MEAT Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠚⠉⠉⠉⠉⠉⠉⠀⠉⠍⠢⣄⣀
⠀⢀⣀⡤⠤⠴⢞⣡⠔⠂⠀⣄⠀⡀⠈⠡⣠⠆⠀⡀⢀⡠⣀⠈⡇
⠀⣏⠀⠀⣠⡔⠋⠑⣤⠀⠀⢈⠿⣅⠠⠀⠀⠀⠠⠈⠁⣀⣠⢴⠇
⠀⠘⡶⠦⢤⣌⣳⣉⡀⢈⣂⣀⣀⡠⠴⠒⠛⣋⣉⣉⣉⡡⠴⠋⢀⣀⣤⣔⣪⣿⣯⣵⣖⡒⣒⢤⡀
⠀⠀⠈⠷⣤⣀⣀⣈⢉⣉⠉⣉⣁⡤⠴⠊⠉⠁⠀⠀⠀⠀⣀⡴⢿⣿⣿⣿⣿⣿⣿⡿⣛⣁⠛⣿⣎⢢
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⠤⣒⣯⣵⣶⢶⢟⣳⣦⣼⣷⣿⣾⣿⣿⡕⣿⣿⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠟⣷⠫⣝⣾⠿⢯⣭⡭⠿⠭⢿⣿⣿⣿⣟⣸⡿⢥⡾
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⣞⢠⣿⡇⠀⣿⣿⡇⠀⠀⠀⢹⣿⡦⢀⣀⡠⠟⠁
⠀⢀⡴⣫⣶⣦⢰⣍⠒⠤⢤⣄⣀⣀⣀⣀⣈⣉⠁⠈⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠛⠋⠉⠉
⠀⣾⣴⣿⣿⣿⠸⣿⣿⣿⠿⢃⣶⣶⣶⣶⣤⣍⣙⠢⣄
⠀⡇⣿⣿⣿⣿⣆⠠⣭⣀⠒⠛⠛⠛⠛⢛⣉⣭⣭⣶⢘⡆
⠀⣇⠹⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣉⣥⣾⡇
⠀⢹⣦⣝⡛⠟⠃⣸⣿⣿⣿⣿⣿⠟⣫⠴⠋⠁⣠⠴⠛⠀⣀⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢿⡄⠉⠛⠒⠬⢭⣭⣭⣭⣶⣿⣿⣤⠔⠋⠀⢀⡔⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠊⠁⠀⠈⠓⠤⡀
⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⣿⣿⠿⠛⠁⠀⠀⢀⠊⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾
⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠓⠤⣀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠔⠒⠉⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠈⠉⠉⠉⠉⠉⠀⣀⣠⡤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠐⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣄⠀⣶⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⠁⣐⡂⠀⠀⠐⢒⢖⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢧⣈⢣⠒⡟⠘⣇⡖⡄⠀⠀⠀⠀⠀⠀⡜⠠⣀⠔⢂⠀⢈⣦⠴⠓⡄⠀⠳⣄⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠌⢱⣠⡄⡂⢀⡸⢹⣠⡆⠀⠀⠀⠀⢰⡧⠀⠙⣦⡴⠚⡍⠀⢱⣤⠞⠣⡀⢈⠦⡄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠾⠞⣹⡻⡁⣄⠙⢉⠘⡿⡄⠀⠀⠀⠸⢈⠀⠈⠉⢶⣀⠝⢿⡁⢈⣧⡔⠉⠣⡀⡨⠶⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣧⣿⣿⣿⣯⠾⡍⢐⡀⢸⣦⠀⠀⠈⠃⣱⣀⠀⠋⠱⣄⡠⠓⣏⠀⢙⣦⠐⠉⠂⣠⢎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠿⠛⠋⠃⡀⠀⢻⠚⠓⠊⠘⠂⠀⠈⢿⢁⡃⢄⣀⣉⠃⠀⠬⠓⠉⢀⣠⠴⠋⠀⠈
⠀⠀⠀⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⠁⠀⠀⠀⣶⢉⡓⣚⣹⠃⠀⠀⢐⠀⠀⠀⠁⠀⠀⢈⡉⢋⠉⠉⠉⠉⡉
⠀⠀⠀⠸⠉⠉⠉⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣴⣦⡃⠀⠀⠘⠀⢱⣦⡦⠘⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀
⠀⠀⠀⠀⣤⣤⣤⣴⠷⠤⣤⣀⣠⣤⣤⡄⢀⣀⠀⠀⠘⠝⠙⠀⠀⠀⠀⣪⡙⣷⡴⢐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣡⠶⠺⣕⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣼⣿⣟⣦⢄⡒⣛⣸⠑⢦⣄⡐⢝⣀⡠⣄⡀⡈⠓⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢾⣧⠈⡀⠈⠹⣿⠘⣇⡀
⠀⠀⠀⠀⠀⠀⢀⡙⠏⠞⠃⠢⠀⡁⠰⡵⣸⡏⠀⣷⢟⠛⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⢋⠴⣊⣬⢿⡮⡅⠀⠀⠉⠀⢇⠄
⠀⠀⠀⠀⠀⠀⠠⠙⠲⠽⣟⡳⢀⣺⣡⣎⠉⠀⠀⢡⡼⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠷⣾⢛⣼⡿⠉⠈⠱⡄⠀⠀⠀⠾⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⢻⡿⠺⢖⣤⡀⣠⣬⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢁⣴⠾⣓⡯⠒⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠙⠮⣭⢱⣎⡀⠄⠚⠁⠀⠀⠀⠀⠀⣀⣤⠔⠋⣉⠠⢚⣩⠔⠛⢉⡴⠀⠀⠌⠀⠀⠀⠈⠀⠀⢀⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⢀⣀⣤⣶⡷⠷⠒⢒⣩⡴⠞⠉⠀⠀⠀⢸⣿⣄⠀⠀⢶⣀⠂⠈⠐⠀⠁⠀⠎⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⣧⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⢀⣾⣿⠓⠀⠀⠀⢹⣇⠀⠀⠀⣨⢀⣴⠇
⠀⠀⠀⠀⠀⣠⠖⠀⠂⠀⠀⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⢀⣀⣀⣰⣿⣯⣻⣿⡶⠦⣦⡄⠀⠄⠠⣀⣺⣟⣾⡿⢷⡄⠀⣀⣼⣟⢃⣠⠜⣡⠞⡟
⠀⠀⠀⠀⠀⠛⠃⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠿⠿⠭⠿⣾⠿⠿⣭⣉⣉⣙⣙⠿⢿⣦⣭⣠⣡⡤⠶⣊⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠿⠿⢶⣭⠴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⢦⡀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠄⡀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⢠⣿⣬⣿⣦⣧⠀⠀⠀⠀⠋⠀⠈⠐⠓⠒⠒⠉⠉⠁⠀⠀⢱⣄⡀
⠀⢲⡻⣷⣿⠀⠀⠀⢀⣀⣠⣿⠀⠀⠀⢸⣾⣇⣿⣼⣅⣿⡾⠀⠀⠴⢄⣀⣀⡀⠀⢀⣀⣀⣠⣴⡆⠀⣾⠇⠁
⠀⢻⣭⣻⣢⡦⢰⡲⣯⡳⣽⡏⠀⠀⣰⢾⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠈⠙⢿⣇⢸⡿⠿⠿⣿⣦⠸⡟
⠀⠀⠙⠻⣿⣷⣶⣿⣮⣽⠟⠀⠀⠀⠈⢢⡻⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢺⠹⣸⠀⠀⠀⢌⡏⢠⠇
⠀⠀⠀⠀⠈⠻⠿⠿⡿⠃⠀⠀⠀⠀⠀⠀⣼⣩⡭⡯⠍⣹⡄⠀⠀⠀⠀⠀⠈⠈⠢⠋⠀⠀⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠁⠐⡘⠀⢤⠀⠀⠀⠀⢰⠋⠛⠤⡷⠞⠋⢳⠀⠀⠀⢀⢾⢀⠀⠀⠀⠀⢀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠀⠀⠀⣧⠀⠀⠸⡧⠀⢠⡿⡟⣿⡄⠀⢠⡴⣻⣭⣵⡽⢿⣷⡆
⠀⠀⠀⠀⣀⡄⡰⠁⢀⢂⣤⣀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠘⠷⣷⠷⠃⠀⣵⡞⣿⣿⡟⣿⣯⠻⣿
⠀⠐⣿⣶⣌⣿⡇⠀⠀⣹⣏⣠⣶⡷⠀⠀⢀⠀⠀⠀⢆⡀⠀⠀⢀⢀⢀⠀⠀⠀⠀⢠⣁⣾⣿⣧⣿⣻⢄⡿
⠀⠀⠈⠉⠋⢻⣿⠀⢠⣿⠏⠙⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠋⠾⣼⣿⣷⣠⡀⠀⢸⡿⣽⣿⣿⣿⠏⣼⠁
⠀⠀⠀⠀⠀⠘⣿⠃⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣛⡀⠀⠀⠤⢤⣴⡏⠀⠀⢠⡷⣱⡿⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠱⣥⣥⠇⠀⠀⠀⠀⠀⠀⠀⠀⠬⢭⣭⠄⠀⠀⠀⠀⢸⡇⠀⠀⠸⣟⣫⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠵⠚⠉⠓⠢⣄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠔⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠹⣧⡀
⠀⠀⠀⠀⢀⡴⠋⡠⠒⠀⠐⠈⠍⣻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⣷
⠀⠀⠀⠀⣼⠁⡎⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡶⠦⣤⣤⣀⣠⣤⣤⣖⡯⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿
⠀⠀⠀⢠⡿⠀⢧⠀⠀⠀⠀⢀⡴⠖⢒⢶⡄⠙⢢⡀⢀⡤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿
⠀⠀⠀⢸⣇⠡⠘⢆⠀⠀⠀⡞⠀⠊⠉⢳⣿⠀⠀⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⣿⡀⢁⠘⢆⠀⠸⡇⡆⠀⢀⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣼
⠀⠀⠀⠀⠸⣧⠀⠣⡈⢣⡀⠳⣝⣒⣋⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡜⣿
⠀⠀⠀⠀⠀⠙⣧⠀⠈⠢⡱⣄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠎⢰⡟
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠈⠪⣳⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠚⠁⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠈⠓⢌⡉⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢚⡡⠔⠉⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⠑⠒⠨⠵⣲⡶⠶⠶⠶⠶⠖⠖⠒⠒⠚⣉⡥⠔⠚⠁⠀⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠉⠁⠁⠀⠀⠀⠀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⠟⠛⠛⠛⠛⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡴⠟⠻⣦⣤⣤⡀
⣼⠀⠀⠀⠁⠀⠈⢻⣄⣀
⠻⣤⣤⣄⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠈⠻⠦⢦⣤⡀⠀⠀⠀⠀⠈⢙⡷⠶⠾⠿⠷⠶⠶⠶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡸⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠶⠶⠾⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⣉⣤⠤⠶⠶⠶⠒⠒⠲⠦⢄⣉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣤⢾⡄⡀⠀⠀⠀⢀⡴⠋⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠙⢦⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⢿⡀⠀⠉⠀⠀⠀⣠⢏⡴⠉⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠈⣧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠟⠁⠀⠀⠸⡇⠀⠀⠀⠀⣴⢣⠏⠀⠀⠀⣠⡤⠶⠶⢤⣀⠀⠀⠀⡴⠊⠉⠀⠀⠈⠉⠙⠢⡀⠀⠀⢸⡆⢿⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣼⢣⠆⠀⠀⣴⠟⠁⠀⠀⠀⣠⠈⢳⠀⢸⡁⠀⣀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡇⢸⡀⠐⢯⣄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⡆⠀⢠⢇⠎⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣠⠴⠃⠈⠧⣄⠈⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡏⢸⠁⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣿⠃⠀⠀⢸⣇⠀⡟⣸⠀⠀⡼⠁⠀⠀⠀⠁⠀⢠⠎⢁⡤⠶⢤⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠾⠃⠀⠀⠀⠀⣿⢰⠇⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⡏⠀⢸⠁⠀⠀⡿⠀⢹⠀⠀⠀⠀⠀⠀⢀⡟⠀⡼⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⢰⡇⠀⢸⡀⠀⠀⠀⠂⠀⠀⢧⠀⠘⠦⠤⠞⠁⢠⡆⠀⠀⠀⠀⠀⢀⡞⠀⡼⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣠⠋⢀⡼⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⣧⠀⠀⠘⢦⣀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠸⡅⠀⠀⠀⣠⠞⠁⣠⠎⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠸⡄⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⠓⠛⠋⠁⢀⡴⢁⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⣉⣙⠒⠒⠶⠶⠶⠒⣒⣊⣩⣥⠴⠟⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣄⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣫⣽⣶⣶⠒⡠⣔⣫⠽⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣭⣾⡿⠛⠛⠛⢁⣦⡉⣓⠾⣭⣿⣓⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣥⠏⠁⡰⣜⢮⡿⣷⣽⣾⣧⠈⠻⢿⣓⣶⠾⢤⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣉⡁⣴⣿⣿⣮⣳⣝⡮⣷⣿⣿⡬⣧⣄⠺⣭⣽⣛⡂⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⣮⣳⡜⢿⡯⣟⢮⣗⢿⣚⠩⣿⣣⢿⣾⣦⠙⢳⢯⣟⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠻⣿⣧⠈⢿⣮⣟⡮⣝⡾⢈⣓⠾⣽⣲⠏⠀⠈⣓⢏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠻⠃⠄⣶⣝⡲⣝⡛⣭⣐⠮⣝⡆⣬⣛⠂⣐⢎⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠃⣼⠀⢿⣿⣝⡲⣽⡷⢮⣙⠾⣱⢰⣝⡷⠌⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣾⣿⠀⣷⣽⡻⣽⣻⢏⡳⣭⣳⣇⡻⣮⠃⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣿⡿⢠⡹⣿⣿⡳⢍⡳⣝⢮⡻⣯⡻⠀⣤⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⣿⢃⣺⣷⣝⢟⢿⡳⣝⢮⣳⡽⠎⢁⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡾⠏⡰⣝⢮⣿⡇⠠⠅⠈⠀⠁⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⣶⠁⣰⣿⣾⣯⠗⠀⠀⠀⠀⣀⣴⢸⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠟⡃⣾⣯⣿⡿⠃⠊⠀⣠⣴⡇⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣎⡙⠙⠛⢋⣀⣠⡆⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣇⣷⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⢶⣳⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄⠠⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢣⡞⠉⣩⣆⠈⠛⣿⣄⠀⠀⠀⠀⠀⠀⠓⢄⠀⣑⡈⣳⣤⣶⡛⠋⠁⠈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⣨⠇⠀⠎⠛⠁⠀⠀⠀⠛⡆⢀⠀⡀⠀⠉⠻⣅⢠⣶⠟⣉⠙⡛⠛⠉⠈⢁⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣶⣄⣀⠀⠀⠉⠲⣄⠀⠀⢲⣄⣼⠦⠶⠃⠄⠀⣈⠀⠀⡀⠈⢉⡟⠉⠛⠉⠙⠋⠀⠐⠋
⠀⠀⠀⠰⠂⠀⠀⠀⠆⠀⢾⣿⢁⠹⣜⢦⠀⠀⠙⠶⣽⡮⠹⣿⣍⣷⠶⠶⠟⠿⠿⢷⣬⡤⠀⠁
⠀⠀⠀⠀⣨⠥⠀⠀⡤⠀⠈⢿⣟⣷⡜⢿⡇⠀⢀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠐
⠀⠀⠐⡤⠿⠔⠀⠀⢅⠀⠀⠸⣿⣿⣿⣛⣣⡄⠀⣀⠀⠀⠀⠀⠀⠀⠠⠄⡦⢀⡀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠡⣀⠀⠄⠀⠀⠀⢄⡀⠈⠻⠿⣿⠟⠁⠀⠁⠀⣀⠤⠀⠀⠀⣠⠠⢲⠇⢰⡆⢀⠀⠀⠀⠀⠀⠈⢶⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⢠⠊⠀⠀⠀⠄⢀⠘⣠⣷⣤⣼⣿⣿⠀⠀⠀⠀⠘⠛⠠
⠀⠀⠀⠀⣀⠀⣈⡄⢦⣤⠀⢄⡀⠀⠀⠀⠀⣰⡤⠀⠀⠀⢀⣠⣾⣿⡿⣿⣿⣿⣿⣿⣷⡦
⠀⢀⠀⣀⡿⠟⣹⢧⣭⣶⣛⣳⠛⡆⠀⠀⠀⠋⢒⠘⠃⡀⣿⣿⣷⣿⡿⣟⠙⠙⠏⠙⠢⣅⣾⠁⠀⠀⠀⢀⠔
⠀⠘⣿⢿⠳⢞⢧⢺⣿⣟⠻⡿⣷⢤⣀⠀⠀⠐⠚⠒⠶⠚⠛⠛⣁⣈⠉⠻⡗⢲⠖⣢⣴⣯⣿⣦⠒⠆⠀⠁⠐⠶
⠀⠀⢰⣾⡖⣸⢋⡧⠌⠹⠇⢰⡜⣧⣌⣧⣄⠀⠀⠀⠀⠀⠀⢀⡿⠉⠀⠀⠀⠘⠓⠴⠼⠿⠟⠁⠀⠀⠀⠂
⠀⠀⠈⢛⣿⣽⡿⢶⠐⣆⡀⢈⡷⣼⠟⣿⣟⣆⠀⠀⢀⣄⣴⠰⣾⡆⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡤
⠀⠀⠀⠨⠙⢛⣛⣛⠁⣬⣿⣟⣿⣯⣯⣿⣿⠷⠂⠀⠀⠙⢿⠂⢩⠃⠈⠉⠉⠀⢧⣤⣡⣶⣿⣿⣧⣄⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⡀⠀⠀⠈⠁⢲⣍⢡⠀⡀⠀⠀⠿⢻⣧⣽⣷⢾⣤⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣾⡓⢸⠂⠀⠀⠀⠼⡦⠀⠁⠀⢀⡄⠈⣿⣿⣿⣸⣯⡐⠀⠀⠀⢠
⠀⠀⣀⠀⢀⡤⠴⠶⠚⠓⣒⠈⢉⠀⠀⠀⠀⠀⠸⣿⣷⠲⡤⣤⡤⠀⠁⠲⠂⠀⠀⠐⢃⡟⣿⣯⣿⣷⣷⡖⠀⡐⠀⡀
⠀⢠⣏⢳⣛⡅⠶⠾⠿⣷⡎⢠⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠤⠂⢉⡭⣿⣇⠀⠀⠀⠩⡰⠛⣿⣿⣿⣿⣷⣵⣤⡟
⠀⠀⣿⣿⢽⣷⣿⣾⣿⡟⠑⠁⠀⠀⡠⠞⠀⠀⠂⠘⠂⠀⡰⣯⣿⠿⠷⠛⠁⢀⠀⠀⠀⠠⠾⣛⠛⠿⢿⣛⠷⠚⠁
⠀⠀⠘⣿⣵⣿⣿⣿⣿⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠈⠐⠀⠒⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢷⠀⠀⠀⠀⠾⢥⣶⣴⠟⢿⣦⡀⠀⢀⠀⠀⠀⠀⢀⡤⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡒⢉⣡⢤⡾⠀⠀⣴⣤⣀⠂⢤⡀⠀⢾⠿⣿⠟⠻⣷⣄⡀
⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⠀⠀⠀⣀⠤⢰⣏⢱⠍⠐⠋⠀⠀⢶⡶⢶⣿⢿⣦⣈⠿⠛⠳⢤⡀⠀⠀⠙⠳⢦⣄
⠀⠀⠀⠀⢤⣉⠂⠠⠊⠀⣠⠶⣯⣵⡿⠟⠋⠀⣀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠻⢿⣳⣦⡀⠙⠧⠀⠈⠀⠤⡉⠻⣿⣶⡆
⠀⠀⠀⠀⠠⢌⠲⢠⣰⡾⣱⣾⡽⠋⠀⠀⠀⢀⠀⠈⢰⣶⣾⣿⡷⠤⡀⠀⠀⠀⠀⠈⠙⠛⠦⠘⢦⠀⠀⣀⣀⡁⠠⠙
⠀⠀⠀⠀⠀⠀⠈⠒⠿⠾⠛⠁⠀⠀⠒⠂⢀⣷⣮⣓⣞⣟⣛⣻⣥⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⡆⠈⠻⠲⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣯⣒⠚⠛⠛⠛⠻⣿⣿⣿⣿⠟⠎⢀⠀⠀⠀⠀⡄⡠⠀⠀⠈⠓⠠⠄⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡟⣾⣿⣷⣷⡶⣞⣾⣶⣾⠗⡶⠁⠀⠀⢀⣻⣴⡌⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⠛⠛⠿⠇⠒⠋

⢀⠀⠀⠀⢀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠚⠂
⠐⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠨⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠂⠀⠐⠀⠀⠀⠀⠀⠤⠤⠄
⠀⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠄
⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠁⠀⠀⠀⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀
⠀⣄⣤⠛⠁⢠⡴⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐
⢄⠀⠀⣁⠔⠀⠀⠀⠀⠀⠠⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠘⠢
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠅⠀⠄⠈⠈
⠀⠀⠀⠀⠅⠀⠸⠒⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠂⠀⠈⠀⠀⠀⠁⠀⠀⡀⠀⢀⣼⠉⡪⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢇⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⡀⠀⠀⢄
⠀⠀⠉⠀⠀⠄⠀⠠⠀⠀⠁⠀⡰⠋⠑⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠘⠀⢠⡀⠊⠀⠀⠀⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⣭⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢸⣿⠂⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡁⠀⠀⠸⣿⠀⠀⣧⡈⠳⣄
⠀⠀⠀⠐⡄⢀⡤⠖⠉⠀⠈⠳⣄⡀⠀⠀⠀⠀⢀⣠⢞⡥⣸⡷⠀⠀⠀⢿⠀⣄⣿⣧⡄⠈⣳⡄
⠀⠀⠀⠀⠘⠾⡀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣤⡴⠋⢠⢸⡇⣿⡇⠀⠀⣠⠀⢰⣯⣿⡟⢃⢞⠝⠁
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⠀⡑⠈⢿⣧⠹⡄⢸⢸⡇⣿⡇⠀⠀⣿⠀⢸⠛⣃⢔⠕⠁
⠀⠀⠀⠀⠀⠀⣈⡆⠀⢐⠆⠰⡁⠘⣆⠈⡿⠛⠛⣆⢼⡇⢹⠀⠀⠀⡟⢀⣠⡾⠗⠁⠀⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣦⡀⢳⡀⢹⣄⣸⠖⠃⠀⠀⠈⢆⠁⢸⠀⢀⣠⡶⢟⣋⠴⢒⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠙⠲⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⢧⣴⡾⠛⠉⠀⢘⠇⠀⠏⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡎⠀⢀⣸⠀⠀⢠⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⠤⢤⣤⣄⣀⣠⡤⠶⠒⠚⢛⣒⣚⠛⢧⠀⢧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣥⣤⣍⡉⠉⣤⣤⣄⠲⣶⣶⣶⣶⣶⣿⢃⣾⣿⡿⣋⣴⣿⢀⡟⣷⠇
⠀⠀⠀⠀⠀⠀⣼⠁⠾⣿⣶⣯⣽⣿⣦⡘⢿⣿⣷⣮⣿⣿⣿⡿⠃⣚⣻⣻⣿⣿⣿⣿⡿⢁⣿
⠀⠀⠀⠀⠀⠶⣿⠗⠲⠶⠤⣭⣛⠿⣿⣿⣌⡛⢋⣭⣭⣭⣍⠐⢋⣭⣭⣭⣭⡭⠀⢀⣴⡿⣿
⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠆⡖⢩⣛⠶⣌⡙⠻⠶⣦⣭⣭⣭⡴⠿⠟⣋⣉⣥⠴⠞⡻⢟⣼⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⢛⡲⠶⣾⣿⣑⠈⠉⠛⠒⠶⠶⠶⠖⠚⠋⠉⠉⠀⠰⠖⣋⡵⢛⡁⠦⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠁⢀⠍⠙⠶⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠛⠋⠇⠀⠢⠀⢁⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠙⠛⠉⠉⣉⠁⠁⠀⢀⠠⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠉⠁⠀⠉⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠖⠒⠒⠒⠒⠒⠛⠋⠙⠛⡒⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠀⠀⠉⢀⠉⠓⠲⣄
⢀⡤⠴⠖⠚⠛⠉⣥⡴⠞⠉⠀⠀⠳⣄⢀⡔⠀⠀⠢⡴⠋⠀⠢⡀⢀⠴⠃⢄⠀⣼
⢸⠀⠀⠀⣡⣴⠛⠁⠈⢢⣤⠀⠀⠀⡨⠻⣄⠠⠖⠁⠀⠀⠀⠤⠈⠁⢀⣀⡤⢾⡇
⠈⣷⢤⣤⣀⡈⠙⢦⡚⠁⠈⠑⠄⠀⠀⠀⢈⣡⡤⠖⠒⠒⠒⠒⠚⠛⠉⣁⡴⠏⠀⠀⢀⣀⣠⠔⣫⣽⣭⣓⣒⠤⠤⠤⢄⡀
⠀⠈⢳⣄⣈⠉⠉⠛⠓⠶⠖⠲⠖⠒⠚⠛⠉⢀⣠⠴⠖⠒⠚⠛⠋⠉⠉⠁⠀⢀⡴⣻⣿⣷⣶⣿⣿⣿⣿⣿⣿⡿⠆⣿⣶⣌⠳⣄
⠀⠀⠀⠙⠛⠷⠴⠤⣴⣤⣤⣤⣀⡴⠴⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⣘⣻⣿⣛⠿⠿⣛⣛⣿⣛⣡⣶⣶⡆⣹⣿⣷⣜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⣫⣽⡶⢿⡿⠿⣟⣩⣿⣿⣜⣿⣿⣿⣿⣿⣿⣿⡷⢹⣿⣿⠏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠻⠿⡗⠮⣍⣲⣟⣿⣯⣭⣭⠭⠽⣯⣼⣟⣛⣛⣛⣿⡿⣃⣿⣋⣥⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⡄⠀⡿⠀⣾⣿⠁⠀⢸⣿⣿⡄⠀⠀⠀⠀⢻⣿⣧⡉⠉⠉⠁⢠⡿⠃
⠀⠀⢀⡴⢒⣛⠛⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠶⠚⠳⠶⣿⣿⣥⣤⣼⣿⣿⣧⣀⣀⣀⣀⣼⣿⣿⠧⠶⠒⠚⠉
⠀⣴⢋⣾⣿⣿⣇⢸⣧⣄⣉⠓⠲⢶⠶⠶⠶⠶⠶⠶⠦⣤⣀⡀
⢸⢃⣾⣿⣿⣿⣇⠸⣿⣿⣿⣿⠿⠋⣰⣿⣿⣿⣿⣶⣶⣤⣉⣙⠶⣄
⣿⢸⣿⣿⣿⣿⣿⡄⠈⣉⣉⠁⠴⠾⠿⠿⠿⠿⠟⠛⠋⣉⣉⣩⣤⠘⣆
⣯⠘⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠟⠋⣠⣿
⢹⣆⠹⣿⣿⣿⣿⡟⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣡⠤⠶⠞⠻⣿⡏
⠀⣿⣷⣬⣉⡛⠋⠀⢸⣿⣿⣿⣿⣿⣿⡿⢛⣡⡖⠋⠁⠀⣠⡴⠞⠋⠀⣀⡤⠔⠒⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠸⣿⡀⠈⠙⠓⠲⠬⣭⣉⣉⣉⣭⣴⣾⣿⣿⣧⣠⠖⠋⠁⠀⢀⡴⠊⠁⠀⠀⠀⠀⠐⠀⠀⠉⠉⠉⠉⠉⠛⠉⠁⠀⠀⠉⠓⢄⡀
⠀⠀⠹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢠⠞⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⣀⣤⠴⠟⠛⠉⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠖⠒⠋⠉⢀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠈⠉⠓⠒⠚⠛⠉⠉⠀⠀⢀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⡀⠀⠀⣠⡤⠴⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⡤⠤⠄⠀⢤⣤⡤⠀⠤⢤⡶⠚⠉⣁⣴⣶⣿⠟⠛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⣠⣶⣯⣴⠶⠞⣿⠟⠁⠀⠀⣰⣿⣡⣴⡾⠿⣽⡿⠁⠊⠀⣀⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⠶⣶⠟⠛⠉⠁⠀⣠⡾⠁⠀⣀⣤⣶⠿⠟⠋⠁⠀⣰⠋⢀⣠⣶⣿⡿⠋⢀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡾⠃⠀⠀⠀⣠⣾⣿⡴⠾⠛⢻⡟⢁⡤⠀⠈⣠⣾⣧⡶⠿⠛⣡⡟⠁⠈⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡋⣀⣠⠤⣶⠾⠋⡁⠀⠀⢠⣼⠏⢐⣁⣴⣶⠟⠛⠛⠁⠀⠀⣼⣿⣥⡶⠶⠶⣿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⣠⠞⠁⠀⠀⠀⣠⣞⣩⣴⠞⠉⣼⠟⠁⠈⠀⢀⣨⣤⠞⠉⠡⠀⠀⢀⡾⠃⣀⣷⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⠻⣄⠀⠀⣀⣠⠞⠛⢩⠄⠂⣠⡜⢁⡠⠔⠒⠋⢉⡼⠃⠠⠄⠂⠈⣠⠎⢀⣠⣾⣿⠀⠈⠲⠦⠞⠀⠴⠺⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣪⡟⠁⠀⠀⠀⣠⣾⣿⣾⣿⣗⣶⣶⣶⣿⣤⣤⣄⣐⣦⣾⣧⣶⣿⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⣿⣿⣧⣠⣤⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢫⣑⣟⣹⠟⣿⣷⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣟⣿⣿⣿⣿⡿⠿⠛⠋⠉⠙⠛⠛⠛⠛⠿⠿⣿⣧⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⣿⠆⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣸⣻⡿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣄⣐⣚⣍⠴⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⣧⡄⠀⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠚⠋⠉⠉⠀⠈⠉⠁⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢠⠞⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠱⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠹⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠁⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⢀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠄⠐⢲⠒⠉⢹⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡭⣽⣿⣆⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣀⠀⣠⡄⠤⢼⠒⠂⠛⠀⠀⠀⣀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣿⡿⣻⣛⡃⣿⣽⣥⣿⣿⣿⡿⢟⡅
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠇⠀⢸⣀⣀⣤⠤⠀⠂⢀⠴⠊⠁⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⣿⡿⠿⠿⠇⣿⣿⣿⣿⢟⣭⣾⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⡴⠃⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⢟⣵⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠸⡀⠑⠦⢄⣀⣀⣀⣀⡤⠔⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣻⠿⣿⣿⣿⣿⠿⣛⣵⣿⠟⠁
⠀⡀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡌⢉⣷⡿⠿⡟⠗⢶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠐⠀⠀⣍⠙⣧⡈⠁⣈⣡⣤⡤⠄⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢁⡿⠁⠠⣶⣿⣤⠀⢨⡿⠷⣄⡀⠀⠀⠀⠀⠀⠀⣩⡙⣆⠀⠤⡄⢨⣽⣿⣿⣇⣀⣀⣐⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⡄⠀⠀⠀⠀⠀⠀⠀⠝⠐⠐⠆⠙⠉⠀⠐⠀⠁⠚⠻⡇⢠⡀⢀⡀⠀⠁⠙⢮⣑⢢⣼⠟⢋⣉⠉⣛⢋⠉⠁⠀⠤⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠿⣧⣦⣄⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠐⣦⡀⣰⣦⣤⣾⠃⣀⠀⠉⠙⠀⠀⡀⠀⠉⠛⠿⠟⠻⠛⠛⠉⠛⠁⠀⠀⠚⠁
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣀⠀⢀⣴⣷⢊⠹⣟⣦⣄⠀⠀⠘⢿⣤⡲⣤⡟⢿⣿⠶⢒⡄⣀⣡⣤⡽⠯⣥⣤⠈⠢⡀⠈⠓⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡎⠀⠆⠀⠀⠀⠈⠀⠈⢻⣿⣸⣆⠈⣦⣹⣆⠀⠀⠀⠉⠙⠿⣥⣉⡶⠷⠾⠛⠉⠉⠁⠉⠉⠙⠙⠻⠻⣃⡀
⠀⠀⠀⠀⠀⣰⣚⠆⠀⠀⣸⠀⠀⠀⢻⣇⣽⣿⣏⣻⣏⠀⠀⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠱⡄⢿⠴⠂⠀⠀⢤⠀⠀⠀⠸⣿⣿⣿⣿⡛⢃⣴⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⢴⠆⣀⡁⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠈⢁⠂⠀⢀⠀⠀⠈⠁⢀⠀⠀⠈⠻⣿⣿⣿⣿⠏⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠀⢀⢀⡔⣴⣁⢢⣦⢈⡀⠀⠀⠀⠀⠀⠀⠀⢧⣦⡀
⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⠴⠟⠁⠀⠀⠀⠀⣰⠖⠣⠀⠀⢀⠀⠑⢩⠀⣶⡁⠀⢘⣷⣴⡇⠀⠀⠀⠀⠀⢶⣶⠀⠩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⢉⣤⣵⣤⣮⣿⣿⣿⣿⣿⣿⡓⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣴⡎⢛⣶⣤⡀⠢⣄⠀⠀⠀⠀⠀⢠⣾⠖⠀⠀⠀⠄⣠⣴⣿⣟⣿⢟⣿⣿⣯⣿⣿⣿⡿⣿⡓⠄⡀
⠀⠀⠀⠀⢀⡿⠿⢋⡹⣤⣭⣵⣞⢏⣳⡛⢧⠀⠀⠀⠀⠘⠥⢚⠹⠋⢁⡠⣿⡯⣷⣿⣿⡟⣿⠉⠋⠙⡏⠉⠓⣤⡀⣿⠃⠀⠀⠀⠀⠀⡠⠂
⠀⠘⢷⣾⣿⡿⣴⢮⣇⢾⣿⣿⠿⣮⣽⣷⣀⡀⠀⠀⠀⢀⣤⣼⣤⣼⣤⣾⣿⠿⠋⠈⠉⠛⢿⣻⠶⣤⡤⢤⣀⣬⣛⣾⣷⢲⢦⡇⠀⠊⠀⢤⣤
⠀⠀⠈⡏⣵⣀⡐⣦⢽⣌⠻⠟⣶⠘⢂⠛⣯⠙⢖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠦⠀⠀⠉⠀⣿⣄⠛⠛⢿⣯⣿⠿⡃⠀⠀⡀⢀⠄
⠀⠀⠀⣿⣿⣯⣴⣧⡼⠛⠓⡀⠉⠁⠹⣧⠸⣿⣷⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠾⡁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠁⠀⠀⢀⠀⠁⣀⠀⠀⠀⠄
⠀⠀⠀⠈⣣⣛⣶⣿⠚⣿⡏⢫⣄⡀⢀⣿⢢⣿⣉⣛⣯⣻⡄⠀⠀⠀⣴⣦⣿⠘⢿⣷⠀⠀⠀⠀⠉⠁⡀⠁⠀⠠⠀⠀⠀⠀⠀⠁⠀⡉⢀⣤⠦
⠀⠀⠀⠀⢈⠙⠛⠛⡟⠿⠃⢨⣭⣿⡟⣻⣯⣥⣭⣿⣿⣿⠗⠂⠀⠀⠈⠻⣷⠂⢈⡽⠀⠈⠉⠉⠉⠐⢧⣤⡉⣤⣶⣿⣿⣿⣤⣄⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠙⠉⠁⠉⠛⠿⠛⠛⠳⠛⠛⠉⠀⠀⠀⠀⣀⠄⠀⠚⠓⢤⡕⠚⢀⠀⠀⠀⠀⠀⣿⢿⣿⣉⡻⣷⣭⣏⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⣄⠀⠀⠀⠀⠛⣯⠛⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⠋⣿⡿⢉⡀⠀⠐⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠘⢿⡄⣁⠋⠀⠀⠀⠀⠙⠻⡗⠀⠀⠀⠀⠠⠞⢀⠸⡿⣿⣿⣴⣿⢷⡄⠀⠀⠀⠀⠘⡄
⠀⠀⢠⣀⠀⣀⣤⠶⠷⠟⠛⠉⢉⡝⠀⢀⡄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⢲⠶⣴⡦⠆⢀⢀⡘⠞⡀⠀⠀⠀⣋⣰⠛⣿⣿⣽⣿⣿⣿⣿⠂⢠⡈⠀⡅
⠀⠀⣾⠟⣷⡟⣃⠴⣶⣾⣿⣿⣶⡞⢂⠎⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠉⠙⠁⣤⠔⠚⠉⣩⣿⣾⡀⠀⠀⠀⠈⣍⣸⠏⣿⣼⣷⣿⣿⣷⢨⣦⣀⣿
⠀⠀⢹⣿⣾⣹⣿⣴⣦⣴⣶⡿⠾⢡⠋⠀⠀⠀⢀⡤⠂⠀⠀⣄⣰⢦⡀⠀⢠⣶⣲⣶⢿⣧⡿⠟⠁⠀⠀⠀⠀⠀⣁⣴⡿⠿⢿⣿⣿⡿⣟⣭⠽⠃
⠀⠀⠐⢿⣟⢽⣻⣿⡿⣿⣿⣿⡄⠄⠀⠀⠀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠿⠟⠋⠉⠁⠀⠀⠀⠶⠀⠀⠀⠀⠈⢩⣴⣦⣌⡉⠙⠒⠎⠉
⠀⠀⠀⠸⣿⠿⠿⠿⠿⢿⣿⢿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢢⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡺⠿⡷⠀⠀⠀⢀⠰⠿⠶⣽⣷⣾⠋⢻⣿⣆⡀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠶⢢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠐⠀⡠⠠⠚⠛⢉⣠⣴⡞⠀⠀⢀⡦⣄⠀⠄⣀⡀⠀⠁⢾⡿⣟⣧⡟⠻⣿⣦⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⣣⠟⠩⠵⠏⠀⠀⣠⣭⣍⣹⡿⣶⣄⠙⠺⣶⣼⡧⣄⠉⠀⠀⠈⠙⠿⣧⣄
⠀⠀⠀⠀⠀⠘⠣⢄⠀⠀⣠⠖⠀⠀⢀⣤⠴⣚⣭⣰⣿⠇⠛⠀⠀⠀⠀⠀⠀⠙⠓⠊⠛⠛⠙⠛⢳⣶⣭⣀⡀⠉⠻⢤⡄⠀⠀⠢⠈⠙⠻⢦⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠱⢤⡀⠀⠱⠂⠀⣠⣾⢋⣵⣿⣿⠟⠋⠁⠀⠀⠤⢀⠘⠀⠈⠀⣐⣄⠀⠀⠀⠀⠀⠈⠻⠾⣭⣛⣦⣀⠀⢹⡂⠀⠀⠀⠙⢢⠀⠙⠿⣾⣿
⠀⠀⠀⠀⠀⠀⢦⡈⠒⢠⣀⣿⠏⣴⣿⡭⠏⠀⠀⠀⠀⢀⣀⠂⠀⠐⣶⣶⣿⣿⣿⢲⢤⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⠦⠀⠻⡆⠀⠀⣠⣀⣈⠁⠠⡌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⣿⣯⡾⠟⠉⠀⠀⢤⠶⠆⠀⢺⣥⣍⠳⡶⢿⡿⠿⡛⣏⣰⣶⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠱⡀⠀⠻⡧⣼⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣰⠦⡤⣤⠿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⡿⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢦⣄⣀⣀⣐⣀⢈⡝⣿⡛⢻⣿⠋⣈⠥⠰⠀⠀⠀⣠⠀⡞⡠⠋⠀⠀⠀⠀⠀⠉⠑⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣋⣻⣿⠿⠿⢿⣟⣻⣽⣿⣿⣿⠟⢰⡏⠀⠀⠀⠀⠤⡽⠷⠷⠐⠲⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠉⠛⠻⠿⠏⠐⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⢿⣿⣿⣦⡀⠀⠀⣈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⢈⢿⣿⣿⣷⣄⠀⠀⠀⠂⠈⠑⢄
⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⢀⠻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠈⢢⡈⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡒⠲⡄
⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆⠀⠐⠐⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠁⠀⣹
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠐⠘⢦⣀⠀⠀⠀⠀⢀⣠⠏⠀⢰⣏
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⢀⡤⣄⠀⠀⠀⢀⣿⣿⠀⡀⠀⠀⡠⠀⠀⠀⡀⠈⠈⠉⠓⠒⠚⡿⠀⠀⠀⠘⠈⠳⡄
⠀⠀⠀⠀⠈⣿⣿⡄⠀⢠⡏⡞⢢⢳⠀⠀⣸⣿⡏⢀⠇⠀⠀⣇⠀⣠⠀⣇⡤⠂⠀⠀⠀⣀⣽⠤⠤⣄⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠘⣿⣷⡀⠘⣇⢧⡼⡸⠀⣠⣿⡟⠀⡼⡠⠊⠀⣿⠞⣁⡴⠿⠒⠚⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠉⠉⣁⣴⣿⠟⢀⣠⠏⢁⡠⢚⣧⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⣟⠥⠒⢁⣮⣔⡫⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⠚⠉⠉⠙⠉⠉⠉⠉⡕⠦⣀
⠀⠀⠀⢀⣀⣀⣤⠞⢉⡠⠄⠀⡀⠀⠀⠀⠒⡀⢀⠄⠀⠀⠀⢠⡈⠉⢳⡀
⠀⢴⠋⠉⠀⢀⡤⠛⢯⠀⡀⠀⠘⢦⡞⠀⡀⠈⠃⠀⢈⠓⠔⠁⠈⣀⣼
⠀⠸⣄⣀⡈⠙⢦⣀⠔⠋⠢⡀⠀⠁⠙⠂⣁⡤⠤⠤⠬⠤⠴⠒⢋⡵⠃⠀⠀⠀⣀⠴⣲⡶⢤⣄⣀⣀⡀
⠀⠀⠳⣍⠉⠛⠒⠞⠶⠦⠶⠶⠶⠶⠚⠉⣀⡤⠶⠒⠒⠒⠒⠛⠉⠀⣠⣶⣾⣿⣷⣿⣿⣿⣿⣷⡦⢰⣮⣕⢦⡀
⠀⠀⠀⠈⠛⠳⠶⠤⠦⢤⣤⣤⠤⠖⠒⠋⠁⠀⠀⠀⠀⠀⢀⣠⠤⣚⣙⣛⣿⡛⠛⢻⣛⣻⣛⣥⣶⡦⢹⣿⣷⡵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣚⣭⣿⣟⣿⣛⣟⣥⣿⣧⣿⣿⣿⣿⣿⣿⡟⣹⣿⠿⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡍⠋⣿⠑⣮⣽⠟⢳⣶⣶⠚⠛⠒⠻⣿⣷⠭⠿⠥⠽⠕⣾⠏
⠀⠀⠀⠀⣠⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⣠⣟⣠⣿⣏⠀⢸⣿⣿⡄⠀⠀⠀⢸⣿⣷⣀⣀⣠⠼⠋
⠀⠀⡴⢫⣶⣿⡆⣶⣉⠓⠦⢤⣤⣤⣤⣤⣤⣤⣈⡉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠛⠉⠉⠁
⠀⣸⢡⣿⣿⣿⡇⢿⣿⣿⣷⡿⠓⣴⣶⣶⣶⣦⣤⣉⣓⠦⡀
⠀⡏⣿⣿⣿⣿⣷⡀⢉⣉⡁⠰⠾⠿⠿⠿⠟⠛⢋⣉⣩⣤⠘⡆
⠀⣇⢸⣿⣿⣿⣿⣧⢸⣿⣿⣿⣷⣶⣶⣶⣶⣿⡿⠟⠛⣉⣴⡇
⠀⢹⣆⡙⢿⣿⡿⠃⣴⣿⣿⣿⣿⣿⣿⡿⢋⣥⠖⠋⠉⣩⡿⠃⠀⠀⠀⡀
⠀⠀⢿⡟⠒⠦⢄⣀⠻⠿⠿⠿⢟⣋⣥⣾⣏⠀⣠⠴⠋⠁⢀⡤⠚⠉⠀⠀⠉⠑⠢⠤⠤⠤⠤⠤⠔⠒⠒⠢⣄
⠀⠀⠘⣷⡀⠀⠀⠀⠉⠉⠙⠛⣻⣿⣿⣿⡿⠋⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠙⠲⠤⣄⣀⣀⣀⣠⡽⠿⠛⠉⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠢⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⢁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠈⠒⠦⣄⡀⠀⢀⣀⣀⡠⠤⠤⠒⠚⠉⣁⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠈⠉⠁⠀⠀⣀⣀⡠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠣⠤⠄⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⣠⡆⠀⢀⠆⠀⢀⠄⠀⡀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠒⠒⠀⠁⣹
⠀⠨⢰⣤⢠⠀⠀⠒⠐⡭⠛⠀⠀⠘⠃⠒⠈⠀⠀⠠⠀⠀⠀⢠⡾⠛⠛⠚⠀⠀⠊⠀⠀⠲⠆⠀⠀⠀⠊⠉⣀⡁⠼
⠀⠀⠙⠁⠘⠀⠐⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠴⡮⠁⠁⠀⠀⠀⢀⡀⠀⢀⢠⠔⠆⠀⠀⠀⠀⠀⠤⠄⢒
⠀⡤⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣷⠸⠁⠀⠀⠰⠀⠐⠀⠀⠐⣂⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⢤⣤⣀⣀⠀⠀⠀⠀⠔⠸⠂⢸⠀⣆⠀⠀⠙⠳⠤⠗⠀⠀⠀⠀⢀⡠⠀⡀
⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠉⠈⠻⠛⠀⠀⡒⠂⡃⠘⠂⢸⣴⣿⠀⠀⠀⠀⠀⠀⢠⢋⠔⠡⠈⡠⠊⡠⠂
⠀⠀⢰⣄⠀⠀⣀⣀⣀⣀⡄⠀⠁⠠⡀⠀⠀⠀⠉⠉⠉⣋⠈⣣⣞⡧⠁⠀⠀⠒⠷⣶⡄⠈⠟⠶⣣⢆⣀⠞⢠⠊⢠
⠀⠀⠈⢿⣯⠁⣯⣹⠁⠣⠃⠀⠀⠀⠈⠁⠀⠀⠈⠻⢷⣾⣮⠿⠈⠀⣤⡄⠀⠀⠐⠁⠙⠃⠀⠀⠈⠈⠛⠛⢷⡶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠁⠀⠀⠈⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠠⠀⠐⡒⠀⢹⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣭⣿⣷⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⠊⠀⣄⠀⡴⠤⠴⠗⠂⠛⠀⠀⢀⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣿⡟⣛⣿⣩⣿⣧⣿⣿⣿⡿⣫⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⡃⡀⡤⠤⠼⠒⠀⠀⡠⠚⠁⢀⡜⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⡟⣟⣛⣹⣿⣿⣿⣻⣵⣿⡿⠁
⠀⠀⠀⠀⠀⠀⡷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⡿⠛⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠉⠒⠒⠒⠒⠒⠉⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣭⣿⣿⣿⣿⣭⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⣀⣀⣀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⢦⡀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠄⡀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⢠⣿⣬⣿⣦⣧⠀⠀⠀⠀⠋⠀⠈⠐⠓⠒⠒⠉⠉⠁⠀⠀⢱⣄⡀
⠀⢲⡻⣷⣿⠀⠀⠀⢀⣀⣠⣿⠀⠀⠀⢸⣾⣇⣿⣼⣅⣿⡾⠀⠀⠴⢄⣀⣀⡀⠀⢀⣀⣀⣠⣴⡆⠀⣾⠇⠁
⠀⢻⣭⣻⣢⡦⢰⡲⣯⡳⣽⡏⠀⠀⣰⢾⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠈⠙⢿⣇⢸⡿⠿⠿⣿⣦⠸⡟
⠀⠀⠙⠻⣿⣷⣶⣿⣮⣽⠟⠀⠀⠀⠈⢢⡻⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢺⠹⣸⠀⠀⠀⢌⡏⢠⠇
⠀⠀⠀⠀⠈⠻⠿⠿⡿⠃⠀⠀⠀⠀⠀⠀⣼⣩⡭⡯⠍⣹⡄⠀⠀⠀⠀⠀⠈⠈⠢⠋⠀⠀⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠁⠐⡘⠀⢤⠀⠀⠀⠀⢰⠋⠛⠤⡷⠞⠋⢳⠀⠀⠀⢀⢾⢀⠀⠀⠀⠀⢀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠀⠀⠀⣧⠀⠀⠸⡧⠀⢠⡿⡟⣿⡄⠀⢠⡴⣻⣭⣵⡽⢿⣷⡆
⠀⠀⠀⠀⣀⡄⡰⠁⢀⢂⣤⣀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠘⠷⣷⠷⠃⠀⣵⡞⣿⣿⡟⣿⣯⠻⣿
⠀⠐⣿⣶⣌⣿⡇⠀⠀⣹⣏⣠⣶⡷⠀⠀⢀⠀⠀⠀⢆⡀⠀⠀⢀⢀⢀⠀⠀⠀⠀⢠⣁⣾⣿⣧⣿⣻⢄⡿
⠀⠀⠈⠉⠋⢻⣿⠀⢠⣿⠏⠙⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠋⠾⣼⣿⣷⣠⡀⠀⢸⡿⣽⣿⣿⣿⠏⣼⠁
⠀⠀⠀⠀⠀⠘⣿⠃⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣛⡀⠀⠀⠤⢤⣴⡏⠀⠀⢠⡷⣱⡿⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠱⣥⣥⠇⠀⠀⠀⠀⠀⠀⠀⠀⠬⢭⣭⠄⠀⠀⠀⠀⢸⡇⠀⠀⠸⣟⣫⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⣸⣽⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠃⢸⣿⡇⠀⢈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠁⠀⠀⠂⢸⣿⠃⠀⢸⣀⡈⠻⣦⡀
⠀⠀⠀⠀⢠⡀⠀⠀⢀⡤⠖⠋⠉⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢞⡁⣿⣦⠀⠀⠀⠀⣿⠀⣆⢸⣿⣇⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠱⣠⡞⠉⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⣠⡴⠋⠀⣿⢀⣿⡏⠀⠀⠀⡀⠙⠀⣿⢸⣿⣿⡿⠀⡰⣻⠏
⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣷⣦⡾⢻⡁⠀⢸⢀⣿⢸⣿⡇⠀⠀⠀⣧⠀⠀⣿⢸⡟⠋⣠⢞⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢠⡀⢈⠃⠈⢿⣷⡈⢷⡀⢸⢸⣿⢸⣿⠃⠀⠀⢠⣿⠀⠀⡏⠉⣠⢞⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⠀⠲⡀⢰⡁⠈⢳⡀⠈⣿⠛⠛⠻⡌⢸⡇⢸⢹⠀⠀⠀⢸⡟⠀⣀⣴⣫⠗⠁⠀⠦
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣌⠀⠱⣄⠀⢳⡄⠀⣳⣤⠟⠀⠀⠀⠹⣄⠃⠀⡾⠀⠀⠀⣈⣤⣾⠟⢋⣠⠴⠒⠒⢿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡷⣄⣙⣦⡴⠟⠛⠋⠀⠀⠀⠀⠀⠀⠘⣆⠀⠃⣠⣴⡾⠟⠉⢰⣴⠋⢀⡞⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠛⠁⠀⠀⠀⢠⠃⠀⠘⡀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣀⣳⠀⠀⠀⡆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣠⡤⠶⠖⠛⠛⠒⠒⠛⠾⣇⠠⢸⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢞⣉⣉⡛⠛⠛⠉⣉⣁⡀⠀⣤⣭⣭⣉⣉⣭⣤⣶⡆⢀⣴⣾⣿⡿⢃⣤⣿⠠⢼⢷⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⣩⣤⣌⡛⠿⠿⣷⣄⠀⢻⣿⣿⣶⣌⠻⣿⣿⣿⣿⣿⡿⢡⣾⣿⣿⣿⣴⣿⣿⣿⣤⡞⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣧⡀⠛⠛⠿⠿⣿⣿⣿⣿⣷⣄⠻⣿⣿⣿⣷⣿⣿⣿⣿⠟⣀⣤⣴⣾⣼⣿⣿⡿⠿⠿⠟⢃⣼⣿
⠀⠀⠀⠀⠀⠀⠸⢽⣿⠛⠛⠛⠶⠶⢦⣍⡛⠿⣿⣿⣧⣉⠋⣡⣴⣤⣤⣤⣅⠈⢋⣤⣴⣶⣶⣶⣶⠆⠀⢀⣴⡿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⡀⣿⣀⠀⠀⠰⠀⡖⢩⣙⠳⣦⣈⠙⠻⠷⣦⣤⣬⣍⣉⣡⣴⡿⠿⠛⠛⣉⣁⣤⠶⢞⣻⣿⢣⡿⢁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡘⡛⠷⠶⠦⣤⣥⣸⣯⡗⡆⠙⠛⠶⠶⣤⣬⣭⣭⣭⣭⠴⠶⠖⠛⠛⠉⠉⣤⠰⠞⣢⡾⠋⢠⡄⠳⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢄⣌⠒⠲⠬⠙⠻⢶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⠶⠚⠉⣄⠃⠀⠌⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠐⠂⠬⠒⠊⣙⠳⠶⢤⣤⣄⣀⣀⣠⣤⣤⠤⠶⠖⠛⠉⢁⡀⠈⠁⣀⠊⠂⡀⠲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡄⢀⠐⠀⠀⠈⢛⣀⠀⠀⣤⢀⡈⠩⢐⢠⠐⢠⡀⢀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠈⠉⠃⠀⠈⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣠⣤⣶⣶⣾⡿⠟⠛⠿⠿⠟⠻⡿⡿⣿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⣠⠊⠉⠟⢋⣿⠿⡿⠿⠿⠻⢻⣿⠉⠀⠀⠀⠀⠀⠀⣈⠥⠀⠒⠛⠉⠁⠀⡀⢀
⠀⠀⢸⣷⣿⣿⣳⣿⠋⡞⠁⣠⠆⣴⠎⢠⠆⣠⣂⠀⠤⠒⠈⠀⢀⣀⠠⠀⠀⠀⠀⠀⠀⣈⠁
⠀⠀⠀⠉⠛⢿⣿⣿⣶⣄⣼⣇⣪⣼⣶⣿⣿⣿⠟⠁⢀⡀⢰⢋⣄⠄⠀⠀⠀⠀⠀⠀⡰⠁⠀⣃⣆⡃⢂
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠉⠀⠼⣏⠁⡄⠀⠀⠀⠀⠀⠀⠰⢃⣄⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⠀⠠⠀⡀⣰⠆⠁⠀⠑⠀⡴⠆⢉⡀⣤⣤⣀⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠁⠀⢿⣀⠀⣨⡴⢷⡀⠁⢀⡤⠀⠀⠛⡛⠷⠾⣿⣿⣉⣀⣀⣠⣄⡀⠤⢶⣤⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣫⡍⢙⢻⡄⠀⠷⢡⣷⡟⢩⠧⢴⡖⢈⢁⠠⠔⠀⠂⢉⣫⣙⣷⣾⣿⣟⡋⣴⣴⠦⠓⢒⠼⠀⢠⡟⠿⢶⡄⠀⢦⡴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡀⠀⠀⠙⣿⣷⣄⡈⠺⠿⠿⣿⠻⣜⠖⣬⡾⣿⣛⡯⣝⠻⣤⣅⠀⠒⠂⠂⣁⣠⣶⣏⡄⡈⠎⢀⣰⣿⡿⡾⠂⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠂⠀⠙⠫⢿⣓⣀⣁⢀⣀⣴⡙⠯⠾⠾⠾⠛⠋⠉⠉⣀⣠⣴⣶⣿⠏⠹⠚⠋⢁⣠⣴⡿⠧⠒⠋⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣄⣀⣀⣉⣁⣀⣀⣤⢤⣴⢶⠾⠿⠿⠻⠛⠛⣉⣁⣠⣤⣤⣶⣶⠿⠟⠛⣉⣡⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠘⠶⠤⠴⠶⠖⠒⠚⠛⠋⠙⣯⣉⣉⣡⣬⡤⠴⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤⣶⡶⣿⠛⠉⠉⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠈⠁⠀⢰⠃⢠⠊⠁⠈⠑⠦⡈⠑⢄⡀
⢠⠖⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⠈⡆⠀⠙⢆
⡏⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢇⠂⠀⠀⠀⢀⠞⢀⠞⠓⣄⠑⡄
⢻⣄⠀⠀⠾⠷⠶⠦⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠙⠲⠤⣤⠏⣠⠋⠀⠀⠈⢦⠘⣆
⢀⡟⠃⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⠉⠒⠢⠤⢍⣉⡐⠒⠒⠚⠀⠘⡄
⣾⠀⠀⠀⣼⠛⠛⠻⠶⠶⠶⠶⠶⠶⠤⠤⣤⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⡀⢳
⠘⠷⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⢄⠀⠀⠀⠀⠀⠀⠀⠱⡌⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⡦⢄⣀⠀⠀⠀⠀⠙⣄⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠀⠀⠀⠀⠀⠈⢣⡘⢦⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠙⢦⡙⠦⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢾⣦⣤⣀⣀⠙⢦⡈⠓⠦⣄⡜⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⢿⣷⣤⣀⣠⠏

⡗⢺⣆⠧⠔⠀⠀⠀⢀⡜⡏⣿⠋⢀⡴⢾⣟⣭⣿⣿⣿⣧⠀⠀⠀⢹⣧⣄⠀⠉⢳⡄⠀⠀⡇⠀⡃⠀⣇⠀⡃
⠟⠊⠁⠀⠀⠀⠀⠀⠈⢷⣿⣯⣤⢻⠽⣴⣻⣮⣿⣿⣿⠏⠀⠀⠀⠀⠉⠙⠻⢿⣿⡇⠀⠀⢧⣄⢰⡀⠱⡀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠊⠓⠒⠚⠻⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠁
⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⣀⣤⣄⣀⠀⠀⠀⠀⣴⣶⣾⣭⣴⣟⠙⢫⡿⠀⠀⠈⠂
⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡌⠃⠀⣴⣿⣥⠙⢦⣭⣏⡘⣆⠀⠀⠉⠻⠏⠿⠯⠽⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⣠⣾⣧⣿⣷⠀⠀⠀⠀⠀⢰⣿⣟⢿⣧⡄⢸⣿⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⡀⠀⠀⢀
⢠⡞⢹⣿⡯⢞⠛⠀⠀⠁⠀⠀⢸⣏⠛⠊⠉⠸⠿⠁⠀⠀⠀⢀⠠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⣀⡿
⣇⢨⡴⠋⠂⠀⠀⠠⠁⠀⠀⢠⣟⢁⡀⣠⡼⠃⠀⠄⠀⣠⣬⣶⣤⡾⢳⡖⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⢮⣾⠃⠀⠀⣾⡜⢻⡿⠉⠢⠀⠀⠀⢰⡟⠀⠿⠋⣼⣷⣶⣏⠽⣷⠄⠀⠀⠀⠀⢠⠏
⣠⣤⠖⣦⠠⣾⠇⡏⠀⠀⠐⢯⢕⡈⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠘⢻⣿⡏⠷⠴⣯⠘⢦⣀⠀⣰⠏⢀⣀⣤⣤
⢏⠁⠀⣀⣠⣡⡟⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢢⡀⠀⠉⢯⣀⣘⣃⠛⢻⣯⣄⣘⠿⣷⡒⣨⣿⠋⢠⣾⡟⠛⠻
⡄⢀⢈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠈⠛⠯⣽⣏⣚⣷⣶⣿⡿⠟⠁⢠⣿⡅⣳⠆⢂
⣶⡿⣫⣿⠏⠀⠀⢆⠀⠘⡄⠀⢢⠀⢃⠘⡄⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⣿⣱⡀⣠⣼
⣸⣿⠟⠁⠀⠀⠀⢠⣡⠀⣇⡀⣸⡆⢨⡄⣱⠀⠀⠀⠰⡈⠿⣷⠶⣦⣄⣀⣀⣀⣀⣤⣤⡀⠀⠀⢿⣿⣆⣩⡌
⠋⠁⠀⣤⡀⠀⣶⡛⢿⣶⡉⡙⡋⠙⣋⣻⣿⠿⠇⠀⠀⠈⢷⣍⠓⢮⣭⣭⣭⠷⣶⣶⣈⣿⣦⠀⠈⣙⣛⣩⣵
⠀⠀⣴⣹⡿⠀⣻⣧⠀⠛⠿⠷⠲⠾⠛⠿⠃⠀⠀⠀⠀⠀⠀⠙⠳⣔⣏⠉⠈⠒⠈⠛⠻⣿⢿⣶⡄⡀⠈⠉⠉
⡼⠋⢉⣽⠁⣰⠋⠋⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢇⠀⠀⠀⠀⠛⠀⣰⡏⣿⡄
⣤⡶⣼⡇⠀⣿⡒⠠⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡀⠀⠈⢳⡀⠀⢀⠐⣊⣼⣶⡿⠁⠀⠀⣴
⣸⢿⣿⠁⠀⠘⢿⣄⣉⣈⠻⢀⣄⣠⣠⡭⡴⠾⠋⠀⠀⢠⣶⡜⠺⢿⣷⡦⠀⠹⣗⣧⣌⣤⠾⠟⠀⠀⠰⠘⣿
⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⢉⡀⠀⠀⠀⠀⠀⢀⣇⠁⠀⠈⠁⠁⣝⣧⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙
⠄⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⣖⣴⣾⡿⢻⣿⣿⣷⡶⣄⠈⠋⠙⢹⣄⡀⠔⢦⡾⠇⠀⠀⠀⡠⠂⡠⠀⡀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⣶⠶⣿⠛⠉⣉⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠁⠀⢰⠃⢠⠋⠀⠈⠑⠦⡈⠑⢄
⣠⠞⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⠆⠀⠑⢄
⣇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠃⠀⠀⠀⠀⠀⠀⠀⢸⡀⢣⡁⠀⠀⠀⣠⠋⡰⠋⠳⡄⠳⡀
⠻⣆⡀⠀⠛⠓⠒⠒⠒⠒⠲⠶⠶⠦⠤⠤⠤⢤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢉⡓⠢⠴⠁⣜⠀⠀⠀⠘⡆⠹⡄
⢠⠏⠁⠀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢪⠈⠉⠒⠒⠬⠭⠉⢉⣉⡉⠀⢱
⢿⡀⠀⢀⡿⠉⠉⠛⠛⠛⠛⠛⠛⠓⠒⠒⠶⠶⠿⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠈⠛⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣵⣄⠀⠀⠀⠀⠀⠀⠙⣆⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡲⠤⠀⠀⠀⠀⠈⢦⡙⢆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠱⣌⠳⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣠⡀⠀⠈⠳⣌⠓⢦⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⠶⢦⣬⣓⣤⡈⠑⠚⢁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢒⢒⢚⡓⠒⣒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣩⡇⣾⢸⣿⡿⠋⢠⣔⠮⢭⣓⣂⣉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⢼⣸⡿⠛⠙⠛⠋⠀⣴⣆⠨⣛⡳⠶⣭⣭⣭⣈⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣥⢿⠉⠀⢠⡘⣮⡻⣝⢷⣝⢾⣿⡆⠙⠻⣿⡷⠶⢶⣬⣥⡘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠛⠛⢀⣴⣮⡷⣝⢮⡙⢮⣻⢮⣛⢿⣷⢐⡦⠀⢩⣛⣛⠶⠶⣯⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣦⠳⣦⠸⣿⣿⡿⣮⡳⢭⡳⢭⣛⠿⣿⣿⡇⣽⣛⡦⠙⢯⣝⡻⢶⣄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢬⣷⣝⢧⠘⢿⣟⢮⣝⠷⣭⣳⢮⡅⣒⠯⣵⣓⠾⣭⣗⠈⠩⣛⢷⡅⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⡻⣝⢷⡄⠈⢿⣷⢭⣛⠶⣭⡿⠀⣭⣛⠷⢯⣝⡾⠁⠀⠀⢩⡳⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⢮⡟⢠⠀⣶⣝⠷⣭⣛⠾⣍⠑⠶⣭⡛⣆⢬⣙⠳⠄⢀⢶⢡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠦⢠⡄⢨⡳⣯⡻⢦⣝⡳⣏⣝⠳⢦⣽⢧⢈⡛⢷⣤⡘⢡⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢡⣷⡇⢠⣝⢾⣝⡷⣬⣻⡧⣍⡻⣦⣍⣾⢠⣙⢷⡍⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⡇⠈⣿⣷⣬⡛⢾⣝⡓⣮⡻⢦⣽⡇⣮⡻⠇⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣿⣿⠇⣠⠹⣿⣷⣝⡳⣬⡛⢦⡙⢦⡙⣷⡝⠃⠀⡄⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⣷⡏⠰⣿⣧⡹⣿⢾⣟⢦⡙⢦⡻⣦⣻⡜⠃⢠⡆⣧⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⢮⠿⢠⡙⣮⡳⣿⣦⠁⠙⠃⠙⠂⠙⠈⠁⢀⣰⡟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡘⡎⢁⣲⣝⢮⡻⣎⠓⠀⠉⠁⠀⠀⠀⣀⣴⢸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢁⣼⡟⢀⢾⣳⣽⣮⣿⠋⠀⠀⠀⠀⠀⣴⢸⡏⣿⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⣋⡰⣎⢷⡽⣿⠟⠠⠊⠀⠀⣴⢸⣧⣿⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣙⠃⠻⠚⠛⠉⣀⢠⣤⢸⡇⣿⣼⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢸⡆⣦⣶⡧⣿⢸⣿⣽⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠿⠿⠷⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⣐⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠀⢠⠂⠀⠀⠀⠀⠀⠂⡈⢉⡀⡅
⠀⠰⣳⠆⡆⠀⠈⠈⠚⠁⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠠⠾⠋⠉⠉⠀⠀⠁⠀⠈⠛⠀⠀⠀⠀⠠⠀⠐⠂
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⢰⡄⠠⡐⠚⠐⡀⢀⠀⠀⢀⠀⢉⡅
⠈⠋⠀⠀⠀⠀⠀⠀⠸⣤⣀⠀⡀⠀⠈⠀⠀⠀⠀⡀⢀⠀⡄⣀⠀⠀⠻⣌⣀⡄⠀⠀⠀⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠙⢻⡷⠂⠀⠠⠄⠄⠤⠠⣆⣼⡄⠀⠀⠀⠁⠀⡠⢀⠔⡠⢋⠄⢡⠀⠁
⠀⠀⢤⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠐⠓⠒⠂⠰⣠⣿⠙⠀⠀⠤⢦⣤⠀⠱⢧⢎⡀⢀⠔⢁⠂⡀
⠀⠀⠸⣿⡗⢻⣫⡏⠫⠃⠀⠀⠀⠑⠀⠀⠈⠶⣦⣧⣼⠯⠉⣠⣄⠀⠀⠐⠉⠛⠂⠀⠈⠘⠻⠲⣧⣴⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣤⠀⢴⡂⣦⠄⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢤⠀⠀⠀⠐⠲⡒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⣄⠁⠎⢎⠈⢥⣖⠆⠀⠀⠀⠀⠀⠀⡏⢠⡠⠔⠀⢀⣡⢄⠊⢣⡀⡘⢦⡀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡤⠡⣶⣆⣁⠀⣜⣹⢵⣆⠀⠀⠀⠀⢸⠇⠀⢙⣦⠚⢻⡀⢈⡷⡖⠉⢦⡀⠕⢦⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⠛⢹⣽⣥⢠⢈⢠⠈⠟⢆⠀⠀⠀⢃⢠⠀⠀⠘⣧⠔⠛⢧⢀⡼⢶⡉⠙⢦⡠⠓⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⣿⣿⠚⠞⣨⢄⡸⢇⢀⠀⠈⢀⣵⢄⠈⠀⢳⣤⠚⠹⣄⢀⠵⠎⠁⢁⡤⠮⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡷⠛⠉⠉⠁⢤⡀⢹⣉⡉⠁⠈⠃⠀⠈⠇⡠⠑⠠⢤⣌⣁⣐⣊⣃⣠⠤⠖⠁
⠀⠀⠀⣠⡶⣤⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⢁⠀⣀⠀⠻⢠⠬⡥⠼⠁⢀⡀⣨⠀⠀⠀⠀⠀⠀⠢⠀⠄
⠀⠀⠀⠙⠀⠀⠀⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⣿⣯⡀⠀⠈⠀⣸⣷⢗⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣀⡠⢀
⠀⠀⠀⠀⠳⠞⠚⠛⠛⠓⠦⣤⣖⠓⠒⣂⡶⣄⡀⠁⠑⢤⡀⠀⠀⠀⠰⣌⠿⠎⡨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⠋⠐⠊⣭⣷⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡞⣻⠟⠻⠮⠇⢮⡱⣷⠃⢸⡷⣮⣙⡢⠔⠂⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢿⠉⢿⡂⠰⠀⠀⠻⠀⣷⠆
⠀⠀⠀⠀⠀⠀⠰⣜⡈⣥⣬⡁⢀⡌⢛⡵⠿⠀⡾⢛⣤⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣛⣣⡖⣥⢾⠎⢷⡛⠀⠀⠀⢠⣽⡦
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠑⠳⣅⢶⣷⣟⣂⠀⠀⠀⠛⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⢉⣉⣹⢇⡺⠛⠁⠀⠀⠹⠀⠀⠀⠈⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠡⣝⠾⢦⣐⠟⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⣡⠶⣋⡥⠞⡉⠀⠀⠄⠀⠀⠀⠆⠀⠀⡅⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠳⠟⠒⠈⠀⠀⠀⠀⢀⣀⣄⣒⣋⡤⠄⣂⣥⠖⠊⠀⠀⣾⡇⠀⠈⢀⡀⢀⠀⢀⡀⠴⠀⢠⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣶⢶⣟⠋⠉⢉⣠⠶⠚⠉⠀⠀⠀⠀⢀⣾⡿⠃⠀⠈⠻⣦⠀⠀⠀⠠⡄⢀⡖
⠀⠀⠀⠀⠀⢀⡤⠤⠀⠄⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢦⣾⣋⣀⣀⠀⠀⠀⢀⠀⣴⣾⣿⣷⡀⠀⠀⢠⣿⡦⠀⢠⠜⣠⣿⠁
⠀⠀⠀⠀⠀⣿⣁⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⢿⣿⣿⣷⣽⣟⣿⣭⣵⣗⠚⠟⠲⠿⣾⣿⣥⣚⠛⠛⣀⣉⡥⠞⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠛⠷⢮⣿⣟⣛⣛⣉⡵⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢋⡉⠉⠙⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠚⠉⠉⠙⠲⣤⡀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⠔⠃⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⣴⠞⢉⡥⠄⠀⠬⢉⡛⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢻⣧
⠀⠀⠀⠀⠀⣾⠃⣰⠁⠀⠀⠀⠀⠀⠈⠁⠚⢿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⠴⣫⠵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿
⠀⠀⠀⠀⢸⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠓⢬⡉⠉⠛⠛⠛⣛⡻⠝⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣿
⠀⠀⠀⠀⢸⡏⡄⢻⡀⠀⠀⠀⠀⣴⠟⢉⡤⣍⢷⡄⠀⠹⡄⢀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠱⠀⢳⡀⠀⠀⢸⡇⡀⠁⠀⢸⡟⣧⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠸⣷⠀⢁⠀⢳⡀⠀⢸⡇⡇⠀⠀⣸⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣼⡀
⠀⠀⠀⠀⠀⢻⣇⠀⠱⡀⠳⣄⠈⢷⣙⠶⠞⣡⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡞⣿
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⠢⡘⢦⡀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⠎⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠉⠢⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣠⠞⠁⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠈⠺⣝⠒⠶⠖⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢁⠤⠊⠁⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠑⠦⢄⣀⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⢊⣡⠴⠚⠁⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠈⠉⠑⠚⠭⠍⣛⣛⡛⠛⠛⠋⠉⠉⠉⣉⡁⠤⠖⠚⠉⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⣶⣶⠶⠶⠶⠶⠾⠛⠛⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠴⠶⠖⠒⠒⠒⡒⠒⠛⠛⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⢀⠀⠀⠙⠀⠙⠲⠶⢤⡀
⠀⠀⣀⣀⣤⡤⠴⠶⣛⢁⣠⡴⠒⠀⠀⢦⡀⠀⠀⡀⠀⠉⣄⢀⡴⠂⠠⡀⠀⠀⢈⡷⣀⠀⠀⣹
⢰⡏⠉⠁⠠⡀⢀⣠⠾⠛⢅⠀⠀⠀⠀⠀⠙⣦⠞⠁⠀⢀⠄⠙⠀⠀⠀⠈⠲⠔⠋⠀⠈⠃⣠⡟
⠸⣇⠀⠀⠠⠞⢫⣅⠀⠀⣨⠷⢯⡀⠀⠀⠐⠉⠓⢤⡘⠁⠀⢀⣀⡀⣐⣋⣀⣀⣠⡤⠶⢋⣽⠃⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⡀
⠀⢻⡗⠶⠦⣤⣀⣈⣳⡎⠀⠀⠀⠈⠂⠀⠀⠀⣀⣤⠴⠞⠋⠉⠉⠉⠉⢉⣉⣉⣀⣠⡴⠞⠁⠀⢀⣀⣠⡤⠖⢫⣴⣾⣶⣬⣭⣝⣒⢒⣓⠲⣄⡀
⠀⠀⠙⣦⣀⣈⠉⠉⠙⠙⠛⠻⠛⠛⠛⠛⠛⠛⠉⠀⣀⣠⠶⠟⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⣠⠞⣫⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⣽⣿⣦⣝⢦⡀
⠀⠀⠀⠈⠛⠿⠦⠤⣤⣤⣦⣠⣤⣄⣀⣠⡤⠶⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠘⠿⢿⣟⠻⠿⠿⡻⠛⢻⠟⢋⣥⣶⣶⡆⢉⣿⣿⣦⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⣻⣧⣴⣿⣿⣿⣿⠟⣹⣷⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⢺⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣶⣦⣛⠛⢯⣵⡿⠷⠶⠶⣿⠛⢿⣿⠷⠹⠿⡿⣿⠿⢿⣿⣿⡇⣼⣿⠟⢉⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⡆⠉⠈⣧⠈⢻⣶⡾⠚⠋⢹⣿⣿⡏⠉⠉⠛⠙⠛⢷⣾⡶⠲⠾⠷⠶⠞⠒⢻⣿⠏
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⣰⡏⠀⣾⣿⣏⠀⠀⣸⣿⣿⣧⠀⠀⠀⠀⠀⢨⣿⣿⣷⠀⠀⠀⢀⣠⠿⠋
⠀⠀⢀⡴⢛⣭⣭⡉⢉⡛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠛⠉⠛⠛⠿⠿⠷⠶⠶⠿⠿⠿⠿⢤⣤⣤⣤⣤⠼⠿⠿⠿⠒⠛⠉⠉
⠀⣴⠏⣰⣿⣿⣿⣷⢸⣿⣤⣀⣉⢛⣓⣒⡒⠒⠒⠒⠒⠒⠲⠶⢦⣤⣀
⢸⡏⣴⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⡿⠛⢡⣾⣿⣿⣿⣿⣶⣶⣤⣌⣉⡛⢦⡄
⣾⢡⣿⣿⣿⣿⣿⣿⡄⠈⠛⠛⠋⠉⣤⣶⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⣋⣀⠙⣦
⣿⠘⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣤⣴⣶⣿⣿⣿⠟⠀⣿
⢿⡀⢻⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢉⣁⣠⣴⣾⣿
⠘⣷⣄⠙⢿⣿⣿⣿⠟⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣡⡶⠚⠉⠉⠉⣉⣿⠏
⠀⢹⣿⡷⠶⣤⣉⡀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⢋⣡⣶⡏⠁⠀⢀⣠⡴⠚⠋⠀⠀⣠⠴⠚⠋⠉⠉⠙⠒⠦⣄⣀⠀⣀⣀⣀⠀⠀⣀⡤⠴⠦⣄⡀
⠀⠀⢿⣇⠀⠀⠈⠙⠓⠶⠤⣭⣭⣭⣥⣤⣶⣾⣿⣿⣿⣷⣠⠶⠋⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠙⠳⢤⡀
⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢠⠟⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠈⠛⠶⢤⣀⣀⠀⠀⠀⠀⣀⣠⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠠⠁⠀⠐⠤⡶⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡴⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠋⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠈⠙⠲⢤⣄⡀⠀⣀⣀⣀⣤⠤⠴⠶⠒⠋⠉⠁⠀⢀⣤⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣤⡀⢀⠀⣠⣤⠤⠞⠛⠉⠁

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⣶⡶⠶⢶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⢀⠔⠉⠟⣻⡷⣶⠾⠿⠟⣿⠋⠁⠀⠀⠀⠀⡰⠄⠐⠚⠉⠁⠀⢀
⠀⠀⢿⣿⣿⣹⣟⠘⠁⡤⠣⡞⢠⣆⣴⣄⠀⠊⠁⣀⠀⠄⠂⠀⠀⠀⠀⡠
⠀⠀⠀⠈⠙⣿⣿⣿⣾⣷⣶⣿⣿⣿⠟⠁⢠⠄⢨⣵⠄⠀⠀⠀⠀⠀⠘⡀⠀⠜⠒⡰
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠋⢀⠀⡀⠀⢈⡑⠀⠠⠀⣀⡔⠄⢀⣑⠑⣀⣀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡏⠀⣿⡀⢀⡨⣯⠐⠀⣠⠀⠁⠶⠾⠾⢿⣿⠋⣁⣀⣀⣀⣠⣤⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢫⠉⣛⡇⠀⠲⣱⣾⠫⠧⢴⣂⣉⡠⠐⠐⠊⣿⣽⣷⣿⣿⣓⠵⡴⠔⢓⠼⠂⣸⠻⡷⡄⠀⣤⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡀⠹⢿⣦⣄⡛⢛⠻⠛⢶⣲⡶⡻⣭⣷⣜⣳⠦⠀⠉⢉⣠⣴⣾⣧⠀⠈⣀⡼⢻⡳⢃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⡀⠙⠛⠢⠤⠖⠛⠊⠛⣉⣉⣁⣤⣤⣴⠶⠿⠛⠋⢈⣁⣤⣴⡾⠏⢉⣁⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠚⠛⢻⣉⣉⣉⣭⠭⠤⠤⠶⣖⠲⠿⣛⣛⣉⠥⠤⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⢠⠚⠙⠦⠦⣄
⣇⡀⡀⠀⠀⠘⠲⢦⣄
⠈⠉⠳⠤⢤⣀⠀⠀⠈⢙⡶⠶⠶⠶⠶⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠈⠓⢦⣤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠪⠀⠀⠀⠀⠀⢀⣀⣀⣈⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣉⡭⠤⠤⠤⠬⢍⣙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⢀⠀⠀⠀⠀⠀⣠⠔⣉⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⣟⠋⠳⠀⠀⢀⢜⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠹⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⢸⡄⠀⠀⢠⠯⠋⠀⢀⡤⠤⠤⣄⠀⠀⣠⠒⠉⠉⠉⠒⠦⡀⠀⢹⢸⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⢏⠂⠀⡴⠋⠀⠀⠠⠆⢱⠐⡧⠀⠄⠀⠀⠀⠀⢱⡀⢸⣾⡇⠘⢦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡦⠀⠀⣿⠀⡎⡆⠀⡜⠀⠀⠐⠀⣠⠞⣉⣄⡈⢳⡀⠀⠀⠀⠀⠀⡇⢸⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠞⠁⠀⠀⢸⢰⢳⠁⢰⠁⠀⠀⠀⠀⡇⢰⠁⠀⡽⠀⠇⠀⠀⠀⠀⡸⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⢸⠀⠀⠀⠀⠀⢣⠀⠑⠊⠁⡴⠀⠀⠀⠀⡰⢁⣜⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢸⡀⠀⠣⡀⠀⠀⠀⡸⠂⠀⠀⠸⡁⠀⠀⣠⠎⢠⢎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢣⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠉⠛⠉⢀⡔⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠑⢄⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠔⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣌⣑⣒⠢⠤⠤⢔⣒⣒⡭⠴⠋⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀⠀⣀⣠⠤⠒⣢⣶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢀⣠⣾⡥⠖⣶⠋⠁⢀⣾⣡⡴⢿⡿⠃⠀⣀⣿⢦
⠀⠀⠀⠀⠀⠀⠀⣸⠗⣻⠋⠉⠀⣠⣞⣡⡤⣶⠟⠋⠁⣀⣞⣠⡶⢿⠟⠠⠈⠇
⠀⠀⠀⠀⠀⠀⠀⣽⣟⣁⡤⣴⠞⠋⠉⢠⡶⢃⣊⣴⠞⠛⠋⠀⣰⣿⡤⠶⣶⢾
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡤⠚⠁⠀⢀⡶⠕⠚⢁⠟⠁⣀⡠⢴⠞⠁⠀⠀⡰⠋⣼⡇⠠⡀⣀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣦⣰⠞⠁⠀⢁⣴⣧⣶⣥⣤⣴⣁⣀⣠⣤⣞⣴⣾⣿⠁⠀⠀⠀⠀⠀⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣟⣫⣽⣽⢻⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠿⠋
⠀⠀⠀⠀⣀⡤⣠⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⢲⠖
⠀⠀⠀⠀⠙⠷⠟⠃⠀⠀⠀⢰⡀⠀⣴⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠶⠒⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠠⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠁



⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⣄⠴⠗⠒
⢘⡁⡄⠀⠐⢀⡚⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠐⠊⠉
⠈⠀⠘⠈⠀⠈⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⣀⡀⠀⠈⠁⠀⠀⠐⣀⠀⠀⠀⠀⠠⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠀⠘⠙⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠂⠠⠀⠀⠀⢀⠀⠐⠤⢀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠀⠠⠂⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠄⠂⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠈⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠉⢁⡀⠀⠀⠀⠀⠈⠀⠀⠸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠈⠀⠀⠀⠀⢁⡀⠀⠒⠒⠒⠂
⠀⠉⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣀
⠌⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠚⠄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⡼⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠁⠀⠀⠄⠀⢀⢀⠰⠓⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠁⠀⢐⡼⣾⠉⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠁⠀⠀⢩⡅⠀⠐⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⢀
⠀⡸⠖⠛⢀⠈⠂⠠⠗⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⡈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠎⡀⢀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠄⠒⠢⣄⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠐⠚⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠁⠀⠀⠀⠀⠐⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⠠⠖⠊⠁⠀⠀⠈⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠒⠀⠀⠀⠀⠀⠠⠂⠀⠀⠘⠀⠀⠂⠀⠀⢀⡀⠨⡶⠋⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠤⠤⠀⠊⠀⠀⢴⢀⠀⡀⠀⡁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣀⠀⠀⠠⣴⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⠀⢂⠀⡟⠉⠄⢉⠀⠈⠀⠀⠀⠀⠀⢀⡄⠐⢀⠁⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠄⠀⠂⠀⠁⡀
⠀⠀⠀⠀⠀⠶⠄⠂⠀⠁⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣟⡒⢤⣁⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠈⠄⠈⡀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢁⠒⣌⡔⠛⢁⠀⣘⡌⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠆⠀⢠⠖⠀⠀⠀⠀⠀⠉
⠀⠀⠀⣀⣀⡠⠤⠶⠆⠁⠀⠀⠀⠀⡠⠀⠁⡀⢀⣼⠋⠣⡀⠀⡳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠀⣄⠀⡆⠰⠀⠀⠐⡄⠰
⠠⠀⠈⠀⠀⠀⢀⠄⠀⠀⢰⠀⠀⠀⠁⠀⠀⡿⠋⠉⠒⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢀⡤⠛⠉⠀⠀⠀⠀⠃⠐
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⢆⡠⣞⣀⠀⡄⢀⡴⠁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠁⠀⠰⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⡈⠂⠴⠓⠁⠀⠀⠀⠀⠀⠀⠱⡤⠄⠠⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⡀⠀⢀⠀⠀⠥
⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣁⣀⣸
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡗⠤⣿⡄⢿⣄⣳⠄⠀⠀⠀⠀⢀⡜⣛⢙⣿⢟⠄⠀⣠⣼⣿⡿⢫⣍⣿⣾⣿⣿⣧⠀⠀⠀⠀⠈⢻⣿⣅⠀⠀⠉⠛⣶⣆⠀⠀⠀⠸⡇⠀⠸⠇⠀⣸⠂⠄⠻
⢷⣀⡾⠓⠊⠉⠀⠀⠀⠀⠀⠀⢺⣇⣽⣾⠿⢀⠀⣾⠧⠬⣙⣓⡫⠿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠉⠻⢿⣷⣦⣤⣦⣿⣿⠀⠀⠀⢸⡁⠀⠘⣇⡀⢻⡃⠀⣷⡀
⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⣿⣿⡿⠶⠇⣽⡘⠹⡧⠤⣿⣷⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⣿⡷⠂⠀⠀⠘⠷⢦⣀⢻⡄⠀⢳⣄⣨⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠛⠒⠒⠋⠛⠻⠷⠿⠟⠿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠆⢐⣶⡞⢣⡶⢶⣶⡀⠀⠀⠐⢢
⠀⠀⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠰⡖⡄⠀⠀⠀⣠⣴⡶⣶⠲⣶⣶⣦⡄⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⣯⣥⠄⣀⣾⠟⠁⠀⠀⠀⢀
⠒⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠋⠀⢀⣾⣿⣧⣄⠉⠶⣦⣭⣿⣁⠘⣧⠀⠀⠀⠀⠉⠳⠯⠼⠷⡶⠬⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠠⠤
⠀⠀⠀⠀⣠⣶⡆⣻⣷⣤⠀⠀⠀⠀⠊⡐⠀⠀⣼⣿⡿⣿⣿⠆⠀⠀⢸⣽⡿⠀⠸⠁
⠀⠀⡠⠾⣯⣹⣷⢛⡿⣻⠀⠀⠀⡜⠀⠁⠀⠀⣿⣿⣷⣌⡻⣿⠏⣤⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠀⠒⠒⠒⠛⠓⠒⠒⠦⠄⢀⠀⠠⠒⠉⣡⣄
⢀⣾⡁⢰⡿⣼⡧⢻⡙⠋⠀⠀⠌⠠⠀⠀⠀⢸⣿⣇⠉⠉⠁⠀⡐⠛⠟⠁⠀⠄⠀⠀⠀⣀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢀⣤⡾⠋
⡯⠨⣉⣤⡞⠡⠂⠀⠀⠀⠀⠎⠀⠀⠀⠀⢀⣾⠷⠁⠀⢀⢀⣩⠟⠁⠈⢀⠄⠀⢀⢐⣨⣤⣄⡠⣶⠖⣶⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡾⠁
⠛⠓⠋⠁⠀⠀⠀⠀⠠⣠⢎⣤⠃⠀⠀⠀⣼⠛⣓⣺⡷⠾⠿⠃⠀⠀⠀⠀⠀⠀⣴⡟⢉⣿⣽⠟⣠⡤⡝⢃⣾⠟⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢿⠃⠀⠀⠀⢸⣯⡿⠈⣽⠟⠃⠈⠑⠀⠀⠀⠀⠀⣸⡏⠡⠀⠛⠁⢲⣟⣿⣿⣿⣯⣈⠻⣿⠇⠀⠀⠀⠀⠀⠠⢄⣼⠃
⠀⣠⣴⠖⣴⣤⠠⣾⣿⠣⡟⠀⠀⠀⠀⠸⣯⣥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠐⠛⣟⣫⣿⡐⠧⣤⡼⣟⡄⠙⣆⢀⠀⠀⣁⣾⠁⠀⢀⢀⣠⣤⣄
⣾⠟⠃⠀⠈⠉⠀⠛⣡⣾⠁⠀⠀⠀⠀⠀⠊⠒⠛⠁⠀⠀⠀⠐⣄⠀⠀⠀⠘⠖⠴⠆⢀⠠⣤⠤⣿⢿⣿⡇⠀⠀⢸⣿⣶⣠⣍⢙⢟⣰⡿⠃⠀⣴⣿⣿⣟⣿⣿
⢙⠄⠀⢠⣤⣦⣞⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠳⡄⠀⠀⠀⠙⠶⣤⠶⣖⣖⣈⣐⣿⣿⠗⣴⠶⠉⣽⣭⡅⣰⣾⡟⠁⢠⣾⡿⣯⠅⠉⠡⠋
⡄⠀⠀⠀⠹⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠉⠛⠦⣥⣿⣍⣛⣛⣗⣻⣿⡿⡿⠟⠉⠀⠀⣾⣯⡦⢙⣿⠁⠃⣀
⣠⣴⣿⡞⣴⣿⡿⠁⠀⠠⡀⠀⠀⠻⡀⠀⠈⢧⠀⠀⢣⠀⢦⡀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢿⡟⢹⠸⠄⢠⠀⣯
⢟⡿⣫⣾⡿⠋⠀⠀⠀⠀⠑⠤⠀⠀⢻⠀⠀⠘⡇⠀⠘⡆⠈⣧⠀⠀⠀⠀⠀⣈⠳⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⢗⢀⣿⠏⣷
⣨⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣄⢆⠀⣼⣡⠀⣤⣿⡆⢠⣾⠀⣸⡀⠀⠀⠀⠀⠹⣄⠘⠻⣟⠻⢟⣶⣦⣄⣀⣤⣤⣤⢤⠤⣤⣤⣄⠀⠀⠀⠸⣿⣿⣾⣀⣴⣦⠠
⡿⠋⠀⠀⢀⣀⡀⠀⢀⣶⠞⢿⣿⣦⡛⡙⠛⠛⡙⠻⠟⠙⢻⣿⣷⣿⡆⠀⠀⠀⠈⢻⣆⠙⠻⢦⣬⣝⣛⣛⣛⣭⣶⡶⣶⣄⠈⢿⣷⡄⠀⠀⡉⠻⠿⠛⢋⣩⣴
⠀⠀⠀⠀⡩⢞⣷⠀⠸⣿⣖⠐⠙⣿⣷⣼⣆⡛⣀⣤⣾⣺⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢉⣿⠿⢿⡛⢎⣭⠻⣿⣿⣶⣾⣿⣿⣀⠀⠈⠉⠻⢿⣿⡿⠟
⠰⢄⡰⣿⣅⢿⡟⠀⢠⣽⣿⠔⢠⠀⠙⠛⠛⠉⠛⠋⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣮⣳⣄⠀⠀⠈⠉⠁⠈⢉⡙⢿⣏⢻⣿⣿⡀⡄
⣴⠟⠋⠋⣹⡿⠁⠀⡿⠙⠋⠉⠈⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡉⠁⠀⠀⠀⠀⠈⠛⠀⠀⢘⡏⢸⣷⡈
⢉⢀⣀⡘⣿⠁⠀⣾⣧⣀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⠀⠀⢀⣠⠚⢸⣃⣾⣿⠃⠀⠀⠀⢀⣤
⠿⣷⣙⣿⡿⠀⠀⢻⣯⡀⠠⣑⡄⢀⣋⣄⠀⠀⠀⠀⠀⢀⡀⢀⠀⣀⣼⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⣀⠀⠀⠘⢷⣀⠀⢠⣤⣤⡤⠶⣻⣛⣿⠃⠀⠀⠀⢀⣾⡛
⡈⠿⣿⣿⠇⠀⠀⠀⠙⢿⣆⣄⣠⣀⠻⠟⢀⣆⣀⣤⣠⣤⣭⣼⡻⠟⠁⠀⠀⠀⠀⣾⣷⡜⠛⠳⢮⣝⣿⡦⠀⠈⠻⣟⣳⣦⣤⣠⣾⡷⠟⠁⠀⠀⠀⢷⠘⣿⠷
⠿⠶⠟⠃⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠐⠻⠍⠋⠓⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠳⠀⠀⡑⠊⠙⢂⠚⣷⣄⠈⠁⠙⠛⠁⠒⠈⠀⠀⠀⠀⠀⠈⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄⡤⢶⣤⣤⣀⣀⣄⠀⠀⠀⠀⣿⡷⢠⣄⣀⠀⠀⠂⠈⠻⠞⣿⡆
⡆⠀⣎⠁⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢘⣋⣓⣿⡿⣶⠏⣻⣿⣿⣿⢳⣿⢓⣤⡀⠘⠃⠈⠀⢷⣠⣀⠐⠈⠿⣾⠿⠇⠀⠀⠀⠀⢀⡞⠁⢀⡖⠀⢠⠂⠀⡴

⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠔⣥⣾⣷⣮⢢⡑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⣾⣿⣿⣿⣿⣿⣿⣿⣖⢄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠐⣋⠅⠊⠉⠉⠚⠀⠐⡧
⠀⠀⠀⠀⢀⠬⣤⣻⣤⡝⣧⢇⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⣏⡿⢿⣿⣿⣧⣭⣽⣷⣿⣽⣿⣸⠀⠀⠀⢀⠖⢋⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡅⠁
⠀⠀⠀⢠⢢⣶⣮⣻⣻⢋⣥⡌⢳⣄⢀⠢⡷⣤⠒⠉⡃⠀⠀⠀⡏⠙⠳⣆⠙⣻⡿⣿⣿⣿⣿⠟⢹⠀⠀⠀⡎⠀⠀⠀⠀⠀⢀⡄⠀⠊⠀⠀⠀⢀⡅⣦
⠀⠀⠀⡃⣿⣿⣿⣿⣿⣿⠿⣴⣇⣿⣾⠿⠛⠛⠓⠤⠇⠀⠀⠀⠈⠲⣄⠈⠍⣙⣿⡾⠟⠃⣁⠔⠁⠀⠀⠀⢻⡀⠀⠀⠀⡰⢃⡇⢰⠁⠀⠀⠀⢈⡿⠋
⠀⠀⠀⠈⠊⠉⢉⣻⣋⣿⣿⣵⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠹⣦⣴⠙⣁⠾⠗⠛⠳⠖⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠥⠤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣦⣔⡢⡀⠀⠀⠀⠀⠘⠢⣄⠈⠂⢄⣀
⠀⠀⠀⠀⢀⣠⣤⣠⣤⣪⣵⣿⣾⢿⣿⣷⣝⡀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣈⣁⢀⣀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠦⠤⣀
⠀⠀⠀⣼⣿⣿⣿⣿⣯⣿⡯⢍⣶⣿⣿⣿⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡿⢁⣿⢛⣡⣟⡠⢷⡀⠀⠀⠀⠀⣠⣯⣉⠙⡆⠀⠀⠀⠑⠤⡀⠀⢀⡀⡀
⠀⠀⠀⡟⢿⣿⢶⣅⢻⣾⡿⢻⣿⡿⢟⣵⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣈⡿⣫⣽⣿⡭⣼⡇⠀⠀⠀⠀⠯⣶⣷⠙⠛⢦⡀⠀⠀⢀⣀⣀⣀⠤⡇
⠀⠀⠀⣟⣦⣍⣛⣭⣖⡂⣴⣿⣭⢾⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⡈⢻⣿⣼⡿⠷⡿⠀⠀⠀⠀⠀⠠⣝⠟⠔⣡⣿⣃⡤⠞⠉
⠀⠀⠀⠉⠻⢿⣷⣼⣌⣦⡧⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⠿⣧⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢫⡭⠍⣹⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⠠
⠀⠀⠀⠀⠀⠀⣠⡞⠛⣷⢦⣄⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⢠⠁⣠⡾⠋⠀⣀⣀⣼⣿⣝⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢺⠿⠿⢿⠻⣯⡑⢮⢳⡀⡇
⠀⠀⠀⢀⡴⠊⠉⠳⣲⣶⣶⣽⣷⡀⠀⠨⠛⢦⠀⠀⠀⡰⣃⣞⡝⢁⣴⣾⣿⣿⣟⡛⠻⢿⣶⣽⡄⠀⠀⠀⢀⡔⠋⠉⠙⢉⡻⡍⢢⢀⡇⣿⢃⣾⡿⠔⠁
⠀⠀⠀⠸⣄⠀⢠⣾⡵⠛⠁⠈⠙⠿⣷⣲⣤⡎⠀⠀⠀⣷⣟⣩⡴⠞⣵⠟⠻⢿⣿⣿⣶⣦⣬⣽⠇⠀⠀⠀⢮⣀⠘⢉⣴⣦⡤⢣⣾⡞⡜⠛⠉
⠀⠀⠀⠀⠈⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠻⠿⠶⠛⠁⠀⠀⠀⠀⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠉⠉⠛⠓⠚⠓⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡬⠥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⢀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⡀⣀⣴⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⣀⡸⠣⠴⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠛⠻⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠞⣛⠩⠀⠀⣀⣀⣄⣀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠭⠭⠅⠐⠒⠂⠉⠁⠀⠀⠀⢸⠁⠀⠀⠀⠙⡆⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣠⣴⠟⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠾⠟⠛⢉⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠳⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣀⣠⠤⣔⠊⠉
⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠙⢦⡙⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠂⢄⠀⠉⠓⢄
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⠤⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⢴⠖⠉⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢳
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⠤⠬⠭⠭⠭⠭⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠤⣀⣈⠁⠒⠂⠠⠤⠄⠀⠤⠠⠤⠀⠠⠤⠀⠐⠒⠈⢁⣀⣠⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⣾⡿⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠆⢸⠀⠀⠀⠀⠀⠀⠠⠍⠃
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⢰⣿⢂⡄⠀⢀⡀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⣤
⠀⠀⠀⠀⢿⡿⠀⢰⣶⡆⠀⠀⢃⡄⠀⠀⠀⢀⡀⠀⠉⠐⠃⠀⢿⠿⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠘⢿⠿⠀⠀⠙⠛
⠀⠀⢠⣶⡆⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠆⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠉⠁⠀⣄⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠙⠛⠁⠃⠸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⢠⣶⣆
⠀⣤⣤⠀⠀⠀⠈⡝⠁⠀⠀⠀⠀⠀⠀⣤⣤
⠀⢻⡛⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡄⢠⣤⣶⣦⣴⣦⠀⠀⠀⠀⡀⠀⠸⢸⣇
⠀⠈⠀⠀⠀⠀⢐⠳⠄⠀⠀⠀⠀⠀⠀⣾⣇⣀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣼⡟⢸⣿⠀⠀⠀⠀⢰⢀⣀⠿⡿⠂
⢀⣤⡀⠀⠀⠀⠀⠁⠸⠇⠀⠀⠀⠀⠀⠛⢟⣋⠀⠀⠀⠀⠀⠀⠀⠟⠀⠻⠇⠸⢃⠀⠀⠀⠀⡀⠢⠉
⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢉⠀⠀⠀⠀⠀⠀⠲⠶⠶⢶⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛
⠀⠀⠀⠀⣀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡄⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠃⠛⠀⠈⠉⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⡧⠀⠀⠀⠀⠀⠙⠿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢉⣂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⣠⠀⠈⠋⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿
⠀⠐⣠⠄⠀⣴⣶⡄⠀⠀⠀⠚⢀⣤⡄
⢠⣤⡀⠀⠀⠆⠾⠀⠀⠀⠀⠀⠘⠻⠋⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⠀⢀⣤⡀⠀⣠⣤⡀
⠘⠛⠃⣘⠳⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠈⠘⠃⠀⠀⠀⢀⠄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢱⠀⠙⠛⠁

⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⣾⣟⡯⠍⠓⠒⠒⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⣽⣿⣧
⠀⣿⠈⠉⠻⠓⠒⢶⣶⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⣶⣶⣶⡶⠟⠛⠉⢸⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⣠⣤⣤⣀⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠟⠋⣉⣉⣉⣙⠻⣷⣄
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⠟⠋⠉⠉⠛⣷⡌⢿⣆
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠃⠀⠀⠀⠀⠀⠘⣿⠸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠃⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿
⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡟⢸⡿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢀⣼⡟⢠⡿⠃
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣠⣾⠋⣰⡿⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣠⣴⡾⠋⣡⡾⠋
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠛⠋⣁⣴⡿⠋
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣴⡾⠛⠁
⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠁
⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡾⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣽⣭⣬⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠲⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣚⡭⠵⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠲⠤⣍⣓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⢀⣀⣠⣤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⢤⣄⡀⠀⠉⠛⢦⡙⢦
⠀⠀⠀⠀⠀⢠⡏⡏⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⠀⠉⠙⠲⣄⠀⣿⢸⡆
⠀⠀⠀⠀⠀⢸⣇⢣⡞⠉⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠋⠀⡤⠤⡄⢹⡀⠀⠀⠀⠀⠀⢀⡿⣡⢾⡇⣀⣤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⢸⡈⠳⣌⡳⢤⣀⡀⠀⠀⠸⡄⠀⠀⢰⠖⠒⠚⠁⠀⢹⢸⡇⠀⢀⣠⡴⢞⣡⠞⣡⠞⠋⠁⠀⣀⣀⣀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠓⠮⣍⣛⠒⠶⠿⠦⢤⣬⣷⣤⣤⣤⠤⠿⠾⠒⢋⣩⡤⠖⠉⠀⠀⣧⣀⡤⠖⠋⠁⠀⠉⠙⣦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⠶⠶⠶⠶⠶⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢧⣄⣀⣀⣀⣀⡤⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠈⠉⠉⠁⣀⡠⠔⠋
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠻⣍⠛⠋⠉⠉⠁
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⢸⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⢦⣌⡙⠲⠦⢤⣄⣀⣀⣀⣀⣀⣀⡤⠤⠖⠚⣋⡥⠞⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠤⠤⠤⠴⠶⠒⠋⠉⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠙⠪⣝⡲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⣚⡭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠺⠭⣍⣓⣒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠴⣖⣒⣻⠭⠷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣄⣀⣠⠴⠒⣒⣒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⣀⣤⠤⣤⡴⠋⠛⠋⠉⠁⡴⠋⠀⠈⠉⠉⠉⠙⠓⠦⣌⠳⣄
⠀⠀⠀⠀⣼⠱⠂⠀⢸⠁⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣈⣧
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠙⠯⢽⡲⣄
⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⢠⠪⠀⠀⠀⠀⠀⢀⣠⣤⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁⠀⠀⠀⡠⠚⠁⠀⠀⠈⠻⠝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠂⠀⠀⡠⠴⠖⠒⠶⢾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠠⠤⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠙⠛⠋⠓⠒⠛⠲⢤⣤⣀⣠⡾⠵⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢠⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⠁⠀⡾⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣰⠇⠀⣸⠃⠀⢠⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⡟⠀⢰⠇⠀⢠⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠖⣾⠀⢸⠃⠀⡿⠀⢠⡟⠀⢀⡏⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣭⠴⠒⠀⠀⡏⠀⣿⠀⢸⡇⠀⢸⠁⠀⣼⠀⠀⠠⣌⣙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢾⣿⣶⣞⣋⣥⣤⣿⣤⣿⢠⣼⡇⣤⣼⢠⣤⣿⣀⡀⠒⠦⣭⣛⠆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠈⢉⣠⠟⢑⣶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣍⠓⠂⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⣉⣀⣤⡴⠾⣛⣵⡆⣚⣛⠧⣄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⢋⣉⣉⣭⣴⡶⠿⣛⣭⠀⠉⠉⠙⣾⡇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠃⢸⣟⡳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣉⣉⣥⣴⡶⠿⣛⡅⠈⠑⣂⡴⣻⡣⢠⡏
⠀⠀⠀⠀⠀⢀⣤⠞⣋⡥⢒⡠⠄⠀⠀⢻⣟⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣉⣉⣉⣩⣭⣴⠶⢟⡋⢡⣞⣛⣽⣾⢟⣵⠟⢶⣄
⠀⠀⠀⠀⣰⠟⣡⡞⢡⡞⢉⡴⠊⡀⠀⠈⢿⣟⠒⠄⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣭⣭⣴⡶⢿⡛⢡⡿⢿⣛⣭⡶⠛⠁⣀⢢⡈⢷⡄
⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⣿⡀⢾⡀⠀⠀⠀⠻⣯⡒⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣭⣥⣤⠴⠶⣞⡛⣱⣿⣿⣿⣿⣿⢿⡌⢳⡈⣆⢳⠈⡇
⠀⠀⠀⠀⠻⣆⠹⣧⣌⠳⢮⣛⠶⣬⣁⠀⠀⠀⠈⠻⣥⣄⣀⠀⠀⠀⠀⠀⠤⠤⠶⢶⣶⣾⣿⣿⣿⣿⣿⢸⣿⡜⣿⢎⡇⢸⠇⡏⢀⡼⠃
⠀⠀⠀⠀⠀⠈⣳⣮⣙⠷⣦⣍⣙⠓⠦⠄⠀⠀⠀⠀⠈⠙⠻⢯⣿⣛⣛⣛⣛⣛⣯⣽⡿⠿⢛⣫⣵⡿⢋⣾⠟⣰⠟⠘⠁⢋⣠⣶⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠨⠽⠿⠶⢮⣭⣙⣓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠽⠷⠶⠶⠶⣚⣛⣋⣭⡵⠾⠛⢁⣈⣡⣤⣶⣾⣿⣿⠿⠭⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣉⣹⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣛⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣟⣛⣛⣛⣛⠛⠛⠛⠛⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⣩⣭⣭⣭⣭⣭⣭⣉⣉⡉⠉⠉⠉⠉⠁

⠀⠀⣴⠶⠄⣤⢀⣄⠀⣠⡀⣶⢀⡀
⠀⠀⢬⣹⡆⣇⢸⣺⠡⣗⣇⣼⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠲⣀⡀
⠀⠀⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢠⡔⠒⣆⠐⠂⠚⠻⣿⠓⠶⠤⠻⢧⣄⣀⣤⣤⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣠⠼⣿⣆⣸⡀⠀⠐⠄⠘⢷⡀⠀⠀⠈⢷⣄⠉⠉⣻⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠿⠲⢾⣇⡐⠀⠩⡈⢿⡟⠒⠲⠶⠮⣷⣄⣀⡁⣀⣙⣦⣴⣿⣧⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣄⠀⠀⠀⠀⢿⡟⠛⢶⣤⣼⣧⠀⠀⠀⠀⠈⢳⡌⠉⠉⢹⠛⠻⠔⣹⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⡎⠛⠓⠶⢤⣄⣸⡇⠀⠀⠉⠑⢿⣷⣤⣀⠀⡀⢀⢿⣦⡄⣴⢠⡀⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⡀⠀⠀⣇⠀⠀⠀⠀⠈⠉⣿⣦⣄⣀⠀⠀⢹⡆⠉⠛⠛⠻⠯⣛⣲⡿⢰⡿⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⣾⣧⣀⠀⠀⣤⠀⢸⣿⠉⠉⠛⠲⠾⣷⣄⣀⠀⠀⠀⢸⡿⢀⣿⣦⠀⠀⠀⠀⢀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⠀⡐⠲⡀⠀⠙⢿⡻⠶⢦⣤⡈⣿⣀⠀⠀⢠⠀⠈⢿⡙⠛⠃⢠⣿⣴⣿⣿⠃⠀⠀⠀⠀⣿⣻⠧⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣇⣻⣤⣭⣄⣀⠀⠣⠀⠳⡀⠀⠈⠉⢻⣿⢦⣀⠈⠀⡄⠘⣧⢀⣴⣿⡿⠜⣿⡇⠀⠀⠀⠀⢀⡘⠁⣰⣿⣷⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣧⣤⣀⣧⣀⠀⠀⠀⢿⠀⠈⠛⣷⣤⣴⣾⣿⡿⠿⠃⣰⣿⠀⠀⠀⠀⠸⣿⣦⡞⠉⠛⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⢿⣮⣻⣿⠛⣿⣿⣷⣷⣤⣼⣧⣴⣿⣿⡿⣿⣿⣿⡧⢀⣿⢻⠃⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⣱⣌⢿⣿⣻⠿⣿⠟⣻⣟⣿⡟⠉⢑⢢⣴⣿⣥⠾⠛⠁⠀⠀⠀⠀⣀⠀⡈⠉⠘⠷⠞⠀⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠛⠻⠾⠿⢿⣶⣿⣿⣿⣿⣿⣧⣴⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⣀⡀⠈⢈⡛⠊⠀⠆⠈⠁⠘⢡⠆
⠀⠀⠀⠀⠴⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣟⠀⠈⠀⠀⣯⠁⠾⠌
⠀⠀⠀⠀⠈⣁⣈⣙⣒⢄⣤⠄⡀⢂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣴⡉⠳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⠿⠿⣟⣛⣻⣿⡿⣿⣶⣶⣤⣶⣶⣤⣄⠄⢤⡀⠀⠀⠀⠀⠀⠀⢤⣿⡿⠛⠿⣷⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠨⠽⠻⠖⠛⠋⢻⠗⠉⠛⢻⣿⣿⣿⣗⣲⣶⣤⣄⣀⠀⠉⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠛⠒⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣠⣤⣤⣤⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣠⣤⡤⠤⢤⡶⠶⠖⠛⠛⠛⢛⡛⠋⠉⠁⠈⠁⠈⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣏⡕⠀⠀⠀⠀⢀⠀⠉⠁⠈⠉⠁⠀⠀⢀⡀⠈⠀⡀⠀⠻⣿⣿⣇⠀⠀⠀⠀⠀⡠⠊⠐⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣯⠄⠀⠒⠂⠀⢀⠀⠠⠤⠐⠒⠂⠀⠀⠀⠀⠒⠀⠀⠀⠛⠿⠾⣿⠀⠀⠀⢀⠔⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣀⠀⠄⠀⠀⠤⠤⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⠀⠈⣰⣿⡿⠀⠀⠀⠁⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠤⠐⠂⠀⠀⡉⠉⠀⠀⠤⠀⠂⠀⠀⠄⠀⠀⠀⠀⢠⣰⣿⡇⠀⠀⠳⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠒⢂⠉⠀⠀⠀⠒⠂⣀⡉⠉⠀⠀⡂⠀⢀⠀⣠⠀⣀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠈⠉⠄⠀⣀⡀⣠⣽⢖⣭⠳⣭⢄⡠⠄⠓⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣃⣤⣤⣽⣷⣶⣶⣾⡿⠿⠿⠿⢤⣄⣽⠉⠉⠀⢸⣆⣰⣼⠏⣻⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⣬⡿⢿⣟⣻⣬⣽⣶⣿⣿⣟⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡻⣿⣿⣿⣿⣷⣶⣶⡶⣶⣶⣾⣿⣭⣿⣷⣿⡻⡿⣿⣿⣿⣲⡽⣟⠻⠿⠙⠊⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣝⢯⣽⢷⡶⣿⢿⣿⣿⣿⣿⣿⡿⠷⡛⠟⠟⠫⠙⠂⠈⠁⠀⣠⣶⣦⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣹⢸⠙⢷⣌⡳⣝⠦⢵⡚⠯⠍⠓⠀⠉⠁⣀⠀⢀⣔⣶⠀⠘⠃⠀⠙⢿⠏⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⢠⠈⠟⢻⣿⠈⢁⢠⣄⠀⣮⣿⡆⠀⠙⠀⠀⠛⢋⡀⠤⠀⠂⣾⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⠁⠀⠑⠢⢼⣿⠀⠻⣿⡿⠀⠈⡉⠀⠀⠄⠐⠚⣿⡅⠀⠀⠀⠀⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠈⢻⣠⠤⠘⠂⠉⠀⠀⠀⠀⢀⣀⣤⢼⣧⠤⢀⠀⠀⠀⢿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢟⢀⣀⣠⡤⠴⠒⠒⠉⠁⠀⠀⠀⣿⡀⠀⠀⠀⠄⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⣀⣀⠤⠤⠖⠆⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠠⠀⠈⢷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣝⠫⢉⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠒⣾⡁⠀⠀⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠁⠈⠀⠠⢀⠀⠤⢀⢀⡀⠀⢀⠠⠄⠐⠀⠀⠈⠀⠀⠀⠀⠀⠻⠧⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⠀⠀⠈⠑⠊⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⢀⠠⠄⢠⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣠⣀⡀
⠀⠀⠀⢘⡧⣦⠴⣠⣀⣤⠴⠴⡋⡁⣢⠇⠀⠀⠀⢧⡄⠀⠀⡾⣿⣿⣿⣾⣿⣿⣷⣿⣿⡷⡄⠀⠀⠓⠒⠒⠦⠤⢄⣹⡾⢯⣾⠛⣶⡟⣓⣄
⠀⠀⠀⠘⢾⠴⡴⢠⣆⠸⣸⣤⡷⠞⡏⢡⠄⠀⠀⠈⣧⠀⠀⠳⣌⠻⠿⣻⣿⢷⣿⣿⣿⣾⠇⠀⠀⡴⢖⣤⠀⠀⠀⣸⡇⡿⢣⣽⠏⣴⠟⣸
⠀⠀⠀⠀⠐⠪⣴⣵⣃⣆⡔⢠⣆⣤⠟⠋⠀⠀⠀⠀⠘⡄⠀⠀⠈⠓⢀⣏⣿⣿⣿⡿⠛⠁⠀⠀⠀⠥⣏⠠⠀⠀⠀⠻⢿⣿⣾⣧⣤⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⢀⣀⡀⠀⠈⠉⠛⠛⠋⠁
⠀⠀⠀⢠⠒⠲⡄⠀⢀⣠⣤⣀⡀⠀⢀⡀⠀⣀⢀⡆⠀⠀⣀⣀⣶⢤⣤⡤⣤⠠⠠⣴⣴⣢⠤⠤⠀⠀⢠⢒⡥⢒⣒⠠⡉⢢⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠑⠤⠊⠰⣿⣽⣿⣿⢹⠀⢀⡝⢠⠛⠋⠀⠀⠠⠌⠈⠙⠚⠻⠀⠛⠂⡃⠉⠁⠀⠀⠀⠀⠀⡇⢸⡀⢯⣲⣃⠄⢸⠀⠀⢻⣜⣝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠙⠸⠋⠀⠀⠀⠠⣲⢦⡠⣆⣶⣖⢰⣶⣊⡟⣧⡴⠖⠂⠀⠀⠀⠹⣄⡙⠒⢭⣤⡴⠋⠀⠀⠀⠈⠉
⠀⠀⠀⢀⢄⡲⢂⡵⠒⣢⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠁⠀⠈⠉⠈⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⣀⣄⣀⠀⠠
⠀⠀⠀⣾⠏⣴⠃⣰⠊⣡⣄⡾⢁⢠⠂⠀⡀⢤⣇⣨⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠦⠋⠀⠀⡠⠉⢀⡨⠂
⠀⠀⠀⠈⠓⠓⠶⠶⠚⠉⠁⠀⠀⠉⠀⠀⠧⠾⣿⠹⡀⠀⠀⠀⣀⠴⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠊⠀⠀⢀⠀⡐⠀⠀⢀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡠⠊⠀⣀⣀⡠⠄⠀⠀⠒⣒⡷⠀⠀⠀⠀⠀⠀⠹⡀⠀⢀⡠⣿⢠⡆⠀⠀⣸⠟⠃
⠀⠀⠀⠀⠔⡤⠋⣩⠕⣋⠽⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⣸⣴⣶⣾⣯⣥⣤⣤⡴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠋⠶⠋⣉⡹⠊⠉⠁
⠀⠀⠀⠘⢻⢧⣾⣥⣞⣵⠊⣡⡖⠛⠉⣉⣷⠄⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢀⡰
⠀⠀⠀⠀⠀⠉⠳⠾⣹⣷⢻⣿⣶⣤⡞⠉⠀⠀⠀⠀⠀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠋⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⡠⠒⢠⠀⠀⠀⠈⠋⢡⡜⠋⠀⠀⠀⢀⣀⣀⡀⠀⠈⠙⢯⣉⠉⠉⠉⠉⡉⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠃⠤⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⢴⣮⣿⡛⢿⣇⠀⠀⠀⠀⡟⠉⠀⠀⡎⢻⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⣟⡿⣦⣄⠀⠀⠀⣼⣿⣯⣽⣿⠿⠃⠀⠀⠀⣼⣂⣬⣤⣤⣤⣼⣷⠄⠀⠀⢰⠀⠀⠀⠰⡀⢐⠀⢰⣶⡂⠀⠘⠁⠀⠀⣷
⠀⠀⠀⠀⠀⣴⣿⡾⣦⣏⡋⣳⠂⠀⠿⣿⠯⠞⠁⠀⠀⠀⠀⡼⠀⠀⠈⠉⢉⣟⣦⣼⡄⠀⠀⠈⠆⠀⠀⠀⡇⢸⠀⠀⢯⠇⠀⢸⢿⠀⠀⠛
⠀⠀⠀⢀⡼⢳⠿⣤⣔⣽⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡸⠁⢀⣀⣤⣔⡚⢿⡿⠏⢻⠀⠀⠀⠀⠀⠀⠀⡇⣾⠀⠀⠸⡄⠀⢸⢸⠀⠀⣠
⠀⠀⠀⠈⠙⢷⣦⡊⠙⡟⠁⠺⢦⣄⡀⠀⠀⠀⠀⠀⢿⣿⣷⠟⠓⠲⠶⠤⠬⠭⠶⠶⠿⠄⠀⠀⠀⠀⠀⠀⢸⡜⠀⠀⠸⡇⠀⢸⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠊⠀⠀⠀⠀⠉⠂⠄⣄⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⠀⣼⠀⡆⠀⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⣤⣾⣸⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠰⠟⠙⠾⣷⣶⣮⣾⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠶⠀⠀⠀⠈⢿⣿⡟⡘⠂⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣋⡭⠀⠀⠀⡀⠀⠀⢠⣠⣀⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣠⠚⣽⡁⣴⣿⠏⠁⠀⠀⣠
⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠹⣿⣭⣽⠁⣠⠔⢩⡇
⠀⠀⠀⠲⣄⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣯⣿⠋⠀⢀⡨⠃
⠀⠀⠀⠀⠘⠷⠟⠛⢷⣤⡄⣢⡀⠀⣀⠀⢀⡜⢁⡔⠀⢀⠀⢰⣶⢬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⢇⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠿⠿⠿⠿⠛⠚⠿⠿⠞⢧⣀⣏⣠⡦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⡉⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⢤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠁⠀⠀⠈⠓⢄⡀⠀⠀⠙⠀⠒⢦⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⠈⠒⢤⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢡⡀⠉⡐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠉⠀⠠⢀⠀⠉⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠈⠸⣧⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢠⡀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣧⠀⡀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠠⠀⠈⣳⠀⢿⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⠀⠀⣷⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠸⡄⠀⠀⣹⡆⢀⡀⠀⠠⠀⠀⣸⠀⠀⡎⠀⢀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⡟⣰⡇⠀⠐⣿⠁⠀⡇⠐⠁⠠⣰⠇⢠⣿⣧⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⠷⠷⠶⣿⣶⣾⣷⢂⣀⣴⣏⣠⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⣴⣾⡆⢠⣶⣶⠄⣠⣴⡄⠀⣠⣦⠀⣠⣶⣄⢀⣠⣴⡄⠀⠀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⢠⣾⣿⡿⣰⣿⡿⢃⣴⣿⡿⢁⣾⣿⣿⢾⣿⣿⣷⢿⣿⣿⠃⠀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢃⣴⢻⣿⣿⠁⣿⡿⣡⠾⣿⡟⣠⣾⣿⡿⠃⣿⣿⠟⠁⣿⣿⠃⠀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⢁⣾⣿⠃⠀⠙⠿⠋⠀⠉⠛⠁⠈⠛⠀⠀⠙⠁⠀⠀⠙⠃⢠⣤⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢛⣿⣿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⣄⠘⢻⡟
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣠⣾⡿⠃⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⣀⠀⢸⣦⠞
⠀⠀⠀⠀⠀⠀⠀⠙⠷⡾⠿⠋⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⣀⣤⠴⠋⠀⣠⣴⣃⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⢀⣀⣉⣿⣤⡀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠈⠀⠢⡠⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢯⡌⠉⠙⠉⠁⠀⠀⠀⠁⣲⣄⣤⠶⠿⣷⣿⣿⣿⠃⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⣾⣿⣷⣦⣷⣄⠀⠀⠀⠀⠀⡀⠭⣽⠟⠁⠀⠀⠀⣽⡿⢿⣆⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢸⣿⣿⣻⠛⠡⠱⡄⡠⢀⢀⡤⠔⠋⠀⠀⠀⠀⡄⠛⢈⣴⣿⡇⠀⡀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⣤⣤⣤⣎⠴⠐⠋⠁⠀⠀⠀⠀⠀⠀⢦⣤⣄⣸⣷⣆⣬⠾⠃⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠽⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡑⣜⣿⣿⡟⠁⣀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢜⠛⢏⡟⢠⣿⠇⣴⣾⣿⣦
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣂⣼⡷⣿⡏⡾⠛⡹⣽⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠢⠀⠀⣀⡯⢙⢫⣿⡿⠁⠊⠀⠉⠁⠀⠱⡀
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⣀⢄⡡⠘⠉⢀⣾⠟⠄⠀⠀⠀⠀⠀⠀⠀⢰
⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡄⠄⠊⠏⠀⠀⣠⣿⡿⠊⠠⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡦⠏⠉⠁⠀⢠⣾⠟⠁⠄
⢼⣤⢤⢀⡄⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢅⠃⠀⠀⠀⣀⣴⣿⣧⣀⠄⠀⠀⠀⠀⠔⢀⠄
⠈⢿⣷⡆⣈⡀⠀⢀⠀⠄⠀⠀⠀⠀⠈⠀⠀⡀⣄⠀⠀⡐⠘⠌⠃⠀⠈⢀⣠⣾⢿⣿⣿⣿⠟⣋⡐⠁⠀⠀⠀⣠⠔⠀⠀⠀⠂
⠀⠀⠻⣿⣿⠼⢦⡁⡄⣌⡔⠤⠈⠦⠹⠀⠗⠅⠘⣀⠃⠀⠀⠀⢀⣤⣶⠟⠋⠀⠀⠙⢿⣿⣶⢂⠏⠖⠘⠐⡀⠠⢞⠴⢀⢀⣾
⠀⠀⠀⠈⠻⢷⣤⣈⡀⠃⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣮⣼⣩⡛⣵⠠⣂⣺⣽⠟⠁
⣀⣀⣀⣀⣀⣀⣉⣙⣛⣿⣿⣿⣿⣶⣿⣿⣿⣛⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣿⣿⣿⣿⣟⣋⣁⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⣠⣾⣿⡇⠀⠀⣼⣷⠀⢰⣶⠀⠀⢀⡀⠀⠀⢀⣴⠿⣷⣤⣾⠿⣷⡀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠛⠁⣿⡇⠀⢠⣿⣿⠀⣾⡇⠀⢀⣿⣿⠀⣿⣾⡏⠀⢻⣿⠇⠀⣿⡇⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠘⢿⣴⡿⢻⣿⠀⢿⣇⢀⣾⠿⣿⠀⣿⣿⠀⠀⣿⡟⠀⠀⣿⠇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⢀⡀⠀⠀⢀⣾⠏⠀⠈⠻⠟⠋⠀⠿⠀⠘⠃⠀⠀⠙⠁⠀⠀⣿⣤⣶⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠈⠻⠷⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⢀⣤⣶⡶⠿⠿⠶⣶⣶⣦⣤⣤⣄⣀⣀⣀⡀⠀⠀⣀⣀⣀⣠⣤⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉




⣶⠒⢲⡶⢶⢶⣶⡶⣶⠶⣶⠶⡶⣶⡶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠐⠀⠀⠀⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⡰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢿⠧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⣀⣀⣀⣀⡀⢁⠶⠭⣓⠦⡀⠀⠙⠔⢅⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠖⠀⠠⠖⠛⠉⢉⡔⠣⢌⢙⠣⣍⡒⢌⠳⡄⣄⠀⠀⡀⡀⡢⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌⠀⠴⠐⠁⠀⠀⠀⠀⠀⠑⠰⢯⡈⠑⠠⣁⠡⡄⢉⠀⢣⠙⣌⠳⣄⠀⠘⠄⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠑⠈⠢⠑⡇⠘⡄⠠⡀⠑⢬⣑⠲⣄⠀⠂⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⡀⠀⠘⡀⠡⡘⢤⡘⠂⡀⡀⠂⢤⡑⣌⢣⠁⡀⠈⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⠂⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡁⠑⣄⠳⡜⢆⠱⢌⠢⢤⡂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⡙⢅⠈⢢⠙⠀⡑⠦⡽⠊⠀⢦⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠣⡀⢡⠈⠢⢌⡀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⡐⠀⠌⠀⠀⠀⠈⠃⠠⡄⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢆⡈⠑⢢⠍⠀⠀⠀⠀⠐⠀⠑⡀
⠀⠀⠀⠀⠀⠌⠀⠘⠄⠀⠀⠀⠀⠠⠀⢳⡀⢀⠀⠤⢤⠤⠂⠀⠲⠂⠰⠂⠐⠢⠄⢀⣀⠀⠀⠀⢠⠏⠢⠌⠛⠀⠀⠀⡀⠀⠀⠀⠀⠄⠈⠄
⠀⠀⠀⠀⠌⠀⡐⠀⠘⡀⠀⠀⠀⠀⡡⠤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠰⣏⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠈⢂⠈⠂
⠀⠀⠀⠐⠀⡐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠈⢄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣿⢲⡷⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢯⣨⣀⣸⣿⣤⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠙⢿⣿⠿⣶⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣿⣿⣿⡆⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢾⠿⠿⠟⠛⣛⣛⣻⣍⣩⣿⣼⡗⡼⢀⣦⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠗⣉⣀⣀⡒⠀⠀⠀⠤⠤⣿⣿⣿⣿⣟⣡⣿⣿⡿⢟⣿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠐⠂⠉⠁⠈⠁⠀⠀⠀⠐⠚⠿⠋⠙⠋⢉⡋⠙⢛⣽⡧⠄⢀⠀⠀⢉⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⢀⣼⣿⡿⠂⠀⠀⠀⣀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠁⠈⠙⠁⠀⠀⠀⠙⠲⢌⠳⣤⣙⡏⠻⣷⣄
⠀⠀⠀⠀⣰⣿⡻⠋⠀⠀⠀⠀⠊⠁⠀⠈⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠶⣌⠻⢷⣤⡀
⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢻⣷⡄
⠀⢀⣿⣏⡎⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠹⣿⡄
⠀⣾⡟⡉⠀⠂⠀⠀⠀⠀⠀⡾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠂⠻⣿⡄
⢸⣿⡏⠁⢠⠁⠀⠀⠀⠀⠌⠰⠁⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣧⠘⡆⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢰⡹⣷
⣿⣿⢿⠀⡦⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⣿⣿⣿⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿
⣿⣿⠁⠀⢀⠁⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠂⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠀⣸⣿
⢸⣿⡜⡰⡄⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⣸⣿
⠘⣿⣿⣷⣿⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠹⣿⣿⣝⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠹⣿⡟⣎⢿⣿⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠔⠀⡀⣿⡟
⠀⠀⠀⠙⣿⣮⢳⢿⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠠⠂⠁⢀⠄⢠⡿⠁
⠀⠀⠀⠀⠈⢿⣿⣆⢻⣷⡐⣤⣤⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⠖⡡⣪⣊⣤⠉⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣧⣹⣿⠹⣿⣷⡀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⣿⡷⣡⣾⢟⣭⠟⣡⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣿⣿⣷⡘⣷⣶⣴⣅⣀⣀⣀⣀⠀⠀⡠⠄⠀⠀⠀⠀⠀⣰⡽⠖⣫⢏⣴⣯⣴⣿⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣤⣭⣝⣻⡶⠤⣤⠤⠤⠴⠤⠶⠖⠚⠒⠀⢰⣋⣴⣾⣿⣿⣟⣫⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⣝⣿⣿⣿⣿⣿⣿⡼⠿⣿⣿⣶⣶⣶⣶⣾⣷⣖⣲⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⣾⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⡉⠙⠛⠛⠙⢿⡍⠉⠉⠁
⣿⣿⡿⠻⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣡⠶⠋⡹⠋⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣠⠞⠂⠀⣼⠁⢠⠀⠀⠀⢈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠛⠀⠀⣴⢃⣠⡦⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠂⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⡤⠚⠀⣠⣦⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢾⠆⠀⠀⠀⠀⠀⠀⠌⠃⠀⠁⠈⠉⢒⡆⠀⠙⣿⠀⡆⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⣤⢀⡉⠻⠀⢇⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⠇⠀⢸⡿⠀⠀⢠⠗⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠱⣤⠦⠈⠢⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠏⣰⡿⠚⠁⠰⠟⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠉⠻⠆⠐⠶⠚⠾⠃⠀⠀⠶⠀⠀⠒⠛⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠠⡟⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⣦⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠀⠀⠈⠁⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⣧⠀⠀⠶⣦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠶⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠀⠀⠀⠀⠀⠀⠐⠒⠲⠚⠀⠀⠀⠀⠀⠀⢀⣤⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢤⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣶⣶⣶⣶⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘






⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣾⣶⣶⣶⣾⣶⣾⣶⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣻⡟⢿⡿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠉⠁⠀⠀⠀⠀⠉⠉⠙⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣡⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠟⠛⠛⠿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢾⡏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣾⢁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣠⣿⠀⣰⡟⡇
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠉⠻⣿⣻⣧⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⣀⣼⡿⣷⣤⣴⣾⣿⣷⣾⣿⣠⠇
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⣾⣿⢿⣦⣶⣤⣶⣾⣿⠟⠛⠛⠛⠛⠉⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢸⣇⣀⣤⣶⣶⣤⡀⠀⠀⢸⣿⣿⣶⣶⣾⡿⠟⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠈⢻⣿⠋⠉⠙⠻⣷⣤⣾⡿⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣄⡀⣀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠒⠒⠒⠒⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⡀⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠈⠉⠓⢦⡀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣸⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⣀⣀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡿⠀⢀⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⠉⠉⠙⠳⢦⡉⠳⡄⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡼⠃⢀⡼⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠙⣦⠙⠆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⣤⡤⠴⠛⠓⠛⠻⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣸⣧⣄⣠⣤⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡴⠶⠦⢭⡀⠀⠀⢀⡿⠦⠤⠴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⠀⠀⢹⣦⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣹⣄⣀⣳⡖⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡑⠀⣸⠙⠛⠉⠋⠟⡴⠃⠀⠀⠀⢀⣀⣀⣀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣀⠀⣀⠀⢀⣤⣇⢸⣷⠀⠀⠀⠀⢀⣤⣤⡷⣶⣻⣏⣽⣯⠾⣷⣟⣛⣏⢹⡗⢦⡀
⠀⠀⠀⠀⢠⣎⠪⠀⡟⢤⠘⡆⠟⠉⠈⠀⢠⡾⠿⠿⠿⠿⣿⣻⣯⡿⣞⣛⣿⣭⣿⣭⣿⣝⣿⢿⠿⠆
⠀⠀⠀⠀⢘⣌⠣⠢⠉⠀⠁⠐⣀⡀⠀⢀⣟⡦⡐⠀⠀⠁⠀⠠⠈⠉⠻⣯⣽⣧⣾⡏⣽⠛⠇
⠀⠀⠀⠀⣓⠤⢵⣦⣤⣶⣾⣦⣿⣷⣶⠿⠿⠿⣷⣮⣄⣀⠀⠀⠀⠀⠀⠀⠹⡟⠘⠙⠉⠀⠀⡠⢴⡇
⠀⠀⠀⠸⠵⢀⣠⡾⠫⠁⠈⢿⣿⠋⠀⠀⠈⠂⠄⠈⠙⠿⢿⣶⣤⣰⣆⣘⣻⡇⠀⠀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠰⣧⠀⣈⣠⣿⣟⣣⢀⠀⠀⠐⠂⠀⠀⠀⠀⠉⢉⠻⢿⣿⣿⣿⣷⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠨⣯⢷⣬⣿⣿⣝⣿⣿⣉⡒⢦⣌⣀⣀⠀⠀⢀⠈⣲⣾⣿⣛⡗⡽⢷⠀⠀⢀⡠⡀⡖⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣽⣿⣌⢤⡙⢛⠻⣿⣿⣷⣤⣷⣾⣿⣿⠶⢻⡧⣋⣁⣑⣼⠃⣜⣉⣹⣧⡇⢠⠜⢻
⠀⣠⣄⠀⣀⣤⣤⣄⡐⠶⠈⠉⠛⠿⣿⡒⠄⢤⣤⠭⢽⣿⣿⡷⣼⡟⠇⡇⠈⠋⠉⠀⠐⡇⠀⢹⣿⣽⠇⢤⣴
⠀⠈⠁⣾⠯⠬⠤⣀⠙⢦⡀⢠⣤⡀⠈⠙⠲⠾⣧⢤⠤⠟⠋⠀⠀⠙⣦⠋⠀⠀⠀⠀⠀⢢⡢⢃⣀⣻⣄⡀⢈
⢀⢀⠈⠻⢤⣄⣀⣀⣈⣑⡷⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣮⣶⡾⠃
⠙⠛⠀⠀⠦⠆⠉⠉⠁⠸⠃⠀⠀⠀⢀⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡠⠀⡀⠀⢀⠢⢤⡀⠀⠀⡏⠀⢻⢀⡤⣠⠊⠉⠀⣀⠤⠄⠀⢞⢡⠀⠑⢆⠀⠐⠤⡀⠀⠀⡞⢳⡄⠀⢠⠟⠓⠦⣔⠒⠓⠦⣄⣀
⠀⠀⢀⠰⢀⠑⣄⢻⠀⡠⠓⠀⠀⠈⠓⠁⠸⡀⣀⠇⠀⢸⣇⠀⠀⠀⠈⠉⠁⠉⠁⠤⠤⠒⠳⡄⠀⣧⣸⠃⢀⡟⠀⠀⠀⠈⠙⠒⠶⢤⡈⠉⠉⢱⡀
⠀⠀⠀⠂⢢⣽⣿⡻⠋⡠⠔⠒⠉⠉⠓⠦⣀⠀⠀⠀⠀⢸⣧⠤⢄⣀⡀⠀⠀⠀⠈⢉⣀⡈⢽⡃⠀⠼⠁⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⢀⡟⡄
⠀⠀⠀⢊⢀⠔⠁⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⢸⡅⢀⡀⠀⠈⣈⡉⢉⣉⣀⠀⠀⣰⠇⠀⠀⣠⠞⠀⠀⠀⠀⠁⠀⠀⣠⢎⡵⠁⠀⡼⡸
⠀⠀⠀⠀⠁⢀⡎⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⣇⠀⠙⠒⠦⠬⠤⠤⠄⠀⢀⣀⠠⠤⠘⠁⠀⠀⣧⣀⠀⠀⠐⠀⠀⠀⣰⢣⠏⠀⠀⢠⢇⡇
⠀⠀⠀⠀⠀⠸⣇⠠⡀⠀⠀⢀⡞⠁⠀⠀⠙⢦⠀⠀⡿⠀⡴⠾⠶⡆⠀⠀⠀⠀⠀⠠⢤⣄⣀⠀⠀⠀⠘⠫⢽⣲⡦⣄⠀⢠⠇⣾⣀⣀⣠⡼⢸⠁
⠀⠀⢀⣤⣄⠀⠙⢦⡀⠀⠀⠘⣇⠀⡀⢀⡄⣸⠀⢠⠇⢸⠁⠀⣰⠏⠀⢠⠒⠒⢦⡀⠀⠀⠈⣧⣀⣀⣀⣀⣀⡈⠙⠲⣝⣫⠞⠓⠶⠶⠶⠖⠋
⠀⠀⣾⠁⠈⠳⣄⠀⠉⢦⡀⠀⠈⠳⠬⠵⠞⠁⢠⠟⠀⠘⠒⠚⠁⠀⠀⠘⢮⣀⣠⡇⠀⢰⣯⣽⣏⣀⣠⣤⠤⠽⠇⠀⠀⠀⠀⠀⢠⠖⠒⠲⡄
⠀⠀⣿⡄⠀⠀⠈⠑⢤⣀⠑⢄⡀⠀⠀⢀⣠⡴⢋⣠⠴⠖⠒⢛⡓⠒⠲⢦⡀⠀⠀⠁⠀⢸⠀⠀⢹⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠏⠀⠀⠠⢹⡀
⠀⠀⠸⣷⡀⠀⢀⣤⣤⡈⠑⢦⡈⠉⠉⠉⠁⡴⢯⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠲⢄⡀⠀⠸⣄⠤⠤⢷⡤⠤⠤⣤⡇⠀⠀⠀⢠⣾⠀⠀⠀⠀⠃⢦⡀
⠀⠀⠀⡟⣧⠀⢻⠀⠀⡝⣦⠀⢹⡄⠀⠀⡞⠀⠈⣧⠈⠀⠀⠀⠀⠀⠀⡎⠀⠀⢠⢙⡆⠀⣟⠒⠒⠚⣷⠚⠛⢉⡇⠀⠀⠀⠸⡍⢙⠋⠙⠉⠉⣩⠁
⠀⠀⠀⢿⠘⣇⠀⠳⢤⣤⠟⠀⢰⠇⠀⢰⠳⣕⠀⠘⣆⢀⣒⣒⣀⣀⣸⣇⣀⠐⣡⠞⢳⠀⢻⠀⠀⠀⠸⣆⠀⢸⠃⠀⣀⡀⠀⠙⠾⣷⣤⡤⠞⠁
⠀⠀⠀⠈⢧⣈⠳⢤⣀⣀⣠⣴⠟⠀⠀⢿⣐⣨⠷⠞⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⢧⣀⣸⠇⠈⢧⣀⣀⣀⣉⣠⠜⠀⢯⠉⢧⠀⢀⣠⡴⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⠋⣁⣀⣀⡀⠀⢉⣁⡀⠀⠀⠀⠸⡳⣶⣄⡇⠀⠀⠀⠀⠈⠁⢀⣴⣦⣍⡉⠉⠀⠀⠀⠀⠈⠓⠻⣄⣈⡓⠶⠴⠶⠞⠃
⠀⠀⠀⢰⡒⠦⣴⡒⠋⢉⣉⣀⠠⠀⠉⢶⡁⠀⡝⠀⠀⠀⣠⠽⠾⢿⣧⣠⠤⠤⣄⡀⠀⢸⣝⡻⠷⣯⣭⣍⣉⣉⣉⣛⠲⢤⣀⠈⠉⠓⠦⣄
⠀⠀⠀⠘⣧⠀⠱⢭⣒⣤⣀⠀⠠⠄⠀⠀⠱⡀⢨⠆⠀⣼⢡⠊⠈⠀⠈⠁⠀⠀⢠⠹⡄⠈⠻⢿⣷⣶⣶⣯⣭⣭⣽⣛⡻⠶⣌⢳⡄⠀⠀⠘⣆
⠀⠀⠀⢰⡿⡦⢤⡘⣿⣭⢻⢷⣦⣄⡀⠀⠀⠑⣌⠆⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠶⠤⢤⣉⣉⣉⣉⡉⠛⠻⢷⣼⠋⠀⠀⠀⠀⣿
⠀⠀⢀⣀⠓⢬⣒⢵⡀⠛⠉⣟⠀⠛⢿⢗⠦⣄⡈⣧⠀⠸⡆⠀⠀⠀⠏⠘⠆⠀⠀⢰⠁⠀⢸⠭⠶⠶⠶⠶⠶⠦⣭⣍⡓⠤⡉⠀⠀⢰⠋⠉⠉⣹⠂
⠀⠀⠀⠈⣳⡀⠈⠑⠮⣓⢦⣈⠉⠳⡀⢀⡙⠲⠽⠇⠀⠀⠹⣔⠤⡀⠀⠀⠀⠄⣠⠏⠀⠀⠘⠿⢿⣒⣒⣒⣒⣒⠢⠌⡙⣾⠀⠀⢠⡟⡶⠒⢲⣿
⠀⠀⠀⠑⢿⣧⣤⡠⠴⡌⠑⢬⡓⢦⣉⣁⡸⠂⠁⢀⣀⠀⠀⠈⠓⠒⠛⠓⠒⠛⠁⠀⣀⣤⠴⠶⠲⠶⠦⣄⡀⠈⠙⢦⣄⣷⠀⠀⠸⠶⢷⣬⣿⣸
⠀⠀⣠⢾⡿⠋⠛⠒⠋⠀⠀⠀⠈⠓⠖⠋⢀⡠⠞⠉⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⢉⢳⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠈⠁
⠀⠀⠉⢁⡤⠤⠤⠄⢤⣄⣤⠤⠤⡀⠀⠀⡿⠤⠤⠤⠤⢤⡴⢦⣄⣈⣓⠲⢄⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⡰⡠⠃⢀⡴⠞⣻⠋⠉⠉⠉⠓⢦
⠀⠀⡎⢹⠉⣿⠛⠛⠛⠛⠛⠛⠛⡏⠀⠀⣛⣷⠀⠀⣀⠀⠙⠋⠀⠀⠉⠉⢻⠀⠈⢧⡳⢤⡀⠀⠀⠀⠀⠀⢸⠁⢀⡞⠀⢎⠉⣠⠶⢶⣄⠀⠈⢧
⠀⠀⠹⣼⣧⠘⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣯⡀⠀⠸⠽⠂⠀⣠⣤⣄⠀⠀⢳⠀⠀⠀⠙⣆⢷⠀⠀⠀⠀⠀⠘⡇⠸⡇⣠⠊⠘⣇⠀⣸⣽⠀⠀⣸
⠀⠀⠀⢠⡛⣦⣀⡀⠀⠀⠀⣠⠗⡄⠀⠀⣾⠁⠀⠀⠀⠀⠀⠛⠶⠟⠀⠀⢸⠀⠀⠀⠀⢿⢸⡆⠀⠀⠀⠀⠀⢹⠀⢷⡸⢄⣠⣌⢉⣩⣁⡖⣲⠏
⠀⠀⠀⠀⠉⠛⠿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠙⠒⠒⠒⠒⠒⠚⠙⠒⠒⠒⠒⠚⠀⠀⠀⠀⠈⠳⣧⣄⣀⣠⣤⡤⠞⠀⠀⠙⠶⣤⣈⣉⣠⡭⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠞⢣⣤⣀⣀⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠈⠙⠲⣏⠉⠉⠁⠀⡼⠁⠀⢸⡄⠀⢀⡴⠋⠀⡞⠉⠓⠲⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠹⡇⠀⠀⠀⠈⢣⡤⠤⢴⡇⠀⠀⢸⠗⢶⠋⠀⠀⢰⣇⣀⣀⠀⠀⠀⠈⠙⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⣀⣠⢧⠀⠀⠀⢀⣀⡵⠀⠘⠗⠒⠲⠎⠀⠻⠤⣀⣠⠏⠀⠀⠈⠉⠙⠒⠦⢤⣀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣠⠴⠋⠁⠀⠀⠓⠖⠋⠉⠀⠀⣠⢖⣒⣒⡦⣄⠀⠀⠀⠀⠁⠀⣀⠤⠤⣤⣄⠀⣤⡄⠈⠙⠲⣄⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⢃⡴⠊⠁⠀⣠⠴⣖⠲⡄⠀⠀⠀⠀⡜⠕⠣⣐⣀⠜⢪⢣⠀⠀⠀⣴⣛⠶⠒⠉⠁⠀⠀⠉⠁⠀⠙⡖⢦⡙⢦⡸⡆
⠀⠀⠀⠀⠀⠀⠀⣽⠞⢀⣀⠀⡸⠓⢄⠈⡶⠋⠀⠀⠀⠀⣧⡃⢀⢿⠻⡀⢸⣸⠀⠀⠘⠁⠁⠀⠀⠀⣔⢶⢓⣆⠀⠀⠀⠈⢧⢱⡀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠚⠀⣷⠤⠤⢿⠀⠀⠀⠀⠀⠀⠙⢾⣣⡸⢄⢜⡵⠃⠀⢀⣠⠤⢤⣄⠀⠀⠈⣹⠛⠾⠁⠀⠀⠀⠈⢧⡇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠘⡿⣆⠀⠀⠈⠢⣴⡋⣹⠒⢲⡒⠈⣃⠀⢀⣈⠉⠉⠁⠀⠀⣰⠋⣇⡰⠋⣀⣹⡄⠀⠀⠀⠀⣠⠔⢲⠄⠀⠈⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠳⣄⡀⠀⠀⠈⠉⠉⠉⠉⠉⡡⠖⢋⡬⠃⠀⠀⠀⣰⢉⠖⠉⠙⢏⣁⣀⣧⠀⣠⠴⣋⣠⠖⠉⠀⢀⡴⠋⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠙⠲⢤⣀⠀⠀⢀⡴⣏⣳⠖⠉⢀⣀⠀⠀⡰⠉⡇⠀⠀⠀⠘⠤⠴⠃⠸⠽⠗⠋⠀⣀⡴⠞⠉⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈⠙⠓⠶⢷⣍⣀⣀⠀⠘⠚⠀⠀⠙⠚⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠚⠋⠁⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠓⠒⠒⠒⠓⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠤⠤⠤⣤⣤⣀⣀⣠⡤⠤⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢁⠈⠀⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠂⠀⠈⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠶⢺⡀⠀⢁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⣹⠀⠀⣏⠉⢵⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢋⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠒⠡⠐⢧⠀⠱⣀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣀⡀⠀⡈⡗⢤⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡜⢀⢊⠐⠁⠀⠀⠀⠑⢤⡈⠑⠂⠷⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠴⠊⠉⠀⠀⠀⢀⠻⠲⡐⠄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⡼⠉⠑⠂⠠⠀⢤⡀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠡⠁⠀⠀⠀⠀⠀⣠⠞⠀⠃⠀⢨⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠙⢒⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀⣁⠤⠖⣒⠼⠶⠶⠦⠤⠶⠚⠁⠀⠀⠀⠠⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠠⠆⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⣠⠴⠶⠒⢉⣡⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠌⡁⠂⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⠀⠔⠊⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠴⠒⢈⠠⢐⡮⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⠀⠀⠀⠈⠉⠁⠒⠒⠀⠤⠤⠤⠤⠤⠤⠠⠤⠤⠶⠶⠾⠛⠛⠓⠊⠉⠉⠠⠀⠂⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠃⠐⠒⠒⠒⠒⠒⠒⠒⠂⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⢀⣤⣀⠀⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣴⣿⣿⣿⡆⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠒⠤⣀⣀⣀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠛⠿⠿⠛⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⡼⠀⠀⠀⣀⣤⡤⠤⠶⠤⣄⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⢺⡆⠀⠀⠀⠀⠀⢨⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠐⠒⠒⠒⠒⢠⡀⠀⠀⠀⠀⠀⢀⡤⠼⠧⠤⠤⠤⠤⠤⠼⠼⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠃
⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⢀⣀⡠⠤⠔⣊⡅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠐⢬⠭⠉⠉⠭⠭⠁⠤⠐⠂⠉⠀⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠘⡄⢀⣀⣀⡀⠤⠤⠀⠐⠒⠈⢹⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⢀⣀⣀⣀⠀⠻⣃⣟⠘⠃⠀⣠⠒⠢⠤⠭⠍⠉⠐⢲⠋⡇
⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⣤⣦⣄⠀⠀⠀⢸⠀⠀⠀⠀⠓⠐⠒⠀⠀⠀⠐⠒⠒⠒⠒⠒⠐⠀⠐⠃⠀⠀⡇⠀⠀⠢⠠⠐⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⢱⠀⠀⠰⣿⢿⣽⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠁⢀⡀⠀⠠⠤⠤⠄⠚⠃⢰
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⠻⠟⠋⢀⣀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⣔⡉⣑⢏⡠⢶⠀⠀⠀⠀⠘⠁⢠⢰⡴⠄⠶⠦⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠐⢖⠒⠒⠂⠈⠉⠀⠀⠀⡄⠀⠀⠀⡠⠠⠤⢊⠭⠠⠕⠀⣈⠔⠨⠥⠄⠒⠤⠀⠀⠀⠀⢰⣈⡡⡀⠤⠤⠤⠤⠒⠃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⠀⠀⣀⡠⠄⠊⠀⠀⠀⠀⡗⢒⡒⢁⠔⠒⠊⠀⠀⠀⠀⣀⠀⠀⡀⡀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢰⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⠂⠑⢊⣐⢂⡮⠭⠱⢞⢄⡨⠮⡀⢴⠀⠀⠀⠀⠆⠀⠐⠒⠂⠀⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠐⠠⣀⠀⠀⠀⠀⠀⢸⣁⢉⡈⡡⠤⠤⠊⠑⠒⠉⠑⠀⠒⠢⠼⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀⠤⡐⠜⠋⠁
⠀⠀⠀⠀⢀⠴⠈⠠⠤⠤⠄⠤⠤⠤⠤⠄⠤⠤⠤⠬⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠧⠤⠶⡈⢁⡀
⠀⠀⠀⠀⠘⣷⣮⡴⡤⠰⢶⣤⣤⣄⣄⣠⣡⣤⣐⣲⣶⠃⠀⠀⠀⠀⠀⠀⠀⣔⠈⡡⠄⠰⡀⠠⡤⢀⣐⠢⠤⣄⡉⢲⡄
⠀⠀⠀⠀⠀⢏⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⢋⡁⠃⠀⠀⠀⠀⠀⠀⠀⢙⠛⠠⠬⢵⣏⣘⣏⣁⣬⣡⠦⠶⠚⢛⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠀⣀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⣀⡤⠤⠄⠀⠀⠄⠄⠀⠀⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⢴⣞⣿⣩⣭⣿⣿⣭⣀⣀⣠⣤⡤⠦⠒⠶⠶⠶⠖⠤⠤⣶⣾⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣖⣚⣩⠅⢰⡾⠗⣫⡵⢛⣩⣤⣶⣾⡏⢉⣁⡄⠈⠅⠀⠀⠀⠀⡚⠀⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⣀⡤⡔⠀⣰⡛⢋⣉⠉⠀⢿⣿⠏⢉⣽⣶⡾⠿⢟⣛⣋⣭⣥⢶⠾⠟⣩⣾⣿⣿⣿⣿⡿⣞⣋⠥⠖⠋⠉⠉⠉⠁
⠀⠀⠀⠀⡠⣢⣅⣾⣽⠦⣻⢟⣍⣁⣐⠿⠛⠋⡴⠾⢿⡽⣿⣋⣉⢩⠁⠺⣡⡤⣲⣾⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⡐⢰⣓⣈⢿⠋⣶⡸⢆⢉⣭⠿⠴⠶⠚⠃⠋⣟⠓⠶⠤⠶⠿⢮⡐⠟⢱⣿⣿⣿⣿⣿⣿⡏
⠀⠀⢰⠁⣏⣼⣿⣎⣳⣝⣇⣘⣉⢠⡋⠀⢀⠊⠀⠸⡄⠀⠏⠈⡃⣽⠈⡿⡆⣿⣿⢿⢿⣿⣿⣿
⠀⠀⠘⣠⣿⣾⣻⠁⣽⢉⢳⠀⡀⢐⠠⠀⠸⠀⣤⢓⢰⠃⠀⢰⠁⠸⢸⣬⢫⡿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⡇⠈⡟⢧⣴⢣⠫⣺⣴⡿⠛⡉⠉⠳⣄⣘⠪⠈⠂⢐⣀⣳⣱⡞⢁⣿⣾⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢠⠄⡀
⠀⠀⠀⢙⠲⣌⢿⣇⣾⣿⣯⡞⠁⡄⢀⢤⡇⢨⠻⡛⣭⣭⠭⢍⡄⠁⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣻⠀⣹⠀⣀⠀⣀⢠⡏⡔⠁
⠀⠀⠀⠈⠀⠈⢧⠻⣿⣿⣿⣵⠘⢅⣀⢾⣀⠰⠵⡁⣢⡬⠞⣉⣴⣿⣏⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣧⣾⠙⣿⠉⣾⢣⡀
⠀⠀⠀⠸⡀⠀⠈⢦⡈⠻⠿⣏⣠⣰⡾⠾⠜⣂⣉⣥⣴⣶⣯⣿⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢉⠒⠁⠈⠀⠉⠀⠘⠔⠁
⠀⠀⠀⠀⠑⢄⠀⠀⠈⠒⠂⠬⠉⢥⣴⢚⠹⠃⣼⢈⡟⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠠⢣⡆⡄⠀⠒⠛⠀⢰⡆⣀
⠀⠀⠀⠀⠀⠀⠱⣆⢀⡤⠀⢠⠀⠮⠋⢘⠀⣾⣷⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠴⢾⠋⣿⠉⣶⢃⡏⣿⡞⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠯⣿⣤⣸⣴⣀⣾⣽⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠾⠀⠘⠆⠳⠀⠻⠚⠇⠻⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠚⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⡄⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠁⣀⠴⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⡆⠘⠀⡀⡂⠀⠀⣠⡳⣸⠞⠁⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀
⠀⠀⠀⢀⠎⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⠀⠀⠀⠀⠀⣰⢣⡟⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠀⢀⠞⠊⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⡸⢁⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠒⠋⠉⠑⠲⢌⣇
⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⢃⠎⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢢⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡜⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢣⠀⠀⠀⠀⢿⠀⡀⠀⢸⠃⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠄⠄⡀⠒⠈⢯⣆⠀⠀⠀⢸⠘⡡⠀⢸⠀⣠⠀⣰⠃⠀⠀⠀⠀⠀⠈⠠⠠⠄⠀⣀⣀⠀⠀⠀⠉⠣⠀⠀⠘⣆
⢧⠁⠤⣔⡒⡛⠛⣡⣠⠤⠖⠛⠚⠚⠛⠛⠶⠤⣬⣦⣘⣇⣼⠀⠁⢀⠃⠀⠀⠀⠀⠀⢀⣤⠶⠒⠋⡩⠩⠍⠉⠭⠝⠛⠻⣄⡀⠀⠸
⠈⠓⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣸⠇⠂⠀⠀⠀⢠⠖⣩⠤⠒⠚⠉⠉⠉⠐⠓⠒⠢⠤⣀⡀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠄⡀⡄⣴⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⢺⣹⠋⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣾⠇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠴⠖⠒⠒⠲⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠒⠦⣄⠙⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⢘⢧⠀⠀⠈⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⡀⡇⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀⠐⠀⠂⠔⠠⠈⠐⠠⢀⠀⠀⠀⠀⢠⠇⢠⠇⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⣊⣠⠾⠖⠋
⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣤⣶⡿⠆⠰⣿⡿⠷⠀⠀⢠⣴⡶⢆⣠⣴⠄⠰⣿⣶⡆⠀⣀⣴⣾⡶⠦
⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⣿⡇⠚⣿⡟⠀⣿⡇⠛⣿⣿⡄⣰⣿⣿⠀⠚⠉⢁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣿⡇⢰⣿⡇⠀⣿⣿⢀⣿⡿⣿⣿⢿⣿⡀⠘⠛⠛⠿⠿⣿⠃
⠀⠀⠀⠀⠀⠀⣤⣶⡆⠀⠉⣿⡇⠘⣿⣇⣠⣿⣏⣸⣿⠁⠉⠉⣈⣿⣧⣄⠳⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣼⡿⣡⣄⠘⣻⣟⣋⡛⠿⣿⡆⠀⢈⡛⣛⣭⣥⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣈⡉⠉⣾⣿⡏⠀⠛⣿⡿⣃⣤⣤⠀⣿⣿⠃⢻⣿⡏⠀⠐⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠓⠀⢻⣿⣄⣀⣿⡇⠛⣿⡟⠀⢹⣿⠀⣾⣿⣿⣤⣾⣿⡏⠁
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠟⠛⠀⣉⠛⠛⣿⡇⠀⣿⣇⠀⢸⣿⢠⣿⡏⠻⡿⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠋⠉⣤⣶⠟⠹⣿⣟⠃⢀⣿⡇⠀⠹⣿⣶⣾⠟⣸⣿⣀⠀⠀⠸⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠙⠿⠿⠿⠛⠀⠀⠀⠈⠉⠁⠘⠟⠛⠛⠁⠀⢲⡿⠿⠟




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠉⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⣿⣿⣿⡋⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠠⠼⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⠿⠿⡿⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠧⢶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀⣀⣀⣀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡝⠳⢤⠤⠤⢼⠴⠖⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⢿⣿⣿⣿⣿⣿⣿⣯⣽⣟⣛⢻⠻⢯⡉
⠀⠀⠀⠀⠀⠀⠀⠰⠀⣾⣳⣆⠀⠀⡆⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⢶⣿⣭⣽⣛⣻⠿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣀⣀⠃⠀⠙⠿⠋⠀⠀⢯⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⣿⣷⣿⣾⣶⣿⣿⣿⣷⣿⣿⣿⡿⠃
⠀⠀⠀⠀⢸⠉⠙⣷⠀⠀⠈⠀⠀⠐⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠑⠚⠋⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣀⠅⠀⠙⡏⣀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣎⡀⠀⠀⠀⠀⢗⠛⠟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠛⠷⣤⢤⣤⣤⣬⠴⠟⠚⢇
⠀⠀⠀⠀⣿⣹⠀⠀⠀⠀⣠⣴⣶⣿⣷⣶⣦⡀⠀⠀⠀⠀⣴⣴⠀⠀⠀⠀⠀⠀⠀⢠⠃⣠⣶⣷⣆⠀⠀⠘⡆⠀⠀⠈⢆
⠀⠀⠀⠀⢻⡞⠀⠀⢀⣾⢷⣿⢿⣿⣿⢿⣿⣟⣦⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⢀⠎⠀⢹⣿⣿⣿⢃⠔⠠⢷⠀⠀⠀⠘⡆
⠀⠀⠀⠀⢰⡇⠀⡏⣿⢿⢸⠰⢸⡆⢎⡇⡎⡾⣿⡏⡏⠀⢸⠀⠀⠀⠀⢀⣠⣤⣼⡀⢀⠀⠉⢋⠅⠀⠈⠀⣹⡂⠐⠠⠀⠘⡄
⠀⠀⠀⠀⢸⡇⠀⡇⢿⣺⢸⠘⣸⠇⣃⡇⢃⣷⣿⡇⠇⠀⢸⡄⠀⠀⠀⢸⠁⠀⢹⣷⠀⠄⠀⠁⠀⡀⠁⠀⠰⣇⠔⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠈⢿⣦⣶⣶⣿⣿⣯⣕⣣⠟⠀⠀⠀⢸⡇⠀⠀⠀⠈⠳⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠚⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄



⣿⣶⣾⢶⢷⠶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⠶⡶⣶⣶⣾⣿⣶⣶⣶⣶⠶⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⢿⣿⣄
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡆⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠈⣿⣿
⠀⠀⠀⣠⣾⣿⣿⡿⠛⠉⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⢀⣼⣿⣿⠟⠁⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢸⣿⣿⠛⠛⣿⡆⠀⠀⠀⠀⠀⣼⣿⠇⢀⣴⠶⣶⣶
⠀⣾⣿⣿⣧⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⠿⠆⠀⣿⠇⠀⠀⠀⢀⣼⣿⡟⠀⣿⡏⠀⠀⠙
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣠⠞⠁⢀⣀⣤⣾⣿⡿⠟⠀⣸⣿⠁⠐⢶⣾
⠀⠀⢀⣤⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⡾⠃⣠⣾⠿⠛⣛⣋⣁⣀⣤⣾⡿⠃⣠⣶⣤
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡿⠋⠀⣼⢁⣾⠏⠠⠖⠋⣉⣭⣭⣭⣉⣉⡀⢀⣩⡿⠋
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠋⠸⡇⠀⠀⢺⣦⡙⠧⠀⠀⠉⠉⠉⠉⠁
⢸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠁⣰⣷⠘⣿⣿⣦⠈⢦⡀
⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⢀⣤⣶⡶⢀⣠⣾⣿⣿⠀⣿⣿⣿⡇⠘⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⡀⢠⣿⣿⡏⢰⣿⣿⡿⠛⠁⢀⣿⣿⣿⣿⠀⣿⠃
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⣀⣾⡇⣾⣿⡿⠀⣿⣿⡟⣴⠟⢠⣾⣿⡿⢻⡏⠀⠁
⠸⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⢀⣤⠾⣿⡿⠠⠿⠋⠀⠀⣿⣿⣿⠏⢠⣿⡿⠋⢠⣾⠃
⠀⢻⣿⣯⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠸⣿⡀⠰⠟⠁
⠀⠀⠻⣿⣏⣩⣽⣿⣿⣿⠿⠛⠉⠀⠀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣈⣀
⠀⠀⠀⠙⠛⠛⠛⠉⠁⢀⣤⣴⣶⣿⣿⡿⢿⣿⠟⣿⡟⠛⣿⡏⢹⣿⠉⢻⡏⠉⣿⠋⢹⣿⠉⣿⡟⠻⣿⡿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢿⡏⠉⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⢸⡇⠀⣿⡄⢸⣿⠀⣿⡇⠀⣿⡆⢸⣿⠉⣿⡿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠀⣿⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⢸⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣿⣇⡀⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⡇⢸⣿⠀⣿⣧⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣾⣿⣦⣿⣧⣤⣿⣇⣸⣿⣀⣼⣇⣀⣿⣅⣸⣿⣠⣿⣧⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣭⣝⣛⡛⠛⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠛⢛⣛⣫⣭⣽⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠀⠀⢰⣶⣦⠀⠸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⠀⠀⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣀⣀⣈⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣶⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠟⠻⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⡀⢠⠄⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣮⣛⣛⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠙⣼⠀⢀⣼⡞⠀⠀⠀⣀⡤⠤⠶⠶⠦⣤⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⢸⣽⣦⡀⢹⣷⣌⣽⣇⢀⣢⡾⠉⠀⠒⠈⠙⠒⢦⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠶⣌⢻⣿⣧⣠⣿⣿⣿⣿⣿⣿⣷⣶⣞⣭⠄⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠠⢾⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣫⣴⡖⠀⠀⠈⣒⢌⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⣁⠀⠲⣿⣿⡟⠚⠋⠉⠁⠀⠈⠙⠛⢿⣿⣿⡟⠀⠀⠀⢀⣸⣷⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⠤⠀⡸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠻⠀⠀⠒⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠐⣄⠀⠀⢀⣠⡄⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⣦⢠⢟⣳⠿⠛⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠀⠀⠜⠛⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠰⠖⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠂⠀⢶⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠰⠾⡿⢁
⠀⠀⠰⠆⣶⣤⣄⡴⠄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠴⠤⠤⣄⣽⡿⠛⠃⠀⠐⠰⢦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠓⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢒⠀⠀⠀⠀⠀⢀⣤⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢹⡁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣤⡀
⠀⠯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠈⠳⣔⠤⣀⠀⡀⠀⠀⠀⠀⠀⢀⠾⠟⠛⡄⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠸⠋⠀⢀⠀⠀⠀⠀⠀⠀⠰⣄⢀⡉⠲⠶⠶⠚⠀⠀⣀⠰⠊⠁⠀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠀⠀⠀⠀⠀⣠⣄⣀⡴⠐⠋⠁⠀⢀⡀⣠⠰⠟⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠈⠉⠁⠀⠀⠀⠀⠀⠉
⠀⢀⠀⠀⢀⣼⣧⣠⣖⣀⠀⠀⠀⠉⠁⠀⠀⠀⢀⠤⠂⠁⠈⢁⣀⠀⣸⠶⣦⠌⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢑⡻⠶⣤⡀
⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠄⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣩⠔⠊⢁⠈⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠁⡠⠊⠀⡀⠔⠒⢦
⠀⠀⠀⠛⠤⢤⡤⠤⠤⠿⠟⠶⠤⣤⣤⡴⠎⠉⠱⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⣴⣿⣷⡤⠺⣤⡀⠀⠀⠀⠀⠊⠀⠀⢀⠊⠀⠰⢊⡤⠂⢈⣩⡀
⠸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄⡏⠀⠀⠀⠀⠀⠀⠰⢤⣼⣿⣿⣹⠃⢁⠘⠻⣶⠟⢥⣿⡀⣀⣠⠁⡠⢊⠴⠋⠀⠀⠈⠐⠶⣦⡄⠒
⠀⠘⢦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠂⢠⣤⣜⣲⣀⣠⣌⠿⡾⠧⣀⣸⣦⡴⠠⣿⠛⠿⠾⠶⠇⣴⡶⠶⠶⠞⠂⠀⠉⢓⠀⠁
⠀⠀⠀⠉⠛⠛⢻⣇⣀⣀⠀⣀⣀⣼⠟⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠀⠠⠤⠬⠤⠄⠐⠈⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⣤⠦⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣺⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠛⠉⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣪⡤⠗⠋⠺⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠄⢀⠊⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠔⠁⠀⢄⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⢹⣤⡀⠀⠀⠀⠀⠀⣀⣴⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰⠣⢦⡀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠱⡀⠀⠀⠒⢠⡶
⠀⠀⠀⠑⠿⣗⠶⠤⢒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⢦⡄⠀⠀⠀⠀⢨⡄⢀⡀⠠⠀⠀⣠⣴⣿⣧⣤⠊
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡀⢀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠤⠤⠀⠙⠳⠦⡤⠦⡀⠤⡀⢀⣠⣐⣫⡴⣛⣡⢤⠄⣷⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⣀⣉⣐⣋⣙⣉⣀⣉⠁⠀⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠒⢤⠜⢀⠀⠀⠀⠀⠀⠀⠀⡔⢶⣶⣄⠐⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠸⣷⣤⣀⡀⠀⠉⠉⠀⠒⠐⠂⠤⠀⠀⠑⠀⠋⢠⠇⣾⠀⠀⠀⠀⠀⠀⢰⣇⡀⠀⠀⠒⠤⠀⠀⠀⠀⠂⠀⠀⠀⠙⠢⣀⠀⠐⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠏⠛⠒⠤⣀⠀⠠⠄⠀⠀⠠⠄⠔⠐⣫⣾⠃⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡌⠀⠀⠐⠶⠀⠀⢀⠰⠂⠄⡀⠈⠓⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⡀⠀⠀⠀⠠⣤⣤⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣶⣤⣄⣠⣴⡯⠄⠒⠀⠃⠀⠀⠒⠈⣙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣶⡶⠦⠴⠶⠶⠀⠀⠰⠒⠟⠛⠉
⠀⠀⠁
⠀⠐⠒⠃

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⡟⡇⣶⢠⡆⣤⢠⣴⣴⡆⠀⠀⠀⠀⣄⢀⡄⡀⢠⣄⡀⢀⡄⠀⠀⠀⠀⠀⡷⣿⣴⣿⣧⣿⣷⣿⢱⡶
⠀⠀⠀⠀⠷⢻⣾⣧⣏⣾⢸⣿⠁⠀⠀⣘⣒⠘⣾⠀⡇⢸⢹⢧⡼⡇⢖⣡⠀⠀⣄⠻⣟⡈⣛⡙⣋⢻⡁⣬⠉
⠀⠀⠀⣾⣅⣼⢇⡷⡾⠶⡖⢿⣼⡇⠀⠐⠊⠁⠟⠀⠳⠼⠸⠈⠁⠇⠑⠒⠀⠀⣿⠬⣿⠿⣿⢿⢿⡿⡹⠃
⠀⠀⠀⠉⠛⠋⠛⠛⠛⠛⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⠀⠀⠀⠀⣁
⠀⠀⠀⢠⡘⡾⣧⣤⣦⣶⣼⣡⠀⠀⠀⢀⡐⢿⡾⣦⢰⣴⣷⣷⠀⣞⣅⠀⠀⠀⣾⣇⣾⣿⣴⢸⡏⡟⣹⠇
⠀⠀⠀⢶⡙⢏⣻⢛⣋⢙⡈⢶⡂⠀⠀⡬⢟⢸⣇⣹⢛⢙⣋⣿⣥⡭⡍⠀⠀⠀⣨⠏⣸⠙⠛⠚⠃⢱⡏⢀
⠀⠀⠀⠋⡿⠸⠿⡿⠿⢻⠇⠇⠀⠀⠀⠾⠁⠀⠻⠏⠘⠾⠛⠛⠛⠃⠀⠀⠀⠸⣿⣶⡟⠀⢄⣈⠄⢻⣷⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠂⠀⠀⢏⡀⠒⠂⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⠀⠀⠈⠀⠀⠉
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⡀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⡴⠀⠀⠰⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡎⠀⠀⠀⢀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠠⠀⡘⢡⡷⢤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⡄⢖⠀⣠⣴⣶⣶⣿⡿⢷⣼⣿⣿⣿⣿⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠷⢂⢹⣧⣤⣬⠍⠉⠉⠁⠉⠉⠉⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠑⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠠⠐⠞⠻⣦⡀⠀⠀⠀⠀⠀⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠷⣤⣍⣓⣶⣶⡤⠴⡋⠁⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣌⠻⣿⣿⣿⣿⣦⣌⣉⣉⣉⣁⡀⠔⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠀⢠⡴⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠋⠉⠉⠀⠀⠀⣀⣀⣤⡔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⠉⠁⠀⠈⠁


⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠤⠀⠆⠰⠀⠂⠀⠂⠐⠂⠐⠂⠆⠀⠄⢠⡀⢀
⠀⠀⠀⠀⢀⡄⠦⣉⣠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠲⠢⠤⢈⡡⠒⣠⡀
⠀⠀⠀⠔⠓⢠⡞⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⡦⠐⢂
⠀⠀⠐⣯⠀⠘⢀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⣎⣿⣷⣿⣷⣾⣇⣙⣷⡖⡦⢤⣤⣤⣴⢧⣀⠟
⠀⠀⠀⡘⢶⣄⣠⠈⠂⠦⠤⠄⣄⣀⣀⣀⢀⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠷⣳⡏
⠀⠀⠀⠶⣶⣿⣿⣿⣷⣿⣤⣴⣄⣤⣈⣤⣈⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢱⡿⠁
⠀⠀⠤⠤⠬⢽⣷⣕⢝⢝⢿⡻⣻⡿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣷⣿⣿⣿⣾⣿⣶⣆⣰⡇
⠀⠀⠀⠀⠒⠚⣿⡪⠂⠑⠕⢍⠪⡪⠪⠛⢕⢝⣿⣥⣭⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠉⠙⠷⣬⡂⡨⡢⢀⢄⠀⠠⣠⣾⣿⣿⣿⣿⣯⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠷⢵⣾⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣛⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠄⠤⣄⣀⣀⡀⢠⠔⠀⠀⠠⠤⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠔⠊⠀⢀⡀⠐⣄⣀⣀⣀⣀⠭⠷⡤⠀⠊⠀⠀⠀⠀⢤⣀⠠⠼⣆⠰⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠊⠀⠀⠀⡀⠒⢶⠛⠛⠛⠉⠀⠈⠻⢦⣄⠀⠀⠀⡈⢑⣶⣤⠀⠀⠀⠉⢳⣶⡏⠀⡄⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠀⠀⠂⠀⠲⣤⣀⡀⠀⠀⠀⣈⣲⣦⣄⣠⣤⣼⠶⠾⠷⢾⡛⠛⠋⠁⠈⠓⢶⣤⣀⡊⠌⢧⠀⠁⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⢀⠐⠁⠁⠀⠀⢄⠀⠀⠀⢠⣀⣤⡩⠿⢶⡛⠛⠛⠁⠈⠻⢤⣄⡀⠀⢀⠀⠈⠉⢓⣦⣀⣤⡔⠚⣋⣹⠁⣼⣇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠼⠗⠺⣏⡉⠁⠀⠀⠀⠌⠛⢧⣈⠐⠄⠀⢀⣈⣙⣶⡤⠶⠛⣛⢋⣽⣿⣿⡿⠁⠀⣨⠿⠋⠙⠻⠦⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⢀⣠⣀⢠⣩⠷⣶⣛⠋⠁⠀⠀⣀⣰⣿⠃⠀⢻⠋⢸⡁⣠⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠒⠐⠚⠛⠻⣯⡉⠁⠀⠐⠀⠀⠙⢻⣦⣀⣸⣿⠽⡿⣷⣤⡬⠤⠒⠋⠁⠀⠉⠒⠠⢀⠀⠀⠀⠀⠀⠰⠁⡌
⠀⠀⠀⢀⢁⠹⡉⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⢀⣙⣣⣄⣤⠶⠾⠛⠋⠉⣿⣿⡏⠀⢷⠿⠋⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡰⠁⠔
⠀⠀⠀⠀⠉⠇⠹⢲⢦⣄⣀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠛⠉⠉⠉⠳⣦⣀⠀⣰⡾⠷⢾⣿⠋⠁⠀⠀⠀⠑⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠌
⠀⠀⠀⠀⠀⠘⠦⠀⡘⠙⠛⠏⢉⠁⠀⠠⣀⣲⠲⠦⣄⡀⠀⢀⣤⡤⠛⠏⢹⠃⢢⠴⠛⢤⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠠⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⡀⠀⠀⢳⠀⠘⠀⠀⢠⡇⠈⠈⢊⠈⠻⣏⠀⣿⣤⠇⠀⠀⠀⠉⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠬⣀⡀⣶⢹⠞⠿⠋⠀⠀⠀⢴⣿⣿⣿⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠠⢤⠐⠐⠄⠀⠜⠉⠉⠀⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡈⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠄⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠐⢄⠊⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠦⣍⣘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⣸⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣠⠔⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⢀⢤⣶⠿⠛⠉⠀⠀⢀⢠⠃
⠀⠀⠀⠈⢇⠀⣈⣉⠐⠠⣀⢮⠢⡐⠄⠀⠀⠀⠀⠀⠙⠓⠲⢦⡴⢒⡵⠟⠁⠀⠀⠀⠀⢄⡴⠃
⠀⠀⠀⠀⠀⠑⠄⡀⠁⠲⣈⣧⠦⠬⠱⡾⣽⡴⡦⡄⣀⣰⣯⣥⣔⣉⡀⡀⠠⢀⡂⠬⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣉⣗⣾⡿⣿⠬⡯⢑⢔⡉⣀⠛⣧⣶⡖⡖⠒⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠤⠠⡿⠿⠹⣞⣛⢯⡉⣯⣽⣿⣘⢣⢻⣯⢏⣿⣾⣤⢀⡠⡀
⠀⠀⠀⠀⠀⠀⠜⠀⢘⢐⣝⡭⠨⣽⢛⠖⢚⣿⣓⠾⠏⠉⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠤⠂⠈⠑⠋⠉⠒⠃⠘⠯⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡷⢿⣿⣿⣷⣶⣶⣦⡿⠏⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⢾⣫⢻⡇⠸⡿⡿⠿⠻⡆⠈⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣡⣶⣶⠶⣬⣿⣿⣄⣄⠘⠛⠟⠁⢀⣼⣿⣿⣷⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡭⠙⢻⣿⣿⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣿⣍⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣄⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡄⠀⣺⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣽⣿⣿⣷⡀⠀⠘⠻⣿⣿⣿⣿⣿⡟⣿⣮⣹⣶⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⢃⠘⠙⣿⣿⣿⣤⣤⣤⣤⡈⢉⣩⣽⣿⢸⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⠿⠗⠀⠠⠄⠛⢋⣿⣿⣿⣿⣿⣷⣿⣿⣿⣻⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣴⣯⡽⣯⣿⣿⣛⣵⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠃⢰⣿⣿⣿⣿⣿⡯⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⢸⠁⢀⣴⠟⠃⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⢠⠚⢿⣿⣄⣈⣙⣿⣏⠎⠀⠀⠀⠀⠀⠈⠐⠻⢷⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⡀⠀⠀⠂⠀⠐⠀⠀⠀⠀⡎⢠⣿⣿⣿⡘⠿⠿⣻⣤⣓⣶⣦⣄⡀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠘⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⣷⡆⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢟⠷⣿⠟⡄⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⡿⠿⠟⠛⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣅⡄⣹⠿⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⣯⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣰⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣶⡌⠁⠀⠀⠧⢀⠀⠀⡀⠀⠀⠀⠀⢼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣁⣤⣴⠖⢢⣤⣤⠀⣀⠀⠀⢤⣅⡰⢟⣷⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠂⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣅⣀⠀⠀⠀⠀⣩⡏⢁⡀⠀⢠⡟⣿⣿⣿⣷⡀⠅⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣷⣄⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⢷⣦⠄⠀⠈⢈⣷⣄⣽⣿⣿⣿⣿⣿⣷⡀⢦⡀⠀⠀⠀⠀⠀⢸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣄⣀⡀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⣎⣛⠀⠀⠀⠀⢠⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠐⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣄⣠⡦⢀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠰⡼⠋⠙⠹⠿⣴⠀⠀⠀⠀⠀⢠⡈⣿⣧⣽⡄⠀⠀⠀⠰⣄⢱⡄⠀⠀⠀⣾⣿⣿⡿⠋
⠀⠀⠀⠤⢿⠹⣿⣿⣿⣿⣿⣿⣿⡧⠤⠀⠀⢸⣷⣿⣿⣿⠇⡀⠀⠀⠀⠈⠉⠳⣄⢀⡴⠋⠉⠉
⠀⠀⠀⠀⠘⢧⣻⣿⣿⣿⣿⣿⡿⠁⠀⠀⣿⣶⣿⠛⠇⠹⣷⣿⠀⠀⠀⠀⠀⢀⡤⠻⣄
⠀⠀⠀⠀⠀⠀⡟⠛⠿⠿⠛⢏⠀⠀⠀⠀⠘⠏⠈⠀⠀⠀⠘⠃⠀⠀⠀⢀⡴⠋⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⣠⡋⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠈⠀⠀⠀⣰⡀⠀⢀⣆⠀⠀⠈
⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣴⡿⠛⠉⠉⠛⢿⣦⡀⠀⠀⠀⢸⣿⣿⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⠄⣾⡟⠀⠀⠀⠀⠀⠀⣿⣧⢠⡄⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⡠⢳⣶⣿⣿⣿⡿⠃⠀⠀⠀⠘⠂⢻⣧⠀⠀⠀⠀⠀⠀⣿⡟⠘⠃⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⢱⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⣀⣴⣾⠟⠁⠀⠀⠀⠈⠛⠛⠀⠙⠛⠃
⠀⠀⢸⡜⣿⣿⠋⠀⠠⠰⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁
⠀⠀⠀⢳⡜⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣦⣄⣀⣀⣀⣠⣴⣦⠆
⠀⠀⠀⠂⢻⡉⣿⣿⣿⣿⣿⣿⣿⡟⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠈⠳⣽⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣀⣰⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠆
⠀⠀⠀⠀⠀⠀⡞⠙⠛⠛⠋⢣⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⣸⢀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⣆⢀⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡍⢉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣘⡧⠃⠀⠀⠀⠀⠀⠀⠡⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⢦⡤⣠⣽⠙⣿⠮⣇
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠠⢠⣏⠱⠘⠒⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢠⣞⡠⠈⡻⠷⢲⣍⣮⣛⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⠓⠐⠥⣥⣴⡞⠁⢘⣨⠄⠉⠍⣰⣶⡶⠂⠀⠀⠳⢄⠚⠁⠀⠶⠃⠰⠓⡽⠊⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⠂⠠⠀⠀⠈⠉⠁⠉⠛⠃⣶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⠐
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⡄⢀
⠀⠀⠀⠀⡄⠂⠀⠠⢀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢴⠀⠅⡀⠡⢨⠆⠟⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢵⣄⡀
⠀⠀⠀⢀⠀⠑⠈⠑⠖⢂⠠⠄⠘⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢨⠀⡈⠦⠁⢘⣴⠾⠆⠀⠀⠀⠀⠀⡮⠀⠀⣠⣴⣴⣦⡀⢡⣍⠀⠐⡀
⠀⠀⠀⠀⠑⠀⠁⠐⠈⠁⠤⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠆⠀⠫⠷⠇⠈⢍⠠⠀⠀⠀⠀⠀⠀⣀⠀⠤⠯⣘⢻⣹⡿⠄⠨⠂⡐⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠠⠀⠀⠀⢁⠀⠠⠀⠀⠀⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣇⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠿⣿⣿⣿⣷⣶⣤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⠿⠟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣿⣿⣇⣐⠃⣼⣿⣾⣿⣿⣿⡿⣿⣮⡻⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠅⠊⣟⠃⠛⠀⢿⣿⣿⣟⣾⣿⠇⣨⣽⣿⣾⣯⠹⣿⣿⣿⢿⣿⣿⣶⣦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡆⢠⣼⣿⠂⠐⠤⠰⣿⣟⢿⣿⣇⠀⢿⣷⣿⠋⠀⠀⣿⠛⣌⢛⡿⠓⢹⣿⡏⣛⣿⣶⣾⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣿⣿⣿⠄⢸⣿⣍⠁⠀⠀⠁⣀⡶⢿⣿⣿⣆⣴⢻⡧⢀⠀⠀⢈⠉⠀⠀⠈⠂⠸⣿⢃⠤⢺⣾⡉⢿⣿⣿⢷⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠘⢿⣿⣿⣧⣤⣿⣏⣫⡀⠈⠀⣀⢹⣿⣿⢿⣟⡁⣙⠂⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⡿⣡⣿⡇⡘⠻⠿⣮⣿⠛⣶⣿⣅⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⠉⠉⣸⣿⣿⡿⣷⣀⠀⢀⢀⣀⠀⠛⠀⠉⢀⠄⠐⠂⠀⡀⠀⠰⢶⣄⣀⣺⡟⢰⣿⢛⣅⣈⠀⣠⣠⢤⣴⣿⡿⢏⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⢿⣿⣿⣶⣾⡿⠗⢤⣤⣽⣿⣷⣶⣾⠀⢠⣶⣿⢣⣤⣤⣶⣶⣾⣿⡿⠁⣿⣿⣿⡿⠿⠶⠦⠙⡚⣉⡥⣎⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣷⡤⢄⡙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣾⣿⣷⠺⠿⠿⠿⠛⠛⡉⡀⠂⠨⠭⠄⡒⠈⠉⠀⣁⡠⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣀⠀⠈⠁⠒⠠⠄⣉⣉⡉⠉⠉⣉⣁⣀⣀⣀⠀⠠⠤⠤⠤⠒⠒⠉⣾⠁⠀⠀⢀⣀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⣄⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⣀⣀⣤⠴⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠶⠤⢤⣄⣠⣴⠤⠴⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⣄⡀⣀⡀⣠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢸⠷⣿⢸⣟⣠⡿⣧⢸⡇⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠒⠒⠂⠒⠒⠒⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⣤⣤⣄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⣀⣀⠤⠄⠒⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡔⢲⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⠓⠲⢄⠀⡀⠤⠒⠒⡲⠦⣀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠋⠀⠑⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠁⠀⠀⠀⠀⠀⠘⣦⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⢠⠀⠀⠀⢻⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠠⠀⠛⠺⡄⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣽⠀⠀⠀⠀⠀⠀⠀⠀⣈⠃⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⡀⣀⡀⠻⠗⠀⣣⠈⠀⠀⠀⠑⠄⣀⢀⠀⠀⠀⠀⠀⣴⠋⠊⢉⠉⠣⣀⠀⠀⡀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠐⠁⣠⢄⠹⢶⠤⠾⠃⠦⡀⠀⠀⠀⠀⠀⠈⠑⠂⠄⣰⣾⡀⠀⠀⠈⠀⠀⡤⠑⠛⠀⠀⠀⠀⠀⠀⠈⠓⠄⣀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⣀⡩⠞⣼⠁⠀⣠⠀⠘⢂⡤⠐⠒⠀⠐⠒⠠⢄⣹⡞⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠇⠀⡠⠄⠠⣄⠸⡀⣶⢌⢁⣠⡞⠉⣠⠀⠐⠀⡒⠀⠀⠀⠉⢧⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⢸⡆⢧⣅⣀⢀⡀⠀⠀⠈⠣⠈⠀⠁⠔⠁⠀⠀⠈⣳⣦⠤⢀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠥⠎⠀⡟⠁⠀⠈⡇⠀⣀⡈⠁⠀⠀⠀⠐⢂⠨⠓⠝⠋⠀⠀⠀⠀⢘⠀⡠⠂⠀⠀⠉⠒⠤⠤⣀⠀⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣇⠸⠖⠀⣣⠀⠀⠤⠂⠀⡐⠠⡜⠁⠀⠀⠀⠀⠀⠀⠀⢀⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠾⠀⠓⠒⠒⠙⢄⠀⢂⠀⠀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣡⠀⠀⠀⠤⠐⢞⠈⠑⠤⣀⣀⠀⣀⣳⡀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢿⠁⠁⠀⠠⣄⠀⠀⠀⠀⠀⢨⠀⠀⠈⠢⠵⠀⣠⣄⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢀⡀⠀⠉⠉⠳⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠈⠐⠂⢤⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⢀⣀⠤⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⠉⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠐⢄⣀⣀⣀⠠⠜⢗⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣁⠠⠊⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⣠⠴⠖⠒⠉⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠉⠉⠁



⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣶⣴⣿⣷⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠋⡉⠀⠀⢠⠀⠉⠙⠻⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⣤⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⠙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢴⠈⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⣿⣿⢳⡆⠀⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢼⠀⠘⣿⣿⣿⠛⢦⡀
⠀⠀⠀⠀⠀⠀⢸⡁⠀⣿⣿⣿⢸⡇⠀⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⣿⣿⣿⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣿⣿⣿⣼⠇⠀⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⣸⠀⢠⣿⣿⣿⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀⠀⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢼⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠋⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⣠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡹⢿⣷⣦⣄⣁⠀⠀⠘⠀⣀⣠⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢭⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢦⠹⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⡄⣿⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢰⠃⣿⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢀⠏⣸⠇⠀⢰⢿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟⠀⠀⡞⢈⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣾⠁⠀⡼⠁⣼⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣰⠏⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⠶⠖⠚⢻⡀⠈⠁⣿⠀⠈⠀⠀⠀⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣋⣥⣤⠴⠶⠶⠒⠈⠻⠦⡘⠙⢧⡘⠛⠳⠶⠶⠤⢤⣀⠉⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣥⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠾⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⣀⣼⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡇⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢹⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠖⢋⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⠶⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⡁
⠀⠀⠀⢀⣀⣤⣶⠾⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀
⠀⢀⣴⡿⠋⠉⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⡶⠶⠖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄
⠀⢸⣿⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢤⣄⠀⠀⠀⠀⠀⣹
⠀⠈⢿⣦⡀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⠿⣶⣄⣈⠙⠿⢷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠞⠋⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⣶⣤⣬⣉⣉⡙⠛⠛⠛⠻⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠉⠉⠁⣀⣀⣠⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠋⠉⠉⠉⠁

⣦⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣭⣭⣿⣵⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣶⣶⣾⣷⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⢰⣷⡇⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢸⢸⣿⡇⠀⢠⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡥⠀⠀⠘⠀⢿⡇⠀⢸⣧⡄⠙⢷⣄
⠀⠀⠀⠀⠰⡀⠀⣠⠴⠊⠉⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣥⢸⣿⡆⠀⠀⠀⢸⡇⢠⢸⣿⣿⣠⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠱⣾⡅⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢀⣠⠴⠋⢠⢸⡿⢸⣿⠀⠀⠀⢠⠀⠀⣸⢸⣿⡿⠟⣠⢞⠝⠁
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠠⡄⠻⣿⣏⢹⡄⠀⢸⢸⡇⣸⣿⠀⠀⠀⣼⠀⠀⣿⢸⠏⣡⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⠀⢀⠦⠐⣄⠀⠹⣿⣦⣽⣆⠏⣸⡇⡿⡏⠀⠀⠀⣿⠀⠀⠋⣠⣞⠕⠋⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣿⡀⠆⢨⡃⠈⢧⡀⠘⣆⠀⣿⠁⠀⠘⣆⠛⡇⠃⡇⠀⠀⠀⠏⣀⣴⡿⠛⠁⣀⣀⣏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⡀⠹⣄⠈⣷⣤⠼⠛⠁⠀⠀⠀⠈⢧⠀⠀⠃⠀⣠⣴⡾⢛⣉⡴⠚⣩⠄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠙⠶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⣶⠿⠛⠁⠀⠈⡿⠀⠀⠃⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣸⠁⠀⠀⢾⠀⠀⠀⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠷⠤⠤⠿⣾⣆⠀⢰⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠲⠶⠶⠖⠛⠋⠉⢉⣙⣛⡛⠛⢛⣛⣩⣤⠀⢠⣤⣶⣶⡖⢀⣽⢀⣸⣧⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣉⣀⠿⢿⣿⣦⡀⠀⢿⣿⣷⣌⡛⢿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣫⣶⣿⣿⣀⡾⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡅⠘⠿⠿⣿⣷⣶⣿⣿⣶⡀⢻⣿⣿⣿⣮⣿⣿⣿⣿⠏⣠⣬⣭⣿⣿⣿⣿⣿⣿⠿⠃⣼⡿
⠀⠀⠀⠀⠀⠀⢶⣿⡿⠛⠲⠶⠦⣬⣙⠻⠿⣿⣿⣦⡙⠛⣩⣥⣤⣭⣭⡁⠘⣋⣭⣤⣤⣤⣤⡤⠀⢀⣠⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠠⠆⢠⢉⣛⠶⣤⣉⠙⠿⢦⣬⣭⣍⣛⣛⣡⣾⣿⠿⠛⢛⣉⣡⣴⠶⣻⣿⢣⡿⠁⠂
⠀⠀⠀⠀⠀⠀⠈⠀⢛⠻⠶⠦⣤⣩⣸⣹⡵⡀⠙⠻⠶⢤⣤⣭⣉⣉⣭⡭⠴⠶⠒⠛⠋⢉⣡⢰⠮⢇⡾⠛⢡⡀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⡐⠒⠤⠉⠛⠷⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣴⠴⡚⠉⣄⠃⠀⠬⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂⠬⠐⠊⣉⠛⠶⢦⣤⣤⣤⣤⣤⡤⠤⠶⠖⠛⠉⠁⠒⠈⠉⣀⠀⠁⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠠⠄⠀⠐⠒⠀⠈⣁⡀⠀⠢⠄⡁⡠⠀⡁⠄⣒⡀⢀⠌⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠋⠉⠁⠀⠉⠉⠉⠛⠛⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⣠⡾⠏⡇⡈⡿⣦⡀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢀⣤⠞⠉⠙⠀⠀⠃⠃⠀⠙⠷⣤⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣀⡠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠠⡇
⠀⠀⠀⠀⠀⠀⡀⣴⣖⣦⠠⢄⣀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⢰⣶⡷⠀⠀⠀⠀⠀⠉⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠄⠊⠠⠘⠛⢉⠀⠀⢄⣑⠀⢿⣀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠈⠁⡠⠲⡄⠉⠀⠀⠀⠀⢀⡴⣞⢿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣒⠷⣖⣉⡽
⠀⠀⠀⠠⢞⠂⡤⢄⡀⠠⠄⠒⠋⡏⠀⣸⢹⠋⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡧⠊⠀⠀⠀⠀⢀⣴⠟⠀⠘⣾⡄⠀⠀⠀⠀⠀⠴⠿⡒⠄⠁⠀⣹
⠀⠀⠀⠀⠐⡙⠄⠀⠑⠀⠀⢀⣠⠇⠀⣧⡏⠀⠀⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠯⠙⡯⠈⠀⠀⣠⢔⣣⠖⠀⢀⣤⡾⠁
⠀⠀⠀⠀⠀⠈⠻⡙⢷⡀⠛⢁⡿⣈⢱⡯⣘⠀⠀⠀⠀⠉⠛⠷⠶⣤⣤⣤⠀⠀⠀⡖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠗⠉⡇⠀⢸⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠉⠃⡜⠁⢀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣷⠀⣀⠀⠀⠀⣀⡤⠴⠖⠒⠒⠒⠂⡄⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡉⣘⠃⠐⠈⠉⠀⣀⣉⡆⠀⢨⢶⣶⣾⣿⠿⠿⠛⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⠄⠒⠶⠚⠀⠀⠀⠀⠀⠑⠲⠶⠤⠴⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒⠦⢄⡤⠖⠚⣛⣉⣉⡉⣉⣉⣉⠉⠙⠛⠒⠲⢤⡀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡴⣋⠤⠒⠀⠀⠀⠀⠀⡞⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡠⠖⠊⠉⠁⠐⠊⠭⢝⡒⠤⣀
⠀⠀⠀⠀⠀⡞⠘⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠪⠙⠢⣄
⠀⠀⠀⣠⠖⡇⢸⠀⡀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢀⡞⠁⠀⢻⠈⡆⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢀⡎⠀⠀⠀⠸⡄⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⢎⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⡠⠤⢔⣒⣉⣭⠤⢤⣷
⣼⠀⢀⠀⣀⡀⢧⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⢠⢣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢔⣪⠵⠒⠋⠉⠀⠀⠀⠀⠀⠀⠙⣄
⡇⠀⠔⢋⣠⠴⠚⢦⣀⣠⡤⣤⣀⣧⠀⠀⠀⠀⢰⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡭⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠳⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣄⠀⠀⠈⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠹⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢕⣦⣔⡀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⢭⣓⣂⣀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠤⠤⠤⠄⣒⣒⡒⠠⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⠔⣡⠤⠖⠒⠒⠛⠉⠉⠉⠉⠉⠉⠉⠙⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⠓⢤⡀⠀⠀⠀⠀⢀⡴⢃⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⣉⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡠⣠⠞⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠠⣤⣦⣶⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣯⣶⣦⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢴⠐⠄⠀⣀⣁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⣾⠁⢴⣷⡄⠉⠿⢦⣀⠀⠀⠀⠀⣩⣰⡀⠬⠄⣿⣶⣯⣉⣀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡦⡀⠀⠀⠀⢠⣌⠀⠉⠉⢀⠀⢀⡈⢹⠀⠂⠠⠀⠀⠘⠢⣵⣿⣭⣤⡭⣥⠀⠀⣠⠏
⠀⠀⠀⢀⠀⠀⠀⠀⠀⣉⣯⠖⣦⣄⠀⠀⠳⣄⣠⣸⢷⣮⠽⣉⣐⣀⣴⣤⣀⠢⠀⠐⠅
⠀⠀⠀⠊⠠⠀⠀⢈⠀⠹⣿⣶⣈⣦⣧⠀⠀⠉⠙⠻⡬⠿⠒⠋⠉⠉⠉⠉⠉⠛⠫⢄⠀⠀⠀⠀⠀⡀
⠀⠀⢀⢰⣟⠁⠀⡘⠀⠀⢸⣿⣿⣧⡿⢀⠀⠂⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠉⠲⠄
⠀⠀⠐⡃⠁⠀⠀⠈⠂⡀⠀⠻⣿⣿⣶⠟⠀⠘⠀⠀⢀⠀⠀⠀⠀⡨⣴⠉⣤⢀⠀⠀⠀⠀⠀⠰⣤
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠑⠢⠶⠚⠃⠀⠀⠀⡐⠋⠀⠀⡄⠀⠁⣰⣅⣠⣿⣾⠀⠀⠀⠀⠲⠆⡈
⠀⠀⠀⠀⠀⠀⣁⠤⣀⠀⢀⡀⠀⠀⠀⢀⣤⠄⠁⠀⢀⣠⣾⣿⢿⣿⣿⣿⣿⣿⡦
⠀⠀⢀⣰⠿⢋⠯⣬⣷⣟⣲⠻⡄⠀⠀⠘⠑⡒⠂⠀⣾⣿⣾⣿⣿⡋⠋⠻⠙⠣⣅⣾⠁⠀⠀⠀⡠
⠀⠘⡿⡟⠓⡳⡞⢿⣟⡙⡞⣷⠤⣀⠀⠀⠒⠓⠚⠚⠛⠋⣀⡀⠙⠻⢣⡖⠰⢶⣿⣿⣯⠛⠃⠈⠀⠒⠂
⠀⠀⣶⣿⣠⣏⡽⠂⠈⠃⢸⡌⣷⣬⣷⡄⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⠈⠉⠒⠒⠛⠋⠀⠀⠀⢀
⠀⠀⠈⣭⣵⣿⣻⡏⢷⣄⣠⣗⡾⣩⣽⣽⡀⠀⠀⣶⣾⡘⢷⠀⢀⣀⡉⠀⠁⡀⠀⣀⣀⡀⠀⠈⠀⠔
⠀⠀⠀⠀⠉⠭⠉⠥⠰⠿⣷⠿⣿⠷⠟⠛⠁⠀⠀⠀⠹⢄⢺⡠⠀⠀⠀⠘⣿⣶⡿⢿⣟⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣴⠀⡀⠀⠀⠙⣦⠃⠀⠀⠀⠈⠘⣿⣿⣟⣹⡞⣁⠀⠂⠄
⠀⠀⠀⠀⠀⣀⣀⣠⠄⣀⠀⠔⠀⠀⠀⠀⠈⣷⣌⣀⣀⢀⠈⠙⠃⣀⠀⠀⠊⠄⣻⣿⣷⣿⣿⣠⡀⠀⠈⠆
⠀⢠⡷⣴⠟⣡⣤⣦⣶⡿⢄⠎⠀⠀⠀⠀⠀⠙⠛⠚⠋⣙⠡⠖⣲⣤⠀⠀⠀⢒⢃⡿⣟⣿⣿⣿⣠⣀⣰
⠀⠘⣷⣏⣿⣴⣶⣶⡟⠃⠊⠀⠀⢠⠄⠀⠠⠠⠦⠀⣠⣖⣶⣾⡽⠟⠁⠀⠀⠀⢡⣴⠿⠿⣿⡿⢿⡭⠃
⠀⠀⢹⣧⣿⣿⣽⣿⣿⡀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠰⠿⠒⠂⠉⠁
⠀⠀⠀⠁⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢷⠀⠀⠀⠠⠯⣴⣦⡾⢿⣦⡀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⣈⣩⠤⡾⠀⠀⠴⣤⡀⠠⢄⣀⢸⠟⠿⠞⠻⢷⣄
⠀⠀⠀⠀⢄⡀⠀⢀⡀⠀⠀⣀⡠⡒⣼⡅⠎⠀⠉⠀⠀⠾⠗⠾⠿⠷⢄⣙⡛⠛⠦⣄⠀⠀⢉⠓⢦⣀⢀
⠀⠀⠀⠀⠦⡌⠐⠊⢀⡴⢚⣷⡿⠛⠋⠀⠠⢀⠐⠀⢀⣠⡀⠀⠀⠀⠘⠛⠿⣷⣄⠈⢓⠀⠀⠀⠢⢈⠻⢿⡇
⠀⠀⠀⠀⠘⠂⣁⣶⡟⣴⠿⠊⠀⣀⡀⢠⡄⣄⣨⣿⣿⣿⠏⢒⠀⠀⠀⠀⠀⠈⠉⠑⡈⢇⠀⢰⡦⢦⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢀⣨⣄⣀⣾⣿⣶⣷⣾⣿⣶⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠈⠢⡆⠀⠉⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⣦⣤⣤⣤⣨⣿⣟⣻⡏⣨⠀⠂⠀⢠⣄⡦⠈⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠛⠋⠙⠻⠿⢿⣿⡟⠠⠇⠀⠀⠀⠙⠉⠉⠈

⠀⠀⠀⠀⠀⣠⣠⣤⣤⣤
⠀⠀⠀⠀⠀⣿⠛⠻⡏⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢦⡀
⠀⠀⠀⠀⠀⢈⠀⠀⠁⢀⢀⢁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢿⣿⣿⣿⡿⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣾⠇
⠀⠀⠀⢠⣄⣈⣀⠀⠀⠈⣈⣀⣠⡄⠀⠀⠀⠀⠀⠃⠀⠀⠻⠁⠀⠸⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⡙⢛⠉⠀⠀⢀
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠁⠉

⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⣀⣤⣤⣀
⠀⠀⢀⠀⠀⠀⢸⠿⠿⠻⡏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⣿⣿⠠⣿⣉⣉⣽⡆
⠀⠀⣽⣿⠒⢶⡞⡶⡦⠶⢳⠒⡒⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠙⠿⠿⠋
⠀⠀⠀⠉⠉⠉⠛⠓⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠶⠖⠈⣳⣾⠃⠴⠆⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠊⠈⠈⠳⡄⠀⠀⠁⠀⠀⠀⢸⣏⣀⣈⣿⠀⣯⣅⣀⣽⠆
⠀⠀⢠⡀⠀⠀⠸⡿⢿⡿⠷⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠈⠛⠛⠉
⠀⠀⠾⣿⣉⣿⣛⡓⣞⢛⣛⣋⣍⣿⠿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⢰⡖⠀⠀⠀⢶⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⣇⣀⡀⢰⣿⠀⢸
⠀⠀⢀⡀⠀⠀⠰⣿⣿⡿⠟⠁⠀⠀⣀⠀⠀⠀⠸⠀⠀⠐⠛⠂⠀⠀⠘⠛⠁⠀⢸⠀⠀⠀⠀⢰⣿⣿⣧⢸⣿⠀⢸
⠀⠀⠾⣿⠓⡶⣒⡟⡷⠶⠒⠒⣒⣿⠿⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⡟⠸⣿⠀⢸
⠀⠀⠀⠈⠉⠉⠉⠙⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠸⠿⣿⡇⠀⢿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣩⣽⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠁⠀⠀⠀⢀⣀⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠒⣶⡾⠯⠥⢐⣲⣶⣦⡒⣒⣲⣶⣶⡒⣠⣄⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡭⠶⣖⣻⣿⣿⣷⣛⣿⣿⣿⣖⣛⣿⣿⣷⣖⣚⣯⡤⢟⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣟⣉⡽⠖⢚⣿⢿⣿⣖⣻⣿⡿⢷⣒⣻⡿⠿⠶⢚⣋⣩⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⣶⣾⣿⣷⣶⣶⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠿⣿⣿⣿⣿⣿⠿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣿⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⠶⠶⠶⣿⠶⠶⠶⣶⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠈⠙⠛⠶⣤⣿⣤⠶⠛⠉⠁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃⠀⠀⠀⠀⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠛⠛⠲⠶⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠉⠉⠉⠛⠛⢒⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠈⠉⠛⠳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⢀⣀⣀⣠⠶⠛⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠚⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠴⢶⡟⠋⠉⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢠⠟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⢠
⠀⠀⠀⠀⠀⣼⠟⠛⠉⠁⠀⠀⠀⠀⠀⡠⠋⠀⠀⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⢀⡾
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢠
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢸
⠀⠀⢰⡏⠀⠀⠀⠀⣀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⣿
⠀⠀⣿⡴⠖⠚⠛⠚⠷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀⣴⠞⠉⠀⠀⠀⠀⢰⡏
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⣀⣠⣤⠤⠶⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⣀⡴⠊⠀⠀⠀⣰⠏
⢠⡿⢶⣄⠀⠀⠀⠀⠀⠈⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡧⠶⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁⠀⠀⢀⣤⠾⠋
⣼⠃⠀⠈⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣤⠶⠋⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⢿⡀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠶⠶⠞⠛⠉⠁
⠈⠻⣦⣀⠀⠀⠉⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⠉⠉
⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠉⠳⣦⣀⢀⡇⠀⣀⣠⣤⠶⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠠⣶⡦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢔⣚⢣
⠀⠀⠀⠀⠀⠀⠀⡟⡎⠓⠒⢮⣝⠢⣄⠀⠀⠀⣀⣀⣄⣀⢀⠀⠀⢀⢄⣺⣵⡬⠖⠚⠉⠸⣼
⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠺⠷⢾⢷⡌⠳⠚⠉⢭⠉⢫⡏⠉⠉⠛⢴⡿⣿⣿⠾⠯⠂⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣧⢏⠀⠀⠀⠐⡆⠁⣤⡉⢻⡻⠅⠧⣾⠟⣣⡄⠈⢠⣏⠁⠀⠀⠀⡞⡿
⠀⠀⠀⠀⠀⠀⠀⠘⢎⢣⡀⢀⣴⡇⣼⣯⡡⡆⡇⠀⠀⠀⡶⢫⡶⣺⣄⣷⠤⠀⢠⡞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣹⣦⣴⡆⡀⠟⠇⠋⠁⠀⠀⠀⠁⠸⠹⠛⡄⣿⣿⣚⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁⢨⡧⠙⠊⡤⠙⡠⠀⠀⢠⡀⠲⠈⠉⠁⣿⢻⡀
⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⣀⣾⡇⠀⡘⠁⡰⢃⣀⣄⣀⠣⠀⢦⠀⠀⢿⡄⠙⣦⡤⠂⠀⠒⢤⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⣿⣿⡇⠰⢣⢂⣞⣉⠒⠓⢀⣑⡆⠈⠐⠀⣼⣧⡞⠁⠀⣆⠀⣸⢀⡇
⠀⠀⠀⠈⣿⣿⣿⣧⣀⣃⣀⣿⣿⣷⣦⣿⣌⣯⡉⠁⠀⢉⣡⢇⣮⣴⣾⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⠿⠿⠿⠿⠿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣚⣽⡶⠾⣷⣶⣬⣍⠓⢤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢚⣉⣀⣀⡀⠀⠈⠉⠻⣿⣦⣝⢦⡉⠲⢤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠚⠉⠉⠀⠀⠉⠙⢿⣶⣄⠈⠙⢿⣧⣳⡄⠀⢳⡀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⣃⠀⠐⠶⣤⣀⡀⠀⠀⠈⠻⣷⣦⡀⠙⣿⣿⠀⠀⢷⡄⠀⢿⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣴⠾⠋⠙⠛⠶⣿⣿⣻⣧⡀⡈⠱⡄⠈⢻⣿⣦⠸⠟⠀⢠⢸⡇⠀⣸⣧⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⠍⠋⠁⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⣷⣿⣿⡿⣿⣶⣌⠀⣈⣀⣿⣼⡾⣿⣷⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡵⠓⠛⣓⠶⢶⣤⣄⠀⢄⣀⠀⠈⣿⣿⣿⡞⢖⣯⣷⣻⣿⣿⣿⣿⣿⡿⠃⣠⣏⣡⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠉⠉⠙⠳⢾⣿⣷⣬⣛⣹⣷⣾⣿⣿⣿⣾⣽⢯⢿⡿⢟⣿⣿⡿⠶⠏⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣠⣲⣽⣿⣿⣾⣿⡻⡟⣿⣿⠙⠖⢛⣸⣿⡿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡣⠖⠛⣉⣉⣙⠺⣷⣿⢻⢿⣿⢟⣟⣀⣿⣷⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣴⣶⡶⣾⣿⢿⣺⣷⣯⣴⡾⠿⠝⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣙⠿⠞⣛⣵⣾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣠⣤⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠁⠀⠀⢚⠁⠀⠀⠈⠉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠀⠄⡤⠸⠂⠀⠢⠤⠶⠟⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢤⠀⢠⣀⢀⡀⠀⠀⠀⠀⡰⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠒⠒⡁⠘⡇⢈⣿⠟⠻⠄⠶⡏⠰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠃⠘⣠⠰⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠘⠃⠀⠀⡶⠆⠀⠀⠀⠀⡠⠚⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠀⠘⠎⠀⠀⢈⡂⠂⠐⠁⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⠀⠀⠀⠀⠀⡠⠒⡀⠀⠀⠀⢳⡀⣤⠙⢶⠊⠋⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢠⡧⠖⠚⠛⠃⢤⠈⠈⠐⡋⠀⠀⢈⠗⠀⠀⡛⠀⢿⣷⠒⠉⠠⠎⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣠⡀⠀⠀⠈⣿⣿⠛⣡⠖⢀⠁⠀⠀⠀⠀⠒⠛⠉⠀⠀⠚⠓⢤⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠉⢲⠀⠀⠀⠸⠖⢛⠀⢈⠀⠀⠀⠈⢹⠁⠀⠀⠠⠒⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡂⠀⠀⠀⡵⠒⠦⡀⠘⠁⠀⠁⢦⠀⠨⡀⢇⠀⢸⣏⡀⢠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠖⣾⠁⠀⠀⠀⠀⠀⠀⠀⣨⠂⠀⠈⠉⠑⠛⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠶⠤⠴⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣟⡗⣶⣶⣶⣿⣿⣫⢑⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣵⣿⣾⣋⣹⣋⡉⢉⠝⢉⠍⢉⠟⢻⣹⡻⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢪⣴⢹⠋⠈⢰⣛⡥⣽⣉⣿⣦⢠⠄⠃⠀⠁⠿⠹⢷⡈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣳⡿⠉⠈⠀⠀⢀⣿⡷⣷⣄⣌⡱⢊⡀⢀⠀⢀⠇⠀⡞⢿⣿⣷⡀
⠀⠀⠀⡄⢠⣀⣠⣿⣳⣀⣀⣄⣄⣤⣾⣻⣶⡟⠳⣟⣗⣾⣷⣷⢻⠁⢀⡼⠃⠈⣷⡍⣧⣀⢠⢠⡄
⠀⠀⠀⠀⡇⠀⠸⡿⠉⢸⡄⠀⠰⡄⠀⠁⠈⠉⠉⠳⣭⣻⡟⢻⢸⣦⡾⠓⠂⢀⡿⣧⢸⡄⠀⠠⡇
⠀⠀⠀⠀⡇⠀⢸⣿⣦⣄⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣾⣯⣅⣁⣤⠼⡿⣿⢞⡇⠀⠀⡇
⠀⠀⠀⠀⡇⠀⢸⢳⣧⠇⠙⠷⣷⣂⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣀⣤⡠⠤⢿⣶⠃⠀⢰⡇
⠀⠀⠀⠣⠟⠉⠙⣿⡿⠀⠀⠀⠘⢝⡻⣿⡦⣤⣄⣀⠀⠀⣀⣀⣀⣀⣤⣸⣿⠇⢠⣋⡞⠉⠙⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣽⣵⣄⠀⠀⠐⠞⠛⣷⢎⣏⡉⡟⠛⠿⠻⠋⠀⠀⠈⢦⡀⣿⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢿⣰⣤⡖⢠⡀⠂⠹⠋⠙⠗⠶⡦⠀⠀⠀⢠⣰⡼⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢭⣿⣿⢶⣷⣤⣄⣀⣁⣠⣾⣧⣴⣰⣶⣸⣵⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠧⢉⣉⢧⣟⡿⠿⠻⣯⣽⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⢀⠉⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠄⠀⠀⠀⠀⠠⠠⠶⠀⠂
⠀⠀⠀⢇⡁⠀⠀⠀⠀⠀⣄⣀⡀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⣠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣲⠀⢀⠀⡄⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠮⠦⠀⠀⠘⠁⠀⢀⠆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠰⣶⠁⢀⡀⠀⠙⠀⠀⠀⠀⣸⢀⠀⠀⠀⠀⠀⢹⠉⠉⠁⠀⢈⠞⠉
⠀⠀⢈⠙⠀⠀⠀⢀⠤⠤⠀⠀⠀⠀⠀⠈⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠁⠇⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠁⠀⢈⣀⡀⡐⠦⣀⡀⠀⠀⠈⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⡖⠶⠒⠚⠉⠀⠀⠀⠀⠖⠀⠀⠀⡐⠂⠉⠀⢀⠀⠀⠀⡀⢀⡄
⠀⠀⠀⠄⠠⠀⣀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠁⠀⠀⢣⡤⢀⠄⣎⡃⠀⠀⠁⠉⡀
⠀⠀⠐⠀⠈⠁⠘⠛⠍⠉⠁⠉⠋⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠉⠓⠖⠋
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠆⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠃⠀⠀⠀⠰⠈⠀⠀⡳⢀
⠀⠀⠀⠀⠅⠈⠉⠉⠉⢩⠾⠍⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠈⢥⠀⠀⠂⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠠⣄⠄⡔⢘⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠆⢀⢁⠠⠂⠄⠀⠀⠀⠀⠀⠀⠁⠀⠂⠈⠀⠀⠀⠉⠉⠈
⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⡄⠀⢀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⣆⡠⠔⠂⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣴⡶⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠃⠀⠀⠀⠀⠢⠈⠶⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⡤⡀⠀⠀⠀⠘⠀⠀⠘⠁⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠂⠀⢀⡘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠓⡶⢈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠂
⠀⠀⠀⠠⠠⠔⠀⠀⠀⠀⠀⠀⢄⡤⠀⠀⠀⠉⠀⢰⠇⠀⠀⠀⠀⠠⠀⠀⠀⠘⠳⣀⠈⠁⢄⣠⣀⣀⣀⠤⠈
⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⢢⠤⠀⠀⠀⡀⢀⡔⠆⠀⠀⠈⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠠⠀⠒⣠⠤⠆
⠀⠀⠀⠀⠀⠀⠚⠌⠀⠠⡀⢠⡼⡆⠀⠀⠀⠀⠀⠀⣿⠫⠀⠒⠂⠧⣹⡋⠀⠀⠀⠀⠀⠀⠀⠰⠔⠛⠓⠀⠀⠀⠸⢗⠻⣦⠠⠶⠀⠈⠟
⠀⠀⢠⠶⣶⠠⠄⠢⣶⣘⣷⠶⡞⢁⡄⠀⠀⠀⠀⠀⠉⠀⠀⠦⠤⠐⣽⠔⠈⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠇⠀⠚⠉
⠀⠀⠀⠚⠛⠨⡍⢂⠠⣯⠠⢠⠴⠛⠀⠀⠀⠀⣊⣟⡃⠀⣀⡐⣆⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢠⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⣿⠀⠀⡉⠀⢁⢿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⢋⣿⡏⠘⠛⡆⢠⡌⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⡵⠛⠀⣿⣿⣀⡀⢸⡇⢸⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⡞⠀⣴⠆⢹⣿⣿⡏⠐⢠⡟⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠞⠀⠼⠁⡀⠘⣿⣿⠁⡄⠈⠃⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠴⢀⣴⠇⠘⠁⠀⣿⣟⠀⡇⢀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣠⣾⡏⠀⠀⠀⠀⣿⣿⠀⣿⠈⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢰⣿⣿⣧⠀⠀⠀⠀⢻⣿⠀⠛⠀⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⢀⣿⣿⠟⠁⠀⣶⣶⣦⣀⢻⡆⢀⠀⠻⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣿⡿⠋⠀⢀⣠⣿⣿⣿⣿⣿⣷⡀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢰⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⢻⡄⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠠⠟⠀⠀⠀⠀⠈⠻⣯⣛⣿⣿⣿⣯⣿⣧⡅⠀⠄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣽⣿⣷⡙⠃⠀⠸⠆⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣬⢻⣿⣿⣧⣄⡀⠀⢂⣡⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣴⣿⡿⠿⠒⣻⣏⡛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢀⡳⡜⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠇⢾⣿⣯⣤⣻⣿⣿⣿⣿⣤⣟⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⡰⠜⣦⠀⣝⢧⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣶⣿⣿⣿⣿⣿⣷⠿⣮⡀⣄⠱⣽⣦⡀
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠙⢿⣯⣭⣿⣿⣯⣭⣉⠉⣭⣥⣴⣿⣿⣿⣿⣿⣿⣧⡀⢶⣿⣷⣬⠣⠙⣷⠻⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣿⠀⢻⡀⠀⠛⢿⣿⣿⣯⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣻⣝⢿⣷⣦⣌⠻⢆⠁⠀⣙⡓⡲⠶⠶⠤⠾⠳⢶⣄
⠀⠀⠀⠀⠀⢠⣿⠀⠀⢳⡄⠀⣾⣿⣻⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠛⣉⣻⣏⣛⣛⠿⠟⢿⣷⣮⣑⡄⠈⢻⣿⢿⣿⡷⢊⣠⣤⣿⡁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣷⡜⢿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⡟⣿⣷⣄⠙⢻⣿⣿⣿⣷⣿⠷⡌⣉⣿⣶⣄⡈⠿⣿⠛⣴⣿⣿⣿⢸⣷
⠀⠀⠀⠀⠀⠘⣿⠀⠉⢿⣿⣿⣿⣄⠈⠙⠛⠟⢛⣋⣭⣿⣿⣿⢿⡶⠶⢾⣿⣿⣷⡄⠈⢩⡄⣮⣭⣽⡇⣿⣿⣯⣿⡿⠇⣘⠃⠋⠁⣸⡿⢻⡿
⠀⠀⠀⠀⠀⠀⢹⡆⠀⢸⣿⣿⣿⣿⠇⡀⠰⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠛⣯⣭⣴⣞⡙⣹⣟⢛⣻⣶⡿⠿⠋⡉⢰⣾⣿⠷⢀⣼⣟⣠⠿⠃
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⣿⣿⣿⢱⣿⣿⣦⡈⠺⢿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣶⡽⠋⢀⣴⢟⣷⣿⣿⣿⡆⢋⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠻⣿⣿⣿⡟⢿⣿⣿⡆⠀⠐⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⢠⣶⠈⢻⣿⣛⣿⠟⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠙⠋⣭⣾⣿⣷⣞⣻⣿⣦⣀⡀⠈⠻⢿⣿⣿⣿⠿⠛⠈⠉⣀⠀⠀⠾⠈⣥⣶⣿⣿⣿⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡜⠆⢲⣄⠉⠻⢯⢼⣿⣿⣿⣟⣛⣿⣶⣶⣶⣶⣧⣀⣠⡴⢞⣩⡴⢚⣣⣴⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡆⠉⠀⠠⡀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠢⣄⠀⣀⡤⠶⠉⠙⣛⣛⣿⣿⣿⠿⣿⣿⣿⠿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡹⣿⠋⠁⢄⣀⣸⡿⠿⢿⣿⣯⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣤⣀⣀⣈⣤⣶⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉