Vaccine Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣯⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠈⢻⣿⣿⣿⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣆⢀
⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⣿⡇⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠈⠁⠀⠉⠘⠀⣿⡇⠀⣿⡇⠁⠉⠀⠀⠁⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿⣻⣽⣆
⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣿⡇⠀⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡶⠖
⠀⠀⠀⠀⠀⠀⠈⠻⢿⠿⡿⠿⠿⠿⡿⠟⠁⠀⠙⠿⠿⠿⠿⢿⠿⢿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢡⣤⣤⠈⠻⣦⡠⠀⠀⢰⣀⣀⣐⡂⠀⠀⣁⣰⠗⢀⡢
⠀⠀⠀⢀⠔⣀⠘⠒⢶⣦⡀⠈⣿⢿⠏⠀⠀⠀⠀⠘⠿⡿⠟⠻⠀⣸⡟⢁⣼⡿⠛⣉⣩⡃
⠀⠀⠀⢁⣴⣿⣷⣶⣿⣿⣿⡟⠡⢳⡀⠀⠀⠀⠀⢰⡞⠛⡾⣄⣴⢛⠀⣸⣿⢁⣼⣿⣿⣧⠂
⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⡛⠀⡬⠋⠁⣸⠂⠀⠀⠀⠀⠀⠺⠀⠀⠘⢺⣿⣿⣿⣿⣿⣿⣾⠀⡀
⠀⠀⠀⠈⠙⣛⣿⣿⣿⣿⣿⣾⣧⣴⡟⠀⠀⠀⠀⢰⣶⣶⣾⣠⣤⣄⣼⣿⣿⣿⣿⣿⣿⣿⡇⢡
⠀⠀⠐⢰⣿⣿⣿⣿⠏⣾⣿⣿⡿⡩⠀⠀⠀⠀⠀⠙⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⡄
⠀⠀⠀⣀⣻⣿⣿⡸⡄⢿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣏⡀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢇⠀⠀⠀⠀⠀⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠓⠁⠀⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠈⠒⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡄⠀⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⡀⣀⡀⠠⡇⠀⢀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣁⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣁⣠⠀⠀⠠⠆⠒⠈⠁⠁⠈⢈⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠒⠒⠚⠉⠁⠀⠉⠙⠄⠤⠠⠤⠠⠦⠤⠴⠦⠤⠖⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠁⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣽⣯⣽⣿⣭⣿⣯⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣭⣭⣭⣭⣭⣭⠉⠁⠀⠀⠀⠀⣨⣛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣤⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿⣟⣛⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⡀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣯⣭⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠿⠿⠿⠿⠿⠿⠿⠷⣿⡇⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉⠉⣭⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡄⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠀⠶⠾⠿⠷⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠍⢉⣉⡸⠟⢛⠖⠀⣈⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⢠⠠⡒⠐⣐⡇⢀⣀⡀⠀⠤⠤⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠱⣤⡀⢀⢐⣛⣏⣉⣈⡭⣼⣋⠁⠀⠀⠰⠿⠿⠷⠖⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⡰⠊⠉⠀⢠⣤⡤⠶⠶⠤⣄⠀⠈⠙⢪⠬⠬⠭⠝⢻⠀⠉⠁⠐⠒⠒⠒⡒⠒⢂⠪⠡⣠
⠀⠀⠀⠀⠀⠀⡟⠒⠤⠤⠤⠤⠬⠉⡉⠉⠡⠤⠤⠐⡊⣂⢀⣐⣉⡺⡐⠀⠀⠀⠀⠀⡨⡢⠋⢕⢕⠝⡼
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡠⢂⢄⢔⢔⠕⠕⠕⡍⣀⠒⢸⠀⠙⣖⡦⣤⣤⣴⣩⣬⡥⠷⣆⡳⡀
⠀⠀⠀⠀⠀⠀⠀⢅⡀⠀⠀⠀⠠⣊⢔⠕⡠⡡⢊⠪⢦⠗⣾⣉⠀⢀⠞⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⡉⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡽⠷⢖⢦⠤⠥⢬⠬⣤⢕⣶⣝⢷⠒⠒⠒⠚⢸⠀⠀⠀⠀⠀⠀⠀⢀⠄⡠⡢⢸⠀⢽
⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠤⢄⣀⣀⣀⣀⣀⠀⠠⠌⠛⢳⣒⡂⠀⢸⠀⠀⣤⣀⠀⠀⠀⠡⠊⢊⣐⣊⠥⠞
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢄⠔⡠⠢⣪⠔⠃⢸⠤⠁⡆⢸⠀⠀⡧⡄⠉⣉⠉⣩⡉⡩⠁⣤⡀⡇
⠀⠀⠀⠀⠀⠀⡟⠒⠤⣀⣀⠀⠀⠀⠐⠠⠂⠊⠊⢉⣋⢸⠈⣀⠀⣺⠀⠀⡇⢠⡆⠯⡀⡃⠀⡇⠀⢯⠱⡇
⠀⠀⠀⠀⠀⠀⣧⡄⡆⢀⢤⠈⡉⠉⠁⠀⠀⠉⣩⠥⠐⢸⠀⠰⡎⣿⠀⠀⡯⢛⡬⢄⣉⡓⢂⣉⡠⠤⠒⡇
⠀⠀⠀⠀⠀⠀⡟⠫⠻⢸⠀⢸⠉⠁⡇⠸⡄⡇⠷⠀⠐⢸⠀⠈⡄⢻⠀⠀⡏⢱⡯⠧⢴⠒⡒⠀⠸⠭⠀⡇
⠀⠀⠀⠀⠀⠀⣩⣑⠢⢌⣋⣈⠲⠂⠃⠀⠈⠇⣓⡢⠄⢘⢀⣠⣦⢸⠀⠀⢃⣁⡪⠥⡐⢒⠒⠀⢐⠒⠀⣷
⠀⠀⠀⠀⠀⠀⡻⢝⠻⠖⠂⢨⣬⠭⠍⢹⠍⠭⡕⠂⠀⢸⣤⣥⢤⣦⠀⠀⠀⠑⠭⣒⣀⣀⢀⣀⡠⠴⠊⣭⣉⣁⣀
⠀⠀⠀⠀⠀⠀⣹⢶⣭⣑⠀⠐⠊⠉⠀⠸⠀⢹⢏⠀⠐⢸⠂⠀⠀⢨⠀⢀⠀⠀⠀⣠⣬⣬⡽⡢⠢⢀⠄⡟⠀⠀⠀⣐⠀⠒⣒⣒⣂
⠀⠀⠀⠀⠀⠀⡍⠒⢍⣂⠀⠚⠉⠈⠀⢘⠀⠨⠈⢁⠄⢸⠄⠀⠀⠀⠳⣌⠳⣤⣈⠛⣴⢦⣀⣻⣿⠻⣰⣓⣒⣒⣒⡒⠒⢒⠒
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉⠙⠒⠒⠠⠴⠄⠠⠂⣁⡄⢸⠤⠤⠤⠀⠀⠈⠙⠢⠭⣍⣐⣑⣉⣭⣴⣚⣓⣒⣒⡒⠀⠀⠤⠈
⠀⠀⠀⠀⠀⠀⠸⣜⢆⡀⠪⢐⢥⠪⢒⢅⣀⣉⣠⡷⢥⠮⠭⠭⠥⠄⠄⠀⡀⠀⠀⠈⠉⠉⠁⠀⠁⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠲⠤⣤⢆⣑⣨⡧⡚⠗⣡⠎⠈⠉⠉⠭⠅⠀⠐⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⢮⣀⡈⣤⡔⠚⣒⠒⠂⠈⠥⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠒⠂⠨⠍⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⢋⣿⣿⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⡴⢋⣴⠟⣡⡾⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠻⣿⣷⣟⣡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣀⡴⠋⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣤⣤⣤⣤⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⡴⠋⢀⣴⠟⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣌⠳⣴⠟⣡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⢷⡿⢷⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣉⡉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢨⣭⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠶⠆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡟⢹⡇⠀⠀⠀⠀⠀⠀⣿⢘⣛⣃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⢨⣭⡉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣤⣤⣤⣤⣾⠃⠀⠀⠀⠀⣾⠀⠀⢹⡆⠀⠀⠀⠀⠀⣿⠰⠶⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢿⡀⠀⣸⠇⠀⠀⠀⠀⠀⣿⠐⠒⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀⣀⣀⡀⢸⡇⠀⠀⣴⠟⠀⠀⠙⢷⡀⠀⠀⠀⠀⣿⠈⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⡿⣰⠟⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢽⠉⠉⠉⠉⢹⡇⢸⡇⠀⠀⣿⠀⠀⣶⠶⢾⡇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⣿⡀⡀
⠀⠀⠀⠀⠀⢀⣤⡀⠈⠁⠀⠻⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣻⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⠀⠀⣷⠀⢸⡇⠀⠀⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿
⠀⠀⠀⠀⠀⣾⠁⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠻⠦⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⠀⠀⡟⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣄⣀⣀⣀⣸⡇⢸⡇⠀⠀⣿⠀⠀⠷⠶⢾⡇⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣉⣉⣩⣍⣉⣤⡿⠀⠀⠀⠻⣤⣀⣀⣠⡾⠃⠀⠀⠀⠀⣿⣛⣛⣛⣛⣛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⡀⠀⠀⠀⠀⠀⣀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠁⠈⠀⠰⠀⢅⡠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢃⣤⢁⢀⠀⠀⠀⠀⠀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠘⠓⠉⠁⡈⢀⠃⣀⠶⠚⠃⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⡀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡈⠕⠊
⠀⠀⠀⠀⠐⠀⠀⡀⠄⠢⠌⠉⠀⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⣠⠄⠒⠁
⠀⠀⠀⠀⠄⡰⠊⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠢⡔⠀⠠
⠀⠀⠀⠀⠌⠀⠠⡀⠀⠀⠀⣄⣀⣀⡜⣂⠀⠀⠀⠂⠐⠄⠀⠀⡀⠀⠀⠠
⠀⠀⠀⡌⠀⡄⠀⠀⠀⠀⠀⢻⠀⣮⢧⣉⡠⠀⠀⠂⠀⠘⡬⡄⠠⠀⡀
⠀⠀⢀⣲⠀⠓⠒⠈⠀⠀⠀⠈⠳⠤⢎⡤⢀⣀⣀⠐⠀⠁⠃
⠀⠀⠀⡄⠀⠀⡀⡀⠀⢀⠀⠀⡠⠂⠚⢀⡄⠂⠉⠀⠠⠘⡇⠀⢀
⠀⠀⠀⢱⡀⠀⠐⠄⡀⢀⠀⠀⠗⢠⡜⠁⢀⠤⠒⠀⠀⡰⠁⠀⠈
⠀⠀⠀⠀⠑⡹⠂⠀⠀⠈⠀⠀⠀⠃⡠⠊⣡⠄⠀⠀⡰⠁⠀⠀⠁
⠀⠀⠀⢀⡀⠉⠢⣀⠀⠀⠀⠘⡥⠊⠀⠈⠀⢀⡠⠊⠁⠀⠆
⠀⠀⠀⠀⠈⠒⠀⠈⠉⠓⠤⠤⠤⠤⠤⠴⠊⠁⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡀⠀⠀⢀⡀
⣿⣿⡿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠘⠿⣇⢀⣸⡿⠥⠤⢼⣅⠀⣠⣄⠀⠀⠀⠀⠀⢂⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⡀⠀⢀⡴⣫⣾⣿⠟⠁⠈⠁
⠀⠀⠀⠀⣤⣄⡠⢟⣩⣴⣶⠟⠻⣿⣾⣿⣿⣍⠀⣠⣶⡄⠀⠐⠻⣿⣽⡟⠉⠀⠀⠀⠀⢀⣼⠿⣦⡙⢾⣿⠟⠁
⠀⢀⡀⠀⢙⡟⣱⣿⡟⢻⣿⣦⣴⣿⠛⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣴⠟⡁⠀⠈⠻⣦⡁
⠀⠿⠿⠶⡾⣰⠟⢻⣿⡿⠿⣿⣿⣿⣷⡟⠉⠉⣿⣧⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣁⠀⠙⠦⠀⣠⠞⠛⠂
⠀⠀⢀⣀⡇⣿⣶⣾⣿⣧⣠⣿⣯⣠⣿⣿⣶⣾⣿⡯⠉⠀⠀⠀⠀⠀⠀⢀⣴⠟⣁⠀⠙⠷⠀⣠⠞⠁
⠀⠀⠘⠟⢷⣿⣿⣅⣽⣟⢛⣿⣿⣿⡟⢻⣿⣏⣹⡿⠶⣾⡦⠀⠀⢀⣴⠟⣥⡀⠈⠳⠀⣠⠞⠁
⠀⠀⢀⣤⣼⢿⣿⣿⡟⢻⣿⣿⠁⠈⣻⣿⣿⣿⣿⣄⠀⠈⠀⢀⣴⠟⢡⡀⠈⠛⠀⣠⠞⠁⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠙⠟⠁⠀⣹⣿⣿⣿⣿⣿⣷⣶⣿⣿⡿⠃⠙⠋⠀⢀⣴⠟⣁⠶⣾⣯⣶⣶⠞⠁⠀⠀⢸⡇⣾⠀⠀⣾⣾⣾⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠙⠁⠀⣹⡟⠛⢻⣿⠉⠀⣿⣦⡀⢀⣴⠟⣁⠺⣿⣷⣽⣿⠟⠁⠀⠀⠀⠀⠘⠷⢿⡦⠴⠿⠿⠿⢿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠈⠉⠀⠀⠈⢉⣴⠟⣡⡘⢿⣷⣾⣿⠟⠁⡴⢦⡀⠀⠀⠀⠀⣀⡼⠀⠰⠶⠶⠦⠸⣄⡀
⠀⠀⣀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣹⣿⣿⣿⣾⣿⠟⠁⠀⡼⣰⠀⢧⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⢠⣙⣦⡗⣾⣶⣂⣤⠀⠀⠀⠀⠀⠀⠀⣏⣾⣿⣿⣿⣿⠟⠁⠀⠀⢰⠃⠏⠀⠘⡇⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠒⢾⣟⣿⢻⣿⣏⣻⠦⠀⠀⠀⠀⠀⡠⣢⣮⡻⣿⠿⠟⠁⠀⠀⠀⠀⠘⣔⠀⠀⣠⠇⠀⣿⢈⡁⠀⣉⣉⣉⣉⣉⣉⣉⣉⢸⡇
⢈⣻⣿⣿⣿⠻⣿⣿⠓⠂⠀⠀⢀⣪⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡁⠀⢧⡀⠀⣿⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠈⠁⠚⢻⠟⡿⢻⡌⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠘⣁⣀⣀⠹⡄⣿⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⣿⣿⣿⡇⡇⣿⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⡇⡇⣿⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡇⣿⣿⣿⡇⡇⣿⢸⣇⢠⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⣿⣿⣿⡇⡇⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣛⣛⣛⣛⡡⠃⠘⢦⣍⣉⣉⣉⣉⣉⣉⣉⣉⣭⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠹⣧⢀⣿⠇⢰⡿⣧⠀⣰⡿⠛⠻⢂⣾⠟⠛⠓⢸⡇⢸⣿⣦⣸⣿⠀⣰⡿⣧⠈⠛⣿⡟⠃⣿⡇⣰⡿⠛⢻⣦⢸⣿⣦⣸⣿
⠀⢻⣿⠏⢠⡿⠷⢿⣧⠻⣧⣤⣤⠈⢿⣤⣤⡤⢸⡇⢸⣿⠙⢿⣿⢠⡿⠷⢿⣧⠀⣿⡇⠀⣿⡇⠻⣧⣤⣼⠟⢸⡿⠙⢿⣿
⣶⣾⠿⠶⠿⠷⣶⣶⣿⣷⣾⣿⣷⣶⣶⣿⣿⣶⣾⣿⣾⣿⣶⣾⣿⣿⣷⣶⣾⣿⣶⣿⣷⣶⣿⣷⣶⣾⣿⣿⣶⣾⣿⣶⣾⣿⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⣠⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢷⣄⣴⡾⠋⠁⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠈⠻⣯⡀⠀⣀⣴⠿⠋⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⣡⣴⠿⢿⣦⡀⠘⢿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⢉⣠⡾⠟⠹⠷⠀⠙⣿⠀⠀⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⢁⣴⡿⢻⣧⠀⠀⢀⣴⡾⠋⣠⣴⠟⠋⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⣠⣾⢿⣇⠀⠀⠀⣠⣶⠟⢉⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⢁⣴⣿⡏⠀⠀⠉⢀⣴⡾⠛⣁⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣶⡟⠉⠈⠛⠀⣠⣴⠟⠋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⢠⣿⠁⠀⣾⡏⠙⠷⠀⢀⣠⡾⠟⢁⣴⡿⠋
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢰⣿⣿⡆⠀⣶⣶⡄⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣧⡀⠈⢻⣦⣀⣴⡿⠋⣠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⠀⠈⠻⣿⡇⠀⢻⡿⠀⠀⣿⣿⠟⢡⣴⣄⠀⢀⣴⡿⠛⣿⣆⠀⠉⠛⢁⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡄⠀⠀⢹⣧⣠⣬⣧⣀⣠⣿⠃⢀⣿⠿⠋⠰⣻⣯⣤⡾⠟⠙⢷⣤⣶⠟⠉
⠀⠀⠀⠀⢀⣴⣤⡀⠀⠈⢻⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡁⠀⣠⠞⠋⠈⠉
⠀⠀⠀⠀⠈⠻⠿⠷⣤⣰⣿⣿⡿⠛⠿⣿⣿⣤⣼⣿⣿⣿⢟⡵⣋⠁⢀⣴⣾⣶⡀
⠀⢀⣴⣶⣦⣄⠀⠀⣸⣿⣿⣿⡀⠀⠀⣿⣿⡟⠁⠀⠀⠀⣡⣾⣿⣿⠟⠿⠿⠟
⠀⠈⠿⣿⣿⠿⠿⢶⣿⣿⣿⣿⣿⣶⣾⣿⣿⣇⠀⠀⠀⢠⣿⡿⢿⣿⣄⣤⣴⣾⣿⡷
⠀⣠⣴⣦⣤⣤⣀⣸⣿⣿⣿⣿⣿⡋⢹⣿⣿⣿⣿⣶⣾⣿⣿⣷⣾⣿⣇⣀⣀⣤⣭⣤⣄
⠀⣿⣿⣿⣿⠿⠛⠛⢿⣿⣿⣿⠿⣿⣿⣿⣿⠿⢿⣿⣿⠛⠻⣿⣿⣿⠛⠛⠿⣿⣿⣿⣿⠇
⠀⠈⠛⢫⣥⣤⣴⠖⠛⢿⣿⣿⣤⣿⣿⣿⣇⣀⣸⣿⣿⣷⣾⣿⣿⠷⣦⣤⣤⣌⡙⠛⠋
⠀⠀⠀⠘⠿⠋⣀⣀⣴⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⣰⣿⣿⣿⠋⠀⠀⢀⣿⠛⢿⣿⣿⣿⣿⠿⣿⠋⠉⠻⣧⣄⡀⠀⠈⠉
⠀⠀⠀⠀⠻⠿⠟⠁⠀⢀⣴⣿⡇⢠⣾⡀⣸⣿⠀⠀⠸⣆⠀⠀⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃⢿⣿⢧⣿⣿⣇⠀⠀⣿⣷⡄⠈⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠆⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠐⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠠⢀⡔⢁⡤⠤⣤⠄⣀⡀⠀⠂⢄⠀⠀⡀⠤⠄⢀⢀⢠⣀⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⡀⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠉⠡⠀⠀⠈⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⠌⠀⠀⠀⠐⠀⠀⠂⣀⣀⣀⡀⡀⣀⣀⡀⠤⡐⠀⠈⠂⠠⠤⠤⠤⠤⡤⠤⠠⢄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣢⣀⠠⡄⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠋⠊⠈⠊⢑⣀⢐⠉⢁⠐⠒⠀⠠⠤⠀⠀⠀⡂⠀⢸⠀⠀⠒⠂⠀⠤⠀⠑⠒⠊⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡴⠡⠁⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠁⠘⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢐⠜⠄⠀⠀⠀⠀⠀⠀⠀⢸⠆⢀⣄⡴⠀⠈⠀⠀⠀⠠⡇⠀⢰⠀⠀⠀⠈⠢⠌⢁⠬⠁⠀⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠀⡠⠁⠸⠀⠀⠀⠀⠈⡇⠀⠘⠀⠀⠀⠀⠄⠃⠈⠀⠈⠀⠠⡇
⠀⠀⠀⠀⠒⠄⡀⢀⠂⠱⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨⢀⡂⠀⠀⠀⠀⠀⠀⠈⡁⢀⢸⠀⠀⠀⠀⠀⠤⠠⠄⠀⣒⠚
⠀⠀⠀⠀⠀⠀⠨⠑⠬⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠉⢙⠘⠀⠀⠐⠂⠤⠬⡥⡤⠤⠰⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣡⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⡐⢀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠈⠀⠈⠀⠀⠈⠀⠁⠈⠀⠀⠀⠁⠒⠤⡤⢌⡡⢨⠤⠐⠈⡀⠀⠳⠤⢀⡀⠀⠊⠂⠐⠁⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠉⠉⠑⠉⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠤⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣾⡟
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠃⣀⡴⠛⣷⡀⠀⠀⠀⠀⠀⢸⡏⠛⠛⠛⠋⢹⡇⠀⠀⠀⠀⠀⢀⣾⠛⢦⣀⠘⠛⠻⣦
⠀⠀⠀⠀⠀⠀⢸⣯⠴⠖⠋⢁⡤⠂⠘⣷⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣿⠃⠰⢤⡈⠙⠲⠦⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠴⠊⣠⠘⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⠃⣄⠑⠦⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⢋⡤⠆⠈⢿⡄⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢠⡿⠁⠰⢤⡙⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠴⠚⣡⠈⢿⡄⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⢠⡿⠁⣌⠑⠦⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠉⠀⠀⢈⣿⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⣿⡁⠀⠀⠉⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⢀⣠⣴⣿⡁⠀⢸⣇⠀⠀⠀⠀⣸⡇⠀⢈⣿⣦⣄⡀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⢿⡏⢷⡹⣷⡀⠈⠛⠛⠿⠿⠛⠋⠁⢀⣾⢏⡾⣹⡿⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢿⣎⢷⡘⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⢋⡾⣱⡿⠁⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠛⠷⢶⣤⣄⠀⠈⢻⣎⢷⡘⣷⡶⢶⡆⢰⡶⢶⣾⢃⡞⣱⡟⠁⠀⣠⣤⡶⠾⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⠀⠀⣠⠾⣿⠀⠀⣠⣿⡆⢳⣨⣵⠟⠁⠈⠻⣮⣅⡞⢰⣿⣄⠀⠀⣿⠷⣄⠀⠀⢾⣏
⠀⠀⠀⠀⠀⠀⠀⣼⣯⣀⣠⣾⣥⣀⣿⠀⠘⠿⠶⠾⠛⠋⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠿⠃⠀⣿⣀⣬⣷⣄⣀⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⢈⠙⠛⠻⠷⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠾⠟⠛⠋⡁⣿⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣤⠇⡞⡆⣴⠒⠆⡴⠢⣩⣭⣍⢩⠉⣭⡀⢶⠒⣆⡴⠉⣧⡟⢦⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣭⣘⡉⠹⠻⢤⠖⢧⣠⢼⣿⡏⢾⣰⠷⢧⢸⠀⠯⠻⠴⢃⣃⣬⣵⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠷⠶⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⠾⠛⠛⠋⠉

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⠿⠿⠏⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠛⠿⣶⣤⡀⠀⠀⠀⠀⢰⣷⠀⠀⢾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢈⣻⣿⣦⡀⠀⠀⣸⡏⠀⠀⠈⢻⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⣰⠾⡉⣯⣿⣷⣦⣄⣾⠋⣀⡿⠾⠿⣦⣀⠀⠀⠀⠀⠈⢳⡄⢀⣾⢋⣥⡄⣾⣿⡄⠀⢿⡈⠉⠉⣙⡋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣷⠿⠾⢷⣿⣟⡉⢩⣾⣿⡜⣡⣶⠶⠺⠶⣽⣧⠀⠀⠀⠀⠰⣿⣿⡟⠈⠉⠁⣼⣿⠇⠀⠘⢻⣷⣾⣧⣿⣯⠴⢶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣇⣰⡶⠟⠙⣹⡟⢘⣿⣈⣿⣿⠁⢠⡆⢰⣿⣿⠂⢠⠤⣶⠶⢛⣻⣧⡠⠤⢌⣩⣿⣀⣠⣴⢟⠻⢞⣷⣿⡷⣶⣿⣛⠧⢤⣀⣀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⡯⠀⣿⡞⣿⡇⣨⣿⣹⣿⡿⠆⡀⠰⠿⣸⠃⠀⣾⠋⣠⠞⠋⠉⠙⢿⣄⣾⣤⡙⢿⣿⣿⢃⣥⣤⠟⠀⣇⢱⣿⠈⣿⣥⠔⠘⠩⠟⠲⠶⢤⣿
⠀⢸⡟⠀⠀⠀⠶⠿⣝⣫⡝⠛⠻⡿⣦⣉⡿⠾⢧⣄⠀⠈⠳⡏⠴⠀⢤⠀⠀⠀⢹⣦⣋⣈⣿⡿⠛⠉⠀⠀⠀⣿⢺⢿⡀⠿⣟⠙⠛⠶⣤⡷⢊⣡⣿
⠀⢸⡿⣝⣆⣐⣶⢚⣡⡿⣤⡀⣠⢠⠀⠩⢦⣴⣿⡿⢷⡀⠀⣧⠀⠀⠶⣻⣧⡀⠸⡅⣽⣿⣧⠄⠀⣠⠀⠀⠀⣿⢸⣿⣧⣰⣿⡒⢤⣀⠀⠀⠉⠐⣿
⠀⢸⡇⠀⢉⡽⢋⡭⠒⠈⢀⣈⠙⢿⣧⣀⣸⣟⡈⢻⣿⣻⣦⣻⠀⡍⠷⣿⣿⡿⠾⢿⡉⠁⢬⣿⠉⠉⠀⠀⠀⢹⣾⣿⡿⠿⣿⣷⣵⣮⡑⠢⣀⠀⣿
⠀⢸⡆⢰⣟⠰⠇⣴⠚⠏⢩⣭⣿⡮⢻⣙⠉⡹⢇⣜⠁⢟⣯⢹⣆⢣⡈⢋⣼⣤⡄⠈⢷⣶⣿⡀⠀⠀⠀⢀⡴⠋⠉⠉⢶⣶⣦⠉⠹⣿⣿⣳⣤⣀⣿
⠀⢸⣎⢿⣿⣿⣾⡿⠘⠛⣁⡿⢿⣿⣟⣹⣦⣷⣷⣿⡶⢦⣤⡞⣿⢶⣥⣤⣬⣥⣬⣠⣬⣿⣿⡇⠀⠀⡰⠟⠻⣷⣾⡟⠋⣀⣩⣿⣟⣄⡉⠛⣿⣼⣿
⠀⢸⡇⢈⠀⠉⣹⠇⠢⣀⣙⠳⠃⠈⠛⢻⣿⣿⣟⠉⠀⠀⠀⠀⡟⢸⡴⠀⠀⠀⣿⠁⢩⢹⣿⡇⠀⢰⣷⣦⣴⡑⠉⢻⣿⣏⠉⠷⢽⣿⣿⡻⡉⠙⣿
⠀⢸⣷⢤⡾⠛⠛⢦⡀⠀⠛⡿⣷⣤⡀⡀⣷⢸⣿⣆⠀⠀⠀⢰⡇⣸⡜⣧⠀⠀⢸⣇⣾⣴⣿⣗⡲⢦⣿⣽⣿⣷⣤⠋⣿⠡⣮⡿⢳⢶⣾⣭⣿⡿⣿
⠀⢸⣿⢠⡇⠀⢀⠀⠈⠳⡎⠲⡾⠋⣻⣧⣯⡤⣿⠻⠃⠀⠀⣼⢃⣿⣚⣞⣧⣤⣾⣵⡾⣷⣄⣀⣴⠶⠛⠋⠉⠀⢹⣌⣉⡀⠈⠀⣼⣆⣸⠟⣡⣀⣿
⠀⢸⡏⠈⢧⠀⣼⡇⠀⣀⣠⡄⢙⡦⡉⢻⠿⣿⣿⡤⣴⡒⠲⠿⠧⣬⣥⣭⣭⣉⣉⠿⠟⠛⠿⠷⠂⢤⣴⠶⢯⡾⠛⠉⢉⣛⣛⣟⣉⢉⣭⢽⣽⡿⣿
⠀⢸⣇⣀⣠⡆⠘⣿⠚⢩⣮⡙⢯⣀⣌⣻⣤⣼⠯⢴⣎⠙⠳⣐⣶⣆⠉⢻⣽⠾⠓⠒⠒⠒⠛⢉⠍⠒⣢⠖⢉⣠⠀⣿⣟⣇⣠⣴⠋⠀⢈⡆⣑⠀⣿
⠀⠐⡏⢀⣤⣳⣄⠙⠷⣽⡻⢤⡈⠈⠟⢌⠻⢶⡓⣜⣎⣂⣒⣀⣙⣿⣳⡾⠁⢀⡀⢿⠄⠀⠀⠀⣠⠞⢁⣤⣪⣵⣾⣻⡿⢳⡿⣷⣤⠶⣿⣤⠿⠟⣿
⠀⠀⣧⢼⣷⠁⢠⡿⣳⣦⡙⠦⡙⢦⣈⠄⠙⠢⣙⣿⠋⢉⣩⣩⣭⣽⣤⣧⣴⣋⠀⠀⠀⢀⣀⡴⢁⣴⡛⢁⣼⣿⠟⣡⠖⢿⠀⠈⢯⡛⡳⠖⠚⠋⣿
⠀⠀⣿⢰⡿⡆⣿⡇⢩⣧⠜⣆⡸⣧⢁⣎⠀⠀⠈⡛⢰⠉⠉⠀⠀⠀⢀⣀⠸⠿⢉⣋⣤⡽⢋⠔⠋⠀⠀⣀⡜⣡⡞⣹⣄⠀⢳⠀⢀⠙⣇⠀⠀⠀⣿
⠀⠀⣿⣾⠁⣿⣄⡇⠸⡃⢀⣼⠛⣿⠿⠹⢟⣿⣿⠁⡌⠀⢐⡶⣖⣫⣥⣤⣤⢄⣀⣭⡽⢿⣤⡖⠎⠘⠺⣯⠞⠋⠩⠭⠉⢷⡈⠀⣿⣷⢸⣶⢿⡏⣿
⠀⠀⣿⢹⠀⡏⢚⡇⠈⣧⡜⠫⣰⡇⠀⢃⠋⣼⠃⢀⣡⣾⢿⢛⣭⠟⠋⢉⣴⢓⡈⠻⣦⠟⣉⡤⠶⢚⣋⡀⠤⠄⠲⠲⠶⣿⡇⢸⡿⠁⢸⡙⣿⡁⣿
⠀⠀⣏⢸⠀⡷⠊⡇⠀⢇⠈⣾⢿⣷⣶⣶⣶⡇⢀⡿⢼⣥⠞⠋⠀⣠⣾⣽⠷⡾⠟⣋⠨⣛⠵⣒⣛⣭⣵⡶⢶⣞⠋⠐⢰⣇⣤⣼⢷⠀⢸⣟⢿⡇⣿
⠀⠀⡟⣿⠀⡧⠰⡇⠀⢞⠀⢸⣿⣿⠉⠙⠛⣿⣿⣃⣤⣿⣆⣠⢛⣉⣡⣤⡶⣻⠀⣿⢴⠲⠟⣙⡋⠭⡉⠃⠠⠤⠍⣁⣙⣓⣦⣹⡄⠀⢸⡍⢪⣻⣿
⠀⠀⣷⣿⠀⣷⠀⡇⠀⢽⡀⠀⣙⣿⠛⠿⣿⣿⣿⣿⠿⠷⠿⠭⠿⠿⠻⠛⣚⣿⠀⣿⠰⢙⠋⠭⠅⣒⠂⠭⣍⣙⠲⠶⠤⠤⠤⠤⠿⢦⣸⡏⠭⠁⣿
⠀⠀⣿⡿⠀⣿⠀⡇⠀⢾⠄⠀⠻⢿⣷⢞⠻⠭⠤⠤⢤⣤⣉⣉⣉⡭⡭⠥⠨⡗⠀⣿⡠⡴⠦⠞⠓⠈⠉⣁⣀⣤⣤⣴⡶⠶⠶⢤⣏⡺⠿⠴⠚⠓⢻
⠀⠀⣿⠇⠀⢿⠀⡇⠀⢉⠀⢀⣙⠈⣿⠈⢉⣑⣒⣒⡰⠦⢀⢀⣀⣀⣀⢀⣘⣇⡀⢸⣤⣤⡶⠶⢶⣛⣻⠯⠭⡅⠐⠀⠀⠀⠒⣀⣀⣤⣤⣶⣒⡒⢻
⠀⠀⣿⠃⠀⠸⢠⠇⠀⠀⣠⣿⠛⣏⣿⠤⠖⠀⠀⠀⠒⠚⠛⠙⢃⣀⣤⣾⣟⠛⢭⣙⠋⠤⠜⠛⠊⠉⣁⣀⣤⣤⣶⢶⣿⡟⠿⠟⠛⠛⢂⣈⣈⣉⣹⠇
⠀⠀⣿⡄⠀⠀⣾⠀⢀⣼⣇⢸⠀⣯⡤⠤⠤⣐⠶⠶⣶⠾⢿⣟⠿⠯⠙⠒⠂⢉⣁⣠⣤⣴⠶⣶⣿⠿⠟⢛⣋⣩⣭⠤⡤⠤⢶⣶⣿⣿⣿⣿⣿⣗⣸⡇
⠀⠀⣿⠀⣾⢙⣿⣀⠋⣯⠻⣸⠀⣯⠙⠳⢦⣤⣶⠒⠈⠉⣀⣠⣤⣤⣶⠶⠛⠛⠉⠛⣒⣉⣩⣤⣴⣶⣿⣻⠙⣏⠙⣧⠀⠨⠌⢿⡀⠄⠀⠹⣧⠋⢹⡇
⠀⠀⡟⢐⣧⢿⣈⣿⣧⠏⢰⡟⠀⡟⠀⠀⡀⠈⠙⣿⠻⢿⡉⢉⣡⣤⣴⣶⣶⣛⣛⣻⡯⢿⠉⠩⣭⣭⣭⣵⣶⣩⠖⠚⡖⠐⠒⠘⠳⡀⠐⠐⠛⢧⣘⡇
⠀⠀⣷⡞⠙⣔⢡⠟⠁⢀⣿⠁⡞⢠⢺⢿⣿⣄⠀⣿⠁⢠⡾⠿⢾⡏⠛⠓⢻⠉⠉⠉⣃⠸⠦⢄⣉⣛⣛⡿⠉⢧⠀⠀⠘⣆⠀⠀⠀⢳⡀⠀⠀⠀⢻⡇
⠀⠀⣿⣠⠷⢿⢯⣀⡴⠚⡷⠚⠳⠾⣼⡾⢷⣿⡇⢹⠀⣸⠧⠤⢾⠆⠒⠒⢺⠖⠶⠶⠿⠶⠒⠖⠺⡇⠐⠒⠲⠺⡗⠛⠉⠙⡏⠉⠉⠉⢻⡉⠉⠉⢹⡇
⠀⠀⡏⢀⡔⠁⠀⠉⠀⠾⠁⠀⠀⢀⡜⠀⠀⠙⢿⣾⠶⠃⠀⠀⢼⠀⠀⠀⠈⡇⠀⠀⠘⠇⠀⠀⠀⠇⠀⠀⠀⠀⠹⠀⠀⠀⠹⡄⠀⠀⠀⢳⡀⠀⢸⡇
⠀⠀⠻⠿⠴⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠾⠿⠾⠿⠿⠿⠿⠾⠿⠷⠶⠞⠳⠿⠶⠾⠾⠿⠶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠙⢿⡂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣛⣻⣧⠀⠀⠀⠐⠒⣦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⣾⣿⣏⣟⣿⣿⡇⠀⢰⡶⢚⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣀⣼⣿⣿⣿⣽⣿⣿⣧⣤⣾⣵⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⢫⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⡟⠙⠻⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠘⠿⣿⠟⢽⣿⣿⣶⣶⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠞⢹⣿⣿⣿⣟⢿⣿⡿⣿⣿⣇⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⣿⠆⠂⣸⠻⣿⣿⣿⣿⣶⡿⣳⣿⠿⠂⠀⠰⣿⣄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠻⣵⢶⣯⣐⠤⡝⠛⠻⠖⢛⣯⠔⣢⣿⡶⣮⠟⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠐⠻⣿⣷⣦⣍⣀⣙⣠⣴⣾⣿⠟⠒⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⡿⢯⣛⣛⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢙⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⡀⡀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⢦⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⢉⣀⣤⡶⠶⠲⠶⢤⣀⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢡⠖⢉⡉⠉⠉⠉⠉⠙⠒⠤⠀⠁⠙⠍⠫⡉⠉⠉⠛⠛⠳⠶⢤⣄⣀⣼⠃⠀⣀⢀⣴⠋⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠏⢠⣷⠾⠋⠙⠳⣠⡀⠀⠀⠀⢀⠠⡀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠈⠙⠣⠶⢶⣿⣿⠇⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⠚⢙⢀⡾⡁⠀⠀⠀⠀⠘⢻⡷⣤⡀⠀⠙⢷⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠫⣄⠀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⢸⢱⠁⠀⠀⠀⠀⠀⣀⡀⠀⣈⣶⣄⠀⠑⠀⠀⡀⠘⣦⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠈⠳⣄⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⢸⣆⣠⡄⠀⠒⣉⣤⣭⡲⣌⠉⠻⣧⡀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⣟⠉⣲⣦⡀⠀⠉⢻⣿⡿⠈⢀⣿⣿⣷⠀⠀⢠⠀⠀⢿⣷⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢻⡟⢹⣷⣄⠈⣉⠀⢀⣾⡏⢻⣿⡇⠀⢼⡄⠀⢸⡿⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⣤⣤⣄⠈⠀⠁⠀⢘⣹⣿⣿⣆⢻⣿⡇⠀⢸⡇⣸⠈⠃⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡈⣿⣿⣿⣆⣀⣠⡶⠉⠉⣻⠿⠿⠈⢿⡇⣰⡿⠁⣿⠀⠀⣸⡿⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣌⢿⣽⣿⣿⣿⠃⠀⣠⠁⠀⢀⣴⠟⠡⠋⠀⢸⡏⠀⢰⣿⣹⣿⣿⡉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡹⢧⡙⠿⣿⣿⠐⠋⠀⠀⣰⣿⠋⠀⢠⠀⠀⠘⠀⢠⣿⣳⣿⡏⠙⠳⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣭⣁⡈⠻⣇⣴⣄⣼⡿⢁⡎⠀⢸⠀⠀⠀⢠⣿⣷⠋⣿⠁⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠿⣿⣿⣯⢻⣮⡙⠿⣿⠃⣾⠁⠀⢸⡇⠀⠀⣿⠏⠁⠀⣿⠀⠀⠀⢳⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⢸⡇⠃⢹⣺⡎⢻⣆⡟⢸⣿⡄⠀⢈⠆⠀⣼⡏⠀⠀⠀⠿⠀⣆⠀⠘⠀⠀⠀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⣇⠀⠀⠀⠀⢸⢻⣇⣼⣿⡁⢸⣿⡇⢀⡎⠀⣼⣟⡄⠀⠀⠀⠀⠀⡿⠀⠀⢀⠀⠘⠇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀⢹⡄⠀⢀⠀⠐⠋⡻⢣⠴⠂⠐⠲⢦⣞⢠⣾⡟⣾⠀⣸⡇⠀⣠⣸⡇⠀⡼⣸⡆⢀⣤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⡇⢸⣣⡴⠃⠀⠀⠀⠀⡏⠀⠀⡀⣷⢷⣌⠻⣟⣼⣿⣶⣿⣷⣬⣵⣿⣷⠀⢡⣿⣷⣾⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡭⠶⠉⠀⠇⠀⠀⠀⠀⢀⡱⣄⢀⣾⠹⢦⡝⢱⣆⢾⣿⣿⢿⣿⣿⠿⣿⣿⣿⣿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⢻⣿⣮⣘⠻⣦⡽⢣⠈⢀⡘⠏⠀⠀⢠⠀⠈⠉⠉⢁⠀⠈⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⣇⢠⣷⢀⠀⠀⣼⣧⣀⡇⡄⠀⠈⣤⠑⠞⠙⠀⠀⢸⡇⠀⠀⠀⠻⠃⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⡿⠻⣦⣥⡾⠛⠙⠛⢡⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⢀⢤⠀⠀⠈⡳⣞⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⣈⣏⣀⡀⠀⢀⣄
⠀⠀⠀⠓⢾⠲⠚⠉⠁⠈⠉⠓⠖⡷⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡷⢊⣭⠀⠶⠀⣭⠲⡞⠉
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣴⡄⣡⡰⠷⠀⣭⠰⡿⡄
⠀⢞⣱⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣏⡇⠀⠀⢿⡦⠖⢹⣏⠀⠉⠉⠲⢴⡿⠀⠀⠐⠿⠒⣇⣤⣀⣍⠰⠷⢠⡍⢴⡆⡗⠺⠗
⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⠀⠀⠀⡜⢁⣀⣈⠀⠀⠀⠔⡟⠈⢣⠀⠀⠀⠀⠀⠹⡉⣬⡏⢀⠿⢊⠱⠶⣰⠃
⠀⠀⠀⡤⢼⠤⢄⡀⠀⠀⢀⡠⢬⡧⢤⠀⠀⣤⣴⠃⢈⡛⠆⠀⡤⣄⠀⠀⠀⠈⣦⣤⠀⠀⠀⣤⠟⠪⢄⣛⣃⣸⠷⠚⢣⣤
⠀⠀⠀⠙⠚⠀⠀⠈⡵⢏⠁⠀⠀⠓⠁⠀⠀⠀⠈⣷⡯⠄⠀⠀⠑⠋⢠⣙⣶⣴⠁⠁⠀⠀⠀⠉⠀⠀⠀⢠⣧⠀⣀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⣹⡴⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⢠⠖⡆⠀⠉⠈⡧⡞
⠀⠀⠀⠀⠀⣤⠀⠀⠀⢈⡁⠀⠀⠀⣤⠀⠀⠀⠀⠛⠋⠙⠒⠿⣧⠤⠒⠋⠙⠛⠀⠀⠀⠀⠀⠀⠈⢙⡝⠚⡽⢻⡄⠙⠢⣄⣰⣙⡆
⠀⠀⠀⠀⠀⠀⣳⢔⡒⢩⣭⠒⡢⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢠⠏⠀⠀⠉⠏⠆⠀⠀⠀⢺⠉⠁
⠀⠀⠰⢤⣀⡜⣁⠘⠋⣨⣍⠘⠚⣨⣧⣀⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠘⠧⠟⡅⠀⠀⠀⠀⠀⠀⠀⠠⣀⣘⡇⣀⡀
⠀⠀⠀⠀⢸⠉⡛⠘⠟⣨⣅⢘⠝⣩⡉⡇⠀⠀⠀⠀⠀⣀⡀⠀⣀⡠⠴⠃⠧⢤⣀⠀⢀⣀⠀⠀⢸⡀⢀⣠⣣⠄⠀⠈⠙⣧⢾⠚⢧⠍
⠀⠀⢀⣀⠼⣟⠛⣺⠗⢨⣦⠛⣣⡍⢱⠧⣄⡀⠀⠀⠀⠙⡎⠋⠁⠀⠀⠀⠀⠀⠀⠙⢫⡞⠀⡖⢾⠥⣌⠲⠼⠀⠀⠀⠀⣴⠃
⠀⠀⠈⠀⠀⠘⢦⡛⠠⣶⠀⣶⣌⡵⠃⠀⠀⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠙⠊⠀⠀⠉⣦⡔⠒⠊⠉⡞⣲
⠀⠀⠀⠀⠀⣤⠋⠈⠉⠒⡖⠉⠁⠙⣤⠀⠀⠀⢀⡤⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⡀⠀⠀⠘⠥⠕⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀⡤⢅⠀⠀⠀⠀⠉⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡒⣆⠀⢀⣀⡧⣞⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⣠⠄⢰⠄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢉⠏⠉⣝⣛⠆⠈⠑⠦⢴⣉⡧⠀⣴⡣⢤⣀⠀⠀⠀⠀⠀⢀⡤⢌⣦⠀⠀⡀⠚⡧⢜⠓⢉⣙⠚⠦⣜⠂⡀
⠀⠀⠀⠀⠀⡤⣄⣠⠋⠀⠀⠈⠏⠓⠀⠀⠀⢸⡀⠀⠀⠈⠁⠀⠈⠉⠓⣤⡒⠉⠁⠀⠀⠁⠀⡌⢹⠋⠀⠻⠃⣨⡋⠸⠽⢈⣳⡉⡀
⠀⠀⠀⠀⠀⠳⠚⢦⠀⠀⠀⠀⠀⠀⢀⡰⣤⣄⢇⣠⣄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢈⣉⡇⠻⠇⢖⡦⣙⡁⠰⡲⠙⠋⣏⢉
⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡤⣧⠂⠀⠀⠈⠧⣺⠚⠢⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢉⡧⣶⠆⣴⡄⢛⡃⣖⠆⠫⠇⡏⢙
⠀⠀⠀⠀⠀⠀⣖⢺⠖⢤⡑⠚⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣹⣄⢰⡮⢀⣘⠊⣀⡺⠟⣼⠉⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⡷⡔⠛⠉⠉⣇⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣨⠗⢤⣘⣚⣀⣳⠷⢎⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠀⠼⠄⠘⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⡄⠙⢿⡟⠁⣴⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⠇⠀⢸⡇⠀⢿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠠⠶⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠉⠉⠿⠶⢻⣯⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⡟⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⠛⠾⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋⠱⣋⠀⢸⡃⢈⣻⠋⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠁⢸⠀⠘⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣀⠀⠀⢸⠀⠀⢀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣦⣼⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⢻⠛⠻⢷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣷⣾⣶⣿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣸⣤⣤⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡛⢻⠛⣻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣼⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣤⣄⣜⣰⣿⣿⣿⡃⣠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⢿⡟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢋⠹⡍⠉⣻⣿⣇⣼⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠈⣓⣲⡝⣿⡟⠿⠻⠿⣦⣌⠢
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⣠⠄⢀⣠⣴⣻⣵⣀⣠⡙⣧⢀⠀⠀⠀⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠴⣤⡤⠘⣡⠖⠋⠋⣩⣾⣿⣿⣿⢷⡘⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⠜⣳⡀⢀⣠⣴⣾⠿⣫⡿⣼⣿⡏⣷⢸⢈⠑⢕⣄⠀⠀⠀⠀⠀⢃⡄⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠐⢳⠈⡭⢿⣾⡯⢩⣴⣿⣿⡇⣿⣿⣿⢸⡇⢃⠳⠊⠀⡀⠡⠴⡊⠉⠉⠛⠛⠴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⣦⠙⣿⣿⣦⣴⢿⣿⣿⣿⣿⣿⣿⣴⣷⣾⡏⠀⣁⣤⡴⠖⠋⠉⠀⠘⠁⠈⠃
⠀⠀⠀⠀⠀⠀⡂⣿⣵⣿⣿⣿⣿⢶⣿⣿⣿⣟⣻⣯⠏⢯⣿⠹⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⡉⢸⣿⣿⣿⣿⡏⢸⢿⣿⣿⣻⡟⣶⡖⠐⠉
⠀⠀⠀⠀⢠⠔⢁⣼⠛⣿⣿⣿⢀⢸⢸⣿⣇⢳⣅⠘⢿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠈⢠⣥⢾⡿⠃⠠⣿⣿⣿⣸⠄⢸⣿⣿⡼⣿⣾⢎⢿⣮⡑⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠠
⠀⠀⠛⠳⠚⠃⡃⠀⣠⣛⣿⢻⣿⣇⠸⣿⡟⣿⣦⣉⣷⡀⠈⢻⢷⡦⢅⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣟⠺⣿⣿⣷⣹⣿⣾⣿⡿⣮⣙⢦⢠⡘⠛⣆⠀⠤⣀⠀⠀⠐⠁⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⡀⢛⠛⡿⢠⣿⠿⣿⣷⣿⣿⣯⣼⡻⢿⣆⢋⠻⢦⣿⣷⠀⠢⢅⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⠰⣬⣴⣾⣿⣿⣿⢿⡍⠻⢷⡸⡿⣿⣶⣶⣿⢧⢸⡴⣄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⠿⠿⠀⠀⠀⠈⠙⠚⠛⠷⡼⠈⠿⠙⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⢈⠆⠀⠀⠀⢣⡼⠃⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢪⢧⠀⠀⢀⠽⡇⠰⡁⢀⡸⠀⠀⢀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⠀⠀⠀⢀⠤⣄⡠⠇⢷⠒⠊⠚⠓⠒⢾⣾⢄⣀⠀⣺⣠⡸⢈⣁⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⠸⡀⠀⢀⠼⠊⠁⠀⠀⣘⠣⡀⠀⠀⠀⡈⣹⠀⠀⠉⠓⢧⡔⡽⠃⡎⢀⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠑⡌⠓⢡⠒⠉⣹⠀⠀⠀⠁⠐⢀⣀⠈⠠⠾⡥⡂⠀⠀⠀⠈⡳⢶⡕⠓⠒⠁
⠀⠀⠀⠀⠀⠀⢀⡠⠤⡄⠘⠧⢼⣦⠖⠀⠀⠀⠉⠙⢿⣇⠀⠀⠀⡜⠁⢀⠈⠦⠀⠀⠂⠀⠠⠔⣫⡞⠉⠙⣷⠀⠀⣠⠒⢆
⠀⠀⠀⠀⠀⠀⠸⡀⢢⣌⣲⢄⣰⠁⠀⠀⠀⠀⠀⠀⠈⣏⠓⡀⠀⠈⢢⣩⡽⠃⠀⠀⠀⢠⠒⠢⢍⠐⠃⢈⠃⠉⡭⠓⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⢓⠮⣕⠒⠀⠀⠀⠀⠀⠀⠒⢬⠖⠒⠒⢤⡻⢌⣊⠀⠀⢀⡆⠰⡤⡸⠀⡠⢣⢎⠆⣡⠀⠔⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣤⡨⠯⣝⠀⠀⠀⢠⠤⡀⠀⠀⠀⠀⠿⡀⠶⠆⣢⠇⠀⠈⠀⠀⠙⣚⠳⠞⠁⠀⠔⡱⠏⠙⢾⠋⠛⠛⢁⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡇⠀⠀⠀⡇⠠⣬⠼⢻⠦⠀⠀⠈⠙⠳⢗⠀⡔⠒⠒⣄⠀⠀⠀⠐⣂⣶⠟⡗⡦⠥⣾⠿⠭⠽⠟⢶⣦⠆
⠀⠀⠀⠀⠀⢀⡤⣀⠀⢀⣈⣇⡠⠄⠀⠈⠒⠁⠀⢐⠀⠀⠀⠀⠀⠀⠂⠘⢄⣒⣑⡼⠀⢀⡠⣊⠔⡑⣊⡬⠴⠒⠉⠑
⠀⠀⠀⠀⠀⢸⠀⢨⠥⠐⠶⡴⠒⠒⠀⠀⠀⠀⢀⡔⣠⣤⠀⠀⠀⠀⡆⢠⠈⢕⡩⠊⢀⣥⠾⠓⢦⣔⠍⡹⢉⣙⠿⠎
⠀⠀⠀⠀⠀⠀⠉⠁⠀⡠⣄⡸⠀⠀⢀⣠⣀⠔⡥⠊⡭⡪⠄⠀⡠⠒⠁⢿⡀⢔⡨⠀⢛⠝⣆⠴⢀⡡⢊⣴⡕⡉⣲⣤⡖⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡴⠊⢢⠠⣏⢀⠀⡞⠀⠀⢒⢂⡤⠀⢣⡈⣲⠜⠂⠀⢔⡳⢤⠊⢝⡋⢁⠔⠉⢫⢎⢰⢁⣀⠙⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣼⠷⡈⠛⠊⢀⡀⠤⣳⡫⠤⢒⡩⠜⡡⢔⡪⢐⢕⠝⣣⣑⠒⡔⠁⣂⣘⠦⣡⠛⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣴⠇⠀⡘⣡⠆⣸⣄⢐⡸⠐⡪⠕⣂⠭⢂⠁⡰⣁⠅⡊⢼⡉⠓⠓⠂⠺⣟⣿⢾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡪⠞⠁⠑⢆⡁⠔⣊⠁⠐⡩⢒⠩⢈⣧⡇⠐⣊⠔⣫⣔⣉⢄⣿⡽⠋⠁⠀⠙⡏⡉⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⡏⠀⠀⠀⣠⡿⠋⠑⡞⣰⣋⠭⠐⡨⢒⣞⡝⢦⠔⣪⣖⢩⣵⠏⢣⢧⠀⠀⠀⠀⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⣁⡼⠃⠀⡰⣱⠃⠈⠁⢶⡶⠘⠦⠥⠊⠙⣯⡀⠀⠀⠀⠀⢫⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢱⠃⠀⠀⢠⠾⡳⡄⠀⠀⠀⢠⣭⣎⡃⠀⠀⠀⢸⣐⢄⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡸⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠈⠙⠊⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⢀⡀⠀⡠⠀⡀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣂⠤⢒⡠⠔⢊⡠⠔⢊⡠⠔⣊⠄⢊⡡⠔⣉⠤⢊⡡⠔⣉⠡⢒⡡⠔⡩⠔⣉⠔⣊⠔⡪⢊⠵⠫⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠁⠀⠊⠁⠔⠊⠡⠒⠉⠤⠊⠁⠒⠉⠠⠒⠁⠔⠊⠀⠒⠁⠐⠉⠐⠊⠁⠊⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠔⣠⡐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠐⠐⠉⠀⠈⡘⢠⠀⠀⠔⠈⠉⣦⠀⠀⡄⢃⠁⠀⠉⠂⠂⢤⣀⡀
⠀⠀⠸⢄⣉⠉⠛⠓⠒⠚⠋⠉⢉⠋⠁⠀⠀⠀⠀⢀⠜⠠⠃⠀⠈⣄⣀⣴⡿⠁⠀⠘⠄⠱⡀⠀⠀⠀⠀⠈⠉⡉⠉⠉⠛⠓⠒⠚⠉⢉⡠⠃
⠀⠀⠐⢍⡉⠉⠉⠀⠀⠀⠈⠉⠈⣁⣀⠀⠀⠀⠀⢬⠀⠂⡀⠀⡀⣸⢿⣿⡇⢀⡀⢀⡀⠀⡅⠀⠀⠀⠀⢀⣈⠈⠉⠀⠀⠀⠀⠉⠉⣉⡩⠆
⠀⠀⠀⠀⠠⡛⠉⠁⠀⠀⠉⠉⠁⠈⣠⣤⠀⠀⠀⠈⢳⣂⣬⣁⢤⢼⠀⣿⡧⣤⣈⡭⣐⡞⠁⠀⠀⠀⣤⣄⠁⠈⠉⠁⠀⠀⠈⠙⢛⠄
⠀⠀⠀⠀⠀⠀⠉⠐⠒⣖⠐⠂⠈⢁⣀⠠⠖⠂⣠⠀⣈⠀⢈⠁⢀⣻⠀⣷⣗⡀⠈⠁⠀⣁⠀⣄⠐⠒⠄⣀⡈⠁⠐⠂⡰⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠛⠐⠚⠋⠉⣁⣉⡉⢹⠀⣿⡏⢉⣉⣈⠉⠙⠓⠂⠛⠀⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢉⣹⠶⡃⠀⠀⣿⠅⢘⠶⢯⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠨⣿⠟⣋⡷⠀⣿⢶⣙⠻⣿⠅⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡈⠙⠫⠤⣀⣿⡤⠝⠋⢁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⡀⠉⠓⢤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢈⡛⣦⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠘⢧⣿⠃⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⡀⠉⠲⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣨⡟⣷⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠻⡅⣿⠟⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⠻⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⡷⣄⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⣿⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⢣⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣽⣆⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹⡏⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣫⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣠⣤
⠀⠀⠀⠀⠀⢀⣶⢿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣶⡀⠀⠀⠀⠀⠀⠀⠿⣿⠿⠀⢿⡿⠁
⠀⠀⠀⠀⠀⢻⡿⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⢾⣿⡇⠰⣿⣿⡇
⠀⠀⠀⠀⠀⠐⢶⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣶⣯⣦⢰⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⡦
⠀⠀⠀⠀⠀⢾⣿⣧⠀⢘⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢸⣿⠃⢰⢸⡇⡇⠀⠀⠀⠀⠀⠀⢀⣄⣠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢈⢻⢿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠈⠉⠀⠸⠸⠇⠇⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠇⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠀⠁⢰⣿⣿⠄⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⣶⣂⣀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣶⣷⠘⠙
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣽⣯⡀⠀⠀⠀⠀⠀⠀⠂⠉⢹⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⣸⣿⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛

⠀⠀⠀⠀⠠⠤⢄
⠀⠀⠀⠘⠤⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⢀⡀⢠⡄⢠⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⡀⢠⡜⠻⡅⠀⠀⠀⠠⣄⡀⠦⠀⣴⠀⡀⠀⠀⢴⡤⠾⠚⠠⢼⡆⠀⠀⢈⢞⡋⡁⠙⢲⡆⠀⢠⣆⠢⣼⢗⠚⠲⣏⣀
⠀⠀⢠⠏⣆⠀⠐⠀⡀⠀⢰⢳⠀⠀⢰⠉⠉⠇⠀⣂⡜⢰⠐⠶⠓⠎⣷⠂⠙⡯⡁⠆⠈⡆⠠⡋⠁⢨⡅⣞⣉⣭⣭⠀⠈⠦⠄
⠀⠀⣽⠀⠂⠐⠒⢉⣁⠀⢈⠿⢴⠖⠖⠙⢋⠖⠀⠀⣨⣀⡁⠈⢁⣰⡏⠀⠘⢹⣵⣚⡂⣀⣰⣉⠀⠐⣿⡝⠛⠋⠁⣀⡼⠳
⠀⠀⠘⢢⣀⣀⠠⡌⠀⠀⠀⠟⠐⢶⠒⠐⠛⠀⠀⠀⠉⠁⠸⠽⠀⠉⠀⠀⠀⠈⠀⠘⠁⣙⠀⠀⠀⠀⠀⠉⠹⠏⠹⠃⠉
⠀⠀⠀⣀⡀⢀⢀⠀⠀⠀⠀⣄⠀⢰⣀⢀⡄⠀⠀⠀⠀⡰⠛⣑⡒⢤⡀⠀⠀⠀⣀⡷⠫⣭⣿⢶⠀⠀⠀⣄⡤⠖⢺⣄⡀
⠀⣠⢂⠿⢁⡔⡑⡜⠃⠀⠀⢉⠏⡔⠮⠉⠑⡶⠀⠀⡸⢁⣨⣵⣱⣇⣹⢆⠀⠀⢩⣦⢀⡬⠶⠋⡀⠀⠀⣷⡴⢩⡉⣾⠁
⠘⠿⢠⣘⡼⠝⣿⠷⢷⡆⠈⠉⢢⠈⠄⠅⢀⠛⠂⢸⣃⠐⠸⣿⣿⡟⠛⡿⠀⠀⠈⡿⠀⠉⠙⢻⡁⠀⠀⠘⡜⣦⣾⣿⠃
⠐⣾⡛⠒⠒⠉⠀⠀⣱⡄⠀⠸⠚⡐⠂⡒⠟⠁⠀⠀⠉⠒⢤⣀⣥⠔⠊⠀⠀⠀⠘⠻⣤⣄⣀⣤⠋⠀⠀⠀⠙⡿⠿⠃
⠀⠀⢸⠢⠤⠒⠐⠒⠁⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢦⡆⠀⣎⠄⠀⠀⠀⣀⣄⣠⣁⣤⡅⡄⠀⠀⠀⠀⠀⢸⡄⣤⠆
⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⣠⣂⡹⣅⣭⣿⣓⡄⠀⠠⣀⣴⣓⣞⣣⡄⠀⠀⣘⡻⢺⡴⢆⠟⠣⣞⡀⢀⠀⢘⢦⣿⣿⢿⣗⣆⢀⡀
⠀⠀⢠⠊⠁⠀⢀⠁⢆⠀⢬⠜⢦⣌⣉⣉⣭⡄⠀⣠⢇⣻⠿⠿⢿⢫⡁⠠⢼⠀⢀⡈⠀⢀⡄⣸⠆⣈⡛⠁⠈⠁⠉⠀⠉⠀⢹⠤
⠀⠀⢸⠀⢂⣚⣗⠃⠾⠀⠻⣀⠂⠈⡉⢆⢴⠃⠀⠐⣇⡀⠀⠀⠐⣾⠂⠀⢛⡀⠉⠀⠀⠀⢠⡏⠀⠀⣩⠆⠀⠀⠀⠀⣠⠤⡞⠆
⠀⠀⠈⠬⣈⣁⣀⡰⠚⠀⠀⠁⠛⠒⠲⠈⠀⠀⠀⠈⠀⢉⠋⡈⠃⠀⠀⠀⠈⠙⠋⠓⠲⠲⠉⠀⠀⠀⠀⠰⠋⢲⠒⠺⠈⠀⠁
⠀⠀⠀⠀⢀⡀⣄⢀⢀⡀⠀⠀⢠⠞⣔⠒⠆⠀⠀⠀⣀⣜⣾⡥⡆⢤⠀⠀⠀⠀⢀⡶⠢⡀⢠⢀⠀⠀⠀⢠⡀⠈⠀⣠
⠀⠀⠉⡞⢐⠛⠓⣾⠄⣣⠂⠀⡜⠤⢛⠄⢳⡀⠀⡌⠀⠈⠈⢂⣀⠀⠀⡄⠀⢰⡒⠂⣢⡎⢡⢘⠇⠀⠰⣾⠽⡇⢛⠶⣿⠃
⠀⠀⠘⣇⣘⣍⣾⢩⡶⠏⠀⢎⡁⡄⠀⠼⢧⡆⠀⡅⠀⠀⠂⠀⠈⠀⢀⡇⠀⠈⠳⠂⡥⠝⠻⠈⠶⠀⠀⢩⡵⡬⢧⡰⣬
⠀⠀⠈⠀⠙⠉⠋⠈⠃⠀⠀⠻⠴⢃⣬⠤⠚⠀⠀⠘⠒⠤⠄⠠⠄⠤⠈⠀⠀⠀⠛⡄⡤⠎⠲⠥⠚⠀⠐⡎⠈⠚⠟⠃⢈⡇
⠀⠀⠀⢠⣀⡐⣢⢤⣀⡀⠀⠀⢀⡀⠀⡃⡀⡤⢀⠀⠀⣀⢀⠷⠴⢧⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠩⠦⣤⢀⡤⠺
⠀⠀⠀⣈⢸⠜⠢⠣⢬⣡⡄⢠⢄⡱⣬⣧⣀⠷⡋⠀⢤⢜⣍⡂⣐⣠⡹⡆⠀⢾⢁⡾⢦⢷⡓⡂⠀⠀⠀⡠⢤⣼⠢⠶⣤
⠀⠀⢀⢀⠁⠀⢀⠀⠀⣉⠄⢀⣠⡍⢶⣂⡡⢃⠇⠐⠚⠀⣡⡀⣀⣤⠌⡅⠠⠊⠇⠈⣉⡁⢸⠙⠄⢠⡤⠹⠅⢧⡺⠗⠊⣢⡄
⠀⠀⠈⠩⣠⢜⠀⢩⣁⢯⡁⠀⠁⡼⠒⠤⡄⠈⠒⠂⠀⠷⡀⠀⠈⠀⢀⡁⠈⠒⡎⠲⠠⠔⠚⡞⠁⢀⣧⠂⢘⣿⣗⣠⢀⢸
⠀⠀⠀⠈⠀⠋⠉⠓⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⡴⠐⢲⠍⠀⠀⠀⠉⠈⠉⠑⠊⠀⠀⠈⠀⠉⠟⠈⠘⠞⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠣⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⠀⠀⠀⠀⣀
⠀⠀⠀⢸⣁⡸⣤⠤⠒⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠴⣦⠴⣚⠉⣩⠍⠀⡀⠀⣸⠇
⠀⠀⠀⠀⢉⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣃⠘⠧⠀⡽⠂⠸⠅⢀⡄
⠀⣠⢤⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡤⠼⠿⠦⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⣰⠏⠧⠀⣞⠀⣳⠀⢰⡃⠙⠂⠀⠀⣠⣄
⠀⢿⣀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡿⠋⠁⢿⣆⠀⠀⠀⣠⡜⢽⠂⠀⠀⠀⠳⠞⢹⣶⣃⢠⡖⠈⢳⡠⣞⠁⠿⠅⠀⠙⠁
⠀⠀⠁⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡀⣀⠀⠀⠀⠀⠘⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣴⠋⢀⡻⠄⣀⠺⠅
⠀⠀⠀⢠⠴⢏⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣼⠁⠀⠓⠯⠁⢀⡤⠄⠀⠀⠀⠀⠀⢠⡶⡄⠀⠀⠀⠰⡛⠮⣀⡻⠄⠀⠧⠀⠀⠢⠴⣄
⠀⠀⠀⠘⠦⠼⠀⠈⠙⢳⢀⡂⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣄⢰⣎⠀⠀⠈⠣⠀⠀⣈⣳⠶⡄⠉⠁⠀⠀⠀⠈⠛⠁⠀⠀⠉⠹⣍⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠠⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⠀⠀⠀⢰⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠚⠰⡉⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠦⣄⡀⢈⣿⠀⠀⠀⠐⠾⢳⠄⠀⠀⠀⠀⠀⠀⢾⣀⡸⣤⠤⣤⣖⠃⠣
⠀⠀⠀⠀⠀⠀⠙⣦⠀⣀⡀⢸⠁⡀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠁⠀⠀⠉⠙⠷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠸⢁⢀⠁
⠀⠀⠀⠀⠀⠀⢀⡼⢫⠄⠀⣖⠀⣠⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⣦⡼⠁⠀⠀⠀⠘⠈⠃
⠀⠀⠰⠒⠶⣤⣏⠄⠈⡃⢰⡞⠀⢙⠂⡴⠀⢀⣠⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢢⠀⠀⠀⠀⠀⠀⠳⡴⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠦⠀⡀
⠀⠀⠀⠀⠀⣾⢘⡃⠘⠧⢠⡅⠀⣻⠄⡩⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣀⠤⠤⠎⠈⠄⠄⢀⠀⠀⠀⠀⠀⠀⢷⠀⠀⡀⠸⢢⠄⠀⠀⠉⢁⣠⠟
⠀⠀⠀⢀⣠⣽⡿⠄⢾⡁⠈⡥⠸⠅⣨⠓⠀⢀⣀⠀⠀⠀⠀⠀⠘⢦⡙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⢠⠶⠞⣁⡀⢧⣄⡥
⠀⠀⠘⠋⠀⠈⢧⡰⣏⠀⡤⠙⢀⠤⠓⠂⠀⠀⠉⠉⠃⠀⠀⠀⢠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠦⡼⠃⠉⠒⢤⡀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⡽⠲⠤⣙⣀⠈⠓⠀⢦⡀⠀⠀⠀⠀⠀⢀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢇
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⣿⠀⠀⠀⠀⠱⠆⠀⠀⠀⠸⣏⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠒⠀⠀⠀⡖⠐⡄⠀⠀⠀⠀⠈⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠉⣇⣀⣀⣤⠕⠀⠁⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠐⢦⠀⢸⡇⠀⡶⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⠈⠡⡞⠉⠑⠀⠀⠐⠀⠀⠀⠀⠀⢠⠟⣠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⠈⣱⠖⡊⠉⣭⠉⠐⠃⢠⡞⠀⣀
⠀⠀⠀⠀⠀⢀⡤⣄⣠⡞⠀⠀⠀⠰⠎⠧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠉⠑⠒⣆⢠⠒⠀⠀⠀⠀⠀⠀⢀⠈⢩⠟⠁⢾⡀⢀⣓⠂⢸⣁⠀⡀⠴⠋
⠀⠀⠀⠀⠀⢸⣀⣤⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠀⠀⠀⠀⠀⠀⠀⠈⢙⡏⣞⡁⢠⠆⠘⠆⠀⢠⠆⠸⠅⠀⠶⠚
⠀⠀⠀⠀⠀⠀⠈⠀⢸⡀⠀⠀⢀⡆⡀⠀⠀⠘⠓⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢺⣅⡤⠀⢈⡃⠰⡉⢀⡬⠃⣴⠂⠀⠠⠴⠆
⠀⠀⠀⠀⠀⠀⠀⣠⣨⠇⠀⢮⡀⢘⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠾⣏⠓⠀⡫⠄⠀⣖⠈⠓⢠⠌⠁⠀⢤⣠
⠀⠀⠀⠀⠀⠀⠀⢧⣀⡞⠙⠢⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⢦⡸⠥⠀⡴⠈⢠⡖⠈⠃⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠲⡤⣍⣀⠀⣹⠀⠲⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠷⠀⠘⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠄⠠⠤⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢠⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠈⡦⢤⠄⡀⠠⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠠⠠⠀⠀⠄⡺⠃⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⡀⠠⡀⠉⠒⢢⠖⢚⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠂⠴⠂⠁⠀⠃⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠀⠂⠊⠡⠈⠒⠂⠄
⠀⠀⠀⠀⡰⠄⠀⠑⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⠠⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠈⢄⠀⠀⠈⢀⠈⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⢀⠸⣄⠠⠁⠀⢱⠀⠑⢄⠀⠀⠀⠀⠁⠀⠁⠀⠀⡠⠈⠓⠀⠠⠴⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡻⡀⠸⠠⠀⠰⠓⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠈⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣄⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⣿⠉⠉⠉⠉⣩⠭⠤⢄⣀⣀⣈⡀⠀⠀⢠⡀⡀⠈⠉⠉⠉⠉⢩⣍⡉⠉⠉⠉⠉⡩⡭⢥⡉⢉⣉⠏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⠙⠉⠉⠉⠉⣿
⠀⣿⣤⣤⡀⠘⠁⠀⢀⠈⠉⠙⢛⣷⣶⣶⣾⣿⠀⠀⠀⢢⣄⣀⣀⣶⠙⢯⢻⡶⡶⢥⡮⡦⢥⣤⠤⠄⠀⠀⠤⠤⠄⠰⠒⡀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⡿⢿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣷⣶⠂⠺⡅⠄⢻⣥⣷⡈⠃⠙⢸⣦⣵⣄⠀⠀⠀⠁⠄⠠⠟⠢⠠⠈⠄⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⣿
⠀⣿⢥⡄⠘⢿⣿⣿⠟⢿⣿⣿⣿⣿⠟⢿⡝⠿⠉⢿⠓⠄⠾⠈⢆⠱⢌⠃⢀⠱⡀⠑⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢇⠀⠀⡇⠈⢧⡀⠈⢦⠀⠀⠀⣿
⠀⣿⣷⣽⣦⠀⠈⢿⣄⠀⠸⣿⠛⡈⠀⣾⠀⡀⠀⢶⠀⠀⠰⡌⠀⠀⠁⠀⠣⡁⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡇⠀⠀⠱⡄⠀⠳⡀⠀⣿
⠀⣿⣷⣄⠹⣷⡀⢈⠻⣆⠀⠛⠹⡆⠀⢸⠃⠡⠀⠀⢂⠀⠀⠸⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⢄⠀⠀⢠⠄⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⢆⠀⠙⡄⣿
⠀⢿⣿⣿⣷⣿⢅⡞⡉⣿⣆⠀⠀⠀⠀⠙⡗⠀⠀⠀⠠⣄⠀⠀⠡⠀⠀⣠⠀⠁⠀⡼⠀⠀⢐⡃⠀⠂⢠⣞⠂⠀⠀⠀⠀⢘⠀⣀⡀⠀⠒⠈⠣⠀⠀⣿
⠀⢺⣿⣎⠛⣿⣿⣿⣿⢿⣿⣿⣷⣄⡇⣰⣤⣤⡀⠸⠀⢤⡐⠄⠈⣤⠊⠄⡤⢴⡶⢡⡄⠁⡈⢐⡋⠀⠄⢩⡋⢉⠓⠀⢠⠏⠀⢸⣏⠁⢘⡀⠀⠀⡺⢻
⠀⢸⡝⣧⣤⢾⢿⡟⠋⠀⣟⠀⠻⣾⡿⢿⡏⠁⠙⠁⠀⠽⡿⣿⣙⢃⠰⡢⢧⣈⡄⢈⣵⡇⣿⢺⡇⢨⠖⠉⠫⡼⠓⠲⠀⢣⢦⡰⠿⣦⣬⡃⠀⠀⣧⣿
⠀⢸⣿⣿⣽⠇⢀⠠⠰⢶⣿⣷⡧⢮⣱⣤⠇⡰⠿⢠⣧⠾⣀⣽⣿⣷⣀⣱⣐⣍⠠⠋⢀⠟⠻⠿⢻⠏⠁⠀⢠⡇⠀⢸⠑⡾⠀⠉⠀⡈⠉⢳⣄⡦⡜⣿
⠀⢸⡟⠉⢻⣧⠀⠐⣺⣴⠟⠛⢿⣦⠠⠤⢏⣴⠒⠈⣛⢁⡏⠀⣿⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⣼⠀⢀⣱⡀⢱⠀⢠⣴⢃⣶⡃⢀⣃⠀⠀⡿⣞⡶⣹
⠀⢸⣛⣿⣶⠟⠃⢠⠛⠁⠈⣩⡷⠂⢀⣠⣶⠿⡦⠀⣄⢈⡷⣦⠈⣥⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢿⣾⣄⢈⠷⡦⢀⣉⣿⡽⠿⡿⢧⣼⣶⠴⣿⣿⣿⣿
⠀⢸⣳⡟⠁⠀⠀⠀⠰⣤⣀⣾⣧⠞⠉⠂⣼⠵⠀⢤⠿⠛⠿⣿⣻⠛⠋⠈⠉⠑⠀⠐⠀⡄⠠⢠⣄⣿⣟⠫⢄⣻⣻⣯⣽⡀⣦⠠⠶⠎⢹⢰⣿⣿⣏⣻
⠀⢸⡟⠙⠶⠒⠀⠃⠀⠀⢹⡅⠀⡄⠀⠀⢙⣿⠖⠣⠀⠔⠀⠈⣤⣀⣰⡄⠀⢀⠠⠔⠚⠁⠐⣶⣿⣿⠃⠀⢸⣧⡟⢨⡛⢷⡄⠀⣶⠄⢹⣞⣏⠋⢿⣿
⠀⢸⣿⡦⠦⡿⠒⠒⢰⡺⠏⣁⢤⢇⠀⣼⣧⣧⡬⠀⡤⣀⠁⠀⠀⠉⠯⢉⡿⢥⣀⠀⠀⠻⠁⢘⣿⣿⣆⡴⣮⣿⢾⠾⣅⠽⡷⣴⣟⢛⣮⣿⡛⢿⢫⣿
⠀⢸⠛⣷⡤⠛⠀⡶⠚⠳⢏⣹⢀⠛⢠⣿⡿⠛⢳⣄⢀⡇⠈⠁⠀⠀⠀⢹⡟⠀⠀⠙⠀⠀⠀⡙⠍⢿⡏⠀⣿⠅⠈⢀⡬⠁⡁⢈⣽⠿⣆⠠⣄⢸⡿⢿
⠀⢸⠀⢿⣇⣴⣥⡀⣰⡛⠚⢇⡀⢢⡟⡳⣾⣦⣀⠋⠉⠀⠀⠀⠀⠀⠀⣸⣿⠐⡀⠀⠀⠀⢀⢷⡀⠾⣛⢲⣿⣶⣶⣻⣿⣤⠁⣨⣷⡢⣿⣿⠆⠘⣇⢼
⠀⢸⠀⣼⣿⡏⠙⣿⣽⠋⠑⠶⡆⠀⠑⢺⣿⢿⣯⡿⠚⠋⠀⠀⠀⠀⠀⢹⣿⣆⠀⢀⠙⠳⠿⣚⡥⡀⠨⢟⡽⠧⠤⢭⣙⣋⣉⣉⣿⢞⢡⣃⣠⢾⡏⢾
⠀⢸⣰⣿⣿⡄⢠⠸⣆⠀⠀⠀⡇⠀⠀⣼⠏⠈⠈⠀⡄⠀⣴⣦⣤⣀⠀⣸⠿⣿⣶⠬⢙⡅⢻⣝⠦⠄⠷⢾⡟⠀⢸⣷⡾⡭⠭⣶⣽⣂⡿⠉⢹⠂⣶⣾
⠀⢸⣿⣿⣿⠁⠀⡆⠹⠀⠀⠐⠃⣶⣶⣷⠆⡸⠶⠦⣤⡶⠀⠅⠪⣿⣶⣿⣦⡤⠛⠲⣾⠢⠤⢍⣗⠾⣿⣿⡁⠀⠀⡛⣍⠀⠀⠈⠙⡇⠤⠄⢈⠀⠀⢻
⠀⢸⣽⣿⣿⠀⠀⢻⡀⢳⡀⣞⣀⣾⣟⣿⣿⣅⠈⠁⠘⢷⣄⠈⠁⠙⣿⢿⣶⡗⠀⠀⠙⣆⠀⠉⣙⠃⠟⠿⠠⣴⣦⣄⠀⠀⠀⠀⠀⠻⢄⠀⠀⠀⢄⣬
⠀⢺⣿⡏⢸⡇⠀⠸⢧⡀⠿⠏⣵⣿⣿⣿⣿⢿⣿⣦⡎⠈⣿⣿⣶⠲⢉⠀⢩⣤⠀⠀⢰⡟⠓⡾⠀⠛⠀⠁⠂⠠⠀⠈⣁⠲⡀⠠⠀⠀⠉⠀⣜⠀⠀⢼
⠀⢸⣿⡟⣺⡇⠀⠀⠀⢹⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⢿⣿⠁⡶⡏⠀⠘⣶⡂⠀⣾⢠⠈⠁⠀⠀⢠⡁⢄⠐⢧⣀⠉⠀⡘⠀⢀⣀⣄⠀⠀⡀⠀⢸
⠀⢸⣿⣿⡾⠷⠖⣾⡗⠘⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠼⣿⣿⡗⠙⢶⣼⣿⠇⣰⠇⠁⠠⢀⣢⠤⡤⢀⣡⣀⣀⣀⣀⣮⠥⠲⠦⠀⠀⡀⠀⡷⠀⢸
⠀⢸⣿⣿⢈⡀⠀⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠆⣿⣻⡅⢴⣬⣿⣿⣿⣺⣷⣦⣤⡦⠄⠀⠀⣯⠀⠉⠉⠉⠈⢩⣔⠂⠀⠀⠀⠈⠁⠀⠀⢸
⠀⢸⣿⣿⣿⡇⠰⠃⣄⠀⠟⠛⠛⢟⣿⣿⣿⣿⣿⣿⣿⣿⡯⢹⣎⡑⢶⣾⣿⠿⢿⠿⣿⣿⠿⣿⣫⡄⢠⡟⢠⡄⠀⢀⢀⠔⠁⠈⠈⠉⠁⠀⣸⠀⠀⢨
⠀⢸⣿⣿⣿⣤⣆⣘⣨⣷⡄⠀⡄⠠⣀⣀⣭⡉⠀⣈⣿⣯⣄⣾⡿⢿⠿⠯⠭⠉⠉⠈⠁⠀⠈⠁⠀⠀⠘⡀⡀⠃⡀⠀⡄⠀⠀⠠⣄⡈⣠⠀⡯⠀⠀⢸
⠀⢹⣈⣛⣛⣏⣛⣛⣉⣉⣀⣃⣀⣀⣀⣀⣀⣙⣂⣈⣀⣙⣁⣀⣀⣘⣛⣓⣚⣛⣂⣀⣈⣁⣀⣛⣂⣀⣀⢈⣁⣀⣀⣀⣈⣁⣁⣀⣀⣀⣀⣀⣀⣂⣀⣸
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄⡀⢠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⢸⡿⠀⠀⠀⢠⡇⠀⠀⠀⢸⣦⣤⣤⣀⡀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠿⣿⣿⡇⠀⠠⣤⣼⣿⠋⢹⣿⣿⡂⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠾⣿⣿⣿⣿⠿⠿⠿⠿⠟⠓⠲⠿⠿⢿⣶⣿⣿⣿⣧⣤⣶⣶⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠉⠋⠉⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠧⢾⠋⠿⣆⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⣀⠀⢸⣿⣧⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡟⠀⣀⠐⡄⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠈⣿⡠⣻⣾⣷⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠐⠀⣼⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⡀⣠⣿⡇⠀⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣦⣶⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣵⣾⣿⣇⣿⣿⡄⠀⣾⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢹⣿⢘⣿⣿⣴⣿⣿⢁⠽⡻⢿⢿⣿⡿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡍⡈⠛⠛⠉⠀⠀⠀⠱⠤⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⣧⠐⠁⠀⠀⠀⠐⠄⣗⡘⠉⠓⠒⠉⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣆⢿⡻⣤⣐⣤⠤⠀⠀⣽⣶⣦⣥⣀⣀⣤⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣯⠙⠌⠳⢬⣟⣁⠀⠀⢠⣿⣤⣦⡈⠉⠉⣉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡀⠀⠀⢐⣽⣶⣤⣬⣵⣶⢟⢅⡠⣰⡴⣞⣻⢮⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣤⣶⣾⣿⣿⣿⣿⣿⣯⠺⣿⣶⢟⡽⠛⢉⡛⣿⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠳⣤⡄⢰⣆⣭⣩⢦⡿⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠧⠉⠘⠛⠁⠈⠓⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⢰⣯⠀⠀⡘⢆⠀⠀⠀⠀⠀⣠⠀⠀⡀⡀⠀⠐⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⣿⣟⣡⢸⡵⡇⡠⢸⡏⠀⠀⠀⠀⠀⠘⣿⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⣩⠓⣿⡿⠿⢞⣿⣴⢇⣨⣭⡤⢤⠂⠆⠀⠀⢈⣿⣳⣾⣿⣿⣆⡀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⣾⣿⣇⣿⣱⣿⣷⡶⢞⣯⣄⣦⣤⡄⣉⠀⢀⣀⣴⣿⣿⣿⣟⢿⣿⡄⠀⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⣙⣿⣿⣿⡉⠙⣻⣯⠿⠿⢶⠶⡤⢼⣤⣀⣠⠛⣉⣽⠃⣸⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠁⣠⣿⣨⠿⣷⡿⢿⣷⣾⡟⠀⠀⢀⣨⣅⣀⠀⠐⠉⠄⠈⠃⣠⡞⠉⣏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⢩⢄⠉⠽⣪⣤⡾⢿⡟⣄⠠⠔⠻⢿⣿⢿⣿⣿⠚⠛⠉⠉⠙⣿⣤⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⠘⣿⠦⣶⠶⠚⣯⡀⠀⣾⠟⠁⢠⡄⠀⠀⢹⡌⢹⠟⠀⡀⣀⣠⠞⠛⢿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⢹⣶⠟⠚⠛⠁⠋⢰⣿⡆⢤⣌⣧⠀⠀⠈⣿⡾⠷⠟⠛⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠙⢿⡼⠟⢭⠓⠂⠐⠶⠋⢹⡿⢨⣿⣿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢣⠄⣧⡏⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡾⢟⠼⡃⠀⢶⠶⢀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⠟⢀⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⢀⣀⡀⠀⠀⢹⣶⠃⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣈⣏⣀⠀⠀⢀⣀
⠀⠀⠀⠺⢴⡣⠔⠊⠉⠀⠉⠓⠦⢼⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡦⢚⣭⠡⣶⠌⣩⡢⡜⠋
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣤⡈⣉⠰⣶⠀⣉⢡⣺⡆
⠀⣰⢳⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡚⡆⠀⠀⢶⢦⡤⢒⣮⠉⠉⠓⠢⣤⢶⠆⠀⠀⢴⡶⢴⢁⣉⠈⡛⢱⡦⢀⡛⢡⣄⢸⠴⣶⠄
⠀⠈⠋⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠙⠁⠀⠀⡸⠃⠀⠀⠋⠃⠀⢠⢶⠎⢳⡀⠀⠀⠀⠀⠀⢏⠛⣘⡛⠀⣶⠟⢫⣮⢁⠎
⠀⠀⠀⢀⣨⣆⡀⠀⠀⠀⠀⠀⣀⣸⣁⡀⠀⠀⣀⣰⠃⠐⠿⡟⠁⢀⡀⠀⠉⠀⠀⢧⣀⡀⠀⠀⢀⣈⠷⣝⡁⢾⠆⢶⣦⡥⢏⣀
⠀⠀⠀⠹⠬⠇⠈⠙⠲⣰⠒⠋⠁⠹⠥⠃⠀⠀⠛⠺⣤⣔⡃⠀⠀⠫⠼⠀⠐⢤⡄⡿⠛⠃⠀⠀⠘⠋⠀⠀⠉⠙⣟⠉⠁⠀⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠀⢲⢀⡄⠈⠉⠛⡻⠃⠀⠀⠀⠀⠀⠀⢀⣀⡀⠘⠛⠰⣏⣱⠄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣶⠀⠀⠀⢀⡀⠀⠀⠀⠀⣾⠝⠢⣄⣺⡿⣀⣀⡤⠺⣵⠄⠀⠀⠀⠀⠀⠀⠺⢴⢧⡤⢤⣴⡋⠧⣄⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠉⢧⣀⣀⠤⠿⠤⣀⣀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠧⡴⡃⠀⠀⠙⢚⠗⠚
⠀⠀⢀⡀⠀⢀⡴⠫⣶⠄⠯⠇⣴⡍⢢⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢛⣦⠞⠀⠀⠀⠀⠁⠉⠀⠀⠀⠈⡆
⠀⠀⠘⠙⠒⡞⢶⡆⣤⡔⠿⠂⡤⡔⠿⢷⠒⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⠀⠀⠀⠀⠀⠈⠉⠈⡇⠀⠀⠀⡀⠀⠀⠠⢌⡶⢲⣓⣤⢲
⠀⠀⠀⠀⠀⣇⣤⣀⣍⠑⠿⢢⣬⠱⡶⢸⠀⠀⠀⠀⠀⠀⢴⠦⣠⠴⠚⠉⠁⠈⠉⠒⠦⣀⡴⡦⠀⠀⣹⠀⢰⡚⢶⠁⠀⠀⠀⠓⡻⠁⠉⠊
⠀⠀⢰⠖⠊⠳⡈⣨⡍⠀⠻⠃⡁⢶⢆⠞⠑⠲⠆⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠘⣍⡷⠙⠢⣍⡉⠀⠀⣀⣤⣸⣁
⠀⠀⠀⠀⠀⠀⢙⠮⣅⣻⢇⣘⣿⠴⡋⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⠵⣏⠉⠀⠀⢯⠴⠃
⠀⠀⠀⠀⠀⠲⠏⠀⠀⠀⣽⠀⠀⠀⠹⠖⠀⠀⠀⠠⣖⣚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⣲⠄⠀⠀⠈⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢠⠒⢦⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣎⡹⣀⣀⣀⡼⠳⣅⠀⠀⢀⣀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢀⠀⢴⠂⢰⡂⢰⡦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠈⠰⣏⣽⠂⠀⠙⠢⢞⠤⠗⠀⢴⡡⠲⢄⣀⠀⠀⠀⠀⣀⡤⠶⣝⡦⠀⢀⡄⢉⠶⣊⡉⢩⢭⠉⣒⢾⡉⢠
⠀⠀⠀⠀⠀⡴⠦⣠⠎⠀⠀⠀⠘⠈⠃⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠉⡇⣞⠉⠁⠀⠀⠀⠀⢠⣌⡸⢃⠀⠓⠃⡬⡅⠘⠿⢀⣽⡅⣡
⠀⠀⠀⠀⠀⠑⠚⢳⡀⠀⠀⢀⠀⠀⠀⢀⣲⡤⣄⢧⡠⢤⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣌⣨⡇⠳⠃⢾⠆⣉⡁⠰⣷⠈⡊⢹⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣠⢼⡴⠂⠀⠀⠘⠴⡿⠋⠲⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢸⡷⣿⠢⣶⡄⢛⡃⢞⠆⠸⠟⣸⠉⡃
⠀⠀⠀⠀⠀⠀⢰⠛⡵⠢⣌⠒⠋⠀⠀⠀⡀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣱⡄⣴⡮⢀⣘⠚⣀⡘⠝⣠⡋⠙
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢙⢦⡖⠛⠉⠉⣎⡹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣨⠦⢄⣘⣚⣀⣳⠧⢾⣁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠾⠀⠽⠅⠸⠆⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠉⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣀⠀⠀⣀⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠛⠃⣀⣴⠟⠹⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⠏⠻⣦⣀⠘⠛⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⡤⠶⠛⠉⣀⡴⠂⠘⣿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⣿⠃⠐⢦⣀⠉⠛⠶⢤⣬⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠈⢉⡠⠖⠀⠘⣿⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⠃⠀⠲⢤⡉⠁⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⢠⠖⠋⣀⠘⣿⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⠃⣀⠙⠲⡄⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠰⠛⣁⡄⠘⣿⣆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣿⠃⢠⣈⠙⠆⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠚⣁⡤⠆⠘⣿⣆⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⣰⣿⠃⠰⢤⣈⠓⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠉⠀⠀⠀⢨⣿⠂⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠐⣿⡅⠀⠀⠀⠉⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⣀⣤⣾⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⣷⣤⣀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⡶⣿⡟⢿⡌⢿⣦⠀⠈⠻⠿⣶⣶⣶⣶⠿⠟⠁⠀⣴⡿⢩⡿⢻⣿⢶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⢻⡌⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢡⡟⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠘⣿⣆⢻⡌⢻⣧⠀⣀⡀⠀⠀⢀⣀⠀⣼⡿⢡⡟⣰⣿⠃⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡈⠉⠛⠛⠿⣶⣦⠀⠀⠘⢿⣆⢻⡄⢻⣿⢛⣿⡇⢸⣿⡛⣿⡟⢠⡟⣰⡿⠃⠀⠀⣴⣶⠿⠟⠛⠉⢁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡷⠀⠀⢀⡴⢻⣿⠀⠀⢠⣾⣿⡆⢻⣠⣵⡿⠋⠀⠀⠙⢿⣮⣄⡟⢰⣿⣷⡄⠀⠀⣿⡟⢦⡀⠀⠀⢾⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣏⡀⢀⣴⣯⣀⣸⣿⠀⠀⢿⣷⣶⡾⠟⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣶⣾⡿⠀⠀⣿⡇⣀⣽⣦⡀⢀⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⠉⠛⠻⠿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⠿⠟⠛⠉⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣇⢠⡆⣰⡄⠉⣩⣍⠛⢛⣛⠛⠛⠛⠛⠛⠛⠛⡛⢛⣛⣛⣉⠉⡥⢤⡄⣶⡄⢾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢻⡾⢠⣧⣽⣸⡇⠈⢣⡏⠉⠓⣼⢸⢷⣸⡇⣼⢳⡈⢹⠁⢸⢸⡇⢀⡟⣿⠙⡿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢷⣶⣬⣄⣈⡋⠓⠶⠋⠳⠤⠗⠿⠸⠀⠻⠧⠟⠛⠇⠸⠀⠘⠈⢛⣋⣠⣤⣶⡾⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠘⠛⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⣴⣿⣿⣿⣿⣿⣿⡤⠤⠴⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠚⠛⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠏⠉⠀⢀⣉⡘⠂⠀⣠⣖⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡍⠀⠀⠀⠀⠈⠛⠁⠀⠀⠛⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢽⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣼⣿⣿⡇⠻⢷⣤⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣭⣴⡆⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⡐⢰⡟⡆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢿⡉⠻⡋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠚⠉⠀⠀⠈⠲⠌⠓⠄⠀⠀⠀⠀⠱⠀⢀⠀⠀⠀⠀⠀⠈⠀⢰⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢸⠀⠀⠰⡆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣⡀⣼⠀⠀⠀⠇
⠀⠀⠀⠀⡰⠋⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⣸⣧⡖⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⢷⡏⠀⠀⠀⢰
⠀⠀⢀⡞⠁⠀⠀⠀⠀⢙⣦⡀⠀⠀⣀⢴⣿⣟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⡞⠀⠀⠀⠀⢸
⠀⢀⠏⠀⠀⠀⠀⢀⠔⢯⣻⣷⣰⠞⠁⠈⠻⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡸⠀⡇⠀⠀⠀⢸
⠀⠊⠀⠀⠀⠀⠀⠑⠦⡀⠉⠻⠏⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠤⠤⠆⠈⠀⠀⠀⣸⠀⣠⣇⠀⢻⡀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⢩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡏⠻⢧⠀⠁⠀⠀⠇
⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⣀⠀⠀⠀⠉⠒⢄⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⠀⠀⠈⠒⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⢻⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢗⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣽⣿⣟⣛⣿⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⢿⣮⠿⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⡀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⠀⠀⡈⠀⢠⣠⣄⣀⣠⠄⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠲⠶⠖⠊⠀⠀⠀⠈⠉⠉⠉⠀⠀⠱⢄⣀⣠⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠑⠒⢤⣀⣀⡤⠎⠀⠀⠀⠀⣀⣤⡤⠶⠒⠒⠒⠒⠶⢄⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⣠⠞⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⡩⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠀⠀⡞⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⢰⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⣀⡇
⠀⠀⠀⠀⠀⠰⠖⠶⢏⠁⠀⠀⠀⠈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠁⠁⠉⠓⡄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⣀⠀⠆⠀⠠⢀⡀⠀⢀⠄⠒⠈⠉⠑⠢⠜⠀⠀⠀⠀⠀⠀⠀⠘⠋⠐⠒⠬⣕⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠈⡆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁
⠀⠀⠀⠀⢀⠀⠀⡠⡢⢂⠔⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⢂⠀⠐⠁⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⠄⢀⠔⠁⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⢠⠒⠈
⠀⠀⠀⠀⠀⠀⠁⠒⠂⠈⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠐⢁⠔⠁⠔⠁⠀⢀⠄⡡⠊⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⠀⠀⣀⠶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠒⠄⠀⠀⠀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠤⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣲⢾⠆⠀⠻⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⡋⣽⣿⣥⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣯⣾⢟⠕⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡠⣪⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⡴⣫⣾⢟⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⠟⠁⡠⣺⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢙⣿⣟⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡦⠙⠦⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠳⡏⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣌⢰⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿






⠀⠀⠀⠀⢀⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠋⠛⠛⠛⠛⠛⠛⠛⢧⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣠⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⡷⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⣇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡷⠀⠀⠀⠀⠀⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠖⠒⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⡗⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠁⢀⢀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂








⣤⣤⢠⡤⠄⣤⢤⡀⣄⢠⡄⣠⠤⠀⠀⢠⡀⢠⡄⢠⢠⣤⣄⠀⢀⡤⣤⢀⣤⣄⢠⡀⢠⢠⣤⡄⢠⠀⣤⡄⣤⢀⡤⣄⢠⡄⢠⢀⣤
⢀⡃⢸⣓⣀⣟⢻⣁⣟⢛⡇⣨⣷⠀⣠⡟⢷⣸⡙⣿⣸⣧⡞⠀⠸⣧⣄⢻⣤⡾⣸⡙⣿⢸⣤⡞⣸⡀⣾⠀⣿⢹⣤⡾⢸⡋⣿⣈⣙
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣠⠄⠀⠀⠀⠀⠀⣀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠁⠀⠀⢤⠉⠀⠀⠀⠀⢠⡾⠋⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠃⠈⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⣠⡟⠀⠀⢀⡾⠁
⠀⣠⣤⠴⢤⣄⠀⠀⠀⠠⠦⠶⠶⠶⠛⠉⠀⠀⠀⠀⢀⣤⠾⣿⠁⣤⠶⠋
⣸⠋⣠⣤⣄⠈⢷⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠛⠋⠁⢰⡇
⣿⠀⠿⠀⢸⡇⢸⡟⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠘⣇
⢹⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⢸⡇⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠙⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠉⠁⠀⠀⠀⣤⣀⠀⣀⣀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠰⣄⠀⠙⣿⠋⠉⠛⠋⠁⠀⠈⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⢷⡄⠈⠓⠀⠤⠀⠀⠀⠀⠀⠀⠀⣀⣘⠋⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⣤⣄⣠⣴⠟⠀⠀⠀⠀⠀⠀⠻⣆⣠⠶⠛⠀⠀⠀⠀⠀⠀⠈⠉⠉⢳⡄⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⢀⡼⠁⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠁⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠⣤⣤⣤⠞⠋⠈⠛⠐⠂⠘⠁
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣀⣀⡴⠃⡴⢲⡀⢳⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣫⠾⠛⠛⠉⠉⠀⠀⠙⠋⠀⠀⠈⠛⠛⠳⠾⢷⣿⣯⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢧⡤⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣽⡳⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣫⡇⠀⡖⢒⣶⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⡱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣽⡇⢀⡇⣞⣹⠆⣼⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⣿⢞⡵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡇⢸⡗⠋⠙⠒⡿⠀⠀⠚⠛⠛⠛⠋⠀⠀⠀⣷⢯⢞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣽⡇⠘⠓⠒⠒⠒⠃⠀⠀⠲⠶⠶⠶⠶⠀⠀⠀⡿⢣⢞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣻⠀⠀⣤⣤⣄⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⢽⠀⠀⣧⣷⡾⠁⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⡵⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣡⢾⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣼⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣟⡵⣿⠀⠀⣟⣀⣼⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣯⣫⢾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢞⣿⠀⠀⠈⠉⣠⠆⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⣿⡷⣻⠀⠀⠀⠀⠀⠀⣠⠶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡷⢋⡏⠀⢰⡯⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣾⣿⠀⠀⠀⢀⡤⠞⢁⣴⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣫⡇⠀⠈⠛⠛⠁⠛⣻⠷⠛⠒⠒⠒⠒⠒⠀⠀⢸⣿⡵⣿⠀⣠⠴⠋⣠⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡷⢫⡇⠀⢰⣯⣻⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢏⣴⡿⠞⢁⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣻⡇⠀⠈⠛⠛⠁⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣸⡿⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡿⡽⣫⢏⡽⣹⢋⡿⡿⢻⢟⡿⣻⢻⢞⡖⡷⣻⠟⠁⣠⠶⡏
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠽⠧⠯⠼⣵⣯⣾⣴⣧⣾⣼⡵⠯⠾⢾⡿⣇⡴⠿⠷⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠂⠈⠉⠉⠉⠉⠉⠉⠉⠓⠒⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⠀⠉⠓⣄⠀⠀⠀⣿⠀⠀⠀⡇⠉⠙⠲⡄⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣆⣀⡠⠏⠀⠀⠀⣿⠀⠀⠀⣇⣀⡤⠔⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠿⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠈⠳⣤⠀⡄⠀⡏⠠⠄⠰⠇⠀⠤⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢆⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠁⠈⠄⣀⣀⣀⣁⣀⣀⣀⣀⣒⣀⠀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠤⣄⠀⠀⠀⠊
⠀⠀⠐⠲⡄⠀⢀⠟⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠒⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠂⢀⡤
⠀⠈⢷⡄⢠⡆⡸⠀⡀⠀⢠⡀⠀⠀⢀⣀⠢⢀⠐⠀⢇⢈⡀⠀⠀⠀⣄⠀⢀⠄⠀⠀⡠⠒⡀⢠⠀⠀⠀⢇⢔⠜⠁
⠁⠀⠘⡘⠎⠂⠀⠨⠴⠀⠠⠱⠰⠄⡈⠀⠙⠀⠹⠇⠂⠀⠛⠒⠀⠀⠈⠀⠑⠈⠠⠊⠄⠀⠣⠀⠃⠀⢀⠀⠀⠒⠃⠀⢀
⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠒⠶⠶⠤⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠲⠶⠤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠶⠶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢠⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⣿⠁⠀⠀⠉⠉⠉⠛⠛⠛⠲⠶⠶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠷⠶⣶⣦⣤⣤⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣼⡇⠀⠀⠀⠀⢀⣴⠞⠛⠉⠉⠙⠛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡿⠁⠀⠀⣰⡟⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⢀⣤⢾⣿⡀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣿⠁⠀⠀⠀⠀⢸⣷⡟⠁⠀⠙⠿⣦⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠙⣷⡀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣾⣿
⠀⠀⠀⠀⢰⡇⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣸⠃⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢰⠏⣿⡀
⠀⠀⠀⠀⣼⠁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⣼⣿⡁⣠⡏⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣾⠀⢹⡇
⠀⠀⠀⢀⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣼⡟⠙⠿⠟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣤⣶⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⣤⣾⡇⠀⠸⣷
⠀⠀⠀⣿⠁⠀⠀⠀⣿⠀⠀⠀⢀⣀⣠⣤⣤⣴⣿⠁⠀⠀⢸⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⣴⣿⣿⡿⣧⠀⠀⣿
⠀⠀⢠⡟⠀⠀⠀⢰⡏⢀⣴⠾⠛⠉⠁⠀⠀⣼⠃⠀⢀⣠⠞⠉⠉⠛⢷⣦⡀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠃⣿⠀⠀⣿⡄
⠀⠀⣸⡇⠀⠀⠀⣸⣿⠛⠁⠀⠀⠀⠀⠀⢸⣿⣶⠾⠟⠁⠀⠀⠀⠀⠀⠙⢷⠀⠀⣠⣾⣿⣿⣿⣿⢿⡏⠀⢻⡀⠀⢹⡇
⠀⠀⣿⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠛⠁⣾⣁⣀⣸⡇⠀⢸⡇
⠀⢸⡏⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠏⠀⠀⢰⡟⠉⠉⠛⠃⠀⠸⣿
⠀⣸⠇⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣾⠿⠷⠶⠶⠶⢶⣶⣿
⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠟⠉⢰⡇⠀⠀⠀⣸⠃
⢰⡟⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⢀⣿⠀⠀⠀⢠⡏
⢸⡇⠀⠀⠀⠈⠛⠛⠛⠛⠷⠶⠶⠶⠶⢦⣤⣴⣶⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⣸⡏⠀⠀⠀⣾
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠿⠿⠿⠁⠀⠀⣸⡇
⣿⣦⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠻⠷⠶⢶⣦⣤⣤⣤⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡾⠍⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠁⡒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣼⣿⣷⠀⢸⠡⢀⠀⠐⢈⡇⠀⠀⠀⠀⠀⠀⠀⣆⣀⣀⣄⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣷⡁⣿⣿⢀⣄⢺⡔⣇⢀⠄⠉⡓⠀⠀⠀⠀⠀⠀⠀⠸⠟⠿⢿⣶⣽⣿⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠂⠋⠉⣙⢣⣿⣿⣸⡿⠘⣿⣆⡼⠠⣀⣉⡀⡀⢀⠀⠀⠀⠀⠀⠈⣲⡇⣶⡼⣿⣿⣧⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡄⣿⢇⣴⣶⣬⣅⣬⡴⠏⠉⡄⠈⢀⡑⠒⠆⠀⠀⠀⢨⣿⡾⠿⠿⢶⣿⣿⡎⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠻⢿⡖⣿⢸⡿⠿⢿⣭⣡⢦⣿⣿⣛⠋⠉⠑⢠⠀⢀⠙⠚⣿⠿⢿⣟⡛⢷⡽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣤⣶⡷⣟⡘⢧⡀⠌⣠⡿⠟⠛⠉⢻⡉⠹⠓⠲⠶⢦⡤⠴⢁⡤⠴⠀⢀⡾⣧⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⢀⣠⣾⡈⠞⠷⠂⠥⢣⣎⣷⡿⡏⠁⠀⠀⣀⣀⣧⣀⠀⠀⠀⠀⠠⠀⠀⠃⣀⣶⠋⠐⠇⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⣴⡟⠉⡀⠈⢙⡫⢊⣄⣴⡿⢿⡦⠀⢀⡠⠼⠿⣿⣿⠿⣯⣷⣿⠶⠚⠚⠊⠉⠉⢻⣧⣤⣄⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠉⣽⣐⣁⣈⢉⣠⣴⡟⠃⠁⢀⣸⠶⠖⠁⠀⠀⠀⠈⠻⣆⠻⢟⡽⠄⠀⠀⠀⠀⣀⡴⢿⡾⠾⠏⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠂⢹⣏⢨⣜⣈⣤⣼⠗⠢⠀⢸⣟⠈⠀⠀⢿⠀⠀⠀⠀⢿⡄⠺⢁⣀⣤⣤⣤⣞⠁⠀⠀⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣿⢿⡋⠉⠉⠀⠀⠀⢰⡿⣿⡅⢲⣦⣼⡇⠀⠀⠀⢸⡷⠖⠛⠉⠁⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠉⠿⣠⡾⠛⢆⠉⠒⠀⠐⠒⠉⠀⣿⡟⠠⣽⣿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢃⡀⡻⢸⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡟⣸⠆⠈⢋⣀⣀⡀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛⢩⣴⠉⣰⠀⠈⠛⠉⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡿⠁⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⢹⡟⠀⠀⠀⢀⣀⣤⣤⢤⡄⠀⠈⠛⢿⣶⠖⠋⠁⠀⠀⠀⢠⡖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣰⣟⠀⠀⠀⢸⣿⠿⠋⣀⣤⢏⣀⣀⣀⡞⡎⠀⠀⠀⠀⠀⠀⠈⣿⣷⣞⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠚⠧⣌⢦⡀⢀⣀⣽⠿⠟⠛⣻⠟⠉⠉⠙⢣⡇⠻⠶⢤⣴⠶⢤⣴⡿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡍⠋⠙⠷⣦⡤⠞⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿⡀⠀⠹⣧⠀⠀⢀⣾⠳⣄
⠀⠀⠀⠀⠀⣠⡾⡆⠀⠀⠀⣠⣿⣥⡄⠀⠀⠀⠀⢳⡇⠀⢀⣤⠶⠶⠶⣶⡄⠀⠀⡼⣷⠦⠶⠟⠳⣤⣸⣿⡄⠸⣇
⠀⠀⠀⠀⢰⡟⢰⣧⣀⣀⣼⠋⠀⢠⠇⠀⠀⠀⠀⠀⠀⢰⣻⣅⣀⣀⣴⠟⠁⠀⠘⠱⠁⠀⠀⠀⡠⢊⠄⠻⢷⣤⠏
⠀⠀⠀⠀⢻⣧⠟⠓⠀⣿⡇⠀⣠⠿⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡽⠇⠀⠀⣠⠶⢤⡀⠀⠀⠀⠰⠃⡀⠻⣆⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠋⠉⠳⠬⠁⠀⢀⣴⠾⢿⠄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢻⣦⠀⠀⠀⣼⠛⢷⠈⠤⠴⠚⢻⣷⣼⡿
⠀⠀⠀⠀⠀⢀⡶⢆⠠⡇⠀⠀⠀⠀⠀⠀⢠⣿⠁⢠⣞⠀⠀⠀⠀⠀⠀⠀⠀⢺⠿⠶⠾⠛⠀⠀⣠⣿⡀⢸⡇⠸⣿⠁⠈⠛⠛⠁
⠀⠀⠀⠀⠀⣾⠀⢸⣗⣒⠒⠒⠂⠀⠀⠀⠸⢿⡶⠛⠛⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠿⠁⠀⣿⠀⢀⣀
⠀⠀⠀⠀⠀⣿⣀⡞⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠈⢻⡦⠀⠀⠀⢀⡴⠒⣦⠀⠀⠀⢀⠀⢀⣿⣠⡿⠏⣷
⠀⠀⠀⠀⠀⠈⠉⠀⢾⡇⢰⡟⢿⣦⠤⠖⠋⠀⠀⠀⠀⠀⠀⢸⡻⣦⣤⡾⠁⠀⠀⣸⣿⠁⠀⣸⠀⠀⠀⠨⠭⠭⣭⣿⠃⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⡿⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠉⠀⠀⠀⠀⠈⢿⣀⣰⠃⠀⣀⣄⡀⠀⢀⡏⠘⢷⠟⠁
⠀⠀⠀⠀⠀⢀⣀⠀⠘⣇⠸⣦⣼⠃⠀⠀⣠⢲⡶⢶⣿⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠉⢿⡿⠁⢨⠇⠀⣼⠇
⠀⠀⠀⠀⠀⣿⡉⢳⣤⡿⢂⠀⠀⠀⠀⠀⢻⣌⢳⡀⢹⣿⠀⠀⠀⠀⣠⡿⠛⠛⢻⣷⠀⠀⠀⢸⣃⣠⠏⠀⣴⡟
⠀⠀⠀⠀⠀⣿⡀⠀⣿⠟⢯⡀⠀⠀⠀⠀⠀⠙⠓⠛⠛⠋⠀⠀⠀⢰⣇⣀⣠⡴⠟⠁⣬⢷⡀⠀⠉⠠⣄⡘⠋⣠⠴⣦
⠀⠀⠀⠀⠀⠘⢷⣴⠏⠀⠀⢙⣦⡀⣠⢏⡀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠈⠁⠀⠀⠀⢀⣽⠞⢛⡿⢈⡷⢦⣙⣿⠃⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⠛⢷⣎⠀⠀⠀⠀⣴⠏⠉⠉⠙⢲⣤⠀⡀⡀⠀⢾⣧⣤⣮⠴⠋⠀⠀⢹⡇⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⢻⡛⠳⢦⣄⡙⠶⢤⣤⣶⠾⠛⠀⣿⣷⠶⠶⠚⠉⠀⠀⠀⠀⠀⠸⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠃⠀⢀⣨⣿⣿⠛⠛⠛⠉⠉⢉⡿⠿⠖⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣈⣱⡄⠀⠀⢰⣇⣀⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⡀
⠀⠀⠀⣼⡿⣿⡿⢿⣿⣷⡀⠀⠀⠀⠀⣾⣿⡿⣿⣿⢿⣧⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⢿⣿⡿⣿⡄
⠀⠀⠀⠙⣀⠀⡀⠸⠿⣿⠁⠀⠀⠀⠀⢻⢏⣭⠀⢤⠸⡏⠀⠀⠀⠀⠀⠀⠫⡌⠉⡀⢸⢿⡿⠀⠀⠀⠀⠀⠘⡇⢠⠈⢨⠀⡿
⠀⠀⠀⠀⠈⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠘⠀⠴⠀⠀⠁
⠀⠠⠄⢀⣀⠚⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠓
⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣿⡀
⠀⠀⠀⣿⠛⠋⠿⠛⢻⠇⠀⠀⠀⠀⠀⠸⡇⢠⠈⢩⠀⡿⠀⠀⠀⠀⠀⠀⢿⠁⡀⠉⣁⢸⠏⠀⠀⠀⠀⠀⠸⡏⢠⠈⢩⠀⡿⠁
⠀⠀⠀⠘⡀⠁⠀⠁⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠗
⠀⠀⠀⠀⣃⠤⡤⢜⠀⠀⠀⠀⠀⠀⠀⠀⡈⡂⣄⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⣈⣭⣍⣉⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⣬⣤⣀⡁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀
⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠘⡏⠈⠉⠉⣿⣿⡟⠀⠀⠀⠀⠀⠸⡟⢉⠙⠛⠉⡿⠁⠀⠀⠀⠀⠀⢿⠁⡄⠉⡅⢸⠇⠀⠀⠀⠀⠀⠸⣏⢠⠈⢩⠀⡿⠁
⠀⠀⠀⠉⢁⠈⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠑⡈⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡌⠀⠈⡈⠁
⠀⠀⠀⠀⠐⠒⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐
⠀⠀⠀⠀⡠⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠘⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⢉⣓⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⠂⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡇⠐⠣⣀⣘⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡉⠙⠇⡈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡴⠛⡆⣠⠁⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⢀⡔⣁⠤⠼⠒⠲⠤⣄
⠀⠀⠀⢀⣞⣀⣠⡇⠘⢦⣀⣠⠞⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢰⣀⠞⠊⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⣠⠊⠉⢰⠃⠀⠀⠀⣀⠯⠂⠀⠀⠈⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⡴⠁⠀⠀⢀⡱⠖⠂⠉⠀⣀⠀⣀⡄⠀⠀⠈⢆⠀⠈⠀⡗⠂⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⡼⠳⣶⠖⠈⠀⠀⠀⣀⠔⠊⠁⠙⠳⠇⠀⣏⠛⠢⢦⠀⠀⠛⠀⠀⠐⡎
⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⣦⠐⠀⠀⠈⠣⣀⠀⠀⠀⢀⠇⠀⠀⠂⠀⢸⡎
⣀⣀⡠⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡌⠂⢄⣀⡼⠓⠢⣀⡸⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠈⡦⡀⠀⠨⡃⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⢠⠃⠈⠢⢄⠘⠤⣀⢀⡎⠈⠒⢄⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⠀⡼⠀⠀⠀⠀⠑⠒⠈⠉⠀⠀⠀⠀⠈⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢠⢲⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⢀⠹⡆⢀⠀⠸⣆⠱⣾⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠈⠀⠀⠀⢡⣾⣇⡀⠻⠑⠈⠋⠿⠇⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⢀⣠⣄⣤⣴⣶⣿⠗⠒⠀⢤⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠙⣿⡿⠛⠁⠀⢠⠀⠜⣻⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣠⡴⠆⠰⢀⣿⡇⡇⠀⠘⡄⢧
⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠂⠘⣡⡎⢀⢼⣿⣷⣿⠀⡇⣧⢸
⠀⠀⠀⠐⢆⡴⠲⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⠀⠐⠘⢿⡀⢉⣾⣿⣿⣿⡇⠙⠳⢸
⠀⠀⠀⢐⡬⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⡄⠀⠄⣼⠇⣼⣿⣿⠇⢿⠇⠀⠀⠈
⠀⠀⠀⠘⡇⠀⠀⣸⣿⣿⣿⡷⣤⣄⠰⠄⠐⠒⠀⠀⡀⠀⠀⠀⠑⠄⠀⠠⢠⣿⣿⢿⣿⠀⠈
⠀⠀⠀⠀⠁⠀⠀⠿⢻⣏⣩⠀⠀⣟⠀⠀⠀⠀⠀⠸⢋⠄⠀⠄⠀⠀⠀⠀⣀⢵⣿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠘⠀⣶⣶⣶⠶⡶⣄⣈⣈⣴⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⢳⡀⠃⢉⠙⣡⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠁⠂⢘⣛⣫⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⡇⠀⠀⠀⠀⠀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⢸⣷⡆⠀⠀⠀⠀⠀⣹⣿⠟⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠒⠚⠀⣻⣿⣋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣐⣿⣿⣥⣇⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠈⢢⡀⠀⠀⠀⠀⠀⢰⡄⢀⣠⣼⣿⣿⣿⣭⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢢⠀⠀⠙⣷⡵⣦⣀⡀⣸⣿⣿⣿⣿⣿⣿⡿⠟⠛⠻⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡆⠀⢧⣠⡖⣿⣿⡏⢿⡇⢻⣿⢃⣿⣷⢿⣿⣷⡆⢀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⣿⠇⢸⡀⣿⣾⡷⠞⠛⣾⣿⠿⣦⡈⢿⡇⢼⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⢨⠆⢿⣿⠁⠀⢰⠟⠁⠀⠀⠛⢿⡇⢀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁

⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⢦⣄⡀⠀⠀⠀⣽⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⢀⣀⡀⠀⠀⠀⣠⢤⣤⣄⠀⠀⠀⠀⠰⡏⠀⠀⢀⣻⢿⣄⠀⠀⣿⠀⠀⢘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣸⣏⣔⣺⠿⠷⣼⣁⠴⣛⣛⡳⣄⠀⠀⠀⠘⣦⣴⡧⠶⠖⣿⡇⠀⢧⣄⢤⢌⣼⢃⣀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⢻⢉⡤⠹⠿⡶⢻⡏⣷⣾⠉⣤⢩⣿⡄⢀⣠⣤⠿⣿⡀⣀⡐⣻⣁⣀⣤⠟⢿⣿⣿⣥⣽⢯⣀⠀⠀⠀⠀⠀⢸
⠀⢺⡼⠀⡶⢺⣇⣸⣟⣻⡷⢀⠐⠞⡍⠀⢾⢋⡴⠛⠉⠻⣄⣿⢉⢻⣿⣗⣡⡬⠏⢽⢻⣏⢹⣌⡍⡛⠷⢲⡦⣾
⠀⢺⣁⠀⣐⣛⠮⡷⠉⠙⠛⠦⠜⠛⣲⣄⠀⢻⠐⠀⣂⡄⠀⢹⢪⣴⡟⠋⠁⠀⠀⢸⢸⣿⠸⣏⠉⠑⠛⠷⠊⣿
⠀⢿⠑⠳⡾⢗⡊⠛⠒⠶⣶⡀⢹⡟⠻⣽⣦⡘⠌⢠⣭⣿⣦⣼⡞⠛⢣⡶⠒⠀⠀⢸⣿⣿⣾⣿⣅⣱⠢⣀⠀⣹
⠀⣼⠀⣞⠴⢡⠔⠂⣪⣷⠜⣛⠛⡱⣠⠋⢻⠟⣇⠁⡈⣉⣅⡀⠱⣴⣾⠁⠀⠀⣠⠞⠛⠹⣦⣮⠛⢻⣿⣦⣅⣾
⠀⣼⢽⠿⣿⡟⠈⠋⣜⡻⢿⣯⣶⣿⣿⠷⠲⠴⢻⢳⠶⠤⠥⣤⢤⣿⣿⠀⠀⡜⠋⣻⣾⣏⣠⣬⣿⣧⣌⣹⣷⣿
⠀⣰⣀⣠⠴⡇⠉⠠⣬⣅⠀⠈⢿⢻⣅⠀⠀⠀⡼⢸⢣⠀⠀⣷⢠⣘⣿⣄⣄⣿⣷⣯⢀⢿⡷⣀⣣⣘⣻⣇⢄⣼
⠀⣿⠁⠁⠀⠀⠑⢤⠈⡽⠛⣶⣌⣼⡿⠇⠀⠀⡇⢸⣺⣣⣀⣾⣭⣯⡉⣁⡤⠟⠛⠉⢹⣘⠃⠉⣁⣧⣛⠿⡉⣹
⠀⣿⠀⢣⠠⣧⢀⣤⣄⣰⠦⣙⠻⢷⡧⠶⣓⠚⠳⠾⣭⣭⢭⡭⠛⠛⣻⠒⢤⡒⢮⡵⠛⣉⣉⠽⢢⡍⢭⠿⡟⣿
⠀⢸⠄⠒⡆⢹⣁⣼⡇⠱⢤⡤⢷⡾⢙⡳⠈⡁⠾⣦⣀⡶⠋⠀⡀⠀⠀⠀⢁⠔⢁⠴⣠⣯⣷⣠⡞⢀⣠⡀⣄⣼
⠀⢸⣀⡟⠋⣲⢬⣙⠦⡂⠀⢂⠂⢝⢷⡿⠓⢸⣛⣻⣻⣀⠔⠀⠁⠀⢀⠔⣁⠖⢛⣿⠟⢋⢵⡋⠻⣿⢭⠭⠥⢾
⠀⢸⣧⡟⡸⣿⠹⡅⢣⣸⣆⣠⣀⠀⢹⠀⠊⠉⠁⢀⢀⠠⠟⢁⣤⡽⡋⠚⠁⢀⣤⢉⣴⣣⡀⢡⠀⡘⢇⠀⠀⢸
⠀⢸⢿⠀⡿⣾⠀⠀⣼⢋⡟⠩⢫⢿⠇⠀⢀⣽⡟⣷⡶⠶⣖⢶⡾⣻⡶⠋⣀⡹⠟⠉⠉⣁⣹⡄⢰⡿⢸⣿⡻⣽
⠀⢸⢰⠀⡅⣿⠀⡟⢬⣿⣧⣀⣠⡏⢠⣖⣋⠷⠚⢁⡴⣾⣱⡦⠈⡯⠒⠩⣄⣒⣦⣭⠕⠀⣼⣃⣼⡇⢸⣯⡷⣿
⠀⠸⢺⠀⡇⣿⠀⡆⠘⣿⣏⠛⢛⣿⣏⣬⣿⣄⣔⣋⣡⡤⣿⠨⡷⠶⢞⠛⢍⠙⠠⠌⠉⠐⠓⠈⢋⠀⢸⡇⣝⣾⠇
⠀⠰⡷⠀⡇⣿⠀⡯⠀⢸⣿⣛⣿⣿⠿⠟⠳⠯⠩⠉⠓⢒⣾⢸⡖⠛⠈⢁⣐⠤⠍⠒⠒⠂⣀⣀⡀⠑⣾⠇⣉⣹⠆
⠀⠀⡇⠀⡇⢿⠀⠏⠀⠨⢹⡏⠓⠄⠅⢀⣁⠀⠀⠉⠬⠤⡏⠀⡗⢘⣉⣁⡤⠤⣶⡒⠛⠛⠉⠩⠍⠙⣉⡉⠥⢼⠇
⠀⠀⡇⠀⢇⡄⠀⠀⣴⠿⢺⡧⠨⠉⠉⠡⠒⠒⠛⣉⣡⡾⠟⣓⠛⠉⠓⠂⠉⣉⣀⣬⣤⣴⡶⠶⠿⠛⢉⣉⣁⣘⠃
⠀⢰⡇⢀⣸⡄⣠⣾⡄⡀⣧⣄⠒⠒⠶⣶⠿⡟⠿⠛⠓⠈⣁⣤⣤⣒⠶⠿⠓⣛⣫⣭⣥⡴⠶⢲⣾⡿⣿⣿⠿⢽⡆
⠀⢸⠃⢟⡼⣷⡄⡇⣧⠃⡇⠈⠛⠻⢯⣩⣤⠦⠔⠚⣋⣉⣩⡥⣤⣶⠶⢾⣿⣟⣃⡹⣌⢧⡀⠀⢱⡄⠀⠘⢦⠺⡇
⠀⢠⡴⢋⠂⡹⠏⢀⡏⡸⢡⢴⣶⡄⠘⡃⢩⡶⣾⡟⠿⠻⠓⠚⢻⠘⣄⡘⢻⢿⡟⠓⠁⠀⢤⠀⠀⠑⡀⠀⠀⢳⡇
⠀⢸⡧⠾⡿⢅⡴⢻⠞⠓⠦⡷⠷⢿⡀⡇⣼⠶⠶⠶⠖⠲⡶⠶⠲⠶⠶⠖⠿⠉⠐⠚⡗⠛⠛⠇⠉⠉⠙⡀⠉⠘⡇
⠀⢐⣀⣊⣀⣀⣀⣃⣀⣀⣰⣁⣀⣈⣿⣏⣁⣀⣸⣀⣀⣀⣧⣀⣀⣡⣀⣀⣰⣀⣀⣀⣸⣀⣀⣈⣄⣀⣀⣱⣄⣠⡇
⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠳⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⣽⣿⣾⠟⣿⣆⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠈⠉⠛⠿⣧⣈⠙⠻⢿⡝⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠸⣆⡀⢀⣠⠶⠛⠛⣳⣾⠷⢤⡀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⡾⠃⠀⠀⠙⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠆⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⣹⠿⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄⠀⠐⠋⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡞⠛⠶⣤⣀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⣩⣿⠶⡍⠛⣆⠀⠀⠀⠈⠁⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⣸⣇⣀⣠⣿⣤⣤⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⣴⠟⠉⠙⠋⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⢀⣼⠏
⠀⠀⠀⠀⠀⠀⢀⣾⠃⢠⡾⠃
⠀⠀⠀⠀⠀⢠⣾⣁⣠⡿⠁
⢠⣴⢶⣶⣄⠈⠉⠉⠉
⠈⠛⠛⠛⢁⣾⢿⡄
⠀⠀⠀⠀⠸⣇⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤
⠀⠀⠀⠀⣠⣷⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣉⡽⠛⠃⠀⠀⠘⣽⣏⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣀⠋
⠀⠀⠀⠀⣿⣿⠲⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠠⡄⢠⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⡿⡛⠀⢀⣀⣀⣀⣀⠀⠀⠀⣸⠛⢀⠀⠀⠀⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⢺⣿⠉⠀⠀⠀⠀⠀⣼⢟
⠀⢀⣀⠈⠀⡇⠨⢸⣿⣷⣿⣿⠀⠀⢀⠁⠙⠀⠀⠀⠀⠀⠀⠠⠁⡇⠀⠀⣀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠁⠹⠈⠄⠀⢺⣿
⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠈
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣲⣶⠄⠀⠀⠀⠀⠀⠀⣠⡀⢶⢤⡄⠀⠀⠀⣶⣶⣲⣶⡆
⠀⠀⠀⢠⣟⡻⠋⠀⠀⠀⠙⠛⠛⠘⢤⡄⠀⠀⠀⠿⠯⠿⠿⠻⣤⡆⠀⠀⠀⠀⠉⠁⠀⠳⣦⠀⠀⠀⠿⠿⠿⠿⠃⠀⣄
⠀⠀⢠⣿⡯⡀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠀⠀⠀⠀⠀⠀⠐⠤⢺⣿⠀⠀⠀⠀⠀⠀⠐⠒⣿⡇⠀⠀⠀⠀⠀⠀⠀⠒⣿⣯
⠀⠀⢀⡿⡇⠈⠁⠀⠀⠀⠀⠀⠀⢀⢸⢿⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⡘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧
⠀⠀⡘⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠃⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⠀⠀⠀⠀⠀⠀⠀⡖⠉⠀⠀⠀⠀⢀⣤⠀⠂⠒⠉⡀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⡀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⢀⣀⣀⡀⠀⠀⠀⢿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠛⠘⠀⠀⢠⡀⠀⠀⢠⣄⢰⡶⠤
⠀⠀⠀⣴⣿⠒⢾⣿⣿⠖⠀⠀⢰⣾⡃⠺⣿⣿⡷⠂⠀⠀⢸⠁⢀⣤⣄⡄⠀⠀⠀⣀⣀⣀⣀⠀⠀⢠⣾⣏⠀⠀⠈⠀⠐⠊
⠀⠀⠀⣿⡏⠁⠀⠈⠀⠀⠀⠀⢸⣿⠉⠀⠀⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣏⠀⣹⣿⠀⠀⢸⣿⠃⠁
⠀⠀⠰⠏⠡⠀⠀⠀⠀⠀⠀⠀⠾⠍⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⠀⠀⠄⠀⢤⠀⠀⠉⠉⠉⠉⠀⠀⠾⠉⠄⠀⠀⠀⠀⡀⠀⡀
⠀⠀⡁⢸⠀⡀⠀⠀⣀⠀⠀⢐⠈⡇⢠⠀⠀⢀⡀⠀⠀⢀⡿⣦⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⢈⠀⠀⠀⢸⡏⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠈⠀⠈⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⠙⠋⠉
⣿⣿⠛⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣜⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣟⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⠛⠛⡛⠛⠻⠛⠛⢻⣟⠛⡟⢿⠻⠛⠛⠛⠛⠛⢻⠛⡟⠛⠛⠛⠛⢛⢛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢉⣉⣭⣭⣭⣭⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⢉⡉⠉⣉⠉⠉⠉⠉⣉⣉⣉⡉⣉⣉⣉⣉⣭⡉⣉⣉⠉⠉⠉⣉⣉⣉⡉⣉⣉⣍⡉⠉⠉⠉⠉⠉⠉⠉
⠀⠉⠿⡿⢿⣿⡿⠿⢿⣿⠀⠀⠀⠀⠀⢀⣴⣾⠿⣿⣿⣷⣤⡀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠀⠀⠐⢻⣟⣧⣶⣮⣉⠛⠁
⠀⠀⠀⠀⠀⢿⠀⠀⠈⠛⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠘⠉⣿⢿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠐⠛⠻⠿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠻⡿⠿⠟⠟⠛⠃⠀⠀⠀⠀⠀⠺⢿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⢀⣰⣅⠀⣠⣶
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠢⠟⠐⡖⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢈⠛⠛⣒⠛⠛⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡄⠀⠀⠀⠀⠈⠢⣸⣿⣇⠐⠁
⠀⠀⠀⣀⣠⣤⣤⣤⣤⣆⠄⡄⠀⠀⠀⠴⠤⠴⠦⠆⠤⠶⠶⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢠⠾⢿⡾⠷⠄⠀⠀⠀⠠⢠⠄⢀⡄⣄⡄
⠐⢧⠼⣿⣿⣿⣿⣿⣿⡥⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠿⠿⠿⡿⠿⠿⠿⠀⠀⠀⠀⠀⠠⠟⠻⠄
⠀⠰⠄⣭⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡀⠀⠀⠀⠠⠤⡄⠄⢤⠤⡤⠀⠀⠀⠀⣀⣀⣀⣈⣁⣀⣀⣀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠈⠈⠛⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⣿⣿⡄⠀⠀⠀⠀⠀⢀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⣾⣾⡀⠀⠀⠀⠀⢘⣿⣿⡀⠀⠀⠀⠀⣨⣷⣿⣿⣻⣦⡀⠀⠀⠀⠀⠀⠀⢀⠤⠶⣤⣄⡀⠘⠻⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣢⣀⣠⣤⣿⣿⠟⠀⠀⠀⠀⠚⢻⣿⣿⣿⠿⣿⡟⠂⠀⠀⠀⢀⣾⣿⣶⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⠿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣧⣤⣿⣆⣼⣇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠐⠒⠚⠚⠐⠚⠒⠂⠀⠀⠀⠀⢠⣀⢠⣀⣄⣠⢀⣄⣠⡄⠀⠀⠀⢀⣠⣼⣿⣿⣿⣷⠸⣿⣦⣄⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⣍⢨⡀⣠⡄⣀⣄⡀⠀⠀⠀⠀⠙⠻⠿⠛⠛
⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⢠⣶⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⢤⢤⣤⡤⠤⢤
⠀⠀⠀⠀⠀⠈⠀⠀⡆⠀⠘⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠉⢿⠃⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠰⠆⠒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣾⣿⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣿⣿⣇⠀⡀⠀⠀⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠔⠉⠀⠀⢲⣦⣄⠀⠀⠘⣿⣦⡀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠛⠛⠛⠉⠙⠛⠁⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣀⣶⣰⣷⣠⣤⣿⣀⣀⠀⣸⣿⣿⣷⣶⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠚⠙⠘⠛⠛⠛⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡟⠉⠁
⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠠⡤⣤⡀⡄⠀⣤⠀⣤⢠⣤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠃⠈⠉⠁⠀⠈⠀⠙⠈⠙⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠂⠂⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠉⠁⠉⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠉⠉⠉⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠉⠉⠉⠉⠉⠉⠲⢄⡀⠀⠀⠀⠀⠀⣇⢠⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⡟⠋⠹⠬⡑⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⡀⢠⠀⢦⡀⢰⡄⢰⣄⠀⠀⠀⢣⠀⠀⠀⠀⠇⠀⣀⠀⠀⠀⠀⢣⠤⢤⠀⠈⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠦⠼⠙⠚⠈⠉⠉⠉⠉⠉⢤⡉⠉⠀⢈⠀⠀⠀⠀⣠⡀⢿⡇⣀⠤⢶⠁⠀⢸⠃⠀⢰⠀⠀⢰⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠐⠉⠁⠀⠀⠀⠀⠀⣀⠀⠁⠀⢰⡼⢄⠀⠀⠸⡀⠈⠉⠉⣀⠄⠸⠀⢰⡇⠀⠀⠘⠀⠀⢸⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢾⣄⠀⠀⢾⣧⠀⠀⠀⠀⣿⡆⢀⡠⠚⠃⢨⠀⠀⠀⠑⡄⠉⠉⢀⡠⠀⢳⠆⠀⢸⠀⢸⠀⣠⢃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣈⡀⡗⠒⠚⣓⣊⡉⢉⣉⡉⠍⠀⢡⡰⠒⠁⠀⠀⠀⠀⠱⣈⠉⠁⢀⠴⠃⠀⢿⠈⣠⢆⡰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠤⠤⠤⡤⠤⠔⠒⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⡲⠀⣠⠚⢳⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠶⢖⣶⣖⣲⡂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⠟⢁⣠⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⡏⠉⠈⠙⡀⡠⢖⡒⣀⡶⠴⠤⡀⠀⠀⠀⠀⡜⣨⠃⢒⠁⠞⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠖⣍⣭⣵⡀⣀⠷⣛⣩⣽⠋⠑⠢⠀⠀⡏⠑⠒⢴⡱⢁⢔⡥⠐⢦⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠐⣤⠁⠀⠈⠻⡓⢻⠁⠙⠯⠿⠖⠒⠉⠰⣅⡀⠀⣾⠔⡡⠊⠀⠀⠈⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠢⡀⠀⠀⠱⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢤⣤⡖⠘⠦⣀⠀⠀⢀⣀⡤⠤⠖⠒⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⣯⢕⣂⠀⠀⠉⠁⢙⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⢦⠙⠒⠒⢎⠉⠧⠄⠒⠒⠂⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣖⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣒⣟
⠀⠀⠀⠀⠀⠀⣿⢉⣉⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⣿⠐⠒⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⢨⣭⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠐⠒⠒⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⢻
⠀⠀⠀⠀⠀⠀⣿⠨⡭⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢭⠅⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⣿⠐⠒⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠒⠂⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⠤⠤⢤⠟
⠀⠀⠀⠀⠀⠀⣿⠨⠭⠤⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠭⠥⠄⢸⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⠲⣄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⢰⣒⣛⣛⣛⣛⣛⣛⣲⡆⠀⠀⠀⠀⠀⣖⣚⣛⣛⣛⣛⣛⣓⣶⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠙⠋⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣈⣀⣉⣁⣀⣿⣿
⠀⠀⠀⠀⠀⠀⣤⣧⣼⣧⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣤⣧⣼⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢄⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠙⢻⣟⣆⠀⠀⠀⠀⠀⠀⠠⠆⠌⠀⠊⠰⠀⠀⠀⠀⠀⠻⣿⢮⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣦
⠀⠀⠀⠀⠀⠙⠻⠗⠀⠀⠀⠀⠀⠶⠁⡀⡠⠈⠠⠀⠀⠀⠀⠀⠈⠊⣿⡇⠀⠀⠀⠀⠀⢾⣷⣾⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠐⠑⠈⠄⠠⠑⠀⠀⠀⠦⣄⠸⡇⠰⠛⠁⠀⠀⠀⠀⠈⠑⠀⠈⠀⠀⠀⠀⠀⠀⣺⣿⣿⣷⡄
⠀⠀⠀⠐⡐⢀⠂⠄⡡⠀⠀⠀⢀⣠⠆⢰⡆⠘⢷⠄⠀⠀⠈⡳⠀⢀⢀⠈⠀⠀⠀⠀⠀⠈⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠄

⠀⠀⠀⢰⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⠀⢀⠀⠀⠀⠀⠀⠀⢀⣼⠇⡀⠀⠀⠀⠀⠀⠀⡀⢤⡀
⠀⠀⠀⠸⣻⡏⠙⣦⠀⠀⠀⠀⠐⠄⡀⠃⠘⠃⢀⠀⠀⠀⣠⢾⣟⠁⣸⠇⣠⠆⠀⠀⢀⡎⢁⣂⠂⠓⡀
⠀⠀⠀⠀⠙⠿⣿⣟⠦⠀⠀⠠⣂⢀⡀⠀⠀⠰⡄⡂⠀⠘⠁⠀⠁⣠⡾⠿⠃⠀⠀⠀⠀⢦⠈⠉⢁⡘
⠀⠀⠀⠀⠀⠐⠻⠋⠀⠀⠀⠀⠀⢀⠀⣘⡆⡀⠀⠀⠀⠀⠀⠀⠺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣤⣀⠄⠀⠀⠀⠀⢢⣶⣿⣶⣴⠀⠀⠀⠀⠀⠔⠀⢀⠀⠀⠀⠀⠀⠀⢠⢖⡄⢀⠀⠀⡀
⠀⠀⠀⢨⣿⡿⠻⣷⡆⠀⠀⠐⣿⣿⣷⣿⣿⠇⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠘⢨⢢⣒⣀⡀
⠀⠀⠀⠈⢹⣷⣾⠛⠁⠀⠀⠀⠸⠻⢯⠟⠻⠦⡀⠀⠀⠀⠀⠀⠁⠒⢷⠀⡀⠀⠀⠀⠀⠀⡀⠁⠚⠥⡃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⣠⣤⣄⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣼⡇⠀⠻⣾⠟⠀⢸⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠛⠉⠉⠉⠉⣽⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⠛⠛⠛⠛⠛⠛⠻⢶⣄⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢀⣀⣀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠟⠋⠉⠉⠻⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠛⠛⠛⠛⢷⣆⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿⠀⠀⠀⣀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⢠⡾⠛⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⢠⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⣰⡿⠁⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠈⠻⠷⠟⠁⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⣿⠀⠀⠀⢸⣷⡟⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡟⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⢰⣿⠉⣻⡇⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⣿⠀⠀⣴⠟⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠙⠛⠋⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⠀⣼⠏⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠏⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠸⣿⣀⣽⠇⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠾⠾⠶⠆⠀⠀⢀⣿⠏⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⣼⠏⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⢀⣴⠶⣶⡄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣾⠃⢀⣾⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠘⢿⣤⡾⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣡⣾⡃⢠⣾⠃⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⡇⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢻⣿⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣿⡇⠀⠀⢸⣿⠀⢠⣾⠛⢿⡆⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡾⠟⠁⣿⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡀⠀⣿⡇⠀⠀⢸⣿⠀⠈⠻⠶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⣇⠀⢸⡇⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⣷⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣎⡲⣴⢟⢦⣂⠄⠀⠀⣀⢤⣶⠃⣶⠀⡀⠀⠀⠀⢢⣴⣶⣾⣤⠂⣀
⠀⠀⠀⠀⢰⠷⠊⠃⠀⠀⠀⠀⠀⠀⠀⣀⣱⣾⣷⠐⢀⡫⣷⣃⠀⢰⣿⣿⣿⣍⣙⣿⣅⠀⢠⢨⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠠⠼⠀⠀⠀⠀⠀⠀⢀⠀⠀⠖⣯⣿⣿⣷⣶⣶⣻⡀⠀⠸⣿⣿⣾⣝⣶⣿⡟⠀⠠⢀⠹⣿⡅⠁⠀⠈⠀⠁
⠀⠀⠀⠀⠀⡌⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠊⡝⠿⡿⠷⡛⠃⠀⠀⠀⠘⠻⠿⠿⠿⠋⠀⠀⠀⠈⢠⠋⠀⠐⡌⠂
⠀⠀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡩⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃
⠀⠀⠀⠀⢸⣾⣷⡆⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠁⠠⠂⣃⢁⡀⠁⢁
⠀⠀⠀⣤⢽⣿⣿⣯⣄⠀⠀⢐⣟⠂⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⢸⠛⣿⣿⣿⡇⠀⠠⠀⢠⡚⠡⢟⠈⠃⠀⡆
⠀⠀⠀⠀⠸⢹⣟⠇⠀⠀⠀⢀⣿⠇⠀⠀⠀⣾⡥⢿⣿⣭⡄⠀⠀⠀⠛⡶⠻⡿⠟⠃⠀⠀⠀⠀⢂⡀⠢⠖⢠⠙⠁
⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠆⣿⡄⠀⠀⠀⠸⠗⠾⠿⠟⠁⠀⠀⠀⡠⠋⠀⠹⠀⠀⠀⠀⠀⠂⠀⠄⠋⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⢸⡟⠃⠀⠀⠀⡊⣿⡆⠀⠀⠀⣂⣀⣀⣀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⢸⣧⠄⠀⠀⠀⡇⣿⡅⠀⠀⠸⡟⠛⠛⠿⠿⣟⠀⠀⢠⣶⣦⣖⣾⣷⣶⠀⠀⠀⢀⣀⣦⣴⣢⣶⠀⡀
⠀⠀⠀⠀⠀⢸⡋⠆⠀⠀⠀⠏⣯⡄⠀⠀⠰⠆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠉⠀⠈⠀⠀⠠⢤⣿⣿⡿⢏⢛⣎⡁
⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⣀⣦⣿⣇⣀⠀⠀⡄⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢒⣯⣿⢕⡡⢁⣛⡒⠂
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠙⣿⠀⠉⠀⠀⣇⣀⣀⣀⣼⣿⠀⠀⠀⣄⣀⠀⣀⣁⡀⠀⠀⠈⢉⢿⣿⣮⣥⡣⣟⠉
⠀⠀⠀⠀⠠⠟⠓⠀⠀⠀⠀⠾⠛⠦⠀⠀⠀⠋⠛⠻⠟⠛⠋⠀⠀⠀⠛⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠃⠘⠀⠁

⣶⣾⠿⣶⠶⢶⣶⡶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢒⣋⣭⡭⣍⡓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠰⣍⡀⠀⠀⣀⡼⢀⠌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠀⠀⠭⠭⠭⠑⠒⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗⠦⢤⣀⣀⣀⣠⠤⠔⠚⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠯⢭⣙⠲⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⡀⠈⠳⡌⢣
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠔⢋⡩⢝⣦⠀⢹⡘
⠀⠀⠀⠀⠀⠀⢸⡈⠳⢤⣀⣀⡀⠀⠀⣀⣠⠤⠒⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⣡⢔⣿⡾⢋⠝⠀⢘⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠑⠢⡄⠀⠉⠉⠉⠀⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⢤⣀⣠⠖⢁⣤⣮⡿⠛⣡⠔⢧⣀⣀⠼⠚
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⢁⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠀⠀⠉⠓⣌⢳⣾⡿⠟⢁⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠈⡇⢄⡀⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸⠔⠚⠉⠉⠉⠓⢄⠈⢧⢻⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠗⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠔⡿⠀⠀⠀⠀⣠⡴⠎⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠠⣉⠒⠒⠦⠤⠤⠖⠒⢋⡡⠚⠁⠀⣠⠴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠒⠉⠁⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠙⢦⠀⠀⠀⢠⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡤⠚⠙⢢⠀⠀⠀⣧⠀⠀⣨⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⡈⠆⢀⡼⠀⠀⢀⣇⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⠞⠓⠚⠡⠀⠀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡴⠊
⠀⠀⠀⠀⠀⢀⡠⢖⡽⠊⠁
⠀⠀⠀⣠⠖⣩⠖⠁
⢀⡴⣪⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠴⠶⠶⣶⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣞⠋⢹⡯⠀⠀⠀⠀⠀⠿⣿⡜⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢀⣴⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢰⣿⣿⣿⣿⡯⠈⢢⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠀⢀⣄⣤⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⣾⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣶⠞⠛⠙⣿⣿⣿⣫⣿⣿⡟⣛⣿⣿⣇⢀⡀⣴⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⡎⠀⢸⡅⠂⢼⣿⣿⣻⣿⣿⣿⡿⢉⣽⣿⣿⣿⡷⢹⣯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢧⡤⢸⡇⢰⣿⣿⣭⣯⢿⣿⣧⠀⢈⣿⣿⣿⣿⣥⡙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢀⣷⣄⢿⣾⣿⣿⣿⣿⣡⣌⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣄⣹⣟⠿⠿⢿⣿⣇⣀⠀⡿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡛⠛⢿⣆⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠛⠻⠿⠿⠿⢿⣿⡿⣿⣿⠻⣿⣿⣿⡏⣿⠏⣿⢰⣖⣿⣯⣭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠈⠋⢰⣿⣿⡄⢿⣿⣿⢠⣿⠄⣿⣸⣹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⠈⢧⡀⠀⠀⠀⣰⣿⣿⡿⢳⣜⣿⣿⣿⣿⡀⢹⡿⣸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣹⣄⣄⢁⡶⠆⠀⣨⣿⣿⡇⢸⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣜⡇⢳⣄⣠⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⡿⠈
⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⠶⣦⡉⢿⠋⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⠈⠻⠿⠟⠿⠥⠇
⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⢸⠏⠘⢷⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⡀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⢹⣿⣷
⠀⠀⠀⠀⢰⠟⠉⠛⠃⠀⠀⠙⠛⠳⠶⣶⣿⣿⣿⣿⣿⡿⢿⣿⣿⣧⡀⣼⠃⢿⣿
⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⣤⠟⠁
⠀⠀⠀⠀⠀⠙⠷⡆⢠⡞⠀⡀⢀⠀⢡⣿⣿⣿⣿⡿⢻⠀⠀⠀⠉⢻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣱⡄⠻⠋⡰⠁⠀⣿⣿⣿⡅⠀⠀⡄⠀⠀⠀⠉⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⢀⡴⠁⠀⢠⣿⣿⣿⣷⣾⠀⠀⠀⢰⣄⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣴⡿⠟⠉⠻⣿⡟⣀⣶⢀⡼⣿⣥⣤⡀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠰⠓⠽⠯⢼⠀⠀⠉⠛⠻⠷⠶⠟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣰⣆⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⡿⠿⢿⣿⡿⣿⣦⡤⠀⠀⡴⣄⠀⢀⡽⣮⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣧⣀⣼⣿⣷⣿⣿⣇⠀⢀⡝⢮⡳⣏⠔⠁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⡏⠉⢹⠟⢁⠔⢁⠐⠀⠙⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⠖⢁⠴⡉⠢⠀⠀⢀⡴⢋⣉⣁⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⢁⠔⠁⠄⠈⠀⢀⡴⠋⢀⣾⡿⣿⣁⣹⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠣⡀⠁⠀⢀⡴⢋⠀⠀⠉⠹⣷⣾⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢦⡀⢀⡴⠋⠀⢻⣷⡒⠀⠈⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠓⠊⣉⠉⣴⣷⣤⣾⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣸⣿⣇⣀⣸⣿⣏⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡻⠿⣿⣿⠿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠠⣄⣼⣤⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⡾⠛⠻⡆⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠚⠓⠂
⠀⠀⢺⢿⡿⠋⠀⠀⢿⣿⣿⠗⠀⠀⠀⠁⠀⠀⠁⠀⠀⡰⠋⠀⣭⠀⠀⢠⣶⣷⡆⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣈⣭⣿⣿⣿⠟⠉⠈⠉⠉⠉⠉⠁⠉⠻⣿⣿⣿⣭⣁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣠⣴⣾⣿⣿⣿⣿⣿⡿⠋⠀⣀⣤⣤⣶⣶⣤⣄⣠⣤⣶⣶⣤⣤⣀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣦⣄⣿⣿⣿⡇
⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠩⠄⠀⠀⠀⠀⠀⠍⡛⠻⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠈⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⣤⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣠⡀⠀⠀⠺⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠻⣿⣿⠟⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡗⠀⠀⢠⣶⡆⠀⠀⢸⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣷⡀⠀⠀⢹⡆⠀⠀⣾⡏⠀⠀⢸⣿⡆⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⣿⣤⣤⣾⣿⣿⣿⣿⣷⣤⣠⡟⠁⠀⠀⢀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣤⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣾⠟⠋⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠿⢶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠶⠞⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠴⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢷⣶⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠘⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣴⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠻⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠁⠀⠀⢀⣼⠛⠛⢿⣿⣿⣿⣿⡿⠛⠛⣿⣄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠸⣿⡇⠀⠀⣸⡿⠀⠀⠸⣇⠀⠀⠈⢿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠸⠿⠃⠀⠀⢼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣾⠉⡇⣿⠀⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠱⢌⠀⠀⡇⢠⡇⡿⠠⣧⢠⡤⠠⠶⠛⠛⠛⠓⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡾⢷⡀⠀⠢⢀⣇⠈⣷⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⣿⣿⣷⠆⠙⣆⠀⠻⠇⠀⠈⠁⠀⠀⠀⠀⠀⣄⣴⣠⠴⣶⢦⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡸⠂⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣽⢦⣶⡶⡒⠊⠁⢠⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡿⣿⣿⠓⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢿⣿⡿⠛⠁⠀⣀⣀⣵⣾⡟⡂⠠⠒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢹⣿⡿⠃⣿⣏⡁⢀⣀⠀⠂⣌⢰⡜⢿⣿⣿⡟⠟⠃⠀⢠⣴⣿⣿⣿⣿⠟⠉⠀⠛⠓⢁
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⢗⣾⢿⡱⣆⠘⢿⣟⡼⣻⣇⣸⣿⣿⣣⣾⣿⠇⠃⠀⠀⢠⣿⣿⡿⠟⢻⣿⣿⣧⣶⣦⣻⣟⠁
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡋⠈⠳⢈⢀⠘⡿⡆⢸⣟⣛⣿⣾⣿⣿⣿⣿⣏⠀⠀⠀⢁⣸⣿⠟⠀⠀⠀⠈⣿⣿⠋⠛⢉⠻⠟⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⡀⣄⣖⣠⣐⠞⠀⣼⣿⣿⣿⠾⠄⠀⢻⣿⣿⣦⣀⣴⣿⣾⣃⡀⢤⡄⠀⠀⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⣿⣹⣿⣿⡓⣤⣷⣮⣿⠗⣰⣿⡟⢻⢽⡂⠦⠠⣎⠿⣿⣯⣿⣿⣿⣿⠟⠃⠀⠉⢦⡸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣍⠺⠿⠿⠛⡩⠴⠞⠉⠀⠀⠀⠀⠈⠠⠿⠈⠀⠀⠘⣿⣿⣿⠋⠀⠀⠀⠀⠀⠹⣿⠇
⠀⠀⠀⠀⠀⣿⣿⡏⣻⣿⠿⡏⢞⡧⠀⠀⠀⠐⠀⠀⠀⢸⡆⢟⢡⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢠⣙⣿⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⠈⢭⠁⣿⡀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠛⠑⠁⢸⣿⣿⡄⠀⠐⢸⠷⣶⣼⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣿⣗⣿⣿⣄⢮⣔⡎⢹⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠈⢠⣼⢻⣿⣷⡄⠀⣦⡈⠁⠏⢉⡝
⠀⠀⠀⠀⠀⢻⣿⣟⣿⣿⣇⠈⢁⠀⡸⣄⠀⠉⠀⠀⠀⣁⡩⠴⠖⠚⠉⢀⣿⣿⣿⣿⣦⣼⣿⣶⣾⣿⣇
⠀⠀⠀⠀⠀⢸⣿⣯⣿⣻⡿⡂⠀⠁⠀⠁⠀⠁⠀⣠⢚⠁⠀⠀⠀⢠⠒⣼⣿⣿⣿⠿⣿⠹⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⣿⣿⣺⣧⢿⡙⠆⠂⠀⠀⠀⠀⢚⠵⠋⠀⠐⢀⠀⠈⣦⣿⣿⣿⢻⣇⠘⠳⢞⡋⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢿⡻⣿⣿⣿⣿⠆⠀⠀⢀⡔⠃⠀⠀⠀⠀⡠⠐⣸⣿⡿⣿⣗⠈⣿⢦⣌⣉⠁⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆⢀⠔⠋⠂⠂⡀⠀⠀⠀⣰⣾⣿⠿⣼⣿⡅⠀⣿⠎⡿⢯⡙⢾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⡶⣉⡀⠜⠣⡁⠿⡩⠡⠚⣿⣿⢻⡶⣿⡄⠙⢷⡿⣷⠄⡀⣻⣶⣿⡿⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣝⣧⢫⡂⠀⢸⡀⢠⠰⠃⣾⠇⡸⢲⡿⢿⣦⣉⠛⠛⣏⣛⣿⣿⣿⠋⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⣿⣿⣾⠲⣤⡀⡀⠀⣇⡎⠀⣰⣿⣤⣇⢨⣿⣦⣿⣿⠷⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣢⡌⢠⡄⢙⣧⣀⣿⣳⣿⣘⠛⣷⣯⡛⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⢢⡇⢸⠻⣿⣧⣿⣻⣿⠃⠀⢈⣲⡈⢹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⢻⣾⣿⣿⣿⣶⣟⣧⣲⣿⣿⣷⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣄⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠋⠍⠁⠀⠀⠀⣈⣻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⠿⠉⠀⠀⠆⠀⠀⠀⠀⠀⢱⡀⠈⠛⢻⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣼⠿⠋⠁⠀⠉⠀⠘⠳⢄⡀⠀⢀⠊⠀⠈⢹⣯⣍⣙⣿⣿⣆
⠀⠀⠀⠀⣠⢾⡁⡠⠶⠤⠴⠒⡳⢶⡀⠀⠈⠉⢙⡆⠀⠀⣿⣥⠾⢿⢿⣿⣿⡄
⠀⠀⠀⢠⡣⢣⡍⠀⠀⢀⠜⠐⠒⢸⡄⠀⢀⡴⠊⠀⠀⢸⣿⣿⣦⣶⣾⣿⣿⣆⠈⠐
⠀⠀⠀⣾⢡⠎⠀⣀⠀⠀⠀⣠⡤⢄⢻⡆⢸⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣧⣀⣾⣷⠋
⠀⠀⢰⣷⢸⠀⠀⠀⣤⠀⠀⢁⡤⠚⠀⢻⣼⡇⢰⣶⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⡿⢾⣳⣄
⠀⠀⢸⣿⡙⡧⠁⠀⠘⠓⢤⡀⠀⠀⠈⠁⠙⣿⣿⣿⣀⣤⣄⠀⢀⣿⣿⣿⣿⣿⣴⣿⣿⣿⠄
⠀⠀⠀⢿⣇⣇⠀⢀⠅⠀⢀⠻⠽⠶⣞⡶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠈⢿⣿⡷⢧⣔⠐⡀⠀⢀⡴⣟⡙⡋⢻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⢿⡿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⢻⣿⣷⣿⣿⡿⡞⠋⣔⣿⣷⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⠋⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⠹⣧⣿⡇⡆⣿⣿⣿⣿⣿⣿⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⡿⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⠁⣿⣿⣿⣿⣿⣿⡿⢷⡐⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠤⢲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡗⣿⣿⣿⣿⣿⣿⣿⣿⣌⡻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣫⣿⣿⣿⣿⣿⣥⣿⣿⣿⣎⢻⣿⣷⡀
⠀⠀⠀⠀⢠⣾⠞⠉⠻⠏⠩⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣜⢿⣿⣷⡙⣿⣧
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⣰⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⡿⢿⣿⢻⣿⣿⣿⣿⣿⣿⣷⡿⣯⢷⡌⢻⡄
⠀⠀⢠⡆⠐⠀⠀⠀⡀⣴⣿⣿⣿⡀⢰⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣯⣿⣿⣿⣷⣦⣄⡀⠙⢿⣿⣿⣧⠸⣿⣿⡴⡅
⠀⠀⠘⢂⠀⢸⣧⣾⣿⣿⣿⡟⠉⠁⢸⣿⣿⣿⣿⣳⣿⣿⣴⡏⠀⣿⣿⡿⠻⣿⠛⢯⣙⢳⣤⡙⣿⣿⣷⣼⣿⢷⡜
⠀⢀⠂⠈⠈⠉⢻⡿⣿⣿⣿⣧⠀⢀⣿⣿⣟⣿⣾⠙⣿⡟⠻⣿⡄⠛⠋⠀⠀⡿⠀⠀⠉⢷⣽⣷⣌⠙⣿⡅⢻⣏⠇⢶
⠀⢸⠀⢀⠀⢀⣿⣇⣿⣿⣿⣿⢧⡘⣿⣿⣿⣿⣿⣀⣯⣴⣶⣾⠇⠀⠀⠀⠈⣇⠀⠀⠀⠀⢻⣿⣿⡄⠛⣿⠆⣿⣆⠘⡃
⠀⢸⡀⠹⣖⣿⣿⣧⣾⣿⣿⣿⠈⢃⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣷⠀⠀⠀⠀⡾⠀⠀⠀⠀⠠⢑⣿⣿⠆⢺⣷⠸⠿⠁
⠀⠀⠉⠉⠛⠿⠍⢰⣟⣿⣿⡿⢦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⢰⣿⣄⣤⣴⣿⣷⣢⣥⣴⣾⣿⣿⣿⡿⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠯⠸⢿⣿⣿⠟⠋⠙⠛⠿⠿⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠿⠛⠋⢻⣿⣷⠀⠀⠀⠀⠁⠀⢤⠄⠀⠉⠀⠀⠀⠀⠀⡿⠛⠉⠙⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⣠⣶⣶⣶⣶⣶⣾⡄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⣰⣿⡿⠛⠋⠁⠹⣿⣿⡆⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣼⣷⡄⢠⡿⠋⠀⠀⠀⠀⠀⠈⠛⣿⠀⠀⠀⠸⣿⡏⠉⠉⣹⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡕⠀⠐⡌⣻⣿⡇⢸⣿⣶⡀⠀⠀⠀⠀⢠⣶⣿⠀⠀⠀⠀⢙⡿⠶⠾⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠀⢸⣦⣼⠁⠸⣿⡃⠀⠻⣿⣷⣄⠀⢀⣠⣿⣿⠟⠀⠀⠈⢾⠉⠑⠀⠀⠊⠙⢧⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠃⢀⡠⠈⠛⢹⠄⢰⠏⠻⠀⠀⠀⠀⠀⡀⠱⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⠀⠄⠌⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⡆⠀⠀⣁⡀⢘⠀⠈⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠐⠀⠀⠀⠸⠗⠀⠀⠐⠐⡇⠀⠀⢻⠃⢸⠀⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⣶⣶⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠄⠂⠒⠤⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⡦⠤⠤⠤⠤⠤⠤⠼⠀⢘⡰⠂⠀⠀⠀⡀⣀⠀⠀⠀⠸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠰⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⠀⠀⠈⠁⠉⠁⠀⠀⠃⠦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡤⠤⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⢱⠀⠀⠘⠀⠀⠀⠀⠂⠀⡇⠰⠀⠘⠀⠀⠀⠀⠈⠀⠀⠐⠀⠐⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠈⠛⠛⠛⠛⠛⠒⠒⠒⠒⠁⠘⠒⠒⠒⠒⠃⠙⠛⠛⠛⠋⠀⠈⠛⠛⠛⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠡⠔⠒⠊⠉⠉⠙⠒⠒⠤⣤⣀⡀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⣠⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⡞⡛⠉⠛⠋⠉⣉⣨⣏⢹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣿⡿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⠀⠀⠀⢀⣾⣯⡀⠈⠉⠀⣀⠤⡀⠀⠰⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠉⡅⠄⠀⠀⠀⠀⠸⣿⣷⠺⠧⠀⠀⠀⠶⠶⠀⠀⢸⠻⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⡀⢹⠏⡐⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣼⣿⣿⢻⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⠘⠂⡞⠀⠀⠀⠀⠀⠀⢠⣿⡀⢀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣾⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢸⠀⠀⠀⡜⢀⠀⠀⠀⠀⠀⢠⣾⠻⢷⡈⣯⡿⠆⠀⠀⠀⣀⢠⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠉⠁⠀⢀⠊⠀⠀⠀⠀⠀⢘⠻⢶⣿⣧⠀⠀⠀⣀⣴⠞⠁⢼⣏⣏⣏⠉⢹⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⢿⣿⠛⠁⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠖⠊⡩⠜⠆⡀⠔⠀⠀⠀⠀⢽⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠉⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⣤⣾⡷⠒⠁⠀⠀⠀⠀⠀⠀⠈⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⣠⡾⠛⠁⠠⠄⠈⠁⡀⠐⠒⠓⠂⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠤⠊⠀⡠⠟⠁⠀⢀⣠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣹⣿⢿⣿⣿⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢇⠀⠀⣔⡊⠉⠀⠂⠈⣀⡠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠞⠉⠁⠂⠠⠔⠉⡁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠉⠛⠒⠆⠀⠀⠐⢶⡇⠀⠀⠀⠀⠀⠀⢀⠞⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⠤⢶⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢠⠂⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣤⣤⠀⣀⣀⣀⣀⣀⣤⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⡆

⠀⠀⠀⢀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢀
⠀⠀⠀⠀⢡⠒⢶⡈⢂⠀⠀⠀⠀⠀⠀⠘⢈⠬⣌⠐⠁⠀⠀⠀⠀⠀⠐⣔⠍⢢⠦⣄⠀⠀⠀⠀⠀⠀⡤⠊⠀⠷⡄⠁⠂
⠀⠀⠀⠐⡈⠢⠔⢁⠂⠀⠀⠀⠀⠀⠀⠀⢌⣀⣉⠄⠀⠀⠀⠀⠀⠀⠈⠨⡞⠁⠀⠘⢲⠀⠀⠀⠀⠀⠑⢒⠒⠒⠁⡐⠄
⠀⠀⠀⠈⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠊⠀⠘⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠄
⠀⠀⠀⣴⣾⡟⢕⡤⣄⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠸⠤⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠧⣀⠀⠀⠀⠀⠀⡎⠁⡠⠞⠉⢣⣀
⠀⠀⠘⠛⠁⠀⠣⠤⠬⠃⠀⠀⠀⠀⠈⢴⠒⠀⠀⠤⠐⠁⠀⠀⠀⠀⢜⠉⠀⠈⢑⠄⠉⠀⠀⠀⠀⠀⠈⠁⠧⡀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉


⠀⠀⠀⡠⢴⠒⠲⡦⡀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠉⢰⡲⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⢔⡢⠀⠀⠀⠀⠀⠀⠀⣎⡇⢤⡀
⠀⠀⢰⢅⣆⢠⡄⣧⢨⡄⠀⠀⠀⠀⣬⠀⢰⠊⠂⠀⢀⠀⠀⠀⠀⠀⡌⠀⢰⡉⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⣿⡇⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⠒⠂⣠⡿⠀⠀⠀⠀⠀⠱⡀⠈⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⢰⡟⣿⡄
⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣺⣿⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⠿⣿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⢿⡁⠻⡶⠋⠀⠀⣀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢯⣴⣿⣆⠘⠧⠴⢞⡟⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⣤⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠭⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣛⠿⠿⠽⢯⡟⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡅⢠⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠉⠱⣄⣤⣶⣶⣾⡇⣀⡀⠂⠹⣄⠀⠀⠀⠀⠀⠀⠀⡼⡆⢳⣿⡄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣷⣿⣿⣦⣦⢹⠷⣄⠀⠀⠀⠀⠰⢁⢹⣎⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠳⣄⡀⢠⠃⢻⣿⣿⣿⡷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣦⣼⣭⣁⣶⣿⣿⣿⠟
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⠀⠀⣀⢀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣐⣝⠿⠃⠿⠋⣨⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⠿⠿⠷⣿⣌⣣⣤⣂⠈⣿⠿⣿⡇⠈⠉⠛⠻⠿⠟⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣦⣄⣠⣶⣾⠉⠉⠉⠘⢡⣾⣿⣿⡀
⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⠻⣿⠻⠿⡿⢿⡿⢿⡿⡿⠿⠋⠛⠁⠘⢹
⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⡆⠀⡄⢰⠖⠀⠀⠐⣴⣿⣦⠂⠀⠀⢸⣿⣿⣿⣿⣿⣏⢹⣟⢻⡏⣹⣿⣿⣿⠋⠈⢹⣿⣿⣿
⠀⠀⢀⣀⠐⢦⣹⣶⣷⣿⡠⠂⢀⡌⠹⠛⠅⠁⠀⠀⢸⣿⣿⡿⢿⣯⡹⠂⠉⠉⠰⢟⣽⡿⢿⣶⣤⣾⣿⣿⣿
⠀⠀⠈⢉⣙⣿⣿⣿⣿⣿⣿⣟⣉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠶⡦⠀⠀⠀⠀⠀⠀⢴⠶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣈⡤⢿⣿⣿⣿⣿⡿⢧⣉⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⢛⡀⠀⠀⠀⠀⢀⡛⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠁⠰⠎⣹⠛⡟⢿⠙⠆⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣏⣥⢁⣤⣠⡸⣮⣻⣷⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⠇⠈⠃⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣸⣷⣾⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⣦⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠀⠰⢿⡗⠀⣠⣶⣶⣤⣴⣶⣶⣄⡀⠀⠀⠀⠸⠷⣶⡟⠀⠀⠀⠀⠀⠀⠀⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣿⡁⠼⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⣠⡀⠀⢸⣿⣿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢭⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠈⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⣿⠀⣾⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⡿⠃⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⢛⣓⣻⣿⣿⣿⡅⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡋⠉⠩⠹⣿⣿⣿⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢢⣬⡔⠋⠉⢻⡀⠀⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⠉⢉⡽⢛⠫⡉⠉⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢲⡄⠀⠀⠘⠛⣿⣷⣾⣛⣿⠀⠘⢷⣏⡀⢨⢀⣨⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠉⠛⠉⠀⠀⠀⠈⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠛⠛⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠲⣦⡈⠻⣦⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠋⠀⢀⡈⠻⣦⡈⠻⠂⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⢀⡴⠋⠀⠀⣤⡈⠻⣦⡈⠓⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⠈⠛⣶⡋⠀⠀⠲⣦⡈⠻⣦⠈⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠛⣦⡀⠀⠈⠻⠆⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢀⣤⡶⣦⣀⠀⠈⠻⣦⡀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣰⠞⠙⣷⡄⠀⠈⠻⣶⡏
⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⠻⢷⣤⡞⠁⣠⡾⠋⣿⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠉⢻⣾⠋⣠⡾⢋⡀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⢹⣿⠋⠀⠈⠻⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠠⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠠⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴⠡⠖⠒⠒⠒⠼⠭⠁⠉⠐⠀⢀⠄⠂⠀⠰⠺⠝⠶⠆⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢄⠀⡐⠁⠐⠀⡠⠂⢀⠈⠀⠀⠀⠀⠀⢠⠊⠀⢅⠀⢠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢈⠂⠀⠀⠔⠀⠀⠀⠐⠲⠢⠄⠔⠖⠒⠆⠀⠢⡀⠀⡠⠒⠒⠂⠂⠂⠂⠂⢀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣱⣗⠦⣙⢤⡀⠀⡸⠄⠀⣀⠀⠀⠀⠀⠠⠀⡀⠆⢠⠁⠀⣀⠀⠀⠀⠀⢄⣀⡠⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠪⠀⠀⠀⠑⠚⠂⡆⠐⠂⠀⠠⢨⡍⠁⠀⠀⡇⠀⢸⠀⠀⡀⠠⠄⠩⢅⡁⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡶⠡⠁⠀⠀⠀⠀⠀⠀⡇⠐⠀⢀⡀⠠⠄⠀⠈⠀⡇⠀⢈⠀⠀⠀⠠⡀⠢⠔⠰⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐⠞⠔⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠛⠺⠀⠀⠀⠀⠀⠀⡇⢰⢸⠀⠀⠃⠀⡀⡅⠵⢈⠀⠀⠲
⠀⠀⠀⠀⣀⠀⠀⢀⠮⠰⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠣⢒⠭⠀⠀⠀⠀⠀⠀⡇⢸⢸⠀⠀⠀⠀⠁⠀⢀⡀⢀⡀⠬
⠀⠀⠀⠀⠐⠁⠢⢮⡒⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠒⠀⠠⠤⠠⠄⠀⠀⠃⢺⢸⠀⠀⢄⡀⠀⠁⠀⠐⠒⣒⠤
⠀⠀⠀⠀⠀⠀⠈⠒⢍⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠠⡠⠂⠈⢸⠀⠀⠀⠀⠀⠈⠉⠉⠩⡉⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠐⠀⠀⠂⠐⠂⠀⠀⠒⠈⠈⠗⠤⣀⡐⠀⠒⠀⣊⢁⡑⠋⠘⢤⣀⠀⠀⢀⢄⠀⡨⠒⢠⠞⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠁⠂⠐⠀⠀⠁⠊⠝⠒⠒⠴⠢⠖⠀⠀⠐⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠉⠉⠉⠐⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠑⢦⣀⠑⠀⣠⣤⣄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠴⠂⠈⠳⡌⠙⠛⣋⣉⣤⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠺⠃⠰⣶⠀⠀⠋⢀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⢀⡆⠈⠂⠀⠂⠗⡓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠖⠋⠀⠀⣱⡄⠀⠀⡇⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐⠂⠈⠄⠙⠀⠀⢻⠊⠁
⠀⠀⠐⡐⢤⡀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⣀⣀⡀⠀⣬⠑⠢⠤⠶⡁⠠⠀⠱⠀⠀⠀⠀⢀⡠⠙⠦⢤⣄⣀⣀⣤⣤⠴⡞⠳⢄⡀
⠘⠁⠠⠀⡶⠁⣎⠁⠀⢠⠃⠀⠀⠀⠀⠀⡠⠊⠁⠀⣀⡠⠀⡏⢡⠝⣴⠀⢠⠁⠀⠀⠙⠢⡀
⠊⠀⠀⠘⣁⠀⠈⢠⠴⢫⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⢃⠀⠀⢰⠉⣣⠃⠀⠁⣂⠤⠀⠁⠀⠉
⠀⠆⠀⠀⠀⠀⠀⢻⠀⢸⠀⠀⠀⢠⠊⡼⠃⠀⠀⠀⠀⠣⡀⢸⠀⡛⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡆⢀⡴⢁⠜⠄⠀⠠⠄⣀⠀⠀⠑⣌⣤⠐⠁⠀⠀⠀⠀⠉⠍⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢷⠊⣠⠊⠐⠀⠠⠀⠀⠀⠀⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡠⠞⠁⢀⠃⠀⠀⠀⠀⠈⠐⠀⠀⠀⠇⠀⠀⠀⠀⠢⠀⠀⠠⠂⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠲⠖⠤⡤⠤⠤⣤⣧⣤⠤⠤⠤⠤⠶⡶⠲⡖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣌⠙⠒⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⢀⣠⠔⠊⠳⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠙⠢⢤⣀⣈⣉⣙⡿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷⣶⣾⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⡑⢈⣠⢀⠑⢋⠀⠀⠀⠀⠀⠀⠀⠀⢤⠰⠘⠐⢀⠄⠀⠀⠀⠀⠀⠀⠀⠄⡥⣤⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⡆⠈⠐⣁
⠀⠀⠀⠒⠀⢺⣿⣷⡇⠀⠂⠀⠀⠀⠀⠀⠀⣈⠂⣴⣿⡝⠆⢊⡀⠀⠀⠀⠀⠀⠀⠄⠻⣻⣼⣿⣿⢄⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡷⠀⠐
⠀⠀⠀⠀⡵⢀⠉⠉⡐⠌⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣤⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⢀⠲⠍
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⣠⣤⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣿⣏⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡿⢫⣶⣤
⠀⠀⠀⣞⢿⠗⠉⣤⣾⣿⣠⠀⠀⠀⠀⠀⠠⣄⣀⡛⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣠⣼⣿⣷⢙⢿⡶⠆⠀⠀⠀⠀⠀⠘⠿⢯⣾⣿⣿⣿⣵⡆
⠀⠀⠀⠉⠉⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐⠏⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣛⡛⣛⠛⠁



⠀⠀⠀⠀⢀⣤⣴⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⣶⣶⣄⢤⡀⠀⠀⠀⠀⠀⠀⢀⡤⣖⣶⣶⢤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⢠
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣾⣆⠀⠀⠀⠀⠀⠀⣰⣿⡿⣛⣻⣷⣽⣅⠀⠀⠀⠀⠀⠀⣰⣿⡟⣛⣻⣮⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢘⣉
⠀⠀⠀⠋⠙⠋⢹⠋⠙⠋⠛⠀⠀⠀⠀⠀⠀⣿⣿⣟⢿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⡇⣧⠈
⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡀⣸⡇
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣋⠄⠀⡀⠈⠉⠉⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⡏⠀⣼⣋⣹⠀⠀⡛⡧⣩⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣷⢄⡘⠏⠟⢀⠠⣭⣿⣿⣿⣷⣤⣤⣤⣤⣤⡄⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣏⣓⡀⣧⠠⠀⡼⣸⣿⣿⣯⠻⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢻⣿⣿⣶⡦⠇⡜⣿⣿⣿⣷⡄⣹⡟⣿⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣿⣿⡟⣿⣿⣸⣿⣿⣿⣿⢿⢻⡇⠸⢏⣼⣿⢿⣿⣿⣿⢈⡟⣰⠙⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣿⣟⠁⠉⠉⢹⣿⣿⣿⣿⢸⢸⣿⢲⣷⣿⣿⣷⣿⣿⡩⠞⣧⡛⠂⠿⠿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣿⣿⣦⣧⣼⣿⣿⣿⡛⣧⢺⢸⣯⢸⡿⣾⡋⠛⠛⠻⣷⠼⣿⣃⣀⣤⣾⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣿⣿⣿⠿⢿⡟⣿⠿⣧⣿⣿⡸⡏⣿⡇⣿⣷⣦⣟⣴⣀⣴⣿⣿⣿⣶⣶⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣿⣿⣿⣷⡌⠃⣭⣿⣟⣵⡿⢧⡙⠿⡇⠿⠾⣿⣿⣿⣿⡿⢛⣫⡴⡛⠻⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣿⣿⢿⣛⢷⣿⠾⠇⡾⢣⠆⢩⣵⣙⣿⣾⣿⡛⣿⣿⣿⣿⣿⣷⣿⣿⣶⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⣿⣿⣆⡸⠛⣧⠀⠾⢿⣻⣿⣿⡿⣿⣿⣿⣟⣉⣩⣿⣿⣾⣿⣿⣿⣮⣸⡇⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠿⠻⣿⣷⣦⣿⡶⢿⣿⠃⠀⠻⣾⣿⣿⣿⣿⣿⡞⠉⣉⣿⠏⢡⣼⢿⣿⡇⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⢀⠤⣀⠈⠙⠿⠿⠴⠿⠧⠌⠀⠙⠶⠿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠟⠟⠟⠛⠃⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣿⣿⣿

⠻⣦⡀
⠀⠈⠻⣦⡀
⠀⠀⠀⠈⠻⣦⡀⡀
⠀⠀⠀⠀⠀⠨⣿⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⠻⢦⣄⣴⢶⣄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⢀⣴⠟⠁⣠⡟⠃⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⣠⡾⠉⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣍⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡾⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⡴⠋⠀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣴⠟⠁⣠⡀⠀⢀⣤⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⠋⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣟⠁⣠⠆⠀⢀⡴⠂⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⣴⠟⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣠⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡾⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣴⠋⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣠⡾⠋⣠⡾⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡾⠋⣠⡾⠋⠀⠀⠀⠈⠳⣄⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⣠⡾⢯⡀⠀⠀⠀⠀⠀⠀⢈⣷⠟⢁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⡾⠋⠀⠀⠙⢦⡀⠀⠀⢀⣴⠟⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠙⢦⣴⠟⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣴⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣶⣛⣁⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣿⡇⢀⣤⣄⠀⢠⣤⣄⠀⢠⣀⡄⣴⣧⡄⠀⢸⣿⠉⠙⣷⡀⣭⡁⣠⣤⡄⢀⣠⣤⡀⢀⣤⣤⡀⢀⣤⣤⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⣿⡃⣿⡷⠾⠇⣴⠖⣿⡇⢸⡏⠀⢸⣏⠀⠀⢸⣿⠀⢀⣿⠃⣿⡇⠻⢷⣄⢸⡷⠾⠿⢠⡶⢺⡇⠘⠿⣦⡰⣿⠶⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠛⠁⠈⠛⠛⠁⠙⠛⠛⠃⠘⠃⠀⠈⠛⠃⠀⠈⠛⠛⢋⠁⠀⠛⠁⠛⠛⠃⠈⠛⠛⠃⠈⠛⠛⠃⠘⠛⠋⠀⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄⠀⠀⢠⣊⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⢤⠷⠻⣿⡄⢠⣼⠿⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⢺⡀⢦⡄⡔⠀⠀⠀⠀⠀⠀⠐⣎⢤⠆⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣼⣿⡿⢛⣛⡤⠀⠀⠀⠀⠀⠀⠀⡳⣜⡠⠙⠲⣴⣿⣿⢀⡤⠷⠣⣀⡴⠋
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣿⣿⣿⣦⣴⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠄⣾⣿⣿⣿⣷⣼⣿⠀⠄⠀⠀⠀⠀⠀⠶⣞⣿⠗⠛⠻⢿⣟⣿⣿⡞⠛⠳⢿⣟⡵
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠁⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿⣿⡿⣾⣿⡉⠁⠀⠀⠀⠈⠀⣸⢷⡀⠀⢀⣾⣿⣿⣿⠇⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠜⡷⢻⡄⣸⣿⣿⣿⡿⠀⠀⣰⢿⠟⠌
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣾⣿⣿⣿⣷⣀⢾⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣤⠄⢐⣭⣾⣷⣶⣶⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⢾⣷⣬⣧⣼⣿⣿⣿⡿⢿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠐⢚⣦⡄
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣦⡂⠈⢿⣯⣿⣿⣿⡏⠅⢼⡆⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⣛⣿⣶⣦⡀
⠀⠀⠀⠀⢠⢃⣾⣿⣿⣿⣿⣿⣿⣦⡸⣿⣿⣿⣿⣜⣆⢀⣇⣹⣿⠿⣇⠀⠀⠀⠔⠀⠀⠀⠸⣿⣿⣷⢸⡀
⠀⠀⠀⠀⣟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢿⣿⣹⣿⣿⣿⣿⠟⠀⡹⠁⠀⠀⠀⠀⠀⢀⣤⡏⢹⡇⢠⠇
⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣀⠈⠋⢉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣣⣿⡅⡼
⠀⠀⠀⠀⠈⢫⣿⣿⣿⣿⣿⣿⣿⣿⠁⣐⣾⠛⠁⠀⠚⠻⠷⢶⣴⣤⣀⣤⣠⣖⣾⠿⣿⣫⣿⡿⠉⢸⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⣿⣿⣿⡏⠀⠈⡇⠀⠂⠠⡀⠠⠀⢠⣿⣿⣿⡿⠟⠛⠛⠾⠋⠛⢻⣗⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠙⠋⠡⠀⠀⢸⣆⣀⢉⡀⣀⣤⣾⠟⠉⢀⣤⣷⡄⠀⠀⠀⠀⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠋⠁⣠⠔⠁⠘⠋⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠑⠀⠀⢺⠻⡗⠀⠀⠀⠀⠂⡀⠀⠈⣀⣀⣤⠖⠁⠀⠀⠀⠀⠀⠀⣠⡿⢿⡃
⠀⠀⠀⠀⠔⠀⠄⠀⠀⢸⠀⠈⠲⣦⡀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⢀⢀⡄⢻⣿⣶⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⢴⠿⠛⠒⢫⣀⣀⡄⡀⣀⣤⣤⠾⠷⣦⣾⡟⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠠⠀⠄⣸⣄⣀⣶⣶⣧⡼⢻⣿⣿⠟⣛⣀⠚⢷⣿⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠈⠙⣿⣿⣭⣿⠁⢀⣻⣁⣠⣟⣹⣾⣌⣹⡀⢀⣠⠄⡀⣠⣃
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣼⣏⣇⣠⡀⣿⣉⣡⣌⣉⣿⣉⣩⣿⡇⣿⠷⡎⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⡉⣠⣿⣿⣿⠀⣼⠿⢉⣭⣤⣤⣤⠨⣭⠹⡷⡏⠀⠽⢷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠛⢛⠛⡎⢻⣶⠃⠀⣻⠿⠿⢶⣷⡤⠀⡀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢋⣠⣾⡿⢾⣯⣼⡟⢀⣶⡇⢠⣾⣿⣿⣷⣼⣟⡄⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠛⢿⡽⠟⠹⠟⠋⣿⣄⠀⢻⣀⣿⣿⠟⠋⡩⡎⠀⠀⠀⣺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠦⠶⣶⠀⣠⣾⠟⣉⣅⣴⡗⠻⠃⡠⠃⠘⣁⡀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠟⢤⡿⠿⠋⠀⠀⠉⢪⣷⡿⠟⠳⠴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣾⡟