Boater Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡲⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⣉⡉⠳⠯⢒⣒⣑⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⢒⡍⡆⠉⢀⡀⣀⡠⠬⡳⡄⠀⠀⠀⠀⣆⠩⡑⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠯⣓⡤⠜⢸⡼⠈⣿⡀⣘⡜⠤⠼⠛⢦⠀⠀⠀⣇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⠶⡂⠽⡥⠤⡤⠖⢖⠛⠳⡿⣌⣩⡤⠤⠤⠖⠒⠒⠒⠈⡑⢄⠀⣿
⠀⠀⠀⢀⣠⠴⠒⠉⠁⠀⣟⠉⠀⢀⠁⠀⠇⠀⠈⡄⢀⡸⠽⣀⢂⠔⠰⠱⠀⠪⠇⠘⠜⠀⣙⣝⠽⢦
⠀⠀⣖⡉⠀⠀⠀⠀⠀⣀⣀⣉⣁⡾⠶⠖⠛⠛⠉⠉⠁⠀⠀⠈⠉⠉⠙⢛⣛⢛⡭⣍⡭⠩⡅⠀⠀⢀⢷⣄
⠀⠀⠈⠛⠭⣉⡉⠉⠉⠉⠁⢶⡒⠒⠉⢳⠠⠴⠦⣰⠖⣆⣖⣛⣉⣉⣽⠸⠼⠚⠖⡚⣉⣉⣥⣤⡶⢾⠿⠱⠃
⠀⠀⠀⠀⠀⢀⡟⠓⠶⢶⣤⣀⠈⠂⠤⠒⠃⠑⠚⠉⠋⠉⢈⠁⠀⢀⣤⣤⣤⣴⣾⣿⡛⢋⠴⠃⠈
⠀⠀⠀⠀⠀⠀⠉⣶⠀⠀⠀⠈⢙⢲⣤⣤⡦⠴⠶⠶⢶⣾⢿⣿⣿⠽⠧⠤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠛⠉⠛⠁⠘⠒⠙⠚⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠀⠀⠀⠀⠀⢀⣀⡤⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠠⠤⠴⠒⠋⠉⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⢾⠀⡎⠀⠀⢀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⢚⣋⣉⠠⠤⠬⠒⠓⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⡄⠀⠀⠀⠀⠒⠒⠒⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣀⡄⢠⡤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⡀⣀⠀⠀⣀⣀⢠⠤⠤⠒⠂⠉⠈⠈⠀⠈⠀⢋⢼⠁⠀⠰⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡤⠤⣔⢒⡃⢈⡏⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠑⠊⠁⡴⠀⠀⡇⠀⡄
⠀⠀⠀⠀⢠⣀⣤⣤⡶⠒⢛⣛⣉⣁⣀⣀⣀⡤⠤⠤⠤⠞⠚⠃⠓⠚⠂⠀⠈⠉⠀⠉⠀⠀⠄⠀⠒⠂⠈⠀⠀⠇⠀⢰⠀⢠⠇
⠀⠀⠀⠐⠚⠛⠻⠿⠫⠭⠉⠉⠉⠉⠐⠐⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⡞⠀⠀⡌⠀⡎
⠀⠀⢈⡈⠀⠀⠁⠀⠐⠒⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠠⠤⠤⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢴⠁⠀⡰⠁⡜
⠀⠀⠈⡍⠉⠁⠀⠀⠂⠀⠐⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⡎⠀⡰⠁⡼
⠀⠀⠀⡋⠉⠁⠀⠀⠀⠀⠀⠐⠀⠀⠤⠄⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠂⠀⡺⠁⢠⠁⡼⠁
⠀⠀⠀⠘⡒⠒⠀⠤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠇⡜
⠀⠀⠀⢀⠱⠄⠀⠀⠀⠀⡀⠀⣀⣈⡁⠐⠒⠒⠂⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠋⠀⠲⣾⡜
⠀⠀⠀⠀⠉⠉⠉⠀⠈⠃⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⠄⠒⠂⠀⡠⠂⠀⠂⠄⠀⠀⠀⠐⠠⠂⠀⠀⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠖⠒⠛⠀⠀⢀⣀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⣠⣶⣤⣀⣀⣀⠀⠀⠠⠤⢀⣀⢀⡀⠋⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠑⢿⣯⣿⣉⣋⣉⡩⠍⠁⡀⣸⠿⠗⠈⢁⠠⠀⠊⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⢉⡠⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠆⠄⠀⢰⠀⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡛⣝⢟⣉⡛⣻⣛⣹⡟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠐⢂⡈⠉⠒⠤⣀
⠀⠀⠀⠀⠠⠁⠀⠀⠈⠑⠠⢄⡀⠉⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢠⣀⠀⠀⠀⠀⠉⠒⠤⣀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⢀⡄⠀⢠⠀⠉⠒⠤⣄⡀⠀⠀⠀⠉⠒⠤⣀⠈⠐⠦⢀
⠀⠀⠀⠀⠀⠟⠀⢸⣆⠐⢲⠠⣄⡉⠑⠠⠀⠀⠀⡤⠄⠚⠀⠄⡀⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠈⣦⠈⠉⠙⠀⠀⢀⡀⢀⠀⣄⠀⠀⠠⢐⠒⠒⠾⠀⠁⠂⢄⡈⠑⠄
⠀⠀⠀⠀⠄⠀⠹⣠⡹⢦⡀⠀⠀⠐⠄⠀⠈⠓⢤⡀⠀⠉⠒⠄⡀⠀⠀⠀⠈⠒⢄⡀⠶⠠⡀⠀⠀⢪⡀
⠀⠀⠀⠀⠀⠀⠀⢄⢉⡐⠈⢤⡀⠀⠀⠑⢄⠀⠀⠙⠲⣄⠀⠀⠀⠀⠤⣀⢀⠀⠤⠁⠰⢄⡀⠘⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⠃⣂⡄⢀⠐⠆⠀⠑⠀⠐⠀⠈⠉⠀⠀⠀⠀⠋⠀⠀⢀⣀⣀⡄⠈⠒⢄⡀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠻⢿⣅⣐⡠⢄⡐⠠⠄⠀⠐⠀⠐⢌⠃⢄⠀⠀⠀⠈⠛⢄⡀⠀⠀⠈⠐⢤⡀⠁⠰⢦⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠙⠻⢦⣤⣀⠀⠀⢀⡀⠠⠄⠈⠢⡈⠒⢄⠀⠀⠀⠘⠆⠈⢸⠁⠀⠀⠀⠀⡎⠀⡸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠈⠉⠓⠦⢄⣭⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡇⠀⢸⠀⠀⠀⠀⢰⠁⢠⠃⠀⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠈⠙⠻⠾⢶⣦⣄⡀⠀⠀⠀⠀⠁⠀⢆⠀⠀⠀⢀⣖⣀⣾⣤⣬⣭⣍⣂⣀⣰⣂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⢤⣀⡀⢀⣄⣤⠽⠶⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⣀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⣈⠑⠢⣄⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠬⣣⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠀⡂⠒⠀⠀⠐⠂⣀⠀⠂⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠶⣶⣶⣶⣆⣤⣤⣤⠤⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⣹⢯⣈⢀⡗⣿⡿⣿⢾⣷⣩⣟⣾⣷⣧⣾⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⣿⡿⢛⡁⣿⣸⣋⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣻⡑⣩⣿⣿⣿⣷⣆⠀⠀⣉⣿⡇⡿⢞⢻⡃⣿⢸⡏⣯⣿⣿⠧⢱⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡙⢿⡗⣟⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡾⠶⠿⡞⢿⣟⠿⣟⡁⣾⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣤⣌⡁⡀⢈⠃⠌⠂⡅⢝⡲⡷⣎⠤⢐⣓⣋⣿⣯⣿⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠶⠿⠾⠿⠿⠟⠛⠛⠋⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠄⠀⠂



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⣥⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠁⢐⡒⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡘⠋⠤⠤⣤⣀⣀⡀⠀⠈⠉⠉⠉⠉⠉⠁⢠⣀⣒⣶⣿⣿⠷⠿⠛⣃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⢉⣉⣩⣩⣥⣤⣶⣶⣶⣾⡏⡚⣥
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢸⢸⢺⣸⣿⣿⣿⣿⣿⢷⣧⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣷⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣸⣀⣈⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀
⠀⠀⢀⡴⠚⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⢸⡀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿
⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠟⠛⠉⠁⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠉⠐⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠒⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠒⠒⠒⠀⠨⠉⠉⠉⠉⠙⠈⠀⠐⠐⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠠⠜⠛⠂⠈⠀⠈⠛⠠⠀⠊⠤⠲⠄⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠂⠁⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠄⠰⠆⠲⡴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⣠⣾⣿⣿⣧
⠀⠀⠀⠀⢀⣠⣄⢶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⢀⣾⡿⢿⣿⣿⣿⣦⣀
⠀⠀⡠⠖⠉⡀⢢⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⢸⣴⣄⡼⢿⣾⣿⣿⣿⣿⣽⡳⣦⣀
⢀⠚⢁⠀⢛⠀⠽⣿⣿⣿⣿⣿⣤⣀⡐⠄⠀⢀⣀⣤⣴⣦⣈⣢⣰⣵⣼⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣳⣄
⡦⠀⢄⡈⠃⢀⠈⠿⣿⣿⣾⣿⣿⣿⣿⣿⣾⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡠⣽⣿⣿
⢁⠂⠂⠐⠢⠈⡰⢀⠉⠙⡟⠻⢿⣿⣿⣿⣿⣿⣬⣾⣝⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⠿⢿⣿⣥⢭⡿⣟⣺
⠀⠰⢄⠠⢠⠀⠠⠀⠈⠦⣈⠢⠬⠈⠉⠟⠻⢟⠻⡿⡟⠛⠛⠛⠛⠙⡙⠋⠛⠃⠧⠏⠉⡟⣿⣿⣽⠧⢱⡼⠋
⠀⠀⠀⠁⠦⣀⡄⠀⠁⠠⠀⠀⠀⠀⢀⠐⠉⡄⠄⠀⠀⢀⠔⠀⠀⠔⠄⠠⠈⠑⠀⡀⠦⠹⡒⣻⣾⠞⠋
⠀⠀⠀⠀⠀⠀⠁⠒⠠⠀⠀⢀⡀⠀⠀⠅⠀⡀⠀⠀⠀⡀⠀⠀⠀⡀⠄⠀⠀⢂⠈⣰⣼⠿⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣭⣥⣾⣷⣶⣤⣴⣤⣼⣤⣤⣦⣴⣶⣶⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣭⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⢄⡀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠉⠓⠪⢝⣓⠤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠈⠉⠒⠮⢕⣢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣙⣓⣒⠢⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠉⠉⠉⠛⠒⠒⠲⠬⠭⢽⣒⣒⣒⡤⠼⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⢤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠈⠉⠁⠒⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢞⣫⠔⢳⡀
⠀⠀⠀⠀⠀⠀⣐⠳⠶⠦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣀⣸⣀⣀⣤⣤⠴⠶⢚⣛⡩⠴⠚⠉⠀⠀⠈⣇
⠀⠀⠀⠀⠀⢰⠉⠉⠓⠲⠦⠤⣤⣀⣀⣈⣉⣉⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣥⠤⠤⠖⠒⠋⠉⠁⣀⠤⠐⠈⠀⠂⠀⣿
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠒⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⣤⠤⠤⣴⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣰⡇⢠⣀⡀⠀⠀⣿⢸⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠼⠏⠉⠛⠲⢿⢼⡀⠙⢯⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⠀⠀⠈⢳⡀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠀⢳⡀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⢸⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⡼⢻⣾⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⢰⡇⢸⢹⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⣾⠀⢸⣿⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⣻⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⣿⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⢻⠀⠀⢠⠏⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠶⠒⠒⠒⠲⠶⠤⣤⣀⡀⢻⡄
⣀⣀⣀⣀⠀⣾⠛⠋⠉⠉⠉⠙⠛⠓⠲⠶⠤⣤⣀⡀⠀⠸⡆⢸⢻⠀⣰⠏⢀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣉⣛⣿⣄⣀
⢿⡖⠶⣍⣉⣛⠒⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠿⣿⢸⣾⠷⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢒⣫⡵⠿⠾⣷⣿⡿⠿⠿⠿⠿
⠀⠉⠛⢿⣈⣉⣛⣦⣈⠳⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣡⡴⠛⠛⠛⠙⠉⠉⣉⡿
⠀⠀⠀⠈⢯⣉⠉⠉⠙⠳⠤⠤⣭⣭⣉⣙⣛⣛⡒⠒⠒⠲⠶⠾⠿⠶⠦⠤⠤⠤⠶⠶⠶⠒⣚⣉⣤⠞⠛⠒⠒⠛⠛⠛⠋⣉⡿
⠀⠀⠀⠀⠈⠻⣍⠛⠛⠓⠒⠶⠶⠶⠶⠤⠭⠭⠭⠭⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠖⠒⠚⠛⠛⢩⡟
⠀⠀⠀⠀⠀⠀⠈⠻⣟⠒⠲⠶⠦⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠴⠶⠶⠶⠚⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠖⠶⠶⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠶⠶⠒⠲⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡤⠄⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢤⠤⠀⠤⡄⠀⠶⠖⠒⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢢⣀⠀⣇⠀⡀⢀⣀⠀⡀⠀⢈⠀⡁⡀⣀⠀⢐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠜⣡⡇⠇⠁⠘⠀⢇⡀⠤⢆⠸⠀⠇⡇⢸⢀⣂⡁⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠔⠀⠀⠀⠈⠀⡆⠍⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠉⠁⡎⠀⢀⡇⠒⠌⠌⠉⣵⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡀⡄⠄⠄⢁⡰⠌⢫⣅⣀⢰⠄⣼⣤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣽⣾⣶⣤⢤⣬⣤⣭⣤⣩⣤⣾⣮⣵⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣽⣻⠿⣟⡙⡛⠛⢿⣿⡝⣩⣟⣹⣿⠾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⣾⣿⣇⠀⠀⠈⠉⠉⠉⠉⢙⣿⢩⣭⣿⡻⣿⣰⣿⣷⡎⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠘⠿⢿⢿⣄⣀⠀⠀⠀⠀⠀⢸⣿⠀⣼⣳⠷⣿⣿⠟⢛⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣒⡤⠀⠀⠈⠉⠓⠒⠒⠲⠺⠟⠾⠍⠀⠉⠈⢄⡢⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠻⠧⠤⠤⠤⠴⠥⠤⠍⠥⠔⠐⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢇⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡂⠀⠈⠀⣠⢾⡋⠡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡛⠀⠀⠈⠡⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢷⠀⠀⠠⠀⡁⠀⡀⠤⠀⠚⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢓⠀⠀⠀⠘⡆⠀⢐⡠⠔⡁⠄⠀⠊⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⡀⠀⠀⠀⣡⠔⠈⠀⠀⠀⢀⠀⠄⠀⠐⠀⢀⡀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠄⠀⠠⠈⢡⠐⠀⠀⠀⠑⢀⡠⠊⠀⠀⠀⠠⠀⡈⠠⠀⣁⡠⠄⠛⠊⠉⠀⠀⠀⠈⢄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠔⢁⠀⠄⠠⡀⠀⣠⠤⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠈⡀⡘⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠡⣸⠀⠀⣀⢴⠀⠔⡂⠥⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢺⠁⠤⠀⠥⠈⡃
⠀⠀⠀⠀⢐⣰⠖⠚⢡⡿⠗⠋⡁⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠂⠀⠀⠀⠀⠰⣘⡴⠀⠀⢰⠀⡇
⠀⠀⠀⢀⠈⢀⠠⠘⡁⠀⠂⠁⠀⠀⠀⠄⢂⡀⠠⠄⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣠⠁
⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⠀⠄⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢂⠃
⠀⠀⢠⠊⠤⠂⠈⠉⢀⠀⢄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠦⢤⣤⢥⢀⣁⣄⣁⣀⣀⣀⣀⣀⣀⡀⣀⡀⡀⠀⠀⠀⢀⢀⡀⢀⠀⢀⣀⡀⠤⠔⠚⠋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⢦⡀⠀⠀⢸⡏⠉⢙⣿⠀⠀⣿⠉⠉⠉⠁⠀⠀⠤⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣶⣶⣦⣤⡤⠀⠀⠀⢸⡷⠲⠾⣅⠀⠀⣿⠒⠒⠂⠀⠀⠀⠤⠴⠶⢶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣤⣤⣤⣤⠀⠀⠀⣸⣇⣀⣀⣿⠃⢀⣿⣀⣀⣀⡀⠀⠀⠠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢠⢴⣤⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⣩⢭⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⡦
⠀⠀⠀⣶⡀⡇⠀⢻⡇⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⠶⡄⠀⢀⣀⠀⠐⠛⠀⢰⣇⠜⠰⠆⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠲⣾⠿⣄
⠀⠀⠀⢸⣧⡇⠀⢸⣇⢠⠖⢻⡆⠀⣼⢋⣦⠀⢸⡇⣠⣤⣶⠎⣿⠀⢸⡇⢀⣼⡏⢸⡇⢀⣾⠏⠘⢀⣶⡳⢀⡴⠋⣸⡇⣼⡏⠀⠉
⠀⠀⠀⠀⢻⣿⠀⠈⣿⡏⠀⣾⠃⢰⠛⢹⡇⣠⠛⢛⠿⡇⠙⠀⣿⣰⠛⠧⢞⣸⣧⡘⠦⠚⣟⠀⣠⢾⣏⡴⠛⠃⢀⡿⢠⡿
⠀⠀⠀⠀⠀⠙⠃⠀⠈⠁⢰⡟⢀⠏⠀⠸⠟⠁⡴⠁⢸⡇⠀⠀⠈⠁⢠⠞⠁⢸⡇⣸⠂⠀⠙⠚⠁⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠋⠀⠀⠀⠀⠀⠧⡤⠞⠀⠀⠀⠀⠀⢺⡀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣣⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡾⢋⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⣀⡴⣫⡾⣫⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢃⠀⠀⠀⠀⣠⠞⣫⢾⣫⡾⣻⠿⣋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠀⣠⡴⢛⡵⢚⡵⢟⡥⢞⡡⢞⡥⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⣟⣡⠖⣫⠞⣩⢖⡫⢖⡩⢖⣡⠞⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢻⡴⢚⢵⡪⢒⣩⠖⣋⡴⢛⡵⣫⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠀⠀⠀⠀⠀⠀⠀⢺⡕⣫⠖⢋⣴⠞⡩⢖⣭⢞⣩⢞
⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣻⣵⡟⠀⠀⠀⠀⠀⠀⠀⠀⢊⣠⠾⣋⠔⣫⠖⣫⠶⣫⣴⣿
⡶⠮⣍⡛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠟⣃⡭⠾⣋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⢞⡵⣊⡵⣻⡵⠞⣉⣤⣴
⣉⡛⢦⣍⡳⢬⡙⠷⣦⣀⠤⠤⠤⠤⠶⣶⡿⠿⠛⣉⡡⠔⢚⣉⣥⣶⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣾⡿⣺⣵⠾⠛⢋⣩⣴
⡯⣝⡓⠮⣝⡶⣌⡳⢬⡙⢷⣦⡤⠭⠵⠖⣚⣋⣭⣴⣶⡿⠿⣛⣩⠵⣒⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣈⣠⣤⠶⠟⣛⣽⣿
⣭⡓⠭⡓⢮⣙⠦⣝⠶⣝⠦⣈⠻⣿⡛⠉⠝⣓⡫⠭⠔⣒⣩⠥⠒⠋⣉⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣡⣴⡶⢟⣿⣟⡵
⢌⡙⠷⣦⡑⠈⢑⠈⠳⢾⣿⡮⣓⠌⠳⣌⠡⠶⠒⠓⢉⣁⣤⠴⠞⣛⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⣠⣴⡿⣫⠞⣡
⠶⢦⣅⡈⠙⠷⣦⣈⠢⣄⠙⢿⣮⣷⢤⡈⢣⡐⢒⠈⠡⣁⣢⡼⢟⣋⡁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢛⡿⢋⡼⢡⡞⢁
⣿⣷⣮⣝⡳⢦⣄⠉⠻⣮⣓⢷⡿⣛⣷⣹⣤⡴⠶⢾⣛⣉⠙⠲⠦⣬⣉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⠔⢉⡴⣫⡶⠋
⠭⢍⣛⣻⣿⣷⣬⣽⡤⠶⠿⠛⠛⠯⠭⣉⡓⠲⠶⣤⣍⡙⠛⠶⣤⣀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣘⣥⠞⣫⡾⢋⣤⠾
⢷⣶⣤⣐⠲⢤⣤⣀⣀⡀⠐⠛⠛⠛⠒⠶⠦⣭⣝⣒⠾⠿⣷⣦⣤⣉⡛⣿⣀⣀⠠⠤⠄⠀⠀⢴⣢⡼⠛⣁⡴⣛⣡⠆⠙⠴⣋⡵⢛
⢦⣌⠓⠮⣽⣒⠶⠬⣭⣝⣛⣛⡛⠛⠛⠒⠲⠶⠾⠽⠿⠷⣶⣾⣿⣿⣿⣿⡋⡏⠉⠉⠉⠀⠀⠒⠒⠒⣈⠡⠬⠭⠭⠥⠵⡯⠵⠚⢋
⡳⢭⣓⠦⣬⣉⡛⠶⢤⣮⣭⣭⡭⠿⢿⣿⣶⣖⣒⣒⣲⣶⣶⣶⣯⣽⠟⠉⣇⢱⠀⠀⠀⠀⠀⠀⠭⠀⠀⣒⣠⣤⣶⡶⠶⠷⠖⣛⣋
⣍⣓⠮⣭⣛⣿⡿⣿⣦⣶⣤⣍⣩⣭⣭⣧⣤⠤⠶⢿⣿⡿⣟⣫⣭⠿⢷⣾⣿⡘⣤⣶⣤⣀⣠⣴⣶⡾⠟⠛⢉⣁⣤⣴⡶⠿⢛⣛⣉
⣚⢿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣶⣶⣶⡶⠿⢛⡩⠥⠴⠶⣖⣒⣛⣉⣽⠿⣱⣭⣿⡿⠟⠋⢉⣠⣤⣶⣿⣿⡿⠿⠿⠛⠛⢉⣉⣉
⠿⣿⣶⣾⠭⢭⣭⣭⣽⣿⠿⠿⣛⣋⣩⣴⣶⣬⣥⣬⠄⠩⠭⠭⠭⢖⣪⣶⠿⠛⢉⣁⣤⣶⡿⠿⠛⣋⣉⣡⠤⠴⠒⠚⠛⠛⠉⠉⠉
⣽⣶⣶⣿⣿⣿⣯⣭⠶⢂⣉⣡⠤⠤⠒⣒⣒⣒⣒⣒⣒⢒⣿⡿⠿⠋⣁⣤⡴⠾⢛⣉⣥⠴⠖⠚⠉⣉⣀⣀⣤⣤⣤⡶⠶⠶⠶⠟⠛
⣭⡭⠭⣷⢒⣪⡭⠖⣛⣩⠥⢖⣛⣋⣉⣭⣭⣭⣉⡭⠟⢋⣡⡴⠾⠛⣉⡥⢖⣚⣩⣥⡶⠶⠟⠛⣛⣉⣭⣭⣴⠶⠶⠶⠶⢞⣛⣛⣛
⢖⡶⠟⣋⡭⢄⣒⣭⣶⡶⠿⠿⠿⢯⣤⡤⢞⣫⣭⠶⣛⣩⣥⣶⣾⡿⠶⢛⣋⣩⣥⣴⠶⠾⢛⣛⣉⣭⠤⠴⠖⢒⣛⣉⣉⣉⣉⣉⣉
⠛⠚⠛⠑⠚⠛⠻⠟⠛⠻⠿⠷⣶⣿⣿⣛⣫⠵⣖⡿⠿⢛⣋⣭⡴⠶⠛⣛⣭⡭⠶⠖⢚⣛⣭⡭⠶⠶⠚⣛⣛⣉⣭⣭⣭⣥⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⢘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣡
⠀⠀⠀⠀⠀⢺⡿⠃⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡹⢋
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠐⢀⣀⣀⣀⡀⠠⠀⠀⠀⠀⠀⠀⢂⣀⣀⣀⠢⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⠀⣄⣀⣴⠞⣠⡌⠹⣿⣟⠿⢷⡀⠀⠀⠀⠀⣾⠻⢟⣿⡟⠁⠈⣷⢦⣤⣀⡠⠁
⠀⠀⠀⠀⠀⠘⡿⡁⠘⢻⡇⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣷⠈⡀⣿⡟
⠀⠀⠀⠀⠀⠀⡷⠁⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠸⣌
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠐⠄⠒⠀⠀⠀⠀⠀⢠⡟⠀⠀⢠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⢠⣴⣶⣶⣤⠀⠀⠀⣠⡿⠁⠀⠀⠂⢀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢄⡈⣷⣄⡀⠀⠐⠒⠂⠀⠀⣠⣶⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⡿⢿⣦⣄⣀⣠⣴⣾⣿⣿⣷⣿⠀⢀⣀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠢⣶⣦⣾⣿⣿⣿⡈⠻⣿⡿⣿⣿⠋⡸⢽⣿⣷⣎⣥⣶⠞
⠀⠀⠀⠄⠀⠀⠀⢀⢀⣽⣿⣿⣿⣿⣷⡍⠀⠘⣦⠟⠀⠀⡀⠈⠽⣿⣿⣿⣥⡄
⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⡋⠉⠿⠀⠈⢫⡀⠀⠀⠀⠀⡀⠀⠀⠺⠁⠉⣩⣿⣷⣶⣀⠀⢀⠀⡀
⠀⠀⠀⠀⣸⡽⣾⣿⠯⣊⢥⠆⠀⠀⠀⠀⠑⠀⠂⠀⠂⠀⠀⠀⠀⠐⢾⣶⣿⣯⣭⣿⣿⣿⣿⡀
⠀⠀⢠⣾⣿⣿⣚⣛⣡⣷⣾⣿⡀⠀⠀⠀⠀⠂⠀⡐⠀⠀⠀⠀⠀⣼⣿⣿⣿⢟⣩⡿⢫⣿⣿⣽⣦
⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣦⣤⡀⢀⡀⢠⠌⠄⠀⣠⠀⣤⣴⣿⣿⣿⣿⣿⣿⣟⣳⣼⡯⠗⠛⠁
⠀⠀⠀⠈⠑⠛⠿⢿⣿⣿⣿⣿⣾⣟⣷⣦⡿⣃⠀⣆⣿⣯⣿⣯⣽⣼⣿⣿⡿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣷⣾⣶⣿⣿⢿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣽⣿⣵⣿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣴⣖⠛⠛⠻⢗⡲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣋⣉⣤⣤⣿⣿⣷⣶⣶⣦⣽⣷⣾⣿⣿⣷⣯⣭⣍⣛⣓⣒⣒⡶⠦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣀⡐⢈⡉⡉⠉⢉⣩⣭⣫⣭⡽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢤⣄⣉⡩⠿⢗⣚⡽⢞
⠀⠀⠀⡰⢎⠻⣿⣿⣇⠀⠈⠐⠺⣟⠛⢿⣿⣿⣿⡆⠘⠛⣿⣿⣾⣿⣿⣷⣾⠿⠿⠿⢛⡛⠋⠩⢄⡐⠒⠌⢙⣩⣵⠖⠉
⣰⠂⣎⢿⣄⡁⣬⠛⢿⣷⣦⣤⣀⣀⣀⠙⣏⠻⠝⠻⠾⠉⠉⡉⠉⠡⠄⣒⣒⠠⠭⠉⠐⣀⣁⣠⣤⣶⢶⣿⡿⠛⠁
⣏⢏⠙⢣⡙⢿⣯⣉⠑⠢⢝⠻⠟⠛⠉⠁⠄⠐⣐⣂⣭⣤⣴⠶⠖⢛⣉⣡⣤⣶⣶⡿⢿⣛⣻⣭⡷⠟⠋⠁
⠹⣦⡑⢄⡉⢰⣽⣻⢿⣶⣦⣄⣤⣴⣶⠾⠿⠛⣛⣫⣭⣴⡶⠾⣟⣛⣿⣭⣿⣷⣾⠿⠟⠛⠉
⠀⡯⣹⣷⢬⣝⡒⠭⠟⠒⣈⣨⣥⣴⠶⣾⣛⣻⣭⡽⢷⣶⣿⠿⠿⠛⠛⠉⠉⠁
⢠⢷⣿⣿⣷⣭⣝⡛⢻⣿⡭⠷⢶⣛⠻⠭⠍⠓⠂⠉⠉
⣬⣽⣿⣿⣥⣤⣭⣽⣿⣾⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⠛⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠙⠁⢸⡇⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⣿⠀⠸⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡀⠀⠀⠀⠀⣿⠀⠀⠠⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢴⡿⢴⠀⠀⠀⣿⡇⠀⠀⠀⠀⠣⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⡀⠀⠀⠚⠁⠀⠀⠐⠀⠀⠑⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣴⣠⣶⢆⣠⠅⢀⣴⡀⠀⢂⣀⣠⣽⣾⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡿⢿⣯⣿⢾⣛⣭⣶⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠠⠀⠉⠀⢀⣥⣶⣿⣩⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠐⣶⡆⣤⠶⠚⢉⣩⣻⢟⣿⣿⡟⢿⣿⣿⠿⠛⠋⠉⠉⣹⢿⣿⣫⣔⡲⢶⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⣷⣒⣩⣤⣾⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⢠⠀⡴⠻⣺⣿⣿⣟⢏
⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣶⣶⣿⣿⡿⠿⠟⠛⠉⠀⢀⠀⠀⠀⣀⣥⠄⠀⠈⣀⢰⣟⠀⠀⠈⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠤⣄⣶⣤⣀⣀⠀⠀⠀⠀⢀⣠⣦⢾⣿⣿⣿⣿⣟⡛⠁⠀⣰⣿⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣿⡿⠉⣠⣬⣿⣿⠟⡡⠋⢡⠾⠿⠛⠻⠷⠢⠀⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣝⠿⡷⢽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣶⣶⠒⠒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣠⣿⣶⣷⣾⣿⣶⣿⣶⣿⣾⣷⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣲⠒⣒⡒⠒⠒⠒⠒⠒⠲⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢠⠃⣸⢿⣿⣦⠀⠀⢀⠀⣸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠒⠂⠾⠾⠿⠿⠿⡿⠦⢤⢿⣿⣿⠿⣤⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠿⢯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠠⠐⠦⠄⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠐⠒⠒⠀⠈⠀⠲⠶⠤⢤⣤⣀⣀⡀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠒⣶⣦⣤⣄⠀⠒⢬⣽⣭⣉⣩⣶⣶⣶⣦⣤⣶⣤⣤⣤⣤⣀⣙⣿⣿⡿⠿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣴⠚⠉⢁⣨⡏⢿⣛⣁⠀⢭⣉⡉⣩⣥⣾⣶⣆⣒⡒⠛⠒⠉⠉⠁⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠳⠖⠭⢉⠥⠄⠁⠊⠐⠻⡿⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠛⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⢺⠒⠒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⢻⠀⣀⡼⠀⠀⠀⠠⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢸⡀⣿⣤⣬⣯⣦⣥⣧⣄⣢⣽⣶⡶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣟⣠⣼⣿⣿⡏⣠⣤⣶⠿⣟⣿⣿⣿⣿⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣟⣿⣟⣿⠟⠉⠀⢰⣿⣿⣿⣿⡏⢻⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣤⡀⠀⠀⢉⢿⣿⣿⣿⣹⠇⢀⣤⡾⠟⠉⣼⣿⣿⢀⣼⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣹⣿⣿⣿⡿⠀⢀⣰⡿⢛⠁⠈⣩⡶⠋⠁⣀⣸⣿⣿⣿⠇⣾⣿⡯⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣼⣿⣷⡾⢛⡅⢸⣿⣿⠛⠁⣴⣾⣿⣿⣿⠟⠛⣿⣷⣿⣿⠟
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠍⣽⣿⣎⣾⣿⣿⣾⣿⣭⣅⣠⣼⡟⠛⠁⣀⣅⣴⣿⣿⣿⣿⣿⣦⣀⠀⠖
⠀⠀⠀⠈⢿⣦⣵⣦⣀⣤⣀⣶⣶⣦⣌⠙⠿⣿⣿⣤⣿⠏⠙⠻⢿⠿⠟⠛⠉⠀⠀⠀⠈⠉⠃⠀⠙⠛⠋⠻⣿⣯⣅⣀
⠀⠀⠀⠀⠀⠉⠋⠉⠿⣿⠿⣿⢿⣿⣿⣿⡿⠿⢿⣿⡿⠟⠁⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣋⣉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠱⠔⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠉⠁⢹⡿⠛⡟⢻⣛⡟⣿⡿⡻⣿⢻⡟⣿⣿⢛⢿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠣⡀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠑⣄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠂⠒⠀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⣀⠀⠰⡧⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢢⡄⠀⠈⠀⢀⡇⠀⣀⠠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢍⣭⡝⢛⣒⢦⣈⣀⡀⠀⠀⠀⠀⠀⣿⠐⠀⠀⠛⠀⠄⢠⡄⢈⣑⣠⣴⠎
⠀⠀⠀⠀⠀⠠⠤⠶⣤⣤⣤⡀⣀⣀⣤⣀⠘⣿⣿⣾⣿⣶⣬⣭⣽⣏⣉⣾⣀⣐⣃⣀⣶⣤⣤⣽⡶⠶⠛⢻⡟⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⢐⣶⣶⣶⣮⣿⣟⣻⣿⣿⣿⣿⠿⣿⣿⠫⣿⣿⣧⠖⢦⣭⣭⣭⠨⢭⣥⣶⣢⣤⣿⣟⠛⢻⣶⣶⠏⢀⣒⠂⠀⠀⠁
⠀⠀⠉⠀⠀⠚⠤⠀⠙⠛⢛⡛⠡⠐⠺⠿⠶⠦⣀⣬⠛⠛⠿⠿⣒⡾⠛⠉⠠⣾⣿⣥⢂⠀⢩⡍⠀⣉⠉⠒⠂⠉⠉⠀⠀⠘⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠛⠀⠲⠂⠘⠛⠛⠻⠤⠀⠀⠐⠒⠀⢲⡀⠉⠉⠩⠤⠄⠀⠀⠉⠥⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣭⣭⣽⣤⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣶⣶⣾⣷⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠷⡶⣤⣄⣀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⡇
⠀⠄⢤⠶⢤⡀⢠⣷⡤⢴⡆⣸⣇⣠⣤⡄⣐⣀⣠⣀⣠⣀⡀⡏⡀⠀⠀⣠⣰⡞⢁⠀⠀⠀⡄⠀⢀⠀⢸⣖⠀⠀⠀⠀⠀⠀⠀⢀⡁⣀⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠠⠹⣀⣼⣤⣬⣿⣦⣌⣤⣹⣯⣬⣭⠉⠿⣿⠙⣩⣹⣿⡗⢭⣶⣶⣖⣾⠻⣿⣿⣤⡼⠯⠷⣶⢸⠂⠛⠛⠠⠀⠤⠀⠀⡀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇
⠀⠀⠀⣭⡇⠀⠀⠉⢙⠋⠉⠋⠁⠿⡟⠛⠛⣿⠒⠾⢿⣿⠟⠛⠛⢛⣛⣿⣿⣿⣿⣭⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠂⢈⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡇
⠀⠂⠠⠴⠄⠠⠄⠀⠈⡈⠈⠀⠀⠀⠀⠀⠀⣄⡀⡤⠌⠙⡟⠛⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠇⠀⡇
⠀⠀⣼⢻⠂⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⡟⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡤⠀⠐⠇
⠀⠤⠾⢿⣯⣭⡁⠈⠁⠀⠀⠀⠂⠀⠀⠀⣶⡶⣄⢰⣶⠦⠀⠀⠀⢠⣤⣧⣀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⣻⡎⠀⣀⣀⣀⣀⣀⣀⣤⡤⣤⣤⣶⣲
⠀⠀⠀⠀⢼⣿⡟⡉⣠⣀⣄⡀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣴⣶⣦⣀⡀⠀⠀⡈⠉⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠿⠿⢭⣭⣶⣶⠿⠿⠿⠿⠿⠛⡟
⠀⠀⠀⠀⠈⠛⠻⣤⣾⠿⢿⡾⠚⠛⠉⠉⠀⢴⠟⠛⠛⠉⠉⠁⠈⠉⠃⡤⠀⠀⠀⢀⣠⣿⣿⠿⠟⠓⠀⠀⠀⣀⡀⠀⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣶⣶⣶⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣠⣴⣾⠯⠉⠛⠛⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⠫⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⡟⠛⠿⠟⠋⠀⠀⠀⠀⠀⢻⡏⠉⠁⠀⠈⠉⠀⠐⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠉⠛⠩⢤⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠒⠒⠒⠶⠤⢴⣿⠠⠤⣄⣻⠀⠀⠈⠁⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⣄⢀⣸⣄⣤⣤⣶⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠰⢶⣶⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀
⠀⠀⠄⠁⠀⠛⣒⣲⣦⣤⣬⣿⣛⣿⣿⣽⣿⠿⣿⣿⣿⣿⣿⡿⣛⣉⣭⣤⣶⣿⣿⣾⣿⢶⡄⠀⠀⠥⠤⠤⠭⠄
⠀⠀⣠⣏⣉⡉⢙⢻⣯⣵⣮⣭⣿⣿⣿⡿⠯⠼⣿⣟⣁⣍⣿⣿⣿⣿⣿⣿⣯⠿⣿⣿⣿⣯⣭⣥⣤⠀⠐⠂
⠈⠉⣉⡙⠛⠛⠛⠛⠿⣉⣉⣁⣚⡉⠛⢿⣿⣟⠋⢽⣿⣿⡿⠛⠒⠒⠉⠉⠭⠯⠭⠿⢿⣿⡇
⠀⠀⠈⠉⠉⠉⠀⠐⠲⠦⢭⣉⣉⡀⠀⠈⠉⠉⠥⠬⠽⣿⠧⠀⠈⠙⠛⠚⢛⣋⣉⡭⠭⠉⠁⠀⠚⠛⠛⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠛⠃⠀⠀⠉⠉⠒⠒⠀⠀⠀⠈⠉⠙⠓⠚⠛⠋⠁

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠿⠟⠛⠛⠛⠻⣿⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣉⣢⡀⠀⠀⠀⠀⠈⠃⢈⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠙⢆⠀⠀⠀⠀⠀⠀⢠⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣁⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣀⠀⡬⠐⠛⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠛⠻⠿⠴⠆
⠀⠀⢀⣀⣀⣀⣈⣁⣠⣤⣤⣤⣭⣤⣭⣤⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣧⣄
⢀⣠⣞⠿⣿⣿⣿⣿⠛⠛⠛⠛⠉⠉⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣷
⠀⠉⠈⠀⠈⠻⢿⣷⣶⣶⣶⠶⠦⠤⠤⣤⣤⣤⣤⣀⣠⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⠶⠶⠶⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣄⣀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⡿⠿⠿⠿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠉⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⡚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣆⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠘⣿⣿⣷⣽⡄⠀⠀⠀⠊⢳⣄⠀⠀⠀⣠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠹⣿⣿⣿⣽⡄⠀⠀⠀⠀⢉⡥⠉⠀⢀⣤⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢀⣤⣿⡿⠛⠁⢀⣀⣤⣤⣴⣾⠁⠀⢀⠏⢡⠀⠀⠈⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢠⣾⣿⣿⣤⡶⣾⣿⣿⣿⣿⠛⠏⠀⠀⢸⠀⠀⠛⠲⠦⢤⣀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣬⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠉⠻⣦⣻⡄
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠛⠉⢀⣤⣿⣿⡏⠀⠀⠀⣶⣮⣄⡀⠀⠀⠀⠀⠀⠈⢿⠇⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣟⠑⠀⠀⣀⣨⣿⣿⡟⠀⠀⠀⡸⠀⠀⠈⠙⠲⣤⡀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢃⣴⣾⣿⣿⣿⡿⡏⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⡿⠞⠁⠀⠀⣴⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢰⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠴⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⢿⡿⠋⠀⠀⠀⣠⣾⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣠⣾⣿⣏⠀⠌⠻⣦⣄⠀⡀⠀⠀⠀⠀⠀⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠋⣀⡤⢀⣴⣿⣿⣿⣿⠷⣦⣄⣈⡑⢿⣿⣦⣧⣄⣀⣠⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠸⢿⣿⡇⢘⡟⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⢛⣉⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣾⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣰⣾⠿⣿⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⣿⣿⣿⣿⣿⣯⣿⠟⠀⣿⣿⡜⠻⠛⠒⠉⢹⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢻⣿⣿⣿⣜⣿⡇⠘⣿⣧⠀⠀⢀⣼⣭⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡇
⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣯⠀⢠⠈⠟⠀⢠⠟⠉⣨⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⡇
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⠐⠐⡆⡄⠀⠓⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡻⣿⣿⣿⣿⣼⣿⡏⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⠙⢻⣿⣿⣿⣿⣿⣦⡆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠈⠻⢿⣿⣿⣏⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⣯⡛⠻⢿⣿⣿⣧⠀⠀⠀⠀⠀⠙⢿⣽⣿⣷⢀⡆⠀⢰⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣾⣿⣎⣿⣄⠁⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣝⣷⡱⣿⣿⣇⠀⠀⠀⠀⢀⣀⣀⡀⠈⢻⣿⣿⣿⣿⣆⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠁⢹⣿⣿⣿⡿⣿⠄⢀⡤⠶⣟⠉⠙⣿⡄⠀⠙⢿⣿⣿⣿⡆⠁⡀⠀⢰⣌⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣃⣀⣠⣼⣿⣿⣿⣷⣾⣾⣟⣄⣀⣙⣄⠀⣈⠿⡄⠀⠈⢿⣿⣿⣿⢦⣤⠄⠘⠿⢆⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⠀⠀⠀⠉⠁⠉⠉⠉⠁⠀⠀⠁⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣀⣀⣠⣄⣀⣠⣤⣄⣀⣤⣄⣀⣀⣀⣠⣤⣀⣀⣀⣀⣠⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠣⠌⢀⣈⡋⠀⠀⠀⠄⠂⢀⠠⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠤⠤⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠆⡀⠁⠀⠀⠀⢄⠀⠰⣄⠀⠀⠀⢀⣀⡀⠔⠊⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠁⠀⠃⠀⠁⠀⠀⠀⠐⢄⡑⠂⢈⣂⣠⠍⠊⠁⠀⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠒⠉⠉⠂⢐⡛⣀⠐⢄⠀⢀⡠⠴⣶⡾⠋⠉⠛⠉⠀⠀⠀⣀⠠⠔⢊⠕⠁
⠀⠀⠀⠀⠀⠸⣅⡀⠀⡠⠤⠂⠁⢂⣈⣤⢔⠊⠡⠖⠂⠉⠁⠀⠀⠀⣀⡠⢴⡲⠭⣒⠤⠚⠁
⠀⠀⠀⠴⠊⠁⠀⣀⠄⠁⣠⠖⠉⠉⠑⠒⠉⠀⠀⢀⣀⠠⠄⢒⠈⠍⣂⠬⠐⠊⠁
⠀⠀⠀⠀⠀⠠⠚⠁⢠⠞⠁⠀⠀⠀⠂⠀⠤⠒⢈⠁⠀⣠⠔⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠄⠀⢀⡠⠄⠀⡠⠊⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠊⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣄⡀⠀⠀⠀⠰⣶⠶⠶⣦⡄⠀⠲⣶⠶⠶⠶⠆⠀⠀⠀⢀⣤⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠉⠙⠂⠀⠀⠀⣿⡀⢀⣼⠏⠀⠈⣿⣀⣀⣀⠀⠀⠀⠐⠉⠁⠀⣀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠓⠒⠂⠀⠀⠀⣿⠉⠉⢻⣦⠀⠈⣿⠉⠉⠉⠀⠀⠀⠐⠚⠛⠛⠛⠛⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⡶⠶⠂⠀⠀⢠⣿⣤⣤⡼⠟⠀⣠⣿⣤⣤⣤⡄⠀⠀⠐⠒⠶⣶⣶⣦
⠀⠀⠀⠀⠀⢠⠖⢶⣆⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷
⠀⠀⠀⢶⡆⣾⠀⠘⣿⡀⠀⠀⠀⠀⠀⢀⣠⡶⠀⠀⣰⠖⠦⠀⠀⣠⣄⠀⢘⡋⠀⢸⣧⠞⣐⠇⠀⣠⣤⡀⠀⠀⠀⠀⠀⣀⣤⡀⠒⣾⡟⢦⡀
⠀⠀⠀⠘⣷⣿⠀⠀⣿⡇⡰⠋⣻⠄⠀⡾⢃⣶⠀⢸⡏⢠⣦⣼⡿⢡⡿⠀⣸⡇⢀⣼⡇⢸⡏⠀⣼⡟⠀⠁⣴⣟⡇⣠⡼⠁⣼⡇⣼⡏⠀⠉
⠀⠀⠀⠀⠹⣿⡄⠀⢸⣿⠁⢀⣿⠀⢰⠛⢹⡇⢠⠞⢛⡿⣿⠈⠃⢸⣇⡴⠻⠧⢏⣸⣧⣘⠧⠞⣿⠀⢀⡴⣿⣁⠜⠙⠁⢠⡟⢠⣿
⠀⠀⠀⠀⠀⠈⠛⠀⠀⠉⠀⢸⡇⢠⠏⠀⠸⠷⠃⣠⠋⠀⣿⠀⠀⠀⠉⠀⣠⠞⠁⢸⡇⣨⠆⠀⠘⠓⠋⠀⠀⠀⠀⠀⠀⢸⠃⠸⡇⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠋⠀⠀⠀⠀⠀⢧⣀⡴⠃⠀⠀⠀⠀⠀⢯⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁







⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣷⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿

⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣄⡀⢈⣛⡛⠿⣿⡿⠛⠛⠻⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣽⣿⢿⣿⣿⣏⢻⣷⣶⣿⠐⠂⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⡗⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠘⠿⠲⠛⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣠⣿⢷⣿⣿⣿⣿⡏
⢸⣿⣿⣿⣧⢀⡉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡟⠿⠃⣸⡿⠿⠟⠹⠗
⢸⣿⣿⣿⣿⣾⡿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣬⣿⣿⣟⠿⡿⠛⢻⣿⣿⠀⠀⠁
⢸⣿⣿⣿⣯⣽⣀⡈⠏⠉⠀⣤⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢠⡦⠤⢴⣿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠤⢤⣿⠇⢀⠀⠀⠀⠄⣀⣀⣀⣴⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⢻⡿⠽⡿⠿⠛⠛⠉⠉⠉⠉⢸⡿⠀⡈⠀⠀⢀⣺⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣾⣇⣶⡇⠠⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⣀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⢰⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⢀⣉⣀⣀⣸⣃⣼⣿⣿⣿⣹⣽⣯⣽⣿⣿⣿⣿⣯⣽⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⣉⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡉⠉⠉⠉⢩⣉⣉⣭⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⡀⡀⢠⣐⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣧⡀⠀⠀⠲⣟⣻⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣂⣿⣿⣽⣟⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣀⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⡀⠀⠀⠈⡏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⡇⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣆⠀⡇⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⡄⠀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⡀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣷⡀⠀⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣧⠀⠸⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣧⠀⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠙⠻⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⠛⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣦⣼⣤⣤⣤⣤⣤⡄⠀⢀⣀⣤⣀⠠⠤⠤⠴⠶⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣶⣤⣤⣤⣤⣤⣿⣿⣯⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣟⣿⣿⣿⣿⣟⣻⣿⣭⡛⢻⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣟⣿⣿⣿⣿⣿⡿⠋⠙⠛⠛⠻⣷⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢹⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢻⣿⡷⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠌⠻⠀⠀⠀⢤⡼⣗⠀⢀⣤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⡄⡄⠀⠀⠀⠀⠉⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣿⣿⣿⠟⣷⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠛⣿⠀⠀⠀⠀⠀⢋⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣘⡿⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣽⡧⠀⠀⠀⠀⠀⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣿⣿⣿⣿⣿⣿⠛⠒⠠⠄⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠲⣁⣿⣿⣿⣿⣿⠻⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢻⡾⣋⣿⣿⣿⣿⣿⠀⣩⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠿⣿⣿⣿⣿⣧⠟⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡟⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢦⠶⢒⠤⢤⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⣿⣿⣿⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡗⠂⠀⠀⠀⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣷⣿⣿⣟⢀⡀⠀⢀⢦⢧⡀⠠⠆⠀⠀⢠⠀⢀⣤⣼⢘⣧⠀⠀⠀⠀⠀⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣇⠀⠉⢛⢷⣟⣷⠟⠉⠁⠰⠀⡀⣾⡅⣨⠿⠀⠇⡀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣦⣄⠀⢉⠫⠀⠐⡠⠶⢀⠁⠀⠻⠏⡄⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣿⢸⡟⠍⠁⠓⣤⣡⠖⠁⠀⠍⠥⢰⣤⣴⡿⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⡇⠘⡇⠀⢠⠢⣿⣽⠄⠀⠀⠀⢲⣦⣼⣿⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠇⠀⠛⠀⠚⠛⠉⠀⠁⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⢠⠂⡰⣂⠀⠀⠀⡀⠀⠚⢿⣿⣿⣷⣶⣶⣦⣷⣬⣥⡿⠿⠗⠽⠯⠿⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣧⣏⣠⠊⠀⠀⠀⢤⡀⠈⣿⣿⣿⣿⣿⣿⣻⣿⣧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠶⠞⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⠉⠀⠀⠀⠀⠀⢀⣀⣀⣬⣿⣦⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣠⣤⣤⠶⠶⠞⠛⠛⠛⠻⠷⠶⠶⠶⠿⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠻⠶⢶⣦⣤⣄⣀
⠀⠀⢀⣴⣾⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⠟⠁
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠈⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠷⠶⠶⠶⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠶⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⡀⠐⠢⢄⡀
⠀⠀⠀⠀⡀⠀⠁⠁⠢⠄⡈⠑⠢⠄⡀
⠀⠀⠀⠀⠀⠈⠠⢄⡀⠀⠀⠈⠐⠠⢄⠉⠐⠠⡀
⠀⠀⠀⠀⢆⠀⢀⠀⠈⠑⠢⢄⡀⠀⠀⠈⠒⠠⣀⠙⠂
⠀⠀⠀⠀⠨⡀⠸⠧⡌⠃⠒⠂⠀⠓⠀⠀⢘⠀⠀⣘⡖⠠⡀⠐⠠
⠀⠀⠀⠀⠀⢿⠀⢁⠀⠂⠀⠢⡈⠉⠢⣀⠀⠁⠠⢀⠀⠀⠀⠁⠢⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⡝⠂⠄⢂⡀⠀⠈⠢⡀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢀⠙⠠⢀⠁⠢⣀⠉
⠀⠀⠀⠀⠀⠀⠈⢌⠓⣀⡥⢀⡰⠄⠘⠂⠀⠀⠉⠁⠀⠀⠀⠋⠀⠀⠀⣀⡄⠑⠢⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠙⠻⣥⣠⠀⠄⠐⠂⠀⠀⠀⠂⢁⠀⡀⠀⠀⠉⠢⣀⠀⠀⡉⠢⠄⠀⠰⡤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠉⠛⠲⢦⣀⡀⠠⡀⠒⠀⠑⠄⠑⠀⡀⠀⠘⠁⠀⡇⠀⠀⠀⢠⠁⠰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠈⠙⠓⠾⢦⣀⣀⠀⠀⠀⠀⠀⠐⠀⢀⠁⠀⠀⠀⡀⢀⡇⢀⣤⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠈⠉⠛⠳⠦⣄⣀⠀⠀⠀⠀⣂⣴⠶⠿⠟⠛⠛⠓⠛⠋⠉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⡀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢀⠀⠑⢄⡀⠀⠀⠀⠀⠀⢀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢈⡑⢄⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠳⠊



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄⠐⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠤⢸⡞⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢂⡇⢳⡐⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡅⢬⢧⠀⠀⠀⠀⠤⠤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢸⠀⠀⠀⠡⠀⢰⠀⠁⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⠘⠀⠀⠀⠀⠐⠈⡇⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠒⠒⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠤⠆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⠭⠉⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠠⠄⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣠⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠙⢧⡀⡉⢉⢉⣉⢉⡁⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢂⠼⠘⡇⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠈⢳⡌⠿⢻⠃⣫⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠵⠃⠖⣽⠀⠀⠀⠀⠀⠀⠘⢻⢳⡀⠀⠀⠀⠀⠑⡄⠁⠊⢁⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠄⠘⠀⡄⠀⠀⠀⠀⠀⠀⠈⢎⢿⣄⣀⠀⠀⠀⠘⢦⠀⠈⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⢀⠾⡇⢸⣀⣀⣀⣀⣀⣀⣠⣸⣮⣙⣛⠿⠷⣦⣠⣀⣳⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣚⣒⣒⣛⣒⣓⣘⣋⣿⣿⣿⣿⣿⣙⣉⠙⠛⠻⢿⣭⣽⣿⣷⣶⠾⠛⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠿⠿⠿⠿⣯⣿⣿⣿⣿⣿⣿⣟⣛⣻⣶⠶⢶⣶⣶⣶⣾⡿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣽⣿⠏⠉⠡⠤⠿⠿⠿⣿⣿⣿⣶⠤⡬⠉⠉⠉⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠒⠀⠀⠒⠚⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⢠⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣵⡟⠀⠀⠀⡌⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣴⡿⠀⠀⠀⢀⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢰⣿⠇⠀⠀⠀⡘⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⣿⡟⠀⠀⠀⠀⡇⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣿⡇⠀⠀⠀⢸⠁⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣯⠀⠄⢰⣿⡇⠀⠀⠀⡘⠀⠀⡅⢸⡟⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠘⣿⣧⠀⠀⢀⠇⠀⢀⠀⢸⣇⢹⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣿⣿⡀⠀⢸⠀⠀⠈⠙⢸⡿⠀⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠈⢾⣿⣧⠀⡇⠀⢀⠃⠀⢺⣿⡄⢿⣿
⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⣼⣿⣿⣶⠇⠀⢸⠀⠀⣼⣿⡀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣷⠄⠀⠀⠀⠈⠾⣿⣿⡄⠀⡆⠀⠀⠈⣿⡇⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡓⠢⡉⠁⠀⠀⠙⣿⣿⣀⡅⠀⠀⠀⣿⣷⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⡀⠀⡀⠘⣿⣿⡀⠀⠀⣐⣿⣯⠀⠀⢈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠉⠲⣿⠆⡙⣿⣷⠀⠀⢽⣿⣿⠇⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠙⠻⢟⣿⣧⠀⠂⢸⣿⣷⣆⠀⢀⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠆⠀⠀⢀⠰⣿⣿⣿⣿⡷⡄⠘⣿⣿⣿⣶⣶⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⢀⣠⣾⣿⣿⡿⠟⠃⠀⠘⡿⠛⠛⠛⠛⣛⣛⡛⠛⠛⢣⣶⣶⠂⠠⠄⠄⠀⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣯⣥⣤⣴⣦⣤⣤⣧⡄⠀⠰⠾⠿⣿⣿⣦⣴⣿⣿⣷⣤⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣉⡉⠛⠿⠻⢿⡻⠿⠭⣭⣿⣟⣾⣟⣸⣯⣿⣭⣭⣭⣽⣯⣽⠿⠿⣿⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠿⠷⣶⣿⣿⣷⣾⣷⣷⣶⣦⣼⣧⣤⣤⣈⣋⣤⣤⣬⣡⣤⣤⣥⣤⣈⣀⣆⣙⣿⣾⣷⡶⣶⣶⣆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⢛⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠋⠉⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢼⡇⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⢸⡇⣷⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⣿⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣀⣠⡤⠶⠚⠛⠉⠉⠉⠀⠀⠀⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣥⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠖⠚⠛⠋⠉⠉⠉⠉⢹⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⡿⣆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣀⣤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡇⣿⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⠋⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡾⠁⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡇⣿⠀⠀⢀⣴⠏⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⢇⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⣿⠀⣰⠟⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣷⠞⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⡇⣿⠓⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⣀⣬⣿⠦⠴⠶⠤⠤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠦⣤⣤⣀⣀⣀⣤⣤⣤⣤⠴⠶⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠲⠶⠶⠶⠶⠿⠓⠒⠛⠛⢳⡆
⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠻⠶⠴⠶⠶⠒⠒⠛⠓⠒⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠻⠿⣿⣆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣤⣤⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣣⣿⣧⣤⣤⣤⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣾⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⢆⡀⠀⢸⣀⠀⠉⠛⢶⣄⢠⣤⠤⠶⣀⠤⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣠⠴⡋⠄⡈⠢⣄⢀⠹⣵⣏⣹⠓⠦⣄⢠⠸⣄⠲⠬⣑⠢⠍⠉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⠲⡶⠚⠓⢦⣝⢦⡈⠀⠰⡀⠈⠑⢆⢣⣑⣲⣖⡚⡄⣀⣤⠏⠤⠓⠊⠉⣀⠤⠔⠊⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠠⠤⢐⠢⠞⢤⠛⠞⠙⠧⠤⠀⠘⠂⠡⠔⠒⠃⠉⣁⣀⠤⠔⠒⠋⠉⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⢀⠂⠀⠀⠀⢀⣠⠴⠢⠌⠀⠀⠀⠀⢉⣀⣀⣤⣤⡒⢒⣉⠁⠀⢀⣀⣀⡀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡌⠀⢀⡴⠊⠁⢠⠀⠠⠀⠀⠐⠂⣩⠔⠊⠁⠀⠀⠀⠀⠀⠉⠒⠉⠀⠀⠈⠉⢉⡝⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢊⡧⠒⠚⠒⠉⢉⠓⣀⡠⠒⠢⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁
⠀⠐⠀⠁⠀⢀⣀⡀⠉⠀⢀⡠⡄⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⠴⠒⠛⠒⠒⠒⠋⠉⠉⠉⠉⠉
⠀⠀⠈⠀⠉⠀⠀⠉⠒⠈⠁⠀⠈⠒⠢⠔⠚⠉⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⣿⡷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⣆⠀⠀⣀⣤⡴⠞⣛⠭⠗⠊⠉⡿⠉⠓⠭⣛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠙⣷⣟⣫⠵⠚⠉⠀⠀⠀⢀⣸⣧⣀⠀⠀⠈⠙⠪⣝⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣀⡤⠖⠊⠁⢸⡇⠈⠙⠢⣄⡀⠀⠀⠙⠢⢝⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⢻⣆⠀⠀⠀⠀⠀⠀⠻⣾⠉⠀⠀⣀⡤⠴⣿⠲⢤⣀⠀⠀⠉⠲⢤⡀⠀⠀⠉⠢⣝⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣉⡴⠞⠋⠙⣧⡀⠀⠀⠀⠀⠀⠙⢷⡚⠉⠁⠀⢀⡿⠀⠀⠈⠑⠦⣀⠀⠀⠉⠓⢤⡀⠀⠀⠙⠪⣝⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢋⡴⢺⣍⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠠⠀⣸⣇⡤⢴⡒⠉⠉⠲⢤⡀⠀⠀⠈⠑⠦⣀⠀⠀⠉⠳⢄⡀⠀⢨⢏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⡡⠞⠁⣠⠔⠛⢷⣦⡀⠀⠀⠀⠻⣆⣀⢠⡀⠑⣿⡁⠀⠀⠉⠲⢄⠀⠀⠈⠓⢤⡀⠀⠀⠈⠓⢤⡀⠀⠀⠉⢢⠏⣼⠃
⠀⠀⠀⠀⠀⠀⢠⡾⢋⡴⠋⣤⣴⣮⣥⣤⣄⣠⣜⣻⣷⣄⡀⠀⠉⠉⠛⠓⣄⠘⢮⠳⢄⡀⠀⠀⠙⠢⣄⠀⠀⠉⠲⢄⢀⣀⣀⣈⡳⣄⢀⡞⣰⠏
⠀⠀⠀⠀⠀⣴⠟⡰⠋⢀⡼⠿⠦⣤⣤⣤⣤⣭⣍⣉⣉⣉⣉⡛⠛⠛⠛⠓⠚⢦⠀⠳⡶⠽⠦⠤⢤⣤⣈⣑⣦⣀⣀⣴⠟⠉⠉⠉⠛⠛⠛⠛⠿⠶⠶⠶⣦
⠀⠀⠀⢀⡾⢃⠞⠀⡰⠋⠀⢀⡴⠊⠀⣀⠴⠋⠙⢯⡉⠉⠉⠉⠻⢿⣟⡛⠛⠚⠳⡄⠙⣶⠤⠤⠤⢤⣤⣄⣀⣀⣀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⢀⡾⢡⠿⣄⡞⠁⢀⡔⠁⢀⡠⠚⠙⢦⡀⠀⠀⠈⠲⣄⠀⠀⠀⠙⠻⣦⣀⠀⠙⣆⠈⢧⡀⠀⠀⠉⠲⣄⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⣾⢁⡏⠀⠈⠓⢴⠋⠀⡠⠋⠢⣀⠀⠀⠉⠢⣄⠀⠀⠈⠑⢦⡀⠀⠀⠈⠛⢷⣤⡈⢷⡀⠳⣄⠀⠀⠀⠀⠑⢦⡈⠛⠶⢶⠶⣶⡶⢦⣤⣤⣤⣤⣴⠏
⠀⢸⡇⡸⠀⠀⠀⠀⠀⠙⢾⡁⠀⠀⠈⠳⣄⠀⠀⠈⠓⢄⡀⠀⠀⠉⠲⣄⠀⠀⠀⠙⠻⣦⣿⣄⠙⣦⠀⠀⠀⠀⠀⠈⠳⡴⠃⣴⠏
⠀⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠑⢤⡀⠀⠀⠙⠦⡀⠀⠀⠈⠑⢤⡀⠀⠀⠈⠙⢿⣦⡈⢷⡀⠀⠀⠀⣠⠞⢀⡾⠃
⢸⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⣄⠀⠀⠀⠉⠢⣄⠀⠀⠀⠉⢳⡄⠻⣆⢀⠞⠁⣴⠟⠁
⢸⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠲⣄⠀⠀⠀⠑⢦⡀⠀⠀⠈⠑⢤⡀⠀⠀⠹⣆⠙⣧⣠⡞⠁
⢸⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠈⠳⢄⠀⠀⠀⠉⠢⣄⠀⠀⠀⠙⣢⠴⠚⣙⣷⡈⢿⡄
⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠑⢦⡀⢀⣀⣨⡷⠔⠒⢉⣡⡴⢾⡿⠋⢻⣄⠹⣦
⢸⡇⢳⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠤⠽⠶⠒⠒⠒⠊⢉⣉⣠⣤⡴⠶⢛⣫⣥⡾⠋⠀⠀⠀⠹⣦⠘⢷⡀
⣿⣀⣀⣠⣭⣭⣭⣭⣍⣉⣉⣡⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠞⠛⠛⣋⣉⡥⠴⢒⣪⣭⠾⠋⠀⠀⠀⠀⠀⠀⠘⢷⣌⣿⠄
⠈⠙⢿⣥⣠⣄⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣤⠤⠤⠤⠤⠒⠒⠚⣉⣉⡥⢤⣶⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠈⠻⢶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠤⠖⠒⢒⣉⣭⣥⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⠶⣤⣤⣤⣤⣤⡴⠶⠶⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⢸⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⢀⡴⠋⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⣴⠋⢀⣴⠒⢺⣟⣛⣻⣉⣯⣉⣏⣩⣏⣹⣿⣓⡖⢦⣄⣀⣠⣤⡶⠶⠿⢻⠛⠛⡟⢉⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣇⠀⢻⣴⣫⢿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣤⡼⡿⠛⡟⠉⠁⢰⠀⣀⣀⣼⣤⣴⠿⠿⠿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡰⣿⠟⠁⢸⡆⠀⠀⠀⠀⢀⣠⣤⠶⠟⢛⣉⠀⠀⣇⣠⡧⠴⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣾⠟⠁⠀⠀⠀⣇⣀⣤⡶⠟⢋⢹⠠⣀⣠⡿⠶⣛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠀⠀⢀⣠⣶⣿⠋⠉⣆⣠⣤⠾⠛⠋⢉⠡⠄⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⣠⣴⢾⠋⠁⢐⣿⡴⠞⠋⠉⠠⣐⠀⠀⠀⠁⠉⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⢘⣧⡶⠛⡏⠀⣸⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠋⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠰⣟⣿⢰⣠⠷⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣁⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⡼⠄⢀⡾⢠⡟
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⠥⠩⠑⢁⡞
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠘⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⣠⣐⣭⣿⣤⡽⣶⣶⣮⣍⣉⣉⣙⣦⣀⣖⣍⣉⡉⣳⡤⠒⠂⠉⠉⠁⠐⠂⣄⣀⣤⡷⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠈⠀⠘⠓⠉⠉⠉⠉⠉⠉⠙⠋⠙⠋⠛⢛⣻⡿⠟⠛⠟⡻⢭⣵⣤⣦⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠈⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⠀⠀⠁⢀⠀⠁⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⣀⠀⡀⢠⠀⠶⡏⠚⢨⣧⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠐⠋⠈⠂⠀⠂⠀⠀⠁⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡤⠔⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⢤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠀⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⠀⠐⠺⠀⢁⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⣤⠖⠋⠁⠀⠀⠀⠀⠉⣄⠀⡉⠀⠔
⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠠⡀⠐⠒⠤⠂⠂⢑⡒⠚⠛⠛⠛⠛⠛⣻⣾⣥⡀⠀⢀⡄⠠⡼⠱⢨⠃⠀⠀⠆
⣠⣴⠯⠒⠀⠀⠀⠂⠠⠀⠁⠀⠀⠀⢀⣀⣤⣎⡀⠀⡀⢋⣁⣤⣾⣿⣿⣿⡿⠀⠀⠀⠀⠑⠂⠼⢀⡀
⠀⠀⠈⠒⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠠⣀
⠀⠀⠈⡒⠒⠛⠀⠀⠀⠒⠤⢄⣀⠀⠀⠀⠐⠂⠤⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠈⠉⠀⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠒⠒⠐⠂⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣿⣿⣿⣿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡞⠀⣿⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣀⣠⣿⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠀⡀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠿⠿⠿⠛⢿⠀⢀⠀⡀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣶⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠙⠛⠛⠛⠛⠛⠉⠉⢸⡇⠀⠀⠀⠀⢀⠀⠡⡀⠀⠈⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣁⣀⣀⣀⣀⣀⣀⣤⣤⣤⣾⡇⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡤⠴⠒⢛⣿⠏
⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⡇⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠴⠶⠒⠛⠋⠉⠁⠀⠀⠀⣰⣟⣥⠶⠒⠒⠶⢤⣀
⣀⣀⣀⣀⣠⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠴⠷⠶⠒⠚⠛⠉⠉⠉⠀⢀⣀⡠⠴⠒⠂⠀⠀⠀⠀⢀⡼⠟⠛⠛⢷⡄⡠⠤⢄⠙⣦
⠙⢿⣿⣿⣿⡛⠛⠛⠉⢉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠐⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⣾⣡⡾⢻⠀⠤⠼⠃⣿
⠀⠀⠙⢿⣿⣿⣦⠠⠤⠼⢦⠤⠤⠤⠄⠒⠒⠒⠒⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⣀⣠⠤⠖⠂⠀⠀⠀⣀⡤⠾⠷⠖⠲⠶⠶⣤⣤⠤⢼⣦⣤⣤⠾⠗⢦
⠀⠀⠀⠀⠉⠻⢿⣷⣄⠀⠈⢧⡀⠀⠀⠀⠀⠀⣠⣤⣤⠤⠤⠤⠴⠒⠒⠋⠉⠀⠀⠀⠀⢀⡤⠖⠉⠁⠀⣠⣤⣄⡀⠀⠀⠀⠙⣷⡄⣀⡉⢳⡄⠘⠾
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠊⠁⠀⠀⠀⢀⡾⠁⢀⣬⣷⠀⠀⠀⣰⡿⠱⣧⣠⠾⠁
⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⢤⣈⣉⠛⠓⠲⠶⠤⠤⠤⠤⠴⠶⠖⠒⣛⣩⠽⠛⠉⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠉⢁⣠⣴⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠤⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣧⣤⣶⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⡏
⠀⠀⠀⠠⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠁⠀⠀⢠⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠈⠓⠶
⠀⠀⠀⠀⠰⠀⠀⠐⣶⠇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⡴⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⡄⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠍⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠋
⠀⠀⢀⠀⠀⠰⠀⣸⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠤⠶⠶⠒⠲⠖⠶⠶⠤⣤⣤⣤⣤⣤⡀
⠀⢠⣦⡄⠠⠁⢨⠅⠈⠉⠁⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠁
⠀⠀⠉⠃⠀⣸⣾⠀⠀⠀⠀⠓⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠄⠠
⠐⠂⠀⠀⢠⣿⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⢠⣀⣰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠠⠄⠀⠀⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢋⣷
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠖⠀⠀⡠⠛⠉
⠀⠀⠀⠀⠐⠒⠂⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣤⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠒⠒⠂⠒⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠋⠒⢿⡷⢀⣀⠀⣠⡶⢾⣿⣿⡋⠁⢀⡹⠧⠤⠖⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠃⠿⠏⠠⠟⠟⠛⠛⠛⢁⠀⣀⠤⠄⠀⣀⣀⣦⣤⣶⣶⡖⣶⣶⣶⣶⣶⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠉⣀⣀⣀⣠⣶⣾⣭⣷⣶⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⢀⣖⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⢿⠿⠿⠿⠛⠉⠉⠙⠉⠉⠂⠁
⠀⠀⠠⠀⠀⠉⠛⠛⠻⠿⠟⠻⠿⠟⠿⠿⠛⠛⠛⠃⠉⠉⠬⠤⢄⣠⣴⣴⡤⠭⠹⠓⢂⣁⡤⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢾⡛⠛⣛⠛⠉⠁
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⢀⡀⠀⠀⠀⡀⠀⢀⠀⠥⠅⠠⠄⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠘⠃⠠⠀⠀⠀⠑⠐⠠⠤⠈⠁

⠀⠀⠀⠀⠀⣠⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⣿⣿⡆⠀⠀⣟⠓⠂⠀⠀⢰⣿⣿⣿⣷
⠀⠀⠀⣼⣿⣿⣿⡿⣥⡀⠀⠈⠫⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠋⠀⠀⠀⠈⠀⠀⠀⠀⠈⠿⣿⣿⡿
⠀⠀⣼⣿⣿⣿⣿⣇⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⢀⣿⣿⣿⣿⡟⣽⣿⣿⣇⠀⠀⠀⠀⢠⡴⠟⢷⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⡄
⢰⣀⣀⣤⣼⣤⣼⣭⣭⣿⣿⣦⡀⠀⠀⠀⢀⣤⣾⣿⡟⢁⠀⠀⠀⠀⠀⠘⢷⣶⣶⣶⣶⣷⣿⣷⣤⣴⣶
⠈⢙⣋⣉⣿⣉⣛⣉⣝⣋⣙⠉⠀⠀⠀⣓⣫⣝⣛⣍⣛⣫⣑⣊⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠉⠉⠀⠉⠉⠁⠈⠉⠁⠀⠀⠀⠀⠈⠀⠈⠁⠀⠉⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤
⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⠀⠀⠛⠛⠋⢩⠟
⢠⣄⡀⣿⡟⠁⠀⠀⣽⠀⠀⠀⠀⣀⣤⡄⠀⠀⣸⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣿⣄
⠀⠙⠛⢿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠀⠀⠀⡟⣿⣾⠀⠀⠲⣶⣶⣤⣴⣶⣶⣴⣿⣿⣶⣶⣶⣶⣶⡶
⠀⠀⠰⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣬⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⡁⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣇⣀
⠀⠀⣾⣿⣷⡀⠀⠀⣤⣭⣥⠀⠀⠀⠀⠀⠬⠴⠮⠭⠴⠮⠤⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⠙⠛⠟⠀⠀⠀⢻⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡦
⠀⠀⠀⠀⢘⡃⠀⠀⣿⣿⣿⠀⠀⠀⠀⢀⣐⣚⣍⣓⣚⣉⣚⣀⠀⠀⠀⠀⠉⠉⠙⠃⠀⠀⠀⠀⣠⡆
⠀⠀⢀⣴⣿⣿⣦⡀⣿⣿⣿⠀⠀⠀⠀⠀⠈⢁⣀⣉⡉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣾⣿⡤⠴
⠀⠀⠀⣿⡏⢹⣿⠀⣿⣿⣿⠀⠀⠀⠀⣀⣀⣰⣿⣭⣿⣿⣿⣿⡆⠀⠀⢲⣶⣶⣶⣶⣯⣥⣤⣤⣭⣥⣤
⣀⣤⣤⣿⣷⣾⣿⣀⣿⣿⣿⣧⣄⠀⠀⠈⠿⠿⣿⣿⣽⣤⣧⣤⡇⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠿⠛⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠘⠁⠀⢸⣿⣷⣾⡏⣹⣿⠃⠈⠉⠓⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣄⣀⠀⠀⠀⠀⠀⢸⣿⣿⠇⣠⣿⣿⣿⣩⣴⠟⠁⠀⢀⣠⣤⠤⠄
⠀⢀⣄⢤⣴⣶⣶⠊⠀⢀⣠⣤⣷⣷⣦⡀⠀⢸⣿⣿⣡⣿⣿⣿⣿⢏⣥⠶⠿⠟⠛⠋⠁
⢠⡾⠟⠉⠀⠀⠀⣀⣀⡀⠉⠙⠻⢿⣿⣷⡄⢸⣿⣿⣿⣿⣿⣿⣿⠃
⠘⢠⡴⠞⣿⠋⠉⠉⠁⢀⣀⣤⣽⣷⣾⣿⣿⣼⣿⣿⡿⣻⡿⠟⠋⣀⣀⡀
⢰⠟⠀⠀⠈⠁⠀⣀⣴⣿⣿⣿⣿⠿⠛⠛⢻⣿⣿⣿⣵⣿⣶⣶⣿⣿⠿⠿⠿⠷⣶⣦⣄
⣠⡀⠀⠀⠀⢀⣾⠟⠛⠸⠟⠋⠁⢀⣠⣾⣿⣿⣿⣿⣯⠉⠙⠿⣿⣿⣿⣿⣦⣄⡀⠈⠻⢷⣤⡀
⠹⠿⠤⠄⠀⠀⠁⠀⠀⠀⠀⣀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠙⠋⠉⠙⠷⢾⣿⣷⣦⡀⠹⣷⡀
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣉⣽⣿⣿⣿⡟⢙⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠋⠻⣦⡹⠃
⠀⠀⠀⣼⠀⣀⡀⠀⠀⠀⠀⣿⣿⣿⠋⠀⢸⣿⣿⣿⣿⣤⣤⡀⠀⠀⣴⣾⣿⠀⠀⠀⠀⠀⢉⣧
⠀⠀⠀⠻⠟⠋⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⣿⣿⣿⡇⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⢀⣿⣿⡿⠟⠻⣿⣿⣿⣦⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣸⣿⣿⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣦⣿⣿⠇⠀⠀⠀⠈⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⣴⡖⢤⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⣦⡀⠀⠀⠀⢸⡆⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡷⠘⣏⠑⢄⡀⢸⡇⠀⠀⠀⣿⡿⠁⠢⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠘⠀⠘⡀⠀⠙⢾⣧⠀⠀⠀⡏⠁⠀⠀⠈⢂⣄⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⡏⠀⠀⢀⠄⠀⠀⠀⠀⠀⢉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡆⠀⢸⠀⠀⠀⠀⠀⠀⠋⢹⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢼⡇⢀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢹⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣾⢷⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⡿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠃⠀⠀⠀⠀⠀⠁⢈⣷⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢻⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⡛⠠⠁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿⢿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣼⠀⢣⠀⠀⠀⠀⠀⠀⠈⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡞⠀⠀⠀⠀⠀⠀⠀⠒⠻⣿⠟⣋⠴⣶⠶⠦⠄⠀⠀⠀⠀⠀⠾⢶⡁⠀⠀⠀⠐⠂⢀⣠⣶⡟⠁
⠀⠀⠀⠀⠀⠀⢀⡾⣟⣀⠠⠘⣁⡀⣀⣀⣀⠤⣈⣿⣿⡿⢿⣿⠀⠒⠒⠛⠀⠀⠀⠀⠿⠿⠿⠶⢀⣠⣴⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠐⠛⠐⠊⠤⠤⠤⠄⠀⠀⠀⠀⠉⣻⣻⡿⣾⣿⡆⠀⠀⠀⠀⠀⢠⡤⣟⣭⣴⣾⣿⣿⣿⣿⣿⣿⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡃⠀⣸⣿⠏⢀⣀⣤⣶⣾⣿⡿⢿⣿⠿⠛⠃⠀⠈⠙⠃⡠⠎
⣀⣀⣀⣀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣤⣼⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣬⣭⣶⡿⠛⠀⣴⡆⠀⣺⣿⠂⢀⣄
⣿⣿⣿⣿⣿⣖⠀⠑⠪⠄⡀⢀⣤⣀⠀⠛⢿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠿⣟⣻⣿⣭⠉⠁⠉⠉⣴⣶⣾⣿⣿⡇⢸⣿⡿⠗⢸⣿⡷⣄⣀⡀⠀⢀⡠⠀⡀
⠛⠛⠛⠳⠆⡀⠀⠀⠀⢰⣬⣀⠉⠙⠛⢶⣶⣭⣭⣭⣭⣭⣿⣿⣭⣤⣼⣿⣿⣿⣿⣷⣦⣿⢿⣿⣿⣿⣿⣿⡟⠘⢿⠀⠀⢾⣿⣿⣏⠻⢿⣿⣷⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣤⣄⠈⠉⠻⠿⠟⠻⣯⣝⣛⡛⠛⠋⠉⠀⠀⠈⠉⠙⠷⠀⠉⠉⠉⠀⠀⠀⠀⠀⠆⠸⣿⣿⣛⠂⠀⢨⣝⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⣄⡀⠀⠙⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢀⣤⣶⣄⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠈⢿⣿⣿⣿⣿⣿
⠀⠘⣿⣿⡿⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣸⣿⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣬⡀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣠⣼⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣿⣿⣿⣿⣿⡇
⠀⠀⠐⠀⣠⣶⣮⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠈⠉⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡿⠟⠛⠛⠃
⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⣀⣴⣶⣿⣿⣷⣶⣤⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢿⣿⣿⣧⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⣼⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣼⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠉⢀⣤⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠁
⠀⠈⠛⠻⢿⣿⣛⡉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⢶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⢀⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⢀⣴⣿⡿⠟⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠀⠀⠀⠘⠻⠛
⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠿⠿⠿⠿⢿⣿⣿⡄
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠶⢢⣴⠄⠀⠀⠀⠀⠀⠀⠘⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⢸⡀⠀⢀⡀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢿⠢⣀⢸⡇⠀⢸⣿⡆⢄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⠈⠆⠈⣻⡇⠀⢰⠟⠀⠀⠱⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠀⢸⠐⠀⠀⠀⠾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⡈⠀⠀⠀⠀⠀⡿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠥⢧⠁⠀⠀⠀⠀⠀⢿⢏
⠀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⠀⠀⣸⣯⡝⠀⠀⠀⠀⠀⠀⣸⡇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⢅⣿⣿⠀⠀⠀⠀⠀⠀⠀⣹⣧⢰⠀⠀⠀⠀⢀⡄⠋⠁⠀⠁⠁
⠀⠀⠀⠀⠀⢠⡶⠃⠀⠀⠀⠐⠶⣿⠿⡚⣥⡤⠄⠀⠀⠀⢸⣿⡈⠀⠠⠄⢀⣤⡿⠁
⠀⠀⠀⠀⢠⡿⢧⠖⠡⠤⠤⠤⠦⣿⣿⣿⣿⠀⠐⠃⠀⠐⢚⡻⢛⣤⣴⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⢻⠛⢻⣦⠀⢀⣠⣼⣿⣿⣿⠿⠛⠛⠿⠁⡸
⣤⣤⣄⡀⢀⠀⠀⢀⠀⠀⢀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡵⠾⣂⣰⡆⣸⣿⠀⣤
⣿⣿⣿⠛⠁⠈⢑⡀⠿⠦⣬⣛⣿⣿⣿⣿⣿⣟⣻⣿⣿⣧⣀⣤⣾⣿⣿⣿⡅⣿⠋⢹⣿⣷⣦⣤⣤⣖⠅
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⣄⡙⠻⠿⠿⢿⣟⣿⠿⠛⠉⠉⠙⠻⠍⠙⠛⠋⠁⠈⠀⠘⣿⣿⡇⠉⣿⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠄⠈⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠚⣿⢸⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⢀⣿⢸⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⢸⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣴⡇⣿⢸⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣀⡤⠶⠚⠋⠉⠁⠀⠀⠀⠀⢸⠃⣿⢸⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠗⠋⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠶⠖⢾⠀⣿⢸⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢹⣄
⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⣀⡤⠶⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠘⢧⡀
⠀⠀⠀⠀⠀⣸⠁⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⢸⠀⠀⠀⡴⠋⠀⠀⣠⠋⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⢠⣇⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⢸⠀⣠⠞⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠟⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⢸⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀
⠀⢿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⣠⠴⠒⠛⠋⠉⠉⠙⠛⠓⠒⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠶⠒⠒⠒⠒⠒⠶⠦⢤⣤⣀⣀⣀⣀⣠⣶⣋⣤⣤⣄
⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⡞
⠿⣤⣤⡤⠤⠶⠶⠶⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠉

⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠈⠐⢄⡈⣁⣒⡒⠤⠤⢄⣀⣀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠫⡉⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢻⡿⠛⢻⣿⡟⢀⢠⣠⣦⣶⣿⣿⣿⣿⣧⣻⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⣼⠃⠀⣿⣿⣟⣻⣟⣛⣛⣟⣛⠛⢻⠟⠛⠛⢻⠟⠛⠛⠛⡿⠛⠛⠛⠛⡽⠯⠭⠭⣍⣽⡋
⠀⠀⠀⠀⠀⢀⣴⣿⣧⣤⣿⣤⣾⡟⠉⠻⠿⠟⠛⠛⠛⠋⠚⠃⠀⠀⠀⠁⢀⣀⣀⣈⣀⣀⣀⣀⠈⠤⠤⠤⠤⠁⠒⣊⡷
⠀⠀⢀⡴⠋⠉⠉⠉⠉⠁⠒⠚⣒⣒⣒⡒⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠠⡶⣖⣀⠀⠀⣾⣟⠠⠤⠤⠤⠶⣲⠟⠁
⢀⣀⡞⠀⠀⣀⣀⠤⠤⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠟⠁
⠈⠑⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠊⢀⢜⣅⠤⠔⠂⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡠⠂⢀⠴⢋⣉⣀⣠⣴⣚⣋⠉⠁⠀⠉⠙⠛⠛⠁⠉⠉⠒⠒⠒⠢⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⢴⠒⠊⠉⠁⠀⠀⠀⠀⠀⢈⣉⣉⣉⣉⣭⣭⣭⣥⣤⣤⣶⣿⠿⠿⠛⣻⠏
⠀⠀⠀⠀⠀⠀⣀⠄⡤⠒⠊⠁⠀⢼⠀⣀⣀⣤⣤⣶⣶⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⣠⠞⠁⠀⣀⣀⣠⣤⣶⣾⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠟⠋
⠀⠀⠸⢛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣶⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠴⠶⡒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣴⣶⣿⣶⣶⣿⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⢤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠘⠛⠋⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠟⢂⢀⣉⣀⣀⠀⠀⢐⡤⢀⣀⣀⣉⣑⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣥⠞⣻⣿⣿⣿⠟⢀⣾⣿⣿⣿⣯⠤⠤⠤⠈⠀⠀⣈⣒⡒⠀⠒⢒⡒⠂⠫⠉⢩⠉⠉⡯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠊⢐⣛⣛⣛⣉⠀⠉⠉⢡⠀⣠⣄⣶⣲⣆⣴⣶⡧⠤⠤⠤⠤⠤⠤⠤⠄⣙⣛⣓⣒⣦⠷⠖⠂
⠀⠀⣀⣄⠀⣀⡤⣖⠯⠤⠤⠛⠓⠒⠒⠛⠛⠉⠉⠉⠉⠉⢉⣀⣒⣒⣒⣒⡂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠤⠋⠁⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠈⠉⠉⠙⠻⣦⣶⣤⣤⣠⣤⣤⣤⣴⠷⢶⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠤⠀⠠⠤⠀⠀⠤⣄⣀⠠⣄⣀⡀⣀⣀⣤⣄⣀⣠⡀⢀⣀⣄⣀⣉⡷⠘⠁⠀⠀⠀⠈⠋⠉⠉⠻⣦⠤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⠶⠛⠛⠿⠿⣶⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣭⣭⣭⣭⣭⣿⣿⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣛⣉⣥⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣡⣴⣾⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⣾⣿⡿⠿⠛⣋⣉⣤⣤⣶⣶⣶⣶⣿⣷⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⠉⢉⣉⠉⢉⣉⡉⢉⣁⡀⢀⣈⡉⠉⠉⢉⣉⠉⠉⣉⣉⠉⢉⠉⠉⡉⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣻⡏⣵⠿⠟⣳⣿⡛⣡⣿⡛⣱⣟⣽⠟⠀⣴⣟⣿⢫⣞⣹⢟⡴⢿⡟⢠⡿⠉⠁

⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣿⣿⠛⠛⡛⠛⠛⠛⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⡇
⠀⠀⠀⣿⣿⠀⠈⣧⠀⠀⣴⠉⣳⢳⣠⣄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⣿⢀⣤⣍⠉⢉⣉⣉⣁⣀⣁⣨⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⣯⠀⣷⠼⣶⡏⢈⡷⢸⡇⢀⡿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⣿⠀⠛⠛⠉⠉⠉⠀⠈⠁⠜⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠅
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⣄⢸⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠌⢉⣀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⠠⠌⡛⢹⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠄⠀⠒⠓⠒⠂⠚⠿⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢸⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢤⠠⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠆⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠸⠀⣀⣤⣄⠀⢳⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠈⠃⠠⣄⣼⡏⢻⡷⠠⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠒⢸⣿⠙⣹⣧⠌⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⠁⠀⠀⠈⠁⠀⠀⠉⠙⠋⠉⠀⠈⠁⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣿⣶⣦⣤⣀⡀⠀⠀⠀⠐⠀⠀⠠⠀⠀⢀⣠⣤⣶⣾⣿⣏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣿⣟⠂⠉⠙⠛⢷⣄⠑⠀⠀⠀⠀⣠⡾⠛⠉⠉⠐⣻⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠻⡍⠁⡠⠒⠲⣄⠙⠆⢰⠒⡆⠀⠋⠀⢠⠖⠲⣄⠀⡶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠤⣤⠰⡇⠀⠀⢸⠀⠀⡼⢿⣁⡄⠀⠀⢸⠖⠒⢧⠀⡷⠒⠒
⠀⠀⠀⠀⠀⠀⠔⣦⠀⠓⠤⠖⠛⠀⠑⠦⠴⠋⠀⠀⠳⠤⠛⠂⢀⠄⠘⠦⠴⠚⠀⠷⠤⠤⠀⠀⠀⣠⡄
⠀⠀⡀⠀⢹⡄⠀⢸⡇⠀⢀⠀⢀⡴⡂⠀⡔⠁⢀⡀⣤⠀⣄⠀⣿⢀⡄⠀⡤⠆⠀⣀⠘⠀⢨⡄⣰⠏⠀⠰
⠀⠀⠈⠁⠈⢷⡀⠈⣧⠀⣸⠁⠘⢸⠇⡐⠃⢺⠙⠃⡏⡀⢇⠀⣿⠸⡅⢾⠁⢀⣾⢈⠜⠃⣸⢡⡏
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⡏⠀⠀⠘⠂⠠⠀⢸⠀⠀⠈⠀⠔⠁⣿⠨⠀⠈⠒⠁⠈⠀⠀⠀⠇⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠢⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠘⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠂⠈⠂⢇⡀⢀⣸⣷⣠⠀⡀⣠⣇⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣇⠘⠒⠇⡀⣀⢡⣤⡤⠀⠶⠌⠓⠛⠚⠛⠛⠘⠈⠉⠉
⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠈⢿⣿⣛⠛⠓⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠁⠈⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠐⣆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠤⠤⢤⠀⠀⠀⠐⠂⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⡄
⠄⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⡀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣰⣤⠀⠀⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒
⠀⠀⠉⠀⡀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠀⠈⠀⠀⠀⠠⠀⠁
⠀⠀⠀⡄⠀⢀⠈⠉⡁⠀⠀⡌⢁⠀⠀⠀⠀⠀⠀⠚⠓⠶⠶⠦⠄⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣐⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡻⠿⢛⠿⢿⢿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀
⠀⠀⠀⠀⠀⣴⣷⢻⣧⠀⢰⡀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣰⣧⣧⣀⠀⠀⠀⠀⠀⠀⣰⡏⢸⣿⣧
⠀⠀⠀⢠⣾⣿⣿⢸⣿⣧⢸⣷⡀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⢠⣤⣤⣴⣾⣿⣿⣿⣿⣶⣤⡀⠀⠠⠾⠿⠇⠸⠿⠿⠆
⠀⠀⠀⠀⠈⠻⠷⠾⠿⠿⠷⠴⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠞⠀⠀⠀⠈⠳⠶⠿⠿⠷⠶⠂
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠲⣦⣤⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠀⠀⢤⣤⣴⣶⣶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣈⣛⠛⠃⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⡿⠿⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠀⠸⠿⠟⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣦⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣦⡀⠀⠀⠀⠀⢀⣿⣿⣿⣗
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢠⣿⠿⠀⠀⠀⠀⠀⠈⢻⣿⣷⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠂⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢨⣴⣾⣷⣤⠀⠀⠀⠀⠉⠁⠈⠉⠈⠉⠀⠉⠁⠁⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠀⠀⣇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⢸⣽⣧⣿⣯⣿⡄⠀⠀⠀⠀⢀⡀⢠⣄⡀⢀⣤⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣼⡇⣷⣸⡄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⢤⣤⣤⣥⣬⣽⣿⣾⣿⣿⣤⡄⠀⠀⠻⠟⢛⡛⠛⠀⠀⠀⠀⢀⣾⣿⡇⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠟⠛⠁⠀⠀⠀⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠠⠬⠥⠭⠼⠭⠥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠒⣒⣋⣉⣉⣉⣀⣉⣉⣧⣤⣤⣉⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⣠⣤⣤⣴⡿⠿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣯⣿⣿⣿⣟⣻⣿⣦
⠀⠀⠀⠀⠈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⢘⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠈⠉⠙⠉⠀⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠈⠈⠙⣧⠴⠶⡄⠀⠀⠸⠷⠞⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠱⠀⠀⢠⣤⣤⣤⣿⣭⣭⣿⣭⣭⣭⣯⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣭⣭⣭⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢉⣿⣿⣿⡉⣻⣿⣿⡏⢙⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⠶⠿⣶⣶⣶⣾⣿⣭⣿⣛⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠿⠿⣿⣿⣶⣾⣯⣭⣭⣭⣿⣿⣿⡭⠽⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⢠⣀⡀⢱⣶⡦⡇⠀⠀⠀⠀⠀⠀⠀⠐⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⡰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠠⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢈⣼⣄⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⣴⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⢴⣿⣷⣾⣶⣤⣄⣀⡇⠀⠀⠀⠀⠀⢸⣶⣿⡇⠀⠀⠀⣀⢀⠀⠭⣶⣾⡟⣬⡇
⠀⠀⠀⠈⠉⡤⠀⠠⠄⠈⢻⡯⠿⣛⣿⣿⣿⣿⣿⣶⣦⣤⣼⣿⡿⠟⠐⢐⢀⣉⣡⢤⠔⠀⢻⡿⠁⠟⠁⠀⠀⠄⠀⣀
⠀⠀⠀⠀⠀⣐⢀⣊⡭⣀⣁⣹⣿⣷⣾⣿⢭⣍⣉⣙⣉⣛⢉⠉⣍⣉⢡⠉⠤⡒⣨⣵⣚⣟⣼⣇⠂⠸⡀⠀⠴⠶⢀
⠀⠒⠂⢒⣖⣒⣒⣤⣀⣀⣘⠚⠛⠻⠿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⠿⠿⠿⠜⡂⢠⡧⠌⠄⠀⢉⡉⠁
⠀⠀⠀⠥⠤⠶⠦⠀⠉⠹⠿⠿⠶⠤⠤⠀⠤⠈⠉⠩⠉⠙⠛⠋⠙⠻⠟⠛⠻⠏⣉⡄⣤⡤⠴⠢⡌⠀⡉⠉⢉⡀⢀⣉⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠲⠖⠋⠁⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣯⣽⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣴⠖⠛⣛⡉⠉⡉⠉⠉⠉⠉⠁⠩⠍⠉⠉⠉⠩⠙⠒⠒⠒⠒⠲⠤⣤⣄
⠀⢠⠶⠒⠾⠛⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡨⠟⠁
⠀⠀⢳⠒⠓⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠋⠉⠉⢉⠉⠉⠍⠉⠉⢀⡠⠟⠉
⠀⠀⠘⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠉
⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠁




⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⣠⠀⢀⡀
⠀⠀⠀⠀⠀⠁⠽⡜⠙⢛⠒⠤⣄⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠓⢤⣀⠀⠠⠔⠐⠚⠛⠋⠉⠉⠉⠋⠙⠻⠶⣤⡀
⠀⠀⠀⠀⠀⣠⡴⠉⠀⠄⠀⠒⠂⢀⣀⣀⣤⣤⣤⣤⣶⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡠⠀⠠
⠀⠀⠀⠀⠈⠙⠛⠛⣷⣂⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⡶⠦⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠰⠞⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠓⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠐⠂⠈⢥⢈⡄
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⣶⣶⣶⣶⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠁⠉⠉⠀⠀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⠟⠋
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣀⣠⣠⣤⣽⣭⣭⣿⣿⣿⠟⠁
⠀⠙⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠈⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠍⠀⢈⡻⠟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢠⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⡀⠀⠀⣸⣿⠖⠋⠉⢛⣛⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢠⡏⠉⠙⠛⠛⠉⠙⠛⠛⠋⠉⢉⢍⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡿⡇⢀⣼⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢨⡎⠀⠀⠀⠀⢀⣀⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⠙⠚⠚⠛⠉⠉⠁⠀⢀⢢⠃⠀⠀⠀⠀⠀⠀⣈⡟⠉⠉⠉⠉⠉⠉⢉⢧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠃⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⡀⢤⠄⠀⢀⢣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠧⠤⠤⠤⠴⢶⣦⠀⠒⠂⣊⠜⢀⡐⠒⠒⠒⠂⠉⠉⠁⠀⠉⢸⠋⠀⠀⣌⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢀⠀⠠⢄⠀⠈⡀⠀⡸⠀⠀⣀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠐⠌⢫⠉⠉⠙⠛⠋⠀⠀⠀⠐⠒⠈⠂⠀⡨⢻⠊⠀⠉⠀⢠⣷⣿⣿⣟⠿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠃⠀⠀⠀⠀⡸⠀⣀⡠⠔⠋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⣀⣀⠤⠒⠋⠉⠀⢀⡠⢺⠇
⠀⠀⠀⠀⠀⠀⢶⡒⠛⠒⠦⡉⠓⢤⣀⡴⠃⠀⠀⠀⢀⣀⣀⡀⠀⠤⠤⠄⠒⠒⠉⠉⠁⠀⠀⣀⠠⠔⠊⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠘⢏⠉⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠒⠒⠈⠵⢦⠀⠀⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠈⢷⠀⠀⠊⠀⠀⠀⠀⢀⣄⠈⠁⠉⠀⢠⣞⡆⠀⠀⠀⣀⠑⠉⠂⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠣⣀⠀⠀⠀⠀⠀⠑⠘⢀⣀⡀⠤⠤⠀⠐⠒⠈⠉⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡒⠀⠀⠀⠒⠒⠒⠒⠂⠈⠉⠉⠉⣁⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⠤⠤⠤⠤⠤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠖⠒⠒⠶⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡾⠛⠻⣆⠀⠀⠀⣿
⠀⠀⠀⠀⣠⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠷⣤⡼⠃⠀⠀⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠲⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠚⠛⠛⣷
⠀⠀⠀⠈⠛⢷⡶⠶⠶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣿
⠀⠀⠀⠀⠀⠈⢷⣦⣤⣤⣤⣀⣀⣈⣉⣉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⣉⣩⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣈⡉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⣿⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣍⡉⠛⠛⠛⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠤⠄⠔⠸⠿⠿⠿⠿⢛⣿⣿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣶⣶⠶⠶⠶⠶⠶⠶⠾⠛⠉⠻⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡙⠛⠛⠛⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⣿⣯⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡄⠀⠀⠀⠀⣀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢇
⠈⠻⣿⣿⡿⣫⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⠈⠺⢄⡄⢸⠣⡑⢄⠀⠀⠀⠀⠀⠀⠐⠂⠒⠒⣒⠒⠒⠤⠤⠚⠁⠧⣤⣷⣄⠔
⠀⠀⠀⠉⢿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⢀⠔⠈⠌⠀⠈⠢⠀⠀⠳⡏⠣⠀⠀⠑⢄⠀⠌⠀⡉⠕⣓⡀⠉⠀⠀⠀⠀⠀⠀⠀⣠⠜⠛⠋
⠀⠀⠀⠈⠀⠹⣿⣿⣿⣿⣿⣿⣿⡦⠊⠀⠀⡐⠀⠀⠀⠀⠐⢄⠀⠘⡄⠀⠀⢀⠎⡐⠠⠈⠴⢋⡁⠄⠀⠀⠀⣀⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⠠⠄⠀⠐⠀⠀⠀⠀⡀⠔⠊⠀⠀⠹⡀⢀⠞⠀⠁⣠⣈⡔⠁⠀⠒⠡⠂⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠀⠀⢡⠎⠀⡌⠀⠀⠈⠁⠀⠖⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠁⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡴⢠⡶⣦⡠⣶⠆⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠘⠷⠾⠁⠿⠀⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠭⠵⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠓⠒⠒⠒⣿⡇⠠⠤⠤⣿⠀⠀⠀⠀⠀⠀⢀⣠⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣦⡀⢀⣸⣀⣤⣤⣴⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡅⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⡀
⠀⠀⢀⠒⠂⠰⠶⠦⢤⣅⣤⣌⣻⡿⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣛⣁⣤⣶⣿⣶⣿⣯⣤⡀⠒⠂⣐⣂⣀⣘⣃⡀
⠀⠀⠀⠴⠶⠒⠶⠿⣿⣟⣻⣛⣛⣿⣿⣿⣿⣿⣟⣠⣿⣯⠕⠹⢷⣿⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣷⡾⠗⠂⠀⠀⢤⣤⣤
⠠⠤⠽⠿⢿⣶⣶⣶⣯⣿⠛⠛⠛⣛⠿⠿⣿⣾⣭⣭⠿⢟⣿⣿⣿⣟⣿⣛⡛⠛⠻⠻⢧⠶⡿⣿⣿⣿⡛⠋⠉⠉⠀⠀⠠⠤⠄
⠀⠀⠒⠦⢤⣤⠤⠀⢀⣉⠓⠶⠶⠶⠖⠀⠀⠻⠿⡧⠤⠸⠿⣿⣿⠁⠀⢀⣀⣀⣀⣈⡭⠭⠭⠭⠽⠿⠇⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⠶⠀⠀⣀⣀⣀⣈⠉⠙⣻⣿⠁⠀⠀⣀⣀⣀⠠⠤⠴⠖⠛⢁⣒⣀⣀⣀⣀⣠⣄⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣈⣉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡇⢻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⠈⡏⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡇⠀⢹⠈⢟⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡇⠀⢸⡄⠸⡜⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡇⠀⠘⡇⠀⢧⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡇⠀⢀⡇⠀⢸⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⡇⠀⢸⠇⠀⢸⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸⡇⠀⣸⠀⠀⢸⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢸⡇⢀⡇⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣾⡇⡼⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣀⡀⠀⢀⣿⣿⡥⠤⣤⣀⡀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢿⣿⡶⠿⠿⠛⠛⠟⠛⠉⢉⣉⣉⣽⣿⣿⡿⠟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣒⣩⣭⣤⣤⣤⣶⣶⣶⡾⠿⠿⠿⠟⠛⠋⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠉⠉⣉⣉⣉⣉⣁⣠⣤⠤⠤⠤⠖⠒⠒⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠲⣄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢷⡦⠤⠤⠤⠀⠀⢀⡀⠀⠘⢆⠀⠈⢆⠰⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡀⠀⠀⠀⠀⠀⠉⠓⠢⠄⣀⠀⠀⢨⠁⠒⠮⢧⡀⠀⠓⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠒⠓⠒⠒⠛⠛⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⠛⠛⠛⠛⠛⠳⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠑⣄⡤⠞⠁
⠀⠀⠀⠀⠀⠠⠒⠉⠀⢀⡠⠴⠒⠛⠛⣉⣉⣙⣛⣓⠲⠮⢭⣟⣳⠶⢦⣤⣄⣀⣀⣀⣀⣠⣾⡷⠖⠛⣉⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠀⠀⠀⠉⠉⠙⠓⠲⠭⣍⣓⠲⠦⠭⢭⣭⣭⠤⠤⠶⠚⠋⣁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⠶⠶⠒⠒⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣤⡀⠀⠀⠀⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠙⢿⣿⣿⠀⠀⠀⣾⠀⠀⣿⠀⠀⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⣿⣤⣤⣿⣤⣤⣧⣤⣼⣿⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢀⣤⣤⣤⣼⣧⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡴⠶⠟⠛⠉⡇
⠀⣤⣤⣼⣿⣷⣤⣼⣧⣤⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢰⠃
⠀⢧⣤⠤⠤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠐⠒⠀⠉⠀⡸
⠀⠘⣦⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠂⠀⠒⠒⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⠇
⠀⠀⠘⢦⠂⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠻⣶⠶⠲⠶⠶⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⢼⣤⣤⣮⣤⠾⠤⠴⠒⠐⠒⣀⡐
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠿⠇⠀⠀⠉⠉⠉⠈⠁⠀⠀⣤⣴⣶⠶⠟⠛⠉⠑⠦⠄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣉⣉⣷⣶⣒⣂⣀⣤⣀⣀⣤⣄⠄⠠⣤⣤⣤⣤⣤⣤⣤⡄⣤⣤⣤⣄⡀⢐⠲⠶⢶⣶⡶⣶⣶⣶⡶⢶⠖⠁
⠀⠀⠀⠀⠀⣠⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠓⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⠛⢛⣓⡛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣠⠞⣛⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠞⠋
⠀⠀⠀⠀⠻⡏⢹⣯⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁





⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⡩⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠀⣇⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⠀⣿⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡟⣾⠀⢸⡀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡇⣿⠀⢸⡇⠀⠀⠀⠀⠱⣄⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢰⡇⣿⠀⢸⡇⠀⠀⠀⠀⠀⠘⣆⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢸⡇⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⢸⠁⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢸⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⣸⠀⢸⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣰⠃⠀⢸⠀⣿⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⡆⣿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣤⣀
⠠⠄⠐⠛⠓⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠾⢣⣿⢸⣧⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⢸⡇⠀⠀⠀⢀⡼⠻⠟
⠀⠀⠀⠀⢠⡶⢤⣤⣠⣖⣲⡖⠲⡄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠈⠉⠉⢉⣉⣉⣥⡤⠶⢖⡿⠁
⠀⠀⠀⠀⠈⣷⣤⣤⣄⣉⣉⣉⠙⠛⠛⠓⠚⠻⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠛⠛⠛⠛⣛⣉⣉⣭⣥⣴⣶⣾⠟
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠂⠈⠑⠢⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠈⠝⠀⠀⠀⠀⠀⠀⡠⢡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⢤⣤⣤⡌⠀⠀⠀⠀⠀⠠⠔⡰⠁⠀⠀⠉⠉⠁⠒⠂⠠⠄⠤⢄⡀
⠀⠀⠀⠀⠀⠀⡠⠐⠮⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠖⢂⣀⠐⠓⠂⠀⠀⠀⠀⠀⠀⠀⢊⡡⢖⢫⠆
⠀⠀⠀⠀⠀⠘⡤⠀⠉⠀⠒⠀⠀⠀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠉⢈⣠⢶
⠀⠀⠀⠀⠀⠀⠙⠛⠙⠓⠲⠦⠤⢤⣀⣀⡀⠀⠀⠈⠉⠀⠒⠐⠂⠀⠒⠀⠤⠤⢀⡠⢎⠩⣀⣤⢾⢋⢃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠲⠢⠤⣤⣀⣀⣀⣀⠀⠐⠂⣋⣤⠖⢫⢁⠣⢃⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⠀⠀⠠⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⡏⢌⡺⡡⠡⠡⠣⡱⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠸⡼⡄⢀⠏⡞⡔⣥⢬⡳⡄⢚⣄⣆⣒⠠⡄⡤⣄⣀⣀⠀⠠⣇⡾⡑⡑⡡⡡⡱⡱⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣧⣇⡜⡸⢹⢹⠁⠀⠧⢹⠀⠘⣏⡏⠙⠃⣯⡖⠺⠿⠀⢰⡟⠐⡜⡔⡐⡐⡐⣱
⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡸⣡⠃⢸⢻⠀⠀⣐⣠⠁⢸⣗⡇⠀⢠⣿⣇⣦⡀⠀⢸⣕⣌⣌⣜⡔⡰⣱⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠏⠀⠘⣌⠧⠔⢁⠞⠀⠸⠛⠃⠀⢘⢻⠋⠉⠁⠀⢸⠙⣏⣞⣞⣜⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠸⣶⡆⠀⢼⠾⡷⡶⠤⠀⢸⢶⢾⠾⠞⢞⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⣴⢦⢦⢦⢦⣮⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠆⠤⠤⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⢸⢀⣢⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠸⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢀⠀⠂⠔⡠⢀⠄⡠⡢⠀⠐⡂⢂⠔⡠⢂⠔⠀⡠⢀⢀⠄⡀⢀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠐⠀⠄⠁⡨⢂⠄⡨⢊⠔⡡⢊⠔⡁⠢⢊⠔⡡⠡⢊⠔⡡⡀⢊⠔⠁⡡⡪⠢⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠁⠊⠐⠁⠁⠊⠐⠀⠁⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢸⡇⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠸⠁⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⢀⡼⢃⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⢀⠀⠀⣠⣾⡿⠋⠁⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢠⡞⢀⡼⠋⠁⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣿⡶⠋⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⡟⣹⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣰⡟⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡼⡿⠁⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⡼⠁⠘⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⡼⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⡼⠁⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⡼⠁⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⡾⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⣡⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠁⠀⠀⠛⠉⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⢀⣴⠟⣁⣠⠀⠀⠀⠀⠀⣸⣷⠏⠀⠀⠀⠀⠀⠀⠸⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⢀⣴⣿⣿⣿⣯⣥⡤⠶⠶⢻⣿⣯⣤⣤⣄⣀⣀⣀⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠃
⠀⠀⠴⠿⠟⢋⣉⣉⣀⣀⣀⣀⣴⣋⣀⣀⣭⣭⡭⣭⠭⠤⡌⣿⠗⠒⠠⠤⠤⠀⠀⢀⣀⣀⣠⣀⣠⣤⣤⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⡻⡟⠻⠿⣯⣭⣀⣀⠘⠂⠸⢧⣌⣙⣀⣀⣀⠀⠀⠀⠀⠤⢼⠆⠈⠉⣯⠉⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢰⡶⠛⠛⠛⠿⣿⣿⡷⠶⠶⢤⣤⣬⣭⣭⣭⣭⣭⣥⡴⠶⠖⠋⢁⣠⡴⠁⡀⠴⠒⣒⣩⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣾⠁⠀⠀⠀⠙⢿⣷⣄⡀⠀⠚⠉⠉⠙⢻⣿⣿⣿⣿⠿⠿⠛⠉⣀⣠⣴⠾⠛⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠤⠀⠀⠀⠀⠙⠻⣿⣛⡒⠚⠀⢀⣠⡿⠈⠛⠻⠿⠿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃⡟⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⢸⠇⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏⢰⡇⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⠁⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⡏⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⠃⠀⣸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⡿⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⡟⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⠇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⡇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⢃⣿⠃⢰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠛⠛⠛⠛⢻⣟⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠶⠾⠛⠛⠛⢛⣛⣛⣛⣛⣛⣩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣛⣉⣉⣥⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡟⠉⢻⡟⠿⠿⠿⠻⢿⠻⠿⠿⢿⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⣿⣿⣿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⡇⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠈⠇⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⢀⡾⢁⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⠀⣰⣿⣶⠟⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⣰⠀⢠⣾⠟⠋⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣏⣼⣃⡴⠋⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢫⣿⠃⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢡⣿⠃⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠰⣫⠇⢠⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⣰⠏⠀⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣠⡏⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣰⠏⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⣰⠏⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⢉⣠⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠀⠀⠀⠋⠉⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⡀⢀⡾⠃⠀⠀⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⢀⣤⡞⣉⣠⠄⠀⠀⠀⠀⣀⣿⡟⠁⠀⠀⠀⠀⠀⠀⢳⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⢀⣴⣿⣿⣿⣯⣥⣤⠶⠶⢚⣿⣿⣥⣤⣤⣤⣀⣀⣀⣀⣤⠼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠴⠿⠿⣯⣭⣍⣁⣀⣀⣀⣠⣏⣀⣈⣭⣭⣭⣭⡭⠭⢥⣙⣿⠶⠶⠦⠀⠀⠀⣤⣤⣤⣄⣤⣤⣤⣤⣤⣴⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⡻⠿⣿⣯⣉⣀⠀⠳⠀⠾⣤⣉⣓⣀⡀⠀⠀⠀⠠⠤⠤⢼⡦⠈⠉⣻⠉⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⢈⣶⠾⠛⠛⠿⢿⣿⣷⠶⢶⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣥⠴⠶⠛⠁⣀⣴⠃⢀⡤⠴⢒⣚⣡⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣤⡖⠀⠀⠀⠉⠻⣷⣦⣀⠀⠀⠟⠉⠉⠛⢿⣿⣿⣿⣶⡶⠿⠿⠛⠁⣈⣤⣶⠿⠛⠉⠁⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⡀⠀⠀⠈⠻⢿⣷⡶⠶⠞⠀⠀⣨⡿⠙⠻⠿⣷⣶⣶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠚⠛⠋