Kopi Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠐⠂⠀⠀⠐⠂⠀⠀⢀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢂⣀⠘⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢏⣈⠹⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣬⡑⠤⢵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠒⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠙⠳⠶⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡠⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣤⠔⠊⠀⠀⢀⠇
⠀⠀⠀⠀⣄⠀⠀⠀⢀⡀⠀⠈⢄⠀⡀⠀⠀⠀⠀⠈⠳⢦⣀⣀⠀⠀⢀⠄⣪⡿⠋⠁⠀⠀⠀⠀⢄⠜
⠀⠀⠀⠀⠘⢄⠒⠦⠄⣈⠑⣮⠣⠑⠬⡂⣤⢀⢀⠀⠀⠀⠀⣉⠽⠛⣡⠞⠁⠀⠀⠀⠀⠠⣀⠕⠁
⠀⠀⠀⠀⠀⠀⠑⠢⠤⣀⡳⣾⣯⣒⣆⣹⣖⠳⣞⠓⣖⡒⠺⠗⠲⠊⠤⢀⣒⠀⠤⠕⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⡼⢿⣻⣷⣚⣖⣽⣨⢀⡬⣁⣾⢯⣏⢉⠉
⠀⠀⠀⠀⠀⠒⢀⠚⢹⡃⢻⣥⡽⣦⡘⡷⠾⢿⣌⡱⢹⣷⣧⣿⣿⡷⠀⢄⡆
⠀⠀⠀⠀⠀⠀⠘⠁⠠⣅⡥⠥⠚⠲⡏⠝⣉⡯⣿⣭⠻⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠞⠀⠀⠉⠁⠈⠉⠁⠈⠻⠖


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢤⠄⣀⣐⡲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⣠⣌⣀⡛⣀⣂⡀⠀⠠⠄⣠⡄⠉⠻⠶⣿⣲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⠁⠠⠰⢆⣤⡙⣿⣹⣦⡀
⠀⠀⠀⠀⠀⢠⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡈⠋⠑⢘⣿⣿⣧⢀⡀
⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣨⣛⣿⣿⠏⣸⡄⣠⡶⠛⠛⠳⣆⠀⡀
⠀⠀⠀⠀⠀⠸⣷⣤⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⢀⣿⣿⠋⠀⠀⠀⠀⣿⣧⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣍⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠄⣼⡏
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⣙⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣽⡦⡁⠠⡍⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢜⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡬⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⠘⢇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣧⣌⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣾⣿⡀⠀⢦⣄⣀⡀⠀⠀⠀⠀⠀⡀⢤⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠾⠶⠿⣿⠷⣼⠤⠤⠶⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣣⣋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠊⠉⠉⠁⢁⠃⠀⠈⠉⠉⠙⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⢀⣀⣠⣤⣤⣼⣤⣤⣤⣤⣤⣀⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠞⢹⣤⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⣀⡤⠒⠹⡇⠀⠙⠻⢯⣟⣻⡿⠿⣿⣿⣿⣿⣿⡿⠿⣿⣻⡯⠷⠋⠁⠀⡟⠑⠢⣄⡀⢨⣿
⠀⠀⢀⡠⠞⠁⠀⠀⠀⣷⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⣹⢏⡟
⠀⣠⠎⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣠⡤⠖⣋⡴⠋⠈⢦⡀
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣩⡤⠖⠋⠁⠀⠀⠀⠀⢷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠶⢶⠶⠶⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠶⠶⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢀⣠⠴⠃⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⣠⠞⠋⣡⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⡇⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠈⢳⡄⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢀⣴⠇⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡞⠁⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⣫⠖⣀⣠⣤⣴⣶⣶⣶⣞⣛⣛⣛⣛⣒⣒⠶⠦⣤⣄⡀⠲⣌⠙⣆⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⡾⠃⣟⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠁⢀⣠⡟⠙⢳⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣾⣿⣿⠁⠀⣸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣹⠃⣀⡼⠃⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠚⠉⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠙⣧⠶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠥⣤⣴⠖⠋
⠀⠀⠀⠀⠀⢠⡞⠁⠀⢀⡾⠁⡼⣡⣿⠧⠴⠤⣤⠀⠀⠀⠀⠀⢀⣀⡴⠞⠁⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⡇⡼⢿⣷⣖⣒⣛⣭⣿⠧⠴⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠦⣌⣻⠷⠾⢯⣍⣉⣡⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠦⠤⠤⢤⣤⣤⣤⡤⠤⠤⠴⠶⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⡄⣗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠸⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⢰⠇⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢃⡞⠀⢸⢻⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣼⠁⢀⠃⣼⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⠀⣰⠃⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠛⢛⣉⠹⡄⠉⣻⡀⣀⣀⣀⠒⠒⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣾⣛⠉⠉⠁⠀⠈⠁⠀⠑⠀⠈⠉⠉⠉⢒⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠛⠛⠛⠒⠶⠶⠶⠶⠶⠖⠒⠒⠛⠋⠉⢁⣸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣤⠶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⠔⣫⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⠊
⠀⠀⠀⢀⣠⣤⡴⠶⠞⠛⠛⠋⠉⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣠⡾⠋⠉⠀⠀⣀⣠⣤⣤⡶⠶⠖⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦
⠀⢿⡄⠀⠀⢰⡾⠋⠉⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⠀⠀⠀⢠⠃
⠀⠈⠻⣦⣀⠘⠷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠀⣀⡴⠋
⠀⠀⠀⠀⠉⠛⠷⢶⣬⣭⣙⣛⡛⠛⠷⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠴⠶⠶⠖⠒⠒⠛⢉⣉⣀⡠⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣯⣽⣿⣽⣭⣿⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⢒⡚⠛⠛⠉⠉⠉⠉⠛⠛⠓⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣤⣼⣿⣿⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀⠀⠉⣳⣀⡠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⠫⠀⢀⣀⣀⡀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠈⢣⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡾⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⡶⠞⣋⠔⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠘⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⡀⣤⡒⠊⠁
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠈⠢⣝⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠂⢄⠈⠓⢄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢖⡒⠠⠤⠤⠤⠤⢤⣰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⣁⣐⠂⠀⠀⠤⠀⠀⠀⠀⠀⠠⠄⠀⠀⠒⣀⣀⠠⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠳⢤⡀⠀⠀⢸⣃⢀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⣦⣶⡄⠀⠘⢿⠢⣧⡄⠀⠸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⢱⡿⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠀⠀⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⣶⡦⣬⠐⠄
⠀⠀⠀⠀⠀⠀⠀⣤⣰⡀⠀⠀⠀⠀⠀⡰⣿⣿⣿⣿⣟⠉⠁⠀⠀⣈⣉⣁⣬⣤⣴⣤⣤⡈⠙⢛⣛⣿⣿⢇⡤
⠀⠀⠀⠀⠀⠀⠘⢿⣆⠹⣆⠀⠀⠀⠀⠑⠉⡙⠛⠛⠫⠾⠿⢄⣤⣈⣉⢙⣿⣽⣄⡙⠿⠿⠖⠛⠛⣋⣤⣾⣷⡾⠧⠯⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⢧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠶⢦⣶⣶⣿⣿⣿⢿⡿⠓⠒⢄⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⣿⣿⣿⣿⢁⡾⠀⠀⠀⠨⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠋⡄⢿⡃⠀⠀⣠⠇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⠋⣼⣿⠿⠏⢀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⡍⢰⣷⣉⣠⣬⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⣦⡈⢤⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⣉⣑⠈⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡴⠞⠉⠙⠻⢦⡑⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⡒⢀⣴⣿⣿⣿⣿⣿⣩⠙⠲⢄⠀⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡾⠁⠀⠀⠀⠀⡴⡀⠈⠺⣿⣿⡆⠤⢄⣀⣀⣀⠀⠀⣐⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⠰⠋⠀⠀⢡⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠘⠐⠀⠘⠅⠀⠀⠀⠰⣎⣰⣋⡠⣀⠈⠻⣏⠢⠤⠍⠛⡉⠻⠿⠟⠛⠛⠿⠟⠛⠛⠋⠈⠁⠀⠀⠀⢀⠠⠃⣴⣟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠓⠆⠀⠀⠙⠲⠖⠃⠀⠀⠸⢆⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣔⡻⠇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀⠀⡀⠀⠀⠀⠀⠀⢄⣶⣦⣄⣀⣀⢹⠟⠒⠂⠀⠀⠀⠀⠁⠀⠀⡠⠤⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠛⠛⠛⠟⡻⢿⠿⣯⠾⠿⠟⠃⠀⠘⠛⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⠀⠀⡸⠀⠀⠈⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠉⠀⠀⠐⠂⠐⠚⠑⠁⡀⠀⠊⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠄⠀⠀⠀⠀⠀⠠⠀⠀⠒⠂⠀⠀⡇⢀⢀⡠⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡛⠋⠩⠶⠂⡀⠀⠀⠀⠄⠠⠭⠥⠤⠄⢀⡁⠂⠈⠀⠉⢠⣖⡭⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⠿⣿⣿⡶⢶⢶⣦⣥⣤⣭⣭⣭⣭⣤⣤⣭⣤⣾⣶⣶⣾⣦⣽⠖⠉⣀⠤⠒⠊⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠍⠉⠿⣿⣿⣿⡿⠅⢺⠅⢸⠅⢀⡤⠖⠒⠶⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢁⣭⡭⢭⣿⣖⣾⠅⢐⡡⢸⡴⠋⠀⠀⠀⠀⠸⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⣈⣭⡭⠵⣖⡋⠀⠊⢸⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠥⠀⠐⣒⣛⣩⡭⠄⢈⠁⣾⠀⠀⠀⠀⠀⣠⠞⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣈⢋⣩⡭⠥⠶⣒⠀⠄⠀⣟⠀⠀⣀⠤⠚⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣒⣚⣛⡭⠅⠒⡁⠀⡯⠌⠀⢀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠭⠭⠷⢖⣚⣋⠀⠂⣸⣡⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⢘⣫⣭⣭⡷⠠⠎⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣳⣌⠠⠄⠀⠀⠀⠀⡀⢀⣀⠒⣀⣐⣲⣿⣿⠯⠀⣩⣾⣁⣀⣀⣀⣀
⠀⠀⠀⣠⡤⠶⠒⠚⠉⠿⠉⠀⠀⠤⠤⠤⠿⣗⣦⢄⠀⠀⠀⠐⠀⢛⣀⣿⣿⣿⠿⠭⣴⣾⡿⠿⠿⠿⣯⣭⣍⣉⠛⠛⠲⠶⣦⣤⣀
⠀⠀⢸⡁⣄⣠⡀⠀⠰⠅⠀⠀⠀⠀⠐⠄⠀⠈⠓⠤⢤⣄⣀⣀⣀⣉⣍⣱⣶⣶⣦⣴⠶⠋⣠⡤⠂⠀⠠⠀⠀⣈⡭⠆⢀⡄⣈⣷⣽
⠀⠀⠀⠉⠳⢯⣝⣓⠒⠶⠦⠤⠤⣥⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣭⣤⣄⣀⣀⣤⣶⣶⣶⣯⡤⠤⠶⠶⠒⢚⣉⣭⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠦⢤⣤⣤⣤⣄⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣠⣤⣤⣤⠤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠄⠀⠒⠶⠶⠶⠶⠤⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠑⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣩⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢬⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⣎⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣤⣄⡀⠀⠀⠐⠲⣶⠶⣶⢶⣦⣤⣄⡀⠀⠀⠀⡁⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠳⢄⠀⠈⠲⣄⣀⣴⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⢀⣤⣿⣷⣿⣿⣿⣿⣿⣿⣷⡤⠚⠀⢀⡠⠚⣧⠖⠛⠛⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠙⠲⠦⣄⣉⠛⠻⠿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣁⣤⠴⠚⠉⠀⠀⡇⠀⠀⢀⣀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⢤⣭⣍⣉⣁⣀⣀⣀⣀⣈⣉⣉⣭⣭⠭⠴⠶⠒⠛⠉⠁⠀⠀⠀⠀⠀⢀⡧⠀⡜⠙⠻⡁⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣠⠃⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠉⠀⠀⢀⡜⠁⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢀⣠⠴⠚⠁⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠉⠀⠀⠀⣀⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡀⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⡊⠑⠓⠦⣤⣀
⠀⠀⠀⢀⡠⠔⠛⠉⠁⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠓⠚⠁⠀⠀⠀⠀⠈⠙⠦⣄
⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠳⢄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠓⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⢀⠀⠙⠶⣤⣄⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⠶⠋⠀⢄⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠈⠙⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠄⣀⣀⡀⠉⠑⠛⠫⠭⠿⠿⠿⠝⠛⠛⠉⠀⠀⣀⣠⠴⠄⠒⠁⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⣀⡤⠤⣰⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢝⡛⠳⠶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣿⡿⠿⠟⣫⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣉⣉⡉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⢉⣉⣠⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠒⠒⠉⠉⠉⠉⠁
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠐⠒⢂⡄⠠⠌⠀⠤⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢂⡤⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠒⠒⠶⠤⣀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡞⠉⠀⠀⣠⢀⢤⣤⣤⣤⣤⣀⠠⠘⠃⡥⠀⠶⠂⠀⢠⠀⢀⡀⠘⢯⣤⡑⢄⢀⠄⠀⠀⠛⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣧⢀⣶⣿⣿⡀⢛⣽⣿⣿⣿⣟⠱⠀⠠⡄⠀⡘⠟⠀⢄⣲⣼⡏⢠⡀⣿⡇⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣈⠻⣿⣿⣿⣿⣶⡀⡼⠟⡿⣧⠀⠀⠶⠀⢖⣨⣴⣿⣿⣯⣿⣿⣼⣷⠟⠁⠀⢀⡤⡀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠻⣷⣦⣌⣉⠛⠻⠿⠿⣷⣾⣿⣿⣿⣶⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠀⠀⠐⠒⠉⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣼⡆⠈⡍⡛⠛⠛⠻⠷⠶⣶⣤⣤⠬⣭⣥⠤⠤⠄⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⡀⢠⣶⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⢄⡂⢄
⠀⠀⠀⠀⠈⠔⣽⣿⢟⣯⣽⢿⣿⣿⣄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢦⡑⢄
⠀⠀⠀⠂⠠⠀⠾⣁⣈⣿⣛⣫⡿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣆
⠀⠀⠸⠀⠸⠀⠀⢿⣿⣛⣿⣿⣯⣷⡙⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢿⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣛⣿⣿⡀⠂⢸⣿⡿⡟⢿⣶⠶⠤⣤⣤⣤⠤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡿⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⣤⣶⣦⣼⣿⣿⣷⣶⡟⠋⢑⠈⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⣄⣠⣤⠀⠀⠀⠀⢠⡞⣡⠊
⠀⠀⠀⠀⠀⠀⠐⠈⠂⢀⡀⠀⠈⠻⣿⣿⣿⡆⠀⠛⠻⠓⠚⠉⠀⠀⠀⠀⠀⠀⠘⠿⢯⠷⠀⠀⠘⠻⠿⠿⠶⠈⠀⢄⡤⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣢⠴⣊⠥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⠶⠒⠛⠉⠅⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⢒⡀⡠⠄⠨⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠀⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⣥⣶⣿⡻⢕⣃⠰⠄⠙⠂⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢍⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣿⣿⣿⣷⣾⣷⣾⣙⣄⢀⣐⢂⡤⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⢼⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠷⣌⡻⢿⣿⣿⣭⣭⣭⣿⣧⣴⣶⣍⡉⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⠜⣣⣞⣿⢀⣍⣡⣍⣀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠉⠓⠪⢝⡛⠿⠿⠿⣯⣿⣽⣿⣿⣿⣷⣦⣵⠶⠤⠶⢚⣛⣋⣭⠶⠛⡫⣰⣿⡿⠋⠈⠉⠙⢲⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⢉⡋⢩⣭⡵⢶⣻⣽⠾⣻⣿⡇⠀⠀⠀⠀⣸⢿⡇
⠀⠀⠀⠀⠀⠀⠀⡠⣤⣒⢚⣿⡀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠒⠦⣍⣒⣭⣲⣽⣿⣻⣿⣿⢛⣲⠖⢁⡞
⠀⠀⠀⠀⡠⠂⢴⣪⢏⣩⣿⣼⣧⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠼⠿⣛⣻⣿⣿⣿⣿⣿⡿⠟⢋⣥⣾⣿⣯⡔⠄
⠀⠀⠀⠈⢔⠖⢚⢱⡟⢀⢿⠻⣿⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢁⣚⠿⣿⣳⣿⣿⣿⣛⣩⣴⣾⣿⣥⠪⠁⠣⢸⡄
⠀⠀⠀⡀⡘⠸⢣⠓⠁⠁⠁⠉⠀⠛⢷⡈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣾⣿⣿⣿⣿⣳⠟⠋⢝⠷⠛⠛⠏⡡⠁⠀⡐⠰
⠀⠀⠀⠠⡘⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠉⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢉⡙⢯⣜⢺⣹⣿⡻⠃⠔⠀⠀⠀⠀⠀⠀⠀⠀⠊⡠⠃
⠀⠀⠀⠀⠈⠢⢑⠰⢭⡢⡀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⢀⡀⢂⣀⣂⣠⣥⡶⠴⣾⡿⠊⠐⠀⠄⠂⠀⠀⢀⡠⠐⣉⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠢⢌⡀⠠⠀⠀⠀⠀⠀⠀⠑⠄⣀⡈⠉⠁⠀⠉⠁⠉⠈⠉⢁⣀⡤⠖⠋⠈⠀⠀⠀⠀⠀⠀⢂⣡⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢦⠤⢀⡀⠒⠀⠀⠄⠉⠉⠉⢉⣉⡀⠉⠉⠉⠉⡁⠀⠠⠄⠐⢂⡈⠤⡐⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄⠀⢀⣀⡀⢒⠒⠒⠀⢀⣀⣀⠀⠠⠄⠐





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⣶⠶⠟⠛⠋⠉⠉⣙⡛⠻⠽⣖⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⣯⢟⡵⢊⡡⠤⠆⠀⢀⣤⠀⠄⠀⠉⠓⠦⢍⡓⢽⣷⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⢞⡽⡵⠋⢐⡡⠚⠉⠀⠀⠠⢝⡔⠒⠀⠀⠀⠈⠲⣍⠢⡝⢿⣦⡽⠖⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡱⠊⢀⣾⡟⡠⢠⠋⠀⠠⠖⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠈⢢⠙⡎⢿⣧⣠⡴⠆⢹
⠀⠀⠀⠀⠀⠀⠀⡴⢃⠞⠀⠀⡼⡸⢰⢁⠃⠀⠀⠰⠣⠀⠂⠀⢀⣄⠠⢐⠁⠘⠀⡆⡀⠀⠀⠀⠘⡟⣿⠁⣠⣞⣆
⠀⠀⠀⠀⠀⠀⣼⠁⠋⠀⠀⠀⡇⡇⢸⠈⠀⡄⠀⠀⠀⠳⣄⠀⡈⠁⡔⠁⠘⢠⠀⠇⡄⠀⠀⠀⡀⡇⣿⡴⠉⠉⠉⠒⠀⠀⠠⣀
⠀⠀⠀⠀⠀⢰⡇⡄⠀⠀⠀⢰⡇⢻⠸⡈⡄⠰⡐⡄⠀⠀⠈⠉⠚⢾⠊⢠⡦⢀⡔⡰⠃⠀⠀⢠⠃⡇⣿⣿⣿⣶⢦⣤⣀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠂⢻⡘⣆⠣⠘⠌⢿⣬⣲⣤⣀⡀⠀⢤⡠⢏⣔⡩⡊⠑⠀⠀⢠⠎⡼⢱⡿⣾⡄⠘⠀⢩⣿⠉⠛⠋
⠀⠀⠀⠀⠀⡟⠰⠀⠀⠀⠀⠀⠸⣷⡘⣆⠀⠀⠀⠉⠛⠯⢭⡛⠷⣦⢠⡾⠟⠉⠀⠀⠀⡰⢋⡼⢡⡿⠃⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣌⠳⣄⠈⠲⢤⣀⠀⠀⠑⠎⢻⠃⠀⠀⠀⠐⢊⡴⢋⣴⣿⠁⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣳⣮⣛⠦⣤⣈⡉⠒⠂⠀⠀⢀⣀⣠⠴⢚⣩⡶⣫⡞⠁⢀⠂⠀⢠⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢄⠘⠿⣶⣍⢛⠲⠮⢭⣭⣭⣭⣭⣭⠴⠶⠚⢉⣥⠞⠋⠀⢂⠊⠄⠀⠄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠈⢆⢣⠘⢌⠻⢷⣷⣤⣤⣄⣠⣠⣤⣤⠶⠞⠋⠁⠀⠔⡠⠁⠊⢀⠎⡀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⢄⠙⠢⣈⠀⢊⠉⠁⠀⠀⠀⠀⠀⠀⠠⠐⡨⢀⠔⡠⢃⠌⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢑⠬⡂⠅⠒⠄⠍⠉⣀⡀⠀⠀⠐⢂⠥⢚⡥⢊⠔⠁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢅⡂⠤⠄⠀⠀⠀⡀⠂⠈⠀⢀⠡⠒⠁⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣪⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣗⡦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢖⣫⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠾⢯⣿⣛⣒⣒⣶⣶⣒⣚⣛⣭⡭⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⢉⡛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⠇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⣁⠴⠒⠉⠁⠀⢀⣀⡉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⡞⠁⠀⠀⠀⠀⢰⠏⠁⠉⣧⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⣴⡏⠀⠀⣀⣀⡀⠀⠸⡄⠈⠛⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢰⣿⠀⢠⡿⢉⣍⡙⣆⠀⠙⠶⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⣿⡀⢸⡁⡇⠀⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠹⣧⠈⢧⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠹⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⠀⠋⠀⠀⠀⠀⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⣴⡞⠋⠁⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠙⣦⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⠙⠓⠀⠚⠉⠁⠀⠀⠀⣀⣀⣀⣀⣀⣤⠤⠾⠋⣠⡾⢿⣶⠖⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠴⠒⠋⠁⠀⣸⠟⠋⠛⣦⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢠⡖⠀⠉⠲⠄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⣦⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⡀⠀⠀⠈⠛⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠶⠛⠁⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠦⢤⣄⣀⣀⣀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠤⢤⣄⣀⣀⣀⣀⣀⣀⡀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⣠⡞⠁⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⢰⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠈⢿⣦⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠈⠻⣷⡀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⢹⣧⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣸⠏⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣘⣁⣀⡀⡀⠐⠋⠀⣀⣀⣀⣘⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠂⣉⣉⣁⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣈⣉⣉⠑⠒⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠁⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠘⡀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡴⠃⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⠒⠉⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⣀⣀⣀⠤⠖⠉
⠀⠀⠀⠀⠀⣠⠤⠔⠒⠂⢉⣉⣥⣤⣤⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣥⣤⣤⣬⣉⡉⠐⠒⠢⠤⣄
⠀⠀⠀⠀⠸⢄⡀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠀⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠠⠤⠌⣉⣉⣙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⣋⣉⣉⠡⠤⠔⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⢍⣉⣉⣉⠉⠉⠈⠛⠛⠛⠛⠛⠛⠊⠀⢉⡉⣉⣉⡩⠥⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⣒⣚⣫⣭⣭⣭⣽⣷⣶⣶⣶⣾⣿⣭⣭⣭⣽⣓⣒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⣭⣶⣞⣯⠿⠛⠉⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡯⠭⣭⣿⣛⡻⢶⣮⣍⡓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣴⣿⡿⣫⣾⢋⡴⠊⣀⣤⣶⡶⠶⠶⣦⣝⣿⣿⠟⠋⣁⣀⣀⡒⠒⠦⣌⣙⠳⣮⡛⢿⣷⣮⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⣿⣿⢱⣿⠁⣿⠁⣾⣿⣟⠁⠀⠀⠀⠀⠉⢻⡷⠊⠉⠀⠀⠉⠟⣿⣶⣌⠻⣷⡌⣿⣎⣿⣿⣿⡆⢳
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠾⣆⠹⣿⣿⠘⣿⡀⠻⣧⡈⠻⢿⣷⣤⣤⣀⣀⡀⠀⠀⠀⠀⠘⠀⣀⣴⣿⣿⡿⢠⡿⢃⣿⣿⣿⣿⣿⡇⣸
⠀⠀⠀⣾⣿⡅⠀⠉⠉⠉⣛⢻⣷⣌⠻⢷⣬⣑⡀⠈⠙⠳⠶⢤⣭⣭⣉⣉⣙⠂⠀⣠⣴⠶⠿⠿⣛⠫⠥⠚⢉⣰⣿⣿⣿⣿⡿⢋⣴⡿⢶⣄⡀
⠀⠀⠀⣿⣿⣿⠿⡿⣶⣶⣼⣼⡿⣿⣿⣶⣬⣙⠻⠿⣿⣷⣶⣶⣶⣬⣭⣭⣉⡓⢸⣿⣭⣭⣥⣶⣶⣶⣾⣿⣿⣿⡿⠟⣋⣥⣾⣿⣿⡇⢅⠫⠻⣦⡀
⠀⢠⡾⣿⡿⣿⣧⠀⠪⡻⠿⢿⡃⣿⣿⣿⣿⣿⣿⣶⣶⣬⡭⣉⣛⣛⣛⡛⠻⠿⠿⠿⠿⠛⢛⣛⣛⣛⣋⣭⣭⣶⣾⣿⣿⣿⣿⠟⣿⠢⠑⠅⠀⠈⠻⣦⡀
⢠⡿⡀⠙⣷⡹⣿⣷⣤⡀⢈⣴⡿⡎⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⣰⣿⣮⡢⠀⠀⠑⡀⠘⣷
⣾⢡⠂⠀⠈⠿⣮⡻⢿⣿⣿⣷⣦⣧⠹⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⣰⢏⢻⣿⣷⡀⢀⠁⠀⠀⠸
⣿⢸⢄⡀⠀⠀⣼⣿⣶⣬⣙⡛⠿⠿⠷⣷⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠬⠐⠀⠀⠀⣴⠋⠀⠂⣿⣿⡇⠈
⢻⡌⢇⠈⠉⠒⠪⠝⣿⠻⠿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠍⠃⠀⠀⡠⢾⠃⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⢀
⠈⢿⡌⢦⡀⠀⠀⠀⠸⣄⠈⠀⠒⠉⠩⢝⡻⣿⣿⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⠁⢀⣠⣴⣿⡇⢸⠀⠀⠀⣸⡿⠃⠀⠀⠀⠀⢀⣾
⠀⠈⠻⣦⡑⢄⡀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠱⡘⣿⣿⣿⣷⣾⣭⣿⣛⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⠊⠀⢀⡼⠋⠁⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠈⠻⣦⣉⠢⣄⠀⠀⠈⠓⠦⣄⡀⠀⠙⠮⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⠄⠁⠀⠀⠉⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠙⠷⣦⣍⡒⠄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠭⣝⣛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⣋⠩⠄⠀⠀⠀⠀⠀⠀⢀⠠⢐⣨⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠀⠂⠀⠀⠉⠉⠁⠀⠀⠀⣀⡀⠠⢄⣒⣭⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⢶⣤⣤⣄⣀⣀⣀⡀⠀⠤⠤⠤⠄⠀⠀⣀⣐⣒⣂⣭⣥⠶⠶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⣾⣟⡯⠍⠓⠒⠒⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⣽⣿⣧
⠀⣿⠈⠉⠻⠓⠒⢶⣶⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⣶⣶⣶⡶⠟⠛⠉⢸⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⣠⣤⣤⣀⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠟⠋⣉⣉⣉⣙⠻⣷⣄
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⠟⠋⠉⠉⠛⣷⡌⢿⣆
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠃⠀⠀⠀⠀⠀⠘⣿⠸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠃⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿
⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡟⢸⡿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢀⣼⡟⢠⡿⠃
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣠⣾⠋⣰⡿⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣠⣴⡾⠋⣡⡾⠋
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠛⠋⣁⣴⡿⠋
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣴⡾⠛⠁
⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠁
⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡾⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣽⣭⣬⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⣽⣽⣯⣯⣗⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⣮⡾⠛⠹⠻⠟⠏⠛⢷⣵⡱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠗⡄⠀⢀⣀⣤⣤⣴⣧⣿⡋⠀⠀⣠⣽⣮⢄⡀⠀⢙⣿⣼⣦⣤⣤⣀⡀⠀⢠⠺⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣾⡟⠛⠉⠍⠉⠉⠙⢿⡶⢝⠜⢯⡽⠣⡫⢶⡿⠛⠉⠉⢩⠉⠛⢻⣷⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⡿⣯⣢⠸⡈⠀⡇⡰⢭⠃⣀⣡⣎⣀⠘⣭⣧⢘⠀⠁⠇⣔⣽⢿⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠄⣈⡚⠿⢇⣣⣤⢧⣽⣷⠞⢋⣉⣉⡉⠻⣾⡧⡼⢬⣜⡸⠿⢓⣁⠠⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣿⡇⠀⠂⠨⠍⣾⣶⣻⣟⡿⢱⣞⣭⠴⠦⣭⣳⡎⢿⣫⣟⣦⢷⠉⠁⠐⠂⢸⣿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⣵⡿⠻⣷⣄⢉⡨⣉⣷⣿⡽⡡⢎⠻⠬⢣⠎⠱⣜⠥⠟⠹⢬⢯⣾⣾⣩⣅⡉⣠⣾⠟⢿⣮⢕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⣾⠋⠀⠀⣜⢏⠧⠛⣽⠟⣱⠶⡛⡆⠠⡢⣬⡐⢂⣥⣔⠄⢰⢛⡶⣎⠻⣯⠛⠼⡹⣣⡀⠀⠙⣿⡞⣆
⠀⠀⠀⠀⣠⣀⣄⣠⣗⣿⣷⡖⣄⣞⡾⣁⠄⣸⡇⢰⢃⡟⡩⠖⢂⠻⡿⣫⣝⢿⠟⡒⠲⢍⢻⡘⡆⢸⣇⠠⣨⢷⣳⣠⢐⣾⣿⣺⣄⣠⣀⣀
⠀⠀⠀⠀⠙⠉⠋⠙⡯⣿⡿⠅⠋⢿⠷⡋⠊⢹⡇⠸⡌⣧⣑⠦⠌⣴⣷⣝⣫⣾⣦⠥⠴⣊⣼⢡⠇⢸⡏⠑⢉⡾⡽⠙⠬⢿⣿⢽⠋⠙⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣿⣄⠀⠈⢫⣎⡖⣤⣻⣦⡹⠾⣭⠇⠐⠝⢻⠡⠌⡟⠫⠂⠸⣬⠷⢏⣴⣟⣤⢶⣱⡝⠁⠀⣠⣿⢯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢕⡻⣷⣴⣿⠋⣈⣙⣋⡿⡿⣳⡑⣆⣰⢒⡜⢆⡰⢣⡒⣦⡰⢎⣞⢿⢿⣉⣚⣉⠙⣿⣦⣾⢟⡪⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣿⡇⠀⠄⠐⣀⢿⠻⣽⣫⣷⡸⢯⣛⠲⠖⣛⡽⢇⣾⣝⣫⠟⡾⣀⡀⠠⠀⢸⣿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠒⢉⢥⣶⡎⡝⠛⡖⣺⡿⣦⣈⣉⣉⣁⡴⢿⡟⢲⢚⢫⢱⣶⡬⡉⠐⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣷⣟⠝⢰⢁⠁⡇⢰⣛⡄⠉⡹⢏⠉⢠⣟⠞⢸⠀⠀⠆⠫⣻⣾⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⢿⣧⣤⣀⣂⣀⣀⣠⣾⠷⣪⢦⣞⣳⡴⣝⠾⣷⣄⣀⣁⣘⣀⣤⣼⡿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡦⠃⠀⠈⠉⠛⠛⠻⡟⣿⣅⠀⠈⠑⣺⣟⠊⠁⠀⣨⣿⢻⠟⠛⠛⠉⠁⠀⠘⢴⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⢟⢷⣤⣸⣴⣦⣇⣤⣾⡻⡱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢭⣻⣻⣿⣟⡯⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠲⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣚⡭⠵⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠲⠤⣍⣓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⢀⣀⣠⣤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⢤⣄⡀⠀⠉⠛⢦⡙⢦
⠀⠀⠀⠀⠀⢠⡏⡏⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⠀⠉⠙⠲⣄⠀⣿⢸⡆
⠀⠀⠀⠀⠀⢸⣇⢣⡞⠉⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠋⠀⡤⠤⡄⢹⡀⠀⠀⠀⠀⠀⢀⡿⣡⢾⡇⣀⣤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⢸⡈⠳⣌⡳⢤⣀⡀⠀⠀⠸⡄⠀⠀⢰⠖⠒⠚⠁⠀⢹⢸⡇⠀⢀⣠⡴⢞⣡⠞⣡⠞⠋⠁⠀⣀⣀⣀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠓⠮⣍⣛⠒⠶⠿⠦⢤⣬⣷⣤⣤⣤⠤⠿⠾⠒⢋⣩⡤⠖⠉⠀⠀⣧⣀⡤⠖⠋⠁⠀⠉⠙⣦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⠶⠶⠶⠶⠶⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢧⣄⣀⣀⣀⣀⡤⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠈⠉⠉⠁⣀⡠⠔⠋
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠻⣍⠛⠋⠉⠉⠁
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⢸⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⢦⣌⡙⠲⠦⢤⣄⣀⣀⣀⣀⣀⣀⡤⠤⠖⠚⣋⡥⠞⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠤⠤⠤⠴⠶⠒⠋⠉⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠙⠪⣝⡲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⣚⡭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠺⠭⣍⣓⣒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠴⣖⣒⣻⠭⠷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢠⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⠁⠀⡾⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣰⠇⠀⣸⠃⠀⢠⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⡟⠀⢰⠇⠀⢠⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠖⣾⠀⢸⠃⠀⡿⠀⢠⡟⠀⢀⡏⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣭⠴⠒⠀⠀⡏⠀⣿⠀⢸⡇⠀⢸⠁⠀⣼⠀⠀⠠⣌⣙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢾⣿⣶⣞⣋⣥⣤⣿⣤⣿⢠⣼⡇⣤⣼⢠⣤⣿⣀⡀⠒⠦⣭⣛⠆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠈⢉⣠⠟⢑⣶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣍⠓⠂⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⣉⣀⣤⡴⠾⣛⣵⡆⣚⣛⠧⣄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⢋⣉⣉⣭⣴⡶⠿⣛⣭⠀⠉⠉⠙⣾⡇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠃⢸⣟⡳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣉⣉⣥⣴⡶⠿⣛⡅⠈⠑⣂⡴⣻⡣⢠⡏
⠀⠀⠀⠀⠀⢀⣤⠞⣋⡥⢒⡠⠄⠀⠀⢻⣟⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣉⣉⣉⣩⣭⣴⠶⢟⡋⢡⣞⣛⣽⣾⢟⣵⠟⢶⣄
⠀⠀⠀⠀⣰⠟⣡⡞⢡⡞⢉⡴⠊⡀⠀⠈⢿⣟⠒⠄⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣭⣭⣴⡶⢿⡛⢡⡿⢿⣛⣭⡶⠛⠁⣀⢢⡈⢷⡄
⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⣿⡀⢾⡀⠀⠀⠀⠻⣯⡒⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣭⣥⣤⠴⠶⣞⡛⣱⣿⣿⣿⣿⣿⢿⡌⢳⡈⣆⢳⠈⡇
⠀⠀⠀⠀⠻⣆⠹⣧⣌⠳⢮⣛⠶⣬⣁⠀⠀⠀⠈⠻⣥⣄⣀⠀⠀⠀⠀⠀⠤⠤⠶⢶⣶⣾⣿⣿⣿⣿⣿⢸⣿⡜⣿⢎⡇⢸⠇⡏⢀⡼⠃
⠀⠀⠀⠀⠀⠈⣳⣮⣙⠷⣦⣍⣙⠓⠦⠄⠀⠀⠀⠀⠈⠙⠻⢯⣿⣛⣛⣛⣛⣛⣯⣽⡿⠿⢛⣫⣵⡿⢋⣾⠟⣰⠟⠘⠁⢋⣠⣶⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠨⠽⠿⠶⢮⣭⣙⣓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠽⠷⠶⠶⠶⣚⣛⣋⣭⡵⠾⠛⢁⣈⣡⣤⣶⣾⣿⣿⠿⠭⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣉⣹⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣛⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣟⣛⣛⣛⣛⠛⠛⠛⠛⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⣩⣭⣭⣭⣭⣭⣭⣉⣉⡉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡐⠒⢦⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠰⡀⠀⠁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠸⣦⠤⠤⠤⠴⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠦⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠉⠙⢦⡀⠹⣄⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉⣳⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⣄⣀⡤⠒⠻⣄⠈⢯⡁⠀⠀⠀⠀⠀⠀⣀⠴⠤⢄⣀⡀⣀⠀⠀⢀⣀⣀⣭⠶⠚⣿⠚⢉⣀⣀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠉⠙⠛⠛⠓⠒⠛⠒⠒⠒⠒⠒⠒⠛⠒⠒⠛⠛⠛⠉⢯⠉⠉⠁⠀⠀⠀⠀⡿⠚⠉⠁⢹⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⡼⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣼⠀⣠⠖⠉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⡇⣸⠃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡼⠛⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⢰⣷⣚⣋⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⡉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⣠⡴⠛⠉⠁⢀⣀⣠⠤⠤⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠉⠉⠉⠉⠒⠠⠀⠀⣹
⠀⠀⢠⡾⠁⢀⠠⠐⠉⠉⠀⠀⠀⠀⠀⠉⠳⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠏
⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⢻⡛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠖⠛⢉⡴⠋
⠀⠀⠀⠙⢯⡙⠒⠲⠶⠤⠤⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣱⠤⣤⣤⣤⡤⠴⠶⠖⠒⠛⠋⠉⠉⠁⣀⣀⣤⠤⠶⠚⠉
⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⡟⠀⠀⠀⠈⠳⣄⣤⡤⠴⠶⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⡀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡴⠶⠖⢲⣖⣶⣦⣄⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⠟⠋⠉⠖⠂⢸⡷⠈⠙⠟⠉⠙⣾⣿⣿⠿⢛⣿⣷⣶⣤⣀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠁⠀⡎⠀⠀⠀⢺⣷⡄⠀⠀⣀⣴⡿⠟⠁⠀⢰⣿⠋⠛⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠡⡀⠀⠀⠀⠈⠙⠃⢼⠋⠁⠀⣀⣠⡴⠞⠃⠀⢠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⠻⣿⣿⣟⡀⠀⡀⠈⡀⢉⠀⠒⠲⣆⢀⣴⠾⠛⠉⣁⣀⣠⣤⣴⣾⣿⣿⣿⢟⣴⣦⠄⠤
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣤⣚⣻⣛⠛⠛⢿⣾⣟⣡⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠑⠾⢿⡗⠀⠠⠦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠷⠚⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠠⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠃⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣀⠤⠊⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⠃⠀⠉⠁⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⣇⢀⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠟⠿⠷⠾⠿⠿⠿⠟⠛⠁

⣿⣿⣦⣀⠀⣹⠜⣶⣛⠚⣧⠆⠉⣲⠻⣯⣿⣿⣿⢿⣏⣽⣯⣯⠙⠀⠀⠿⠁⠆⠈⣿⠈⠛⠈⠛⢿⣶⡦⠟⠀⢰⠏⠀⣹
⣿⣿⣿⣿⣿⣷⣤⢼⠉⠀⠈⠛⠆⢸⠓⠿⡳⢿⣿⣿⠻⡿⣵⣿⠦⠏⠉⣸⣁⡀⡀⠆⠀⠀⠀⠀⠈⣿⠁⠀⠦⢀⢠⠤⠛
⣿⣿⡿⠿⠷⣬⣿⡦⢻⠧⡶⠄⠀⠂⢐⡷⢿⡙⠻⠏⠛⠻⠷⠎⠉⠀⣶⠻⢿⡿⡿⡀⠀⠆⡀⠀⢤⠈⡀⠊⠀⣸⣦⣴⠿
⣿⣿⣶⣶⣾⢁⣿⣧⡾⢞⣀⣤⠀⠠⠮⠗⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣨⠄⠼⠘⠀⣡⡉⠀⠀⢂⣠⣄⠀⡀⠉⡿⠁⠀⣠
⣿⣿⣿⣿⣿⣿⣿⣶⣶⣟⣛⢭⠽⠛⠛⠠⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠟⡂⢴⢈⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣹⢁⣀⡀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣠⣿⠏⣿⣾⣆⠀⠀⠰⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣾⣟⣀⡀⣠⡀⠀⠀⠀⠠⠴⠦⠄⠀⠀⠀⠀⠦⠀⢀⣻⡬⢹⡛⡿⢹⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄⠀⠀⠀⠀⠶⠀⠀⠀⠀⠘⠀⡀⠐⠦⠀⠀⠘⣃⡤⡾⠀⠀⠂⠀⠀⠰⣦⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⡴⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⡦⠲⣀⠀⠁⠀⠀⠀⠀⠀⠰⠞⢫
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡶⣣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠒⠎⢀⠀⢀⣄⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⠅⠀⠀⠀⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢂⣶⣯⣀⣳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠸⠋⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡷⠦⢀⠀⠀⠀⠀⢀⣬⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠸⢫⣮⣤⣀⡘⣟⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⢤⣴⡇⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⡀⠨⣿⠑⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢺⣥⡾⢿⣧⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⢉⣳⣾⠯⣷⣶⠟⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡄⠀⠀⠀⠀⠀⠀⣴⠙⢻⣌⣾⡟⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⣴⡶⠆⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣤⣶⣶⣤⣠⣀⠀⣀⠀⡀⡀⢠⣀⠀⢀⢀⣀⠤⣠⠀⣄⣨⣴⡍⢀⠐⢀⣲⡛⠻⣿⣿
⠀⠀⠀⠀⠲⣞⣿⣿⣿⣿⣿⣿⣿⣿⢻⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠮⠊⠁⠀⠀⠐⢄⠙
⠀⠀⠀⣤⢟⣿⣿⣿⣿⣿⣷⣿⣿⠻⣷⣾⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡔⠀⣠⠔⠀⠀⣼⣷
⠀⠀⢀⣿⣿⣿⡿⠿⠟⠙⠛⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣴⡞⣀⠾⠱⢳⣤⣼⣿⣿
⠀⠀⠀⠹⣿⣿⣿⠎⠹⠿⣿⣿⣿⣿⣿⢿⣷⢏⣭⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⣿⣴⣥⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠽⣿⣿⠠⡖⣀⣀⡀⡀⣀⢀⠀⣼⣿⣿⢃⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠠⠴⠛⠛⡀⠁⠉⠈⠀⠁⠁⠉⠀⠘⡡⠾⢿⠿⢏⡀⠀⠐⠱⣄⣈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠣⡀⠀⠀⠀⢐⣄⠀⠀⠀⠀⠀⢀⠜⠉⠀⢸⠁⠀⠀⠩⡀⠞⠢⣄⣈⠐⠒⠄⢀⣈⠙⠛⠿⠿⣿⣿⣿⣟⣿
⠀⠀⠀⠀⣁⣢⣄⣀⠈⠈⠉⠉⠁⠈⠉⢁⣀⣀⣸⣀⠓⠀⢄⡁⠈⠧⢤⣈⡁⠒⠂⢄⣉⣉⠛⠠⢢⠂⠀⠉⡹⠛
⠀⠀⠀⠀⠙⠛⠻⠿⠷⠶⠶⣶⠶⠶⠶⠾⠿⠿⣿⣿⣿⣶⣤⣈⠑⠠⢤⣈⠉⠓⠖⢠⣦⡉⠙⠒⠆⠀⣠⡾⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⢿⣶⣤⣁⡈⠉⠐⠤⢤⣌⠛⢀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠢⠤⢌⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣡⠶⠋⡹⠋⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣠⠞⠂⠀⣼⠁⢠⠀⠀⠀⢈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠛⠀⠀⣴⢃⣠⡦⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠂⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⡤⠚⠀⣠⣦⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢾⠆⠀⠀⠀⠀⠀⠀⠌⠃⠀⠁⠈⠉⢒⡆⠀⠙⣿⠀⡆⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⣤⢀⡉⠻⠀⢇⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⠇⠀⢸⡿⠀⠀⢠⠗⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠱⣤⠦⠈⠢⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠏⣰⡿⠚⠁⠰⠟⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠉⠻⠆⠐⠶⠚⠾⠃⠀⠀⠶⠀⠀⠒⠛⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠠⡟⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⣦⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠀⠀⠈⠁⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⣧⠀⠀⠶⣦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠶⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠀⠀⠀⠀⠀⠀⠐⠒⠲⠚⠀⠀⠀⠀⠀⠀⢀⣤⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢤⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣶⣶⣶⣶⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘






⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣾⣶⣶⣶⣾⣶⣾⣶⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣻⡟⢿⡿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⣄⣠⡏⣦⡤⠋⢩⠠
⠀⠀⠀⠀⢀⣀⡨⢉⠄⠀⠀⠀⢀⠲⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⣀⣤⣀⣀⡀
⠀⠀⠀⢰⠍⣭⣀⡘⢁⣦⡉⢢⢀⣠⡔⠀⠀⠀⠀⠠⠊⠛⢦⢤⡭⠭⠭⠶⢖⣃
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣤⡀
⠀⠀⠀⠀⣴⠀⡀⠀⠀⠀⠀⠀⠀⣿⠛⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠈⢿⡀
⠀⠀⠀⠀⢸⠢⠈⠀⠀⠀⢀⣀⠀⡏⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢠⣄⡀
⠀⠀⠀⠀⢨⠁⠀⠂⠀⠓⠐⠅⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣠⠌⠀⠀⠀⠀⠿⢀⠙⠂⠉⠁⣶⣊⣼⣟⣷⣀
⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⣰⠜⠃⠐⠀⠀⠀⠀⠸⣿⣶⠶⠶⠶⣶⣶⡿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠁⣶⠟⠉⠹
⠀⠀⠀⠀⠀⠈⢓⡦⠤⠤⠴⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣓⣒⣟⣻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⢝⣒⣚⣻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠭⣭⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢀⡏⢉⠉⠀⠀⠠
⠀⠀⠀⠀⠀⣤⡔⢀⠀⢪⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⢠⣤⠆⡀⠐⣮⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⣠⣾⣷⡿⠛⠁⠀⠐
⠀⠀⠀⠀⠘⠿⠂⣁⣄⣀⡀⠉⠛⠐⠀⠀⠀⠀⠀⠀⠚⠟⢀⣁⣀⣀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠐⠒⠲⠦⢤⣤⣤⣾⣿⣿⠿⠿⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⡤⡀⠄⠶⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠦⠴⠂⠚⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠤⠄⠒⠂⠉⠁⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣀⣾⠿⢤⠶⠏⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠃⣽⠟⠛⠖⠀⠀⠀⠀⠀⠀⠀⠎⠉⠙⠛⠛⡿⢿⣶⠀⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⡿⠤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⣰⣇⣤⡤⠈⠀⡀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⣠⠴⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢀⣀⣀⣀⣀⣤⣿⣿⣿⡯⠤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣷⠯⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠠⠄⠰⠐⠂⠓⠊⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠲⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣾⣷⡿⢿⣿⠾⣾⣿⣿⣿⣾⡿⣷⣷⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣿⣿⣷⣶
⣿⣿⣦⣷⣴⣶⣧⣾⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣀⡀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣀⡀
⠀⠐⠿⠋⢶⣶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠻⠚⠁⠀⠻⠿⠀⠀⠀⠐⠿⠋⢶⣶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠻⠛⠁⠀⠻⠿
⠀⠀⣴⣶⠄⠉⠀⠀⠀⠀⢠⠠⠀⢀⠀⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⠠⣴⣶⠄⠉⠀⠀⠀⠀⢠⢠⠀⢀⠀⠀⠀⠀⢰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠠⠌⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠄⢥⠀⠀⠀⠀⠀⠈⠽⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢭⡁⠀⠀⠀⠀⠀⠿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⣤⣤⣀⣤⣤⣤⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠰⣶⡄⠀⠀⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄⠀⢈⣿⣿⣿⣿⣿⣿⠄
⠀⣀⣤⣶⠀⠀⠀⠀⠘⠹⣿⣿⣿⣿⠇⠀⠀⠿⡷⠆⢠⣶⠄⠀⠀⢀⣤⣦⠀⠀⠀⠀⠘⠹⣿⣿⣿⣿⠇⠀⠀⠿⡷⠆⢠⣶⠄
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⣠⣠⣤⣤⣴⣦⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀
⠀⠀⠀⡐⣷⣴⣾⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣷⣶⣾⣭⣵⣯⣽⣟⡻⠿⣶⣤⣀⡆⡀⠀⣀⠄⠀⠄⡀
⠀⠀⠀⣿⠎⣛⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⣴⣷⣿⡥⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠯⠉⠥⣶⣿⡗⣿⣿⡟⠀⠀⠀⠀⠸
⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡿⠉⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁⠀⢀⡠⢂⠄
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⠞⣉⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠛⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢤⣤⣤⡤⣤⣴⡶⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⢉⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡤⠃⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⣀⡠⠤⠤⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠔⠋⠀⠀⠀⣠⠶⠦⡌⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢁⡾⠃⠀⠀⠀⠀⠀⡏⠀⣄⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣾⠃⢀⣴⠶⠶⣄⠀⠱⣄⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡀⢸⢃⡎⠳⠈⡇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⣧⠘⣎⠳⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠻⣆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢻⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⠠⠃⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣶⠋⠁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠂⠘⠋⠀⠀⣀⣀⣀⣀⠤⠤⠤⠴⠒⢋⣠⠾⢻⣷⣒⡑⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠊⠉⠀⠀⣸⠋⠉⠙⡇⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠐⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢀⡄⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢀⠀⠀⠀⠀⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣠⠴⠞⠁⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣤⣀⣀⡈⠉⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⣭⣉⣉⣁⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢁⠈⠀⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠂⠀⠈⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠶⢺⡀⠀⢁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⣹⠀⠀⣏⠉⢵⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢋⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠒⠡⠐⢧⠀⠱⣀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣀⡀⠀⡈⡗⢤⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡜⢀⢊⠐⠁⠀⠀⠀⠑⢤⡈⠑⠂⠷⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠴⠊⠉⠀⠀⠀⢀⠻⠲⡐⠄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⡼⠉⠑⠂⠠⠀⢤⡀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠡⠁⠀⠀⠀⠀⠀⣠⠞⠀⠃⠀⢨⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠙⢒⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀⣁⠤⠖⣒⠼⠶⠶⠦⠤⠶⠚⠁⠀⠀⠀⠠⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠠⠆⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⣠⠴⠶⠒⢉⣡⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠌⡁⠂⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⠀⠔⠊⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠴⠒⢈⠠⢐⡮⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⠀⠀⠀⠈⠉⠁⠒⠒⠀⠤⠤⠤⠤⠤⠤⠠⠤⠤⠶⠶⠾⠛⠛⠓⠊⠉⠉⠠⠀⠂⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠃⠐⠒⠒⠒⠒⠒⠒⠒⠂⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠤⠀⢀⠀⠀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠈⠀⠀⠀⠐⠊⠀⠀⠀⠈⠂⠀⠀⠀⣐⠤⠤⠄⠠⠄⠱⣀⡀⠀⠀⢧⡋⠸⢰
⠀⠀⠠⣔⣛⣈⣡⠀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠂⠆⠶⠐⠀⠄⠀⠈⠃⢠⣶⣶⣴⣦⠀⠍⠃⠀⢀⣀⣢⣔⡃⢀
⠀⠀⠄⠙⠛⠛⠋⠀⠀⠀⠀⠓⠀⠀⠉⠀⠀⠂⠀⠀⠀⠀⠐⠀⠐⢀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠋⠀⠀⠀⠀⣆⠐⠷⠶⠂
⡀⠀⠀⠈⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢄
⠀⡀⠀⠤⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣠⡢⠀⠁
⠀⠀⢰⣶⣤⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣬⣤⣵⠆
⠀⠀⠄⠉⠛⠋⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠔
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠐⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣿⠗⢺⣯⡿⠃
⠀⠀⠀⣴⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠂⠀⠀⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⠏
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠄⠀⠄⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂
⠀⠁⠀⡤⢶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠲⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣀⠀⣀⠐⠄⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠡⠞⠶⠁⠀⠀⠀⠀⠀⠀⠴⠲⠆⠀⠀⠀⠀⠈⠀⠀⠳⠞⠉⠀⠁⠀⠀⠀⠒⠒⠐⠒⠂⠀⠀⠀⠀⠀⠐⠂⠀⠒
⠀⠰⠆⠁⠀⠀⠀⠶⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠐⡀⠀⢠⡆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠂⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠰⣆⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣰⠏⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠏⢠⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢻⡄⠀⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⢉⣉⣉⣠⣶⣿⠁⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢁⣴⡿⠿⠟⠻⠿⠿⠁⠀⠀⠀⣠⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠿⠃⣰⡇⠀⠀⠀⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⣉⣡⣤⣤⣶⣿⡿⠀⠀⠀⠀⠀⢙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⣾⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠛⣵⣦⡙⠻⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⠋⣴⣿⣿⡗⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣩⣭⣍⠻⣿⣿⣿⣼⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⢣⣾⣿⣿⡿⢟⡀⣹⣿⣿⣿⣿⣿⡿⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇⢋⣥⣶⣶⣶⡿⣡⣿⣿⣿⣿⣿⡿⢡⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣶⣶⣶⠙⢿⣿⣿⣿⣷⣜⡛⢛⣛⣡⣾⣿⣿⣿⣿⣿⡿⣡⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣭⣍⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢠⣿⠟





⣶⠒⠲⣶⠶⠶⣶⢶⢶⣶⠶⢶⢶⡶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⡶⢶⣶⡶⣶⣶⠶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⣤⣤⣤⣤⣤⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣖⣮⣽⣶⠟⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣭⣗⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣱⣾⣿⡿⢫⠖⣠⣴⠶⠶⠶⣽⣿⢟⡉⠠⢄⣐⡲⢬⣑⢮⡻⣷⣮⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡅⣿⣿⢺⡇⢿⡘⢿⣧⣀⡀⠀⠀⠙⠁⠀⠄⠀⣹⣿⡦⣹⡆⣿⣿⣿⣷⢹
⠀⠀⠀⠀⠀⠀⠀⢰⣾⡉⠉⠉⢛⢻⣮⡛⢦⣕⣈⠙⠲⠬⢭⣽⣛⣓⠀⣠⡴⠶⠿⠛⠩⢒⣩⣾⣿⣿⠟⣡⡿⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠛⠿⣿⣾⢿⣿⣷⣮⣝⣛⠿⠿⣷⣶⣶⣿⣅⣿⣶⣶⣿⣿⡿⠿⢿⣛⣭⣶⣿⣿⡇⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⡞⠙⣿⢿⣷⣄⠈⣹⣮⢿⢿⣿⣿⣿⣿⣷⣶⣶⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⠟⠉⣾⣆⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢸⢵⠀⠈⠳⣝⡻⣿⣷⣾⣘⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠋⠁⠀⠀⡼⠙⣿⣇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⡸⡒⠦⢼⣿⣿⣶⣮⣭⣍⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⠞⠁⠀⣿⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⢷⡱⡀⠀⠀⠹⡀⠉⠉⠙⢛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⣀⣤⣾⠁⠇⠀⣰⡿⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠈⠳⣌⡢⢀⠀⠙⠦⣀⠀⠀⠢⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠴⠋⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣅⡂⠀⠀⠉⠀⠀⠈⠒⠩⢛⣻⠿⠿⣿⣿⣿⡿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⡀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⡴⠀⠀⠰⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡎⠀⠀⠀⢀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠠⠀⡘⢡⡷⢤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⡄⢖⠀⣠⣴⣶⣶⣿⡿⢷⣼⣿⣿⣿⣿⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠷⢂⢹⣧⣤⣬⠍⠉⠉⠁⠉⠉⠉⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠑⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠠⠐⠞⠻⣦⡀⠀⠀⠀⠀⠀⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠷⣤⣍⣓⣶⣶⡤⠴⡋⠁⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣌⠻⣿⣿⣿⣿⣦⣌⣉⣉⣉⣁⡀⠔⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠀⢠⡴⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠋⠉⠉⠀⠀⠀⣀⣀⣤⡔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⠉⠁⠀⠈⠁


⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠰⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣰⠋⢡⡆⣰⢠⡆⢰⠊
⠀⠀⠀⠀⠀⢠⠁⠐⠀⢣⡀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠈⠀⢸⠀⡇⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⡀
⠀⠀⠀⠀⠀⠘⠀⠈⠂⠸⢱⠀⢀⠄⠈⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠘⡀⠀⢡⠀⠆⣦⠁⠀⢠⠊⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠙⣄⠨⣧⠘⣠⠀⣠⠋⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠢⠄⡈⠁⠘⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢘⠀⠁⠀⡸⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠇
⠀⠀⠀⠀⢃⢀⠀⢀⠁⠀⢤⠀⠫⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⡇⠀⠀⠀⠀⠐⠀⠤⠀⠀⠀⠀⣀⠠⠄⠀⠁⢒⡀
⠀⠀⠀⠀⠈⠀⠡⡀⠁⠈⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠁⠉⠉⠒⠂⠀⠀⠀⠀⠂⠀⠐⠀⠉⠉
⠀⠀⠀⠀⠀⢁⠀⠐⠄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠐⠀⠀⠈⠦⠀⠑⠜⠀⢀⠀⠀⠄⠰⠐⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣶⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠢⣀⠀⢲⡐⠆⢀⠀⠒⠀⣀⣠⠴⠊⢁⠀⠀⠀⠀⠀⢀⠋⠂⠀⠀⠀⢀⡤⠚⠻⠝⢿⣿⣿⣿⣿⣿⣿⣿⡜⡄⢀
⠀⠀⠀⠀⠀⠀⠉⠀⠛⠂⡠⠶⠞⠋⠁⠀⡠⠄⠀⠀⠀⠀⠇⢠⠆⠀⠀⠠⠀⡰⠃⠀⠀⠀⠀⠐⠉⠛⠛⠛⠛⠛⠉⠈⠁⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠤⠀⠒⠁⠀⡇⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢤⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠁⠠⠀⠀⠀⠐⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢠⠀⡀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉
⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡸⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠉⡆⠀⠀⠀⠐⠢⢀⠀⠀⠁⠐⠀⠈⠁⠀⠃⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠼⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⡶⠚⠂
⠀⠀⠀⠰⡀⠀⢀⡀⡀⠁⠒⠠⠀⠀⠀⠀⠤⠄⠒⠁⠐⣃⠀⠀⠀⠀⠀⡀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠤⠨⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠒⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠂⠀⠀⠉⠉⠉⠉⣉⡉⠉⠉⠉⠉⠉⠑⠐⠒⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢑⠈⠡⠀⠒⠒⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠒⠒⠀⠄⠁⡒⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠉⠉⠉⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠢⡀⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⡀⠀⢸⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣹⠀⠀⣏⣉⢵⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠹⡑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡡⠔⡉⠄⢧⡀⠹⣄⠠⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⣀⠀⢀⢻⠦⡈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⡜⢁⠎⠀⠈⠀⠀⠀⠑⢤⡈⠑⠂⠵⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡡⠖⠉⠁⠀⠀⠀⠸⠗⢌⠢⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠒⠠⠄⢠⣀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠎⠀⠀⠀⠀⠀⢀⡴⠃⠐⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠙⠖⠉⠐⠒⠢⠤⠤⠤⠄⠀⠐⠂⢈⡠⠴⢒⡲⠷⠶⢤⣤⠤⠖⠋⠀⠀⠀⢀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠠⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠢⢤⣤⠴⠒⠂⣉⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢑⠢⠌⡑⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⠈⠁⢀⠠⠒⠋⠁⠀⠀⠀⠀⣀⣀⣠⣄⣤⠤⠤⠒⡁⠤⣲⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡁⠀⠀⠠⠍⠉⠐⠒⠒⠂⠤⠤⠤⠤⠤⠠⠀⠤⠤⠤⠶⠶⠶⠛⠛⠛⠉⠉⠡⠄⠐⠀⡉⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣼⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡞⠀⢰⠇⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣼⠁⢠⡏⠀⢠⠏⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⠇⢰⡇⠀⡾⠀⢠⠏⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣛⡋⠉⣼⠀⣸⠀⢸⡇⠀⡾⠀⢠⡏⠙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⣾⣫⡥⠄⢀⣿⢀⣹⢀⣸⡄⣀⡇⠀⣸⠀⠀⢬⣙⠲⣬⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣦⣬⠙⠃⣰⠯⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡙⠢⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⣉⣁⣤⠶⣛⣵⠰⠿⢿⣉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠛⣛⣉⣭⣥⡶⠿⣛⡥⠘⠛⠛⣾⡇⢸⡇
⠀⠀⠀⢀⣀⣤⠶⠚⠋⠸⣿⡳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⣉⣩⣭⣶⠾⢟⣫⠄⠉⣒⡴⣿⡷⣸⠇
⠀⣠⡶⣛⡵⢒⡭⠀⠀⠀⢻⣝⡓⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣉⣩⣭⣥⡴⠶⢟⣫⠅⣾⣿⣵⡾⣫⡼⠋⠶⣄
⣼⠋⣾⢋⡾⢁⡶⢋⡤⠀⠀⢻⣭⣁⠀⠀⠀⠀⠀⠀⠠⠤⢤⡤⠤⠴⠶⠾⣛⣻⢏⣾⣛⣯⣷⣞⠉⡀⢌⢳⡘⣷
⣿⠘⣿⡘⢧⡘⢧⣘⠦⣄⠀⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠐⠒⠒⢒⣚⣛⣿⣽⣧⣾⡿⣿⡿⣿⡽⣧⢹⡌⡇⠇⣼
⠈⠳⣌⡻⣦⣝⠳⠯⣟⣒⠀⠀⠀⠀⠙⠷⣶⣤⣤⣤⣤⣭⣭⣭⣭⣿⣿⣿⡿⣻⣿⡇⣿⡇⣿⢧⠏⡼⠀⣡⡾⠃
⠀⠐⠚⢛⣶⣭⣟⡳⠦⢬⣁⠀⠀⠀⠀⠀⠈⠉⠛⣛⣒⣒⣚⣛⣿⣭⣭⠶⢾⣻⣵⠾⠋⠜⢁⣀⣤⣶⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠐⠚⠛⠛⣷⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣒⣒⣲⣶⣶⣲⣟⣿⣯⣭⣴⣶⣾⣿⣿⣿⣿⡿⠿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⡭⠭⠭⠭⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠶⠶⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣭⣭⣭⣭⣽⣯⣭⣭⣍⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣅⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠆⠀⠀⢀⣴⣿⣿⣿⣠⣾⣿⣿⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⣿⠟⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⠛⣥⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣧⣴⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⢀⣠⡴⠶⠶⠆⠀⠀⠀⠈⠙⣿⣿⡿⠛⡉⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢻⣩⣿⣧⠀⠉⠉⣤⢤⣤⣄⠀⠀⠀⠀⠈⠛⠋⢀⡆⠈⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡀⠀⠀⠀⠀⠀⡁⢸⡟⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⣟⠀⠀⡀⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡀⠀⠀⠀⠀⠀⠹⠟⢛⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢂⣀⠀⡄⢀⣀⣀⣀⣀⣐⣿⣥⠦⠖⠒⠀⠀⠀⠀⠀⠀⠀⢀⡬⠁⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣟⡛⠃⠀⠚⠛⠛⠋⠉⠉⠛⣦⠄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⢀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡠⠗⣸⣇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⣠⠂⠀⠀⠀⠘⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣟⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⣶⣿⡽⠟⠒⠀⠀⠠⠎⠁⠀⠀⠀⢀⠀⠀⠈⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠛⣿⣻⣿⡿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣿⣦⣀⣠⡔⠊⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠿⠟⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠀⣸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠴⢾⣿⠟⣿⠁⠀⠀⠀⣀⣤⡤⠴⠶⠛⠁⠀⣰⣿⣿⡇⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠘⠉⣰⠋⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠘⣻⣿⡟⠁⠀⢠⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⢀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢳⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣤⡖⠂⠀⠀⣀⡀⠀⠉⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠉⠉⠀⠀⠸⡇⠀⠀⠀⠙⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡦⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠊⠑⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⢿⣿⣿⣿⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠋⠻⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣁⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠀⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⡾⠋⠁⠀⠈⣿⣿⣆⠸⣿⣿⣿⣿⣿⣷⣦⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⡀⠀⢀⡿⠁⠀⠀⣠⡾⠀⠀⠸⣿⣿⣆⠈⠙⠻⢿⣿⣿⣿⡇⢸⡟
⠀⠀⠀⠀⠀⢹⣦⠈⠃⠀⣠⣴⣿⠃⣀⣀⣴⣿⣿⣿⠀⢀⠀⠀⠙⢿⡿⠀⠈⠀⣀⡤
⠀⠀⠀⠀⠀⣸⡿⢁⣴⠟⠛⠉⣴⡿⠿⣿⣿⣿⣿⡿⠀⠀⢻⣦⠀⠸⠁⣴⣿⣿⠿⠁
⠀⠀⠀⠀⠀⣿⠁⠈⠻⠀⠀⠈⠁⠀⠀⠀⠈⠉⠉⠀⠀⢀⣰⣿⠀⠀⠘⠛⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⣠⠀⢀⣴⣶⡄⠀⠰⣶⣆⠀⣾⣿⡿⠛⠋⠁⠰⠿⢿⣿⣶⡶⠿⠋
⠀⠀⠀⣰⠿⠛⠻⣿⡆⠈⢀⣿⡇⠀⠀⣿⣿⠀⠹⢿⣿⣦⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠁⠀⠀⢀⣿⣷⠀⣾⣿⣇⢀⣴⣿⠃⠀⠀⢀⣩⣿⣷⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⣀⣤⣶⣾⣿⠏⠀⢿⣿⣿⠿⣿⣿⡄⠀⡾⠛⠛⠛⠋⠀⠀⠀⠠⠾⠋
⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⣀⣀⣠⣤⡤⠤⠥⠴⢶⡶⠶⠶⠶⠶⢤⡤⢤⣤⣄⣀⡀
⠀⠀⠀⢀⣠⣤⠖⠒⣿⣿⠁⠀⢠⡏⢀⣤⣤⣤⣼⡇⠀⣤⣤⣤⣾⡇⠀⢀⣀⠈⠙⠳⣦⡀
⠀⣤⣾⣿⡿⠁⠀⠀⠘⠏⠀⢠⡟⠀⠈⠉⠛⣻⣿⠀⠀⠉⠉⣿⣿⡇⠀⢸⣿⠿⠀⠀⣸⣧
⢸⣿⣿⠏⠀⣠⣾⠀⠀⠀⢠⡿⠁⠀⠾⠷⠶⣿⡿⠀⠸⠿⠿⠿⣿⡇⠀⠈⣀⣠⣴⣾⣿⣿⣤⣤⣤⣄⡀
⣿⣿⣿⣶⣾⣿⣿⣇⣀⣠⣿⠃⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣤⣤⣿⣷⣾⣿⠿⠟⠋⢹⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⡟⠋⠙⢿⣿⣿⠿⣿⣿⡿⠿⠿⠿⠿⣿⠿⠿⠿⠟⣿⠛⠛⠉⠉⡇⠀⣀⣤⣼⣿⣿⠀⠀⠈⠹⣿⣿⣷
⣿⣿⣿⠀⢰⣷⠀⣿⡇⠀⠹⣿⡇⢀⣤⣤⣤⣿⠀⣤⣤⣤⣿⠀⣶⣶⣿⡇⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿
⢿⣿⣿⠀⢸⣿⣦⣿⡇⢸⠀⢿⡇⠘⠿⢿⣿⡇⠀⠿⠿⣿⡟⠀⠿⢿⣿⡇⢀⣁⣿⣿⣿⡿⠀⠀⠀⢀⣿⣿⣿
⢸⣿⣿⠀⢸⣿⣿⣿⡇⢸⡄⢸⡇⠀⣶⣾⣿⡇⢠⣶⣶⣿⡇⠀⣶⣾⣿⡇⢸⣿⣿⣿⣿⣷⣀⢀⣠⣾⣿⣿⣿
⠀⣿⣿⠀⢸⣿⣿⣿⡇⢸⡇⢸⡇⠀⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠘⣿⡄⠘⣿⣿⠸⡇⢸⡇⢸⡇⠀⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⡇⣾⠿⣻⣿⣿⠿⠿⠿⠿⠛⠉
⠀⠀⠘⢷⡄⠙⠏⢸⣧⠀⡇⢸⡇⠀⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⠃⠀⣠⠟⠉
⠀⠀⠀⠀⠹⢦⣄⣼⣿⠀⠃⣼⡇⠀⣿⣿⣿⠀⢸⣿⣿⣿⡇⠀⠿⠛⢻⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣇⣠⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⣿⡇⣀⣤⣾⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⡿⠿⠛

⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠢⡀⠀⣀
⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠁⡉⠍
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⠀⢀⠀⢀⠀⢀⠀⣤⠀⢘⡇⠀⢀⠀⢀⠀⠀⠀⢠⡄⠀⣉⠀⡀⠀⢀⠀⠀⠀⡁⠖⠀⠁
⣿⡿⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢧⠀⢧⠀⠀⠃⠈⠲⢄⠀⣿⠀⢸⡇⠀⢸⡆⢾⠀⠀⠃⢸⡆⠀⣿⠸⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠑⠂⠀⠈⠀⠈⠀⠈⠀⠈⠁⠀⠈⠁⠈⠑⠀⠀⠀⠁⠀⠉⠀⠉⠒⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠐⠀⠀⠿⠀⠀⣟⠀⠈⡇⢠⠁⠀⢹⠄⢻⠀⡇⠀⠣⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠛⠀⠀⠛⠀⠀⠛⠈⠃⠀⠘⠀⢻⠀⠓⠀⠂⠀⠃⠀⠀⠰⣶⣿⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⡀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠸⣶⣝⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣷⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⡀⣀⣿⣿
⠀⢠⣀⠀⠀⠈⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡿⣿⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠌⠳⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣜⣭⣙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⣨⣭⢂
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣴⣿⣿⣿⣯⡆
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢠⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⠿⠿⠀⠘⠒⣶⡖⣶⣶⡖⣶⡶⠐⠁⠐⠿⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⢿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠛⠁⢻⣿⠇⠘⠃⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡇
⠀⠀⠀⠀⢸⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡇
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠻⠄⠀⠀⠀⠀⠀⠰⠋⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠹⡄⠒⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠒⣰⠃
⠀⠀⠀⠀⠀⠙⢶⡛⠛⠀⠀⠀⠀⠀⠀⠀⢤⣠⣤⠶⣄⣀⠄⠀⠀⠀⠀⠀⠀⠈⠛⣳⡾⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⢣⡀⠀⣠⠎⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⣶⠶⠦⠤⠽⠿⠥⠤⠶⠶⡖⠚⠋⠉⣰⠏⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡆⠀⢠⠀⠀⢠⡀⠀⡀⢱⡀⠀⣰⠏⠈⠁⠀⣿⣟⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢳⣤⢮⣴⣶⣼⣧⣤⠇⠀⡇⢠⠏⠀⠀⣠⠖⢋⣽⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⡿⠃⣀⡴⠋⠁⠀⠈⢀⣤⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠸⣿⣿⣿⣿⡿⠀⣀⡼⠓⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠒⠠⠤⠤⠤⠤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠁⠀⠀⣀⡀⣠⣤⢠⣤⠤⠤⠤⣤⢤⣤⣤⣌⣀⠀⠀⠀⠀⢰⡦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠂⠉⠀⠄⠁⢠⡤⠀⠀⠀⠀⠀⡶⠿⠿⠟⠿⣿⣿⣿⣶⣧⣄⡈⠁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠦⠀⠀⠀⠀⢠⠀⠰⣖⣀⣠⣤⣤⡶⠂⠀⢀⠀⠉⠉⠙⠃⣭⠉⠻⣿⣿⡇⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢦⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢁⠀⠁⢙⠂⠔⠚⢛⣻⣦⣀⠀⠙⢀⣀⣿⣿⠇⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡙⠒⠤⣀⣀⠀⠀⠀⠛⠁⠈⣉⠑⢀⡀⠀⠀⠶⠿⠿⡶⢶⡯⠟⣁⡴⢳⣾⠃⢀⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⢶⣦⣮⣍⠁⠀⠀⠀⠀⠠⠶⠿⠷⠤⠰⠿⠒⢛⣍⣡⠀⠀⣿⣿⣰⣿⡟⠉⠉⠻⣖⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠃⠀⠀⠀⢀⣰⢶⠗⠀⠈⠉⠛⡀⠘⠃⣿⢿⣿⠁⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣼⣏⣀⣠⣴⠿⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣰⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣿⣅⣉⣉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠷⠊⠙⠉⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣿⣟⠿⠿⢷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⢠⠠⠃⠀⠀⠀⠀⠀⠴⣻⣷⣵⣆⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⣰⠽⣿⡿⣋⡬⣿⠀⠈⠀⡈⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢴⡁⢹⣧⣠⣹⣿⣆⢀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠁⠀⢀⣩⠾⣿⡾⣥⣾⣷⠤⡀⠀⠐⠀⠘⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠉⣹⣭⣿⡋⠥⣺⠏⣻⣷⣷⣦⣄⣀⡀⠀⣀⠀⡂⢀⣉⣀⣴⣾⣿⣱⢊⣽⣓⣺⣿⣾⡷⠏⠀⠁⠀⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢛⣡⠀⢀⠉⠉⠛⠻⠿⠿⢿⣿⣿⠿⣿⡿⠿⠿⣿⣻⢍⣫⡙⣌⠛⠉⠀⠀⠀⠀⢀⠠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣥⠬⠋⠀⣴⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⠋⠛⠉⠀⠀⢀⣥⢴⢊⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⣀⣀⣀⣀⠠⣄⣀⠀⡀⠤⣶⢎⢃⠈⠿⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⠂⠀⠂⠁⠁⠀⠀⠀⠘⠓⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠻⠿⠟⠛⠉⡀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡶⠄⠀⠀⠀⠀⠀⠀⠀⣴⢿⠿⠂
⠀⠀⠀⢀⣀⣤⣤⡀⠀⠐⠾⣿⣿⣾⣿⣿⣶⣖⣒⣠⣤⣶⣾⣿⣧⡤⠶⠖⠛⠀⠀⣠⣶⣿⡆
⠀⠀⠀⠈⣿⣿⣿⣿⣶⣤⣀⣀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠉⠀⣀⣀⣀⣤⣶⣿⣿⣿⡯
⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠿⠟⠟⠛⠋⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⠒⠚⢛⣉⣉⣀⣀⣀⡀⠀⠉⠉⠛⢶⣄⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠉⢁⣠⠴⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠳⢤⣀⣀⣽⡇⣠⠖⢉⡤⢤⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠿⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠚⠋⠉⠈⣿⢃⠔⠁⠀⠀⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣦⣀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⢠⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⠻⣟⠻⢯⣉⡉⠉⠉⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡴⠃⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠳⢦⣄⡉⠉⠓⠲⢤⣀⠀⠈⠙⢦⡀⠀⠀⠀⠀⡇⢀⡴⠊⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠙⠓⢦⣄⠀⠈⠙⢦⡀⠀⢻⡀⠀⠀⢠⡗⣁⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠱⡄⠈⡇⠀⠀⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⡇⢰⠇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣧⠏⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠈⢳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢏⣠⣾⠣⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢾⡁⠉⠙⠛⠛⠲⠶⠶⠶⠶⠾⠛⠋⠁⠈⢳⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈⠙⠶⢤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠤⠴⠚⠁⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠤⠤⢀⠀⠐⡇⠀⡇⠀⠠⡤⠠⠤⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⣀⠒⠈⠉⠉⠁⠀⠀⠀⣄⣃⣀⣇⡀⠀⠀⠀⠀⠈⠉⠐⢢⡉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠲⢄⣈⡙⠓⠀⠴⠦⠤⠤⠀⠀⠉⠈⠁⠀⠀⠀⠄⠀⠀⠒⢛⣉⡠⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠤⠀⠉⣉⠁⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⢉⠁⠤⠔⢂⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠉⠐⠒⠀⠠⠤⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠒⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡠⠄⠒⠒⠂⠀⠉⠉⠀⡇⠀⡇⠉⠉⠉⠀⠀⠒⠒⠠⣄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠲⠄⠀⠆⢠⠀⡀⠀⢀⣀⢀⣀⣀⣤⣄⡄⠀⠰⠒⠉⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣶⣶⣤⣤⣤⣀⣤⣄⣀⣠⣀⣀⣠⣀⣀⣤⣤⣤⣴⣶⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⣿⣿⣿⣿⣿⡿⣿⣯⣿⣿⣏⣶⣷⣶⡾⣿⣿⣿⡇⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣯⣿⣷⣯⣝⣻⣿⣭⣽⣷⣾⣿⣿⣿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⢿⣻⣿⣻⣿⣿⣿⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⣿⣿⣿⣿⣾⣯⣭⣽⣾⣿⡼⣿⣿⣿⣽⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⢟⣯⣵⡟⣿⣿⣮⣿⣿⣾⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢿⣿⣟⣽⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣞⢿⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⣿⣿⣷⣝⣿⣿⣫⣷⣿⣿⣿⣿⣿⣿⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠂⠤⠤⠀⠀⢀⣀⠀⠀⠠⠤⠄⠒⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠒⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣶⣤⡀⠀⠀⣤⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⣿⣆⣾⣿⣿⣥⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢀⣀⡀⠀⠀⢀⣾⠿⠿⢿⣿⣿⠟⠻⣿⣿⣟⠛⠛⠳⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠈⠻⣿⣧⠶⣶⣤⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠈⢿⣦⣴⣿⣿⣿⣴⣶⣶⣤⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣷⠻⠄⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠙⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⣀⣀⣀⣘⣁⣀⣀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⢿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠈⠉⠁⠙⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡼⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣠⠏⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⠀⡼⠃⠐⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢹⢷⣇⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⣹⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢹⡿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀⣀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢇⣼⣦⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣬⣍⣛⡻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠀⠀⢸⡏⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠑⠒⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢋⠀⢠⡀⠀⠀⣠
⢰⡆⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⣿⣿⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠘⢣⠛⠃⢶⡄⠉
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠙⢿⣷⣌⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢳⣺⣮⠁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣮⡙⠳⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⣼⢛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣟⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣀⣠⡶⢯⣼⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠿⢿⣵⠆⢲⡎⠁⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠿⠿⠛⠛⣿⢶⣶⡠⢴⠆⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣼⣦⣤⣬⣛⣡⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣟⣻⣋⣜⣛⣟⣛⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣭⣭⣽⣿⣯⣭⣭⣭⣭⣽⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠸⡏⠀⢜⡤⠠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⡴⠋⣑⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠐⠏⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⣀⡠⠂⠀⠀⠀⠀⢀⠤⠂⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⣠⣴⣶⣶⣶⣶⣤⣄⠀⠄
⠀⠀⠀⠀⠀⢀⠦⡤⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⡸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣸⠄
⠀⠀⢀⠄⠀⠐⠈⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠑⠬⡻⢿⣿⣿⣿⣿⣿⠿⢫⠎⠐⠢⡀
⠀⠀⢈⢀⠎⠇⠀⢀⠔⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠂⠀⣌⠀⠀⠑⡂
⠀⠀⠀⠑⠒⠒⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⢀⠀⣸⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠈⠑⠂⠀⠀⠀⠈⠁⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠃⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠵⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢀⡠⡺⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠔⠡⠊⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣈⠖⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⢶⢶⡒⢒⣲⣞⣶⣶⢦⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠉⡀⠚⠉⢸⣿⠁⠈⠋⠀⠈⣿⣿⡿⠋⢩⣻⣿⣷⣶⣄⡀
⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠸⠀⠀⠀⠘⠿⣦⡀⢀⣤⠾⠛⠁⠀⠀⣼⡟⠀⠙⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠠⠄⠀⠄⣤⠀⠁⠘⠁⣀⣤⡤⠶⠛⠉⠀⢀⣼⣿⣿⣿⡟⡁
⠀⠀⠀⠀⠀⠙⠿⣿⣶⣄⡀⣄⣤⣤⣤⣤⣭⣳⣤⠞⣋⣡⣤⣤⣴⣶⣶⣿⣿⣿⡿⢟⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠾⢷⣧⣤⣄⣙⣿⣯⣾⣿⣿⣿⣿⡿⠿⠿⠛⠋⠡⠀⠉⣹⡟⠀⠐⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡠⣿⠃⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠃⠀⠀⠀⡠⠔⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣺⠃⠀⠐⠒⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠁⠀⣠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⢶⣶⣶⣶⣶⣾⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠄⠤⠤⣴⣾⣧⣤⣤⣤⣤⠤⣤⣽⣿⣽⣿⣿⣤⣀⣐⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⣿⣿⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⣿⣟⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⣸⡿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣟⢠⡟⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣯⡿⠋⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⣿⣿⡟
⠀⠀⠀⠀⢀⣀⡤⢐⢦⣴⣤⣶⣶⣶⣿⣷⣦⣤⣀⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠈⠈⠉⠀⠀⠃⢰⣝⡟⠿⠛⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠆⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣓⡿⠿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠐⠊⠀⠘⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣶⣶⣦⣴⣦⣷⡘⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⢁⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣟⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢋⣾⢿⣿⣿⣿⣿⣿⠉⢿⣿⣿⣿⡷⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣇⢈⡉⠛⢿⣿⠁⠀⠀⢹⣿⣿⣟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠉⢭⢍⣉⣛⡃⣰⣿⣔⣰⠀⠛⠋⢻⣿⣿⠀⢸⡟⠈⠀⠀⢸⣿⡀⠀⠀⣸⣿⣿⣿⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⢰⢰⣶⡦⡇⡤⡤⠤⡤⠀⢻⠁⢸⣿⣿⠀⢸⣦⠀⠀⠀⠋⢸⣿⣷⣿⣿⣿⣿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠠⣿⠷⠇⣿⣿⣟⣿⠀⢸⡇⢸⣿⣿⠀⠺⠿⣧⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⢸⡄⣿⣷⡧⡀⠀⢀⢀⠀⠸⠃⠸⠃⠀⠀⢂⣁⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣋⣤⣤⠂⣭⣸⣍⡉⣀⣞⡇⠀⠀⠠⠀⢨⣭⣭⣭⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠛⠃⠙⠺⠃⠀⠀⠈⡃⠄⠀⠀⠀⢐⣓⢶⣶⠿⣿⢿⣿⠟⠁⠀⠀⠉⠉⠉⠻⢿⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠛⢳⠀⠀⠈⠁⠀⠀⠀⠀⠀⠿⠬⡧⠀⠉⠀⠻⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠖⠀⠂⠂⠀⠀⢛⣛⢠⠀⠀⠀⠀⠄⠀⠀⠀⣛⠐⡓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠒⠲⠆⠀⠆⣭⣭⠀⠀⠀⠀⠀⠑⠀⠀⠀⣭⠬⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⠀⠴⠶⠀⠀⠀⠀⣶⢄⡀⠀⠀⢛⢒⣿⠆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡐⠤⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣇
⠀⠀⣾⡏⠈⢙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠂⠌⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠹⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡦⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢠⢀⣀⠴⠊⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡌⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠚⠃⠀⠈⠙⠋⠂⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣀⣟⠶⣤⣶⣾⣿⣿⣿⣿⠶⠴⢞⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⣤⡜⢶⣛⢇⡀⠩⠘⣿⣷⣶⠆⠀⠀⠀⠀⠻⠛⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡶⠋⠐⠾⠿⠿⠋⠶⢶⣤⠀⠳⡄⠀⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠔⢉⣤⣶⣤⢦⣤⣤⣶⣶⣾⣿⠀⠀⡇⠀⣻⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠉⠜⠉⠀⢀⣤⣄⢀⣀⣤⣽⠀⠀⣷⠀⠈⣿⡟⠅⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠴⠾⠟⠛⠛⠻⠿⣿⡏⠛⠁⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⠴⠶⠶⢶⣶⣦⣼⡿⠄⠀⠀⠀⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⣤⡤⠶⠛⠁⢠⣤⣤⣤⣤⢀⠉⡯⠙⢦⣄⡀⠀⠀⠐⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠂⣀⣠⣁⣀⢠⣴⣦⣭⡿⠿⠿⠟⠛⠁⠀⠀⠈⠉⠉⢻⡁⠰⣿⣿⡃⠀⠀⠀⠀⠀⠹⢗⡇
⠀⠀⠀⣴⠞⠋⠁⠈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠿⠷⠦⠤⠤⠤⠴⠖⠚⠉


⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣠⠴⢒⣚⣩⣭⠭⠭⠭⠍⠉⠩⠭⡍⢉⣓⡒⠦⢄⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⠔⠋⢉⣀⣠⣤⣤⣤⠤⣤⠤⣤⣤⣤⣀⡀⠉⠙⠶⣍⢦
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠰⡇⠻⣴⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⣄⠞⢸⡟⠉⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⣌⣙⠛⠿⠿⢿⣿⣿⣷⣤⣾⣿⣿⡿⠿⠿⠛⢋⣡⠔⢹⠟⠉⢹⡆⠈⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⠉⠓⠒⠲⠲⠦⠀⠀⠠⠤⠴⠖⠒⠒⠋⠉⠀⠀⣸⠀⢀⡼⠁⣰⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠖⢋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠠⠂⠂⠀⠀⠀⠀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠉⠳⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢴⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⡇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠈⠓⠦⣄⣈⡉⠓⠒⠒⠒⠒⠒⠚⢉⣁⣠⠴⠊⠁⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣬⣉⣉⣉⣉⣉⣉⣉⣉⣉⣠⡤⠤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣠⣄⢠⡤⣄⢠⠤⡄⢠⠀⣤⠀⣠⠄⠈⣄⡤⢤⢀⡶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡤⠸⠧⠿⢸⡧⠞⢸⡤⠏⠸⠦⠿⠀⠳⡤⠀⠟⠇⠸⠻⣄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠃⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠠⠆⠉⠈⡈⠉⣁⠉⠀⠀⠀⠀⣼⣿⣿⠿⣽⣶⣶⣶⣮⣉⠙⠁⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠴⠀⠀⠀⠀⢀⠀⢲⣂⣀⣤⣤⣶⠂⠀⡀⠀⠁⠘⠛⣩⠉⠿⣿⣿⡆⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠄⠀⣸⠃⠀⠀⠀⠀⠀⢈⡀⠁⣙⡂⠐⠛⢿⢷⣄⡀⠈⠃⣬⣿⡿⠁⢘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢌⣙⠀⠤⣄⣀⠀⠈⠉⠀⠩⠁⣠⡀⠀⢐⠛⢛⣛⡻⠞⠋⣤⡞⢾⣏⣀⣄⣈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠻⣶⣶⠄⠀⠀⠀⡀⠐⠓⠛⠓⠂⣙⡋⠩⣷⣴⡀⢠⡿⣧⣿⡟⠁⠀⠙⣯⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠁⠀⠀⠉⠄⠈⢸⠏⣿⠇⠀⠀⣠⡿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⠶⠶⠛⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠷⠟⠛⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡴⣿⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢀⣶⣿⡿⢟⣳⡄⠈⠉⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢘⣿⣉⣻⣷⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⠀⠀⣛⡾⣿⣟⣩⣿⣥⡀⠀⠱⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢁⣯⣽⡟⠉⣻⠏⣿⣾⣧⣤⣀⣀⠀⣀⢀⠀⣀⣁⣠⣶⣿⣫⢊⡽⣒⣿⣿⣷⠞⠀⠀⠀⢘⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⣥⡀⣌⠉⢙⠛⠻⠿⠿⠿⠿⠿⡿⠿⠿⣿⣟⢍⣯⣱⣝⠋⠁⠀⠀⠀⢀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠞⠁⠰⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠉⠉⢁⠀⢀⡮⢶⡋⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠰⠤⠦⠤⠤⠤⣄⣢⣤⣄⣤⡐⠿⠯⠁⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⠀⢀⣔⡄
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠠⠽⠚⠁⠀⠀⠀⠀⠀⠀⢰⢢
⠀⠀⢀⡀⠣⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢤⣀⣀⣀⣀⣀⡀⠀⠀⢀⡴⢂⠭⠯⠭⢭⡔⢄⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠓⠒⠃⡬⠤⡀⠀⠀⠀⠀⠀⠀⠰⣉⣀⣀⣀⣀⣀⣀⣀⣈⡆⢀⣎⢐⣣⠒⠒⠔⠋⠙⠢⣧⠀⠀⣀⣀⣀⣀⣀⣘⣀⣀⡀
⠀⠀⠀⢰⠒⠁⠀⠀⠀⠀⠀⠈⠹⢤⠀⢅⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡤⠭⠭⠭⠭⠭⠭⠭⠭⡼⠀⠸⡄⠤⠤⠤⠼⠼⠤⠤⠤⡜
⠀⠀⠀⠀⣏⣉⣉⣉⣉⣁⣀⣉⣹⠇⠀⢸⠘⠐⠒⣲⣒⡒⠐⢻⠀⠀⡧⢤⣤⣀⣀⢀⣠⡤⢤⡇⠀⠀⡗⠒⠒⠒⡖⠖⠒⠒⢚⠃
⠀⠀⠀⠀⡆⠀⠀⣠⣀⣀⠀⠀⢸⠀⠀⢸⠀⠔⡞⠛⢛⠠⠀⡈⠀⠀⢰⠀⠀⠠⠄⣤⠀⠀⢰⠀⠀⠀⢃⢖⠉⡆⡇⡀⡁⣀⢸
⠀⠀⠀⠀⢠⠀⠀⠣⣉⠜⠀⠀⠸⠀⠀⠸⠀⡀⡄⠇⢠⢀⠀⡇⠀⠀⢘⠀⡄⣬⣥⣄⡄⡆⠘⠀⠀⠀⢸⠈⡝⢀⡇⡇⠤⠘⡀
⠀⠀⠀⠀⠸⠤⠀⢀⣈⡀⠀⠤⡄⠀⠀⠀⡆⠀⠉⠉⠉⠁⠰⠂⠀⠀⠀⡤⢉⣁⣀⣀⣁⠠⡆⠀⠀⠀⠀⡆⡈⠀⡇⡇⠀⠀⡇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢇⠀⠀⠉⠁⠅⢁⠃
⠀⠀⠀⠀⠀⠈⠀⠐⠒⠒⠀⠉⠀⠀⠀⠀⠈⠀⠐⠒⠂⠀⠁⠀⠀⠀⠀⠈⠁⠀⠒⠂⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⡠⠄⠶⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢚⣉⣉⣉⣩⣍⣉⣉⣉⡓⠲⢤⣀⠀⠉⢕⢆⡀
⠀⠀⠀⠀⠀⠀⡰⢊⣴⠖⠛⣙⣠⣤⣤⣴⣦⣴⣄⠠⠉⠙⠒⢬⡓⣴⠀⠀⠉⠙⠲⡄⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠙⣦⠀⠀⠀⠙⣾⡗⢦⡂⢀⡤⠟⠛⠚⠁⣀⡀⠀⠀⢀⡼
⢀⡠⠴⠶⠶⣾⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⡞⠁⠄⠌⢈⡴⣽⣷⠏⠉⠉⠀⡖⣠⠴⠾⠇⠀⠀⠀⡈
⡎⢠⠟⠉⠳⣾⣿⠇⠩⠛⠻⠿⣿⣿⣿⣿⣿⣿⣽⡧⠤⠖⣛⣥⣦⣿⠀⢀⡔⣡⡼⠿⠧⠤⠀⠁⢀⠤⠚⠳⢤⡀
⢧⠘⢆⣠⠴⠛⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣾⣿⣿⣿⡿⠻⣾⠟⠛⢤⡀⠀⠀⢀⣰⡧⠤⣦⠥⣤⡭⠑
⠈⠳⣄⡙⠲⣤⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⡧⢿⣿⣿⣿⠇⠐⣮⢳⡄⣒⣀⣠⠞⢧⡀⠓⠄⠀⠀⠀⢉⣀
⠀⢸⠁⠻⠒⠦⣽⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⣧⣵⣿⣿⡟⠀⣤⠗⣕⢻⣯⣤⣍⡓⣤⣁⣀⡤⠖⠒⠒⠋
⠀⢹⠐⠈⠀⠀⠀⠙⢆⠀⡠⣄⣀⣠⢰⡄⣧⣶⣼⣼⣿⣿⣿⣿⣿⡧⣀⣤⡿⣸⣧⣈⣹⡿⠋
⠀⠈⢧⠀⡀⠀⠀⠀⠀⠙⠾⢿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣏⣡⣧⣥⠐⣠⣿⣏⣷⣸⡇
⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⡿⣿⣿⡿⠭⠉⠉⠉⠁⣠⣾⣿⠟⢩⣿⣟⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠉⠲⢤⣍⣃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⢛⣻⣿⣗⠀⠘⢿⣋⣉⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠲⢖⠶⡶⠶⠟⠿⠻⠟⢷⡾⠿⢛⣻⣿⣴⠶⣎⠉⢀⣤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠑⣿⠹⣷⡘⣇⢾⠛⠿⠷⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣙⡷⠏⠈⠙⠛⠛⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⠰⠶⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠖⣊⣉⠭⡤⠤⢤⠤⠬⢭⣙⡒⠦⣀⠀⠻⠷⣀⣀
⠀⠀⠀⠀⠀⡰⣡⣶⣋⣭⣥⣴⣶⣶⣶⣶⣧⣔⠀⠈⠑⠪⡻⣄⠀⠀⠈⠑⣄⣀⣠⠄⠀⠤⠄⠠⢄
⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣺⠗⠀⠀⠀⡹⣏⣹⠦⠤⠚⡁⠀⣀⡴⠂⠈⠀⠠⠊
⡴⢊⡭⠭⣙⣿⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⢿⡇⢀⣠⠤⣚⣹⡟⠋⢀⠄⣸⣷⣚⡉⠉⠀⠀⣠⣆
⣇⢸⡀⢀⡼⣿⣿⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⣍⣴⣿⣿⣿⢧⣤⡷⢞⣉⠁⠀⠀⠀⣴⣉⣠⣄⣙⡢⢄
⠘⢦⡙⠯⣄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣿⠀⣌⢧⡀⠭⠀⢠⢶⡉⠻⠀⠉⠈⠉⠂⢘
⠀⣼⠙⡷⠦⣽⣟⠣⠀⠀⠀⠀⠀⠀⠀⠀⣌⠙⣿⣻⣿⣿⠇⢀⡿⣧⢿⣯⣭⣓⢦⣉⣀⣠⠤⠤⠶⠋⠉
⠀⢿⠠⠂⠀⠀⠙⢦⡀⣠⣄⣀⣄⣠⣀⣦⣿⣼⣿⣿⣿⣿⣶⣄⣠⡿⣻⣧⣩⣿⠟⠁
⠀⠘⢆⠠⠀⠀⠀⠀⠙⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⠤⠆⣱⣿⣍⣧⣽⣀⡀
⠀⠀⠈⠓⢤⡀⠀⢀⡀⠀⠀⠈⠙⠛⠿⠟⠿⠿⠟⠛⠉⠉⢀⣠⣾⣿⠛⢽⣿⣭⣿⢿⡆
⠀⠀⠀⠀⠀⠈⠓⠲⠤⣍⣁⣀⣀⣀⢀⣀⣀⣀⣠⣤⣶⣞⣩⣿⣿⣿⢀⡘⠿⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠛⠉⠉⠉⠈⠿⣯⠭⠶⣿⢿⣝⢧⣀⣾⣯⣽⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣎⢿⣼⠏⠳⠬⡅⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠶⢲⠋⠀⠀⠀⠀⠈⠁⠒⠲⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣿⡉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠚⠛⠋⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠦⢤⣉⣙⣓⣒⡒⠒⠒⠒⠒⢒⣒⣚⣋⣩⡤⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣶⣤⣀⣀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣠⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣯⣭⣭⣿⣛⣒⣒⣒⠒⠲⢬⣭⣽⠟⠛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣻⣿⣿⡇⢿⡇⠇⠀⠀⠀⠀⠀⠀⡸⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠛⠒⠒⠂⠠⠤⠤⠤⠤⠤⠶⠗⠒⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢤⡠⣌⢳⢳⡀⣀⠒⢢⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⣤⣄⣈⣉⡉⠈⠣⠣⠑⠓⣚⣀⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⣿⣿⣿⣿⣻⠉⠉⠉⠉⠉⢩⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⡏⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠃⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⣿⣿⣿⠿⡁⠀⠀⠀⠀⢀⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠛⠛⠒⠀⠀⠀⠀⠀⠊⠙⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣤⡤⡄⠤⠤⠤⢤⡤⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣁⣀⣀⣀⣨⡥⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⠀⠀⡸⠀⠀⠈⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠉⠀⠀⠐⠂⠐⠚⠑⠁⡀⠀⠊⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠄⠀⠀⠀⠀⠀⠠⠀⠀⠒⠂⠀⠀⡇⢀⢀⡠⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡛⠋⠩⠶⠂⡀⠀⠀⠀⠄⠠⠭⠥⠤⠄⢀⡁⠂⠈⠀⠉⢠⣖⡭⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⠿⣿⣿⡶⢶⢶⣦⣥⣤⣭⣭⣭⣭⣤⣤⣭⣤⣾⣶⣶⣾⣦⣽⠖⠉⣀⠤⠒⠊⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠍⠉⠿⣿⣿⣿⡿⠅⢺⠅⢸⠅⢀⡤⠖⠒⠶⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢁⣭⡭⢭⣿⣖⣾⠅⢐⡡⢸⡴⠋⠀⠀⠀⠀⠸⡆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⣈⣭⡭⠵⣖⡋⠀⠊⢸⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠥⠀⠐⣒⣛⣩⡭⠄⢈⠁⣾⠀⠀⠀⠀⠀⣠⠞⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣈⢋⣩⡭⠥⠶⣒⠀⠄⠀⣟⠀⠀⣀⠤⠚⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣒⣚⣛⡭⠅⠒⡁⠀⡯⠌⠀⢀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠭⠭⠷⢖⣚⣋⠀⠂⣸⣡⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⢘⣫⣭⣭⡷⠠⠎⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣳⣌⠠⠄⠀⠀⠀⠀⡀⢀⣀⠒⣀⣐⣲⣿⣿⠯⠀⣩⣾⣁⣀⣀⣀⣀
⠀⠀⠀⣠⡤⠶⠒⠚⠉⠿⠉⠀⠀⠤⠤⠤⠿⣗⣦⢄⠀⠀⠀⠐⠀⢛⣀⣿⣿⣿⠿⠭⣴⣾⡿⠿⠿⠿⣯⣭⣍⣉⠛⠛⠲⠶⣦⣤⣀
⠀⠀⢸⡁⣄⣠⡀⠀⠰⠅⠀⠀⠀⠀⠐⠄⠀⠈⠓⠤⢤⣄⣀⣀⣀⣉⣍⣱⣶⣶⣦⣴⠶⠋⣠⡤⠂⠀⠠⠀⠀⣈⡭⠆⢀⡄⣈⣷⣽
⠀⠀⠀⠉⠳⢯⣝⣓⠒⠶⠦⠤⠤⣥⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣭⣤⣄⣀⣀⣤⣶⣶⣶⣯⡤⠤⠶⠶⠒⢚⣉⣭⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠦⢤⣤⣤⣤⣄⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣠⣤⣤⣤⠤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠠⠶⠶⠦⠄⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠒⠊⠉⠀⠀⠀⠀⢀⣀⣠⣤⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿
⡤⠤⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠒⠈⠉⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⣿⣿
⠀⠀⢀⣴⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿
⢀⣘⣻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣆⠀⠠⠄⠀⠤⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠹⣿⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠙⠿⣿⣿⣶⣬⣽
⣿⣿⣿⣇⢹⣧⣤⡤⠀⠀⠀⠀⠀⠀⢠⠀⡾⠀⢀⣀⣀⣀⣄⣀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿
⣿⣿⣿⢿⣟⠛⠉⠀⠀⠀⠀⠀⠀⣀⣘⣒⣛⣤⣿⣭⣭⣿⣿⣶⡶⠿⠿⠷⠆⠁⠀⠀⠴⠶⠾⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣄⡁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣽⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⡀⠀⠀⢀⠀⢀⡀⠀⠀⡀⠀⠀⡀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⢸⣷⣤⣴⣷⠀⠇⠳⠢⠎⢡⢰⠉⠦⠴⠉⡆⣸⣷⣤⣾⣿⠀⠰⠹⠖⠴⡍⡄⠀⠀⡘⡡⠤⢼⠱
⠀⢺⣿⣿⣿⣿⠀⠀⡄⠀⣤⢸⢰⢰⡆⢠⡖⡆⣿⣽⣿⣯⢹⠆⢀⢠⠀⢠⡄⡇⠀⠀⠃⣤⠀⣠⠈
⠀⠘⣿⣿⣿⠟⠀⢦⡀⠒⢀⠌⠘⣌⠑⠊⣡⠃⠹⣿⣿⣿⡿⠀⠹⣄⠀⠀⡰⠃⠀⠀⢧⠀⠀⠈⡸
⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠙⠉

⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠶⠶⠶⠶⣶⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣺⣟⠿⢋⡁⠄⢀⡠⡀⠉⠁⠒⠬⢍⡚⢗⣶⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠖⣿⡟⠁⣢⠔⠊⢀⠀⠈⣣⠐⠀⠀⠀⠐⢈⡳⣝⢷⣬⠷⠒⠢⣄
⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⣾⠏⠄⡜⠁⠠⠚⠀⠀⠀⠁⠀⠀⡀⢠⢀⠀⠑⠈⠎⢿⣦⣴⠖⢸
⠀⠀⠀⠀⠀⢀⡞⡔⠁⠀⢸⣼⠈⠐⠀⠀⠀⠁⡀⠀⠀⢴⠀⡠⢁⠘⠀⢠⠀⠀⠀⠸⣼⡇⣠⢾⣧⡀
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣿⢿⢀⠀⠰⡀⠀⠀⠈⠢⠄⢄⡌⠀⢈⠄⡀⡸⠀⠀⠀⠀⣏⣿⣥⣀⡀⠀⠀⠀⠐⢆
⠀⠀⠀⠀⢰⠃⠃⠀⠀⠀⢻⡸⡌⠄⢆⢱⣔⢄⡀⠀⠀⢀⠁⡴⡣⣒⠔⠁⠀⠀⡔⣸⣹⣿⡋⠹⠉⠛⡖⠦⠴⠃
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⣧⡹⡄⠀⠀⠙⠻⠿⣛⠳⢦⡁⣴⠿⠋⠀⠀⢀⠜⡰⢣⡿⠉⠁⠀⠀⠘⣇
⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠘⣷⣜⢦⡈⠓⠤⣀⠀⠉⠢⢹⠇⠀⠀⠀⢐⡡⢞⣴⡿⠁⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⢮⣓⠦⢤⣉⣁⣀⣀⣀⠤⠴⣒⣩⠖⣫⠟⠁⡀⠀⢀⠂⠀⠀⡟
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠠⠈⠻⡷⣮⣍⡓⠒⠒⠲⠖⠲⠚⢋⣡⡴⠛⢁⠐⠀⠀⠀⠄⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠡⡡⡈⠢⡙⠻⠛⠟⠒⠒⠛⠛⠉⠁⠀⠀⠁⠈⡀⢠⢊⠄⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠪⡢⢀⠕⠀⠄⡀⠀⠀⠀⠀⢀⠠⢀⠠⣊⠔⡡⠃⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢂⠀⠄⠀⠀⠀⠀⠐⠈⠀⠈⠐⠉⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢽⣶⣦⣤⣀⣀⣀⡀⠀⣀⣀⣀⣀⡤⢴⣒⡭⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠯⠽⠿⠯⠿⠷⠒⠛⠉⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠋⠉⠈⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣀⣤⣤⣶⠏⠐⠲⠦⢤⣀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠞⠛⠛⠛⠂⠀⠀⠀⠐⠛⠋⠛⠳⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡄⠀⠰⠞⣿⠲⡄⠀⠀⠀⡤⢲⡟⠶⠄⠀⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣇⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠈⠁⠋⠀⢀⡿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠀⠀⠀⠀⠛⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣘⡲⡄⠀⢀⣴⣲⡲⣄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢳⢳⠀⡐⣌⢁⠀⢀⠈⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡏⣇⢳⠘⣮⣉⣩⠞⠀⠀⡜⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢸⣸⡆⢳⡀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠃⠀⢳⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣄⣀⡀
⠀⣰⣾⠿⢟⣛⣫⡭⠭⠭⠭⠒⠒⠒⠒⠒⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠩⠉⢛⣛⡻⢿⣶⡄
⠀⣿⡟⠻⠧⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠿⠟⣿⡇
⠀⣿⡇⠀⠀⠁⠀⠂⠉⠿⠭⠭⠽⠿⢿⣛⣛⣛⣛⣛⣛⣛⣻⠿⠿⠯⠭⠭⠿⠛⠚⠃⠀⠁⠀⣿⡇
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠛⠉⣉⣉⣉⡉⠛⢿⣶⡄
⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⣾⠿⠟⠛⠛⠿⣷⣄⠙⣿⡆
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⣾⡿⠁⠀⠀⠀⠀⠀⠈⣿⣇⢹⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢠⣿⡇
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣸⣿
⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⢰⣿⠇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⣰⣿⠋
⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣴⣿⠟⣡⣾⡿⠁
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣠⣴⣾⠿⠋⣡⣾⡿⠋
⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠟⠛⠉⢀⣤⣾⠟⠋
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣾⡿⠋⠁
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⣿⣷⣶⣶⣶⣶⣾⣿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣸⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣰⡿⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⢠⣾⠋⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢠⡟⠁⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⡟⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠒⠚⠋⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠐⠒⠠⠄
⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠠⠴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠦⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⣧⠈⠛⠲⠤⣄⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⠉⠉⠙⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣀⡀⠀⢀⣀⣴⠟⠃
⠀⠀⠀⠀⣀⣤⣶⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡡⠔⠛⠛⠻⢉
⠀⢀⣴⣾⠟⠋⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠁
⣰⣿⡿⠁⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡿⠋
⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢍⣛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁
⠀⠈⠻⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣽⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣷⡿⠓⠀⠉⠁⠚⢷⣶⠀⠀⠀⠀⠀⣷⣾⠗⠂⠀⠀⠒⢿⣶⡇⠀⠀⠀⠀⢺⣾⠶⠒⠀⠀⠒⠾⣷⡇
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠟⠁⢀⠀⠀⢀⡀⠀⠙⠇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠆⠀⠻⠟⠀⠘⠿⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠲
⠀⠀⠀⠀⣂⠀⠀⠀⣭⠀⠀⠀⣈⠀⠀⠀⠀⠀⢠⠀⠀⠀⣬⡄⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢈⡁⠀⠀⢀⡂
⠀⠀⠀⠀⠈⠓⠠⢀⣀⡀⠄⠚⠁⠀⠀⠀⠀⠀⠈⠓⠠⢀⣀⣀⠠⠐⠃⠀⠀⠀⠀⠀⠀⠙⠢⢀⣀⢀⡀⠔⠋


⠀⠀⠀⠀⣿⡿⠒⠀⠀⠀⠒⠿⣿⠀⠀⠀⠀⠀⣿⣿⠖⠂⠀⠀⠒⢾⣿⡇⠀⠀⠀⠀⢺⣿⠗⠂⠀⠀⠐⠶⣿⡇
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠥⠿⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠸⠁⢀⠀⠀⠀⡀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠑⠀⠀⠀⠂⠀⠆⠀⠀⠀⠀⠀⠀⠸⡾⠀⠀⢷⠇
⠀⠀⠀⠀⣀⠀⠀⠀⣭⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢬⠄⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢨⡅⠀⠀⢀⠂
⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠈⠓⠠⠀⠀⠀⠠⠐⠋⠀⠀⠀⠀⠀⠀⠙⠢⠀⠈⠁⠀⠔⠋


⠀⠀⠀⠀⣷⡷⠓⠀⠉⠀⠒⢾⣿⠂⠀⠀⠀⠀⣷⣶⠖⠂⠀⠀⠒⢶⣶⡇⠀⠀⠀⠀⢲⣶⠶⠒⠀⠀⠒⠶⣶⡆
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠟⠁⢀⠀⠀⢀⡀⠀⠙⠇⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⡞⠀⠘⡿⠃⠀⡀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠉⠁⡀
⠀⠀⠀⠀⣂⠀⠀⠀⣭⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢈⠀⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢈⡁⠀⠀⢀
⠀⠀⠀⠀⠈⠓⠠⢀⣀⡀⠄⠚⠁⠀⠀⠀⠀⠀⠀⠑⠠⢀⣀⣀⠠⠐⠃⠀⠀⠀⠀⠀⠀⠙⠢⢀⣀⣀⡀⠔⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣾⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⡿⠿⠿⣿⣿⣿⣿⡿⣿⣛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣶⣭⣭⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠈⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠓⠿⠷⠀⠀⠀⠐⠛⠗⠈⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⣶⣦⠀⠀⠀⠐⣿⠆⠀⢸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠈⢙⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠐⢒⣲⠀⠀⠙⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠰⣏⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠈⠋⠁⢀⡀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠙⠛⠛⣇⣿⡇⠀⠀⠘⢷⡶⠶⠶⠖⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⣀⣿⡟⠁⠀⠀⠀⠀⠙⠶⠴⠞⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠳⣤⣀⠀⠀⠀⠀⢀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠸⡏⠛⠛⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠘⡇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿