Asian carp Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀
⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣶⣤⣤⣤⣄⣀⡀
⠀⢸⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⢻⣿⣿⣿⣿⣦⣀⣀⡠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣴⣿⣿⣿⣅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠛⣟⠛⢺⣿⣿⣿⣿⣿⣯⣵⣿⣿⠇
⠀⣸⣿⣿⡿⠟⠋⠁⠀⠀⣭⣿⣿⣿⣿⣽⢉⢿⣿⣿⣻⣧⣧⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠿⠿⠟⠛⠙⠁
⠀⠉⠉⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠟⠛⠿⠿⠿⢿⡿⣻⣿⣿⡿⠿⠿⠿⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠲⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠄⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠁⢀⠔⠀⠀⠀⠀⠀⠆⠀⠀⠛⠄⠐⠀⡂⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠓⠂⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣀⠂⢠⠈⡁⢀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠠⠰⠀⠈⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢁⡄⠀⠀⠔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⣄⢠⠆⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⡏⠁⠀⠀⠀⠀⠀⢸⡿⠹⡟⢻⡇⠀⠀⠀⢸⡜⢃⠜⠉⢈⠐⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⢸⣷⣶⣇⣾⡇⠀⠀⠀⠘⠀⠈⣠⠀⢏⡀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠘⣹⣿⣤⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⠎⠒⡶⠠⠸⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣕⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢉⠈⠃⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣦⢻⣳⠮⡶⣤⡀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠼⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⣿⢯⢫⡚⡎⡍⠯⡓⢦⡑⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡿⣿⡡⠰⠞⠓⣅⠥⡔⣈⣲⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢻⣗⢖⢚⣯⢅⢤⢘⡴⢬⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡕⡞⡦⢒⢊⡎⢝⢓⠈⣉⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣠⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⠈⠫⣝⠽⠀⡅⠧⠂⠈⠁⠘⢄⠲⢄⣠⠀⠀⠀⠀⠀⠀⠀⢄⠀⠓⠒⠂⠠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢀⠀⠀⠀⠀⠀⢙⠢⢅⡀⠀⠀⠲⠖⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⢶⣶⠶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⢦⣄⡉⠈⠛⠱⠶⣦⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡤⠄⠐⠂⠤⠐⠩⠗⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠀⢀⡀⠀⠁⠶⣤⡀⠠⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠀⠀⠀⠤⠞⠛⠓⠒⠛⠁⣀⠀⠀⢈⡀⠀⣀⠙⠷⡄⠀⡈⢢⡀
⠀⠀⣠⠶⠛⠛⠂⠀⠀⠀⠀⠒⠀⠀⠉⠉⠀⠀⠀⣀⠁⠀⠀⡉⠀⡑⢆⠈⢂⠈⢄⠙⢦⡀
⡠⠚⠁⠀⠀⠈⠁⠀⠀⠀⠀⠐⢠⡄⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠐⣈⣠⣁⡀⣡⠈⣠⢈⣿⣷⣄
⢱⠆⠀⠀⡠⠂⠈⠀⠉⠀⠂⠄⠀⠐⠢⠀⠂⠄⣀⠀⠐⠒⠀⣤⣶⠟⠹⢋⣽⣭⣼⣯⣽⢯⣻⣿⣷⣄
⢠⡀⠀⠀⠀⠀⢠⣄⣀⣀⣀⡀⠀⠐⢂⡤⠀⢀⣄⣀⣀⣐⢤⠔⢚⠾⣿⣏⣩⣴⢿⣿⣿⣿⣿⢯⡿⣿⣦
⠈⠓⠶⢄⡠⠞⠋⠀⠀⠀⠀⠀⠉⠓⠦⠄⣈⡉⠈⠉⠉⠉⠀⠁⣀⡠⠴⠞⠿⠿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⡀⠙⢿⣿⣿⡁⠄⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠘⠧⡌⣿⣿⡁⠀⠁⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡄⠀⠀⠿⣿⡏⠋⠐⠁⢾⣻⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⠄⠀⢄⣿⡟⠆⠐⣦⠿⣻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⣸⣿⠶⡀⢰⢟⣛⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢡⣼⣿⡇⠈⢠⡟⣏⣡⣯⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣧⣀⡶⣾⡿⠈⠁⠀⡿⠫⠵⢾⣴⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠁⠉⢡⠀⠀⠀⣀⠽⡖⠒⠒⠶⢶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠸⠀⠀⣀⠹⢒⣟⣹⠛⢳⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢰⠀⢠⡀⠀⣄⠰⠏⢚⣹⣡⣝⣛⡿⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⠀⠀⠀⡄⠁⠀⡛⠤⠘⣩⢭⣼⣽⣿⣤⢋⠋⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣁⢙⠀⠀⠀⠀⠰⠄⠱⠖⢋⠡⡬⣭⣿⠛⠁⠌⠰⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣁⣈⠉⠙⣃⠀⠀⢀⣴⢈⢠⡈⠃⣀⡴⣼⡏⠀⢀⠐⠁⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡍⡙⠉⠛⠶⢭⡄⠀⣈⡥⠘⢤⠐⠟⠉⣡⣿⡅⢀⠠⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠻⠹⠦⠆⡄⢁⠸⢆⠘⠋⢰⡈⠁⡴⠞⠁⣹⡇⢀⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⡶⢶⢰⣄⡔⢁⠼⠰⡎⠡⠿⢄⡈⢡⠈⡴⢯⢻⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣭⣥⣄⣀⠊⡠⠋⠛⠐⡊⢀⠔⠠⡔⠓⢤⠸⠁⣠⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⢉⠁⡙⠈⠓⠸⠤⡎⢀⠙⠀⠀⠈⢲⡈⠢⠨⢏⠂⣿⡄
⠀⠀⡔⠲⣀⠤⠤⢞⠁⠉⠛⢶⣄⡀⢸⡿⠟⠻⠺⣇⢷⠴⡁⡄⠁⡡⠤⠔⠁⢘⣌⢁⡆⠓⢌⡟⢹⡇⠀⢀⠤⠤⠞⠛⠓⣀⣀⡀
⠀⡰⠧⠀⠉⠤⡀⠀⠁⠀⠀⠀⡙⠁⢿⣧⣤⡆⠀⠀⣠⣇⠀⠿⠚⢁⡐⠋⠀⠀⠀⠀⠈⠁⠾⠀⣼⡷⢴⠎⠀⠀⠀⠀⠠⠉⠈⠋⡳
⠀⢷⣄⠈⠐⢀⠈⠐⠀⠈⠂⠀⠀⠀⣬⢫⡟⠀⠀⢰⠋⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣠⠄⢀⠀⡀⠀⠀⡠⠈⠔⢀
⠀⠀⠑⢌⡐⠠⠈⠐⠀⠀⠁⠚⠡⠶⢽⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠒⠈⠉⠉⠙⢆⠀⠀⡼⣧⠖⠋⠀⠀⠀⠈⠐⣡⠔⠋
⠀⠀⠀⠀⠉⠲⢬⣀⠂⠀⠀⠀⠀⠐⠂⢙⠀⠀⠀⠸⠀⢀⡠⢤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡧⣷⡀⠀⢀⣐⠤⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂⠀⠤⠤⠌⠛⢧⠀⠀⠠⠑⣮⣑⡀⢠⠙⠇⡠⠄⠀⡀⠤⠠⠄⣀⢾⣄⣿⣤⣿⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢰⣟⡿⠀⣸⠆⠀⠸⣧⡜⠀⠀⠀⠀⠸⣿⠛⣯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢈⠉⣁⡀⠀⡀⠀⠄⠀⠀⠀⠀⠀⡄⠀⣯⡧⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⡿⠋⠀⠁⢈⠠⡼⠀⠀⠀⠀⠠⣧⠄⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⡀⠀⠰⠎⠠⠾⠤⢦⣤⣤⣤⠷⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣆⠰⢻⣤⠒⣶⣦⣤⣤⣤⢶⡼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⡤⠶⠾⣦⣈⠛⠛⠛⢋⣰⠷⢌⣙⠲⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠞⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠱⣼⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⢀⣀⣠⡾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡔⠁⠚⢳⠈⠦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢛⠀⡄⠀⠁⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢀⠀⠀⠀⠀⡄⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠀⠀⠀⠀⠀⠀⠦⢱⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⠀⡀⠀⠀⠠⣔⡒⢰⡄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⠍⡣⢦⣾⠦⣅⢰⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡏⢨⣁⡎⢭⣾⠓⢦⡝⠒⠀⠀⢀⢠⡀⡀⡀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢧⢗⡤⣉⣆⠸⣵⠻⢟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⡒⠍⡾⢟⡀⠐⢑⠶⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⡴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠹⣲⢆⣹⠆⠘⢬⠯⣁⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠀⠀⣻⢑⠀⠀⠀⠠⡰⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠙⣰⣭⣟⡀⠰⢶⠹⠆⠀⠀⠀⠀⢀⠀⡀⠄⠙⠃⠀⠀⠐⠍⢳⡄⢀⠄⠀⠀⠀⣀⣀⠀⣾⡔⠁
⠀⠀⠀⠀⠀⠀⠄⡀⠖⠀⠀⠀⠀⠀⢀⢹⣾⡾⠷⠀⢡⣟⡻⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡏⣾⢆⡖⠀⢰⠏⣨⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢜⠈⢡⣝⣗⠈⣳⣽⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢻⣮⠁⢠⡎⡼⣿⡿⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠛⡺⡤⣟⡞⣾⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⡗⣀⠀⢠⠞⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠿⣭⣟⣶⠿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠈⣫⠁⣠⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠋⠘⢾⡿⡿⣶⣤⣀⣀⣀⠴⠀⠀⠀⠀⢀⣰⡿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⠿⠯⣟⣿⣿⣿⠧⠤⠀⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠒⠒⠒⣲⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣿⠏⠀⠀⠀⠀⠀⠀⠀⠙⠁⡀⠬⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠓⠦⢤⣴⡿⡾⣽⣽⡄⠀⠀⠀⠀⣀⢀⡠⢐⣡⠒⢉⠜⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡠⣎⣗⢯⣷⡷⠛⠉⠢⣄⠀⠨⠒⠐⠾⠟⠓⠈⠉⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣰⡯⣟⣿⡽⠋⢠⠂⠀⠁⠈⠰⠒⠒⠀⠀⠀⣂⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣇⡯⣲⠏⠀⠀⡌⠀⠀⡀⠔⠁⡀⠄⢲⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣾⣏⣿⠃⠀⠀⠀⠀⠉⠀⣠⠴⡇⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣿⢷⡏⠀⠀⠀⠀⠀⢀⠞⠁⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢽⠀⠀⠀⠀⠀⣰⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣻⠀⠀⠀⠀⢰⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⡠⠄⠀⡏⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⡀⠀⠈⠀⠈⣿⢦⡀⠀⠀⠀⣀⡤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠃⠤⢾⢽⣝⣦⣄⡞⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠋⠲⣄⡀⠀⠀⠉⠻⠾⣯⡆⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣀⣀⣀⠇⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⡀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠓

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠒⠒⠚⠒⠒⠒⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⠀⠀⠀⠀⣠⡀⠀⠀⠹
⠀⠀⠀⠀⢀⣴⠾⠻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⣯⠀⠀⢈⠃
⠀⠀⣠⠴⠟⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⡀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠠⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡂⠀⠀⠀⠛⠷⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠈⠀⠀⢤⠠⠀⠄⠀⠀⠀⠀⠉⠚⠛⠃⠀⠀⠀⠀⠀⠀⠐⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⡴⠶⠿⣛⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣧⣤⣀⣀⣴⣦⡄⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣠⣶⠋⠁⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢛⣉⣯⠀⢿⣿⡟⢭⢷⣶⣦⣤⡤⠾⢷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⢠⣿⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠀⠀⠀⠘⠻⠿⠿⠇⠀⣈⡛⠃⠀⠈⠉⠀⠀⠀⠀⠀⠸⣿⣿⣷⡄⠀⠀⠀⠀⢰⣿⠿⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠀⣀⢀⣠⣿⡟
⠀⠀⠀⣴⢿⣢⣿⣿⣿⣿⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⢽⠋⠉
⠀⠀⠀⠀⠈⠙⢻⡇⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⠖⢒⠻⠛⢛⠿⡟⢷⣶⣦⣀⠀⠀⠀⠀⠀⠀⠸⣷⣼⣁⣀⣴⠦
⠀⠀⠀⠀⠀⠀⠘⢿⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠈⠁⠀⠀⠀⠀⢀⡴⢉⣭⣥⣥⣤⣤⣤⣶⣖⣠⣿⣿⣿⡿⠀⠀⠀⠀⠀⣁⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠀⠀⠀⠀⢶⣦⣰⣶⠀⠉⠘⠀⠀⠀⠀⠀⠀⡎⡴⠋⠉⠀⠀⠀⠀⣉⠙⡿⠛⠙⠉⠁⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⣀⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⢿⣤⠆⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠾⣯⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⡀⠘⠛⠻⠦⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡏⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⡀⠀⠀⠀⠀⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢸⡇
⠀⠀⠀⢠⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠇⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣇
⠀⠀⠀⠀⠛⠟⠛⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡖⢂⣀⡀⠀⠀⠀⠀⣀⣤⣀⣿⣿⣿⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣄⡀⠀⢠⣦⣤⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣿⣿⣿⣿⠉⠉⠉⠀⠀⠀⠈⠉⠛⢻⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠁⠀⠈⠛⡜⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠀⢻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⡁⢀⣈⣳⡄⠀⠀⠀⠀⠐⣶⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣰⣿⣧⡀⠀⠀⠀⠀⠀⠚⠛⠋⠉⠛⠉⠋⠁⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢸⣏⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣄⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⣤⡀⣸⣿⡆
⠀⠀⢀⣠⡶⣿⣿⣿⣿⣿⠿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠈⢿⣿⡷⢦⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇
⠀⠀⠈⠙⠛⢹⣿⣿⣿⣿⠀⠉⠁⠀⠀⢰⣷⣶⣷⣶⣶⣅⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣯⣱⣄⠀⣠⡤⠀⠀⠀⠈⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠟⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⠾⠃
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⡻⢿⣿⣿⣿⣟⠿⠿⣦⡀⠀⠀⠀⠀⠀⢙⣿⣿⣣⡝⣿⡄⠀⠀⠀⠀⠀⠀⡇⡿⠀⢀⣀⣤⢞⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠉⠛⠻⠿⠿⣦⣥⣄⠀⠀⠀⠠⠟⠛⠉⠻⢿⣿⡷⠀⠀⠀⠀⢀⣠⠿⢷⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⡞⠁⠀⢀⣿⡿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠚⠉⠉⠂⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠴⢶⠶⡶⠶⣶⣦⣤⣤⣄⡀⠈⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤
⠀⠀⠀⠀⠀⢀⣀⡤⠔⢾⣿⣿⣿⣷⡗⠿⠟⢾⠿⠶⠗⠴⢦⢩⠉⠋⠻⢿⣷⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢻⣿⣻⠁
⠀⢀⡠⠖⠋⠁⠀⠤⡞⠀⠙⠻⣿⠅⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣤⣴⣾⡣⠶⣦⣾⡟⠁
⠰⠋⠁⣤⣦⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿⣷⣿⡿⠋
⠀⠈⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠷⠤⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠌⠀⠈⠀⠐⠲⣿⡿⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠐⠒⠦⢄⣀⠀⣿⣿⣤⣾⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢴⣶⣻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠍⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿

⠀⠀⠀⠀⠀⣤⣀⣀⠀⠀⢀⠀⠀⠀⠀⢀⣤⢀⣀⣀⣀⣤⣀⡀⠀⣀⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣤
⠀⠀⣠⡶⠋⢻⣬⣭⣷⠞⠛⣄⠀⢀⣴⣿⣿⣿⣿⣿⡿⠿⠟⠉⠉⠉⢁⣀⠀⢰⣿⣿⣀⣳⣀⠀⠀⠀⠀⣰⠎⠉⠀⣸⣿⣶⣶⣤⡀
⠀⢾⡅⠀⠀⠀⠀⠀⠛⡓⣆⣼⡇⠸⣿⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠛⠿⠛⠛⠷⠦⣤⠀⠀⠀⠀⠀⠉⠁⠙⠟⠿⣧⣍
⠀⠈⠃⣠⣄⠀⠀⠀⠀⠁⠀⣄⣀⣀⡀⠁⠀⠀⠀⠀⠀⢠⣄⢠⣤⣀⡀⣠⡴⠲⠀⢤⡄⠀⠀⠀⠀⠀⢁⣦⣀⣀⠀⠀⠀⠘⠓⢠⣿⣿⡇
⠀⠀⣼⣿⣿⡆⠀⠀⠀⢠⣾⣿⣿⠋⠉⢳⡶⠦⠀⠀⢠⣤⣤⣶⣿⡟⠁⠋⠁⠀⠀⢨⣷⣶⠂⠀⣴⣿⣿⣿⣿⣿⣿⣦⡤⠀⠀⢠⠙⠛⠁
⠀⠴⢿⣿⣿⡿⠆⢀⣤⣿⣿⣿⠁⠀⠀⠀⠁⠀⢘⣣⣾⣿⣿⠟⠿⠀⠀⠀⠀⠀⠈⠉⠙⣿⠇⠀⠈⠛⠛⠻⠏⠉⢹⢿⣿⡄⠀⢸⠃
⠀⠀⠘⣿⣿⡇⠀⠈⣿⣿⣿⣿⡆⠀⠀⠀⢀⣴⡿⠋⡍⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⣤⣿⡀⢀⣠⣄⠀⠀⠀⠀⠀⠀⢻⠇⣤⣿⣀
⠀⠀⠀⢸⣿⣷⠄⠀⠹⠿⠿⠃⠀⠀⠒⣿⠟⠁⠀⠀⠀⣴⡾⠛⠿⠷⣾⣿⡀⠀⠘⠉⢻⡟⣣⠸⣿⡋⠁⢲⠀⢠⣼⣿⠟⢀⣿⣿⡏
⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⢀⣦⠀⠀⠐⠋⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠙⠉⠱⣄⠸⣷⣿⠀⠁⠀⠹⣿⣦⣼⣆⠀⠀⠀⢀⣼⣿⣿⠇
⠀⢠⣶⡆⠀⠀⠻⠷⠀⠀⢸⡅⣀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⡿⠃⠀⠀⠀⠀⡟⢧⠀⠀⠀⠀⣠⣀⠀⢹⣿⣁⡙⠀⠀⢹⣿⣿⣿⠖
⠀⢋⡈⣷⣼⡃⠀⠀⠀⠀⠸⣯⣩⣤⣤⣤⡄⠀⣀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋⠉⠉⠀⠹⡿⠃⠀⠀⠘⢿⡿⠋
⠀⠟⠛⠛⠿⣷⣄⡰⠂⠀⠘⠉⠃⠨⣿⡿⢿⣶⣿⡄⠀⠉⠉⠀⠀⠀⣀⣤⣤⣴⠞⢿⣧⠀⠀⠀⠀⠀⢀⣴⠇⠠⣤⣠⣤⣤⣷⣤⣄⡀
⠀⡀⢠⠄⠀⠙⠏⠉⠛⠐⠲⣾⡃⠀⠀⢠⠄⣿⣿⠗⠀⠀⣄⠀⣠⣾⣿⣿⣿⡿⠀⠘⠛⠀⠀⠀⠀⠀⠚⠋⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⢿⣿⣿⠿⣿⠅⠀⠀⠀⠀⠀⠀⠀⢠⣴⡎⠉⠀⠛⠁⠀⠸⠏⠉⠉⠉
⠀⠀⠸⠧⣤⠀⣀⣶⣶⣿⣶⡄⢀⣤⠀⠠⠶⠀⠀⠀⣤⣀⣾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⣿⡆⢀⠀⠀⠀⠀⢀⣤⣄⣠⣤⣄
⠀⠀⠛⠢⣿⣿⣿⠛⢻⣿⠿⠃⠈⢧⡄⠴⣶⣿⡷⠾⢛⠿⠀⠀⠙⠻⢦⣤⣤⣘⠃⢻⣷⣶⣶⡏⠙⠀⠼⠷⢤⡞⠈⢿⡿⠿⠁⢹⣿⣿⡆
⠀⠀⠀⠀⠈⠉⠋⠰⠆⠀⠀⠀⠀⠀⠀⠶⠈⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠻⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠋⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡄⠀⠀⠀⠀⠸⣻⡅⠀⠀⢠⠖⠓⠂⣤⡀⠠⡒⢲⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣹⣶⣶⣦⡀⠀⣹⠃⣀⠀⣈⣺⡀⠐⡌⢷⣀⡶⠢⣽⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣟⣯⣿⣿⣒⢨⡏⠁⢯⠅⠈⣩⠓⡆⠻⠀⠹⡗⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣟⣿⣟⣀⡀⠀⠀⢸⣷⡅⠀⠀⠀⠀⠻⢳⣦⣴⣶⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡘⠺⡿⣿⣿⣿⣿⣾⣞⢇⣨⣿⣇⠀⠢⠀⣤⣾⣽⠎⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠺⠯⠾⠷⠚⠀⠉⠶⣤⡝⠿⣿⣯⣿⣿⠟⠀⢫⣂⠀⢰⣿⣿⣿⢷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣿⡿⠀⠀⠀⠁⢙⣹⣿⣿⣷⣶⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⢀⣱⠶⠦⣖⣿⣳⣳⡽⡟⣿⣿⣿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣾⡿⠋⠀⠒⢒⣓⣼⢿⣿⣿⣝⢼⡿⢃⡿⠟⣀⣀⣤⣘⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢻⡿⣿⣁⠀⠀⠐⢹⣿⣟⡏⢿⣿⡿⠉⣰⡎⢀⣀⢭⣝⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⠰⡏⢀⡴⠿⠀⠀⠳⣿⣯⢷⣾⣿⠋⣠⠟⠁⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠒⠘⠺⡁⣷⣀⠀⠈⢿⣯⣻⣷⣿⡟⠜⠀⠀⠀⠀⠀⠆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡎⠁⠈⣡⡚⢛⣏⣓⠀⠘⢗⣿⣟⣿⡇
⠀⠀⠀⢀⠒⢒⠶⡤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⢁⡄⠈⡙⢦⠀⢓⡙⢿⣹⣦⣄⣀⡀
⠀⠀⠀⠈⢡⣏⠀⣖⡍⢰⡄⣀⠀⠀⡤⠒⢠⠖⠚⢷⢤⡅⣴⠓⠺⣌⣿⢿⣿⡿⡿⠛⠀⠀⠀⡀⠠⠀⠄
⠀⠄⠀⠉⠀⢰⡂⢩⠥⣎⠉⣩⣭⣤⡉⠀⠈⠁⠀⠀⠃⠙⠀⠀⠀⢵⡄⣀⡀⠛⠀⠀⠅⠀⠈
⠀⠀⠀⠀⠃⠪⠁⠀⠀⠈⢲⠁⠀⠰⡿⠤⠀⠂⠀⠀⠘⠀⠀⠀⠀⢠⣶⢟⠇⠀⠀⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⡎⡎⡔⠄⠀⠀⠀⠰⢴⠄⠰⠀⠐⠈⠀⠀⠀⠀⠠⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡦⢢⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢓⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣯⣯⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⢶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⣠⡔⠂
⠀⠀⠀⣀⣠⣴⡶⠟⢿⠏⠩⣸⣥⠎⠘⢁⡋⡛⠶⠈⠞⠻⣿⠿⡟⣿⣶⡶⢲⠦⠤⢤⢤⣤⣤⣤⣤⣤⣶⣿⣟⣭⣶⣟⣣⠈
⢠⣶⣿⣿⠛⢻⠀⠀⠈⠃⠉⠠⠃⠌⠀⠡⢿⠉⠀⠀⠀⠀⠁⠈⠀⠀⠉⠀⠀⠀⠀⠀⠈⠘⠋⠙⣹⢸⣿⣿⣏⣽⣿⠋⠁
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡁⣀⣀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄⠀⠀⠀⠘⠻⢿⣿⣿⣲⢯⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣤⠀⠀⠀⠀⠀⠙⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⣥⣐⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⠀⠀⠈⠻⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣿⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⣀⣬⣴⣾⣿⡶⠶⠶⠶⠶⠶⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⣧⣶⣿⣯⣥⣶⣶⠿⠷⠶⠶⢿⣿⣿⠛⣒⣶⣦⣭⡓⠦⣀
⠀⠀⠀⠀⠀⠀⢀⣴⠞⣫⣷⠟⠋⠉⠀⠉⠉⠀⠀⠀⣠⢴⣶⢲⠙⣶⣾⣿⣿⣿⣿⢇⣤⣈⠑⠢⣄
⠀⠀⠀⠀⠀⢀⡞⣡⣾⠿⠁⠀⠀⠀⠀⡀⠀⠀⠀⣔⠻⣆⢿⠸⠐⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⢷⡌⠗⢦⡀
⠀⠀⠀⠀⢀⣿⣾⠟⠉⠀⠂⠀⠀⠀⡀⠀⠀⠀⢰⣏⠳⡝⠆⡆⡞⣿⡟⠛⣹⡟⣾⣿⣿⣿⣿⣦⣥⣶⣶⣾⣦⣄⡀
⠀⠀⠀⠀⣸⣿⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣭⠓⠀⠀⡀⣯⣻⣄⠀⠁⠀⣿⣿⣿⠿⠿⣿⣿⠿⠿⠿⠟⠓⠿⣆⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⢀⣿⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣤⣾⠏⠒⠐⠂⠁⢸⡜⢶⣤⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣴⠂⣴⡶⠾⢠⠞⠀⠀⢀⡜
⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠿⠛⠀⠀⠀⠀⠀⠀⠀⠑⢌⡳⣄⣤⡤⣶⣐⠓⠀⠀⠀⠀⠰⠇⣤⠈⠃⠀⠘⠦⠤⠤⠞⢸⣿
⠀⠀⣤⣿⠶⣶⣶⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠉⠙⠃⠀⠀⠐⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⣿⠀⠀⠉⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣤⢀⡒⡒⣚⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠶⡆⠈⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠚⠳⠆
⠀⠀⠀⣿⡀⠀⠀⢀⣴⡾⡟⡽⡴⢈⠀⡇
⠀⠀⠀⢹⣷⣤⣴⣿⠟⠀⠹⣴⢡⡟⣠⠃
⠀⠀⠀⠈⢿⣿⣿⡃⠀⠀⠀⢈⠻⠧⣾
⠀⠀⠀⠀⠘⣿⡇⣿⣦⣴⠒⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣤⠀⣠⡄⣠⡄⠀⣤⡄⣤⢀⣤⡀⢠⣤⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠘⢿⣿⣿⢻⢦⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⠀⣿⠘⢿⡄⢸⣷⣶⣿⡇⣿⠀⣿⢿⣼⣿⢸⡟⠁⢩⡄
⠀⠀⠀⠀⠀⠀⠀⠙⣯⠞⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠿⠰⠾⠃⠺⠇⠀⠿⠇⠿⠀⠿⠀⠻⠿⠈⠻⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⢟⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣓⢭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠅⠳⡄
⠠⣤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⢁⣀⣤⣴⠖⠖⠻⠏⠛⡓⠲⠶⣤⣀
⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠖⢻⣿⡶⡖⠿⠛⠀⠋⢐⣊⠅⠄⠌⠠⠄⢀⠀⠈⢌⣵⡦⢄⡀
⠀⠀⠈⠳⡄⡀⠀⣸⠎⠉⠃⠉⠉⠀⠀⠀⠄⠀⠀⠐⠐⢒⠀⢀⠀⣀⣍⡤⠇⣼⠊⢁⣃⠙⣻⣷⣤⡀
⠀⠀⠀⢀⣶⠆⣴⣾⡀⠀⠀⠀⠀⠂⠀⠁⠃⠀⠄⠀⠀⠈⠉⣁⠀⠁⢁⠀⠄⠏⠀⠀⠋⠀⠋⠹⣭⢯⡇
⠀⣠⠞⠉⠁⣀⣼⠿⠗⠁⠈⢉⠲⣶⣤⣤⣄⠀⠀⠀⠁⠁⠀⠐⠒⠀⠀⠀⢀⢁⡢⢔⣢⣄⡤⠴⠯⠋
⠰⠷⠆⠒⠈⠁⠀⠀⠀⠀⠀⢘⡀⠀⠕⠁⠈⠉⠁⠀⠀⢨⣤⡀⠀⠀⠀⣦⡸⣽⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠂⢾⢩⠇⠀⠀⠀⢻⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣤⣤⢦⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⣀⡀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣠⣶⣮⠔⠸⠀⠀⠀⠀⠀⠀⠈⠀⠠⠋⢩⣟⣿⡶⢤⣼⣿⣿⣿⣿⡿⠿⠯⡕⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⣋⣥⠀⢠⠀⠀⠀⠀⠀⠀⠈⢶⡦⠀⣿⡟⠛⠁⠀⠻⣿⣿⣿⣿⣿⣄⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡾⢋⣤⠆⠸⢄⠀⠀⠀⠀⠀⠀⠀⠑⢆⢨⣧⣄⠀⠉⠀⠀⠙⢯⡉⠉⠛⢧⠤⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣦⠀⠳⢄⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡙⢷⣴⣶⡀⠀⠹⣦⡸⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⣿⣿⣿⣿⠟⠛⠛⣛⠃⠀⠒⠈⠉⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⢝⠷⣦⣲⣟⠙⣟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⣈⢹⣿⣿⢃⠈⣇⣼⣿⠤⠀⠀⠀⠀⠀⠀⡆⣲⣾⡧⢈⣙⠏⠑⢮⡃⠑⢌⡻⢿⣷⡘⣧⣉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣸⣿⠋⠃⡰⣄⠀⠀⠀⠀⠀⠀⠠⠴⠟⠁⠘⠛⠁⠀⠢⣌⠳⣄⢹⣶⣽⣿⣿⣿⣧⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣤⣰⣧⡌⠂⠀⠀⠀⠀⠀⠀⠠⣴⣤⣄⡀⠀⠀⠀⢈⣧⠘⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⡟⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⢦⡀⠀⠀⠉⠀⢻⣧⣹⣿⣿⣿⣿⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣼⣿⣿⣿⣿⣾⣿⣿⡀⠀⠀⠀⠀⠀⢀⡀⢂⣶⠓⢂⢹⠀⠀⠀⠀⠾⣿⢯⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⣿⣈⡙⠲⣤⣀⠀⣾⣿⣄⣼⣿⠧⡹⢠⠀⠀⠀⠘⠶⠉⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡟⢿⣶⣬⣍⡂⢀⠈⠑⠾⣿⣿⣿⡁⠀⠈⠁⠀⠀⠀⠀⠀⠀⢺⠙⢏⣿⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠙⣦⡙⠻⠉⠛⢶⣤⣢⡀⠈⠙⠻⣥⠀⠠⠄⠀⠀⠄⣀⡀⢀⣈⣨⣾⣿⣿⡇⠁
⠀⠀⠀⠀⠀⢀⡔⠊⠀⠀⠀⡀⢂⠀⠀⠀⠈⢎⠙⠦⣄⠀⠀⠈⠉⠙⠓⠈⠽⡏⠱⠦⡀⠈⠢⡀⢻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⣙⣧⣤⣀⠀⠀⢀⣴⣺⡄⠀⠙⠢⡀⠐⠂⣸⠿⢛⣽⣿⣿⡿
⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠙⢻⣿⣿⣿⣿⣿⣷⢤⣿⣿⡿⢁⡀⢀⣀⠇⠜⠀⠰⠾⢛⣽⣿⣿⠃⢸⡀
⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⡜⣀⡠⢤⣴⣾⣿⣿⣿⣿⠃⠀⡠⠈⠉⠉⠉⠁⠀⠀⡦⠤⣶⣿⠿⢛⡿⠋⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⠻⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠁⠀⣠⠞⠁⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠐⠊⠤⠤⠤⠀⠒⠈⠀⠀⠀⢀⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣴⣾⣤⣤⣀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⠿⣿⡿⣿⣿⣿⡿⡿⢿⣿⣿⣽⠿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⢀⠇
⠀⠀⢀⣀⢤⣶⣿⡿⠟⠋⠉⠉⠉⠁⠀⠙⠀⠈⠠⠑⠑⠛⠿⢿⠉⠋⠾⠋⠹⡇⠻⠷⠝⠻⠿⠏⣷⡶⣤⠄⣀⣀⣀⣀⡠⠄⠈⠀⠀⢀⡰⠋
⠀⠀⠈⠉⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣿⠂⠀⠀⢶⡏
⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣀⣀⣠⣤⣤⢤⣤⡀⠂⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⡀⠠⠄⣀⡶⠀⠀⠈⠁⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠉⠻⢿⣅⣀⣠⣤⣶⣏⡀⠀⠀⠀⠀⠀⣀⠀⡠⠒⠋⠁⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⠄⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠑⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠑⠠⠠⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠠⠀⠸⠷⠶⠦⠦⠴⠤⣤⣌⣈⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⣠⢔⠋⠀⠨⠅
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠛⠁⠀⢤⠍
⠀⠀⠀⠀⢀⣴⣿⣿⠁⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡈
⠀⠀⠀⠖⠛⠻⠋⠀⠀⠀⠀⢨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠲⠄
⠀⠀⠐⢤⣀⡀⠀⠀⡠⠀⠀⠀⠠⠄⠒⠒⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢒⠀⠀⠀⠀⠀⠈⠛⢆⠀⠈⠡⡀
⠀⠀⠀⠀⠀⠉⠉⠁⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣀⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠻⠅⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠚⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣽⣯⣮⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠐⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣴⣾⣶⣶⣿⠿⢤⣶⣶⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢀⠛⠛⠛⠛⠛⠟⠒⠆⠀⠀⢀⡀⠀⠀⠀⢀⣤⣾⣿⡯⠄
⠀⠀⠀⠀⢀⣠⣴⣾⠟⠉⠻⣇⠹⢻⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⢺⣾⠃⢠⣤⣿⣿⣿⣧⠀⠀⠀⠿⠿⠁⠀⠀⠀⡿⠿⠛
⠀⠀⠠⠾⠋⠳⠈⠁⠀⠀⠈⡄⠀⠀⠹⣿⣿⠻⠈⠈⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⣄⠀⠀⠀⢀⠀⠀⡴⢁⡄⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠋⠁⠀⠀⢀⡀⠀⠀⠀⠀⣼⣿⠓
⠀⠀⠀⠈⠀⠒⠒⠁⠀⠉⣴⡾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠤⠤⠀⠒⠂⠀⠀⠈⠁⠉⠢⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⡠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠁⠙⠑⠰⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠿⠾⠷⠶⠶⠤⢤⣄⣈⠈⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣖⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢃⢂⠂⢂⢂⠂⢂⢂⠂⠂⢀⢀⠂⢋⢋⠓⠦⢄⠀⠀⠀⠀⠀⣀⣴⡫⠐⠨⠒⠂
⠀⠀⠀⠀⠀⠀⢀⣴⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠭⠅
⠀⠀⠀⠀⣠⡾⢛⡋⠑⠆⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠈⠈⠀⠈⠈⠀⠀⠀⠠⠤⠥
⠀⠀⡴⠾⠱⠉⠀⠃⠀⠀⣀⢀⠀⢂⢂⠂⢀⠂⢂⢀⢂⠂⢀⢂⢂⠀⠀⢀⠂⠀⠀⠂⠂⠂⢂⡄⠂⠨⠭
⠀⠈⠳⣀⠀⠀⠀⡀⠀⣴⢃⠂⠂⠂⠂⠂⠂⠂⠂⠂⠂⠂⠂⠂⠂⠂⠀⠂⠀⢀⠄⠀⠀⠀⠻⣾⠔⠈⠙⣒
⠀⠀⠀⠈⠙⠒⠺⠗⠋⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠈⠁⠀⠡⣲⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠀⠀⠠⠤⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢠⣠⠤⠒⠃⣡⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠒⠂⠠⠄⢠⠊⠀⡀⢀⡴⠋⠁⠀⣀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣼⠏⠀⢀⡤⠞⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠤⠴⠞⠢⠴⠯⠀⣀⣠⡤⠔⠊⠉⠉⠉⠉⠉⠐⠂⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠟⠒⠚⠀⠀⠈⠛⠓⠒⠒⠂⠵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠀⠀⠀⠀⠀⠀⣀⣀⣠⣎⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢀⠀⠀⠄⠀⠀⠀⠠⣤⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢀⡴⠚⠉⣥⣶⠐⠐⠂⣤⡭⢙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡐⡀⠀⠲⣶⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⢋⡖⠁⡀⠐⠀⠁⠚⠉⢮⠙⠂⠀⢱⣬⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣹⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⡎⡇⠈⠀⢤⣄⠠⠄⠀⢀⡂⢀⡈⠀⢀⡹⣎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢈⠀⢉⠰⠆⢀⡀⢁⡀⠐⠊⠁⣼⠨⠀⠉⣀⠈⠳⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢃⡈⣤⠈⠀⠀⠈⠙⠀⢀⠉⠁⠀⠀⠻⠁⢀⣄⠀⢹⡀⠀⠀⠀⠀⢀⣠⣮⡟⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡻⣌⢷⡀⠀⢣⠀⠶⠀⠤⠠⠀⠺⢇⠀⠀⠐⠒⠤⡾⢸⡇⠀⠀⢀⣠⣾⣿⣿⣳⢧⣬⣀⠀⢢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣹⢦⡙⠦⢠⡀⠀⢰⣇⣀⡈⠁⠈⠓⠀⢀⣀⡔⢁⣾⠀⣠⣠⣿⣿⣿⣻⠟⠃⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡙⠳⣤⣙⠲⠀⠤⢬⣤⣤⣀⠹⠥⠚⣉⣴⣿⣿⣷⣿⣿⠿⠟⠋⣠⣄⠀⠀⠀⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣾⠉⠛⠲⠶⠤⠤⠤⠴⠶⠒⣋⠽⢯⣉⣉⠁⠀⠀⠀⠀⠘⣧⡼⠃⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠽⠿⢿⣶⣤⣮⣤⣤⣤⡴⠞⠋⠁⠀⠀⠈⠉⠛⠛⠃⠐⠚⠛⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣠⣄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠿⣿⠿⠿⠿⠿⠟⢛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠘⠿⠿⢖⣶⡶⠶⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⢤⠀⠩⠘⠃⠶⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠠⢀⡀⠘⠢⣄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡤⠤⠤⠰⠋⠀⠀⠀⠀⠀⠉⠲⣄⠀⡑⢄
⠀⢀⣤⠖⠒⠉⠀⠀⠀⠤⠀⡀⠒⠀⠠⠀⡀⠂⠄⡐⠀⡂⠈⠃⠀⢀⠳⣄
⡔⠊⠀⠂⠀⣀⠀⡀⠀⠐⠠⡄⠈⠀⠀⠀⠀⠀⠀⠀⠐⣌⣄⣀⢀⣄⣠⣈⣿⣦⡀
⠸⠁⠀⠐⠀⠀⠀⠀⠈⠂⠀⠈⠂⠌⠂⠀⠀⠉⠀⠔⣲⢋⣋⣵⡷⢺⣷⣟⣪⡿⣿⣦
⠰⡀⡀⠀⢀⡴⠂⠒⠒⠦⠠⣉⠐⠂⠤⠲⠴⠶⠄⠊⠁⠛⢛⣶⣿⣯⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠶⠤⠶⠄⠚⠉⠉⠀⠉⠙⠻⣿⣿⡟⡉⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠱⢆⡹⣿⡇⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢿⣿⡙⠀⠡⢿⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠠⠀⢸⣿⠑⠈⣦⣛⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢠⣾⠏⠄⣸⡫⠷⡶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⣀⣸⡟⠻⠀⢰⠧⢂⣟⣛⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠈⠛⠃⠈⠁⠀⣐⡮⠠⠤⣭⣬⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠂⠀⠀⠀⠀⣀⠗⡺⣸⠒⢷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡇⠰⠀⠠⠀⢠⠐⢁⣩⡯⣼⣹⠏⣀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠿⡄⠀⠀⠀⠀⡤⢀⡨⠖⠶⢾⣿⠞⠌⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠻⢶⡇⠀⠀⢀⡻⠌⠡⠖⠞⣛⡿⠋⠈⡠⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠿⠶⢠⢚⡀⠀⢚⠱⢊⠡⠴⠛⣹⡇⠀⠀⠀⣂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⠛⠳⢰⢠⠈⠱⠀⠛⠂⡜⠣⠄⠚⢙⡇⢈⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠖⠶⢰⢤⣀⠜⢰⠉⠼⠃⡄⠡⣁⡞⢙⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣴⣠⣀⡌⢈⡄⣄⠁⢂⣉⣂⡁⣤⣞⣿⡀
⠀⠀⣀⠀⢀⣀⡠⠀⢤⣀⠀⠀⣿⠟⠩⠙⠏⠞⠹⠛⠷⢶⣉⡿⢥⡽⢧⣽⣟⣮⣿⡇⠀⢀⣀⣠⣤⣄
⠀⣸⡀⠙⠋⠉⠑⠀⠀⠈⠷⠢⣿⡤⣤⠤⠀⢤⣄⣤⣴⣏⡽⠷⠖⠛⠚⠷⢼⣿⣾⣧⣴⠟⠉⣽⠀⣸⠛⣻⢶
⠈⣦⡀⠠⢀⠐⠠⢈⠢⡀⠁⠠⣿⣧⠀⠀⣤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⢠⠀⠹⣏⣿⠆⣰⣋⠴⢋⡴⣣⢏
⠀⠈⠢⢄⠀⢈⠐⠠⠀⠐⠁⠾⢽⡅⠀⠀⠁⠀⠀⠀⠀⠀⠈⡠⠐⠉⠉⠉⠛⠄⠀⠹⡥⠛⠉⠀⠀⠉⣈⠕⠋
⠀⠀⠀⠀⠑⠢⠌⢀⡀⠀⠀⠀⣀⡄⠀⠀⠀⡀⣤⠲⠀⢤⠀⠀⠀⠀⠀⠀⠀⠈⠐⣷⠦⡀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡄⠀⠀⣸⣷⡄⢰⡀⠑⣦⠠⠀⠁⠀⠀⣼⣷⡾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠙⠚⠀⠊⠁⠀⠘⡋⠀⠀⠀⠀⠉⠀⢟⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢱⠊⠀⠊⠀⢠⠄⠀⠀⢀⢈⡀⠫⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡀⠀⠐⠁⠞⠠⠦⣤⣤⠘⢴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⠱⡌⢿⣶⣶⣶⡖⡦⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠴⠊⠉⠙⠢⠬⢭⠭⠔⠋⠓⠀⠁⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣰⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠈⠁⠂⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⢀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡶⣿⡿⠿⠿⢿⠿⢻⠿⢿⡿⣿⡷⣷⣦⣦⣤⣀⡀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣟⠿⠹⠸⠻⠸⠐⠂⠆⢃⢇⠃⠒⠘⡱⠘⠘⠀⠈⠇⠏⠏⠟⠷⠶⢤⡀⠀⠀⠀⠀⠀⠀⢀⡠⢂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⣠⣶⠟⠛⠻⢿⣄⢃⢣⢣⢁⡉⡈⡌⡌⡌⣉⢡⢡⢡⢡⡁⡉⡜⡜⡌⡈⢀⢀⢃⢃⢋⢛⠒⡒⡒⡐⡒⣿⣮⠅⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠔⣋⠁⠀⠀⠀⠀⠀⠹⡆⠆⠂⠠⠰⠰⠰⠰⠐⠄⠆⠆⠆⠦⠠⠰⠰⠰⠀⠀⠄⠆⠎⠄⠄⠠⠠⠡⠱⠡⠡⢼⣿⠖⠂⠀⠀⠜
⠀⡠⠈⠀⠀⠻⠇⠀⠀⠀⠀⠀⠆⣿⠀⢃⢃⢃⠁⠛⡘⡘⡘⡘⡘⢈⢃⢃⢃⢃⠛⡘⡘⡘⡘⣈⢈⢃⢃⢃⢃⠃⡙⡘⣸⣿⠀⠀⠀⠈
⠐⠇⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣰⡷⠀⢆⠂⠂⠰⠰⡰⡰⡰⡰⡐⠀⢀⢆⠎⠆⠠⠡⠡⠡⠡⠡⠄⠎⠆⠆⠆⠂⠰⠰⢘⡗⠀⠀⠀⠀⠡
⠀⠀⠘⠃⠠⠤⣀⣠⣔⣡⣴⣪⠽⣵⡈⠠⠠⠡⠡⠡⠡⠡⠡⠡⠁⠈⠎⠌⠌⠈⠡⢡⢡⢡⢡⠀⠌⣨⠔⠂⠀⠀⠀⠙⠻⣅⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡖⠚⠉⠁⠃⠃⠃⠁⠡⠡⠡⠡⠩⠈⠎⠌⠌⠀⠁⠃⠃⠂⠰⣶⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢄⢀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠉⠉⠉⠀⠒⠒⢲⠒⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢰⡏⢀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⣿⣥⣿⣷⣴⣄⠀⠀⠀⣠⢠⠄⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⣿⣿⠙⣝⢿⣿⠂⠋⠁⠀⠀⠀⠀⣿⢢⠀⠀⠀⠀⠀⠀⠀⠼⢅⡀
⠀⠀⠀⠀⠀⢸⡿⠇⠀⠈⢑⣿⠂⢠⠀⢰⠀⢦⠀⢹⣇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠶⢤⡦⢄
⠀⠀⠀⠀⠐⢸⡇⠀⠀⠀⠀⠹⣇⢈⣆⣨⣦⡼⢀⣾⣿⣷⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⢠⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠆⡸⠻⠿⠛⠉⠙⠶⣄⠠⣼⡗⠀⣰⠀⢸⠀⡐⢠⢹⠀⠀⠘⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣮⣷⣤⡀⠴⡏⢀⠎⡠⢁⢆⠜⠀⣀⡠⠴⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣙⢿⣿⣿⣿⣶⣧⡼⠂⠁⠹⡇⠀⠸⢟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣻⣷⣬⠛⠛⠛⠛⢷⣦⡀⢸⠇⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣷⡀⢀⢀⡀⡤⣹⠋⠣⠀⠀⠀⠀⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣶⣶⡶⠛⠁⠀⠀⠀⢀⡠⢞⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠁⠀⠀⠀⣠⠴⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣤⡀⠀⠀⠀⣠⠞⢉⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⡶⠛⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠤⠤⠤⠤⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠲⢄⣀⣀⢀⡴⠖⠀⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⣳⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣳⠄⣣⣘⠛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⢹⡆⠉⣿⠄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣾⠃⠀⡿⠈⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠀⠀⠹⢦⣬⠃⠀⠀⡀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠉⠖⣚⣓⣦⡴⠋⠀⠀⠀⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⡠⠂⠼⠀⠂⠀⠀⠀⠀⠠⠤⠤⠤⠈⣀⣰⣾⡛⠃⠀⠹⠀⠀⠀⢀⡾⠀⠀⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡥⠄⠋⠀⠀⠈⠑⠃⠀⠀⠀⠀⠤⡤⠴⡾⢻⠁⣀⠁⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⡀⠀⣀⠔⠳⠄⣀⠘⠀⠞⠁⠀⠰⠷⢀⣱⠄⣷⠏⡘⠁⠀⠀⣠⢶⣤⠊⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠤⠐⠈⠁⠀⠀⠀⠀⠀⠀⡈⢸⡿⣀⢄⠀⢠⣀⣀⠐⢦⠀⠚⣄⠀⢠⡀⢸⡃⠒⠀⣤⠾⢧⢺⡍⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠴⠋⡀⠀⠀⠁⠀⡀⠀⠀⠀⠁⠀⡅⠀⠀⠀⢙⠁⠈⡇⠀⠸⠀⠀⠃⠀⠸⠉⠙⡉⠂⣠⣀⢴⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠇⠀⡖⠉⠰⠈⠁⡤⠃⢠⡄⠁⢶⠈⠑⢲⠒⠀⠛⣉⣁⠜⠶⠁⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠛⠃⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠀⠀⢀⣰⠂⠘⠊⢱⠄⠢⠀⠀⣥⠀⠀⢱⣄⠀⠓⠶⠊⠁⠀⠀⠈⠉⠐⠒⠀⠒⠀⢀⣤⡄⠀⠀⠀⠀⠀⠲⠁
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣴⠉⠆⠞⠂⠘⣤⢀⠘⢙⠀⠁⠞⠀⠠⣄⢀⡀⠀⠀⢀⣀⡈⣿⣿⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠉⡈⠀⠀⠀⠀⠐⠀⠄⠀⡌⠉⣇⠀⠰⠀⡔⠀⠀⠀⠀⠀⠄⠛⢀⠄⠼⠸⠅⡈⠒⠐⢋⠉⠉⡙⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠
⠀⠀⢀⠞⠁⠠⠀⠁⠀⠀⠀⠀⠺⠁⢲⠀⢈⡇⠀⢠⣤⠘⠒⠆⠒⢄⡀⢸⡀⠀⣱⠂⠁⣤⡜⢢⠤⠦⠀⠄⠀⠀⠠⠐⣤⠀⠀⠐⠂
⠀⠀⠂⠠⠂⠀⠐⠀⠠⠂⠀⡴⠇⢴⡏⠸⠃⠀⠒⠀⢠⡀⠠⢆⠀⣸⠁⠀⡏⠈⠁⠐⠰⠀⠰⠃⠀⠀⠀⠂⠴⠄⠐⠦⠁
⠀⠀⠀⠀⠄⢀⠔⠀⠀⠂⢐⡷⠀⢠⡇⢰⠦⠄⠛⠄⠚⡁⠀⢈⡁⠀⠐⠀⠉⠂⠚⢃⣀⡾⠃
⢀⡇⠠⠀⠀⠀⠀⠀⠀⠀⣨⠀⡇⠸⡀⠀⣀⠀⠠⢄⡀⠼⠃⠀⠟⠂⢸⡦⠀⢳⢀⣈⠙
⠈⡀⠀⠀⢀⠀⠀⠀⠀⠀⡙⢰⡇⠰⡇⠀⠙⠀⠀⣨⠀⢀⣶⠀⠋⠀⡊⠀⣴⣀⡀⠁
⠀⠙⠴⠂⢀⠆⠀⠀⠀⠸⣇⠠⣇⠀⣴⠀⠼⠋⠈⠃⠀⠀⣴⡤⣾⠃⣸⢦⡤⠉
⠀⠀⠀⠀⡏⠀⠀⠀⠈⢤⡀⠀⡈⠀⢸⡀⠀⡶⠀⠈⢛⠀⠄⢠⢿⡆⠛⠁
⠀⠀⠀⠘⠀⠀⠀⠀⠁⠈⠁⠆⠉⡶⠘⡋⠀⠰⠄⠴⠉⢙⣤⠂⠉⣿⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡀⣰⠀⠧⠀⡈⠈⢃⣀⠐⠉⠟⡩⠀⠀⠀⡇⢠
⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⠉⠁⣄⢠⡏⠁⢠⡄⠉⠆⠀⠀⠡⡀⠀⠃⣀
⠀⠀⠀⠀⢢⠀⠀⠀⢀⡘⢷⡄⠛⠀⠘⡖⠈⠐⠒⠀⠀⠀⠀⡆⠀⡀⠛
⠀⠀⠀⠀⠀⠀⡄⠀⢀⠑⠈⣴⡄⢀⠋⢱⠆⢱⡆⠀⠀⠀⠀⠱⠞
⠀⠀⠀⠀⠀⠀⠀⢶⡈⠶⠛⠹⣄⡈⢤⢴⠂⢂⡹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠐⢀⡁⠙⠳⢶⡖⢊⠁⣞⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢎⠁⠘⣆⡀⢠⡈⢡⡀⢙⠛⠷⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡓⠰⡌⠉⡁⠉⠁⠙⣀⣳⠚⠢⣜⢻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢾⣀⠉⡆⠁⡔⠣⠀⠲⢔⠀⢱⡈⣷⡂⠀⠄⠒⠂⠀⠒⠒⠰⠦⠔⠒⠋⠙⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠼⠉⢡⠄⠑⣆⠀⡸⢂⡀⠛⣇⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠁⡔⠁⢸⡀⣬⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠉⣞⣉⣁⣰⠾⡁⠂⣁⠠⠀⠀⠀⠀⠀⢀⡀⠀⡒⠲⠦⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠉⠹⠋⠹⠆⠁⠂⡀⠀⢀⠀⠀⠀⠀⣤⡀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⠄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀⠂⠈⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠀⠀⠒⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⣀⡀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠦⠶⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠛⠛⠋⠙⠛⠛⠿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣴⢿⣿⠟⠛⢻⣇⣹⠟⢻⢻⣷⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢸⣷⣴⣿⣌⣁⣼⡿⠻⣆⣀⣼⡿⣷⡀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⣾⡏⢠⠘⣿⡿⠿⢯⣥⣿⣿⠿⣦⣾⣿⡄⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢠⣿⠓⣶⡾⢿⣔⠂⣸⣧⣼⡀⠃⢸⣿⠿⢿⡄⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⢟⡉⢻⣧⣣⣿⣿⡏⣻⡿⢿⡟⢻⣿⣄⣼⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿⢮⣥⡼⠛⢿⠁⢠⢹⣿⣌⣠⣿⢋⡉⢹⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣷⡾⠛⠳⣥⣾⣷⡶⢿⡟⠛⠻⣿⣧⣥⣾⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠸⣧⣋⣠⡟⠉⢉⢳⣿⣄⡃⣀⣿⠿⠛⢿⣴⡿⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠻⣟⣙⣧⣀⣙⣴⣿⣿⣿⠿⣿⡀⠀⣻⣿⠇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠈⠻⣿⣿⣿⣏⢤⠀⣻⣶⣿⣿⠿⠛⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠙⢿⣿⣶⣾⡿⠟⠉⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⡀⠀⠀⠉⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⠶⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠰⠀⠀⠈⣄⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣷⣾⣿⣶⣶⡿⠛⢶⣷⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡯⠐
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣀⢉⡉⠉⠋⣋⡋⣛⡂⠀⠀⢀⣀⣀⡀⠀⠀⢴⣿⣿⣿⠂
⠀⠀⢀⣠⢤⣶⡿⠉⠈⠻⡆⠛⢹⣿⣿⣿⣿⢿⣿⠋⠃⠀⠀⠻⡿⠁⢰⣶⣿⣿⣿⣷⠀⠀⠀⠻⠛⠀⠀⠀⠀⣿⣛⡋
⡀⠔⠛⠉⠃⠀⠀⠀⠀⢸⠀⠀⠀⢻⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⠂
⠀⠠⡄⠀⠀⠀⡠⠀⣀⠞⢠⠆⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⢴⣿⢓⠂
⠀⠀⠀⠈⠒⠋⠀⠉⠀⣶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⡤⠴⠦⠒⠒⠀⠀⠁⠈⠁⠀⠉⠂⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡖⠂⠴⢤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠁⠈⠉⠒⠰⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⠶⠿⠻⡟⠻⠿⠿⠿⡿⣿⣷⣶⣶⣦⣤⣄⣄⡀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠛⣩⣍⢳⠀⠀⠀⠸⢃⡿⢀⡷⠀⡇⡿⢸⠀⠀⠀⠈⣅⢹⡙⢛⠻⠷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡟⠇⣌⢋⠌⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠁⠀⠀⠀⠀⠀⠃⠸⠁⡼⠃⡷⢸⡌⣋⡓⠢⢤⣀⣀⠀⣀⣀⣤⣞⠵⣊⠁
⠀⠀⠀⠀⢀⣤⠾⠿⠟⠁⠀⠈⠙⢾⡟⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠁⠃⠃⠀⠀⠀⠀⠀⠶⣤⢙⣿⣛⢁⣀⠀⠀⠀⡞
⠀⠀⢀⠔⠉⣴⡂⠀⠀⠀⠀⠀⠀⠀⢻⡀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡅⡈⣿⡟⠉⠀⠀⠀⠘
⢀⠔⠁⠁⠀⠛⠛⠀⠀⠀⠀⠀⠀⠐⢸⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⢂⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠠⠃⠘⠁⣿⡇⠀⠀⠀⠈⠄
⠺⢄⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣠⢟⠇⠈⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠀⠀⠀⠀⠀⢘⣿⠃⠀⠀⠀⠀⠈⠆
⠀⠀⠛⠆⠤⣀⣀⣀⣠⢞⣁⡤⣞⣥⣫⡆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠂⠀⠀⠀⠘⠻⣏⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⢫⣠⠤⠿⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠛⠀⠀⠐⠂⠀⠀⠀⠀⠈⠑⠦⣀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠲⠶⠤⠤⠤⠤⢀⣀⣀⣀⣠⠤⠤⠠⠔⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣀⠭⠭⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣁⠀⢰⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣇⣠⣽⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡟⢿⣿⣿⣋⠀⢈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢿⡿⠿⠋⠀⠀⠉⠉⣿⣿⣿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠻⣷⡼⣷⠀⣄⠀⠀⠃⢸⣿⢿⣿⣟⣹⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠛⣶⡄⠈⢷⡾⠷⣼⣳⣦⣴⣾⣿⠀⢷⡿⢿⣍⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⣿⣿⣏⣽⠆⠀⢦⣴⣿⢿⣿⣽⣧⣸⡆⢸⣧⣾⠟⠁⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⣤⣿⠛⣶⡄⠈⡙⣧⣴⣿⡟⢿⣈⡇⢸⣿⣿⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣿⣀⣿⡉⣸⣿⡇⠀⣿⠛⣯⣸⣷⠾⡟⠁⢸⣿⠉⠳⢶⣿⠂⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⣿⡛⣿⣁⣽⣿⡅⠀⣹⣿⢿⡏⣻⣠⣿⠆⣈⣿⠂⠀⠀⢹⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡿⠿⠛⠉⠉⣿⠇⠀⣽⣷⣾⣿⢿⡋⣧⠀⣿⣿⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠡⠀⠀⠀⠀⢰⣿⣏⠀⠘⣫⣹⣧⣼⡿⡟⢀⣿⡏⠀⠀⠀⢸⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⣿⣿⣿⠀⣼⣿⣿⣏⣿⣷⣿⠚⣿⡇⠀⠀⠀⡼⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠈⠙⢯⢀⣿⣿⣿⣿⣿⣿⣿⢃⡿⠁⠀⠀⣞⠁⠀⢀⣼⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣙⣆⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟⠈⢁⠀⠀⠀⢁⠴⢚⡌⣷⣀⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠤⠅⠒⠂⢉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⠳⣄⠀⠀⠀⠈⠛⠛⠋⢁⣠⣴⣿⣦⣄⠀⠀⠂⣈⣈⣽⠿⢓⠊⡟⠀⢀⣀⡠⠤⠔⠒⠛⠋⠁⠀⠀⠀⠀⠀⠂⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⠳⣦⣍⡀⠄⡀⠀⠀⠰⠿⠿⢿⣿⣿⣿⣆⣰⣿⡿⡟⠋⠁⢠⡔⠲⠛⠉⠀⠀⠀⠀⢀⣼⠖⠂⠁⠀⠀⠒⠶⣾⣇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣬⣿⣾⣿⣿⣷⣦⣄⣀⠀⠀⠀⠰⣆⠘⠃⠈⣿⠟⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⣀⣀⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢠⠖⣡⡤⠘⣻⡏⢀⡃⠀⠀⠀⠀⠰⠴⠒⠚⠉⢁⣼⡟⠀⠀⠀⢀⢀⣤⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣶⠾⠟⠁⠘⣿⣁⣠⣤⠶⠖⠉⠁⠀⠀⠀⠀⣠⢋⠛⡁⠀⠀⢀⣴⣿⣾⡏⠀⠀⠀⠀⣾⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⢾⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣤⣤⣤⣥⣶⣿⣿⣿⣿⣿⣆⠀⠈⠈⠀⠀⠁⢀⡈⠏⠀⣇⠀⠀⠈⢿⣿⢻⡟⠀⠀⠀⠀⣿⣶⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣴⣮⣭⣭⣿⣿⣿⣿⡿⠁⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠄⠀⠀⠀⠉⠀⠀⠰⣆⠀⠀⠀⠉⠋⠁⠀⠀⢀⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠸⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⡈⢻⣿⣿⣿⣿⣿⣿⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢉⣽⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠉⢷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡶⠿⠿⣿⣿⡏⠀⠀⠀⠀⡀⠀⠀⣀⠀⡠⠀⠸⣿⣿⣿⣶⣦⠘⣄
⠀⠀⠀⠀⠀⠀⢠⡟⠙⠿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⣤⡜⣦
⠀⠀⠀⠀⠀⣠⠟⣠⡀⠼⣿⣿⣿⡇⠀⠀⢠⠔⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣶⣆⡘⣆
⠀⠀⠀⢀⡜⠁⠀⢿⣿⣷⢿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⣻⠭⠯⠁⢸
⠀⠀⠀⡼⠀⠀⠀⠀⠙⠋⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡍⠐⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⡖⢾⣏⣉⣉⣩⣉⣉⣉⣿⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⢀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢸⣆⣠⣿⣤⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⠔⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⣷⡄
⠀⠀⣼⠋⠃⠀⢸⣤⣾⣦⣶⣿⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⣿⣿⣿⣿⣿⣿⣥⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢄⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠘⠆⠀⣼⣾⢟⡟⠈⣹⠞⡿⢿⣧⡤⠤⢤⣄⡀⠐⣄⠀⣠⣟⢧⣿⡙⣼⣻⣿⣿⡿⡃⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⣾⡄⠀⠀⠀⠀⠀⠸⣿⣤⣤⣾⣿⠃
⠀⠀⠀⠀⠘⢿⣿⣧⣷⠞⡿⢿⠟⠿⣿⣧⣀⡴⠟⠁⠀⠘⠻⢿⣛⢟⣧⣟⣀⣙⣽⣿⣧⡠⡀⠀⠀⠀⠀⠀⣠⢾⠁⠘⡀⢣⡀⠐⣦⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⣶⣂⣹⣿⣿⣼⠅⠛⠖⡉⣨⣷⣶⠀⠀⠀⠀⠀⠀⡄⠐⢻⣿⣾⣯⣼⣿⣿⠿⠿⠀⠙⢮⡆⠀⠀⢀⡾⠁⡇⡆⠀⡇⠱⣿⣆⠈⠳⠦⣤
⠀⠀⠈⠀⠀⢀⣿⣿⣾⣷⣄⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⢠⢀⡟⢻⡿⣿⣿⠛⣿⠇⠀⠀⠀⠈⠁⠀⠐⠋⣀⣼⢀⡇⣸⠇⠀⠙⠻⢷⣦
⠀⠀⠀⠀⢠⣆⢀⠟⠁⠸⡿⠛⡇⠈⠉⠙⠛⠀⠀⠀⠀⠀⢀⣀⠞⠰⢋⡟⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⡾⠁⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠐⠃⠀⡀⢔⣠⣶⣲⣾⠆⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠓⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣄⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠟⠛⣋⣡⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣤⣾⡿
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡣⡄⣦⠋⠀⣀⣰⠃⠀⠀⠀⠀⠀⣀⠀⠀⠙⠿⣿⣉⡁⠔⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢒⡤⢙⣛⣵⢿⣦⣶⡽⠛⣠⡟⠀⠀⠀⠀⠉⠙⡄⠀⠀⠀⢀⡠⡄⠀⠀⠀⠀⠀⢀⣸⣿⣿⣶⣄⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠌⣻⣦⢿⡉⠛⠛⠋⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⢠⢫⠊⠃⠀⠀⠀⠠⠶⠻⠟⠟⠛⠿⠿⠷⠤⠾⢿⣿⣿⠛⠛⠉
⠀⠀⠀⠀⠀⠱⣠⢤⣄⡀⠀⡀⠀⠀⠀⠀⠙⢼⠃⠹⠀⠀⠀⢠⣶⣦⣄⡀⠀⢸⢸⡀⠀⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⣿⠛⣾⣿⣷⡟⢧⡙⢻⡆
⠀⠀⠀⠀⠀⠀⣧⠀⢹⣿⠛⠁⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⠀⠈⠛⢷⣄⠘⢸⣷⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣼⢰⠿⢹⣹⡌⠙⠳⠿
⠀⠀⠀⠀⠀⠀⠙⡇⢴⣏⠁⠒⢤⣤⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⢾⣿⣿⣧⣄⣀⣀⣀⠀⠀⠀⠀⠀⠘⠊⠀⢰⢃⡟⠀⢸⡏⠇
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⣫⡴⠶⠋⠀⠀⠉⠀⠀⠀⠉⢻⣇⠀⠀⠀⠀⠀⠰⣿⡜⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣤⠾⠟⣡⡾⠷⠋⣠⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢛⡴⠚⢁⣤⡴⢻⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣿⣿⣷⣾⢟⣏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⢿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⢠⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⠞⠉⠨⣰⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠰⣧⠀⠀⠙⡂⠀⠛⢏⠑⠛⣠⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⣶⣿⣿⣿⣶⣴⣻⣜⢺⠏⠀⠠⢄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⠷⢤⣿⣯⣿⣿⣿⣿⣆⣻⣿⣿⣤⡟⠑⠛⠉⠀⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣄⡀⠛⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⣤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣗⠀⠀⠑⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢈⢣⣶⣿⢿⣿⣿⣿⠟⢿⣧⣀⣀⣴⡿⠏⠉
⠀⠀⠀⠀⠠⠄⠒⠲⠖⠲⢦⠤⣤⣄⠀⠀⠀⠀⣀⡀⣾⢁⡀⢸⡿⠋⠻⣧⠀⠀⠉⢋
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠳⣤⡽⠙⢣⡀⠀⢿⡄⠘⠆⠁⣿⡇⠁⢀⠿⢷⡁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⣹⠟⢦⡈⢿⡄⠀⠙⢸⡇⢀⠈⠀⠀⠀⠁⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣤⣤⣠⣵⠀⠻⣆⠀⢸⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣜⣷⣼⡄⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣉⡉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⠟⠛⠛⠋⠛⡻⠟⣿⣿⣿⣍⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠉⠀⠀⠀⠀⣴⣶⣿⣿⣾⣿⣿⣿⣿⣼⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠃⠀⠀⠀⠀⠀⣴⣷⣾⣾⣿⣿⡿⠈⢿⣿⣿⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇⠀⠀⠙⣿⣗⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢻⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⣇⠈⠙⢟⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠎⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣡⢣⢊⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⢰⣿⣱⣱⣳⣣⢪⢈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⡍⠻⡉⢷⢩⡝⣝⣫⢽⢿⣿⣷⣷⣣⡣⡠⢊⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠁⠀⠀⠀⠄⣱⡌⢏⢺⠣⡦⣩⢉⠻⠾⣆⣹⠺⢽⣷⣵⡷⡱⣪⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠄⢸⡀⣏⢳⢣⡷⡹⣙⢛⠷⠾⣽⣝⠳⡦⣟⠻⣽⡵⡣⣺⢧⡀
⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢽⢹⠉⡞⢾⢣⢇⢇⣏⡛⠲⠼⣽⣻⡳⢶⣿⣎⠺⣞⣵⠍⣻⡄
⠀⠀⠀⣠⠾⢏⠁⠀⠀⠀⠀⢰⣿⣟⣿⡄⠀⠀⠀⠀⠀⠈⢰⡉⡏⢱⢻⠜⡼⡨⡸⣙⠟⢳⠷⣧⣝⡛⠾⢾⡟⣎⡿⣟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠈⣿⡗⢦⠠⡀⠀⠀⠀⠀⠳⠿⠛⠀⠁⠀⠀⠀⠀⣸⠁⡏⢹⠉⠞⡶⡧⡴⡧⣍⣹⡉⢰⠶⢶⣸⣙⣿⠲⠌⣽⣹⣷⣄⡀⠀⠀⠀⠀⣠⣴⡾⢟⣛⡿
⠀⠀⢠⣿⠓⠻⠧⡔⢄⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠸⣁⡏⣸⢉⠆⡖⡷⡷⡷⣧⣬⣼⣙⣹⣿⢿⠶⢿⢤⣿⣓⡲⣷⠽⣭⣝⡗⡶⢶⣻⡿⣓⠮⠥⡾⠃
⠀⠀⠀⡏⠙⢯⣜⣹⠷⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠰⡃⡜⣰⢋⠏⠟⠗⡗⠗⠶⠶⢾⣬⣭⣽⣩⣛⣻⡓⡶⢮⢭⣭⣟⣚⣺⢿⢥⣵⣚⣿⠥⠤⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠳⠶⣤⣄⣀⣀⣀⣤⣤⠄⣎⠰⡥⡛⡞⢛⣛⣛⣛⣋⠙⠛⢛⢛⣶⡾⢻⢽⣽⣭⣿⢿⣻⡺⢗⣛⣽⣹⢨⢽⣚⣸⣟⠯⢽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠾⠠⠞⡇⠷⣦⣜⣯⣭⣭⣭⡴⠶⢶⣛⣋⠁⢻⠟⢛⣛⣈⣂⣬⠶⢿⠀⠀⠉⠓⠚⣰⢟⢴⡩⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠧⣮⣵⣠⣬⣩⣀⠂⣀⣐⢤⣯⣤⠴⠶⠒⠚⠉⠁⠀⢻⡄⢢⡏⠀⠀⠀⠀⠀⠙⢿⣗⢭⡲⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⢉⣇⢂⣓⡦⣄⠀⠀⠀⠀⠀⠀⠀⠙⠾⠃⠀⠀⠀⠀⠀⠀⠀⠛⢧⣉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣞⣿⢜⣗⢭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⢿⣮⣪⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⢸⡄⢸⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠓⠀⢸⡇⡸⡇⢸⡇⠀⠉⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣄⣀⠀⢸⡇⡁⡇⢸⠇⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⣟⣒⣶⣶⣒⣛⣻⣯⡭⠶⠿⠶⠶⠾⠶⢿⠇⢸⡇⠃⣇⣼⠀⠿⠿⠿⠯⣭⣿⣛⣒⣶⣶⣒⣚⣿⣫⣭⠽⠿
⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢠⠤⠄⠀⠀⠀⠈⣧⡨⣾⡿⠀⡀⠀⠀⢀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⡿⢯⠀⢿⡀⠹⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡾⢧⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⡼⠁⠈⢷⣘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣭⡝⢷⡀⠀⣼⢡⢮⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟⠀⢻⢸⡇⠀⣿⣿⡀⢻⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣧⣀⡿⣸⠇⠀⠘⣯⠳⠾⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢮⣉⡽⠋⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠤⢤⣾⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⡠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠋⠁⡰⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢀⡞⠀⠀⠀⠀⠈⠄⠀⠐⠊⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠒⠒⠶⠦⠤⣄⡀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠴⠚⠉⣀⣠⠤⠴⠶⠦⠤⢤⣀⣀⠈⠑⢦⡀
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠚⢁⡤⠖⠋⢁⣀⡠⣤⣤⣤⣀⣀⠀⠈⠙⠦⣄⠙⢦⣄
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣠⠎⡠⠚⢁⣤⡒⡿⣫⠉⡷⢡⡄⣣⡞⢉⠙⡖⢄⡀⠈⠳⣄⠙⢦⡀
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠃⡜⠁⣴⢋⡄⣹⠱⠏⠜⡇⠛⣡⡟⢰⡟⣰⠗⠋⡿⢦⡀⠘⢆⠀⢳⡀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠼⠀⡸⡇⠺⡁⢻⣤⣎⢀⡷⣶⠁⣇⣀⠘⠏⠸⢃⡥⠖⢷⡄⠈⢧⠀⢻
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡃⠀⡄⢰⠇⡷⢶⢀⠎⣤⢸⡎⣴⠌⣧⢊⡉⡿⣇⣠⡞⣰⠆⡸⢹⡄⠘⡆
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠁⢠⠁⢸⡟⣰⡆⡿⠘⠋⣾⠀⠉⢸⡇⠛⣡⠟⠙⣶⡀⢀⣴⡷⢦⡇⠀⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⠃⣿⠀⢸⡀⠸⡇⠈⣐⣷⣤⢋⡜⠷⣩⠼⡧⢺⠃⡿⢠⣫⠟⢳⣾⣴⠘⡇⠀⡇
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠁⠀⠀⡴⠁⠀⢸⠀⠘⡇⠀⡟⢺⢋⡄⢸⡎⣴⢀⡇⢠⡘⣾⠀⢀⢆⠏⡶⢸⡇⠈⢸⡇⠀⡇
⠀⠘⢦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⣇⠀⢻⡀⠸⣾⠘⠃⣾⣁⣠⡶⢾⡌⠃⣏⡷⠛⣾⡀⣠⣯⣿⣤⢾⡇⢰⠁⢰
⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠸⣆⠈⢧⡀⢻⡤⢾⠋⣝⢶⡀⣦⠻⠦⠿⢡⡆⣏⡝⣩⠄⡇⢀⡞⢀⠇⠀⣾
⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠳⣄⠙⢦⡱⣍⠈⢿⡌⠀⣦⡆⠀⣠⡟⠀⢀⣞⡴⠋⣠⠏⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡈⠳⣄⠉⠛⠓⠻⠼⢶⣿⣧⡴⠥⠷⠞⠋⢁⣠⠞⠁⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣙⠒⠲⠤⣤⣀⣀⣠⡤⠤⠴⠒⠚⠉⣀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⣤⣤⣤⡤⠴⠖⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠠⣿⠤
⠀⠀⠀⠀⠀⠀⠀⢰⠚⢦⢰⡖⣲⠀⠰⡇⠐⣇⠀⡇⠸⣛⠆⠸⠻⠆⠀⠀⠐⡟⠀⣦⠖⢠⡦⣆⠀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠀⠉⠈⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢞⣩⠭⣭⣙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⡟⠁⠀⠀⠙⣆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣾⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⡀⠀⠀⠀⢀⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢷⣄⣀⣠⠞⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡌⠉⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠶⣤⣤⡟⠀⠀⣀⣤⠶⠚⠋⣩⡴⠞⠛⠛⠛⠲⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣾⡥⢾⡋⠉⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⢸⠇⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⡇⣀⢠⡏⠀⠘⣧⠀⢀⣀⡀⢿⣄⠀⠀⠀⠀⣀⣴⠟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⠛⠛⠓⠶⠶⠦⣤⣶⣶⣛⣋⠉⠀⢸⡟⠛⢹⡷⠶⠖⠛⠛⠋⠉⠙⠷⣭⣛⣓⣚⣛⣩⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⣿⠁⠀⣿⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡟⠀⠀⠀⠉⠉⠙⠛⠛⠲⠶⠶⢤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿
⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠒⠶⠶⢤⣤⣤⣀⣀⡀⠀⠀⣼⠃⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⡿⠀⣼⢧⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠞⠁⠀⠀⠀⠈⠉⠉⠛⠛⠓⠶⠶⢤⣤⣤⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⢰⣤
⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣲⣤⡀
⠀⠀⢀⣠⣿⣿⣿⣧⡀⠀⣀⠀⠀⠀⠀⠀⢰⣫⣾⣿⣿⣦⣤⣶⣶⣶⣾⣿⣿⣿⡿⠿⠟⠻⢿⡿⣿⡿⠂
⠤⣚⣽⣿⣿⣟⠉⠉⠛⠿⠿⣿⣷⣿⣿⣿⣿⢿⡿⠿⠿⠿⠟⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠤
⠀⠀⠁⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠒⠒⠚⠒⠒⠒⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣀⠀⠀⠀⠀⣠⣀⠀⠀⠸
⠀⠀⠀⠀⢀⣴⠾⠻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⣏⠀⠀⢈⠃
⠀⠀⣠⠴⠟⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠠⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡆⠀⠀⠀⠛⠣⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠈⠀⠀⢤⠠⠠⠄⠀⠀⠀⠀⠉⠒⠛⠃⠀⠀⠀⠀⠀⠀⠐⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣤⣄⣀⣀⠤⢔⣒⣶⣞⣁⣙⣷⣲⣒⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⡿⢛⣭⣶⣿⣏⣿⠿⡿⢿⢿⡿⡻⣋⣙⢷⣾⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⣡⣾⣿⣿⣿⠇⢸⠁⠀⢠⣳⣿⣿⡇⢧⣬⣿⣽⣿⣿⠟⣓⣚⣷
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢋⣾⣿⡟⢻⢘⣿⠠⣿⡀⢰⠀⣿⡇⠀⠙⠓⠀⠀⠀⠸⣡⢞⢕⣭⣤⣴
⠀⠀⠀⠀⠀⠀⢠⣿⣿⢃⣿⣿⢿⠁⠋⠁⠁⠈⢙⣿⡛⢧⡘⢯⣄⠄⣀⣀⣠⠴⠞⢛⡷⣋⣩⣭⣽
⠀⠀⠀⠀⠀⠀⢨⣿⡟⣼⣿⡟⡀⠀⣠⠤⢶⣶⣿⣧⣍⢻⡿⣦⣬⣭⣭⢭⣶⣾⣛⣛⣿⡽⠋⠀⠃
⠀⠀⠀⠀⠀⠀⠘⢿⡇⣿⣿⠃⣴⣾⣿⡿⠁⣀⢞⡿⠀⠀⠱⢬⣉⠩⢒⣋⣤⣶⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⣞⣉⣿⣷⣶⣯⠷⠛⠁⠀⡀⠀⢠⣶⣿⠿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠃⠈⠈⠉⠉⠀⠀⠀⠄⠀⠀⣡⡿⠋⠁⠀⠈⢿⣇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣀⣤⢖⣿⠀⠀⠀⡀⠀⢰⠃⠀⣀⣠⡤⢶⣛⣻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢟⣵⣟⣿⣿⡀⠀⠂⠀⠈⠿⠛⣿⣿⡿⢛⣻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠈⠙⢿⣿⣿⣾⣿⣷⣴⣄⣴⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⣿⡿⠛⠛⠻⣿⣿⣿⣿⣯⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠈⠳⣝⢿⣿⣮⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⠴⣽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠛⠉⠉⠉⠙⠛⠒⠤⣄⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠖⠋⣁⡤⠴⠒⠚⠉⠉⠛⠓⠲⠦⣄⡀⠙⢦⡀
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⠀⡠⠞⣡⠔⠋⣁⣠⡤⢶⣲⠖⢶⠒⡶⢤⣀⠀⠉⠲⣄⠙⢧⡀
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡞⢡⠎⢀⡴⠛⣴⢣⡖⣸⠇⡿⢠⡏⣠⡄⣟⣙⣦⡀⠈⠳⡀⠙⣆
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⢠⠃⢠⡏⣼⠢⣏⢈⠔⠹⣆⡠⢻⠀⠛⣴⠋⣤⠘⢙⣦⡀⠙⣄⠘⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡟⠀⡁⢠⡟⣇⡡⠂⠼⠛⣦⡜⣉⢧⢈⠷⢦⣿⠀⢡⠟⣉⢈⢷⠀⠹⡄⠘
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⢿⠇⠘⠃⢸⣰⢉⠹⡜⢰⠇⡾⠸⠏⣼⢣⡞⣠⡽⣞⡏⠘⢋⣜⣀⡇⠀⣇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢠⡏⣼⠀⢸⠀⢸⡇⠛⢸⣇⣀⠎⣧⣠⠎⣫⣀⡴⢣⡆⡸⢧⣴⣩⢋⡈⣿⠀⡟
⠸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢹⠀⢸⡆⠈⡧⢴⠟⡉⢧⡌⡄⢹⡏⠙⢧⡇⠈⢠⢳⢣⡀⣿⠘⠃⡿⠀⡇
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠘⡆⠀⢧⠀⢧⡌⠸⠃⣿⠘⢁⣞⣇⠿⠸⢧⡴⢇⡇⠙⣐⢹⡀⣸⡇⢠⠇⢀
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢹⡄⠘⣄⠘⣧⣀⣼⠛⢶⣹⢠⡽⣦⣸⡏⣠⢸⢳⠞⡋⢹⠉⣱⠃⡼⠀⣸
⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⢦⡈⢧⡘⢌⠓⠹⣎⠃⢱⣸⠃⠉⣸⠏⠈⣡⢋⡴⢁⡼⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠶⠶⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠙⢄⠉⠒⠷⠦⣏⣦⣤⣿⣀⣜⣹⡤⠾⠋⢁⡠⠋⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣙⠲⠤⢤⣀⣀⣈⡉⣁⣀⡀⠤⠔⠚⠉⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⢤⣬⣭⣍⣀⣀⣤⠤⠶⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡧⠄
⠀⠀⠀⠀⠀⠀⠀⡶⠒⡆⣴⠒⡆⠀⢾⠀⢾⡀⣾⠀⣏⡇⠰⠞⡆⠀⠀⠐⡗⠀⣦⠖⢰⣠⡆⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣟⠉⠁⠈⠉⠁⠀⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣻⣛⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣿⣿⣿⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢆⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣈⢂⡂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠸⣿⣷⠰⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⢠⣿⠀⠀⠀⠈⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⢀⣐⣾⠏⠴⠀⠀⠀⠻⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢸⣿⣛⡄⠛⣿⠛⣤⠂⠀⠀⠀⠔⠛
⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⣠⠀⠀⢨⠀⢈⣡⣾⡷⠾⢫⠄⣛⠀⠀⠀⢀⣤⣶⡾⢷⠆
⠀⠀⠀⠀⠀⠠⠁⢀⣄⣀⣨⡄⢸⣿⠶⠛⠉⠅⠂⠁⢁⠈⣧⢠⡀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡿⢻⡿⢿⣢⠋⠀⠐⠂⠀⠀⣀⠈⠠⢀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠎
⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠁⠀⠀⠊⠧⡴⢋⣍⢠⡰⣚⠥⠚⠂⠀⠁⠀⢀⣀⣀⣀⡀⠀⠂
⠀⠀⠀⠀⢛⣄⣀⠀⢠⡄⠉⠀⠀⠂⢰⠛⠠⢁⡸⠛⠁⠙⢻⣿⡿⠓⠈⠛⠉⠉⠉
⠀⠀⠁⢡⡿⢿⡿⠂⠉⠀⠀⠀⠀⢤⡃⠈⠀⣿⢀⠀⠀⠀⠀⠥⠐⠀⡄⢀⠇
⠀⠀⠀⠀⠀⠐⠄⣼⠁⠀⠀⠀⠀⠘⣿⡓⠺⣡⣦⡤⠀⠰⣄⣷⣄⠶⠛⠁⠈⡀
⠀⠀⠀⠀⠀⢀⣠⠉⠆⠀⠀⡀⠐⠀⠘⢳⠄⠈⠙⠁⠀⢠⣿⡿⢉⣀⠀⠀⠈
⠀⠀⠀⠀⠀⣻⣯⡞⠀⠀⠀⠉⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠋⠉⠂⠀⠀⣰⣶⡆⠀⠈
⠀⣀⠀⠀⠀⠉⠛⣿⠈⢄⠀⠁⠑⠂⠀⠀⠈⠷⡀⠀⠀⠀⠀⠀⠀⠀⡃⢀⣰⡟⠛⠁⢀⣴⣀
⠀⠈⠀⠀⠀⠂⠐⠈⠃⡀⠈⠈⠠⠀⠃⠐⠀⠀⠈⠠⠄⡀⠀⠀⠀⠀⠿⣿⣿⣄⢀⣶⣾⣿⣿⣶⠀⠀⠀⠀⠀⣠⣤⣴⣇⣀⣴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⠀⠀⠀⡠⠀⠂⠀⠀⠀⠀⠐⠄⠙⠆⢤⢀⡀⢈⠙⢛⣿⡽⢿⣿⣿⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⠿⠋⣤⣾⣿⠛
⠀⠀⠀⠀⠀⠀⠂⢸⡃⠀⠀⠊⡌⠑⠀⠀⠐⠀⠀⡘⠐⠀⠀⠀⠀⠈⠀⠀⠄⡈⠛⠠⣾⣿⣿⣷⣤⣾⣿⣿⡿⠛⢻⡿⠿⠀⠛⠯⣉⠙⡅
⠀⠀⠀⠀⠀⣬⣷⣼⣿⠀⠮⠀⢤⣀⠐⠀⠀⠄⠀⠁⠂⠀⠀⠀⢀⠀⢀⠀⠀⠁⠀⠀⠸⢿⣿⣿⣿⣿⣿⠿⢶⡗⠀⠠⠀⠐⠀⢰⠋⠀⡄
⠀⠀⢰⣦⣤⠈⠙⠛⠉⠀⠀⠀⠀⠉⠳⠀⠂⢀⠐⠁⠀⠀⠀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣿⠀⠀⢀⠀⠀⠀⣀⣢⣤⡚⠁⢠
⠀⠀⢂⣽⠇⠀⠀⠀⠀⠀⠀⢀⢀⡄⡀⠀⠠⠀⠀⠀⠀⠀⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡏⣀⢐⣀⠀⠒⠀⠀⠀⠉⠉⢉⡗
⠀⢰⣿⣿⡁⠀⠀⠀⠀⠀⠀⢠⣾⣿⣄⠂⠐⢤⡉⠈⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣿⢻⠇⣀⣼⣿⣯⣶⣶⡄⠀⢀⣴
⠀⠈⢉⣿⡇⠀⠀⠈⠀⠀⢤⣿⣿⣿⣿⣷⣔⠸⡇⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⠀⢹⣿⣿⣿⣯⢾⣿⣥⣿⠿⠇⠈⠉⠶⠊⠁
⠀⠀⠈⠙⢿⡄⠀⠀⠀⠄⣀⠛⢷⡌⢻⣿⣿⣿⣇⠀⠁⠀⠀⠀⠀⠀⢐⢶⣧⠀⠀⢄⠀⠀⠀⠀⠈⢡⣿⣿⣥⣴⠷⢾⡅⠤⢄⡀⠀⠀⠈
⠀⠀⠀⠀⣸⣿⣅⡀⠀⠀⠀⠀⣆⡻⢸⣿⠟⢀⣿⣷⣤⡆⠀⠀⠀⠈⠚⠛⠃⠀⠐⠀⠀⢀⡀⠀⠀⣠⣶⣯⣵⠶⠾⣛⠁⣀⠀⠙⡄
⠀⠀⠀⠀⠙⠛⣿⣷⣄⢀⠀⣸⣿⣶⢟⣵⣶⣿⣿⣿⣿⣿⣄⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠀⠀⠘⢁⠀⠈⠃⠀⡇
⠀⠀⠀⠀⠀⠀⣿⡿⠻⠿⠧⠟⠘⠋⠀⢡⡶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⣀⣦⣤⣠⣤⣴⡾⠿⠿⠃⠀⠀⠀⠀⢀⠀⠀⣀⡴⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠿⠷⠎⢿⡏⢉⡿⢏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⣀⠀⠀⠀⠴⢿⣿⣿⣯⣹⡛⢿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠉⠉⠹⣿⣁⡼⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠁⠀⠀⠀⠀⠀⠙⢿⣤⣶⣶⣾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⢬⣟⣛⡛⣛⡛⣛⠛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣄⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠈⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠋⠙⠛⠛⠛⠛⠛⡟⢻⢛⡟⣻⢻⡟⣻⢟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠃⠞⢰⠃⡞⣱⢟⣽⢿⡿⣿⣿⣿⣿⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢃⠜⢡⠟⣽⢋⣿⣿⣿⣿⣿
⢠⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀⠈⠐⠁⠞⣡⢟⣵⢏⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠀⠀⠙⢦⠀⠀⠀⠁⠞⡱⢋⡴⣿⣿
⢸⣿⡏⠉⠉⠉⠉⠛⢻⣿⣿⣿⣿⣷⣶⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣏⡁⠙⠻⣿⡿⠁⠀⠀⠀⢈⡇⠀⠀⠀⠈⠠⢋⡴⣿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⣿⣶⣦⣤⣀⣤⠴⠶⠶⠶⠶⣤⣤⣀⠀⠀⠈⣿⣦⣄⣀⣉⢳⣦⣤⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⠝⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡶⣾⠏⣹⡏⠘⠛⠦⠀⠰⡻⠂⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠉⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⡼⡝⢁⡿⣄⠛⠇⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⢀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢶⣠⣆⣴⣴⣾⡇⠀⠀⠀⢀⡼⠃⠀⠀⠀⡼⠀⠁⣸⠃⠘⣿⣶⠖⣼⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣠⡞⡀⠀⠀⠀⣸⢃⡀⠀⡏⠑⠾⢿⣡⠞⠁⣼⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣾⣿⡇⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢸⡏⠉⠉⠹⣿⣁⣠⣴⣿⣟⠄⠀⠀⠠⡿⠛⠻⠀⡇⠀⣠⠞⠁⠀⣸⣇⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠈⢳⡄⠀⠀⠈⠀⣼⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣧⠞⣡⣄⠀⣰⣿⣿⣅⠀⠈⢿⣦⣀⡀⠀⠀⠀⠀⠀⢸⣿
⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣠⣦⢠⡇⠀⠀⠀⢰⣿⣿⣿⣯⠂⠀⠀⠀⠀⠀⢀⡼⠃⠚⠉⢹⣴⣿⣿⣿⡷⠀⠀⠀⢻⡝⠻⣶⣄⡀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣿⣿⣾⡇⢀⣀⡀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⢠⠏⠀⣞⢻⡓⣾⣿⣿⣿⣿⣿⣒⠀⠀⠀⢿⡀⠈⠛⢿⣷⣄⣿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠻⡇⠘⡏⢹⠈⢿⣿⣿⣿⣿⡳⡀⠀⠀⣰⠋⠀⠀⢹⡈⣇⢸⣿⣿⣿⣿⣿⣷⣅⠀⠀⣘⣧⠀⠀⠀⠙⢿⣿⣿
⠀⣿⡇⠀⠀⢰⣄⠀⡀⠀⠀⠀⢸⡇⠀⡁⠀⢹⠈⡇⢸⣿⣿⣿⣿⡧⢡⠂⣰⠃⠀⠀⠀⠈⣧⢹⡀⢿⣿⣿⣿⣿⠟⣣⠜⠋⠉⢹⡄⠀⠀⠀⠀⣽⣿
⠀⣿⡇⡀⢶⣬⣿⣿⣿⣆⣰⣆⣼⢳⠀⡇⠀⠈⡆⢱⠀⢿⣿⣿⣿⣿⣧⣾⡟⡖⡀⠀⠀⠀⢹⡌⣇⠸⣿⣿⣿⢏⡾⠁⠀⠀⠀⠀⢷⠀⠀⠀⠀⣿⣿
⠀⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣷⣠⠀⢳⠘⡇⢸⣿⣿⣿⣿⣿⣿⣾⣿⣼⡌⣆⠀⣴⣧⢸⣤⣷⣙⢿⣿⣀⣠⣤⣤⠤⢤⣬⣧⡀⠀⠀⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡆⢸⣿⣶⢘⡄⢻⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠿⠾⠿⣿⢋⣥⡀⠉⠻⠟⠉⠁⠀⠀⠀⠀⠀⠈⠻⣄⠀⣿⣿
⠰⣿⣿⣿⣿⣿⣿⠹⠟⣹⡿⣿⣿⣿⡽⣆⠻⣿⣿⣧⠸⣶⡏⠉⠉⠀⠀⣀⣀⣀⣀⣀⣀⣠⡴⢟⣿⣿⠟⠀⠀⣠⢤⣤⣴⣶⣶⣤⣄⣀⡀⣿⠀⣿⣿
⠀⣿⣿⣿⣿⣯⠙⠀⠀⠀⠸⠛⣿⣿⣿⠾⠷⠞⠿⠿⠦⣿⡙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡩⣔⣿⣿⣵⣤⡶⣏⠙⠓⠶⣭⣛⠿⢿⣿⣿⢏⡿⠀⣿⣿
⠀⣿⣿⡷⠈⠁⠀⠀⠀⢀⣤⠾⠋⠁⠀⢀⣀⣒⣃⡀⠀⠀⠉⠛⠻⠿⠿⠿⠿⢛⣩⣶⣿⣾⣿⣿⠉⢷⠘⣇⢹⡄⠀⠀⠀⠉⠛⠻⠶⠖⠋⠀⠀⣿⣿
⠀⢻⣿⠀⠀⠀⠀⠀⣰⠟⢁⣀⡴⠖⠛⠋⣿⠉⠉⣿⣿⡟⣶⠦⣤⣤⣤⣶⣺⣿⢿⢻⢿⣿⡿⡟⣇⠸⡆⢻⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⠀⠀⠀⠀⠀⡿⠖⠉⠀⠀⠀⢀⢀⡟⠀⢀⣿⣿⡇⢸⡀⣿⣤⣶⠀⠻⠘⠘⠘⠈⠀⠀⠀⠸⡄⢻⡘⣇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣠⡄⠀⠀⠀⢰⣦⣀⣶⣼⣧⣾⣿⣷⢆⡾⠉⠿⠿⡇⣿⣿⣿⣿⣿⣟⣀⣀⣀⣀⣀⣀⣀⣠⣧⣼⡇⢻⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣿⣿⡇⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣴⠟⠀⠀⠀⠀⠛⠩⠿⣿⣿⣿⣿⣯⡍⠉⢁⣀⣀⣀⣀⣀⣀⣿⠘⣧⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠚⢻⣿⣿⣿⣿⣟⣉⠉⠉⠙⣛⣽⣿⠁⠘⣇⢹⡄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣯⣤⠀⠀⠰⢟⣿⠀⠀⢻⡈⣧⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠻⢛⣿⣿⣿⣿⣿⣏⡁⠀⠀⢘⣽⡏⠀⠀⠘⣇⢹⡄⢹⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣽⣿⣿⣿⡉⣁⡀⠀⠠⢺⡇⠀⠀⠀⢻⡀⣷⠈⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣿⣿⣿⣿⣿⡟⠒⠒⠛⠛⠛⠛⠛⠛⠛⣇⠸⡆⢸⡆⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣥⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⠿⠷⠶⢶⣶⡶⠒⠒⠒⠖⢿⡄⢷⠀⢿⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣾⣷⣶⣶⣶⣶⣾⣷⣾⣷⣾⣷⣶⣶⣶⣶⣶⣿⣿
⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣿⣿⣯⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣬⣭⣭⣿⣿⣿⣯⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣷⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡄⢠⡬⠙⠫⣅⣆⡞⣷⣽⠟⣶⢶⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⣁⡘⠖⣣⣞⣽⠿⠛⢟⣧⣿⣿⣷⡵⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠺⢏⣿⠿⢿⠟⣹⡿⠠⣖⣶⣾⡿⠛⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢿⣿⣿⣿⢿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡀⣀⠀⠀⠀⠀⠀⣀⣀⢀⡀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⣄⣀⣀⣀⣠⣀⡀⠀⠀⠀⠀⢀⠄⣠⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⠾⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠻⡳⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣷⣶⣶⣦⣦⣬⣀⣈⡉⠚⢠⠟⡷⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣻⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣷⣦⣁⡀⢀⣾⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣟⣿⣋⠙⡳⢷⢛⠛⢡⠛⡜⡙⠙⡙⠛⠋⠻⢯⣿⢿⣷⣿⣿⣾⣧⣄⣀⣀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣠⣼⢿⡿⣿⡿⣏⠉⠑⡀⠀⠀⠁⠑⠀⠒⠐⠐⠐⠀⠀⠀⠀⠀⠠⠅⠉⡀⠃⠁⠃⠋⠉⠛⢻⠿⡻⣿⡿⣿⣿⣿⡿⣿⠟⠋
⠀⠀⠀⠀⣾⣿⡟⠀⠘⠋⠁⣿⠀⠀⢱⡀⣀⣤⣴⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⢸⣮⣤⣃
⠀⠀⠀⠀⠶⣿⣿⣷⣤⣄⡈⠉⡀⠀⠈⠉⢛⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⢇⡀⠀⠀⠉⠉⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠤⢤⣴⣾⡤⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠀⠁⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣮⠅
⠀⠀⢀⣀⢤⡤⠶⡿⠿⠛⣟⣿⠟⡿⣿⡿⢿⠿⣿⠿⠿⠛⠿⢻⠿⠻⠟⠻⠻⣿⣟⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣆⡀⠀⠀⢀⣠⣴⣿⣿⣿⡇
⠀⢿⠙⠛⠁⠀⠀⠀⠛⠀⠹⠀⠈⠙⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠁⠈⠈⠁⠙⠉⠚⠋⠉⠙⠟⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣄⣿⣶⣶⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⠤⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣶⣾⠁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⣾⣼⣷⣿⣼⣧⣶⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠋⡇⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠄⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣄⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠤⡄⡂⠀⢀⢀⣀⢄⡀⡀
⣀⠀⠀⠘⠆⠀⠀⠀⠰⣀⠀⠀⢢⣤⣤⠀⠀⠀⠀⠸⡜⠀⢀⡔⠀⠀⠀⢸⣙⣯⣸⣶⣠⡼⡈⠆⣄
⠈⠁⠀⠀⣀⣢⣄⣠⠁⠉⠤⠄⠀⠉⠁⣀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠉⠘⢳⢸⢟⡹⠿⠂⢫⡷⣄
⠀⠀⠀⠰⣿⢿⡿⣿⡷⠀⠀⠀⠠⠀⠚⠁⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠰⠂⠘⠀⠀⠋⠣⠒⠂⠞⠣⠼⣉⣦
⠒⠒⠈⠙⢻⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⢻⡖⣊⡥⣘⠃⠐⠿⠷⠘⢴⣿⠂
⠀⠀⠀⠀⠐⠛⠛⣧⠀⠒⠲⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠊⠥⠶⡼⣅⠀⢀⠂⣰⣆⣼⠋
⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠈⠁⣿⠧⠼⠛⢙⡹⠃
⠀⠀⠀⠀⠀⡀⠀⢠⣀⡘⠀⠀⠀⠀⠀⠐⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣄⣀⣀⣁⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣧⠜⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠁
⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⣤⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠤⠤⡴⠂⠀⠀⠀⠀⠀⡆⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠠⡼⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠁⡈
⠀⠀⠀⠈⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⠀⠀⠈⠓⠀⠤⠔⠉⣄⣁
⠀⠀⠀⠀⠀⠀⠀⣸⠄⡠⠤⣒⠨⠁⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠂⠀⠀⠁⠂⠉⠐⠉⡀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⣠⣾⣿⣿⠟⠛⠉⠁⠀⠀⠀⣹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⠖
⠀⠀⣰⠟⠛⠛⠁⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣆⠀⠀⠀⣠⣤⢤⣾⣿⣿⣿⣿⣿⠟⠉
⠀⠀⣿⣷⠄⠀⠀⠀⠀⠀⠀⠋⢉⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠿⢟⣿⣿⣿⣿⣧⣴⠆
⠀⠀⣿⣷⣤⡀⠀⠀⠀⠀⠙⢿⡿⠿⠋⠁⣸⣿⣿⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣧⣀⡀⣀⣤⣶⡟
⠀⠀⢻⣿⣿⣿⣂⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦⣠⣿⣶⣄⡀⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠈⠛⢿⣇
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⣶⣾⣿⣶⣴⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⣤⣤⡬⣅⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣤⣽⣷⣬⢳⡀
⠀⢀⡞⠋⠁⠀⠙⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⣟⡿⡯⣧⡟⣧⠰⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣯⣿⣏⣻⢀⣿
⠀⣾⡇⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣦⣟⣷⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢜⢿⣿⣿⣿⡶⢋⣾⣿⠁
⠀⣿⣯⡀⠀⠀⠀⣠⣾⣿⡟⠀⠀⣼⠷⣭⣓⠶⣤⣥⣀⡀⡀⠠⠘⢮⡷⣿⣽⣯⠏⡘⠀⠀⠀⠀⠀⠀⠀⢬⡳⢿⣯⡵⠞⢋⡽⠋
⠀⠘⢿⣿⣿⣾⣿⣿⣿⠟⠁⠀⠀⢻⠀⠈⠉⢻⡶⣶⣦⣴⡾⠀⠐⠢⠍⠉⠩⠅⠊⢀⠦⠀⠀⠀⣴⠟⢉⣾⠃⠀⠉⠉⠉⠁
⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣵⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⢸⡀⠀⡇⡀⠋⢠⡾⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⣀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠸⢷⣼⡅⠀⣰⣯⠈⠁
⠀⠀⠀⣤⡠⠄⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣆⡀⠀⠀⠶⢋⣾⠃⠀⠀⠈⣿⣿⣿⣿⣁⢁⠀⠀⠀⠐⣲⡶
⠀⠀⠀⢳⠙⢷⡂⠀⠀⠀⣾⠃⠀⠀⢀⡴⠛⣉⠄⠘⠉⠛⢿⣷⣤⡄⣠⡿⠁⠀⠀⠀⠀⠛⠙⣿⣿⣭⣤⣤⠤⠴⠛⠋
⠀⠀⠀⠈⠃⠀⠻⣧⣰⣷⣿⠀⠀⢠⣿⣴⣿⠉⠀⠀⣀⣤⣾⣭⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠁⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢄⣮⣿⣿⡏⠄⠀⠸⡿⠻⣯⣀⣰⣾⣿⠟⡛⡛⠛⠻⢷⣶⣶⢦⣿⡀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⢿⣶⣄
⠀⢠⣤⣤⠤⠔⠒⠛⠋⠉⠂⠀⠀⠀⠹⣦⡈⠛⢿⡿⣇⢺⣽⣻⣿⡟⣦⣬⣫⡀⠀⠀⠀⢀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠛⠻⠿⢇⠘⠷⣍⠻⠷⣿⣿⣻⠇⠀⠀⠀⠀⠈⠁⠸⣿⣿⣿⡄⠀⠀⠀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠈⢻⣶⣶⣶⣶⡞⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠭⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠟⠛⠁

⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⠀⢠⠁⣰⡁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣴⣿⠟⠀⡀⠆
⠀⠀⠀⠂⠀⠀⢀⠀⠀⣆⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⢈⠀⢀⣼⣿⠃⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠈⡄⠀⢹⡄⠀⠐⣄⡀⠀⣇⠀⠀⠀⠀⠀⣆⡠⠾⠟⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈
⠀⠀⠰⠀⠐⠄⠀⢇⠀⢈⡷⡀⠀⢢⡈⠢⠜⠀⠀⠀⠀⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡐⠀⠀⠈⠄⠘⡆⠀⢇⠙⢄⠀⠁⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠄
⠀⢀⠇⠀⠀⠀⠈⠂⡹⣄⠈⡄⠀⠱⠄⢦⠆⢠⠂⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠤⠃⠀⠀⠀⠀⠐⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠁⢢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⢃⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⠁⠒⠂⠠⠀⣀⢀⡀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠶⠤⠤⢤⣤⣄⣀⣀⣀⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠒⠤⠤⠤⢤⣀⣀⣀⣀⣀⣀⡀⠉⠉⠉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠒⠲⠆⠀⠠⠎⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠀⡀⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣐⣢⡉⣣⢠⡁⣀⠙⡙⠆⠈⠓⠀⢀⣀⠀⡀
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣯⠿⣘⣏⣧⣤⣶⣷⣾⣿⣿⣿⡿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠙⢿⡟⢷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣏⣣⣥⣾⣾⣿⣿⣿⣿⣿⡿⠿⠿⠙⠻⠃⠉⠉⠙⠻⠛⠽⡋⡻⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠈⠻⣟⣷⠄⢠⢆⣿⣷⣶⣦⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⠻⣫⡹⠁⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⣿⣿⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢙⣿⡓⠀⣾⣿⣿⣿⣿⣿⣟⣿⠹⠟⢯⠑⠉⠋⡀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣀⠄⡄⠀⠀⣤⠀⠛⢤⡉⠳⠦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⠄⠀⢿⣿⣿⡿⠿⠿⠿⣯⣷⣶⣠⣤⣁⡠⠑⠀⢈⠀⡀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⣲⣿⣽⡃⠘⠢⢀⡷⠀⠀⣠⡌⢹⣇
⠀⠀⠀⠀⠀⣴⣿⣯⠯⠴⠶⠛⠉⠀⠀⠀⠀⠀⠠⢛⣼⣛⠻⢿⣿⣿⣿⣟⣾⣷⣿⣶⣿⣆⠀⢰⡾⠿⠶⠒⠀⠠⠲⠾⠿⠿⠿⢿⣷⣶⠷⠖⠚⠛⠛⠉⠁
⠀⠀⠀⠀⠾⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠉⠉⠉⠉⠙⠙⠛⠛⣯⢻⣼⡧⠀⠀⠒⠂⠖⠒⠒⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠋⢡⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠽⠶⠿⠿⢿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡰⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢿⣿⣷⣄⠀⠀⠀⣠⣮⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠳⣿⣿⣿⣦⣴⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡱⡹⣾⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⣤⣹⣿⣿⣿⣿⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⢘⡿⢿⣿⢣⠹⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣹⡗⠈⠘⣄⣃⣀⣾⡿⠿⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢎⠀⠓⠀⠀⠀⢀⡀⠀⢀⠀⠈⢀⣀⡴⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⢆⡑⡄⣠⢀⠀⠀⠀⠦⠀⢠⣰⠞⠀⠀⠀⢀⡤⠀⠠⣶⠀⠀⠀⡀⠀⣀⠀⠀⠈⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣾⣮⣵⣼⣄⣩⢄⣄⣴⡏⠀⠀⠠⠚⠉⢁⣹⠷⠿⠀⠾⡦⢳⢦⣙⣃⣠⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠹⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠲⠤⠀⡀⠉⠁⢈⣤⡑⣄⠊⣴⠛⣥⡀⢀⠘⡆⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡉⠻⣟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢐⣆⠘⢋⣥⡖⠀⢨⣍⠁⢉⡆⣠⠀⠉⠓⠉⠀⠛⠄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⢶⣾⣿⣭⣽⠯⡤⢾⣿⣦⠀⠈⡉⡄⣠⣅⣰⠤⠺⢶⡤⣬⠿⠷⠾⢿⡧⠔⠲⠞⡤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣿⣿⣷⠀⠙⠿⠿⣋⠀⣴⣻⠆⢸⡇⢠⡴⠄⢀⣤⣶⡧⢠⠇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣣⡀⠀⠈⠻⣷⡎⡋⢁⠘⠋⢠⣽⡂⡀⠁⠀⣠⠎⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣢⣀⠀⠀⠁⠉⠛⠷⢦⠀⠈⡀⠀⠀⠀⠉⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣿⣿⣿⣧⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡻⣿⣿⣿⣿⣿⣶⣶⣦⡤⢤⠤⠤⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠨⢿⣿⣿⣿⣿⣅⣀⡄⠐⡼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡀⠨⡉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠿⠶⠷⠷⣵⣥⣓⡈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠵⢲⡴⠾⠚⠚⠂⠳⠃⠇⢆⠍⡟⡻⢶⣤⡊⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋⣰⣿⡷⣝⢮⡹⠀⣀⢀⡠⢄⠀⠠⡄⣀⠈⠈⠀⠩⠳⣧⣈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠩⠶⠂⠠⡙⠂⠀⠈⢷⡰⡌⢧⠹⡈⣷⢠⠹⡈⡏⠃⠠⠀⠀⠈⠻⣟⢃
⠀⠀⠀⠀⠀⠀⠀⡴⠁⢸⠥⠀⠀⠀⠘⣆⠀⠀⢸⡇⠏⠘⠠⠃⠃⠔⠐⠥⢃⡄⡸⢁⠆⠃⡀⠈⢷⡀
⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⢠⢲⡇⢤⠤⠤⣤⣐⣒⣷⣦⠀⠀⠀⠁⠼⡰⡡⢂⠈⢧
⠀⠀⠀⠀⠰⣭⣥⠀⢭⣽⣀⣀⣀⡤⠴⠋⣖⣡⠞⠀⠾⢃⢤⣒⠲⠖⠀⣸⠀⢀⠀⣠⠤⢞⡒⠒⣦⠘⡆
⠀⠀⠀⠀⠀⠀⠯⠴⡖⠳⠶⠶⠶⠾⠟⠛⠭⣅⡀⠀⠀⠀⠑⠪⠭⠤⠞⠁⠀⠀⡿⣚⠋⠡⢄⡔⡩⠂⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠡⢟⢷⢦⣄⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣉⠉⠀⠐⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣉⣠⠥⠖⠒⠚⠁⠈⠉⠉⠉⢿⡻⠻⣩⠃⠀⠀⢹⠙⢎⠙⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣖⠥⠤⠤⠤⡎⠀⠘⠀⣦⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠉⠘⠛⠋⣧⡀⢠⡾⣅⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣀⡤⡀⠙⠛⡅⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⡔⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⠃⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⠎⠼⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⣠⠞⠕⡥⠋⡪⢛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠛⠋⠉⠙⠛⠯⠛⠿⠿⠿⣟⠿⠿⠶⣾⣭⣭⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⡿⠛
⠀⠀⠀⠀⠀⣠⠖⠋⠉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠻⠛⠷⠶⠶⢶⣾⢟⢭⢜⣿⠟⠉
⠀⢀⡤⠂⠱⠶⡄⢰⠀⢿⣇⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⣀⣀⣠⢐⣾⣿⣿⣿⣿⣧
⢰⣁⠀⣀⠂⠀⠀⣠⠖⣹⡷⠚⢟⡟⢻⠛⠉⠀⠀⠠⠀⢀⡠⢀⣄⣠⣤⣴⣶⣾⣷⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣾⣾⣿⣿⣧⣄
⠀⠉⠙⠛⡗⠒⠛⠓⠒⠛⠿⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠟⠛⠛⠛⠋⠈⠛⠻⢿⣿⣷⡀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠑⢦⣤⣤⡤⠋⠉⠀⣀⡥⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣤⣶⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠖⠂⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢻⡿⠛⠃⠀⠒⠀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡈⡁⠨⠙⠦⠀⡴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡇⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣠⣴⣿⣷⡀⠀⠀⠀⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠉⠉⠱⣼⣦⡙⢇⢠⣤⠤⢺⢟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢠⣤⡆⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠘⠛⠁⠀⠀⠀⠀⠉⠳⢦⡄
⠀⠀⠀⢀⡔⠋⠀⠀⠀⢤⡀⢀⠀⣀⠄⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣐⠆⠸⡄
⠀⠀⡰⠋⠀⠈⠉⠙⢲⣀⡗⢰⣠⣽⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⡇
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠹⡦⠛⠉⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸
⠀⠀⠘⣦⠀⢰⠀⠀⠀⠀⠙⢦⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠐⠀⠀⢸
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠈⠻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⠜⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠀⢠⡟⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣿⡇⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢡⣄⠈⠀⣄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣠⣀⣿⢀⡁⠀⠀⠀⠀⠀⠀⠐⠀⢁⠀⠀⠁⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⠀⢿⣈⣿⡦⠀⠀⠀⢶⣤⡶⢰⠟⣶⢄⠀⠀⣰⠏
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿⠾⠂⠀⣿⣽⢆⣿⣄⢠⠚⣁⡀⣤⣈⠘⣈⣿⣿⠃⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠉⠩⣶⣼⢷⣦⠈⠻⣮⡻⣿⣦⡀⠈⠙⠾⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠐⠀⣤⠈⢿⣦⣝⠻⠦⣶⣴⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠘⠷⢸⣿⣿⣿⣷⣿⣿⣿⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡦⢠⣾⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠒⠛⠛⠋⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡄⡄⡠⡀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣰⣧⣴⣤⣠⣰⣸⣘⡘⡘⠔⠔⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠩⠙⠛⠶⠤⣌⣀⢄⠄⠀⠀⠀⠀⠀⠀⢀⡤⠶⠾⠗
⠀⠀⠀⠀⠀⣀⣤⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⠁⠁⠈⠀⠀⠀⠉⠉⠓⠊⠀⠀⠀⣠⣶⡟⠉⣐⠮⠁
⠀⠀⣀⣴⣾⣿⡇⠀⠈⢣⢀⠀⡀⡀⢀⢀⢀⡀⡀⢂⢀⠀⡀⡀⢀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⢠⣾⠍
⠀⠚⠛⠛⠁⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠹⣿⡀
⠱⢤⣀⠀⠀⠀⡠⠀⠀⢀⢃⡠⠤⠤⠀⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠠⢤⣾⣤⠀⠭⣖⡀
⠀⠀⠈⠁⠒⠈⠰⠶⢆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡠⢖⡉⣸⡆⠀⠀⠀⠀⠈⠳⢄⠂⢽⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠲⠶⡄⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠔⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⡈⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢹⣧
⣿⣿⣿⣿⣿⣿⣿⠟⣋⠁⢰⣿⣇⠀⠙⠻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠃⠀⢻⣷⣦⣤⡀
⣿⣿⣿⣿⣿⣿⡇⣾⣿⠇⣼⣿⣿⡀⣿⣷⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡾⠉⠙⣿⠀⠀⠸⣿⠀⠈⢿
⣿⣿⣿⣿⣿⣿⣷⣬⣭⣼⣿⣿⣿⣷⣌⣉⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢧⣀⣴⠇⠀⠀⠀⠘⠷⠴⠟⠑
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠊⠀⠀⢀⠐⡩⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣠⢊⠄⡐⡡⠊⠐⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣴⣷⣧⣮⣞⣔⡠⣀⠄⡀⠀⠒⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⢿⢳⣿⣿⣿⣿⣿⣿⣿⣾⣶⣠⡴⠒⠉⠰⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⣿⢟⠋⢛⡋⣻⠞⠸⢻⡳⡯⠽⠿⣿⣻⣿⣿⣷⣧⣄⡀⠈⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⢿⢿⠙⣌⢧⣿⠱⠀⠛⠀⠈⠑⠀⠐⠺⢣⢒⠃⢱⠞⢩⡟⣿⣿⣿⣿⣷⣶⡶⠊⠑
⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣟⣿⣻⠙⣈⠂⣾⠛⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⡁⠈⠣⠀⠄⠙⠫⠿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⣚⣿
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠣⡍⠉⠉⠈⠀⢀⠀⠂⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈⠈⠈⠂⠤⠒⠠⡄⣈⡑⠉⠙⡛⢛⢻⣿⣷⣦⣤⣤⣤⣤⣴⣾⣿⡿⠞⣋⣥⠂
⠀⣠⣿⢋⣽⣿⣯⢻⣿⠃⠈⠳⡔⡁⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠠⢈⠒⠒⠆⠠⡁⢻⠯⠭⢝⠻⣻⣿⣿⡿⣓⣚⡭⡵⠂
⣾⡟⠿⣝⣛⣭⠹⠘⣿⡇⠀⠀⡀⣨⡀⠠⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠐⠂⠀⠈⠀⠁⢉⡘⢂⠀⠄⠩⠄⠐⠨⠁⣼⣿⣿⣿⠿⢿⡿⠁
⠸⣿⣄⢄⣭⠐⠁⠀⣿⡀⢀⡄⣁⣿⢈⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⢐⣥⣤⣶⣿⣿⣿⣷⣷⣾⣾⣿⣿⣟⠿⣭⣟⣻⠆
⠀⠀⠙⠺⣍⢓⣶⢶⣿⣥⢎⣠⠟⣑⣿⣿⣿⣿⣿⣽⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢄⠐⠂⡰⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠻⢿⣻⣗⠮⣕⠩⣛⠆
⠀⠀⠀⠀⠀⠉⠛⢿⣾⣿⣿⣿⣇⡠⡉⠛⠻⠛⠿⠺⠺⠉⠀⠀⠀⠀⠀⢈⣀⠒⠀⠀⣀⣡⣦⣴⣿⣿⣿⣟⣳⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠝⣮⡛⢮⣹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣾⣶⣷⣶⣞⣤⣠⣴⣶⣶⣶⣬⣷⣾⣷⣿⡿⠿⠿⣿⣿⣿⡯⠻⢭⣙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣽⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⣿⡿⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠻⣿⡍⠺⠐⢴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣣⣴⣾⠿⢛⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣾⣿⡟⣡⣾⣿⡿⠟⢛⣛⣓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⡿⣴⣿⡟⣡⣾⣿⣿⠿⠿⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣶⣄⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠷⠿⠿⣿⣶⣦⣥⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⢟⢻⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⣏⡽⣿⣷⣶⠾⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣒⣒⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢿⠸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣟⣯⡗⢾⠀⠹⣿⣽⣿⣿⢛⣿⣛⡟⣯⢞⡷
⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⣛⣿⣷⠿⠛⠉⠀⢸⡆⠀⠈⠻⠿⠟⡟⠿⢿⠟⠋⣸⠇
⠀⠀⢀⣤⣶⣿⣿⣿⡿⠟⠻⡆⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠅⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡆⠀⠀⠀⣸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠸⡇⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣷⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢿⠀⠀⢀⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣤⣴⣿⢏⣤⣶⣶⣶⣶⣷⣆⣶⣷⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣁⣲⢞⣿⡿⣵⣿⣿⣿⣿⡿⠽⠛⢭⠟⠉⠉⠙⢿⡿⣫⣥⣲⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠘⡗⣿⢻⣼⣿⣿⢋⣿⣿⣿⠏⠀⠀⠀⠀⠉⠀⠠⠰⠾⢿⣷⡟⡃⠛⡟⠻⣿⢁⡀
⠀⠀⠀⠀⠀⠐⠃⣶⣼⣿⣿⡇⣸⣿⡿⠈⠀⠀⡢⠆⠼⡆⠀⠀⠀⠀⠀⠀⠉⠛⠁⠛⠀⠀⣀⠙⠒⠶⢄⣄
⠀⠀⠀⠀⠀⣴⢾⡟⡟⠉⣿⠁⣿⣛⣀⡀⡀⠉⣀⣶⣀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⢴⣶⣤⣽⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠐⢋⣾⣟⢿⡄⢹⣇⢻⣿⡗⢩⣑⠀⣇⣿⡇⡄⠈⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡁⡸⢗⣠⠤⡭⢿⣿⣿⣿⣿⢆⡀
⠀⠀⠀⠀⠀⠀⠻⡟⣄⣷⡄⠛⢷⣌⠓⠊⢠⣍⡁⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⡞⠋⣴⡶⠛⠋⠉⠉⢻⣿⠹⠁
⠀⠀⠀⢰⣶⠆⣠⠎⠉⠀⠀⠀⠀⠉⠀⠀⢈⡿⣿⣿⣿⣤⡀⢦⣤⣄⣨⣿⣷⡶⠶⠶⠾⢉⡙⣧⡆⢰⣻⣌⣤⣤⣴⠶⠚⠛⡎
⠀⠀⠰⠟⠋⣁⣠⣤⠴⠖⠊⠀⠀⠀⠀⠀⠄⣀⠛⣯⣭⣛⣛⣳⣦⣭⣥⣤⣤⡴⠾⡿⠾⠿⠿⠾⡅⣮⣅⣉⣉⠉⠀⠀⠀⠀⡷⡆
⠀⠀⠀⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠉⠭⢀⣘⣻⡿⠋⠛⠛⠛⠛⠓⠒⠛⠛⠃⠙⢯⡉⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣶⣶⣶⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡄⠂⢰⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣇⣸⠈⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠛⠓⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡿⣿⣰⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣟⠠⠖⣣⡀⠀⣆⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢻⡄⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣿⠿⣿⣿⠇⢠⣿⣴⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠉⠉⠻⣿⣯⠸⡸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢶⡌⠣⣄⡀⠀⢸⠟⠿⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣄⣤⡋⢿⣴⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⢦⣠⠌⠙⠻⠀⠀⠀⠈⠛⠲⠦⢤⣤⠖⢱⣦⣶⠦⠀⠀⠀⢹⣿⣷⣄⣿⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣶⣤⡽⠀⠀⠀⡈⢀⡀⣀⣠⣤⠴⢶⡴⠶⠆⠉⠀⠀⠀⠀⠀⠘⠛⣛⣿⣿⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟⢿⠿⣿⡟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣴⣾⣿⣿⡸⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠐⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣉⡇⡧⠀⢠⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣽⡟⢋⣼⢣⣿⢰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣷⡇⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠐⢺⡵⣿⣿⣿⣤⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⣿⠃⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠠⣴⠋⠀⠘⣿⣿⣿⣇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣿⣿⣿⡷⡏⡘⠇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⣴⠃⠀⠀⠀⠘⢿⣿⣿⡞⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢧⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⣻⣿⠸⢠⠙⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢷⠀⠀⠀⠀⠀⠀⠀⢰⡼⣷⣿⣿⢀⢸⠁⠀⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⡇⠀⠀⠀⠀⠀⢀⣞⡷⠉⢋⣷⠈⢸⡇⢀⠀⢹⣆⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⠀⠀⣠⠾⠟⠀⠀⢟⣿⡀⠀⠑⢸⠀⠀⠀⠙⢡⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡄⢀⡴⠏⠛⠀⣀⣰⢦⣾⡉⢡⡀⠀⠻⡄⠀⠒⠒⠐⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠧⣄⣠⣦⣶⠼⠿⠓⠛⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣧⣤⣤⣤⡀⠀⠀⠤⠤⠶⣾⠿⠿⢿⣿⣿⣶⡾⠛⠉
⠀⠀⠀⣀⣤⣤⣴⣼⣿⢿⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠈⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢂⢀⢁⠃⠫⢣⢆⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣼⣦⣦⣦⣦⣡⣡⡠⠈⠊⢖⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠢⠩⠡⠠⠀⠀⠄⠄⠄⠌⠈⠉⢩⢳⠳⠧⣤⣀⠄⠀⠀⠀⠀⠀⠀⢀⣴⢎⡣⢑⣫⠅
⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⢂⢃⢃⠂⠀⠐⡐⡐⡐⠐⠀⢂⢂⢂⢃⠃⠀⠐⡘⡙⠐⠀⠀⠀⠠⠶⣿⠯⣑⠨⠴⠒
⠀⠀⠀⠀⣠⣶⡿⣄⡀⠍⠌⠌⠌⠄⠀⠡⠡⠡⠡⠁⠀⠌⠌⠌⠌⠄⠀⠠⠡⠡⠡⠁⠀⠄⠄⠄⠁⠀⢀⣨⠭⠁
⠀⢀⣴⠿⡇⢱⠀⠈⠑⢜⡘⡐⢐⢀⢂⢂⢂⠂⡂⡀⡐⡐⡘⠐⢀⢀⢀⢂⢂⠂⠂⡀⡐⡐⡐⠀⠀⡀⠐⠿⡁
⣴⠋⠃⠛⠀⠀⠀⠀⠀⡼⠀⠀⠀⠄⠄⠄⠀⠀⠀⠠⠠⠀⠀⠀⠀⠄⠄⠄⠀⠀⠀⠀⠀⠁⠀⣠⣴⢀⠭⢙⣒⠂
⠙⠧⣀⠀⠀⢀⡆⣀⣼⣣⡥⠀⠈⠘⠈⠈⠄⠀⠀⠡⠡⠡⠁⠀⠈⠈⠌⠌⠀⠀⡠⠂⠀⠀⠀⠉⠻⣷⡪⢔⡪⢭⡀
⠀⠀⠀⠉⠙⠛⠋⠉⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠤⠄⠀⠀⠀⠀⠀⠁⠣⠪⢗⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠦⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡩⠷⠒⠒⢼⣏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠊⠀⠀⠀⠀⠀⠀⠱⡜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡄
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢸⠳⣄⠀⠀⡎⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢧
⠀⠀⠀⠀⠈⠓⠬⣓⠢⢄⡈⢣⡈⠓⢶⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀
⠀⠀⠀⢀⡤⠤⠤⠀⣉⣦⠈⠉⠁⠀⠀⠙⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⠌⡃
⠀⠀⠀⠀⢀⣉⣉⠉⠠⠦⠀⠀⠀⠀⠀⠀⠘⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⡟⠛⠛⠶⡒⢤⡀
⠀⠀⠀⠀⠙⢒⣒⣐⡒⠂⡄⡴⠒⠒⠒⠒⠒⠛⠒⣲⣦⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⣧⣠⣤⣴⠡⠾⠇
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⠛⠛⠛⠛⠉⠉⣩⣿⡇⣯⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠷⣿⡟⢏⠉⣲⠛⠦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠈⣿⠁⠀⡰⠋⠲⢄⡀⠀⠀⠈⣧⠤⠤⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠞⠀⠀⠀⠀⢘⣦⠞⠉⠀⠀⠀⠀⠀⠀⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡜⡇⠀⢀⡤⠤⠄⠀⠤⠤⣄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣠⡄⢰⡇⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⡖⠒⠤⠤⠄⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣠⠎⠀⢠⠘⠇⡇⠀⢻⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡴⠁⠀⠀⢸⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣸⠁⠀⠀⠀⢸⠀⡄⡄⠀⢸⠀⠀⠀⠀⠀⠀⡸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸⠀⡇⡇⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡆⠀⠀⠀⠀⡌⠀⢰⢁⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡁⠀⠀⠀⢠⠃⠀⢸⢸⠀⠀⢣⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢇⣀⣀⡠⠋⠀⠀⢸⠘⡀⠀⠀⠙⠒⠒⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⢫⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢇⣀⠀⠀⠀⠀⠀⠀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠙⠋⠀⠀⠈⠉⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡖⠁⠀⢠⢤⠀⠀⠀⠀⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢚⠀⠀⡜⢀⣤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣭⠃⠀⠙⠋⢁⠀⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠸⣦⣀⡼⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡞⠀⠀⠀⡀⠀⡉⠑⠒⡄⢀⠀⠀⠀⡰⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢒⠏⠻⠤⠕⠀⢾⣿⣿⣿⢖⠝⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣴⣶⣤⣴⣿⣿⣿⡷⡍⣴⡁⠀⠀⠀⣀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣯⢟⠞⠱⡄⠈⠓⣖⠉⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡻⢿⣿⣿⣿⡿⡣⢯⠶⠚⠙⠦⣤⡼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣹⠊⠀⢀⢞⠞⡔⠛⠿⣬⡉⠁⠀⡶⠒⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠰⣇⢀⡰⡻⠃⠀⢸⠀⢀⣨⣿⡆⣸⠃
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠙⢦⠌⡐⣟⠀⠀⢠⠃⠀⢻⣿⡿⠙⠋
⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸⣢⠾⠁⠀⡰⠁⠠⣾⣿⡟
⠀⠀⠀⠀⢠⡏⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠜⠁⠀⢀⠜⠀⠀⣤⣾⣿
⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡡⠋⠀⠀⢠⣾⣷⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠘⣇⠀⢸⠀⠀⠀⠀⢀⠤⠒⠤⣾⠝⠀⠀⠀⠐⠛⠛⢻⠉⠙⣯⠉⠙⣇
⠀⠀⠀⠀⠀⠈⠂⠀⠂⠀⢀⠔⠡⢄⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠈⡀⠀⡇⢇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⠄⠀⠑⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢣⢀⡇⠘⡄⢸
⠀⠀⠀⠀⠀⠠⠴⠚⣉⠄⢀⠅⠀⡠⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⢣⢸
⠀⠀⠀⠀⠀⠀⠠⠭⠒⠭⡀⢄⣪⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠈⠟

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⢛⣿⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠶⢤⣀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⢠⠆⢀⡄⠀⠀⠀⠀⠀⢀⡀⠈⠙⠢⣀⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠸⡄⠀⠀⠀⠀⠀⠈⣋⣀⠀⠀⠈⢷⣤⣤⡼⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠉⠀⠀⠀⠀⠀⣞⠁⡴⠲⣶⣶⣏
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢀⣸⠭⠥⠖⡂⢀⡝⠀⠀⢤⣀⣀⣀⣹⣄⣈⣀⣉⣭⣵⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢾⣉⠉⢉⣉⣴⣋⣠⣤⣶⡶⠿⠯⣽⣿⠻⡄⠈⠣⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢀⡤⠀⣠⣽⣿⣟⠋⠉⠉⠀⠈⠁⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢠⣿⣶⣿⠙⢿⡿⣙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠘⠙⢹⠹⡄⠀⠙⠦⢭⣙⣓⣦⣤⣤⣀⣀⣴⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣼⣧⡙⠦⣄⣀⠀⠀⠉⠉⠉⠉⠉⣹⣿⣬⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢬⣭⣉⣓⣒⣾⠉⠉⠉⠙⠻⢿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡁⠀⠀⠀⠀⠈⢀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢀⡀⠀⠀⠬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠛⢢⠛⠁⠀⠀⠀⠀⠀⢄⠀⢰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⡀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠰⡄⠀⠀⠀⢣⠀⠀⠀⠀⠀⢠⠄⠀⢀⠤⠙⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠁⠀⢀⢶⠀⠀⠠⣄⡌⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠁

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠈⠀⠐⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠂⠀⢀⡞⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠃⠀⠀⠀⠀⠀⠀⠋⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠺⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢰⡔⠋⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⡰⠚⡅⢱⣄⢀⣀⡠⠠⣤⠤⠴⠶⡖⠒⠒⠐⠢⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠉⠓⠉⠀⠀⠀⢀⠖⠚⠠⠤⢀⡀⠈⠓⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠂⠀⠈⠆⠠⠘⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠤⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⢀⡀⠀⠀⢀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠠⠤⠐⠛⠋⡀⢀⣠⠆⢀⡆⠄⠀⠀⠀⠠⠓⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⡀⠈⡆
⠁⠀⠀⠔⠁⠌⣉⠉⠸⠗⠂⠀⠀⠀⠈⠉⠍⢀⣀⣀⡤⢄⣴⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⣀⢠⠦⠴⠞⠲⢄⠠⣤⣄⠀⠀⢀⢳
⠀⠀⠀⠇⠀⠀⠐⠷⠐⠀⠀⠀⠀⠀⠀⠀⠘⠚⠛⠿⠀⠔⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⠒⠒⠊⠉⠀⠀⠁⣀⣀⠀⢀⠀⠠⠈⢉⣿⠋⠀⠈⡾
⠀⠀⠀⠀⠀⠀⡈⠉⠂⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⡒⠲⠖⡈⠁⠀⠤⠄⠤⠤⠔⠂⠉⠉⠉⠉⠀⠀⢀⣀⣀⠀⠀⠁⠀⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠂⠀⠀⠀⠋⣠⣤⣤⡔⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⡀⠜⠁⡜⢀⡴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣷⣒⣉⣀⣀⣀⣉⣁⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠵⠄⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢱⢃⣰⣶⡦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⣨⣿⣿⣿⣒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣶⣶⣦⣄⣠⣤⣴⣶⣶⣤⣤⣦⣤⣤⣀⡀⠛⠿⡏⢿⣿⠃⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡶⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⡿⠋⡁⣀⣠⣤⡌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⡟⣿⣿⣿⣿⣿⣿⢿⣿⣯⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡅⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠖⢿⣿⡛⣿⠁⠛⠿⣿⡿⢿⣽
⠀⠀⠀⠀⢀⣤⣶⣾⢃⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠫⠋⠐⠀⠀⠀⠀⠄⠄⢀⣠⣀⣴⣒⠀⠈⠦⠁⠊⣅⡀⢴⣿⣴⣻⠇
⠀⠀⠀⠀⠈⢉⣾⣿⣿⢿⣿⣿⣿⣿⣿⠿⠟⡻⠛⠙⠉⠅⠁⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠤⠐⡚⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡿⣽⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⢠⣈⣁⣾⠞⢊⣡⣿⠟⠋⣿⣿⡿⠷⠿⠷⠦⢤⣤⣤⣤⡤⢤⣴⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣳⣼⣿⣹⠟⠉
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠙⠛⠛⠛⠋⠿⠛⠹⠋⠙⠛⠋⠛⢿⠿⠿⠟⠉




⣿⣿⡟⠛⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣿⡤⠤⢤⣶⣶⣶⣶⡶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃⡼⠓⣛⣿⣯⡙⡍⣉⠉⡍⢍⠙⣓⠺⣝⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⣸⠿⢭⣤⣈⣏⣀⣉⣔⣲⣌⣭⡾⠤⢗⣫⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣰⣯⣍⣙⡒⠒⠲⠶⠶⠶⠶⠖⠒⠒⣋⣉⣤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⠏⣇⠙⢮⡉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⡟⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞⠀⢻⠀⠀⠉⠲⣄⣀⠀⠀⠀⠀⠀⢀⣠⠼⠗⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢾⡁⠀⠘⡇⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠺⠿⡶⠚⠁
⠀⠀⠀⠀⠀⠀⡄⠀⢀⣤⣤⣀⠀⢠⣞⣉⣙⠺⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⣀⠀⢙⡶⣫⠶⠦⣌⣳⢚⣹⢹⡟⣿⣷⠦⣤⠼⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠂⠀⠀⣯⣍⡭⠞⡥⠖⠶⢤⡼⢋⡡⢫⣷⣿⣅⡠⢬⡛⣩⠴⢭⣳⡀⠀⠀⠀⠀⣀⡼⢣⡄⢀
⠀⠀⠀⠀⠙⢦⡤⠞⣱⠚⣹⡏⣰⢋⡶⢡⠋⡏⣽⡖⠲⠭⠗⣲⡶⠏⢝⢟⢛⡛⣫⠝⡴⡢⠢⠀⠑⠂
⠀⠀⠀⠀⠀⢀⣝⡻⢥⣞⡴⢧⣹⣼⣿⣜⣦⡳⢝⠻⣏⠙⣻⡥⣀⠀⠀⠐⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠐⠂⠁⠀⠒⠆⠉⠙⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⠰⠀⡠⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁⢀⡠⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣄⠀⠀⠠⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⡠⠊
⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠸⡄⠀⠀⢦⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡂⢀⢾⡿⠃⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠒⠀⠠
⠀⠀⠀⠀⠈⢄⠀⠐⡆⠀⠘⠳⡀⠀⠈⡈⠂⢄⡱⠀⠀⠀⠀⠀⠊⠛⠁
⠀⠀⢠⠀⠀⠈⠀⠀⢳⠀⠀⡇⠱⡄⠀⠙⡄⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡈⠀⠀⠀⠀⢂⡈⢧⠀⠘⡄⠈⢢⡀⠀⡀⠀⠀⢉⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⢧⠄⠡⠀⠀⠙⠒⠳⠀⡴⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠂⠐⢄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠣⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠉⠀⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠲⠶⠦⠤⢤⣤⣤⣄⣀⣀⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠒⠂⠤⠤⠤⢤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠀⠀⠤⠴⢡
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣛⣛⣛⣛⣒⡒⡒⠶⠶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣴⡏⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⣚⣒⣤⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡠⠚⠙⢣⡾⠒⠉⠉⠉⢻⡿⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⡟⠻⡶⠞⠋⠀⠀⡇⢰⢀⢎⡟
⠀⠀⠀⠀⠀⠀⣴⣶⣦⡟⠀⡀⠀⠀⠀⠀⢀⡀⠀⢨⢪⡇
⠀⠀⠀⠀⠀⠀⠛⣉⠜⣇⠠⢍⣀⣀⣠⠴⢿⡀⢀⠃⠀⠣⣀
⠀⠀⠀⠀⠀⢠⣾⣵⢢⡈⠀⠀⠀⣴⠀⠀⣾⡽⣇⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⢺⣿⣿⡿⡇⠀⠀⢸⡏⠀⠀⠹⣷⣻⣿⡶
⠀⠀⠀⠀⠀⠀⠙⠋⠀⢻⡀⠀⢸⣧⠀⠀⠀⠙⣷⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢻⣄⠀⠀⠀⠈⠙⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢹⣦⡀⠀⠀⠀⠀⠀⠭⣟⣓⡦⠀⠀⠀⣒⣪⣾⠛⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠉⠙⠛⣛⣛⣭⡭⠩⠥⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣽⡿⠟⠋⠀⠀⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠒⠒⠒⠒⠷⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⡴⠋⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⢠⠞⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢰⠃⠀⠀⠀⠐⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠦⠤⠤⣀⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠔⠋⣉⣠⠤⠔⠒⠲⠦⠤⢄⣀⠉⠳⣄
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠀⠀⡠⠊⡠⠔⢋⣁⣠⣤⢰⠶⣴⢤⣤⣀⠈⠑⠢⣌⠳⣆
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠎⡴⠋⣠⠶⢭⢯⡆⣸⢣⡖⣸⢃⣤⢸⣛⣦⡀⠈⢢⡈⠳⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⡜⠀⣼⢱⡆⣾⢈⠐⢹⣀⡴⡏⠘⣣⡟⣡⢨⢉⣦⡀⠳⡄⠹⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⠀⡼⢣⣠⠊⡜⠛⢦⢞⡙⣆⡹⠺⣴⡀⢁⠞⣡⢈⢷⠀⢱⠀⢹
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⠀⠸⠀⣧⢋⡌⣶⠰⢇⡟⠘⢃⡟⠸⢃⡯⢿⣝⠈⢁⣮⣌⡇⠀⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣾⠀⢂⠀⣿⠘⢁⣷⡤⢊⠷⣔⣡⣧⢴⠏⡶⢸⣹⠶⣥⢣⡌⡇⠀⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣠⠃⠀⢻⠀⢸⡀⢸⠓⢏⡤⢱⢧⡆⣸⠁⡌⣾⠀⢠⢻⢣⡆⣿⠈⢡⡇⢠⠃⢰
⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⣇⠀⢇⠈⣾⠈⢃⣾⣀⢴⠳⣧⠁⢸⡟⠛⣾⣀⡼⢼⢦⢾⠇⡸⠀⣸
⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠹⣆⠘⣆⠘⣗⠻⡁⣽⢾⡸⡞⡗⢻⠰⠇⣧⠋⠖⡸⢀⠞⣰⠃⢠⡏
⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⣤⣤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠳⣌⠳⢜⣦⣈⡷⡀⢸⡇⢀⡼⣇⣠⡾⠗⢁⡴⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠳⢤⣀⣉⠉⠙⠛⠛⠋⠉⢉⣁⡤⠖⠉⣀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢬⣉⣙⣒⣋⣉⣉⣁⣤⠴⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⢰⠒⣦⢰⣒⡆⠀⣮⠀⣮⢠⠆⢾⡷⠀⠟⡆⠀⠀⢺⡗⢰⡴⠂⡆⡆⠐⡆⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈⠉⠉⠉⠉⠛⣷⣦⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡤⢻⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠄⣠⠀⠀⠀⠀⠀⡀⠠⡌⠟⡻⠁⣽⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣨⣿⣧⠇
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣷⣖⣀⡀⠈⣀⣀⣄⣜⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⣻⣿⠿⢋⣠⠤⠒⠒⠀⠀⠙⠓⠲⠠⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠻⢯⣽⣟⣿⣛⣥⠆⠙⠉⠉⠀⣴⠋⠀⠠⠖⣤⡗⢠⣠⡀⢤⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⣀⡈⠃⢠⣄⡥⠽⢣⣍⡐⡄⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠷⠈⡚⣀⢰⣀⣲⡦⣤⡄⠀⣍⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠈⠛⣋⡠⣶⢸⡅⠔⠀⠤⠶⡸⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⠈⠛⠰⠦⢄⠛⠀⠂⠘⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣷⣦⣄⣀⢀⡀⠀⣀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣿⣿⣋⡉⠱⠟⠋


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

⠀⠀⠀⢀⣠⣶⣶⣦⡄⠀⠉⠀⠐⠀⠀⠀⠀⡁
⠀⠀⠐⠻⣿⣶⣿⠀⠀⠀⠀⣄⣀⣴⡆⠀⠀⠛
⠀⢀⣤⣶⣶⣿⠄⠀⠀⢠⣾⠿⡛⢻⣿⠈⠀⠀⠀⠀⠀⠠⠀⠰⣆
⢰⡿⠻⣿⣿⣥⠾⠦⣤⡀⠉⠀⢿⡾⣧⣶⣴⣶⣶⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠛⠀⠛⠀⠀⠈⠀⠘⠠⠀⠀⠀⣀⣼⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⢿⣿⣿⣿⣧⣤⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠰⠿⠿⢿⣿⠿⣿⣿⣿⣷⣤⣤⣀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠰⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦⣶⡄⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⠙⠿⠿⠟⠛⢻⣿⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠱⡄⠀⠁⠀⠀⢾⣿⣇⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠸⣍⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠑⢠⡀⠀⠈⢿⣿⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡜⠀⠀⠈⠀⠀⢤⣿⣄⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⣀⠀⠀⠀⡤⠀⡰⠄⢀⠁⠀⠀⠀⣀⣤⣾⣯⡻⣦⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⣄⠀⠀⠀⠀⠂⠀⠂⠀⡁⠀⠀⣤⣿⠿⣼⣿⣿⣿⣇⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠘⠁⠄⠀⠀⠀⣀⣤⣀⠣⡒⠲⢠⣤⣄⠀⢠⣄⣴⣿⣿⣿⡽⠿⠛⡋⠹⡀⠀⢸⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠙⠿⠏⣀⡄⠁⡿⣆⠐⠟⠿⠟⠛⠉⠀⠀⠀⠀⠀⠁⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠂⠂⠚⠉⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⢀⣾⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣴⡦⠀⠀⠀⠀⣠⢣⡾⢃⣤⣶⠶⣦⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣷⣦⣾⣷⣶⣶⣶⠀⣴⠃⢸⣧⣞⡁⠘⢿⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣼⣿⣿⣿⣽⣟⣷⣿⣿⠝⣿⢋⣿⠶⢿⣿⠀⠀⠞⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡜⠷⠞⠛⣁⣩⡶⠁⠾⠇⠀⣼⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⣡⡄⠀⢀⣴⠋⠀⠂⠀⠀⢀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⣻⣦⣄⣸⡇⠀⣤⡤⠀⠰⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⢠⣽⣓⢿⣿⠁⢸⡿⠶⡶⠀⠉⠢⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠁⠨⣿⣿⣿⠁⢼⣄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⡀⠀⠀⠉⣹⣧⢨⡀⠁⠐⣤⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⣿⣿⣧⣀⠀⣠⣁⡀⠀⠀⠈⢻⣿⠓⠈⣿⠛⠁⣤⣧⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠈⢿⡆⢰⣿⡿⡜⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠟⠋⢩⣿⣿⣷⡄⠀⠀⡖⠀⢳⣞⣿⣿⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠛⠛⢿⠃⡄⠉⢀⣼⠟⢻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣈⠀⠀⣤⣶⣾⢿⣦⣿⡇⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⣠⣼⡿⢫⡼⠃⣿⣧⣭⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠹⣍⠙⢀⣶⠶⣾⠟⠷⢮⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠳⢀⡀⣈⣿⣦⡙⠶⣶⡄⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣠⠤⠀⠂⠀⣿⣶⠆⠀⠀⠀⠀⠀⠈⢀⣞⣠⡿⣿⡉⢻⣦⡉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠄⠀⠻⡟⠀⢀⠀⠀⠀⠀⠀⢸⡿⢉⣿⡿⠃⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⢾⣿⣷⠀⠀⠀⠠⢀⠇⠀⣿⣷⡟⢻⣆⠀⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠰⠀⡀⠀⢰⣿⣿⡄⠐⠀⠀⣿⡄⠀⣿⣿⣷⡈⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠘⠃⠀⠀⠀⠈⣿⣷⠀⠀⢠⢸⣧⠀⣹⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢾⣿⡆⠀⠀⢸⣿⣷⣿⣎⡀⢹⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢀⠀⠄⢻⡿⢿⡿⢿⣿⣥⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢀⠃⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠃
⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡟⠁
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠿⢿⣾⣿⣆
⠀⠈⠛⢿⠿⠿⠿⠟⠿⠛⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉⠀⠈⠉⠛⢿⣿⣦⡀⠀⠀⠀⠀⠈⠛⢿⣷⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠉⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣻⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠉⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣚⣋⡉⠉⠉⠉⣉⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣯⡟⢻⣶⠏⠀⠻⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠁⠀⣀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠖⢻⣧⠘⢦⠀⠀⢀⣴⣿⠙⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠀⣿⠐⢧⡙⣦⣤⣤⡴⠟⣹⠏⠀⣷⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢿⠀⣿⠀⠀⣧⣘⣿⠤⣠⡾⠋⢠⡴⠿⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⣼⠧⠾⠦⡄⠈⠉⢻⡷⠯⠶⠾⠟⠀⠀⢀⣤⠄⠀⠀⢹⡇
⠀⠀⠀⠀⢀⡾⠋⠀⢠⡞⠁⠀⠐⢚⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠘⣧
⠀⠀⠀⢠⡟⠀⢠⣴⠋⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣿⡤⢶⡟⠁⠀⠀⣠⣴⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⢀⡿⠁⠀⢸⣷⣀⣴⣿⣿⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠆
⠀⠘⡇⠀⠀⠸⣿⡿⢹⠏⢹⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢣⡀⠀⠀⣀⠔⠁⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠸⣿⠀⢿⣄⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡟⠛⠂⣠⣤⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡧⠾⠗⠚⠉⠁⠀⠈⣿⡄⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠀⠀⠀⠀⠀⢀⣤⣴⣿⡇⣀⣤⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⣀⣴⠟⠉⠉⠛⠿⠿⠿⠟⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⣴⣿⠟⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠉⣿⠀⠀⢠⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⠀⠀⣼⠃⠹⣇⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⣿⠀⢹⡆⢰⡏⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⠀⢸⣇⡿⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡇⠸⣿⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣾⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣯⠞⣹⣿⡇⠀⠀⠀⠀⠀⠀⡴⠚⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢰⡏⣰⣿⣿⣷⣶⣶⣶⣶⣶⡞⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠇⠀⠀⠀⣾⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⡿⠛⢛⣿⣆⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣤⡶⠟⢻⣿⣿⣿⣟⣛⣛⠛⠛⣿⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠇⠀⢺⣿⣿⡀⠀⠀⠀⠀⠀⡾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠶⣶⣾⣿⣿⡷⣄⠀⠀⠀⠀⢳⣶⣶⣶⣤⣤⣤⣤⣤⣉⣉⣉⣿⣧⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣟⣮⠳⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿⣿⡇⢹⡟⣿⢹⣿⣿⠸⣧⢹⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣼⣿⡇⢸⡇⣿⡘⣿⣿⡇⣿⣾⣿⠀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢿⣿⣿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇⢸⣧⢸⡇⠿⢹⡇⣿⢻⣿⢸⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⡿⢿⣿
⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⠇⢸⡿⢸⣿⢰⠺⡇⣿⢸⡏⠀⢹⣿⣿⡟⠋⠉⠙⠛⠛⠛⠛⠿⣿⣿⣆⠾⠛⠁⠀⠈⠙
⠻⠿⠿⠿⠿⠿⠿⠛⠻⠿⢿⡟⣠⣼⠇⣸⣿⣾⡇⣇⣻⡘⣧⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣾⣥⣿⣾⡿⠿⠿⢿⣿⣿⣷⣿⡧
⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣶⣦⣐⣀⣀⣠⣄⣆⣄⣀⣠⣀⣐⡀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣶⣶⣦⣤⣴⣥⣰⣄⣄⣀⣈⣦⣠⣁⣄⣀⡏⠭⠌⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢾⣿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣏⣩⣿⣿⣟⣶⡯⢿⣃⣨⣿⣿⣏⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣻⣿⣿⢿⣏⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣿⣯⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣛⣻⣿⣯⣿⣿
⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣲⠄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⣫⣭⣥⣽⣶⡶⢶⠿⣿⣿⡿⠿⠟⠛⢋⡩⠭⠖⠒⠿⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠛⢉⠩⠝⠒⠀⠉⠁⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⡫⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣤⣀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠂⠉⠀⣀⡤⢂⣄⠀⠀⣠⠀⠰⠞⠻⠻⠛⠛⠛⠃⠁
⠀⠀⠀⢀⡴⠟⠉⠉⣀⠀⣠⣶⣿⠏⣠⡿⠷⣻⡽⠁
⠀⠀⣰⣿⣿⣈⣷⣠⣯⣧⣿⣿⣷⡾⠿⠷⠿⠿⠟⠃
⠀⣼⣇⣀⣴⣿⣽⡿⠿⠟⠛⠉⠁
⠀⠛⠶⠾⠏⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣺⣿⣿⣷⣄
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣲⣤⣴⣶⣾⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣶⣦⢤⣄⣀
⠀⠀⠀⠀⠘⣿⣿⣿⣷⠶⢛⠿⠭⠅⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠘⠛⠻⢿⠶⠄
⠀⠀⠀⢀⣴⣿⣿⣟⣉⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠉⠀⠀⣀⣠⣤⣔⡋⠉
⠀⠀⠀⠈⠁⠉⠉⠁⠀⠠⠈⠉⠉⠉⠀⠀⠀⠀⢀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠐⠄⠀⡀⠀⠋⠁⠀⠀⠀⠀⠀⢀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠒⠄⠐⠺⠿⠦⠀⠀⠀⠨⠟⠛⠛⠉⠋⠉⠁⠀⠀⢤⢶⡶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠻⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣃⣆⣠⢪⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⢛⣋⣉⣉⣭⣭⣛⣛⡳⢶⣤⣉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⣴⣶⣌⠙⠟⠉⠉⠡⣁⠠⠉⣍⠓⠈⠙⠻⣮⣌⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣯⠬⣛⢿⠙⠋⢷⡀⠀⠛⢠⠌⣰⢄⢧⢃⠀⠀⠀⠈⠙⢿⡖⡀
⠀⠀⠀⠀⠀⠀⢀⠞⢡⣮⠟⠀⠈⢳⡀⠀⢸⣇⠀⠈⠖⢀⡙⠸⠴⡋⠂⠂⢀⠀⠀⠀⠳⣆
⠀⠀⠀⠀⢀⡔⠵⠚⠉⠁⠀⠀⠀⠀⣷⠀⡌⡿⢠⡀⠀⠠⠴⣒⣶⡄⠀⠀⠀⠐⠐⠀⠀⢹⡄
⠀⠀⠀⠀⢼⣥⡆⢠⣰⣄⣀⣀⡤⠞⣹⣊⡼⠃⠼⢇⢭⣻⠯⠒⢀⡇⠀⢠⡤⠴⠲⡒⣂⠜⣷
⠀⠀⠀⠀⠀⠘⠖⠶⠓⠖⠒⠲⠿⠛⠛⠳⢤⣄⣀⣀⠉⠚⠭⠴⠋⠀⠀⠘⠛⠓⠦⠖⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣁⣪⠮⠽⠋⠉⠛⠛⠛⢿⣿⡟⢻⠛⠲⣶⣤⢄⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣨⠏⠀⠀⠸⠈⠃⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠭⣭⣥⡇⠀⠀⣤⢷⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣤⠔⠠⠛⢦⣾⠛⠳⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣯⠂⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢃⡧⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⢀⡿⣦⡀⣄⡤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⡀⡀⠠⣄⡛⣴⢏⠀⠙⠛⠛⠃⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⡶⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⡀⠰⢄⠐⢶⣄⢶⣄⠠⣀⢛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠥⠭⠀⣘⣇⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢉⠻⠾⠿⠧⠴⢤⣰⣤⡀⠀⠀⠀⠀⠘⠷⠀⢰⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠹⢄⠀⠀⠀⠉⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠼⠃⠀⡀⠖⠀⡀⠀⣄⠀⠈⣷⣄⠀⠐⢀⠀⠐⣻⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠆⠸⠢⣧⣱⠿⣤⣏⣀⠟⠀⠀⣾⡿⠛⣉⠉⠑⣚⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠄⢐⣿⡿⠶⢤⡉⢠⡆⠀⡰⣽⡛⣹⣤⣴⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠄⢀⣬⠹⣦⠨⣿⠎⢀⠄⢀⡿⠃⢿⣩⠭⠭⡿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡼⢄⣲⠝⠁⣀⠄⠁⣴⡟⠃⡟⣤⠄⡀⠄⠂⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⢀⣀⡤⠞⣁⣠⡴⠛⠙⠿⣦⢾⠆⢻⡀⠐⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠓⠚⠋⣿⣇⢈⠄⠀⠀⠙⠿⣧⣀⠠⠄⠓⠀⠈⠁