Biplane Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏⠀⠀⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠟⢁⠀⠲⡡⠐⣀⠀⢳
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣔⠀⠀⣀⣀⣠⡞⠋⣏⠁⠀⣵⡈⣀⣠⣴⣿⣿⣿⡆⠀⠇
⠀⠠⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣭⡀⢀⣷⠐⠈⢀⣾⠥⡿⣟⣭⣽⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢾⡷⠞⠁⠼⠀⢸⠁⣡⣾⣿⡿⠛⠉
⠀⠀⠀⠀⢀⠀⠁⠐⢄⠀⠀⠀⠀⡀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣲⡶⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠗⠊⠁⠀⡾⣿⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⠛⠀⣀⣀⣠⠤⠔⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠂⠴⠿⠺⢗⡛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⡹⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣧⢿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣻⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡿⠗⢟⠉⠐⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⠿⠿⢿⣿⣿⣮⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠸⡀⠀⣣⣠⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⢀⣠⣤⣶⣦⡙⢿⡿⠁⠀⢀⣀⣤⣬⣍⡙⠻⠿⡏⠀⠀⠈⡜⠛⠉⠀⠀⠀⠀⠀⠀⣀⣤⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣴⣿⣿⣿⣽⣿⣿⣿⡦⢲⣴⣆⢈⠀⠀⠀⡍⠉⠛⠶⢬⣀⠀⠀⡇⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠙⣿⣿⣿⣾⣿⣿⣿⡏⠻⠷⠣⡈⠳⣦⣤⣅⡀⠀⠀⠀⠈⠙⠲⢤⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠈⠻⣿⡿⠿⠟⠛⠳⣤⣤⣶⣦⣄⡈⣛⣿⣿⣷⣤⣄⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁
⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⡟⣿⣿⢿⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡿⣣⣿⣿⣷⣹⠿⠛⢿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣶⣤
⠀⠀⠀⠙⠿⢿⣿⣿⡿⠿⠿⢿⢿⡿⠟⠙⠛⠋⢉⣉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠷⣶⣤⣄⣀⡀⠀⠀⠈⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⠛⠉⠉⠉⠈⢳⣄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠙⢷⡒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠢⣀⣀⣠⣤⣖⣶⣭⣿⣿⣾⡿⠿⠟⠛⠋⠉⡁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣶⣾⠿⠾⡿⣿⠿⠟⠛⣋⣉⣁⠀⠀⠀⠀⠀⠀⡜⣱⡄
⠀⠀⠀⠴⢶⣾⣷⣿⡿⡿⠿⠛⣻⠟⢦⠀⣶⣦⣤⣬⣍⡍⣛⣛⡀⠀⠀⠀⠠⢼⣠⣿⣿⣄⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣶⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡩⣫⣿⣟⣿⣷⣿⣵⣶⣾⣿⡿⠿⠿⠿⠿⠛⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠠⠠⢴⣶⣷⣷⣶⢾⣻⠟⠛⠛⠋⠙⣃⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣧⠀⠀⠀⠀⠀⠙⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⢀⣀⣀⣤⡤⢤⢤⣴⣶⡄⣾⠷⣴⢶⡶⣶⣶⠶⣶⠶⠶⢶⣤⣤⡰⠤⠔⣶⠆⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠉⠁⠘⣷⠈⠀⠰⠇⠠⣅⠆⢸⠸⠙⡿⡏⠋⠀⠀⠀⠉⠉⢻⠀⠀⠐⡟⠀⠊⠀⠙⠛⠛⠓⠦⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡄⠰⣤⡄⠀⠈⠳⡤⣌⣙⣳⣶⡀⠀⠀⠀⠀⠀⠘⡆⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣇⣀⣯⣿⠀⠀⠀⠃⠀⠀⠈⠉⠙⠒⠿⢤⣄⡀⠀⢳⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣻⣿⣿⣷⣄⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢚⣦⡧⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠶⢤⣤⣿⣵⣹⣿⣽⣿⠦⡀⢠⣤⣴⣥⣤⣤⣦⣤⣦⣤⣼⣿⠷⠦⠶⠿⠖⢶⣶⣶⣶⣶⣶⣦⣤⣄⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢨⠋⠈⠉⢹⠟⠙⢛⠟⠛⠛⠓⠾⠶⠶⠦⠤⠬⢭⣍⣀⣈⣀⣀⣈⣉⣿⣿⡿⠟⠛⠋⠉⠉⠁⢸
⠀⠀⠀⠀⠀⣰⣛⣿⣄⠀⠈⡸⠿⠷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢿⡻⢿⠏⠀⠀⢷⡄⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠚

⠀⠀⠀⠀⣠⣤⡖⣾⣿⣿⣷⠶⢦⣤⣀⣀
⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⣾⣷⠴⣿⣛⣏⣿⣿⣷⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢡⡏⠉⠙⠛⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠙⢿⡐⠻⣿⢿⠽⣿⣿⢿⣷⡲⢦⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠠⠈⠂⠀⠀⠀⠀⠈⣧⠂⢹⣀⣠⣤⠷⢞⣿⣧⡚⠑⠙⠪⠭⠝⣻⡛⣿⣓⣲⣦⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡞⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣶⣿⠟⣙⠃⢸⡙⠿⣿⠃⠀⠀⠀⠀⠀⠀⢀⡏⠉⣹⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⢯⣼⠟⠁⠚⠉⠀⢸⣿⣶⣿⡁⡀⠀⢒⡤⠀⠀⡼⠀⢠⡇
⣠⣶⡞⢛⡻⢛⢿⣿⣷⣶⣿⣴⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠠⣀⠔⢻⣷⣫⣕⠂⠀⠀⠀⢀⣼⣬⣭⣭⡝⠵⡁⡅⠀⠠⢰⢃⡄⡼
⠀⠈⠉⠉⠉⠛⠛⠛⠛⠻⠫⠤⢤⠌⠀⢉⡽⢿⣿⣷⠶⢶⣶⣤⣯⣬⣄⣻⣿⠀⢀⣠⣾⣿⣿⣿⢿⠋⠀⠘⠀⠀⠀⠀⡞⠈⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⡩⢽⢞⣻⢻⡟⡿⠿⠿⠿⣿⣿⣷⣿⣿⣿⣿⣿⡝⣾⣧⣶⣶⣶⣶⣤⣶⣷⣦⣿⣤⣤⣤
⠀⠀⠀⠀⣠⣤⣴⣦⣦⣤⣤⣤⣷⣵⣤⠞⠋⠀⠸⣿⣀⣈⡥⠴⠶⠚⠛⣻⠯⡉⠯⢰⠃⠀⠀⠙⣄⡇⡛⠦⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠿⠿⢦⠔⠂⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡏⣤⡷⢰⡿⣤⣤⣄⣠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣹⢷⡎⠀⠀⠀⠀⠀⢺⣿⡿⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣴⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⣿⣽⣿⡿⣿⣮⣖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢯⣾⣿⣿⣿⣷⣅⠀⡀
⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠿⢿⣿⣿⣮⡢⢄
⠀⠀⠀⠀⠀⠀⡿⠿⣷⣶⣤⡀⠄⠀⠤⠤⠠⣶⣿⣿⣷⣶⣤⡺⢧⣤⡁⠀⠀⠲⠿⠿⢿⣿⣿⣶⣕⢄
⠀⠀⠀⠀⠀⠀⠳⢦⣬⣛⠛⠛⠁⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣶⣌⠁⠙⣀⠴⠊⠁⠀⠀⠀⠉⠉⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠒⠶⢤⣤⣤⣀⣀⣀⣙⣻⣿⣾⣯⣿⣿⣦⣌⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠂⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣷⣄⠤⣤⣤⣴⣶⡿⡅⠀⢙⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡿⠈⣹⣝⠉⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⣥⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⠼⠛⡷⠿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠏


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠄⠀⠰⡎⠀⠀⡀⠀⠀⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠈⠁⠀⠀⠀⠀⣀⣿⡤⠶⣟⡛⠳⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠴⠿⣿⠟⠓⢄⠀⣀⣠⣤⣴⢖⣋⠭⠥⡗⠻⡈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠠⠖⢿⣶⣿⠯⢥⢤⣀⡀⠄⠂⠀⢄⣽⡿⡖⠛⠋⠉⠁⠀⠀⠀⡄⠀⢃⢀
⠀⠀⠀⣠⣴⠖⠚⠒⠲⡍⠉⠀⣀⣀⣀⢤⣤⣻⢿⣿⡷⠀⢠⣶⣌⣉⣡⠚⠣⣀⣽⣷⡄⠀⠀⠀⢀⠂⠷⠾⠋⠉⢀⣂⣴⠉⠙⣂
⠀⠀⠀⠙⠻⠖⠀⠠⢶⢿⠮⠉⢿⠂⠈⠀⠉⣿⢳⣾⣗⡆⢠⢾⠁⠀⠁⡀⠤⣝⣛⠭⠕⠒⠈⠀⠀⣀⣠⠤⠖⠛⣉⠭⠛⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠀⠀⠀⠙⣙⡾⠿⠄⣻⠏⡀⢪⣵⣞⡋⠁⠀⠀⠀⠠⢴⣞⣋⡩⠄⠲⠤⠐⠁⠀⠀⢸⡀⣀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣄⣶⠤⠤⠦⠈⠛⢻⣿⠛⢃⣽⣿⣾⣧⣷⣿⣾⣾⣣⢤⣴⠖⠋⠁⠀⠀⠀⠐⠮⢍⣁⡀⢀⣠⡤⠐⠃⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⠾⠭⠅⠒⠚⠉⢟⠝⠛⡉⠛⠛⠿⠿⢷⣶⣦⣤⣤⣀⣀⠀⠀⠉⠉⠛⢛⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠚⠛⠋⠉⡩⠆⣾⡁⠀⠀⣀⠀⠀⠰⣎⠠⠆⡅⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠙⠛⢿⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠼⠟⡏⠀⠀⠀⠀⠀⠀⠙⠳⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚

⢀⣀⡀⠄⢠⡆⠀⠀⠂⠰⠀⠀⠺⠁⠀⠚⢳⢀⢱⣶⡒⠒⠂⠐⠂⠐⠒⢶⡆⠀⠂⠀⠀⠀⠀⠤⠄⢀⣀
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠼⠀⠀⢄⡀⠀⡅⠙⠈⡟⢿⠉⠁⠀⠀⠀⠀⠀⠈⢏⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⠙⠛⠒⠂
⠀⠀⠀⠀⠀⠸⡄⠀⢦⣶⡄⠀⠀⠀⢠⠧⢄⣉⣙⢺⣄⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢇⢠⣸⣬⣇⠀⠀⠀⠘⠀⠀⠀⠀⠈⠉⠙⠛⠛⠤⠄⣀⠀⠀⢧⠀⠸
⠀⠀⠀⠀⠀⠀⠘⡌⣿⢿⣿⣷⣄⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠻⣦⡧⠀⣀
⠀⠀⠀⠠⠤⠴⢤⣭⣧⡌⣾⣿⣭⢹⠇⠀⠀⢠⣤⡤⢤⣤⠤⣤⢤⡤⠤⣤⠤⠼⣟⠓⠢⠤⠾⠭⠔⢲⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⠈⠉⠈⡅⠋⠉⢛⠟⠛⠛⠛⠒⠲⠲⠦⠤⠤⠤⠤⠀⠄⢀⣀⡀⢀⣀⣀⣁⣹⡿⠋⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢠⡞⣻⣷⣀⠀⠀⢀⠟⠛⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢻⡹⠿⣿⠃⠀⠀⢸⡄⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⣆
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠙⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣿⣷⣶⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣶⣤⣀⢀⡀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠀⠀⠹⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠉⢛⣻⣿⡿⢟⠛⠃⣿⣻⣿⣿⣿⡿⠿⠿⢶⠠⠄⠀⣸⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣤⣠⡄⣶⠰⠿⠦⠶⠛⠉⠁⠈⠉⠈⠙⠁⠉⠁⢠⣤⣶⣶⠶⠾⠟⠀⠀⠀⠈⢉⡀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⠿⢿⣿⣿⠟⠛⠋
⣠⣴⣶⣾⣿⣿⣿⣷⣶⣦⣤⠀⠀⣀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⣀⣰⢀⣶⣤⣶⣶⣿⣿⡿⠛⠻⠁⠀⠀⡿⣿
⠈⠛⠿⠿⠛⠻⡟⠛⠛⢋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣛⣿⣿⡟⠉⠉⠙⠚⠁⠀⠈⠉⠟⠛⠁⢠⢀⠌⠀⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⡿⠿⠟⢛⣿⣮⡻⣼⠀⣐⣶⣀⣄⣀⣤⡄⢸⣶⣤⣼⣃⡀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠐⠘⠛⣩⣬⠉⠉⠉⠉⠙⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⣮⡿⠛⢻⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠛⠀⣿⣟⠻⢻⡟⠿⣿⣟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣶⡾⠟⠉⢡⣬⣁⣴⡌⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣝⣛⣽⡿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠉⠙⠓⠒⠢⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⢸⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⢸
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⡇⡷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⡸⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰
⡇⠈⣆⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⡇⡟⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡙⢶⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠠⡀⠈⢦⠀⠀⢀⡴⠚⠙⠫⡉⠉⠳⡶⠤⠴⠷⠷⠤⠤⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⢆⠀⠀⠀⠀⠀⠀⠳⠤⠤⡆
⠀⠱⡀⠈⢳⣀⠎⠀⠀⠀⠀⠰⠀⠀⠘⡍⠭⠍⠀⠀⠀⠈⢳⠒⠓⢄⠀⠀⣀⡤⠤⠤⠤⠤⢤⣌⣐⠳⣄⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠙⢦⡀⢙⣦⣄⠀⠀⠀⠀⡇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠷⢾⣁⣀⣀⠀⠀⠀⠀⣀⣈⣉⡚⢷⡄⠀⠀⠀⠀⡎
⠀⠀⠀⢠⠟⠉⠁⠈⢧⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⡍⠉⠉⠉⠉⠹⠉⠉⠉⠉⠉⠉⠀⠀⠀⢰⠁
⠀⠀⠀⠘⠦⣄⣎⠉⠛⣄⠀⠀⡆⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⢣⠈⢧⠀⠈⢂⠀⠇⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠈⢣⠀⠑⡄⠀⠀⠀⠀⢀⠏⠀⢠⠔⠉⠁⠀⠀⠀⠀⠈⠉⢐⣿⣤⣀⡀⠀⠀⠀⠠⠀⠀⠀⣀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠓⠦⣌⣆⠘⣤⣤⠞⠚⠛⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠹⡍⠀⠰⣰⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡵⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⣨⡏⠀⠙⡌⠙⣆⠠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢄⠈⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠛⣇⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣴⠚⠉⠉⠉⠙⠒⠒⠲⠤⢤⣝⣦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣤⣄⣀⣠⣄⣀⣀⣐⣒⣊⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠉⠉⠉⠉⠉⠛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣛⠛⢛⣻⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠙⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⣐⣀⣀⣀⣠⠟⠛⠛⢧⣾⣁⣀⣁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠀⢀⠀⠉⢹⢣⡖⣤⡞⣧⢻⠉⠁⠀⠀⠘⠠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊⠀⠀⠀⠀⠀⠀⢸⣯⡿⠯⡵⢋⣾⠀⠀⠀⠀⠀⠀⠈⠂⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⠿⣯⡿⠛⠻⣯⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠘⠇







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣲⣿⡭⢿⣷⢿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⢿⣉⣛⣲⣀⣈⣈⣫⣹⣯⣿
⠀⠀⠀⠀⠀⡞⠏⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢿⡿⣿⣿⠁⢿⣙⣿⡟⣿
⠀⠀⠀⠀⢘⣀⣤⠄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢰⠃⣸⣷⣿⣿⣿⣿⣿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣸⢀⣸⡆⠀⠹⠤⠤⠖⠒⠂⠐⠋⠉⠉⠉⠉⠀⠀⢈⣿⣿⣀⣟⣿⣟⢩⣽⣏⡼⡗⠀⠈⢇⠈⠀⠀⠂⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠐⠳⣦⣶⣶⣖⣀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⡟⣻⠟⢿⣻⣯⣿⣿⣿⡿⣹⠣⠀⡀⢈⠅⠀⠀⠀⠀⠀⠼⠟
⠀⠀⠀⠀⠀⠀⠉⠉⠹⠟⠛⠛⠛⠛⠻⠿⠿⠿⠿⢷⣷⣷⣶⣶⣶⣷⣷⣿⣿⣿⣿⣿⣿⡿⢲⣶⡀⢀⡀⠀⠀⢀⡀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⢈⡏⣽⠿⠉⠉⠉⠿⠃⡤⠞⠙⠋⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⢋⣁⣀⣀⣠⣤⣤⠤⠴⢿⣁⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠹⢦⣹⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣮⣙⠒⠀⠐⠒⠶⠒⠤⠤⣤⠤⢄⣀⣀⣀⣀
⠀⠀⠀⠀⠈⠂⠀⠀⡀⠀⢀⠀⠂⢰⣶⣿⣐⣶⣤⠖⠄⠀⠀⣤⠀⣿⠠⣀⢀⣈⣉⡭⠿⢿⣗⠒⠀⢠⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢹⠁⠈⠁⠉⠀⠈⠁⠻⠀⢹⠉⠀⠀⣠⢛⠐⠟⢻⣬⠷⠆⠘⠁⢠⣼⢿⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸⡀⠀⣴⣁⣾⣾⠉⠀⠀⠘⣄⣲⣴⣿⠋⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣾⣦⣈⣂⣤⣷⣾⣟⣹⣁⣽⡆⡀⠀⠔⡵⣿⠿⠛⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠴⢦⣤⣤⣤⣬⣬⡥⣤⣤⣤⣤⣭⣭⣭⣿⣽⣿⣿⣿⣷⠶⡶⠾⣾⣷⣶⣿⣷⡶⠶⠷⠶⠾⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠁⠁⠈⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⣰⣤⡆⠀⠑⢤⣷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⢸⠗⠀⠀⠈⢯⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠈⠉⠛⠿⣿⣿⣿⣿⢿⣿⣿⣿⣶⣤⡀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣤⣄⡤⠴⠦⠖⠚⠙⠁⠐⠈⠉⣭⣭⠵⠿⠇⠀⠈⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣿⣿⠿⠗
⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣶⣤⠀⢀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⡤⢤⣴⣶⡾⠟⡛⠁⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣭⣿⣿⣿⢿⣿⢿⡀⢀⡈⣀⣀⡈⣅⣠⡄⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⡍⠉⠉⠙⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⢓⡿⡛⢿⣿⣿⣴⣿⣿⣿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣥⡾⠋⣽⢻⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⡽⠃

⠀⠂
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⡀⢠⣴⣀⢠⣤⡄⢤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠰⢶⣶⠛⢹⣿⠘⠋⠛⢙⡟⠛⡸⠗⠉⢋⡇⡿⣥⣾⣿⣭⡭⠃⠉⢀⣀⣺⣻⣿⣂⡀⠀⢠⣽⠉⢐⠿⠱⠶⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠈⢻⠆⠐⠀⠘⠃⠀⠳⣤⡁⢀⠁⠁⣀⡏⢯⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢈⡏⠀⠀⠈⠀⠀⠁⠉⠉⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠘⣇⠀⢢⣴⣦⠀⠀⠀⠓⡼⣤⣈⡉⣛⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢻⠀⠨⣏⣿⠀⠀⠀⠀⠇⠀⠀⠈⠉⠙⠛⠳⠶⢷⣀⡀⠀⠀⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⡾⣷⣿⣿⣧⡀⠀⠀⢆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠦⢬⣇⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢷⣼⠟⢽⣿⡿⣦⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢿⣿⠓⠒⠢⢤⣀⣀⠀⢀⢀⣀⣀⣀⢀⠀⠀⠀⠐⠀⠀⠀⢸⣿
⠀⠀⠀⠘⠛⠻⠿⢷⣷⣶⣸⣿⣷⣾⣻⠳⢢⣀⣶⣶⣖⣶⣶⠶⣶⠶⠶⠿⠷⠿⠾⢿⣿⠛⠛⠛⠛⡒⠋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢨⠝⠉⠉⢙⠟⠛⠛⠛⠒⠲⠷⠶⠦⠤⠤⠭⢥⣄⣀⣀⣀⡀⢀⠀⠀⠁⣹⣿⣿⣿⠻⠯⠛⠉⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣄⠀⠠⠊⢸⣾⣷⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⠥⠤⠄⣏⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⢧⣍⡼⠃⠀⠀⠀⢻⣟⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠑⢄⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣢
⠀⠀⠑⣄⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠠⠄⠚⠉⠀⠀⠀⠀⠀⠀⠀⢣
⢀⠤⠒⠉⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠁⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⣀⠠⠤⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠒⠄⡀⠀⠀⠀⠃
⠀⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠉⠒⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⡤⠤
⠒⠄⠀⠀⠀⠀⠀⠀⠀⢐⣀⠤⠤⠒⠊⠉⠀⠁⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄⣀⡀⠤⠐⠒⠉⠁⠀⠀⠀⠑⡄
⣀⣀⢠⡠⠤⠴⠒⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢈⢃⠀⠒⠂⠈
⠢⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⠈⠀⠀⢀⠠⠀⠀⠉
⠰⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⢀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠖
⠰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⠀⠀⠈⠁⠀⠐
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢤⠀⠀⢀⠙⡴⠢⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠐⠀⠀⠀⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⣶⠒⢂⠀⠀⠈⣠⡴⠂⡀⠂⠀⠀⢤⠄⠠⠀⠀⠀⠐⠀⠥⠤⠎⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣡⣦⢠⡀⢸⠀⢸⠀⠀⡀⠓⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢁⣀⠄⠰⠫⠄⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣬⣭⣛⠺⡁⠀⡙⠎⠀⠄⠀⢀⡸⠠⣀⠈⠀⠀⠀⠀⡄⠀⠀⠦⠀⠊⢡⣈⠀⠆⠀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠉⠀⠀⢸⠏⠙⠖⡄⠀⠀⠁⠐⠀⠀⠁⠀⠀⠀⢀⠀⠄⠰⢅⠀⠀⠀⣠⠦⠀⠀⠍⠛⡐
⠀⠀⠀⠀⠀⠀⠀⣼⠉⠄⠘⠵⢦⠋⠀⠀⡀⣿⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠎⠀⠀⠀⠁⠒⣦⠀⣾⢁⠃
⠀⠀⠀⠀⠀⠀⢸⡸⠀⢀⡠⠚⠁⢀⡠⢊⢴⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠂⠀⠋⢊⠃⠄
⠀⠀⠀⠀⠀⠀⡈⠠⣄⡠⠭⠭⠀⢐⣈⢔⣣⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⢲⠂⠀⠀⠀⠀⠀⣇⠌⡐
⠀⠀⠀⠀⢀⠎⠘⢆⢤⣉⠭⢿⡯⠭⢀⣻⣹⠄⠐⠂⠀⠁⠀⠀⠀⠀⠀⡐⣱⠒⠒⠚⡆⠀⠀⠰⣿⣆⡼⣰⠃
⠀⠀⢀⠀⠧⠤⢀⣀⣁⠀⠀⠀⠉⠒⡊⢁⠟⡀⠀⠀⠀⠀⠄⠀⠀⡀⣘⠢⣸⣐⣛⣙⣃⠛⠀⡀⣛⣛⢃⠏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⢓⠂⠄⢤⡆⢀⡀⠠⠠⠐⠐⠀⠉⠁⠀⠈⠢⡈⠢⡀⠀⡏⠀⠀⣀⢧⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡈⠢⡂⠀⠫⢄⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃

⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⡀
⠠⡀⠀⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠠⠊⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠉⠑⡆⠂⠉⠀⠒⠒⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠂⠉⠁⠀⠈⠐⠤⡀
⠐⡀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠓⠤⣀
⠀⠐⡄⠀⠀⠀⠈⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁
⠀⠀⠈⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠑⢄⡀⠀⠀⢈⣆⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡁⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠑⢄⠘⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⣠⠦⠆⠐⠐⠒⠒⠒⠒⠒⢄⡀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⠁⢀⡔⠉⠉⠂⠆⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⠀⠀⠁⢆⠀⠀⠀⠀⠀⠀⡠⠊⠈⠀⠀⣀⠠⠀⠘⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⣀⠔⠁⠀⠀⠀⡀⣀⠠⠤⠄⠐⠓⢤⡀⠀⢠⢞⠘⠀⠀⠀⠈⠄⠀⠀⠀⠀⠈⠢⡀
⠀⠉⠉⠁⠀⠀⠀⠒⠂⠤⠔⠠⡀⢀⣣⣤⠮⠤⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⢈⠟⠀⠈⠛⠁⡱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠎⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢰⠀⢀⡀⠠⠠⠀⠒⠂⠁⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⡠⠔⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢀⠊⠐
⢀⠄⠀⡒⠨⠆⠒⠉⠁⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠠⠃
⠐⠈⠁⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠆
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠉⢩⡍⠉⠉⠙⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡟⠁⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⣿⡇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠙⢿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢀⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣤⣤⣀⣈⣁⣀⣤⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⢄⠀⠀⠑⠀⢀⣀⣀⣇⠀⠂⠁⠀⠀⣠⠀⠀⢠⠃⠀⠀⡴⠁⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢀⡠⠷⡊⠉⠉⠀⠀⣀⣈⣦⣀⣀⡀⡔⠁⠀⢀⠎⠀⢀⠞⠀⠀⢈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣏⡠⠔⠊⠁⠀⠀⠘⡤⠒⠒⠉⠁⠀⠀⠂⠀⠈⢻⠒⡤⣋⠀⣠⠋⠂
⠀⠀⠀⠀⠔⠚⣇⡇⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⡀⠀⠈⠀⠀⢜⣔⠁⢀⡽⠧⣀⡀⢀⢂⠀⢀⣀⡠
⠀⠀⠀⠀⠉⠐⠪⠄⠀⠄⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⢄⢀⡴⠋⠈⣱⠮⣄⡀⠀⠈⢸⠁⠉⠄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⣠⠜⠫⡀⢀⡜⠉⢣⠀⠉⠓⢦⣞⣀⣀
⠀⠀⠀⠀⣀⣤⢤⠧⠴⠦⡄⢀⣀⣰⠚⡟⠛⠋⠉⠉⠀⠀⠀⡼⠋⠀⠀⢨⡀⠀⢀⡜⠀⠀⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡏⠉⢇⠀⠀⡇⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣸⠈⠑⠼⡀⠀⠀⠀⠀⢀⠢
⠀⠀⠀⠀⡀⠤⡔⢳⢊⣹⣇⣠⡼⡤⠤⢵⠴⠶⠶⠖⠚⠁⠀⠈⠠⠀⢠⠃⠀⡀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⣱⠋⠉⠀⠳⡀⠀⢱⠀⢸⣀⣀⣀⣀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠠⠌⠀⠈⠣⡀
⠀⠀⠀⠀⠀⢠⠟⠦⣀⠀⠀⢘⣤⠔⠇⠀⠀⠀⡜⠀⠉⠒⠤⣀⣠⠃⠀⠀⠀⠐⠀⠁⠀⠀⠀⡀⠘⠢
⠀⠀⠀⠀⢠⠏⠀⠀⠈⢢⡔⠁⠈⠣⡘⠀⠀⢸⠀⠀⠀⠀⠀⢠⠋⠑⠢⠄⣀
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠳⣤⣄⣀⡙⡇⠀⠈⡄⢀⣀⣀⣠⣃⣀⣠⣤⣤⡤⡭⠶⠤⣤⡶⠖⠒⠛⠉
⠀⠀⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠉⣏⠉⢹⠋⠉⠉⢁⠏⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠉⠢⢄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⡤⠤⠤⠶⠖⠒⠒⣒⣚⣛⣛⣉⣉⣉⣉⠉⠉⠩⠍⠩⠭⠭⠄⠀⠠⠤⡀⠙⡆
⠀⠀⠀⠀⡼⠁⡖⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢻
⠀⠀⠀⢸⡇⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠋⠙⢲⡄⠀⠀⠀⠈⡄⢸⡄
⠀⠀⠀⠀⡇⢃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣴⠉⠀⢀⡀⠀⠀⠇⠈⡇
⠀⠀⠀⠀⢧⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠇⠀⠀⠘⠲⠴⠚⢳⠀⠀⢰⠀⢳
⠀⠀⠀⠀⢸⠘⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⣠⠆⠀⢈⡀⠀⠀⣠⠞⠀⠀⠀⠇⢸⡄
⠀⠀⠀⠀⠈⡇⡇⢇⠀⠀⠀⠀⠀⢀⡴⠶⠶⢤⠖⡡⢔⠕⠁⣠⠖⠉⠙⠒⠉⠁⠀⠀⠀⠀⢸⠀⣇
⠀⠀⠀⠀⠀⢷⢠⢸⠀⠀⠀⠀⢰⠃⢀⣀⠀⠀⠀⠓⢁⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢸⡸⠈⡄⠀⠀⠀⠸⡖⠉⠉⢹⠄⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡄
⠀⠀⠀⠀⠀⠘⡇⡆⢃⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⡇⠇⢸⠀⠀⠀⠀⠀⠀⠉⠓⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠆⢸⠇
⠀⠀⠀⠀⠀⠀⢹⠀⠈⡆⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠤⠴⢴⡶⣶⣒⣚⡫⠭⠥⠤⠴⠖⠚⢯⡀
⠀⠀⠀⠀⠀⠀⠘⢷⣄⣘⠒⣒⣒⣒⣛⡩⠭⠤⠔⠒⠒⠂⠉⠉⠉⠉⠀⠀⣀⣀⡤⠤⢴⠒⠲⣄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢠⠿⡉⠉⠉⠁⣀⣀⣀⣀⣀⣠⠤⠤⠶⡒⠒⠺⡉⢉⣻⣍⠨⠦⡄⠒⠳⣌⣉⣳⣄⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠱⡀⠀⢏⠀⠈⢣⣀⠀⠬⢦⠀⠒⠛⡌⠉⢙⣄⠀⠸⢦⠀⠛⣎⠉⣉⢧⡤⠬⠳⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠹⡄⠀⠻⡉⠀⢱⡄⠤⠄⢷⠀⠒⢻⣉⣁⠹⣤⠀⠐⢳⣂⣉⣧⣤⠤⠓⠐⠂⢉⣱⡄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠘⡄⠀⠙⣍⣀⢧⠄⠒⠚⢳⢉⣉⣹⡤⠤⠞⠓⣛⣉⡭⠤⠖⠚⠓⠚⠉⠉⠉⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠘⢆⠀⠘⢤⣔⣓⣂⡩⠽⠭⠕⠒⠒⠒⠛⠉⠉⠀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⡄⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠚⠉⠉⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠠⠤⠴⠒⠚⠉⠁⣀⣀⠠⠤⠤⠶⢒⣒⣊⣩⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠤⠔⢒⣒⣢⡤⠭⠥⠶⠖⠚⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣬⣓⣒⣂⣈⣉⣉⣥⠤⠤⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠤⠤⠶⣒⣒⡚⠫⠭⠭⠦⠄⠒⠒⠒⠤⠍⡒⢤⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣶⢊⣼⣟⡚⠻⠭⠭⠩⢝⣀⠒⠒⠀⠉⡽⠉⠀⠀⠀⣿⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠑⠊⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣴⡿⣫⣿⠒⠲⠴⣤⣤⣶⣻⣿⠀⢀⣀⡞⠁⠀⠀⠀⠀⣇⣀⣀⣸⣿⣠⠤⠤⠤⠄⠒⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣮⢾⠁⡏⣠⠔⠚⠉⠀⡼⢉⠏⠘⢳⡾⣭⣷⣏⠉⠉⠉⢹⠁⠀⡅⢸
⠀⠀⠀⠀⠀⠀⠀⢀⢞⡟⠚⢸⠃⠀⠀⠀⢠⠞⠀⢸⠃⢀⡞⠀⠀⠈⠻⢶⡶⢄⣸⠀⢠⠀⠘⡄
⠀⠀⠀⠀⠀⠀⣴⠟⡏⢙⣯⣉⠓⠄⡀⠰⠏⠀⠀⡜⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⡦⡀⠀⡇⠀⠀⠀⠀⠀⣀⢤⡀
⠀⠀⠀⠀⠀⢸⢃⡜⠒⠿⡗⣾⠷⢤⣈⣵⠢⢤⡀⠇⢸⣤⡤⢤⠀⣀⣀⣀⣀⣀⠀⣿⠁⠈⠑⣷⣀⠀⣠⠔⠋⢸⠀⢇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⢤⣯⡁⣀⡾⠛⡶⠦⠬⣥⣖⣝⣣⣌⠋⠀⠐⠒⠒⠒⠠⠭⠩⠝⠳⠿⢦⣍⡐⢷⣦⣼⣀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡕⣹⠿⣿⠉⠀⠀⢺⢖⣚⠻⡀⠉⠉⠉⠙⠓⠒⠶⠶⣦⣶⣤⣦⠤⠤⠤⠤⠤⠝⠦⠙⠲⢯⡭⢛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢟⣿⢶⡶⠤⣬⣭⣏⠎⢰⣷⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠴⠦⠴⠦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠈⣧⠦⣹⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋

⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠢⠂⠃⠂⠘⠞⠚⠘⠂⢀⠠⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠈⠁⠈⠀⠈⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠼⠔⠀⠉⠁⠈⠀⠀⠐
⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⠀⠂⢄
⠀⢀⠄⠪⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⢸⠀⠀⠀⢀⠄⢀⡀
⠀⠈⢄⠀⠀⠀⠈⠐⠢⠤⡀⡀⠤⠐⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠘⡀⠀⠀⠀⢸⠠⠐⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⣰⡤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠠⠀⠈⠢⡀⠀⠀⠀⠀⢀⠤⠐⠈⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⣴⡄⠀⠀⠈⢢⡤⠐⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠂⠁⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⠄⠀⢀⠠⠐⠈⠀⠀⠠⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡁⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠁⢙⠁⠀⠈⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⡀⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣿⣿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠟⠛⠛⠛⠛⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⢁⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣀⡉⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⡏⢹⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⣴⣿⣿⣿⣿⣿⠟⢋⣥⣶⣶⣿⠃⠘⣿⣶⣶⣬⡙⠻⣿⣿⣿⣿⣿⣧⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣷⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣾⣿⣿⣿⣿⠏⣴⣿⣿⣎⠛⢿⣿⡇⣀⢀⢸⣿⡿⠛⣱⣿⣿⣦⠹⣿⣿⣿⣿⣷⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣧⡀⣸⠃⠈⠓⠘⣇⢀⣼⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣿⣿⣿⣿⠥⠿⠟⠛⠛⠉⣍⡁⠀⡖⣡⣌⠢⠀⡬⠉⠉⠛⠛⠻⠿⠬⣿⣿⣿⣿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣿⣿⣿⣷⡒⣶⣤⣤⣄⣀⣉⠉⠀⢦⣛⣛⡴⠀⠓⣀⣀⣠⣤⣤⣶⢲⣾⣿⣿⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⡿⠁⢹⡄⢠⡀⢠⡏⠈⢿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢿⣿⣿⣿⣿⣆⢻⣿⣿⢟⣠⣶⣿⡇⠐⠁⢸⣿⣶⣄⡻⣿⣿⡟⣰⣿⣿⣿⣿⡿⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠈⢿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⡿⠁⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⢻⣿⣿⣿⣿⣿⣦⣌⡛⠿⢿⣿⡄⢠⣿⡿⠿⢛⣡⣴⣿⣿⣿⣿⣿⡟⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣇⣸⣴⣶⣾⣿⣿⣿⣿⣿⣿⡿⠋⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡈⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⢁⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⣤⣤⣤⣤⣤⣤⠶⠶⠟⠋⠉



⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⡿⠿⣶⣦⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⡇⢰⣶⣶⡶⠀⣠⣶⣶⣶⡆⢸⣿⡇⣿⣏⢉⣿⡇⠀⠀⠉⠙⠛⠻⠿⠿⢿⣿⠟⣻⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣁⣘⠉⠉⣿⣿⡇⢸⣿⡇⣿⣿⢿⣿⣇⡀⠶⠾⠿⣷⣦⣄⣀⣀⣤⣼⣿⣟⣿⠟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⠏⠀⠀⣿⣿⡇⢸⣿⡇⣿⣷⣤⣿⡇⠑⡆⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠟⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣆
⠀⠀⠀⠀⠀⢸⣿⣇⣈⣉⣉⣁⣀⣀⣈⣉⣉⣁⣸⣿⡇⣿⠿⠿⢿⣿⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠛⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⣿⡖⠒⠚⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⣉⣉⣉⠉⠉⠉⠉⠉⠉⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣿⠋⠉⠉⠉⠉⣉⣉⣉⠉⠉⠉⠉⠉⠉⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⠿⣽⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⠉⠀⢙⣿⣻⣿⣿⣇
⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣽⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⡤⠖⠚⢹⡇⠉⠒⠦⣄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⢷⣄⣀⠤⣾⡇⠠⢄⣀⣴⠟⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢈⠿⣿⣦⣿⡇⢠⣾⡿⢏⠀⠈⡆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⣘⣁⣀⣈⣀⣈⣉⣻⣇⣼⣿⣷⣾⣦⣤⣼⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠹⡉⠛⢻⠛⣻⣿⠋⣿⣦⣠⡀⢠⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢈⣿⡿⠛⠀⣿⡟⠻⣿⣎⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⢟⠉⠈⠙⠀⣿⠇⠉⠈⢙⢷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠒⠤⠄⣿⠤⠴⠚⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣴⡶⠋
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣦⣄⣀⠀⣀⣠⣤⡀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠈⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠿⠟⠛⠋⠉
⠀⠀⠀⢀⠀⠈⢿⣿⣿⣿⣷⣦⣄⣀⣀⡀⠀⠀⠄⠀⠀⢀⣀⣤⣴⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣭⣅
⠀⠀⠀⠈⠻⢿⣾⣿⣿⣿⡿⠟⠛⣉⣁⣐⣤⣴⣶⣶⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠉⠀⠀⠀⣄⢺⣿⣿⣿⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠎⡉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠉⠒⠠⣍⡒⠤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣠⣤⣄⡀⠀⠈⠑⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣄⠀⣤⢄⣀⡜⠁⢀⡴⠋⠁⠀⠈⠙⢦⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢥⡇⠀⠀⠀⠀⡔⠉⢆⠀⠀⠀⠀⠀⡠⡫⠊⠁⠀⢀⡠⢘⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⣠⠔⣉⠉⠭⣉⣿⡲⡀⠀⠀⠹⠛⡟⠀⠀⠀⠀⠀⣿⠁⠀⢀⠔⠁⢀⠎⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⡾⠁⠀⡌
⠀⠀⠀⠀⢰⠁⡜⠁⠀⠀⠈⡇⢸⠄⠀⡠⠃⡰⠁⠀⠀⠀⠀⠀⣾⠀⢠⠋⠀⠀⡌⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠋⠀⠀⡰
⠀⠀⠀⠀⣾⠟⢆⠀⠀⠀⡠⠃⡜⢆⠉⣀⠔⠁⠀⠀⠀⠀⠀⠸⡆⢠⠃⠀⠀⡸⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⠀⢠⢻
⠀⠀⠀⠀⠑⠵⢤⣉⣉⣉⣤⠮⠒⡌⢷⠻⠼⢖⢄⠀⠀⠀⣠⡳⢁⠏⠀⠀⢀⣇⡀⠀⠀⠀⡰⠁⠰⠁⠀⠀⠀⠀⢀⠇⠈⡄
⠀⠀⠀⠀⠀⢀⠊⢣⠰⡊⣴⣺⢷⢘⡌⢆⠀⠀⠑⠵⣖⠮⠛⠁⡘⠀⠀⣠⠋⠑⡦⠕⡒⡴⠁⣰⠁⠀⠀⠀⠀⢠⠋⠀⢠⠁
⠀⠀⠀⠀⠀⢸⠀⡇⠱⡑⣌⠥⠒⢉⣠⡼⣆⠀⠀⠀⠀⠀⠀⠀⢧⠀⢠⠁⠀⠀⠉⠒⠼⠁⢠⠏⣒⠤⣀⠀⡰⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠘⡄⠸⡄⢱⣠⣴⠾⠛⠉⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢰⠁⢠⠋⠒⠢⠟⠒⠿⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠘⣶⠛⢹⠻⡀⠀⠀⠷⣄⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⠀⠀⢠⠃⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠱⡀⠀⠸⡈⠢⡀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢃⠀⠘⣦⠜⠓⢒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠜⢆⠀⠈⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡡⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠒⠈⠉⠉⠁⠒⠤⠠⠤⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⠐⠆⠁⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⠦⣀⠀⠀⡄⠀⠀⠀⠀⢀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠘⢋⡉⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⡲⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣑⠁⡠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢱⣊⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠂⠀⠁⠀⠀⠀⠀⢀⠀⡂⠀⣀⡠⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀⠀⠀⠀⠈⠀⠀⡃⠀⠈⣆⢀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⡀⠁⠐⠃⠀⠀⢀⠂⠡⡠⠓⠒⠃⣤⠎⢀⠕⠁⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠀⢑⠀⠀⠀⠀⠈⠟⣠⢀⠀⣌⡵⡴⠧⡰⠉⠑⠀⠀⣷⣀⠀⢸⡃⠤⠴⢢⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⢀⠃⠀⡇⠀⠉⠙⠀⠬⠈⠀⠀⠠⠄⠄⠚⡕⠀⠐⠠⠄⡀⠔
⠀⠀⠀⠀⠀⠀⢠⠒⠄⡀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⡀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢦⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⠀⡈⢉⣑⡃⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠐⠀⠀⡀⠠⣄⠀⠂⣀⡀⠀⣀⠀⣆⡀⠐⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢲⠘⡁⠀⠀⡎⠀⠀⠠⠁⠀⡀⠁⠂⢄⡀⠉⠈⡢⠀⠃⢠⠜⣫⡠⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⢃⠀⠀⠃⠀⠠⠁⠀⡾⣼⠓⠲⠀⠈⠰⠤⠁⢠⢔⣁⣀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠰⡀⠀⠁⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠰⠠⠧⠧⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠲⠀⠀⠐⠰⠐⠐⠀⠀⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠂⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⣀⣲⣤⡤⠤⣴⡆
⠀⠀⠀⠀⠂⢰⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠤⣼⣿⣿⣆⠀⠀⠀⣤⣄⠀⠶⠶⠒⠈⠛⠛⠉⠉⠀⠀⠀⠀⠠⠿⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢾⣿⣿⡷⠀⠀⠀⡧⠐⠂⠀⠀⠀⠀⠀⠀⠘⢻⣿⣧⣄⠀⢀⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠚⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⠉⠀⠈⠙⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣦⣶⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄⠀⠀⣠⣤⣤⣤⣤⣤⡀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⢹⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠹⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏⢀⣴⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠛⠋⠁⠰⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣤⡀⠀⠀⠙⢿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⢀⣈⣉⡉⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠚⢛⣩⡭⠿⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠋⠀⡴⠋⠁⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠼⢧⣯⣀⣸⡇⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣀⣉⣩⣿⣶⣦⣤⡤⠴⠶⣾⠁
⠀⠀⢠⠲⢦⣀⣀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠋⠸⣵⣀⠟⠰⣶⣀⣼⡄⠀⠀⠘⣏⣣
⠀⠀⣡⡀⠀⡻⣄⠙⣦⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⢠⠖⠋⠀⠀⠁⢰⠖⠀⠀⠀⣿⠁
⠀⠀⡇⠻⣍⣡⠟⠀⣸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⣦⣀⣴⠆⠀⠞⠀⠀⠀⢰⡟
⠀⠀⠙⢦⡉⠀⡀⠠⣏⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠉⠀⠀⠴⠆⠀⠀⣠⠟
⠀⠀⠀⠀⠉⠉⠙⣆⠘⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠙⡶⢤⣄⣀⣀⣀⣤⣴⠞⣿⢦⡀
⠀⠀⠀⠀⢀⣀⠀⠘⢶⢋⠉⢉⡹⣆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡇⠀⡇⠀⠀⠈⠉⠁⠀⣿⠀⣿⢦⠙⢆
⠀⠀⠀⡞⠉⠈⢳⠀⢨⠗⠉⣩⠷⡌⣧⢤⣀⣀⠀⣀⡴⠋⠀⣸⡇⢰⡇⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⠳⣌⢳⡄
⠀⠀⠀⠳⣄⣠⠞⠀⠈⠻⣭⡄⢊⣳⣼⣀⡀⠉⠙⣉⣀⡴⠞⢹⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀⠀⠘⡆⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡭⠽⡄⢣⡈⠉⠉⠉⠉⠁⠀⠀⢸⠀⢸⠧⠤⠴⠶⠶⠶⠒⢛⣀⢸⡇⠀⠀⢹⠀⢳⡀⠀⠀⠀⠀⢠⢊⡩⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡦⠙⠋⣛⢲⡀⠀⠀⠀⢸⣶⣩⠆⠀⠀⠀⠀⠀⠀⢧⡼⠹⡇⠀⠀⠸⣇⠀⣳⣄⠀⢀⡠⠞⣈⠳⠖
⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⡇⠀⢰⡃⢠⡚⠙⣆⡇⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡷⠋⠀⠀⠙⢯⡔⠋⠈⠙⠋
⠀⠀⠀⣏⠀⠀⢹⠀⠀⠀⠀⡇⠀⠀⠳⢤⣱⡄⠈⠁⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣼⡠⠾⣆⠀⠀⠀⢹⠄
⠀⠀⠀⠈⠳⠖⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠲⠖⢋⡤⢞⣋⣠⡴⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠸⠷⠚⢦⠐⡏⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⢿⡀⠀⠀⠀⠀⢠⣔⣁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠘⡇⠀⠀⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣾⠀⣷⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⣀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢀⠠⠤⢺⣇
⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⣀⣤⠤⣴⠋⠉⠉⠉⠛⠛⢻⠉⠀⢸⠛⠛⠒⠒⠛⠋⠭⠚⠋⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⢀⣤⣖⣉⣹⠀⠀⢼⠲⠶⢿⣦⣄⣀⣀⣀⣠⣤⡴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠭⠝⠒⠒⠺⠿⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠉⠁⠐⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⢀⡀⠤⢠⢐⡀⡆⣂⢂⣐⣐⢒⡐⠂⡀⣄⡀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⡀⠄⢂⣡⡴⠼⠋⠉⢀⣀⠀⠠⠤⠌⠉⣍⠹⢓⠶⠬⣓⢤⡀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⡠⠢⣀⠶⠚⠓⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠉⠙⠺⣆⣀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡰⠎⠡⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠐⠀⠀⠀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡔⢀⡶⠉⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠎⣿⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠂⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡚⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠄⡬⠀⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠌⢠⠃⣀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠭⠥⢄⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠁⠨⢁⡫⠤⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠨⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣓⠂⠀⠀⠀⠀⠀⠄⠈⠀⠁⠀⠀⢠⠀⡨⠋⠘⢲⡌⢽⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠥⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢌⣡⡞⠫⢽⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⠠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠰⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠱
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⣨⡄⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⢀⠞⠀⠀⠀⠀⡠⠀⠐⠁⠀⠀⠀⠀⠠⢤⠄⠀⠀⢠⠃⡙
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡯⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⢀⠒⠈⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⠐⡶⢯⣇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠘⢥⠀⠁⠀⠀⠀⠀⠀⠀⡀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠠⠊⠾⠹⠜⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠢⡀⠀⠀⠑⠀⡀⠀⠀⠀⠀⢠⡧⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠀⠔⢢⠶⠚⠆⡶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣈⠲⢄⡀⠀⠀⠁⠂⠤⢀⣈⡁⠀⠀⠀⢀⣀⠠⠄⠒⠛⠉⠁⠀⠀⠀⢀⡄⢂⠁⠈⢠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠙⠂⢀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠡⠐⠀⡢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠈⠉⠒⠢⠠⠄⡄⡀⢀⠄⡤⡤⠤⠒⡈⠡⠀⠀⠣⠀⣠⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣄⣀⡀⠂⠀⠀⠀⠂⠠⠈⠀⢂⠈⣀⣥⠰⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡉⠈⠐⠒⠓⠒⠒⠃⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠄⠊⠠⠂⠀⠐⢀⠄⠂⠡⠀⡠⠀⠀⠔⠀⠠⠂⠈⠄⠂⠀⠐⠀⡠⠒⢁⠔⠈⠄⠀⠔⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⡠⡔⣒⣁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡄
⠀⠀⠀⠀⢀⡀⠤⢠⠀⣀⠀⠀⠀⠀⠰⠒⣟⣏⢭⠅⠃⠀⠀⣀⣀⣤⡤⠚⡁⠀⠊⠁⢲⠶⢤⠤⣄⡀⠀⠀⢍⠀⠈⠁
⠀⠀⠀⢸⢼⢶⢺⠻⠉⠉⣀⠀⠀⣀⡄⠀⠀⠀⠀⠀⢠⡖⠋⠹⣉⡀⠀⠀⠁⢀⢀⣀⣉⠤⠖⠁⣢⣿⠂
⠀⠀⠀⠘⠚⠉⠈⠀⠀⠀⠘⢻⡿⢛⠁⠀⠀⠀⠀⠀⠈⠻⠶⣤⣄⣈⣉⣉⣉⣉⣡⣤⣤⣤⡶⠾⠋⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⣤⠆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠉⠉⡤⠑⠀⠀⢀⡀⡔⢢⠀⠻⠇⠀⠀⠀⠀⠀⠠⠀⠀⡤⡤⣦⢤⢤⡀⠺⠿⣀⡖⡗⢟⢻⢹
⠀⠀⢀⡠⡤⠴⠀⠀⠀⡈⠀⠀⠀⠁⠀⠀⠀⠀⠛⠀⠈⠐⠃⣠⠀⣇⢀⢹⣸⡼⠋⠀⠀⠏⠳⠶⠼⠼⠼⠄
⠀⠀⠀⠀⠀⠀⣀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠼⠀⠠⡻⠀⠈⠉⠉⠀⠀⠀⢐⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠄⠖⠈⠀⠀⠀⢀⢀⢀⠀⠀⣈⠋⠀⠀⠀⠸⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠈⠑⢦⣀⣷⣦⡀⠀⠀⢠⣶⣶⣄⣴⠀⢰⣷⠿⣿⡟⠃⣿⣯⣩⢹⣿⣹⠗⠀⠀⠀⠈⡗⠒⠂⢰⠂⠐
⠀⠀⠀⠠⢤⣀⣀⠀⠀⠀⠉⠃⠀⠀⣿⣁⣿⡏⢿⣧⣼⠏⠀⣿⡇⠸⣯⣤⣤⠸⡿⠙⠷⠀⠀⠠⠀⢹⠋⠓⣾
⠀⠀⠀⠀⠀⠈⠃⠀⠀⡤⠶⣄⠀⠀⠘⠛⢉⣤⣄⢩⡤⢤⡀⣴⢶⡀⢀⣴⢶⢠⡶⠖⡂⠈⠀⠀⠀⠾⠲⠶⠻⣤⠐⠐⠂
⠀⠀⠰⠖⠶⢶⡀⠀⠀⢧⣀⠞⠀⠀⠀⠀⠘⠳⢦⣼⣷⠾⠛⣿⠾⣧⢼⣇⣠⠜⣟⢋⣁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠻⠷⠖⠀⠀⠀⠀⠀⢈⠠⠂⠈⠛⠛⠁⠛⠀⠀⠉⠀⠀⠀⠉⠁⠀⠛⠉⠉⠀⠀⠀⠀⠉⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀⡰⠀⣔⢈⠀⢀⡀
⠀⠀⠀⠀⠉⠰⠶⠀⠀⠀⠀⢀⠖⠉⢠⠄⡀⠀⢁⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢦⣀⡀⠉⠀⠄⠈⢁⣸⠔⠋⠀⠀⠇⠀⠀⠄⠀⠈⡷⠀⠀⢀⠀⠄
⠀⠀⠀⠃⣄⡀⢀⡀⠤⢾⡁⠀⠀⠀⠉⠋⠍⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠁⠀⠀⣌⠀⠄⠀⢀⡀⠀⢠⠐⢀
⠀⠀⠀⣴⠇⠀⠈⠇⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠋⠀⠐⢦⣤⠄⠒⠾⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢟⣬⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣾⣿⣧⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣙⣛⣫⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⣛⠻⣿⣿⣿⣿⠿⣛⠻⢿⠇⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⢘⣋⣵⣿⣿⣿⣮⣛⣛⣵⣿⡟⣵⣶⣶⡌⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⢟⣩⣤⣍⠻⣿⣿⣿⢏⣾⡟⣩⡝⣿⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣶⣿⣿⣿⣿⣿⣶⡿⢣⣾⣿⣇⢿⡇⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⡟⣵⣿⣿⣿⣿⢸⢇⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⠿⠿⢿⣿⢏⣼⣿⣿⣿⣿⣿⡮⢼⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣜⣸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣮⣭⣭⣭⣭⣥⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⣿⣿⠯⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⣿⣯⣿⣿⢷⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣽⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣯⣿⣾⣿⡿⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣶⣿⣿⣿⣿⣿⣻⣏⣹⣉⡃⣶⣇⡀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣌⣿⣿⣿⣷⠹⠉⢻⣛⣻⣏⠉⠙⠻⡿⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⢠⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠙⣷⣶⣿⣿⣶⡀⢸⣿⠁⢈⡀⠀⠾⠶⣤⣤⠍⠙⣛⣩⣤⣾⣷⣶⣶⣶⢾⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣝⣿⣿⣿⣯⣾⣿⣿⡿⠙⣒⣬⣤⣶⣾⣿⡿⢿⡿⠯⠟⠋⠘⠓⡛⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠉⠙⠁⠙⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⣯⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣨⣤⣶⣶⣿⣿⢿⡿⡿⠛⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠿⠿⠟⠛⠻⠋⠉⢻⡶⠀⠀⠀⠀⠘⣦⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣿⣶⠃⠀⠀⠀⠀⠙⠿⠯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠈⢆⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠱⠀⠀⠀⠀⠀⣸⡀⠀⠀⠌⠀⠀⠀⢸⡀⠀⠀⠀⡞⠀⠀⠀⢠⠞⠁⠀⢀⠜⢶
⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢳⣀⠤⠔⠒⠒⠊⠉⠉⠈⢣⡀⠀⠀⠀⠀⢀⠞⠀⠀⠀⡰⠃⠀⠀⣰⠋⠀⠀⠀⠁
⠀⠀⢀⡷⠀⠀⢀⣠⠴⢒⠁⠙⢆⠀⠀⢀⣀⣤⠤⠖⠒⠻⠉⠒⠲⠦⣏⣀⠀⠀⡴⠁⠀⢀⣼⢁⠀⠠⠎⠐
⠀⠀⢸⣠⡴⠚⠉⠀⠀⠀⠀⠀⠈⢷⡊⠉⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢸⡀⢩⠾⢄⡀⢠⡞⠀⠁⢄
⠴⠚⠹⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠐⠠⠄⣀⠀⠀⡊⠀⠂⠄⣜⣡⠋⠀⠀⡽⠻⢄⡀⠀⠀⠑⠀⠀⠀⠀⢀
⠒⠒⠦⣞⢁⠀⢀⡀⠄⠀⠀⠉⠀⢸⡀⠀⠀⠀⠀⠈⠁⠀⠀⠄⠀⣠⠞⠉⠙⢦⣞⡁⠀⠀⠉⠉⠉⡏⠉⠙⢉⠉⠁
⠀⠀⠀⠈⢩⠂⠁⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠂⢙⣦⠞⠁⠀⢀⡴⠣⡈⠙⠳⢦⣄⡀⢠⠇⠀⠀⠀⠑⠄
⠀⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⣀⣀⣀⣠⡤⠖⠋⠀⠱⣄⣠⠟⠀⠀⢱⠀⠀⠀⠈⢙⡿⠶⠦⡧⣤⣄⣀
⣠⣤⠤⠴⠷⠒⠒⠲⣆⠀⣀⣀⣹⠉⢿⢉⢉⠉⠁⠀⠐⠒⢂⠀⢠⡞⠃⠠⠀⠀⢼⣅⠀⠀⢀⡞⠀⠀⢀⠁
⠀⠀⠀⠀⡆⠀⠀⠀⡏⠉⢹⡀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⡵⢏⠀⠀⠀⠀⠀⢸⠀⠑⠢⣞⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⣀⣧⠤⠴⢻⠃⠀⠀⢧⠀⡀⢘⣀⣀⣀⣀⣀⣀⣴⠟⠀⠀⠑⡀⠀⠀⢀⡇⠀⠀⠈⠈⠑⢦⣠⠀⠀⠀⢠⠃
⠒⠉⠩⣅⣸⡤⠶⠻⡖⠚⠛⠛⡏⠉⠉⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠞⠀⢑⠾⢤⣀⠀⢶⣞⠳⢦⣴⠇⠂
⠀⠀⠀⡼⠀⡆⠀⠀⠘⢦⠀⠀⢱⣀⣀⣇⣀⣤⠤⣤⣀⡀⠀⠀⠀⠀⠠⢠⡍⠠⢤⣉⠀⠠⠈⢹⠾⢄⠙⠦⠙⢦⠨
⠉⠉⣹⠛⠶⣄⠀⠀⠀⠀⣳⡤⠔⢇⠀⠀⠀⠀⢰⠃⠀⠉⠑⠢⢄⣀⢀⠎⠀⠀⡀⠀⢀⠂⠀⠁⠀⠀⠐⠄⡀⠀⠑
⠀⣰⠃⠀⠀⠈⠳⣄⡠⠚⠁⠙⢄⠸⡄⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢈⠟⠒⠤⣄⡀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⣰⠇⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠈⠣⡇⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠈⠉⠛⠢⢤⡀⠀⠀⠀⢀⣀⣀⣤⣤
⣏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠲⠦⢤⣽⣀⣀⣀⣧⣤⣤⡤⠤⢤⡾⠴⠶⠶⠶⠒⠺⡖⠛⠛⠛⠛⠻⢭⡉⠉
⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠉⠰⠇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⠑⠦⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠿⢶⣤⣀
⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣄⠀⠀⠀⢀⣈⣙⣿⣶⣦⣤⣤⣤⣄⣀⣀⡀
⠀⠀⢸⡟⠋⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⡿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠻⢶⣄⡀
⠀⠀⢸⡇⠀⠀⠀⠙⠻⣷⣦⣀⡀⠀⢀⣀⣤⣶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⣄⠀⠀⢸⣿⣦⡝⠳⢶⣤⣀
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⢉⣥⣄⠀⢠⣶⡆⠀⣾⣷⠀⠸⣿⠇⠘⠿⠟⠀⠛⠟⠀⠀⠈⠉⠉⠁⠀⠀⠈⠙⠻⣷
⠀⠀⢈⣿⣧⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⢀⣾⠏⠀⠀⠀⠉⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⡿⠋
⢀⣾⠃⠀⠀⢀⣠⣾⡿⢿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⠿⠟⠋⠁
⢾⣷⣶⣶⣾⣿⡿⠋⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⢛⣛⣿⡿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠛⠛⠋⠉⠉⠉
⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠛⣿⣿⠃⠀⠀⠀⠀⠀⢠⣾⣿⡇⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡏⠀⠀⠀⠀⢀⣴⣟⣉⣿⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⣠⣴⡿⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣾⣿⠟⠋
⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠋⠉
⠀⠄⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢲⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠰⠀⠇⠀⠤⠄⠀⠀⠀⠀⠀⠤⠄⠇⠰⠇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠆⠤⠀⣼⠁⠀⠀⢹⣄⠀⠤⠶⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠉⠉⠁⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠀⡠⢴⣶⢶⣤⡀⢠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠴⠀⠘⡄⣠⠟⠋⠛⢆⡁⡜⠀⠰⠤⠤⠤⠤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⣠⠀⠀⠀⠘⠈⠂⠀⢀⡀⡤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠈⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠐⠒⠂⠲⠀⣶⡇⠀⠀⠀⠀⠀⠀⣷⡆⠰⠂⠂⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣾⣿⡷⠒⠒⠒⠒⠒⠲⣿⣿⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⠿⠁⡇⠀⠀⠀⠀⠀⠀⡏⠹⢯⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠲⡞⢣⣤⣧⠤⠤⠤⠤⠤⢤⡧⣤⢛⡶⠒⠒⠒⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠾⠉⡈⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⡘⡹⢦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⢞⡥⠊⠀⠇⢠⣇⣀⣀⣀⣀⣀⣀⣇⠀⢇⠈⠢⣙⢦⡀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠴⠋⢀⠔⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠈⠑⢄⡈⠓⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⠔⠉⣠⠜⠁⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⢤⡀⠓⢌⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠔⠁⠀⢸⠀⠀⠀⠀⠀⠀⢈⡯⠉⠉⣩⡉⠉⠨⣏⠀⠀⠀⠀⠀⠀⢨⠀⠀⠑⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⡎⠀⠀⠀⣿⡇⠀⠀⠈⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠛⠛⠛⠛⢻⠛⠛⠛⠛⠛⠛⡇⠀⠀⠀⣿⡇⠀⠀⠀⡟⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠲⡞⠀⠤⠄⠀⣀⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⡇⢀⣀⠀⠤⠄⠐⠶⠂⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠤⠤⠤⠬⡇⢹⡶⠶⢾⣿⡗⡶⠶⢺⠃⡻⠤⠤⠤⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢁⠀⣿⣿⣿⡿⢿⣿⣿⣿⠀⠃⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣰⣄⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣴⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠉⠉⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠉⠉⠉⠉⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⠀⠀⠄⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠄⠤⠤

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

⠀⠀⠘⢲⣲⡦⠤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠂⠀⠛⠒⠀⠚⠀⠀⠀⠀⠀⠠⠇⠀⣀⣰
⠀⠀⢘⠉⠀⠀⡆⠀⠀⠀⠀⠀⠐⡄⠀⡀⠠⠒⠒⠐⠊⠉⠀⠐⠐⠂⠰⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠚⠀⠒⢨⠁⠀⠀⠁⠀⡀⠀⠀⠊⠁⠀⣠⣀⠀⣠⢀⢀⡄⠀⠀⠀⠀⡇⠀⢀⡤⠀⠒⠁
⠀⠀⣴⣦⣢⣼⠀⠀⠀⠀⠀⠀⠘⠲⠀⠀⠀⠉⠉⠀⠋⠛⠉⠀⠀⠀⠀⢰⠗
⠀⠀⠉⠉⠉⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠄⠢⡀⢠⡀⠀⡀⠀⠄⠴⠂⠃⠸⡟
⠀⠀⠀⠁⠀⠁⠀⢀⠀⠀⢠⠀⠀⢀⡀⠀⠀⠀⣤⣀⠈⠁⠀⠰⢠⣀⠁⢀⡀⣁⠀⢂⣀⣀⣀⡀
⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⡈⠀⠅⠀⠀⠀⠀⠀⠍⠉⠁⢀⠀⠀⠀⠅⠉⢸⣇⢹⣷⠀⠑⠚⠐⠐
⠀⠀⠀⠀⡀⠀⠀⡀⠣⠀⡇⠀⠀⡤⠄⠀⣿⠀⠀⠀⡀⢸⡆⠀⢠⠀⠀⣠⢉⠙⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⡇⠀⣀⠱⠀⠊⠁⠀⠀⠀⠁⢸⡇⠀⠀⠀⢐⣈⠀⠀⠋⠀⡤⠇⠆⣀
⠀⠀⠀⠰⠀⠰⠃⠂⠒⢲⠃⢀⠄⢠⠀⢀⠀⠀⡄⠀⡀⠈⠇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡆⠈⠐⠂
⠀⠀⠀⡄⠀⣸⣤⠠⣴⠚⠻⣤⣼⢅⢠⡀⠠⠇⠁⠀⠀⠀⠀⠐⠀⠀⠀⠃⠀⡀⠁⠀⠀⠀⡆
⠀⠀⢀⠀⠀⠄⠁⠀⠃⠀⠀⠁⠀⠄⠈⠀⠀⠁⢀⠀⢀⠀⢀⠸⠉⠀⠀⠁⠀⠁⠀⠀⠀⠁⠁
⠀⠀⠸⠀⠀⠀⠀⠀⠀⣀⠀⢀⡄⠀⠞⠃⠘⠀⠸⠂⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠄
⠀⠀⡂⠀⣠⠴⢼⠀⠀⠋⠀⠀⢠⠀⡴⠆⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐
⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⠴⠚⠠⢠⠔⠋⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠁⣄
⠀⠀⠀⠀⡀⠤⠰⠚⠉⠀⠠⡴⠷⠶⠴⠾⠷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡐⣢

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⣀⣼⣷⠶⣄⡀
⠀⠀⠀⠀⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠛⣡⠞⠁⠈⠢⡀
⠀⠀⠘⠤⡐⢂⣉⡵⠦⡀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⣀⣠⣴⣾⡿⠛⠉⡠⠞⠁⠀⠀⠀⣠⣾⣆
⠀⠀⠀⠀⠈⠢⢔⢛⣮⣍⠶⣄⠀⠀⣜⣀⣠⣤⣶⣾⣿⠿⠛⠋⠁⠀⠔⠉⠀⠀⠀⠀⣠⣾⡿⢋⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠑⢶⣊⣻⣭⠵⠴⡿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢋⣴⠟⠁⠘⡀
⠀⠀⠀⠀⠀⠀⠄⠒⠒⠻⣦⣀⡛⣮⠡⠤⣐⢖⠐⡤⢄⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⣁⠴⠋⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⠟⡆⠘⢸⡟⣿⠃⡇⢸⠚⣀⣤⣴⣾⣿⠿⠛⣁⠤⠊⠁⠀⠀⠀⣠⡾⢋⡅
⠀⠀⠀⠀⢀⡇⠤⣐⣒⠆⠀⢰⣿⢀⣧⣥⣬⠷⢷⡛⣿⡿⠛⠋⣁⠤⠐⠈⠀⠀⠀⠀⢀⡠⠞⢁⣴⣿⠃
⠀⠐⠽⠋⠉⠹⡀⠀⠀⠀⠀⡼⠉⠀⢛⣉⣁⣲⣉⠽⠦⣚⣺⡉⠀⠀⠀⠀⠀⢀⡠⠒⠉⢀⣴⠟⠋⠎
⠀⠀⠀⠀⠀⠀⠈⠢⠄⠤⠚⠁⠀⠀⠀⠻⡉⠁⠈⠒⢍⡩⠷⣜⣓⣄⡠⠔⠊⠁⣀⣤⠾⠋⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣤⠴⠷⠮⡩⠕⠐⠛⢢⠶⠛⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣤⣶⠾⠒⠋⠉⠀⠀⢀⡀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠛⠀⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣤⡏⣿⡉⣾⠹⠏⢹⠉⣿⢹⡆⣿⣹⠆⡇⢉⡟⢸⣉⠀⠀⣯⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⠁⣿⠁⢿⣠⡆⢹⠀⣿⢸⡇⣿⢻⠀⣇⣼⠁⢸⠉⠀⠰⣟⣶⠀⠀⠀⣿⣿⡿⠛⠛⠛⠛⠛⣿⣿⡿
⠀⠀⠀⠉⠀⢉⣁⡀⣉⠀⣈⣀⠈⣉⠀⣁⢈⡀⠁⣉⢉⡈⣉⣀⠀⠉⠁⠀⠀⠀⣟⠛⠛⠋⠊⠉⠒⠋⠋⠛⡗
⠀⠀⠀⠀⠀⢸⠁⠓⣿⠀⢸⣄⢠⣿⡄⣿⣾⡇⠀⣿⢸⡇⣿⣼⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣤⡗⢿⣀⢼⣀⣸⠛⡇⣿⠸⡇⠀⠻⣸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠓⠛⠧
⠀⠀⠀⠀⠀⠀⡄⢰⣤⢠⣄⠦⣤⠀⡄⣤⣤⡄⣤⣤⣤⣤⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠘⠁⠘⠛⠚⠙⠋⠋⠈⠙⠉⠛⠁⠋⠙⠉⠉⠙
⠀⠀⠀⠀⠀⢸⡏⣷⢸⣏⡧⢸⣿⠈⣷⣿⣼⢸⡇⣷⣼⢰⡏⠻
⠀⠀⠀⠀⠀⢸⣇⣿⢸⡟⣇⣸⢿⡆⣿⠏⡟⢸⡇⡟⢿⠸⣇⣿⠀⠀⠀⠀⠀⠘⠁⠀⠀⢸⠀⠀⢸⠇

⠀⠀⠀⠀⣰⣆⣤⣢⣤⣶⣦⣀⣉⣥⡘⠋
⠀⠀⠀⠀⠛⠛⡿⠿⣿⠟⠉⢩⣝⠛⢲⠛⠂
⠀⠀⠀⠀⠀⢸⣧⠉⠁⠀⠀⠂⠂⠀⠌
⠀⠀⠀⠀⠀⠸⠋⠘⠀⠀⠐⠂⠀⠀⠂
⠀⢀⠀⠤⠀⠀⠀⠀⠐⣤⡀⠀⠐
⠀⠀⠀⠊⠁⠀⠀⠀⠠⠿⢿⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣠⣤⣶⣶⡾⠖
⠀⠀⠀⠀⠀⠀⠀⢀⡰⡗⠀⠉⠑⠉⠠⡐⠀⠀⠀⢀⡀⢀⣸⣼⣭⣶⣶⣿⣿⣿⣿⣿⣉⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢰⣷⠀⠀⠀⠀⠀⠐⢆⣀⣴⠺⣿⣿⣿⣿⣿⢿⢿⣯⠵⠖⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢠⢸⣯⡭⠁⠛⡛⠛⠋⠙⠆⠈⢨⣀⠀⠀⣀⠂⢀
⠀⠀⠀⠀⠀⢀⢴⠀⣿⣿⡇⠀⠐⠀⠒⣞⠀⠄⠀⠀⠒⠀⠀⠠⠶⠶⠒⠒⠛⠓⠛⠛⠃⠓⠂
⠀⠀⠀⠀⠀⠁⢸⠀⣿⣿⡇⠀⠀⠀⠀⡿⠀⠐⡀
⠀⠀⠤⠄⠀⠀⠌⠀⣿⣿⣷⠀⠀⠀⢀⡃⣠⠤⢧
⠀⠀⠀⠀⢀⣠⠄⠀⣿⣿⡟⠀⠀⠀⣌⢀⣀⣀⠘⠒⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⣋⣉⣉⢉⡓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣩⣥⣍⣉⠉⠙⠻⠾⣦⠈⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⣭⣭⣙⠛⠾⢭⣓⠶⣤⣀⠀⠀⠧⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡸⡾⣿⣿⣿⣷⣤⡀⠈⠙⠲⢭⣛⠶⣴⡖⢤⡈⠓⢄⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠳⡹⣿⣿⣿⣿⣿⣷⣤⡐⢤⣈⠛⢿⣝⠳⢭⡳⢄⠙⢦⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⢎⠻⣿⣿⣿⣿⣿⣿⣾⣌⠓⢦⣈⠛⢦⣉⠳⣽⡢⣍⠢⣄⠀⢀⠀⠀⠀⠀⠀⢠⣿⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠑⢮⡻⣿⣿⣿⣿⣿⣿⣿⣦⣌⠳⢦⡈⠓⢄⠙⠾⣗⡬⡳⣄⠁⠀⠀⠀⢠⣿⡷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣝⠿⣿⣿⣛⣿⣿⡻⢷⣦⣈⠀⠀⠀⣴⣿⣿⣿⣿⣷⣶⣶⣒⣻⢿⣓⠧⣤⣤⣤⣄⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⣤⣶⣶⣯⣷⣮⣛⢿⣿⣿⣿⡶⣍⡻⢷⣤⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣹⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿
⣤⣤⣤⣶⣶⡶⣫⣬⣝⢿⠛⠶⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣮⣛⠿⣿⣾⣿⣶⣿⣿⣷⣔⣩⣿⣿⣿⣿⣿⣿⣜⢿⣶⢟⣼⣷⣄⣼⠉⠹⢿⣇
⠒⠓⠛⠛⠻⢸⣿⣯⣹⢏⣧⡀⠘⡿⠿⠿⣿⣟⣿⣿⣿⣿⣿⣿⣷⣮⣝⡻⣽⡻⠮⢝⣻⣿⣿⣿⡉⢛⢿⡛⠛⠻⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠳⣝⣛⣯⣾⣿⣿⣾⠀⠀⠀⠈⠙⠿⠀⠀⠀⠀⠀⠀⠈⣿⣹⠒⠭⣟⡻⣿⣛⣿⣿⣻⣷⣷⣝⣷⣤⡈⠻⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣿⣿⣀⠀⠀⠈⢿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠀⠀⠙⠿⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⢿⣍⣹⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡿⠟⠛⠉⣉⡠⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠏⠛⠁⠀⣄⣠⢦⣿⣿⣿⣿⣿⣿⡿⠿⢛⡛⢿⣿⣿⣿⣿⠋⠄⠠⠀⠀⢽⣦
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠊⢴⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡆⡛⢿⢣⣻⣿⡏⠃⠀⠀⣤⠀⠁⢸⣿⡇
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠿⢿⣿⣷⡤⡺⣿⣿⣿⣿⠟⠋⠉⠈⠀⢀⠀⠀⠃⢧⠈⡆⢿⣿⣇⣄⠀⠃⢬⢀⣸⣾⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣥⣠⣼⣿⣿⣿⡿⠿⠛⠉⡏⢰⠻⣦⠀⠀⢸⡀⠀⠰⡘⣤⣇⢸⣾⣿⣮⣻⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣸⣿⣿⣏⣽⣿⡟⢻⠟⠋⣹⣀⢀⣀⣬⣧⣺⡟⢿⣴⣴⣾⣷⣤⣤⣵⣌⣛⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⣿⢻⣿⣿⣷⣾⣷⣿⣿⣿⣤⣿⣿⣿⠟⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢭⡏⠈⠢⠤⠌⠻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⢿⡿⠫⢉⡉⠉⠉⠛⣿⣿⢿⣿⣿⣿⣿⡏⣾⣛⡄⢉⣿⣿⣦⢹⣟⢇⣠⣿⣯⠂⠀⠀⠀⠀⠀⠄⢻⠿⣿⣿
⠀⠀⠀⠘⠟⣽⢛⠀⠉⠉⠀⠂⠀⠀⠀⠻⣷⡽⣿⣯⣿⣿⣿⣿⣇⡸⣿⣿⣿⣷⣬⡠⣉⡛⠿⢦⣥⣀⣀⠀⠀⠀⠘⠓⠈⠙
⠀⠀⠀⠀⣼⣛⣊⠈⠁⠀⠀⠀⠀⢠⡾⠟⣉⢹⣽⣿⡟⢯⠍⠛⠾⠦⠉⣽⠛⣿⣿⣿⣿⣿⡗⠲⣤⣍⣙⠛⠻⠿⠁⠀⠘⠀⣱⡄
⠀⠀⠀⠀⣿⣅⣈⣀⣤⣤⣤⢶⣦⠟⠁⠀⠁⣼⣿⣿⣿⢋⣶⣶⣶⣶⣶⣶⠐⣿⣿⣤⣿⣿⣿⣤⣀⣠⣍⣝⣿⣿⣶⣤⣶⠛⠓⠚
⠀⠀⠀⠀⣿⡆⠛⠮⠙⠷⠟⠋⠁⠀⠀⠈⣠⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⡅⢿⣿⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣉⣿⣿⠏
⠀⠀⠀⠀⠘⢧⣤⡀⣀⠀⠀⠀⠐⢀⣤⢾⣵⣿⣿⣿⢃⣷⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⡿⡃⠀⠁⣉⣾⣿⢻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠳⠶⢆⣀⣂⣈⡿⢞⣿⣿⣿⣿⡏⡸⣻⣿⣻⣿⣿⣿⣿⣿⡆⣿⣿⣟⣫⣅⣤⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⣶⣿⡿⣾⣿⣿⡿⢃⣷⣿⣿⣿⡟⣿⣿⣿⣿⣧⡸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡷⣻⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣷⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢹⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠉⠉⠉⠉⢧⡙⠉⠉⠉⠉⠈⢿

⣿⣿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⡀⠀⠀⢀⣀⣤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣀⡀⢻⣿⣏⡀⠀⣻⣼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠣⡀⠈⠙⠛⠁⠈⢛⡫⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠖⠒⠚⠉⠉⠉⠛⣲
⠀⠀⠀⠀⢀⠠⠐⠾⣟⡷⣦⣄⣠⣤⣼⠆⠀⢀⠀⠀⠀⠉⠻⢶⣭⣔⡢⠤⣀⡀⢀⣀⣠⠤⠴⠖⠚⠋⠉⠁⠀⠀⢀⡴⠪⠉⠳⣆⣠⡾⠋
⠀⠀⠀⠀⠸⠦⠶⠖⠚⠛⠉⠉⠉⠈⢷⣄⠀⠸⣙⠲⣄⡀⠀⠀⠈⠙⠻⣿⣶⣬⣝⡓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣮⣾⢀⣠⣾⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⠈⠲⣄⡛⢲⣄⡀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣭⣓⠦⣄⠀⠀⠀⢰⣥⣿⡿⠿⠁⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠟⠛⠶⣄⡙⠾⣄⢹⠳⣤⡀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⡹⣦⣶⡿⣟⢛⠀⢀⢘⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠙⠳⣬⡛⢤⣭⠉⠳⣄⠀⠀⡴⠒⠒⠒⢦⡐⡇⠈⢏⠀⠈⠓⠤⠜⠋
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⣠⠔⢫⠉⢓⢦⡀⠀⠀⠀⠀⠙⠳⣭⣢⣄⢹⡀⠀⣧⢀⠀⠀⠀⢱⡹⣷⠾⡄
⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⣰⣱⣵⠈⠀⠐⡅⢳⣀⣀⣠⣤⣴⣶⣿⣿⣿⡙⢁⡀⠸⡾⣶⡇⣀⡴⢛⡿⡛⢿⡉⠛⢦⡀
⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣧⣤⡶⣶⣿⣿⣿⡿⠿⠿⠛⠛⢿⣉⠛⠿⠋⠙⢷⣘⣦⡞⠁⢠⢏⣮⠎⢈⣹⡲⣅⢳
⠸⣿⣤⣀⣀⣀⣤⣤⣴⣶⣾⡿⡿⡿⡟⠛⠛⢉⣿⣽⠀⠀⠀⠀⠀⠀⠈⣿⡷⢦⣀⠀⣀⣍⠁⠀⠀⡞⣾⣿⡾⣿⣿⣷⣧⣻⡇
⠀⠉⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠹⡜⢬⡂⢰⢈⡽⠃⠀⠀⠀⠀⠀⠀⠀⡸⠷⣄⠉⠛⠳⠿⣦⣄⣀⣇⣿⣿⣷⣶⣿⠿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡆⢻⡇⠀⠀⠀⠀⠘⠿⢿⡙⣋⠯⡉⡀⠸⢛⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠱⢌⡓⠀⣛⣬⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠀⠈⠉⠉⠉⠉⠁⠂⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⣀⡀⠠⠤⢀⣀⣀⡀⠤⠄⢀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠉⠙⢿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡟⣿⣿⣧⣄⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣄⠙⢿⣿⣯⣝⣛⠛⠻⠿⠿⠟⠛⣛⣫⣽⣿⡿⠋⢠⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣉⠀⣿⣷⣄⠉⠛⠿⠿⣿⢿⣿⣿⡿⢿⠿⠻⠛⢉⣠⣶⣷⠀⣈⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⣿⣿⣿⣿⣿⣸⠑⠒⠂⠤⠄⠄⠀⠤⠤⠐⠒⠋⣿⣿⣿⣿⣿⣿⢿⡋
⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⢷⠸⠘⡀⠈⠙⠻⢧⣖⣀⣀⣀⣀⣀⣀⣀⣀⣲⡬⠟⠋⠁⢀⠃⠆⡾⠉⠒⠢⢄⡀
⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⢸⠀⡆⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⢠⠀⡇⠀⠀⠀⠀⠙⢢⡀
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢠⡇⠀⠀⠀⠀⠀⠀⠹⡄
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡔⠉⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠉⢢⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⣀⣤⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⢺⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠘⠁⠸⠿⣿⣿⣿⣿⣧
⠲⠶⣶⣶⣦⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠙⢿⡿⠟
⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠶⠿⠟⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠂⠐⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⢀⣀⣠⠤⢇⡀⢠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢎⡀
⠀⠀⠀⠂⠉⠁⠀⠀⠀⠈⠣⣀⠀⠀⠉⠉⠉⠛⠶⠶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣸⡶⢋⣴⣖⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣤⣀⣀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠁⠁⠀⠻⠿⠛⠋⢻⠈⢉⠁⢸⠓⢮⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠈⠉⠒⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠤⠤⠶⢷⣾⣃⣄⣀⣀⣀⡀⢀⣤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⣀⢠⠠⠤⠔⠐⠒⢶⣿⠶⢚⣩⡅⣥⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠂⠀⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠺⣯⡙⠛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣐⠤⠤⠆⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⣿⣿⠿⠿⠿⠿⠀⠺⠿⠿⠿⢿⣿⡆⠀⢀⣿⣿⡆⠀⠀⢸⣿⡿⠿⠿⠿⠇⠀⠿⠿⠿⠿⠿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⣧⠀⠀⢸⣿⡇
⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡟⢸⣿⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⢸⣿⡿⠿⠿⠿⠃⢰⣿⡇⠘⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⣼⣿⢠⡆⣿⣿⠀⢸⣿⡇
⠀⠀⠀⣶⣶⣶⣶⣿⡿⠀⢸⣿⠀⠀⠀⠀⠀⣿⡿⢸⡇⢸⣿⡄⢸⣿⣷⣶⣶⣶⡆⠀⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡛
⠀⠀⠀⠀⠀⢀⣤⣄⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⣤⠀⢰⣶⣾⣿⣿⣷⣤⣄
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣧⣤⣀⣠⣄⠀⢀⣤⣀⣸⣷⠀⢀⣀⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⣶⠶⠆⢲⣄⡶⢰⡶⢶⡆⣶⠀⠀⡶⠶⣦⢰⡶⢶⡄⢰⣶⠀⠶⣶⠖⢰⠀⣶⠶⣶⢰⣆⢰⡆
⠀⠀⠀⣿⣦⠄⠈⣿⡃⢸⣇⣸⡇⣿⠀⠀⡇⠀⣿⢸⣧⡼⠇⣼⢻⡄⠀⣿⠀⣼⠀⣿⠀⣿⢸⣿⣼⡇
⠀⠀⠀⣿⣀⡀⣸⠟⣧⢸⡏⠀⠀⣿⣀⡘⣧⣀⣿⢸⡏⢷⢀⡿⠺⣇⠀⣿⠀⣿⠀⣿⣀⣿⢸⡇⢿⡇
⠀⠀⠀⠈⠉⠁⠉⠀⠈⠈⠁⠀⠀⠉⠉⠀⠉⠉⠁⠈⠀⠈⠀⠁⠀⠁⠀⠁⠀⠈⠀⠉⠉⠁⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢾⡻⣿⣿⡿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋⠁⠈⠙⣿⣼⡸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣼⡿⠛⠀⠐⠚⠆⢐⣿⣿⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠲⠶⢶⡆⠀⠶⠠⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⠁⠀⢀⡄⠀⠀⢀⠾⠿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠘⠓⠶⠦⣤⣄⡀⠀⠙⣡⠀⢠⡄⠀⠀⢠⠊⢠⡄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡖⠒⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⠋⠀⠀⠰⡀⠀⠉⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⡀⠉⠒⠀⠤⠔⢣⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠈⠀⢀⡴⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢣⠀⠀⠀⢀⠠⠊⠁⠀⠀⠀⠀⡀⠀⣀⣀⣄⠠⢀⠤⠶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⣆⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⣀⣍⣹⡏⡠⠋⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠘⢆⠀⠀⠀⠀⠀⠀⡀⣠⣴⡿⢿⡟⠀⠀⠁⠀⠀⠀⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠱⣄⠈⠧⣀⠠⢐⣻⣭⣿⣻⣅⡀⡜⠁⠀⠀⠀⠀⠀⠠⠁⠀⠈⢣⣀⣀⢀⡠⢒⡶⢁⢋⢙⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠈⠳⢦⣤⣶⣿⣿⣿⣿⣿⣦⣉⣶⣶⣄⡀⢀⣤⣔⣒⣂⣉⠉⠀⠀⠀⠈⠂⠀⠓⠓⠛⠈⢉⣉⣩⡴⠾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⣤⡐⠒⠀⠀⠀⠀⠀⢹⠉⠉⠙⠛⠛⠛⠛⠛⠉⢉⣻⡇⠀⠈⠉⠉⠉⠉⠛⠛⢻⠒⠲⢶⡶⠚⠋⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡁⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⢸⠀⠀⠀⠀⠀⠀⠀⠀⠯⣧⠎⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣠⣤⣠⣤⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠈⡇⠈⡆⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠠⠜⠣⠤⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⡏⠀⢹⡀⠀⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡖⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀
⠀⠀⠀⠀⢿⡠⡇⢠⡀⢙⣦⠈⣿⣿⣿⣿⣿⡅⠈⢛⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠁⠐⡆⠒⠂⢿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⣁⡀
⠀⠀⠀⠀⠸⡟⠒⠚⡏⣿⣿⣧⣸⣿⣿⣿⣿⡟⢉⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⡠⠲⠝
⠀⠀⠀⠀⠀⢻⡀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⠠⠧⠄⠀⢸⠂⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠙⠶⠇⠘⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠛⢷⣶⣶⣶⣶⣶⣤⡤⠤⠤⢿⣤⣀⡐⠂⠈⠁⠀⠀⠀⠀⠀⢸⡆⠀⠀⣀⣠⡴⢶⣟⢠⡀⠌
⠀⠀⠀⠀⠀⢀⡠⠠⡀⢀⣀⠹⢿⣿⣿⣿⣿⣿⡇⢰⠀⠀⡀⠀⠉⠻⢿⣿⣿⣿⡧⠀⠀⡜⣠⠛⠉⠉⠉⣙⡟⠓⠒⠶⠶⡾⡷⠚⠉⠁⡔⠉⡟⠘⠓⠃⡇
⠀⣠⢄⠀⣔⡩⠀⠈⢧⡠⠎⠉⣇⣹⢿⣿⣿⣿⠇⠊⠚⠋⠁⠞⠙⣆⡞⡿⠛⠋⢁⣤⡀⠉⢑⠨⠄⠂⠪⠿⠟⠐⡏⢩⢤⣁⣸⣄⠀⠀⣠⠊⠓⠒
⠀⠈⠢⠌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⣩⠥⠀⡠⠒⠑⠿⢤⠈⠁⠀⠡⠘⠀⠃⠈⠋⠚⠁⠀⠀⠀⠀⠘⠓⠀⠈⠀⢀⠠⡡⠇⠀⡎⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⡉⠉⠉⠉⠉⠁⠀⠐⠒⠒⠢⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢆⠕⠐⢀⣀⡠⡄⢴⢲⠰⡤⡤⢤⣄⣀⡀⠀⠀⠀⠀⠈⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⠁⢀⡠⡒⣉⣦⡷⠾⠾⠟⣓⣓⣛⠺⠞⢲⡿⢇⣲⣠⡄⣀⠀⠀⠀⠀⠈⡒⢤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⡠⢆⣩⡶⠛⠋⣁⢥⡞⢁⣨⠿⠽⠯⠯⠭⠭⣽⣟⣗⡾⢟⣿⢭⣗⡦⣀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⡴⠁⠀⣠⢎⡱⠞⢃⡤⢾⣯⠟⠚⠉⡁⠐⣶⣹⠀⠀⠀⠀⠞⣯⣝⠙⠲⠤⣽⡷⣿⣻⢷⣤⡀⠀⠕⡈⢦⡀
⠀⠀⠀⠞⠀⢀⡴⣩⡾⠁⡰⢋⡴⠊⠀⠀⠈⠀⠀⠀⠛⠛⠀⠀⢀⣾⢄⡗⠈⠀⠂⡠⠀⠉⠛⢾⡹⠫⡟⢦⣈⢔⡤⠙⣦⡀
⠀⠀⠌⠀⢀⠾⣵⠃⢀⡾⡷⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢯⠴⢸⣓⠀⠀⠀⠀⠀⠀⠠⠄⠉⠢⡈⢢⡛⡬⢊⠕⡈⢷⢄
⠀⡜⠀⠀⡼⣹⡛⢠⢿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠃⣂⡣⣼⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡿⣜⡄⡪⢒⠈⣟⣆
⢠⠁⠀⢰⠳⡏⢠⢃⡞⠀⠁⠀⠀⢀⠀⢀⠤⠂⠀⠀⠀⣾⢃⠾⠩⠁⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠹⣮⢻⡔⠥⡢⠸⠬⣇
⠨⠀⠀⡧⣹⠂⣾⡿⠀⠀⠀⠀⠐⠩⠒⠁⠀⠀⠀⠀⣰⢡⡶⣚⢭⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⡘⡆⢣⠪⢐⡀⡗⢼⡀
⡆⠀⢠⣳⡇⢰⢋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣲⣹⢡⡤⢙⣯⢸⣿⠀⠀⠤⢤⣤⣤⣤⡄⠀⠀⠀⠈⡟⠋⠇⢸⣿⢐⡥⢂⢸⠳⡇
⡇⠀⠸⣽⠅⢈⣸⠀⡁⠀⠀⠀⠀⠀⠀⢀⣀⠤⢛⡇⠘⠷⠶⠿⠛⣾⢶⣶⡲⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡧⣊⠅⢸⣝⡇
⡇⠀⠀⣿⠅⣻⣿⠀⢆⣀⠀⠀⠠⠶⠾⠿⠾⠿⢿⠀⠀⢀⡴⢀⡾⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠇⡆⢕⠭⠸⢬⡇
⢡⠀⠀⢿⡀⡸⢿⠀⣻⡿⠄⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢠⣾⡏⡟⠛⠿⣦⣄⠀⠀⠀⠀⣀⢀⠔⠂⠀⠀⠀⠄⢀⣦⢁⠅⡊⡟⢾
⢸⠀⠀⠘⣆⢸⠈⡆⠩⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠘⣠⡟⢻⣧⢁⠀⠀⠀⠈⠀⢀⠤⠊⠖⠁⠀⠀⠀⠀⠠⠁⣸⡟⢴⡪⣰⡙⡎
⠀⣧⠀⠀⠘⣌⣧⣴⡈⠁⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣰⡓⠀⠀⢿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⢋⠗⡩⢰⢇⢹⠁
⠀⠘⣧⠀⠀⠘⢾⣯⣷⣄⠠⠀⡀⠀⠀⠀⠀⡟⠀⣴⡓⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠒⣰⣧⢞⣯⢶⠷⣌⡇
⠀⠀⠡⡱⣄⠀⠀⠙⢷⣎⡳⣄⠐⠠⠀⠀⢀⠇⡼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⢀⣘⣛⠛⣋⢠⡟⢦⡟
⠀⠀⠀⠹⣜⠣⡀⠀⠀⠙⠻⣿⣗⢦⡈⠀⢼⡞⡌⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⢀⡴⢋⢜⡥⡋⣠⠣⣘⠎
⠀⠀⠀⠀⠈⠣⣍⢳⢄⠀⠀⠀⠉⠳⢌⣓⣬⣐⡈⠀⠐⠀⠠⠀⠤⠀⠀⢻⣥⠀⠂⠘⠀⢀⣠⢴⣻⣶⢗⡵⠊⣴⠙⣦⠏
⠀⠀⠀⠀⠀⠀⠈⠺⣌⡝⠦⣄⠀⠄⢀⡈⡉⠓⠻⠿⢶⣒⡒⠂⢤⣤⣤⣤⣤⠰⢒⣒⡩⠿⢚⠟⡫⣶⠕⣫⠚⣌⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⣸⡹⠲⡮⣌⣐⡡⠢⠔⡤⣂⠄⡩⢍⡩⢉⠍⣉⠍⣠⢔⡠⣺⠕⢉⡪⡔⢫⠈⡧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⢥⣸⡉⡏⠖⢲⠠⠬⡤⣭⢤⢥⡈⣵⢯⠤⡥⢾⠓⡉⢏⢣⡸⠖⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠚⠚⠂⠦⠵⢼⣌⣆⣣⣘⣄⣧⡴⠼⠖⠛⠉⠁
⠀⠀⠐⠭⢪⢕⡩⢒⡩⢐⡡⢔⡢⢒⠤⢂⠤⣢⠔⡠⢒⠔⣠⠔⡠⢀⠤⢀⠤⢀⠄⣀⠔⣂⠔⣠⠔⡠⢂⠔⡢⢒⠔⡪⣪⢔⡡⣪⢞⠔⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⡠⠔⢪⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⠿⠿⠷⠶⠶⠶⠾⣦⣴⣿⣥⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣛⠟⣿
⠀⠀⠀⠀⣠⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠶⢶⣤⣄⣤⡾⠋⡰⠊⠀⢠⡿
⠀⠀⢠⣾⣏⡀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠶⠾⢶⣶⣶⣾⡇
⠀⠀⣿⡀⢹⡟⠓⠲⠶⠿⠦⠤⠤⢤⣤⡤⢤⣤⠤⢤⣤⠄⣶⡦⠠⣤⡤⢤⣤⡤⣤⡤⠤⢀⣀⣀⣀⣀⣸⣿⡇
⠀⠀⠘⢷⣼⠃⠒⠒⠒⠂⠀⠤⢤⣼⠿⠷⠾⠿⠿⠿⠿⢶⠶⢶⣶⣚⠓⠒⣛⣒⣛⣓⣶⣾⣿⣍⡫⢍⠉⠙⣷⡄
⠀⠀⠀⠀⠉⠛⠶⢶⣤⣤⣤⣀⣀⣹⣷⣄⠀⠀⠀⠀⠀⠀⠑⠤⡈⠙⢿⣟⡋⠉⠉⠉⠉⠀⠀⠈⠙⠻⠿⢿⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⢻⣦⣄⡀⠀⠀⠀⠀⠈⠑⠤⢌⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣭⣛⣶⣤⣀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠷⢶⣤⣄⣀⣀⣀⣈⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠠⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⣀⡀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⣀⠀⠀⠀⠠⡴⢦⡚⠟⠛⠛⠋⠁⠉⠙⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⢉⢄⠠⠄⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠦⣼⢇⣨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣯⡇⠁⢘⡆⢀⠀⠀⡬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠈⠉⠉⠋⠙⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠆⠦⠼⢀⠀⢠⠃⠐⠃⠀⠀⠰⠃⠀⡠⠈⢠⠰⢿⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡜⠂⡌⡿⠃⠀⠀⠀⠀⡀⠀⠐⠈⠃⠀⠘⢸⡄⢣⢹⡀⠀⡤⠂⡄
⠀⠀⠀⠙⠀⠀⠀⠀⢠⡄⠉⠙⠛⠋⠁⢀⣀⡠⠄⡎⡠⣶⡄⠀⠀⡇⠀⠀⡎⢧⣸⠀⡇⠀⠉⠈⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⠉⠀⠀⣰⣆⢷⠶⣧⣣⣪⣼⣄⢀⣘⣌⠻⡦⠇⠀⢀⠀⠀⡀⠆⡀⣀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⡶⠿⠷⣿⠿⠿⠿⠾⣇⢽⠟⠛⣿⠋⠛⠟⠟⠛⠈⠀⢠⢿⠉⠁⠀⠈⠁⠘⡁⠈⠀⣸⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠉⠉⠀⠀⠀⠀⠀⣄⠀⠀⠈⠓⠓⡟⠃⠀⠀⠀⠀⢀⣠⣴⣿⣭⡀⠀⢀⣠⡴⢟⣣⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢹⢿⠁⠀⢀⣄⣰⡗⠆⠀⠀⠀⣴⣿⣿⡁⠹⠿⢗⣛⡩⡵⠞⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣤⡀⠀⠀⠀⠀⠀⠈⠛⢹⡇⣁⠀⠀⠀⠈⠀⠉⠉⠁⠈⠛⠳⠦⠒⠊
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠠⠆⠀⠀⠀⠁⠛⣷⡄⠀⠀⢀⣀⠶⢿⣛⡈⠓⣄⡀⠀⠀⢀⣤⠐⠀⠄
⠀⠀⠀⠀⠀⠀⢰⣏⠤⠀⠀⠀⠀⠀⣾⠋⠤⢻⠀⠀⠘⢭⣕⢠⣬⡍⢘⡏⠀⠀⢀⣿⠃⠀⡀
⠀⠀⠀⠀⠀⠀⣿⡏⠒⢀⣀⣤⣶⡶⠃⠀⠀⣺⡇⠀⠀⣸⡏⢸⣿⠇⠈⣿⠀⠀⠀⢿⣧⣠⡁⢀
⠀⠀⠀⠀⠀⠀⢸⣏⡾⠯⠛⠟⠋⠀⡀⢐⣴⡿⠀⠀⠄⡿⠁⢸⣿⠀⠀⢹⡄⠀⠀⢠⡙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⡀⢴⣮⠟⠀⠀⠀⣸⠇⠀⢸⣿⣀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠘⠚⠉⠀⠀⠀⠀⢀⢃⣠⠞⠁⣿⠈⠳⣄⢸⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣮⠿⠁⠀⠀⠀⠀⠀⠈⠹⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠀⠀⠀⠀⣼⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠇⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⡤⢖⣒⡛⠋⠉⠉⠭⠩⠭⠥⠤⠀⠐⠒⠒⠒⠒⠒⠀⠀⠀⠈⠑⢆⠈⣇
⠀⠀⠀⣸⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠈⠄⢹
⠀⠀⠀⢹⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢠⠞⠀⠀⠀⠀⠇⠘⡆
⠀⠀⠀⠸⡈⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠻⢄⣠⠶⡄⠀⠸⠀⣇
⠀⠀⠀⠀⡇⡅⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⣤⠀⠀⠀⠀⠀⣠⠇⠀⠀⡅⢸
⠀⠀⠀⠀⢳⢀⢠⠀⠀⠀⠀⠀⡠⠴⠦⣤⠔⡡⣀⠉⢀⠴⠋⠓⠒⠋⠀⠀⠀⠀⢰⠘⡆
⠀⠀⠀⠀⢸⡘⠘⠀⠀⠀⠀⡞⢀⣀⡀⠀⠀⠋⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⡇⡄⡂⠀⠀⠀⠻⠉⠀⢹⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸
⠀⠀⠀⠀⠀⣧⡄⢃⠀⠀⠀⠀⠀⢷⣁⣀⣀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸
⠀⠀⠀⠀⠀⢸⠁⠘⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠒⠒⣊⣡⡞
⠀⠀⠀⠀⠀⠘⣆⠀⠣⣀⡀⠤⠤⠤⠔⢒⣛⣉⠩⠭⠭⠍⠙⠒⠒⠋⠉⢉⣉⣉⣀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠈⣷⠒⠒⠒⠂⠉⠉⠉⠀⢀⣀⣀⡤⠤⢤⠤⠒⢖⠒⣫⡉⠬⠦⡄⠓⣄⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠐⡇⢣⠀⠰⡉⠉⢫⡉⢉⠹⣄⠀⠐⠖⠂⢳⡁⣉⢡⡤⠴⢖⠂⠙⣌⡉⠵⢄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠱⡄⢣⡀⠙⡏⠉⢱⡄⠠⠰⡆⠐⠚⣏⣉⠹⡄⠀⠒⣖⣈⣧⣡⠬⠂⠒⠈⣑⡄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠱⡀⠈⢯⠠⢷⠀⠒⢘⣏⣁⡨⠦⠴⠚⣛⣉⠭⠤⠒⠚⠃⠉⠉⠉⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠱⡀⠀⢲⡚⠭⠥⠴⠓⠒⠒⠊⠉⠉⣁⣀⡠⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠁⠀⠀⢀⣀⣨⠆⠀⠀⠀⠀⠀⢀⣀⣀⡠⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣂⣉⡉⠀⠤⠀⣐⣒⣒⡬⠭⠭⠴⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡑⢤⡄⠠⠤⠄⠒⢒⣂⣀⣭⠥⠔⠒⠚⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠈⠈
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠐⠉⠁⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⠀⠀⠀⠀⢀⠀⠀⣀⠀⣰
⠀⠀⠀⠀⠉⠁⠚⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀
⠀⠀⠀⠀⣸⣿⡀⠈⡄⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⡀⣠⣭⣽⣿⡿⣿⢿⣿⣏⢻⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⣀⣀⣾⣾⣃⠀⢸⣄⣀⡠⠤⠀⢒⠀⠉⠀⠅⡧⠝⠻⠟⠛⠃⠛⠒⠿⠿⠍⠋⠛⠻⢿⣟⠟⡍⠛⠁
⠀⠀⠈⠉⢻⣿⣿⣿⣃⣀⣠⣀⡀⠀⢀⡀⠀⠀⠀⣆⠀⠀⢬⣦⣀⣀⣀⣠⣧⣤⣶⣾⣧⡿⠶⢡⠦
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⡼⠄⠀⠀⠀⢀⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠉⠉⢿⡉⠉⠉⠉⠉⠀⠀⠸⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣴⠎⣦⠀⠀⠀⣾⣣⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠙⠛⠋⠀⢀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠈⠀⠁⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁
⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠉⠉⠟⠻⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⠋





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠛

⣤⡄
⣿⡇
⣿⡇
⣿⡇
⣿⣇⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀⢀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢆⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠒⠀⠘
⠀⠀⠀⠀⡀⠄⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠘⠂⠂⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠛⠒⠂⠒⠒⠂⠈⠉⠉⠉⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⢀⠀⠁⠒⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⣈⠉⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢶⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⠀⠀⠈⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣤⡀⠀⠀⠐⠂⠀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠈⠉⠀⠀⠠⠀⢀⣀⣀⠀⠠⠤⠄⠀⠐⢰⣶⣧⠑⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠄⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠙⠦⠤⠴⠄⠐⠒⠒⠀⠈⠉⢹⡟⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠄⣀⠀⠀⠀⠘⠛⠋⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⠄⣀⣀⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠉⠉⠐⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⣀⡪⠔⠒⠂⠀⠈⠁⠒⠒⠠⢄⡀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡊⣤⠄⠀⠀⠀⠁⣀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢵⠂⠀⠀⠀⢀⡖⠂⠈⠉⠐⠯⠉⠒⢤⡀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⠀⢣⠀⠀⠀⠀⢀⠀⡀⠂⠈⠑⠤⡀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⠀⠀⠀⢀⢀⡎⠀⠠⠀⢄⠀⠈⠒⢄⠱⡦⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠏⠀⠀⠀⠡⠀⠀⢘⡀⠀⠠⠀⠀⠀⠀⠑⢵⡈⠢⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⡤⠀⠀⠶⠄⠀⠀⠚⠉⠁⠘⠁⠀⠀⢀⡀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢄⠀⢀⠃⠀⢰⠀⠀⠄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠎⠈⡄⣈⠀⠀⠄⠀⠄⠂⠀⢠⠎⠘⢄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡇⠀⡤⠒⣹⢿⠀⠀⠀⠀⠀⢀⡔⣁⠀⠀⢘⠣⡀⠀⠀⠀⠀⢀⠞⠀⠀⡘⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠊⠀⠀⠀⡅⠀⡖⡈⢹⠒⠢⡀⠀⡀⠴⠌⠁⣨⣙⠀⢸⠀⠈⠐⠢⠄⠒⠁⠀⠀⠀⠃⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠒⠞⢱⡄⢣⠀⠀⠀⠀⠀⠀⢀⠀⣠⠋⠈⠛⠀⠀⠈⠁⠀⠀⢆⠨⠓⠉⠂⠜⠀⠰⠦⠴⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠰⠀⠰⠁⠀⠀⠑⣄⠀⠀⠀⠀⠈⠑⣌⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣎⠀⡆⢀⣤⣤⠤⠤⠤⠤⠄⢀⠀⠀⠀⠀⠰
⠀⠀⠠⠤⢃⠗⡂⠀⠀⠠⠐⠉⠑⠠⣀⠀⠀⠈⠈⠣⣌⡱⠐⡤⠀⠀⡀⠀⢆⠸⠔⠁⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⡀
⠀⠀⠀⠑⠊⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡒⠂⠀⠄⢫⠄⢌⡑⠄⡘⠀⠰⢊⠀⠐⠂⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⠁⠀⠁⠁⠈⠒⠀⠀⠀⢀⠵⣀⣀⣄⣎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠂⠀⠀⠀⠀⠀⠢⣄⡀⢠⠔⢠⡿⡹⠺⠧⠥⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⢇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⡙⠋⠀⠀⡈⠀⠀⠀⠀⢀⡠⡁⢢⠀⠀⠀⠀⠀⠤⠑⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠠⢈⣦⠞
⠀⢀⠄⠀⠐⠶⡄⢄⣀⡀⠀⠀⢠⠁⠀⠀⢀⣃⠀⠀⠀⠀⠈⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠀⠀⠪⠀⠐⠄⠀⠀⠀⣇⡴⠋⠁
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠺⠉⢈⣐⣠⠀⠀⠀⠀⠀⠀⠀⢀⢤⠤⣶⣶⠒⡞⢳⣶⣶⢲⡖⣶⣶⣶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣽⣿⡀⣷⢞⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡜⣿⣿⡇⠷⠸⠿⠷⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠰⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⢁⣀⣀⣀⣨⣬⡀⠉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠏⠁⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⣀⣚⣶⣂⡀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡬⠵⠚⢛⣿⣉⣤⣯⣤⣉⣿⡛⠒⠮⣅⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡵⢾⣧⠴⠖⠋⠉⠘⣀⣠⢿⣄⣀⠋⠉⠙⠲⢤⣹⣷⢭⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⡵⢋⡴⠊⠉⣀⢤⣲⡬⠷⠓⠛⢸⡗⠒⠮⢥⣖⠤⡀⠋⠳⢄⠙⢮⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣣⣿⣴⠋⠐⣦⣮⠞⠉⠀⢠⠀⠀⠃⢸⣧⠀⢀⠄⠈⠙⠪⣵⣴⠊⠳⣄⣷⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⠃⡼⠉⢠⢮⠞⠙⠷⣄⡀⠀⢧⠀⠀⢸⣿⢀⡞⠀⠀⠄⣠⠞⢣⡳⡐⠙⢏⠙⡎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⠃⡸⠁⢠⣧⠏⠀⠀⠑⠌⠻⣶⣜⡷⠀⢸⣿⣿⠁⠂⣠⡾⠃⠀⠀⢹⡵⡀⠈⣆⠸⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣿⢶⡧⠀⣆⡏⠀⠐⠲⢤⣈⣂⠈⠻⣿⡀⢺⣿⠃⣰⣿⣟⣠⡤⠖⠂⠀⢳⡱⢀⣸⣶⣧⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣺⣇⠇⠀⡇⢸⣀⣰⣼⣥⣤⣤⣶⣶⣿⣿⣷⣦⣬⣷⣼⣷⠾⠛⠉⠉⠑⠂⠀⠀⠀⠼⢯⣀⣸⡇⢹⠀⠀⣿⡓
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠀⠋⡏⢻⠉⢹⢯⠍⠩⠭⠉⣉⡉⠉⠉⣉⣭⢿⡿⣟⠿⣿⣿⣿⣿⣿⡿⠿⠿⢿⡿⠛⢻⡏⢹⠩⠀⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⡇⣷⣾⡖⠀⡞⡇⠀⢀⡠⠴⠚⣻⣿⠟⢡⣾⡇⠹⣷⣄⠙⢛⠳⠦⣄⡀⠀⣸⢳⠀⢼⣷⡾⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣹⡀⢧⠀⠰⡿⡄⠀⠀⢀⣼⠟⠁⢔⣿⣿⡇⠀⣿⠻⣷⣄⠢⠀⠀⠈⢰⢧⠃⠀⡞⢨⢇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢳⡈⢧⢀⠱⡿⣄⣠⠟⠁⠊⠀⡼⠹⣿⡇⢠⠙⣇⠈⠙⠷⣄⢀⡰⢫⢂⢀⡼⢀⡏⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠳⡿⠣⡀⢈⡾⡷⣄⡀⠀⠰⠁⠀⣿⡇⠘⠀⠘⠄⠀⢀⡨⢿⢷⡁⣠⠿⣷⢏⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣝⢦⡙⢦⡀⢈⠚⠽⣲⠦⣤⣀⣹⡇⣃⣀⣤⠴⣞⠭⢃⠁⣠⠜⢁⡴⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢝⢦⣾⠗⠦⣀⡀⠉⠘⠃⢻⣧⠋⠘⠂⠁⢀⣠⠴⢿⣧⠴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⢑⡢⢤⣉⣹⣿⠒⠶⡿⠖⢺⣷⣋⣉⡤⢖⡫⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠭⠩⠟⠛⠿⠛⠫⠭⠭⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣴⣤
⠘⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣦⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠈⢻⣿⣿⣿⣦⣄⡀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠀⠀⠘
⠀⠠⢶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢽⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⢡⣾⡿⠻⡿⡍⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠒⠒⠒⠒⠒⠒⠂⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣝⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢔⡣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠑⠤⣉⡁⠒⠚⠟⠭⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠛⠒⢉⣁⡴⠋⠒⢤⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡠⢤⣀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠲⠤⠐⠒⠒⠒⠒⠒⠊⠉⠁⠀⠀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠀⠸⢍⡤⠤⠒⠒⢉⣢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⠃
⠀⠀⠀⠀⠀⠀⠑⢶⡊⢉⣳⣤⡬⠗⢭⡭⢑⣒⠂⠴⠤⠤⠄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠠⠤⠤⠴⣒⣒⡊⠭⠭⠒⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⠤⠛⢶⣈⣙⣦⡀⠀⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠉⠉⠁⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢖⡉⣉⣷⣤⠜⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠲⢽⣦⡠⠬⠷⣶⣎⣉⣫⡲⢶⡒⠒⣤⣤⡤⠤⠤⠤⠤⠄⠖⠒⠚⠉⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠈⢟⢾⡉⠙⡍⣤⣖⣶⣽⠝⡄⠘⡀⣱⣀⣀⣀⣠⣀⠠⠤⠴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⡎⣷⠀⢸⢹⡏⠻⣆⠀⡇⢀⠋
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⢒⣊⣩⠇⠀⠀⢰⣿⠀⣸⠤⠄⢒⣚⠿⢿⡓⢒⣄
⠀⠀⠀⢸⣹⠞⠓⠊⢫⠉⠀⠀⠀⠀⠀⠀⣾⠋⠉⠀⠀⠀⠈⠲⣖⠊⣙⣧⣤⠝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡤⠼⠳⣎⡉⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡂⢉⣙⢦⡤⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠤⠛⠒⠉⠉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠰⣿⡿⢃⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢦⠰⠂⠀⢀⣶⡾⠟⢛⣯⣵⡄⠟⣡⣎⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣵⣞⡁⠀⠀⠀⠀⠁⠀⠀⠈⣩⣷⣯⣾⣛⡣⣀⣻⣦⡤⠖⠊⠉
⠀⠀⠀⣴⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣠⣀⣀⣀⣀⣀⣀⡀⣦⣶⣿⣿⣿⣿⣿⡿⠿⠛⠉⠹⣷⡀
⠀⠀⠀⠉⠙⠛⠿⠟⠛⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠙⠛⠛⡿⢿⣿⣿⣿⣿⣟⢸⣯⣿⣿⣿⠿⠿⠛⠉⠉⠀⠀⣗⣀
⠀⠀⠀⠀⠐⠶⢶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⠟⠫⠟⢉⣽⠍⠁⠀⠀⠀⠀⠀⠀⠠⣟⠉⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣉⡟⠉⠀⠉⠉⠀⠀⠰⣴⣃⣿⣦⠇⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠂⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢰⣿⣷⣷⡄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⣀⣤⣤⣤⡀⠀⣀⣤⣤⣤
⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⡟⣽⣿⡏⣧⠀⠀⢠⣤⣤⣿⡟⣿⣿⣿⡿⠃⠀⠀⠻⠿⠾⣿⣿⣤⣟⡿⠶⠿
⠠⣿⣿⣿⣿⣷⠀⠀⠀⠀⢠⡿⢸⣧⠀⠀⠀⠘⠛⠋⠹⠇⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃
⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠸⠇⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⣲⣶⣶⣄⠀⠀⠀⠀⠰⣿⣷⢀⣀⣄⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣷⣦⡴⣶⡦
⠀⠀⠈⣿⣏⣹⣿⠁⠀⠀⠀⠀⠈⢱⣿⡿⢃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡶⢿⣿⣷⡩⢿⠃
⢀⠄⠀⠿⢿⠿⠿⠂⠠⡀⠀⠀⠀⠪⠉⠰⣿⣷⠆⠀⠀⠘⢿⣿⣿⣿⡿⠏⠀⠀⠀⠸⡟⣧⣜⣻⣿
⠈⠁⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠄
⠀⠀⠀⡀
⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⢠⣾⡂⢀⡀⠉⠉⠀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣿⡆⠀⠠⣄⡀⢀⣠⣤⣀
⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⣇⢻⣿⢾⣃⡤⣶⠀⠀⠀⠀⣸⣿⣿⡿⠿⠻⠛⠁⠀⠀⠀⠑⣾⣿⣿⣿⡆
⢀⡾⢿⡿⢷⡆⠀⠀⠀⣿⣿⡿⠉⠀⠉⣾⣿⡷⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠻⠷
⠘⠁⠘⠇⠈⠇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⡀
⢀⣴⣿⡾⣶⣄⠀⠀⠀⠀⢠⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣷⣦⣀
⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⢿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣠⣼⣿⣟⣟⣧⣄⡀⠀⠀⠀⠀⠀⠈⠿⠈⠉⠑
⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⡟⠿⠿⠗⠀⠀⠀⠙⢻⠿⠿⡿⠿⠿⣟⠋⠀⠀⠀⠀⢀⠂⠂⢂

⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠺⠿⠻⣿⡇⠀⢰⣦⣼⡇⠀⢸⣆⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠒⠒⠤⠤⣀⡀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⢾⣶⣶⣿⣿⣿⣿⣷
⠀⠀⠒⢶⣶⣿⡇⠀⠀⢹⡟⠀⠐⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡀⠁⠀⠐⣿⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣴⣶⣦⣄
⠀⠀⠀⣸⣿⣿⣧⠀⠀⢸⡇⠀⠀⠈⣿⡇⠀⠀⠀⣰⡶⣦⡀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠻⠟⠻⢿⣿⣿⠟
⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⢰⣿⡇⠀⠀⢀⣿⣧⣿⣷⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⢸⣿⡇⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⡀⡀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⣿⣿⠈⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇⠀⠘⣿⣿⣼⣿⣿⠀⠀⢠⣧⣷⠀⠀⠀⠀⣿⣸⡧⠀⣶⣶⣶⣶⠄⢠⣶⣶⣶⣶⣶⣶⠀⣤⣤⣴⣶⣶⣄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡇⠀⠀⢹⣿⣿⣿⡇⠀⠀⣨⣿⣿⡀⠀⠀⠀⢿⣿⡇⠀⠀⢸⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠉⠀⢹⣿⡏⠉⠣
⠀⠀⠀⣿⣿⠀⠀⠀⠀⢌⡙⠀⠀⢸⠟⠇⠀⠀⣸⡿⢿⣿⡇⠀⠀⣼⠇⢿⡄⠀⠀⠀⣾⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⣿⡇⠀⣿⡇⠀⢸⡟⠀⠘⣿⠀⠀⢠⡟⠈⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⡇⠀⣿⡇⠀⢸⡇⠀⠀⣿⠀⠀⢸⠃⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠈⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠃⠁⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡄⠀⢸⠃⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡃⠀⠀⠀⠀⠛⠀⠀⠸⠃⠀⠀⠻⠀⠀⠈⠇⠀⠟⠀⠀⢸⡇⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢠⣤⣤⣤⣤⣤⣤⡄⠀⠀⡀⠄⠒⠈⠈⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⢰⣦⡆⠀⠀⠀⠀⣶⣶⡆⠀⠀⣷⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⢰⢿⠀⠀⠀⢸⠿⡇⠀⠀⠀⠀⢿⣿⠇⠀⠀⣿⠀⠀⢸⠁⠀⢰⡇⠀⠀⠀⡀⠀⠀⠀⠘⠛⠃
⠀⠈⠛⢿⣿⣿⠟⠋⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⡎⠀⢇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠘⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⢀⢀⡀⠀⠀⣼⣿⡄⠀⠀⣏⠀⠀⢈⣄⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⡇⢸⡟⣽⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢤⣤⣤⣀⡀⠀⠀⠀⢀⣠⣅⡤⠇⠀⠀⢿⣿⡇⠀⠀⣿⠀⠀⢀⣯⠀⢸⡇⠀⠀⣠⣇⠀⠀⠀⡇⠀⡇⢸⡇
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣽⣿⣿⣷⣶⣏⡤⢤⣄⣀⠀⠀⢸⣿⡇⠀⢰⣿⠀⠀⢸⣿⠀⢸⡇⠀⣾⣿⣿⣷⡄⠀⡇⠀⡇⢸⡇
⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠑⠀⠀⠘⠁⠀⠸⠿⠀⠀⠸⠿⠀⠀⠸⠿⠀⠸⠟⠀⠻⣿⣿⠿⠁⠀⠁⠀⠁⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢏⣵⠶⠶⣦⡹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣷⣾⣷⣾⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣿⣿⣿⣿⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠿⠛⠛⠛⠛⠿⠷⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⢀⣀⠤⠤⠄⠄⠀⠄⠤⠤⣀⡀⠈⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⡤⠊⠁⠀⢠⡀⠀⠀⠀⣀⣀⣰⣆⣀⣀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⡴⠋⠀⠀⠀⠀⢸⣷⡶⠉⠁⠀⠀⢸⡇⠀⠀⠈⠉⠒⠤⡀⠀⠀⠀⠀⠙⢦⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⠎⠀⠀⠀⠀⡠⠚⠁⠁⢣⡀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠱⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⠋⠀⠀⠀⢀⠞⠀⠀⢠⡀⠀⠱⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⡄⠀⠀⠳⡀⠀⠀⠀⠙⡄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡎⠀⠀⠀⢀⠊⠀⠀⠀⠀⠙⢦⣄⡱⡖⠚⠉⠉⠓⠲⢄⣀⡴⠋⠀⠀⠀⠀⠑⡄⠀⠀⠀⢱⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢰⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⡨⠋⣠⣾⣿⣿⣿⣿⣶⣄⠙⢅⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⡆⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠐⢺⠗⠻⠿⠿⠿⠿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠿⠿⠿⠿⠿⠺⡗⠂⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢡⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡌⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠘⡀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢸⣦⡈⠛⠻⠿⠿⠟⢟⢁⣴⡇⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⠇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢣⠀⠀⠀⠈⢣⠀⠀⠀⢀⡴⠋⠀⠈⠑⠒⣤⣤⠒⠊⢣⠀⠙⢦⡀⠀⠀⠀⡜⠁⠀⠀⠀⡜⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠈⢧⠀⠀⠀⠀⠳⡄⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢣⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⡴⠁⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠳⡀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢳⣤⣠⠖⠁⠀⠀⠀⢀⠞⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠈⢦⡀⠀⠀⠀⠀⠉⠒⠦⠄⣀⣀⣸⣇⣀⣀⠠⠄⢾⣿⣿⣆⠀⠀⢀⡴⠁⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡦⡀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠿⠀⠀⣠⠔⠁⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣢⢄⡀⠉⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⢀⡠⣖⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣮⣗⡢⠤⣀⡀⠉⠉⠉⠈⠁⠈⠉⠉⠀⣀⠤⢔⣺⣽⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⢮⣭⣭⣤⣄⣠⣤⣭⣭⡵⠶⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠃
⠀⠀⠀⠀⢀⠀⠤⠤⣤⠐⠛⠀⠢⡄⠀⠀⠀⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠘⠶⠦⣄⠠⠌⣁⡐⠀⢀⣄
⠀⠀⠀⠀⠔⠈⠁⠀⡏⠀⠀⣾⣷⡇⣆⣁⠠⠄⣑⠮⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣘⣣⣝⡁⢤⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠜⠀⠀⠉⠑⢓⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣀⠄⠂⣀⠀⢀⠀⠀⠉⢱⡈⠁⠻⣋⣀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠠⠄⣀⣠⠀⠀⢸⠀⠀⠀⠸⠚⠖⠁⠀⠀⠀⠀⠀⠀⠜⠠⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡂⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠚⠛⠛⠙⠛⠉⢁⠀⡑⢄⣺⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡁⠤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⢀⡞⡈⡠⠒⠀⠁⠀⠈⣽⡄⠀⠀⢀⠀⠀⠀⠄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⡌⢰⠄⠀⠰⠆⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⡐⡵⠁⠌⠀⠀⣠⠄⣱⣴⣷⠀⠠⠀⠀⠀⠀⠀⠀⠘⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠇⠃⠠⠲⠞⠀⠀⠀⠀⠄⠀⠀⠀⢀⠄⠂⠀⠀⠢⠀⠐⢁⠀⣡⣶⡏⠂⣠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠈⢀⠀⠀⣀⠀⠀⠀⠀⡐⠀⠀⠀⠄⠈⠐⠑⢃⡼⡿⢄⠁⢸⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠬⠁⢄⠀⡀⠄⠠⠔⢠⣂⣙⣝⡃⣄⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢙⡿⠛⣒⢲⣦⣶⢾⡶⠗⠚⠃⠉⠉⠉⠁⠛⠚⠃⠀⠈⠀⠀⠀⠀⠄⠠⠀⠀⠤⡀⣐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⡀⠀⡐⠈⢁⣀⠉⠙⠉⣻⠃⠀⠀⢀⢀⡀⣀⠀⠄⠀⠀⠄⠀⠀⠀⢆⠠⠀⠈⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢧⠄⠁⠰⣯⣁⡹⠀⢰⠁⠀⠐⠒⢀⣼⠿⢄⠓⠂⡶⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⡄⠀⡄⢈⠉⠀⠀⠀⢆⠰⠘⡘⣿⠿⠲⡎⠄⣰⡎⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⠁⠀⠁⠀⠀⠀⠉⠉⠀⠀⠀⢱⣮⡀⠀⠀⠂⠀⠰⠂⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠀⠀⣏⠀⠩⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⢆⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡀⠘⢠⠀⠀⠀⠀⢹⡇⠆⡀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠠⠀⣀⡧⠈⠀⠀⠀⠐⡦⡄⠔⢹⠀⠀⠀⠀⠁⠀⠁⠀⠒
⠀⠀⠀⠀⠀⠀⠜⠛⠄⠀⠀⠀⠀⠈⡰⣆⣀⡈⠑⡎⣙⠁⠁⠀⠀⠀⠀⠀⣈⡧⠀⠇⠀⠀⠀⠁⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⠫⡁⠈⠀⠜⠛⠙⠙⠛⠙⠓⠋⠹⡇⠞⠀⠀⠀⠘⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠢⠀⢶⣿⣶⣴⠶⣶⣿⡴⢚⣁⣤⣤⣶⠀⠀⠀⠀⠀⣸⡶⢲⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢹⡏⠉⠙⠉⠛⠁⢈⠓⡋⠈⠉⠙⠇⠀⡀⠀⠀⡈⠁⠀⢹⡃⣀⣤⣴⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣼⣳⠀⠀⠀⠀⠀⣤⣀⡄⠀⠀⢀⣼⣆⣄⣠⣼⣧⡦⢠⡾⣿⠛⠉⠁⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⢸⠇⠀⠀⠀⠀⠀⠈⠩⠀⠀⠀⠈⠁⠉⠉⠉⠈⠋⠁⠉⠁⠉⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠓⢀⠞⠀⠀⠀⠀⠀⠀⠘⡇⠆⠀⠀⣀⠤⠼⡄⠀⠀⠀⠈⠀⠈⠱⠎⠀⠀⠀⠀⠀⠀⠈⠻⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠐⠒⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡦⡀⠄⠀⢂⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐⠋⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠐⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡈⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠠⣉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⠋⠉⠀⣖⡉⠥⠀⠀⠀⠉⠙⢦
⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢒⣤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⣩⠀⠀⠯⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⣠⠖⠊⠹⣀⠀⢀⢩⠝⠒⠂⠄⠀⠀⠀⡖⠀⠀⠓⠒⣀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡸⣁⣀⠀⢀⡴⠋⠁⠈⠢⠤⠀⠀⠀⠀⠀⠈⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠲⢤⣀⠸⣄⡀⠄⠀⢀⡠⠔⢊⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠁⠂⠀⠀⢀⡠⢜⡑⠢⠤⠔⢊⣡⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠠⠐⢩⠀⠘⢂⡠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⠆⠀⡀
⠀⠀⠀⠀⣀⠀⠀⠉⠓⠒⡂⢀⡨⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠂⠀⡞⠁⠀⣈⡶⣤⠈⢢⡀
⠀⠀⠀⠀⠈⠉⠒⠲⠒⠋⠀⠁⠀⠀⢀⣀⣀⣀⠤⠤⠒⠊⠉⠁⠀⠀⠀⠈⠒⢭⡉⢼⣁⡀⠈⠊⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢭⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠄

⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⣀⣁⢨⡭⣄⣁⣂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⢈⣤⣊⠽⠓⠛⠉⠉⠙⠛⢚⠿⣷⣪⡐⠄
⠀⠀⠀⠀⠊⣴⡵⠋⣡⡤⣴⣒⢲⠢⢤⢄⡀⠀⠈⠙⢿⡌⠄
⠀⠀⠀⠀⣼⣟⣴⣯⠶⡋⣁⡼⣿⡿⡿⢳⠿⣶⣄⠀⠀⠙⢀
⠀⠀⠀⢰⣿⡍⠇⠀⠀⠣⡙⠾⠿⠛⣡⠎⠀⠀⢙⡧⠀⠀⡘
⠀⠀⠀⢸⣷⠉⠓⠶⣶⣦⡬⣝⣲⣯⣥⡔⠾⠛⠉⠀⠀⠀⡄⢀
⠀⠀⠀⢌⠻⣆⠀⠤⢄⣀⡀⠈⠀⠀⠀⠀⢀⡀⠔⠃⠀⡜⠀⢸
⠀⠀⠀⠈⠳⡈⠑⠦⢄⡀⠉⠉⠉⠉⠉⠉⠀⠀⢀⠄⠃⠰⠃⢨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⠭⢍⣛⢒⠩⠍⠖⠋⣠⠂⠠⠋⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⠀⡐⠐⠁⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠁⠀⠈⢀⠜⠁⠀⡔⠡⠀⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠀⣀⣴⠓⢃⠀⠀⠀⠀⠠⠐⠀⠂⠒⢂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠠⢄⡀⠀⠀⠀⠒⠀⠁⠢⠰⠤⠦⠾⠖⠛⠉⠀⠀⣼⠀⠀⠀⠈⠀⠀⠉⠈⠉⠀⠀⠀⠀⢀⢠⠤⠐⡀
⠀⠀⠀⠰⠀⠐⠀⠈⠀⡀⠂⡁⠐⠢⡀⠐⢄⠀⠀⠀⠀⠁⠀⠈⣼⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠓⠒⠚⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⣀⣀⣐⣤⣾⢻⡄⢄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠋⠀⠁⢰⠇⠶⠌⣁⠠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠁⠀⠀⠀⠀⠀⣤⢠⡤⡈⠢⡶⠶⠛⣦⣴⣀⣶⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡥⢼⠔⣵⣄⣴⡿⠋⠁⠱⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⠛⠾⠁⠀⠘⠀⠠⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⠤⠤⠤⠤⠴⠶⠟⠁⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢰⢶⡶⣾⢿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡟⣟⢻⢳⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⢸⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⢸⢸⡇⣿⢸⢸⡆⠀⠀⠀⠀⠰⣿⣛⡑⠀⠀⠀⠀⠀⢸⣻⡇⣿⣘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣦⡀⠀⠀⠀⠀⣼⢸⡇⣿⣸⣇⣧⠀⠀⠀⠀⠀⣼⠙⣧⠀⠀⠀⠀⠀⠘⠛⣻⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⠀⠀⠀⣠⣿⡟⠉⠉⢿⡉⠁⠀⠀⠀⠀⣰⠏⠀⠈⢷⣄⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠛⣠⣦⡺⣿⡟⠀⠀⠀⠀⠙⢦⣤⣄⣤⡾⠃⠀⠀⠀⠀⠉⠛⠲⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣾⣿⣿⣿⠿⠀⠀⠀⢀⣠⣴⡾⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣴⣶⣿⣷⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣦⣍⡛⠻⠿⣿⣿⣿⣿⣧⡗⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣴⣿⣿⣿⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⣀⣻⣿⣤⣀⣀⣀⣀⣀⣀⣠⣿⣷⣤⣀
⠀⠀⠀⠀⢰⡿⠉⠉⠉⢻⣏⠉⠉⠉⢹⣿⠉⠉⠉⠹⣧
⠀⠀⠀⢀⣿⣃⣀⣀⣀⣀⣿⣄⣀⣀⣾⠃⠀⠀⠀⠀⢻⡆
⠀⠀⠀⣼⠏⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣴⣶⣶⣶⣶⣼⣿⡀
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢸⣷⡟⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⢠⣿⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣽⣇
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣿⡍⠉⠉⠉⠉⠉⠉⠉⠉⢿⡄
⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿