Rosary Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣠⡴⢐⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣶⠳⣴⠋⢉⣛⠋⢙⡛⠉⠿⣤⢴⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣙⡿⠙⢻⠷⡴⠏⠛⠋⠈⠛⠋⠻⠖⠛⣏⣨⣿⣾⣷⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣺⡟⠛⡞⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⢼⡙⡓⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠈⢹⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠈⣻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡼⣶⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢢⣤⣤⢄⠀⠀⠀⢾⢀⠇⢻⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢣⠴⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠀⠈⠃⣾⡀⠀⠀⠰⡇⠀⢿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⣶⠟⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢀⡛⠃⠀⠀⠀⠸⣅⠁⠀⠀⢠⣯⠛⠵⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣡⣔⠛⡄⠀⠀⠀⠀⠀⠀⠛⠿⢿⣾⣶⣍⡀⠀⠀⠀⠠⣿⢛⢂⣀⢀⣿⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⠈⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠖⠻⠿⠟⠀⠀⠀⠀⠙⠻⠅⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠽⣅⣴⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣄⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠲⣬⠗⢄⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠔⢉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣟⢶⣧⣀⠀⠀⠀⠀⣰⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣳⠀⣹⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣨⡻⢤⣄⣀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣺⡏⠱⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣠⣻⣬⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢲⡟⢛⡁⣲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢫⣀⣽⣳⠶⢷⣇⣑⣤⡞⣻⡶⣤⡼⠋⣹⣁⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⠷⠾⢿⣟⢛⡿⣯⣠⡿⠷⠶⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠰⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠲⠶⣞⠉⠱⢆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⣠⠟⠉⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠈⢃⢀⡼⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣄⠀⢀⡼⠋⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠑⢦⠀⠀⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠇⠀⠴⠊⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠾⢦⡀⠀⢠⠶⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠶⢶⠉⠙⠶⠖⠑⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⡳⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠰⢶⡉⠉⢢⡀⠈⡃⠀⣠⠃⠀⡰⠟⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠳⣄⢀⡤⠀⣤⠃
⠀⠀⠀⠀⠠⠎⠁⠉⢳⡄⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠹⡆⣸⠷⠞⠉⠉⠉⠑⠖⢧⡀⡠⠖⠉⠳⡄⠀⠀⠀⠀⠀⠀⠈⡏⠀⢀⣠⣇
⠀⠀⠀⣠⡴⢦⡀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⠉⠉⢢⡀⢠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⣠⡀⢧⡀⠀⠀⠀⠀⠀⠀⠱⠞⠁⠀⠈
⠀⠀⠘⠃⠀⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠱⢆⣰⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⢲⣤⣀⣤
⠀⢠⠼⠷⢦⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠱⢦⢀⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠆⠀⠾⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈
⠀⢉⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠄⣤⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣤⡀⢀⡟⠀⠀⠀⠀⠀⢀⡸⠶⠶⣴
⢠⡞⠉⠙⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠖⠢⣀⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠈⢋⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⢸
⢨⣄⣤⣀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣤⡀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠋⠉⢢⡴⠃⠀⠀⠀⠀⠀⠀⢀⡷⠒⠲⢏
⡎⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⠹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣠⣄⣀⠇⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⢸
⢣⡴⠒⠲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠙⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠁⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠰⠋⠉⠱⠶⠉
⢠⡀⣠⣤⣼⡃⠀⠀⠀⠀⠀⠀⠀⢀⠿⠈⠑⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣤⠷⠦⡀⢠⡜
⠀⠻⠁⠀⠀⣱⠀⠀⠀⠀⠀⠀⠀⢸⡤⠠⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡔⠉⠙⢣⡀⠀⠀⣤⠾⢤⡀⠀⢸⠃
⠀⠘⢦⡞⠉⠉⢶⡀⠀⠀⠀⠀⠀⣊⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⡞⠉⠉⠻⡀⠀⠀⢹⠴⠋
⠀⠀⠰⣆⠀⣀⣘⠃⠀⠀⠀⠀⡴⠉⠉⠲⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⣆⠀⠀⣠⠷⢠⠰⠋
⠀⠀⠀⠈⠉⣿⠛⣷⣤⣄⠀⠀⣆⠀⣀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣸⡝⠃⠉⠙⢛⣾⢻⣿⠉⠁
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠈⣥⡿⠋
⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠈⣿⡒⢀⣠⣶⠟⠁
⠀⠀⠀⠀⢰⣿⣽⡿⠉
⠀⠀⡴⠒⠛⢬⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠢⣄
⠀⠸⡀⠀⠀⢀⠇⢀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⢀⠀⣼⠁⠀⠀⠸⣧⡀
⠀⠀⠙⠒⠛⢯⡴⠉⠉⠳⠋⠉⡝⠁⠀⠀⠿⠎⠉⠀⠈⠓⠦⠔⠂⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⢆⡀⠀⣲⡀⠀⢸⣦⣄⣤⠗⠀⠀⠀⠀⠀⠀⣀⣤⣴⢷⣽⢷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡁⠀⠈⠉⠀⠀⢈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣦⡀⠀⠀⠀⠻⣶⡀⠀⢠⣶⠟⠀⠀⠀⣠⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠈⠛⠷⠶⢦⣤⣼⠇⠀⢾⣦⣤⡶⠶⠞⠋⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⢀⣀⣤⣤⣤⣤⡄⠀⣤⣤⣤⣤⣤⣀⠀⠀⢀⡼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡴⠛⠉⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠈⠉⠻⣦⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁⠀⠘⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡤⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡁⠀⠀⠀⠀⠀⣈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣦⣀⣴⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠳⣦⣠⡴⠛⠛⣷⣄⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠲⣿⠀⠀⠀⢸⣿⣇⠀⠀⢈⣿⡏⠀⠀⠛⣷⡶⠛⢶⣦⠀⢀⣀
⠀⠀⠀⠀⢀⣴⠶⣶⣿⠁⠀⠀⢹⣷⠦⢶⠾⠋⠹⠷⠶⠟⠻⣧⣄⣀⣼⣿⠀⠀⠀⢸⡿⠋⠛⢷⣆⣀⣀⣀
⠀⠀⠀⢰⣿⠁⠀⠀⣿⡿⠦⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠙⠷⠦⠶⢿⡇⠀⠀⣠⣿⠋⠉⠙⣷⣀⣀⣀⡀
⠀⢀⣤⣼⣿⣦⣤⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠾⠿⣧⡄⠀⢀⣼⠟⠉⠉⢳⡆
⠀⣿⠋⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠛⢿⣄⠀⢀⣼⣷⣤⣄
⠀⢻⣤⡀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡏⠀⠀⠙⢷
⣤⡞⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣀⣴⣶⣄
⣿⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠁⠀⠙⣷
⠈⢹⣷⠾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⣠⡿
⠀⣿⠀⠀⠈⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠏⠙⢷
⠀⠻⣦⣤⣴⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀
⠀⠀⢨⣿⠁⠀⠈⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠻⣿
⠀⠀⠈⠻⣦⣀⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡀⠀⠀⣿
⠀⠀⠀⢠⡿⠛⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⣶⣾⠛
⠀⠀⠀⢾⣄⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡆
⠀⠀⠀⢈⣿⠶⢶⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⣿⣦⣠⣶⠟⠁
⠀⠀⠀⣾⠉⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠈⣿⡏
⠀⠀⠀⠘⢷⣤⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡼⢿⣤⣤⣴⠟⠁
⠀⠀⠀⢀⣼⠟⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⠀⢹⡏
⠀⠀⠀⠸⣿⡀⠀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣶⣤⣴⡾⠋
⠀⠀⠀⠀⠙⠻⣶⣿⠿⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠈⠻⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣄⣰⠿
⠀⠀⠀⠀⠀⠀⠘⠿⣦⣤⣾⠟⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⡿⢿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠋⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⣤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠷⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣦⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠈⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣾⣧⣄⢀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣦⣴⡟⠋
⠀⠀⠀⠀⠀⣸⡟⠀⠀⠻⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠈⣿
⠀⠀⠀⠀⠀⠹⣦⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣷⣄⣀⣴⠿
⠀⠀⠀⠀⠀⣰⡿⠛⠿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣼⡟⠀⠀⠈⣿⠋⠁
⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠁⠉⢹⣿⣤⣤⡾⠏
⠀⠀⠀⠀⠀⠹⣿⣤⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠓⠻⢿⣤⠀⣠⣼⠋⠈⠉
⠀⠀⠀⠀⢀⡾⠃⠀⠈⢿⡆⠀⠀⠀⠀⣠⣤⡀⣠⡾⠛⠛⢿⣯⡀⠀⠀⣸⠟⠁⠋⠁
⠀⠀⠀⠀⠈⢹⣦⣀⣠⡿⠃⢀⣀⣀⡿⠃⠀⠙⢿⣄⠀⠀⣠⡟⠛⠶⠟⠋
⠀⠀⠀⢀⣴⠛⠛⠻⢿⣀⣴⠟⠛⠛⣷⣀⣀⣴⡿⠛⠶⠾⠛⠁
⠀⠀⠀⠸⣿⡀⠀⢀⣼⣯⣤⡀⠀⢠⣾⠛⠛⠋
⠀⠀⠀⠀⠈⠻⢶⣿⠟⠉⠻⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⢼⣇⠀⣸⡗
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣤⡟
⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠛⢧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡆
⠀⣤⣤⣤⣤⣤⣼⠇⠀⠀⠀⠠⣦⣤⣤⣤⣤⣄
⢈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇
⠈⣷⣦⣤⣤⣤⣤⡀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣾⠃
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⣫⡀⠀⠀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣈⣷⣶⣶⣶⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠉⠁⢈⠉⠉⢁⠈⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣠⠀⣤⠂⡖⠉⠑⣒⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣘⠆⠊⠁⠁⠉⠑⠂⠐⠁⠈⠀⠙⠉⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⠤⣀⠀⠀⣀⠤⣀⠤⣀⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢠⣤⡖⠺⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⡠⠟⠛⠤⠋⠒⠉⠒⠑⠐⠣⠸⢏⣱⡆⣲⠒⣴⠒⡎⠉⠂⠀⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡊⠣⠊
⠀⠀⠀⠀⠀⠀⣠⣴⣐⠶⠛⠒⠉
⢀⠄⠉⠢⠔⠁⢛⠄
⠀⠑⡤⠀⠀⠀⢎⡀
⢀⠔⠁⢀⠤⢄⠀⡘
⠣⡀⡠⠁⠀⠀⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡴⠊⠰⡞⠙⢱⡞⠉⢶⠶⠶⣄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣤⢴⡋⠉⢻⡄⢀⡷⢄⠴⠷⠤⠞⠳⠤⠾⣤⣠⣌⠁⢈⡷⠂⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣄⠀⠀⠙⣆⡷⠶⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣤⣾⢁⡼⠓⠰⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢞⠀⢀⣗⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢧⡀⣨⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣴⠋⠈⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢧⡀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⡉⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣬⣀⣀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢞⠉⠉⣦
⠀⠀⠀⠀⠀⠀⢰⡋⠀⣙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠙⢦
⠀⠀⠀⠀⠀⢠⠋⠉⢲⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡤⠻⣤
⠀⠀⠀⠀⢀⡜⠓⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣀⣼⡃
⠀⠀⠀⠀⢸⣧⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠈⣹
⠀⠀⠀⠀⣏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠚⠻⡄
⠀⠀⠀⠀⡞⠑⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡤⢴⡁
⠀⠀⠀⠀⣳⠤⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⣀⠷
⠀⠀⠀⠀⡷⠒⠛⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠙⡆
⠀⠀⠀⢼⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠓⢾
⠀⠀⠀⠈⠷⣴⠛⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠉⠓⣿
⠀⠀⠀⠀⠀⠙⢦⡴⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⢤⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⢧⣀⡴⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⣄⣀⡷⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣀⣴⠛⠛⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠛⢧⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢶⡏⠀⢹⡶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠶⣾⠁⠈⣧⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠻⣄⣠⡟⠉⠙⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠶⡞⠁⢙⣧⣀⡼⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⡅⠀⡹⠶⢦⠀⠀⠀⠀⠀⢀⣠⣜⠉⠛⣤⣠⠟⠲⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢧⣀⡼⠖⣦⣄⣀⣴⢏⡀⢘⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡜⠁⠀⠀⠹⣄⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣀⣄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢣⣀⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣼⣀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⢹⡟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠗⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣀⣤⠼⠁⠈⠉⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠉⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⡄⠀⠀⠀⠀⠀⠀⠘⠣⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⢀⣀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢠⠛⠋⠁⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⡟
⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣝⣷⣾⣠⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢾⣃⣸⣯⣹⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠀⢀⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠑⢦⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⢸⡇⠀⣠⣤⣤⣤⡈⠳⣌⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠹⣦⢻⠀⣀⣀⣀⣄⣠⣤⣄⠀⢀⣀⠸⠄
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠀⠈⠃⠈⣿⢧⠉⠃⠁⠈⣧⠴⢶⡄
⢀⡤⠞⠋⠀⠀⠐⠒⠛⠁⠀⠀⠀⣶⠛⠒⠀⡴⠚⣾⡗⠀⠁⡄⢶⡄⢀⣤⠾⠗⠋⢻⡄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠃⢀⢠⡄⠚⠉⠛⠛⠀⢙⡛⢡⠀⢀⣄⠐⠚⠋⢀
⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠷⠶⠆⡈⠋⡀⠀⣰⠀⠐⠤⠞⠷⡼⠃⠘⠾⠇⢠⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣶⣀⣷⣀⣿⠋⠹⣄⡀⠳⠾⢤⣀⣀⡦⣈⣷⣄⣹⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⢰⡏⢀⣾⢦⣄⣽⠋⠀⠉⠁⠉⠁⠀⠀⠀⠀⢣
⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢄⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣇
⠈⠉⠛⠛⠛⠲⣦⡀⠀⠀⠀⠀⠀⠳⢤⣤⣀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢀⠀⣹⢂⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠖⠛⠋⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠘⢦⠿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢨⡇⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⠞⠋⠀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠇⠀⠀⣼⠉
⣿⣿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠘⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠛⠉⠛⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠦⠤⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⣀⡀
⢠⡤⢤⣀⣀⣀⣀⣠⣤⠤⠤⠼⢿⣥⠤⠄⠀⠈⠙⠿⣿⣇⠀⠀⠀⠀⣾⣿⡿⠉⠁⠀⠤⢤⣾⠟⠦⠴⢶⣶⠤⠤⠤⠴⠒⣊⠟
⠀⠈⢲⣤⣄⣀⣀⠈⠙⢿⣖⣒⡀⠉⠻⢿⣍⣁⡤⠀⣈⠏⠀⠀⠀⠀⣿⠁⢀⣀⣈⣻⠿⢋⣁⣀⣚⡿⠟⠁⠀⠤⠴⢒⠟
⠀⠀⠀⠈⠲⣦⠤⠤⠄⠀⠈⠙⠿⢧⣤⣤⢀⣉⣉⡽⠋⠀⠀⠀⠀⠀⠈⠻⣖⡛⠉⠠⣤⣤⡽⠟⠁⠀⠀⠒⢒⡲⠟⠃
⠀⠀⠀⠀⠀⠀⠉⠳⠶⣒⣒⣀⣀⣀⡨⠭⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠯⠭⣁⣀⣀⣀⡈⠭⠟⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠑⣶⣤⡀⢰⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣿⣷⣼⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣛⣛⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢁⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⣀⣤⣤⣴⣦⣤⣤⣄⣀
⣿⣿⣿⣿⣿⡇⠀⢀⣀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⣿⣿⣿⣿⣿⡇⢀⢸⣿⠃⠆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠈⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣦⣴⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠀⠠⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⡀⢠⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠸⠀⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⡸⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⢸⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⡇⠀⢸⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡏⠀⢸⣿⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋
⣿⣿⣿⣿⣧⡀⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣭⣤⣤⣄
⣿⣿⣿⣿⡿⣾⣿⣿⣿⣏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠀⠀⠀⢸⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣦⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡀⠆⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡼⠃⠀⠀⠀⠀⣾⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣌⡙⠿⢿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠠⠈⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣷⣦⣤⣄⣀⡀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠻⠿⢿⣿⣿⣿⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⡐⠀⠀⠈⠔⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣇⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠇⠀⠀⠀⣀⢀⠀⠀⠀⠀⡞⠀⠃⡀⠀⠆⢀⠖⠁⡀
⠀⠀⠀⠀⠀⠀⣠⠖⣄⠗⠚⡡⠐⠀⢘⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣷⣤⡀⣼⠃⣦⡥⣦⣥⣔⣃⡴⠿⠋⠉
⠀⠀⠀⠀⠀⣰⣫⣴⣋⣉⡀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡷⠒⠉⠓⠃⠀⠀⠀⠈⠀⢀⡑⣾⠟⠓⠚⠻⢷⢝⡇⠁
⠀⠀⠀⠒⠈⠉⠉⠉⠉⣠⣄⠀⠀⠀⣇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠰⣆⠑⠤⠔⢈⣿⡓⠃⠀⠤⢶⢤⣄⡀⢣⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢸⠀⣉⣿⠙⡎⣄⣀⣡⠀⠀⣀⣀⡟⠀⢾⡁⠀⠀⠀⠀⠘⢦⠤⠴⠾⠛⢿⣦⡀⢀⢠⣟⠊⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠈⠉⢹⣯⠀⠹⡌⠧⣿⠀⢀⡩⢋⠀⠀⢸⠋⠁⠁⠀⠀⠀⠀⠀⣠⡀⠀⠀⠉⠻⠷⢿⣀⠈⠀⠀⠒⠲⠦⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠠⡀⠀⠀⠀⢉⣁⡠⢂⢄⢿⠀⣄⠀⠀⠀⠀⠀⡼⣧⠡⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠶⠶⠶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠋⠻⣄⠘⠦⣀⡀⠀⢀⣴⢏⡞⢸⠀⢹⡀⠛⠀⢀⡸⣿⡇⠀⠐⡀
⠀⠀⠀⢸⠟⠫⠀⠀⡿⠀⠀⢹⢷⠀⣤⣉⣡⠀⠐⡇⠀⣼⠃⠀⢳⡀⢀⠀⠀⠀⠻⣆⢰⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⢰⡇⠀⢀⠈⠁⠀⠊⠁⠀⡟⠛⠻⠀⠈⠳⠂⠋⠀⠀⠀⠑⠾⣬⣀⠬⣷⣌⡋⠁⠀⢠⡞⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢹⠷⢄⣢⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⠋⠁⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⢰⡷⠾⠷⠶⠾⠿⠛⡄⠀⣀
⠀⠀⠀⠀⢡⠖⠠⢌⠳⡈⠆⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⢫⣶⣶⠄⢠⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣆⠀⡀⠀⡀⢶⠃
⠀⠀⠀⠀⠁⣴⣶⣄⡽⠁⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⢀⠥⠀⣲⠏
⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⡼⠀⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠈⣿⣟⣻⠁
⠀⠀⠀⠀⠀⠢⠀⠄⠙⢻⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⣱⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠶⡀⠀⠀⠀⠀⠀⠸⣿⡅
⠀⠀⠀⠀⢀⢦⠀⠀⠀⢸⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⣬⣛⡻⢿⣿⣦⣄⠀⠀⠀⠀⣀⠴⠚⣁⡤⠶⠿⣧⠀⠀⠀⠀⠀⣀⡇⣀⣀
⠀⠀⠀⠀⠼⢦⣤⣙⠲⢿⠗⠑⠀⠀⡇⠀⠀⠀⠀⢠⣿⣀⣠⣭⣭⠿⡻⠿⢷⠄⠠⠊⠀⠈⣠⡴⣶⡮⠶⣶⣂⠀⠀⠀⠘⠯⠁⠑⠋⡁
⠀⠀⠀⠀⠀⠈⠀⠀⠚⠋⠀⠀⠀⠀⠇⠀⠸⠀⠀⠿⠷⠶⠒⠛⠛⠛⠛⠛⠓⠶⠷⠒⠛⠛⠛⠛⠛⠛⠻⠿⠿⠆⠀⠀⠈⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣄⠀⢠⠏⠹⡄⠀⣠⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣸⠀⠈⠳⠋⠀⠀⠙⠞⠁⠀⣇⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠙⠲⠏⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠸⠖⠋⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡒⠚⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⠓⢒⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡚⠉⠀⠀⠀⠙⠿⡿⣿⣿⣿⢿⠿⠋⠀⠀⠀⠀⠉⢓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⣀⣤⣤⣿⣿⣿⣤⣤⣀⠀⠀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣼⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣿⣿⣿⣿⣿⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⣈⡉⠿⣋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠉⠁⠀⠀⠈⠛⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⣛⠀⡒⡄⢀⢖⡀⢹⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠈⠁⠨⠀⠈⠁⢸⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣿⣟⡀⠀⣸⣾⣄⠀⣺⡟⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡿⣿⣟⠟⡨⢧⡘⣿⡿⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣼⠀⣿⡟⢷⣼⣇⣵⡾⣧⣳⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣾⡿⠃⠀⠀⠈⠁⠀⢿⠙⢿⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠟⢛⣿⠋⠀⠀⠠⠤⠤⠤⢶⣾⢰⠀⠘⡄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠘⡄⠀⣇⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⡇⠀⢹⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡇⠀⢸⠀⠀⠀⢸⡀⠀⠀⡠⠤⠀⣠⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣰⡃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⡇⠀⢸⡄⠀⠀⢤⣧⢀⠔⠐⠃⠐⠙⣹⠂
⠀⠀⠀⠀⠀⠀⢀⡤⡌⠆⠀⢠⣿⡓⠦⣀⠀⠠⣼⠀⠀⠀⠀⠀⠀⠀⢠⠎⢹⠇⠀⣧⠀⢸⡇⠀⣠⠔⠋⣤⠀⠘⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠡⠌⠳⠓⠀⠁⠀⠉⠢⡅⠢⣻⠀⠀⠀⠀⠀⡠⠖⢁⣀⢼⠀⠀⣿⠀⢸⣇⡴⠁⠜⠀⡿⠀⣴⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⡇⠀⠀⠀⢹⡀⣿⣧⡠⠤⢒⠩⠄⠊⠁⠀⢸⢈⠀⣿⠀⠀⣿⠃⠀⠀⠀⣇⣰⠟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢰⠸⣇⣿⣓⣒⣒⣁⣀⡤⠔⠒⠲⢿⠸⠀⣿⠀⠀⡇⠀⠀⡆⠀⣿⣿⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⡇⠀⢀⢸⠀⣿⣿⣿⡉⠉⠀⠀⠀⠀⠀⠀⣼⢠⠀⣿⠀⠀⠃⠀⢰⡇⠀⡿⠇⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⢸⢸⠀⣿⡇⢿⡙⠲⠶⠒⠉⠀⠀⢠⠿⣿⠀⣿⠀⠀⠀⠰⢸⠄⠀⡇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡹⠀⠀⢸⠀⠀⣿⡅⠈⣇⠀⠀⠀⣀⠤⠒⠁⡄⡏⠀⣿⠀⠀⠀⡆⢸⠀⢠⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⣿⣿⡇⠀⠈⠂⠀⠋⠀⠀⠀⣸⠇⡇⠀⢿⠀⠀⢰⠃⢸⠀⠀⣿⢠⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠸⠀⣿⠟⠙⠀⠀⠐⢦⠀⠀⠀⠰⡿⠀⠇⠀⢹⠀⠀⠘⠀⠸⠀⠀⢿⠃⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠀⠉⠙⢿⣿⣦⡀
⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢁⣴⣾⣿⣿⣿⣿⣿⣶⣄⠙⣿⣇
⢠⣿⣏⣿⢻⣧⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢺⣿⡿⠋⠉⠛⠿⠿⣿⣿⡆⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡄
⣼⣿⣹⣼⣆⣿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡧⣶⣶⡀⣴⣶⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⢣⣿⣿
⢿⣏⠿⠟⢻⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠈⠉⢹⣿⠉⢹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣷⣯⣟⡾⣿
⠈⣿⣆⠀⠘⠁⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣰⡾⠿⢿⣷⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠘⢿⠁⠈⠉⣼⡏
⠀⠘⢿⣦⣴⡆⣾⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠈⢁⣠⣼⡿⠁
⠀⠀⢀⣿⣿⣷⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡟⢀⣼⣿⡏
⠀⢀⣾⣿⣿⣿⣿⢻⡏⣿⣿⣆⠀⠀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣍⡻⠆⠀⡿⠃⣹⣿⠿⣿⣿⣿⣶⣦⣤⣄⡀⠀⠀⠀⣾⡟⣿⣿⣷⣾⣿⣿⡇
⠀⣾⣿⣿⣿⣿⣿⠘⠃⣿⣿⣿⣿⣿⡿⢁⠈⢿⣿⠁⠈⢿⡟⢿⣿⣿⣷⣤⣴⣿⣿⣿⠀⣿⣿⣿⡟⢻⣿⠿⢿⣷⣶⣶⣿⡇⠉⢿⣿⣿⣿⣿⣇
⢸⣿⣿⣿⣿⣿⣿⠀⠀⣿⠋⣿⠁⣿⡇⣾⠃⠘⣿⠀⠀⠀⠀⢸⣿⡿⢻⣿⣿⣿⣿⡇⠀⣿⡟⡿⠀⣿⡏⢠⣾⣿⠉⣹⣿⠃⠀⢸⣿⣿⣿⣿⣿
⢸⣿⢻⣿⣿⣿⣿⠀⠀⣿⡇⣿⣷⣿⡇⣿⠀⠀⠉⠀⠀⠀⠀⠀⣿⣇⠸⣿⠟⣡⣿⡇⢸⣿⡇⠀⢸⣿⢰⣿⣿⡟⢰⣿⣿⠀⣀⢸⣿⣿⣿⣿⣿⡆
⣿⣿⢸⣿⣿⣿⡇⠀⠀⣿⡇⢻⣿⣿⣿⣿⡄⢀⠀⠀⣤⠀⠀⠀⣿⣿⠀⣠⣴⠟⣹⠃⢸⣿⡇⠀⣼⣿⣿⢻⣿⡇⣾⣿⣿⠀⣿⢸⣿⣿⣿⢿⣿⡇
⣿⣿⠘⣿⣿⣿⡇⠀⢰⣿⣿⡸⣿⣿⣿⣿⣷⢸⣇⠀⣿⡇⢸⡆⣿⣿⣾⠟⠁⣠⣿⠀⢸⣿⡇⠀⣿⣿⡿⢸⣿⣿⣿⣿⣿⠀⣿⠸⣿⣿⣿⢸⣿⡇
⢿⣿⡆⢿⣿⣿⠁⠀⣸⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣾⣿⣿⠟⢁⣤⣾⣿⣿⠀⢸⣿⡇⢀⣿⣿⡇⣸⣿⢸⣿⡿⣿⡇⣿⠀⣿⣿⣿⢸⣿⡇
⠸⣿⣿⣬⣿⠏⢸⠇⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⡿⣋⣥⣶⠿⣿⣿⣿⣿⠀⢸⣿⡇⢸⣿⣿⡇⢿⣿⡄⠉⠁⣿⣷⣿⡆⢻⣿⡟⣸⣿⡇
⠀⠹⣿⣿⣋⣴⣟⣼⣿⣷⣿⠿⠛⠉⢁⣼⣿⣿⣿⣿⣿⡿⠿⠛⣉⣴⣾⣿⣿⣿⣿⠀⢸⣿⡇⢸⣿⣿⡇⢸⣿⣿⣦⣾⣿⣿⣿⡇⢸⣿⠃⣿⣿⡇
⠀⠀⠈⠙⠻⠿⠿⠛⠛⠛⠁⠀⠀⠀⢸⣿⣧⣉⣉⣉⣀⣤⣶⠿⢛⣿⣿⣿⣿⣿⡟⠀⣸⣿⡇⢸⣿⣿⡇⣸⣿⡟⠿⠿⣿⣿⣿⣿⡌⠋⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠛⠛⠛⠋⢉⣠⣴⣿⣿⣿⣿⣿⣿⡇⠀⢩⣍⡁⢠⣯⣿⡇⢸⣿⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣶⣶⣶⣾⣿⣿⡿⠟⢉⣾⣿⣿⣿⡇⠀⢽⣿⣧⢸⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⡉⠉⠉⠉⢁⣠⣴⣿⣿⣿⣿⣿⡇⠀⢸⣿⣷⣼⣿⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣍⠉⠉⢉⣡⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡻⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⢿⡦⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⢀⣩⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⡇⢹⣿⣿⠀⠀⣿⣿⣿⣿⡇⢿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⡇⢻⣿⣿⡇⢸⣿⣿⠀⠀⣿⣿⣿⣿⡇⢸⣿⢸⣿⡇

⢠⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣆
⣿⠃⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢻
⢿⡀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣾
⠈⢷⡀⠀⠀⠈⠛⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠾⢻
⠀⠈⣿⣄⡀⠀⠀⠀⠀⠙⠻⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⣼
⠀⠀⢿⡌⠙⠦⣀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⡿⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⢸⡇⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠠⢻⡇
⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠀⠀⠉⠰⢦⣀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠈⢻⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣀⣤⣾⠇⠀⠀⣴⠟⠱⣦⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⣾⠟⠉⠁⠀⢀⡾⠃⠀⠀⠙⠳⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⣸⠃⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠈⢷⣄⠀⢻⡄⠀⢀⣠⠞⠳⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠄⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⢹⡇⢸⡇⠀⡉⠀⠀⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢠⣤⠆⠀⠀⠁⠾⠁⠎⣴⠇⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠙⠛⠻⠶⣤⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⡀⠀⢠⡿⠀⠂⢺⡟⢷⣶⡀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠉⠛⠋⠑⠀⠤⢸⣷⠀⠘⠛⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠻⢦⡀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠙⠶⠤⣤⠴⠶⠶⠶⠛⠛⠛⠿⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠐⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠛⢻⣿⣿⡟⠛⠿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠁⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠈⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⢀⣴⣿⣿⡿⠋⢸⣿⣿⡇⠙⢿⣿⣿⣦⡀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏
⣴⣦⣄⢇⣰⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣴⣿⣿⡿⠋⠀⠀⢸⣿⣿⡇⠀⠀⠙⢿⣿⣿⣦⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣆⡸⣠⣴
⠙⠻⠿⢿⡿⠛⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡋⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢙⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠿⠟
⢠⣤⣤⣀⢣⢠⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣀⣀⢸⣿⣿⡇⣀⣀⣤⣶⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡄⡜⣀⣤⣤
⠘⠻⠿⠿⠿⢿⠛⠉⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠉⠛⡿⠿⠿⠿⠟
⠀⢠⣤⣤⣄⣈⢧⢰⣿⣿⡿⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢿⣿⣿⡆⡼⣁⣤⣤⣤⡄
⠀⠹⠿⣿⣿⣿⡷⠿⡟⠉⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠉⢻⠿⢾⣿⣿⣿⠿⠏
⠀⠀⠀⠀⣠⣤⣄⣀⡘⢦⡸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢇⡴⢃⣀⣠⣤⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡿⢿⡋⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⢿⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠙⠺⣿⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⢺⣷⣀⠔⠋⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠟⠛⢻⣷⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⣠⣤⣄⠀⣿⣭⡿⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠲⣶⣾⣋⣠⣿⣧⣿⠇⢰⣿⠛⣿⠇⠰⣿⡀⠋⣀⣿⠿⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠉⠉⠉⠻⠷⣦⡶⠿⣷⣾⠿⣧⣤⡿⠛⠛⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠉⢀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢠⡀⠀⠀⠀⠀⠀⠀⠈⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠂⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠮⢀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢿⣤⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠘⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠸⠀⢀⣘⡋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣠⣤⣀⣀⠀⠠⢀⣄⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣐⣒⡂⠐⠢⣐⣢⣵⣦⣶⣀⣬⡙⣷⣤⡢⢄⡑⠢⣥⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠖⠀⠂⠀⠈⠓⠢⡀⠙⠿⣯⣻⢿⣿⣾⣿⣿⣷⣭⡲⣴⡙⢳⣟⢿⣷⣆⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈⠛⠛⠿⠿⣿⣯⡻⣟⢾⣝⢿⠻⣿⣧⣿⣷⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⠢⠙⢎⢷⢩⠻⣏⣿⣧⢌⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠈⣿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠠⡀⠀⣀⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠄⠰⢀⣡⢰⣿⣼⣼⣆⣿⣷⣿⣿⣿⣷⣶⣦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣌⢇⣳⣿⢷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢳⣷⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠿⣿⣿⣿⣿⣻⠿⣿⣿⠛⠻⠿⢿⣿⣿⣿⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⡋⠹⠤⠚⣿⣽⣿⣿⣾⠀⠰⣿⠀⠀⠀⠀⠈⠉⠛⢿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣭⣿⣿⣿⣿⠿⠟⠁⠀⠘⣆⠀⠀⠀⠀⠉⠙⠓⠁⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠂⠒⠂⣤⠄⠀⠐⠘⠦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⠂⠔⠂⡀⠀⠈⠉⠛⠓⠒⠛⠓⠒⠚⠛⠉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⡀⠀⠀⠈⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢇⣈⣻⣷⣦⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠿⣴⣦⣴⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠏⠀⠀⠈⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⢳⣿⠏⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⡀⠀⠀⣩⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠹⣿⡉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠻⠟⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠂⠈⠀⠘⣷⣿⣧⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⣤⣶⣒⡛⡻⣾⣿⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠶⠿⢿⣟⣿⣿⡟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣲⣿⣿⣿⣿⣿⡿⠟⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⣀⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣠⠀⠀⠀⠀⠀⠀⢀⣠⣤⡿⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠄⠀⠀⢀⣠⣾⣿⣿⣿⢦⣶⣬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⣿⡄⠀⣴⣿⣿⣿⣿⣿⡿⠇⠩⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢹⣷⣜⣿⣿⣿⣿⡟⡛⢁⣠⣌⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣸⣿⣿⣿⣷⣿⣿⣿⣭⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⣬⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣿⡁⠈⠷⣮⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⣿⣿⣿⡟⠉⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢧⡹⣿⣿⣿⣧⣀⣸⣦⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣃⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⣿⠟⣡⣤⣤⣤⣶⡦⠀⢲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⣿⢺⡿⠟⣡⣶⡿⢟⣀⣫⣬⣹⣷⣾⣿⣿⣦⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠾⣿⣾⣶⣾⣿⠅⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⡈⠽⠿⣋⣁⣤⣠⣤⣀⠀⠀⠀⠂⣀⣽⣿⣿⣿⣿⣿⣿⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣏⡴⡦⠖⢾⣷⣿⣟⣻⡿⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣠⣀⠙⢿⣿⠟⠛⠓⠶⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣃⣨⢇⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠘⠻⣶⣿⣿⣿⡾⠋⠞⠁⠀⠀⢠⠀⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⡈⠀⣰⡆⠀⠀⠀⠘⢁⣁⣴⣾⣿⡾⠛⠃⠀⣠⣶⢾⣿⣿⣿⣿⣷⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢧⠀⠹⣷⣶⣶⡄⠀⠀⠀⠉⣉⣭⡿⠀⠀⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠍⠒⠀⢸⡦⠛⠄⠉⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠤⣿⣿⣿⣿⣿⣿⡛⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣷⣾⣿⣿⣧⡄⣀⠀⠐⢠⡀⠀⠀⣿⣆⢀⣴⣾⣿⠟⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢭⣶⣿⣿⠹⣧⡇⠹⠛⠉⢀⡀⠈⠙⢳⣶⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢙⣷⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠠⠽⠿⢿⣿⠏
⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣉⣁⣀⡙⣿⡀⠀⠘⢷⡄⠀⠀⠈⠂⠀⠀⢶⣇⣾⡏
⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⢻⣦⡄⠀⠀⠀⠀⣸⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⡈⣿⣿⣶⣿⣿⣿⡿⣿⣿⣿⣿⣶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠙⢾⣿⣿⣿⣿⣿⡇⠈⢻⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠹⠿⢫⣿⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠈⠛⣿⣆⠟⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣿⣆⠈⠙⠻⣿⣿⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠉⠁⠀⠀⢈⡿⣿⣿⣿⣿⣿⣿⣴⣦⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠛⡟⠑⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡐⠁⣈⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣋⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡠⡀⠤⢀⣐⠀⢀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣋⠑⡟⢉⣛⠧⠤⠍⠰⣼⣿⠆⡔⣻⠁⡫⢄⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠈⡰⢷⣹⢆⡊⠀⠀⠀⠘⠁⠀⠚⡴⢷⡎⢟⠲⣷⢀⡄⠲⠫⠄⢄⡀
⠀⠀⠀⠀⠀⣠⣖⢽⡆⢭⣤⡁⢉⡐⠀⠊⠈⠀⠀⡀⠀⠀⠈⠀⠀⠒⠚⠂⡦⡄⠗⠀⢈⡡⣀
⠀⠀⠀⢀⠀⣽⣟⣏⡱⣇⠏⠁⠀⠀⠀⠀⠀⠸⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⠍⣀
⠀⠀⠀⠎⠀⣴⡭⡩⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⢂⡰⠀⠀⠄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢠
⢀⡯⠦⠱⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⢀⣀⣍⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠋⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⡀⢣⠄⢠
⠀⠀⠂⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠈
⠀⡀⠀⠥⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⡄⠠
⠀⢋⠁⠄⡜⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠰⠁⠀⠀⠳⠂⠀⠠⠘⢃⠀⠄⠀⠀⡄⠀⠀⠀⠠
⠀⠈⢧⣀⠀⠉⠶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⣾⠃⠀⠀⠀⠟⠠
⠀⠀⠀⠂⠀⢡⣡⢀⣠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠋⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠟⡇⠆⣸
⠀⠀⠀⠀⠄⠀⢻⠟⡚⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⢜⢰⡀⠀⠀⠀⠀⠀⠠⠂⠂⠄⠳⢁⣠⡧⠄
⠀⠀⠀⠀⠈⠀⠨⣀⡀⠀⠀⠀⠀⠉⠀⠠⠄⠀⠀⠤⠷⠇⡠⠀⠀⠈⠀⢀⠈⠘⢃⡑⠀⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠐⠍⡀⡀⠘⡂⠂⠁⠀⡴⠆⠀⠀⠂⠀⠐⠀⠈⡀⠹⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢈⣀⠸⠋⢣⠠⠀⡕⠀⠀⠓⠀⣩⠇⠈⠁⠌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠲⣀⣉⢀⣤⢀⠂⠃⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⣿⣿⣿⣭⡭⢝⣛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣟⢷⡮⡳⢔⠭⣊⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⣰⣿⣿⣿⣿⣕⡭⣊⠕⠊⠔⠪⣦⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠙⠀⠀⠀⢸⣿⣿⡿⣿⣝⡻⣮⢉⠓⠦⣤⡐⢿⣿⣾⣯⣉⣉⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⡷⣝⢪⡲⣕⢦⡀⡳⣝⣦⣽⣿⡯⠓⣤⣄⡑⣷
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣮⡳⡱⡌⢳⡑⢈⢻⣿⣿⡟⡠⡺⠛⢽⣿⣾
⠀⠀⠀⠀⢠⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠙⠋⠉⠻⣗⠔⡙⣄⠙⢦⡹⣿⣿⣿⠈⢀⠀⣿⣿⡃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⠘⣎⢪⣳⣽⣿⣿⣿⢸⢨⢦⢫⢸⠃
⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⡳⣌⠿⣾⣿⣿⣿⣿⣿⣿⣊⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⡝⣝⢮⣓⣚⣫⠵⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡻⣝⠮⣕⣒⣲⡴⠞⡽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣕⠮⣟⣒⣒⣦⠴⢋⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡫⠖⢒⣒⣬⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡄⢻⣾⡇⢀⡄⠀⠀⢀⣀⠀⠀⠀⢀⣴⣶⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠄⠙⠃⠼⠛⠀⣼⡟⠉⣷⠀⢤⣼⣿⠧⠄⠀⠀⠀⠀⠀⠠⣏⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠻⡆⠀⣠⣾⠟⠳⣶⣶⢴⣿⡇⠀⠀⠀⢰⣿⡟⠀⠀⠀⣴⣾⠟⢳⡄⠉⡟⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣠⠞⠁⢰⣿⡟⠀⢰⣿⡟⠈⣿⣷⠀⠀⠀⣿⣿⠁⠀⠀⣸⣿⣏⣤⠞⠀⣸⠃⢸⣿⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⡾⣿⠁⢀⣾⣿⠁⣸⠻⣿⣇⣀⣼⣿⡇⠀⠀⢠⣿⣿⠁⠀⠀⢠⠏⠀⣿⡿⠀⡼⠁⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⣀⡞⠁⠉⠛⠉⠸⣏⡰⠃⡴⢻⣿⠁⢸⣿⠀⠀⢠⠟⠘⣿⡀⢀⣠⠟⠀⠀⢻⣇⡼⠁⠠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠋⠀⠀⠻⠦⠶⠋⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠐⣶⡖⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⣿⣷⢶⣦⢸⣿⠀⣴⡞⢶⡄⣤⣶⠶⡆⣠⣴⠶⣶⢀⣶⠀⢐⣿⣧⣤⡀⠀⢀⣤⣿⣿⠁⣤⣶⠶⡖
⠀⠀⠀⠀⠀⠀⢸⣿⣃⣼⡿⢸⣿⢸⣿⣤⠾⠹⣿⣤⣀⠀⢿⣧⣄⡀⠸⣿⠀⢸⣿⠇⢸⣷⢠⣿⠋⣼⣿⠘⣿⣥⣀
⠀⠀⠀⠀⠀⠀⢠⣿⠿⠉⠀⠸⡿⠀⠻⡷⠖⠀⠀⢭⣿⠇⠀⢬⣽⠗⠠⡿⠷⠸⠿⠶⣼⣿⠸⢧⠾⢹⣿⠀⠀⢭⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠠⡄⣄⠀⣄⠀⣀⠀⣀⠠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠵⢛⣼⢇⣨⡷⣈⡷⢈⣿⢎⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠈⠀⢄⠀⠄⠀⠠⠤⡄⠀⠀⠀⡀⠀⠀⠀⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠂⠀⠀⠀⠒⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⣀⣀⣐⣀⣀⣀⣀⣁⡉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠙⠀⠀⠠⠠⢄⠀⠀⠀⠀⠀⡇
⠀⠀⡇⢀⠤⠀⠀⢄⠀⢠⠔⠢⡀⡔⠒⢄⠁⠀⠀⠀⠀⠀⡎⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⡸⠂⠀⠀⠀⠀⠁
⠀⠀⡇⡎⠀⠀⠀⠀⢡⠘⢄⠤⠃⠣⣠⠎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠶⠲⡄
⠀⠀⡇⠇⠀⠀⠀⠀⠸⢰⠁⠉⡄⡋⠈⠁⠀⠀⠀⠀⠀⠀⡉⡈⣁⠀⠀⣀⣀⡀⠀⢀⠀⢠⠀⠀⠀⠘⢄⡀⡀
⠀⠀⡇⠈⠢⢄⡠⠔⠁⠈⠒⠊⠀⠁⠒⠁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠉⠈⠉⠉⡉⡉⠁⠀⠀⢀⠀⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠁⠀⠀⠀⠡⠀⡀⠀⠀⠀⠀⠀⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢩⡋⠽⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⡀⠇⡄
⠀⠀⢰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⢀⠀⠀⣇⠆⠀⢹⢠⠎⡗⠀⡔⠉⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢀⠀⠀⠀⠘⢤⣠⣤⣎⡫⠃⠀⢇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠢⠈⠁⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⢀⠀⢀⡀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⢡⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠛⠉⢲⣄
⣤⣤⡤⠤⠤⠤⠤⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⡿⡡⢄⣒⡟⣋⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⠀⠀⠒⢒⣒⠒⠒⢲
⠉⠛⠿⢭⣭⣭⣀⡀⢀⡀⠀⢀⡉⠉⠒⢦⡀⠀⠀⣰⢡⡎⠀⠀⢀⠌⠹⡍⢩⡕⠀⠀⠀⣠⠖⣉⡤⢠⠤⠀⣀⠆⠀⠤⠶⠶⠒⠋⠁
⠀⠺⢶⡂⠤⠤⠀⠀⠠⣜⡃⠀⢱⠮⠈⠀⡹⠀⠠⢤⠆⣷⠚⠀⠀⠀⢀⣿⣆⣱⣄⠀⠀⢧⡀⠈⢚⠷⠄⠀⠽⡃⠀⠀⢘⣚⠭⠚⠁
⠀⡤⠤⢤⠉⠀⠂⠀⢀⢀⡀⢰⡖⠀⠀⡟⠁⠀⠀⢻⡶⣿⣧⣄⣀⡠⠈⣿⠉⢥⣿⡄⠀⠀⢹⠀⢈⣻⠖⠐⢊⠁⠈⠁⣀⡠⠤⣶⠆
⠀⠀⠁⠐⠂⠀⠀⠠⡜⠒⠒⢊⣽⠅⡀⢳⡀⠀⠀⠈⢧⣜⢻⣿⠉⠀⠀⠹⣿⣷⣿⡅⠀⢀⡎⢀⣌⡵⣀⡉⠠⠒⠒⠒⠒⠈⠉
⠀⠀⠠⢖⡛⠓⠒⢂⡰⢦⠄⠀⢀⠻⢥⠄⠙⢦⣀⡔⢻⠾⡋⠁⠀⠀⠀⣀⣼⠋⠙⢳⡴⠋⢀⣤⠃⡖⠉⠤⡔⠃⣀⣀⠠⠭⠿⠂
⠀⠀⠀⠀⢤⠒⣊⠉⠀⠀⠾⡋⠀⡀⠈⠈⣤⣀⡟⠀⡞⠉⠀⠙⠱⡶⠋⠀⠃⠀⠀⠈⢷⡴⠽⠉⠀⢤⣀⡩⠭⠄⡈⢉⣙⡶⠂
⠀⠀⠀⠀⠀⢀⣔⣊⠄⠀⢀⢉⡀⠉⢫⡄⡈⣿⠁⣸⡇⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠘⣧⣴⠴⠖⠨⡀⢤⣀⠀⢄⡢⣄
⠀⠀⠀⠀⠰⣏⣭⠖⠀⡠⠄⠘⠋⣠⣇⡔⣻⢇⠆⣿⢣⠀⡀⠀⢀⠀⠀⢄⠀⢀⣰⠀⢰⠸⣏⣷⣼⣇⠙⠲⢬⠙⣒⠊⢿⠗
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣒⣩⠔⠒⣹⢿⡿⠛⠤⣋⢻⣦⣁⡿⣄⣁⣤⣄⡀⣱⣰⣿⠀⠀⣀⣨⣿⡟⣌⠲⣤⣐⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠝⠈⣱⡮⠂⣠⢾⣷⠀⠀⣾⣿⡿⠋⠉⠓⡷⠲⡞⠹⡛⢿⣿⣤⠋⠀⠀⣬⣷⣼⡢⣤⣄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡾⠗⡿⠁⠠⡑⣼⣿⡇⠀⠀⠀⠀⠀⢣⡆⠃⢸⣿⣿⠀⢁⡎⠀⠹⣿⣿⣄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⢠⣷⡿⢸⡇⡇⠀⠀⠀⠀⠈⢀⠄⠘⢿⣿⣧⣄⣥⡀⠀⠹⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡞⠀⣴⣿⣿⠇⢸⠃⢹⡀⠀⠢⢀⣒⠩⠀⠀⠻⣿⣿⣿⣿⣿⣆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⣾⠻⣿⣿⠀⢸⡀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣫⣷⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢦⡟⣤⠻⣿⠀⢸⡇⢀⢻⡙⠷⢀⡄⠔⠀⠀⠀⣿⣿⣿⡿⢱⣿⢿⣷⣼⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠋⢠⣿⡇⣿⠀⢸⣷⠈⡜⣿⣶⡞⠀⠀⠀⠀⠀⣿⣿⣿⠇⠸⣿⣾⣿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢰⣷⠘⣿⡄⢹⡆⢸⡇⢇⢳⡘⣿⠀⠀⠀⠀⠀⢰⣿⣿⡿⣧⠀⠀⣿⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⣿⣦⠈⣴⣿⣧⠈⣿⠘⣎⣷⠈⠁⠀⠀⠀⠀⢸⣿⣿⠃⣿⣧⠀⡿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⣧⠀⣿⣿⣿⠀⢿⣇⠹⣾⡁⡞⠀⠀⠀⠀⣾⣿⡇⢰⣿⡟⢀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠙⢿⣿⣧⠸⣿⡄⢻⣷⡇⠀⠀⠀⠀⠻⣿⣿⣦⠟⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣸⣿⣆⢻⣷⠆⢿⡿⡀⢻⡟⠀⠀⠀⠀⣤⣿⠟⢡⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⠛⠋⣾⣿⡎⡘⣿⠣⠀⢳⡰⡄⠘⢰⡿⣿⣦⡀⠈⠷⠖⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢛⠋⠀⣠⣿⠻⣿⡀⠸⣧⠁⠀⠻⣌⠀⠉⢀⣿⢿⣿⠆⠀⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠽⢒⣬⡅⢰⡌⢙⣀⢻⣧⠀⠹⣧⠀⠀⠈⠳⠶⠿⢏⣞⣿⡀⢽⣀⣡⡀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⣟⠸⣆⢻⡀⣿⢿⣧⣿⣇⠀⠙⣧⠀⠀⠄⠀⠙⠛⢭⣾⣿⣄⣰⠌⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠉⢩⣿⠟⣸⣿⣿⣿⣆⠀⢌⢧⢰⣶⠶⣠⠖⠒⣮⠻⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣁⢸⡋⣿⡿⣻⠿⣎⢦⠁⠀⠽⠟⣡⠴⠂⠬⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠉⠰⠷⣽⡄⠠⡖⠈⠓⠀⢀⠴⠢⠤⣰⣉⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠄⠄⢨⢚⡇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⡟⠫⣄⡠⢖⠒⠲⢾⡆⠰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠷⠊⠐⠒⢿⡧⠉⣝⠳⡄⢫⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢳⢲⣿⡋⢩⣀⢙⣶⠭⠃⡅⣸⣧⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠌⢱⣷⣤⠘⠃⣸⣬⣦⡩⡄⠈⣿⡏⠩⣹⠋
⠀⠀⠀⠀⠀⠀⢠⠆⠁⠙⠿⢦⣗⣶⡉⣺⣷⣌⣀⢛⡗⡻⠀⢀⡠⠝⢣⣾⡏
⠀⠀⠀⠀⠠⠀⠙⠘⡂⣄⠀⠀⠈⠀⠀⠟⢛⡁⣩⠯⢷⠺⣿⣿⣤⡂⣀⡟
⠀⠀⠀⣰⢀⡘⡇⢸⣱⡿⢳⣴⣷⣽⡷⣾⣟⢾⣿⣳⣦⠀⠀⠸⡯⢙⡟
⠀⠀⠀⢡⣦⣱⡆⢳⡟⢝⣘⠋⠀⠀⡏⣨⡻⣘⣧⠞⡙⣀⢘⣦⣿⡟
⠀⠀⠀⠀⠹⠏⢓⣰⣼⠛⠨⣬⣲⢾⣷⡟⠺⣧⠥⡴⠻⠗⠿⠟⣠
⠀⠀⠀⠀⡀⢀⢋⣥⢄⣤⣈⣼⣗⡾⠚⣷⣼⠃⠀⣵⠈⢸⠛⣿⠃
⠀⠀⠀⠀⠂⠾⠾⣼⢿⠁⠼⢃⠆⠀⠀⠑⢙⠤⣠⢋⠘⠆⠀⣧
⠀⠀⠀⠀⠀⠀⠑⠰⠴⠛⠁⣼⢈⠁⢲⠉⣩⣒⠷⠦⣀⣷⣻⠟
⠀⠀⠀⠀⠀⢠⠀⢀⣐⡃⠐⠈⠁⢈⡿⠥⠁⠄⢶⡤⣄⣤⠕
⠀⠀⠀⠀⠀⠠⣴⠺⢦⣌⠠⣌⠉⣁⠀⠀⠀⠀⣹⠁⠉⠁
⠀⠀⠀⠀⠀⠈⢿⡀⠀⠽⣷⠛⠂⠈⠓⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠙⠂⠊

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⢿⣦⣤⡀⠀⠀⠀⠀⣤⠖⠀⠛⠛⢀⡀⠀⠀⠀⣀⠀⡼⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣻⣿⣽⣿⣇⠀⠀⠐⣿⠈⠊⠀⠀⠀⢄⠀⡀⡀⣬⣮⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣾⣟⣿⣦⢂⠀⠿⠒⡀⠶⠀⢱⡔⢀⠓⣾⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣝⣿⣿⣷⡿⣶⣌⡃⠀⢠⢂⣀⣣⠒⢫⣿⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⠋⠁⡀⠈⠐⠒⠁⢀⠙⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣰⡾⠆⠀⡚⠋⢀⡿⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢡⠀⠂⣀⣤⠛⢛⠡⢄⠲⠢⠶⢇⡀⠀⠈⠉⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡔⣾⡿⣿⣡⠏⠉⠓⣿⣁⡅⠐⢯⢳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⡇⠀⠀⢻⣻⡀⠀⠸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⡀⠘⠻⢧⡀⠀⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠁⠀⠙⡃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣦⠀⠀⠀⠀⠀⠸⣶⣤⢁⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢀⣀⠘⡈⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⣀⣙⣛⠳⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣀⣴⠟⠉⣉⠙⠻⠷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠉⠀⢀⣴⠾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠶⣶⣶⣶⣶⣶⡶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠐⠷⠶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠛⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⣼⡏

⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠐⠲⣄
⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⠀⠙⡆⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⣀⡀⠀⠀⣼⡗
⠀⠀⠀⠀⠀⠀⠘⡿⠀⢰⠋⠉⡟⢻⠛⠛⠋⠀⠀⠀⠀⠀⣀⣀⣠⣤⡀⢀⣀⣈⠑⢲⣤⠾⠿⠿⠷⠚⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠁⠈⡇⠀⠀⠀⠀⠀⢠⣶⣿⣿⠙⠉⢿⣿⡿⣿⣷⣅⠀⡇⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣴⣆⠀⠀⠀⢻⢏⠁⠈⣇⠀⠀⡋⢀⡵⢾⢿⣷⡏⠙⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢾⡿⣿⣿⣯⠀⢠⣶⠙⠃⠀⠀⢸⣀⣀⡼⠋⠀⠐⠋⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣏⠀⠉⠉⠁⠈⠁⣸⣯⣤⣀⠠⠄⢾⣿⡿⠁⣀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣆⠀⠀⠀⠀⠀⢻⣟⠉⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⣽⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠶⡄⠀⠀⠀⠘⢿⣷⣄⡀⠎⠀⢀⡇⠀⠘⢶⣼⡿⠋⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡖⠚⡀⠙⣆⠀⣠⣄⣀⠉⢻⣿⣠⣤⣾⣿⣦⣶⣿⠧⣤⣴⣟⣉⣙⣖⣢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⣈⣶⣿⡿⢿⡟⢻⡀⠈⣹⠛⠛⡿⠟⡟⠛⢋⣴⣾⣿⣿⣿⣋⠻⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠀⢠⣾⡻⣏⠉⠀⠸⠃⢀⡆⢰⠃⠀⠠⣤⠊⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡤⠄⠁⣤⣄⣷⣿⣧⡾⠀⠀⡼⡇⠀⣴⠀⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣬⣭⣤⣀⣸⣿⣿⣿⣿⡇⠀⢸⠁⠸⣀⡿⠃⠀⠀⠉⠛⠛⠻⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣡⣤⠤⡼⠿⠧⠽⣿⣷⣄⠘⣄⠀⣸⠶⣄⠀⠀⢠⣶⣶⡤⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⡠⣀⡏⠙⢢⠀⠹⠉⠙⢎⢣⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣷⣀⣀⣸⣇⠤⢎⠀⢣⣠⣄⣸⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⠈⡆⢸⣿⠻⠇⠀⠻⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣧⡀⠛⠡⡀⠙⣶⣄⣙⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢻⠁⢀⠀⢱⣄⣸⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⡄⠀⠀⠀⢠⢿⣿⣾⣯⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠙⠋⠁⢀⠇⠀⠀⢹⡀⠀⠀⢘⣨⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⡸⠀⢠⡟⠲⣧⠀⡠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠛⠀⠀⢸⡞⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠰⢻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⣴⠶⣿⣿⣿⣅⡀⠀⢀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡰⠋⠀⣠⣿⢉⣧⡙⡙⢷⡀⢀⠢⡀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⣀⡴⢡⣿⠘⣿⣧⢹⡌⢻⣄⢣⠘⢦⣀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣀⣤⣤⠶⠛⣁⣴⠟⢹⡆⠙⠛⠦⣷⡀⢻⣎⢧⠀⠙⠷⣤⣌⣙⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣾⡿⠛⢉⣠⣶⠿⠛⠁⠀⠀⢿⡀⠀⠀⠸⣷⠄⠹⣷⡀⠀⠀⠀⠀⠉⣉⠬⠙⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠙⣧⡀⢉⠻⢦⣤⡤⠖⡋⠁⠀⠀⢀⣤⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⣴⣿⣿⣿⠒⠶⣄⠀⠀⠀⣠⡴⠞⢿⣄⠀⠀⠈⠻⢦⣿⣶⡶⢖⡉⠄⠀⢖⣤⣿⣗⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣾⠁⣸⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⡴⢾⣿⣿⣮⢳⣦⣤⣤⣤⣴⣶⠿⣷⣶⣾⣿⣎⢿⣏⠳⠄⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣿⣿⣿⣿⡏⠉⠉⠀⣰⠀⠀⠀⠈⠈⠉⠉⠘⢿⣽⣿⣿⣿⣿⣿⣷⡘⢆⠹⣷⡹⣟⢿⣷⣶⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⣿⡿⢹⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡜⣿⡿⢹⡟⡞⡄⢹⣷⢹⣶⣽⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠟⢀⣿⣿⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⡿⠸⢧⡿⠁⣸⠧⠀⣿⢸⣿⡇⣿⡆⢿⣆⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣾⣿⣿⣷⡄⢠⣶⣶⢶⣤⡄⠀⠀⠀⠀⠀⡸⢁⣴⠟⠁⣰⠏⡀⡰⡏⢸⣿⣷⠘⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣾⢫⣶⣿⣿⣿⣷⡈⠛⠷⠖⠉⠀⠀⠀⢀⣤⣎⣴⠟⠁⣠⡞⠁⣰⠟⣸⡇⣼⣿⣿⠀⡿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⠁⢠⣿⠙⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⣴⣿⡿⠋⢀⣠⠾⢋⣠⠞⠁⢰⣿⠁⢿⣿⣿⡾⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡏⢸⣌⢻⣿⣿⣿⣷⣶⡾⣿⣿⡿⠋⣀⣴⠟⣡⠶⢋⣤⠶⠖⠛⣸⣢⠸⣿⣿⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠘⣷⠀⠻⣷⣝⣿⣿⣿⣿⠇⣿⠟⢠⣶⢟⡵⠞⠁⣠⡯⠔⣲⣯⡴⠛⠻⣦⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⢳⡀⠘⣦⠑⠸⡏⢻⣿⣿⠏⣰⡏⣰⣿⢣⡞⠁⣤⠞⠁⣠⠞⣽⡟⠀⠀⢀⣼⣦⣿⣿⣯⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⠇⢸⣷⡀⠘⣧⠀⢰⣼⠟⠁⣴⣿⣀⣿⢣⡟⢀⣾⠁⠀⣴⠏⠘⣿⠀⠀⢠⠟⠁⠀⢿⡄⢧⠻⣮⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⡏⢠⣿⣿⣷⠀⠘⣇⠈⠁⣠⣾⢻⣿⣿⣿⠸⡇⣼⡇⠀⢸⡿⠀⢸⡟⠀⠀⠰⠄⠀⠀⠈⢧⠈⢧⡈⠻⣦⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢸⡇⢹⣿⠀⡀⢿⣠⡾⢿⡿⢸⣿⣿⣿⣇⢸⣿⠁⠀⣿⡇⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⢻⣦⡈⠛⢦⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⣿⣠⣷⠿⠋⣠⢪⠃⣼⣿⣿⣿⣿⡌⣿⠀⠀⣿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣿⢦⣄⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢀⣷⡴⠟⠋⠀⢀⡴⢡⠎⣰⣿⣿⡟⢸⣿⣷⣿⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠈⢳⡍⠳⢦⡻⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠟⠁⣀⣀⡤⠔⢁⣴⢋⣴⣿⣿⣿⡇⡄⣿⣿⣿⠀⠀⣿⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡻⣄⠀⠙⢶⣄⡉⠻⠄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠗⠛⣉⣉⣡⣤⡶⢟⣵⣿⣿⠏⣼⣿⢰⡇⣿⣿⣿⠀⠀⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣍⠳⣦⣀⠉⠻⢶⣤⡿
⠀⠀⠀⠀⠀⠀⢠⠞⣁⣤⣶⣿⣿⡿⡿⠿⠿⣿⣿⣿⠋⣼⣿⠇⣼⢡⣿⠏⣿⠀⠀⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣿⣿⣷⣌⠙⠿⣿⠛⢹⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡿⢛⣭⣥⣴⠾⡷⢶⣶⣞⣋⣡⣾⣿⠟⠠⢣⣾⡟⢰⣿⠀⠀⣿⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⣻⡿⣿⠈⣷⣀⡿⢀⠆
⠀⠀⠀⠀⢠⡾⢋⠏⢰⣿⣿⡿⣡⢫⡆⢾⣿⠉⣿⣿⡿⢋⣀⣴⠿⡿⢁⣿⡇⠀⢀⡇⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⣀⣴⣾⠟⠀⠿⣳⡿⢋⡴⠃
⠀⠀⠀⠀⢻⣅⠈⢧⠘⣿⣿⡇⣿⢸⣧⡈⠉⢶⣮⣛⣻⣛⣋⣥⡞⢁⡾⣿⠁⠀⣸⣷⠟⢧⣀⣀⢀⣀⣠⠴⠒⠋⠉⢉⠾⠋⠁⢀⡰⣶⣯⠖⠋
⠀⠀⠀⠀⠀⠙⠳⣤⣙⠺⣿⣿⣼⣆⠻⠦⣭⣜⠛⢿⣟⣿⡿⠋⢀⠞⣽⠃⠀⣰⣿⡇⠘⣷⣬⣉⣉⣁⣠⣤⣤⣀⣤⣤⡤⠶⢚⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣍⡙⠛⠿⠶⠦⠽⠷⠾⠛⠉⢀⠴⣋⡾⠁⢀⣴⡿⣡⣯⡾⠛⠛⠛⠋⠉⠀⠀⣽⣿⣿⠙⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣭⣙⠲⠤⣀⡀⠀⠀⠀⣈⣤⠞⢋⣠⡴⠻⣿⠀⣿⠁⠀⣀⠔⠒⠒⢒⣒⣚⡋⠭⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⡋⠛⠻⢿⡿⣷⣶⠞⠋⠹⡇⠰⠙⠦⠬⢉⣉⠴⠖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠛⠛⠉⠀⠀⠀⠀⠈⠑⠒⠻⠛⠋⠁

⠀⠀⠀⠀⣠⠖⠋⠉⠉⠓⢦
⠀⠀⠀⡼⠁⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢠⣤
⠀⠀⠀⢳⡀⠀⡇⠀⠀⠀⣰⠃⠀⠀⠀⡟⠺⡆
⠀⠀⠀⠀⠉⢑⣇⣀⡤⠞⣅⡀⠀⠀⢠⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠉⡏⠁⠀⠐⠀⠹⡆⠀⢸⡇⢸⠃⠀⠀⣠⡄⢀⡶⣄⠀⢠⠦⡀⠀⠀⠈⡁⠀⣀⡀⣀⡀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠸⣧⠏⠀⠀⢀⡇⡇⠈⢧⣿⠀⠈⠣⣧⡀⠀⠀⡇⠼⠃⣿⢋⡇⢀⡞⠁⣇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡼⠁⠀⡰⣯⠀⠀⠀⠸⡿⠁⢀⣴⡛⢦⠀⢠⣏⡙⡆⢠⡇⠀⢠⡏⢸⠃⢸⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠒⠶⠒⠋⠠⠴⠊⠁⠘⠢⠄⠠⠞⠳⠭⠉⢯⣩⡿⡵⠛⢯⣹⠿⠁⠙⠤⠸⠁⠘⠦⢞⠓⠖⢺⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣞⣺⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⣤⠐⠲⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠃⢠⠀⠣⣄⠀⢀⣠⣴⣮⣁⣴⡞⣡⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⣠⣴⣿⠟⢀⣤⣾⣀⣀⠔⠾⠿⠿⠿⠿⠷⠆⠀⠀⠀⠀⢀⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⠄⠙⣿⣯⣦⣤⣿⣴⠞⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⣊⣤⣾⣿⢿⣿⣿⠀⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡉⠛⠛⠀⠛⠋⠀⠸⠟⠻⠏⠟⠉⠁
⠀⠀⠀⠀⣠⣾⡿⠙⠛⠉⠀⡙⠻⠀⠰⠙⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢲⠀⠀⢀⠛⠀⠀⠰⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠂⠀⠀⠀⠁⠀⠀⢀⡀⢀⠀⡀⢠⣠⣬⠀⢠⠀⢀⣤⠤⡤⢬⠭⢭⠉⠀⠀⠀⠀⢃⠀⠀⢚⢇⠀⠀⠃⠀⠀⠰⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢒⡄⠀⢈⠀⠀⠀⣵⠀⠀⢸⠇⠸⠀⠇⠸⠀⠏⠀⠀⢀⣉⣹⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⡅⠀⢀⣸⢰⠀⠀⡌⠄⠀⢀⣅⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢀⣮⠧⠀⣰⣿⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣹⢀⣄⡧⢇⠀⣰⡳⠇⠀⣸⡾⠀⢠⡧
⠀⠀⠀⠀⠀⠀⠈⠻⠁⠈⢻⠁⠀⠈⡏⠁⠀⠀⠀⢐⣒⣒⣛⣛⣛⣛⣦⣶⠒⠒⠒⠒⣶⠒⠒⣛⣆⣠⣈⠺⠏⠁⠙⡏⠁⠀⠘⡟⠁⠈⢺⠋⠀⠐⠟⠁
⠀⠀⠀⢀⡀⠀⢀⣀⣀⠤⠤⠐⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠙⠋⠉⠛⠐⠀⠢⠤⠀⠤⠤⢄⣀⠴⢆⣀⣠⠀⠀⢁
⠀⠀⢀⣠⠜⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢉⡓⠉⠲⠂
⠀⠐⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣀⣤⣤⡶⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠾⢵⣾⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⢸⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⡄⠀⠈⣁⠀⢀⡴⡄⠀⠀⡘⠁⠀⡰⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⣤⣤⢿⣗⣚⣒⣛⡒⠻⠧⣤⣤⠁⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠟⠋⠉⣿⡟⣩⡻⡝⡏⠙⠒⠿⢿⣿⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠃⣼⣷⣦⣤⣤⣯⣧⣿⣵⣷⣥⠤⣤⣶⣟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠉⠁⠸⠟⢻⡷⢲⡒⡶⠲⣿⠋⠳⠇⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣰⠂⠀⡈⠀⠀⠙⠁⠀⢀⠀⠀⢰⡀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡇⠀⠀⢸⠀⠀⠈⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⡿⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢀⡇⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢸⠁⠀⠀⣾⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⢀⡏⠀⠀⢰⠇⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡜⠀⠀⢀⡞⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡰⠁⠀⠀⡼⠁⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠐⠀⠀⢀⡜⠁⠀⢀⡞⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠠⡘⠆⢠⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⠎⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣤⡙⠆⠰⢫⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡐⠁⢀⣾⣿⡏
⠀⠀⠀⠀⠀⢀⡈⠳⢄⠠⠏⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⣸⢃⣴⠾⠿⢷⣶⣄⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠟⠀⠒⠂⠀⠈⠻⣷⣄⠠⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠈⢿⣦⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠹⣷⡈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢀⡶⢶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠈⠻⣌⣿⠀⠀⠀⠀⠀⠀⠀⢐⣿⠏⢀⡾⠁⠸⠁⠠⠞⠉⣩⡿⠛⠳⢶⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⡞⠀⠀⠀⠀⠀⠐⠋⢁⣠⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠠⠏⠤⠔⠊⠀⠀⠀⠀⣀⡠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⠀⠀⠀⣀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠿⠛⠉⠉⠉⠉⠉⠙⠻⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠄⠊⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣶⠂⠁⠠⣤⡄⠀⠀⣤⡤⠀⠁⠐⣤⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠉⠉⠉⠙⠿⣦⣀⠀⠀⠐⠀⠀⢀⣠⡾⠟⠉⠉⠉⠙⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⡀⠀⠀⠀⠀⠈⠻⡿⠃⠀⠂⢺⡿⠋⠀⠀⠀⠀⠀⣀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣅⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢀⣽⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀⠀⠀⢰⠇⠀⠀⣼⡄⠀⠀⢧⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣤⠀⡞⠀⠀⠀⠿⠇⠀⠀⠘⡆⢠⣤⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⡀⠀⠀⠐⠒⠒⠀⠀⠀⣀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀⠀⠀⠀⠀⠀⣄⠀⠀⢤⡈⢑⡐⠂⡠⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡏⡌⢲⠤⠄⠤⡴⡫⡗⠾⣹⠑⢒⣀⣀⡙⠆⠀⠈⣡⠈⣇⡈⠃⠰⡇⣠⣭⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢄⡏⡊⢾⠚⠐⠿⠘⡓⠙⣋⣉⠇⣰⣾⠟⣀⣈⣬⣤⡆⠒⠈⣭⣇⣼⣱⣛⢿⣟⡊⠙⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠏⠁⠀⢃⣥⡶⢤⣶⢜⡫⠔⢈⠡⣥⡎⠙⠉⠉⠹⣆⢀⠀⠻⢤⡀⢨⣕⢈⠧⠝⣱⠋⠀⣄⠧
⠀⠀⠀⠀⠀⠉⠰⠅⣄⡣⡤⠪⠉⣷⣾⣡⣾⠀⡾⢁⢀⡿⢃⡤⡾⠋⢷⡘⠲⠶⡀⠀⠉⢯⣠⠿⠿⣾⡏⢀⡜
⠀⠀⠀⠀⠀⠀⢀⣼⠶⡘⢙⣦⠃⡾⢱⢾⡏⢀⠃⣀⢊⣠⠏⠆⢀⣴⠄⢹⡶⠲⢽⠀⡀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⢀⣴⢴⡻⣏⣖⢿⡧⢘⠻⡀⢻⣾⣿⢨⠰⠛⢉⡄⡀⡀⠉⠛⠂⠈⢉⠀⣀⠈⢀⣰⢲⠾⠶⣞⡻⠒⢶
⠀⠀⠀⠃⠽⠁⣺⠞⠇⠉⠰⠠⠠⠀⡜⣿⣿⠀⣠⢆⡴⡟⠃⠻⠄⠖⣰⠟⠻⣿⡇⢀⠊⣵⡠⣄⣆⣀⡡⠖⡍
⠀⠀⠀⢀⣼⣶⣷⣴⢶⣴⣿⣼⣷⣶⠝⣿⢿⣯⠉⠉⠀⠙⠲⠴⠤⡊⢊⠳⡀⠚⠀⡈⢼⠀⠳⡝⢓⣉⡉⢻⣠⣀⡀
⠀⠀⠀⢹⡾⠻⠿⣿⣾⡿⢾⣿⣸⣷⠂⣸⠘⣿⣦⡀⢀⣄⡀⢠⠬⠔⣎⣤⢘⣦⠤⠾⠫⠀⠁⠌⠉⠉⣧⣀⡫⠜⠃
⠀⠀⠈⠿⢿⣿⣿⡿⠛⠦⣿⡟⠛⠈⠛⣠⡦⠹⣷⣶⠟⠁⡋⢉⠄⡄⠈⡌⢉⣉⠢⢊⡆⠀⢰⡀⢀⡤⢣⠌
⠀⠀⠈⡅⡠⠮⢴⡄⠀⣼⣿⠃⠀⠀⠀⡲⢃⣤⡙⣟⢛⠉⢡⡳⡆⢷⢰⡝⡀⡏⡟⣶⠉⣂⣯⠛⡟⣧⠴⢺⡔⡀
⠀⠀⠀⠰⠉⣼⠯⠕⡶⣿⣿⠀⠀⣀⢤⢆⣼⣷⣿⠇⠘⣠⠄⣇⢛⠸⠇⠉⠉⠉⠁⠈⠉⠁⠉⠁⠙⠀⡀⢸⠇⡄
⠀⠀⠀⠀⠀⠉⠘⠬⠋⡈⢡⠉⡋⢈⣸⣀⠟⢿⠏⠀⠀⠓⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠘⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⢟⣠⣭⣖⡠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⠀⠉⠉⠐⠢⠽⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡖⠤⡀⠀⠈⠯⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠋⢱⡙⢲⣄⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⢢⠞⠘⢿⠳⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡿⣘⣄⠠⠀⣤⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣤⢀⠀⠀⡴⢡⡷⠀⡞⢅⢸⣄⠀⡿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢸⠇⠀⠒⠋⠉⣀⣾⢹⣿⡇⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢸⣷⣆⠀⠀⢀⠟⣵⡾⢭⠖⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠘⠀⠀⠊⠉⣁⡀⠀⠈⢛⣥⣿⣻⣰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣤⡄⡀⠀⠀⠠⠁⢉⠀⡄⣵⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⡀⠀⠙⢎⢳⡄⠀⠈⠀⠈⢠⣧⣿⢿⣿⣿⣿⡹⣷⡖⢄
⠀⠀⠀⡀⠀⠀⠐⢰⠆⠀⠠⠈⠀⠀⠀⠳⣝⣄⠂⠐⠁⢹⡟⠃⡾⣿⠀⣿⡇⠀⠻⣄⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⢮⣦⣀⣦⢘⠀⢸⠇⠸⡇⠘⣦⠀⠀⠙⢇⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣝⢾⡿⠇⡏⠑⢲⡏⠀⢻⡆⠀⠀⠈⢇⠀⠹⡏⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡑⢌⠩⢳⠏⡆⢶⠀⢣⠂⠀⠀⠸⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠱⣌⠹⣦⠀⢃⣽⡄⠈⢣⠆⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠈⠳⡌⠳⣤⢉⣷⠀⠀⢆⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠱⢧⣧⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡄⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠸⣿⣷⢀⠀⠀⠀⠀⠀⢘⠁⠉⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⣹⣿⣿⣿⣦⣀⣶⣄⣘⡷⠶⠦⣾⣿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⠻⣷⢲⢾⣿⡥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡶⢹⣿⠋⠉⢻⣧⠀⣿⣿⣿⣿⣕⣿⠋⠀⢀⡄⠀⢴⣀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢶⡄⢾⡉⣾⠟⡛⢿⠃⠉⠀⠀⠀⠠⣿⣾⣿⣿⣿⣿⠾⢿⣦⣄⣴⣿⠃⢈⣛⣸⠧⠦
⠀⠀⠀⠀⠀⠀⠀⢘⣁⡂⠠⠙⡄⠑⠻⠀⣴⣟⠀⠀⠀⠀⠀⠀⡻⠛⢳⣽⣿⠟⠂⠀⠀⠀⡠⠻⢿⣯⣁
⠀⠀⠀⠀⠀⡠⠊⣀⣽⠗⠤⢼⠿⠀⠀⠀⠈⠹⠤⣐⡀⠀⠀⠉⠀⠉⠈⠿⣷⣤⣄⠠⠖⠲⡄⠠⢶⣽⢬⣱⡐⢄
⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⢠⡗⣠⡾⠋⣩⠀⠀⠀⠂⠀⠀⠈⢳⣶⣬⣀⠀⡈⣿⠤⡰⣧⠈⣿⣷⡄
⠀⠀⢀⡄⠀⠀⢀⣀⣴⠋⠙⠶⢶⣾⣷⣿⣴⣿⡧⠀⣁⣀⣀⣀⠄⠀⠀⠀⠀⠙⠿⣿⣯⣮⡣⣀⠀⠚⠀⣿⣗⣧⡀
⠀⠀⠞⢀⡠⠶⠋⡟⠁⠀⣠⡿⠉⠀⣼⣿⣿⣿⣷⠎⠉⠀⠉⠛⢂⡒⠈⠢⢄⡀⠀⠹⣿⣿⣿⣬⣿⡿⣾⣿⣮⣿⣭⡵
⠀⠀⣠⣻⠀⠀⠀⡏⢀⣾⡟⠀⠀⢠⣿⣽⢿⣿⣿⡂⠀⠀⠀⠀⠀⠐⢶⡖⠂⠀⠉⠠⣽⣿⣿⣷⣜⣻⠶⣿⣿⣿⣿⣿⣦⣤⣤⣤⣶
⠀⠘⠉⠋⣄⠀⠀⢻⣿⣿⠁⠀⢀⣿⣿⠟⣻⣿⣿⣿⣶⣠⡀⠀⠀⠀⢸⣿⣤⠀⠀⠀⠸⣿⣿⣷⣍⡙⠦⡘⠮⢻⡿⣭⣥⡀⠀⣴⠇
⠀⠀⠀⠀⠹⣦⡀⠸⣻⡟⠀⢠⣿⢟⣱⣿⣿⡿⢺⣿⣿⣿⣿⣯⣔⣆⠘⣿⡿⢿⣷⣤⣤⣌⡉⠛⠿⣿⣦⠀⠑⠢⢭⣭⣛⣿⡤
⠀⠀⠀⠀⠀⠈⠙⢯⠻⡻⣶⣯⣶⣾⡿⢿⣿⣷⣿⣿⢿⣿⣿⣿⣷⡉⣩⣿⣿⣦⣿⣿⣿⣍⠑⠀⠀⠀⠻⢧⡒⢤⡤⢈⣩⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢯⡷⠮⠟⠿⣅⣸⣿⣿⠟⠻⢿⡀⠚⠈⣻⣟⠻⣷⣭⣉⣉⣛⠉⢛⣹⣷⢤⣄⣀⡒⠊⠓⠒⣒⢾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠹⠏⠙⢿⣿⣿⣿⣿⣛⣉⣉⣉⣉⣭⣽⣿⣷⣾⡉⢥⣤⣆⡌⠑⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠂⠒⠁⠉⠀⠈⠻⡟⠈⠛⠛⠛⠈⠙⢿⡀⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠙⠄⡒⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⢀⣀⣀⣤⡤⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠖⠉⡐⠒⢲⣿⣿⣮⣍⠉⠛⠶⣬⣣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⣫⣠⠞⠛⠉⠭⢭⣿⣿⣶⣬⣿⣿⣄⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣵⢿⡀⢰⣏⢄⣢⡴⠶⠛⠉⠉⢹⣿⣿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⣪⢏⣼⡇⢸⣛⠉⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⢿⡻⡀⠑⠝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡾⣷⢿⣿⣇⡌⣉⠁⠀⠐⠚⠒⠲⢤⠀⠹⣿⢿⡀⠹⣌⠳⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⡞⣵⣿⣿⠟⠁⠙⠁⠀⠀⠀⠾⠳⠄⠀⠀⢻⣷⣝⢦⡈⢗⢌⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⡜⡜⣟⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⡧⡠⡙⠢⢭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢸⢸⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡋⢇⣹⣌⡛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠈⢀⢂⣿⠀⠁⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡆⡆⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡏⣰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢥⣙⡻⣿⣿⣦⣤⣤⡤⠠⠀⠂⠀⠀⠀⠀⢸⠀⢧⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣺⠟⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣜⠌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⣡⢀⠜⠋⠻⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠛⠁⠀⠠⣺⢿⠈⣿⣹⢿⡆⠀⡐⠁⠀⠀⠀⠀⠀⠀⠉⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣡⡞⡱⣾⠀⣼⢋⡞⣠⢣⢟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠻⣧⡁⢳⡀⣯⠞⡰⡱⣣⣣⣿⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⣻⣿⡇⠁⡸⠐⣱⠿⡫⢾⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣇⡇⡇⡇⡏⠞⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⢻⣌⣄⠡⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣪⠿⣿⡴⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣿⣿⡇⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠀⠹⣿⡇⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⡞⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢿⣿⣷⣄⠀⢻⡇⠀⣸⣿⠟
⠀⠀⠀⠀⠀⠀⢸⡿⠀⡟⠀⠀⣿⠀⠀⠀⠀⠀⢀⣴⠋⠁⠙⡆⠀⠀⠀⠀⠀⠀⠈⠉⠳⠈⠃⠐⠋⠁
⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⠀⢰⡿⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡇⢰⣿⡛⢰⣿⣧⢾⣟⡛⠻⣿⡿⢇⣿⣛⢳⣿⣿⠀⠀⢀⢀⣶
⠀⠀⠀⠀⠀⠀⢸⣷⢸⠀⢠⣿⡥⠤⣄⠀⠀⢰⡏⠀⠀⢀⡞⠀⠸⢿⡥⠿⠟⠟⢠⣼⠟⡀⢿⠃⠘⢿⣭⠸⡟⠿⠀⠻⣆⢸⠏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠰⠛⠁⠀⠀⠘⣷⠀⢸⡇⢀⠴⣫⢦⠀⣠⡖⣦⠀⢠⡖⣦⠀⢈⠃⠀⣀⠀⢀⡴⣦⠀⠀⣠⡤⣄⠀⠒⠛⢁⡴⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⣿⡇⢸⡏⠁⢰⣿⠼⠀⣻⡾⠟⡠⢻⡿⠟⡠⣾⠁⢰⣿⢀⡏⢀⡿⠀⢰⡟⢀⡜⠹⣧⠤⠺⣧⠿⣡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⠇⢸⡇⠀⡼⣷⠀⡰⠉⣻⣎⠀⠀⢻⣎⠀⢿⣀⡜⣿⢸⠁⢸⡇⢀⣿⣇⡞⣠⠞⣿⠀⠀⢹⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⣰⠟⠀⠀⢿⡼⠁⠈⠉⠀⣴⠁⢻⡆⣴⠃⢻⣆⠀⠉⠀⠈⠋⠀⠘⢧⠞⠀⢩⠞⠁⢠⡿⠀⢠⠏⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣸⡇⢀⡟⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣰⠟⣧⣀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣰⡟⠀⠈⢷⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠞⠋⠀⠀⠀⠀⠈⠁

⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣄⡀⢤⣶⠄⠠⢶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣠⣤⡄⠀⠀⠀⠀⠀⣠⣤⢀⠀⠀⠀⠀⠀⠋⠙⢿⣿⣧⠀⠀⠀⠀⣸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣶⠈⡷⣤⡴⣾⣿⣿⠂⠀⠀⠀⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡃⠀⠀⠀⠀⠁⠀⠉⠙⠋⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠛⠉⠉⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣍⣷⡀⠀⠀⠀⠀⣼⣿⣷⡀⠀⠀⣠⣴⣾⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⣿⣷⣾⣿⣦⡀⠀⢸⣿⣿⣿⡇⣠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢿⣿⣿⣿⣿⣿⡛⠛⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⢈⣯⡀⠀⠉⠉⠁⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢸⣿⣿⣿⣿⣦⣤⣤⣀⡀⠀⠀⠀⠀⠈⢹⣿⣿⢏⡍⠀⠀⡀⠈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⢸⣟⡈⠯⣷⠆⣿⣧⢈⣽⣇⣄⣀⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠀⢨⣭⣦⡀⣠⣶⣿⣿⣾⣿⠃⢀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⣾⣿⣿⣿⣿⣿⣿⡿⠟⣁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣅⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣽⣿⣿⣶⣶⣾⣭⣝⡲⣬⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣝⣛⡻⠿⣷⣯⡛⢦⡙⢎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠀⠉⠛⠿⣷⣮⣻⣿⡴⡍⡊⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢸⣦⡙⣿⣿⣷⢻⡜⣆⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢷⡘⣿⣿⢸⣿⢸⣿⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⣿⡏⣾⣿⢸⣿⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣾⣿⣯⣿⢋⣾⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣵⠟⠡⠞⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢫⣾⣻⠇⣼⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⡏⣿⣼⣯⠁⠐⣃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡸⣯⡘⢿⣿⣄⢠⣞⣚⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣮⡳⠌⠻⣶⣝⢻⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠠⣾⣿⣦⡈⣷⡝⠧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢉⣵⣿⠿⠃⠐⢵⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢠⢤⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠯⠖⠾⠶⠿⠼⠶⠮⠽⡶⠀⠰⠿⠴⠾⠾⠯⠀⢰⠵⠾⠧⠗⠾⠷⠦⠾⠶⠾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠋⠁⠀⠀⣿⣿⠀⠀⠈⠙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣿⣿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⢀⣴⣿⣿⠟⠁⠀⣿⣿⠀⠈⠻⣿⣿⣦⡀⠀⠀⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣀⣴⣿⣿⠟⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠲⣶⣷⣆⣀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⡀⠀⠀⣿⣿⠀⠀⢀⣠⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⣠⡴⠶⠶⣤⡀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣹⡆⠀⠀⢠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣴⠋⢀⡜⠷⣀⣹⡄⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⢲⡈⢇⢰⠉⢦⣰⡏⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡧⢜⣁⡀⣀⣈⢻⣷⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⡆⠶⠚⢇⢸⠿⠃⠀⡏⠁⢰⢏⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⡿⠐⠛⠁⡜⠛⢹⣿⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣯⣄⣀⠀⣺⠀⣿⠀⠀⢧⠾⣟⡈⠽⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠣⡀⠘⣿⢃⠀⣠⠲⠀⣼⡟⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⢇⡀⢹⡿⢿⣀⣹⡄⠴⣦⠴⢏⣤⡲⠆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠈⠢⣽⡝⠳⣴⣦⠿⣿⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠉⠓⠬⠲⣎⡉⠤⠶⢁⠠⠚⠋⠀⣤⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⢻⢻⣿⢧⣀⠀⠀⣀⢿⡟⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢇⠀⠀⠀⠀⠀⠠⡁⠀⠀⢀⡇⠀⠀⠀⢀⢻⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣏⠀⠀⢸⢸⣿⡄⠈⠙⢉⡀⣸⡇⠀⠀⠀⡟⣄⠀⠀⠀⠀⠀⠀⠀⢰⡇⠘⢆⠀⠀⠀⢀⡴⠀⠀⠀⡼⠁⠀⠀⣠⠃⢸⡇⠀⢿
⠀⠀⠀⠀⠀⠀⣰⠃⢸⡆⠀⢸⠸⣿⡇⠰⣶⣞⠿⠉⠉⠲⢄⣀⡇⠈⢧⡀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⢤⣀⡚⠀⠀⢀⡜⠁⢀⣠⠞⢁⠔⢹⠃⠀⢸⡆
⠀⠀⠀⠀⠀⣰⠋⢀⣾⠀⢀⡇⠀⣿⡇⠀⠻⣤⣤⣀⣀⣀⣀⢨⠏⡖⠬⢷⡀⠀⠀⠀⠀⣾⢱⠀⢀⠀⠀⠉⣉⣽⣿⣿⣿⣿⣧⡀⠀⠀⠀⣸⠀⠀⡸⡇
⠀⠀⠀⠀⢰⡇⢀⠼⣏⣤⠊⠇⠀⢸⣷⠀⠀⠀⢈⡿⠀⢀⠉⢻⠀⡇⠀⠈⢷⡀⠀⠀⠀⣿⠈⡆⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⡴⠃⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⡿⠀⡉⠈⠻⠀⠆⠀⣿⡆⠀⠀⣸⡇⢀⠂⠀⡸⠀⠅⠀⠀⠸⣇⠀⠀⠀⣿⠀⢰⠀⠀⣿⠉⠙⠛⠛⠛⠻⢿⣿⣿⣷⠀⠀⡄⠀⢰⠀⣿
⠀⠀⠀⢰⡇⡘⢃⣀⣵⣴⣶⣤⣄⠀⠸⣿⠀⢀⡿⢀⠀⠀⠀⠇⠀⡆⠀⠀⠀⢿⠀⠀⠀⢿⠀⠀⢣⠀⠘⢧⣶⡈⢻⡷⠀⠀⠛⠉⡟⠀⠀⠀⠀⠆⠀⢿
⠀⠀⠀⢸⠏⢠⣾⠟⠛⠛⠻⠿⠛⢧⠀⢹⣇⠘⠁⠀⠀⠀⠘⢀⣀⣄⣠⣤⣤⣾⣄⠀⠀⢸⠀⠀⠀⠳⠀⠸⡏⢁⠀⠀⠀⠀⠰⢾⡀⠀⠀⠀⠞⠀⠀⣸
⠀⠀⠀⣾⠀⣿⣿⡇⠀⣀⣀⢀⣀⣨⡆⠀⢻⡆⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⡆⠀⠀⠀⠀⠀⠹⡌⢉⠉⢢⡠⠃⡸⠟⠲⣤⣄⣀⠀⠀⡟
⠀⠀⠀⣿⠀⢿⡿⠃⠨⠿⠆⠘⣶⠆⡇⠀⠀⢻⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⢹⠀⠀⠀⢳⠀⠀⠀⢀⣠⡴⠛⢾⡷⠋⣠⠞⢡⠃⠀⠀⠀⢉⠙⢶
⠀⠀⠀⣿⠆⠈⢷⣄⠀⠀⠀⣠⠇⠀⡇⠀⠀⠀⠙⠆⣾⣿⣿⣿⣿⣿⣿⡟⠁⢴⡿⣇⠀⠀⠀⢣⣀⣴⠏⠁⡄⠀⣤⢻⡟⢀⡴⠃⢄⢴⣠⠞⠁⠀⠈⣇
⠀⠀⠀⣿⠈⢄⠀⣿⠣⣀⠚⢳⣀⡾⠶⠶⠶⢶⣤⡀⠘⢿⣿⣿⣿⣿⠿⠇⠀⠀⠁⢈⡧⠀⠀⣠⠟⠁⢺⠈⠁⠀⣧⣾⣄⡜⠁⠀⢀⡾⠁⠀⠀⠀⠀⢹⡄
⠀⠀⠀⣿⣀⡴⠟⣿⡀⠈⠙⠛⣻⡇⠀⠀⠀⣀⠀⠙⢦⡀⠙⣿⠟⠋⢦⣀⠀⠀⠐⢾⣃⣤⠞⠁⠀⠀⢈⡖⠂⠀⣿⣿⠏⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⢀⡼⠋⠀⠀⠉⠛⡶⢴⡶⢻⠃⠀⠀⠀⡩⠀⠀⣠⢝⡾⠥⣙⠒⠦⣈⡶⠶⠴⠴⠋⠀⢀⡠⢤⠲⣏⢹⡀⠀⣿⡇⠀⢀⠄⡰⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣼⠃⠀⠀⣠⠷⠛⠙⡆⠀⡸⠀⠀⠀⡘⠀⠀⣼⣟⡎⠀⠀⠀⠈⠉⠚⠓⢶⣖⠒⣂⠉⠁⠀⠀⣇⡿⠐⢧⠀⣻⡀⠀⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⡀⣿
⠀⢠⡏⠀⠀⣼⠁⠀⠀⡱⣡⠖⠓⠴⠦⠶⡅⠀⢠⣞⡿⠀⠀⠀⠀⠀⠀⠀⡐⠀⢨⠟⠉⠳⢦⣀⡴⠋⠀⠀⢸⣆⣯⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣿
⠀⣸⠇⠀⢀⡇⠀⠀⣼⠑⣇⠠⡁⠈⠀⠀⠘⣦⢤⡟⠁⠀⠀⠀⠀⠀⠀⢘⠁⠀⣿⠀⠀⢂⡤⣍⡛⡟⠛⡛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⡶⣦⠴⢋⣠⠀⡿
⠀⣿⠀⠰⠚⢃⢀⡼⠁⣤⠋⠑⢦⣤⣀⣀⣀⡿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠙⠳⠖⠁⠀⠀⠙⠛⠶⣧⣀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠰⠷⠒⢋⡎⣸⡇
⠀⣿⠀⠀⠀⠈⢏⠀⡠⠙⣄⠔⠁⠀⠈⠉⠀⡋⢘⣧⠀⠀⠀⠀⠀⠀⡎⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠶⢶⠿⡋⠀⠀⠀⠀⣸⢠⣿
⠀⣿⠀⠀⠀⠀⠀⠻⣄⠊⠈⢦⠀⠀⠀⠀⣸⠃⠘⣿⠀⠀⠀⠀⠀⡎⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣱⣀⣠⡤⠞⢁⣹⡏
⠀⢿⡈⢄⠀⠀⠀⠀⠈⠳⢄⡀⠑⢄⠀⡴⠋⡐⠀⢸⡀⠀⠀⣠⠊⠀⠀⠀⠀⡠⠋⢹⠒⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠋⠉⠀⠀⠀⢠⠆⡿
⠀⠸⣧⠀⠢⣀⠀⠀⠀⠀⠀⠉⠲⢤⣙⣁⠈⠀⠀⢸⡇⡀⠜⠁⠀⠀⢀⣠⠞⠁⠀⢸⠀⠀⠀⢀⡜⠉⢉⡟⡟⠛⠛⠛⠉⠉⠁⠀⠀⠀⢀⡴⠋⣸⠃
⠀⠀⢻⣆⠀⠈⠁⠂⠤⠤⠀⠀⠀⠀⠈⠉⠛⠛⠶⢾⣧⣀⠤⠀⣀⣤⠉⠐⠂⠁⠀⡿⠀⣀⡴⠋⠀⢀⡞⢰⡇⠀⠀⠀⠀⠀⠀⢠⠤⠖⠉⠀⣰⠏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠈⠉⠉⠀⠀⠀⠀⠀⠘⠟⠋⠉⠀⠀⣠⠎⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠓⠦⠤⠄⣀⡀⠠⠤⠀⠨⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠇⠀⠀⢀⣴⣿⣿⠆⠀⢀⠀⢀⢄⠘⠻⡂⣠⡞⢻⣿⠇⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣩⣾⣿⣆⢸⣿⣾⠿⣄⡀⣘⣄⣨⠝⠳⢄⣹⡝⠛⠛⣧⠸⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢦⣥⣬⡽⠂⣶⡎⣽⢂⣬⡅⢂⣶⣦⡁⠾⡛⠦⣈⠃⢀⢸⡅
⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⢀⠲⢀⡠⠔⠁⢀⡈⠀⢄⠀⠀⢫⠘⣯⠀⣾⣯⣻⣷⣞⣴⣶⡈⠓⢈⢠⡶⠂⣈⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠛⢁⡁⡤⠐⢉⡀⠘⠀⣈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⠠⣘⠋⠀⠃⣀⢀⡄⡌⠛⠛⣿⠏⣁
⠀⠀⠀⠀⠀⠀⣀⠴⠃⢪⠀⠀⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡉⠀⠀⠀⠀⠸⢦⡀⢲⣄⡸⠆⣿
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡔⠀⣩⣴⢄⡀⠑⢀⣈⡁⠀⠹⠇⢄⠠
⠀⠀⠀⠀⢠⠁⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠘⠃⠈⠤⣬⡭⡄⠠⡀⠐⠈⠟
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢰⠂⠀⠺⠗⠂⠀⡀⠀⠇⠈⢹⣶⣿⣿⣧
⠀⠀⠀⠀⢸⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠠⠀⠀⢠⡾⠷⠄⠠⣆⠸⣿⣥⣿⣿⠆⠀⠀⠁
⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⢤⡀⢀⡄⠀⡐⠄⠙⠀⢈⡏⠛⠷⢰⡄⢶⡾⢶⣄
⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠾⣀⠀⣀⣀⢻⡿⣷⡘⡀⠲⡄⠁⢀⠇⠰⣽⡇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⢙⡿⢿⡿⠉⠀⠈⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡘⡧⠿⣧⣹⡏⢬⣦⠀⢰⡫⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣧⣝⡫⠸⣷⣼⣇⣀⣴⣾⠯⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠌⢀⠄⠠⠙⠁⠌⠉⣠⠈⢹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣟⠲⢾⣯⡁⠀⣿⣿⣿⣿⣿⣥⣴⡒⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⡄⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡷⢀⡼⠷⠒⠘⢻⡩⡛⠛⣯⣉⣃⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⣿⠇⠀⠀⠀⡼⠀⠀⠀⠀⠀⠄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⡷⢌⠃⢀⢠⡄⢸⣿⣿⡰⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣿⡇⢸⣁⣠⣤⣄⡀⣶⣿⣿⣦⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⢿⣯⡿⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⠙⠦⡿⢶⣾⠿⠋⢸⣿⣼⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡅⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠀⠰⠦⣀⢳⠈⣤⢟⠁⠈⢄⣰⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣿⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣥⣴⡅⠂⠈⢌⠀⠈⠢⠀⠀⢄⢈⣿⣧⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢻⣿⢀⣦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣟⡣⠀⠀⠈⠁⠀⠀⠠⠀⠀⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣟⣸⠃⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⢀⠿⡧⣀⠀⠀⠀⠀⠀⠈⠠⠀⠀⡀⠀⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡢⡆⢯⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢋⣿⣷⢢⠀⡌⠑⣀⣀⠀⠘⡑⠁⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣡⠞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢤⠀⠀⠟⠁⣥⣠⣅⣼⣿⡉⢛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠑⣢⣤⠆⠀⣿⠻⣿⣏⣻⣿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠯⠉⠀⢀⣿⣠⠟⠻⠿⠟⠧⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠘⣤⠀⠀⢊⠙⠗⠶⠄⠁⠀⢄⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠐⠀⠢⠀⠠⠀⠀⢰⣦⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠂⠀⠀⠐⠀⠈⠙⠤⣄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠒⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠈⠀⠐⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠢⡀⠀⠀⠀⠀⠠⠊⢀⠄⠂⠀⠀⠐⠠⡀⠑⠄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠈⠒⢄⡀⠀⠃⠠⠃⠀⠀⡠⢄⠀⠀⠈⠄⠘⠀⢀⡠⠒⠁⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠚⠀⠃⠰⠒⠈⠀⠀⠁⠒⠆⠸⠀⠓⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠟⠀⠀⡀⠀⠀⠀⠀⢠⠁⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡄⠀⠀⠀⠘⢄⡀⠀⠀⠘⠤⣀⣀⠔⠁⠀⠀⢀⡠⠃⠀⠀⠀⢠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡨⠋⠀⠀⠀⢀⡀⠀⠀⠀⠙⢅⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⠀⢀⠌⠀⠀⠰⠁⠀⠰⠆⠀⠈⠆⠀⠀⠣⡀⠀⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠘⠂⡤⢤⡴⠤⠀⠁⠀⠀⠀⠱⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⡆⡰⢆⢰⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠋⠀⠀⠙⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡤⠤⠀⣀⣠⠀⠀⣄⣀⠀⠤⢠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠐⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠂⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠤⠄⠀⠀⠀⠂⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⡒⢢⠤⠤⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄⣒⣩⠭⠓⡚⢁⡿⣅⣐⡠⡤⠀⠀⠘⡀⠀⣠⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⢖⡶⣿⣿⠿⠟⣛⣯⣥⣖⣊⣭⠖⣩⢒⡕⣩⢿⣍⡣⠦⡜⠀⣰⣿⣿⣿⣿⣿⡧⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣖⡺⠿⢖⣊⠍⠍⣈⡥⠴⢒⣻⠵⢚⡽⠛⣻⠖⠿⢕⣿⣞⡵⢋⣴⢻⣆⣇⠀⠸⣯⡽⢟⣩⣭⡶⠟⠁
⠀⠀⠀⠀⠀⠘⣺⠓⠉⠠⠔⣊⣉⠤⠔⣋⠁⠀⡟⠉⡀⣚⠁⠔⠋⡄⢚⡰⢋⡄⠛⣞⣥⠞⡁⠛⣾⠒⢘⣻⣿⣿⣿⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠁⠀⠈⠀⠀⠀⠁⠁⠈⠈⠀⠁⠀⠈⢓⠀⣈⠐⠲⢚⣧⣦⠷⣿⣿⣥⣷⣰⢋⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢊⡶⢣⡿⠟⣩⠟⣻⣝⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠀⠀⠀⢀⠀⠠⣤⣤⣴⣄⢀⠀⡁⢋⠔⠣⢞⣴⠣⢠⡅⡻⣴⠆⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠀⣉⠿⡫⠕⠊⠀⠁⠀⣽⣿⡹⠖⡷⢶⠟⢛⠉⣭⣶⣛⠛⠉⢟⣂⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢛⠩⠀⠀⠀⠀⠀⠀⢀⣠⡤⢛⠄⢁⠀⠂⠀⠀⣼⠷⢦⣀⣈⣶⠶⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡿⢋⢀⠅⢀⠌⠞⡜⠒⢞⠀⢺⠏⡈⠡⠖⣸⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡽⠈⣰⡡⠔⠻⣖⠁⠀⢆⣀⣶⠊⠋⢁⡤⣸⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠌⠀⠁⠀⠩⠁⠀⠀⠀⠀⠀⡠⢎⡞⠡⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠠⢀⣤⠌⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠢⢔⣴⡎⠁⡡⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠐⠤⢀⡀⠀⢀⡀⠠⠀⠴⠟⠁⠀⡀⢠⠊⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠈⠉⠒⠤⠀⠈⢙⢛⣂⢀⡀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⠀⠀⠀⢀⠀⠀⠠⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠒⠒⡄⢉⠛⠛⠉⠀⣀⠀⠀⠀⠀⠀⠀⣀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢆⡐⠉⠀⠐⠂⠀⢀⡀⠠⠄⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⢀⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠂⠀⢀⡠⠤⣀⠀⠀⠀⡀⠀⠀⢀⢰⠐⣀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⠀⠀⠀⠀⠀⠀⢁⠄⢀⠀⢁⠀⠀⢀⡠⠄⠠⠄⣀⠀⢀⠋
⠀⢂⠀⠀⠀⡀⢠⠁⠀⢀⠈⢂⠀⢀⢀⡀⡀⢄⢡⠁⠀⢁⠔⠉⠀⠀⠉⠑⢄⠢⢀⠠⠊⡀⠄⠒⠂⢄⡐⠀⠠⠄⣀⠆⣠⠊⠁⠀⠀⠀⠀⠀⠑⢌⠒⠂
⠀⠀⠈⠉⠁⠀⢂⠀⠀⠄⠀⡀⠀⠀⣤⠀⠠⡤⡸⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠊⠀⠀⠀⠀⠀⠑⡈⠉⠉⠀⢠⠃⠀⠀⡀⠀⠀⠀⠀⠀⠈⡄
⠀⢀⠊⠁⠈⠑⡄⠑⠠⠤⠊⠀⠀⠀⠀⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠆⠀⠰⠆⠀⠶⠄⠀⡅⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⡇
⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠢⡀⠀⠢⡀⠂⠀⠒⠁⡠⠊⣀⣀⠀⠀⠀⢫⣀⠀⠀⡠⢁⠔⠂⠒⠜⣆⠀⠀⠀⠀⠒⠂⠀⢀⠜
⠀⠈⠂⠤⠶⠒⡱⢊⠁⠀⠁⢀⠀⠀⠀⢀⡀⠀⡀⢰⠀⠀⠀⠉⠀⠈⠁⠀⠈⠀⠀⠃⠐⠢⠤⠀⠤⠒⠁⠸⡀⠀⢀⠀⠈⠒⠄⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠈⠀⠘⢃⠀⡁⠀⠀⠀⠐⡄⠠⠠⢂⠌⠀⠀⣀⡀⠀⠀⠄⠀⠈⠂⠒⠁⠀⡀⠀⢄⠀⠀⢀⡀⠤⠍⠀⠠⠄⡀⠀⠀⠀⠀⠂⠈⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⢀⠜⠀⠀⠀⠀⠁⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡠⠀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠣⡀⢰⠀⡰⠂⠀⠶
⠀⠀⡠⣄⡠⠤⢤⡀⠁⠉⠐⠊⠀⠀⠉⢂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠎⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⡅⠈⢄⠀⠲⠽⠁⡠⠃
⠀⠀⠑⠋⠀⠀⠀⠘⠄⠈⡏⢉⠸⠠⠲⠀⡆⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⢠⠀⠈⠁⣀⠉⢀⡀
⠀⠀⠸⣀⡀⠀⢀⣢⠀⠐⡀⠀⠐⠂⠀⡠⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠠⠀⠤⠆⠀⢀⠌⢀⠀⠁⠀⢊⡀⠀⡀⠈
⠀⠀⠀⠈⠓⠒⠊⣁⠀⠀⠈⢀⠒⠒⠈⡠⠤⢄⢃⠀⠀⠐⠐⠀⠀⠂⠀⠂⠐⠀⠀⠀⠀⡀⠢⠤⠈⠢⣀⡀⠀⠀⢀⡀⠔⠁⠀⠀⠒⠁⠠⡈⠗⢠⣽⡴⠁
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠙⢄⠈⢀⠀⠀⡇⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠊⠀⠀⠀⠀⢉⣭⣠⠄⠀⠀⠀⠖⠒⠐⠤⡀⠉⢉⡥⢤
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠈⡌⠂⠤⠈⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠂⠀⠐⢑⠀⠀⠐⠉⠉⠉⠓⡄⠘⠮⠿⠀⡠⠀⠀⠀⠀⠀⠀⠈⠢⡀⠥⠼
⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠉⠀⠀⠀⡇⠀⠠⠚⠉⠉⠒⠤⡀⠀⠀⡰⠒⠈⠓⠠⡀⠁⢠⠁⡄⠀⠀⠀⠀⢈⠀⠀⠀⢀⠁⢠⠀⠀⠀⠀⠐⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⢁⡹⠀⡰⠀⠀⢠⣦⠀⢠⣶⠀⢱⠀⠐⠀⠀⠀⠀⡀⠈⡄⢰⠀⠀⡄⣠⠀⠀⠨⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⣀⠀⠀⡘
⠀⠀⠀⠈⠢⡀⠀⠁⠀⠀⠀⢀⠔⠁⠀⢀⠀⠉⠀⠌⠁⢀⠌⠀⢠⠀⠁⠀⠈⠄⢀⠃⠀⠑⠄⡈⣀⠀⠔⠡⠊⠉⠉⠂⠱⣀⠀⠈⠉⠉⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠂⠁⠀⠀⠀⠀⠐⠠⠤⠄⠐⠁⠀⠀⠀⠑⠠⠤⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣤⡬⠂⠀⠀⠁⠂⠂⠐⠈
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠠⠀⠀⠀⠀⠀⡀⠐⠶⠀⠀⢘⠃⠀⠀⢄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣯⠢⡀⠀⠀⠾⠳⡄⠀⠀⠀⢀⠘⠀⠀⠀⠐⠁⠠⡀⠀⠀⠀⠀⠀⣤⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⣽⡆⠈⠲⣏⡀⠀⠘⡂⠄⠀⠀⠀⠀⠀⠀⠀⡠⠴⠳⡖⠂⠙⠆⠠⢼⣟⣋⣉⠐⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⠐⠀⠀⠀⢠⡄⢹⡀⠀⣉⣉⠀⠀⣿⡌⠠⣠⠀⡠⢀⠄⠈⠀⠐⠣⡁⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠄⠀⠀⠀⣹⠦⢿⡈⠀⢁⢼⡌⠉⠻⢆⢿⣄⡿⣳⠆⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠈⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠀⠴⣷⡿⠁⠀⠘⣑⠀⠀⠀⠛⠦⠀⠪⠞⠋⠷⠁⠀⠘⠀⠄⠀⠀⣿⣤⡼⠀⠀⢀⡤⠶⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠦⢿⠓⠖⠉⠀⠀⠀⠀⠘⣶⡴⢦⠐⠀⠀⠀⠀⠀⠀⡁⠐⢃⡀⠀⠀⠀⢻⣿⣿⣇⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⣀⠤⠤⠀⠐⠂⠈⠹⠋⠙⠛⠒⠒⠂⠀⢴⠇⠀⠀⠑⠀⠀⢠⠀⠀⠜⠀⠀⠀⠏⠀⠀⠀⠹⡿⢿⣷⡆
⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⣀⠣⠄⢀⠀⠈⠄⢢⢸⡀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⡘⢻⠈⠁⠀⠢⡄⠠
⣠⠞⠁⠀⠀⣦⠄⠀⢀⡐⣈⠅⠀⣀⠀⠀⢴⣽⡶⠂⠀⠀⠀⠀⠠⣼⣌⣿⣷⣾⠦⠀⠀⠈⠀⠀⢁⠀⠀⠜⠀⠈⡀⠀⠀⠀⣼⡃
⠀⠀⠀⠀⠰⣈⣆⣱⡉⠙⠓⠒⢈⠀⠐⠚⠋⠉⠁⠑⠀⠀⠀⠀⠀⠈⢻⠿⣿⢧⠶⠛⠂⠀⠀⠀⠉⠀⠤⠤⠀⠀⠓⠂⠐⠈⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠏⠛⠁⡁⠀⠀⠀⣾⠻⣶⡤⠄⠤⠤⠤⠠⣔⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⢀⡀⠐⠂⠀⠉⠁⠀⠠⣄⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠃⠀⠀⠹⣏⡀⠀⠀⠈⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⢏⠒⠚⠀⠀⣎⣀⢀⡀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⡋⢄⣤⠂⠁⠘⣿⣿⣶⠁⠠⠙⠖⠀⠀⢢⣤⣼⠖⠀⠀⢀⡴⠁⠀⠐⣶⣿⡻⠿⠛⠓⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠠⣴⠄⠠⢤⠔⠒⠈⠈⣟⠉⠀⠀⠀⠄⠉⠿⡷⠀⡀⠘⠀⠀⠀⡘⠙⠉⠀⠀⠀⡘⠁⢠⣾⣿⣿⡏⠃⠀⠀⠈⠁⠐⢀⠁
⠀⠀⠷⠊⠹⡶⠰⠁⠀⠀⢀⣾⠋⣠⣠⠀⠀⠀⠀⠀⠴⢻⡅⣰⠀⢀⡼⠁⠀⠀⠀⠀⠠⠀⡠⠁⠸⠟⠋⠙⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢃⡾⠁⠣⣀⠀⣠⠎⠀⠀⢸⣷⡏⢀⠚⡅⢢⡐⠀⠀⢀⣀⠜⠁⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣔⡈⠹⠊⠀⠀⠀⡩⢺⡁⠀⠀⠀⠊⠙⣦⠧⠄⠠⠟⢳⣶⡆⡅⢸⡀⢀⡀⠀⠀⠀⠀⠹⡄⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⡶⢚⣿⡤⠔⠉⠀⠀⠀⠀⠀⠉⠘⠓⠠⠛⠁⠀⠇⠁⠀⡰⠉⠀⢁⠀⠀⡉⠀⡶⠂⠒⠦⢍⣁⣀⣀⢈⣼⠋⠀⢠⣀
⠀⠀⠀⠀⠀⢀⡾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠀⠀⠀⠀⠀⠀⢆⠐⡿⠢⡀⠀⠀⠀⠀⠀⠀⢹⡁⠀⠉⠉
⠀⠀⠀⠠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡔⠀⠀⢀⣄⢀⣯⠑⠀⠤⠒⣵⡀⠱⡄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠈⠃⠀⢘⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⣼⣽⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣴⣾⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠛⢿⠁⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣀⠀⠀⢸⢹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁⠀⠸⠆⢀⣾⣿⣿⡇⠀⠀⠀⢀⣿⣿⣷⠀⠀⢸⡇⠹⣿⣿⣿⣇⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⢀⣾⣿⣿⡏⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⣰⣿⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⢀⣾⣿⣿⣿⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣇⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢿⣿⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟⠁⠈⠛⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⠀⢻⣿⣿⣶⣤⣤⣤⣴⣾⣿⣿⣿⠿⠃⠀⠀⠀⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣆⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⢀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣀⠀⠈⠙⠉⠉⠁⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⠿⠋⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠈⠉⠉⠙⡏⢿⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡏⠀⠐⠀⠶⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣰⣌⣡⡿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⣄⠀⠀⠀⠈⡀⣴⣿⣿⠾⣟⣶⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⡈⠁⠀⠘⣦⢃⡀⠸⣿⡿⠛⠁⣼⡟⠛⠙⠋⠈⠳⡄⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⡠⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣶⣤⣟⣾⣿⣴⣟⣡⡴⠟⠉⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣠⡎⢀⣤⠞⠁⠘⠉
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣀⣤⣴⣦⣶⣿⣿⣿⡿⢋⠁⠀⠀⢀⣀⢠⣜⠛⠃⠀⡸⠀⠀⠀⢠⣾⣋⣀⡟⠁⠰⠊
⠀⠀⠀⠀⣠⠊⠀⠀⣴⣾⠿⠛⠉⣉⡤⠀⢠⡞⠁⠰⣎⠲⣄⠀⠀⠉⠀⠀⠀⠀⢀⡏⠀⠀⣠⣿⣿⣿⣿⠿⠗⠒⠉
⠀⠀⢀⢀⡁⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⣰⡿⢁⠀⠀⢻⣦⠈⠛⣄⠀⠀⠀⠀⠀⣸⠃⢠⡜⠛⠉⠉⠁
⠀⣠⢻⣿⡃⠀⠀⠀⠀⢰⣿⣅⠀⠀⢰⣿⠃⢸⠀⠀⠀⠈⠠⣾⣿⡃⠀⠀⠀⢀⠋⣰⠋
⠠⢛⡏⢻⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠿⠇⠀⡇⠀⠀⠀⠀⢲⣿⣿⡆⠀⠀⠀⣮⠾⠁
⠀⢻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣿⡃⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣰⠏⠀⠀⠀⠀⠀⠀⠀⢛⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⢄⡀⠀⠀⠀⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⡿⡟⠁⠀⠀⠀⠀⠁⢦⠀⢸⡇⢰⠃⠀⡀⠲⢦⡀⠀⠀⠀⢰⣿⣿⣧⠠⡤⠆
⠀⠀⠀⠀⠀⠀⢀⡤⠴⠾⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⢇⢸⢁⣼⠀⠊⠁⠀⠀⠈⠀⢸⣾⣿⣿⣟⠉⠐⠒
⠀⠀⠀⠀⠀⠀⣀⠄⢂⣠⡟⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣾⡟⠁⣃⠀⠀⠀⠀⠈⠁⣿⣿⣿⣿⡏
⠀⠀⠀⢀⣀⠌⣴⣾⡿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢿⣿⣿⣿⣿⡧⠀⠀⠀⢀⠀⠙⣿⣿⣿⣿⣿⣶⠄
⠀⠀⢠⠠⠋⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⡿⠁⠀⠉⠀⠲⠚⠉⢂⢿⠀⣾⠟⣿⣿⡿⣿⡇
⠀⠀⠀⠃⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣴⡅⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⣾⠐⢿⣿⣾⡿⠃
⠀⠀⣨⣾⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢹⣿⠁
⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡄⠀⠀⠀⠀⢀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⢸⡇⢀⣤⣶⣟⠋⠉⠁⠀⠉⠙⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⣄⢸⣧⣿⣿⣿⣿⣆⡀⢀⣀⣀⡀⠄⠀⢉⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣾⣷⣿⣸⣿⣿⣋⡭⠉⠉⠙⠻⠿⢤⣴⣶⣶⣿⣿⣿⡿⢦⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠉⠻⣦⠱⢆⠠⠶⠀⠀⠀⢈⣉⣻⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣰⣾⣿⣿⣿⣛⣿⠻⣿⠀⠀⠈⠑⠚⠐⠒⠒⠚⠛⠛⠛⠻⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠋⠀⠀⠀⠀⠀⠀⣿⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⣾⠟⠋⠀⠀⠀⢀⡄⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠁⢠⡞⠁⠀⠀⠀⠀⠀⣼⣇⠀⠀⡇
⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⣾⠤⠤⠤⠤⠤⠤⣴⠃⠘⢦⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢿⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠙⠶⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣤⡶⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢷⡀⢀⣴⠟⠁⠀⠀⣀⡴⠖⠛⠉⠁
⠀⠀⠀⠀⢠⡞⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⣠⠞⠋
⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠻⢮⣷⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⡀⠀⠀⣀⣠⠞⣽
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⡴⠚⠛⠛⠶⣤⣀⡀⠀⠀⢀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠘⣦⠀⣾⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣿⠀⠀⣾⡁⠀⠀⠈⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢰⣇⠀⠀⣿⠀⠀⣿⢷⣤⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡹⣆⠀⣿⠀⠀⠙⣆⣰⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢻⡄⡿⢠⠀⠀⠘⠃⠀⡀⠀⣼⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣸⠃⣿⣿⠀⠀⠀⠀⢀⣿⣸⢿⡍⣿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠷⠋⠀⣿⢹⡆⠀⠀⠀⢸⡟⣿⣦⡉⠛⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣧⠀⠀⠀⢠⡏⠀⢷⡀⢠⣆⠀⢷⠈⢿⡟⠲⢤⣄⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢻⡀⢰⡄⢸⡇⠀⠈⢷⣼⡟⢧⣸⣧⠀⠙⠷⣄⠈⢿⡿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠻⣟⠻⣜⣧⠀⠀⠀⠙⠿⣄⡙⠛⠧⠀⠀⠈⠙⢮⣷⡄⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡤⣦⡄⠀⢀⣾⠉⠗⢸⡇⠀⢸⡇⣿⢸⡇⠀⣶⣠⠒⣶⠀⣤⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣶⠚⠉⢸⣿⠀⣸⠇⠀⠘⣿⠀⢀⠼⣷⢤⣸⠇⡟⢸⡇⠀⣿⠃⢰⡿⢸⡿⠁⣸⡿⢐⣶⠦⢤⡄
⠀⠀⠀⠀⠀⠀⢶⡄⠀⢿⡄⢻⡤⠄⠀⢿⠘⢧⣀⠀⠀⠛⠦⠟⠀⠃⠀⠘⠀⠃⠼⠧⠴⠗⠤⠟⠁⣼⠱⣞⠋⠀⣼⣧⠀⠀⣾⡧⢒⣦
⠀⠀⠀⠀⠀⠀⢈⣷⡤⠬⢷⠈⠷⠖⠂⠈⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡰⠿⢤⠀⣸⠏⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⣶⠀⢠⡦⠠⠶⣦⢀⡶⠀⢠⡶⠒⠄⢠⡶⠢⠀⠀⣴⠤⣤⡞⢻⠲⠶⢤⣤⣀⣴⣿⠛⢒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣀⠀⣾⠁⡰⣿⢀⠼⣧⠂⠀⠀⢸⡇⢀⠜⢷⣤⠔⢻⡏⠀⡰⠁⣿⢠⠞⢰⢋⠸⡇⢀⡖⡄⠉⣽⣷⠋
⠀⠀⠀⠀⠀⠀⠀⡄⣿⣴⡛⠉⠀⠈⠋⠁⠉⠁⠀⠉⠀⠀⠀⠀⠙⠁⠀⠎⠽⠀⠈⠑⠈⣠⠞⠁⣿⢤⣅⣘⡃⠻⠼⠐⠁⣼⣁⣿
⠀⠀⠀⠀⠀⠀⣄⣿⠷⡖⠀⠀⠀⠀⡀⠀⠀⡤⠀⠀⢄⢠⠀⢠⡄⠀⠀⡄⢠⢀⡄⢠⡄⢭⠙⠓⠾⠿⠁⠉⠉⠛⠓⠶⠾⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠻⢿⡞⡁⠀⠀⣠⣶⠖⠓⠀⡇⠀⣀⣼⢸⠀⢸⠁⠀⠀⡇⢸⢸⡆⢸⠆⠋⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢹⣦⣷⠇⠀⢀⠻⡂⠀⠀⠀⠉⠀⠀⠈⠈⠀⠈⠀⠀⠀⠁⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣤⡄⠘⣟⠁⠀⢀⣤⣤⣤⡀⠀⢄⣀⡀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠉⢀⣀⣤⠶⠋⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⠷⢼⣾⣄⠀⠀⠀⠙⣛⣛⣦⣄⣉⣙⣳⣄⣀⢿⣿⣾⡷⣀⣴⡎⠉⠀⠀⠒⢒⣒⣉⣩⣥⣤⠶⠖⠋⠉⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢷⡞⠋⢩⠾⠋⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣉⠉⠁⢀⣴⢆⡀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⢋⠔⣿⠀⠀⠀⠀⣾⠗⢹⡆⣶⠀⢠⣤⣤⡄⣀⢀⠀⣀⢤⡀⣤⠴⠖⢰⡜⠙⣧⠀⣫⠵⠋⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⣴⠏⢀⣿⠀⠀⠀⢸⡇⣸⠏⢰⡇⠀⢰⣇⡀⠸⣇⠸⠓⢧⡀⠃⢸⡧⠄⠘⣿⠀⣿⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠠⠞⢛⣩⠎⢹⡆⠈⢡⣤⣀⣀⣀⣈⡉⠀⠼⠤⠤⠼⠧⡄⠀⡌⣷⠀⢠⣙⡆⠨⠷⠖⠀⠉⢈⣡⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣰⠇⢻⣦⡘⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠲⠶⠦⢤⣬⣤⣤⣤⣤⣤⣤⣤⣶⣶⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠟⢰⣇⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠩⡛⠷
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣶⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠃⠀⠀⢹⠀⡟⠉⠛⠛⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣭⣧⣿⣄⣠⡾⣹⠃⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⡇⢸⣇⠀⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠤⠀⠀⠀⢀⣴⣦⣤⣿⡇⣸⡃⣈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⠀⠀⠀⠈⠀⢀⣈⣼⣀⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠀⣸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡟⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣦⣿⣻⣿⡿⡿⢡⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣿⣿⡏⠸⣿⢿⣋⣄⣾⣷⣼⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⣶⣿⣿⣿⠁⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⡿⠿⠀⣿⡷⡿⣾⣿⣛⣿⣿⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⣠⣴⣿⡄⠀⣟⠁⢸⣿⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⣿⡿⡄⠀⣏⠀⠸⠇⣰⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣩⣉⡅⠀⢸⣯⠀⠀⠀⣻⣿⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠿⠿⠀⠀⢸⡟⠋⠀⢾⡟⣛⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⢸⣷⣶⠀⠈⠁⠏⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣾⣿⣈⣋⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠠⠆⡀⠀⠚⠻⣯⣤⣤⣤⣤⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠐⣀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⡐⠧⠀⠀⠠⢤⣸⣿⡛⣿⣿⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣥⣀⣀⣀⢀⣀⡀⠙⣿⡡⣷⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⣿⡿⢃⣿⣿⡃⠉⢉⣉⠉⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣴⡿⠿⠿⠟⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠈⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⢸⣿⣿⣹⣿⣧⠀⣀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⢻⣿⣿⢻⣿⣿⢸⣿⡿⣻⣿⣿⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢿⣿⣧⢿⣿⡟⣿⣿⣻⣿⠇⣿⣿⢣⣿⣿⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⣠⡿⣿⠮⠟⠁⡘⠉⠈⠋⠀⠻⢣⣿⠿⣁⢺⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣫⣿⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠴⠛⠙⠛⠿⣶⣦⠀⠀⠀⠀⣴⣶⠿⠛⠋⠛⠦⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢯⡀⠀⣀⣤⣶⣶⣄⡄⠁⠀⠀⠀⠀⠈⢠⣠⣶⣶⣤⣀⠀⢠⡹⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⢺⡇⠀⠙⠣⠴⣿⠭⠿⠆⠀⡁⠈⠀⠰⠿⣭⣿⡦⠜⠋⠀⢸⡗⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣂⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡾⢐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢂⠀⠀⠀⠀⠀⠠⡵⠦⢄⣠⠴⢮⠄⠀⠀⠀⠀⠀⠰⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⠀⣨⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⣄⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⠰⠶⠫⠤⠤⠤⠤⠽⠲⠆⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠦⣄⣀⣀⣠⠴⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠦⣄⣀⣀⣀⣀⣠⡴⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣀⣤⣤⣶⡾⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢷⣶⣤⣤⣀⡀
⢶⡿⠿⠟⢛⡿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣼⣿⢿⡙⠻⠿⢿
⠘⣷⡘⢃⡞⠺⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⡟⠻⢳⡤⢠⡶⢶⡄⢤⡾⠛⢻⡍⣿⢾⣿⣿⣿⣿⣿⣿⣿⠛⠗⢳⡘⢃⣾
⠀⠈⢿⣌⢡⡍⢉⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠳⣤⣼⢧⣽⡦⢠⣿⡼⣧⣤⠞⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⡉⢩⡌⣡⡿⠁
⠀⠀⠀⠻⣦⣀⡮⠔⠚⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠈⠑⠚⠁⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⡗⠂⢵⢀⣴⠟
⠀⠀⠀⠀⠈⠻⣄⠛⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣠⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣦⣿⣿⣿⣧⣿⡇⠀⢸⣟⡙⢋⢻⣿⡻⢟⣿⡿⡙⠫⣹⣇⡈⢘⣿⣿⣿⣿⣿⣷⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣧⣿⡏⠠⠤⠃⣤⠈⣌⡕⢁⡈⢪⣡⠁⣤⠈⠲⠅⠙⣿⣯⣽⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠛⠷⠶⣤⣤⣄⣙⣀⣈⣁⣀⣉⣠⣤⣤⠶⠾⠛⠋⠁⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣋⣉⣿⣀⣬⣉⣭⣉⣸⣟⣩⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⣿⠀⠀⠀⠀⡴⠒⠒⠒⠢⢄⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠐⠃⣠⢤⠀⡀⢉⠀⣀⢀⡀⠀⣸⠏⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⢀⣾⠉⡟⣺⢃⠾⣠⡿⢩⡟⢠⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢀⠞⠙⠚⠀⠈⣉⠀⠀⢁⡸⠇⢺⣇⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠁⠀⠀⠧⣶⢴⢻⣶⣴⣨⣧⠄⠈⠉
⠀⠀⠀⠀⠀⢀⣤⡄⣠⠔⢢⠀⠀⠀⠀⢀⡶⠀⣔⡟⠈⠀⠀⠀⠁⠯⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠈⢸⣿⠁⢀⣾⠤⣄⡀⣠⣾⠃⠈⠋⣠⡄⣴⡆⠀⣸⠀⡄⣠⢖⡆⣠⡶⢛⣆⠀⣿⠞
⠀⠀⠀⠀⠀⠀⢸⣿⣰⠟⠁⠀⢈⣿⡅⡿⢰⡾⠂⡽⣧⠙⣧⠊⣿⣸⣿⠃⣾⢃⣿⡠⢺⡷⠚⢻⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡴⠁⣸⠟⣇⡾⠧⠊⢠⠿⢰⣹⠃⠈⠁⠉⠘⠷⠃⡠⠖⣿⠁⢮⣸⠟
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡜⠀⢠⠟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠴⠋




⣿⡏⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⣶⣶⣦⣌⣝⣿⣿⣟⣉⣤⣶⣶⣦⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣛⣛⣻⣿⣿⣟⣛⣛⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣶⣶⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⣛⣚⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠿⠛⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠙⠐⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉
⠀⠀⠀⠀⠈⠠⢈⠉⠑⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⢄⠀⠀⠀⣰⣹⡄⠀⠀⠀⠠⠁⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠀⠀⠈⠂⢸⡗⠉⠌⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠈⠳⣄⠀⠘⡎⠁⠄⠀⢠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢁⠄⠨⠀⠾⣄⠀⠁⠀⡀⠈⠀⢻⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢱⣄⢠⢰⠀⠄⠀⠀⠀⠱⣠⠀⡤⠠⡒
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠘⠹⠂⠀⡄⠀⠀⢀⣀⠀⠈⠕⢾⠄⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠚⢃⠀⠈⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠄⠀⠀⠀⠀⠀⠀⡰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢧⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠑⠄⠀⠠⠆⡄⠀⠆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠠⠀⠀⠀⠀⠐⠊⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢤⣄⠀⡀⡰⡀⠀⠀⠀⠠⠤⣀⠠⠤⠄⠒⠒⠒⠉⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⢀⡔⢋⡏⠁⠀⠀⠀⠀⠈⠀⢀⢻⡢⣁⠃⢠⠀⡀⠀⡀⠂⠈⠀⠄⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⢀⠂⠁⢠⠖⠒⠉⢯⡀⠀⡄⠀⠀⠀⠀⠀⡠⠈⠉⢁⢑⠿⢜⠛⠀⠀⠀⠀⢈⠀⢸⡤⠶⠚⠊⠁⠀⠀⠀⠀⠀⣀⣀⣠⡴⠮⠋
⠀⠀⠀⠀⠀⠀⠘⣠⠶⡋⠀⠀⠀⠈⠈⠳⠐⣰⠀⣀⠜⠉⡀⠀⠀⠀⠀⠀⠀⠉⠑⡦⡄⠀⠤⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠁
⠀⠀⠀⠀⠀⠨⡞⠁⠉⠉⠀⠁⠒⠩⣈⣃⠀⠃⡎⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣼⠁⠚⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠔⠙⢄⠀⠀⠒⠈⢄⠀⠀⠀⢀⠜⠃⠀⠀⡄⠀⠀⠀⠀⠀⡠⢂⠀⠙⠣⡞
⠀⠀⠀⠨⡀⠁⢀⠲⣄⠀⠀⢀⠈⠀⡄⠀⢀⡀⠀⠀⠕⠤⠀⠡⡀⠈⠂⠀⢸⡒⠀⠀⠀⠁⠀⠀⠀⠀⣠⢠⡏⡔⠀⠀⠘⢄
⠀⠀⠀⠀⠀⢉⣲⡿⠖⢲⠃⠀⠁⠠⢃⠜⡂⢀⡀⠀⠰⡀⡀⠀⠑⢄⢀⣀⠃⡷⠛⠉⠁⡷⢁⡀⢠⠔⠚⢺⡏⠁⠀⠀⠂⠈⡆
⠀⠀⢀⠄⢩⣰⠁⠀⠠⢊⡀⢢⠼⠉⠉⠀⠁⠉⠉⠑⠒⠁⠁⠁⠂⠌⠤⠿⡤⠁⠀⠀⡠⠃⠈⠀⠞⢀⠀⡃⡐⢖⠀⠄⢠⡜⠁
⠀⡴⠉⣀⡾⠁⠀⢀⠁⢀⡔⠃⠘⠆⠀⠀⠐⣀⠀⠠⠀⠀⠀⠀⠀⢰⠖⢻⠀⠀⠀⠰⣃⣹⣧⠁⠀⠀⠆⠑⠡⠈⠲⡀⠸⡁
⠈⠇⠘⠁⠀⠀⠀⠐⡀⠞⢀⡤⠟⠃⠒⠈⠉⠑⡀⠀⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠐⠹⠟⠉⠙⠚⢑⠖⠀⠒⠂⢁⡐⠙⠲⢵⣄
⠀⠀⠀⠀⠀⠀⠀⢠⢏⠔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠤⠁⠸⠄⠀⠒⠸⢃⡜⠁⣀⠄⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣾⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣿⠿⢿⣿⡟⠿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠞⠁⠀⠀⠙⢿⣷⣦⢭⣑⣒⡾⠤⣬⣙⠮⡻⡿⣿⣿⣿⣿⣿⣿⣭⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠃⠀⠀⠀⠀⠀⠀⠈⠛⠳⣶⣤⣍⣙⡓⢮⣿⣾⣾⣾⣿⣿⣿⣿⣿⣿⣿⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠀⠀⠉⠛⠛⢿⣿⣿⣿⣿⡛⠛⢻⣿⣿⣿⣿⣟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⠀⠀⠀⣀⠤⠐⠛⣿⣿⣿⣷⡀⡀⣿⣿⣿⣿⣿⡣⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣤⠶⠆⠀⡤⠂⠀⠒⢿⡿⠛⠚⠁⠀⠀⠀⠀⠸⡏⠹⣿⢻⣶⣿⣿⣿⣿⣿⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡤⢤⣀⢤⣠⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣟⡧⠔⠏⢺⠉⢢⡀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢿⢹⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⡇⠀⠠⠘⠁⠀⠉⠒⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡱⠢⠤⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠘⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠢⠀⢀⣠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢟⢆⠀⠀⠀⠑⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⢻⣿⣿⣿⣿⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⡈⢧⡀⠀⠀⠀⠠⡀⠀⠀⠀⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀⢘⠻⣿⣿⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢥⠈⠳⣄⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢰⢷⡀⣸⠧⠞
⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠶⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠉⠉⠙⠳⣦⡀⠘⠾⣳⢏⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠶⣶⣦⣄⡀⠀⠀⠀⠀⣼⠋⠀⠀⠀⢠⣦⠀⠈⠻⣦⣤⡏⠻⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⠀⢰⡟⠀⠀⠀⠀⠈⠉⠀⠀⣰⠏⢸⢣⡴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣴⡶⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠉⠛⠛⠻⠶⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣠⣴⠾⠋⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢠⡞⡉⠁⠈⠙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢰⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⡶⢶⡄⢠⡟⠉⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⠶⠶⠛⣡⡶⠶⣦⠀⢠⡾⠋⠉⣼⡆⠀⢰⡏⠀⠀⠀⢿⣤⠾⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣟⣀⣴⠏⠀⣿⡇⠀⣠⣿⣦⣠⡼⢷⣤⣤⠶⠛⠛⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠈⠻⣭⣤⡴⠞⠋⠙⠛⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⣿⣿⡿⠻⠟⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣀⡀
⠀⠀⠀⠲⣀⣀⣀⠀⠀⠆⠀⠀⠒⡀⠄⠊⠂⠐⠀⢀⠀⣭⢁⣤⣒⣤⠑⡄
⠀⠀⠀⠀⠀⠁⠒⠶⠀⣀⣀⡄⢊⠀⡣⢬⣽⢿⡷⢖⣍⠤⢢⡶⠿⢎⢀⣇⡀⢄⣠⣤⣀⣀⣀⡄⣀⣀⠐⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣋⣀⠁⡘⣷⢧⣞⣿⣋⡭⠿⠶⣋⣴⣦⡾⣏⠩⢧⡌⢘⠟⠁⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡀⠀⠒⢲⣆⠄⣒⠩⢟⣚⣽⣥⣿⡿⠿⣀⠀⠀⠀⠀⠀⡉⠁⡠⠞⠈⢀⠖⠁
⠀⠐⠂⠰⠖⢢⣭⣑⡊⠽⣶⣮⣅⡸⢿⡟⠁⢡⢦⣽⣯⣍⡉⠤⡄⣿⣎⣡⠖⡰⡄
⠀⢀⠠⠀⠀⠒⢂⡪⠭⠍⠭⣍⡡⠤⢼⣤⣀⠄⠞⢸⡻⢽⣿⣗⡴⠋⡠⠏⢰⣷⣾⠆
⠀⣤⠈⢩⠟⢊⢡⣴⣤⡴⠊⠁⠒⠀⣀⠩⠥⢀⣀⣾⣿⣿⣏⠻⢍⡉⠀⣠⣫⡟⢻
⠀⠈⠁⠈⠈⠁⢌⠙⠋⠁⠀⢀⣠⢚⣥⠤⠀⠀⢹⣿⡿⠉⠉⢓⢶⢖⡚⠁⠙⠀⠸
⠀⠀⠀⠀⠀⠀⠘⠷⢶⣶⢶⠿⠟⠋⠀⢀⡠⣤⡿⠣⢃⠀⣚⠁⣀⠾⣯⣲⣁⠄⠁
⠀⠀⠀⠀⠀⢠⠟⠠⣼⠿⡀⠀⠀⢀⣴⣽⠞⣼⠕⠁⢸⡿⣁⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⠤⣆⣠⣤⠾⡟⣦⡴⢻⡿⠃⠜⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢰⡇⣠⣴⠾⠋⠁⠀⠞⡠⠊
⠀⠀⠀⠀⠈⠁⠁⠎⠀⠠⠄⠈⠒⠉

⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠂⠠⠀⠠⠈⠀⠀⠀⠂⠀⠀⠀⡀⠠⠀⠀⠄⣒⠀⠀⠐⠀⠀⠀⠀⢡⠀⠀⢠⠐⠀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⣁⣀⣈⡀⠀⠀⠀⠠⡀⠀⠀⡠⠀⠀⠀⠀⣈⣀⣈⡀⠀⠀⠀⠸⣭⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣉⣀⣈⠁
⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠁⠈⠀⠨⠁⠀⠀⠀⠐⠨⠈⠠⠄⠀⠀⠀⠙⠈⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉

⠀⣠⠦⠤⣤⠤⠵⣄⠀⠀⠊⢀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠀⡀⡀⠀⠠
⠀⠹⠐⠗⠉⠺⠢⠟⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠃⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠘⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣍⣉⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⠄⠀⠠⠀⠀⠈⣁⠀⣀⠀⢀⡀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⡀⠠⡀
⠀⠀⠀⠘⠀⠁⠀⠀⠸⣏⢀⠀⠀⠀⡀⢸⡷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⣀⠂⠀⠀⠀⠀⠘⠃⠀⠛⠀⡀⠀⠀⢠⠉⠛⠉⠆
⠀⠀⠀⠘⠲⢃⠀⠀⠀⠈⠂⠐⠀⠐⠠⠊⠀⠀⠀⢀⠉⢁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠙⠀⠀⠈⠃⡀⠉⠀⠜⠁

⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁⠚⠁⢄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⢀⡀⠠⠀⠀⠀⠀⡀⠀⢈⠀⡀
⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠀⠀⠀⠀⠃⠐⣣⠀⠀⠀⠀⠻⠏⠈⠿⠃⠀⠀⠀⠘⠃⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠁⠁⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠋⠛⠃⠄⠀⠀⠀⠄⠈⠉⡋⠐⠀⠀⠀⠄⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⢀⡀⠀⠀⠀⡀⠤⠤⠄⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠨⠖⠠⠤⠐⠇⠀⠀⠀⠕⠀⠄⠐⠊⠀⠀⠨⠀⠀⠀⠀⠢⠁
⠀⠀⠀⠆⠀⠰⠀⠀⠀⠀⠀⣿⠀⢸⡄⠀⠀⠈⠻⠿⠏⠻⠿⠟⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠠⠄⠀⢤⠀⠁⠀⠀⠀⠆⠀⠴
⠀⠀⠀⠃⠒⠒⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠂⠀⠀⠃⢀⠈⠀⠀⠀⡀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠗⢀⠀⠀⠀⠀⢌⠲⠶⠞⡡
⠀⠠⠀⠈⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠐⢆⡀⠉⢀⡠⣀⠒⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⡀⢄⠀⠀⠀⠀⢀⣴⣾⣿⣶⣄⠀⠀⠀⡐⠀⠁⠐⢀⠱
⠀⠀⣴⣀⠀⣀⣶⠈⠀⠀⠀⠐⠀⠀⡀⣠⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠨⠱⠦⡁⠀⠀⠀⠀⢀⠈⠉⢀⡀⠁⠀⠈⠘⠂⠀⡐⠀⠁
⠀⠀⠙⢭⣁⠩⠍⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠰⣶⣶⣷⣶⡶⠀⠀⠘⢄⠪⠈⢒⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠⠀⢰⠀⡀⠀⢀⠄⠀⠀⠀⠐⠀⣤⡄
⠀⠀⠀⠀⠀⡿⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⣾⡇⠀⠀⣠⡄⣧⣴⠇⠀⠀⠀⠀⡀⣠⡿⠿⠂⠀⠀⠀⢠⣶⠾⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠐⠀⠈⠉⠁⠝⡀⠀⢁⠉⠁⠀⠀⠀⠀⠐⠈⠀⠈⡆⠀⠀⠀⢐⣿⢿⣆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢒⣄⠀⠀⠀⠀⣀⣠⡀⠀⠀⠂⠈⢡⡖⠀⠀⠀⢀⠀⠰⣿⣆⠀⠀⠀⢀⠾⠀⠀⠛⣄
⠀⠀⠀⠀⠀⠈⠁⡴⠤⠀⠀⠉⠶⠄⡀⠈⠹⢿⠃⡀⠉⠀⠀⣾⡿⠊⠓⠰⠘⠠⠀⠈⢠⣀⡀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠃⢀⣀⡀⠡⠀⠀⠀⣈⠦⡄⢸⠂⠀⠀⠀⠀⠉⠁⢀⠠⣶⣾⠆⠀⠈⠀⠀⢙⣻⣦⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠸⠿⠃⠐⠀⠀⠀⡀⠸⠂⠈⡼⠟⠧⠀⠀⢀⡀⠁⠠⢈⡗⣠⡆⢀⡀⠐⠉⠙⠃⠀⣤⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣄⡀⠟⠀⠀⠀⡼⢃⡸⠃⡀⠘⠃⠀⢀⠀⠈⠁⣈⣡⡀⠉⠁⠀⠀⡠⠞⠋⢿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⠓⢀⠀⠀⠣⠀⠀⠈⠁⠀⠀⣀⣨⣀⠀⡤⠉⠀⠄⠀⠀⠀⠀⠀⠀⢀⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⡀⠈⠛⠋⠁⠠⠀⢀⡾⠃⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⢗⡿⣛⡒⠒⣤⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⡶⣶⣯⣿⣾⣟⢾⣧⣿⣄⣴⣿⢬⣙⠶⣺⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢶⣄⣈⡯⠽⣫⢦⣽⣿⣿⣭⣽⢻⢷⡶⣯⣭⣽⢵⣾⣝⠯⣿⣁⣠⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡓⠢⣶⡿⠿⣳⣈⣭⣭⣭⡍⣸⣧⣭⣭⣿⣯⣸⡻⢿⣿⣶⡄⢶⠁
⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⣷⢲⣿⣿⣷⡏⠙⠻⠉⡟⠉⠉⢿⠉⡹⠞⣿⣿⠷⣤⣭⡼⠿⠿⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣔⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⣿⡇⠒⣒⣲⣶⣦⠀⢰⣶⣶⣒⠒⢺⣿⡾⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡯⣾⡟⠛⢛⣻⡗⠿⠹⠀⣿⡿⢖⣟⡛⠛⢻⣷⢀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣿⣷⠀⠀⠉⠁⠀⠀⠀⣿⠀⠀⠉⠀⢀⣾⣿⣮⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣽⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⢀⣾⣿⣿⣿⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣵⣿⣿⢿⣿⡙⡁⢀⡠⠜⣛⣶⣟⣛⢦⣀⣼⢟⣿⣟⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⢻⡿⠁⢸⣿⣿⣼⢟⣴⠟⢛⣛⣛⡛⢳⣼⣿⣾⣿⣿⣶⣔⢮⡇
⠀⠀⠀⠀⠀⠀⠀⡾⣻⣾⣷⣾⣿⣿⣿⣿⣿⣿⠆⠀⢀⠈⢰⣿⣯⣿⣿⣿⢿⣟⢛⣻⣧
⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⣌⣿⣿⣿⡇⠘⢿⣅⢷⣜⣽⡸⢯⡸⣿⣿⣿⣿⣞⣿⣿⣿⡿⢳
⠀⠀⠀⠀⠀⠀⢠⢾⣟⣷⣿⠟⢻⡏⠁⠀⠈⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢷⣯⣭⢻⣿
⠀⠀⠀⠀⠀⠀⠳⣏⢿⣿⡽⣯⠾⢄⠀⠀⠀⠀⢻⡟⠉⠙⢻⡿⢡⠿⠟⣿⠿⣾⣿⣿⣿⣮⠂
⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠽⠿⣷⣤⣉⠒⠤⢄⣈⣙⣀⣀⣉⡁⠤⠖⢋⣤⣾⠿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢷⣶⣤⣤⣤⣤⣤⣴⣶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⣿
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢻⡀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⡇
⣿⣷⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⢷⡀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠘⢿⣿⢸⣿⡄⢠⣄⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢣⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⣠⣤⣬⣾⣿⡟⢸⣿⣷⡀⠀⠀⣀⠼⠤⣄⡀⠀⠀⢠⠞⣡⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠈⠛⠿⣿⠟⢧⡘⣿⣿⡇⢀⠌⠁⠀⠀⠀⠹⣦⣰⠋⡼⠁⠀⠀⠀⠀⠠⠴⢶⡍⠁
⠀⠀⢀⣤⣶⣶⣾⣿⡿⣤⣸⢤⠸⠗⠀⠀⠀⠘⠃⠸⠀⠀⠀⠀⣀⣠⡤⠴⢋
⠀⠀⠙⠿⣿⣿⡿⠋⠀⠀⠗⠚⠗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣀⡈⢳⡀⠀⠀⢀⣀⠴⠊⠁⠘⠀⡄⣧⣹⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡏⠉⠉⠀⠀⠀⠀⠀⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⣷⣶⡆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢳⡄⠀⣰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⢰⠂⢀⣾⣿⣿⡏⠀⢸⠇⢠⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠸⣆⢸⣿⣿⣿⠁⣠⠎⠀⢸⣿⣿⠁⢀⣠⣤⣀⠀⠀⣤⣀⠀⠀⣿⣿⠀⠀⣶⣤⠀⠀⠀⠀⠀⢀⣀⡀⠀⣿⣿⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠋⠀⠀⠀⢸⣿⣿⢰⠋⠀⣿⣿⡇⠈⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⣇⡀⠀⠀⣰⣿⡏⠀⠀⣿⣿⡆⠀⡞⢻⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⡀⠘⣿⣿⡇⠀⠀⢹⣿⡇⠀⣿⣿⡀⡠⢿⣿⡇⢀⠂⠙⢻⣿⣧⠀⣿⣿⣷⠀⣠⢿⣿⣧⢸⠁⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⢸⠇⠀⠻⠿⠇⠀⠀⠸⣿⣿⡰⠀⠉⠉⠀⠈⠛⠟⠁⠀⠀⠈⠻⠿⠞⠉⠻⠿⠚⠁⠀⠻⠿⠋⠀⠀⢿⣿⣧⢠
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡤⠴⠋⠹⡉⠉⢉⣭⡉⣉⢁⡉⣏⡁⠉⠉⢉⢉⡉⣩⣍⢉⡉⠉⣭⢉⡉⢉⢉⣭⠉⣩⣍⢉⣩⡉⠉⢉⠏⠈⠻⠿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢸⣀⡇⣿⣸⡇⡷⡇⠀⠀⢸⡿⡇⣇⣸⠈⡇⠀⣿⢸⡷⣾⢸⡖⠀⡓⣤⢈⡳⡅⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠤⠤⠭⠤⠬⠭⠤⠥⠭⠤⠤⠤⠤⠥⠬⠥⠤⠭⠤⠭⠤⠥⠭⠬⠭⠥⠭⠭⠬⠭⠥⠄⠄⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⣀⣤⡌⡃⢠⣀⣀⣶⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢚⡛⠺⠛⠀⠀⠀⠀⠘⠋⠀⠀⢀⣤⡙⣟⣠⣤⡄⢐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠴⠟⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠠⡿⢦⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡔⠑⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⢹⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠈⠉⠀⠀⠀⣀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢸⠉⠀⢀⡯⣩⡟⠓⠃⠀⠀⠀⠀⠀⢀⣄⠀⢀⣄⠀⠀⠀⢀⣄⠀⠀⠀⠉⢠⣅⠰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢚⠁⠀⠀⠀⠀⠈⠓⠒⣸⢣⡿⠃⠀⠀⠀⠀⠀⡀⠀⣾⡋⢠⣿⠋⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠉⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠀⢾⠇⠀⠀⠀⠀⢠⠀⢠⡏⣼⢿⡇⣴⣿⣟⣼⢟⣟⣼⣟⢁⣾⠥⣾⣼⣣⡟⡁⠀⠀⠀⠀⠀⠀⠀⣸⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⢀⣁⣘⣃⣈⣉⣋⣉⣈⣋⣩⡬⠹⣿⠀⠤⢬⣁⣉⡀⠀⠀⠀⠀⠀⠀⠀⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢠⣤⣀⠀⣠⡩⣭⣭⡅⣤⣤⢠⣥⠄⣤⠀⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢸⡇⣽⢀⣟⡇⠀⡇⠀⣿⡀⢸⡇⠀⡿⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⣤⠀⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⢻⢸⣿⡅⢸⡟⣿⠀⡇⠀⣿⠃⢸⡏⠀⣿⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠘⢃⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠘⠃⠛⠚⠃⠹⠀⠇⠀⠻⠶⠘⠃⠀⠙⠾⢃⡿⠶⠀⠀⠀⠀⠀⠀⠀⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡷⠀⢠⠞⢱⠀⣠⣤⣤⣀⠀⠀⠀⠀⡰⠋⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠾⣇⢠⠏⡰⠃⠘⠁⠀⠀⠈⠉⠒⠆⣴⠗⠚⠁⠀⠀⠀⠀⠀⠀⠀⢴⣾⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡇⠰⠇⠀⠀⠀⠀⠀⠀⣸⣅⣄⣽⣾⠋⢡⡖⣠⣏⣁⣞⢰⡾⢀⣶⡏⠀⢀⠀⠀⠀⠀⠀⠀⠰⠧⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢱⡴⠆⠀⠀⠀⠀⠀⢏⣀⠴⠃⠺⠴⠋⠉⠑⠟⠙⠟⠉⠉⠉⠉⠳⠴⠃⠀⠀⠀⠀⠀⠀⣰⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠍⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢋⣛⠔⠶⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠑⠠⣄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣄⣠⠤⠾⠁⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠏⠰⢢⣄⣀⣠⣦⢀⣘⢁⣀⢀⣤⠚⠙⠃⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⢠⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⡤⠤⢠⠀⠀⠀⠠⠀⢀⠀⣠⠀⢀⠀⡀⣀⠀⡀⠀⠀⠰⠀⠀⠤⡠⢀⡀⠄⠠⠀⢄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⣀⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣩⣴⣾⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⢀⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢱⣾⣿⠋⢫⡀⣿⣿⠃⡜⠀⠀⠀⠐⠑⠃⠀⠀⢀⣤⣉⣦⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⢠⠊⠐⠶⠋⠀⠑⠀⠸⠄⣻⡏⢰⠁⠀⠀⡇⣠⠔⢀⣀⠀⠈⢻⠻⣿⠀⠀⢠⠃⣀⠆⢡⡐⡄
⠀⠀⠀⠀⠀⡔⠃⢶⢀⠀⠀⠀⠈⠂⠂⣬⣿⠁⡇⠀⠀⠘⢛⡇⠀⠈⠁⠴⣶⢬⣧⣿⠀⠀⡄⠠⠃⢠⡘⠁⡰
⠀⠀⠀⠀⠾⢀⠀⠀⠁⠀⠀⠀⠀⠃⠲⣠⣿⡀⡇⠀⠀⢀⣻⣇⠀⢀⡒⠀⢠⣼⡿⠃⠀⡐⢠⡄⣸⠜⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠈⠀⠁⠐⢳⢿⣧⠘⣄⠀⠈⠩⠈⢦⣬⣤⣶⣿⡿⠂⢀⠄⣰⣼⠗⠁⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠘⢀⣉⠿⣭⣤⡤⠁⠀⠀⠀⠀⢙⣿⡁⣁⣠⣴⣡⣾⠏⠈⠖⢀⠀⠁⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢁⠈⠠⠀⠀⠊⠉⠀⠀⠀⠀⠀⠈⠲⠿⢻⢿⣿⣿⣿⡟⡀⠌⠀⠠⠀⠀⠠⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠲⠦⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⣿⡿⢇⠀⠀⠃⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠈⡚⠀⠀⠀⠠⡔⠀⠀⠁⠀⠀⢀⠀⠀⠀⠸⠤⢈⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⣤⠞⠀⢀⣣⠄⠀⠀⠀⠀⠀⠀⢰⣄⠀⠁⠀⢀⡀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⠔⡟⠁⠀⢠⣟⠋⠀⠀⠀⠀⠀⠀⢀⣿⢸⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢸⠃⠀⣠⣿⣭⠁⠀⠀⠀⢀⡠⠴⠋⢹⠐⠄⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣴⣿⣿⡀⣀⠀⠀⠴⠋⠀⠀⠀⢸⠀⠑⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢀⣀⣶⠞⠓⠒⠒⠲⠠⠄⢠⠇⠸⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⣾⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣽⣿⣿⣿⣣⣤⣴⣿⡄
⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠙⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢼⣿⣹⣿⣿⣿⣿⣿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣦⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⡷⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⣷⣦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⣰⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣮⣿⠂⠀⠀⠀⠀⠀⢀⣀⠀⠚⠋⠉⠐⢿⡿⢾⣿⣿⣿⣍⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⢿⣿⡌⢦⠀⠀⠀⠊⠩⢿⣤⠟⠀⠀⠀⠈⠁⣈⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⣈⣻⠟⠈⢃⢀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⢟⣿⠟⣻⡛⠞⠃⠈⠻⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡛⠋⠀⠀⣿⡟⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠤⢤⣴⣶⣿⡄
⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠀⡀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣾⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⣿⣦⡀⠀⠀⠀⠀⠀⢦⣀⣾⣿⣿⣿⣿⣿⣿⣿⠁⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡇⠀⠀⠀⠀⡤⠸⣿⣿⢿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⡿⣿⠄⠀⠀⠀⠀⠀⠀⠀⣾⠻⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⡼⠃⠀⠀⠀⣀⠀⠀⢀⣀⡏⠳⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⣷⡌⠿⡿⢿⡋⠉⢳⠀⠀⠀⠀⣸⡿⢿⠿⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⢀⠀⢀⣤⣆⠀⠘⠉⠙⠛⠛⠉⠒⠒⠸⢿⣦⣀⣄⠀⢀⠀⢻⡷⣄⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠄⠀⠀⠀⠛⠉⠀⠀⢀⡤⣤⡤⠄⢐⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣿⣬⢽⣿⣿⣿⡇⠀⠀⢀⠔
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠰⠿⠿⠿⠿⠿⠿⣿⠿⠿⠻⠿⢉⣿⣿⣿⠿⠟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢄⣀⣩⣷⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠼⣯⣍⣁⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡛⠋⠀⠀⢀⠄⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿⣷⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⣿⣿⣿⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣴⣾⡿⢿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⠄⠘⢁⡠⠀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⣼⣷⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⠟⣿⣿⣿⣿⣿⡁⢻⠟⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⡠⠄⠒⠠⠤⡀⠀⠀⠀⠀⣀⠤⠀⠂⠤⢄⠀⠀⠀⠀⢀⡠⠄⠒⠠⠤⡀⠀⠀⠀⠀⢀⠤⠄⠒⠠⢄⠀⠀⠀⠀⠀⡠⠤⠐⠂⠤⣀
⠀⠀⠀⢠⠋⠀⠀⠶⠆⠀⠈⢢⠀⠀⡜⠁⢤⠄⠀⢤⠄⠑⡄⠀⢠⠊⠀⠀⠤⠄⠀⠈⢢⠀⠀⡔⠁⠀⠀⡄⡄⠀⠑⡄⠀⢀⠎⠀⠀⢀⡀⠀⠈⢣
⠀⠀⠀⡆⠀⠀⠠⠀⠀⠊⠀⢈⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠌⠀⠀⠀⠀⠀⠀⠀⢈⠀⠰⠁⢠⡄⠀⠀⠀⣠⠄⢶⠀⢸⠀⠀⠀⠈⢁⠀⠀⠀⠄
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⡜⠀⠀⢧⠀⠀⠀⠀⠀⠀⣠⠃⠀⠸⡄⠀⠀⠀⠀⠀⢀⡜⠀⠀⢣⡀⠙⠟⠛⠿⠋⢠⠇⠀⠘⣄⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠈⠒⠤⠤⠤⠔⠋⠀⠀⠀⠀⠑⠢⠤⠤⠤⠚⠁⠀⠀⠀⠈⠒⠤⠤⠤⠔⠋⠀⠀⠀⠀⠙⠢⠤⠤⠤⠒⠁⠀⠀⠀⠈⠓⠤⠤⠤⠔⠊

⠀⠀⠀⠀⢀⠔⠊⠉⠁⠒⢄⠀⠀⠀⠀⡠⠒⠈⠉⠐⠢⡀⠀⠀⠀⢀⠔⠂⠉⠁⠒⢄⠀⠀⠀⠀⡠⠒⠈⠉⠐⠢⡀⠀⠀⠀⢀⠔⠂⠉⠁⠒⢄
⠀⠀⠀⠰⠁⠀⠰⠀⠰⠀⠀⢳⠀⠀⡎⠀⠠⡆⠀⡤⠀⠘⡆⠀⢰⠁⠀⣀⣒⣂⣀⠀⢳⠀⠀⡎⠀⠀⠀⠶⠀⠀⠈⠆⠀⢰⠃⠀⠀⠀⢠⠀⠀⢱
⠀⠀⠀⢇⠀⢿⣤⣤⣤⣼⠃⢠⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢂⠐⣿⣿⣿⣿⣿⡗⢠⠀⠐⡀⢰⡤⢠⣤⡤⢤⠆⠹⠀⢸⠀⠀⢀⠀⠀⢀⠀⢀⠂
⠀⠀⠀⠘⢆⠀⠙⠒⠚⠁⢀⠎⠀⠀⠳⡀⠀⠀⠀⠀⢀⡰⠁⠀⠘⢆⠈⠙⠛⠛⠉⣀⠎⠀⠀⠱⣀⠀⠀⠀⠀⠀⡰⠃⠀⠈⢆⡀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠉⠒⠒⠒⠊⠁⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠀⠀⠉⠒⠂⠒⠊⠁⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠀⠀⠀⠉⠒⠂⠒⠊⠁
⠀⠀⠀⠀⠀⣀⠤⠤⠤⣄⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢤⣀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠑⢄⠀⠀⡰⠋⠀⠀⠀⠀⠈⠣⡀⠀⢀⠞⠁⠀⠀⠀⠀⠙⢆⠀⠀⡰⠋⠀⠀⠀⠀⠈⠳⡀⠀⢀⠞⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⡎⠀⠀⠀⣠⡀⠀⠀⠘⠀⢠⠁⠀⢀⣬⣽⣄⡀⠀⡇⠀⠌⠀⠀⠀⠚⠃⠀⠀⠘⠀⢠⠁⠀⠀⠰⠶⠀⠀⠀⣱⠀⢸⠀⠀⠀⠚⠃⠀⠀⠈⠄
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢇⠀⠿⠋⠉⠛⠷⢀⠇⠀⠰⡀⠸⠀⠀⠀⠀⠇⡸⠀⠀⢇⠀⠀⠀⠀⠀⠀⢀⠎⠀⠸⡀⠀⠀⠘⠃⠀⠀⡰
⠀⠀⠀⠀⠙⠦⢄⣁⣀⠤⠊⠀⠀⠀⠈⠓⠤⣀⣀⡠⠔⠋⠀⠀⠀⠙⠢⢄⣀⣀⠤⠊⠀⠀⠀⠀⠓⠤⣀⣀⡠⠔⠋⠀⠀⠀⠈⠢⢄⣀⣀⠤⠚⠁

⠀⠀⠀⠀⢀⠤⠒⠒⠒⠦⣀⠀⠀⠀⠀⣠⠔⠒⠒⠒⢄⡀⠀⠀⠀⢀⡤⠒⠒⠒⠢⣀⠀⠀⠀⠀⣀⠴⠒⠒⠒⢤⡀⠀⠀⠀⢀⡠⠒⠒⠒⠢⢄
⠀⠀⠀⢰⠃⠀⠀⣤⡄⠀⠈⢢⠀⠀⡜⠁⠀⠀⠀⠀⠀⠙⡄⠀⢠⠃⠀⠀⡤⠀⠀⠈⢣⠀⠀⡜⢁⡄⠀⠶⠆⠀⣙⡄⠀⢠⠋⠀⠀⠶⠆⠀⠀⢳
⠀⠀⠀⡇⠀⣄⠀⠀⠀⢀⡆⢈⠀⠰⡀⠀⠀⠐⠛⠀⠀⠀⡇⠀⠂⢀⣤⣶⣾⣿⣿⣶⢈⠀⠐⡀⠠⣿⣿⣿⣿⣶⡄⡼⠀⢸⠀⠀⠀⠀⠐⠀⠀⠀⠆
⠀⠀⠀⠘⣄⠈⠛⠉⠹⠋⢀⠜⠀⠀⢣⡀⠀⠀⠀⠀⠀⡠⠃⠀⠘⣄⠻⠟⠛⠻⠟⢁⠜⠀⠀⠣⡀⠉⠉⠙⠿⠛⡠⠃⠀⠘⢄⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠈⠑⠢⠤⠤⠒⠁⠀⠀⠀⠀⠉⠒⠤⠤⠔⠊⠀⠀⠀⠀⠈⠑⠢⠤⠤⠒⠁⠀⠀⠀⠀⠈⠒⠤⠤⠔⠊⠁⠀⠀⠀⠀⠑⠢⠤⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣇⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠐⣄⠳⣤⢡⢠⢃⠎⣰⢀⠞⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢙⡫⡢⡑⠿⣦⣰⢸⣼⣃⢔⡨⠪⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡒⠶⣬⣶⣭⠛⠩⡬⠙⣵⣶⠶⠖⠫⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠚⠡⠿⣭⣿⣟⠃⠀⣻⣾⣟⡪⠴⠦⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠷⠾⢿⣿⣧⣸⢗⣨⣿⢟⠻⣶⣬⡁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣺⡢⣿⣿⣿⢋⠹⢿⣿⡂⡵⡐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡩⣼⣿⡿⠡⠘⡜⡸⣿⣯⢪⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⡏⠀⠆⢸⠃⠇⢹⣿⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣠⣤⣄⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣿⣿⣿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⣸⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢋⡜⡸⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡱⢡⠊⡜⢠⢎⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠀⣹⣼⡇⢠⣧⢣⠼⡉⠁⣐⣂⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⣀⣠⣶⢺⣣⣝⣮⣿⣇⣰⣾⣍⣛⣻⣿⣟⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⣶⣾⣿⣿⣿⣿⣆⠈⣬⡄⣿⡿⠛⠏⡟⣽⣿⣿⣿⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠛⠿⠿⠻⠿⠿⣿⡏⢈⣘⣳⣼⣷⣎⢀⡃⢹⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣶⣶⣶⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠚⠹⠿⠿⠿⡿⣿⣿⡿⢿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢋⣡⣤⣤⣉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⣤⡀⠀⠀⠀⠀⠀⠀⠘⠁⣿⠟⠙⠛⢿⣧⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⠇⣧⠀⠀⠀⠀⠀⠀⠀⣼⣿⠙⠃⣿⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⢰⣏⡀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠙⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⣿⣷⡀⠀⠀⣀⣀⣾⣿⣿⢿⠿⣿⢻⣿⣿⣄⣀⠀⠀⠀⢀⣶⣿⣁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠛⣿⣿⣶⡟⡙⣿⠉⠿⢻⣷⣦⣵⣾⡇⣿⣿⢻⡿⢷⣤⣾⡏⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⠀⣿⢻⣼⢸⡇⠙⠀⠀⠸⣿⢹⠟⣿⠁⣿⠉⣼⣱⣿⢣⣿⢀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡏⠀⣿⣸⣿⣿⡇⡄⢰⡄⡀⣿⣤⠞⣹⠀⣿⢀⣿⣿⣿⣼⣿⢸⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡜⣿⠇⢰⣿⣷⣿⣿⣿⣿⡜⣷⣿⢟⣡⣾⣿⠀⣿⢸⣿⣿⣏⢿⢿⣼⡟⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⣜⣾⣿⠿⠛⢩⣿⣿⣿⠿⢟⣫⣿⣿⣿⢸⣿⢸⣿⣿⣿⣦⣾⣿⡇⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠀⢸⠷⠶⠶⠚⣫⣿⣿⣿⡏⢸⣿⢸⣿⡿⣿⠉⠙⢿⣿⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠶⠶⠿⠟⣩⣿⣿⡇⢸⣿⣸⣿⡇⣿⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣷⣶⠶⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣶⣴⢾⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠀⢴⣾⣿⣿⢿⣿⡇⢸⣿⣿⢻⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⢿⣿⠈⣿⡇⢸⣿⣿⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠘⠛⠘⠛⠀⠛⠃⠘⠛⠛⠘⠃⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⡋⣳⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠇⠀⠉⠁⠀⠿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⢲⠀⠀⠀⠀⠀⢰⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⡼⠁⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠹⡄
⠀⣸⠁⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠹⡀
⢠⠇⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢧
⣼⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢸⡆
⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢸⡇
⣇⠀⠀⠀⠀⠀⢻⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⠃⠀⠀⠀⠀⢀
⢹⠀⠀⠀⠀⠀⠘⡆⣠⠞⠁⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀⡞⠀⠀⠀⠀⠀⢸
⠈⣇⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⢀⡴⠋⢸⠀⠀⠀⠀⠀⢸⠈⠳⣄⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢀⡏
⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⢷⣾⣯⡳⣄⠰⣤⡐⠒⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⡿⢉⡴⣆⣾⣯⠛⠉⠂⠁⠛⣿⢷⡜⢿⡶⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⢏⣀⣾⣷⣿⣿⢯⣳⡄⠀⠀⠀⠸⡄⠳⠘⣇⠙⠁⠢
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⣡⣼⣿⣿⣿⣿⣿⢿⡻⣗⢄⠐⠀⠀⠑⢄⠀⠹⣆⡆
⠀⠀⠀⠀⣠⢖⣿⣿⣿⣿⠏⠈⠉⢽⣿⣿⣿⣿⢍⠣⠙⢮⡀⠀⢀⠀⠀⡀⢀⠀⠹⣾⣦⣷
⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠉⠙⢿⣷⣷⣄⢦⣷⣄⢠⣵⣄⢴⡌⢳⠀⠈⣿⣿⠂
⠀⠀⠀⠸⣿⢫⣿⣿⣿⣧⠀⣴⠋⠁⠀⠀⠀⠀⢻⣿⣻⣿⣿⣿⣿⣿⣿⢸⣦⡌⢣⣷⣙⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡸⠇⢄⣠⠆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⡏⣿⡇⢨⣿⣿⣟⠗
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⠈⠉⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⡟⣋⣠⣿⣃⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣼⢟⣾⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣴⣖⠀⠀⠀⠀⠀⠀⠀⠋⠉⠈⡽⠁⠀⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⠶⠆⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢸⢿⢻⠹⣿⣿⡿⣤⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠈⠃⠈⠀⠀⠉⠛⠈⠄⣀⠤⠔⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⣠⠤⠄⢠⡶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⢻⡄⠀⠐⠇⠀⢀⣼⠃⠀⠀⣀⡀⠀⢀⣾⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠁⠀⣴⡖⠚⠛⠁⢠⣶⡆⠀⡟⠀⠸⠃⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠠⡀⠀⣰⡿⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⠈⠀⠈⠓⠋⠀⢀⡈⠙⡧⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⡳⠾⠦⠤⠔⢀⡀⢠⠀⢘⠓⠺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣶⢠⠀⠀⠇⢸⠀⡞⠀⠀⢸⢰⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⡇⠀⠀⠀⢸⠀⠀⣷⠞⠀⠀⠈⢀⡐⠀⠀⠀⠁⢸⡇⠀⠀⠀⣤⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⠃⠀⣠⣤⢸⡀⢀⠿⠶⠖⠋⠉⠀⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢡⣠⡎⠁⠀⠇⠀⠀⠀⣶⡀⠉⠁⠀⢀⣴⠆⠀⢸⠁⠿⣇⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⠈⠛⠀⠀⠀⠀⣠⡄⠀⠀⠁⠀⠇⠉⡀⠀⠀⠀⠈⠀⠀⠸⣶⠃⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⣼⡏⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⣿⡄⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠾⠋⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠶⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠉⠑⠲⠶⢦⣴⣶⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⣠⣶⠿⠿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁




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

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⣀⣀⠀⣀⠀⠀⢀⠀⣀⣀⠀⠀⠀⠀⢀⡀
⠀⠄⠀⠁⠀⢠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠲⣊⡖⠂⠀⣯⡽⠂⣿⠍⢰⣿⣧⢰⡟⠀⣿⠍⠀⠀⢿⣦⣿⠃⠠⠄⠀⠉⠀⠙⠀⢴
⠀⠭⠀⠀⠀⠃⠀⠀⠀⠀⠀⣤⣖⠗⠀⢀⣀⣤⣈⠀⠁⠀⠀⠛⠀⠀⠛⠛⠘⠛⠓⠈⠛⠃⠛⠛⠀⡂⠀⠻⠏⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠡⡀⠀⠀⠀⠀⠐⠀⡶⠉⠀⠀⡴⢋⠀⡇⣀⡙⡄⠈⠁⠀⠀⠀⣶⣀⠀⠢⠀⠀⠀⠦⠀⠀⢰⡷⠀⠀⠀⠰⠆⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠈⠒⡄⠀⠀⠀⠀⠀⢀⡀⠀⣇⢀⡴⣷⢤⣀⡟⠀⠀⢀⠤⢼⠉⠸⠇⠀⠀⠀⢠⡄⠀⠀⠀⠃⠀⠀⢠⡠⠀⠠
⠀⢤⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠯⣄⣿⡠⠞⠁⠀⠀⠀⣀⣀⣀⡀⠀⠠⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⢶⠄⠀⠀⠀⡀⠀⢾⣿⣷⣴⣿⣿⠀⠀⠀⠀⡄⠀⣠⣾⣿⣿⣿⣿⣿⣷⣤⠀⠀⢶⡶⠂⠀⠀⠀⠀⠀⢶⣶⠆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠂⠀⠀⠑⠈⠻⣿⣿⠟⠁⠐⠁⠀⠀⠀⣼⣿⣿⠛⢟⠛⡋⡏⢿⣻⣷⠀⠀⠀⠀⠀⠼⠧⠀⠀⠀⠁⠀⠤
⠀⠐⠢⠤⠀⠀⠀⠃⡀⠀⠀⠈⠃⠀⠀⠀⠘⠀⣴⠀⣿⣿⣿⣶⣷⣿⣿⣶⣾⣿⣿⠀⠙⠀⠛⠙⠋⠓⠋⠛⠠⠆⠀⠀⠀⠀⠀⠂⠀⠶
⠀⠀⣀⡤⢶⡶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⠂⠀⢿⣿⣿⢲⣔⣃⣛⣃⣿⣿⣿⠁⣴⢦⡀⠐⠃⣀⣀⠀⠀⠠⠀⠀⠀⠀⠠
⠀⡼⠁⠀⢸⡇⠀⠈⢧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣽⣿⣿⣻⣿⣿⡿⠃⠀⣧⠀⠙⣶⠛⠉⣹⠂⠀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⡇⠀⣠⢼⣧⣀⠀⢸⠀⠃⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⣀⠈⠙⠛⠛⠛⠋⠉⠀⡀⠀⠘⣶⢺⡿⣦⡴⠋⢀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠹⣏⠁⢸⠇⠈⣳⠏⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣶⣿⠏⠈⣷⣄⠀⠈⠓⠀⠈⠻⠟⠉⠀⠀⠈⠀⠀⠀⠀⠀⣀⣀⣀⠀⡠
⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⡀⠀⢿⣟⠙⠀⣀⣿⣭⡆⢀⠀⠀⠀⠂⡀⠀⠀⠂⠀⢀⣀⣠⣬⠿⣿⣯⣍⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈⢁⠀⠈⣷⡀⢘⢻⡷⠀⢿⣿⠟⢁⣜⠁⠀⠐⡿⢇⠀⠀⠀⠀⠀⠛⠽⢫⠟⠛⠱⠋⠀⠆
⠀⠀⢀⠄⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢃⡄⠉⠈⠠⠜⠉⢠⣘⠂⠀⠀⠀⠀⠀⢠⣤⡄⣤⠀⢠⠀⢠⡄⣤⡀⢠⡤⡀
⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠘⠃⠀⠀⠘⠻⠉⠛⠀⠸⠟⠿⠁⠿⠧⠸⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣄⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠘⠇⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⡀⠀⠈⠢⣄⣀⣀⣠⣤⣤⠶⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⡠⠀⣰⣿⡃⠀⠀⢹
⠀⠀⠴⢦⣦⡤⣄⠀⠀⠀⢀⣿⡏⠀⠠⠞⠀⠀⣀⣴⢤⡀⠀⣠⡀⠀⢀⡀⠀⣀⠀⣠⣤⡀⠈⣽⣿⠉⠀⠀⢸⠀⠀⠀⢀⣤⡖⠂
⠀⠀⠀⠀⢳⠙⢦⠱⢄⠀⠘⣿⣇⠀⠀⠀⠀⣸⣿⠿⣄⣷⣰⣿⠃⠀⣿⡇⠀⣿⣿⠉⣿⡇⠀⣿⡇⠀⠀⠀⡟⠀⠀⡔⣽⠹
⠀⠀⠀⠀⠀⠳⡈⠳⣸⠂⠀⢻⣿⣆⠀⢀⣠⢿⣿⡀⢀⡟⠸⣿⣀⡼⣿⡇⠀⣿⡇⠀⣿⣇⠀⣿⡇⠀⢀⡞⠀⠀⢰⠀⡍⢰
⠀⠀⠀⠀⠀⠀⠈⠁⠙⠂⠀⠀⠉⠛⠛⠉⢡⡄⠙⠛⠋⠀⠀⠉⠉⠁⠙⠿⠞⠛⠃⠀⠈⠛⠛⠙⠷⢖⡋⠀⠀⠀⠸⣴⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣄⡻⠀⠐⢦⣰⢁⡔⠛⠓⠆⣰⠀⠀⡆⣏⡩⠽⠂⠀⠰⠟⠁⠀⠀⠀⠀⠊
⠀⠀⣾⣷⣰⠛⢿⣦⠀⣠⣾⠛⢳⠀⢀⣀⣭⠭⠀⠀⢠⠇⠘⢧⣀⣠⠆⣧⣀⡼⡇⡏⠒⢤⣀⠀⣘⡛⠛
⠀⠀⣿⣿⠃⢠⣿⠟⢠⣿⠇⢀⡞⠀⠈⠉⠀⠀⣀⣀⠀⠀⠀⣀⣀⠀⠀⢴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⣀⣀
⠀⠀⣿⡟⠸⠿⣷⣄⢸⣿⠛⠁⠀⣤⣶⢦⡀⣾⣿⠛⣷⠀⣾⡟⠛⣷⠀⣶⡆⠀⣶⣄⡴⣶⡆⠀⣠⣶⠖⢶⣶⠀⣾⡿⠛⣷⠀⢀
⠀⠀⣿⡇⣀⣀⣹⣿⣼⣿⠀⠀⢸⣿⠃⣸⠁⣹⢿⣄⢁⣠⠟⢿⣌⢁⣠⣿⡇⠀⣿⡿⠀⣿⡇⠀⣿⡇⠀⣼⣿⠀⢹⢿⣄⢁⣠⠞
⠀⠀⢿⠿⣅⣠⣿⠏⠈⣿⡇⠀⣸⢿⣏⣁⡴⠃⡼⢻⣧⠀⢀⡼⢿⣇⠀⢿⣇⣠⣿⡇⠀⢻⣷⣠⠿⣷⡼⢹⣿⠀⡜⡼⢻⣧
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠛⠟⠋⠀⠈⠛⠟⠃⠀⠀⠉⠁⠀⢀⡀⠀⠈⠁⠀⣠⠤⢾⣿⠋⠀⠛⠛⠋
⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠒⠒⠒⠒⠒⠲⠤⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠯⠛⠀⠀⠀⣸⠁⢀⣿⡏⠀⠀⠀⠀⠀⡰⡿
⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⡀⠀⠀⢀⡄⠀⣀⣀⠀⠀⠀⠈⠉⠙⠒⠲⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠘⠶⠾⠋⠀⠀⠀⠀⣠⠞⠀⡇
⠀⠠⣞⠁⠀⢶⠚⠉⠁⠀⠹⡄⢀⡾⠀⢀⡏⠉⠉⠁⠀⣶⠒⢒⣦⠀⢀⠀⠀⣀⠉⠉⠙⠓⠒⠒⠒⠶⠦⠤⠤⠶⠒⠋⠀⠀⠀⡇
⠀⠀⠈⠑⢦⠸⡶⠒⠉⣀⠀⠹⡾⠁⠀⢸⡋⠉⠉⠀⢠⡿⢍⠉⠁⠀⠈⢧⡴⠃⠀⡾⠒⠢⣄⠀⠀⡶⣆⠀⠀⢤⡀⢰⠇⠀⠀⡇
⠀⠀⠀⢀⡞⠀⠓⠚⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠁⠀⠑⢦⠀⢀⡾⠁⠀⣸⠁⠀⢀⡸⠂⣸⠥⠿⡖⠀⠀⠙⡿⠀⠀⠀⡇
⠀⠀⠀⡜⠀⢀⣠⠤⠒⠛⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠦⠤⢤⣄⣀⣀⡀⠀⠀⠀⠉⠋⠉⠉⠀⠐⠃⠀⠀⠹⠀⠀⠘⠃⠀⠀⢠⠇
⠀⠀⣸⣡⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠲⠦⠤⢤⣀⣀⣀⣀⠀⠀⠀⣀⣀⡠⠖⠁
⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⣤⣀
⠀⠀⠀⣠⡴⠋⠉⠀⠀⠀⠀⠈⠉⠳⣤⡀⠀⠀⠀⠀⠀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⢠⠴⠋⠁⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢀⡞⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢳⡄⠀⠀⣴⠋⠀⠰⣄⠀⠀⠀⠀⢀⡴⠂⠀⠙⣆
⠀⣸⠃⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢻⡀⠀⣾⠀⠀⢈⣷⡦⣤⣤⣤⣤⢴⣾⡁⠀⠀⢻⠀⢰⠃⠀⠀⣠⣌⠓⠀⠀⠀⠛⣠⡄⠀⠀⠸⡆
⠀⣿⠀⣤⠀⠛⠃⠀⠀⠀⠀⠀⠛⠃⢠⡄⢸⡇⠀⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⢸⠀⢸⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⡇
⠀⢻⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣸⠃⠀⢿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⡾⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠈⢧⡀⠹⣄⡀⠀⠀⠀⠀⠀⣠⡾⠁⣠⠏⠀⠀⠈⢧⡀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⢀⡼⠃⠀⠀⠳⡄⠀⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⣠⠏
⠀⠀⠀⠳⢦⣺⠿⠓⠲⠶⠒⠛⠁⣠⠾⠁⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⢀⣠⠞⠃
⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠛⠉
⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡤⠞⢯⣀⠀⠀⠀⢀⣀⡽⠳⣄⠀⠀⠀⠀⠀⢀⡴⠞⠉⠁⠀⠀⠀⠈⠙⠳⢤⡀
⠀⢀⡞⠁⠀⣤⣤⣤⠀⠀⢠⣤⣤⡄⠀⠹⣆⠀⠀⢠⡞⠁⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⠈⢳⡀⠀⠀⣠⠏⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠻⡄
⠀⣼⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣤⡄⠀⠀⢹⡄⠀⡾⠀⠀⢀⣤⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢷⠀⢰⠇⠀⠀⢈⡉⠁⠀⠀⠀⠈⣉⡁⠀⠀⢹⡄
⠀⣿⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸⡇⠀⡇⠀⠀⠈⠙⠁⠀⠀⠀⠀⠈⠋⠁⠀⠀⢸⠀⢸⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⡇
⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠳⡄⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⢀⡴⠃⠀⠉⠙⢧⡀⠀⠈⠳⠦⣤⣤⠴⠚⠁⠀⢀⡼⠋⠉⠀⢳⡀⠀⠀⠴⠞⠋⠉⠉⠳⠆⠀⠀⢀⡞
⠀⠀⠀⠙⠲⣄⣀⠀⠀⠀⠀⢀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠤⠶⠖⠚⠉
⠀⠀⠀⠀⢀⡤⠶⠚⠛⠛⠓⠲⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠦⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⢤⣄⣀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣄⣀⡈⠳⣄⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⢠⡏⠀⡤⠞⠃⠀⠀⠀⠀⠀⠈⠉⠀⠘⣧⠀⠀⢠⡞⠁⣀⡴⠆⠀⠀⠀⠈⠉⠛⠂⠀⠳⡄⠀⠀⣰⠋⢀⣠⠤⠀⠀⠀⠀⠒⠚⠛⠀⠙⣆
⠀⣾⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⣶⡆⠀⠀⢸⡆⠀⣾⠀⠈⢁⣤⡄⠀⠀⠀⠀⢀⣤⡀⠀⠀⢿⠀⢰⠃⠀⠉⣡⣄⠀⠀⠀⠀⠀⣠⡄⠀⠀⢸⡆
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⣶⠀⠉⠁⠀⠀⠀⠀⠈⠉⠁⢰⠀⢸⠀⢸⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⡇
⠀⠸⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⣀⠀⠀⠀⡼⠀⠀⢿⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡾⠀⢸⡄⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠙⣆⠀⠀⠀⠉⠛⠓⠚⠋⠁⠀⢀⡞⠁⠀⠀⠈⢧⡀⠙⢦⣀⠀⠀⠀⠀⣀⡴⠏⢀⡼⠃⠀⠀⢳⡄⠀⠀⠀⠈⠉⠙⠓⠶⠀⠀⠀⣠⠏
⠀⠀⠀⠈⠛⠦⣄⣀⣀⣀⣀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠉⠛⠛⠋⠉⣀⡴⠋⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠶⠒⠚⠋⠁