Baby talk Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠉⠉⠉⠑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢆⠀⢸⣄⣠⣇⠀⡤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢈⡷⣚⡥⠳⢄⡀⠈⠉⠳⢎⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠉⠀⠒⢄⠉⡛⠓⠆⢦⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⠀⠀⠀⠀⠀⠀⠈⡎⠀⠀⠀⠈⠙⢦⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⡴⠛⣄⠀⠀⠀⠀⠀⣠⢇⠠⠀⠀⢀⡄⠀⠑⢬⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣥⠞⠀⠀⠈⠳⠤⢤⣤⡞⠡⠌⢑⡒⠒⠋⠀⠀⠀⠈⠳⣼⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡟⠁⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠘⢿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⡀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⡟⠀⢸⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⠃⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠴⠖⠋⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠘⡄⠀⠀⠀⣴⠚⠓⢲⠆⠀⡜⠓
⠀⠀⠀⠀⢼⠀⠀⠀⠃⠀⠀⠶⠯⠤⢉⡆⢸⠀⠀⠀⠀⠀⠀⣿⣷⣶⣤⣀⣀⣀⡂⠐⠂⢐⣀⣀⣀⣤⣶⣿⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢄⠈⠉⠁⠚⣤⠔
⠀⢀⠀⠀⠀⠦⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠻⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢠⣧⣀⠤⣤⡤⠊⠀⢀⡶⠚⠁
⠐⠃⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⢻⢦⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⠏⠀⠉⢦⡸⡆⠀⢀⠞
⠀⠀⣀⡀⣀⡴⠃⠀⠀⠌⠀⠀⠀⠅⠀⠀⠀⡌⠸⠶⢄⡀⠀⠀⠈⠻⢿⣿⡿⠉⠀⠉⠁⠀⠙⣿⣿⡿⠟⠁⠀⠀⢀⣠⣞⣡⠤⠤⠤⠤⢽⣷⠒⠁
⡠⢶⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⡼⠀⠀⠀⠀⠙⡗⠒⠤⣤⣀⣀⣁⣀⣀⣀⣀⣀⣀⣉⣀⣠⡤⠤⠒⠊⠉⠀⠀⠀⠀⢀⣀⡀⡀⣯
⢇⢸⠀⠀⠀⠀⣤⠀⠀⠀⠁⠀⠀⠀⣠⠚⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠉⠳⢤⡀⠀⢸⡅⠀⢀⡤⠚⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠥⠤⠤⠤⠀⠀⠀⠀⠀⡇⠀⠁⠂⢀⠀⠙⢦⠎⢡⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠈⠳⣄⣄⣀⣠⡠⠤⠚⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⣤⠊⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢶
⠀⠀⠀⠀⢱⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⣠⠴⠋⠁⠀⠀⠀⠀⠈⠓⢦⣄⣀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠉⠀⠀⢸
⠀⠀⠀⢰⠃⡴⠋⢉⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠙⣅⣰⠋⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠙⠦⠤⠛⠒⢦⡀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢵⠶⡶⣒⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⡤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠉⠉⠉⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡦⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠤⢴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠉⢹⠓⠚⠓⠛⠓⠚⠛⠛⠓⠚⡏⠉⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣤⣤⣤⣤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⣤⣤⣤⣤⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣿⣿⣿⣟⣻⣿⣿⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣿⣟⣻⣿⣿⣿⣿⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀⠀⡠⠖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⠔⠉⠀⠀⠈⠙⠌⠉⠀⠀⠀⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠠⢞⠁⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠑⢦⣀⣀⢧⡀⣠
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢰⠁⣀⣀⠤⢄⠈⠁⢷⠤⠝⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡛⠁⠀⠀⢣⠀⢸⡖
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢠⠒⠒⠄⠀⡆⢠⣀⣶⡄⣸⠀⢸⡇
⠀⠀⠀⠀⠀⠰⢖⣊⠀⠀⠢⣘⣁⢰⠀⠡⠈⠀⠀⠀⠉⠀⣻⠃
⠀⠀⠀⠀⠀⠀⣤⣜⠁⠀⠀⢨⣥⡌⢠⠃⠀⠀⠀⠀⠀⡜⠀⠑⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣺⣠⠀⣸⠁⠈⠁⠸⣐⠶⢀⣠⠞⠫⡉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠀⠐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⣊⠜⠀⠀⠀⢤⠐⠚⠉⠉⠒⠒⠚⠁⠀⠀⠀⠀⠀⢠⠖⠁⣀⡠⠴⠒⢻⠁⠀⠈⠳⠟⠴⢦⡤
⠀⠀⠀⠀⠀⠀⢶⠐⠀⠀⠙⠦⣀⡀⠀⣸⠦⣤⠤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⠁⡴⠊⣁⡐⢢⠀⠘⡄⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠈⠉⠁⢈⡖⡰⢁⡀⠈⢢⠀⠀⢸⠀⠀⠀⠀⠀⠀⡟⠛⢻
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⠒⢻⡈⠁⣀⣨⠀⠀⠈⡇⠀⠀⠀⠀⢀⡇⠀⣸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣠⠤⠤⠴⠤⡙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⠀⠉⠂⠀⡸⢁⠤⢤⠀⠀⡼⠀⠀⡟
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢀⡤⣀⠀⠀⠀⠀⠐⢂⠇⡴⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠔⠁⠀⢈⠑⠆⠀⠀⠙⠦⣸⣤⠝⠀⠀⢀⣠⣎⠡⠊⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢄⡀⠀⠂⠉⠀⠒⠒⢣⠧⠔⠂⠀⠐⠐⠉⠉⣓⠒⣢⠀⠀⣠⣩⣟⣉⢉⣉⣧⣀⡀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠠⠐⠀⠂⠤⠤⠬⠉⠝⠀⡸⠀⠀⠉⠛⠛⠋⠁⠀⠙⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⠓⠢⠤⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠁⠀⠀⠀⢸⠀⠈⡇⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡘⠤⡀⠀⠀⡌⠀⠀⣿⠀⠀⡇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠇⠀⠈⠙⠚⠁⠀⠀⡟⠀⠀⣇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠹⢤⡤⠤⠤⠴⠶⠒⠒⠒⠒⠶⠾⢤⣄⣀⣀⣀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠀⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠚⠖⠶⠶⠦⠤⠤⠤⠴⠤⠤⠤⠼⠤⠤⠾⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣏⢽⣟⡛⣛⣛⡛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠂⠈⠉⠀⠉⠀⠀⠀⠀⠀⠁⠀⠉⠀⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠁⠀⠀⠐⠒⠒⠒⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣄⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠮⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠟⠛⠿⣷⡄⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠛⠉⠉⠉⠀⠀⠀⠀⠀⢻⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠊⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⣾⣽⠆⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢣⡀⠈⠁⠀⠀⢤⢀⡄⠀⠀⡠⠊⠙⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⢀⡦⠉⠧⠤⠊⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠘⠁⠀⠀⠀⠉⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⡄⠀⡀⠀⠉⠉⠀⠐⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣄⣀⣀⣀⣀⣀⣠⣤⣴⣶⣿⣿⡿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣀⣀⡤⠄⣰⣾⣹⣷⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣷⡐⠋⢹⣯⡁⣴⠛⣾⠛⣼⠋⣴⡄⠀⣠⣤⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⡀⠀⣿⣿⣿⣿⣿⣿⣷⠹⣿⣇⠀⠸⠿⠃⠀⠀⠀⠀⣴⣦⣬⣄⠀⢱⣷⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠻⠟⢃⣸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⡄⠀⠀⢀⣀⣀⠸⠉⡿⠛⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟⢿⣷
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⣿⣿⣃⡀⢰⣿⣿⣿⡇⠀⠀⠀⢶⣾⣿⣿⢍⣿⣿⣿⣿⣿⠿⠟⢿⣦
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣤⡄⠀⠰⣿⣿⣿⠇⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⡏⠀⠀⣶⣿⣦⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡽⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠉⠁⠀⠀⠀⣰⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠆⠘⠃⠀⠀⠴⠀⢨⠉⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⣿⣿⣿⣇⡈⠁⠘⠀⠀⠀⠁⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣑⣓⣛⡛⠋⠀⠀⢰⣶⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣷⣆⠀⠀⠀⠈⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠃⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡆⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠟⠋⢹⠏⣿⠿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠄⠀⠀⠀⢀⡀⠀⠀⠀⠠⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠂⠀⠀⠀⢠⠀⠀⠀⠀⠰⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⠦⣄⣀⠀⠀⠀⠀⣀⣠⢴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠉⠉⠈⠁⠄⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠐⢹⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢼⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⣀⣀⣀⣙⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⢋⣴⣤⣀⣀⣀⣀⣀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿⠉⠛⠿⢦⡀⠀⠀⣶⣶⠶⠾⠛⠉⠁⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏⠸⣿⣿⣿⡟⠀⠀⠀⠈⠙⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⣿⣿⠿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠙⠛⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠀⠀⠀⠀⢶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡶⠀⠀⣶⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠾⠴⢯⣤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠸⢅⡘⡶⢁⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢠⠤⠤⠤⠽⡿⠯⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⠔⠉⠉⠉⠲⡄⠀⣠⠊⠉⠹⡕⢦⠀⠀⠘⡆⠀⠀⠀⡇⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠀⡎⠀⠀⢠⡄⠀⢱⣀⡇⢠⡄⠀⢳⣀⡇⠀⠀⣧⠤⠤⠤⡷⠤⠤⠤⢼
⠀⠀⠀⠀⠀⠀⠀⣏⢈⡇⠀⠘⢄⡀⠈⢁⡠⢏⣀⠳⣌⠁⠀⣸⡽⣹⠀⠀⢹⠀⠀⠀⣷⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡀⠀⠀⠉⠉⣩⠤⣄⣀⣀⡤⠉⢩⠏⠉⢡⠤⠤⠼⠦⢤⡤⡾⠤⠤⠤⠧⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠙⠦⠤⠤⠖⢀⡰⠋⠀⠀⢸⠀⠀⠀⠀⠸⣷⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠶⢤⣤⣤⣤⣤⡖⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠈⠀⠐⠘⢷⣄⢺⡟⠶⢤⣄⣀⣠⣾⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⠀⠁⠀⠀⣠⣝⣿⠷⣤⣀⣀⣀⣀⣼⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣈⢧⣤⣴⣾⡿⢹⡏⠀⠀⠀⠉⣉⣉⣹⣀⣀⣀⣀⣀⣟⣆⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟⠛⠛⢻⡟⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢷⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣏⣉⢺⣴⣚⣱⠆⠀⣀⣀⣀⣀⣸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⡽⢻⡛⡍⠉⠉⠉⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠸⣦⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⣀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⢫⡤⢹⣿⣿⡟⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⣭⣴⣿⣿⠟⠀⠀⠀⠈⠻⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣈⣙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⢀⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⣿⡏⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⡛⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣃⡻⣿⣿⣿⣿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⣦⣭⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⣼⣿⣿⠿⣛⣭⣴⣶⣌⠻⣫⣾⣿⣿⣿⣿⣏⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡃⣋⣭⣶⣿⣿⣿⡿⣋⣍⡘⢿⣿⢋⣭⣭⣭⣭⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢡⣿⣿⣿⣷⣿⣜⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⢸⣿⣿⠙⠻⢿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠩⠄⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⡄⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡀
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠙⣀⠀⠀⠀⠀⠀⡀⠀⠀⠸⡔⠲⢄⣀⢠⡼⢃⠂⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⣀⡌
⠀⠀⠀⠀⢈⡠⠦⣀⣀⠜⠁⡠⠖⠂⠉⠐⠞⠉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢐⣶⢟⠁
⠀⠀⠀⠀⠉⠀⠀⠈⠉⠲⡌⠀⠀⣀⠐⠋⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠁⢹⠞⠉⠹⢿⠦⣤⡀
⠀⢀⡄⠀⠀⠀⠀⠀⣆⠀⠱⡈⠉⢸⠛⠲⠶⠦⠀⠤⡶⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠁
⠀⢈⠀⠇⠀⠀⠀⠀⠸⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢄⠐⠀⢀⠆⠀⠀⠠⡆⢣
⠀⠈⢹⠀⠀⠀⠀⠀⠀⠱⡀⠈⠳⠘⠓⠲⣤⠄⡠⠄⠟⢢⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠆⠺⡄
⠀⠀⠘⠗⠀⠀⠀⢠⡀⠀⠊⠢⠀⠼⠀⠀⢀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄
⠀⠀⡄⠀⠀⠀⠀⠀⠸⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠓⢬⠆
⠀⠀⠈⠢⠑⠀⠀⠀⠀⠃⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠔⢛⢀⠘⠄
⠀⠀⠀⠀⣠⣄⣤⡤⠀⡄⠀⠀⠀⠀⠢⡤⠤⣤⣤⡤⠖⠂⠀⠀⠀⠀⠀⠀⠑⠚⡄⠀⢀⡴⠊⢀⠀⠰⠂
⠀⠀⠀⠀⣇⠀⡟⠀⢀⠁⠀⠀⠀⠀⠀⡇⠀⡽⠇⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠈⠀⠀⠡
⠀⠀⠀⠀⢡⠀⡃⠀⡐⠀⠀⠀⠀⠀⠀⢸⠄⣶⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⡀
⠀⠀⠀⠀⢸⠒⡟⢶⠃⠀⠀⠀⠈⠁⠈⠻⠶⠛⠛⠛⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣗⣀⡃
⠀⠀⠀⠁⠸⣤⣅⣌⣑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢶⠄⠼⠇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠑⠀⠀⠀⡈⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⣠⠀⠀⠠⢀⡀
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢀⡐⠀⠀⠀⢀⠀⡔⠲⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠄⠀⠘⠆
⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠁⠁⠁⡀⠔⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢰⡀⠠⠀⠀⡄⠍⠒⠂⠀⠺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠃⡐⠂⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠊⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠐⣀⠀⠀⣀⣈⣁⣂⡀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⡀⠈⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠰⠀⢄⠣⢐⡠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣟⣻⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⣿⠏⠈⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⠏⠀⠀⢻⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⣿⠿⠿⠿⢿⣿⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠀⠸⣿⣿⣿⣿⣿⣧
⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠙⣿⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⢈⠛⠋⠉⠁⠀⢸⣿⣿⣿⣿⣀⠀⠀⠠⠤⠀⠀⠀⠐⠃⡀⠀⠘⢻⣿⣆⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⢠⣄⡀⣤⣶⡄⠀⣿⣿⣿⣿⡆⠁⠀⢡⡶⠀⠀⠀⠘⠿⠀⠀⣴⣿⣿⣿⣦⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿
⠀⠘⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠎⠈⠛⠛⠛⠛⠃
⠀⢶⡀⣠⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠴⠋⠀⠀⠌⠀⠀⢠⣶⣶⣆⠁⠀⠄⠀⠀⠀⠉⠁
⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⣄⣰⠟⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠣⢀⠀⠀⠀⡙⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⠂⠐⠒⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠴⠁⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠤⢀⡀⢀⡀⠀⠀⠀⠀⠀⢀⠴⠛⠁⠀⠀⠉⠑⡄⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠢⠄⣀⠐⠂⡐⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⢀⠀⠀⠧⠀⡟⠌⠀⢠⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⡀⢘⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠁⠀⠀⠀⠀⠀⠀⠀⠐⢄⠑⠄⠀⠠⠒⠀⠀⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠔⣹⠆⠤⡤⠀⠠⠐⠇⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣦⡐⢐⠱⠀⠁⠛⠀⠢⡁⠀⠀⠀⠀⠀⠀⠀⠀⣬⠉⢀⣀⠀⠀⠈⠉⠀⡌⠐⢄⠘⢌⡢⢀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠧⠉⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠈⠉⠀⠀⠀⠚⠀⠃⠀⠈⢣⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠑⠒⠈⠀⠀⠠⡀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⣄⠐⠂⠃⡐⠀⠐⠀⠀⠀⠀⠀⠙⠆⡀⠀⣀⠀⠀⠀⠰⠇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡁⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠁⠀⠐⠂⠀⡀⠷⣔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠁⠈⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠐⠑⠄⠀⢚⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠐⠁⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⠆⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠸⠵⠃
⣿⠿⣿⠿⡿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⡀⣶⡀⠀⣀⡀⢰⣶⡶⣆⢠⡞⣷⠀⠀⢀⣴⣉⣁⡀⠀⠀⠈⠉⢲⡄
⠀⠀⠀⠀⠀⢸⡿⡿⠳⠸⣧⡷⠘⠃⠃⠙⠈⠉⠉⠀⠐⡷⠖⠒⠂⠠⠍⣙⡒⠦⣼
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⡤⠀⠀⡁⡀⠀⠀⠀⠀⠈⠩⢳⠏
⠀⠀⠀⠀⠀⠀⣠⠤⠤⠴⠒⠚⠙⣆⣀⠀⠀⠀⠀⠀⢀⠕⠘⠓⠀⠀⠀⠀⡠⠾⣤⣤⣄⣤⣤⣀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⣀⡼⠁⠀⠀⠀⣠⡤⠤⠤⣞⠀⠀⠀⠀⠀⠈⠑⠄⠀⡶⠀⢠⠎⢠⠾⠁⣀⣄⠀⠀⠀⢏⣹⠋⣩⠿
⠀⠀⠀⡞⠁⠀⠀⠀⠀⠘⣯⠀⠂⣠⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡞⠁⠀⣰⠁⠧⢠⣴⢾⠋⠉⠓⠟
⠀⠀⢀⡿⠂⠀⠀⠀⠀⢼⡁⠀⠸⣏⡻⠈⠓⠤⠤⡄⠀⠀⠀⠀⠀⠀⠈⣧⠀⣠⣏⠀⠀⠀⠀⢸⡇
⠀⠀⠘⢧⡀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⡴⢋⡼⠛⣇⣹⠗⠒⠒⠒⠚⡇
⠀⠀⠀⣀⠇⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⣤⠒⠊⠀⠀⠀⠀⣠⢖⣫⠔⠋⢀⣠⡞⠁⠀⠀⣠⡴⠋⢳
⠀⠀⠰⢧⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡟⢀⣀⡤⢶⣯⠴⠋⠀⣠⡴⠛⠹⢧⡶⠚⠋⠁⠀⠀⡼
⠀⠀⠀⣨⠇⠀⠀⠀⠀⠘⣳⣠⣋⠒⣒⣛⣋⡭⠵⠚⠉⠀⢀⣤⠞⠉⠀⠀⠀⠘⡇⠀⠀⠀⢀⣴⠁
⠀⠀⢾⡁⠀⠀⠀⢀⡤⠞⠉⠉⠉⠉⠉⠁⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⢷⣀
⠀⠀⣠⠇⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢄⣠⠟⠈⡇
⠀⠀⠙⠒⠒⠒⣿⠀⠀⠀⠀⠀⠀⣠⠾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⣇⢀⡼⠁
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠚⠁⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣲⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⣙⠛⠛⠋⠁⠀⠀⠀⠀⣀⠀⣾⣿⠿⠿⢿⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣟⠻⣷⠀⠀⠀⠀⠀⣴⣿⡿⠀⠹⣅⣋⣝⣣⡟⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡛⢿⣿⣶⣤⡀⠀⠀⠀⠈⢉⣀⣠⠀⣈⣭⣭⣁⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣮⠙⠿⣿⣷⣶⣶⣿⡿⢋⣴⢿⣿⣿⣿⣿⣿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠉⠋⠉⠁⣴⠟⠁⢸⣿⣿⣿⣿⣿⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⣀⣀⣀⣠⣄⡀⠀⢸⣿⡷⠀⢸⣿⣿⣿⣿⣿⠀⢰⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠱⠿⠿⠿⣿⣿⠇⠀⠀⠉⠁⠀⢸⣿⣿⣿⣿⣿⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠹⣿⣿⡄⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣶⣿⣿⣷⡀⠀⣿⣿⠇⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠟⠁⠀⠿⠿⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠹⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠉⠻⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠈⠛⠃⠸⠿⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦
⠀⠀⠀⣠⣶⣶⣶⢄⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⢰⣿⣿⣿⣟⡈⠃⠀⠀⠈⠃⠀⠠⢠⠀⠛⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠈⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⠀⢀
⠀⠀⠀⠀⢹⣿⡿⣧⡀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠆⡠⠤⣄⠀⠀⠀⠀⠀⡔⠊⠍⠀⠒⠂⢀⠈⢹
⠀⠀⠀⠀⢸⣿⣇⣌⣻⠃⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠃⢰⢠⠒⠖⠀⠀⠀⠀⠃⠐⠲⠒⠂⠀⠘⠰⠚
⠀⠀⠀⠀⠈⠛⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣭⣤⡄⡀
⠀⠀⠀⠀⠀⠀⢹⣿⡅⠀⠀⠀⠀⠀⠀⠙⣻⠙⠛⠛⠋⢱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⢉⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠠⠖⠂⠠⡄⠘⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢐⠔⠁⡠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠒⢆⣮⠭⠄⠂⠀⠀⢠⠀⠀⠀⠀⣀⢠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⠪⡢⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢄⡲⣶⠖⠀⡀⠉⢸⠀⠀⠀⠰⠐⠃⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁⠀⠈⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠘⠀⣀⠈⡢⠀⠀⢀⡴⡁⠀⡠⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠚⡀⠀⢀⣈⠑⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⠋⣓⠴⢗⢄⡀⡠⠔⢁⣄⡒⠲⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡅⡡⠬⡑⠒⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢴⠠⠀⠑⣜⣆⢀⣰⠜⠗⠉⠛⠛⣋⠀⠀⠀⠀⠀⠀⠀⠀⢎⢉⣉⡆⠀⠀⣔⡟⢠⠁⠀⡸⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⣂⠊⢆⠀⠀⠘⠛⠛⠁⠉⢀⣀⡤⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡍⠉⠳⢀⣼⠼⠥⠊⣀⠴⠋⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠬⠠⢂⠓⢀⣀⣀⡠⠄⠊⠀⠍⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⢀⡷⣤⠀⣀⣀⣠⠤⠚⠭⠊⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠄⢀⡄⠈⠓⠒⠒⠒⠒⠤⢀⠠⠆⠀⠀⠀⠐⠂⠐⠂⠒⠒⣌⡡⠌⠒⠋⠉⠁⠀⢁⠤⡖⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡀⢠⠮⠀⠀⠀⠀⠀⠀⠀⠀⠌⢢⠀⠈⠛⠻⣿⣿⠿⠋⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⣈⠐⠐⠐⠀⠀⠀⠀⠀⠀⠸⢀⠈⡆⠀⠀⠀⡟⡇⠀⠀⢀⠀⠀⠀⠙⠓⠂⢠⡒⠖⠴⠲⠚⠒⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣈⣉⣉⡓⢒⠒⡄⠀⠸⠀⢀⢏⡀⠸⠀⠀⠀⡁⡇⠀⠀⠸⡀⠀⠈⠁⢡⣦⢸⣷⡆⠀⠀⠐⠒⠒⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽⠀⠀⠀⠀⡙⠠⠁⠠⠀⠀⣚⠀⠄⠀⠀⠀⠀⠆⡇⠀⠀⠀⠣⠀⠰⠞⡋⡥⠀⢯⡇⠀⠀⠀⠀⢀⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠾⠀⠀⠀⡘⠀⠀⠀⢀⣈⠁⣀⡈⣈⡂⠀⠉⠉⠱⡅⠀⠀⠀⢰⠀⠀⠈⡐⠠⠸⣽⡇⠀⠀⠀⠀⢰⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢨⠀⠀⠀⠐⡄⠀⠀⡇⠀⠀⠀⠀⠀⢩⢖⡴⡂⣱⢋⢀⣀⣀⠎⣀⠔⢡⠃⠀⠁⢘⠄⠀⠀⠀⠀⠘⠂⡂
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠴⢘⠒⠠⣄⠠⢽⣦⣀⣰⠄⡄⠀⢘⠀⢉⡛⠫⢚⠃⠳⢚⠚⢫⡦⠠⣔⣁⣤⠴⢻⣭⡖⠢⢐⠀⢠⢀⣦⣇⢀⡀⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠉⠀⠉⠀⠈⠀⠈⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠀⠀⣤⡴⠶⠶⠛⠛⠛⠛⠛⠿⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠶⠶⠶⠶⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠳⣤⣤⣤⡴⠶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠈⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠉⠙⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣶⣶⠶⠶⠶⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⢠⡞⢉⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⢀⡠⠤⠴⠶⠶⠶⠶⠶⠶⢤⣤⣀⠀⠀⠀⠀⠙⢧⡉⠀⢋⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠻⣆⠀⠀⠀⠈⢷⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢰⡟⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⡈⢱⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡟⠁
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣄⠘⣤⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠈⠉⠈⢳⣤⣴⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢠⡟
⢻⡄⠀⠀⠀⢠⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠀⠀⠀⠀⠀⠀⠻⢯⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠂
⠈⢷⡄⠀⠀⠈⢧⡀⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠔⠉⠀⠙⢳⣄⠀⢀⡏⠀⠀⠀⠀⢀⣴⡿⠛⠁
⠀⠀⠙⢦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊⠉⠀⠀⠀⠀⠀⠀⠹⡆⢸⡇⠀⠀⠀⣠⡿⠉
⠀⠀⠀⠀⠉⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠓⠀⠐⠶⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢹⣤⠶⡄⠀⠀⠀⠀⢹⠀⡇⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⡴⠋⠉⠀⠟⢁⡾⠁⣀⠀⠀⠀⠈⡄⢃⠀⠀⣾
⠀⠀⠀⠀⡞⠛⠟⠉⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣼⠁⠀⠀⠀⠀⡾⠷⠛⣩⠇⠀⠀⠀⠃⠈⠀⠀⡟
⠀⠀⢀⣸⠃⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠠⠴⠟⢳⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢰⡇⠙⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠶⠚⠋⠁⠀⠀⠀⠀⢀⡴⠋⠉⢷⡄
⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⢳⡈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠠⠀⠀⠁
⠀⢿⡀⠀⡀⠀⠀⠀⠈⠀⠀⠈⠷⡀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠘⣧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⠶⣄⣀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠈⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠙⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢛⣿⣿⠟⢛⣿⣿⠛⠻⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠟⡿⡿⣿⣦⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠜⠁⠀⠀⠙⠛⠻⡇⠀⠀⠀⠀⠀⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠇⠀⠀⠀⠀⠀⣀⡀⣾⡂⠰⣦⠀⠀⢈⣿⣿⣳⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣾⣠⠰⢧⡀⠀⠈⠽⠗⠈⠋⠐⠦⠀⠀⠸⣿⠻⣮⣿⣿⣿⣿⣿⣿⣿⣷⣶⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣵⣿⡇⠀⠀⠀⠀⢘⠂⠀⠀⠀⠀⠈⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⠖⠀⠀⠀⣀⡨⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢽⣿⣦⣀⠀⠀⠀⠀⢰⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣆⠀⠀⠘⠃⠀⠙⡿⠿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠀⠀⠀⠀⠀⠀⢀⢂⢀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠛⠻⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣦⢀⢀⡀⠠⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠨⡝⠂⠀⠀⠀⠀⠀⣑⣤⣤⡶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠠⢀⠀⠉⠁⣔⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢍⠀⠀⠀⠀⠀⠀⣹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⠔⠀⠀⠀⠀⢾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⣻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣇⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠈⠙⢀⠠⠂⠀⠀⠀⠀⢀⡀⣿⣿⣿⡇
⠀⠈⠰⠀⠀⠀⠀⠀⠀⠐⠐⠲⣾⠀⠀⠀⠀⠡⡄⠀⢀⡐⠄⠘⠀⡈⢀⠀⠀⠀⠀⠀⢈⢡⢹⣿⣿⡇
⠀⢤⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠉⠀⠂⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⢸⣾⣾⣿⣿⣇
⠀⠈⢯⣵⣀⣿⣄⠀⠀⠀⠀⠀⣼⡷⠀⡠⠤⣀⠀⠠⠐⢷⡺⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣼⣿⣿⠟
⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⣰⢿⠥⣠⠒⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡌⠀⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣿⢸⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠀⠀⠀⠠⠤⣀⠀⠀⠀⢀⡠⠐⠊⠉⠀⠀⠀⠉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡴⠁⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠘⠠⡀⡌⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠰⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⡠⠐⠒⠒⠢⠆⠉⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠉⢢⠄⢀⣀⣀⡀⡴⠀⡄⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢸⠀⠀⠀⠀⠘⡄⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠸⠀⠐⠂⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠢⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠁⢨⡳⠤⠤⣀⢸⡀⠙⡾⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠺⠉⠑⢄⣀⣀⠇⠀⡰⠀⠀⡠⠄⣀⠀⠀⠀⢀⡀⠤⠒⠁⠀⠀⠀⠁⠒⠠⠀⠤⠠⠄⠒⠂⠤⠤⠱⢀⠀⠀⠀⠀⠙⡜⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠑⠾⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡤⠂⢀⠰⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣆⠉⢁⣺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣋⢁⡠⠣⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢄⠈⠉⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡠⠠⡈⢆
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⢀⠀⠀⠀⠀⠈⡎⢢⠀⠀⠀⠀⡀⠴⡛⢽⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡔⠀⠀⠐⠈⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠱⡀⡠⠊⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡘⠮⠐⢄⠀⠀⠀⠀⠀⡇⢀⠃⠀⠀⠀⠐⠀⠇
⠀⠀⠀⠀⠇⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⡑⠂⠄⠀⡀⡇⡘⠀⠀⠀⠀⠄
⠀⠀⠀⢰⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠁⢣⠀⠀⠀⡜⠀⠀⢀
⠀⠀⠀⠘⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡤⠀⣂⠀⡔⠀⠀⠀⠘
⠀⠀⠀⡈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣧⣄⠈⠻⠟⠛⠿⢿⡿⠿⢿⣿⡿⠃⠀⠀⠈⢻⣿⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣷⣀⠀⢀⣀⣀⣀⣠⣤⣄⣠⣾⠟
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢀⡀⠀⠀⠀⠀⠀⠀⡀⠰⣤⣤⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠙⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⡿⣿⠄
⣿⣿⣿⣿⣿⡆⠀⡆⠀⣦⡀⠹⢶⣿⣿⠿⠟⢿⣿⣿⣿⣿⠀⠀⠀⠀⣀⣄⣀⠀⠈⢉⣰⣿⢿⣿⡏⣿⣿⠁
⣿⣿⣿⣿⣿⡀⠀⠀⠀⣿⣿⣦⡾⠋⠀⠀⠀⠀⠘⠛⠻⣿⢀⣤⣤⣾⣿⣿⣿⣿⣷⣌⠛⠁⠈⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠛⠿⣶⡀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⣴⣾⣿⣿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠈⠀⠀⢠⣤⣤⣀⣴⣶⣾⠈⠛⠙⠿⠟⠛⢻⣿⣿⣾⣿⣿⡿⠋⠁
⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣤⡀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡗
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡀
⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠊⠉
⠀⠀⠀⠀⠀⢀⣀⣤⣴⠛⠉⠙⠳⠶⠦⣄⠀⠀⠀⠀⠀⢉⠍⡠⠊⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⣰⠟⠉⠉⠉⢻⠒⠚⣳⣶⡀⠘⡆⠀⠀⠀⢀⣎⠜
⠀⠀⠀⠀⣿⠀⢀⡤⠤⠜⠀⢸⠋⠀⠙⣄⢱⡄⠀⢀⡾⠁
⠀⠀⢠⡞⠁⠀⢾⢀⡴⠦⠦⠚⠀⠀⠀⢸⣸⠁⠀⠈
⠀⠀⣿⠀⠀⠀⢀⣻⠁⠐⣒⡒⠀⢀⣩⣽⣧⣀
⠀⠀⠘⢦⡀⠀⢺⡘⠀⠉⠉⠁⠀⠀⠉⢹⠀⣿⡆
⠀⠀⠀⠀⠈⡦⠈⣻⡆⠀⠀⢀⣀⣀⠀⣾⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠳⣴⣿⣯⠀⠀⠈⠙⠋⢼⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠜⠫⢄⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⡀
⠀⠀⠀⠀⠀⢀⠞⠀⠄⡀⠀⣉⠳⣤⠜⠒⢤⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡄
⠀⠀⢀⣴⠊⣁⡀⠀⠀⠈⠒⠁⠀⠘⠄⠀⠀⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⡅
⠀⠀⠸⠁⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠈⠢⡀⠀⠑⡀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠘⠀⠀⢐⡀⠂⠀⠀⠄⠁
⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⢈⠁⠀⠊⠀⠀⠀⠌
⠀⠀⠀⠈⣎⢣⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⠟⠛⠄⠀⠀⢀⠂
⠀⠀⠀⠀⠘⣄⠱⣄⠀⠀⠀⠀⠀⣷⠁⠁⠱⠀⡘⣄⡠
⠀⠀⠀⠀⠀⠈⢆⠈⠢⣄⠀⠀⠀⠘⠀⠀⠀⢀⠅⣾⠁
⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠑⠤⡀⠀⠀⠀⠀⡠⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠈⠐⠒⠂⠁⠀⢨⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠉⠢⣄⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡠⠤⡀⠀⠹⡀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⢞⡉⠀⠀⣘⣆⠀⢣⠀⠀⠀⠈⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠞⠁⠠⠄⠀⠀⠄⢸⠀⢸⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠑⡄⠀⢠⠬⡄⠀⡜⠀⢸⠀⠀⠀⠀⣸⣁⡤⠖⠛⣆⠀⠀⠀⠀⠀⠀⢠⠖⠦⢤⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣃⠀⠀⠈⡦⢤⡉⣥⠚⠁⢰⡃⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠑⠦⠤⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⢶⠟⠀⠀⠀⠘⠷⣮⣉⠁⠀⠀⠀⠀⠀⢀⣀⣼⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠑⠦⠤⠤⠤⠊⠁⠀⠙⢦⠀⠀⠀⠀⡎⠉⠈⢉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢀⢤⡠⣤⡀⠀⠀⠀⠀⠱⡀⠀⡠⣇⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡄⠀⡴⠁⠋⢪⣿⡇⠀⠀⡄⠀⠀⠱⡜⠁⠀⢉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡧⠒⠣⣀⢀⡠⠋⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⠊⠀⠀⠀⢀⠝⠀⠀⠀⠀⠀⡟⢄⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⡇⠀⠙⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⢠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠔⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⡀⠀⠀⠀⠈⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⠔⠈⠀⣀⠄⠂⠀⠴⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀⠀⡰⠁⣀⠔⠊⠀⡠⠔⢠⠂⠘⢆⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⠜⢠⠞⠁⡠⠔⢉⡠⠖⠁⠀⠀⠀⠠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠄⡜⡰⢁⢴⣯⡖⡚⠁⠀⣀⡀⠀⠀⠀⠆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⢸⠰⢡⢢⠞⡩⠊⠀⠠⣀⡀⠀⡀⠒⡚⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡰⠃⠃⠠⠚⠋⠙⠲⣄⠀⠀⠀⠀⡇⢩⡇⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⣁⡴⠁⠀⠀⠀⠀⠀⠈⣧⠀⣀⠒⣣⠏⢇⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⢀⣬⡇⠀⢀⠁⠀⠀⠀⢾⠎⣇⡡⠼⠃⠠⣜⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠙⣇⡀⡇⠀⠀⠀⠀⠀⠀⠁⠙⠉⠉⢆⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢿⣀⢀⠀⠀⠀⠂⠀⠀⢰⠀⠀⠘⣂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢿⣱⡖⠈⠒⠶⠴⡁⠸⠀⠀⢀⡏⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢧⠀⠀⠀⠀⠀⠀⠀⠻⡂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⢧⠀⠀⠀⠀⠀⠀⠀⠹⡏⠉⢹⡏⢉⠉⣹⠃⠀⠀⠀⡞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢳⡀⠀⠀⠀⠀⠀⠀⠹⣾⠛⠁⣨⠟⠁⠑⢄⠀⡼⠁⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠱⣄⠀⠀⠀⠀⠀⠀⠹⣄⠞⠁⠀⠀⠀⠈⣾⣇⡪⣍⠢⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠘⣦⠀⠀⠀⠀⢀⢜⢿⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⢧⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡸⣊⡷⡄⠀⠀⠈⢸⠘⡀⠀⠀⠀⡼⠁⠀⢀⣀⣀⣼⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢽⡴⣏⠦⡀⠀⠈⠀⠀⠀⢠⠞⠉⠉⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠉⠀⠈⠢⣄⢀⣠⠖⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⠙⠛⠻⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⡿⢟⣋⣙⠛⠿⣿⣿⣿⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣄⠈⠻⣿⣧⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣦⠀⠈⠙⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣆⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⠛⠁⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⠄⠈⡀⠀⠀⣀⡀⠤⠀⠤⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠊⠉⠉⠑⢆⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⠴⠆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⢦⠖⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢛⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀⠀⠀⠀⠈⢿⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣥⡖⣿⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢣⠀⠀⠰⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢧⠀⠀⠠⣋⡤⣬⡆⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⡆⠀⠀⢿⣷⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⢱⡀⠀⠀⠉⠉⠀⠀⠀⠀⣄⠤⢄⠀⠀⠀⠀⠀⢀⡴⠋⠹⣖⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠻⣄⠀⠀⠀⠀⠀⠀⠀⡟⣀⣼⡃⠀⠀⢀⡴⠊⠀⠀⠀⠈⢦⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⡀⡀⣠⠼⢀⠀⢷⠤⠞⠋⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠿⣟⠢⠀⢀⠌⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢸⡍⠢⡖⠁⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⡇⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢈⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡆⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⡠⠤⠐⠒⠘⠣⢤⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠁⠀⠀⠀⠀⢠⢔⠊⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⠋⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠈⠒⠤⣀⡀⠀⠀⣀⡠⠜⠁⠀⠀⠀⠀⠀⠰⠘⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣌⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢒⡢⠍⠁⡀⠤⠍⠒⠛⠂⠤⠤⣀⡀⠀⣀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣤⠎⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠳⢀⡛⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠛⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠓⠂⠠⠤⠤⠤⠤⠤⠤⠄⠐⠒⠒⠈⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⢿⠿⠿⠿⠻⠿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⢻⣿⠛⣛⢻⣿⡿⠿⡿⢿⡿⣿⣿⠟⣻⡿⢿⡿⢟⡻⣿⡛⠿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣶⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠸⣿⣷⣿⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠙⠋⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠒⣶⣶⣦⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⠀⣰⣾⣟⠉⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠻⢿⣿⣷⣦⡀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠃⠀⠸⣿⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣰⣏⡀⠀⠀⠀⠀⣀⣤⣬⣽⣿⣄⣀⣀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⠛⠛⠛⠛⣿⡆⠀⠀⢸
⠀⠀⠀⠀⢠⣤⣠⣶⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣿⡇⠀⢠⠃
⠀⠀⠀⠀⠀⢹⣿⠏⠀⠀⠀⠱⡀⠀⠀⠀⠀⠘⢧⣴⣴⣤⣤⣠⣠⣠⣄⡀⠀⠀⢀⣀⣀⣄⣢⣷⣿⣿⣟⠀⠀⠂⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣴⠶⠋⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠻⣿⣷⣀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣆⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⡴⠞⠛⠛⠛⠳⠷⠾⠾⠶⠾⠦⠶⠷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⠀⠀⢀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣾⣶⣦⣦⣴⣶⣶⣦⣤⣦⣤⣴⣾⣶⣷⣦⣴⣶⣾⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠛⠛⠛⠛⠋⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⠒⠒⠒⠒⠒⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢤⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠁⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⢄⡐⠒⠂⠑⠒⢓⢊⣡⠞⠀⠀⠀⠳⠄⠲⠀⡰⠓⠒⠒⠒⠉⡁⠀⣤⣸⣷⣿⣷⣶⣿⣿⣯⣶⠦
⠀⠀⠀⣠⠊⠀⠀⠀⢀⠜⠁⢀⣀⢷⡄⠀⠀⠀⠀⢄⠜⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠋⢉⠉⣿⣿⣿⣿⣿⡇
⠀⠀⠠⠁⠀⠀⠀⠀⠇⠀⡼⠉⡁⢄⡇⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠊⠈⠙⡀⢿⣿⣿⣿⡟⠃
⠀⠀⠰⠀⠀⠀⠀⠀⣇⠀⠑⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⡀⠀⠠⠃⠘⠗⢡⡿⠧⡀

⠀⠀⡰⠀⠀⠀⠀⠀⠈⢆⠀⠰⠴⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡂⠀⡠⠠⢄⡠⠊⠀⠀⡇
⠀⠈⠒⠖⡆⠀⠀⠀⠀⠈⡄⡖⠊⠉⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢍⠒⠊⠁⠀⢦⠀⢸
⠀⠀⠐⠪⢚⡁⣠⠆⠀⢠⠇⢳⣼⠒⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡎⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠁⠀⢠⡔⠁⣴⠔⢆⡹⠂⠔⢠⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇⠀⠀⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⠀⠣⠼⡍⠀⠒⢤⣰⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡀⠀⠀⠀⢰⣑⠒⢾
⠀⠀⠀⠀⠀⢀⠠⠟⢶⠞⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠓⠒⠒⠒⠺⣰⠴⠃
⠀⠀⠀⠀⠀⠐⢄⣀⡘⡄⠀⠀⣀⣬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠈⠻⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠋⠩⡜⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠰⡇⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠒⢤⣴⠼⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢠⠋⠻⡳⣦⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⢈⣧⣀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣾⣀⠀⢰⣈⣁⣈⣼⠇
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣉⣉⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⢯⣽⠙⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣛⣛⠡⢿⣷⣍⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⠋⠉⠉⠉⠦⠄⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠊⣴⡄⠀⠀⣶⡄⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⡇⠀⡻⠇⠀⠀⠿⡁⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠈⠠⠌⠤⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠶⣶⣶⠟⠉⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠈⠉⠀⢠⡀⡴⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠯⣽⠶⣴⡇⠀⠀⠀⣤⠿⣿⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆⣾⣷⣦⣤⣾⣿⣦⢿⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡖⠦⣤⡟⠀⡀⣬⣿⡿⣯⣉⣉⣹⣿⣿⣿⣷⣒⣺⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡣⡀⠀⠛⢿⣽⣯⣴⣻⡿⠀⠈⢿⡿⣥⡽⡍⠉⠉⣀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢭⣷⣚⣛⣿⣭⣥⣤⣤⣤⣤⣽⣿⣿⣇⣶⣮⠿⠋⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣀⠀⠀⢰⣦⣀⣤⣤⣴⣿⣿⣤⣄
⠀⠀⠀⠀⣠⣶⡿⢿⣿⡷⠀⣴⡿⠂⠈⠛⢫⣤⠀⢀⣀⡀⠀⠀⠀⠈⣡⣯⣾⣿⣁⠀⠘⠛⠛⢻⣿⡿⠉⠉⣻⣿
⠀⠀⠀⠀⠿⣿⣄⠚⠛⢡⣾⣟⣵⣶⠄⣠⣤⡍⢠⣤⡍⢀⣴⣴⣶⡆⢠⣾⡿⢁⣤⣶⢦⠀⣠⣿⡟⠀⠀⢠⣿⠏
⠀⠀⣠⡾⠁⢠⣿⡇⣰⣿⣿⡿⠿⣋⣴⡿⢋⣴⣿⠟⣡⡾⣻⣿⢏⣴⣿⡟⢀⣿⡿⠶⣋⣴⣿⡟⠀⠀⣰⡿⠋
⠀⠀⣿⣷⣶⣿⠟⣱⣿⠟⣿⡇⠺⣿⣿⣷⢿⣿⣿⡾⠋⢰⣿⣿⡿⣿⣿⡶⢿⣿⣶⡾⠛⣿⣿⠁⣠⣾⠟
⠀⠀⠈⠛⠉⠁⠀⠙⠉⠘⣿⣷⠏⠈⠉⠀⠀⠉⠁⠀⠀⠈⠉⠁⠀⠈⠉⠀⠈⠉⠁⠀⠀⠻⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡈⠑⠒⡲⠒⠒⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢐⣥⡾⠁⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣆⠀⢀⡀⠀⢀⡀⠈⠓⢄⠠⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠿⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢿⡙⠦⠜⢁⣴⢞⡄⠀⠀⠀⠑⠢⢍⣙⣓⣒⣒⣒⡽⠋⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡇⠀⠀⠈⠉⢻⣤⡤⠔⠦⠀⠀⠘⡀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⣀⠤⢤⣀⡁⡀⠀⢀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣸⠟⠉⠀⠀⠀⠀⠉⠛⠉⠢⡀⠀⢀⠀⠀⠀⠀⠀⠀⡸⠋⠙⢦⡀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣃⡀⠀⠀⠀⠀⠀⠁⠀⢀⡴⠣⢤⣀⠀⠸⢀⣀⡀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⢠⠞⠋⠉⣉⠉⠒⠂⢠⠎⠀⠀⠍⠀⢠⡀⡠⣲⠶⠋⢳⡏⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠁⠀⠀⠀⠈⠉⠁⡞⠁⠀⠀⡼⠛⠦⡠⠟⠋⠁⠀⠀⠁⠙⣄⢀⣀⣀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠤⢸⣄⣀⣀⡀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⡭⠽⢿⡆⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⣲⡶⠃⢁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠠⠤⠤⡴⠒⠚⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠄⣀⣀⠀⠀⢀⡀⣀⣀⣀⡈⡛⠶⠠⠀⠉⠉⠒⠮⠁⣀⠤⠔⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣢⣤⣴⣄⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣽⣦⣴⣶⣾⣿⣶⣬⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡱⠈⢻⣿⣿⣧
⠀⠀⠀⣠⠆⣴⣶⡿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣇⡇⠀⢸⣿⣿⣿⡀
⠀⠀⣼⣵⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⡀⠀⣸⣿⣿⣿⠁
⠀⣼⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⢠⣿⠟⢻⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡏⠁
⢸⠏⠀⠘⣿⣿⣿⡿⢿⣶⣤⡄⢠⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⡿⣿⠁⠀⣀
⠸⡄⠀⠀⠘⢿⣿⣿⣄⠀⠀⠀⣿⣧⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣼⣿⠦
⠀⢧⠀⠀⠀⠀⠙⠻⠿⢷⣦⠀⠈⠉⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢰⣿⠛
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣿⠁
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⡎⠈⠓⢦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⡏
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢧⠀⠀⠸⣄⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠄⠈⠧⠀⠈⠉⠀⠀⠀⠀⠁⠢⡄⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⣼⡷
⠀⠀⠀⠀⢸⡀⠀⠀⣠⠧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢣⠀⠀⠀⠀⠀⠀⠀⡼⢴⣧⡞
⠀⠀⠀⠀⠈⡇⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣇⡀⠀⣀⣠⣴⡚⡱⡠⡉⠁
⠀⠀⠀⠀⠀⣇⠀⠀⠙⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡐⡱⡅⣠⠄
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣷⣜⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⡿⠛⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣟⣿⣟⣻⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠉⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⠁⠀⠀⢀⣠⣤⠶⠶⣶⣤⣄⡀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠋⠀⠀⢀⣴⠞⠋⣡⣴⣾⠟⠋⠉⠉⠛⠓⢼⣇
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⣠⠔⢋⣠⣶⠟⢋⡿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⢀⣠⡶⠟⠋⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇⠀⠐⠋⠁⣀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⡶⢶⣴⡿⠋⠁⠠⠖⢛⣛⡻⠷⠄⠀⠀⠠⠶⠿⠶⣼⠇
⠀⠀⠀⠀⠀⢸⣿⣿⡏⢸⡇⠀⠉⠀⠀⠀⠀⠘⠛⠿⠛⠂⠀⠀⠀⠰⠿⠗⢰⡟
⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⣿⣿⡇⢸⣿⡷⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⢀⣾⠁⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⡇⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⣈⡉⠁⠀⢠⡞⠁⣠⣴⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢠⣿⣿⣿⠀⢸⣿⠇⢠⠈⢿⣦⡀⠀⠀⠀⠈⠛⠻⠛⠃⠀⡾⢡⣾⣿⣿⣿⣫⠥⠔⠒⠂⠀⠀⣹⡇⠙⢿⣿⣦⡀
⠀⠀⠀⠀⣸⣿⣿⣿⠀⣾⣿⠀⠘⡇⠈⠻⣿⣶⣄⡀⠀⠀⠀⠀⣠⡾⢱⣿⣿⣿⣟⣩⠤⠤⠐⠀⢀⣤⠞⠻⣧⡀⠀⠹⣿⣷⡀
⠀⠀⠀⢀⣿⣿⣿⡇⢀⣿⡿⠀⠀⣷⠀⠀⠈⠙⠿⠿⠷⢶⣶⠞⠋⢀⣿⣿⣿⣋⣥⣤⡤⠴⠶⠞⠋⠁⠀⠀⠈⢻⡄⠀⢹⣟⣷
⠀⠀⠀⣼⣿⣿⣿⠀⣸⣿⡇⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣿⢹⡇
⠀⠀⢰⣿⣿⣿⠇⢀⣿⣿⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⣿⣿⣿⠿⠇⢀⠤⠶⠦⠀⠀⠀⠤⠶⠦⣤⠀⢸⠷⢶⣿⢸⡇
⠀⢠⣿⣿⣿⡟⠀⣼⣿⠇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⣆⠀⠀⠘⢿⠆⠀⠀⠀⠀⠿⠟⠀⠀⠀⢠⣿⣿⢸⡇
⠀⣾⣿⣿⡿⠀⣸⣿⡟⠀⠀⠀⠀⢸⣿⢀⠀⠀⠀⠀⠀⡔⠀⠀⠀⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣸⡇
⢸⣿⣿⣿⠁⢰⣿⡟⠀⠀⠀⠀⠀⣾⡿⠀⣷⠀⠀⠀⣼⠁⠀⠀⠀⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⣉⡀⠀⠀⠀⠀⣼⣿⣿⣿⣇⣿
⣾⣿⣿⡇⢠⣿⡿⠁⠀⠀⠀⠀⣼⣿⠃⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠘⣿⣿⠿⢿⣦⠀⠀⠈⠉⠉⠀⠀⣠⢾⣿⣿⠿⢟⣼⠃
⣿⣿⣿⠁⣼⣿⠇⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈⠛⡧⠀⢻⡆⠑⠦⣤⡤⠶⠛⠁⢸⣿⠀⢰⡟⠁
⢿⣿⣿⡀⣿⣿⡄⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⢹⠀⡿⠀⠀⠀⠀⠀⣀⢀⣠⡾⠁⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⡀⠻⣦⡀⠀⢀⡀
⠘⢿⣿⣷⡸⣿⣧⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠸⡇⡇⠀⠀⠀⠀⠀⠿⣿⣷⣶⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⣾⣿⣿⡿⠃
⠀⠀⠙⢿⣿⣿⣿⣷⣄⠀⠸⣿⡄⠀⠀⠀⠀⠀⢷⡇⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀⠘⣷⣄⡀
⠀⠀⠀⠀⠉⠛⢿⣿⣿⣷⣄⡹⣿⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⡶⠶⣖⠟⢿⠯⠻⢗⠶⡶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠿⢈⣙⠀⠤⣤⣤⣂⡀⡠⢍⢀⡱⢽⠓⠦
⠀⢐⣓⣓⣚⣓⣛⣛⣛⣛⣛⣛⣛⣒⣒⣒⣒⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣚⣓⣒⣾⢿⣿⣭⣄⠠⠒⣝⢶⡻⠦⡉⠯⡨⠹⠏⠫⠙⢟⣥⡀⡀⢙⢫⣭⣭⣭⣹
⠀⢈⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣭⣽⡏⣶⠋⠁⠀⠀⢠⡀⠑⠝⠩⠁⠠⠀⠀⠀⠁⠀⠀⠀⠀⠉⠘⠄⠠⡻⣭⣭⣭
⠀⠨⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⠁⠀⠀⠀⠀⠀⠙⠶⣄⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢢⢹⢸⣿⣿
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢅⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠐⠅⠠⠀⠁⡀⠀⠀⢀⠀⡀⠀⠸⣔⡰⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⠿⠙⠋⠉⠀⢆⡄⠑⠒⣍⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠠⣶⣦⣤⡀⠀⠀⠀⠘⠀⠙⠒⢣⣤⢊⡀⡄⡎⠠⢀⠀⡟⡏⢹⣿⣿
⠀⠀⣿⣿⣿⣿⡟⠅⠀⠙⠒⠢⠀⠀⠀⠀⠀⠲⠦⣴⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⡴⠀⠈⠳⣄⠀⠀⠀⡯⠂⠘⡄⠻⠆⠁⠀⠀⠠⠀⠀⠀⣁⢸⣿⣿
⠀⠀⣿⣿⣿⣯⠸⠀⡀⠀⠀⡤⠦⢤⡜⡏⠓⠄⣜⠁⠀⠹⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠶⢾⣧⣤⣤⠀⠘⠂⠀⠠⢁⠀⠀⠀⠀⢃⠁⠰⠀⠀⠂⢀⢀⠋⣸⣿⣿
⠀⠀⣿⣿⣹⣏⠣⡀⠑⠤⠙⠟⠆⠀⠜⢁⣀⡴⠃⠀⠀⠀⢻⣿⣿⣿⣿⡁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⠃⠀⠁⢠⠄⠊⠀⠈⢀⣼⡀⠀⢀⠂⢣⣿⣿
⠀⠀⣿⣿⣿⡀⠀⠀⠃⠧⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣤⣸⣿⣿⣿⣿⣿⣧⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠈⠺⡆⠇⡔⠠⢁⣷⢿⠐⢊⣀⣾⣿⣿
⠀⠀⣿⣿⣿⣇⠓⠠⢤⣑⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠞⠉⠁⢻⣿⣿⣿⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⣎⣇⣷⡌⣇⠀⢀⡆⠽⣀⢠⢃⠔⡣⠿⠾⠻⣿⣿⣿⣿
⠀⠀⣿⣿⣿⠟⣀⢄⡶⠬⡽⠃⠀⠀⠰⠋⠋⠁⠀⣀⠀⠦⠀⠈⡿⢿⣿⣿⣿⡚⠛⠁⠀⠀⠀⠀⠀⠀⠈⠻⢾⡛⢌⡾⣶⣙⣮⣦⡛⠙⡼⠁⠀⢠⡤⣺⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣏⣸⡈⢉⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠆⠀⡇⢸⠉⠛⠻⢥⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣮⣩⠭⠆⠲⠰⡇⢀⣦⠼⢃⣽⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⢿⣿⣶⠬⠉⣔⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠞⠀⢘⠤⠖⢂⣀⣥⡬⠿⠶⣤⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣽⠞⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⡿⣛⣭⣷⠿⠋⠁⠀⠀⠈⣗⢤⡠⢀⣀⠠⠤⠶⢖⡏⣀⣊⣁⣤⠚⠉⠁⠀⠀⠀⠀⠀⠙⢿⡛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣕⣾⣾⣿⣿⣿⣿⣿⣿
⠀⠸⢟⣼⡿⠉⠀⠀⠀⠀⡀⠀⠀⢌⠂⠉⠒⠂⠒⠚⡑⡁⠀⠀⠈⠙⢦⡉⠁⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡻⣿⣿⣿⣿⣿⣿
⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠈⠑⠒⠶⠦⠀⢻⣰⠀⠄⡀⠀⢀⡗⠀⢠⣄⣶⣤⡴⠶⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⡻⣿⣿⣿⣿
⠀⢘⡇⠀⠀⠀⠀⠀⠀⣸⣁⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠈⠙⠓⢛⣉⠿⠖⠛⠉⠉⠙⠲⣍⡓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⣻⢿⣿
⢀⡟⠐⠊⠉⣉⡥⠶⠴⠦⠤⢤⣤⣄⣀⠀⠆⠶⠦⠤⢴⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣝⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣝
⣼⠁⠀⣠⠊⠁⠀⠀⠀⠀⡔⠁⢀⠈⠠⣙⢢⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠙⢦⣀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠃⣀⣓⣬⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⠋⠭⠛⣛⡶⠶⠶⠓⠲⣵⡾⠗⠲⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠁⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠴⠚⠉⠁⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠀⠀⠀⠀⠈⠙⢦⣠⣴⣶⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣞⣠⠖⠒⠶⠒⠊⠁⠈⠻⢿⣿⣿⣿⣿⣿⢿⣿⣦⣀⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠘⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⡿⠃⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠔⠋
⠀⠀⠀⠀⢠⡞⠛⠛⠛⠛⠛⠛⠛⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⢽⠀⠀⡄⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠈⠧⢼⡃⠀⢸⠀⠀⢹⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⡟⠀⠀⡇⠀⠀⡷⠞⠁
⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡇⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⢰⠒⡛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠀⣇⠀⠀⠈⠉⠉⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠔⠢⠤⠞⠀⠀⠀⢰⠋⠉⠉⠉⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⢬⡷⠋⠉⢢⣀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢴⡇⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠉⠀⠀⠀⡤⢄⣀⠀⠀⠀⠀⢀⡀⣀⡯⢿⣄⣀⡀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⡦⢈⠥⠃⠀⠀⠀⠙⡼⠁⠲⠒⠂⡠⠃⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣀⠴⠒⢲⠉⠳⡇⡞⠀⠀⠀⠀⠀⠀⣇⢀⣀⣀⣏⡀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠚⠛⠒⠒⠚⠓⠚⠓⠛⠛⠛⠛⠉⠋⠉⠉⠉⠉⠉⢳⡝⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢰⣇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⣽⠉⠘⢦⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⢲⠀⢰⠛⢆⠀⡰⠋⢳⠀⢠⠚⡆⢀⡞⡂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡄⡄⢸⠀⢸⠀⠇⠀⠀⡇⠇⠀⠀⢸⠀⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢃⢠⢸⠀⠘⡀⠀⠀⠀⠃⡂⠀⡇⡄⠀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠨⠘⡀⡄⠀⡇⠀⠀⠀⢠⠁⠀⡇⡇⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡄⡇⠆⠀⢀⢠⠀⠀⡎⡀⠀⠁⡇⡜⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢃⢠⠸⡀⡘⠘⣄⢀⠃⣇⣸⠀⣇⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠁⠀⠙⠁⠀⠈⠁⠀⠈⠁⠀⠈⠀⣛⠛⠒⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠛⠙⠛⢛⣓⣛⣉⣉⣉⣉⣉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠛⠶⠶⠶⠒⠒⠒⠒⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠒⠈⠉⠀⠉⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⣁⠤⠀⠂⠀⠘⠓⠤⡑⠄⠀⠀⠀⠈⠐⢄⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠳⣀⠀⠲⡄⠀⠀⠑⢌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⠆⡀⠈⢢⠀⠀⠀⢡⠂
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠀⠀⢅⠈⠀⠀⠀⠀⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠘⡀⠀⠆⠀⠀⠀⢏
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢀⣀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⢄⠁⠀⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⣠⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⣀⡇⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠄⠑⠀⠤⠄⠀⠤⠤⠄⠒⠋⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠆⠀⠀⠀⠀⢠⠆
⠀⠀⠠⠀⠂⠈⡆⠀⠀⠀⠀⡀⢄⠈⢜⠀⠀⠀⠀⢀⠀⢠⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⠀⠐⠈⠁
⠀⠀⠈⠦⣀⡴⠁⠀⠀⠀⠀⠠⡈⢂⠀⠢⣀⠀⠀⠀⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠈⠳⣞⠳⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⡀⠁⠀⠀⠉⠂⢄⡀⠀⠀⠀⠀⠀⠀⠠⠔⡩⠂⠀⠀⠀⠀⠠⠊⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠌⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠔⠊⠉⠉⠉⣑⠀⠀⣠⠔⠁⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⣀⠤⠊⠁⣉⠪⡀⠀⠀⠀⡠⠤⢄⡃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢫⠉⠁⠀⣀⠀⠀⠋⣁⠈⠢⣀⠊⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠑⡄⠀⠉⠁⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣙⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⢫⠀⣀⣀⠀⢀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠌⣶⠒⠁⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢸⠁⡜⠀⠈⠁⠢⣄⣀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠒⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠁⠀⠀⣂⣠⠜⠋⠀⠀⠀⣸⡀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡀⢀⣄⣧⣀⡠⠖⠉⠀⠀⢀⣀⠤⠚⠁⠉⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡱⠃⠀⠀⠀⠀⢀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠬⡖⡄⡀⡔⠊⠙⠛⢛⣶⠂⠤⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⢺⡋⠹⢦⡄⣀⣀⣻⣧⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⠀⠀⠈⡰⠊⠑⠂⡤⢿⠋⠉⡡⠃⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⠀⢃⠀⡠⠾⡁⠰⡖⠊⠀⠐⢄⢸⡅⠀⠀⠀⡀⠔⠋⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠎⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠒⠈⠉⡁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⠀⢠⠃⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠤⢰⠆⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠉⠢⠀⠀⠀⢂⠀⠀⠀⠀⠠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡜⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠀⠀⠀⠁⠢⠀⣠⣖⡲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡰⢶⠩⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⡔⡐⠰⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⡰⠋⠀⠸⡑⢝⠦⣙⠢⢄⡠⡀⡀⠀⠀⠀⢸⠔⠇⠤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠁⠀⠀⠀⠈⢓⡳⠦⡭⣙⠻⣌⠮⠒⠀⠀⠀⡇⢀⣐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠀⠀⡀⠀⠀⠀⠀⠄⠐⠀⠈⠙⣶⢽⡇⠁⠀⢠⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡑⢾⡁⠀⠀⠀⠀⣶⡷⠆⠀⠀⠈⣿⡗⠈⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣿⡾⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠪⠤⢀⡠⠻⠀⠀⠀⢀⠉⢶⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠉⠑⠀⠀⡠⠃⠀⠈⠆⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡯⡆⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠤⠐⢊⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠁⠀⠀⠠⠈⠰⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠀⠀⠀⠀⠀⡠⢑⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡠⠀⠀⠀⠀⠀⢶⣶⠐⣄⠀⠀⡠⠴⠻⣡⢀⢔⠣⠄⠀⠠⠔⢈⠔⢁⠠⠌⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣿⡆⠀⠲⠀⠁⠀⠙⡠⡊⠀⢄⡀⠠⠂⠀⠉⠀⠀⠀⠈⠀⠐⠁⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠈⠡⡀⠴⢨⠆⠀⢀⡳⠊⠀⢢⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣡⠄⠁⢀⣀⣂⠛⣆⡀⠀⠀⠈⢉⡀⠀⣠⣏⠤⠤⢤⡓⣤⠀⠀⠠⠤⢤⣤⡤⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢠⡁⠉⠉⠁⠉⠑⠖⠒⢶⣊⠭⠛⠉⠉⠉⠀⠈⠑⢄⠀⠀⠉⠒⠂⠀⠀⠀⢀⠀⣀⢰⢀⣀⠄⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⢀⡀⠀⢀⠴⢓⢟⠕⡠⠀⠀⢀⣀⡀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠘⢭⠉⠉⠠⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠉⠀⠓⠑⠨⠔⠁⠀⠀⡀⠀⢄⠀⠀⠀⠈⠐⠠⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠢⠄⠀⢤⠀⢀⡞⣠⠂⠀⠑⢄⠀⠀⠀⠀⠀⠈⠑⠂⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣂⠀⠀⠈⢃⠀⠀⣼⢠⠃⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡀⢌⡢⢌⣢⡇⢸⠀⢀⡠⠄⠀⢀⠁⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⠂⠉⠷⡌⠁⠀⢰⠋⠀⢀⡠⠊⠀⠀⠀⠀⠀⠁⠂⠠⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠘⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠖⡿⡻⣳⢶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣿⡿⡣⢚⢌⢞⠕⡱⣫⢊⠜⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠁⢸⣮⢀⠔⡥⡣⡫⢚⣞⣕⣫⣊⣟⠝⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⠀⠘⢇⡵⡫⡺⢊⢴⣷⡋⠁⠁⠀⠙⠟⠚⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣻⠏⠀⠀⠈⠙⠺⠶⢷⣫⠟⠁⠀⠀⠀⠀⠀⠀⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢄⠀⢰⡿⣿⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠱⣽⡆⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠘⡆⡾⣺⠇⢠⠟⠛⠻⡞⠃⠀⠀⠀⠀⠲⠞⠉⠙⠻⠄⠀⠀⠈⡻⠁⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢱⡟⠁⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠇⡜⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢢⠈⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⣀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⠐⢿⠉⢨⠇⠀⠀⠀⠀⢀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠑⠒⠲⣄⣸⠀⢸⠀⣀⠴⠒⠈⠁⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡓⠒⠀⠒⣾⠀⠈⡄⠤⠄⠐⠂⠁⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠛⠓⠦⢽⡶⣤⡔⠛⠀⠀⠈⢦⠔⠒⠐⠈⠉⠀⠙⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠜⠣⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⢲⠖⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⡸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⠞⢹⠁⠀⠀⠀⠀⠀⠘⡶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⡰⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⠹⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⡼⠁⡠⠋⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⣄⣀⣠⣤⠀⠀⠀⠀⠀⡽⢢⣄⠀⠀⠀⠀⠀⠀⣩⡔⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⡜⠀⠔⠀⡴⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢀⡴⠁⡼⠈⠷⢄⡀⠀⠀⠀⣿⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⡜⠀⠀⠀⠈⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡠⠔⠋⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠛⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⣰⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠁⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣤⣤⣤⣤⣤⣤⣤⣴⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣿
⠀⠀⠀⠀⣠⣾⡿⢻⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿
⠀⠀⠀⣰⣿⣟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿
⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠈⠁⢈⣍⡙⢻⣿⣿⣿⡇⣿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠈⣿⣿⣿⠃⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣜⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⠙⣷⠀⠿⣿⡟⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀
⠀⠀⠀⠈⣿⠿⠻⢿⣿⡇⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⢈⣸⠇⠀⢸⡇⣠⠞⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡇⠈⠙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⢸⠋⠀⠀⠈⠛⠁
⠀⠀⠀⠀⢙⣦⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⠀⠀⢸⣷⣶⣦⣤⣤⣀
⠀⠀⣠⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣤⣤⣤⢤⡤⠴⠚⠁⠀⠀⣾⣿⣿⣿⠃⠀⣽
⠀⣼⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⣍⠛⠛⣿⠀⠀⠀⠀⠀⣸⣿⣿⠛⠳⣤⠴⢧⡀
⢺⣿⣿⣿⣿⣿⣿⣿⠏⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⠀⠀⠀⣠⡾⠛⢿⡇⠰⢤⡏⢀⠞⢻⡆
⠙⠿⢿⣿⡿⠿⠛⠁⠀⠀⠀⠙⣳⡦⣤⣀⣀⣀⣀⣀⣀⣀⣤⠴⠺⣧⡀⠀⠈⣷⠀⠈⠙⠻⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠁⠀⠀⠀⠉⠛⠻⠿⠿⠛⠁⠀⢀⠀⢙⣦⡞⠁⠀⢠⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠹⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠟⠀⠀⠀⣰⠏
⠀⠀⣀⣤⣤⣠⡤⠞⠁⠀⢀⡴⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⣤⠞⠁
⠀⠸⣇⠀⠀⠀⠀⠀⠀⡴⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠸⣟⣛⣻⢿⡄⠀⢹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠀⠀⠙⠛⠋⠀⠀⢻⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣶⡖⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠚⠛⢻⡇⡟⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⡼⢧⣤⣤⣼⠀⣷⣤⣤⣴⢧⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⣙⣃⣛⣁⣀⣀⣸⣆⣇⣀⣀⣈⣛⣘⣋⣉⣳⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣶⣶⣶⣴⣶⣶⣾⣋⣷⣶⣶⣶⣶⣶⣶⣶⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠛⣿⣷⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠟⣽⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣽⣷⣾⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⣷⣾⣟⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣍⢻⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⢯⣽⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣾⣼⢿⣿⣻⣿⣵⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣋⣾⣿⡿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⡝⣿⡏⢿⣿⣿⣿⣿⣟⣿⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣷⣟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣿⣹⣿⣿⣾⣿⣿⣿⡽⢿⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⡷⠟⠛⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⢿⣿⣿⡿⣿⣽⣟⣿⣿⣯⣿⣿⣿⣷⣻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠙⢷⣿⣷⣹⣯⣿⣿⣿⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡽⠃⠀⠀⠀⠀⣤⠄⠀⠀⠈⠛⠋⠛⠶⢦⣄⣀⡄⠀⠈⠻⣿⣿⣾⣿⣿⣿⣿⣿⢿⣟⣽⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡟⠁⠀⠀⣀⡠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣏⣿⠃⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠂⠀⠀⠀⠀⠐⠒⠁⠉⠀⠐⠂⠀⠀⠀⠀⠈⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⣿⡟⡇
⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⢰⠀⠀⠀⣿⣧⣿⣿⣿⣿⣿⣿⣧⣾⣧
⠀⠀⠀⠀⠀⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⠀⠀⡘⠀⠀⠀⢠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣸⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆⢀⠁⠀⠀⠀⢸⠆⠀⢴⣶⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠇⠀⢠⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢸⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠀⠣⣄⡀⣤⠈⠃⠀⠀⠀⠀⠀⠘⠛⠃⠀⣷⣀⣀⣀⣀⠖⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣀⢻⣿⣿⣿⣿⡀⠀⢤⣶⡄⠀⠀⢀⣾⣿⣿⣿⣿⡏⠀⠀⣾⣿⡼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣧⠼⣛⣁⣿⣿⣿⣿⣿⣦⣤⣤⣤⣶⣿⣿⣿⣏⢿⣿⣶⣧⣾⣿⡿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡝⣧⣿⣿⣿⢛⡾⠋⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⢏⠙⡃⢿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡿⣸⣿⠃
⠀⠀⠀⠀⠀⠀⣼⣿⣯⣿⣿⣿⣯⣿⣯⣿⡏⣿⣿⡇⣧⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣪⠻⣏⣸⡿⠙⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣶⣟⣿⣿⣷⡿⣯⣿⠀⢧⣺⣳⡶⠐⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⠟⡏⠑⠻⠿⠛⠁⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡽⣿⣿⣏⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⢸⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⡿⣯⣿⣿⣯⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣷⣿⢿⡿⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣯⢿⣼⣿⣏⣿⠃⠀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠿⠿⣧⣀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣹⢿⣷⣦⣿⣿⣿⣿⠟
⠀⠀⠀⠀⣰⣿⡿⣿⣟⣿⣿⣿⣼⣿⣿⣿⣿⠀⠀⠀⡼⠃⠀⠘⢷⣼⣿⣿⣿⣿⣿⣿⡧⣤⡶⠒⠶⠔⠷⣄⠀⠀⠀⢸⣿⢿⣯⣿⣿⣿⣻⣮⢿⣿⣿⣇
⠀⢠⠾⣿⣿⣽⡿⣿⣭⣿⣿⣿⣧⣿⣿⣽⣿⣷⣄⣀⡗⠀⢀⠐⠛⠎⠙⢻⣿⣿⣿⡏⠀⠀⠘⣰⠇⠀⠀⠀⠂⠀⢠⣟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡿⢦⡄
⢠⣟⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⣿⣏⣿⠁⠀⠸⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠋⠀⠀⠀⢸⡶⣶⣯⢻⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢺⣿⡀
⠸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣾⣿⡋⠻⠷⠀⠰⠀⠀⠀⠀⠈⣿⣿⣿⣿⡀⠀⠰⠁⢀⣀⣠⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⡇
⢰⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣯⠀⢀⠀⠀⠄⠁⡀⠀⠀⢀⣿⣿⣿⣿⣷⠶⠃⠐⠀⠀⡤⣿⣟⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿
⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣤⣤⣶⣦⣤⣶⣿⣟⣻⣽⣿⣿⣿⣿⣿⣄⠀⠀⠈⠀⢰⣦⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻
⢸⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⣿⠏⠁⠈⠉⠉⠙⠛⢿⣿⡿⣷⢶⣿⠿⣿⣿⣾⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠘⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣀⣠⣤⣦⣀⣀⡈⢻⣿⣯⣿⣽⣾⣿⣽⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⡟⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣷⡟
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠇
⠀⠀⠘⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠘⢷⠟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣞⣡⠾⣧⠟
⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⡿⠿⣫⡿⠋
⠀⠀⠀⠀⠀⠀⠉⠻⢯⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣹⣿⢷⣿⡿⢿⣿⣻⡶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡿⣿⣿⡿⡿⢿⣶⣿⡿⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠿⠷⠶⠷⢿⣿⣿⣿⡿⠷⠾⠾⠟⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠰⠶⡶⠶⠦⠶⠶⠶⠶⠦⠶⠀⠘⢠⣄⣶⣀⣶⣶⣖⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⢿⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣶⣿⣷⣶⣾⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⡤⠶⠶⠶⠶⠶⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⠤⠤⠤⠤⠤⢤⡄⠀⣤⠴⠖⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠞⠁⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢧⣴⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣟⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢴⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣰⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡏⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠉⢹⡏⠉⠉⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢿⡄⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⢿⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠀⠀⠀⠘⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⢩⣿⣿⣿⣿⣿⣿⠿⣽⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⡿⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⢀⣿⣿⣿⣿⣿⣿⡟⠀⠈⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⢀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⡀⠀⠀⠀⠀⠀⣴⣿⠿⠯⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⢰⣿⠇⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠁⠀⠀⠀⠈⠂⢤⠀⢠⠞⠁⠀⠈⠐⠋⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⣼⡿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠃⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⡇⣼⣿⣿⣿⣿⣿⡿⣯⣿⣿⣿⡿⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⡟⠁⠀⠀⠀⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢯⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⣤⠂⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣣⣿⣿⣿⣿⣿⠁⠀⠀⢀⠄⠠⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠒⠒⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣿⣿⣿⣿⡇⠀⠀⠔⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣿⣿⣿⣿⣷⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠘⢯⠿⣿⣿⣿⣿⣿⣧⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣾⡇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⣠⣾⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠷⠦⠤⠾⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⠀⠀⢀⣠⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣄⡀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⣿⣿⣿⣿⣿⣿⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠉⠙⠛⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠙⠁⠀⠀⠀⠀⠀⠉⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠻⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣿⣠⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠈⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣄⣀⣩⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠻⣿⣿⣿⠛⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢀⡀⠀⠈⠛⠁⠀⠀⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠸⠟⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠂⠓⠀⢀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣀⣈⡝⠆⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣻⣶⣤⣀
⠀⠀⠀⢀⣾⡀⠀⠀⠄⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢨⣧⡳⣤⣀⣁⣠⣤⠤⠶⠿⠿⣿⣿⣿⣿⡿⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣧⠙⠲⠬⡁⠀⢀⡤⢶⠿⠛⡻⠿⠿⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣏⠙⣷⣄⠀⠀⠁⠀⠀⠆⢠⡎⠀⠀⠀⠀⠈⠀⠀⡿⠛⠛⠛⢿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣦⡈⠙⠒⠀⠀⠀⠀⠀⠀⣤⣶⣤⣀⠀⠀⣀⣼⠃⠀⠀⠁⠀⠙⢄⠈⠉⠀⢨⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⢸⠃⠙⢿⣒⡀⠀⣤⣤⣤⡀⠀⠉⠛⠿⢿⣿⣿⡿⠁⠀⠙⠢⠤⠀⠀⢀⠄⠀⡀⣻⡟⠿⢿⣿⣿⠇
⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠞⠃⠀⠀⠀⠹⣋⠀⠀⢰⠟⠛
⠀⠀⠀⠀⠈⠙⠳⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢁⣩⠗⠀⠀⠁⡿⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠻⣭⡉⠉⠛⠛⠻⠷⠶⠶⡶⡿⠛⠀⠊⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⡽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠃⠎⢰⣿⣿⢿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⢿⣏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠡⠃⠃⠛⠉⠀⠁⠿⠟⢹⣿⡟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠹⣯⣿⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡹⠀⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠉⠉⠉⠙⠟⠁⡀⠀⢰⡿⠁⢀⣀⡀⠀⠙⣿⡄⠀⠀⢸⡟⠀⠀⢀⠞⠁⠀⠀⠈⢳⡄⠀⢿⡄⣿⠀⠀⣎⠀⠀⢀⡷⠀⣿
⣦⡀⠀⠀⠀⣯⣉⠂⢼⡇⢠⡏⠀⠉⣷⠀⢸⡇⠀⠀⢸⡇⠀⠀⠸⡀⠀⠀⠀⠀⣸⠇⠀⢸⡇⠸⣇⠀⠘⠶⠶⠋⠀⢀⡿⢀⣀⡴⠷
⠈⣿⠀⠀⠀⠀⠁⠀⠸⣧⠀⢧⣤⡴⠋⠀⣼⠇⠀⠀⠈⣿⠀⠀⠀⠳⣄⣀⣠⠔⠃⠀⠀⣾⠁⠀⠙⢷⣦⣀⣀⣠⡴⠋⠁⢸⡏⡖⠙
⣡⡟⠀⣀⣠⠶⠶⢶⣤⣈⠻⢶⣤⣤⣴⠞⠋⠀⡔⠈⠳⡈⢷⣄⡀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⢀⣬⣩⣁⣀⠀⠀⠀⠘⣧⡳⣤
⠋⢀⡴⠋⠀⠀⠀⠀⠀⠙⠿⣷⠈⠀⠀⠀⠀⠀⠙⠶⠞⠁⠀⠉⠻⠷⢦⣤⡤⠴⠛⠉⠀⠀⢀⡴⠟⠋⠉⠁⠉⠙⠻⢦⣤⡀⠉⠙⠻
⢠⡿⠀⠀⢀⡴⠛⠙⠓⢦⠀⠘⣷⠀⠀⣠⠶⠲⣦⡀⠀⠀⢀⣴⠞⠛⠛⠿⣦⣀⠀⠀⠀⣴⠟⠀⠀⠀⠀⢀⣀⡀⠀⠀⠙⣦⡀
⢸⡇⠀⠀⣾⠀⠀⠀⠀⢈⡇⠀⣾⡆⠀⢻⣄⠀⣸⠇⠀⣰⠟⠀⢀⡤⢤⡀⠀⢻⣇⠀⢸⡇⠀⠀⠀⣠⠞⠉⠉⠉⠓⣄⠀⠘⣷
⠘⣷⠀⠀⠹⣄⣀⢀⣠⠞⠁⠀⣿⠃⠀⠀⠉⠉⠁⠀⠀⣿⠀⠀⣏⠀⠀⣹⡆⠀⣿⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⢸⡆
⠀⠹⢷⣀⠀⠈⠉⠉⠁⠀⢀⡼⠋⣀⣠⠶⠶⢶⣤⣄⠀⢹⣇⠀⠘⠓⠚⠋⠀⣰⡟⠀⠘⣧⠀⠀⠀⠱⣀⠀⠀⣀⡴⠋⠀⠀⣾
⠀⠀⠈⠻⠿⢦⣤⡤⣴⣾⠉⠀⣶⠏⠀⠀⣀⠀⠈⠙⣦⠀⠻⢷⣤⣀⣀⣤⣴⠋⠀⠀⠀⠛⣦⡀⠀⠀⠈⠙⠋⠉⠀⠀⢠⣾⠇
⠟⠛⠿⢷⣦⡀⠀⠀⠉⠉⠀⣼⡇⠀⢠⠏⠉⠙⣦⠀⢹⡇⠀⠀⠈⠉⠉⠓⠁⠀⠀⠀⠀⠀⣨⣿⣶⣦⣄⣀⣀⣀⣠⡴⣛⠁⠀⠀⣠
⠀⣀⡀⠀⠈⠻⣆⠀⠀⠀⠀⠙⣧⠀⠈⢧⣤⠴⠋⠀⣸⠇⡠⢤⡀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠈⠙⠻⣯⠉⠉⠀⠘⠿⠀⢠⡾⠁
⠋⠀⠉⠳⡄⠀⢿⡄⠀⠀⡠⢤⡘⣧⣄⠀⠀⠀⣀⡴⠏⠘⢧⣠⠇⢀⣀⡀⠀⠀⢀⡟⠁⠀⣠⠖⠶⢤⡀⠘⣷⠀⠀⠀⠀⠀⣿⠁
⣄⣀⣀⡼⠃⠀⣾⠃⠀⠐⢧⣀⡼⠀⢉⣛⣛⣋⡉⠀⠀⠀⠀⠀⡴⠋⠉⠙⢷⡄⢸⡇⠀⠀⡇⠀⠀⢈⡷⠀⣹⠀⢀⣀⡀⠀⢻⡄
⠈⠉⠁⠀⢀⣴⠏⠀⠀⠀⠀⢉⡴⠟⠋⠉⠉⠙⠻⢷⣤⡀⠀⢸⣇⠀⠀⠀⢸⠇⠈⣷⡀⠀⠙⠲⠚⠋⠀⢠⡟⠀⠸⠦⠃⠀⠈⣷⣄
⠶⢤⠴⠾⣋⣥⢤⣄⠀⠀⣰⠟⠀⠀⠀⣠⠤⢤⣄⠀⠙⢷⡀⠀⠙⠳⠶⠶⠋⠀⠀⠈⠻⣶⣄⣀⣀⣠⡶⠋⠀⠀⠀⠀⠀⢀⡀⠈⠛
⠀⠀⠀⢰⡏⠀⠀⠈⣷⢀⡟⠀⠀⢠⠟⠀⠀⠀⠈⢷⠀⠘⣷⠀⠀⠀⠀⣴⠄⠀⠀⣠⠞⠉⠈⠉⠉⠉⠛⢷⣄⠀⠀⢠⡞⠋⠙⢳⣄
⠀⠀⠀⠘⢧⣀⣀⣠⠟⠐⣧⠀⠀⠘⡄⠀⠀⠀⠀⣼⠁⠀⣿⠃⠀⢀⣠⡉⠀⢠⡞⠁⠀⠀⠀⣠⣤⣤⣀⠀⠘⢷⡀⢸⣄⠀⠀⢀⡿
⠶⠖⠶⢶⣤⣈⠉⠀⠀⠀⢹⣆⠀⠀⠱⢤⣤⠴⠚⠁⠀⢰⡟⠀⠀⠸⢄⠅⠀⣾⠁⠀⠀⢠⠞⠁⠀⠀⠉⣦⠀⠘⣧⠀⠙⠓⠚⠋⣤
⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠸⡀⠀⠀⠀⢀⡟⠀⠀⣿⠀⠀⠀⣠⠟⠁
⠶⠋⠉⠙⢲⡄⠈⢷⡀⠀⠀⠀⠀⠙⠛⣿⠿⠛⠉⠙⠻⢶⣄⠀⠀⣠⠤⣤⡀⢸⣇⠀⠀⠀⠙⠶⠶⠖⠋⠀⠀⢰⡟⠀⠀⣼⠃
⠀⠀⠀⠀⠀⣿⠀⢸⣇⠀⣠⠤⣄⠀⢸⠏⠀⣠⠒⠲⢤⠀⢻⡄⠀⣇⠀⢈⡿⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⣿
⣄⠀⠀⣀⡴⠃⠀⢸⡏⠀⠻⣦⠞⠀⢿⡀⠀⢧⠀⠀⣸⠇⢸⡇⠀⢈⣽⣋⣁⣀⠀⠀⠙⠛⠷⢦⣤⡤⠖⣋⣁⠀⠀⠀⠀⢿⡀
⠈⠙⠋⠉⠀⠀⢰⡾⠀⠀⠀⣤⠄⠀⠈⢿⣄⠈⠙⠋⠁⣠⣿⡴⠛⠋⠉⠉⠉⠙⠷⣦⡄⠀⠀⠀⣠⡴⠛⠉⠙⠛⢷⣄⠀⠸⣷⡀
⣀⡀⠀⢀⣤⡶⠋⠀⠀⢀⣠⡶⠶⠶⣦⣌⠙⠻⠶⠶⠛⣥⠟⠀⠀⠀⢀⣤⣤⣀⠀⠈⢻⣄⠀⣸⠏⠀⢀⡴⠦⣤⡀⠹⣧⠀⠈⠛⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠂⠀⠀⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⣀⠠⠄⢠⣀⡠⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⢱⠀⡎⠀⠀⠀⠀⢈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢩⠑⢢⡤⠠⠤⠔⠁⠀⠀⠀⠀⠀⠀⣀⡀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢇⠀⠀⠀⠀⢸⠂⢤⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⢴⠁⠀⢐⢂⠃⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇⠉⠉⠐⠢⣼⠆⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⡏⠈⠢⢀⣠⡨⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠐⠒⢶⠐⠒⠈⠑⠂⠀⠓⠀⢇⣀⡀⠀⠀⢸⢃⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣄⡀⠈⢢⠇⠀⠀⠀⠸⠀⠀⠀⣉⣉⣉⣉⣉⣉⣉⣁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣊⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠈⠉⢹⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⡠⠔⠓⢄
⠀⠀⠀⠀⠀⠀⢀⢔⠥⡀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⡜⠀⢀⠤⢀⣀⣀⣀⣀⠀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⢣⠁⠀⠈⢢⠀⠀⠀⠀⢣⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡀⠈⠁⠄⠀⡆
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢣⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣧⠀⢀⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡐⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⡠⡮⣋⠔⠁⠀⠀⠐⡌⢷⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⢸⢘⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠀⠀⠀⠀⡠⠒⠁⠞⣫⠊⠀⠀⠀⠀⠀⠈⠪⡛⠀⠑⢄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠄⠀⢸
⠀⠀⠀⠀⠀⢸⠀⢆⠀⠀⠀⠀⠀⡆⠀⠀⢀⡀⠠⠄⠊⣀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄⡁⠀⠂⠄⢸⡐⠀⠀⠀⠀⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠀⠈⡄⠀⠀⠀⠀⡰⠒⠈⠁⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠈⠄⠀⠀⠠⠁⠀⠀⠘
⠀⠀⠀⠀⠀⢠⠀⠀⠘⡄⠀⠀⠰⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠂⠠⠄⠠⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠤⠤⠗⠒⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡖⠒⠤⠒⠉⠉⠉⠒⢒⠁
⠀⠀⠀⠀⠀⠀⡯⠥⠤⠄⠈⠉⠉⠉⠉⣩⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢀⠁⠣⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠀⠤⠤⢀⡀⠀⠀⠀⠀⡀⠄⠂⠀⠉⠉⠉⠁⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⢰⠊⠉⠉⠀⠀⠈⠤⡀⠀⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢠⠁⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠃⠀⡠⠀⠀⠠⢤⠔⠒⠂⡄
⠀⠀⠀⠀⠀⠋⢦⡀⠀⠀⠀⠀⢀⠄⢸⠀⠀⠀⣁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡆⡊⠀⠀⠀⠀⢠⡀⠀⠀⡘
⠀⠀⠀⠀⢸⠀⢀⠀⢨⠃⠩⠌⢘⠀⠘⢄⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡓⢄⣠⡀⠀⢸⠉⠲⡼⠆
⠀⠀⠀⠀⠀⡄⢅⠁⠘⡀⠀⢁⣰⠀⠀⡜⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠉⠒⠄⠠⢀⣀⡀⠠⠤⠤⢄⡀⢘⡄⠀⠀⠀⠀⠑⢾⡱⢠
⠀⠀⠀⠀⠀⠈⠺⡀⠀⠈⠹⢙⠃⠀⠘⠒⠒⠊⠁⠉⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⡢⠀⠀⣸⠕⠃
⠀⠀⠀⠀⠀⠀⢀⡏⠐⠂⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡖⠈⢨⣀
⠀⠀⠀⠀⢀⣔⡁⡈⠒⠒⠊⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠊⠁⠀⠱⡀
⠀⠀⠀⢠⠏⠁⠀⠈⡀⠀⠀⠀⠈⢛⢦⠀⠀⠀⠀⠀⢀⡠⢾⢷⡴⠏⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡐⠁⠀⢡⠰
⠀⠀⠠⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠱⡀⠀⢀⠔⠁⠈⢢⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡐⠀⠀⠀⠘⠀⡇
⠀⢠⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡄⠀⠐⠔⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠽⣓⠄⡀⠀⠀⠀⠀⠀⢰⠀⢠⠁⠀⠀⠀⠘⠀⡇
⠀⡌⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⡀⠈⠢⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡌⠀⡇
⠀⠁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠈⠀⠐⠰⢤⠁⠀⠀⠀⠰
⢸⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⡀⠀⢁⠀⠀⠠⠁⠀⠀⢰
⠈⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⠆⠔⠁
⣧⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⣁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢄⠋⠀⡠⠂⠢⠀⠀⠀⠀⡀⢀⣠⠀⠀⠀⠀⠤⢀⠀⣀⡀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠈⠀⢀⡤⡀⠀⠀⠐⠁⠀⠀⠀⠤⡔⠀⠀⠀⠊⠀⢀⡅⠀⠀⠀⠀⢰⠉⢠⠈⠃⠀⠀⠀⢙⣡⠤⠐⠳
⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠀⠈⠲⠋⠂⠊⠕⠀⠀⠀⠠⠗⠀⠣⠠⠁⠀⠀⠀⠃⠓⠀⠲⠀⠀⠀⠀⠀⠘⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠌⠁

⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢐⠁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣘⢩⡀⠀⠀⠀⠀⠀⢠⠄⢠⠀⠀⠀⠀⠀⢠⠄⠠⡐⡈
⠀⠀⠀⠀⠰⠒⠂⠔⠒⠀⠀⠀⠰⡑⠋⠐⠁⠀⠀⠀⠌⠀⠀⣰⡀⠀⠀⠀⠀⣤⡊⣁⠔⡀⠀⠀⠀⠁⠀⠀⡴⢄
⠀⠀⠀⠀⠀⠐⠂⣠⠈⠀⠀⠀⠐⠊⠒⠊⠀⠀⠀⠀⠈⠂⠚⠀⠀⠀⠀⠀⠀⠈⠪⠇⠀⠀⠀⠀⠀⠀⢠⢌⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⢰⢒⣀⠀⡀⠀⠀⠀⢀⠤⠀⠔⢃⠁⠀⠀⠀⠀⡄⢖⠂⠀⠀⠀⠀⠊⠁⠀⢁⠀⠀⠀⠀⣐⡠⠠⠁⡠⡀
⠀⠀⠀⠀⡴⠒⠁⠀⠈⠀⠀⢀⡬⡀⠀⣠⠆⠁⠀⠀⠀⠰⠂⠘⠀⠀⠀⠀⠀⠑⠠⢐⠋⠃⠀⠀⠀⢋⠀⡀⡠⢌
⠀⠀⠀⠀⠁⠀⢅⢀⠊⠀⠀⠀⠀⠥⠴⠑⠀⠀⠀⠀⠀⠈⢀⡐⠂⠀⠀⠀⠀⠀⠀⠈⠄⠃⠀⠀⠀⠀⢁⠘⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⠶⠶⠶⠦⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⢀⠀⣀⣀⠔⣆⠠⠤⠤⠤⠊⣹⠀⠀⠀⠸⡄⠀⢀⡀⠀⢀⣤⣀⣀⣀⣤⠤⠖⠒⢋⠁⢠⣄⣰⣦⣄⣀⣼⣧⣶⠇⣀⡀
⠀⠀⠀⠀⠀⢑⠆⠀⠀⠀⠀⠀⡠⠄⠐⠺⡁⠀⠀⠀⠀⠈⠉⠁⠀⢠⠊⠀⠀⠀⠀⠀⠀⣦⣀⣸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣠⠞⠀⢀⣀⡤⠿⣦⠀⠀⠀⠀⠀⡘⡰⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠟⣁⡀⣀⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⡜⠀⠀⠀⠀⠀⢠⠃⠐⢴⠋⠀⠀⣀⢸⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠑⠋⠀⠈⢻⠀⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠘⡄⠀⠙⠦⠀⠁⠈⣹⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠂⠀⣸⠀⠸⣧⠐⣽⣿⣀
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣀⢀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡼⠁⠀⠀⢁⡀⠀⠀⣠⠏⠀⠈⡇
⠀⠠⣔⠃⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠙⣉⠴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣐⣁⣈⠵⠚⢁⠀⠀⢠⠃
⠀⠀⠀⠉⢩⠇⡀⠀⠀⠀⠀⠀⢳⢸⠉⢀⠀⣈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠁⠀⠀⠀⠹⠀⠀⡼
⠀⠀⠀⠒⠷⢚⡇⢀⡴⡆⠀⢀⡟⠚⢷⣾⠛⠉⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠉⠁⠀⣇⡤⠋⢠⣦⠼⡌⢱⠆⣠⠜⣠⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⡇⠀⠀⠀⠀⠀⣸⠃⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⡎⢳⣀⣱⠗⠤⠤⢄⡈⢩⡽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡼⠁⠀⠀⠀⠀⢀⠯⢄⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⢠⡇⠀⠉⠈⣧⠀⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢦⣄⣀⣀⣀⣘⣒⡂⣀⢑⡄
⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⣿⠉⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣀⡉⠉⠉⡉⠳⡿⠊
⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⡘⣆⠀⠀⠀⣀⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⡓⠒⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡏⠉⢍⡜⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣀⡀⠀⡇⢸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⣶⢧⣽⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠙⢶⠷⠋⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⢰⠁⠀⠱⡙⡷⡦⢤⡤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣘⣧⣄⣀⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠿⣄⡀⠀⢱⣤⣤⣤⣴⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡟⠁⢰⡀⠀⡏⠈⠉⣽⣗⣒⡶⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠴⠖⠒⠋⠉⠀⠀⣀
⠀⠀⠀⠀⠀⡧⠂⠀⠀⠠⣼⠉⢀⣿⣷⣾⣿⣄⣠⣏⣀⣉⣉⣛⣛⣒⠲⠶⠶⠦⠤⢤⣤⣤⠤⠤⠴⠶⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠠⠽⡄⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣙⣛⠋⠭⠿⢿⣿⣿⣿⣷⠶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠚⠳⠤⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⢙⣿⠿⣿⣿⣿⣿⣶⣶⣄⡲⣤⣬⣭⣉⠩⢭⣥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣉⣿⣿⡿⢿⣿⣿⣧⣾⠋⠀⢠⣿⠟⠛⣿⣿⣷⣤⣌⡉⠉⠙⠛⠓⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠈⣻⠿⠛⢻⡇⠈⠀⠈⢻⡄⠀⠰⠿⠃⢀⣸⣿⠃⢀⣹⡿⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠏⠀⣰⠏⠀⠀⡿⠀⠀⠀⠀⠈⢿⠦⣄⠀⠀⠈⠉⠁⠀⠚⠛⠁⠀⠈⠉⠉
⠀⠀⠀⢀⣠⠞⠁⠀⢠⡟⠀⠀⢰⠇⠀⠀⠀⠀⠀⠈⢧⠈⠙⠢⠄
⠀⠐⠒⠉⠀⠀⠀⠀⡾⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⣿⣿⣿⣿⡏⠛⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠓⠀⢀⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⢿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡆⠉⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡠⠀⠀⠀⠀⠏⢀
⠀⠀⠀⠀⣄⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⢻⠂
⠀⠀⠀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⣀⠀⠠⠠⠀⠄
⠀⠀⠀⢂⠀⠀⠀⠘⡂⠀⠀⣀⡈⠉⠉⠙⠐⠀⣸⡄
⠀⠀⠀⠀⣣⣄⠀⣀⣠⠀⠀⢠⡄⠀⠴⢄⡀⠊⠙
⠀⠀⠀⠀⠀⠀⠛⠀⠱⡘⠀⠄⠀⠈⢠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⣴⣿⡋⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⣿⠿⠟⠻⠋⠱⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠛⢔⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠢⣰⡟⠈⠒⢄
⠀⠀⠀⠠⠀⠂⠀⠁⠀⠀⠀⠀⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠐⠢⡀⠈⠐⠐⠀⠀⡀
⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠬⠐⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠱⠀⠄⢀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢁⢊⠝⢀⠀⠈⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠉⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢀⡤⠄⠤⠳⠄⡀⠀⠀⠀⠀⠑⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠁⢀⠀⡀⠀⠀⠈⠢⡀⠀⠀⠀⠈⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⡀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠂⠀⠤⠀⠀⣀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠶⠦⣄⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈⢶⣠⠖⠛⠁⠢⣱
⠀⠀⠀⠀⠀⠀⢀⠄⠒⢍⠑⠋⠈⠀⠀⠙⠢⣰⠂⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⣨⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢀⠠⠀⠧⢀⣀⣀⣁⣀⣀⣤⡤⠥⠤⠋⠀⠀⠀⠠⠤⠤⠀⠀⠀⠒⠒⠒⠒⠒⠈⠀⠀⠂⠀⠀⠀⢀⠆⢀
⠀⠀⠀⠈⠀⠀⠀⠒⠒⡠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢛⠛⢆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠐⠀⡸

⠀⡀⠌⠀⠀⣒⠀⠀⠠⠀⠄⠀⠀⢒⠄⠀⠀⠀⠀⡄⠀⠀⠀⠡⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠁⠀⡀⢀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⢀⠤⢀⠀⠀⢀⠀⡀⠀⢀⠄⠀⡀⠀⠣⡜⠈⠂⠀⠑⠂⠀⠐⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠄⠀⠁⠀⠃⣀⡠⠂⠀⠐⠤⠂⠀⠈⠤⠂⠉⠉⠐⠄
⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠈⡆⠀⢀⢀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠰⠀⠀⠐⠈⠀⠄
⠐⠄⠔⠀⠀⠀⢄⠀⠀⠀⠀⠧⠀⠈⠈⠀⠀⢀⣀⡀⠀⢀⠅⢃⠀⡐⠄⠀⢀⠀⢁⢀⠀⠐⠠⠄⠁
⡄⠀⠀⠀⠊⠉⡆⠀⠐⡈⠀⢀⠁⠀⠄⠄⠀⠂⠀⢈⠀⠀⠀⠘⠀⠐⠄⠠⠂⠀⠀⠐⠀⠀⠀⠀⠀⣐⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠁⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠒⠢⠀⠀⠠⣀⡰
⠠⠂⠀⠐⠄⠀⠀⠂⠀⣂⢀⠀⠀⠀⡀⠀⠐⠄⠄⠀⠀⠆⠀⠐⠀⠀⠀⠀⠠⠁⠀⠡⢀⠔
⢀⠀⠀⠀⡘⠀⠀⠀⡎⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠄⠀⠠⠂⠀⠀⠀⠠⠀⢄
⠀⠁⠒⠈⠀⢀⠀⠀⠁⠄⠀⠔⠀⠀⡀⠀⠐⠁⠀⡨⠂⠀⠀⢄⠀⠤⠔⠀⠀⠀⠀⠀⠀⠆⠆⠀⠠⠀⠐
⠀⠱⢄⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠌⠀⠀⠀⠀⠀⠡⠀⠀⡐⠁⠁
⠐⠥⠔⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⡄⠀⠀⠀⡠⠄⢤⠀⠀⠐⠀⠐⠁⠀⢁⡨⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠜⠐⢢⠀⠀⠀⠀⠀⠠⠔⠀⠀⡀⠈⠤⠀⠀⡀⠀⢀⠂⠀⠠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠊
⠈⠒⠀⠀⠀⠄⠀⠔⠠⠂⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠁⠀⠀⠉⠀⠀⠂⠂⠀⠀⠀⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣄⠀⠈⠊⠀⠀⠀⠀⠢⠀⠀⠀⠐⠀⠀⠠⠀⠈⠁
⠒⠪⡈⠀⠀⠀⠄⠀⠈⠠⠀⠀⠠⠊⠀⠀⠈⠘⠐⠀⠀⡀⠀⡀⠀⠀⠀⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠑
⠀⠀⠘⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⢆⡠⠀⠀⠀⠀⠆⠀⠸⠀⠈⠐⠐⠈⢀⠀⡐⠄⣀⡀⠀⠁⡄⠀⠈
⢠⠊⠉⠂⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠔⠀⢢⠀⠀⠂⢀⠀⠀⠀⠣⠀⠠⠁⠀⠐⢄
⠈⠑⠐⠁⠀⠀⢀⠀⡀⡌⠀⠀⠂⠀⠀⠀⠑⠐⠁⠀⠀⡀⠑⠠⡊⡸⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⡢
⠔⠒⢄⠀⠀⠀⠐⠀⠂⢈⠂⠐⠁⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠐⠀⠀⠈⠒⠁⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠖⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠈⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠑⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠁⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⡀⠌⠀⠀⠀⠠⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠐⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⠟⠉⠁⠀⠀⠉⠻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣤⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠘⠻⠟⠛⠉⠁⠀⠐⢶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠰⡆⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡏
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⣤⣤⡀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⠁⠀⠀⠀⠉⠔⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⡀⠄⠒⠀⠀⠒⠠⡀⢂⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠜⠀⣠⠤⠤⠤⢀⡀⠈⢳⡀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⡠⠋⠀⠀⠀⠠⠄⠈⠑⡀⣇⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢻⡧⠊⠁⠀⠀⠀⠀⢀⡀⠈⣼⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⢣⠤⠤⠀⠀⠀⠉⠁⠀⠀⡟⠀⠀⠀⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠄⠀⠀⠢⠀⢀⠀⠀⢀⠃⠀⠀⢀⠇⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⣢⣀⠀⠐⠀⢀⣠⠎⠀⠀⢀⠌⢀⠎⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⠁⠀⢀⣈⠙⣒⠞⢧⣎⣀⣀⣔⣥⠖⠥⠤⠤⠄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡰⠄⢞⠩⠠⠔⠢⠴⡢⣀⡀⢀⡠⠔⠈⠁⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡌⠀⢰⠃⠀⣁⠀⠀⢁⡀⠑⡊⠁⢀⡠⠄⠒⠒⢍⡉⠂⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢨⠀⠈⠀⠀⠆⠀⢁⠀⠏⠀⠀⡇⠀⠀⠀⠀⠱⡄⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⢄⡤⢄⡠⠄⢚⡐⠁⠀⠀⡇⠀⠀⠀⠀⠀⢠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠉⠚⠤⡀⠀⠉⠐⣄⢰⠀⠀⠀⠀⠀⠆⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⡴⡀⣠⠀⠀⠀⠀⠀⠀⠑⠌⢢⠀⠀⢀⠏⠀⠀⠀⠀⠘⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⠷⠑⠴⠡⠀⠀⠀⠀⠀⠀⠀⢀⣣⠴⠋⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⡗⡖⠠⠤⠄⠒⠒⠉⠀⠀⠀⠀⠀⢀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄⣀⠀⠀⠀⢀⣀⣠⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄⣤⢠⡤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⡴⣦⣿⢯⣛⣷⣾⣿⣿⣿⣾⣦⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣵⣻⣿⣿⠏⣤⣾⣿⠿⠛⢋⣻⡯⠙⢦⣽⣿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣻⣾⣿⣿⣶⣶⡿⠿⠥⠵⣞⠛⠋⠀⠀⠈⠁⠉⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢻⣿⠻⣿⣿⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹⣿⣿⡟⠀⣿⣿⣟⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣤⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣦⣼⣿⡟⠀⠀⠀⠀⣉⣭⣭⡛⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⣿⣿⣿⡇⠀⠀⠀⠔⠻⢿⡿⠃⠀⠀⠈⢿⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣼⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣇⢸⣿⠔⣲⡤⢶⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⠻⣿⡟⠁⠘⠋⣠⠞⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡀⢄⡀⠀⠀⠀⠀⠀⠘⠛⠂⠟⠀⠀⠀⠀⠁⠰⠛⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⡿⠇⣀⠙⠓⠲⠤⠀⠀⠐⣾⠃⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠑⠤⡀⠸⠤⣆⠀⠀⢀⣸⣿⣇⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠈⠑⠊⠵⢶⣾⣶⣶⣶⣶⣛⣿⠀⠀⠀⠠⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠈⠉⣳⣦⣀⢀⠀⠀⠀⠈⠉⠀⣤⣽⡗⠦⡶⢲⢰⣿⣿⡿⣛⣶⣲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⢀⣀⣤⠖⠉⠁⣀⣀⢈⣹⡆⠀⠀⠀⠀⠘⡶⠶⠚⠛⠛⢳⣬⣹⣿⣿⣿⣶⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡴⠉⠀⠀⠀⠀⠀⢿⣿⣼⣿⣿⣆⠀⠀⠀⢰⠇⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣰⠃⠀⠀⣄⢰⣤⣀⣼⠛⠛⣻⣿⣿⠣⠀⠀⡞⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠉⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠠⠶⣿⣯⣿⠟⠁⣰⠟⠉⠻⢮⠙⠷⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⢞⣿⣟⣧⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣠⡀⠀⣴⢫⠇⠀⣴⠃⠀⠀⠀⠀⠀⣷⠘⢧⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡛⢿⣿⣿⣷⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⠙⠿⢶⠇⠎⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⡀⢮⣳⡀⠀⠀⠀⢹⣿⣿⣿⣿⣷⡾⣿⠋⠀⠓⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡗⠒⢦⡀⢠⣏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠃⠀⠀⠀⣸⣿⣿⣿⣿⣿⣫⢟⣫⡤⠤⠀⣻⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠛⠲⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⣠⠿⢹⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⣸⣿⡮⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠈⠻⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠁⣠⡿⠟⢉⡶⢿⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⢶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣷⣶⣶⣶⣶⡿⠋⣀⣴⡟⢁⣾⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⣿⣿⣩⡇
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡀⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣚⡄⣤⡈⠀⠀⠈⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠀⠀⠀⠶⢀⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢤⠀⠀⠀⠀⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠂⠀⠀⠀⢀⡎⠧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠁⠤⠀⠤⠾⠆⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣥⣀⡀⢂⠀⠀⠠⡀⠀⠀⢀⠀⠀⠄⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠈⠙⢛⠀⠀⠀⠄⠀⠰⠀⠀⢰⠀⡞⠁⠘⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡄⠀⢨⠀⠀⠂⠀⠁⠀⠒⠀⢨⡄⠈⠀⡄⠀⢂
⠀⠀⠀⠀⠀⠀⢀⠇⠀⡘⠀⢠⢆⡁⠀⠀⠀⠁⠀⢀⣀⣸⣇⣀⠀⣇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣘⠀⠐⠀⠀⣾⠄⠽⠓⡂⢒⡚⠛⠛⢙⠁⡀⠀⠑⠺⡄⠀⠃
⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⡻⡴⠀⠀⠀⠀⢉⣗⡴⡗⠛⢀⠔⡀⢰⡄⣄⠀⡐⠠⢀⡀⠀⠀⠀⢀⠄⠄
⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠱⣇⢀⣴⡋⠀⠀⠛⢅⡀⠉⠁⠶⡤⠤⠤⢿⣉⡁⠀⠀⠈⠉⠒⠂⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠠⠄⡀⠀⠈⠐⠃⠀⡽⠀⠀⠀⠀⠈⠁⠀⠀⠠⠄⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⡀⠌


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣡⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣤⣄⡀⠀⢠⡾⣫⣾⣿⣿⣿⣿⠿⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠀⣽⣿⣿⣦⣿⣿⣿⣿⠟⠋⠁⠀⠀⢀⣤⣀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣾⣿⡄⠀⠀⠀⠀⠀⠀⠉⢻⣷⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠛⠿⢿⣿⣿⣿⠟⠋⠉⠉⠉⠁⠀⠀⠀⣴⣿⣷⣄⠀⣿⣧⣿⣿
⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⣠⣾⣿⡿⠁⠀⠀⠀⠀⠀⢠⣶⡄⢰⣿⣿⣿⣿⠸⠟⠛⠉⢻⣧
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⣴⣿⣿⣿⣶⣿⣶⡄⢰⣾⣷⣼⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⢀⣿⠁⠀⠀⠀⢀⣾⣿⡟⠀⣿⠿⡿⠆⠟⣸⣿⡿⠿⠿⠋⠉⠁⠀⠀⠀⠀⢀⣠⣴⣦⣄⢻⣧
⠀⠀⠀⢸⡟⠀⠀⠀⢀⣾⣿⣿⠀⠀⢻⡟⠂⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⣁⣈⡉⠈⣿⡄
⠀⠀⠀⣸⡇⢰⡆⠀⣼⣿⣿⠇⠀⢀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⠁⣾⣿⣿⣿⡆⢸⣧
⠀⠀⠀⢹⣷⢸⡇⢀⣿⣿⡿⠀⠀⣸⠃⣿⠀⠀⠀⣠⣴⠿⠟⠛⠛⠂⠀⠀⠀⠀⣿⠀⠿⠏⠛⠛⠀⠘⣿⡀
⠀⠀⠀⠸⣿⡈⠇⢸⣿⣿⡇⣰⡆⠟⣼⡇⠀⠀⠀⠿⠁⠀⣀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⢻⣇⠀⢸⣿⣿⣿⣿⠁⣼⡿⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⢻⣧⡀
⠀⠀⠀⠀⠈⣿⡆⢸⣿⣿⣿⣿⢰⣿⠁⠀⠀⠀⠀⠀⠀⠘⣿⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠙⣷⡄⠀⠀⢀⣤⣀⠀⣀
⠀⠀⠀⠀⠀⠘⣿⣾⣿⣿⡿⠟⠘⣃⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢴⠟⠈⠛⠃⠀⠀⠀⢸⣷⠀⢀⣿⠛⣿⢸⣿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⢰⠋⣤⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠁⠀⠀⠀⠀⠀⣤⡀⠀⣾⡏⠀⢸⣿⠀⣿⠈⢿⣆⠙⣿⡀
⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢺⡄⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⢃⡀⠀⠀⣀⣠⣼⣿⣧⣼⡟⠀⠀⢸⣧⠀⢿⡆⠀⣿⠀⢸⡷⣦
⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⣌⣿⣌⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠘⠻⢿⣿⣿⠿⣿⣇⠈⠛⢷⣦⡀⠘⣿⠀⢸⣷⣼⠏⢀⣾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⢿⡄⣴⠾⣾⣷⣿⣿⣿⣷⣦⣤⣤⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⣿⣿⣦⣄⠈⠛⢿⣿⡇⠀⠙⠋⠀⠘⠋⠀⣼⠃
⠀⠀⠀⠀⣰⠄⠀⠈⠁⠀⠀⠈⠛⠿⠿⠻⣧⢀⡄⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⠈⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⢰⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠛⢸⣿⠀⠀⠈⠛⠷⢶⣤⣤⣤⣤⣶⡶⣶⣶⣶⣶⣿⣟⣋⣀⣀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠸⣿⡄⠀⠀⠨⣿⣦⣀⡀⠀⣀⣴⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⣿⠟⠛⠙⠛⠛⠷⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠹⣿⣦⡀⠀⠈⠻⢮⣽⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠻⠟⠿⣿⡷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⡆⠀⢠⣾⣿⣿⣿⣿⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠸⣷⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠃⠀⠙⠻⠶⣦⣤⣀⣀⡀⠀⠀⢀⣾⡟⢀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠶⣤⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⣿⡿⢶⣤⣀⣀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠉⠛⠻⠛⠛⠁⠀⠠⠀⠀⠀⠀⠀⣄⢴⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠾⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⢈⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠨⠛⠿⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⠈⢸⣿⠀⠀⠀⠀⡀⠀⠀⠠⢀⣴⣧⡀⠀⠀⡰⠀⠰⠃⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⢻⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣆⠀⠀⠂⠀⠆⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⡀⠘⡆⠀⢀⣠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢣⡀⢃⠀⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠐⣢⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠐⡀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄


⡿⠿⠿⠿⠿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⢿⣿⢿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠶⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢸⠁⡎⣼⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⢡⡴⠒⠋⠀⠀⠀⠘⢄⠱⠜⠂⠘⢦⠀⠀⠀⠀⠀⠀⢀⡶⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠘⠯⣍⣉⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⡤⠤⠼⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⡀
⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳



⣶⣶⡶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⡻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠋⢲⢰⡇⠀⠀⢹⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣥⣶⡝⡏⣾⣿⣿⡎⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠘⠦⠤⠞⣀⠑⠦⠴⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣧⣙⣛⣵⢿⣮⣛⣫⣼⣿⣿⣿
⠀⠀⠀⠀⠀⢠⠞⣫⡥⠤⢭⡛⢦⠀⡏⠉⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⣡⠶⣛⣛⣲⣮⡙⣿⢰⣶⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⡏⠘⠁⠀⠀⠀⠀⠈⣧⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢸⣧⣾⣿⣿⣿⣿⣷⢸⣷⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡼⠷⢤⡀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣃⣘⡻⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡏⠈⠀⠀⢳⡀⣠⡤⢤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢰⣿⣭⣽⡬⣿⣟⣻⡟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⢤⣤⡴⠋⠉⣇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⣛⠟⣒⣴⣶⠲⣶⡾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠄⠀⠀⠁⠠⢲⡃⣠⠶⢀⢠⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢄⠀⠀⠁⢠⡀⠀⢀⠼⢋⣔⠔⢈⣤⠆⠐⣂⠰⠂⢀⣼⠀⡆
⠀⠀⠀⠀⠀⠀⠠⠄⠒⣒⣀⠒⠒⠦⠵⠀⠀⠀⠛⠀⢀⠀⠉⠀⠀⠉⠀⠀⠌⠠⠄⢊⠵⠃⢠⡃⣀⡀
⠀⠀⠀⠀⠀⣠⠔⡊⠩⠤⠤⣄⡀⠀⢢⣤⠖⠶⠀⡰⠋⢠⣀⠉⠙⠃⣀⡄⠀⠐⠚⠹⠓⠊⡜⠉⠉⠉⠉⠙⡆⠀⠀⢡
⠀⠀⠀⠠⠞⠋⠁⡀⠀⠀⠀⠈⣁⠔⠋⠀⠴⠗⠊⠀⠀⠀⠉⠁⠈⠉⢠⢴⠀⠈⢁⣠⣈⠭⢍⡉⠹⡇⠀⠀⣿⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⠀⠄⢂⠞⢁⠜⣡⣴⠆⠀⠀⠀⠀⠀⠐⠛⠛⠻⢷⡄⠈⢠⣶⣰⣁⢀⣀⣀⠀⠳⡄⠉⠀⠀⡏⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠘⠉⠀⡰⠁⡰⠃⠞⡁⠤⠀⣀⠀⠀⠀⠀⠀⠀⠒⠂⢤⡀⠀⠀⠁⢁⠟⠉⠀⠀⠁⠀⠀⠀⠀⠀⣧⠀⠀⠀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⢠⣷⡸⠁⠀⠀⣠⠖⠉⠙⡄⠀⠀⢀⠉⠉⠲⢤⣀⢳⠀⠀⠀⠀⠒⠂⠐⠂⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠾⣹⠃⠀⠀⠎⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⢤⣄⠈⡇⠀⠀⠀⢐⣠⠀⠀⠘⣄⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⡠⣿⠀⠀⠀⢀⠀⠠⣿⣦⠧⠄⠀⠀⠢⢈⡻⠯⠊⠀⠀⠀⠀⠘⠁⢠⠺⠗⠉⠉⠇⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⡜⠁⢸⠀⠀⠀⠀⠈⠀⠙⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⡴⠃⢸⠀⠀⠀⠀⠀⠀⠳⡄
⠀⢀⠀⣀⠀⢧⡀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⢾⡀⠀⠀⠀⠀⠀⠀⠙⣆
⠐⠓⠈⠉⠉⣀⣉⠿⠷⣖⠢⡀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⣉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⣠⠒⠺⡁⠈⢱⡄⠘⡄⣇⡰⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢯⡼⢃⡲⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⡇⢀⠀⠘⠀⢐⠃⡰⠶⠛⠁⡤⠂⣁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⠖⠂⠛⠺⠵⠿⠼⢆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣷⠈⠀⠈⠐⠊⢣⠀⠀⢀⡜⢑⣤⣤⣀⣀⡀⠀⠀⣀⣀⣀⣤⠴⠚⠉⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡠⢎⡴⠋⡌⠀⠀⠉⢹⡋⠉⣉⡵⠋⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⢀⡝⠉⠁⠀⠈⡩⠔⠋⠀⡼⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⣠⠋⠀⠀⠀⠀⠀⡄⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠖⠛⠉
⠀⡜⠁⠀⠀⠀⠀⠀⠀⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢲⣾⡛⠋⠉
⡜⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⢩⡿⡇
⡇⠀⠀⠀⠀⠀⠀⢀⡜⣡⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⡼⠁⣧
⢱⡀⠀⠀⠀⣀⣴⠿⠚⠁⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠆⢻
⠀⠙⠲⠖⠚⠉⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢼⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⠁⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠄⠀⠚⠛⠀⠀⠀⣀⠀⠀⠀⢀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⡤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠔⠒⡖⠋⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠘⡶⠶⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣹⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⢾⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⣉⣉⣁⣦⣀⡈⠉⣜⣭⡿⢏⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣵⡾⠛⠿⠿⠿⢿⣿⣧⡾⠋⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⣦⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠈⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣠⣴⣦⣤⣀⠀⠀⠀⠀⣤⣾⡿⠿⢶⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠘⠁⠀⠀⠈⢛⠀⠀⠀⢠⠉⢁⠀⠀⠀⠀⢸⣿⢄⠀⠀⠠⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⣿⠁⠀⠀⠀⣠⡴⠽⣆⠱⡄⠀⠏⣠⠷⣿⠙⠆⢠⢸⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡓⣜⠃⠀⠁⠘⠈⠛⠁⠚⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠈⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢹
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⡙⠇⠀⠀⠀⠀⠀⠀⢀⡄⢧⠀⠀⠂⡷⢦⡀⠀⠀⠀⡷⠋⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⠀⠀⠀⠀⠀⢰⠘⢉⡡⡒⢋⢭⠀⠀⢱⠀⠀⠀⣿⠀⠀⠉⠹⠗
⠀⠀⠀⠀⠀⠀⠸⠽⠀⠀⢠⣶⠸⡆⢠⠀⠀⠀⣸⠔⠙⢐⠟⠃⢸⡗⠂⠃⠀⠀⠀⡏⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⣇⣠⠞⠁⢀⠔⡁⠀⡠⢏⡩⡆⠀⠀⢠⣿⠇⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⠊⠁⢀⠔⠁⠀⣠⠞⠁⢀⠞⢁⣀⠴⣫⠎
⠀⠀⠀⠀⠀⢐⡒⢦⠀⠀⢀⡴⢺⠟⠀⡐⠒⠁⠀⢀⠞⠁⠘⣴⣋⣉⣉⠔⡖⠁
⠀⠀⠀⠀⠀⣼⠀⢠⣇⠔⠋⠀⠀⠀⠘⠀⠀⠀⠈⠁⠀⣠⣊⡭⠞⠉⠈⡸⠃
⠀⠀⠀⠀⠀⣿⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠴⠛⠁⠀⣠⠴⠋
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⣉⣉⣩⣿⢷⣲⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣀⣐⣒⣪⣝⢷⡝⣫⠲⡟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠉⠀⠀⠀⠀⠙⡎⢿⢜⢷⠌⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁⠑⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣴⣒⣀⡀⠀⠀⣀⣈⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡩⠿⢿⣤⡀⠺⠿⠇⢀⠀⠀⠀⠀⠉⠉⡡⣖⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣧⣤⠏⠀⠉⠓⠶⠖⠃⠀⠀⠀⠀⠀⠜⠁⠀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠾⠤⠂⠐⠂⠀⠀⠀⠀⡀⢀⠦⡀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢠⡤⠀⠀⠀⣀⢸⡍⢻⡀⢹⡦⡀⢀⠔⠁⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠈⠳⡝⢧⠀⢣⠀⡇⡇⠀⢠⠇⠀⠀⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⡤⠄⠀⠀⡿⠈⠀⢸⠀⡗⡿⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠑⡆⠀⢠⠃⠀⠀⠀⠀⠁⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡡⠀⠀⠀⠀⢰⠰⡇⣠⠃⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⠃⠀⠀⠀⠀⠀⡏⠀⠐⠁⠀⠀⠀⠀⠀⠠⠃⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⢸⠀⠀⠀⠀⠀⢸⠁⣀⡤⠴⠒⠒⠀⠀⢀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠀⢸⡀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⣀⠀⠄⠀⢳⡀⠀⠀⠔⠁
⠀⠀⢀⡠⠴⠒⠋⠉⠀⠀⠀⠀⠀⡄⠀⠀⠀⣇⠀⠀⠀⢺⡀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⢱
⠠⠊⠁⠀⠀⣂⢄⠀⠀⠀⠢⡀⢸⠀⠀⠀⠀⢸⠀⠀⠀⡇⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄

⠀⠀⠀⢀⠔⠂⡠⠐⠇⢀⡂⣀⡀
⠀⠀⢠⠁⠀⠔⠀⠀⠀⠀⠀⠐⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠈⠁⠀⠉⠁⠒⠠⢀
⠀⠀⡄⠀⠀⠀⢀⡄⢀⡀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⡠⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠇⠀⠠⠀⠀⢠⡁⠂⠀⠀⢀⢀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢌⠁⠒⠃⠒⠂⠀⠐
⠉⠐⡄⠀⢐⠀⠀⠻⡘⠋⠒⠊⢫⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠤⢀⠀⢠⠌⡄⠀⠀⠀⠀⠀⠀⠘
⠀⢪⠀⠀⠈⠀⠀⠀⠀⠀⢀⠚⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⠁⣕⠀⠐⠊⠀⡘⠅⠀⠀⠀⠀⡤
⠀⠂⢴⠀⠀⠀⠀⠀⠀⣾⣦⠅⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⠉⠀⠀⠄⢀⡾⡀⠀⠀⢀⠃
⠀⠀⠈⢧⡀⠀⠀⠀⠢⠉⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠐⠢⠴⠋⠠⠁⠀⢠⠎
⠀⠀⠀⠀⠙⠢⢄⣀⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⠀⠀⠀⢀⣈⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠄⠉⡌⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠔⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠰⡀⠂⠀⠀⢪⠀⠈⠀⢄⠠⠀⠄⠀⠀⠐
⠀⠀⠀⠈⠀⠀⠀⠀⡀⣠⠀⠄⠀⢀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⡀⠀⠸⠂⠀⠉⠰⠠⠤⡘⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢂⠂⠀⠀⠀⠀⠀⠀⢰⠀⠠⠂
⠀⠀⠀⡇⠀⠒⠀⠀⠀⠀⠤⠃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠤⠀⠀⠰⠇⠀⠂
⠀⠀⠀⢘⠁⠐⠂⠀⠀⠀⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⢸⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠐⠀⠉⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⣰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡃⠀⣈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠀⠀⠀⠀⠠⠤
⠀⠀⠀⢀⢗⠐⠒⠅⣀⠀⠸⣉⠀⠈⡐⠂⠐⢦⠀⠀⠀⠀⡀⠀⠈⠀⢀⡀⣄⠀⢈⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢳⠨⠤⠀⢠⠁⠀⠀⢸⠃⠒⠃⠄⠀⠘⠆⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡤⠔⠒⠀⠉⠉⠉⠉⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⢦⡀
⠀⣠⠞⢁⠀⠀⠀⠀⠀⠀⢠⣔⡄⠀⢴⡿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⢱
⢰⠁⠠⣁⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠁⠀⠀⠀⠀⢀⡤⢖⣒⣖⣖⣶⣲⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾
⡇⠀⠀⠀⠀⠐⠀⣰⣯⣾⣿⣿⣿⣿⣿⡫⠋⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⢁⠀⡴⣾⣿⡟⢻⣿⣿⣿⣿⣿⡏⢩⢠⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⡄
⠈⠸⠃⠉⠉⣧⣬⣟⠛⠛⠛⠋⠀⠸⡟⢳⠀⠀⢀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠙⠃⠚
⠀⠀⠀⠀⠀⠙⢦⠙⢄⣀⣀⣀⣀⣤⡫⠃⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠁⣈⢳⠀⠀⠀⠙⢷⣤⣀⣀⣀⣠⠴⠻⣿⣶⢀
⠀⠀⠀⢀⣠⠴⠒⠒⠺⣵⡟⢫⡿⢤⣄⣀⣀⡀⠀⠉⠉⠉⠀⠀⠀⠉⠒⠁⠀⠀⢀⡤⠤⢄⡀
⠀⢀⡴⠋⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⠤⢤⣤⣀⣀⡀⠀⣿⢤⣤⢤⡇
⢀⡞⠀⠀⠀⠀⠀⣾⣟⣹⡟⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠈⠉⠙⠛⢿⡇⣞⠁
⢸⡇⠀⠀⠀⠀⠀⠙⢺⢿⠳⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⣶⡇⡏⠓⢦⡀
⠸⣧⡀⠀⠀⠀⠀⠀⡿⢞⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠂⠌⢄⡿⠇⡇⠀⠀⠙⣆
⠀⠙⠿⣶⣤⡤⠀⣀⣳⣬⣕⣃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢂⡞⣯⠉⢃⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠿⠯⠥⣤⣐⣂⡤⠤⢀⡀⠀⠀⠀⠀⠸⡛⢷⣆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠚⠻⠷⢶⣤⣄⣙⡒⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠶⣶⡶⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠉⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⣠⠶⠛⠛⠛⠛⠻⠿⠶⠶⠄⠀⠀⠀⠀⠲⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠈⢳⣤⡙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⢀⣤⣶⣶⣿⣿⣷⣶⣶⣶⣶⡶⠿⠋⠀⠀⡆⢿⡧⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣴⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⠇⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⢀⡞⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣤⣤⣷⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⡟⠀⠀⠀⠀⠀⢀⣾⡷⣄⠀⠀⠀⣄⣶⠟⠋⠀⠀⠀⠀⠀⠙⢳⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣾⠁⠀⠀⠀⠀⠀⢸⣿⠁⠈⠲⢤⣴⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⣼⡇⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠳⢦⡀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⢧⠀⠳⣤⣤⣀⣠⣤⣴⣾⣿⢹⡏⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠈⢧⠀⢀⣴⠟⠁⠀⠀⠀⢹⣾⡇⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠈⢳⣼⠋⠀⠀⠀⠀⢀⣾⡟⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠉⠀⠀⠀⢀⣴⣿⠟⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⣠⣤⡾⢛⡿⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠋⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠊⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⢀⠄⠠⢀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠔⠀⠐⠨⡐⢄⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠄⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠑⠒⠀⠒⠀⠀⠑⡀⠀⠀⠀⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠐⠠⠤⠐⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠂⠁⠐⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠔⢂⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⡀⠀⠁⠂⠁⠀⡀⠂⢠⢠⡠
⠀⠀⠀⠀⠀⠀⠀⠰⢁⠆⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⡀⠄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠐⠁⠀⠀⠀⠐⠀⠂⢄⠂⠀⠢⢀⣀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⠂⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠁
⠀⠀⠀⠀⠀⠀⡂⠈⠂⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⡆⢤⠔⠁⠀⠈⠀⠀⠐⠉⠐⢄⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠈⠀⠠⠀⠀⠀⠠⠀⠈⠀⠀⠀⠠⠐⠰⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀⠔⠀⠣⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⡴⠶⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠀⠀⠙⡇⠀⠀⠀⠈⠷⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⠞⠳⣤⣤⠞⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠉⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠻⣤⡤⠶⠶⢤⡀⠀⠀⠀⠀⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠉⢳⠏⠀⠀⠀⠀⢧⡤⠂⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠶⠶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⢀⣤⠶⠒⠲⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠞⠉
⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢳⣤⣄⠀⠀⠀⠀⠀⠀⣠⣀⣀⡴⠗⠛⠦⠤⣤⣤⠄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⣁⡼⠃⠀⠀⠀⠀⠘⢧⣼⠋⠀⠶⠒⠞⠀⢀⡜⠀⠀⠀⠀⣀⣠⡾⠋
⠀⢰⠟⠃⠀⠀⠀⢀⣠⣀⡤⠶⣄⡸⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢠⠖⠋⠀⠀⢰⡟⠛⠉⠁
⠀⢹⣄⠀⠀⣴⠛⠉⠀⠈⣷⠀⠈⣻⢀⡏⠀⠀⠀⠀⢀⣀⣀⣀⣀⣧⣀⣤⣴⣠⣼⡶⢤⣀⣤⠟
⡴⠞⠛⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⢳⡄⢿
⢷⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇
⠘⣇⣀⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⣤⢤⣤⡼⢿⡀⠹⣆
⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠻⢦⣜⣷⡀
⠀⠀⢸⡇⡴⢦⠀⠀⢠⠖⢦⡀⠀⠀⣴⠚⠲⣆⠀⠀⡴⠶⣄⠀⢀⡴⢦⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠈⣇⢻⠈⡆⠀⣿⠀⠈⣇⠀⢸⠃⠀⠀⢹⠀⢸⠁⠀⢿⠀⡼⠁⢸⠀⢰⡏
⠀⠀⠀⣿⢸⡄⢱⠀⢻⠀⠀⢹⠀⢸⠀⠀⠀⢸⡇⡘⠀⠀⣾⠀⡇⠀⢸⠀⢸⠇
⠀⠀⠀⢹⡀⡇⠘⡄⢸⠀⠀⢸⡀⢸⠀⠀⠀⠀⡇⡇⠀⠀⣿⢸⠀⠀⡾⠀⣿
⠀⠀⠀⢸⡇⢱⠀⣇⠸⡆⠀⠈⡇⠀⠀⠀⠀⠀⡇⡇⠀⠀⡟⢸⠀⠀⡇⢀⡟
⠀⠀⠀⠀⣇⠘⡀⢸⠀⣇⠀⠀⣇⠀⡇⠀⠀⠀⡇⡇⠀⠀⡇⢸⠀⢸⠁⢸⡇
⠀⠀⠀⠀⢿⠀⠇⠈⡆⢹⠀⠀⢸⠀⡇⠀⠀⠀⡇⡇⠀⠀⡇⣼⠀⡼⠀⣸
⠀⠀⠀⠀⢸⡆⢸⠀⣇⢸⡇⠀⢸⠀⣧⠀⠀⢸⠁⡇⠀⢰⠀⣿⠀⡇⠀⡿
⠀⠀⠀⠀⠀⣷⠈⣆⡾⠀⢷⣀⡼⠀⠘⢦⣤⠞⠀⠹⡤⠏⠀⠹⠶⠁⢰⡇
⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⠉⠉⠛⣷
⠀⠀⠀⠀⠀⠈⠛⠳⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠒⠛⠛⠛⠛⠛⠋⠀⠀⠀⠈⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣤⣤⣤⣤⣴⠶⠶⠤⠤⠶⠶⠶⠶⢶⡶⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠐⣿⣷⣄⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣴⣦⡀⠈⢻⣿⣦⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠙⣿⣷⡄⠀⢻⣿⡆⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠈⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⣸⣿⡏⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⣴⣿⡟⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣀⡠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠺⠇⠀⠀⡼⢟⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠒⠦⢄⡉⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⣀⠀⠀⠀⠀⠀⠉⠢⡀⢖⠟⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⢹⡀⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠥⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢦⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⣈⡉⠑⠉⠉⠉⢀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠻⠯⣉⠓⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠙⠓⠒⢲⡋⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡌⢀⡠⠃⠀⠀⠉⠉⠁⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⠇⠀⢠⠂⠀⠀⢸⢧
⠀⠀⠀⠀⠀⠀⠀⡍⠁⠀⠀⢠⠄⠀⠰⠂⠀⢸⠣⣄⡀⠀⠠⠀⡤⠀⠀⠀⠀⠀⣠⠃⡜⠀⠀⢠⠀⠀⠀⢸⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢄⠀⠀⠀⠀⢀⡀⡸⠀⠀⠈⠉⠘⠛⠂⠀⠀⠀⠀⡰⠃⢀⠇⠀⠀⢀⠀⠀⠀⢸⠀⠘
⠀⢀⡠⢶⠀⡠⠀⠀⠠⡀⠀⠙⠒⠒⠒⠉⣠⠃⠀⠀⠀⠀⠁⠀⠁⠀⠀⢀⠐⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠐⢄
⠀⠌⣶⡾⣋⡀⠀⠀⢀⠀⡗⠀⠀⠤⠀⠘⢅⠀⠀⠀⠀⠀⠀⠀⣀⢠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢱⠖⠒⠢⣄
⠀⠐⠁⢧⠔⠁⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡞⢃⡐⣷⠀⠀⠀⠀⠤⠀⠀⠀⠊⠀⠀⠀⠀⠀⠁⡀⠀⠹⠤⠗⠤⠬
⠀⠀⠀⠀⠀⠀⠀⡎⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⣟⠚⢁⡝⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⢀⣇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⢀⢊⡴⡤⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⠔⠒
⠀⠀⠀⠀⠈⠉⠀⠀⢰⡗⠂⠠⠤⠀⠤⠄⠀⠐⢊⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠉⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠁⠈⠓⠀⠒⠂⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⢤⠴⠀⠒⠒⠐⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠈⢩⠞⢳
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⣇⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⡄⠀⠀⠀⢠⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠙⡄⠀⠀⠀⠀⠓⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡋⠀⠈⠙⣻⠄⠀⠀⠀⠓⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠠⠂⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣿⣿⡇⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⢀⣿⠇⠃
⠀⠀⠀⠀⠀⠘⠀⠀⣈⣻⣿⣿⣿⣿⣿⣿⣟⠻⠉⠉⠉⠉⠁⠀⠀⠀⢱⣾⠇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠈⣿⢡⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⠀⣠⠞⢛⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⡰⢋⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣶⣤⡴⠏⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠉⢽⣿⣿⣿⠟
⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⠀⠀⠘⠿⢿⣇
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⢋⠁⠀⠀⠀⠀⠐⢄⡀⠀⠙⣿⡟⠁⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠘⡗⠀⠀⠀⠀⠀⠀⢘⣿⡆⢠
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⣀⡀⠀⠀⢀⣼⡿⣷⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⢸⡆⠀⡀⠀⠀⠀⠈⠉⠛⠛⠛⠉⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⣿⣿⣿⣷⣶⣤⣴⣿⣃⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⢀⡠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣄⡀⠀⠀⠀⠸⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠃⠀⠈⠉⠁⠰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠸⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠲⣾⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠐⢦⣀⣿⡆⠙⠻⣿⣿⣿⣿⡀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⣥⣧⠻⠦⠿⡾⢹⣿⡻⣿⣶⣾⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⡧⢀⠀⠀⠻⣿⣿⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⡇⠘⣿⣤⣤⢸⡍⢀⣹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣿⡿⠁⢸⣥⣼⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣴⣿⡟⢀⣸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡟⣳⣽⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣾⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠁⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠟⢩⣿⣿⣿⣿⣿⣿⠿⣽⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⡿⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠈⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠠⡀⠀⠀⠀⠀⠀⣴⣿⠿⠯⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢰⣿⠇⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠁⠀⠀⠀⠈⠀⢤⠀⢠⠞⠁⠀⠈⠐⠋⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⣼⡿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⡇⣼⣿⣿⣿⣿⣿⡿⣫⣿⣿⣿⡿⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⡟⠁⠀⠀⠀⡦⢀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢏⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⣤⠁⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣣⣿⣿⣿⣿⣿⠁⠀⠀⢀⠄⠠⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠒⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⣿⣿⣿⣿⡇⠀⠀⠔⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣿⣿⣿⣿⣷⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢯⠿⣿⣿⣿⣿⣿⣧⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⣽⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⣠⣿⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠷⠦⠤⠾⠿⠛⠛⠋⠉