇⠀⠀⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠑⠤⠀⠸⢄⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠐⠉⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠒⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⡠⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⣀⠠⠶⠋⠈⠈⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢐⠶⣈⠁⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠉⠒⠒⠒⠒⠂⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⡀⠀⠀⠀⠀⠀⠀⠀⣀⡎⣀⠄⠒⠀⠁⠈⢩⠏⠉⠉⠐⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠈⠑⠂⠤⠤⠴⠖⠛⢹⠊⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⣀⠤⠄⠼⠬⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠔⠉⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀⣀⣀⣀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠳⢄⡀⠀⠀⠀⠀⠀⠀⡠⠜⠣⡀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠁⠉⠉⠉⠙⠓⠒⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⡆⠈⠁⠒⠒⠐⠊⠉⠀⠀⠀⢈⢲⠤⢀⣀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠎⠀⠀⠀⠱⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡠⠊⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠊⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠄⣀⡀⠀⠀⠀⣀⡠⠄⠊⠁

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

⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠘⣇⠈⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣸⠀⠀⠀⠀⠀⠀⠀⠰⠋⡇⣦⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠰⡆⣨⢛⢥⡶⠀⠀⠀⠀⠀⠀⢸⢸⣄⣧⣠⠀⠀⠀⠀⠀⢀⠀⣲⠛⡏⠀⠀⠀⠀⠀⠀⡾⡿⠜⡄
⠀⠀⠀⣷⣼⢀⡾⣤⠦⡼⠀⠀⠀⠀⢸⣾⠈⡟⠙⢻⠀⠀⠀⠀⠀⠘⢟⢫⣿⣀⡀⠀⠀⠀⢀⠠⢇⠠⣤⢷⡄
⠀⠀⠀⠘⠦⠾⠇⠘⠃⠁⠀⠀⠀⠀⠀⠁⠰⠛⠖⠁⠀⠀⠀⠀⠀⠀⠀⠱⠐⢯⠃⠀⠀⠀⠘⣷⠛⠘⢆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠜⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀
⠀⠀⠀⠀⠘⡿⡏⡅⠀⠀⠀⠀⠀⢀⢀⠀⣿⢣⣀⡀⠀⠀⠀⠀⠀⠀⠎⣼⣿⢰⡀⠀⠀⠀⠀⠀⡄⣾⡼⣃⡋
⠀⠀⠀⠀⠸⣧⠛⣇⠀⠀⠀⠀⠀⠈⠉⢷⠾⡌⠛⢛⡁⠀⠀⠀⠀⢰⣜⣇⢹⡾⣧⡀⠀⠀⠀⠀⡁⢏⢁⢹⠠⠇
⠀⠀⠀⠸⣆⣿⢶⣼⠀⠀⠀⠀⠀⠀⣻⡛⠀⡿⣆⠛⠇⠀⠀⠀⠀⠸⣿⠹⢰⡀⠻⠀⠀⠀⠀⠠⣼⡾⢿⡘⢾
⠀⠀⠀⠀⠀⠰⠏⠋⠀⠀⠀⠀⠀⠀⠈⠃⠻⠅⠙⠋⠀⠀⠀⠀⠀⠀⠙⠘⠋⠃⠀⠀⠀⠀⠀⠀⠈⠃⡎⠻⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠘⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⢺⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠻⣿⡄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⡀⣰⣿⡿⠃⠀⢠⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡼⣿⠁⠀⠀⠸⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠘⣿⣿⣷⠈⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣦⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⢤⡀⠀⢀⣿⣷⡀⠀⠀⣸⣿⣿⡟⠀⢤⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⢋⣼⣿⣿⣿⣿⡇⠀⠀⠀⢱
⠀⠀⠀⢤⡀⠀⣴⡿⠀⡀⠙⣿⣿⣿⣿⡆⠀⠀⢸⣷⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⠁⠀⠈⣿⣷⡄⠀⠀⠙⠻⣿⣿⡿⠁⢸⣿⣿⣿⣿⣿⣷⣤⣀⣤⣿
⠀⠀⠀⠈⣿⣶⠟⠀⠀⣿⣦⣾⣿⣿⣿⠇⢀⣴⣿⠟⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⣿⣿⡀⠀⠀⠀⠀⠉⣰⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⣇⠀⢀⣿⣿⠀⡀⠀⠘⢿⣿⣿⣿⢟⣴⣿⣿⠋⠀⢠⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⠀⣿⣷⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟
⢿⣿⣿⣿⣿⣇⣀⣽⣷⡀⠈⠙⠿⢋⢸⣿⣿⣿⣷⣤⣾⡇⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⣿⣌⣿⣿⣿⣿⣿⡿⢡⣾⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣠⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⢠⠀⠙⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⡟⠀⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣷⣶⣿⣿⣿⣿⡟⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣘⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⡟⠛⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠙⠢⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣁⢸⠉⠁⠀⠀⠈⢂⡔⠋⠩⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡴⠊⠀⠘⡄⠀⠀⠀⢀⡎⠀⠑⠀⠐⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡌⠀⠀⠀⠀⠱⡀⠀⠀⡼⠀⠀⠀⠐⣀⡥⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠢⣄⣀⠀⠀⠀⠀⡹⠶⠖⣇⠀⣠⠖⠉⠀⠀⠄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠔⠁⠀⠈⠉⠓⢲⠋⠀⠀⠀⢘⡟⠁⠀⠀⠀⠆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠸⡇⣀⡠⠔⠊⠉⢩⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡙⠢⢄⣀⣀⣀⡴⠛⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⣠⢤⡇⠀⠀⣀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⢀⡠⢤⢒⠣⡁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠖⠋⠀⠈⠊⣦⠸⠥⠖⠧⠄
⠀⠀⢀⣤⡔⢠⠣⠂⢸⠁⠀⠙⠦⡀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠈⣧⣠⠎⠀⠀⠀⠀⣠⠊⠀⠀⢠⢄⢰⡞⡆
⠀⢰⠁⠀⠀⡎⠀⠀⢸⠀⠀⠀⠀⠈⢑⠞⠁⠀⠀⠀⠀⠀⠀⠰⢯⢿⡀⠀⠀⠀⢰⣇⠀⠀⣠⠀⠘⠀⠁⢑⡄
⠒⠚⠲⣤⡤⢧⣀⣀⣨⣧⣄⣀⣠⣴⢿⠒⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⣿⢲⣤⣀⡠⡟⠢⣀⣻⠤⣀⡠⠔⢚⣒⣒⢂
⢀⣀⣠⣤⣽⣋⣀⣙⣷⣇⣐⣒⣓⣒⣿⣤⣤⠤⠮⠭⢍⣛⣛⣫⣤⡭⠬⠭⣤⣬⣷⣿⣴⣲⠾⠯⠟⠿⠿⠿⠿⠿⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⡆⠀⠀⢀⠀⠀⠁⠀⠀⠐⠀⡀⠀⠁⠀⢀⠀⠀⠀⢁⠑⠄
⠀⠀⠀⠀⠀⠀⠀⢂⠄⢐⡀⢀⡴⠁⠸⠀⠀⠀⠀⠄⠀⠐⠀⠀⠘⡄⠓⣄⠀⠈⠁⢁⠐⠄
⠀⠀⠀⠀⠀⠐⠁⡀⠀⠘⡀⡼⠯⠁⢀⡀⠀⠀⣠⡆⠀⢸⠁⠀⠀⠇⠈⠿⡆⠈⠃⠀⠀⠘⠄
⠀⠀⠀⠀⠀⡄⠀⣁⠂⠀⠁⠧⠈⠀⠀⠉⢢⠜⠘⠀⠀⢨⡤⠚⠁⠀⠀⢂⠿⠘⠀⢂⠘⠀⢀
⠀⠀⠀⠀⠀⡀⠈⡇⢀⠆⡼⡄⠀⠀⡐⢰⡏⠃⠀⠀⠀⠙⠁⢀⠠⠂⠀⠀⣰⡄⢦⠀⠹⠀⠈
⠀⠀⠀⠀⠀⢡⠀⢠⠂⢄⢷⡙⢄⢀⠠⠄⣧⣀⠀⠀⡀⢀⡄⠀⣠⢁⠀⣨⢃⡟⣀⣡⠀⠠⠘
⠀⠀⠀⠀⠀⠸⠀⠈⠇⣈⠹⡎⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠃⠃⠀⠘⠈⠐⣩⠗⠀⠴⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠉⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡐⠉⠒⠋
⠀⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⠀⠀⠀⠀⢠⠀⣀⠐⠂⠀⠀⢀⠀⠀⠀⠘⠁⠀⠀⠀⣠⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠈⢠⠇⠀⠵⠲⠉⠀⠀⠓⣁⣶⠀⠂⠀⣶⡠⢀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠀⠈⠲⣤⣀⠁⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⠀⠀⠀⠀⠀⠀⢁⡄⣹⠁⠘⠀⠒⣆⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠸⠃⠁⠀⠄⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢻⡟⠛⠛⠛⢳⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠀⠀⢸⡇⠀⠀⠀⢸⠇⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢿⡀⠀⣀⣤⠷⠶⠶⠦⣿⡀⠀⠀⣼⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠈⢷⡾⠋⠀⠀⠀⠀⠀⠈⠻⣦⣰⠏⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⣠⡞⠁⢻⣄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⢀⣼⠛⢧⡀
⠀⣀⣤⣤⡴⢾⡏⠀⠀⠀⢙⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡟⠁⠀⢠⡟⠷⣤⣤⣤⣄⣀⡀
⣾⠉⢿⣼⠇⠀⢷⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠟⠀⠀⠀⡿⠀⠀⠉⢹
⠻⣄⣀⣀⣀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣾⣃⣀⣀⣤⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠓⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⢀⡠⠴⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠲⢤⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠈⢦⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⡟⠒⠂⠤⣤⣀⣀⡀⠀⢹⠀⠀⠀⠠⠔⠂⠀⠀⠀⢸⠀⠀⣀⣀⣠⡤⠐⠒⠚⡇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣇⠀⠀⠀⠉⠀⠀⠈⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠁⠀⠀⠉⠀⠀⠀⢠⠇⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣠⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠁⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⣠⠺⡄⠀⠀⠀⠀⢙⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠚⠁
⠀⠀⠀⢀⣼⠁⠀⠹⡀⠀⠀⢠⠎⠀⣻⡉⠙⠒⠶⠤⢥⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣤⠤⠴⠖⠚⠉⣸⠁
⢀⠴⢒⠉⠀⢡⠀⠀⢱⣀⡰⠃⠀⡔⠁⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠁
⡏⠀⠘⡀⠀⠀⠡⣀⠞⠀⠙⣤⠎⠀⠀⠘⣆⠙⠲⠦⢤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠤⠖⠋⣡⠏
⠱⣄⡀⣃⣀⡤⠞⠁⠀⠀⠀⠈⠓⠤⣀⣰⠈⠳⢤⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣠⠔⠁
⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣒⣛⣛⣓⣲⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣖⣒⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣋⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⢲⡀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⡼⠉⠉⠀⠀⠳⠴⠞⠓⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡖⠒⠋
⠀⠀⠀⠀⠀⠀⣼⣁⣀⣀⣀⣀⣤⠇
⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠉⠉⠛⢦⣄
⠀⠀⢀⠞⣵⣮⠶⠦⡀⣀⠤⢤⣦⡄⠀⠹⣦
⠀⢀⠁⢴⢻⣧⢰⡷⣼⢱⣶⢸⣿⢧⣤⠀⢸⡇
⠀⡆⠠⢿⣦⡉⠙⠋⠉⠓⠶⠚⣡⡜⡏⠁⣸⡇
⠀⠹⣄⠘⠚⢻⡷⣖⣲⢖⣺⠿⠟⠉⠁⣠⣿⠇
⠀⠀⠈⠳⣦⣀⡀⠈⠁⠀⠁⣀⣠⣴⣾⣿⣿⡄⠀⠀⠀⠀⠀⣠⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠙⠿⣿⣟⣋⣉⣙⢛⣫⢵⡷⢹⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣍⠉⠉⠡⢒⣿⣇⢸⠀⠀⢀⡞⠀⠀⠀⠈⣷⡀⠀⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠙⠓⠚⠉⢘⡏⢸⠀⢀⡞⠀⠀⠀⣀⠈⢻⣿⣴⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣬⡍⢸⣶⡞⠁⠀⠀⠀⠘⣶⣿⣿⣿⣗⡠⠂⢸⠀⠀⠀⣠⣖⡉⠉⢹⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠐⠃⢸⣿⡇⠀⠀⢤⠀⢰⡿⠃⣿⣿⡟⠀⠀⣿⠀⠀⣰⡃⠀⠈⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠒⠒⠊⠁⠀⠈⠻⠀⣆⠀⢘⣷⡟⠁⠀⣿⢿⣏⣀⣠⢿⣆⣠⠯⡀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣄⣀⡤⠊⠀⠄⠀⣿⢀⡼⠋⠀⠀⠀⢻⣟⠂⠀⠀⠈⠣⠙⠂⠈⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⣚⣀⣴⠿⠋⠀⠀⠀⠀⠀⠘⢿⡖⠒⠈⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣶⣥⡤⠞⠋

⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⡰⠊⠐⠀⠂⠀⠀⠒⠒⠉⠢⡀
⠀⠀⠀⠀⡼⢁⠖⣭⣵⡄⠀⠀⣴⣭⡕⠄⠹⡀
⠀⠀⠀⢰⡇⠈⢎⣿⣿⠃⠀⠀⢿⣿⣧⠆⠀⣧⠀⠀⣀⣠⠤⢤⡤⢤⣀⡀
⠀⠀⠀⠸⡆⠀⠀⠀⠀⢾⠖⣶⠂⠀⠀⠀⠀⣿⠖⠉⠀⠀⠀⠀⠱⡄⠀⠉⠑⠦⣀⠀⠀⠀⢀⣠⠤⣤⣤⣀⡀
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⢷⣤⠚⠁⡰⠊⠀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠳⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠈⠳⣼⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡇⠈⠓⠤⢄⣀⣠⠤⠖⠋⢸⠀⠀⠈⢧⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡆⠀⠀⠈⠙⠒⢶⠆⠀⠀⠀⠀⢹⠤⠤⠦⢤⡀
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⠀⢸⠀⠀⢀⣀⣠⣏⣀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡾⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⢁⣼⠔⠊⠉⠀⠀⠀⠀⠉⠓⠧⣄⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⣼⣁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⢀⣀⣤⣪⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢄⣀⣠⣃⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣠⡇⠀⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠋⠀⢀⣾⠟⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⢀⣴⣿⡏⠀⠀⣠⣴⡏⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⢳⣿⢀⠠⠾⢻⣿⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⣿⡿⣱⣿⡿⠃⠀⠀⣼⣿⣼⠠⠄⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣽⣿⣿⣿⣿⣆⣠⣾⣿⢿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢟⣿⡿⣿⣿⣿⣿⣿⢣⣼⣿⡎⢷⣇⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⢏⣾⡟⣸⣿⣿⣿⣿⡇⡾⣿⣿⢻⡌⠻⣆⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢀⢠⠁⣿⠟⣿⡿⣰⣿⣿⣿⣿⣿⡘⣇⢸⣿⣿⣿⠀⠙⣷⣄⠠⡱⣄
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣿⣾⣿⡇⣿⢸⣿⣿⣿⡿⠧⠀⣼⡈⣿⣿⡇⢢⠘⣿⣧⠙⣿⣦
⠀⠀⠀⠀⠀⠀⢸⡀⣼⢿⣿⣿⠀⣿⣼⠳⠏⢠⣷⢀⣼⣿⠃⣸⢹⡇⣸⡄⠸⣿⡇⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣧⢹⡜⣿⣿⡄⢹⣿⠀⠀⢸⣾⡈⠻⣿⢀⡟⣹⠇⣿⣥⠀⣿⣷⠀⢻⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠹⣿⣿⣿⣦⡿⣷⣄⡈⠻⠏⠀⡹⣾⣱⠏⣼⣿⣧⠀⣿⣿⠀⢸⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⣻⣯⣛⣿⣿⡝⢷⣂⠠⣼⣷⣿⢏⠼⠁⢸⡇⢰⣿⡟⠀⣸⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠀⠈⠋⣉⣿⣿⣿⣷⡙⠿⠿⠯⠁⠀⢀⠴⠋⣴⣿⠟⠁⣰⢃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⣿⣿⣿⣿⣷⣦⣤⣤⣤⣷⣼⣿⣿⠗⢀⡴⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣹⣿⣿⣿⣿⣿⣿⣿⠿⢛⣡⠴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣧⣴⣿⡭⠤⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠿⠟⢻⠟⢿⠿⢻⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢇⣤⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⢠⠟⠙⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⡀⢠⡆⠀⠀⠀⠈⣧⠄⠐⠸⠀⡤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠀⡰⠉⠷⠀⠐⠑⠀⠃⠀⣤⡠⠞⡷⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠸⢤⡇⠀⠄⠀⠀⠀⠀⠀⠀⠉⠀⠐⡁⠘⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠈⠁⡐⠂⡰⠁⠀⠀⠀⠤⠂⡳⠀⠇⠆⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣦⡱⡄⠲⠇⠘⠀⠀⠀⠀⠀⠀⢀⠀⠞⠀⠔⢁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠌⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠊⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠈⡤⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢥⡀⠀⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠮⠴⠖⠖⢤⣼⢶⡿⣷⣾⣷⠋⢻⣿⣷⡖⠦⢤⡄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣤⣤⣴⠞⠋⠀⣸⠁⢀⣿⢻⣆⠀⠙⢯⣟⠉⣷⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢹⣁⣀⣠⣶⡇⠀⠸⣿⠄⣻⡆⠀⠈⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠈⣿⠷⠏⠛⠉⠘⠶⠶⠞

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⢡⡿⠟⢳⡄⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⢀⣀⡀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠺⠁⠀⠀⠀⡇⠀⠀⠀⠀⠓⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⠡⠄⠈⠁⠉⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⢐⡏⢀⠀⢄⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠊⠁⠐⠄⠀⠈⡋⠀⠇⠀⠃⠀⡀⠀⠀⠀⢀⡀⠋⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢭⣈⡁⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠘⠂⠁⠀⠃⠀⠀⠀⠚⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⡀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠀⠰⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⠆⠀⠲⠅⢢⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡤⠀⠼⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡘⠁⠀⣿⠃⠸⠅⠀⠀⠀⠀⠆⣠⠂⠃⠈⠡⠄⡤⢤⣿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠄⠃⢹⢇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡘⠒⣠⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠄⠀⠀⠐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠸⠄⢪⡀⢀⣀⡤⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⢜⣄⠀⠀⠀⠀⠠⠻⣀⡿⠛⠋⠉⠋⠙⠉⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⣀⢤⡄⠠
⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⣶⢟⣄⠀⠀⠀⠐⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⢦⣀⡀⠀⠒⠃⣀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠓⠾⢡⡀⠀⠀⠀⠀⠉⠉⠙⢫⣥⡄⡀⠀⠀⠀⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠑⠀⠀⠀⠞⠣⠂⠀⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠑⢴⡠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣰⣷⡏⠹⡷⣦⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠐⠴⠟⠂⠀⠀⢄⡀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⡶⢂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢒⣘⠯⡓⢈⣐⣻⣆⠀⠠⣀
⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⠀⠔⠀⠁⠀⠀⡀⠀⠀⠀⠀⠋⠻⠆⢈⠁⠀⣠⠤⠶⡀⠀⡀⠀⠍⠀⠀⠙⠀⠉⠀⠀⠀⠚⣯⡏⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠘⣡⡄⠀⠀⠀⠀⠀⠐⢞⡂⣈⠖⠔⢂⠘⠈⢡⠀⠀⠀⠀⠀⠀⠀⠒⠸⠃⠠⡿⣇⠀⡀⢳⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⢀⠀⠀⡘⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠘⢄⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠏⠀⢐⣈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠛⠀⠀⠛⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⡰⠋⠉⢶⡿⠋⠙⣏⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⢶⡟⠋⠙⡍⠙⢦⡀
⠀⠀⡇⢰⡅⢸⠑⣤⠂⣸⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡄⣸⠀⣬⠂⣿⠀⠀⢳⡀
⠀⢰⡟⠦⠴⠋⠧⣭⡵⠋⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠦⠖⠉⠦⠭⠴⠃⠀⠀⠀⡇
⠀⢸⡄⠀⠀⠀⢰⣷⣆⡀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢐⠶⠀⠀⠀⠀⢀⠇
⠀⠀⠙⠶⣤⣶⣿⠷⢻⡟⢋⣓⡻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣤⣴⡮⠴⢺⠁⢀⡀⡘⣆
⠀⠀⠀⠀⣀⣼⠁⣴⣾⣿⣿⣿⣿⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⢀⡴⠊⡱⣇⠲⢺⢸
⠀⠀⠀⠰⠿⣿⠈⠛⢹⡇⠙⢛⣫⡿⣤⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⣿⠈⠂⢹⠀⠈⠉⠁⠛⢤⡀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣻⠀⢀⣼⣷⡆⣠⣶⣦⡙⡿⠛⠛⣿⠋⢉⠙⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠜⢒⠄⢠⢔⡦⡀⠉⠉⠉⠋⠁⠀⠙⢉⣓⡄
⠀⠀⠀⠀⢾⠿⡄⠛⠛⢻⡀⢻⣿⣿⠃⢡⣶⣶⠀⠐⣿⡷⣸⡿⠇⠀⠀⠀⠀⠀⠀⠀⣎⢽⡀⠓⠒⢯⠀⠰⣌⡡⠂⠀⡖⠶⡄⠀⣏⡇⣈⡾⠃
⠀⠀⠀⠀⠀⠀⢻⡄⠀⣠⣿⣿⣀⢀⡀⠈⠛⠋⣰⣆⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡸⠋⣵⠀⠀⠀⠀⠉⠊⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠿⠟⠢⣟⡋⠈⢛⣿⡿⠶⣶⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠟⠦⣗⡉⠈⢓⠎⡱⠤⣤⡶⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⠛⠛⠚⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣭⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣰⡆⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣤⣿⣿⣤⣶⣶⣶⣯⣴⣷⣭⣿⡆⠀⠛⢲⡀⢛⣒⣶
⠀⠀⢀⢸⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⡏⢿⣿⣿⠀⢀⢀⡐⢾⢈⣦⣠⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⣰⣶⠟⣤⢿⡏⢈⡷
⠀⢀⣾⣿⣿⣧⣷⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠋⠀⢀⡌⣦⣠⢷⣏⣿⣯⣯⣧⡐⠀⠀⠉⢈⣍⣉⣩⣍⣭⣥⣤⡄⠀⣜⣿⣶⣼⠟⢹⣏⣸⡅
⠀⣼⣿⠏⠀⢿⣿⣇⠀⠀⠸⣿⣿⢿⢿⣿⣿⠃⠀⠀⢀⣜⣾⣷⣿⣿⣿⣛⣿⣿⣼⣿⣿⣷⣦⡰⣼⣿⣍⠓⢺⣟⡋⠉⠉⠀⠘⠿⠞⢿⣶⣿⠿⠛⠁
⠀⠻⢧⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⣁⣸⣿⣿⡟⠛⠈⠑⠀⠀⠀⠀⡀⠀⠘⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⡀⣴⢠⡀⠀⠀⠀⠀⠀⠀⠆⡐⠂⡆⠐⠀⠀⠀⠀⢰⠀⠀⠀⣿⣿⣿⣿⠇⢀⠀⠀⠀⠀⣠⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⣸⢼⣠⢧⣟⣳⢼⡰⣅⣷⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠉⠁⡄⣑⣘⣄⠀⣀⣙⣦⡴⣃⠀⠀⠀⢠⡀
⠀⠀⠀⢸⡆⠀⠀⣯⣻⣾⣾⣿⣿⣿⣾⣷⣿⣷⣿⡄⠀⠀⠀⠀⠀⢀⣼⣧⣶⡤⣄⠀⠀⠀⠀⢀⠙⠐⠀⠑⠿⠟⠀⢠⣞⣿⡻⣞⡽⠁⠀⢠⣤⣽⠆
⠀⣰⢠⡿⢁⡀⠀⠈⠛⠛⢉⠉⠉⠉⠙⠛⠛⠛⠋⠀⠀⠀⠀⢠⣼⣽⣿⣿⣿⣿⣿⣤⠀⠀⣀⡴⠛⢦⡞⠀⠀⠀⠀⠀⣹⣟⣿⣏⠀⠀⢰⣛⣦⣹⣦
⠀⢿⡈⢷⣼⣄⠀⠀⠀⣠⢿⣆⣤⠀⠀⠀⠀⣦⣷⡄⢣⣀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣞⠀⡄⡟⢷⡜⣮⣷⣴⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⢈⣽⣿⠁
⠀⢸⡿⢸⣿⣿⠇⠀⣰⣯⢾⣿⣏⣦⠀⠀⠲⣬⡟⣷⣦⡙⣷⠀⠁⢸⣿⣿⣿⣿⡟⠃⠸⣧⣿⣿⣵⣿⣟⣿⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⣿⡟⠋
⠀⣾⣷⣾⣿⣿⠀⠀⣿⡇⠀⠋⢹⡿⠀⠈⣧⣿⣷⣿⣼⣷⣡⣧⠀⠻⣿⡟⢽⣿⡅⠀⠀⣽⣯⣿⣿⣿⣿⣻⡇⠀⠀⣀⣤⣴⣆⡀⠀⠀⠀⠉⢻⣿⣦
⠀⠈⠛⠛⠛⠋⠀⠀⠈⠛⠀⠐⠋⠁⠀⠀⠙⠛⠉⠋⠋⠙⠛⠋⠀⠀⠘⠀⠘⠋⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠃⠀⠾⠿⠿⠿⠿⠟⠛⠗⠀⠚⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣾⠃⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢀⣿⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠙⣦⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣤⡾⠿⠛⠻⣷⡀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠈⣣⠀⠀⠀⠀⠙⠻⣷⣤⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛⠛⠁⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠙⠿⣇⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⠏⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠂⠀⠀⠈⠛⣦⡀⢀⣆⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⢀⡴
⠀⠀⠀⠀⠠⡀⠀⠀⠸⣦⡀⠄⠀⠀⠈⠣⠀⠀⠀⠀⠹⣿⣅⠀⠀⠀⠀⠈⢿⣴⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠙⢦⡰⣄⠀⠂⠀⣤⡀⠀⠀⠈⠙⢧⡀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠘⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⠀⠀⠛⠾⣟⠷⣤⣌⣙⣦⡀⠀⠀⠀⠻⣦⠀⠀⠀⢸⡀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢠⠈⠳⣄⠉⠛⠛⣷⠶⢶⣤⣤⣿⠃⠀⠀⢨⣇⠀⠀⣠⡟⠁⠀⠀⠀⣠⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠉⠓⢾⣄⠀⠈⢷⡄⠀⣿⡀⠀⠙⠻⣿⣦⡆⠀⠈⢿⣤⣾⠋⠀⠀⠀⢀⣼⣁⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⡀⠀⠀⠀⠉⠳⡄⠀⠉⠲⣤⡙⢤⠀⠀⠘⢇⢻⡄⠀⢸⡿⠀⠀⠀⣀⣴⣿⠿⠛⠉⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠹⣆⠀⠐⠛⠛⠻⢷⣦⡀⢸⡀⢿⡄⢸⣷⠀⣷⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⣀⣈⠑⠿⠷⣦⣄⡀⠀⠀⠈⠛⢿⣧⠸⣷⠈⣿⣄⣿⣿⠟⠷⣤⣀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡻⠛⠶⣄⡉⠻⣍⠳⢶⣤⣀⠀⠹⣆⣻⡇⠈⢻⣿⣇⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣀⠀⠈⠀⠀⡠⠌⡙⣷⣮⡑⠂⢤⣙⠿⣷⣽⣿⣷⡀⢀⠻⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠟⠛⠛⠛⠛⠛⠛⢛⠻⠿⣷⣤⣈⣳⣦⣍⣿⣿⣷⣌⣷⣿⣿⣶⣦⣤⣤⣶⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡤⠖⠊⠉⠉⠉⠙⠛⠛⠶⣶⢶⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠛⠋⠉⠉⢁⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠠⠄⠒⠉⠀⠀⢒⣠⣴⣿⣿⣿⣿⡿⠟⢛⠿⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠤⠶⠛⣉⣁⣴⣿⣿⣿⣿⣶⡞⠋⠀⠀⣀⣤⣄⣟⠀⣿⡗⢀⡀⠀⠀⠀⠀⠤⠤⠜⠓⠂
⠀⠀⠀⠀⠀⠀⣠⠶⡞⠛⠙⠛⠁⠀⠀⠀⠀⠛⣉⣽⣿⣿⣿⣿⣿⣿⠁⠀⢆⣴⣿⣶⣿⣧⠀⣿⠀⠈⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⢀⣤⡤⠐⣂⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣷⡟⠀⠀⠘⠀⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠔⠂⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢻⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠘⣿⣇⠀⠈⡁⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢻⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣙⣿⣿⣿⣿⣿⣿⠶⠆⠀⣄
⠀⠀⠀⠀⠀⠀⠐⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠿⠛⠃
⠀⠀⠀⠀⠀⡠⠊⢁⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣠⠖⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⡔⠁⠠⠚⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠈⠀⠀⠀⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠛⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⡟⠉⠘⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠉⠀⠀⢠⠟⠠⠋⢠⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⠟⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠠⠀⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⠄⠑⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠢⡀⠀⠀⠠⠀⠀⠠⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠌⢀⠄⠁⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠑⠄⠀⠄⠀⠁⠀⠘⠀⠀⠃⠀⡀⠀⠠⡀
⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⡀⠑⢀⠰⠁⠈⠁⠁⠀⠀⠀⠀⠀⢂⠀⠄⠀⠀⠀⠠⡀⠂⠀⠀⠀⢀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⢀⢘⢁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠢⢀⠀⠁⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠈⠀⠀⢁⡤⠀⠀⠀⠀⠀⡠
⡇⠀⠀⠀⠀⠇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠣⡀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠄⠀⡀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠂⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠈⢄⠈⠠⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠐⢠⠆⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠑⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⢀⣀⣀⠀⠀⣀⡀⣀⡈⣀⣀⣀⣀⣀⣀⡀⠀⣂⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡀⣀⡀⢀⢀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣥⡴⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⣀⣤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠙⣳⠞⠁⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢠⡏⠀⠀⢀⣀⡀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⡷⠶⢺⡃⠀⠀⠀⠀⢸⡇⠀⢰⡃⠀⢹⠄⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣡⡟⠀⠀⢘⡧⠴⠶⠶⢤⣀⠳⣄⠀⠙⠒⠉⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡴⠋⠀⠀⠀⣀⣀⡈⢳⡈⠓⠶⠦⠴⠖⠋⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠁⠀⣹⠀⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠙⠚⠁⢀⡏⠀⠀⠀⠀⣀⠀⣸⠂⠀⠀⠀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣠⠞⠳⠦⣄⡀⠀⠙⠛⠁⠀⠀⢀⣴⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠋⠁⠀⠀⠀⠈⢳⣄⣀⣀⣀⣤⠴⠛⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠟⠀⠀⠁⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠈⠳⣤⣀⠀⠀⠀⢀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡽⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⡀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀⣀⣤⠞⠁⠈⠙⠓⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠋⠉⠀⠀⠀⠀⠈⣯⠁⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠚⠛⠉⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠛⣦⡤⠶⠖⠒⠒⠒⠶⠶⠿⠟⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠦⠴⠞⠋
⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣏⢹⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⡀⢻⡿⠿⡟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣀⢷⣾⣿⣇⢠⣾⣷⣅⠀⣀⣀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⠾⣿⣢⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣛⣿⣿⣿⣿⣿⣿⣋⣿⣿⠿⠿⣿⣿⣿⣿⣻⡏⠀⣰⣿⡄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣷⣾⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠿⣿⣿⣿⠀⠿⣽⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣧⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⢋⡷⠊⠽⢿⣿⣿⣿⡿⡟⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠸⠛⠿⣿⣿⣿⣿⣿⣿⢟⣿⢟⡟⠿⢿⣿⠗⠀⢻⣶⡿⢁⠀⠀⢪⣿⣿⣿⣾⡻⢟⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣀⠀⡿⠁⠙⠷⠦⣼⣿⣟⣿⣿⢿⣩⣽⠇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⢔⣣⠞⠁⠀⠀⠀⢰⣿⣿⣿⣿⣯⣶⣤⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢁⣤⣷⠿⠁⠀⠀⠀⠀⢴⣿⠟⢏⡉⠀⢽⡟⠀⠉
⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⢟⣉⣡⣶⣶⣦⣤⣶⣿⡟⠁⠀⠀⠀⠉⠉
⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠿⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣀⡀⢤⣠⣶⣿⣿⣿⡿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠋⠘⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠻⣿⡾⡇⠀⠀⠀⣈⣛⡙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⢹⣶⣄⣤⣶⣾⣿⣷⣶⣶⣿⣿⣿⣻⣿⣟⣯⡳⣲⣤⣤⣤⡀
⠀⠀⢀⣀⣤⣾⣿⣿⣿⣷⡿⢷⣿⣿⣷⣽⣿⣯⣥⣉⡙⠻⣿⣷⣟⣿⢿⣿⡿⢳⡄⡀
⠠⠿⠿⢟⣿⡿⣿⣿⠟⢋⠀⠉⠉⠛⠿⠿⣿⣷⣭⡷⣿⠀⠀⠙⠛⠿⡷⡛⠚⡌⠁
⠀⠐⠒⠛⠛⠋⠉⠀⠀⠀⠀⠀⠘⠓⠒⠂⠈⠻⢿⡟⡏⠀⠀⠀⠀⠀⠀⠁⠈⠁


⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⢀⣼⣇
⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣤⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⢰⣷⣿⠇⠀⢻⣷⡆⠀⠀⢰⣷⣿⠿⣿⣿⣆⡄⠀⠀⠀⠀⢻⣿⣿⣿⣶⣇⠀⠀⠀⠀⢸⣿⣿⠻⣿⣶⠀⠀⠀⠀⡄⣿⣿⠏⢿⣿⡆
⠀⠀⠸⣿⠁⣰⣦⠈⣿⡇⠀⠀⢸⣿⠿⠀⠘⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠐⣶⡿⠁⡀⠛⣿⠀⠀⠀⠀⣿⡿⠁⠀⢸⢿⡿
⠀⠀⠀⠙⠦⡙⢛⠴⠋⠀⠀⠀⠀⠻⠄⠀⠀⠠⠟⠀⠀⠀⠀⠈⠻⢿⣿⡿⠟⠁⠀⠀⠀⠀⠹⠄⢾⡿⠀⠋⠀⠀⠀⠀⠙⠧⠀⠀⠀⠜⠁

⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣿⣷⠀⠀⠀⠀⠀⠀⣆⢸⣿⡄⡄⠀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀
⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀⠀⠀⠀⢠⡄⣾⣿⢿⣿⣀⡄⠀⠀⠀⠀⣶⣻⣿⣷⢀⡄
⠀⠀⠀⢸⣿⣿⣿⣇⣷⠀⠀⠀⢈⣿⣿⠏⠻⣿⣿⣶⠀⠀⠀⣼⣿⣿⠋⢻⣿⣧⠀⠀⠀⣿⣿⣿⠟⠈⣿⣿⣿⠀⠀⢰⣦⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣷⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⡇⢀⣦⠀⣿⠇⠀⠀⠀⢻⣿⠁⠀⣼⢿⡟⠀⠀⠀⠸⣿⠉⢀⣆⠉⣿⠏⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⠛⠻⠟⠛⠁⠀⠀⠀⠀⠈⠁⠘⠟⠁⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠛⠟⠀⠉⠀⠀⠀⠀⠈⠙⠚⠛⠛⠋

⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⢀⢀⣼⣿⣷⡄⠀⠀⠀⠀⡀⠈⣿⣧⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⠀⡀
⠀⠀⢀⣿⣿⣿⢿⣿⣿⡄⠀⠀⢀⣿⣄⣽⣿⡟⢀⡀⠀⠀⠀⡀⢻⣿⠛⢿⣿⣆⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⡀⣿⣿⣿⣾⣿
⠀⠀⠸⣿⡟⠉⠀⠹⣿⡇⠀⠀⢸⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⣿⣾⠋⢠⡀⢻⣿⠀⠀⠀⢠⣸⣿⡏⠿⣷⣾⡆⠀⠀⢸⣿⣿⣿⠛⢿⣿⣿
⠀⠀⠀⠻⣧⡀⠀⣰⠟⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⢿⣄⣿⣧⡼⠃⠀⠀⠀⠀⠻⡏⠁⠐⢻⡟⠀⠀⠀⠈⢿⡿⠷⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠓⠓⠚⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠤⢄⠀⢀⣀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⣶⣦⣤⣤⡤⠂⡜⠀⠀⠀⠀⠀⠀⡰⠋⠉⠉⠉⣻⠀⠀⠀⣀⠔⢢
⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⠿⣿⠿⠿⡗⠐⡇⠀⠀⠀⠀⠀⠀⠙⢶⠒⠚⠋⡙⠀⢠⠞⠁⡔⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⣀⠁⠀⡼⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⡇⡴⠃⣠⠞⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⠉⠉⠀⠀⣀⡇⠀⠀⣀⣀⣀⠀⠀⠀⠈⡆⠀⣗⣱⠟⠁⠀⣠⣯⣤⠶⠶⠛⠛⠛⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠒⠒⠂⠉⠀⡷⠚⠉⠉⠉⠉⠛⠳⢦⣄⣱⠀⣏⢛⣠⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠠⠂⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣿⣩⣟⡁⠀⠀⠀⠀⠀⢄⠀⠀⢀⣀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠂⠈⠀⠀⡠⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⠵⢞⢡⡀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⡀⠔⠊⠀⢀⠝⢦⡐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢀⠔⠁⠀⠀⣸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⠢⡄⠁⠀⠀⢀⡰⢃⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣈⠒⠒⠚⡉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⡀⠀⠀⠄⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⠽⠯⠟⠛⡙⠓⠶⠶⠴⠶⠞⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠉⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠂⠀⠒⠦⢸
⠀⠀⠀⠀⠀⠀⢰⠁⢠⣀⣀⡀⢀⡠⢸⠀⠀⠀⠀⠀⢀⠤⠴⠒⠲⣄⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢸⡂⣾⣿⣿⡿⣿⠀⡏⠀⠀⠀⠀⠀⠘⢦⠤⠴⠖⡇⠀⣠⠚⢡⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡇⢯⡈⠁⢀⠆⠀⡇⠀⠀⠀⠀⠀⠀⠈⢆⠀⢠⢃⡔⢁⡴⠃⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠉⠉⠁⠀⣀⡇⠀⢀⣀⣀⣀⠀⠀⠈⡆⢸⣎⡴⠋⠀⣠⣮⡤⠶⠖⠚⠛⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠀⡗⠋⠉⠉⠉⠉⠛⠶⣤⣧⠈⣏⣡⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠔⠊⣹⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠿⣾⡥⠀⠀⠀⠀⠐⠄⠀⣀⠤⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡀⠔⠉⢇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⡄⢆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠒⠈⠀⠀⠐⠁⠑⢦⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠣⡀⠀⠀⠀⠀⣠⢇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢈⠢⠤⠤⠚⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⡀⡀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢛⡻⣿⠿⠟⠻⠥⣤⣤⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⠀⠈⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⢀⠄⠀⡆⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠰⠳⣄⡎⠀⠀⠁⠠⠀⡜⢆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠈⠆⠀⠀⢀⠀⠰⠁⢸⠈⠰⠃⢰⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡏⠲⢦⠀⠀⠀⠀⠀⠈⢄⠁⠀⢸⠀⠀⠀⡌⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⡘⠘⡐⢚⡇⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⡀⠀⠳⠉⠀⠌⡰⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢤⣄⡀⠀⠀⠀⠀⠂⠄⠐⠂⠀⠀⠁⠄⠂⠀⠀⠤⠊⠉⠈
⠀⠀⠀⠀⠀⣼⠹⣭⢍⣠⢴⠚⢠⣀⣤⡤⡖⠢⣤⢼⣐⠒⣦⠀⢲⢢⡀⠘⠛⣻⣻⡄⢠
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠻⠼⠛⠛⠊⠳⣾⡴⠃⠰⣌⣠⡼⠷⣤⣸⡖⠲⠮⠯⠟⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⣠⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠉⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠤⠤⣤⣤⣄⣀⠀⠀⢠⠞⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⡟⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠀⠀⠀⡞⠉⣽⣯⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣀⣠⣤⢤⣄⠀⠀⠀⠀⠀⠀⣴⠶⠖⠛⠋⠉⠁⠀⠘⠧⠀⠀⠀⠻⠶⠶⠋⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⣧⠀⠀⢶⡖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠙⠛⣻⠂⢰⡏⣀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠙⢦⡀⠘⠳⢦⣤⣤⣤⣤⡤⠴⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⣠⡤⠤⣤⣉⠀⣸⠃⠉⠙⠛⠛⠒⠒⠚⠛⠁⠙⠳⠄⠙⢦⣀
⠀⠀⠀⠀⠀⠀⢸⠃⠀⣰⠟⢡⡴⢦⣄⠙⢧⢉⡀⠀⣀⡀⣀⣤⣀⠀⢠⠀⠀⣀⢠⣤⣀⠁⠀⠀⢀⣤⣤⡄⣠⣤⣤⠀⣄⠀⢀⡀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢸⡆⠀⢿⡀⢿⡄⠀⣸⠇⢸⡇⠹⣶⠋⣼⠋⠀⠙⣧⢸⠀⠀⣿⢸⣀⣹⠇⠀⠀⠀⢸⡇⠀⣿⣤⣄⠀⠙⣶⠏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠈⠳⢦⡴⠞⠋⢀⡼⠁⠀⣿⠀⠘⢦⣤⡴⠋⠸⣦⣴⠏⢼⠉⢻⡄⠀⠀⠀⢸⡇⠀⣿⣤⣤⠀⡾⠋⢷⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⣤⡤⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠀⠀⢿⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠙⢦⣌⢷⣄⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠙⣆⠙⠷⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢛⡓⠦⡀⠈⢷⡀⠀⠀⠀⣿⠀⠀⠘⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⡄⠀⠻⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⣰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡆⠀⣠⠟⠀⠀⠀⠹⣄⡀⢻⠀⠸⣇⠀⠀⠀⠀⠀⢰⡇⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⠟⢰⡏⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠘⢷⣄⠀⠀⠀⠀⠻⣄⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⡉⠀⠀⠻⢦⣄⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠉⠳⢦⣄⠀⠀⠈⠳⡄⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠀⠀⠈⠉⢻⠓⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠃⠀⠀⠀⠀⢠⣸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⢠⡞⠀⠀⠀⢀⣤⠟⣿⠃⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠀⠀⢹⡆⢀⡤⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢁⡖⠉⠀⣀⡴⠞⠋⠀⣰⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠘⡇⢸⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠘⢧⣠⠞⠁⠀⠀⠀⠘⠁⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢰⠇⢸⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠁⠀⠀⠀⠀⠀⢠⢀⠞⠁⠀⣰⠋⣠⡾⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠃⢀⣠⠄⣀⣴⠆⢀⠀⠀⠀⠞⠀⠀⠀⠀⠟⠉⠁⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠰⣶⣤⣛⠀⠈⢁⣾⣿⣷⣿⣿⠃⠀⠀⣀⠀⠖⢀⣀⡀⠀⠀⠀⠀⠀⣰⣯⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⣍⣻⣦⣄⡀⠀⣀⠐⣀⣈⠛⠻⣟⣻⣿⣿⡿⠟⣋⠀⠀⣠⣤⣤⣤⡶⠿⠛⠁⠀⠀⠀⣀⡼⠯⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠴⠶⠛⠻⢶⣤⣄⣈⣁⣰⣿⡿⣯⣥⣤⣀⡛⣿⣿⣶⣬⣍⠀⠻⠷⠶⢶⣆⣤⣴⣾⣿⣶⣶⣶⠤⠴⠶⠖⣶
⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⣤⡤⠤⠶⠶⠚⠋⠉⠉⠉⠉⠁⣀⡤⣿⣿⣿⣿⣿⣿⣿⡿⠿⢶⠲⠶⣿⣿⣭⢤⣭⣍⣛⡛⠛⠶⠦⢤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠋⢑⣻⡿⠟⠋⢁⣀⡈⠉⠉⠉⠋⠉⠛⠒⠾⠭⣅⣀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⣯⣉⣀⠤⠖⠚⠉⠉⠀⠠⠖⠛⢉⣀⡙⠚⠙⠛⠶⠶⢦⣄⣠⣤⡴⠶⠯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠒⠀⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⣀⠀⠀⠀⠀⠀⠈⢉⣑⡊⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⡾⠋⣁⣀⡀⠀⠀⠀⠀⡎⠀⠈⠒⡌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⠀⠜⠁⠀⠉⠢⡀⠀⠀⢡⠞⣿⡆⠈⡌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠈⠀⠰⣶⣦⠀⢱⠀⠀⠀⢿⠿⠏⠀⡤⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠰⣿⣿⠇⠀⡆⠀⠀⠈⠢⠤⠈⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠓⠒⠺⠥⡀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢠⢦⣀⣄⣤⣤⣤⠤⠴⠖⠋⢹⠇⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⠞⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠤⠴⠚⠁⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠢⢄⡀⠀⠀⠀⣀⣠⠔⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠒⠒⠲⠤⢀⣿⠦⠀⠀⠀⠀⠈⠉⠉⠉⢀⣠⠞⣽
⠀⠀⠀⠀⠀⢀⣴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠐⠒⠒⠒⠉⠉⠀⠀⣿
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠓⣄⠀⠈⡄⠀⢠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡀⢢
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⡆⢠⡇⢀⡞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⢀⣠⠂⠀⡆
⠀⠀⡟⣠⠂⠀⠀⠀⠀⠀⢀⣀⣀⢳⣷⣼⡤⠛⠀⠀⠀⠀⠀⠀⠒⠤⠤⠤⠴⠒⠋⢹⠉⠉⠀⠀⠀⣾
⠀⢸⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠠⠀⠀⢸⠇
⠀⠈⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠸⢀⡴⠋
⠀⠀⢹⡄⠀⡇⠀⠀⠀⢀⠁⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣤⢴⠟⢀⣠⠴⠛⠉
⠀⠀⠀⢳⡀⢇⠀⠀⠀⠸⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠉
⠀⠀⠀⠀⠙⢬⣆⠀⠀⠀⢣⡀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣄⣑⣤⣀⣀⣀⣀⣈⣉⣷⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣾⣿⠿⠷⠶⠿⠯⣟⡷⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡼⣯⢿⡿⠑⠄⠁⠀⠀⠀⠀⠉⠳⣶⠞⣦⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠀⠐⠁⣂⡄⠐⢤⠀⠀⠀⠀⠀⠈⢳⡚⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠁⡉⠀⠀⠀⠀⠀⠋⢻⣤⠜⠲⢤⠀⠀⠀⣠⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⠁⠀⠀⠀⠀⠀⡆⣼⣶⡎⠁⢹⠃⠀⠀⠈⡠⡴⠚⣽⠧⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⡰⠁⠀⠀⠀⠀⠀⠈⠁⣾⣤⣿⠀⠘⠀⠀⡠⢊⣼⠁⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⣀⣀⠀⠀⠀⢠⠀⠀⠀⠀⣤⠌⡟⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⡺⠇⠀⠞⣠⠞⠁⣬⠻⠶⢤⠶⢒⣲⠖⡜
⠀⠀⠀⠀⡢⠀⢀⣫⠦⢒⢶⣿⣿⣠⣤⡈⠀⠄⠀⠀⠀⠠⠔⣰⡇⢄⡀⠀⠀⠀⠀⠐⠓⠀⢻⣻⡦⣀⠞⠃⠀⠀⠀⠀⠀⣿⠀⠀⠉⣀
⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠛⠃⠀⠚⢛⠿⣶⣄⣀⢠⠠⢀⠔⠋⠒⠉⠐⠚⠛⠃⠀⠀⠀⠀⠸⡿⠟⠁⣤⠄⠀⠀⠀⠀⢨⡎⠛⠒⣿⠋⠁⠵
⠀⠀⠀⠐⠀⠀⠠⠄⠰⡆⠀⠀⣠⠀⠦⣤⣶⠟⠛⢿⣤⢣⠂⡀⠀⠀⠀⠀⠀⠀⠀⡠⠌⢠⡄⢷⡀⠈⠁⠀⠀⣠⡦⠀⠀⠀⠀⣔⢻⣍⡀
⠀⠀⠀⠀⠀⢁⠀⢰⣠⣵⣄⣠⢀⣤⡾⠊⣠⣤⣤⡀⠛⢬⠈⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠶⣄⢀⡀⠡⠀⢀⣤⠄⢈⢉⠁⠤⢉⡟⠓⠄
⠀⠀⢠⡐⠀⠢⣪⡿⢿⣿⣿⣥⠟⠉⠀⣼⠛⣠⠿⠷⠀⢢⠈⠁⠀⠀⠀⠀⠀⣴⠄⠂⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⡀⠈⢁⠀⣈⡀⠠⢔⢚⣧⠀⡃
⠀⠀⠀⠑⢀⡞⠁⠞⠁⠒⣿⡟⠀⡀⠄⠑⢾⣟⣆⠰⢷⡀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠑⠲⣤⣉⠁⣀⠀⠀⡈⠑⢦⡃
⠀⠀⡀⠐⠀⠀⠐⢂⣀⡴⢀⡄⠀⢣⡢⣄⠀⣊⣥⠴⠾⠿⣦⣤⣦⡙⠩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠠⠉⡀⠝⠶⣤⠾⠗⠀⣠⣤⢽⡄
⠀⠘⠏⠀⠀⠐⢠⢾⣷⢃⠾⣡⠀⣸⣳⡿⣛⣛⡵⢶⣯⣠⣶⣿⡻⣿⣆⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡅⠐⠂⢠⡄⣗⢉⢧⡀⣽⠾⣽⡀⢿
⠀⠀⠄⠀⠐⠂⠈⢰⣷⣱⣼⣵⠄⡁⣿⣿⠟⡛⡷⣾⢿⡿⠋⣿⠿⢻⣿⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠔⠁⠃⢙⡐⠀⠉⣚⢆⢹⡆⡀⠸⢷⡸⡄
⠀⠀⠀⠐⠐⠠⠒⢿⣿⠾⣿⠋⠀⣿⢫⡅⣼⠀⠹⠇⡘⠉⠉⠁⠋⠈⠂⠚⣛⣿⣷⣦⣄⣀⠀⠐⠚⣀⠀⢏⡴⠂⠀⠢⡈⠁⠀⢘⢨⢸⢍⣷⠟⠳⠓⣟
⠀⠀⠀⡀⠀⠀⣠⠘⣿⢄⡇⡀⠀⠙⠳⠺⠉⠣⡐⠀⣠⣶⠶⠾⣿⢿⣷⣶⣔⠀⠻⣶⣿⣿⣟⢶⣦⣤⣙⣦⡤⡼⣶⠶⠾⠧⢴⣭⣀⠉⠘⣿⣈⣷⡀⢸
⠀⠀⠀⠀⢀⠀⠴⢀⡘⢾⠇⠀⠀⠠⣄⢩⠂⠀⢠⣞⡿⣶⠟⢉⡛⢻⣾⣻⣷⢟⣦⡱⡼⣭⢹⣙⣍⣽⡷⠖⠋⠉⠉⠉⡁⢀⣄⣈⣹⡳⣦⣈⡛⡄⢣⠈⠁
⠀⠀⠀⠀⠀⠀⠠⡂⢹⣇⢠⠀⣧⣤⠛⡃⠀⢰⣿⣿⢫⡆⣴⢻⢿⣷⣏⣟⣿⠆⠩⢷⠳⣾⠉⠀⠀⠀⠀⠀⠀⢠⡦⡎⠉⡁⠀⠁⠀⡅⢾⢳⣏⡑
⠀⠀⠀⠘⢄⢩⠄⢁⣸⡹⣄⠀⣪⢻⠷⢠⢈⢸⣿⡛⠈⠀⠻⡾⠶⣋⢾⣟⣿⠙⣰⠋⠻⢿⣷⣄⠀⠀⠀⢀⣼⣟⣿⢀⠑⠀⠀⢣⣀⠀⡹⣆⣏⢳⣇
⠀⠀⠀⢀⠀⠀⠀⠀⠛⠀⠀⢳⣿⣿⡷⣤⠀⠸⣿⣿⣶⣶⠷⠈⠉⡁⢹⣿⣏⣹⠏⠀⠀⠀⠈⠉⠛⣒⣀⣾⣿⣾⡿⠠⠁⠄⠀⠐⠀⠀⡞⣿⣹⢸⣇
⠀⠀⢀⠈⠈⠰⠂⠀⠀⢠⣶⠘⣋⣿⣿⣌⠀⠄⠹⣷⣦⣤⡤⠁⠐⣠⣬⢟⣤⠏⠀⠀⠀⠀⢀⣴⣾⡿⣽⣿⣿⡿⠉⠄⠀⢀⢀⠀⠀⡚⣇⣿⣪⣼⡏
⠀⠀⠈⠁⠀⠀⠀⠀⠀⡟⠛⠘⠛⢿⣿⣧⡝⠂⡀⠙⢿⣯⣤⣴⡷⢛⡥⠚⠁⠠⠀⠀⠐⣴⣿⣿⡳⣻⢧⣿⠟⠁⠨⠘⠌⠌⠀⡤⢻⢁⣾⢯⢇⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡜⠃⣀⡂⠙⢿⣿⣱⠖⡀⠀⠙⣻⢟⣴⠻⠁⠂⠀⠀⠀⢀⣼⣿⣳⣯⣾⡿⠟⣁⠠⠨⠠⢀⢂⡀⢛⠟⣲⣿⡿⠋⡤⠚⡸
⠀⠀⠀⠀⠀⠀⠔⠂⠀⠘⢣⣎⢅⠐⠙⠀⢹⣿⣯⡘⣉⡴⣣⠋⠁⠀⠀⠄⠀⢠⡶⢿⡿⡟⡛⠋⠉⠄⡀⠈⠈⡤⣤⢌⣉⠓⣯⣾⣿⡿⠅⣠⢴⡀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣆⢄⠸⣧⠤⢀⡶⣠⣾⣿⣿⣿⠞⠜⠁⠀⠀⠀⣀⣴⡟⢉⣁⡀⠀⣀⠀⣄⣴⣴⠈⣺⠛⠲⠚⣨⣥⣾⠟⢡⠉⣦⠔⣁⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠎⠳⣙⠷⣶⣾⣵⡯⠾⠛⠁⠀⡀⠀⣠⣴⣾⣇⣀⣛⣋⢉⢳⠔⠋⠷⢞⢃⣈⣛⣥⣴⣾⠿⢋⣻⠄⣤⣈⣵⡼⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⣼⡓⢬⡑⠒⠂⠲⠛⣄⡚⠉⣀⠿⢿⠿⢿⠿⣿⣿⣿⠟⠻⠟⣛⣿⣿⡿⠟⣉⡀⠰⠆⠈⠉⠀⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠬⠥⠤⠄⢠⡆⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⠆⠀⠀⠀⠀⠈⢩⡍⠛⠻⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢘⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠀⠀⠀⠀⠀⣾⠄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣄⠀⠀⢿⢻⡇⠀⢀⣠⣤⡆⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣟⡳⠀⠘⡘⠀⠰⣛⣿⠟⠀⠀⠀⠀⢀⣠⣿⡿⠿⣛⣻⣭⣭⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢆⣇⡜⠉⠁⠀⠀⠀⠀⠀⢰⡏⣵⣶⣾⡿⢿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡿⢿⡸⢏⡿⢿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⡻⡿⢻⣦⣤⡀⠀⠀⠀⠀⢸⡇⣿⠺⢎⣿⣿⡱⢇⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡟⣿⡿⡾⣿⡟⣿⣿⣄⠀⠀⠀⠈⢧⡻⢿⣿⣿⣿⣿⡿⣫⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠷⠾⠿⠾⠿⠾⠿⠶⠿⠿⠃⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⠚⠋

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣤⡴⢳⢣⣤⡄⠀⠀⠀⢀⠔⡊⠡⠌⡒⢢⠀⠀⠀⠨⠵⠒⡀⣠⠀⠰⠴⠀⠀⠀⠀⠀⡀⣯⡓⠃⠀⠀⠀⢀⠓⠐⣋⠀⣙⠀⡋⡆
⠀⠀⢼⠃⠠⡀⠀⡔⠀⠄⠀⠀⠜⠘⢀⣔⢒⠈⡌⠢⠀⠀⣰⡂⢺⠕⢊⠤⣄⡢⠀⠀⠀⠠⠺⠇⠃⠟⠢⠀⠀⠀⠘⢂⢈⠨⠀⠅⠃⡘⡇
⠀⠀⢸⡅⠀⠀⠀⡗⠒⡇⠀⠀⢱⠐⢄⠁⠚⡠⠠⠋⠀⠀⠁⠁⡇⠙⠥⢴⢲⠀⠀⠀⠀⠗⠸⠂⠀⠺⠐⠇⠀⠀⠰⡒⠀⠘⠶⠁⡄⢂⠇
⠀⠀⠈⠚⠦⠴⠤⠃⠀⠁⠀⠀⠀⠑⠒⠄⠤⠒⠊⠀⠀⠀⠀⠘⠝⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠇⠠⠿⠊⠀⠀⠀⠀⠀⠘⠈⠉⠠⠃

⠀⠀⠀⠖⠀⠄⠍⠑⢤⠀⠀⠀⠀⠀⠠⠤⠤⠀⣈⡢⠀⠀⠠⢀⡀⢀⠀⠉⠉⠁⠀⠀⠀⣀⣀⡄⢠⠤⠢⢄⠀⠀⠀⠀⡔⠰⠭⠐⢆
⠀⠀⠃⢀⠄⠷⠧⠄⠀⠁⠀⠀⢎⡉⠠⠤⠄⠀⢈⠀⠀⠀⠀⠀⠀⣠⠢⠐⡂⡠⠀⠀⢀⢤⣠⢤⣯⢯⣽⢯⠀⠀⠀⠀⡀⢀⠤⡀⢈
⠀⠀⡄⠸⣀⠀⢀⡀⠀⠄⠀⠀⢀⡀⠀⠨⠭⠭⢭⠕⠀⠀⠀⠘⠃⠓⠒⠃⠀⡆⠀⠀⢨⢾⣨⢯⠏⡑⠂⡁⠀⠀⠀⠀⢁⠬⡪⠤⡀
⠀⠀⠈⠢⠄⢀⠀⠐⠎⠀⠀⠀⠣⡤⠬⠭⠀⠀⠀⠀⠀⠀⠈⠀⢶⡀⣠⠦⠈⠁⠀⠀⠀⠙⠚⠉⠈⠁⠀⠀⠀⠀⠀⠀⠸⣤⠀⡤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠤⠤⣢⠀⠀⠀⠀⠀⠀⠀⡚⢲⠀⠀⠀⠀⠀⠀⢀⡀⠠⡦⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⢠⠁⠈⢆⠀⢀⠊⠈⠂⠀⠀⣌⠀⠀⡀⡀⠀⠀⡁⠀⠀⠀⠀⠖⠊⠑⠒⠀⠀⠀⠀⠀⡈⠃⠄⡃⠄⠋⡀⠀⠀⢀⠧⠧⠠⠶⠀⠥⠤⡀
⠀⢀⡃⡠⢪⠷⣯⠠⡀⣀⠀⠀⠀⠁⠒⢿⡗⠐⠁⠀⠀⠀⢠⠂⠳⢄⡠⠞⠐⠄⠀⠀⢈⡉⠁⠚⠜⠀⠉⡋⠀⠀⠘⠆⠴⣎⠉⡱⠦⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠂⠄⠀⠀⠈⠒⠄⢠⠄⠠⠀⠁⠀⠀⠈⠉⠀⠢⠡⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠠⢒⡀⣐⢢⠀⠀⠀⠀⠀⠀⣎⡰⢎⡰⠀⠀⠀⠀⠀⣠⢒⠂⣘⣖⣤⠀⠀⠀⠀⠀⠀⠀⣆⡾⠆⠀⠀⠀⢠⣴⡤⠔⠒⠆⢠⣤⡆
⠀⠀⠀⠀⢆⠀⢈⠀⠀⠀⠀⠀⠀⡞⠀⢠⠀⠀⠇⠀⠀⠀⠈⠀⠘⠙⡇⠯⠀⢑⠀⠀⠀⠀⠀⠠⠣⠀⠀⠀⠀⠀⠈⡉⠁⠀⠀⠀⠈⠃⠁
⠀⠀⠠⠘⢄⠉⢁⠎⠤⠀⠀⠀⠀⢻⠀⢸⠀⠠⠆⠀⠀⠀⠉⠉⢉⠁⠊⠑⠈⢹⠀⠀⠀⠀⡐⠁⡂⠳⡀⠀⠀⠀⠀⢡⠀⠀⣀⠀⢀⠄
⠀⠀⠀⠈⠄⠲⠃⠈⠀⠀⠀⠀⠀⠀⡀⢐⢸⢐⠀⠀⠀⠀⠈⠲⢙⡁⢈⣋⠖⠁⠀⠀⠀⡔⠀⠁⠃⠁⠐⡀⠀⠀⠀⠀⠁⡞⠞⡢⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⢀⡰⠴⢢⠀⠀⠀⠀⠀⡠⠐⠂⠀⠰⣀⠀⠀⠀⣦⢶⢀⣀⣀⡀⣴⢦⠀⠀⠀⠠⠔⢒⡀⠐⢄⡀⠀⠀⠐⢒⢂⠐⡖⠀⣶⠐⡄
⠀⠀⢪⠀⠈⠀⠒⢘⠈⠂⠀⠀⡌⢀⠴⠊⡒⠦⡈⠂⠀⠀⢉⡍⢀⠤⠤⡀⠉⠉⠀⠀⠰⠃⢆⠵⠲⢕⢤⠵⠀⠀⠰⠆⣀⠔⡄⠆⣀⠐⡇
⠀⠀⢸⠀⣃⣉⣁⡁⡀⠀⠀⠀⡀⢅⡉⣠⠀⡈⠝⡀⠀⠀⠀⠀⠨⡢⢔⠄⢠⡄⠀⠀⢀⠀⡸⣂⣄⡸⡐⢰⠀⠀⠘⢉⢤⠉⢟⠈⣬⠘⡄
⠀⠀⡆⠑⣄⣤⣤⠄⠑⡆⠀⠀⠘⢢⡀⠀⠈⣀⠔⠁⠀⠀⢾⣳⠠⠤⠤⠄⣷⣾⠀⠀⠈⠢⣋⠐⠱⣊⡠⠊⠀⠀⢰⠎⠉⠆⣘⠦⠉⠴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠁⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⠶⠶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣧⣤⣄⣀⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⢹⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢹⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢰⡏
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠷⣦⣴⠟⠋⢹⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢹⡇⢀⡿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⣸⡇⣸⡇⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⢀⣿⢀⣿⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⣦⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⡛⠛⠛⠛⢛⣛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡏
⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠙⠛⠛⠛⠛⠉⠉⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠛⣿⠁
⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⢀⣾⠛⠛⠷⣶⣶⣾⣧⣀⣀⣿
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢸⡇⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⢸⡇⠀⢀⣾⠋⠀⠀⠈⠉⠉⠛⠛⠻⠶⠶⣤⡀
⠀⠀⣾⠃⠀⠀⠀⣶⢶⣄⠀⠀⢸⣇⣠⡿⠁⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣿⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢠⡿⠀⠀⠀⠀⣿⡀⠙⢷⡄⣸⡟⢉⣤⡶⣦⣄⠀⠀⠀⢸⡇⠀⠀⢸⣏⣴⠏⠀⠀⢀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⢸⡇⠀⠀⠀⠀⠈⠷⣦⣼⣇⣿⣴⠟⠷⡶⠾⠃⠀⠀⠀⠈⣿⠀⠀⣾⡿⠃⠀⠀⠀⣾⠃⢸⡏⢠⡆⠀⠀⠀⠀⢰⡟
⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣴⠿⠷⠶⢶⣤⣠⡿⠀⣿⠁⣼⣇⣀⣀⡀⠀⣼⠇
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠙⢷⣴⣿⡶⠟⠉⠉⠉⠛⠛⠿⢦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⢹⣧⣤⡶⠶⠾⠿⠷⠶⠶⣦⣤⣤⣤⣤⣤⣴⠶⠟⠛⠋⠉⠙⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠈⣿⣀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⣬⣭⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣀⣤⣾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡏⣹⠟⢁⣼⣶⠶⢾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠽⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣜⠿⠷⠒⡿⠋⠙⢳⣎⠉⠛⣯⣀⡀⠀⠀⠁⠈⠀⠄⠀⢆⣀⣀⠤⣄⣒⣤⣷⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⡴⠾⢁⣠⣃⠉⠻⣷⡄⠀⣫⡤⢀⡼⠁⠀⣀⠀⠊⢩⣴⣾⣿⣿⡿⠿⠁⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⡿⣿⠟⣿⣿⡀⣸⠀⠀⢸⣷⡁⢂⠔⠁⢀⣶⣿⣿⡿⠛⠋⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢆⣀⡠⢄⣤⠟⠋⠁⣶⡿⠉⢀⠃⠀⠀⣽⠟⡥⠏⣠⣾⣿⣿⣿⣟⣈⣀⣀⣀⣀⣀⠘⠿⢤⡀
⠀⠀⡿⠐⣀⡟⣩⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣀⣹⡭⠿⠋⠀⠀⢈⣿⣿⣦⢸⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠈⠈⠉⡿⠉⠙
⠀⠀⠉⣓⢸⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠊⠉⠈⠁⠀⠀⠀⢾⣾⣿⣟⠙⡟⢻⣿⣿⣿⣿⡤⢀⡀⠀⠀⠀⠀⠀⡌
⠀⣗⣘⠚⠆⠓⠦⣄⠀⠀⠀⠀⠀⣴⠗⠀⠀⠀⠀⠀⢀⣀⡴⠚⠛⠛⠻⠟⣉⣵⣾⣿⣿⡤⢅⡀⠀⠀⠈⠓⠦⣄⣐⣇
⠀⠈⠉⠉⡋⢺⠳⡈⠑⢤⣄⡔⠉⠀⠀⠀⠀⢠⠖⠋⣉⢛⣰⣾⣷⡿⢿⣿⣿⡿⣿⣶⠀⠁⠀⠈⠢⡀⠀⢠⠆⠉⠉⠉⠃
⠀⠀⠀⢀⡷⠜⠀⠈⢶⣴⠋⠀⠀⠀⠀⠀⠔⢣⡾⠟⠉⠀⠁⠀⠘⡁⠈⠀⠀⠁⠀⠈⠂⠠⡀⠀⠀⢁⢐⡏
⠀⠀⠀⠀⠀⠀⠀⣠⣸⣿⣄⠀⠀⠀⠀⢀⠾⠋⢀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠹⡆
⠀⠀⠀⠀⠀⠀⢀⣸⣷⡿⠍⠓⠂⠀⠀⠀⠀⠀⢿⡯⢟⣿⢿⣿⣷⣶⣤⣶⡀⣀⣤⣤⣴⣂⡲⢜⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣟⣴⣾⡆⠀⠀⠀⠀⠀⠀⠛⠀⠀⡈⠀⠀⡈⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣾⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡻⣾⣧⠀⢰⡀⠀⠀⠀⠀⢠⣦⣽⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⣵⢿⣇⡧⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣾⡉⢻⣟⡿⢿⣣⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡸⠉⠙⠛⠷⣜⡓⣄⣹⠏⠛⠛⠛⠋⠁⠀⠈⠀⠙⣿⡋⢹⡈⢸⡏⡏⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⠄⠰⠦⢤⣄⣣⠸⢿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⣏⢨⡇⠐⣧⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⢀⣀⣀⠀⠀⠀⠉⢆⢈⣦⡈⠁⣀⣀⠀⠈⠉⠐⣄⣸⣉⣩⠄⡀⣸⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⣞⡽⠛⠚⢲⠦⢀⠀⠈⡄⢿⠟⢀⣭⣟⠉⠛⣷⡀⠀⢿⠄⠁⣠⢣⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡏⠀⠀⠀⠛⠒⣦⡥⠟⠀⣴⣾⠋⠁⠀⠉⠉⠛⠳⡆⠘⣷⣾⢣⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢱⠄⠀⠀⢀⣼⣦⣴⠏⢰⣿⣷⣤⠄⢀⠀⣀⠀⢠⠇⢸⡿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠉⠡⢿⣿⣿⣿⡾⠟⠀⢨⡟⠛⠿⠿⢵⠶⢋⡴⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠓⠒⠚⠋⢁⠤⠀⢄⣚⡷⠶⠶⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠴⠷⢶⣾⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧

⠀⠀⡔⠛⠲⡴⠒⠲⡖⠛⠛⠛⠲⢤⡀
⠀⢘⢰⣿⢰⠁⣶⡄⢸⠀⠀⠀⠀⠀⠙⢆
⠀⢹⠘⠟⢸⡀⠿⠃⡸⠀⠀⠀⠀⠀⠀⠈⢧
⠀⡠⠓⠶⠊⠱⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈⡆
⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢻
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠸
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠛⠛⠒⢆
⠀⠙⢆⡀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⡞⠙⢦⡀
⠀⠀⠀⠙⠲⠤⠤⡤⠖⠋⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡱⡄
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡘⠀⠹⡀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⠓⠤⣄⣀⣀⣀⠤⠀⡎⠀⠀⠀⠀⢰⠁⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⠀⣧⠀⠀⢠⠞⠁⢀⡤⠛⠲⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣈⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⣼⠀⢠⠧⠤⣀⣌⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡏⠉⠲⣄⠀⢹⣀⣀⡤⠖⠊⢳⠀⠀⠀⠀⡘⢸⠀⢸⠀⠀⢸⡈⡦⠒⠲⠄⠃
⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⢰⠀⠀⠀⠈⠆⣘⠉⠀⠀⠀⠀⠘⡄⠀⠀⢠⠁⠈⡄⢸⡠⠚⠉⢻⠇⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢹⠈⠉⠛⠋⠉⠀⠀⠹⡀⠀⠀⣠⠛⠢⡀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⠀⠀⣇⠈⣆⠀⠀⡼⢤⡀⠀⠀⣇⣀⣀⣠⢤
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠁⠀⠀⠘⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⡈⠳⠼⣶⠊⠀⠀⠱⡀⢰⠋⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠘⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⠁⠀⠀⠈⠆⠀⠀⠀⢠⠏⠀⠀⠀⠰⡠
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣀⡴⠋⠀⠀⠀⠀⢰⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡰⠳⢤⣤⣤⣴⣾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠁⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⣼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣀⣠⣴⣿⣷⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⢠⣃⣠⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠠⠀⠀⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠈⠀⠀⡁⠢⠀⠀⠀⠀⠀⠒⠡⠀⠈⠀⠀⠀⠀⠀⠀⠢⡀⠄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠃⠂⠄⠀⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢵⠉⣆⠰⠘⠟⠰⠀⡒⢢⣢⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⡀⢶⣤⠛⢆⠠⠄⢀⡆⠏⣠⠇⡣⠀⢃⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠁⠀⠈⠀⠁⠐⠀⠀⠡⠈⠁⠐⠃⠀⠸⠦⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⢈⠀⠀⠁⠀⠠⠀⠀⠀⠀⠠⠊⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⢀⠈⢢⠄⠀⠀⠀⠠⡄⠔⠁⠀⠀⢠⡈⠀⠀⢄⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⠀⠀⢀⠂⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠑⠄⠀⡀⠌⠀⠀⠀⠀⠑⠀⠉⣖⡤⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠞⠱⠠⡀⠈⠁⠀⠤⠀⠀⠖⠑⠀⠀⡇⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⠀⠠⠄⠀⠀⠀⠀⠀⠀⠄⠀⢸⠀⠀⠄⠠⠃
⠀⠃⠈⠖⠈⠖⠀⣆⡤⠤⢄⡤⠤⠤⣤⠤⠤⠤⠤⠤⡠⠤⠤⠤⡤⠤⣀⡤⠤⢄⡤⠤⠤⢄
⠀⠘⡄⠀⣄⠀⠠⢋⠀⠂⢨⡇⠀⠒⢾⠀⣶⠀⣶⠀⡧⠀⣶⠀⣶⠀⣯⠀⠂⢨⡇⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡞⠋⠉⠉⢻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣻⣟⣏⣿⣿⣟⡿⡟⣿⣿⢿⣿⣿⣿⣷⣿⣇⠀⠀⠀⣼⣿⡿⣻⠷⣤⣀
⢶⣿⣿⣿⣿⣿⣿⡽⠿⠛⠋⠉⠉⠛⠛⠷⠷⠧⠷⠿⠞⠛⠋⠉⠁⠀⠉⠛⠛⠚⠛⢷⣝⢡⢞⡵⠋⠛⢦⡀
⠀⠈⠉⠉⣩⡤⠴⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡼⠀⠀⠀⠀⠹⡄
⠀⠀⢀⡞⢡⡞⡛⢦⣤⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⣀⣶⡆⣿
⠀⠀⢸⠀⣧⠙⢶⢿⣆⣻⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣙⠟⣀⢼⡁
⠀⠀⠘⢧⣈⠳⢭⣁⣈⡥⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠶⠃
⠀⠀⠀⠀⠈⠙⠲⢬⡉⣠⣾⣿⣄⣀⣀⣀⡀⠀⢀⣠⣴⣞⣏⣿⣿⣿⣿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⢿⠟⠉⠉⠉⣿⣿⢹⣸⣿⣿⣿⠿⠿⣿⣿⡿⣾⡿⣶⣤⣤⣠⣤⣤⣶⣶⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⠃⠀⠀⠀⣰⢻⣿⣿⠽⠛⠉⠀⠀⠀⠀⠙⢾⣽⣻⣻⢿⣷⣷⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢀⡤⡶⠶⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⣴⣋⣸⢿⣿⡧⠀⠙⢦⣤⣀⣀⡀⢀⣠⡤⢴⣾⡟⣽⣿⣾⡎⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠹⣍⣥⣤⡬⠀⢀⣀⣸⣿⣹⣿⣿⣿⠀⠀⠀⢿⣻⣿⠿⠛⠛⠛⠛⢯⣿⣻⣿⣷⢶⣶⡶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠈⠛⠒⠖⠚⠋⠁⠀⠉⠉⠛⠛⠛⠳⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠾⠿⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⡴⢶⣄⣈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠛⣛⢿⣿⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⣶⣶⣿⣿⣶⣶⣶⣤⣤⣤⣀⣠⡤⠴⠶⣶⣿⣿⣧⠀⠀⣿⣠⡖⢀⡼⠃
⣠⣴⣶⣾⣿⣿⣿⣿⢻⣿⣿⢿⢿⡃⣿⣟⣿⣽⣿⣼⣿⣿⣿⠿⣿⡇⠀⠀⠀⢿⣿⣿⠿⠶⠶⠮⠵⠞⠋
⢛⣛⣛⠉⠉⠉⠉⠙⠛⠛⠛⠚⠛⠛⠉⠉⠁⠀⠀⠀⠈⠉⠙⠛⠿⢦⣤⡴⠖⠛⠉
⢸⣿⡿⠭⠭⠭⠥⠭⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣛⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡏⢀⠤⣄⠀⣿⠀⠀⠀⠀⣠⠖⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⣸⢰⡃⢀⠈⢳⢸⡀⠀⠀⣼⠁⣴⠛⢷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢾⠸⡀⠀⠆⠘⡏⢧⠀⠀⡇⠀⡇⠀⡜⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⣇⠑⢤⣠⠼⠁⠈⢣⣀⠳⡀⠑⠊⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠄⠉⠫⣷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠠⠊⠀⠀⠀⠀⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⢀⠞⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣤⡤⠚⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡇⢻⡄⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢺⠁⣸⠃⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢸⠀⣿⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣿⠀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⣿⠀⣿⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣿⠀⣿⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⠒⠀⠤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⡆⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⣟⡴⠁⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⢦⣎⠁⠀⠀⣰⠋⠙⠛⠲⠲⠟⠲⢤⡀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡉⠛⠲⢤⣧⣀⠀⠀⠀⠀⠀⠀⣀⣹⡦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⢙⠛⣲⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⣼⣀⠟⣢⠄⣂⣁⡀⠀⠀⠀⠀⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣾⠁⣠⠊⣴⣿⣿⣿⣿⣷⡀⠀⢠⣿⣿⣵⡃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⡶⠟⡴⠁⢸⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣧⣽⡆
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⡇⣸⠁⠀⠀⣿⣿⣿⣿⣿⣿⣀⣀⣈⣿⣿⣿⣿⡏⠃
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢸⡇⣷⠀⠀⠀⠙⡏⣿⠟⢿⣿⣿⣿⣿⡅⠙⣿⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⣧⠸⡄⠀⠀⠀⢠⣿⠀⢿⣿⣿⣿⣿⡇⠀⣭⠀⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢿⠀⠙⢦⡀⠀⠘⢧⡀⣿⣿⣿⣿⣿⣅⠴⠃⠀⠀⢀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⡇⠀⠀⠉⢓⣦⣤⣬⣹⣿⣿⣿⣿⠥⠴⠶⠒⢻⠏⠁⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠁⠀⣠⣶⠋⠀⠀⢸⠀⢻⣽⡏⡟⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠈⢳⡄⠀⠘⣧⣈⡅⠀⣟⠀⣀⣀⣀⣯⠤⠤⢤⣀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣷⠀⠀⠈⣻⠛⠛⡟⠻⢭⣿⣽⠁⠀⠀⠀⠙⣷⠁⠀⢀⣤⡊⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣠⡴⢶⠟⢒⣿⠟⢻⡀⠀⠧⠀⠀⠀⣧⠀⠀⠀⢀⠜⠃⣀⡴⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣰⡞⠉⢠⡏⠀⣾⢯⣤⠌⢳⣖⡀⠀⠀⠀⢸⡄⢀⡴⠿⡶⣡⠿⢧⡀⠀⠀⢸⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠈⣷⣶⣿⣶⣟⢷⣀⡴⠻⣟⠛⠛⠻⣷⣿⣷⠜⣱⠃⠀⠀⠻⣆⢀⣼⠃⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⡿⣿⣾⠋⠀⠁⠙⢷⣄⣀⣀⡈⢀⣸⡇⢀⣠⡤⡴⠟⠋⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠳⣮⣯⣞⢫⡏⠀⠀⠀⠀⠘⢧⡉⠉⠙⠿⢽⣿⡿⠟⠀⠀⠀⠀⣀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢺⣿⡎⠛⢷⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣤⣤⣤⣤⡤⠴⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠿⠿⠿⠿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣄⠀⣀⣤⣿⡇⠀⠀⠀⠀⠀⠀⣿⣧⣄⡀⢀⣤⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠈⠻⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠋⠈⠻⣷⣄
⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠿⠿⠿⠿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⣠⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⡀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⢨⣿⠃⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⢻⣯
⠀⠀⢀⣀⣀⣿⠇⠀⢀⣾⠏⠀⢀⣤⣤⣤⣤⣤⣤⣤⣿⠿⣿⣤⣤⣤⣄⠀⠘⢿⣆⠀⠀⢿⣇⣀⣀
⠀⠀⣿⡟⠛⠋⠀⠀⣸⡟⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⢿⣶⠿⠋⠉⠉⠉⠀⠀⠘⣿⡀⠀⠈⠛⠛⣿⡇
⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣠⣴⡿⢿⣦⣠⣀⣄⣀⣀⣀⣀⣄⡀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠉⠉⠉⠻⣷⣾⠟⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⢿⣧⣤⣄⠀⠀⢹⣷⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡿⢿⣦⣀⣀⠀⠀⢰⣿⠁⠀⢀⣤⣤⣿⡇
⠀⠀⠀⠉⠉⣿⡆⠀⠀⢿⣧⠀⠈⠙⠉⠙⠛⠛⠛⠉⠉⠻⣷⣶⡿⠛⠉⠀⢠⣿⠇⠀⠀⣾⡏⠉⠁
⠀⠀⠀⠀⠀⢘⣿⡄⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⣼⣟
⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣶⣶⣶⡾⠿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⣴⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣄⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠉⠙⣿⡇⠀⠀⠀⠀⠀⠀⣿⡟⠉⠁⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⣶⣶⣶⣶⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣶⣶⣶⠶⣶⢖⠚⠂
⠀⠀⠀⠀⠀⠀⣈⠀⠀⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢫⠏⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠁⡠⠲⡇⣀⠦⠄⠀⠀⠀⠀⠀⡤⣶⣒⠤⣄⣀⢀⣀⡤⢒⣿⡷⠿⠲⠦⣀
⠀⠀⠀⠀⠀⢠⣴⠀⠀⢀⢙⠀⠀⠀⠀⠀⠀⡨⠺⠋⠿⠻⠶⣯⣽⠁⣴⣧⠧⠤⣤⣤⡀⠈⢣
⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣾⡁⢰⠂⠀⠈⢿⣌⢄⠀⢀⠀⠐⠦⣵⠷⠤⡀
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣿⠿⠇⠀⠀⠀⠀⠈⠻⢧⡉⢺⣄⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠢⠒⠀⠀⠀⠀⠀⠀⡷⠄⠙⠷⠶⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢇⢦⠀⡇⠀⠀⠀⢀⣠⣴⣶⣶⣤⡀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⢲⠤⠀⠀⠀⡈⢂⠀⠀⠀⠀⢠⣟⠿⡿⡿⢿⡿⣿⣦⠀⠻⣿⢷⣦⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⣆⣦⣤⣾⠇⠀⠀⠀⡄⠀⣾⣿⣿⣿⣿⠿⣿⣾⣿⠄⠀⠸⡆⠹⣧⠀⠀⠀⠟⠻⡧⠀⠀⠀⢀⣤⣤⣤⣤⣄⠒⠁
⠀⠀⢐⣿⡿⠛⠁⠀⢀⠀⠀⠃⠀⠸⣿⣿⣿⣯⣶⣿⣿⡿⠀⠀⠀⢳⡈⣻⡧⡀⢼⠀⢶⠀⠀⠀⣠⣿⣏⣉⣭⠏⠉⠁
⠀⠀⠘⣿⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠈⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠙⢥⠔⣼⡿⠀⡴⡴⠖⠾⡏⠀⢈⡵⠋
⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣄⠉⠁⠒⠸⠦⢮⣉⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⢀⢀⣀⡀⠀⠀⠀⢴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠄⣀⣀⣀⠍⠑⠲⠶⠤⠂
⠀⠀⠀⠀⠠⠶⣿⡷⢰⡿⠻⢫⣞⠁⠀⢹⣿⠁⠢⠀⠈⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠋
⠀⠀⠀⠁⢑⡆⣸⠁⣾⠃⠂⣿⣿⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⢴⢶⣦⣄⡀⡀
⠀⠀⠀⣰⣾⠒⣥⣾⡿⠟⣸⣿⠏⠀⠀⠘⠂⠈⠂⠀⠀⠀⠀⠀⠀⣟⠛⠻⣟⠛⠛⠫⠰⡄⠀⠀⠀⠀⣄⣀
⠀⠀⠀⡟⠁⠀⢀⣀⣤⣴⣿⡿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠄⠈⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣥⢄⡀
⠀⠀⠀⡇⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⠦⠄⠀⠴⠀⠀⠰⡏⣷⡘⣷⡆
⠀⠀⠀⠱⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡉⠫⠠⠐
⠀⠀⠀⠀⠐⢀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⡏⠀⠐
⠀⠀⠀⠀⢀⡸⣀⠀⣍⡓⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠄⠀⠀⠀⠀⡀⠀⠀⢰⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠙⠀⣿⡙⣫⡍⢩⡍⠹⠉⣟⢹⠟⣭⡟⡍⢹⠉⣿⣿⢹⣶⡏⠍⢻⠟⢭⡟⡉⢹⡍⣩⡏⣿⢹⡏⡍⣻⠻⢽⠛⢽⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣼⣴⣧⣽⣧⣿⣧⣼⣧⣼⣤⣿⣼⣦⣹⣧⣦⣼⣤⣽⣷⣘⣻⣧⣷⣼⣤⣿⣥⣦⣼⣧⣽⣧⣭⣼⣧⣧⣽⣌⣿⣝⣼⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⡴⠶⡄
⠀⠀⠀⠀⠀⣰⣗⡈⠁⠀⠀⠀⠀⠀⠀⠘⠳⠄⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉
⠀⠀⠀⠀⢀⣭⣌⣁⠀⠀⠀⠀⠀⠲⠖⠀⢰⣆⣸⣇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠘⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠻⣏⠋⣀⣀⠀⠀⠀⠀⣀⠀⠠⣿⣿⣿⣾⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠀⡹
⠀⠀⠀⠀⠀⠀⠈⢹⣿⠉⠀⠀⠀⠉⠁⠀⠀⠉⢻⣿⡟⢿⣿⡿⣆⣀⡤⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠙⡿⠛⠡⡏⢻⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠐⠿⠟

⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠀⠀⠀⢰⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡷⠶⢾⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡀
⠀⠀⠀⠀⠀⠀⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⣸⣧⣠⣸⣿⣦⣀⠀⠀⠀⠀⠀⢈⣻⣿⠟⢻⡇
⠀⠀⠀⠀⠀⠀⡿⠈⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⣿⢿⠉⣽⢿⠉⠁⠀⠀⠀⠀⠀⠀⣠⣤⣤⠿⠃
⠀⠀⠀⠀⠀⠸⠃⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠀⠘⠚⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠭⠃

⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠠⠾⠿⣿⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠛⣆
⠀⠀⠀⠀⠀⠀⠐⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠈⠁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣀⠋
⠀⠀⠀⠀⠀⠀⣀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣦⣤⣤⣤⣤⡀⠀⠀⠀⠀⠙⢠⣴⣷⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⡠
⠀⠀⠀⠀⠀⢰⣯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠻⡟⠛⢿⣿⡈⠇⠀⠀⠀⠀⠀⢾⠛⡿⠛⢻⡿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⡆
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠌⠀⠅⠀⠀⠀⠀⠀⠀⠈⠀⣰⠀⠠⠁⡸⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡿⠋⠉⠙⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣤⡀⠈⠛⠁⣮⣿⡆⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣾⣿⠀⠀⠀⠙⠟⠃⢀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠉⠁⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣄⣀⣀⣀⡤⢂⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⠋⠉⠉⢠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿
⣿⣿⡟⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⣿⠿⠃⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢻
⣯⠸⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⣷⡀
⣿⣿⣶⣦⣤⣤⣴⣶⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠐⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠸⢿⠿⡿⠋⠀⠀⠀⠀⠀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⢰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⡀⠀⣀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣶⣶⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣠⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣄⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣴⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠋⠉⠉⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⣀⣀⡀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣷⡀⠀⠀⣀⣀⡀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣾⣿⣿⣿⣶⣶⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⢀⣀⣤⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠉⠉⠉⠛⢻⣿⣿⣿⣿⡄⠀⠀⠀⢀⣀⣠⣶⣶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣆⠀⠀⠘⠛⠻⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢀⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠟⠋⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠉⠛⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣻⣟⣀⣀⣀⣀⣀⣀⣀⣘⣿⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣻⣩⣛⣿⣹⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠞⡶⢶⡄⠀⠀⢠⡴⣶⢛⣟⡛⣻⢛⠿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⣀⣤⠶⢾⣧⡶⢻⠀⠀⢸⡇⣿⠉⠙⠛⠛⠛⢻⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣠⡾⠋⠀⠀⠀⣿⡷⣾⠂⠀⢀⣷⣸⡆⠀⠀⠀⢀⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢰⡟⠀⠀⠀⠀⣼⣿⢶⡟⠀⠀⠈⣿⢛⠛⣿⢷⡶⢾⠷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⣿⠃⠀⠀⠀⢰⣿⡷⣸⣧⡄⠀⠀⢻⡿⣿⠿⠾⠷⣿⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⣿⡀⠀⠀⠀⣬⠿⠛⣉⣬⡗⢀⣀⣼⣷⠟⠀⠀⠀⣿⡟⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⣽⡇⠀⠀⠀⠘⠖⢿⡍⠈⢿⡋⣉⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢻⣇⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡼⣷⠀⠀⠀⠀⠀⠀⣿⣦⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠶⣶⡟⢻⣆⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢿⡆⠀⠀⠀⠀⢠⣿⣾⣥⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠎⣿⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⣹⣇⠀⠀⠀⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣋⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⠀⠀⠀⠀⢸⡇
⢀⡴⠛⣩⣤⣴⣾⣥⣟⣙⡷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⣿⠀⠀⢸⣿⣿⣿⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⣼⡋⠀⠀⠀⠈⠉⠻⢿⣯⣍⠳⣦⡀⠀⠀⠀⠀⠀⢸⡇⢸⡏⠀⠀⣸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠓⠒⠂⠀⠀⠀⢸⡇
⢻⡄⠸⣧⡀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣼⠿⣦⣀⠀⢀⣠⠟⢠⡿⠁⠀⠀⣿⣿⣿⣷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠻⣦⡈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡉⠙⠋⢁⣴⡟⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⡀⠀⠀⢀⣿
⠀⠀⠈⠙⢷⣝⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠙⢷⡙⣆⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⢿⡿⣿⠿⣿⠿⣷⡶⣦⣤⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡌⣷⣄⠀⠀⢸⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣟⡹⢟⣷⣄⠀⣷⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠄⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⣱⠟⣧⣹⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣾⡹⠟⣿⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠶⠶⠄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣡⣾⠛⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣭⣥⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢯⡙⢦⡀⠀⠀⠀⠀⠈⣿⣦⣤⣤⣤⣤⣬⣭⣭⣍⣉⣉⣙⣛⣛⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣝⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⡳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣝⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣙⠲⣤⣀⠀⠀⠀⠀⢰⣦⣤⣄⡀⠀⠘⠉⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣭⣍⣛⣛⣛⣩⣷⡴⠋⠁⠀⠀⠲⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠒⡦⢺⡪⢗⡷⢶⣦⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⢁⡠⠭⠭⣀⠀⢸⣿⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣏⠀⡴⠁⢀⡤⣄⠀⠱⡄⢈⣙⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢞⡎⡇⡇⠀⣇⠀⢀⡇⠀⡇⢸⢞⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠯⣄⠱⣄⠈⠉⠉⢀⡴⠁⠈⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⣿⡁⠀⠉⣐⣋⠁⢀⣴⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⠶⢸⢝⡯⡇⠸⠬⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢠⢴⠠⣄⣤⣠⠀⡄⡄⣀⡤⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁⠋⠈⠈⠁⠁⠁⠉⠈⠈



⣏⣈⣯⣭⣿⣯⣿⣹⣽⣽⣉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠢⢤⡨⠔⠒⠈⠒⠒⠠⠤⡤⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠑⠢⢌⡑⠢⣄⡀⠀⡐⠚⠒⠒⠒⠒⢢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠉⠒⠢⢌⡉⠓⠢⢄⣀⠀⠀⠸⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡩⠳⢄⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⣉⠒⠠⠤⢛⣦⡠⠒⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡱⢀⡠⠼⠃⠂⠀⠟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠃⠀⠀⠠⢀⡀⢀⠂⠄⠀⠀⠀⠀⠀⠀⢀⠊⡠⠻⠀⠀⠀⢸⠀⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⡴⠁⠀⢀⠀⠂⠈⣐⠈⡀⠀⠀⠀⠀⠀⠀⡠⠃⡔⠁⠀⡇⠀⠀⢸⠀⡄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠠⢋⡴⠋⠀⠀⠀⠀⢀⠈⠀⣉⣠⠤⠈⠁⠀⠀⠀⢠⠁⡞⡅⠀⠀⠁⠀⠀⢸⠀⡇⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⣹⠓⠀⠀⠀⠀⠀⠀⠰⠛⡉⠠⠈⠁⠀⠀⠁⢀⠃⡼⠐⠠⠄⠀⠀⠀⠀⢸⠀⡇⠀⢠⠋⡄
⠀⠀⠀⠀⠀⠀⡰⢀⠇⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠈⠀⠐⠁⠀⠀⡜⢰⢧⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡀⠀⢣
⠀⠀⠀⢀⡤⠊⡠⠊⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠎⠠⢠⠈⠀⠀⠀⠀⠀⠀⡆⠃⢀⠀⠐⠈⢦
⠀⠀⠀⠫⣀⠛⠤⢀⡀⠀⠁⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⢀⠜⢡⠞⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠉⠒⠢⠤⢉⣁⠒⠂⠠⠤⣀⡀⠀⠀⠀⠀⢠⠋⡔⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⠈⣀⣀⣙⡖⣌⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⢲⠠⢌⡑⠢⢄⡀⠸⠰⠧⠄⠀⠼⠤⠐⣒⣉⠥⠔⠂⠀⠒⠂⠰⠁⠀⠀⡖⠈⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠉⣉⠵⠦⢌⡁⣸⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠺⠔⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠜⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠰⠂⠀⠀⠀⠠⣶⡟⠃⠀⠀⠀⠀⠀⢸⡇⢰⣾⣿⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠈
⠀⠀⠀⢲⣿⣿⣶⣶⣿⣿⡿⠛⠁⠀⠀⠀⠀⢉⣒⣙⡛⢷⣿⣅⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠛⠋⠅
⠀⠀⠀⢾⠻⡟⠛⣿⢿⡟⠻⠷⠀⠀⠀⠀⠰⣿⣏⣉⣛⣛⣻⣿⠆⠀⠀⠀⠀⢼⡦⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠓⠉⠀⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠈⠉⠐⠄⢠⠀⠀⢀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠤⣼⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⡄⠀⡀
⠀⠀⠀⠀⠀⢀⠟⣿⣿⠧⣀⠀⠀⠀⠀⠀⣼⡟⣦⣦⣄⣀⣼⣛⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣏⠀⣿⡿⣄
⠀⠀⠀⠀⢀⣾⣿⡿⢿⡆⠘⠀⠀⠀⠀⠤⠿⠃⣹⣿⠻⠿⢿⡿⠳⡄⠀⠘⠋⣰⣾⠿⣿⣿⣿⣿⣷⣿⣗⡀
⠀⠀⠀⠠⠟⢱⠛⠁⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⢏⠀⠀⢸⠀⠐⠇⠀⠀⠀⢯⡸⢰⣏⡘⢯⠙⣻⡋⠙⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂⠀⠘⠂

⠀⠀⠀⠀⠀⢰⣿⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⠀⠀⢀⣴⣷
⠀⠀⠀⡀⢠⣾⠟⠋⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠐⠹⠆⠀⠀⠀⠀⠘⣿⣇⣰⣏⢠⣿⡿
⠀⠀⠀⠀⣴⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣶⣲⣶⣤⣬⣧⡈⡀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠟⠋⠁
⠀⠀⠀⠀⢿⣿⠁⣀⡀⢠⠀⠀⠀⠀⠀⢠⣾⣿⡞⣿⠉⠛⢿⡟⠻⠀⠀⠀⠀⠀⠀⠀⠚⢳⣦⣤⣄
⠀⠀⠀⠀⠈⠻⣧⣿⡽⠸⠷⠂⠀⠀⠀⠈⠉⠀⡄⠸⠄⠀⠈⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢠
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠒⠶⠤⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠤⠄⠉⢛⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠱⡄
⠀⠀⠀⠀⠀⠐⠂⢀⣤⡀⠈⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠔⢂⣀⡐⠂⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠀⠀⠀⠀⠀⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⡜⠃
⠀⠀⠀⠀⠀⠠⡀⠓⠀⠒⢀⠄⠀⠀⠀⠀⠀⠀⡀⠒⠒⠀⠨⣦⡀⠀⠀⠔⠁⢀⠔⠂⠉⠁⠉⠉⠀⠒⠤⢀⠀⠀⠀⠀⠀⠜⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠀⡀⠀⠀⠀⡖⠂⠄⣑⠀⠀⠀⣀⠈⠃⠈⠀⠀⡔⠁⠀⠀⠀⠠⣄⠠⢆⣀⠀⠀⠘⠢⡀⠀⡔⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠒⠒⠒⢆⠰⠆⠀⠀⠀⢰⡀⠀⡤⠻⣆⡉⠠⢀⡜⠂⠈⡝⠀⠀⠀⠀⠀⠀⠀⣶⡌⢫⣿⡶⡀⠀⠈⠖⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙⠁⠀⠀⠀⢸⠀⡠⠄⢴⠋⠀⢀⣼⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢱⠟⣄⣦⡀⢅⠃⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⡅⠁⠈⠆⠘⠇⠀⠀⠀⠀⠏⢀⠔⠒⡀⡠⠃⠀⢁⠀⠀⢀⠔⠊⠉⠉⠉⠑⠠⡱⠊⠀⣩⠀⠀⠀⡶⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠰⡁⠀⠀⠀⢠⢀⠈⠀⠀⢱⡄⠀⠀⡘⠀⠀⣌⠤⠠⢄⣀⡠⠤⠄⣰⠀⠀⣨⠶⠜⠆⠀⠀⠈
⠀⠀⠀⠀⠀⠠⡅⢤⠤⢤⠄⠃⠀⠀⠀⢘⠀⡀⠀⢀⠆⠑⢤⡴⠁⠀⠀⢣⡀⠀⠀⠀⠀⠀⢀⡘⠀⠀⣨⡛⠱⡑⠀⠀⢠
⠀⠀⠀⠀⠀⢀⡂⣀⣀⡀⢀⡆⠀⠀⠀⢸⠀⠈⠐⡎⠀⠀⠀⢠⠀⠀⢠⠀⠙⠂⠠⠤⠄⠀⠪⣛⣠⣤⡬⡀⠀⠀⠛⠀⠫⡀
⠀⠀⠀⠀⠀⠀⠖⠀⠐⠃⠀⠅⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⠈⡄⠀⠈⡀⠀⠀⠀⠀⠀⠀⣷⢍⡀⣅⣠⢞⠀⠀⠀⡜⠆⠙⢄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠶⠈⡄⠀⠀⠀⢈⠓⠀⠀⠺⣆⠀⠀⠠⠾⠦⡄⠷⡀⠀⠀⠀⠀⣀⣈⠛⣥⡬⠿⠁⠀⣠⠞⠀⠈⡄⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠒⠢⠤⠂⠂⠀⠀⠀⠀⢸⡀⣀⠤⠚⢄⠉⠚⠀⠀⠀⠈⠱⠑⢄⠀⠀⠀⠀⠀⠙⠀⠀⠠⠄⠊⠀⠀⠀⠀⠈⢂⠀⠑⣄
⠀⠀⠀⠀⠀⠠⠊⡠⠄⢄⠉⠆⠀⠀⠀⠸⠉⠡⣄⣀⡀⣆⣀⠦⢀⠀⠀⢀⠃⢀⠝⠲⢤⡤⢤⡄⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⢃
⠀⠀⠀⠀⠀⠐⠦⠈⠀⠉⠤⠃⠀⠀⠀⠰⡤⠊⠁⢀⠱⣷⢋⠀⠈⠩⠝⠉⠉⢁⡠⢃⣼⠃⠀⠐⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⣠⠆
⠀⠀⠀⠀⠀⠐⠒⠙⠙⠋⠒⠆⠀⠀⠀⠀⣁⣀⣤⠪⠛⢤⡈⠀⠁⢀⡀⠠⠐⢉⠀⡫⠛⠢⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⡅⠀⠈⠓⠈⡂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠓⠠⠤⠦⠬⠐⠒⠃⠀⠄⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠃⠒⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣰⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣸⣁⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠶⠒⠒⠒⠒⠒⠶⠞⢰⡯⠉⠀⠀⠀⠀⠈⠉⠉⠛⠷⣶⣤⣀
⠀⠀⠀⠀⢀⣠⠶⠛⠉⠠⠀⠉⠁⠀⠀⠀⠀⠀⢠⣟⢘⣿⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⢦⣀
⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⢼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣦
⢀⡼⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉
⠙⠧⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠈⠳⣦⡀⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣰⠋⣹
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⢆⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡴⠃⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠈⠣⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠹⣡⠀⣿⠀⡀⠀⠀⠀⠀⢸⣇⠀⠈⣣⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡘⠁⣠⡶⠚⠚⠋⢀⡿⣤⠋⠀⠀⠀⠀⢸⣿⠀⠀⠿⠂⠀⣷
⠀⠀⠀⠀⠀⢀⠁⠀⠈⠀⠀⠀⠀⡾⠙⠈⡀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠃⠀⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⢼⠀⠘⡀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠸⣄⣀⠀⠐⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢳⡔⠶⡄
⠀⠀⠀⠀⠀⠀⠈⣉⣉⡿⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠃⠄⢰⡀
⠀⠀⠀⠀⠀⣴⢟⠫⠽⠀⡀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣁⢀⣠⣴⠾⠫⠟⢷⣦⣤⣄
⠀⠀⠀⠀⠀⢿⣀⠴⠛⠁⠀⢀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠹⣏⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠈⠻⣶⡀⠀⠊⠉⠀⠀⠀⠤⠴⠀⠐⠃⠀⠀⠀⠀⠀⠀⠐⠋⢀⣽⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⡿⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⡀⠀⠠⠀⠀⠀⣀⡤⠴⠖⠋⠁⢤⠶⠉⠩⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣦⣤⣤⣮⣤⠤⠤⠶⠚⢿⣦⣄⣀⣀⣐⣡⡤⠖⠛⠁
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠀⢸⠇⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠿⠀⠀⠀⠀⡆⢸⠇⠀⠀⢠⡀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠘⠀⠀⠀⢰⠃⠉⠀⡼⠀⠀⣇⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠁⠀⢠⡏⠀⠙⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇⣶⠀⠀⢰⡀⠈⠁⢠⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢀⡟⠀⠀⠈⣷⠀⠀⢸⡄⠀⠇⠀⠀⣀⢠⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠸⠇⢄⠀⡄⠀⠀⠀⠈⠁⠀⠀⢠⠀⠛⠈⠀⣧⣰⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⠀⠀⢀⠈⠀⠁⡀⠀⠀⠚⠀⠀⣆⠛⠀⠀⡇⢸⢿⠋⠀⡀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⠋⢸⠃⠀⠀⠃⠀⢰⣆⠑⡆⣼⠘⠀⠀⠙⠀⠀⠄⢠⠀⠈⠀⠠⡇⠸⠁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠈⣷⢈⡁⡀⠀⠀⢀⠘⠛⠀⠁⠻⠀⠀⢠⡀⠀⠀⣄⠀⠰⠀⠀⠀⠃⡿⠀⣆⠸⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠃⠀⠐⡋⢻⠃⣷⠀⠀⠚⡀⠀⠀⠀⠀⠀⢰⡌⡅⠏⢠⠟⡄⠀⠄⡀⠀⠀⣀⠠⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢸⡸⠇⠑⠴⡄⢡⣲⠐⣄⡇⠀⢀⠀⡄⢈⠀⡀⠿⠀⡾⠠⣠⡇⠸⠇⢀⡜⠃⠀⠀⢰⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢸⠂⠓⡆⣠⡄⠀⢧⣴⣾⡿⣶⡔⡌⠇⠀⠘⢠⡇⠀⠀⠀⠀⠋⡠⡖⣠⣸⣧⡘⠁⠀⠈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡦⠘⣄⠀⠈⠛⠈⠃⠙⢟⠃⣿⠿⣦⠀⠶⠀⠈⠓⣦⠀⠀⠐⠈⣷⠏⣻⢁⣼⡿⠋⢀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣤⣌⠀⡮⡄⢈⣗⣦⡼⢆⢠⡄⠻⠃⠀⠀⠊⡰⣿⠄⠈⠀⠰⢷⠠⠉⠈⠈⣴⢒⡊⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢈⣛⣠⠤⣿⣄⡧⢴⣾⣿⡹⠇⢸⣧⢱⣄⡶⠀⠆⠁⠁⠁⢠⣄⠀⢁⣈⡀⢻⣧⢀⣿⡌⠃⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠞⠁⠀⠀⠀⢹⣿⣆⡈⠯⠀⣀⣀⠀⠈⠛⢡⣿⣧⠤⠓⠠⢾⣿⣦⡈⠿⠃⢘⠡⢜⣿⡛⠁⠀⢉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⣀⣀⣠⣾⡾⠁⠀⠀⠀⣨⡿⠂⠒⠒⢿⠏⠀⠀⠀⠀⠀⠙⢿⡟⠈⠁⠙⣄⠀⠈⠙⢿⣯⡭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⡽⠯⠽⠿⣵⣠⢤⣄⣾⡿⣇⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⣠⣿⣷⣤⣠⢔⣿⡆⠀⣠⡴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠐⠾⠶⠛⠿⠧⢯⡥⠿⣿⣓⠠⠄⠂⠾⠛⣋⠿⠷⠿⠿⠳⢿⠶⠾⠾⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⠀⢀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⢰⣿⡿⠂⠀⢱⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠸⢃⣤⣤⡔⠛⠁⠀⠀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣨⠀⠈⠛⡃⠒⡦⢶⡀⠀⠀⠀⠀⡿⣿⣷⣷⡀⢀⠀⠀⡠⢴⡠⠃⣾⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠰⢤⡤⢁⣼⡆⠀⠀⠐⠀⠀⠛⠛⠋⣍⠖⠒⢺⠀⡘⢁⠀⡀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡀⡀⠀⡀⣀⣾⠋⠿⣿⣦⠀⠀⠈⠲⢤⣀⡀⢸⢾⡆⡼⠊⢴⡜⢫⡄⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠴⠁⣸⣁⡉⢏⣉⣿⡿⢾⣤⣽⠟⢠⠀⠀⠀⠀⠈⠿⠃⠊⣯⡷⠀⠈⠐⠏⡃⢠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠰⠦⠤⠤⠔⢢⡾⠀⡦⠓⠁⠀⠀⠀⠀⠄⠁⡔⣣⣤⣷⣤⣄⣀⡤⠞⢥⣀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⣿⣿⣿⣶⣶⣾⣿⣿⡿⠽⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⡹⣿⣶⣶⣶⣶⣾⣿⣳⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣾⣿⣿⣿⣿⢿⢿⣿⡿⡿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⢿⣏⡷⢮⣻⣭⡞⡝⣧⠵⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣄⠀⠀⣿⣿⣸⡿⣫⣽⡿⣽⣧⣣⠿⢬⡯⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⡳⡀⠉⣡⠏⣞⡽⣟⣼⣇⣣⡿⢬⣼⢹⣎⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢫⣶⣶⣿⣿⣕⣪⣿⣜⣻⡼⢷⡞⡝⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡻⢿⣿⣜⣼⣯⣚⣒⣽⣮⣵⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣝⣫⣿⣿⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣣⣿⣽⣯⣿⣜⣿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣶⡿⠿⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣶⣶⡶⠶⠶⠶⠶⠶⢶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠉⠉⣉⣉⣉⣉⣉⣉⡉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣾⣿⡿⠟⠛⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠁⠀⢀⣠⣤⠄⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⡿⣓⣀⣴⣿⣏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢀⠽⠿⠋⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣟⣟⣛⣛⣛⣛⣻⢟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠦⠥⠤⠤⠤⠤⠜⡠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⣪⠅⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⠕⣾⢽⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣥⣛⠰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢚⠬⣏⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠿⢠⡬⠛⣤⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⣛⣻⣿⣿⣿⣿⣨⡟⢳⣼⠛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⡖⢲⣖⣶⠒⣾⢶⣶⣶⠆⠀⠀⠀⢸⣿⣿⣿⣏⠉⠉⠉⠉⠉⠉⠉⣩⡍⠹⢩⣯⢉⡍⠉⢉⣽⣿⣿⣿
⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⣾⣅⣽⣿⡿⠟⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠈⠁⠻⠃⠀⣀⣠⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠈⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⠀⠀⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀
⠀⠠⠤⢤⣤⣤⣤⣤⣤⣾⡿⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠙⣿⣷⣤⣤⣤⣦⣤⠤⠄
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⢰⣿⣿⠛⣿⣽⣿⣿⣟⣿⣿⣿⠛⣿⣿⣧⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠿⢻⣿⣿⣿⣿⣿⣿⣟⠿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣹⣿⡍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⣄⡀⠀⣸⢀⠏
⠀⠀⠀⠀⣉⣦⠽⠞⠒⠒⠒⠲⠦⣤⡀
⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⡀⣿⠀⠀⠀⠀⠀⠀⣀⡀⠙⢳⡄⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⠉⠈⢧⠀⠀⠀⢹⡀⠉⠙⢶⣄
⢸⡇⢸⠀⠀⠀⢀⡾⠋⠉⠉⠛⢦⡀⠀⠀⠀⠀⠹⣄⣀⡠⠔⠚⠁⠈⠳⡄⠀⠀⠀⠘⣇⠀⠀⢸⡇⠀⠀⣼⠈⠻⣄
⠘⠇⢸⠞⠉⠛⢾⠃⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠉⣀⣀⠀⠄⠀⠀⠘⣆⠀⠀⠀⣿⠀⠀⢸⡇⠀⢀⡟⠀⠀⣿⣆
⠀⠀⢸⡀⣰⣄⠀⢸⣿⡆⠀⠀⣸⠃⠀⠀⢀⣠⣴⣶⣿⣿⣿⣧⠀⠀⠀⠀⢸⡆⠀⢀⣿⣤⣤⣾⣄⣀⡾⠁⠀⡼⠃⠙⣦⡀⠀⠀⠀⠀⠴⣻⣯⡛⢶
⠀⠀⠈⢧⠈⠁⢠⣄⣀⣀⣠⠾⠁⣀⣤⠾⢿⣿⣿⣿⡟⠋⢉⡿⠀⠀⣦⢀⣠⠷⠞⠋⠁⠀⠀⠀⠀⠙⢧⣀⡾⠁⠀⢠⡟⠙⣦⣤⣤⣤⣴⡟⠹⣿⡆
⠀⠀⠀⠈⠛⠶⠻⣄⣈⣁⣤⠶⠛⠉⠀⠀⠀⢈⡟⠑⠃⢀⡾⠁⠀⣼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⢠⡟⠀⢠⡇⠀⠀⢱⠀⢳⣄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⣠⣤⣧⠴⠖⠋⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⠏⠀⠀⢸⠁⠀⠀⠈⢧⠀⣿⣿⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⡀⠀⠸⡄⢀⣀⣤⠾⣿⣿⣿⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⠀⠾⣿⣥⠶⠟⠁








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡒⠒⠶⠶⠶⠦⢤⣤⣤⣤⡤⠤⠶⠶⠶⠖⠒⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⢦⣄⠀⠀⠀⠀⠀⣠⡾⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢷⣄⠀⢀⡾⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⠁⠀⠲⣟⠛⠛⠛⠛⠶⠶⣤⣀⠀⠀⢠⡟⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠻⣆⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠙⠳⣄⢸⡇⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢹⡆⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣀⠀⠀⠈⢿⡇⠀⠀⣶⠀⠀⠈⡇⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣁⣤⣤⣄⠙⢷⡀⠀⠈⣷⠀⠀⣿⠀⠀⢠⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢹⡆⠀⠈⠉⠛⠶⣄⠀⠀⣿⠀⢰⡆⠀⡿⠀⠀⣠⠶⠛⠉⠁⠀⣸⠇⠀⠀⠹⣆⠀⢹⡆⠀⣿⠀⠀⣼⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⡀⠈⢷⡄⠘⣇⢸⡇⣸⠃⢠⡞⠁⣀⠀⠀⠀⣰⠏⠀⠛⢦⣄⠙⣧⣸⣿⣄⡿⢀⡾⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠙⠷⣄⣻⡄⠙⣾⣿⠃⢠⣟⣠⠾⠋⢀⣠⠾⠃⠀⠀⠀⠀⠉⠛⠿⣿⠀⢻⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠾⠿⣧⡀⢸⡇⢀⡾⠿⠷⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣼⣧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠛⠁⠠⡄⠀⠀⣰⠄⠈⠛⠲⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠙⠶⠶⠋⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣀
⠀⠀⠀⠀⠀⠀⣠⠾⠃⠀⠀⠀⢻⠀⠀⢀⣤⣤⡀⠀⢠⡟⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠖⠛⠋⠁⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠐⠳⠶⠋⠀⠀⠟⠀⠈⠿⠀⠀⠛⠶⠞⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠿
⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠛⠉⠉⠁
⡤⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠
⠧⠤⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⡀⠀⠀⠀⣀⡀⠀⡀⠀⢀⢀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⣀⠀⣀⢀⠀⢀⣀⡀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⢸⠁⠈⡇⢸⠀⣯⠀⣸⣱⠀⣯⢬⡃⣿⠀⠨⡥⠄⠀⠘⢥⡀⠈⡇⠀⣧⣨⢢⡏⠀⢹⠀⣧⡋⢸⡥⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣥⣤⣸⣦⣴⣁⣼⣀⣯⣠⣇⣈⣇⣷⣤⣃⣼⣤⣬⣦⣤⣀⣠⣤⣇⣀⣇⣀⣇⣈⣆⣳⣤⣎⣠⣇⣱⣌⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣫⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⡠⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣧⡀⠀⠀⠀⠀⠀⢠⡾⣧⣸⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢩⠏⠳⡄⠀⠀⠀⣿⠀⠈⠁⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⡏⠁⠀⠀⢹⡄⠀⠀⡟⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠙⣦⠀⠀⠀⣻⡄⠀⣇⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠈⢳⣄⠀⡙⠷⠶⠿⢰⣧⣽⠷⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⢀⠎⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠹⡈⠁⠀⠀⠀⠊⣱⡏⠀⠀⢹⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣦⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⠄⢄⡄⢠⣿⣶⢮⣼⡟⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⣀⡀⠀⠀⠀⠀⢴⠛⣿⣿⡿⢆⡐⠀⠀⠳⠤⠔⢘⡟⣼⡟⠉⠀⠀⠀⠈⢘⣿⣿⣟⣘⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡎⣿⣿⣦⠀⠧⠀⣾⠀⠙⠛⣿⣨⠁⠀⠀⠀⠀⢠⣿⣼⣧⣷⡆⢨⣾⢶⡾⠟⠻⠛⣿⠉
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢠⠀⠀⣤⡤⢵⣠⣭⠹⠃⠀⠂⢼⡤⠚⠋⣠⡇⠈⡄⠀⠀⢀⣠⡿⠋⠙⣿⣿⠿⣽⣶⣤⣬⢥⠞⠃
⠀⠀⢀⠀⠀⠀⠀⠀⠀⣆⣀⣄⢀⠀⠈⠿⣾⡀⠹⣶⠶⠀⠀⠀⠀⡰⢶⣹⠁⢀⣧⣤⡾⠟⡉⠐⣤⣞⣷⣾⡿⠟⠋⠁
⠀⠀⠀⠀⢀⢀⣾⢶⠚⠙⠉⠁⠈⡀⠀⣶⣭⣷⣴⣀⣤⣴⡎⢹⢉⡁⡜⣃⡴⢿⡟⠃⠀⣠⢴⣾⣿⠟⠋⠁
⠀⠀⠀⢤⣾⡟⠁⠘⠛⠛⠛⢁⣠⠄⠀⠀⠀⠀⠉⠉⢛⡹⢿⡿⠟⠛⢻⡉⠙⠈⠁⣠⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠂⠐⠦⣤⡄⠉⠉⠐⢆⡀⠀⢀⡤⠖⠉⠀⣾⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠛⠋⠀⠀⠀⢺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⠀⠀⢀⣠⣶⠾⠟⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠶⠶⠶⠶⠾⠻⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠞⢋⣉⠀⣄⣴⠏⠀⠀⠀⠀⠀⠀⠀⡄⠀⣤⣉⢿⣶⡄⠀⠄⢰⠟⡀⠀⠀⠀⠀⠀⠀⠀⠐⢢⣾⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠊⠁⠈⠉⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⢹⣭⣲⣿⡻⢷⣴⡟⡀⠁⣀⡰⣀⣼⣷⣶⠾⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢠⡏⣤⠀⢀⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⡏⣏⣾⣿⣶⣶⣿⣿⣦⣿⣼⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣦⣀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡜⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⢋⡤⢨⣤⣤⣀⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣶⣿⣿⣿⣿⣿⣿⢠⠜⣛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⢻⡶⢿⣿⣿⡇⣶⣔⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣠⣼⣧⣾⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣾⣿⣾⣿⣯⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡾⣿⣭⣼⣿⣿⣿⢿⢿⣟⢸⣿⣿⣿⣿⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢺⣿⡿⢿⣽⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣐⡛⣴⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢶⡶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⢿⡀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢧⣄⠀⠀⣀⡀⠈⣷⠀⢀⣀⠀⠀⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡆⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠙⢷⣌⠧⠜⠀⢸⡇⠻⡬⠃⢰⡏⠈⢻⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣸⠃⣰⠇⠀⠀⠀⠀⠀⠀⣼⡁⠀⢀⡤⣄⠹⣦⠀⠀⠀⣷⠀⠀⢀⡿⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣰⠟⢧⣾⣁⣠⠟⠻⢶⡟⠻⡟⠳⣦⠀⠀⢰⡏⠙⠷⣌⠲⠋⠀⠘⣧⡴⠞⠛⠲⣤⡾⠃⣞⣙⠄⣼⣷
⠀⠀⠀⠀⠀⣿⢓⡿⣄⣨⡏⠙⠛⢺⠗⠒⢻⠉⢹⡇⢀⣾⠀⢀⡤⡘⢷⡄⢀⡼⠋⠀⠀⠀⠀⠈⢻⡄⠀⢁⣴⠏⢸⡆
⠀⠀⠀⠀⠀⠘⢿⣥⣀⣿⣉⣛⡒⡾⠒⠶⢾⠒⣿⣧⠞⠙⢷⡜⠶⠃⠀⢻⡾⠁⠀⠀⠀⠀⠀⠀⢸⣇⣴⢟⡥⡄⠈⣷⡀
⠀⠀⠀⠀⠀⢰⠋⢁⣉⢻⣉⡭⠿⠿⣶⠶⣿⣞⠙⣧⠀⠀⡀⠹⣆⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠘⠲⣫⡾⠉⠙⣷⢦⣄
⠀⠀⠀⠀⠀⠸⣄⡈⣛⡼⢻⠰⡷⠀⢸⠟⠁⠸⠇⠘⣧⢯⣹⠂⢹⡆⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡴⠟⡵⢦⠀⣰⠏⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢸⡏⠱⣄⡼⠳⠤⠴⠋⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠁⠀⠀⠙⢊⣴⠟⣎⣳⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣴⠟⢁⡀⣈⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⠀⠀⢠⡄⠀⠀⠀⢸⣇⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠛⠒⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠙⠛⣿⠒⠒⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠏⢹⣿⣆⠀⠀⢠⣿⣏⡛⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⢩⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣤⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣬⣛⣿⣿⠋⠘⢿⣿⣇⠡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⣤⣶⣶⣶⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⣷⣷⣤⣄⣀⠀⠀⢠⡀⠠⣜⣶⡀⠀⠞⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣭⣤⣴⣶⣶⣿⢿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⢹⣝⠿⣿⣿⣿⣶⣤⣿⣶⣞⣷⣤⡆⠀⠐⠲⠶⣶⢄⠄⠀⠀⠀⢠⣾⣿⣿⣿⠋⡛⣋⡉⠉⣉⣑⣦⣿⣿⣿⣿⣷⣬⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠈⠒⢾⣿⣿⣿⣿⣿⣿⣿⣟⣿⠀⠀⠀⠀⢠⣴⠚⠁⠀⠀⠀⠀⣿⣿⡦⠃⣄⣉⣁⠈⠛⠀⣉⠀⠉⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠿⣿⣿⣿⣿⠏⣿⣤⠄⠀⠀⠉⠛⡷⠐⠒⠆⠀⣴⣿⣍⡍⠀⢦⢐⣉⣋⠻⢍⠉⠑⠢⣄⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡘⠛⣿⣯⡜⡟⢉⣿⠂⠂⣠⠞⣷⣀⣠⣴⣾⣟⣿⡏⢠⡷⠒⠤⢁⠉⢁⡠⡀⣠⣄⢉⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣯⡙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣿⣻⣯⣷⣾⡇⠀⠀⠁⢼⢁⣻⣿⣟⣹⣿⣟⣿⣎⣿⣶⣶⣄⣦⣠⡿⢾⣿⡙⠻⢿⣿⡄⠹⢿⣿⣿⣿⣿⣿⣧⡁⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⣝⠟⢣⣤⣆⢓⠍⣉⣴⡟⣿⣿⣶⣺⠿⣿⣿⣿⣽⡟⣿⠙⢿⠀⠙⠁⠀⠀⠈⠁⠀⠈⠛⢿⣿⣿⣿⣿⣷⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⠇⢛⠂⠀⠋⠹⠉⠀⠁⠀⠸⠏⢻⡩⣧⣵⡿⠟⠀⠀⠈⠉⠲⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠟⠻⣿⣿⣿⣿⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⠀⢠⡄⠀⢀⡀⣠⠀⠀⠤⣤⣀⣾⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⢙⢷⠠⣄⠀⠉⠠⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠘⢀⠁⢀⡀⠞⠺⠛⣄⢀⣠⠇⡿⣝⢿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⢦⠐⡤⣱⡐⠲⢲⣟⠋⢻⡟⠻⠁⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⢰⣙⣄⡀⣠⠗⠀⣴⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠒⣽⠷⠶⢾⣇⡾⢿⣤⣈⣁⡃⢰⠞⠛⠛⢿⠇⠀⠀⠀⢀⢠⠀⠀⠘⠁⠁⠀⠀⠺⢍⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⣶⠁⠀⠀⠘⠋⠀⢈⡹⣾⠏⠀⠘⡄⣠⣀⠸⠀⠀⠀⠀⠀⠻⡗⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⠂⠀⠀⠀⡀⡠⢔⡯⠚⠁⠀⠀⠀⠙⠙⢯⡀⠀⢰⡆⠀⠀⠀⠰⡈⠑⣤⡄⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠆⡀⢠⣀⣦⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣾⠁⠀⠀⠀⠀⠘⠆⠻⢧⣤⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡏⣜⣠⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⣰⣶⡦⣶⣾⡷⣯⣱⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡶⡯⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣿⡄⠀⠀⠀⢠⣤⣤⣄⣄⠸⠁
⠀⠀⠀⠀⠀⠀⠻⡟⠗⠛⠁⠀⣿⣿⡾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⢶⠶⡇⠀⠀⠀⠉⠀⠉⠍⠩
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠸⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣤⣤⣶⢶⣶⣶⣦⣶⣶⣶⣿⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⠟⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡰⢦⣿⣫⣿⣿⣿⣿⣿⣿⣶⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠂⠠⣤⡀⢀⣠⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣾⠿⠷⠛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⡳⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⢺⠁
⠀⠀⢠⣤⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⠀⣀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⢸⠁⠀⠰⠆
⠀⠀⠈⠻⠛⢿⣿⣿⣇⢰⣀⠀⣶⣶⣀⡀⡌⢿⣧⣷⣼⣿⣾⣿⣿⣿⣷⠿⠟⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⢐⣀⠈⣀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⣿⠿⣿⣤⣌⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⣿⣿⠿⠄⡶⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠉⠉⠙⠛⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡏⠀⠀⢸⡀⠀⠀⢀⣵⣴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣗⠤⡅⠀⢸⠀⣠⣦⣿⠟⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⡿⣟⠙⢿⣤⡀⠐⣼⣿⣿⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠙⣿⢿⣿⡟⠀⢸⠘⢻⣿⠟⢿⠻⢯⣷⡄⣭⣹⣿⣿⣟⣛⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠿⠀⠀⢸⠀⢸⣻⠀⢸⠀⠀⢹⠉⠙⠉⠙⠁⠉⠉⠁