Muscular Emojis & Symbols

⠀⠀⠀⠀⠀⠀⣀⠄⢊⡔⠊⠁⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠉⠐⣀⠃⢤⡀
⠀⠀⠀⣠⡤⠟⢡⢖⡿⠋⠧⣤⣤⡀⠀⠀⠀⠾⣿⣶⣾⡿⠆⠀⠀⠀⢠⣤⡤⠏⠻⣟⢢⠙⠷⣤⡀
⠀⢀⣾⠁⠀⠀⣿⣿⡀⠀⠀⠀⠁⠀⠀⠀⠀⡣⣄⠩⢉⠤⡇⠀⠀⠀⠀⠁⠀⠀⠀⣸⣿⡧⠀⠀⢹⣦
⠀⢘⡻⠄⠐⢶⡟⠋⠀⠀⣶⡀⠀⠁⣦⣀⣀⢷⣄⣠⣀⣴⢧⣀⢠⡖⠁⠀⣰⡆⠀⠈⠙⣷⠖⠂⠼⣟
⠀⠀⠙⢦⣄⡩⢿⣄⠀⠀⣹⣿⣶⠀⠀⠈⠉⠐⢾⣿⣿⠖⠈⠉⠀⠀⢠⣾⣿⡁⠀⢀⣼⠯⣁⣤⠜⠁
⠀⠀⠀⠀⠻⣄⠀⠙⣿⣿⣿⠭⢿⣦⣀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣠⣾⠯⢭⣿⣿⡟⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⢁⡆⠈⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡏⠀⣆⢹⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣴⣣⠋⣿⢋⣀⣀⣽⣇⣀⣈⢻⡏⢳⣱⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣏⣿⢰⡟⠉⠉⠉⣿⡏⠉⠉⠙⣷⢘⣯⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠠⣷⡶⠖⠺⣿⡿⠒⠶⣶⡧⢨⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⠀⠀⠀⣿⡇⠀⠀⢸⡁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠘⡗⠒⠚⠻⠛⠒⠒⡿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣧⡀⠀⠀⠀⠀⢀⣧⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠛⠓⠚⠋⠁

⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⢀⣠⡤⠿⢧⣤⡀⠀⠀⠀⣀⡠⡎⢹⢄⣀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⢸⣿⣿⡆⣿⣿⣿⠀⠀⠸⣿⣷⡇⢸⣾⣿⡇⠀⠀⠀⢰⠶⠜⡫⠶⣆⠀⠀⠀⣰⣶⡇⣜⣶⡄
⠀⠀⠀⢸⣿⡁⠁⡉⡿⣿⠀⠀⢈⠹⣿⠇⠸⣿⠏⡁⠀⠀⠀⢸⢄⣀⣆⣀⣯⠀⠀⠀⠿⢫⡁⢉⡋⡇
⠀⠀⠀⣾⢀⣿⣧⣷⡇⢻⠀⠀⢸⡇⡿⠀⠀⢽⠸⡇⠀⠀⠀⣾⢸⣿⣿⣇⢿⠀⠀⠀⡆⣸⠃⠸⡇⣶
⠀⠀⠀⣇⢺⣿⣿⣿⣿⢸⠀⠀⢘⠀⣷⣤⣤⣾⡀⠃⠀⠀⠀⡇⣾⣿⣿⣿⡼⠆⠀⢰⠇⣷⣦⣴⣶⢸
⠀⠀⠀⠁⠘⣿⡆⣿⢟⠀⠀⠀⠈⠘⢿⡟⢻⡿⠃⠁⠀⠀⠰⠂⣿⣯⢹⣿⠀⠆⠀⠰⠈⣻⡟⣻⡟⠀⠆
⠀⠀⠀⠀⢸⣿⠁⣿⡿⠀⠀⠀⠀⠀⣿⡇⢸⣿⠇⠀⠀⠀⠀⠀⣿⡏⢸⣿⠀⠀⠀⠀⠀⣿⡇⢹⣿
⠀⠀⠀⠀⢠⢸⠀⢩⢀⠀⠀⠀⠀⠀⣿⡇⠸⣿⠀⠀⠀⠀⠀⠀⡀⡄⢸⢀⠀⠀⠀⠀⠀⣿⡇⢸⡿
⠀⠀⠀⠀⢸⡟⠀⢸⡟⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡇⠈⣾⠀⠀⠀⠀⠀⣿⠅⢸⣿
⠀⠀⠀⠀⠈⠇⠀⠘⠇⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⠃⠀⡟⠀⠀⠀⠀⠀⢉⠀⠈⡉
⠀⠀⠀⠀⠞⠁⠀⠀⠑⠀⠀⠀⠀⠀⠃⠀⠀⠈⠂⠀⠀⠀⠀⠀⠞⠀⠀⠰⠀⠀⠀⠀⠀⠂⠀⠀⠁
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠠⠐⠒⣒⣲⡀⠈⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⠔⠒⠠⠤⠄⠀⠒⠲⣄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠀⡲⢶⠀⠀⢀⠒⡂⠀⣽⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⠉⠀⠀⠀⠁⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠠⠄⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠐⠦⠤⠔⠀⠀⡡⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⣧⡀⠀⢀⡤⢚⠃⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠶⠴⠮⢄⠀⠀⠃⠉⠛⠉⠀⠀⠀⠀⠀⠰⡄⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⡤⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⡤⠶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠁⠀⠠⠒⠚⠉⠉⠉⠛⠛⠛⠉⢸⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⢰⠃⠀⠐⠲⢄⠀⠀⣠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⠃⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⡀⠀⠈⠉⠙⠂⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠐⢿⡀⠀⠀⠀⠀⡏⠀⠀⠈⠉⠓⠲⠤⠀⠀⠀⠙⠳⢶⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠒⢄⠀⠀⠿⡗⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠑⢎⢰⠁⠉⠉⠒⠒⠒⠹⠉⠉⠉⠉⢉⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠠⡀⠀⠀⠀⡃⠀⢀⡠⠞⠁⣀⠀⠀⠀⠘⢦⣀⠄⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠉⠑⠒⠤⣄⠀⠀⠉⠉⠀⠀⠀⠀⢀⡤⠚⢁⡠⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠢⣄⠀⠀⠀⠀⠀⠈⠙⢶⡤⠄⠀⠀⠀⠀⠀⠀⠐⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣉⠙⠓⠢⢄⠀⠀⠀⠉⠀⠀⠒⠒⠲⢄⡀⠔⠋⠀⠀⠀⠀⣀⠀⠀⠀⢠⠊⢀⡠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣠⠤⢄⡀⠀⠀⠀⠀⠀⡀⠀⠀⢹⠀⠀⠀⢠⠔⠋⠁⠀⢀⣔⣡⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣎⠁⠀⠀⠀⠈⢆⠱⠘⡄⠰⣄⠇⢀⡀⠤⠤⠤⠤⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠭⠦⠜⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⣼⠁⠀⢰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⣴⡟⠀⢱⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣀⣤⠔⠀⠀⠠⠴⣛⠁⠀⠈⠷⣿⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠘⣿⣆⠀⠀⠀⠀⢻⣿⠀⠀⠀⢠⠘⣧⡜⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣈⣓⠀⠤⣀⠙⠿⣾⡶⠂⣻⣿⣄⠀⠀⢈⣷⠙⣷⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣷⣦⣈⢷⡀⠙⢻⣿⣯⣽⣿⡆⢠⣠⠁⠑⣦⠀⣰⠖⢬⡥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⢿⣿⣿⠿⠿⠻⢄⣶⣿⠇⣼⣿⠄⠘⣻⣇⠀⠈⠷⣏⠀⠀⣼⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠙⣆⠀⠀⠈⣿⠁⢠⣿⠿⠀⡈⠀⠘⢷⣀⢳⢳⡀⠘⢃⣞⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣜⣦⢀⣸⣿⣷⡿⠇⢹⠀⠁⠀⠀⠀⠙⠻⢮⡃⢐⣩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣬⣿⠿⠈⢃⠀⡌⠀⠀⠀⠀⠀⡀⠀⠀⠉⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⢀⣿⡗⠀⠀⠀⣀⡤⣂⡉⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠁⠀⠀⠀⣺⣿⡷⠀⠀⠀⡡⢖⣉⡀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⡐⢿⠟⠀⠀⠀⢠⠶⢿⣤⣀⠑⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⣽⣿⡇⠀⠀⠐⠀⠀⠔⠋⠡⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⡏⠀⠀⠀⢈⠀⠀⠀⠀⠀⠘⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣹⣿⣿⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⢰⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠉⠉⠂⠀⠀⠀⠀⠐⠀⠀⠀⠠⠃⡠⠀⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠐⠀⠀⠀⠁⡜⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠿⠿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠭⢻⠘⠛⠍⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠼⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⠀⠐⣒⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠄⢻⠀⠠⠄⣤⠤⠔⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠀⣀⡁⠀⠐⣄⠀⡠⠀⠀⠙⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠙⠃⠀⠀⠈⠁⠀⠁⠉⠛⠒⠺⠷⠗⠚⠋⠉⠀⠉⠙⠓⠐⠢⠤⢀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣦⣄⣀⠀⠀⠀⠀⠺⣿⠒⠒⠀⠀⠀⠀⢬⡤⠔⠀⠀⠀⠠⣼⡀⠀⠀⠠⢴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡙⠉⠀⠀⠠⠀⠈⠲⡀⢣⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠢⡈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠐⠒⠳⠾⠿⠷⠶⢦⣤⣼⣧⠤⠆⠐⠲⢾⣇⠀⡀⠀⠀⠀⠑⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠀⠀⠀⠀⠀⢀⡀⢀⣴⣶⡬⠗⠋⠁⠀⠀⠀⡄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣄⠀⠀⠒⠚⠋⠀⠀⠀⣠⡤⠶⠖⢋⣽⠞⠛⠉⠁⠀⠀⠰⠶⠶⠶⠾⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠀⠐⠶⢊⣀⣀⣠⣀⣤⣶⡿⠛⠁⠀⠀⠀⣀⣀⣀⣤⠴⢂⣀⣴⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣾⣿⣿⣿⣿⣿⠟⠛⠛⡛⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣷⣶⣶⣶⣿⠶⠶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢸⣿⡏⠀⠀⢱⠀⠀⠀⣼⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢄⠀⢻⣿⠀⠀⠀⠀⠀⢀⠇⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠙

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣄⠀⠀⣤⣤⣤⣤⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⣿⠏⠁⢏⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠰⣿⣇⡀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠑⠀⠀⠀⣿⣿⣧⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠀⣿⠻⣿⣿⣿⡇⠆
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠈⢹⣿⣷⠠⡀
⠀⢰⠶⣿⣻⣟⣿⡿⠿⠛⠃⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡙⣷⣤⣿⣏⣄⠱⣄⣀⣀⣀⣀⣀⢀⣠⣤⣴⠶⠂⠀⣠⡀
⠀⣼⠿⠟⢛⣿⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣼⣶⡏⠻⣿⣽⣦⣽⣉⣩⠽⢓⣷⣿⡿⠟⠀⠀⡠⠀⢀⣽⣆
⠀⣴⣦⣾⣿⡛⠉⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣹⣿⣧⡀⠈⣿⣿⣿⠤⣤⣾⣻⠟⠁⠀⠀⠀⠟⠀⠀⣠⣿⣿
⠀⠈⠀⠙⢟⣿⣀⡀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢸⣿⡆⠈⣻⣾⡟⣹⡗⠟⢋⣡⣴⣾⣷⣞⡶⠀⠀⠀⣼⣿⣿
⠀⢀⣀⣀⣼⣿⣟⠁⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣸⣿⣷⣄⡙⢻⣧⣽⣷⣰⣿⣿⣿⣿⣿⡿⣿⣄⠠⣴⣽⣿⡇
⠀⠀⠙⠛⠋⠉⠉⠙⢿⣿⣿⡿⣿⣿⣆⠘⠲⠶⠆⠀⠒⠒⠲⠶⠶⠀⠐⣿⣿⣿⣿⢿⣿⣿⡇⢹⣿⣿⣿⣯⣿⡿⠋⠀⣸⣿⣿⣮⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠡⠀⠈⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣯⡿⣿⣿⡇⢸⣿⣿⣿⡟⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢪⣿⣿⡸⢱⣿⠟⣿⣸⣿⣿⣿⣄⣠⡾⣫⣾⡿⠋⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⣀⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠏⣧⠃⢺⡿⠀⣿⣿⣿⣿⣿⢹⣿⣾⣿⡟⠁⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢤⢸⢸⣏⢦⣿⡇⠀⠁⢨⣍⣿⠿⣄⣿⠛⢿⠀⢀⣠⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣆⠹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣤⢸⣿⣏⣞⣿⡇⠾⠴⣿⣿⣿⣶⣿⡇⠀⢹⣶⣼⠿⠋⠁⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢧⠈⢎⠱⣝⣷⡄⠀⠀⠀⠀⣤⢼⣿⡿⠛⠉⣡⣤⠼⢿⣿⣷⡼⣾⠁⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⢣⠈⠳⡜⢎⠻⣄⠀⠀⠀⣤⠘⡅⠀⠀⠀⢉⣠⣶⣿⣯⣽⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠁⠄⠀⠘⣦⠀⠀⣤⠀⡇⠀⠀⢰⣽⣿⣿⣏⢹⡆⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⡄⠀⡇⠀⠀⠴⣿⣿⣿⣯⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠱⡈⠻⢀⡷⠀⡗⠀⣠⣾⣿⣿⡇⣿⢾⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡗⢸⣷⡿⠋⠀⣕⣿⠅⣿⣟⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣧⠀⢠⡀⠀⠘⠀⠀⠀⢈⠀⡈⠙⣇⠀⣷⡏⣿⠀⣿⣯⢹⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢿⣼⣼⣷⠀⠀⠀⠀⠀⠘⠀⠃⠐⠈⢆⣿⢸⡏⠀⣿⣿⡈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡇⢼⡿⠿⠻⠀⠀⠀⠀⠀⠀⠀⠘⣄⠠⡌⠻⣿⣇⠀⣴⣿⣷⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢠⠀⠀⠈⠿⣿⣷⣿⣶⠷⠶⠾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⣈⣽⠟⠉⠎⣵⣷⡀⠀⠀⠀⠈⠀⠀⠀⡜⢡⣼⡿⣟⠳⣦⡀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⠟⠁⠀⠀⠀⠈⠛⢿⡀⡀⠀⠈⠀⠀⣾⣰⡟⠉⠀⠋⠀⠂⠝⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⡀⠀⠀⣼⣷⣟⡷⠀⠀⠀⠀⠀⠀⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣼⣾⣿⡟⠁⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⢿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⢢⠸⣯⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡆⠀⠀⠀⠀⠀⠀⡴⢹⣿⡿⣽⣿⢿⣿⣿⣿⣦⣦⡀⢀⢣⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣤⣾⣄⣄⣠⡜⢠⣿⣿⣿⡿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⠟⠿⠛⠻⠿⠛⠟⠻⠿⠿⠛⠻⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⣀⠀⠀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣷⡀⠀⣿⣿⡟⠻⡻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⣿⣿⠃⠀⣿⡏⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠀⠀⣿⣀⠀⠘⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠙⠀⠀⠀⣿⣿⣧⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠃⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠘⢻⣿⣧⠁⡀
⠀⢠⣴⡶⣾⣿⣿⣿⣷⡶⠦⣀⡤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠹⣶⣄⢻⣿⣄⠙⡄⠀⠀⢀⣀⠀⠀⠀⣀⣤⣤⡤⠔⠀⣀
⠀⣨⣧⣶⠾⣿⡿⢷⡦⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣿⣝⠻⣿⡿⣦⣹⡟⢛⣟⣻⢷⣴⣿⣿⠟⠁⠀⣀⠈⢉⣷⡀
⠀⢹⣁⣤⣾⡿⠟⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠹⣿⣏⠀⠘⣿⣽⣿⣉⣉⣩⣶⣿⡿⠋⠀⠀⡠⠞⠀⠀⢘⣿⣷
⠀⠘⠛⠹⣿⡿⠄⠀⠈⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢸⣾⣏⠀⢮⣽⣿⢻⣿⣾⡿⠟⣁⣠⣶⣤⡮⠀⠀⠀⠀⣬⣿⡟
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠉⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣿⣯⡀⢺⣿⣟⢸⣿⡀⣰⣾⣿⣿⣿⣿⣷⣀⠀⣀⣸⢾⣿⡇
⠀⠀⠺⠿⠿⠿⠛⠠⣴⣶⣶⣶⣿⣿⣦⠀⠤⣄⣠⣄⠤⠀⣤⣤⣤⣦⡀⢠⣿⣿⣿⣿⣿⣷⣮⣿⢻⣿⣷⣿⣏⡹⣡⡿⠋⠈⣿⣷⣮⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⡉⠙⠏⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⢈⣹⣿⣿⣿⢹⢿⣿⣿⠈⣿⣿⣿⣧⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠁⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⣻⣿⠻⡏⣾⡿⣿⠀⣿⣿⣸⡟⠀⢀⣴⢟⣽⣿⡟⡟⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣱⡿⣟⣷⢰⣿⠇⢿⣾⣿⣿⣟⡿⣾⣿⣴⣿⡿⠉⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣴⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⣿⣤⢾⡿⠀⠸⢿⣿⣻⣿⡇⢺⣿⣿⡟⠁⠀⢀⣠⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡾⣝⠿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠋⣿⣼⡿⣹⣾⡗⢰⣄⣨⣷⣿⣏⣻⣾⠃⠈⣿⠀⣶⣿⡿⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⢷⠈⠟⢿⡻⣷⡄⠀⠀⠀⠀⠀⠋⣿⣿⣷⣿⡾⠗⠘⢛⣿⣿⣿⠻⢻⣿⠀⠀⢹⣿⡟⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⢣⠀⠃⠙⣌⠻⣆⠀⠀⠀⠀⡞⢿⡿⠋⠁⠀⠠⠟⠋⣩⣿⠿⡿⢻⡇⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢣⠀⠙⣌⠃⠙⢦⡀⠀⠀⠗⢸⡇⠀⠀⢀⣠⣶⣿⣿⣿⡗⢷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠂⠀⢈⣷⡀⠀⢶⠀⡇⠀⠀⢘⣯⣿⣿⣿⠆⣟⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠃⠀⠀⠀⠀⠀⠀⡀⢻⣷⠈⢿⠀⡇⠀⠀⢋⣹⣿⣿⢿⡆⢏⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠙⡄⠁⠙⢾⠀⡇⣠⣴⠿⢿⢹⡏⣸⡅⣿⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⡇⢸⢿⡟⠁⠀⠀⣸⡇⣿⣿⢿⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⡀⣶⡀⠀⠀⠀⠀⠀⢠⠀⢸⠈⢷⠀⣸⡆⣿⠁⣽⣿⡼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⠛⣿⣷⣿⣇⠀⠀⠀⠀⠀⠈⠀⡄⢠⠀⠳⣽⣷⣿⠀⠻⢿⣇⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣸⣿⣶⣿⣟⣋⡙⠀⠀⠀⠀⠀⠀⠀⠘⢦⡰⣄⡈⣿⣿⠀⣴⣿⣿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢗⠀⠀⠀⠀⠀⢰⠀⠀⠀⠿⣿⢿⣿⣷⡚⠛⠛⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏⢀⣼⠿⠉⠐⠙⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⡜⢡⡾⡿⣯⡛⢶⣤⣀⡀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡿⠃⠀⠀⠀⠀⠘⢻⣧⡀⡀⠀⠘⠀⠀⣼⣌⠛⠀⠈⠀⠉⠀⠘⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡄⡘⠀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣷⣽⣶⣿⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⣻⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠠⠸⣄⣿⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡸⠀⠠⠀⠀⠀⠀⢀⡴⣹⣿⣿⣾⡿⣿⣿⣿⣿⣤⣤⣄⢀⠺⣴⣳⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢀⣤⡿⣀⢠⣠⠞⣴⣻⣿⣷⡿⣱⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣯⣾⣿⣻⣾⣹⣶⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠻⠏⠛⣿⠉⣻⠛⢻⡛⠏⣛⣿⢟⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣿⣿⣶⣿⣶⣾⣷⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡴⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠯⠀⢠⡆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢢⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠶⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⡀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠄⠀⢀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⣠⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠈⠉⠀⠈⠁⣀⠀⠀⠀⠰⠀⠀⠀⢀⣰⡄⡀⠀⠀⠀⡀⠀⠀⠀⢈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⡀⠠⠾⠆⠀⠀⠀⠸⣷⡝⢲⠖⠛⠁⠀⠂⠒⠛⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣀⠀⠀⠠⣌⠙⠛⠀⠂⠀⠀⠀⠀⣤⣴⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣼⣤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡉⠙⠛⡀⠠⠀⠀⠘⠛⠻⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⢄⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣾⣶⣷⣄⠀⠀⠀⠀⢉⣴⠶⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⣤⣷⣶⣶⣾⡿⣫⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠟⠛⠓⠞⠛⠉⠐⠛⠉⠥⠶⠌⠛⠛⠛⠿⠶⢿⡛⠟⢫⣽⣾⡿⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡛⣙⢟⣉⠛⣻⣻⣹⡟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢻⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠸⣿⣿⣿⣿
⠀⠀⠀⢀⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣾⣿⣿⣷⣦⣀⣀⢠⣾⣿⣿⣿⣿⣷⣄⣀⣀⣴⣶⣿⣿⣷⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣷⣝⠻⠿⣿⣿⣉⢿⣿⣿⣿⣿⣿⣿⣿⣏⣩⣿⣿⣿⣿⣿⣿⣿⡿⣉⣿⣿⠿⠟⣫⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣿⣿⣿⣶⣦⣤⣤⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣤⣤⣴⣶⣿⣿⣿⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⢿⣿⣿⣿⣄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⣿⣿⣿⡿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣬⣻⣿⣿⣿⣧⣼⣿⣿⣿⣟⣥⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⣿⣿⣷⣼⣿⣿⣍⣩⣿⣿⣧⣾⣿⣿⠈⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡌⣿⣿⠟⠻⣿⣿⢡⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⣿⣿⣿⣿⣿⣿⢟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢿⡇⢸⣿⣧⣼⣿⡇⢸⡿⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠿⠾⠿⠿⠿⠿⠷⠿⠁⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⣀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡎⠀⣠⣦⠈⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠈⢀⣦⡀⠈⣲⣤⡀
⠀⠀⠀⠀⢀⢄⣠⡼⠟⢁⡴⢫⣿⠛⠹⣧⣄⣀⣀⡀⠀⠀⠀⠀⠀⡿⠿⠧⠤⠤⠿⠿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣤⡼⠙⠻⣿⡳⣄⠙⠻⣤⣀⢀
⠀⠀⠀⣔⡵⠋⠉⠀⢀⣾⣴⣿⠁⠀⠀⠀⠉⠛⠟⠁⠀⠀⠀⠀⠀⡄⠙⠛⠗⠿⠛⠁⣼⠀⠀⠀⠀⠀⠙⠻⠋⠁⠀⠀⠀⠹⣷⣜⣇⠀⠈⠉⠱⣄⡄
⠀⠀⢸⣿⠁⠀⠀⠀⠨⢾⣿⡥⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡏⠇⣬⠕⠬⡄⠏⣃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⣽⣿⠮⠃⠀⠀⠀⢸⣿
⠀⠀⡘⠿⣧⡀⠒⠢⢶⡟⠋⠀⠀⠠⣘⣶⠀⠀⠀⠈⢲⣄⠀⠀⢸⢿⣆⢀⣉⣉⢀⣾⢿⡀⠀⢀⣴⠊⠁⠀⠀⢰⣶⡠⠀⠀⠈⠛⣿⠦⠒⠂⣠⣾⠟⡂
⠀⠀⠙⠻⣄⠀⠀⠐⢺⣗⠤⠀⠀⠀⠈⣿⣿⣧⣀⠀⠈⠉⠛⠛⠾⢠⣝⣿⣿⣿⣛⣥⠸⠞⠛⠊⠉⠀⢀⣠⣿⣿⡟⠀⠀⠀⠀⠴⣿⠓⠂⠀⢀⡼⠟⠁
⠀⠀⠀⠀⠈⢻⣷⠮⠍⠻⣷⣦⣄⣀⣠⣿⣿⣟⣶⠨⠀⠀⠀⠀⠀⠀⢙⣿⡿⣿⣟⡁⠀⠀⠀⠀⠀⠨⢴⣞⣿⣿⣧⣀⣀⣤⣶⡿⠋⠭⠶⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠀⠛⣿⣿⣿⣷⡦⠼⡿⣷⣤⡀⠀⠀⠀⠀⠈⣿⣟⣿⣏⠄⠀⠀⠀⠀⣀⣴⡿⣿⠥⣶⣾⣿⣿⡟⠋⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢶⣿⠿⠿⡿⠁⣰⠀⠹⣿⣿⣾⣷⣶⣶⣶⣶⣿⣸⣸⣷⣶⣶⣶⣶⣿⣶⣿⣿⠁⢰⡄⠙⡿⠿⠿⣷⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣰⣿⠀⢀⣾⣻⣿⣿⣿⡿⠿⠿⠯⣽⡭⠿⠿⠿⣿⣿⣿⣿⢻⣇⠀⢸⣿⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⣾⣁⡟⠁⣻⠋⢀⣀⣀⣀⣿⣇⣀⣀⣀⡀⢻⡇⠙⣧⣘⣦⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠉⣽⠁⢰⣿⠟⠋⠉⠉⠻⣧⡿⠋⠉⠉⠛⠻⣷⠀⢸⡏⢹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⣿⠀⠸⣇⠀⠀⣀⣀⣠⡟⣷⣀⣀⡀⠀⠀⣿⠀⢸⣟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠄⠲⣿⡿⠟⠛⠉⠻⣾⣿⠋⠙⠛⠿⣿⡷⠆⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⢸⠁⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⢹⠁⢠⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠂⠸⣧⣀⣀⣀⣼⣷⣿⣤⣀⣀⣠⣿⠂⠚⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⠀⢿⠉⠉⠀⠀⠹⠁⠀⠀⠉⢹⠇⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣶⣤⣤⣴⣤⣤⣴⡾⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠛⢿⣿⡟⠋⠙⠛⠇⠀⠀⠀⠀⢀⡄
⠀⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⡁⢀⣶⡟⠑⠾⠋⠿⢇⣰⣄⣤⣴⣿⠇
⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣻⠠⣀⣈⢀⣙⠛⢛⡋⠉⠉
⠀⢿⣿⣿⣿⣿⣿⣆⢠⠀⠀⠀⠀⠀⠀⠀⣠⡜⠟⢉⠴⠖⠀⠁⢻⡧⠀⠉⠛⠻⠀⠀⠀⠀⠠⠶
⠀⠘⣿⣿⣿⣿⣿⣿⡄⣆⠀⢀⣄⡀⢠⣿⣿⢸⣞⣿⡈⠀⠀⠀⣻⡇⠀⠀⠀⢠⣰⡶⠶⠶⣷⣦⠀⡀
⠀⠀⠹⣿⣿⣿⣿⣿⡇⣿⣷⣾⣿⣿⣿⣿⣟⣄⣽⣿⣷⣀⡀⣠⢿⣤⠀⠀⠀⢠⡇⣀⡀⣀⡉⠛⢀⡀
⠀⠀⠀⠙⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠟⠉⠋⠉⠉⣿⢉⡿⠤⣦⠉⠛⢷⣿⣿⡿⠟⠟⠉⠀⢀⣼⢿⡤
⠀⠀⠀⠀⠈⠻⢿⡿⠣⣿⣿⡿⠛⠋⠀⠀⠀⠀⢀⣤⣿⣺⠿⠰⣟⡓⣚⠹⠃⠁⡀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣏⣧⠀⠆⠀⠈⢁⣴⡢⠞⠁⠠⠞⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠛⠛⠿⣿⣿⣶⣶⣶⣶⡿⠯⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡏⠀⠀⠀⠙⣿⣿⣿⡋⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢹⡇⠀⠀⠀⠀⢸⢿⣿⣿⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⡶⠀⠀⢷⠟⠸⡏⠻⣧⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⣿⡇⠀⠀⣸⡄⠀⣇⠀⠙⣧⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠹⠿⡦⢶⠟⠀⠀⠹⣧⡀⠈⣿⡀⢀⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢢⣄⣤⠄⢦⡀⠀⠀⠀⠈⠻⠾⢿⡟⠋⠀⠁⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡿⣷⡆⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⣶⡆⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⢸⣿⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⡈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣮⣿⣿⣿⣿⣥⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠻⠿⠿⠟⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡴⠶⠆⠀⠀⠰⠶⠦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠈⠀⢰⡀⠀⠀⠀⠀⠀⢲⡀
⠀⠀⠀⢀⣀⣠⠖⠀⠙⠻⣾⡁⠀⣀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣀⠀⢈⣷⠟⠋⠀⠒⣄⣀⡀
⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠙⢟⠛⠛⠛⠻⣧⡀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠛⠛⠛⡻⠋⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠉⢻⣿⣿⡟⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡀⠀⠀⠀⠀⠀⠐⣿⡿⠀⠀⠀⠀⠀⠀⢀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢠⣿⣦⠀⠀⣀⠀⣰⣦⠀⠀⠉⠉⢻⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⡟⠉⠉⠀⠀⣴⡆⠀⣀⠀⢀⣴⣿
⡏⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣰⣦⣄⣀⣠⣴⣾⣿⣿⣷⣦⣄⣀⣠⣴⣆⡀⠀⢀
⠸⣄⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠻⣿⣿⣿⠟⠛⠟⠿⣿⣿⠿⠻⠛⠻⣿⣿⣿⠟⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠈⠉⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⢀⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡀
⠀⠈⠻⣾⣿⣿⣿⣿⣿⣦⣴⣤⣶⠶⠾⢷⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⡾⠷⠶⣶⣤⣦⣶⣿⣿⣿⣿⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠙⠛⠿⠿⢛⡿⠿⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠿⢿⠛⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠘⠉⠙⠻⣶⣤⡀⠀⠀⢀⣤⣶⠟⠋⠉⠃⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠟⠉⠉⠻⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣾⣦⣀⣀⡴⣿⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣿⢦⣀⣀⣴⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⣯⠀⠈⠉⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠉⠁⠀⣽⠿⣄
⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠙⢦⣤⠞⠀⠙⢇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠀⠳⣤⡴⠋⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⢲⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡆⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⢀⣤⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⡀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠠⣅⠀⠀⠀⣀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣀⠀⠀⠀⣨⠄
⣤⣤⣤⣤⣤⣤⣤⣽⣶⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣶⣯⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠾⠛⠷⣤⡐⠿⠀⠺⠇⠐⠿⠀⣴⡆⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠉⠁⢀⣠⡶⢿⣤⣾⠃⢠⣶⡿⢿⣦⣤⣤⣤⣀⠈⠉⠀⠿⠃⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⢠⣤⣶⡛⠁⠀⠀⠉⠁⣠⣿⡿⣱⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⠁⢼⡷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠻⣧⡀⠀⠐⢮⣝⢿⣦⣀⠀⠶⣾⣿⠿⠁⣿⣿⡿⢿⣿⠿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⠋⣀⡀
⠀⠀⠀⠀⠀⢀⣄⠘⠋⠀⠹⣷⡀⠀⠀⠉⠀⠉⠙⢿⣦⡀⠀⢀⣴⡿⠿⠁⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⣤⡀
⠀⠀⠀⠀⣠⣌⠛⠀⠀⠀⠀⠘⢿⣾⡆⣀⠀⠀⠀⠀⣻⣷⣾⠛⠋⢰⣄⣀⠀⢀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣡⣄
⠀⠀⠀⣀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣜⢷⣄⠀⣰⣿⣿⣿⣦⣄⣀⣻⣏⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠙⢉⣄
⠀⠀⣀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠶⠶⣦⣀⠀⠀⠀⠀⠀⢺⣏⠁⠙⣿⡄⠀⠀⠀⠈⠋⣀
⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠉⠛⠓⢦⡹⣧⠀⠀⠀⠀⠀⢙⣿⠆⣸⡇⠀⠀⠀⠀⠘⠛
⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠃⢹⣧⣄⡀⠀⣴⠟⠁⢠⡿⠀⠀⠀⠀⠀⠀⠸⠟
⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠘⠁⠉⠻⣿⠃⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣶
⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣿⠀⠈⠙⠻⣿⣿⡿⠟⠷⣦⣙⠶⣤⣀⡀⠈⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣤⣀⡀⠀⣾⠇⠀⠀⠀⠈⠙⠻⠶⣯⣭⣅⣀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠋⠁⠀⢸⣿⠀⠀⠀⠉⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠁⢀⣴⠃⠀⢸⣿⢷⣄⡀⠀⠀⠀⠀⠀⠉⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢰⡿⠁⠀⠀⠸⣿⠀⠉⠛⢃⣴⠆⠀⢀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⣿⠿⠿⢿⣿⡟⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠉⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠘⣷⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠉
⠀⠀⠀⠙⣁⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣼⣿⡀⢻⡆⠀⠀⣼⡏⠀⠀⠘⣿⣧⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⠉
⠀⠀⠀⠀⢹⣿⠿⣷⣶⣶⠶⠾⠿⠛⠋⠉⠉⠉⠉⠛⠻⠆⠁⠀⢠⣿⠁⠀⠀⠀⢹⡿⣆⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡙⠋
⠀⠀⠀⠀⢹⡇⠀⢠⣤⣀⣀⣀⣀⣀⠉⠛⠛⠛⠛⢛⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⢸⡇⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠙⠁
⠀⠀⠀⠀⢸⣷⠀⣿⠋⣩⣯⠉⠛⠛⠻⠿⠷⣶⣦⣾⣷⣤⣤⣾⠟⠀⠀⠀⠀⠀⢸⣿⠀⣼⡇⠀⠀⠀⠀⠀⠀⢴⣦⠉⠁
⠀⠀⠀⠀⠀⣿⡀⢻⣆⠈⠁⢾⡗⢀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣾⡏⠀⠘⠻⠷⣦⣄⡀⢺⡷⠀⠁
⠀⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠘⠋⢠⣶⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣶⣶⣶⣦⣤⣈⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠸⠟⠀⣶⠆⢰⡦⠀⣶⠆⠻⠇⠈⠛⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⣶⠀⠘⠓⢨⣧
⠀⠀⠀⠀⠀⢀⡼⣿⡆⠀⢶⣿⡟
⠀⠀⠀⠀⢰⣷⣀⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠈⢿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠘⣏
⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⣾⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡶⠈⢳⡄⠀⠀⠀⠀⢠⣾⠛⢿⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠀⢻⡀⠀⠀⠀⡾⠃⣀⡀⠀⠈⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠈⠷⠢⣄⠀⠇⠀⠉⠀⠛⠓⣩⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣵⣦⣼⡂⠀⠀⠙⣆⡀⠺⠟⠿⠆⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⣿⡇⠀⠀⠀⠘⢿⣦⣀⣤⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⡇⠀⠀⠀⠀⠈⠛⣿⣿⣿⣥⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⡇⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣿⣿⣧⡀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡈⠟⠛⠷⣶⡾⢋⣤⡀⠀⠀⠀⠠⢤⣤⠀⠀⠘⠛⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠘⠿⠃⠈⠻⣿⣶⣶⣶⣾⣿⣿⢷⣌⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⢀⡠⡴⠶⣦⣀⠈⠙⠻⣿⣿⣿⠏⠀⠙⢷⣤⣀⡀⡀⠙⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠁⠀⠈⠛⣻⣿⣿⡿⣳⠏⠀⠀⠀⠀⠈⠛⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢼⣿⠿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡷⠀⠀⠐⠂⠀⠀⠚⢳⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠁⠈⠉⠒⠛⠉⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⣠⣄⣀⣀⣀⣠⡤⣶⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⢽⣿⣿⣿⡟⣃⡸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀⢻⣿⣿⣿⣹⣧⣹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⢿⣿⣿⣿⡿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠹⣿⣿⡿⠼⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠀⠀⣀⣀⣤⣍⣙⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⢀⣴⠿⠛⠛⠛⠛⠛⠛⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠖⢁⣉⣘⠂⢰⣯⣤⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠙⣿⡀⠀⠀⠀⠈⠉⠀⠈⢁⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠘⠳⠄⠀⠀⠀⠀⠀⣀⠜⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⢹⡿⠿⢶⣶⣶⡶⢾⣿⣿⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠑⠀⠀⠀⠈⠁⠘⣿⣿⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢿⣿⣷⣄⠀⠀⠀⠀⢀⣠⣦⣽⣿⣷⡶⠶⣦⣤⣤⣽⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣈⡿⠿⠿⡦⢠⣷⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⠋⢁⣤⠾⠋⠁⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢠⣾⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣄⠀⠀⠀⢀⡿⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡰⣿⡇⢃⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠋⣼⣿⣿⠷⠦⠤⠚⠁⠀⠀⠀⠿⠁⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢸⣿⡈⠀⠀⠀⣀⣠⡄⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⣇
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠈⣿⣷⣠⡴⠛⢉⣠⠶⠛⠉⠛⠿⢶⣦⣤⣤⣴⣶⣶⣿⣿⡿⠋⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⣿
⠀⠀⠀⠀⢠⠏⠀⠀⣴⠀⠀⠀⠀⠘⣿⣿⣶⣾⠛⠁⠀⢠⡇⠀⠀⠀⠀⠈⠉⠻⣿⣿⠉⠉⠀⠀⠀⢿⣿⣆⠀⠀⠀⢀⡄⠀⠀⢿⣷⣀⣿
⠀⠀⠀⠀⡞⠀⠀⠀⣿⡀⠀⠀⠀⣰⣿⣿⣿⣧⠀⠄⠀⢸⡦⠶⠖⠒⠛⠂⠀⠀⠘⣿⣇⣠⡴⠂⠀⠈⢿⣿⣷⣄⠀⠀⣷⠀⠀⠘⣿⣿⣿⣷
⠀⠀⠀⢨⠃⠀⠀⠀⢹⣷⡀⠀⣰⣿⠏⢹⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⣇⣀⡴⠀⠀⠀⣻⣿⠛⢷⣾⣿⠀⠀⠀⠈⢻⣿⣿⡄
⠀⠀⠀⣾⠀⠀⠀⠠⣬⣿⣿⣿⡿⠁⠀⢸⣿⣧⠀⠀⠀⢸⠗⠊⠉⠉⠉⠀⠀⠃⡇⠀⣿⣿⡉⣠⠀⠀⠀⣿⡇⢀⡴⠛⠛⠳⠀⠀⠀⠀⢻⣿⡇
⠀⠀⢀⡇⠀⠀⠀⠀⠈⢿⣿⣿⠃⠀⠀⠀⣿⡇⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⢀⣧⡇⠀⢸⣿⡟⠁⠀⠀⢰⣿⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⣿⣷⡠⠤⠀⠀⣶⠒⠀⠀⠀⠀⠿⣼⠁⠀⢸⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢀⢸⡇⠀⢸⣿⡇⠀⠀⣠⡟⠙⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠟⠁
⠀⠀⠀⣷⠀⠈⠲⣄⡀⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢽⢸⡇⠀⢸⣿⣇⣀⡴⠋⠀⠀⠀⠀⣀⣤⠾⠋⣱⣿⠿⠃
⠀⠀⠀⢸⡄⠀⠀⠈⠛⢿⣿⠃⠀⠀⠀⠀⣿⣿⣶⠶⠶⣾⣿⣤⣤⣤⣤⣤⣧⣿⣤⣤⣼⣿⡿⠋⠀⠀⠀⡤⠶⠛⠉⠀⣠⡾⠟⠁
⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⠒⠀⠀⠀⢀⣐⣶⣶⣶⡿⠿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠘⣆⠀⠀⠀⣨⣧⠴⢤⣀⠀⢠⣿⣿⠟⠓⠀⠀⠀⠀⣻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢻
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣀⢉⣻⣿⣿⣿⠀⠀⠀⠀⢠⡾⢅⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣇⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠉⢀⣸⣿⡿⠀⠀⠀⠀⠸⣇⣀⣀⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏⠀⢻⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣶⡀⠀⠀⣀⣤⡶⠿⢿⣿⣿⣿⣆⠈⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⡿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠟⠛⠉⠀⠀⠀⠀⠉⠻⢿⣆⣸⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⢰⣿⠀⠈⣁⣾⡆
⠀⠀⠀⠀⣤⠛⣿⡇⠈⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⡇⠀⢹⣿⡇
⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⢻⠁
⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢰⣇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⠃⠘⣆⠀⠀⠀⠀⣾⠏⠹⠟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠘⣆⠀⠀⢸⠉⠠⠄⣀⡀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣁⠀⡄⠉⠙⢦⠈⢀⣤⣤⡀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⢿⡇⠀⠀⠈⢿⣦⠈⢈⣡⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡏⠀⠀⠀⠀⠻⣿⣿⣿⣉⠣⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⣧⠀⠀⠀⠀⢀⠉⠈⠀⠀⠀⠈⠉⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢻⣿⣦⡀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠙⠉⢛⣶⠋⣴⣄⡀⠀⠀⢐⣶⣆⠀⠘⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⠈⠉⢀⠈⠻⣿⣿⢿⣿⣿⡏⢷⣄⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠒⠏⠙⢷⣦⣤⣬⣿⣿⡟⠀⠀⠙⠻⢶⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⢘⣽⣿⣯⠟⠀⠀⠀⠀⠀⠀⠻⠶⣶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠀⠀⠰⢯⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠖⠊⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⢠⣄⣀⣄⣠⡄⢶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢺⣿⣿⣿⣡⡜⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣄⣿⣿⣿⣾⣿⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠻⣿⣿⣯⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠉⠉⣿⣿⡏⠀⠀⠀⠀⠀⠘⠻⠆⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣾⣿⡇⠀⠀⠀⠀⠀⣤⣾⠟⠛⣿⣾⣯⣤⣤⡄⠀⠀⠀⠀⠀⠀⠘⠓⠲⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢼⢻⣲⡄⠀⠀⠀⢸⡇⢹⠀⠼⣿⣿⣩⠽⣿⠇⠀⠀⡠⠂⠐⠊⠀⠴⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣯⢼⡯⣬⠇⠀⠀⠀⠀⣇⢸⣤⣾⡏⠉⠠⠞⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⢸⡀⠊⣿⠦
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⢿⣶⠀⠀⠀⠀⠀⠸⡘⢇⣸⢿⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠀⣮⢀⡆⠋⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⣟⠉⠀⠀⠀⠀⠀⠀⢻⡿⢿⣭⣇⡃⠀⠀⠀⠀⠀⢀⠀⠠⠂⣷⣾⡇⢀⠃⣼⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣸⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⣿⢻⡇⠀⠀⠀⠀⠀⠈⢀⣼⣆⠸⣿⠄⢀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣿⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣿⣶⡇⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⢿⡔⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡜⢿⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⣦⣻⠃
⠀⠀⠀⠀⠀⠀⡔⣴⡔⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠛⣇⡫⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠉⠛⢳⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠭⠀⠘⠿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⢀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠉⠁⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠈⠉⠉⠀⠘⠦⠄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠀⠀⠋⠀⣀⣀⠀⠁⠀⣀⠀⢀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠂⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣆⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⣿⡟⠷⠤⠤⠖⠋⠈⠓⠲⠶⠶⠾⠏⢡⣧⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡏⢹⡀⠀⠒⠒⢀⠀⠛⠛⠉⠀⠀⢀⡞⠙⢦⡀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠂⢀⡼⠁⢸⡀⠀⠒⠀⠈⠠⠤⠤⠖⠀⠠⡟⠀⠀⠀⢁⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⠃⢠⠀⠀⠉⠀⠀⠀⢰⠀⢀⡇⠀⠀⠀⠸⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇⡼⠀⠀⣼⣀⢸⡆⠀⢸⠀⠀⢀⡟⣀⣸⣇⠀⠀⠀⠀⠹⣆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡼⠁⠀⣼⠓⠍⡛⠛⠓⣾⣟⣛⠛⣛⠉⠡⢿⡄⠀⠀⠀⠀⠹⡆⠀⢿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡼⠁⠀⠀⠁⠒⠚⠁⠀⠈⢿⠉⠁⠀⠘⠓⠒⠘⡇⠀⠀⠀⠀⠀⠳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡆⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠰⡄⠘⣇⠧⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡗⢠⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠪⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣿⠿⠿⠷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡾⠀⣀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠙⡁⠀⠈⠉⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠂⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣤⣄⣀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡄⠈⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⠁⠀⠀⠀⠈⠀⠂⠱⣸⠃⠻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠢⠀⠽⡀⠀⠀⠿⠢⣄⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⠳⣄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⠀⢢⣤⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠈⣦⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠈⠓⠄⠉⠉⢹⣧⡀⠠⠄⠀⠀⠀⢀⣰⠿⢍⣀⡀⠀⢠⣼⡏⢻⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣿⠻⢿⣿⣿⠿⠛⢋⠉⠀⢠⡤⠈⣉⣿⣿⣿⡇⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠎⠀⠀⠀⠁⢀⣾⣧⡀⢶⣽⣄⡀⠈⢀⡀⠀⢀⡄⢀⣨⢿⡿⣿⣇⠀⠀⢂⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢶⡶⠟⠈⠛⢿⣿⣿⠉⠛⣦⠀⡉⠉⠁⠀⠉⠋⢺⠇⠈⠻⣷⣶⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠹⡄⠀⠠⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⡁⠛⠒⠐⠃⠚⡑⢿⠀⠀⠀⢸⣇⠀⠀⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢐⡆⠀⠀⠀⠀⠀⡿⢦⡀⠀⢇⠀⠀⠠⠀⠀⡇⢸⠀⠀⠀⢸⡇⠦⠀⢀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠳⡀⢰⠎⡀⠀⠀⠀⠀⢸⡟⠀⠿⡅⠀⠁⠀⠀⠀⣰⣃⠨⡆⠀⠀⠸⣇⠀⠁⡜⢱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠑⣌⢰⠃⠀⠀⠀⢀⣾⠖⠀⠀⠄⣶⠄⠀⠀⠀⢩⣱⡆⠵⡀⠀⠀⠙⣇⠀⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⠀⠀⠀⠀⠀⢸⣿⠁⠀⠉⣚⠈⣷⡀⠀⠀⠟⠁⡈⢔⣇⠀⠀⠀⠈⢃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⣼⣯⡂⠀⠀⠉⢀⢏⠳⣄⠀⠀⣠⠁⠀⣸⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠤⠢⢷⡘⣠⠞⠋⠀⠀⠻⡇⠀⢀⠠⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠬⢿⠀⠀⠀⠀⠀⠀⠀⢀⡟⠠⠴⠒⠀⠀⠀⣧⠀⠀⣤⣤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠿⠖⠀⠘⠀⠀⠀⠀⠀⠀⠰⠻⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣦⡀⠀⠈⠷⠿⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠋⢁⣼⠀⠀⠀⠿⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡌⣶⣸⡏⠀⣠⡄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡼⠛⠀⢸⡏
⠀⠀⠒⢢⣄⠤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠐⣷⣀⠀⠀⠀⠀⠀⢀
⠀⠀⡾⠟⠉⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡣⢈⣶⠀⠀⠀⠀⠙
⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣬⣀⣀⠵⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢷⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣀⠤⢤⡀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠐
⠸⠿⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⡄⠁⠀⠈⠁⣠⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣿⠃⠀⣴⠿⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⣴⠁⠈⠋⣔⡛⡀⠈⠛⠛⠛⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⢀⠘⠁⣠⠿⢣⡀⠀⠀⢸⡇
⠀⣠⠄⠸⡇⠀⠀⠀⠀⠀⢀⡟⢀⣠⣶⣛⠁⠘⠷⣦⣤⣀⡁⠀⠀⠋⠀⠀⢀⡀⠀⠈⠁⠘⠀⠰⠋⣠⠞⢉⡤⠶⠻⡇
⠀⠿⠀⠀⠙⠡⡄⠀⠀⢠⡈⢠⡞⠁⢸⣿⣿⣦⡀⢀⠉⣹⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⣴
⠀⠀⣰⡀⠀⠀⠸⣆⡀⠈⠑⠆⠀⠀⢈⣿⣿⡋⠉⠀⠛⠉⠁⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠂⠀⢿⡇⠀⠀⠁
⠀⠀⠛⣁⡄⠀⠀⡘⠃⠀⠀⠂⡀⢀⠙⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⢰⣿⠁⠀⢀⡀⠀⢀
⠀⠀⠈⠃⠐⢷⣄⠙⠃⠀⣰⣾⠷⠌⠀⠀⠘⣿⢄⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠰⡆⠻⠿⠆
⠀⠀⠀⠀⠀⠘⠿⠃⠠⠊⠁⠄⠁⠀⠀⠀⠀⢠⡌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⠀⠀⠀⠀⠀⢠⡀⠀⠀⠶⠀⠐⠀⠀⠀⠀⠐⡿⠀⠀⠀⠀⢸⠇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⠀⠀⠀⠀⠿⠀⠀⡟⠉⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⢿⠏⠀⠀⠀⠀⠁⢶⠀⠀⡀⠺⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡄⠀⠐⢃⡀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠛⠁⠀⠀⠊⢉⡦⠀⢀⣴⡆⢨⡟⠃⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠲⠄⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠀⠘⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢀⣶⠀⠀⢀⣴⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣘⠃⢠⡟⢾⣿⣀⣿⣤⣤⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢠⣼⣿⠃⢸⠇⠀⠙⣿⣧⣀⠀⠀⠀⣧⡀⠀⠀⣹⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⣰⠋⣹⠇⠀⠀⠀⠀⠀⣿⠋⣿⡆⠀⠀⣿⡇⠀⠀⠟⠀⠀⠀⠀⠀⠀⠈⠁⢤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡏⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⢻⡇⠀⠀⣿⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠙⠀⠀⡆⠀⠀⠀⠀⠀⠘⠇⠀⠀⣼⠃⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠸⠀⠀⠀⠀⠀⣦⠀⠀⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢾⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠰⠀⠀⢸⣿⡄⠀⠀⠀⠰⡇⣰⡿⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠰⠂⣀⡀⢸⣿⠁⠀⠀⠀⢸⣟⢹⠇⠀⠠⠆⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠈⠿⠇⠀⠁⠀⠀⠀⠀⠀⠋⣁⡀⠀⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠙⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢰⡇⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⢠⠀⣿⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠂⢠⠀⣾⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠐⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠶⣄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢰⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢰⣦⠀⠠⠄⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣬⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣵⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠀⠀⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣾⣶⣶⣾⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠉⠉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢣⡀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠉⠁⠀⠑⠄⣀⠐⠘⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢞⣫⠭⢍⡁⠀⢀⠀⠀⠀⠀⠀⠤⣄⡀⠀⠠⢤⣝⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠏⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠈⢳⡄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠀⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⣿⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠠⠴⣾⡿⠿⠻⠷⢶⣤⣤⠶⠾⠛⠋⠀⠙⠲⠤⣶⠏⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠠⡶⠚⠉⠉⠉⠙⡏⠀⠈⡿⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⢀⡟⠀⢀⠀⢸⡟⠛⠛⠙⢿⣇⠀⠀⠀⠀⣠⡇⠀⠲⢤⣄⣠⣿⠤⢾⠷⢴⠞⠀⠀⢹⣧
⠀⠀⠀⠀⠀⢸⢇⠀⠀⣷⠀⣇⠀⠀⠀⠈⠻⣧⡀⠀⠀⠙⣦⠀⢀⣀⣀⣴⣏⣠⡏⠀⢸⡇⠠⡆⠀⢿⡄
⠀⠀⠀⠀⠀⢸⡆⠀⠀⡟⢠⡏⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⡇⠀⠀⠀⠀⢸⠇⢸⡇⠀⢸⡄⠀⢳⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠹⡆⠀⠇⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡈⠀⠀⠀⠀⠈⠀⣸⠃⠀⠀⠻⣄⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠈⠉⠓⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠈⠃⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⢚⠀⠀⠀⠀⠀⠀⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⢀⠎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⣀⡀
⠀⠀⠀⠀⠀⠉⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠉⠘⠒⡄
⠀⡘⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⣀⡀⠤⠤⠤⠤⠀⠀⠀⠒⠒⠂⠠⠤⢀⣴⠋⠈⢣⠔⢢⠖⡄⠀⠀⡜⠀⠀⠐⠀⠀⠀⠀⠀⠀⡏
⠀⠇⠀⠀⠀⠀⠐⠀⠀⢳⠀⢠⢀⠈⠀⡅⢰⠁⠸⠃⠁⠀⢀⣀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠤⢄⡄⠬⣆⠀⢚⡀⠃⠼⠃⠒⣶⠁⠀⠀⠀⠀⠀⡀⠀⠀⢸⠃
⠀⠐⠀⠀⠀⠀⠀⠆⠀⢸⠊⠙⣄⠢⠄⠁⠘⠄⢼⠶⠚⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡈⠁⠉⠀⠁⠶⠠⢤⡏⠀⠀⡐⠀⠀⢠⠁⠀⢀⣏⣀
⡠⠒⡂⠀⠀⠀⠀⠈⠀⠘⡖⠉⠀⠀⢰⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣤⠊⠀⠀⢸⠁⠀⢠⠁⠀⠀⠂⠀⠀⡼⠀⠈
⠁⠀⢳⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⡥⡴⠶⠏⠁⠀⠀⠀⠀⢀⠤⠒⠒⠒⠀⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⠀⠀⢀⠇⠀⠀⠀⠀⠀⠈⠀⠀⢀⠟⠲⠖
⠀⠀⠘⡀⠀⢐⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⣠⡴⠂⠀⠐⠀⢄⠑⡀⠀⠀⠀⣸⠀⠀⠀⠘⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢇⠀⠀⠀⠀⢀⣠⠴⠃⠀⠀⠀⠀⠀⠀⢸⠀⢀⡠⠴⠁⠈⢠⠊⠉⢱⠀⠀⠀⠓⢄⠀⠀⠃⠀⠀⠀⡄⠀⠀⠀⠈⠓⠲⠤⠤⢄⣀⣀⡴⠃
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢳⡄⣀⠀⠀⣀⣴⠶⡴⢋⠖⢢⣄⢨⡟⠀⠸⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⢂⡏⠀⠚⠋⠉⠀⠙⠻⢴⣋⠜⠀⡗⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⢃⠄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢃⠝⠛⠁⠀⠀⠀⠈⠛⠄⠖⠉⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⣔⡠⠋⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠈⡘⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠣⡀⠀⠀⠀⠀⠀⢠⠃⢠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢧⡀⣀⣀⡤⠂⠀⠙⠲⠦⠤⠄⠊⠁⠀⠀⠑⠄⡀⠀⣤⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡓⠒⣶⠋⠑⣶⠚⠛⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠉⠙⠦⠐⠋⠉⠉⠀⠈⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣀⣀⠤⠴⠄⣀⡀⠀⠀⠀⠀⠈⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡰⠋⠁⠀⠀⠀⠀⠀⠀⠈⣉⠇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠙⠢⢄⡀⠀⠀⠀⢠⣴⣋⣁⣀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠢⠶⠶⠾⠉⠀⣄⢴⠁⠞⠙⠟⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠤⠤⠠⣤⠀⠴⠊⣀⣀⡀⠀⣀⣀⡤⠼⠙⠄⠀⢀⡤⠤⠤⠾⠀⣠⣤⠀⣀⣀⣀⠀⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣥⣤⣾⣭⣭⣥⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⠆⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣿⠀⢀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⢀⣻⠁⠀⠈⣆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠴⠤⢤⣤⠶⠋⠁⠀⠀⠒⣩⣄⠀⠀⠀⠈⠪⡆⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣨⡉⠀⠀⠀⠀⢿⣖⠁⠀⠀⠀⠈⣲⡿⠀⠀⠠⠐⡄⢻⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣡⣦⣉⢀⠐⠤⣀⠈⠻⢿⣶⠂⠈⣽⣿⡀⠀⠠⠀⡝⠄⠹⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣷⣤⣈⣷⡀⠈⠛⢻⣿⢩⣙⣿⡄⢠⡀⡀⠀⠢⡄⠀⣴⢒⣈⠥⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠦⠾⠿⣿⣿⣿⠿⠾⠿⣄⣠⣼⣿⠀⣽⣿⡀⠀⢻⣧⠀⠀⠹⢄⡇⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠻⡄⠀⠀⠀⢻⡟⠀⢠⣿⣿⠀⠐⠀⠈⢷⡄⠐⡌⣧⠀⠠⠏⣰⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠹⡄⠀⠐⣼⣷⣰⣾⠏⠤⠀⠁⠀⠀⠀⠙⢷⣦⡘⠄⣀⠖⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠙⠂⣶⣿⡟⢿⠃⠀⠒⠀⠁⠀⠀⠀⠀⠀⠈⠙⠆⠐⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⡿⠟⠁⠀⢡⣀⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠁⠀⠀⠀⢤⣻⡇⠀⠁⠀⠠⠖⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣻⡿⠃⠀⠀⠀⠤⢊⣥⠠⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⣀⠘⠟⠃⠀⠀⠀⠀⠖⠋⣶⣦⡄⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠠⣿⣿⠀⠀⠀⠐⠀⠀⠐⠋⠀⠀⠀⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡖⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢹⣿⣾⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠛⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠰⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⢉⠉⠙⢦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠶⠚⢧⠀⠈⠂⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⢠⠞⢃⡀⠈⠛⢦⡜⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠊⠀⠨⣍⣲⠀⢀⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠆⡀⠲⡄⠀⢀⢰⡏⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢱⡀⠁⠀⢤⡟⣠⢾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⠙⣆⠀⠙⠲⠤⣤⡴⠃⠀⠀⠉⠛⠲⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠏⠁⢤⣼⣀⣬⠶⠶⠖⠶⠶⠶⠶⢦⣄⠀⠀⠀⠙⠋⠀⠈⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢠⡎⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢤⡆⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣿⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⡀⠀⢀⡀⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢿⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⡋⠉⠉⠉⠉⠙⣷⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⢤⡿⠃⠐⢂⣛⢉⡑⠲⠶⠶⠶⠶⠶⠶⠶⠞⠋⠁⠀⢷⡀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡌⢻⠀⣰⠋⠀⠀⠈⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠾⠳⣤⡀⠀⠀⢠⠇⠀⠀⠀⢿
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣀⣧⠀⠀⣀⡼⠁⠈⢧⡈⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⢉⣿⠇⠀⠀⣴⠂⠀⢸
⠀⢀⣀⣿⠸⣄⠀⠀⠀⣠⠞⠉⠀⠈⠉⠉⠀⠀⠀⣀⡽⢦⠀⡀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⠞⠁⢻⣀⣰⡇⠀⠀⠘
⣰⠯⠉⣹⠆⠈⢷⡤⠞⠁⠀⢀⣀⣠⠤⠀⠀⢀⡴⠋⠀⠈⣾⣧⠀⠀⠀⠀⠸⠓⢤⡀⠀⠀⢀⣠⣤⠖⠛⠁⠀⠀⠀⢹⣧⡗
⢳⡶⠒⣿⠀⠀⠀⠀⠀⠀⠀⠬⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⣿⢸⠇⠀⠀⣴⠃⠀⠀⠀⠀⠀⢈⣠⠏⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣰
⠀⠙⢇⣼⣇⡰⠾⠶⠖⠒⠒⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⣿⠘⠀⠀⡞⠁⠀⠀⢀⡀⢀⡴⠛⠁⠀⠀⠀⠀⠀⣀⣀⣤⠟⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣄⣤⣤⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⠞⡉⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠐⠿⢊⡿⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢿⣿⣆⠀⠀⠀⣠⠖⣫⣖⣤⣀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠛⠊⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣹⠟⠋⠀⠀⠀⠀⠀⡾⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣀⣤⠴⠞⠉⠀⣠⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡴⠋⠀⠀⠀⣠⠞⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡏⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢷⣄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢷⡄⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢦⡀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⣿⣦⠀⠀⠉⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡏⠘⣆⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⢸⡇⠀⢹⡆⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠈⣧⠀⠈⢿⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠛⠁⣀⣠⣤⣀⣀⡿⠀⢠⡟⢀⣠⡿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣯⣤⣦⣤⣤⣶⣶⣾⣿⣿⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠁⠙⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠚⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣻⣿⠵⡇⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡾⠇⠀⣀⡼⢿⣿⣿⣭⣩⡿⢤⣤⡀⢀⣀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠉⠁⠀⡴⠋⠉⠀⠀⠈⠉⣋⠁⠀⠀⠀⠀⠙⠁⠀⠀⠀⠈⠉⠘⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⡄⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠉⠉⣿⣉⠉⠀⠀⠀⠀⠤⣤⣴⣷⠆⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢿⣾⡇⠟⠙⠁⠻⣟⠁⠀⠀⠀⠀⠈⠛⣿⡅⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠘⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡶⠉⠃⠀⠀⠀⠀⢻⣷⣦⣤⣤⣤⣴⠞⠿⠳⣶⣤⣤⣤⣤⣴⡇⠀⠀⠀⠀⢷⡀⢦⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠆⣤⣼⡄⠘⠀⠀⣠⣾⣿⠟⠛⠃⠈⠁⠀⣶⡄⠼⠀⣙⠁⠸⣿⣿⣄⠀⠀⠀⣹⡇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠈⢹⣿⣶⡶⠛⠁⠈⠙⣇⠀⠘⠃⣄⣀⣴⣄⢠⡜⠉⠀⡖⠉⠀⠀⠙⢳⠶⠋⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⢤⠀⣏⠁⠀⠀⠈⠀⢹⣧⣤⢤⣤⣀⣸⣇⡀⠈⠁⠀⢸⡇⢀⡄⠀⢺⡇⠀⠀⠀⠀⠸⣄⠀⠁⠀⠀⠑
⠀⠀⠀⠀⠀⣠⣌⣉⣧⣾⣴⣿⣾⡄⠋⠠⠀⠠⢦⣴⡿⢋⣶⣾⣧⣼⣿⣍⢡⣤⣷⡀⢠⣄⠸⠇⠀⣀⠙⣦⠀⠀⠀⠈⠳⣿⡶⠀⠀⢀
⠀⠀⠀⠀⢤⣠⣸⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠀⣸⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣜⣍⡿⣉⣼⠷⠠⠀⣉⠣⠘⣧⡀⠀⠀⠀⠀⣷⣴⣇⣨
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡇⢀⠀⠀⢉⢢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⠠⡆⡆⠹⡄⠀⠈⢳⡄⠀⠀⠀⠀⣻⠿⣿
⠀⢘⣃⡀⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠇⣤⣥⢰⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣹⣏⣸⡄⢳⡀⣷⠀⡀⠀⢷⠀⠄⠀⠘⠁⣤⣿
⢰⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡇⠀⠹⣇⣼⠁⠀⢾⠀⠀⠀⠠⠀⠐⠋
⠀⠀⠁⠀⠀⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⢷⣼⠁⠻⣿⠀⠀⠀⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠽⠛⣿⣿⣿⣿⠿⠆⠀⠀⠀⠰⠿⣿⣿⡿⣿⡿⣿⠛⠈⢁⡞⠁⠀⠀⠈⠀⠀⢠⡿⢻⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⢓⡀⣜⠛⠿⠀⠀⠀⠀⠀⠀⠀⠋⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠨⠉⠁⠘⢣⠀⠀⠀⠀⠀⠀⠚⢧⣀⣴⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⡇⢸⡀⢈⡇⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠘⢡⠈⢃⡼⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢸⠀⣿⠁⠀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠈⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⠚⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠛⣙⢻⡋⣉⠛⢛⣟⠋⣿⡿⣻⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⠀⠀⠀⠀⠀⣠⣶⣶⣤⡀⠀⢀⣴⣶⣶⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠈⠻⠛⠁⠹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠂⠀⠀⠀⢀⡀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣷⣦⣤⣤⣄⠀⣠⣿⣿⣿⣿⣯⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠀⠈⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⠀⠘⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠈⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣯⣭⣧⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣶⣶⣿⣿⣷⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡄⠀⠀⢀⠀⢀⠀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠤⠐⢊⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⣶⣾⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢦⣼⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⡇⠈⠉⠛⠛⣻⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⣼⢫⣿⣿⡿⠋⠁⠀⠀⠀⠈⢻⣿⣿⣿⣿⠟⣥⣄⣀⣀⣀⣻⡁⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⢸⣿⣿⣿⣿⡿⠁⢀⣾⠃⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠁⠀⠈⠛⠉⣿⣿⣿⠿⠂⣿⣿⡆⠙⢿⣿⣿⣿⣿⠟⣻⠋
⠀⢸⣿⣿⣿⡟⠀⡴⢋⡎⣼⣿⣿⣿⡞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣽⣿⣿⢆⣴⣿⠍⠀⠀⠈⠻⣿⠟⣡⠺⠋
⠀⢸⣿⣿⣿⠀⠌⠀⣾⢡⣿⣿⣿⣿⣿⣼⡇⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⣷⠞⢻⣿⣿⣷⣿⣿⡇⠁⠀⠀⠀⠀⠀⠚⠂⠁
⠀⢸⣿⣿⡇⡼⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⢶⣖⣛⣛⣻⣷⣤⣊⣥⣾⣇⣹⣿⣿⣿⣿⣿⢣
⠀⢸⣿⣿⣿⡇⠀⢸⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣉⣛⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⡟⣼⣤⡀
⠀⠸⣿⣿⣿⣧⣀⠈⠯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠉⠉⠉⠛⠛⠿⣷⡻⢹⣿⣦⡀
⠀⠀⢝⢿⣿⣿⣄⠀⢐⡲⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠙⠇⣄⠉⠻⢟⠂⣤⣶⣶⣄
⠀⠀⠀⠑⢝⣿⣿⣷⣤⣈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠈⣎⢷⡄⠀⠁⠈⠻⣝⣿⣄
⠀⢠⠀⠀⠀⠊⡻⣿⣿⣿⣿⣷⣶⣶⣖⠉⠿⠿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣿⣼⣿⣆⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠑⣄⠀⠀⠈⠪⣻⣿⣿⣿⣿⣿⣿⣷⡆⠐⠛⠁⠀⢀⣿⣿⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣄⡀⠀⣤⣿⣿⣿⡇
⠀⢰⣶⡌⠳⡄⠀⠀⠐⢝⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⡟⣻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⢸⣿⣿⣦⡘⢦⡀⠀⠀⠑⢝⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⡿⠛⠛⡉⢳⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢸⣿⣿⣿⣷⣄⠑⣄⠀⠀⠀⠪⡻⣿⠟⠀⠹⡄⠀⢸⣿⠀⠀⠀⣝⣿⡟⢷⣿⣿⣿⠿⠿⢿⣾⣿⣷⣝⣿⣿⣿⣿⣿⣣⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⢸⣿⣿⣿⣿⣿⣷⡈⠳⡀⠀⠀⠈⠀⠀⠀⠀⠙⢦⣻⣿⠳⢤⣀⣽⠿⣄⣻⣿⣿⣿⣿⣿⣦⣿⣟⠻⣿⡟⣸⣿⣿⣿⡟⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠉⠀⠹⣿⣿⣿⡿⠿⠿⣿⣿⣿⣷⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢢⡀⢀⡀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠳⣿⣿⣿⣿⣿⣷⣼⣿⣿⣻⡟⢿⣿⣿⣿⣿⡓⠀⢀⣼⣿⣿⣿⣿⣿⣿⡅
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⢦⡙⣆⠀⠀⠀⠀⠀⢸⡿⣄⡀⠀⠀⣿⣿⣿⣿⡿⣿⣿⡿⠿⢹⡀⠀⢾⣿⣿⣿⣿⣆⡀⢹⣿⡿⠿⢿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠳⣌⠳⣄⠀⠀⢀⡼⠧⣌⣻⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣷⣧⣾⣿⣿⣿⣿⡟⣴⢉⡰⣬⡛⢮⣻⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⢷⣌⢷⣴⢯⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⠙⠿⢿⣿⣿⣿⣾⣿⣿⣎⣿⣶⡟⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠙⢶⡟⠈⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⡀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣄⢤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠘⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣶⣦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠛⠁⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢩⠈⠿⠀⠀⠚⠟⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠑⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢄⡀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⡀⠀⠀⠀⠙⣷⣶⡒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⣠⠊⣼⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠯⠤⡀⠀⠀⠀⠀⠀⠀⢸⡵⠠⠛⠿⢿⠿⠷⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠐⠸⠃⠐⠈⠉⠉⠑⠄⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⢤⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡾⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠁⠀⠀⠀⢸⣿⣦⡀⠀⠀⣀⠀⠀⠀⢀⡼⡀⠀⠀⠀⣀⡀⣼⡁⠸⡀
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢸⡇⠀⠙⠒⠦⠤⠤⠤⠖⠋⠀⠙⠲⠤⣤⣭⣾⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡿⣷⡀⠀⣀⠀⠢⡀⠀⠀⠀⢠⠀⠀⠀⣠⠀⣸⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡼⠁⠈⠳⡄⢈⠁⠠⠜⠀⠀⠐⠉⠐⠒⠚⢿⣆⣿⣿⣇⣰⢣
⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⣤⠞⠁⠀⠀⠀⢹⡄⠀⠀⠀⡀⠀⠀⠀⡇⠀⢠⡿⡏⣿⣿⠟⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⠑⠒⠋⠑⠒⠋⠀⠁⠈⠉⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠒⠀⠒⠚⠈⠀⢨⣿⣿⣿⣷⣎⠉⠉⠒⠂⠈⠁⠀⠀⠉⠉⠐⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢿⡻⣷⡀⠀⠀⠀⠀⠀⠈⠳⡶⢃⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣐⠲⣷⣤⠀⠠⠴⠘⡓⠾⢿⡎⣉⠈⣁⡉⠻⢒⣃⠀⢀⣀⣤⠾⡁⠈⠹⠁⠀⢠⣸⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣸⣷⣄⣹⣶⠀⠀⠀⣡⠍⠑⠀⠳⣄⣀⣼⠁⠂⠉⠁⡀⠀⣀⣀⣼⣷⣶⣇⠀⠀⣼⣷⣷
⠀⠀⠀⠀⠀⠀⠀⣇⢿⡾⠟⠉⢿⣿⠛⠻⣿⣿⠋⠀⠀⠀⠀⠙⢿⡃⠀⠀⠀⠀⠹⣿⣿⡿⠉⠹⣿⣿⣶⠀⠁⡟⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⣤⣴⠿⠃⠀⠀⠙⣟⠶⣤⣀⣀⣀⣠⡼⢦⣀⣀⣀⣠⣾⣯⡟⠁⠀⠀⠙⢿⣿⠀⡄⠙⣇
⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠙⠤⡀⠀⠀⠀⠀⠈⠑⢺⣿⢹⣯⠀⠰⡆⠀⢀⣾⣾⡟⠉⠀⠀⠀⠀⠀⣀⠟⡀⠇⡄⣹
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⢀⣿⣽⣿⡀⠀⡇⠀⣨⣯⣿⣧⠀⠀⠀⠀⠀⢸⠄⠀⠙⠃⠀⣹
⠀⠀⠀⠀⠀⠀⢘⡗⠀⠀⠛⢳⣾⣷⡆⠀⠀⠀⡠⠎⠈⣻⣿⣷⣶⣷⣶⣿⠋⠉⠛⢇⠀⠀⠀⣶⣿⣶⠲⠖⠀⠀⣿
⠀⠀⠀⢰⡟⣋⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⠊⠀⠀⠰⣾⣿⣿⣿⣿⡏⡇⠀⠀⠀⠀⠣⡀⢰⣿⣿⣿⣷⣶⣶⣾⣿⣛⠛⣶
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡇⠀⠀⢀⡞⣻⢛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣇⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⡿⠛⣿⠋⢹⡍⠀⣷⠀⢸⣇⣿⣛⣛⣛⣉⣉⣭⣭⣭⣭⣭⣭⣭⣍⣛⣛⣿⣄⡿⠀⢸⠇⠉⣿⠙⢻⡟⠿⣿⠁
⠀⠀⢈⢡⣬⣇⠀⢹⡀⣈⣧⣠⣼⣤⣾⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣦⣿⣤⣸⣇⡀⣿⠀⢰⣯⣍⢱
⠀⠀⠘⠿⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠿⠿⠇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣭⣴⣮⣷⡀⠀⠀⠀⠀⢠⡏⠁⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⣾⣧⣤⣭⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠉⠋⠉⠉⠁⠀⠀⠀⠀⢸⣿⣷⣧⣷⣾⣿⡇⠀⠀⠀⠀⠈⠉⠉⠙⠋⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⡟⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⢹⡄
⠀⠀⠀⠀⠀⢸⣠⣿⣿⣿⣿⣾⠟⠓⢦⣤⡶⠶⢤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣶⣶⣤⡴⠞⠛⣷⣿⣿⣿⣿⣄⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣷⣄⠀⠀⠈⠉⢿⣿⣿⣿⣿⠟⠋⠉⠁⠀⣠⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⠇
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣾⣟⣿⣿⣻⣶⣦⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⠿⠋⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣬⣤⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⣸⠇⡅⢀⣼⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣳⡿⠀⢥⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⠄⠈⢉⡠⠿⠃⠀⠘⣧⣿⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠉⠉⣿⣯⡥⠀⠀⠀⢶⣿⠃⠀⡀⢠⢹⣄⡳⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣒⡤⢄⣈⠛⢷⣶⡤⠄⢺⣿⡄⠀⡀⢘⢦⢻⣇⠀⠀⠀⠉⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣈⣷⡀⠽⠿⣿⣟⣛⣿⡄⢀⢈⠃⠳⣍⠀⣴⣶⣚⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⡿⠿⠿⣄⣤⣿⡿⣽⣿⡄⠈⣿⣇⠀⠹⢦⡇⠀⢀⡾⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⣧⠀⠀⠈⢿⠋⢀⣿⢿⠅⡂⠀⠙⢦⡈⡞⣧⠀⠘⣡⣯⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣬⢧⣀⣘⣿⣷⣿⠏⢹⠀⠁⠀⠀⠀⠛⠿⣜⠂⢚⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣬⣿⡿⠏⢫⠀⡜⠀⠀⠀⠀⢀⡀⠀⠈⠡⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⢀⣿⡷⠀⠀⠀⣠⠴⣊⠉⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣺⣿⠷⠀⠀⢀⠤⢺⡁⢀⠁⡁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⣀⢿⠋⠀⠀⠀⠠⠞⣿⣦⣤⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢠⣾⣿⠇⠀⠀⢘⠀⠀⠚⠁⠶⠀⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⠀⠀⠀⠠⠀⠀⠀⠀⠅⢈⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣼⣿⡿⠁⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⢻⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠁⠁⢀⠀⠀⠀⠈⠀⠀⠀⡘⢁⠔⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⡄⠀⠈⠀⠀⠐⢀⠏⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⣤⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠠⠀⠄⠄⠠⠀⠀⠄⠄⠠⠠⠀⠤⠄⠤⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠠⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⣤⠀⠀⣀⡀⡀
⠀⠀⠀⠀⡇⠒⠀⠀⠀⠙⠀⠀⠀⠀⠋⠀⠀⠀⠒⢸⠀⠀⠀⠀⠀⢸⠐⠀⠀⠀⠈⠃⣼⣿⣇⠘⠁⠀⠀⠀⠂⡇
⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠐⠒⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿⣿⣿⣿⣿⡇⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠸⠿⠿⠿⠿⠿⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠘⠃⠀⣿⣿⣿⣿⣿⡇⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⣤⡀⠀⣤⠀⢠⣤⣤⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡉⠉⢉⠉⠉⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠛⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠀⠸⠇⠀⠺⠿⠇



⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠴⠶⠒⠛⠋⠙⠛⠒⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⠚⠛⠙⠛⠓⠲⠶⠄
⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣍⣿⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠶⣿⠀⣸⠷⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣿⢿⣝⣃⠈⣴⡿⢿⣿⡏⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠃⣸⡆⢸⡆⠛⢻⣏⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣽⡿⣿⣴⣮⣿⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣶⣶⣇⣷⣾⣿⣶⣾⣟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⣿⣿⡹⠟⢳⠛⢿⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⣿⡟⢻⡇⣸⠛⣿⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢸⠏⣿⣿⠀⢹⠂⢺⣿⠋⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢸⠃⣿⣿⠞⠙⠙⢾⣿⡇⢻⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡼⠀⣿⢽⡉⠸⠁⣙⢽⠀⢰⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡾⠀⣿⣿⠀⠀⠀⢸⣿⡇⠘⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠃⠀⣿⡿⡁⠉⠀⠁⢿⡇⠀⢻⣍⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠁⠀⣻⣧⣄⠀⢀⣠⣽⡇⠀⠹⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠁⠀⢸⣿⣄⢳⣴⣶⢀⣾⣇⠀⠀⠻⠧⠀⠀⠀⠀⠀⠀⢸⢻⠁⠀⢠⣿⡟⠻⣷⡿⠛⢿⣧⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⡦⣦⡄⠀⠀⣾⠋⢻⣿⠉⣻⡿⠋⢿⡀⠀⠀⡄⠐⠆⠀⠀⠀⣴⢆⡄⠀⠀⢈⣿⣄⣠⣿⣷⣀⣼⡏⠀⠀⠀⣦⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⡄⢻⡇⣿⢁⢠⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⢾⣼⣿⣿⡏⣿⣿⣯⣸⡇⠀⠀⠀⠚⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣿⣸⠃⣿⣾⣄⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡟⣿⣿⡇⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠹⠀⢸⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠹⡇⠀⢸⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠸⠃⠀⠀⠇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠄⠀⠀⠸⣿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡄⠀⠀⣦⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⡄⠀⠀⣠⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⢹⡇⠀⠀⣿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⣷⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠁⠀⠀⠏⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢈⠃⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠶⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⠀⠀⠀⢻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢀⠔⠂⠀⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢀⠀⣀⠀⠀⠇⠀⠀⠀⠯⡀⠀⠀⠀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠂⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠔⠋⠈⠲⠖⠂⠀⢰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢰⡄⠀⠀⢀⠡⢀⡆⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠎⠀⠀⢀⠄⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠰⠋⠈⠑⠂⠂⠀⢸⡇⠀⠀⠀⠤⠀⢂⡃⠀⠀⠀⡀⠀⠊⠁⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣀⠤⢄⡀⠀⠄⠀⢠⠞⠀⠀⠰⠀⠀⠀⠈⠀⠀⠀⢰⠇⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠔⠀⢀⡼⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠤⠦⢄⡀⠀⠀⠀⠀⠀⠈⣦⠀⢀⡴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣁⣒⠀⢀⣀⠠⠄⠀⠀⠀⠈⠠⠄⠈⠀⠠⠞⠁⠰⡅⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⠀⠀⠀⢄⠀⠀⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠇⠀⢡⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠲⠶⠤⣄⠀⠀⢘⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⠠⠐⠚⠛⠺⣟⠂⠼⢻⠃⠁⠉⠉⠉⠉⠉⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⡿⠶⠖⠚⠀⢀⣈⣩⣭⠥⠤⠤⠤⠀⠀⠀⢑⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⣸⠁⠀⠀⢀⡥⠞⠉⠀⠀⠀⣀⣀⠤⢔⣢⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠠⠃⠀⢀⡜⢰⠇⠀⠀⣀⣀⠤⢀⣠⡤⠀⠩⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢰⠇⢀⠆⠀⢠⠏⢠⠟⡀⠀⠀⢡⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡉⠲⣤⣤⠀⠀⠒⠚⠉⠁⠀⠉⠁⠀⣸⣠⠏⢀⡴⠃⣠⠏⠀⠈⢢⠀⢸⠇
⠀⡀⠀⢀⠤⠚⠉⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠞⢀⡴⠃⠀⠀⠀⠀⢣⠀⠀⢀
⠀⠀⠠⠯⠤⠤⠤⠤⠄⠒⠒⠒⠀⠈⠉⠉⠉⠛⠒⠲⠶⠶⠤⠤⠭⠭⠬⠿⠋⠀⠀⠀⠀⠀⠀⠤⠷⠶⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢰⠋⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠃⠈⠹⡆⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢲⠀⠀⠀⠀⠀⢀⡀⡀⠀⣠⠷⡦⣤⡾⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠘⠒⠉⠑⠒⠋⠁⠀⠈⠉⠛⠂⠀⠀⠀⠘⠛⠉⠀⠈⠙⠒⠚⠉⠒⠇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⣀⣀⠤⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠦⠤⣀⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⢦⣀⣀⣀⣤⣴⠟⢧⡀⠀⠀⢀⡀⠠⣀⠀⠀⠀⣀⠼⢷⣄⣀⣀⣀⣠⠴⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⡄⢉⢛⣿⠿⠄⠀⠼⠿⣿⠋⠉⢦⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣧⢈⡁⠀⠀⠀⠀⠀⢸⢡⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡘⠋⠁⠀⠀⠀⠈⠛⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⠒⠀⠀⠐⠲⠗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢷⣼⣆⣀⣄⣠⣴⣶⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠘⢻⣿⣿⣿⣿⡿⠛⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⢸⡏⢻⡿⠉⡇⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇⠀⢸⠀⠀⡇⠀⢻⠀⠀⡟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢠⣇⠀⠀⠀⢀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢰⡄⠀⢰⠏⠈⣦⠀⠀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠃⠀⣸⠀⠀⢹⠀⠐⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⠀⣿⠀⠀⢸⡄⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⢰⣏⠀⠀⢸⣧⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⠀⠀⣾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠸⠀⠀⢻⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣠⠃⠀⠀⠘⣦⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠏⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⢶⡶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⣦⣤⣴⣦⣠⣴⣌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠈⠙⠛⠛⠋⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣤⡀⠀⠀⢀⣠⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢄⡈⠛⠋⠉⠛⢋⣠⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⢈⢷⠈⠃⠈⠲⡖⠁⠘⠁⣾⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⢠⣿⠘⣷⡀⠀⢚⡓⠀⢀⣾⠃⣳⡄⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠃⠀⠒⠂⢤⣶⠟⣿⡄⢈⣿⣿⣿⣿⣿⣿⠃⢀⣿⡿⣦⣀⡀⠤⠄⠹⢤⠤⠄⢀
⠀⠀⢀⡤⣾⣿⠃⠀⠀⠀⠀⠀⠀⢀⡄⡟⠋⠉⠉⠘⢿⣿⣿⣿⣿⣿⡇⠚⠛⠃⠻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡄
⠀⠀⡎⢰⣿⣇⢰⡀⠀⠀⠀⠀⣠⣾⡷⠀⠀⠀⠀⠀⠀⠉⢹⡟⠁⠀⠀⠀⠀⠀⣀⣷⣄⠀⠀⠀⠀⠀⢀⠀⠀⣿⡎⢆
⠀⠀⣷⣸⣿⣿⣌⣿⣶⣶⣴⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣀⣼⣇⡤⠄⠀⠀⠀⠀⠀⠙⣿⣦⣀⠀⠀⣠⣾⠀⢠⣿⡇⢸
⠀⢀⠞⢻⠟⠛⠉⢻⡏⠁⠀⠀⠈⠙⣿⡆⠀⠀⠀⠀⠀⠈⢻⠃⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⠿⠿⣿⣿⣯⣴⣿⣿⣣⡞
⢀⠃⢠⡁⢀⡠⠒⠋⠉⠑⠢⢤⣀⣀⣿⣿⣦⣀⠀⠀⢀⣀⣾⣄⣀⡀⠀⠀⠀⣠⣴⣿⠏⠀⠀⠀⢀⡿⠛⠛⠻⡿⠛⠳⡀
⢈⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⢈⣛⣛⣻⣿⠿⠿⠿⠿⠿⠿⣿⣿⡟⠛⣿⣿⣿⣿⣿⡀⢄⣀⠤⠼⠥⢀⣀⣀⣷⠀⠀⢱
⠸⣾⣿⣿⣷⣦⣶⢶⣿⣿⣟⣛⣉⣁⣀⣀⣀⣀⣀⣀⡠⠀⠀⠀⠉⠄⠀⠀⠀⢹⣿⠿⠋⠁⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠹⣿⣿⡉⣙⡉⠉⠉⣉⣭⣥⣤⣴⠶⣚⣋⣉⡁⠀⠀⣠⣴⣾⣶⣶⣄⣹⠖⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⢀⣴
⠀⠀⠹⣿⣷⣌⣙⣛⣿⣿⣿⡿⠛⠉⠉⠉⠁⣀⡠⠶⣿⣿⣿⡿⢉⣀⣀⣀⣀⣀⣩⣽⣿⣿⡷⢶⣶⣶⣿⣿⢿⣿⣿⣿⡿
⠀⠀⠀⠻⡻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣟⣿⣫⣽⣿⣿⣿⣿⣿⣶⣶⡤⡤⠤⠤⠄⠤⠀⠒⠒⠊⠉⢁⣀⠀⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠈⠁⠉⠉⠛⠿⣿⡿⠿⠿⣿⡻⠟⠛⠿⢿⣿⢿⣿⣿⣿⣶⣶⣠⣦⣄⣠⣴⣶⣶⣶⣿⣿⣶⣿⣿⠟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠯⢕⡻⠿⣿⣭⣿⣶⣶⣿⣿⣿⣿⣿⡭⠭⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠪⠭⠭⠋⠫⠭⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠙⠻⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠖⠀⠀⠉⠉⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡼⣿⠀⢿⢶⣤⡄⢠⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢡⠀⠀⠀⠀⢀⠀⢰⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⠀⠀⠀⠠⡤⠟⠟⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠃⠘⣷⣄⡀⠀⠲⠶⠮⢰⣿⣿⢷⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡋⠉⠙⢷⣄⠀⠉⠻⣤⣤⣤⡤⠞⠈⣿⠀⠻⣿⡉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠉⠀⠉⢻⣆⣠⣬⣿⣷⣤⣄⡈⠻⣷⣀⣤⠴⠿⢤⣄⣻⣧⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠀⠙⢷⣼⡿⠁⠀⠀⠀⠈⠙⣿⡀⠀⠀⠘⣿⡀
⠀⢠⣶⣿⣿⣶⣶⣶⣶⣿⠷⠟⠛⠶⣴⣿⠃⠀⣀⣀⣀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣏⡀⠀⠻⢿⣦
⢀⣿⠻⢻⡟⡿⣿⠏⢹⣿⣄⣠⣤⣄⣼⣯⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀⢻⡇⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⣿⣷⣄
⣼⡏⠀⣼⠏⠀⣿⢷⡟⢀⡼⠁⣸⠋⢈⣿⣶⡟⠹⣿⣿⠟⣿⠏⠹⣇⣿⣿⣄⣀⣀⣠⣴⣾⣿⣿⣦⡀⢀⣤⣾⡏⠈⢻⣆
⣿⡇⠀⣿⠀⢸⣿⡞⠀⡞⠀⢠⠃⠀⣼⣇⣿⠁⠀⣾⠀⢸⡟⢠⠀⡿⠛⠋⠙⠛⠛⠛⣫⣿⣿⡿⣿⣿⣿⣿⣿⡆⠀⠈⣿
⣿⣇⡀⣿⣀⢸⣿⣄⠀⡀⠀⠀⠀⠀⣿⣿⣿⠀⢀⡇⠀⣾⡇⣾⢰⡇⠀⠠⣤⣤⣶⣿⣿⣿⡿⠁⠸⣿⢹⠁⠀⣁⠀⠀⣿
⢹⣿⣷⣿⣿⣾⣧⠙⣷⢿⣄⡀⢣⣀⠉⠉⣿⠀⣸⡇⡀⣿⡇⠛⣸⠇⢸⡇⣀⣠⣴⣿⣿⠋⠀⠀⠀⠻⣮⡂⠀⠉⠀⣠⡿
⠀⠛⠿⠿⣿⣿⣿⣿⠏⠀⠈⠙⠛⠛⠛⠛⢿⣿⣿⣷⣿⣿⣇⣠⡟⢀⣼⣷⣈⣩⣽⣿⠇⠀⠀⠀⠀⠀⠈⠛⠷⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⠟⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣂⣀⢠⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡙⠛⠈⠙⢩⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠠⣿⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣭⣿⣵⣧⠂⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⠉⢷⣿⣿⣿⣿⠋⠀⠀⠐⠛⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠔⠚⠉⠉⢿⠿⠁⠀⠀⠀⠀⢄⠀⠀⠈⡑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⣦⣵⣤⠆⡜⣎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣷⣿⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡠⢞⣽⣿⣿⣿⡷⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⠋⢹⣷⣄⣀⣀⣬⠵⠛⠶⣦⣤⣥⣾⣿⣿⣿⠀⠇⡇⠀⢣⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡿⣿⡿⠿⠙⠀⠀⣤⣈⣽⠟⣿⣿⣿⣿⡄⠀⠁⠀⢠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣄⣀⣼⡇⢸⣧⣀⠀⠀⠀⠀⠈⠀⡚⠱⣫⣿⢿⣧⣀⡶⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠘⣿⡟⠀⢸⠸⠉⠉⠉⠴⠒⠚⢛⣿⢿⣿⢿⠈⠀⢻⠁⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⡇⠀⡜⠀⣇⠀⠐⠲⠤⣤⣾⠇⣹⠋⣾⠀⠀⢸⠀⢠⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⢠⡇⠇⠀⢳⡄⢉⠀⠀⠀⡠⢞⡟⢀⣵⣾⣧⡇⠀⠂⠀⠊⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⣧⠀⢀⣾⣷⣤⣷⣦⣤⣤⣾⣦⣿⣿⣿⣿⡇⠀⡀⠀⠴⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⢛⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⠃⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠙⣄⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⡡⢊⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣮⣇⠀⠙⠻⢿⣿⣿⣿⣿⡟⠉⠀⠀⢼⣦⣾⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣼⡇⠛⠀⠀⠀⠀⠈⠛⠿⠛⠀⠀⠀⠀⠋⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠠⡔⠀⠘⢿⣿⣿⣿⣻⣷⣿⣿⣿⣿⣿⣗⣦⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄
⠀⠀⠀⠀⠀⣰⡄⣿⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⡂
⠀⠀⢰⣾⣿⣿⡅⠟⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⠆
⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⣉⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⢄⡀
⠀⠀⢀⣿⡿⠿⠇⠀⠀⢀⣴⣶⣷⣾⣶⣿⣿⣿⣿⣿⣿⣷⠄⠀⠙⠉⠘⠃⠈⠁⠉⠉⠀⠍⢉⡉⠛⠧
⠀⠀⠀⠁⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⢀⣀⣤⣤⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⣼⠛⠁⠀⠠⠼⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡶⣶⣶⣿⣿⣿⣿⣷⣦
⠀⠀⠈⠛⠶⠶⠶⠶⠶⠿⠛⠉⠉⠉⠉⠀⢀⣩⣵⣾⣶⣿⣿⣿⣯⣭⣿⣿⣿⣧⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠒⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠿⢿⠀⠀⠀⣀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡑⠦⠀⠀⠉⠉⠉⠟⠛⠉⠁⡀⠀⠀⠀⠠⠤⢳⠀⢘⣩⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠡⠶⠖⠒⠛⠻⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⠶⡿⠿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣢⣧⣾⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠧⣄⡄⣤⠀⣾⠀⣄
⠀⠀⠀⠀⠀⠀⢠⠾⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠇⣶⠄⢠⣶⠀⣿⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠸⡄⠩⢧⡀
⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠨⡀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠛⢳⡀
⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⠀⠊⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⢠⣴⣿⠿⠒⠒⠂⢄⣀⠄⠂⠒⠤⣄⡀⠀⠀⠘⢷⣤⣤⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣇⠀⠀⠼⡆
⠀⠠⣽⣿⣧⡐⣄⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠙⢦⡄⠋⠈⠛⡏⢉⢱⣤⣤⠖⠊⠉⠁⠈⠁⠖⠁⠀⠀⠈⠙⠳⣿⣿⡇
⠀⠀⠀⠈⠙⠇⣈⠉⠢⣤⣤⣶⣿⣦⡀⠀⠀⠀⠀⠈⠀⠢⣄⠁⢀⡩⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⡀⠐⢳⠿⠟
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠿⠿⠿⢿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠸⡄⠈⣻⣷⣶⠶⠾⣆⠲⠶⣶⠾⠿⠃⡿⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣮⠛⠀⡀⠀⠀⠀⠀⡉⠀⠀⢉⡀⠀⢠⠇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠐⠓⠀⠀⢚⡁⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⣾⣄⣀⠾⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠀⠀⡖⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⣦⣤⣤⣤⣤⣶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠉⠉⠈⠉⠉⠀⠈
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠛⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣶⣿⡇⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⡿⢿⠇⠀⠀⠀⠀⠀⠀⠙⣙⡛⠋⠛⢀⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣿⣿⣷⣤⣶⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢷⡙⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣧⠈⠙⠛⠋⠀⠀⠀⣠⣶⣾⣛⣓⠂⢠⣾⣿⣿⣿⣿⣷⣬⣙⣛⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣛⣘⣿⠷⠀⢰⣧⠀⢀⡀⢻⣿⣿⣿⣿⡷⢰⣾⣿⡿⢟⣭⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣶⣍⠰⣿⣿⣾⣿⣿⣿⣿⢃⣍⢩⣴⣾⣿⣿⣿⣿⠍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⡙⢣⣾⣿⠈⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣻⣿⠿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠿⡼⣿⣿⣿⡇⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⣵⣾⣿⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⢀⣜⣿⠿⠏⣼⣿⠏
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣠⣤⣤⡴⢠⣾⣿⡿⣻⣿⣿⣿⣿⣿⣿⡻⠋
⠀⠀⠀⠀⠴⣿⣿⣿⣷⣮⡙⠋⠁⠀⠀⠀⣰⣿⡟⠋⣴⣿⣿⢏⣸⣿⣿⣿⣿⣏⣿⣿⡇
⠀⠀⠀⢀⣶⣮⣿⣿⣿⣿⣟⡂⣀⠀⠀⢴⣿⣿⣧⠘⣿⣿⣿⣿⣷⡻⣿⣿⣿⣿⢸⣿⡗
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣮⣓⠦⣀⠈⣿⣿⡄⢿⣿⣿⣿⣿⡗⠈⣩⣭⣥⣌⣭⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣶⣥⡘⠻⠇⢸⣿⣿⣿⣿⣇⠀⢿⣿⣿⣿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣮⣟⢿⣿⣷⡈⣉⣉⣉⣘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⢾⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠟⠁⠀⠉⠉⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠁⠀⠉⠛⠲⢤⣄⣀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣄
⠀⠀⠀⣤⠼⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠔⠦⣄⡀⣊⣈⣑⡄
⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠠⠴⠒⣒⣒⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠉⠉⠓⠶⢤⣀
⠀⠀⣴⡣⠂⠀⠀⠀⠀⠀⠀⠠⠖⠛⠋⣉⣉⣁⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡬⢕⡠⠀⠀⠀⠈⠙⠂⢄
⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢉⡠⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠓⢤⠹⠁⠀⠀⢀⣤⣄⠀⠑
⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢋⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⠀⠈⠉⠓⢾⡀⠀⠀⠀⠉⠉
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁
⠀⢿⢳⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠦⢀⡄⣠⠄⣀⠄⡀⣠⠖⠒⠋⠉⠘⠁
⠀⢠⠈⠬⡇⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡤⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⣀⡀
⠀⢸⡄⠀⡇⠋⠀⡖⢠⣠⢀⡠⣠⣤⡠⣀⣤⣤⠖⠖⠖⠐⠉⠙⠻⠿⣅⣱⡛⠛⠂⠀⠒⠦⠴⣶⣂⣈⠩⣙⣻⣻⣶
⠀⠘⣇⠀⠘⢄⠀⠀⠀⠀⠐⠶⠶⠶⠒⠒⠒⠛⠒⠒⠒⠒⠲⢭⣻⡝⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠂
⠀⠀⠘⢷⡀⠀⠑⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠤⠄⢀⣐⣒⣒⣒⣒⣀⣤⢄⣀
⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠈⠉⠐⠒⠂⠀⠀⠀⠐⠒⠒⢒⣂⣉⣭⡤⠴⠶⠒⠛⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠛⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠒⠒⠶⠖⠒⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⡙⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢡⠤⠤⠤⠔⠒⠒⠲⠾⠌⠉⠉⠙⠶⠒⠒⠂⠤⠤⠤⢤⠃
⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⢀⣀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠄⣀⡀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⢦⣀⣠⠤⡶⠛⢄⡀⠀⣀⠄⢠⣀⡀⣀⡠⠿⣦⢄⣀⡠⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⢀⢉⠿⠛⠂⠀⠛⢿⡁⡀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⡦⠄⠀⠀⠀⢴⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢤⠤⠀⠀⠤⡤⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⣬⣄⣠⣀⣠⣤⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢹⣿⣿⣿⣿⠉⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢸⡇⢻⠃⢻⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢻⠀⠈⠀⢸⠀⠘⠀⠀⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄⠀⢠⠎⢣⡀⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠛⠀⡸⠀⠸⡇⠀⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⠀⡧⠀⠀⣇⢀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠰⣇⠀⢀⡿⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⢸⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠟⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢻⣿⣿⣿⣿⣿⣿⠏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠃⠀⢸⣿⣿⣿⡿⣿⣿⣿⣿⡀⠀⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡼⠋⠐⠚⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠘⢷⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢷⣤⣠⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠷⠒⠒⠢⠴⠂⠉⠉⠙⠛⠳⠄⠀⠀⠀⠸⠖⠊⠉⠉⠐⠦⠖⠒⠒⢾⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣀⣀⠀⠁⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣤⣤⡀⠈⠁⢀⣀⣼⢿⡄⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⣠⣯⣀⠀⠈⠉⠀⣠⣤⠴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠘⣇⢀⡜⠲⠶⣶⣾⡁⠀⣹⣶⣶⠶⠚⢁⢀⡟⠉⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣆⠞⠉⠀⠀⠀⠀⠀⠙⢢⢢⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠸⠶⠀⠀⠀⠀⠐⠾⢆⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⣦⡤⠀⠀⠀⢤⣦⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⣩⡀⠀⡀⠀⢀⣁⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢹⣿⣿⣶⣷⣾⣿⢿⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡀⠀⢸⡿⣿⣿⠿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠃⠀⢸⠃⠘⡏⠀⣿⠀⠀⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠈⠀⠀⡇⠀⠘⠀⠀⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣇⠀⠀⢀⡼⢳⡀⠀⠀⢀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠹⡇⠀⢸⠁⠀⢿⠀⠀⠿⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡿⠀⠀⢸⡀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡤⢀⡿⠀⠀⢸⡇⢠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠼⣧⠀⠀⢸⡿⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢹⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠸⠀⠀⢹⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣤⠁⠀⠀⠀⢣⡄⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠃⠀⠀⠀⠀⠈⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠰⣿⣇⡀⠀⠀⠑⡀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡀⣀⣿⡷⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠤⡶⠋⠉⠛⠿⠷⠦⠂⠀⠀⠀⢰⡏⠙⢛⠃⠛⠃⠀⡐⠀⠀⠀⠀⠢⠶⠿⠟⠋⠉⠳⡦⠄⠀⠀⠁⠢
⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣘⣒⡀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠈⠢
⠀⠀⠀⣰⣟⠀⠀⠀⢀⣠⣾⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⡿⠚⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣦⣀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⣟⢷⣤⠀⠀⢉⡭⠁⠀⠀⠀⠐⢶⡀⠀⠀⠀⢤⣴⡞⠙⠀⠘⣟⠒⠒⠉⠀⠀⠃⠀⣤⠄⠀⠀⠀⣱⠖⠀⠀⠀⠀⠹⡍⠀⠀⢀⣴⢟⡇
⠀⠀⠀⠈⢷⣭⣀⡀⠙⠘⢦⠀⣀⣀⠀⠈⢻⣏⠀⠀⠀⠀⠉⠁⠠⢀⣈⠦⢔⣁⠀⠀⠈⠁⠀⠀⠀⠀⡀⠁⠀⣀⣀⡀⢠⠞⠀⠀⣀⣨⣵⠏
⠀⠀⠀⠀⠀⠻⣷⣬⡀⠀⠈⠑⠚⢛⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠿⢛⣛⠛⠒⠉⠀⠀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣶⣶⣾⣿⣿⣿⡿⠋⠻⣷⣄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠈⠉⠙⣿⣿⣶⣶⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣿⣿⠁⠀⠀⢸⣿⣷⣶⡶⠴⡶⠖⠋⠓⠶⠶⢴⣶⡶⠿⣻⠀⠀⠀⠀⢠⡏⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⢀⢾⣥⡜⠀⣴⠋⢀⡀⢸⣧⣄⠀⠀⠀⠀⠀⣽⣆⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⢈⡷⡜⠀⢠⡿⠛⠉⠀⢠⡍⠉⠳⢦⠀⠀⠀⡴⣏⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣤⣴⡏⠀⠀⣷⣄⡀⠀⢸⣧⣀⠀⠀⡀⠀⠈⣷⣥⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠘⣟⠉⠉⠀⠀⡏⠉⠀⠀⠀⠀⠀⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⣿⡶⣤⡀⢘⣧⣤⡤⢤⠀⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⠀⠡⠀⠀⠀⠀⠀⢀⠀⢠⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢸⣿⠀⠀⢠⠹⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠲⣌⠙⣿⣥⡽⢿⡄⠀⠨⠣⡹⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣦⣀⣽⡗⣻⣏⠘⣾⡀⠘⢆⡏⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢣⠀⠀⢹⡏⢀⣿⠇⠀⠀⠹⣄⠒⣧⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡳⢠⣽⣿⡿⠃⡇⠀⠀⠀⠈⠛⠮⡀⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⠟⠁⠰⣴⡄⠀⠀⠀⡀⠤⠀⠐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢰⣿⠇⠀⠀⡈⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣀⡿⠃⠀⠀⠠⢶⣧⡤⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣴⣿⠇⠀⠀⠅⠀⠉⠐⠀⠜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠀⡀⠀⠀⠀⠀⣄⢡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣄⣀⠀⢀⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⠀⢸⣿⠏⠟⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⢸⣿⡄⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠃⠀⠀⢸⣿⣷⣅⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⠈⢻⣿⣿⣧⠁
⠀⠀⣀⣤⣴⣶⣦⣤⣤⣤⣤⡀⠀⢀⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢷⣄⡘⣿⣿⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⢸⣻⣿⣽⣿⣿⣿⣉⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⢤⡻⠷⣾⣷⣿⡄⢳⣶⣶⡶⣶⣖⢢⣴⣾⣿⠟⠉⠀⢀⡞⢶⡄
⠀⢾⠛⢋⣩⣿⣥⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣿⣿⡇⠈⢻⣿⣿⣾⡷⠶⠞⣉⣽⣿⡿⠛⠁⠀⣠⡖⠁⠀⢺⣿
⠀⠼⠿⣿⣯⣭⡄⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣮⣿⡗⢀⣀⢾⣿⣿⣟⣲⣿⣿⠟⠉⣀⡀⣀⠆⠉⠀⠀⠰⣾⣿
⠀⠀⠀⠈⠙⣿⣧⣤⠀⠀⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⢸⣿⡇⠀⣹⣿⣿⢹⣿⠛⠋⣠⣴⣿⣿⣿⣿⣟⠀⠀⠀⣶⣿⣿
⠀⠀⣤⣤⣼⣿⣯⣁⡀⢀⣀⣀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⣿⣦⣌⢹⣿⣼⣿⣧⣾⣿⣿⢿⣿⣿⠟⢿⣤⣀⢶⣿⣿⣿
⠀⠀⠈⠋⠉⠉⠁⠀⠻⣿⣿⣿⠿⣿⢿⣧⠈⠛⠖⠒⠂⠀⠒⠚⠛⠿⠗⠀⠸⣿⣿⣿⣿⢿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠟⠁⢀⣼⣿⣿⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠄⠀⠈⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣾⢻⣿⣿⠀⣿⣿⣿⣿⠟⠁⠀⣠⢞⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢺⣿⣿⣹⠃⣿⡿⣿⡄⣿⣿⣿⣿⣀⣠⡞⣣⣿⡿⠛⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⢀⡀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⡟⣯⡏⣸⣿⠀⢿⣿⣿⣿⣿⡟⣻⣷⣿⣿⠟⠁⠀⠀⢀⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⢾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⢰⡟⣦⣻⣟⠀⠘⠉⢛⣹⣿⣧⣈⣿⠻⣿⠀⠀⣠⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠶⣍⢻⡿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡁⣾⣿⣿⣼⢿⡏⢰⣄⣴⣿⣿⣧⣼⣿⠃⠀⢿⣤⣼⣿⠟⠛⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⣆⠙⣝⢫⡻⣷⡀⠀⠀⠀⠀⠀⡉⣿⣿⣿⡿⠿⠋⢁⣠⣿⣾⣿⣙⣏⣿⠀⠀⠈⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠈⣆⠈⢦⡹⣎⢻⡄⠀⠀⠀⠀⠋⠿⠟⠉⠀⠀⠛⠋⣉⣶⣿⣛⢿⢹⡇⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠈⠀⠀⠱⡈⠂⠙⣆⠀⠀⠈⠃⢸⡇⠀⠀⢀⢴⣿⣿⣿⠿⣟⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠈⠀⠀⢼⣷⠀⠰⠓⠀⡇⠀⠀⢘⣿⣿⣿⣿⠂⣿⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠃⠀⠀⠀⠀⠀⠀⢠⡈⢿⣧⠸⡇⠀⡇⠀⠀⣋⣹⣿⣿⣿⡆⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠱⠀⠉⢰⡇⢈⡇⣠⣴⠿⢿⣿⡟⢺⣿⣿⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠇⣸⣿⡟⠁⣀⢸⢹⡇⢽⣿⣹⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠰⡆⠀⠀⠀⠀⠀⠀⢠⠀⠁⠈⠇⠀⣿⡇⣾⠃⣸⣿⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⢻⣿⣧⣿⠀⠀⠀⠀⠀⠀⠈⠀⡃⠀⠀⢳⣿⣇⣿⠀⢹⢿⣷⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣼⣧⢾⣿⡿⠛⠇⠀⠀⠀⠀⠀⠀⠀⠹⣄⠰⡄⠙⣿⣯⠀⢨⣿⣿⣗⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣿⣿⡿⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠿⣿⣿⣿⣷⡶⠶⠶⢾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⣀⣿⠟⠋⠳⢱⣿⡄⠀⠀⠀⠀⠈⠀⠀⢀⡜⠁⣼⢿⢝⠻⣦⣄⠀⢙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⡟⠁⠀⠀⠀⠀⠛⢿⣆⠀⠀⠀⠀⠀⠀⣼⡰⠟⠉⠁⠋⠑⠢⢙⢿⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣆⠀⠀⠀⣼⣿⡟⣧⠀⠀⠀⠀⠀⠀⠈⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⢦⣰⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣯⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⢠⣆⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⡿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣤⣠⣮⢷⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⣀⠢⠀⠀⢀⡞⣱⣿⣿⣽⣿⢋⣿⣿⣿⣿⣿⣿⣷⣔⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣿⣿⣧⣡⣶⠏⣴⢫⣿⣿⣻⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋⠉⠋⠉⠉⠉⠙⠛⠉⠉⠉⠙⠋⠉⠋⠙⠛⠛⠉⠉⠉⠛⠉⠙⠋⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⡷⠀⠀⣀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣯⣤⣾⣿⣷⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠿⢿⡟⣿⣿⡇⣿⣿⣿⣿⣿⣿⠈⠛⠙⡉⠙⢽⣿⣻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣥⣶⣦⣤⣬⣿⣿⣴⣿⣿⣿⣿⣿⣿⣏⠀⠀⢨⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡤⣾⣿⠃⢠⣹⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢠⢿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⣿⣿⡂⣧⣿⣷⣄⣬⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⣠⣄⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠳⠈⠟⠋⠉⠁⢉⣉⣉⣩⣿⣿⣿⣯⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡟⠻⠿⢛⣻⠟⠻⣿⣧⡟⠛⠛⢻⣿⣿⣷⣮⡔⠒⠂
⠀⠀⠀⣀⠤⠤⠶⠤⠄⠀⠠⢽⣟⣟⢿⡿⣿⣿⣿⣿⢿⠿⠟⠋⠉⠉⢠⣛⣻⡷⣽⣾⠿⠿⡴⠶⡮⠽⠾⠷⢶⣾⣿⣿⣯⣿⣙⣛⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⠉⢽⢧⠤⠶⣤⡤⠴⠦⢤⠤⠤⠤⠤⠠⠄⠀⠉⠉⠉⠙⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣂⠀⠤⢤⣴⣖⠒⢶⠶⠶⠤⢶⠦⠴⠶⡶⢪⡷⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠉⠉⠈⠉⠉⠁⠀⠈⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⠿⠿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠈⠭⢻⠘⠛⠏⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠼⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⣒⡂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⢻⠀⠠⢀⣤⡤⠔⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⢀⠁⠀⠐⣄⠀⣠⠀⠀⠙⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠒⠃⠀⠀⠈⠉⠁⠀⠙⠛⠒⠺⠷⠷⠚⠛⠃⠉⠉⠘⠓⠂⠢⡄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣦⣀⣀⠀⠀⠀⠀⠶⣿⠒⠂⠀⠀⠀⠲⢼⡤⠤⠀⠀⠀⠠⣴⠀⠀⠀⠠⢴⡃
⠀⠀⠀⠀⠀⠀⠀⠈⡛⠉⠀⠀⠀⠀⠀⢢⡈⢣⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⠀⠶⠷⠾⠿⠶⢶⣤⣤⣼⣧⡤⠄⠀⠰⢾⣇⠀⡀⠀⠀⠀⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣦⣭⠗⠋⠁⠀⠀⠀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠒⠚⠋⠀⠀⠀⣀⣤⠴⠶⢛⣭⡶⠛⠛⠉⠀⠀⠤⢤⣤⣴⠾⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠀⠰⠴⢊⡀⣀⣀⣀⣠⣴⡿⠟⠁⠀⠀⠀⠀⣀⣀⣤⠴⠒⣀⣤⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠴⣶⣿⣿⣿⣿⣿⠟⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⣿⣷⣤⣤⣶⣿⠶⢦⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢸⣿⡏⠀⠈⢻⠀⠀⠀⣼⠁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄⠀⢻⣿⠀⠀⠀⠀⠀⢀⡇⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠙⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢰⣶⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⠄⠿⣿⣻⣯⣿⣿⣷⣦⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⠿⠃⠙⣿⡿⣻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣏⠀⠀⠀⣽⡿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⢚⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠻⡿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡀⠀⠀⠀⠀⠠⠀⡰⠀⠀⢀⠀⠚⣿⣿⣿⣿⣯⣾⣿⠀⠀⠀⠀⠀⢻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣿⠀⠀⠀⠀⠀⡇⠈⠀⠁⢀⠈⠻⣴⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣠⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠂⠀⠀⠀⣷⠀⠀⠀⠘⠇⠀⠸⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⢴⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠚⢷⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⣻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢀⠀⠀⠀⠀⢸⣿⡟⣿⣯⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠐⠀⠀⠀⠈⠿⣿⣾⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠟⠿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⡣⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⠋⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣐⠋⠐⢿⠀⠐⢉⠭⠓⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡐⢞⣻⣿⣠⣀⠈⠀⡄⠊⣶⣄⡀⠠⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣣⣼⣿⠞⢻⠁⠀⢰⠀⣹⣿⣿⡦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣹⠃⠀⡿⠋⠀⠀⣿⠀⣴⠇⣠⣽⣿⡟⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠏⠀⠘⣿⡧⠀⠀⣧⣼⡏⢴⠤⣾⣿⠃⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣴⠏⠀⠀⠀⣧⣇⢀⣴⣿⠁⢀⣤⣴⣿⡟⠀⠀⠀⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣤⣴⠏⠀⠀⠀⢠⣸⠟⠋⢹⡿⠀⣸⠉⣽⡿⠀⠀⠀⠀⢱⡀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⡇⢸⣿⡇⠀⠀⠀⠀⠈⣿⣷⠿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣾⡿⠃⠀⠀⠀⠀⠀⠀⢸⠇⠀⡸⢠⡟⠀⠘⣿⡇⠀⠀⠀⠀⠀⠘⣿⡠⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠊⡱⠋⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⠀⣧⠊⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠈⢷⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠀⠀⠐⣿⣤⣉⣻⡷⠍⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡇
⠀⠀⠀⠀⠐⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠁⠈⢻⣿⣿⡇⠀⣄⣀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⠁⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⢿⣿⠀⠀⠀⢻⣿⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⢸⡏⠀⠀⠀⠘⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⠀⠀⠀⠀⠸⣿⣷⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣿⠀⠀⠀⠀⠀⢟⢹⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢩⡇⠀⠀⠀⠀⠀⠘⢶⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣾⣇⠀⠀⠀⠀⠀⠀⠸⡆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⣿⡇⠀⠀⠀⠀⠀⠹⡿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⠀⠀⠀⠀⠀⠀⠀⢣⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡏⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠚⠛⠛⠉⠁⠀⠀⠀⠀⠈⠋⠛⠒⠶⠦⠄⠶⠖⠒⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡠⠒⠖⠉⠉⠊⠑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⠊⠉⠑⠒⠢
⡇⠀⠠⠀⠀⠐⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠐⠀⠀⢠
⠛⠀⠊⠐⠲⠓⠄⠀⢨⠀⠀⠀⠀⠀⢀⣀⡀⢀⠤⠒⠒⠒⠂⢄⣠⠁⠴⠶⠀⠰⠶⠄⢩⣀⠄⠒⠒⠒⠢⢄⡀⣀⣀⠀⠀⠀⠀⠀⢨⠀⠀⠔⠳⠒⠈⠂⠘
⠘⣄⠀⠀⠀⠀⠀⢤⠊⢀⣀⣀⡠⠊⠁⠀⠈⠣⠀⠀⠀⠀⠀⠂⢱⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠠⠋⠀⠀⠉⠢⣀⣀⣀⠈⠢⠄⠀⠀⠀⠀⢀⡔
⠀⢸⠀⠀⠀⠀⠀⠀⠓⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠃⠀⠀⠀⠀⠀⢸
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠦⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠜⠂⠤⠤⠤⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠠⠤⠤⠄⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢲⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣠⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠀⠀⠉⠉⠹⣿⡿⠛⠛⢄⠀⠀⠀⠀⠀⠀⠀⣠⠀⢢⡀⠀⠀⠀⠀⠀⠀⢀⠞⠛⠻⣿⡿⠉⠉⠁⠀⠈⠉⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⡀⠀⠑⠢⠤⠤⠤⠔⠊⠁⠀⠀⠉⠒⠤⠤⠤⠤⠒⠁⠀⣀⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠈⠉⠍⡉⠀⠀⠀⠈⡉⠍⠉⠁⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢰⠈⠉⠉⠀⠀⠀⠈⠉⠉⢱⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠘⠲⣄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣧⣤⣤⣤⣤⣤⣤⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣣⣤⣤⣤⣤⣤⣤⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⢀⣄⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣾⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠉⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠈⢃⡀⡤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣦⣤⣤⣶⣰
⠀⠀⠀⠤⣤⣴⠁⠀⠀⣀⠀⠀⠀⠀⣴⠁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⡄
⠀⢰⣤⣀⠀⠉⠀⠀⠀⠁⠀⠈⠀⢴⡼⠋⠁⠀⠀⠜⢒⠂⠀⠀⠀⢀⣴⠶⠒⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⠷⠁
⠀⠀⠙⢿⣦⣄⣀⣴⣦⡅⠀⠀⠀⢠⠀⠀⠀⠀⠀⡴⠿⣄⣀⣀⣀⡸⠃⠀⠀⠀⠀⠀⠉⠹⠿⢿⡿⠿⠋
⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⠀⠀⠀⠘⠷⠀⠀⠀⠸⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡈⠓⠁⠀⠀⠤⠄⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⡁⠀⠀⠀⠀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠔

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣭⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠉⠈⠁
⠀⣊⣼⣷⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡼⢣⡄⠰⣷⣄⣰
⠀⠀⠘⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠿⣦⣯⣩⣿⣟⡽⠃
⠀⠀⠀⠀⠻⣿⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣹⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣷⣤⣤⣤⣴⣶⣶⣦⣶⣿⣟⣿⣶⡟⣿⠿⣿⣿⣿⡿⠟⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⢿⣾⣿⣿⣿⣿⣿⣿⣘⣿⣿⢦⣿⠟⠁⠴⢶⠿⠛⢳⣾⡟⢛⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠿⠿⠿⠿⠿⠿⡿⠷⢼⣿⡿⡦⠃⢰⡿⢶⣾⣛⣧⣬⣿⣵⣮⣤⢸⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡏⣌⠋⢑⣾⢿⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⣼⣁⣙⣛⣿⣿⣣⣽⣷⣿⣿⠙⢯⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢱⠺⡟⢏⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠈⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⠀⣿⣄⠠⡴⠶⢾⣿⣿⣿⡟⠁⠸⣶⣿⣿⢿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⠐⢿⣿⣖⢒⣲⣯⡿⡿⠋⠀⠀⠀⠈⢻⣿⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣿⠀⠘⢻⣿⣷⣿⡟⠠⠇⠀⠀⠀⠀⠀⠀⢽⣠⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⠀⢺⠋⣿⣿⣿⣧⣜⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠟⠠⣦⣿⡿⢟⠋⠑⠏⣆⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⢚⡛⣼⡇⢀⡰⣿⣟⡴⢦⣴⡏⠐⣿⡆⠀⠀⠀⠀⠀⢹⣿⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣋⣡⣴⣾⣿⣿⠏⣴⠢⠀⠐⢻⠆⢻⣿⡇⢰⢿⣷⠀⠀⠀⠀⠀⢸⠟⢰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡟⠉⢻⣽⡿⠁⠄⠀⠀⠀⠀⠊⠀⠘⣥⠁⠀⢸⣿⠀⠀⠀⠀⠀⠿⠆⠘⠄
⠀⠀⠀⠀⠀⣀⢾⣿⣿⣿⣿⣿⣿⣷⣴⡿⢿⡾⢷⣶⡟⠲⣶⡆⠀⣎⢳⡏⠀⠘⣿⠃⠀⠀⠀⠀⠘⢀⡀⣤⢂
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⣟⡽⢿⣿⣷⠌⠛⠉⠀⠀⢀⣿⢀⣰⠟⠉⠁⠀⣼⠃⠀⠀⠀⠀⠀⠰⠋⠙⠾⢻
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣥⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠸⣿⢸⡇⠀⠀⢸⣦⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠉⡛⣯⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⡴⠀⣸⣯⠇
⠀⠀⠀⠀⠀⢿⠹⣿⣟⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢀⣿⣿
⠀⠀⠀⠀⠀⠸⡆⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⣿⣾⡿⣿⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢳⣼⣿⣶⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⡟⢻⢧⣿⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠛⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⣾⢺⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⣏⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣡⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⣠⡿⣜⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⣀⢤⣴⣎⣭⣾⣿⣿⡿⢄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⡇
⣤⣻⣵⣶⠿⠟⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⠁
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠉⡯
⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠲⣌⣸⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠓⠛⣽⣋⠽⠿⠿⣦⣤⠤⠤⠤⠤⢤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠂⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⢀⣠⡆⠉⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣿⣿⣼⠈⣿⣇⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⠘⣯⡿⠟⢃⣸⣿⣿⣧⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣾⠌⠳⣶⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢋⣾⣿⠆⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠻⣿⡿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣯⡉⠠⢇⣿⣿⣿⣿⣿⣄⡘⠳⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡀⢈⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣨⣷⣯⠻⠓⠺⣯⡙⢿⣶⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠚⠛⠘⠛⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣿⣿⣿⡒⢷⠀⠀⠐⠍⢩⣷⣽⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣠⣰⣄⠀⠀⢻⠿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⠻⠀⠀⠘⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣦⠤⡅⢹⣤⣴⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣏⣿⣿⣾⣿⣿⣿⣧⠒⠾⠞⢿⣿⣿⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣟⠟⣿⣿⣿⣇⡀⢠⣸⣿⣿⣧⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠢⠦⣿⣿⣿⣿⣷⣄⠻⠿⣏⣽⣾⡝⠙⠙⣜⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡼⠐⠲⠤⠈⣽⣯⣉⣉⣳⣾⣿⣷⣃⣙⣆⢤⣝⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠙⠛⠛⠿⣿⣷⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣠⣄⣀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠙⠛⠛⠛⠛⠒⠚⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢹⣤⣾⠀⢸⡆⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⡀⠀⢀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠙⣿⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣦⣄⣙⠅⡸⠂⢐⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣁⣀⡀
⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡿⡀⢀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁
⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢏⠔⠋⣿⣿⣿⡟⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢃⣆
⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⠈⠑⠄⠀⠀⠀⢀⠀⠘⠁⠀⢰⣿⣿⠀⢫⡀⠘⠦⢀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠈⠉⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠙⢿⣿⣿⣦⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠈⣦⣄⣠⠠⠄⡠⠂⢀⠀⢀⣇⣾⡏⢹⡆⠀⠙⠲⠄⠀⠀⠀⠀⠉⢦⠂⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠉⠛⠂
⠀⠀⠀⠀⠀⠸⣿⣟⡛⠛⠉⠉⠀⢿⣆⠀⠀⠔⠒⠉⠁⠀⠀⠀⠀⠀⠈⠀⣼⣿⣿⠃⠘⣿⡄⠀⠀⠀⠀⠠⣤⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⣇⠀⠈⠻⣿⡇⠀⠀⠀⠀⢠⡆⠀⠀⡰⠁⢠⣿⣿⡿⠀⠀⣿⡏⡄⠀⠀⠀⠀⠙⠿⣿⣿⠇⠀⠀⠀⠀⢀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⣷⣹⣆⣆⠀⢿⣧⣠⡄⠀⠀⣸⡇⢀⡜⠁⣠⣿⣿⣿⠇⠀⠀⢸⣿⡸⣦⣀⠀⠀⠀⠀⠙⣿⡆⠀⣀⣴⣾⣿⣿⠿⠃⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⣄⠻⣿⣇⠀⢰⣿⣷⣾⣶⣿⣿⣿⣿⡟⠀⠀⠀⠀⢿⣧⡹⣿⣿⣯⠄⠀⠀⣸⢧⠾⠟⠛⢛⣭⠗⠀⠀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠘⢿⣷⣌⠻⣿⣶⠶⠞⠁⠀⠀⠀⢰⡿⠁⠀⣀⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣀⠀⠀⡀⢀⡴⣫⢀⣠⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣿⣟⡵⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⢿⢿⣻⡿⡿⡿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣟⣛⣻⣿⣛⣛⣟⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⢰⡆⡀⢀⣰⡗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⣿⠀⢇⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠊⢁⣿⠇⠀⠸⣆⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠒⠒⢰⣶⣞⣉⠀⠀⢘⣭⣷⡆⠀⠀⠉⢻⡌⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣄⠀⠀⠻⢿⣦⣀⣀⣂⣾⣿⡀⠀⠃⠸⡄⢷⣇⠀⠐⠀⠐⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣍⡙⠲⣄⠉⢿⣯⣴⡿⢿⣧⠀⠁⠐⠿⢦⡟⠀⢀⣀⡠⠖⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣾⣿⣮⠀⢘⣿⡟⢻⣿⠇⠘⣾⡆⠀⠻⡄⡾⠃⠚⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣟⢿⡉⠉⠉⠳⣿⠟⢃⣿⣿⠇⠀⠛⢻⣄⠀⣝⣗⠀⢰⠟⣠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢷⡀⠀⢢⣿⣄⣴⡟⢭⠁⠂⠀⠀⠙⢶⣾⡹⣆⣀⠶⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⢲⣾⣿⠿⡟⠁⠸⠀⠀⠀⠀⠀⠀⠈⠛⠦⢴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⡿⠋⠁⠸⣄⣇⠀⡀⠀⠀⠀⡠⠤⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⣰⣿⣏⠀⠀⠀⠔⣻⡶⠀⠄⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⣊⣵⠖⠀⢄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣦⣦⡝⠁⠀⢠⠀⠘⢉⡿⠟⠒⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣰⣿⡿⠃⠀⠀⠠⠀⠀⠁⠀⠘⢠⠎⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⠃⠀⠀⠀⠔⠀⠀⠀⠀⠂⠘⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡾⠿⡛⠀⠀⠀⠀⢈⠀⠀⠀⢀⡆⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠢⡀⠀⠀⢀⠀⠀⠀⠜⣠⠎⠁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡆⠀⠈⠀⠀⠈⢀⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠧⠀⠀⠀⠀⠷⠈⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⡠⠊⠀⣨⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠠⠔⠂⠒⠀⠀⠀⠄⠉⠑⠄⠀⠀⠈⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠊⠰⠦⠀⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⣸⢆⡀⠀⠁⢀⠀⠄⠀⠀⠀⢠⠊⠁⠘⠀⢀⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠁⠀⠀⠀⠀⡸⠀⢀⠀⠛⠠⠆⠀⠀⣤⢀⣀⣠⠴⠂
⠀⠀⠀⠀⠀⠀⠀⢠⢋⠄⠀⡴⠁⠈⠀⠄⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⢈⠀⠂⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠈⣢⠀⣠⠄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠠⠐⠀⠀⠀⠀⠐⠀⠀⠚⠁⠰⠄⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠐⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⠀⠀⢀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⢒⣒⣄⡀⢸⣦⢀⣀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠔⠈⠀⠀⠉⡇⠈⠹⠀⠀⠀⠀⠀⠀⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢸⠷⠖⠚⠀⣀⣭⠭⠥⠤⠄⠀⠀⢑⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢀⢀⡏⠀⠀⠤⠚⠁⠀⠤⠤⠔⣒⡦⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡘⠀⢠⠎⡼⠁⠀⠖⣲⠶⠛⠁⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡠⠖⠒⣄⣀⠀⠀⠴⠒⠋⠓⠒⠀⣏⡴⠁⡴⢃⡜⠀⠢⡀⢸⠇
⠀⠀⢀⠔⠊⠉⠀⠀⠀⡄⠀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠀⣞⣠⠎⠀⠀⠀⠑⡀⠠
⠀⠀⠓⠒⠒⠒⠂⠉⠉⠁⠀⠀⠈⠉⠉⠉⠙⠛⠛⠓⠒⠛⠁⠀⠀⠀⠀⠀⠙⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⣀⠀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠤⣤⢀⡿⠿⠛⣿⠒⠖⠻⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠈⠁⣷⣾⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⢀⠀⢀⠀⠀⠀⠀⢽⣯⡉⢹⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠿⠿⠦⣤⣤⣤⣤⣴⠾⠟⢻⣮⡡⣼⢾⣿⣿⣶⣤⣷⣿⣿⡿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⡀⠀⠀⠀⣀⣬⡥⠿⣿⣿⣿⣿⣿⢿⠿⠛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉




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