Ships Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢀⡠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠳⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡀⢀⣠⡤⠄⠀⠀⠀⠀⠀⠂⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠒⠒⠋⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢉⠀⠀⠀⠀⠀⠀⠀⣸⣀⡤⠴⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠖⢺⡏⠉⠉⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢴⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⣀⠤⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⣀⡠⢴⡗⠛⠂⠉⠁⠸⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠀⠀⣸⠁⠀⠀⠀⠀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣇⠀⠀⠀⢠⡋⠀⠀⠀⠀⠀⠁⠀⢳⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣴⠟⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⣀⡴⣶⠚⠍⡀⣸⡋⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣰⡸⡤⠒⠛⠈⠗⢂⣀⣤⣶⣗⣧⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠉⠀⠀⡣⠀⢂⣤⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣒⣭⣴⣾⣿⣿⣿⣟⠛⡿⣿⣿⣿⣃⣀⣀⣀⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⣀⠀⣄⠀⣙⣛⡛⠋⣩⣭⣍⣛⣹⣷⠏⣿⢿⣯⢙⣿⣿⣿⢛⡻⢶⣾⣿⣿⣍⣁
⠀⠀⠀⠀⠉⠁⠀⢠⣌⢉⣡⣼⠯⣬⣿⣿⣞⣛⣿⣤⣶⣿⣿⡟⠛⢋⣥⣶⠈⠠⣭⣛⣉⡙⠻⣿⠿⠷⠾⣿⣿⡟⠁⠤
⠀⠀⠀⠀⠈⠉⠉⠀⠁⠭⠤⠤⠀⠽⣏⣉⠀⠠⠼⠉⣉⣉⡑⠂⠁⢼⣿⡋⠀⠈⢶⣬⣟⣢⠴⠶⢶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠷⠀⠀⠉⠀⠀⠀⠀⠉⠉⠀⠀⠈⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡴⠖⠋⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢦⣄⠀⠈⠑⠲⣄⡀⠀⠀⠀⣾⣷⠆⠀⠀⠀⠀⠀⠐⢊⠁⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠙⠢⣀⠶⣿⠟⠑⠂⢀⠠⣿⠀⠀⠀⠀⠀⠀⠀⢴⣷⣀⡤⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⡆⠀⠁⠀⠀⣹⣇⠀⢀⣤⡴⠖⠖⠊⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠳⣄⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣶⡀⠀⠀⠀⠀⠀⠀⢹⡄⠈⠑⢆⡀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⡩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣿⣧⣶⣾⣷⡀⠀⠀⠀⣷⠀⠀⠀⠙⡆⢻⡀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⠨⠀⡠⢊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣵⣾⣿⣿⣾⡔⣦⣖⣏⣆⣴⠂⢨⡇⠘⣧⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⣨⣤⣤⢾⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⢿⢿⣿⣿⣯⡍⢩⡷⠚⢫⡉⢩⡍⠡⡄⠉⣴⠠⠀⠀⠀⢀⡤⠞⣻⠟⣛⣉⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣤⣬⢈⣈⣄⣉⣀⣾⣿⣅⣤⣿⣯⣤⣼⣯⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⡃⠀⠈⠀⠀⠀⠴⣉⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣼⣿⣿⣿⣿⣿⣿⣿⡗⠆⢐⠤⢤⣴⣶⣶⣾⣿⡥⣵⣟⣦⠝⠿⠻⠷⠛⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣬⣟⣻⣃⣒⡀⢄⣒⣚⣛⡛⠭⣭⣥⣼⣷⣎⣳⣶⣾⠓⢶⡒⢦⠀⠀⠑⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⣍⣉⣭⠛⠿⠛⠈⡀⢘⡷⠶⠾⣒⣀⡍⠉⠥⣼⡭⠭⠭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠤⠄⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡖⠶⠔⠂⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠉⠉⠉⠑⣀⣘⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣛⣓⣼⣔⣶⣒⣤⡇⡃⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢣⠀⠀⠀⠀⣼⡧⠤⢶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠷⠶⣐⣾⠤⢤⡤⠤⣄⡇⠀⠀⠈⠯⡒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡲⠤⣼⠀⠀⠈⠫⠉⠉⠉⢭⣽⠷⠶⠶⣶⣤⣙⣦⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠞⠒⠋⠑⠒⡇⠀⠀⠀⠈⢆⠀⠀⠈⠻⡲⢄⣣⠀⠀⢈⣑⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢱⠠⢼⣇⣾⣻⣟⣥⡲⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡼⠀⣀⣀⣀⠀⡇⣠⢏⣘⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢸⢃⣠⡴⢒⠾⡟⢻⣿⠿⣽⣽⠗⣋⢭⣿⣌⡝⣻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡾⣛⡛⡓⡚⣶⣷⣾⣇⣀⣾⣿⣿⣿⣿⣾⣷⣷⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣵⣓⣐⣐⣸⡿⢿⠿⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⠄⠠⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣂⣐⡿⢿⣿⣿⣿⣷⣾⣿⣿⡿⠿⠿⠟⠋⠉⣭⣩⢉⣁⢻⡑⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠭⠭⠥⠷⠒⣦⡀⠙⠛⠓⠠⠤⠍⠀⠐⠂⠀⠴⠚⠃⠼⠃⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠈⠀⠀⠉⠁⠀⠒⠒⠒⠂⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⢤⡽⣍⣛⡉⠁⠀⣠⣄⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠘⣤⣀⣅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣋⣓⣸⣧⣲⣖⣒⣤⡇⢱⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠁⠉⢣⠀⠁⠈⠉⠉⣩⣏⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣧⣤⠤⣼⢀⣀⣀⣀⡀⠉⢧⠀⠀⠉⠻⣟⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠉⡇⠉⠙⠻⣿⠫⠥⠿⠭⠭⠽⣿⣀⣀⣀⡀⠈⠷⣌⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠉⢛⡇⣀⣀⣀⡈⡇⠀⠀⠀⠙⢿⡋⠛⠛⠛⢿⣷⡒⠞⢦⠀⠈⠑⠦⣀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⠉⠁⠀⠀⠉⢹⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠙⡝⠲⢌⣧⠀⠀⣠⢤⣷⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡆⠦⣻⣠⣞⣳⣿⣳⣞⠲⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡗⠀⣼⢓⡞⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠷⢀⡇⠀⣀⣤⠤⣴⡷⠶⣶⣾⣟⣻⢲⣧⠔⢚⣙⣯⡉⡓⠷⣾⣷⣾⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⣸⡗⣻⣿⡤⢬⣀⣧⣸⣿⠃⢸⣽⣿⣶⣽⣆⣿⣻⣧⣯⣼⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⢦⢶⠐⡘⡂⢻⣿⣿⣿⣶⣾⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣭⣥⣥⣴⣾⡛⠻⣟⣻⣭⣷⣮⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣁⡑⠤⣠⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⣡⣀⣿⠻⠿⢿⣿⣿⣿⡿⢿⡽⣿⡖⠛⠿⠿⠉⠀⠈⡦⠤⠎⣤⡼⢥⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡬⠭⠤⠽⠖⢫⣠⡀⠙⠛⠛⠁⣠⠤⠬⠁⠀⠒⠀⠠⠶⠞⢋⠠⠟⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠐⠒⠂⠀⠈⠉⠉⠠⠤⠤⠢⠴⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⢹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⡾⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⡇⢀⡰⠁⡀⠀⢤⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠀⡀⠀⢫⣸⣧⠶⠿⠶⣾⣷⣤⣾⣾⡟⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣧⣴⣿⣿⣿⣤⡶⠟⢻⣿⣷⣿⣿⣁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⡿⣿⠁⠀⣠⡾⠿⣿⣿⡏⢸⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⢲⣦⣾⣷⡄⠀⢈⣼⠿⠋⠷⣃⠴⠛⠉⣥⣾⣿⣿⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣏⣉⣽⣿⣤⠖⡛⢱⣾⡷⠞⢁⣴⣶⣾⣿⠿⢿⣷⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢉⣿⣯⣹⣿⣿⣶⣾⣯⣁⣠⡿⠛⠋⣉⣁⣶⣿⣿⣿⣿⣤⡀⠀⠆
⠀⠀⠀⠹⢷⣴⣦⣄⣄⣰⣶⣶⣤⢙⣿⣿⣧⣾⡟⡉⠹⢿⠿⠛⠋⠀⠀⠀⠈⠉⠁⠈⠛⠋⠻⣿⣯⣤⡀
⠀⠀⠀⠀⠀⠁⠀⠻⢿⣻⠟⠿⠿⠿⠟⠛⠿⠿⠛⢁⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢯⣤⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠠⣄⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠉⠀⠀⠀⠀⠠⠄⠒⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣀⠦⠔⠒⠀⠀⠀⢰⢀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠉⠀⠀⠀⢠⣄⡀⠠⠙⠚⠉⠉⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠶⠟⠛⠂⠀⢀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠔⠒⠈⠁⠀⠀⠀⣶⠀⢸⣀⣠⡤⠤⠤⠤⠘⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡘⠺⠐⠛⠉⠀⠀⠀⠀⢄⣀⣰⡿⠀⠚⠉⠀⠀⠀⠀⠀⢦⡀⠛⣦⣀⣀⡀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡤⠇⠒⠿⢉⡉⣹⠆⠀⠀⠀⠀⠀⠀⠀⡛⣷⣀⠀⠉⠉⣁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣐⢲⠩⠀⠀⢸⡏⢉⠁⡀⠀⢀⣀⠤⡶⢶⡞⢛⣉⣭⣤⣶⡦⠁
⠀⠀⠀⠀⠀⠀⢰⡟⠉⠈⡁⠀⠎⠀⠀⠀⠀⣇⠀⢊⣵⠊⠀⠀⠀⣧⣴⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢸⠀⢘⣃⣤⣴⣶⣾⣿⡿⢻⠿⣿⣿⣿⣵⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢶⠧⢾⣧⣤⣾⣷⠔⢙⣉⣽⣿⣿⣿⣿⣿⣿⡿⢿⣵⣴⣦⣅⠘⣿⣿⣿⣦⠄
⠀⠀⠸⢿⣶⣤⣄⡀⠈⠿⠟⣠⣶⡲⣿⣿⠿⠿⠯⠭⣭⡽⠛⠛⢉⣧⣾⣿⣿⣿⣿⡇⣿⠿⢿⡧
⠀⠀⠀⠈⠉⠈⠉⠛⠻⠿⠟⠏⠉⠁⠹⠤⠀⠦⠒⠀⠴⠶⠷⠾⠟⢿⣿⣿⡿⣿⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢲⠀⠁⠀⢈⠃⠀⣷⣼⣶⣿⣥⣿⢂⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⣀⡤⠚⠁⠀⠀⠈⡆⢀⣀⣸⠷⠒⢛⣿⣿⣿⣿⣯⡾⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⡛⠲⠶⠶⠶⠒⠚⠋⠉⠁⠀⣰⠊⡀⠛⠛⠿⠿⠿⠛⢿⣇⢀⣦⣄
⠀⠀⠀⠀⢀⠐⢒⣛⣿⡟⣻⣿⡿⠿⢯⣥⣤⣤⠴⣤⣄⣰⣯⣾⣿⣧⣶⣴⣶⣠⣶⣎⢻⣿⡿⠏
⠀⠀⠀⠀⠛⠛⣛⣿⣟⣻⣭⣭⣴⣾⣾⡿⠛⢡⣤⣼⡻⠧⠀⠀⣬⣍⠛⠻⠿⢿⣿⣿⣧⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⣥⣀⣀⣠⢴⣶⣶⣶⠾⠭⠁⠀⠀⠰⠿⠿⠿⠂⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠀⠉

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣬⣙⣟⣛⠀⢀⠀⠀⠐⠀⠠⠴⠶⠛⠒⠒⠒⠒⠒⠀⠀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠉⠉⠙⠉⠉⠉⢉⠿⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣛⠒⠀⠚⠓⠐⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣾⡆⠀⣠⣶⡶⠀⠀⠀⠀⠀⢠⣴⡿⠿⠀⠀⠀⠀⠀⢠⢄⡀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣀⢠⣤⣴⣾⣿⣿⢿⣉⢿⣧⣾⠟⣛⠓⢀⣴⣶⣖⣋⡉⣉⣀⣀⣀⠀⠀⠤⠴⠿⠋⠻⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡾⢀⣦⢭⡥⣭⠿⠿⠿⠿⠿⠿⠿⠛⣻⡿⠟⠸⠿⠛⠉⠀⠀⠀⠈⠈⠉⠉⠈⠘⢿⣿⣿⡶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢀⣴⣿⣧⡾⣫⠟⠁⠀⠀⠀⠀⠀⢀⣀⣨⡟⠉⠁⠀⢘⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠿⠈⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣿⡿⢡⣿⣿⣅⡀⠀⠀⣀⣠⣠⣶⣾⣿⣿⠀⠀⠀⠀⠀⠛⠉⠓⣀⠀⡀⠀⢀⣐⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⢉⣡⡴⠿⠋⣿⠟⠟⠻⠟⡿⠛⢻⠿⠛⠛⠙⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠁⠙⠋⠀⠀⠀⠁⠠⢸⡧⠴⡿⠏⠄
⠀⠀⠀⠀⠀⠀⠀⠲⠤⢤⣀⣀⣀⣉⣩⣍⣬⣡⣤⣤⣤⣤⣷⣼⣶⣶⣿⣿⣿⣾⣿⣿⣷⣥⣤⣤⣤⣧⣤⣦⣤⣤⣤⣴⣾⣶⣷⣷⣿⣷⣶⣷⣶⣮⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣿⣿⠿⣿⡿⠿⠿⠿⣿⣿⡿⠿⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⠤⣤⣬⢅⡶⣶⣟⡛⠛⠻⠛⠿⠽⠷⠲⣖⡒⣟⢋⡁⠀⠈⣠⠤⠔⢂⣀⣁⠀⢉⡐⣻⣗⣾⣿⣿⣿⣯⣿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠾⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡤⣥⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠈⡄⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⡇⢰⣁⣰⣦⣄⣿⣀⣄⣦⣴⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣆⣠⣿⣿⣏⣠⣴⠾⢿⣿⣽⣿⡏⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⡿⠋⠁⣀⣿⢿⣿⣿⠉⣧⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⣤⣰⣶⡄⠀⢐⣽⡿⠛⡷⢃⡴⠞⠋⣠⣼⣿⣏⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⢛⣻⣿⣡⡴⠞⣩⣶⣦⠞⢁⣤⣴⣾⣿⠿⣿⣼⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣻⣯⣭⣿⣾⣷⣞⣛⣁⣠⡟⠛⢉⣉⣡⣶⣿⣿⣿⣧⡀⠀⠄
⠀⠀⠀⠻⣦⣴⣤⣠⣄⣴⣶⣦⡙⡻⣿⣯⣼⡿⡉⠻⢿⠿⠛⠋⠀⠀⠀⠉⠋⠈⠙⠛⠙⢿⣯⣤⡀
⠀⠀⠀⠀⠀⠁⠘⠻⣟⡻⠻⠿⠿⠟⠛⠻⠿⠛⢁⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⣥⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡾⠎⣲⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⠀⠀⠀⠀⢣⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⢻⡶⢄⠀⠀⣿⠀⠀⠀⢹⣷⡦⢀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⡆⢹⠀⠙⢦⣿⡀⠀⠀⢸⢻⠈⠀⠱⣄⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⡁⠈⠁⠀⠤⣿⡇⠀⠀⢽⢈⠀⠀⠀⠘⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣿⣑⠀⠀⡇⠈⠀⠀⠀⠀⠞⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⡄⢠⠃⠀⠀⠀⠀⠀⠀⢻⡺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⡇⡸⠀⠀⠀⠀⠀⠀⠀⢸⣦⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢠⣏⣇⢱⠃⠀⠀⠀⠀⠀⠀⠀⢘⣯⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⡷⣧⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⡇⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⡁⢼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢇⢸⠀⠀⠀⠀⠀⠀⠈⡀⠜⠉⠃⠑⠊⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣥⠈⠀⠀⠀⠀⢀⣈⣸⣿⣿⣤⠤⠐⢂⠀⠀⠀⠀⠀⢘⣷⣾⠀⡇⠀⠀⠀⠀⠀⠘⣽⡤
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⡀⠀⠀⠈⠉⠉⢻⣿⣩⣔⣻⡟⠛⠚⣉⠀⠀⠀⠀⢻⣿⣄⠃⠀⠐⢉⣠⣴⣾⡟
⠀⠀⠀⠀⠀⠀⣞⡹⢷⡖⠈⠤⠤⠤⠴⠶⠦⢶⣿⣿⣟⣿⡇⠉⠉⠉⠁⠀⠈⠈⣛⡛⠛⣣⣤⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⢻⠛⠻⣷⣤⠀⠀⠀⢀⣰⣮⣷⣾⣿⣿⡿⠿⠟⢿⣿⠀⢹⡆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣾⣤⣴⣿⣯⣤⣶⣾⣿⣿⣿⣏⠻⠟⣩⣤⡀⢠⣀⠁⣰⣭⡄⠀⠁
⣶⣿⣶⣶⣭⣔⠢⣄⢄⠀⠀⣀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⡿⠿⠟⠿⢟⣛⣡⣦⣼⡿⠀⣾⣿⣄⢰⣷⡄⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⡙⠃⠀⠀⢉⣓⠀⠻⠷⢦⣬⣙⣻⣿⣟⣛⣻⣿⣿⠏⣩⣿⣿⣿⣯⣄⣠⣴⣾⣿⣿⣿⣿⣧⠀⣿⠉⠀⣸⣿⣿⣷⣶⣶⣶⣮⣡⠆
⠀⠀⠀⠀⠉⠂⠀⠀⠀⠙⠿⣶⣤⣄⡉⠙⠿⢿⣿⣿⣿⡻⠿⠿⢿⠿⠛⠋⠙⠛⠻⢿⡍⠛⠛⠛⠛⠋⠀⠘⠀⡀⢹⣿⣿⣷⠈⠙⣛⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠂⠀⠀⠀⠈⠙⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣗⡀⠀⠙⢿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⠶⠷⠶⠾⠾⣤⡄⠀⠈⠀⠙⠉⠉⠉⠈⠉⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⠠⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⣿⣶⣶⢀⣰⣶⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⣠⣀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠾⢤⣶⣾⣿⣿⣿⣦⣿⣿⢩⣬⠤⠟⠛⠡⠤⠤⠤⢶⣤⣴⣈⣋⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣴⣇⡼⡹⠋⠉⠀⠈⠀⠀⢐⡖⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠉⠉⠉⣥
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡿⢣⣿⣯⣀⣀⣀⣀⣤⣶⣿⣿⠀⠀⠀⠀⠋⠛⡄⢀⣀⣠⣰⠇⠀⡀⠀⡀⢀⣰⠟
⠀⠀⠀⠀⠀⢀⠀⠀⠉⠠⠾⢟⠡⠟⠉⠉⢹⣋⣝⣋⣭⣭⣄⣠⣄⡀⠀⠀⡃⢀⠀⠈⠀⣀⣀⣀⣨⣗⣚⣁⣀
⠀⠀⠀⠀⠀⠀⠉⠙⠲⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣈⠩⠭⡭⣽⣿⡟⠛⠻⠛⠯⠟⢲⡒⣾⢋⣈⣈⡤⠥⢀⣀⣀⠈⠐⢿⣺⣿⣿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆⠃⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⣠⣤⠤⠗⠘⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠰⠂⠀⠀⠀⠀⠀⠀⠀⡄⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠔⠛⠈⠉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠖⠚⠉⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢀⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢈⢀⠀⠀⠠⠴⡖⠂⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣤⡤⠶⠒⠹⡏⠀⠀⠀⠀⢠⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠨⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⣤⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⣤⡾⠃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣤⠴⢷⠟⠍⠒⠂⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣤⡼⣶⠉⠋⡁⠘⢂⣠⣴⣶⣾⣷⣧⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⠀⠀⡹⠄⢈⣤⣾⣿⣿⣿⣿⣿⣿⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣥⣴⣾⣿⣿⣿⣿⡛⢻⢿⣿⣿⣟⣁⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀⡀⢀⣀⢀⣶⠶⣙⣛⣛⠋⢉⣽⣭⣙⣋⣿⡶⢻⣿⣿⡍⢻⣿⣾⡿⢛⡻⢦⣶⣶⣶⣯⣁⡀
⠀⠀⠀⠀⠉⠁⠀⠀⠤⠉⣉⣠⣽⢥⣽⣿⣿⡟⢛⣿⣥⣶⣿⣿⣟⠛⠛⣩⣴⡆⠁⢬⣙⣛⣉⠙⢻⣿⡿⠷⢾⣿⣽⣛⠋⠠⠤
⠀⠀⠀⠀⠀⠐⠒⠀⠉⠉⠭⣤⠄⠠⢿⡛⠋⠁⠤⠴⠛⡋⡉⠉⠐⠃⢠⣿⢟⠁⠀⠰⣦⣭⣿⣠⠴⠶⣶⣾⣶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⠆⠀⠁⠉⠀⠀⠀⠈⠉⠉⠁⠀⠐⠲⠭⠭⠁

⣿⣿⡿⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⢠⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣸⡄⠀⠀⠀⠀⠀⠀⠀⠈⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⢄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠳⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⢧⡀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠃⡀⠀⠀⠀⠀⡞⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢷⡒⢤⠼⠯⣙⠠⠀⠀⠀⠀⠀⠠⡇⠀⠀⠼⠀⠀⢀⠰⠇⢀⣠⠉⠁
⠀⠀⠀⠀⠀⣀⣤⣀⣄⠀⣀⣀⣀⠘⣿⣦⣿⣮⣟⣓⣶⣒⣶⣀⣰⣃⣠⣄⣠⣴⡤⠽⢶⡿⡿⠁
⠀⣀⡀⠀⣶⣶⣶⣿⣟⣻⣿⢿⣿⠿⣿⡟⢻⣿⣯⠟⣭⣭⣭⠩⢭⣽⣇⣤⣿⣟⢓⣶⣶⠎⣐⡒
⠀⠀⠀⠀⠉⠂⠀⣛⣛⠉⢀⡘⠛⠓⢢⣤⣍⡙⠛⠛⠓⠛⢉⠰⣿⣿⡆⠀⠰⢆⣀⠀⠀⣀⠉⠀⠀⠨⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠓⠀⠀⠉⠀⠘⠓⠀⠀⠒⠂⠀⠀⠒
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⠟⡋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⠿⠿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣔⣈⣀⣸⣀⣈⣑⣂⣄⣀⣀⣀⣀⠀⢀⠤⢺⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣶⣶⣶⣶⣶⣶⣶⣿⣍⣉⣭⣍⣈⣁⣀⣸⣇⣀⣁⣢⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠿⠿⠿⣿⠿⠿⢿⡿⢫⣾⣿⣿⡿⠿⠟⠛⣿⠿⠿⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢛⣟⣛⣛⣛⣛⣛⣛⣛⣿⣛⣛⣻⡿⠛⠛⠛⠛⠛⠛⠃⠀⣿⠀⠀⠀⠀⠙⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣯⣭⣭⣭⣭⡩⣭⣭⣭⣭⣭⣭⣭⣯⠈⠙⣿⣿⣻⣛⣛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⣿⣿⣿⡜⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠘⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣀⡤⠚⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⡏⠀⠀⠀⣠⠶⣫⣵⢋⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇⠀⠀⠺⠿⢿⣿⣿⣿⣿⣿⢸⣿⣿⡟⠀⢀⡴⢞⣵⣿⡟⣡⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⢿⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⢸⣿⡟⢀⡴⣫⣾⣿⣿⡟⣰⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⡇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⢃⣿⢏⡴⣫⠾⠿⠛⠛⠏⠰⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣿⠀⠀⣬⣝⣿⣿⣿⣿⣿⣿⢏⣾⣵⣫⠤⣖⣾⣯⢩⠍⠀⠀⣹⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣺⡇⣠⣾⣿⠿⠿⣿⠟⠛⢛⡣⢞⣫⣵⣾⠿⣛⣩⣥⡟⢀⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗⣷⠟⠋⠁⠀⠀⢰⣿⠴⣪⣵⣾⣿⣻⣽⣶⣿⣻⡏⣼⠁⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣻⣎⣀⣀⣀⣀⣀⣺⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⠇⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢞⣛⣭⣥⠶⠶⠿⢧⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡏⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⣿⣧⣼⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡟⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⠎⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡵⠃⢀⣼⣿⣋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠊⠀⠚⠛⠛⣋⣉⣉⣉⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠤⢤⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣉⣉⣉⣤⣤⣤⣤⡤⠀⠀⠈⠉⠙⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠩⠿⠿⠿⠿⠿⠿⠿⠿⠭⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡄⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠂⠀⠀⣠⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠢⠀⠀⠀⠀⠂⢺⠁⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣷⡲⠦⡤⡿⣅⣑⠠⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢠⡟⠀⠀⠀⠠⢇⡀⢐⣨⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠘⣿⣧⣴⣶⣚⣓⠶⣖⡦⢤⣄⠀⢰⠗⠀⣤⠀⣀⣆⣠⣭⣴⣶⣾⡿⠁
⠀⠀⠀⠀⢠⣬⣭⣍⣻⣿⣷⣶⣿⣿⣿⣦⣿⣿⡿⣿⣿⣟⣛⣛⡛⠛⡛⡚⠛⣛⡛⠉⢹⣿⡿⢦⣤⣤⣥⠎⢡⠄
⠀⠐⠒⠀⠠⠿⠋⠙⢿⣷⡶⠟⠉⢽⣿⣥⣉⠛⠀⠿⠿⣿⣧⣼⣻⡿⠗⢈⣭⣿⣋⠉⠙⣛⠂⠐⠛⠛⠅⠒⠿⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠥⠄⢤⡄⠰⠿⠿⢶⣀⠀⠀⢶⠄⠀⣤⠘⠛⠛⠓⣀⠀⠉⠑⠒⠀⠀⠠⠀⠀⠀⠀⠈⠉⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⣛⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠽⠓⠒⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⢋⣙⡆⣿⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠯⠟⠛⢿⣿⢀⣀⣤⡤⠶⠿⠶⠶⠶⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣠⠴⣾⣿⠭⠭⠴⠶⠖⠒⠒⣒⣒⣒⣒⣒⣒⣛⠒⠶⠦⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠼⠷⠶⣿⠀⣿⣖⣶⣟⣿⠯⠿⠿⠿⠿⠿⠿⠛⠛⠿⠿⠿⠿⠿⠷⠄⡇
⠀⠀⠀⠀⠀⠀⡏⠍⠍⠀⠐⣀⣀⣡⢭⡇⠀⣿⠤⠶⠶⠒⠒⢛⣛⣛⣋⣉⣉⣉⣉⣉⣉⣛⣛⠛⢛⡏⠁
⠀⠀⠀⠀⢰⣒⣓⣯⣭⣭⣭⡶⠶⠾⢿⠇⠀⣿⣶⡞⡏⣿⢹⠉⡏⣟⣸⣸⣏⣿⣇⣏⣿⣹⣿⣿⣿⡧⠤⢤⣄
⠀⢀⣀⡴⠾⠙⠐⠒⠒⠒⠒⡈⢉⠉⡅⠠⠀⡿⠷⠓⠓⠛⠉⢉⣉⣉⣭⡭⠥⠾⠟⠛⠋⠉⠉⠀⠀⠀⢀⣀⣿⣷
⠀⠸⢤⡤⠬⠤⠥⠥⠬⠤⠤⠴⠶⠒⠒⠚⠉⠙⠒⠒⠚⠛⠉⠉⠁⠀⠀⠀⣀⣀⣠⡤⠤⠶⠒⠚⠋⠉⣉⣹⡿⠁
⠀⠀⠸⣇⣀⣀⣀⣀⠀⠐⠐⠐⣐⣀⣀⣀⣁⣈⣠⡤⠤⠤⠴⠖⢒⣚⣉⣉⣭⠤⠤⠶⠒⠲⣿⡿⠉⠉⠉⡿
⢀⣠⣄⣿⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠓⠒⠒⠒⠒⠒⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠢⣙⣤⠤⠀⢸⠃
⣀⣀⡈⠛⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⣀⠤⠚⢿⡷⠟⠉⢻⣿⣖⣒⡲⢶⡤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⢼⣟⣒⠚⠉⠙
⠀⠀⠀⠀⠈⠒⠢⠟⠀⠈⠛⠭⣉⣠⠞⠉⠙⠓⠺⠤⣉⣛⣮⣭⡟⠉⠻⢭⣉⣹⡍⠻⢭⣉⠶⢯⡝⠻⡄⡈⠳⠶⠖
⠤⠤⠟⠑⠒⠠⠤⠤⠄⣀⡠⠖⠦⢤⣀⡀⠀⠀⣰⠶⠦⣤⣀⡀⢀⡶⠤⣄⡉⠉⠀⢀⡤⢍⡛⠚⠁⢀⡏⠉⠓⢦⡀
⠀⠀⢀⣠⡴⠒⠂⠒⠒⠤⢄⡀⠀⠀⠀⠉⠉⠉⣀⡠⠔⠢⠤⢭⣉⡀⠀⠀⠈⠙⠛⠛⠀⢀⡨⠽⠿⠭⢄⣀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢛⣋⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠽⠒⠒⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⣉⣉⣧⣸⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠛⢻⣿⣀⣤⡤⠶⠖⠛⠒⠒⠒⠒⠒⠒⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢰⠖⢾⠿⠷⢒⣒⣚⣛⣛⣉⣉⣭⣭⣭⣭⣉⣙⣛⣒⡒⠶⣦
⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠴⠖⢛⠛⣿⠀⣺⠯⠿⠯⠿⠓⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⡿
⠀⠀⠀⠀⠀⢸⣓⣐⣀⣠⠥⠤⢴⣒⣿⡀⣿⢒⣛⣋⣉⣉⣭⣭⡥⣤⢤⢤⢤⢤⣤⣤⣭⣭⣭⣿
⠀⠀⠀⠀⣿⠽⠶⠶⠶⢾⣟⣛⣛⠿⠇⠀⣿⡏⡇⣇⣻⣸⣴⣧⠧⠿⠼⠼⢾⣼⣼⣿⡿⠿⠿⠛⠛⠓⣦⡀
⠀⣠⠶⠏⠉⠩⠉⠉⠅⠀⠠⠀⢂⣐⣀⣠⣿⣉⣉⣉⣀⣤⠤⠴⠶⠒⠛⠛⠉⠉⠁⠀⢀⣀⣀⣤⡤⢤⣯⡿
⠀⠈⢻⠛⠛⠛⠛⠛⠋⠉⠉⡉⡉⡉⠀⠠⠠⠉⠉⠀⠀⠀⣀⣀⣀⣤⠤⠴⠖⠒⣛⣉⣩⣥⣠⠤⠤⢾⠟
⠀⠀⠘⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⣒⣒⣒⣚⣫⣭⠭⠭⠤⠶⠒⠒⠛⠉⠉⠉⣅⢻⡟⠁⠀⢀⡟
⠐⠚⠳⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠋⠀⢸⠁
⠒⢒⣢⣼⣗⡶⠦⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣤⣠⠤⠤
⠚⠁⠀⠙⠢⣄⣀⡞⠛⠶⣖⠶⠋⢩⠿⠿⢭⣿⠛⠲⢶⣒⠒⡶⠶⣖⠒⠖⢶⠶⣖⠒⢒⣾⡿⢽⡟⠓⢤⣀⣀
⡀⣀⡶⠤⢄⣀⣀⠀⠀⠀⣠⣍⣛⠉⠀⠀⠀⢀⣭⣉⡒⠛⠋⢁⣄⣈⠙⠒⠚⠀⢈⡙⠾⠭⠟⠀⢷⠒⠦⣀
⠀⠀⠀⢀⣠⠤⠤⠭⣍⡉⠁⠀⠈⠙⠒⠒⠒⠊⢀⣀⣉⣙⠒⠚⠀⠈⠉⠒⠦⠴⠟⠈⠑⣒⣦⣖⠋⠀⠀⠈⠳
⣠⠴⠞⠋⠁⠀⠀⠀⠀⠈⠉⠒⠠⠤⢤⡤⠴⠚⠉⠀⠀⠀⠉⠉⠒⠒⠤⠄⣀⣀⣤⠴⠊⠀⠀⠀⠈⠉⠒⠢⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠚⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡀⣠⡤⠴⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠶⠒⠛⠋⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣰⣄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⢴⡶⠒⠚⠋⠁⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⠚⠋⠉⠁⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣻⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⢀⣀⣴⠶⠀⠈⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣀⣤⡤⣤⠗⠋⠀⠀⠀⠀⡹⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠒⠛⠉⠉⠀⢀⡿⠀⠀⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⢷⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡇⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠰⠀⠀⠀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⢡⠀⣠⡾⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡠⢔⣈⣴⠏⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣠⣤⠶⣋⢁⠀⣽⠃⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⡤⣤⠶⠎⣻⡁⠈⠁⣀⣠⣤⣸⠋⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠆⠠⠶⠟⠛⢹⡁⠀⡠⠒⢁⣤⣶⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⡀⠼⠓⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣬⣴⣶⣿⣿⣿⣿⣿⢟⣉⠹⠟⣿⣿⣿⣏⣀⡤⢤⣤⣄⣀⣤⡤⠀⣀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⣐⠶⠶⣙⣛⣛⠋⠉⣩⣯⣄⢛⣋⣿⣧⠙⢿⣿⣿⡍⢻⣿⣿⣿⠝⢛⠻⣦⣶⣶⣶⣿⣍⣀⡀
⠀⠀⠀⠀⠀⠛⠉⠀⠀⢀⠘⠛⠉⣨⣿⣋⣩⣿⣛⣷⠞⠻⣿⣥⣤⣾⣿⣟⡻⠿⠟⢛⣡⣦⠉⠀⣍⣛⡛⠛⠛⠿⣿⣿⣿⣶⣮⣿⣿⣻⠟⠉⣠⣄
⠀⠀⠀⠀⠀⠀⠤⠶⠄⠚⠒⢚⣛⣉⡁⢈⣿⡿⠿⠛⢁⣉⣹⠾⠿⠛⠛⣋⠡⠆⠈⣿⣿⠟⠀⠀⣮⣉⠛⣿⣀⣀⣈⣭⣤⣴⣦⡌⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠡⠤⠉⠒⠒⠂⠀⠀⠀⠐⣶⣶⣶⣤⣄⡀⠈⠭⠓⠃⠀⠀⠈⠛⠿⠿⠒⠂⠀⢌⣛⡛⠛⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣿⡿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⡐⠊⠀⣿⡀⠉⠒⠀⣀⣀⠀⠀⠀⠀⠀⢀⠔⡷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡉⠉⣉⣉⠁⠊⠁⢀⣇⠀⠑⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣿⣿⣿⣿⣿⣿⣭⣭⡍⢽⣭⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠶⠯⠭⠭⠤⠤⠤⣿⠤⠬⢭⣥⣾⠿⠿⠯⠭⠤⠤⠀⣸⡇⠀⠈⠉⠛⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣟⣛⣓⣲⡛⢷⣶⣶⣶⣶⡦⠴⠤⠤⠙⠧⠤⠤⠤⠤⢤⢿⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣮⠻⣿⣿⣿⣿⣿⣿⣿⠂⠀⠘⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⢿⣿⣿⣿⣿⣿⣿⣷⠹⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠉⠙⠛⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⡴⣚⡍⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡏⠀⠀⢀⡴⣛⣵⣾⢋⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⢀⣉⣩⣽⣿⣿⣿⣿⣿⢸⣿⣿⡟⠀⣠⠾⣫⣾⣿⡿⢡⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣻⠇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⠏⣠⢞⣵⣾⣿⣿⡟⢠⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣺⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⢁⡿⢫⠾⠱⢛⣛⣉⣭⡍⠀⢻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⣽⠇⠀⣠⣶⣾⣿⣿⣿⣿⣿⡿⢣⣾⣵⠥⣒⣯⣽⡷⢦⡆⢀⠀⣠⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢭⢯⢁⣼⡿⠿⠛⢻⣿⠋⢉⡻⢕⣋⣵⣾⡿⢟⣫⣵⡖⡾⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⡟⠋⠁⠀⠀⠀⣸⡿⣞⣥⣾⣿⣿⣯⣷⣾⣿⣿⣿⣸⠃⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣛⣿⣅⣀⣀⣀⣀⣀⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡏⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢚⣋⣭⣥⡶⠒⡒⠒⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡟⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡷⣼⣿⣤⡿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡞⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⠞⠀⢠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡳⠋⢀⣴⣿⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠊⠀⠔⠛⠛⠛⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣈⣽⣿⣿⣿⣿⣟⣛⡛⠛⠛⠛⠛⠛⣉⣁⣀⣀⣀⣀⣀⠀⠉⠙⠛⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⢻⣿⣿⣿⣿⡿⠿⠿⠿⠭⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠟⠱⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢢⣀⠀⠀⠀⢸⡄⠀⠀⢦⣄⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⢻⡓⢤⡀⢸⡇⠀⠀⢸⣿⢗⠢⡀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢹⠀⣇⠀⠙⢿⣇⠀⠀⢸⠫⠀⠁⠈⣣⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠀⠀⠀⠐⣽⡟⠀⠀⢸⠠⠀⠀⠀⠀⢩⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⢹⣧⡂⠀⡆⠀⠀⠀⠀⠄⠉⣿⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⢇⢠⠁⠀⠀⠀⠀⠀⠀⢸⡚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡼⣵⠼⣼⠀⠀⠀⠀⠀⠀⠀⢸⣷⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⣛⡏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠆⠀⠀⠀⠀⠢⠀⣿⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣧⠀⠀⠀⠀⠀⠀⠄⠀⡤⢤⠢⣀⠄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠐⠮⣿⣿⡀⠀⢀⠠⠀⠀⠀⠀⠀⢠⣿⣧⢸⠀⠀⠀⠀⠀⠀⠔⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠄⠀⠐⠺⠷⣿⠿⣋⠭⣷⣶⢦⠒⠀⠀⠀⠘⣿⣗⠀⡆⠀⠠⠖⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⡴⣿⣡⡤⢚⣁⣀⣀⣀⣤⣈⣿⣿⡿⣾⣗⠒⠂⠚⠂⠀⠠⠤⠾⠿⠿⢄⣀⣤⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠛⠐⠚⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⡿⢿⣿⣀⠀⠀⠀⠀⢠⣴⣯⣵⣶⣿⣿⣿⣿⣿⡟⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣨⣇⣀⣿⣟⣀⣤⣴⣾⣿⣿⡟⣿⡿⣛⣁⠀⠀⠙⠁⣔⣋⠀⠐
⣤⣶⣤⣤⣀⠠⢄⠀⡀⠠⠀⠀⠀⢀⣶⣶⣶⣾⣿⣿⣛⣿⣿⣿⣿⣿⣿⡿⣿⠿⣶⡾⠿⣉⣁⣼⡇⢈⣿⣏⡀⣴⡄
⣿⣿⣿⣛⠛⠍⠀⠉⠒⠦⠸⠷⣤⣈⣙⠿⠿⠿⠿⠿⣿⣿⠿⢛⣽⣿⣿⣷⣀⢠⣤⣽⣿⣿⣿⣿⡇⢸⡟⠋⢁⣿⣿⣶⣦⣤⣤⣔⣂⡤
⠉⠁⠈⠉⠑⠀⠀⠀⠘⠿⣶⣤⣄⠈⠛⠿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⠛⠻⠿⠿⣯⡛⠻⠿⠿⠟⠃⠈⠓⢀⠘⣿⣿⣿⡌⠛⠻⠿⣿⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠦⠄⠀⠁⠀⠈⠛⠿⠷⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⣟⠂⠀⠘⢷⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠒⢄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠑⠄⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⡠⠧⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⢠⢀⣀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠄⡶⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡏⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⣁⣀⡀⠤⡔⠂⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡤⠴⠒⠂⡿⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⡤⠦⣾⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣠⠄⢴⠚⠛⢁⣀⣸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠰⠚⠋⡇⠀⢀⣠⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣐⣊⣤⣾⣿⣿⡿⢻⣿⣿⣿⠋⠀⠀⡀
⠀⠀⠀⢀⡀⠀⠀⣀⣀⢀⣀⣀⣠⡀⣩⣛⡛⢉⣭⣭⣙⣽⡶⠹⡿⣿⣙⣿⣿⣟⣛⡶⣾⣷⣯⣁
⠀⠀⠀⠀⣀⣀⠰⠤⠤⠾⠓⣾⣿⡿⠭⠿⣶⣿⡿⠿⡉⢩⣾⡇⠀⡲⠯⣥⠉⠛⢛⣛⡛⠿⠿⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠤⠛⠒⠂⠈⠉⠤⣶⣦⣄⡀⠚⠟⠀⠀⠙⠿⠿⠊⠉⢙⣛⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠠⠆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠏⠀⠉⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡆⢿⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠸⠀⠀⠂⠀⠉⠉⠀⠀⠀⠠⠖⠠⠞⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⣀⣤⣤⣶⣄⠀⠈⠢⡼⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⣴⡾⣿⢹⣇⡉⠉⣄⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣥⠶⠗⠛⠋⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠊⠀⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⣶⠆⠀⠀⠀⣶⠤⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⢸⠀⠀⢀⣤⣶⣫⣤⠼⣧⣬⣁⣸⣆⣠⣤⣬⣧⣼⣧⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⣾⣠⡶⠟⢛⣛⣤⣿⡤⠿⡷⠛⠛⠋⠀⠀⠀⢸⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣠⣿⠵⠖⠚⠋⠉⠀⠀⠀⠀⠁⠀⠀⢀⠀⠰⡆⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⣐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣤⣼⣿⣄⣙⡄⠉⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⣠⡄⣦⠀⢸⣿⢿⣿⣿⣿⣿⣯⣍⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣴⡞⠋⢻⡇⣻⠿⣯⡰⢿⣻⣿⣿⣿⣧⣭⡛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠈⠡⠀⢀⣿⡿⣯⢶⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣵⣮⠗⢛⣿⣿⠿⣿⣿⣿⣿⡿⠿⠿
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣯⠀⠀⠀⢀⠀⠀⠙⣿⠏⠀⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠶⢤⣤⡶⠴⠖⠒⠲⢦⣶⣤⣭⡿⠿⠿⠁⣤⣤⡀⠀⢀⡐⣦⣿⠀⠀⣄⠘⠃⠲⡶⠖⠀⠀⠀⠀⠐⠶⠶⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣴⣾⡖⣲⣶⣶⣤⣤⣤⡄⠀⠰⢶⣻⣽⡯⠉⣩⣾⣿⣿⢃⠀⠻⣿⣿⣿⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠃⣤⣭⣉⠛⠷⠀⣠⠼⢿⣿⠇⠀⠸⢿⠿⣋⡘⢿⣷⣤⠉⠉⠉⠀⠀⠀⠉⠀⠲⠆
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠘⠛⠛⠀⠈⠀⠀⠀⠀⠚⠋⠙⠃⠀⠙⠿⠧⠶
⠀⢀
⠀⠨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣽⠳⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠛⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠆⠃⠀⠀⠀⠈⠁⠀⠀⠀⠂⠐⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣀⣤⡶⣿⠻⠷⠀⠀⠙⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣚⣹⡭⠧⠟⢳⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⠀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⣀⡤⠰⡞⢛⠟⢀⠀⠀⠻⡉⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⣀⣴⡾⠷⢻⡉⣿⣗⣶⡾⠒⠛⠚⢻⡿⠓⠒⠒⠒⠒⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣸⣟⣭⠴⠖⠚⠉⠉⠁⠃⠀⠀⠀⠀⡄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣴⣶⣿⣤⣽⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⢀⣀⠤⣶⢹⣤⡼⢛⣽⣿⣿⣿⡷⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡰⠛⠁⢀⣿⡼⣫⢿⣬⣷⣿⣿⣿⣿⣷⣦⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣼⣶⣾⣯⡾⠷⣿⣿⣿⣿⣿⣿⡿⠿⡇
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⡇⠈⠉⢀⠀⠙⢿⠟⠀⠂⠀⠂
⠀⠇⠀⠀⠀⠀⠰⠶⣤⡶⠖⠒⠓⠲⢶⣦⡯⠿⠿⠋⢤⣦⣀⣀⣀⣲⣾⡀⠠⣄⣛⡐⠶⠖⠀⠀⠀⠐⠒⠓⠒⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠶⠿⢏⣽⠿⢶⣶⣶⡆⠀⠚⣯⣾⠇⠠⣿⣿⣿⢡⣀⡛⠿⣿⣥⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⢿⡶⠌⢩⣤⡞⠫⠟⠁⠀⢉⣻⢵⡎⠻⣿⣦⢀⡀⠀⠀⠀⠀⠈⠉
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⠀⠈⠉⠙⠲⠀⢠⣧⣼⣤⣾⣏⣠⣦⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⡠⠖⠀⠀⠀⢹⠀⠀⠀⠀⣷⣤⠤⡻⣿⣿⣿⣿⣿⣇⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣦⣞⣉⣀⣀⣀⣀⣀⣤⡧⠴⠖⠛⠉⣀⡠⠚⣿⢿⣿⣿⣿⣿⣧⣾⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣹⣦⣤⣭⣭⣭⣭⠍⠀⠀⠀⠀⠀⢀⣰⠏⣠⣤⠄⢀⠉⢛⡉⠃⡉⢹⣷⣤⣾⣷⡆⠀⠂
⠀⠀⠀⠀⢀⣤⣈⣭⣤⣭⣭⣭⣽⣿⠟⢋⡳⣾⣿⣿⠿⠛⢿⣲⡶⠾⠿⠻⢿⣷⣿⣾⣿⣧⣿⣿⣷⡙⢿⣿⠋
⠀⠀⠀⠀⠈⠉⠩⣽⣛⣿⣭⣥⣶⣶⣾⣿⣿⡿⠛⠁⠴⣶⣮⡝⠛⠂⠀⠰⣶⣮⣙⠛⠻⠿⣿⣿⣿⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣠⣤⣀⣀⣠⢶⣶⣶⣶⣿⠶⠯⠅⠀⠀⠀⠾⠿⠿⠿⠃⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠃⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⣷⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠤⠒⠊⠉⠀⠀⠀⣸⣄⣰⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣰⣶⠐⠂⠉⠁⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⠀⠶⠛⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡤⠌⠀⠀⠀⠀⠀⢀⡟⢀⡤⠶⠒⠚⠉⠉⢁⠈⢣⡀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⣄⣀⣵⠶⠞⣧⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠙⠲⠶⠆⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠎⠁⠀⢩⡶⠶⠛⠀⠀⠀⠀⣀⣠⣤⠴⠿⠷⣀⣀⣀⠚
⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠒⡈⠀⠀⠀⠈⡇⠀⠰⣀⠄⠒⠉⢹⣈⣿⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠉⠀⠀⢰⠀⠀⠀⠀⠀⠀⢻⠀⠹⠅⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⢤⣴⣸⣆⣀⣴⣦⡠⠲⠚⣲⣾⣿⣿⣿⣿⣿⡷⣾⣅⣭⡛⠙⣿⣿⣷⣄⡀
⠀⠀⠴⣶⣤⣄⡀⠀⢳⣶⢊⣩⡽⣿⣶⡿⣿⣿⣿⣛⡻⠿⠟⢻⣬⣿⣿⣿⣿⡆⣿⣿⣿⣇
⠀⠀⠀⠊⠉⠙⠛⠿⠶⠿⢛⠉⠋⠹⣍⠠⠤⠄⠀⠤⠶⠶⠾⠿⣿⣟⣿⢿⣿⠃⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠂⣏⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠠⠀⡀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠟⠲⢄⡀⣽⡶⠄⠀⠙⣟⠀⠀⠀⠀⠀⣠⡄⢀⡀⠤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣇⠀⠀⠀⢹⡀⠀⠤⠤⠒⠺⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⠑⠦⡀⠈⣧⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢤⠖
⠀⠀⠀⠀⠀⠀⠢⢽⣇⣠⣤⠀⠀⠈⣇⠀⠉⢢⢸⡄⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢀⠔⠇
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣷⣰⣶⣯⣤⡆⣸⠀⢷⠀⠀⠀⠀⠀⢹⣀⣠⣤⠴⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀⠊⠩⠷⠸⠆⣰⡄⢆⣤⠀⣠⣖⣫⣿⣭⣵⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡗⠈⠃⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠠⢤⣼⣿⣿⠿⠿⠿⢗⣂⣆⣤⡶⡶⣿⠿⢴⡷⠝⠛⠻⠚⢻⠆
⠀⠀⠀⠀⠀⠀⢤⡤⠿⠿⠯⣤⣴⠤⠼⢯⣑⣞⠻⢿⠮⢛⡿⠭⠯⠗⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⢀⠀⠉⠀⠀⠈⠁⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⡟⠻⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠹⢿⣷⡀⣀⣀⣤⣤⣿⠶⠶⢠⡄⠀⢀⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⣤⠀⠛⣿⣿⠋⠀⠀⠁⠀⠀⢀⣿⡃⠙⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠂⣀⣉⡁⠀⠀⠀⠀⡀⠀⠈⢿⡖⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⣀⣡⣍⣿⣾⣷⣺⣿⣿⣿⣛⣳⠈⠁⢡⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣿⡟⠙⠫⣯⠙⠙⠈⠻⠟⠀⠀⠋⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡏⠀⠀⢘⣿⡆⠀⠀⠀⠀⠀⡼⠻⠞⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣇⠀⠀⠀⡀⠁⠀⠀⢸⣧⢷⡆⠀⠀⣾⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⡄⢀⣁⡷⡶⠀⢚⠛⢻⣎⣿⡄⠠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠂
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣹⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠟⠃
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡟⠁⠉⠛⠳⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⠿⠯⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡆⠀⠀⠀⢠⣀⣤⡤⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⢴⡖⠈⠃⠛
⠀⠀⠀⠀⠀⠀⣦⣠⣴⡾⠵⢿⡄⠀⡀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊
⠀⠀⠀⠀⠀⢀⠟⠉⠘⠒⣴⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣄⠀⠀⠀⣀⣀⣠⣤⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⡄⣀⠤⢺⣿⡌⠣
⠀⠀⠀⠀⠀⣻⢀⣀⠴⣶⣟⣛⣋⣛⣋⡴⠚⢻⣿⡇⠠⡄⢀⣈⡗⠂⠀⠀⠀⠀⠀⢹⡿⠻⣻⣿⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠠⣿⣟⣫⣄⣱⣼⠟⠛⠭⠃⠄⢦⢤⣦⣶⠶⠟⠛⠁⠀⠀⠀⣀⡀⠀⠀⠞⠴⢄⠀⠉⢢⡀⠀⡀⢰⣆⠀⠈⣦⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⠀⠀⠈⠃⣿⣸⣟⣿⣆⣀⣀⣤⣤⣴⡿⠋⢿⣧⠠⠀⠀⠈⢳⣤⣾⣷⠷⠿⠊⠉⠄⠀⣿⣿⣀⡠⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣾⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠁⠀⠀⣺⣿⣠⠀⠉⠳⢼⣿⣻⡇⠀⠀⢀⣀⡀⡸⠛⣛⣻⡿⡶⠖⠛⠉⠉
⠀⠀⢲⣶⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⢼⣿⣷⣶⣶⣾⠿⡶⠦⠜⣿⣗⣞⣛⣛⣿⣿⣻⣿⣶⣾⣿⣿⣿⣿⣿⣿⠏⠂⠒⠃
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡀⠀⣠⣐⣷⡀
⠀⠀⣒⣛⣓⣚⢒⣾⣿⣿⢿⡟⢿⣛⣿⡽⢋⣉⣠⣴⣠⣉⣉⡩⣄⠩⠻⠿⠿⠷⠉⠉⠿⠉⠍⠋⢍⠛⠁⠀⠈⠉⠉⠉⠃⠘⠋⠙⠉⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠉⢍⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣷⣶⡖⠂⠘⡓⣤⣤⣀⡄⠀⢀⣤⣴⣶⣦⣧⣤⣬⣥⣌⣀⡀
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠀⠐⠓⠤⠬⣬⠈⣙⣉⣩⣽⣿⣿⣿⡿⢿⣧⣽⢿⡿⣿⣿⣿⣿⣿⣤⣿⣷⣼⣬⣿⣯⣹⣷⡶⠶⣤⠴⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠚⠛⠾⢕⣯⣼⣦⢮⣶⣤⣾⣿⠲⢿⢦⡉⣉⠩⡉⠈⠉⠉⠻⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠛⠉⠛⠓⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⣶⣼⠿⠛⠂⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢶⣼⡿⠟⠃⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆⠀⠀⢰⣦⣽⡙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠆⠀⠀⠀⠀⠑⣦⣿⡈⠙⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢸⠃⠠⣄⡀⠀⠀⠀⡤⠀⠀⠀⢸⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⡆⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠁⢀⣀⢀⢀⡀⠰⠒⢾⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡇⢄⠹⣷⣄⠀⠀⠀⠀⠀⠀⣘⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⠏⠀⠠⣜⣾⣿⣿⣿⣷⠘⠻⢿⣿⣿⣿⡇⠈⢧⠹⣿⣷⡄⢀⣀⣠⣜⣻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠿⣿⣹⣿⣿⣿⣷⣦⡥⠄⠘⣧⠛⠛⢻⣨⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠰⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⠁⠘⣿⣿⣿⣿⡿⢶⣄⢸⣧⡠⠊⣹⣿⣿⡿⢃
⠀⠀⠀⠀⠀⠀⠘⡇⠀⣰⣦⣶⣶⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣆⣹⣿⠿⢿⡇⠀⣹⣿⣿⣤⣾⣿⣿⠟⢁⡟
⠀⠀⠀⠀⠀⠀⠀⢻⡀⣽⣟⣿⣵⣯⣿⣿⣯⣿⠏⣿⠏⠈⠻⠃⠛⣷⣾⣤⣼⣷⣿⣿⣿⡿⠛⠛⠉⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⢨⣿⣿⣿⣼⣿⣿⣿⣿⣷⣿⣶⣶⣀⣀⣶⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣠⣏⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⢟⣿⣿⣿⣿⣶⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⡭⠭⠯⠥⠥⠭⠤⠼⠾⠛⠛⠛⠛⠛⠛⠻⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠈⠀⠀⠈⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⢀⡞⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⣠⣯⠾⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢃⡞⢀⠾⠋⠁⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣾⡶⠉⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣻⡟⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣱⡟⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠈⡜⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⡜⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⡞⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⢀⡞⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢀⡞⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠛⣡⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣠⠞⢁⠀⠀⠀⠀⢰⣤⠏⠀⠀⠀⠀⠐⣄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣠⣾⣿⣟⣭⡤⠴⢖⣾⣯⣁⣤⣀⣀⣀⣀⡤⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟
⠀⠐⠛⠛⠉⠭⠤⢤⣤⣤⣮⣤⣤⣤⠄⢠⡐⢲⠽⣟⠒⠂⠀⠀⠀⠀⠤⠤⠤⣤⣴⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠝⠉⣙⣿⣶⣦⣤⣅⣈⣓⣒⠢⠤⠤⠤⠤⠤⠤⣙⣀⡤⠇⢀⡄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⠄⠀⠙⠻⣿⣍⠉⠉⠿⠛⠻⢿⣿⣿⣭⣤⡴⠶⠋⢀⣨⣥⠶⠚⠋⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠄⠀⠀⠈⠻⢿⣶⠶⠂⠀⣀⡿⠙⠻⠷⠶⠶⠞⠋⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠒⠒⢒⣿⡀⠤⠤⡇⠀⠀⠀⢀⣀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣳⣤⣤⣷⣴⣶⣾⣿⣿⣿⠁
⠀⠀⠈⠀⢀⣀⣀⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣋⣯⣟⣿⣆⠀⠀⠀⠀⠐⠦⠄⠐
⠀⠀⠠⠴⠶⠶⣾⣟⣛⣛⣿⣿⣿⣿⣏⣽⣿⠟⠿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠉⢉⡉⠁
⠐⠚⠛⠿⠷⠶⠾⣟⠛⠛⠹⠛⠿⣷⣿⠟⢻⣿⣿⡿⠿⠽⠛⠛⢛⣖⣿⢿⣿⡟⠉⠉
⠀⠀⠉⠛⠛⠀⠰⠦⢭⣍⣉⣁⠀⠈⠛⠋⠬⠽⣿⠅⠀⠒⠒⠒⠒⣂⣉⡭⠉⠉⠀⠒⠒⠒⠢⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠛⠁⠀⠈⠉⠑⠒⠀⠀⠀⠉⠉⠓⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠙⠛⠶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠛⠛⠛⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⠀⢿⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣿⠀⠘⣇⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⠀⠀⢻⡀⢻⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣿⠀⠀⠸⡇⠈⢧⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿⠀⠀⠀⣧⠀⠸⡆⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢻⠈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠈⡇⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣧⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡏⠀⠀⠀⣿⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠇⠀⠀⠀⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⡆⣿⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⡇⣿⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣀⣀⣠⠄⠀⢀⣀⣧⣿⣿⠴⠶⠶⢤⣤⣀⣀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⣯⠭⢷⣶⣾⣿⠯⠽⠶⠿⠚⠛⠛⠛⠉⠉⠉⣉⣹⣿⣿⣿⣿⡿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡥⠶⠒⠋⠉⠁⠀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿⠿⠟⠛⠉⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣶⣶⣶⣾⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠀⣀⣀⣀⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠉⠉⠉⠉⠉⢀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠴⠶⠶⠒⠚⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⡀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠒⠒⠒⣿⡂⠤⠤⣯⠀⠀⠀⠀⣀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣦⣤⣼⣴⣶⣾⣿⣿⣿⠏
⠀⠀⠈⠁⢀⣀⣀⡀⠙⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣋⣯⣽⣻⣇⠀⠀⠀⠀⠀⠲⠄⠐⠒
⠀⠈⠡⠴⠶⠶⣾⣟⣛⣛⣿⣿⣿⣿⣟⣽⣿⠟⠿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠈⠍⣉⠁
⠒⠚⠛⠿⠷⠶⠾⣟⡛⠛⠫⠛⠻⣷⣾⠿⢛⣿⣿⣿⠿⠯⠟⠛⢛⣷⣛⡿⣿⣿⠉⠉⠁
⠀⠀⠉⠛⠛⠀⠐⠦⠭⣍⣉⣁⠀⠈⠛⠋⠥⠽⣿⡧⠀⠐⠒⠒⠒⣒⣀⡭⠉⠉⠀⠐⠒⠒⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠛⠃⠀⠀⠉⠑⠒⠒⠀⠀⠈⠉⠙⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢷⠀⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣦⠀⢿⠏⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣽⡄⠈⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⢨⣿⡄⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠄⠐⠒⠀⠈⠉⠀⠀⠀⢸⣃⣾⠟⢳⡄⠀⠐⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⣿⠄⠀⠀⢀⡄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢿⢸⡆⠀⢫⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣶⠾⠿⠯⣿⢿⣦⣀⣀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠃⠒⠒⠶⠤⠤⣤⣴⡇⠀⣠⣾⣦⣴⣯⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠉⠙⠿⠿⢯⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⡿⠰⢿⢰⣶⠤⣤⣤⣤⣀⣀⡄⠀⠀⢀⣀⣀⣤⠤⡶⠃
⠀⠀⠀⠀⠀⠰⣠⣰⣶⣿⣿⡏⠉⠉⠀⠀⠀⢀⡀⣀⣒⣒⡐⠺⣥⡿⣷⠶⠞⠛⠫⠉⠁⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⡿⠿⣿⡤⣦⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠨⠽⠷⠶⣾⣧⠠⠔⠒⠤⠄⣀⣀⣤⠤⠀⠀⠤⣤⣤⣤⡴⠛⠛⠁⢠⣤⠤⠤⠄
⠀⠀⠀⠀⠈⠙⠛⠚⠛⠋⢉⠉⠀⣀⣀⣀⣀⣉⣉⣉⣀⠀⠀⠀⠀⠀⣉⣡⣤⣄⡓⠛⠉
⠀⠀⠀⠀⠀⠀⠠⠔⠛⠋⠉⠀⠈⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢸⡇⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠃⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⢀⡼⣃⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠇⢀⠀⠀⣠⣾⠿⠋⠁⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢠⡟⣠⡾⠋⠁⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡟⣹⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣼⡟⢀⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⠜⡿⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⡼⠁⠈⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡼⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⡾⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⡾⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⢉⣴⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠉⠁⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢠⡀⣰⠏⠀⠀⠀⠀⠀⢀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⣀⣴⣋⣡⡤⠀⠀⠀⢀⣀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠘⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⣠⣾⣿⣿⣿⡷⠶⠶⠚⠛⣹⠿⣷⣶⣶⣦⣤⣤⣤⣤⣴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠁
⠀⠀⠚⠛⠋⠡⠤⠤⢤⣤⣤⣤⣾⣥⣤⣤⡤⠤⠤⢶⠒⠒⡆⢿⣋⠛⠒⠒⠒⠀⠀⠠⠤⠤⠤⢤⣤⣴⣶⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡹⠯⠛⠻⣷⣦⣤⣅⣈⣁⠘⠳⠦⠬⢥⣤⣀⣀⣀⣀⣀⣐⠺⠗⠀⣀⡵⠀⠀⡈⠉⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡟⠋⡉⠙⠛⠿⣿⣟⡛⠛⠲⠶⢶⣶⣶⣶⣶⣶⣶⣖⣚⣋⣉⣠⣤⠞⠀⠤⢒⣋⣭⣤⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠈⠻⣿⣦⣄⡀⠋⢀⡀⠈⠹⣿⢿⣿⣿⣛⣛⣉⣁⣤⣴⠾⠛⠉⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠀⠀⠀⠀⠈⠙⠿⣯⣍⣉⣀⣠⣴⠟⠀⠉⠙⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡨⠦⠡⡀⠀⠀⢢⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠢⣢⠄⠀⢰⠦⠀⢀⢀⡝⣄⠛⠐⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⡇⠀⠀⠀⠀⢀⠃⠀⠀⠀⠈⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠤⠄⠈⠁⢠⡐⢄⡈⠀⠀⠀⠀⢸⡎⠙⢢⡐⠄
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⡀⠀⠨⠈⠀⠤⠤⠤⠤⠄⠀⠆⢸⡦⣕⢄⠀⠀⠀⠸⠀⠌⣲⠉⠈⠀⠄
⠀⠀⠠⠤⠀⢀⣀⡃⠀⠀⠀⠀⠀⢰⠄⠀⠠⠤⠤⠤⠤⠤⠤⠄⠄⠀⡌⠩⣀⠷⣥⠄⠀⡆⠀⠀⡆
⠀⠀⠐⠠⡀⠀⠀⠀⠈⠉⠉⠐⠂⠒⠤⠤⣄⢀⣀⠀⠀⠀⠀⠀⠠⠀⠉⢄⠈⠳⡜⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠑⢌⠉⠉⠉⠉⠁⠐⠒⠒⠂⠤⠄⣉⡉⠓⠂⠤⣀⡀⠘⢀⠨⠑⢄⠈⠢⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⠌⣉⠑⠢⠬⣠⠑⢄⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠉⠐⠂⠠⠤⢈⣁⠒⠠⢍⡓⠷⣤⡈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠈⡉⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⠬⠉⠒⡂⠭⠒⣩⠶⣌⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠁⠄⣐⠒⠠⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣆⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⣀⣘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠋⣃⢸⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⡐⠴⠖⣲⠚⠋⠋⠛⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⠨⢯⣀⣀⣤⣤⠤⠶⠞⡒⡉⡩⠻⠢⠊⠉⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⣀⣀⣀⣀⡀⡁⠀⠀⠀⠩⠛⠉⠀⠀⢀⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⢀⡀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣴⡤⠄⣀⡀⠀⢇⠀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢄⠈⠴⣿⠖⠂⠈⠉⠑⠊⣿⣿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠘⣿⡃⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠸⠟⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⢿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢋⡁⠀⠀⠀⠀⠀⢠⡴⠒⠀⠠⡿⠿⠿⠿⠿⣿⣿⡟⠉⠀⢀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡠⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⢢⡀⠀⠀⠀⣿⣿⣇⠘⠷⡼⠋⢻⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⠀⠀⠀⣁⣀⣀⣠⣀⣘⣶⡄⠀⠀⣿⡿⣟⡦⣸⡧⡶⢸⡿⠛⣋⣉⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡴⠾⢿⣷⢀⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⣱⣶⣿⣿⣯⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠜⢧⡤⠐⠞⠋⠉⠉⠉⠁⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠙⡿⢻⣿⣿⡎⣻⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⣧⠘⠛⠛⠛⠛⠛⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠀⠀⠶⠀⣀⣀⡀⠀⣠⣄⣀⣀⣀⡀⠀⣸⣆⡀⠀⠙⡄⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⣴⠖⠚⢙⠿⣿⡇⠀⣸⡟⣿⣿⣿⠿⠿⠿⠿⠧⣤⣀⣵⡀⠀⠀⠀⠘⠛⢿⣿⣿⡇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡿⠃⠀⠀⠐⠚⠛⠋⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠈⠁⠀⠀⠀⠀⠀⠀⠹⣿⣧⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠂⠀⠀⠀⣤⠄⠀⠀⠀⢹⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⣀⡀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠉⣀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠻⣎⡄⠀⠀⢱
⠀⠀⠀⠀⠀⠙⢦⠀⠀⠹⣆⠀⢀⣠⣶⣾⣋⠁⣸⡇⠀⣿⣿⣿⣿⣿⠟⠛⢛⣷⣦⣶⣿⣄⠀⢀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⣣⡀⢀⡙⣿⣾⣿⣿⣿⡿⡿⠿⠧⠴⠿⠿⠿⠷⠶⠶⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋⠉⣻⠇
⠀⣀⡤⠶⠚⠛⠉⠉⠙⢦⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⣀⣀⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠛⠂
⠉⢹⡇⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⢀⠔⡡⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠛
⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⢀⡀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡐⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟
⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣈⣿⣿⣷⣄⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠂
⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⢠⣶⣟⢻⠟⠻⣿⣿⣇⣬⡃⠀⢿⣶⣤⡀⠀⠀⠀⢠⣤⣾⣏
⠀⠀⠀⠀⠈⢷⣄⢠⣇⣴⣿⡿⢫⢻⣶⡿⠛⠋⢹⣿⡰⢆⣬⠛⢿⣿⣶⣄⠀⠸⣿⣿⠋⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⣉⡶⠚⠻⣯⡀⠀⣤⣸⣿⡇⠀⠈⠙⠢⣬⡛⢿⣿⣦⣿⡇⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡠⢞⡷⠞⠁⠀⠀⠀⠈⢿⣀⣸⣿⣼⡇⠀⠀⠀⠀⠈⠙⠶⣌⡻⢿⣧⠀⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠀⠀⠀⠀⠻⣴⡟⠀⠀⠀⠀⠀⠀⠀⠰⣆⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣝⢿⣿⢭⡟⠀⠈⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠸⣏⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⡛⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣦⣀⡀⠀⠀⠀⠀⠘⠂⠉⠉⡆⠀⠀⠀⠀⠀⠀⢠⡀⢠⡀⠀⠸⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣿⣿⣿⣼⣦⠀⠀⠀⠀⠀⢰⠱⡀⠀⠀⠀⠀⣾⣶⣿⣾⣧⣴⣦⡄⢹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠛⣻⣿⣿⣷⣄⠀⠀⡆⠸⣄⣿⠀⣠⣄⣰⣿⡿⣿⣿⡍⢻⣿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⢫⣽⣿⣿⣟⣿⡄⣿⠀⣿⣿⣆⣿⣿⣿⣿⣿⣦⠙⠃⠈⠋⠈⠻⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠻⢿⣿⣿⣿⣿⡄⢹⣿⣿⣿⣿⣿⣿⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣿⣿⣿⡇⢸⣿⣿⣿⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⡇⢸⡿⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⡏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢸⠂⢐⢧⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⢸⡌⢀⣤⡞⠐⠃⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠐⡂⢀⠔⣸⣿⡽⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⡉⣰⡿⣿⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣤⠾⢂⣾⣿⠀⡟⣧⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⢈⠘⠁⠀⣾⣏⡏⠀⡇⢿⡀⠀⠀⠐⠒⠀⠀⠀⠸⠆⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡤⠁⠀⠀⣾⣿⣿⡇⠀⡇⢸⡇⠀⠀⠀⠀⠑⡀⠀⢠⠅⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⢎⡀⠀⢀⡿⣿⣿⣿⠄⠠⡇⠀⣿⡂⠀⠀⠀⠀⠈⠆⢀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣤⣉⣀⠀⡀⠀⡇⠀⢻⣏⠀⠀⠀⠀⠀⠀⠸⡷⡶⢦⣤⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦⣤⣼⣶⣤⣤⣤⣃⣀⣀⣀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⢠⣴⠶⠾⠿⡭⣤⣒⡚⠛⣛⠛⠛⠻⠿⠟⠛⠻⠿⠿⠿⣿⣿⠟⠿⠿⠿⠿⣿⣿⣿⠁⠤⠤
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠃⠀⠤⠖⠊⠉⠩⠵⠖⠛⢳⡖⢀⣤⡴⠶⠶⢟⣩⠴⠤⣤⣴⣶⡆⠨⢤⡤⠉⠀⢀⢀⡤⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠴⠤⠈⠭⢉⣉⡥⠴⠶⠒⠒⠒⠲⠄⠈⠉⠉⠀⠀⠀⠉⠛⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢾⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢸⡏⢹⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⢸⡇⣰⠌⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⢸⡏⠁⠨⠶⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣾⠇⠀⠀⢸⡇⠀⠀⢐⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢠⣿⠀⠀⠀⢸⣿⢶⣞⣯⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⣿⠀⠀⠀⢸⡇⢻⡅⠞⠃⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⢸⣿⡀⠀⠀⢸⡇⠀⢂⣠⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢸⣿⠁⠀⠀⢸⣗⡀⠈⣿⡯⠖⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⡇⠀⠀⠈⡶⣒⣈⠄⠀⠈⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠸⣟⢿⡄⠀⢸⣯⣀⣤⠀⠺⡿⠂⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠄⠀⠀⠀⠀⠀⠀⣿⢘⣷⡀⢸⣿⣫⣥⡀⠀⠸⠀⠀⠀⠀⠀⢀⣱⡀⢀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠡⠀⠀⠀⠀⠀⠀⠠⠈⢿⣿⣷⣸⣿⣿⣿⡁⠀⠀⣡⣤⣶⠾⠛⠋⠙⠉⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⣀⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⡷⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠐⢒⣖⠠⣤⢠⣿⣿⣤⣶⠾⠿⠟⠛⠛⠛⠛⠉⠉⡉⠉⢻⣿⠀⠀⠀⠀⢀⡀⣀⣀⣀⡀⠀⠀⢀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⢿⠾⠶⠶⠶⠶⠶⠶⠷⠾⣿⠿⠿⠿⠿⠿⠶⠶⠷⠟⠿⠿⡟⠿⠷⠶⢶⡶⠶⣿⠷⣦⠔⠃
⠀⠀⠀⠀⠀⠠⠤⢤⣤⣵⣤⠈⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠛⠋⠛⠒⠒⠀⠀⠙⠛⠋⠀⠀⠛⠛⠓⠈⠛⠛⠛⠾⠟⠛⢋⣷⣤⣤⣤⡤⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⣷⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡤⠤⠤⠴⠶⡾⠿⠯⠥⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠬⠉⠙⠛⠖⢛⣻⣿⣟⣙⠛⢻⣯⣭⣄⣉⡙⢛⣯⣭⣥⣭⣭⣥⠶⠾⠟⠛⠛⠛⠛⠒⠶⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢉⡙⣛⣉⣉⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣄⣘⡛⠻⠿⢿⣿⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⢉⣉⠁⠀⠤⠖⠒⠒⠛⠒⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣤⣤⣶⣶⣶⣾⣻⣂⣀⣀⣀⠀⠄⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠉⠉⠉⠉⠉⠉⠥⠤⣤⡀⠉⠑⠓⠲⣶⢶⣶⣶⡶⣶⣶⣶⣦⠐⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⠖⢛⣩⠤⠀⠀⠐⠀⠐⠛⠛⠛⠛⠛⠛⠉⠁⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠈⠉⣀⡀⢀⣀⠈⠃⣀⡉⠀⠀⠀⠀⠒⠒⠂⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡠⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠱⠄⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⢦⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠂⢾⡷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣿⡇⠙⠆⢆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠸⢻⡏⠀⡇⠸⣀
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠁⡌⡇⠀⢧⠀⢩⣄
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⢸⠇⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢘⡀⠀⡈⢿⣆
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⣇⠀⠦⠈⠟
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢻⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⢀⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⡑⢀
⠀⠀⠀⠀⣸⣿⢦⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠷⢀⠀⠀⠀⠀⢀⡹⢦⣶⡀
⠀⠀⠀⢀⣿⣿⣦⠂⠀⠀⠀⠀⠀⠀⠐⣼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠲⠶⣽⣾⣿⣿⣦⡀
⠀⠀⠀⠈⠛⠛⠷⠾⠷⣤⣤⣤⣀⣀⣸⣻⣰⣀⣀⣠⠀⡀⠀⠀⠈⢂⠀⠀⠀⠀⠈⠉⠙⠛⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣄⣼⣭⣭⣿⣿⣡⣶⠰⣧⣧⣤⠧⣀⣸⣦⣤⣴⠶⠒⠒⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣴⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠙⠹⠟⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣸⣿⣿⣟⣛⣉⣭⣤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠾⠟⠛⠛⣛⠒⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠿⠿⠿⠿⠟⠛⠉⠀⠒⠒⠒⠉




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣍⣤⣬⣽⣏⣴⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣼⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠒⡇⠀⠀⣄⣀⣀⣾⡇⠀⠀⠀⠀⠀⠀⠠⣤⣀⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣼⣄⣰⣇⡀⣀⠀⠀⣸⢻⠋⠉⢨⡅⠀⢀⡄⠀⠀⠀⠉⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⣜⡏⡀⠀⠀⠠⠃⢸⠀⠀⠠⠀⠀⠀⠄⠀⢀⠀⠀⡏⢠⡀⢀⣀⣀⣀⣠⣤⠴⠛⠉
⠀⠀⠀⠀⠀⢀⣀⣈⣳⡓⠀⣐⣛⣀⣸⣐⣄⣨⣤⣶⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠃⠀⠀⠀⠁
⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣯⠽⠙⠯⢽⣭⣯⣽⣶⣿⣿⣶⣷⣾⠶⠷⠾⠾⠿⠿⢿⡿⣴⡶⣆
⠀⠀⠀⠀⠩⠉⠉⠉⠉⠉⢋⠉⠁⠀⠀⠀⠈⠛⢫⣭⠍⠀⡨⠀⢋⠀⠴⠿⠧⠶⣿⡿⢿⣿⣿⡿⣤⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂⠀⡀⠀⠀⠤⠤⠈⠀⠀⠑⠂⡈⠀⠀⠒⣒⠈⠉⢆⣀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⠀⡀⡄⠠⠒⠚⠀⠠⠀⠠⠄⠐⠢⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⢉⡅⠀⠘⢉⢈⣁⣀⣐⠐⣠⡄⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡆⠸⠀⠀⡀⠂⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠧⠤⠀⠀⠀⠀⠰⠇⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⡆⢸⠀⢃⠀⠀⠈⠈⠁⠈⠈⠁
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⢸⠀⠀⡇⠀⠇⠀⡇⢸⠀⢸
⠀⠀⠀⠑⣖⠠⠤⠔⠒⠊⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠔⠊⢡⠀⠉⠱⣀⣸⠀⠀⡇⢀⡀⠀⣀⢸⣀⣈⣀
⠀⠀⠀⠀⠈⠻⡐⠢⠄⣀⣀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠨⠄⢂⣸⣉⣥⣸⠤⠠⠤⠤⠤⠤⠤⢤⣤⣄⣀⣀⣸⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠉⡉⠉⠉⠑⠒⠒⠒⠒⠛⠚⠒⠛⠛⠛⠒⠚⠛⠛⠛⠉⢉⣉⣉⣉⣉⣉⠀⠠⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠇⣀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢤⣤⠶⢒⣊⣩⠵⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⠛⢶⣄⢐⣠⣤⣄⡤⠦⠴⠶⠖⠚⣻⣛⡉⡉⠩⠔⠒⠈⠊⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠂⠚⠒⠂⠎⠖⠲⠔⠒⠓⠋⠉⠉

⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉
⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⣈⣿⣇⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠛⢛⢛⣿⡟⠛⠛⠛⠛⠉⠉⢙⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⢿⣿⣿⣷⣶⣶⣷⣶⣤⣾⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣼⡿⠿⠛⣛⠛⠛⠛⠛⠿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣿⠂⠀⠀⠀⢀⢠⡀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣀⣀⣀⣀⣿⣿⣿⢦⣤⡄⠤⣬⠤⠤⠤⠤⠴⢦⣤
⠀⠀⠀⠀⠀⠀⢾⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠉⠉⢉⡉⣭⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⡸⢿⣦⡀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠚⠋⠀⣽⡿
⠀⠀⠀⠀⠀⢠⢡⠘⣿⠘⢦⠀⠀⠀⢀⠞⠃⠀⠀⠀⠀⣰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢡⡤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⢘⠀⠸⣧⠀⠙⠠⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣾⣆⡀⢀⡀⠀⠀⠀⠀⠀⠸⣿⣿⣇
⠀⠀⠀⠀⢀⠀⢸⠀⠀⢹⣇⠀⣰⡟⢦⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣿⣶⣍⣷⣶⣿⣿⣿⡄
⠀⠀⠀⢀⡄⠀⢨⠄⣴⣶⣿⣶⣿⠸⣦⡐⢤⡀⠀⠌⡍⢠⣿⣿⣿⠀⣿⡏⠉⠻⢿⣿⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢀⣾⠓⠁⢨⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢦⣈⡴⣹⣿⣿⣿⣶⣿⣧⣀⣀⡀⠈⠁⠮⣟⣿⣿⣿⣿⣷⡀⠀⣀
⠀⢰⣿⠃⠀⠀⣼⡿⢟⣿⡿⠻⣿⣿⣿⣿⣿⠿⣴⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⢹⣿⠙⠒⠒⠮⣭⣽⣻⣿⣿⣷⡀⢹⣦
⠀⣾⠇⣀⠀⠈⢹⠠⠞⠋⠀⠀⠘⣯⢻⠿⠟⣠⣿⣿⣿⡿⢛⢹⣿⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠈⠙⠛⠛⢿⣷⣾⣿⣧⡀
⠀⠋⠀⠉⠀⠿⢻⣿⣾⣿⣿⣿⣷⢿⣿⣾⠿⠿⠿⠿⠿⡿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠻⠛⠛⠛⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠋⠉⠁⠀⠀⢠⣿⣿⠀⠀⠀⢀⡔⠀⢢⠁⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⠀⠀⡄⠁⠀⠀⠀⠓⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠶⠦⠶⠦⠶
⣆⣠⣀⣀⣀⣀⣸⣁⣀⣴⣴⣶⣶⣶⣾⣿⣶⣴⣶⣴⣶⣶⣄⣀⣀⣠⣄⣀⣌⣡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢻⣿⡀⠀⠀⠙⠁
⢛⠉⠉⠉⠉⠉⢹⠛⡉⠉⠉⠉⠉⠈⠁⢹⣿⣏⠀⢠⣜⣍⠀⠀⠀⠀⠀⠀⠀⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⡃
⠈⠄⠀⠀⠀⠀⠘⣧⣽⡴⣤⡄⠀⣠⡀⣠⣿⣿⣾⣃⣽⣏⣤⣤⣤⣤⣤⣤⣤⣮⣥⣤⣀⣀⣠⣴⣂⣤⣄⣼⣇⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠿⠟⠙⣿⣿⡿⠿⠛⢻⣿⠿⣿⣿⠟⠛⣿⣿⣿⡿⠿⣿⣿⣿⣿⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠋⠉⠀⠀⠀⠀⠘⣿⡄⠀⠀⣿⣿⠀⠀⠈⠀⠀⣿⢻⠏⡀⠀⠀⠉⠀⠀⠀⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣶⡆⣿⣿⣀⠀⢀⡀⢀⣿⠸⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣦⠀⠀⣰⣮⣹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣧⣾⣿⣄⣦⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡉⠉⠉⠙⠛⠉⠉⠙⠲⠤⣤⣀
⠀⠀⠀⠀⢠⣤⣤⣴⣶⣶⣶⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣶⣦⣤⣤⠄
⠀⠀⠀⠀⠀⠙⡟⠛⠛⣋⣠⣤⣤⣤⡈⠉⠉⠉⠉⠙⣛⣻⣿⣿⣟⣙⣛⣛⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠹⢿⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣾⡿⠛⠉⠉⠉⠙⠛⠛⠿⠶⠾⠟⠛⠛⠛⠛⠛⠉⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠓⠒⣺⣟⡐⠒⠤⢸⠀⠀⠀⠀⠀⣀⣠⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣳⣄⣀⣼⣤⣴⣶⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠲⠤⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢟⣛⠿⣷⡄⠀⠀⠀⠀⠀⠠⢄⡀⠠⠤
⠀⠀⠀⠁⣀⣈⣉⣽⣶⣶⣶⣿⣭⣽⣯⣭⣿⡿⠿⣿⢿⠿⣿⣛⣛⣫⣥⣤⣴⣶⣿⣿⣿⣿⣖⣦⠀⠀⠠⠤⠤⠤⠤⠄
⠀⢀⣀⣭⣍⣉⣉⡻⣯⣥⣶⣾⣿⢟⣿⣿⡿⠧⠾⡿⣏⣠⣭⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣭⣥⣤⠤⠐⠒⠒
⠈⠉⣉⡉⠛⠛⠛⠛⠻⢌⣉⣉⣐⣋⠉⠻⣿⣿⡛⠉⣴⣿⣿⡯⠛⠓⠊⠀⠉⠩⠭⠭⠭⠽⣿⣿
⠀⠀⠈⠉⠉⠉⠁⠀⠲⠦⢬⣍⣉⣁⡀⠀⠉⠉⠉⠥⠬⢽⣿⠧⠀⠀⠛⠓⠒⠚⢊⣉⣩⡭⠉⡉⠀⠐⠚⠛⠓⠒⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠒⠒⠂⠀⠘⠿⠃⠀⠀⠀⠑⠒⠂⠨⠁⠀⠀⠉⠙⠒⠲⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠉⠉⠁⢸⡀⠀⠀⡇⠀⠀⢀⡀⠤⠤⠶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡧⠄⠀⠛⠈⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠋⠉⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠀⠀⠀⠀⣀⣀⣀⣤⣤⣥⠀⠀⠾⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣤⣶⠿⢿⠋⠉⢹⣇⣀⣰⣦⠶⠒⠁⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢁⣸⣤⣼⣶⣶⡿⠛⡟⢹⢻⡄⠀⠀⠀⠀⡈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠉⠁⠔⠋⠉⠁⠀⠘⠀⣾⡀⠌⢤⣤⣾⣿⣿⣿⣢⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⡄⠀⠀⠀⠀⣀⣀⣤⣤⡤⠆⠀⠀⠀⠀⠈⢥⣤⠤⠔⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣣⣴⣶⣿⣿⣿⣏⡉⢀⣠⡄⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡄⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⢏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠉⠉⠉⠉⠉⠋⠙⠛⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠐⠌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠿⠧⣈⣉⣙⡛⠿⣶⣶⣆⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣡⠀⠀⠀⠀⠀⣶⣿⡇⢠⡇⠀⠀⠀⣼⣿⣦⡀⠉⠉⠉⠛⠓⠤⢠⣴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⣧⠀⠀⠀⠀⢿⣿⡇⠈⠁⢰⣶⣦⡬⠿⢿⣿⣦⡀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠇⣿⡄⠀⠀⠀⣾⣿⣿⢀⠀⠈⢻⣿⣷⡀⠀⠀⠈⢝⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⠓⠤⠀⣿⡄⢻⣷⡀⠀⠀⣿⣿⣿⣾⣤⠤⠼⢿⣿⣿⡤⢊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⠴⠾⢿⣷⣤⣤⢿⡿⠛⠛⠻⠀⢒⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⣿⣤⠴⢶⣶⣾⣶⠶⠞⠋⢁⠀⠀⠊⠉⠀⣾⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣿⣿⣿⣿⣷⣦⣤⡴⠊⠁⠀⠀⠀⠿⠀⣾⣿⡇⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⠿⠛⠙⠿⣿⣿⣯⣂⠀⣠⠈⠓⠁⠀⡐⠈⠀⠦⠛⠛⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⠶⠺⠶⠂⠺⠿⠆⡀⢈⣩⠤⣴⣴⡆⠶⢶⣾⣷⣾⢿⣷⠺⠛⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠟⠁⠈⠡⠛⠻⠓⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⣀⠀⡀⠀⠒⣨⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⣄⠀⡠⠄⠈⠈⣿⣿⠁⢰⣿⣿⣿⣿⡶⠂⠉⠛⠿⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⣿⣶⣶⣶⣤⣶⡿⠃⠀⣨⣿⣿⣿⣿⣷⣦⣥⣬⣖⣤⡙⠻⣿⣿⣿⣿⣿⠋⠀⠁⠀⣀⣀⣤⣾⣿⣿⣿⣿⡿⠛⠉⠙⠛⠋⠻⣿
⠀⣠⠔⠋⠉⠁⠀⠐⠛⠻⣿⣿⣿⣿⠟⠋⠉⠹⣿⣿⣿⣷⡌⠻⣿⣿⣿⣀⣀⠀⢸⣿⣿⣿⣿⡟⠩⠊⠉⠡⠖⠂⠀⠀⢀⡀⣿
⣿⣷⣶⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣦⣒⣥⡀⢀⣴⣿⣿⣿⣦⠘⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣀⡀⠉⠁⠀⠉
⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⡿⠀⢰⣿⣿⣿⣿⣿⡟⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠐
⣴⣶⡆⠈⠉⠻⢿⣿⡿⢋⣥⣶⣶⡶⠂⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣷⡙⠁⢠⡄⠙⠻⢿⣿⣿⣷⣄⣀⡀⠀⢀⣀⠠⠔⢒⣴⣿
⣿⡟⠀⠀⠀⠀⠀⠛⣰⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠔⠊⠘⠻⣿⣿⣿⣿⣿⣞⠉⢀⡤⠄⣾⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠉⠉⠀⡀⠀⠀⠀⠀⢂⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿
⣿⣇⠀⠀⠀⠀⢀⡀⠻⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡌⠀⠀⠀⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣭⡍⠀⢀⣀⣤⣤⣶⣶⣤⣭⣍⣉⡁⠀⢤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⣿⣿⣿⣟⣋⠉⣉⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠃⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⢿⣿⣟⣻⡥⠄⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⠏⠈⢻⣿⣿⣿⣿⠀⣾⣿⣿⡟⢀⢀⠤⠚⠋⣸⣿⣿⣿⣿⣿⣿⠟⣁⠤
⣧⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣷⣄⠀⠀⣀⣴⣿⣿⣿⡿⠛⠋⠁
⣿⣿⣦⣍⡛⠿⠿⠿⠿⠿⠟⢛⣋⣥⣾⣿⠟⠉⠀⣹⣿⣿⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠟⠁⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿
⣿⠛⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⣿⣿⣿⠿⢋⣥⡀⠀⠀⠀⠀⠈⠙⠛⠀⠤⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⠋
⣍⠀⠀⠀⠀⠁⠈⠛⣿⣿⣿⣿⣿⠅⠉⠀⢐⣋⣭⣴⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣭⣭⣭⣭⣥⣴⣶⣾⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠀⠀⠀⠈⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠰⠤⡤⠤⢤⣀⣀⣀⣉⣃⠀⠀⠀⠀⠐⠂⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠤⠀⠀⢘⠁⠀⠀⠙⠲⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠘⠛⠷⠾⡖⠒⠖⠂⠀⠀⠾⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠤⠐⠢⠴⡤⠄⠀⠐⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⠛⠛⠓⠲⠶⠶⠷⠤⢤⣤⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⣀⣀⡔⠀⡄⠀⣀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣤⣄⠀⠀⠖⠚⠛⠙⠋⠉⠉⠀⠙⠀⣿⠁⠀⠉⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣀⢠⡐⣫⠦⠴⠿⠷⣤⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠐⠷⣾⣶⣼⣟⣉⡉⠀⠀⠀⢀⣧⣀⡀⣤⣤⣿⣦⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢄⠀⠀⠀⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⠉⠉⠛⠻⠶⠾⢿⣯⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠦⣀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣟⣀⠀⠀⢀⣤⣤⣅⣈⣁⣰⣆⣙⣈⡓⠮⠤⠐⠀⠹⡆⠴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠴⠦⣤⣶⣀⣠⡄⠛⢋⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠥⠅⠀⠀⠀⠀⠀⠱⠄⠈⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠈⠐⢀⠀⠉⠉⠉⠓⠲⠶⠶⠤⣤⣤⣄⣸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠚⠒⠿⡀⠂⠤⣀⡀⠘⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠗⠁⠀⡽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⢃⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡔⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠲⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⠀⠀⠀⢨⠀⠀⠀⡀⠀⣈⠇⠀⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠹⠀⣧⠀⠀⣤⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢰⡏⠀⠀⠀⡟⠀⠀⠀⠀⢀⡜⠁⢀⡽⠆⢈⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡿⠊⡠⠂⠀⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠻⠇⠀⠀⡸⠁⠀⠀⠀⠀⠙⠂⣀⠀⣤⠀⢸⠇⠀⠰⠀⠀⠀⠀⢀⠀⠀⠀⠀⢺⡄⢀⠀⠀⠀⠀⠊⠀⠚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠈⠀⠼⠀⠀⠠⠁⠀⢀⡠⠄⢰⡷⢉⢹⣶⠀⠀⡌⢀⢰⠆⢀⣰⠶⠶⠛⠀⠀⠀⠀⠀⠇⠀⠀⠠⠄⠀⠔⠀⢀⠖⠉
⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣴⡎⣿⡀⠀⠈⡟⠸⡾⠁⢀⡜⣁⡴⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⣰⠆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡴⠐⠉⠀⢺⡇⢿⡇⠀⠀⣧⣐⣁⡴⠛⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠞⣡⢾⣀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⡇⠘⠃⠀⢀⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⢰⠅⡀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⡀⣸⡇⠀⠃⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡉⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣤⣿⣄⡀⣈⣓⣲⣤⣀⡀⠀⣄⣉⣁⣬⡷⠁⣀⣷⣆⡄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣍⢉⣿⣿⣿⣿⣿⡛⣂⣿⣌⣩⣿⣴⠟⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⢻⣻⣟⡻⣟⣿⠛⣻⣻⣟⠛⡟⣯⡉⣹⣽⣅⠀⠀⠀⡁⠈⠋⣠⣾⡏⣿⣾⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⠐⠿⣿⠇⢿⡿⠯⠸⠿⠿⠀⠿⠿⠃⢀⣀⣤⣴⣶⠟⠫⣽⡇⡌⠙⢳⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠒⠒⠒⠒⣖⢲⣶⢒⣖⣒⣒⣲⣒⣒⠖⠒⠋⠉⠉⠥⠀⠂⠠⡀⣼⣷⣆⣠⠍⠻⠧⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠸⠋⠉⠛⠛⠛⠛⡛⠛⠉⠈⠉⠁⡀⢠⡀⠀⠀⠀⡀⠀⠀⠀⠘⠻⠿⠛⢁⡀⠰⠦⠓⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠁⠀⠠⠄⠀⠀⢀⡀⠒⣀⣀⠈⠁⠀⠀⠀⠈⠀⠑⠚⠛⠁⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠑⠢⣼⣆⣀⣀⣀⠀⠀⠀⠀⠀⣄⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠉⠒⠤⢀⠀⠀⠁⠲⠢⠬⣦⡀⠘⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠐⠀⠑⠐⠒⠓⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣱⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠉⣉⣉⣉⣉⣙⣛⡲⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣈⣦⠖⠋
⠀⠀⠀⠀⠀⠉⠀⠀⠔⠚⠉⣉⡠⠤⠤⠤⠬⠭⣽⣓⡻⠽⣟⣓⡶⠶⠶⠶⠶⠶⠛⣋⣩⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠪⠭⢭⣙⣛⣛⣋⣉⣡⠤⠔⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠛⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⡗⠀⠀⠀⠀⠀⠀⣀⡀⠼⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⢀⡀⠋⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣹⠶⠛⠊⠉⠀⠀⠀⠀⢀⣧⣄⣲⡬⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⢀⣶⣬⠄⠒⠘⠉⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠀⠀⠀⠀⣠⡲⠟⠛⠛⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠔⠂⠈⠁⠀⠀⠀⠠⣷⠀⢨⢀⣀⣀⠤⠤⠤⢦⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⠦⠖⠒⠂⠀⠀⠀⠀⢀⠀⠀⣸⡿⠀⠚⠋⠉⠁⠀⠀⠀⠈⢆⡀⠸⢷⡀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠆⢠⣇⠤⢶⡞⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⡀⠀⠉⠛⠛⠃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠍⢂⠂⢉⡴⠷⠖⠋⠄⢀⠀⠀⠀⣀⣠⣄⡤⠷⠿⠿⣇⣀⣀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠶⠖⢂⠍⠁⠀⠀⠀⢻⠀⠠⢰⢂⡠⠐⠒⠋⢹⡄⢛⣷⣴⣶⣿⣿⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢰⠀⠈⠀⠀⠀⠀⠀⠸⡆⠀⢷⡛⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣆⠀⡀⢀⡀⠀⣀⣀⣷⢀⣨⣵⣾⣿⣿⣿⣿⣿⣛⣹⠛⠿⣿⠿⣷⣷⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣳⠾⠿⠶⠶⠿⠿⢏⣉⣡⣤⣽⣿⣿⣿⣿⣿⣿⣿⠛⣿⣾⣶⣷⣦⡀⢹⣿⣿⣿⣷⠆
⠀⠀⠘⠿⣿⣶⣦⣄⣀⠈⠻⠿⢏⣠⣞⡻⣿⣿⡿⠿⠿⠯⠮⣭⡭⠛⠛⠉⣩⣷⣾⣿⣿⣿⣿⣿⢸⡿⠿⢿⡷
⠀⠀⠀⠐⠉⠉⠉⠉⠻⠿⠿⠿⢟⠍⠉⠁⠹⣄⠀⠤⠤⠆⠀⠤⠶⡷⢶⠿⠿⣿⣿⣻⣿⢿⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠙⠁

⠀⡀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠲⢄⣠⠤⢄⣀⣀⣠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠤⠤⢌⡀⠀⠀⢁⠔⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡖⠀⠐⠲⣼⠖⠊⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⡀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢨⣄⣀⡀⢀⡷⣆⠀⠀⢱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢾⣯⣿⣿⣿⣿⣽⡿⣿⣿⣶⣶⣯⣽⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⣠⣤⣤⣤⣇⡀⠀⠈⠉⢋⠉⠀⠀⣏⠉⢹⣏⣥⡴⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣻⣶⣤⣴⣧⣼⣽⣌⢂⣸⣤⠤⣶⠿⢛⣉⡽⠒⠋⡁⠼⡷⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡿⣿⣇⣃⠩⢙⢻⡏⢻⠿⢲⣩⡷⠞⠋⢁⡄⠰⠀⣁⡴⢟⠽⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣷⣷⠚⣻⠛⠿⣶⣼⣧⡴⠞⠋⠁⠀⡦⡆⢀⣥⣶⠟⣫⠞⠅⣠⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠷⢉⣡⢽⣾⡉⠜⠓⡲⢤⣎⣿⡀⢸⠀⠁⣠⢶⣫⠕⣋⡴⠋⠠⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⢁⣨⠖⢋⠡⠒⠉⠄⢂⠡⠐⠈⢈⣟⡟⠻⣒⡭⠖⢉⡤⠚⠁⠀⣀⠔⣫⠂
⠀⠀⠀⢠⡠⢀⡴⠚⠉⣐⣤⠜⠋⠄⠂⡁⠐⡈⠔⠈⠀⢀⣄⣺⠷⠞⠛⠉⢁⡤⠞⠉⢀⠀⣠⠾⠓⠊
⠀⠀⠀⠈⢷⠥⠤⠬⠴⠯⠤⠤⠤⠴⠦⠴⠶⠒⠚⠛⠉⠉⠀⠀⣀⡠⠔⡊⠁⠀⢀⢄⣵⡚⠍⠂
⠀⠀⠀⠀⠀⠑⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠚⠉⠁⠀⠀⠀⠀⢊⣴⠋⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠫⣍⠉⠙⠋⠉⠉⡹⠛⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡋⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣄⡀⠳⣶⡗⠀⠀⠀⠀⠀⢈⠀⣰⡞⢙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣻⣿⣉⣑⣦⣴⣾⠶⠯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣶⡾⠛⣿⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢻⣿⣾⣮⡠⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡿⠟⢿⡀⠀⠀⠀⠈⣿⣧⡿⠿⠛⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠋⠉⠁⠀⠠⠘⣧⠀⠀⠀⠀⠹⠋⠀⠀⠀⠀⠀⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢟⠲⠶⠾⢿⣦⣄⣄⣀⡀⠐⡖⠶⠤⢴⣤⣷⣀⠈⢷
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠀⠀⠀⠀⠀⠉⠉⠛⠳⠷⠦⢤
⠀⠀⠀⢀⣠⣴⡾⣷⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠉⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣸⣷⢤⡀⠀⠀⠀⡘⠀⢀⣀⣠⣄⣀⠀⠀⠀⠞⠀⠀⠐⠀⠶⢲⡀⣶⠆⠀⠀⠀⠐
⠀⢛⠶⢤⣄⣀⠀⠙⢧⡜⠂⠀⠀⣜⣁⠀⠀⠀⣽⢿⣏⣤⠀⠀⠀⠀⠿⠙⠛⠛⠚⠷⠿⠶⠤⢤⣤⣥⣤⣤⡀
⠀⠈⠓⠂⠼⠿⠓⠲⢾⣧⡀⠀⡀⠉⢏⠉⠛⠛⠛⠿⠿⢷⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣙⡳⢷⣤⣀⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠉⠉⠒⠘⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠑⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣠⣴⡶⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣠⣠⣬⡿⢏⠉⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣿⣠⣼⣼⡾⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢺⡟⣿⣻⣯⣤⣴⢾⢿⡿⠋⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠰⠍⠺⢿⣿⣿⣿⣿⣥⢴⣶⣺⣿⣿⣿⣧⣀⠐⠚⠀⠀⡀⢀⣼⣇⣽⣿⣿⣯⣶⣶⠏⠀⠀⠀⠀⠤⠎⠀⠤⠤⠤⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣿⣭⣷⣾⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉

⡀⠜⣰⡶⣦⡀⠀⠀⢀⣴⢿⣦⠀⠀⠀⠀⠀⠀⠀⠹⠧⠐⢡⣶⠿⣆⠀⠀⠀⣰⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣆⠻
⠀⠀⢿⣧⣼⠃⠀⠀⠈⢧⣼⡷⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡜⢿⠤⠏⠀⠀⠀⠹⠦⠿⠂⠀⠀⠀⠀⠀⠀⢀⣾⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡿⠁
⠀⠀⣰⡶⠂⠀⠀⠀⢀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣾⣿⡶⠒⢀⠀⠀⠀⣰⠛⠀⣀⠀⠀⠀⠀⠀⠀⠈⠿⠾⠿⠁
⠀⠀⠉⠀⠀⢠⠶⢶⡌⢇⣄⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡞⠿⢧⡙⠤⠄⢃⣴⡶⣦⡀
⠀⠀⠀⠀⠀⠙⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠾⠁⠀⠀⠈⢿⣄⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠰⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣿⡟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢸⡀⢠⢠⣵⠶⣶⠤⠠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠸⠷⠦⠆
⠀⠀⠀⠀⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⣘⡄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠛⠋⠉⠁⡀⠀⠐⠀⠀⠀⠐⢚⣢⠇
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣎⣹⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠋⢿⠿⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⣠⡦⠤⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄⣀⣀⠀⠀⠀⠀⠀⢀⡠⠄⢘⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠤⠶⠤⠤⠤⠄⠶⣾⡇
⠀⠀⠀⠀⠀⠀⢿⡷⣦⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣌⣀⡁⠀⠀⠀⠘⣀⣤⠀⠀⠀⠀⠀⠐⠾⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠠⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠠⠐⠘⠀⠀⠀⢀⠀⠁⠈⠣⣽⠄⠀⠀⠀⠀⠀⠀⠠⠀⠉
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠒⠂
⠀⠀⠀⠀⠸⠀⠀⠀⠀⢶⢆⡀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⢀
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠉⢧⣖⡀⢤⡄⠀⠈⢀⣠⣿⡁⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⠯⠁⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⣸⣿⣿⣷⣯⣀⣀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠙⠁⠘⡇⠀⠀⠀⠀⠘⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢱⡞⢷⠇⠀⠀⠀⠀⠤⢋⡿⣠⠾⣠
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢈⠛⠏⠀⠀⠀⠀⠀⠀⣸⠄⠁⡪⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡆⠀⠀⠀⠀⠀⠀⠑⢄⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠉⠀⢀⡠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣀⠀⠀⣿⢣⡀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢂⠀⠀⠀⢀⡇⠀⠀⠠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣭⣏⢛⣖⣦⣀⣀⣀⡀⡀⠀⡟⠀⠀⠘⣀⠀⢤⣈⣱⣤⠖
⠀⠀⠀⢀⣐⣒⡲⣶⣶⣤⣤⣶⣤⣸⣿⣿⣿⣿⢶⡾⠷⠾⠧⠼⠷⠶⠗⣶⣿⣏⣉⣙⡫⢋⣀
⠀⠐⠂⠀⠿⡛⠻⣿⡷⠿⠛⣽⣯⣍⠛⠡⠿⣿⣯⣽⣻⠿⠆⣹⣿⣓⠛⢛⡓⠐⠛⢻⠁⠶⠂⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⠒⠀⠤⠀⠻⠿⢦⡀⠀⠨⠀⠠⣆⠙⠛⢃⡀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⠀⠀⣀⣐⣃⣀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⣠⣡⡄⠀⠙⣿⡏⠉⠀⠀⠀⠀⠀⢠⡁⣀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢻⠋⠉⣉⡙⠉⠙⣿⡟⠉⠁⠀⠶⠓⠒⠶⣶⠭⠄⢀⣀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢆⢸⠁⠀⠐⠛⠡⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⣿⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣦⡄⠀⠈⠀⢹⣇⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡇⠀⠄⠐⢸⠈⢢⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⠀⠀⠀⠸⠠⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢃⠈⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡠⢾⣿⣿⣿⣶⠲⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣹⣽⣡⣮⣄⡼⠊
⠀⠀⠀⠀⠀⠀⣼⣀⣮⣭⣬⣽⣽⣆⣶⢤⣄⣀⠀⠀⠴⢄⡂⠒⠒⠀⠘⠤⠀⣸⢻⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣿⡿⣿⠿⢶⣶⣬⣍⠿⠄⠈⠈⠉⢉⡁⠈⣁⠀⠤⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡳⠿⠷⠿⢷⣧⠰⠄⣤⠈⡉⢷⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣷⣶⣿⣿⣦⣀⣀⠀⠀⠀⣶⡄⠀⠀⢤⡄⠀⠀⠸⠂⠀⠉⠀⣴⠂
⠀⠀⠀⠀⠀⠀⣄⣨⣿⣯⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠚⡃⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠂⣹⢹⣟⣹⣿⣏⣻⣿⣿⡟⢻⣿⠿⠍⠡⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⠋⠘⠿⠡⠀⠲⠿⠃⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡿⠀⠀⠀⠀⡤⣤⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠻⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⡜⠛⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢾⠆⢗⣷⣿⡇⡄⠀⠠⠴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡈⣷⠀⡸⡿⣼⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⢂⣴⣼⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠘⠀⠀⣧⣸⣿⣤⠄⢀⣀
⠀⠀⠀⠀⠀⢡⡈⢉⠈⠈⡸⠀⠈⠀⠀⠀⠀⠀⠀⠢⢾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣅⡺⢿⣿⣁⠴⠃⡆⠻⣿⣾⣿⠏
⠀⠀⠀⠀⠀⣼⣷⣶⣤⣄⠀⠀⠀⠀⠀⣧⣴⣿⡇⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⢳⣦⣆⣀⠈⠬⠿⢿⡿⣴⣿⣿⣿⣯⣽⠿⡷⠆⡄⠀⠀⠀⣿⣿⣿⡿⣿⢫⣭⣿⡍⠻⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣾⢿⣿⢉⢿⡶⢆⢸⣿⣼⣿⣿⡏⠙⣿⣿⣾⡐⢦⣠⠀⠀⣿⠀⡿⢑⣧⠘⢫⣷⣶⣾⣿⡿⠿⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣬⣿⣯⣽⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣷⣿⣀⣶⣜⣿⣇⢸⣿⣴⣿⣿⣿⢁⣴⣤
⠀⠀⠀⠀⠠⠿⣿⣿⣿⣿⣿⣿⣦⣵⢁⠨⣿⣿⣿⣿⣿⡿⠋⠘⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣯⣥⣤⡀
⠀⠀⠀⠀⠀⣠⣤⣉⣿⣟⠋⠉⢉⣛⣝⢿⣿⣿⠟⠛⣃⣤⣴⣦⣴⣬⣟⣿⣿⠿⢷⣾⣿⣿⡿⠿⠟⡛⠿⠉⠈⢙⣛⣿⠿⠭
⠀⠀⠀⠀⠀⠙⠻⠛⠛⠛⠂⠀⢫⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣮⠥⠼⠿⠟⠛⠉⠀⠠⠄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠚⠉⠛⣛⡛⠠⣬⠉⠉⠉⠛⠉⠛⠛⠛⠛⠛⠻⠇⠐⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⣷⢄⡀⠀⣿⠀⠀⠸⣿⡆⢀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢷⠘⡄⠙⠲⣿⡀⠀⢀⡟⠍⠀⠑⣄⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠁⠀⢒⣿⠁⠀⢨⠀⠀⠀⠀⠉⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⡇⠀⡸⠀⠀⠀⠀⠐⠉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣿⢸⠀⡇⠀⠀⠀⠀⠀⠀⣿⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢀⡿⡎⣦⠁⠀⠀⠀⠀⠀⠀⠽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⢸⣿⣇⠇⠀⠀⠀⠀⠀⠀⠀⢻⡿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡷⠋⠀⠀⠀⠀⡁⣸⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⡀⠀⠀⠀⠀⠀⠀⠰⡒⠮⠢⠂⠘
⠀⠀⠀⠀⠀⠀⠀⣨⡅⠀⠀⠀⠀⢀⣈⣹⣿⣧⡤⢄⣀⠀⠀⠀⠀⠀⢺⣾⡀⡇⠀⠀⠀⠀⠀⣫⡤
⠀⠀⠀⠀⠀⠀⣴⡿⠀⣀⠄⠁⠀⠀⠉⢹⣯⣥⣮⣟⢛⣑⣨⡁⢀⠀⢸⣿⣦⣠⠀⠈⣡⣴⣾⡟⠁
⠀⠀⠀⠀⠀⠸⠝⢻⢟⣀⡒⠀⠒⠒⠛⠻⣿⣿⣽⣿⠀⠀⠀⠀⠀⠀⣩⢝⣉⣵⣶⣿⣿⣿⡿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⡏⠉⣿⣶⠂⢀⣠⣤⣿⣾⣿⣿⡿⠟⠙⠉⠻⠃⡠⠃
⣀⣄⣀⡀⠠⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣾⠟⡂⣰⡆⢘⣿⡇⢀⣄
⣿⣿⣿⠿⠿⠀⠑⠓⠄⡠⣦⣄⡘⠻⣿⣿⣿⡿⣿⣿⡿⠟⣿⣿⣿⣄⠉⣈⣱⣶⣿⣿⣿⡃⢸⡿⠛⢸⣿⣷⣤⣄⣀⡤⠐⡂
⠉⠉⠉⠑⠂⠀⠀⠘⠷⣦⣤⣈⠙⠻⢿⣾⣶⣿⡿⢿⣷⣾⣿⠿⠛⠻⠿⣿⡟⠿⠿⠿⠿⠃⠘⠃⡀⢸⣿⣿⣎⠛⠿⢿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠦⠄⠈⠁⠈⠙⠳⠶⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡋⠀⠘⢷⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⠐⠒⠀⠀⠀⠀⠀⠚⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣀⠀⠀⠛⠀⠀⠀⣀⣠⣀⣀⣀⠀⠂⠀⢀⡰⡆⠀⠨⢀⠠⠤⠌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣶⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠉⠁⠀⠁⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠐⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠲⠊⠁
⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠂⣸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠩⡧⢑⡏⠀⠀⠀⠀⠈⠀⠐⣂⡀⠠⠀⠹⣀
⠀⠀⠀⢸⣛⣻⠶⡶⠦⠶⠆⣆⣰⣴⣶⣤⣤⣼⣄⣜⣁⣀⣠⣦⣤⣤⣤⣷⣿⣴⣶⣷⠾⢷⣦⣴⣶⣾⣿⢷⣿⡿⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠉⠛⠛⠲⣶⣦⣤⣤⣤⣎⣃⣇⣁⣀⣉⣉⣉⣉⣉⣉⣁⣖⣐⣜⣐⣐⣒⣉⣭⣭⣭⣭⠤⠖⣂⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⡛⠿⠟⠛⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⢿⡿⠿⠿⠿⠿⠿⠿⠷⠴⠀⠴⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠀⠈⠉⠐⠈⠉⠓⠾⠟⠋⠘⠷⠖⠀⠉⠉⠙⠓⠒⠛⠂⠂⠉⠀⠂

⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦⡀⠀⠀⠀⠹⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠙⢿⣿⣿⣿⣄⠀⠀⠀⠘⣿⣿⣿⣿⡆
⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣄⠀⠀⠀⠻⣿⣿⣿⣦⠀⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⠀⠀⠀⠙⠻⣿⣿⣷⣄⠀⠀⠘⢿⣿⣿⣧⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣶⣤⡀⠀⠈⠻⣿⣿⣧⡀⠀⠈⢿⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿
⣴⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⠉⠛⢿⣿⣷⣄⠀⠈⢻⣿⣷⡀⠀⠈⢿⣿⣿⡀⠀⠀⢸⣿⣿⣿⡀
⣿⣿⣿⠟⠋⠁⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣷⣦⣄⡀⠈⠻⣿⣷⡀⠀⠹⣿⣷⡀⠀⠸⣿⣿⣇⠀⠀⠸⣿⣿⣿⡇
⡿⠋⠁⠀⠀⠀⣠⣴⣿⣿⣿⡿⠛⠋⠉⠀⠀⣀⣀⣠⣤⣄⣀⣀⠈⠙⠻⢿⣦⡄⠈⢻⣿⡄⠀⢹⣿⣧⠀⠀⢿⣿⣿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢀⣴⣾⣿⣿⡿⠋⠁⠀⢀⣠⣴⣾⣿⠿⠟⠛⠛⠛⠛⠛⠻⢷⣦⣄⠙⢿⣦⠀⠹⣿⡄⠀⢿⣿⡀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⠁
⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⣠⣾⣿⡿⠛⠉⢀⣠⣴⣶⠾⠿⠿⠿⠶⣦⣌⠙⢷⡄⠹⣧⠀⢻⣷⠀⢸⣿⡇⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿
⣠⣾⣿⣿⡿⠁⠀⠀⣠⣾⣿⠟⠉⠀⣠⣾⡿⠛⢁⣠⣴⠶⠖⠒⠒⠶⣌⠳⣄⢻⡄⢻⡇⢸⣿⠀⢸⣿⡇⠀⣸⣿⡟⠀⠀⣸⣿⣿⡇
⣿⣿⣿⠏⠀⠀⢀⣼⣿⡿⠋⠀⣠⣾⡿⠋⢀⣴⠟⢉⣴⠞⣋⡭⠭⢝⡲⡱⡜⡆⣷⢸⡇⢸⣿⠀⣸⣿⠀⢀⣿⣿⠃⠀⢠⣿⣿⡿
⣿⣿⠃⠀⠀⢠⣿⣿⡟⠁⠀⣰⣿⡟⠀⣰⡿⠃⣰⠟⣡⠞⡵⡫⡛⣕⡺⢰⢣⢇⡏⣼⠃⣼⠇⢠⣿⠇⠀⣼⣿⠏⠀⢀⣾⣿⣿⠁⠀⠀⣰
⣿⠃⠀⠀⢠⣿⣿⡿⠀⠀⣰⣿⡟⠀⣰⣿⠃⢰⡟⢠⡏⣸⢹⢳⢇⡈⠧⢅⡫⢎⡴⢃⣼⠋⣠⣿⠋⢀⣼⣿⠏⠀⢀⣾⣿⡿⠁⠀⠀⣰⣿
⠏⠀⠀⠀⣾⣿⣿⠁⠀⢠⣿⣿⠁⠀⣿⡏⠀⣿⡇⢸⡇⢿⠸⣌⢮⡪⢵⣓⣊⣭⠶⠋⣡⣾⠟⠁⣠⣾⡿⠃⠀⣠⣿⣿⡟⠁⠀⠀⣴⣿⣿
⠀⠀⠀⣸⣿⣿⡏⠀⠀⣼⣿⡏⠀⢸⣿⡇⠀⣿⡇⢸⣧⠘⣧⡙⢦⣙⠲⠶⠶⠶⠶⠛⠋⣀⣴⣾⠟⠋⠀⣠⣾⣿⡿⠋⠀⠀⣠⣾⣿⣿⡿
⠀⠀⠀⣿⣿⣿⠇⠀⠀⣿⣿⡇⠀⢸⣿⡇⠀⢻⣧⠀⢻⣦⠈⢷⣤⡉⠛⠶⠶⠶⠶⠾⠿⠛⠉⠀⣀⣴⣾⣿⠟⠋⠀⠀⣠⣾⣿⣿⣿⠋
⠀⠀⢠⣿⣿⣿⠀⠀⠀⣿⣿⡇⠀⠈⣿⣷⠀⠘⣿⣧⠀⠻⣷⣄⠉⠻⠿⣶⣤⣤⣤⣤⣤⣶⣾⣿⠿⠟⠋⠁⠀⢀⣤⣾⣿⣿⡿⠋
⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣷⠀⠀⢻⣿⣇⠀⠘⣿⣷⡀⠈⠻⣿⣦⣄⣀⠈⠉⠉⠉⠉⠉⠁⠀⠀⣀⣤⣴⣾⣿⣿⡿⠟⠉⠀⠀⠀⢀⣤
⠀⠀⢸⣿⣿⣿⡆⠀⠀⢹⣿⣿⡆⠀⠈⢿⣿⣦⠀⠈⢻⣿⣦⣀⠀⠙⠻⠿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⢀⣠⣶⣿⣿
⠀⠀⠈⣿⣿⣿⡇⠀⠀⠈⣿⣿⣿⡀⠀⠈⢿⣿⣷⡀⠀⠙⠻⣿⣷⣦⣄⡀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⡿
⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣷⡀⠀⠀⠻⣿⣿⣦⡀⠀⠈⠙⠿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠹⣿⣿⣷⡄⠀⠀⠙⢿⣿⣿⣶⣄⡀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠹⣿⣿⣿⣦⡀⠀⠀⠙⠻⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴
⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠘⢿⣿⣿⣷⣄⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿
⡆⠀⠀⠀⠀⠹⣿⣿⣿⣿⣆⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢰⡎⠹⠶⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⠛⠉⠉⠀⠀⠶⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⣀⣀⠀⠀⠀⣀⡠⣿⠆⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢒⣿⠍⠉⠀⣀⣪⡭⠐⠖⠛⠋⠹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠐⠋⠉⠹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣦⣤⣤⣀⠀⠀⠀⠀⣂⡀⢤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⡶⢶⣄⢘⡿⠋⢉⣁⣿⣷⣟⣳⣤⣀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃⠀⠀⣻⠟⠁⣀⣾⣷⣤⣬⣿⡟⠉⠉⠉⠁⠀⠀⠘⠻⠋⠀⡃⠀⠀⠀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⢻⣿⠁⢀⣀⢤⠞⠚⠋⠉⠁⠀⠈⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣯⡴⠶⢺⣿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⡁⠀⠀⠺⢗⣺⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⣄⢀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⠟⡶⢞⠃⣠⡴⠃⠀⢸⣀⣀⣤⣤⣤⣤⣤⡀⢿⠿⠛⡿⠿⣿⣷⣦⣤⠀⠘⣿⣻⡇⠀⠀⢼⡇
⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⢸⣿⣯⣿⣴⣷⣟⣿⠁⠀⣠⡿⠟⠉⠙⣯⣿⣿⣿⡛⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⡇⠀⠀⢿⡧
⠀⠀⠀⠀⠀⠈⠳⢤⣄⠀⠘⣿⣿⣽⣿⡶⠿⠿⣤⣼⠥⠶⠖⠒⠛⠛⠛⠛⠛⣿⣿⣿⡿⠛⠛⠛⠉⠉⠛⠻⠿⣿⣿⣿⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡈⠻⣿⠏⠀⠀⢠⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⣃⠀⠀⢸⠀⠀⣀⠀⣀⣄⠀⠀⠀⠀⠀⢸⣿⣇⣤⣀⡀⠀⠀⠀⠀⠆⢰⣿⣿⣿⣄⣤⣼⣿⣷⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⡈⢿⣶⡿⠿⠿⠿⠿⢶⣶⣄⡀⢸⡿⠿⠿⢿⣿⣷⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣷⣾⣄⡀⣤⣤⡀⢐⣿⡟⢿⣦⡁⠀⠀⠀⢘⣿⡟⢿⣶⣄⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣧⣀⡙⠻⣦⣀⠀⠀⣿⡇⠀⢻⣻⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣞⣋⣵⡿⣿⢟⡿⣿⣽⣟⣛⢿⡶⣶⣯⣶⣄⣿⣷⣤⣤⣷⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠿⣿⡹⣿⠝⣳⣌⠀⢸⣿⣯⢿⡿⢶⣿⡤⣟⣉⣁⣛⣛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢻⣶⡤⣽⣿⣿⣿⣿⣧⣷⣘⣟⠇⣸⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣨⠐⠲⣜⣿⣾⠍⡓⠿⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠾⣿⣿⡿⣿⠛⢿⣿⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠦⢶⠓⠒⠉⠉⠉⠄⠙⠀⠂⠒⠅⠀⠊⠉⠳⠌⠛⠛⠻⠗⠐⠚⢉⣃⠤⠄⠉⡒⠳⡯⠛⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠀⠉⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠳⠒⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⢸⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠂⠤⠤⢀⡠⠀⠜⢀⡈⠀⠀⣕⠈⣄⠀⠀⠘⠘⡌⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣴⣥⣶⡄⣼⣧⠀⠀⢸⠀⢸⡄⠀⠀⠀⢻⡀⠑⠝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⣧⠈⠉⠁⠉⠉⠀⠀⠳⣄⢸⡇⠀⠀⠀⠈⢷⠀⠸⣦⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠛⠛⠛⡄⠀⠀⠀⠀⠀⠀⠈⣿⣿⡣⠠⣦⡤⠀⢈⣇⠀⠈⢷⡀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠈⠉⠉⠂⡄⠀⠈⣷⡀⠀⠀⠹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣼⠇⠀⠀⠀⠀⢠⠀⢀⣴⣷⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⡓⣲⣶⣖⠀⠭⠵⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠂⢀⣼⣿⠀⠀⢀⣂⣀⠈⣒⣿⣯⡇⢠⠃⡄⠀⠀⠀⠀⢘⡏⢉⠞⠑⠒⠒⠛⢷⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠐⠈⠉⢹⡷⠚⠋⣿⠀⡇⠀⠀⠀⠹⠁⢒⣡⣴⡖⣶⠊⠁⣹⣣⢃⣀⣠⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣟⠻⠿⠶⡾⣷⢶⡶⣅⣀⣧⣀⣠⣤⣧⣬⣤⠿⠿⠷⢾⣷⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠤⠀⠂⠐⠐⠂⠂⠐⠀⠂⠂⠀⠀⠒⠊⠉⢁⣀⣬⣽⣶⣴⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠁⠘⠛⠉⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣼⡏⠀⣰⠀⠀⠀⣠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢹⣆⠀⠀⣾⡇⠀⢸⣿⣀⣼⡟⠀⢠⣿⠀⣠⡾⠋⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢻⣦⢄⣿⡇⠀⢸⣿⣿⣿⠁⢀⣾⣿⣿⠟⠀⣠⣾⠁⢀⣠⡤⠖
⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣤⡀⠀⠀⠘⣿⡄⠀⠈⣿⣿⣿⣇⡀⢸⣿⣿⡿⣦⣼⣿⣿⣧⣀⣼⣿⣿⠾⠛⠁⣀⡄
⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠈⠻⣶⣤⣄⣿⣷⠀⠀⠸⣿⣿⡿⢿⡿⣿⣧⣘⣯⣿⣧⣼⣿⣿⢟⣿⣇⣠⣶⣿⣃⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠈⢻⣿⣿⣿⣶⣤⣶⣿⣿⣧⣮⣼⣿⣿⡿⠿⠿⠟⠿⠿⢿⣿⣿⡿⢿⡿⠋⠉⢉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠙⢿⣿⣿⡇⢺⠏⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣉⡀
⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⢻⣿⣀⠀⠀⢀⣿⣿⣿⣶⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣄⣈⣉⣉⠙⠒
⢄⡀⠀⠀⠀⠈⠙⠻⢷⣾⣍⣻⣿⣿⣿⠋⣽⡏⢽⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⠀⠀⠀⢹⣿⣿⠛⢿⣍⡁
⠀⠙⠷⣦⣀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣏⡀⣃⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠾⠁⠀⢀⣿⣿⣿⣷⣦⣌⠙⠢
⠀⠀⠀⠈⠙⢿⣶⣄⡀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣄⣤⣴⣿⣿⢿⣦⡘⢧⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣀⣠⣾⠋⣿⠈⢿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢻⢟⣿⠿⣿⢹⣿⡈⣏⠉⠀⠑
⠀⠀⢀⣀⣀⡀⠀⠲⠾⠿⣿⣿⣿⣷⣶⠀⠁⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⡞⠙⢸⠀⠁⠘
⠀⠀⠉⠉⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⢿⡏⠹⣿⣾⣿⣇
⠒⠲⠦⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⣰⡇⠈⠟⠳⠀⣽⣿⣿⣦⡀
⣤⣤⣤⣤⣤⣬⣿⣿⣿⣶⣶⣤⣾⣿⣶⣾⣤⣤⣤⣿⣿⣯⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣟⣻⣙⣛⣿⢛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣏⣻⣿⣿⣿⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠉⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡷⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠉⠀⠘⠻⠓⠒⠂⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠄⠀⠀⡔⠁⠀⠀⠀⠀⢀⠀⢠⣄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠞⠀⢀⣤⣶⡷⢶⣿⣿⣿⠛⠻⢿⣦⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⡠⠃⢠⣴⠿⠥⠤⠄⠞⠻⠟⠋⠀⠀⠒⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠁⢀⣼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⢀⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⡏⢠⡾⠁⠀⡄⢠⣀⣀⣀⡀⢀⡀⣀⡀⢧⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣇⡿⠁⢠⣧⣴⠟⠛⠛⣿⣿⣿⣿⠟⠻⡟⠿⢿⣯⣸⣶⣦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣷⡟⠀⠀⣼⡿⠤⠴⠦⠴⠿⠟⠛⠛⠃⠘⠛⠚⠛⢿⠛⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⢱⡟⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠇⢰⡏⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢋⠛⣷⣦⣤⠤
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡘⢃⠈⡿⠀⠀⠀⠀⣸⣀⣤⣤⣤⣤⣴⣶⣶⣶⣾⣶⠶⣤⣀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⠁⠀⠀⠀⠀⡿⠛⠉⠉⠋⢉⣽⣿⣿⣿⣿⣀⡀⢛⣛⠻⣷⣤⡀⠰
⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠁⢸⡏⠀⠀⠀⠀⠀⡆⠀⠲⠤⠖⠛⠛⠛⠛⠛⠛⠋⠉⡹⠛⢛⣛⠉⢀⠀⢠⣦⡀
⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⠃⣾⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⣿⣷⣦⡾⠷⠦
⠀⠀⠈⠻⣀⠀⠀⠀⠀⢰⣼⣿⣧⣄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠁⠀⠀⠉⠀⠌⢠⠆
⠀⠀⠀⢀⣯⠀⠀⠀⠀⣸⣯⡉⢻⣿⣧⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⡈⠀⠀⠃
⠀⠀⡠⠊⠙⣧⠀⠀⢀⡟⠁⠉⠂⠙⢮⡙⢷⡄⢠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⠀⠁⣄
⠀⢠⠀⠀⠀⠸⣇⠀⢸⠇⠀⠀⠀⠀⠈⠻⢆⠹⣮⡟⠙⢻⣿⣶⣶⣦⣤⣀⣀⠀⠀⠁⠀⠀⠀⠛⠷⠶⠦⡨⢶⣄
⠀⢈⠀⠀⠀⠀⠸⣆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣴⠉⢴⣿⣿⣿⣿⣿⣛⠻⣦⣀⠀⠀⠀⢸⠗⣴⣤⣤⣄⣉⣡⡴⢦
⠀⠘⣄⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣾⣿⢿⣿⣿⣆⠉⢻⣾⣿⡗⠶⣄⠀⠀⠃⠀⠀⠀⠀⠉
⠀⠀⠘⣇⠀⢀⠎⣸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠈⠛⣿⡟⢻⣿⡆⢰⣄⠙⣿⣄⠀⠈⠀⠀⠀⢰⠁
⠀⠀⠀⠹⣠⣿⢠⡟⠌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣆⡀⠀⠀⣿⡇⠈⣿⡏⠀⢿⣦⡙⣉⣧⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠏⠈⠿⣀⣾⣿⣇⠀⠀⠀⠀⣀⣴⣿⣶⣦⣼⣁⠀⢰⣿⣷⠀⢸⣷⡀⢠⣼⠟⢹⣿⠷⣠⣆⠀⢸⣆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆⠀⠀⢰⣿⣿⡟⠠⠛⠛⠿⣷⣿⣿⣷⣀⣼⣿⣏⣸⣿⠀⢀⣿⡄⠘⣿⣧⡀⣝⡳⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⢸⡿⡿⣷⣶⣶⣤⣄⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣀⣯⠙⢈⡙⠳⢄⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⢟⣿⣾⣿⣷⣟⣠⠀⠀⠈⠉⠛⠻⣿⢗⣦⡆⣈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⣿⡟⢨⠁⠀⢠⣤⣄⠀⢀⡀⠈⠙⠻⢿⣷⣦⣌⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢛⣿⡿⢀⠀⠀⠈⠉⠋⠀⢀⡼⠿⣦⣀⡄⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣼⣿⣿⠀⠀⠀⠀⠀⠀⣼⣯⣴⣾⡿⠛⢹⡿⠋⠰⣀⣠⣼⣧⡤⠯⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⢉⣿⣿⠀⠀⠀⢀⠄⠀⡸⠟⠛⠁⠀⠀⠀⠀⠀⠀⠉⠁⠸⢿⣟⢃⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠛⢀⣀⠀⢀⣠⣤⠤⣀⣀⣀⣀⣀⣀⣴⣿⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⣹⣿⣿⣿⡆⠀⢀⣰⠟⠙⠿⠿⢿⣿⣷⣿⡿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣾⣿⣿⣿⠧⠔⢃⣽⣶⣶⣶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣿⣿⣿⣧⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢸⠀⠀⢀⡠⢔⠝⠉⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⢸⣠⡖⣭⠊⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠕⠀⠀⠀⠀⢸⢹⣷⣤⣑⢤⡀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⣼⠘⣿⣿⣿⣿⣾⣶⣄⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⡁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣆⢼⡄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⡇⣸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠷⠾⣄⣀⣀⡀
⠀⠀⠀⢠⣿⣿⡿⠿⠟⠃⠀⠀⠀⠀⠀⡿⢷⡄⡇⠀⠀⠀⠀⠀⠀⣶⣶⡒⢖⠒⡖⠂⠀⡤⠦⢤⠀⠤
⠀⠒⠛⠛⠒⠒⠒⢲⣦⣴⣶⣶⣤⣤⣴⣧⣼⣧⣯⣭⣉⣉⣹⣇⣀⣿⣿⣿⣿⣷⣿⣶⣶⣷⣶⣼⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠴⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢰⠟⠻⠦⠀⠠⠏⠀⢠⠀⠀⠀⠐⠂⠋⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⢰⠀⠉⠀⠚⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠀⡠⠀⠀⢀⣠⠂⠀⢀⠄⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢒⣋⠀⠀⣻⠻⢷⣷⣶⠚⠠⣮⡀⠠⣄⣤⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠙⣯⣻⡿⠤⣀⠈⠻⠀⠀⠀⠀⠀⠠⠙⠻⢷⡚⠿⠉⠂⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⢀⣾⣿⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⡀⠈⢣⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣞⢻⣿⣿⣦⡾⣿⢁⡾⠀⠀⣀⣀⡀⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠰⠄⠴⢄⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡛⣷⣯⣿⣿⣿⣿⣶⡞⠀⠀⠉⠈⣹⡇⣏⡁⠀⣀⣰⠋⢠⡄⢠⠄⠀⠀⠀⣸⠀⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡾⠛⢿⣿⣿⣿⢿⣴⣶⣶⣦⣿⣠⣿⣗⣾⣿⣿⣶⣾⣷⣿⢧⣄⢀⠀⣒⣛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣇⡒⠠⣬⡛⠁⠈⠛⠛⠻⠟⠉⠋⢉⣿⣿⣿⣹⣯⡉⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠩⠛⠋⠿⠊⠡⠍⠛⣿⠦⣤⣤⣀⣀⣀⣎⣀⣀⣀⣀⣀⣼⣨⣽⣾⣿⣿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠊⠚⠋⠹⠏⠉⠉⢩⡉⠉⠉⠉⢁⣉⠼⠳⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠀⠀⠅⠀⠙⠁⠀⢄⠁


⣀⠀⣀⣀⣀⡀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠷⠾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢦⣄⠀⠀⠠⡇⠀⠀⢹⣦⣀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⣯⠑⢦⣀⣇⠀⠀⠸⣿⡷⡑⢢⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡃⠸⠀⠀⣹⣿⠄⠀⢸⠇⡇⠀⠀⠻⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢼⣿⠀⠀⠸⠁⠀⠀⠀⠀⡼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⣿⢇⠀⡚⠀⠀⠀⠀⠀⠀⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⣷⢸⠀⠇⠀⠀⠀⠀⠀⠀⢹⣜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣻⡋⡞⠀⠀⠀⠀⠀⠀⠀⢸⣻⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⡄⠀⠀⠀⢀⠀⣸⢾⣶⠃⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⣈⢂⠀⠀⢢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⣅⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡅⢣⠀⠀⠀⠀⠀⠈⣀⠚⠉⠃⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠀⠀⠀⢀⣠⣤⣽⣿⡧⠖⣂⣉⣀⠀⠀⠀⠀⠸⣿⣧⢸⠀⠀⣀⡀⠀⢈⣵⡖
⠀⠀⠀⠀⠀⠀⣰⣿⠁⣀⠔⠀⠀⠀⢀⡈⢻⣷⣶⣭⣿⠭⠍⣡⠄⠀⢀⣈⣿⣿⣬⠁⠀⣁⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠨⠷⠩⠿⣁⣈⠉⠉⠉⠉⠙⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣠⣶⣃⣤⣶⣿⣿⣿⣿⡟⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⡏⠉⣿⣷⠂⢀⣠⣤⣾⣿⣿⣿⣿⠿⠋⠉⠙⠿⢀⠴⠃⠀⠄
⣠⣄⣄⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣴⡿⢓⢀⣷⠀⢸⣿⡇⢀⣆
⣿⣿⣿⠿⢿⠂⠉⠚⠦⡀⣶⣬⣀⠛⢿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⣿⣇⠉⢈⣉⣶⣾⣿⣿⣿⡀⣿⡿⠛⢸⣿⣦⣤⣀⣀⣠⠤⢢
⠉⠉⠉⠛⠢⠀⠀⠀⠻⣦⣤⣈⠉⠛⢿⣷⣾⣶⣷⣿⣿⣷⣾⣿⡿⠿⠿⠿⣾⣟⠿⠿⣿⡿⠿⠁⠹⠆⠀⢿⣿⣿⣏⠛⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⡤⠌⠉⠉⠉⠛⠶⡿⠍⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠘⢿⣿⡯⠀⠘⢷⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠲⣤⠶⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣴⠀⠈⠉⠉⠙⠻⠂⠀⣼⡆⣾⣤⣾⢟⣅⣴⣵⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⡤⠄⠀⠈⠀⢹⡀⠀⠀⠀⢀⣷⣀⣀⠽⣛⣽⣿⣿⣿⣿⣿⢇⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⣇⣀⣠⣤⠼⠛⠋⠐⣩⢷⣿⣿⣿⣿⣿⣿⣾⣟⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠛⠳⠶⠶⠶⠶⠶⠶⠛⠛⠋⠉⠀⠀⠀⢀⣺⠛⠁⠀⠛⠙⠻⣿⣿⢿⡏⠿⢿⣿⠀⢀⣴⣄⣀
⠀⠀⠠⠀⠀⠀⠀⠠⢀⡤⠶⣴⣶⣦⣿⣿⣿⣿⣿⣾⠿⣧⣀⣀⣀⣀⣀⡀⠀⠠⢬⣃⣴⣿⣖⡁⣠⢄⣠⣤⠀⣠⣤⡐⢾⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⢠⣤⣤⣰⣦⣶⣶⣷⣤⣼⣿⠿⠛⢩⣉⣿⣿⣿⣿⠟⠋⠛⣛⣿⡿⠿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣆⠻⢿⣟⠉
⠀⠀⠀⠀⠀⠈⠉⠉⢭⣝⣛⣛⣩⣥⣴⣶⣶⣿⣻⣿⣿⠿⠋⠁⢴⣶⣶⣮⡝⠛⠂⠀⠀⢶⣶⣮⠙⠛⠛⠿⠿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⣁⡀⠀⠀⠀⣥⣤⣴⣦⣶⣶⣮⣭⣛⠁⠀⢀⠀⢤⣾⣿⣿⣿⠗⠀⠀⠀⠛⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠟⠿⢿⣿⣷⣶⣿⠯⠍⠹⠯⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡤⠞⣻⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣥⠤⠴⠶⠒⣲⠶⠤⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢡⠋⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⡽⠁⢠⠃⠀⡠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⡜⠀⠀⡎⠠⢾⡶⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⡼⠀⠀⠀⠇⠠⠤⠤⣤⣤⣤⣤⣬⡿⠿⠷⢷⣶⣿⣄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣠⣝⠇⢰⠃⠀⡜⠉⠉⠓⠲⠤⠤⠒⠚⠉⢹⢇⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⡤⠊⠐⠲⢄⠪⡏⠀⡆⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⢐⠇⢀⣀⣀⡀⠸⡄
⠀⠠⢤⡔⢁⠀⠀⠀⠀⠀⠙⣄⢸⠀⠀⡇⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣱⠏⠀⠉⠉⠙⠛⠒⠶⠤⣤⣀
⠀⠀⠀⠈⠓⢭⡁⠀⠀⠀⢀⠈⢿⠀⠀⡇⢀⠤⠶⠲⠤⣄⡀⠀⠀⠀⠸⣠⠏⠀⠀⣀⣀⣀⣀⣀⣀⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠈⠉⠐⠂⠰⠥⠤⠤⠤⠤⠤⠭⠷⠶⠤⠞⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢹⡄
⠀⠀⠂⠀⠀⠀⠀⠀⠙⢶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠴⢤⠈⣇⠀⠀⠀⠀⠀⢀⣀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣦⡧⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢰⢰⢸⣀⣿⣤⣤⣤⣬⠥⠤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣿⡿⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣤⠤⠶⠶⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠤⠤⠠⠤⠤⠔⠒⠒⠋⠉⠩⡥⠤⠄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⠀⠈⠉
⣶⣶⡶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠞⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣀⣼⠃⡇⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡋⠙⠷⠠⠀⠀⠐⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⣀⡀⢀⢸⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠤⢬⣿⡀⠀⡧⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣸⠂⠈⠉⠛⢶⠿⢤⣑⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡄⠀⠀⢀⡤⠀⠀⢀⢿⡏⠀⠀⠀⠀⠈⣇⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣘⡂⠘⠀⠦⣽⠇⢀⣴⣾⣿⣁⣀⣀⢀⡀⠀⢹⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠉⠙⠛⠻⠿⣿⣿⣿⣾⣿⣿⡏⠉⠙⠛⠿⠤⣀⣿⠚⡇⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⠀⠈⠈⠉⠛⢻⠃⠀⠀⠀⠀⠀⠀⠈⠙⡟⠲⠦⢄⣱⡀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⠏⡐⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠇⠀⠀⠄⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣣⠠⢂⢀⣄⣠⣀⣤⢄⣴⣿⠋⠀⣥⢀⡄⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⢀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣉⢐⣱⠷⣿⢿⣿⣿⣿⣿⣿⣿⣡⣤⣤⣤⣤⣀⡀⠀⠀⠀⣠⣿⠇⣀⡽⠛⠦⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡛⠛⠿⣿⣿⣿⣧⣽⣿⠟⠽⢻⡿⣿⣿⣿⣿⣯⣅⣀⣈⣷⣿⣿⣴⣯⣤⣀⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⡿⠤⣌⡙⠛⠛⠀⠀⢹⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣀⣴⣞⣛⢿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠏⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣀⡜⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⢸⡛⠹⢿⣾⣿⣿⣿⣿⣿⣿⢋⣥⠶⠦⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠰⣻⣭⣴⣶⣷⣾⣿⣿⣿⣿⣿⣿⣩⣾⣿⠄⠀⠀⠛⠿⡻⡛⠛⢛⣇⠘⠛⠀⠀⢀⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠈⠳⢦⣬⣿⡿⠛⠟⢛⣻⡿⠿⠿⢿⡿⢭⣍⣛⠧⣀⠀⠀⢀⣐⣛⢦⣘⠿⣿⣦⣴⣦⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠈⠉⠴⣶⣿⣿⣿⣷⣦⡌⠛⢷⣬⡀⠐⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⡶⠶⢟⣉⣙⣿⣿⣿⣿⣿⡧⢀⣌⠻⣄⡀⠢⢭⣯⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠋⠙⠿⣿⣿⣟⠹⠇⢀⣿⣦⣀⣙⠛⠲⠶⠾⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠹⠿⠷⣤⡐⠈⠙⠻⠿⠛⠛⠉

⢀⠌⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠔⡩⢂⠥⠂⢁⣠⣴⣶⣭⣶⣯⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠠
⠂⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢂⠔⡡⠂⢁⣠⣾⣿⣿⡿⢟⣻⣽⣿⣿⣿⣿⣿⠟⠋⠹⣿⣿⣿⣏⠀⠘⢿⣿⣿⡟⠉⠹⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⢀⠔⠁⠈⢀⣴⣿⣿⠿⣛⣵⣾⣿⣿⡿⠋⠹⣿⣿⣿⣦⣄⣀⠈⠙⠛⠻⠷⣦⣀⡈⠛⠿⣦⣀⠙⠻⣿⣯⠀⠹⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠐⠁⠀⠀⡰⠡⢊⢀⣴⣿⡿⢛⣥⣾⣿⣿⢿⣿⣿⣄⣀⣀⣀⣈⣉⣉⣉⣉⣉⣛⠓⠲⠶⢤⣌⣙⠳⢦⣄⡙⠳⢤⡈⠛⢷⡄⠘⢿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠐⠐⡡⡿⡻⢋⣴⣿⣿⣿⡿⠁⠙⠛⠛⠛⠛⢉⣉⣉⣉⣭⣭⠭⠍⣉⣉⣙⠓⠲⠦⣭⣙⠲⢬⣙⠳⣤⡙⠦⣀⠙⢦⡀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠌⣸⠏⠀⣠⣿⣿⣿⣿⣿⣧⣀⣤⣤⠶⠞⢛⣋⠭⠥⠒⣒⣒⣂⣐⣒⡒⠲⠬⢍⣁⠒⠨⣙⠲⢬⡛⢦⣙⢷⣌⠳⣄⠙⢦⠈⢳
⠀⠀⠀⠀⠀⠀⠆⠀⠀⢜⠋⠀⣼⣿⣿⣿⠏⢿⠿⠿⠛⢉⣤⠶⣚⡭⠖⣚⡩⠥⠤⢄⣒⣒⡴⠦⠭⢍⣒⠠⠍⡓⠦⣉⠢⢍⠳⢌⡳⣌⠳⣌⠳⣄⠳⣄
⠀⠀⠀⠀⠀⡘⠀⡈⠌⠊⠀⣼⢳⣿⣿⣿⠀⠀⣀⣤⠾⢋⡴⢞⡥⢚⡩⠔⣂⡩⠤⠴⠖⠚⠶⠬⠭⣕⣒⠬⣑⠢⢀⠀⠙⠢⡙⢦⡙⢮⠳⣌⢳⣌⢦⡘
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣸⠇⣾⣿⢻⣿⣷⡿⠛⢁⡾⢋⠔⣩⢖⡡⢒⡭⠖⣛⠭⠁⣀⣀⣀⣀⡀⠀⢈⠙⠒⢝⡢⢄⠀⠀⠈⠀⠙⢦⠱⣌⢳⡙⣦⠳
⠀⠀⠂⠀⠀⠀⡌⠀⠀⢀⡟⠀⡍⣿⠘⠛⠋⢀⣴⠟⡴⢫⢞⡵⢣⠞⣩⠖⣩⠔⣊⣥⠤⠶⠶⠶⠮⢭⣖⡮⣓⢴⣬⡂⠑⠄⠀⠀⠀⠀⠱⡌⢦⠹⡌⢧
⠀⠀⠀⠇⠀⠀⡇⠀⠀⢸⡇⠀⡇⣿⡀⣀⣤⡾⢁⡾⢡⢏⡞⡴⢣⡞⣡⠎⣴⠞⣩⣴⠞⠋⢉⣉⣙⣓⡲⢭⣛⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠘⣄⢣⠹⡎
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⢁⢹⣿⣿⠟⠀⡼⢁⡏⡞⣸⢁⡏⣰⠃⣾⠃⠀⣿⡇⠀⢸⣿⡏⠉⠉⠙⠻⢮⣷⣮⡻⣿⣿⣦⠀⠀⠀⠀⠀⠘⡆⢂⠹
⠀⠀⠀⠀⠀⠈⡇⠀⠀⢸⡀⠠⠸⡄⠉⠁⠀⣼⠇⣸⢁⡇⣿⢸⠀⣿⠀⢿⣆⡀⠙⢿⡄⠈⢿⡇⠀⢸⣿⠿⢶⣍⡻⣽⣌⢿⣿⡥⠀⠀⠀⠀⠀⠸⡀
⠀⠀⡇⠀⠀⠀⢇⠀⠀⢈⣇⠀⢇⢻⣦⣀⣼⡿⠀⣿⠀⣇⣿⠸⡆⢻⣆⡈⠙⠛⠶⣄⠙⣦⠈⣷⠀⢸⡟⠀⠀⠈⠻⣟⣮⢎⣯⠣⠁⠀⠀⠀⠀⠀⢣
⡆⠀⢡⠀⠀⠀⢸⡀⠀⠀⢟⣂⢿⣧⡙⠿⠿⠃⠀⣿⠀⢿⠘⢧⠙⣦⣉⠛⠿⠷⣦⣌⢳⡌⢷⠸⡇⢺⡇⢀⣾⣿⣷⣌⢿⠎⢹⣇⠀⠀⠀⠀⠀⠀⠘⡄
⢣⠀⠈⡆⠀⠀⠀⢧⠀⠌⣮⣯⢻⣻⣿⣦⡀⠀⣸⣿⡄⠸⣷⠈⢳⣄⠙⠻⣶⣄⠀⢻⡆⢳⢸⡇⡇⢺⠇⢸⡿⠁⠈⠻⡎⢇⠀⣯⠀⠀⠀⠄⠀⠀⠀⡇
⠘⡆⢂⠸⡀⠀⠀⠈⢇⠈⢼⣿⣧⡹⣝⣾⣝⠻⠿⠿⠁⠀⣿⣧⠀⠹⣷⡄⠈⢻⡇⢸⡇⣼⢸⡇⡇⣿⢀⡿⠁⢀⣤⣄⢻⠘⠀⢹⠀⠀⠀⢀⠀⡇
⣧⢹⡌⢆⢳⡀⠀⠀⠈⠢⠀⠪⣿⣷⣮⠛⢿⣻⠶⣤⣤⣴⣿⠟⠀⢀⣿⠇⢀⡾⢃⡞⣰⠇⡞⣸⢡⠇⡼⠁⣰⣿⢿⣿⡄⠀⠀⣾⠀⠀⠀⠘⠀⡇
⡘⣇⢳⡈⢆⠱⡀⠀⠀⠀⡑⠀⠈⠻⣮⣻⢷⣬⣝⣒⠦⠤⠤⠴⠶⢛⣩⡴⢛⡵⢋⡴⢃⠞⡰⢃⠎⡼⢁⣼⠋⠀⠀⢸⡇⠇⠀⣿⠀⠀⠀⡇⠆⡅⠂
⢳⡜⢧⡹⣌⢣⡙⣄⠀⠐⢌⠢⡀⠀⠨⣈⠑⠯⠓⠫⠍⣙⣛⣛⣉⣉⠤⢒⡩⠔⣫⠔⡣⠞⡡⢣⠞⣡⠞⠁⣠⣶⣶⢸⣷⡼⢰⡇⠀⠀⢠⠁⢨⢁⠀⠁
⢦⠻⣌⢳⡝⢦⡙⣌⠣⡄⠀⡑⢬⡲⢄⡈⠑⠲⢤⣀⡀⠀⠀⠀⠀⠀⠬⠥⣖⡫⠕⡊⠔⣪⠔⣡⠞⢁⣴⠾⠟⢿⣿⣾⣿⢣⡿⠀⠀⠀⡎⠀⡜⠀⠐
⡈⢳⡘⢧⡙⢦⡙⣮⡳⣌⠒⢌⠢⢌⡒⢭⣓⠦⣤⣀⡉⠉⠙⠛⠛⣿⠭⠭⢖⣒⣭⠔⣪⡵⢚⣥⠾⠋⠁⣀⠀⢸⣿⣿⣏⡾⠁⠠⠠⡰⠁⠰
⢳⣄⠹⣆⠙⣦⡙⢦⡙⢮⣙⢦⣁⠢⢉⠓⠬⣍⣒⠢⠭⢉⣉⣓⣒⣒⣈⣩⠭⠴⢖⣫⠵⠚⣋⣠⣴⣾⣿⣿⣧⣿⣿⣿⡿⠁⣀⡵⡱⠁⠠⠁
⡀⠹⣦⡈⠳⣄⠛⢦⡙⠶⣍⡳⢮⣙⠲⢬⣀⠒⠨⠭⣉⣒⣒⡒⠒⢒⣒⣒⣈⣭⣥⠶⠶⠛⠋⠉⠁⠈⣿⣿⣿⣿⣿⠋⡀⣰⡟⡴⡡⠡⠁
⣿⡄⠘⣿⣄⠈⢳⣄⠙⠲⣌⠙⠶⣬⣙⠲⢬⣍⣛⠲⠶⠦⣤⣬⣬⣭⣭⣥⣤⣤⣤⣶⣶⣶⣾⣶⢄⣴⣿⣿⣿⢟⣡⣪⣿⢏⠞⡐⠑⠁⠀⠀⡔
⠻⣷⣦⣿⣿⣷⡀⠙⢷⣦⡈⠙⢦⣄⡉⠛⠶⣤⣉⡙⠛⠒⠶⠦⢤⣤⣤⣤⣄⣀⡀⠀⠀⣹⣿⣿⣿⣿⡿⣋⣵⣿⣿⢟⠕⡣⣪⠈⠀⠀⢀⠞
⠑⠌⠻⢟⠿⢿⣿⣄⣨⣿⣿⣦⡀⠉⢿⣷⣦⣀⠈⠙⠿⣶⣦⣤⡀⠀⠉⢻⣿⣿⣿⣴⣾⣿⣿⢿⣛⣵⣾⣿⣿⢟⠕⡫⢊⠔⠁⠀⠀⡰⠋
⠄⠀⠈⠂⠑⠢⠈⠉⠛⠿⠿⢿⣿⣤⣴⣿⣿⣿⣷⣀⣤⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣽⣶⣿⣿⣿⠿⠫⢊⠕⡪⢞⠅⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠛⠻⠟⠛⠝⠊⠙⠈⠔⠊⠕⠫⠒⠁⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠀⣀⣀⣷⠀⠀⠀⣀⡠⢤⠀⠠⠤⠤⠀⣸⣇⠀⠀⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠤⣀⣀⣀⣀⡀⡄⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢴⠁⣤⣤⠄⠑⡤⠇⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡔⠚⢢⣬⣩⢤⠚⢒⡢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠣⠃⠉⠀⠁⠑⠊⠀⢸⠀⠀⣘⣀⡀⠀⢀⣀⣀⣀⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠤⠤⠤⠤⡤⠤⠤⠀⠀⢀⡗⠂⠩⠤⠀⠀⢀⣇⣀⣀⣀⡈⠉⠤⠒⡆
⠀⠀⠀⠀⠀⠀⠰⠒⠠⠤⠤⠀⣀⣀⣇⡀⠀⢀⢀⣀⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢄⠀⢀⢤⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠒⠒⠒⠒⠒⡦⠀⠀⠠⠤⠄⢀⣆⠀⠀⢺⣻⠿⢯⡉⡙
⠈⠑⡀⠑⢧⡀⠃⠉⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠉⠀⠐⠚⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠳⣤⣴⠟
⠀⠀⠘⡄⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡠⡪⠃⠀⠀⡘
⠀⠀⠀⠰⡀⠀⠀⠈⠉⠛⠛⠓⠒⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠛⠛⠓⠉⠀⠀⠀⢠⠁
⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣒⣒⣒⣒⣂⣒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⣆⠀⠀⠀⣀⡤⠖⠲⣄⠀⠀⢀⡴⠒⠶⣄⠀⠀⢀⡴⠒⠶⣄⠀⠀⠀⣴⠖⠒⢦⣀⣀⣀⡰
⠀⠀⠀⠀⠀⠸⡀⡀⠀⠹⣇⠀⠀⣸⠂⠀⢸⡀⠀⠀⡽⠀⠀⢸⡀⠀⠀⣽⠀⠀⠘⣇⠀⠀⣨⠇⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢣⠤⠀⠀⠉⠓⣚⣁⣀⡀⠀⠙⠒⠋⠁⠐⠒⠀⠙⠓⠋⠁⢀⣀⣀⣈⡙⠋⠍⠀⠤⡌
⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠤⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠤⠄⠒⠊⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠃⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠀⠉⠙⢻⣿⣿⣿⡏⠙⠟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⢿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣷⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠿⠽⠿⠛⠿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣸⠟⣿⡿⠋⠘⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣬⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣏⠉⠉⢀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣧⣬⣜⣓⠤⡦⠶⠶⠤⣄⣚⣶⠈⠹⠹
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡤⠈⠙⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠐⠀⡴⣤⣴⡄⢘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣸⣿⣗⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⠳⠆⠶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢾⠋⢻⡆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⠓⠒⠶⠾⣧⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠁⠉⠛⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠐⡐⠒⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠐⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⣠⣤⣶⣶⣶⣾⣿⣖⣆⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠺⠟⠿⠿⠯⠭⠼⢿⣟⣿⣿⣻⣇⣀⣀⣀⠀⠀⠄⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠞⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⢓⡿
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠰⠦⠒⠒⠒⢾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡠⠴⠐⣿⡏⠉⣉⣁⣂⣿⣧⣤⡤⡀⠤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠶⠷⠒⠛⠋⢻⣿⠉⠉⢉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣦⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡻⣿⣿⣿⣿⣯⣾⣿⢻⣷⡄⠙⢤⣶⢖⣖⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠇⣛⣭⣭⣿⣶⣾⣿⠷⣄⣰⣿⣮⣷⠿⠋⠉⠁⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢈⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⠛⠻⣿⣯⡉⣻⡿⣿⣿⡟⠀⣼⣿⡇⣿⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣧⣜⡀⠚⠻⣃⡁⠀⠈⠻⢧⡶⣿⠿⢶⣿⣆⢷⣄
⠀⠀⠀⠀⠀⢀⣤⣿⡄⢈⡁⠉⠿⠛⠛⠛⠦⡀⠉⠑⠙⣀⡀⠀⠀⣉⠀⠀⠻⣿⣷⣭⡛⠒⠀⣀⣠⣤⣀⣀
⠀⠀⠀⠠⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡐⠦⠉⠉⠻⣶⣄⡉⠛⠛⠛⠋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣷⣶⣤⣤⣬⣭⢤⣀⠐⠳⢿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⣇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⠋⢶⣿⡤⠊⠀⣰⠄⡰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⡄⠋⢰⡇⠘⠟⠋⠀⣤⡾⠴⠟
⠀⠀⠀⠀⠀⡰⠋⡤⠀⠂⠀⠀⣠⣶⡆⠀⠀⠠⣄⡈⢾⡿⠃⠀⣰⡆⠀⢁⠄
⠀⠀⠀⡾⠀⡀⠘⠀⠀⠀⠀⠀⠙⢻⠀⠀⠀⣽⣿⣿⣬⣍⣓⡂⠉⠀⢰⢏
⠀⠀⠀⠁⠀⣿⢂⠀⠀⠀⠀⠀⠀⠈⠀⠀⣄⣹⣿⣿⢹⣿⣿⡛⠉⠉⠀⠀⠁⡀
⠀⠀⢰⣷⠀⠇⠠⠀⠀⠀⠀⠀⠀⠈⣧⠀⠻⣿⡿⡟⢸⣿⣿⣷⣆⡀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⠀⠀⢆⠀⠀⠀⠀⠀⢘⣿⣧⣀⣠⣴⣤⣼⣿⣿⣿⣿⣇⣠⣦⠀⠀⢸⡀
⠀⠀⠀⠁⠀⠸⣷⡘⣮⣴⡶⡀⠀⠈⠿⣿⠟⠁⠉⠁⠀⠉⠉⢻⡿⡍⠀⠉
⠀⠀⠀⠀⣀⠀⠉⡇⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢱⣕⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠻⣧⠀⠁⠀⠈⠻⣷⡀⢀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢇⢔⣄⠀⢠⣴⠶⠚⣯⢄⡄
⠀⠀⠀⠀⠀⢈⡑⠀⠀⠠⣤⡈⠛⢿⣶⠿⢁⡀⣴⡤⣦⣤⣤⣤⣶⣿⣫⣄⠈⠙⣜⣻⣿⣯⣏⣽⡟
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠙⢷⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠳⡄⠀⠀⠉⠀⣶⡌⢿⣿⣿⣿⣿⣿⠿⠛⠙⠛⠛⠛⠉⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀⠀⠈⠹⡌⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⣤⡐⣦⣠⣆⠐⠿⣿⠷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠄⠀⠀⠀⠀⠀⢀⡞⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠊⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⢠⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣴⡥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣾⣟⣿⣤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣶⣶⣶⣤⣄⡀⠀⣻⠀⠀⠂⠀⠄⠀⠀⠀⣷⣾⣿⡇⠀⠀⠀⠒⠀⠀⠀⡀⣤⢦⣴⣾⡿⢈⣩⡅⠀⠀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠤⠤⠤⠀⠀⢒⣀⡁⠙⢯⣟⣻⠿⣿⣿⣿⣿⣿⣧⣤⣦⣁⣄⠀⠀⢰⣾⣿⣿⣿⣁⣈⠤⠄⠖⠚⠁⠉⠁⠀⢸⣿⣿⠃⢸⡟⠁⠀⠀⠀⠠⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠚⠛⠂⠈⠭⠅⠀⠜⢾⣭⣑⣒⠶⠝⢯⢿⡧⣿⣿⣿⣿⣿⣿⣿⣿⢏⣩⣤⠤⠤⠄⠲⣒⠺⡋⠙⣁⣐⡀⣿⠇⠀⠈⡄⠀⡀⠀⠉⠀⠐⠶⠆
⠀⠀⠀⠀⠀⠀⠠⠭⠤⠤⠴⠦⢤⣤⣤⣻⣿⣿⣿⣿⣳⣲⡦⠴⠤⣤⢤⢤⣤⣤⣄⣐⣀⡀⠀⠤⠌⢚⣓⣲⣋⢯⣝⣶⣴⡿⢄⡁⠀⢳⠀⣀⠘⠛⠋⣀⣀⣀
⠀⠐⠒⠀⠒⣒⣒⣒⣒⣂⣀⣀⡀⠀⠒⠚⠚⠻⢿⣿⣿⣿⣿⣿⣿⣿⣾⣴⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠇⡔⠂⠀⣼⠤⠤⠄⠀⠀⢀
⠀⠀⠀⠀⣉⣉⣩⣭⣉⠉⠙⠛⣻⣿⣿⣧⣤⣀⣀⡀⢀⣉⡉⣛⠛⢛⠙⠛⠿⠿⠟⠛⠿⠿⢿⡿⢛⣻⠟⠛⢋⠀⠀⣀⢀⡤⣄⠙⠀⠙⠛⠓⠒⠒⠀⠀⠙⠀⠒
⠀⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠈⠉⠀⠶⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠂⠙⠉⠀⠀⠉⠃⠀⠰⣆⣀⣉⡳⠦⠔⢛⠓⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁
⣶⣶⣴⣤⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣤⣦⣶⣦⣴⣴⣶⣤
⣿⣿⣏⣀⣁⣨⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢳⠀⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡜⣧⠀⢻⠟⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣽⡄⠈⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡀⢨⣿⡄⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠠⠐⠒⠁⠈⠉⠀⠀⠀⢸⣧⣼⠟⢻⡄⠀⠰⡀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⣿⣄⠀⠀⢀⡄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠀⠀⠀⠀⢿⢸⡆⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣠⣀⣶⡾⠿⠯⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡀⠀⢠⡾⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠉⠛⠃⠒⠶⠶⠦⢤⣤⣴⡇⠀⣠⣾⣦⣴⣯⣀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠉⠙⠻⠿⢯⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⠸⢿⢰⣶⡤⣬⣤⣤⣀⣄⡄⠀⠀⢀⣀⣀⣤⠤⡶⠃
⠀⠀⠀⠀⠀⠰⣬⣵⣾⣿⣿⡏⠉⠉⠀⠀⠀⣀⡀⣀⣒⣒⣴⠺⣥⡿⣿⠶⠞⠛⠫⠉⠑⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⡿⠿⣿⣦⣦⣤⣤⡤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠬⠽⠷⠶⣿⣿⠠⠔⠒⠤⠄⣀⣀⣤⠤⠀⠀⠤⣤⣤⣤⡼⠛⠛⠁⢠⣤⡤⠤⠄
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⣉⡉⠀⢀⣀⣀⣀⣉⣉⣉⣀⠀⠄⠀⠀⠀⣉⡥⣤⣄⣓⡙⠉
⠀⠀⠀⠀⠀⠀⠠⠔⠛⠋⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠒⠒