Rocks Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠓⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠁⠀⠀⠀⠠⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠑⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡘⡷⡀⢠⡄⢠⡟⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⣀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣏⣷⣽⣿⣤⡟⠀⠀⠀⠀⠀⠀⠃⠹⡀⠑⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡏⠉⠀⠈⠀⠀⢸⠧⡀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢿⣠⡾⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢣⠀⠀⠀⠀⣰⡟⠉⠉⢢⡀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⡇⠀⠀⡀⣀⠀⠀⠸⡿⣡⠔⡪⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡑⢶⢤⣏⣀⠀⠀⠀⠘⢦⡒⠀⣴⠋⠀⠀⠀⠀⣣⡤⠶⠶⣤⡀⠀⡼⢧⡀⠀⢀⣴⠟⢻⣉⠉⠉⠉⠙⠛⠿⣅⡾⡶⣃
⠀⠀⠀⠀⠙⢶⣄⣻⢮⣿⠀⠈⠉⠒⠲⠒⣶⢿⠾⠷⢤⣄⣀⡴⠛⢅⣀⠤⣾⡥⠿⣾⣀⡤⡷⢶⡞⠁⠀⠀⠑⢄⠈⠀⠀⠀⠀⠈⠛⣶⡏
⠀⠀⠀⠀⢀⣀⣙⣽⠿⡿⢤⣄⣀⡀⠀⢸⡏⠙⢳⠀⢠⠻⣇⠀⠀⠀⠣⣰⡯⠤⠤⠬⣿⣀⡇⡿⡆⠀⠀⠀⠀⠈⣳⢄⡀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠈⣹⠞⠁⠀⠱⡄⠀⠈⢹⣷⠚⢷⣤⣤⣧⠃⠀⣹⡦⠤⠶⠖⠛⠷⣤⣠⠶⠛⠙⢻⡇⠘⣄⠀⢀⣀⡰⢁⣀⣉⠒⠤⣉⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢷⡒⠒⠒⠋⠉⢓⠒⣿⠀⢣⠀⠀⠀⣿⢀⡾⠉⠓⠒⠤⡤⡶⠒⠊⢿⢳⡀⠀⠈⣷⣠⣨⡞⠋⡟⣿⣉⡏⢉⠟⣦⠈⡏⠉⣹⠇
⠀⠀⠀⢀⡀⠈⡛⠲⠶⢶⡶⠛⠛⠛⠲⣧⣤⣤⣤⠟⣿⠁⠀⠀⠈⢰⠁⢸⠀⢠⣿⣛⠛⢲⣾⡍⣹⠋⠀⠀⢱⠖⠋⣷⣫⡶⣻⠛⢷⠞⠋
⠀⡀⠀⠀⣤⠶⠶⣤⠀⠾⣄⣀⠉⠀⢂⣼⠃⠒⠂⠀⠈⠟⠫⠛⣛⠺⠶⠾⠟⠋⠻⢮⣭⣥⣤⡿⠻⢧⣀⣠⣧⣤⡴⠏⠙⢷⣡⣤⠞⠀⠉⠛
⠀⠀⠀⠈⠳⠿⠶⠞⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣴⠞⠛⢳⡦⢤⡀⠂⠀⠀⠀⠈⠭⠄⠀⢀⣭⠤⡄⠑⠄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣷⣾⣭⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠌⢣⡀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠛⠛⠛⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡆⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢊⡠⠠⣓⠵⠟⢷⣆⡀⠀⠀⠀⢰⠋⠄⠊⠀⢰⡿⣿⣿⣿⣿⣿⢢⡀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣇⢸⡆⠀⠈
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠠⡀⢔⠬⠀⠀⠄⣾⣿⠃⠀⠀⢰⠋⠀⠀⠀⠀⡾⠀⠉⠛⠙⢿⡇⠱⣽⣷⡄⠀⠀⠀⠸⣿⣿⠃⣙⣿⠡⢁⠀⠞⢲⡀
⠀⠀⠀⠀⢀⢞⡠⢖⠠⠀⢀⠤⠐⢉⡠⡐⢋⣿⣟⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠄⠂⠠⢼⣿⣞⡚⢏⣿⠀⠀⠀⠀⣽⣿⡎⠀⠘⣧⠄⣼⡆⠀⠁⢂
⠀⠀⠀⡴⠟⣍⠤⠢⠑⠈⡐⢔⠪⠑⣁⣾⣿⣿⣍⠄⠀⠀⠙⣦⠀⠀⢀⣸⠇⠀⠄⠀⢀⡴⠏⠏⣉⣛⣯⠄⠀⠀⠀⠙⠿⣧⡀⠈⠉⣷⠸⣯⣆⠀⠀⢣
⠀⠀⢰⠗⠎⠠⠀⠜⠪⠑⡬⣒⡶⣾⠩⠯⠙⣉⠎⠀⠀⠀⠀⠈⢳⣤⣽⣧⣤⣕⣈⡵⢾⣷⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⠒⠠⣄⢀⠁⢈⣓⣴⡾⠁
⠀⠀⡜⠀⣠⠒⣼⣾⣷⣾⣯⠿⠷⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠯⠒⠋
⠀⠀⠑⠠⢁⠂⠾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠛⣷⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣼⣯⡟⠁⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠉⠙⢿⣿⣶⣶⠤⣀⡀⠀⠀⠀⠀⠘⣿⣿⡿⣾⣽⣟⡁⠽⠟⠒⢦⡀
⠀⠀⠀⠀⢴⣯⣟⡋⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢷⣦⡄⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⣿⣽⣶⣿⣿⣆
⠀⠀⠀⠀⠈⢿⡍⠙⢶⣄⠀⠀⠀⠀⣠⣿⣧⠀⠀⠀⠀⣰⠁⠀⢀⢀⡀⠰⡄⡀⠀⠀⠀⠀⠀⣀⡶⠀⠀⢰⡿⠟⠿⢬⣩⣛⡙⡻⠿⢿⣿⣿⣨⣷
⠀⠀⠀⠀⠀⠀⠘⠦⣴⢿⡀⠀⠀⣰⣿⣿⣿⠁⠀⠀⠀⠙⡄⣶⣦⣈⢻⣶⣄⠙⠷⢤⣀⣀⡰⠏⠀⠀⠀⠀⠳⣀⠐⠀⠈⠩⠉⠙⠳⠾⣵⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣁⣠⣾⡽⣿⡿⠋⠀⠀⠀⠀⠀⠘⢮⣿⣿⣷⣌⡬⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⣹⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠠⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣽⣯⣙⡓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⢘⢿⣧⡀⠀⠀⠀⠀⠀⢸⣇⠙⢦⠈⠙⠛⠭⢷⣦⡀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣬⣐⡩⣟⣲⣄⡀
⠀⠀⠀⠀⠀⣠⠔⠒⠃⠠⠐⠁⢘⣦⣡⡀⠀⠀⠀⠀⠈⣿⣿⣯⡑⣄⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠸⡌⣧⢾⡭⢿⣳⣯⣝⣛⡲⢦⣄⡀
⠀⠀⠀⠀⠐⣝⠲⣤⡀⠀⢴⠾⠾⢚⡿⣇⠀⠀⠀⠀⠀⣿⣾⣾⡶⠎⢳⠀⢀⣰⡾⠃⠀⠀⠀⠀⠀⠀⠱⡈⠓⢭⡤⣉⡻⠶⣭⣟⢷⣦⣯⣓⡶⣤⢤⡀
⠀⠀⠀⠀⠀⠈⢦⡀⢉⠛⣦⡀⠐⠋⠀⠈⡀⠀⠀⠀⠀⣿⣿⣿⣿⢋⡏⣠⣾⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢀⠑⢮⣙⣷⣮⣝⣷⠶⠟⠛⠛⠛⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠺⡇⡇⢹⣀⣀⡴⠦⠖⠀⠀⠀⠀⠘⣷⠿⠖⣿⣴⣟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡜⠀⠀⠙⠫⠟⠋⠁⠀⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠻⠭⠇⠀⠀⠀⠀⠀⠀⠀⠘⠻⠯⡿⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠠⠀⠀⢀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉⠉⠀⠀⠈⠲⣌⠳⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡍⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣴⠃⡀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⣆⣠⠴⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⢀⠟⠁⠀⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢰⠀⠀⠙⣄⠀⢀⣠⣄
⠀⠀⠀⠀⠀⣀⠴⠚⢏⠉⠛⠒⠲⢤⡑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⠀⠀⠀⠈⢳⡁⠀⡼
⠀⠀⢀⡠⢴⠁⠀⠀⠈⢣⡀⠀⠀⠀⢹⣇⣀⣤⣶⣶⠖⠒⠒⠒⠒⠲⢤⣀⠀⠀⠀⢸⠇⠀⠀⢀⣀⣬⣇⣀⡇
⠀⠀⣼⠀⠘⣆⠀⠀⠀⡠⠔⠲⢄⣠⢼⠏⠢⠀⠀⠈⠉⠙⠓⢦⡀⠀⣀⠬⠽⠛⠒⠚⠲⣄⣰⠋⢆⠀⠀⠈⢳⡀
⠀⠀⠈⠓⠒⠘⠓⠒⠘⠣⢤⣤⠼⠤⠺⠄⣀⣀⣀⣀⠀⠒⠒⣁⢗⠶⢧⣀⠀⠀⣀⣀⣀⣀⠟⠒⠚⠶⠤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⢧⣀⣠⠏⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣭⣯⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠄
⠀⠀⠀⠀⠀⠇⠀⠂⡀⠀⠀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢩⡲⣶⠃
⠀⠀⠀⠀⢈⡄⠀⠀⣽⠀⡆⠀⠰⡄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠳⠅⠀⠀⡀⢈⠑⣄⢠⠂⡀⢀⣈⣴⠞⣿
⠀⠀⠀⡀⠼⣇⡀⡘⣇⣸⠃⠀⠀⢁⡥⠃⢠⠐⠞⠳⡀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣦⣺⣴⣿⣦⣀⣨⣾⠿⣛⠛⣧
⠀⠀⠒⠛⠓⠛⠓⠛⠛⠻⠄⠀⠐⠚⠛⠓⠛⠀⠀⠐⠛⠒⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠋⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣖⣀⣀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣶⠦⢵⠟⢻⡿⠟⣟⢐⣶⣧
⠀⠀⠀⢦⣁⡀⠠⣤⣴⠶⣿⣦⠠⠔⠐⠂⠦⣄⣤⡴⠖⠚⠚⠻⢿⣿⣿⣻⣿⣴⣆⣀⣰⣿⢸⣿⣿⣦⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠉⠛⠙⠲⠟⠙⠿⠲⠶⠶⠶⡈⡘⠿⠾⡵⠤⠶⠾⠿⠻⢿⣿⣯⡬⠻⠿⠿⡿⠿⠿⠿⠿⡻⠓⠂⠒⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣦⣥⣿⣵⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠙⢻⡛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠀⠙⡎⠀⠀⠀⠀⠹⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠑⠢⢤⠴⠚⠛⠉⢛⣦⡀⡀⠘⠄⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠙⠳⡥⠶⣤⣤⣖⡥⠞⠉⠉⠙⠛⠷⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢹⣄⣊⣀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⠉⠉⠉⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠤⠀⠀⠀⠈⢧⢀⣀⣠⡤⠶⠶⢴⣏⣧⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠻⣄⠀⠀⢸⡇⠀⠀⠀⢀⣠⣤⡤⠤⢤⣀⡼⠋⠀⠀⠀⠀⠀⠀⠙⣇⠙⣆⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣔⡋⠀⠀⢸⠉⠉⠉⠳⣄⠀⠀⡎⠁⠀⢀⣀⣀⣸⡖⠳⣤⠴⠒⠶⠦⢤⡀⣸⠀⠈⠻⡒⠒⡿⠠⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⢒⣲⣧⣤⣀⡀⢸⠛⢒⡷⠖⠛⠉⠉⠁⠈⠙⣾⠁⠀⠀⠀⠀⠀⠉⠳⣎⠉⢳⠚⢺⠁⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢀⠼⣄⣀⡴⠋⠁⠀⠀⠀⠉⢻⡄⢸⠧⣄⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠳⣄⠀⠶⠬⣧
⠀⠀⠀⣀⣀⠀⢈⢀⠄⣹⠀⠀⠀⠀⠀⠀⠀⠀⡗⣾⠀⠀⢉⡷⢲⠞⠉⠉⠛⠛⢲⣄⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠘⣟⣀⣀⣸
⠀⢠⡞⠉⠣⣉⠉⠉⢒⠇⠀⢠⡒⢒⣆⣀⣀⣀⣅⣈⣦⣤⡼⢤⣼⡀⠀⠀⠀⠀⠀⢹⣆⣤⣤⡤⠶⢯⣄⠀⠀⠀⢀⡟⠀⠀⠀⠹⡄
⠀⡿⠀⠀⠀⠈⠳⠦⣌⢲⡞⠉⠉⠉⠉⢡⠞⠋⠉⠉⠙⠲⢤⣄⣀⣙⣷⣄⣀⣤⠴⠋⠀⠀⠀⠀⠀⠀⣸⠧⢤⣤⠚⠀⠀⠈⠀⠀⣷
⢰⡇⠀⠀⠀⠀⢀⣤⠞⠛⠋⠉⠛⠛⠛⠛⢦⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠈⣯⠳⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢀⣴⠟⠢⣄
⢸⡀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣀⣀⣸⠁⠀⠀⠀⢀⡏⠀⢹⠀⠀⠀⠀⠀⠀⢠⡿⣦⠴⠚⠛⠛⠲⣾⡭⠖⠚⠉⠙
⠈⢧⣄⣀⡀⢸⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠒⣿⠀⠀⠉⠉⠉⢳⡄⠀⢀⣸⡃⢀⡞⠒⢤⣬⣡⡵⢤⡜⣱⠁⠀⠀⠀⠀⠀⠘⣇⣀⣀⡤⠾
⠀⠀⠀⠉⠉⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢸⡧⠖⠋⣠⠟⠉⠉⠛⠉⠁⠀⠀⠀⣷⢿⡀⠀⠀⠀⠀⣀⣼⠧⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠤⠤⠴⠶⠾⣅⡀⠀⠀⢀⣀⣠⠾⠁⠀⠐⣇⠀⢀⣤⣤⡤⠤⠤⠤⣴⡃⠀⣙⠳⣞⠛⠛⠁⢃⠀⠀⢁⣸
⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠿⢭⣭⡉⠉⣠⠤⠖⠲⣦⠈⢱⠛⢦⣀⣀⣀⡀⠀⠀⠙⢦⡙⠶⠚⠀⠀⠀⠀⠉⠉⠉
⣰⠋⠉⠉⠉⠉⠙⢲⡄⠀⠀⠀⠘⠧⠤⣀⣀⡀⠀⢀⡷⠀⢧⣤⣤⠴⠛⠀⢸⡀⠀⠀⠈⠉⠉⠉⠉⠉⠁⡿
⢧⣀⠀⠀⠐⠒⠂⢠⡇⠀⠀⠀⠀⠈⠒⠶⠶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣤⣤⣀⣀⣀⣤⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣾⣿⣿⡄
⠀⠀⢀⡀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⡴⠄⠠⠤⠄⠀⠀⠀⢠⡀⠺⣿⠟⢀
⠀⠀⢠⣗⣀⡃⠀⠟⠋⠁⠈⠙⠁⠀⠀⠀⢿⠀⠀⠀⠚⠳⡿⠷⠀⠀⠒⠤⠄⠀⠈⠉⣀⠀⠀⠁⢠⢶
⠀⠀⠈⠛⠛⠅⠀⢀⣤⣤⣤⣶⣿⠷⣄⠠⠀⠁⠀⠀⠀⠀⠛⠓⠶⠦⡄⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠘⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣇⠁⠀⡀⠃⠀⠀⠀⠀⠀⠀⠹⣿⡄⠹⠷
⠀⠀⠀⠀⠤⣄⠀⠀⠠⣤⠀⢰⣶⣴⠎⠀⠀⠀⠀⡇⠒⣿⣦⣌⡃⠀⠀⠀⠀⠀⠀⠰⠈⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠘⠓⠀⠸⠏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠳⢀⣿⣿⡟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠆⠀⠲⣾⡇⠀⠀⠀⠀⣿⠪⢻⡄⠀⠀⠀⠀⢸⢠⣴⡄⠂⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣦⠹⡄⢳⠁⠀⠀⠀⠀⠘⠸⡯⢳⠀⠔⠀⠀⠀⠀⣷⣤⣤⣀⡀⠈⣧⣠⣨
⠀⠀⠀⠰⠤⠄⠴⠚⠀⠀⠀⠀⠿⢷⠂⣼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⣿⣿⣿⣿⣇⣄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡀⠛⢿⣷⣟⡷⠲⠤⣤⠀⠀⢀⣿⣿⣿⣦⣶⡆⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠔⠚⠳⣶⡐⢫⣙⣉⡩⠄⣂⢵⡖⠂⠈⠉⠀⠀⢿⣇⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣴⣤⣄⠀⠈⠑⠒⠂⠠⢬⠀⠸⣯⣩⣷⣦⣄⣀⣠⣌⣉⣹⣷⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣏⣹⣶⣦⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠧
⠀⠀⠀⠀⣀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉⠙⠛⢻⣏⣉⣉⠉⠉⠉⠉⠉⠩⠭⠉⠉⠉⠻⠿⠾⠿⠷⠶⠶⠂⠀⢠⡄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠊⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠒⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⢹⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠙⠒⠤⣀⠈⡆⠀⠀⠀⢠⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣀⣠⣤⣤⣄⡀⠉⢣⠀⠀⣰⠃⠀⠀⠙⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠚⠋⠉⠉⠀⠀⡇⠀⠙⠲⡄⠳⡰⠁⠀⠀⠀⠀⠘⣏⢉⠟⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠓⠛⠦⣀⠀⠀⠀⣠⡃⠀⠀⠀⢻⠀⣷⡀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⢳⡔⠋⠀⠉⠢⠤⢤⣸⣶⠀⢳⡀⠀⢀⣠⠴⠟⠒⠾⣍⠉⠉⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠒⠚⠛⠻⡆⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⣸⠿⠶⠶⠛⠋⣹⠉⠳⣄⠀⠀⠈⠳⣄⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠦⣇⠀⢀⣀⣀⣀⡟⠲⣤⣀⡀⠀⠙⣄⣀⣠⡤⠞⠛⠶⣤⣠⠒⠉⠉⠉⠉⠉⢷⠀⠀⠀⠘⢧⡇⠈⢷
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢸⣶⣋⣉⡽⣿⠀⠀⠀⠉⠻⢍⣉⡉⠉⣀⣀⣀⡤⠤⡔⠻⡆⠀⠀⠀⠀⢀⡼⠧⣀⠀⠀⠈⡹⣶⠺⡇
⠀⠀⠀⠀⣀⣤⣿⣀⡀⠀⠀⣸⣤⠤⠶⠿⠿⣤⡀⠀⠀⠀⠈⢣⢨⠋⠀⠀⠀⠀⠀⠈⢆⢻⠀⠄⢀⡴⠋⠀⠀⠀⠉⡖⢺⠓⣿⣠⣇
⠀⠀⠀⣾⣋⣼⣿⠭⣿⡶⠚⠉⠀⠙⠢⣀⣀⣀⡙⢧⡀⠀⠀⢀⡿⠞⠓⢦⡀⠀⠀⢀⣼⠟⠛⠛⠛⠓⠶⠤⣄⣁⣸⠀⣾⡴⠯⣄⣈⣙⣦
⠀⠀⠘⠷⠴⢾⡿⣏⣁⠀⠀⠀⠀⡠⠊⠹⡄⠀⠀⠉⢳⠀⡼⠛⢄⣀⣀⠀⠙⣦⢀⡞⡹⠦⣄⣀⣀⣀⠀⠀⠀⠉⢻⢒⡿⠶⣴⣧⣤⣤⡼⠃
⠀⠀⠀⠀⠀⢸⡏⠁⠀⠉⠉⠉⠙⡄⠀⠀⠘⡄⠀⣠⣼⢶⡇⠀⡼⠁⠈⠉⠉⠙⣿⣷⠁⠀⡼⠁⠉⠉⠉⢉⣩⠿⢋⠸⠯⠭⠿⠁⠠
⠀⠀⠀⣠⡶⣦⠓⢦⣤⣤⣤⣀⣀⣸⣤⣤⠶⠾⠛⠛⠶⠭⠽⠿⣤⣀⣤⣤⡤⠞⠁⠈⠙⠺⠷⠶⠒⠒⠛⢩⣵⢶⡄
⠀⡀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠈⠉⠰⣾⣭⠿⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠋
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠃
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡒⣇⠈⠂⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⡁⠘⣆⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⣠⠋⠈⠘⡆⠀⠈⣐⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⢢⡃⠀⠀⠒⠙⠶⢰⠃⠀⠁⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣄⠠⠏⠀⠀⠀⡬⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⠁⠈⠐⠂⠇⠀⠂⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠸⠀⠀⠀⠤⡀⢸⠉⠀⠀⢨⡁⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠄⠀⠀⢣⠜⡇⢠⠃⠸⠀⠅⠠⠀⠀⢁
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠅⠀⠀⠀⢀⠁⣼⠀⠨⡀⠁⠈⠐⠡⠀⠂
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠂⠀⡁⠀⠈⡅⠀⠀⠀⡀⠀⢸⠀⢀⡗⠠⠀⠅⡈⠀⢰⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠂⠀⠂⠠⠀⠈⡹⠃⢀⠘⡀⢐⠁⠂⠀⠘⠛⡄
⠀⠀⠀⠀⡀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠸⠨⣐⠀⠀⠀⡖⠀⠀⠌⠀⣂⢄⢈⠂⠀⡀⢀⠈⢆
⠀⠀⠀⠀⡄⢸⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡨⠍⠀⠀⠀⢸⠈⠨⠀⠈⠉⠠⡩⠍⠀⠄⡯⢗⠍⢸⡄
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⢠⠆⠀⠀⢨⢁⠂⠅⠀⡬⠐⠈⠀⡘⠀⠀⠚⠒⠒⡖⠢⠒⠔⠊⣻⠓⠂
⠀⠀⠀⠀⢧⡻⠆⠀⠀⠁⠀⠀⣔⠁⠀⠀⠀⡬⠀⠀⠁⠆⠙⠂⢀⠜⠀⢀⠀⢀⠀⠀⡇⡈⠀⠈⠀⢻⡇
⠀⠀⠀⠀⠀⠙⢢⡐⠀⠀⢘⠀⡇⠠⠀⢀⡴⠁⠀⠀⣀⠀⢈⠔⠠⠀⠀⠀⠀⠀⠂⠂⢹⡀⠀⠀⣠⠼⠀⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠⠥⣁⡁⠀⠀⠊⠀⠀⠀⢀⣀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⠠⠞⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠒⠦⠒⠆⠬⣥⢉⡀⠀⡤⠬⠤⠤⠤⠀⠤⠊⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⡀⡀⠀⠀⠀⠀⠀⠈⠉⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣜⢆⠐⡀⠀⠀⠀⠣⡀⠀⠀⠋⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢾⣦⠻⡄⢀⠀⠀⠀⠈⠢⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢳⣬⣫⣇⠁⢸⢧⠀⠀⠀⠀⠀⡀⠁⢈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡿⣾⢻⣿⣿⣧⠀⣧⢳⡀⠀⠀⠀⠠⠀⠹⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠿⣄⣿⣯⣿⡄⠈⢮⠃⠀⠀⠀⠠⣇⠀⠀⡈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠸⣭⣻⣾⣷⠀⠈⢣⡀⠀⠀⢠⢻⠀⠀⢀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⡀⠀⠙⢻⣿⣿⣇⠀⠀⠁⠀⠀⣴⣽⡇⠀⠈⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢠⣷⡦⣼⣿⣿⣿⣆⠀⠀⠀⠀⢿⣾⣿⠀⠀⢀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠈⠉⠀⠹⣿⣿⣿⣿⡟⢷⣶⣄⣀⠀⠁⠀⠀⠈⠄⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡾⣟⢿⣿⣿⣦⡀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣀⠀⠀⠀⢰⡹⣼⣿⣿⣷⣜⢮⡳⣝⢿⣿⣷⣄⠀⠀⠠⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⢂⠀⠀⠺⡿⣿⣿⣿⣯⣾⣷⡝⢮⡳⣝⢿⣿⡆⠀⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⡿⠾⢶⣿⣿⡿⢿⣿⢿⣿⣮⡀⠙⢮⡳⣽⣿⡄⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠝⡉⠈⠀⣐⠉⠪⣙⣇⡀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⡶⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⠀⠀⡰⠁⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⣀⡤⠼⠤⣀
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠒⠈⢳⡀
⠀⠀⠀⢠⠏⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢸⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢄⡀
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⠀⠀⠀⢀⡴⢂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠰⣄⡈⢳⡄⢀⣴⡏⠀⠈⢳⢄⢠⡾⢡⣻⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⡇⠙⢶⣄⡳⣏⡏⢂⣹⣊⣃⣀⣀⣀⡀⢨⢃⡷⢃⡟⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⣇⠐⠲⣝⢝⢿⣿⠋⠁⠀⠀⡰⠁⠙⣿⠞⢸⡇⡜⣸⡟⡅⠀⠀⠀⢠⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠘⢦⠖⠚⠻⣆⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⣤⠗⠛⠓⣶⠒⢦⠀⢸⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠘⠢⠤⠔⠋⠙⠲⠤⠒⠒⠒⠒⠋⠉⠙⠋⠙⠦⠤⠔⠃⠐⠛⠒⠚⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠛⠛⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠙⣧
⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠹⣇
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⡇
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⣀⣸⣧⣤⣤⣤⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠁⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠐⢤⣀⡀
⠀⠈⣷⠟⠛⠛⠉⠉⠉⠉⠉⠉⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⣠⣤⠴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠉⠁⠀⠀⠀⡇
⣿⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⣀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⢻⡀⢸⡗⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⣤⣤⡿⠶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⠶⠞⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠈⢷⡈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⣀⣉⣭⣭⣭⢭⠽⠿⠿⠟⢻⢶⣄⠀⠈⢷⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠙⠻⢦⣤⣀⣀⣠⣤⠶⠞⠛⠋⠀⠀⢀⡴⢟⡉⠉⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⢹⡆⠀⠈⠙⠶⠦⣤⣤⣄⣀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠙⠒⠶⠶⠤⠦⡤⠶⡞⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⣀⣀⣀⣤⣴⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡴⠖⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣄⠀⠀⠈⠀⠀⠀⠀⢀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠖⠋⠁⠀⠀⠀⠀⠀⣀⣤⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⢀⣀⠚⠃⠀⠈⣡⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⢀⣴⠀⡀⠀⠊⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⠟⠉⣸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢄⠠⠀⠀⠀⠀⠀⠀⣤⣩⣤⣠⠠⣾⣿⣿⣿⢟⢯⡟⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠘⣿⣄⠀⠀⠀⢴⣿⣟⠿⣻⡏⣾⣿⣿⡻⢅⣤⣿⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠺⠛⠛⠓⢦⡀⠀⣀⠀⢀⣤⣴⣿⣻⣿⣥⢿⣵⣾⣿⣿⣿⡇
⠀⠀⠀⠾⠒⠚⠓⠒⠒⠒⢿⠖⠉⠐⠐⠻⠀⠀⠀⠀⠈⠀⠀⠠⠤⠀⠀⠈⣿⡆⡇⠃⣆⢁⣣⣿⢟⠝⣷⠟⣳⡾⢙⣿⣿⡏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⢰⣴⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⡇⢸⡉⠈⣫⡴⢫⠞⠁⠎⠾⠞⠗⣻⣿⣤⣀
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡀⢀⣐⣰⣿⣿⣇⣰⡾⣱⢯⠟⠺⢟⡾⠙⣽⣿⣿⣥⣤⣷⣿⠿⠋⠉⠀⠀⠀⠀⣠⡾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣆⠀⢀⠀⠀⠀⣩⣹⢿⣿⣿⡏⠀⠽⠏⠐⠀⠈⠰⠺⣿⣟⣻⣿⣿⣿⠂⠀⠀⠀⠀⠀⢀⡞⢛⡶⠫⣹⣿⣿
⠀⠀⠀⠀⢀⡀⠀⢹⡿⣅⣴⣾⣿⣿⣵⣾⣿⣿⣦⠾⠶⠆⠴⣶⣶⣾⣿⣿⠿⠿⣿⣿⣿⣶⣥⠀⢫⣷⣶⣿⣽⡏⠀⠈⣸⡛⠋
⠀⠀⠀⠈⠩⠀⠀⠀⣶⢾⡿⠆⠀⠀⢛⡉⠁⠀⠄⢐⡷⠲⣦⣤⣭⣭⣠⡄⠤⠤⠖⢠⡿⠿⠟⠛⢛⠉⠡⠄⢠⣄⣀⣀⣀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠿⣿⠁⣶⡎⢻⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣄⡀⣾⠉⢿⡄⠲⢤⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⣺⣕⡦⠴⠛⠻⣿⣶⣦⡝⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠮⢽⠯⢟⣫⣤⣔⠊⠙⠛⢦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣽⡞⠀⢾⣷⠄⣸⠿⣟⡠⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⢩⣿⡥⠌⣿⣇⣀⡤⣿⡻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣖⣠⣤⣖⣮⣟⣳⣽⣳⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢈⡹⠿⡛⠯⢟⠮⠿⡷⠽
⣿⣿⣿⣿⣿⣿⣿⡿⢁⣤⣾⣿⠞⢠⣤⠤⠘⠳⣾⣧⣄⡹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⠉⣭⡿⢛⣻⣿⣍⠉⠙⠻⣧
⣿⣿⣿⣿⣿⣿⣿⠃⠛⢿⣿⣿⣄⠐⠿⠟⠀⣠⣿⣿⠟⠃⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣥⡀⠀⠈⢿⣯⣕⣤⣿⠟⠀⠀⣠⣼⡆
⣿⣿⣿⣿⣿⣿⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢚⣿⣟⣛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣋⣭⠤⢶⡺⠉⢠⣤⠈⠳⣌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣛⡩⠔⠚⠉⠀⠀⠀⣈⣇⠀⠈⣽⡄⠀⢿⠜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⠕⠊⠁⣀⠀⠄⠀⠀⢀⠀⠈⠘⣧⣦⣄⡐⠼⠋⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡥⠔⠉⠀⣀⠀⠀⠉⠀⠢⡄⠁⠰⠀⠄⠀⠀⠉⠁⠀⠀⠀⢃⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⠰⠂⠁⠀⠀⠀⠀⢠⠀⠰⣠⠀⠎⠸⡀⠊⡀⠈⠀⠀⠀⠀⠀⠀⠀⣘⣧⠘⣧
⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠶⠖⠛⡩⠄⠒⠈⠀⠀⠠⠐⠀⢀⠀⠀⠀⡇⠂⠀⢣⠐⡀⢣⠈⠘⡄⢀⠀⠀⠀⠀⠀⣠⠽⢿⡄⠹⣆
⠀⠀⢸⡇⢡⡥⠼⠒⠶⠦⣄⡈⠁⠀⢀⠀⠀⠀⢀⠀⢀⡁⠘⡘⡆⠄⠹⠘⡀⠈⣇⢡⠘⠶⠃⢉⡡⠀⠀⠀⠀⠀⡇⢸⣦⠹⡆⢹⡄
⠀⠀⠀⣷⠀⡇⠀⠲⣶⡆⠘⡇⠠⠀⠠⠀⠀⡄⠰⡀⣋⡀⠀⡇⢡⡌⠄⢃⣉⠕⠘⠋⢠⣀⣠⣌⠀⣀⣀⠤⠔⢒⣙⣲⣴⣞⣁⣈⣧⡀
⠀⠀⠀⠘⣇⠙⢄⠀⠼⠧⣠⣷⠂⠀⠘⡄⠠⠃⠐⣷⠀⠄⢀⡇⠀⠘⠶⠀⠀⠤⣀⠤⠛⢉⡠⢈⣈⣥⣴⠶⠿⣻⣛⣛⠛⠛⣯⠉⠁
⠀⠀⠀⢀⣘⣦⠀⠙⠒⠚⠟⠉⠀⠀⠀⢳⠀⠶⡀⢹⣇⢉⡠⠷⠊⠀⢀⡠⠔⠉⣀⠤⣊⣥⠶⠿⠿⠁⢲⣲⠟⢉⣭⡉⠳⡄⢹
⠀⠀⣠⢺⡇⢹⡎⠠⢱⠄⠀⠀⠀⠀⠀⠀⣇⡀⠱⠖⠁⠀⠀⢀⡠⠒⠁⡀⢔⣪⣷⡟⠫⠀⠀⠀⠀⠀⣈⡁⠀⢘⣻⡀⠀⣿⢸⡀
⢀⡔⠻⢸⡇⢨⢿⡀⠀⢾⡄⣀⣀⡀⠈⠀⠈⠠⠀⠀⢰⠦⠂⠁⠠⢐⣯⡶⠛⡡⠀⠐⠂⡤⠀⠀⠖⠄⠐⢷⣄⡀⡉⣄⡴⠋⠘⡇
⣾⢐⠀⠈⣧⠸⣌⣧⠀⡼⠉⠠⣤⠙⣦⠀⠀⠀⠀⣀⠓⢀⣰⣴⢾⠋⠀⢃⠰⠐⢲⠐⠊⢱⠀⣰⠃⠁⠀⠈⠙⠉⠉⠁⠀⢇⠀⣧
⢻⠘⡇⠀⢹⡀⠈⠛⣶⠀⠳⣄⠙⢂⡾⢠⠀⣂⣭⣶⠞⠋⠁⣼⠘⠂⡄⢸⠀⠀⢸⡆⡀⢸⠀⡴⡄⢰⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿
⢸⠀⡇⠀⠘⣇⠀⢘⡜⠶⠶⠶⠿⠿⠖⡟⠛⣏⠈⣷⠘⣁⣀⡇⠀⠃⠐⢸⠀⠂⡀⡇⠃⠸⠆⢃⣃⠈⠀⢀⣀⠀⠀⠀⢒⣿⡇⢺⡇
⢸⠀⢹⠀⠀⣿⠀⠀⣶⠀⠀⠀⠀⠆⠀⡧⡤⢸⢠⠟⠠⠤⠼⠧⠆⠈⠓⠈⠉⠉⠁⠙⢉⡴⠋⠁⠀⠀⠀⠀⠀⠉⠒⢎⣭⠙⢧⠘⡇
⢸⠀⢸⠀⡀⣿⡀⢀⡝⢉⣉⠑⣄⠀⠀⠀⠀⠀⠀⣄⣠⣀⣄⠀⡀⠠⠤⠤⠤⠤⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡬⠭⠥⣝⣰⠷
⢸⡀⠀⡀⠁⢸⣇⠈⣇⠀⢿⠀⣿⠀⠀⠀⢀⣀⠫⠉⠁⠀⠀⠀⠤⠤⠤⠤⢤⡄⢰⠁⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠸⡇⢣⠃⠸⠀⣷⣀⣈⣱⣰⣾⣥⣯⣤⠤⠴⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠉⠃⡆⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣧⠘⠈⠀⡶⠭⠭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠸⡄⣣⠿⢥⣄⡤⠤⠤⠶⠖⠒⠂⠀⠉⠉⠉⠉⠀⢀⣒⣒⣒⣀⣀⣀⣈⣥⠥⠽⣦⠀⠀⠀⢸⠁⠀⠀⠇⡄⠀⠀⠴⠀⠀⠘⠂⠘⠀⠄⠀⠸
⠀⠀⠱⠏⠠⠤⠤⠴⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⢸⠀⠄⠘⠀⡇⠀⣐⠀⠀⡀⠀⢰⠀⠀⡂⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡆⠀⠁⠈⠈⠁⠙⠀⠉⠈⠉⠈⠀⠁⠙⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣶⣶⣯⣥⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⣤⡶⠂⠈⠉⠀⠉⠁⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⠿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣠⣤⣄
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣦⣤⡤⠲⢖⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠐
⠀⠀⠈⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⣋⢀⣠⣀⢈⣉⡉⢉⠛⠛⠷⠖⠐⠄⣩⠙⠀⠀⠀⠀⠀⠀⠘⠃⢘
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣼⣿⣿⣿⣾⣿⣟⣿⣧⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⡀⢀⡃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣭⣀⣘⣲⡄⠀⠈⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⣄⠘⡇⣸⣶⣠⢼
⠀⠀⠀⠀⠑⢤⡌⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⣶⣶⣬⡇⢶⡇⠀⠀⠁⠀⠀⠀⡀⡻⣿⣶⣿⣿⣿⣟⣾⣴⣦⣖
⠀⠀⢀⣀⣸⣾⣿⣦⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⠆⠀⠔⠀⠝⠳⣼⣿⣿⣿⣿⡿⠟⣛⠏
⠀⢠⡽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣰⣄⣤⣴⣿⠟⠀⢉⣿⡁⠀⠘⡇
⠀⠀⠚⠉⠛⠛⠛⠛⠛⠛⠛⠛⠿⠟⠛⠛⠻⠟⠟⠛⠻⣿⡟⠛⠛⠻⢿⠟⢿⠿⠿⢛⡿⠻⢿⡿⢿⠿⠿⡿⠿⢿⢿⣿⣶⣾⣿⣿⡿⣿⠶⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠚⠿⠙⠓⠆⠀⠀⠀⠀⡀⣀⣤⣄⣀⠀⠀⠈⠀⠁⠂⠉⠀⠉⠉⠉⠀⠀⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⣀⠀⠀⣀⣀⣀⣀⡤⣤⣠⢤⣄⣀⣀⣠⣤⣤⣤⠀⡔⡖⣿⣱⣽⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡼⠿⠘⠁⠀⠓⠈⠐⢛⠛⣫⠏⡇⢟⡕⠙⠟⠛⠉⠉⠉⠉⠉⠁⠉⠩⠤⠀⠉⣼⣯⣿⠅
⠀⠀⠀⠀⠀⠀⢀⢔⡕⠁⠀⢀⢤⡠⡀⠀⠀⡰⣡⣾⡟⡼⢱⢿⡇⠀⡆⠀⠀⢠⣠⣤⣤⡄⠀⠀⠀⠀⡇⢩⣏⣿⡄
⠀⠀⠀⠀⠀⡠⡣⠋⠀⠂⠀⠀⠂⠑⠁⠀⠀⣿⣿⣧⢊⡆⣿⣿⡇⢠⠃⠀⠀⠊⠛⠛⠛⠀⠀⢀⠀⡜⡀⣾⣿⣷⠃
⠀⠀⠀⠀⡔⣵⣅⣤⣤⢠⢂⡀⡴⠴⢂⣀⣀⣰⡿⢃⢪⡇⣿⣫⣄⣁⣀⣀⠃⣀⡀⢃⣐⣛⣫⠭⡜⣸⡞⣼⣿⠃
⠀⠀⠀⢬⣬⣭⣿⠿⣶⠶⣶⣮⣭⣥⣭⣍⣙⣻⣣⣇⣬⣇⣿⣭⣿⣿⣿⣿⣿⣿⣿⣯⡭⠼⠶⠶⡾⣻⣿⣿⠁
⠀⠀⠀⢸⢻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣗⣒⣶⠿⣿⣄⣀⣈⣭⣙⣶⣖⣶⣿⣿⠇
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⢯⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⢀⣤⣄⣀⣀⡴⢦⣀⣠⣿⡿⠛⠉⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⣠⠞⠁⠈⠉⢁⣠⣶⡿⠟⠉⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⡾⠁⠀⠀⣠⣴⣿⠟⠉⢀⣠⠶⣤⣀⣀⣠⡄⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⡴⠋⠀⣀⣴⣿⠟⠋⢀⣠⠶⠋⠁⠀⠀⠉⠉⠁⠹⣦⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⣀⣴⣾⠿⠋⠀⢠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣤⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⣤⣾⡿⠛⠁⣀⠀⠀⢻⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣠⣶⡿⠛⠁⢀⣴⠞⠉⠀⠀⣼⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡿⠟⠉⢀⣤⠞⠋⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⢠⣾⣿⣿⠟⠉⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢀⣠⠶⠋⠁⣠⣴⡿⠟⠉
⠀⠀⠀⣰⣿⠋⢸⣿⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣤⣾⡟⠀⠀⠀⠛⠁⣠⣴⣿⠟⠋
⠀⠀⠀⢿⣧⡀⠀⠹⣿⣄⠀⠠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠈⠻⢿⣦⣄⠘⢿⣧⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠀⢀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⠻⣷⣄⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠙⠛⠉⠙⠳⣦⠞⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣆⣀⣠⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠴⠒⠂⠀⠉⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠉⣠⠔⠊⣀⣀⣈⢉⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠞⣁⡴⢧⣄⠠⠶⠶⠤⠰⠶⣄⡀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡾⢁⠀⢀⣿⠷⠆⠀⠤⣭⠏⢴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠸⠋⠀⣀⣉⣫⡥⠩⣟⣓⡉⠙⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⠈⢸⠘⠀⡄⠀⢀⠐⢒⡂⣄⣓⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠸⠰⣇⠋⠰⠈⡀⠡⠍⠰⠸⢀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠁⡀⡀⠃⠘⠧⠙⢳⣤⣤⣤⣤⡾⢋⡹⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⡀⢃⡅⢰⡘⠦⠠⠀⡿⠁⠈⢻⠧⠠⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⣸⣿⣄⠉⠂⡴⠀⡇⠀⠀⠨⠗⠄⠞⢃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠻⣷⣾⠁⠀⠐⢦⡴⠂⠀⣸⣾⠛⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣾⣿⡉⢦⡀⠈⠳⢦⠀⠚⠃⢠⡴⣋⣠⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡟⢳⡄⠁⠀⠀⠈⠛⠾⠟⢩⡾⠋⢉⡡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠘⠻⣦⡀⠠⣀⣀⠀⠠⠏⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣄⡀⠀⠙⢦⡈⠙⠛⠃⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠢⡄⠁⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢰⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣈⣠⡿⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠡⠀⠐⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠛⠀⡄⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢇⠀⠤⠖⠒⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠆⡠⡅⣸⠀⠀⠀⠀⠀⠀⢈⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢹⣟⡄⠀⢀⡔⢹⣷⣿⣇⠢⠀⢦⠔⠀⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠆⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢲⣎⠀⢸⣿⠘⣰⢎⢀⣿⣿⣿⣿⠀⠀⡟⢠⢻⣾⠁⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠸⣿⣿⠦⢀⣾⣿⣿⣿⣿⣧⣀⠀⡿⣾⣏⣳⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⢰⣿⣿⡀⠈⣿⢿⣟⣉⡽⠿⠛⠚⠛⠛⠻⠖⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⣿⣿⣧⡀⠘⢿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⢁⠀⡀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣇⣿⣿⣿⣿⡖⠋⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠈⡴⠤⢤⣶⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠃⢘⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠐⠐⠲⢄⡀⠀⠀⠀⠀⠀⣼⣀⣠⠾⠟⠁⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠠⠴⠾⠋⠈⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠿⣷⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣄⣀⣤⣤⣴⠶⠛⠋⣀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣁⣠⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣾⠿⠟⠛⠻⠿⣶⣤⡐⢶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡉⠙⠛⠻⠿⠿⠿⠏⣰⡿⢋⣤⣶⡿⠿⢿⣶⣄⠙⢿⡆⠹⠿⠿⠟⠛⠋⢁⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣤⡄⢸⡿⢡⣾⣿⡏⢰⣶⣦⣽⣿⣷⡀⣴⡄⢠⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⣿⡇⢸⣿⣿⣷⣬⣈⡙⢿⣿⣿⡇⠘⠃⢸⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣤⣄⢸⣷⡈⢿⣿⣏⠻⠿⠟⣸⣿⡿⢁⣿⠇⣀⣀⣤⣤⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣷⣌⠛⠿⣷⣤⣾⠿⠋⣠⣾⠏⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣧⡌⠛⠿⣶⣦⣤⣴⣶⠿⠛⠡⠾⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠋⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠤⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠙⣆
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠸⣆
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡇
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣤⠴⠷⠒⠒⠒⢦⣀
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠦
⠘⣧⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢠⠎⠀⠀⠀⠀⠢⣄⣀
⠀⣸⠟⠉⠉⠉⠉⠉⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠤⠴⠒⣿
⡾⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢻⡀
⢧⠈⡗⠀⠤⠤⠤⠤⢤⣤⣤⣀⣼⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⢾⡆⠉⠀⠀⠀⠀⠀⢸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠘⢦⣻⠀⠀⠀⠀⠀⢀⣀⣠⠞⠁⠀⣀⣬⠭⠭⠿⠿⠟⠛⢛⠷⣄⠀⠻⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠉⠓⠲⠶⠖⠚⠉⠉⠀⠀⢠⡞⠓⠤⣀⣀⣀⣀⣀⣀⡞⠀⢸⠀⠀⠈⠙⠛⠒⠲⠤⡤⠤⠶⠶⠒⠛⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠈⠉⠉⠉⢹⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⣤⠴⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⡌⠂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⣸⠀⠈⠙⠃⠀⠐⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⣍⡆⠀⠀⠀⠠⠀⠀⠈⠀⡏⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠉⢸⠆⠀⠀⠀⢁⡦⠀⠀⢇⡏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡤⢿⡷⠐⣏⣠⣖⡒⠉⡁⠀⣠⠊⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⢠⡾⢋⠅⡼⢟⣡⣄⠄⣰⠃⡀⠀⡀⠉⠳⣄⠐⠁⠀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣮⡰⣇⠈⢧⠸⢊⣴⢋⡜⠁⣼⠃⢸⠀⠀⠈⠛⢆⠈⠢⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡲⣆⠹⣿⠇⢸⢠⢺⡎⢸⢀⢸⡎⢀⡟⠀⠈⠉⠒⠄⠀⠀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡉⠟⣸⠇⠠⠟⠈⢯⠃⣟⢸⠸⡆⠸⠁⠀⠀⢀⠀⠀⠀⠴⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠓⣀⠀⠈⣦⠀⠀⠀⠈⢧⠙⠮⠧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣤⣦⠀⠶⠆⠛⠛⠋⠀⣤⣤⡤⠀⠙⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢀⣄⠤⠠⠖⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣮⣿⡷⠶⠤⣤⣤⣠⣴⣶⠧⠚⠛⠉⠀⠀⠈⡀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⢿⣉⡉⣉⠙⠟⣳⡮⣗⠒⠖⠻⠿⠳⣦⣤⣤⣄⡤⢀⣀⡟⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠉⠉⠁⣠⠏⠀⠀⢹⡆⣌⡀⠀⠿⠛⠉⠉⠉⡠⠜⠛⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠉⠀⠀⣰⣿⣗⣻⡿⠇⠀⠄⠒⠛⢁⠀⠀⠀⠒⢛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡖⠂⠄⠒⠀⠀⠄⠐⣦⣬⠟⠋⠁⠀⠀⠀⠀⠀⠖⢀⢴⡾⠃⢺⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠷⢿⠟⢁⣀⡀⢀⡀⣠⣤⠄⣽⠇⠴⠟⠲⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⡤⠶⠛⠂⠈⠴⣉⣀⠻⠜⠓⠒⠠⠄⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢀⣦⣶⣤⣤⠙⠀⠀⠤⠄⠀⠬⢁⣀⡐⠠⠀⠀⢄⠉⠀⠀⠀⠀⠀⢀⣼⡷⠋⠁
⠀⠀⠀⠀⠀⠀⡀⣬⡆⠀⠀⠀⠒⠀⠀⠀⢶⡶⠟⠁⣠⠆⠀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⢀⠀⠉⠉⠀⠦⠀⠐⣾⡆⣠⠿⣁⣤⠀⠈⠉⠁
⠀⠀⠀⠀⢀⡶⠈⠀⠀⠀⠀⠀⠠⣀⠀⡀⠈⠙⣄⠠⠬⠀⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⡷⠦⠀⠀⠀⠀⠀⠀⢀⡀⢀⠈⡀⠀⠀⠀⠀⢠⣄⣤⣰⡄
⠀⠀⠀⠈⠙⠁⠀⠀⠀⠠⣀⣀⠴⠟⠐⠚⠂⠚⠃⣀⠀⠀⠀⠀⠐⠒⠚⠃⠀⠀⢤⣶⢾⡁⣀⣀⠀⢀⣨⣧⠀⠀⢻⡥⠘⠃⠀⠀⢠⡀⠀⠾⠛⠛⠋⠁⠁
⠂⠀⠀⣀⢠⣴⣶⠀⠀⣰⣟⣀⠀⠤⡀⣤⣤⣤⣤⠽⠆⢄⠀⠀⠐⠦⣤⣤⣤⠄⢀⣄⡀⢀⣀⠈⠀⠘⠁⠀⠀⠀⠀⠀⠐⠀⠐⢨⣸⣿⡆⠀⠀⠀⠄⡀
⠀⠀⠀⠛⠛⠉⡁⠀⣴⡟⢈⡀⣠⠔⢛⣻⡏⠀⣀⣀⠠⠴⠃⠤⣀⢴⠄⠉⠁⠀⢸⡿⠀⠀⠉⠀⠀⣴⣿⣦⠀⡀⣀⣀⣀⣤⣤⣾⠋⠀⠀⠀⠀⣀⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠁⠚⠋⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠠⠈⠉⢻⠛⠂⠀⠀⢙⡟⠉⠙⠛⠀⣡⡈⣶⣶⣿⣿⣿⣟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣠⣿⣟⠃⢀⣴⣶⡆⠀⠀⢙⣶⣶⣽⣌⢩⡌⠛⠻⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⢾⡛⠉⠁⠀⠀⢠⡓⣚⣁⣠⣾⢾⠿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⣀⠀⡀⠀⢁⠍⠉⢉⣷⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡄⣀⣠⣾⡿⢀⠔⠁⠀⢠⣿⡿⠟⠉⠀⠀⠀⠀⠀⣆⢀⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣤⣶⣾⣿⣟⠟⠋⠚⠁⠀⢀⡴⠋⠁⠀⠀⠀⢀⣀⣀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⣡⠇⡀⠀⠀⣤⡴⠋⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⡿⢿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠀⢀⣼⠃⠀⠀⠀⠀⠀⠶⢤⣶⣿⣿⠿⠟⠟⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢴⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠃⠀⠰⠤⣤⣄⣰⣄⣤⣠⣴⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠟⠛⠛⠓⠶⢤⡄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣾⣿⣁⡀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣾⣶⡿⠿⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⡆⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠙⠻⣟⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⠀⢀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⢩⣍⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣰⣿⣏⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⣿⣯⣿⣿⡄⠤⠀⠀⣀⡤⠤⣄⠀⠠⠤⠤⠄⣾⣿⣿⣿⠛⠿⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠛⣿⣿⡅⠀⣶⣦⠀⠀⠀⣠⣴⣾⣷⣦⡀⣿⡟⠙⢯⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⠒⣿⣿⠧⣼⣿⣧⣤⣤⣾⣿⣯⣍⣩⣿⣿⣿⣷⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠇⠈⢿⣿⣿⣿⠿⠁⠉⣉⣽⣿⠏⢽⣿⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣀⣀⣀⣉⣉⣡⣤⣤⣀⠈⣛⣁⣀⣿⣿⡄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠉⠁⠀⠀⠀⢸⣇⠀⠉⠛⢤⣀⠀⣹⢹⣧⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⡀⠀⣰⠞⠶⣀⢿⣄⠀⠀⠀⠈⠉⠻⢀⣽⣇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠴⠶⢾⣿⠇⠀⣿⣦⠀⠈⠃⠙⣷⣦⣀⠀⠀⠀⠀⠙⣿⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⣄⠀⠀⠈⠆⠘⠏⠳⢤⣀⣀⡄⠘⣷⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢷⣤⣤⣤⣿⣶⠀⠀⠀⠐⢶⣿⣾⣿⣻⣿⣦⠈⠇⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠁⠙⢏⡀⠠⣧⡀⠀⠀⠀⠙⢿⣿⣿⡙⢧⣄⠈⠀⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠁⠺⣿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣄⣽⣧⠀⢠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⢀⡀⢀⠈⠉⠛⢿⣷⡄⠐⠀⠀⠀⠀⠋⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢠⣾⣿⣧⣄⠀⠀⠀⠉⠑⠲⣆⡀⠀⠀⠀⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣾⠿⣷⠻⣿⢿⡄⠀⠀⠀⠀⢻⣅⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠀⢠⢜⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢲⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠿⢿⣿⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠠⠴⢬⣭⠷⢶⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠳⢦⣶⣦⣀⠂⠠⠍⢸⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠙⠓⠚⠂⠿⠿⠿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣄⣀⣀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣛⡿⢿⣿⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⢛⠹⠁⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⠀⣤⣤⠄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠮⡥⠉⢡⡽⠷⡜⠁⣤⢾⣷⡈⠟⡃⣸⡷⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⢈⣧⣶⢶⡰⣟⣽⢳⣧⢻⣿⢯⣶⣷⣾⡽⣿⡿⠈⠽⠛⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⣏⡬⢿⣿⠛⠓⠈⠀⠀⢀⣀⣠⣤⣀⣉⡉⠁⠻⣿⣃⣴⣿⣼⠏⢤⡀
⠀⠀⠀⠀⠀⠀⣼⣃⡏⢰⣿⣶⠊⠁⠀⢀⣤⢞⡟⢋⣹⠯⣌⣿⣷⣿⣂⣦⠀⠐⡍⢻⣟⠁⢃⠷⡀
⠀⠀⠀⠀⠀⢸⡩⢮⡕⠆⣻⠁⠀⣠⡾⡹⢿⣭⡽⣶⣿⣿⡶⠟⡛⠈⠀⠁⠀⠀⠐⠀⠘⣿⣯⠚⡧
⠀⠀⠀⠀⠀⢺⣶⢄⡟⢈⣿⠀⠀⣿⣟⣷⢭⣍⡁⠻⣿⣿⣷⣋⢽⣀⡀⠀⢀⡔⠀⠈⠀⢸⡷⢾⡶
⠀⠀⠀⠀⠀⢨⡹⠾⣷⡈⠿⣦⡴⣿⡋⠀⠠⣄⠴⠷⣌⠳⣤⡘⢦⣈⠙⠓⠋⠀⠀⠃⠀⠐⡷⢶⡁
⠀⠀⠀⠀⠀⠀⢻⡄⠾⣛⠛⣫⣽⡏⠂⢀⣲⡃⠀⠀⠀⠙⠚⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⣼⡯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠨⢙⠘⠛⠛⠊⠁⠀⣤⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣸⡷⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠆⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡞⣩⣹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣻⢯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⠆⠳⣿⣵⡸⠏⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣤⣴⣴⢶⣾⣾⠉⢿⣿⣣⣾⣾⣿⣿⣿⣿⣿⡶⢦⢤⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢠⡤⣦⢦⣼⠿⢛⡻⠟⣷⣀⣪⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣟⡷⢼⠯⠇⢀⣦⣄
⠀⠀⠈⢀⣤⣶⠿⠿⠳⣷⣿⠝⠓⠉⠉⢁⡴⢻⡟⣿⣿⡿⠟⠋⠉⡁⠀⠠⠠⠠⠠⠡⠉⠛⠷⣤⣶⢿⣮⣛⣉⣡⡄
⠀⢀⠶⣿⠯⣍⢿⣿⠟⠉⠀⠀⠀⠀⢠⣯⣷⣮⣛⡿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⡃⠛⠋⢩⣿⡆
⠀⣼⠲⠞⠧⠙⣼⡏⠀⠀⠀⠀⠀⠀⣾⡌⢻⣿⣿⣇⡀⠀⠀⢀⣀⣠⣤⡔⢲⣖⣶⣶⣶⡄⣀⠀⠀⠈⡿⣿⣿⡷⣿⢛
⠀⣿⠛⡿⢷⡞⢿⠃⠀⠀⠀⠀⠀⠀⢺⣵⡆⢥⡄⣿⣷⠶⢋⡽⠛⢛⣯⠒⣩⡭⣬⠉⣷⠀⣾⣗⣄⠀⣷⣮⣁⠸⢿⡞
⠀⢹⣺⢳⣿⢏⡐⣜⡦⠀⠀⠀⠀⠀⠈⢷⣶⣗⣲⢿⡶⢃⣴⣲⣮⣅⣀⡼⠿⠿⠯⢿⣯⠈⢀⣡⣥⣿⣭⣻⣭⣧⡍⣠
⠀⠈⢿⣿⣥⢾⣿⢩⢮⢀⣷⣄⡀⠀⠀⠀⠉⠻⠿⢮⢡⣼⠽⠿⠛⠉⠁⠀⠀⠀⠀⢀⣨⣿⢮⡽⢿⣿⣆⣫⣭⠟⢠⠇
⠀⠀⠀⠻⣿⢛⣏⡘⢫⡠⣽⡶⢶⣝⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣯⠿⣉⡧⠔⠘⣇⢻⡟⠋⣰⠃
⠀⠀⠀⠀⠈⠙⡯⣿⣌⣱⣮⠿⣑⡛⢈⡳⢠⣾⢻⣿⠱⡴⢇⣾⣉⣻⣻⣿⣿⣿⣿⣿⣷⣯⣼⢀⡜⣿⣶⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠍⡙⠓⠾⢿⣿⣌⣡⣿⡴⢟⣙⡃⢀⣾⣟⣻⣿⣓⡿⣿⣿⣿⣿⠿⠞⢹⠌⢻⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⢀⡉⠉⠁⠙⠛⠻⠿⠛⠿⠟⠟⠇⠃⠛⠉⢉⡀⠄⠒⡧⢀⣼⡃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⢜⡑⢿⢐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠃⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⢦⣤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢀⠶⣆⣰⣯⡿⠟⢻⣽⣿⣦⣅⠀⢀⡠⢲⣭⣵⣟⣻⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢽⣿⢿⣁⣀⣀⣀⣀⣩⣥⡿⣸⡆⢣⣵⣟⣻⣾⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣻⣖⣎⣝⢣⠮⣝⣿⣿⡆⡾⣷⡸⣃⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⣉⡛⣩⣟⡿⠛⠉⣿⣿⢚⡷⠟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⣀⣼⣫⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠒⠂

⠀⠀⠀⢀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣿⣿⣿⣿⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠀⠀⠀⣶⣶⣶⣶⣶⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⣿⠉⣿⠉⠉⠉⡏⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⣿⣀⣿⣀⣀⣠⣧⣸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠟
⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⠀⠀⠄⠀⠠⠀⠀⠄⠀⠠⠀⠀⠄⠀⠀⠆⠀⠀⠄⠀⠀⠀⠆
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠷⠿⠿⠿⠿⠿⠿⠿⠀⠀⠦⠤⠤⠠⠤⠄⠤⠤⠠⠤⠄⠤⠤⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠄⣀⡴⠖⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⡶⠞⠁⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠚⠛⠲⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⡟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠲⣶⣦⣤⠁⢸⡀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⡹⣿⣇⠘⡇⠰⡇⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠻⣧⣿⣶⣤⣤⣄⣀⢀⠠⠔⠊⡀⢻⣿⡄⢳⣼⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡀⠘⣿⣿⣾⣿⡷⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⡛⢿⡇⠀⠀⢠⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣞⡝⣷⠈⣿⢀⣴⣿⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡕⣃⢹⢰⣿⠷⠲⣦⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⣿⣅⠀⢿⣟⠀⢶⡏⣸⢛⡿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠌⣻⣶⠋⠉⢛⣾⡷⣷⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⢳⣦⡘⠦⢸⣇⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢋⣤⣠⣾⣟⠛⢿⣿⠉⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣴⣶⡤⣤⣞⡃⠀⠀⠀⠙⣿⡧⠾⡛⠛⠛⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣻⣿⣿⣿⣿⣿⡇⢿⣿⡃⠀⠀⢀⣼⠟⠳⣶⣿⣶⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⡇⢿⡋⢡⣄⡀⣀⠏⠀⣧⣉⣰⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠈⠉⣿⡃⢀⣾⢿⣿⡛⢻⠉⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣿⣿⣿⣿⣿⣿⣿⣇⢸⣇⣀⣴⣿⠂⠀⣼⡿⠈⣷⠈⢲⣶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣵⠃⠀⢰⣿⠃⠀⠉⠄⢡⢻⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⢃⠎⠀⡼⠂⠀⠀⠀⠀⠈⢆⢻⡍⢌⢷⣀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⡿⢡⠏⠀⣼⣤⡶⢾⡷⠶⣤⣄⠈⠊⢿⡌⢣⡻⣷⣄⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠛⢰⠋⠀⣴⣿⣿⡀⢸⡇⠀⠈⢻⠀⠀⠈⢻⡄⠀⢹⣾⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢻⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣹⡇⢀⣤⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡌⢿⣕⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣿⠀⣞⠇⡐⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣷⢀⠀⡄⠀⠀⠈⣿⣿⢞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢸⣿⣿⡃⠀⢠⠀⣄⠀⠀⠸⣿⠀⢻⣿⡟⠈⢀⣡⢀⠁⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣾⣿⣿⣧⠀⣞⡀⢿⣷⣤⣠⣿⣤⡿⠋⠀⠀⢿⡟⢸⣼⣈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢠⣿⡇⠀⠀⠀⠉⠛⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⣿⣿⢳⣼⣇⢠⢰⢠⣰⢠⣧⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣧⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⢶⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣀⣀⣼⠃⠀⣄⠠⠤⢶⢀⡉⠀⠀⢀⣦⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⣿⠀⢰⣿⠀⠀⠀⠋⠈⠁⠈⡤⠟⢻⣿⡇⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡀⡙⣄⢼⣏⡀⠀⠀⠠⠑⢲⡁⠠⠾⠾⢿⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠸⡇⠀⠘⣿⣦⣤⡀⠀⠨⠦⠀⠄⠀⠀⠉⠙⠁⠉⡏⣶⠂⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⣭⣇⠀⢷⡀⠀⠘⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣤⣄⠀⠀⠀⢀⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⢸⣿⣤⣀⠹⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢀⣠⡾⠟⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠄⠀⠈⠛⠷⣝⣟⠻⣷⣟⣷⣤⡀⠀⠂⠁⠀⠀⢀⣠⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠈⠛⠿⣿⣿⣏⣉⠀⠀⠀⢀⣠⣴⣿⣿⠋⠀⢀⣤⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠑⢦⣀⡀⠈⠛⢿⣿⣷⣤⣀⣀⣀⣴⡿⢁⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠁⣽⣦⣤⣤⠛⣩⣿⡿⠿⠿⠛⠂⠉⡉⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠙⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠚⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢠⡞⠁⠐⠂⠬⢍⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⠭⠄⠂⠈⠙⣦
⠀⠀⠀⢸⡖⠚⢋⣉⣓⡒⠦⢝⡮⠷⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠴⠭⣞⠥⠖⣚⣋⣙⠛⠒⣿
⠀⠀⠀⠈⢷⡉⣉⠀⠨⢭⣍⣻⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣼⣋⣭⠍⠀⣈⡉⣸⠇
⠀⠀⠀⠀⠘⢧⡠⠤⠖⠒⠒⠒⠛⠦⣄⠀⠀⠙⢦⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠞⠒⠒⠒⠒⠦⠄⣰⠏
⠀⠀⠀⠀⠀⠈⢷⣐⠊⠉⢉⣉⣉⡙⠛⠛⠒⠦⣄⡳⣐⣢⡴⣁⡤⠖⠛⠛⠛⣉⣉⣉⠉⠉⢒⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠧⣤⣤⠬⣭⣭⣭⠯⠭⢽⣏⢘⣿⠭⠭⣯⣭⣭⡭⢤⣤⡬⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣁⠤⠄⠐⣶⠤⠒⠊⡝⣿⢾⡿⡑⠒⠤⢴⡒⠀⠤⢄⣉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⣉⠤⠒⢈⡏⠀⠀⣠⢾⢀⡏⡘⡇⠹⢆⡀⠀⠙⣏⠐⠢⢌⠓⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠤⠂⡩⠳⡴⢺⠁⠀⣸⢀⢃⢹⠸⠀⢻⠲⡴⠫⡁⠢⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⢠⠊⠀⡜⠀⠇⠀⢰⠇⠈⠃⠈⣿⠀⠈⡄⠑⡀⠈⢦⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠀⡘⠀⠀⡆⢀⡞⠀⠀⠀⠀⠸⣆⠀⠇⠀⢸⠀⢈⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠶⠚⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠶⠚⠉

⠀⠴⣶⣦⡀
⠀⠀⠈⠛⠿⣆⠀⢠⣾⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀
⠀⠀⠀⠀⠀⠸⡄⢺⡟⢡⠀⣷⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠸⡇⠀⢀⣴
⠀⠀⠀⠀⠀⠀⢳⣸⡀⠈⣒⡁⣀⠀⢀⡀⠀⠀⠀⠉⢿⣿⣦⠀⠀⣴⣦⣼⣿⡄⠀⠀⠀⣀⣸⣷⣶⣷⣿⡁⠀⣇⣀⠠⠄⠚⠉
⠀⠀⠀⠀⢀⡠⠤⢿⣷⣤⣼⣿⣿⣷⣿⣧⠀⠀⠀⠰⣼⣿⡟⠀⠀⣿⣿⣿⠿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣣⣶⣾⠇⢀⡒⠄
⠀⠀⠀⢰⡁⢤⠀⠰⣾⣿⣿⣿⣿⡛⠉⠉⣷⣶⡄⠀⠈⠙⢧⠀⣿⡿⠋⢀⡀⠠⠶⢦⣭⣭⣽⣿⣼⣿⣿⣿⡇⠀⠀⡑⠤⠃
⠀⠀⠀⠀⣩⣼⣶⣾⣿⣿⣿⣿⣏⣿⡦⢀⣽⣿⣿⣦⡀⠀⣬⣿⣿⣤⣶⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⣧⣼⣿⣧⢀
⠀⠀⠀⢠⣿⣿⣟⣿⣿⣿⣿⣿⣟⠁⠀⠀⠉⠀⠘⠻⠿⢷⣿⡁⢙⣿⣿⣥⣀⡀⠀⠉⠀⠀⠉⠛⠛⠿⣤⣤⣼⣯⣿⣿⡏
⠀⠀⠀⠚⠻⣿⣿⣿⠿⠛⢻⣿⣿⣿⣶⡤⣾⣶⣶⡶⠂⠽⢿⣿⣿⡟⠉⠙⢿⣷⣶⡿⠂⠠⠀⡽⠀⠀⠉⠙⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠁⠘⢧⠾⠿⢿⣭⠀⠀⠀⢀⣾⣿⠁⢤⡀⠀⠀⠻⣿⣿⠆⠀⣘⣥⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡷⠻⠁⠀⠀⣠⣶⣿⣿⠉⠀⢸⣿⠢⣄⠀⠀⠈⠛⣝⠻⣅
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠊⠄⡞⣀⠤⡀⠀⠀⢻⣿⢿⣿⠄⠀⢿⣿⣆⠈⠀⠀⡀⢤⠈⡆⠈⠳⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠙⠻⠿⠛⠉⠀⢰⡆⠳⣘⣦⡏⠀⠀⢸⢃⠜⠃⠀⠀⠀⠉⠻⠻⠀⠀⢇⣀⣠⢃⣀⠀⠈⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⣽⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡴⠶⠶⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⡰⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣤⠒⠋⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠬⠳⡀
⠀⠀⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⣠⡇⠀⠲⣄⠀⠀⣠⡟⣡⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣇⢠⣀⠀⣀⣹⢦⡀⠻⣄⣴⠟⡇⠀⠀⠹⡳⡜⡝⡴⢹⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣿⢀⣙⠻⡦⣹⣾⣌⡦⠾⠴⠴⠣⠶⣦⣄⣸⢠⡷⢁⠟⣿⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢹⡀⠉⢳⣝⣜⢿⠏⠀⠀⠀⠠⠊⠀⠈⢷⡇⢸⡇⡜⢸⣿⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢱⠛⠍⠉⢻⣾⡀⠀⠀⠀⠀⠀⠀⣀⣼⣇⣸⣷⠋⠉⠉⣹⠉⢹⡄⠀⡇⠀⢀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁⠀⠉⠑⠚⠉⠉⠉⠉⠀⠀⠈⠁⠈⠓⠒⠚⠉⠀⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⡀⡀⡀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⠉⠉⠉⠙⠊⠁⠀⠀⠀⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡥⠿⠂⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⣀⣤⠀⠐⡾⠟⣯⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⢀⠀⢀⣠⠤⠤⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣠⡴⠃⠠⠐⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠲⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⠿⠋⢁⣽⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⣤⣉⣄⣀⡀⢴⣿⣿⣿⣿⡟⡝⣼⢯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠓⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⣠⣾⠿⣿⢿⡟⢱⣿⣿⡿⣿⠑⣰⣼⣧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠠⣴⣿⡿⠷⣄⠀⠁⠈⠋⠉⠀⠈⢉⣼⡿⣫⣾⣣⡾⣫⣹⣿⣿⣿⣿
⠀⠀⠀⢠⣤⣤⣼⣧⣠⣤⣤⣤⢄⠤⠤⠤⢤⣼⠁⠀⠀⠀⠠⠀⠀⠁⠈⠀⠀⠀⠈⢳⣆⢠⡇⡄⣌⠛⡹⢻⣿⡟⡿⣫⣾⠟⣩⡿⣿⣿⣿⣇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⠀⠀⢁⢀⡀⠀⠀⠀⠀⠀⠀⠈⠉⣀⣀⣤⣼⣿⣾⡇⠠⡿⡄⣷⠟⣫⡮⣼⠟⣡⢞⢟⡵⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⣸⣾⡄⠐⢀⣀⢠⣤⣼⣻⣟⢿⣿⡿⣿⣿⣿⠁⠘⡇⣀⣸⣞⣅⠻⠁⠘⠃⠉⠉⠉⣩⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠠⠀⢠⣥⣾⣿⣿⣿⣠⡿⢃⣞⡿⠡⠙⢋⡞⡑⣙⣿⣿⣿⣶⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⢀⣠⡾⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀⠀⠀⠈⣀⣤⡼⢹⣿⣿⡟⠀⠀⠻⠁⠁⠀⠀⠐⠛⢿⣿⣭⣿⣿⣿⣿⠃⠀⠀⠀⡀⠀⢀⢠⣛⢉⡵⠫⠪⣾⣿⣿
⠀⠀⠀⠀⠀⣀⠀⠀⢻⡿⣆⣸⣶⣿⣟⣯⣴⣾⣿⣿⣷⡴⠿⢶⠂⠶⣶⣷⣿⣿⣿⡿⠿⠿⣿⣿⣿⣷⣧⡄⠘⣸⡷⣿⣿⣾⣏⠀⠠⢁⣛⡙⠁
⠀⠀⠀⠀⠉⡉⠀⠠⠠⣶⢾⡿⠧⠀⠈⠘⠛⠋⠉⢀⡀⠀⣲⠶⢶⣤⣬⣭⣍⣉⣉⢠⣄⣤⠄⢀⡿⠿⠿⠷⠶⢛⣋⠁⠄⢀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠄⠈⠀⠠⠄⠀⠈⠉⠁⠀⠈⠉⠉⠉⠉⢋⠀⠀⠠⠄⠀⠻⠶⠶⠶⠀⢀⣀⣤⡤⠖⠉⠛⠛⠒⠳⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠲⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠐⣌⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢩⣽⠄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣠⡁⠀⠀⠀⠀⠀⢸⣿⣇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠙⠉⠁⠀⠀⠀⠀⠀⠰⠿⠿⠖⠋⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣥⣤⣀⢀⠀⠀⠀⠀⠀⠀⠀⣻⠶⠀⠀⠀⠀⠀⠀⢾⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢳⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⡿⡉⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⡖⠹⣷⠋⠉⠪⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣳⡀⠀⠀⠀⠀⠀⠀⠀⡰⣾⣿⡿⠀⠀⠀⠀⠀⠠⠀⠀⠀⣻⣶⣦⣠⡇⠀⠀⠀⢳⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⢹⠀⠀⠀⠀⠀⠀⠀⢠⢻⣻⡿⠀⠀⠀⠀⠀⠀⣶⡅⠀⢠⣾⡿⠿⠿⣅⡄⠀⠀⠸⣏⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⣿⠲⠃⠀⠀⠀⠀⢀⠀⡷⣠⣤⣿⠋⠀⠀⡀⠙⢷⡀⠀⠀⢸⡄⢰
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⠀⠀⢀⢠⡶⣶⡿⠋⠁⠈⠉⠄⠀⠀⡆⠀⠘⡇⠀⠀⢸⡠⣿
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢰⣿⠇⠀⠀⠀⠅⠲⣾⠃⠀⠀⠀⠀⠀⢀⣼⠅⠀⢀⡷⠀⠀⠀⡄⣽
⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⣸⣯⠀⢠⣤⣿⣶⠟⠁⠀⠀⠀⠀⠀⠀⣾⣟⠀⠀⢸⡿⠀⠀⠘⡇⣿
⠀⠀⠀⠀⠀⢳⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⡀⡒⢵⢷⡿⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠘⣧⠀⢸⠓⡇
⠀⠀⠀⠀⠀⠀⠱⢦⣿⣿⡗⢄⡀⠀⠀⢀⣀⠀⣿⣿⢿⣥⣤⣭⠿⠁⠀⠀⠀⠀⢀⣠⣴⣯⠶⠂⠀⢀⢀⡴⣿⡧⡸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⢄⣀⣈⣙⣀⣹⡿⠛⠉⠁⠙⢦⣠⡀⣤⣠⣾⣿⣿⣿⣆⣤⣒⣰⠻⡏⠰⠟⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠚⠿⠿⠿⠯⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠈⠳⡀⠀⠈⠉⠻⢿⡛⠁⠙⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠛⢭⠀⠀⠀⠀⠙⢦⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⢆⡀⠀⢐⡀⠀⠀⣠⠖⠛⠩⣄⡬⠟⠛⠛⢦
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣠⡇⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠙⠆⠘⡇⢀⡴⠁⠀⠀⢠⣿⣏⠀⠀⠀⣼⡆
⠀⠀⠀⠀⠀⠀⡞⠀⠈⠹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⡘⣿⣿⣿⠀⠀⠀⠀⣸⣿⡠⠀⠀⢐⡟⢧⡀
⠀⠀⠀⠀⠀⢀⡧⡀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠟⠁⢁⣻⡟⠀⠀⠀⠀⢸⣿⣿⣤⡶⠁⢀⣾⢷⡀
⠀⠀⠀⠀⠀⣸⠀⢉⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣦⡄⣰⣿⣿⠇⠀⠀⠀⠀⣸⣿⡏⢻⡇⠀⠨⠒⣾⠁
⠀⠀⠀⠀⠀⠹⠶⢾⣿⠀⠀⠀⠀⠀⠀⠀⣠⣔⠀⠀⣐⣮⣤⣤⣄⢸⣿⣿⡷⠟⠛⢻⠂⠄⠀⠀⠀⢿⣿⣂⣸⠧⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠶⠶⠒⠚⠿⣍⢙⣲⡿⠁⠀⣿⡇⠘⣿⠋⠁⠀⠀⠀⠈⠉⠙⠛⠛⠒⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣀⣀⡀⠸⣧⠀⣘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡍⠠⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⠔⠢⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠤⡐⠊⠐⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⡠⠀⠀⠀⠐⠤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠆⠀⠸⠃⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠈⠢⡀⠀⠀⠀⠀⠐⠾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣑⣀⠈⠢⢀⠄⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⢠⠔⠀⠀⡀⠀⡀⠀⠼⢠⠐⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠠⠀⠀⠀⠀⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠑⠄⠀⠄⠀⠀⠀⠴⠀⠢⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠤⡀⠀⠀⠀⠀⠀⠁⠀⠬⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠚⠁⠘⠡⢀⠀⠀⠀⠠⠀⡆⠠⠀⠀⠀⢀⡠⠚⠀⠉⠒⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠖⠒⠚⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂⠀⠀⠁⠀⠀⠠⣀⠀⠀⢀⠉⠀⠀⠐⡀⠄⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠐⠂⠀⠀⢀⠁⠢⡄⠁⠀⠂⠀⠀⠀⠈⠤⠆⢈⠀⠀⠀⠐⠀⠀⠈⠳⣱⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠠⠃⠀⣀⠀⠀⠀⠀⠀⠈⠀⠁⠀⢀⠀⠀⠀⠀⢀⠀⠘⠠⠈⠀⠀⠀⠀⡐⠀⡀⠀⡻
⠀⠀⠀⠀⠂⡀⠀⠀⠀⢈⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⠠⠄⠁⠀⠁⠁⠀⠀⣿⠆
⠀⠀⠀⠀⠘⢠⡄⠀⠀⠠⠂⠊⠀⠁⢀⠀⡀⠀⠐⡀⠀⣴⠀⠀⡀⠀⠀⠀⠀⠐⠰⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠈⠂⠈⢀⠀⠄⠀⡠⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡄⡆⠃
⠀⠀⠀⠀⠀⠀⡈⣁⠀⠀⠀⠀⠀⡱⠊⠂⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⢉⣃
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠄⠀⠀⠀⠀⢀⠁⡀⠀⠉⠀⠂⢀⡀⠀⠀⠁⠀⠐⠒⠀⠐⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⡀⠂⠀⠀⠀⠊⢛⣄⠘⠸⠀⣬⠛⠃⠀⠀⠀⠀⠠⠀⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⢀⠔⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠐⠀⠈⠢⠀⠀⠀⢠⣀⣀⣀⠀⠀⠀⠈⠀⠀⠐⠢⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠐⠀⡀⠀⠀⠣⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠄⠀⠀⠈⠀⠈⠄⠁⠘⠀⠀⠀⠴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡨⠉⠀⠔⠀⠎⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⠔⢀⠐⠁⠀⠀⠀⠀⠀⢶⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠑⠀⠀⠐⠀⠐⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠒⠡⠄⢠⠀⠀⠀⡠⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⠢⠔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣋⠐⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣖⠚⠋⠉⠉⠉⠁⠀⠀⠉⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢳
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢀⣀⡀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣼⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠈⢧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠚⠛⠒⠦⣄⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠬⡗⠶⢤⣄⣀⠀⠀⠀⠀⣀⣀⡀⢸⡇⠀⠈⠓⢦⡀⢀⣀⡀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠸⣅⣂⣀⠟⠀⠀⠈⠉⠉⠛⢻⠋⡁⠈⠹⡾⣇⠀⠀⠀⠀⡷⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠛⠛⠲⣜⢄⣈⢀⣀⠗⠙⢦⣄⣀⢀⡇⠀⠀⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠳⢄⠀⠈⢧⠈⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠠⠄⠄⠄⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⠀⠉⠉⠉⠀⠁⠈⠁⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⡀⠀⡀⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠈⠀⠈⠉⠉⠉⠉⠉⠉⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢤⢀⣀⣀⣀⢤⢊⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⣀⣤⢤⠤⣤⣤⣤⣀⣠⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠀⠈⠁⠉⠉⠑⢶⠲⠶⠆⠄⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠠⠂⢈⠀⠀⡀⠀⢀⠀⠑⢢⣀⡀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞⠀⠀⢀⠀⡀
⠀⠃⠅⢈⠀⡀⠄⡀⠀⠀⠀⠀⠂⠈⠙⠒⠲⠖⠶⠠⠰⠒⠒⠒⠓⠛⠉⠩⠑⠉⠉⠁⠐⠁⠀⠠⠂⢈⠀⡀
⠀⠀⠢⠀⠑⠠⠀⠀⠀⠀⠂⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠
⠀⠀⠈⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠌⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡯⣀⠀⠀⠀⠀⠀⠀⡼⠷⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⢷⣦⡀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⣿⣿⣿⣾⣿⣇⠘⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠴⣻⢿⢿⠿⡟⡏⠁⠉⠛⠛⠛⠛⠻⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⣯⡟⡼⠈⡀⠐⠂⠀⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠠⠤⠐⠈⠀⠀⠀⠀⠀⠀⢘⡟⠠⠁⢸⣧⣀⡰⠼⠟⠓⠓⠓⠛⠃
⠀⠀⠀⣠⣦⣾⣷⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠀⠀⢀⡉⠉⠀⠀⣸⠀⠀⠀⠀⣠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⢀⣤⣿⣷⣤⣥⠶⠛⠛⠛⠛⠛⢣
⠀⠀⠸⠿⠁⠋⠛⡉⣿⢹⠀⢸⠛⠁⠀⠀⢠⣴⣿⠏⠀⠀⠀⢼⣿⢹⡥⡍⠀⠀⣹⣀⣀⣀⣀⣄⡇
⠀⠀⣤⠀⣦⣄⣦⣷⣿⢸⠀⠈⣿⡿⣷⣚⣻⣿⡏⠀⠀⠀⠀⠘⢿⣾⣇⣿⠷⠾⠟⠛⠛⠛⠛⠛⢇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢸⠀⠈⣟⠀⠀⢸⠇⠀⠀⠀⠀⠀⣤⣤⢸⡇⢹⠉⣿⡆⠀⠀⠀⠀⢀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡆⣇⡀⠘⠀⠀⢸⣳⣄⠀⠀⠀⠀⠘⢿⣾⣧⣼⣶⣿⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⣀⣿⣿⣷⣦⡄⠀⠀⠀⠹⠏⠻⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠙⠉⠉⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⡀⠀⠠⠀⠄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣽⠇⠀⠀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⡀⠀⠀⠀⠦⠹⠴⠋
⠀⠀⠀⠀⢷⡶⠃⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠻⡆⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠖⠀⠀⠀⠀⣴⡀⠀⠀⠓⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠉⠉⠤⠄⠈⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠟⡆⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⡰⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠐⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⢄⣶⣿⣿⣤⣰⣿⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣟⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣿⣿⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⡭⢥⠑⠂
⠀⠀⠀⡠⢄⡄⠱⣤⢐⢦⡔⢦⣞⢤⢐⠢⣴⠢⡄⠂⠤⠀⠄⢀⣤⣶⣤⣰⣶⣶⣠⡶⣦⢸⣷⣾⡷
⠀⠀⢰⣇⠈⠇⣤⢉⣹⡇⠇⡘⠇⠉⠜⢧⣍⡀⡅⢯⡇⢳⡤⢸⣯⣭⡿⢻⣯⣿⣿⡇⠿⢸⣧⣿⠇
⠀⠀⠈⡇⢸⠀⣿⠸⢠⡿⢠⢰⡅⣿⠶⣦⢻⠀⡃⢶⡄⠉⡄⢸⣟⠙⣿⢸⣟⣿⢿⣇⣷⣿⡟⢻⣶
⠀⠀⠀⡇⠘⡇⣻⢠⣄⢣⣴⠾⠧⠿⣇⢀⣼⢰⣤⣌⡃⣷⣀⣼⣿⢀⠛⢈⡙⡟⢈⠻⠋⠹⠁⠘⢿
⠀⠀⠀⢗⣢⢏⢉⠣⣤⣭⣤⣴⣯⣓⡌⠁⠀⠀⢀⠛⠙⠝⠞⢹⡏⠈⠀⡈⠁⠄⢀⡀⠀⣴⠐
⠀⠀⠀⠈⢁⣿⠟⢋⣛⣿⢻⣿⣷⣿⣟⣶⣶⠾⣿⣿⡆⠄⢆⡘⡇⠀⠀⡿⢻⠟⢻⠁⠀⠻⣃⣤⡆
⠀⠀⠀⢰⣿⣿⠠⣟⠛⣻⠘⢿⢋⡙⡇⢶⡛⢀⣻⣿⣗⣬⡼⢷⢶⣶⠂⠀⠁⠈⠁⠚⠻⢿⣿⡟
⠀⠀⠀⣾⢿⣿⣦⣈⣁⣥⣸⣼⣦⣴⣛⣸⡇⣾⣿⣿⢟⡄⢑⣟⣦⡁⢰⣧⣶⣤⣸⠇⠀⢸⡏⠒⠂⠀⠁
⠀⠀⢠⣿⣶⣿⡏⢠⣽⢹⣿⣿⣿⠛⣿⢿⠟⣿⡦⣮⣄⠁⢘⠹⣻⣷⡛⠉⠀⠀⠉⠀⠀⣾⣇⠉⡃⡀⡄
⠰⡄⢲⣿⣿⣿⠿⢈⡇⣈⣙⠰⠌⡁⢀⠈⣰⣿⣿⢿⣿⣄⠈⡒⣻⣿⣷⢦⣤⠀⠤⣀⣰⣿⡇⠘⡨⠁⠃
⢀⣀⠜⣿⣿⣿⣶⣿⣷⣿⡿⣶⣾⣧⣿⣼⣿⣿⣧⣼⣿⠟⠰⠌⣿⣿⣿⣿⡿⣶⣦⢘⣛⠟⢣⢊⡀⡆
⠀⣉⡀⠘⢻⣿⠃⢻⣧⠈⠄⠌⠙⢿⣿⣬⣿⠻⡿⠋⠁⠀⠀⠀⢻⣿⣿⣏⡁⠈⠁⢿⠿⣶⠀⡀⠖⢁
⠀⠈⠁⢸⡄⣿⠃⠈⠯⠖⠀⠀⠑⠀⣽⠃⢟⠐⠆⠀⠀⠀⠀⠀⢘⣿⣿⡇⣿⣿⡀⢠⠀⡉⢀⠇⢀
⠀⠀⠀⠀⣤⢻⣆⠀⠀⢶⠄⠂⡟⠂⠻⠇⠨⠒⡆⠂⠀⢰⠠⠀⠉⠉⠁⠀⡙⠻⠿⡦⠀⠔⠀⢀⡄⠧
⠀⠸⢿⠂⠈⢨⣋⠀⢀⡀⠀⠀⠃⠐⠀⠀⠄⠀⠤⠀⠀⠁⠀⣀⠀⠸⠿⣊⠉⢀⣠⡠⠄⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠘⢰⠟⣡⣥⢀⡈⢠⠌⠌⣀⠐⢠⠀⠀⠀⠈⠍⠀⢀⡴⣒⣷⠉⢿⣷⣶⣶⢷⠀⠀⡘
⠀⠠⠀⢣⠀⢈⠳⣤⢎⠩⣤⡶⡇⠁⣂⢄⠋⡤⠤⢐⡖⠲⡄⢀⡼⠩⡀⢻⣿⡦⠄⠘⠉⠀⢀
⠰⠇⠀⠠⠤⠌⠀⠂⠐⢠⠈⠰⣶⠘⣽⠆⢡⡁⣠⠀⢀⢀⡇⣼⠀⠀⠙⡈⡉⢢⠘⠃⢠⠀⠀⠀⠀⡀
⠀⣤⠀⠄⠀⠀⣳⣿⣶⣦⠀⠂⠹⠂⠘⠁⠀⢛⠛⢄⣸⡞⡇⢹⡀⡐⢛⡟⢀⣌⣡⣶
⠀⢸⣧⠀⠐⠀⡈⠿⣿⡟⠀⠠⠀⠀⠀⠀⠀⠈⢆⠀⠈⠈⢠⠄⠹⣷⣼⡶⢾⠻⢁⠀⠀⠀⢀⡁
⠀⠘⠟⣳⡤⠀⠁⠀⠸⡿⠆⠀⠡⠀⠀⠀⠀⠀⠀⡃⠀⢀⠠⡀⠀⠈⠀⠀⢀⠳⣖⣶⢤⡵⡤⡌⠃
⠀⢈⣿⡿⠟⠓⠀⠁⠀⢸⡝⣠⣄⠀⠐⠀⠁⠈⠀⠀⠀⠋⢁⡤⣶⢲⡆⠀⠈⠂⣩⠉⠉⠈⠇⡣⡀⢀⣀
⠀⣾⣼⣷⠀⠀⠀⣹⣈⣥⣿⢯⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠁⠄⢠⠂⡝⠀⠀⠘⢹⣿⠔⠀⠃
⠀⠈⠋⣽⣶⣤⣀⣿⡟⠛⠁⠐⣣⠀⠀⠀⠀⠀⠃⠐⠦⣀⠀⠀⠀⠈⡠⠀⠙⠊⠀⠀⠰⠅⡀
⠀⠀⠚⠿⢿⡿⠿⠟⠛⠀⠀⠀⠘⠁⠀⠀⢖⡿⠲⡱⠀⠈⠛⢦⡤⠎⣤⠖⠎⠈⠉⠐⡒⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠖⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠐⠒⠊⠁⠀⠀⠀⠀⠐⠂⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⡙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠙⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⠀⠀⠀⠀⠀⠀⠀⣀⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢰⠶⠚⠛⣿⡿⢶⡆⢠⠂⠀⠀⢼⠛⢡⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣼⡀⠀⢸⣿⢱⡆⡀⢈⡤⢠⡚⡦⠀⠀⠄⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣀⣿⠿⣿⢻⣿⠚⢉⣃⢪⣬⡭⣁⠀⢀⣠⡀⠀⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣖⡿⣿⣿⡿⢿⣿⣿⠂⣰⣶⣇⣼⣿⣬⣽⡿⠣⠆⠈⠀⠤⣀⠀⠈⠀⠀⣬⣿⣶⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡧⢼⣿⣃⡽⣿⣛⣷⣿⢿⡾⣿⣿⠿⣿⣿⠂⠈⠂⠉⢻⢋⡈⠘⠀⢀⡌⢍⠿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣱⣾⣿⣿⣤⢾⣟⣺⣿⣿⡇⡼⢷⣦⡼⠛⢠⣷⡖⢆⠀⢤⠧⠀⠀⣾⠣⢌⣿⣏⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⣿⡿⢿⣟⣿⣿⠟⡿⣿⣿⣿⢀⡿⢱⣿⣿⣿⣷⣆⣰⣼⣶⣦⠘⣿⣿⣿⣟⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡶⠒⣿⢟⣹⣃⣁⣿⣿⡟⣈⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢼⡿⡋⠻⣿⣿⢹⣿⡿⢾⣿⣿⣿⣿⣏⠹⠿⣛⢿⣿⣟⢿⡿⢓⣦⣲⣟⣭⡿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⡟⢻⡻⢿⣏⢙⣿⣶⣿⡿⠛⠿⣿⡗⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⣷⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢵⣈⣿⣶⣿⠿⣿⣹⣿⣀⣀⣀⢾⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠐⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⡟⢿⡟⠻⢋⠊⠁⢉⢻⣿⠿⠛⠉⡍⣸⡇⢈⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣟⠛⢻⣿⣿⣿⣿⣿⡿⠻⠟⠻⢿⣷⣾⣷⣤⠼⣤⡤⠹⣿⣿⡤⡄⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣿⣿⣿⣿⣏⣤⣤⣴⣦⣬⣿⣿⣿⣷⠀⣩⣐⣳⣿⣿⣇⣧⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣋⢻⣿⣟⣻⣿⣿⣿⣿⣇⣷⠞⢷⣾⠛⠉⠁⠙⣢⠤⣌⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣯⣿⣿⣿⣯⣰⣾⣷⣴⣿⣴⣶⣠⣿⣶⣶⣟⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠛⠛⠉⠛⠋⠋⠉⠉⠉⠋⠉⠛⠛⠋⠉⠙⠛⠉⠉⠉⠋⠙⠉⠛⠋⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⠤⠶⠒⠲⠶⠖⠒⠋⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠓⠬⣳⣄⠀⠐⠒⠖⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠉⠦⣀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠉⠙⢦⣤⣤⣴⣶⣶⣾⣿⡟⠧⢄⣀⡀⣀⣠⣤⣄⣤⣬⣵⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠀⠀⠀⠀⠈⣿⣿⣿⡿⣫⣵⢟⣧⠀⠀⠈⠙⣿⠉⢉⣿⢻⣿⣇⣧
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⠀⠈⠀⢿⢟⢥⣾⣿⡷⢋⣿⡆⠀⠀⠀⢇⢳⡸⣿⡏⢻⣿⡘⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⢸⡿⣿⠟⡫⣲⣿⡿⣷⠀⠀⠀⢸⣸⣇⣿⠇⣿⣿⢷⢱⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣿⡇⠀⠀⠈⡏⣵⣽⣾⣽⣿⣿⣫⡇⠀⠀⠀⢹⢸⠹⠀⣿⢻⠈⡆⢳⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢹⡇⠀⠀⠀⢿⣩⣿⣿⡿⣿⡿⠗⠻⡄⠀⠀⡎⠈⠂⠀⢸⠈⠀⢱⠀⠱⡄
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣾⣧⠀⠀⠀⢸⠨⠟⠵⠊⣁⡠⠴⡄⠀⠀⠀⢸⡀⠀⠀⢸⣶⡄⠈⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣰⣿⣿⣆⠀⠀⠈⡇⠀⠀⠀⢁⠄⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⣧⠀⢀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣄⠀⠀⢧⠀⠀⠐⠁⠈⠀⠼⠁⠀⠀⠀⡁⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⢀⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢾⣄⣀⣀⣠⣤⠤⠤⠤⠤⠶⠶⠧⠬⣦⣄⣀⣀⣀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣮⡾⣿⢣⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣇⣦⡀
⠀⠀⠀⢀⣠⣀⡤⣮⡾⣫⠞⢡⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⣾⢏⣿⢠⡀
⠀⢀⣴⣿⣿⡗⣺⢫⡾⠋⣰⡿⣻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⡤⣄⣴⣾⣿⣿⣿⣷⣿⣼⣇⣾⡿⡇⡀
⣴⣿⣿⣿⣿⣗⡿⣸⡇⣾⡟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣷⣧⢻⡆⣤
⣿⣿⣿⣿⣿⣿⡇⣿⢱⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢷⣝⠷⣝⡻⣿⣿⣿⣿⣿⣏⣾⣿⢻⣮⣻⣿⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣧⡣⡻⣿⣿⣿⣿⣿⣿⣿⣿⣯⡝⢷⣍⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣙⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⢣
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠄⠀⠀⠀⠀⠸⡁⠔⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠔⠁⠀⠀⠰⠀⢠⠀⠀⠂⠀⠀⡀⠀⠀⠀⢸⠁⡀⠀⠀⢀⠀⠂⢉⡁⠈⠤⠀⠀⠃⠀⠀⠈⡐⠀⢠⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠄⠀⠀⠀⢀⠀⠀⢀⠁⠀⡐⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⡈⠄⠈⠃
⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠈⠀⠀⢠⠌⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⠎⡉⠉⠐⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠁⠁⠀⠀⠘⠀⠀⠀⠀⠤⠄⠀⠀⠀⠊⠀⠀⠀⠘⠓⠀⠒⠂⠂⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠉⠀⢲⠀⠀⠀⠀⠐⠈⠐⡀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢠⡀⠀⡄⠀⠀⠀⠀⠤⠆⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠚⠀⠐⠉⠀⡀⠀⠀⡄⠀⠐⠠⠐⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠅⠀⠀⠀⠄⡀⠀⠀⢸⠁⠀⠀⠀⢠⠂⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠘⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠐⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉


⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠠⠠⠤⠀⢀
⠀⢀⣰⣶⣟⠲⢶⢖⠳⣘⣫⣶⠿⠿⠞⠛⢓⢹⠄⠂⡈⣡⢰⣺⣦⣷⣠⡄⠒⠚⠚⠇⡴⡂⢄⣃⣀⡀⢀
⠀⣠⡼⢖⣀⣤⣯⣥⠴⠂⣿⠃⠀⣀⠀⣄⣾⢻⠀⣀⠝⠨⠧⢸⡽⣏⠉⠳⢄⣄⣤⢤⡀⣤⣴⣛⣃⢼⣌⡙⣤
⢀⠏⡀⠈⠁⢈⢇⣘⣆⢀⡼⡆⠒⠉⢣⣀⢯⡘⡆⠀⠠⠤⠀⠈⢃⡷⠲⠆⠀⠀⠐⠊⠏⠁⠀⠉⠹⡀⠀⢈⣛⠁
⠊⢿⡾⠠⣿⣾⣹⡽⣝⢧⣀⣈⣀⠀⣉⡠⠀⠘⣇⠀⠐⢠⣯⠴⣟⢲⣄⢀⣐⣆⡀⢀⠶⠂⠈⠛⠂⠀⠃⣀⠉
⠀⢀⠇⠀⠀⡞⡩⠙⣶⡾⠟⠛⠻⡛⠻⣟⡣⢾⣿⣭⣶⣿⠋⠛⠉⣠⣟⠦⢌⠿⣻⣿⣾⠛⡯⡀⠠⠂⠀⠀⢀
⠀⢨⠔⠊⡰⣠⠇⢡⣿⠷⣶⡶⠀⠹⢷⡟⠙⠷⢼⣮⢻⣻⣧⠄⠞⢻⣯⣿⣃⣼⠇⠀⠈⠛⣧⣼⡁⠀⠀⠀⢨⢰⡀
⠰⠖⠶⠂⠤⠀⠀⠈⣩⡗⠈⠀⢴⡄⠀⡓⢊⠒⠶⠶⣾⡽⡈⠋⣄⣀⡏⣩⣷⡋⠀⢀⣀⣀⢸⣿⣖⡆⢸⣄⠱⣣⠂
⠀⠰⠀⠀⢘⠄⠀⠀⠀⠱⢤⣀⠤⡄⣰⡀⠙⢂⠀⠀⢪⡄⠇⡠⡤⠀⣤⢸⡂⠛⠃⢀⠁⢉⠾⣟⣿⠇⠀⠈⣄⠉
⠀⠃⣈⠉⠩⠓⠒⢲⡞⢁⠘⢿⣟⣻⠷⠖⢶⣿⣦⣐⣿⣧⣤⣀⢀⡼⠗⠛⠗⠢⡄⠛⠋⠜⠃⣁⣀⡀⠀⣀⡄⢀
⠀⢶⠉⡚⡶⠞⠶⡖⡽⠎⠃⠼⢸⣵⠴⠚⠛⠛⠿⢏⡙⠛⡛⠛⣻⠆⠀⠀⠀⠀⠙⢶⣵⣧⣼⡿⣭⣙⢷⣄⣀⠆⣃
⠀⠈⢻⠉⠀⠀⠃⠞⠀⠀⠀⠀⢈⢻⣾⣷⠄⠺⣿⡆⣿⡇⠉⣉⣿⠆⠂⠰⠶⠈⠐⢀⣏⣽⠻⠋⣹⡛⠿⠻⡆⠀⣹
⠀⠀⢩⠁⠀⠀⠀⠀⣀⡀⡵⠖⣶⠞⣇⠀⢀⣀⠀⢀⡋⠀⠘⠛⠣⡄⠷⠀⠀⠀⠀⠊⠉⠈⡆⣀⣭⣩⠀⠜⠀⠙⠃
⠀⠀⢨⡇⠤⢖⠨⠱⢠⠀⠀⠀⠀⠀⣨⣁⠠⣠⠀⡈⠀⠀⠀⠀⣰⣬⠀⠂⢀⠀⠀⠀⠀⠀⣙⢏⣁⠤⠀⠰
⠀⠀⣚⠦⣿⡴⣴⠂⠟⠀⠸⠁⠀⠀⣸⣭⠓⠻⢺⣾⡃⢤⢠⢤⣼⣿⣲⢶⠃⠘⠿⠗⢀⣶⠳⠾⠲⣄⠀⠒⠿⢻
⠀⠀⠀⡞⢿⣠⣤⠠⠒⠓⡆⠈⠀⣜⢟⣁⡈⠚⠓⢋⣷⠻⠇⡆⠨⡿⢿⠀⠀⠀⠀⠀⠀⣹⠈⠁⠰⠟⠛⡖⣺⣿⢧
⠀⠀⠀⢀⠈⠎⠘⢲⣥⠀⢈⢶⣺⢶⠊⠛⡛⡁⠸⡈⢻⠀⠀⠓⠼⠀⢸⡏⣡⠶⠄⠤⣸⠛⠟⢀⡄⠀⠀⢰⠿⠛⠈
⠀⠀⠀⠉⢠⡔⠓⠂⣶⠒⡡⡶⠀⠛⠀⠀⢀⡁⠀⢉⢠⠀⠀⠚⠻⢂⡞⠼⢿⡷⢶⣟⡹⢠⡀⠀⠀⠊⡠⠀⠆⠄
⠀⠀⠀⠀⠠⠱⡤⠀⠄⠀⡄⠀⠉⣥⠀⡠⢬⣁⣀⡠⠈⠀⠀⠤⠀⠈⠊⠉⠸⠉⠓⠛⠀⠀⠘⠃⠄⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠿⣿⣿⠁⠀⠀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⢀⣀⠀⠀⠀⢸⣿⣆⣀⣀⡴⠲⢤⣀⠀⠉⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠉⠁⢀⣀⣀⣤⡤⠤⠶⠖⠚⠛⠋⠉⠙⢦⣀⡀⠀⣿⣟⠉⠁⠀⠀⠀⠈⠙⠓⠦⣄⣀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⡀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢲⠋⠀⠀⠀⠀⠀⢠⡞⠛⢷⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡆⠀⠀⠀⢠⡞⠉⢳⡄⠀⠀⠀⠀⠸⣇⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡆⠀⠀⠀⠀⠀⠘⠧⡤⠞⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣇⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠀⠀⠀⢹⡇⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⠀⢻⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣄⡀⠀⠀⢸⡄⠘⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡞⠀⣸⣿⣧⣀
⠀⠀⠀⢀⣠⣶⡿⠟⠉⢹⣿⡀⠸⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⠀⠈⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡾⠁⣰⣿⠋⠙⠿⣷⣦⣀
⠀⣠⣶⡿⠟⠉⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⣾⣿⡿⠋⣿⣿⠁⢀⣀⠈⢿⣿⣇⠀⢿⠀⢹⣿⠋⣀⡀⠙⣿⣿⡇⠀⠀⣼⠁⢠⣿⠏⠀⠀⠀⠈⠙⠿⣷⣦⡀
⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⠀⢸⣿⡿⠁⠰⢟⡋⠀⣭⣭⠀⢰⣿⣿⠀⢸⡄⠘⣿⣷⣭⡃⠀⣿⣿⡇⠀⣸⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡇⠈⣿⣿⣆⠈⠉⠁⢠⣿⣇⣠⣾⣿⡟⠀⠈⣇⠀⣿⣿⣀⣠⣾⣾⡿⠀⣰⠃⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿
⣿⡏⠙⠻⢿⣶⣄⡀⠀⠀⠈⣿⡇⠀⣧⠀⠘⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⢻⠀⢸⣿⣿⣿⡿⠋⠀⣰⠏⢀⣾⡟⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⣿
⣿⡇⠀⠀⠀⠉⠻⢿⣶⣄⡀⢻⣿⠀⢹⡀⠀⠀⠉⠻⠿⢿⣿⠿⠿⠛⠁⠀⠀⠀⠀⢸⡆⠘⣿⡏⠁⠀⠀⢠⠏⠀⣾⡿⠁⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣷⠀⠀⢠⡏⠀⣼⡿⠁⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⢠⡴⠲⣤⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿⡀⢀⡞⠀⣼⣿⣷⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⡀⠀⠀⠀⠀⠀⠸⣄⣀⡼⠀⠀⠀⠀⠀⠀⣸⡆⠈⣿⣇⡾⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢿⣿⠁⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠛⢦⡀⣀⣀⣠⣤⠤⠴⠶⠒⠚⠛⠉⠁⠀⢀⣀⣾⡿⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠉⠉⢀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⡿⠛⠋⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣶⣿⠿⠿⠿⠛⠛⠛⠉⠙⠿⠟⠉⠙⠻⢿⣷⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠙⢿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⡀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⠟⠃⠈⠓⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡒⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉⠀⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠓⢤⠀⠀⠀⠀⡀⢠⠀⠀⠜⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⡀⠀⠓⠚⠉⠀⠁⠁⠀⠁⠂⢀⡤⠀⠀⡔⠢⣀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⡀⠀⠠⠔⠚⠀⠀⠄⠫⠤⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⢀⠀⠁⠀⠀⠀⠀⣰⠶⠡⠀⠀⠀⠀⠀⡀⠀⠶⠦⠈⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠅⢐⡠⣌⠀⠤⡀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠠⢭⠐⠂⠀⠀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠁⡌⠙⣆⣆⡤⠞⣓⠒⠢⢄⠀⠀⣀⠀⢠⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣉⢉⢉⣩⠞⠁⢰⠒⣏⠑⡢⣠⣇⡎⣡⠤⠶⣐⠒⡒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠁⡴⣴⠋⡜⣡⢋⢧⠀⠸⡯⠤⠴⣂⠑⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⢠⡿⣌⣛⡻⢢⠤⠻⠂⠂⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡈⠳⣄⢀⣠⣾⣷⣦⠀⠀⠀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠏⢰⣈⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣬⣁⣤⡈⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢉⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⠁⠙⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠸⠀⠁⠈⠀⠀⠀⠀⠡⠀⠀⠀⠀⢠⠃⠀⠄
⠀⠀⠀⢠⣄⣀⠀⢪⣩⣆⢠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⡇⠰⠀⠀⠀⠐⠀⡔⠀⠁⠀⡁⠀⠆
⠀⣀⠐⢦⡀⠙⣷⣘⢳⠏⢰⢀⠏⠰⠀⠀⠀⠀⠀⡔⠡⡐⢤⠤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⠀⠀⠈
⠀⠸⣆⠀⠙⠄⠈⣿⢼⠇⠰⣎⡴⠓⠄⢀⢔⣤⡞⠀⢀⠇⠈⡆⢠⠔⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠈⠑⢄⢸⣆⢻⣶⡆⠐⣽⠁⠀⠀⢈⠘⣿⡇⠀⢨⡆⠀⢿⢀⡟⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⢦⠻⡈⣻⢃⢣⡏⠀⠀⠀⠀⠄⠻⣷⡄⠸⣆⣀⡞⣼⢁⠇⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⡄⠀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠣⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠣⠼⡿⠿⠋⠜⡿⠑⠁⠀⠀⠀⠀⠀⠀⠈⠯⣤⠀⢳⠆⠀⢰⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠘⠀⠀⠀⠀⠀⠀⡴⠒⠂⣴⠶⢄⡀⢳⣼⣇⣼⣿⠷⠚⠃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠑⣄⡀⠈⠷⣇⡙⢶⣍⢯⠉⣤⣤⡶⠿⢛⡋⡳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⠀⠹⢦⣦⡭⣷⣶⠽⠷⠦⣤⠤⡤⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣉⣀⣀⣑⣀⣐⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⣟⡛⢻⡟⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⣤⣤⢤⣤⠄⠀⠀⢠⣤⡀⠀⠀⢠⠤⣤⠄⠀⢠⠀⠀⠀⠀⢀⣤⠤⣤⣤⠀⠀⠀⠀⣤⡀⣀⣀⣀⣤⡤⠀⠀⠀⠀⠀⠠⢤⣤⣤⣀⣤⣤⣄
⠀⠘⣿⠁⠀⠉⠀⠀⠀⠀⢿⣿⣷⡶⠁⠀⠀⠀⠀⠈⢷⣠⡤⠀⠀⠘⠀⠀⠀⠈⢷⣞⠉⠉⠉⠉⣩⠃⠙⡇⠀⠀⠀⡀⠀⠉⠀⢻⣟⠀⠁⠙⠃
⣠⣴⣿⡆⠀⠁⠀⠀⠀⣀⣿⣿⣿⢷⣤⣀⠀⠀⠀⠀⠈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⠄⠀⠀⠰⠒⣤⣴⣧⠀⠀⠀⠉⠀⠀⠀⢸⣿⡀
⠁⠈⠉⢻⣦⣤⣶⣾⠿⠏⠉⠁⠀⠀⠉⢻⣿⠶⠶⠶⠶⢷⣤⣄⣠⣼⣷⣤⣶⡤⣴⣿⣶⣴⣦⣶⡟⢻⣿⣿⣧⡀⠀⣀⡀⣀⣠⣾⣿⣅⣀⣀⣀
⠀⠀⡀⠈⠻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠉⣿⡿⠛⠉⠉⠛⠿⣿⡿⣋⠀⠀⠉⠀⠀⠉⠛⣿⣿⠛⠉⠉⠉⢿⣿⡿⠛⠉⠉⠉
⣿⣾⣶⣦⡴⠟⠛⣿⣿⣶⣾⠇⠴⠶⢾⣟⠀⢄⡀⠀⠀⠀⣰⣿⡁⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢁⠀⠀⠀⠘⣿⠁⠀⠀⠀⠓⢸⣿⣇⣤⣤⣄
⡟⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⠘⣿⣶⡤⠌⢁⣴⣿⠋⠁⠁⠐⠦⢤⣶⣿⡿⠓⠖⢠⣴⣿⣿⣶⣶⣶⣿⣿⣿⣶⣦⣤⡾⠋⠉⠉
⣶⠶⢤⣤⣤⠤⠫⢿⣿⣀⣀⣀⣀⣀⣠⡿⠋⠀⠀⠀⠉⣁⣀⣀⠀⠀⢀⣈⠛⠉⠀⠀⠀⠀⠀⢹⡿⠀⠉⠉⠀⠈⠉⠉⠻⣯⣀⡀
⣿⣤⠎⠁⠀⠀⠀⠸⣿⣿⡿⠛⠋⠁⠀⠁⠀⠀⠀⢀⡔⠉⠉⠉⠛⠛⠻⠿⣧⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠉⠳⣆⢀⣶
⡋⠇⠀⠀⠀⠀⣀⣠⣿⣿⡀⠀⠀⠀⢠⣧⣤⣀⣴⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡶⠖⠀⠚⠛⠿⢷⣶⣷⡾⠒⠳⡿⠁⠀⠀⠀⠀⣀⡿⠋
⣷⢶⣶⣶⣿⣿⡿⠛⠁⠀⠉⢳⣄⣀⡼⠟⠋⠉⠀⢰⣿⠟⠛⠛⠒⠄⢠⣄⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⢀⢀⣀⣀⣴⣿⣿⠁
⠃⠀⠄⠀⠐⠹⣧⡀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢘⡇⠀⠙⣀⣀⣀⡀⠀⠀⠀⠀⣽⣤⠀⠀⢀⣿⠟⠋⠉⠀⠘⢶⡶⠓
⣄⣩⢤⠀⠠⣄⣼⣿⣷⣶⣶⣶⣾⣦⣀⣀⣀⣀⣀⣸⣆⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡿⠉⠙⠛⠛⠿⠿⣿⡟⠀⠀⠈⢛⠻⣧⠀⠀⠀⠀⠈⢀⣀
⠈⢻⣿⣯⣷⡿⠟⠋⠁⠀⠀⠈⠹⣿⣿⣿⡿⠟⠻⢿⣅⣷⣀⠀⠀⠀⠀⣾⣷⡶⢤⣖⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⣙⣷⡴⠶⠶⢶⣾⡟
⠀⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇⠀⠀⠀⠀⠈⠈⠙⠷⣤⣤⠞⠋⠉⠀⠀⠉⢄⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠈⠻
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⣷⡶⠶⠛⠛⠉⠉⠓⠒⠈⠉⣿⡀
⣤⡾⠇⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⡀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀
⣴⡦⣄⠀⠀⠀⠀⠀⠉⠉⠉⠃⠀⠉⢛⠀⠀⠴⠂⠰⠶⠒⠂⠹⠻⠿⠿⠿⢻⣶⠶⠖⠶⣿⣧⠀⠀⠀⣀⣀⣠⣴⣶⣿⠛⢿⣀⣀⣀⣠⣄
⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣉⣈⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣋⣉⣁⣉⣀⣀⣀⣉⣉⣙⣀⣘⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠠⡀⠀⠀⢤⡀⠀⠂⢦⠀⠀⠚⠀⠀⠀⠀⢹⡆⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠄
⠀⠀⠀⠈⠢⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇⠀⠒⠰⠻⡧⢀⠀⠰⠖⠓⠀⢠⣀⠁⠈⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠒⠄⠀⠈⠁⢈⠴⠚⠁⠀⠀⣀⠁⠀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠰⠁⠀⠀⠀⠉⠠⠐⠊⢀⠄⠈⠩⠵⠂⢀⣄⠄
⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⢸⣶⣆⠀⢆⠀⠀⠈⠀⠉⢁⣀⡤⠀⠒⢉⠜⡠⢀⠃⠐⠂
⠀⠐⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⢸⣿⡏⠀⠀⠁⠀⠀⠀⠀⠠⡶⠿⠷⠗⠚⢊⣴⣧⡀
⠀⠀⠀⠀⡁⠀⠀⡁⢘⠀⠀⠀⠀⣿⠀⠀⠀⠀⢺⣿⡇⠐⠀⠀⠀⢠⡆⠀⠀⠈⠓⠶⠠⠶⠻⠀⠙⠛⣠⠄
⠀⠀⠀⠀⡄⠀⠀⠁⢈⠀⠀⠀⠀⠸⠀⠀⣀⠀⠼⠛⠛⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠹⣿⣄⣀
⠀⠀⠀⠀⡀⠀⣀⠆⠀⢀⠤⠀⠄⠉⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⢸⡇⢈⢀⡤⠐⠂⠀⠀⠁⠀⠈⢘⣿⠟⠋
⠀⠂⠄⠲⠀⠀⠀⠀⠐⠭⠿⠿⡖⢂⡀⠀⠀⠀⢠⣄⡀⠅⠀⠀⢨⣿⡏⢠⡠⠧⠖⠀⠀⠰⠏⡀⠀⠼⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠄⠩⢈⣀⣋⣉⣁⡄⡀⠀⠀⠈⣿⠧⠴⠓⣺⣉⠲⣄⠉⢳⠀⠀⠀⠤⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠟⣻⣥⢬⣴⣿⣽⣷⠂⠀⠀⢘⣿⣿⣅⠀⠀⠀⣨⠟⢀⣀⣤⡴⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠡⠖⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣅⠀⡀⠀⣰⠍⠩⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣦⣶⣶⣴⣦⣴⣿⣴⣷⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣄⣀⠀⢀⣠⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣾⠀⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠙⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡴⣛⣭⣭⣮⣭⣭⡓⢦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⢫⣾⣿⡿⠛⠈⠻⣿⣿⣦⡹⡎⢻⣿⡿⠿⣿⣿⠿⣿⣟⣯⣵⣿⣿⣿⠁⠙⣿⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⡅⠺⢸⣧⣼⣿⣿⣷⢹⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣧⣀⠈⠀⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡿⣷⢠⡉⢻⣿⣿⣿⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⢷⡹⣿⣿⣧⣈⠈⣁⣼⣿⣿⢣⡟⣴⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠳⣬⡻⠿⣿⣾⡿⠿⣛⡵⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠛⠉⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⢿⣀⣮⣤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠛⠛⢻⠋⣷⣄⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢣⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣄⠙⣿⣦⡄
⠀⠶⣿⣼⣿⣿⣿⠟⢀⣠⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣿⣿⣿⣿⣿⠁
⣶⣶⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠿⠛⢯⣿⣿⣿⣿⣿⣿⣧⠀⣦⣾
⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠏
⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣬⣉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣤⣀⣀⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣽⣿⣿⣿⣶⣶⢈⠉⠉⠙⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋⠉⣉⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣿⣿⣿⣿⣿⣿⣧⡸⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠿⢿⣿⣿⣿⡿⠋⢻⣿⣿⣿⣿⣷⡌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣴⣿⣿⣿⠏⠀⠀⠈⣿⣿⣿⠈⢿⣿
⠟⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠈⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⡿⠃⠀⠀⠀⠀⣿⣿⣿⠀⠀⢿
⠀⣾⣿⣿⠟⣿⣿⡿⢿⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡆⠀⠸
⠸⡿⠻⣿⢰⣿⢰⡇⠈⢿⠀⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⣸⡇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⢰⣿⣿⣆⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⢻⡀
⠀⠀⠀⢀⣿⠘⣻⣿⣿⣿⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢦⣾⣿⣿⣥⣄⢸⡇
⠀⠀⠀⢸⡏⢀⢼⣿⢿⣿⡿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣯⡥⢼⡇
⠀⠀⠀⣾⠇⠀⠈⠁⢸⡏⠀⠈⠃⢀⣤⣿⣿⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⡄⠀⠰⡛⠛⢹⢿⡏⠙⠿⠀⣷
⠀⠀⢀⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⡾⢿⡿⠶⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⠀⢴⢾⣿⣿⡅⠀⠁⠀⠚⠈⠁⠀⠀⠀⣿
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣼⣿⡗⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠂⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⣀⣀⣿⡇⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⣸⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠠⡈⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢘⣳⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠂⠀⠀⠀⠀⠀⢸⣷⣆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠻⠛⠃⠀⠀⠀⠀⠀⢈⣿⣿⣦⡾⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠠⠔⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠠⢠⡀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣾⡥⠀⠀⠀⠀⠀⠀⠘⢳⠄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⠓⠀⠀⠀⠀⠀⠀⠀⣧⣂⣀⡤⣧⣠⠶⢶⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣙⠀⣹⠃⠀⠀⠱⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢷⣶⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡋⠀⠀⠀⠀⠀⣀⡅⠀⠀⣸⡷⣿⡶⡿⠀⠀⠀⠀⣷⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠌⣿⣚⡏⠀⠀⠀⠀⠀⠀⣿⠃⠀⣰⣿⠟⠉⠋⠻⣮⠄⠀⠀⠘⢦⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠈⢀⡏⠊⠀⠀⠀⠀⢀⣄⣤⣯⠴⠾⣿⠁⠀⠀⢰⠀⠈⣷⠀⠀⠀⣸⢀⡅
⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠠⡜⢏⣻⡟⠀⠀⠀⠀⠁⠀⠀⡰⠀⠀⢸⠀⠀⠀⢳⠊⡏
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡿⠀⠀⠀⣄⡂⣙⡿⠁⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⣼⡃⠀⠀⢠⢂⡇
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣷⠀⢰⣶⡾⣿⠋⠀⠀⠀⠀⠄⠀⠀⢸⣿⠄⠀⠀⠿⣇⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠘⣆⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡄⠅⣾⣴⡟⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⢸⣧⠀⡾⠹⡇
⠀⠀⠀⠀⠀⠀⠈⠳⣾⣿⡍⠀⠤⠀⠀⠀⣠⡄⠠⣿⡿⣻⣴⣤⣦⠟⠁⠀⠀⠀⠀⣀⣤⣶⠿⠾⠊⠀⠀⢤⡴⢺⣿⣇⣗⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠳⠤⣄⣉⣋⣤⣿⠟⠉⠁⠀⠈⢳⣰⡀⣶⡠⢶⣿⣿⣯⣵⣤⣄⣠⣎⣛⠀⠾⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠚⠛⠛⠒⠚⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡏
⠀⠀⠀⠀⠀⠀⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠒⠿⠂
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠂
⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⠾⡿⢧⠀⠈⠁⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣏⡀⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣛⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⢋⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⣿⣀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣤⣀⣀⣀⣤⣴⣶⣿⠿⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠦⣴⣤⡀⠀⠴⠔⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣹⣿⣿⣿⣶⣾⠲⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⣿⣿⣦⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⡿⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣭⣿⣿⣿⣿⣾⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣽⣿⣯⡿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣥⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢷⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⣹⡇⣸⣿⣧⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢿⣿⣏⠀⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠻⠟⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣧⣀⠟⢾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣧⠴⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡟⠻⢛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣇⣤⡈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠻⣿⣃⡟⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠀⠈⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠉⠛⠳⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⢸⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣧⠀⠀⠀⠀⠀⢰⣞⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠸⣿⡀⠀⠀⠀⠀⢸⠟⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⡳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⢿
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣟⢣⠀⠀⠀⢠⢏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠶⣕⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⣯⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢻⠀⡆⠀⠀⡏⡼⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⡘⢹⡟⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⢤⣿⡀⢰⣧⡟⡀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⠀⣿⣷⢸⣹⡟⣳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢄⠀⠀⠀⠀⠀⢰⠀⣾⡅⢸⡇
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠙⠀⣸⡿⢿⡹⣷⣷⠀⠘⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠄⠀⠀⢀⠇⢠⣿⡇⠸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠀⠸⣷⢹⣿⣀⡴⠊⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡆⠀⠀⡾⠲⣿⡁⣏⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⢠⡇⠀⠀⠀⢫⣼⡿⠃⢠⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣇⠀⣸⠁⠀⣿⡇⠁⠀⣧
⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⠟⠀⠀⠀⠚⠈⣿⠀⠀⡸⠀⠀⠀⠀⠸⣾⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡟⢩⣻⣿⣦⣀⡿⣿⡄⣤⣿
⠀⠀⠀⠀⠀⠀⠘⠷⣄⣸⠀⠀⠀⠀⠀⠀⢹⣄⠈⠀⠀⠀⠀⠀⠀⠸⡄⢿⣦⠀⠀⠀⠀⢀⠀⠀⣴⠋⠀⠀⠽⢩⣿⣿⣿⣿⣿⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢧⠸⣿⡇⠀⠀⠀⠰⠀⠀⣯⠀⠀⠀⠀⡸⠚⣟⠳⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⠀⠀⢸⣀⣀⣀⣀⠀⠀⠀⠀⠘⣆⡽⠓⠒⠒⠒⠚⠒⠛⠙⠒⠒⠲⠴⠷⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠚⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣔⣂⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣣⣾⣀⣴⡅
⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⢀⠆
⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⡻⣿⣿⣿⣿⣿⡶⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢟⣽⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⡎⡻⣿⣿⣿⣶⣶⠶⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢿⣿⣿⣿⠿⣿⣿⣷⢸⣿⣿⣿⣷⡝⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⠏⠀⣸⣿⡿⣼⣿⣿⣿⣛⣋⣈⢻⡿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡷⢠⣿⣿⠿⣛⣿⣿⣿⣿⣿⣿⣧⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⢀⣤⣤⢄⣀⣈⠉⠁⣰⣾⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣄⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣄
⠀⠀⢀⣠⣴⣾⣿⡟⢫⣍⣙⣿⢿⣿⣿⣷⡼⠛⣩⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣟⣿⣍⠛⢻⣿⢿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⢠⣾⣿⡟⠉⠀⣼⣿⣿⣿⠿⢿⣷⣾⣭⣭⣥⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣮⢻⡆⠊⠙⠀⠀⡿⠸⢹⣿⣿⡿⣿⣿⢻⣿⣿⣿
⢻⣿⣯⠁⣠⣾⣿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⠀⠀⠀⠜⠁⠤⠟⣿⣿⡇⣿⡏⣼⣿⣿⣿
⡞⢡⡟⢠⡛⢻⣿⠆⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⢹⡿⣿⢠⡿⢹⡏⠁
⠻⠞⠀⠀⢹⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠩⣿⣿⣿⣎⠻⡼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠃⠑⠝⠦⠓⠈⠷⠄
⠀⠀⠀⠀⢸⡋⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣶⡶⢦
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣷⣯⣿⣿⠿⢿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣧⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣀⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⡏⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⡗⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⡏⢸⠹⠿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠭⠴⠛⠉⠉⡩⢲⠎⢩⡏⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⣉⣀⡀⠀⠀⠀⠀⢛⠈⣢⣉⣴⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠺⠑⠒⠀⠈⠉⠒⠚⠳⠦⣀⡀⣠⣾⣿⣫⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⢟⠒⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡿⠧⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣒⠭⠒⠁⠀⠈⢄⡀⠀⠐⠭⠩⢛⣶⡦⣄⢀⢆⢆⣿⣿⣿⡿⠏⣏⣠⣷⣿
⠀⠀⠀⠀⠀⠀⣀⠠⠔⣊⣥⢖⠙⠓⡀⠀⠀⠀⠀⠀⠉⠒⠒⠚⢈⡡⣪⣾⣿⠀⢫⣾⣿⡿⠟⢁⣶⢿⣿⢻⢫⡇
⠀⠀⠀⠀⠀⢀⣿⢴⣢⣉⡁⢦⣀⠑⠒⡀⠀⠀⠀⠀⠀⢀⡠⣒⢵⣾⣾⣿⣿⠁⣿⣿⡟⠟⠚⠻⣿⣕⡕⣵⠋
⠀⠀⠀⠀⠀⠀⡏⠑⢶⡬⣉⡓⠶⣭⣕⣲⠊⣩⡽⠟⡩⢵⣺⣵⣿⣿⣿⣿⣯⣖⠛⠈⠀⠀⣀⡄⣿⣮⣾⠋⠱⠶⢤⣀
⠀⠀⠀⠀⠀⠠⡏⠀⠀⠙⡑⠪⢝⣒⢿⣾⣽⣓⣞⣿⣾⢿⣿⣿⣿⠿⠿⠏⠓⠒⣦⣤⣄⢦⡛⣄⣿⡿⣙⣑⣦⠀⠩⠃⣩⡕⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢭⣛⡲⠬⣅⡂⠉⢘⠛⣿⣶⣿⣧⣟⠛⠈⠤⠤⠔⡪⣻⣯⡟⢺⠁⣹⠟⠋⠉⡀⠺⢕⡶⢄⣠⣶⣷⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠳⠦⢌⣙⡋⣼⡿⠟⠋⠬⠍⠀⠀⠀⢀⢞⣵⢿⣿⡏⢃⡽⠓⠀⢀⠔⠁⠀⠀⠈⣉⡚⠻⢿⠷⢇⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⢉⠴⠉⢰⡁⠒⠂⢬⣙⣋⠉⠁⠀⠀⣀⣀⣀⢔⣕⣟⡷⣿⣿⣦⠯⠄⠀⠒⠁⠀⠀⠀⠀⠤⠀⠀⠀⡜⠀⢄⢨⠉⣱⡆
⠀⠀⠀⠀⠀⠀⢐⡯⠬⠥⠉⣀⣀⡒⠊⠭⠭⠶⣢⡤⠤⠊⢁⠖⣃⢴⡿⡯⣾⣾⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⠂⣸⢄⣠⠴⢊⠥⢻⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠤⢂⣐⡒⠒⠀⠭⠭⠤⠐⢀⣀⣀⡨⡭⠍⠐⣫⣺⣿⣿⣏⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⡚⠭⢒⡩⣴⣫⣵⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢅⣒⣒⣒⡰⠯⠭⠭⠥⠔⢛⣒⣐⡤⠭⡍⡭⣿⣿⠿⡟⠙⠿⣿⢤⣘⠓⢬⠑⠲⣔⡢⠭⣐⣻⣵⣶⣿⢿⣿⠿⠛⡋⢬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢕⠬⡛⢌⡙⠩⢋⢟⡹⡋⢞⠑⡋⠛⠝⡒⢛⢷⣕⡢⠀⠀⠀⠹⢦⢽⣒⣺⣽⣕⣛⣭⣶⣿⡯⠟⢛⠩⢔⣊⣥⠔⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠉⠢⢤⣄⡀⢀⣁⣈⣀⡁⠈⣈⣰⣥⡿⣻⣄⠀⠀⠀⠈⠃⢰⣿⣿⠿⢟⢚⣉⣥⡒⠥⢒⠮⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠨⣛⢥⡀⠴⠚⣋⣽⢿⣿⣿⡿⣿⣟⠝⠒⢄⡀⠀⠐⣗⣖⣯⣽⠿⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠸⡩⡉⠹⢿⣻⣷⣟⣯⡿⠟⠊⠀⠀⠀⠀⠈⠁⠊⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⡀⠈⡆⢐⣿⡿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢐⠀⠂⠐⡠⠂⠔⡠⠂⡀⠀⠄⡠⢀⠀⠀⠀⠠⢀⠄⠀⠀⡠⢂⠐⠀⠀⠠⢀⠄⠀⢀⠔⡠⡠⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠊⠔⠥⢂⠔⢈⢔⡠⢊⠔⢊⠔⡠⢂⠑⡨⠐⠡⢊⠐⡡⢂⠀⡠⡠⢊⠄⠁⢒⡑⡠⢂⡔⢃⠔⡡⢪⠊⠔⠑⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠘⠁⠊⠁⠊⠐⠁⠊⠐⠡⠂⡠⢊⠔⠡⠪⠊⠀⠁⠊⠀⠋⠀⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠿⠿⠟⠻⠿⠿⠿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⢁⣀⣤⠤⠶⠶⠶⠶⠤⣤⣀⡈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿
⠀⠀⠀⠀⠀⠀⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡈⠉⠛⠒⠶⠶⠶⠶⠒⠛⠉⢁⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣶⣦⣤⣤⣶⣶⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⣴⣆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⢠⣶⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣴⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣫⡖⣦⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⠵⣯⣼⣿⣿⢸⣧⡴⢞⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡔⢻⣶⣿⣿⡿⣟⠈⢷⠄⠙⠃⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣼⢋⣴⣉⣿⣿⠿⠋⣵⠄⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣴⠟⣵⣧⣼⡿⢋⡅⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣿⠀⠀⠀⠀⢀⡾⢫⣾⣿⠟⠉⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡇⠀⠀⠀⣠⡟⣱⣿⡿⣁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⣰⡟⣼⡿⢫⡄⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⢸⣟⢱⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣧⠀⢰⡟⣼⠟⣥⡌⠀⠀⠀⠀⠀⣠⣾⡿⠿⣿⣿⣿⠿⣿⣿⣶⣄⠀⠀⣿⣿⣋⣣⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡻⣤⣿⢱⠏⣦⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⢸⣿⣿⠀⠈⢻⣿⣿⣦⣼⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡴⠛⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⣦⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠛⡟⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢈⣿⣶⡄⠈⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣟⢛⡿⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⣷⣦⣤⣟⢷⣭⡻⢶⣦⣤⣤⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡋⣻⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⢻⣿⣿⣷⣬⡛⢶⣭⣛⠿⣍⣽⣿⠛⣿⠟⣿⠟⣿⣏⣹⣧⠼⣛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆⠈⠙⠻⠶⣮⣭⣽⣛⣓⣛⣛⣫⣭⣭⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋

⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⢠⡀⠀⠀⠠⡀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣤⡄⠀⠀⠀⣀⣤⠀⠀⠀⢤⠀⠀⠀⠠⠄⠀⠀⠀⣀
⠀⠀⠀⠙⠃⠀⠀⠈⠉⠉⠀⠀⠀⠀⠸⠓⠀⠀⠛⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠸⠓⠻⠀⠀⠐⠀⠃⠀⠀⠀⠂⠀⠀⠀⠐⠶⠀⠀⠀⠉
⠀⠀⠀⣠⡀⠀⠀⠀⠠⢶⠀⠀⠀⢀⣴⠀⠀⢠⣤⠀⢰⡒⣶⣦⣀⠠⣤⣀⡀⠀⢀⣠⣤⡀⠀⠀⣴⡆⠀⢰⣷⠆⠀⠀⢰⠶⡆⠀⢀⣶⣷⣦⠀⢀⣠⣤⡀
⠀⠀⠈⠛⠛⠀⠀⠀⢦⡸⠀⠀⠐⠿⠟⠀⠀⠻⠋⠀⢸⣷⣿⣿⠏⠀⠀⠚⠟⠂⠀⠙⠋⠡⠆⠐⠋⠀⠀⠻⣿⠏⠀⠀⠸⠾⠃⠀⠺⠿⢿⠃⠘⠛⠛⠿⠃
⠀⠀⣀⠀⠀⠀⠀⢀⣤⣶⡄⠀⠀⣤⣤⡀⠀⣀⣀⡀⠀⢐⣦⡀⠀⠀⢰⢶⡆⠀⢀⣤⣤⡀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⡠⠶⢶⣦⠀⠀⠀⠀⠀⢤⣤
⠀⠀⠈⠛⠳⠆⠀⠹⢟⡉⠀⠀⣼⠏⠁⠀⠀⠿⠷⠇⠀⠸⣿⣷⠀⠀⠀⢾⡇⠀⠈⠛⢿⡇⠀⠛⠃⠀⠀⠉⠛⠛⠀⠾⣁⣀⡀⠏⠀⠈⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠉
⠀⢠⣀⣒⣛⡳⠆⣴⣿⠷⠀⠀⠀⣿⡷⠀⢸⠿⣀⡀⠀⠺⣲⡀⠀⠀⢶⡶⠦⠀⢸⣿⣿⠆⠀⢴⡀⠀⠀⢺⣿⡇⠀⠀⢀⣴⠆⠀⠀⢶⣿⠄⠀⢰⣗
⠀⠀⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠛⠙⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣀⠈
⠀⠀⠀⣴⡄⠀⠀⣤⡰⣶⡄⠀⢤⣶⠾⠆⠀⣴⣿⣆⠀⣾⣿⣶⠄⠀⠖⣻⡷⠀⠀⣠⣤⠄⠀⡠⠄⠀⠀⠀⡸⠀⠀⠀⢹⣷⣄⠀⠀⠠⡀⠀⣿⣿⡟⣶⠆
⠀⠀⠀⠙⠁⠀⠀⠈⠋⠉⠀⠀⠀⠁⠀⠀⠘⠛⠛⠃⠀⠛⠟⠋⠀⠀⠈⠙⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠻⡿⠟⠁
⠀⠀⠀⢀⡄⠀⠀⠀⣠⢦⡀⠀⠀⣀⣄⡀⠀⣠⡤⠄⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠠⣄⠀⠀⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠐⢦⠀⠀⢀⠀⠀⠀⣶⣦⣀
⠀⠀⠀⠟⠀⠀⠀⢠⠸⠊⠀⠀⠘⠿⠿⠀⠀⠙⠶⠆⠀⠀⠊⠁⠀⠀⠈⠛⠁⠀⠀⠊⠋⠀⠀⠛⠂⠀⠀⠻⠟⠛⠁⠀⢨⠿⠚⠃⠀⠘⠃⠀⠐⢿⣟⠛
⠀⠀⠀⢀⣀⡀⠀⠀⡤⢄⠀⠀⠀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣴⣿⡌⡟⢠⡎⠀⠀⠆⠀⢰⣿⡿⠀⠀⣜⣁⡄⢀⣴⡞⣼⠃⠀⢰⣹⡇⠀⡈⠳⠀⠌⠀⣄⠆⠀⠀⠀⠠⠀⠀⠀⣼⣷⡆⠀⠠⣻⣿⠆⠀⣾⡉⠶
⠀⠀⠘⠃⠁⠀⠀⠉⠀⠰⠎⠀⠀⠋⣀⠀⠀⠟⠉⠀⠀⠙⠒⠛⠀⠀⠈⣛⠁⠀⠛⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠈⠉
⠀⠀⢴⣦⣤⡀⠀⠀⣠⠤⠀⠀⣾⣿⣿⣷⠀⠀⣀⡀⠀⢠⣶⣶⠀⣴⠾⠛⢳⡄⠀⢠⠀⠀⠀⣀⡀⠀⣴⡖⠶⣄⡀⠀⣤⣦⣤⠀⡐⢶⣤⣄⠀⢠⣶⡄
⠀⠀⠘⠿⠟⠀⠀⠘⠛⠉⠀⠀⠙⢿⣿⡟⠀⠘⠉⠀⠀⠨⠉⠁⠀⠀⠨⢍⣛⣿⠀⠈⠂⠀⠀⠀⠁⠀⠛⠁⠀⠈⠙⠀⠛⠛⠃⠀⠉⠂⠙⠉⠀⠀⠐
⠀⠀⠀⠀⠀⣀⠀⢤⣄⡀⠀⠀⢀⣀⡍⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢉⠁⠀⢠⡀⠀⣀⣤⣀⠀⠀⠀⠀⡀⠀⠀⢀⠤⡀⠀⠀⠀⢀⠀⠀⣤⣄
⠀⠠⣶⡟⢛⠋⠀⠙⣿⣿⡄⠀⠘⣦⣮⠀⠀⣸⡏⠀⠀⢖⣟⠉⠃⠀⣠⠂⠀⠀⠀⣿⣿⠈⢻⣿⣿⡇⠀⠀⢵⠇⠀⠀⢸⣷⡟⠀⢠⣶⡟⠀⠀⠈⠙⡄
⠀⠀⠀⠀⠉⠀⠀⠀⠘⠛⠁⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠉⠋⠀⠀⠀⠈⠀⠀⠀⠈⠛⠁⠀⠀⠈⠀⠀⠀⠀⠛⠛
⠀⠀⢀⣠⣶⣄⠀⡐⠶⣦⡀⠀⢠⣆⡀⠀⠀⡴⠀⡀⠀⠐⠢⢤⠀⠀⣿⣶⣄⠀⠀⣀⣆⠀⢰⡆⢄⠀⠀⠀⣸⡧⠀⣠⣶⣶⢦⡀⠀⢀⡀⠀⠀⢠⡴⠆
⠀⠀⠿⠿⠚⠃⠀⠸⠖⠛⠃⠀⠈⠛⠓⠀⠀⠘⠚⠃⠀⠙⠋⠀⠀⠀⠈⠩⠟⠀⠀⠙⠛⠀⠘⠿⠃⠀⠀⠀⠙⢷⠀⠙⠛⠃⠉⠁⠀⠈⠀⠀⠀⠘⠒⠂

⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀
⢤⡀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠈⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠂⠀⢀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣀⣀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠃⠠⢻⡧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀
⠉⠙⠛⠈⠉⢧⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠼⠁⠀⠀⠀⠀⠀⠀⢻⣥⠤⠖⠶⠚⠋⠉⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣠⣒⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠖⠒⠒⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⢀⠞⢱⣤⣀⠀⠈⠁⠀⠀⠀⠀⠀⣯⠀⠀⠀⠠⣿⣦⣄⡀
⠀⣀⣀⠠⠤⠀⠂⠈⠁⠀⠀⠀⡰⠁⠘⣿⣿⣿⡆⠀⢀⠃⠀⠀⠀⠀⠙⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣼⣿⣿⣿⣿⡄⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠹⣿⣷⡂⠀⢸⡄⠀⠀⡀⣀⠀⠀⣿⣿⣿⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢸⣿⣿⣇⠀⣽⡿⠀⠀⠈⡇⠀⠀⢱⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⣿⣾⣿⠇⠀⠀⠀⡇⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠙⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠹⣄
⠀⠀⠀⠀⢀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠈⢧⡀
⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⢧
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡇
⠀⡼⠁⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⣿⣀⣀⣀
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠉⠉⠁⠀⠈⠙⠦⣤⣀
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⣠⠂⠀⠀⠀⢀⠀⠀⠀⠈
⠀⡹⠚⠛⠉⠉⠉⠉⠛⢢⡀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⣠⠋⠀⠠⠤⢴⡇⠀⠀⠀⠀⠀⠉⠉
⡾⣅⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣇
⢧⠸⡆⠒⠲⠤⠤⠤⠤⠤⣴⠷⠦⣤⣤⣤⣤⣤⣤⣤⡴⠖⢻⡀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣠
⠈⠳⣧⣀⠀⢀⣀⣤⠤⠞⠁⢀⡤⠖⠒⠚⠛⠉⠉⢩⠛⣦⠈⠳⣄⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠞⠁
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⡏⠈⠑⠒⠶⠤⠤⠴⡇⢀⡟⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣄⠀⠀⠀⣀⣠⡷⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠒⢤⣄⢀⣀⡀⠀⠀⠀⠀⠈⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠊⠀⠀⠀⠀⠀⠀⠁⠠⣷⡄⢀⡀⠀⠈
⠀⠀⠀⣠⣾⠙⡉⠉⠉⠉⠑⠀⠃⠠⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠛⠁
⠀⠀⠀⠹⢿⣦⡦⡀⠀⠀⠀⠀⠒⠗⠀⠈⠐⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠁
⠀⠀⠀⠀⠀⠉⢣⠰⣄⡰⣄⣀⢠⠀⠀⡀⠬⣤⣘⠀⢤⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠪⣍⠳⢷⡀⢁⡑⠰⠡⡥⠚⢰⡊⢳⣶⡆⠠⢀⣠⠞⠀⣀⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠥⣤⣿⣥⣏⣽⣿⣟⣿⣧⣥⡼⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⣿⢻⡟⣿⢻⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣿⣼⣧⣿⣼⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⢻⡟⡟⣿⢹⡟⡟⣿⢻⡟⡟⣿⢻⡟
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⣷⣷⣿⣾⠷
⠀⠀⢸⢹⡏⣿⢹⢹⡏⣿⣽⡿⢟⣛⣭⣭⣭⣭⣛⡻⢶⣼⣏⡏⣿⢹⡏⡏⣿⢹⡏⣿⣿⢹⡏⣿⣿⢹⡇
⠀⣀⣸⣾⣷⣿⣾⣾⣿⢟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡻⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣷⣿⣿⣾⣅
⠀⣿⢹⣿⡏⣿⢹⡿⣡⣿⣿⣿⣿⠿⠛⠀⠀⠛⠻⢿⣿⣿⣷⡌⢿⣿⢹⡿⡏⣿⢸⣿⡏⣿⢹⡏⡏⣿⢸
⠀⠙⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⡏⠀⠀⢰⣦⣤⣄⣾⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛
⠀⠀⣿⣸⢸⣿⡏⣾⣿⣿⣿⣿⣷⣄⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣧⢹⡇⣿⣸⢸⡇⣿⣸⢸⡇⣿⢸
⣤⣤⢿⣿⣿⣿⣇⢿⣿⣿⣿⣿⡿⢿⣿⣶⣦⠀⠀⠘⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⢿⣿⣿⣿⢿⡿⣿⣤
⣇⣿⣸⣇⣇⣿⣿⡜⣿⣿⣿⣿⣅⡀⠀⠉⠁⠀⣀⣼⣿⣿⣿⣿⢣⣿⣸⣇⣇⣿⣸⣇⣇⣿⣸⣇⣿⣿
⠉⡿⣿⢿⣿⡿⣿⢷⡜⢿⣿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⡿⣣⣿⣿⢿⡿⣿⣿⢿⡿⣿⢿⢿⡿⡏⠉
⠀⣇⣿⣸⣿⣇⣿⣸⣿⣦⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣾⣟⣿⣿⣸⣇⣿⣿⣸⣇⣿⣼⣸⣇⡇
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⠶⣮⣭⣭⣭⣭⣵⠶⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⡋⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠰⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⢀⡇⠀⠀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⢤⣄⡀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠉⠳⡀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢀⣙⡲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠐⢤⡀⠀⠀⠀⠀⠀⣷⠋⠁
⠀⠀⠀⠀⠀⠀⢞⠩⠄⢈⠆⠈⠙⠓⠒⠶⠶⡶⠚⠙⢲⣼⡃⠀⠀⠀⠙⡆⠈⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⣠⠶⠖⠲⠤⣜⢀⠀⠀⢀⡏⠳⣄⠀⠀⢀⠃⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠒⢄⠀⠈⢧⠉⠛⠉⠀⠀⠈⠉⠙⠛⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠁⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠧⠀⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠠⣄⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠁
⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠈⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⣷⡀⠁⠀⠀⠀⠰⠇⠀⣀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢠⠁⠀⠀⠀⠀⢧⣄⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠎⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘
⠀⠀⠠⡆⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢐⣶
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠈
⣿⡿⠻⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣠⡶⠶⢦⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠾⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢠⡏⠀⣠⠜⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⢠⠟⣁⣀⡀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠻⢬⣥⠴⠇⠀⣿⡇
⠀⠀⠀⠀⠀⢠⡟⣾⡁⠀⢿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠒⠒⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠈⣿⠿⣝⣶⠟⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠠⠀⠀⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢹⣆⠤⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠸⡟⠀⠉⠁⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡈⣿⠀⠀⢀⣤⠞⠻⠛⠛⠲⣤⠀⣠⡶⠛⠛⠛⢳⣦⡀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠄⢸⡇⢠⡟⠁⠊⠀⠀⠀⠀⣘⣷⣿⠈⠂⠀⠀⠀⠈⣿⣤⡇⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡌⠀⠀⠀⢀⡀⢻⣾⢱⡄⠀⠀⠀⠀⠀⠀⢻⡗⣶⡶⢤⡀⠀⠈⠘⣿⣤⣄⡀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢡⡾⠋⠸⡇⠀⠀⠀⠀⠀⠀⣼⠟⠁⡙⢦⣽⡶⣤⣀⣿⠈⠙⠻⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠐⠁⠀⠀⠁⠈⠀⣧⠀⠀⠀⠀⠀⠸⡿⠁⠀⠈⠂⠉⠒⣂⣭⡿⣦⣀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠘⡷⠢⣤⡀⠀⠀⠀⠀⠈⠁⣦⠙⢷⡄⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢄⠀⠀⠀⠀⠀⠀⣿⡇⠆⠀⠀⠀⠀⠻⡶⣌⢳⡀⠀⠀⠀⠀⠀⠈⠁⠰⣿⣄⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠠⠀⠀⠀⠀⠀⡀⠻⣷⡀⠆⢴⢢⡀⢰⣿⣿⣉⡇⠀⢀⠀⣀⣤⠀⠀⠀⠙⢿⣆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⡠⠆⠀⢀⢌⡀⢀⣈⣿⠶⢾⣮⠷⠾⠏⠈⢛⡛⠿⣿⣾⣿⡟⠀⠀⠀⠀⢀⠽⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠐⠁⠀⠀⡽⣡⠖⠋⢀⠀⠀⠀⠟⠀⠀⠀⠁⠀⠀⢸⣷⣶⡿⠷⢆⠀⠀⠀⠊⠁⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⠂⠀⠀⠈⡼⠁⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⣿⣿⡖⠔⠂⠀⠀⠀⠀⠀⠐⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡎⠂⠀⢸⣷⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣾⢋⣿⡻⠃⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢸⣿⣿⣠⠖⢋⠥⠄⢚⡒⢄⣀⡟⡾⣻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⡠⣤⡎⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⢷⣻⣯⣟⣴⣶⣞⣷⠞⠋⢹⡝⣣⣿⡗⣩⠀⠀⠀⠀⠀⠀⠀⠀⠄⣾⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠆⡀⠙⠻⠿⠟⠛⠉⠀⠀⠀⢸⡷⣿⡿⣿⠀⠄⠀⠀⠀⠀⠀⠀⡬⡩⢋⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡆⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡨⣽⣿⠦⡀⠀⠀⠀⠀⠀⠀⣀⣹⢡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠿⣿⡿⢤⣞⡵⢂⡀⠀⠠⠊⡳⣽⣿⡆⡉⠀⠀⢀⠠⠴⢖⣤⣰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡛⢷⣶⣾⣶⣏⠙⢒⡀⠀⠀⠈⠂⠍⠋⠀⠀⠤⠴⠶⢖⣿⠛⡃⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠒⠍⠀⠈⠁⠀⢿⣓⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣰⠞⣿⡊⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⣘⡥⠂⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠉⠉⠉⠛⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠠⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠸⣆
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⣀⣿⣤⣤⣤⣄⡀
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠋⠉⠀⠀⠀⠀⠈⠙⠶⣤⣄⡀
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⡠⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠈⢷⡴⠶⠶⠶⠶⠶⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢀⣀⣼⠃⠀⠀⠀⠀⠀⠙⠒⠒⠂⠀⠀⡷
⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠉⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⢸⡏⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⡀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠈⢧⠘⡏⠉⠙⠛⠒⠒⠒⠒⠒⢲⡾⠛⠳⠶⠶⠦⠤⠤⢤⣤⣴⣶⡚⠋⠹⣆⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠈⠳⣷⣀⠀⠀⠀⣀⣠⣤⠶⠋⠀⢀⣤⠶⠚⠛⠋⠙⠉⠉⠉⡼⠙⣦⠀⠙⢶⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⡏⠈⠓⠢⠤⣤⣤⣤⣤⣼⠃⠀⣿⠀⠀⠀⠈⠉⠉⠙⠓⠒⠚⠛⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⢸⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣤⠶⠖⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠁⠁⠀⠀⠐⠀⠀⠈⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⡠⠴⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣓⣶⣦⣄⠀⠀⠀⠀⠀⠀⢠⣤⣤⡴⣲⡖⠉⠉⠓⣄
⠀⠀⠀⠀⢠⣇⢾⣅⡬⠽⡛⠲⣄⡀⠀⠀⠀⠀⠀⠀⡾⠙⠿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣰⢳⢆⠛⡟⠉⠛⠶⣄⠀⠈⢒⣄
⠀⠀⠀⢀⡯⣭⢭⣿⣭⢷⣶⣷⡞⠛⡆⠀⠀⠀⠀⣴⠃⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣋⢄⠤⠂⠳⡄⠀⠀⠀⠉⢿⡿⣽
⠀⠀⠀⣼⣷⣿⣿⡷⠿⣿⡟⠓⠒⣰⠁⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣿⣹⣿⣿⣿⡟⠁⠀⠀⠈⢧⣄⣢⡾⢰⡇⠀⠀⠀⠀⢸⢻⠇
⠀⠀⠀⢣⡻⣿⣶⣿⣻⣾⠿⠟⠻⠋⠀⠀⠀⠀⠀⠀⣹⠠⠰⢀⢀⣟⣩⣭⣽⣿⠇⠀⠀⠀⠀⠀⢹⣽⣿⣿⣅⡀⠀⠀⠀⢸⡜
⠀⠀⠀⠀⠑⢄⣈⣉⢀⡀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⠤⣥⣹⣟⣼⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⡀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⢴⣤⡤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣤⣄
⠀⠀⠀⢀⡤⣤⣶⣿⣿⣷⡖⠒⠶⢦⡀⠀⠀⠳⡀⠈⠛⠷⣬⣿⣿⡿⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣧⡀
⠀⠀⢀⣼⡾⠷⣿⣿⣿⣿⣥⣤⠴⠚⢣⠀⠀⠀⣧⠀⠀⠀⡠⠉⠻⣅⣸⣿⣿⣿⣿⣷⠦⡄⠀⠀⢸⣿⣿⡿⢿⢿⠟⠻⠿⢿⡀
⠀⠀⢾⡋⢧⡁⠸⣿⣿⣿⣿⢀⡄⡰⢠⠃⠀⢸⣿⣣⡄⠀⠀⠀⠀⠉⠻⣾⣻⢻⡿⣿⡇⢹⠀⠀⢸⠉⠉⠀⠀⠀⠀⠀⠂⡸⠁
⠀⠀⠘⣟⢶⣭⡂⢻⣿⣿⣏⢈⡼⣡⠋⠀⠀⠈⠛⠻⣿⣶⡀⠀⠀⠀⠀⠀⠭⠛⠳⢾⣅⡏⠀⠀⠈⢆⠀⠀⠀⡀⠀⠀⠜⠁
⠀⠀⠀⠘⣦⡙⠛⠺⢿⣿⣿⣾⠾⠁⠀⠀⠀⠀⠀⠀⠈⣿⣹⡀⠀⠀⠂⠁⢠⠦⠤⠤⠞⠀⠀⠀⠀⠀⠳⡄⠀⢀⣠⠤⠃
⠀⠀⠀⠀⠈⠙⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢄⣀⣀⠠⠏⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⡠⡴⣖⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠹⢿⠝⠛⠂⣠⢤⣄
⠀⠀⠀⠀⢀⡾⠀⢸⠈⠉⠦⣼⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⢸⣆⠀⢀⡴⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠌⠄⢀⢸⠀⠀⠀⢹⡌⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠶⢶⣿⢯⣵⡞⠁⠀⠘⡎⠒⠦⠤⣤⣤⣿⡿⣇
⠀⠀⠀⢸⠀⢁⢸⠰⡆⠀⠀⢀⣷⣄⡷⠀⠀⠀⠀⠀⠀⡔⠉⠀⣀⣀⡀⠀⢰⣿⣼⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠘⢷⢻⡀
⠀⠀⠀⣎⠀⠈⢸⠠⢠⢠⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⣧⣄⣀⡘⢟⡿⣦⢿⢿⠟⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠈⢧⢣⡀
⠀⠀⠀⠘⣆⠸⣼⠀⡇⣦⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣩⣿⡿⢩⠛⢯⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠈⣷⢷⡄
⠀⠀⠀⠀⠘⣆⢻⡇⣸⢃⣛⣻⣯⠏⠀⠀⠀⠀⠀⠀⠘⣿⣟⡴⣷⠄⣘⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠘⠲⠠⣤⣼⣔⡽⣄
⠀⠀⠀⠀⠀⠘⣞⣷⢷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠮⠾⠷⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣿⣷⣾⣶⣿⣦⣬⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣾⡭⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠿⠻⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⢴⠖⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣾⢸⠀⠀⠀⠈⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢸⣿⠎⠠⠀⠀⠐⠒⠺⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⡆⠀⢸
⠀⠀⠀⠀⠀⢀⣼⣿⣠⣤⣤⣴⣾⣶⣾⣧⣌⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣧⣷⣀⣤⠶⠚⢤⡀
⠀⠀⠀⠀⠀⠚⠛⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠧⠄⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠷⠶⠽⠷⢶⠤⠷


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⣤⠤⠶⠒⠒⠲⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣦⣾⣷⣶⣆⣄⠀⠀⠀⡀⢹⡧⣭⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠿⠶⠟⠛⠋⠉⠉⠈⠉⠉

⠀⠀⠀⢸⣘⡯⠯⣭⡟⠋⠁⠀⠀⠀⠈⠑⢻⣭⠽⢽⡋⡇⠀⠀⠀⠀⠀⠀⢸⢘⡯⠯⣽⡟⠊⠁⠀⠀⠀⠈⠑⢻⣯⠿⢽⡃⡇
⣀⣀⡀⠈⠽⣧⣽⠦⠉⠀⡀⣠⣿⡀⢀⠀⠉⠴⣯⢼⠿⠁⢀⣀⣀⣀⣀⡀⠈⠿⡧⣽⠦⠉⠀⣀⣠⣟⡄⢀⠀⠉⠴⣯⣼⠯⠁⢀⣀
⡶⡼⠾⡧⡅⠉⠈⠀⣀⣦⡹⡞⣽⢹⢏⣴⣀⠀⠁⠉⢨⣼⠿⢦⢶⡷⡴⠿⣧⡅⠉⠈⠀⣀⣦⡽⡏⣯⢻⢏⣴⣀⠀⠁⠉⢨⢼⠿⢧
⠟⣧⠳⡟⠂⠀⠀⠀⢮⣧⣥⢮⣿⡽⣬⣽⡵⢀⠀⠀⠐⢻⠿⣴⣸⣅⣦⠳⡟⠂⠀⠀⠀⠮⣧⣥⢧⣿⡵⣬⣼⡵⠀⠀⠀⠐⢻⠞⣼
⡿⠉⠃⠀⠤⣼⠮⣿⡅⠉⠓⠒⠚⠒⠚⠉⠨⣼⠷⣧⠄⠀⠘⠑⢟⡻⠈⠃⠀⠠⣼⠾⣧⠅⠉⠓⠒⠛⠒⠚⠉⢨⣼⠵⣧⠤⠀⠘⠋
⠁⠀⠀⣈⣳⠮⡥⣟⢹⠀⠀⠀⠀⠀⠀⠀⡇⣻⢌⠵⣾⣅⠀⠀⠈⠁⠀⠀⣨⣷⠯⡡⣟⢸⠀⠀⠀⠀⠀⠀⠀⡏⣻⢌⠵⣞⣅
⠀⠀⠛⠾⢗⣛⣛⡿⢺⠀⠀⠀⠀⠀⠀⠀⡇⢿⣻⣛⣻⠷⠛⠀⠀⠀⠀⠛⠼⣟⣛⣛⡿⢸⠀⠀⠀⠀⠀⠀⠀⡟⢿⣛⣛⣺⠗⠛
⣆⠀⠀⠐⢉⣵⣂⣿⡞⠀⠀⠀⠀⠀⠀⠀⢳⣿⣰⣮⡙⠃⠀⠀⡰⢆⠀⠀⠘⢋⣵⣆⣿⡞⠀⠀⠀⠀⠀⠀⠀⢳⣿⣰⣾⡙⠃
⡟⣶⢃⡄⠀⠘⠑⠃⣂⣦⣭⣩⣭⣍⣽⣶⣐⠘⠋⠃⠀⢠⣘⣯⢿⡿⣵⢃⡄⠀⠘⠙⠃⣂⣶⣯⣩⣭⣍⣭⣶⣐⠙⠊⠃⠀⢠⡘⣶
⣟⣅⣋⣟⡅⠀⠀⠀⠵⣧⣆⡕⣿⢺⡰⣾⠯⠀⠀⠀⢨⣿⣛⣸⣺⣗⣅⣛⣿⡅⠀⠀⠀⠵⣷⢆⡇⣿⢪⡰⣼⠯⠀⠀⠀⢨⣻⣙⣸
⠭⠬⠟⠃⣂⣶⣴⣀⠀⠁⠞⢿⢾⡞⠷⠈⠀⣀⣦⣶⣐⠘⠻⠥⠭⠯⠬⠟⠃⣂⣶⣴⣀⠀⠁⠞⢱⠷⡿⠳⠈⠀⣀⣦⣶⣐⠘⠻⠥
⠀⠀⠀⢰⠻⣇⢛⣧⢖⠀⠀⠀⠛⠀⠀⠀⡲⣼⡛⣸⠟⡆⠀⠀⠀⠀⠀⠀⢰⠻⣇⢛⣥⢖⠀⠀⠀⠛⠀⠀⠀⣢⣬⡓⣸⠟⡆
⠀⠀⠀⢸⢼⡷⡷⢞⣿⡱⠆⠀⠀⠀⠰⢎⣿⡳⢿⢾⡧⡇⠀⠀⠀⠀⠀⠀⢸⢸⡷⡷⢾⣿⡷⠆⠀⠀⠀⠰⢎⣿⡰⢾⢾⡗⡇
⠀⠀⠀⠸⣴⡏⢬⡟⠮⠀⠀⠀⣤⠀⠀⠀⠵⢻⣥⢹⣦⠇⠀⠀⠀⠀⠀⠀⠸⣴⡏⣬⡟⠯⠀⠀⠀⣤⠀⠀⠀⠴⢻⡥⢹⣦⠇
⣒⢒⣦⡄⠍⠿⠻⠉⠀⡀⢦⣾⣾⢧⡶⢀⠀⠉⠟⠿⠩⢠⣴⡒⣒⣒⢒⣦⡄⠍⠿⠻⠉⠀⡀⢢⡼⡶⣷⡴⢀⠀⠉⠟⠿⠩⢠⣰⡒
⣯⡏⣍⣯⡃⠀⠀⠀⣲⡟⠎⡣⣿⢼⠱⢿⣖⠀⠀⠀⢘⣿⣭⢙⢽⡯⡋⣭⣿⡃⠀⠀⠀⡲⡿⠎⡇⣿⢜⠹⢻⣖⠀⠀⠀⢘⣽⣩⢙
⣷⠿⡌⠃⠀⢠⡠⣄⠍⠟⣛⣙⣛⣋⣻⠻⠩⢠⢄⡄⠀⠘⢩⢿⣾⣷⡻⡍⠃⠀⢠⡠⡄⠍⠿⣛⣙⣛⣋⣻⠿⠭⣠⢄⡄⠀⠘⢡⢿
⠏⠀⠀⢠⣌⡻⠍⣿⢧⠀⠀⠀⠀⠀⠀⠀⡞⣿⠹⢟⣡⡄⠀⠀⠹⠏⠀⠀⢠⣌⡻⠏⣿⢧⠀⠀⠀⠀⠀⠀⠀⡼⣿⠹⢿⣡⡄
⠀⠀⣤⢶⣯⣭⣯⣷⣼⠀⠀⠀⠀⠀⠀⠀⡇⣾⣽⣭⣵⡖⣤⠀⠀⠀⠀⣤⢶⣯⣭⣯⣷⢸⠀⠀⠀⠀⠀⠀⠀⣧⣾⣭⣭⣵⡶⣤
⡀⠀⠀⢉⡽⢞⡑⣯⣸⠀⠀⠀⠀⠀⠀⠀⡇⣽⢊⣲⢿⡋⠀⠀⢀⡀⠀⠀⢉⡿⣖⡑⣯⢸⠀⠀⠀⠀⠀⠀⠀⣇⣽⢊⡳⢿⡋
⣵⣀⡄⠀⠒⢻⢶⡿⡃⣀⡤⠤⣤⠤⢤⣀⢐⢻⡶⡟⠂⠀⠠⡠⣮⣵⢀⡄⠀⠐⢻⢶⡟⡃⣀⡤⠤⡤⠤⢤⣀⢐⢻⡲⡟⠒⠀⢠⣀
⣧⡟⡴⣧⠄⠀⠀⠀⢞⡟⡛⢞⣿⣳⢛⣻⡳⠀⠀⠀⠠⣼⢶⠻⣹⣎⠟⣶⣧⠄⠀⠀⠀⢞⣟⡛⡞⣿⡳⢛⢻⡳⠀⠀⠀⠠⣼⢦⢻
⠷⠳⡶⡟⡃⣀⢀⠀⠉⠟⣱⢧⣻⣸⣞⠻⠉⠀⡀⣀⢘⢺⣶⠞⢾⡷⠳⣶⡟⡃⣀⢀⠀⠉⠟⣳⣏⣿⡽⣎⠻⠉⠀⡀⣀⢘⢺⣶⠞
⠉⠉⠁⢀⣶⡗⣻⠖⣀⠀⠁⠘⣽⠁⠉⠀⣀⠲⣟⢺⣶⡀⠈⠉⠉⠉⠉⠁⢀⣲⡗⣻⠖⣀⠀⠉⠘⣯⠋⠈⠀⣀⠲⣟⢻⣶⡀⠈⠉
⠀⠀⠀⢸⣨⣗⣖⡛⣧⢄⡀⠀⠀⠀⢀⣠⣼⣛⣲⣺⡅⡇⠀⠀⠀⠀⠀⠀⢸⢨⣗⣔⣻⣯⢄⡀⠀⠀⠀⢀⡠⣾⣟⣶⣺⡅⡇