Steak Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠈⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠱⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠚⠓⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⣀⣠⠤⠤⠶⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⢿⢟⣛⣟⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡄⣀⣸⣠⠄⡧⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⢹⣿⡟⣿⣿⣿⣿⡿⡃
⠀⠀⠀⠀⠀⠀⣰⠁⠀⢸⠀⠈⠉⠀⣸⣀⠀⠁⠀⡠⠒⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢸⣿⣶⣿⣿⢾⣿⣿⣿⣿⣟⣭⣾⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠐⠘⠉⠉⠉⠀⠀⡠⠋⠀⣠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣽⣿⣶⣶⣷⣿⣿⣿⣫⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⢿⠑⠦⢀⣀⣀⣀⣀⣀⠤⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣝⡿⣿⣿⣿⣿⣿⠿⣻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣶⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉


⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣠⠴⠚⣹⣇⣤⣾⣤⣨⡏⠀⠀⠂⠲⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢠⡤⡀
⠀⠀⠀⣠⠊⡁⠀⣸⣿⣿⣿⣟⣿⡿⠟⣇⠀⣀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣐⣤⣤⣄⡀⠙⣩⡄
⠀⠀⡼⠁⠐⢻⠉⣿⡿⢋⣷⠶⠺⡟⢠⢿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢀⣴⣿⢯⠿⡁⠉⠻⣿⡍⣶⠆
⠀⢠⠁⠀⠠⢾⠞⠛⡏⠀⢹⡘⢐⣧⠄⠸⠂⢉⣡⣤⠾⠁⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⢀⣤⣿⣿⣿⠁⠀⠈⢳⣄⣿⣿⠈⢡⡀⡀
⠀⣬⠀⠀⠀⣸⡤⠀⢿⠂⢻⡇⠀⣽⠛⠛⣟⠯⠙⠋⠁⠀⣀⣤⡾⠟⠒⠛⠚⠻⠷⣶⣀⡸⣿⣿⣻⣦⣀⣀⣼⣽⣿⣿⠁⠀⠻⠑
⠀⣷⠀⠁⠀⠈⡇⠀⢠⡄⠀⢇⣼⡇⠀⠎⠁⠀⢀⣠⣴⡟⠉⡀⡶⠠⠶⠤⣾⠤⣴⠤⣍⠳⣼⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢔⢀
⠀⠹⢤⡀⠀⠈⠋⠁⠈⠂⢀⣾⣽⠏⠀⠀⡰⣺⢉⣿⡏⠀⠲⢿⡧⠤⡦⠬⣷⣆⣼⣀⣸⣄⡛⢷⣾⣿⣿⠛⣋⢽⡿⠛⠳⣦⡀⡠⠘⢋⣠⡀
⠀⠀⠀⠈⠑⠲⠶⠦⠴⠞⠻⠚⠀⠀⠀⠀⢧⢿⣿⣿⡀⠀⠀⢴⡄⠀⡇⢀⣏⢉⣼⣿⣿⣿⣷⡇⠙⣿⣷⠓⢲⠙⣷⡀⣸⣿⡇⠀⠄⠊⠷⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣛⣿⣿⣦⠀⣸⡀⣀⡄⢃⣇⠀⡏⠻⣿⠁⢹⠋⠉⡇⠙⢶⣟⣾⣿⣿⣿⠟⠀⠀⠀⠀⠔⢃⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡼⣿⣿⣧⣌⠀⢈⣁⣬⣭⣬⣭⣭⣭⣡⣙⡃⠘⠷⠂⠀⣙⡟⠋⠁⠀⠀⠠⠐⣩⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⢿⣿⣿⣷⣿⢟⣻⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⢿⡿⠀⡀⠄⢊⣡⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢾⣿⣿⣽⢿⣿⣿⣿⣿⣿⢓⡀⡽⢘⣿⣿⠿⠋⢒⣡⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠻⡉⡀⠛⠛⣉⠥⠚⣿⣶⣥⣤⡾⠋⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣻⣤⠖⠈⠀⠀⠀⠾⠛⣛⣭⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣌⢄⣀⠀⢀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠶⣒⣂⣠⡶⠒⠀⣲⡞⢉⣠⣾⣿⠏⠉⠑⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣤⣤⠾⠟⠋⢉⡵⠋⠀⣀⣴⡿⠾⠛⢉⡿⠋⣀⣤⣾⠟⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⠞⠁⠀⢀⣴⣿⠵⠖⢛⡿⣁⠄⢀⣠⣾⡴⠾⢋⣽⠁⠈⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠤⢖⡾⠛⠁⠀⢀⠶⢋⡨⢴⡾⠋⠉⠁⠀⣠⡿⠷⠞⠛⣿⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣴⠋⠀⠀⢀⡴⠟⠚⠉⣠⠏⣀⡠⠤⢒⡞⠁⡀⠀⢀⡜⠁⣰⡇⠀⢦⣀⡄⢀⣤⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣧⢖⡟⠁⠀⠈⣠⣾⣷⣾⣦⣤⣴⣏⣄⣀⣠⣴⣯⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣛⣵⣯⣟⢻⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣟⣿⡿⠛⠋⠁⠀⠀⠈⠁⠈⠉⠙⠻⠿⢿⣿⡿⠟⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣴⠄⠈
⠀⠀⠀⠀⠸⣿⣴⣿⠃⠀⠀⠀⢀⡀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢄⣐⡭⠐⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠼⠋⢀⣄⡄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣀⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈



⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢿⡂⡖⣴⡆⢴⡆⣿⡔
⠀⢀⣓⣃⣓⣙⣃⣛⣃⣛⣛⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⡤⣤⣤⣖⣲⣶⡀⠀⡈⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣼⣧⣘⡆⠀⠠⠈⢷⡀⠀⠀⠻⡍⠛⠛⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠗⠶⣯⣀⠀⠉⢹⡞⠒⠲⠦⢷⣄⣀⣀⣘⣦⣼⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⣦⣀⠀⠀⢹⡏⠙⠷⣦⣷⡀⠀⠀⠀⠹⣄⠀⠉⠉⠛⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢼⠀⠉⠙⠓⠾⣧⡀⠀⠀⠙⣿⠲⠤⣦⣤⣿⣇⣼⢒⡤⣸⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢤⣼⣆⠀⠀⢀⠀⢹⡟⠓⠶⢤⣼⣆⠀⠀⠀⠉⣾⡏⣾⣅⠁⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⢄⠀⡀⢄⠈⠙⣿⠶⠦⣤⣘⣧⡀⠀⠀⠀⠹⡟⠛⠂⣰⣯⣾⣿⠃⠀⠀⠀⣟⡿⣤⡆
⠀⠀⠀⠀⠀⠀⠹⣼⣦⣶⣤⣀⡑⠀⢣⠀⠀⠈⢻⡷⢦⣀⠀⠀⢻⣀⣾⣿⠓⣹⠇⠀⠀⠀⣠⠈⢀⣿⡿⣴
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣶⣼⣧⣄⡀⠈⣇⠀⣩⣿⣶⣿⣿⡏⢋⣴⡟⠀⠀⠀⠈⠻⠟⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣬⠉⢪⡛⣼⣿⣿⣿⣿⣿⣿⣿⠛⢝⣛⣿⣃⣼⠗⠁⠀⠀⠀⠀⠐⠆⢰⣤⠀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠿⠷⠿⣿⣦⣿⣾⣿⣿⣧⣤⡷⠟⠛⠉⠀⠀⠀⠀⣀⠈⢁⡳⠆⠠⠐⠂⠘⣨
⠀⠀⠀⠔⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡇⠀⠁⠠⠍⠰⠏
⠀⠀⠀⠀⠤⠤⠭⠦⡶⣤⣄⣤⣤⣄⣀⣀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣍⡳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⢒⠺⢿⣽⣿⠿⢿⡟⠿⢷⣤⣷⣤⣄⣀⡀⠀⠘⠻⠋⠈⠹⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠶⠀⠈⠙⠛⠛⠛⠛⠉⠉⠒⠒⠦⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣩⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⢻⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⣄⡔⠁⠀⠀⠀⠀⠀⠀⠊⢀⣠⡤⠺⢛⠿⡀⠀⢸⠋⠁⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⠄⢀⡔⠉⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⣠⠞⣸⠁⠀⣾⠁⡅⠀⠉⣠⣤⡿⢋⡺⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠁⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢲⠋⠀⠐⢁⣀⣤⣾⢟⣻⣯⠟⠹⠿⣿⣶⣄⣭⣖⣤⡄
⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠀⠀⠁⠠⣤⣽⡛⠻⠷⠛⠛⠀⢀⣜⠀⠀⠀⠸⠁⠀⢡⢸⣷⡄
⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⣠⠆⠀⠀⠐⠚⠛⢛⡟⠃⠀⢠⠃⠀⠀⠽⠟⠂⠀⣾⣀⠀⢈⣿⣭⣿⠀⢀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⡜⠁⡀⢠⠟⠁⠀⢠⡞⠀⠀⢰⠋⠀⠀⣶⠧⠀⠀⣰⠛⢲⣶⣟⠿⣿⣿⢀⡌
⠀⠀⠲⢭⣀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣦⡋⠀⠀⣸⣏⣀⣀⣤⣟⣀⡀⣠⠟⠀⢤⣾⡏⠀⢦⣾⣿⢽⣏⣧⣿⣿⢟⣯⣾⡇
⠀⠀⠀⠀⠈⠉⠓⠒⠦⠄⠠⠂⠀⠁⢄⣄⣉⣻⣉⠉⠉⠉⠟⠛⠟⣿⣿⠻⣙⣷⣬⣿⣦⣀⣉⣙⣿⣿⠛⣁⣴⠿⣩⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠁⠀⠀⠀⢀⡀⠈⠀⠀⠉⠛⠃⠚⠳⠦⠽⠿⠓⠛⠛⠋⢁⠀⠈⠉⢈⡥⠒⠛⠁⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠈⢄⠀⠈⠳⣤⡶⣶⠟⠉⠀⠀⠀⠀⠀⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⡠⠞⠁⠴⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⡟⡟⠓⠤⢀⠀⠀⠀⠀⠀⠁⠀⠀⡀⢀⣀⣀⣀⣀⣀⡤⠖⠋⠁⠀⣠⠴⠒⠉⠀⢠⡦⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠛⠀⢀⡤
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠒⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⢤⣤⡑⣄⡐⠀⢤⣀⣌⣒⣒⣶⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠉⠁⠈⠈⠉⠉⠀⢿⠿⠟⠛⢛⣿⡿⢟⡫⢟⡿⠿⢉⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠡⠚⠁⠀⠁⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢶⢚⣫⣩⡩⠉⡉⣟⣒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣥⣾⣿⣼⣿⠟⢁⠲⢭⣒⡾⠭⣤⣭⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡵⡞⠿⢛⡁⣤⣠⡀⣄⡿⣇⠰⣮⣽⣛⣻⡶⠶⠌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣵⡧⠛⣀⣤⡳⣝⢮⡳⣽⡻⣽⣿⡆⣌⠙⠿⢭⣽⣟⣓⡦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡠⣤⡀⢾⣿⢿⣝⢮⡳⢝⡳⢽⣷⣿⣷⠺⢧⣤⠳⣶⠮⣭⣝⡂⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡻⣎⢿⡈⢿⣷⣽⡳⣯⣳⢭⣓⠋⣽⣛⡨⣿⣲⣦⠘⠛⠶⣭⡃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⢾⡷⣅⠀⢻⣶⣝⡳⢭⣓⣾⠀⡓⠾⣽⣳⢮⡟⠀⠀⢙⠮⢱⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠳⣝⠏⡀⢰⣝⠾⣝⡳⢾⣙⠢⣭⣛⢶⣉⡛⠶⡄⠀⣮⢃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠐⡄⢠⠁⣝⢯⣳⢮⣝⠷⣏⣙⠶⣍⡿⠁⣙⠷⣦⡘⢃⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣿⠁⣬⡳⣿⡳⣮⣻⡧⣝⡳⣮⣑⡇⣝⠷⡎⠁⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠀⢹⣿⣮⣝⡳⣭⣓⢮⡻⣮⣿⢰⣝⠟⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⣿⡿⢠⣎⢿⣿⣮⣝⠦⣝⢷⣝⢦⡹⣧⠙⠀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣸⣿⠁⡸⣿⣮⡻⣝⢮⡳⣌⢳⣝⣦⠻⠉⢀⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⢧⠋⡰⣝⢮⡻⣿⠀⡠⠅⠈⠁⠀⠀⣀⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢰⣎⠃⣰⣿⣜⢷⣹⠎⠀⠀⠀⠀⢀⢠⡆⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣴⡿⢃⣴⣝⣿⣿⠟⣁⠆⠀⢀⢠⡄⣿⢸⣧⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠺⡎⣷⠻⠞⠃⠀⣀⢠⡆⣿⢸⣷⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⢠⢠⡄⣤⣴⡇⣿⣸⣧⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣼⣧⣿⡾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠀⠤⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡀⠀⠀⠀⢁⠀⠀⢲⡀⣀⣀⣤⠠⠤⣤⠀⠈⠓⠀⢈⠀⠀⠐⠦⣐⣳
⠀⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⠀⠈⠻⠃⠤⠀⠀⠉⢄⠀⠀⠀⠉⠠⡄⠀⠀⠀⢓⣴⣧⣄⡠⠄⠒⢀⣠⣼⣿⠂⠒⠀⢀
⠀⠀⠀⠀⠀⠀⠒⢆⡀⠈⣙⡑⠀⠀⠀⠠⡐⣀⡶⠀⠭⢶⣀⠘⠗⠒⡒⣆⠉⠀⠐⣶⣚⢍⣻⡍⠩⢍⣉⣿⠤⠈⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠴⠋⠁⠀⠀⠀⠉⠉⠑⢄⡀⠀⠀⠀⠐⢦⣀⡤⠄⠈⠳⢊⣩⠀⣿⠟⡟⢢⡴⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⠀⠀⠀⠜⠲⣞⠙⠈⠁⠈⠒⣄⣀⣴⡿⠷⣄⡈⠠⠓⠁⠀⠀⠠⠄⠀⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠢⠀⠀⠀⢠⡵⢦⡖⠚⠉⢀⣿⣿⡀⠾⠋⠀⠀⠀⠀⠀⠀⠀⡈⣂⡠⠃⠛⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠓⠶⠚⠛⠁⠠⡤⣄⡀⠀⢀⣉⠷⠖⡯⢉⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠆⠈⢀⠀⣏⡁⠈⠘⢆⣰⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠙⠆⠀⠀⠀⠁⠀⠂⠰⠿⠋⠉⠀⡀⠀⠿⠟⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠛⠛⢦⡀⠀⠀⣀⣠⠤⠤⠖⠒⠛⠋⠉⠉⠉⠀⢀⡴⠶⠶⠶⣤⣄⣀⠀⠈⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⡟⠮⠉⠀⠀⠀⠀⡿⠉⠉⠉⢥⠤⣔⣘⣒⣦⣀⣀⣀⡴⠛⠛⠁⠀⠀⠀⠠⢼⡍⠛⢶⡄⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣩⡄⢹⣧⣀⣠⡤⠶⠞⠛⠓⠶⢤⡴⠟⠚⠛⠉⠉⠉⠉⠉⠁⢀⠀⠀⠠⠄⠀⠀⠀⠀⢀⠀⢸⡏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⠆⠀⢴⣽⠟⠁⠀⠀⠀⠍⡉⠉⠁⠀⡩⠉⠉⠈⠡⡛⠛⠀⠀⠀⠀⠖⠠⠀⠀⠀⠀⣠⢊⣴⠏⢀⣀⡀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠳⣤⣈⣿⠃⢀⠀⠂⠒⠂⠀⠁⠀⡀⠋⠀⠈⠁⠀⠒⠒⠀⠀⠀⠤⠀⠀⠀⠐⢂⢔⠽⣩⡾⠷⠦⠾⢡⢈⡗⡐⡝
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣡⠽⢿⣇⠀⠀⠈⠀⠀⠐⠂⠀⠒⠙⠀⠀⠀⠖⠀⠀⠙⠀⠠⠂⠀⢀⠀⠀⢀⢨⢀⡾⢋⡴⠀⠀⠀⠀⢸⢀⡴⠁
⠀⠀⠀⠀⠀⢰⡟⠀⠀⢠⠋⠀⠀⠀⢻⣥⡄⠑⠤⠤⠤⣀⠀⠺⠦⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠵⠃⢈⡇⠗⠈⠀⠤⣤⢤⣴⠏
⠀⠀⠀⠀⠀⣿⠇⠀⠀⢻⡔⠀⠀⠀⠐⣿⣿⣷⣄⡀⠀⠄⠀⠉⠑⢖⠒⠒⠲⠤⠤⠤⠠⠄⠤⠤⠔⠊⠁⣠⠞⠻⣄⣲⡷⢦⣼⡷⠛⠁
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⡉⠓⠶⠶⢺⡿⠿⠉⠛⠻⡿⢶⠶⠶⠤⠤⣬⣀⡀⠀⠀⠂⢀⣀⠀⣤⣕⣤⠾⠋⢀⡠⢀⣨⠴⠛⠉
⠀⠀⠀⠀⠀⠘⢷⡀⠠⡀⠀⠀⠀⠀⠛⠶⣄⠠⠤⣞⣁⡾⠈⠉⢀⡀⠀⠈⠉⠛⠛⠋⠉⠉⠛⠟⠉⠀⣀⣡⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣬⣀⡒⠒⠢⠤⠀⣈⣉⣉⣉⣉⢀⣀⣀⣀⣠⡤⠤⠤⠔⢒⣒⣠⣤⣴⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠒⠒⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣤⣀⣠⣄⡀⢀⣀⣀⡤⠤⠤⠀⣠⣀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠖⠉⠀⠀⠀⣄⠀⠀⠀⣀⣙⣷⣥⠀⠤⠀⠀⠀⠀⢤⣈⠉⢩⣷⡀⠀⠁⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⢤⡴⠶⠾⠓⠛⠉⠙⠻⣭⡀⠀⠀⠀⠐⠲⣠⣄⡈⠀⠀⡙⠳⢦⣼⠇⢠⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠉⠌⠠⣄⡀⠀⠀⠁⠀⠈⠰⢤⣄⡀⣀⣄⣰⣦⣽⣧⣤⡶⠿⠞⠛⠉⠳⢤⣤⣀⠠⠋⡻⡇⠀⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠐⠈⠁⠀⠀⠀⠀⠀⡀⠀⡙⣻⣧⣤⠶⠶⠿⠋⠙⢿⣉⡉⠉⠁⠀⠀⠘⠓⠦⣄⠀⢀⣀⠤⠽⠛⡿⠀⣴⠁⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠠⢌⣷⡤⣴⠖⠛⠋⠉⠁⠀⡹⢷⡄⠂⠤⡀⠀⠈⠙⠶⣄⣀⣤⣴⠶⠶⢛⣿⣿⣿⡔⠀⠉⢳⣼⠟⢷⣆⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠙⠳⣄⣤⠶⠶⠛⠛⠉⠋⢤⣤⡞⠉⢹⡿⠋⡟⠁⡠⠞⠉⠀⠀⠀⠀⠉⠑⠀⠀⠀⠸
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⢠⡽⢷⣤⡖⠛⢏⠛⠉⠀⠙⢶⡦⠄⠀⢀⣾⢺⣿⣥⡀⠈⠃⣀⡷⠖⠁⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⠃⠰
⠀⠀⠀⠀⢈⢢⠤⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡄⠠⠠⣄⣤⣤⠾⠛⢷⣾⣿⠋⢫⣽⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⢀⠊⠠⠁
⠀⠀⠀⠐⠠⣆⢳⣀⣀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠀⢠⠴⠶⠿⠿⣟⡉⠀⠀⢀⣠⣿⣻⢿⡄⠼⠋⠀⠐⠪⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡠⠁
⠀⠀⠀⠀⠀⠸⡈⠉⠹⡷⣶⡶⠿⠿⠛⠓⠂⢀⣀⣀⠀⠀⠀⠀⠀⠉⠻⣶⡤⢾⡏⣉⢙⣿⠁⠀⠀⠀⠀⠀⠀⠁⠒⠠⡀⠀⠀⠀⠀⠀⡠⠊⢀⠔
⠀⠀⠀⠀⠀⠀⠉⠀⠒⢄⢀⠋⠀⠂⢠⡀⠑⡶⠃⠉⢹⡗⠤⢀⠼⢻⣭⡄⠁⣼⠀⡸⠋⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⣠⣀⢸⣥⣤⠀⠡⠀⠙⣧⣤⣿⡿⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⢬⣇⡟⠉⠋⠀⠀⠀⠀⢻⣿⠿⠟⢥⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⡉⢉⠒⠒⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡴⠀⠀⡔⠀⠀⠢⠀⢀⠀⠀⠀⠀⠀⠀⢀⠈⣄⠀⠀⢠⠄⠁⠀⠀⠀⠀⠰⠒⠒⠀⠀⢹
⠀⠨⢰⣤⢠⡀⠀⠔⠐⣪⠟⠁⠀⠀⠓⠐⠀⠁⠀⠀⠀⠀⠀⠀⣠⡞⠛⠓⠋⠀⠀⠃⠀⠀⢶⠆⠀⠀⠀⠒⠁⢈⣁⢸
⠀⠀⠙⠁⠸⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠦⣴⡈⠈⠀⠀⠀⠀⡀⠀⠀⢀⣠⠐⡀⠀⠐⠀⠀⠀⠀⠤⠴
⠀⣤⡄⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⡇⣍⠂⠀⠀⠀⠀⠀⠀⠀⠄⠒⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣄⣠⣄⣀⡀⠀⠀⠀⠂⢠⠀⡦⠤⡇⢰⡄⠀⠈⠛⠦⡤⠆⠀⠀⠀⠀⠀⣀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠉⠈⠑⠉⠻⠿⠃⠀⠠⠒⠐⠀⠦⠀⣇⣾⡧⠀⠀⠀⠀⠀⠀⣰⠂⡔⢁⠜⢠⠂⡠⠄
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡀⠀⠀⠀⠐⠚⠃⠸⠂⠸⣠⢿⡙⠁⠀⠢⠴⣶⣤⠀⠸⢿⡴⣁⡄⢀⠖⢀⠔⢀
⠀⠀⠘⣿⣓⡂⣿⢛⡏⢋⠗⠀⠀⠀⠈⠒⠀⠀⠀⠲⣾⣬⣧⣺⡏⠉⢀⣄⠀⠀⠀⠘⠋⠳⠆⠀⠈⠉⠚⠷⠶⣧⣴⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀⣀⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠐⠶⣻⣭⣥⣾⡿⠷⣤⣠⣤⣤⠶⠖⠒⠒⠟⠻⠓⠒⠒⢼⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⢀⣤⣄⣀⣀⡠⠤⣤⣶⣚⣭⡭⠖⡰⠟⣃⣵⢖⣫⣤⣶⣾⣿⢁⣀⠤⢀⣐⣀⣀⣀⠀⠌⠉⠀⠰⠟
⠀⠀⠀⠀⠀⢀⡤⣶⣆⢀⣿⣉⣅⡄⢀⣘⣿⡏⠀⣴⣾⣟⣛⣻⣭⣥⢽⠶⠿⠟⢋⣵⣿⣿⣿⣿⣿⣳⣯⠴⠒⠉⠉
⠀⠀⠀⢀⢔⡵⢦⣟⡜⠢⣗⠵⣤⣤⡝⠋⢉⣺⣛⣻⣿⠻⠭⠤⢔⣀⡘⣤⡖⣿⣿⣿⣿⣟⣿⡿⠋⠁
⠀⠀⠀⡆⡾⢱⣬⣿⠀⣧⠘⣦⡴⢞⡑⠛⢋⠁⠀⡸⠉⢙⠒⣛⢛⢳⣬⡋⣸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢸⢀⣧⢿⢟⠦⢷⣾⠤⠥⢄⠪⠀⠀⡌⠀⢀⠳⡀⠉⠀⢃⢺⢀⡻⣧⣿⣿⣿⣻⣿⣿⡟
⠀⠀⠈⡞⢿⣿⠟⢠⢏⠌⡇⢠⣤⣬⣄⣀⡃⠰⣎⠀⡅⠀⡘⣄⢠⣨⡗⣹⣋⣿⣿⣿⣿⡿⠃
⠀⠀⠈⣇⡈⢯⣷⡏⣾⣤⡷⣿⠁⠐⠀⣈⠻⢦⣥⡦⣤⢤⡶⠼⠾⠋⣰⣷⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⡖⠢⡄⠀⠀⠀⠀⣰⠆⡆
⠀⠀⠀⡦⠙⢢⡻⣽⣿⣿⣿⡁⡼⠀⢆⡏⠀⢪⡻⠘⠿⣋⡵⠖⣠⡼⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡇⣴⢃⡴⣆⡤⣤⣯⠊
⠀⠀⠀⣂⠀⠀⢱⠉⢻⣿⣷⠏⠀⣒⣎⡗⣄⠣⠧⠐⣋⡩⣴⣻⣿⣷⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡸⠛⠇⠸⠅⠸⡋⡇
⠀⠀⠀⠈⢄⠀⠀⠑⢤⣉⠙⠷⠷⢛⣉⠥⡶⠲⠟⢿⣟⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⢢⢀⠀⠀⣀⣤⡄⠀⣀
⠀⠀⠀⠀⠈⠑⣀⡀⠀⡀⠈⡁⢨⠽⠃⠆⣡⣜⣶⣯⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡀⠀⣁⣿⢜⡥⢄⡤⣴⢸⣯⠖
⠀⠀⠀⠀⠀⠀⠘⢧⣶⡇⡀⣇⡈⢁⡶⣷⣿⣿⣿⣷⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⡇⠉⣇⢸⡀⢼⡥⣏⢸⡿⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠷⠿⠿⠿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠐⠒⠒⠠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣈⡡⠴⠒⠒⠀⠉⠉⠉⠉⠉⠉⠐⠢⢭⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⡩⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣊⠴⠊⠁⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⢦⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡥⠊⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⢇⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡡⠊⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢞⡭⠞⠁⠀⠀⠀⠀⠒⠀⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠄⠀⣀⡀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⣩⠴⠚⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠁
⠀⠀⠀⠀⠀⠀⣠⢞⡡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠖⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠠⠄⠀⠀⠀⢠⢃⠏
⠀⠀⠀⠀⢀⡼⣱⠋⠀⠀⠀⠀⠀⣀⣀⡐⠒⠀⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠚⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢡⠎
⠀⠀⠀⠀⣼⢱⠃⠀⠀⠀⠀⠰⠋⠡⣀⠉⢆⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⢀⡠⠖⡋⠔⡡⠂
⠀⠀⠀⠀⣟⢸⠀⠀⠀⠀⠀⢀⠘⠤⠜⢀⡌⠀⠀⠴⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⢊⣩⣴⡊⠐⠈
⠀⠀⠀⠀⠻⠸⡄⠀⠀⠀⠀⠀⠓⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢖⣒⣒⣚⣒⣒⣒⣛⣛⣛⣋⣩⣭⣥⠤⠶⠒⠊⠙⠉⠀⠀⠀⠄
⠀⠀⠀⠀⡄⢣⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⡥⠖⠋⣉⠡⠤⠄⠒⠒⠀⠉⠉⠉⠭⠔⠒⠂
⠀⠀⠀⠀⢰⡀⠑⢝⠲⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⡴⢞⣩⠴⠋⠁⠀⠈⠄⠐⠀⣀⣀⡀⠀⠀⠤⠤⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠱⣄⠀⠉⠲⠬⣍⣛⣒⣒⣒⣚⣯⠵⠞⠉⠀⢀⣠⠤⠖⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠤⠤⠭⠭⠭⠁⠀⣀⡠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⠒⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⡠⠖⢀⣀⠀⠀⠀⣤⣍⡋⣴⣲⠶⢚⣉⣥⣿⡟⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠁⠀⢠⡼⠉⠀⠀⠀⠀⠀⢨⠈⠏⣼⣿⣿⡷⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠠⡄
⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⣴⣒⣶⠟⣼⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⡤⢢⣀⢀⠀⠀⠉⠒⠏
⠀⢠⡀⠆⣂⣁⠀⠀⡀⣠⣾⣖⣿⣷⡽⢻⣿⣿⠁⠀⠀⠀⠀⡀⠀⣡⠤⣤⣿⣿⣿⠜⣹⣆
⠀⠀⠃⠀⠹⠃⠀⠼⠃⣼⠿⠞⠋⣡⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⢬⣡⣵⣿⣯⡿⠿⠯⠴⠟
⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⣠⣴⣿⣿⣿⣿⡿⠁⠀⠀⠀⡀⠀⠀⠀⢀⣤⣴⣿⠻⡛⠲⠐⡄
⠀⠀⠀⠀⠀⠀⠤⣤⣠⣄⢸⡿⠿⠛⠉⠁⠀⠀⠀⡄⠀⠐⠂⠀⠀⠀⠈⠛⠛⢷⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢀⡄⣠⢖⠀⠀⠠⠀⠄⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠠⠴⠥⠤⣤⠎⢀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠄⠿⢱⡧⠡⠀⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⢠⣽⣧⡀⠀⠀⠀⠤⠶⠤⢤⣤⣎⠁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⡄⠀⠄⠠⡎⠀⣡⣾⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠑⠥⡀⠀⠀⠀⠀⠀⠀⠈⠀⢠⢀⠰⣶⡐⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⢀⣾⣿⠐⡶⡲⠂⣀⢘⠉⠋⠀⢱⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠠⢦⠀⠀⠁⠀⠀⠀⠳⣄⠘⠻⣧⡆⣤⣤⢤⣦⣶⠀⠀⠀⠀⡀⠀⢔⡼⡿⢋⣾⠈⢆⡠⣘⣿⣿⡿⡛⠁⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣽⠸⠟⠁⠀⠘⣻⢗⡍⢿⡟⠫⣿⠋⠀⠀⠀⠀⠷⠀⠞⠉⠐⠃⢙⠆⠘⠭⠋⠹⠉⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⡾⠜⢴⡉⡔⢁⠊⠛⠀⠀⠀⠀⠀⠂⠘⠂⢤⡤⢀⠀⠐⠒⠀⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣘⡍⠋⠀⢢⠠⠿⡶⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠁⠰⠒⠲⠴⠁


⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣠⣶⣶⣤⣤⢴⣯⠶⢾⡿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠐⠉⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣶⣦⠶⠦⠼⠖⠶⢶⣶⣤⣄⡀⠀⠀⠀⠀⣀⡀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠈⠁⠂⠀⢐⣂⡝⠋⠉⠉⣉⠀⠀⠀⠀⠐⠀⠀⠀⠀⢹⠁⢀⠍⡀⣀⡉⡁⠀⠀⠀⠀⠈⠉⠋⠙⢻⣞⠻⠤⠈⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠭⣛⠛⠂⠀⠀⠐⠗⠂⠀⠀⠘⡎⠔⠚⠉⠁⠈⠈⠀⠄⠀⠁⠀⠀⣰⠖⣻⣿⡏⢁⣠⠆
⠀⠀⠀⠀⢒⠒⠛⠲⠖⢶⢶⡷⠿⣟⣶⠦⠀⠀⢀⠀⠲⠊⠉⠀⠐⠶⠚⠛⠓⠀⠀⠀⠀⡀⠠⠖⢀⣐⢀⣐⡘⠠⠀⣄⠀⢀⡀⠀⠙⢻⣷⡆⠛
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠆⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣭⣤⠶⣤⡤⠐⠂⣤⣥⠞⣳⣿⣇⣀⠤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⣶⣦⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⢸⣂⡠⠚⠛⠛⠠⢷⣶⣶⣾⣛⣀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⢤⣄⣻⠟⠧⣤⣄⡀⠀⠁⠁⠀⠂⠀⠉⠀⠀⠀⠀⠈⢍⡛⠻⡿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢄⠐⠻⢤⡀⠀⠀⠠⠍⠛⠳⠆⣀⡤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠑⠓⣁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠁⠀⢐⣟⣀⣠⡄⠀⠉⠀⠂⠠⠀⡀⠀⠀⠉⠑⠢⠤⠀⠀⠀⠀⠀⠐⠄⠀⡀⠀⠀⠘⠛⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⠀⠀⢰⡏⠉⠉⡓⠛⡾⢧⣤⡄⠀⠀⠙⠃⠲⠦⢀⠀⠀⠀⠀⠐⠂⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣏⡇⠛⠋⠀⡈⠱⠤⠀⠰⠮⠀⠰⠟⠉⠙⠋⠓⡦⠀⠀⠀⠀⠉⠂⠠⢤⣀⡀⠀⠈⢑⠤⠆⠠⠄⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣩⡄⠈⠁⣜⠣⠈⠀⠀⠀⠘⠛⠓⠯⣅⣤⣠⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⢀⡰⢛⡭⡴⡥⠀⠀⠀⡣⢄⠈⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢱⢾⡴⠂⣼⡋⠀⠘⠻⠒⠤⢒⣀⣄⠠⠀⠈⠉⠑⠲⠆⣠⣤⣀⣠⣴⠒⣂⣩⣶⠶⠚⡩⢖⡥⣪⣞⡀⠁⠸⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢘⣿⠷⠶⠙⢿⣤⣖⡦⠤⠀⠀⠀⠁⠘⠲⣟⢓⠤⠴⠊⠙⠛⢿⢿⣟⠉⠐⢊⠴⣣⢞⠕⡩⢊⡹⢻⣵⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⢿⣿⣦⡤⠁⠋⢉⣰⣶⣦⣬⡷⠢⠀⠀⠀⠀⠀⠀⠀⠘⠉⡡⢀⣈⠁⠔⠱⡳⢋⠔⣉⣴⣿⣿⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⣿⢦⡟⢿⣿⣶⣶⣿⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠋⠀⢀⡡⢔⡁⣢⣮⡾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⢿⣇⣧⡖⣳⣿⡿⠏⠊⠠⠀⠠⡀⠀⢀⣀⠀⡀⠀⠀⠀⠀⡴⠀⠈⡰⢀⣔⣩⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⣧⣿⣿⣿⣿⣿⡇⠐⠆⠀⠈⠂⠀⠀⠀⢁⣸⣀⣀⣀⣄⣴⣷⣤⣤⠴⠛⠛⠉⠁⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡃⠀⡤⢶⠔⣀⣤⣶⣿⣿⠿⠿⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠦⠀⠀⠈⠒⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣻⣿⡿⠿⠟⠋⠁⠿⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠤⠀⠙⠀⠀⠀⠀⡠⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠄⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⢤⣦⠄⠀⢀⡔⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⠉⠁⠀⠁⠉⢀⡴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⢀⣴⣫⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣠⣤⡄⠀⢀⣀⢀⣀⠀⠀⣀⣹⣏⣠⢴⠶⢶⣶⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⠋⠉⢓⣶⣏⣩⣹⣶⣏⣉⢙⡿⢭⣁⣩⠟⠻⣝⠚⠛⠛⠛⡉⠠⡞⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⡉⠛⠍⣿⠿⠥⣭⣿⣿⣿⣿⣿⡿⣦⣤⣿⠛⠓⣶⡿⣿⡟⠛⠉⠁⢔⣃⡺⣧⣀⢠
⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠋⠣⡉⢲⣜⢀⠲⣄⣾⡁⠉⢹⡿⠯⢄⣠⡟⠉⠙⣿⠿⣄⣸⡇⠀⣀⣸⢎⣯⣴⠛⡇⠲⢁
⠀⠀⠀⠣⡀⠀⠸⣷⡦⣄⠠⠀⡙⠀⠈⠳⣅⠈⠛⢯⣿⡿⠦⣤⣤⣟⠉⣑⣶⣯⣴⣿⡿⠗⠋⠩⠑⣙⣣⣭⣶⣾⣯⡿
⠀⠀⠀⠀⠈⠢⡀⠈⠻⣟⣷⢄⡀⠀⡃⠲⢬⡛⠿⣄⡈⢒⣶⣾⣿⣿⣯⡵⠞⠛⢋⣉⣠⠴⣶⣌⣭⣤⠷⠾⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠻⣤⡽⢶⣄⣀⠈⢿⠂⠋⠙⠛⠋⣉⣡⣭⣀⣐⣒⣻⡷⠶⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠀⡀⠙⠚⠃⢿⠿⠥⠀⣀⣀⣠⣭⠭⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆⠤⣤⠦⣠⣀⡀⢀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠀⠉⠀⠀⢿⡇⠸⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠤⢤⣤⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠓⠈⠓
⠀⠀⠀⠀⢴⣚⣺⡭⣷⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⣾⡹⡜⡖⣃⣾⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠾⠷⠝⠛⠛⢛⣉⣤⣀⡀⠀⢀
⠀⠀⣾⣿⣟⡏⢵⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠉⠂⣠⣤⡤⢶⣶⣷⣶⣿⣿⣶⣤⣝⠛⢥⠖
⠀⠀⠘⢿⣷⣿⣿⣿⠿⣿⢿⣿⠿⣿⣿⠿⠿⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠤⣤⣤⣤⣶⣶⡶⠶
⠀⠀⠀⠀⠈⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⡹⢶⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣚⣿⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⣿⣿⡿⢿⠿⠏

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠷⢿⣿⣿⣿⣿⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⢿⡷⣄⢀⣾⣾⡟⣿⠿⣧⡟⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡅⢀⣯⠀⠈⠛⣅⢾⣿⡀⣺⣿⡧⠈⢻⡷⠹⣿⡿⢿⡖⠦⣤⣠⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣸⣇⠀⠀⠀⣀⠤⣿⣧⢨⠻⠁⠀⠈⠑⠁⠈⠀⢸⠇⠀⣩⠉⢿⡿⢶⡄
⠀⠀⠀⠀⠀⠀⠙⠿⠿⣴⣿⣭⠀⠀⠀⠘⢻⠛⠇⠀⠀⠀⠀⠀⢀⠀⠐⣶⢣⣾⠇⠈⠑⠺⢏⣾⣷⢀⡄
⠀⠀⠀⠀⠀⢹⣄⠀⢰⣿⣮⣈⡷⠄⣀⣲⣦⣤⡄⠀⣤⡆⣀⣀⣬⣷⣶⠇⣾⣷⡶⠦⠀⠨⠛⣋⢡⡞
⠀⠀⠀⠀⠀⠀⢿⡷⠀⠉⠛⠻⠷⣾⣿⣿⣿⡟⠁⠼⠷⠖⠛⠛⠛⠋⣅⠂⠉⠁⠄⠀⠀⠀⠔⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠦⣀⠀⠀⠐⠀⠀⠀⠀⠐⠂⠐⠀⠀⠀⠀⠀⣀⠧⠤⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠂⠤⠤⠄⣀⣠⡤⠄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠿⠇⠼⠧⠹




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠚⠉⠉⠉⠉⠉⠉⠀⠉⠍⠢⣄⣀
⠀⢀⣀⡤⠤⠴⢞⣡⠔⠂⠀⣄⠀⡀⠈⠡⣠⠆⠀⡀⢀⡠⣀⠈⡇
⠀⣏⠀⠀⣠⡔⠋⠑⣤⠀⠀⢈⠿⣅⠠⠀⠀⠀⠠⠈⠁⣀⣠⢴⠇
⠀⠘⡶⠦⢤⣌⣳⣉⡀⢈⣂⣀⣀⡠⠴⠒⠛⣋⣉⣉⣉⡡⠴⠋⢀⣀⣤⣔⣪⣿⣯⣵⣖⡒⣒⢤⡀
⠀⠀⠈⠷⣤⣀⣀⣈⢉⣉⠉⣉⣁⡤⠴⠊⠉⠁⠀⠀⠀⠀⣀⡴⢿⣿⣿⣿⣿⣿⣿⡿⣛⣁⠛⣿⣎⢢
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⠤⣒⣯⣵⣶⢶⢟⣳⣦⣼⣷⣿⣾⣿⣿⡕⣿⣿⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠟⣷⠫⣝⣾⠿⢯⣭⡭⠿⠭⢿⣿⣿⣿⣟⣸⡿⢥⡾
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⣞⢠⣿⡇⠀⣿⣿⡇⠀⠀⠀⢹⣿⡦⢀⣀⡠⠟⠁
⠀⢀⡴⣫⣶⣦⢰⣍⠒⠤⢤⣄⣀⣀⣀⣀⣈⣉⠁⠈⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠛⠋⠉⠉
⠀⣾⣴⣿⣿⣿⠸⣿⣿⣿⠿⢃⣶⣶⣶⣶⣤⣍⣙⠢⣄
⠀⡇⣿⣿⣿⣿⣆⠠⣭⣀⠒⠛⠛⠛⠛⢛⣉⣭⣭⣶⢘⡆
⠀⣇⠹⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣉⣥⣾⡇
⠀⢹⣦⣝⡛⠟⠃⣸⣿⣿⣿⣿⣿⠟⣫⠴⠋⠁⣠⠴⠛⠀⣀⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢿⡄⠉⠛⠒⠬⢭⣭⣭⣭⣶⣿⣿⣤⠔⠋⠀⢀⡔⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠊⠁⠀⠈⠓⠤⡀
⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⣿⣿⠿⠛⠁⠀⠀⢀⠊⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾
⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠓⠤⣀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠔⠒⠉⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠈⠉⠉⠉⠉⠉⠀⣀⣠⡤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠐⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣄⠀⣶⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⠁⣐⡂⠀⠀⠐⢒⢖⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢧⣈⢣⠒⡟⠘⣇⡖⡄⠀⠀⠀⠀⠀⠀⡜⠠⣀⠔⢂⠀⢈⣦⠴⠓⡄⠀⠳⣄⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠌⢱⣠⡄⡂⢀⡸⢹⣠⡆⠀⠀⠀⠀⢰⡧⠀⠙⣦⡴⠚⡍⠀⢱⣤⠞⠣⡀⢈⠦⡄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠾⠞⣹⡻⡁⣄⠙⢉⠘⡿⡄⠀⠀⠀⠸⢈⠀⠈⠉⢶⣀⠝⢿⡁⢈⣧⡔⠉⠣⡀⡨⠶⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣧⣿⣿⣿⣯⠾⡍⢐⡀⢸⣦⠀⠀⠈⠃⣱⣀⠀⠋⠱⣄⡠⠓⣏⠀⢙⣦⠐⠉⠂⣠⢎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠿⠛⠋⠃⡀⠀⢻⠚⠓⠊⠘⠂⠀⠈⢿⢁⡃⢄⣀⣉⠃⠀⠬⠓⠉⢀⣠⠴⠋⠀⠈
⠀⠀⠀⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⠁⠀⠀⠀⣶⢉⡓⣚⣹⠃⠀⠀⢐⠀⠀⠀⠁⠀⠀⢈⡉⢋⠉⠉⠉⠉⡉
⠀⠀⠀⠸⠉⠉⠉⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣴⣦⡃⠀⠀⠘⠀⢱⣦⡦⠘⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀
⠀⠀⠀⠀⣤⣤⣤⣴⠷⠤⣤⣀⣠⣤⣤⡄⢀⣀⠀⠀⠘⠝⠙⠀⠀⠀⠀⣪⡙⣷⡴⢐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣡⠶⠺⣕⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣼⣿⣟⣦⢄⡒⣛⣸⠑⢦⣄⡐⢝⣀⡠⣄⡀⡈⠓⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢾⣧⠈⡀⠈⠹⣿⠘⣇⡀
⠀⠀⠀⠀⠀⠀⢀⡙⠏⠞⠃⠢⠀⡁⠰⡵⣸⡏⠀⣷⢟⠛⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⢋⠴⣊⣬⢿⡮⡅⠀⠀⠉⠀⢇⠄
⠀⠀⠀⠀⠀⠀⠠⠙⠲⠽⣟⡳⢀⣺⣡⣎⠉⠀⠀⢡⡼⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠷⣾⢛⣼⡿⠉⠈⠱⡄⠀⠀⠀⠾⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⢻⡿⠺⢖⣤⡀⣠⣬⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢁⣴⠾⣓⡯⠒⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠙⠮⣭⢱⣎⡀⠄⠚⠁⠀⠀⠀⠀⠀⣀⣤⠔⠋⣉⠠⢚⣩⠔⠛⢉⡴⠀⠀⠌⠀⠀⠀⠈⠀⠀⢀⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⢀⣀⣤⣶⡷⠷⠒⢒⣩⡴⠞⠉⠀⠀⠀⢸⣿⣄⠀⠀⢶⣀⠂⠈⠐⠀⠁⠀⠎⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⣧⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⢀⣾⣿⠓⠀⠀⠀⢹⣇⠀⠀⠀⣨⢀⣴⠇
⠀⠀⠀⠀⠀⣠⠖⠀⠂⠀⠀⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⢀⣀⣀⣰⣿⣯⣻⣿⡶⠦⣦⡄⠀⠄⠠⣀⣺⣟⣾⡿⢷⡄⠀⣀⣼⣟⢃⣠⠜⣡⠞⡟
⠀⠀⠀⠀⠀⠛⠃⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠿⠿⠭⠿⣾⠿⠿⣭⣉⣉⣙⣙⠿⢿⣦⣭⣠⣡⡤⠶⣊⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠿⠿⢶⣭⠴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⢦⡀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠄⡀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⢠⣿⣬⣿⣦⣧⠀⠀⠀⠀⠋⠀⠈⠐⠓⠒⠒⠉⠉⠁⠀⠀⢱⣄⡀
⠀⢲⡻⣷⣿⠀⠀⠀⢀⣀⣠⣿⠀⠀⠀⢸⣾⣇⣿⣼⣅⣿⡾⠀⠀⠴⢄⣀⣀⡀⠀⢀⣀⣀⣠⣴⡆⠀⣾⠇⠁
⠀⢻⣭⣻⣢⡦⢰⡲⣯⡳⣽⡏⠀⠀⣰⢾⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠈⠙⢿⣇⢸⡿⠿⠿⣿⣦⠸⡟
⠀⠀⠙⠻⣿⣷⣶⣿⣮⣽⠟⠀⠀⠀⠈⢢⡻⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢺⠹⣸⠀⠀⠀⢌⡏⢠⠇
⠀⠀⠀⠀⠈⠻⠿⠿⡿⠃⠀⠀⠀⠀⠀⠀⣼⣩⡭⡯⠍⣹⡄⠀⠀⠀⠀⠀⠈⠈⠢⠋⠀⠀⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠁⠐⡘⠀⢤⠀⠀⠀⠀⢰⠋⠛⠤⡷⠞⠋⢳⠀⠀⠀⢀⢾⢀⠀⠀⠀⠀⢀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠀⠀⠀⣧⠀⠀⠸⡧⠀⢠⡿⡟⣿⡄⠀⢠⡴⣻⣭⣵⡽⢿⣷⡆
⠀⠀⠀⠀⣀⡄⡰⠁⢀⢂⣤⣀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠘⠷⣷⠷⠃⠀⣵⡞⣿⣿⡟⣿⣯⠻⣿
⠀⠐⣿⣶⣌⣿⡇⠀⠀⣹⣏⣠⣶⡷⠀⠀⢀⠀⠀⠀⢆⡀⠀⠀⢀⢀⢀⠀⠀⠀⠀⢠⣁⣾⣿⣧⣿⣻⢄⡿
⠀⠀⠈⠉⠋⢻⣿⠀⢠⣿⠏⠙⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠋⠾⣼⣿⣷⣠⡀⠀⢸⡿⣽⣿⣿⣿⠏⣼⠁
⠀⠀⠀⠀⠀⠘⣿⠃⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣛⡀⠀⠀⠤⢤⣴⡏⠀⠀⢠⡷⣱⡿⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠱⣥⣥⠇⠀⠀⠀⠀⠀⠀⠀⠀⠬⢭⣭⠄⠀⠀⠀⠀⢸⡇⠀⠀⠸⣟⣫⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠔⣥⣾⣷⣮⢢⡑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⣾⣿⣿⣿⣿⣿⣿⣿⣖⢄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠐⣋⠅⠊⠉⠉⠚⠀⠐⡧
⠀⠀⠀⠀⢀⠬⣤⣻⣤⡝⣧⢇⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⣏⡿⢿⣿⣿⣧⣭⣽⣷⣿⣽⣿⣸⠀⠀⠀⢀⠖⢋⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡅⠁
⠀⠀⠀⢠⢢⣶⣮⣻⣻⢋⣥⡌⢳⣄⢀⠢⡷⣤⠒⠉⡃⠀⠀⠀⡏⠙⠳⣆⠙⣻⡿⣿⣿⣿⣿⠟⢹⠀⠀⠀⡎⠀⠀⠀⠀⠀⢀⡄⠀⠊⠀⠀⠀⢀⡅⣦
⠀⠀⠀⡃⣿⣿⣿⣿⣿⣿⠿⣴⣇⣿⣾⠿⠛⠛⠓⠤⠇⠀⠀⠀⠈⠲⣄⠈⠍⣙⣿⡾⠟⠃⣁⠔⠁⠀⠀⠀⢻⡀⠀⠀⠀⡰⢃⡇⢰⠁⠀⠀⠀⢈⡿⠋
⠀⠀⠀⠈⠊⠉⢉⣻⣋⣿⣿⣵⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠹⣦⣴⠙⣁⠾⠗⠛⠳⠖⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠥⠤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣦⣔⡢⡀⠀⠀⠀⠀⠘⠢⣄⠈⠂⢄⣀
⠀⠀⠀⠀⢀⣠⣤⣠⣤⣪⣵⣿⣾⢿⣿⣷⣝⡀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣈⣁⢀⣀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠦⠤⣀
⠀⠀⠀⣼⣿⣿⣿⣿⣯⣿⡯⢍⣶⣿⣿⣿⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡿⢁⣿⢛⣡⣟⡠⢷⡀⠀⠀⠀⠀⣠⣯⣉⠙⡆⠀⠀⠀⠑⠤⡀⠀⢀⡀⡀
⠀⠀⠀⡟⢿⣿⢶⣅⢻⣾⡿⢻⣿⡿⢟⣵⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣈⡿⣫⣽⣿⡭⣼⡇⠀⠀⠀⠀⠯⣶⣷⠙⠛⢦⡀⠀⠀⢀⣀⣀⣀⠤⡇
⠀⠀⠀⣟⣦⣍⣛⣭⣖⡂⣴⣿⣭⢾⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⡈⢻⣿⣼⡿⠷⡿⠀⠀⠀⠀⠀⠠⣝⠟⠔⣡⣿⣃⡤⠞⠉
⠀⠀⠀⠉⠻⢿⣷⣼⣌⣦⡧⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⠿⣧⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢫⡭⠍⣹⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⠠
⠀⠀⠀⠀⠀⠀⣠⡞⠛⣷⢦⣄⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⢠⠁⣠⡾⠋⠀⣀⣀⣼⣿⣝⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢺⠿⠿⢿⠻⣯⡑⢮⢳⡀⡇
⠀⠀⠀⢀⡴⠊⠉⠳⣲⣶⣶⣽⣷⡀⠀⠨⠛⢦⠀⠀⠀⡰⣃⣞⡝⢁⣴⣾⣿⣿⣟⡛⠻⢿⣶⣽⡄⠀⠀⠀⢀⡔⠋⠉⠙⢉⡻⡍⢢⢀⡇⣿⢃⣾⡿⠔⠁
⠀⠀⠀⠸⣄⠀⢠⣾⡵⠛⠁⠈⠙⠿⣷⣲⣤⡎⠀⠀⠀⣷⣟⣩⡴⠞⣵⠟⠻⢿⣿⣿⣶⣦⣬⣽⠇⠀⠀⠀⢮⣀⠘⢉⣴⣦⡤⢣⣾⡞⡜⠛⠉
⠀⠀⠀⠀⠈⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠻⠿⠶⠛⠁⠀⠀⠀⠀⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠉⠉⠛⠓⠚⠓⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠵⠚⠉⠓⠢⣄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠔⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠹⣧⡀
⠀⠀⠀⠀⢀⡴⠋⡠⠒⠀⠐⠈⠍⣻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⣷
⠀⠀⠀⠀⣼⠁⡎⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡶⠦⣤⣤⣀⣠⣤⣤⣖⡯⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿
⠀⠀⠀⢠⡿⠀⢧⠀⠀⠀⠀⢀⡴⠖⢒⢶⡄⠙⢢⡀⢀⡤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿
⠀⠀⠀⢸⣇⠡⠘⢆⠀⠀⠀⡞⠀⠊⠉⢳⣿⠀⠀⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⣿⡀⢁⠘⢆⠀⠸⡇⡆⠀⢀⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣼
⠀⠀⠀⠀⠸⣧⠀⠣⡈⢣⡀⠳⣝⣒⣋⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡜⣿
⠀⠀⠀⠀⠀⠙⣧⠀⠈⠢⡱⣄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠎⢰⡟
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠈⠪⣳⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠚⠁⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠈⠓⢌⡉⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢚⡡⠔⠉⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⠑⠒⠨⠵⣲⡶⠶⠶⠶⠶⠖⠖⠒⠒⠚⣉⡥⠔⠚⠁⠀⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠉⠁⠁⠀⠀⠀⠀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⠟⠛⠛⠛⠛⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡴⠟⠻⣦⣤⣤⡀
⣼⠀⠀⠀⠁⠀⠈⢻⣄⣀
⠻⣤⣤⣄⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠈⠻⠦⢦⣤⡀⠀⠀⠀⠀⠈⢙⡷⠶⠾⠿⠷⠶⠶⠶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡸⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠶⠶⠾⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⣉⣤⠤⠶⠶⠶⠒⠒⠲⠦⢄⣉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣤⢾⡄⡀⠀⠀⠀⢀⡴⠋⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠙⢦⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⢿⡀⠀⠉⠀⠀⠀⣠⢏⡴⠉⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠈⣧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠟⠁⠀⠀⠸⡇⠀⠀⠀⠀⣴⢣⠏⠀⠀⠀⣠⡤⠶⠶⢤⣀⠀⠀⠀⡴⠊⠉⠀⠀⠈⠉⠙⠢⡀⠀⠀⢸⡆⢿⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣼⢣⠆⠀⠀⣴⠟⠁⠀⠀⠀⣠⠈⢳⠀⢸⡁⠀⣀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡇⢸⡀⠐⢯⣄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⡆⠀⢠⢇⠎⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣠⠴⠃⠈⠧⣄⠈⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡏⢸⠁⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣿⠃⠀⠀⢸⣇⠀⡟⣸⠀⠀⡼⠁⠀⠀⠀⠁⠀⢠⠎⢁⡤⠶⢤⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠾⠃⠀⠀⠀⠀⣿⢰⠇⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⡏⠀⢸⠁⠀⠀⡿⠀⢹⠀⠀⠀⠀⠀⠀⢀⡟⠀⡼⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⢰⡇⠀⢸⡀⠀⠀⠀⠂⠀⠀⢧⠀⠘⠦⠤⠞⠁⢠⡆⠀⠀⠀⠀⠀⢀⡞⠀⡼⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣠⠋⢀⡼⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⣧⠀⠀⠘⢦⣀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠸⡅⠀⠀⠀⣠⠞⠁⣠⠎⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠸⡄⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⠓⠛⠋⠁⢀⡴⢁⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⣉⣙⠒⠒⠶⠶⠶⠒⣒⣊⣩⣥⠴⠟⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣄⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣠⣤⣤⣤⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣠⣤⡤⠤⢤⡶⠶⠖⠛⠛⠛⢛⡛⠋⠉⠁⠈⠁⠈⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣏⡕⠀⠀⠀⠀⢀⠀⠉⠁⠈⠉⠁⠀⠀⢀⡀⠈⠀⡀⠀⠻⣿⣿⣇⠀⠀⠀⠀⠀⡠⠊⠐⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣯⠄⠀⠒⠂⠀⢀⠀⠠⠤⠐⠒⠂⠀⠀⠀⠀⠒⠀⠀⠀⠛⠿⠾⣿⠀⠀⠀⢀⠔⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣀⠀⠄⠀⠀⠤⠤⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⠀⠈⣰⣿⡿⠀⠀⠀⠁⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠤⠐⠂⠀⠀⡉⠉⠀⠀⠤⠀⠂⠀⠀⠄⠀⠀⠀⠀⢠⣰⣿⡇⠀⠀⠳⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠒⢂⠉⠀⠀⠀⠒⠂⣀⡉⠉⠀⠀⡂⠀⢀⠀⣠⠀⣀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠈⠉⠄⠀⣀⡀⣠⣽⢖⣭⠳⣭⢄⡠⠄⠓⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣃⣤⣤⣽⣷⣶⣶⣾⡿⠿⠿⠿⢤⣄⣽⠉⠉⠀⢸⣆⣰⣼⠏⣻⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⣬⡿⢿⣟⣻⣬⣽⣶⣿⣿⣟⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡻⣿⣿⣿⣿⣷⣶⣶⡶⣶⣶⣾⣿⣭⣿⣷⣿⡻⡿⣿⣿⣿⣲⡽⣟⠻⠿⠙⠊⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣝⢯⣽⢷⡶⣿⢿⣿⣿⣿⣿⣿⡿⠷⡛⠟⠟⠫⠙⠂⠈⠁⠀⣠⣶⣦⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣹⢸⠙⢷⣌⡳⣝⠦⢵⡚⠯⠍⠓⠀⠉⠁⣀⠀⢀⣔⣶⠀⠘⠃⠀⠙⢿⠏⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⢠⠈⠟⢻⣿⠈⢁⢠⣄⠀⣮⣿⡆⠀⠙⠀⠀⠛⢋⡀⠤⠀⠂⣾⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⠁⠀⠑⠢⢼⣿⠀⠻⣿⡿⠀⠈⡉⠀⠀⠄⠐⠚⣿⡅⠀⠀⠀⠀⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠈⢻⣠⠤⠘⠂⠉⠀⠀⠀⠀⢀⣀⣤⢼⣧⠤⢀⠀⠀⠀⢿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢟⢀⣀⣠⡤⠴⠒⠒⠉⠁⠀⠀⠀⣿⡀⠀⠀⠀⠄⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⣀⣀⠤⠤⠖⠆⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠠⠀⠈⢷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣝⠫⢉⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠒⣾⡁⠀⠀⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠁⠈⠀⠠⢀⠀⠤⢀⢀⡀⠀⢀⠠⠄⠐⠀⠀⠈⠀⠀⠀⠀⠀⠻⠧⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⠀⠀⠈⠑⠊⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⢛⣛⣛⣉⣉⡛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢿⣿⣿⣿⣶⡛⠀⠈⢳⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⠤⣿⢻⣽⣿⣿⣷⣟⠀⠀⢸⣏⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⣠⠴⣲⡿⠋⠉⣠⠖⠋⠀⢀⣿⢻⣽⣿⣿⣿⣟⠀⠀⣸⣯⠟⠉⣩⠟⠉⠉⣹⠷⠦⣄⡀
⠀⠀⠀⠀⣾⣡⠞⠁⢀⡴⠛⠁⢀⣠⠞⢉⡇⠻⠿⠿⠿⠷⢟⣠⡾⠛⠁⣠⠞⠁⠀⣠⠞⠁⢀⡴⠋⣷
⠀⣀⣀⣀⣿⠓⠦⣶⣋⡀⣀⡴⠋⠀⣠⠾⠛⠛⠋⠉⠉⢉⡽⠋⠀⣠⠞⠁⠀⢀⡼⠃⢀⣴⣯⠴⠚⣿⣀⣀⣀⡀
⡞⢩⡤⠤⢤⣄⠀⠀⠈⠉⠙⠛⠒⠺⠷⠶⠤⠤⠤⠤⠴⠯⠤⠴⠾⠥⠶⠶⠖⠛⠛⠋⠉⠁⠀⠀⣠⣤⠤⠤⣬⠙⡆
⠹⣌⡛⣋⣩⣭⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣭⣉⡓⣋⡼⠃
⠀⠈⠉⠉⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⣉⣓⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣚⣋⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢩⡏⠀⠀⠀⠉⠻⡯⠭⣿⠛⠁⠀⠀⠹⡏⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼⠀⠀⠀⠀⠀⠀⡇⠀⣏⠀⠀⠀⠀⠀⢷⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⡇⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⢸⡆⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢷⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠸⡆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⡷⠦⢤⣀⡀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⢀⣀⣠⠴⠾⡆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼⠶⢤⣄⣀⠉⠙⠒⠶⢤⣧⡀⣿⠴⠖⠛⠉⢁⣀⣤⠴⢷⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⠀⠀⠀⠉⠙⠓⠶⢤⣄⣀⡽⣿⡤⠴⠖⠋⠉⠁⠀⠀⠸⡆⠘⣇
⠀⠀⠀⠀⠀⢠⣾⡿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⢿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣄⣠⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⣨⣿⠇
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⡼⠃
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⢀⠠⠄⢠⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣠⣀⡀
⠀⠀⠀⢘⡧⣦⠴⣠⣀⣤⠴⠴⡋⡁⣢⠇⠀⠀⠀⢧⡄⠀⠀⡾⣿⣿⣿⣾⣿⣿⣷⣿⣿⡷⡄⠀⠀⠓⠒⠒⠦⠤⢄⣹⡾⢯⣾⠛⣶⡟⣓⣄
⠀⠀⠀⠘⢾⠴⡴⢠⣆⠸⣸⣤⡷⠞⡏⢡⠄⠀⠀⠈⣧⠀⠀⠳⣌⠻⠿⣻⣿⢷⣿⣿⣿⣾⠇⠀⠀⡴⢖⣤⠀⠀⠀⣸⡇⡿⢣⣽⠏⣴⠟⣸
⠀⠀⠀⠀⠐⠪⣴⣵⣃⣆⡔⢠⣆⣤⠟⠋⠀⠀⠀⠀⠘⡄⠀⠀⠈⠓⢀⣏⣿⣿⣿⡿⠛⠁⠀⠀⠀⠥⣏⠠⠀⠀⠀⠻⢿⣿⣾⣧⣤⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⢀⣀⡀⠀⠈⠉⠛⠛⠋⠁
⠀⠀⠀⢠⠒⠲⡄⠀⢀⣠⣤⣀⡀⠀⢀⡀⠀⣀⢀⡆⠀⠀⣀⣀⣶⢤⣤⡤⣤⠠⠠⣴⣴⣢⠤⠤⠀⠀⢠⢒⡥⢒⣒⠠⡉⢢⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠑⠤⠊⠰⣿⣽⣿⣿⢹⠀⢀⡝⢠⠛⠋⠀⠀⠠⠌⠈⠙⠚⠻⠀⠛⠂⡃⠉⠁⠀⠀⠀⠀⠀⡇⢸⡀⢯⣲⣃⠄⢸⠀⠀⢻⣜⣝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠙⠸⠋⠀⠀⠀⠠⣲⢦⡠⣆⣶⣖⢰⣶⣊⡟⣧⡴⠖⠂⠀⠀⠀⠹⣄⡙⠒⢭⣤⡴⠋⠀⠀⠀⠈⠉
⠀⠀⠀⢀⢄⡲⢂⡵⠒⣢⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠁⠀⠈⠉⠈⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⣀⣄⣀⠀⠠
⠀⠀⠀⣾⠏⣴⠃⣰⠊⣡⣄⡾⢁⢠⠂⠀⡀⢤⣇⣨⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠦⠋⠀⠀⡠⠉⢀⡨⠂
⠀⠀⠀⠈⠓⠓⠶⠶⠚⠉⠁⠀⠀⠉⠀⠀⠧⠾⣿⠹⡀⠀⠀⠀⣀⠴⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠊⠀⠀⢀⠀⡐⠀⠀⢀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡠⠊⠀⣀⣀⡠⠄⠀⠀⠒⣒⡷⠀⠀⠀⠀⠀⠀⠹⡀⠀⢀⡠⣿⢠⡆⠀⠀⣸⠟⠃
⠀⠀⠀⠀⠔⡤⠋⣩⠕⣋⠽⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⣸⣴⣶⣾⣯⣥⣤⣤⡴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠋⠶⠋⣉⡹⠊⠉⠁
⠀⠀⠀⠘⢻⢧⣾⣥⣞⣵⠊⣡⡖⠛⠉⣉⣷⠄⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢀⡰
⠀⠀⠀⠀⠀⠉⠳⠾⣹⣷⢻⣿⣶⣤⡞⠉⠀⠀⠀⠀⠀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠋⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⡠⠒⢠⠀⠀⠀⠈⠋⢡⡜⠋⠀⠀⠀⢀⣀⣀⡀⠀⠈⠙⢯⣉⠉⠉⠉⠉⡉⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠃⠤⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⢴⣮⣿⡛⢿⣇⠀⠀⠀⠀⡟⠉⠀⠀⡎⢻⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⣟⡿⣦⣄⠀⠀⠀⣼⣿⣯⣽⣿⠿⠃⠀⠀⠀⣼⣂⣬⣤⣤⣤⣼⣷⠄⠀⠀⢰⠀⠀⠀⠰⡀⢐⠀⢰⣶⡂⠀⠘⠁⠀⠀⣷
⠀⠀⠀⠀⠀⣴⣿⡾⣦⣏⡋⣳⠂⠀⠿⣿⠯⠞⠁⠀⠀⠀⠀⡼⠀⠀⠈⠉⢉⣟⣦⣼⡄⠀⠀⠈⠆⠀⠀⠀⡇⢸⠀⠀⢯⠇⠀⢸⢿⠀⠀⠛
⠀⠀⠀⢀⡼⢳⠿⣤⣔⣽⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡸⠁⢀⣀⣤⣔⡚⢿⡿⠏⢻⠀⠀⠀⠀⠀⠀⠀⡇⣾⠀⠀⠸⡄⠀⢸⢸⠀⠀⣠
⠀⠀⠀⠈⠙⢷⣦⡊⠙⡟⠁⠺⢦⣄⡀⠀⠀⠀⠀⠀⢿⣿⣷⠟⠓⠲⠶⠤⠬⠭⠶⠶⠿⠄⠀⠀⠀⠀⠀⠀⢸⡜⠀⠀⠸⡇⠀⢸⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠊⠀⠀⠀⠀⠉⠂⠄⣄⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⠀⣼⠀⡆⠀⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣴⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠻⠿⣿⣿⣿⣯⠉⠽⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢊⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠍⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⡉⠉⢻⡏⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡁⠀⠀⠀⠀⡛⠻⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⢤⣤⣤⣤⣤⣤⣤⣴⡶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠛⢦⠤⠤⠤⢷⠖⠛⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⠿⣿⣿⣿⣿⣿⣿⣾⣷⣦⣽⣭⣹⣛⣛⠿⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⣾⣳⣦⠀⠀⢸⠄⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣦⢭⣭⣛⣛⠻⠿⡿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠿⣿⠿⠁⠀⠘⣗⠀⠀⢀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢟⣻⣿⣴⣿⣯⣭⣿⣿⣷⣾⣭⣭⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢠⠶⠾⣧⡀⠀⠀⠠⠀⠀⠀⠠⢿⠄⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⠋⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠸⣄⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠺⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠟⠻⠛⠿⣿⡿⠛⣏
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠤⠂⠀⠀⢸⠇⣤⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣇⠀⠀⠀⠀⠀⠈⣎⠙⢛⠛⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠽⠷⣤⣄⣀⣀⣀⣨⡤⠾⠗⠻⡅
⠀⠀⠀⠀⢠⡿⢻⡇⠀⠀⠀⠀⢀⣠⡤⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣠⣤⣾⣄⠀⠀⠀⠸⡆⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⣿⣹⠁⠀⠀⠀⣴⣏⣷⣿⣿⣿⣿⣿⣾⣹⢦⡀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣿⣯⣿⣿⣧⠀⠀⢀⣷⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⢹⡏⠀⢤⠤⡼⣷⢹⠓⠣⣷⠛⠹⡩⠚⢝⣾⢷⠤⣤⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠚⡿⣿⠿⠃⠂⡀⠈⢽⣃⠔⣀⠈⠀⢳⡀
⠀⠀⠀⠀⠀⢸⡇⠀⢸⢰⡇⢿⢸⠀⡃⣿⠘⢣⡇⣿⢸⢺⢸⡇⡇⠀⠀⣿⠀⠀⠀⠀⠀⣠⡶⢿⣿⣤⠈⠔⠂⠐⡡⠈⠀⠀⡐⠂⠼⣇⠀⢁⠐⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠸⠄⣧⣿⢸⡀⢃⣿⢘⣣⣇⠋⣘⣸⣸⠇⠧⠀⠀⣿⠀⠀⠀⠀⠐⣏⠀⠄⠘⣷⡇⠐⠁⠀⠠⠈⣀⠀⡀⠂⠠⢯⡴⠂
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢯⡦⣶⣶⣶⣷⣾⣯⣵⢛⣸⠏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠶⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠛⠯⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉



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

⢀⠀⠀⠀⢀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠚⠂
⠐⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠨⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠂⠀⠐⠀⠀⠀⠀⠀⠤⠤⠄
⠀⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠄
⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠁⠀⠀⠀⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀
⠀⣄⣤⠛⠁⢠⡴⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐
⢄⠀⠀⣁⠔⠀⠀⠀⠀⠀⠠⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠘⠢
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠅⠀⠄⠈⠈
⠀⠀⠀⠀⠅⠀⠸⠒⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠂⠀⠈⠀⠀⠀⠁⠀⠀⡀⠀⢀⣼⠉⡪⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢇⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⡀⠀⠀⢄
⠀⠀⠉⠀⠀⠄⠀⠠⠀⠀⠁⠀⡰⠋⠑⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠘⠀⢠⡀⠊⠀⠀⠀⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⢶⣳⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄⠠⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢣⡞⠉⣩⣆⠈⠛⣿⣄⠀⠀⠀⠀⠀⠀⠓⢄⠀⣑⡈⣳⣤⣶⡛⠋⠁⠈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⣨⠇⠀⠎⠛⠁⠀⠀⠀⠛⡆⢀⠀⡀⠀⠉⠻⣅⢠⣶⠟⣉⠙⡛⠛⠉⠈⢁⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣶⣄⣀⠀⠀⠉⠲⣄⠀⠀⢲⣄⣼⠦⠶⠃⠄⠀⣈⠀⠀⡀⠈⢉⡟⠉⠛⠉⠙⠋⠀⠐⠋
⠀⠀⠀⠰⠂⠀⠀⠀⠆⠀⢾⣿⢁⠹⣜⢦⠀⠀⠙⠶⣽⡮⠹⣿⣍⣷⠶⠶⠟⠿⠿⢷⣬⡤⠀⠁
⠀⠀⠀⠀⣨⠥⠀⠀⡤⠀⠈⢿⣟⣷⡜⢿⡇⠀⢀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠐
⠀⠀⠐⡤⠿⠔⠀⠀⢅⠀⠀⠸⣿⣿⣿⣛⣣⡄⠀⣀⠀⠀⠀⠀⠀⠀⠠⠄⡦⢀⡀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠡⣀⠀⠄⠀⠀⠀⢄⡀⠈⠻⠿⣿⠟⠁⠀⠁⠀⣀⠤⠀⠀⠀⣠⠠⢲⠇⢰⡆⢀⠀⠀⠀⠀⠀⠈⢶⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⢠⠊⠀⠀⠀⠄⢀⠘⣠⣷⣤⣼⣿⣿⠀⠀⠀⠀⠘⠛⠠
⠀⠀⠀⠀⣀⠀⣈⡄⢦⣤⠀⢄⡀⠀⠀⠀⠀⣰⡤⠀⠀⠀⢀⣠⣾⣿⡿⣿⣿⣿⣿⣿⣷⡦
⠀⢀⠀⣀⡿⠟⣹⢧⣭⣶⣛⣳⠛⡆⠀⠀⠀⠋⢒⠘⠃⡀⣿⣿⣷⣿⡿⣟⠙⠙⠏⠙⠢⣅⣾⠁⠀⠀⠀⢀⠔
⠀⠘⣿⢿⠳⢞⢧⢺⣿⣟⠻⡿⣷⢤⣀⠀⠀⠐⠚⠒⠶⠚⠛⠛⣁⣈⠉⠻⡗⢲⠖⣢⣴⣯⣿⣦⠒⠆⠀⠁⠐⠶
⠀⠀⢰⣾⡖⣸⢋⡧⠌⠹⠇⢰⡜⣧⣌⣧⣄⠀⠀⠀⠀⠀⠀⢀⡿⠉⠀⠀⠀⠘⠓⠴⠼⠿⠟⠁⠀⠀⠀⠂
⠀⠀⠈⢛⣿⣽⡿⢶⠐⣆⡀⢈⡷⣼⠟⣿⣟⣆⠀⠀⢀⣄⣴⠰⣾⡆⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡤
⠀⠀⠀⠨⠙⢛⣛⣛⠁⣬⣿⣟⣿⣯⣯⣿⣿⠷⠂⠀⠀⠙⢿⠂⢩⠃⠈⠉⠉⠀⢧⣤⣡⣶⣿⣿⣧⣄⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⡀⠀⠀⠈⠁⢲⣍⢡⠀⡀⠀⠀⠿⢻⣧⣽⣷⢾⣤⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣾⡓⢸⠂⠀⠀⠀⠼⡦⠀⠁⠀⢀⡄⠈⣿⣿⣿⣸⣯⡐⠀⠀⠀⢠
⠀⠀⣀⠀⢀⡤⠴⠶⠚⠓⣒⠈⢉⠀⠀⠀⠀⠀⠸⣿⣷⠲⡤⣤⡤⠀⠁⠲⠂⠀⠀⠐⢃⡟⣿⣯⣿⣷⣷⡖⠀⡐⠀⡀
⠀⢠⣏⢳⣛⡅⠶⠾⠿⣷⡎⢠⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠤⠂⢉⡭⣿⣇⠀⠀⠀⠩⡰⠛⣿⣿⣿⣿⣷⣵⣤⡟
⠀⠀⣿⣿⢽⣷⣿⣾⣿⡟⠑⠁⠀⠀⡠⠞⠀⠀⠂⠘⠂⠀⡰⣯⣿⠿⠷⠛⠁⢀⠀⠀⠀⠠⠾⣛⠛⠿⢿⣛⠷⠚⠁
⠀⠀⠘⣿⣵⣿⣿⣿⣿⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠈⠐⠀⠒⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢷⠀⠀⠀⠀⠾⢥⣶⣴⠟⢿⣦⡀⠀⢀⠀⠀⠀⠀⢀⡤⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡒⢉⣡⢤⡾⠀⠀⣴⣤⣀⠂⢤⡀⠀⢾⠿⣿⠟⠻⣷⣄⡀
⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⠀⠀⠀⣀⠤⢰⣏⢱⠍⠐⠋⠀⠀⢶⡶⢶⣿⢿⣦⣈⠿⠛⠳⢤⡀⠀⠀⠙⠳⢦⣄
⠀⠀⠀⠀⢤⣉⠂⠠⠊⠀⣠⠶⣯⣵⡿⠟⠋⠀⣀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠻⢿⣳⣦⡀⠙⠧⠀⠈⠀⠤⡉⠻⣿⣶⡆
⠀⠀⠀⠀⠠⢌⠲⢠⣰⡾⣱⣾⡽⠋⠀⠀⠀⢀⠀⠈⢰⣶⣾⣿⡷⠤⡀⠀⠀⠀⠀⠈⠙⠛⠦⠘⢦⠀⠀⣀⣀⡁⠠⠙
⠀⠀⠀⠀⠀⠀⠈⠒⠿⠾⠛⠁⠀⠀⠒⠂⢀⣷⣮⣓⣞⣟⣛⣻⣥⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⡆⠈⠻⠲⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣯⣒⠚⠛⠛⠛⠻⣿⣿⣿⣿⠟⠎⢀⠀⠀⠀⠀⡄⡠⠀⠀⠈⠓⠠⠄⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡟⣾⣿⣷⣷⡶⣞⣾⣶⣾⠗⡶⠁⠀⠀⢀⣻⣴⡌⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⠛⠛⠿⠇⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⣭⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢸⣿⠂⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡁⠀⠀⠸⣿⠀⠀⣧⡈⠳⣄
⠀⠀⠀⠐⡄⢀⡤⠖⠉⠀⠈⠳⣄⡀⠀⠀⠀⠀⢀⣠⢞⡥⣸⡷⠀⠀⠀⢿⠀⣄⣿⣧⡄⠈⣳⡄
⠀⠀⠀⠀⠘⠾⡀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣤⡴⠋⢠⢸⡇⣿⡇⠀⠀⣠⠀⢰⣯⣿⡟⢃⢞⠝⠁
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⠀⡑⠈⢿⣧⠹⡄⢸⢸⡇⣿⡇⠀⠀⣿⠀⢸⠛⣃⢔⠕⠁
⠀⠀⠀⠀⠀⠀⣈⡆⠀⢐⠆⠰⡁⠘⣆⠈⡿⠛⠛⣆⢼⡇⢹⠀⠀⠀⡟⢀⣠⡾⠗⠁⠀⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣦⡀⢳⡀⢹⣄⣸⠖⠃⠀⠀⠈⢆⠁⢸⠀⢀⣠⡶⢟⣋⠴⢒⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠙⠲⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢧⣴⡾⠛⠉⠀⢘⠇⠀⠏⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡎⠀⢀⣸⠀⠀⢠⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⠤⢤⣤⣄⣀⣠⡤⠶⠒⠚⢛⣒⣚⠛⢧⠀⢧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣥⣤⣍⡉⠉⣤⣤⣄⠲⣶⣶⣶⣶⣶⣿⢃⣾⣿⡿⣋⣴⣿⢀⡟⣷⠇
⠀⠀⠀⠀⠀⠀⣼⠁⠾⣿⣶⣯⣽⣿⣦⡘⢿⣿⣷⣮⣿⣿⣿⡿⠃⣚⣻⣻⣿⣿⣿⣿⡿⢁⣿
⠀⠀⠀⠀⠀⠶⣿⠗⠲⠶⠤⣭⣛⠿⣿⣿⣌⡛⢋⣭⣭⣭⣍⠐⢋⣭⣭⣭⣭⡭⠀⢀⣴⡿⣿
⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠆⡖⢩⣛⠶⣌⡙⠻⠶⣦⣭⣭⣭⡴⠿⠟⣋⣉⣥⠴⠞⡻⢟⣼⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⡲⠶⣾⣿⣑⠈⠉⠛⠒⠶⠶⠶⠖⠚⠋⠉⠉⠀⠰⠖⣋⡵⢛⡁⠦⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠁⢀⠍⠙⠶⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠛⠋⠇⠀⠢⠀⢁⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠙⠛⠉⠉⣉⠁⠁⠀⢀⠠⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠉⠁⠀⠉⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣤⣤⣤⣄⣀⣀⣀⠀⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⢿⣿⡷⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢋⡀⠤⠖⠋⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡄⠀⠀⠀⠀⣀⠔⠊⡡⠒⠉⠀⠀⠀⠀⠀⠀⣴⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⢿⣿⣿⣶⣤⡀⠀⠀⠀⠀⢀⠀⣠⣴⣾⣿⣿⠟⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣠⠖⠁⠈⠛⠿⠿⠓⠀⠀⠀⣰⡏⠐⠻⠿⠟⠋⠀⠑⢦⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⣾⡀⢠⣿⡇⠀⠀⣠⠀⠀⠀⠀⠀⠙⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢹⠇⣼⣿⣇⠀⠰⣿⠀⢠⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⡄⢠⣷⡿⠀⠀⠀⣿⣿⣿⡄⠀⠃⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡄
⠀⠀⠀⠀⠀⢀⣴⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣿⣿⡇⢸⣆⢰⣿⣿⣿⣷⢠⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣦⣄
⠀⠀⠀⢠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣼⡇⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦
⠀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⠿⢿⣿⡟⢻⣿⠟⣛⡛⣿⢛⣛⠻⡟⣛⣛⣿⢛⡛⢿⠻⣿⡿⡿⠿⠿⣷⣶
⠀⠀⠀⠀⢤⣴⣶⣶⣶⣶⣶⣿⣿⠀⣛⠠⣿⠱⠆⢿⡇⣭⠡⣿⢨⣭⠘⡇⢨⣽⡇⢺⣿⢾⠐⣿⡇⡇⢘⣻⣿⣿⣶⣶⣶⣶⣶⣶⡤⠄
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣀⣛⣥⣧⣿⣿⣼⣷⣿⣷⣿⣶⣶⣾⣷⣶⣶⣿⣦⣵⣾⣦⣭⣼⣧⣘⣛⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠴⠿⠿⠿⠟⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⠿⠿⠦
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣹⣄⣀⣳⡖⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡑⠀⣸⠙⠛⠉⠋⠟⡴⠃⠀⠀⠀⢀⣀⣀⣀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣀⠀⣀⠀⢀⣤⣇⢸⣷⠀⠀⠀⠀⢀⣤⣤⡷⣶⣻⣏⣽⣯⠾⣷⣟⣛⣏⢹⡗⢦⡀
⠀⠀⠀⠀⢠⣎⠪⠀⡟⢤⠘⡆⠟⠉⠈⠀⢠⡾⠿⠿⠿⠿⣿⣻⣯⡿⣞⣛⣿⣭⣿⣭⣿⣝⣿⢿⠿⠆
⠀⠀⠀⠀⢘⣌⠣⠢⠉⠀⠁⠐⣀⡀⠀⢀⣟⡦⡐⠀⠀⠁⠀⠠⠈⠉⠻⣯⣽⣧⣾⡏⣽⠛⠇
⠀⠀⠀⠀⣓⠤⢵⣦⣤⣶⣾⣦⣿⣷⣶⠿⠿⠿⣷⣮⣄⣀⠀⠀⠀⠀⠀⠀⠹⡟⠘⠙⠉⠀⠀⡠⢴⡇
⠀⠀⠀⠸⠵⢀⣠⡾⠫⠁⠈⢿⣿⠋⠀⠀⠈⠂⠄⠈⠙⠿⢿⣶⣤⣰⣆⣘⣻⡇⠀⠀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠰⣧⠀⣈⣠⣿⣟⣣⢀⠀⠀⠐⠂⠀⠀⠀⠀⠉⢉⠻⢿⣿⣿⣿⣷⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠨⣯⢷⣬⣿⣿⣝⣿⣿⣉⡒⢦⣌⣀⣀⠀⠀⢀⠈⣲⣾⣿⣛⡗⡽⢷⠀⠀⢀⡠⡀⡖⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣽⣿⣌⢤⡙⢛⠻⣿⣿⣷⣤⣷⣾⣿⣿⠶⢻⡧⣋⣁⣑⣼⠃⣜⣉⣹⣧⡇⢠⠜⢻
⠀⣠⣄⠀⣀⣤⣤⣄⡐⠶⠈⠉⠛⠿⣿⡒⠄⢤⣤⠭⢽⣿⣿⡷⣼⡟⠇⡇⠈⠋⠉⠀⠐⡇⠀⢹⣿⣽⠇⢤⣴
⠀⠈⠁⣾⠯⠬⠤⣀⠙⢦⡀⢠⣤⡀⠈⠙⠲⠾⣧⢤⠤⠟⠋⠀⠀⠙⣦⠋⠀⠀⠀⠀⠀⢢⡢⢃⣀⣻⣄⡀⢈
⢀⢀⠈⠻⢤⣄⣀⣀⣈⣑⡷⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣮⣶⡾⠃
⠙⠛⠀⠀⠦⠆⠉⠉⠁⠸⠃⠀⠀⠀⢀⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠌⠓⠖⠰⠆⠾⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣋⣡⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠿⠿⣭⣓⠶⠶⠚⠓⠒⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡛⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣈⠺⢆⠀⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠄⣀⣀⡀⡀⠀⠀⢠⣭⣛⣿⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣤⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⢀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠠⢚⣋⣨⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠝⠁⠀⢡⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠆⢀⣴⣾⠿⠿⠿⣟⡓⠶⠶⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠠⣿⠇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠁⠀⠻⣟⡓⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣰⠋⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠁⠀⠀⢸⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠲⣆⠀⡾⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠇⠀⠀⠀⠉⠛⠢⢄⡀⠀⠀⢀⠀⠀⢨⣝⣛⣷⣦⣿⠁⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠖⠒⠒⠒⠒⠒⠛⠋⠙⠛⡒⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠀⠀⠉⢀⠉⠓⠲⣄
⢀⡤⠴⠖⠚⠛⠉⣥⡴⠞⠉⠀⠀⠳⣄⢀⡔⠀⠀⠢⡴⠋⠀⠢⡀⢀⠴⠃⢄⠀⣼
⢸⠀⠀⠀⣡⣴⠛⠁⠈⢢⣤⠀⠀⠀⡨⠻⣄⠠⠖⠁⠀⠀⠀⠤⠈⠁⢀⣀⡤⢾⡇
⠈⣷⢤⣤⣀⡈⠙⢦⡚⠁⠈⠑⠄⠀⠀⠀⢈⣡⡤⠖⠒⠒⠒⠒⠚⠛⠉⣁⡴⠏⠀⠀⢀⣀⣠⠔⣫⣽⣭⣓⣒⠤⠤⠤⢄⡀
⠀⠈⢳⣄⣈⠉⠉⠛⠓⠶⠖⠲⠖⠒⠚⠛⠉⢀⣠⠴⠖⠒⠚⠛⠋⠉⠉⠁⠀⢀⡴⣻⣿⣷⣶⣿⣿⣿⣿⣿⣿⡿⠆⣿⣶⣌⠳⣄
⠀⠀⠀⠙⠛⠷⠴⠤⣴⣤⣤⣤⣀⡴⠴⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⣘⣻⣿⣛⠿⠿⣛⣛⣿⣛⣡⣶⣶⡆⣹⣿⣷⣜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⣫⣽⡶⢿⡿⠿⣟⣩⣿⣿⣜⣿⣿⣿⣿⣿⣿⣿⡷⢹⣿⣿⠏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠻⠿⡗⠮⣍⣲⣟⣿⣯⣭⣭⠭⠽⣯⣼⣟⣛⣛⣛⣿⡿⣃⣿⣋⣥⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⡄⠀⡿⠀⣾⣿⠁⠀⢸⣿⣿⡄⠀⠀⠀⠀⢻⣿⣧⡉⠉⠉⠁⢠⡿⠃
⠀⠀⢀⡴⢒⣛⠛⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠶⠚⠳⠶⣿⣿⣥⣤⣼⣿⣿⣧⣀⣀⣀⣀⣼⣿⣿⠧⠶⠒⠚⠉
⠀⣴⢋⣾⣿⣿⣇⢸⣧⣄⣉⠓⠲⢶⠶⠶⠶⠶⠶⠶⠦⣤⣀⡀
⢸⢃⣾⣿⣿⣿⣇⠸⣿⣿⣿⣿⠿⠋⣰⣿⣿⣿⣿⣶⣶⣤⣉⣙⠶⣄
⣿⢸⣿⣿⣿⣿⣿⡄⠈⣉⣉⠁⠴⠾⠿⠿⠿⠿⠟⠛⠋⣉⣉⣩⣤⠘⣆
⣯⠘⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠟⠋⣠⣿
⢹⣆⠹⣿⣿⣿⣿⡟⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣡⠤⠶⠞⠻⣿⡏
⠀⣿⣷⣬⣉⡛⠋⠀⢸⣿⣿⣿⣿⣿⣿⡿⢛⣡⡖⠋⠁⠀⣠⡴⠞⠋⠀⣀⡤⠔⠒⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠸⣿⡀⠈⠙⠓⠲⠬⣭⣉⣉⣉⣭⣴⣾⣿⣿⣧⣠⠖⠋⠁⠀⢀⡴⠊⠁⠀⠀⠀⠀⠐⠀⠀⠉⠉⠉⠉⠉⠛⠉⠁⠀⠀⠉⠓⢄⡀
⠀⠀⠹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢠⠞⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⣀⣤⠴⠟⠛⠉⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠖⠒⠋⠉⢀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠈⠉⠓⠒⠚⠛⠉⠉⠀⠀⢀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⡀⠀⠀⣠⡤⠴⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣤⡴⢖⣚⣛⣉⣉⣉⣙⡓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⣠⣴⠟⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⣴⣟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡟
⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⠟⣽⠆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢋
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⣫⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⣠⡾⠁
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⣻⣼⠟⠁⢀⣴⣶⣶⣶⣦⣄⠀⢰⡏⢠⣾⡿⠁⣠⣴⣶⣿⣿⣿⣷⣦
⢀⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡿⣫⣾⠟⠁⠀⢀⣿⣿⡿⠛⠛⠿⣿⣷⡈⢠⡿⠋⣠⣾⣿⣿⣿⠟⠉⠀⠀⠉
⣼⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣵⡿⠋⠁⠀⠀⠀⠈⢿⣿⣧⣠⠀⠀⢹⣿⡇⠘⢁⣾⣿⣿⣿⠟⠁
⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣭⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣸⡟⠀⣠⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣫⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣀⣾⣿⣿⣿⠟⠁⣀⣤⡴⠶⣒⣀⣀
⢹⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⢟⣯⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣠⣴⣾⣿⡿⠿⠋⣡⣴⣿⠟⠉⠐⠊⠉⠉⢙⣿
⠈⢿⣿⣿⡿⣟⣋⣭⡶⢟⣻⣵⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠩⢭⣤⣴⡾⠿⠛⠋⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠈⡿⠿⠿⣛⣯⡵⠞⠛⣉⣀⣠⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠉⠛⢋⡩⣴⣶⡿⠿⠛⠛⣩⣟⠛⢁⣤⠖⠀⣠⡞⠀⢠⡾⠠⠀⣶⠀⠈⢯⡉⠉⢿⡟⠿⠿⢷⣦⣤⡀
⠀⠀⠀⠀⣏⢼⡋⠀⠀⣠⡴⠛⠁⢀⡴⠟⠁⠀⣴⠏⠀⠀⣼⠇⠀⠀⣿⡅⠀⠈⢷⡀⠀⠙⣦⡀⠀⠈⠙⢯⡂
⠀⠀⠀⠀⠹⣾⣿⣴⣿⣁⡀⣠⡼⠋⠁⠀⢠⡾⠋⠀⠀⢰⣿⠀⠀⠀⢹⡇⠀⠀⠘⣷⡄⠀⠈⢻⣦⣠⣴⣿⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷⣶⣦⣤⣾⣧⣤⣤⣤⣾⣷⣶⣶⣶⣾⣿⣿⣿⣿⡿⠟⢛⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣾⣭⣽⣛⣛⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⣾⡇⢸⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡾
⠀⠀⠀⠀⠀⠀⠹⣎⢿⣿⣿⡉⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠻⣿⣿⣦⣬⣭⣭⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠈⠉⠉⢹⣿⡏⠉⠉⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣤⣶⣶⣶⣿⣿⣇⣿⣿⣸⣿⣿⣿⣷⣶⣶⣦⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠈⠻⠿⣿⣿⣛⣿⣿⢻⣛⣻⣛⠿⠍⠁⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣄⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⢀⣶⣿⣿⣿⣿⢧⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⣿⢿⣿⡛⣵⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⣤⣶⣶⣶⣽⣻⣴⣿⢿⣿⣿⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢿⣿⣿⣿⠉⣨⢣⣯⢿⣿⠇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣴⣿⡿⣿⣯⡧⠉⢻⣿⡿⢸⠔⢴⣷⣽⢟⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠈⢠⣿⠻⣿⣯⣿⣿⠂⠁⡻⣃⣹⣶⡿⠿⠿⢟⣃⠠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣶⣸⣿⣿⢟⣻⣿⣿⣧⣶⣞⣯⣶⣶⣿⣿⣿⣿⣿⣆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⠁⢡⣾⣿⣿⣿⣿⣿⠿⢟⣛⣙⣋⣩⡭⢿⢿⣿⣿⢂⣈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⣿⣇⠻⠿⣿⣯⣶⣿⣟⠽⠹⠀⠋⠡⢓⢈⣽⣿⠏⣸⣿⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⡟⢿⣤⡉⠉⠉⢿⣧⣬⣴⠄⠤⠖⢛⣿⣿⢰⡋⣽⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⠿⣧⢰⣦⣄⠉⠛⠿⣷⣶⣶⣴⣿⡇⣿⢜⡅⢻⣿⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣽⣿⣿⣆⢲⣤⣤⣉⠻⢿⡟⢸⣻⣿⣿⣾⣿⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠾⣿⣿⣿⣷⣦⣄⣘⠻⢿⣿⣿⣿⡧⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠛⣿⣿⣿⣿⣾⣿⣯⣝⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠣⣾⣻⣽⣠⣀⣠⡤⠴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣺⣿⣟⡋⣿⣿⣷⡶⠬⠿⠀⣠⠤⠶⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣻⣿⣿⣝⡉⠉⢋⣩⠤⠐⠒⠛⠁⠀⢀⡠⢰⣶⡀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⢘⠛⠉⣉⡭⠬⠉⠙⡂⠀⠀⠀⠀⠐⠚⠋⠀⢠⣾⣿⡟⢀⣾⣏⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣥⣄⣀⣀⣤⣖⡋⠁⠀⢀⣈⡥⠖⣻⣿⣿⠟⢉⣼⡿⠃⣾⡿⢾⡖⠂⠄
⠀⠀⠀⠀⠀⠀⠠⠈⠻⣿⣿⣿⣿⣿⢻⣿⡙⢿⣿⣿⡶⣶⠾⣿⣿⡏⢀⣾⣿⠃⣰⣿⢛⣴⡿⠀⢀⣶
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣉⣉⣙⣋⣉⠓⠾⢿⣾⣦⡿⠶⠟⢻⣿⣿⡿⢿⣾⣿⠗⠚⠋⢁⡴⢁⠏
⠀⠀⠀⠀⠘⡗⠤⣀⡀⠀⠤⠤⠤⣬⣭⣭⣃⣀⣀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⢺⡿⠃
⠀⠀⠀⠀⢀⣓⠤⣀⡉⠃⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⠿⣿⣶⣤⣤⣂⡀⠀⠀⢀⠠⢞⡤⠋
⠀⠀⠀⢰⣿⣏⠑⣄⠉⡑⠢⠤⣒⡤⢤⣄⣀⣀⠀⠀⠈⠀⣒⣸⡯⠙⢛⠻⡿⠷⢶⣶⠁
⠀⠀⠀⠸⡹⢟⣻⣻⠗⠐⠀⠀⢤⡉⠑⠲⠤⣉⡙⠛⠷⣟⡛⠛⠿⠿⠿⣿⣶⣶⣦⣴⠆
⠀⠀⠀⠀⠈⠒⠚⠁⢾⣥⡴⠃⠀⠐⠓⠀⠂⠀⠈⠙⠲⠤⣈⠙⠒⠤⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠢⠼⠋

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

⠀⠀⠀⠀⠀⢀⡀⢠⠄⣠⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣼⣱⣿⣆⠫⣆⠀⠀⠀⣰⣿⣿⢦⡀⠀⠀⢀⡾⢡⡀⠀⠀⠀⣠⣾⠆⠀⠀⠀⣠⡾⡀
⠀⠀⠀⠀⢿⡿⣮⡈⡼⢢⢟⡆⠀⠰⣏⡻⣴⡾⡷⠀⠀⢼⠿⣆⣿⠀⠀⠘⢿⣏⣠⠇⠀⠀⢿⡱⢹⠆
⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠖⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⢀
⠀⠀⠀⠀⢀⡖⣩⣦⣭⠲⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢷⢀⣹⣆⢾⡄
⠀⠀⠀⠀⠈⢰⣿⣿⣿⣗⠘⠀⠀⠀⠀⠀⢀⣠⣾⣟⣿⣦⣀⠀⠀⠀⠀⠀⠀⢠⣿⢈⡸⢿⢻⢸⣷⣠
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⣰⣿⡿⣿⡿⡿⣿⣿⣷⠀⠀⢀⣀⢀⣟⣻⣎⣻⣆⣾⣾⣹⣟⡁⢀⡀
⠀⠀⠀⠐⣒⠀⠀⠀⠀⠉⣙⣓⠀⠀⠀⣿⣯⡇⢺⡇⢸⢱⢸⣿⡇⠀⠀⠘⠻⡍⢿⢹⣿⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢻⣷⣧⣵⣯⣜⠎⡰⣿⠃⠀⠀⠀⠀⠙⢮⣣⣻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠤⠠⠩⠉⠉⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣏⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢈⠟⠉⠉⠉⠛⡃
⠀⠀⠀⠀⠀⢰⣾⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⡜⠒⠒⠲⠶⠒⢳
⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠠⢤⣿⢾⣿⣦⠄⠀⠀⠀⠀⠀⠀⢠⠄⠐⠀⠠⡀⠀⠀⠀⠀⢰⡉⢹⡆⠄⠀⠀⠀⠄⠀⠸⣄
⠀⠀⠀⠀⠀⠈⡿⠿⢿⠁⠀⠀⠀⠀⠀⣠⠔⣡⣶⡶⢶⣦⡈⠢⡀⠀⠀⠀⠉⠉⠀⠀⢀⣀⠀⠀⠀⠀⠉
⠀⠀⠀⡆⠀⠰⢱⣿⡍⠀⠀⢰⠀⠀⢘⠡⣴⣿⣿⣤⣼⣿⣿⣲⠉⠂⠀⠀⠀⠀⠠⡀⢸⣭⢀⣤⡀
⠀⠀⠀⠣⢠⣃⡈⠀⣉⣻⡇⠜⠀⠀⠲⣄⠙⣿⡏⠏⠻⣻⡿⢚⡴⠂⠀⢀⡀⠀⣀⣙⢦⣥⢚⣋⠀⠀⣀
⠀⠀⠀⢰⣰⣮⡅⠄⢸⣅⣦⡦⠀⠀⠀⠈⠀⣈⠛⠛⠛⢋⡠⠊⠀⠀⠀⠈⠉⠉⠉⣩⠞⡟⢺⣍⠉⠉⠉
⠀⠀⠀⠈⠀⠒⠶⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠇⠀⠙⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⢿⣿⣄
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡆⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠈⣿⣿
⠀⠀⠀⣠⣾⣿⣿⡿⠛⠉⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⢀⣼⣿⣿⠟⠁⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢸⣿⣿⠛⠛⣿⡆⠀⠀⠀⠀⠀⣼⣿⠇⢀⣴⠶⣶⣶
⠀⣾⣿⣿⣧⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⠿⠆⠀⣿⠇⠀⠀⠀⢀⣼⣿⡟⠀⣿⡏⠀⠀⠙
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣠⠞⠁⢀⣀⣤⣾⣿⡿⠟⠀⣸⣿⠁⠐⢶⣾
⠀⠀⢀⣤⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⡾⠃⣠⣾⠿⠛⣛⣋⣁⣀⣤⣾⡿⠃⣠⣶⣤
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡿⠋⠀⣼⢁⣾⠏⠠⠖⠋⣉⣭⣭⣭⣉⣉⡀⢀⣩⡿⠋
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠋⠸⡇⠀⠀⢺⣦⡙⠧⠀⠀⠉⠉⠉⠉⠁
⢸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠁⣰⣷⠘⣿⣿⣦⠈⢦⡀
⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⢀⣤⣶⡶⢀⣠⣾⣿⣿⠀⣿⣿⣿⡇⠘⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⡀⢠⣿⣿⡏⢰⣿⣿⡿⠛⠁⢀⣿⣿⣿⣿⠀⣿⠃
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⣀⣾⡇⣾⣿⡿⠀⣿⣿⡟⣴⠟⢠⣾⣿⡿⢻⡏⠀⠁
⠸⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⢀⣤⠾⣿⡿⠠⠿⠋⠀⠀⣿⣿⣿⠏⢠⣿⡿⠋⢠⣾⠃
⠀⢻⣿⣯⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠸⣿⡀⠰⠟⠁
⠀⠀⠻⣿⣏⣩⣽⣿⣿⣿⠿⠛⠉⠀⠀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣈⣀
⠀⠀⠀⠙⠛⠛⠛⠉⠁⢀⣤⣴⣶⣿⣿⡿⢿⣿⠟⣿⡟⠛⣿⡏⢹⣿⠉⢻⡏⠉⣿⠋⢹⣿⠉⣿⡟⠻⣿⡿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢿⡏⠉⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⢸⡇⠀⣿⡄⢸⣿⠀⣿⡇⠀⣿⡆⢸⣿⠉⣿⡿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠀⣿⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⢸⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣿⣇⡀⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⣿⣧⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣾⣿⣦⣿⣧⣤⣿⣇⣸⣿⣀⣼⣇⣀⣿⣅⣸⣿⣠⣿⣧⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣭⣝⣛⡛⠛⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠛⢛⣛⣫⣭⣽⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠀⠀⢰⣶⣦⠀⠸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⠀⠀⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣀⣀⣈⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣶⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠻⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⡀⢠⠄⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣮⣛⣛⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠂

⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣀⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠙⣶⡆⠈⠉⠚⡿⠟⣿⣿⣿⣖⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣧⣭⡀⠀⠈⠰⣿⣿⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠈⠻⣷⣶⣤⠿⣿⣿⣿⡿⢦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⠘⠿⢿⣄⡠⠤⠆⠘⠻⢷⣾⣿⣿⣿⠳⠶⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡶⠀⠀⠾⢹⡇⠀⠀⠀⠀⠀⠙⢻⣿⠋⠀⠀⠀⢨⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣏⣤⠄⣸⣿⣿⣶⣀⠀⠀⢶⠀⠀⠀⠀⠀⣀⡔⢾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣿⣿⣟⣡⣀⣈⠻⣿⢶⠚⠏⠀⠰⣦⣠⡟⢿⣏⠀⢻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣼⣆⣠⣤⣶⠁⠙⠃⢸⡄⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⣿⣿⡟⠉⠘⠿⢿⠉⠁⠀⠀⢠⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡏⠀⠀⠀⣰⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣟⡿⠛⠢⠨⠥⣼⣿⣿⣿⡆⠀⣸⡇⠀⢀⣸⣿⣿⣿⠂⡀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠏⠳⠖⠀⢶⣶⣶⣿⣿⣿⣿⢀⣿⣧⠀⣾⣿⣿⣿⣿⣡⣿⣇⢀⢨⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣄⣁⣀⠀⠀⠤⠀⠚⠻⠟⣋⢙⣿⣿⣿⣿⣾⣿⢹⣿⣿⣿⣿⣿⣿⡇⡁⠄
⠀⠀⠀⠀⠀⠀⠄⡼⣯⡷⠶⠄⠁⠀⠀⠀⠀⢀⢻⡿⢿⣿⣿⣿⡿⣏⢸⣿⣿⣿⣿⡟⣿⡇⣤⡆
⠀⠀⠀⠀⠀⢨⣽⣿⠿⡖⠀⠀⠐⠀⠀⠀⠀⠘⣜⠛⣿⣿⣿⣿⣿⣿⠞⣿⣿⣿⠿⠃⢸⣧⠯⠇
⠀⠀⠀⠀⠀⠸⣿⣿⣟⠳⠶⣢⣀⠀⠀⠀⢀⠠⡌⢹⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣧⠀⡿⠀⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣙⣿⠟⢻⣟⡀⠀⠀⢶⠀⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣁⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣻⡄⠸⣿⡿⠶⣴⡇⢠⣿⣿⡿⠿⠛⣽⡇⠀⢯⣥⣄⣸⠓⠙⠁⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣧⣀⣳⡀⢀⣙⣏⡒⣽⣿⠓⢠⣼⣿⣉⠀⠈⠿⣿⣿⣧⢾⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠰⣾⡋⣽⣿⣿⣿⡿⢄⣽⣿⣇⣼⣿⣧⣿⡟⣩⣟⡀⠖⢿⣿⣻⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠏⠤⠡⠀⠀⠉⢹⣿⣿⣾⣿⣧⣿⣿⣿⣆⠈⢻⣿⣽⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣜⢿⣿⣿⣿⣿⣿⠄⠶⣿⣮⣍⣙⠻⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠾⣷⣶⣄⣠⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⢿⣿⣯⣿⣷⣾⣿⣿⠧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣄⣠⡦⢀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠰⡼⠋⠙⠹⠿⣴⠀⠀⠀⠀⠀⢠⡈⣿⣧⣽⡄⠀⠀⠀⠰⣄⢱⡄⠀⠀⠀⣾⣿⣿⡿⠋
⠀⠀⠀⠤⢿⠹⣿⣿⣿⣿⣿⣿⣿⡧⠤⠀⠀⢸⣷⣿⣿⣿⠇⡀⠀⠀⠀⠈⠉⠳⣄⢀⡴⠋⠉⠉
⠀⠀⠀⠀⠘⢧⣻⣿⣿⣿⣿⣿⡿⠁⠀⠀⣿⣶⣿⠛⠇⠹⣷⣿⠀⠀⠀⠀⠀⢀⡤⠻⣄
⠀⠀⠀⠀⠀⠀⡟⠛⠿⠿⠛⢏⠀⠀⠀⠀⠘⠏⠈⠀⠀⠀⠘⠃⠀⠀⠀⢀⡴⠋⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⣠⡋⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠈⠀⠀⠀⣰⡀⠀⢀⣆⠀⠀⠈
⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣴⡿⠛⠉⠉⠛⢿⣦⡀⠀⠀⠀⢸⣿⣿⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⠄⣾⡟⠀⠀⠀⠀⠀⠀⣿⣧⢠⡄⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⡠⢳⣶⣿⣿⣿⡿⠃⠀⠀⠀⠘⠂⢻⣧⠀⠀⠀⠀⠀⠀⣿⡟⠘⠃⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⢱⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⣀⣴⣾⠟⠁⠀⠀⠀⠈⠛⠛⠀⠙⠛⠃
⠀⠀⢸⡜⣿⣿⠋⠀⠠⠰⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁
⠀⠀⠀⢳⡜⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣦⣄⣀⣀⣀⣠⣴⣦⠆
⠀⠀⠀⠂⢻⡉⣿⣿⣿⣿⣿⣿⣿⡟⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠈⠳⣽⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣀⣰⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠆
⠀⠀⠀⠀⠀⠀⡞⠙⠛⠛⠋⢣⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⣸⢀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⣿⣿⣿⣷⣶⣤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡿⠿⠞⠛⠿⠹⣾⣿⣿⣿⣿⣿⣿⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⣿⣿⣂⣀⠂⣸⣻⣾⣿⡟⣿⡟⣿⣦⡻⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⠅⠊⣛⠃⠉⠀⢿⣿⡛⣉⣮⣿⠃⢨⣹⣿⣎⠅⠹⣿⣿⡿⠿⣿⣷⣶⣦⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⡿⡄⢠⣼⣷⠂⠀⠄⠰⣿⠜⢿⣿⡇⠀⢿⣷⡿⠋⠀⠀⡿⠛⢄⢛⡿⠓⢹⣿⡏⢛⣿⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣽⣿⣿⣏⠀⢸⣧⡄⠁⠀⠀⠀⣀⠰⢼⣿⣿⡆⣴⢻⠇⢀⠀⠀⠀⠉⠀⠀⠀⠂⠸⣿⢃⠄⢐⡶⠉⢿⣿⡿⣷⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠘⢿⣿⣿⣅⣴⣿⣏⢉⡀⠀⠀⠀⢹⣷⡿⠟⣛⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠒⢦⣴⡞⣡⣿⡇⠘⠛⠱⢬⣿⠘⣶⣿⣄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠉⠉⣸⣿⣻⡟⢡⣄⠀⠀⠀⣀⠀⠋⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⢶⣄⣀⣸⡟⢰⣿⢛⡁⠀⠀⡂⢀⢤⣴⣿⡿⢏⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⢿⣿⣷⣦⣼⠿⠗⢤⣤⣭⣾⡷⣶⣎⠀⢀⡾⡿⢁⣤⣤⣶⣦⣾⣿⡿⠁⣿⣷⣾⡟⠿⠶⠤⠐⠒⠉⠥⠊⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣷⠤⢄⡙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠠⣮⡿⠶⠘⠿⠿⠻⠛⠛⡉⡀⠂⠬⠥⠄⡒⠈⠁⠀⢀⡠⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣴⣀⠀⠈⠁⠒⠠⠄⣉⡉⠉⠉⠉⣉⣁⣀⢀⠀⠀⠤⠤⠤⠄⠒⠀⠈⣾⠁⠀⠀⢀⣀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⡤⠴⠚⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠲⠶⠤⠤⣄⣤⡴⠤⠴⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⣤⠀⣀⡀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢿⢸⣟⣠⡿⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠐⠂⠐⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠿⣿⣿⣿⣷⣶⣤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⠿⠟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣿⣿⣇⣐⠃⣼⣿⣾⣿⣿⣿⡿⣿⣮⡻⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠅⠊⣟⠃⠛⠀⢿⣿⣿⣟⣾⣿⠇⣨⣽⣿⣾⣯⠹⣿⣿⣿⢿⣿⣿⣶⣦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡆⢠⣼⣿⠂⠐⠤⠰⣿⣟⢿⣿⣇⠀⢿⣷⣿⠋⠀⠀⣿⠛⣌⢛⡿⠓⢹⣿⡏⣛⣿⣶⣾⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣿⣿⣿⠄⢸⣿⣍⠁⠀⠀⠁⣀⡶⢿⣿⣿⣆⣴⢻⡧⢀⠀⠀⢈⠉⠀⠀⠈⠂⠸⣿⢃⠤⢺⣾⡉⢿⣿⣿⢷⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠘⢿⣿⣿⣧⣤⣿⣏⣫⡀⠈⠀⣀⢹⣿⣿⢿⣟⡁⣙⠂⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⡿⣡⣿⡇⡘⠻⠿⣮⣿⠛⣶⣿⣅⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⠉⠉⣸⣿⣿⡿⣷⣀⠀⢀⢀⣀⠀⠛⠀⠉⢀⠄⠐⠂⠀⡀⠀⠰⢶⣄⣀⣺⡟⢰⣿⢛⣅⣈⠀⣠⣠⢤⣴⣿⡿⢏⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⢿⣿⣿⣶⣾⡿⠗⢤⣤⣽⣿⣷⣶⣾⠀⢠⣶⣿⢣⣤⣤⣶⣶⣾⣿⡿⠁⣿⣿⣿⡿⠿⠶⠦⠙⡚⣉⡥⣎⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣷⡤⢄⡙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣾⣿⣷⠺⠿⠿⠿⠛⠛⡉⡀⠂⠨⠭⠄⡒⠈⠉⠀⣁⡠⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣀⠀⠈⠁⠒⠠⠄⣉⣉⡉⠉⠉⣉⣁⣀⣀⣀⠀⠠⠤⠤⠤⠒⠒⠉⣾⠁⠀⠀⢀⣀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⣄⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⣀⣀⣤⠴⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠶⠤⢤⣄⣠⣴⠤⠴⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⣄⡀⣀⡀⣠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢸⠷⣿⢸⣟⣠⡿⣧⢸⡇⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠒⠒⠂⠒⠒⠒⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢾⢉⣤⣴⣷⣶⣶⣶⣤⣀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣭⣽⣥⣄⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⢄⣉⣠⠆⠀⠀⠀⠀⠀⠀⠀⣾⣧⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⢸⣿⣿⣿⠿⠛⢋⡅⢠⡆⢸⣾⠃⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⢚⣋⣉⣀⣀⢶⣾⢿⠿⡷⢾⡿⢿⣿⣂⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⢖⡞⣻⠏⡹⠋⡽⢋⡿⢛⡶⢦⣤⡆
⠀⠀⠀⠀⠈⢿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠋⠹⣿⣒⣷⠾⠷⠾⠴⠾⢶⣟⣺⡏⠙⠃
⠀⠀⠀⠀⠀⠈⢷⣌⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠿⠿⠿⠿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠙⠛⢛⡛⠋⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠸⡇⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣷⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢀⣸⡿⠉⠉⠉⠉⠉⠉⠉⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡶⢲⢞⡿⢻⣿⡳⢲⠖⣾⣧⡀
⠀⠀⠀⠀⠀⣿⣛⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠶⠧⢮⣴⣥⣿⡴⠧⠾⠖⢿⡁
⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠋⡉⠀⠀⢠⠀⠉⠙⠻⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⣤⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢴⠈⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⣿⣿⢳⡆⠀⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢼⠀⠘⣿⣿⣿⠛⢦⡀
⠀⠀⠀⠀⠀⠀⢸⡁⠀⣿⣿⣿⢸⡇⠀⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⣿⣿⣿⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣿⣿⣿⣼⠇⠀⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⣸⠀⢠⣿⣿⣿⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢼⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠋⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⣠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡹⢿⣷⣦⣄⣁⠀⠀⠘⠀⣀⣠⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢭⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⣟⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠛⠉⠀⠀⠀⠀⢀⣀⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢀⣀⣀⠤⠔⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡭⢝⣛⣳⣬⣽⣿⣿⣯⣴⣾⣿⣯⣽⣒⣲⡦⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣫⠽⣗⣾⡿⣿⣷⣾⡿⣿⣷⣺⡽⣿⣷⣺⡿⠽⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣿⣭⣿⣿⣯⣭⣵⣿⣯⣭⣽⣶⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠟⠻⠿⣿⠿⠟⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣿⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⣤⣿⣤⣤⣤⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⠙⠓⠦⣤⣿⡤⠶⠛⠉⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇

⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠀⠀⠀⠀⠀⠀⠄⠴⠆⠐
⠀⠀⠀⢯⠀⠀⠀⠀⡀⢠⡤⢀⢈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⢤⠆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡆⢀⠀⢤⠀⢀⡆
⠀⠀⠀⡀⠀⠀⠀⠐⠳⠆⠀⠀⠁⠀⠀⡜⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢠⡄⠀⠘⠀⠀⢀⣀⡧⠤⠄⠀⠀⠀⠀⡏⠀⠀⠀⠀⠏⠁
⠀⠀⢈⡉⠀⠀⠀⢰⠲⠂⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⠈⠩⢃⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠁⠀⠬⠀⣈⣘⣂⡤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠛⠋⠉⠀⠀⠀⠀⠘⢃⠀⠀⢀⠉⠀⠀⠠⡄⠀⠀⣄⣠⠄
⠀⠀⠀⠆⣐⡠⢄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⢖⣂⠖⣲⡤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠈⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⢄⠀⠀⠀⠈⠈⡉⡀⣀
⠀⠀⠀⠀⠤⣤⠄⣀⣀⣀⣀⡀⢠⡠⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⢩⡠⠄⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠃⠐⢀⢨⣱
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠊⠉⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠐⡀⠈⡀⠈⠀⠀⠀⠀⠠⠂⠀⠀⠀⡀⠐⡆⠀⠳⠞⣬⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⡀⠀⢈⡀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠂⠀⠀⠀⠂⠈⠄⠀⠀⠀⢠⠂⠀⠀⠀⢀⠠⢀⠀⠀⢤⣀⡤⠷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠛⠫⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⢀⠄⠀⠀⣀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡛⠲⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠺⠝⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠲⠀⠀⠀⠀⠀⠘⠉⠆⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠉⠈⠁⠀⣴⠀⠀⢀⠀⠛⠁⠀⠀⠀⠀⠈⠃⡀⡀⠀⠀⠈⠀⠀⢀⡄⠈⠒⠂⠀⠓⠚⠒⠒⠊⠁⠀⠀⠀⠀⠀⢀⡀⢀⠀⢠⠀⠐⡄
⠀⠀⠀⠀⠀⠀⡴⠣⠀⠀⠀⠛⣆⠀⠀⠀⠀⠀⠀⢰⡿⠭⣀⠀⡄⢦⠜⠃⠁⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⠀⠀⣴⣶⣄⢁⣀⡀⠡⣤⠜⠈⠁
⠀⠀⢀⣀⣀⢀⠣⣁⣀⢱⣄⣛⠕⠀⠀⠀⠀⠀⠀⠻⠀⠀⣤⠀⠠⢿⢃⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠡⣃⣟⠀⡡⠴⠀⠁
⠀⠀⠀⠼⢗⣢⠶⠉⣳⡛⠉⠉⡠⠟⠀⠀⠀⢠⣢⠤⠀⠀⢀⡌⠀⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠀⠀⠠⠀⠀⠀⠀⢸⠛⠕⠒⠈⠓⢽⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠚⠲⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⢿⠿⠿⡿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⠾⣿⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠉⢢⡀
⠀⠀⠀⠀⠀⣀⣠⣴⡿⢠⣿⠇⠀⠠⢶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠀⠀⢀⠇⠀⠠⢶⠒⠦⢄
⠀⠀⠀⣠⣿⣿⡏⠀⠐⢿⣯⣤⣤⣴⠿⢹⡿⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠋⠀⠀⠀⠰⠯⠤⠤⠐⠉⠀⠀⠘⠄⡀
⢀⣠⣾⣿⣦⣿⣶⣶⣶⣷⡟⣳⣾⣿⣶⣶⣰⣾⣮⠇⣶⣠⣠⡀⢀⣀⢀⠋⠀⠀⠠⠒⠂⠐⠐⠠⠄⠀⠀⠀⠐⠀⢀⠅⠀⣀⣀
⠀⠈⡟⠙⠻⣿⡿⠟⠛⠛⠃⠛⠻⠿⠿⠿⠿⣿⠏⠉⣿⠳⠴⠃⠀⠈⠎⠐⠒⠀⠀⠂⠂⠀⠂⠐⠀⠠⠀⠠⠤⡀⠈⠈⠞⠁⠤
⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⣿⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣶⣤⡀⣿⠀⠀⠀⠀⠀⢐⠀⠀⡤⠆⠨⡁⠀⠀⠐⣂⡉⣭⠙⠃⠐⠠⡀⠅
⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⡿⢿⣿⣷⠏⠀⠀⠀⠀⠀⠐⢦⡈⠤⣄⠀⢅⠤⠀⠀⠀⠊⡀⢂⠀⠀⠆⢰
⠀⠀⠈⠙⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠌⠑⠪⠄⠀⣀⣁⣀⣀⣀⡁⠤⠥⠒⡝⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⢀⠄⠀⠀⠀⡀⠀⡠⠊
⠀⠀⠀⠀⠀⢠⣯⡽⠿⢿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡙⠲⢼⢨⣤⡤⢴⣶⡋
⠀⠀⠀⠀⠀⣾⠋⠛⠛⠾⣿⢿⠟⠛⠋⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠐⠀⠸⢸⠭⠅⠘⠀⠁⣆
⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣿⢸⡆⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⡂
⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⢆
⠀⠀⠀⢰⡏⠀⠀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⡆
⠀⠄⠠⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁
⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⢿⣿⣿⣦⡀⠀⠀⣈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⢈⢿⣿⣿⣷⣄⠀⠀⠀⠂⠈⠑⢄
⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⢀⠻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠈⢢⡈⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡒⠲⡄
⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆⠀⠐⠐⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠁⠀⣹
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠐⠘⢦⣀⠀⠀⠀⠀⢀⣠⠏⠀⢰⣏
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⢀⡤⣄⠀⠀⠀⢀⣿⣿⠀⡀⠀⠀⡠⠀⠀⠀⡀⠈⠈⠉⠓⠒⠚⡿⠀⠀⠀⠘⠈⠳⡄
⠀⠀⠀⠀⠈⣿⣿⡄⠀⢠⡏⡞⢢⢳⠀⠀⣸⣿⡏⢀⠇⠀⠀⣇⠀⣠⠀⣇⡤⠂⠀⠀⠀⣀⣽⠤⠤⣄⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠘⣿⣷⡀⠘⣇⢧⡼⡸⠀⣠⣿⡟⠀⡼⡠⠊⠀⣿⠞⣁⡴⠿⠒⠚⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠉⠉⣁⣴⣿⠟⢀⣠⠏⢁⡠⢚⣧⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⣟⠥⠒⢁⣮⣔⡫⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣀⣀⣀⣀⣠⣤⣤⠴⠶⠶⠗⠛⠛⠛⠛⠛⠉⠛⠉⢙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⡿⢛⡉⠩⢥⠀⠀⠉⠁⠐⠂⠀⠒⣂⠈⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⡠⠊⠄⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢃⣀⡀⠠⠀⠐⠀⠉⠁⢀⣀⠀⠤⠀⠀⠀⠀⠀⠉⠭⠁⠀⣴⣿⣿⣿⠀⠀⠀⠀⢀⠔⠀⢠⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠂⠠⠤⠀⠀⠀⠀⠈⠉⠀⢀⡀⠠⠄⠀⠀⠂⠠⠤⠄⠀⠀⠐⢩⣭⣿⠀⠀⠀⡰⠋⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡶⠀⣉⡁⠀⠀⠉⠉⠄⠀⠀⠀⢀⣐⠐⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡟⠀⠀⣀⠠⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢟⣋⢉⡀⠀⠀⠤⠄⠀⠀⠤⣀⠡⠤⠄⠀⠈⠉⠐⠒⠒⠀⠀⢻⣿⣿⡇⢀⠄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠉⠐⠂⣀⠀⠀⠉⠀⠀⠐⠀⠀⠀⣉⣀⣠⣔⣂⠤⠒⠤⢢⣰⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣍⡮⠉⣀⣀⠀⠀⠀⣐⣂⣐⣤⣄⠔⠲⢭⣶⣬⡳⠯⣔⠊⠀⣩⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣴⣤⣾⣷⣶⣿⣿⡿⠿⠛⠛⠛⠛⠲⢦⣼⡄⠀⢀⢀⣿⣤⣾⡿⢃⣿⣿⣿⣷⣖⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣶⢾⣿⣿⣿⣭⣿⢶⡶⣿⣿⡿⣽⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣟⢟⣿⣿⡿⣿⣷⣶⠶⡶⣖⣺⣺⣭⣿⣷⣾⣿⣟⢟⡿⣻⣟⣿⡷⣜⡫⠿⠛⠉⠑⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣷⣝⡻⣽⡶⢶⠾⣿⢿⡿⣿⢿⣿⣿⣿⢿⠿⢏⣛⠿⠍⠚⠊⠁⠀⠀⢀⡶⣶⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡇⠇⡏⣿⣮⡙⢾⡑⠮⢷⣙⠯⠏⠝⠉⠐⠃⠈⠀⠀⣀⣶⣶⠀⠘⠃⠀⠘⢷⡿⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣦⢇⡌⡏⡟⣾⡟⠉⣀⣄⣀⠀⣴⣻⡆⠀⠘⠃⠀⠘⠿⠏⢀⠀⠄⠄⢴⣖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠓⠧⣀⢿⡇⠀⣿⣼⣿⠀⠙⠛⠁⢀⣀⠠⠄⢲⣶⠁⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡽⠀⠀⠀⠀⠈⠹⡇⠀⣈⡙⠥⠐⠂⠀⠈⠀⠀⠀⠀⣀⣿⣆⡀⠀⠀⠀⠀⢻⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢀⣀⡤⠤⠖⠚⠋⠉⢹⣿⠐⠍⠀⢀⠠⢸⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⠀⠀⠀⠀⠀⢀⣀⡄⣷⠶⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⢈⠐⢗⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣧⠴⠖⠒⠋⠉⠉⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣃⠀⠀⠀⠀⠀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠭⣒⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠒⠈⠁⠈⣿⡄⠀⠀⠀⠀⠀⠘⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠉⠒⠤⣁⠒⠄⣴⣠⡠⠄⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣟⡗⣶⣶⣶⣿⣿⣫⢑⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣵⣿⣾⣋⣹⣋⡉⢉⠝⢉⠍⢉⠟⢻⣹⡻⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢪⣴⢹⠋⠈⢰⣛⡥⣽⣉⣿⣦⢠⠄⠃⠀⠁⠿⠹⢷⡈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣳⡿⠉⠈⠀⠀⢀⣿⡷⣷⣄⣌⡱⢊⡀⢀⠀⢀⠇⠀⡞⢿⣿⣷⡀
⠀⠀⠀⡄⢠⣀⣠⣿⣳⣀⣀⣄⣄⣤⣾⣻⣶⡟⠳⣟⣗⣾⣷⣷⢻⠁⢀⡼⠃⠈⣷⡍⣧⣀⢠⢠⡄
⠀⠀⠀⠀⡇⠀⠸⡿⠉⢸⡄⠀⠰⡄⠀⠁⠈⠉⠉⠳⣭⣻⡟⢻⢸⣦⡾⠓⠂⢀⡿⣧⢸⡄⠀⠠⡇
⠀⠀⠀⠀⡇⠀⢸⣿⣦⣄⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣾⣯⣅⣁⣤⠼⡿⣿⢞⡇⠀⠀⡇
⠀⠀⠀⠀⡇⠀⢸⢳⣧⠇⠙⠷⣷⣂⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣀⣤⡠⠤⢿⣶⠃⠀⢰⡇
⠀⠀⠀⠣⠟⠉⠙⣿⡿⠀⠀⠀⠘⢝⡻⣿⡦⣤⣄⣀⠀⠀⣀⣀⣀⣀⣤⣸⣿⠇⢠⣋⡞⠉⠙⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣽⣵⣄⠀⠀⠐⠞⠛⣷⢎⣏⡉⡟⠛⠿⠻⠋⠀⠀⠈⢦⡀⣿⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢿⣰⣤⡖⢠⡀⠂⠹⠋⠙⠗⠶⡦⠀⠀⠀⢠⣰⡼⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢭⣿⣿⢶⣷⣤⣄⣀⣁⣠⣾⣧⣴⣰⣶⣸⣵⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠧⢉⣉⢧⣟⡿⠿⠻⣯⣽⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⣿⣿⢘⣏⠀⠰⣷⣍⣐⠂⠄⣉⠀⠖⢈⠀⠀⠀⠀⠉⠐⠀⠀⠀⠈⠀⢈⠉⢉⠉⣉⣉⣉⣉⡉⣉⣉⠉⠉⠉⠛⢛⣛⣛⡛⢙⡛
⣿⡿⢸⣇⠀⠀⢄⣬⠉⠛⠒⠦⢬⣏⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡈⠙⠻⣍⢿⡉⢫⣍⣋⡙⠛⠳⠶⠶⠶⠶⠶⠦⡷⠀⡇
⣿⠓⣽⡏⠑⢦⣄⡘⢉⠁⠐⠂⠀⣍⣙⠑⠒⠤⠄⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠙⠻⣵⣉⢲⣌⡻⣿⡉⠉⣉⡉⣿⠈⣿⠀⡇
⣏⡶⢸⠃⠀⠀⠈⠙⢧⡀⠀⠀⠀⡄⢈⣀⠐⠀⠠⠀⣀⠀⠀⠀⠀⠀⢀⡇⠲⠤⣀⠀⠀⠙⠣⣊⢝⡦⣝⠳⣍⣁⡿⡇⣿⠀⣇
⠏⣤⢸⡆⠀⠀⠀⠀⠀⠉⠑⠢⣀⡁⢠⣄⣀⣉⠀⠒⠀⠀⠉⠀⠀⠀⠀⠄⠈⠀⠈⠡⠒⢄⡀⠀⠉⠫⢶⡝⣦⣝⠃⡇⠳⠀⣿
⢸⠉⣼⠁⠀⠀⠀⠀⠀⠀⡄⠀⠈⠓⢨⣀⠉⠉⠙⠛⠒⠶⠤⠤⣀⣀⠀⠘⠐⠂⠠⠄⣀⠀⠀⠁⠒⠠⢀⡰⢕⡪⣛⣣⢸⠀⣿
⢸⠆⣹⠀⠀⠀⠀⢠⡄⢄⣱⣀⠀⡀⢀⠀⠑⠶⠔⠂⠀⠀⠀⠀⠀⠀⠈⢹⡶⣤⢤⣄⣀⣈⠁⠐⠂⠤⠬⣉⣛⣷⣮⣜⣾⠀⣿
⡈⢰⠟⠀⢀⠀⠀⠀⠀⠈⣥⠉⡈⠁⠈⠀⠀⠀⠘⠳⣆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠉⠉⠙⠛⠒⠒⠀⠤⣄⠀⣉⡹⠿⣤⣿
⣧⣼⡁⠀⠈⠀⠀⠀⠀⠀⠸⡄⢣⡖⠶⠒⠒⠒⠒⠂⠀⠛⠖⠴⢄⠀⠀⠀⠦⢤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣩⣷⣿⡛
⣌⡉⢻⡓⢾⣦⣤⠄⣖⣂⠛⠑⠊⣿⡛⠉⠉⠀⠈⠁⠁⠀⠀⠀⠈⠳⣆⠀⠒⣾⢣⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠠⠤⠤⠠⢸⡏
⣬⡉⢻⢮⣼⡉⢫⠉⡿⢷⣤⣉⡉⣯⡵⠦⠤⣀⣀⣀⠀⠀⣀⠉⠁⠉⣹⠋⣿⠛⣀⢇⡠⡭⠅⠒⠒⠀⠈⠉⠀⠐⡀⠾⣩⣿⡃
⡠⡙⣾⠃⢼⡛⠻⠴⣯⣉⠲⢉⠽⠻⢶⣄⣀⠀⠀⠀⠛⠋⠥⢾⡛⢻⡟⢐⡟⠀⠋⡜⡤⠤⠐⠒⠀⠀⠀⠀⠀⠉⠀⢰⠀⠀⡅
⢷⡌⡿⠀⠐⠌⠲⠄⠘⠛⠿⣶⣬⡶⣿⠤⢈⡙⠲⢦⣍⣀⠰⡎⠀⣾⠁⣹⠂⣧⠀⢠⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢹⡄
⢂⠥⣿⢀⠱⠀⠑⢄⠙⠃⠀⠀⠉⣟⣻⢶⣤⣀⠀⠀⣸⠉⣻⠁⢹⡏⢸⠧⡜⢻⡄⠀⠆⠰⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⢰⡇
⠸⢎⣷⠀⢠⠀⠀⠀⠙⢂⠀⠈⠀⡆⠀⠈⠙⠻⢶⣤⡾⢶⡟⢨⡟⢀⡿⢠⠁⣘⢿⠀⠘⡀⢀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⢸⡇
⢸⠈⣿⠀⠰⠅⠀⠀⠀⢄⠐⢄⠀⠘⠛⠠⡀⠀⠀⢹⠁⣘⠃⢴⠏⣾⠃⡌⠀⠋⠘⠇⠀⢱⡈⠀⠀⠀⢠⣾⣿⣿⣿⣿⠀⢸⡇
⠇⣿⣶⠀⠰⡄⠠⣆⠀⠀⠂⠀⠀⢦⠀⠀⠀⠑⢠⡌⢀⠇⢠⡏⢠⡟⠠⠀⠂⢰⡄⠾⣀⠈⠡⠀⠀⣰⣿⣿⣿⣿⣿⠏⠀⢸⡇
⡀⣿⡇⠀⠰⠛⣄⢁⠀⠀⠄⠀⠀⡄⠘⢤⡀⠀⢀⠓⡼⠀⣾⠁⡼⢁⠆⠐⢀⠟⣠⠀⢃⠀⠀⠆⠈⣿⣿⣿⣿⣿⠏⡰⠀⠸⡇
⡌⣿⣿⡄⢨⡄⠘⠇⠀⠈⠂⠀⠀⠐⢢⠀⠉⠢⡜⠀⣧⢸⡇⢰⠃⡌⢀⠁⡜⠀⢛⠀⠘⡀⠀⠐⡀⠘⣿⣿⣿⠏⠀⣰⠀⠸⡇
⣱⣿⣿⣿⣀⡃⡀⢰⡀⠀⣠⠀⠀⠳⠄⠈⢂⣠⠇⣼⠃⡾⢀⡞⠰⠀⠈⢰⠁⠀⠸⡄⠀⢱⡀⠀⢁⠀⢹⣿⠇⠄⢠⡿⠀⠰⡇
⡟⠻⣿⣿⣿⣿⣇⢪⠃⡀⠘⠂⠀⢃⠘⢀⠀⢏⡸⠇⢰⠃⣼⢁⠇⡀⢀⠇⠀⠀⠀⢇⠀⠀⢡⠀⠈⠄⠈⠋⠌⣠⡓⣽⠀⠀⡇
⠁⠰⡏⠻⣿⣿⣿⣿⠀⠁⠀⡀⠀⡉⠛⠀⢰⠏⠱⠀⡞⢠⡏⡘⢠⠁⡜⠀⠠⠀⠀⠾⠀⠀⠈⡄⠀⠀⠀⠈⢬⢧⡙⣾⠀⠀⡇
⠄⡐⠆⠀⠈⠿⢿⣿⠀⠀⠀⡇⠀⠁⡀⠀⡎⢠⠆⢰⠁⡾⢠⠃⠆⣸⠁⠂⠀⠀⠀⠘⡆⠀⠀⢡⠀⠀⠰⠀⢪⢳⣽⣿⠂⠀⡀
⢤⡼⠧⠤⢀⡓⠀⠙⣦⡀⠀⠑⠀⡄⠹⣾⠃⡤⠀⡏⢰⠃⡈⠘⢠⠃⠀⡈⠀⠀⠀⠐⣇⠀⠀⠈⠄⠀⠀⢃⠀⣻⣿⣿⠀⠀⠁
⠂⣾⠒⠂⠠⣷⠀⠀⡟⢡⠀⠸⠀⢃⡙⡘⢠⠀⣼⢀⡏⠈⢀⠃⡎⠀⠀⠀⠀⠀⠀⣰⢻⠀⠀⠀⠠⠀⠀⠀⡄⠘⣿⣿⣶⠀⣱
⠀⣬⠀⠀⠀⣯⠐⠈⡏⠠⠐⣦⣀⡛⠀⠃⡌⢠⡇⣸⢡⠀⠈⠸⠀⠀⠀⠀⠐⠀⢠⠇⢨⡄⠀⠀⠀⠀⠀⠀⠰⠀⢹⣿⢿⠠⠛
⠀⠉⠉⠉⠈⣉⢉⠉⡏⠉⠙⣿⠋⠀⡿⢠⠀⡾⢠⠇⡄⠀⢀⢇⢂⠀⠀⠀⠐⢀⠇⣀⠣⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⢿⣧⡆
⠀⣀⡀⠠⠄⢳⠀⠀⣣⠐⣾⡟⡄⠰⠃⡌⢰⠇⡾⠰⠠⢀⡘⡈⠆⠀⡀⠀⠠⠏⠐⢬⠁⣬⠀⠀⠀⠀⠰⠀⠀⠈⢁⠘⡏
⠁⠀⠀⠲⠊⢛⣀⣤⡏⣴⣯⠾⡅⡄⢠⠁⡎⢸⠇⠂⠆⡌⠁⠘⠀⠠⡀⢀⠟⠰⣌⠢⣱⢾⡁⠀⠀⠀⠀⠇⠀⠀⢰⡆⠀⡀
⠀⢀⠀⢀⣠⣾⣿⡟⣹⡟⡋⠆⣵⠃⡈⢰⢀⡟⠘⠐⢠⠀⢄⠁⢀⠠⢀⠎⢀⡑⠌⣱⢯⡳⡗⠀⠀⠀⠀⠸⡀⠀⠀⢠⡀⠃
⠂⣠⣴⣿⣿⣿⠏⡰⢏⣾⠛⢠⡟⣤⠀⠇⡼⡀⠀⠃⢆⠐⠀⠀⢀⠁⡜⢀⠂⢀⣲⣏⢢⣿⠀⠀⠀⠀⠀⠀⢳⠀⠀⠠⠇
⣿⣿⣿⣿⡿⢩⡞⢡⢈⣵⠛⣼⠃⠇⡸⢠⢃⠁⠈⡘⠀⠀⢀⡁⠦⡙⠀⠎⢐⣾⣟⢦⣿⡟⠀⠀⠀⠀⠐⣀⠚⢀⠐⠁⠈⡆
⣿⣿⣿⠟⠀⡎⠀⢸⡿⣷⢋⡟⠀⠸⢇⡜⡌⡄⢠⠃⠀⠀⠠⣉⠆⢠⠉⢀⣼⡟⢮⣿⣿⠀⠀⠃⠀⠀⢠⠃⢠⠟⡀⠀⠀⠁⡀
⣿⣿⠋⡠⠁⣤⠀⣈⡾⣴⢺⠇⠃⡎⣨⢣⢡⠀⡌⠀⠀⠀⠂⠌⢀⠣⣴⣾⡓⣬⣿⣿⡇⠀⠀⠠⠀⠀⠀⣰⠁⠈⢁⠀⠀⠀⣱
⠿⢃⡜⢁⣼⣿⠀⡟⢮⣧⡿⠐⠰⢠⡟⡄⠇⢴⠀⠀⠀⡄⠚⢁⢢⣠⣿⡓⣼⣿⣿⣿⠀⠄⠀⠀⡴⠀⠐⠁⠆⠀⠀⡄⠀⠀⠁
⢀⡞⢀⣾⣿⣿⠀⣿⣎⣹⠁⠀⠆⣾⢣⠸⠰⠆⠀⠀⡄⠈⠀⢊⣶⣿⡳⣽⣿⣿⣿⠇⠀⠀⠀⠀⢠⠆⠀⠀⠀⣀⡼⢳⠂⠀⣼
⢋⣠⣿⣿⣿⣿⢸⡝⢿⡇⠀⢰⢰⡝⣄⢃⠓⠄⢀⣶⠀⠀⣴⣾⣿⠣⣽⣿⣿⣿⡿⢀⠁⠀⠀⠀⡼⠂⠀⢀⡴⢋⠠⠉⣤⣾⣿
⣃⠉⠙⠛⠻⠿⢿⣿⣿⠙⠀⠆⣆⢣⠈⡜⠇⢀⣶⣷⣄⢳⣿⣟⢦⠹⣿⣿⣿⣿⡇⠄⠀⠀⢀⠜⠀⢀⡴⠋⢀⠚⠀⢠⡿⣾⣿
⠉⠙⠛⠓⠶⠶⠤⢤⣭⠁⠓⠚⣺⣬⣡⣙⣀⣿⣿⡿⢿⣕⡺⢿⡃⠀⠘⣿⣿⣿⠀⠀⠀⢠⠎⠀⡰⠋⠀⠔⠁⠀⣰⣿⣿⡿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠛⠻⡿⣿⣿⣾⠷⣦⣹⢦⣄⠘⣿⡇⡀⠀⡰⠃⡠⠊⠀⣠⠊⠀⠀⣴⣿⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡜⢀⡴⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⣠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⡄⠸⡰⡄⠀⠀⠀⠀⠀⠀⢀⡌⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⡦⠐⡈⡆⠀⠀⢀⢀⠀⡊⢀⠤⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⡠⠃⠀⠀⣀⣋⠄⠑⡼⣸⡰⡀⠀⠀⠀⠀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠰⡀⠀⠀⠀⢽⣅⠀⢀⢌⠔⡱⢈⠀⠀⠀⡐⡕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡷⠀⠀⠀⣀⣀⢀⡸⠄⣙⣷⡦⣤⣘⣀⠀⠁⠊⢰⠱⡁⠀⠀⠀⠑⠌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⢛⣴⡀⠠⠼⠶⣦⠽⠿⠛⠛⠓⠓⠓⠓⠛⠛⢋⣒⠤⢤⣘⠀⠀⠀⠔⡥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⢐⠌⡆⠀⢸⡌⢯⣭⡻⠋⠁⠀⠀⡀⠀⠀⠀⡀⢀⢀⣠⢠⣠⣀⣀⣦⣙⣶⣄⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠘
⠀⠀⠀⠀⠀⠸⡀⢎⠀⠀⠀⠀⡆⠏⢀⡐⠻⢄⣾⠿⠷⠖⠒⠒⠚⠛⠛⠛⣛⡛⠙⠋⠉⠩⠛⠐⠀⠤⡄⠄⠀⢱⣾⣭⣒⡢⠄⣀⠀⠀⠀⠀⠀⡠⡦⠃⠐⡲⠒
⠀⠀⠀⠀⠀⠀⠈⠊⠑⠀⠀⠀⠀⠉⠟⠿⠷⢶⡎⠀⠀⠀⠉⠉⠀⠀⠀⣄⣀⣀⣀⣀⣀⣀⣀⣠⣀⣄⣀⣄⢠⢌⢌⣷⡫⠿⠛⢷⠿⣿⣤⣷⠤⠧⠣⠀⠠⠕
⠀⠀⠀⠀⠀⠀⡠⡣⠊⠀⠀⣴⣮⣇⡚⠯⠽⣾⡧⠶⠶⠾⠒⠗⠓⠛⠉⠩⠤⠀⠤⠀⠈⠈⠁⣬⠈⠉⢉⡭⡿⣿⣾⢿⣿⣥⣟⣶⣈⡥⣏⣿⡿⣷⣬⣐⡂⠄
⠀⠀⠀⠀⠀⠀⠐⠀⢀⡲⢶⣞⢷⡿⣿⡿⠶⣿⡀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠠⡀⣄⣀⣠⢠⣠⡖⠉⠀⢓⢿⢵⡿⠭⠛⣻⣿⣭⣽⣟⡷⠾⢟⣁⠉⢯
⠀⠀⠀⠀⠀⠀⢀⣀⣘⣙⣻⣶⣭⣓⡺⢿⣟⣽⠗⠒⠒⠚⠛⠛⠉⠉⣉⡉⠉⠉⠉⠉⣩⡽⠉⠉⡏⣿⠖⡀⠀⠀⣻⣿⠿⣍⡿⣽⠙⣯⣿⣿⣷⣦⣌⣙⠳⠶⡂
⠀⠀⠀⢀⣀⣽⠷⣮⣝⣒⠽⣗⣶⣿⠟⢿⣿⡄⠀⠀⠀⡀⢀⠀⢀⠈⠈⢉⠉⠀⡄⠀⢀⢠⣰⣾⠥⠉⢸⣔⣽⡚⣛⣛⣿⣷⡟⣃⣼⡿⣛⢹⡟⣀⣽⢿⡻⣳⣬
⠀⠀⠐⢶⣭⣑⠻⠷⣺⣻⣿⡶⣬⣵⡾⣿⡿⠷⠶⠾⠷⠚⠛⡛⢚⠓⠛⢛⣉⡉⠉⠉⠉⢉⠝⠑⠗⣺⢾⣷⣶⣿⣙⣛⠻⢇⣌⠙⣍⠟⡍⣿⣔⣻⣯⢾⣥⠤⠉
⣀⡦⣬⣁⡈⢹⣿⣷⣦⣉⡛⠶⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⡀⢉⢉⠀⠀⠀⢀⣀⣀⣠⢔⢅⢒⣱⣢⣰⣿⣽⣧⣭⡿⢛⣷⣦⣤⣉⣂⣤⣹⣟⣿⣅⣿⣶⣦⡉
⠀⠽⠳⣮⣝⣛⠻⢻⣟⣙⣿⡿⣧⣼⠹⡒⠒⠓⠒⠓⠒⠒⠒⠛⢒⣚⠋⠉⢉⡭⠋⢁⡀⠀⣤⠾⣟⣉⣥⣬⣭⣟⣒⢻⣿⣿⣿⣿⣿⣿⣾⣽⡻⡿⣿⡿⠇
⠀⡢⢦⣊⢋⣹⠿⣶⣬⣙⡓⠿⣤⡟⢠⣕⢤⣐⠒⠒⠀⠀⠀⢀⣈⣠⠤⢲⡛⢄⡢⡣⡊⠀⠈⢊⣿⣭⡍⣠⣿⣿⣭⣤⣤⣍⣉⣭⣭⣽⣿⣿⣿⡿⠦⠁
⠈⠉⠒⢬⣙⠣⠾⣿⡉⢭⡿⠷⢶⡇⢸⣟⠀⠈⢩⣉⠛⠛⢉⣭⢥⡀⠀⠈⢩⡷⣟⣹⢆⡴⣾⣿⢿⣾⣯⣙⠢⠟⣻⣏⠗⠶⠿⡷⣦⣍⡛⠶
⠀⠀⠀⠀⠀⠉⠓⠦⣭⡛⠯⣭⣉⠻⣬⣧⠀⠀⣱⡒⠒⠀⢀⠿⣛⡀⠀⠀⠵⣫⣴⣶⣿⣿⣿⡷⢶⡞⢋⡭⠻⢦⣤⣉⠛⠯⣔⣫⣈⠭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢭⣛⠫⢝⣳⢬⣹⣶⣄⠀⠀⠘⢻⣤⢶⠴⠿⠷⣿⣝⡷⠲⣽⣟⡶⣮⣙⠲⢤⣌⢉⣩⠛⠶⠦⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢼⣙⠻⢯⣽⣛⠛⢻⣯⣿⣿⡦⣶⢛⣿⣽⣿⣳⢤⣍⡑⠢⣍⣛⠶⣮⣍⠒⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠮⣍⡲⠦⢖⣚⡿⢛⣶⡏⣹⣿⠾⣿⡒⢛⣛⡷⢶⣬⡛⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣭⣚⢿⣟⡅⠛⡤⠦⣿⠽⣺⣿⣞⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣝⡒⠤⣥⢠⢞⣹⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⣿⣿⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⢿⣿⣦⠀⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠈⢻⣿⣇⠀⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣿⣿⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⢹⣿⡆⠀⠈⣿⣿⠀⠀⠀⣾⡇⠀⣿⣿⠀⢀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠹⣿⣧⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⣿⡟⠀⠈⣿⡇⠀⠀⢹⣿⠀⠀⠘⣿⣧⡀⠹⣿⣧⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠘⣿⣇⠀⠙⣿⣦⠀⠀⠀⠀⢰⣿⠇⠀⢰⣿⠇⠀⠀⣾⡿⠀⠀⠀⠈⠻⣿⡄⠙⣿⣇⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⢸⣿⠀⠀⢹⣿⠀⠀⠀⢠⡿⠃⠀⠀⣾⡏⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⣹⡇⠀⢸⣿⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⣸⡏⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣼⡟⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢠⣿⠁⠀⠼⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⢠⡿⠁⠀⠼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⠶⠟⠛⠛⠛⠉⠉⠉⣉⣉⣉⣁⣁⡠⣤⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⠙⠛⠛⠻⠷⢶⣦⣤⣀⡀
⠀⠀⠀⠀⢀⣠⣤⡶⠟⠛⠉⣉⣁⡤⣴⠶⢂⣶⣾⡿⠛⠻⣿⡿⠛⢉⣩⣿⠟⠁⣉⡼⠛⢿⡉⢉⣭⡯⠟⠉⢀⣿⡾⠷⠢⣤⣤⣌⡉⠛⠿⣦⣄⡀
⠀⠀⣠⡶⠟⢋⣡⠤⠐⠛⣡⣽⡯⠛⢁⣴⡿⠛⠻⣷⣄⢀⡼⣠⣴⠿⢋⣡⡴⠛⢿⣦⡀⢠⣿⠿⠋⠀⣠⡴⠛⠿⣧⡄⢀⡼⠛⠉⡹⢿⣶⡀⠉⠻⣦
⢠⣾⠋⢀⡴⠋⠁⣀⣤⣾⣽⣧⣴⡞⠉⠛⣷⠆⠀⢀⣤⣿⣿⣿⣿⠛⠛⢻⣷⠀⠀⣉⣼⣟⣡⣴⡚⠛⢻⣷⠄⠀⢈⡴⠋⢀⣤⣾⠟⠁⠀⢱⠀⠀⢸
⣾⠇⠀⠸⣄⣠⣾⠿⠋⣹⠁⠀⠹⡿⠂⠀⣁⣤⣾⣿⢿⠟⠁⠈⢻⠷⠀⠈⣁⣤⡾⠋⠝⠁⠉⢹⡿⠀⠀⣁⡤⠞⠉⢀⣴⡿⠛⠁⢀⣀⠴⠃⢀⣠⣾
⣿⣆⡀⠀⠙⠛⠧⣤⣴⠿⢄⣀⣀⠤⠞⢋⣤⡿⠛⠁⠈⣦⣀⣀⣀⠤⣴⣿⠟⠁⢀⣴⡷⣄⣀⣀⣤⡶⠟⠁⠀⢀⣴⣿⡟⠤⠖⠛⠉⣀⣤⣶⣿⣿⣿
⢻⣿⣿⣷⣦⣄⣀⡈⠈⠙⠛⠛⠣⠴⡾⠿⠃⣀⣀⣤⣾⣋⣁⣀⣤⣾⣏⣁⣠⣾⣟⣁⣀⡀⣴⠿⠿⠤⠶⠐⠚⠛⠋⣁⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⣀⣀⣀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠈⠉⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⠋⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢠⣿⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⣿⡄⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⢻⣿⠛⢻⡿⠻⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢀⣿⠁⣀⣀⣤⡤⠶⠛⠛⠉⠉⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠉⠙⠛⠶⣤⣄⣀⠀⠀⠸⣷⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣿⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠦⣿⡆⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣾⠇⢀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⠿⢿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⢰⡿⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠘⣷
⠀⠀⠀⠀⠀⠀⣾⠇⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢻⣇
⠀⠀⠀⠀⠀⢰⡟⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠈⣿⡄
⠀⠀⠀⠀⠀⣾⠃⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠸⣧
⠀⠀⠀⠀⢰⡟⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻⣆
⠀⠀⠀⠀⣿⡃⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⣿⡀
⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣦⠉⠀⠀⠚⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⣺⣧⣄⢀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⣿⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⠀⠀⠈⠘⠋⢹⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⡇⠉⠑⠺⠁⢸⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠁⢀⠠⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⠁⠀⠀⣡⣶⡖⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⢻⣿⣷⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣵⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡖⠅⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⢭⡭⢭⣛⠀⠲⡀⠘⡄
⠀⠀⠀⠀⠀⢀⡠⠖⢚⣅⠴⢄⠲⣾⠂⠒⠠⡿⣶⣌⢀⠘⢆
⠀⠀⠀⢀⡔⢩⡆⠀⠙⠀⠀⠀⠀⠉⠳⠪⠀⠀⠈⢻⣦⡀⡈⢳⡀
⠀⠀⠀⡿⢐⠈⠉⠀⠀⠀⠀⠀⠀⣠⣀⡰⣶⡒⠆⢉⠈⢻⣌⢄⠙⣆
⠀⠀⠀⢳⢹⡆⠀⠀⡀⠰⠂⠀⠀⣙⣿⣥⢉⡁⠀⡨⠃⠈⡹⣧⡁⠈⠳⣄
⠀⠀⠀⠀⠞⣷⢀⠀⠉⠀⠀⣤⡌⠋⠲⣄⠠⡔⠂⠀⠀⢰⡘⠜⢿⣄⢢⠈⠢⡀
⠀⠀⠀⠀⢺⢻⣚⠄⠀⠀⠈⠻⠄⠂⠠⡙⢗⠀⢀⡐⠞⠃⠀⣠⠀⢙⣦⣘⢄⠈⠢⣄
⠀⠀⠀⠀⠈⢯⢿⣄⠀⠀⠀⢰⣄⠀⣤⠃⠀⠀⡘⠃⠀⠀⠠⣿⡎⠀⠘⠻⠶⣕⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠈⢻⡿⡶⢗⣈⠐⠉⠵⢂⠀⠀⠴⡿⡄⠀⣴⢂⡹⠁⠀⠀⠂⠀⢻⡓⢄⠠⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠿⢿⣦⣤⣒⡈⠀⢠⡤⠤⣀⣘⣗⠁⠀⠰⠊⠀⠀⠀⠈⠙⠚⢷⣤⡑⢄⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣤⢈⢉⣀⠈⢩⡍⠻⠯⡅⣄⡐⠐⠀⠀⠀⠀⠀⠈⠛⢆⣉⡠⠬⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠿⠿⢿⣿⣄⣀⠊⠉⣠⣰⠡⠄⠀⣔⠦⠄⣀⣀⠉⠛⢶⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⢖⡉⢰⣾⡆⠒⠉⠀⠀⠉⠉⣛⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣯⣬⣥⣀⡀⠀⡀⠀⠀⣠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠺⣿⡷⣲⣠⣴⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠛⠁

⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣠⣤⣶⣶⣾⡿⠟⠛⠿⠿⠟⠻⡿⡿⣿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⣠⠊⠉⠟⢋⣿⠿⡿⠿⠿⠻⢻⣿⠉⠀⠀⠀⠀⠀⠀⣈⠥⠀⠒⠛⠉⠁⠀⡀⢀
⠀⠀⢸⣷⣿⣿⣳⣿⠋⡞⠁⣠⠆⣴⠎⢠⠆⣠⣂⠀⠤⠒⠈⠀⢀⣀⠠⠀⠀⠀⠀⠀⠀⣈⠁
⠀⠀⠀⠉⠛⢿⣿⣿⣶⣄⣼⣇⣪⣼⣶⣿⣿⣿⠟⠁⢀⡀⢰⢋⣄⠄⠀⠀⠀⠀⠀⠀⡰⠁⠀⣃⣆⡃⢂
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠉⠀⠼⣏⠁⡄⠀⠀⠀⠀⠀⠀⠰⢃⣄⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⠀⠠⠀⡀⣰⠆⠁⠀⠑⠀⡴⠆⢉⡀⣤⣤⣀⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠁⠀⢿⣀⠀⣨⡴⢷⡀⠁⢀⡤⠀⠀⠛⡛⠷⠾⣿⣿⣉⣀⣀⣠⣄⡀⠤⢶⣤⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣫⡍⢙⢻⡄⠀⠷⢡⣷⡟⢩⠧⢴⡖⢈⢁⠠⠔⠀⠂⢉⣫⣙⣷⣾⣿⣟⡋⣴⣴⠦⠓⢒⠼⠀⢠⡟⠿⢶⡄⠀⢦⡴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡀⠀⠀⠙⣿⣷⣄⡈⠺⠿⠿⣿⠻⣜⠖⣬⡾⣿⣛⡯⣝⠻⣤⣅⠀⠒⠂⠂⣁⣠⣶⣏⡄⡈⠎⢀⣰⣿⡿⡾⠂⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠂⠀⠙⠫⢿⣓⣀⣁⢀⣀⣴⡙⠯⠾⠾⠾⠛⠋⠉⠉⣀⣠⣴⣶⣿⠏⠹⠚⠋⢁⣠⣴⡿⠧⠒⠋⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣄⣀⣀⣉⣁⣀⣀⣤⢤⣴⢶⠾⠿⠿⠻⠛⠛⣉⣁⣠⣤⣤⣶⣶⠿⠟⠛⣉⣡⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠘⠶⠤⠴⠶⠖⠒⠚⠛⠋⠙⣯⣉⣉⣡⣬⡤⠴⠒⠛⠉⠉

⠀⠀⠀⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠻⣦⣬⣙⡛⢷⣶⣤⣤⣶⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⠞⣴⣤⣤⢀⣤⣤⣶⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣶⣶⣶⣿⣿⣿⢰⡟⣛⣻⣛⣻⡇⣿⣿⣿⣿⢸⣟⣷⡄
⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⣛⣿⡻⠏⣼⣿⣿⢏⣾⠛⠛⠛⠛⣿⡇⣿⣿⢰⡎⡏⣹⢠⡌⣿⣿⢸⡿⣛⣛⣿⣯⡭⠬⣭⢯⣭⡭⣭⡅⣿⣿⣿⣿⣜⢿⡏⣻
⠀⢠⣶⣿⣿⣿⣿⣿⡧⡿⠫⣩⡿⣸⠣⢬⡏⣼⣯⣽⣻⣣⣜⣿⡇⣿⡏⢸⢣⡇⣿⢸⠇⣿⣿⢸⣧⣭⣭⣙⣛⣛⢒⣛⣛⣛⣛⣛⡃⢿⡟⣭⣭⢻⣷⠙⣛
⠀⠀⣭⣭⡥⠾⠿⣭⡄⠳⢾⡿⢁⡟⠉⡿⢰⣿⡧⠴⢶⣾⣭⣿⡇⣿⡇⣶⢸⡇⣿⢰⡎⣿⣿⢸⡷⢇⠱⠉⣭⡟⠺⠦⠾⠤⢭⣥⡿⣸⣷⡶⢚⣻⣿⠀⢹
⠀⠀⠈⠉⠉⠙⠛⠋⣼⣿⣶⡆⣸⣷⣾⢃⣿⣿⠓⢲⣶⠤⢼⣿⡇⣿⣇⣽⡘⡇⣿⠸⠇⣿⣿⢸⢷⢖⠶⢷⣿⡇⣿⢿⢿⠶⣷⣶⡆⣿⣷⣶⡿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠻⣿⡏⠾⠿⠿⠿⠷⠶⠶⣾⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⡷⢸⣬⣽⣼⣬⣼⢸⡿⣼⣿⣿⣎⣼⠇⣿⡯⠭⢭⣾⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣿⡿⠿⠿⠿⠿⠷⣶⢶⡄⣶⣶⠶⢶⠶⡶⡶⣶⡆⣤⣭⣭⣛⡛⠟⣼⣛⣀⢿⠙⠙⣿⢸⣿⠜⣛⣃⣿⡟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠠⠃⡸⣿⢸⣧⢹⡟⣀⣿⠸⣿⣟⣿⡇⣿⢧⣏⡟⣻⢓⢦⣍⡛⠿⣿⣾⢇⣿⣇⣛⡸⣾⣿⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣷⣷⣿⡿⠾⠿⠿⠎⠷⠿⠶⠶⠶⣮⣿⢡⣿⣼⣽⣷⣇⣸⣰⣿⣿⣷⡦⡉⢼⣿⣿⣿⣿⣿⡟⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⡶⡖⡖⡞⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠋⠉⠱⣾⣟⡳⢶⢶⠽⡿⣶⣦⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡟⠘⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣤⠀⠀⢰⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠈⣭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⢠⣿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⢠⡖⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣤⣶⣿⣿⡿⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣳⡶⠟⠉⠁⠀⠉⣡⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢃⣼⠋⠀⠀⠀⠀⣨⣾⣿⣿⣿⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣾⡇⠀⠀⠀⠀⣸⠉⠉⠛⠛⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⠆⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⢧⡟⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⣠⡾⣁⡀⠀⢤⣄⢀⣀⠀⠀⠀⡠⠐⠀⢤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⢿⡿⣻⣛⣿⡀⢘⠃⡈⠀⠀⡈⢰⣿⣿⣷⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣹⣯⢟⣵⣫⣾⣿⣿⢿⠇⣸⣎⠹⡋⢸⡷⣫⣞⡴⣩⠛⣷⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣿⣿⣿⣿⢄⡴⣻⢞⣯⢎⡼⣫⣞⡵⣫⢞⣿⣿⢞⡴⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣉⠛⠳⠿⠾⠷⠿⠵⠯⠾⠿⠿⠾⠓⢛⣩⣵⡿⣥⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⠻⣓⠀⠲⠖⠒⠒⠒⠒⠀⠙⠉⢛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣆⡁⠀⠁⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⣿⣿⣿⣿⣤⣄⣦⣴⣶⣶⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠃⢻⣿⣿⣿⡯⢹⣿⣿⠃⢻⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠈⣿⡉⠀⠀⢤⣿⡟⠛⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠙⠧⣤⣤⣤⠟⠁⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣀⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣼⣡⣿⣯⠙⣶⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⢀⣠⡴⠞⠋⠛⠷⠿⠛⠙⠲⠤⣀⡀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣂⣍⣻⣦⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠄⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⠿⣿⣥⣦⣤⣤⣤⠤⠤⠤⡤⣤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠛⠿⣏⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠟⢿⡿⢛⡿⢿⢿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣾⣶⣾⣷⣾⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⡤⠀⠀⢀⣌⣩⣉⡁⢴⠀⠀⠀⠀⡔⠀⠀⠈⠙⠀⠀⠀⣀⢶⠀⡹⠆⠀⠀⠀⠀⠀⢾⡉⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠀⠀⢾⠖
⠀⠀⠈⣄⡁⠘⣟⠀⢹⠀⣉⢀⣿⠀⠀⠀⠰⠆⠀⠀⡠⠋⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠻⣦⣀⣹⣦⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⢀⠄⠀⢀⡤⠋
⠀⠀⠀⠈⠉⠈⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣌⣱⡶⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠐⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠈⠒⠚⠾⢞⣫⠔⠁⠀⠀⠀⠀⠀⠀⠀⠂⠈⠂⠀⠀⠋⠀⠀⠀⠀⠀⣆⠀⣧⠀⡆⠀⠀⠈⢟⡉⡄⠀⠐⢤⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤
⠀⠀⠀⠀⠐⠒⠛⠉⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣐⣂⡀⣇⠨⡅
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠰⠀⠀⠄⠐⠶⣈⠳⢄⠀⠀⠀⠀⠀⠩⠰⢖⡀⣶⠆⢴⡿⠀⠀⠀⠀⠀⠀⡠⣔⡨⠀⠐⠄⠀⠀⠀⠀⠂⠡⠉⣨⡩⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠳⠄⠀⠀⠀⠀⠀⠈⠑⠊⠉⠛⠁⠀⢀⡀⢀⣤⢴⣩⣐⢃⣘⣡⣴⡇⠀⠀⠀⠠⣀
⠀⠀⠀⠠⠒⠒⠛⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣶⣾⣿⣭⣿⣈⣴⠿⠟⠁⠀⠀⠈⠓⠛⢵⣶⣷⡆⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣄⠀⠀⡀⠀⠀⣡⠀⠀⠀⠀⣠⡄⠤⠴⡂⠐⢻⣯⣢⣤⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⢰⣖⣿⣷⣶⣿⣿⠟⠀⠀⠀⠘⣏⢲⣦⣄⣀⠀⠀⠀⠀⠀⢿⡀⠀⢀
⠀⠀⠀⠀⠀⡀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣯⠽⠛⠋⠀⠀⠀⠀⠀⠙⠳⣕⣦⢙⠲⢶⣤⡀⠀⠈⠻⣆⣌⠀⣄⡀
⠀⠀⠀⢠⣄⣀⢖⠢⢠⡜⣀⣀⣬⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢴⣌⡑⠂⣤⣑⡂⠀⠀⠈⠻⠀⠙⠋⠱⠶⣄⡀
⠀⠀⠀⠈⠓⠻⠭⠱⠈⠽⠃⠚⠋⠁⠀⠀⠀⠀⠀⣀⡀⢂⠠⠀⠀⠀⠀⠰⢦⣄⣤⣤⣠⡴⣤⠤⠶⠀⠀⠀⠀⠀⠉⠖⢦⣄⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠶⠶⠶⠖⠒⠒⠒⠒⠲⠶⠶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⠛⠫⠍⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠒⠉⠉⠛⠳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠛⢶⣤⣀
⠀⠀⠀⠀⢀⡾⡡⠀⠀⠀⢀⡠⠤⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⠀⠀⠈⠂⡀⢀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⡾⠱⠁⠀⠀⢠⠃⢠⠖⠀⠄⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡄⠂⠀⠀⠈⠂⣀⡠⠀⠊⠙⠄⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⣸⢧⠃⠀⠀⠀⢸⠀⠫⣀⠠⠃⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⠤⠄⠒⠀⠀⠀⠀⠈⢷⡄
⠀⠀⢀⡿⠨⠀⠀⠀⠀⠀⠑⠤⣀⡀⠀⢀⣀⠜⠀⠀⠀⠀⠀⠙⠢⢀⠀⣀⠄⠊⠙⢄⠀⠀⠀⠀⠷⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⡄
⠀⠀⣼⠷⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠆⠀⠀⠀⢀⡱⠆⠀⠀⠀⠀⢄⠀⠠⠔⠚⢏⠀⠀⠀⠀⠀⢸⣷
⠀⠀⡿⠀⠀⠉⠐⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠐⣤⠀⠁⠀⠀⠂⡀⠀⠀⠈⢣⠀⠀⠀⠠⠦⠀⠒⠀⠀⠀⢻⡄
⠀⠀⢿⣤⠀⠀⠀⠀⠉⠀⠒⠂⠀⠉⠁⠒⠒⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠄⠊⠡⡀⠀⠠⠐⠻⠁⠀⠀⠀⢂⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠙⠻⠷⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠠⠛⡁⠀⡠⠀⠙⠀⠀⠀⢀⣠⠤⠐⠀⠉⠃⠀⠀⢠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡉⠲⢄⡀⠀⠀⠀⠀⠸⠁⠀⠀⢀⠰⠖⠈⠀⠀⢃⠀⠀⠀⠀⠀⠀⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⠀⠀⠀⠀⠀⠀⠐⢄⠉⠲⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠀⠀⠘⡃⠉⠉⠉⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⠁⠢⡀⠙⢦⡀⠀⠀⠀⠀⠐⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡄⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠠⠤⠤⠤⢐⣲⠿⠊⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⡀⠀⢀⠀⠀⠀⠀⠈⠉⠐⣒⡂⠉⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢷⣤⣀⡀⠀⠀⠀⠀⠛⢃⣀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣌⠀⠀⠀⠀⠀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢀⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢈⡃⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠲⡀⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠀⠸⠀⠄⠀⠁⢠⠀⠘⠠⠆⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⠀⠐⠆
⡶⠄⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⣀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠉⠸⠀⠘⠀⠀⠛⠂⠀⠀⠀⠈⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠰⠆⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠐⡀
⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡄
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠰⠀⠀⠀⠀⠸⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡗⢺⣆⠧⠔⠀⠀⠀⢀⡜⡏⣿⠋⢀⡴⢾⣟⣭⣿⣿⣿⣧⠀⠀⠀⢹⣧⣄⠀⠉⢳⡄⠀⠀⡇⠀⡃⠀⣇⠀⡃
⠟⠊⠁⠀⠀⠀⠀⠀⠈⢷⣿⣯⣤⢻⠽⣴⣻⣮⣿⣿⣿⠏⠀⠀⠀⠀⠉⠙⠻⢿⣿⡇⠀⠀⢧⣄⢰⡀⠱⡀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠊⠓⠒⠚⠻⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠁
⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⣀⣤⣄⣀⠀⠀⠀⠀⣴⣶⣾⣭⣴⣟⠙⢫⡿⠀⠀⠈⠂
⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡌⠃⠀⣴⣿⣥⠙⢦⣭⣏⡘⣆⠀⠀⠉⠻⠏⠿⠯⠽⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⣠⣾⣧⣿⣷⠀⠀⠀⠀⠀⢰⣿⣟⢿⣧⡄⢸⣿⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⡀⠀⠀⢀
⢠⡞⢹⣿⡯⢞⠛⠀⠀⠁⠀⠀⢸⣏⠛⠊⠉⠸⠿⠁⠀⠀⠀⢀⠠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⣀⡿
⣇⢨⡴⠋⠂⠀⠀⠠⠁⠀⠀⢠⣟⢁⡀⣠⡼⠃⠀⠄⠀⣠⣬⣶⣤⡾⢳⡖⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⢮⣾⠃⠀⠀⣾⡜⢻⡿⠉⠢⠀⠀⠀⢰⡟⠀⠿⠋⣼⣷⣶⣏⠽⣷⠄⠀⠀⠀⠀⢠⠏
⣠⣤⠖⣦⠠⣾⠇⡏⠀⠀⠐⢯⢕⡈⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠘⢻⣿⡏⠷⠴⣯⠘⢦⣀⠀⣰⠏⢀⣀⣤⣤
⢏⠁⠀⣀⣠⣡⡟⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢢⡀⠀⠉⢯⣀⣘⣃⠛⢻⣯⣄⣘⠿⣷⡒⣨⣿⠋⢠⣾⡟⠛⠻
⡄⢀⢈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠈⠛⠯⣽⣏⣚⣷⣶⣿⡿⠟⠁⢠⣿⡅⣳⠆⢂
⣶⡿⣫⣿⠏⠀⠀⢆⠀⠘⡄⠀⢢⠀⢃⠘⡄⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⣿⣱⡀⣠⣼
⣸⣿⠟⠁⠀⠀⠀⢠⣡⠀⣇⡀⣸⡆⢨⡄⣱⠀⠀⠀⠰⡈⠿⣷⠶⣦⣄⣀⣀⣀⣀⣤⣤⡀⠀⠀⢿⣿⣆⣩⡌
⠋⠁⠀⣤⡀⠀⣶⡛⢿⣶⡉⡙⡋⠙⣋⣻⣿⠿⠇⠀⠀⠈⢷⣍⠓⢮⣭⣭⣭⠷⣶⣶⣈⣿⣦⠀⠈⣙⣛⣩⣵
⠀⠀⣴⣹⡿⠀⣻⣧⠀⠛⠿⠷⠲⠾⠛⠿⠃⠀⠀⠀⠀⠀⠀⠙⠳⣔⣏⠉⠈⠒⠈⠛⠻⣿⢿⣶⡄⡀⠈⠉⠉
⡼⠋⢉⣽⠁⣰⠋⠋⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢇⠀⠀⠀⠀⠛⠀⣰⡏⣿⡄
⣤⡶⣼⡇⠀⣿⡒⠠⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡀⠀⠈⢳⡀⠀⢀⠐⣊⣼⣶⡿⠁⠀⠀⣴
⣸⢿⣿⠁⠀⠘⢿⣄⣉⣈⠻⢀⣄⣠⣠⡭⡴⠾⠋⠀⠀⢠⣶⡜⠺⢿⣷⡦⠀⠹⣗⣧⣌⣤⠾⠟⠀⠀⠰⠘⣿
⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⢉⡀⠀⠀⠀⠀⠀⢀⣇⠁⠀⠈⠁⠁⣝⣧⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙
⠄⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⣖⣴⣾⡿⢻⣿⣿⣷⡶⣄⠈⠋⠙⢹⣄⡀⠔⢦⡾⠇⠀⠀⠀⡠⠂⡠⠀⡀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤⣶⡶⣿⠛⠉⠉⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠈⠁⠀⢰⠃⢠⠊⠁⠈⠑⠦⡈⠑⢄⡀
⢠⠖⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⠈⡆⠀⠙⢆
⡏⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢇⠂⠀⠀⠀⢀⠞⢀⠞⠓⣄⠑⡄
⢻⣄⠀⠀⠾⠷⠶⠦⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠙⠲⠤⣤⠏⣠⠋⠀⠀⠈⢦⠘⣆
⢀⡟⠃⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⠉⠒⠢⠤⢍⣉⡐⠒⠒⠚⠀⠘⡄
⣾⠀⠀⠀⣼⠛⠛⠻⠶⠶⠶⠶⠶⠶⠤⠤⣤⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⡀⢳
⠘⠷⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⢄⠀⠀⠀⠀⠀⠀⠀⠱⡌⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡦⢄⣀⠀⠀⠀⠀⠙⣄⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠀⠀⠀⠀⠀⠈⢣⡘⢦⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠙⢦⡙⠦⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢾⣦⣤⣀⣀⠙⢦⡈⠓⠦⣄⡜⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⢿⣷⣤⣀⣠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡷⠶⠤⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⣴⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠅⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⡀⠞⠁⡄⠀⠀⣀⣤⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠃⠀⠀⠈⠈⢹⠟⠉⠀⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠠⠞⠶⣆⠀⠀⠁⢛⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠈⠁⠒⠒⠒⠂⢤⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣴⣏⣽⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠄⢸⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣧⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠢⢄⡀⠀⠀⠄⠂⠐⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣸⡧⡀⠀⠠⠐⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡥⠂⠒⠣⠤⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡿⠀⠙⢄⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠘⠿⣤⢸⠀⠀⠀⠀⠀⠠⡀⠀⢸⠁⢸⠁⠀⠀⠀⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠄⠀⠀⠀⠀⡇⢸⡆⡠⠊⠀⠀⠀⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⢠⡤⠤⢤⣤⢀⣀⢀⣀⢀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢚⡇⠁⠀⠀⡠⠊
⠀⠀⣠⠴⠚⡃⢀⡹⣿⣿⡿⣿⣷⠆⠶⠶⠶⠶⠾⠟⣃⡉⠁⠘⠳⢤⡀⠀⠀⠀⠀⠀⠀⣏⡯⡀⣀⠴⠋
⠀⠘⣷⣔⣿⣿⣿⡿⠿⣀⣀⣤⣴⣴⣶⠷⠀⠀⠾⣿⣿⣿⠿⠆⢀⡴⡇⠀⠀⠀⠀⠀⢸⡿⣄⣸⣿⡀
⠀⠀⠃⠈⠉⠙⠓⠲⠶⠾⠭⠭⠥⠤⣤⡤⠤⠤⠦⠶⣒⣒⣾⣭⣷⢆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡇
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⣼⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡇
⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣸⣿⣿⣿⢟⣼⡿⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠈⠲⣄⣤⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢟⡿⠋⠸⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠓⠯⢿⣿⣶⣶⣶⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠉⠉⣻⡏⠉⠀⢻⡆⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣀⠤⠿⠇⠀⢀⣀⣿⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢰⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⢿⣿⣷⣆⣀⣀⣀⣬⣿⣷⠀⠀⠀⠈⠙⣾⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠑⠸⢿⣿⣿⣿⣿⠿⠛⠁⠸⣇⠀⠀⠀⠀⣿⠹⠿⢿⣷⣶⣦⣤⣤⠇⠀⢇
⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⢹⠀⠀⠀⠹⡏⠉⠉⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠈⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⡀⢂⢰

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡄⠀⣠⠀⣤⠀⣤⠀⣴⠠⡄⡤⠀⠀⡀⠀⢀⡰⠀⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠠⠘⠋⡷⠀⠙⠇⠿⠀⠿⠀⠽⠺⡅⠯⢠⠤⡍⠓⢬⠹⠢⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⡀⠀⡀⠀⣀⡀⣀⠀⡀⢀⢤⢀⢀⣀⢀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣿⠀⣽⠀⠀⡛⢶⠀⣿⠀⣿⠀⡇⢿⠐⡀⢸⡇⢸⡇⠂⣘⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠁⠉⠀⠀⠁⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠁⠉⠈⠁
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣿⠋⠉⠉⠉⠉⠉⠉⢉⡉⣩⣽⣿⣿⣿⣭⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⣿⡗⠾⠀⠀⣤⠂⠀⠁⢰⣿⣿⣿⣭⣿⣿⣿⠿⣷⣤⣤⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⣷⠲⠰⠄⠩⠉⠠⡿⢺⣿⣿⣿⣿⠟⢉⣠⣾⣿⡿⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣤⣀⣀⣀⠂⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⣿⢿⠀⠀⠀⠀⠀⠀⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣽⣿⣿⣿⣿⣿⣷⠀⠀⠀⢈⣿⣿⠀⠁⠀⠀⠀⢀⡀⡀⣿⡇
⠀⠀⠀⣿⣏⣁⡒⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣍⣓⣘⣻⣿⠛⠿⠿⠟⠁⠀⠀⠀⠸⣟⣿⡧⠀⠀⠀⢴⡋⣰⠇⢸⡇
⠀⠀⠀⣿⡶⠿⢥⡭⢬⣁⠚⡿⣿⠟⠛⠛⠛⣽⣟⣼⣯⢿⣛⢿⡟⣿⣇⣀⠀⠀⠀⠀⠸⠀⠀⣽⡿⢷⠀⠀⣀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⢿⠍⢉⠁⣿⣿⣿⣿⣿⣆⠀⠀⠛⢻⣿⣷⣿⠀⠀⣽⡷⠂⠀⠀⣿⡇
⠀⠀⠀⣿⠃⠁⠀⠐⠀⠀⠀⠀⠀⠀⠈⡻⣯⣿⡿⢙⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠿⠅⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⣻⡇⢻⣷⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠰⠿⠇⠘⠛⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠿⠏⠈⠉⠀⠒⠦⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠂⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠑⠷⠒⠀⣿
⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⣠⢁⠄⠀⠀⠀⠀⢸⣿⣿⣿⡜⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⣰⢃⠌⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠆⠀⠀⠀⠀⠰⣯⣾⠀⠀⠀⠀⠀⠀⠙⣿⡟⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⢀⣾⣿⠟⠛⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⡀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠇⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢿⣿⣿⣿⡀⠀⠀⠀⠤⠔⠒⠶⠶⡀⠐⠦⠤⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⡄⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣝⣿⣿⣇⠀⠀⠀⠀⠀⠀⠑⠒⣉⣙⠓⠿⣋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢛⡓⠒⠛⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⣿
⠀⠀⠀⠻⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⢀⣀⣤⣶⣤⣤⣤⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠄⣀⣤⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠛⣋⣉⣉⣩⣭⣥⣤⣥⣤⣤⣤⡶⣤⣶⣶⡤⠶⠶⣶⣾⣿⣿⠿⠿⠟⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⣀⡀
⠀⠀⠀⠀⠓⠒⠀⠂⠀⠀⠒⠢⠀⠀⠀⠒⠂⠠⠤⢀⣀⠠⠤⠤⠄⠠⠤⠐⠒⠢⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣿⠁
⠀⢀⡶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣄⠀⠆⠀⠀⠀⠀⣄⣘⡛⠓⠦⣤⣈⣉⣉⣓⣀⡄⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠠⠀⣐⠋⠙⢶⢤⣀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠻⠿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠢⣿⣿⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠁⠻⢢⡄⠀⠀⠀⠀⠠⣄⠀⠈⠀⠀⠀⠀⢿⣿⣶⢶⡶⣶⣦⣤⣤⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠃⠐⠀⡀⠀⠀⠀⠀⠘⠿⢶⣤⣬⣝⣛⠳⠶⢶⣯⣭⣋⣛⢁⢀⠀⢀⣠⣶⠖⠒⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠘⣦⠀⠀⠄⠀⠈⠁⠀⠃⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠿⠿⣿⢶⡶⠋⠉⠉⠁⢠⠟⠁⠐⠒⠐⠲⠒⣦⣀⣤⠈⠻⡷⡀
⠀⠀⠀⠈⠻⣦⣀⡀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠳⠶⠶⡶⣶⣶⢶⠤⠶⠶⠖⠀⠀⠈⣴⠀⠀⠀⠀⠀⠀⠀⠉⠉⢙⣷⣏⣺⡄
⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⢀⣰⣽⡥⠀⠰⡖⠀⠀⠀⠀⠿⠷⣦⣤⣤⣦⣤⣤⣤⡶⠖⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡀⠀⠀⠉⠒⠛⠛⢫⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⣼⣿⣷⣦⣶⡴⠃⣾⡁⠀⠙⠳⠶⠦⢤⣭⣉⣩⡭⠵⠛⠁⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢹⣅⡀⠀⣠⣴⣷⠀⣾
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠉⠻⠿⠿⠿⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⣰⠿⠀⠛⠟⢫⣿⢰⡟
⠀⠀⠀⠀⠀⠘⣧⣄⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠖⢮⠷⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠣⠀⠁⢠⡋⠰⣢⠚
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠷⣤⣤⣤⣤⣤⣴⠏⠀⠀⠀⠀⠀⣈⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠁
⠀⠀⠰⡶⠒⠒⢶⡄⠀⠀⠀⠀⣤⣶⣤⣄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠆⠠
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠐⠋⣈⠉⡉⣄⡉⠀⠀⠀⠀⠀⡠⢃⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⢇⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⡞⠁⠈⠀⠀⠀⠀⠀⠀⣼⠟⣻⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣧⣴⣤⡇⠀⠀⠀⠀⠀⠀⠀⠈⠲
⠀⠀⠀⢠⡥⠠⠀⢿⣄⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠁⠀⠀⡠⢤⣀⠀⠀⡀⠀⠀⠉⠙⢿⣿⠎⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⣤⢴⣿⣶⣦⣭⣅⣀⠀⠒⠒⢚⡟⣡⠏⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢹⠆
⠀⠀⢀⣹⡾⠋⠋⠉⠻⣯⠚⠟⣦⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠽⠇⠀⡄
⠀⠀⠼⠟⡜⠀⠀⣀⣐⠇⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⣄⡈⢣⡀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠈⠁
⠀⠀⠀⠀⠀⠀⠠⠭⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠶⣿⣿⠄⡀⠀⠀⠀⠀⠀⠋⠀⠀⠶⠂

⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⢀⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠆⠁⠀⢀⣴⠇⠀⠀⠀⠀⠺⢿⣿⡿⢦⡀⠀⠀⠀⠀⢀⣠⣿⣻⣷⣦⡄⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⢀
⠀⠀⠀⠘⠿⣻⢿⣿⣦⠀⠀⠀⠀⠐⢤⣤⣶⠿⠋⠀⠀⠀⠀⠀⠐⠐⠿⠌⡛⢳⡄⠀⠀⠀⠀⣻⣿⠿⠽⣿⣵⠆⠀⠀⠀⠘⠿⠿⢿⠏⣥⠉
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁

⠀⠀⠀⣤⡴⢆⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣴⣄⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⣶⣿⣿⣯⡿⣿⠀⠀⠀⠀⠀⢀⣾⣏⣷⡄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣴⡇
⠀⠀⠀⠀⠐⠹⢏⢛⣿⠏⠀⠀⠀⠰⠍⣉⣿⣿⠿⠃⠀⠀⠀⠀⣰⣿⣿⣞⡿⠋⠀⠀⠀⠀⠈⠹⣿⣿⠾⣋⠅⠀⠀⠀⠀⢰⢺⣏⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⠿⡻⡿⢛⠿⢿⠿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠷⠤⡗⠂⡗⠒⢦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢛⣶⣲⠓⠒⡗⠒⢺⠒⠒⣗⢒⣦
⠀⠀⠀⠀⠀⠶⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠃⠸⠁⡾⢳⡖⡷⠒⠾⡖⠒⢻⠶⠾⡷⠤⠄
⠀⠀⠀⠀⠀⠒⠂⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠿⣦⡤⣏⠠⠼⣧⠤⣽⣷⠃
⠀⠀⠀⠀⠘⠂⠄⢀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠻⠁⠈⣘⠏⢩⢏⡿⡇
⠀⠀⠀⠀⣀⡀⠀⠀⠹⠄⢀⢀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠓⠾⠘⠚⠉
⠀⠀⠀⠀⡈⠀⠈⠑⢢⡄⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠷⠤⠀⠀⠀⠁⠀⣤⣖⣛⡛⠛⡉⠛⠓⠍⡳⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣄⣀⠐⠌⢻⠷⡯⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣬⣭⠍⡩⢭⠍⠁⠀⠀⣀⣔⠜⠀⠀⠀⠀⡟⢲⣦⠤⠲⠓⡿⢸⣿⣿⣿⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠛⢷⠤⠤⠤⠤⢂⠈⠀⠘⠀⠀⠀⠀⠀⠀⠃⢸⢿⠀⠀⢸⣯⠁⠁⣥⡜⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠈⠘⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⡀⠀⢑⠀⠀⠀⠀⢸⣀⣡⡀
⠀⠀⠀⠀⠀⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠁⠉⠁⠀⠀⠀⠀⣰⡉⠅
⠀⠀⠀⣴⠟⢻⣿⣤⣍⠙⢶⣄⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠐⠀⣢⡄⡀
⠀⠀⢸⡁⠀⠀⠼⠧⣀⡁⠀⢹⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠸⡿⠷⠿⢀⣀⡟⠙⣧⠶⣇⠈⢳⢤⡀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡏⠃⠀⣴⠒⠁⠀⠀⣴⣠⡇
⠀⠀⠀⠁⠀⢀⠊⠃⠁⠀⠀⠀⠈⠙⠛⠛⠓⠤⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⣈⠒⠱⠒⢓⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠠⠤⣀⠠⠀⠁⢀⠀⠨⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠉⠻⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⣔⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠿⣛⣫⣭⣭⣥⣭⣭⣭⣛⠻⢿⣿⣿⣷⣴⠞⠋⢺⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣫⣴⠿⠛⣩⡵⠛⠁⣠⠘⠛⠙⣻⣶⣍⠻⣿⣿⣷⣄⠀⠙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢋⣾⠟⣁⣴⠟⠉⣀⣴⠞⢡⢇⣴⠞⢋⢄⣽⢷⣌⢻⣿⣿⣦⡴⠟⠛⠃
⠀⠀⠀⠀⠀⢀⣿⣿⡿⢡⣿⣥⠞⠋⢀⣤⠾⠋⢀⣄⣼⠈⢀⡀⡞⠈⢁⣠⣿⣧⢻⣿⣿⡆
⠀⠀⠀⠀⠀⣸⣿⣿⢃⣿⠋⣱⢯⣙⡒⢧⡠⡶⠛⠁⡙⣷⣶⠞⣡⡴⠛⠁⣠⣿⡆⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⢸⣯⢤⣿⣓⠲⢭⣿⣷⡀⠴⠟⢡⣏⣴⠞⠉⢀⣴⠞⠋⢸⣿⢸⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⢸⣏⢈⡓⠦⢭⣽⠿⠋⠉⢛⠲⣾⣬⣄⣀⡚⠋⠀⣠⠾⢻⡿⢸⣿⣿⠇
⠀⠀⠀⠀⠀⢹⣿⣿⡌⣿⡋⢁⣠⡶⠊⢀⣠⡶⠋⣼⢇⡶⠉⠛⠿⢿⣿⡦⣠⣾⠇⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣷⡘⣷⡟⠁⣠⡴⠟⠁⡀⣸⣿⠀⣀⡴⠟⠁⣀⣼⠞⣽⡟⣼⣿⣿⠇
⠀⠀⠀⣤⢤⡾⠛⢿⣿⣷⣌⢿⣾⡉⢀⣴⠞⢱⣿⡏⠞⠋⢀⣤⠞⠋⣠⡾⢋⣼⣿⣿⠏
⠀⠀⠀⠹⣷⡷⡀⠈⢻⣿⣿⣷⣝⠻⣿⣤⣠⡶⠉⢀⣠⡾⣋⣡⣴⠾⣋⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠘⢿⣿⣄⣴⠟⠻⣿⣿⣿⣶⣭⣝⣛⣛⠛⣛⣛⣛⣭⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤⣀⠠⣤⠀⠀⠀⠀⠀⢳⣶⣦⣴⣶⠀⠓⠀⢦⣄⣠⣀⣀⠀⢀⣦⣾⣷⣶⣷⣶
⡿⡄⢸⣿⡧⠁⠀⣰⣶⣶⣶⣶⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣶⣬⣤⣤⣤⣾⣿⣿⣿⣿⣿⣧⣀⣀⣈⣻⣻⣗⣾⣶⣾⣿⣿⣿⣿⣿⣿
⡁⠉⠟⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣦⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⣿⣿⣿⣿⡿⠻⠋⢙⡉⠻⣿⣿⣿⣿⣿⡏⠉⠉⢹⢋⣍⠉⠻⡏⠻⣿⠉⢻⠻⢿⡿⢧
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠛⠋⠉⠀⠀⠀⠀⠈⠉⠀⠀⠉⠛⠛⠛⠁⠀⠀⠉⠛⠻⣿⣟⣻⠚⠛⠓⠀⠀⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⣀⡀⠀⠀⠀⠀⠀⠀⠐⠐⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢻⢿⡦
⣤⣤⡄⢰⣤⠤⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣀⡀⠀⠈⠻⢷⡄⢠⣤⡶⠶⢒⣊⣉⣽⣆⣀⢀⡀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣴⣾⣶⣦
⠛⠿⠅⣽⣧⣤⣦⣤⣤⣤⣤⣦⣴⣇⣄⣀⣀⣀⣿⣿⣿⣿⡿⠿⢟⣇⣾⣿⣿⣿⣿⣿⣿⣿⣏⣭⣭⣭⣭⣉⣍⣉⣼⣯⣭⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣅⣄⣰⣾⡿⠛⠛⠻⣷⣮⠛⢿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣭⣭⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠉⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⣤⡀⠄⠀⢠⣤⣤⣤⡄⡀⠀⠀⠀⢀⣀⣀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠶⠂⠀⠀⠀⠀⠀⠸⠿⠿⠿⠶⠶⢾⣿⣿⣿⣿⣿⡿⠿⠶⠶
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣷⣶
⣴⣶⣿⣿⡿⣄⡀⢀⠀⠀⢀⣀⣀⢀⣀⣠⣠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣤⡤⠔⠿⠂⣷⣿⡀⠀⠀⠀⠀⢀⣤⣾⢻⣿⣿⣦⣄⡀⠀⠀⠀⠀⣠⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠆⠸⣟⣰⡶⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠷⢧⣰⠀⠟⣿⣇⡻⠿⠿⣛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣚⣻⡇⣰⠘⣹⣿⣷⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠰⡿⠋⡙⠀⢸⠟⣛⣭⣶⣿⣿⣿⣿⣶⣦⣄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⣿⣿⠃⠠⢼⡇⠀⣺⢻⣷⠽⣛⠛⣿⣶⣝⡛⣛⣓⠹⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⣄⣄⠈⠻⣆⡐⠈⡿⣠⣥⣶⣂⠘⠏⠸⠿⣯⣶⡀⢤⣽⣿⣷⣶⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠿⣿⣾⣦⡘⡧⣷⢰⣎⡉⢡⣄⣀⣤⡐⢻⣿⣿⣷⡾⣿⣿⣾⠿⠁⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢤⣌⣶⣤⣘⢷⣯⣶⣸⣿⣿⣶⡌⣉⢉⣉⣽⡻⣿⣿⣧⣀⣥⠄⠀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠝⠊⠉⣛⣛⣻⣿⣿⣿⣷⣝⡃⢰⡏⣍⣉⡉⠙⠛⠁⠀⣔⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⠶⣾⣿⣿⣿⣿⡿⠋⠁⠈⠙⠓⠒⠼⣛⣿⣿⣿⣿⣿⣡⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⡶⢿⣿⣛⣻⡿⣿⠿⠁⣤⡀⠀⠀⠀⠠⣶⠄⠈⠉⠛⠛⠛⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠈⠉⣴⠏⠀⣿⠀⠀⠉⠁⠀⠀⠰⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈