Penny Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠
⠀⠀⠀⠀⠀⢣⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠑
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣶⣾⢷⣄⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⠾⠷⡿⢟⡾⣿⠧⠂⣃⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇⠀⠀⣠⠾⠻⠋⠠⠿⠻⡷⠆⡀⢀⡸⣻⡼⡧⣀⡗⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠀⠀⠘⣔⢎⠟⢠⣾⣽⡷⠶⠶⠒⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠒⠐⠀⠀⠈⠙⠛⠛⠛⢉⣠⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⢀⣤⡄⠀⠀⠀⡰⠄⠀⠀⢀⣾⣻⣿⣿⣿⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠣⡄⠀⠀⠀⠀⠀⢠⠄⠀⠀⠁⠀⠀⠀⢸⣿⣿⢿⣿⢏⢾⠇
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣄⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠒⠀⠓⠯⠧⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡶⠿⠿⠿⠿⠿⠿⠿⠿⢶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠁⠀⠀⠀⠀⠐⠂⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠀⠀⠀⠂⠀⠀⠀⠈⠀⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⢐⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⢤⠀⠄⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠻⣷⡄
⠀⠀⢠⣿⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⠀⠘⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠄⠀⠹⣿⡄
⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢤⠠⠄⠠⡔⡲⠒⠒⠒⠒⢲⠒⠒⠒⠒⢲⣒⣖⡒⠒⣒⣒⠒⠒⠒⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠹⣿⡀
⠀⣾⡏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⣀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢹⣷
⢰⣿⠁⠀⠀⠀⠂⠐⠀⠀⠙⠀⠛⠛⠒⠒⠒⠒⠒⠒⠒⠐⠒⠀⠐⠂⠀⠐⠂⠀⠐⠂⠀⠐⠂⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿
⣼⡏⠀⡀⠀⠁⠀⠀⠀⠀⠰⣤⠒⠆⠐⠓⠆⠖⠓⠠⠖⠳⠀⠞⠒⠰⠒⢳⣠⠒⠶⠰⠒⢶⣴⠒⡦⠖⠒⡆⠒⢲⠔⠀⠀⠀⠠⠄⠀⠈⠀⠀⢹
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⡆⠀⠀⡸⠀⠀⠀⠀⠀⠘⠀⢸⠇⠀⢸⠀⠀⢸⣿⠀⣷⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡄⠀⠀⠘⠠⢸⠀⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⡇⠀⢸⠸⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⣇⣀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠸⣄⠀⠈⣇⠀⢸⡀⠀⣿⠀⠀⣇⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠄⠉⡹⠁⠈⠀⠀⠀⠀⠁⠈⠀⠁⠀⠈⠁⠀⣆⢘⠇⠒⠒⠶⠳⠶⠶⠶⢄⢀⠀⠀⠀⠀⠀⠀⣸
⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠢⡄⠀⣄⡐⠀⠤⡄⠀⠀⠀⠀⠀⠀⢀⣿
⠀⢿⣇⠀⠀⠀⣐⣲⣦⣦⣤⣀⣠⣁⣐⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣈⣉⣒⣒⣓⣂⣒⣂⣀⣤⣂⣐⡀⠀⠀⠀⣸⡿
⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠐⡡⠂⠀⠠⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⡀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⢀⠔⣠⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠃⠀⠑⠈⡀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠈⠻⣷⡄⠀⠈⠀⠀⠀⢈⢀⠆⠀⢀⠐⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠠⠡⠀⠀⠀⠈⠒⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠠⡠⠀⠀⠠⠀⠂⢀⠁⠂⠀⠁⠀⠀⠠⠀⠠⠴⠀⠁⠄⠀⡁⠡⠑⡀⠅⡓⠐⡀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠠⡠⠀⠀⠀⠔⠀⠔⠀⠀⠀⢠⠀⠀⠀⠀⠠⠈⠐⠈⠀⠀⡀⠀⠀⠀⠀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⡀⠀⠀⠘⠠⠁⠢⠀⠀⠀⠀⠄⠢⠀⠀⡄⠀⢂⠒⠀⠐⠀⠀⢀⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⢀⣀⣠⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠷⣶⣶⣶⣶⣶⣶⣶⣶⠾⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡉⡉⢍⣉⣉⢉⣋⠀⠒⠒⠦⢤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⠀⢰⠀⠖⠆⠀⠐⢐⠀⠬⠉⠉⠓⡒⠦⣔⡁⢐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⢡⣆⠀⠈⠛⠁⠁⠀⣀⢀⡀⠀⡀⠉⠀⠀⠍⢁⡆⠄⠉⠲⢄⠢⡑⢄
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠃⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠤⢬⣰⠀⠀⠀⠈⠈⢛⣲⢀⡑⢌⠳⣕⣄
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⡂⠀⠀⠘⡀⠀⠀⠀⠀⠀⠁⠞⠁⠀⠑⣌⢲⡣⡀
⠀⠀⠀⠀⢰⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⢈⠅⠀⣠⣦⡇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠈⠄⠱⣳
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⣶⡊⠀⣤⡀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠀⢹⣧
⠀⠀⠀⠀⠅⠠⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠣⡥⠀⢘⡄⠀⢠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⡟⡆
⠀⠀⠀⠀⡅⢠⡨⢴⡦⡖⢰⡔⡆⢠⠆⢘⠀⠀⠐⢦⣀⡀⢀⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡇
⠀⠀⠀⠀⣇⢉⣉⠉⠁⡉⠉⠁⠁⠈⠀⠌⠀⢁⡢⣀⡬⣷⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡇
⠀⠀⠀⠀⠰⡈⠀⢁⢔⠀⡂⠀⠀⢀⠜⢤⡀⠐⢝⠲⣄⣙⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖
⠀⠀⠀⠀⠄⠠⡢⢑⠕⠨⡂⠐⡤⠂⠀⣀⠈⠲⣌⠑⠨⣥⡤⣼⠀⠀⠀⣋⠠⠇⠀⡇⣤⠀⠀⠀⠀⠀⠨⠍
⠀⠀⠀⠀⠈⣌⢃⢑⠁⡪⡠⣰⢁⠀⠒⠬⡙⢦⠘⢦⠀⠈⢿⡮⣧⡀⠀⠀⠃⠈⠀⠁⠘⠁⠀⠀⠈⢠⡛
⠀⠀⠀⠀⠀⠈⢂⠓⡠⠨⡢⡇⠀⠀⠀⠀⣌⢢⡁⠨⣧⠀⠀⢳⠈⢫⢦⠀⠂⠀⠀⠀⠀⠀⡠⢀⡔⠝
⠀⠀⠀⠀⠀⠀⠀⠑⢌⠘⠬⠡⢤⣀⠀⠀⠈⠃⢱⡄⢻⢑⡄⠀⢳⠘⠃⠑⡄⠀⢀⢁⠴⠊⣠⡾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢌⠲⢈⣑⠢⢦⣀⣈⣓⠘⢁⣀⣀⣀⡥⠤⠖⠚⠀⠂⡠⣴⠝⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠍⣒⠂⠉⠬⠩⠍⠭⠉⠀⠂⣀⣄⣤⠶⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢉⠙⠛⠛⠛⠛⠛⠋⢉⠉⡁⡀⠀⠠⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠀⠈⠀⠐⠀⠀⠀⠂⠁⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⡒⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢴⠶⠒⠦⠤⣄⡀
⠀⠀⢀⡴⠊⠁⠀⣠⣴⣶⣤⣄⡀⠀⠨⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠩⡐⠘⠊⠘⠈⠃⠀⠂⣨⠑⢦⡀
⠀⣠⠋⠐⠀⢠⣿⣿⣿⣿⣏⠉⠋⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀⠞⢅⠫⠀⠀⠀⠁⠀⠀⡀⠀⠀⠈⠙⡢⠙⣄
⢰⠃⠀⠀⠀⠘⣿⣿⢻⣿⡇⠠⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢀⡏⢨⠀⠀⠐⠀⠂⠂⠀⠂⠂⠂⠒⠒⠀⠀⣛⠜
⡞⠀⠀⠀⠀⠀⠹⣿⠗⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠈⠋⠀⠁⠚⠛⠚⠚⠚⠚⠚⠚⠚⠚⠘⠀⠈⣅
⣇⠀⠀⠀⠀⠀⢀⣿⣦⣈⢻⡿⠟⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣠⣀⣀⣀⣀⡀
⢹⡀⠀⠀⠀⣠⣿⣿⣿⣿⣷⡓⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠸⡄⠠⠤⠤⠮⠥⠡⠀⠀⠀⠀⠀⠡⠹⠦⠤⠤
⠀⢳⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡄⠠⠀⠀⠀⢀⠎⠀⠀⠀⠀⠱⡀⢰⢋⣆⡀⠀⠀⠀⠀⠀⠀⠀⡰⠧⡀⢀⡼
⠀⠀⠙⢦⡻⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠙⢦⡁⠊⡷⢱⡒⢀⡔⠰⡆⠻⡚⠆⡠⠎
⠀⠀⠀⠀⠉⠒⠬⢍⣛⣛⣛⣛⡭⠵⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣍⣀⣀⣓⣀⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⠀⠁⠂⢠⠈⠀⠀⠲⡲⠂⢠⠌⠀⢐⠈⠀⡂⢄
⠀⠀⠀⠀⠀⠀⠀⡠⢐⠅⠊⣴⡄⠩⠅⠘⠁⡀⠄⠀⠠⠠⢉⠀⠀⠠⠈⣐⠄⢑⠨⡂⢄
⠀⠀⠀⠀⠀⡠⢊⠔⣅⠀⠀⠈⠁⠀⡠⢐⠣⠐⠁⠔⣀⡀⠀⠈⠁⠢⡀⠈⠁⠶⢀⣈⡢⡑⢄
⠀⠀⠀⠀⠔⠔⠑⢄⠒⠁⠀⠀⢀⠊⠀⢈⠐⠪⠀⡀⠀⡂⠀⠙⠢⡠⠁⠀⠀⠀⠈⠋⡑⠌⠢⠢
⠀⠀⢀⢊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡉⠄⠀⠇⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡑⡀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢐⠀⠀⡄⣚⠰⠈⠒⣂⡠⢀⣄⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠄⠘⢀⠺⠀⠀⠀⢄⠀⠀⢨⡡⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⢤⣤⠁⠈⠀⠄⡀⠀⠠⠊⢑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⣁⠀⠈⠄⠂⠠⠀⠒⠠⠺
⠀⠀⠀⠂⢰⢒⣔⣶⢲⠖⡐⠂⠂⢀⠂⠁⠀⠅⡂⠄⡈⠐⠢⡀⠲⠨⠂
⠀⡄⡄⠉⠉⠉⠈⠉⠉⠁⠁⠈⡠⠒⠄⡀⠀⠀⠈⠐⢄⠑⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠑⢄⠀⠀⠀⢇⡜⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⡀
⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡂⠄⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠘⠑⠱⠄⢀⡀⡬⢰⠸⢀⠶⢃⠀⠀⠀⠐
⠀⠀⠈⢌⢀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠈⢂⠀⠠⠀⠀⠀⠀⠈⠄⠑⡈⣧⡀⠀⠀⠀⠀⠀⠀⡠⡡⠁
⠀⠀⠀⠀⠂⠢⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⠑⡁⠀⠀⠀⠈⠖⢌⢊⢳⡀⠀⠀⠀⢀⠔⠐
⠀⠀⠀⠀⠀⠑⢌⠢⡀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢄⠀⠀⠀⠘⡀⠑⠅⡱⠀⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠑⠨⡂⢄⡁⠂⠄⡀⠀⠀⠀⠐⠀⠀⠀⠡⡀⠀⡀⠴⠐⠈⡠⢐⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠀⡀⠄⠀⢀⠀⠀⠁⠀⠀⠈⡁⠀⠀⢀⠀⠅⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣒⣀⣀⣀⣀⣀⣀⣒⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠴⠦⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠶⠤⠤⢤⣀
⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠁⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠐⠀⠈⠙⠲⣄
⠀⢀⡴⠋⠀⠀⠀⠠⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠳⡄
⢀⡞⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡇⠀⠀⢁⠰⠘⠛⠍⠉⠉⠉⠉⠉⠉⠋⠉⠋⠉⠋⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠈⡍⠋⠙⢰⠘⠐⡀⠀⠈⠁⠐
⣇⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⡀⠒⠀⠀⠇⠇⠠⡸⡀⡄⢅⢀⢀⠀⡀⠀⠀⠀⠀⡀⢀
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠀⣇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣁⣁⣈⣈⣈⠁⠀⠀⠀⣸
⠀⢳⡀⠀⠀⠐⠀⠀⠀⠀⠀⠘⣆⠠⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠙⢦⡀⢄⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠈⠁⠁⠀⠉⠈⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠙⠲⢤⣀⠠⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠒⠒⣛⣛⣛⡓⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢚⡉⠀⠀⠂⠈⠁⢠⣴⣴⠀⣶⡆⠀⠈⠀⢐⣀⠠⢉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⢀⣥⡀⢿⣹⡄⠀⠀⠈⠛⠛⠐⠻⠆⠀⠀⠀⢨⠏⠀⣴⣬⠐⠠⡉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠊⡴⣦⠘⠦⠇⠈⠉⠀⡠⠔⠊⡍⠀⠉⢙⠑⠲⣤⣀⡈⠀⠐⠻⠇⢀⡖⣠⠑⠠⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠐⣁⠀⠀⠙⠛⠁⠀⠀⣀⠴⢋⠀⠀⠊⠀⠀⠀⠀⠀⢴⡄⠀⢨⢢⠀⠀⠀⠈⠛⠁⣠⣶⡌⠢⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢠⠎⡀⣀⠐⢿⠷⠀⠀⠀⠀⢀⣼⣥⠂⠘⠀⡀⠀⠀⠀⡀⠡⠃⠈⠙⠃⢀⡴⠀⠀⠀⠀⠀⠀⠛⠃⠀⣶⡀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⣰⠃⠜⠀⠈⠳⠌⠀⠀⠀⠀⠀⢸⠛⡁⠀⠀⠀⢠⣆⠀⠉⠁⠀⠂⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠂⠘⣄
⠀⠀⠀⠀⠀⣰⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⠀⠘⠉⠼⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⡆
⠀⠀⠀⠀⢰⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⠄⠘⠀⣸⠖⠡⠀⣦⠰⠀⠂⠀⠀⠠⣿⠋⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡄
⠀⠀⠀⠀⡏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⡀⠀⢽⡀⢰⠀⠔⠀⠰⢀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢸⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⠃⠄⠀⠙⣆⡠⠦⠘⠀⠈⠑⠄⠀⠀⠦⣤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡆
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠐⢀⠀⠱⣄⠀⠀⠀⠀⠀⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡷⣄⠀⠀⠀⠘⢷⡖⢦⡜⣀⠁⠀⣸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷
⠀⠀⠀⡇⠈⠀⢀⠀⡀⣀⡀⣀⡀⢀⣀⣀⣀⣀⢀⠀⠀⡜⠉⠢⢅⠂⠄⡀⠀⢢⠀⣷⣌⢏⠁⢰⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡟
⠀⠀⠀⢷⠀⠀⢯⡄⡯⢿⠿⢿⡇⠽⢿⠸⡇⠸⡏⠀⣰⠁⠀⠀⠀⠀⠀⠌⡐⠤⣱⡅⢀⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠑⠢⢄⠀⠀⠀⠀⠈⠢⣀⢙⣿⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠇
⠀⠀⠀⠀⣇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⢀⢨⢣⡤⠰⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠸⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠘⢄⠀⠘⡀⠈⢳⡤⣌⠳⣄⠀⠘⣻⠃⡟⣷⠘⡇⢸⣛⡆⠀⠀⡘⢰⠃
⠀⠀⠀⠀⠀⠹⡀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⣞⠀⠐⢧⠀⠀⠹⡆⠀⡈⠱⣌⠉⠁⠉⢁⡀⠁⠈⠉⠀⠀⠐⢠⠇
⠀⠀⠀⠀⠀⠀⠱⡄⢂⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡀⠀⠀⠀⠀⠈⠂⡀⠘⣆⠀⠈⢧⠀⠀⠙⡆⠀⠀⠈⢢⡀⠀⣿⣽⠀⠀⠀⠀⠔⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠑⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢀⠈⠀⠈⡆⠀⠈⢧⠀⠀⠸⡆⠀⠀⠁⠑⠄⠀⠀⠀⠀⡠⠊⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠢⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⡌⢆⠀⠘⡆⠀⠀⠹⡀⠀⠀⠀⢈⡆⠀⠀⠔⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠂⢈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⢃⠀⠣⡀⠃⠀⠀⠀⢇⠀⢀⡤⠊⡡⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣁⠂⠌⡑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⠀⢀⣀⠤⠜⢋⠁⠐⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢬⣀⠀⠀⠌⢉⣐⠒⠒⠒⠂⠐⠒⠒⣂⡉⠁⠀⠀⣈⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠤⠤⢬⣭⣭⣭⠤⠤⠤⠶⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣶⠾⠿⢿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣍⣿⡿⣷⡀⠀⢸⣿⣿⣿⣽⣿⡏⠈⠉⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣿⢷⣾⣿⣹⡿⠿⠟⠀⠀⠀⠛⠙⠋⠛⠻⠃⠀⢀⣾⢏⣽⣿⣿⢻⣶⣄
⠀⠀⠀⠀⠀⠀⣠⡿⡛⠁⠀⠿⣿⡿⠋⠉⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣶⣦⣤⡠⠉⠘⢿⣿⣽⡿⣯⡿⣷⣄
⠀⠀⠀⠀⢠⣾⢿⣬⣿⣦⡄⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠿⢿⣿⣿⣿⣿⣯⣿⣧⠀⠀⠈⠻⢿⣿⣿⣿⠻⣷⡀
⠀⠀⠀⣰⡿⠻⣷⣿⣿⡏⠅⠀⠀⠀⠀⢰⡿⣥⣶⣶⢶⠾⣿⡿⠃⠀⠈⠛⢿⣷⠿⠃⠀⠀⠀⠀⠻⠿⣣⣿⣿⢿⣄
⠀⠀⣰⡿⠁⠀⠀⠉⠋⠘⠠⠀⠀⠀⠀⣿⡇⢉⣐⣂⣄⣤⣿⢿⣆⠀⢀⣠⣬⣿⣇⠀⠀⠀⠀⠀⠀⠘⠛⠁⠉⡜⢿⣆⢠
⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣿⡿⠟⣻⣿⣿⣯⣶⣾⡏⠀⠛⢫⣭⡍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⣿⡄
⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡇⣸⣿⣿⡀⣉⣿⣾⣧⠀⠀⠙⠿⡃⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⢸⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢹⣿⡿⣶⡿⣟⣻⣿⣿⡇⠀⠀⠀⢰⣦⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡿⠰⠀⠀⠀⢀⢠⣄⣀⠀⠀⡄⠅⠰⠃⡹⣷⡻⣿⣿⣿⣿⠸⣿⣀⠀⣤⣶⣾⡿⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢼⡇⠀⣀⡀⣈⣒⣋⣉⣂⣈⣘⣘⣀⢀⡆⡐⣼⣿⣿⣤⣿⡿⣷⣬⣝⠻⠿⠿⢿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣧⠐⢹⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱⢸⣿⣅⢹⣿⣿⣿⣿⣿⣿⣷⣶⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⢸⣿⠀⠈⠉⢁⣍⣉⠈⠀⠀⠁⠁⠈⣩⡶⣿⣿⣿⣿⣌⡻⣿⣻⣿⣿⣿⣷⣤⡀⠀⢀⠀⠠⡀⢀⡐⠀⠀⠀⠠⠀⠀⠀⣿
⠀⢿⣇⠄⣤⢻⣿⣿⣤⣤⣄⡠⢀⣼⠿⠟⠛⠋⠙⠿⢿⣿⣾⣿⢾⣿⣿⣿⣿⣿⣄⠐⠀⢹⣿⣿⣿⣟⣿⢟⣾⣿⡂⢸⣿
⠀⠺⣿⡄⡝⢻⣿⣿⢛⠛⢛⣼⣿⣡⣶⠿⢟⠛⡻⢿⣮⣟⠈⠻⣷⣝⣛⣿⢿⣿⣿⣷⣄⠺⢿⣽⣿⣿⡿⠈⠛⠿⢃⣿⠇
⣦⣠⠹⣿⡡⠁⠀⠀⢉⠂⢸⡟⣼⢿⣶⠨⡄⠀⠀⢀⡌⠻⣷⣷⡀⢻⣟⣻⣦⡙⠿⣿⣽⣷⣌⠭⢡⠀⠀⠀⠀⢄⣾⡏⡐
⣿⣋⠐⡹⣷⡄⡲⠂⠈⢠⣿⣿⡏⣾⡗⢌⣿⣶⣶⣤⣭⠗⣹⣿⣷⢀⡙⢿⣾⣻⢷⣽⡻⣿⣿⣶⡂⠀⠀⠀⢀⣾⠟
⣿⣿⣷⣅⣙⢿⣆⠫⠀⡸⣿⣿⡇⣿⡏⠖⣿⣿⣿⣿⣿⠏⣿⣿⣿⠘⣿⣦⠨⠛⢷⣽⡻⣾⣿⣿⣷⡀⠀⣠⣿⢏⠈
⣿⣿⣿⣿⣿⢀⡻⣷⣅⣸⣿⣿⡇⣿⡇⢼⣿⣿⣿⣶⣶⡧⢼⣿⣿⠀⠘⣿⣧⣨⢲⠍⠻⣮⡻⣯⣿⣧⣾⡟⡁⢈⠂
⣿⣿⣿⣿⣿⣏⣿⣬⡛⢿⣿⣿⣧⣟⠀⠔⠀⣿⣿⣿⣿⣷⢸⣿⣿⡇⢤⠸⣿⣿⣭⡂⢔⣽⣿⣿⣿⢟⡡⣉⣿⣿⣇
⠉⣿⣿⣿⣿⣿⣿⣶⣷⣷⢩⡻⢿⣿⣿⣶⣮⣽⣛⣇⡋⠀⠸⣿⠸⣷⣐⣀⣩⣥⣿⣿⣿⡿⢟⣋⣶⣽⣵⣧⣭⣿⣿⣷⣶
⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣳⣰⣯⣛⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⡾⢾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⡿⠿⠿⠟⠛⠛⠻⠿⠿⢿⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠛⣉⡁⣤⢤⡀⠀⢲⡶⡴⢰⠶⠀⠀⠠⣌⠉⠛⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⣠⢤⡘⢅⠕⠘⠊⣀⣤⣤⣤⣤⣬⣍⣀⣀⠘⠁⠾⡵⢀⠄⠙⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⡀⠀⠘⠒⠁⠀⢀⣴⣾⡿⠛⠛⠋⢥⣩⡍⡉⠙⠻⣷⣆⠀⠛⠜⣡⡤⡈⠻⣿⣆
⠀⠀⠀⠀⠀⢠⣾⠟⠁⣜⠻⠛⠀⠀⠀⢀⣴⣿⣿⣿⡁⠀⣴⠄⠀⠈⠃⠛⠷⢦⣼⣿⠁⠀⠀⠉⠋⣠⠦⠈⠻⣷⡄
⠀⠀⠀⠀⣠⣿⠋⠀⠀⠈⠁⠀⠀⠀⠀⢸⣿⣿⠃⢀⡀⡀⡈⠾⠄⠀⠀⠀⠀⠘⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣄⣌⣃⢉⢐⣦⡄⠀⠀⠀⠀⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠤⢸⣯⣯⠪⠏⠁⠀⠀⠀⠒⢏⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠙⡈⣯⠙⠆⠀⠀⠲⣤⣀⠀⠈⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⣳⠈⠓⣦⡀⠀⠀⠈⠿⣧⢘⣹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣼⡇⠀⠘⢷⣦⣀⠀⠀⠋⠉⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠸⣿⡄⠀⡄⢠⣤⡤⡤⢠⣤⢠⠄⣤⠀⠀⣰⣿⢷⣄⠀⠀⠀⠹⢿⢷⣦⣤⣤⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⣿⡇⠀⠑⠈⠉⠉⠁⠈⠈⠈⠀⠈⠀⢰⣿⠏⠀⠉⠻⢦⣄⠀⠢⣄⡼⠋⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⠳⣦⣀⠀⠠⣌⠳⣤⣨⠷⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠀⠀⠀⠙⢷⡄⠘⣦⢿⢤⡖⠻⣅⠀⠀⠀⣄⢠⢄⠠⡤⢠⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⠅⠀⠀⢄⡀⠲⣤⣻⡄⠸⣆⠀⠙⢦⠘⢷⣄⠀⠋⠘⠋⠘⠁⠘⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⣾⣏⠙⠀⠠⢤⡀⠙⠢⡌⢻⣇⠀⢽⣧⠀⠀⠀⠀⠙⢷⡀⠀⢰⢲⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠈⠂⠀⠸⣆⢻⣆⠀⠘⣧⠀⢣⠀⠀⠘⣿⣆⠈⠉⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠸⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⣿⣧⠀⢹⠀⠀⢣⠀⠀⠏⢿⣧⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠙⠳⢦⣄⡀⠀⠀⠀⠀⠈⠃⢿⡈⢷⡘⠀⠀⠈⣷⣤⠶⠋⣡⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣈⡉⠛⠲⠶⠤⢤⣤⣾⡧⠤⠿⠖⠚⢋⣁⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣷⣶⣶⣶⣤⣤⣴⣶⣶⣾⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⢰⡔⠇⠋⠀⠈⠁⠉⠀⠘⠴⢀
⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠚⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠀⠀⠀⠀⠀⠆
⠀⠐⠴⠷⠿⠟⠛⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣎⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠆⠀⠀⠠⠂⣀⠰⠀⠦⠱⠸⠀⠇⠐⢀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡐⢦⠘⠀⠴⠰⠦⡶⡶⡶⠶⠆⠈⠋⢶⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡑⠀⣀⣀⣀⣈⣋⣋⣋⣋⣀⣀⣀⡀⠀⢋⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡒⢀⣄⣛⣟⣛⣛⣻⣻⣻⣻⣿⣟⣟⣳⣤⠀⢐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠅⠀⠁⡇⠁⠉⠈⠈⡡⡌⠈⠈⠁⠁⠁⠉⠀⠈⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠇⢤⡤⠠⠠⠁⠡⠰⠠⣤⠄⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠲⠋⠙⠁⠀⠀⠀⠀⠀⠀⠙⠋⠷⠶⠶⠔⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡶⣿⠿⠿⠿⠿⠿⠿⢿⢷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠒⠶⡄⢬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⣦⡀
⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠁⠈⢣⠀⠀⠀⣌⠀⠀⠀⢀⡀⢘⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⣾⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠘⡄⠀⢠⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠘⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣼⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠇⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠠⢀⢀⣀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⣀⡸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠙⣆⢘⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢀⠀⠘⠆⠀⠀⣿⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⣀⠤⠀⠸⣷⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠹⡆⠀⠀⠹⣆⠀⠀⠐⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠸⡄⠀⠀⠹⣇⠀⠀⠐⡀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣗⣄⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢹⡄⠀⠀⢻⠀⠀⠀⠰⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⣠⣺⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠘⢆⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣷⣦⡀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢢⠀⠀⠀⠀⠀⠀⠀⠐⢀⠤⠊⢁⣴⣾⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣀⠀⠉⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠑⠀⠀⢀⣀⠤⠒⠉⣀⣤⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣤⣀⡉⠉⠛⠓⠒⠂⠀⠠⠀⠒⠒⠒⠊⢉⣁⣤⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠷⣿⣶⣶⣶⣶⣶⣶⣾⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠊⣉⣭⡭⠿⠷⢶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⣀⠔⣋⠃⠀⠀⣴⠞⡩⢄⡀⠀⠀⠀⠀⠉⠛⢮⣕⢤⡀
⠀⠀⠀⢀⠞⡁⠀⠀⠀⠀⢸⣇⠘⢰⠕⠊⡉⠀⠀⠀⠀⠀⠀⠙⣦⠉⢢⡀
⠀⠀⡰⠁⠁⠀⠀⠀⠀⣴⣛⣉⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠙⣄
⠀⡜⢁⠌⠀⠀⠀⠀⣸⣷⣆⡉⠃⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠘⡆
⢰⠁⠀⠀⠀⢀⣴⠟⠉⣿⠥⠛⠀⠘⣄⢇⢰⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠅⠸⡀
⡆⠀⠀⠀⠀⢼⣁⣀⡀⠈⠀⠀⠀⠀⠈⠙⠾⢦⣀⣀⠀⡀⢠⣰⠇⠀⠀⠀⠐⡀⡇
⡇⠐⠀⠀⠀⠀⠉⣿⠁⠀⠀⠀⢨⠃⠀⠀⢀⣀⠜⠁⢀⣧⣾⠏⠀⠀⠀⠀⠀⠄⠰
⡇⠐⠀⠀⠀⠀⠈⣿⠒⠂⠒⠚⠁⠀⠀⠀⡸⠳⢄⣴⣾⠿⣇⠀⠀⠀⠀⠀⠀⠄⡎
⢸⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⢀⣀⣠⢞⡁⠤⠴⡇⡇⡇⢹⡀⠀⠀⠀⠀⠸⢀⠃
⠈⢇⠀⠀⠀⠀⠀⠉⠙⣯⢉⡭⠔⠊⠉⠀⠀⠀⠀⡇⠆⠀⢸⠇⠀⠀⠀⠐⠂⡎
⠀⠈⢆⠀⠀⠀⠀⠀⡾⠋⠁⠀⠀⢀⣀⡤⠔⠒⠊⠓⢄⣠⡟⠀⠀⠀⠀⢀⠜⠀⠀⣀⡤⢴⣲⣾⠙⡿⣿⡛⣗⣶⡤⣄⡀
⠀⠀⠀⠳⣄⠀⠀⣼⠁⣠⠤⠒⣨⠋⠁⠀⣀⠤⣤⠀⠀⠙⣦⠀⠀⠀⡠⠊⣀⡴⡯⣿⡆⢙⣉⣭⣤⣥⣍⡉⠙⠑⠁⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠈⠳⢤⡟⠉⠀⠀⠀⠇⠀⠀⠈⠐⠛⠓⠀⠀⠀⢸⣆⡠⠊⢀⣞⣁⠿⠋⣡⠞⠋⠁⠼⣦⠶⠦⢽⡀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠚⠁⠀⣰⠋⠛⠋⠀⢰⡇⠀⠀⠀⣹⣢⠀⣴⠆⢻⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⣇⠀⠀⠀⢘⠏⠀⠰⠿⠏⢷⡀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠹⣆⣔⡚⣼⠀⠀⠀⠀⠰⣤⡿⠂⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢻⣗⣭⡏⢆⠀⠒⠤⠤⠼⣇⠀⠀⠄⠙⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡟⢄⣉⠉⢁⡍⣉⡉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⠀⡄⢂⠰⠀⢰⡟⠤⠤⢭⣍⣛⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡐⡄⠞⠊⠃⠁⠀⠀⢠⡟⠀⠈⠁⠲⣌⠑⠺⣿⢿⣦⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣼⡵⠊⠉⠒⠦⡈⠒⢲⠈⢣⡙⣿⣄⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⡟⠀⡿⠟⠋⠀⠘⡆⠈⠲⢄⠱⡘⢿⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⣵⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⢧⣀⣀⠀⠀⠀⢀⣘⣄⡤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠠⠤⠤⠄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⣉⡉⠤⠤⠐⠒⠒⠒⠒⠒⠒⠀⠤⠤⣉⣉⠒⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⠥⠒⠊⢩⢄⠀⠰⢆⡽⠂⢸⠼⡋⢠⠿⡹⠆⠀⣀⡀⠉⠑⠒⠬⣉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⡡⠔⠋⢠⣒⡽⠀⠘⣾⠧⡀⠸⣧⠀⢸⠮⣁⢠⣵⣌⠂⠎⠖⡏⡆⡔⡢⣤⠀⠙⠢⢌⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⠔⠋⢰⣫⣓⡀⠈⢿⡄⠀⡷⠙⠓⠀⠉⠀⠈⠁⠁⠈⠉⠁⠀⠓⠯⠔⣜⡔⠻⠁⣠⣴⡱⠀⠑⠢⡙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⠔⣡⢤⣄⡀⠑⣒⢞⠆⠈⢩⣤⠀⠀⣤⣀⡄⢠⣤⢠⣤⣠⡄⣤⢀⣤⣄⣠⡌⠀⠀⠾⠧⡻⠃⣠⣞⣇⣈⠢⡉⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠊⡴⢁⡀⠳⣧⣙⡿⠀⠈⠉⠀⠀⠸⠿⠛⠛⡝⠩⠧⠼⠿⠺⠿⠭⠇⠿⠟⠿⠿⠷⠇⠀⠀⠀⠈⠃⠼⠿⣿⣧⠎⢀⡈⢦⠑⣄
⠀⠀⠀⠀⠀⡴⢁⠞⠰⣥⠗⡆⣈⠳⠊⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣲⡇⣞⡾⢰⣷⡇⢸⣾⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⣡⣔⢻⡮⠆⠱⡈⢦
⠀⠀⠀⠀⡜⢠⠋⡜⣆⠀⠙⠿⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠛⠙⠈⠛⠁⠛⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠧⠉⡠⣔⡢⠙⡄⢣
⠀⠀⠀⡸⢡⠃⠘⠋⠙⠻⣦⠀⠀⠀⠘⣿⣿⢻⡿⢻⡟⢻⡟⣿⡟⣿⠛⣿⠻⣿⢻⡿⢻⡟⢻⡟⣿⡟⣿⠛⣿⠻⣿⡟⠀⠀⠀⠀⠐⢿⣦⠿⠝⠀⠘⡌⢇
⠀⠀⢰⠁⡎⠘⠯⣖⡤⣀⠀⣀⣀⣀⣀⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣷⣀⣀⣀⡀⠀⠘⣃⢤⣶⠽⠃⢱⠈⡆
⠀⠀⠇⢸⠐⢶⣦⡤⢌⡉⠀⢙⣾⣉⣉⣉⣭⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣈⣉⣱⣞⠁⠀⠀⠉⢁⣤⣤⡤⡀⠆⢸
⠀⢸⠀⡇⠺⠤⠿⣷⣍⠀⠀⠸⣿⡏⠸⣿⠇⠹⣿⠇⢻⣿⠏⢻⣿⠇⢿⣽⠇⠁⢻⣽⠇⢿⣽⠇⢿⣽⠇⢿⣽⠇⢿⣽⠃⢿⣿⠀⠀⠀⠀⢇⣏⠀⠼⠏⢸⠀⡇
⠀⠸⢰⠀⠶⢦⣤⡤⡀⠀⠀⠀⡇⡁⠀⠀⠀⠀⠐⠀⢰⢰⠀⢸⢸⠀⢸⢸⢀⢤⠀⢲⠀⠀⠀⠀⠀⠠⠀⢸⠐⠀⢸⠐⠀⢸⠐⠀⠀⠀⠀⠰⣖⣠⡤⢤⠀⡄⠇
⠀⡇⢸⠐⠶⠶⠶⠭⠃⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸⠀⢸⢸⣿⠴⡷⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠠⠯⠞⠙⠉⠀⡇⠈
⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⢀⠀⢸⢸⠀⢸⢸⠀⢸⢸⠳⠀⠇⠸⠀⠀⢠⠀⠀⢠⠀⢸⠐⠀⢸⢀⠀⢸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰
⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⢀⡤⠷⠧⠔⠶⠦⠔⠾⡷⢴⣶⠠⠼⠾⠄⠸⠾⠤⠄⠔⠾⠤⠴⠶⠤⠰⣶⡤⣿⠾⠤⠘⠶⠤⠺⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇
⠀⢸⠀⡇⠀⠀⠀⠀⢰⠒⠛⠒⠒⢒⣒⣰⣶⣒⡞⠻⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠓⠛⣖⣰⣶⣒⣒⠒⠒⠚⠓⢲⠀⡀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⡇⢸⠀⠀⠀⣠⣮⣷⡶⣛⠶⢻⡙⢤⡬⠓⠳⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠇⠊⣤⠜⣹⡷⢶⡳⣶⣯⣦⡙⢿⡄⠀⠀⡇⢸
⠀⠀⠸⡀⢇⠀⠘⠛⠓⠛⠚⠛⠓⠚⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠛⠛⠚⠓⠛⠛⠀⠀⠀⡸⢀⠇
⠀⠀⠀⢱⡘⡄⠀⠀⠀⠀⠀⣠⠔⣒⣓⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢩⠂⠀⠀⠀⠀⠀⠀⠀⢠⢃⡎
⠀⠀⠀⠀⢣⠘⢄⠀⠀⢀⠞⡵⠋⠀⢈⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣏⠴⢌⡲⢄⠀⠀⠀⠀⠀⣠⠃⡜
⠀⠀⠀⠀⠀⠱⡈⢆⠀⠸⡘⢄⣀⠔⣫⠜⢀⠜⢙⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⡜⢆⠀⠀⠉⠢⡑⢦⠀⠀⡰⢁⠞
⠀⠀⠀⠀⠀⠀⠘⣄⠳⡀⠉⠒⠒⠈⠁⢠⢋⡾⠸⣠⢋⠏⡰⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⢲⡈⢦⠨⣕⠬⣆⠣⡀⠀⠀⠈⠁⢀⠞⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣈⠢⡀⠀⠀⡰⡡⠋⡇⡿⡡⠃⢠⢣⣞⠒⠰⠃⠀⢀⠞⡩⢌⠱⡄⢣⢰⣈⡡⣄⠀⠱⡙⢍⠪⢅⣹⠄⠀⢀⠔⣁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣌⠢⣄⠈⠁⠰⣁⡜⠁⠀⡎⡤⠤⢩⠇⠀⠀⢸⢰⠁⠀⠉⠁⠈⡆⢡⠐⠊⠀⠀⠘⢌⡦⠀⠀⣠⠔⣡⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡑⠢⣄⠈⠀⠀⢸⣀⠓⠠⢄⠀⠀⠀⠸⡈⣄⢀⠖⡆⠀⠘⡈⠒⣋⡹⠆⠀⠀⣠⠔⢊⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣉⡒⠤⢄⣀⠉⠑⠋⠀⠀⠀⠀⠑⠤⠥⠔⠁⠀⠀⠉⠉⣀⡠⠤⢒⣉⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⣉⣁⠒⠒⠀⠤⠤⠤⠤⠤⠤⠐⠒⠒⣉⣉⠤⠔⠊⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣁⣐⣒⣀⣀⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡒⠫⠩⠀⠂⠂⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⢔⠕⠁⡠⢤⠀⠀⠰⠶⠯⣧⡤⣄⠀⠀⠙⡳⣖⡶⣤⢤⠤⡤⡶⢶⣂⣀⠤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⠆⢀⠈⠀⠀⠈⠲⠄⠀⠤⠈⠹⠶⣯⣵⡄⠘⢪⡳⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⠪⢕⡦⡀
⠀⠀⠀⠀⠀⣸⠘⠀⠘⠃⠢⣄⣀⡺⠀⠀⠀⠀⠀⠀⠀⠉⣯⣵⡄⠉⠘⡆⢰⡦⠖⠊⠀⣀⡀⠀⠀⢀⡉⢆⢆
⠀⠀⠀⠀⠀⣿⠠⠀⠀⢸⣷⡀⠀⣀⠢⢀⡀⠀⡀⠤⡄⠁⢀⠿⣥⡀⠀⠜⡟⠛⠀⠀⠉⠀⢠⠀⠈⡋⢉⠀⠎⡆
⠀⠀⠀⠀⠀⢸⣧⠂⠀⠈⢏⡽⡀⠁⢀⠄⠠⠀⢀⡘⠀⠀⠈⡛⣷⠁⠀⡀⡇⢀⢀⡀⢠⠀⠀⢱⠀⣢⢀⠀⡉⣿
⠀⠀⡄⠈⠀⠀⢻⣧⡀⠀⠈⢮⡽⣤⠄⠀⠉⢠⠈⠀⠀⢀⣤⣴⣿⣲⢖⢠⠇⠒⠉⠁⠉⠀⠀⠈⠀⢀⢀⢄⡴⠃
⠀⠀⠳⣄⠀⣴⣦⣿⣷⣔⠀⠀⠳⠽⠈⢯⡢⣤⢤⣤⡤⣼⠁⣼⠁⢀⢄⡎⠀⠠⠄⠠⠠⠤⠄⠐⣃⣭⠶⠋
⠀⠀⠀⠘⠢⣿⣿⣿⣿⣿⣿⣠⢄⡀⠈⠉⠙⠫⠿⠮⠗⠉⠀⠁⢐⣵⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠛⠿⠿⢾⣧⣶⣦⣤⣤⣤⣤⣤⣶⠾⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢉⣉⣭⣭⣯⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠫⠁⠀⠀⣠⠾⢋⣉⠀⠀⠀⠀⠀⠉⠙⠻⢯⣓⠦⡀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠂⠀⠀⣼⠋⣰⢫⢘⡭⠤⠄⠀⠀⠀⠀⠀⠀⠙⢷⡈⠳⣄
⠀⠀⠀⡰⠫⠀⠋⠀⠀⠀⠀⣠⡿⠆⠉⠙⠁⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠈⢣⡀
⠀⠀⡼⠡⢀⠀⠀⠀⠀⠀⣼⢋⠶⣦⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡀⠹⡄
⠀⡼⠑⡔⠀⠀⠀⠀⢀⣸⣿⣿⢤⡀⠁⢀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢀⢀⠱⡀
⢰⠃⠀⠀⠀⠀⢀⡴⠋⠁⢸⡯⠒⠃⠀⠀⢧⡸⡀⡏⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠡⠂⢣
⡜⠈⠀⠀⠀⠀⣾⣁⡀⣀⠀⠉⠀⠀⠀⠀⠀⠙⠳⠼⣄⣀⣀⠀⡀⠀⡴⣼⠃⠀⠀⠀⠐⠒⠘⡄
⡇⠀⠀⠀⠀⠀⠈⠉⣻⠋⠀⠀⠀⠀⢸⠇⠀⠀⠀⡀⢀⡼⠁⠀⣇⣴⣿⠃⠀⠀⠀⠀⠀⣀⠀⡇
⡇⢠⠄⠀⠀⠀⠀⠐⣿⣤⣀⣀⢤⠴⠋⠀⠀⠀⠀⡿⣍⠀⢀⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⠀⡇
⢱⠉⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢉⡏⣯⣇⢹⡆⠀⠀⠀⠀⠀⠀⡀⢰⠁
⠘⡆⠀⠀⠀⠀⠀⠈⠳⠤⣤⠀⠤⠤⠴⣶⡶⠿⠖⠛⠉⠉⡇⢰⠘⠀⣿⠀⠀⠀⠀⢀⡈⠀⡎
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣸⣇⠤⠒⠉⠀⠀⠀⠀⠀⠀⢀⡇⠛⠀⢀⡿⠀⠀⠀⠀⠀⠈⡼⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠹⡄⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⣀⣠⠴⠖⠚⠉⠉⠙⠢⣤⡾⠁⠀⠀⠀⠀⢀⡜⠁⠀⢀⣠⠶⣞⣿⣯⡏⠙⡟⣿⣍⣭⡝⣷⠶⢤⣀
⠀⠀⠀⠈⢦⡀⠀⠀⣸⠇⢀⡠⠔⠚⣩⠋⠁⠀⢀⣀⠤⣤⠀⠀⠈⢷⡄⠀⠀⠀⡠⠊⢀⣤⢾⢭⣿⡇⠘⣋⣡⣥⣤⣤⣤⣈⠉⠁⠋⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠙⢦⣀⡿⠚⠁⠀⠀⠠⠇⠀⠀⠈⠁⠶⠟⠓⠀⠀⠀⠈⣷⢀⡠⠚⠁⡴⢋⠻⡿⠋⣡⡶⠛⠉⢩⢗⣾⡲⣬⣹⡆⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠉⠀⢠⠞⠽⡟⠃⠀⢸⡏⠀⠀⠀⢒⡾⡀⠀⣠⡌⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠶⠤⠤⠤⠤⠤⠴⠒⠒⠉⠁⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣷⠇⠀⢉⣴⡼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠈⣷⠀⢀⣀⡀⡎⠀⠀⠈⠉⠁⡈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⡇⢤⢽⣗⠀⠀⠀⠀⠘⠛⣿⠋⠀⠀⢀⢀⡄⡔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣶⣾⠌⢄⡀⠉⠐⠒⢚⣻⣆⠀⠀⠎⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢄⣈⠉⢉⡍⣡⣠⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣀⢠⡄⢆⠲⠀⠠⣾⠧⠤⢬⣭⣙⣛⣻⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠠⡀⠮⡇⠫⠈⠁⠀⠀⠀⣼⠋⠉⠑⠒⢤⡉⠓⠬⣯⡽⣧⣄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠁⠀⠀⠀⠀⠀⠀⠀⢰⣏⡠⠤⠤⢄⡀⠹⠤⢤⣈⠳⡈⢿⣷⡀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⣿⠋⣠⣤⣤⠄⠈⢢⡀⠀⢇⡀⠙⣄⢻⣿⡀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣿⠀⠘⠉⠁⠀⠀⠀⢱⠀⠀⠉⠢⡘⢆⠙⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠧⣄⣀⣀⡀⠀⠀⣀⣀⣧⠤⠶⠚⠉
⣶⣶⣶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠀⠄⠀⠀⠀⠀⠀⡀⠂⠀⠄⠆⠀⠈⠉⡓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢋⠁⠀⠂⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠁⠁⠀⣈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣠⡴⡆⠋⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠔⠁⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣟⠛⠀⠀⠀⠀⠀⠀⠀⠀⡉⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠨⠀⠀⠀⠀⠀⠀⣠⣾⣿⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡂⠠⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢆⡙⢦
⠀⠀⠀⠀⠀⠄⢀⠜⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰⣿⡈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠂⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡍⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠇⠀⠀⠠⠀⠀⠄⠈⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⡆⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⠋⠀⠀⠈⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧⡀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠈⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢧⠀⠀⠀⠀⠀⠁⠀⠂
⠀⢸⠀⠀⠀⠁⠀⠀⠀⡁⠀⡁⠀⢈⡁⠈⡀⠀⠉⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢃⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠈
⠀⠀⠈⡄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠒⠃⠀⠂⠀⠊⠂⠀⠀⢠⠇
⠀⠀⠀⠰⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠐⠀⠀⠀⠀⠴⠇⠀⠠⠔⠀⠠⠀⠔⠤⠀⢀⡎
⠀⠀⠀⠀⠱⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠱⡄⠙⣄⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀
⠀⠀⠀⠀⠀⠀⠈⢆⠈⠢⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠀⠹⠀⠐⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠢⡀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠲⢄⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠉⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⣈⠉⠒⠂⠤⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠉⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠄⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢀⡤⠴⠒⠋⠉⠉⠉⠙⠓⠲⢤⣀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⠉⠉⠙⠒⠦⢤⡀
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⢀⠞⠁⠀⠀⠀⢀⠊⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠳⡞⠁⠀⠀⠀⠀⠀⠂⠀⠐⠐⠂⠀⠀⠀⠀⠀⠈⠳⡀
⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣠⣠⣀⠀⢀⣀⢈⣤⣤⡄⠄⠀⠀⠀⠀⠹
⡾⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡒⣀⢀⢂⢀⡀⣀⣀⢀⣀⣀⣀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇⠀⠀⠠⠀⠀⡇⠀⢸⢸
⣇⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠔⠀⠀⠢⠂⠀⠀⠀⠂⢁⠂⠜⠌⠄
⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠄⠈⠀⠒⠀⠒⠒⠒⠒⠒⠒⠒⠓⠑⠈⠂⠀⢠
⠀⠳⡀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠹⡀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠙⢦⡀⢁⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⡰⠋⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠙⠢⣄⡀⠠⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠄⣀⣠⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⢉⡭⠥⠄⡀⢀⠀⠀⣀⠀⠀⠤⠬⢉⡉⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⡡⠔⢚⠉⢴⠀⣧⡈⠇⠹⠂⢈⢑⠀⠀⢔⢂⠖⠈⠉⠛⡦⢄⡙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣡⠔⠋⠀⠀⠈⠒⠂⠒⠁⠁⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠺⠁⢴⠈⠲⢌⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⢞⠁⠫⣉⡂⠀⠀⠁⠁⡠⠀⠀⢠⢄⠀⠀⠀⣄⠀⣀⡀⠀⢀⡀⠈⠈⠘⠈⠱⠖⣑⢄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡜⣅⠙⠦⡔⠈⠀⠀⠀⠀⠀⣯⡀⠈⠘⠀⠃⠐⠈⠁⠂⠫⡈⠐⠀⠘⠀⠀⠀⠈⠃⠘⠛⡀⠱⣄⠱⡀
⠀⠀⠀⠀⠀⠀⢀⠌⢠⠎⠤⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠏⠀⠋⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠑⠙⣆⠙⣄
⠀⠀⠀⠀⠀⢀⡎⢠⠏⣬⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢙⠏⠘⣇⠘⡆
⠀⠀⠀⠀⠀⡞⢠⠏⠈⠢⡁⠀⠀⠀⠀⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠸⡄⢹⡀
⠀⠀⠀⠀⢰⠁⡾⠈⠁⠊⠀⠀⠀⠀⠀⣿⠀⠀⠐⢒⣒⣒⣒⣀⣒⠒⠒⣒⣒⢒⠒⠉⠉⠉⠀⡉⠓⠒⠒⠒⠀⢸⡇⠀⠀⠀⠀⠀⢳⠀⢇
⠀⠀⠀⠀⡎⢰⡇⠀⠀⠀⠀⠀⠀⣾⣉⡉⢩⡍⣩⠉⢉⣀⣀⡠⣭⣭⣭⣭⣭⣬⣭⣭⣭⢍⡭⣭⠍⠉⡉⣀⠉⠉⣭⣇⠀⠀⠀⠀⠸⡆⢸
⠀⠀⠀⢰⡃⣸⡇⠀⠀⠀⠀⠀⠀⡎⢩⠀⠀⠃⠃⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⡇⠀⠈⠀⠀⠀⠀⡇⢠⠀⠀⡆⠘⠀⠀⠀⠀⠀⡇⠈⡇
⠀⠀⠀⢸⠁⣿⣷⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⢸⠀⠀⣇⠀⠀⠀⠀⠸⠀⠀⠀⡇⡇⠀⠀⡀⠀⠀⠀⡇⢸⠀⠀⠀⡸⠀⠀⠀⠀⠀⣷⠀⡇
⠀⠀⠀⠸⠀⢿⢻⡀⠀⠀⠀⠀⠀⠃⢸⠀⠀⢸⢸⠀⠀⢋⠀⠀⠀⢠⠀⠀⠀⠀⠃⡇⠀⠀⡇⡇⠀⠀⠇⢸⠀⠀⡆⡁⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⡆⢸⣿⡇⠀⠀⠀⠀⠀⢠⣸⣀⣀⣸⣸⣀⣀⣸⢀⣀⣀⣸⣀⣆⣀⣀⣀⣁⣀⣀⣇⣀⣀⣀⣇⣈⡀⢀⣄⡇⠀⠀⠀⠀⠀⡟⢀⠃
⠀⠀⠀⠀⢣⠈⣿⣷⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢠⡇⡸
⠀⠀⠀⠀⠀⠆⠹⣜⣇⠀⠀⣼⣁⣤⣤⣀⣀⢀⣀⡀⠀⠀⠀⣀⡀⠀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣠⣌⡆⠀⠀⡼⢠⠇
⠀⠀⠀⠀⠀⠸⡄⢻⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡾
⠀⠀⠀⠀⠀⠀⠹⡄⢻⣿⡄⠀⠀⠀⡀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⣰⠃⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢄⠹⣿⡄⠀⠈⠀⠀⢈⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠡⠀⠀⠀⢀⡔⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡈⠻⣦⠀⣀⠠⠊⠀⠃⠀⠈⠀⠖⠒⠀⠀⠀⠀⠄⠄⠀⠠⠀⠂⠀⠒⠠⢄⡈⡀⠀⠱⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⡓⢤⡀⠐⠀⢠⠌⠀⡰⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄⠀⠀⠀⡄⠀⠈⠙⣀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⡑⠢⢄⡀⠰⠄⢀⠀⠀⠀⠀⠀⢄⣀⡠⠀⢀⡀⠠⠀⣀⡤⠔⢚⡡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢉⠒⠢⠤⠄⣀⣀⣀⣀⣠⣤⡤⠤⠴⠒⢉⣡⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠒⠒⠤⠤⠤⠤⠀⠐⠒⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠂⣀⡩⡥⠤⠤⡄⣶⡔⡤⠤⢤⢭⣀⣒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣋⡥⡖⠛⣍⢰⣷⡃⠾⢸⡿⠄⠧⠇⢙⡣⠸⠞⢨⠉⣒⠦⣍⡓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣉⠴⣚⠁⢎⣁⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠞⣤⠍⡳⢌⡑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⢞⡗⠈⠞⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠾⠴⠂⣟⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠞⣡⡞⠱⣄⡀⠁⠀⠀⠀⠀⠀⢀⡀⠀⢦⡀⠀⠐⣄⠀⠆⢰⠀⠀⠀⣀⣠⠶⠀⠀⠀⠀⠉⢹⠃⣜⢦⡙⢆
⠀⠀⠀⠀⠀⣠⠋⣴⣯⠏⠁⠈⠀⠀⠀⠀⠀⢠⠀⡇⠀⠈⠀⡇⠀⢀⡈⢣⡇⢸⠀⠐⠀⣁⣶⠀⠆⠀⠀⠀⠀⠀⠀⠈⠳⡑⡌⢳⡀
⠀⠀⠀⠀⢰⠃⣼⡛⢈⡀⠄⢂⠀⠀⠀⠀⠀⠈⠀⢧⡀⠀⢀⡇⠐⢰⡏⢦⠀⢸⠀⢠⠀⡏⠁⠀⣀⠀⠀⠀⠀⢰⠁⠢⣄⠘⠙⣆⢳⡀
⠀⠀⠀⢠⠃⣼⣿⣁⠊⠀⠀⠈⡃⠀⠀⠀⠀⠀⠈⠢⠤⠴⠊⠁⠰⠴⠃⠀⠳⠼⠀⠐⠂⠤⠤⠴⠟⠀⠀⠀⠀⢨⠀⠀⠀⢳⠀⠘⡆⢳⡀
⠀⠀⠀⡞⢸⢞⡃⠃⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⡄⠀⠀⠀⢃⠀⢸⡈⣧
⠀⠀⢰⠁⡟⡈⢡⠀⠀⠀⠐⠁⠀⠀⠀⠀⢠⠶⠶⠇⠀⠃⠶⢒⡋⠀⠀⠀⠰⡀⠀⢸⡆⠀⠓⠲⠀⡖⠛⠀⠀⠀⠘⡄⠁⢀⢀⡄⠈⡇⢸⡄
⠀⠀⢸⢠⢷⠃⡌⢀⢒⠀⠰⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⡶⠚⠃⠀⠀⢸⢦⡙⠆⠸⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢡⢁⡊⣄⡀⠀⢸⠸⡇
⠀⠀⣼⢸⢸⠀⣇⡎⢂⡀⡃⠀⠀⠀⠀⠠⣈⠃⠀⣤⠀⠀⠓⠀⠀⣤⢀⢸⠆⠱⣄⣸⡇⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⣼⢸⠑⣹⡇⠀⢸⠀⣧
⠀⠀⢹⢸⣼⠀⢏⡃⠾⢱⠃⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠈⠉⠉⠉⠉⠈⠉⠀⠀⠈⠉⠀⠀⠀⠈⠉⠁⠇⢀⠀⠀⠀⢻⢀⠶⡩⡆⠀⢸⠀⡏
⠀⠀⢸⠀⡇⠇⠸⡗⢐⠉⡖⠀⣤⢠⡄⢠⡄⠄⢠⠤⣤⠄⣴⠆⢴⢤⡀⠀⣤⠦⠰⣶⠆⣰⡄⢶⡶⠀⣶⠂⣴⠂⠠⢆⡏⡖⣁⠡⠀⠸⢸⡇
⠀⠀⠈⡇⢡⠀⠀⢸⡀⣶⢋⡄⢸⣠⠇⢸⠙⣷⢸⠀⡿⠀⣯⣁⢺⣨⠇⠀⣉⣷⠀⣿⢠⠟⢻⢸⠇⠸⣯⡄⣬⡷⢸⡜⠀⢉⠠⠀⠀⠁⣼⠃
⠀⠀⠀⢱⡘⣏⠀⠪⢯⡛⡌⣨⣆⠀⠀⢀⣀⠀⣀⣀⠀⠀⣀⠀⢀⠀⡀⠀⣀⡀⣁⣀⢀⢀⣀⡀⢀⡀⠂⠀⠀⡰⠯⢷⣋⡥⠀⠀⠀⢠⡏
⠀⠀⠀⠀⢣⠸⡄⠀⠳⣥⠀⢤⡾⡦⠀⡟⠸⡇⡿⠆⠀⣰⣽⡀⣾⠿⢿⢠⡷⠆⡷⣿⢸⢸⡁⠀⣾⢿⡀⠀⡔⡉⢳⣨⢍⠅⠁⠀⢀⡿
⠀⠀⠀⠀⠈⢧⡙⣌⠀⠘⢕⣖⡃⡽⢁⠈⠛⠁⠁⡆⠀⠁⠈⠁⠁⠀⠈⠀⠋⠁⠁⠉⠈⠀⡉⠁⠁⠈⡅⢰⠎⠺⠐⢮⠑⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠳⡌⢦⡀⠀⠙⠻⠍⡞⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣂⠀⠀⢁⠈⠛⠉⠀⠂⢀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢄⠀⠀⠀⠀⠉⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠨⠀⢁⠠⠐⠁⡀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⠄⡀⠀⠀⠀⠀⠈⠑⠒⠒⠆⠀⠀⠀⠀⠀⠀⠀⡇⠐⠂⠀⡊⣠⢸⠀⠂⠁⢀⠄⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⡀⠀⢀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⡀⢰⠀⠆⠀⠰⠀⠀⠋⠀⡀⠄⢊⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣍⡀⠂⠠⠀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄⠀⣀⣡⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠦⢤⣤⣤⣭⣤⣤⣤⠤⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣒⣒⠒⠒⠒⠶⠀⠀⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡴⠖⡋⠉⡉⢨⢤⠉⠉⠩⠉⡉⢒⠒⠒⠦⢤⣍⣀⠕⡂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⡀⢂⣤⢞⡁⠀⠘⠦⠎⠁⠀⠀⠀⠀⠀⠒⠀⠿⠀⠀⣰⠒⢀⢉⠙⠢⣕⠢⡁⠢⡀
⠀⠀⠀⠀⠀⠀⠄⣰⠟⠰⡝⠛⠀⠀⠀⠀⠀⡀⠔⠀⠈⠉⠁⠐⠠⢄⠀⠀⠀⠛⠆⣴⣤⣈⠑⢎⠲⢌⠦⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠐⠈⣄⡀⠉⠹⢻⡁⠀⠀⠀⠀⠀⠲⠟⣰⣄⠑⢄⠳⣕⣄
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠠⣨⡄⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠈⢁⢀⠀⠀⠳⡈⢦⢣
⠀⠀⠀⠀⢠⣏⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠀⠂⠄⢤⠰⠚⠀⠔⡷⡎⠀⠀⠀⠀⠀⠀⢄⢔⠔⡠⡂⠀⠑⠀⢫⣧
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⣿⢿⠁⠀⣖⠄⠀⢀⠈⡄⠀⠀⠀⠀⠀⠀⠁⠂⠀⡨⣂⠀⠡⠀⢿⣇
⠀⠀⠀⠁⡏⡨⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠏⢢⠄⠀⢘⡆⠀⢠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⢺
⠀⠀⠀⡀⢱⠊⣆⠱⢶⣢⡂⢶⡔⣖⢰⡶⠀⡁⠀⠀⠱⣤⣀⣀⣀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭
⠀⠀⠀⠃⢸⡈⢙⡉⢝⠁⡉⠉⠉⠈⠀⠁⠰⠀⠌⣒⢤⣀⡼⣶⠚⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣛
⠀⠀⠀⠀⠀⡬⡫⠐⢑⠕⡢⠀⠀⠀⠀⡠⢣⡀⠀⠌⠑⢬⣀⠛⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡺
⠀⠀⠀⠀⢀⠩⡨⢀⠕⡡⡊⢔⠡⣂⡜⠀⠀⠈⠳⢄⠑⠆⢙⣉⣀⣹⠀⠀⠀⠐⠂⢰⠆⢠⠂⣖⠂⠀⠀⠀⠀⠀⣓⠃
⠀⠀⠀⠀⠀⠆⠲⠕⠩⡪⢔⠅⣪⠎⠀⠠⣈⠓⢦⡈⢧⡀⠀⠹⣽⣜⣦⠀⠀⠚⠛⠛⠚⠘⠃⢹⠇⠀⠀⣰⠃⣼⡞
⠀⠀⠀⠀⠀⠈⢆⠡⣪⢐⢕⠀⡚⢀⠀⠀⠀⠙⢢⡙⠀⢻⡄⠁⠈⢦⠝⢗⢤⠀⣄⢄⣄⡀⡀⢀⢀⢀⡄⢁⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠡⡈⠓⣕⠪⢇⠀⠐⠐⠀⠘⢦⠹⣀⠈⣿⣄⠀⠘⣆⠘⣎⠣⡀⠀⡠⠂⢀⢔⡠⠊⡠⡲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢌⡑⠬⡛⠲⢥⣀⢀⠀⠁⢹⡄⢻⠴⠃⠀⠈⠆⠀⠂⣙⡌⠈⡔⠕⢉⣤⡺⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠬⣑⠢⢬⣉⡓⠶⠴⢥⢤⢤⡤⡴⠶⢖⠛⠋⠍⠐⢈⡠⣖⠯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠢⢬⣅⣐⣒⣐⣒⡒⣀⣉⣀⣠⣲⠶⠟⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⠀⠠⡂⢐⠄⠢⡀⢄⠀⡉⢉⠍⡉⠉⡩⠉⢌⠄⡠⡀⢔⠠⠀⢄⠄⠠⠀⢄⢀⡠⠠⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠈⠀⠈⠀⠁⠈⠀⠁⠀⠈⠀⠁⠈⠀⠀⠁⠈⠀⠁

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

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

⡿⠿⠿⠿⡿⣻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣟⣻⣿⣿⢿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠶⠘⠂⠋⠈⠁⠁⠈⠈⠁⠙⠘⠂⠦⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠑⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠶⣀
⠀⠀⠀⠀⠀⠀⣀⠶⠉⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⡀⠀⠀⠉⠶⣀
⠀⠀⠀⠀⠀⠶⠉⠀⠀⠀⠀⢀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⡠⠊⠀⠀⠀⠩⠆
⠀⠀⠀⠠⡬⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠒⠂⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥
⠀⠀⢀⡒⠂⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠊⠠⠀⠀⠀⠀⠀⠀⠄⠑⠄⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠐⢒⡀
⠀⠀⢤⠅⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⠨⡤
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒
⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇⠀⠠⠀⠀⠈⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁
⠀⠰⠆⠀⡆⠁⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⠐⠄⠀⢰⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⠀⡀⠁⠀⠰⠄
⠀⢘⡃⠀⠈⠐⠂⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⡃
⠀⠈⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠁
⠀⠀⠲⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⢉⠁⠀⢈⠀⠈⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠖
⠀⠀⠈⢥⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠐⡀⠀⢠⠀⠀⠣⠀⠈⠆⠀⠘⠀⠀⠀⢤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠠⡌⠁
⠀⠀⠀⠐⢓⡀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒
⠀⠀⠀⠀⠈⠷⡀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠃⠀⠀⠀⠀⡐⠎⠁
⠀⠀⠀⠀⠀⠀⠉⠦⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡀⠦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢠⢄⡀⠀⠀⠀⠀⠀⠀⠀⢃⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⠶⢠⡄⣀⢀⡀⡀⣀⢀⡀⣀⢠⡄⠦⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⢈⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢘⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢨⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⣤⡤⠴⣿⣀⡀
⠀⠀⢀⣀⣀⡤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢨⠀⠑⢄⡀⠀⠠⠤⠤⠀⠠⠀⣿⡞⠁⠉⠀⠀⠀⠀⠀⠀⡠⢴⡇
⠀⠀⠀⠨⠀⠀⠀⠐⠤⠄⠀⠀⠀⢀⠀⠀⠀⡀⣰⣀⣸⣀⡄⠂⢠⡄⠐⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣧⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⠂⠉⠁⢸⡃⠀⠀⠀⠀⠀⡀⣀⠠⠠⠀⠂⠀⠂⠁⠘⠀⠀⠀⡇⠎⡇
⠀⠀⠀⢸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣸⠏⠐⠀⠀⠀⠀⢑⣀⣀⡀⢀⣀⣀⣐⣀⣠⣂⠀⠀⣇⣀⡇
⠀⠀⠀⢰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡆⡀⠀⠀⠀⠀⢀⠀⠉⢁⢸⡿⡿⢿⠿⣿⡿⣧⠀⣿⣿⣧⣤⣄
⠀⠀⠀⠚⠐⠒⠒⠒⠺⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣾⠧⡆⠀⠀⠀⠀⠐⠒⠒⠖⠲⠒⠒⠒⠒⢲⡖⠾⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠟⠀⠣⠄⡀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠾⠉⠁⠁⠂⠀⠀⠀⠀⠀⠐⠀⢀⣠⣤⣤⢴⡖⠒⠉⠍⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⡀⠀⢀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⣻⠟⠓⠶⠾⠷⠴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣙⣦⠠⡟⠦⣄⣙⣷⠐⠶⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⡙⣿⣧⣣⠀⠀⠠⢋⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣽⣟⡿⣿⣦⣴⢶⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⡹⠫⠋⣼⠇⠘⠀⠙⢦⡈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⣩⢠⡟⣡⠀⠸⠋⡀⠀⢠⣀⠀⠉⢂⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⣱⣾⠇⠀⠰⠟⠀⢀⣼⣿⣷⣾⣿⣷⣤⡄⠀⠀⠐⢿⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⣺⡟⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⠉⣿⣇⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣏⡁⢠⣤⡀⠀⠀⠈⢿⣿⣿⣿⣞⣿⡿⠿⣷⣄⠀⠀⠀⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣸⣿⣿⣾⣿⡇⠀⠀⠀⠀⠉⣿⡟⠉⣿⡇⠀⣼⣿⠀⠀⠰⣿⡆⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠉⢿⣿⣿⣿⣇⠀⠀⢠⣤⣀⣿⣇⣸⣿⣧⣾⣿⡟⠀⠀⠀⠿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⢻⣿⡟⡿⣆⠀⠀⠉⢻⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢄⠈⠁⠃⠁⠈⠧⠀⠀⣼⡿⠀⠿⠋⠀⠀⡀⣀⣾⣷⡄⠀⡔⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣷⢦⣤⣀⣀⣀⠀⠀⠈⠁⠀⠀⢀⡴⠞⠿⣿⣿⣏⣄⣼⣳⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠿⠿⢿⣿⣷⣤⣤⣀⣀⡤⠾⠶⠶⢦⣼⡯⠾⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⡬⣥
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠔⠄⡈⠁⠠⡤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠀⠘⠁⢤⠁⠀⠀⠀⣓⠀⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠐⢨⣖⠋⠠⠤⠄⠀⠀⠈⠓⠉⠁⠀⠀⠀⠉⠑⠌⠀⠀⠀⠀⠀⠉⢐⠢⠠⠀⠄⠀⡀
⠀⠀⠀⠀⢀⠀⣠⣆⠀⡈⠛⠀⠀⢰⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⣶⡀
⠀⠀⠀⠀⡁⡨⠋⠀⠐⠤⠀⠀⠀⠈⠀⠀⠀⢠⠄⢀⣴⢆⡴⣊⡉⣉⡀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠇⢀⠸⡣
⠀⠀⠀⠀⡜⡀⣠⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⣿⠼⠵⣿⢧⢿⡿⡼⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡜⡞⠈⣄⠒⠀⠀⠀⠀⢠⡖⢰⢲⣷⠞⣉⡁⣀⡉⣸⣢⡇⢀⣘⡀⡆⣚⣁⢆⣀⠀⣀⡀⠀⠀⠀⠀⢠
⠀⠀⢰⠋⠀⠀⠉⠀⠊⠂⠀⠀⠀⢀⡟⠞⠿⠾⠼⠿⠹⠷⠛⣦⠿⢿⣼⠿⠿⠝⠿⠟⠹⠾⣵⠇⠀⠀⠀⡰⠋
⠀⣆⠨⠀⢆⢀⣚⣐⠒⢤⠀⢀⢐⠮⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⣰⠃
⠀⢹⢰⠀⣴⠥⠒⠒⠂⠊⠂⡀⠡⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠊⠁
⠀⢸⡄⢈⡵⣊⠠⠀⠀⠀⠀⣉⢄⠤⠀⠄⠀⠀⠀⢠
⠀⠈⠾⢯⢚⡥⠀⠀⢀⡠⠚⠁⣠⣞⠛⠉⠻⢿⣕⠓⠢⢤⣁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢀
⠀⠀⢸⢣⠊⠀⠀⠠⠂⡴⠯⡢⡀⠀⠑⠄⠀⠀⠈⠳⣄⠀⠈⠳⣈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡘
⠀⠀⢈⠆⡐⢀⠌⠠⠊⠰⠒⠆⠪⠲⠒⠀⠑⠀⠀⠈⠳⠡⠀⠀⢉⣶⠇⠀⠀⠀⠀⠀⢆⠀⣀⠀⡇⢀⠀⢠
⠀⠀⢸⠀⠁⠘⠀⢁⠈⡒⠙⢂⠨⢎⠰⡄⠀⠁⠀⠀⠠⠱⢀⡀⠂⠀⡄⢄⠀⢀⡀⠀⠈⠡⢩⢀⠂⢈⠰⠋
⠀⠀⠈⠰⠀⡗⠾⢀⠀⠀⠀⠘⠒⠺⠐⠳⡞⠃⠒⢅⠈⠰⡮⢙⠂⠻⠋⡘⢕⣁⡖⠈⢠⡀⠒⠐⠂
⠀⠀⠀⠰⣇⡦⠽⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⢁⡀⢀⡴⠀⣀⡇⢸⠨⣤⡌⢻⣠⣶⡤⣤⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⣿⠭⡀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠁⠀⠀⢸⢇⣼⣻⣀⣒⠶⡌⣤⣈⣁⣠⢃⡰⠓⠦⠒⠁⢄⣄⣂⠤⠄
⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⡇⠀⠀⠀⢸⠆⠀⢰⢡⠏⢠⡏⠀⢹⡄⠘⡆⢋⠏⢛⡔⢿⣗⠤⠂⠔⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⣄⠀⢁⠀⠀⠀⣸⠀⠘⣩⠙⡆⠸⡇⠀⢸⠇⠀⡗⡁⢲⡰⠆⠶⢿⠿⠆⠚⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠺⣀⠴⡰⢮⡁⠈⠍⠀⠻⠆⠷⠀⠼⠁⠼⠃⠹⢄⢈⠡⡶⠊⠀⠀⠀⠈⠁

⠀⠀⠀⡴⠛⠛⠛⠛⠛⢳⣄⠀⠀⠀⠀⠀⣠⡤⠶⠶⢶⣤⣄⡀⠀⠀⠀⠀⢀⡤⠶⠖⠲⢿⡛⠒⢦⡀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢀⠜⠻⢶⣄⡀
⠀⢀⠞⠀⠀⠀⠀⠀⠀⢸⠈⢳⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠑⢍⢢⡀⠀⢠⠏⠀⠀⠀⠀⠀⠈⠣⡀⠙⡆⠀⡼⢉⡉⠹⡈⡇⣰⠋⠀⠀⠀⣱⠙
⣴⣋⡀⢀⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢳⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢹⠀⢳⣈⣁⡰⠟⡽⠁⠀⠀⢀⠔⢠⠞
⠉⠓⠬⢥⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⡇⠀⠈⠁⢀⡞⠀⠀⠀⢠⠂⣠⠋
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⡇⠸⡇⠀⠀⠀⠰⢢⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⣔⡆⠀⠀⠀⠀⡇⢠⡇⠀⠀⢠⠏⠀⠀⠀⡰⠁⡴⠃
⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⠀⠈⡇⠀⣇⠀⠀⠀⠈⠉⠀⠀⠀⠀⣸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⣼⠀⠀⣰⠇⠀⠀⢠⡞⢀⡞⣥⠞⠛⣷
⠀⣀⣀⣀⣿⠀⠀⠀⠀⠸⠀⠠⢧⣄⣹⡀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣰⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠏⠀⠸⠃⠀⠀⢠⠏⣰⠏⠀⣿⡘⠛⣸
⢻⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⢳⡀⠀⠀⠀⠀⠀⢀⠎⡴⠃⠀⠘⢦⡀⠀⠀⠀⠀⠀⣼⣡⠏⠀⠀⠹⣄⠀⢠⢋⡴⠉⠀⠀⠈⠙⠚⠋
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠀⠙⢦⣄⣀⣀⠴⠿⠋⠁⠀⠀⠀⠀⠙⠒⠶⠤⠖⠛⠉⠁⠀⠀⠀⠀⠈⠶⠓⠋

⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠚⠛⠉⠉⠉⠛⠓⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠚⠛⠉⠉⠉⠛⠓⠲⢤⣄
⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⢀⡞⠁⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠳⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣠⠏⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠒⠒⠒⠂⠒⠒⠒⠂⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠹⡄
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠉⠠⠁⢌⠤⠡⠄⠤⡠⡄⢤⡠⡀⢄⢀⠀⠀⠀⠀⠀⣷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠃⠀⢸⠀⠀⢸⢸⠀⠀⠀⠀⠀⢸
⣧⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠂⠐⠀⠀⠀⠀⠂⠐⠈⠀⠆⠰⠌⣀⠀⠀⠀⠀⣼
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣀⢐⣀⡂⣀⢀⣀⢀⣀⠀⠀⠀⠰⠤⠦⢲⠐⣀⡀⠀⢀⡏
⠀⢷⡀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠈⢳⡄⠀⠀⡄⠀⠀⠀⠀⠈⠂⢁⠈⢧⠈⠀⠀⠀⠀⠀⠀⢀⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠙⢦⣀⠳⢄⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠉⠓⢦⣄⡂⠀⠤⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁

⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⡠⣤⠀⠀⠀⠴⣤⡀
⠀⠀⠀⠈⠦⣤⠈⢁⡆⠐⢀⠀⠠⠷⠬⠠⠠⠄⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠰⠞⠀⠸⡷⣶⣡⣂⡼
⠀⠀⠀⠰⡶⢦⣷⣞⠁⠉⠁⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠑⢤
⠀⠀⠀⠈⡦⠂⢻⡟⠛⠋⠉⠁⠀⣀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⡀⠛⠀⡿⠑⢳⣃⡀⠬⢁⠀⠀⢆
⠀⠀⠀⠀⢉⡼⣿⡏⠡⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠒⠠⠤⠄⠀⠀⠀⠒⠀⠠⡤⠀⠐⠂⠉⠳⠤⠀⠲⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⢶⠾⠏⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣠⣁⣰⡶⡻⠖⠠⠆⢐⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠉⠙⠁⠀⠀⠀⠁⠈⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⡀⣏⡺⢦⣤⡤⠤⠤⠤⠄⠄⠀⢀⡀⠀⠀⠀⠾⠤⢤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣾⡆⠀⠀⢃
⠀⠀⠀⢀⡀⢠⡴⠟⢷⣄⠀⠉⠀⠀⠒⣀⠀⠠⡤⠔⠀⠀⠐⠆⢄⠰⡂⠀⡁⠐⣤⠴⡄⠲⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣄⠆⡮
⠀⠀⠀⠀⠉⠛⣇⢠⡈⠉⢻⣦⣀⠀⠀⣨⢓⠘⢠⠀⡈⠻⡳⣄⡏⠘⡉⠄⠸⠋⠀⢡⡠⡛⣀⠄⠒⠀⣼⣻⣿⣿⣿⣿⣿⠠⠁
⠀⠀⠀⠀⠀⠀⢉⡐⢁⣄⣀⣭⢿⣧⠉⡅⠀⠀⠈⠁⢠⠎⠁⠿⡿⠋⠀⠀⠀⣠⣐⠦⣽⣾⡫⠤⢃⠢⠛⢒⣎⣛⠛⠻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠀⠐⣈⢁⣤⠧⠇⠈⠀⢀⢋⠜⣃⣼⡷⠃⠒⣤⡾⠟⢁⡄⣸⡉⠁⢈⠺⣀⡀⠓⠈⡙⢮⡰⣊⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠧⠰⠀⠏⠀⠂⠁⡉⣽⢢⡴⢌⠃⠂⠉⠉⠑⣦⣐⣹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠀⠀⠀⠀⠁⠀⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⡯⣥⣀⠀⣠⣤⣤⠚⠿⠖⠒⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣤⡤⠴⢒⡿⠛⣧⠘⡏⠉⣛⣶⠛⠛⡹⢃⡤⣴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠢⢼⡉⠉⠉⠉⠉⠉⢁⣾⠊⣰⢗⠕⠊⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡬⢷⣄⣻⡀⠀⠀⠀⣠⡏⡧⡼⢳⢤⡔⢲⣪⠋⣔⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠋⣭⡴⠊⣠⠋⢓⡮⠥⠤⠬⠷⠿⣟⣋⠹⣟⢦⡉⠳⢄⢀⡼⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣼⠋⠀⣰⢃⡴⡋⠀⠀⠀⠀⠀⠀⢀⡜⡱⠓⠷⣮⣒⣐⣫⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⢰⠇⣠⠞⠁⠚⠋⠁⢠⡄⠀⠀⣤⠀⠘⠋⠁⠀⠀⠀⠁⡄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠃⠀⠀⣼⡾⠋⠀⠀⠀⠀⢀⣸⣧⣤⣤⣿⣄⣀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠰⠞⠉⠀⠀⠀⠀⢀⣴⠿⣿⠋⠉⢩⡿⠛⢿⡷⠀⠀⠀⣠⡟⠀⠀⠀⣀⠈⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⣿⠀⠀⢸⡇⠀⠀⠁⠀⠀⠀⠉⢀⣤⠴⠚⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣾⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⣿⠛⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠙⢦⡀⠀⠀⢠⣶⡀⢸⡇⠀⢰⡏⠀⢀⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠉⠀⠀⠈⠙⠿⣿⣶⣶⣾⣷⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠖⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣹⠇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠚⠉⡆⠐⢲⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣄⣀⣠⣤⣞⣉⣉⣀⣀⣀⣨⡵⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠉⠑⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡠⠔⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⡠⢊⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⢄
⠀⠀⠀⠀⢀⠌⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠳⡄
⠀⠀⠀⢠⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣩⠣⡄⠀⠀⠀⠐⠄⠀⠀⠠⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⣀⣀⣉⢘⡀⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⡈⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣛⠓⠒⠂⠂⠀⠉⠈⠀⠀⠀⠀⠀⠀⢩⡟⠀⠀⠀⠀⠀⠀⣀⠀⠠⠬⡇⠀⠀⠀⠀⠀⠘⡀
⡇⠀⠀⠀⠀⢹⡉⠀⢫⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠸⠟⠉⠉⠉⠛⢃⡉⢭⡽⠋⠉⠉⠹⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠠⠁⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠸⢢⠀⠀⠈⠴⠲⠆⠀⠀⠈⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠠⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⡇
⢡⠀⠀⠀⠀⢠⠀⠀⡇⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣰⡤⣼⠀⠀⣤⣤⠀⠀⣤⣄⢀⣀⣃⣼⣄⠀⢦⠀⠀⠀⠀⡋⠁⠀⠀⠈⠀⠇⠀⠀⠀⢀⠃
⠸⠀⠀⠀⠀⠤⠤⠤⠀⠀⠤⠤⠼⠤⠤⠤⠤⠤⣼⣏⣋⣉⣉⣉⣉⣁⣀⣀⣀⣈⣈⣉⣁⣀⣸⣦⡤⠬⣤⣤⣤⣄⣤⣤⡤⠼⠤⠤⠤⠼⠂⡘
⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠙⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠁
⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠃⠀⠀⠀⠆⠄
⠀⠀⠘⡄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠊
⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡐⠁⠂
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⡠⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⢊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠐⠂⠉
⡿⢿⡿⠿⠿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣿⣿⠿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠒⢒⠲⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠴⠶⠤⠤⣄⡀
⠀⠀⠀⢀⠴⠋⠁⠀⢀⣤⣤⣬⣀⠀⠀⠈⢙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡉⡆⠦⠡⠸⠈⠆⠀⠄⠉⡓⢦⡀
⠀⢀⡴⠁⠄⠀⢠⣶⣿⣿⣿⣿⠛⠻⠂⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢀⡴⡋⢖⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠈⠳⡧⢙⢦
⠀⡞⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⡀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⡞⡐⠍⠂⢀⣀⣀⣀⣀⣉⣀⣁⣀⣀⣀⠀⠈⢡⡄⢳⡀
⣸⠁⠀⠀⠀⠀⠘⣿⣿⣜⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣼⢠⡾⠁⠤⣠⣠⣤⣄⣄⣄⣤⣠⣠⣠⣤⢆⠄⠀⡮⠄⣧
⡏⠀⠀⠀⠀⠀⠀⠘⣿⠁⠹⣿⣤⣠⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠬⠄⠀⢠⠸⢈⠀⠄⡅⡀⠀⢈⠀⠀⠀⠁⠁⠀⠸⠒⢸
⢷⠀⠀⠂⠐⠀⠀⣼⣿⣿⣦⡼⢿⠋⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⠀⠀⠀⠀⢴⣞⢦⡤⠧⠷⠤⡴⠴⡤⢂⡂⠀⠀⠀⠀⣸
⠸⡄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢹⡀⠀⠂⠀⠓⠀⠊⠀⠀⠀⠀⠀⠀⠉⠀⠛⠂⠒⠂⢀⠏
⠀⠹⣄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠂⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠳⡀⢰⢋⡵⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢄⠀⢀⠞
⠀⠀⠈⠢⣘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠜⣿⠎⣖⠂⢠⠔⢰⣂⠹⡗⠷⢀⡴⠋
⠀⠀⠀⠀⠈⠓⠮⣍⣛⡛⠿⠿⢛⣛⣫⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣈⡓⠀⠘⠦⠀⢓⣂⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣀⣀⡀⢀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠒⠚⣆⢀⡏⠧⢴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠃⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡓⠶⠖⣒⣫⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣋⡦⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠶⠒⠒⠒⢾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⡂⠁⠘⠋⠀⠀⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⠐⠘⣧⠀⠀⢻⡻⣦⠀⠀⡶⢃⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⣠⣾⡾⣶⡄⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⢾⡠⠷⠛⠀⠸⡏⢳⡀⠀⠱⣿⠀⣀⠔⠁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡄⠀⠀⢿⣮⣇⣷⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠂⠱⡀⠀⠀⣀⠙⠲⢷⡄⢀⠸⡞⢁⣠⣔⡦⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⠀⠀⠀⠀⠙⡟⣿⣦⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⢆⠐⣔⠙⠉⠡⢶⣮⡟⠀⢠⡿⢿⡛⠉⣠⡴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠃⠀⠀⠀⣤⣀⡇⣿⣿⡇⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢂⠢⡈⣇⢖⣻⠽⠟⠓⠚⣏⡀⠀⢹⣻⣥⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⡾⠀⣷⠀⠀⠀⠀⠀⠀⢳⡙⠛⢅⠀⠠⠒⣊⡩⢹⣇⣀⠜⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠰⢄⠀⠀⠀⠈⠀⠁⠀⠀⡀⠀⠃⠀⠛⢆⠀⠀⠀⠀⠀⠀⠙⢆⣸⣶⣶⣯⠷⠚⠉⠉
⠀⠀⠀⠀⠀⠀⢰⣇⣀⣠⣄⣀⣠⣄⣀⣀⣀⣀⣀⣭⣴⠤⠴⠦⣄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⣀⠤⠒⠚⠉⠉⠍⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠩⠉⠉⠓⠒⠤⣀
⠀⠀⣠⠚⠁⠀⣀⣴⣾⣿⣶⣦⡄⠀⠐⠉⠢⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⣀⠀⢀⠀⡀⢀⣀⠐⠈⡓⣄
⠀⡜⠁⠉⠀⢸⣿⣿⣿⣿⡇⠀⠁⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣰⢃⡀⠀⠀⢾⠀⡇⢸⢧⡇⢸⠤⠀⠀⠀⡈⢣
⡼⠀⠀⠀⠀⠘⣿⣿⠻⣿⡅⠘⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢰⠃⢽⡇⠀⢀⣈⠉⣁⡈⣈⠁⡈⣉⣀⠀⠀⣿⠎
⡇⠀⠀⠀⠀⠀⠹⣿⠉⢿⣧⡀⣀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣼⢨⣾⠀⠀⢸⠀⠀⡗⠂⣟⢧⡇⢸⡁⠀⠀⢻⣄
⡇⠀⠀⠀⠀⠀⣰⣿⣶⣄⣻⡟⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⢻⢸⣿⠀⠀⣀⢉⢀⢉⢁⢁⢈⢁⢈⠀⠀⠀⣼⡟
⢹⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠂⢰⠃⠀⠀⠘⡆⢿⣧⠀⢉⢁⠈⡀⣀⢀⢀⢈⠀⡀⠀⣰⣿⢃
⠀⠳⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠘⣌⠿⣇⠀⠈⠀⠁⠉⠈⠈⠈⠀⠉⠀⡿⢋⠞
⠀⠀⠘⠦⣙⠿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠴⠃
⣀⣀⣀⣀⣈⣑⣲⣬⣭⣍⣩⣭⣵⣒⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣤⣤⣤⣤⣤⣤⣖⣋⣁⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⠉⠓⠋⠷⣄⡀⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡀⠀⠀⢹⡀⠀⡇⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡴⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⣹⡀⠄⠀⣷⠞⠁⠀⠀⠹⣤⣀⡀⠀⠀⢠⠶⠴⠋⠀⠀⠀⠀⠛⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡆⠀⠀⠀⠀⣹⠀⡖⠋⠀⠀⠀⠠⠀⠀⠸⡏⢻⣠⠟⠀⠀⠀⠀⢀⠄⠀⠀⢳⡄
⠀⠀⠀⢀⣼⡟⠲⠶⠞⠹⣄⣹⡶⢦⠀⠀⢈⡆⢿⠀⠀⠀⢠⣘⠠⠀⠀⠙⣶⠀⠀⠀⢀⠀⣼⠋⠉⠀⠀⣻⠆
⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠈⠁⠀⠸⡄⠀⢀⣱⡟⠀⠀⡄⡸⢁⠀⠀⠀⠀⢻⡄⢀⠀⢰⡼⠕⠂⠂⠀⠰⢧
⠀⠀⠀⠻⡄⠀⠀⠢⣠⡀⢀⠀⢠⠀⠙⣳⣿⠁⠀⠠⡀⢰⢋⣅⣀⡞⢧⣤⠟⡇⠘⣄⡜⢀⠤⠀⠀⠀⣠⣧
⠀⠀⠀⢰⣇⠀⠀⠈⠀⢈⠹⢆⣸⢀⠇⡟⢁⣄⢀⣄⠐⣶⠾⠃⠁⠀⠀⠀⠈⣇⡀⡟⢉⣄⣀⣠⡴⢋⠁⠘⠦⠤⣄
⠀⠀⠀⠀⠈⢹⣆⠀⠀⠀⠐⠒⣷⣎⡶⠛⠋⠈⠉⠙⢦⣽⠀⠀⠀⡴⠀⠀⠀⢈⡽⠟⠉⠀⠉⠀⠉⠙⠦⡤⣤⡄⠘⠓⣶
⠀⣠⠼⠛⢒⠶⢭⡙⢦⢶⣀⣄⢹⡆⠀⠀⠀⣤⠀⠀⠀⡏⠀⠀⢸⡀⠄⠀⢠⡟⠀⠀⠀⡐⠀⣀⠀⠀⠀⠀⣏⠀⠀⣨⡧
⣼⣃⠀⠀⠠⡭⠶⣽⡟⠀⠉⠈⠛⢛⡇⠀⠀⠀⠙⠣⣤⣇⠀⢸⠧⠀⢂⣀⣨⣧⣤⣤⡼⠶⣋⠁⠀⠀⢀⡾⠛⠚⠛⢷
⠉⢽⣄⣀⣀⠀⠀⣇⠀⠀⢀⣀⣀⠛⠒⢦⣀⣀⡼⠓⠚⠛⢦⣏⣠⣖⣋⠡⠄⠄⠀⠀⠯⣄⠀⠀⠀⢾⣉⣔⡀⠀⠀⠘⠧
⠀⠀⠀⡰⠋⠴⢾⡟⠀⠀⠀⢀⣈⣽⡖⢠⡏⠁⠂⠀⠁⣲⡞⠌⣯⠉⠓⢦⣤⠄⠀⠀⠀⠘⢷⣶⣾⣏⣉⣭⣄⠀⠀⢀⡞
⠀⠀⣼⡁⠀⣴⠚⠁⠀⠀⠀⡠⠛⡀⠀⢿⡀⠀⠀⠀⣼⠁⡀⠀⢸⡄⠀⠈⠈⠗⢆⠀⠀⠀⢰⡋⢿⡒⠄⠀⠈⠳⣾⠉⠁
⠀⠀⠉⡇⢀⡼⠀⠀⠉⢉⠽⠀⠀⠁⠀⢠⡇⠀⠠⡎⠘⠀⠀⠀⣸⣇⠀⠀⠀⠀⠈⠧⠀⠀⠈⢧⠀⠹⡗⠀⠀⠀⢷⡀
⠀⠀⠀⣧⠾⢧⠀⠀⠀⠋⠀⠀⠀⢀⣰⠋⠧⡄⠀⠀⠀⠀⣠⣼⠁⠈⢧⠤⣄⡀⠀⠀⠀⠀⢠⣯⠀⠀⠀⠓⠀⠀⢸⡁
⠀⠀⠀⠀⠀⢸⣀⣤⣤⡀⣀⡤⣤⡏⠁⠀⠀⣿⡴⢦⠞⠙⠃⠀⠀⠀⠀⠀⠀⠻⠤⠤⣤⣀⡏⠈⢧⣀⣄⡀⠀⠐⡏⠁
⣀⣀⣀⣀⣀⣈⣁⣀⣀⣙⣁⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣁⣀⣀⣀⣈⣙⣉⣙⣿⣀⣀
⣿⣿⣉⣩⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠁⢀⠀⠀⠀⡀⠈⠁⠀⠀⣠⠀⠀⠁
⣿⣿⣿⣿⣿⣯⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⣠⣾⣯⠀⠀⡀
⣿⣿⣿⣿⠀⠀⠀⠈⠻⣿⣿⣿⡇⠈⠙⢿⡇⠀⠉⠻⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⣠⣶⣿⡇⢀⣴⣿⡇⠀⠀⠄⣠⣾⣿⣿⣿
⣿⣿⣿⣿⣧⡀⠀⠀⠀⠈⠻⣿⣇⣰⣦⡀⢷⣤⣦⡄⢹⣿⣿⣿⣿⠀⠀⠀⠐⢰⣿⣿⣿⣧⣿⣿⣿⡷⠀⣤⣾⣿⣿⣿⣿⠟⠂⠀⠂
⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠈⠛⠛⠛⠿⠚⠛⠿⠿⠺⣿⣿⣿⣿⠀⠀⠀⠀⣽⣥⣽⣿⣿⣭⣽⣿⣷⣾⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⡤⠀⢤⡀⠀⠀⠈⣿⣿⣿⠀⠀⣠⣾⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⠟⠁⡀⠀⠁⢀⠀⠉
⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣦⡀⠀⠀⢿⡀⠿⢂⡿⠀⠀⡄⢻⣿⣿⠀⠀⢻⣿⣿⣿⣼⡘⠿⢀⣸⣿⣿⣿⠟⠁⠀⠀⢀⡀⣀
⣿⣿⣿⣿⠒⠛⠒⠛⠛⠛⣿⣿⣦⡀⠈⠛⠛⠁⠀⠀⢀⣿⣾⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣥⣤⣴⣶⣤⣿⣵⣿
⣿⣿⣿⣿⣧⣄⣀⣀⣀⣀⣀⡙⢿⣿⣦⠀⠀⢀⣀⣠⣾⢇⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡟⠷⢿⣿⡿⣿⣿⣿⡿⠿⠋⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣮⡙⠛⠋⢁⣼⣿⣿⣿⢯⡄⠀⠘⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣷⠀⣠⣾⣿⣿⡿⢯⣒⣐⠂⠀⠈⠻⣿⠃⢀⠀⠐⠀⠀⠀⠀⡀⠈⢀⢼⣀⡽⠀⠀⠁
⣿⣿⣿⠉⠻⡿⣟⠈⣿⣿⣿⣧⡀⣠⣿⣿⣿⣿⣾⣿⣿⣿⣟⣳⣟⠬⠡⠤⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣾⣦⣀⣤⣿⠀⠄
⣿⣿⣿⣷⣄⠷⣯⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⣿⣿⣿⡿⣧⡾⣉⢋⣈⠁⠀⠀⣾⡄⠠⠂⠀⠠⠀⠀⠀⠄⢠⣿⣿⡿⠟⠋
⣿⣿⣿⣿⣏⣟⣿⠀⠀⢙⠿⣿⣿⣿⣿⡿⠋⠀⠀⢻⣿⣿⣿⣯⣽⡒⠖⠀⠀⠀⢰⣿⣿⣦⣤⡀⠀⠀⠀⣤⣴⣿⣿⡿⠑
⣿⣿⣿⣿⣯⣭⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⣿⣿⣷⣞⣳⠭⡬⠉⠀⠀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈
⣿⣿⣿⣿⡷⣷⣿⣿⠙⢿⣶⣦⡀⢀⣤⣶⣿⣿⡇⠀⢸⣿⣿⡿⢯⣓⣐⠒⠀⣸⣿⣿⠀⠈⣿⠛⢿⡿⠛⣿⠁⢀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣷⢿⣿⣇⣠⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣟⠆⠱⠄⢀⣿⣿⣿⠀⠀⢿⡀⠀⠁⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣷⣾⡭⠉⠀⣼⣿⣿⣿⠀⠀⠙⢷⢶⣴⠾⠋⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣽⣒⠒⢠⣿⣿⣿⣿⠀⣀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣿⣿⣷⡀⠈⠁⠀⠀⠀⠀⠀⠁⢁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣻⠡⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈
⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣴⠆⠀⠀⢻⣷⣋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠐⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣀⠐⠒⠒⠛⠛⢛⣋⣾⠀⠀⠈⣯⢖⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠘⣼⣿⣿⣿⣷⠆⠀⠀⠐⠀⠀⢀⣴⡿⠛⠂⠀⠀⠂⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣶⣶⣶⠖⣏⠉⠉⠉⠁⠀⠀⠀⠀⢀⣴⡿⠋
⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠙⢿⣿⣿⣿⣿⣿⠏⠀⣿⣧⠁⢀⠀⠉⠀⢀⣼⣿⣯⣤⣤⣴⣤⣤⣤⣤⣥⣤⡀⠈⠀⡀
⣿⣿⣿⣿⠁⠀⠀⢀⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡟⠀⠀⣿⣿⡇⠀⠀⡄⠀⠉⠉⠉⠉⢉⠉⠉⠉⠉⠁⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣼⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠒⠛⠚⠒⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠤⣻⣿⣿⠀⠀⠂⠀⠀⠐⠀⠀⠐⠀⠀⠀⠂⠀⠀⠂⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⡚⣛⢯⣿⣿⠙⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⠻⡟⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠰⠖⣿⣿⡟⢀⠀⠁⢀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⣠⣆⣤⣤⠈⠀⡀
⣿⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠩⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠾⠿⠿⠿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡿⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡀⠀⣀⣀⣤⣤⣦⣤⣀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠾⠛⠛⢋⣉⣉⣙⣛⡛⠛⠋⠉⢉⣴⣟⡥⠶⠟⠛⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⠛⠛⠛⣫⣿⡿⠟⢿⡆⢀⣠⣿⣽⣶⣄⣀⣠⣤⣴⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠹⣿⣶⣶⣾⣿⠿⠛⢉⡼⠉⠉⠉⠉⢉⡿⠋
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠙⢿⣄⠀⠀⠀⢀⡾⠁⠀⠀⢀⡴⢋⣀⣤⠶⠶⠶⢶⣶⣦
⠀⠀⠀⠀⣼⢋⣿⣿⡿⠛⠛⠛⠋⠙⠉⣍⠉⠀⠀⠀⠀⠀⠀⣀⣤⣽⣛⡷⢶⣿⣧⣤⣤⣄⣠⣤⣤⢴⣿⣿⣿⣧⣴⣶⣶⡿⠿⠋⠁
⠀⠀⣠⣴⣿⣾⡟⠁⠀⠀⠀⠀⠀⠀⣶⣾⣷⡄⠀⢀⣠⣶⡿⠟⠁⠈⠙⠛⠛⠿⠿⠿⠿⠿⠿⠾⠿⠛⠻⢦⣄⡀
⠀⣸⡿⢋⠉⠉⠀⠀⢀⣀⡀⠀⠀⠀⣈⣀⠈⣷⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄
⠀⢹⣷⣿⣷⡄⠀⠠⣿⠟⠁⠀⠀⠀⠃⣡⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠙⢿⣿⡿⠀⠀⠈⠀⢙⡥⠂⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⡀
⠀⠀⠀⣼⢋⣽⣇⠀⠀⠀⠛⢀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢆⣿⡀
⠀⠀⠀⠹⣦⡀⠋⢀⡄⠀⣰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣩⣼⣇
⠀⠀⠀⠀⠈⠛⠛⠙⢷⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⣈⣷⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢡⡶⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⡿⠟⠛⣿⠉⠉⠛⠿⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⢿⡏
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢻⡀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⢰⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣷⣤⣴⣶⣿⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣴⢟⣵⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣾⠁⠘⠿⠿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⡏⠀⠀⠘⣧⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⣿⠀⠐⠿⣾⣙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠈⣷⠀⠀⠀⢸⣧⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⣄⠀⢒⣶⣦⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣾⣷⣶⠿⠿⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠙⠷⣮⣭⣿⠃⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣻⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣙⡟⢿⠿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⡟⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣻⣿⣷⣿⣾⢷⡿⣾⣿⠋⠉⠙⠛⠻⣿⣿⣛⣻⣟⣻⣿⢿⣽⢿⣿⣯⣿⣄⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⣶⣛⡼⢏⣽⣩⡟⣿⣿⠏⠀⠀⠀⠀⠀⠈⠻⣿⣷⣾⡻⣾⠿⣽⣷⣟⣿⠷⣿⣦⡾⠋⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⡟⣷⣟⣱⢿⣼⣯⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣧⡿⣿⣷⣿⣧⣴⣿⣟⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣧⣟⣳⣾⣹⣿⡷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⢿⣿⣿⣿⡿⠃⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠻⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣿⣿⡿⢦⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣾⣷⠄⠀⠀⠀⠰⣷⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣦⣌⠻⣿⣄⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⣿⣧⠄⠀⠈⠿⠧
⠀⠀⠀⠀⠀⢀⠀⡆⢀⢸⠀⡇⠀⠀⠀⣿⣿⡍⠠⠤⢤⠍⠉
⠀⢐⠂⡀⢰⢸⠀⡇⢀⢸⠀⡇⠀⠀⠀⣿⣿⠷⣀⣲⣊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠄⠤⠤
⠀⠀⠀⠀⢀⠀⠀⣶⢸⡇⠀⣶⠀⠀⠀⢀⣤⣤⣤⣶⡄⠀⣀⣀⣀⣀⠀⠀⢸⡏⠁⠀⠀⠀⠩⠭⠥⠤
⠀⠀⠀⣀⢸⡇⡀⣿⢸⡇⡇⣿⠀⠀⠀⠿⣿⡿⠟⠁⠻⡾⠋⠙⢿⣿⠀⠀⢸⣿⣿⣆⠀⠀⠈⠤⢤⠄
⠀⢀⡀⣿⢸⡇⡇⣿⢸⡇⡇⣿⠀⠀⠀⣶⣄⣴⣶⣿⣶⣴⣶⣶⣶⣤⠀⠀⢸⣿⡿⠉⠀⠀⠀⠲⠚⣆⡀
⠀⠀⠁⠉⠈⠀⠁⠉⠈⠀⠁⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⠭⠭⠭⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠀⡿⢸⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣭⣭⣭⣤⣤⣤⡤⡄⡀⠀⠀⠀⢸⡇⡇⢻⢠⠘⡇⠁⠘⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣉⣉⡉⠁⠀⠀⠸⠁⠃⣀⢨⡀⡀⣦⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠁⠉⠈⠉⠁⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⡇⡀⢸⠀⠀⠀⢠⣄⡀⠀⢀⣴⣶⣦⣄⢀⣄
⠀⠀⡀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡇⡄⣶⢸⠀⡇⡇⢸⠀⠀⠀⠘⠻⣿⣶⡿⢿⣿⣿⣿⣿⡿
⠀⠐⠒⠒⠒⡐⠒⠒⠐⠒⠂⠀⠀⠀⠀⢸⢸⡂⡇⣿⢸⠀⡇⡇⢸⠀⠀⠀⢠⣄⡀⢉⣤⣤⣉⣤⣄⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠉⠈⠀⠁⠁⠈⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠁⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠆⠁⠤⠐⠇⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡈⠀⠀⠀⢠⠅⠀⠠⠁⠀⠈⠄⠀⠀⠀⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⠀⠄⠀⠀⠀⠸⠀⠀⡃⠀⢀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠐⠂⠀⠀⠀⢸⣅⠠⠖⠀⠐⠂⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⡀⠀⠣⠀⠐⠂⢀⡤⠐⠂⠒⠁⠚⠒⠂⠤⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⠀⢱⠀⠚⠀⠄⠃⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⣸⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠜⡄⠀⠁⢀⠃⠀⠀⠀⠀⢪⠜⣁⠀⡠⢲⠏⠘⠨⡇⠘⠣⣸⣆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⠲⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠈⠁⠁⢠⣿⢯⡵⢚⠏
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣝⣦⣄⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⢴⡿⠵⠪⠚⠻⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣤⠄⣤⣤⣤⡤⠀⠘⠻⠀⠀⠀⠀⠀⢠⢠⠐⠖⠁⠀⠁⠀⡄⠦⠀⢠⡀⠀⠈
⠀⠀⠀⠀⠀⢀⡞⠛⣉⡴⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡸⠘⡇⠀⠤⠀⡀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠓⠞⡄⠐⣖⠀⡃⠰⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠰⠄⠀⠀⠈⠘⠀⠀⠀⠐⠢⢉⡐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠠⠄⠔⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⢰⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⢸
⠀⠀⠀⠀⢰⠀⠀⠀⢰⣱⠀⢀⠤⠒⠈⠹⠉⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⢸⣼⢠⠃⠀⠀⠀⢸⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⢸⢺⣧⠀⠀⠀⢀⣤⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⢡⠀⣀⠔⢩⠃⠀⠀⠀⠀⡌⠀⠬⠱⣄⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⣄⠀⠆⠀⠀⠀⣀⠜⠀⠀⣀⣀⡇⣱⣔⢨⠀⠢⠑⢄⡀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠉⠀⠀⠀⠀⠤⠤⣯⠤⠤⣿⠒⠒⢠⠤⠡⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣟⣂⣐⣻⡀⢨⢸⠭⢹⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠟⠤⠤⠻⠤⠤⠘⠤⠼⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⣂⠈⠉⠉⠉⠉⠉⠉⠀⣐⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢈⠠⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣦⠄⡉⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠂⡀⠐⠈⠀⠀⢀⠄⠂⠁⠀⠘⠋⠀⠀⠀⠀⠀⠀⠐⠒⠀⢶⡥⣀⠞⠑⠂⠀⠁⢤⡀
⠀⠀⠀⠀⢀⡴⠃⡠⠺⠻⣀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠘⠐⣁⣼⠷⢢⡈⠢⡀
⠀⠀⠀⡠⠊⣠⡚⠿⣲⠄⠉⠀⣰⠄⠒⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⠣⣴⠎⠈⣀⠘⢄
⠀⠀⡔⠀⠜⠦⣍⡶⠀⠀⠀⠰⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⢠⠔⠳⠑⠀⠣
⠀⠀⠀⠾⠔⣦⡆⠀⠀⠀⠠⠃⠀⠀⡠⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢠⠴⠻⠔⡀⠁
⢀⠁⣼⡶⢠⡌⠀⠀⠀⠀⠐⡄⠀⠀⠐⢄⠀⠀⡘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢠⠀⢡
⠈⢀⠁⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠁⠈⡀⠌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⡄⠄
⡇⠘⣶⢶⣴⠀⠀⠀⡀⢃⣀⣀⠀⠀⠀⠐⠀⠁⠀⠀⢀⠱⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⣁⢋⡁
⡇⠘⣤⣠⣄⠀⠀⠀⠉⠀⠒⠈⠀⠀⠀⢀⠀⢀⠐⠈⠀⠀⠀⠀⠁⠉⢉⡈⠹⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠣⠤⠂
⣷⠀⡜⠉⠁⠀⠀⠀⠀⠀⠞⠀⠀⠀⠰⡁⠔⠁⠀⠀⠀⠀⢀⠠⠐⠂⠁⠄⠐⠈⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡉⢐⢀
⣿⡆⠰⠀⡤⢤⡀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⣀⡀⠔⠚⢉⣡⠰⠐⠊⠁⠀⠀⠀⠀⠱⡀⠀⠀⠃⠀⠀⠀⠠⠄⢭⠁⠌⢰
⣿⣿⡀⠹⠓⣨⢥⡀⠀⠀⠈⠉⠁⠀⠀⠀⠐⠂⣀⡶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠑⡀⠀⠀⢈⠒⠊⠌⢠⣿
⣿⣿⣯⢄⠘⡧⠖⢡⣄⠀⠀⠀⠀⠀⠀⠀⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⢄⠀⠑⢄⠀⠈⢀⠎⢠⣿⣿
⣿⣿⣿⣧⡢⡀⠺⡦⠚⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠁⠀⠀⠀⠈⡄⠈⢑⠖⢁⣴⣿⣿⣿
⣿⣿⣿⡟⡥⡪⡠⡈⠂⢤⡜⡧⠒⣠⠀⢈⠀⠀⠀⠀⠀⠀⠀⢀⣠⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢁⡴⢊⠝⢹⣿⣿
⣿⣿⣿⣯⡺⡭⠂⠀⠐⠄⠈⠃⠶⢁⡠⢺⠀⠀⠀⢀⠠⠐⠈⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠐⠑⠀⠠⣕⣽⣿⣿
⣿⣿⣿⣿⠯⠂⠠⠄⠔⠠⠂⠅⡂⠄⣉⠙⠂⠠⠖⠁⣀⡀⣰⠁⢀⣀⠀⠠⠤⠐⠂⢁⠠⢐⠨⠀⠄⠠⠢⡲⡩⣿⣻⣿⣿
⢻⣿⣿⣟⢇⠁⠀⢅⠀⡀⢐⠠⢀⠠⠀⡀⠨⠄⢒⣒⠂⠀⠀⠀⠀⠀⣒⡂⠠⠄⢀⠀⠀⡀⠀⡀⢀⠀⢠⡋⠪⡸⣿⣿⣿
⠘⣿⣿⡿⣷⡣⡀⠀⡈⢀⠀⡈⠀⡀⠀⢀⡈⠀⠄⠂⢀⣀⣀⣀⣀⠀⠐⠠⠀⢁⠀⠀⠁⠀⢁⠈⡀⢁⠄⢈⠮⢼⣿⣿⣿
⠀⠹⣿⣿⡯⡋⠪⠈⠂⠐⠁⠂⠁⠂⠉⠐⠀⠨⠁⠀⠐⠒⠒⡒⠒⠂⠀⠨⠁⠐⠂⠁⠐⠈⠐⠀⠊⠀⠁⠅⠙⣕⣽⣿⠃
⠀⠀⠙⣦⠃⡢⠈⠢⢐⠈⡈⠐⠨⠐⠀⠅⢐⠂⠀⠽⠁⠀⠀⠀⠀⠨⠭⠄⢐⠀⠥⠐⠀⠀⠂⠐⠀⠠⠠⠃⢐⣿⡼⠃
⠀⠀⠀⠈⢿⣔⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⢐⠈⠀⠀⠀⠀⠀⠀⠀⣂⠀⠤⠀⠀⠄⠀⠄⠀⠀⠀⠠⠀⢨⡞⠁
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⢐⠀⠀⣤⡭⠌⠡⣬⣅⠀⠐⠂⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠠⠀⠀⠀⡀⠠⠁⢀⣂⠒⠂⠀⠀⠀⢒⣒⠀⠩⠀⠀⠀⠀⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢀⠀⠀⡈⠀⠄⠃⠀⣀⣒⣒⣀⠀⠀⠠⠀⢁⠀⠀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣐⣂⣈⣉⣀⣐⣒⣒⣒⣒⣀⣀⣉⣁⣐⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡛⠄⣁⠔⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡀⠀⣨⠞⠁⠀⠀⠀⢇
⠀⠀⠀⠀⠀⣀⣀⣀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⣤⠛⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠯⢄⣤⠖⠒⠋⢳⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡽⡉⢱⠀⠀⠀⣠⠤⣈
⠀⠀⠀⠀⢀⠸⣀⠨⡇⠀⠀⠈⠷⠶⠞⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⠂⡌⠀⠀⠀⡇
⠀⠀⣠⡚⠁⠁⣠⠼⠃⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠉⠈⠊⠑⣵⡇⠀⠀⠀⣟⡁⢆
⠀⡔⠑⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⣠⡤⠤⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢄⠀⣡⠶⠚⠉⠉⠀⠀⠀⠀⠉⠉⠓
⢰⡂⠀⣰⠏⠀⠀⠀⢀⡄⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠠⢀⡴⠋⠁
⡆⠀⢀⡏⠀⠀⠀⣴⢋⢹⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠄⢠⡞⠁⠀⠀⢀⣠⢴⡆⠀⠀⠀⡖⠒⠦⣴
⡇⠐⢸⠁⠀⠀⢰⠟⡐⠸⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀⢢⡟⠀⠀⠀⣰⠿⡡⣸⠁⠀⠀⢰⠇
⡧⠀⣹⠀⠀⠀⢸⠀⡅⠀⣧⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢈⡾⠀⠀⠀⣸⠃⠆⣈⣿⠀⠀⠀⣼
⢓⠀⢼⡀⠀⠀⢸⡄⢃⠀⢽⠀⠀⠀⢸⣄⡠⠲⠤⡀⠀⠀⠀⢰⠁⢨⡇⠀⠀⠀⡏⢸⠁⢠⡇⠀⠀⠀⡇
⠘⣄⠘⣇⠀⠀⠀⠳⣼⠐⢺⡇⠀⠀⠀⣇⠀⣠⡴⠛⢦⠀⠀⠈⡑⢼⡇⠀⠀⠀⡇⡼⡀⢺⠀⠀⠀⢸⠇
⠀⠐⣄⠙⣆⠀⠀⠀⠈⠳⠬⡇⠀⠀⠀⠹⠚⠁⠀⠀⠈⢧⠀⠀⢏⠸⣇⠀⠀⠀⠹⡇⢈⡏⠀⠀⠀⣼⠤⠔⢦⡀
⠀⠀⠈⠢⣝⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠈⢧⠹⡄⠀⠀⠀⠈⠺⠁⠀⠀⠀⡗⣀⣢⡴⠾⡄
⠀⠀⠀⠀⠈⠑⠻⠷⠦⣤⣀⣀⡀⠀⠀⠀⢴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠑⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡀⣷⠀⠀⠀⢸⡔⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⢤⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠼
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠖⠛⠢⣽⠀⠀⠀⠘⠧⠴⠤⢽⠂⠀⠀⠀⠀⠀⠀⠀⠸⣦⣸⠃⠀⠀⠀⡞⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢵⠚⠋⠁⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⢤⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⢀⠞⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠐⠘⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠌⠰⠳

⠀⠀⠀⠀⠀⣀⠀⠀⢀⣼⣤⠀⠀⣼⣧⠀⠀⢀⣰⡀
⠀⠀⠀⠀⣴⣿⣆⠀⠊⣿⡏⠃⠘⢻⡏⠃⠀⠜⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⢠⣾⣄
⠀⠀⠀⠀⢩⣿⠈⠀⠀⠛⠃⠀⠀⠘⠃⠀⠀⠀⠿⠇⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠁⠀⠋⣿⡏
⠀⠀⠀⠀⠀⠉⠀⠀⢀⣶⡀⠀⠀⣰⡀⠀⠀⣠⡀⠀⠘⢻⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⣤⠀⠛⠃
⠀⠀⠀⠀⣼⣧⠀⠀⢟⣿⠳⠀⠼⣿⡿⠄⡼⣿⡿⡄⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⢧
⠀⠀⠀⠘⢹⡏⠃⠀⠘⠿⠀⠀⠀⠿⠇⠀⠀⣿⡇⠀⠀⣀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⠃⠀⠀⣰⣆⠀⠀⠀⢠⠀⠀⠀⢠⠄⠀⠰⢻⣿⠣
⠀⠀⠀⠀⣰⡀⠀⠰⢿⡟⠆⠀⣰⣿⣷⡀⢰⣿⡿⡄⠀⠸⠿
⠀⠀⠀⠸⣿⡟⠆⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⢠⣆
⠀⠀⠀⠀⢿⡇⠀⠀⣀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠀⠠⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣦⠀⠀⢠⣿⣄⠀⢠⣾⣦⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠁⣿⡏⠁⠀⠋⣿⡝⠀⠈⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠛⠃⠀⠀⠙⠃

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣄⡀
⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣄
⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣀
⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠈⠙⠿⣿⣿⣦⣄⡀
⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣦⣄
⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣀⠀⣰⣆
⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⣿⠆
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣼⡇⠈⠁⠈⠁⠈⢹⡇
⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢼⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⢼⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⢼⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢼⣇⣀⣀⣀⣀⣀⣸⣟⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣿⣃⣀⣀⣼⣇⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠈⢳
⠀⠀⠀⠀⢀⣠⡀⡅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠞⠉⠁
⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤
⠀⠀⠀⠀⠀⠐⠊⢃⠀⠀⣤⠙⢂⡦⣆⡈⠀⢀⠁⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⢴⠁⠇⠈⠁⠄⠂
⠀⠀⠀⣆⠀⠀⢀⡀⢤⠆⠘⣗⡣⠜⢡⡜⡠⢬⠀⣡⠀⠀⣭⠁⠂⠈⢀⠄⠀⠀⠀⠉⠊⡈⠙⠠⠊⢉⡐
⣠⠖⡰⠞⠛⠓⠚⢅⠈⠀⠀⠈⠳⢎⡡⠄⢁⠶⠀⠹⠢⡐⠀⢈⠖⠉⠀⠈⠐⠀⠀⠀⠀⠈⠄⠀⠘⡁⠀⠑⡀
⠁⠀⠡⠀⠀⠀⠀⣢⠀⠐⠦⡀⠄⠈⠳⢖⣁⠾⡄⠀⠔⠁⠀⠃⠀⠀⠀⠀⠀⠀⠑⠀⠀⠄⠚⠒⠂⠌⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢤⢀⣀⣀⠀⠈⠠⡀⠒⠶⡿⡀⠉⡎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠉⢈⠃⠰⠂⠢⡰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠉⢂
⠀⠀⠀⠀⠀⠔⠁⡏⠀⠀⠈⢶⡠⡄⠦⠈⠌⢦⡙⠇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⡀⠀⠀⠰⠤⡀⢠⠀⠁⠑⢄⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠄⣳⢐⡀⠀⠀⠀⠡⡀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⢢⣀⠀⠈⡊⠀⠀⠀⠈⠁⠑⢤⠀⡀⠤⠞⠁⢄⠐⠀⠈
⠀⠀⠀⠀⠢⠀⠠⢠⡶⠤⢀⡔⠄⠀⡀⠀⠠⢂⠔⣆⠑⠤⠀⠀⠀⠘⠢⡀⢀⠔⠀⠀⠀⠀⠀⠀⡈⠀⠑⡤⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠢⠀⠑⠀⢡
⠀⠀⠀⠀⠀⣸⢢⠀⠉⠑⢐⠁⠀⡈⢱⡄⠀⠈⠻⠀⠀⠁⠀⠘⠂⠀⡀⠋⠀⠀⠀⠀⠀⠀⢀⠈⠐⡀⢀⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠘⠈⠃⠀⠀⠢⣁⠀⢁⢀⡄⠐⢀⠀⠀⠀⣆⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⣐⠁⠢⡀⣈⠃⠀⠠⠀⠀⠀⠄⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠡⡀⠀⡠⠐⢣⣈⠾⠌⠑⡸⠤⣍⣘⠁⡐⠉⠐⠓⠊⠣⣄⣀⣀⣠⠔⠆⠘⢄⠀⠈⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠠⠁⠀⣐⠠⠊
⠀⠀⠀⠀⠀⠀⠀⡸⠡⡌⠈⡣⢈⠦⡚⠎⠀⠀⠘⠃⠀⠈⠂⠀⠀⠀⠀⡤⠀⠎⢀⠀⠀⢀⠠⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠠⠊⠀⡠⠁⡀
⠀⠀⠀⠀⠀⠤⠅⠓⡤⢍⠈⡤⡔⠢⠞⠂⠀⠀⠀⠀⠀⠀⢠⠤⠠⡡⢾⣥⡀⡀⠀⡐⠐⠀⠀⠀⠔⠁⠀⡠⠀⠀⠀⡀⠄⠀⠠⠊⠁⠁
⠀⠀⠀⠀⠁⠠⢃⠤⠢⠾⠙⠀⠒⠐⠁⠑⠤⢀⡀⣀⡤⠚⠁⠀⠀⡸⢈⠳⠀⠘⠃⠀⠀⠀⠊⠀⠀⠐⠉⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⡀⠀⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣶⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠀⠠⠤⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠉⠁⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠌⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠡⠂⠁⠐⠀⠁⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠔⠁⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⡠⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⣄
⠀⠀⠀⢀⠔⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢒⡡⠈⡓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⡀⠤⣐⣊⠥⠒⠉⠀⠀⠈⠓⠮⣝⣒⠤⢄⣀⣀⣀⠀⠀⣀⠀⠀⢀⠀⠈⠆⠱⡀
⠀⠀⠎⠀⠀⠁⠄⠀⠀⠀⠀⡿⠦⠤⠤⠶⠒⠒⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠿⠍⠉⠈⠀⠀⠀⠁⡀⡀⠢⠱⡀
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠂⠀⠀⠀⠇⠀⠂⠀⠀⠆⠀⠀⠨⡀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢳
⢰⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⣇⠉⠀⠘⠁⠘⠀⠀⠀⠂⠃⠀⠐⠀⠈⠂⠐⠃⠀⠀⠀⠂⠂⠰⠀⠸⠡⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡆
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⣤⠤⢤⡤⣄⣀⣀⣀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠀⠄⢱
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⡏⠉⡋⠉⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠉⢻⠙⡶⠁⠀⠀⠀⠀⠀⠐⠄⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⢂⡐⠀⠀⠀⢸
⢡⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠐⡄⠀⢻⡇⠀⠀⡘
⠸⡀⠀⠀⠀⠀⣰⠟⠉⢀⣀⡀⠀⠀⠀⠀⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⢸⠁⠀⢀⠃
⠀⢣⠀⠀⠀⠀⣿⠀⡀⠀⠀⢻⢀⠄⠹⣆⠇⠀⠀⢀⣀⠀⠀⠀⠀⠐⠀⠂⠠⠀⠒⠒⠀⠀⢀⠀⠀⠀⠀⠒⠀⡎⠁⠀⡟⠀⠀⡘
⠀⠈⢇⠀⠀⠀⢹⠀⠙⠦⠤⠵⠃⠻⠆⠘⠶⠀⠡⢤⣤⡄⠀⠀⢧⡀⠀⠀⠀⡄⠚⠛⠀⠀⠈⢳⡀⠀⠀⠀⣀⡇⠀⣼⠁⠀⡰
⠀⠀⠈⢆⠀⠀⠀⢧⢠⡖⠂⠐⠆⠤⠤⠤⠤⠤⠤⣀⣀⣀⣀⡀⠀⠉⠉⠉⠈⠓⠒⠒⠀⠐⠂⠀⠛⠂⠀⠀⠉⣁⠼⠃⠀⡰⠁
⠀⠀⠀⠈⢣⡀⠀⠈⢣⡁⠀⢀⠄⠀⠀⠀⠀⢳⡀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠓⠒⠲⢶⡶⠒⠒⠒⠒⠈⠉⠉⠉⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠙⢄⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡄⠀⠀⠀⠂⢀⣆⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⢰⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠐⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⢤⡔⠚⠳⠤⡤⡴⣛⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡉⠛⠳⣿⡥⣴⣌⣾⣧⠽⠯⢭⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡹⣧⠈⢸⠋⠩⣟⣀⡠⠴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣌⡻⠤⡿⣇⣸⡿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡻⢶⠗⢞⣹⣭⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⢷⡿⠻⠻⠙⢟⠳⡈⠉⢓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠠⠋⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈⠂⠈⡙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠔⠈⠀⠀⣀⣀⣀⣿⢝⡆⠀⠀⠀⠀⠀⠀⠀⠐⠡⠂⢈⠦⣄
⠀⠀⠀⠀⠀⡠⠊⠀⠜⠀⠀⠀⣠⠍⢀⣰⣒⠊⢕⡝⠶⣄⠀⠀⠀⠀⠀⠀⠀⢑⡡⢊⢌⢦⡀
⠀⠀⠀⡠⠊⢀⠠⠀⠀⠀⠀⠀⣇⣴⡏⠀⡿⣦⡟⠒⢶⡿⠃⠀⠀⠀⠀⠀⠀⠀⢐⠅⡠⢊⢗⡄
⠀⢀⠎⢠⠆⠂⠀⠀⠀⠀⠀⠀⠻⣯⣿⢦⢟⣕⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠔⡡⢣⡻⡄
⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢯⢙⣿⡷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠔⡔⡑⣷
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣾⢛⡟⠺⣕⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⢊⠌⢾⣾
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡯⠻⣆⠀⣷⣾⡇⠀⣹⣫⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⢔⢕⢕⢕⢼
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣼⢻⣿⠿⣳⢞⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⢕⠕⡡⡣⡠⣫
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣧⣾⠗⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡨⢔⠵⡪⡪⢊⢔⠇
⠀⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢊⠔⡡⡪⢊⢔⢵⡿⠤⠤⠤
⠀⠀⠙⢄⠃⠄⠀⠀⠀⠀⠀⠀⠀⢀⢆⠔⠀⡀⢀⠄⢀⢀⠄⡠⡡⡣⠢⡲⢡⢊⢜⠔⡡⡡⠾⠥⠤⠭⠤⠼⠯⠗⠒
⠀⠀⠀⠀⢱⢄⣉⠐⠐⠀⠀⠂⠃⠃⠃⠊⢘⣀⣁⣈⣀⣁⠊⠊⢔⢔⡕⡔⡡⣡⣢⡞⢋⣉⠁⡈⠉⠉⠉⢉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⢛⣒⡶⠆⠨⠻⠿⠿⣿⣿⢿⣶⠶⠖⠒⢤⠤⣥⣯⣾⠾⠛⣉⣐⠒⠀⠔⠒⠒⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠚⢋⠁⠀⠐⣦⣀⡐⢂⣀⣀⣀⠛⠋⠙⠉⠛⠋⠀⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢖⣚⠭⠭⠭⠭⠭⠭⢝⣒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣫⠔⢫⡱⡦⠀⠐⠚⠛⠀⠀⠖⣠⠉⠲⢍⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⢊⡁⠓⠊⢁⠴⠊⠉⠉⢙⣉⡉⠢⡀⠘⠂⡶⢙⢮⡳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡜⠑⠋⠁⠀⢰⠁⠀⠠⠂⠀⣸⠄⠈⠹⠁⠀⠀⠀⠁⠀⠱⡝⡄
⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⠀⠀⠀⠀⢸⡦⠐⡴⣲⡄⢸⠀⠰⠂⡇⠀⠀⠀⠀⠀⠀⠹⡸⡄
⠀⠀⠀⠀⠀⠀⢸⢱⠃⠀⠀⠀⠀⠀⠈⢂⠒⢰⣥⠂⠘⡔⠄⢠⣸⠄⠀⠀⠀⠀⠀⠀⣇⢧
⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠈⢲⣎⠀⠳⣄⡈⠳⠼⡁⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⢸⢸⠆⠴⡶⠖⠆⠆⠖⢠⡃⠭⡻⢶⠬⣿⡶⠒⠁⠀⠀⠀⠀⠀⠀⠀⡸⣸
⠀⠀⠀⠀⠀⠀⠘⡎⡆⠀⠀⠀⠀⠀⡠⠗⠉⠒⢕⢄⠱⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠱⡹⡄⠀⠀⠀⡼⠒⠂⠠⢍⠠⡗⢢⡈⢎⢻⣦⠟⠫⠻⠇⠇⢀⡞⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡝⢄⠀⢰⠁⣀⡐⢄⢢⢳⠱⡈⢷⠈⢖⢷⡐⣀⠀⠀⣀⢞⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡓⢜⡦⣉⠉⠒⠲⢄⣏⠷⡜⠀⠘⡄⣱⠼⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢭⣒⠭⣖⣲⣤⠼⠤⢼⣖⣒⠭⢗⣫⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣮⣭⣭⣥⣶⣶⣿⣿⣿⣿⣿⣿⣷⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⠢⠀⠀⠀⠀⠂⠀⠧⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠁⢀⣠⠴⠲⠐⠒⠒⠂⠀⠀⠈⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⣠⡶⠛⠱⢀⠀⠀⠠⡸⠀⠀⣀⢠⠀⠀⠀⠀⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⣾⣿⠁⡀⡰⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⡰⠀⠠⠆⠤⡁⡐⠀⠀⠠⠤⡶⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡄⠀⠀⠀⠀⡥⢀⢀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠀⠀⠀⠤⠀⠂⠀⠀⠈⠂⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠂⡀⡀⣠⠄
⠀⠀⡀⢰⢀⠀⠀⠀⠁⠈⡀⠐⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠉⠃⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠠⢀⠀⠀⠀⠈⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠁⠢⠀⠀⠀⠈⠁⠠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠀⠀⠢⢍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢀⡀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠀⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⣀⠀⠀⠀⠈⠀⠀⠳⠀⢀⠀⠀⠀⠀⢀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠦⣤⣀⣀⣰⣆⡀⠱⠄⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠔
⠀⠀⠀⠀⠀⠀⠀⣢⣤⡼⠿⠂⠶⣀⠀⠀⠀⢀⠀⠂⠐⠛⠛⠁⢦⡀⢀⢀⡀
⠀⠀⠀⠀⠀⠐⠉⠁⠈⠁⠀⠀⠀⠌⠠⠀⠐⢡⠐⠆⠀⡀⠀⠄⠈⣳⣼⣟
⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠠⠀⢁⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⡉⠉⢹⠏⠙⠓
⠀⠀⠀⠀⠀⠀⠀⢠⠙⠂⠠⠄⢈⠀⠁⠀⠀⠀⠀⢀⡄⡤⢀⠅⠂⠘⠀⠃⠀⡆⢠⣆⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠐⠓⠠⠀⠀⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠈⣹⡷⠓⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⣼⡄⢘⡋⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢆⡆⡤⢀⠄⡤⠦⣼⣿⢿⡃⠘⠃⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡞⢃⡻⡶⣁⡘⠇⠀⠀⠉⠀⢂⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠋⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⣄⠀⠀⠀⠀⠈⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠘⣯⠄⡄⠤⢀⣀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠘⠘⡀⠀⠰⠭⠀⠛⠏⠑⠛⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠋⠙⠻⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⡌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠛⠛⠛⠳⠶⢦⣤⡐⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣿
⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠹⡇
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣀⣠⣤⠶⠶⠶⠶⠶⠶⠶⢦⣭⣀⢹⡄
⠀⠀⠀⠀⢰⠏⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣆
⠀⠀⠀⢀⡿⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠹⣦
⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣴⣞⣿⣉⣹⡇⣠⡴⣮⡉⠛⣷⣤⠼⢷⣄⠀⠀⣠⡤⠶⢺⡿⠀⠀⢀⣠⠄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣾⡋⢼⡋⢹⣿⣿⠶⣿⣄⣼⠇⣼⠋⠀⠀⠀⢙⣷⠟⠁⠀⢠⣾⣁⡀⠐⠋⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⣟⠈⣿⠈⣿⠋⠁⠀⠀⠀⠈⢳⡄⢿⡀⠀⢀⣴⠟⣡⡾⠋⠀⢈⣉⣭⡿⠃
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡏⠙⢻⣏⣸⡇⠀⠀⠀⠀⠀⣠⠾⠛⠛⠛⠛⠿⣧⡞⠋⠀⢀⠐⠿⣯⣄⡀⠀⢀⣤⣤⡤
⠀⠀⠀⠈⣿⠀⠀⠀⠀⣾⠻⢶⡾⠋⠉⣷⡀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠘⣷⠶⠟⠋⠁⠀⠀⣨⠟
⠀⠀⠀⠀⣹⣧⡴⠶⠶⠿⢦⣼⣇⠀⠀⠈⠳⢦⣤⣤⣿⡁⠀⠀⢀⠀⠀⣸⠇⣿⠶⢦⡾⠛⠛⠻⠷⣤⡀
⠤⣤⣄⠀⠻⢷⣄⡀⠀⠰⢤⣤⣙⢻⣦⠀⠀⠀⣼⠀⠘⢷⣤⡾⠛⠴⠾⣋⣼⠇⠀⢸⡇⠀⠀⠀⣠⡌⣻
⠀⠀⠀⠀⠀⠈⣿⡿⠃⠀⠀⠀⠙⢷⣽⡷⠶⠛⠻⢦⣄⢰⡟⠛⠓⣶⡟⠋⣡⡴⠶⠚⠛⠻⠶⣬⣭⠾⠋
⠀⠀⠀⠀⠀⣴⡏⠀⠒⠒⠓⠀⠀⣼⠋⠀⠀⠀⠀⠀⠙⣿⠶⠛⠛⠛⠻⣾⡉⠀⠀⠀⢸⣇⠶⢘⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠛⠛⣻⡟⣀⣠⣴⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣠⡘⣧⠀⣠⣤⠄⢛⣷⡾⠁⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⢿⣍⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⣤⠶⠶⣾⠛⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢙⣿⢷⣄⣀⣠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠛⢠⡿⠀⣴⠛⠛⣷⣤⠾⣧
⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠘⣧⣀⣽⠿⣏⠀⠻⣄⠀⠀⢤⣤⣴⠖⣠⡾⠻⣦⡿⣦⣤⠾⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⣿⠀⢹⡷⢶⣿⠛⠶⠦⠤⠴⠞⠋⠀⠀⠻⣤⣼⠷⢦⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠋⠀⠸⣶⣋⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⠤⠤⠖⠒⠒⠛⢚⣛⢛⠓⠛⠒⠒⠲⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢉⣁⠤⠔⠒⠚⡉⠉⠉⠉⠁⠏⢉⢉⡉⠉⠓⠒⠢⢤⣈⡉⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠖⢋⡠⠔⠋⠉⠀⠈⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠤⠉⠛⠢⢄⡙⠲⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢉⡤⠚⠁⠈⠁⠀⠀⠀⠀⠀⣀⠀⠁⠁⠀⠀⠀⠐⠒⠒⠂⠀⠀⢀⠀⠐⠀⡊⠀⠀⢠⡈⠓⢤⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠉⠀⠀⠀⠀⠀⢄⠁⠀⠀⠀⠀⠀⠀⢤⠒⠈⠴⠂⠀⠀⢈⣩⠶⡌⠁⠀⠰⡅⠀⠀⠀⠉⠀⢵⡙⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠞⢁⠔⠁⠓⠀⠀⣀⡴⣃⠀⠀⠀⠀⠀⠀⣠⠶⠶⠧⠤⣆⣠⢤⣄⠠⡀⠈⠠⡄⠀⣀⠀⡀⠀⠀⠴⠀⠀⠀⠈⠀⠈⢢⡈⠳⡄
⠀⠀⠀⠀⢀⡰⠃⡴⠃⠀⠀⠀⠀⠠⠤⠊⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⣹⠀⠉⠀⠀⢀⡀⠀⠲⢦⡑⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠙⢦⠘⢆
⠀⠀⠀⢀⡜⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡸⠛⠀⢦⣀⣀⠀⠈⠀⠪⠙⠛⠀⣸⡈⣆⠀⠀⠀⠀⣄⠀⠀⠠⠌⠳⡈⢧⡀
⠀⠀⢀⡞⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⣼⢧⠀⠀⠀⠀⠉⠥⢤⡀⣄⠲⢤⣖⡺⢿⡿⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢳⡀
⠀⠀⡜⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣧⣤⡃⠀⠀⠀⢿⡂⠢⠄⠀⠑⢦⠀⠀⢈⠈⠳⠄⠀⠤⠦⣽⡤⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢣
⠀⣴⠃⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠈⣛⠯⡀⠀⠀⠀⠱⠀⢀⠀⠳⣄⠢⠴⣤⠱⣄⢀⣒⣒⠒⠦⣇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⣦
⠀⡇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⢀⠏⠀⠁⠀⠀⢠⢠⠀⠀⢕⠦⠀⡴⢠⠄⠤⠬⠁⠒⠒⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢸
⢸⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⠀⡀⠀⠀⠀⠀⠀⠀⢈⡿⠰⠆⢀⠀⢸⡁⢧⡀⠀⠰⢤⡙⣦⠀⠀⢀⡘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈
⣼⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠈⠀⠀⠀⠀⠀⠀⠀⣇⠀⣤⠌⢠⣶⣅⢀⢀⡠⠀⠀⡅⠈⠀⢢⠂⢻⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣾⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡑⠓⢀⣀⠃⠈⢿⣿⣗⣁⡼⠃⠀⡠⠘⢀⠃⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⢿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⡭⠉⠀⢘⠛⠯⠛⢻⣿⣤⣖⡰⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⢻⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠘⠁⠀⠀⠀⠙⠲⣶⡈⢳⡙⠈⣡⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣠⠦⠄⠀⠐⠯⢶⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⣻⠀⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀
⠀⡇⠸⡀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⡄⠉⠉⣡⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸
⠀⠻⡀⢣⠀⠀⠓⠑⠄⠀⣤⡀⣯⠀⠡⠲⠄⠀⠠⠂⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⢀⣷⠀⢀⣠⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠟
⠀⠈⢣⠈⣇⠀⠀⠀⠀⠐⠙⠃⠈⠁⠛⠠⠠⠀⠁⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⠈⡶⢋⠀⣁⢃⠀⠀⠀⠀⠀⠀⠀⣸⠁⡜
⠀⠀⠈⢧⠈⢆⠀⠀⠀⠀⠀⠁⠁⠀⠂⠲⠂⠰⠀⠀⠀⠀⠀⠀⣷⡄⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⠥⠀⢀⠚⠋⢀⢠⠀⠀⠀⠀⡰⠁⡼⠁
⠀⠀⠀⠈⢣⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⢀⡈⠗⡼⠈⠀⠀⠀⠀⡼⢁⡞⠁
⠀⠀⠀⠀⠈⠱⡄⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⢹⠄⠀⠀⠀⠀⢠⠞⢠⠎⠁
⠀⠀⠀⠀⠀⠀⠘⢦⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠉⢻⣧⠇⠀⠀⠀⢀⠔⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠢⣀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠓⠲⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣴⣺⣯⠿⠋⠁⠀⠀⠀⠁⠀⠀⣠⠔⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣌⠓⢤⣀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠲⠀⠀⠈⠉⠍⢭⠉⠉⠉⠉⠉⠒⠀⡄⠀⠀⠀⠀⣀⠤⠚⣁⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣌⠑⠦⢄⣀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠠⠀⠀⣀⣠⠔⠊⣡⠴⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣌⡉⠒⠢⠤⢄⣁⣀⣀⣀⠀⡐⢐⣊⣀⣀⡠⠤⠔⠚⢉⣁⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠦⠤⡤⣤⢬⣭⣩⣤⣤⠤⠤⠴⠒⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠶⡒⠋⠙⢉⠙⢉⠙⢙⠒⢖⠲⢤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⠚⠫⡑⠄⠈⢢⣈⣢⣬⣦⣴⠶⠶⠦⢶⣤⣵⣤⣐⣈⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⡉⠑⠤⣀⣦⣼⡾⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⣤⣀⡑⢄⡀
⠀⠀⠀⠀⠀⢘⡴⠫⡉⠒⢌⣲⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠤⠀⠠⠀⠠⠄⠀⡄⠀⡀⠀⠀⠀⠀⠉⠛⢮⣝⢦
⠀⠀⠀⠀⣰⠫⣍⠒⢄⣵⣿⠟⠁⠀⠀⢀⡀⠰⠂⠈⠁⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠆⣀⠀⠀⠀⠙⠳⣷⣄
⠀⠀⢀⡾⡁⠑⢆⣹⣿⠟⠁⠀⠀⠠⠆⠈⠀⠀⠀⠀⠀⠀⡗⢌⣁⣈⣉⡦⡄⠀⠀⠀⠀⠀⠀⠈⠁⢤⡀⠀⠀⠈⢿⣦
⠀⢀⡞⠀⡀⢑⣾⡿⠃⠀⠀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠈⢳⠀⠀⠀⢸⠖⠢⢄⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠹⣷⡀
⠀⡤⡀⠀⢀⣿⡟⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠑⠈⣱⡽⠀⠀⠀⠈⠛⠛⠒⢳⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠹⣿⡆
⢰⠃⠄⠑⣾⡟⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⢢⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠀⠸⣷⡀
⣸⠀⠀⣼⡟⠀⠀⢀⠈⠀⠀⠀⡀⠀⠀⢠⠇⠀⢁⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⢤⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢿⣧
⡏⠀⢢⣿⠃⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⡞⠀⢁⡾⠁⠀⠀⠀⣠⢾⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠘⣿⡆
⡇⡀⢹⡟⠀⠀⠐⠁⠀⠀⠀⠀⢇⠀⢸⡁⠀⣼⠁⠀⠀⢠⣾⠁⢼⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡅⣿⡇
⡇⠀⣹⡇⠀⠀⢰⠀⣀⣀⣀⠀⠈⠀⣾⢌⣰⡇⠀⠀⠀⡾⢰⠄⣐⡇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠁⢹⣷
⡇⠀⣿⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠐⠠⢸⡇⠀⠀⠀⡇⢸⠐⢬⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⣿
⢹⡀⢾⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⣇⢸⡀⠰⡇⠀⠀⠀⣿⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⣿⡟
⠈⢇⠘⣷⠀⠀⠀⠢⠀⠀⠀⠀⢸⠀⠈⣇⠒⣧⠀⠀⠀⠘⣾⠈⣘⡇⠀⠀⠀⣿⠉⡁⠈⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣿⡇
⠀⠘⡎⢻⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠌⠘⢇⠺⣆⠀⠀⠀⠈⠓⠮⡇⠀⠀⠀⢻⡴⠖⠋⠁⠘⡆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣸⡿
⠀⠀⠘⣌⢿⣆⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠈⠳⣜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠄⠀⠀⠀⠀⢠⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠘⢟⢿⣆⠀⠀⠀⠁⣀⠌⠀⠀⠀⠀⠀⠀⠙⠛⠷⢤⣀⣀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁⠀⠀⠀⢀⡀⠃⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣧⡀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⡷⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⢀⡀⠊⢁⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠉⠫⣷⣀⣀⣀⣸⡇⠀⠀⠀⠀⢀⡄⠉⠀⠀⠘⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⠀⠐⠂⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠂⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠻⢷⣦⣄⣀⠀⠀⠀⠀⠉⠀⠐⠀⠀⠒⠀⠐⠃⠀⠁⠀⠀⠀⣀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⡶⠿⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣅⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣴⣿⣿⣗⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠐⢲⠆
⠀⠀⠀⣸⣿⣿⣿⣷⣿⡿⠿⠛⠃⠀⠀⠀⠀⠀⢸⢸⣆⢀⡀⢀⡏⢷⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀⠏
⠀⠀⠀⢺⣽⣿⣟⣽⣛⣄⠀⠀⡤⠀⠀⠀⠀⠀⢸⢸⠙⠎⠙⠾⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⠈⢀⡌
⠀⠀⠀⠈⠹⣾⣿⣿⣿⣭⣢⠔⠁⠀⠀⠀⠀⠀⢸⢸⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠐⠓⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡄⠀⠀⡾⣄⠀⡞⠀⠹⠄
⠀⠀⠀⠀⡤⠖⠋⠉⠁⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢻⠀⢸⠛⠙⠾⠃
⠀⠀⢰⡎⠀⠀⠀⠀⠀⠀⡇⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⡏
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠁


⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⡧⡆⠀⠀⢠⠀⢰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⣒⠈
⠀⠀⠀⣗⡇⢰⡆⢸⠀⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⣿⣉⣉⣉⣉⡃
⠀⠀⠀⡧⡗⢸⡇⢸⡀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢠⠊⠉⠒⠂⠀⠀⠀⠀⠀⠀⣿⣭⣭⣭⣭⣭⡭⠭⠭⠅
⠀⠀⠀⠧⠧⠼⠧⠼⠧⠼⠧⠼⠦⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣬⣥⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣋⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⠀⠀⠀⢀⣴⠟⠋⠀⠀⠉⠛⠷⠶⠶⠾⠛⢋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠶⠶⠟⠁⢰⡆⠀⠀⠀⠀⢀⣴⠄⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠸⣷⠀⠀⠀⣰⡟⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⣠⡄⣤⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣠⣴⣿⣷⣿⣦⣄⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⢸⣿⠏⣿⡇⣿⠛⠿⠇⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⣻⣿⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠰⢿⣷⣿⣧⣿⣶⣿⠟⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠈⣿⡟⣿⠉⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣤⣄⣀⣀⣀⣀⣤⣤⣴⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢻⡿⠋⠉⣿⠋⠀⠀⣠⡴⠟⠶⢦⣄⡀
⠀⠀⠀⠀⣠⡴⠿⠍⣳⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣻⡟⠁⢸⠃⠀⣸⡇⠀⣀⡾⠋⠀⣀⣤⠶⠾⠿⣦⣄
⠀⠀⠀⣠⠿⠄⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇⠠⣿⠀⠀⣿⣧⠖⠛⠛⠋⠉⠙⠓⠛⠛⠷⣤⣤⣤⣤⣝⣧
⠀⠀⣾⣃⠀⠖⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣿⠀⠀⣿⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠈⠙⠳⣄
⠀⣰⠟⢡⠄⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⢹⣇⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣤⣹⣇
⠀⢿⡶⢂⡀⠺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢿⡀⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠉⢳⡄
⢠⣿⠀⠀⢀⡰⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠷⠦⣵
⠈⣿⡟⠀⠉⠀⠀⡽⠳⣦⣀⣀⠀⢀⣀⣤⣤⣴⡾⡿⠉⡏⠘⣧⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠈⢷
⠀⣿⠀⢀⡀⠒⠃⠀⢀⠚⠋⠉⣻⠋⠀⢠⠏⠀⠀⠁⠀⣧⠀⠈⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⢦⣸⠇
⠀⠹⣧⣾⠁⠀⠀⢠⠟⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢦⣄⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⢹
⠀⠀⠈⣿⠀⠀⣠⠀⠀⠀⢰⠇⠀⢀⡾⠀⠀⠀⠀⡀⠀⢿⡄⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⡄⢸⡆
⠀⠀⠀⠻⣦⢰⣿⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢡⠀⢸⡿⢦⣤⣹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⢻⡞
⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠘⣧⠀⢷⡀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠙⣷⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢻⣆⠀⢠⡀⠀⠀⣴⠀⠀⣿⠀⠸⡇⠹⣆⠀⠛⢶⣤⡤⣬⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠻⣷⠀⢸⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⣇⠀⠀⢻⡄⠀⢻⡆⠀⣿⠀⠹⣆⠀⠀⠻⣦⡀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠃⠀⣿⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠘⣷⡀⠘⣷⡀⠘⣇⠀⠹⣦⠀⠀⠈⢹⡷⠦⠴⠿⠶⢤⣀⣀⣀⡀⠀⠀⢀⣀⣠⡴⣿⠃⠀⢹⡇⠀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⣽⣷⡄⠹⣷⡀⠹⣆⠀⠘⢷⣄⠀⠀⠙⢦⣤⣀⣠⡴⠟⠉⠉⣽⠟⠉⢉⡿⠁⠀⣿⠀⢠⣿⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣦⣽⣷⡀⠹⣧⡀⠀⠙⢶⣄⠀⠀⠙⠻⢦⣤⣀⣤⠞⠁⠀⠀⣼⠇⠀⣰⣏⣰⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣦⣈⣿⣦⣀⣀⣹⠷⣦⣄⣀⣠⡴⠛⠳⢶⣤⠶⠞⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡶⢾⣿⣿⣿⣿⣿⣿⡷⢶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢿⣿⡯⠽⣶⣿⣿⣿⣿⣿⢿⣿⢓⣽⣿⣿⣿⣾⣩⣽⣻⡿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⣟⣭⣾⣯⣻⣷⣹⣾⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣫⣿⣿⣟⢲⣭⣛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣽⣾⢿⣿⣿⣯⣽⣿⣿⣿⣻⣽⠟⠉⠣⠈⠍⠉⡉⠀⠛⠛⢿⣿⣿⣿⣎⣿⣟⣿⣷⣯⡻⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⢟⣵⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⠟⠋⢰⠀⠆⠀⠀⢀⣂⠷⠀⠀⠀⠠⣝⣿⣿⣿⣿⣿⣽⣿⢿⣿⣾⡻⣷⣄
⠀⠀⠀⠀⢀⣴⣟⣵⣿⣝⢤⣽⣿⣿⣿⣿⡿⣽⢿⠋⠀⠀⢢⡄⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣦⡀
⠀⠀⠀⢠⣾⢯⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⢿⣻⣿⢠⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⡽⣷⡄
⠀⠀⢠⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⡄
⠀⢠⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠐⠌⠂⠀⠠⠚⡐⢆⠅⡀⠁⠀⠀⠀⠶⢺⠷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡄
⠀⣾⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀⠁⠀⠀⠀⠀⠆⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣷
⢰⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠌⠃⠀⠀⠀⣀⠀⠀⡀⠀⠀⠈⠳⠀⠀⠀⢀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿
⣼⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠓⠀⠀⢀⡀⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣦⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡤⠐⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⣿⣿⣿⣽⣿⣽⣿⣯⣿⣯⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠐⠒⣺⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣯⣿⣿⣷⣿⣾⣿⣿⣿⣾⣿⣷⣿⣿⣾⣿⠿⠯⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹
⠸⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠑⠢⡀⠀⠀⠀⠂⠀⠀⣤⠄⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿
⠀⢿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠄⠀⠒⠠⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿
⠀⠈⢿⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠲⢤⡀⢆⠀⠀⠂⠀⠈⠢⢄⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠁
⠀⠀⠘⢿⣞⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠐⠀⠐⠠⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣿⠃
⠀⠀⠀⠈⢿⣾⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠐⡄⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⣷⡿⠁
⠀⠀⠀⠀⠀⠻⣷⡿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠠⡀⠀⠀⢢⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⢿⣾⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣮⡻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⢢⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣟⢿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢳⠀⠡⡀⠀⠀⠀⠀⠀⠂⠀⢀⣠⣴⡶⣫⣷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣾⣝⡿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⡄⠀⠘⢦⡀⠀⣀⣤⣴⣾⢿⣻⣷⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣾⣽⣟⣿⡿⣽⣿⣷⣶⣤⣴⣷⣶⣦⣼⢿⣿⣻⣯⣿⡾⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣻⢿⣿⣷⣶⣾⣷⣶⣾⣿⡿⠟⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠾⠟⠛⠛⠛⠛⠻⠷⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⡀⢠⡄⠀⢀⠁⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⡅⠀⠈⠁⡀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠈⢀⣀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠀⠀⠀⠀⠀⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠁⠀⠀⠤⠤⠒⠁
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⠀⠱⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠐⠤⠈⢀⠈⣷⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠢⣀⠘⢆⠀⠘⣇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠙⢷⣄⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⣄⠀⠸⡇⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢚⢆⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠈⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠢⠀⠀⠀⠀⠀⢀⠠⠂⣡⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣈⡑⠒⠤⠤⣀⣀⠀⠘⢀⣀⠀⠤⠄⢂⣉⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⢶⣤⣤⣤⣤⣤⣤⡶⠶⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣠⣟⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⢀⠀⠀⠀⠀⠀⣼⠁⢀⣴⠿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢿⣷⠏⠀⠀⠀⠀⣼⠃⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⡇⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢀⣼⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣬⡇⠀⠀⠀⣠⠟⠁⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢻⣿⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡇⠀⢀⡴⠋⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡞⠁⠀⠀⠀⢠⡞⠁⠀⠀⠀⣶⣤⠀⢰⠏⠀⠀⠀⠀⠀⢠⡾⠛⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠏⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠙⢿⣷⠏⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠉⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣠⡾⠛⠛⠲⠶⠤⣤⣀⣀⣴⠋⠀⣀⣀⣤⣾⣏⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢀⡴⠋⠀⠀⣀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉⠀⠀⠈⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣸⣆⠀⠀⠀⠀⠀⢸⠇⠀⢠⠞⠁⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣾⠀⣰⠋⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣿⡄⠀⠀⠀⢹⡿⠀⠀⠀⠀⢀⣇⡼⠃⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣀⣠⡶⠛⠁⠀⠀⠀⠀⢀⣀⡀
⠘⢿⡄⠀⠀⣾⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣰⣯⣤⠤⠶⠶⠶⠚⠛⠉⠉⠛⠲⠶⠶⠶⠶⠞⠛⠿⣿⣷⡆
⠀⠀⠹⣆⠀⡿⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⡼⠋⠁
⠀⠀⠀⠘⣦⣧⠀⠀⠀⠀⣿⠀⠀⠀⣠⣶⡋
⠀⠀⢶⣶⣿⣿⠀⠀⠀⢰⡇⠀⣴⠞⠉⠿⣿⣦
⠀⠀⠀⠙⠛⠻⣇⠀⠀⢸⣧⠞⠁
⠀⠀⠀⠀⠀⠀⢻⡄⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠈⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⣀⡾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⡀⢜⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⣄⡐⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢁⢀⢸⢴⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠞⡑⢁⠊⣯⢺⢻⠟⠉⠃⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡴⣞⣯⣯⣯⣮⣾⣖⣻⣐⠓⢿⣜⣄⣀⣥⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣧⣿⣿⠿⠋⠁⠀⠀⠸⠈⠙⠻⢯⣢⡄⠨⣿⣿⣿⣿⣇
⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⡿⠇⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠳⡳⠸⠿⠋⣿⡿⠟⠁
⢻⣿⡿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣿⣤⠀⣿⣃
⠈⢙⢿⡐⠒⠀⠀⠀⠀⠀⠠⢀⣤⣦⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⡧
⠀⠁⠀⣨⣥⣐⣶⠐⠀⠀⠀⡼⠿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⡿⡛⠹⢿⣿⠃
⠀⠀⠀⣽⣿⣿⠅⠀⠀⠀⠀⠠⠂⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢈⠠⣳⠄⢆⠄⠀⠀⠀⠰⣿⢡⠅
⠀⠀⠸⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⡏⠀⢤⠆⢒⡣⠽⠁⠀⢈⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢸⠠⠙⢉⠒⠀⠄⠙⢺⠉⡃⠂
⠀⢀⣀⣤⣤⣤⠄⣠⢤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠀⠀⠀⠁⠉⠁
⠀⠉⠉⠉⠉⠀⠊⠉⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠒⠒⠂⠀⠀⠀⠀⠐⠒⠂⠠⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⣀⡠⠄⠒⠀⠉⠉⠀⠀⠀⠀⠉⠉⠁⠒⠠⠄⣀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢁⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣶⠂⠒⢰⢂⡖⠠⠒⡆⠈⠓⢄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡠⠊⠀⢀⠀⠠⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⡘⣒⡀⢓⠁⠂⣑⢐⣂⢀⡀⠀⠀⠑⢄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠜⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠁⢸⠃⢀⡇⠈⠀⠀⠀⢴⡲⠀⢀⠣⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠄⠠⠐⠀⠇⠀⠐⠀⠠⠀⠀⠀⠀⠐⡷⠃⢀⠘⡄⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠂⠀⠘⡀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡆⠀⠰⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠃⠈⡀
⠀⠀⠀⠀⠀⠀⠸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣖⡙⠂⡈⡉⠐⢛⡠⢤⡀⠀⠀⠀⠀⢉⣄⠀⠸⠀⠃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⢈⡅⠀⠀⠉⡉⠀⠰⢄⠀⠀⠲⡀⠀⠀⠘⠊⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⠒⠀⡰⠌⠁⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⡾⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢰⠀⡆
⠀⠀⠀⠀⠀⠀⠈⡀⢠⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⡄⢀⠁
⠀⠀⠀⠀⠀⠀⠀⢃⠈⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠂⠴⠯⠽⠤⠀⠐⠂⣀⡀⠀⠀⠃⠀⠀⠀⠰⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠁⠀⠀⣲⠀⠀⢠⠃⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠃⠀⢠⠃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢦⠀⠀⠀⠮⡆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠑⠀⠦⠀⠀⠀⠀⢤⠋⠀⠀⠔⠁⠀⡰⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠑⢄⠂⢀⠈⠒⢅⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠢⠀⣠⡀⢀⠀⠮⠀⠈⠀⠀⡠⠊⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠑⠤⡀⠀⠈⠁⠰⠽⢁⡠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡈⠐⠤⣀⠀⠀⠀⠉⠀⠨⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠉⠐⠂⠤⠈⣀⣀⠀⠀⠀⠀⣀⣀⠀⠤⠐⠂⠉⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠂⠠⠤⠄⠀⢀⡀⠀⠠⠤⠄⠀⠒⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡾⠻⢷⣦⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⣛⣉⡉⠉⣀⣤⣤⡴⠶⠶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣀⠉⠛⠋⠁⠀⠀⠀⢠⣾⣯⣴⠶⠿⠛⠛⠿⠶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣷⣄⠀⠀⢀⣀⣰⣿⡿⠿⠶⠶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠟⠛⠿⠿⠯⠽⠟⠙⢿⣦⣄⡀⠀⠀⠉⠛⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⠀⠙⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣆⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣸⣿⣄⡀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠉⣿⡟⢿⣶⡄⠀⠀⠀⠀⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠘⣿⡆⣿⡇⠀⢹⣿⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣿⡇⣿⡇⠀⠾⠋⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡿⢾⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢹⣿⡻⣷⣦⡀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⢸⣿⠀⠈⣿⣇⠈⢻⣷⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⣿⡇⠀⢹⣿⡀⣸⡿⠀⠀⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣹⣿⣀⣀⣿⣷⠟⠁⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡟⠉⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⢠⣿⡇⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠙⠿⢶⣶⡶⠿⠿⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⢀⣰⠟
⠀⢠⣿⡝
⢀⣿⣯⢧
⢸⣏⣹⡌⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⣒⣒⣶⣖⠒⡶⠤
⢸⣏⠀⠹⢾⣵⣄⠀⠀⠀⣀⣠⡄⢃⣼⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⡀⠄
⣿⡾⣦⠰⣻⣷⣏⠿⣿⢿⣿⡏⣰⠟⠉⠀⠀⠀⠀⢀⢙⣙⣿⡇⢈⡞⠀⠀⠀⢠⡄
⠛⣀⠙⣶⡘⣳⣿⣿⣿⡾⣿⡠⡏⠈⠀⠀⠀⠀⠀⠐⢋⣡⡑⢷⣸⢀⣀⣴⠶⣺⢀
⠈⡿⢿⠋⡿⢧⣀⣽⡇⡇⣼⢉⡧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠈⣷⣯⠚⢁⣴⣿⢸⣀⣠⣤⠤
⠀⠐⣼⢧⡠⠀⡾⡻⢧⣧⣾⡅⢣⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣠⠟⣟⣁⣟⠫⢉⡀
⠀⠀⠈⠎⢸⢠⣄⠀⠘⠈⠈⣇⣸⠠⠀⢀⠄⠀⠀⠀⠀⠀⠁⢰⢇⠹⡏⠄⡼⣵⠚⣦⣜⣴⣾⣷⣷⣶⣶⣤⣀⠀⡀
⠀⠀⠀⠘⠦⣨⣿⣧⣄⣄⣾⠏⠀⠀⠀⠋⠀⠀⠀⣠⠾⠄⣌⢆⡓⠊⡰⢼⣹⢟⣣⣿⣿⣿⡿⠟⠹⠟⣿⣿⣿⣿⣋
⠀⠀⡀⠀⠀⢈⠋⠀⠀⢹⣿⠀⠀⠀⠀⠘⠅⠀⠌⠃⠀⠃⢘⣳⠛⣖⢻⣟⢷⡶⣋⣛⣙⣛⣇⠀⠀⠀⠛⠛⠛⠉⠋
⠀⠀⠁⠀⠀⢠⣦⠀⣠⡼⢟⢀⠀⠀⠀⠀⡀⠁⠀⢈⡀⠀⠜⠄⠈⠛⠐⠉⠌⠃⠙⠉⠁⠈⠈⠁
⠀⠀⠀⠀⠀⢀⡀⠊⠀⠀⡾⢃⠤⠤⠄⠚⢛⠋⠁⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠌⣨⡴⠀⠀⠀⠀⠀⠀⠀⢀⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⠐⢮⠈
⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠢⡨⠂
⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠈⠀⠉⠀⠀⠀⢀⢀⢠⡴⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⡀⠀⡀⢀⢀⣀⠂⠕⠔⠁⠀⠀⠀⡴⠒⣉⣔⡄⠐
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢀⢀⡴⠞⠋⠁⣀⡀⢀⡀⠀⠀⠀⡷⠤⣷⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⣠⣠⠴⠀⠑⠒⠁⠀⠀⠀⣘⡒⢠⣾⣼⠀⠀⠀⢣⣈⣟⣿⡇
⠀⠀⠀⠀⠀⠀⢀⢠⠨⠤⠞⠉⠀⢀⣤⠤⠀⢀⣈⠀⠀⠣⠤⢿⣿⣿⡄⠀⠀⠰⠆⠷⠶⣧
⠀⠀⣀⠠⡰⠜⠋⠁⠀⠀⠀⠀⠀⠀⠤⠬⠯⠿⠿⠆⠀⢘⣆⣘⣟⣿⡇⠀⠀⣸⣩⣭⣿⡟
⠀⠀⠀⠊⠀⠀⠐⠋⠉⢡⡬⣴⠀⠀⣏⣈⣛⣛⣿⠀⠀⠐⡦⢤⣶⣾⡇⠀⠀⡲⠤⡿⠾⡇
⠀⣠⠤⠀⡤⡀⠀⡏⢘⣛⣿⣽⠀⠀⢱⠁⠽⠷⣿⠁⠀⢸⣉⣚⣛⣻⡆⠀⠀⠯⠁⣟⣿⡷
⠈⠤⢰⡾⡯⣧⣀⡦⠴⠶⠶⣆⣠⣄⡨⢉⣹⣭⣽⠀⢀⣸⡊⢽⣭⣿⠃⠀⢀⣟⢒⣲⣶⣆⣀⡀⢀⡤⠀⠄⣀⢤⡀⠀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠒⠒⠒⠦⢤⣄⡀
⠀⠀⠀⠀⠀⢀⠴⢋⠁⠒⠋⠉⠉⠉⠉⠀⠰⣄⠻⣷⣤⡀
⠀⠀⠀⠀⡴⢃⡜⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⡈⠃⠈⢿⣿⣆
⠀⠀⠀⢰⢃⠉⢀⠴⠶⢦⡀⠀⠀⠃⡴⠶⠶⠿⠀⠹⡌⠛⣿⡇
⠀⠀⠀⡟⣼⠠⣁⣾⠙⠀⡿⠀⠀⠀⣩⣤⣄⠲⣄⠀⣧⢸⣿⣿
⠀⠀⠀⣇⢿⠀⠈⠁⠀⣸⠃⠀⠀⢀⣀⠀⢈⠇⣹⠆⠁⠀⣲⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⢹⡀⡀⠀⡌⠀⠓⠀⣐⣆⠸⣄⣉⣡⡴⠏⣠⠇⣼⣿⡿⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⣋⣭⠭⠭⣭⣉⣛⠛⠒⣦⣤⣀
⠀⠀⠀⠀⢳⡙⠄⠙⠛⠛⠋⠉⠁⠀⠈⠉⠉⢀⡴⢋⣼⣿⣿⠃⠀⠀⠀⢀⡴⠟⣩⠴⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠈⡛⢿⡛⣦⡀
⠀⠀⠀⠀⠀⠙⢦⣐⠦⣄⣀⠀⠀⠀⣀⣠⠀⣉⣴⢿⣿⠟⠁⠀⠀⠀⢠⠟⠀⠊⠁⢰⠖⠶⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⠯⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠬⣭⣥⣤⣤⣶⣿⠿⠟⠋⠀⠀⠀⠀⠀⢠⠏⡴⠃⠀⠀⡎⢠⡟⠓⢦⣼⡇⢀⠄⠀⠁⠒⢦⡀⠈⢧⠈⣹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠁⠀⠀⣸⠁⠸⢇⡀⠈⠛⠃⠂⢠⣾⠟⢆⠈⣿⡆⠈⡇⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⡾⠀⠀⠀⠘⠻⢶⣄⠙⣦⠀⡈⠀⣾⡏⠀⣸⠀⣿⡇⠀⠛⠘⡿⠯⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⢠⠞⢲⡀⠀⠀⣸⠀⣿⠀⡇⢠⣿⠀⢀⡏⠀⣿⠇⠀⢰⠀⢋⣛⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⡘⢦⡀⠙⠒⠊⢁⣴⠟⠀⡇⠘⣿⣀⡼⠁⣼⡏⠀⠀⡼⠀⣔⣒⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠁⡀⠉⠛⠻⠛⠛⠁⠀⠀⢳⣀⠉⠉⣠⣾⠏⠀⠀⡴⢣⣟⡓⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠠⠞⣴⣏⣒⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⡑⠦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠂⣴⣾⠭⣟⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣅⣀⣈⣙⣛⣛⣉⣩⡤⣶⣲⣤⡭⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠓⠚⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⡟⢰⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣧⠸⣿⣿⣿⣿⣿⢁⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⡿⢟⣡⣾⣧
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⡏⣸⣿⣿⣿⡿⢟⣩⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠻⠟⢰⣿⣿⠟⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠠⠟⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣦⣭⣭⣭⣭⣭⣤⣤⣈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣌⡙⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⡙⠛⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠲⠤⠤⢄⡀
⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⣄⣤⡴⠴⠦⢤⣹⢢
⠀⠀⠀⠀⡴⠋⠀⠀⢠⠞⠁⠀⠀⢠⡆⣨⠇⡘
⠀⠀⢀⡞⠁⠀⠀⠀⢹⡆⠀⠀⠀⠈⠛⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⣆
⠀⢀⣾⠀⠀⠀⠀⠀⠈⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢱⡋
⠀⣸⠁⠀⠀⠀⠀⠀⠐⠀⠘⠉⠓⠲⢴⢦⠄⠠⠄⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢻⡟⢻⠄
⠀⣿⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣦⣤⣈⡙⣲⡤⠴⠖⢒⡾⣃⣾⣷⠒⠢⠤⣀
⠀⣿⡸⠁⠀⠀⠀⠄⠀⠀⣠⠶⢿⣛⣻⠿⠻⢿⡿⠉⠉⠀⢨⠟⠁⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠙⢆
⠀⢷⢄⣴⠀⠀⠀⠀⠀⡴⠁⠀⣠⠏⠀⣈⣠⡧⠤⣤⡈⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠸⣾⡃⠄⠀⠀⠀⢸⠃⠀⢾⣿⣻⣾⣟⣛⠋⠛⠻⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⣠⣄
⠀⠀⣿⡧⡣⡄⢸⠁⣺⡄⠀⣾⣿⣯⡛⢷⣄⡉⠻⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠤⢿⠘⠛⢷⡆
⠀⠀⠙⣷⣷⡧⢸⡦⠸⡟⠂⠛⠻⣿⣷⣤⣌⣿⣷⠞⠛⠻⣧⣀⣤⡀⠀⠀⢠⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠀⣼
⠀⠀⠀⠈⣿⣿⣮⣿⡦⣿⡥⣀⣀⠾⣿⠟⢿⡿⠟⠲⢤⡎⠁⠀⠀⠈⢳⠴⠒⠚⠒⠖⠐⣧⣤⣄⠀⠀⠀⡻⠓⠀⠀⠀⠰⡻
⠀⠀⠀⠀⠈⠙⠿⣿⣿⡿⡄⠀⠃⢰⢧⠀⠀⠃⠀⢀⠌⠁⠈⠐⡄⠀⣸⣀⣀⡀⡐⠀⠀⠀⠉⠈⠉⠉⠙⠲⠒⠤⢤⡥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠻⢧⠣⣀⠀⢘⡌⢆⡀⠀⠀⡄⠀⠀⠀⠀⠘⠈⠻⣯⠙⢹⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠉⠈⠓⠚⠒⠚⠳⠦⡀⢀⢤⣀⣀⢠⣿⣤⣠⡽⢻⡿⠛⠛⠓⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠃⠀⠈⠉⠙⠙⠛⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⢻⣿⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⡻⣿⣿⣿⣿⣿⣷⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡗⠈⠀⠈⠛⢿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣟⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⢢⡀⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠋⣽⠁⢠⣤⣶⣶⠂⠀⢠⣿⣿⡄⠻⣿⣿⣿⣿⡿⣿⡻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠛⠋⠁⣰⠂⣾⣿⣿⣷⠀⢻⣿⣿⣿⣿⣞⢿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡏⣼⣿⣿⣿⣿⡄⠈⢿⣿⣿⣿⣿⣷⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢸⣿⣰⣿⣿⣿⣿⣿⡇⡇⠈⣿⢻⣿⣿⣿⣿⣮⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣸⣿⣿⣿⣿⢿⣿⣿⣿⡇⢠⣿⠀⣿⣿⣿⣿⣟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⢠⣿⣿⡿⠛⠁⠘⣿⣿⡿⣰⡿⠁⠀⣿⣿⣿⣿⣿⡎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣣⣀⠀⢠⢿⡿⠋⠀⠀⠀⠀⢻⣿⣷⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⢠⣧⣿⠁⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⢠⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣡⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣠⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⡠⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣴⣿⡀⢀⡖⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣴⣾⡿⣽⣿⣿⣿⣿⣿⣿⡿⣷⡞⠫⡏⠉⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⣿⣇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⠥⠀⡀⠄⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⢸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠳⠶⠚⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠀⠠⡿⠿⠿⠟⠛⠋⠉⠉⠀⠀⢀⣀⣤⣴⡖⡀⠀⠀⠀⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⣤⣤⣶⣶⣿⣿⣿⣿⠟⢩⣗⠠⠀⢰⣷⠈⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⢠⣀⣿⠀⠀⠋⢀⣤⠀⣀⣍⠴⣍⡛⢿⣿⣿⡿⡟⢿⠿⢿⣿⠛⠍⠀⣼⣿⡀⠒⣸⣿⠀⣤⠆⢀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣼⡿⠞⠀⢈⠗⠲⣂⠄⢈⡅⠉⢁⡀⠆⣉⡩⠛⠉⠁⢀⠐⠂⡄⠑⠐⠭⣢⡀⡀⢨⠑⠣⠛⠻⢇⠁⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠊⠀⠀⠠⠖⠧⢀⡄⠀⠀⡀⠀⠁⠘⠋⠈⠁⠀⡀⠄⠂⠙⡀⢰⣷⠀⠠⡀⠀⠀⠈⠻⠗⡤⠀⠀⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⠃⠀⠀⢀⠂⠠⢸⣇⣾⠀⠈⠁⠀⠀⠀⢠⡆⠈⢀⠀⠂⢠⣇⣸⣿⡀⣀⣀⠄⢀⠅⠀⠁⢠⡀⠀⢀
⠀⠀⡆⠀⠐⠀⠀⠀⠤⣀⣀⠀⢀⣠⣤⡤⣬⣍⠀⠀⠴⠄⠀⣠⡄⣺⣇⢒⠶⠖⠠⡚⠛⠙⠉⠉⢀⠄⠊⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠁⠨⠑⠠⠄⠉⠀⢀⠀⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠐⠆⠨⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠂⠈⠘⢿⣿⣶⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡰⢠⢤⣄⡄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠉⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⠉⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⠋⠁⠀⠀⠀⢀⣀⠀⠀⣾⣿⡆⠀⠙⢿⣆⠀⠀⠀⠀⢀⣾⠟⠁⠀⣴⣶⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠀⠻⣷⡄⠀⣴⡿⠃⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣤⣤⡄⠈⢿⣾⠏⠀⣀⣀⡀⠀⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣶⣶⡆⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠙⠛⠁⠀⠙⠛⠁⠀⠙⠛⠁⠀⠙⠛⠁⠀⠙⠛⠁⠀⠙⠛⠁⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡗⠀⢀⣀⡀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⢷⣤⣙⣷⡀⠻⡟⢠⢄⠙⢅⢤⣞⢀⡷⠾⠯⣭⣬⣙⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢁⡚⢉⠀⣈⡑⢿⠿⡿⠃⣙⣍⡷⣿⣤⠀⠀⠈⢹⣿⡆⠸⡇
⠀⠀⠀⢀⣤⣴⡾⡟⠛⣩⡍⠋⣀⣐⡀⢒⢒⠎⠿⡛⡋⢁⣍⢱⠫⣿⠉⠑⢦⣦⠟⠀⣖⠁
⠀⠀⠀⣉⣁⣀⣠⢤⣟⡉⣭⡭⣭⣭⢝⣗⣳⣏⣾⣷⣴⢴⣬⣽⣦⣼⣉⣺⡿⢋⣀⣷⡿⡄
⣴⠾⠛⠙⠉⣉⢵⡾⢋⡩⠞⣁⣋⢳⣹⣾⣼⡏⠠⣏⣤⣄⣠⠈⣻⡍⠟⠉⣤⣼⠻⣅⠳⡘⠆
⠀⠀⠀⠀⠘⠁⠈⠀⠊⠔⠋⢹⠟⠒⣅⣠⠌⡷⣄⣤⣽⣿⣿⡇⢡⣤⡿⠺⣯⡙⠢⡈⠳⠌⡢⣷⣢⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠸⢱⢯⢧⡏⠳⠉⠉⠉⠉⠛⠳⢦⣍⠓⣬⢴⣪⣷⠾⣿⣦⣄⣀⠀⠀⠀⠀⠀⡨⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣾⣗⣦⢭⣞⣿⣿⣿⣿⣿⣛⣽⡷⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠤⠶⣦⡀⠀⣀⣠⠴⢺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⠁⠀⠙⢿⣛⡉⠅⠀⠀⣷⠉⣹⠋⠁⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠙⣦⠀⠀⠀⢸⢀⡟⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠀⠀⠀⠘⠉⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣽⣦⡴⠶⠶⠖⠒⠒⠚⠛⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠷⣿⣛⣁⣀⣀⣀⣀⣄⣤⠴⠖⠛⠉⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⣿⠟⡿⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣻⠟⠁⡼⠁⠀⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⡼⠁⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⠔⠊⠀⡀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠈⠀⠐⡇⠏⠢⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠈⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠦⡀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⡄⢨⢀⡅⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠈⠚⠈⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣀⣀⠀⠀⣀⠀⠀⠀⢀⡀⠀⢀⣀⣀⣤⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣽⣿⣿⣯⣿⠇⢠⡏⢁⡶⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣿⣿⣿⣯⣻⡷⣻⣷⣾⡿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡟⠃⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡦⠀⠲⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⡮⠝⣿⣯⣹⠯⠔⣈⣻⢶⡆⣪⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⣿⣿⠇⠈⢉⡸⠿⢋⣼⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⡙⢢⡬⠧⢤⣤⣾⡷⠂⠉⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢩⣄⢹⣮⣿⠛⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣻⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡘⠧⠉⣠⣽⣼⠇⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⡳⠐⢻⣿⠟⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⠀⠉⠁⠀⠀⢂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠀⣀⣀⠀⠻⣿⣿⡿⠛⠁⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣿⣿⣿⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢠⠿⠟⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣦⡟⠀⠀⠀⠀⢀⡎⠀⢠⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣁⣀⡀⠀⢠⣞⠀⣤⣰⣿⡿⠿⠟⠻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡿⠀⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⢐⠀⢀⡄⣸⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⡛⠛⠓⠒⠳⣾⠛⠏⠭⠉⠉⢹⣿⣿⡿⢿⣿⣿⠟⢛⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠛⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⣒⣤⡤⠬⠭⠭⠭⠭⠭⠌⠀⠒⡠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣪⠵⠚⣋⣉⣀⣴⠾⡟⠛⢶⣾⠻⠦⣄⣀⡀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣮⠞⠻⣶⠖⠛⠀⠈⠉⠁⡘⢠⠀⠀⠙⢃⠠⢄⡠⢝⢦⠤⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⢠⢔⡵⣋⠐⣃⠈⢟⣊⡼⠃⢀⣠⣀⣈⡂⠃⠀⠀⠀⠀⢁⣀⣀⣈⣁⠠⡄⠀⠀⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⢀⢔⡵⣻⡟⣀⣡⠿⣐⣲⡿⠒⠒⠲⠟⠻⡿⣷⣯⣵⣶⣶⡦⠵⠚⠉⠋⠁⠀⠳⢦⡀⠀⢀⢔⡴⠂⠈⠢⡀
⠀⠀⠀⢠⣻⠏⣼⣟⡠⠉⣇⠇⠚⠋⡀⠀⠀⠀⠀⠀⠐⠁⠋⠈⡔⠠⠶⠶⠤⠀⠀⠀⠀⠀⠈⢳⡀⠀⠸⣶⣿⢧⠐⠝⡄
⠀⠀⢰⣿⠃⢠⡿⠋⡠⣴⠟⠄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⣀⡄⣠⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⢵⣤⠾⠈⢎⢆
⠀⢠⣿⠁⣰⠋⢠⠞⣸⠁⡸⢠⣆⢠⠘⡆⠀⠀⠀⠰⣄⠀⠀⠀⠀⣾⢸⠫⠚⠃⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢁⠚⢷⡘⣎⡄
⠀⢮⠃⡜⠁⠔⡁⣴⡟⢀⠇⢸⣿⢧⢧⠉⠀⡆⠀⠀⠘⢧⣸⡆⠀⠻⢬⡀⠀⠀⠀⣀⣠⣴⡶⠶⣿⣿⡇⠀⠀⠀⢡⠌⢁⠸⡴
⠘⠘⠈⠀⣤⡾⣟⡿⢠⡞⢰⣾⡟⠀⠀⠁⠀⣷⡀⠀⠀⡀⠻⡗⡄⠐⣄⠀⠀⠀⠀⢦⣭⣶⣶⣶⣿⠈⢦⡀⠀⠀⠀⠾⠷⡴⣷
⡄⠀⠳⠾⠋⠀⡟⢀⠧⡁⣼⡟⠀⠀⠀⠀⠀⠘⠻⣆⠀⢹⡄⢣⡇⠀⠈⠳⠀⠀⠀⠀⠀⠙⠛⡆⠁⠀⠀⠑⠀⠀⠀⠢⢯⣻⢽
⡇⠀⠀⠀⠀⣸⠁⠌⠀⣷⣿⡅⠀⠈⢦⣀⣀⠀⠠⣌⠃⢸⠃⡏⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⢠⡇⠀⣰⡇⢀⣿⣷⢷⡤⣀⠙⠻⢿⣦⠙⢷⠈⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⠴⢀⣀⣀⠀⣸⠀⠀⠀⠀⢸
⠃⠀⠀⠀⡸⠀⣰⠞⣧⣾⣿⡙⢦⡀⢀⡉⠢⠀⠻⡇⠸⠀⡇⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⣠⡞⠈⠛⠋⢹⠿⠿⠋⠀⠀⠀⠀⣾
⢰⠀⠀⠀⡇⠀⠃⠀⣹⡏⠉⣿⣦⣽⠿⣷⡀⣀⣀⣬⣤⣤⣷⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⢀⡏
⠀⡆⠀⢸⠁⠈⢇⣴⠿⠷⠖⠹⣷⠀⠀⠘⠿⣾⣿⣿⣻⠿⣋⣽⣿⡇⠀⠀⠀⠀⡀⠀⠀⣠⠾⠒⠒⠒⠲⠁⠀⠀⠀⠀⠀⣼⢱
⠀⠸⡀⠘⣆⣠⡾⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣿⣧⠔⢳⡟⠛⢿⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⢰⢃⠇
⠀⠀⠸⡀⠈⠉⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠀⠀⣿⣿⣦⣄⠇⠀⣸⠈⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⢃⠆
⠀⠀⠀⠘⣄⢄⠀⠀⠀⠀⣠⡶⠿⠛⠛⠛⣓⡂⢸⠃⡏⠳⣌⠁⠉⢿⠀⠀⠀⢨⡟⠲⠤⠀⠄⣀⠤⠞⠁⠀⠀⠀⡴⣣⠃
⠀⠀⠀⠀⠈⠢⡣⡀⠴⠛⠁⠀⣤⢀⢠⡴⢤⡾⠛⠐⠁⠈⡻⡷⣤⡘⣄⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠔⠁
⠀⠀⠀⠀⠀⠀⠘⢪⡢⡀⠀⢠⣿⣬⢸⣯⢸⢸⡇⢷⣰⡄⢻⢦⣤⡀⣿⣄⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣠⢞⠕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣗⢌⡀⠀⠈⠂⠊⠀⠘⣌⣷⣅⠘⢧⠙⢬⡉⢸⣇⡀⠏⠀⠀⠀⠀⣀⠴⡊⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠷⣦⣄⣀⠀⠀⠈⠛⣿⢗⠀⠇⠀⠈⠀⠉⠠⣴⣤⡤⢖⡯⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠿⠒⠚⢛⣘⣃⣰⣖⣒⠒⠛⠯⠙⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠤⠄⠀⠀⠠⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡑⢄
⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢂⠥⠂⠀⠪⣐⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢢
⠀⠀⠌⠀⠠⠀⠀⠀⠀⠀⣴⣂⣀⣀⣒⣀⠬⠵⠒⠉⠀⠀⠀⠀⠀⠈⠉⠓⠮⠭⣖⣒⣒⡂⠐⠲⠀⠀⠠⠀⠈⢀⢳⡀
⠀⡈⠀⠀⠁⠀⠂⠀⠀⠀⡏⠉⠉⠉⡀⡄⠀⢠⠀⢠⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢣
⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠰⠁⠄⠀⠀⠠⠠⠀⠠⠀⠠⠀⠠⠀⠀⠀⠠⠃⢠⠀⠐⢂⠀⠀⠀⠀⠀⠠⠀⠀⠀⡆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢡
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠲⡟⠉⡏⠙⠁⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⡏⢻⠒⡾⠀⠀⠀⠀⠀⠀⢠⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡇⠀⠀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠒⢈⠂⠀⠀⢠
⢁⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠰⠄⠈⡟⠀⠀⠘
⠘⡀⠀⠀⠀⢠⡞⠁⢀⠤⣄⠀⠀⣀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢀⡇⠀⠀⠇
⠀⢡⠀⠀⠀⠸⡇⢠⠀⠀⠸⡀⣄⠘⣖⠀⠀⠀⠠⠆⠀⠀⡀⠈⠀⠁⢰⠀⣉⡁⠀⠠⡄⠀⠀⠀⠉⢸⠀⠀⣼⠀⠀⡈
⠀⠀⢣⠀⠀⠀⢇⠀⠙⠒⠊⠀⠉⠁⠈⠋⠀⠓⠒⠒⠀⠀⠱⢤⣄⡀⢸⣈⣉⡀⢀⣀⠘⣆⡀⠀⠀⡼⠀⣰⠃⠀⡐
⠀⠀⠀⢣⡀⠀⠈⢆⠯⠀⠈⠉⠀⠀⠀⢲⡀⠒⠒⠲⠶⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣁⣀⣀⠀⠠⠤⠄⠚⠁⠀⠔
⠀⠀⠀⠀⠱⣄⠀⠈⠢⢤⡴⠀⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⡄⠀⠀⠀⠀⢀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢠⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⠤⢄⣀⣀⠀⠀⠀⠀⣀⣀⡀⠀⠄⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡸⣿⣿⣿⣿⣿⣿⣿⣣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠿⠿⠿⠹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣶⣦⢰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠟⠋⠁⠈⠙⠻⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⠀⣴⡇⢰⣦⠀⢘⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡄⠀⠙⠃⠸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡄⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⣿⡇⢸⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠙⠃⠘⠋⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣏⣉⣻⣿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣯⣭⣽⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⢾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⢿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⣏⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡻⣿⣷⣷⣻⣿⣷⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⣼⡟⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢰⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣾⢿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⣿⣿⣟⢩⠚⡽⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⣏⢻⣷⡒⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⣿⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣏⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⣟⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣛⣿⣿⣿⣺⣿⣿⣿⣿⡯⢝⡲⢝⢮⣾⣷⡿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡻⢟⢯⣦⡳⡽⣿⡭⣺⣝⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⣵⣿⡷⣝⡶⣝⢮⡻⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡻⠿⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢝⣮⣳⢱⣻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠄⢍⡋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⣧⡹⣿⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣏⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠈⠂⠝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠿⣽⣿⣿⡿⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠒⢦⡀⠈⠎⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⢻⣟⣟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠀⣸⠇⠀⢐⡼⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣽⡜⣾⡟
⠀⠀⠀⠀⠀⠀⢸⢸⣽⣻⣿⣿⣿⣿⣿⡿⠉⠀⣠⠼⠿⣿⠛⠛⢶⣞⣠⣟⠋⢉⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⢇⣿⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠷⠦⣼⣇⡀⢀⣸⠀⣀⡾⠿⠧⣤⠜⠃⢄⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⢸⣽⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⠟⠛⠛⠓⣆⢀⣼⠏⠉⠉⠉⣿⡯⠤⣄⣠⣿⣆⠀⠀⢣⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⡿
⠀⠀⠀⠀⠀⢰⡿⣿⣿⣿⣿⣦⣀⣠⠴⠿⡿⠛⠳⣤⡶⠶⣿⠀⠀⢸⡿⠃⠈⢳⡄⠀⠀⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠁
⠀⠀⠀⣀⣀⣠⣧⣿⣿⣋⣁⣈⣿⠁⠀⢀⣿⣀⣀⡾⠀⠀⢹⠦⠴⠋⢻⣤⣤⣾⣧⠀⠃⢿⡿⣿⣿⣿⣿⣿⣿⣿⡏⣿⠏
⠀⣠⡾⠛⠉⠉⠙⠿⠿⠋⠉⠉⠛⠿⣶⡋⠉⢻⡿⠋⠉⠉⢿⣄⣀⣴⣿⣿⣿⣿⣿⣿⠘⠈⣿⣿⣿⣿⣿⣿⣿⣿⣽⠁
⣰⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣾⣇⡀⣀⣤⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⣿⣿⣯⣤⣤⣄⣀⣀⡀
⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣭⣉⡁⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣾⣿⠿⠿⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⢻⡽⣷⠿⣿⢿⣿⣿⣶⣶⣤⣥⣤⣤⣤⣤⣤⣶⠶⠿⠛⢻⠟⠛⠋⠉⣀⣴⡾⠋
⠘⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣭⣵⡞⢧⠟⢿⣻⣿⣿⣟⣻⣿⣿⣿⣿⠟⠋⠉⢁⣤⠤⠤⠴⠋⣠⣤⣶⣿⠿⠋
⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠈⠙⢿⣿⣿⣿⣶⠾⠿⠿⠛⠛⠛⠉⠡⠤⠤⠤⠴⠞⠋⢀⣠⣴⣾⡿⠿⠛⠉
⠀⠀⠀⠉⠛⠻⠿⠛⠻⠶⠾⠛⠉⠀⠀⠀⠀⠀⠈⠙⠻⠷⢶⣾⣿⣷⣦⣤⣤⣶⣶⣶⠶⠶⠶⠿⠛⠛⠉

⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢁⣀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⢸⠀⠀⠀⢀⣠⡴⠞⠉⣤⣤⢸⣿
⠀⣿⠀⠀⠀⠀⠀⠀⣠⣀⡴⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡜⢢⠀⣀⠀⡰⣦⠃⠀⡀⠀⠀⢸⠀⠰⠛⢉⣥⠀⣿⡇⣿⣿⢸⣿
⠀⣿⠀⡠⣄⣀⣤⠞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣁⠞⢄⡠⠫⣀⠟⠜⠀⠱⠜⠁⠀⠀⢸⠀⣶⣶⢸⣿⠀⣿⡇⣿⣿⢸⣿
⠀⢿⣎⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣟⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⠀⠀⢸⣄⣸⣟⣸⣿⣀⣿⣇⣹⣟⣸⣿⣀⣠


⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡧⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢺⣅⠀⣀⢰⣶⠀⠀⠀⢸⡀⠀⠀⡀⠀⠀⢈⠀⠤⣀⣄
⠀⠀⡷⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡁⣴⡆⣿⢸⣿⠀⠀⠀⢸⠆⠐⠀⡀⡄⠁⢀⠠⠒⠀⠝⠈
⠀⠀⡟⠿⠿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣤⢸⡇⣿⡇⣿⢸⣿⠀⠀⠀⢸⠀⠫⠒⠄⠉⠅⠀⠀⠁⠀⠀⢀
⠀⠀⡟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⣿⠆⣿⢸⡇⢿⠇⣿⠸⡿⠀⠀⠀⢸
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆
⠀⣿⠀⠀⢀⡄⠀⠀⠀⢀⠔⣁⡀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣤⡄⣿⡇⣿⡇
⠀⣿⠠⠚⢡⡌⢆⡠⠖⢡⡄⢻⡇⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⡀⣚⡂⣛⡀⣛⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⣾⡆⣿⡇⣿⡇⣿⡇
⠀⣿⢰⣶⢸⡇⣠⡀⣶⢸⡇⢸⡇⣿⠀⠀⠀⢸⡇⠀⠀⢠⠄⢬⠅⢬⠄⢭⠄⢭⠀⠀⠀⠀⢸⠀⣶⡆⣿⡇⣿⡇⣿⡇⣿⡇⣿⡇
⠀⣿⣘⣿⣸⣇⣻⣃⣿⣸⣇⣸⣇⣿⣀⣀⠀⠸⣇⣸⣂⣸⣂⣺⣂⣺⣂⣺⣂⣺⣀⣀⡀⠀⢸⣀⣿⣇⣿⣃⣿⣃⣿⣃⣿⣃⣿⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⡨⢖⣨⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣊⣢⣥⣵⣶⣶⣾⣿⣛⣟⣿⣿⣶⣶⣦⣥⣏⣉⣈⣡⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣐⡀
⠀⣿⠉⠉⠀⠀⠈⠁⠀⠈⠉⠉⢉⡡⠔⣋⠭⠄⠛⡉⣉⠉⠉⢥⣩⢩⠉⣩⠭⠉⠉⢛⠪⠭⣙⠣⢌⡈⠀⠀⠉⠉⠉⠉⠉⠉⠉⠹⡇
⠆⣿⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⡡⢒⡉⠀⡞⢳⠀⢧⠼⠀⢀⠘⠃⠛⠀⠻⠅⠀⠀⢸⠋⢀⡦⡍⠒⢬⡒⢄⡀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⣿⠀⠀⠀⠀⠀⢀⠔⣫⠖⠁⠀⢯⣘⠆⠉⠉⠀⣀⠞⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠁⡜⢀⠌⠲⣍⠢⡀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⢀⡔⣡⠊⢤⡼⣦⠀⠀⠀⠀⠀⣤⣚⠁⡤⢄⣸⠷⢶⠟⢦⡷⢤⣤⣦⢤⣼⡀⠀⠀⠙⠋⡀⣏⠂⠑⣌⢢⡀⠀⠀⠀⡇
⠀⣿⠀⠀⣠⢊⠜⠁⠳⣄⠑⠀⠀⠀⠀⠀⠞⠀⢿⢷⣅⡇⠉⠃⣠⡄⠀⠀⠀⠀⠈⠙⡏⠀⠀⠀⠀⠀⠈⠁⣠⠞⠊⠣⡑⡄⠀⠀⡇
⠀⣿⠀⡰⢡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡲⣤⡈⠣⣈⠓⠉⠉⠀⣩⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢆⠐⡇
⠀⣿⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠁⡄⢄⣠⡞⢁⠷⠟⠁⠀⠀⠀⢠⠶⠆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢶⣧
⠂⣿⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣗⠒⠦⠸⢯⢴⠙⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣘⣏
⢀⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⢲⡷⠈⢣⡉⠁⠀⠀⠀⠓⢦⣤⡄⠀⢄⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⢀⡢⠀⠉⠃⠀⠀⠀⠀⠀⠉⠁⠂⢀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠖⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⣿⣿⠀⢀⠀⢀⢀⣀⢀⣠⢀⣀⣀⣀⢀⢀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠈⠉⠡⡶⣀⢧⣀⢠⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⣿⣿⠀⠸⠄⠸⠸⠸⠸⠂⠸⠺⠁⠇⠈⠏⠀⣴⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠻⣮⠗⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⢠⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⡈⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠦⠅⠚⣧⠀⠀⠀⠠⢲⣄⡔⢦⠠⡆⠐⡦⠀⠀⠀⣸⣿
⠂⣿⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⢄⠀⠀⠀⠈⠢⡀⠀⠴⠯⠘⠧⠼⠀⠇⠬⠽⠀⠀⢰⢩⡏
⠄⣿⠳⡘⡄⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠉⠉⠑⠀⠀⠈⢯⠉⠙⠛⡆⠀⠱⡀⠀⠀⠢⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⢃⠾⡏
⠀⣿⠀⠱⡘⣄⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⠀⠀⠳⡀⠀⠙⠀⠀⠹⡆⠀⠀⠐⣄⠀⠀⠛⠃⠀⠀⠀⣠⢃⠎⠀⡇
⠀⣿⠀⠀⠑⣌⢢⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡢⣻⡄⠀⠑⡄⠀⠐⠀⠀⠹⡄⠠⡀⠀⠻⣦⡀⠀⠀⢀⡴⣡⠊⠀⠨⡇
⠀⣿⠀⠀⠀⠈⠢⡑⢄⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⢆⠀⠀⠀⠀⠹⡀⠈⠐⢀⡸⠋⠀⡠⢊⠔⠁⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠈⠢⡑⢦⡀⠀⠀⠈⠻⢷⣦⣄⡀⠀⠀⠀⠈⠋⠀⠀⠀⠈⢣⡀⠀⠀⢀⣠⡴⠞⠉⢀⡤⢊⠔⠁⠀⠀⠀⠀⠀⡇
⡀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⡓⠤⣀⠀⠀⠈⠉⠛⠛⠶⠶⠤⣤⣄⣀⣠⡤⠤⠕⠒⠊⠉⠁⣀⠤⢒⡡⠚⠁⠀⠀⠀⠀⠀⠀⢀⡇
⡁⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣑⣢⣍⣒⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣒⣩⣔⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡧
⠀⣈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⢋⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠞⠻⠛⠓⢆
⠀⠀⠀⠀⠀⠀⠰⠛⠉⠓⣦⣷⠾⢣⢀⡀
⠀⠀⠀⠀⠀⣰⡖⢀⣠⣤⣾⡅⠀⠘⠋
⠀⠀⠀⠀⠀⣿⣰⠟⠁⠀⣠⡤⠶⠦⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⠇⠀⣰⠟⠁⠀⣠⠴⠛⠉⣠⡴⠞⢋⡉⠭⠤⠤⠤⠭⠭⢉⣛⠲⢦⣄
⠀⠀⠀⠀⠀⢹⠀⣼⠃⠀⠀⡴⠁⠀⢠⠞⡡⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⡓⢄
⠀⠀⠀⠀⠀⢸⣰⠇⠀⠀⡘⠀⠀⣰⢃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠄⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠀⢰⠁⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠉⠀⠀⠀⠀⠈⠀⣀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠂⠀⣀⠘⠀⠀⠀⠀⠶⣷⠀⣉⡀⢰⡏⠠⠤⠖⠢
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠊⠝⠠⡄⠀⠀⠒⠒⠢⠀⠀⠉⠀⢀⢀⡴⠚⠋⣷⠟⠉⢉⡥⢧⡴⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢧⠄⠀⠀⠀⠀⠓⣂⠀⠀⢀⣀⠀⠀⢀⡀⠙⠀⠀⣰⠃⡾⠁⠀⣼⠁⠀⢠⡏⠀⠀⢿⠀⢲⡈⠂
⠀⠀⠀⠀⠀⢲⣤⢈⣠⣀⡀⠛⠀⠉⠀⠙⠋⠀⠉⠉⠉⠙⠭⡀⠀⢰⡇⠀⡇⠀⠀⡇⠀⠀⣾⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠑⣘⢏⡖⠛⢑⠀⠠⡄⠐⢦⠀⠀⠀⢀⠌⠴⠩⠛⢺⡇⠀⡇⠀⠀⡇⠀⠀⡟⠀⠀⠀⢸⠁⠀⡇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⠊⠉⠉⠀⢀⠀⠁⠀⠀⠀⠐⢚⠷⠿⠷⠫⢔⡡⢊⠹⣇⠀⠹⡀⠀⠇⠀⠀⡇⠀⠀⠀⡿⠀⢸⠇⠀⡎⢀⣀⣑
⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠆⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣤⣅⣀⢻⡄⠀⢣⠀⠀⠀⠀⣷⠀⠀⢰⠇⢀⡾⠀⣰⣃⠉⠁⠉⠂
⠀⠀⠀⠀⠀⢄⣳⡈⠀⠀⠀⠸⠻⠖⠋⠀⠈⠁⠀⠀⠐⠉⠀⠀⠀⠑⠈⣿⣄⠀⠁⠀⠀⠀⢸⡀⢀⡟⢀⣾⣅⣴⠃⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⢤⣤⣀⠀⠀⠀⢀⡜⠛⠳⠲⢀⡀⠀⠀⠁⢠⣴⣿⠁⠀⠈⠓⠀⠐⠂
⠀⠀⠀⠀⠀⠀⣤⣤⣄⣷⠤⢄⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠑⠂⠀⠠⢀⣠⠛⠉⠒⢀⠤⠀⠀⠀⠀⠀⣄⠀⠀⢀⣠⣤⡀⢀⣡⠤⠂
⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⠂⢊⣩⠖⣢⠟⠒⠒⠶⣤⠀⠀⠀⢀⠌⠑⠒⢢⡈⠙⢋⡀⠖⠀⣔⢒⣖⣉⣨⡉⠉⠉⠀⠀⠁⠀⠈⠙⠄
⠀⠀⠀⠀⠀⢴⣿⣿⣟⣩⣵⣮⠔⠈⠀⠀⠀⠀⠀⠀⠁⢀⠔⠁⢈⠉⠒⢢⡌⠓⠒⠤⠀⠀⠸⡿⠉⣿⣅⢰⠄⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠐⠿⠟⠛⠻⠿⠋⠁⠀⠀⢀⠀⠀⠀⢀⡠⠂⠁⠀⠀⢈⣉⠭⡑⠶⠏⠉⠀⠀⠀⠀⠨⣏⠁⠈⡅⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠒⢉⡕⠒⠀⠀⠀⠀⠀⠀⠑⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⢌⡁

⣶⣶⣷⢶⣶⠶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣷⣷⣾⣶⣾⣶⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠲⠤⠆⠤⠀⠀⠀⠀⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠀⠀⢠⠶⠤⠞⢈⠓⡆⢀⣤⠀⠇⠔⠒⠂⠘⡠⢃⡀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠐⡄⠀⠀⠀⢸⡠⠶⠬⣇⢀⣶⣿⣀⡖⠞⢣⣍⢫⣈⠕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣶⡿⠶⠿⠷⠾⠷⠶⠶⠷⢎⣠⠤⠖⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣶⣶⣶⣴⣤⣤⣤⣤⡤⢴⣶⣶⣶⡶⠶⠚⠛⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠉⠏⢿⡇⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢷⣴⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡶⠶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠄⠀⠀⠤⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠂⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⡠⢊⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⡁⠈⡐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠤⢔⣂⠥⠒⠉⠀⠀⠈⠓⠬⣕⡂⠤⢄⣀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠆⠱⡀
⠀⠀⠎⠀⠀⠁⠀⠀⠀⠀⠀⡿⠦⠤⠤⠴⠖⠒⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠶⠉⠉⠉⠈⠀⠀⠀⠁⠀⠀⠠⠱⡀
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠆⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢳
⠰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣧⠈⠀⠐⠁⠀⠄⠀⠀⠂⠃⠀⠀⠀⠀⠀⠐⠃⠀⠁⠀⠂⠂⠠⠁⠠⠱⠀⠄⠀⠀⠀⠀⠂⠀⠀⠈⡆
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⠤⠤⠤⠤⠤⠤⣤⠤⢤⣄⣀⣀⣀⣀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠺⡏⠉⡏⠉⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠏⠻⠛⡶⠁⠀⠀⠀⠀⠀⠀⡄⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠂⡒⠀⠀⠀⢰
⢡⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠰⣀⠀⢻⡇⠀⠀⠸
⠸⠀⠀⠀⠀⠀⣠⠞⠉⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠁⠀⢀⠃
⠀⢣⠀⠀⠀⠀⢟⠀⡀⠀⠀⢳⠠⠀⠹⣆⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠒⠂⠠⠀⠀⠤⠀⠀⢀⠀⠀⠀⠀⠒⠀⡎⠁⠀⡟⠀⠀⡈
⠀⠈⣆⠀⠀⠀⢸⠀⠑⠤⠀⠵⠃⠻⠆⠙⠦⠀⠠⣄⣀⡀⠀⠀⢆⠀⠀⠀⠀⡄⠒⠒⠀⠀⠈⢣⡀⢰⠀⠀⢀⡇⠀⣼⠁⠀⡰⠁
⠀⠀⠈⢆⠀⠀⠀⢇⢠⡔⠂⠀⠤⠤⠀⠀⣀⢀⣀⣀⣀⣀⡀⠀⠀⠉⠛⠃⠀⠓⠒⠒⠀⠐⠂⠀⠓⠚⠀⠀⠛⢁⡴⠃⠀⡰⠁
⠀⠀⠀⠈⢢⡀⠀⠈⢆⠑⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠉⠉⠉⠛⠛⠓⠒⠒⠲⠶⠶⡶⠒⠒⠒⠒⠂⠀⠉⠉⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠑⣄⠀⠀⠙⠒⠊⠀⠀⠀⠀⠀⠀⠳⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡀⠀⠀⠀⠀⢀⣆⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢐⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⠀⣀⣀⠀⠀⠀⠀⢀⣀⠀⠤⠠⠐⠂⠁
⡶⣶⢶⠶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⣶⠶⣶⣶⣶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⢀⠀⠀⠄⠀⣀⠘⣿⣿⣿⠦
⣀⣀⣀⣤⣤⣴⣐⠢⢀⡀⠀⠀⢠⣄⣽⢃⣀⠀⠀⣠⣄⠀⢀⣈⣙⣿⣾⣿⣷⡎⢿⣲⣆⣻⣦⣲⣄⣤⣤⣶⣶⣦⣶⣦⣴⣴⣶⣾⣿
⣷⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⢻⣾⣿⡾⣿⣿⣿⣿⣷⣟⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣯⡿⢿⣿⣿⡿⢟⣻⣿⣯⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣋⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠆⠐⡐⠖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠣⢄⢣⡰⠂⣙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠣⣠⠳⠀⢠⡸⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢄⡀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠁⠢⡛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠈⠢⣄⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠑⡀⠀⠑⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠈⢢⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢃⠀⠡⡀⠀⠱⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣺⣯⠭⠭⣑⠢⣆⠀⠑⠄⠀⠐⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣡⣴⣶⣿⣿⣿⣶⣤⡉⠢⣖⡊⠁⠑⢄⠁⠀⠀⡄⠀⢧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⡥⠘⢻⢿⣾⢿⣿⣿⡿⠿⠛⢄⠑⣀⠔⣊⠉⢢⠀⠀⠘⡰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⠇⠮⢇⢈⣀⠓⠈⣿⠐⢿⣿⠀⠀⢸⡄⢹⠛⣁⡤⠾⡇⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣀⡾⣼⡴⠈⢿⡇⣤⠀⠀⠀⠈⡿⡗⢰⣿⡇⠀⣜⡥⠔⢉⠁⠀⢰⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠢⡍⠀⠹⠀⣺⣑⣄⠀⣰⢆⡺⢴⠍⢂⣦⡧⠀⠈⣤⠖⢛⢁⣁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⢈⠋⠀⠀⠸⣜⣃⣿⠻⡴⣽⢯⠟⠞⢧⣴⣼⡏⠰⣘⣤⣢⡬⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠼⠃⣂⣾⡀⡀⢀⢊⠄⡡⢈⡏⡟⣯⣿⢬⠠⣼⡣⠸⠃⠀⡩⠚⠁
⠀⠀⠀⠀⠀⢐⠂⠂⠀⠈⣿⣻⣿⣾⡘⢠⠁⢰⠁⡇⢸⢿⠿⢽⣛⣄⢀⢶⠈
⠀⠀⠤⣶⡝⡌⣇⣰⡀⣸⣚⣚⣻⣯⣳⣼⣄⠎⢠⢁⣾⣳⠀⠸⢹⢏⠝⠀⠁
⢀⣀⠀⢀⠀⠊⠘⢹⣭⢳⣻⡊⠈⢉⣹⡼⡏⣞⣯⣿⣿⣿⡄⠈⠰⠊
⢨⠀⠁⠂⠁⠀⢘⣾⢈⡈⢙⣿⢤⣀⢀⡞⠸⡻⢹⣭⠩⠄⠳⡄⠀⠄
⠈⠑⠒⠊⠁⠐⠘⠺⠛⠁⠘⣹⡿⠁⠀⠁⠀⠀⠈⠀⠀⠀⠀⠈⠓⠻⠂

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠖⠒⠶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠴⠦⠤⢤⣀⡀
⠀⠀⠀⢀⡴⠚⠁⠐⢈⣠⣤⣬⣁⡁⠀⠀⢍⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⠱⡰⠦⠇⠇⠹⠀⠀⠄⢉⡳⢤⡀
⠀⠀⡴⠋⠄⠀⢀⣶⣿⣿⣿⣿⠟⠻⠆⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢠⠞⠡⣢⠀⠁⠀⠠⠀⠤⠄⠄⠤⠀⠙⠳⣦⠙⢦
⠀⡼⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⠆⢀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢠⢏⠮⠓⠀⣀⣀⣀⣀⣀⣁⣈⣁⣀⣀⣀⠀⠘⢩⡌⢧
⣸⠁⠀⠀⠀⠀⠘⣿⣿⣍⣻⣿⠀⠉⡀⠄⠀⠀⠀⠀⠀⠸⡄⠀⠀⢀⡏⣨⠕⠠⣄⣍⣡⣩⣩⣍⣍⣍⣭⣩⣩⣩⣄⡄⠐⡺⠌
⡿⠀⠀⠀⠀⠀⠀⠘⣿⠇⠙⣿⣄⣀⣆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⠐⠀⠀⡇⡏⢩⢨⠈⠁⠁⡁⠉⢸⠸⠀⠁⡇⠀⠱⠆
⣿⠀⠀⠄⠠⠀⠀⣠⣿⣿⣦⡸⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⠀⢀⣅⣄⣨⣼⣤⣤⣆⣧⣤⣼⣼⣄⣄⣇
⠸⡄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣾⠓⠀⠀⠀⠐⠀⠀⠂⣸⠃⠀⠀⠈⣇⠀⠴⠤⠴⠿⠵⠾⠶⠶⠶⠶⠶⠶⠷⠿⠿⠤⠴⠆⢀
⠀⠳⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧⡈⢢⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠘⣆⠀⡴⢓⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡀⠀⢀⡞
⠀⠀⠙⢦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠄⢀⡴⠁⠀⠀⠀⠀⠀⠀⠘⢦⡙⢋⠞⣧⢆⡤⠄⢀⠤⢠⠄⢲⠼⣆⠈⣠⠞
⠀⠀⠀⠀⠙⠦⣍⡛⠿⠿⢿⣿⡿⠿⠟⣃⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡁⠸⠍⠀⠸⣀⠘⠧⠀⣁⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠖⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠢⣄⠀⠀⠀⢀⠤⠒⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠑⠤⠴⠁⠀⠀⠀⠀⠀⠣⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢌⣣⡀⠀⢄⠀⠀⠀⠀⡠⠃⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠷⠤⠄⠓⠾⠥⠾⣗⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣀⡠⠤⣤⡤⢀⣀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠃⢇⠀⠀⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢠⡞⠁⢸⠁⡏⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⢧⡀⢸⢠⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢾⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⠃⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠰⣇⣰⡇⣾⣀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢹⠻⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠴⠢⢄⣀⣀⣀⣀⣀⠤⢆⣀⣀⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⣄⢸⡇⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⣠⡀⣠⡾⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠰⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀
⠀⠀⠀⠀⣶⢸⣿⢸⡇⣿⢸⣿⠀⠀⠀⠀⢀⣴⡾⠋⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠑⠖
⠀⠀⠀⠀⠛⠈⠋⠘⠃⠛⠈⢋⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀


⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣾⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠁⠀⠀⠀⠀⠀⢀⡄⡆⣶⢠⢸⣰⡇
⠀⠀⠀⠀⣴⣾⣶⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠈⠁⠁⠁⠉⠈⠁⡀


⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠉⠀⠀⠀⣀⡀⣄⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡏
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣭⣭⡭⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠅⠤⡇⡆⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⢀⣀⡀⠀⣁⡀⡀⡃⡀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠂⠀⠃⠃⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⢠


⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⣀⡀⠀⠛⠛⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⡷⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠛⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡄⠸⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⠁⢸⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣤⠿⠉⠲⠶⠂⠀⠀⠀⡇⠘⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠘⠀⠉⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⡿⠛⡿⢿⡿⢻⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢰⡏⠉⠉⠉⠉⠉⢹⠀⠀⢀⡴⠋⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢯⢉⡁⠀⠀⠘⢆⠀⣧⠀⠀⠀⠀⡐⠺⣤⡴⠋⠀⠀⠀⠀⢀⠝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⢆⠁⠀⣀⣀⡈⢢⢸⣀⣀⠀⠰⠂⠀⠀⢙⠲⣕⠢⣀⡠⠀⢠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡀⠈⣧⠜⠑⠂⠈⠀⢻⠿⣧⡤⣀⠀⠀⠐⠪⡱⢙⠆⢹⣀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⣷⡔⠀⣠⡤⠶⠾⢻⣅⠀⠀⠉⠲⢵⡄⡠⡠⣡⢿⣠⣾⣿⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠻⣧⣴⠁⣴⣄⠔⠁⠘⢯⣐⣄⡀⠀⠙⢷⡵⠟⢋⣥⠶⠋⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠒⢤⣙⠲⢿⣥⣤⣤⣀⣀⣀⣠⠼⠧⠤⠼⠗⠛⠋⢁⣠⠶⠋⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠁⠀⠠⠤⠄⠀⠠⠄⠀⠀⠤⠶⠖⠒⠚⠋⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⢤⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠓⠦⣤⣀⡀⠀⠀⠀⢀⣠⡤⠾⠗⠒⠛⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⠘⠂⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠔⢃⡀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠁⠀⠀⠉⠙⠒⠦⠤⠤⢀⣀⣀⣀⣀⠤⠤⠖⠋⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⢶⣤⠀⠀⣤⡶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠷⠶⠶⠶⣶⣶⣶⡶⠶⠶⠾⠛⣿⠀⠀⣿⠛⠷⠶⠶⢶⣶⣶⣶⠶⠶⠶⠾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠙⠛⠷⢶⣤⣄⣀⣠⣴⣿⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢀⡀⠀⠀⠀⠀⠀⠉⠉⠉⢸⡟⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢁⣠⣄⡀⠀⣀⠀⠀⠀⢸⡇⠀⠀⢸⣧⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣧⣼⠇⢀⡉⠀⠀⠀⢸⡇⠀⠀⢸⡏⠀⠀⠀⠀⠉⠛⠷⠶⣦⣤⡶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣀⠀⢰⣟⣻⡆⠀⠀⣿⠁⠀⠀⠈⣿⠀⠘⠃⢠⡶⣦⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠉⠀⠀⠉⠉⠀⠀⣠⣿⠀⠀⠀⠀⣿⣄⠀⠀⠹⠷⠿⠁⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠶⢶⣤⣤⣤⣴⠶⠟⢻⡇⠀⠀⠀⠀⢹⡟⠻⠶⢦⣤⣤⣤⣴⠶⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣄⣀⣀⣀⣠⣤⡶⠟⠁⠀⠀⠀⠀⠈⠻⢶⣤⣄⣀⣀⣀⣠⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉






⣶⣶⢶⠶⠶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⡶⠶⢶⡶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠡⠶⣀⣠⣤⣴⠶⠶⠶⢶⣦⣤⣌⣛⠷⣮⡙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣡⠀⣠⣶⠟⠋⠉⠀⠀⢀⣀⣀⠀⠀⠀⠉⠙⠻⣶⣄⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣱⠟⣱⡾⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⠀⣈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣼⢏⣼⢣⣾⠋⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠙⣷⡌⠁⠹⣧
⠀⠀⠀⠀⠀⠀⣸⡏⣼⢣⡿⠁⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⠀⠈⠙⠻⣿⣷⡄⠀⠀⠀⠀⠈⢿⡄⠀⢹⣇
⠀⠀⠀⠀⠀⢠⣿⢰⡇⣾⠃⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠀⠀⠀⠀⠀⠘⣷⠀⠀⣿⡄
⠀⠀⠀⠀⠀⢸⡇⣾⢰⡟⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣄⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠛⢸⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⢸⣧⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⡿⢸⡇
⠀⠀⠀⠀⠀⠘⣿⠀⠀⢿⡄⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠂⠀⠀⠀⠀⢠⡿⢸⠇⣿⠃
⠀⠀⠀⠀⠀⠀⢹⣇⢠⡘⣷⡀⠀⠀⠀⠀⠙⣿⣿⣦⣀⠀⠀⠀⠀⣀⣴⣿⡿⠃⠀⠀⠀⠀⢀⣾⢣⡟⣸⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣎⢡⡜⢿⣄⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⣠⡿⣣⡟⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡹⣦⡻⢷⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⢏⣴⢋⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡻⢦⣙⠿⣦⣄⣀⠀⠀⠈⠉⠉⠀⠀⠀⣀⣠⣴⠟⣫⡴⢛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣍⡻⠶⠉⠙⠛⠿⠶⠶⠶⠾⠟⡛⠋⠉⠶⢛⣩⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣀⣀⣀⣀⣙⣋⣭⣤⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣿⠀⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢦⠀⡿
⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢻⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⣤⠤⢤⣸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⡏⠉⢹⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⣶⠒⢲⣾⡇⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⢸⠉⠉⡇⡇⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⣿⠀⢸⣿⡇⠀⢸⠀⠀⠀⠀⡇⠀⢸⢰⠒⠒⣾⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⢸⠀⠀⡇⡇⠀⢸⣿⡇⠀⣿⢸⠉⠉⡇⣿⠀⢸⣿⡇⠀⢸⢠⠤⠤⡄⡇⠀⢸⢸⠀⠀⣿⡇⠀⢸⣤⠤⢤⡄⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⢸⠀⠀⡇⡇⠀⢸⣿⡇⠀⣿⢸⠀⠀⡇⣿⠀⢸⣿⡇⠀⣿⢸⠀⠀⡇⡇⠀⢸⢸⠀⠀⣿⡇⠀⢸⣿⠀⠀⡇⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⢸⠀⠀⡇⡇⠀⢸⣿⡇⠀⣿⢸⠀⠀⡇⣿⠀⢸⣿⡇⠀⣿⢸⠀⠀⡇⡇⠀⢸⢸⠀⠀⣿⡇⠀⢸⣿⠀⠀⡇⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⢸⠀⠀⡇⣇⠀⢸⣿⡇⠀⣿⢸⡀⠀⡇⣿⠀⢸⣿⡇⠀⣻⢸⠀⠀⡇⡇⠀⢸⢸⠀⠀⣿⡇⠀⢸⣿⠀⢀⡇⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⢾⣿⣿⣷⣿⣿⣿⣾⣿⣿⣷⣾⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣷⡿⣿⣿⣾⣿⣿⣷⣿⣿⢿⣿⣿⣿⡷⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⣿⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⣿
⠀⠀⠀⣿⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣿
⠀⠀⠀⠻⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡟
⠀⢷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⢀⣾⡧
⠀⠈⢣⡈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠉⠀⣠⠴⠋⣼⣥
⠀⠀⠻⣝⠲⣄⣀⠀⠉⠉⠓⠒⠒⠦⠤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣶⠶⠶⢲⡆⣀⠴⠚⠉⢀⣀⡤⠴⣒⣋⠁⣠⠞⣫⠇
⠀⠀⠰⣾⢷⣄⠉⢉⣓⣒⡲⠶⠤⠤⣤⣀⠈⠑⠦⡀⠀⣴⣾⠟⢋⣵⣿⣷⣿⠿⠟⣿⣾⠞⠁⣠⠴⠚⠉⠀⠀⣀⠴⠊⠀⢁⣴⡁
⠀⠀⠀⠈⠳⣌⠙⠀⠑⠢⣤⣄⠀⠀⠀⠀⠉⠲⢄⠈⢮⡻⢤⡀⢜⢯⡉⠀⠀⠀⠀⣿⠋⣠⠞⠁⠀⠀⣀⣀⡩⠵⠚⠠⠖⣫⠟⠁
⠀⠀⠀⠀⠘⠿⣟⠶⠤⠉⠒⠒⡶⢖⣂⣀⡀⠀⢀⡑⠄⠹⡄⠉⠲⣵⣿⣷⡀⡆⠘⣿⣰⠁⠀⠀⣤⠔⠒⠒⢉⣠⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⣦⣤⠄⠈⠀⠀⠉⠲⢯⣽⣶⣦⣽⣄⣀⣰⣿⣿⣿⠿⣶⣟⣿⡀⠈⠉⢀⣀⣤⣤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⠿⠶⠤⠤⢤⣌⣩⣽⠿⢿⠿⣿⣿⣿⠛⣟⢿⡈⠻⣶⠖⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⣥⣤⣴⣶⣿⠟⢱⠃⠀⣿⢆⠳⡄⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠙⠺⠛⠉⠀⠀⠀⠀⠀⠈⠀⠀⠙⠄⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣅⣀⡀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⡻⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠶⠿⠟⠛⠋⠉