Nucleic acid Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣀⣀⣀⣀⣀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣟⠛⠛⢛⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⣠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⣻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃⠀⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣈⣉⣉⣉⣉⣉⣉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⠿⠿⠿⠿⠿⠿⠷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⣉⣉⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⣤⣤⣤⣤⣤⣤⣤⡄⠀⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⣴⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⢀⣤⣄⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠹⣿⡿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⢶⣶⣶⣶⣶⣶⣶⡶⢶⡦⠀⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⠀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢼⣿⣿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠈⠙⠉⠀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣷⡆⠀⣠⣤⣤⣤⣤⣤⣤⡀⢀⣶⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⡀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⣠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢆⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⢶⣦⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠌⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⣌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠅⠄⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⡚⠖⢣⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠮⠤⠐⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⠀⠀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠂⢠⡤⢂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡘⢕⡈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢦⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠚⠃


⠀⠀⠀⠀⠀⠀⠀⠀⢄⣄⡀⡀⣀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡩⣀⢀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠃⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣍⡁⠀⣈⢥⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡔⠒⠒⠒⠒⢤
⠀⠀⠀⠀⠀⠀⢀⢀⢕⠉⠀⠀⠀⢐⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⣢⠤⠤⡚⡂
⠀⠀⠀⠀⠀⠀⠀⠑⢁⠋⠀⠉⠈⠑⠝⠢⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠕⠷⠂⠐⠒⠕⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡔⢁⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠖⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠈⠀⠀⣱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢑⠷⠂⠐⠲⢴⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠈⠆⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠠⠤⠤⠤⠤⠤⠤⠄⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠰⠶⠶⠶⠶⠆⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⡒⢒⣢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠻⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣈⣉⣉⣁⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣈⣉⣉⣉⣉⣁⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠠⠤⠤⠤⠤⠤⠤⠤⠤⠄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠖⠶⠶⠒⠒⠒⠒⠶⠶⠶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡐⠒⠒⠒⠒⠒⠒⠒⠒⢂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⢉⣉⣉⣉⣉⡉⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡠⠤⠤⢄⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠫⠤⠤⠝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠐⠒⠒⠒⠒⠂⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣘⣛⣉⣋⣉⣉⣙⣃⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣈⣉⣉⣉⣉⣉⣉⣉⣉⣁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠸⠇

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣿⠃⡇⠀⣀⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⢰⡗⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠆⠀⠖⠓⠢⢄⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⣀⠀⠀⠐⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⢀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠉⠛⠶⣤⣀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣡⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣆⠘⢿⣷⡄⠀⠀⠉⠙⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⡇⠸⣆⠀⠻⣿⣦⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠹⡄⠀⠈⣫⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⢠⡄⠀⠀⠀⢸⡷⠀⢀⣹⣴⠾⠋⠱⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⡄⠀⠀⠀⠉⠢⣄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠈⠉⢀⣀⡀⣾⡇⠘⠛⠉⠀⠀⠀⠀⠈⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣨⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠠⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡜⢿⣿⡇⠀⠀⠀⠀⢻⡇⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣝⣷⣝⠂⠀⠀⠀⠀⠈⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⡀⠀⢀⣿⣿⣿⡍⠳⣦⡀⠀⠀⢠⣿⢃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠑⡄⢼⣿⣿⣿⠀⠀⠀⠙⢆⣴⡿⠁⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠈⢿⣿⡇⠀⠀⠀⣠⡾⠋⠀⠀⠀⠙⠂⠀⠀⠀⠀⢠⣾⣶⣷⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⣼⡇⠀⠀⣾⣄⠀⠹⠀⣀⣴⠿⢋⣀⣀⣠⣤⣄⢀⣤⣤⣄⣀⣻⣨⡿⠿⠛⠉⠁
⠀⠀⠀⠀⢸⠀⠄⠤⠀⠂⠀⠀⠀⠰⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇⠀⠘⢋⣉⣤⡴⠟⣋⣵⠞⠛⠻⠿⣿⠿⠋⠘⠿⠛⡫⠟⠁
⡴⠶⠞⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠁⠀⠀⠰⣿⠁⠚⣛⣫⣭⡄⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠌⠀⠀⠉
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⠀⣾⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⣿⠃⢰⡆⠀⠀⠀⠀⠔⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠚⠛⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠠⠤⠤⠤⠤⠤⠤⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⣤⣤⢤⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠈⠉⠉⠉⠁⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠉⠉⠉⠉⠉⠉⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢠⣀⣀⣀⣀⣀⣠⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢀⣀⣀⣀⣀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠛⠛⠛⠁⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠘⠛⠛⠛⠛⠛⠛⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⣀⣀⡀⢀⣀⠀⡀⠀⢀⡀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⢸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠃⠀⣿⣿⣿⠀⠸⠿⠿⠿⠿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⠉⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠋⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⢿⠇⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⢀⣶⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣷⣄⣀⣠⣴⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⠀⣿⣿⣿⡇⢠⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠀⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠺⣿⣿⣿⠿⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠈⠋⠁⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣦⡀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡆⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠏⡆⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢦⡀⠀⠀⠀⠀⠀⠀⡜⡎⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⢆⠳⡙⢆⠀⠀⢀⡤⡞⡜⠀⠚⠂⣿⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⠀⢀⡱⠬⠖⠛⠉⠈⣻⣰⡥⠤⠖⠊⢫⢳⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠴⠗⠒⠉⠁⣀⡠⠤⠖⠒⠉⠉⠈⢷⡀⠀⠀⠀⢇⢇⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠉⢁⣀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⢀⣀⡬⣧⢀⣀⠤⢾⣿⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡳⣤⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⡺⠯⠗⠓⡏⠀⠀⠀⢸⣼⣿⡿⠟⢡⡾⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⠼⢄⡀⠳⠗⠀⠀⠀⠀⠀⣀⡤⡴⣖⡹⠷⠛⠋⠁⠀⠀⠀⠀⣇⣀⣠⣴⣾⠟⠋⢀⡴⠋⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⣾⣷⡜⠌⠓⢖⣉⡹⠀⢀⣤⢴⣲⣹⠓⠛⠉⠀⠀⠀⠀⣀⡠⠤⠔⠒⠻⡍⠳⡜⢯⠁⣠⠴⠋⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠚⠉⠁⠀⢀⣀⡤⠴⠒⢎⠉⠀⠀⠀⠀⠀⠀⠈⢦⡈⢢⡙⣅⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣽⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⠀⢀⠀⢀⣀⣀⡠⠤⠶⢶⣯⣭⣭⣭⣭⣭⡭⠭⠶⣒⡤⠄⣀⠀⠀⠀⠑⣄⠙⣌⣳⡀⠀⣀⣴⣿⣿⠟⣡⣾⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠁⠒⠫⠵⠶⠶⠮⠍⠚⢉⣤⣾⣿⡿⠋⣁⣾⠟⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠉⣠⡾⠋⠁⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠉⣀⡴⠚⠁⠀⠀⠀⠀⢀⣴⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠋⠁⣠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣤⠿⠋⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠋⠉⢀⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠉⠉⢁⣀⣠⡤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⣠⣶⡿⠋⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⣀⣴⡾⠛⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⢀⣠⣴⠾⠛⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣶⡶⠶⠞⠛⠋⠁⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣀⣀⣀⣤⣴⣶⡾⠟⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠥⠄⣀⣀⣀⣀⠠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⠻⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⣿⠈⠻⣦⣈⣻⣶⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠀⣷⣄⠙⢷⠌⠉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠻⣯⡙⠃⣿⡏⠻⠿⠛⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣄⠰⣶⣾⡄⢻⣤⣤⣤⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠻⣦⡈⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢶⣄⠙⢷⣌⠻⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡈⠻⣦⣨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⢷⣄⠙⠷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣤⣶⣤⣸⣿⢠⣌⣻⣦⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠛⠛⣧⠘⡿⠿⠆⠙⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠿⣯⡉⠻⣦⡀⣿⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣀⡐⢷⣄⠙⢿⠀⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡙⢷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣍⠀⠁⠀⢂⠲⢀⡀⠀⠀⠐⡡⢂⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢊⠈⠁⠀⠀⢠⡀⣋⠀⠈⠉⠚⠠⠅⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠐⠐⠀⠄⢀
⠀⠀⠀⠀⠀⡀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⣼⢤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⡀⠉⠀⠂⠀⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠀⠈⠈⠐⠘⠀⢰⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡐⠈
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢇⠀⠀⠀⣠⡿⢛⡿⠈⠐⡠⢄⣠⠀⠀⠀⠀⠀⠈⠉⠚⠃⢇⡖⡤⣀⠀⠀⠀⠀⠀⠑⠀⣀⠔⢢⠜⢠⡀
⠀⠀⠀⠀⡠⠸⣽⣿⠿⠗⢛⣴⣵⢄⢘⡶⣶⡇⢀⠄⡀⠀⠀⠉⠀⠀⠒⠠⠀⡀⠀⠀⠀⠀⠁⠙⠽⢃⢂⢤⠀⠀⠄⠠⠃⠀⠂⣴⣿⣦⡀
⠀⠀⠀⢠⠽⡺⢁⡰⣢⣿⡶⢣⡇⠘⡻⢀⣍⡗⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⢄⣀⡀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡆
⠀⠀⠀⡾⠄⢡⣳⠉⠹⡿⠂⢴⡟⢲⠁⠀⢖⠅⢀⡐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠠⠀⠀⠀⠀⠠⠿⠿⠿⠛⠁
⠀⠀⢰⣄⠀⣘⠉⢠⡔⠂⢹⣿⠰⠈⠓⠊⠀⠁⠀⠀⠐⠀⠀⠀⠉
⠀⠀⢸⠀⠩⡇⢐⡺⠋⠸⠃⢐⣤⡟⡄⠀⠀⠈
⠀⠀⠀⠀⠴⠇⢈⡟⢶⣤⠶⢦⣟⣴⠇
⠀⠀⠀⠈⠠⢺⡞⢛⠻⠮⡶⡾⢞⣁⠈
⠀⠀⠀⠀⠘⠿⠗⠀⠀⠒⢃⣍⣄⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠘⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠄⠀⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣡⡀⢠⠴⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠞⠁

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

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⡐⠀⠀⠀⠐⡃⠄⠀⠀⠀⠠⢰⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠘⠀⠀⠀⢀⠘⠐⠀⠀⠀⠀⠀⠠⡑⠀⠒⡐⢊⠄⠀⠀⠀⠀⠈⠉⠌⠈⠁⡁⢁
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠩⠐⠁⠀⠀⠀⠀⠀⠀⠀⡈⠑
⠀⠀⠀⠀⢀⡈⠁⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠴⠠⠤⠔⠆⠂⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⡈⢂⠁⠀⠀⠈⡂⠀⠀⠀⠀⠀⢘⣐⣐⣒⣒⣒⡃⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠘⠈
⠀⠀⠀⠀⢁⠄⡀⠀⠀⢀⠥⠀⠀⠀⠀⠀⢨⠤⠤⠤⠤⠤⠅⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⢀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠂⡱⠒⠒⠲⢎⠄⠀⠀⠀⠀⠀⠈⠐⠀⠀⡄
⠀⠀⠀⠀⠀⢀⠄⠊⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⣩⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁
⠀⠀⠀⠀⢠⠂⠄⠀⠈⢠⠂⠀⠀⠀⠀⠀⠔⠴⠤⠤⠤⠌⠂⠀⠀⠀⠀⢀⡀⠐⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡖⠠⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠘⠒⠒⠒⠒⠚⠂⠀⠀⠀⠀⣐⠈⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠈




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠙⢦⡈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠳⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢦⡈⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⣾⠿⠿⣿⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠻⣄⠘⢦⡈⠳⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣄⠙⢦⡈⠱⣄⠙⢦⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠢⡈⠳⣄⠉⢦⡈⠱⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠑⢄⠉⠢⡈⠳⣄⠙⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠢⡀⠑⣄⠈⣦⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣾⣷⠶⠿⠶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⢟⠉⠣⡀⠑⢄⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢏⠉⠢⡀⠑⢄⠈⠢⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠣⡀⠑⢄⠈⠢⡀⠑⢄⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠑⢄⠈⠢⡀⠑⢄⠈⢢⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠢⡀⠑⢄⠈⠢⣀⣱⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⢄⠈⢢⣀⣵⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⣶⠶⢾⡿⠾⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠻⣅⠙⢦⡈⠣⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣄⠙⢦⡈⠳⣄⠙⢦⣼⡇
⠀⠀⠀⠀⠀⠀⢀⣿⠛⢦⡈⠳⣄⠙⢦⡈⠳⣤⡿
⠀⠀⠀⠀⠀⠀⢸⡏⠳⣄⠙⢦⡈⠳⣄⠙⣦⡿⠁
⠀⠀⠀⠀⠀⠀⣿⡟⢦⡈⠳⣄⠙⣦⣼⡿⠋
⢀⣀⣠⣤⣤⣤⣿⣧⣶⣿⡶⠾⠟⠛⠉
⣟⠙⢯⡈⠳⣄⣿
⡈⠳⣄⠙⢦⣨⣿
⠙⢦⡈⠳⣄⣽⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢦⠀⠀⠀⠑⢦⣁⠚⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⠀⠘⡐⠓⡀⠀⠀⠀⢻⠻⣄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠂⠀⠂⠀⠋⠀⠀⠀⠀⠀⢿⠏⠳⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡔⠈⠂⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠠⠀⠀⠀⠁⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢰⣄⠀⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⣱⡀⠜⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠓⢀⠀⠀⠈⠉⠰⠆
⠀⠀⠀⠀⠀⠀⢉⠀⢄⠀⠀⠀⠀⠀⠀⠉⢀⡀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⢸⡀⢈⡆⠀⠀⠀⠀⠠⡄⠀⢛⣆⠀⠸⣅⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠇⠈⠁⠀⠀⠀⠀⠈⢠⠄⠘⢀⠱⡀⠀⢰⠄⠀⠀⠀⠀⢐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠙⣷⠀⠀⠐⠁⠀⠁⠈⠀⠀⠄⠀⠀⠲⣄⠀⠀⢐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠳⣮⣾⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠙⣆⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣴⠏⠳⣄⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣈⣁⣀⣀⣀⣀⣠⣚⣁⣀⣀⣌⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠏⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⣼⣿⣿⣿⣿⣷⣦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣼⣿⣿⣿⣿⣿⣿⡿⣿⣿⣧⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣷⡀
⠀⣼⣿⣟⠃⢛⣿⣿⣿⣿⣿⣿⣿⣿⡫⠴⠒⠒⠒⠒⠒⢒⣩⣾⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⠃⣻⣿⣧
⢰⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⢟⣩⠶⣚⣛⣉⣙⣛⣛⣭⠵⠞⣱⡿⢈⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿
⣾⣿⣿⣿⣿⣿⡿⠿⠿⠟⣿⡿⣫⠖⠋⠉⠉⠀⠀⠀⣀⣠⣴⣿⠟⣡⣾⢋⣿⡟⠿⠿⠿⢿⣿⣿⣿⣿⣿
⣿⡿⠛⠉⠁⠀⠀⣠⣴⣶⣿⡟⣁⣴⣾⣿⣿⣿⣿⣿⠿⠛⠉⣠⣾⢟⣵⠟⣻⣷⣶⣦⣄⠀⠀⠈⠉⠛⢿
⣿⣄⠀⠀⠀⠀⠀⠛⠿⢿⣿⣴⣿⣿⣋⣉⣀⣀⣀⣀⣠⣄⣼⣿⣽⣟⣡⣾⣿⣿⠿⠿⠛⠀⠀⠀⠀⢀⣠
⢿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⡀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿
⠸⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣛⣛⣿⣿⡿⠛⣿⠟⠛⣿⢟⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣛⢉⣉⣤⠞⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⡟
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡏⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣷⣤⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢄⣣⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠟⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣶⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠷⢶⣤⣄⣀⠀⠀⠀⢤⣍⣙⠛⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣴⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠉⠙⠻⠶⣦⣼⣿⣿⣿⡆⠀⠈⠉⠛⠛⠿⠛⠿⠿⠻⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠛⠻⠶⠦⢄⣀⡀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠛⠶⠦⠄⢀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⣀⣀⣀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠁⠐⣦⣾⠇
⠀⠠⠒⠉⠐⠀⠁⠐⠠⢜⣿⣿⣿⣿⣿⣿⣷⣶⣤⣬⣉⡛⠃⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠉⠉⠉⣙⣋⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠦⠔⠒⠒⠢⠴⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⣠⢤⣔⣒⣒⣈⣉⣭⣭⣭⣀⣒⣒⡢⠤⣄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠉⠙⢻⣿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣻⣿⣿⡀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⠘⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣀⣥⣭⣭⣅⢉⣾⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⣷⣄⠙⠻⢿⣿⣷⣶⣤⣤⣄⣀⣀⣠⣤⣤⣶⣿⣿⣷⡧⠄⣠⣴⠬⠶⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣄⣀⣿⣿⣷⡄⣀⡉⠙⠿⢛⣻⣿⣿⣿⣿⣛⡛⠿⠛⢉⣀⣴⣾⣿⣿⣀⣠⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠿⡿⣿⣿⣿⣿⣗⡏⠉⠑⠒⠲⠶⠤⠤⠤⠤⠶⠒⠒⠊⠉⢹⣿⣿⣿⣿⣿⢿⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢻⡆⠇⢱⠀⠀⠙⠿⢿⣴⣒⡠⠀⠀⠀⠤⠄⠀⠀⠀⢄⣒⣾⡿⠿⠋⠀⠀⡎⠸⢰⡟⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢸⡇⢰⠈⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠀⡇⢸⡇⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢸⠃⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠸⠀⠀⠀⡰⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⢦⠀⠀⠀⠇⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⡉⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢀⠰⠀⠒⠄⠀⠀⠀⠀⠀⠠⠰⠀⠀⠄⠆⢀
⠀⠀⠀⠀⠀⠀⠸⠤⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⢃⢀⠞⡍⠌⣂⠄⡀⠀⠀⠀⠀⠐⢯⢨⠭⠅⠭⠝⠹⡆
⠀⠀⠀⠀⠀⠓⠖⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⡤⢁⡀⣤⠀⠀⡖⠒⠀⠀⠀⠀⠀⢀⠉⠲⡠⠔⣇⡄⠴⠀⠀⠀⠀⠀⠇⡀⠀⢆⠂⠀⠠⠇
⠀⠀⠀⠀⠀⠰⠉⠈⠉⠡⠀⠀⠀⠀⠀⠀⠀⠨⠀⠨⠤⠄⠅⠀⠑⠀⠀⠀⠀⠀⠀⠈⠚⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠱

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠀⠀⠈⠀⠠⢀⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡉⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠄⠀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠊⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠰⣤⠚⡖⣰⠒⠀⠀⠀⠀⠀⠀⢠⣾⠒⢠⡘⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠠⡢⠀⣠⠀⡔⠄⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠀⠀⠈⢬⡱⡀⠰⠹⠀⠀⠀⠀⡔⠁⠀⠈⣤⡀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣈⠉⠀⡀⠡⠂⠀⠀⠀⠀⡄⠀⣠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⢀⠁⠐⢄⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢦⠀⢤⠂⠀⠀⠀⠀⠀⠀⢀⡔⠓⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢶⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⢇⠈⢧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣧⠈⢇⠈⢆⠈⢇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠠⣦⠘⢇⠘⣇⠘⣇⠙⢧⠈⢆⠈⢆⠈⢆⠈⣦⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣈⣿⣿⣿⣿⣿⣿⡙⡇⠘⣄⠈⣆⠘⣆⠈⣆⠸⡧⠸⠇⠸⠇⠸⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠛⠋⠈⠛⠛⠿⣿⣿⣿⣿⣧⠘⡄⠈⡄⠘⡄⠈⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢋⣬⠐⡄⠘⣄⠸⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⡷⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡠⠞⠋⠁⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⣀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⢶⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⢂⣴⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣴⣾⡟⣼⡇⣾⣿⣿⣿⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠚⢿⠟⠂⠀⠀⠀⠀⠀⣀⡤⣷⣿⣿⣿⣇⣿⣇⢿⣿⣿⣿⣿⣦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢖⣵⣿⣷⣾⣿⣿⣿⣿⣿⡸⣿⡌⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⣠⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣦⡙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⠿⠶⣤⠄
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢿⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣌⠻⣿⣿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣾⣿⠀⠀⢀⣰⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⠀⣶⣿⣿⣿⣿⣿⣿⡿⠟⢩⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⡀⠙⠟⠉⠁⠈⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⣼⣿⡟⠀⠈⠙⠉⠸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣰⣦⣀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣧⠹⣿⣆⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠹⣿⡆⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⢻⣦⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣶⣶⣾⣿⣶⡶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠀⠘⠀⠘⠃⠀⠛⠀⠀⠀⠸⠀⡂⠀⠀⢀⠀⠁⠘⢀⠀⠠⠀⠀⠐⠂⢐⠀⠀⠀⠐⠟⠀⠘⠃⠀⠃⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣿⣽⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⢤⣶⣴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠔⠂⠉⣿⡈⠣⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡠⠚⠈⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠐⢲⣶⣦⠊⠁⠀⠀⠀⠀⡄⠀⠰⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢣⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠻⡃⠛⢦⠀⠀⠀⠀⠀⠘⠀⠠⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡾⣧⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠈⠑⠂⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠐⠀⣤⠳⣮⣶⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠠⠄⢀⠤⣰⠘⣟⠣⣼⡀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠘⠀⣇⡤⠀⣼⣴⢸⡇⠘⢷⣷⣼⣦
⠀⠀⠀⠀⠁⠀⡤⠒⠀⢿⣭⠘⣿⠄⠠⣿⡿⢾⡗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⠀⣀⠁⠈⠻⢧⡉⠀⠀⠋⠙⠲⠠⠦⡳⣄⠀⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠇
⠀⠀⠀⠀⠘⠧⠀⠀⠁⠀⠌⠠⠀⠠⠘⠐⠈⠁⠀⣀⠐⡌⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠀⣀⠀⠀⠀⠀⠈⣤⠰
⠀⠀⠀⠀⠀⠀⠈⠐⠢⣤⡴⠟⠓⠚⠋⠘⠃⠉⢘⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⢉⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡁⠀⠀⠈⠀⠀⠛⠂⢘⡗⣧⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠃⠀⠀⠀⠀⠀⠛⢂⠀⠑⣼⣧⣾⡗⠈⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠤⠧⠛⢿⣧⢤⣰⣯⢫⠁
⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢸⠀⣤⢀⡉⣅⠀⠒⢲⠀⣀⢤⣀⣀⠀⠀⠀⢀⣀⡾⠇
⠀⠀⠀⠀⠀⠀⢠⠘⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠘⠀⠉⠈⠁⠉⠀⠀⠘⢚⠑⣧⠿⠇⡃⢀⣤⡽⠋⠀⠀⠀⠀⠀⠌⠣⡄
⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⢀⣀⣀⡀⠀⠀⠀⠀⣀⠀⢠⠂⣦⡇⠳⡀⠀⠀⠀⠠⠎⠀⠎⠐⠀⣴⣿⡏⡌⠀⣤⣀⠞⠋⠀⠛⠛⠶⠓⣆⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠇⣄⠀⢘⠁⡇⠀⠀⠀⢠⣭⡄⢸⠀⡝⡇⢠⠇⠰⠖⠀⠰⣶⣧⠘⠀⠈⠉⠙⠻⢿⡓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡉⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠌⢧⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠠⠀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠚⠀⠃⢀⠀⠀⠀⠀⠀⢠⣘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⠣⣜⣆⣀⠉⠀⣀⠀⠀⢉⣉⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠰⠂⠰⠠⠔⠇⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢹⠑⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠊⠓⠒⠄⠘⠀⠒⠀⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⡀⡀⠀⠀⣄⠀⠀⠀⡀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠁⠁⠈⠂⠀⠉⠈⠊⠁⠀⠁⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠘⠋⣡⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢈⣤⡀⠈⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⠟⠿⣿⡀⠈⠑
⠀⠀⠀⠀⠀⠀⡛⠁⠀⠀⢠⣤⣤⡌⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠁⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣈⣥⡦
⠀⠀⠀⠀⢀⡀⡁⠀⠀⠠⠿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣄⠋⠁⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⡂⠀⠀⠀⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⢰⣶



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣋⣍⣹⣟⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣀⣙⠁⣼⡟⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⣈⠁⢰⣿⡀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠁⠀⣾⠏⠙⠶⣄⡀⠀⠀⠙⠳⢦⡀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠙⠳⣤⡀⠀⣠⡿⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠈⠙⢶⡟⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡿⠲⣤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣤⣸⠇⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢸⣧⡀⠀⠉⠛⠳⢤⣀⠀⠀⠀⣠⡾⠋⣄⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⠙⠳⢤⣀⠀⠀⠉⠓⢶⡿⠋⠀⠃⠈⢀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢿⡄⠀⠀⢉⣻⣦⣤⣀⣀⣹⠄⠀⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣧⠀⠀⢸⣷⠶⠟⠛⠉⠁⠀⠈⢉⣡⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠟⠛⠉⠉⠘⣿⠀⠀⠘⣿⠀⠀⢀⣀⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠟⠛⠉⠁⠀⠀⠀⣀⣠⣤⣾⣿⠀⠀⠀⣿⡶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⢀⣤⣤⣴⣶⠟⠛⠋⠉⠁⢠⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⣿⡃⠀⢀⣼⠷⣄⡀⠉⠛⠳⢦⣄⠀⢸⡟⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠰⠆⠀⣴⠟⠁⠀⠀⠙⠳⢦⣄⡀⠈⠙⣿⠃⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢀⣤⣼⣏⡀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣾⡟⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣅⠀⠈⠙⠳⣤⣀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡟⠶⣤⣀⠀⠉⠛⠦⣄⣀⣴⠟⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢸⣧⠀⠀⠉⠛⠶⣄⣀⣬⡿⠋⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⢿⡄⠀⢀⣀⣤⡾⢟⡉⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣀⣠⣼⣿⠾⠛⠋⣁⢲⡘⢧⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠈⠁⢀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⡶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢀⣠⣴⠶⠿⢿⣟⠉⠉⠀⠹⣷⡀
⠀⠀⠀⠀⢠⡾⢋⡲⡝⢀⣴⣾⣟⣉⠀⠀⠀⠀⠻⣧⡀⠀⠀⠘⢿⡄
⠀⠀⠀⣰⡟⠁⠈⢁⣴⣿⣁⠀⠀⠉⠛⠲⢦⣄⡀⠹⣷⡀⠀⠀⠘⣿⡀
⠀⠀⣼⠟⠀⠀⣠⡾⠋⠀⠉⠙⠲⢤⣄⡀⠀⠈⠙⠳⠿⣇⡀⠀⠀⢹⡇
⠀⣾⠏⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣀⡀⠈⠙⣦⠀⠸⣿
⣼⠏⠀⠀⣰⡟⠙⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠒⠋⠀⠰⣿
⢿⣦⣠⣼⠟⠙⠳⢦⣤⣀⠈⠉⠛⠲⠦⣤⣀⠀⠀⠀⢀⣿⠀⠀⠀⢸⣿
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠶⢤⣀⡀⠀⠈⠙⠓⠦⣼⡏⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣄⣀⠈⢹⡆⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠋⠁⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⣤⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠈⠉⠉⠉⠐⠒⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⢋⣷⢄⡀
⠀⠀⠀⠀⢀⣀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠛⢁⣴⠟⠁⠀⠉⠢⡀
⠀⠀⠀⠻⣥⠄⠒⠒⣉⣱⢤⡀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠉⢀⡴⠋⠀⠀⠀⠀⠀⣠⣿⣄
⠀⠀⠀⠀⠀⠑⢯⣁⡩⠶⣔⠛⢢⣀⠀⠀⠀⠀⡜⠀⠀⠀⣀⣠⣤⣶⣿⣿⡿⠟⠋⠁⠀⡠⠞⠁⠀⠀⠀⠀⠀⣠⣾⣿⡿⣩⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣖⠊⣛⣥⡨⠷⢄⠀⣼⣶⣶⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⢉⣼⡿⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡤⠝⠶⣊⠙⣻⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⢁⣴⠟⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠉⠙⠿⣖⣒⡭⢿⠤⠤⢐⣒⡺⡍⠑⡔⢢⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠋⣠⠶⠋⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠙⡹⣆⠈⠆⢿⠿⣿⠛⢸⠀⢱⠴⠁⢀⣠⣤⣶⣿⣿⠿⠛⠉⣀⠴⠊⠁⠀⠀⠀⠀⢀⣼⠟⢉
⠀⠀⠀⠀⠀⠀⢈⠀⣀⣀⠤⢄⠀⠀⢡⢻⠀⢸⠘⢃⣉⣶⣼⣤⣾⣶⣿⣿⡿⠟⠛⠉⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⣠⡶⠋⢁⣴⡿
⠀⠀⡠⢤⣤⡲⢯⠭⠔⠒⠊⠉⠀⠀⢸⡿⠄⢺⣿⣿⠿⣉⡩⠷⢤⡚⢫⣄⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⣴⡿⢛⠇
⠀⠀⠑⠉⠀⠀⠀⠣⡀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢣⣤⣤⣼⠶⣖⠊⢙⣧⣤⠵⢤⡀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⣠⣶⠟⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠖⠁⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠙⢤⡤⠜⠳⣎⣉⡦⣤⠔⠚⠉⠀⢀⣠⣴⠟⠋⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣤⣤⣤⡼⠶⣉⣉⠹⠤⠘⠓⢤⣴⠾⠛⠉⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢍⣀⣀⣀⣬⣷⣤⠴⠖⠚⠉⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠫⠥⠀⣀⣀⣀⠠⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠶⠒⠒⠛⠋⠁⠀⠀⠀⣀⣀⣀⡠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣣⠤⠤⠒⠒⠒⠛⠋⠉⠉⠉⢁⣀⣀⣀⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⠦⠤⠶⠒⠒⠒⠛⠉⠉⠉⠉⠉⢁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠒⠲⠖⠒⠒⠒⠋⠉⠉⢉⡽⠋⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡒⠓⠒⠚⠛⠊⠟⠉⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠲⠋⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢀⡠⠂⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⢀⡤⠞⠓⠒⠓⠶⠒⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡠⠾⠿⠶⠶⠶⠶⠶⠶⠶⠲⠶⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠶⠓⠲⠒⠒⠲⠦⠦⠦⠤⠤⠤⠤⠦⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⢤⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠠⠦⢤⠤⠤⠤⢤⣤⣤⣤⣤⣤⠤⠊⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠤⠴⠤⡤⠤⠤⣄⡤⠖⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⣤⠦⠐⠉⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⡀⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⣤⣆⣉⣄⣀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣀⣴⢮⣅⣀⣀⣀⣀⠉⣌⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠾⠥⢄⣀⣀⣀⣀⣀⠀⠈⠉⠙⠉⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠤⠤⣤⣀⣀⣀⣀⣀⣉⡉⠉⠉⠉⠉⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⠄⢀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⡀⠸⢀⣄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⣀⣴⣾⡯⠤⠾⠛⠁⠀⠀⠀⢠⣿⣥⣿⣿⡷⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠛⠋⠂⠈⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣇⠄⠈⣾⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠏⠈⡄⠀⡠⠔⢏⠉⢹⣇⣇⣦⠖⠁⠀⣴⣿⣿⣿⡆⠀⠀⠀⠤⣿⡀⠃⣼⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⣢⣇⣴⠿⣦⣤⣼⣿⠾⣷⣆⠀⠹⣿⣿⡿⠃⠀⠀⠀⡖⣿⣏⣹⣿⢸
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠄⠄⠀⠀⠈⠁⢻⣄⣽⣿⡿⠧⠤⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⢸⣿⣹
⠀⠀⡤⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⢰⠀⠀⢻⣥⣽⣷⠀⣠⣴⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⡷⡿⡇⢸⢿⢾
⠀⠀⣤⣀⣴⡇⠀⠀⠈⠉⠛⠶⢶⣾⣄⣤⡀⠀⠈⢿⡏⠀⣿⡓⣶⣵⡀⠀⠀⠀⠀⠀⠀⣿⠃⡇⢸⡎⢿⡀
⠀⢾⣿⠋⠙⠳⣄⠀⠙⠓⠶⠟⠋⠀⠈⠉⠉⠀⠀⠈⢿⣤⣴⡝⢿⣟⠁⠀⠀⠀⠀⢀⣾⣹⡿⢹⣾⢹⣿⣷⡄
⠀⢠⣿⣦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠁⠘⣉⠀⠀⠀⠀⠀⠀⠁⣟⣷⢿⣿⢾⣘⠆⠁
⠀⠉⠀⠀⠉⠳⢴⣄⣀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠻⢥⡑⣄⠀⠀⠀⠀⠘⠁⠈⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢂⠀⠀⠉⠁⠀⠀⠀⠀⣠⡖⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠈⠙⠲⢕⢄⠀⠀⠀⠀⠀⠀⠀⢹⣤⡆⠀⠀⣰⠃⠀⠘⠋⠉⠠⠂⡀⠐⠀⣆
⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⢀⢈⠀⠀⠀⣄⠁⠰⠆⠀⡀⣐⣰⣿
⠀⠀⠀⠀⠘⢠⣤⡀⠀⣠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣤⠶⠷⠛⣨⠀⠀⠈⣶⣴⣰⣾⣥⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠛⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡤⠚⠋⠀⣀⣴⣾⣯⣄⣀⠀⠈⠙⠛⠛⠛⠛⠁⠀⡀
⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠚⠋⠀⢀⣤⡼⢟⣋⠥⠭⠯⠥⠭⠵⠖⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠈⠁⠀⠀⢀⠐⠁⠀⣠⢾⠀⣀⣴⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠔⠂⠄⠀⠄⠒⠐⠁⠀⠰⠿⠷⠛⠋⠉⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠴⠘⡘⢃⡄⣀⠠⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣏⣻⣿⣬⠿⡿⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⡏⠀⢠⣆⠀⢈⣍⠈⠑⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⡿⣿⣿⠋⢿⣿⠶⣿⣿⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠊⠁⠈⠁⠉⣿⣿⣌⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣾⣷⡄⠀⠹⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠈⠛⣷⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣦⡌⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠁⣗⠁⠀⠀⠀⡸⠗⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⡏⢿⣿⣿⣿⣿⣨⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⡠⠑⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠠⡛⠿⣦⣜⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡘⠛⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠠⣡⠓⠀⠀⠈⠹⣇⡔⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣯⣿⣿⣿⣦⡈⢟⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⠀⢀⢀⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢾⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢁⠙⠿⣶⣤⡘⠒⠀⠆⣠⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠈⠛⠻⣭⣿⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⡠⢀⡔⠟⠙⠻⣷⣔⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣧⣶⣤⡈⠛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣇⠃⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⣆⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣀⠀⢠⡾⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⡈⠙⠿⣿⣿⠟⢼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠋⠊⡟⢻⣿⣮⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢷⣿⡦⣤⡀⠉⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠤⠙⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧⣟⣶⡌⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠑⠤⡀⣼⠁⠀⠀⠀⢻⡠⠆⠔⠂⠀⠀⠀⣿⣿⣿⣿⣿⣝⣿⡏⣿⣿⣿⣿⣧⢼⣽⣿⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡅⢿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣼⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠠⠴⣶⠖⠢⠄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⣠⠶⠛⠒⠆⡀⠐⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠒⠒⠁⡄⢠⡄⢠⠈⠒⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠌⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⠀⢀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠁⡌⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢰⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠈⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⠔⠁⠀⠀⡠⠂⠀⠀⠋⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠘⢤⡀⠀⠈⠢⡈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀⠔⠁⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠢⡄⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠂⠁⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠈⠐⢄⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠊⠁⡠⠊⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠧⠤⠄⠠⠤⠤⠠⠤⠤⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄⠈⠑⢦⡀
⠀⠀⠀⣰⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠀⠠⠠⡔⢢⠤⠄⠀⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠢⡀⠙⣆
⠀⠀⠀⢹⠀⠃⠒⠀⠒⠒⠒⠀⠒⠂⠀⠒⠒⠒⠒⠒⠒⠂⠖⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠲⠐⠀⠒⠒⠒⠒⠀⠐⠂⠒⠂⠒⠒⠒⠂⠒⠘⠀⡏
⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠚⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠃⢤⣀⣀⡀⣴⢿⡿⢠⣀⣀⣀⠠⡞⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣀⣭⣹⣯⣨⣅⣀⣈⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡿⠋⣍⣀⣀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⣠⠤⠔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⢠⡤⣼⢾⢚⢚⡹⠀⠀⠀⠀⠀⢀⣀⠠⠆⠀⢠⠛⠠⣤⣄⣀⣀⠀⠀⠀⠀⠙⠃⠀⠚⠁
⠀⠀⠀⠀⣆⣏⣥⡦⠶⠒⠒⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠀⢀⣠⠴⣺⠟⠋⠉⠀⠒⠀⠀⠀⠠⠞⢈⣹⠎⠍⣳⡄⠀⠀⠁
⠀⠀⠀⠀⠏⡧⠖⠗⠁⠀⠀⠳⣶⣤⣶⡏⠀⠀⠀⠀⠀⠀⢺⣧⣀⠈⠓⠒⠖⠒⠶⠒⠶⠒⠚⠋⢉⣀⣤⡾⠟⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠔⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⣀⢤⠶⠋⠁⠀⠀⠀⠀⢀⡀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⢀⠠⠀⠫⡽⢀⠘⠋⠀⣠⣀⠀⠀⠠⡀⠀⠀⣖⢻⢻⠚⡗⣦⡄⠓⠋⣼⡏⠇⢹⠸⠸⡇
⠀⠀⠀⠴⠶⠚⠃⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⣵⠀⢸⣸⣸⡤⠷⠋⠀⢀⠘⠀⠙⠛⠚⠒⠗⠃
⠀⠀⠀⠀⠀⠀⣀⣬⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠀⢀⠝⠀⠈⠈⠀⠀⠀⠀⠀⣸⣣⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠔⠐⠒⠀⠀⣀⡀⢀⣀⣀⡀⠀⣀⣘⠁⠀⠀⠀⠸⡀⠀⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠉⠓⢤⣸⣷⣦⠀⠀⠀⢀⣴⣶⣦⣰⡆⠀⣾⣷⠿⣿⡟⠃⣾⣯⣭⡍⣿⣏⣽⠇⠀⠀⠀⠀⡗⠒⠒⠀⡖⠀⠂
⠀⠀⠀⠀⣤⣀⠀⠀⠈⠀⠀⠙⠣⠀⠀⢼⡇⣸⣿⠹⣿⣤⣿⠇⠀⣿⡇⠀⣿⣥⣤⡀⣿⡟⠻⠦⠀⠄⣀⠀⢹⠚⠓⢺⡃
⠀⠀⠀⠀⠀⠈⠙⠡⠀⠀⣠⢤⡀⠀⠀⠘⠛⠛⢁⣀⡈⣩⣥⣄⠀⣬⣤⡀⠀⣠⣶⡄⣠⡴⠶⠆⠀⠀⠀⠀⣼⠦⠤⠼⣇⡈⠄⠤
⠀⠀⠀⡤⢤⣤⡄⡀⠀⢸⡁⢀⡽⠀⠀⠀⡀⠀⠿⣤⣅⢹⣧⣾⠗⣿⡾⣇⢸⣏⠀⡀⢿⡷⠚⠃⠀⠀⠀⠀⠀⠁⠒⠀⠉⠁
⠀⠀⠀⠀⠠⢶⣷⣤⡄⠀⠉⠉⠀⠀⠀⠁⡀⠀⠶⠶⠿⠸⠿⠀⠀⠛⠀⠉⠈⠛⠛⠁⠸⠷⠚⠛⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠘⠈⢀⡄⠂⡂⠠
⠀⠀⠀⠀⠰⠄⣀⡉⠀⠀⠀⠀⠀⣀⠤⠆⠚⠀⠈⠒⠂⠀⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⡁⠀⡀⠧⠖⠀⠀⠤⠀⣀⢀⡼⠀⠀⠆⠂⠀⢀⠀⡐⢶⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠑⠦⠦⣄⣀⡑⠠⠤⠠⠚⠉⢀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⠀⠰⠀⠑⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠈⣶⠆⠀⢤⠒⠈⠳⣤⡀⠀⠀⠀⠀⠐⠒⠘⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠑⠀⠀⠸⣦⣄⡒⠀⠠⠤⠀⠀⠂⣀⡜
⠀⠀⠀⠸⠇⠀⠀⠘⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠳⠶⠖⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠁⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡄⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⠀⠀⠀⠀⠠⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡤⠌⠀⠀⠀⠈⠋⠐⠉⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠘⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠚⠦⠠⢮⠭⠄⠠⠀⢀⢰⠀⠀⠀⠀⠀⠠⠂⠀⠈⣁⠒⠐⠲⢄⡀⠁⠀⠀⠀⠀⠀⡄⢀
⠀⠀⢠⡆⠀⢸⡄⠀⠜⠃⠠⠀⠂⠀⢀⠔⠀⠀⠀⠀⠐⣠⣤⡄⠀⠈⠀⠀⠀⠀⣇⠀⠀⠀⠀⣀⡀⢨⣟
⠀⠀⠀⡀⠀⣀⡄⠀⠀⠈⠂⠰⠀⠈⠀⠀⢀⠀⠊⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠈⠁⢀⢀⢀⣒⣂⣞⠈⣷⢒
⠀⠀⢰⡅⠀⠁⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⣀⣀⠄⠲⠀⢼⡀⢎⠑⡀⠀⠀⠀⠀⠀⠈⠢⠀⣀⡀⣁⣠⠂⣥
⠀⠀⠀⠐⠀⠀⠀⢠⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠐⡀⠀⠀⠙⠲⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠉⠺⣇⠦
⠀⠀⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠢⠂⢠⠀⠉⠂⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⣀
⠀⠠⠀⢀⠠⠤⠀⢸⠆⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠑⠄⠀⠄⠊⠢⠀⢀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠢⠥⢄
⠀⠀⠀⠈⠀⠀⣀⣈⡇⠀⠰⠒⠂⠀⢂⡀⠀⠀⠀⠘⠦⣰⣠⣤⣤⠀⡩⠂⢉⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⢤⡀⠀⠀⠀⠈⠁⠀⡇⠀⠀⢈⣠⠤⡀⠀⠇⠀⠁⢘⣷⣤
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡏⠀⠁⠋⠈⢿⡉⣄⡀⠀⠀⠀⢰⡃⣨⡷⠋⢀⣁⠇⠈⠀⠀⠸⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣤⡤⠄⠀⠈⠟⡙⠂⠀⠀⠀⡉⠋⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⣐⠀⡄⠀⠀⠀⠀⠀⠀⢩⡙⠓⠀⠀⠀⠌⠀⠀⠀⠈⡀
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢣⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡷
⠀⠈⠁⠀⠀⢀⡀⠀⠀⠁⠀⠠⠀⠈⣏⠖⠊⠉⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠣⠈⢈⠀⠀⠀⠉⠀⠀⣀⠀⠀⠠⢠⠀⠀⢂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⢀⡀⠀⢀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⡄⠀⠀⠀⠁⢠⡄⡖⡺⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠄⠁
⠀⠲⠐⢀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠑⠀⠠⠁⠀⢀⡂⠩⠅⠹⠈⠁⠀⠀⠀⠁⠀⢀⣀⣠⡴⠤⠤⠤
⠀⠀⠀⠀⠀⠄⠀⢀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠁⢸⣀⠀⠤⢤⠌⢠⢀⣀⣠⠶⣚⣩⠥⠒
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⠒⠈⠀⠀⠐⠀⠀⠀⠀⠈⡰⢯⠄⠒⠀⠀⠢⠀⠀⠀⡘
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⠀⣱⢄⠀⠀⠀⢀⠆⡼⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣌⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠓⠒⠢⠤⠀⠀⠀⢠⠀⠀⢄⣀⠹⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡿⠃⠀⠠⠀⠀⣀⡀⠀⠀⠉⠒⠈⠁⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⡽⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⣀⡄⠐⠚⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⠐⠊⠉⠉⠁⠛⠰⡄⠀⠤⠭⠱⠲⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡜⠄⠀⠀⠀⠀⠤⠈⠐⠒⠒⠒⠲⠶⣶⡁⠀⠈⠃⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠂⠒⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠓⠂⠒⢆⡈⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠐⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀⣎⠉⡁⠀⠀⢀⠀⠀⠐⡀⠀⠀⠀⣀⡠⠔⠁⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⡀⠁⡁⠚⠛⠲⠄⢀⡠⠄⠈⢁⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⣗⠶⠒⠉⠁⠀⠀⢀⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠒⠛⠁⠈⠛⠖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⣀⠀⠀⠉⠉⠈⠉⠙⢣⣄⣁⣀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠒⠑⠐⠒⠂⠀⠀⠀⠃⠀⠀⠀⠢⡤⣤⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠀⠀⠀⠀⠠⠤⠄⠠⠤⠀⠀⠀⠠⠀⠀⢠⠟⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⢲⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⡉⠛⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡴⠎⡃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠄⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠈⢛⣍⣉⣉⣉⡁⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣲⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⣀⣀⣤⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠒⠀⠿⠦⠾⠉⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡀⠈⠁⠁⠉⠉⠎⠰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⣃⡀⠀⠀⠀⠀⠀⠀⠉⠒⣤⠀⡀⠈⡓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢧⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⢌⡚⠘⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⢀⠧⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⠧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢾⠚⣲⠆⠂⠒⠂⠃⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠿⢟⣛⣛⣿⣷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⣉⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢀⣼⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠊⢩⠽⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣤⠀⠀⠀⠀⠀⠠⠖⢛⣹⣿⣿⣿⣿⣿⣿⡆⢠⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠁⠀⠀⠀⠀⠀⠀⣉⣩⣭⣿⣿⣿⣿⣿⡿⢡⣿⠟⠁
⠀⠀⠐⣦⡀⠀⠀⠀⢰⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⡠⠔⠒⢋⣉⠥⣶⣾⣿⣿⣿⣻⣿⣷⣶⣇
⠀⠀⠀⠉⠛⠀⠀⠀⢸⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢉⡠⠔⠂⠉⢁⣒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣷⣿⣿⣿⣿⠋⣹⣿⣿⣿⣿⣿⣿⣦⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⢸⣿⣿⣿⡿⠿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣾⣿⣿⣿⡁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡄
⠀⠀⠀⠀⠀⠀⡀⢠⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣜⣷⣽⣿⣿⣿⣿⣿⣿⣏⠀⣠⣾⣿⣿⣿⡟⡡⢤⣤⣤⣬⣻⣿⣿⣿⣿⣿⣿⡟⢉⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣋⣹⣼⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠐⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣻⣿⣿⣿⣿⡿⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡀⠀⡠⠠⣀
⠀⢀⠜⠀⠳⣤⣶⣼⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⠈⠉⠉⢉⣭⡄⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠞⠁⢠⢻⣦
⠀⠊⣀⡠⣄⣀⠉⠁⢤⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣧⣀⣀⣠⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⢀⣤⣜⠷⠍⠃
⠐⠚⠻⠿⠾⣤⡉⢙⣤⠙⣂⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣼⠉⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⢀⣠⣤⣉⠛⠛⠛⠒⠒
⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⡈⠉⠛⠻⠦⢄⡉⠛⠻⠿⠛⠉⠉⠛⠧⢤⠿⡄⣼⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⡀⠤⠖⠊⠉⠉⠁
⠀⠀⠠⠴⠶⠮⠽⠫⢭⡛⠿⠾⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣠⣀⠀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠻⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢀⣤⣦⣄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⣀⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠂⠀⢀⠀⠀⠀⠀⠀⠘⢣⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠉⠉⠀⠂⠀⠀⠠⢄⣀⠤⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢤⣀⠡⠄⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠒⠛⠂⠐⢢⣀⢈⢁⡒⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠭⠀⠀⠀⠀⠄⠐⠀⠒⠐⠶⢶⢎⠀⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡉⠉⠙⣫⣍⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⡦⠄⠠⠠⢠⣄⡀⠀⠀⠀⠀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣔⣤⣀⣩⠥⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠦⠉⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠚⠋⠁⠐⠶⣀⠂⢀⢙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⠥⠌⠀⠀⠀⠸⠀⠀⠈⠠⡥⡍⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡠⠟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢹⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⣃⣥⡠⠤⠤⠀⠀⢀⡤⠔⠋⢠⠤⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⣔⣓⡶⠒⠒⠀⠀⡄⠄⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⢀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠰⣯⡽⠇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠓⠒⠐⢊⠔⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢉⢣⡀⠀⠀⠀⠀⢀⠈⠒⠄⠠⠀⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠒⢆⠀⠀⠀⠀⠀⠀⠁⠀⠲⢜⡀⡃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣚⡂⣀⣀⢀⣀⡀⠀⠀⠀⢙⡓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠕⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡀⠃⠐⠒⠀⠀⠀⠀⠀⠒⠘⢀⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠒⠒⢰⣏⣀⣀⣹⡤⠀⠒⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⢀⠀⢘⠠⢨⡿⠭⡓⠆⠃⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠈⠁⠁⠰⠆⠌⠈⠁⢀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⢹⡖⠂⠐⠂⠀⢲⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠰⢀⣾⡇⠀⠀⠀⠀⢸⣷⡀⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⢿⡃⠐⠀⠀⠀⢘⡻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣰⡏⢸⡇⠀⠀⠀⠀⢸⡅⢹⣆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⠘⢙⡏⠉⠉⠉⠉⢹⡋⠃⢿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⢖⠁⠁⡆⢘⡃⠀⠀⠀⠀⢘⡁⢸⠈⠨⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠕⠁⡠⠊⠀⢨⡇⠉⠉⠈⠉⢸⡇⠀⠑⢄⠈⠪⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠔⢁⠄⠊⠀⠀⠀⢸⣇⢀⡀⢀⡀⢸⡇⠀⠀⠀⠁⠠⡈⠢⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⡡⠂⠀⠀⠈⠀⠀⠀⠀⠀⠸⠃⠀⢴⡦⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢳
⠀⠀⠀⠀⠀⠀⠀⣧⣉⣉⣉⣉⣭⣉⣉⣍⣉⣩⡇⠀⠀⢸⡇⠀⠀⢸⣍⣉⣉⣉⣉⣭⣉⣉⣩⣉⣼
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⠀⢀⡤⠀⣀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣀⠀⢠⡀⠀⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠤⣤⠤⠬⡆⢲⠤⢤⣾⡷⣤⠤⠾⢰⠏⠤⣬⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢁⠈⣿⣿⡿⠿⣿⣿⡇⠘⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣶⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣶⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡘⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣷⡄⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢠⣶⣾⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⢀⡀⠹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⠿⠟⠛⠉⠀⠀⣀⣠⣴⣶⣿⣿⣷⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⡿⠿⠛⠋⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡀⢻⣿⣿⣿⣿⠿⠛⠛⠉⠀⢀⣀⣤⠄⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄⠙⠉⠁⠀⣀⣠⣤⣶⣿⣿⡿⠃⣠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⡀⠾⣿⣿⣿⣿⣿⠿⠋⣀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣌⡛⠉⠀⠀⣠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣶⣄⡈⠻⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⣄⠙⠻⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠋⠀⠀⣀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⢁⣤⣶⣾⣿⣿⣷⣄⠈⢻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠁⣴⣿⣿⣿⠿⠟⠋⠉⠀⠀⣀⠙⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠾⠛⠉⠁⠀⣀⣠⣤⣶⣾⣿⣿⣆⠘⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠁⢀⣠⣤⣶⣾⣿⣿⣿⣿⡿⠿⠛⠛⠉⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⢿⣿⣿⣿⠿⠟⠛⠉⠁⠀⣀⣠⣤⣶⡆⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠘⠉⠁⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⡿⠇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠰⣾⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣆⠘⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠭⣽⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠟⠋⡁⢄⣤⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⡛⢉⡁⢀⣢⣤⢶⣿⣿⣿⣿⣿⠿⠟⠟⣩⣝⢿⣿⣿⣿⠟⠡⠀⠂⣀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣏⠰⠟⣡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠀⠀⠀⡝⢏⢷⣻⣿⠰⠀⢀⠰⡆⠀⣂⣿⡇
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠛⣿⣿⣷⣨⣿⣿⣿⣿⠛⣉⡀⠀⠀⢀⠀⠀⡄⣇⢸⡄⣏⣿⣷⣄⣈⣐⣆⣬⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣶⣿⣿⣿⡟⠋⠉⠁⣌⢰⣉⣽⢀⠀⢸⡄⠀⠸⡜⢾⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⣿⣿⣴⣿⣿⠉⣟⣡⣴⣧⣤⣤⣶⣿⣼⣇⢹⣿⣿⣿⣿⣶⣶⣾⣢⣭⣿⣿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⢏⠉⠛⢿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣽⣿⡿⢻⣿⠟⠿⣿⣿⡻⣿⣿⠛⢻⣿⢍⠚⠀⠈⠉⠑⠈⢻⣿⣿⣿⡄
⠀⠀⠀⢸⡿⣫⡟⠡⠤⠖⢀⠀⠀⠘⢻⣯⣻⣿⡿⣿⣧⣽⣶⡇⢰⣿⣿⣷⣜⠪⠐⢾⣿⣇⠠⠀⠀⠀⠀⠉⢈⣭⡛⢿⠁
⠀⠀⠀⠈⣱⣏⠀⠠⠀⠀⠀⠀⠀⣀⣤⠹⠷⢻⣷⣿⣿⠿⢿⣳⡊⠛⢿⣿⣿⣷⣧⣶⣦⣍⡛⠳⠶⢦⣤⣤⡄⠀⠐⡀⢄
⠀⠀⠀⢠⣿⠒⠖⠀⠀⢀⢀⣀⣠⡟⠈⢘⢸⣿⣿⣷⡝⣅⣀⣭⣭⣁⣟⢡⣿⣟⢻⣿⣿⣄⠙⠛⠳⢶⣦⣬⣤⣀⢀⣠⣾⣫
⠀⠀⠀⠸⣿⣳⣖⡻⢿⣿⡿⠟⠋⠀⠁⠀⣺⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠘⣿⣿⢿⣿⣿⣿⣿⣶⣿⣿⣿⡟⢿⣿⡿
⠀⠀⠀⠀⢻⣅⡀⠉⠀⠉⠀⠀⡀⢀⣢⣞⣿⣿⣿⡇⡌⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣷⠛⠛⠛⢻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠿⣤⣀⠀⠠⠀⢶⣯⢷⣿⣿⣿⣿⢡⢿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⠿⣛⡀⠀⣸⣿⣿⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢬⣭⣭⣭⣶⣿⣿⣿⣿⡏⣌⣾⣿⣿⡿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣿⣿⣿⢶⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢦⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡟⣼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣧⢻⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠙⠛⠛⠛⢇⠛⠛⠛⠛⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠛⠀⠀⠀⢘⠃⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⡦⠀⠀⠀⢸⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠀⠀⠀⠈⠈⠓⢄
⠀⠀⠀⢄⠀⠠⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠃⠀⣶⠀⠹⡀⠀⠀⠀⡁⠀⠀⡑⡀
⠀⠀⠀⠈⠐⢌⠲⣅⢄⡀⠀⠀⠁⠀⠉⠑⠒⠒⠒⠒⠒⡊⠉⠁⠈⠒⠀⡚⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠐⢬⡓⣎⣢⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠴⣤⣤⣿⣤⢤⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠹⡶⡞⠀⠄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠛⠛⠛⠿⠿
⠀⠀⢀⡤⠴⠀⠀⠃⠀⠿⠐⠀⢞⠻⣕⠢⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠊⠀⠀⠀⠀⠉⠂⡙⢭⡣⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠈⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠉⠉⠉⠉⠛⢻⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⢿⣿⡍⠛⠿⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠈⠙⠻⢿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠉⠙⠛⠻⠿⣿⣷⣶⣤⣀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣤⣀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣧⣀⡀⠀⠀⢀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣾⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠉⠉⠉⠉⠙⢻⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⡍⠙⠻⢿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠈⠙⠻⢿⣿⣷⣶⣤⣀⣀⣀⣀⣤⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄
⣤⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠉⠛⠛⠿⣿⣿⣶⣤⣀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣦⣄⡹⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣧⣀⣀⠀⠀⣀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡿⢿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠃
⣶⡶⠒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣾⣿⡶
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣛⣟⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣦⣄⡀⡀⠀⠀⣠⣶⣿⠿⣷⣦⣄⠀⢠⣴⣶⡿⢿⣶⣦
⠀⠀⠀⣼⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⡟⢸⣿⣿⣇⢹⡇⠀⠀⠀⣀⣤⣤⣸⣿⡿⢻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠙⠿⠿⠿⠿⠿⢿⣤⣿⣛⠿⠿⠿⠿⠛
⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠁⣿⡏⢹⣿⠈⠁⠀⠀⠀⠹⣿⡿⠟⣿⣿⣸⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠏
⠀⠛⠛⠿⣿⡿⠛⠛⠃⠀⠀⠀⠀⠀⠘⡿⠁⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⡅⠀⠀⠀⣤⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠛⠂

⠀⠀⠀⠰⣀⣀⢰⣂⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠙⣤⡄⠀⢀⣀⡀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠫⣿⡯⠋⣠⣤⡝⠆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢘⣶⣿⣿⠛⣿⣿⡛⡄
⠀⠀⠀⠈⢻⣿⣅⣽⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⢠⣾⣿⣿⠇⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⡿⠽⣿⣿⣏⣀⠿⣿⠃
⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⠀⠃⠀⠀⠀⠀⠀⠰⣜⠿⠟⢁⣾⣿⣦⣄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢈⣿⣧⣞⣙⠿⣿⣿⠖⠁
⠠⠮⠀⠀⠉⠉⠭⠍⠉⠉⠀⠀⠥⠄⠀⠀⠀⠀⠈⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠈⠛⠿⠿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠈⠛⠴⠿⠟⠂⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗

⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⡀
⠀⠀⠀⣍⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣯⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠘⠳⣦⠄⢠⣴⣶⣶⣦⡀
⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡷⠘⣿⣿⠷⣟⢀⡤⣾⡇⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠈⠑⣿⣿⢿⣿⣿⣿
⠀⣤⣾⣿⣻⣿⣦⡄⠀⠀⠀⠀⣰⣿⣿⣯⡻⠿⠤⠼⢟⣶⣿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠯⣰⣤⣀
⠀⣿⠋⢸⣿⠈⢻⡇⠀⠀⠀⠀⠙⢿⣿⠟⠀⠀⠀⠀⠀⠻⣿⡿⠇⠀⠀⠀⠀⠀⠘⠛⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉
⠀⠃⠀⠀⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣷⣦⣤⡀
⢀⣾⣿⣿⣿⣹⣿⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣿⣿⢿⣶⣤
⠸⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⢿⣟⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⢀⣀⣴⣹⣛⡟⢻⣛⣏⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢧⠀⠈⠉⠙
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉⢻⣽⣽⢿⣿⣿⠄⠀⠀⠀⠀⠿⣿⣿⣷⣿⣿⣿⣷⣾⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⡸⠠⡄⢆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣼⣧⣬⡉⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠉⢹⡏⠉⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⣀⣤⡾⠛⠉⠉⠉⠉⠉⠉⠉⣉⢉⣟⡷⠋⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⣤⣾⡟⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⡿⠋⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡎⡏⣾⣁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠋⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠸⣿⡀⢻⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢦⡓⣄⣀⠤⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠉⠀⠀⢀⣠⡶⠂⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡤⠶⡟⠥⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢀⣠⡶⠛⠙⢦⣄⠁⢾⠿⣿⣿⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⢀⣤⣶⠟⠉⠀⠀⠀⠀⠘⠉⠉⠛⠉⠉⢙⠛⢿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣔⡝⣽⣁⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⠀⡀⢡⡈⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⣾⠠⡛⠒⠒⠀⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⢒⣶⢀⢧⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣵⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣷⣠⣄⣤⠠⣄⠀⣤⡀⠀⠀⣠⡶⠿⠋⠉⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⡷⠈⠁⢯⣭⡷⠟⠉⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠁⠀⢀⡠⠴⠻⢿⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠁⠀⠀⠠⡄⠐⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⢀⣠⡴⠿⠩⠅⠀⠀⠀⠀⠀⠐⣿⠀⢁⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⣤⣶⠟⠡⠤⠄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣮⠜⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡠⣺⣿⠯⠥⠤⠬⠤⠤⠌⠒⠒⠒⠒⣲⣶⠿⠛⠛⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠁⠁⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠁⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⡘⡀⠹⣄⠀⠀⠀⣀⣤⠾⠛⢉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡤⠈⡷⠖⠊⠉⠀⠠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⢀⣠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠻⣯⠾⣷⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⣤⡶⠟⠉⠈⠙⠶⠄⠳⠌⠛⠾⠟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣠⣾⡿⠉⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⡌⢻⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⢮⢫⡟⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⢤⡤⠴⠄⡷⡇⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢸⢿⠘⠛⠛⠛⠛⠛⠛⠛⠛⠐⠒⠒⠒⠒⠒⠒⣾⡰⡵⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠾⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⢺⠓⡀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠊⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡉⢀⢬⠝⠀⠀⠀⠀⠀⠀⠤⠤⡀⣃⠀⢠⠔⢓⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠾⡄⠀⠀⠀⠀⠀⠀⠈⠢⢻⣟⡔⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡚⣇⠴⠃⠀⠀⠀⠀⠀⠀⢀⣧⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⢏⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣖⣲⠖⢒⣒⡒⠲⣲⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢒⠲⣆⠀⠘⢆⠹⣤⠤⠤⠤⢤⣤⠋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠿⠃⡸⢀⣠⡈⠓⢌⡻⣦⠼⢛⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡒⠛⢧⡏⡹⠁⠀⡴⢛⡴⢎⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠤⣿⠃⢠⠞⣴⡯⠠⢤⣵⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⢼⣤⠋⣼⢉⠉⠉⢉⣉⣙⣆⢹⡀⠀⠀⢀⣨⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢨⠓⠛⠛⠒⠒⠒⠚⢻⡄⡗⢦⣀⠜⣡⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⡇⠤⠒⠨⠀⠀⠲⣾⠃⡇⠸⢡⣾⣩⣾⢄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠱⣤⠤⠤⠤⢤⣴⠇⡼⡡⠖⠣⡀⠳⠵⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⡙⢮⠽⢋⡤⠞⠊⢱⡂⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡥⢆⠛⢄⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣠⠿⠅⠨⠷⣌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣼⣓⣲⠖⠒⢲⡾⣆⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣸⠁⠀⠀⠀⠀⠀⠀⠘⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢲⠾⡀⠀⠀⠀⠀⠀⠀⠀⠰⠟⡴⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣦⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠊⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠦⠦⡍⠃⠀⠀⠀⠀⠀⠀⠀⠘⢢⠤⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣂⠓⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠘⠁⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢠⠏⣀⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⡰⠚⠓⢦⠀⢀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⡄⠀⣇⢀⣀⣸⡦⠞
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡉⢀⣠⠀⡀⢀⡤⠀⠀⡀⠀⠀⠠⣄⡀⢀⡈⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠛⠛⠙⠯⡥⠞
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢟⣋⣥⣶⣿⣿⣿⣵⡞⣱⣿⣣⣾⣾⣿⣦⣱⣵⣮⣿⣶⣿⣾⣷⣦⣭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢸⣿⣿⠿⣟⣫⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣟⣻⣿⣿⣿⣿⡇⠀⠀⠀⠈⠁⠈⣹
⠀⠀⠀⠀⠀⢈⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠚⠓⠒⠒
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠿⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠱⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠰⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠠⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢡⡃⠀⠉⠉⠀⠀⠀⢹⠁⠀⣧⠁⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⣰⠏⠈⢧⢸⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠂⠀⠀⠘⡇⠀⢰⠏⠀⠀⠀⢿⠀⠀⡟⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢀⠃⠀⢀⣀⠀⠀⠆⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⣀⣀⣀⡇⢸⠰⢩⡿⠯⢏⠡⢸⠀⣇⢀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⡀⠁⠱⠦⠦⠄⠁⢹⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡍⠙⡖⠒⠒⠒⠒⠒⣿⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠒⠒⠂⠒⡀⣾⡇⠀⠀⠀⠀⠀⣿⡆⡗⠐⠂⠒⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡀⣠⠀⠖⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡿⠈⡇⠀⠀⠀⠀⠀⣷⠸⣧⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⠃⢤⠀⠒⠀⠈⣀⡈⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠁⠈⢈⡟⢳⠚⡗⠒⠒⠒⠒⠚⣟⢲⠛⣅⠁⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠑⠦⡉⠁⣙⢶⣤⠄⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠴⡫⠀⡀⢈⡇⠀⠀⠀⠀⠀⣿⠈⡈⠩⠳⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠌⠳⣶⡀⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠞⡡⠊⢀⠔⠁⠸⡏⠉⠉⠉⠉⠉⣿⠀⠑⢄⠈⠢⡑⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠴⠈⠛⢷⣀⠈⡡⢄⠀⠀⠀⠀⠀⢀⡴⢊⡵⠋⡠⠒⠁⠀⠀⠐⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠤⡈⠢⣝⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠲⡂⠈⠉⠳⣄⣀⠱⠆⠀⢀⡴⣋⢔⣁⠀⡎⠀⠀⠀⠀⠀⢸⡇⠀⣀⣄⢀⠀⣿⠀⠀⠀⠀⠀⠈⡆⠈⠑⢌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠈⠉⠻⣧⣉⡡⠬⢷⠦⠄⠐⢚⣁⡙⠦⠀⠐⡗⠲⣖⠒⢒⢒⡎⠀⠀⢸⣿⠀⠀⠈⣖⠒⠒⠒⠒⠒⠓⠒⡒⠒⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠐⡉⢷⡀⠀⢂⢻⣿⡷⡾⠟⠃⢃⠀⢡⠤⠋⠉⠉⠉⡇⠀⠀⢸⣿⠀⠀⠀⡏⠉⠉⠉⠉⠉⡏⢉⣉⠉⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣧⠀⠸⠀⠷⠀⠛⣤⡀⡘⢀⡜⠓⠀⠠⠤⡄⣇⣀⣀⣸⣿⣀⣀⣀⡧⡄⠤⠀⠐⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠤⠀⠒⣈⡉⠆⠀⠀⠀⠀⡿⠀⢰⠤⠤⠀⢒⣊⠹⢧⣝⠛⠫⣯⠉⠉⢃⠸⣯⣭⣿⣿⣿⣭⣽⢀⠉⠈⠹⠉⠉⠹
⠀⠀⠀⢀⠀⡶⠖⠙⠒⢂⠈⠉⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠀⠀⠀⠀⠙⢤⡄⠒⠙⢻⣄⡈⠵⢄⣘⠀⠉⠉⠉⠉⠉⠉⠉⢘⣀⣀⣀⡀⣀⣀⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣉⠁⢈⠻⣦⡄⠙⠢⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠙⠋⠛⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⡀⡀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⣻⣌⣀⠀⠙⠶⣄⠋⠓⣄⠀⠀⠀⠀⣃⣀⣀⣀⣀⣀⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡤⠀⠛⠻⠎⢀⣑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢖⠀⠉⠀⣀⣀⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁