BLT Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⣿⣿⣿⣷⣶⣶⣤⣭⣭⣥⣔⣒⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣕⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢈⣁
⠀⠀⠀⠀⠀⠀⢸⣷⣌⠙⠿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣭⣥⣽⣶⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣤⣄⠈⠀⠉⠙⠛⠷⢿⣿⣿⢿⠛⠉⠉⣀⠀⠭⠉⠀⣀⣉⣥⣤⣶⣶⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢰⣶⣶⣍⠓⠭⢙⠻⢷⣦⣴⣤⣄⣤⣀⣀⣀⣬⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠯⠭⠁⠒⠒⠒⠓⣈⣉⠉⠉⣉⠉⠤⠒⠶⠶⢂⡄
⠀⠀⠀⠀⠀⠓⠉⢛⣉⣉⠀⠀⢄⣀⡉⠛⠿⢿⣿⢹⣿⣯⣿⣿⡿⠿⠻⠛⠋⠉⠉⢉⡩⢤⣤⠖⠀⠙⠛⠶⣯⣭⡭⢉⡠⠒⡡⠶⠖⠒⠒⢠⠌⠁
⠀⠀⠀⠀⠀⠀⠀⡘⠷⣾⣯⣤⠈⠻⣛⣣⣴⣤⠄⡉⠛⠉⠉⠀⠤⣤⣲⠶⠶⠞⣋⡥⢚⣡⣤⣴⣤⣤⣤⣄⡀⣀⣀⣭⠴⠶⢂⣠⣤⠉⣨⣤⣼⣶⠂
⠀⠀⠀⠀⠰⠖⠳⣤⣀⡀⠀⠈⠁⠀⠙⣉⣥⡴⢊⣴⡿⢟⡻⠭⠥⣤⣞⣉⣉⣉⠤⠄⠉⠙⠛⠿⠶⡒⣊⠡⠶⠦⣤⠤⠤⠶⢒⠋⣁⡞⣄⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⣈⡉⢉⠉⠛⠳⠄⠀⠀⠴⠾⠛⠉⠀⣤⣤⣁⣀⣀⣀⣀⣤⣴⠶⢛⣁⣶⣿⣷⢿⠿⢮⣿⠃⠀⠆⠲⣶⣽⠏⣈⣁⣤⣴⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣮⣭⣤⡸⠶⠟⣣⣴⣶⢾⠿⢷⣄⢬⣉⠉⢉⡉⡉⢤⠔⢾⠋⠁⠈⠛⠁⣀⣀⣈⣫⣤⣴⣶⡶⠦⠶⠿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣾⣷⣬⣉⣀⡐⠂⠆⠙⠛⣁⡚⠈⣁⣀⣠⣤⣤⣶⣶⣾⣿⣿⣿⠿⠿⠍⠁⠀⠀⠐⠂⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠉⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠸⠿⠏⠛⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄

⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⠒⢒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⣶⣿⣿⣿⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⢨⡇⠀⠀⠀⢸⣿⣿⡿⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⣽⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣹⣿⣆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⠛⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠺⣿⣿⣆
⠀⠀⠀⠀⠀⢧⡀⠀⢀⠉⠐⢄⠀⠹⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⡜⡄⠀⠀⠀⠀⠀⢙⣿⡜⡆
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠋⠀⠀⠀⠀⠀⣠⢊⡿⠳⠄⠀⡀⣀⡶⣿⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢻⣾⣿⣷⣦⡴⠒⠈⠁⠉⢉⠻⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠴⣏⣁⣶⣿⣿⣽⣯⡍⠁⠐⠛⠀⠙⢻⠁⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢯⠚⠁⠀⠀⠀⠉⠑⠢⠤⢤⣤⣤⣤⠤⠔⢲⠛⢻⣻⣲⠖⣻⣧⣿⡿⠻⠿⠟⠓⠀⠀⠀⠀⠀⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⡨⢿⣦⣉⠛⠟⣳⣇⣠⣷⣤⣶⣿⣷⣿⡅⣸⣿⣶⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⠀⠀⠀⠀⠀⠈⣿⡯⠁⠀⢹⠷⠯⣡⣿⣽⠟⣿⣏⢸⡟⠀⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣴⢁⡄⡀⠀⣰⠏⠀⠀⡜⠓⠀⠀⣈⠉⠛⢴⢿⣿⣦⠁⠀⠀⢹⠀⢀⣤⣶⣶⣦⣀⣠⠴⠢⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⠿⣷⣤⡾⠛⠀⠠⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠻⠟⠀⠀⣀⡼⣰⣿⣋⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡆⠻⣿⣇⡤⠀⠀⠀⠄⠀⠀⠀⠀⢷⡀⠀⢀⣈⠉⠃⠘⡟⠀⢹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⡶⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣾⠛⠿⣤⣴⡇⠃⠀⠈⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣶⣿⣿⣧⡀⠈⡙⠊⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⣴⡿⠛⢿⣿⣿⣿⣿⢿⡿⠷⣠⣦⣄⠀⢀⣿⣿⡃⠀⠀⢀⡄⠀⠀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⡀⣰⣿⡏⢀⣀⠈⠙⠿⣿⠷⠋⠀⠄⢿⣿⣿⠛⠉⢠⡌⠁⠀⠈⠀⠀⠠⠀⠀⢫

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠋⠛⠛⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠆⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠿⠶⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀⣼⣿⢀⣴⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠋⠩⡄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⡶⠟⠉⣿⠟⣫⢆⢌⡝⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠙⠻⢶⣤⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⣛⠉⠉⠉⠁⠀⠀⢀⣠⡿⠠⢖⣒⣩⣴⠟⠁
⠀⠀⠀⠀⠀⠀⢰⡟⣫⠼⣿⣤⣐⠠⠈⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠁⠀⠙⣂⣠⣤⣤⣴⣶⠾⣟⠋⢠⣾⣟⣋⠉⠁
⠀⠀⠀⠀⠀⠀⠸⣷⣍⡧⢖⣭⣿⣿⣶⣤⣀⠀⠀⢉⠛⠷⣶⣤⣀⣀⣴⡾⠟⠉⠀⠀⢀⣤⡶⠟⠋⣉⡵⢹⡿⠷⣶⣶⡶⠟⠉⢉⡛⣷⡀
⠀⠀⠀⠀⠀⣀⣀⣈⣙⣿⢿⢍⡡⠴⠚⠯⠛⠻⢶⣦⣄⡀⠀⠉⢛⣫⠁⠀⠀⢀⣤⣾⣿⡩⣀⠶⠶⠃⣠⡿⠃⠐⠠⠀⢀⣤⣬⣉⣴⡿⠁
⠀⠀⠀⠀⠀⣿⡛⠛⠛⠅⣠⢄⡔⢉⣭⣭⣉⣡⣴⡿⢋⠻⢷⣦⣐⣂⣠⣴⣾⠛⡉⠄⡌⢿⣶⣶⠾⢟⡫⠤⠤⠤⠴⢃⣿⢿⡏⠉⠁
⠀⠀⠀⠀⠀⠈⠻⣦⠉⠩⠖⢋⣰⣿⣏⢭⠩⠉⠙⢿⣌⠒⠶⡈⠛⣛⣉⡩⢻⣧⠘⡖⠘⠈⣻⣦⡀⠀⣴⠟⠛⠻⠿⠟⠋⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⠛⠩⠄⣿⣌⣓⣩⣭⣀⠙⠷⣦⣬⣭⣤⣤⣶⠿⠻⣷⣬⣾⠟⢛⣉⣡⣼⡟⣈⠥⠄⠀⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣭⣀⠀⠉⣯⢍⡉⢻⣆⠀⡀⠈⣉⠉⢁⣠⣤⣜⣈⠙⣡⣾⠟⣉⣩⣥⡶⠶⠾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠄⢻⣦⣉⣩⣤⡾⣛⠉⠉⡉⠛⠛⣫⣵⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⣈⠉⠉⠱⠶⠒⠀⠀⣠⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣌⣉⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⠼⠆⠀⠀⠀⠂⠀⠢⠤⣄⣀⣀⣀⣤⠤⠤⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠠⠔⠇⡶
⠀⠀⠀⠀⠀⠁⣀⡠⠤⢝⡂⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠄⠒⠚⠋⢉⠉⠉⠁⡀⠔⠍⠀⠀⠉⠲⠈
⠀⠀⠀⠀⠀⠀⠮⡣⢀⡀⠑⢦⠀⠈⠉⠉⣲⠶⠶⠶⠉⠉⠀⠠⢐⣠⠔⠋⠁⠀⠉⠙⠿⠋⣀⡁⠄⠂⢀⠖⠈⠀⠐
⠀⠀⠀⠀⠀⠀⢣⣌⣑⡂⠛⠺⡶⠄⠠⠊⢀⣀⠤⢀⣤⣤⡤⠭⠍⠀⡰⠒⠒⠀⠂⠃⢠⠀⠀⠄⠒⡊⣡⠌⠆
⠀⠀⠀⠀⠀⠀⠈⡀⠉⠈⠛⠉⣉⠑⠒⠋⠉⠑⠤⠤⣤⠤⢤⠀⡂⣡⡤⣄⡠⠲⠧⠀⠋⠛⠑⠂⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣀⡀⠈⠐⠒⠋⠭⠱⠤⠶⠐⠚⠉⠘⠈⠁⠀⠀⣀⡀⠀⠀⠀⠤⠔⠒⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⣀⡀⠀⠀⠀⠀⡀⠢⠤⠐⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡆⢸⠀⠀⡇⠀⠰⣙⢂⢤⠀⣤⢄⢀⢤⡇⣀⠀⣠⡅⢠⠄⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠟⠸⠀⠀⠇⠀⠰⠼⠛⠤⠧⠏⠸⠜⠤⠧⠙⠛⠃⠣⠣⠀⠇⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⣼⣼⡿⣦⣿⣧⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠉⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣼⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡶⠂⠲⠶⠼⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣝⣻⠿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣽⣿⣷⣶⣭⣍⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠢⠀⠀⠈⠁⠄⠀⣀⠀⠀⠄⠀⡤⠀⠌⠁⢰⢤⠦⠶⠿⠶⠶⠶
⣶⣄⣀⡀⠀⠀⠀⣀⣠⣤⣠⣤⣴⣶⣤⣄⠀⠀⠀⣀⡠⠀⠀⣦⡀⠀⠀⠉⠓⠯⣛⣶⣀
⣿⣄⠙⠻⣶⣶⣾⣿⣿⡿⠛⣏⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⠿⠞⣷⣶⡀⠉⠉⠁⠄
⠈⠹⠖⢀⠀⢉⣉⣉⣉⣉⡐⠠⣛⢻⣿⣿⣿⣟⣛⠛⠛⠟⠋⠉⠁⠦⢿⢿⣿⠇⠀⠈⠀⢰⡀
⡀⣬⠀⢰⣿⣿⣿⡏⠙⠁⠉⠑⢮⢻⣿⣿⣿⣿⣿⣦⣆⣀⡠⠴⢀⣤⣄⠀⠀⢀⡄⠀⠀⠀⡁
⠿⡘⠂⠘⠁⢹⣟⣤⣄⠀⣁⢠⣀⢠⡍⢙⠿⢿⠷⠶⠈⠫⠾⠃⡈⠉⠛⠁⢀⡐⠤⠂⠀⠐⠁
⢼⣟⠻⢷⠖⣼⠻⢟⣅⠰⢀⠄⡄⠈⣧⣽⡶⢘⣲⠄⠀⠤⠀⠀⢀⣤⣀⠿⠭⠒⠁
⢾⣿⢗⣘⠀⠟⠀⠈⢋⡿⣈⢠⠗⢶⣈⣉⡠⢻⣳⣅⠥⢊⠉⠛⢮⠀⠉⠁⣦⠈⢀
⠒⠈⠁⠊⠓⣒⡒⠈⢉⠉⠉⠙⠂⢀⠉⢀⡀⠀⢸⣧⠐⠎⠸⠂⢠⡀⠐⠀⠀⠊
⠈⠀⠐⢒⠂⠈⣀⠀⠈⢁⠑⢇⣀⡀⠀⠠⠁⠀⠚⠛⠊⠈⠉⠁
⠑⠂⠠⠤⠤⠐⠈⠁⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉⠓⢛⠛⣂⣀⣀⣀⡀
⠀⠀⠀⠐⠒⠒⠚⠻⠯⢽⠿⠖⠒⠂⢀⠤⢤⢤⠤⢴⣣⣴⢤⡄⢠⡄⢨⣷⣈⣘⣩⣿⡯⢤⠴⣶⣒
⠀⠀⢀⡀⠐⠀⠋⠛⣻⣯⠦⠰⢷⣿⣿⣤⣤⣤⢤⠼⣿⠛⢛⣛⣛⣫⣭⡭⢷⣿⡶⣷⣿⣿⠿⠻⠓⠒⠂
⢐⢽⣭⠇⠀⠀⠀⠀⠘⠋⠈⠸⠼⠻⣿⠛⠉⢉⠙⠩⢅⣭⣙⠛⠻⠽⠋⠉⠉⠉⠉
⠈⠑⠙⠂⠱⣀⡀⠤⠋⠐⡒⠴⡄⢨⣬⠀⠀⠀⠀⠑⠠⠬⠠⠀⠀⢀
⠠⢳⢤⣀⠈⠐⠀⠀⡣⠤⢤⢀⣈⣉⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠚⠰⠂⣠⠄⢀⡀
⠀⠘⢸⣿⡍⢓⣦⢤⣀⠀⠀⠀⠠⠀⠈⠑⠤⣀⡄⠄⠞⠃⣀⡠⠤⠟⠻⠗⠀⠄⢀⣀⡁⠐⠉⠄⣒⣠⡀
⠀⠀⠈⠁⠀⠀⠘⢸⡿⠛⠒⠪⡅⣐⠀⠀⠀⠀⠍⠙⠛⠉⠑⠒⠀⠉⠐⠊⠓⠂⠀⠈⠙⠒⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠈⠃⣿⠿⠗⣶⣤⣔⣀⠑⢀⡀⠀⢀⣤⣀⡀⠀⠽⠁⠠⠨⡅⠀⢀⣠⣾⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢃⣿⣟⡲⠦⣄⡀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⣨⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠓⠨⢽⢲⣤⣼⣿⣿⣶⣄⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠤⠬⣍⣛⡃⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⠛⠛⠛⠿⢿⣿⣷⣶⣤⣀⡀⠀⠀⠀⣠⣶⠾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢠⡄⠀⠀⠀⠀⢈⡉⠛⠻⣿⣿⣿⣶⡾⠟⠁⣀⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⡀⠀⠀⠀⣈⣁⣤⡾⠋⠉⠀⣀⣀⣠⣾⠛⠁⢹⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠦⠀⠈⢁⣴⡾⠛⠛⠛⠉⠀⣠⡾⠟⠛⠛⠉⠀⣠⡾⠟⠻⠿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢁⣤⣶⡶⠶⠟⠋⢀⣤⣶⠶⠶⠟⠋⢀⣠⣶⠶⠾⠟⠋⠀⣠⣴⣦⡄⠉⠙⠛⠿⢿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣴⠿⠋⢀⣀⣤⣤⣴⡿⠋⢀⣀⣤⣤⣴⡿⠋⠀⣤⠀⣠⣶⣾⠏⠀⠈⣿⠀⠀⣤⠀⠀⠈⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠉⠉⠀⣀⣴⡿⠋⠉⠉⢀⣀⣴⡿⠋⠉⠉⠀⠀⠀⠀⢀⣸⡏⠀⠀⣀⠀⠘⢻⣦⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠸⣿⠛⠁⢀⣴⠾⠛⠛⠛⠁⢀⣴⠾⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠃⠀⢀⡉⠀⠀⢠⣿⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⢹⣷⡾⠟⠁⣀⣤⣶⣶⡾⠟⠁⠀⣀⠀⠀⠰⠆⠀⠀⠀⢀⣤⣾⡇⠀⣤⠀⢀⣿⠃⠀⠀⢿⣧⠀⠀⠠⣤⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢻⣧⣠⣶⠟⠁⢀⣀⣠⣄⣀⡀⠉⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⢻⡇⣼⡏⣀⣤⡄⠀⣿⠇⢀⡀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⢀⣴⡾⢛⣉⣭⡉⠉⠛⢷⣦⡀⠘⠃⣠⣴⠾⠟⠀⢴⡆⠀⢸⣿⣿⡾⠟⠁⠀⢾⣯⡀⠈⠁⢠⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⠇⢠⣾⠋⣴⣻⣼⣷⠇⢠⣖⣦⡙⣷⡀⢰⣿⠁⠀⠀⠀⠘⣿⡀⢸⣿⠟⠁⢀⣀⠀⠀⣽⡇⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⣸⣿⡟⠀⣼⡇⢀⣈⡙⠛⠟⠠⣿⣿⡴⣧⢸⣧⢸⣿⠀⠐⣶⡀⠀⢿⣇⣾⢏⣠⡾⠟⠉⠀⢴⣟⠀⠀⣸⣿⡟
⠀⠀⠀⠀⢠⣿⡿⠀⠀⢿⡇⢿⠾⣿⣷⠀⣤⣌⣉⠛⠋⢸⡟⠘⣿⠀⠀⢹⣧⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠈⣿⡄⢰⣿⡿
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠘⣿⣌⠿⡽⠃⣸⢿⡟⣯⠗⣠⡿⠃⠀⢿⡇⠀⠈⣿⡀⣿⣿⠏⢀⣤⡶⠟⠃⠀⣠⡿⢁⣿⣿⠃
⠀⠀⠀⠀⠈⢿⣿⣆⡀⠀⠈⠻⣷⣤⣀⣘⣛⣋⣥⣾⠟⠁⠀⠀⠸⣿⠀⠀⣿⣧⣿⢋⣴⠟⠋⠀⠀⣠⣾⠟⠁⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠛⢿⣿⣷⣶⣤⣀⡉⠙⠛⠛⠋⠉⢀⠀⠰⠦⠀⠀⢿⡇⠀⣿⣿⣿⡿⠁⠀⢀⣴⡾⠋⠁⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣶⣦⣤⣀⠙⠀⠀⢀⣄⠀⢸⣷⢀⣿⣿⠏⢀⣤⡾⠟⠁⢀⣄⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣷⣶⣤⣅⡀⠀⣿⣼⣿⢃⣴⠟⠋⠀⠀⠀⠀⠁⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠶⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠟⠿⢿⣿⣷⣶⣤⣤⣤⣴⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠛⠛⠛⠛⠛⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢦⣤⣀⡀
⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡝⢆
⠀⠀⢰⡋⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⣼
⠀⠀⢸⡟⠲⢬⣙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢚⣩⠴⠋⢹
⢀⡠⠴⣇⠀⠀⠈⠙⠲⢭⣓⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣫⠵⠚⠉⠀⠀⠀⣼
⡞⠀⠀⠙⢤⣀⠀⠀⠀⠀⠈⠙⠲⢭⣓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⡥⠖⠋⠀⠀⠀⠀⢀⡠⠞⢹
⠙⢦⣴⠶⡄⠈⠙⠲⠤⣀⠀⠀⠀⠀⠈⠙⠲⢭⣓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣚⡥⠖⠋⠁⠀⠀⠀⠀⡠⠴⠚⢉⡠⠴⠚⠳⡄
⠀⠀⠈⢙⡿⡟⢧⠀⠀⠈⠑⠂⢤⣀⠀⠀⠀⠀⠈⠙⠲⢍⡓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⡩⠔⠋⠁⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⢠⡇⣀⡀⠀⣠⠟⠦
⢀⡤⠚⠉⠀⡟⢸⡀⣀⣀⡀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠈⠑⠦⣍⡓⠦⣤⣀⣠⠤⢖⣩⠴⠚⠉⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⣸⢧⠃⠈⠉⠁⢀⣠
⠫⣄⠀⠀⠀⠹⢄⣉⣁⠀⢹⡀⠀⢀⡀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠒⠉⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⢠⡖⠒⢺⠁⢀⣀⡤⠖⠊⣯
⠀⢀⡽⠒⠦⣄⡀⠀⠈⡇⠀⠙⠚⠉⠙⢦⣀⣤⢤⡀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣋⣁⠀⠀⠀⣠⠖⠲⣄⡀⠀⡼⣓⡶⠿⠚⠉⠀⠀⣀⡜⢻
⠀⢸⠀⠀⠀⠀⠈⠙⠲⢿⣄⣠⠔⠋⠙⢦⠀⠀⢀⣹⡄⠀⠀⠀⢀⡬⠛⠲⢤⣤⡤⠴⠾⡭⠞⠁⠀⠈⣳⣦⠖⣣⠞⠓⢦⡽⠟⠋⠁⠀⢀⣀⣤⠴⠚⠁⣠⡏
⠀⠸⡷⣄⣀⠀⠀⠀⠀⠀⠀⢉⡷⠤⣄⠀⠙⠒⠉⠀⠈⠓⠶⠶⠿⠒⠋⠙⡏⠙⠒⢤⡀⠀⠀⠀⢀⡜⠁⠈⠉⣡⢴⡾⡟⠒⠒⠒⠚⠉⠉⠁⣀⣠⣴⣪⣇⢳
⠀⠀⣳⣄⠉⠙⠓⠒⠒⠒⠒⠺⣧⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠙⠦⠤⠔⠋⢀⣠⢔⣫⠵⠋⢠⣷⣤⢤⡴⠖⠒⣂⣭⣿⣿⣿⣿⡯⠞
⠀⠀⣷⣌⠻⢷⣲⣦⡤⠤⠤⠼⣇⠑⠦⣄⣀⠀⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣀⡤⠶⠛⠛⠉⢀⣠⣶⣿⡿⠗⠋⢀⡠⣾⠟⢹⢻⣋⡽⡇
⠀⠀⢻⠈⠙⠲⢬⣙⠻⢷⣦⣄⡈⠳⢄⣀⠈⠉⠛⠓⠒⠚⠛⠢⣄⡀⠀⠀⠀⢠⣿⣦⣖⣋⣁⣀⣤⣤⡶⠚⢹⠷⠋⠁⠀⢀⣴⡻⠞⢁⣠⣞⡞⠁⢀⡇
⠀⠀⠸⡄⠀⠀⠀⠈⠙⠲⢬⣙⠻⢿⡲⠬⣉⡓⠒⠶⠦⠤⠤⠤⢦⣿⣳⣤⡴⠟⠋⠉⠉⢁⣠⣖⣿⣋⣀⣤⣾⣬⣤⣤⣺⣵⣿⠿⠿⠛⠛⠉⠀⠀⡼⠁
⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠀⠈⠙⠲⢬⣑⠢⢭⡙⠓⠦⣄⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⢾⡿⣿⡿⢻⢿⠟⠋⠉⠉⢩⢟⡗⠉⠀⠀⠀⠀⢀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠈⠙⠲⢬⣓⠦⢄⡙⠲⣄⣠⣯⡎⠁⠀⠀⣠⣮⠞⠁⢉⡾⢾⣦⣤⣴⣾⠗⠋⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠈⠙⠲⢬⣓⠦⣍⡳⢷⣶⡶⠿⢛⣡⠴⢚⡥⠖⠋⠉⠉⠁⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠈⠙⠲⢍⣓⠒⠒⣚⣩⠴⠊⠁⠀⠀⠀⠀⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⢀⣠⠖⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣶⣶⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣈⣄⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠋⠉⠉⠀⡉⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠉⠉⠉⠀⠀⠲⠶⠶⠂⠀⠛⠻⠿⠿⠿⣦⣤⣤⣀⢀⣀⢠⡀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠⣄⠔⣶⣶⣶⣶⣤⣤⢤⡠⣠⢄⠀⠀⠠⠀⠀⠈⠉⠁⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣷⣼⣯⣽⣿⣿⣿⣟⣛⣧⣳⣼⡆⡓⠄⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⣸⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣟⣻⣟⣛⣿⣿⣿⣿⣵⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⣩⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠹⢯⣽⣿⡿⠿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣭⣽⣿⣿⡛⢛⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠓⢦⣄⣰⣏⣈⣻⣿⣿⣿⣿⠟⠛⠻⠛⠉⠉⠱⣛⠍⣠⣾⣿⣿⣿⣿⠷⠀⠀⣨⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣵⣬⣿⡇⢨⣤⣯⣭⣿⣾⣛⣎⢡⡤⠖⣃⣭⣵⣾⣿⣿⣿⡿⠟⠁⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣟⣿⡁⠀⠀⠀⣿⣿⣿⣿⠿⠟⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⣿⣟⣯⣁⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣛⣭⡙⣿⣿⢻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡿⢫⣼⣿⣿⢸⣿⣿⣿⣶⣶⡦⣰⣿⣿⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣴⣶⡶⢸⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠟⣫⣴⣿⡇⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⣤⣶⣶⣶⣾⣿⣇⣿⣿⣿⣿⣿⣷⣾⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠒⠒⠒⠋⠉⠉⠉⠉⠈⠃⠒⠒⠤⠤⣄⣀
⠀⠀⠀⢀⣀⡠⠤⠤⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠤⠤⣀⡀
⠀⢰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⢄⣀
⠀⢻⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢑⣆
⠀⠘⡄⠀⠀⠀⠉⠛⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡪⢻
⠀⠀⠙⠢⢄⡀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣴⡲⠬⠛⠊⠉⢀⠏
⠀⣀⠤⠔⠊⠙⠓⠦⣀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⣖⠶⠝⠛⠊⠉⠀⠀⠀⠀⠀⢀⠞⠂
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠙⠻⢖⣤⣀⡀⠀⠀⠀⠀⢀⣀⢤⣔⡲⠮⠛⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠁⡠⠒
⠈⠊⠉⢇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠈⠙⠛⠫⠭⠭⠛⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠈⠀⠀⠀⠀⡠⠤⠃
⠀⠀⠀⢰⢏⣉⣃⠀⠀⡤⣀⡀⠀⠀⠀⠀⢈⠗⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠉⠁⠀⠀⠀⠀⠀⢠⠤⠤⠾⡀
⠀⠀⠀⠈⢙⡶⢪⠶⣴⡢⠤⣈⢳⠀⠀⡰⠁⠀⠀⠀⠉⠒⠤⠄⠀⠀⠠⠀⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⠉⣙⡶⠴⢇⣒⣒⢭⡟
⠀⠀⠀⡴⣾⠦⣕⡦⠒⣿⣉⣲⠾⣀⡸⢵⣤⠴⠙⢆⣰⠦⣲⡄⠀⠐⡍⠀⠀⠉⢣⠀⢾⣙⣢⣄⣠⣤⣯⠕⣶⣮⣽⡭⠭⠒⣩⢇
⠀⣠⠒⢗⠮⢷⣦⣤⣍⣉⣉⣉⣩⡽⡈⢗⣭⡳⢶⡒⢁⣠⣾⠮⡷⢶⣛⠢⢤⡶⣖⡳⠾⣪⣵⡾⣻⣦⣤⣤⣤⣤⣤⣴⣶⣏⣠⢴⡷⢤
⠀⡟⠷⣦⣍⡒⠒⢤⡬⠭⠿⡢⠤⣱⣌⡑⠢⠼⣷⣦⠶⣛⣒⣉⣧⣬⡷⠬⠽⠚⡱⠶⠿⠇⠒⣉⡥⠯⠶⠶⠭⣭⣭⠭⠭⠐⣲⢟⠝⣠⠃
⠀⣇⠀⠀⠉⠚⠷⣦⣌⡑⠢⠬⣉⡲⠒⠸⠷⠶⢤⣬⣶⢤⣤⠤⣤⣤⠤⠒⠚⣻⣶⣶⣶⡾⠿⣛⠝⠉⢩⡿⡯⢹⣋⣉⠩⣿⡿⠟⣪⣿⡇
⠀⠘⢄⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⠙⠢⢄⡀⠀⠸⡦⠤⢤⣶⣿⠿⠿⠟⣛⠭⢾⣭⡥⣲⣶⠿⠤⠤⣔⣟⠘⠥⠚⢤⣒⣭⣥⡲⠽⠊⢁⠃
⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠈⠙⠻⣶⣄⡈⠑⠤⡈⠉⠁⠒⠀⣉⢉⠉⠀⡰⠃⡠⠓⢷⡗⢒⣂⣭⡿⣭⣔⠦⠝⠓⠉⠉⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠀⠉⠛⠷⣦⣌⠑⠦⣀⡊⠁⣈⡩⠽⠓⣉⣀⢤⣴⡺⠭⠛⠂⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠈⠉⠻⠖⣢⣍⣍⣤⣒⠮⠝⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⢀⣀⠤⠔⠒⠋⠁
⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣀⣀⣤⡤⠤⡆⠀⢸⠦⠤⣤⣄⣀⢀
⡔⠩⠊⠁⠀⠀⠀⣤⣆⠀⢸⣤⡀⠀⠀⠈⠙⢎⠲⡀
⡗⠬⣢⣄⣀⣀⠀⠉⠉⠛⠉⠉⠀⣀⣀⣠⢴⠢⠞⡇
⢣⡀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⣠⠃
⢠⡈⠉⠐⠒⠂⠤⠤⠤⠤⠤⠤⠤⠄⠒⠂⠉⠁⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⣀⡀
⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡤⠖⠛⠉⠉⠉⠉⠉⠙⠓⠶⢤⣄⣀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⢤⣀⣀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⡤⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣞⢍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⣴
⠀⢸⠀⠀⠀⠀⠀⠀⢀⡏⠑⠪⢕⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣐⡭⠐⠉⡘
⠀⠀⡄⠀⠀⠀⠀⡜⠁⠹⢄⡀⠀⠈⠑⠪⢕⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠔⠊⠀⠀⣀⠤⢻
⠀⠀⡇⠀⠀⠀⠀⠈⠲⠽⣄⣈⠑⠢⢄⡀⠀⠈⠑⠪⢕⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡊⠕⠊⠁⠀⢀⡠⠔⠉⠤⠒⠉⢱⡀
⠀⠀⠃⠀⠀⠀⠀⢀⠤⠊⢹⠺⡀⣀⡀⠈⠑⠢⢄⡀⠀⠈⠑⠪⢑⠠⢀⠀⠤⣒⠅⠊⠁⠀⢀⡠⠔⠊⠀⠀⠀⢀⠴⠉⠂⢁⡨
⠀⠀⢰⠀⠀⠀⠀⠙⢤⣀⠀⠓⠠⡄⠡⣀⠤⡄⠀⢈⠑⠢⢄⡀⠀⠉⠉⠉⠉⠀⠀⣀⠔⠊⠁⢀⣀⠀⠀⣼⣉⣇⠤⠶⠙⣹⡀
⠀⠀⢸⠀⠀⠀⠀⠀⡎⠀⠛⠓⠢⠴⣀⡤⠒⢪⠉⠉⣧⠀⠀⢈⡵⠢⣄⡀⠤⠖⠝⠁⢑⣤⠄⠁⠀⡵⠶⠃⣉⣀⡠⠔⢋⣴⠁
⠀⠀⠸⡄⠀⠀⠀⠀⢸⠒⠤⢨⣀⣀⣀⡟⠒⠤⡉⠉⠀⠉⠘⠋⠉⠉⢇⠈⠢⡀⠀⡠⠊⢀⠠⣲⢛⡋⠉⢉⣉⠠⣤⣶⣿⣽⠄
⠀⠀⠀⠈⠙⠒⠢⠤⢼⣑⠶⣤⣤⠤⠤⢏⠢⢄⣋⡑⢦⣀⠀⠀⠀⠀⠈⢆⠀⠀⢁⠤⠖⠋⢉⣠⣾⠿⠋⢀⡴⠻⣟⡙⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠉⠒⠭⣓⠦⣬⣑⠠⢈⣉⡉⠉⠙⠂⣄⡀⢀⣾⠷⠪⠀⣴⡶⠋⣹⡏⢁⣠⣶⣏⣤⡴⠋⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠉⠂⠭⣛⠴⡒⠠⣌⠈⠉⠉⣉⣥⣠⡤⣴⣿⠿⣶⠟⠛⢛⣿⠟⠉⠁⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠑⠢⠥⡢⢍⡢⢬⣏⣀⣠⡴⠏⡠⢝⡿⠦⠖⠛⠁⢀⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠤⣀⠀⠀⠑⠢⢕⡻⠉⠥⣒⠥⠂⠁⠀⢀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⣀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⣄⣄⣠⣄⣀⣄⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣍⣦⣤⣤⣎⣁⣄⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠅⣀⠤⠈⠐⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠠⠄⠀⠀⠄⠒⠂⠒⠒⠀⠂⠂
⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠬⠀⠈⠄⠐⠀⠀⠀⠀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠒⠚⠦⠤⠤⠄⠂⡀⣂⠀⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠈⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠢⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⣀⠀⠄⠄⠠⠒⠂⠀⠀⠀⠀⠈⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⠀⣤⡀⠤⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣈⣈⣁⣀⡀⠄⠈⠀⠀⠀⠀⠀⠘⠀⠀⢰⠆⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠁⠁⠐⠒
⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠉⠤⠤⠄⣠⠂
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠶⢤⠠⣄
⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠨⠙⠒⠠⢴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠈⡀⠈⠉⠝⠀⠢⠄⠨⠁
⠀⡀⠀⠂⢀
⠔⠐⠀⠲⠤⣍⣠⣄⡀⠀⠀⠀⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠄⠀⣠⢀⠀⠀⠀⣀⡀⠍⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠩⠘⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠼⢀⡀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⠤⠂⠔⠐⠀⠖⠀⠚⠉
⣀⢠⠤⠴⠒⠊⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠑⠢⠐⠤⡀⠀⠀⠀⠀⠀⠠⠀⢂⣀⣀⡤⠤⠄⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀
⠀⠀⠀⠀⠀⠐⠏⠁⠈⠂⢤⡀⠄⠀⠀⠈⠉⠒⠒⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠵⠶⠒⠒⢒⡶⠖⠊⠭⣅⡀⠡⡋
⠀⠀⠀⠀⠀⠀⠑⣶⠚⠙⠓⢄⡈⠂⠐⢢⢄⣀⠀⠀⠀⣀⣀⣤⠔⠶⠴⠒⡚⠉⣀⡤⠤⠔⡦⣀⣀⡠⢀⡡⠊⢀⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠳⢕⣂⠄⢄⠑⡄⠀⠀⠀⢈⠕⠛⠋⠉⠀⠀⠀⠀⢀⣩⠔⠉⠀⠀⠀⠀⢀⠀⠋⠤⠤⠔⠂⠉⣠⠜⡀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠈⢳⣤⣈⢉⣉⡑⣿⣒⠄⠒⢁⣀⡤⠔⠄⠀⠶⠦⠤⠒⠒⠈⠜⠋⠉⠀⠀⠈⠉⣲⢒⡒⠂⠉⡈⠀⠦⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠙⠉⢉⡉⠙⠉⠉⢉⠑⠖⠤⢤⠤⠤⡔⢒⠈⣠⡰⢤⣤⡶⠯⠤⠐⠒⠋⠉⠒⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣄⠀⠀⠁⠒⠚⠯⠍⠱⠤⠴⠁⠒⠋⠉⠊⠉⠁⠀⠀⠀⢀⣀⠀⠠⠤⠤⠤⠔⠒⠒⠊⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⣐⠢⠤⠐⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⣤⠀⠀⡤⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠄⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠺⡄⣿⠀⠀⡇⠀⠀⡓⣄⡔⢲⠀⡷⢰⢀⡆⣿⢸⣆⣀⠖⡇⡔⠓⣾⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠁⠛⠀⠀⠃⠀⠀⠛⠋⠙⠊⠉⠁⠈⠉⠓⠉⠁⠉⠉⠀⠉⠙⠀⠉⠈⠉


⡿⠿⠿⠿⡿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣴⣿⣷⣶⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠛⠛⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡞⠒⠲⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠒⠊⡆
⠀⠀⠀⠀⠀⠀⠀⠧⢤⣀⣐⠀⠀⠈⠩⠉⠉⠛⢲⠏⢀⠀⠁⣱⠈⢉⣉⣉⡀⠤⢤⣔⣢⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⣯⣠⣄⣀⣴⢟⠒⠒⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣼⡏⠈⠉⢿⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣸⡿⠀⠀⠀⠘⣿⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢠⢷⠇⠀⠀⠀⠀⢹⣱⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠘⣼⠀⠀⠀⠀⠀⠘⡏⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⠀⠀⡏⠀⠀⠀⠀⠀⠀⣷⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡠⡇⠀⠀⠀⠀⠀⠀⢹⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠈⠀⠀⠀⠀⠀⠞⠁⠀⠐⠂⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠊⠀⠀⠀⠄⠀⠀⠉⠀⠀⠀⠂⠀⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⣀⠄⠀⠀⠀⠀⠀⠠⢀⡄⠁⠐⠒⢠⠤⢀⡀⠀⠀⠀⠀⠀⢀⡠⢂⡡⠔⢋⠤⠇
⠀⠀⠀⠀⠈⠡⡀⠐⡀⣐⡐⡐⠀⠐⠂⠤⣄⡂⠐⠐⠀⠐⠐⠂⠀⠈⠀⠀⠀⠀⠀⣎⠙⠂⠆⠠⡀⣒⠀⠀⠠⠉⠀⡐⢒⠉⠔⣩⠤⠮⡥⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⢤⡀⢀⡀⠐⠉⠀⠀⠈⠀⠐⢇⡀⡀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠀⠈⢦⡐⠂⠑⢒⠴⠤⠴⠮⠤⠂⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠢⠀⠄⠠⠐⡁⢀⡀⠄⠄⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠒⠁



⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠐⠀⡀⠂⠀⠀⠂⠀⠀⢒⠀⠀⠀⠐⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠝⡄⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠲⠀⠀⢁⣄⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠀⠀⠒⠒⠂⠀⠀⠀⠒⠆⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠄⡀⠸⡀⠁⠀⠀⡀⠄⣂⠄⠂⡀⣄⠰⠒⢁⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⡔⣴⡈⢧⠀⠀⠩⠖⢉⠀⠐⠴⠣⠃⡀⡀⠘⡄⠀⠀⠀⠀⠀⠀⠘⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⢠⣤⡪
⠀⠀⠀⠀⠀⠀⠀⠇⠘⠷⣾⣆⣤⠔⡰⠈⠉⠀⠋⢀⠀⠁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠠⡓⠀⠠⠴⠆⠚⠉⠃⠀⠤⠴⠶⠒⠋⠉⢐⠛⠩⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⡃⠀⠂⠐⠀⠀⠈⠀⠁⠀⠀⠀⠀⡀⣀⡀⣠⠄⣛⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡄⡐⠂⠀⣀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠰⠤⠒⠙⠆⠘⠏⠉⠔⠉⠀⠙⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⡇⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠾⠻⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠄⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⢛⡋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣧⡼⠟⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣤⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣠⣤⠶⠶⠶⠟⠛⠋⠁⢸⡏⣚⡹⠔⣨⡟
⠀⠀⠀⠀⠀⢠⡾⡛⣿⣀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠰⠎⣀⣀⣀⣤⣤⠾⠛⣡⣶⠞⠛⠋
⠀⠀⠀⠀⠀⠹⣮⣳⢬⣩⣿⣶⣤⣀⠀⠉⠛⠳⢦⣤⣀⣠⣴⠾⠋⠁⠀⣠⣴⠾⠛⣉⠍⣿⠷⣦⣥⠾⠋⠙⡻⣦
⠀⠀⠀⠀⣠⣤⣈⣽⠿⢯⠔⣒⣚⣉⡛⠷⣦⣄⡀⠈⠹⠍⠀⠀⣠⣴⠿⣯⡤⠖⠒⣡⡼⠋⠀⠐⠀⣠⣤⣭⣴⠟
⠀⠀⠀⠀⢻⣌⡉⢵⠴⡢⢋⣴⠶⠶⡶⢾⡏⢌⢙⠷⣮⣥⠶⣿⡉⢴⢒⡜⢿⡟⠛⠩⣒⣦⣒⣊⣰⠟⣿
⠀⠀⠀⠀⠀⠸⣧⣤⣭⡴⠞⢻⡎⠦⠤⠄⠻⣦⣍⡓⠤⠒⣒⣨⣿⣌⢃⣤⡾⠿⠆⣰⡏⠤⠉⠉⠁⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣅⣒⠀⠛⢛⡛⠛⣷⡀⠉⠛⢛⠛⢋⣁⣀⠙⠿⣡⡶⠟⣛⣫⣬⣭⡶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣄⡡⠘⢷⣍⣉⣥⠾⡛⠉⢙⠛⢛⣫⡶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣉⠉⢑⡒⠂⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠶⠞⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠆⠤⠶⣄⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠦⠈⢄⠠⠀⣤⡶⠿⠀⣪⣍⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣱⠄⠩⣠⡀⠀⠈⠉⠓⠾⠛⠂⠀⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠉⠛⠧⡀⠀⠀⠀⠀⠀⢤⣌⣡⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⣀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⢰⢤⣴⣴⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠜⠉⠀⠀⠺⢷⣌⠻⢷⣦⣬⣀⠀⠀⠀⢁⢀⠸⠀⠀⠀⠙⠛⣿⣿⢿⢻⣳⠂⣾
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠓⢹⣿⠛⠛⠛⠛⠒⠂⠉⠹⠀⠀⠀⠀⡀⡙⣿⣅⣾⢱⣼⠇
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠐⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢐⢰⠰⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠶⠛⠁⠀⠀⠀⢶⣶⡄⡄⣄⢠⡠⡀⣈⢑⣂⣛⠥⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣆⣦⠤⢮⢹⡜⣿⣺⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡿⠯⠽⠞⠓⣿⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⣿⡿⢿⡟⢿⡟⢿⡿⠻⣿⡿⢻⣿⠟⣿⡿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⣠⡶⣿⡟⠻⡆⠹⡆⠹⡇⠘⠃⠸⠇⠸⠁⠀⠟⠀⡼⠁⢰⠏⢀⣿⠟⢉⣿⠿⣿⣿⣦⣄
⠀⠀⠀⠉⠳⡈⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣤⡄⠀⠀⠀⠀⠘⠁⠀⠊⠀⠘⠁⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡙⣿⡝⢧⡘⣷⡀
⠀⠀⠀⢠⠐⣆⠹⣏⠉⢷⣄⠀⠀⠀⠈⠛⢷⣌⣻⣮⡳⣼⡻⣶⢄
⠀⠀⠀⣨⣷⡈⠣⣌⢑⣦⡙⢷⡒⢦⣄⠀⠀⠈⠻⣿⣧⢻⣇⢹⡌⢄⡀⠀⣀⠤⣴⠤⣤⣤⣶⣆
⠀⠀⠀⠀⠉⠛⠶⣿⣿⣯⣻⣆⠳⣌⠻⣦⣀⣀⣀⣈⣻⣦⣍⠓⢯⠱⡄⢹⡵⠚⢉⡭⠯⠭⠤⠼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣏⣽⣾⣿⡿⣍⠻⣝⢮⣧⡬⠿⣷⡈⢧⡙⠦⢰⣞⢩⡴⢋⣩⣻⣋⣩⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⣉⡽⠿⣿⣿⡷⡜⢷⡙⠷⠾⠿⠷⢶⡭⣻⣆⠹⣆⢳⡾⠿⠿⠛⠉
⠀⠀⠀⢰⡿⢛⣩⠽⠿⠭⠤⠞⢉⣠⠞⠋⠙⣷⡘⠦⠭⠭⠭⢍⡙⢦⡙⢮⣻⣦⡙⢮⣷⢄⣀
⠀⠀⠀⠰⢿⣿⣴⠞⠋⡉⣉⡓⠛⢁⣄⣼⡿⠿⠟⠻⢿⣿⣷⣦⣝⢦⡙⢦⣈⣽⡟⠲⣌⠓⢮⣍⣝⠖
⠀⠀⠀⠀⠀⠈⠻⣮⣿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣥⡿⠿⠛⠛⢾⣆⣸⡿⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠈⠀⠀⠀⡀⠀⠈⠁⠈⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠖⠒⠉⢉⣩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⠙⠻⡏⠀⠀⠀⠀⢀⣀⣀⠀⠀⠓⢢⣄⠀⠈⠑⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⡀⠀⠀⠉⠲⢤⡀⠈⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⢀⡀⠁⠉⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠉⠁⠀⠀⠀⠈⠙⠲⠶⠶⠶⠶⠶⠶⠛⠛⠉⠉⠀⠀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠸⠇
⠀⠀⠀⠀⠘⠿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡐⠲⠃⡠⠀⠀⠀⢹⠇
⠀⠀⣦⣄⠀⠀⠀⠉⠙⠛⠫⠧⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠐⠀⠀⠀⣀⠤⠚⠀⣀⢸⠀⠀⠸⠗⠀⠀⠀⣾
⠀⢄⡉⠛⠋⠶⠴⣦⠀⣿⠒⠒⠞⠋⠀⠀⠀⣀⣀⡀⠉⠂⠀⠀⠀⡀⠤⠒⠉⠀⠀⠀⡜⠉⠀⠳⣴⠁⠀⠀⠀⡔⠁
⠀⠈⠛⠷⠶⢠⣤⣄⠠⠏⠀⠀⠀⠀⠀⠒⠂⠀⠀⢁⣀⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⢀⡤⠈⠀⠀⠐⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⣀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢹⣆⠻⠀⠀⠀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠉⠀⠀⠀⢀⠀⠀⠀⠐⠒⠤⠄⡀⠀⠀⠀⠀⢡⣾⡏⠁⠀⠀⡹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠘⠓⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⡀⠀⠀⢻⣿⣿⣿⠷⢤⣀⠀⠀⢀⠀⠀⢚⡄⠀⠀⠀⠀⠀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠉⠛⠿⣶⣬⡟⠃⣀⡠⠤⠟⠀⠀⠀⡠⠀⢌⡁⠀⠀⠀⠘⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠠⠛⢛⡶⢶⡊⠀⢹⡄
⠀⠀⠘⢦⣀⣀⣀⡀⠀⢀⣀⣀⣠⡴⠃⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣠⠀⠀⠀⠘⡏⠀⠀⣸⠁
⠀⠀⠀⠀⠉⠙⠛⠓⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠤⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠹⢄⣄⣀⣤⡑⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠈⠉⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠁⠀⠀⠀⢀⡠⠤⠖⠓⢒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⣀⠴⠊⠁⣀⡠⠴⠚⠋⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡠⣪⣴⣂⠔⠋⠁⠀⠀⠀⢀⣠⠞⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⢄⡴⢞⠵⠋⠀⠀⢀⣠⠴⠒⠋⠉⠀⠀⠆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⡴⡥⣫⠔⠁⢀⣠⠴⠚⢁⣠⡃⣜⡧⠟⠶⠄⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠊⠀⠀⠠⠊⢀⠞⠁⣠⠔⢋⢰⣬⢰⡸⢸⠷⠏⣁⡠⠴⠒⠛⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠐⢁⡴⠁⡠⠞⠡⣰⣯⣧⣿⠻⢃⡤⠖⠉⢀⣀⣤⠤⠴⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠈⢠⠎⢠⠞⣥⡛⢖⠓⠉⢀⡠⠚⠁⢀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠠⠃⡰⠃⠀⠁⢀⠈⢁⡴⠋⢀⣴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠦⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠈⠀⡠⠁⡼⠁⢀⣤⡤⠀⡠⠊⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡩⣴⣶⡿⠿⠿⢓⣮⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡰⠁⢰⡧⠞⣻⡟⠁⠜⠁⢀⡔⠁⠀⠀⠀⠀⠀⠀⣠⢖⣩⣴⠌⠑⣊⠥⠒⢊⢉⣭⡉⠙⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡜⠀⠀⠉⠀⡰⠋⢠⠊⠀⡰⠊⠀⠀⠀⠀⠀⢀⣴⣫⡶⠟⠋⣠⠔⣋⡤⡀⢿⣿⣿⣷⠠⢀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣠⠞⠁⡰⠁⢀⠞⠁⠀⠀⠀⠀⣠⣶⠏⡡⠊⢀⡤⢮⣥⣝⠿⠿⠿⠷⣏⣙⣛⠀⠀⠓⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠄⣀⣠⠖⠁⠀⠀⠀⠴⠋⠀⠔⠀⣠⠋⠀⠀⠀⠀⣠⡾⢿⡿⠋⣀⠔⣣⡝⠮⢟⣩⠵⠒⠊⠉⠉⣀⣈⣉⠉⢲⠎
⠀⠀⠀⣀⠀⣀⠀⡀⠈⠀⠀⠠⠀⠀⣼⠓⡽⠁⠀⠀⠀⢀⡰⠦⠊⢀⡔⠁⠀⠀⢀⢴⣿⣯⠔⠂⡠⠚⣁⠶⠈⡠⠖⠉⠀⢀⡠⠔⠒⠉⠉⠉⠉⠉⠋
⠀⠀⠸⣄⢀⡉⠈⡀⠀⠐⣠⣶⣿⠛⠁⠚⠀⠀⠀⢀⡴⠏⠔⠀⢀⠌⠀⠀⡀⢂⢕⡛⠚⣁⡤⠊⠀⠀⣈⠔⠈⠀⢀⡤⠚⠁
⠀⠀⠀⠁⠀⠀⠀⢚⠽⢷⣿⡉⠡⠔⠀⠀⠀⠀⣔⢂⠐⠁⠀⠔⠁⠀⠠⣪⠞⠕⠊⠠⠚⠁⠀⢀⠀⠊⡠⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠈⢦⣀⣀⠊⣹⠁⢸⡀⠀⢤⠶⡓⠂⠀⢀⠔⠀⠀⢀⡡⠊⠉⠀⠀⠀⠀⠀⠀⠔⠁⠤⣊⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠎⠀⡆⢀⢨⡙⠷⣯⣥⣴⣇⣾⣡⡀⠄⠚⠉⢀⠀⠀⠀⢀⡠⠔⠊⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇⢸⣷⢼⣤⣤⡠⠄⢀⣀⡀⠤⠔⠒⠈⠁⠀⠀⢛⡥⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣦⣾⣿⣶⣦⣤⣐⣤⡲⠖⠛⠂⡀⠤⠚⠁⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⠟⠿⠋⠉⠉⠈⠀⠀⣈⣡⠤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢍⣃⣀⠠⠤⠒⠉⠁



⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠋⠙⠛⠛⠛⠛⠻⠿⢿⣿⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡭⠶⠶⠶⠶⠶⠶⠦⣤⡀⠀⠀⠀⠀⠀⠉⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁⠉⠁⠀⠀⠀⠛⠛⠛⠲⠶⠶⣤⣤⣀⣀⠢⣄⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢋⣁⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠉⠁⠀⠐⠒⠛⠲⠶⢦⣤⣈⣉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢼⣶⣄⡀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠚⠉⣭⣉⣙⠛⠲⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠶⣬⣍⣛⡒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠉
⠀⠀⠀⠀⠰⠋⠠⠖⠚⠲⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⣶⡄⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⢀⠀⠀⠀⠘⠓⠶⣤⣀⠈⠙⠳⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣄⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠷⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⣀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠳⢦⡀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢀⡴⠛⠛⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢰⡏⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠙⢷⣄⠀⠀⢀⣀⣀⡀⠀⢀⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⠟⠀⠀⠀⠀⢰⡆⠀⠀⠀⠹⣦⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠉⠻⣶⠋⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠙⠷⣤
⠀⠀⠈⣧⠀⠀⠀⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢠⡀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢀⡿⠀⠀⠀⠀⢻⡄⠀⠐⣦⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠈⠻⣦⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢠⡶⠛⠁⠀⠀⠀⠀⠀⢻⣄⠀⢹⡄⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠉⠉⢿⡀⠀⠀⣰⠇⠀⠀⠀⠀⣿⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⡾⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣼⠃⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠞⠁⠀⠀⠀⠀⣸⠇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠸⣧
⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⡀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢰⡏⠀⠀⠀⣀⠀⠀⠀⠀⠘⣷
⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⣀⢻⣄⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢿⡀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡿⠶⠶⠶⠛⠉⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠸⣇⠀⠀⠰⠶⣤⣤⣤⣤⠶⠿⣦⡀⢹⡆⠙⠳⣦⣄⡀⠀⠀⠀⠈⠙⠛⢻⡇⠀⠀⠀⠀⡀⠀⣷⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣷⠀⠀⠀⠉⠛⢶⣄⠀⠀⠀⠀⢸⡇⠀⠀⢀⣼⠃⠀⢹⡿⠁⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠘⢷⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⣇⣤⡶⠟⠁⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⣦⡀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢸⡇⠀⠀⠀⢸⡏⠀⠀⠀⠀⣰⡟⠀⠀⢀⣀⣴⠄⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢠⣤⣤⡴⠶⠛⠛⢷⣄⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣼⠋⠛⠛⠛⠋⠉⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣿⡀⠀⠀⢸⡇⠀⠀⣸⠇⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠦⣤⠴⠶⠶⠶⣤⡀⠀⠀⢹⡇⠀⠀⢸⠇⠀⢰⡟⠀⠀⠀⢀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀⢿⡀⠀⣿⠀⢠⡟⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⣄⣿⣰⠟⣀⣤⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⡘⠁⠁⣀⠤⠥⠖⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡴⠁⠀⢰⠁⠀⣀⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣋⠥⠚⢀⡤⠤⠊⣼⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⣪⠅⠒⠊⡡⠤⠴⠊⠀⣀⡤⣣⠀⣠⠤⣤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡔⢉⡁⠴⠒⢉⢔⠶⠆⠉⠛⠋⠈⢠⠒⢁⠊⠁⠀⣀⠭⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠞⢠⠃⠠⡦⠶⠛⠁⠀⠀⠀⠀⠀⠀⡄⢀⠞⠀⠀⡎⠀⠀⡰⡟
⠀⠀⠀⠀⢀⡠⠖⠒⠤⠤⠤⠠⠖⠁⣠⠎⢰⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⠖⣊⠤⠃⢀⡠⠔⠡⡶⠛⠊
⠀⢀⠔⠁⠈⢀⣀⣀⠤⠤⠠⠔⠊⣩⠔⠒⠁⠀⢠⣾⠇⢀⠀⠄⠒⡩⠭⠔⢊⡀⠠⠔⠁⠀⣠⡔⡇
⠀⠸⠀⠀⡜⠉⢠⠔⠒⣒⠒⣒⡉⠀⢀⣾⠍⠭⠛⠁⢀⠇⢠⠒⢉⡠⠔⠒⡡⡦⠶⠦⠉⠙⠓⠉
⠀⠸⠤⠼⠥⠴⣁⣀⣯⠖⠭⠕⠒⠛⠒⠁⠀⠀⠀⡔⠉⡠⠃⡜⠁⣔⠤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡤⠠⠤⢀⢄⣀⡤⠎⠀⡼⢡⡜⠀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠉⢀⣀⣀⢤⣀⡠⠖⢊⡩⠔⠊⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠊⠁⡠⠄⠠⠐⠒⡒⠉⠀⣰⡢⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠰⠇⡠⠮⠤⡞⠀⣸⡭⠭⠭⠒⠁⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⢚⡔⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠖⢤⠏⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠾⣤⠎⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠐⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⢄⡔⢲⠁⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠤⠄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⢃⠾⣡⠎⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⣠⢐⡖⠁⡀⢡⠴⠾⡧⠖⠓⠀⠀⠘⠉⠉⠀⠀⠀⠀⢀⡀⠀⠀⠈⠉⠈⢁⣨⠤⠁⢢
⠀⠀⠀⠀⡸⡱⠊⡀⠊⠀⠈⠂⠀⠠⠤⠠⠒⢶⠲⠚⠙⣻⣻⡯⠉⠫⣭⠫⠼⢏⠙⠿⣯⠜⢛⡷⠮
⠀⠀⠀⠐⢻⢠⠊⠀⠀⠀⠐⠀⠁⠊⡁⠀⠀⠀⢳⣤⡴⠏⠁⠛⠻⡹⠊⠁⠠⠼⠷⠈⠤⢿⣛⣽⠤
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⣿⡟⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠿⢿⣿⣿⣶⣦⣄⣈⡇⣿⢹⣿⠃⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣟⢿⣿⣿⣧⢿⣸⣿⣿⡿⣟⣻⣭⣭⣭⣍⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⠿⢿⣶⣝⡿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣟⣾⣿⣟⣽⣾⣷⢻⣿⣿⣿⣶⣽⡻⣮⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣼⢟⣽⣿⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣮⡋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠿⢿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⡇⡿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⢠⠔⠈⠹⡄
⠀⠀⠀⠀⢸⣏⠢⣄⠀⠀⠀⢀⢿⣆⡀⠀⢣
⠀⠀⠀⠀⢸⣿⠂⠘⠲⠖⠒⠺⠈⢿⣾⡅⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠤⠤⠀⠀⠀⠀⠂⠀⠀⠠⠤⠀⣀⡀
⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠟⠉⠘⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⡀
⠀⠀⠀⠀⡻⠁⠀⠀⢲⣾⣷⡀⠀⠐⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦
⣀⣄⣤⡊⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⣿⠗⣿⠏⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⠿⢶⣟⣛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠈⠙⠓⠒⢶⣀⠀⠀⢀⠄⠀⢀⢠⠴⡄⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⡀⠁⠀⠌⣠⣁⣌⡀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠋⣡⡀⠁⠀⠀⠀⠀⢀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣾⣧⣽⣿⣿⡘⠧⠀⠀⠀⠀⢰⡟⠈⣖⣡⣴⣾⣥⣤⡄⠀⠀⣠⣤⣿⣿⡄⠀⠀⠀⢀⠊⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿⣿⣿⠻⠂⠀⠀⠀⡰⠛⠛⠿⠿⠿⠟⠿⢿⣿⣴⡷⠟⠛⠉⢻⣿⡇⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠿⢀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⡇⢇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⣨⠀⢸⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢠⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⡼⡟⢀⠃⠀⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⡾⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠟⠁⢠⣾⣦⡾⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⣿⣧⣠⠾⠇
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠠⠖⠿⢗⠀⠀⡐⠀⠂⠀⠀⠀⠀⠈⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡴⠄⠀⠀⠀⠤⠀⢀⠀⢀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠠⠀⡀⠀⠀⠕⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠁⠈⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⢀⠀⠀⠈⢀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡸⠄
⠀⠀⠀⠣⡀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡀⠤⠂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠟⠃
⠀⠀⠀⢆⠈⠂⠤⣀⠀⠀⠀⠉⠐⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠬⠌⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠆⠊⠁
⠀⠀⠀⠀⠑⠢⣄⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⠀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠈⠐⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⢤⠖⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⣀⠀⠈⠁⠐⠂⠀⠀⠀⠐⠂⠀⠀⠀⢀⣠⡤⠴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠦⠤⣤⣤⠤⢤⣄⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠮⢤⡀⢈⠖⠉⠁⠈⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠁⠀⢠⡇⠸⣇⠀⠀⠀⠀⠀⠣⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠞⠋⠀⠀⠀⠀⠀⢇⠀⠙⢧⢰⣠⠞⠉⠉⢻⠉⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠰⠀⠀⠈⠣⣴⠛⠋⠓⠀⠀⠀⠘⢗⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢽⡄⠀⠀⠀⠀⠀⠈⠘⠓⢄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⡀⣡⠊⠀⠀⠀⠀⠀⢷⠀⠀⠀⢠⠀⠀⠀⠀⢸⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢇⠃⠀⠀⡀⠀⠀⢀⡞⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⣾⠘⢀⡴⠋⠀⠀⠀⣸⠀⠀⠀⠀⠀⣧⢠⠄⠐⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠳⡀⡇⣴⠊⠀⢀⠀⠀⠀⢱⡇⠀⠀⡄⠀⢳⠇⠀⠀⠘⡆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠱⡇⠃⢀⡔⠕⠈⠀⠀⢾⡀⠀⠀⠃⠠⠘⢀⠆⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠐⡄⡇⣇⠎⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠆⠖⠉⠀⢀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠘⡇⡿⢀⠔⠋⠀⠀⠀⢯⡀⢰⣄⠎⠌⠀⡀⠸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⡄⣧⢱⠎⠀⡀⠀⠀⡤⠚⠃⢠⢏⠌⡠⠎⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⡰⠁⠀⠀⢳⠀⡠⢃⠞⢉⡶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠸⢸⡜⠁⠀⢠⣢⢞⠟⢡⠋⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣴⠘⡇⠀⠰⡿⣠⠊⡰⠛⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⡇⢶⢺⠟⠁⣼⠕⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡏⠀⢇⣸⠏⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣇⣠⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⣀⣸⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠛⠓⢶⣤⢀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣡⡀⠀⠀⠘⠁⣕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⠄⠀⢻⡌⠦⡰⠀⢼⣟⣀⣀⣀⣀⣀⡀⣀⣀⣠⣤⣤⡤⠶⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠀⠀⠻⣧⣀⠀⢿⣴⡇⢨⣾⣷⣡⡤⠴⠚⢛⣩⣵⣖⣻⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣀⣠⣤⡤⠤⣤⣌⣻⣷⢽⣿⡀⣾⣿⣴⣦⣶⣿⣿⣟⡛⠋⠉⠉⠉⠛⠂⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠉⠉⠉⠉⠙⣻⠷⠶⣾⣯⣿⣯⢽⣿⣿⣿⣾⣷⣿⣿⣷⣾⣿⣷⣶⣶⣷⣴⣠⣌⢠
⠀⠀⠀⠀⠀⠠⠊⠜⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠛⠉⠁⠀⠀⠀⠙⣿⣷⡄⠀⠉⠙⠈⠀⠈⠉⠉⠙⠛⠻⣿⣿⣿⣷⡶⡤⠄
⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠉⠹⢷⣷⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⢧
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣷⣃
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⣿⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣯⣿⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣿⣿⠃
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢠⢀⣷⣿⡄
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣻⣿⢳⠁
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢿⣿⡟⡞
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣩⡿⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⣘⣿⣿⣟⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣾⣽⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⣠⣔⣸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢬⣀⡀⠁⠀⠀⠀⡀⠀⡀⠀⠐⠀⡀⠁⣴⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠓⠶⠴⠶⠶⠶⠷⠶⠦⠷⠾⠛⠋⠁

⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⢶⣿⣿⡿⠛⣿⣿⠛⠛⢷⠶⠶⠶⢦⣤⣀⡀
⠀⠀⣠⡶⠻⠉⠈⠀⠀⠠⠆⠁⠀⡀⣀⠉⠀⠀⡀⣿⠿⠓⠒⠶⠤⠉⠛⠷⣦⣀
⠀⣸⠟⠀⠪⠀⠀⡀⠀⠀⠀⠀⠈⠀⠉⠀⠠⠨⠀⠀⢀⣤⡴⠶⠛⢛⡛⠛⠛⠛⠿⢶⣄
⠀⣿⡌⠀⠀⡁⡜⡍⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠞⡩⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠠⠙⣧
⠀⢻⣷⣾⡆⣠⢰⢸⢰⢾⡜⡎⢀⢀⢬⠡⠠⣸⡏⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢻⠟⠿⢿⣿⣾⣿⣿⣿⣿⣧⣿⣾⣀⠸⣿⡇⠀⠀⠀⠀⠅⠀⠁⠀⣀⣀⣀⣀⡀⠀⣾⠃
⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⢿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣠⣶⣿⠿⢛⡉⠉⠉⠙⠻⣄
⠀⠀⣸⠇⠂⠀⠀⠀⠀⠐⠀⠄⠐⠂⠦⠄⡈⠉⠙⢷⡰⡠⢠⣿⣿⠟⠁⠀⠀⠁⠀⠀⢀⣠⣽⡷⠟⢛⡉⠉⠩⠛⠳⣄
⠀⢀⣿⢀⡀⠰⠀⠃⠰⠀⠐⠀⠄⠀⠀⠀⠀⡄⠀⢨⣷⠀⣼⠿⡇⡄⠠⠀⠀⠀⢉⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⢂⠀⢸⡇
⠀⢼⣯⣼⡗⣆⢀⢀⠈⢡⠀⠀⠀⠀⠀⠀⢰⢀⡖⢾⢹⠀⢸⡆⢡⡱⠀⠀⠀⠀⣾⢻⡇⠀⢀⡀⠀⠂⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠙⠿⢿⣿⣿⣿⣷⣼⣰⡄⣴⡇⣀⠀⢸⢨⢻⣿⣿⠀⠀⠻⣿⣿⣶⣤⡐⠀⢿⡸⡄⠄⠠⠀⢀⠀⠀⠀⠀⠈⠐⠘⣇
⠀⠀⠀⠈⠉⠉⠛⢿⣟⣿⣿⣧⣿⣧⣿⣷⣧⣾⣿⣿⣯⠀⠀⠀⠈⠻⡏⠙⡇⠀⡈⠻⢿⣦⣄⡀⠀⠀⣀⠀⠁⠒⠂⢀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠩⠭⠭⠭⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⡹⠀⠀⠀⣀⣠⣿⣦⣵⢣⡀⠀⠀⢻⠛⢻⠀⠀⠒⠃⠀⠀⠀⠀⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠒⠒⠒⠒⠲⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣷⣝⣂⣀⣸⣷⣼⣇⢀⠀⠀⠀⠀⠀⠀⠀⠐⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⣛⣻⡟⠛⠛⠛⠛⠛⢻⣿⣿⣦⣓⡀⠀⢀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠐⠒⠀⠀⠀⠶⠰⠒⠆⠐⠐⠂⠐⠐⠰⠀⠀⠂⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣰⡷⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣁⡴⠛⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣤⡦⢂⣥⡞⣻⡍⠀⢺⠟⠄⠀⢠⣤⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣀⣤⣠⣀⠀⢀⠒⠚⠉⢨⡥⠶⠟⠋⣀⡴⠋⣰⢟⢴⠒⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⠀⠠⠤⣤⣤⠄⠠⢀⡀⢀⢀⠐⢛⠁⠴⠚⢁⣾⡇⡀
⠀⠀⠀⠀⠀⢀⣤⠤⠀⠿⠛⠙⠛⠷⠶⠶⠂⠄⣊⣭⡾⢛⣥⢞⡩⢄⠀⣀⡤⠟⠈⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢸⣟⡙⠛⠓⠒⠒⠀⠂⠀⠀⠀⠉⠉⠤⠶⢛⠕⣹⠯⠗⠋⠀⠀⠀⠠⠤⠚⠁⠀⠀⣨⠞⣻⣦
⠀⠀⠀⠀⠀⢸⣿⠿⢯⣤⣀⣉⣉⣉⣀⡩⢹⣡⣬⠵⠖⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⠁⢸⡟⠛⢳
⠀⠀⠀⠀⠀⠸⠑⠀⠀⠀⠉⠙⠛⠋⢉⡀⠭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠴⣊⣤⠟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢠⠠⢤⣀⣀⡄⠤⠄⠴⠶⠶⣚⠉⢀⠔⠁⠺⠛⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠶⠦⠴⠂⠈⠉⣿⠛⠋⡡⡬⠞⠛⠁⠒⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⡤⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⣀⠀⢀⡔⠈⣰⠃
⠀⠀⠀⠀⠀⠀⠠⠄⢀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠉⠉⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠘⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠉⠉⠑⠲⢤⣀⣀⣀⣀⠀⠀⢀⡴⠒⠒⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡖⠉⠀⠀⠀⠀⠀⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⢄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠐⠈⠀⣀⡀⣈⡦⣄⣀⣠⣄⡀⠀⠀⠐⡀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠓⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⢀⠴⠒⠒⠚⠋⣁⡤⠤⣥⣀⣀⣉⠛⠋⠉⠙⠳⣄⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠢⢈⡼⠋⠀⠀⣠⠖⠋⠀⠀⠀⢰⠊⠉⠙⠦⣄⣀⠀⠀⠈⢧⠀⠀⠀⠀⢹⠙⢦⡀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⡠⠖⠉⠀⢀⡴⠋⠁⠀⠀⠀⣠⠤⠊⠀⠀⠀⠀⠐⠉⢳⡄⠀⠸⡆⠀⠀⠀⠘⡄⠀⠻⡄⠀⠀⠸⣥⣀
⠀⠀⠀⣠⠞⠀⠀⠀⢠⠏⠀⠀⠒⣲⠋⠀⠀⠀⠀⠀⢰⠃⠀⠀⠑⠀⠀⠀⠀⠀⠀⢳⡀⠀⢳⡀⠀⠀⠀⠙⠢⢤⡘⣆⠀⠀⠀⠈⠳⡄
⠀⣠⠞⢁⡴⠤⣄⠀⡏⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠘⠤⡀⠀⠀⠀⡀⠳⡄⠀⠙⠲⣄⡀⠀⠀⠀⡇⢹⠀⠀⠀⠀⢠⠇
⠀⡇⠀⢸⠀⠀⠈⠳⣷⣠⠤⣴⡃⠀⠀⠀⠀⡤⠒⠉⠀⠄⠀⡀⠀⠀⠀⠈⠂⠀⠀⠰⠀⠙⣆⠀⠀⠀⠉⠳⣄⠀⠱⣸⠀⠀⠀⠀⣾
⠀⢣⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⡖⠀⠀⠀⠀⠘⡆⠀⢸⠀⠀⠀⠀⢻
⠀⠈⣆⡼⠀⠀⠀⠀⡀⠔⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠘⡀⠀⠀⠀⠀⣷⠀⠀⠀⢀⡴⠃⠀⢸⠀⠀⠀⠀⠈⢣⡀
⢠⡞⠉⠀⠀⢀⠂⢁⠀⠀⠀⢀⡟⠀⠀⢀⠴⠁⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠡⡐⠀⠀⠀⠀⡏⠀⠀⣴⠋⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠹
⣸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⡟⠀⠀⢰⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢠⡐⠀⢰⠧⡀⠀⢧⠀⠀⠀⠀⠀⡇
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠠⠍⢢⡟⠀⠈⠢⢘⡆⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣠
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄⠀⠈⢣⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠓⢀⡴⠚⠀⠀⠀⠀⡞⠔⡁⠊⠀⢠⠏
⢹⠀⠀⡇⠀⠀⠀⠠⡀⠀⠀⠀⠂⠈⠛⠲⢤⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠞⠀⠀⠀⠀⢀⣾⠁⠈⠀⠀⠀⢸
⠘⡇⠀⢠⠀⠄⠀⠀⠀⠀⠂⠠⠄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠒⠒⠒⠲⠦⠤⣄⣀⣇⣰⠃⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⣸
⠀⢱⡀⠀⢂⠈⡀⠀⠀⡀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠤⠄⠀⡀⠀⠀⠀⠀⢀⠠⠂⠉⠙⣇⠀⠀⠀⠀⣠⠟⠂⠈⠀⠀⠀⠀⢀⡇
⠀⠀⠳⡄⠀⠡⡈⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠁⠒⠀⠐⠁⠀⠀⠀⠀⡟⠀⠀⢀⠴⡉⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠙⢦⡀⠈⠀⠀⠀⠀⡇⢀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⢀⡤⠞⠛⠛⠉⠁⠀⠁⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠇⠀⠠⢀⡀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⡰⠋⠀⣀⣀⠀⠀⣀⣠⠤⣶⠶⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠘⠀⠀⠀⠀⠁⠒⠒⢂⡴⠒⠲⢤⡀⠀⠀⠀⣠⢞⡴⠚⠉⠉⠉⠉⠉⢉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣤⣤⣤⡤⠤⠖⠋⢠⠏⠉⠲⣝⠲⠴⠚⡵⠋⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⣏⠀⠀⠀⠈⠓⠶⠚⠁⠀⢀⣀⡤⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣲⣦⣤⣤⣴⣶⣒⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⡠⠤⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⣤⣄⣀⣀⡀
⠀⠀⠀⢰⡸⠭⠇⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠉⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⣀⠠⠤⢺
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠇⠠⡤⠔⠃
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⡄⠠⠦⠠⢦⠁⣤⢟⢧⣷⢠⠴⠀⣤⢤⠤⠤⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡀⠁⠞⠁⠐⡒⠄⠙⠸⠜⡧⠓⠯⠥⡈⠠⠭⠄⠀⠀⠠⠬⠐⠴⠤⠉⠂⠀⠾⠂⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⡸⣆⠀⠠⢸⠋⠒⠢⠤⠄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠢⢠⣀⡀⠀⠊⠀⠀⠃⡀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠂⡴⠁⠸⡄⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢸⠙⠀⠀⠀⢃⠀⠀⠉⠐⠒⠂⠶⠬⣀⢒⡞
⠀⠀⠀⠀⠀⠀⠀⡆⠈⣼⠀⠀⠀⠱⡄⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡎⠳⡄⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⢧⠀⠀⠀⠀⠱⡈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⣆⠀⠘⣄⠚⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠓⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠉⠉⠉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠂⣉⣠⡀⠀⠤⠖⡒⣀⠌⠉⠿⠛⢋⣰⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠉⠁⠀⠀⢠⠤⠦⠤⠤⠀⠀⠀⠒⠋⠁⠀⠀⢀⠀⠀⠀⢀⡤⢀⣴⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⠄⠀⠐⠛⠀⠀⠊⠀⠀⠀⣀⣤⣾⠁⠀⠙⠛⠷⣶⣿⣧⣭⣉⣟⣿⠟⠁
⠀⠀⠀⠀⠀⠊⢀⣰⡴⠚⠋⠁⣀⣤⣶⣶⡦⢰⣿⡿⠿⣿⣿⣿⣿⣿⠇⠀⠊⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢁⣴⣶⣾⠿⠿⣛⣥⣤⣶⣶⣶⣿⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡻⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢤⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣰⣴⡠⢀⡀
⠀⠀⠀⠀⠀⣠⠶⢛⣽⣳⢄⣾⢿⡛⢶⡀⠀⠀⠀⣴⣾⢿⣟⣣⣀⣿⣿⣿⣦⡀⠀⠀⠀⣴⠶⣶⣶⣿⣯⣭⣽⣶⣦⡄
⠀⠀⠀⠀⢠⡏⣴⠈⠁⠀⠀⠛⠀⠀⢢⣿⠀⠀⣸⡏⠀⠊⠁⠀⠘⣿⣿⣿⣿⢷⠀⠀⢰⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠈⣸⠀⠀⢻⣧⠀⠀⠀⠀⣠⣿⣿⣿⡿⠘⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢻⣜⠆⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⢿⣿⣶⣶⣾⣿⣿⣿⠟⢁⠎⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⣤⡤⠶⠛⠁⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣯⠥⠒⠁⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣜⣂⣿⣚⣠⡴⠁⠀⠀⠀⠀⠀⠀⠠⣤⢤⣶⣤⠄
⠀⠀⠀⠀⠀⢠⡴⠿⢹⡿⣿⣿⣭⠶⣄⠀⠀⠀⠀⣠⡶⠾⠛⠛⠛⠿⢿⣷⣦⡀⠀⠀⠀⢀⠀⢠⣞⠧⢴⡿⠦⠀⢀
⠀⠀⠀⠀⢰⡏⠀⠀⠘⠀⠈⠃⠈⠃⠘⣧⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⢠⡏⠀⠈⠀⠀⠀⠃⠀⠀⠈⣇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠙⠳⠦⢤⡤⠴⠞⠋⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣾⣿⡿⠟⠋⠀⠀⠀⠀⠀⠙⠿⢿⣷⣶⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⢀⣤⠤⡴⢲⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⣄⡇⡄⢀
⠀⠀⠀⠀⢠⣶⢿⡿⠗⣀⠀⣙⢲⣶⣦⡀⠀⠀⠀⠀⡀⠀⢰⠟⢿⠻⣄⠐⣦⣄⠀⠀⠀⠀⣀⡤⠙⠷⣾⣿⠶⠋⢤⣄
⠀⠀⠀⢠⣿⢤⣼⣷⣿⣿⣷⣿⣿⣿⣿⣿⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣶⡹⡆
⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⢀⣾⣿⡗⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢹⣿⣿⣦⣤⣤⣤⣶⣿⣿⣿⠇⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⡾⣱⠇
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠙⠷⣤⣀⣀⣀⣀⣬⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⠲⠀⠀⢠⣶⣦⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠈⠀⠘⣶⣦⠀⣼⢹⠁⠠⡆⠀⡰⠷⠰⢀⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⢻⠀⠀⠀⠘⠁⠀⡿⠞⠀⠀⠹⠞⠃⠀⢀⠄⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠉⡦⠀⠀⢹⡆⠀⠀⣶⠄⠀⠀⠀⠕⠀⠀⢰⢀⡋⠀⠻⣷⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠄⠁⠂⠀⠀⣿⠀⠠⢿⣦⠀⠀⠠⠀⣆⢰⡼⠊⣀⠔⠘⢧⠞⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠸⠀⢀⠘⠒⠀⠀⠈⠀⠘⠃⠀⠀⠂⡄⠙⡌⠀⠊⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡴⠂⠀⠀⡇⡼⠁⠀⢀⢴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠉⠀⢤⠀⠀⠸⠃⢀⡤⠟⠋
⠀⠀⠀⠀⠀⠤⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠏⠀⠀⠀⡀⠀⢀⠄⢀⣬⡆
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠠⠀⠠⠒⠀⡀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠄⠀⠀⡄⣠⠂⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡞⡧⠀⠀⠈⠀⣀⢤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡅⠄⠘⠀⠌⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠐⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠠⡄⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢻⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⣇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠈⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠄⠀⠀⠂⠀⠀⠂⠀⠀⠀⠉⠈⠙⠉⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠄⠨⠀⠀⠀⠀⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢶⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢋⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢾⡀⡏⢸⣉⡉⠛⠛⢳⡶⠖⢛⣛⣿⣿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣯⢭⡤⣄⣸⡇⢡⠞⢑⠏⣠⠶⢃⡤⠚⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢉⡥⠤⠤⠴⠶⢍⠲⢍⣠⠃⠀⡾⢞⡅⣠⣏⡀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⣑⡄⠀⠀⢰⡏⠀⠉⠀⠠⢍⡓⢤⡀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠉⠀⠀⠀⠀⠀⡠⠎⢁⣀⣤⡀⠡⠐⠂⠋⠓⠒⠶⣌⡂⢹⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⡞⠄⡼⠉⠀⠸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠱⡜⠀⠀⣀⣤⠶⠛⠛⠛⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣷⡜⠁⠀⠀⠀⢱⣰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⣀⢀⠞⠁⠀⡀⠀⠠⠤⢤⡀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣀⡠⠤⠤⣿⣊⠜⠀⠀⠈⠀⠐⠀⠀⠀⠙⡄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⠞⠁⢀⠀⠄⢀⠀⠀⠀⠔⠀⡔⢅⠶⡀⠒⠄⡘⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡜⠁⢠⠖⠁⢀⠐⡂⠑⢄⠀⣄⠀⡄⠀⢙⣃⠔⢊⡀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡼⠀⢠⠗⢉⠁⢠⠜⠉⠱⢬⠇⠈⠈⠀⠀⠈⢄⡔⠉⠁⢤⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣝⠋⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡇⠀⢸⠄⣂⠐⠓⢠⡄⠀⠀⡠⢄⠀⡖⠐⠄⢀⣼⠢⠀⢸⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣂⣈⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢣⠀⠀⢧⣀⡈⠵⢊⣠⡄⠀⣠⢼⡆⡻⡤⡈⠟⠈⠠⢠⠏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡉⠩⣤⢭⣓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣹⡆⠀⠈⣆⠀⠀⠀⡴⠈⠈⠊⠱⠒⠣⠀⡣⠁⠘⠦⠴⠒⠓⣉⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⢶⡌⠉⠛⢛⠃⠈⣙⠛⠃⠈⢉⠩⠝⠛⠉⠉⠉⠀⢹⣆⠀⠘⣆⠀⠀⠳⡜⠠⠠⠀⡅⣠⠖⠁⢀⣠⣤⣴⣶⣟⣉⣙⣛⠛⡷
⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠞⠋⠁⠀⠀⠀⣀⠀⠀⠙⠒⠋⠀⠀⢀⠄⣀⡤⠶⠒⣾⠈⠳⣤⡈⠣⡀⠀⠉⠒⠢⠴⠖⠁⢀⣴⠋⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⣠⠞⢉⡀⠤⠒⠒⠈⠉⠉⠙⠁⠀⠀⠀⠀⠌⣠⠄⡰⣡⠞⠁⠀⠀⣠⠏⡀⠔⢹⠙⢳⡾⣷⣤⣄⣀⣤⡤⠶⠋⠁
⠀⠀⠀⣴⣧⣬⡥⠤⠤⢤⣤⣤⣀⣀⣀⣀⣤⠖⣃⣠⠞⣯⣼⠞⠋⠀⢀⡴⣛⣥⣤⠀⠀⡦⠀⡀⠳⣄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢼⠷⠛⠉⠀⠀⠉⠀⠀⠀⠀⠘⢋⡵⡚⣁⠀⠩⠁⡀⣿⠳⢼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣹⠏⣠⢋⣼⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⡷⠛⠁⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⠘⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⢤⣀⠜⠀⢰⠂⠁⢸⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⠀⡏⠉⠁⠀⠀⠉⠈⠉⠀⢠⠒⠉⠋⠀⢠⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⡟⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠄⠀⢀⠃⠀⠀⠀⡉⠉⠓⠋⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢼⠀⠀⠒⠇⠀⠀⠀⠀⠀⠈⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⠀⡁⠉⠀⠀⠀⠂⠀⠀⠀⠀⠸⠀⠀⠎⠘⡀⠀⠀⠠⠀⠀⠀⡀⠂⠀⢀⠀⣹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⠲⠇⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠄⡀⠀⠠⣇⡀⠀⠀⠀⣠⠀⡄⠀⠀⢠⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⠀⠠⠀⠀⠒⢤⠠⡀⢈⠀⡄⠐⠂⠘⢂⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠆⠘⠀⡀⠀⡀⢀⢀⠈⢀⠂⢀⠒⠀⠀⠀⠠⠀⠀⠀⠈⠊⡠⠀⢠⠀⠀⠀⢀⣈⣙⠆
⠀⠀⠀⠀⠀⠀⠈⢧⢀⡀⠀⠀⠀⠀⠀⢙⡋⠒⠁⠈⠁⡆⠀⠊⠀⠀⠹⡀⠀⡐⠀⠀⠀⠀⠀⠀⠁⡀⢸⣯⠵⠚⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠧⡄⠀⠀⠀⠀⣯⣄⠀⠀⠀⡠⠔⠀⠀⠀⠀⠴⡁⠰⠃⠀⠀⠐⠀⠀⠃⠁⠀⢨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠀⢰⠶⠞⢺⡉⠀⠀⢱⠄⢀⠄⠀⠀⡠⢠⡀⠀⠀⠀⢄⠀⠀⢀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣧⠀⠀⠘⡂⠠⡂⡤⢈⢀⠀⠣⡀⢀⡤⠊⠐⠀⠀⠐⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣤⡀⠉⠂⢘⠀⠀⠀⣀⡴⠁⠀⢀⠐⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣄⠀⠀⠉⠞⢀⠁⠑⡀⠀⠒⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠻⣄⢀⣠⡀⣑⠀⠁⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⡇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠄⢸⡛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠙⢀⡵⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⣴⣀⣴⢂⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣴⠎⢹⢧⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠙⠓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⡇⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⠀⠈⠙⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠘⠀⠀⠀⠀⣸⠀⠀⠀⠀⠟⠀⠀⠳⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⣀⣤⡤⠴⠶⠖⢻⡟
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠒⠛⠉⠉⠁⠀⠀⢀⡎⣠⠏
⠀⠀⠀⠀⠀⣠⠞⠁⠠⠂⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠚⠛⠉⠉⠀⠀⠀⠀⢀⡄⠀⠀⢰⠂⣀⣺⡴⠃
⠀⠀⠀⢀⡾⠃⠀⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⣁⣠⡤⠶⠚⠋⠉⠉⠓⣦⣄
⠀⠀⢠⣏⠰⠂⠀⢀⡤⠀⠀⠈⠁⠀⢀⣀⣀⣤⡤⠶⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠾⢏⡉⠀⠀⢀⡴⠖⢺⡏⠉⠉⠁
⠀⠀⢸⡏⠓⠲⠶⠶⠶⠶⠖⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢴⡶⣿⣭⣭⣤⡤⠤⠤⠴⠶⠖⠚⠋⠑⠀⣠⠷⣤
⠀⠀⠈⡇⣇⠀⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠒⠛⢛⡟⠛⠿⠶⠴⠶⠋⠀⠈⡆⠀⠀⠀⠀⠀⠀⠄⢀⣀⡤⠶⠛⣡⣾⠁
⠀⠀⠀⠳⢦⣤⣬⣁⣀⣀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠖⠛⠉⠉⣇⠀⠈⠆⠀⢀⡼⠁⠀⠀⡆⠀⠰⠀⠀⠀⠁⠀⠂⣀⣤⡤⠶⠚⠉⣁⣠⠴⠚⠁⠘⣇
⠀⠀⠀⠀⠸⣇⠰⠁⠉⠉⡍⠉⠉⠉⠉⠉⠉⠉⠛⠓⠒⠶⠶⠾⠶⠶⠶⠖⠋⠁⠀⠀⠀⠀⣀⣀⣠⡤⠴⠖⠛⣉⣁⣤⡤⠶⠚⠋⠉⠀⠀⢰⠀⠀⣿
⠀⠀⠀⠀⣰⠟⠳⢤⣥⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠴⠶⠶⠒⣒⣚⣛⣋⣭⣭⠥⠴⠖⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⠄⠀⠈⣠⡼⠃
⠀⠀⠀⠀⠷⣤⣤⠤⠶⠶⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣤⠶⠛⠁
⠀⠀⠀⠀⠀⡟⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠘⣇⠸⠀⢰⡀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠤⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠈⠛⠒⠶⠷⠶⠶⠶⠶⠖⠒⠒⠒⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣧⢀⣀⣠⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢹⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠛⢶⣾⠁⠀⠙⠷⣤⡀
⠀⠀⠀⠀⢰⣿⡿⠛⡛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠻⣄⠈⠛⠷⣦
⠀⠀⠀⠀⢸⣿⣧⡀⠈⣨⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡳⣴⠿⣦⠀⠀⣿
⠀⠀⠀⠀⢸⣟⢿⣟⣛⢉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠳⢤⣄⣀⡀⠀⠀⢸⢇⡴⠆⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠉⠉⡹⣦⠘⠷⣴⠏
⠀⠀⠀⠀⠀⠹⣦⡌⠙⠋⠉⠙⠛⠛⠛⠛⠛⠛⠟⢛⡟⠀⠀⠀⠀⠈⠉⠉⠉⠉⠠⢦⡄⠀⠀⣠⠞⠉⠀⠀⠀⠀⠉⠳⣦⡿⠘⢷⣄
⠀⠀⠀⠀⢀⣾⡋⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⡟⢠⠞⠁⠀⡴⠋⢷⠀⣾⠳⢄⠈⠳⡄⠀⠙⢷⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣼⠁⠀⣿⠀⠹⡄⠈⡇⠀⠹⡄⠀⠈⣿
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠒⠈⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠹⣤⡴⠃⠀⠀⠉⠛⠁⠀⢠⠇⠀⠀⣾⠁
⠀⠀⠀⣿⠂⠁⠀⠀⠠⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣰⠏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣄⠀⠀⠀⠈⠉⠓⠲⠶⠶⠶⠖⠊⠁⠀⠀⢀⡾⠋
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣦⣤⣤⣤⣤⣤⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠐⣤⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡤⠀⣀⣤⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⡏⠉⠓⣦⣤⣤⣀⣀⣠⡇⠀⠀⠀⣼⡗⠋⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣦⣉⣽⡄⠀⠉⠉⠉⣧⠀⠀⢰⡿⣷⠄⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢙⣧⠉⠉⠀⠀⠀⠀⠀⢻⣅⠀⢸⡇⠿⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⣛⣛⣭⣭⣛⣛⣒⠶⢤⣄⡀⠀⠀⢀⣠⡤⠶⣒⣛⣛⣭⣭⣛⣓⡲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠵⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢭⣓⣚⡭⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠉⠓⠮⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡄⠀⠀⠀⠈⢦⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡾⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣉⠉⠀⠀⠀⠺⠿⠆⠀⠀⠀⠀⠈⠈⠀⠘⠀⠀⠀⠘⡇⣷
⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠙⢆⠀⢀⠤⠤⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣟⢹
⠀⠀⠀⠀⠀⠀⠀⣷⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣎⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠛⠀⣇⠀⠃⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⡗⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⣀⡔⠹⣄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠉⠛⠋⠉⠀⠀⠈⠙⠛⠉⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢙⠇⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸
⠀⠀⠀⠀⠀⣀⡤⢶⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡶⢤⣀
⠀⠀⠀⣠⠞⠁⣠⢼⠇⡇⠀⠀⠠⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡧⣄⠈⠳⣄
⠀⠀⣼⠁⢀⡞⠁⢸⢰⡇⠀⠀⠈⠒⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⡇⠈⢳⡀⠈⣧
⠀⢠⡇⠀⢸⠀⠀⢸⢸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⡇⠀⢸⡄
⠀⠘⡁⠀⠸⢦⡀⢸⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⡇⢀⡴⠃⠀⢈⠃
⢀⡞⠁⠀⠀⠀⢧⣀⣳⡹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣞⣠⡼⠀⠀⠀⠈⢳
⣼⠀⢀⣴⡦⣄⣀⣠⠇⠙⢮⣑⡢⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⢔⣫⡵⠋⠸⣄⣀⣠⢴⣦⡀
⡇⠀⢸⠇⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⣶⠶⡶⢶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⣾⠸⡇
⠙⠲⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⣿⠀⡇⢸⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠈⠉⠙⠓⠿⠀⡇⢸⠀⠿⠚⠋⠉⠁⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡗⢺⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡟
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠛⠋⠙⠛⠉⠉⠉⠉⠉⠙⠛⠉⠛⠋⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⠀⠀⢀⣐⣋⡉⠀⠀⠐⠿⠇⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢠⠎⠀⠀⠉⢢⢀⠄⠂⠠⡀⠀⠈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡎⠀⠀⠀⣀⠀⡏⢀⠀⠀⢱⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⢱⡀⠀⠀⠀⣰⢇⠀⠀⠀⣸⠀⠈⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⡀⠀⠙⠶⠤⠚⠁⠈⠳⠤⠖⠁⠀⣠⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠉⠀⠒⠒⠒⠂⠈⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣇⣀⣀⡀
⠀⠀⠀⠀⠀⣠⠖⠋⢁⣀⣀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⠉⠓⢦⡀
⠀⠀⠀⠀⡜⠁⢠⠞⠉⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⢆⠀⠱⡄
⠀⢀⡤⠞⠁⠀⢮⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢨⠆⠀⠙⠦⣄
⣰⠋⢀⣤⡄⠀⢼⡀⠀⠀⠀⣼⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⣸⠄⢀⣤⡀⠈⢣
⢧⣠⡞⡸⠘⠦⣤⠇⠀⠀⠀⡟⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⡇⠀⠀⠘⢧⡤⠞⠸⡘⣦⣠
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠦⠤⠤⠦⣤⢤⣤⣤⢤⡤⠦⠤⠤⠦⠤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣿⠀⡇⡇⢸⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠉⠉⠙⠓⠿⠀⡇⡇⠸⠖⠛⠉⠉⠉⠉⠉⠓⢤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣗⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣃⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⡔⡻⢶⢀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠙⠓⠒⠦⢄⡀
⠀⠀⠀⠀⠳⣌⣊⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⢄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣵⢤⠤⠐⠒⣻
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠫⠼⠀⢤⡤⠞⠁
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠆⠀⢀⠸⠸⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⠀⢀⠐⠂⠀⠀⠀⠈⠀⠀⠎⠃⠂⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⣰⡆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡊⠀⣴⠀⠀⢐⢑⡶⠤⣄⣈⠀⠀⠀⠁⠀⡀⠀⢀⠀⢠⠀⠀⠀⢀⠀⡀⠅⠀⠀⠄⢀⠀⢠⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⡰⠉⣧⠀⠀⣼⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠒⠒⠒⣶⢆⠀⠀⠈⢰⠋⠙⠒⠤⠀⣀⡀⢀⡑⠄⢀⡌⣻
⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢀⡜⠁⠀⠸⣄⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⣄⠀⠈⣆⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠁⠀⠀⠀⠘⡆⢀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠇⠈⢣⡠⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢺⡀⠀⠀⠀⠀⠙⣌⣚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠘⣦⠀⠀⠳⡌⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⡀⠀⠠⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠈⡆
⠀⠀⢀⠀⠀⠓⠒⠢⡀⢑⣀⣇⠀⠀⠀⠀⠀⠀⠀⢠⢃⡿⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠈⣅⠀⠑⡄⠆
⠀⠀⢀⠵⣴⠦⣀⡂⠘⠮⠿⣈⡿⣄⠀⡄⠀⠀⠀⠈⢸⣇⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢽⠢⢀⡿⡆
⠀⠀⠁⢄⠀⠂⠙⢛⣾⡀⡀⠈⠻⢉⢄⡈⠀⠀⠀⣧⢈⣿⠟⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡄⠸⠛⠸⠑
⠀⠀⠀⠀⠻⢦⣀⠀⠠⡟⠛⠀⡾⢉⠀⢈⠠⡀⠀⡿⣅⢃⣼⡿⠀⠀⠀⠰⣆⠈⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠈⠑⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⣀⠈⠈⠉⠋⠀⠀⠀⠈⠀⡙⣦⡀⠂⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠘⠷⠁⠀⠀⢀
⠀⠀⠀⠀⢀⠀⢀⢀⠀⣠⠀⣴⠀⠀⠀⠀⠀⠀⠀⠠⠤⠷⠀⠤⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠒⠒⠁
⠀⢀⠀⡀⠹⣄⡻⠰⡔⠃⠀⠞⡇⠀⠀⠀⠀⢀⡀⢀⡔⠀⠤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠈⠠⡇⠰⠈⢇⢸⡃⠀⡀⢪⡅⠀⠀⠀⠘⠀⠀⠐⡇⢀⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠟⢀⠄
⠀⠀⢀⣇⠀⢠⠈⡯⠱⠞⢁⡯⠁⠀⠀⠀⡀⢥⠀⠀⡙⠀⠤⢤⣤⣴⡴⠏⡜⠀⠀⠀⠀⠄⣷⠞⢉⠧
⠀⠀⠈⢻⢀⠘⡙⠓⠀⣼⣿⠃⠀⠀⠀⠀⠳⡨⢠⠞⠁⢠⠖⠋⠁⠀⠀⢰⡀⠀⠀⢀⣴⠞⠯⠀⣄⡀
⠀⠀⠀⠸⡏⣼⡡⠌⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣉⣤⣾⣀⣆⡀⠐⣴⠟⠀⠀⢠⠊⠁⠀⠀⠀⠈⠈⣿
⠀⠀⠀⠀⡏⡬⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉⠛⠛⠚⠋⠁⠀⠀⠀⠂⠀⠀⠀⢠⠁⠹⣶⡃⡀
⠀⠀⠀⠀⠳⢣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠂⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⡋⡁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡐⠀⣌⠏⢰⠿⣭⡠⠆⠀⠀⠀⠈⢡⣀⣄⡐⠀⠤⡀⠀⠀⠀⠀⠀⡀⠰⢁⡆⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡻⣆⠠⣔⠰⢖⡼⢐⢒⡀⠀⠀⠀⢀⡈⠙⢻⢢⣤⡠⠀⠀⠀⠀⠨⠆⠀⢸⡦
⠀⠀⠀⢀⡃⠀⢤⣞⡈⣩⡏⣁⣖⠒⠒⠊⢠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⣿⠛⠤⡀⠀⠀⠀⣷⢀⡔⢀⡇
⠀⠀⠀⠀⠂⠀⠀⠃⠀⠀⢂⣀⠀⠐⠒⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠛⢊⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣦⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣧⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣷⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠋
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣧⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠥⣄⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠆⠀⠉⠙⠓⠒⠒⠒⡶⢯⣤⣔⣙⣉⠑⣂⠀⠀⠀⠠⢀⡠⣤⣾⣿⠟⠉⢃⣸⣶⡤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⠊⣷⣕⡒⠋⠀⢠⡶⠒⠋⡡⠞⠉⠀⢀⣀⠽⠟⠛⠛⣶⡶⠶⠗⠓⣿⡹⠄⠊⢀⣾⠏⣿⣧⠁⠘⢿⣒
⠀⠀⠀⠀⢀⣴⣿⣶⣿⣟⣿⣷⠀⢸⣷⣚⣛⣀⠀⠤⠒⠅⡉⢓⢒⡟⢋⣙⠉⢙⣙⣋⣡⣤⣴⣶⣿⣯⣴⣿⣿⡄⠀⠼⠁⣰⣄⢒⡛⠂⣤⢄⡀
⠀⠀⠀⠀⡽⣿⣿⣿⡉⠛⠷⣀⠀⠈⠉⠛⠚⠿⢷⣦⣤⣤⣌⣭⣵⣾⣿⡟⠃⠸⣿⡿⣭⣽⣿⡟⠓⠞⢩⣠⠄⠁⠀⠀⡌⠋⢸⠆⢀⠀⠀⠀⠀⢢⡄
⠀⠠⣄⣀⠀⣬⣿⡇⠙⠓⠲⡦⠀⠐⢀⡀⠀⠀⠀⠈⠉⠛⠛⠋⢉⣠⠟⠀⠀⠀⠀⡠⠋⣤⠈⠹⠃⠀⠈⠀⠀⢀⣴⠋⠀⢠⠓⠠⠞⠉⡀⢀⣴⡿
⠀⠀⠀⠈⠉⠙⣿⣿⠀⠀⠀⠀⠈⠙⣉⠛⠢⠄⠀⣀⠀⠀⠈⠉⠀⣀⣀⠀⢒⣐⣿⡺⣿⣿⣃⡀⠈⠀⢀⣠⠚⠁⡇⠀⠐⠇⣄⢀⠀⡼⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠑⠒⢛⣋⣉⣥⣽⣿⣷⢶⣿⣿⡏⠉⠀⣠⣴⠎⠋⠀⣰⠷⠃⠀⣷⢐⡴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⡿⠿⣿⣿⠿⡿⠟⠛⠂⢰⡃⠛⣿⡀⠉⠱⠍⣀⣀⣴⠋⠀⠐⠄⠈⢃⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠛⠲⢬⡝⠛⠤⠀⠀⠿⠅⠀⠈⢷⡀⠐⠈⡋⠙⢿⣾⣿⠷⢆⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣛⠓⠂⢤⣄⡀⠀⠀⠙⢦⡀⣼⣶⣴⡿⠓⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠐⠢⠄⠙⠉⠉⠁⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⠀⠀⣿⡏⠉⢻⣿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠑⠀⠀⣰⣿⡇⢀⣾⣋⣾⡏⢈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣰⣟⡿⠀⣾⢣⣏⣿⠃⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⣼⣿⡟⠁⣼⣏⣾⣿⣿⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⡀⠀⢀⣾⣿⠏⠀⣰⣿⣿⣻⣿⠇⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠰⠀⢀⣞⣿⠃⢀⣼⣿⡻⣫⣿⠋⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢀⣿⣼⠇⣴⡏⡵⢨⣿⡟⠁⠀⣰⡿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢸⣿⡟⢸⣿⣝⡆⣺⡟⠀⢀⣾⢟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢠⣿⣿⠁⣾⣿⣿⣷⣿⠁⣰⡟⣡⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢀⣴⢿⡿⠃⣰⣿⢿⣿⣿⠏⣴⣫⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢀⣴⢟⣡⡟⠁⣰⡿⣿⡟⣹⠏⣰⡟⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⢀⡴⠟⣡⡾⢋⣤⢾⡿⣿⢋⡽⠋⣠⣿⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⣴⣯⣾⡾⠋⣴⣟⣽⣋⠀⣡⠟⠁⠐⣱⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⣾⣿⣿⠫⠀⣼⢿⡟⡾⢃⣾⠋⢀⣴⠟⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣸⣿⣿⠃⠀⢰⡟⣿⡟⢀⡾⠁⢀⣾⠃⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠈⠀⢠⣿⣿⠏⠀⢀⣾⣿⣿⠖⣼⠇⠀⣼⠇⣾⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⣴⣿⣿⠏⠀⢠⣾⢿⣿⡷⣻⡟⠀⢸⡿⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⣰⣿⡿⠋⠀⢀⡴⢛⠏⡂⣿⣿⠟⠀⠀⣼⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣼⣿⠏⠀⠀⣴⠏⡀⠤⢊⣾⠟⠁⠀⢀⡼⣳⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠰⢢⣾⣿⠋⠀⠀⣼⣿⣬⣻⣿⡟⠁⠀⠀⣴⢟⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠛⠑⢁⣠⣿⡿⠁⠀⣠⣾⡟⣿⣿⡿⠿⢂⢀⣢⣞⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⣠⣶⣿⡿⠋⣁⣴⣿⣿⣿⣿⣿⣥⡤⠶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⡀⣠⣾⢻⡾⠋⢀⣴⠟⢰⢿⣿⡿⢿⣿⣷⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣯⡉⢀⣴⣿⣿⠟⠁⣠⣾⣫⣴⣽⣿⣭⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣏⣀⣤⣾⡿⣯⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠹⡄⢸⡆⠀⠀⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡆⠀⡇⠘⡇⠀⠀⡇⡇⡠⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣏⡛⠒⠲⠶⣻⠄⡟⠉⡇⠹⠛⠓⠶⢁⡏⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠁⠉⠢⣁⢿⡰⡡⢠⠀⡦⠀⠈⣡⠾⡛⠍⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡼⡁⠀⠀⢠⠜⠊⠉⠉⠈⠉⠛⠻⠚⠉⠉⠉⠉⠛⢣⢤⠀⠘⢛⢦
⠀⠀⠀⠀⠀⠀⣼⠓⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠱⡀⠪⣮⣧
⠀⠀⠀⠀⠀⢀⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⠀⠀⠀⠀⠂⡀⡔⢔⢅⠀⣹⣿⡇
⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡇⠀⠀⠀⠀⠺⠱⠰⢈⣎⠃⢿⣿⣷⠀⠀⢀⣀⣠⠤⢦
⠀⠀⠀⠀⠀⢸⡏⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣠⣀⡤⠼⠶⠶⠴⠶⠗⠒⠚⠛⠛⠋⠉⠉⠐⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠈⣇⠀⠀⢰⠀⠀⠀⠀⠀⠀⡀⠀⣼⣿⠋⠁⠀⠀⠀⠒⠒⠀⠄⠀⠀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠹⣤⠀⠘⠀⠀⢀⣤⠀⠸⠀⢰⠏⢀⡀⠠⠀⠐⠒⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣶⡷⠄⠀⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠐⢄⠃⡜⣿⡀⡶⠄⣿⠀⠠⠰⡀⠀⢀⡀⣟⢛⡬⠇⣴⢦⡄⣤⢄⠠⡰⣶⣦⠏⢰⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡉⠂⢷⢟⢽⡇⠂⣿⠀⠀⠣⠳⣄⠀⠙⢮⡚⠲⠊⠊⢸⡁⣿⣾⠘⣷⢘⡷⢠⠊⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⣘⠒⠁⢈⡾⣇⠀⢑⠵⡜⢮⡓⢀⠙⠲⢦⣜⣁⠙⣉⣠⣛⣉⠥⠚⠁⠰⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢤⣬⣧⣀⢈⠚⣷⣿⣦⣀⠰⢦⠀⠈⠁⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢭⣛⣿⣿⣿⣽⡶⣤⠶⠀⠴⠦⢀⣠⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠛⠛⠚⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠉⠉⠉⠉⠉⢛⣓⣲⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠉⠀⣀⣀⡤⠤⠼⠓⠒⠒⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠖⠋⠁⣀⣠⠤⠖⠒⠋⠉⠀⠀⢀⣀⣠⠤⠴⠒⠒⠒⠒⠒⠛⢓⣢⣤⣀
⠀⠀⠀⠀⢸⡇⠀⠀⠠⠾⠕⠒⠚⠉⠁⠀⠀⠀⠀⣀⡴⠖⠋⠉⠁⠀⠀⣀⣠⠴⠖⠚⠉⠉⠉⠉⠉⠉⠉⠙⠒⢤⣀
⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⢀⣀⡠⠴⠒⠊⠉⠁⠀⠀⠀⣀⣤⠤⠖⠒⠒⠛⠛⠛⠒⠒⠮⠷⣄
⠀⠀⠀⠀⠀⢻⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠽⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠙⢷⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⣀⣀⣠⠤⠤⠖⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠶⠶⠶⠦⠤⠤⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠯⣤⣀⠤⠀⣠⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⣀⣀⣠⡤⠤⠤⠤⠤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢶⣇⣔⣁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⠶⠶⠒⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣥⣔⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠧⣄⣤⣀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠺⠷⠶⠶⠶⠾⠗⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⠤⢢⠤⢀⣄⣀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣢
⠀⠀⠀⠀⠃⠀⠰⠏⣠⣀⣸⣷⣎⣉⣀⡿⣷⣼⣟⠋⣼⣿⡿⣿⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠉⠉⠉⠉⢉⣉⡁⣤⡤⢾⣛⣿⡟⠀⡹⠃⢀⣤⣞⣯⣴⢅⣠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠿⣧⠀⣠⣴⣾⣿⣿⣿⣷⣿⣿⣽⣿⡿⢿⣶⢁⣾⣿⣿⣏⣿⣁⡀⠘
⠀⠀⠀⠀⠶⣤⣤⣤⣿⡛⠋⠉⣉⣽⣛⣛⣺⡿⠋⢡⣶⣾⣿⣾⣿⣿⢧⣿⣏⣹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⢛⣩⠙⢿⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣭⣿⣟⣿⣿⡏⣾⡏⢙⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡿⡿⣹⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠘⠃⠐⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡷⣿⣿⣿⣿⣿⣷⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣙⣏⣿⣿⣾⡿⣻⣿⣯⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠯⡝⠠⣿⢻⣿⣿⡏⠁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠋⠀⠘⠚⠯⡼⢻⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠙⣷⣶⣦⡀⣦⣀⢀⡀⠀⠀⠀⠀⠀⢼⣿⢘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣴⣦⡀⠀⠀⠀⠀⣠⠸⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⡗⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢟⣵⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡌⣿⢾⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⢎⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⣭⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⢸⣿⣿⡟⣿⣿⣷⠀⠀⢸⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⣼⣿⣿⢠⣿⣿⣿⠀⠀⢸⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠿⠟⠿⠛⠛⠛⠛⠿⠅⠛⠿⠀⠒⠈⠛⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡐⣤⣤⠀⡎⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⣹⣷⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⡀⢽⡄⢀⠆⠀⠀⡤⠄⠀⠄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⢀⣂⣫⡾⣶⡋⠀⠀⠁⢠⠀⠀⡀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠈⢹⡿⠟⠿⣟⠉⠀⠀⣾⣷⣎⠁⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠈⠂⢠⠀⢈⢹⠧⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⡾⡀⢸⡇⢀⣽⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⣀⣴⠏⠀⢧⡾⠉⡍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠋⣉⣰⣼⣾⣡⡦⢈⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠀⣨⠠⠵⢔⣢⡀⠀⠀⠀⠉⠛⠻⢿⣷⡼⠇⠀⠰⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⢂⡠⠠⠔⠀⣿⢿⡃⠀⠀⠀⠀⠀⠀⠀⠉⡷⠹⣶⡀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⠙⠛⠓⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢁⣀⠹⢿⣒⠖⣒⢦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠒⢿⣿⠾⠋⠉⢳⣤⣿⣏⢁⣉⠋⠉⣉⣉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠉⣹⡇⠀⠀⠀⠀⠀⠐⡏⠁⠀⠈⠙⠉⠙⠛⠛⠉⠻⡛⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠽⠾⠁⠀⠀⡠⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡁⣳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡇⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⣶⠫⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢡⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄⢀⣴⢋⠄⠀⢰⠒⠋⠁⠀⡽⠀⢠⠁⠀⠀⠀⠀⠀⣀⣴⡆⢸⡁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⡼⢁⡎⠀⠀⢸⡇⠀⠀⠀⠁⡠⠃⠀⠀⠀⣰⡾⠛⠉⠁⠀⠈⡛⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⡇⣼⠀⠀⠀⣸⡇⠀⠀⢀⡜⠁⢀⣀⣀⣠⡏⠀⠀⠀⠀⠀⢠⣿⠂⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⡇⡏⠀⢯⠈⠉⠀⠀⣠⠎⠀⠀⠁⠀⠀⢈⣀⡀⠀⠀⢀⣠⣿⠟⠋⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠷⣠⠎⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢠⠽⠛⠶⠟⠋⠀⠀⠀⣏⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠀⣸⠀⢀⣴⠟⠀⢀⣤⠴⢶⣤⣄⡀⠀⣸⠆⠀⠀⠀⠀⠀⡖⠛⣉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠠⠟⠀⠞⠏⠀⠀⠀⠀⢀⡾⠀⠀⠉⠛⠁⠀⡀⠀⠀⠀⠳⣽⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣤⣀⠀⠀⣀⡀⣠⣤⠖⠚⣛⣀⣀⣀⣠⣴⠞⠋⠀⠀⠀⢀⣸⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⣷⣈⣩⡽⠿⠛⠛⠓⠛⠿⠿⠿⠿⠻⠟⠛⠲⢦⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠚⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠠⠠⣀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠉⠉⠛⠛⢛⠒⠢⠤⡀⠀⠀⠀⠀⠀⣠⡤⠶⠲⢤⣄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⣀⣠⣤⣨⡾⠳⠦⣄⠀⣏⢮⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣐⣈⣆⠀⢀⡤⠻⢿⡄⠀⠀⢸⡿⠓⠦⡀
⠀⠀⠀⠀⠀⠀⢿⡦⡊⠁⠀⠉⠟⠛⠀⡀⢀⠸⡆⠸⣷⠇⠀⢠⣴⣟⣗⡆⠀⠀⢨⣽⣿⠋⢀⡾⠆⠀⠀⠃⠀⠀⢸⠀⠀⣠⡿⡄
⠀⠀⠀⠀⠀⠀⠈⣷⡇⠀⠀⡸⢀⠀⠀⠈⠳⣿⠂⠀⢨⠀⠀⠀⠈⠏⠁⠀⠀⠀⠀⣹⠏⠀⣎⠀⣈⡶⠤⠾⠶⠶⠛⠒⠾⢧⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⣠⣜⣇⣑⣐⣀⣤⠟⠀⠀⠘⠳⢤⣤⣴⠀⢀⣀⣠⣤⡬⠏⠀⠀⠰⠜⠁⠀⠀⠀⢀⣀⣀⣀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣴⣾⠛⢻⣿⣮⡵⣄
⠀⠀⠀⠀⠀⢀⡤⠤⡤⠀⡶⢤⣤⠖⠒⣶⢶⣶⠶⠋⢉⣩⣟⡲⠚⠉⣹⢒⣳⣶⠾⠉⠓⠒⢆⠀⢠⠓⠒⠉⠉⢻⣿⡟⠉⠙⠇⠀⢷
⠀⠀⠀⠀⠀⣏⡀⠠⠴⠋⠛⠁⠤⠒⠛⠛⠉⠀⠐⠚⠙⠉⡉⢀⣒⢉⡋⢉⡉⠀⢀⠀⣀⢄⡼⠀⠸⡀⠀⠀⠀⠟⠀⠙⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠙⠛⠛⠒⠓⠒⠓⠻⠿⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠈⠒⠶⣤⣤⣀⣤⣤⣴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⢒⡲⠶⠶⠖⢒⣢⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣠⠤⣤⣀⣀⣀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣎⢀⣷⡦⣰⣯⣄⣀⡴⣮⢬⣉⣀⣀⣀⣱⡄⠀⠀⢀⠤⠂⠉⢡⣫⠟⠋⢁⣴⡫⠿⡋⢀⡴⡿⠟⠋⠁⠢⡀
⠀⠀⠀⠀⠀⠀⣾⢸⡟⠻⣆⠙⠽⢧⡌⠛⠷⣌⣀⠈⠳⢤⡈⠉⠙⢦⠀⢺⣠⠀⠀⠟⠁⠀⠀⣸⡿⠁⠀⠀⠚⠉⠂⠀⠀⡀⠩⡿
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣄⣘⣤⣄⠈⠻⡀⠀⠈⠛⢅⡂⠀⣙⣦⠴⠋⠀⠀⠙⠷⣆⣀⠀⢀⣤⡅⠀⠀⢀⣀⣀⠀⣀⣤⣯⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠓⠒⠚⠑⠒⠛⠚⠉⠉⠉⠁⣀⣀⣀⣀⣠⣤⣀⡈⠉⠉⠙⠛⠒⠚⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⡠⠒⢗⠖⢛⠍⡭⠊⢡⠔⠂⠈⠛⡆⠀⠀⠀⠀⠀⣀⣤⡤⠀⡒⢢⣤⢄⡀
⠀⠀⠀⠀⠀⠀⣰⢇⡴⠊⠛⢄⠙⣆⠀⠀⠀⢀⠎⢰⠫⠊⡠⣣⣿⣴⠞⠁⠀⠀⠁⢠⠃⠀⠀⢀⢔⠟⢛⣽⠛⡄⡦⢠⢘⡦⠙⣆
⠀⠀⠀⠀⠀⠀⡏⣼⢇⣀⡤⠤⠶⢿⡀⠀⠀⣾⢾⡇⣾⠸⡃⡇⠘⣷⠀⠂⠀⠀⠂⠘⠀⠀⣴⠃⡆⢠⡋⣂⡬⢔⡥⠂⢼⡗⠀⢼
⠀⠀⠀⠀⠀⠀⣇⠸⡃⣌⣉⣓⣶⠶⡈⢦⠀⠹⣌⠣⡱⠄⠳⡘⢷⡝⡆⠀⠠⠀⠀⢰⠀⢸⠁⡀⠑⢀⣀⣀⣠⠵⠂⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠘⢦⢷⣅⠙⠈⠀⠀⢸⠋⢧⠀⠈⠱⡌⣎⢆⠘⡀⠃⡇⠀⠀⠀⢁⡎⠀⠘⣷⡈⠲⣦⣤⣤⣤⣤⡤⢴⣾⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡁⠗⠰⠖⢁⡰⠃⠀⠀⠀⠸⣌⣘⡄⡇⡄⢿⡄⠀⡠⠊⠀⠀⠀⠈⠑⠲⠼⠿⠾⠷⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⡉⠑⠒⠶⠤⠤⣠⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠂⠀⠰⠟⡉
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠉⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⠋⠁⠀⠀⢀⣠⠴⢎⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠈⢄⠀⠀⠉⣿⡩⢙⠶⠦⣤⣀⡀⠀⠀⠀⠀⠉⠐⠂⠠⠄⠀⠀⠀⠀⠐⠂⠁⠈⠿⠀⠀⠴⣶⡉⣀⣀⣀⣡⠀⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒⣲⣾⣷⣌⣀⠀⠀⠎⠉⠛⠶⢤⣄⣀⠀⠀⡀⠀⠀⠀⣶⠀⠀⠀⣀⣤⣶⠁⠀⣀⣀⢙⠉⢉⣽⡃⠀⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⢀⡈⠙⠻⢯⣶⣖⣶⣶⣾⠿⣿⣿⡟⠶⠦⣤⣀⣀⣈⡤⠶⠻⠁⢠⡷⠺⠫⢉⡍⣾⡾⠟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠉⠲⣄⠀⠀⢀⣀⠀⠀⠀⢉⣉⣻⣶⡀⠺⠀⠉⠡⢀⡀⠄⣠⣏⡠⠤⠚⠋⠰⠁⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢄⡀⠀⠈⠑⠢⠤⠀⡀⠉⠁⠀⠚⠂⠙⠻⣷⣤⣤⣤⡤⢶⡮⠭⠤⢤⣤⣤⠄⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣁⡀⠈⠐⠠⢄⡀⠀⠀⠀⠙⠢⣄⠀⠀⢀⡀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣯⣿⣿⢷⣜⣿⠞⠛⠳⣤⣀⠀⠀⠈⠷⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢻⡍⠛⠿⣏⣿⣴⠶⠤⣄⣀⠉⠲⢦⣄⡀⠀⠀⠀⣀⠀⠀⣀⣤⡀⠀⠀⠈⠀⢀⡰⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠀⠀⠀⠀⠉⠻⠦⢀⣄⠉⠓⢦⣄⠈⠛⠓⠒⠶⠾⣶⠿⠮⣙⣿⠷⠶⠶⠚⢋⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠓⠄⠄⣈⠳⣤⡀⠀⠀⠀⠀⢀⠠⠀⠛⠛⠃⠒⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⠀⠀⠀⠀⠀⠉⠘⠛⠂⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠒⠦⢤⣀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⢀⣠⣿⡿⠓⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠋⠓⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⣾⡿⠟⣉⡤⠞⣠⠟⠉⠛⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣀⣤⣾⡿⠋⣸⠛⡶⢤⣀⡀⠀⠀⠀⡟⠀⢀⣿⡏⢀⡞⣡⠖⠋⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣼⡿⠋⣡⠴⣚⣥⠞⠁⠀⠈⣹⠀⠀⡴⠃⠀⣸⣿⠁⣼⢰⡏⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢰⣿⠃⣰⢃⡞⠉⠀⠀⠀⣠⠖⠃⡤⠚⠁⢀⣴⣿⠋⢠⠇⣼⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⢀⣾⡟⢀⡏⣼⠁⠀⠀⠀⣰⠃⢀⠞⠁⢀⣴⡿⠋⢁⡴⢃⡼⠃⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⢀⣴⣿⠟⢀⡜⢡⠏⠀⠀⠀⠀⡇⠀⡟⠀⢀⣾⡟⠁⡴⢁⡴⠋⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣰⣿⠏⢀⡔⢋⡴⠋⠀⠀⠀⢀⡼⠃⣰⠃⠀⣸⣿⠀⡼⢡⡟⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⣿⠃⣠⠋⡴⠋⠀⠀⠀⣠⠞⢋⡥⠞⠁⢀⣴⣿⠃⢠⠇⣾⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⢀⣾⡟⢀⡇⣼⠁⠀⠀⠀⡼⠁⢠⠏⠀⢀⣾⡿⠛⢁⡴⢋⡼⠃⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⣴⠚⠁⢀⣴⣿⠟⢀⡜⢠⠇⠀⠀⠀⢸⡇⠀⡞⠀⢀⣿⡟⢀⡜⢡⡶⠋⠀⠀⠀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⣼⠁⠀⣴⣿⠋⢡⠔⣋⡴⠋⠀⠀⠀⢀⡞⠀⡼⠁⠀⣼⣿⠁⡼⢠⡏⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⢰⠇⠀⢰⣿⠃⣰⢃⡼⠉⠀⠀⠀⣠⠞⠉⡴⠋⠀⣠⣼⣿⠃⣰⠃⡾⠀⠀⠀⢀⡏
⠀⠀⠀⢀⣠⠏⠀⢀⣿⡟⢠⡇⣼⠁⠀⠀⠀⡞⠁⠀⠀⠓⠦⣾⣿⠋⣡⠖⣃⡼⠁⠀⠀⠀⡼⠁
⠀⠀⢰⣏⡀⣠⣶⡿⠟⢀⠞⣰⠇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⣅⡀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠙⠻⠿⣤⡞⣡⠖⠋⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠻⣇⡤⠞⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠁⠀⠀⣿⠁⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⢠⡏⠀⡀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⢸⡀⠀⠀⠀⢀⣤⠶⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⣠⣾⠁⠀⢀⡶⠛⠁⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⢹⣶⠶⠚⠃⠀⢠⡾⠋⠻⢷⣶⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⠋⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⡄⠀⠀⠀⢸⡇⠀⠀⢀⣮⡤⠴⠄⠀⢰⠟⠀⣠⠀⣼⠃⠀⠀⠀⠀⠀⣾⣿⣦⡀⠀⣠⡶⠛⠁⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⡏⢳⣀⣤⡀⠈⣧⠀⢀⣞⣁⣀⡀⠀⢠⡟⢰⢋⣿⢾⠃⠀⠀⠀⠀⠀⣤⣿⠿⠿⠟⠉⠁⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⢰⠀⠉⠈⠀⠀⠿⣦⣻⠀⢸⠋⠉⠁⠀⢀⣼⠁⣾⠟⠁⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠘⠷⠶⢶⡄⠀⠀⠀⠹⣦⢸⣷⠞⠁⣠⣾⣿⢠⡟⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠙⣦⡀⣶⡄⠿⠸⠃⠀⣰⣿⣿⣿⣼⡇⠀⠀⠀⠀⢀⡶⠛⠇⠀⠀⠀⠀⠀⢰⡾⠔⠀⠀⠈⣴⣿⣤⣄
⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⢀⣠⡶⠟⢉⡴⠋⢉⡇⠙⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⡿⠉⢹⡇
⠀⠀⠀⠀⠀⠈⣿⡁⠀⠀⠀⠀⢦⣄⡄⠸⣿⣶⠟⢁⣠⣾⣿⠃⣾⠋⠁⠀⠀⠀⢠⡟⠀⠀⠀⢀⠀⢀⣀⣿⣯⣤⠀⠄⠀⠀⣸⣡⣴⠿⣧⡴⠶⠶⠶⣦⣤
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠹⣷⠀⢻⣷⣴⡿⢿⣿⣿⣠⠿⠀⠀⣤⣤⣤⣾⠃⠀⠀⠀⢸⠀⣼⠟⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣾⡿⡄⠀⠀⠀⠈⢳⣜⡇⢸⣿⠋⠀⣼⢻⣉⡿⠂⠀⡼⠁⢠⠀⠈⠀⠀⢀⠀⣸⡿⠥⠤⠄⠀⠀⠀⠀⣰⠃⠀⢀⠀⠀⡐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠱⣦⣝⣿⠸⣿⠀⢰⢏⣿⣇⠀⢀⣰⡇⢀⣸⡇⠀⢀⠀⢸⡾⠇⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠂⢀⡴⠛⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠈⢦⡘⢿⣿⠀⣿⣠⡏⡜⢸⡟⠀⣾⠛⠃⣼⠏⠁⠀⢸⣦⠎⠀⠀⠀⠀⢀⣴⠿⢶⠟⠁⠀⠀⣸⡴⠋⠀⠀⠀⠀⣀⠀⢀⣿
⠀⠀⠀⢠⡾⠋⣿⡀⠀⠀⠲⣄⠈⢿⣌⣿⢰⣿⠋⡐⠀⡘⠁⢀⣿⠀⠀⡇⠀⠀⣶⣞⣉⠀⠐⠀⠀⢀⡾⠁⠀⠀⠀⢠⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⢻⣾⠋
⠀⠀⣰⠏⠀⡜⣿⠗⠀⠀⠀⠹⣧⠘⣿⡟⢸⡟⠀⠀⠠⠁⢀⣿⠁⠀⣤⡇⠀⠀⠈⠉⠉⠉⢀⣠⣴⠟⠀⢰⣀⣤⠶⠛⠳⠄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⢀⡟⠀⢸⠇⠹⣆⠀⠀⠀⢰⡹⣷⣽⠃⣼⠇⠀⠀⠀⠀⣸⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠋⣹⣡⡴⠖⠻⠯⠄⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣰⡎
⠀⢸⠁⢀⠏⠀⠀⠙⣧⡀⠀⠀⢿⣿⠏⢠⡿⠀⠀⠀⠀⢀⡏⠀⣲⠶⠀⠀⠀⠀⢰⣶⣫⠵⠚⠻⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⢳⡆
⢀⡇⢀⡏⢀⡀⠀⣼⠋⢻⠀⠀⢸⠟⠀⣼⠇⣠⠂⠀⣠⠟⢁⠞⠁⠀⠀⣠⣶⣄⣸⡟⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⣠⡴⠟⠁
⣼⠀⢸⠁⢸⣅⢠⡏⠀⢸⡇⠀⠈⠀⢠⣿⡴⠃⢀⣾⠋⠀⠀⠀⠀⢀⡾⠋⠀⠉⠉⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠛⠛⠉⠉⠁
⣿⠀⠸⠀⠀⠙⣿⠁⠀⣼⠂⡠⠀⢀⣾⣉⣠⡶⠟⠁⠀⠀⠀⠀⢠⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣤⣤⣤⣴⡾⠟⠋
⣿⡄⠀⠀⢀⡼⠟⠀⠀⣿⡾⠁⢀⣼⣿⠟⠋⠀⠀⠀⠀⠀⠀⣰⣿⡿⢿⡿⠿⠶⣶⣦⣤⣤⣤⣤⣤⣤⣶⣿⠟
⢸⣧⠀⠀⠸⣧⡄⠀⢸⣿⡷⠾⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⢉⣴⣿⣀⣶⡶⠿⠃⠀⠉⠉⠉⠉⠁
⠀⢿⣆⠀⢰⡟⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⣼⣇⣠⣴⠿⠋⠉⠉⠙⠁
⠀⠈⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣠⡴⠒⠒⠚⠋⠀⢀⣽⣿⠟⠁
⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠋
⠀⠀⠙⠿⣷⣤⣤⣤⣴⣾⡷⣶⡶⠿⠿⠟⠛⠉
⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣶⣶⡆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⠒⢲⢀⡆
⠀⠀⠀⠀⠀⣀⣤⣾⣿⣶⣿⣿⣷⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢯⢳⣺⠀⢸⢊⠧⣀⣀⡀
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡜⠻⠶⣖⣚⡀⢸⢀⡇⠮⢔⠽⠛⢦⡀
⠀⠀⣰⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⠎⡀⠀⠀⠀⣀⢎⠤⡉⣄⠐⠥⣀⣀⡀⢳⡀
⠀⢀⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡸⢰⠁⠀⠐⠛⠛⠁⠀⣷⠀⠉⠑⠒⠋⠁⠀⣇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢳⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⣆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⠀⠀⠀⠠⠤⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⣾⡿⡫⣄
⠀⠀⠰⣌⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⡉⠀⠀⠘⠿⠞⠁
⠀⠀⠀⠈⠉⠉⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⢩⡿⠁⠒⠲⠀⠈
⠀⠀⠀⠀⠀⠀⢻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣦
⠀⠀⠀⠀⠀⠀⠸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡤⠀⠀⠀⠀⠀⡐⠺⠦⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠉⠀⠀⠀⠀⠀⢠⣴⡾⢏⣀⣀⡀⠠⠤⠀⠐⠠⢀⠀⠀⢀⢁⣨⡋⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠿⣷⣶⣾⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣷⣻⣿⣿⣿⣩⣿⣛⠀⠂⠀⠀⠄⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⣾⣿⡿⠉⠉⠙⠛⠛⠛⠿⠿⠿⠛⠛⠛⠛⠿⠿⠿⠿⢇⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡂⠈⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠁⠀⠀⢾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣸⣁⠀⠀⠸⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⣠⡀
⠀⠀⠀⠰⠒⠒⠲⠆⠐⠒⠎⠛⠉⠛⠚⠃⠈⠉⠉⠉⠙⠉⠙⠓⠂⠀⠀⠀⠉⠛⠉⠁⠉⠉⠉⠉⠉⠀⠈⠛⠒⠒⠒⠢⠤⠤


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⢤⣤⡤⠐⠒⣒⣶⣤⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⡠⣺⠉⣀⠔⡪⢁⡠⢤⣤⣤⣄⣀⣀⡒⠋⢉⣉⠤⠬⢭⣶⠤⣀
⠀⠀⠀⠀⡀⣼⢃⠞⡱⠊⡔⠁⠼⡳⠄⢒⣒⣒⣚⣛⡳⠦⣄⣐⠀⠀⣀⣀⣀⠱⡀
⠀⠀⠀⠀⢇⠀⠁⠊⠀⢸⠀⠀⠤⠒⠉⠁⠀⠀⢀⡠⠤⠤⠤⠬⠍⣒⣤⣀⠀⠀⡇
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠸⠀⠀⠀⠀⠤⠶⠤⠤⠒⠒⠉⢉⣉⣉⠀⠠⢀⣈⡑⠦⡁
⠀⠀⠀⠀⢀⠤⢀⠔⠀⠐⠢⣀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠠⠤⠒⠒⠉⠉⠉⠛⠋⠺⠶⣄
⠀⠀⠀⠀⠇⢠⡁⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠈⠁⠀⠀⠀⢀⡠⠤⠀⠀⢈⣉⣁⣒⠢⢬⡳⡄
⠀⠀⠀⠀⢗⡸⣈⠑⠢⣄⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠘⠋⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠙⠚⢆
⠀⠀⠀⠀⠀⠑⢨⡛⠓⠺⣗⠤⠤⠤⠤⣔⡇⠠⢄⡀⠀⠀⠀⠀⠀⠀⠶⠶⢀⣀⣈⣉⡁⠒⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠈⣟⠿⠿⢭⣴⠞⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⠔⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠
⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀
⠀⠀⠀⢀⡴⠻⢛⡳⢤⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣟⠃⠐
⠀⠀⠜⢡⣤⠶⠦⠁⠨⣿⡇⠀⠹⡆⠀⠀⣀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⡙⡄⢀
⠀⠀⠀⠀⠀⢠⣤⣠⣄⣀⠉⠉⠙⢿⡟⠛⠛⠦⣍⠙⠓⠒⠀⠀⠀⢐⣶⠂⠀⠉⠉⠉⢛⣓⠺⠛⠉⠝⠓⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣰⡀⠰⠻⡍⠻⣍⠑⠦⠷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⣀⣀⣐⣚⡋⠂⠀⠀⠀⠀⠠⠄⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠻⣷⡶⠂⠈⠲⣤⣌⣑⡖⣶⣄⠀⠀⠀⠀⠀⠀⢀⠴⠺⠿⢿⣯⣭⠬⣬⣟⠛⣛⡶
⠀⠀⠀⠀⠀⠀⠙⢅⡾⣷⠆⠠⠞⠋⠭⠉⠛⠳⣄⠀⠀⢀⣴⠧⠀⠀⢴⡀⠀⢀⡀⠀⡀⠀⠀⣀⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠁⠀⠈⠁⠀⠈⠂⠰⠧⠀⠀⠀⠀⠀⠈⠧⠀⠀⠠⠷⠗⠂⠋



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢱⠃⠀⠀⠀⢀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡙⠘⠦⠶⢮⣽⠟⠋⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣄⡀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠙⠳⡦⠤⣴⣾⠉⣠⢰⡦⣼⡧⠤⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠒⣶⣒⣒⡚⠛⠉⠉⠛⠋⠛⢶⣄⡴⣻⡥⠼⠟⡠⠀⠈⠀⠀⠀⣞⢠⢿⠀⡇⠀⠈⠳⢤⣶⠆⠀⠘⣦
⠀⠀⢀⠔⠁⠀⠀⠈⠉⠛⠓⠶⣶⣶⡷⢢⠋⠸⣤⡤⢤⣴⡃⠀⠀⠀⠀⠀⢯⠊⠈⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⢀⡤⠋⢳⠀⠈⠹⠷⣌⡻⡄⠀⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠰⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡤⠚⠉⠀⠀⠘⡆⢰⠀⠀⠘⢧⣻⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣇⠎⠀⠀⠀⠈⠻⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⠀⠀⠘⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠈⠑⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠈⠀⠀⠁
⠀⠀⠀⠀⠈⠑⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣤⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡤⡤⠤⣀
⠀⠀⠀⠀⠀⡀⢤⠒⠋⢹⡖⠂⠈⢳⡌⢂⠙⣿⠣⡀⠀⠀⠐⡟⠡⠔⣛⣫⣶⣦⠀⠀⢀⠎⠀⣀⠤⠜⠀⠧⣀⠀⠡⡀
⠀⠀⡠⠒⣭⢄⠸⡍⠳⡈⢳⡀⠀⠀⢻⡌⡄⠘⣄⡇⠀⢸⠀⠃⢠⠊⠉⠉⢉⣈⣁⠀⣸⡀⠚⠀⠀⡄⢠⠀⠀⠱⠀⡧
⠀⠰⠁⠀⢿⠀⠂⠹⣆⢱⡈⢷⡀⠀⠘⡇⣧⣶⠟⠀⠀⠘⢧⡀⠐⠔⠒⠉⠉⢉⡟⠀⢿⣧⡀⠀⠀⠸⠂⠀⠀⣀⣴⠇
⠀⢠⡀⠀⠈⠓⠄⣦⣘⢦⣧⣀⣷⣶⡶⠟⠋⠁⠀⠀⠀⠀⠀⠙⠲⠤⣤⣤⠴⠚⠁⠀⠈⠻⣿⣷⣶⣶⣶⣾⣿⡿⠋
⠀⠀⠉⠛⠓⠒⠛⠛⠛⠛⠛⠋⠉⠀⡠⠒⠀⠁⠀⠠⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢉⡩⠭⠟⠫⢤⣀⣀
⠀⠀⣠⠒⠫⣅⠉⠉⠱⣶⢤⠀⠀⡌⠀⠀⠀⠀⠀⠀⠳⠀⢀⠄⠀⠀⠙⡿⠉⠱⡀⠀⠀⢀⣴⡙⠀⠀⠀⠀⢨⠀⢨⣉⠵
⢀⣞⠣⠄⢈⢮⡀⠀⣠⡇⢠⠳⡄⡇⠁⠀⠀⠀⠀⠀⣸⠀⣸⠶⢦⣄⡰⠁⠀⠀⣷⠀⣰⡋⠀⠘⢎⢂⠀⠀⣸⠀⠊⠁
⠠⠁⠀⠉⣷⣿⣿⡾⠛⠶⠾⢤⠁⠘⢄⣀⣀⣀⣠⠴⠃⠀⡇⠀⠀⠘⡇⠀⠤⠞⡿⠠⠀⠬⢷⣤⠼⠿⠶⠛⠁
⠘⠦⠒⠛⣉⣁⠀⠄⡠⠠⣀⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠰⣤⡀⠀⠰⣄⠀⠚⠀⠀⠀⠀⠤⠶⠾⠥⣄⡀⠈⠙⠲⣄
⠀⠀⣴⠍⠁⠀⠀⠀⠀⠀⠛⣯⠀⠀⠀⠀⠀⠀⠀⠤⡤⢤⣤⣤⣭⣉⠉⠁⠀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⠙⢲⡀⠀⠈⡆
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣤⠈⠢⡀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢲⣧⡤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡴⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⡱⠀⠀⠠⠁⠀⢀⠀⠀⠀⠀⠀⢹⠁⠀⢸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⠦⠄⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⡾⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣸
⠀⠀⠀⠙⠢⠤⠴⠄⠂⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠒⠒⠒⠋⠀⠀⠀⠀⠘⠓⠢⠤⠤⣄⣀⡀⠀⡼⠖⠉
⡶⠒⢶⢶⡶⠶⠶⣶⢶⠶⠶⡶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⢶⣶⠶⠶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⣷⡀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⢀⣴⠟⠋⣉⡉⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣻⣧⣀⣠⡾⠋⠀⠈⢻⡄⠀⠀⢸⡇⠠⡏⠉⡟⠀⠀⠘⠛⠶⠶⠞⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠧⠀⠊⠀⠈⠉⠉⠃⠀⠀⠀⢸⡿⣦⣤⠾⢃⣀⡿⠸⣄⠀⢀⣀⡀⠀⣀⣄⡀⠀⠸⣇⣠⣀⣀⣠⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢸⡅⣰⠋⠉⠀⢠⡇⠀⢸⠏⣇⣠⡏⢠⡷⢁⡞⠋⠀⠉⠉⠁⠀⢹⡇⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣦⠀⠀⠀⠀⠀⠤⢦⠀⣰⠃⠀⠀⢸⣀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠈⠁⠀⠸⢦⡿⢀⠀⢰⠚⢶⠂⠀⡈⠷⠶⠛⠁⠀⠈⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⡾⠁⠀⠘⠢⣝⡄⠁⠀⠘⠶⢤⡄⠀⠉⠙⢦⠀⠐⢄⡀⢀⢀⡄⢀⡴⠶⠶⠶⢃⡞⠛⠋⠀⠙⠲⡾⠃⢠⣀⠀⢰⡟⠛⢲⣏
⠀⠀⠀⠀⣀⣤⠾⠃⠀⠘⢦⡀⠀⠀⠀⠈⢻⡀⠀⠠⠀⣠⠏⠀⠀⠀⠈⠛⠲⢶⣍⠲⣮⠀⡞⠀⣠⠶⠶⠛⠀⠀⠀⠀⢠⠎⠀⣰⠋⠹⠤⠼⠃⠀⠀⠙⣷
⠀⠀⠀⣼⠋⠀⠀⠀⣤⣦⣴⠃⠀⠀⠀⠀⠸⡇⣠⢃⠀⠙⠶⣎⠀⣀⣤⣄⠀⠀⠙⡇⢁⣠⠇⠀⣏⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⠋⠀⠀⠀⢀⣠⠾⠶⠶⠋
⠀⠀⠀⠻⣦⠀⠀⠀⠈⢧⢨⠀⠀⠀⢢⡳⣄⣧⢡⡎⠀⠀⢰⠇⠀⣿⠀⠈⣷⠀⢸⠁⡿⢁⠀⢀⣽⠀⠀⡀⠀⠀⢠⠀⠀⢀⠄⠀⠀⠀⠀⠈⣿⣀⣀⡀
⠀⠀⠀⢀⡼⠇⠀⠠⠴⡞⠈⣏⠀⠀⠀⠑⢌⠻⡆⢃⠀⠀⠘⠧⠴⠋⠀⠘⣧⠀⠈⠛⠧⡜⠉⠉⠁⠀⠀⢣⠀⠀⣸⠃⡴⠃⠀⠀⠀⠀⠀⠀⠈⢉⣩⡗
⠀⠀⠀⣿⠁⠀⣰⣄⠀⡟⠀⠙⢦⣤⡀⠀⠀⠑⢿⡌⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢶⣀⡟⠀⠀⠀⠀⠀⢸⡇⡼⢃⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⡍
⠀⠀⠀⠻⣄⣰⡟⠉⠻⣇⣀⠀⠀⢀⡧⠀⠀⠀⠀⢳⡈⠀⠀⠀⠀⢀⠃⠀⠀⠀⡴⠋⡭⠀⠀⠀⠀⠀⠀⢸⡿⠁⡾⠀⠀⠀⠀⡀⠀⠀⠠⣀⡀⠀⣿
⠀⠀⠀⠀⠈⠉⠀⠀⠀⢩⡟⠀⠀⣿⡀⠀⠠⡀⠀⠀⢧⠀⠀⠀⣠⢎⠀⠀⠀⠠⡇⠀⠳⡄⠀⠠⠀⠀⠀⢘⠇⣸⠃⠀⢀⣴⠋⠀⠀⠀⢸⡟⠳⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡄⠀⠘⠦⢦⡄⠈⢓⠦⣌⡄⢢⡜⣡⠃⠀⠀⠀⢠⡟⠀⣤⠃⠀⠀⡇⠀⠀⢸⠀⡏⠀⢠⡿⢹⣆⣀⣸⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⡷⠀⠀⠑⢌⡃⠈⠰⠁⠀⠀⠀⠀⣿⠀⡞⠀⠀⠀⢰⡃⠀⠀⡌⢀⠇⠀⢸⣿⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠸⡇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢆⠀⠀⠀⢿⡀⢠⠁⣸⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⢻⠀⠀⠀⠀⢠⠀⠀⠀⠀⣠⢀⡞⠁⠀⢀⡎⠀⠀⠀⠘⢰⡆⢠⡯⠂⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⠀⠈⢦⣄⡀⠠⣌⠀⠀⢀⣰⠃⠘⣧⠀⢸⡁⠀⠀⠀⠀⢠⡎⢠⠏⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠈⢳⠀⠈⡃⠀⠈⣇⠀⠀⢘⡇⢀⡷⠀⢰⠀⢀⠎⡰⠃⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣄⠀⢧⡀⠘⢆⠀⢻⠀⠀⠈⢧⡞⠁⠀⢸⡴⠃⡰⠁⣴⠷⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠙⢦⠈⢂⠈⡆⠀⠀⣠⠃⢀⡼⠋⣠⠞⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡘⣇⠀⠣⠀⠀⠸⡇⠀⠞⠀⣸⣥⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠖⠛⠓⠀⠀⠓⠶⢤⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣶⣶⠶⠶⣤⣤⣀⣀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⣶⣶⣶⣶⣶⡾⢷⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣾⢷⣶⡿⠿⠿⠶⣶⠶⠾⠷⡶⢿⣶

⠀⠀⠀⢠⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⣦⣄
⠀⠀⠀⣿⠀⡉⠙⠛⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢠⠞⢡⠊⠳⣄
⠀⠀⠀⣿⠢⢄⠀⠉⠉⠀⠒⠒⠚⠷⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⢶⣦⠾⢋⠔⠁⡰⠁⢀⠔⠉⠳⢤⡀
⠀⢀⡴⢏⡀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠁⠒⠫⢍⠋⠭⠹⢛⡛⢲⣦⠖⠋⠁⠀⢀⡴⠋⠄⠈⡠⠔⠊⠀⡠⠃⠀⣠⠴⠖⠛
⠀⠛⠷⢶⡈⠉⠒⣂⣨⠥⠤⢤⣀⡈⠑⠂⠤⣀⡈⢐⣒⣤⠶⠋⡁⠄⠤⠤⢠⣾⠕⠁⢀⡎⠀⢠⠔⠊⠀⡴⠋⠁
⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠉⠓⠲⣶⣤⠾⠋⠁⠀⠐⠁⠀⠀⣀⣠⡿⠃⠀⢀⡜⠀⢠⠃⣀⣴⠾⢅
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉⢉⣉⣛⣛⣉⠭⠤⠀⠂⣀⡀⠤⠔⢒⣭⡿⠋⣠⠔⢒⡉⠤⠔⠁⡼⢥⣀⣠⠏
⠀⠀⠀⠀⢀⡴⠋⢀⡠⠒⠉⢁⣀⠀⠠⠤⠐⠒⠈⢁⣀⠠⠤⣲⡯⠊⠀⢠⠁⢰⠁⣀⣤⣴⠞⢧⣀⣰⠃
⠀⣀⣠⠴⠋⢀⠴⢋⠄⠊⠉⠀⠀⠀⡠⠒⢁⡤⠤⠤⠤⣤⠖⡫⠤⠒⠂⠁⠀⢤⠞⠁⠀⠈⠻⢦⣼⠏
⢻⡉⢀⡠⠊⢁⡔⢁⠄⠀⠀⠀⠀⣉⣤⣴⠏⢀⠤⠒⢉⡤⢊⠀⠐⠒⢀⣀⡠⠏
⠀⠻⣤⡤⠔⠉⡐⠁⣰⠞⠛⠛⠋⠁⣰⢋⠞⡠⠔⠈⠁⠀⠀⠀⡴⠋⠉
⠀⠀⠘⢷⣤⠞⢀⡾⠁⠀⠀⠀⣠⠞⢁⠎⡜⠀⠠⠂⢀⣀⣠⡼⠁
⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠚⠥⣤⣋⡐⠀⠀⡄⠀⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣴⣀⡴⠋

⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠈⢿⣿⢷⣍⠑⣄⡀⠄⠒⠓⠀⠂⠀⡠⠂⢁⣴⡿⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⠠
⠀⠀⠀⠀⠐⢿⣿⣧⠌⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⣹⡏
⠀⠀⠀⠀⡀⠀⠹⡟⢀⡀⠀⠀⠀⠀⠀⠀⣀⢸⡿⠏⠀⡀⠀⠀⠀⠀⢀⠁⢀⣀⠀⠻⠠⠊⠀⠀⠂⠀⠀⠐⠢⣄⢀
⠀⠀⠀⠀⠀⠀⠒⠃⠘⠛⠀⠀⠀⠀⠀⠘⢻⠘⠃⠐⠀⠀⠀⠀⠀⠀⢞⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣠⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠺⢖⠀⠀⠀⠀⠀⠃⠀⠀⢀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢄⣤⣄⡤⠂⠀⢠⠃⠀⠰⣶⣤⠶⣿⡇⠀⠀⠀⠉⢫⢁⠀⠀⡀⠀⢸⠇⠀⢀⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⡲⢤⣀⠀⠉⢽⣂⡤⠖⠁⢀⣀⠀⡹⠛⢁⡌⠉⢻⣿⣶⣤⠀⢠⠹⠋⠃⠀⡼⠓⠒⡟⡇⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠠⣉⠁⠀⠉⢉⠀⠀⠀⣸⡟⠀⠉⠣⢾⣧⣾⣀⣿⣿⡟⠀⠈⠀⡀⠁⢸⠀⠀⣔⣱⠇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣶⣤⣶⡏⠀⠀⡰⠋⠀⠀⠀⠀⠈⣿⡟⠉⣻⣻⠀⠀⠘⠚⠀⠷⠊⠀⠀⠉⠙⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠉⠁⠀⠐⠁⠀⠀⠀⠀⠀⠈⠙⠁⠈⠁⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⠀⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡠⠊⡠⢊⠐⢀⠀⠈⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⢊⠔⠈⠔⢁⠔⡡⠂⡀⠂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠴⠐⠁⠀⡡⠀⠀⠊⡠⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠉⠢⡀⠑⠀⡀⠀⠀⡠⠊⠀⡠⠈⠉⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈⠢⡈⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢄⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠐⢄⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠄⠁⠀⠀⠁⢄⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠁⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠤⠀⠀⠒⡂⠐⠒⠦⠤⣋⣉⠉⠩⠉⠙⠒⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣚⣋⡀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠉⠓⢦⣀⣀⠀⢄⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠌⠠⠒⠔⠲⠈⠍⠩⠉⠙⠛⠛⠛⠛⠒⠒⠤⣌⡀⠠⡀⠉⠁⠀⠈⠙⠲⣄⢀⠀⡴⠐⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡈⣀⢈⣥⣤⣀⡀⣀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠼⠄⠀⠀⠀⠀⢠⡈⠛⠀⡠⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣞⠽⠚⠁⠉⠀⠀⠐⠉⠍⠉⠉⠙⣛⠓⠚⠣⠦⣄⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠠⠀⠀⢀⠀⠀⠀⠨⢆
⠀⠀⠀⠀⠀⠀⡰⡷⠁⠀⠀⠂⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠈⠑⠫⣢⡀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠈⠀⡀⠀⠅⢪⡀
⠀⠀⠀⠀⠀⠀⠸⣧⡈⠀⠠⠀⠤⠉⠀⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠅⠀⠀⣀⠊⠀⢈⡹⣷⣌⠂⠄⠀⢁⠀⠀⠆⠄⠆⠀⢁⡨⠀⠀⠀⠀⣿⣓⣒⡀⡀
⠀⠀⠀⠀⢀⣀⣴⠿⠿⣦⣄⣈⡀⠀⠀⠀⠄⠠⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠻⣦⠀⡀⢂⡴⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⣴⡋⠐⠂⠠⠀⢀⡀
⠀⢀⣴⡾⠛⠉⠉⠉⠨⠉⣉⠉⠈⠛⠀⠛⠷⢶⢤⢬⣀⡀⠁⠀⠀⠠⠀⠐⠂⠀⠀⡀⢂⠂⠠⢳⡧⠐⠃⠀⠀⠐⠀⠂⠀⠀⢀⢀⡨⠞⢁⣀⠀⠐⠀⠀⠉
⢠⣨⠁⠀⡀⠀⠀⠈⠁⠀⠠⠀⠀⠀⠀⠅⢀⠀⠀⠨⠍⠐⠛⠿⢶⣤⡀⠀⠀⢀⠀⠀⠀⡩⠂⣾⠷⠔⠐⠶⠳⠶⠾⠶⣶⣷⣶⣾⣿⣶⣶⠶⣶⣶⣶⣷⣶⠖⠂
⢺⠉⠙⠳⢦⣤⣐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠡⠀⢛⢶⡀⠈⠡⠀⠀⢀⣴⡽⠲⠲⠒⢒⠳⠶⠖⠛⠛⠻⠿⠏⢻⣿⣿⣿⣿⣿⠟⠋⠉
⠈⠉⠁⠶⢄⡀⠉⠛⠶⢤⣀⠀⠁⠀⠐⡀⠀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠄⢳⣿⡻⣷⠖⣞⣿⢫⠾⠟⠛⠛⠿⠏⣛⣛⠛⠛⠛⠒⠒⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠉⠒⠲⣄⠈⠉⠓⠶⣠⡀⠀⠀⠀⠀⠀⠀⠆⠈⠤⠀⣀⣎⣼⠤⠩⠀⠀⠉⠑⠒⠒⠢⠥⠒⠒⠠⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⣤⣀⣙⠣⠗⠶⠒⢒⣲⣖⣼⡿⡛⣡⡮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠴⠶⠤⢴⣂⣲⣧⣴⠟⠛⡁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣄⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡠⠶⠂⠀⠀⠀⠀⠀⠀⢠⣧⠆⠀⠀⠀⠀⠀⣠⠤⠀⡇
⠀⠀⠀⠀⠀⠀⡏⠀⠈⠣⠄⠠⡆⠀⠀⠐⠀⠀⡠⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⢘
⠀⠀⠀⠀⠀⢠⠀⠀⠐⠦⠀⠀⣷⠀⢀⠀⠀⠘⠁⠀⠐⠋⠀⠀⠙⡆⠀⠀⠀⠀⢸⣠⠂⠀⠀⢀⠀⢀⠔⠊⠁⠸
⠀⠀⠀⠀⠀⠀⡆⠀⠀⢈⠀⠀⠛⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘⠐⠁⠀⢀⡀⢰
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⡀⢀⡤⠀⠀⢧⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢸⢀⡄⡀⠀⠰⣦⡤⠊⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠉⠻⠷⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣀⡆⠀⠀⢿⣇⣀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠻⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⡄⢤⡠⠄⠀⣀⡀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠐⢄⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠠⠀⠀⠰⠆⠀⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⡀⣠⠀⠀⠀⠀⢷⣄⡀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⣿⣿⢛⠿⠛⠛⢿⠟⣻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠶⣶⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠶⢦⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣹⡟
⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣽⣟⣛⣲⡄
⠀⠀⠀⠀⠀⠀⢸⡿⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⢀⣼⠟⠈⢙⣻⡄
⠀⠀⠀⠀⣠⠶⠛⢷⣄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⢤⣤⣀⣀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠀⢀⣴⣿⡟⠛⣻⠟⠉
⠀⠀⠀⠀⠛⢶⡶⠂⠉⠛⠒⠲⠶⠦⠤⣤⣤⣄⣀⣀⣀⠀⠀⣸⣿⣤⣤⡴⠞⠋⠁⠀⣀⣤⠞⢋⣠⡤⣿⢿⣇
⠀⠀⠀⠀⠀⠘⢿⣅⣠⣤⣶⣆⣤⡀⢐⣀⣀⣀⡈⠉⠉⠉⠛⢻⡏⠉⠁⠀⣀⣠⡴⠞⠋⠑⠀⠈⢛⣺⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠉⠉⠛⠒⠿⢿⡏⠉⠉⠻⣥⣶⠿⣟⣂⡹⠿⠟⠛⠻⢥⣄⣰⣏⠭⠛⠛⢉⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣤⣈⡁⠀⠸⣇⣀⣠⠶⠋⠉⠛⠻⠷⠶⠶⠖⠚⠛⠛⠉⠉⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⠾⢿⣤⣤⣤⣄⣀⣀⣀⣦⡀⠀⠀⣂⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⢉⡏⡈⠟⠻⠶⠶⣚⡍⢙⠷⢌⠘⢆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⣶⠄⠀⠋⠀⣠⡶⠠⠯⠀⠐⠀⢁⣁⠀⢀⣶⣶⢦⡉⠉⢦
⠀⠀⠀⠀⠀⠠⣶⣤⣴⡟⠀⠀⠀⠠⡀⢠⢤⠚⠀⢄⠀⠀⠀⠨⠶⠫⡀⠅⠈⢧⡀⢸⠃⡄
⠀⠀⠀⡠⠒⠁⠨⣿⣿⠢⡀⠀⢇⠀⠀⠘⣎⠃⠀⠉⠲⠉⠀⠀⡤⠛⣁⡴⠥⢼⣷⡟⠀⠸⢆⠐⠀⡤
⠀⠀⠀⠀⠀⠠⣠⣿⢿⠀⠰⡀⢹⣄⠀⠀⠈⢢⠀⠀⠤⠠⢄⣾⣿⡀⡀⢞⣨⠿⢋⣀⣴⣷⣿⣾⣀⡟⣶
⠀⠀⡄⡄⠀⢀⣽⠃⠈⡆⠀⢱⡄⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠙⡆⠉⢉⣿⣿⣿⣿⣿⣧⢻⡟⠻⣶⡂⡞
⠀⠀⡇⠀⠀⢺⣿⠁⠀⠹⢶⡄⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⡘⣆⣼⢿⣿⣿⣳⡿⠋⠀⣽⣶⣄⠉⠂
⠀⠀⠃⠀⠀⠰⢿⡄⠀⠀⢼⡁⠀⠀⢀⠀⠈⠴⠃⣠⠔⢋⣄⣸⣅⢿⣿⣿⠟⠋⠁⠀⢀⣾⣿⡞⠟⠁⠀⠃
⠀⠀⠀⢆⠀⣠⠎⠉⣶⠀⠲⠷⡶⣤⣌⣅⣘⣗⣸⣟⣱⣯⣿⣷⠶⠛⠋⠙⠲⠾⣿⣶⣿⣏⠏⠀⠈⠀⣈⣼
⠀⠀⠀⠀⠘⠁⠀⠀⠐⢳⣦⣤⣀⣀⡛⠘⢘⡿⣷⠞⠟⠉⠉⢱⣄⠀⠀⠀⠀⠀⢌⣼⣿⡥⠽⠢⠤⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠛⠛⣿⣿⡖⠾⠛⠋⠀⠀⠀⠃⡀⠙⠒⠌⣁⣤⣶⡿⠋⠁⠀⠀⢠⠤⢂⡤⠂
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠉⢹⢿⣧⣀⣀⣠⢀⣈⣀⣀⣀⡀⢤⣾⠛⠙⠋⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⡤⠯⠿⠛⠁⠀⠀⠉⠁⠀⠀⠀⠈⠁⠀⠐⠒⠂⠀⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢰⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠓⣆⠀⠻⣿⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠆⠀⢸⣿⡇⠀⠀⠸⡆⠀⢻⡆⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣷⠀⢸⣿⠁⠀⠀⠀⠀⢿⣧⠀⠀⢀⡇⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣧⠸⣿⡇⠀⠀⠀⠀⠈⠻⣦⣠⡾⠁⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⠋⣸⣄
⠀⠀⠀⠀⠀⣤⣴⠞⢷⡆⠀⠀⠀⠀⠀⠀⠀⢿⣧⡴⢿⡆⠹⣿⣄⠀⢀⣤⠀⠀⠈⠁⠀⠀⠀⠀⣀⣴⣾⣿⡿⠿⠛⢁⣤⣾⠛⠉⡆
⠀⠀⠀⠀⠀⢻⣇⠀⣸⣇⣠⣤⣄⣀⡀⠀⠀⢸⣯⠀⢸⣿⡀⠈⠛⠻⠟⠁⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠟⣁⠤⠤⠴⢞⣹⠁⠀⣀⡇
⠀⠀⠀⠀⠀⠈⣿⡾⠛⠉⠉⠉⠉⠻⣿⣷⡀⢸⣿⠀⠀⢿⡧⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣛⡽⠋⣠⠾⣗⡼⠤⠵⠋⢀⣠⣾⡿⢷
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⢹⣿⠇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⠫⢿⠟⠛⠛⠋⢀⡼⣧⠤⠞⠀⢀⣠⣶⣿⠿⠃⣠⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⣠⠿⠋⠀⠈⠀⠀⣀⣤⣶⡶⡾⣟⢣⣼⣮⠟⠁⠀⣠⡄⣴⣿⡽⠃⣀⣴⣿⣿⠟⠋⠀⡠⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡴⠖⠋⠁⠀⠀⣀⣀⣴⣾⣿⠿⠛⠛⠛⠋⠉⠉⢀⣀⣀⣼⣿⠷⠋⣡⣶⣾⣷⡷⠛⠁⢀⠔⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣀⣤⣴⡶⣿⣿⣿⣿⣿⠿⠁⠀⠀⢀⠔⠢⠤⢲⣻⠿⠛⢁⣠⣴⣿⣿⠟⠋⠁⢀⣤⡶⠕⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⣻⠻⣷⣼⠿⠃⠛⠋⠀⣀⡠⠔⢾⣩⣄⡤⠚⢉⣤⠤⢶⣿⡶⠟⠚⠋⣀⡴⠞⠉⠁
⠀⠀⠀⠀⠀⢀⣾⣿⣷⣿⣭⠜⠋⠀⠀⠀⣤⢶⢁⠁⣠⠴⠋⢀⣠⣴⣾⣿⡿⠿⠋⠁⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⣤⡖⣤⡉⠁⣤⡶⠍⢡⣥⠴⠛⠟⢛⢉⣠⣶⣿⣿⣿⠟⠋⠀⣀⣤⡶⠟⠊⠁
⠀⠀⠀⠀⠀⣿⣿⣜⣩⠿⡿⠖⠋⠉⢀⣠⢴⣾⣏⢾⡿⠓⠾⠋⣀⣤⡶⠞⠋
⠀⠀⠀⠀⠀⡄⡇⠀⢀⣀⣠⣤⣶⡞⣛⣳⠾⠷⠛⠋⢀⣠⡴⠞⠊⠁
⠀⠀⠀⠀⣼⣿⠋⢩⣿⣿⣿⠿⠋⠉⠉⠀⢀⣠⣴⠞⠋⠁
⠀⠀⠀⠀⢯⡏⠉⠉⠀⠀⠀⠀⠀⡀⠄⣂⠕⠊
⠀⠀⠀⠀⠘⡴⡤⠄⠠⢄⣐⠠⠁⠒⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⣶⢸⢰⡆⡆⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⣠⣴⣶⣶⣄⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⡏⢸⢸⡇⡇⣠⡤⠤⣸⣿⣿⠠⠤⠄⣴⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⡇⡇⠀⠀⠀⣿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⡇⢸⢸⡇⣿⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣇⣸⢸⡇⣿⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⢦⡀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢻⡄
⠀⢀⡟⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⢻⡄
⠀⣾⠁⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠁⠀⠀⣾⣿⣿⣿⠀⠀⠀⠈⠙⢻⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠈⣷
⢰⡇⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢸
⣼⠁⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢈⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢠
⠸⣇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢸
⠀⢻⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⡟
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣇⣀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⣀⣸⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⠉⠙⠓⣿⣿⣿⠐⠛⠋⠉⠁⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⠲⢶⣶⣤⣤⣤⣤⣤⣤⣶⣶⠒⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠉⠉⠉⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⠷⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠉⠛⠛⠳⢾⣽⣷⡄⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠶⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⣀⣭⣽⣶⣿⣭⣭⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣀⣀⣀⣤⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⣆⣈⣰⠅⢤⠐
⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⠀⠛⡾⣿⣇⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⣠⢀⣤⣄
⠀⠀⠀⠀⠀⠀⣰⣟⠃⠀⠀⠀⢈⣽⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠲⣴⠊
⠀⠀⠀⠀⠀⢰⣿⣿⣯⢀⣀⣀⠨⣿⣿⣿⡀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⠄⠀⠀⠀⠊⣰⡿⣿⣿⠟⠤⠶⠶⠄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠃⠀⠀⠀⣀⡀⠈⠠⣿⡿⠀⠂⠂
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦⡠⠀⠀⠀⢠⣰⣿⡯⠀⠀⠀⠀⠀⠈⣿⡄⣼⡏⠀⢀
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⢱⣿⣿⣿⡇⠉⠛⠻⠷⣦⡂⠠⢙⣿⣷⣶⢂⣤⣾⣿⣿⣾⣿⣿⣿⡀⢸⣷⣧⣖⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⢊⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⡧⠌⣿⣛⣿⣿⣿⠉⠀⠀⠉⠉⠉⠋⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣼⢸⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⡏⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡇⢨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠈⢿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠛⠛⠛⠛⠲⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢤⣄⣀
⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣶⢄
⠀⠀⣞⠻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣸⡆
⠀⠀⢯⠑⠮⣝⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⣊⠵⠚⢉⠇
⡠⠒⠻⡀⠀⠀⠉⠓⠮⣕⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⠭⠒⠉⠀⠀⠀⣼⡀
⢧⣀⣀⡉⠒⠤⣀⠀⠀⠀⠉⠓⠮⣕⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⡭⠖⠋⠁⠀⠀⣀⡤⠖⢋⣠⢇
⠀⠈⠒⢾⡖⡄⠀⠉⠒⠤⣀⠀⠀⠀⠉⠓⠬⣕⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⡫⠕⠊⠁⠀⠀⢀⡠⠔⠊⠁⢠⠊⠉⠀⢀⠧⣄
⢀⠴⠊⠁⡗⢧⣀⣀⣀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠉⠒⠭⣓⠦⢄⣀⡤⠔⣪⠵⠚⠁⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⡸⠼⠉⠉⠉⣀⡬
⠳⢄⡀⠀⠘⠦⠤⡀⠘⣄⣀⣠⡀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠉⠒⠒⠒⠉⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⣰⠋⢹⢁⣠⠴⠒⠉⣇
⠀⡰⠉⠙⠒⠤⣀⣸⡀⠀⢁⣤⣙⠒⠒⠲⡄⠀⠈⠙⠲⣤⣀⠀⠀⠀⣀⡤⢲⠟⠲⢄⡀⣀⢞⣉⡓⢤⣤⡯⠝⠋⠁⠀⣀⡤⠊⡽
⠀⢳⣀⠀⠀⠀⠀⠉⠑⠺⣅⡀⠈⠦⠤⠔⠛⢤⣀⣠⣾⠥⠼⡿⠯⣍⠉⠉⠁⠀⢀⠞⠛⠒⢊⣠⣼⣋⡥⠤⠤⠔⠒⠋⢁⡠⡾⣇
⠀⠘⣌⠙⠒⠦⠤⠤⠤⢼⡅⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠑⠤⣤⠴⠊⢀⡤⣒⡽⠋⣿⣤⣤⠤⠤⢔⣲⣿⣿⣿⣿⠟
⠀⠸⣦⡙⠶⣶⣤⡤⠤⠼⣍⠲⢤⣀⡀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠱⡄⠀⠀⣀⠴⠞⠛⠉⣁⣤⣾⡿⠟⠋⣀⣴⠟⢻⣿⡩⢿
⠀⠀⡇⠈⠑⠪⢝⡻⢷⣦⣌⡳⠤⣀⡉⠉⠉⠉⠉⠒⢤⣀⠀⠀⣴⣿⠶⠭⠤⢤⣶⡶⠋⣹⡿⠁⢀⣠⣾⣟⣁⣤⡿⠋⠀⡿
⠀⠀⠳⣄⡀⠀⠀⠈⠑⠪⢝⡻⢵⣒⠬⢍⣉⠉⠉⠛⠛⠛⠛⠚⢉⣀⣀⣠⣾⣿⣷⣶⣾⠿⠷⢾⣿⡿⠛⠋⠉⠉⠀⣀⠴⠁
⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠈⠑⠮⣝⡲⢬⣙⠢⣄⡀⣰⡾⠛⠛⠛⣹⡿⠛⠣⣿⣇⣀⣀⣴⡿⠋⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠉⠓⠮⣕⡢⢍⡻⢷⣤⣴⠾⢛⡡⢖⡫⠔⠋⠛⠉⠁⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠉⠓⠮⣕⣒⣒⣪⠵⠚⠁⠀⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣤⣀⣀⣤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠈⠳⡒⢄⠰⡎⣢⠄⡲⠋⠀⠉⠙⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣀⢤⣀⣸⠈⠢⠃⡇⢰⡥⠤⠐⢒⡩⠃⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠰⠯⠀⠠⠤⣀⡀⠠⠤⠔⠀⢠⠔⠊⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⡠⠔⠢⠄⣉⡐⢄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣀⠔⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣀⡀⠀⠀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⡴⠶⠿⠶⠞⠛⠋⠉⠁⠀⠀⠀⡀⠀⠈⠉⠈⠁⠛⠛⠉⠙⠛⠛⠛⠻⠿⠿⠛⠷⠲⠦⣄
⠀⠀⠀⠀⠀⠀⣇⣀⠤⠀⠀⠒⠐⠂⢤⠅⢠⡀⡙⠠⣬⠔⡷⢲⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡾
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠂⠐⠂⠒⠶⠘⠛⠋⠈⠉⠀⠀⡀⠀⣀⣈⠉⠂⠈⠥⠀⢀⡲⢆⢘⡀⢒⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢸⣶⣤⡶⣾⣿⣷⣖⣴⣾⢿⣽⣏⣐⣳⣠⣵⣤⣙⢯⣶⡄⡄⣀⠠⠀⠀⣤⣤⣤⣶⡾
⠀⠀⠀⠀⠀⠀⠈⠷⠟⠛⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠈⠉⠛⠿⢯⣶⣯⣃⣘⣳⣿⣿⠟⠳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⢀⡠⢶⠶⠶⠶⠶⠒⠒⠒⠒⠋⠉⣌⡀⣠⡄⢉⡙⠳⠶⢤⣄⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢸⣷⠗⠌⠲⣤⢖⣾⣾⣾⣿⣿⠧⠟⠉⠿⠶⠞⠣⢾⣶⠆⢤⡀⣉⣉⣉⣉⣁⣈⣥⣨⣩⠉⠁⢀⡉⣉⠓⢦⡀
⠀⠈⣿⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⡀⣀⠀⠀⠀⠀⠀⠙⠚⠷⣭⣕⣼⡏⠿⣿⣿⣟⣿⠿⠤⡄⠽⠪⣽⣶⡤⣹⠆
⠀⠀⢿⣫⣀⣥⣘⣰⣣⣄⣶⣚⣻⣻⣭⣷⣶⣤⣁⠂⠢⠄⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠀⠀⢀⡀⢀⠀⠀⠀⣻⡏
⠀⠀⠈⠉⠉⠀⠈⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠉⠛⠳⠾⢷⣽⣕⡢⠄⣀⠄⡠⡤⢰⣖⣿⣿⣿⣿⣶⣃⣲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠟⠻⠛⠛⠛⠛⠉⠀⠀⠀⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠤⠤⣤⣀⣀⠀⠀⠀⢠⠴⠒⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠀⠀⠀⠀⠀⢀⠀⢀⡀⡉⠉⠛⠲⣎⢀⡀⣀⠀⢹⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⣀⣠⣴⠶⠲⠋⠀⠀⠀⠀⠉⠉⠁⠙⠛⢶⣤⡟⠉⠁⠫⣷⡼
⠀⠀⠀⠀⢯⢡⣤⡤⠞⠁⠀⠀⠠⠖⠿⠉⠁⠀⠀⣀⠀⣠⠀⠀⡤⠄⠆⠐⣖⠄⣄⢨⡗⢶⣢⢄⣸⠃
⠀⠀⠀⠀⠈⠳⣳⣶⡠⠦⠛⠉⠀⠀⢠⣤⡤⣞⢯⡀⠄⠂⠒⠚⠛⠋⠓⠈⠻⠕⣻⣿⠿⠻⢾⣾⡇
⠀⠀⠀⠀⠀⠀⠙⣿⢲⡶⢠⣘⢒⣥⠲⠟⠂⠉⠀⢀⣀⣤⣴⠶⠶⠶⠶⢶⣄⣀⠹⠿⣯⣽⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢮⣵⠆⡿⠃⢀⣀⣤⡦⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠾⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⣶⣴⠞⠉⠉⢈⡁⠈⠉⠛⠶⠤⠤⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⢠⡽⣿⣿⡄⠠⠐⠉⠁⠙⠻⠶⠦⠠⠤⠦⣤⣈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⠤⠈⠁⠉⠁⣘⢹⡇⠀⠀⠀⢀⠀⣀⠀⢀⠀⢀⡀⠀⠱⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡀⢀⡴⠞⠁⠀⠀⢀⡀⠴⣻⢾⣷⠤⢧⡐⣀⡀⣈⣡⡅⠀⠈⠁⡘⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⡀⠀⠀⠀⣀⠀⠬⠐⠉⡛⣿⣟⡉⣀⣠⣬⣠⣤⣀⡀⠀⠘⠻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠢⠠⠄⢀⣠⣤⣷⣶⣿⣾⠷⠋⠁⠀⠀⠈⠈⠉⠙⠻⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣁⡶⠛⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⢰⣿⣦⣀
⠀⢻⡧⣉⠙⠛⠓⠒⠲⠦⢤⣀⡀
⠀⠀⠹⡎⠓⢄⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⣿⠀⠀⠑⡄⠀⠀⠀⣀⣠⠤⠤⠽⠶⠶⠶⠤⣤⣄⡀⠀⠀⠀⣀⡤⠶⠒⠚⠛⠛⠒⠒⠶⠦⢤⣤⣤⣤⢾⡆
⠀⠀⠀⢸⠀⠀⠀⢸⡤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣴⡿⠋
⠀⠀⠀⢸⡀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⢀⡞⠁
⠀⠀⠀⢀⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡟
⠀⠀⢀⡞⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡞
⠀⠀⣼⢀⣀⣻⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠟
⢀⡴⠋⠉⠀⠀⠀⠉⠑⢦⡀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠿⠵⠤⠤⢤⣤⣀⣀
⡞⠀⡠⣄⠀⠀⢀⡀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⢧⠈⠦⠞⠀⠐⣍⡽⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⢀⣤⣴⣤⡀⠀⢀
⠈⠳⣄⡀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢠⣿⡋⠉⣿⣷⣠⣿
⠀⠀⠀⢻⡉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡻⢷⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠹⣗⠦⣄⣀⣀⣀⣀⣠⠴⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣧⠀⠉⠉⢽⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠘⣲⠞⢀⡔⠀⠀⠀⠀⢀⣀⡤⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣭⠥⠖⠛⠛⠛⠛⠛⠋⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠳⢄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣈⣇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣼⠒⠒⢻⡏⠁⠈⢧⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⡄⠀⠀⠀⠀⠀⣰⠃⠀⣠⠞⢛⠒⠒⢺⡆⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⡝⠒⠒⣻⠀⠀⠀⠀⢠⠏⠰⢯⣥⠴⠃⠀⢀⣸⣧⣀⣀⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡶⠊⢀⣀⣠⡿⢤⣤⣀⣠⡟⠀⠀⠀⠙⠛⢛⣭⠟⢉⡴⠃⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣭⣟⣋⣠⠏⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠉⠿⣯⣤⠤⠴⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠐⠒⠒⡐⠒⠒⠒⠒⠒⠒⠦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠑⢲⣄⠀⠀⠈⠲⣄⠀⠈⠳⢄⠀⠀⠀⠀⡀⠀⠉⠨⢤⡈⠙⠲⢤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡹⠚⠠⢀⡀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⢗⠀⠉⠙⠲⣘⠂⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢵⡴⠂⠀⠈⠁⠂⠀⠄⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠂⠀⠀⠉⠓⠀⠀⠀⠉⣮⣉⣵⣶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠘⠇⠀⢠⡔⠣⡄⠀⣦⠀⠠⡠⡤⠈⠑⠢⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠶⠖⠒⠉⠉⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⢒⠶⠢⣤⣉⣁⠒⠁⠀⠄⢨⠀⠁⢡⡆⣠⠆⣶⣴⣶⠀⠈⢉⡟⠉⢡⡁⠒⢖⠉⠉⠉⠉⠁⠀⠄⠀⠀⠀⡠⢰⠶⢜⡰
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣀⣀⣀⠀⢈⠛⢿⣷⣶⣤⣤⣤⣌⣀⣁⡀⠋⠋⠀⠀⠀⠀⣠⡿⠘⠁⣼⠏⢀⠀⢀⠠⠀⢠⣤⠆⢃⣻⣧⣴⣞⠉
⠀⠀⠀⠀⠀⠀⢀⠜⠃⠒⠻⢛⣿⣿⡈⢻⠂⣸⣼⣯⡿⠿⠟⠛⢛⠩⢍⠉⠛⠻⣶⠶⣶⣤⣅⣈⣛⣀⣁⣈⣉⣁⣉⣤⣤⡴⡷⠟⠋⠉⢻⠛⠳⣄⡀
⠀⠀⠀⠀⠀⠰⣃⠀⠀⢀⠀⠈⠈⠛⠙⠢⠴⢿⣯⣠⣤⣶⡖⠻⠿⢿⣦⣈⠢⠀⢐⣾⣿⠿⠛⠻⠏⠭⢉⠻⠿⢿⡻⠉⠉⠀⠀⣤⣴⣖⡥⠄⠋⠀⢙⠄
⠀⠀⠀⠀⠀⠀⡻⠟⢳⡲⠆⠈⢍⣁⣀⣐⣂⠀⠀⠉⢉⣉⢻⣄⡀⠀⠀⠈⡱⢧⣷⣿⣿⣾⡛⢻⣷⣶⣶⣕⢦⠀⠙⠀⠀⣠⠟⠏⢉⡉⣀⣠⣤⡶⣯
⠀⠀⠀⣀⠄⠈⣄⠠⣆⣙⣌⠓⠯⣉⣁⠈⠉⣛⣛⣷⡶⠖⠀⠈⠻⠷⠦⠍⠰⠛⠿⠋⢹⠢⡀⠀⠁⣉⢿⡿⣆⠱⠀⢀⣴⡿⢛⢛⡿⢟⣛⣉⠷⠀⠈⡇
⠀⠀⠀⠉⢆⠀⠢⢴⣯⡏⣙⡦⣀⡀⠀⢬⣭⣭⣭⣧⣬⣤⡄⣲⣶⠶⠭⢱⣶⣾⣶⣶⡐⣿⣶⡠⢀⠈⠀⠰⢸⠀⣷⠋⠉⠀⡎⢋⣠⠞⢏⠐⢆⣶⣂⡇
⠀⠀⠀⠀⠀⠉⠉⠁⢮⠴⠲⣿⣷⠾⣍⠐⡿⢻⡏⡫⣉⡏⢉⣕⣠⣤⣤⣤⡩⣠⣀⣹⣷⣦⡈⠛⠓⠨⠀⠀⠡⠞⠷⡦⠽⢾⣠⠞⠀⢤⡘⠇⣾⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢨⣾⣶⠋⠀⢈⣱⢾⣾⡧⡄⠈⡏⢠⣝⣿⣓⠦⣄⣈⠉⠉⠛⠛⠷⠶⠶⠤⠤⠤⣤⣤⡖⠋⠀⢘⠆⣠⣵⢞⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠹⣀⣴⣿⣿⠆⠉⠁⢹⡶⢤⢴⣵⣤⢟⣑⣢⣤⣈⣙⣂⣙⠲⠾⠶⠖⢟⣙⣻⣿⠇⠀⢀⣴⣿⠿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣚⣒⢤⣬⣤⣀⣀⡀⠀⠂⣴⢴⠛⢊⡀⠀⠈⠑⠾⣿⡇⠉⠙⠓⢐⡿⢹⠿⠁⣀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠉⠻⠽⢿⠿⠶⣦⣤⣀⢺⣵⣀⠀⠀⠒⠇⣧⣴⣶⣾⣽⣿⣶⣶⡿⢿⣿⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠺⢿⡿⢿⣿⣿⠷⠖⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠔⠋⠀⠀⠐⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠈⡀⠀⠀⠀⢀⣹⡩⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣿⢿⣧⢻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠑⠚⠁⠒⠢⠀⠀⠀⡀⠘⣹⡅⣺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣛⣛⣿⣿⣿⣘⣿⣿⣿⣿⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠢⢴⡱⣶⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣿⣿⣿⣼⣿⣯⣟⣿⣿⣜⢿⣿⣷⡀
⠀⠀⠀⠀⠀⡖⠈⠀⢅⠀⠀⠀⢀⠀⠀⠀⠄⡀⠀⠀⠀⠉⢎⢻⡌⢧⡀⠀⠀⠀⠀⠀⣴⣺⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣿⣿⣯⡻⣿⣷⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠈⡆⠀⠀⠈⠛⢆⡀⠀⠈⢄⠀⠀⠙⠒⠷⡝⢦⢹⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⡄
⠀⠀⠀⠀⠀⠣⣄⡤⠴⠶⣄⡀⠀⠓⠀⢭⡢⣄⠀⠈⠙⣏⢳⣄⠛⣎⢻⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⡇
⠀⠀⠀⠀⢀⠊⠁⠀⠀⢤⣀⣯⣍⣳⣦⡀⠈⠻⢷⣲⢄⡈⠲⣌⡙⠻⠌⣧⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⢿⣿⣿⣿⣿⣿⡇⢳
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⠋⠙⠿⠟⠷⠿⠶⣮⣵⣦⣬⣉⡛⠒⠀⣹⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣉⡛⠛⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⠤⠒⠒⠢⠔⠒⠉⠓⠒⠶⠴⠴⠶⠛⠋⠑⠲⠚⠁⠀⠀⠀⠀⠉⠉⠙⠿⠿⠟⠛⠻⠷⠛⠋⠙⠛⠻⠿⠿⠿⠛⠋⠙⠒⠛⠃

⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠻⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣟⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⡯⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⣸⣟⠻⣦⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⠿⣿⣶⣾⣿⠖⠛⣹⣿⣧⣠⣶⣾⣿⣿⣶⣶⣶⣿⣷⣆
⠀⠀⠀⠀⣟⢹⣿⠀⣿⣿⣧⣤⣼⣿⣿⠖⠀⢿⡟⠿⠟⣩⣷⣾⣿⣿⣿⣛⣿⣦⡘⢿⣷⣶⣾⣿⣷⣶⡀
⠀⠀⠀⠀⢻⣌⣿⠀⣿⡎⠻⣿⡻⣿⣿⣿⣿⡿⢿⣶⣾⣿⠟⠑⢾⠋⠈⣿⡿⣿⣷⡀⠈⣩⣿⣿⠿⠿⣿⣿⠿⠿⣶⡄
⠀⠀⠀⢠⣾⣿⡟⠀⣿⡇⠀⣿⣧⢹⣏⡀⣀⢄⡀⠉⠈⠀⠀⠀⠀⠀⢸⣿⡾⠋⢹⡿⣶⡳⣿⡇⠀⣠⣶⣶⣄⠀⠘⣿⣿⡿⣶⡄
⠀⠀⠀⣼⣿⣿⡇⢸⣿⣿⡀⠸⣿⡎⣿⣿⣿⣟⣴⢃⠄⣠⠀⠀⠀⠀⠘⣿⣤⣶⡞⠿⠿⠿⠻⠿⠟⠻⠋⠀⢻⣿⣶⣄⠀⠀⢿⣿⣿⣷⡆
⠀⠀⣾⣿⣿⣿⠁⢸⣿⣿⡇⠀⠙⣿⣽⣿⣿⣿⣿⣿⣾⠋⠀⠀⠀⡸⡄⠀⠘⠋⠀⣀⠀⠀⠀⠀⠀⠀⢀⠄⣀⣴⣾⣿⣷⠀⣲⣿⣿⣿⠃
⠀⠀⣿⣿⣿⣿⠀⣾⣿⣿⣷⢀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢱⣻⡄⠀⠛⠫⠗⠀⡴⠁⢠⠆⣠⠏⢐⣿⣿⣿⠋⡿⢰⣿⠋⢠⣿⠀⢀⣴⣶⣦⡀
⠀⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣾⡇⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⢳⣿⣷⡹⣆⠀⠀⠀⡼⠃⢠⣿⡞⠁⠀⣻⣿⡟⠀⢸⡇⢠⡟⠀⢸⣿⣶⡿⠋⠀⠻⠟⣿
⠀⢻⣿⣿⣿⠃⢸⣿⡿⢹⣿⣿⠃⢀⣼⣿⣧⢻⣿⣿⢟⠀⠀⠀⠈⢿⣿⣿⣾⠇⠀⡼⠁⠀⣼⣏⡠⠀⢘⣿⣿⠃⠀⢸⡇⠈⢷⣶⡿⠛⠉⠀⠀⡰⠀⢰⣿
⠀⣸⣿⣿⣿⠀⣿⠟⣰⣿⣿⠏⠀⣾⣿⣿⣿⢸⣿⣷⣏⠀⠀⢦⣆⣸⣿⣿⡟⠀⣼⠃⠀⣰⠟⠋⠀⠀⢨⣿⣿⣄⠀⢸⡇⠀⠀⠀⠀⣰⠃⢀⣼⠃⠀⢸⣿
⣿⣿⣿⣿⡿⠀⠏⣼⣿⣿⠏⠀⠀⣸⣿⣿⡟⣼⣿⣿⠏⠀⠀⠘⣿⣿⣿⡿⠀⢰⡟⠀⢠⡏⠀⠀⠀⠀⠠⢿⣿⣿⣇⢸⡇⠀⣷⠀⣰⣏⣴⠟⠁⠀⣀⣀⣼
⠸⣿⣟⣿⡇⠀⢸⣿⣿⠋⢀⠀⠀⢿⣿⣿⢣⣿⣿⣿⠃⠀⠀⠀⠘⣿⣿⠃⠀⣿⠃⣠⡟⣀⣤⠖⠀⠀⢠⣿⣿⡟⠁⣼⠇⢀⣿⣰⡿⠋⠁⠀⠀⣴⣿⣿⠟
⣰⣿⣿⣿⡇⠀⣿⣿⢃⣴⡏⠀⠀⢈⣿⣿⢸⣿⣿⠃⢀⠀⢦⢠⡀⠘⠟⠀⣸⣿⣴⡿⠛⠉⠀⠀⠀⠀⠠⢿⣿⡇⢀⣿⠀⠸⣿⠟⠀⠀⡠⠀⠸⣿⡟⠁
⣿⣿⣿⣿⡇⠀⣿⣧⣾⣿⠁⠀⣾⣿⣿⣿⡇⣿⣷⠀⢸⡀⡘⢷⣷⡀⠀⢠⣿⡿⠋⠀⠀⠀⣠⠖⠀⠀⠀⠛⣿⡇⢸⡇⠀⣼⡏⠀⣠⡞⠁⠀⠀⢹⣧
⠸⣿⣿⣿⡇⢠⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⢸⣿⡷⠀⠀⢻⣿⣿⣿⡇⠀⣼⡿⢁⣤⣶⣶⣟⣥⠔⠀⠀⠀⠀⠸⣿⣾⠁⣼⣿⣴⠾⠋⠀⠀⠀⣴⡿⠋
⠀⢿⣿⣿⠇⢸⣿⣿⣿⣿⠀⠀⠈⢿⣿⡟⣿⣿⡇⠀⠐⣄⢻⣿⣿⠇⢀⣿⡿⠋⠁⠀⠀⠀⢀⣤⠎⠀⠀⠀⢀⣿⠏⢰⣿⠟⠁⣀⣤⠄⠀⢀⣿
⠀⠸⣿⣿⠀⣼⣿⡿⢡⣿⠀⠀⣾⣿⣿⡇⣿⡏⠀⠀⠀⢸⣾⣿⣿⠀⢸⡿⠀⠀⠀⣠⣤⠞⠋⢀⣠⡔⠀⢠⣾⠏⠀⣼⣿⣶⠟⠋⢀⣴⣾⡿⠏
⠀⠀⢿⣿⠀⣿⡿⠁⢸⡇⠀⠀⢹⣿⣿⡇⣽⡃⠀⠀⡆⠘⣿⣿⡟⠀⣸⡇⠀⢀⣾⣯⣤⣶⠾⠛⠁⠀⢀⣾⡟⠀⢠⣿⡏⠀⠀⠀⢸⣿
⠀⠀⢸⡿⠀⡿⠁⡀⢸⡇⠀⠀⣾⣿⣿⡇⢺⠁⠀⠀⢹⣄⣿⣿⡇⠀⣿⠀⢀⡾⠟⠉⠀⠀⡀⠀⠀⢀⣼⠟⢰⠀⣼⡿⠀⠀⣤⡾⠟⠃
⠀⠀⢸⡇⠀⠀⣰⡇⢸⡇⠀⣆⠘⢿⣿⣧⠈⢀⠀⡀⠈⢻⣿⣿⠃⠀⣿⣶⠋⠀⣀⣤⠶⠋⠀⣠⣶⡿⠋⣠⠟⢠⣿⠃⢠⡾⠋
⠀⠀⢸⡇⠀⣼⣿⡇⢸⡇⠀⣿⢠⠈⣿⣿⠀⢾⡆⠹⣆⠀⠙⡿⠀⢠⡿⠁⢠⡾⠉⠀⠀⠀⢰⣿⠋⠀⢰⡟⢠⣿⠃⢠⣿⠃
⠀⠀⢸⣿⠀⣿⣿⡇⢸⡇⠀⣿⢸⠀⣿⡇⠀⢸⣿⣴⣿⣧⠀⠀⠀⠸⠁⣰⠟⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⣡⣿⡁⢀⣾⠏
⠀⠀⠈⣿⠀⣿⣿⡇⢸⣷⠀⢸⢼⠀⣿⡇⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢠⣿⠃⠀⣸⣿⣿⠟⢡⡾⠃
⠀⠀⠀⣿⡄⢻⣿⠇⠘⣿⡀⢸⣸⡆⢻⣷⣳⡀⠈⣿⣿⣿⡟⠀⢠⠀⣀⣤⡴⠋⠀⢀⣴⡿⠃⠀⢀⣿⡿⢃⣴⡟⠁
⠀⠀⠀⢸⡇⢸⣿⣿⡀⣿⡇⠸⣿⣿⡈⢿⣿⣿⣦⠸⣿⣿⡇⠀⣾⡾⢋⣁⡤⠀⠀⣾⡏⠀⠀⠀⣸⣿⣷⡿⠋
⠀⠀⠀⠈⣷⠈⣿⣿⡇⢿⡇⠀⣿⣿⣧⠈⣿⣿⣿⣷⣿⣿⠀⢠⣿⠟⠋⠁⢀⠄⢸⡿⠀⢠⡇⢀⣿⡟⠁
⠀⠀⠀⠀⢻⡄⢿⣿⣧⢸⣇⠀⢹⣿⣿⠀⢻⣿⣿⣿⣿⠇⠀⣾⠃⠀⢀⡴⠋⢠⣿⠃⢀⡾⠀⣼⡟
⠀⠀⠀⠀⠘⣷⠘⣿⣿⡆⣿⡀⠸⣿⣿⠀⢸⣿⣿⣿⡟⠀⣸⡏⠀⣠⡟⠀⢀⣿⠏⠀⣼⡇⣰⡟
⠀⠀⠀⠀⠀⠹⣧⢻⣿⣷⢹⣧⠀⢿⣿⠀⣾⣿⣿⡟⠀⣰⣿⡇⣰⡟⠀⢠⣾⠋⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣧⣿⣿⣧⣿⣆⠸⣿⣇⣿⣿⠟⢀⣼⣿⣿⡿⠋⢀⣴⠟⢁⡞⢀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣆⣿⣿⡿⢋⣰⣿⣿⠿⠋⣀⡴⠟⠁⣠⡿⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣭⣭⣬⣁⡰⠿⠿⢋⡥⠴⠛⠉⢀⣴⣾⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠿⠷⣤⣀⣠⣤⣴⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣴⣶⣿⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠠⠤⠤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠠⠄⠂⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⣀⢞⠄
⠀⠐⡑⡀⠀⠀⠀⠀⠀⢀⣠⠌⠈⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡨⠄
⠀⠀⠣⡈⠢⣀⡀⠐⢊⢽⠁⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⠆
⠀⠀⠠⡘⢢⠋⠀⠀⠈⠼⣴⣧⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠑⠺⡀⠀⠀⠀⡀⠀⠙⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠈⣃⠀⠀⠐⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⡠⠒⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠉⠀⠀⡀⠐⡅⠀⠀⠀⠀⡜
⠀⠀⠀⢜⠂⠃⣁⣀⠄⠀⠀⠀⠀⠀⠀⠐⢄⣀⠀⠀⠀⠀⠰⠦⠤⠤⠄⠄⠀⠒⠈⠉⠣⡀⠸⡄⢀⡠⠔
⠀⠀⠀⠀⠿⠝⠁⣀⣀⣀⣀⡀⢤⠚⠁⠘⠇⠈⡄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡅⠀⢀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠴⠁⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡘⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢠⡂⢀⠍
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣽⣿⣿⣭⣿⣽⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠓⠲⠶⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠒⠲⠶⠦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⢸⣇
⠀⠀⠀⠀⣀⣤⡴⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⢀⡾⣿⡄
⠀⠀⠀⣾⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠁⠀⠀⠀⢀⣤⠞⡹⠺⢹⠇
⠀⠀⠀⣿⠉⠉⠙⠛⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⣠⡶⠋⣠⢞⠕⢡⡿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⢀⣠⠞⣩⠛⣏⠕⠀⣠⢿⡇
⠀⠀⢀⣽⡶⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⣀⡴⠋⠀⢈⠇⠀⣧⡤⠚⣱⢿⡇
⠀⣰⣏⣀⣧⢀⠀⠈⠉⠙⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠦⣤⢶⡶⠋⠁⠀⠀⠀⠀⢀⡴⠞⠉⠀⢀⡴⠃⠀⠀⣇⣤⢞⡵⠟⠁
⠀⠀⠹⣯⡉⢟⠳⣔⣀⣀⣤⡀⠀⠈⠉⢛⣲⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⡷⠤⠚⣉⣥⢶⡶⠛⠛⢋⣴⠟⠁
⠀⠀⠀⣿⢿⣟⠳⣌⡉⠈⠙⠳⢶⣖⠛⢻⣋⡀⠀⠈⠉⠛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠛⠁⠀⣰⡧⢶⢻⣡⡴⠋⠀⣠⡾⠋⠁
⠀⠀⠀⢿⣄⣈⠙⠲⠭⣓⠶⠤⣤⣉⣙⡶⣌⡙⢦⣀⣀⣀⣀⡤⣤⣿⠙⠛⠲⠦⣤⣄⣰⣧⠞⠋⠀⢰⡷⠚⡡⢒⡵⠋⠁⠀⣠⠞⠉
⠀⠀⠀⠀⠈⠙⠛⠷⣶⣤⣙⠲⢤⣀⡙⠲⣬⣝⣦⣈⠉⠉⠁⠀⣀⣨⡷⠦⠤⢤⠤⢬⣉⢁⡤⠴⠳⠏⢒⣩⣴⠏⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣬⣍⡉⠉⠉⠉⠙⢦⣤⣶⡛⠓⠤⢤⣉⣉⣀⣀⣨⠟⠉⣓⣒⣒⣿⡽⠞⣁⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣄⡀⠁⠈⠉⠛⠲⠤⣤⣤⣤⠤⠖⠛⠉⠉⠉⠀⢀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣤⣄⣀⣀⣯⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣿⡆⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠒⠛⣛⣯⡿⢱⠁⢰⣤⣄⠤⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢿⡶⢶⣤⠤⠶⢿⣇⠈⢆⡾⣻⢳⢆⣼⡼⠙⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣵⣾⣋⣲⣾⡿⣷⡶⢶⣦⣈⡘⣾⡿⢡⣯⣾⣯⣑⡀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠞⢚⣻⣿⣷⣿⠣⠀⠈⠀⠸⠛⠻⣏⠽⠒⣡⣁⣠⡴⠶⠦⣍⡲⢤⣀⡽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⢈⣩⣾⣿⡖⢀⣤⣤⣤⣄⡈⠙⠦⡄⠀⠀⠀⠀⠀⠈⢷⡜⠉⠙⢿⡃⠀⠀⠀⠈⠛⠋⢰⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣼⣿⠟⡩⠋⣴⣿⡏⠋⢠⡄⠩⣷⣄⠘⢆⠀⠀⠀⠀⠀⠀⣿⢂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⣧
⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⠖⣛⣉⣀⠀⠀⣿⡏⣁⠀⠀⠀⠀⠀⢻⣄⠈⡆⠀⠀⠀⠀⠠⠾⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢠⠃⠠⠈⣴⡿⠟⠉⠉⠻⡔⠿⠏⢹⠄⠀⠀⠀⠐⠂⢹⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⢢⡇
⠀⠀⠀⠀⠀⡌⠀⠀⢸⣿⠁⠀⠠⣎⣁⡀⠀⠀⠉⠀⣰⠀⠀⠀⢀⣸⡇⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣸⠃
⠀⠀⠀⠀⠀⠁⠀⠀⢸⡯⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠌⣸⠁⡾⢀⠖⠘⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣧⠏
⠀⠀⠀⠀⠀⢰⡈⡀⠘⣧⡀⠀⠀⠀⠉⠉⢀⡠⠀⢀⠉⠀⠀⠀⣤⠋⢰⢷⡦⠲⢀⡀⠒⠀⠀⠀⢀⠀⠀⡠⠀⢀⣼⡞⠁
⠀⠀⠀⠀⠀⠀⢦⢳⠀⠘⣇⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠛⠋⠀⣠⣿⣾⣲⣆⣞⢚⢈⣀⠤⠀⠃⣠⣠⣧⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠳⡆⠀⠈⠃⠀⢀⡀⠀⠀⢀⣠⠟⠀⠀⠀⣠⠾⣿⣵⣿⣿⣿⣿⣷⣶⣧⣶⣟⣻⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣢⣄⣈⠉⠙⠛⠉⠁⣀⠠⠐⠊⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⣤⣀⣀⣤⣤⣤⣤⣤⣶⣾⣵⣶⣶⡶⠶⠟⠛⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⡉⠙⠛⠉⠉⠉⠉⠉⠉⠉⢀⣀⣰⣴⡿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠶⠶⠶⠶⠶⠶⠛⠋⠉⠙⢛⠛⠁⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⡀⠀⠀⢀⣀⣀⣤⠴⠖⠛⠋⠉⠉⠉⢉⣩⣤⣶⠴⠞⠛⣋⣝⣫⣤⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠉⠛⠛⠉⢩⡕⠶⠤⠤⠤⠤⠶⠶⠿⠛⠛⠋⣁⣺⡥⠖⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣌⠛⠳⢮⣝⡳⠶⢤⣄⠈⠙⠛⠷⠾⢶⣿⠿⢿⠿⠿⠿⠛⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠲⢤⣀⡀⠀⠉⠓⠦⣌⠀⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⡀⠈⠳⢦⡀⠉⠫⠛⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠉⠓⠦⢤⣈⡀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⣤⣍⡉⢛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄