Happy halloween Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠞⢲⣄⠀⠀⠀⠀⠀⠀⠀⣀⠰⠋⢘⠄⢠⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢐
⠀⠀⠀⠀⠀⢠⡟⣾⡼⠋⢳⠀⠀⠀⢀⡴⣪⡼⠃⡠⠋⠀⠈⠐⠂⠀⠄⠠⠤⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘
⠀⠀⠀⠀⢀⣸⣇⠸⣇⡇⠸⠗⢆⢠⡏⢠⡟⠀⠜⡇⠀⠀⠀⠀⣧⢼⠀⡀⣠⣀⣀⣀⢀⠀⢨⡅⡟⢀⡀⡅⢁⢀⡀⣀⢀⡀⢠⢀⡀⣸
⠀⠀⠀⠀⢸⠀⣌⡛⠛⣇⢰⡴⢸⡎⡇⠸⡹⣄⠃⢑⡤⠦⠤⡀⠃⠰⠘⠙⠚⠁⢘⠘⢼⠀⠀⠃⠰⠙⠑⠓⠚⠐⠛⠋⠇⠇⠸⠘⠻⠎
⠀⠀⠀⠀⠘⣆⡛⠛⣺⠋⠻⠤⢚⢃⢇⠠⠱⠌⠹⣏⡤⠖⣚⣛⣛⠲⢦⡀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣤
⠀⠀⢶⣯⣶⠴⣿⡉⡡⠖⠒⢲⡁⠈⠘⠀⠇⢠⢒⣭⣽⠒⠦⣍⠻⣿⣦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠞⠁
⠀⠀⠀⠈⠛⢿⣿⣿⠖⣛⢉⡽⣫⠭⠐⢦⣴⣿⡴⠚⠓⠶⣤⣈⠙⣮⠻⣿⣎⢳⡀
⠀⠀⠀⠀⠀⢀⡞⣡⣾⢃⡞⢈⡵⠋⠙⣿⢰⣆⢻⡄⢴⣶⣹⠟⡓⣮⣷⡘⣿⡌⣧
⠀⠀⠀⠀⠀⣼⢰⣿⠇⡾⢀⣾⠃⣶⣆⢻⡄⣿⡆⢿⠞⣋⣵⡞⣱⠋⣿⢷⡸⡇⣿
⠀⠀⠀⠀⠀⣿⢸⣿⢰⡇⡾⢛⣛⣛⣛⡛⣇⢻⣿⡌⢧⠙⣩⠞⠁⢀⡿⢀⣧⡥⣿⠀⢀⡀
⠀⠀⠀⠀⠀⢻⠸⣿⠘⣿⡟⠦⣍⣛⣛⣃⣿⢘⣋⢭⣴⡿⠁⠀⣠⡾⣿⠉⣤⡆⣼⠛⣉⣽⡛⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠘⣧⢻⡇⢿⠻⣄⠀⠀⠉⠉⠛⠛⠃⠀⠁⣀⣤⡾⢟⣡⣿⣐⣏⢀⣷⣿⡭⠛⢶⡞⡠⠒⡓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠈⢷⣍⡘⣆⢨⡛⠷⢶⣶⣶⣶⣶⣶⠿⠟⢻⣶⠦⠴⢞⠋⡼⣏⡤⠀⣤⠖⠋⣀⣳⡛⣫⠥⣟⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦⣙⠿⠶⢂⣴⢿⣄⡒⠞⣋⡞⠀⠀⠔⠀⠌⡷⢌⡭⠴⠛⠀⠰⡉⡧⠉⢻⣳⣦⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠀⠈⠉⠉⠉⠀⠀⠀⠈⠁⠙⢄⣀⠤⠒⠉⠉⠑⡉⠢⠞⠢⠬⠷⠼⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⡿⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠁⢀⢀⡀⠈⠙⠛⠛⠋⠁⠤⣶⣶⣶⣄⣠⣠⣄⣴⣾⣿⠟
⠀⠀⠀⠀⣾⣿⣦⠀⠀⠑⡄⢸⠚⣣⣤⣰⠂⣶⠤⢰⠀⠈⠛⠿⢿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠁⠈⢀⡉⠀⠉⡄⣥⠈⠽⡤⠀⠀⠀⠀⠉⠉⠉⠀⡄
⠀⠀⠀⠙⠿⡿⠙⠋⠀⡰⠀⠀⠈⠷⠶⠄⡏⢃⡴⢦⡇⣇⡲⡔⡤⣾⢸⢰⢴⠃
⠀⠀⠀⠀⠀⠻⡀⠀⠀⢁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠀⠀⠀⠂
⠀⠀⢀⣀⡀⣀⣀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯
⠀⠘⣿⣿⣿⣿⣿⡷⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣿⣿⣿⠇
⠀⠀⠈⢻⣿⣿⣿⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠛⣻⣿⡟⠛⣿⣿⣿⣿⡝⠿⠃
⠀⠀⠀⠀⠉⢻⢫⣶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣷⣿⣾⡿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡀⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣌⠙⠻⣿⠿⣿⠟⠋⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠳⣄⣀⠀⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣩⣻⣿⣿⣶⣦⣤⣴⣶⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠈⠉⠉⠛⠿⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⢀⣀⣠⢄⡠⡄⠠⡇⡇⣠⡄⡇⡇⢠⠀⡀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡆⡇⡞⠾⠃⢸⠀⢚⠇⠀⠁⠉⠁⠉⠁⠉⠊⠛⠙⠹⠸⡇⣦⡼
⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⣄⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⡄
⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⢾⣿⠂⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢺⣿⠆
⠀⠀⠀⠀⢩⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣨⠁
⠀⠴⣤⡄⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣤⡴⠄
⠀⠀⠈⣡⣈⠉⠉⠀⢀⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣆⠀⠈⠉⢉⣠⡉
⠀⠀⢘⣿⣇⣠⣾⣿⢿⣿⠿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⢿⣿⢿⣿⣦⣀⣿⣟
⠀⠀⢸⠁⢹⣿⣿⢿⣿⣿⣾⡿⣿⣷⢀⣶⠟⠀⠀⠀⠀⠀⠘⢷⣆⢰⣿⡿⣿⣾⣿⣿⢿⣿⣿⠁⢹
⠀⠀⣾⣶⣾⡌⠻⣶⣬⣭⣥⣾⠟⠁⢀⣿⣷⣄⠀⠀⠀⢀⣴⣿⡇⠀⠙⢿⣦⣭⣭⣴⣾⠋⣼⣶⣾⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡦
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⡇⢀⡶⣿⢻⡆⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣷⣀⡠⢤⠀⠀⢠⡀
⠀⠀⠠⠀⠀⠀⠉⠀⣀⠀⠻⣿⡿⠟⢿⠷⠃⠀⠀⠀⢴⠄⠀⠸⣄⡀⠀⠀⠀⢠⣄⡀⠀⠀⠀⢀⣿⣿⡏⠸⠀⠶⠃⠀⠀⠑⠠⠶
⠀⠀⠀⠀⠀⡠⠄⣴⣿⣿⣄⣻⣶⣾⣾⣧⣤⡄⢸⡶⣮⣷⠀⠀⠀⠀⠀⣶⠀⣠⠈⡷⠀⠀⢰⣾⣿⣿⣧⣎⡀⠀⠶⢿⣷⣤⡀
⠀⠠⢤⣀⠀⠰⠀⡿⠀⠈⠉⠛⠛⠻⠿⠿⠛⠛⠛⠿⠿⠿⠖⠒⠒⠶⠶⠿⠷⠿⠿⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⢸⡏⠃⢁⣀⣀⣀⣀
⠲⣤⢾⣿⣷⣑⡾⠁⠀⠀⠀⠀⡠⣴⡖⠀⣶⠦⠀⣠⣦⠠⣤⠀⠀⠀⠇⠂⠠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣥⣯⣷⣦⡀⢠
⠀⠈⠙⢛⣿⡏⠀⡀⠀⠀⠀⠀⠁⢻⡇⠀⡇⠀⣠⡈⢹⠀⡏⠀⠀⡠⡄⠀⠀⡤⣠⠀⢀⡤⡄⠀⡀⢀⠀⠀⠀⠀⠀⢀⠀⠈⣯⢻⡿⠃⡸
⠀⠀⡴⢿⣿⡇⠀⠁⠀⠀⠀⠀⠀⢸⣅⣀⡇⣰⢂⡝⠘⠀⢁⢸⣽⠆⢇⡽⣰⠃⣜⠢⢼⡄⡚⠁⡷⠋⣧⠀⠀⠀⠀⠘⠁⢀⡿⠋⠓⠋⠁
⠀⠀⠙⠉⠙⣿⣳⡄⠀⠀⠀⠀⠀⠸⡇⠀⡇⠓⠸⠌⠀⠇⠉⠐⠃⠀⠾⠃⠙⠀⣇⣲⠈⠓⠃⠠⠇⠀⢿⠀⠀⠀⠀⠀⣰⣋
⠀⠀⠀⠀⢨⣐⠈⣷⠀⠀⠀⠀⠀⠀⢁⠀⠁⠀⠀⢀⣀⣀⣀⣀⣠⣤⣠⣤⣤⣤⣤⣤⣤⣀⣀⣠⣤⣤⣤⣄⣀⣀⣀⢰⡏⡮⠃
⠀⠀⠀⠀⠀⠁⠀⠙⠚⠛⣛⣿⣿⣭⣍⡉⢿⣿⡿⠁⠉⠙⠀⡄⠁⠐⠈⠉⠀⠀⠴⣿⣿⣫⣹⣿⡍⠉⠉⠉⣻⣿⣿⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢻⣿⣏⣻⣿⡆⠀⠀⠀⠁⠀⠄⠠⠀⠤⠀⠀⠀⠀⠰⢿⣾⡿⠛⠋⠀⠀⠀⠀⠈⢻⠛⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣍⣉⣛⣫⣵⡿⠃⠀⠀⠀⢀⠌⠀⠀⢁⠀⠈⠁⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⣷⣦⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠻⠿⠂⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣶⣷⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠜⣿⣿⣷⠥⠀⠀⠀⠀⠛⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣄⣦⣀
⠀⠀⠀⠀⠈⠊⠛⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣤⡄⢰⡿⠀⠀⠀⠹⠿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣦⣄⠀⠀⠀⠀⠀⢀⣤⣾⡿⠃⠀⠈⢀⣿⠀⣾⡇⠀⠀⠀⠀⠀⠙⠋⠙⠟⠿⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠿⠋⠛⠃⠀⠀⠀⣠⣾⣿⠿⠏⠀⠀⠀⠀⣺⡟⢺⣿⣴⠗⣾⢲⡖⣦⣴⡖⣦⣶⠀⣶
⠀⠀⠀⠀⠀⣤⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣰⣿⠁⢼⠇⢿⡜⢯⣿⣃⣿⣿⣀⡿⣿⣰⣿⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠔⢸⡟⢀⣿⠀⠀⠉⢡⣶⢀⣴⡀⠀⢠⡿⠉⢡⡟⠉⠁⠈⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⣸⡟⣤⣀⡄⣾⡟⣼⡗⢠⣄⡙⣡⡀⠙⠁⢀⡀⠘⢚⡁⠀⣀
⠀⠀⠀⢀⣠⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⣿⢿⡇⣾⣳⡟⢠⡟⢠⣿⣿⣇⣿⣶⣠⣤⣿⠃⢰⣿⠃⢸⡟⣻⠆
⠀⠀⠀⠀⠻⣿⣿⣿⣤⡄⠀⠀⠀⠀⣀⣠⣦⡄⠀⠀⠀⢤⣿⠇⠐⠛⠈⠉⠈⠹⠗⠻⡷⠊⠛⠊⠉⠻⢻⡼⠋⠛⠂⠈⠛⠂⠙⢁⣿⠌
⠀⠀⠀⠀⠀⠙⠻⠛⠛⠃⠀⠀⣀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠾⠰⠏⠾⠀⢸⠼⠆⠸⠸⠶⠃⠇⠇
⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣤⣟⣿⣻⣷⠀⠀⠀⠀⣶⣾⡆
⠀⠀⠀⠀⠉⠉⠛⠉⠉⠀⠀⠀⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢠⠶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣄⣀⣀⣀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠿⠟⠛⠛⠿⠿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⢀⣠⣴⣶⣶⣶⣤⣀⠀⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⣛⠿⢿⣿⣿⡟⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⣀⣴⡿⠋⠁⠀⠈⠻⢿⣿⣿⣷⣶⣤⡌⢻⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣍⣛⣛⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢡⣶⣿⣿⣿⣇⣄⣼⠇⠉⠙⣏⠹⢻⣿⣿⡿⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠈⢻⣿⣿⣿⡇⠉⠃⠀⣤⠂⠸⠀⣼⣿⡿⠁⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣾⣄⠈⠉⠛⠻⠿⣿⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠻⣿⣿⣟⠂⠀⠀⠀⠀⢀⣿⠟⡁⠀⠀⠀⠀⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣄⠉⠙⠓⢠⣦⡀⠐⠛⠋⠉⡛⢻⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⠀⠀⢉⣙⡂⣐⡖⠀⢀⣾⣥⡾⠀⠐⢦⣄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠙⢷⣀⣠⣤⣤⣤⣤⣀⣴⠟⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠰⢚⣯⠄⠀⠀⢻⣿⣏⣙⣀⣋⣹⡿⠃⡤⠛⠳⠄⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠁⠙⠛⠛⠀⠉⠁⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠉⠴⣾⠗⡤⠀⠙⠛⠿⠿⠛⠋⡠⣆⠙⠳⠆⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠑⠒⠋⠒⠒⠉⣉⠁⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢼⡁⣼⢠⡞⢳⠲⣼⢧⡇⠘⠚⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣷⣿⣼⣷⣿⣸⡠⣏⡿⣧⣶⢿⣏⢹⣇⣹⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⡛⠋⢀⠉⠀⠅⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠈⣀⠁⢈⡀⡁⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠋⣁⠀⣈⡙⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⡠⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠇⠘⣿⠇⠿⠇⢹⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠐⣿⢷⣄⣰⡦⠀⠀⡗⡇⣸⡀⡧⠃⡧⠂⠚⠀⠀⢴⣆⣠⢿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠰⠀⠀⠸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠉⠤⠙⠛⠤⠄⠀⠁⠁⠁⠃⠃⠀⠃⠀⠀⠀⠤⠤⠛⠉⠤⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢴⢠⡆⢰⣆⠀⡆⠀⢴⠀⢠⡖⡦⢰⢠⡆⣴⢰⢶⠆⡖⡶⢰⡄⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠻⡇⣾⣻⠀⡇⠀⢸⠀⠸⡇⣯⠈⣾⣷⡏⢸⠆⠀⡷⠀⢸⢳⡇⠀⠀⠀⠀⠈⣿⣿⣿⣿⣏⣠⣄⠀⠀⢀⣠⣄⠀⢀⣤⣀⣀⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠺⠀⠇⠏⠹⠆⠧⠧⠸⠸⠄⠧⠿⠀⠛⠸⠁⠸⠬⠆⠧⠷⠸⠀⠇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⠦⢶⣿⠿⢿⣷⠿⠿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⢈⣋⠸⠁⣾⠆⡁⢸⡷⢸⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣭⣭⣿⣽⣯⣭⣿⣿⣭⣽⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣤⣅⣴⣷⣤⣴⣿⣦⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⠿⠿⠿⠛⠉

⣀⡀⣀⢀⣀⣀⣀⣀⠀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠘⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠲⢤⢛⡤⢄⡤⢄
⠀⠀⠀⠀⠀⠀⠈⠙⣷⣴⣶⣆⣀⣠⣴⣾⡷⠦⠀⠀⠀⠀⠀⠀⢀⣠⣼⣷⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠜⠻⠏⠓⣸⣧⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠻⣿⣿⣿⠟⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⣿⣷⡄
⠀⠀⠀⠀⠐⠺⣷⡄⠀⠀⠀⣰⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣷⣾⣟⣿⣾⣿⢟⡇⠀⠀⠀⠀⠀⠿⠇⠠⣿⣿⣯⣹⣿⣷⣾⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠹⣶⣶⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣁⣋⡛⠛⠛⢙⣡⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣫⣋⡛⣉⣠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠉⡁⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⠈
⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢰⣿⣿⣏⢛⣿⣿⣋⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡛⣿⠛⢻⣇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠿⠿⠛⣩⣿⣿⡿⠀⠀⠀⠀⢀⣀⣤⣴⣿⣞⣼⣿⣿⣧⣄⣀⡀⠀⠀⠀⠀⠀⣿⣿⡙⠻⠿⢛⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣾⣶⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠘⣿⣿⣷⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡉⣳⡂
⠀⠀⠀⠀⠀⠈⢷⣄⣀⣤⣤⣄⠀⢰⣦⣤⣶⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣷⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣷⣄⣀⢀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⣾⣟⣉⣿⣟⣉⣻⣿⣿⡆⠀⠀⠀⠀⠀⠀⣼⣿⡿⢿⣿⣿⣿⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡟⢿⣯⣿⡿⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⢿⣿⢷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢺⡙⣇⠀⠈⡏⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣴⣦⣶⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣦⣴⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠖⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⢿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⡾⢵⡟⠸⠿⠋⠀⠀⠀⣤⣀⣄⣠⣤⡴⠊⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠆⠤⠈⠁⠀⠀⠀⠀⠰⣻⠿⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠈⡿⠟⠛⠛
⠀⠀⣾⠀⠀⢀⣀
⠀⢀⣹⡦⢾⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤
⠐⢹⣿⠀⠈⠁⢈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡚⢣⣶⣷⣶⣦⣴⢲
⠀⠸⣇⣀⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⡡⠊⣉⣾⡟⢿⡿⣿⣿⢋⡇
⠀⠀⠹⣿⣦⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡸⠁⠙⠻⠿⠷⠾⢍⡈⣧
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⢳⠄⢠⣯⠀⠀⠀⠀⣴⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⢸⠃⢻
⠀⠀⣴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⡇⢠⠖⢴⢀⡿⢸⡇⢠⣖⣲⣠⡆⣠⢴⡄⣠⢲⠄⡤⢲⢠⡆⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⢠⣀⣨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⢸⡇⢻⡄⢿⠸⣧⢸⡅⢿⣈⠽⠹⣇⢿⡨⠁⢿⣀⠜⢯⡀⢼⡇⢸⡇
⠘⣿⣿⣿⣿⠆
⠀⠈⢻⣿⣿⣆
⠀⠀⠸⣿⡿⣿⡀
⠀⠀⠀⠘⣧⠸⡇
⠀⠀⠀⠀⠘⠀⠉⠀⢠⣶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⢿⣻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡀⠀⢀⠀⠄⠀⢀⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣤⣤⣤⣴⠖⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⠶⠦⠄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠉⠁⠀⣿⣿⠿⠛⠻⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⣀⣀⣀⣀⠀⠀⠀⢸⣿⣿⣿⠟⠛⠻⣿⣆⠀⣰⡟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣀⣀⣀⣈⣟⣉⣁⣀⣀⣀⣀⣏⣙⣁⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⠿⠿⠿⠙⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣛⣛⣛⣿⣟⣛⣟⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠁⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⠃⠀⡞⡠⣢⣠⢠⡤⢠⡴⡰⠀⠀⠀⠀⠀⠀⡀⢼⣿⢀⣶⣆⢀⣤⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡢⢠⠤⠔⠁⠀⢀⠧⡅⢡⠋⣡⠉⠁⡡⠃⣠⠒⢤⢒⢲⠀⢱⣤⣕⣊⣋⡠⣊⣡⣿⠠⣽⠂⣤⠀⠒⡲⣄⠀⡀
⠀⢀⠔⢉⡙⠢⡰⠁⠈⠠⡀⠀⣏⡍⡻⣄⠁⠀⠀⠀⠀⠀⠀⠀⠈⣉⠜⠋⡟⡲⢦⣿⣿⣿⠿⠟⡏⢇⣶⡑⢤⡀⠀⠸⠮⣀⠋⡞⠀⣙⣀⣄⡀
⠀⠘⡀⢫⣴⠆⠀⠀⡠⡤⠁⠀⢀⠜⢁⣶⠍⠑⢒⣲⠒⢶⡌⠉⢡⡾⠀⢄⠘⠛⢻⠒⣠⠰⠄⣶⠚⠆⡙⡄⠀⢪⠅⡰⠚⢛⠛⠀⢴⣁⡊⢰⠑
⡔⠢⠂⠀⠀⠀⠀⠀⡳⣤⡤⠒⡡⠀⠀⡇⠘⣶⠂⠘⡆⠀⡇⠀⣇⢣⣰⡦⡇⠀⡇⢀⠃⠀⠀⡇⠀⠀⢶⠘⡄⢸⢠⠁⠀⠁⠀⠀⠈⠉⠈⠛⠁
⠡⡀⢀⢠⢄⠀⠀⠀⡷⠚⠙⡄⢱⠀⠀⡇⢰⠁⡇⠀⢰⠀⢸⠀⠈⠦⣈⡤⠃⠀⡀⠸⠔⠷⠀⢷⢰⡆⢸⠀⠈⢮⡄⢇⠀⡤⠠⡀
⠀⠈⠉⠁⠀⠑⠜⠉⠁⠀⠀⡇⢸⠒⠒⡇⢸⠉⠹⡀⠀⢇⠈⡇⠀⠀⠰⡀⠀⡀⡇⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠹⣮⡷⢌⡠⠃
⠀⠀⠀⠀⠀⠀⠀⢾⠨⠃⡰⠡⣎⠀⠀⡇⢸⡀⠀⢳⠀⠸⡀⢓⣉⡃⡆⢱⡞⠙⢿⠀⢧⡠⣤⠸⠤⠤⠄⢹⡠⡰⠋⠀⠠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢲⠋⢉⠉⠙⠦⡀⢧⡨⠉⠀⢐⣙⣀⣉⣁⣀⣈⣙⠄⣃⠤⠤⠕⠒⠒⠒⠒⠒⠒⠢⢝⣱⠕⠑⡄⠀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⢠⠒⡜⣌⡴⠒⠉⠁⠀⠀⠀⠀⠀⡠⠌⢳⠡⠶⡄⠀⠀⠀⠀⠀⠀⠀⢀⡄⡽⠶⡪⢂⣤⠌⠛⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣠⠿⣀⡄⠀⠀⠀⠀⠀⠀⠀⠙⠒⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡗⢵⡎⡛⢀⣴⡇⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⠛⣿⣤⣯⢁⢂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⡰⢢⠄⢲⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⠭⠵⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠲⠞⠛⠄⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠟⠛⠛⠛⠛⠛⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣶⣤⣄⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⢀⣤⣤⣼⣿⣿⡿⠟⠁⠀⠁⠀⠀⠀⠀⣹⣿⣷⣠⣀
⠀⠀⠀⢠⣿⣿⣿⣽⠆⠀⠤⣴⣦⡀⣀⡀⣠⣤⣀⠀⠘⢿⣧⠀⠀⢀⣾⠿⣿⠟⠿⠟⠀⣠⡀⢠⡀⠐⠀⣿⣴⣿⣄⣿⣿⣷⣤⣄
⠀⠀⠀⣿⡟⠹⣯⣥⣀⠀⠀⠈⠛⢿⣿⡿⠛⠋⠀⠁⠀⠘⣿⡆⠀⠸⠁⠀⠀⠀⠀⢀⣼⣟⣿⣿⣯⢀⢰⣿⣿⣿⣿⡿⠉⠹⣿⣿⡇
⠀⠀⢸⣿⡇⠀⢀⣬⣽⣦⣤⣤⡀⠀⠉⠀⠀⢠⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⢿⣿⣻⣿⣧⣽⣿⠛⣿⡇⠀⠀⠀⠈⡇
⠀⠀⠸⣿⡆⢰⣿⣿⣿⣟⣿⣿⣷⠀⢠⣾⣿⣿⣿⣿⣶⡀⣼⣿⠀⠀⠀⠀⠀⠄⠀⢸⣿⣷⣽⣿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⢿⣧⢺⣿⠋⠛⠙⢉⣹⣿⠃⢿⣿⢷⣿⣷⣾⣿⣧⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⢋⡟⣿⣿⣿⣿⣇⠀⡀
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠘⢿⣶⣬⣥⣶⣿⣿⡟⠀⠀⠀⠀⠐⠀⠀⠀⢸⣿⣛⣺⣿⡇⣾⣿⣿⣿⠀⣿⣿⣿⡏⠁⠀⡂
⠀⠀⠀⠀⠈⠻⣿⣭⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠐⢶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⠿⢻⣿⣿⣿⣿⣿⣿⡿⠿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⠆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣠⣶⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⣸⣤⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⣿⣷⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀⠀⠀⣿⠟⢻⡇⢀⠄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⢿⣄⣠⠇⢸⠀⢀
⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⠃⢹⡿⣿⠇⠹⣿⣿⣿⣿⣿⣷⠤⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⢀⣼⣇⣿⣿⣗⣿⡄⠀⢿⡇⣀⡆⠐⠁
⠀⠀⠀⠀⣿⡿⠻⣿⣿⣷⣶⣾⣁⣹⣶⣶⣿⣿⡿⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠉⣿⣿⣿⣿⣿⢿⡏⠀⢸⡟⠉⠉
⠀⠀⠀⠀⣿⣇⠀⣿⣿⡀⠉⢻⣿⣿⠛⠁⣰⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢢⣀⠈⡇⢡⠤⣿⣿⣯⣹⣿⣿⡇⠀⠸⣧⡤
⠀⠀⠀⠀⣿⣿⠶⣿⣿⣿⣿⣿⣁⣹⣿⣿⣿⣿⡿⢶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⢀⠉⣿⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢠⡟⠀⠈
⠀⠀⠀⠀⢸⣿⠀⣿⣿⣇⠉⠛⠛⠛⠋⢁⣿⣿⡇⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢻⠀⠀⣿⢼⡇⠀⣿⢸⡇⠀⢸⡇
⠀⠀⠀⠀⣨⣿⣧⣿⣿⣿⣷⣴⣿⣿⣤⣿⣿⣿⣧⣾⣿⡀⠀⠀⠀⠀⠀⠀⢠⣄⣀⢀⣼⣀⣠⣿⣿⣇⢀⣿⣿⣇⣀⣸⣧⣀⣀⣤
⠀⠀⠐⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠓⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠛⠛⠛⠛⠋⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⢰⡀⢠⣿⡿⠟⠁
⠀⠀⢀⠀⠀⠱⣾⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⠶⠖⠂
⠀⣴⣿⣿⣿⡿⠛⢿⡿⠏⠀⠀⠀⠀⠻⢤⣄⡀⠀⠀⠀⣿⠀⢸⡇⠀⢀⡀⠀⢀⢀⡀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏
⢰⠟⠛⠋⠈⠀⠀⠀⠀⢤⡀⣀⡶⠀⠀⠈⠈⠉⠀⠀⠐⣿⠋⢹⡟⠆⠉⢿⠀⣿⢸⡇⢹⠁⣿⠀⢲⡀⡞⢁⣀⣐⣃⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠷⠼⠂⡿⠰⠇⣾⢀⠿⠀⠈⡿⠀⢿⣿⣿⣟⠻⣿⣿⢩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢻⡇⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠻⠀⠤⣀⡴⠁⠀⠀⠙⠻⢿⣿⣶⣾⣬⣵⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⣸⡇⠀⠀⠀⠀⣾⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⡏⠉⢹⡷⣆⠒⣲⠀⣻⠀⣿⢠⣶⢿⣷⢶⣄⢰⡇⢰⡇⢸⠀⡶⢳⡄⣴⢻⡆⢸⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡇⠀⣶⣸⡆⢿⠀⣿⠸⣿⣾⣿⣾⡟⠸⢇⣸⣄⣹⠠⣇⠀⠀⢿⠀⡀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠛⠛⠀⠀⠀⠀⢾⠀⠿⠀⠈⠉⠉⠉⠀⠀⠀⠉⠀⠀⠀⠈⠉⠀⠀⠉⠀⠉⠃⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⣸⣯⣙⣿⣭⣙⣽⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⣠⠀⠀⠀⢻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠐⠸⠁⣾⣀⣿⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠙⢦⣤⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣦⣁⡀⣿⠉⣇⣴⠛⡇⣾⢙⣶⡏⢱⣶⠐⡆⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⢉⣠⠟⢋⢀⠛⠀⠁⠙⠛⠣⣿⢻⣻⡗⠋⡈⢛⡇⠀⠀⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡿⠄⢚⡎⢸⡇⣿⠀⠐⠀⢛⠆⠼⢟⢾⣤⡾⠁⠴⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣼⡇⠀⣀⣀⣼⡃⣿⠀⣀⣀⠼⡀⠀⣐⠀⣀⠀⣀⣄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⡟⠉⣿⢸⡇⣿⣼⠋⢹⡆⣏⢸⣿⠀⣿⢸⢣⡾⠃⡏⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠈⠃⠛⠛⠛⠘⠃⠛⠘⠲⠿⠁⠻⠞⡹⢾⠏⠘⠶⠞⠀⠛⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠐⠁⠀⡸⠟⢘⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠈⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⢀⣴⣶⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠶⠶⠶⢤⣤⣀⡀⠀⠈⠛⠿⣷⣾⣿⣶⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⠀⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠄⠀⠀⢀⡀⡀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣦⣀⡀⢸⣿⣷⣦⣤⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣧⡀⠀⠀⠙⣆
⠀⠀⠀⢀⣀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣍⣻⡿⠁⠀⠈⠙⢻⣿⡇⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠉⣿⣿⣇⢀⣀⢀⣴⣿⣿⠋⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠈⣿⠻⣿⣿⣿⠛⠛⠀⠀⠀⣠⣿⣿⣛⣻⣿⣿⣿⡿⠷⣿⣷⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢀⠀⠀⠀⣿⠀⠈⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⣿⠀⢀⠀⡠
⠀⠀⢐⠂⠈⠸⣿⠀⠰⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⢁⣀⣲⣯⣷⠟⠁⠀⠀⠀⣿⠗⠁⠐⡂
⠀⠀⠁⠀⠀⠀⣻⣙⣦⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡛⠉⠁⠀⠀⠀⣴⣋⣟⠀⠀⠀⠈
⠀⠀⠀⠈⠀⠉⠙⣇⢸⡇⢀⡀⠀⣼⣿⣿⡿⠗⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⡄⠀⢀⡀⢸⡇⣸⠋⠉⠈⠁
⠀⠀⠀⠀⠀⠓⠤⠺⣶⣧⣸⠇⢰⣿⣿⣿⣥⣤⣀⠉⣻⣿⣸⣿⣟⠉⣀⣠⣤⣿⣿⢻⣷⠀⠸⣇⡼⣴⠷⠤⠒
⠀⠀⠀⠀⠀⠈⠀⠀⠘⣦⣹⣄⣸⣿⣿⣿⣿⣽⣿⣿⣿⡿⠉⢻⣿⣿⣿⣿⣿⢿⣿⣼⣿⣄⣠⣏⡴⠃⢀⠀⠁
⠀⠀⠀⠀⢀⡤⡷⠿⢾⣿⣿⣽⣿⣿⣿⣿⡌⠛⠙⠻⠿⢷⣶⡿⠿⠟⠋⠋⠁⢸⣿⣿⣿⣿⣽⣿⣿⠾⠿⡮⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣴⣤⣼⣿⣿⣿⣿⣿⣴⣆⣀⣄⠀⣄⠀⢠⣤⢀⣶⣤⣿⣿⣿⣿⣿⣤⣦⣬⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠋⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣯⣉⠉⠛⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠟⠛⠟⠻⡿⣿⡿⠿⢿⣿⣿⣿⡿⡿⠿⣿⡿⡿⠛⠟⠛⠯⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠉⠃⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⢸⣿⢿⣦⡀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣆⠀⠀⢸⣿⢻⣷⡀⠀⣸⣿⣶⡿⢁⣾⡟⢻⣧⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣤⣿⣿⣆⠀⣼⣿⠛⢻⣷⡄⣿⣿⠀⠀⣼⣿⠻⠟⠃⢸⣿⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠘⢿⠆⠛⠛⠀⠀⠁⠀⠀⠀⠀⠈⠛⠇⠀⠀⣠⣾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣧⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⢰⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠖⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⢿⡇⠀⠀⠀⠀⢀⣠⣴⣾⣿⠀⠀⠀⣠⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣧⣤⣾⣋⣿⢻⡇⠀⠀⢠⣆⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⠳⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣤⣶⣿⣿⣧⠀⠀⠀⠀⢹⣿⣤⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠉⠀⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠘⠿⠻⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡆⠘⠋⠁⠀⠈⣿⣿⡿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣴⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⢿⣿⡟⢋⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⣿⠿⢿⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣉⣻⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⣶⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠸⣿⠛⠉⠙⢿⣿⠀⢀⡛⠿⣿⣯⣭⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⣠⣤⣶⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠇⣠⣿⢟⣤⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⢿⣴⡿⠋⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠻⢿⣿⣏⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⠁⣴⣿⠟⠁⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣾⣟⣫⣄⠀⠉⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣾⣿⣃⠀⠀⣼⣿⠏⠀⣤⣶⣶⣄⢴⣦⢸⣿⣜⣿⠀⠙⣿⣯⣵⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⣼⣿⡋⠀⢸⣿⠋⢹⣿⠀⢻⣿⣿⢿⣿⡇⠀⠈⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡗⠘⣿⣷⣿⠟⠀⠀⠹⡿⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⣀⣀⣀⣀⠀⠰⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⣰⣴⣾⣼⠟⠛⠂⢀⣾⣿⣿⣿⣿⣿⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠸⠿⠿⠟⠛⠀⠀⢀⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢾⡷⠀⠀⣤⣀⣀⣀⣀⣠⣦⡉⠉⠙⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢶⣦⠶⠀⢀⡌⣿⣿⣾⡇⣀⠀⠘⠛⠛⠿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣷⣶⣤⣀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⣀⠀⣾⣿⣿⣴⣿⣿⣿⣇⣠⠀⠀⡸⠀⢠⣏⠛⠻⠿⠛⢛⣿⠛⢻⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣏⢰⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣈⠉⠃⢠⡆⠐⠛⠉⣀⠈⠂⠘⠃⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣿⣿⣷⢘⠁⣿⢀⢹⡇⠦⢹⠰⠈⣇⠛⣴⣿⡀⠙⢧⠴⣶⣶⡶⢦⠞⠁⡸⠐⢭⠖⠸⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣟⣸⣇⣇⣬⣘⣇⣿⣿⣰⣿⣿⣀⣿⣿⣿⣦⣤⣤⣬⣥⣤⣤⣤⣾⣿⣿⣷⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠰⢶⣿⠉⠿⢩⣿⠉⠙⣿⠉⣿⣯⢹⣿⡏⣥⠉⣯⢸⠏⠿⢨⡏⠩⢭⡏⠨⢬⡏⠘⢿⢸⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠔⣷⠸⡟⢸⡄⣿⠀⣿⣿⢸⣿⡇⣿⠀⣿⠀⢠⡀⣸⡇⢰⣾⡇⢰⣾⡇⢰⡀⢘⣿⠤
⠀⠀⠀⠀⠆⠀⠀⣀⣴⣿⣶⣿⣶⣷⣾⣷⣾⣶⣦⣾⣦⣴⣷⣦⣶⣿⣷⣾⣶⣿⣷⣶⣶⣷⣴⣴⣷⣼⣷⣼⣿
⠀⠀⠀⠀⠀⠀⠘⠡⠀⣽⠻⣌⡛⢛⣛⣙⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⡞⠛⠉⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠇⡀⠤⠄⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠈⢨⠀⠈⠻⣍⠓⢦⡶⠴⠾⢈⣉⣁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⡑⢤⠀⠀⠀⠀⠀⠁⠂⠅⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠂



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⢀⠀⠀⢀⣀⠀⣀⡀⣀⠀⠀⢀⠀⠀⠀⣀⣤⣤⣴⡥⣬⡶⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡇⠀⡼⣇⠀⢹⣨⠇⣇⣹⠙⣆⢠⠎⠀⢀⣾⠟⢱⣿⠃⠀⢸⣿⡄⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⡇⣸⠧⠾⡄⢸⠁⠀⡏⠀⠀⠘⡏⠀⠀⢸⣿⠀⢸⡟⠀⠀⢸⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⣤⡄⠀⠘⠀⠀⠃⠃⠀⠀⠙⠘⠀⠀⠃⠀⠀⠀⠃⠀⠀⠘⣿⡀⠘⣷⠀⠀⣸⡿⠁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⠀⢀⠀⠀⠀⠀⠀⢠⡖⠀⣠⡞⠙⠀⠀⠀⠀⠀⠀⢰⠂⡈⠙⠓⠛⠓⠚⠛⠒⠊⢁⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢸⣏⣀⣠⣼⡿⠚⣡⣤⡤⠀⠀⢀⣿⠀⢰⣿⠀⠀⡤⢴⠂⢠⡷⠀⠸⣄⠀⠀⣠⡄⠀⣴⠋⠀⠀⣿⢃⠔⢁⡿
⠀⠀⠀⠰⠿⠛⢻⣟⠉⠁⢸⡇⢰⣿⠃⠀⣤⡆⢸⡟⠀⣸⡟⠀⢸⡇⠈⠉⣸⠇⢸⡏⠈⠙⢾⡇⠈⢰⣿⠐⢀⢘⣿⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⡇⠘⠿⠦⠊⣿⠀⠸⣷⠀⢻⣧⠀⠈⠛⠖⠁⢿⠀⢸⡇⠀⡀⠘⠳⠄⠂⠙⠒⠀⢸⡟⠀⣾⠁⣶
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⣷⠀⠀⠀⠀⠘⠦⠂⠈⠁⠀⠻⠤⠁⠀⠀⠀⠘⠗⠀⠳⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢧⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣧⡈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⠄
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢹⡟⠈⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣿⣶⣶⣶⣶⣶⣿⣿⣶⣄
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡄⠉⠛⠉⣡⣿⣿⠟⠁⠀⠀⠀⢸⣿⣿⣿⡿⠟⠛⠛⠛⠻⠷⠤
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠘⠃⠺⠿⠛⠉⠀⠀⠀⠀⠀⠀⣿⣿⣯⡀
⢰⣿⣿⣿⡇⢹⡅⡽⢀⡄⢩⠀⣤⠙⡄⢠⡈⡇⠹⠁⣼⣿⣋⣰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿
⠈⣿⣿⣿⡇⠘⠃⣦⠘⠃⢸⠀⣩⣼⠀⣨⣴⣿⡄⣰⣿⣿⣿⣿⣿⣷⣶⠆⠀⠀⣤⣶⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿
⠀⣿⣿⣿⣧⣼⣇⣿⣸⣧⣾⣴⣿⣿⣇⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣧⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⠿⡿⠿⡿⠿⠿⡿⠿⣿⡿⠿⣿⣿⠟⠛⠻⡿⢿⡿⢿⡿⠿⡿⠿⠿⢿⠿⠿⠿⡿⢿⡿⢿⣿⣿⣿⠇
⠀⠀⢿⣿⣿⠀⣷⠀⠇⢰⠀⠇⢸⣿⡇⢰⣿⡇⠀⣦⠀⡇⢸⡇⢸⡇⢠⡏⢰⣶⣾⠀⣶⣶⡇⠈⣇⢸⣿⣿⡟
⠀⠀⠈⢿⣿⠀⣭⢸⡇⣩⡀⡇⢸⣿⡇⢸⣿⣿⡀⣿⢀⣷⢸⡇⡈⡇⣼⣗⢠⣥⣿⠀⣬⣼⡇⢰⡄⢸⣿⡟
⠀⠀⠀⠀⠻⣶⣿⣼⣧⣿⣧⣿⣙⣛⣿⣘⣛⣻⣧⣉⣼⣿⣆⣰⣷⣡⣿⣿⣘⣛⣻⣇⣛⣛⣿⣾⣷⣼⠏
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⡀⣄⣀⡀⣠⣰⡢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣝⢿⣷⣿⡅⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡻⣷⣽⣿⣿⣮⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣘⠛⠻⢿⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿⣩⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⣿⡟⠃⠀⠀⠻⣿⣿⣯⡤⠻⣷⡀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡳⣦⣴⣀⣴⢛⣛⡳⣦⡙⣿⡀⢱⣄⡽⣿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣟⣼⣧⠘⠻⢃⣿⣿⢹⣷⠛⠛⡄⠘⠆⠳⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠁⠙⠛⠛⢻⣿⣿⣴⣿⣧⡤⠛⠠⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣄⣄⣠⣬⣵⣾⣻⣿⣿⡗⠀⠱⡀⠅⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣭⣿⣿⣿⣿⣭⣭⡿⣿⣷⠀⢠⠽⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⢿⣿⣿⣿⣷⣿⣈⢉⢈⣦⣷⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡙⢦⣈⠛⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠈⠙⠛⠻⠿⠇⠀⠂⠀⠀⠺⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠎⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡎⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢧⠀⠀⡀⠄⠚⠓⠒⠒⠂⢄⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⠪⠄⠀⠘⠷⣦⡀⠀⠀⢑⣤⡀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠙⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠒⠠⠀⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠒⠒⠂⢠⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⢀⠄⠤⠀⠁⢊⠄⠊⠁⠉⠉⠈⡺⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠂⢄⠀⠀⠀⠀⣸⡦⠀⡀
⠇⢀⣠⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⡟⠓⠠⢀⠑⢄
⠀⠉⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⢄⠠
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠄⠂
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠔⠀⠻⡄
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣤⣄⣀⣀⣀⣀⣈⣐⣂⣀⣤⣀⣀⣀⣂⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠉⠢⢀⠀⠀⠀⠀⢀⠠⠐⠂⠁⠁⠀⠀⠀⠀⠉⠐⠠⢄⠀⠀⠀⢀⡠⠊⠀⢡⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠂⠄⡀⠑⠢⡠⠊⠁⠀⠀⡠⠐⠈⠀⠀⠀⠈⠐⢄⠀⠀⠈⠂⢔⠁⢀⠄⠊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠊⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈⠂⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⠑⠈⢀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠰⠀⠀⠀⠐⠀⠂⠀⠐⢧
⠀⠀⠀⠠⠐⠠⡀⠀⠀⠀⠀⡘⠤⢃⠀⠁⠃⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠰⢀⢁⠉⡀⠀⠀⠀⢀⠔⠀⢢
⠀⠀⠀⢁⠀⠀⠈⠄⠀⠀⠀⠃⠀⠅⠅⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠀⠇⠀⠀⢀⠃⠀⠀⠘⢀⠀⡀
⠐⠁⠈⠐⢦⡀⠀⢰⠀⠀⢀⡤⠐⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠑⠠⢄⠀⠀⢸⠀⠀⡴⠋⠀⠀⠈
⢇⠀⠀⠀⠀⠙⠀⠈⠀⠈⠁⡠⠂⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠢⡀⠈⠁⠁⠀⠈⠀⠀⠀⠀⡠
⢀⠁⠖⠂⠤⠄⠀⠀⢀⠔⠊⠠⠀⡈⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠈⡀⢀⠁⠑⠤⡀⢀⣀⠒⠐⠛⠁⠂
⢀⠀⠀⣀⣠⠂⠀⠈⠑⠠⡀⠈⡤⠁⠰⠀⠀⠀⠀⠀⠘⠁⠒⠠⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠘⠀⠐⡌⠀⡠⠐⠉⠀⠀⢱⠦⠄⣀⡠
⠀⠈⠀⠀⠄⠂⠄⣀⡀⠀⠈⠒⠁⠀⡀⠀⠀⠀⠀⠀⠃⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠊⠀⢀⣀⣠⠴⠡⠂
⠀⠀⠀⠀⢀⠂⠉⠉⠉⠉⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠐⡀⠀⠀⢀⡠⠔⠀⡄⠰⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⢠⠀⠒⢤⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠉⢁⠋⠀⠀⡸⡀⠀⡁⢀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢘⠀⢀⢆⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠠⠔⠀⠀⠁⠀⠈⠀⢀⠉⠒⠠⠄⠀⢀⣀⠀⢀⣀⣀⠀⠀⠤⠰⡊⠀⢁⠀⠀⠈⠁⠀⠑⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡀⠐⠤⣴⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣈⣠⠐⢂⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠀⠀⠀⠀⣀⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⠀⠀⠀⠈⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠢⠀⣀⡀⠌

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠒⠠⡀⠀⠀⠀⠀⢀⡤⠒⠠⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⢀⣀⠄⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⡀⠠⠀⠈⢀⠀⠈⠆⠀⠠⠀⡞⠠⠀⠈⠀⠀⢉⡉⣠⣅⢠⡄⠀⠀⡔⢎⠀⢸⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠅⣾⠀⠀⠀⠀⠀⠀⠍⢠⡌⢡⡄⠀⠀⠀⠻⣬⠑⡯⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠅⠀⢠⠀⠀⠡⡈⠀⠘⠀⠄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢁⠉⠀⠀⠀⠈⠀⠀⠀⠒⠊⠀⠀⠀⠀⢀⠐⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠚⡆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⢀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠐⠀⠀⠀⠁⢸⣏⠀⠀⠀⡀⢀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠆⠐⠤⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⢰⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠀⠐⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⢀⡁⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠈⠃⠀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠠⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠱⣀⢀⣀⣤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⠂⠉⣩⠉⠑⠠⡀⠀⠀⠀⠰⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⢄⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⣐⣀⠔⠁⠀⠀⠠⢄⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠔⠉⡑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣃⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠑⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠃⠂⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠊⡀⠀⣀⠈⠳⣄⠀⠰⠌⡤⠂⡐⡄⠀⠀⠛⠀⢠⡷⢾⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠉⠁⠢⠀⡇⠈⠋⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠇⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠁
⠀⠀⠀⢠⡀⠢⡈⠀⠀⠰⠂⠀⠀⠀⠀⠐⢁⡄⠀⠑⡈⠀⡠⠁⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠈⠒⠴⣦⣿⣔⡾⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡷⠈⠀⡌⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⢀⣤⠀⣔⣃⡠⠤⠤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡀⠀⠀⠀⡴⢦
⠘⢬⡎⠉⢀⡄⡀⠀⠔⠩⢍⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⣦⠎⢱⣠⣄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢱⠀⠀⢰⠇⢸
⠀⠀⡀⢠⢬⡀⠁⣀⣃⣈⣁⣀⠀⠀⠀⠀⠀⢀⡖⠢⠴⠃⠀⠋⠀⠘⠁⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠉⠉⠀⠈⡆
⠀⠀⡇⣷⠀⠙⠫⠿⡼⣇⠧⠟⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⢠⡎⠰⠰⠆⠰⣂⣒⡏
⠀⠀⡇⠙⢄⣀⡀⣲⡐⡤⣂⠔⣆⠀⠀⠀⠀⣘⠰⣍⣱⣀⠝⠀⠯⠮⠝⢺⡴⡄⠀⠀⠀⠀⢸⠘⢞⣣⠛⣤⢣⠟⣜⣧
⠰⣆⡇⠀⠀⠀⠉⠉⠉⠉⠁⠀⣇⣴⠆⠀⠀⡏⠐⠲⠿⠶⠷⠮⠷⠾⠽⠞⠒⠛⠳⡄⠀⠀⠸⡄⠀⠀⠀⣀⣁⣁⣀⡠⠁
⠀⠉⠑⠤⣀⡀⠀⠀⠀⠀⣀⠴⠁⠀⠀⠀⠀⣧⡤⢤⡀⣀⣀⡀⢀⣀⣀⢀⣀⢀⣀⡝⠀⠀⠤⢵⠀⠀⠈⠉⠉⠀⢠⠅
⠀⠀⠀⠀⠀⠡⡌⠉⡽⠏⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠚⠀⠀⠣⠏⠉⠙⠼⠙⠞⠉⠃⠀⠀⠀⠀⠣⠜⠀⠀⠈⠻⠍
⠀⠀⠀⠀⣀⡄⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡐⡄
⠀⠀⠀⡼⠁⣁⠐⢁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢡⠍⢲⠼⠶⠞⠁⢠
⠀⠀⢀⠗⠀⠉⠀⠈⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠑⠶⣾⢁⣄⢀⡄⢀⣀⡙⡽⠄
⠀⢠⠃⡤⢤⡦⡄⢰⠱⢋⠎⡆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⡞⠁⠀⠀⠀⠈⢢⡀⠀⠀⠀⢸⠈⠉⠉⠙⠛⠣⠯⢼
⠀⡎⢀⡉⣉⠉⣁⡀⣩⣁⢄⡇⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠇⠘⠚⠤⠿⠤⠿⠴⠯⠜⢳⠀⠀⠀⠀⠀⠈⡎⢧⡀⢀⠇⠀⠘⡗⠀⠀⠀⠘⠦⡠⠤⠃⠀⠀⠀⠀⠀⠀⢰⠃
⡰⢋⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢉⠆⠀⠀⠀⠀⠙⢄⡉⠁⠀⣠⠇⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠑⠉⠈⢲⠄⣀⣀⡀⢠⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡉⡦
⠀⠀⠀⠘⠜⠁⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠌⠙⡖⢢⠤⠤⠀⠒⠒⠛⢄⠜
⠀⠀⡀⠀⢀⣴⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠉⠀⠀⢀⡄⡀⢠⠀⠤⠤⠤⠤⠤⣄⣄⣠
⠀⠈⢯⠒⢁⡤⡄⠀⢏⣂⠶⣀⡠⢤⠎⢳⡀⠀⠀⢀⠔⠒⢤⠖⢲⠎⣀⠉⢱⡀⠀⠀⠘⠋⠸⢠⡪⠏⣱⠀⡖⠶⠢⢸⠈
⠀⠀⠀⡇⠘⠤⠂⠀⣠⢲⡶⠭⠧⠚⠒⠁⢸⠀⠀⢺⡘⣃⠴⡄⠘⢄⣉⣠⠜⠁⠀⠀⠀⠀⡇⢀⢩⡭⣄⢤⢈⣭⡥⠚
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⡞⣷⢢⡜⡏⠀⠀⠀⠀⠀⢀⠴⠃⠀⠉⠑⠉⠋⢹⣉
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⠀⠀⠀⡰⠋⠉⠉⢹⠃⠀⠀⠀⠀⠀⠷⢹⠀⠀⠀⠀⠀⠀⢸⠳⡇
⠀⠀⠀⠈⢙⢲⢲⠀⡎⠈⢳⢸⠉⡏⡟⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⡀⠀⠀⣀⡼
⠀⠀⠀⠀⠸⡍⠸⠀⠃⠀⠸⠎⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠱⣲⠖⢜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠵⠉⠉⠹⡍
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣤⠖⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⣤
⠀⠀⠀⠀⢠⣿⡇⠐⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⢀⢠⡀⡄⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠙⠶⠞⠃
⠀⠀⠀⠀⣾⣿⠓⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣷
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀⠀⠀⠀⠀⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣸⢀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠿⠿⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣯⣽⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⣿⣷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⡾⠿⠿⠶⡿⠷⣶⡿⠷⢾⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⢒⠺⠓⠢⠒⠘⠏⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⣀⡀⠀⠀⡰⣴⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⡤⠤⠀⠤⡴⠀⠀⢠⠂⠁⠀⢀⠔⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡛⠚⠒⠐⠢⠐⠚⠁⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠠⠇⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠀⠀⠷⠀⠀⠀⢀⣠⠖⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠂⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢳⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠠⡴⠿⠁⠙⠉⠁⠀⠀⠀⠩⠆⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠙⠉⠃
⠀⠀⠀⠀⠀⠀⢐⠀⢓⠀⠀⠀⠀⠀⠘⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠜⡁⡐⠲⠂⠀⠀⠀⠈⠇⢀⣄⠀⠀⠀⠀⠀⠀⢰⡶⣶
⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⡀⣈⠙⠀⠀⢐⣖⣶⡞⠆⠀⠀⠀⠀⢤⠚⣁⡚⣠⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠌⣄⡀⠀⠈⠉⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠯⣣⠅⠀⠀⠀⠀⠀⠀⠀⣩⠃
⠀⠀⠀⠀⠀⠈⠛⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠄⢺⠠⠄⠀⠀⠀⠁⢀⣦⢤⣤⡤⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣠⠰⠿⠒⠛⠋⡉⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢤⡘⡛⡀⠜⠁
⠀⠀⠀⠀⠤⠴⠦⣐⡀⠁⣠⠦⠤⢀⣀⡸⠂⠘⠀⠙⠤⠖⠀⠀⠀⠛⠋⠙⠁⠀⠀⠀⢀⡀⠀⣴⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⡀⠀⠀⠆⣄⠈⠀⣠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⡀⠀⠀⠀⠀⠙⣃⠅⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠑⡐⠋⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠄⠀⠀⠀⠀⣤⢘⠃⠀⡀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⠀⠀⠀⠀⠠⣶⢈⠀⣠⢫⠄⠀⠀⠀⠀⠒⠤⢄⣀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⡌⠰⠀⠀⠀⠀⠀⠲⢿⡆⣀⡌⠀⠀⠀⠀⠀⠀⠐⠘⠛⡀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⣁⢴⡂⠀⠀⠀⠀⠀⢭⣿⠋⠁⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⡀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠄⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣘⠉⠀⣚⠀⠛⣄⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠋

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

⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢠⡄⢀⢄⢠⡀⣠⡀⠀⠀⠀⠀⣀⣤⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⣀
⠀⠀⠀⣠⡾⠛⠉⠛⢷⡀⠀⠀⠀⠐⠷⣤⠶⠿⢿⣧⡀⠀⠀⠀⢀⣈⣿⣿⡾⣾⣿⠯⢄⠀⠀⠀⢰⣟⠀⣦⠀⣹⠇⠀⠀⠀⠀⠀⣀⣠⠞⢿⢻⡿⢷⣶⠃
⠀⠀⣰⡟⠘⠓⠘⠓⣈⢿⠀⠀⠀⠀⢸⡇⢰⣧⡄⣻⣿⠀⠀⠀⠸⣖⣽⡇⠶⢌⣿⣯⢎⠀⠀⠀⣼⠿⠶⠶⠞⢋⢠⣄⠀⢰⡿⣿⢉⣿⡾⣾⣉⡷⣾⠗⢶⣤⣀
⢾⣇⣿⢸⣷⣿⣾⣿⡿⢸⣇⡶⠀⠀⠘⣿⣦⣤⣴⣿⣿⡇⠀⢀⡳⣾⢛⠛⠷⢛⡛⣿⣏⡤⠀⢸⡇⠀⢀⣀⣀⠙⣿⠉⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⡟⠀⠐⠏⠿
⠀⢸⣿⣦⣬⣤⣥⣤⣴⣿⡏⠀⠀⠀⠀⣿⣿⣿⠠⠆⢹⡇⠀⠈⢻⣿⢐⣄⢀⡈⣁⣿⣿⠁⠀⠈⠻⣿⡿⠿⣿⡟⠁⠀⠀⠀⠀⠀⣿⠟⠉⠉⠻⣿⡇
⠀⠘⠿⢿⡿⢿⡿⣿⠿⣿⠇⠀⠀⠀⣰⣿⣬⣿⣷⣶⣿⣷⡀⠀⢸⡿⣿⠿⡿⣿⡟⣿⣿⠀⠀⠀⠀⢸⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠈⠁⠈⠀⠉⠀⠉⠁⠀⠀⠐⠋⠙⠋⠙⠉⠙⠁⠈⠁⠀⠘⠃⠟⠘⠃⠙⠁⠋⠘⠀⠀⠀⠒⠛⠀⠀⠀⠛⠦⠀⠀⠀⠐⠚⠃⠀⠀⠀⠀⠀⠹⠶⠄
⠀⢀⣠⣤⣄⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄
⠀⢺⣿⣿⠿⡆⠀⢠⡾⢿⣿⣿⡄⠀⠀⢼⣗⣠⣤⣤⡽⠛⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⣀⣿⠀⠀⠀⣰⠛⠛⠛⠦⣄⠀⠀⠀⠐⣧⣤⣶⣶⣦⣼⠇
⠀⠀⠘⠁⢀⣽⠶⠾⣄⠀⠉⠁⠀⠀⠀⢀⣾⢟⠛⡛⠿⣦⠀⠀⠀⠀⠀⠀⠈⢻⣿⢿⣿⡿⠛⠋⠀⠀⠀⡟⠐⠂⠰⢠⡏⠀⠀⠀⠀⢸⣿⣐⣂⡿⣿⡀
⠀⠀⠀⠀⣾⠁⢴⡄⢘⣧⠀⠀⠀⠀⢠⣿⣧⣈⠁⢉⣠⣿⣧⠀⠀⠀⠀⠀⠷⣾⠰⠆⢸⣤⠇⠀⠀⠀⢸⣷⣤⣤⣀⣼⠁⠀⠀⠀⠀⠘⡷⢩⢻⣖⣿⣧
⠀⠀⠀⠀⠹⢷⣤⣤⡾⠃⠀⠀⠀⢠⣿⣿⣿⣿⣟⣛⣿⣿⣿⡆⠀⠀⠀⣠⣴⡿⠛⠺⣿⣦⡀⠀⠀⠀⡌⠀⣿⢹⡏⢹⠀⠀⠀⠸⣇⣠⡟⡲⣎⢍⣿⣿⢀⣠⠇
⠀⠀⠀⠀⠀⡴⠁⠙⠦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⠇⢸⣷⣿⣷⡇⠸⠃⠀⠀⣰⣀⢠⡇⢸⡇⢸⠀⠀⠀⠀⠀⠸⣷⣶⡿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⠟⠛⠻⣇⠀⠀⠀⠀⠓⠀⢸⡇⢸⠀⠹⠷⠀⠀⠀⠀⣶⣴⣤⣴⣿⣿⡟
⠀⠀⠀⠀⢀⣼⠿⠻⢷⣶⠃⠀⠀⠀⠠⠏⠩⠏⠹⠏⠹⠍⠳⠄⠀⠀⠀⠀⠛⠀⠀⠀⠹⠀⠀⠀⠀⠀⣀⣤⠇⠸⣤⡀⠀⠀⠀⠀⠀⠈⣛⣿⠿⣟⠋
⠀⠀⠀⠀⢾⡇⠸⡶⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⢈⣿⠶⢶⡾⠋⠀⠀⠀⠀⠀⠀⠸⣏⣀⣀⡾⠃⠀⠀⠀⠀⠀⢰⢀⣠⣤⣾⣄⠀⠀⠀⠀⣆⠉⢀⣠⣤⣤⣀⣀⡤⠋⢀⠄⢄⠀⠠⣉⠌⠀⣥⣂⠅
⠀⠀⠀⠀⣿⡁⠶⠄⣿⠀⠀⠀⢠⡆⠀⣠⣾⠟⠛⢿⡗⠀⣤⠀⠀⠀⠀⠀⠿⣟⢡⠙⣯⣂⠀⣀⠀⠈⢻⣿⠉⠆⠆⢹⣿⠀⠀⠘⠈⠺⣄⢰⡇⢠⡞⠁
⠀⠀⠀⠀⢨⡿⠒⢾⡇⠀⠀⢀⡘⢷⣴⣿⣯⠐⠷⢀⣷⡴⠋⢀⠀⠠⣤⠘⣵⣿⠶⢻⢿⣿⡴⠋⠀⠀⡜⠛⡶⠶⢶⠾⢏⠀⠀⠀⠀⠀⢸⣾⣧⣿⡀
⠀⠀⠀⠀⢸⣧⣛⣼⠇⠀⠀⠘⢿⣄⣼⣿⠿⣷⣶⣿⠿⣷⡶⠿⠁⠀⠉⠉⣿⡇⠃⠀⡄⣿⡇⠀⠀⢘⡁⠀⡁⠀⠈⠀⢨⡆⠀⠀⠀⠀⢀⣯⣐⣨⣧⡀
⠀⢤⣶⡂⠹⣏⣊⣿⠐⣶⣦⠄⠀⠀⠹⣿⣄⡈⠈⣁⣼⣿⠇⠀⠀⠀⠀⢠⣿⣤⣥⣤⣴⣾⡇⠀⠀⠈⠁⠐⡇⠀⢾⠀⠸⠃⠀⠀⣴⠟⠛⣿⠟⠉⠉⠙⢷⡄
⠀⠈⢀⣀⡜⠉⠉⠙⣆⣀⠉⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣾⠟⣿⠟⣿⡟⢻⡇⠀⠀⠀⢀⣰⣧⠀⢸⣧⣀⠀⠀⠀⠻⡀⠀⢿⡀⠀⠀⠀⡸⠃
⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⣹⣥⣤⣤⣼⡋⠀⠀⠀⠀⠀⠀⢀⣄⢀⣴⣧⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⡀⠀⠀⢠⣤⠤⠒⠒⠒⡒⠶⣷⠀⠀⠀⠀⠀⢉⣤⣤⣤⣤⡍
⢶⣄⣾⣿⡿⠿⢿⣿⢷⣤⠦⠀⠀⠀⣠⣈⣿⡿⢿⡿⢿⣿⣀⠀⠀⠀⢸⡏⢍⣩⣭⣩⠏⣿⠀⠀⢸⣧⣀⠻⠀⠘⣛⣠⡿⠀⠀⠀⠀⣰⡟⠩⠄⠦⠙⣿⣆
⠀⣾⡱⢨⣇⠻⢢⣇⣂⣿⠀⠀⠀⠀⢙⣿⡏⠊⢵⣫⢕⢿⡟⠀⠀⠀⠈⡇⢾⣿⣿⣿⠀⡏⠀⠀⠀⠉⠙⠋⠻⡟⠉⠁⠀⠀⠀⠀⠀⣿⣇⠒⠀⠀⣠⣿⣿
⠀⠙⢿⡿⠿⢻⣿⣿⣿⣿⠀⠀⠀⠛⣿⣿⣮⣃⣞⠭⣪⣾⣿⠃⠀⠀⠀⢧⣈⣉⣩⣬⡴⠇⠀⠀⠀⢠⣤⡄⠀⡇⠀⠺⡟⠀⠀⢠⡞⠻⣿⣿⣿⣿⣿⣿⠏⠙⠂
⠀⠰⣶⣾⣷⣿⣿⣿⡿⠉⢢⣀⠀⠀⣿⣿⣿⣿⣿⠿⢟⣿⣿⠀⠀⠀⠀⠀⢸⣍⣥⣼⠀⠀⠀⠀⠀⠈⠉⣄⢠⣿⣄⠴⠁⠀⠀⠀⠀⠀⠈⢻⠟⣿⠛⣧
⠀⠀⣈⡹⠛⠛⢋⡁⠀⠀⠁⠉⠀⢠⣿⣿⣿⣶⣤⣶⣿⣿⣿⣆⠀⠀⠀⠀⠈⣿⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠀⠘⡄
⠀⠀⠈⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠺⠟⠛⠛⠿⠿⠛⠻⠿⠛⠛⠃⠀⠀⠀⠾⠉⠿⠳⠛⠂⠀⠀⠀⠀⠀⣠⠛⠛⠋⢦⡀⠀⠀⠀⠀⠀⠶⠗⢀⣸⣄⠀⠿⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣦⡀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣹⣿⣿⡿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣽⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢤⣤⣄⠀⠀⣀⣸⣿⣿⣿⣿⣤⠀⠀⠀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣦⣤⣈⣴⣿⡿⢿⣿⠟⢻⣿⠿⣿⣿⣧⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⣿⣿⡛⠛⣛⣫⣤⣶⠁⠀⢿⣿⣿⣿⣿⡿⠻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⣿⣷⡿⠛⣛⠛⢻⠇⢁⣀⢀⣈⣿⣿⣿⣿⡇⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠸⣿⣿⣿⠗⠖⠉⠉⣸⠃⣿⡉⠈⣉⣉⣩⣿⣿⣇⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⠛⣿⡦⠞⠛⠁⠈⠀⣿⣇⢀⡀⠈⠛⠛⣿⣿⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⣿⣿⣧⣤⡞⠛⠁⢾⣆⣸⣿⡦⠀⠀⠛⠢⣬⣻⠄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣾⣷⣟⣉⣸⠷⠀⢂⣼⣿⣹⣿⡆⠀⠀⠻⠦⣈⣿⡄⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⣿⣾⣽⡿⠀⠀⠀⣨⣿⣿⣯⣄⡀⠀⠲⢦⣹⣿⠃⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠸⣿⡿⠯⠀⢀⡘⠛⣽⣿⣿⣿⣝⠛⡀⠰⠆⠀⠿⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠸⠷⠶⠴⠛⠛⢉⣉⣿⣿⣟⠉⠙⠛⠢⢤⡴⠆⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⢀⣀⣄⣀⠀⠀⠈⣻⣿⣧⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢻⡏⠀⠀⠀⣴⣿⣿⠿⢿⣷⣄⠈⣿⣿⣿⣿⣠⣶⣿⣿⣷⡆⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠰⣟⠻⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠟⠁⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡶⠤⠛⢿⣿⣿⣿⣿⠿⠻⠿⣿⡿⠆⠀⠀⠀⠀⠀⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣾⣷⣾⡿⢷⣴⣿⡎⢹⣿⣿⣿⠶⣤⣾⣿⣷⣾⢆⣀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⠈⣿⣿⣿⡁⠀⢿⣿⣇⣰⣿⠿⠋⠀⢻⣿⡏⠈⢿⣾⣿⡆⠀⢰⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⠀⠈⠿⠿⠋⠀⠀⠀⠘⠿⠋⠀⠀⣾⣿⡿⠁⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠇⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣠⣿⣿⡿
⠀⠀⠀⠀⣤⠟⠁⣿⠈⠙⣿⡄⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠃⢠⣿⡇
⠀⠀⠀⠈⠁⠀⣾⠉⣿⡇⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣿⡟
⠀⠀⠀⠀⠀⢠⠛⣤⠘⠃⢸⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢰⣿⠃
⠀⠀⠀⠀⠀⠙⠀⢹⠀⠸⠇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡏⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠀⠀⠀⠀⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣿⡇⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⢀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣤⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢀⡀
⠀⠀⣀⣤⡄⢠⠀⢀⣠⠄⢄⡀⠘⠈⠁⠁⠈⠈⠙⠃⠁⠀⠀⠀⠓⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁⠀⠛⠚⠛⠆⠀⠗⠛⠓⠸⠸⠾⠇
⠀⠈⠉⠹⠆⠆⠂⣉⡛⠙⠉⠃⠀⠘⠄⠖⠠⠐⠂⠇⠐⠡⠀⠀⠀⠁⠀⠀⠂⠁⠀⠁⠈⠁⠈⠉⠉⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠠⠤⢄⣀⠀⢤⡤⠠⡄
⠀⠀⠀⠀⠀⠚⠛⠣⠀⠀⠀⠀⠀⠐⢠⠂⡆⠄⢠⠄⠆⠀⠀⠀⠰⠚⠀⢀⣴⣤⠶⠠⣶⠒⠀⠲⡆⢴⠀⠐⠂⠆⠶⠒⠀⠆⠀⠀⢐⣤⠡⡀⠆
⠀⠐⠀⠃⠶⠰⠀⠀⠀⠀⣴⠀⡀⠀⢀⠀⡇⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠶⠂⠛⠀⠄⠦⠟⠸⠀⡤⠀⡦⢢⡗⠰⠀⠀⢀⣀⡀⢉⠀⢀⢀⡀⣀
⠀⠀⠀⠰⡴⢆⢰⠠⡔⠄⠀⠀⠀⠀⢴⡆⠀⠀⡤⢠⡤⣤⠠⠄⢴⣴⡔⠂⠀⠀⢢⠁⡿⠎⢰⠀⠀⠀⠄⠀⢠⠀⣤⠄⢄⠆⠀⠀⠀⡄⡤⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⢀⢀⠀⡀⡄⠀⠀⢀⠀⠀⡀⠀⠀⠀⡀⢀⣤⠀⡀⡀⠀⠀⠀⢀⣀⣀⣀⣤⣀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠘⠇⣤⡆⠶⠢⠁⠀⠀⠙⠃⠂⠁⠈⠃⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠋⠚⠃⠘⠘⠃⠀⠀⠀⠈⠁⡬⣥⢤⡍⠉⠀⠠⠄⠀⠠⠀⠀⠀⠀⠠
⠀⠀⡀⠀⢀⡈⣈⣁⡈⠈⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠶⠐⠠⠰⠀⠀⠀⠀⠲⠶⠶⠆
⠀⠀⠁⠀⠹⠆⠻⠋⠃⠀⠉⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠈⠀⠀⠀⠈⠀⠐⠀⠐⠀⠀⠀⠛⠛⠛⠛⠘⢛⠚⠛⠀⠀⠀⠠⠀⢀
⠀⠀⠐⠆⣴⢠⢰⡄⠀⠀⠀⠀⠀⠯⢾⠀⢶⠆⠹⣁⠴⠀⠀⠀⠀⠀⠀⣤⡠⢒⡒⣠⣰⠀⢼⡠⡷⠛⠁⠀⠈⠉⠙⠉⠉
⠀⠀⡐⠀⠃⠀⠈⠀⠀⠀⠀⢤⠤⠤⠄⢠⠀⢠⢤⠀⠠⠠⢤⠀⢠⠄⠠⢤⠁⢀⣬⢄⣄⣁⣤⢄⣀⠀⣄⡄⠀⢰⣤⠶⠶⠀⠀⠀⡀⠀⠀⢀⠈⠀⠀⡀
⠀⠀⢀⠀⠀⠄⠠⢠⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⢠⢀⣀⠀⡤⢀⠤⡀⠀⠘⠆⠙⠶⠏⠒⠘⠃⠀⠉⢰⣴⣄⣠⣦⣴⣶⣶⠀⠒⠀⠐
⠀⠀⠉⠀⠠⠀⠀⡍⠀⠛⠀⠈⠃⠀⠀⢣⠟⣴⣠⣀⢈⠸⠿⣦⠀⣴⠀⠿⠀⠐⠀⠒⠀⠘⠐⠂⠀⠀⠀⠀⠀⠀⢠⡠⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛
⠀⠀⢀⡤⠠⠠⠄⠀⡄⠀⡄⠀⡄⠀⠀⠠⡄⢠⣠⡄⠄⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠀⠀⠀⢀⠀⠈⠀⠀⢀⡀⢀⡀⢀⠉⢉⢉⡉
⠀⠀⠀⠀⠀⠄⠠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠉⠀⠉⠀⡀⠈⠀⠂⠁⢀⠉⠀⠀⠘⠇⠈⠓⠉⠀⠙⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⡁⠠⠠⠀⠀⠀⠒⠐⠀⠀⠀⠢⠐⠀⠐⠀⠘⠃⠇⠚⠸⠸⡿⠿⠓⠒⠚⠇⢸⠘⠲⠀⠀⠀⠀⠲⠲⠷⠞⠁
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠐⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠲⠲⠶⠼⠇⠰⠀⡦⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡉⡀⠀⠀⠀⠀⠀⠀⠘⢀⣈⡃⠒⠁⠀⠓⠓⠚⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠉⠀⠁⠀⠋⠊
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⠤⣤⠀⠀⠀⠠⠀⠀⠀⠀⠀⡶⣶⣎⡀⠶⠄⠂⠀⠀⠐⠲⠶⠶⠲⠂⠇⠀⠀⣟⣻
⠀⠀⠄⠠⠀⠀⠠⠀⠀⠀⠖⠃⠀⠄⠒⠲⠄⠂⠀⠀⠀⠶⠶⠾⠂⠀⠀⠀⠀⣀⠀⢀⣠⠈⣄⠀⣀⣄⠀⠀⠀⡤⡀⠀⢠⠀⡀⠀⡀⣀⣀⢀⣀⠀⣠⠄
⠀⢀⡤⠤⡤⠀⢤⡄⡀⠀⠀⠐⠴⠀⠦⠶⠀⠀⠀⠀⠀⠀⢀⠘⠓⢛⠒⠂⠀⠉⠈⠛⠙⠀⠃⠘⠋⠃⠀⠀⠰⠂⠁⠀⠘⠃⠐⠀⠇⠘⠚⡘⠇⠸⡧
⠀⠀⠀⠀⠶⠴⠀⠀⠠⡂⡴⢂⠴⠐⠀⠴⠦⠀⠆⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⢠⢀⢇⣄⠀⢠⢠⢄⣀⣰⡆⢰⢂⡄⡄⠀⠀⠀⢀⣴⠒⠀⠁⡤⢰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠾⣿⢀⡌⣅⠼⠟⠿⠅⠬⠼⠿⠀⠤⠀⡄⠀⣤⢀⣀⠋⡀⢈⣠
⠀⠀⠈⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠄⠀⠀⠄⣀⣀⡀⢉⣈⣉⠀⣀⢀⣀⣀⣀⣀⠀⠀⣀⣒⡂⠀⠒⠛⠳⠸⠗⠚⠉⠃
⠀⠰⠖⠿⠻⠷⠷⠾⠆⠀⠠⠀⠤⠄⠀⠀⠀⠀⠀⠀⠄⠄⠠⠄⠄⠄⠠⠄⠀⠉⠛⠁⠈⠛⢙⣀⡛⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠐⠼⠧⠴⠀⠰⠾⠂⠁
⠀⠀⠲⠾⠾⠿⠿⠳⢠⡄⠠⣤⢤⠀⠐⠀⢒⡐⣀⠀⡀⠀⠀⠂⠀⠀⠐⠀⠀⢀⣀⣀⣀⣀⣈⡉⢃⢀⣀⣀⣀⡀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣒⢀⡀⠀⠀⠀⠀⠀⠠⣤⡤⠤⣤⣤⣤⠄⠠⡄⠀⠈⠉⠋⠉⠉⠋⠁⠈⠈⠙⠁⠀⠁⠀⠀⠀⠀⠠⠀⠄⠀⠀⠡⠃⠀⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⠲⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣀⡤⠴⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠳⣩⢷⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠞⢧⣹⠞⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢃⣀⣙⣲⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣖⣋⡀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡞⠋⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠹⡄⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⢠⢤⡖⢲⢀⣀⡀⠀⠀⢹⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡏⠀⠀⠀⠀⠀⢀⣄⡀⡀
⠀⠀⠀⠀⠀⠀⢰⠾⡌⠳⠘⢋⣨⠇⠀⠀⠘⡇⠀⠻⣤⣤⠴⠶⠒⠲⠦⣴⠃⠀⢸⡇⠀⠀⠀⠀⡖⠻⠸⠋⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠲⠶⣄⠀⠈⢧⡀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢀⣽⠆⠀⡴⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠈⠳⢤⡞⠀⠀⣠⣤⢤⡤⢤⡤⠴⣞⠉⡹⠉⠓⡄⠈⠷⠖⠋⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⢠⠎⠀⠳⠞⠳⠞⠑⠚⠉⠉⠁⠀⠀⢸⠀⢀⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⠀⢀⡤⢤⣀⡖⠓⠲⡄⠀⠀⡸⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⢇⠀⢠⡇⠀⠀⠘⠁⠀⢀⣹⠀⢠⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢦⡀⡷⢄⡠⢦⡖⢲⣏⣈⠶⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⠓⠶⠓⠋⠉⠉⠉⠁⠀⠀⠀⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣀⣀⣀⣀⣀⣠⠤⠖⠒⠦⣄⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡼⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⢹⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣀⠲⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⣤⡼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣢⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣴⠀⠀⠀⠠⡄⣤⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⢀⠠⠐⡈⠀⠀⠀⠀⢀⠀⣀⠤⣀⠄⣆⠀⠀⢾⠒⣿⢀⣠⣄⢰⡇⣾⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⣀
⠀⠀⢀⠴⠊⠀⢳⣀⣷⠀⣰⣆⢻⡭⠋⢸⠂⠹⠄⣏⠀⠀⠺⠀⠳⠐⠋⠹⠺⠀⠟⢧⡸⢳⡞⣧⠖⣾⢁⡶⠄⠀⠀⣰⠊⠑⠤⡀
⠀⠠⠋⠀⠀⠀⠚⣇⠘⡦⠿⠈⠃⠁⠀⠸⢄⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠉⠘⠃⠸⠍⡼⣧⠿⠏⠀⠀⠀⠙⠄
⠀⠐⢄⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⢣⠀⠔⠀⠀⠀⠠⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠠⠀⡜
⠀⠀⠀⠀⠫⠀⠄⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠟
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⡓⠀⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣀⠆⡁⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⣸⣿⣦⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠿⠿⠛⡇⢈⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⣰⣧⣶⣿⡟⠁⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⡇⠻⣿⠏⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠑⣤⣾⣿⣿⣾⣿⣿⣿⣿⣾⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⢹⠀⠀⢀⣿⣿⣄
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣶⣶⣾⣅⢠⣿⣿⣿⣿⡆
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣡⣄⣾⣿⣿⣄⣤⣻⣿⣿⣿⣿⣧⣴⣾⣶⣾⣿⣶⠀⠀⠀⠀⠀⠀⠀⢈⡿⠉⠉⠹⡟⢸⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠆⢀⡇⠈⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠈⢿⣿⣿⣆⠈⠛⢻⣿⠿⣿⣿⠛⠉⢀⣾⣿⣿⡏⠀⢼⣿⣿⠇⠀⠀⣀⣠⠀⠀⠀⠀⣿⣇⣀⣀⣰⣷⠀⣿⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⣤⣤⣄⣤⣤⣴⣾⣿⣿⠟⠩⠀⠐⠤⣉⣠⢤⣤⠀⠇⠀⢀⢀⠀⢴⣿⣿⣿⣿⣿⣿⣶⠿⠟⠟⠿⠈⠛
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠻⣿⣿⣿⠟⠻⠿⠟⢋⡀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠂⠈⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇⢀⣴⢀⣿⣿⣿⣿⣆⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⢹⣿⣿⣿⠄⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢴⣶⣿⡇⠀⠠⠆⠄⠤⠀⡀⣾⣤⣤⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠋⢸⣏⠶⠒⠀⣦⡀⠠⢽⣿⠃⢉⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠈⠛⠁⠀⠀⠀⠀⠀⠙⠛⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⣀⣠⣄⣀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢰⣌⠉⣉⡶⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢠⡈⠉⢉⣠⠤⠖
⠀⠀⠀⠀⠀⢀⣠⣄⢀⡀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠶⢂⡉⠀⠈⠀⡶⠞⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⣀⡀⣀⡀
⠀⢠⡆⠀⠩⠓⠲⠶⠶⢖⣒⠂⠉⠛⢦⣄⣀⣀⡤⢊⣀⠐⠶⠦⠖⠿⠛⠋⠛⠴⠴⠶⠂⣀⠁⢤⣀⢀⣠⣴⡚⠉⠀⣒⡲⠶⠶⢶⠚⠯⠁⢢⡄
⠀⠀⢡⡀⣀⡀⠀⢀⣠⢤⡌⢉⡲⠞⢛⡦⣍⣙⠓⠻⠤⠴⠖⠒⣧⣼⠶⠶⢲⡶⠒⠶⠶⠴⠷⠚⢛⣉⢴⡛⠷⠖⡋⢉⣤⣄⡀⠀⢀⣀⢀⡌
⢀⡴⠆⠉⠉⠙⢶⠉⠀⠀⢈⠁⠀⣠⠋⠰⠉⡙⠳⣄⠀⠀⠶⠞⠉⠉⠛⠛⠉⠉⠳⠦⠀⠀⣠⠞⠟⠉⠦⠙⣆⠀⠈⡉⠀⠀⠉⡷⠋⠉⠉⠱⢤
⢸⡇⠀⠀⠀⠀⢸⠋⠀⠰⠁⣠⠴⣿⡀⣇⠤⠗⢒⠉⠙⠲⣆⡀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢙⡂⠰⠤⣰⠂⣿⠣⣄⠈⠆⠀⠘⠏⠀⠀⠀⠀⢀
⠈⢧⠀⠀⠀⣸⢿⠀⠀⠀⢰⢃⡤⠜⣷⣈⠲⣶⠀⠲⠴⠀⠈⢱⡀⠀⠀⠀⠀⢀⡞⠁⠀⠶⠖⠒⣸⠶⢃⣼⠣⣤⡘⡆⠀⠀⠀⡿⣦⠀⠀⠀⡼
⠀⠀⡆⠀⠸⠃⠀⠳⣄⠀⡎⢾⡵⠟⣀⠭⡷⠢⣍⣉⠻⠉⠛⠀⠷⠤⠴⠦⠤⡾⠀⠙⠁⡎⢉⣉⠥⢾⡍⣀⠻⢮⡷⢺⠀⢠⠜⠁⠘⠇⠀⢠⠁
⠀⠀⠇⠀⠀⠀⠀⠀⠘⠀⣷⣌⠷⣎⡁⠔⠛⢂⡀⠀⠛⠃⠀⢈⣷⠀⠀⠀⠀⣾⡁⠀⠘⠛⠀⢀⡐⠚⠆⢈⣡⡾⢃⣾⠀⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢰⠀⠀⠀⠀⢰⣷⠸⡙⢷⣌⡓⠾⠉⠀⠀⠠⢤⣤⡌⠛⣁⠀⠀⠀⠀⢈⠛⢡⣄⡠⢄⠀⠀⠉⠶⠞⣡⡴⢋⠃⣾⡄⠀⠀⠀⠀⡆
⠀⠀⠀⠸⠀⡇⠀⠀⣼⢹⡆⢱⡊⣉⠿⢲⣄⠉⠙⠛⢂⣁⠰⣦⣥⠀⠀⠀⠀⣼⡴⠆⣈⡙⠛⠛⠉⣠⡴⠿⣍⠑⡮⢡⡏⢷⠀⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣽⣦⠙⢶⡞⠁⠀⠠⠶⠞⢋⣉⣡⡶⢣⡄⠀⠀⢠⡜⢶⣍⣉⠙⠲⠶⠅⠀⠉⢓⡴⠋⣠⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢛⡀⢓⣄⡉⠛⠁⠴⣀⣈⠠⠔⣉⣴⠇⣀⠀⠀⢀⠸⣦⣉⠂⠄⣁⣐⠤⠌⠛⠋⣠⡚⢁⡻⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢻⣄⠙⣁⠀⠙⢡⢤⣠⣩⡅⠖⢫⡾⠳⣿⠇⠀⠀⠸⣿⢎⢷⡝⠂⢀⣉⣀⣤⡄⠛⠠⡈⠋⢀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠉⢶⣌⠒⢮⣀⡔⠋⠁⣀⠴⣋⡞⡲⠃⠀⢀⡀⠀⠘⢞⠳⣝⠦⣀⠈⠙⠢⣀⡱⠖⣡⡴⠋⢧⠄⠀⠀⢰⡄
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢰⡙⠇⢈⠿⠦⣀⠐⠀⣀⠀⠘⣩⠟⡀⢟⣴⠉⠙⢦⡹⢀⠻⣍⠃⠄⣀⠐⢂⣈⠴⠾⡉⠸⢏⡜⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⢳⡤⠑⢤⡔⢉⣉⠛⢁⣠⠞⡵⢺⠏⣼⠧⠶⠶⠤⣦⠰⡗⢬⡳⢄⡈⠛⣉⠙⢢⡤⠚⣡⠞⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⡌⢦⠙⢠⡈⡹⠉⠋⣉⠔⠛⣰⢃⡼⡇⠀⢀⡀⠀⠘⢣⡙⣆⠛⠦⢉⠑⠈⢋⢁⡔⠛⣤⢃⡇⠀⠀⠀⠘⠇
⠀⠀⡀⠀⢰⠀⠀⠀⠀⣾⠟⢄⠳⣅⡸⠛⠛⠋⠀⣠⠞⠁⣼⠃⣾⠋⠉⠉⠉⣱⠈⣷⠈⠳⣄⠀⠙⠛⠛⠇⣈⠞⣡⠞⣷⠀⠀⠀⠀⡆⠀⢀
⠀⠰⡇⠀⣼⠀⠀⠀⠀⡟⡆⢎⡱⢌⠳⠮⠤⠴⠚⠁⠀⣸⠇⠀⣏⢀⣤⣤⣀⣻⠀⠙⣇⠀⠈⠓⠤⠤⠴⠖⠁⢎⡱⢠⣿⠀⠀⠀⠀⣷⠀⠸⠆
⠀⡀⠀⠀⣿⠀⠀⠀⠀⢇⣿⡌⢣⡜⠓⠄⡴⠖⣲⠀⣰⠋⠀⠔⢟⠉⠀⠀⠈⢻⠀⠆⠘⣆⠀⣦⠐⠦⠠⠚⢃⡜⢡⣯⡸⠀⠀⠀⠀⣿⠀⠀⣠
⠀⠃⠀⠀⢻⠀⠀⠀⠀⠈⣼⢉⣤⠳⣤⣦⠴⠊⣡⣾⠃⠀⠀⠀⣤⠀⠀⣀⣀⣸⡀⢀⠀⠈⣷⣈⠑⠦⢴⣤⠞⢠⡉⣯⠁⠀⠀⠀⠀⡟⠀⠀⠉
⣸⠀⠀⠀⠀⠰⡀⠀⠀⠀⠛⢾⡱⣁⠠⡤⠶⠛⠁⠀⠀⠀⠁⠀⠏⠀⠉⠋⠉⣠⠏⠠⠀⠀⠀⠉⠛⡶⠤⡤⣘⢌⣧⠻⠀⠀⠀⢀⠄⠀⠀⠀⠐
⣇⠀⠀⠀⠀⢤⡸⠀⠀⠀⠘⡈⢻⣎⣉⣉⡁⢀⠂⠀⠀⠀⠀⠀⣄⣀⣠⣤⣀⣈⠀⠀⠀⠀⠀⠀⡀⠀⣈⣉⣡⡞⢁⠃⠀⠀⠀⢎⡤
⠸⣧⣤⣦⣄⣸⡇⠀⠀⠀⠀⠹⣦⡙⠦⢞⡡⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠑⢌⡳⠤⢛⣴⠏⠀⠀⠀⠀⢸⣇⣠⣴⣄⣼
⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣷⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣾⣷⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣏⣻⡏⠀⠀⠀⢹⣿⣷⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⠿⣿⣷⡀⠀⠀⣸⣿⡇⠀⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣇⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⣿⠿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣄⡀⢤⣄⠀⠀⠀⠀⠀⣾⣿⣿⡿⢿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⢻⣿⣶⣾⣷⡀⠘⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⡿⠆⢻⣿⣿⣿⣷⣀⣶⣶⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣥⣾⣇⠀⠀⠀⠀⠀⠀⢠⣀⠀⣀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣉⣹⣿⣧⣀⠀⠀⠀⠲⠀⠈⢹⣿⡿⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢿⣿⣿⣿⡁⠀⠀⠀⠀⢰⣿⣿⣧⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢣⣤⣾⣿⣿⣿⣇⣠⣴⣿⣶⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣻⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠋⠉⣽⣿⣿⣿⣦⣀⢀⠀⣀⣼⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣛⣭⣿⡍⠛⠿⠿⠟⠛⠛⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣴⠆⢻⣩⣽⠿⣿⣿⣷⣦⣤⣄⣀⣤⣤⣶⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠈⢿⣯⣤⣾⣿⣿⠟⠛⠛⠛⠉⠙⠀⠀⢈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠹⠃⠀⠘⠻⣿⣿⣿⣿⣶⣶⣶⣶⣴⣶⣶⣾⠿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⠟⠿⠛⠛⠋⠁⠀⢀⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠁⠀⠀⠈⠷⢠⣤⣴⣁⣼⣿⣿⣿⣿⠄⡐⢀⢀⣠⣴⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⡤⠈⢙⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣴⡏⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠺⣥⣤⣄⣀⣤⣿⣿⣿⣿⡏⠉⢉⣿⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢻⠿⣿⣿⣷⣿⣿⣿⣷⣾⣿⠿⠷⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣟⠀⡾⢩⣿⣿⠿⠿⣿⣿⣄⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠳⢾⣷⣤⣴⣿⣿⣿⣿⣿⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⡆⠀⠐⠠⠝⠛⢿⣿⣿⣿⢿⣧⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣠⡆⠀⠀⠀⣀⣉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⠉⠛⠷⢾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⣀⡀⠀⠀⠈⣻⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⢠⣶⡆⠀⠀⠀⣴⣶⠆⠀⢀⣤⠀⠀⣶⣶⠀⠀⢀⣶⡶⠀⠀⠀⠀⠀⣼⣷⠀⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⢀⣤⣤⠀⣿⡿⠀⣰⣿⡟⣀⡀⣿⡿⢀⡀⣼⡿⣡⣤⡀⣴⣶⡆⣿⣿⣰⣿⠃
⠀⠀⠀⠀⠀⣀⣼⣿⣧⣾⣿⢈⣭⣿⠀⣿⣇⣴⣿⠋⣾⣋⡟⣿⣧⣾⣷⣿⢻⣯⡾⢻⣿⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠛⠛⣿⠁⢸⡿⣿⣵⣿⢸⣿⠀⠘⠿⠶⠙⠋⠀⣿⣿⢹⣿⠇⠻⠷⠶⢿⣧⣤⣸⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⠀⠀⠛⠸⠿⠿⣿⠇⠀⠀⠀⠀⠈⠁⠘⠟⠀⠀⠀⠀⠈⠛⠋⣹⡇⠘⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⠶⠦⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣶⣄⣀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⠉⡶⢤⣀⡰⠋⠀⠀⠀⠙⢯⡻⣦⣀⡀⣀⣀⣀⡀
⠀⠀⠀⢠⠞⣠⠖⠋⠉⠉⠉⠙⠓⠶⣄⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣳⡌⢻⡙⢯⠀⠉⠉⠻⣦⣄⡀
⠀⠀⢠⢏⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⡀⢧⠈⢣⠀⠀⠀⣷⠀⠙⠳⣄
⠀⠀⣾⡟⠉⠙⡆⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⠸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⢷⡀
⠀⢠⢟⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⡠⡄⠐⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠉⣿⠀⠂⠀⠀⠀⠀⠇⢀⠀⠀⠀⠀⠙⣦⡀
⠀⡜⠈⠒⠒⣿⣿⣿⣿⣿⣿⣿⡇⠁⠈⠈⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢩⠏⠀⠀⣻⡇⠀⡆⠀⠀⠀⠀⠸⠀⠀⠀⠠⡀⢩⠳⡄
⠀⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⠀⣸⠁⠐⠁⣀⠴⠧⣄⠀⠀⠀⠁⠸⠀⢻
⠀⢷⠀⠀⠀⣿⣿⣿⣿⣿⣉⣿⠁⠀⠀⠀⠀⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⡟⣰⠟⠋⠉⠉⠁⠀⠀⠈⣯⠁⠀⡇⠀⠀⢸⡆
⠀⠸⡆⠀⠀⠘⠿⣿⣿⣿⠟⠡⢳⣿⡆⣠⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡧⠀⠀⢸⠃
⠀⠀⢻⣆⠀⡐⠀⠀⠀⠀⠀⠀⠈⢉⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇⢠⠂⡞
⠀⠀⠀⠻⣝⠦⣄⣀⣀⣀⣠⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠐⠀⠀⡼⠁
⠀⠀⠀⠀⠈⢷⣄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢁⡴⠋⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣧⣀⠤⣀⠀⠀⣠⡴⠋⢀⡔⠋⠁⠈⠉⠛⠲⠶⢶⣖⣚⠉⠁⠀⢔⣠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⡀⠀⢀⣀⣀⣤⡤⠿⠛⠁⠀⢀⡎⠀⣴⡯⢤⣄⣀⠤⠀⢀⣀⣤⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⠀⣸⠁⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣌⠳⢤⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠼⠆⠀⠀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⡀⠀⠀⠀⠈⢣⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠱⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡗⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⢴⣾⣛⠛⣲⠶⢦⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣴⠁⢀⡈⢻⡃⢴⣆⣱⠀⠈⣉⣉⠓⠒⠶⠤⣤⣀⣀⣀⣀⡤⠎⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⡀⠤⢷⠀⠀⠀⠻⣄⡘⣷⡟⠙⠾⠛⢁⡴⠋⠁⠉⠛⢦⠀⠀⠀⠈⠉⠉⠁⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⡠⠄⠦⡀⠱⣄⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⠽⢋⣀⠀⣀⡀⢾⡁⣾⣏⡆⠀⢸⡇⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⢀⡤⠖⢾⠁⠀⠀⠘⢄⠈⠛⠒⠚⠛⠉⢹⡆⠀⠀⠰⡅⠈⢉⣀⣉⡁⠸⢧⡈⠉⠀⢀⡼⡁⠀⢀⣠⡖⠋⠁
⠀⠀⣰⠋⠀⠀⢈⡳⠶⠖⠀⠈⠳⣄⠀⠀⠀⠀⠀⡇⢰⠂⠀⣠⠞⠉⠀⠀⠈⣷⣄⠙⠛⠛⠛⠊⢁⣴⠋⠀⠈⠉⠉⢀⡴⠁⢀⠔⠊⠉⠙⠂
⠀⠀⡏⠀⣀⡀⠀⢻⠀⠀⢀⡀⠀⠈⡆⠀⠀⠀⠀⡇⡼⢠⣾⣥⣤⣤⣴⣤⣴⣿⡈⢦⠀⠀⠀⠀⡼⠉⠛⠲⠶⠶⠒⠉⠀⢰⠃⠀⢀⠀⢁⡞
⠀⠀⠣⡜⠉⣟⣠⢞⣀⡀⢨⡟⢦⠀⢸⠀⠀⠀⠀⣷⣦⡿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡇⠆⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⠙⠉⠙⢦
⠀⠀⠀⠀⢰⠁⠙⠺⡄⠗⠋⠀⢸⠀⢸⠀⠀⠀⢰⣿⡿⠇⠂⠀⠠⠤⢤⣀⣀⣿⠋⠻⣿⣸⠀⣸⠣⡀⠀⠀⠀⠀⢀⡴⠃⠀⣠⠄⣀⡴⠃⠀⠀⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⡞⠀⠀⢧⠀⠀⢸⠿⠀⠀⠀⠀⠐⠶⠦⠤⠈⠉⠓⠢⣌⣿⢀⡿⣄⠈⠙⠒⠒⠋⠁⠀⢀⣴⡅⠄⠻⣄⣀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠙⠒⡿⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣿⠸⡇⠉⠳⠦⣤⣤⣤⠤⠖⠋⠀⠙⠦⣤⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣠⡾⠁⠀⣴⠋⠈⡇⠀⠀⣰⠋⠙⢷⣄⠀⠰⠇⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠉⢀⣠⠞⠁⠀⠀⡇⠀⢀⡏⠀⠀⠀⠙⢦⡀⠀⠐⣄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⡴⠚⠋⠁⠀⠀⠀⢸⠀⠀⢸⠃⠀⠀⠀⠀⠀⢻⡄⠀⠘⣦⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠙⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢹⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢐⡇⠘⡆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢻⡄⠀⢳⡀
⠀⠀⠀⣀⡤⠴⠶⠤⠤⣤⡾⠀⠀⣰⣅⣀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸⡀⠀⢸⡃⠀⠀⠹⣄⠀⠙⠦⠤⠠⠒⠒⠒⠒⠦⢤⡀
⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⣠⡤⢄⣀⣼⠄⠀⠀⢸⣀⣀⡀⣀⡤⠼⠧⠀⠘⢧⣄⣀⢀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡗
⠀⠀⠳⣤⣤⡽⢦⣀⣀⠀⣀⣴⣄⣀⣸⣤⠞⠀⣞⠁⠀⢀⡀⠀⠀⠀⠀⠀⢤⡀⠉⠉⣳⠀⢰⡄⠀⠀⠀⠀⢍⣳⡀⢀⣤⡀⠀⣎⣀⡼⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠁⠀⠀⠀⠹⠦⡴⠞⠳⣄⣀⣀⠂⠀⢈⡷⠶⠚⠛⠶⠺⡄⠀⠀⢀⠀⢀⡟⠈⠉⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠛⠒⠚⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⣤⣶⣿⣿⣿⣿⣿⣿⡛⠂⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⣠⣧⣴⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⣰⡟⠉⠙⣷⣦⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢟⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣿⣈⠓⠐⣿⣿⣿⣷⣿⣽⣿⣿⣿⢿⣩⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠘⢿⣷⣶⣿⣿⣿⡿⠽⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⠿⠿⢻⣿⡿⢁⣀⡁⠀⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣷⣶⣤⠚⠿⠿⣿⣿⣿⣷⡜⢃⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡄⠀⠈⣹⣿⠟⠀⠸⣧⣀⠀⠀⠹⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⢹⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠿⢿⠉⠈⠁⠀⠀⠲⢶⣤⣤⢾⣿⠟⣴⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⡂⠒⠒⠢⠜⠿⠇⠀⠀⡼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣤⣤⢤⢄⣀⣀⠀⠀⠂⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠦⣤⣀⡀⣀⣀⣠⣶⠿⣾⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⠖⣾⠟⠋⣰⣾⣷⡗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣨⡽⣿⢻⣿⣿⣟⣵⣟⣴⣿⣿⣿⢡⡇⢄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⣀⣈⣿⣿⣹⡇⣿⣷⣿⣿⣿⣾⣿⣿⡜⣿⢣⣿⣿⣧⣬⡭⠁⣒⠂⢄
⠀⠀⠀⠀⢀⣴⣿⣾⣿⣷⣿⡏⠙⢿⠿⢿⣿⣸⣿⣿⣿⣿⣿⣿⣁⣷⡀⢸⣿⡿⠙⠁⢰⣿⣿⣷⣆⠑⠄
⠀⠀⠀⠀⠀⠉⢾⣿⣿⡏⣠⣀⠀⣠⣶⣦⣿⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣾⢟⣡⣴⣶⣄⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⢯⡷⡎⣿⡽⣿⡿⠛⡛⢿⡇⠹⠋⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠉⠀⣿⣿⣿⣿⡇⢸⣷⣶⣿⣿⣿⣆⢻⣿⠚⠃⣠⣤⣉⠙⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣦⣙⠻⠿⣿⠿⠻⣛⣡⣴⣿⣿⣿⣿⣠⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣯⠉⠲⢿⣟⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠮⠅⠁⠀⠀⠈⠀⠉⠙⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⢫⣧⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⣠⡇⠐⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⣴⣾⠋⠀⠰⠷⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣋⣭⣵⣾⣯⡐⠷⠀⠀⡀⠀⣿⣿⣿⣿⣦⡀⠀⠘⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠛⠉⣿⣦⣄⠈⠁⠀⠀⠉⠛⠿⣿⣿⣶⣶⢟⣿⠇⠀⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠅⠈⠀⠃⡸⠋⡣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣏⣀⣠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣫⣿⠃⢀⠀⠐⠜⠂⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣽⣿⡿⢛⡛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠠⠀⠤⠜⠀⢀⡠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢹⡿⣷⣌⢻⣿⣿⣿⠿⠿⢿⣦⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⡀⠀⠀⠐⠀⠀⠠⠄⠅⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣾⣿⡟⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠵⡀⢄⣴⣶⣿⣿⠥⢀⠀⠄⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⣿⡿⣏⣿⡇⠀⠀⣀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣴⢬⣿⣿⣝⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⢰⣿⣿⠀⠻⣾⣿⣦⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠋⠈⢿⣝⡻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⣿⣟⢇⠀⠀⠉⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠉⠛⣿⣷⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠿⠟⢻⣮⡓⠦⡤⣶⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⣴⡿⠟⠿⣶⡀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣾⣿⠃⣴⢿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠹⠦⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⢠⣾⡿⠙⣷⠀⣀⣤⣤⡴⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⣴⣿⠟⠀⢀⣿⣿⣿⠟⠁⢀⣤⣾⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⡲⠶⢶⣶⣶⠿⠟⠋⠀⢀⣠⣾⡿⠋⠀⣠⣾⣿⠟⠁⣠⣶⣿⣿⡿⠛⠁⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠳⣝⡶⣤⣤⣤⣤⣤⣴⣶⡿⠟⠋⣠⣴⣿⠿⠋⢁⣴⣾⣿⣿⠟⠋⠀⢀⣴⠶⠶⢤⡈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣛⠶⣯⣭⣍⣉⣩⣥⣤⡶⠿⠛⠉⢁⣤⣾⣿⡿⠟⠉⠀⠀⠀⠀⣾⡇⢰⣆⣼⡇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣍⣉⣉⣁⣀⣤⣤⣶⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣉⣉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠾⠀⠀⠀⡄⠀⢀⠀⡄⢀⠀⣀⡄⠀⠀⣷⣣⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣤⣀⣤⣼⠈⠛⠈⠁⠈⠙⠁⢧⣤⣶⠟⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣽⡿⠋⢀⣀⠀⠀⠀⠀⢀⣀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢡⣾⠁⠀⠉⢢⢠⣾⠁⠀⠉⠂⠀⠀⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠻⣿⣄⢰⡷⡸⢹⡿⣄⢾⡆⡜⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⠗⣤⣝⣓⣒⣋⣁⣀⣙⣓⣒⣫⣤⣆⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡀⣿⣿⣦⣿⣦⣿⣼⣿⣼⣷⣾⣿⡿⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⢻⣿⣧⠻⣽⠻⡿⢿⡿⢿⣿⡿⣿⣛⠟⠁⠀⢀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣸⣿⣿⠗⠬⠉⠛⠚⠓⠒⠛⠛⠉⠁⠀⠀⠀⢸⡇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢨⣿⡆⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠸⢻⣷⣠⣤⣸⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣴⣀⠎⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠯⠀⠙⣸⡿⠿⠿⣷⡀⠀⠀⠀⠀⠀⠀⣠⠄⠰⢌⠃⠠⠿⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⢿⣿⣶⣶⣴⣷⣿⣿⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡽⠷⠄⠀⡂⠀⠈⠉⠉⠉⠀⠈⣿⡧⠄⠰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⣶⣷⣾⣿⣶⣷⣶⣶⡶⢶⢶⣶⣾⣿⣿⣿⣿⣿⣷⣦⠤

⠀⠀⠀⠀⠀⣀⣠⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡒⢒⢀
⠀⠀⠀⢠⠖⠉⠀⠈⠀⠉⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⡶⡆⠀⠀⠀⡋⠗⠀⠀⠀⠀⠀⠀⠎⣡⠸⡆⠀⠀⠀⣼⡷⡀
⠀⠀⢰⠇⠀⣤⠀⡄⠀⡄⠀⠈⣢⠀⠀⠀⠀⠀⠀⠀⠈⢙⠆⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⠣⠭⠐⢇⠀⠀⠑⠌⠨⠘
⠀⠀⠧⠀⠀⠀⢀⠀⢀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⢠⠊⠔⢠⡄⢂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠣⡀⠀⠁⠆
⠀⠀⠳⠀⠀⠀⠈⠙⠉⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢⠀⠠⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠙⠉⠈⠣
⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣐⣀⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⠀⢀⣳⠧⣀
⠀⠀⠀⠀⠓⣤⣤⠤⣤⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⠉⠳⠋⠀⠀⠀⢈⣂
⠀⠀⠀⠀⠐⠛⠛⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⣃⠄⠰⣀⡓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠈⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⡄⠂
⠀⠀⠀⠀⠹⠟⠆⠀⠀⠸⠽⠅⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⡄⠀⢀⠀⠈⢢⣀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠺⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣠⠄⢀⠀⡁⠀⠈⠀⡀⠀⣤⠙⡄⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⡂⠀⠀⠀⠘⠀⠸⡀⠀⣶⣷⣾⣾⣶⠇⠀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠰⠳⠀⠒⠒⠒⠒⠀⢀⠀⢋⠀⠀⠀⠰⣤⠇⠰⣄⠈⠘⠀⠃⠋⢠⠖⢽⣤⡀⠀⠀⠀⠀⢀⠀⠀⠄⠂⠐⢄⢀⣄
⠀⠀⠀⠒⠀⠀⠀⠐⠒⠊⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡖⡆⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠁⠤⢀⣤⠀⠈⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⡇⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠂⠀⠄⠀⠠⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠃⠑⠂⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠
⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡄⣀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠖⠂⠀⠲⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠓⣀
⠀⢀⠀⠀⠰⠀⣉⣀⣀⣬⠀⢡⠀⠴⣄⠀⠀⠀⣴⢤⠊⣶⣷⣀⣾⣿⢷⢣⣸⠗⠀⠀⠀⠀⢰⠀⠀⠶⠀⠀⠰⡂⠀⠈⡆
⠀⢪⠠⢈⠁⠀⠙⠿⠟⠋⠀⠀⡀⠂⠋⠀⠀⠀⠀⢾⠀⠉⡉⣀⢳⡿⠋⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢃⠀⢿⢿⣿⡗⢀⠎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠳⣄⠀⠀⡠⢞⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠡⠀⠤⠀⡄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⠀⠑⠊⠀⡈⣄⠀⠀⠀⠀⠀⠀⢆⣀⡀⡀⣀⢀⣀⠀⣀⣀⠠
⠀⠀⠀⠀⠘⠀⠀⠃⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠒⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⢀⠀⠀⢀⡼⠓
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⡄⣤⡈⠑
⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠈⢈⡁⣉⠉⣁⠖⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠟⣁⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠰⣿⣻⠷⢞⣻⡿⠇
⠀⠀⠀⠀⠀⠈⢻⣦⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠐⡫⠊
⠀⠀⠀⠀⠀⣼⠁⣶⠘⣇⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠮⠖⠀⢘⢁
⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢠⡀⣀⣀⣀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣤⣤⣄⠀⠀⢸⡁⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢸⠁⠀⠈⢿⣧⠀⣾⣿⣿⣿⣿⡄⢠⣿⣿⣿⣿⣧⠀⠘⢿⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⢸⣿⢰⡇⠃⠀⠀⠋⡇⢸⠘⠀⠀⠉⢹⠀⡇⢸⡄⠀⢸⣿⠀⠀⡿
⠀⠀⠀⠀⢠⠚⠿⠆⠀⠀⠀⢀⣾⠋⢸⡀⠀⠀⢸⡿⢸⣧⡄⠀⢠⣤⠃⢸⣠⣀⣀⣰⡼⠀⡇⢸⡇⠀⢸⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠤⠴⠊⠁⠀⢸⠉⠒⠶⠛⠁⠀⠙⠿⡿⠿⠋⠀⠈⠻⠿⠿⠟⠁⠀⠇⠘⠷⡄⠈⠿⣦⠖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠟⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠠⠀⠀⠀⠀⡠⠐⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⣀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⠊⠀⠀⠀⠒⠲⣄⠀⠀⠀⠀⠀⠀⡀⠀⠓⠠⠀⠨⠄⠂
⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⡠⠒⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡀⠤⠀⠀⠈⠐⠀⠀⠈⠉⠉⠀⣀⡀
⠀⠀⠀⠀⠂⠀⠁⠉⠉⠀⡀⠠⠔⣈⡀⠀⡀⠀⠠⠄⢂⢠⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⠤⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⢾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠨⠤⢄⣤⣬⣥⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠠⢔⣽⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠄⣀⠀⠉⠐⠒⠀⠄⠠⠤⠀⠀⠀⠀⠤⠄⠐⠷⢤⣤⣀⣀⣀⣀⣐⣐⣨⡧⢀⠠⣊⠔⠒⠉⠉⠉⠙⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⠒⠠⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢁⠀⡀⠀⡴⠃⢀⠠⠀⠀⠄⠠⢌⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⢧⡔⡥⠚⠉⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⠀⠀⠀⠀⠠⠀⠐⡀⠔⡯⠥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣟⠒⠤⠤⣤⠭⢤⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌⡀⠉⠉⠙⠛⠛⠛⠛⠛⢑⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠶⠖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠄⠠⠈⠠⠀⠀⠠⠂⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠮⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠄⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⠀⠀⠈⠢⡀⠀⠀⠀⠄⠀⢀⠎⠀⠀⠀⠀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄⣈⠂⠀⠀⠀⠀⠀⠀⠠⡄⠐⠁
⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⢀⠀⠠⠆⠁⠀⠀⠀⠤⠀⠠⠀⠒⠀⠂⠀⠐⠀⠒⠀⠈⠀⠈⠑⠂⠀⠈⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠀⢉⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⠀⠀⢀⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠉⠀⠐⢦⠁⠐⠀⠁⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠅⠀⠀⠀⠠⠄⡀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈
⠀⠀⠀⠀⠀⠄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣦⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣦⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠛⢤⣶⣿⣿⣦⣝⠁⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⡵⠀⠘⠛⠀⠺⡗
⠀⠀⠀⠀⠀⠀⡩⢙⠹⠉⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠛⢀⠀⠀⠺⢵⠂⠀⠀⢀⣀⠀⠀⠀⢻⠗
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡀⠈⡈⠀⡀⠄⠀⣤⡄⠀⠀⠀⠘⠋⠄⠀⠀⢰⣤
⠀⠀⠀⠊⠀⠁⠀⢐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⡀⠁⠉⠁⠀⠀⠀⠉⠄⠀⠀⠠⠄⠐⠁⠀⠁
⠀⠀⠁⠈⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠠⡀⠀⠁⠀⠂⠃⠀⠀⠄⠀⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⠀⣀⣀⡈⠀⠀⢄⣀⣠⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠠⠾⠇⠸⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠿⠿⠿⠿⠇⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠰⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠂⡀⠀⠀⢐⣂⡂⠀⠀⠀⡄⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢂⠀⠀⠀⢸⣦⣀⠀⠀⠀⠀⠀⢈⣁⣶⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⢺⣿⠟⠀⠀⢀⠀⠈⠀⠀⠀⠆⠀⢀
⠀⠀⢠⠛⠀⠀⠀⠈⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠰⠀⠀⠒⠒⠂⠀⠀⠃
⠀⠀⢼⠀⢂⠀⠐⠠⠐⠊⠁⠀⠠⡇⠀⠀⠀⠀⠈⢀⣀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠐⠂⢲⠂⠈⠑⠂
⠀⠀⠀⠀⠀⠁⣐⠒⠀⣒⡀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠈⠁⠀⠀⡠⠁⠀⠀⠀⠀⠀⠚⠂⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣒⠂⠐⢂⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢸⣿⡟⠉⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⢯⣽⣧⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣺⢦⣤⣤⡿⢉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡋⠉⢩⣿⣿⣿⣿⣶⡝⠛⠳⢦⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⡠⠛⢻⡿⠿⣿⣿⡇⢀⣠⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣰⡿⠁⣀⣨⣛⣴⣻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⣿⣹⣿⣟⣋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⠀⠘⠻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡆⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣯⡁⠀⣭⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠈⠀⠀⠈⣿⣷⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⢀⡀⠔⠊⢀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡖⠐⠒⠂⠈⠁⠀⢀⡠⢚⠢⣀⠀⠀⠉⠑⠒⠒⠒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢢⡤⠀⠤⠐⠈⠁⠀⣸⡀⠀⠉⠐⠀⠄⠀⣤⠎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡄⠈⢦⡀⠀⠤⠐⠈⠀⠈⠓⠤⠄⠀⣠⠜⠁⡆⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⡜⠀⠀⡌⠑⢄⣀⣀⠠⠚⠢⢄⡀⣀⠔⠙⡄⠀⠘⡄⠀⠘⢄
⠀⠀⠀⡀⠀⠀⢀⠤⠊⠀⣠⠎⠀⢀⡜⠀⠀⡌⠱⡀⠀⠀⠀⠀⡠⠉⡄⠀⠘⣄⠀⠘⢦⡀⠈⠳⢄⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠉⠻⡀⠐⠺⡒⠂⢴⠫⠄⢠⣎⣀⠀⠈⠢⡀⡠⠊⠀⠀⣈⣦⡀⠨⢳⠄⠐⠻⠒⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢱⠀⠀⠁⠀⠀⠂⠀⠈⢉⠭⡻⡧⢍⠉⠀⠀⠎⠀⢀⠃⠀⠠⠁⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⣠⠔⠊⠀⢰⠁⢱⠀⠈⠒⢤⡀⠀⢐⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⣠⣔⠈⠀⠉⠢⣀⠇⠀⠀⢃⡠⠂⠁⠈⢑⣼⣀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠗⢁⠀⠀⠉⠢⡄⠀⠝⠉⠉⠉⠙⡅⠀⡠⠊⠁⠀⢀⠑⠶⢄⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠴⠟⠥⢀⠀⠀⠈⠀⠀⠀⠈⣦⠤⠀⠒⠂⠠⢤⡎⠀⠀⠠⠊⠀⠀⢀⡠⠵⠷⢄⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠠⢀⠃⠀⣀⣀⣀⡀⠀⢃⠠⠊⠀⢀⡤⠊⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⡟⠉⠀⠀⠀⠀⠀⠈⠙⡇⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣔⠠⠐⠀⠀⠀⠀⠀⠐⠠⢘⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡀⠀⣠⣤⣤⣄⡀⠙⠏⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢀⣸⣧⡀⠙⠛⠛⠛⠁⠈⠋⠀⣀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠈⢻⡟⠉⠀⠀⠀⠰⠶⠀⠀⠀⠙⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠇⠀⠃⠀⠈⠁⢀⣤⣶⣷⣦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⡀⠀⠀⠉⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⡀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⠋⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠉⠁⠉⠀⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣤⠀⠈⠋⠀⢰⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⣿⣿⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⡿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡀⣀⣤⣤⣄⣤⠤⠀⠀⠀⣀⠄⠘⠛⠉⠹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠈⠛⠟⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⢀⣀⣀⠐⠛⠛⠃⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⢠⠠⣶⢄⣀⣸⣇⡿⢸⣧⣤⣃⡀⢸⡉⠏⠀⠀⣁⣈⡁⡀⡘⠛⠆⣤⣴⠀⠀⠀⠀⠀⠀⠀⡂⣇⠀⡄⢀⡀⠀⠀⣦⠀⠀⠀⠆⠀⠀⢀⡀⠀⠀⠀⠀⣸
⣿⠚⠒⠀⠀⠀⠀⠈⠑⠾⠋⠛⠛⠁⠆⠀⠤⣠⣞⠟⢷⣾⠀⠀⠀⠀⣿⣿⡇⠀⡀⣤⣴⠀⢠⣧⢻⣴⡗⢾⡟⡿⡖⣿⣐⠀⡄⢰⣤⣤⣤⠀⢠⡆⢀⣈⣾
⣿⡀⣦⡀⢀⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⢳⣄⠇⠐⠀⠀⠀⠹⠛⠁⠘⠣⠐⠁⢹⠉⠻⣼⡀⠡⠤⢼⢁⢰⡈⠛⠾⠽⠟⠛⠗⠚⠘⠘⡇⠸⠙⢸
⣿⡄⠓⠀⠠⠠⢠⠀⠁⠀⠂⠀⠰⠆⠀⠀⠈⠋⠩⠁⠀⠀⢠⠀⠤⠀⢀⠀⠸⢸⠇⠈⠀⠀⠀⡀⠀⠀⢀⠀⡀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠇⠀⢠⣼⠀⣰⡦⣴⣾⣄⣀⠀⡀⡀⣀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠈⠁⠀⠙⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⣀⣀⣴⠦⠄⠀⠀⠀⠀⠠⢤⣄⡀⠀⡔⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠔⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠢⡀⠀⡌
⠀⠀⠋⠀⠀⡀⠀⠀⠀⠔⠄⠠⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⢲
⠃⠀⢉⠀⠉⠜⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠢⠁⠆⠂
⠀⠀⠓⠊⠑⠅⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠀⠀⠀⠁⠢⢹
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⠈⠐⠠⡄⠀⢀⣀⠀⣀⣀⡀⠠⠔⠊⠁⠀⠀⠀⠀⡀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⢁⠊⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣀⢄⢀⠠⠄⠀⠀⡀⠀⠀⠀⠘⡀⢀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠅⣀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠤⠒⠊⠁⠀⠀⠀⡄⠀⢀⠀⠡⡀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠑⠐⠒⠀⠒⠂⠒⠠⠤⠄⠀⠤⠐⠂⠀⠤⠤⠍⠁⣀⡀⠀⠠⠤⠐⠒⠀⠒⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⡀⠋⠶⠴⠴⣄⡄⠀⠀⠧⢀⠀⡀⠑⢖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠒⠂⢠⢀⣤⠀⠀⣰⠀⠘⣧⡄⠐⠛⠁⠛⠄⠀⣅⠂⠬⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠠⡔⡖⢻⢀⠠⠒⣄⢼⠝⠏⠋⠀⠀⠶⠋⠑⠒⠚⠷⠂⢀⣀⠀⠂⠰⣌⣆⠀⡀⠈⢹⣄⡄
⠀⠀⠀⠀⠀⠀⢠⠹⣦⣼⣧⣶⡿⠟⠉⠀⢓⡳⠂⠡⣄⠀⠀⠀⠀⠘⣧⠀⠀⢉⡧⢀⠀⢹⡦⣀⠀⠈⢿⡇⠀⡀
⠀⠀⠀⡀⢸⠀⠎⢀⣸⣿⢳⣏⣤⠤⠚⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠲⢇⣆⣷⢀⠁⠘⡧⢸⣦⡀⠎⢷⡴⢻⡆
⠀⠀⠀⣧⠀⡆⠀⡸⢋⡿⣾⣉⣆⢠⢠⡠⣴⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢿⠳⠷⣧⡄⣣⡑⡄⢼⣷⣾⠁
⠀⠀⢀⠸⡔⡉⢶⡟⣹⣱⡟⠓⠁⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠂⠳⣄⣏⠘⠻⢷⣿⣀⣿⡃⣨
⠀⢳⠈⡦⢻⣦⡟⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢸⣿⡝⣿⡷⡴⠂
⠀⠈⢷⡄⠈⢿⣾⢃⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠳⣿⡇⢻⣇⢽⠄
⣄⠀⣸⢿⣆⣼⣟⣱⣿⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢻⡷⣼⣿⣷
⠘⣧⡽⣎⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⣸⣿⣼⣿⠏
⠀⠘⣧⣿⢸⣿⣿⢀⣄⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⡏
⠀⢀⢹⣾⢾⣿⣿⣨⠋⠡⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣀⠀⡓⢾⣿⣿⢳
⠐⢾⡤⣿⣤⣿⣿⡄⠀⢸⣿⣫⡀⠀⢠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⡻⡟⣾⣿⣿⢸⠇
⠀⠸⣿⢏⢿⣿⣿⣧⠀⢒⢸⣟⡆⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡇⢈⣿⣿⡇⢸⠀⠠
⠀⠀⢹⡥⡂⢻⣿⣿⣧⠀⢖⣿⢩⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣿⠥⣼⣿⣿⣇⡏⠀⠏
⠀⠀⠀⠘⢷⡈⣿⣿⣿⣷⡨⣯⣾⣿⢘⣠⣀⡆⠀⠀⠀⠀⠀⠀⠀⢶⣷⣆⡀⠀⠈⢀⣤⡸⣿⣏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⣿⣛⠛⣿⣻⣿⣶⣄⢀⠀⢀⠀⠀⠘⢝⢯⣇⢆⣇⡞⣿⠩⢿⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠛⠻⣿⣷⣾⠼⠿⠿⠿⠯⣄⣠⣤⡄⢠⠴⠓⢉⣩⣍⠙⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠦⠤⣼⣿⣥⠤⠄⣨⡤⠐⠀⠀⢈⠗⣅⠀⣲⣊⣹⣥⣾⣿⣿⣿⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡏⠉⢻⣿⣟⣩⣹⣷⣶⣊⣠⠶⢻⣯⠟⠉⠀⠀⢤⣤⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠿⠏⠀⠊⠁⠐⢤⠞⠂⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⠀⠀⡀⠀⢀⣠⣄⡀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣿⠀⣸⣷⠀⢸⣿⠙⣿⢸⣿⢻⣇⢻⣦⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⣯⢠⣿⣿⣇⢸⣿⡾⣿⢸⣿⣾⡟⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠻⡟⢸⣿⠻⣷⠀⣿⠁⠀⠀⣿⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠇⠸⠃⠀⠙⠄⡏⠀⠀⠀⠇⠀⠀⠼⠁
⠀⠀⠀⢠⣴⠄⢰⣶⠀⣰⡄⠀⢘⠁⣰⠀⢀⣴⣶⡀⣦⠀⢀⢀⠀⠀⢀⣤⣠⣄⢠⣤⣤⡄⣶⡆⠠⣦⡄
⠀⠀⠀⠸⣿⠀⢸⡟⢠⣿⣷⠀⣸⡀⢹⡇⢸⡇⢘⣿⢸⡇⢸⣿⠀⣼⠸⣿⠉⠉⢸⣷⠈⠁⢻⣿⡀⣿⠇
⠀⠀⠀⣼⣿⣶⣿⡇⢺⣿⣿⡇⢹⡇⢹⡇⠘⢿⣾⠟⠘⣧⣸⣿⢸⡏⢰⣿⠟⠃⢸⣿⠛⠀⢾⡿⣷⣿
⠀⠀⠀⠀⣿⠃⢸⡇⣿⠇⠘⣇⣿⡇⠼⢷⣶⡆⠀⠀⠀⢻⡿⢻⣾⠃⢸⣿⣤⣤⢸⣿⣤⡄⢸⠃⢹⣿
⠀⠀⠀⠀⢿⠀⠈⠀⠃⠀⠀⠠⠿⢷⣶⡆⠀⠀⠀⠀⠀⠈⠃⠘⠟⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠀⠘⡇

⢦⣼⠀⡌⣿⣿⡆⢻⡰⣴⣃⡶⠋⠹⢲⡼⠀⢠⡸⠀⠀⢸⡄⢴⠥⣰⣷⡞⢻⡈⢸⣿⣿⣧⢸⣿⣀⢼⢿
⣛⣿⣇⡿⣿⣿⣷⠋⣷⠋⢸⠀⣥⠟⡼⠓⠂⢩⡤⣰⠀⠈⢷⡘⠮⢼⡞⣷⠀⢻⣾⣿⣿⣿⣾⣿⣸⣽⣿
⠘⢧⣸⣧⣿⣿⣿⣼⠉⣧⡯⣂⡴⠚⠁⠀⢐⠘⠆⣬⡳⢦⡈⢷⡤⣉⣿⡜⣄⡄⠹⣿⣿⣿⣿⣿⢉⣿⣿
⡦⣹⡟⠈⣿⣿⣿⣇⣾⣿⡞⠉⠀⢬⠌⣠⠀⠤⠄⠰⠭⠵⢿⣿⣷⣌⣐⠿⣎⠻⣄⣿⣿⣿⣿⣿⣼⣿⡷
⣷⣼⣷⣞⣿⣿⣿⠞⢡⡿⢀⣤⣉⣤⣞⠚⠯⠀⠀⠉⠊⠀⠀⠀⠉⠻⢾⣝⠾⣿⡟⣿⣿⣿⣿⣿⣿⣯⣏
⠸⣏⣿⣿⣿⣿⣿⢸⡿⣷⣩⣠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣿⣿⣿⣿⣿⣿⣦⣾
⠀⢹⣏⣸⣿⣿⣿⣿⣋⡽⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢹
⢀⡾⠙⣿⣿⣿⣿⣏⠿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⢸
⣼⠁⢰⣿⣿⣿⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⡇⢸
⣏⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⣿⣿⣿⣿⣧⢠
⣿⡟⢰⣿⣿⣿⣿⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠤⢻⣿⣿⣿⣿⣿⣿
⣿⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⡏
⣿⣇⢸⣿⣿⣿⣿⣧⡀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡇⠉⣿⣿⣿⣿⣿⣿⡇
⠘⣿⣾⣿⣿⣿⣿⣿⡘⣌⠂⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡇⣴⣿⣿⣿⣿⣿⣿⣇
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⡿⣝⡞⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⢸⡷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢹⣸⣿⣿⣿⣿⢿⣿⣿⡏⣿⣿
⢋⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡕⠀⠀⡀⢀⣀⣀⣠⣤⣤⡤⣶⣷⣿⣿⡟⢻⣿⣿⣆⣿⣿⣧⢻⣿
⣾⠃⣸⣾⢻⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣦⣴⡷⠛⢋⣭⢿⣉⣴⢿⣿⣿⣿⡿⣠⣾⣧⢻⣇⠟⣿⣞⣷⣻
⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣷⡟⢣⣾⣱⣷⣟⣿⣿⣿⢇⣿⣿⣿⣾⣿⣷⣻⣿⡟⣿
⣷⠀⣿⡟⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣷⣿⣿⣿⣿⡏⣽⣿⣿⠸⣿⣿⣿⢣⣿⢹⣿⣿⣿⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡿⠋
⠀⠀⠀⠀⢀⡀⠀⠈⠀⠈⠀⡄⢀⠀⠈⠙⠋⠋⠁⠰⢶⣶⣷⣤⣠⣤⣦⣶⣿⣿⠟
⠀⠀⠀⢠⣿⣿⣆⠀⠈⠱⠀⣟⢛⣴⣆⡖⢲⠶⣰⠀⠈⠉⠻⠿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⣾⣿⣿⣿⣷⠀⠈⠀⠀⣤⠀⠀⢠⢰⠄⠸⡆⡄⠀⠀⠀⠈⠀⠁⠀⡄
⠀⠀⠀⠙⠛⡏⠉⠁⠀⡎⡐⠂⠘⠓⠖⠸⠸⠴⠳⠇⠿⠸⠾⠶⠻⠸⠸⢾⠁
⠀⠀⠀⠀⠀⠳⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠛
⠀⢀⣤⣤⣄⣤⣄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣿⣧⣀⣀⡀⠀⠀⣀⣀⣀
⠀⠈⢿⣿⣿⣿⡿⢇⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠟
⠀⠀⠀⠹⣿⣿⡿⢋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⣅⣿⣿⡇⣝⣿⣿⣿⣿⡘⠋
⠀⠀⠀⠀⠀⣽⠸⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⢹⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⣿⣧⣄⡙⠛⠙⠛⣁⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠴⠖⠒⠂⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣾⠿⠿⠿⣿⣿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⠿⠿⠋⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⠂⡸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⡝⠻⢧⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⡏⢠⡎⠹⡆⠀⠀⠋⠒⢤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠡⡘⢷⣠⡇⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⠸⣿⡀⠀⠈⡢⣝⠛⠧⢀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠷⠾⠮⣅⣉⡒⣤⣀⠁⠓⠺⢅⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⠞⠟⠓⣲⣦⣤⣄⡀⠈⠙⠺⣟⡲⣦⡂⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⢀⣽⡿⠛⠉⠈⠹⣆⠀⠀⠀⠹⡄⠙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⣸⣟⠂⠀⠀⠀⢠⡿⠀⠀⠀⠀⡇⠈⠈⢷⣁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡋⠉⠙⢻⠒⢤⣾⡟⠁⠀⠀⣠⠞⠀⢀⠀⢸⣿⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⢄⡈⢿⡻⠶⠃⠀⠃⢈⢿⠿⢷⣿⣅⡀⠀⠈⢀⣾⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⢦⢹⡿⢽⣶⣄⣀⠀⠈⠁⠀⠀⠀⠈⠉⠓⠶⢮⣈⡴⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠙⣿⠁⠄⠈⠹⣿⣭⡒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣦⡀⠀⠀⠻⣷⡙⢦⡉⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⠀⠀⠀⠀⠀⢈⡄⠀⠈⡆⠉⢳⠢⠄⡘⢷⡼⠟⠋⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢤⣝⢦⠈⠳⢤⣀⠀⠀⠀⢇⡀⠀⢱⣀⣨⠦⠖⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡀⠀⠀⠈⠓⢄⣀⠼⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⢀⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠄⠀⠀⠀⠀⠀⠀⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⡗⣊⣉⣦⢠⠤⠤⠤⠀⠛⠚⠉⠙⠛⠓⠒⠠⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣭⣭⡵⠤⠤⠤⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠐⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠞⠁⢀⠀⣈⡨⢷⣤⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠴⠢⠶⠀⢲⠟⠸⡄⠁⠐⠧⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠞⠁⠀⠀⠀⠀⠲⠀⠂⠘⠁⠀⡀⢁⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠒⢍⣁⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠒⢜⠂⣀⠀⢠⣿⣿⣷⣦⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡶⠈⠀⣠⣴⣌⠳⠀⠀⠀⡉⢀⠸⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠁⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡄⡀⣀⠻⣿⡿⠀⠀⢀⡸⣰⣼⡆⠀⠉⠉⠁⢹
⠀⠀⠀⠀⣼⠀⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠋⠓⠈⠓⠀⠀⡠⠀⠉⠀⠉⠈⠁⠀⠠⢀⠉
⠀⠀⠀⠀⠈⠢⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⠐⣠⠄⠄⠤⠓⠟⠈⠄⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⠉⠁⣀⡠⢼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢠⠘⠀⠀⠰⢤⠀⢀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⣀⣀⡀⢤⠀⢸⡄⠀⢀⡀⠃⠀⣆⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠈⠤⠀⢺⡇⠀⠈⠆⠀⢦⡄⠀⡧⠤⢶⣧⣀⣰⠿⠏⠀⠀⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⣤⢸⠿⡆⠀⠐⣦⠶⠇⠀⡗⡀⠾⠇⠈⢀⡄⠀⠀⠀⠀⠀⣘⠁⠀⠄⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠉⠀⠛⢇⣄⠀⠀⣇⠀⡂⠀⢂⠀⢰⡃⠀⣀⠙⠤⡀⠁⡀⣛⠃⠐⢀⣴⠶⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⡈⠏⠀⠀⡏⠁⣠⠀⢠⠀⢠⡇⠨⠿⣄⠐⠒⠀⡒⠀⣰⠂⠹⠁⠀⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⠀⠀⣿⠉⢉⠀⠈⠁⢤⠄⠐⡇⠈⠏⣉⠅⠁⠀⠀⠐⠀⠀⠈⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣤⠈⠃⠀⠀⠈⠀⢘⠀⠘⠀⣉⠤⢚⠈⠰⠒⠁⠀⠚⠁⠀⢠⡄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢻⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠚⣿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡏⠀⠀⠀⠀⠀⠀⠀⠘⡎⠀⡠⠀⠀⢀⡠⠂⢘⠉⠹⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡞⠀⠀⠀⠀⠀⠀⠀⠂⡤⠟⠋⠀⡀⠀⠁⠀⠀⠸⠅⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡄⠂⠀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣘⣿⡾⣯⣳⣔⣄⡞⣿⡶⣿⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⠀⠀⠀⠀⠂⢠⣀⣹⣿⣾⣿⣿⣷⣿⣿⣿⣹⡭⠿⠿⠷⣯⡀⠀⠠⠀⠀⠰⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠋⠀⠰⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⣠⠞⠉⠁⠁⠉⠀⠈⠛⠛⠙⠉⢻⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠏⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⣀⠀⠀⠀⠈⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠘⣇⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣼⠁⠀⠀⢀⣠⣤⣄⡀⠀⠀⠸⡆⠀⠀⣰⠃⠀⠀⢀⣤⣤⣄⡀⠀⠀⠈⣇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠐⣿⣿⣿⠟⠻⡆⠀⠀⣷⠀⠀⣟⠀⠀⠔⣿⣿⣿⠛⠻⡄⠀⠀⢸⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⠀⢧⠀⠀⠀⢿⣿⣿⣶⣴⡟⠀⢀⡏⠀⠀⢻⠀⠀⡄⢿⣿⣿⣦⣾⡇⠀⠀⡾⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠈⢧⡀⠈⠢⢙⡛⠛⠟⠁⣠⠞⠀⠀⠀⠈⢳⡀⠈⠢⢙⢛⠻⠋⠀⢀⡜⠁⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠉⠳⠤⠤⠤⠤⠴⠚⠁⠀⠐⠛⠂⠀⠀⠉⠓⠦⢤⡤⠤⠴⠚⠉⠀⠀⠀⠀⣸⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⣿⠀⠀⠀⠀⠀⠀⠰⡟⣶⣶⠒⠒⠒⣶⣶⣶⣶⡖⠒⠒⣻⣿⣏⣛⣆⠀⠀⠀⠀⠀⠀⢀⣿⠃⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣤⣾⣿⣿⣿⣿⣷⣤⣴⣿⡿⢿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣽⣿⣦⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡏⠀⠙⣿⣿⣿⣿⣟⣠⠴⠛⠀⠀⠀⠀⠀⠀⣰⣿⣁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢷⡿⠿⠙⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣼⠋⠿⢿⡾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣔⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠁⠙⠻⢻⣿⣶⣤⡄⣲⣤⣠⣄⣄⣀⣀⣤⣬⣴⣶⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡿⠻⣿⠛⠟⠛⠿⣿⠻⠟⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣶⠾⠃⠈⢻⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠹⢶⣶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣿⣾⣁⣀⣀⣀⣼⣶⣶⣶⣶⣶⣶⣶⣾⣇⣀⣀⣀⣸⣮⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⢰⠆⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠃⠀⠀⠀⠀⠀⢤⢰⠀⠀⠀⠀⠀⠀⠀⠂⡘
⠀⢀⡎⠠⠀⢸⡁⠀⠀⠀⠀⠀⣄⠀⠺⢢⠚⠃⢀⡄⠀⠀⠀⠲⠀⣀⣀⠐⠃⠀⠀⠀⠠⠴⠀⣀⠄⠀⠀⠤⠀⠒⠂⠀⠀⠀⠈⡟⢻⠀⢐
⠀⠀⢶⣶⠟⡇⡏⠀⠀⠀⠀⠀⠘⡗⣬⣿⡹⡄⡯⠁⠀⢀⣐⠀⠃⢴⣄⠘⡁⢂⡀⠀⣰⣦⠀⣤⠟⠀⠚⡴⠂⠈⠀⠀⠀⠑⢘⣓⣘⡣⠊
⠀⠀⠨⣐⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠗⠉⠀⠀⠀⠀⠀⠠⠧⣙⣁⠴⠑⠂⠀⠈⠘⠁⠀⠀⠉⣀⡘⠁⠀⠀⠀⢠⠀⠀⠀⠏⠹⠄⠀⠀⢀⡀
⠀⠀⠀⢿⡵⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠘⠖⠀⠀⠀⡀⠀⠀⠀⠀⢭⡥⠀⠀⠀⠀⣡⢣⠀⠀⢀⣀⠀⢠⠌⠙⠆
⠀⠀⠀⣘⢠⠀⠀⠀⢀⡷⢯⡀⠀⠀⠀⢀⢀⠀⠀⣀⠤⠄⡄⡀⠀⠸⠗⠀⠀⡀⠠⠀⡁⠠⠀⠀⠀⢰⡆⠀⢀⠀⡘⠔⢥⡏⠀⠨⠍⠀⢰⢚⠀⠈⠀⠀⡀
⠀⠀⠀⢸⡖⠀⠠⣁⠁⣷⣯⡇⠀⠤⠀⠘⢉⠀⠮⢰⡶⠳⣶⠹⡄⠀⡂⠀⠀⠀⠛⠀⡅⠘⠃⠀⠀⠈⣷⠒⠋⠀⠀⣄⣢⠀⠀⢀⣆⠀⠀⠙⠶⣀⡈⠟⠁
⠀⠀⠀⢰⢺⠀⢄⡅⢰⢾⠿⡇⠤⠄⠀⠸⠚⠀⣇⢹⣦⣤⣟⢁⠃⠈⠋⠀⠠⠀⣀⠈⠂⢀⣀⠈⠀⠀⢸⠀⠀⠀⠀⠰⠖⠀⠀⠘⡞⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢬⡷⣾⠔⠡⠀⠀⢰⣠⠀⠀⠃⠠⡅⠄⠋⠀⠀⠀⠀⠀⠐⠁⠀⡀⠸⠁⠂⠀⠀⡟⠀⠀⠀⠀⠈⡏⠀⠀⢀⢦⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣴⠄⠀⢐⣸⠀⠐⠀⠀⢠⠀⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠰⠇⠀⠀⠀⡏⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢐⠀⠀⠀⠘⠿⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠁⠀⠀⠀⠘⠦⠆⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⡄⡄⠀⠀⠀⡁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⡆⠀⠀⢠⠉⠄⠀⠀⠀⠘
⠀⠀⠀⢸⢉⠀⠀⠀⠖⠐⢠⠁⠀⠀⠀⠀⠇⠁⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠷⡆⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢦⡆⠀⠀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢐⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⡞⠰⠀⢠⢠⠄⢄⠄⠀⠀⠀⠀⠀⠀⢠⢠⠄⠀⢀⠄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠄⠰⠒⠚⠀⠀⠀⠀⠀⠞⠀⠐⠀⠰⠠⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠉⠀⠘⡿⢻⡏⠀⠉⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⣀⠁⠀⢢⡀⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣠⣄⣼⣿⣿⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠉⢹⣟⣻⣿⣿⣿⠛⠛⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⢷⣇⠸⣿⣿⣿⣿⡏⢠⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡿⠿⡿⡿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣍⣟⣛⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣭⣿⣭⣼⣓⣀
⠀⢠⣶⣢⣚⣉⣛⡋⠛⠢⢄⡀⠀⢀⡴⢿⣭⣉⡋⣭⣯⣛⣋⣹⡿⢦⡀⠀⢀⡠⠔⠛⢛⣛⣛⣓⣔⣦⡄
⠀⠀⣛⣛⠿⣟⣭⣝⣿⣴⣾⣭⣛⠲⣶⣶⣶⣆⣙⡠⠜⣁⢰⣶⣴⣶⠶⣛⣭⣶⣴⣿⣫⣽⣻⠿⣛⣛
⠀⣾⣿⣿⣧⣹⣿⣻⡿⡋⢼⠿⣟⣀⣬⡿⠳⣿⣿⣿⣿⣿⣿⡾⢿⣥⣀⣻⡿⡧⢹⢿⣟⣿⣏⣼⣿⣿⣷
⠀⢻⣿⡿⠁⣿⣿⡿⡻⣳⢌⠣⣍⡤⠤⠤⠦⡜⣿⣿⣿⣿⢣⡤⠤⠤⣄⣉⡜⡡⣞⢯⣿⣿⣿⠈⢿⣿⡟
⠀⢸⣿⡇⠀⠈⢹⡧⣇⢵⣺⡷⢈⣹⣷⠒⢾⣿⢨⣥⣬⡅⣿⡷⠔⣾⣏⡁⢾⣓⡮⣸⢼⡟⠁⠀⢸⣿⡇
⠀⠈⣿⡇⠀⠀⠘⢳⢈⡳⣝⡛⠉⠁⠀⠀⢀⣩⣾⣿⣿⣷⣍⡀⠀⠀⠈⠉⢛⣫⢞⣁⡞⠃⠀⠀⢸⣿⠁
⠀⠀⣿⡇⠀⠀⢰⠘⣎⢿⣢⠝⠳⢶⣼⠿⠋⡙⣹⣿⣿⣇⢋⠙⠾⣦⣶⠞⠫⣔⡮⣳⠃⡆⠀⠀⢸⣿
⠀⢰⣿⡇⠀⠀⠘⡄⣼⢷⣄⣀⣤⠶⠒⠋⣉⠔⣹⣿⣿⣏⠢⣉⠙⠲⠶⣤⣀⣠⡾⣧⢠⠃⠀⠀⢸⣿⡆
⠀⢸⣿⡇⠀⠀⢠⠻⣥⠶⠋⠛⢿⠶⠶⠛⢛⠔⢉⣿⣿⡉⢢⡛⠛⠶⠴⡿⠛⠙⠶⣨⠟⡄⠀⠀⢸⣿⡇
⠀⢸⣿⡇⠀⠀⠘⡴⠙⡳⣔⣟⠁⢀⣀⡴⠣⣤⢾⣿⢿⡷⣤⠜⢦⣀⠀⠈⣻⣢⢞⠋⢮⠃⠀⠀⢸⣿⡇
⠀⠘⣿⡇⠀⠀⠀⡝⢾⡋⠈⣙⡻⠛⠉⣀⠞⢡⢊⡿⢷⡵⡌⠳⡀⠉⠛⢻⣋⠁⢙⣷⢫⠀⠀⠀⢸⣿⠃
⠀⠀⣿⡇⠀⠀⠀⢹⡀⣽⢾⣭⣤⡴⠞⠁⣹⠋⢾⣿⣷⣖⠹⣍⠈⠳⢤⣤⣭⡷⣯⢀⡏⠀⠀⠀⢸⣿
⠀⢰⣿⡇⠀⠀⠀⢠⡿⣍⢠⡴⠀⠀⣀⣼⠇⠤⢴⣶⣾⡶⠴⠸⣦⡀⠀⠀⢦⡄⣩⢿⡄⠀⠀⠀⢸⣿⡆
⠀⢸⣿⠀⠀⠀⠀⡌⢇⣼⠓⠶⠶⡾⣿⡟⠀⠀⢙⣭⣭⣇⢀⠀⢻⣿⢷⠶⠶⠞⣧⡰⢡⠁⠀⠀⠈⣿⡇
⠀⣿⣿⡄⠀⠀⠀⠁⡼⢧⡀⢀⣺⡵⠋⠀⠀⠀⠼⠿⠿⠿⠀⠀⠀⠙⢮⣗⠄⢀⡼⢧⠈⠀⠀⠀⢠⣿⣿
⢰⣿⣿⣿⣄⠀⠀⠀⡇⢄⡹⠉⣡⡾⠀⠀⠈⠁⢻⣿⣾⣟⠙⠋⠀⠈⢷⣌⠙⠏⡠⢺⠀⠀⠀⣠⣿⣿⣿
⠘⢿⠿⣿⠍⠀⠀⠀⠙⣆⡭⢉⡿⠁⠀⠀⠀⠀⠉⠉⠍⠉⠀⠀⠀⠀⠈⢿⡉⢭⣰⠋⠀⠀⠀⢩⣿⠿⡿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠱⣌⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠈⢧⠀⠀⠠⠙⣄⠀⠀⠈⠑⠲⠤⣄⣀⣀⣀⣀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠙⠒⠤⣄⣀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠘⣆⡠⢇⠀⠈⠳⢄⡀⠀⠀⠀⠀⣀⡠⠝⠋⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠈⠉⠓⠒⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠸⣆⠀⠣⣄⠀⠀⠉⢓⣴⡶⠋⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠉⠙⠒⠲⠤⣶⢭⣀⣀⣀⠀⠀⠀⠄⠘⡆⠀⣀⡭⡶⠊⠉⣸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⢆⠀⠀⠉⠉⠉⠙⣛⡷⠾⡿⢧⣤⣅⣀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⣀⡠⠖⠋⠑⢠⡀⡇⡠⠙⢦⡈⠉⡻⠲⠤⣄⣀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⣀⡴⠶⠯⣅⣀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠙⣶⠀⠀⠀⠀⠈⠉⠛⠲⠤⢤⣑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣘⣀⡤⠒⠋⠁⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⡇⢀⡴⠂⠉⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⡠⠴⠚⠉⠉⠉⠉⠙⠒⠒⠦⢄⣀⠀⠀⠀⠀⠑⣌⡧⠋⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠓⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠙⣾⡇⠀⣀⡤⠶⠒⠒⠒⠒⠒⠢⢜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⢸⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡏⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠖⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠦⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡋⢙⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣹⣿⣿⣿⣿⣯⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣽⣿⣷⣄⠀⠀⠀⠘⠛⠛⠛⠛⢿⠟⠛⢻⡟⠉⠉⠉⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣿⣿⣿⣿⢿⣿⣦⠀⠀⠀⠀⣀⣀⡀⣼⠀⠀⢸⡇⢀⣴⣿⡿⠟⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣷⣾⣿⣿⠀⢀⣴⣿⠟⣿⣿⣿⡀⠀⢸⣧⣾⣿⣿⣿⣦⣿⣿⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⡍⠙⡏⠉⠉⠀⣴⣿⣿⣷⣦⣾⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢦⣄⠀⠈⡇⠀⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣿⣷⣼⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣾⣿⣦⠀⣷⠀⢻⠀⣼⣿⣿⣿⣿⣯⡸⣿⣿⣿⣿⣿⡇⡿⠛⠛⣻⠛⠛⠛⡟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣙⣿⣟⣡⣿⣿⣷⡰⡇⠘⠀⠘⠛⠛⠛⠻⡿⠿⠿⣿⠿⠿⠟⢡⡇⠀⠀⡏⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⠉⠉⠀⣃⣄⣀⣾⠂⠀⠀⠀⠀⡇⠀⠀⡟⠀⠀⠠⢸⠇⠀⢸⠱⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠁⢣⠀⠙⡆⠀⠀⠀⢰⡇⠀⢠⡇⠀⢀⡀⣾⠀⢀⡏⠁⢀⣴⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⣴⣶⣦⡘⡆⠀⢸⡀⠀⠀⢸⠀⠀⠘⣡⡾⣿⣿⡆⠀⡼⠀⢠⣾⣟⣿⣅⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣿⣿⣷⠀⠈⡇⠀⠀⡞⠀⢰⣾⣿⣿⡿⣿⣿⡌⠁⠀⠘⣻⠛⠿⠿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⣿⠿⡟⠋⡇⠀⢸⣶⣶⡇⠀⢸⡟⣻⠛⢻⠿⢿⠇⠀⠀⢠⠇⡌⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡀⢹⡄⢹⢀⣿⣧⣿⣿⠆⢸⠁⠇⠀⣾⡴⠃⠀⠀⢠⠏⣸⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣄⢧⠘⢷⡘⡆⠀⡏⢸⠁⠀⣿⡜⠀⣴⠋⠀⠀⡠⣠⢳⣌⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣼⣟⠈⢆⠀⢳⣇⠀⡇⢸⠀⢠⡿⠁⢰⡏⠀⣠⢞⡴⠇⡁⠈⢳⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣆⠈⢦⠘⣆⠀⠙⣄⡷⢸⠀⡼⠁⢀⡞⠀⡪⣐⢩⠴⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣄⠀⠀⠈⠣⡈⠧⠈⢧⠀⠈⠃⠘⠋⠀⠀⠚⠋⠉⠈⠀⠀⣠⠔⠋⠀⠀⢠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠆⠀⢸⣿⣷⡄⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠰⠀⠈⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⡀⠀⣼⣿⣿⣿⣿⣷⠀⢀⠀⠸⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⣧⠤⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣄⠀⢅⠀⡼⢀⠇⠀⠀⠀⡜⣄⡰⠁⠀⠑⠄⡠⠈⡇
⠀⠀⠀⠀⠀⠀⢰⠈⢑⡎⠀⢠⠃⠀⠀⠀⢀⣗⠋⠀⠀⠀⠀⡌⠀⡰⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⢉⠀⠀⡆⠀⠀⢀⡔⠁⠀⢱⠀⠀⠀⠀⠈⠊⠱⡀⠀⠀⢠⠋⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⣠⣿⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠃⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢃⢰⠁⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢈⠁⠉⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢀⢀⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⡀⠃⠶⠤⢄⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢀⢀⣀⢠⣀⢀
⠀⠀⠀⠀⠉⢐⠀⠀⠐⠂⠠⢄⡐⢄⠁⠈⢲⣾⣿⠀⠀⠠⠀⠰⠀⠀⠀⢸⠀⠀⣐⠤⠆⠈⠁⠀⠈⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢨⣁⠐⢁⠨⣽⡟⠀⠠⠁⢠⠃⠀⠀⠀⣠⠖⠋⠀⠀⠀⣀⡀⠀⡘⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠀⠉⠓⠛⠋⠀⡰⠁⠀⠎⠀⠀⠀⠐⠀⠀⠀⢀⠔⠉⠀⢇⠀⠥⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢍⠀⠀⠀⢀⠔⠁⢀⠌⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠈⠑⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠓⡤⠒⠁⠀⢀⢎⢀⢀⠀⠤⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠰⠅⠈⠁⠁⠀⠀⠀⣠⠃⡇⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠀⠀⠀⢀⠜⠔⠀⠁⠰⠀⠑⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡆⠀⠀⣀⡴⢁⣀⣀⣀⠆⢇⠀⠀⠀⠐⠂⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠈

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠟⢛⢛⡹⣿⣿⣿⣿⡿⠿⠿⣿⠿⠛⠋⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡷⠆⠂⠈⠉
⡿⠋⡒⠈⡺⢛⢳⡌⠉⠁⣀⣀⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⠉⢻⠿⠟⠋⠉⠁⠈⠉⠀⢀⠠⠀⠀⠢
⠀⠀⠀⠁⠀⠑⢻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣤⠙
⠀⠀⠀⠀⢀⣠⣤⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢿⣿⠿⣿⡆⠀⠀⠀⢦⠀⠀⠀⠀⣴⠿⢿⣿⡿⢷⡄
⠀⠀⠀⠀⣿⣧⣬⣿⣥⣼⣿⠀⠀⢀⡀⠀⠀⠸⣿⡿⣿⣿⢿⣿⡇⣠⡾⠿⣿⡿⠷⣆⠀⣿⣷⣾⣿⣶⣿⡇
⠀⠀⠀⠀⠻⣯⣝⣛⣫⣽⠟⢰⣶⣾⣿⣷⣷⡄⠻⣿⣦⣴⣾⠟⠀⣿⣷⣶⣿⣷⣾⣿⡆⠹⣿⣦⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⣿⣷⣼⣿⣧⣾⣧⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⣶⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣭⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⢀⣠⠴⢻⠳⢤⣀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡤⠤⠶⠚⠋⠁⠀⢸⠀⠀⠉⠛⠲⠦⠤⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠳⣄⡀⣀⣀⡤⠖⢻⠓⢦⣄⣀⡀⣀⡴⠋⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢨⡿⣍⡁⠀⠀⣸⡀⠀⠀⢩⡽⣯⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡼⠁⠈⢻⣶⡚⢻⠛⣲⣶⠋⠀⠹⡄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣼⣁⣀⣀⣾⣁⣀⣠⣾⣈⣳⣾⣾⣋⣸⣦⣀⣀⣹⣆⣀⣀⣹⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠻⡄⠀⠈⢻⣠⡾⢻⠻⣦⣰⠋⠀⠀⡼⠃⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⡀⣠⠾⠛⠲⣾⡶⠚⠛⢦⡀⣼⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣨⡿⣥⣄⡀⠀⢸⠀⠀⣀⣨⣽⣯⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡞⠁⠀⠀⠀⠉⠓⢾⡴⠋⠉⠀⠀⠀⠙⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠛⠲⢦⣄⠀⢸⠀⢀⣤⠶⠚⠋⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣀⠀⣀⣀⢀⣀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⢺⡰⠹⡄⡷⠐⢹⡇⠚⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⣠⡤⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡔⠲⡀⡴⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢰⣿⠁⠀⠀⢳⡀⠀⠀⢸⠁⠀⠀⠀⠀⣸⠃⠰⢹⡇⢠⢃⣄⠀⠀⠀⣠⡄⠀⣀⡀⠀⡜⠀⣠⣀⠀⢀⣠⡀⠀⢀⡀
⠀⠀⠘⠻⣦⣀⠀⠀⡇⠀⠀⢸⠀⠀⡴⢶⡆⣿⡐⠁⣿⡤⠃⠈⠉⠃⠀⠘⠋⠁⡰⣿⢱⡶⡇⣼⣏⡼⠠⣾⣏⡼⠀⣼⡵⣿
⠀⠀⠀⠀⠀⠈⠉⠙⡗⠒⠒⢺⡖⠚⡷⠛⠿⣿⡀⡰⣿⡀⡠⢳⣄⣀⣉⣀⣠⡾⠰⡟⢸⣷⡇⢿⣇⢀⡼⢿⡅⢀⠞⠻⠁⠹⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠘⣇⠀⠀⠀⠀⠻⠋⠀⠻⠛⠁⠀⠉⠹⠿⠿⠉⠁⠀⠀⠘⠛⠁⠀⠉⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠠⠊⠉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⠀⠠⠚⠁

⠀⠀⠀⠀⣠⣤⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠘⡄⢠⢃⡨⠛⠧⣽⡤⠃⠠⠿⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣧⡟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⠯⠤⠤⠔⠊⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠰⢿⠀⠀⠀⣄⠫⠑⠀⡀⠀⠀⠀⠠⠀⠀⠈⠇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠐⠐⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠉⠉
⠀⠀⠄⠀⠀⠀⠀⠾⠆⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠄⢀⣴⣿⠟⠁
⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠁⠤⢘⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠘⠀⡒⠈⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⡀⠀⠀⠀⣾⡇⠀⠀⢠⡶⣆⠀⠀⠀⠛⠛⠛
⠀⠐⠂⠀⠀⠀⠀⠳⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠈⠀⠀⠀⢸⣽⣿⡇⠀⢀⢀
⠀⠀⠀⠀⠀⣀⡀⠳⠆⠀⠀⠀⠈⠀⠁⠀⠀⠀⡀⠀⠀⣠⣶⡧⠀⠀⠀⠀⠀⠻⠓⠀⠙⠛
⠠⠀⠀⠀⠀⢙⡃⠀⠀⠀⠐⠂⠀⢠⣤⡄⠀⠻⡟⠀⠀⠘⠁⠀⠀⡀⠀⢀⡴⡶⢢⠀⢀⣦⡀⠀⠀⠀⠀⢴⡷
⠀⠐⠢⠀⢰⣏⣷⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣤⣤⡄⠀⣾⠿⠀⠘⢷⣏⡿⠁⠈⠰
⠀⢈⠡⠀⠹⣾⡿⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠑⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠾⠮⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠛⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⢰⠀⠀⠆⠀⠀⠈⠀⢀⡤⠒⡻⣏⠑⠢⣄⠀⠀⠀⠀⢒⠀⠗⠀⣀
⣐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣏⣤⣤⡧⠼⣦⠤⢬⣇⠀⠀⠠⠈⠀⠀⠀⠐⠀⠀⢀⠠
⠀⠅⠀⠂⠀⢲⡶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠉⠲⣾⣀⠀⣹⡖⠉⢸⠀⠀⠙⠀⠀⠀⠁⣰⣰⡀⠘

⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣤⣰⠨⠯⠲⢠⡟⠠⠁⣴⠂
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡀⠤⠁⠀⠀⠀⣼⡣⠁⣸⡇⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⣿⠖⣿⠀⣠⠖⠀⠀⠀⣿⠀⠀⣿⠁⣸⠇⡆⣿⢁⡀⢀⡞⢢⡶⢰⡦⢒⡦
⠀⠀⠀⠀⠀⢠⠞⣿⠀⢹⡀⣏⡘⣿⠀⠠⢿⡀⠊⣿⠀⣿⠀⠀⠷⢺⡀⡟⠧⢼⠁⠸⡇⣾⠁
⠀⠀⠀⠀⢰⠃⠀⢸⡇⠘⠃⠈⠀⠻⡤⠁⠀⠀⠀⣿⢠⠈⠁⠀⠀⠈⠉⠀⠀⠘⠓⠀⠀⡇⠀⡀
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁
⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠤⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⢴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢹⡤⢄⣀⣀⣀⣀⣀⣀⣀⣀⡠⢤⡎⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⣠⠃⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⢆⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡔⠁⠀⠀⢳⣀⠀⠀⠀⠀⠀⠀⣀⡜⠀⠀⠈⠣⡀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠠⢤⣀⠀⣠⠴⠊⠀⠀⠀⣀⠔⠋⠀⠀⠀⢀⡞⢫⠉⠉⠉⠉⠉⠉⡹⢳⡀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠈⠓⢤⣀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠑⠲⠤⣤⡔⠊⠁⠀⠀⠀⠀⣠⠊⠀⠀⢧⠀⠀⠀⠀⣰⠁⠀⠑⢄⠀⠀⠀⠀⠈⠓⢦⣀⠤⠖⢂⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢻⠈⠙⠒⠦⣤⣴⠊⠁⠀⠀⠀⡼⢗⠒⠒⢺⢇⠀⠀⠀⠀⠙⢢⣤⠤⠒⠋⢉⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⠀⠀⢸⠈⠉⠓⠢⣤⣞⡀⠘⡆⢠⠇⠀⣑⣦⠔⠒⠉⠁⡏⠀⠀⠀⢸⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠃⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢉⣑⢾⡾⣎⣉⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡇⠀⠀⠀⢸⠀⣀⣠⠤⠷⣌⠁⠀⡞⠹⡄⠈⢉⠾⠦⢤⣀⠀⣇⠀⠀⠀⢘⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⣀⡤⠴⠚⠯⣁⠀⠀⠀⠈⠳⣼⣀⠤⢱⡴⠁⠀⠀⠀⣠⠽⠛⠲⠤⣀⡀⡇⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⠤⠖⠛⠯⣄⠀⠀⠀⠀⠈⠑⢄⠀⠀⡼⠁⠀⠀⠀⢳⡀⠀⢠⠎⠁⠀⠀⠀⢀⡠⠜⠛⠒⠤⣄⣣
⠀⠀⠀⠀⠀⠀⠒⠋⠉⠑⠦⡀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠈⢳⣰⠁⣀⣀⣀⣀⡀⢣⡰⠃⠀⠀⠀⣠⠖⠉⠀⠀⠀⢀⡠⠒⠉⠉⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠈⠢⡀⠀⠀⢠⠋⠉⠉⠀⠀⠈⠉⠉⢧⠀⠀⢀⠞⠁⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠘⢆⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢠⠋⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⢈⠿⠒⠊⠉⠉⠉⠉⠉⠉⠉⠒⠺⣇⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠤⠤⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠤⢽⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣴⣿⣷⣍⣉⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢱
⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡛⢻⣿⣿⣽⣿⣦⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣇⣿⣿⢿⣿⣿⣿⣿⣿⣛⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠇
⠀⠀⠀⠀⠀⠀⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡟⠻⣿⣿⡿⠞⠋
⠀⠀⠀⠀⠀⠀⠹⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠁⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⠛⠋⣿⣿⣿⣿⣿⠇⣠⣶⣼⠇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠅⠀⢰⣿⣿⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣌⡛⠿⣿⠿⠟⠋⠉⠀⠀⠸⠟⠃⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠉⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⡏⢳⣶⡾⠓⣲⣦⣀⣀⠀⠀⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣇⠈⠡⡴⢠⣿⠻⣿⠻⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣿⣿⣿⡆⠀⠘⠘⠛⠀⠉⠛⠗⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠞⠂⠀⠀⠀⠀⠀⡀⢷⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠋⣻⢿⡟⠑⣤⣤⣀⠀⠀⠀⠈⠓⠺⡇⠀⠀⠀⠀⠀⢛⣨⣇⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣾⠀⠀⠀⠀⠀⠉⠲⠦⣤⣀⣠⠜⠀⠀⠀⠈⠁⠘⢯⠉⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⡤⠈⠀⢲⣦⠌⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⣧⡖
⠀⠀⠀⠀⠀⣠⣿⠋⣴⣿⣷⡌⠀⠀⠁⠀⠀⠀⠛⣢⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠈⡇⢸⡇⠀⣼⠂⡒
⠀⠀⠀⠀⡼⠟⢀⣤⣿⣿⣿⣿⡄⠀⠰⣿⣿⣶⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠠⣧⣾⡷⢸⡟⣸⠇
⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣆⠀⣿⣿⠟⠉⢙⣟⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢘⣳⠀⣸⣿⣿⣿⣟⡗⣿⠇
⠀⠀⠀⠀⠀⠀⣽⣷⣿⣿⣿⣿⣿⣿⣷⡈⠁⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣩⡿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣷⣦⣍⣉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢷⣈⣻⡟⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢾⡟⠉⣷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣶⣄⠀⠁⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡟⠇⠀⣼⣿⣿⣿⣶⣤⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠋⠙⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠋⠛⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣥⣤⣥⣴⣤⣼⣧⣴⣿⣥⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⠢⡀⡜⠈⠳⣀⡔⢣⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⢨⣳⠤⠶⠛⠒⠛⠒⠛⠺⠴⢖⣁⢸⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣠⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣄⠇⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣀⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢄⡀⠀⠀⠀⢰⡄⠀⠻⣬⠁⡇⣀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⡆⠀⠀⢸⠃⠀⠀⣈⢺⡅⡆⣠⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⡞⠀⢠⠔⠂⢤⠀⠀⠀⠀⠀⠐⡀⠀⠉⠀⡠⠃⠀⡠⢋⡄⠀⠀⢉⣤⡝⢷⣰⡴⡄⡀⢠⢸⣴⢿
⠀⠀⠀⠀⠀⠀⡇⠀⡃⢠⡄⠀⡃⠀⠀⠀⠀⠀⠈⠑⠒⠉⣀⠴⣚⠀⠀⠀⠀⠀⠀⠉⣁⡲⢙⡧⣯⣵⣆⡤⢿⠈
⠀⠀⠀⠀⠀⠀⣇⠀⠑⠄⠀⠔⠁⠀⠀⠀⠀⢀⣀⡤⠖⠛⠭⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢣⠝⣣⣧⣌⠀⣟⠂⢸
⠀⠀⠀⠀⠀⠀⢹⡀⠀⢀⣀⣀⣀⡤⠔⢞⡋⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡦⢈⣙⡉⣤⠙⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠉⠉⠬⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠞⠉⠙⡄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠛⢋⡼⠉⠀⠀⠈⠙⡆⣠⠖⠚⠋⠉⠉⢱⡀⢹⡀⠀⡼⢁⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⣄⠤⠼⠃⣿⠀⠀⠀⢀⡴⠛⣀⣰⠇⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠁⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠆⠒⠀⠀⠀⠒⠒⠈⠁⠀⠸⠀⠀⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⠤⠺⢄⠀⠀⠀⠀⠀⠉⠁⠀⢀⠔⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠠⠋⠍⠐⠀⠀⠈⠁⠀⠀⡄⠀⠉⠂⠤⢀⡀⠀⠀⡠⠊⠀⠘⡄
⠀⠀⠀⢠⣄⠒⠊⠁⠀⠀⠀⡀⠔⠁⠀⠈⢄⠀⠀⠀⢀⡠⠔⠧⣀⠀⠀⠀⠀⠀⢀⠎⢃⠀⠀⠀⠈⢆
⠀⠀⠀⢰⠀⠁⠢⣤⠀⠒⠉⠀⠀⠀⠀⡠⠉⢍⠉⠁⠀⠀⢀⠀⠀⠁⠒⠀⢀⠴⠁⠀⠀⢢⠀⠀⠀⠀⠁⠢⣀
⠀⠀⠀⢸⠀⠀⠀⡃⠈⠒⢄⣀⠀⠄⠊⠀⠀⢈⢆⠀⠐⠒⢹⠑⠤⠀⢀⡠⠊⠀⢢⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⣉⠐⡦
⠀⠀⠀⡄⠀⠀⢀⠃⠀⠀⠀⡏⠐⠠⣀⠠⠐⠁⠈⣆⣀⡀⣤⡀⠀⢀⠊⠱⡀⠀⠀⠑⠄⡀⠀⣀⠠⠑⡶⠈⠁⠀⡔
⠀⠀⡜⠀⠀⢀⠌⠀⠀⠀⢠⠁⠀⠀⠈⠁⠢⣤⠒⠉⣦⠠⠧⢀⡜⠣⡀⠀⠙⣢⡤⠄⠒⢸⠁⠀⠀⢰⠀⠀⠀⢰
⣠⢊⡀⠀⣠⢊⣀⣀⣀⣠⣃⠀⠀⣠⠃⠀⢀⠃⢈⠃⢌⣢⣧⠂⠱⡖⠺⠋⠁⢸⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢰
⠈⠢⡀⠀⠈⢢⠀⠀⠀⠑⡄⠀⠀⠉⡄⠀⠈⠇⢉⡉⠔⡛⡻⡑⢲⠷⢄⣦⣀⠈⢄⠀⠀⠰⡀⠀⠀⠸⡀⠀⠀⠀⠆
⠀⠀⠱⡀⠀⠀⢣⠀⠀⠀⠘⡀⠀⠀⠀⢀⡠⠷⢍⠈⡾⠠⡥⠘⢣⣠⠃⠀⢀⠔⠋⠑⠒⠀⢴⡄⣀⡀⠑⠄⠀⠀⠈⢆
⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⠇⢀⠤⠛⠣⢄⠀⠀⡝⠉⠀⢲⠊⠉⠙⢄⠀⠃⠀⠀⠀⡠⠊⠀⠀⠀⠀⡩⠝⠓⠂⠠⠄⣡
⠀⠀⠀⢰⠀⠀⢸⠀⠀⡠⠼⠮⢀⠀⠀⠀⠀⢡⣜⠀⢀⡀⠘⢀⡠⠄⠀⠳⣆⠀⠀⡌⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠠⠒⠈
⠀⠀⠀⡀⠀⢀⡦⢔⡉⠀⠀⠀⠀⠈⠢⡀⠀⡐⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠢⡀⠁⠀⠀⠠⠃⠀⠀⠀⢀⠔⠁
⠀⠀⢠⡅⠊⠁⠀⠀⠈⠑⠠⡀⠀⠀⠀⠐⣴⡀⠄⠤⢀⡀⠀⠇⢀⡠⠔⠂⠀⠀⠘⠣⠀⠀⠃⠀⠀⠀⡰⠁
⠀⠀⠀⠈⠉⠐⠢⢄⠀⠀⠀⠈⢢⠀⠀⡐⠁⠀⠀⠀⠀⠀⠁⢲⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⡄⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⢡⡐⢁⡀⠤⠄⢀⡀⠀⠀⢸⠀⠀⢀⠤⠐⠂⠈⠀⠐⠒⠛⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⡰⠉⠀⠀⠀⠀⠀⠀⠁⠂⢄⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣁⡀⠠⠤⠤⠄⣀⣀⠀⠀⠀⠃⠀⠀⣀⠤⠒⠉⠉⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⣀⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠶⠶⠶⠶⠶⠶⢶⡦⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⠛⠋⠉⠀⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⠉⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠁⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡉⠙⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠋⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⢀⡸⠃⠀⠀⠀⠰⠄⠀⠀⠀⠀⢶⣀⠀⠘⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠈⠀⠀⠀⣨⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⢀⡀⢨⣿⠀⠀⠀⢀⠀⡀⢰⠀⣀⢀⣼⡿⠀⡈⠁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⢰⣿⣠⣿⣤⣀⢸⠃⢀⣀⠀⢀⣴⣶⣿⡇⠁⣿⣿⣿⣶⣶⣾⣇⣷⣼⡷⠿⠾⢿⣿⠀⠁⠀⠀⢸⢿⣄
⠀⠀⠀⠀⠀⠀⣠⣟⠁⠀⠀⠀⠀⠃⢸⣿⣟⠛⠛⠻⠉⣷⠙⠋⠹⡏⠉⣹⡿⠀⠀⢹⣿⡉⠋⠈⠉⢹⠋⠁⠀⠀⠀⣿⡿⠇⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⢰⣿⠉⠀⠀⠂⠀⠀⠀⠀⢹⣿⣄⠀⠀⠀⢸⠀⠀⠀⠀⣰⣿⠀⢠⡆⠀⡹⣧⡀⠀⠀⠘⠀⠀⠀⢀⣼⢏⠀⠀⠀⠀⠀⢀⠀⠻⣧⡀
⠀⠀⠀⠀⢀⣿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⢾⡛⣷⣤⣤⣴⣤⣤⣴⢾⠋⠉⠀⠈⠁⠀⠁⢸⡿⢶⣶⣤⣤⣶⣾⠿⠁⠀⠀⠀⠀⠀⠃⠀⠠⣠⣿⡄
⠀⠀⠀⠀⣾⠟⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡹⣿⠿⠷⡦⠀⠀⠀⢰⡄⠀⠀⠀⠈⠁⠀⠳⠾⠿⠁⠿⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠘⢯⣿
⠀⠀⠀⣼⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⢠⠌⠀⣰⣾⣀⣀⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⡀⠀⠀⣼⣿⡆
⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⣰⣦⣿⣇⣴⣿⣿⣿⣿⣿⣿⣿⣶⣆⣀⠀⢠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⣽⣧
⠀⠀⣾⡇⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣷⣿⣴⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣿⡀
⠀⠀⣿⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣧⡀⢻⣿⣿⠛⡿⣿⣿⣿⣿⣿⣧⣤⣄⣀⣀⣠⣴⡾⣶⡏⢹⡏⢡⣴⣇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣿⡗
⠀⢀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⢾⣿⣿⠇⡿⠈⠱⠄⠀⠟⠦⠼⢉⡛⠻⠿⢿⣿⠛⠉⠙⠁⠛⠁⠈⠁⢼⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⢸⣿⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠀⢀⡭⠀⠄⠀⠀⠀⠀⡞⠃⠄⢠⠊⠀⠀⠀⠀⢰⠀⠀⡄⠀⣼⣿⡟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⢹⣿⠃⠰⠂⠀⠀⠀⠀⢀⠀⠀⢻⣹⣿⠏⡀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠿⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⢸⡟⢇⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⢺⣿⡇
⠀⣿⡇⠀⣶⡄⠀⠀⠀⠀⠈⠀⠀⡀⢿⣿⣷⡄⠀⠀⣶⠀⠖⠀⠀⠈⠀⣀⣈⣠⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠈⠀⠀⠀⠀⠀⠰⠆⠀⠀⢀⡴⢊⣿⡁
⠀⣿⣧⠾⠏⠀⠀⠀⠀⠀⠀⣀⠀⠑⣿⣿⣿⠁⠀⠀⠈⠀⠀⠀⡀⢠⣴⣿⣿⣟⣷⣾⣦⡀⠀⠀⠀⠀⠀⠐⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢠⣬⣿⡇
⠀⣿⣯⠀⠈⠂⠀⠀⠀⠀⠀⠈⠂⠀⢩⣿⣟⠃⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠟⠛⠿⣿⣿⡿⣢⡄⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠘⠁⠛⠀⠐⢆⡤⣿⠃
⠀⣾⣿⡞⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⠀⠀⠀⠀⠀⢠⣥⣼⡿⠀⠀⠀⠀⠀⢻⣿⡟⡠⠀⠀⠀⠀⢹⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⣧⣿⠃
⠈⢹⣿⣿⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⢠⠀⠀⠀⣴⣾⡿⠁⠀⠀⠀⠀⠀⠀⢻⣿⣡⠆⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⢠⠀⠀⠀⠁⠘⢠⣷⣿⡇
⠀⢸⣿⣷⡇⣴⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡏⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠨⣿⣷⠀⠀⠀⠀⠀⣿⣿⠂⠀⠀⠀⠘⠀⠐⠀⠀⢰⣞⢣⣿⡇
⠀⠰⣿⣻⡇⢸⢠⡀⠁⠀⠀⠀⠀⠀⠀⢩⣽⣿⠀⠀⠀⣰⠸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠠⣿⣁⠀⠀⠀⠀⠀⣿⡕⠋⠀⠀⠁⠀⠀⠀⠀⢀⣦⣬⢸⣿⡁
⠀⠰⣿⡿⣶⠃⣀⠁⠀⠀⢂⡀⠀⠀⠐⠘⣿⣷⠀⠀⠀⠃⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡀⠀⠀⠀⠀⣿⡗⠀⡀⠀⠀⠀⢀⠀⢀⣴⡇⠸⡈⣿⠁
⠀⠀⢿⣷⣿⣦⣿⠲⡄⢣⣘⢇⠀⢀⢀⡾⣿⣿⡀⠀⠀⢠⣧⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠁⣿⡿⣟⢣⡄⡆⠀⢀⡄⢻⣽⢻⣶⣽⣿
⠀⠀⢸⣿⣿⣏⣩⣶⣷⣮⣽⣯⣵⣬⣿⠿⠿⢿⣿⠗⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠃⠀⠀⠀⠰⣿⣷⣿⣟⡐⠃⣀⡏⣤⣻⣯⣾⣿⣤⣿
⠀⠀⠘⠿⣿⣿⣿⡿⠟⣿⡿⠛⠟⠛⠉⠀⠀⠀⠀⠐⠀⠀⢼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⡿⠛⠻⣿⣷⣿⣿⠶⢿⣿⣿⣿⣿⣾⠃
⠀⠀⠀⠀⠈⠻⡿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢃⠀⠀⢰⠞⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⣿⠋
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠘⡁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠠⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠓⠃⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⢾⣤⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⡠⠠⣄⠀⠈⠀⠀⡶⠀⢀⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢲⠀⠀⠐⠂⠀⠀⢿⠀⠀⣿⣾⣿⡝⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡇⢢⡟⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠻⠦⢠⣤⣀⡀⣠⣀⠀⣨⣇⣴⢫⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣾⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠿⢿⡿⢻⡿⣿⣿⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣧⣩⣴⣶⣶⣧⣀⣠⣤⣀⣴⠁
⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠋⠉⠉⠀⠄⠀⠀⠀⠀⢰⡿⠛⠋⠙⠃⠈⠛⠉⠉⠋⠙⡟⠙⢿⡋⠟⠻⣇
⠀⠀⠀⠠⠔⠋⠁⠠⠒⠀⢀⡤⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣴⣦⣄⠀⠀⠀⣦⠀⠀⠀⣿⠃⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠣⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠘⠉⠀⠀⣤⡤⠐⠛⠉⠁⠀⠈⢤⠀⠀⠈⠁⠀⢠⣿⠀⠀⣴⡟⠻⣶⣄⡀⠀⠀⠠⣶⣀⠘⠃⡀⠙⠆⠀⠙⢦
⠢⣄⡀⠀⠀⠀⠀⣤⠀⠀⠀⠐⠀⠀⠀⣠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⢹⣧⠀⠘⣧⣀⠀⠉⠛⢿⣧⣀⠀⠙⠲⣄⠈⢢⠀⠰⡄⠀⢷
⠀⠈⠀⢀⡀⣴⠋⠀⠀⣰⡆⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⡦⠤⠘⠀⠀⠀⠀⠀⠛⢷⣤⡀⢹⡆⠀⠀⠀⠀⠉⠻⣶⠀⠈⠆⠀⠃⠀⢰⡄⢸⣇
⠀⠐⠚⠉⠁⠈⠀⣀⠴⢏⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⣰⡄⣀⡄⢰⣷⣀⡿
⠀⠀⠀⠀⠀⠙⠃⠉⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⢸⣅⣹⣇⠰⠏⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⡀⠀⠀⠚⠏⠀⠀⠀⣄⣀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠉⠛⠿⣶⣄⠀⠀⠀⠚⠻⠁⠀⠀⠠⠴⢿⡟
⠀⠀⣠⡾⠛⠉⠀⠀⢀⣠⡤⠶⠚⠛⣋⣉⣩⣭⣭⣭⣛⣦⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠁
⢠⣾⠋⠀⠀⢀⡴⠞⠉⢀⡤⢖⣪⠭⠒⠒⢋⣩⡭⠿⠛⠉⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⣿⠏
⣿⠃⡎⣖⡆⡞⠀⢠⠞⡡⠚⠁⢀⣠⠶⠛⠉⠀⠀⠀⠀⢀⣀⣠⣤⣶⠟⠁⠀⠀⢀⡾⣻⠏
⣿⡄⢣⠳⠃⠳⢤⣏⣞⣀⡤⠞⠋⠀⠀⠀⠀⣀⣤⣶⠟⢛⣿⠉⠁⠀⠀⠀⠀⢀⡞⣰⠃
⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠁⣠⢎⣿⡄⠀⠀⠀⠀⢠⡞⣴⠃⢀⣸⣦⠄
⠀⠈⠻⢷⣤⣀⣀⠀⠀⢀⣀⣠⣴⡾⠛⠉⢀⣠⠴⢊⡵⢋⠘⣧⠀⠀⠀⢠⠟⡼⠃⠀⠀⠀⠙
⠀⠀⠀⠀⠈⠉⠛⣿⣛⠿⠿⠯⠗⢒⣒⣉⠥⠔⠊⡥⢖⡡⠂⠹⣆⠀⢠⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠐⡩⠔⣡⠀⢻⣴⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡴⣢⣴⢃⡞
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠚⣵⢣⡞
⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢡⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣀⣠⣤⡴⠶⠛⡿⢡⠏⠀⠀⣰⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⢀⣾⣿⠏⠀⠀⠈⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠃

⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆
⠀⠸⣿⣷⣶⣄⠀⠀⠀⠐⢀⣴⣶⣿⡿⠁⠀⠀⠈⠀⠀⠀⠒⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠄⠈⠉⠛⠻⣷⢻⠿⢳⡾⠛⠋⠉⠀⠀⠀⠐⠄⢀⡀⣴⣶⠄⠀⠀⣀⡀⣰⡆⠀⠀⠀⠘⠓⠋
⠀⠀⠂⠀⠨⠳⣜⠒⠔⠚⡾⠪⠀⢀⣴⣄⠀⢆⣤⣼⣦⣬⣟⡀⢀⣄⡈⠉⠋⢀⠀⠀⠀⠐
⢁⣄⠀⠰⣾⡀⠀⠀⠄⠀⠀⣿⢳⡜⣿⢹⣴⣿⢛⣻⣿⣍⣿⣿⣾⣿⠁⣿⡇⣾⠇⣰⡟⣷⠀⢀⡁⠀⠘⠚⠁
⠙⣿⡄⠀⢻⣧⣀⠀⣴⢶⣦⢹⣇⡇⣿⡞⢻⣝⠻⣿⠾⠟⣡⣿⢻⣿⣰⣿⣅⡿⢴⡿⠴⢋⣴⡟⣹⡇⠀⠀⠀⢠⣀
⠀⠘⣿⣤⡾⣿⡏⠀⣿⡀⣿⣌⣿⡁⣿⣧⡜⠻⢿⣶⣶⠿⠟⠋⠀⠙⠁⠈⠋⠀⠈⠛⠚⠻⣿⣙⣉⣼⢟⣤⣦⠀⢀⡀
⠀⠾⠻⣿⡀⠘⣿⣤⡿⣷⠏⠉⠈⠛⠁⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠟⣿⠏⣱⠂⠯⠆
⠀⠀⠀⢻⣷⠀⠈⠛⠣⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⢠⠄⠄⠀⠩⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠙⢙⡟⢆
⠀⠀⠀⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠃

⣴⣴⣴⣶⣦⣦⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣶⣦⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⢦⡩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠿⡀⠈⠛⠀⠀⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣟⢲⣮⣇⠀⠀⠙⣫⣿⣾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡓⢽⡫⣧⠀⠀⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢿⣿⡧⣼⣿⣬⡺⡿⡟⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣯⢹⣿⣿⣿⠿⣴⣧⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⣾⢿⣿⣿⣿⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⡿⣟⣷⣿⣿⣛⣹⡟⣻⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⡟⠻⠟⢿⣿⠿⣣⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⣻⣿⣟⡟⠙⢻⡁⠠⠀⠀⢹⢆⣽⣽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣿⣻⣤⣀⣸⠲⠿⠄⢳⣦⠘⣿⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣾⣤⣶⣩⣄⠼⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⡿⢿⣿⣿⠿⣿⡙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⣽⡉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⣰⠿⠋⠀⠀⠀⠀⠀⡔⠀⠀⠀⣀⠄⠁
⠀⠀⢀⡠⠼⠿⡉⠻⣠⡄⠀⠀⢺⡟⠁⢀⡾⠃⠀⠀⢠⡦⢀⡤⢚⡵⠛⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠈⢢⠈⢷⡀⢀⣸⡇⣠⣿⡇⠀⢀⡴⢋⢔⡥⠚⠁⠰⠄⠀⢀⠔⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⣦⣹⣟⠹⣿⣧⣾⡿⠞⣡⣿⠂⠀⠀⣀⣤⠶⠛⣀⣮⡿⠿⠒⠾⠛⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡈⢿⣿⣰⣿⣿⣿⣿⣾⣟⢻⣿⣵⣟⣫⣴⠾⠟⠉⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⣿⣾⣿⣿⡿⠴⣿⣿⣿⣷⣦⣍⢿⣿⢟⠷⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⣿⣿⣿⣿⣷⣿⡿⣿⣿⣦⣿⢥⣉⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢈⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠚⠁⠀⠐⣾⣇⠄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⢻⣯⣹⣿⣿⣿⡏⠩⠀⢀⡟⣸⣿⢿⢾⣧⣌⣿⠃⢿⢄⣀⡀
⠀⠀⠀⠀⠀⠀⢲⡟⠃⣨⡯⠚⠛⠿⢻⣿⣷⣶⣿⣧⡏⢅⠗⠯⡼⠇⠻⢧⡜⣁⡠⢷⠂
⠀⠀⠀⠀⠀⢠⠎⠙⢿⣽⠇⠀⣀⠀⣷⣿⣿⣿⣿⣿⢠⣾⣠⣾⡷⣄⠀⢫⣿⣟⣠⠎⠱⡤
⠀⠀⠀⠀⠀⣟⠀⢀⣴⠏⠀⠀⠉⠈⠙⣿⣿⣿⣯⡯⣾⣿⣻⢣⡀⠘⣆⠘⠐⢻⣷⣦⣤⣷⡄
⠀⠀⠀⠀⠀⣏⡠⠐⣷⠀⠀⠗⡀⠀⢼⣿⣿⣿⢹⣇⠿⢻⣿⢻⣇⠀⠘⣆⠀⠀⢠⣿⠉⣹⡇
⠀⠀⣰⠀⠀⠽⢲⡂⡿⠀⡸⠀⠴⠂⣸⣿⣿⣿⡟⢻⢋⣿⡇⠈⠀⠀⠀⠈⠁⠀⠀⠙⠉⣳⡃
⠀⢀⡏⠀⠀⠀⢸⣧⠻⠀⠁⢠⠖⠂⣿⣿⣿⣿⠡⠌⢈⠃⠒⠀⠠⠆⠀⠙⠠⣀⠀⠀⣰⣸⣿⡆⢠⣴⠆
⠀⠸⡇⠀⠀⠀⠀⣝⡧⠀⣤⣄⠀⠀⡻⠛⠳⠂⠈⠋⠡⠀⠀⠀⠀⠀⠀⠘⠃⠉⢠⡿⠜⠉⠋⠁⡜⠁
⠀⠀⣿⠀⠀⠀⢀⡈⠀⡄⠏⡹⣯⠓⠬⠠⠳⣏⠉⠣⠆⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡞⠣⣾⡴
⠀⠀⠠⡀⠀⠀⠸⣦⣥⢰⣌⡷⠋⠀⣴⣁⢀⣀⡷⢤⠌⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⣁⢻⠉⠁
⠀⠀⠀⠀⢐⣤⢠⠏⢿⠜⠁⠀⠀⠈⢠⠟⣛⣩⣤⣤⣥⠀⠀⠀⠀⠀⠐⠉⠁⠀⣼⠛⠀⠇
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠲⠴⠟⠛⢡⣡⠞⢀⣠⠖⠀⠀⠀⠀⠀⠀⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣷⠲⢋⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⠿⢙⣦⣤⡄⠀⠀⢀⣀⡨⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠋⠠⠶⣎⡙⠋⠳⠈⢠⡜⠂⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⣀⣄⣤⣤⣍⣐⣃⠚⠤⣤⢴⣢⣠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠷⠀⣀⠠⠞⣩⣿⣿⣦⣭⣙⠿⠿⣟⠻⣿⡟⢹⠋⢩⣅⡀⠀⣡⠀⠚⠳⠀⡀
⠀⠀⠀⠀⠀⠠⡾⢧⠔⠈⢙⣿⠿⠇⢾⣿⣧⠤⢬⣙⠛⢻⣾⣿⠿⢛⣻⠆⠿⠿⣤⣄⣉⢻⡿⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠾⠛⠛⢛⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢰⣿⣯⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⢾⣶⣷⡾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⣻⣿⠉⠉⢁⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣏⢉⣠⣿⣿⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⣛⣏⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠠⣶⣿⣿⠄⠀⠀⢀⣀⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⣿⣤⣯⣀⣀⠒⣶⡶⠞⠉⠀⠁⣤⠸⠷⣿⣅⣀⣁⣤⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠻⣿⢿⣿⣙⢿⠆⢀⣤⠆⠀⣧⠀⣀⠀⢹⣿⣿⣿⠿⠷⢤⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣍⣻⠿⣦⣤⠀⠂⠀⢩⠀⠩⣶⠂⣉⡿⢿⠇⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠾⠿⢿⣦⣤⠤⠀⠆⠀⢶⣟⠁⠶⠿⣛⣻⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⣦⣬⣤⠀⠀⢠⡀⢿⣯⣀⣤⣾⣿⣿⣾⡇⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠶⠮⢏⡁⠤⣀⢿⣿⣧⠉⡛⠠⣼⡏⠙⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢳⣤⣴⠋⠁⠐⠁⠀⢛⣿⣏⣅⠀⣨⡿⠛⣿⡌⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣤⠜⠉⣀⠴⠂⠠⣟⣿⠄⢈⠉⠀⠰⠛⣿⡄⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⢁⣴⠟⠁⠐⠂⠀⣿⣿⡗⠈⠙⠃⠀⠛⠛⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣽⣍⠀⠀⠀⠀⠀⠀⢰⡄⠀⠈⠻⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠾⣿⠿⢷⣄⣸⣏⣙⣿⡶⠛⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠺⢿⣷⣄⠀⠀⣉⣽⣿⣿⣷⣶⣤⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠋⣵⡆⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠛⣿⠶⠛⠛⢻⡟⠛⠛⠁⠀⠀⡀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠁⢼⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠿⢀⠀⠀⠀⠀⠉⠁⠀⠀⣄⡑⣶⣿⠷⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢠⣤⣾⣶⡄⠠⡄⠀⣴⡄⠀⠙⠿⠀⠈⠀⣾⡟⠀⠀⠀⠀⠀⢡⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠀⠀⠀⠀⠀⣿⠁⠀⠀⠁⠀⠀⠀⠈⠻⢷⣴⠆⠀⠰⣶⣿⠀⠀⠀⠀⠀⠀⠀⢳⣄⣿⡆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢿⡀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠀⠀⠰⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣙⡿⡉⠻⣦⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠃⠀⢸⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣉⠀⣽⣄⠈⢷⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠟⠃⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢹⡟⣿⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣼⣿⣿⠆⠀⠀⠀⠀⠀⠀⠟⠛⠛⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⢀⣌⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⡆⠀⠀⠀⠀⠀⠸⣿⣦⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣾⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠠⣿⣿⠻⣿⠿⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣟⠦⣄⡀⠀⠀⢀⣠⠴⣻⢻⡀
⠀⠀⠀⠀⠀⠀⠛⣦⡀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣿⣄⠈⠉⠙⠋⠉⠁⣠⣿⠈⢳⡀⠀⠀⠀⠀⠀⠀⢀⣴⠛
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡦⠤⠤⠴⠚⠉⠀⢀⡟⣏⠳⢤⣄⣠⡤⠞⣹⢻⡀⠀⠉⠓⠦⠤⠤⢴⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠹⣦⣄⣀⣀⣀⡴⠋⠀⣿⣆⠀⠀⠀⠀⣠⣿⠀⠙⢦⣀⣀⣀⣠⣴⠏⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢹⡟⢯⡉⠁⠀⠀⣠⠟⣯⠙⠒⠒⠋⣽⠻⣄⠀⠀⠈⢉⡽⢻⡏⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⢸⠇⠈⢿⢿⡚⠋⠁⠀⣸⣆⡀⣀⣰⣇⠀⠈⠙⢓⡿⡿⠁⠸⡇⠀⠀⢳⣀
⠲⠦⣤⣤⣤⠴⠚⠉⠀⣀⡴⠋⠀⠀⡼⠀⠹⣦⡤⠴⠋⣧⠉⠉⣼⠙⠦⢤⣴⠏⠀⣧⠀⠀⠙⢦⣀⠀⠉⠓⠦⣤⣤⣤⠴
⠀⠀⠀⠉⠻⣍⠛⢿⣿⠭⣤⣤⣤⠞⠁⠀⢀⡏⠙⢦⣀⣸⡦⢴⣇⣀⡴⠋⢹⡀⠀⠈⠳⣤⣤⣤⡭⣿⠿⠛⣩⠟⠉
⠀⠀⠀⠀⠀⠙⡆⠀⠙⣆⠀⠙⢯⠙⠛⢿⡯⣤⣀⣼⠛⢦⣷⣾⡴⠛⣧⣀⣤⢽⡿⠛⠋⡽⠋⠀⣰⠋⠀⢰⠏
⠀⠀⠀⠀⠀⠀⡷⠀⠀⣹⠀⠀⢸⠆⠀⢈⡇⠀⢈⣿⣩⣿⣿⣿⡯⣍⣿⡁⠀⢸⡁⠀⠰⡇⠀⠀⣏⠀⠀⢾
⠀⠀⠀⠀⠀⣰⠇⠀⣠⠏⢀⣠⣏⣤⠴⠾⣟⠋⠉⢹⣶⡋⡟⢻⢙⣶⡏⠉⠙⣻⠷⠦⣤⣹⣄⡀⠹⣄⠀⠸⣆
⠀⠀⠀⣠⣾⣣⡴⠾⣿⣛⠉⠉⠛⢦⡀⠀⠈⣇⡴⠋⠁⢹⠟⠻⡏⠈⠙⢦⣸⠁⠀⢀⡶⠛⠉⠉⣛⣾⠷⢦⣌⣶⣄
⠶⠚⠛⠛⠛⠒⠦⣄⠀⠈⠙⣆⠀⠀⢻⢀⡴⠟⠛⠲⣄⣟⣀⣀⣻⣠⠖⠛⠻⢦⡀⡟⠀⠀⣰⠋⠁⠀⣠⠴⠒⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢸⡆⢀⣾⠿⠤⢤⣀⠀⢸⠋⠀⠀⠙⡇⠀⢀⡤⠤⠿⣷⡀⢰⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣸⣷⣋⣁⡀⠀⠀⠙⣦⣏⡤⠶⠶⢤⣹⣴⠋⠀⠀⢀⣀⣑⣾⣇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡴⠟⠉⠉⠉⠉⠓⢦⠀⣿⠋⠀⠀⠀⠀⠙⣿⠀⡴⠛⠉⠁⠉⠉⠻⢦⣸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠒⠛⠛⠒⠦⣄⠀⠈⣧⣯⠴⠛⠉⠉⠛⠦⣽⣼⠁⠀⣠⠴⠒⠛⠛⠒⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⣿⠃⢀⣠⣤⣤⣄⡀⠘⣿⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣧⠞⠉⠀⠀⠀⠀⠉⠳⣼⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢰⡆⠀⣠⠀⠴⠾⠿⠁⣴⣦⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢰⡏⢸⠂⣿⢹⡇⠀⠀⠄⡰⡀⡂⢀⠀⠠⠀⠀⣄⣀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⢸⣇⣤⣿⠖⠒⢂⡀⢸⡇⡞⢸⡇⣸⠃⢀⣀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠙⠿⠛⠂
⠀⠀⠀⠀⠀⣠⡾⣏⠁⢿⡄⣾⢹⡇⠸⣿⠀⢸⡷⠃⠀⣼⢅⣇⣶⠀⡄⣿⡗⣴⠲⣠⡖⢰⡶⡗
⠀⠀⠀⠀⠀⠁⠀⢿⠀⠸⡇⠻⠾⣧⡠⠻⡤⠚⣧⣀⠜⠿⠾⠁⠻⡼⢧⣼⠛⢿⠤⠻⡤⠾⠁⣇⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠈⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠠⡄⠀⣀⡀
⠀⠀⠀⢀⡀⡴⠂⠀⠀⠀⣤⣴⣶⣦⡄⠀⢠⡤⠀⠲⠀⠄⠀⠠⠀⠀⢀⠀⠀⢰⢸⠄⠂⠽⠀⠓⠀⠀⠀⠒⣀⢀⡀
⠀⠀⠀⠉⣰⠀⠀⠀⠁⠀⣿⣌⡹⢿⠷⢄⢀⠀⠘⠜⠀⠀⠈⠀⠀⡸⠂⠘⠋⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⡆⠑
⠀⠀⠀⢀⠋⠀⠀⠀⠀⠠⠿⠿⢿⣆⡀⠀⢀⠀⢘⢂⠀⠀⢘⠐⠀⠁⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢬⡟⠂⠀⠠⡤⠀⡀⠀⠀⠂⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠉⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡄⠀⠰⠂⠠⠀⡴⠒⠀⣠⠀⠀⣠⣿⣆
⠀⠀⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠂⠀⠄⠀⠄⠀⢸⣷⣤⣴⣿⣿⡿⣿⠿⣿⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠒⠀⠀⠀⠀⠀⢹⡿⠿⠁⢹⠟⠀⠈⠀⠟
⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠊⠀⠀⠀⠂⢠⠀⠀⠈⠃⠀⠀⠉⡰⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠠⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡐⠀⠐⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢲⠁⡆⠀⣠⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡤⠀⠖⠅⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡈⠀⠂⢴⣶⣦⣄⣀⣤⡴⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⣀⠀⢀⡄⢀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠆⠀⠀⠀⠀⠤⠄⠀⢄⠘⠉⠉⢻⣿⠛⠛⣿⠋⠉⣛⣩⠄⠀⠀⠄⠀⠀⠀⠀⠌⠀⠊⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⡉⠙⠛⠛⠛⠛⠋⠉⠉⠁⠀⢰⡟⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣅⠀⠀⠈⠙⠻⢶⣤⣤⣄⣀⣀⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣹⢷⣖⠒⠶⠶⠒⠛⠋⢩⡏⠙⢷⣤⣀⠀⠀⠀⠀⣈⣭⠟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣰⠇⠀⠙⢷⡄⠀⢀⣀⣠⣾⡀⠀⠀⠈⠙⢛⣿⣶⡟⠉⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣴⠋⠀⠀⠀⢀⡴⠃⠀⠀⠀⢠⠟⢯⡉⠉⢹⠏⠛⢷⣤⡴⠞⠋⠁⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠛⠛⠶⢶⣿⣤⣤⣄⣀⣴⣋⠀⠀⠳⡄⣏⣠⠾⠛⣿⠀⠀⠀⠀⣿⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠙⣧⠀⠀⠈⢻⡍⠉⠉⣳⠾⡯⢤⣤⣀⣹⡆⠀⠀⠀⢸⡇⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⢹⣇⠀⠀⠀⣿⣴⡞⠁⠀⢳⡀⠀⡽⠛⠛⠛⠛⠶⢶⡿⠶⠤⢤⣬⣷⡤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⣠⡴⠛⠛⠛⠛⠶⣤⣧⡞⠁⠀⠀⠀⢀⡶⠋⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢀⣿⣾⣋⣀⡀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⡴⠋⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⠟⠉⠉⠉⠉⠛⠛⠷⢦⣤⣀⠀⢹⡆⢀⡞⠁⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣯⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣿⡾⠁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠶⢶⣤⣀⡀⠀⠀⠈⣷⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⠸⣇⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠒⠒⠂⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⢀⣤⣤⣤⣄⠀⠀⢀⣤⣶⣤⣀⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠉⢸⣿⣿⣿⡏⠀⠀⢹⣿⣿⣿⡿⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠇⠘⠿⠿⠿⢋⣿⣷⡈⠿⠿⠛⠃⠸⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⣀⣀⣀⠀⢸⣿⢿⡇⠀⠀⣀⣀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢿⣿⣀⠀⣀⣀⢀⠀⣿⡟⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⢭⡭⡧⣭⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⠀⠀⠀⢰⡀⠀⠀⠉⠉⠉⠉⠉⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⠁⠀⠀⠀⠀⠈⠉⠓⠤⡀⠙⠤⣀⠀⠀⠀⠀⣀⠤⠊⠀⢀⠤⠖⠊⠉⠁⠀⠀⠀⠀⠁⠒⠠
⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠈⠉⠉⠉⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠸⢳⣾⣿⣿⣿⡆⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⡰⣿⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⣿⣿⣿⣿⠃⣴⣿⣿⣧⠀⠀⠀⠀⠀⢴⢰⣿⡿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢆⠀⠀⠀⢀⠀⠀⠘⠃⠀⠀⠈⠉⠀⣸⣿⣿⠀⠀⠆⠀⠀⠀⠀⢸⠈⠉⠀⠸⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠈⠣⡀⠀⠈⢹⣿⠟⠶⢶⡶⠂⠀⠀⠉⠋⠉⠖⠋⠀⠀⠀⠀⠀⠋⠀⠀⠢⣶⠖⣶⡷⣶⠖⠶⠆⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠈⠐⠒⠛⠻⠀⠀⠸⣇⣀⡀⠖⠖⠒⠒⢲⠀⠀⠀⠀⠀⠰⢤⠤⡤⢴⢻⠀⠀⠀⡁⠀⠈⠠⣄⣀⣀⠠⠤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⠒⠂⠓⠻⠿⠿⢿⠀⠀⠀⠀⠀⢘⡯⠽⠼⠋⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⡀⠠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠲⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠹⣦⡀⠀⠀⠀⠀⠀⢀⣠⠔⠋⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⠉⠿⣏⠒⠒⠚⠋⠉⠀⣠⠖⢩⢿⡄⠙⠦⣤⣄⣀⣀⣤⠤⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠎⢸⠀⡟⢿⡒⠒⠒⠚⠉⢀⡴⠋⣼⠙⢤⣀⠀⠀⣀⣠⡴⢚⡏
⠀⠀⠀⠀⠀⠓⠲⠶⣶⣒⣉⡥⠴⠚⢁⡴⠃⢠⠇⢠⠙⢷⡒⠒⠛⢉⣠⠞⣹⢷⡒⠚⠛⣉⢽⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢯⣤⣔⠁⢀⡞⠀⡘⠹⣯⡉⠉⢀⡴⣿⠀⠈⢉⣽⠃⢸⡅⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠟⠶⣾⣅⣰⠀⡹⢿⣅⡠⣿⢙⣶⠋⢸⡀⠀⢷⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋⠀⣠⠀⠀⡜⠀⢹⠛⠳⠮⣽⣷⣿⢿⣘⣧⣀⣑⣤⣀⣙⣦⣄⣤⣽⣷⣦⡤⠤⠴⠶⠶⠂
⠀⠀⠀⣠⣤⣤⣀⣀⣤⣤⣴⣾⣭⣥⡤⠤⢶⣧⠶⠾⢶⣶⢗⠺⣟⢛⣿⣟⣿⣯⣩⠋⢉⡝⠋⢉⠴⠋⠁⣠⠞⠉
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠷⣔⢤⡀⠈⢳⡄⠀⠘⣮⢣⡼⠿⠤⣼⣇⣎⠿⣄⡞⠀⢰⠋⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠱⡄⠀⣧⠀⢠⡿⠟⠒⠒⠲⣼⡧⠼⣀⡸⢷⡀⣿⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠘⣄⣧⡾⠿⠒⠚⠛⠲⢦⣸⡤⠒⢿⠀⣠⠽⢿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⣴⡿⠕⠒⢋⠉⠙⠓⢦⣀⣿⢀⠤⠼⣿⠁⠀⣀⣽⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠷⠚⠛⠛⠳⠦⣭⣑⠢⣌⣿⠁⣠⠤⢼⣦⠊⠁⢀⡤⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣌⣿⠋⠀⠀⠀⠹⡄⡴⠋⢀⡴⠊⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⠀⠀⠀⠀⠀⠹⡇⢠⠋⢀⡴⠋⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⣿⠀⠀⠀⠀⠀⠀⢹⡇⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡇⠀⠀⠀⠀⠀⠀⠀⢷⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⣀⠀⡠⠀⣀⢀⡀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠒⠀⠀⡤⠀⠀⠀⠀⠠⡤⠄
⠀⠤⢄⠀⡄⠀⠰⡞⠁⠀⠀⠀⠀⠀⠀⢀⠁⠈⠀⡈⠈⢳⣀⠀⠀⠒⠂⠀⠠⠾⢧⣴⣶⣴⠾⢶⠄⠀⠀⣟
⠀⠀⠀⢰⠀⠀⠀⢇⠀⠀⠀⠀⢄⠀⠀⢰⠀⠀⢀⠛⠛⢻⣿⡿⢭⠓⢦⠄⠀⠀⢀⠘⠛⠀⠀⣀⠀⢀⠴⠩⠟⠀⠂
⠀⠀⠀⠈⡀⠀⠀⢸⠀⣮⡄⠈⠀⡇⠀⠂⠀⠀⢸⠀⠀⡏⠀⠱⠌⣱⠈⢂⠀⢀⠀⡇⠸⡖⠂⠀⠠⣄⣠⣀⣦⠐⡄⠠⢄⢀
⠀⠀⠀⠀⡔⠒⠂⢸⠀⠉⢀⡠⠀⠇⠀⠀⠀⠀⢸⠀⠀⠁⡜⠑⠆⢱⠀⠈⣀⡌⠀⡇⣴⠃⣰⣿⣃⡏⣰⣳⣸⡄⣸⡦⠀⣼⡀
⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⢈⠂⠀⡿⠠⠴⠴⠆⠴⠀⠸⠉⠰⠟⢛⣿⡇⣾⣤⣠⣿
⠀⠀⣀⢰⠀⠀⠀⢘⣤⣀⣁⣀⡠⡄⠀⡧⣆⠀⡀⠮⡶⠀⠇⠀⢀⠎⢀⡐⠁⢈⡜⠁⢄⠹⠀⣚⢄⢸⠰⢞⡉⢀⠏⠉⠙⡃
⠠⠸⠿⠆⠐⠀⠠⠴⠗⠀⠀⠈⠉⠒⠒⠚⠃⠰⠴⠾⠄⠘⠒⠒⠁⠀⠉⠉⠉⠉⠉⠀⠈⠙⠉⠘⠛⠓⠒⠛⠶⠧⠄⠠⣄⢛⡘