Undead Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⡀⠀⠀⠀⠀⢸⣿⠀⡱⠀⠀⠀⠀⢀⣠⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⣿⠀⠀⠀⠀⢸⠿⠛⣇⠀⠀⠀⠀⣿⣟⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⢶⡄⠀⢙⣿⣿⣤⣀⣀⣴⠷⢀⠄⠿⢦⣀⣀⣤⠿⣻⡉⠀⢠⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣇⣴⣿⣿⣿⣿⣿⣭⣥⣶⠏⠘⢦⣤⣬⣭⣴⠎⢼⡿⣦⣸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣮⣴⣿⣿⣿⣮⣟⣻⣿⣿⣢⣄⠀⠙⠛⠋⠁⠀⠘⣷⣦⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⠀⠀⠀⢶⡶⠄⠈⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⠓⠒⠲⠦⢤⣄⡀⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣛⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⢿⠿⢿⠿⣿⣶⣿⣿⣷⣝⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⡏⢹⡷⠙⠧⠀⠀⠀⠀⢀⡤⠀⠈⠁⣹⠉⣿⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣥⣤⣭⣀⢲⣄⠀⢈⣶⣏⣀⣤⢆⣀⣤⡄⣹⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣿⣿⣷⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⡙⠻⠿⠟⠫⣿⣿⣿⣌⠋⠛⠻⠿⢛⣥⣟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡥⣤⣴⣶⣿⣦⣺⣿⢿⣿⠄⢠⣶⣞⣂⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⠋⠇⠘⠇⠀⠀⠘⣿⣿⣿⡟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠾⢷⡷⡇⡶⡶⠶⣾⣿⣍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣛⢿⣿⣟⡓⡧⡧⠷⢒⣻⡿⠟⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⣿⣽⣿⣿⣏⠻⡇⠉⠉⠀⢀⣤⠞⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡃⠘⣦⠀⠀⠀⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠟⠻⠶⠖⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⡀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣗⣀⠀⠀⠠⠴⠶⠦⣄⢠⠆⠀⣄⠀⣤⠖⠛⠶⣤⡀⠀⠀⠀⡦⠃⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡘⢿⠀⠀⠀⠀⣢⡀⢈⠁⠀⠀⢩⠀⠀⣤⣄⣀⠀⠀⠀⠀⠀⠇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠇⠸⠆⠀⠀⠀⠉⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠀⠰⠀⠀⠀⠀⠀⠈⠛⠀⠁⡀⠸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⢠⡄⠀⣠⠀⣸⢧⠟⠂⠳⡀⠀⡄⢧⠀⠀⠀⢠⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣾⠁⣰⠟⠀⠤⣈⡆⣹⠀⠀⠀⠀⠀⢠⡏⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⣇⠠⠀⠜⠉⠀⠀⢀⡁⠠⠦⣄⠀⢀⠠⠀⢄⠀⡟⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⣤⠀⠀⠁⠀⢀⠀⣄⠀⠑⠄⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⡀⠀⡅⠈⢙⡄⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠉⠀⠁⠈⡀⠃⠈⠃⠀⠐⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠀⠨⠁⡀⠀⠈⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠈⠉⠁⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣴⡆⢀⣽⣿⡿⢋⣽⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣼⣟⣋⠰⣿⣿⣿⣿⣿⣿⣿⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣟⣻⣧⣴⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣿⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢫⢿⢻⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⠏⠉⠛⠁⠁⡀⠀⠀⡎⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠠⠅⠠⠄⠀⠀⠀⣶⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣭⣿⣿⣿⣿⡇⠀⠠⠤⣶⡖⠂⠀⢰⣏⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⢫⢿⣿⣿⣿⣿⣷⠀⠀⠀⠉⣿⣆⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁⣴⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⡏⠓⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⡀⣸⣿⣳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠉⠻⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠐⠀⣰⡿⠫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣾⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣆⠀⠀⠻⣿⣿⣿⣿⣿⣿⠛⢴⡿⢋⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣻⣄⠀⠀⣀⣉⣉⣉⣁⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠻⣦⣠⣌⣉⠉⠉⣉⣥⣶⣼⣋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⢿⣿⣿⣷⣤⣾⣿⣿⡿⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠟⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡄⠀⣶⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⣄⡀⠀⠀⠀⡄⢀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠐⣤⣤⡄⠘⠠⣶⠄⠀⠀⠀⢀⣦⠲⠤⡄⢠⡀⠀⠀⠀⠀⠐⠀⠀⢠⡿⠶⢠⣄⡀⠄⠀⠀⢀⣀⡀⠘⢦⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠶⠈⠁
⠀⠀⠀⠀⠀⠈⢋⣁⠀⠺⣷⠀⠀⠀⠀⠈⠉⠐⠲⡘⢳⣗⠀⠀⠀⠀⡄⠀⣴⣯⣐⢛⡉⠈⠙⠢⣀⠐⣿⣿⡿⠀⢸⡴⠀⠀⠀⠂⠙⠃
⠀⢠⠀⣤⣶⡆⠻⣿⡷⠲⠀⠀⡀⠀⠀⠀⣰⣾⣷⠛⢧⡹⣷⡄⢀⡀⠀⠏⠉⠿⠉⠉⠙⢶⡀⠙⠂⢡⣬⠉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠠⠀⠘⠇
⠀⠀⠀⠈⠉⠀⠀⠀⢀⣤⠰⣿⣿⣷⣦⣀⣴⢉⡁⠀⠀⠁⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⠌⣉⠀⠀⠈⠁⠠⣤⣴⣦⣴⣶⣆⠀⣶⠆⡀⠀⠀⠐
⠀⠀⠙⠢⣀⠀⠀⡸⢷⣬⣇⡈⠻⢟⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠀⠉⠀⠀⠀⠀⠠⣽⣿⣿⣿⡿⠈⠻⣾⡦
⠀⠀⠆⠀⠻⠳⣾⣿⣿⣿⣿⣿⠛⣩⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⢧⠀⠀⠀⠀⠀⠀⠹⢟⣿⣿⣷⣶⣆⠈
⠀⠀⠀⠀⠀⠀⠶⠉⠙⠿⣫⣵⡀⣽⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡂⠀⠀⠓⢀⣰⠌⢿⡿⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣀⠻⠇⠸⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⣆⠀⢠⣛⠻⣿⠎⠀⠀⣾⠃⠁⠀⠤
⠀⠀⠀⠀⠀⠂⠘⠛⠈⠙⢻⠿⠄⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⡀⢙⡤⠋⢀⣡⠠⣤⡀
⠀⠀⠀⠀⠐⠀⢀⠴⠒⣉⣤⠤⢶⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢀⡴⣀⣴⡆⠀⣿⡿⠛⠀⢀⡇⢰⡄⠈⠁⠀⠀⠀⠀⠀⠀⠰⠄
⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⢦⡀⠭⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⡀⠠⠆⠀⠀⠠⠤⠤⠀⠀⠈⣻⠿⠐⠀⡀⠀⠀⢘⡇⢈⠐⠢⡀⠀⠀⠀⠀⠀⠀⡀
⠐⠃⣀⠀⠀⠚⠀⠀⠺⢿⣿⠂⣹⠀⠀⡶⣼⡏⠀⠉⠲⢶⢶⣄⡴⠃⢀⡀⠰⢾⣇⠀⢀⣥⡄⠚⠀⠀⠀⠈⢿⡀⠈⢸⣇⠈⠁⠀⠈
⠀⣼⣿⣿⡇⠀⠀⠀⣸⣍⣻⡆⠁⠆⡀⠀⠼⢷⣄⣀⣤⠴⢟⠟⠉⠀⢀⠀⢂⠀⠹⠒⠿⣿⠟⠀⡀⠀⠀⠀⠘⠁⢀⣾⠇⣼⣄
⠁⢉⣿⠟⠃⠀⠀⠠⠊⠀⣿⠁⣿⡀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠐⠀⠀⣸⣧⡀⠀⠀⠘⠦⡄⠀⠀⠀⠀⠠⠀⢀⡄⠜⠁⣼⣿⣿⣷
⠀⠈⠛⠁⠀⠀⠀⠀⠀⢢⡙⢃⠹⡝⢦⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣸⡝⡏⠸⣄⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡈⢹⠀⣠⣿⣯⣙⢫⡀⠀⢁⣴⣤⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⣿⣆⠉⠉⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠿⡄⠐⡙⠻⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿⡿⢹⣿⣏⣤⣤⡀⣹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⣦⠀⢹⠓⢦⠀⠀⢘⠀⠀⠀⠈⠀⠔⠀⠑⠠⠃⠀⠀⢀⣸⡆⠀⠀⠀⡁⢠⣿⡗⠈⠙⣿⣿⡿⠣⠌⠁⢤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠿⣿⣿⣷⣼⡆⠈⠃⠀⢸⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣨⣼⣇⠀⠀⠀⠀⣾⣿⡷⠀⢠⣶⡶⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠐⠃⠹⣿⣿⣿⣿⣄⠀⠀⠈⢷⡆⢀⠀⠑⠄⠀⠀⢀⣀⣔⢲⣟⠀⣻⠀⠀⠀⠸⠃⠀⠑⠄⠀⠀⠶⣿⠆⠀⠀⠁
⠀⠈⣤⠀⠀⠀⠀⠀⠉⣠⡄⠰⠟⠶⣿⣿⣿⣿⣿⣷⡄⠀⢸⣿⠟⠻⣆⣈⡆⠑⣏⣠⣿⣿⡯⠀⠽⠀⠰⠀⠀⡇⠀⠀⣈⢰⣶⣦⡀⠑⠀⠀⠃
⠀⠀⠈⠀⠀⣤⣄⠀⠀⠙⠃⠀⣠⡴⠞⢻⡏⠈⠿⣿⣿⣦⠀⠙⠀⠀⠻⢿⣿⠿⠿⠛⢋⠀⠀⠀⠀⠀⣠⠀⠀⣇⢠⠉⠀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠙⠁⠀⠠⠂⠄⠈⠱⠀⠀⠸⡇⠺⢆⡙⠛⠉⡀⠀⠀⠀⠀⠀⢻⠀⠘⠀⠈⠀⠀⢀⠀⠀⠀⠀⣠⠿⣷⡖⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡁⠀⠈⠁⠀⠀⠈⢂⡄⠈⠂⠄⣀⡀⠀⠀⠠⠤⠒⠁⢠⡃⢰⡷⠁⣤⠛⠁⠀⠀⠘⢛⡅⢀
⠁⠀⠂⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠰⢆⣘⠃⠶⠀⠀⠀⠀⢠⠄⠙⡀⠀⠀⢀⠀⡠⠤⢤⡤⠄⠀⠀⠀⠘⡟⠀⠙⣄⠀⠠⠧⠀⠀⠔⠃⠀⠑⢄
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢸⣦⡐⢄⠀⠀⠀⣀⡀⠀⢀⣾⣧⠔⠁⠀⣰⣾⣄⠀⠀⠀⠢⠤⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠋⠀⠀⠀⠶⠀⠤⠀⠘⢻⣿⣾⣿⣶⣿⠀⢹⣿⡏⠂⣄⠀⠀⠐⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⠋⠛⢿⠋⠀⡾⠿⡀⠠⡉⠀⠦⠀⠈⢹⠛⠻⠏⠁⠀⢀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢋⠀⢰⠋⠀⣰⠇⠀⢣⠀⠀⠀⠀⠃⠀⠀⠀⠈⡰⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⠀⠰⡁⠀⢀⡀⠑⠂⠀⠀⠀⠀⠰⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠐⠠⡀⠁⠆⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣀⡅⢀⡀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⢈⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣖⢒⣶⣶⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣿⣿⣿⣿⣿⣦⡰⣶⣦⣬⣭⣓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⢷⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢏⡴⣚⣭⣭⣿⣭⣝⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠡⣷⣾⣿⣿⣿⣿⣿⣿⣿⣶⣮⡛⢿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⡻⣿⣛⠿⠿⠿⢿⣿⣶⣦⣤⣀⡀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠉⢻⢉⣿⣹⣿⡏⣩⣤⣤⣽⣛⠿⢿⢿⣿⣶⣶⣄⣠⣿⣛⣛⠿⠃⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢪⣳⣿⠡⠀⠿⣿⣷⣾⣿⠟⠉⣾⣟⣽⣿⠉⠉⠉⠉⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣤⣄⣈⣩⡥⠀⣠⣶⣾⠙⠽⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⠙⣿⣿⣇⠀⠀⠋⠛⠀⣴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣴⣾⠃⢹⠂⠈⠻⡷⣶⠤⠀⢠⡤⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⠠⡘⢶⣦⠰⢄⢀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⣀⣠⣾⣿⣿⣿⡶⢬⡙⠻⢷⣤⣤⣤⣶⡤⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣿⣿⣿⣿⣿⣷⣻⣦⡀⠀⠑⢮⡙⣿⣏⣦⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣿⣿⣿⣿⣷⡹⣿⣄⠀⠀⢰⣿⣿⣾⣿⣿⣿⣿⣷⣦⡀⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣍⣿⣌⢿⣷⣄⡟⠙⣿⣏⠻⠛⣿⣿⣿⣷⣷⣄⣤⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣝⣿⡁⠀⢿⣿⡄⠀⢻⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣆⣺⣿⣿⣤⣼⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣶⢸⡆⣶⢰⡆⡿⢳⠀⡟⠛⠀⣾⡆⢰⡟⣶⢸⣦⠇⣶⢰⡆⠀⢺⠀⠀⣿⠛⢰⡟⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⣿⡇⡇⢸⠀⡷⠒⢠⣏⣧⢸⡇⣿⢸⣿⡀⣿⢸⡇⠀⢸⠀⠀⣿⠒⢸⡷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠃⠟⠹⠃⠷⠾⠀⠷⠶⣼⣿⣯⣭⣽⣟⢾⡋⠇⠿⠸⠷⠦⠺⠶⠆⠿⠶⠘⠇⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡖⣲⣶⣶⣶⠖⣶⣾⣿⣿⡿⠿⣿⣿⣮⣝⣶⡶⢲⣶⢶⣶⣶⡶⢲⣶⣶⣲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣫⣾⠿⠿⠿⠋⣼⣿⣿⣿⣿⢑⢀⠉⢿⣿⣿⡟⣴⣾⣃⣿⣿⣿⢡⢸⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡇⣾⣷⡆⢺⣿⣿⣿⣿⣾⣦⣿⣶⣿⣿⣿⣇⣿⣽⣿⡿⣿⢯⣿⣦⡥⠸⣛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣳⣿⢟⣴⣿⣧⢸⣿⣿⣿⣛⣛⢻⣿⣟⠟⣫⣽⣿⣿⡿⣟⣹⣫⢮⠏⢠⣾⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⣡⣴⣿⣿⡿⢋⣼⣽⣿⠏⢻⣿⣾⣏⣹⣷⣿⡿⣱⣿⣿⣌⣿⣇⣈⣤⣿⢿⡛⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣻⣖⣿⣿⣿⢿⡻⣌⠙⠛⠾⠿⣾⢿⣿⠋⣿⣿⣿⣿⣷⣽⣿⣟⣿⣣⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡵⣾⣷⣿⣿⢿⣿⣿⣷⣝⢬⡉⠙⢿⣶⣶⣾⢟⣾⣿⣿⣿⡏⣿⣿⣿⢟⡏⣿⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣾⣶⣴⣶⣾⣿⡿⢃⣿⣿⣿⣴⣿⣿⣿⣿⣧⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣽⣿⣿⢻⡿⣯⠻⣿⣝⣿⠿⠋⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣮⣿⢶⣀⣀⣠⣴⣿⣿⣾⣿⣿⣿⡟⣹⣿⣯⣿⣯⢽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣓⣚⣹⣍⣙⣿⣿⣿⣿⣿⣯⣿⠿⣿⣿⣿⣿⣷⣹⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣴⣿⣯⣵⣿⣷⣮⣽⡿⢼⣿⣿⣿⡷⢿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣾⣿⣶⣿⣿⣿⢿⣷⣿⣿⣽⣿⣷⣿⡿⠿⣿⣿⢿⣟⣿⣦⣽⣿⣽⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⣿⣿⣿⣿⣿⣽⣿⣾⣾⣿⣿⡟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣽⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡋⣿⣿⣿⣿⢋⠉⣩⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠿⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠵⠾⠿⠿⠿⠿⢿⣧⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠿⣇⠿⠿⠧⠾⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠦⠴⠙⠂⠴⠠⠰⠆⠂⠆⠰⠀⠤⠆⠰⠠⠴⠷⠀⠰⠐⠆

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠂⠀⠀⠀⠀⠀⠀⠻⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⠀⠀⠸⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⢺⠴⠖⠀⠘⠀⠀⠀⠀⠰⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢂⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⢀⠈⢻⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⣾
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣧⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣀
⡇⠀⠀⠀⠀⠴⠿⠀⠀⡀⠀⠤⠀⠀⠀⠀⠘⠀⡈⠀⠀⠀⠀⠐⠀⠀⠹⢧⠂
⡇⠀⠀⠀⠀⠂⠀⠀⢀⡉⠅⠀⠀⠀⠀⠀⢀⠂⠇⡝⠀⢠⡤⠆⠀⡄⠀⠰⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⡇⠀⠀⠀⠀⠂⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠐
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠐⠒⠠⠄⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢰⠃⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⡀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠍⠓
⡇
⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢟⣛⣛⠛⠚⠻⠿⢆⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢼⡋⢰⡏⠃⠈⠓⠀⠀⠀⠀⠐⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠳⠀⠺⠧⢈⣁⠀⢀⠉⠁⢤⣆⠀⢾⣿⡟⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⢀⠀⠹⠇⠹⣦⠀⣴⠠⠐⠋⠋⢈⣛⡛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⢤⣷⣤⣴⣄⠀⣤⣤⣤⡒⠒⠋⢻⣧⠀⠀⠈⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⢀⣠⣿⣟⠉⢁⣿⣿⣉⣿⡟⠲⠄⣸⣿⣇⡀⠀⣠⠞⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⣤⡴⠿⠿⠟⠛⠲⣿⢟⣿⣿⣿⣷⣄⠀⠀⠈⣻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣶⣦⣈⡛⠛⠋⠉⠉⠛⠉⠀⠉⠀⠀⠀⠀⠀⠀⡆⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠛⣭⢿⣿⣶⣶⣄⣤⣤⡤⠀⠀⠤⣠⠤⠂⠤⣤⣦⣤⣴⣦⡤⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣿⣿⣿⣤⣤⡁⠀⠀⠀⠀⣆⣀⣀⠀⢹⣿⣿⣿⣿⣷⡶⠟⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢿⣿⣿⣿⣿⣷⡄⢀⣴⣿⣿⣿⣷⡌⣿⣿⣿⣧⣬⣛⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢹⡈⢿⣿⣭⠽⠋⡇⠀⠙⠻⠽⠿⠿⠁⣿⣿⠿⠦⡁⠉⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠠⠶⠴⢦⣤⠤⠘⢦⣄⡀⠀⣿⣿⣷⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣄⣠⣤⣦⣤⣤⣤⣀⣤⡽⢸⣿⡿⣄⡘⠠⠀⣀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠙⠿⣿⣿⣛⣿⡿⠟⠉⢀⡿⠋⡳⢄⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣆⠀⠀⠀⠉⠉⠁⠀⢠⣀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡼⣽⣦⣀⡀⠀⠀⢀⣀⡾⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⣷⠙⣿⢹⣿⠗⠛⠉⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠈⠿⣤⣼⣿⣇⠀⠀⠀⠀⠀⣀⠾⠃⣼⡇⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠹⠧⠀⢤⣄⡘⢻⡟⣿⣄⠀⠀⣤⠗⠂⣀⢮⠽⠒⠓⠄⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠛⠷⢸⣿⡀⢻⠆⣠⠁⡐⠋⠉⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣀⠀⣀⣀⣠⣤⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⣽⣟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠹⡉⠈⠋⠀⠀⠉⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣄⣾⣭⡻⠦⢄⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⢀⡤⢾⣿⢿⣿⠟⣿⣿⣦⣀⠁⠈⠀⠀⠀⢸⠏⠀⠀⠰⢿⣿⣷⣶⣤⣤⣤⣤⡼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⣠⡶⠖⣻⠇⠘⣿⠀⡍⠀⠉⡿⢹⣿⡶⣦⣤⡀⢠⠟⠀⠀⠀⠀⠀⢧⠀⡈⠉⠀⠀⠈⠁⠀⢸⡇
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠉⡁⠀⠉⠀⠀⡏⢠⠀⠀⣠⠀⠀⢸⣇⣟⣛⠛⠋⠀⠀⠀⠀⠀⠀⠈⢧⡘⢢⠀⠀⠀⠰⠀⢺⡇
⠀⠀⠀⠀⠸⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⠀⠀⠀⠀⣿⡀⠀⠀⣿⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢀⠘⡄⠀⠀⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠀⠀⢹⡇⠀⣸⠇⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇⢸⠇⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⣷⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⡄⡆⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠸⢇⠹⢿⣶⣧⣿⡄⣠⡴⠟⣿⣇⡄⠀⠀⠀⠀⢀⣠⣴⡿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠹⠀⠀⢿⣿⠿⣿⠟⠴⠷⠿⠛⡏⠀⠀⠰⢴⠾⡿⠛⠁⠜⠀⠦⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣭⣿⣿⣿⡿⠭⠐⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣿⣿⣿⣿⣿⣯⠂⠀⠀⢲⠀⢹⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠀⣾⣿⣦⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣄⠐⢡⣷⣿⣿⣿⣿⣷⡝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣯⢹⣽⣿⣿⣿⣿⣿⣿⣭⣖⡶⢯⣙⠳⠿⣿⣿⣿⣿⣿⣿⣿⣆⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣰⣿⣷⣮⣟⡿⣿⣿⣿⣿⣿⣿⣿⣦⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⣿⣿⣿⣿⣿⣿⡿⢿⠛⠻⠿⠿⣶⣤⣍⣉⡛⠛⠟⠻⠿⠛⠛⣿⣿⣿⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⣿⣿⣿⣿⣿⣿⣷⡶⣴⣶⣦⣤⠄⣤⣤⣾⣿⡝⣿⣶⣶⣶⣾⣿⣿⣿⡇⢇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣿⣿⣿⣿⣿⣿⣿⠃⡻⠿⠿⠟⣴⣞⠿⠿⠿⡂⢻⣿⣿⣿⣿⣿⣿⣿⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⡛⠐⠿⠿⠐⣷⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⣿⣿⡿⢛⣍⢻⣿⣷⣿⢸⣦⠤⠤⠤⣴⡟⢸⣧⣿⣹⠟⣹⡛⢿⣿⣿⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡘⣿⣤⡟⣿⣦⣿⣿⣿⣾⡙⠓⠠⠒⠋⣥⣾⣿⡿⢃⣾⡿⣻⣦⣿⢇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠳⡜⣿⢻⣿⣿⣟⢧⣻⣿⣿⣤⣤⣤⣼⣿⣿⡟⣡⣾⣏⣵⠟⡿⢫⠞⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠿⠼⠾⠄⠭⡅⢹⡷⠽⣝⣿⠿⣟⡻⢿⡿⣫⠾⣿⡇⢩⠭⠤⠷⠯⠿⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣳⣭⣤⢤⣐⣤⡻⣧⡒⠍⣳⣯⣵⡾⠍⢂⣽⢟⣤⣦⡤⣤⣭⡛⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠸⡆⠙⢿⣿⡿⢾⣭⣷⣿⣿⣿⣿⣿⣽⠷⣿⣿⡿⠋⢰⠇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠳⠽⢿⣟⣛⣛⣻⡿⠯⠚⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⣠⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢹⡄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢹⡆⠀⠀⠀⡄⠀⢠⠀⢠⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣀⠀⣷⢠⠀⠀⢇⡀⢸⢇⠀⠸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⢻⡄⠈⢧⡄⠰⢧⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣾⡱⣄⢸⣧⢹⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣀⠐⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣼⠸⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⢈⡩⠀⠀⠊⠁⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇⠹⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠿⣿⣩⣭⣽⠗⡫⢚⣤⣶⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣆⡦⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣸⣿⣿⣅⣠⣀⣩⣤⣀⢛⣟⢩⣽⣷⠾⣾⣿⡿⣟⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⣶⡉⠀⠀⠀⠀⠈⠛⠩⠣⣿⡖⢚⣛⣻⣾⡿⣛⣿⣿⣿⣿⠻⣯⠿⣬⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⡄⢧
⠀⠀⠀⠀⠀⠀⢀⠹⡏⣿⣾⡿⡀⠀⠀⠀⢀⣤⣶⣤⣌⠙⣿⢿⠷⣶⣿⣿⠿⣿⠛⣡⣟⣹⣾⣟⣛⡱⢦⣬⠛⢮⢻⣿⣿⡿⣿⣈⡄
⠀⠀⠀⠀⠀⠀⠁⠸⢹⣿⣿⣿⣗⣀⣠⣈⠛⠿⠿⠿⠋⠀⢸⣷⣿⣿⣿⣿⠛⣻⣳⣏⣾⣋⣉⣉⠉⠙⠷⣌⠳⣮⠃⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠀⠀⡅⢾⣿⣿⣿⣽⣿⣿⢷⡆⢀⠀⠃⢺⣿⣈⣿⣿⣿⠿⣿⣿⣿⣍⣛⣋⣛⣛⠿⣶⣵⣌⢳⡼⡇⣿⣿⣾⢿⣿⣿⡃⢀⣠⣀⡀
⠀⠀⠀⠀⢠⠊⠀⠀⡆⣸⢿⣻⣿⣿⣿⣏⣼⣷⣏⡷⠠⠘⣿⡿⠚⢻⣿⣷⣎⠻⣮⡛⣿⣿⣿⣿⣿⣮⣻⣿⣆⣴⣿⣿⡿⣿⣿⣿⣿⣟⣹⣿⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⣿⣿⣿⣿⣿⣿⣿⣿⢂⣴⣾⣿⣟⣶⣾⣿⣿⣿⣦⣹⣷⢸⣿⡈⣿⣿⣿⣷⣹⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠸⣿⣟⣿⣿⣿⣿⣿⣿⣟⣿⣷⣋⢛⣻⣶⣿⣿⣿⣿⣿⣧⣹⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⣘⣿⣿⣟⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣼⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉⠉⠙⠛⠛⠻⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣻⣯⡻⣏⢻⣻⣷⣿⣏⣉⣈⣉⣥⣴⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣪⡿⠃⠀⠉⠛⠫⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢮⣿⣉⣀⠀⠀⠀⠀⢢⣿⣿⠟⠁⠀⠈⠁⠀⠀⢀⣼⠿⢋⣿⣿⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⠀⠀⠀⠀⣰⣿⠃⣠⣾⡟⣿⡏⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣟⠀⠀⢀⣠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⣾⠟⢁⣴⡿⠏⠀⠋
⠀⠀⠀⠀⠀⢀⡼⠛⠉⠁⠈⠀⣀⣴⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠴⠿⠃⠀⠀⢀⣠⣴⣿⡿⠋⠈⠉
⠀⠀⣴⣏⣀⡄⠀⠀⢂⣿⣿⡿⠋
⠀⣾⣿⣿⣿⡦⠀⠀⠸⣿⣿⣧
⠀⣿⡟⠉⠀⠀⠀⣀⢾⣿⠅⠻⡇
⠀⡿⠀⢀⣈⡄⣀⣿⡈⠟
⢀⠙⡄⣻⣿⠈⣿⠋⢷⡄
⢸⣶⢇⣿⣴⣇⣙⠀⢸⣷
⡐⢩⡆⢹⣶⠁⣿⡀⠀⣿
⠛⣦⡅⢸⡇⠀⠘⠇
⣤⣾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠀⠠⠄⢈⡉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠀⠀⠠⠀⠘⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠃⠀⠀⠀⠀⠀⠀⢰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠒
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠸⢿⣷⡀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⡀⠀⠁⠀⠀⠖⠀⠀⠘⢦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠴⠋⡾
⠀⠀⠀⠀⡀⢳⣄⠘⠀⢀⣠⡾⠁⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠻⣈⢧⣵⣗⢠⡏⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⡅⠂
⠀⠀⠀⠀⠀⠏⠰⣿⠛⣀⠃⠀⠀⠀⠠⢀⡤⠬⠝⠷⢾⣦⣀⠀⣴⡟⠀⠀⠀⡎⠀⠀⢀⣤⠶⠋⠉⠛⠻⢷⣦⣄⠀⠀⠀⠃⠀⠀⢀⣿⡼⠋⠙⠿
⠀⠀⠀⠀⠁⢀⡀⠘⢷⣿⣰⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠙⠳⣄⠁⠂⠀⠀⠷⠇⠀⠞⠁⠀⠀⠀⠀⠀⠀⠉⠣⠀⠀⠀⠀⠀⠀⢸⡟⠀⣰⡆
⠀⠀⠀⢠⠀⠀⢻⡄⠀⠹⣿⠀⠀⠀⠀⡀⠐⠃⢈⣷⡀⠈⢦⡀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠠⢤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠃⢀⣏
⠀⠀⠀⠀⠄⠀⢸⣿⠀⠀⣿⡄⠀⠀⠈⠀⠀⠀⠘⠋⠁⠀⠘⠋⢱⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼⠏
⠀⠀⠀⠀⠠⡄⠀⠋⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣄⡀⠀⠀⢘⡀⠀⠀⠀⣿⠀⣴
⠀⠀⠀⠀⠰⠃⢀⠀⡀⠸⠆⡀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠂⠀⠀⡆⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠆⠀⠛⠀⠀⠀⠀⠟⠀⠇
⠀⠀⠀⠀⠀⠀⠈⢸⠇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠀⢀⣀⠈⢴⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠈⡄⠀⣠⡄⠀⠀⠀⢀⣠⠀⠀⣸⡟⣠⡛⠙⢇⠈⢷⡄⠀⠀⠀⢰⡄⣷⡀⠀⠀⠀⠀⠀⠀⢀⠋⠀⣶⣾⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠛⠀⠛⠋⠀⢀⣴⡾⠁⠀⣰⣿⡇⠘⠃⠀⠈⣦⠀⠙⣦⡀⠀⠸⠃⠈⠳⡆⠀⡀⠀⠀⣠⢿⠄⠀⠻⠿⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⣰⣿⠏⠁⠀⠄⣤⡀⢘⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⡟⣀⠀⠀⠉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⣻⡇⠀⠀⠀⠀⠀⣠⠞⠷⠀⠀⠙⠁⠀⠰⠒⠀⠀⡌⠛⠋⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⢠⣿⡇⢸⠀⢴⡄⠠⠉
⠀⠀⠀⠀⠀⠀⠈⠒⠦⠚⠹⣿⡀⠀⠶⠀⡾⠁⠀⠀⠀⠀⠀⠠⠤⣤⠤⠦⠶⠛⢶⣄⡀⠀⠐⣄⠀⠋⠀⠈⣦⠀⠀⡾⠁⢁⣿⠀⣸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⠳⡄⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⢠⠀⠈⠈⢁⠀⡀⠈⠙⢦⠀⠀⠈⢳⡀⠃⠀⠸⣧⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠠⠀⠀⠀⠀⠄⠀⢠⢀⣸⣀⣼⣧⠙⢄⠀⠀⠑⠠⡀⠀⠂⠀⠀⠀⠷⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠄⠐⠠⠙⡇⠈⢧⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠻⡄⠐⣧⠀⢸⡆⠀⢸⡇⠀⢠⠀⠀⣿⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠃⠀⠉⠀⠀⠃⠀⠈⡁⠀⠿⠀⠀⠁⠛⠀⠀⠀⠘⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣀⡀⠀⠀⠀⣀⡤⠞⢀⡴⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⡀⠀⠀⠀⠀⠠⢤⣀⠁⠈⠁⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⡄⠀⠀⠀⠀⠀⠈⠙⠓⠤⠤⠴⢂⠉⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣼⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⣀⠁⠐⠈⠀⠃⠀⠈⠀⠔⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⡀⡔⠉⠉⡉⠀⠒⠤⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢰⠿⠍⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡅⠀⡇⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⣿⠀⠀⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢸⠀⠰⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢿⣠⠖⢊⠉⠉⢉⡢⡀⠀⠀⠀⢀⢔⡯⠩⡉⠑⠺⣆⡆⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠸⢧⡤⠚⠨⠉⠉⠙⢿⡀⠀⡄⡾⠛⠉⠉⠉⡒⢤⡼⠲⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢆⠐⣇⡀⠀⠀⠀⠀⢸⡗⠂⠛⣇⠀⠀⠀⠀⢸⣿⡇⠀⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⢼⢂⠻⣇⣀⢀⣀⣰⡾⢸⠐⠀⢻⡳⠀⠀⠀⣠⠟⢱⡇⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠈⣦⠈⢉⠛⢻⠏⠐⣸⣀⠀⠀⠙⡟⠚⠉⠁⢴⠉⡟⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢸⠀⠈⠒⠠⠖⠡⠊⡰⣁⢠⡑⡄⢁⡈⠒⠤⠒⠁⠀⣷⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⣆⡀⠀⠀⡰⠂⠀⠙⠋⠈⠃⠚⣄⠈⠳⣄⠀⠀⢀⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡡⣝⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠘⣷⢖⣩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠇⣻⠃⠀⡧⠀⠀⠁⠉⠐⠢⣠⡇⠀⢰⠇⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⢰⣶⠶⠶⠶⣶⣶⠀⢻⠀⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⢸⠘⠿⠧⠶⠦⠿⠟⠀⢸⠠⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡈⠄⠀⠀⠀⠀⠐⢀⠄⠜⢦⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠐⠂⢤⣤⠄⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢈⡁⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⣀⣹⣿⣡⣀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣄⠀⠀⡼⣆⠀⠀⣠⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣹⡇⣹⣫⡕⢪⡿⡏⣸⣋⣠⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢢⡀⠲⢤⣤⠽⢾⣿⣦⣼⣿⠶⢯⣤⠤⢂⢀⡂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢶⡿⣟⠀⠀⣴⣶⣚⣽⣯⣿⣯⣳⣶⣆⠀⠈⣿⣷⢞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⡖⣧⢾⣂⢜⣿⣿⣽⡿⣅⡨⣿⣭⣿⣟⠇⣸⡷⡜⢲⠁⡄⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠢⡠⣁⡏⣜⡏⣾⡟⢛⠻⠿⠾⠿⠿⢛⡛⣿⣧⠿⡄⢳⣡⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣱⡙⠋⣿⢸⡟⣃⠈⠐⠛⠆⢘⠓⠈⠀⡜⣿⢈⣿⠙⢁⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠁⢀⢻⡝⣚⢿⢸⣿⣞⡛⠳⣄⣄⣠⡼⠞⢛⢳⣿⢸⠿⣓⣹⠗
⠀⠀⠀⠀⠀⠀⢀⠀⠹⣾⡙⣮⡖⢸⡿⢿⣷⣦⡀⠀⠀⢀⣶⣿⡾⣿⡆⢶⡽⣱⣶⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⢿⣻⣿⠋⣿⡜⠻⣆⡻⣾⣧⣟⣼⠟⢶⡟⢸⣿⣏⡟⠛⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠰⡦⠒⠺⣯⣗⢶⡎⣧⢰⣻⣟⣿⣿⣻⢻⠄⣾⢱⣶⣺⣿⠗⠲⢴
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠘⢻⠉⢫⠙⣧⣿⢿⣿⣿⣿⣿⣿⣿⢿⣟⣼⠋⠟⢉⡟⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣌⡤⢽⡘⢄⢻⡟⣟⣿⡀⣈⣿⡻⣻⡏⡞⣣⡧⢤⡠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢶⣼⢺⡛⢿⣿⣿⣿⣿⡿⢻⡟⣧⡆⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣯⢸⡌⢽⣧⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠛⠋⡟⡟⢟⠟⠛⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⡒⠦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣟⠋⠁⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡶⠂⠀⠀⠀⠙⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣟⡤⠀⠀⠀⣀⡠⢷⣿⢧⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⡄⣲⡋⢹⡄⠀⣧⣿⣧⠤⣶⣾⡛⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣇⣈⣿⣾⠿⠒⠋⣁⣤⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣟⣛⣛⣉⣩⣭⣤⣴⡶⠞⠋⠙⣿⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣟⣛⠛⠉⠉⠉⢀⣠⣤⣤⣤⣽⠁⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠾⠿⠿⠛⠉⠀⣹⡏⣿⣿⣿⣿⣿⣶⠛⢶⣿⣿⣿⣿⡇⠙⢆⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠰⡋⠀⠈⠛⠛⠛⠋⣿⣾⡎⠝⠛⠛⠛⠁⣤⠌⠀⠀⠀⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⡿⡆⠻⡿⠿⠀⠚⣽⣿⡏⠀⠀⠀⠀⠀⠀⢰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⠀⠀⢀⣤⣿⠹⣿⣵⡇⠀⠅⠀⣀⣠⣿⡟⡀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢧⢀⡀⠠⠀⠀⢀⡌⢿⣿⠡⣙⣿⡀⢠⠀⠠⠠⢐⠟⠀⡛⠀⠀⠀⠀⡄⢀⡀⡼⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢋⢡⡤⣚⣾⣿⣿⣆⠙⢿⣮⣟⠿⠈⠑⠁⠁⠀⣠⣾⣿⣷⣒⡤⣅⡀⠀⠛⣲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢞⢿⣿⣿⣿⣿⣿⣿⣧⡘⠿⣿⢷⣄⣀⣀⣤⣾⡿⣿⣿⡀⠙⠻⣿⣦⡭⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡹⣿⣿⡿⣿⣿⣿⣿⠀⠀⣠⣾⣿⣿⣿⣯⣄⡈⣿⣷⣶⣠⣼⢟⡕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣻⢷⣼⣿⣷⡅⠀⡼⢿⣿⣿⣿⣿⠋⢹⡟⣿⣿⡿⢟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢭⣛⠿⣿⡄⠃⠘⣿⣿⣿⣿⡆⠈⠷⣛⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠺⠿⣿⣿⣿⡿⠯⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠂⠀⣁⠈⣉⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⢤⠴⠤⠤⠀⠀⠀⠀⠈⠙⠻⠿⠟⠧⠴⢢⠄⠀⠒⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠘⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠈⠚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⡔⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀⢀⠀⣀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃
⠀⡐⠀⠀⠀⠀⠞⡁⠄⠂⠁⠀⠀⢀⠻⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠒⠠⠄⠐⠂⠀⠀⠀⠀⠈⢠
⢠⠀⡀⠀⠀⠠⠂⢀⠀⠂⠁⠀⠀⠀⡑⠤⠀⣄⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⣜⣧⣹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢻⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀
⠀⣇⠀⠀⠀⠈⠉⠛⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⡸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠃
⠇⠈⠇⢠⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡽⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⣄⠃⠠⠀⠀⠐⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠐⠖⠒⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉
⡇⠀⠀⠃⢁⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠐⠀⠀⢀⠠⠀⠀⠀⠀⠁⠀⠀⠤⡁
⢳⠀⠀⠀⠘⡆⠀⠀⠀⠀⡀⠀⠸⠀⠀⡀⠀⠀⠀⢄⠀⠀⠀⠄⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠸⡆⡀⠀⠀⠣⠀⣤⠀⠀⠀⠀⠀⠀⠀⡇⠆
⠀⢻⡁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠
⠀⠈⢧⢢⡀⠀⠀⠘⣆⠀⠰⠀⢀⠀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⢀⠉⠄⠀⠠⡤⠀⠀⠀⠀⠄⠀⠀⣼⠃
⠀⠀⠀⠀⠹⢦⡄⣄⢉⠶⠄⠀⠂⢀⡜⠀⠀⠠⢚⣣⠞⣣⡄⠀⠀⠀⢀⡆⠐⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠌⠀⠀⠘⠃⠀⠀⠐⠋⣵⡿⢟⡁⢠⡀⠀⢻⡇⠀⠀⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠔⣫⣿⢿⡇⠀⠈⠀⠀⠄⡐⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠙⣿⠈⠃⠀⠀⢀⠀⠀⡁⠀⠀⠀⠠⠀⢀⠀⠀⠀⠐⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⢦⡀⠁⠀⠀⠀⠻⠄⠀⠀⠀⠀⠘⠐⠃⠀⠐⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠁⢀⠠⠀⢀⠀⢐⠂⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣷⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣿⣿⣮⣙⣿⣋⣽⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣇⣽⣏⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡏⢻⣿⣟⡆
⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⡇⠘⡟⠁
⠀⠀⠀⠀⣼⡟⢰⡿⢡⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⢀⡿⠀⣾⠃⢸⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⠇
⠀⠀⠀⢸⡇⠀⠋⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⣀⣀⣀⣀⠀⠀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⡀⢀⣀⡀⢀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀
⣿⠿⡿⢿⠿⡿⠟⠻⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⣿⣿⣿⠿⠿⠿⣿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠈⠉⠉⠀⠒⠢⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣹⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠙⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⣿⠂⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣿⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡘⠃⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⣷⣅⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢹⣿⣿⣿⣿⣿⣷⣶⣦⣴⢀⣶⣶⣿⣿⣿⣿⣿⣧⢿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠄⣧⣿⢿⣿⣿⣿⣿⡿⣿⠆⠀⠉⣿⣿⣿⣿⣿⣿⣿⢸⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⡘⠂⠐⠛⠿⢛⡛⠃⣀⡀⢀⡉⠟⠉⠻⠿⠟⠁⠈⣥⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠉⠁⠀⢀⣠⣽⣷⣾⣿⣿⣆⠀⣠⣄⡀⠀⠀⣠⣮⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⣶⢟⣿⣿⣿⣿⣿⣿⣽⣿⠀⢛⣯⣭⠶⠶⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⡁⠙⠉⡀⠚⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢱⡀⠀⠀⠀⠀⠘⡻⠿⣿⢿⣿⠳⡀⢄⢁⠀⡠⠈⠀⠀⠀⠀⠀⢀⣴⠿⠀⣈⠇
⠀⠀⠀⠀⠀⠀⢀⣱⡠⢤⡄⠘⠛⠷⣤⣄⡀⠀⠙⠶⣇⡁⠀⢙⠈⠈⡀⣂⠄⠀⠀⠀⢀⣤⣤⣿⠃⢀⣤⠝⠒⢄
⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⣀⣀⣧⣄⡀⠀⠛⠛⠒⠤⢈⡉⠁⠉⠉⠁⢉⣀⡤⠤⡲⠿⠛⠉⠋⠁⢀⣄⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠈⠓⠙⠟⠛⠙⠛⠛⠙⠿⢶⡀⠀⠀⠀⠀⠁⠲⠶⣉⡁⠀⠀⢀⠀⣤⡤⠴⠎⠓⠛⠉⠓⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀⣀⡠⠜⠋⠓⠦⢄⣀⠀⠀⠀⠉⠻⠶⣯⣍⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠋⠛⠻⣿⣿⠟⠛⠋⣀⣤⣤⡴⠖⠚⠙⠉⠐⠲⠤⣀⠀⠈⠉⠛⠻⠿⠶⠒⢒⡖⠛⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠘⠦⣄⣠⣤⡾⠛⠁⣺⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠥⣷⣦⣠⣴⣶⢴⣶⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠃⢈⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣇⠀⠁⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡴⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠈⠉⠁⠀⠈⠉⠉⠛⠒⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠻⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠧⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣿⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⢠⣾⣿⣿⣿⣷⣤⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡀⢻⣷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⣿⣿⣿⡟⠀⣀⡀⠀⢻⣿⣿⣿⣿⣿⣷⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⢿⣻⣿⣿⣿⡿⠁⢸⣿⣷⡆⠘⣿⣿⣿⣿⣿⡿⠀⠙⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠉⠉⣉⣭⡤⠀⣾⣿⣿⣇⠀⢤⣉⠉⠉⠉⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠈⣿⣶⣔⠃⡼⠟⢀⠈⠿⠋⠻⠏⠀⠈⢿⡟⠒⠒⣋⣵⠃⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠁⠀⢈⣷⠀⠀⠀⠀⢸⣿⣿⠑⣤⡀⢀⠀⠀⠀⡀⢀⠀⠀⢀⣷⠛⣿⣿⠇⠀⠀⠀⠀⣾⢁⠀⠈⣷⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⠶⣾⣿⠀⠀⠀⠀⠀⣿⠘⣦⣿⣛⠀⠀⡃⢀⠀⡎⢠⠀⣸⣇⣸⡿⡿⠀⠀⠀⠀⠀⣹⣿⠶⣾⠋⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⣰⠚⠿⠅⠛⠿⠗⢂⡀⠀⠀⣧⠀⣿⣏⢻⠿⠟⢿⠻⠛⠿⠛⠻⣿⣿⠏⠀⡗⠀⠀⠀⡀⠲⠿⠛⠓⠿⠓⢦⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢙⣈⠳⠶⠴⠶⠾⣿⣆⠀⠀⠻⣄⠸⣿⠏⠰⡇⠸⢴⠄⠆⠴⠐⠻⡟⢀⡰⠃⠀⠀⣠⣿⠷⠶⠦⠴⠖⢀⡮⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠳⠮⠶⢶⣤⣈⡛⢷⣄⠀⠈⠳⡌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣠⣾⠟⣁⣤⡶⠶⠴⠞⠋⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣝⡛⠰⠤⡈⠢⢀⣳⡀⠀⠠⠀⠚⠩⠝⢀⡠⠖⢛⣋⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⡉⠢⣄⡀⠀⠀⢀⣠⠄⠈⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣈⠑⠞⢥⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢋⣽⣿⡿⢶⣤⣌⡙⠒⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠚⠭⣀⣠⣾⠿⠛⠁⠀⠀⠈⠙⠿⣿⣤⣀⠩⠓⣢⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢠⡴⠞⠷⢒⣀⡬⢭⠉⠁⠀⣀⡰⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡄⠀⠈⠉⡩⢧⣀⠒⠾⠛⠦⣄⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⢻⡄⠀⣠⣌⣉⠁⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢆⠀⠀⠈⣉⣡⣄⡀⢠⡿⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠛⢿⡏⠉⢀⣰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠈⠹⣿⠟⠋⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣤⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠀⠀⣹⣿⣿⡁⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⣀⣤⣾⡿⢁⠹⣿⣦⣄⣀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠙⠻⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡻⠿⠛⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣄⠀⠀⠀⠀⠀⢻⡿⣿⡿⣿⡿⣿⡿⣿⠃⠀⠀⠀⠀⠀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠘⠇⣼⡇⣿⡇⣿⡇⠟⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠛⠻⠿⣿⣷⣦⣤⣀⠈⠁⠉⠁⠉⢀⣠⣤⣶⣿⡿⠿⠛⠋⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣶⣤⣴⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⠿⠛⠋⠙⠻⢷⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠛⠛⠋⠉⠓⠲⠤⣀
⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⡴⢡⡾⠋⠉⢉⡛⡆⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣽⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⡎⠙⠗⠒⣤⡀
⠀⣷⡀⢠⡤⠀⠀⠈⠍⠦⡄⠀⠀⠀⢀⣼⠋⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠜⢷⣄⠀⠀⢀⣠⡾⠽⡏⠀⡧⠀⠀⠡⢻
⢠⡌⢱⣿⠃⠀⠀⠀⠀⠀⢀⣀⠀⢀⣾⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣿⡶⠾⠋⠉⠀⠀⠙⡀⠂
⠘⣯⡄⢹⣇⡀⠀⠀⠀⠀⠈⠀⠀⢹⣿⣦⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⣿⡇⠀⠀⠀⣻⣆⠀⠛⠂⠀⠀⠀⣀
⠀⠈⠙⠛⠛⠻⠦⠶⠒⠶⠄⢀⣀⣸⣿⡇⣼⡇⢀⣄⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⣏⣿⣿⣷⣾⠿⠒⠛⠻⠦⣶⣀⣠⡄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣟⠗⣿⠇⢈⣣⣤⣤⣤⣴⣭⡉⠀⠀⠀⡄⢠⣴⣶⣶⣶⣦⣬⣟⠘⣷⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡸⢺⡄⢻⢿⣿⣿⣿⣥⣤⣿⠀⠇⢸⠀⣿⣤⣬⣿⣿⣿⣿⢿⠁⣿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡇⠈⣷⣌⡛⣿⣿⣿⠟⢻⢃⠸⣦⣻⣿⣿⣿⠟⠋⣴⡟⠀⣹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠛⠃⠀⠈⠉⠉⠉⠁⢀⠀⣸⣏⢹⣆⠀⡀⠈⠉⠉⠉⠉⠀⠀⠙⡄
⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⠤⠒⠋⣽⣿⣧⡀⠀⠀⠀⠀⠀⠀⢋⣱⡏⣿⣏⢻⡜⠃⠀⠀⠀⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⢰⡿⠃⠈⠂⠶⠒⣋⡉⠀⠀⠀⠀⠈⢟⡷⠶⠤⢤⣀⣀⣀⣀⠀⣛⢤⠇⠳⠬⡋⠃⠀⠀⣠⣤⣤⣿⣿⣿⡀⠀⠈⠑⠢⠤⣀⣀⣀⣤⣎⡀⠛⠏⢢
⠀⢸⡇⠀⠀⣶⠖⠈⠿⠃⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⢻⣿⠈⡎⠠⠈⠀⠀⠀⠀⠀⠀⣄⢠⡿⡇⠀⠙⠳⢤⡄⠀⠀⠀⠀⠀⠀⠁⠘⠻⣿⣿⣄⣤⣾
⢀⡴⠇⠀⠰⠗⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢦⣧⣰⠀⠀⠀⠀⠀⠀⣄⣼⣾⢿⠇⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⢀⣠⡟⣿⡿⠃⢀
⡏⣠⡶⠾⢂⠃⠀⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣄⡋⢹⠛⢿⠃⠻⠉⠋⣹⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣦⡈⢻⠃⢿⠝⢀⠎
⠀⢉⣷⠀⢫⡁⢄⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠐⠚⠀⠒⠚⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⠦⠀⡌
⣤⣬⣽⣷⣦⣼⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣥⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⠾⠟⠛⠛⠛⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠠⠶⣾⡆⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⣀⡀⠀⠤⠶⠶⠄⠀⠉⠉⠉⠀⠀⠀⢸⡇⠀⣶⣦⠀⠀⢠⣿⡇⠀⠀⣀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣀⣀⠶⠶⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⣀⣼⡏⢳⣄⡸⢏⠀⣀⡶⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠃⠈⠻⠁⣸⡷⠋⣠⣾⣍⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⣶⠀⠘⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠶⠶⠶⢿⠟⠳⣶⣆⣀⠀⠉⠀⠀⢛⣉⡙⠛⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡁⠾⠿⣧⡀⢸⣇⠀⠀⠀⠀⠀⢀⣀⣀⠰⠶⠶⠊⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣼⠏⠉⠶⢀⣠⣴⠎⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠂⣼⡿⠀⠀⢨⣧⣼⣿⣀⡀⠶⠶⠖⠈⠉⠉⠀⠀⠀⠀⠀⣀⣀⣠⣴⡶⣶⠿⠋⠉⠁⠀⠀⠀⠈⠛⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⠀⠀⢰⡟⠉⠉⠉⠉⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⡿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠾⠋⠁⠀⠀⢺⣿⢤⣴⠶⠶⢾⣷⣮⠉⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠺⠿⠇⢀⣠⣀⠀⠀⣆⠁⠀⢀⣠⠶⠂⠀⠀⠹⠿⣦⡀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠉⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⡇⠀⢀⡜⠃⠀⢀⡀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠺⠖⠀⠒⠀⠀⠀⠀⠐⠀⠀⠀⣀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠷⣿⡇⠀⢸⡿⠀⠰⣧⣸⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠶⢦⣤⡀⠀⠀⠀⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠆⢈⣷⠀⠈⢳⣄⠀⠀⠁⠀⠀⠀⠀⣸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠋⠁⠀⠀⠀⠀⠉⠛⣦⡀⠀⠀⠀⢿
⠀⠀⣠⣶⣶⣿⠿⣿⠿⣧⠀⢠⣧⠀⠻⢿⠀⠀⠀⠙⠿⣤⣄⣀⣤⣤⡾⢿⡇⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣠⣄⠀⠘⠁⠀⠀⠀⢀
⠀⣀⡙⠀⢸⡿⠀⣿⢠⣿⡄⠈⠻⠿⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣴⠆⠀⠀⢰⣾⠅⢀⡾⠃⠀⠀⠰⣿⣇⠀⠀⠀⠀⠘⠛⠛⠀⣴⡆⠀⠀⠀⣼
⣿⠿⣿⠀⢸⠃⠀⢹⣾⢻⣷⠀⠀⠀⠈⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠁⠀⠘⠛⠛⠀⠀⠀⣉⠿⣧⡄⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⢻⠷⣤
⢿⣄⠙⣴⣿⠀⣷⣸⣿⣸⡿⠀⠀⠀⠀⠘⢻⣦⠀⠀⠀⠀⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⠙⠛⠓⠶⠿⠛⠋⠀⠀⠀⠀⣠⠀⠀⠈
⠘⣿⣄⡿⢻⣿⠟⠋⢸⣿⠃⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⡉⠁⠈⠁⠀⠙⠛⠙⠛⠛⠂⣤⣤⡄⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣴⡿
⠀⠈⣿⣿⡀⣩⣤⣴⣾⠟⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠈⠃⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⢀⡿⠛⢢⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠻⣦⣄⣴⠇
⠀⠀⢹⣿⠟⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⣆⠀⠀⠁⠀⢠⠆⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⠜⠁⠀⠀⠀⠀⠀⢀⡀⠀⣾⠟⠁
⠀⠀⢸⣿⡄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠛⠉
⠀⠀⠀⣿⡄⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠛⢳⣤⡀⠀⠀⣴⣦⣤⣤⢤⣤⣤⣤⣤⡴⠞⠛⠋
⠀⠀⠀⣿⠟⢸⣿⣶⣤⣤⣶⣦⡄⣠⣶⠟⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠹⠇⠀⠸⣿⡤⠀⠀⠀⢀
⠀⠀⠀⠻⣶⣾⠀⣀⣀⣉⣭⡀⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠴⠀⠀⠀⠀⠀⠙⠛⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⢿⣶⣶⣤⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢷⣦⠀⠀⣠⠶⠊⢙⣇⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠷⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣇⠀⠀⠈⠉⠉⢻⡆⠀⠀⠀⠀⠀⠹⢧⡀⠀⠀⠀⠀⠀⣴⣶⣎⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡸⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣶⣄⡀⠀⠀⢿⠋⠉⢶⣄⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⠷⠤⠤⠀⠀⠀⡀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⢶⣶⣿⠇⠀⠀⠻⣧⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣠⣤⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠻⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣠⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⠋⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡅⠀⠀⠀⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⣀⠈⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⣼⠁⣴⠀⣾⠀⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢿⣷⣿⣀⣿⣄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⠐⠂⠘⢿⡿⠛⠉⠙⠻⣦⠞⠓⣤⣤⢤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⠇⠀⠀⠀⢀⡍⠀⠀⢹⡇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡌⠀⠀⠀⠀⢸⡇⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡇⠀⠀⢸⣇⠀⠀⠀⢀⣾⣇⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠛⠁⠛⠛⢿⡆⠀⣸⡟⠛⠛⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠇⠀⣿⡇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢰⡆⠀⣿⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠿⠦⠄⠀⠀⠀⠀⢸⡇⠀⠸⠃⠀⠀⠀⠀⠀⣾⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡋⠀⠤⣄⠀⠀⠁⠀⠀⠀⢸⡇⢀⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠀⠘⣧⡀⠀⠀⠀⠀⢸⠇⢸⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣿⣾⣿⣃⣀⣠⣄⣤⡼⠂⠈⠛⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⡿⠿⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣾⣶⣶⣶⣶⣿⣷⣆⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠉⠁⠀⠈⠀⠘⢿⣿⣿⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢨⠀⡄⠀⠀⠀⠀⠀⠀⣠⣄⡄⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣶⠁⠛⠻⢤⣁⡒⢞⡛⠋⠙⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣤⣶⣶⣶⡍⢻⠉⣶⣾⣿⣿⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⡿⢳⠜⣾⣿⣿⣿⣿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠘⢿⢟⣋⡴⠂⣤⣼⣟⣿⣿⣯⢴⣿⣿⡿⠀⠀⠀⠀⠴⠄⠱⡄
⠀⠀⠀⠀⢀⣟⣠⡤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢠⣴⡴⢰⣿⣿⣿⣷⣶⣶⢿⣿⡟⠀⠀⠊⢁⣠⣶⣦⣤⣅⠀⢰⠐⡀
⠀⠀⠀⡀⢺⣿⠁⠀⠀⠈⠙⠛⠶⢤⣄⡀⠘⠐⢧⡀⢧⢾⣿⣿⣿⣿⣿⣥⡾⠛⣁⣠⡴⠚⠉⠉⠁⠀⠀⢹⣷⣊⣴⠃
⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠴⣁⠀⠃⠸⠙⡉⣿⣿⣿⡷⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣬⣝⣸⣩⣩⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠈⠙⠓⠶⣤⣀
⠀⠀⠀⠀⢠⠀⠐⣄⠁⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⡉⣠⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣾⣶⠦⣀⡟
⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⠋
⣀⡀⠀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⡀⡀⢀⠀⡀⡀⣀
⣿⣿⠛⢛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠒⠛⠛⠛⠓⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡟⣠⣶⣿⣿⣶⡄⠀⠀⣠⣾⣿⣿⣷⡄⢻⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣹⠃⢹⣿⣿⣿⣿⠇⠀⠀⠹⣿⣿⣿⣿⠇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠻⣦⣀⠀⣿⠀⠀⠻⠿⠟⠁⢀⣾⣷⡀⠈⠛⠟⠋⠀⠀⣼⠀⠀⣀⡾⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠰⣯⣤⣤⣌⡙⠳⢾⣦⣀⡀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⣀⣴⣯⠴⠚⠋⢀⣈⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠛⠳⢦⣌⣿⢟⣷⠀⠀⠛⠋⠘⠛⠀⠀⡞⠿⠿⠋⣠⣴⠶⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠘⣿⢶⢦⢤⣤⣤⡤⡶⣾⣿⠂⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡀⠻⣿⣺⠾⡴⡧⡷⣟⣿⠏⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣤⠶⠟⠋⢉⣳⣄⠈⠙⠛⠳⠓⠛⠉⠀⣠⣞⡉⠛⠶⣤⣄⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⣀⣤⠶⠞⠋⠁⠈⠳⣦⣀⡀⢀⣀⣠⠾⠋⠉⠛⠷⣤⣀⠉⠉⢁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠻⣦⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣒⠭⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣫⠕⡊⠁⠀⠀⠀⣀⣤⡼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⢀⡩⠖⠉⠀⢈⡠⣔⠿⠋⠁⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⡚⠉⢀⡤⠞⠉⠀⢀⣤⡾⠝⠊⠁⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣀⠴⣊⠵⢊⠤⠚⡉⠔⣀⣤⡶⠛⠁⠀⠀⠀⠀⠀⠀⢠⠖⠛⢲⡄⠘⢦
⠀⠀⢠⡊⠁⠈⠀⠀⠀⠀⢀⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡴⡿⠀⠈⠳⣄
⠀⠀⠀⠱⣄⠀⠀⢀⣴⡾⠟⠁⠀⠀⠀⢀⠤⠤⠤⣤⠤⠤⣤⠖⣾⠻⣗⡋⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠈⠳⣖⠕⠉⠀⠀⠀⠀⠀⠀⡏⠐⠒⠲⢼⠆⠀⣯⠀⢿⢀⡿⢀⠵⡒⠒⣦⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⢀⡴⡟⠶⣶⠚⡿⢻⢉⡹⡗⠛⡿⢥⣯⣀⣬⠖⣻⣀⣀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢾⢧⠈⢦⢸⡇⢷⠘⣯⣀⡿⠀⡼⣋⣉⣙⣶⠊⣹⠉⠉⠙⡆⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠘⢮⣛⣶⠟⣡⡎⠛⢞⢯⣠⠶⣷⣶⣖⣻⠟⠀⢿⣒⣖⣺⠇⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⢳⣙⠲⢤⣞⠻⡶⣄⠻⣍⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⡸⡋⠉⠀⣹⠈⠺⠶⠋⠀⢠⠤⠤⣄⠀⠀⠀⠀⢀⣄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠱⣌⣓⣲⠟⠀⠀⠀⠀⢠⣇⣀⣀⣼⠀⠀⠀⣪⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢀⡔⢻⠓⣆⠀⠛⠻⠛⠃⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢸⣄⣸⣠⣾⠀⠀⢀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠈⠛⠛⠛⠁⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦
⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⢸⢿⣄⠀⢀⣴⣷
⠀⠀⠀⠀⠀⠘⣷⡝⢳⣄⠸⡆⢹⣤⡟⠁⣿⣄
⠀⠀⠀⣀⣀⣤⣈⣿⣄⠙⢿⣷⠀⠛⠁⠈⠛⣿⣦⣤⠤⠴⠶⠶⠶⢶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠻⠿⠿⠿⠶⢉⡉⠃⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣤⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠣⣤⡀⡀⠀⠉⠉⠙⠻⢦⣤⡀⢀⡀⠀⢀⣾⠿⠛⢩⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⣈⡉⠹⢧⣀⣀⡀⠀⠀⣙⡿⠛⠛⣶⡞⠁⠀⠀⠺⠷⠶⣶⡶
⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣷⣴⠿⠛⠉⠻⣷⠞⠉⠀⠀⠀⠈⢳⡄⠀⠀⠀⢠⣶⠋
⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢻⡏⠁⠀⠈⠙⠛⠶⣄
⠀⠀⠀⣴⠟⠁⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⠟⠀⢀⣴⠟⠋⠉⠉⠉⠻⣦⡀⠀⠀⣿⠀⢀⣠⣤⡤⠚⠛⠛
⠀⠀⠰⣿⠀⢀⠀⠀⠈⠃⠀⠀⠀⠈⠉⠉⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡿⠁⠀⠀⠀⣀⣀⠀⠈⢿⡀⠀⢹⣾⠛⠁
⠀⠀⠀⣿⠀⠈⠀⠀⠀⣀⣤⡶⠾⠿⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠐⣥⡽⠃⠀⢸⡇⠀⢸⣿⠞⣿⠆
⠀⠀⠀⠟⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⢸⠁⠘⢿⠀⠀⠀⢀⠀⢀⡀
⠀⠀⢸⡇⠀⠀⠀⢰⡏⢠⡶⢦⡄⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⣠⣀⠀⠀⠘⡟⠿⣦⣤⣀⣠⣤⠟⠋⠀⠀⠀⣿⠛⠂⣸⡆⠀⢺⡟⣿⠟⣿⣿⢶⡆
⠀⠀⢈⡇⠀⠀⠀⢸⣧⠈⠳⠾⠁⠀⠀⠀⠀⣰⣿⠃⠀⠀⣸⠷⡀⢸⣏⡤⠀⠀⠙⣦⣀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⠶⠿⠟⠀⢀⣼⡇⢹⡄⢸⡇⢨⣇⣀
⣠⣴⠾⢿⠀⠀⠀⠈⢻⣦⣀⠀⠀⠀⢀⣠⣾⠏⣿⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡟⣷⢸⣧⢸⡇⢸⣿⠛
⣿⠀⠀⠈⣤⠀⠀⠀⠀⠉⠛⠿⠶⠾⠛⠋⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣤⣤⢴⣶⠶⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⢸⣇⣿⣨⣿⠘⣷⣼⠏⣰
⠈⢻⡦⠀⠛⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⣀⠀⢤⣤⠖⠛⠛⠋⠉⣿⠀⠈⢃⠈⠿⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠘⣿⡎⠛⠻⣿⣏⡋⢠⣿
⠀⠸⣧⣤⣴⠟⠛⢷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠖⠈⣻⡅⠀⡇⠀⠀⠀⠀⠀⢹⠀⠀⢸⠀⠀⡇⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣶⡇⢈⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠳⠴⠋⠙⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⣰⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣤⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣦⣤⣤⣀⣀⣤⣴⣦⡀⠀⢀⣴⠞⠛⠉⠉⠉⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣿⡇⠀⠸⠿⠷⠿⠿⠿⠟⠛⠛⠉⠉⠉⠉⠻⣶⣤⣤⣤⣴⣤⣤⣴⢾⡿⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠋⠀⠀⠀⠀⠈⠷⠤⢾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠀⠀⢠⣾⠉⢀⣀⣀⣴⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣤⣤⣶⠿⠿⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠉⠓⠶⢄⡀⠀⣴⡿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⢠⡴⠛⠁⠀⠘⢛⣶⣤⠀⠀⠀⠀⠀⢠⡼⠋⠀⠀⠀⠀⠀⠀⠏⠉⠛⠀⠀⠀⣈⠋⠀⣿
⠀⠀⠀⠀⣴⠏⠀⠀⠀⢠⡴⠋⢹⡟⠀⠀⠀⣠⣶⠋⠁⠀⠀⠀⠀⠀⠀⡀⠀⠈⠛⠛⠉⠉⠛⣇⠀⣿
⠀⠀⠀⣴⠋⠀⠀⠀⣰⡟⠁⠀⢸⣷⣶⡾⠟⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣤⠀⠀⣤⣤⡤⠼⢷⣿
⠀⠀⢠⡟⠀⢀⡀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⢸⣇⠀⢬⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣀⣇⢸⡇
⠀⠀⠸⡏⠀⢸⠃⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠛⣷⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⣿⡀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷⠖⠒⠒⠒⠚⠛⠛⠛⠛⠛⠒⠒⠀⠀⠉⢻⡏
⠀⢀⣠⣾⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣾⠋⠈⡀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢸⡇
⠀⠹⣷⣾⣿⠀⣶⠀⣦⠘⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⣿⠀⣿⡄⣽⣶⡿⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡇⠀⠀⠀⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⠴⠦⣤⡾⠛⠙⠻⢾⡷⠆⠒⠀⣤⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠈⣏⠀⠀⢹⡀⠀⠀⠀⠘⣧⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢸⣧⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠿⠖⠒⠛⢻⡇⠀⢠⡾⠟⠛⠒⠒⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⡇⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣿⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣤⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠉⠉⠀⠀⠀⠀⠀⠀⣿⠀⠈⡇⠀⠀⠀⠘⠉⠀⣀⣀⡈⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠠⡇⠀⠀⠀⠀⠀⣶⠉⠉⠘⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠾⠿⠃⠀⣿⣀⣀⣀⣀⣈⣿⣶⣤⣧⣼⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⠒⠶⠶⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⢀⣼⣷⡦
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠛⢿⣷⣆⠨⠿⣿⡀
⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠎⡼⠀⠀⠀⠀⡰⠋⠀⠀⢀⣀⡀⠀⠙⢿⡀⣘⣿⣇
⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣰⠃⠀⣠⣾⠛⠛⠛⠀⠀⠘⣇⠐⣺⣿⠀⣠⣤⡄
⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠇⢠⡾⠿⢿⣿⣶⣄⠀⠀⠀⢹⡀⠽⣿⠟⢉⣀⢹⡀
⠀⠀⠀⠀⠀⠀⣹⣟⠃⠀⠀⠸⠀⠀⠀⠀⠘⣰⠋⣤⡀⠀⠈⣿⣿⡆⠀⠀⢈⡇⣽⠋⣴⢟⡇⢸⡇
⠀⠀⠀⠀⠀⢸⡏⠀⣈⠉⠓⠲⢷⣦⣀⢀⣴⣧⡀⠋⠛⠁⣰⣿⣿⠇⣠⣶⠟⠱⠁⠸⠑⠻⢣⣿
⠀⠀⠀⠀⠀⠈⠙⣿⢿⡑⠒⢛⣿⠟⣿⠛⠋⠙⠿⣶⣶⠿⣿⢟⠟⠸⠋⠁⠀⢐⣶⣤⠞⠏⢻⡅
⠀⠀⠀⠀⠀⠀⠀⣿⡈⠫⠉⠉⠝⢀⢃⣦⣤⣀⠀⠀⠉⠙⠁⠀⠀⠃⠀⠀⣴⠋⠀⠀⢆⠑⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⣾⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⣾⡾⡇⠀⣿⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡙⠋⢀⠀⠀⠀⠀⠀⢀⠀⣿⠀⠻⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠈⠉⣡⡀⣿⠶⣿⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⣦⡿⠛⠃⠀⠀⢿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣟⣛⣟⣛⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⢀⠀⣦⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⢈⡷⢇⣤⠂⠀⠅⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠀⠀⢀⢀⡀⣘⠀⠛⠁⠀⢀⣤⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢃⡠⣤⡼⠒⠈⠐⣱⠀⠀⠈⠀⣄⡼⠇⠀⢀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣶⢠⡴⢂⡾⠀⠀⠀⠀⠘⣅⡀⠀⠀⠈⠻⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣯⣵⣿⠇⠀⠀⠀⠀⠀⠻⢟⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡟⠄⠀⠐⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⡀⠀⠀⠀⠀⢈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠞⣳⡆⠀⠀⣴⠟⠚⠉⡑⢠⠴⠀⠀⠀⢀⠘⣄⡉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠀⡀⠈⠁⠉⡀⣤⠟⠀⠚⠀⠀⠈⠀⢈⠀⠂⠠⠖⠚⠁⠉⠀⢱
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠦⠬⠕⡀⠀⠀⢸⡟⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠔⠂⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠀⡔⠀⠀⠀⢀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣤⡆⢷⡶⠁⠀⠀⠠⣾⠟⢛⠳⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡔⠁⠀⠀⠀⠀⠀⠈⠀⢈⡚⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡠⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠈⠂⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠉⢒⢤⣀⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡔⠒⠏⠁⠀⠀⡠⠟⠛⠛⠛⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠶⣤⡴⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⢿⣿⣿⡿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⠀⠀⢀⣿⣿⡀⠀⠀⢈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣿⠏⠹⣿⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠹⣿⣿⣶⣶⣿⣿⠏⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡀⠀⠀⠙⠛⠿⠿⠛⠋⠀⠀⢀⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣿⣿⣿⣿⣿⣿⣋⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣿⣿⡿⠟⠛⠉⠁⠈⠉⠛⠻⢿⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⡤⠐⠒⡶⠒⠠⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⠉⠇⠁⠨⠀⠠⠃⢀⣀⣀⠞⠂⢉⡉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠉⡣⡀⠀⣤⠄⠀⢦⠤⠾⠒⡉⠀⠀⣀⠀⠈⠀⠀⠊⠈⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡤⠤⠤⠤⠤⡴⢃⣀⡴⠇⠦⠤⢈⣃⡠⠞⠭⠶⠮⣁⣔⠥⠀⠓⡈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⢢⠸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠘⠀⡇⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⡇⠀⣍⠸⠅⠇⢼⡄⡱
⠀⠀⠀⠀⠀⠀⣠⠖⣈⠉⠉⠙⢛⠂⢀⣽⡿⣛⢍⣉⠉⠙⠻⡿⣷⠒⠁⠀⠀⠂⠉⠀⠉⡆⠀⠁⠀⠀⠘⣸⠁⠠⢌⠁⠀⠤⢄⡠⠃
⠀⠀⠀⠀⠀⣌⢙⣿⣷⡕⣆⠀⠀⢱⣯⣠⣿⣿⣷⣳⠀⠀⠀⢱⠙⠀⣶⡀⠀⣤⠀⡀⡰⠀⠀⠀⠀⠀⢠⠏⠘⠁⠒⠄⣁⡂⢒⡦
⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⡗⠀⠀⠸⣿⣿⡿⣿⡿⡼⠁⠀⢀⡎⠀⠀⠀⠙⡄⠛⣿⣧⡇⠀⡄⠀⠀⣰⢋⡤⠠⣀⡀⣀⠀⠑⠊
⠀⠀⠀⠀⠀⠈⢿⣿⣫⠜⠀⡴⠊⠀⠘⠿⣿⡓⠊⢁⣀⠴⢊⠔⠀⠀⠀⢀⠇⠸⢟⡟⠀⠀⠁⢀⠞⢁⢛⣄⠂⡈⠐⡄⢑
⠀⠀⠀⠀⠀⠀⠀⠈⠁⡰⠚⠀⠀⠀⠀⠐⣄⠉⠉⣠⣤⣤⣄⡀⠀⠀⢠⠋⠑⠒⠉⠀⠀⣤⠚⢁⡤⣌⠐⠠⢈⠓⠎⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣀⣀⣤⣶⣦⡤⢋⡤⠞⠉⡉⠉⠙⣿⣷⠚⠁⠀⠈⠀⠀⠀⠀⢱⠀⢸⠒⠠⢙⠒⠠⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⡁⠠⡤⠬⣉⣠⠴⡊⠁⢠⣳⠀⣿⣄⠀⠀⠀⠀⠀⠄⡴⠂⠡⡎⠁⠰⡁⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⠤⠧⢸⠔⠖⠚⠚⣉⠍⢦⠈⠃⢀⣀⣠⣶⠖⡺⠃⡉⡐⢤⡑⢶⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⡖⠾⡏⠀⠀⠠⢎⠡⢈⠄⠀⠄⣀⣀⣴⠃⠢⡀⠣⢄⠈⣆⠈⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠑⠅⠀⠁⠊⢇⠀⠀⢃⠈⠀⠁⠀⢀⣬⠔⠛⢰⠀⠀⠈⢢⡈⠐⢃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⠀⠠⠀⠬⢧⠀⠈⢧⡀⠀⢰⡁⣰⡆⢠⠃⠀⣠⡒⠄⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠲⠄⢀⠷⠀⢸⠀⢀⡈⠳⢻⠃⢧⠤⠚⢁⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠦⣀⡔⠁⠀⠀⣌⣠⡤⠒⠁⠘⢆⢀⡤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠡⠄⠤⠊⠀⠀⠓⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠂⠄⠀⠀⠀⠀⢀⠀⠀⠀⡄⡄⡤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠃⠉⠀⠀⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⠀⠀⠐⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡤
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡄⠀⠀⠇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⣃⠀⣴⣿⣧⠀⠈⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⣀⣤⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣇⡟⠀⣿⣿⠁⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠦⣀⠈⡗⠀⢹⠋⠀⡀⠀⠀⣄⡀⡀
⠀⣴⣤⣀⣠⣤⣶⡆⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⠀⠀⠀⠈⠀⠀⠠⣀⠃⠘⠃⠀⠀⠀⠀⠹
⠀⠈⣿⣿⣿⣿⣿⣷⡄⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠙⢲⡤⠀⠀⠀⠐
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣆⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣀⣀⡘⠁
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠻⣿⠿
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⠢⣶⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀⣤⣤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⡿⠿⠿⠷⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠙⠛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢿⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠠⡽⠻⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠈⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⠁⢀⡦⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣶⠀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇⠀⠀⢈⡿⠂⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⠐⠀⡀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⡠⢾⣶⣦⡴⠊⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠘⠋⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢋⠿⡀⣾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢻⣀⣿⡿⠀⠀⠀⠀⢀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⡦⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⡇⢸⠟⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⢦⣄⠀⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⡤⢠⣄⠹⠏⣿⣄⣸⣿⣄⡀
⠀⠀⠀⠀⠀⢰⣿⡇⠈⠻⣿⣿⠃⠀⣰⣄⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡖⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⣿⣿⠋⠙⢻⣦
⠀⠀⠀⠀⠀⠈⢻⣇⠸⣧⣼⣿⠀⣸⡿⣿⣿⣿⠿⠿⠷⠶⣾⣤⣶⣤⣠⣾⡟⠻⣶⣄⣠⡀⣤⣤⣶⣿⣿⣿⣶⣤⡄⠀⢰⣿⢁⣤⠾⣾⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣦⢺⣿⣿⡆⢹⡇⣿⠈⠻⢷⣤⣤⣤⣾⣿⣿⢿⣿⡃⠀⣄⠙⡟⣽⣿⣿⡉⠀⢀⣸⣿⠟⣻⡿⡀⢼⡇⢸⡟⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡟⢿⣷⣿⣧⠀⠅⣙⣶⣶⣿⠗⣤⣼⣯⡕⢠⡿⡟⢠⣿⣿⢠⡇⠘⣿⣿⣿⣿⣯⡭⠔⡃⠘⣷⠀⠀⣷⣾⡦⢾⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⣿⣄⠉⠀⠰⠿⠿⠿⠏⢙⡿⢹⡿⡴⠋⠁⣶⣿⣛⣿⣷⡌⠀⠀⢙⣿⠷⣦⣤⣊⣵⣶⡛⠀⠘⠛⣿⣇⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣼⣿⣷⣆⣀⣠⡀⠀⠁⠀⠐⢛⡇⠀⠀⠀⠾⠿⠋⠉⠻⠇⠀⠀⢪⠻⠆⠀⠀⠁⠀⢸⡦⢰⣶⠿⠛⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣏⠋⠀⣿⠃⣠⣴⣄⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⡟⠁⣾⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⢦⣿⢀⣿⣥⣾⣿⣿⠈⣣⣄⡀⠀⠀⣀⣦⠀⢠⣀⣀⣀⡀⠀⠀⣠⠀⣠⠎⢰⢡⡆⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⠘⣿⣿⣿⣧⣿⣿⣯⣸⣿⣶⣟⢛⣿⣧⣸⣿⣏⣹⣿⣿⠤⣿⢾⣟⢀⣻⣸⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢻⣿⣇⣹⡿⣿⣯⣭⣴⣿⣿⣿⣻⣿⡿⣿⢿⣿⣿⣿⠿⣿⣿⣾⣷⣾⣻⣿⠟⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣰⠰⣿⠃⢸⣿⣿⢿⠟⠁⡿⠟⠏⠉⠛⠱⠃⠀⠉⠙⢡⠠⠉⠻⠿⣟⡿⣽⡷⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⡟⠀⢈⣿⣿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠼⡇⠀⢸⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⣿⣇⠃⠀⠈⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⡞⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣼⣿⣿⣆⠀⡀⠀⢠⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣾⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣿⣿⣿⣿⣿⣿⡆⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⣿⣿⣿⣿⣧⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⣿⢻⣿⣿⣿⣿⣿⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣭⣟⣀⣼⣿⣿⣿⣿⡇⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡆⣁⠈⣿⣿⣿⣿⡿⢿⣿⣿⣿⣧⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠉⡇⠻⠀⢿⣿⣿⣿⣿⡈⢿⣿⣿⣿⣿⣿⣶⣶⣿⣶⣦⣄⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢰⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⣼⠋⠀⠀⢸⣿⣿⣿⣿⣷⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⢻⡟⣼⣿⣿⡏⠀⠀⠘⣧⢹⣿⣆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣴⠟⠄⣀⣀⠸⢿⣿⣏⠻⣿⣧⠀⠛⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⣵⣿⠟⡐⡏⣰⡟⠃⡼⠀⠀⠀⠀⣹⣆⢿⣿⡄
⠀⠀⠀⠀⠀⢰⣿⠟⠉⠁⠀⡴⠉⠀⠀⠀⣿⠙⠁⠹⣿⣇⠀⠘⢉⣿⣷⣄⠉⠛⣿⠟⣻⣿⣿⠇⢠⡟⣰⡿⠁⠸⠃⠀⠀⢀⠀⠻⠏⢢⡽⣿⣄
⠀⠀⠀⠀⢠⣿⣿⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⠧⠤⠄⠘⢿⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠪⣤⡀⠀⠀⠀⠘⡿⢷⣄
⠀⠀⠀⢠⣾⠋⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠘⣿⡟⠿⠟⠉⠉⢸⡟⠀⢀⡼⣹⠏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⢀⡈⠻⣷⣆
⠀⢀⣴⠿⠋⠀⣬⣿⣿⠷⣶⣶⣦⣀⣀⡀⣀⣄⠀⠀⠠⣤⣀⠹⣿⣷⡌⠻⡄⠀⠀⠀⢈⣠⣶⣿⣷⢃⡄⠀⠀⠀⢀⣀⣠⣤⣀⢀⣤⣴⣶⣟⣁⣀⠀⠙⣿
⠰⢿⣧⣶⡞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢷⣦⣤⣤⣉⣳⣿⣿⣿⣷⣾⣦⣤⣾⠿⢿⣿⣿⣿⣿⣷⣦⣶⣾⣿⣿⠛⠿⠯⠉⠙⠛⠋⠉⠉⠉⢱⣾⠟
⠀⠀⠈⠙⠿⣷⣶⣆⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⢿⣿⠶⠌⢙⣿⠛⠛⠛⠿⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢷⣄⣾⡏
⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⣿⣷⣦⣤⣄⣀⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⣠⡟⠁⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠶⠶⣦⣤⣤⣤⡀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⠶⠿⠿⢷⣶⣶⡿⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣈⣤⣤⣯⣄⣼⣀⣰⣷⣥⣴⣟⣱⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠴⢶⢤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⢻⢛⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠻⣿⠇⣀⣀⣤⣄⠀⢠⡤⢀⣴⣶⣤⣴⣲⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡈⣿⣾⣿⣿⣿⡿⣇⣼⢷⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⢸⣇⡇⢹⣿⣿⣿⠇⢼⣸⢸⣿⣿⣿⣿⣿⡿⢻⣿⣷⡀⠀⠀⠀⠀⡄⠐⣆
⠀⠀⠀⠀⠀⠀⡠⠠⣄⠀⠚⠣⣬⠉⠹⣄⡀⠸⡌⠀⠈⠛⠋⠥⠚⠁⢹⣿⣿⣿⣟⣉⣩⣤⣾⣿⡟⠀⠀⡠⠔⠊⠙⢁⡨⠦⢤⠄⡄
⠀⠀⠀⠀⠀⠐⠀⣰⣿⠶⠛⠻⠿⣷⣦⣄⡉⠐⠰⢦⡐⠌⠁⠘⠂⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠂⠀⣀⣤⣾⡿⠿⠷⣦⣼⠃⣠
⠀⠀⠀⠀⠀⢣⣰⣿⠇⠀⠀⠀⠀⠀⠈⠙⠛⠷⢦⣄⡈⠙⢳⡄⠀⠐⠙⠿⠿⢿⣿⣿⣿⣿⣿⣯⡶⠶⠛⠉⠀⠀⠀⠀⠀⢿⣷⡶⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⡧⣀⡀⢀⣀⣀⣸⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠷⣾⣬⡭⠽⠾⣿⣽⣿⣿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠟⠋⠁⠀⠀⠀⠀⠉⠛⠻⢦⣤⣉⠑⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠔⠀⠀⣀⣠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣄⣀⠀⠀⠀⣰⠮⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠞⣹⣦⠀⣀⣤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⡞⠓⢆⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⢤⣄⢰⣆⣠⠤⣄⣤⡄⠀⠀⢿⣿⣾⣷⣶⣶⡏⢶⣶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⠶⠶⣤⡀
⢼⡤⠋⠀⠀⠉⠒⠈⣹⣿⡀⠀⣾⢧⡉⠟⠋⠉⠀⠀⠀⠀⠉⠙⢦⣀⠀⠀⠀⠀⢠⡊⠞⠁⠀⠀⠀⢹
⡶⡇⠀⠀⠀⠀⠀⣿⣜⣿⣿⠀⣯⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢰⣿⢀⣀⣉⢙⡁⢸⡇
⣾⡇⢠⣎⢁⣀⡸⣿⣿⣿⡟⢰⣿⡷⠀⠤⡀⠀⠀⢰⠀⠀⠀⢀⣧⣼⡇⠀⠀⠀⣸⣿⣿⣿⡟⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⠘⠿⠇⠘⢿⣿⣿⣾⣿⣷⠀⠈⠁⣠⡄⠀⠀⠀⢰⣿⣻⣿⠁⠀⠀⠀⠸⣿⣿⣁⣚⠉⣥⣿⡁⠀⠀⠀⢀⣀⣤⣤⡤⠀⠀⠀⠀⠀⣾
⣿⣿⣟⣩⣾⡇⠺⣷⢸⠛⠛⢿⣿⣇⣠⣾⣷⣶⢦⣶⣦⣤⡀⠙⡿⣿⣆⣤⠒⠀⠀⣾⣿⣿⣿⣿⣿⣋⡽⠳⣤⡶⠛⠉⠀⠀⠀⠈⠁⢀⠀⢠⣿
⠉⣩⢿⣽⣿⣾⣦⡙⠿⢇⣰⣿⣿⡿⣿⣿⣿⣟⡀⠻⣿⣿⣿⣨⡆⣿⣿⡿⢀⡀⢀⣿⣿⢷⣤⡿⠛⣈⡇⢰⡏⠂⡀⠀⣀⠀⡀⠀⣶⣾⣇⠸⣿
⠀⣷⣿⣽⣿⣿⣿⣷⡆⢾⣿⣿⣿⣱⡍⣿⢡⣾⠻⣆⠈⢻⡿⠟⢀⡉⠙⠛⠿⣷⣾⣿⣿⢦⣾⣴⣾⣿⢤⣾⣯⡎⠁⢀⡹⣿⣥⠂⠈⣿⣿⠀⣿
⣧⣿⣿⣏⣿⣿⣿⣿⠷⣾⣿⣿⣿⣿⣷⣿⣿⣿⢧⣙⣠⣿⣙⣶⣿⡘⣄⠀⠀⠀⠙⠿⣿⣾⣿⠇⠙⣃⣼⣿⡏⣾⣿⣿⣿⠯⣿⣿⣷⣿⡇⢀⣿
⠿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⠟⠋⠀⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠸⡄⠀⠀⠀⠀⠈⠙⢷⣤⣼⣿⣿⠿⣷⢿⡿⣻⡷⣤⡘⣿⣿⣿⠡⢸⣿
⠀⠀⠙⠿⣿⣿⣿⣿⣿⡟⠁⠀⠀⡀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠇⠀⡄⠀⠀⠀⠀⠀⠙⠋⠉⠁⢰⣿⣮⣿⣿⣥⣟⣹⣿⣿⣤⠀⠀⠉
⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⠀⠀⠀⢱⣼⣇⢿⢻⣿⣿⣿⣿⡿⣿⣿⣿⠀⢀⣠⡇⣼⠀⠀⠀⠀⠀⠀⠈⢀⣼⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢂⡀⠀⡐⠀⠀⠹⣿⡁⠀⡀⢀⠀⢸⣿⢿⣸⣃⠻⣿⣿⡿⠁⣿⣿⣿⠇⠀⡇⢿⢹⡗⠀⠀⠀⠀⠀⠀⢸⣿⣧⢻⣿⢿⣿⣿⢿⣿⣿⠁
⡞⣀⡾⠀⠀⠀⠀⣿⡇⠀⠹⣮⡄⣾⠿⢿⣿⣿⣾⡖⠒⢷⣼⣿⣿⣿⣦⣀⣧⠀⢸⣇⡌⠀⠀⠀⢠⠖⠀⣸⣿⡾⣿⣾⣿⠏⣹⣿⣿⠀⡀⣼⣤
⣿⣿⡇⠰⢀⣦⢀⡿⢇⠀⠲⣽⡋⢉⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣭⣿⣇⣠⢸⣿⠇⠀⣠⠀⠀⢰⠆⣿⣿⣧⣽⣿⣿⣿⣿⢿⡏⠀⢡⣿⡇
⣽⣿⣱⠇⠀⢡⠈⣠⣾⢷⣼⣿⣇⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠐⣿⡇⢹⣷⣿⣧⡆⢰⣿⠀⡴⢸⠀⠛⣋⡿⣿⣏⣩⣿⢁⣼⣷⣄⣾⣿⣷
⣿⣿⣿⡄⠀⠎⠀⣀⣉⣀⢈⡉⠻⣇⣀⠈⡏⢹⣿⣿⣩⣿⠛⢻⣇⡀⣿⠀⠈⣿⢸⡏⣷⣾⣟⠀⠇⣸⠁⠀⣿⡇⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣽⣿⠚⢿⠿⣉⡉⢮⣻⣷⡏⠈⠀⡇⠘⠇⠘⢻⣿⠻⣿⣽⣟⢻⣾⣦⣬⣿⣇⣿⣿⣿⣴⣿⡿⠀⠀⡈⢁⣿⡟⣾⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠋⠁⠾⢦⠤⢄⣈⠙⢿⡇⢿⣿⣿⡷⠺⠇⢀⣿⡾⢫⣿⠋⠀⣭⣽⢧⡿⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⣿⣿⣥⣾⠇⠐⠒⠚⢻⣿⣮⣻⡌⢙⡏⢠⡶⠀⠉⢹⠁⢸⣿⣦⣤⣿⣿⠎⢀⡾⠟⠻⣿⣿⠟⣡⢂⣠⣄⣴⣌⡹⠿⠛⣻⣿⣿⣿⠉⢻⣿
⣿⣿⣿⣫⡿⢋⡔⠀⠀⣠⣿⢯⣿⠋⠘⣿⠃⣀⣠⠀⣿⢸⣶⠉⢿⠙⠿⠛⣉⠔⢋⣴⣯⣀⠈⠳⣾⣿⣿⡷⠶⠯⢤⡉⠺⣻⣶⣟⠏⠁⠀⣼⣿
⣿⣿⣿⣿⣿⣿⠁⠀⣾⣿⣿⣿⠏⠀⢀⡹⢳⡛⠋⠀⠛⢸⡿⠀⠈⣶⣞⣻⣥⣶⣯⣭⣾⣿⡟⢦⡉⠻⣿⣯⠶⣦⣀⢀⡀⢳⡾⠋⢀⠀⠀⢿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠈⢿⣿⠏⠀⠀⠈⣡⠑⣿⡆⠀⠙⢸⠿⡗⠸⣟⣿⣿⣿⣿⣻⣟⣯⣭⣤⡀⣣⠀⣿⣿⣷⣦⣙⣷⡝⢾⣧⡴⠃⠀⠀⠈⠻
⣿⣿⣿⣿⣉⠤⠄⠀⠀⠘⣿⠀⠀⠀⢰⡿⠆⢸⠃⠀⢠⢠⣤⣿⣦⡉⢿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣿⣆⣾⣿⣿⣿⡿⢿⣿⣿⣿⠖⣀⣴⣿
⣿⣿⣿⣏⣤⣶⠶⣶⣦⡀⢻⣧⠀⠀⠈⠳⠀⠿⠛⣿⣿⣿⡅⣠⣿⣷⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠻⣿⣷⠻⢛⣉⣤⣎⣡⣿⣇
⢹⣿⣿⣿⣿⡇⠰⢿⣿⣿⣦⣿⠀⠀⠀⢸⣷⣜⣋⣩⢿⣽⣷⡟⡝⢿⡇⣀⠀⢈⠛⢻⡿⡿⠛⠿⣿⣿⣿⣿⣦⠈⢿⣿⣹⣿⣿⣿⣿⣿⣿
⢸⣿⣾⣿⣿⢤⠇⣠⣿⣿⡿⠋⠀⠀⠀⣾⣿⣿⣿⣿⣧⣼⣿⣷⣬⣓⣿⣏⣠⣀⣉⣀⣤⣴⢾⣧⡝⠻⣿⣿⣿⣦⡀⠻⣿⣿⣿⣿⣿⣿⡟⠀⠐
⣾⣿⣿⣿⡟⠀⢰⣿⣿⣤⣄⠀⠀⠀⠀⠹⣿⣮⣛⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⡯⠷⡿⣿⣿⣶⣿⣿⣿⣿⣿⣄⠈⠻⣿⣿⣿⣿⢡⡆
⣿⣿⣿⠟⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⡚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣫⣤⣾⡟⠂⣻⡁⣿⣿⣿⣿⣿⣷⣿⣦⣀⠹⣿⣿⡿⣼⠁
⣿⡿⠃⠀⢰⡏⢸⣿⣇⣀⠀⠀⠀⠀⠈⠁⠛⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⣻⣿⣛⣿⣿⣿⣽⣿⠿⣿⠉⠙⢿⣿⣿⣿⣿⣿⣿⡀⠸⣿⡿⠁
⣿⣾⠁⣰⣿⠇⣼⣿⣿⣿⡀⠀⠀⠀⠈⠀⢠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣿⣿⣿⣿⡏⠀⠹⠀⠀⠀⠻⣿⣿⣿⣿⢹⣧⠀⠈⠁
⣱⡟⣰⣿⣿⡄⢹⣿⣿⣿⡇⠀⠀⢠⡀⠀⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣭⣁⣛⣛⣩⣭⣿⣿⡷⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⡎⣿⠂⠘⡀
⣿⣿⣿⣿⣿⠀⣿⣿⣿⡿⠃⠀⢀⡞⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠄⠀⠀⠀⠀⢸⣿⡷⣿⣇⣿⣧⡈⣷⡘
⣿⣿⣿⣿⡟⢠⣿⡿⠋⠀⠀⠠⢿⠁⠀⡀⣿⣿⣿⣿⣿⣿⣿⡆⠀⢻⣿⣿⣿⣿⡿⢿⣿⠴⠎⠀⠀⠀⠀⠀⠀⠀⣿⣅⣿⣻⣿⣿⣿⠸⢱⡆⣠
⣷⠟⣿⣿⣿⣸⣿⣧⣶⡄⠀⠐⡇⠀⣊⣫⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⡏⠀⠀⠀⠀⣠⢿⡏⢿⣿⣿⡇⣿⣴⠏⣼⣿
⣿⠀⢻⣿⣿⣿⣿⣿⣿⣠⡀⢀⣷⣿⣽⣿⣿⣿⣿⣿⡟⠿⠄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠁⢰⣧⢰⡀⠀⠀⣿⣿⣿⣼⣿⣿⡧⣿⡟⠀⠻⣟
⣿⠀⢸⣿⣿⣿⣿⣿⣏⠿⢡⣾⢻⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠏⠁⢠⣿⣿⠏⠀⠀⠰⣿⣿⣿⣾⣿⣿⡇⣻⣿⣤⣄⠈
⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠆⠰⢠⡄⠀⠠⠀⠀⠀⣀⠀⣤⠀⢀⠀⡀⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠀⠄⠀⠀⡤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠁⠈⠀⠈⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⣰⠋⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠠⠀⠀⡀⠀⡠⡷⠀⠛⢀⡀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⢰⠀⣄⠈⣧⣴⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠴⢯⣄⡀⠀⢀⠐⠀⠀⠀⣨⣧⠛⠀⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠈⠀⠈⣟⣿⠐⠄⠃⠀⠀⠀⢆⠉⠛⠁⣿⡅⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣿⣿⣷⡆⠀⠀⠀⠀⠀⢈⡵⢿⣯⣙⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠐⠟⢻⣿⡿⠀⠀⠀⠀⡐⠀⠀⠀⠀⠉⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣀⠋⡟⠁⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⢃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠐⢶⠶⠇⣸⣸⢦⣶⡩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠤⢄⡀⠀⠻⠔⠊⠉⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢁
⠀⠀⠀⠀⠀⡀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣟⢬⣷⠄⠀⠀⠀⡴⠖⠀⣴⡾⠀⢡⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠉⠀⣀⣀⡀⢰⠁⠀⠀⣿⡀⠀⠀⠛⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⢠⣾⠋⠉⠀⢿⠀⠀⠀⠀⠻⢦⡀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠀⠀⠀⠀⠀⠀⠈⣶⡇⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⢀⡄⠀⠀⠀⠀⠀⣿⢳⡀⠀⢆⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⣼⠇⠀⠀⠀⠀⠀⢸⠋⣇⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠀⢨⣦⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⣼⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠃⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠁


⣿⣿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⡿⠿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢀⣀⡀⠀⣀⡀⢈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣦⣤⠏⢧⣤⡾⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡘⠛⢣⣎⠛⠋⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣦⣀⠙⠿⠿⣋⣤⣾⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠘⠿⢷⣄⣀⣴⠿⠋⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠀⢉⣻⡷⣿⣏⠁⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣷⠶⢈⡁⠰⢾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⣶⣾⠟⠁⠀⠀⠙⢻⣶⠦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡋⠉⢠⣶⣿⣿⣷⣤⠉⢉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⠧⠔⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⣀⣀⠀⠀⠀⢈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢉⠒⠶⢤⣄⠀⢠⣀⣤⡼⠟⠿⣄⡉⠇⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠿⢷⣶⡾⠀⠀⠀⠀⠁⠈⠂⠀⠀⠘⡄⢯⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠒⣄⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⠀⣿⢱⡖⠁⠀⠀⠁⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢀⡏⠘⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⣠⠞⠑⠒⢻⡷⠆⠀⠀⠀⠀⡼⠀⠀⢀⣼⠇⠀⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠛⠋⠀⠃⠀⠁⡀⢀⠈⠀⢢⣤⡾⣧⠀⠈⠉⠁⠀⢀⠐⠀⠀⣾⠀⠁⠀⠀⢀⣤⠆⢤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⠄⠀⠀⠈⠦⠀⠀⢀⠁⡀⠀⠀⠑⣔⡁⠀⣀⡔⣇⠀⠀⠀⠀⠈⠁⠀⡐
⠀⠀⠀⠐⠀⠤⠀⠀⣶⠀⠀⠀⠀⠀⣔⢀⠐⠛⢹⡗⢤⣤⣤⡅⣳⡆⠀⡀⠀⠀⠀⢸⡟⠛⠛⠉⠁⣤⠿⠦⡀⢇⣤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢽⠋⠀⠀⠀⢹⣀⣬⠋⠀⠀⠙⡄⢸⣿⣦⣀⢰⣅⣀⣀⣠⣴⡿⣌⠑⠂
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⢀⡀⠀⠀⠘⢦⣄⣡⣤⡾⠟⠛⠀⠀⠀⠀⠨⣿⠋⠉⠉⢿⣿⣿⣯⣁⠀⠘⡌⠑⢀⡀
⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡙⢲⡖⠚⠛⢿⣿⣿⡇⠄⠀⠑⠼⡁⠀⠀⠢
⠀⠈⠀⠀⢀⢠⠭⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠁⠀⢴⠀⠀⠀⣀⣾⣷⢚⡿⢶⣄⠀⠀⠈⠓⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠐⠁⠀⠘⢿⠀⠲⠆⢀⣀⣀⣠⠑⠦⠤⠜⢰⠛⢙⠛⢻⣹⢀⣠⣤⣤⠛⠠⠄⠙⠦⠤⠐⢁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⡀⠘⠀⠀⠀⠀⠀⠈⢡⠀⠀⢠⡄⠀⠀⢸⠀⠰⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠀⠤⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢠⠞⡇⠀⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢑⡤⠋⠁⠀⠀⡀⠐
⠀⠀⠀⣀⠜⠂⠀⠀⠀⠈⠿⠷⡀⠀⠀⠀⠀⠀⠀⢀⠜⠰⠀⢈⠀⠀⠀⠀⠀⢠⣤⠎⠀⠀⡀⠐⠊⠉⠀⠀⠀⠀⠐⠉⠀⠀⢦⡀
⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠛⠡⠖⠁⠀⢀⡀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣦⣀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣏⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠙⣫⣥⣶⠟⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠶⠚⠉⠁⠀⠀⣿⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣂⠁⠐⣮⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠶⠲⠟⠛⠷⠺⣿⠀⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠐⢁⠀⠀⠆⠀⠘⠂⠸⡌⠃⠀⠉⠉⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⢉⣄⠀⠡⠔⢁⡀⠀⢸⠇⠀⠀⡀⠐⢂⠇⠀⠀⠀⠈⠀⡄⠊⠑⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠰⠁⠀⠀⠀⠘⠃⣀⡤⠜⠛⠈⢲⡏⠈⠉⠀⠀⠀⠂⠀⠓⠀⠀⠀⠀⠙⠣⣀
⠀⠀⠀⠀⠀⠀⢀⠄⠈⢀⡤⠞⠓⢀⠀⠀⠉⣋⣿⠀⠀⠀⠀⠻⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢈⠻⡀
⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⢀⡤⠤⣨⡄⠒⢟⡏⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠄⠀⠀⣀⣀⣀⠀⠈⠂⠀⠀⠀⡄⢯⡀
⠀⠀⠀⠀⢣⣾⠋⠓⢲⣤⠒⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⢁⣀⣀⡀⠈⠀⠀⠁⠀⠀⠈⠌⣧
⠀⠀⠀⢠⣿⠛⢶⡏⠉⠟⠀⠀⢠⠀⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠙⡏
⠀⠀⠈⠐⣍⢆⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠀⠀⠠⢄⠀⡠⠀⡄⠀⠀⡀⠀⠀⣀⣤⠤⠄⠀⠀⠀⠀⠁⠀⢛⠀⣿
⠀⠀⠀⠀⠙⢦⠁⠙⢠⠀⠋⠀⠠⢶⣶⣶⣤⠀⠀⠀⠀⠙⢃⢸⠀⠀⠀⠀⢠⠞⠻⠷⠀⠀⠀⠀⠀⠀⢱⠀⢺⣾⠇
⠀⠀⠀⠀⠀⠈⢇⡠⠰⣇⠀⢀⡴⣿⣿⣿⣿⣿⡴⠤⠀⢀⣼⣀⡱⠶⢶⠊⠁⠀⣠⠀⠀⠈⢂⠐⠤⠵⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡿⢠⡦⠀⢘⡄⠛⠋⢋⣛⣿⣿⣶⣺⠿⠭⠹⣧⣼⠺⣄⠀⢆⡙⢃⠄⠀⣸⠗⡤⠚⠉⠁⣴⠇
⠀⠀⠀⠀⠀⠀⡼⠀⢀⣿⠄⠃⠉⢠⠀⠿⠟⠋⠉⠉⠁⠀⠐⡀⠀⠀⠀⠈⠐⠀⠀⠠⠤⠐⠁⢀⡁⠀⠠⣹⡇
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠉⠁⠀⢸⣏⣀⣀⡠⠊⠀⠀⢰⣶⣆⣶⣦⠀⠀⢀⠢⣄⣐⠒⠒⠒⠋⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⢠⡰⣤⠀⠀⠀⠀⠀⠈⠻⠉⠀⡠⠔⠃⠀⣾⣿⠈⣿⣿⡄⠘⡄⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⣀⣴⣿⡾
⠀⠀⠀⠀⠀⠙⢚⠀⠀⠀⡀⠀⠀⠀⣿⠇⠀⡔⠠⠾⣿⣿⠀⣿⣿⠿⢧⠈⠄⢀⠉⠈⠟⠓⠀⠂⡀⠢⡼⡾⠋
⠀⠀⠀⠀⠀⠀⠈⢆⡐⠈⠛⠀⢀⡴⢿⣁⠸⡀⠀⠀⠉⠛⠀⠙⠒⠀⠈⢠⡿⣨⠆⣀⣴⢶⣦⠀⠘⡀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠚⠉⣀⣤⠄⣈⡐⠿⠓⠊⣁⠀⢉⡈⠃⢙⡒⠛⠁⢀⡜⠋⣁⣀⣿⡄⢸⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⡿⠃⢠⡈⢙⠲⣤⡀⠀⠀⠈⠉⠀⠀⢀⣠⢔⠉⢠⠉⣾⠛⣿⣠⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢸⣧⡏⠈⠗⠀⢁⠈⠴⠛⠙⠱⠎⠻⠚⠀⡙⢃⠀⢻⡀⢈⣤⣿⡿⢼⡿⠄⡄
⠀⠀⠀⠀⠀⠃⠐⠒⠢⣀⣿⠀⣿⣷⣶⢦⡄⣸⣿⠀⢠⣶⠁⠀⣾⡇⠀⠸⣮⣠⣴⣿⣿⣿⣿⠃⢛⡤⢚⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣦⣼⣿⣿⣿⣷⣿⣿⣶⣿⣿⣶⣦⣾⣷⡾⠿⠿⠿⢿⡟⠻⣿⠋⠘⠛⢛⣉⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣽⠁⣼⣿⠿⠛⠋⠉⣿⠁⠉⢹⡟⠛⢫⣥⣤⣤⡀⠀⠀⠑⡄⠈⢳⠀⠀⣙⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠹⣯⠡⡀⢳⡔⠉⣸⡆⠀⠟⠀⠈⢟⣛⣿⣿⠀⠀⠀⠀⠀⢸⣦⣤⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡠⠄⣤⠄⠀⣠⡈⣯⢀⣠⣤⣤⣄⣀⡈⢉⣿⠀⠀⠀⢠⠀⢺⡴⣿⠁⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠑⠀⠃⠀⠸⠙⠠⠁⣁⣺⡉⠀⠀⣹⣏⠙⣿⠀⠀⠀⠘⡆⠀⠻⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣦⣄⠀⠀⠚⡟⠉⠙⢿⣶⣿⡿⠛⢻⣿⣧⠀⠀⠀⠹⣄⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠑⠀⠀⠀⢀⣀⠀⠀⠚⢉⣿⣧⡀⠀⠀⠀⠁⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠞⠛⠉⠉⠛⠻⠟⠋⠉⠻⠿⢻⣶⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⡿⣿⢹⠟⠿⠙⠛⠁⠟⠈⠳⠜⠀⠃⠈⣿⢿⡟⠛⢻⡿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠀⠃⠘⠁⠀⠀⠃⠀⠉⠙⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⡭⠄⢀⣀⡀⠤⠤⠤⡏⢹⠤⠤⠤⢤⣀⣀⡸⢯⠂⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⣀⠤⠒⣤⣏⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠀⣵⡞⠓⠦⣄⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⠋⠀⠀⠀⠈⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⢿⣿⣿⣷⢀⣤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣼⣿⣿⣿⣷⢶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡐⣿⣿⣿⡿⠃⠀⠀⠠⣄⣀⣀⣀⣤⣶⣤⣀⣀⣤⣶⣤⣀⣀⣀⣀⡄⠀⠀⠈⢿⣿⣿⣿⣇⣠⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣻⣟⣿⣿⡿⢀⣴⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣧⡈⣿⣿⣿⣻⣟⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣇⢸⢿⣿⣿⣋⣀⣤⣴⡶⢾⣿⣿⡈⠁⣽⣿⡷⢶⣦⣤⣀⣈⣿⣿⣿⣇⢠⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣫⣤⣽⢿⡈⠘⢿⡛⠛⠛⠳⠿⠖⢿⣿⡿⠀⠀⢿⣿⣿⠳⠿⠟⢛⡿⠟⣹⠇⠈⢸⢻⣏⣙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⢿⡿⠀⠀⠀⢈⠁⠀⠀⠀⠀⠴⠋⣸⡇⠀⠀⢸⣧⠙⠢⠀⠀⠁⠀⠈⠁⠀⡀⠀⢸⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣧⠰⡀⠀⠀⠀⠀⠀⠀⣀⢠⠞⠋⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠂⢀⣠⠄⠀⣸⠇⠀⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢦⡹⣧⠙⣦⡘⠲⠆⢀⠀⠠⠿⣰⣾⣷⣤⣤⣾⣷⣆⡿⣏⠀⠠⠀⢻⡟⠃⠀⢠⣟⡠⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢿⢿⡆⠘⡿⣷⠤⠀⠀⠀⠀⠉⠛⠛⢿⣿⠟⠛⠉⠀⠉⠀⠀⠀⢸⠀⠀⣇⣿⠿⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣴⣿⡀⠁⢿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⣿⡧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠚⠀⢻⠀⠀⠈⣤⣼⣦⣤⣤⣧⣾⣤⡀⠀⢀⢠⠀⠀⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠸⠀⠀⠸⠇⠳⢾⡋⠁⠈⣉⣉⣁⡀⠈⠹⠞⠀⠧⠀⠀⢱⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠰⡀⠀⠀⠀⠀⠀⡤⠖⠛⠛⠛⠀⠀⠀⠀⢀⠀⠀⠀⢃⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣦⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⣶⣀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣿⣦⡰⣄⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⢿⣿⣿⣿⣦⣉⠀⠀⠀⢄⣀⣤⣤⣤⣶⣿⣿⣿⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⠀⠈⠣⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠻⠃⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢿⡝⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣱⠇⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⢷⡈⢻⣏⠻⣿⢿⣿⡏⠘⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⢧⡈⠣⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⢸⡀⠀⠀⠀⠀⢸⠟⢴⣾⣿⣿⡟⠂⣀⣶⣿⣿⡇
⠀⠀⠀⠀⠈⠠⠐⠠⡀⢳⣄⣤⣰⣿⡿⠀⠻⣿⣿⠿⠟⢸⣿⣿⠿⣿⠇⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠰⢆⠀⠘⠄⠉⠋⠉⣜⠁⠀⠀⠀⠀⠀⠀⢿⣻⡿⠐⠀⠀⠀⣀⠠⠆⠀⠀⠁⢦
⠀⠀⠀⠀⠀⠀⠈⠀⠂⠒⠓⣤⡀⠸⣷⣦⣤⣶⣄⠀⠀⠀⢀⣁⢸⣿⣾⠋⢀⣀⣀⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡿⠾⠻⠛⠛⠻⠷⠶⠿⣋⠉⠛⠙⠛⠦
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠉
⠀⠀⠀⠀⠀⠠⠄⠤⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⣺⡭⠉⠉⣑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡴⠋⡡⠴⠶⠶⠶⣤⣄⣈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣼⣅⣀⣠⠖⠁⢈⣩⣭⣝⣛⠷⣦⣕⠦⣄⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣷⣤⣾⣍⣈⣉⠀⠉⠉⠛⢶⣏⡀⠀⠈⠉⠓⢦⡙⢿⡓⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⣛⣩⣭⣥⣤⣤⣬⣽⣻⣿⣦⣄⡠⠘⣿⣷⣖⣚⣲⣦⣄⠳⣄⠻⡄⢹⡆
⠀⠀⠀⠀⠀⠀⢀⡴⢞⡩⠒⠉⠁⠒⠈⢉⣩⣽⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣷⡙⡆⠹⡄⣿
⠀⠀⠀⠀⢀⣴⠟⡕⡡⠀⠀⣀⣀⣤⠶⠛⠉⣠⣶⠿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣧⢹⠀⢻⣿
⠀⠀⠀⢀⡾⡟⢈⠆⠀⠈⠉⠉⠁⠀⠀⠀⣴⠋⠀⢀⡾⠋⣹⡿⠛⢯⡻⣌⠙⣌⢿⣿⣿⡇⣿⢸⠃⢸⡟
⠀⠀⠀⢸⠃⢡⠏⠀⠀⠀⠒⠲⠤⢄⠤⠞⠁⠀⢠⠟⢁⣾⢻⡀⠀⠀⠳⠘⠆⠘⡆⢻⣿⡇⡏⡼⠀⣸
⠀⠀⠀⢸⡄⣼⠀⠀⠀⣀⣀⣀⣀⣀⠀⠠⠀⣐⣭⣶⠟⠻⣄⡑⠤⡀⠀⠀⠐⠀⠀⢸⣿⣾⢱⠁⣰⠃
⠀⠀⠀⠈⣷⣿⠀⣰⡿⠛⠛⠋⠉⠉⠛⠛⠿⠍⠉⠀⠀⠀⠈⠛⣶⣄⠀⠀⠀⠀⢆⣿⣿⡏⠇⣰⠃
⠀⠀⠀⠀⠸⣿⡀⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⠠⣠⣾⣿⡟⠀⡰⠁
⠀⠀⠀⠀⠀⢻⡇⠈⡇⡠⠒⠂⢄⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣀⡀⠀⢻⣿⣿⢿⣿⡿⠀⠀⡇
⠀⠀⢀⡶⢛⣻⣷⡾⠛⣛⣩⣿⣷⣮⣾⣼⣯⠀⢬⣴⣶⣿⣯⣭⣍⡀⢸⡟⠛⠻⣹⡇⠀⣀⣇
⠀⠀⢸⠁⢸⣿⣿⢶⣾⢿⡁⠀⠈⡻⣄⡀⠈⢠⣠⣼⡉⠀⠀⣹⡿⠟⢹⣿⣿⠀⡻⣇⢰⠻⣿⣆
⠀⠀⠈⢧⠈⣿⣿⠀⢌⡛⠛⠛⣛⣿⠟⠁⣀⠈⡈⠻⣟⡃⠋⢁⡀⠀⣼⣿⡏⠀⠀⢻⣿⠀⠀⠉⠁
⠀⠀⢀⢜⡷⠜⠿⠀⠀⠉⠉⠙⠉⠠⢂⠎⣀⡀⠀⠀⠓⠭⢭⣁⣠⡾⣿⡟⠀⠀⠀⠀⠙⢦
⠈⠉⠀⠀⠀⠀⢌⡡⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⡄⠀⠀⠀⠉⠉⠁⠀⣿⡶⠁
⠀⠀⠀⠀⠀⠀⠀⣻⠉⢿⣶⣴⣄⠀⠀⠙⡋⢛⠁⠀⠀⠀⢀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣄⠘⢟⣻⠇⢀⠴⣫⢥⣥⣍⣷⣿⡷⠌⠀⠀⣸⠯
⠀⠀⠀⠀⠀⠀⣠⠟⠈⠻⣿⠉⠀⠙⢿⠉⠛⣛⡉⣹⠏⠁⠀⠀⡔⢿⡀
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢹⣷⡀⠀⠈⠓⠶⠶⠞⠁⠀⠀⢀⡾⠀⠈⠻⠆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⠃⠀⠀⢀⣰⢿⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⠙⠷⠤⠤⢤⡤⠴⠛⠁⠸⡇⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣿⡏⠀⠀⠀⢀⣰⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣼⡿⠃⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢻⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠈⣉⣿⠗⠀⢀⡀
⠀⠀⠀⢀⡠⠖⢻⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⠀⠀⠀⠀⣄⠙⠦⠀⡀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢆⠀⠙⠳⣄⣀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠀⣀⡴⠊⢿⣆⠈⣆⠁
⠀⠀⠀⠀⠀⠀⠙⠠⠀⠉⠘⠉⠢⠔⠠⠴⠴⠦⠔⠒⠉⠀⠀⠀⠈⠻⣧⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠯⢭⣭⣭⣭⣍⣉⣉⣉⣉⣉⣭⣭⣿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⠙⠳⠦⢤⣤⣄⣀⣀⣀⣀⣉⣁⣀⣠⣤⡤⠾⠛⢳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢻⡶⢶⣶⣦⣤⣤⣭⣭⣭⣥⣤⣤⣴⣶⡶⢿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣼⣧⣸⣇⣸⣇⣸⣇⣸⣇⣸⣇⣸⣇⣼⣧⣾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢾⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⠆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠰⠷⢶⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⠿⣿⡇⠀⠀⠀⠀⢀⣴⠟⡽⠊⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⣶⣦⣤⣄⡀⠀⠀⢸⣿⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⣿⣇⠀⠀⠀⣰⡿⠛⠳⣄⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣸⠒⠲⠍⣿⠀⠀⢸⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠄⠀⣿⡀⠀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣶⣦⣶⠟⠀⠀⠀⠙⠿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡿⠟⠋⠀⠀⠈⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⢿⡟⠛⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠈⠉⢹⡇⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠚⠛⠛⠉⠉⠉⠛⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⢤⡀⠀⢠⠎⠀⠈⠓⠦⠎⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠈⠀⠀⢀⠤⣄⠀⠀⡰⠃⡀⡀⠀⠉⣲⠃⠀⣀⡔⠰⢄⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣴⠒⢄⡀⢠⠏⠀⠈⠑⠖⠁⡈⠀⠘⡄⠈⠁⠀⢘⣁⡀⠠⠞⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠜⠀⡄⠀⠈⡁⡔⠢⡀⠀⠠⢀⠃⠀⠀⠱⡜⠀⠀⡤⠀⠦⠔⠁⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠈⢛⠁⠀⢃⡇⠀⢡⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⡤⠤⣄⡀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⡄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡟⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠤⠞⠁⠀⠀⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣇⠀⠀⠀⠀⣸⠃⠀⢀⣠⠤⢠⣤⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡤⠥⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⡀⠀⠀⠀⠈⠓⠒⠒⢩⠏⠀⢠⠛⣸⡈⠓⠉⣉⠇⠀⠀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣡⠵⠶⣄⠀⠀⠀⠀⠐⠃⠀⠀⠸⣉⠏⠚⠋⠁⠀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⢠⢦⡀⠀⠀⠀⠀⣀⡤⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣦⠀⠀⠉⠀⢀⡠⠤⠾⠿⠶⢏⡉⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣉⣐⡴⠞⠁⠀⠀⠀⠀⠀⠀⢈⡶⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠀⠀⠀⠐⠒⣿⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠖⣤⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢇⣠⠔⠀⠀⠀⢀⡸⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣀⡤⠖⠊⠁⢀⡀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠎⠁⠀⢀⡤⠞⠉⠉⠙⠦⣄⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣶⠋⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣤⣄⡽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠔⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠂⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡇⠀⠀⠉⠙⠳⢶⣤⡀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⡄⠀⠀⠀⠀⠀⠉⠻⣇⡒⠉⡁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠐⡿
⣰⡂⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢨⡄⠿⣦⠀⠀⠀⠀⣀⠔⠀⠙⠦⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠠
⣏⠀⠀⠀⠈⢆⠀⠋⠉⠀⠁⠀⠰⠇⠀⠈⠳⡀⠀⠊⠀⠀⠀⠀⠀⠈⠣⡠⣀⠀⠀⡆⠀⣀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄
⠘⣆⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡡⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⡹⢾⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣀⣤⠖⠉⠁⠀⠀⠀⡀
⠀⠸⣧⣼⡖⢺⣿⣦⠀⠀⠀⠀⢰⣷⣿⢀⣰⢦⠟⢜⣳⣤⣤⣵⣦⣴⣦⣾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⢛⡁⠀⠀⠀⠀⣀⣴⠇
⠀⠀⠹⣿⣿⡄⢹⣿⣆⠀⢀⠠⠿⠁⠃⠀⠀⠀⠀⠀⠀⠈⡟⠛⠿⠿⠿⡟⣤⠀⠀⠀⠀⠀⠀⢰⠉⣋⣡⣴⣾⣿⣯⣀⡤⠶⠛⠉
⠀⠀⠀⠹⣷⣿⡄⢻⣿⡄⠀⠈⠐⢤⣀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⢡⡇⠀⠀⠀⠀⠀⠈⢷⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠹⣿⣿⡏⠘⠁⡤⠄⠀⠀⠉⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⢀⠑⣼⣿⣷⣀⠀⠀⠒⠒⠂⠚⠉⠁⠀⠀⠠⢾⡿⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢦⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠠⣄⠙⠳⣶⣿⡆⠙⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠛⠐⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠷⢤⣤⣤⠄⠠⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢢⠈⠛⢿⣿⣦⡈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠙⠻⠿⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡀
⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠘⣝⠶⣴⡧⠒⢲⢋⣽⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣶⡄⠀⠀⠀⠀⡴⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣿⠃⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣠⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠲⠚⣋⣍⣽⡷⣄⡀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠀⣀⣀⠀⣐⣺⡥⠚⠼⠛⠷⠛⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⣏⠀⣴⠟⠁⠉⠉⠁⡾⠛⠃⠀⠀⠿⠷⠀⠀⠹⣆⠀⠀⠀⠀⢰⠟⠻⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣷⠹⡾⣇⠀⠀⠀⠀⣸⢳⡆⠀⠀⡰⠆⠀⠀⠀⠀⢸⡄⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠘⢷⡻⠾⠀⠀⠀⡴⠃⠘⠋⠀⠀⠘⠇⠀⢰⣿⣿⣿⣇
⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⢀⣼⠟⠂⠀⠀⢰⠁⠀⠺⠤⣤⠈⢦⠀⠀⠀⠙⢛⡿⢉⠇
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⢸⠀⣷⠀⢤⣄⠀⠈⠃⠀⠀⠀⣬⡴⠋
⠀⠀⣾⠁⠀⠀⢀⣀⣠⠶⠿⣿⠄⠀⠀⠀⠀⠀⠀⠘⣆⠘⣆⠀⠈⠉⠀⠀⠀⠀⠀⣼
⠀⠀⠉⢷⡀⠀⠉⠉⠀⢠⡾⠛⠛⠛⠿⣦⣤⠶⠶⠶⣬⣷⣌⠳⠶⠶⠀⠀⠀⣠⠾⣇
⠀⠀⠀⠀⠻⣦⡀⠀⠀⠋⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⢀⣽⢿⣷⡶⠤⠖⠋⠀⣠⢾⣿⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣴⠟⢁⡿⠈⣧⠀⣠⡴⠛⢡⠏⢸⡇⠈⠙⠛⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⣀⢀⡆⠀⢀⡽⠿⠋⣼⠇⣠⣾⣻⣇⠀⢠⠟⣀⣸⣧⠀⠀⠀⢀⡀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢠⡇⣼⠁⠀⢸⡇⠀⣸⡟⠋⢀⣿⡿⠛⢷⡏⠈⠻⣏⡉⠀⠀⠀⣾⠁⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣳⠃⠀⠀⢸⡇⢀⡿⠁⢠⣿⣿⠇⢠⡟⠀⠀⣠⠏⠀⠀⠀⢸⡏⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⢸⡇⢸⠃⠀⣿⣿⣿⢀⣿⠁⣠⠞⠁⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠇⠈⠀⠸⠟⠻⣿⣼⠃⠐⠁⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠛⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣴⣶⣴⣦⠤⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠈⠉⠛⠉⡁⠀⠀⠀⠀⠀⠀⠀⢀⡀⢘⢳⣿⣿⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣉⣷⣶⣶⣿⣿⣾⡄⣠⣿⣷⣤⣶⣾⣿⣿⣿⣦⣄⠠⣫⣻⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠻⢿⣿⡿⠃⢻⣿⣿⣿⠛⠋⠛⠻⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⢠⣰⣾⠟⠁⠀⣨⡝⠟⣉⠓⠀⠀⢠⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿⢯⣾⠋⣤⣤⡐⢿⣷⣤⠉⠈⠓⠷⣾⠟⣽⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⡀⠀⠁⣸⣿⣿⣿⣶⣌⣿⠇⠐⠀⠀⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⠀⠻⣿⠿⢟⣿⠿⠃⠀⠀⠀⠀⠸⠁⣈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠹⠟⠋⠁⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⡋⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⢴⣶⣆⣴⣶⣄⣠⣤⣤⠀⠀⠀⣴⣿⡿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠸⡟⠃⢹⢻⣿⣿⣿⣿⡿⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⢘⡇⠄⠊⠟⣿⣆⠜⣉⡥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠹⢋⡙⠋⠛⠛⢻⣿⡇⠀⠀⠸⠁⢠⡄⢸⡿⠿⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⣷⣴⣾⣄⣼⡿⣇⣠⢀⡄⠃⢸⣵⡎⠁⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠋⠀⠉⠉⠉⠿⣿⠻⠆⢾⣷⣷⣿⡧⡚⠁⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣶⣦⣼⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⢂⣀⠠⢿⠓⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠀⠈⠉⡵⢿⢏⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠙⢛⣉⠉⠉⠁⠀⠀⠀⢀⣀⣠⠤⠤⢽⣧⣎⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⡿⠏⢤⣶⣶⠢⠤⠐⠊⠉⠀⠀⠀⠠⠀⠶⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⣉⠉⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠸⣿⣦⡀⠀⡟⣧⠀⢠⡿⡇
⠀⠀⠀⠀⠀⠻⣎⠻⣦⣡⠈⣾⠇⠸⣿⣆⠀⠀⠀⣀⣀⣀⣀⡀
⠀⢀⣴⣶⣦⡶⠿⠷⠈⠙⠀⠀⢀⡤⠞⠛⠋⠉⠉⠉⠉⠉⣿⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⡀
⠀⠀⠀⠀⠀⠀⠙⠳⣄⣀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⠙⠉⠛⠛⠓⠶⣄⣀⠀⠀⠀⠀⣠⣿⠷⢻⡿
⠀⠀⠀⠀⠀⠀⣠⡞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢁⣈⡙⠛⣦⣀⣀⠀⠈⢙⡿⠛⢷⣴⠏⠀⠀⢿⡦⢤⣶
⠀⠀⠀⠀⠀⣶⠋⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣷⡴⠿⠛⠉⠷⠖⠉⠀⠀⠀⠙⢆⠀⠀⠀⣴⡞⠁
⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠈⣟⠁⠀⠀⠉⠙⢶⡄
⠀⠀⣰⠟⠀⠀⠀⢸⠀⠀⠀⠀⡠⠤⢀⣀⠀⠠⠆⠀⠀⠀⠀⠰⠋⢀⣴⠟⠁⠀⠀⠀⠙⢦⡀⠀⢙⣃⣤⠶⠚⠛⠋⠁
⠀⠀⢻⡄⠰⠀⠀⠀⠀⣀⣀⣤⣤⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⣢⠀⠸⡇⠀⢸⣏⣤⣤
⠀⠀⣸⠁⠀⠀⠀⣰⠞⠉⠉⠀⠈⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⠀⠀⠀⠉⠁⠀⣰⠇⠀⠸⡏⠰⡎
⠀⠀⣿⠀⠀⠀⢸⡇⢀⠦⣀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣀⠀⠀⢸⡿⢦⣄⣀⣀⣠⠶⠋⠀⠀⣼⠷⠀⣭⠀⢠⡾⣶⠿⣷⣶⣤
⠀⠀⣿⠀⠀⠀⢸⣇⠙⠒⠋⠀⠀⠀⢀⣼⡇⠀⠀⣴⢆⠀⢹⡆⠀⠀⠱⣄⡉⠉⠀⠀⠀⠀⠀⠀⣸⡶⠾⠋⠀⣰⡇⢸⠂⢻⡄⣿⢀
⣠⠾⠛⡆⠀⠀⠀⢻⣦⣀⣀⣀⣀⣤⠞⢱⠄⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢘⡿⠀⠀⠀⢨⣽⠿⣸⡆⢸⡇⣿⠟
⢿⣄⠀⣠⡄⠀⠀⠀⠈⠉⠉⠉⠉⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣤⣄⢤⡶⠶⠶⠶⣶⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⣿⠸⣧⣷⣸⣧⡏⢰
⠀⢹⡇⠁⣸⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣤⠆⢻⡟⠁⠀⠀⠈⠇⠀⢠⡄⠙⡀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠹⣧⣀⣙⠟⢤⣼⡿
⠀⠈⠛⠛⠁⠀⠙⢷⣤⠀⠀⠀⠀⠀⠚⠋⢿⣀⠟⠦⠜⠁⠀⠀⠀⠀⢠⠀⠈⠃⠀⢃⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⢷⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠷⣤⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⣀⣀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⣤⣤⣤⣤⣤⣤⣤⡄⠀⢀⣴⠞⠋⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢘⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⢾⡇⠀⠸⡟⠚⠛⠛⠛⠛⠛⠛⠛⠉⠛⣷⡤⠴⠶⣶⡴⠶⢿⡛⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠛⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⢀⣶⣆⣀⣤⣤⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢄⣀⡀⠀⣤⡶⠶⠟⠛⠃
⠀⠀⠀⠀⢀⣴⠞⠉⠀⠘⣤⣤⠀⠀⠀⠀⠀⣴⠶⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⢉⠁⠀⣿
⠀⠀⠀⢠⡞⠁⠀⠀⣤⠞⠉⣿⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠸⠀⠀⠉⠈⠁⠙⢆⢸⡏
⠀⠀⢠⡟⠀⠀⠀⣾⠃⠀⠀⠿⠶⠟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠳⠤⠦⠀⠠⠶⠶⠈⢿⡇
⠀⠀⣾⠁⢀⠔⣿⡏⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⢻⡇
⠀⠀⣿⠀⠘⠠⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠿⢸⡇
⠀⠀⢹⡀⠘⠲⢿⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠈⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣶⠦⠤⠤⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⢴⡏
⠀⢀⣴⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠠⣟⠁⣀⡀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⡇
⠀⠙⠷⣿⠀⣿⠀⣦⢈⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢿⣤⠿⣦⡿⠟⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⣤⣤⠴⢦⣴⠞⠛⠿⢶⡶⠶⠀⢄⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⢩⠀⠈⣅⠀⠀⠀⢸⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣿⠀⠀⠀⠈⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣠⣿⣦⠀⠀⣠⣴⣄⣠⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⣿⠀⠀⡟⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠇⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⣆⠀⠠⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⠀⠀⠀⠀⠀⢹⡄⠀⣟⠀⠀⠀⢀⡀⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠉⠀⠀⠀⠀⠀⢰⡇⠀⣏⠀⠀⠘⠃⠀⡠⢄⣈⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡁⠀⠀⠀⢀⣀⣀⣀⣀⣸⡇⠀⣿⠀⠀⠀⠀⣾⡀⢠⠈⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⣿⣶⣶⣶⣶⣿⣿⣿⠿⠛⠁
⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠒⠒⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⣤⣀⣀
⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⠶⠋⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣦⡀
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣫⠞⠋⠹⣆
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠸⣆
⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣫⠞⠁⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⣀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⢃⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣠⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠛⠛⠋⠉⠁

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

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

⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠐⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠄⠄⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠄⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠲⢀⠀⠀⠈⠡⠀⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠂⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠒⠆⠀⠀⠀⠀⠀⠐⣿⠙⠉⠐⠖⠄
⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠄⠀⠀⢠⡟⠠⣄⠀⣀⡀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠈⠐⢁⠀⠀⠀⠀⠀⠉⡀⠀⠀⠐⠀⢀⡀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠈⡴⠲⠖⠤⠜⠚⠆⠀⠟⠀⠹⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢑⠀⠀⠤⠓⠆⠀⠀⠀⠀⠉⡁⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠈⠉⠏⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣮⠀⠀⠀⠀⠘⡆⠀⠀⠀⠈⠁⠀⠀⠀⢣
⠀⠀⠀⠀⠰⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣏⡇⠀⠀⠀⠀⢻⠀⠀⢴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣆⠀⠀⠀⠀⠀⢰⢠⠆⡀⠀⠀⠀⠀⢥⣠⠀⣼⣿⠃⠀⠀⠀⠀⠈⠀⠀⢸⡆⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠘⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⡼⠂⠀⠉⠀⠀⠈
⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⠛⠁
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣤⣤⡄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⡄⣶⢸⣇⠀⠀⠐⠀⣀⠀⠻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣠⣄⣀⠀⡤⠄⠀⠀⡀⠀⠀⠀⠠⣠⠁⠀⣿⡇⣿⢸⡟⠀⠀⠀⠰⠈⠅⠀⠀⠙⢷
⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⣄⠀⠂⠀⠀⠀⡑⠀⠀⣀⣰⠆⠀⠀⠀⠀⠋⣸⠁⠀⠀⠀⠀⠀⠁⢀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠘⠂⠃⠒⠘⠀⠘⠀⠀⠂⠀⠀⣀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⡄⠴⠀⠂⠅⠀⠀⠀⠀⠀⠀⠀⠻⣀⡤⡄⣿⠝⠱⠚⠀⢠⡾⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠒⠚⠀⠀⠒⠘⠊⠀⠀⢈⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⠜⢇⡀⢀⡀⠸⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣿⡿⣛⢋⡘⠃⠀⠻⠘⠁⠉⢯
⠀⠀⠀⠀⠀⠀⠀⢤⣶⣆⣀⣀⠀⠀⣰⡄⠰⠯⢄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⡀⣄⢀⡉⣉⠆⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣈⠀⡄⢨⢁⡀⣀⡇
⠀⠀⠀⠀⠐⠀⢻⡅⠀⢻⡀⣀⡀⢀⣀⠀⠀⢹⠁⠁⠀⠀⠀⠀⠀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠴⠀⠄
⠀⠁⠀⢠⠀⠀⢍⣿⣶⣶⣽⣿⣿⣿⣿⣤⣶⣎⡴⠂⠀⠀⠀⠀⠀⠉⠀⠀⡶⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄
⠀⠀⠀⠰⠀⠀⣾⣿⣿⣿⣿⣿⡿⠟⠛⠉⢷⡉⢥⡄⠀⠀⠐⠛⠛⠃⠆⠀⠀⠀⠀⠘⠁⠠⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⡄⠀⠰⣏⣿⣿⣿⣿⣿⣧⣀⡀⠈⠐⢿⣤⡄⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡙⢠⡀⠤
⠀⠀⠀⢠⠀⢀⠀⣩⣿⣿⣿⣿⣿⡭⠅⠀⠐⠶⣿⡰⠄⠀⠀⠀⠀⠀⠓⠀⠈⠛⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⠀⡄
⠀⠀⠀⡄⠀⠈⠈⡿⠹⣿⡟⣻⣿⣷⣄⠀⡄⠁⠀⠑⠄⠀⠀⠀⠀⠰⠄⠄⠠⠠⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣋⣴⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡰⢾⡀⠀⠀⠀⠀⠀⡸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⢀⠂⢰⡄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣴⣶⣶⣶⣶⣾⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣷⣽⡝⢏⢻⣿⣿⡭⠿⠿⠿⠿⣫⣋⣉⣁⣀⣀⠈⠉⠻⣆
⠀⠀⣿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣾⣿⣿⣿⠠⠗⠒⠦⠤⢤⣤⣍⡁⠘⠋⠱⣄⠀⠹⣆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣶⣤⣀⡀⠀⠈⠐⠙⢦⡀⠀⠈⠳⡀⠹⣆
⠀⠀⠀⢉⡟⣹⣿⣿⠋⠉⠉⠙⢷⣜⣟⢿⣿⣿⣿⡿⣗⠺⠉⠙⠛⠿⣿⣿⣶⡄⠀⠀⠙⢦⡀⠀⠘⢦⣹⣆
⠀⠀⠀⣸⠃⠈⠿⣿⠀⠀⠀⠀⠀⠙⠿⠦⠉⠛⠋⠁⠈⠛⠦⣤⣀⠀⠠⠉⠻⣿⣦⠀⠀⠀⠙⠆⠀⠀⠉⢻⣆
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⢲⡀⠀⢻⡄
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣧⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠃⠰⡌⢿⡄
⠀⢸⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⣷
⠀⠸⣏⠉⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢹⣧
⠀⠀⣿⢠⣤⣈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⣴⠏⡄⠉⢿⣿⠶⣄⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⣾⣏⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⢸⡏⠀⠙⠲⠄⠀⢠⡟⠀⠀⡴⠂⢠⣭⣭⣟⣛⣛⠻⠿⢿⣷⣦⡀⠀⠀⠹⣯⡿⢿⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⣿⡇⠀⠀⠀⠀⢀⡿⠁⠀⠀⠁⠀⢠⡘⠂⠈⢿⣿⡿⠶⣄⠈⠉⠁⠀⠀⠀⠻⠏⠾⢁⣼⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⢸⡇⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠛⠶⠆⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠠⣿⠡⣤⡙⢿⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣆
⢸⣿⠀⠀⣴⠟⠁⠀⠹⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢀⡃⢀⡙⢷⣄⡀⠀⠀⠀⣠⡾⠁⢀⣿⣆
⢸⣿⠀⢀⡙⣷⡀⠀⣠⣄⠀⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠅⣠⠟⣵⠋⠀⠀⠻⣷⡀⠀⣴⠟⠀⢀⣾⠋⢻⣆
⢸⣿⠀⠸⣇⠀⠀⠀⠀⠉⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠶⠛⢁⡾⠁⠀⠀⠀⠀⠙⣧⡾⠋⠀⣰⡟⠁⠀⠀⢹⣆
⢸⣿⠀⠀⠘⣿⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠳⢄⡀⢸⡇⠀⠀⠀⠀⠀⢠⣿⣧⣤⣤⡶⠋⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⣾⠋⠀⠀⠀⠀⠀⢻
⠘⣿⠀⠀⠀⠹⣦⣄⣈⡉⠙⠛⣛⣿⡿⠂⢀⡟⠀⣾⠃⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁
⠀⢻⡇⠀⠀⣄⡀⠉⠉⠙⠛⠋⠉⠁⠀⠀⠈⠀⠘⠃⠀⠀⠀⢀⣴⡿⠁
⠀⠘⣿⠀⠀⠀⠉⠛⠲⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠁
⠀⠀⠀⠻⣦⣄⡀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣀⣀⢀⡀⡀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣤⠄⡄⣠⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠉⠋⠙⠛⠁⠋⠁⠀⠀⠀⠀⠀⠠⣿⡻⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠽⠇⠹⠏⠸⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣶⣶⣶⣶⡄⣠⣶⣤⣶⣾⣷⣶⣦⡄⠀⢹⣿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⣻⡟⠁⠙⢿⠿⠍⠀⠉⠻⣿⣿⣶⣏⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣇⢸⠿⢘⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡍⠄⣸⢃⣀⡀⢼⣀⠐⠀⠀⠀⠛⢻⣿⣿⣿⣿⣽⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⠀⣼⣿⣿⣦⣹⠇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣟⠻⣿⠃
⠀⠀⠀⠀⢰⣶⡆⣶⣰⣦⡖⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡷⠀⠹⣿⡭⠟⠉⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠿⣿⡎
⠀⠀⠀⠀⠈⠃⠀⠃⠙⠉⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⣀⣀⣠⡀⢀⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣴⠟⠁
⠀⠀⠀⠀⠀⣠⣄⢠⣀⡄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣿⣿⣿⣿⣿⣯⠄⠀⠀⠛⠁⠉⣿⠿⣿⡿⠫
⠀⠀⠀⠀⠀⠿⠼⠠⠿⠆⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⠿⠿⠿⠿⣿⡿⡂⠀⠀⡄⠀⠀⣹⣇⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣄⣀⣄⣀⠸⠆⡀⠀⠀⢘⣠⠉⢀
⠀⠀⠀⠀⠀⢸⣲⡿⠇⣟⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠏⠁⠉⠈⠉⢿⠊⢀⣿⣠⣿⠁⠀⢨
⠀⠀⠀⠀⠀⠚⠁⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣾⣶⣶⣿⣿⣿⣿⡿⠟⠠⠀⠈⢁⡄
⠀⠀⠀⠀⠀⣴⣦⢰⢦⡄⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠛⠛⠛⠋⠀⠀⠀⡀⣀⠀⣿⣥⠃
⠀⠀⠀⠀⠀⠛⠁⠘⠞⠃⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣠⠟⢻⡿⠐⠢⢀⣴⠀⠀⠀⠀⣤⠀⠰⠆⠐⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡁⢀⣤⣾⠛⠋⢰⠀⠀⠀⠀⠀⠀⠠⠄⠀⠉⠉⣀⡭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⡄⠀⠾⠿⠿⠿⠿⡀⢀⣼⠖⠋⠀⠀⡀⢀⣤⠶⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠠⠀⠀⠂⠂⠠⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠠⡀⠤⣖⠐⠀⠀⠀⠀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠀⠀⠁⢀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢄
⠠⠠⠉⠀⠀⠐⠄⠀⠀⠂⠀⠀⠀⠀⠂⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠂⠀⠙⠁
⠀⠞⠄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠈⠂⠀⠀⠀⠀⠠⠀⠄⠀⠀⡀⠈⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠄⠀⡀⠠⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈⠁⢀⠀⠁⠀⠀⠀⢄
⠀⠀⠀⠀⠂⡄⠀⢀⡠⠁⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠘
⠀⠀⠀⠠⠀⠀⠄⡀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠠⠄⠄⠀⡀⢠⠀⠀⠐⠤⠤
⠀⠀⠀⠀⠀⠈⠐⠂⠠⠤⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⡠⠂⠀⠀⠈
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣯⣝⡛⠻⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣍⠒⢭⣛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡈⠙⠮⣝⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠈⠛⢮⣻⣷⢶⣶⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣴⠟⡡⠀⠙⢷⣌⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠡⠚⠁⠀⠀⢀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⣦⡀⠀⢀⠔⣡⣾⡟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⣴⠟⠈⢻⣦⣠⣾⣏⠀⠹⣮⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣣⠞⠁⣀⣴⠟⠉⠉⠀⠙⢷⣄⠈⢷⡹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠻⣧⡀⢳⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠱⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡁⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡌⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣫⠞⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣴⡾⣫⠞⠁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣌⠁⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣼⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠉⠀⠉⠉⠁⠀⠙⠻⠶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣰⣿⣶⣶⣆⣴⣧⣶⣴⣶⣤⡄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠛⠛⠉⠉⠉⠍⠀⠀⠀⠐⣿⣤⡀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠘⡇⠀⠀⢠⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⣀⠤⠀⠀⠀⠤⢄⡠⡺⠀⢀⡃⠀⡀⢀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠂⠐⠀⠐⠀⠠⢀⣤⣄⡀⠱⠀⢿⣋⠀⠡⢊⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣦⣄⣤⣿⣤⣤⣶⠟⠛⣿⣷⡄⡀⠘⡇⠀⣰⢏⣴⣶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡁⣸⡏⢹⢻⠻⣦⣤⣿⣿⢃⠄⠀⠈⠙⠛⣸⣿⢏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣷⣌⠈⡀⠀⣍⡉⢡⣊⠀⢀⣶⣿⣦⣙⣉⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠛⣠⠈⢦⢠⣾⣽⣿⣶⡿⠏⣿⣿⠟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣁⣀⣀⢋⣜⣿⠿⣿⣿⡟⠁⠘⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣯⡯⠧⣾⣿⣿⡂⠘⡈⢠⠀⣼⢣⣸⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠭⢭⠋⡙⢩⠘⠁⢀⣄⣴⣿⠃⠈⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠈⠈⠀⣴⣿⣿⣿⡟⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣀⠀⠀⢐⢾⣿⣿⣿⡟⠡⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠈⣿⣿⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠛⠓⠾⢟⠆⠈⢣⡔⢹⠇⠀⠀⠀⠀⠀⢀⣀⠴⠞⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡜⡰⠋⡱⠒⠀⠀⠀⠀⠃⣼⠀⠀⡴⠒⠒⠊⠉⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠘⠒⢁⠜⠁⡴⢦⡀⠀⠀⠀⠳⠴⡞⠀⠀⠀⠀⢀⠤⣄⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣁⣠⠞⠛⢦⡙⠦⣄⣀⡠⢴⡁⠀⣠⠔⠊⠁⠀⢹⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣴⡋⠩⡈⠳⣄⡄⠙⢷⣄⠀⠀⠀⠓⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣷
⢀⡤⠄⠠⠤⣴⠋⠀⠑⠄⠙⠂⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⣀⣀⠀⠀⠀⢸⠀⠀⠀⠈⣿
⠻⠀⠀⢀⡞⣿⢃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡜⢁⠈⢣⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⢠⡏⠀⣿⠈⠨⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⢀⡟⠀⣸⣶⡦⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠀⣇⠀⠀⠈⢏⢲⡄⠀⠀⠀⣀⣴⡾⠟⠁⠀⠈⠁⠀⣧⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⣠⣤⡄⠀⠻⢿⣄⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠛⠿⣶⣦⣿⠀⠀⠀⢰⣿
⠀⠀⢀⣤⣤⣿⣿⠈⠻⣦⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⢸⡏
⠀⠀⢻⣏⠻⣯⠙⢷⣄⣹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⠀⠀⠀⢸⡇
⠀⠀⠀⠙⠷⣿⠀⠀⠉⢉⡤⠾⢅⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⣠⠞⠉⠀⠀⠀⠙⠲⣌⠉⢿⣶⣄⠀⠀⠀⠀⠀⢀⠇⢸⡄⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡙⠛⠃⢰⡄⠀⠀⢸⠀⢸⡇⠀⠀⢸⡛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠈⠀⠀⠀⡆⠀⢸⡇⠀⠀⠈⠃⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⢀⣼⠀⠀⠘⣇⠀⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⡜⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠘⢨⣿⣿⣿⣦⡀⠛⡧⠀⠀⠀⠀⠀⣿⣷
⢀⡴⠚⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⢈⣿⣿⣿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠘⣆⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⢀⠟⠀⣿⡟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆
⠀⠘⣆⠀⠀⠀⠈⢦⡤⢦⡀⠀⢀⣀⠐⣧⠈⠳⡄⠀⡠⠋⠀⢰⣿⣷⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠘⢦⠀⠀⠀⠸⣄⠀⠈⠉⠁⠀⠉⠈⠳⣄⠹⠊⠁⠀⠀⢸⣿⣿⡆⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠳⣄⠀⠀⢈⠋⠻⢄⡀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⢀⠞⠙⠖⣺⣄⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠀⠀⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟
⠀⠀⠀⠀⠈⠲⠤⠊⠁⡌⠉⢻⠂⠸⠃⠘⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⣰⣠⠟⠳⢄⣀⣀⠀⠄⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡰⠃⠀⠀⣄⡀⠃⠀⠀⠀⠀⠀⠀⠘⢿⠇⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⢀⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠗⠀⠀⠀⢠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠲⠒⠒⠦⢤⣤⣀⡀⠹⡆⠀⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁⠀⠀⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠂⠐⠒⣶⠀⠀⠀⠀⠸⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⣷⣞⠻⠇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⠗⠈⠉⠀⣠⡄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⠀⣀⡴⠞⠉⠳⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⠔⠋⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡰⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠁⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⠁⠀⠀⠁⠢⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠶⠲⠤⣀⠸⣄⣀⣀⣀⣀⣀⡀⠤⡏⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠂⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢄⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠠⠀⣀⣑⣅⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠻⡀⠀⠀⠀⠀⠀⢠⠏⠀⢀⣠⠴⠒⠋⣉⠝⢓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⣠⠤⠖⠁⠀⢠⠏⠀⢀⠄⢊⠤⠊⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠫⢍⡉⠈⠉⠒⢄⡀⠀⠀⠀⠀⢸⠤⠎⣀⣐⡁⠀⠄⢊⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠁⠂⠌⡑⠢⢄⡀⠙⢆⠀⠀⠀⠀⠑⠒⠶⠶⠶⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠐⠠⠀⢀⡀⢁⣢⡬⠑⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣤⣶⣾⣿⣿⣶⣦⣴⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⡝⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⢀⠛⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣸⣿⡿⣿⢿⡟⣷⡹⣿⣨⢿⡻⣿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣾⣟⣩⡿⣻⠟⣻⣟⣿⠟⠋⠀⠀⠀⣿⣿⣷⡟⡘⢷⡘⣷⣿⣿⠈⣿⠈⢹⣀⡟⠓
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⣿⣿⣿⣿⣿⣿⠾⠋⠘⠁⠀⠀⠀⠀⠀⢿⣿⣿⣧⡁⠈⢧⡹⣿⣿⣇⢻⡆⢸⢟⣷⠄
⠀⠀⠀⠀⢠⡶⢋⣼⣿⣿⣿⣿⣿⢡⡞⢠⠐⠀⠀⠀⠀⠂⠁⠀⠀⠈⢿⣿⡌⢿⡄⠀⢇⣿⣿⣿⣼⣷⡜⣾⣿⠆⠀⣾⠱⣀
⠀⠀⢀⣠⠟⣥⣿⣿⣿⢫⣿⢹⡟⢺⠋⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠈⠻⣿⣎⣿⣴⠸⣿⣿⣿⣿⠻⣷⠹⣿⣤⡴⠋⠀⠈⠱⣄
⠀⠀⠋⣰⡾⣿⣿⣿⣯⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠶⡏⢻⣿⢹⣿⣿⡄⠈⢿⣄⣀⣾⡄⠀⠘⣦
⠀⠀⠰⣯⣾⣿⡿⣿⣿⡿⣿⠟⠡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⠅⠀⢾⢿⣿⣿⣯⣲⣧⣸⡏⢻⣿⣿⣿⡆⠀⠙⣿⡄⠙⠦⠂⣥
⠀⠀⠐⢫⣿⣿⣾⡿⢋⣾⠋⣄⢀⡈⢳⡄⠛⠀⠀⠀⢀⡰⣆⠀⡴⠂⠀⠁⠀⢿⣿⣿⡉⠸⣿⣿⣼⣿⣿⣿⣿⠀⢸⡿⢻⣶⣤⠾⣧⠸⡀
⠀⠀⢰⣿⣻⣿⣿⣷⠟⠁⠀⠈⠻⣿⠏⢸⡆⢠⠀⢰⡿⠘⡏⠀⠀⡀⠀⠀⣠⣾⠿⣿⣿⣀⣹⣿⣿⣿⣿⣷⣿⠀⠀⢴⣿⠀⣿⠀⢻⣄⠃
⠀⢠⣿⣿⣻⣿⣿⣇⣾⣷⣤⡀⠀⠈⠀⣠⣥⣤⠛⣾⣧⡀⠀⠀⠀⣀⣐⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿⡆⠠⡄⣸⡏⠁⣠⡟⢠⠃
⠀⣼⣿⠇⣿⣿⣍⢿⣿⣿⣿⣿⣴⣶⣶⣿⡇⠀⠀⠈⣯⠻⣷⣿⣿⣿⣿⡿⠋⠁⠹⣿⣿⣿⣯⠹⣿⣿⣿⣿⣿⣷⣄⢀⣿⣟⣋⡹⢀⣿⡄
⠀⠻⣿⢸⣟⢸⠹⣿⣿⡟⠉⠙⢿⣿⣿⡿⠁⠀⠀⠀⣿⠀⣿⣿⣿⣿⣏⣇⠐⠂⠀⣸⣿⣿⣿⠟⣿⣿⢿⣿⣿⣿⣿⣿⣷⣾⣿⣷⠋⢸⣇
⠀⠀⡿⠀⣿⢸⣶⣿⣿⣧⣈⣁⣼⣿⣿⣥⣤⡄⡀⠀⡟⠀⠈⢿⣾⣿⣜⣿⣶⣖⢾⣿⣿⣿⣿⠀⠘⣿⠈⢿⣻⣿⣿⣿⡛⠶⣿⣿⣦⡀⠸
⠀⠀⢠⢄⣿⣼⡟⢱⣿⣏⣻⣿⣿⣿⡿⣻⣷⣄⠙⠂⠈⢰⣆⠈⠻⢯⣽⣼⣿⣿⣿⣷⣿⣿⣿⠀⠀⢿⡀⠀⢿⠻⢌⣻⡟⠻⢿⣭⡽⠿⠞⢀
⠀⠀⢰⡟⣷⣿⠀⠀⠉⠛⠉⠹⣿⣿⡷⣿⣿⣿⣧⣀⠀⠀⠹⣆⠀⠀⠳⢿⡻⠉⠁⠀⠘⣿⡇⠃⠀⠀⣏⣀⡠⠁⠀⣽⣿⡷⠄⠀⢀⣴⣶⣼⣿⡄
⠀⠀⣮⣵⣿⣿⡇⢀⣶⣄⡄⠀⠘⠻⡏⢿⣿⣿⣿⣿⣿⣦⡀⢘⡄⠀⢴⣤⡐⠀⠀⠀⠀⢻⣧⠀⣠⣿⡈⣿⠖⡠⠀⢸⣿⣷⠀⠀⣿⣿⡿⠏⠉⠱
⠀⠈⣿⣿⢿⣿⣇⢿⡿⠿⢁⡀⠀⠀⡄⣾⠻⣿⠈⠻⡿⠿⠑⢿⡇⠀⠀⠈⢻⣆⠀⠀⣠⣾⣿⣿⣋⣿⠇⣀⠞⠁⣀⣼⣿⣿⠀⠀⣿⣿⣇⠀⣠⠁
⠀⠀⢻⣿⠸⣇⠹⣮⣷⣴⣾⡇⠀⠀⣹⡟⠀⠈⠀⠀⠀⠀⢀⢀⣼⣦⡀⢠⠘⣿⡆⣼⡿⠀⡾⣉⡉⣩⡶⠃⠀⢻⣿⣿⣿⡿⠀⣾⣿⡿⠿⣇⢹⡇
⠀⠀⠀⢻⡄⠙⣧⣿⣻⣿⣿⢃⠀⡼⠋⠀⢀⠀⠀⣀⣴⣷⣾⠟⠉⣿⣿⣿⡇⢹⡇⢹⣟⣰⣿⣯⢼⡋⣤⡀⣠⡿⢿⣿⡟⠀⠐⣿⡿⢃⡴⠋⢀⠃
⠀⠀⠀⠀⣿⠀⠹⣿⣿⣿⣿⣾⣼⣿⣶⣴⠿⡿⠛⢻⡟⢻⡿⠓⣆⠸⣿⣿⣴⣾⣧⠀⢿⣿⠃⣱⡟⠀⢿⣶⡿⠁⢸⡗⠀⢀⣴⡿⣴⢋⣠⡖⠁
⠀⠀⠀⠀⠻⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⣇⣀⢀⡇⢘⣇⢀⣿⣠⡏⢸⡃⣿⣿⠀⠈⣿⣿⡏⠀⠀⢀⡏⠀⠀⢸⡇⢰⠿⠏⠀⣾⠛⠁
⠀⠀⠀⠀⠀⠀⣸⡿⠘⣿⣿⣷⣿⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⡏⢰⠁⠀⡸⠁⠀⠀⢸⣷⣶⡦⠐⠊⠁
⠀⠀⠀⠀⠀⢀⣿⠃⠀⢹⣿⣿⣿⣿⣿⣿⣤⣜⡠⠆⣴⣀⣨⠉⣷⡀⣧⠀⣿⣶⣿⠀⠀⠈⠀⢸⢠⡞⢡⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⣿⣿⠟⠛⠉⠉⠀⠀⠀⠙⠃⠀⠤⠶⠿⠿⢻⣿⣭⣿⠇⣀⠀⢀⣴⣾⠋⣠⣾⣦⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣦⠙⠳⠞⠛⠃⠛⠃⠐⠀⠀⠂⠐⢶⣦⣐⣀⣀⣤⢂⣴⣁⣴⣿⣿⠏⢸⣿⣞⢹⣿⠟⣿⣿⣷
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠘⣷⣤⡤⠖⠒⠀⣴⣶⠦⠤⢀⠙⢿⣿⣿⣿⡿⠋⢸⣿⣿⠟⣻⣣⣶⣿⣿⣿⠛⠋⣠⣿⣿⠿⣧⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠬⣿⠀⠀⠈⠳⠈⠛⣿⡿⢁⣴⣾⠟⣡⣾⣟⣨⠟⣿⠟⠉⣁⣴⣿⣿⡿⠀⢹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠀⠛⢋⣡⣶⣿⣿⡿⠿⠋⠁⣠⣾⠟⠙⠀⣸⡇⢠⣾⠁⡛⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⠀⠀⠈⠤⠄⢤⣤⠄⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣏⣤⡾⠟⣱⡟⢠⠀⣠⢦⣿⣷⡿⢁⡰⣟⣶⣿⣿⠟⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠖⠻⣷⣶⣤⣽⣟⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣾⣿⣷⡏⢿⡟⣻⣷⣧⣾⣿⣿⡿⠋⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠉⠙⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⢿⣿⣏⡗⢀⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡀⠈⠿⣿⣶⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢡⣾⠟⢛⡁⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⠙⠀⢺⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠿⠁⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠒⠉⣿⣿⣿⣿⣿⣿⡿⠋⠘⠉⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠼⡿⠿⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠛⠉⠉⠉⠙⢿⡿⠋⠉⠉⠉⠛⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⠀⠀⠀⠀⢀⣼⢧⡀⠀⠀⠀⠀⠈⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣤⣤⣴⣿⠃⠈⢿⣦⣤⣤⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⠀⠻⣿⣿⣿⣿⣿⡇⠀⠀⢈⣿⣿⣿⣿⣿⠿⠁⢀⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣆⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠿⢿⣿⣦⣌⡛⠿⠿⣿⣿⣿⣿⠿⠿⢟⣣⣴⣾⣿⠿⠟⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣤⣀⠀⠀⣀⣤⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠟⠻⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣤⣶⣿⣿⠿⠛⠉⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠛⠉⠀⠈⢻⣷⡀⠀⠀⣀⣤⣶⠿⣛⣩⣥⡶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠁⣠⣴⡶⢾⣿⡆⣀⣿⡷⢰⣿⠟⣫⣴⣿⠿⢋⣡⣤⣶⠶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣀⣴⡶⢿⡿⠟⠋⠀⠀⠀⣾⣿⣿⣷⣾⡿⢛⣉⣀⣤⣤⡤
⠀⠀⠀⠀⠀⢀⠀⢠⣿⠇⠀⠀⣠⣾⡿⠩⠶⠾⢷⣆⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⣛⣉⣀⣀⣀
⠀⠀⠀⠀⠀⣿⣷⣌⠻⢷⣶⡾⢿⣿⠁⠀⣠⡄⠀⠙⠀⠀⠀⠀⠶⢶⣶⡄⠈⠹⠿⠿⣿⡋⠉⠉
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣄⠀⣿⡿⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⣀⡄⠈⢻⡀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠏⠀⣹⣷⢸⣿⠁⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠈⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⢠⣿⠇⠈⠉⠁⣠⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⢀⣤⣤⣾⣟
⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⢾⣿⠀⠀⠀⠐⣧⣀⣠⠀⠈⠉⠀⠀⠀⠀⠀⠀⠟⣡⣴⣌⢻⡇
⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠘⣿⣆⠀⠠⢤⣈⣉⠁⠀⣶⠀⠀⠀⠀⠀⣠⡄⠴⠋⠐⢋⣾⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠉⠉⠛⠻⠆⠀⠀⢀⣴⣿⣷⣤⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣤⣤⣶⣿⠟⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠋⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠟⠁⠀⠈⠻⣿⣿⣿⣧⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣶⣶⣷⣴⣶⣿⣿⣿⣿⡇⢸⣿⠿⠟⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⠉⢻⣿⣿⣿⡿⠛⠃⠘⣿⣤⠀⠀⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣼⣿⣿⣿⡇⠀⠸⣿⣿⣿⣧⣤⣤⡄⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠟⠙⠻⠃⠀⠀⠿⠟⠙⠻⠿⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠂⠈⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣀⡤⢄⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠈⢁⠄⣊⣭⡙⣯⡟⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠠⣀⡀⡀⢈⣩⣼⠻⣿⣿⣿⣷⣿⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠟⢿⠔⠁⠈⠟⠁⢹⣿⣿⣿⣿⣱⢦
⠀⠀⢀⣠⠴⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡚⠂⠙⠖⠛⢀⣴⣿⣿⣿⡿⢀⣾
⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛⣿⡗⠂⠀⠀⠀⠈⠻⢿⣿⣇⡸⠃
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⢎⣿⠗⠚⠀⠀⠀⠀⠘⢿⠟
⢨⡳⢤⣄⣀⣀⣀⣀⣀⡠⠄⢀⠀⠜⠀⠀⠀⢀⠀⠀⠈⠛⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠙⠷⢤⡄⠀⠀⠀⠀⠠⠀⠋⠀⠀⠀⠀⠀⠀⠁⣦⡀⠀⠀⠀⡅⠀⠀⠸⠂⠀⠀⠀⠀⠳⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⢤⡂⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠈⠒⠒⠀⠀⠀⠀⠀⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢃⡘⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⠟⠋⠉⠛⢿⣿⣿⣿⣿⣧⡼⠿⠿⠿⣦⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡿⠿⠛⣁⣤⣤⡴⠶⣤⡄⠀⠀⢿⣿⣿⣿⣿⣿⠿⠿⢶⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠀⠟⣉⣤⡴⠶⠶⠶⠄⠀⠀⠀⢹⣿⣿⣿⣧⣤⣤⡀⠙⣷⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡄⢙⣋⣥⠀⣴⣶⣶⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣧⡀⠈⣿⣾⠟⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠿⠋⠁⠀⠈⢀⣉⣙⣛⠀⠀⠀⢸⣿⣿⣿⡶⠶⣬⠉⠀⠹⣿⡀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣴⣾⡿⢷⣴⣿⡿⠉⠛⢷⡄⠀⣾⣿⣿⣁⣠⣤⣤⠀⠀⠀⠈⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠀⢈⣿⠀⠀⠀⠀⢸⣿⠀⢹⣿⣿⠋⣿⠏⠁⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣤⣤⡾⢿⣄⠀⠀⠀⣸⡏⠀⢸⣿⣧⡀⢿⡀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠙⣷⣶⣾⣿⣴⠒⠿⣿⣍⠻⣄⢻⡆⠀⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡍⠉⠀⠀⢸⣯⠈⢁⣿⣿⣿⣿⡟⢷⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠻⣦⠀⠀⢸⡿⠀⢸⣿⣿⣿⡘⢷⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⢹⣇⠀⠀⠀⠠⠎⣿⣿⡍⠁⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⡀⣀⣀⣤⣾⡿⠉⣿⡀⠀⠀⢻⡆⣤⠀⠀⣰⣿⢿⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⠋⠁⠀⠀⢸⣇⠀⠀⠈⣿⠹⣧⣴⡿⠃⢸⣿⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡟⠀⠀⠀⣿⠀⣿⠉⠀⢀⣿⡿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠻⣿⣷⣦⣄⠀⠿⠀⣿⠀⠀⣾⡿⠁⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣿⣉⣉⡁⠀⠀⠀⣿⡀⢸⣿⠃⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⢻⡿⣟⠁⠀⠀⠀⢹⡟⢿⡏⠀⠀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣿⠇⣻⡆⠀⢀⣠⣾⣿⠛⣷⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣏⠙⣷⠻⣾⠟⣿⡿⢿⡟⣿⣿⣧⠏⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠻⣦⠸⣇⣹⣧⣼⣥⣠⣧⣾⠟⠋⣰⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⣦⠀⢿⣤⡿⢻⣿⠉⠉⠉⠉⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢴⡟⠀⣼⠟⢁⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣎⠻⠟⠋⣰⣿⠟⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣌⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢷⣶⣿⣻⣧⣄⣿⣐⡇⣶⢰⡆⣻⡆⣹⣧⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⣿⣾⢿⣿⡟⠛⠻⠿⠿⠿⠿⠿⠟⠛⠛⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⣿⢨⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⢿⣾⠇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⣶⣶⣿⣿⣶⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣄⣀⣠⣤⣤⣴⣿⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠉⠙⠛⠛⠛⠉⠀⠈⠉⠛⠛⠛⠛⠛⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣍⣭⣭⣩⣭⣭⣭⣭⣤⣭⣉⣿⣉⣉⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣬⣥⣤⣤⣥⣥⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠉⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⣀⡀⠀⢀⣀⡀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣀⡽⣴⢯⣀⡹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⣀⡽⢯⣀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠘⠒⡇⠘⠾⠃⢼⠒⠃⡠⣄⠀⠀⠀⠀⠀⠀⣀⡀⠙⠉⣿⣧⣼⣿⡋⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⡈⠣⣄⠙⠲⠶⠒⠋⣠⠜⠁⣈⡷⠀⠀⠀⠀⢾⣿⣷⣄⠀⠙⠛⠛⠋⠀⣀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⣝⣲⡤⢖⣫⠴⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⣠⣴⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣚⡥⠒⢯⡓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣞⡉⢉⡴⠊⠁⠀⠀⠀⠈⠓⢮⠉⢉⡧⠀⠀⠀⠀⣾⣿⡿⠋⠁⠀⠀⠀⠀⠈⠙⢿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣾⣷⡶⠰⡶⣿⣾⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⠟⣿⣿⠿⠓⠺⡏⠹⣾⢷⡼⣿⢿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⣿⠀⢻⡏⠀⠀⢸⡃⠀⡿⣼⠁⢡⣶⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⢿⣷⠈⢳⠀⣷⡀⠀⠈⠡⢠⠃⡉⣤⢺⣷⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣇⠀⠹⡁⢰⠊⣵⠓⣆⣀⠸⠀⠠⢿⣭⣾⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡄⠘⣆⠈⠗⠀⠀⢃⣀⡏⡀⠉⣹⢡⣾⣿⣻⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠈⠁⠀⠀⠀⠀⢠⣿⣿⣿⣶⣤⣾⣿⣿⣿⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢠⣾⣶⣾⡿⣿⣿⣿⡏⠹⣿⣿⣯⣁⣨⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣤⣽⣿⣿⣟⣠⠿⣿⣿⢙⠻⣿⣿⠿⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣁⡁⣿⠻⠿⠟⢛⡁⠈⠛⠀⠘⣿⢻⣾⣿⣿⣛⣷⣷⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⢿⡇⢥⠈⣠⠔⠒⠀⠁⢦⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣶⣶⣶⣤⣷⣿⠄⠢⣉⣽⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡛⢿⣿⣍⠀⣠⣿⣟⢻⣬⣶⢿⠾⡷⢿⣿⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣷⣠⡉⠈⣄⣿⣿⣿⣭⣤⣷⣾⣷⣿⡾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹⣿⣿⣿⣻⣿⠿⠻⣛⢉⣠⣤⣼⣾⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⠻⠏⠙⠛⠻⢛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣶⣦⣷⣶⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢿⣿⣿⣿⣿⢝⣿⣿⣟⡿⣿⣿⠻⢯⣿⣿⣷⡀⠀⠀⠐⠆⡀⠊⠰⠛⢗⠶⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⣻⣝⢿⣷⣿⣿⣻⣿⣧⣶⣿⣷⢿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠁⠀⠃⠀⠸⠦⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⣿⣿⣟⠋⡿⢻⠶⣻⣿⣷⣿⣿⣿⣿⡿⣭⣈⢿⣿⣿⣾⢿⣝⣿⣦⣄⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠙⠻⢯⣽⣛⣿⣥⣤⣴⣶⣿⡿⠿⢿⡙⣿⣿⢋⣿⡿⠼⠿⣿⡷⡞⠿⢿⣿⣿⡿⠋⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠠⠖⠊⠁⠀⠀⠀⠀⠈⠙⠿⣭⣄⣁⣁⡀⢀⣀⠀⠈⣿⣿⠻⠛⣀⣠⣤⣦⣶⣿⣿⠿⠛⠉⠀⠀⠀⠊⠉⠵⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⠻⠧⠤⡙⢁⣴⢿⡏⢩⡿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢿⣿⣇⡺⢶⣿⠀⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢹⣿⢷⣼⡟⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⢈⡷⢇⣤⠀⠀⠅⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⠀⠀⢀⢀⡀⣘⠀⠋⠁⠀⢀⣠⠀⣠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢃⡠⣤⡜⠒⠈⠐⣡⠀⠀⠈⠀⣄⡼⠃⠀⢀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣶⢠⡴⢀⡾⠀⠀⠀⠀⠘⣁⡀⠀⠀⠈⠻⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢯⣵⣾⠇⠀⠀⠀⠀⠀⠻⢟⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡟⠄⠀⠐⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀⡀⠀⠀⠀⠀⢈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠖⣳⡆⠀⠀⣴⠏⠚⠙⡑⢠⠔⠀⠀⠀⠀⠘⣄⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠀⡀⠈⠁⠉⡀⣤⠟⢀⠺⠈⠀⠁⠀⠀⠀⠂⠠⠖⠚⠁⠉⠀⢱
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⠬⠕⡀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠀⡔⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣤⡄⢷⠶⠀⠀⠀⠠⣾⠟⢛⠓⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡐⠁⠀⠀⠀⠀⠀⠈⠀⢈⡚⢨
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡠⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆⠀⠀⣄
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠈⠂⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠀⢒⢤⣀⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡔⠒⠏⠁⠀⠀⡠⠟⠛⠛⠛⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠴⢤⡴⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⢷⠤⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⢤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠠⠐⢲⡞⠉⢳⣠⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⢠⣴⠏⠀⢀⡜⢡⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡀⡾⢠⣿⠋⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⠙⠦⣀⠀⠀⠀⠀⠈⢆⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢶⡀⢠⣿⠃⠀⢠⠋⣰⠇⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠛⢦⣌⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⣾⠃⡜⠀⠀⡴⠟⣀⠀⠀⠀⠐⠚⢿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠙⢷⡀⠂⠀⠀⠀⠀⣱⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⣸⠃⢰⠁⠀⠜⠀⡚⠁⠀⠀⠀⠀⢰⣿⣿⡿⠛⢿⣿⣿⡄⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠩⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣆⣠⡀⠘⠇⢀⡏⠀⠀⠀⠆⠀⡙⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⠙⢿⣿⡄⠀⠀⠀⠀⢹⡄⠀⢀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠁⢻⠀⠀⠀⠁⠀⠀⡜⠀⡴⠁⠀⠀⠀⢰⡿⣷⡜⢿⣦⣀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠰⠚⠀⠰⠂⡀⠄⠈⠀⠀⠀⠀⠀⠸⣇⠹⣿⣎⠙⢯⡒⣤⣿⣿⣆⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠘⠀⠀⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡄⣉⠑⠀⠀⢢⣿⣿⡿⠻⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠘⠻⠿⠛⠉⢠⡾⠿⢂⣤⣹⣆⠁⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⢠⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣄⢀⣀⣀⣀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠠⠼⢆⠀⠀⠀⠀⠱⢄⣸⠐⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠙⣤⡿⠛⢿⣿⠏⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢶⣄⡌⠀⠀⠀⠀⠀⠀⠀⣸⡟⠈⢱⣾⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡇⠀⢹⣧⡀⣸⣏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⣸⠇⠀⠀⢰⡆⠀⠀⢠⣿⡇⠀⣿⢿⠀⠀⠀⠀⣤⠀⠀⣀⡀⠀⢀⣤⣀⡈⠋⠘⠛⠁⠀⠀⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⡙⠀⠀⢀⣼⣧⠀⢠⣿⢹⡆⢠⡏⠀⠀⠀⢀⡀⣀⣤⠀⣀⡀⡈⢠⣆⠀⢹⠛⢢⠀⠀⣿⣆⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⠃⠠⠄⢻⡿⣹⠀⢸⡏⠘⠃⣾⠃⠀⣀⣼⣿⣧⣯⣼⣀⣏⢉⡟⢉⣟⠀⢸⠛⠷⢄⠰⣿⢳⡞⠠⣂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⣀⠀⣿⣿⣿⡇⠸⡇⡀⢰⡟⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡎⠲⣿⠶⠿⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡁⠀⠛⠀⣿⠿⡿⠁⠀⠁⠁⠾⡁⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠱⠀⠀⢰⡟⠀⡇⠀⠀⠀⠀⠐⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⠈⠁⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⡟⠀⠀⡇⠀⠀⠀⠀⠀⢿⢺⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠇⢀⠀⠀⠀⢠⡞⡿⢿⣿⣿⣿⣿⣿⣿⠟⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠄⠀⠀⠸⠀⠀⡀⣾⣁⠇⠀⠹⣿⣿⣿⣿⣿⢠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢀⠀⠀⡜⠀⠀⢸⠀⠀⢀⣿⣷⡿⠀⠀⠠⠿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠈⠀⢸⡇⠀⠀⢠⣷⡀⠘⣿⡿⠇⠀⠀⠀⠀⠛⠻⢿⣿⣥⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠐⠁⠀⠀⠸⣿⣿⣄⠛⠱⠀⢀⠀⠀⠀⠀⠀⠀⠹⣧⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⣿⡏⢙⣦⠀⢠⠏⠀⡇⠀⠀⠀⠀⠀⢟⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣀⠀⠀⠀⠀⣿⣷⣋⡉⠠⢠⠤⢠⠇⣉⠀⢂⠠⠀⣾⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⡟⠀⠰⠀⠀⡇⢴⡀⠀⢉⡄⠰⡈⠰⢿⡀⠀⠀⠸⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢰⣇⣴⡇⠀⢀⠀⢨⡀⢀⢀⡅⠀⠀⡀⠀⠁⠑⠂⠑⢰⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣾⣿⣿⠁⠀⢸⠀⢘⡇⣠⣠⡇⠀⠀⠐⠒⠒⢠⣄⡀⢀⡿⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣯⣿⣿⠀⠀⣸⠀⢸⣷⠘⠋⠡⣄⣀⣀⠄⡀⠀⠈⠁⣿⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡿⢹⣿⠀⠀⡄⠀⢸⣿⡁⠀⠀⠤⠵⠶⢶⣿⣦⣤⣾⣿⡀⠀⠀⠈⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠃⣿⣿⠀⠀⠃⠀⣼⣿⣧⣀⣀⣀⣀⡀⠘⠛⠛⢿⣿⣿⣇⠀⠀⠀⠘⣯⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠇⢡⣾⣿⣿⠀⡀⠀⢀⣿⡟⢹⣏⣉⡉⠉⢁⣀⣀⣀⣤⣿⣿⣿⡈⢄⠀⠀⠹⣿⣌⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⣿⣿⣿⢀⠃⠀⢸⠏⣰⣿⣄⠉⠛⠛⠛⠛⠿⠿⠟⠛⢿⣿⣧⠈⡆⡄⠀⠹⣿⣦⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣿⣿⣿⡘⠀⠀⠀⣼⣿⡏⠘⠛⠛⠓⠒⠀⠀⠀⠀⠀⠈⣿⣿⣆⠻⡘⢦⡀⢻⣿⣷⡘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠋⠋⠀⠈⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠁⠈⠁⠀⠙⠉⠋⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣴⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⣈⣀⣄⡀⠈⠉⠛⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢠⠞⢋⣁⣤⣉⠃⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠀⠀⠀⠀⠞⠉⠀⢀⡈⠻⣆⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠊⠙⠳⢦⠀⢸⠀⠀⠀⠈⠁⠀⣽⢣⡀⠀⢀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠆⠀⠀⢸⠇⠘⢧⣀⠀⢀⣠⡼⢃⡼⠃⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠶⠖⠋⠀⠀⣀⣩⣍⣉⡁⠀⠀⠀⠀⢀⣾⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⡤⡶⢾⢉⣿⢰⣇⡛⠳⣀⠀⣠⡟⠳⡍⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠴⠟⠈⣩⡍⢠⠍⣁⣰⠟⣰⡏⢡⠀⠹⡆⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠚⠳⠷⠾⠚⠋⠁⢀⣿⡅⠀⡆⠀⠹⡄⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣀⣀⠤⣤⠤⠴⠿⣼⣇⠀⢰⠀⠀⠘⡄⢀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⡇⢻⠀⠘⠀⠀⠀⡇⠀⢶⣄⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⡾⠁⢸⡶⣤⣅⣀⣘⠃⠀⠀⠈⠱⡄⠀⠀⠀⠈⢙⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡆⠀⢠⡞⢀⣠⠞⢁⣤⣌⡉⠙⠷⣄⢀⣀⣀⣷⢤⡀⢀⣀⣿⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣤⣏⡴⠋⠁⢰⡟⢀⡤⣻⠄⠀⠀⠙⢧⡀⠀⠀⢻⡁⠀⠀⠉⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⢸⠿⢦⡤⣴⣯⣾⠮⠖⣉⣀⡀⠀⠀⠈⣷⠀⠀⠈⣷⠶⠶⠦⠦⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⣀⡴⠃⠀⠀⠀⣿⠏⠀⣰⣛⣁⡸⠃⢀⡴⠚⣿⡆⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠛⢁⡴⠛⠁⠀⠀⠀⠀⠾⣧⣤⠾⠃⢠⢹⡅⢴⠋⠀⠀⢙⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⠿⠶⢿⢂⠀⣠⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣧⠘⢏⠉⠁⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢸⡄⠀⠀⢳⣾⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⣀⣿⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣼⡴⠦⣀⣀⣸⠀⠀⠀⠀⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡁⢿⡵⠋⣤⠀⢠⡌⠓⠦⣄⠀⠀⠀⠀⠀⠈⠀⠈⢿⣠⠃⠀⠀⠀⠀⠀⢀⡀⢿
⠀⠀⠀⠀⠀⠀⠀⣼⢃⣠⠇⣼⠃⢼⠁⡇⠹⣏⡛⠦⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣆⣠⣤⣄⣰⡋⠙⠏
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠈⢳⡈⢳⠑⣄⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⡈⠁⢸⡍⠋⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣸⠃⠘⢤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢻⡀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣼⠳⣄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⡏⠀⠘⢧⡀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠸⣧⣀⠀⠈⢷⡀⢿⣳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⡶⠚⠓⠌⠘⠓⠈⠉⢲⣨⢧⣸⠈⠋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠉⠛⢶⣧⠠⠤⠀⠀⠀⠠⠀⢈⡳⡏⣁⠀⠀⠀⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠤⣄⣀⣀⠀⠘⠣⣄⣀⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠶⠮⠝⠀⠉⣷⠖⠯⠔⠚⢉⡩⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣸⣧⠐⠖⣈⣥⣤⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⠀⠀⠉⠛⠲⠦⢤⣀⣀⣀⣠⡴⠶⢛⣩⡿⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⢦⣤⣤⣤⣤⣤⣤⣤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠾⠿⠿⠷⢶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣋⣁⣀⣀⡀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠿⠛⠛⠛⠛⠛⠛⠿⢷⣶⣤⣀⠀⠀⠀⠀⢙⣿⡿⢿⣦⡀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣄⣴⡿⠋⠀⠀⠙⠿⠟⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢠⣾⢿⣦⡀⣠⣾⠟⠁⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢀⣴⡟⠁⠀⠙⠿⠿⣷⡄⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢸⡿
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣿⠇
⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⠋
⠀⠀⣼⡟⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁
⠀⠀⢻⣧⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠙⠿⣶⣶⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠁⠈⠉⠀⠁⠀⠀⠑⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠲⠄⠀⠀⠠⠀⠀⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠐⠄⠀⠀⠀⠀⠠⠀⠠⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠐⠂⠀⠀⠀⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠠⠆⠀⠖⠃⠀⠀⠁⠀⠋⠀⠀⣌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠠⠀⡀⠃⠀⠀⠀⠀⠀⠀⠣⠀⠴⠆⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡤⡀⠀⠀⠀⠀⠀⠀⢀⠰⢠⠆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠖⠁⠒⢦⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡜⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠤⡁⠀⠉⠙⠀⡀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠈⠁⠘⠀⣀⠔⠡⠈⠉⠲
⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡻⣤⡽⠄
⢀⡀⠧⠤⠄⡉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢁⠴⡇⠇⠀⠀⠀⠀⠀⠐⠆
⠀⠢⠐⢂⠲⠦⠀⠁⠙⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠐⠁⠀⡇⠰⠀⠀⠀⠀⠀⢀⡲⠄
⠀⠘⢍⡠⠔⠊⠶⠆⠀⠀⠀⡠⠉⢈⣀⡒⠒⠒⠺⠖⠲⠒⠚⢋⡤⠊⠀⠀⠀⠀⡗⠀⡀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠣⡀⠔⠁⠒⠈⠀⠀⠀⠀⠀⠉⠉⠰⠄⣀⠤⠚⠁⠀⠀⠀⠀⣠⡎⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⡀⢉⣠⡤⠤⠤⠀⠒⠒⠈⠁⣠⠏⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠉⠁⠈⠉⠠⠤⠤⠤⠤⠤⢬⠀⠀⣀⡀⠤⠄⠒⡝⠀⠁⠀⠀⠱⡀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠠⠈⣠⠔⢀⢰⠰⠢⠤⠴⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠘⢤⠀⢼⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠃⠘⢨⠀⠃⠸⠀⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠢⡀⠀⡏⠀⠀⠀⠔⢒⡶⠇
⠀⠀⠀⠀⠐⠠⠄⠀⠀⢠⠀⠑⠢⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⢄⣀⠈⠑⠃⣀⣀⠀⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢊⠔⠊⠀⠈⠉⠉⠀⠀⠂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢤⣀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡰⣎⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢦⡀⠌⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⠠⢶⠤⠤⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠘⢩⡄⠀⠀⠑⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢁⣈⣠⡄⠀⠐⠀⢀⣀⣀⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠁⠀⠈⠀⠀⠀⠀⠀⠉⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠀⠀⠀⠠⠤⠤⠞⠧⠤⠬

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠿⠿⠿⠶⠖⠶⠖⠀⠀⠀⠀⠀⠀⢀⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⣀⣀⡀⣄⠀⠀⣀⠀⢀⣀⣤⣄⣀⡀⠀⠀⠹⣷⣦⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⡿⣿⣿⣿⠇⢾⣿⣿⣿⠿⠿⢿⣿⣿⣷⣀⣀⢿⣿⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠁⠀⣠⡿⠋⠀⢈⠛⠟⠑⠀⠀⠀⣹⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠮⠀⢠⡟⢠⣤⡀⠾⣥⡀⠁⠀⠀⠐⠛⣹⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣀⡀⠀⢀⣿⣿⣿⣷⣌⠿⠀⠀⠀⠀⠘⢹⡿⣿⣿⣿⣿⣿⣟⠻⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠈⢻⣟⣩⠟⠋⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣶⣾⣿⣿⣿⣷⣾⣷⠀⠀⠀⠘⡿⠏⢛⢽⣯⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣿⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⡅⠀⠀⠟⣮⠉⠛⣉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠉⠉⠉⠛⠛⢻⣅⠀⠀⠀⠁⢠⠀⢸⠟⠷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣶⠦⣤⠶⣤⣌⠛⡤⠀⡄⠀⣨⡶⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⠀⢀⠀⠀⠙⣉⣁⣘⣯⣴⣿⠂⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠿⠏⠐⠂⠀⠁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠟⠋⠛⠛⠁⠀⠀⠀⡀⢀⣀⣈⣏⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠩⢽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⢾⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠉⢁⡼⣿⣷⣂⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠛⠀⠸⠟⠁⠀⢁⣺⣧⡄⣀⠀⠀⠘⠁⠉⠀⠀⠀⠀⢀⠠⠤⠄⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⣤⣼⣿⡗⠀⠀⢸⡄⣠⠄⠀⠀⠀⠐⠂⠉⠀⣀⣠⡤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣤⣶⠆⠀⠛⠛⠛⠛⠛⠁⣀⣠⡾⠋⠁⠀⠀⣀⠀⣤⠶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⠿⠛⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠟⠋⠁⠀⠀⠀⠀⠻⣿⣦⣤⣴⣶⣿⣷⡄⠀⠀⠀⣴⣿⡟⠁⣀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⠿⠛⠋⠉⠙⣿⣷⠀⠀⠰⢿⡿⠀⢾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⡄⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠿⠟⣿⣷⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠈⠻⠟⠋⠉⠉⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣾⣿⡿⠛⠋⠁⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⡿⠟⠁
⠀⠀⠀⠀⣀⣀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢠⣿⡟⠁⠀⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠙⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⣆⣀
⠀⠀⠀⠀⠹⢿⣷⣤⣀⡀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣇⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⣷⣦⣀
⠀⢸⣿⣧⠀⠀⠉⠛⠛⠃⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀
⠀⠀⠙⠿⣷⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠈⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣾⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢢⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡿⠀⣠⣤⣤⣤⠀⠀⠉⠀⠀⣀⣠⣤⣤⡈⢿⣆⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⣼⣿⣿⣿⣿⡿⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⣿⣿⣿⣿⣿⠇⢰⣿⣦⠀⣿⣿⣿⣿⣿⡇⠸⠿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣣⡀⠈⠉⠉⢉⣁⠀⣼⣿⣿⡀⢨⣉⠉⠉⠉⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⠀⠀⠀⠈⢷⣦⣔⠡⠿⠁⠀⠻⠋⠛⠃⠀⢻⡟⠂⢐⣨⠎⠀⠀⠀⢀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢈⡷⠀⠀⠀⠸⡿⢏⢰⣄⠀⠀⡀⠀⠀⠀⠀⢠⡶⢹⣿⡟⠀⠀⠀⠀⣯⡀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣙⣿⣇⠀⠀⠀⠀⡷⠸⣾⣽⣤⣀⣄⣠⣀⣠⣤⣼⣧⡾⠻⠃⠀⠀⠀⢀⣿⣟⢻⣏⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠋⢉⣀⠉⣉⣠⡆⠀⠀⣷⠀⣿⣤⠉⡏⠉⢉⠉⠅⡉⣽⣿⠁⢀⠀⠀⠀⣦⣉⣉⠁⣈⠉⢱⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣝⣉⣍⡙⠻⣄⡀⠈⠣⣈⠇⠀⠉⠈⠉⠀⠀⠁⠈⢃⡴⠁⠀⢀⣴⠟⢉⣩⣙⣡⡴⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣌⡻⠖⢄⡈⠀⡀⢠⠀⡀⣀⠠⣒⡢⠊⣀⠤⠾⣋⡤⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠑⠢⣅⡀⠀⠀⢀⡤⠐⢉⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣦⣈⠐⠺⢤⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⣩⣽⣿⠿⠶⣦⣌⡑⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣄⡠⠴⠚⠃⢀⣰⠾⠛⠉⠀⠀⠀⠀⠙⠻⢴⣂⡈⠛⠲⠦⣀⣤⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠡⡶⠆⠉⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⡀⠀⠁⠲⠦⠋⠉⠙⣦⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣼⡷⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠢⣿⣶⠾⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⡽⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠒⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣀⣴⠾⠛⠉⠉⠑⢦⣀⠀⢀⡖⢰⠆⠛⠛⠒⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⡔⠊⠁⠀⠀⠀⠀⠀⠀⣀⣉⢿⣿⣇⡿⠀⢀⡴⠋⢀⠘⠂⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠟⠋⠁⠀⠀⣠⣤⠶⠶⢻⣿⡿⠋⠙⠈⠁⠈⠀⠒⣯⠀⢠⠞⠀⠀⠈⢲⣄
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢀⣠⣾⣿⣿⠂⢠⣾⠋⠀⠀⣠⣤⣶⠗⠀⠀⢹⡖⠻⣤⢐⡇⠀⠀⠙⢧⡄
⠀⠀⠀⠀⢰⠁⠀⠀⠀⣠⣿⠿⣛⠉⠀⠀⢸⡏⠁⣴⠻⠟⠻⠛⠀⠀⠀⠘⠓⣶⣌⢻⡙⡀⠀⠀⠀⠹⣀
⠀⠀⠀⠀⠇⠀⠀⠀⢠⣿⣧⣾⣿⣦⣴⡦⠺⠇⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣇⣱⡄⣀⠀⠀⠉⢶⢤⡄⡀
⠀⠀⠀⡿⠀⠀⠀⠀⣾⣿⣿⡿⠉⠙⠁⠀⠀⢀⣡⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠾⠘⠁⠉⢧⣀⠀⠀⠀⣴⠀⠹⡌⠲⢄⡀
⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣷⡶⠶⠶⠆⢀⣨⠋⠀⠀⠀⠀⢀⣤⣤⣴⣄⠀⠀⣼⡄⠀⠀⠈⠙⣋⠀⠘⠁⠀⠀⢘⠓⠛⠛⠛⠳⣦⡀
⠀⠀⢸⡇⠀⠀⠀⠀⢻⣿⣿⡏⢴⢾⡤⠰⣦⠈⠀⠀⠀⠀⠀⠋⢁⣠⡀⠀⠀⡀⣩⠃⠀⠀⠀⠀⠈⠳⣄⠀⠀⣠⢾⡗⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠸⡇⠀⠀⠀⠀⢸⣿⣿⡇⢰⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠈⠃⠆⠐⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢰⡈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⢿⣄⠀⠀⠀⠘⣿⣿⣧⠈⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠙⣦⣄⠀⠀⢹⣏⠻⣿⣇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠤⠀⣠⠿⠀⣀⣠⣤⣀⣤⡄⣠⣴⠿⣤⣤⣴⠶⠶⠤⠤⣀⠀⠀⠀⠀⢀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠙⠛⠷⣿⣿⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⡜⠁⠀⠀⠀⠹⠀⠀⠁⠈⡀⢉⠉⣸⠟⠁⠀⠙⠭⢄⣀⡀⠀⠀⠈⠑⠒⠲⣤⣾⠇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⢁⠀⠀⠀⢣⣄⠀⠀⠀⠀⠀⡁⠀⠀⠘⠻⠶⢟⡦⠀⠀⠀⠀⣷⠈⣰⡿⠀⠀⠀⠀⠀⠈⠉⠉⠛⢶⡠⠄⠀⠂⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡼⠏⠻⣿⡇⢨⠇⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠛⡇⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢢⠀⠓⠀⠀⠀⣠⠞⠁
⠀⠀⡠⠖⢻⡿⠁⠀⠀⠹⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠀⢿⠀⢸⡄⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⢾⡇⠈⡀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⣼⠟⠲⠒⠛⠻⠁⠀⠀⠀⠀⢈⡇⠸⣇⣿⠁⠀⢀⠄⠀⠀⠀⢀⣴⠞⠋
⠀⠀⠀⠀⠈⠷⡀⢁⡀⠀⠀⢠⣿⡦⢀⡀⠀⠀⠀⠀⠴⠎⢠⡿⠐⠲⢶⣄⡀⢣⡀⠀⠀⠀⠈⠃⢠⣿⡟⠀⣴⠏⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⣾⣿⠃⠀⠈⠑⠢⢄⠀⠀⢀⡞⠁⠀⠀⠀⠈⠛⢾⡇⠀⠀⠀⠀⠀⢸⣿⠀⠞⣁⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣾⣿⠃⠈⠁⠒⠒⠒⠀⠈⠑⢿⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣧⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠈⠁⣀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣉⣭⣿⣦⣤⣄⡀⠀⠀⠀⣀⢀⣤⣤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣋⣶⡾⣛⡩⢁⣤⠤⣎⡙⠷⣤⣴⡇⣾⠉⠀⢀⡤⠙⢣⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢟⢫⡵⢶⣭⣤⣿⣷⣶⣶⣾⣿⣿⣿⢷⣍⣀⡴⠋⠀⣴⠀⠀⠘⣄
⠀⠀⠀⠀⠀⢀⣴⠞⡋⢔⡋⢠⣴⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣼⣯⣾⣹⣷⣀⣼⡁⢠⡄⠀⠘⢷⣄
⠀⠀⠀⠀⢀⡿⡁⢺⡵⢏⣼⣿⣿⣿⣟⣿⣹⣿⣿⣋⣵⣿⣿⡿⠛⠛⠉⣿⣯⣽⣿⣶⢿⠀⠀⠀⠙⢧
⠀⠀⠀⠀⣼⢃⣹⠘⣡⣿⣿⣿⣿⣧⣽⣟⣿⣿⣷⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⢧⠀⠀⠀⠈⠲⣄⡀
⠀⠀⠀⣼⢣⡜⡋⠆⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠗⠾⣷⠲⠀⠀⠀⢹⡿⣷⣦⣄⡀
⠀⠀⢀⡿⢇⠼⡿⢤⣿⣿⣿⣿⣴⣦⣿⣿⣯⣿⡿⠁⠀⠀⠀⠀⠀⣛⣐⡀⠀⠀⣙⡇⠀⠀⠻⣷⣤⣤⣴⡿⣧⣜⢿⣿⣿⣦⣤⣄⡀
⠀⠀⢸⡇⢊⣰⣧⠸⣿⣿⣿⡟⣯⣻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠶⠟⠛⠛⠛⠀⠀⠿⡇⠀⠀⠀⠈⠻⣿⣻⣟⠻⢇⣽⣧⣷⣷⣶⣽⡿⣦⣄
⠀⠀⢸⡇⣸⢿⣿⡇⣿⣿⣿⡇⢘⣽⡏⠛⠛⠀⠀⠀⠀⠀⠀⢀⡴⢻⡇⡀⢠⠶⡚⠀⠀⠀⠀⠀⠀⠈⠙⢷⣾⣿⢻⣿⢗⡼⠷⣉⠾⣃⣝⠳⣤⡀
⠀⠀⠈⣷⠉⣎⡻⠆⣿⣿⣿⣇⠸⣭⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠤⠀⠁⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢻⣿⢩⠂⠴⢷⣌⡳⠬⣈⡳⣬⡻⣦⡀
⠀⠀⠀⠘⢷⣌⠈⡣⠘⣿⡿⣿⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡀⠀⣀⣿⡘⢋⣬⣭⣭⣤⣌⠙⠶⠦⠐⣉⣷⠋⢿⣄
⠀⠀⠀⠀⠈⠙⠿⣶⣤⣿⣷⠀⠙⠷⠿⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠇⠈⠉⢀⢿⡁⠴⠶⣿⣻⣷⡿⣳⣾⠟⠿⣿⡿⡟⠛⠏⠛⣛⠷⣶⣭⣤⣧⣼⣷⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡆⠀⠀⠀⣄⠀⠀⠀⠀⠀⠘⡟⢼⠁⢤⣤⣤⣴⣜⢯⡂⢰⠀⣿⢿⣧⣿⡇⣎⢼⣄⠛⢿⣿⣷⣶⣤⣝⣪⣭⣽⠿⠿⢛⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⢰⡄⠀⠀⠘⢿⣄⠀⠀⠀⠀⢇⠀⠀⠀⢨⣭⣴⣿⢤⣧⢸⠀⣿⠈⣿⠿⣻⡏⢀⣤⠉⠰⠒⣿⣎⡿⣿⡛⠿⣦⢔⣴⠞⠁
⠀⠀⢀⣠⣤⣾⣿⣿⣹⣿⠃⠘⠁⠀⠀⠀⠀⠉⠀⠁⠄⠀⠈⠀⠀⠀⠀⠀⠀⣿⢸⡜⢠⠀⣿⡇⣿⡆⣿⡇⡼⠡⠖⣮⠉⢁⠻⢦⣠⠏⣩⣼⠟⠁
⠀⠀⠻⣏⣿⣿⣿⣟⣷⣌⢳⣄⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣠⣤⣤⣠⣴⣶⠿⡘⢣⡞⣤⣿⣇⢹⣧⣿⡇⣼⢁⣶⡎⣛⢯⠴⣊⣵⡾⠋
⠀⠀⠀⠈⢻⣿⣿⣿⣻⣾⢃⠜⣧⣀⠀⠀⠀⠈⠉⠀⣀⡭⠉⣿⣉⣯⣝⡡⢿⣷⣹⠎⣧⢙⣹⡟⢸⣿⣿⠁⢡⣿⢏⡾⣋⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠙⠛⢸⣇⢱⢾⣿⣿⡿⠻⢦⣄⣀⠀⠀⠉⠀⣸⡟⣍⣉⡙⠻⣶⣽⣧⡸⠇⣨⣿⣧⢸⣿⡏⣴⡟⣡⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣼⣿⡟⡷⢾⣧⣬⣭⣟⣦⣄⣴⡟⢰⣌⣬⠱⠕⠪⠛⣿⠀⡄⠙⣿⠉⣿⣿⣿⣽⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠿⠟⠉⠁⠚⠫⠍⢽⣿⣷⣮⣹⣿⡡⠜⠣⠞⠴⠛⠰⢧⡄⣠⣾⠛⣾⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠛⠓⠚⠛⠚⠛⠛⠒⠚⠓⠒⠂⠉⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢦⣤⣤⠤⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠉⠀⠀⠀⠀⠀⣀⠀⠈⠉⠛⠳⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣇⣾⣷⣷⣶⣗⣠⣾⣵⣶⣴⣶⣶⣤⡄⠀⠀⠹⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⠿⠛⠛⠛⣛⠉⠀⠀⠀⠀⠼⣯⣁⠀⠒⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣻⡛⠃⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡆⠀⢠⡇⠀⠀⠀⠟⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠷⠤⠤⠴⠒⠈⠀⠀⠒⠒⠢⡕⢼⠁⠀⣨⣄⡀⢠⠀⡤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠑⠌⢂⠀⠑⠂⠀⢄⣠⣤⣄⡀⠘⠄⠠⢿⣉⠀⠀⠜⢁⣠⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⣄⡀⣀⣸⣧⣠⣤⣴⡿⠟⢿⣿⣿⡄⢀⠀⠸⡏⠀⢀⣾⢋⣴⣶⡎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋⠙⣿⠟⢻⣿⢿⣇⡐⣀⣿⣿⡇⢈⠀⠀⠙⠶⠾⠇⣿⣾⡿⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣵⣿⠀⢧⠸⡌⠙⠻⠿⣿⡿⢡⠆⠀⠀⢀⣤⣀⠰⣿⠿⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣿⣿⣿⣆⠀⢤⡀⠱⠶⢂⣬⣥⡀⣀⣴⣿⣿⣿⣷⣤⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢿⣿⣟⢛⣰⠀⢹⡄⢰⣯⣶⣾⣿⣿⡟⠃⢿⣿⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠟⠈⠀⢐⡺⢩⢛⣿⣿⢿⡿⣯⡎⠁⠈⣿⡇⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣻⣙⣏⣏⣹⣭⡷⣿⠆⠘⠇⠏⢃⣀⣠⡿⡁⣿⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠷⡶⠷⡿⢛⢩⠆⠀⠐⠁⢃⣴⡿⠁⢷⠏⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡛⠹⠀⠐⡄⡫⠀⠀⣠⣾⣷⣿⣿⠇⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠁⠀⠐⢀⣾⣿⣿⣿⣿⡏⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡋⡀⠀⠀⠀⣸⢾⣿⣿⣿⣿⡟⣠⢆⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠷⠶⠛⠁⠈⣿⣿⣿⠏⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠋

⠀⠀⠀⠀⠀⠀⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⣶⠶⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣼⠀⠀⠀⠀⠀⠉⠷⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠈⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⢻⣄⠀⠀⠢⣠⣀⣀⠘⠿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⣼⠘⡇⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠈⠻⢶⣆⡀⠀⢹⣿⢷⣦⣄⠙⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⢛⡄⠀⢀⠔⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠉⠻⢾⣿⣿⣿⣿⣿⣿⣶⣄⣙⣿⣍⡑⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣾⣻⣶⣁⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣳⣿⣟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢤⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⢫⡟⢱⠒⣤⡽⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣾⣌⣹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠾⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣴⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⠀⠀⢠⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠒⠊⢡⠆⠀⠀⠀⢀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣾
⢰⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠉⠀⠀⢀⣠⣶⠟⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠹⡇⠀⠄⣀⡀⢀⢇⣀⣠⣶⣿⡿
⣿⣿⡰⠦⠐⠂⠒⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠰⣤⣀⢀⣀⣀⣠⣶⢔⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⣉⢾⣿⣿⣿⠟⠁
⠙⢿⣷⣤⣀⣈⣡⣤⣤⡤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣦⣭⠼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠘⠋⠉
⠀⠀⠈⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡀⠂⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣡⡤⠶
⠀⠀⠀⢠⠞⠟⠛⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢿⠏
⠀⠀⠀⣇⠀⠰⠚⠋⠀⠡⠀⠀⠀⠀⠀⠀⠀⠘⣧⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⣿⣴⠇
⠀⠀⢸⣯⡖⢾⠉⠹⣧⡀⢡⠀⠀⠀⠀⠀⠀⠀⢿⡼⡀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⢀⣀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⡀
⠀⠀⢸⡿⢶⡤⠦⠐⠛⠯⠸⠆⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠈⣿⠈⣷⠀⠀⠀⠀⠀⠀⢸⣷⣶⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠈⣿⣿⠁⣀⢀⡀⢱⣤⡁⠀⠀⠀⠀⠀⠀⠀⢻⠎⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠘⣿⡶⠿⠿⡛⢽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠽⢶⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠈⠷⡖⠋⠙⡩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⠧⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣀⣀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠲⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠁⠀⠀⠀⠀⢠⡔⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄⠀⠀⠀⠀⢸⠾⡀⠀⠀⠀⠀⠀⠉⠉⠙⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠈⠉⢂⡇⠀⠀⠀⠈⠚⠁⠀⠀⠀⠀⠀⠀⢀⣚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠈⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇⠀⠀⠀⢠⠆⠱⠋⡄⠀⠀⠀⡜⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⠇⠀⡴⠚⠒⣤⠒⠓⢄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣀⢠⠞⣧⠀⠀⠤⠀⠂⣼⠢⡤⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠃⡞⣉⣤⠤⠶⠤⢤⡉⢳⠳⣤⠾⠋⠉⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠴⠒⠒⠓⠤⠟⠁⠀⠐⠀⠀⠀⠈⠻⠤⠛⠐⠚⢦⡀⠈⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠯⢼⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠲⢤⡀⠀⡼⠉⠑⠲⠦⠤⠤⢤⡤⠤⠤⠤⠴⠒⠋⠁⠀⠀⠀⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡔⠋⠉⠉⠳⡌⣾⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠤⠴⠶⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⡖⢰⠀⠀⣸⠇⠀⠀⢸⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠃⣘⠒⠀⡉⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⡁⠨⣉⠀⠁⠀⠀⠈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡌⠀⠀⠀⡗⣲⠾⢆⠀⠀⠀⠀⠀⠀⠀⡠⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢳⡴⢦⣿⡇⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⠀⢹⠷⡀⠀⠀⠸⡅⠀⢠⠇⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣸⡙⠀⠌⢠⠉⠁⠒⠒⠑⠀⠞⠒⠂⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⣈⠀⠶⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⠀⠀⠀⠑⠒⠒⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠷⠋⠀⠀⣀⠀⠓⠒⠶⣀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠉⠈⠉⠀⠐⣏⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣧⣤⡀⠀⢀⣀⣀⣀⣀⣀⣤⣭⡼⠷⠋⠁⠀⡜⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠟⠉⢩⣷⠟⠛⠛⠉⢉⣭⣭⠓⠀⠀⠈⣀⣤⣧⢤⣤⠄⠀⠀⠀⠀⠀⠠⠄
⣀⣀⣀⣠⣴⣶⣦⠀⢀⣤⣰⣿⣿⣁⢁⣴⠿⣏⠀⠀⠴⢾⣿⣯⣽⣷⣶⡾⠿⠛⠉⠁⠈⣿⡆⠒⠃
⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣧⣿⣿⣿⡟⢲⡝⠿⠀⢀⣠⢶⣲⣾⣿⣣⣼⡏⠀⠀⣄⠀⢿⣷⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡴⠋⢱⣶⣮⣿⣿⣿⢿⣧⣿⣿⡿⣷⢀⣄⣹⣆⢠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣟⣻⣿⣭⣾⣿⣿⣿⣷⡿⣿⣛⣷⣿⠷⢿⡗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡼⢀⢸⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠉⠆⣿⡀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣠⣽⣧⡀⢴⡗⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡆⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⢿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⢀⡴⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠈⠳⣄⢀⡴⠋⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠈⠉⠙⠲⣄⠀⠈⠿⣄⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢀⣀⣤⠴⠞⣧⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⡤⠶⠚⠛⠉⢁⣀⣤⠴⢾⠄⠀⠀⠀⠙⢶⣤⣀⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⡤⠶⠖⠛⠉⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣕⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠷⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⣯⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣮⡟⡅⠈⠳⡀
⠀⠀⠀⠀⠀⠀⡴⢧⣴⠊⠈⢂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢶⣤⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⠒⠢⠭⢄⡩⣄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠱⢾⣏⣿⣌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢵⠋⠢⣀⠀⠀⠀⠀⢀⡴⠃⢀⡴⠚⠁⠀⠙⢷⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⢄⠀⣠⠎⠀⣠⠊⠀⠀⠀⠀⠀⠀⠳⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡟⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣴⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡴⠋⠈⠑⢦⣈⣗⣤⡀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⠞⠀⠀⠀⠀⠈⠣⣝⢮⡻⢶⣶⡿⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⠉⠂⠙⢧⡀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠳⠄⠀⠙⠆⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⢿⠿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⠟⠛⢢⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣇⢀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠛⢧
⠀⠀⠀⢠⣳⡀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣸⡏⣸⡇⠀⠀⠀⠀⠀⣸⠃⠀⠀⣸⣣
⠀⠀⢠⠋⡉⢻⣶⠦⠌⠻⣦⡀⠀⠀⠀⢹⣷⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⣿⢁⣿⠇⠀⠀⠀⣠⡾⠋⠤⢶⣾⠋⡉⢣
⠀⠀⠀⢧⣼⣄⣍⠑⢦⣄⠈⠙⠤⡀⠀⠈⣿⡄⠘⢃⣤⠀⠀⠀⠀⠰⠶⠀⠠⠤⠄⠀⠐⠒⢿⢷⣟⡇⢸⡇⠀⠀⡠⠜⠉⢀⣤⠖⢋⣅⣼⣤⠇
⠀⠀⠀⠀⠈⠉⠉⠑⢦⣉⠢⣄⠀⠈⠲⢄⣹⣷⠢⠊⠁⣀⣀⡀⠀⠰⡄⠀⡀⠀⠀⢀⣠⣀⣀⡸⠞⢳⣿⢣⠴⠊⠀⢀⡤⢊⣡⠞⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⡳⢤⡀⠀⢹⣇⣠⣶⣾⣿⣿⣿⣷⣞⠱⠋⠹⣶⣼⣿⣿⣿⣿⣿⣦⡜⢻⠀⠀⣠⠴⣫⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢽⣲⣴⡇⠘⢿⣿⣿⣿⣿⣿⢏⡆⠀⠀⣼⣝⣿⣿⣿⣿⣿⠟⠓⣬⢶⣚⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠁⠙⠛⠛⠿⢏⡨⣧⣾⣷⣽⡇⡙⠛⠛⠛⠛⠁⠤⢒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠄⣀⣀⣂⡰⣿⠀⣿⣿⣿⣇⢵⣶⣦⠀⠀⢀⣀⣠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⠖⠹⠗⠛⣻⠋⠙⠚⠗⠸⠿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢸⣿⣇⡀⡀⡀⢀⠐⢀⠀⡄⠀⠀⣹⣿⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⣝⣩⢩⠎⠋⢹⠉⡏⢿⠉⢻⣿⡟⡂⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢀⣹⡆⢞⢿⣽⠴⢤⠄⠼⠠⠧⡔⠄⡫⠿⢁⣱⣇⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⢁⣠⡶⠋⠈⠣⠀⠀⠈⡉⠘⢐⢔⠐⠲⢱⡺⢅⠀⠁⠃⠈⠳⢦⣀⠙⠲⠤⣀
⠀⠀⠀⢀⠀⠀⣀⣀⡤⠖⠋⣁⣠⣴⡶⠛⠁⠀⠀⠀⠀⠀⠑⡄⠀⠁⠀⠀⠀⠀⠉⣣⡼⠃⠀⠀⠀⠀⠀⠙⠳⣶⣤⣀⡉⠓⠦⣄⣀⡀⠀⢂
⠀⠀⠘⢿⡀⠀⠟⠁⠴⠚⢻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠚⠁⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠛⠲⠄⠙⠇⠀⣸⠟
⠀⠀⠀⠀⠻⢿⠟⠁⠐⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠒⠀⠙⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣥⣀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣤⡁⣩⣤⠽⣾⢁⡆⢀⠀⢀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⣦⠀⢯⠘⢿⣸⣟⡿⣷⣾⣷⣮⣦⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⡴⣦⣶⣿⣿⣸⣿⣻⣿⣿⣿⣿⡇⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡛⢷⡀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣌⢱⡴⣿⣿⣿⣿⣿⠟⠉⠉⠉⠛⠻⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣾⣷⣿⡿⣿⣿⣿⠀⠀⡄⠀⠀⠀⢠⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⣿⣿⡆⠘⢧⡀⡀⠀⠀⣿⣶⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡗⣿⣿⣿⣦⠘⢿⣿⡄⣿⡿⣌⣿⣦⣴
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⠟⢸⣿⣿⣷⡘⣬⡁⢠⣤⣉⢸⣿⡿⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣌⠻⠌⠿⣿⣾⣿⠃⠛
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣞⢻⣦⣻⣿⣿⣧⡐⣦⢈⡉⠙
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⡌⢻⣿⣝⠿⣿⣿⣞⢿⣿⣿⡄⣸⠀⠀⠀⢳⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡘⠛⠛⢀⡿⣿⣧⣿⣆⠀⠀⠈⣷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣄⠘⢃⣮⠉⣿⣿⡄⠀⣦⠹⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⢿⠆⡸⣿⣶⡀⠉⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢑⣃⣈⡛⠃⠀⠀⢻
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⢁⡉⠙⠀⠠⣤⠘
⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡉⠛⠦⢠⣦⠈
⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⠀⠀⠸⠗⢄⣠⣿⣷⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠐⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⢠⠔⢋⡀⠀⠀⠐⠀⢿⡇⠉⢃
⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠶⠞⠟⠻⡏⣹⣿⣿⣿⣿⣿⣿⣿⠍⣠⣶⡿⢿⣆⠀⣀⠰⡜⣿⣿⣉
⣟⠷⢀⣉⢡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⣴⣦⣛⣿⣿⣿⣿⣿⣿⣽⡷⠘⠿⠉⠀⣸⣿⣄⠹⣷⡄⠉⠹⠿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡁⣀⣈⣿⣿⣿⣿⣿⡿⠟⢁⣩⣉⢁⡔⠀⢠⢀⣿⣿⣿⢧⡉⠉⡀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣶⣿⣿⣿⡿⣻⣿⣿⣿⣶⣿⣿⠟⠙⣿⣥⣶⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡛⠋⣹⣿⣿⡠⡧⢸⣿⠹⣿⡿⠋⠀⠀⣇⣴⣿⣿⣿⣿⣷⠆⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⣿⣿⣿⣿⠻⣇⣠⣌⠻⠇⢠⣀⢁⣤⣾⣿⣿⣿⣿⣿⣿⣧⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠹⠟⠛⠃⠈⠙⠿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢻⡄
⣇⠀⠀⠀⠀⡀⠀⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⣀
⡿⠗⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢾⣋⠄⡀⠈⠉⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡠⣄⣰⡿⣿⣿⢿⡿⠿⠿⢣⢀⣴⢧⣿⣿⣵⡆
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡸⠟⠻⠟⠃⢻⣷⠂⠀⠀⠀⠛⠈⠁⠀⠀⠉⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⠋⠉⠁⠀⠈⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠇⢀⡀⠀⠀⠤⣤⣀⡀⠀⠀⢀⣀⡠⠤⠀⠀⠀⢀⡾⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢼⠁⠀⣀⣀⣠⣠⡄⠈⢁⡀⠉⣠⣤⣤⣀⣀⡘⣾⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣰⣿⣿⣿⣿⣿⣿⡄⢘⡇⢰⣿⣿⣿⣿⣿⣿⣾⡆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠇⢿⣿⣿⣿⣿⣿⡿⡏⣉⠉⡿⣿⣿⣿⣿⣿⣿⡏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⡈⠻⢿⠿⢿⠿⣣⢱⣿⡇⢣⣈⠻⠿⣿⣿⠟⢁⢨⠗
⠀⠀⠀⢀⡤⣤⠤⣄⠀⢳⣀⣸⠛⠛⠛⠓⢺⢧⣿⣿⣿⡄⡏⠛⠛⢢⢖⣾⣅⡼⠀⠀⢀⣤⠶⠦⣄
⠀⠀⠀⣏⠀⠀⠀⢸⡆⠈⠛⠉⠉⠉⢳⡀⢸⡼⠿⠛⠟⢃⣧⠀⠐⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸
⠀⠀⢠⠜⠒⠦⢤⡈⠳⣄⠀⠀⠀⠀⢸⣤⡈⠀⠀⠀⠀⠀⠘⠀⡓⠀⠀⠀⠀⠀⣀⡼⢁⠎⠀⠚⠓⢦
⠀⠀⣧⠀⠀⠀⠀⠑⠄⠈⠙⠢⣄⡀⠀⠣⣜⣙⡛⢺⢲⣲⣙⡭⠗⢀⣠⠴⠚⠉⠁⢀⡀⠀⠀⠀⣀⡜
⠀⠀⠈⠳⢤⣤⠤⠖⠒⠲⠦⣄⠀⠉⠒⠦⠤⣀⡈⠁⢁⣀⡤⠖⠚⠉⡀⣠⠴⠖⠊⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⣀⣠⠤⠛⠋⠩⠒⢀⣠⣴⣛⢉⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⡠⠶⠒⠲⠤⠤⠤⠖⠋⠁⠀⠀⣀⡤⠶⣞⡉⠁⠀⠉⠉⠀⠀⠀⠀⠈⢹⠄
⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⠀⢀⠖⢁⡴⠖⠋⠁⠀⠀⠀⠈⠑⠢⣀⠀⠱⠄⠒⠶⣶⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠓⡖⠀⠉⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⠴⠃