Winter cold Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢚⡒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠉⠈⠈⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣀⣀⣄⣀⣄⣀⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡠⠖⠀⠉⣴⢹⡇⠁⠐⠦⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠚⠁⠀⣠⡴⣋⠛⠁⣘⣲⣄⠀⢈⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⣼⠋⣛⡋⠀⠀⠈⣉⠉⠳⡄⠊⠶⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⣯⠈⠛⠃⡠⢤⠘⠛⠃⠀⠇⠀⠀⠐⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⡄⣠⡆⢀⣄⣹⣦⢀⡼⠻⣼⢦⡀⢀⠜⠦⢄⠀⠀⣀⠠⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡞⠡⢡⠀⠎⠀⠛⠁⡏⠀⢠⢧⠀⠈⢣⠀⠀⠈⡇⠀⢹⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠞⠘⠘⢀⡽⡆⠀⠀⠈⡃⠀⠇⠀⡅⠀⣞⠁⠀⠀⠘⡆⠺⢧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠰⣄⠴⡼⠁⠀⡼⣤⣆⠀⠘⣤⠴⢄⡰⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢏⡀⠀⡔⠁⠀⠈⠂⢼⠀⠀⢠⠷⠊⠀⠀⠀⠷⢎⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⢈⠋⠀⠀⠀⠀⠀⡘⠁⠲⡞⢇⠀⠀⠀⠀⠀⠈⠇⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡴⢇⠀⠀⠀⠀⠀⢠⣅⣀⣀⣡⠈⡇⠀⠀⠀⢀⠌⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⠳⢄⡀⠀⢠⠏⠀⠀⠀⢹⠀⠈⢆⣀⠔⠋⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣈⡱⠋⠀⠀⠀⠀⡀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⡤⠜⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣳⠤⠤⠰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠉⠒⠤⠤⣤⡀⠀⠀⣀⢠⡇⠴⠖⠒⠂⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢀⣀⠠⠄⠚⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠉⠐⠒⠒⠢⠤⠤⠤⠄⠾⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠂⠠⠤⠀⢀⣀⣀⣀⡀⢠⡅⠠⢤⡤⠤⠤⠀⠀⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⣒⣴⣶⣖⣆⢄⣀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠂⠀⠄
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⢈⠛⡇⢠⡆⠈⠉⣛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠲⢦⡵⠈⠑⡎⠙⠾⠯⠎⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⣸⠁⠀⠆⠀⠐⠀⠀⠀⠀⠀⠀⠐
⠀⠐⠀⠀⢕⠣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡇⠀⠦⠀⠀⠀⠠
⠀⠀⠀⠀⠠⠀⠀⠔⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⢀⠠⢔⠀⢠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠢⠤⠤⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠲⠶⠦⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⢄⣰⡀⠀⠌⠛⡀⠀⣄⠐⣀⠘⢦⡀⣦⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠈⠀⣠⣦⣶⣆⠀⠀⠀⠐⠀⠲⠞⠀⠀⢠⡄⠓⠿⢴⠰⣤⠙⠦⣵⠀⠀⢈⣺⣧⡾⢦⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⣰⣿⣿⣿⣿⣿⣦⣄⠠⠀⠀⠀⢀⠀⠀⠀⠡⠘⠓⠒⠻⢿⣗⡀⡘⠧⠀⠉⠛⣿⡇⡀⢳⡀⢀⡸
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠛⠭⠄⠈⠐⠦⣄⣠⣿⠋⡁⠀⠳⣼⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣶⡄⠀⠀⣢⠀⠀⠀⠀⢀⠀⠀⠀⢙⢦⡀⠀⢻⣿⡆⠱⣄⠀⣿⡄⡄⠀⡆
⠀⠀⠀⠀⠀⣠⣴⣷⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠈⡁⡄⢀⣀⠐⠈⠉⠂⠖⠈⠉⠉⠛⣿⣷⡀⣿⡀⠸⣷⠁⣼⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣪⣈⣙⣀⣛⣀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣯⠙⢷⣄⣿⣧⣿⡤⠆
⠀⠀⠀⢐⣿⣿⣿⠿⠋⠡⡽⣯⣭⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢉⠉⠉⠉⡷⠶⣦⣤⣴⣿⣿⣏⢹⣿⣿⠁⠀⣿⣿⣿⣿⠥
⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⢨⢻⣿⡄⠀⡸⣹⣻⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⠇⠐⢊⣁⣴⡟⠁⠀⠈⠉⠉⠛⣿⣿⡆⠸⢻⣿⣿⠇
⠀⠀⠀⣠⣏⣀⣤⣲⡄⣠⣤⣼⣿⣿⣿⡆⣸⣿⠋⠌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣌⢆⣐⣢⠀⣂⣀⣳⡄⠰⣄⣿⣹⣿⠀⡞⢿⡃
⠀⠀⠀⠈⠉⠁⠐⠛⣟⣛⠛⠛⠟⠛⠉⠀⠁⠁⠃⠈⠘⠚⢋⣠⣽⣍⣭⣿⣭⣤⣤⣤⣥⣤⣈⣁⠀⠀⠀⠙⣶⣄⣼⣿⡏⠙⣤⣃
⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⠿⡃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣠⣶⣿⣿⡿⠛⠃⠀⠹⠏⠁
⠀⠀⠀⠀⠀⢠⣴⣦⣶⣉⣀⣤⣄⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣤⣿⣿⣿⠿⠟⢋⡟⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⡭⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⢉⠁⡙⡛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡴⢦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⠓⠇⠸⠞⢉⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⣞⠳⢼⠃⢿⠔⢛⡆⢀⣀
⠀⠀⠀⠀⠀⠀⣾⢻⡆⢀⣠⠬⣑⡆⢰⣞⡥⢤⣀⢠⡟⢳⠀⠀⠀⠀⠀⠀⢀⡤⣄⡟⢨⠇⣈⣓⢦⡀⣠⢖⣋⡀⣯⠸⣆⡤⣄
⠀⠀⠀⠀⣏⠑⠟⢸⡇⡏⢹⠦⣌⡁⢈⣡⠴⣿⢸⣆⡇⠸⠚⣹⠆⠀⠀⠀⠈⣲⡌⠁⢸⣸⠁⣈⡙⠁⠙⣉⡀⢹⣸⠀⠉⣴⡋
⠀⠀⠀⢠⠼⠛⣀⣄⡙⠃⢸⣀⠀⡇⢸⠁⢀⡿⠘⢋⣠⣄⠙⠧⣄⠀⠀⠰⣏⣡⠴⣒⡦⠉⠀⣇⣹⠆⢾⣁⡇⠈⢡⣔⡲⢤⣉⡷
⠀⠀⠀⠘⠒⠋⡵⠋⣡⠴⢦⣉⠛⠇⠸⠛⣉⡤⠦⣌⡙⢢⠙⠒⠋⠀⠀⡴⢦⡀⣎⠁⢤⠖⢲⠈⠀⠀⠀⠈⢰⠒⢦⠄⢙⠆⡠⠶⡄
⠀⠀⠀⠀⣀⠀⣟⠙⠧⣄⣠⠞⢋⡀⢀⡘⠳⢤⣠⠴⠛⣹⠀⣀⡀⠀⠀⢙⣦⠙⠮⠷⠀⠙⢋⠀⠀⠀⠀⠀⢈⠛⠁⠰⠯⠞⢡⣞⠁
⠀⠀⠀⠸⣍⡙⠲⠽⠂⠀⢠⠞⠉⡇⢸⠉⠓⣦⠀⠐⠯⠕⢋⣡⠟⠀⠸⣅⣤⡄⢠⢶⠆⣸⣉⡷⢀⣄⡀⠰⣏⣱⡀⢶⢦⠀⣤⣌⡷
⠀⠀⠀⠀⡴⢋⣀⢰⡞⡇⢸⣠⠴⠃⠘⠦⣄⣿⢸⡟⡆⢀⡙⢳⡄⠀⠀⠀⢼⢀⡏⠺⣤⣤⡄⠀⠘⣤⠟⠀⢀⣤⣄⡼⠘⣆⢹
⠀⠀⠀⠀⠙⠋⣿⢸⡇⠓⢾⣡⢶⡇⢸⡶⢤⡽⠚⠡⡇⣸⠉⠋⠀⠀⠀⠀⠈⠉⡴⠒⠒⠚⠁⢾⢣⡀⣰⢻⠄⠙⠒⠒⢲⡌⠉
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣠⠖⢋⡀⢀⣉⠓⢤⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣽⢁⣶⡈⢣⡉⣱⠋⣰⡄⢻⡉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⢧⣸⠉⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠙⠚⠁⠓⠚⠁⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣀⡤⣄⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠯⣝⠛⠘⣋⡽⠂⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣲⢬⡁⣩⢖⣫⣄
⠀⠀⠀⠀⠀⠀⣯⡻⠃⣿⡶⣞⡻⠷⠸⠟⣻⠶⣼⡘⠟⣹⠂⠀⠀⠀⠀⠀⠀⣞⢹⢰⢲⡄⠳⢬⡙⠃⢛⣩⠶⠋⣤⢦⣸⢹⡄
⠀⠀⠀⠀⠀⣏⡭⣥⣤⠛⢰⠇⠉⣦⢰⠋⠘⡇⠛⣡⣤⢭⣹⠆⠀⠀⠀⠀⣯⠋⠸⣼⢸⡧⠖⣋⣥⡆⣦⣍⡓⢦⣿⢸⡽⠘⢹⡇
⠀⠀⠀⢀⡤⡄⣸⢯⠴⠒⢮⡳⣄⣿⢸⣠⠞⡡⠖⠦⡭⢧⢠⢤⡄⠀⠀⣎⡥⣞⡷⠀⠀⠰⢯⡁⢸⡇⡧⢀⣩⠷⠀⢀⠲⣗⡦⣍⡇
⠀⠀⠀⡤⠷⠘⠿⠎⠳⠤⠤⠽⠮⠃⠘⠡⠮⠤⠤⠼⠡⠯⠏⠼⢥⡀⠀⠀⠘⠧⠞⢻⢸⡟⠲⣌⡛⠃⠛⣋⠴⠚⣿⢸⠓⠦⠽
⠀⠀⠀⠓⡶⢠⣶⠆⡴⠒⠒⣲⠖⡀⢠⡰⣖⠒⠒⢢⡰⣶⣄⢶⡚⠁⠀⠀⢴⡲⢤⣸⢸⣧⠴⢚⣡⡄⣤⣙⠲⢤⣿⢸⣠⠴⣦
⠀⠀⠀⠘⠳⠋⢻⡞⠶⠤⠞⡡⠊⡷⢸⠙⢮⡳⠤⠶⢓⡞⠘⠾⠃⠀⠠⣞⠲⢽⡶⠌⠈⠡⣞⡉⢸⡇⡇⠈⣙⡦⠁⠈⠴⣾⠗⣛⡆
⠀⠀⠀⠀⠀⣞⣓⠻⠟⣠⠸⡇⣀⡗⢸⣀⢠⡇⣤⠻⠟⣒⣳⠆⠀⠀⠀⠈⣟⣀⢰⣿⢸⡗⠦⣝⠺⠇⠿⢊⡥⠖⣿⢸⢳⢀⣘⡇
⠀⠀⠀⠀⠀⠀⣟⣡⡌⣟⠧⣏⣥⡆⢰⣬⣹⠴⢻⢡⣮⣻⠄⠀⠀⠀⠀⠀⠀⢿⣸⠙⠾⢁⡤⠞⣡⡄⣬⡙⠦⡀⠻⠼⢸⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠳⠞⠀⣖⣋⡅⢠⣝⢲⡄⠻⠞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡯⠟⠁⡙⠿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠻⠞⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠋⠳⠋⠓⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢣⠒⠊⠀⢋⠁⠀⢠⡭⠡⡅⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢆⠀⠖⠀⠀⠀⡕⠄⢨⣝⠲⠖⢂⣁⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠀⠀⠠⠋⠆⠀⢀⠠⠜⢒⢉⠠⠔⠈⠉⠀⠵⣦⣁⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠊⠀⠀⠳⢒⠀⠏⠀⡄⢲⠀⠀⠀⠒⠀⠉⡟⠁
⠀⠀⠀⠀⠀⠠⠈⡁⠄⠐⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠉⠀⠀⠂⠸⠦⠅⣇⡠⠀⠠⠄⠀⠁⠐⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠊⠀⠄⠂⠀⠀⠀⠂⠱⡄⠀⠂⠀⠀⢸⣀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠛⠛⠓⠛⠀⠈
⠀⠀⠀⠀⠀⠂⠄⠀⣀⠐⠀⣀⡤⠀⠢⡈⠄⠀⠀⣔⠨⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⣀⠈⠀⢀⡁⠉⠇⠀⠞⠁⡊⠄⢂⡀⠀⣐⠨⢹⡀
⠀⠀⠀⠀⠀⠘⣆⢀⠀⠀⠋⠀⠀⠘⠃⠀⠃⢰⢎⢊⠔⡛⠒⠀⠐⠓⠠⠄⡇⠀⠀⠀⡠⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠈⢢⡢⡀⠀⠁⠈⠀⡠⠂⠀⠀⡆⠀⠀⠀⠧⠀⠀⣴⠂⠀⠁⠀⠠⠊⠀⠀⠀⠀⢢⡀
⠀⠀⠀⢰⡆⠉⢀⡓⣮⣉⡉⣉⠏⠀⠀⠀⡀⠀⢙⠀⠐⠀⠀⣒⡂⠀⠀⢸⢀⠔⠩⠁⢐⡂⠤⣐⠯⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣇⣾⣆⡀⠀⠀⠀⠀⠀⢂⡘⡒⠦⠤⠠⠄⣊⠀⣮⠀⢈⡣⢂⠀⠀⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⠀⠈⠦⠤⡄⡔⠿⠃⠀⠀⠀⠀⢀⢂⠎⠉⠁⠐⠈⠢⣸⡄⠸⠓⠢⠈⠁⠒⠂⢃⠀⠀⡀⠀⢠⠁
⠀⠀⠀⣴⠂⠚⠠⢁⠀⣠⠁⠧⠃⡀⠀⠀⠀⠀⠰⠾⠀⢀⣈⣙⠄⠆⠚⡻⠉⠁⢀⠀⠰⢻⠉⡍⢲⠋⠉⠉
⠀⠀⠀⠚⠒⠀⠐⠈⠀⠀⠑⢆⣠⠗⠆⠀⠀⠀⠀⠀⢄⠀⠀⠬⢴⠀⠀⡃⠀⠀⡀⠴⢙⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⠃⠈⠀⠀⠀⠀⠀⠀⢰⠄⢰⡆⣼⠀⠀⠡⡀⠀⠀⠀⠀⠬⠿⠇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠻⠒⠘⠂⠀⠀⠈⠢⠤⠄⠀⠀⠰⠪

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠶⠶⠶⣦⣤⣴⡶⠿⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢀⣠⣤⣴⣶⠶⣶⣤⣤⣀⣠⣴⠶⠿⠛⠻⠷⢶⣤⣄
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣠⡾⠛⠉⠁⠀⠀⠀⠀⠀⣽⡟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⣠⣴⣿⠁⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⢀⣾⠏⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢨⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⢶⣼⣇
⣿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⢿⣦
⠹⣧⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠙⢷⣄⡀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⡇⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠉⠛⠻⠿⠿⠛⠋⠻⢶⣤⣄⣀⣀⣀⣤⣴⠾⢻⣧⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢭⣉⡍⣩⠁⡀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣸⠀⠀⢹⡟⠁⠀⣿⡤⠀⠀⠹⢷⣄⡀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⣠⣼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠙⢦⡀⢸⡇⢀⡴⠋⠉⠉⣀⠀⠀⠙⠛⠷⠶⠶⠶⠶⠿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠻⠶⠶⠶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⣀⣀⣙⣶⣷⣏⣀⣀⣠⣼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠟⠛⠛⢛⣿⣿⡿⠟⠛⠻⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⢀⡴⢛⡟⢦⡀⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣀⢀⡴⠋⢸⡇⠙⢦⡀⣀⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣶⠋⠀⢸⡇⠀⠙⣶⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⢿⠀⠀⢸⣇⠀⠀⡿⠯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠀⣠⠾⡷⣄⠀⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠶⠻⠏⠳⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠈⢻⡷⠃⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠛⠻⣄⢸⡇⣠⠞⠛⢓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡦⠤⠬⣿⣿⠧⠤⠴⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣀⣠⠞⢹⡏⠳⣄⣀⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⣼⣧⡀⢸⠛
⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣷⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⢸⣿⣧⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣾⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣽⣿⣇⠀⠀⠀⣰⣿⣷⣶⣼⣿⣿⣤⣤⣤⣶⣶⣆⠀⠀⠀⢀⣿⣿⣴⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣛⣿⣿⣿⣿⣆⡀⠀⣿⣿⠋⠛⢿⣿⣿⠟⠛⠛⢻⣿⣿⠀⢀⣠⣾⣿⣿⣿⣿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠺⡿⠿⠿⠟⠛⠻⢿⣿⣿⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣾⣿⡿⠟⠋⠉⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣶⣤⣿⣿⣀⣠⣴⣾⣿⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠙⠻⣿⣿⣿⣿⣿⡿⠋⠉⠙⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣶⣄⡀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠛⠁⣿⣿⡇⠈⠙⠻⢿⣿⣿⣿⣇⠀⠀⢀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣿⣿⣿⣿⣿⠿⢿⣿⡄⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠁⠀⢸⣿⣷⣶⣶⣿⣿⣿⣶⣦⣤⣰⣿⡏⠀⠈⢿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠁⣿⣿⠃⠀⠀⠀⠙⠛⠛⠛⢻⣿⡟⠙⠛⠿⣿⣿⠇⠀⠀⠈⣿⣿⡍⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⣿⣿⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡖⠀⠀⠈⢻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠧⠤⠤⠤⠬⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡎⣿⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿⡎⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢸⠀⣿⡀⡤⠒⠈⠋⠁⠀⠈⠁⠀⠀⠙⠙⠒⢦⣄⣿⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢬⡴⠋⠀⠀⠀⠀⣤⠴⠂⠀⠀⠀⠀⠀⠠⢤⣀⠀⠙⢷⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠖⠚⢧⣄⠀⠀⠀⠀⢁⠂⠉⠉⠃⡤⠊⠉⠁⠢⡝⠀⠀⢀⡟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠤⣄⠀⠀⠀⠀⠞⠀⠀⠀⠀⠹⠇⠖⠈⠉⠁⠀⣤⣄⠀⠀⠀⣤⡄⠀⠘⠰⡦⠞⠙⢇
⣤⣄⡀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠈⠙⣆⠀⢘⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠐⢀⠉⠁⡀⠘⠁⠺⣅⠠⠂⠀⠀⠠⠀⢸
⢹⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠊⠑⢤⣀⣀⣠⣮⡜⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢈⡆⣀⣠⢧⡄⠀⣸
⠸⡆⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠶⠿⠿⠷⢶⣧⠀⠀⠀⣈⢩⡏⠀⠈⠐⠄⡀⠀⠉⠛⢿⣦⣤⣠⡤⠊⠉⠁⢀⠀⢷⡞⠁
⠀⡇⠀⠀⠀⣀⣴⣾⡿⠛⠁⡀⠀⠀⠀⠀⠀⠘⣦⠔⠋⠁⠀⣧⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠁⠀⠐⠘⣧⣄
⠀⠳⠖⠒⠋⠉⢿⠁⢀⡴⠋⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢠⡾⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣼⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠘⣦⠋⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⣰⠏⠀⠀⠀⠙⠳⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⠴⠞⠋⠙⣧⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠺⣿⣷⣤⣀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣠⡾⠋⠲⢤⣀
⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠈⠙⠻⠿⠿⠿⣶⠛⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣤⣤⣶⡿⠛⠁⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣠⣴⣶⠀⠀⠀⢸⣾⣧⠉⠛⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢻⣦⡟⠉⠁⠀⣤⣄⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠹⢦⣤⣤⡴⠞⠉⠀⢸⡀⠀⠀⣸⠇⠙⢦⣄⡀⠀⠈⠉⠙⠛⠲⠶⢦⣤⣤⣤⣥⣤⣤⣤⡤⠤⠶⠚⠛⠉⠀⠀⣼⠿⡇⠀⠀⠀⡇⠀⠉⠳⢦⣤⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠚⠋⠀⠀⢸⡏⠙⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢾⠁⠀⠙⠦⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠛⠓⠶⠶⠤⠤⠤⣤⣤⠤⠤⠶⠖⠚⠋⠉⠀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⡴⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⠓⠠⠄⢀⣀⣀⣀⣀⠀⠤⢻⡗⠠⠤⠤⠤⠤⠀⠛⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠠⠄⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠣⢴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣔⡤⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣘⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠐⠆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣁⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠟⠡⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠉⢹⠉⢻⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠤⠤⠚⠓⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡤⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠈⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣀⣀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠂⠲⢤⡀
⠀⠀⠀⠀⠀⠀⢠⠔⠋⠉⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣀⣀⣀⣀⣨⣷⢀⡀
⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠀⠀⠀⣠⠴⠞⠋⠉⠁⠀⢠⠤⡀⠈⠉⠳⢦⡀
⠀⠀⢀⠔⠉⠀⢀⣀⢠⡤⣤⠠⢤⣤⣤⣤⣤⣄⢀⠈⢣⠀⠀⠀⣠⠚⠁⠀⠀⢀⠠⠀⠀⠘⠂⠀⢀⠄⡄⠓⠻⡄
⠀⠀⣎⡴⠖⠛⢫⡁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢈⠟⠺⡄⠀⣰⡕⢀⣀⠤⠴⠚⠒⠛⠛⠋⠉⠛⠉⠛⠛⠐⠦⣵⢄⡀
⢠⠞⢩⠀⠀⠀⠀⡇⠀⠀⣀⣀⣀⣀⣀⣇⡀⠠⠄⠀⠀⢸⣤⡷⠚⠁⠀⠀⠀⢀⣀⠠⠤⠤⠤⠀⢀⣠⠞⠀⠒⠌⠂⢱
⢸⠀⠀⡄⠀⣠⠴⠛⠉⠁⠈⠁⠀⠀⠀⠈⠙⠳⣀⠀⢠⠎⣥⠀⠄⣀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢢⣍⣂⢀⣼⡇
⠘⣅⠀⣰⠞⠁⠀⠀⠀⠀⠺⠃⠘⠛⠀⠀⠀⠀⠈⢶⠋⢰⡇⢠⡞⠏⠀⠀⠀⠀⢀⢴⡦⠠⡶⢦⡀⠀⠈⠑⢲⡶⢋⠏
⠀⢣⣼⣁⣀⣀⣀⣀⣀⡤⠂⠉⠉⢛⠓⠢⠤⠤⣤⡼⣧⠃⢱⣸⣀⣀⢀⣀⣀⣀⣈⣀⣀⣠⣤⣤⣤⣀⡀⠀⣀⢹⠇
⠀⠈⢧⡤⠄⣉⣈⣯⠀⠀⠀⠀⠀⢘⠂⢀⣀⣀⠼⢧⠃⠀⠀⢻⡍⠋⠹⣉⣉⠋⠉⠉⠁⠸⣀⡀⠀⠈⠉⠉⢏⣹⠇
⠀⠀⠘⢦⡀⠈⣧⠀⠀⠈⠉⠉⠉⠉⠁⢸⡇⠀⢀⠜⠀⠀⠀⠈⢧⠤⡀⠀⠀⠀⠀⡀⢄⠀⠀⠀⠀⢀⢀⠀⠀⡟
⠀⠀⠀⠀⡇⠀⢸⡀⢀⣀⣀⣴⣀⠀⠀⢸⡇⢠⡇⠀⠀⠀⠀⠀⠈⢷⣾⡤⡤⠤⠴⢦⣴⠥⣤⣀⣀⢤⣠⡶⠚
⠀⠀⠀⠀⠛⡶⣟⠉⠁⢸⠁⣹⢹⡟⠉⠉⢻⡟⢧⡀⠀⠀⠀⠀⠀⠘⠊⠻⡷⢶⠾⠳⠟⠳⣿⡅⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠃⡇⠂⠐⢛⢀⡇⢸⠀⠀⠀⠀⡇⠀⠉⠳⣄⡀⠀⠀⠀⠀⡼⢁⠋⠀⠀⠀⢸⡟⡄⠀⠀⢸⣇
⠀⠀⠀⠀⢸⢀⠁⠀⠀⢸⠨⡇⢸⠀⠀⠀⠀⢰⡀⠀⠀⠀⠙⠳⣤⣀⡰⠀⡌⠀⠀⠀⠀⢸⣷⡇⠀⠐⢬⡇⢷
⠀⠀⠀⠀⡆⢸⣄⡤⢤⣄⠀⣧⠸⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⢹⠃⠰⠀⠀⠀⣀⡀⢸⡿⠄⡀⠀⠀⠇⠘⣧
⠀⠀⠀⣼⠀⡌⠀⠀⠀⢺⡄⠻⣄⠀⠀⠀⠀⠀⢨⠙⠳⢦⣄⣀⠀⡜⠀⡇⡾⢁⠉⠀⢸⡇⣇⠄⠀⠀⠤⢐⡆⠘⣷⡀
⠀⠀⡰⢧⣠⠃⠀⠀⠀⠐⡄⠀⢻⡄⠀⠑⢄⡀⠈⡀⠀⠀⠈⠉⣻⠀⢠⠀⢡⠀⠀⠀⠘⡇⢳⣄⠀⠻⣤⣘⡟⠀⣸⣧⡀
⢠⣞⠈⢠⡏⠉⠒⠠⠤⢤⣷⠒⠘⡇⠀⠀⠀⠉⠀⡇⠀⠀⠀⡼⠃⠤⣼⠀⠸⣄⠀⣀⡴⢀⠀⠉⠉⠁⠉⠸⡗⠉⠀⣨⡇
⡞⠀⠙⠻⢷⣶⢤⢤⢤⡤⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⢞⣛⠤⣀⣾⠤⠤⠤⠈⠡⡤⣼⢧⡤⠤⠤⠤⠚⣧⣴⠚⡃⠛⢆
⠳⣀⠀⢀⡨⣿⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⢸⢾⡝⢀⠐⢢⠜⢿⠄⠀⠀⠤⠤⠤⢼⡤⠄⠀⠀⠐⠲⡟⠢⣩⣀⣌⡹⠃
⠀⠀⠉⠁⠀⡿⠛⠲⠒⠂⠀⠤⢼⠧⠤⠤⠤⠔⠒⢻⠀⠉⠋⠉⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢰⡇⠀⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠹⠆
⠀⠀⠀⠀⣏⠀⠀⠀⣀⣀⣀⣀⣘⣀⣠⣀⡀⢀⣀⣀⢈⡇⠀⠀⠀⢠⣧⠤⠤⠤⠶⠶⠶⢤⣷⠴⠦⠀⠒⠒⠐⢦
⠀⠀⠀⠀⠘⡋⠉⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣀⡰⠤⠀⠀⠀⠀⠸⡷⠀⠒⠀⠀⠐⠢⢼
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡏⢱⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡏⢸
⠀⠀⠀⠀⠀⢰⣀⣀⣀⣀⣀⡀⠀⡧⠄⣀⣠⣤⣤⣴⣧⣀⡀⠀⠀⠀⢸⡼⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠱⡾
⠀⠀⠀⣠⠶⠛⠉⠉⠀⠀⠉⠙⢦⡷⠞⠉⠁⠀⠀⠀⠀⠈⠙⢦⠀⣠⠴⢛⣈⣛⠛⠲⣤⡀⢸⠀⢀⣤⠴⠖⣒⠛⠶⢦⣄
⠀⢀⡾⠁⠐⠀⠀⣀⡀⠀⠀⠀⠘⡃⠠⠀⠀⢀⠀⠀⢀⡀⠀⠀⢟⠁⠈⠁⠀⠀⠈⠀⠈⢷⣌⡼⠋⠐⠉⠀⠀⠈⠀⠀⠙⣦
⢀⠼⢄⡀⡀⠀⠀⠀⠀⠀⢀⡀⢠⡄⠀⣀⡀⠀⠀⢀⠀⢀⣀⣀⣼⡤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⣂⣀⣀⣀⣀⣀⣀⣀⣀⣼⡄
⠈⠒⠂⠀⠬⠬⠥⠤⠌⠁⠐⠒⠊⠁⠀⠀⠀⠠⠤⠠⠤⠀⠀⠀⠚⠛⠛⠛⠓⠒⠚⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠂⠐⠀⢁⡈⣁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⣄⠀⢀⡴⠒⠒⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⡆⠈⢳⣼⠁⠀⣤⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠑⠲⣌⠇⢀⡞⠉⠈⠙⣷⡀⠀⣀⣤⣄⡀⠀⠀⠀⠘⠷⠆⢨⡄⢠⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⡀⠀⠘⣿⡾⠀⠀⠺⡤⠞⠁⣾⠋⡀⠀⠙⡄⠀⠀⠀⠀⣠⠞⠀⠘⢧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⠙⠳⠞⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⢠⡇⢀⣠⡴⠞⠛⠛⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠋⢙⣳⠀⠀⠀⢠⠟⠉⣉⣳⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡯⠖⠉⠀⠀⠀⠀⣤⠄⣠⡇
⠀⠀⠀⠀⠀⡴⠶⣆⠈⣇⡀⠈⠁⠀⠀⠀⠸⡄⠀⠙⠁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣀⣤⠶⠟⠛⠛⠒⠶⣄⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠻⠶⠏⢀⣈⣹⣶⠤⣤⣀⣀⣀⡙⢦⣄⡀⠀⠀⠀⠀⠀⢀⡴⠛⠳⢦⡴⠚⠉⠀⠀⠀⠀⢀⣄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢰⠋⣁⡀⠀⠀⠀⠀⢀⠴⠚⠛⠛⠛⠛⠲⠶⠶⠶⣾⡁⠀⠀⢈⡷⠀⠀⠀⠀⠀⠀⠘⢦⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠶⣽⠇⠀⠀⠀⢰⠃⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡤⠞⠣⣄⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣈⣹⠇⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠘⠷⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢉⣤⠤⣄⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠘⢷⣦⣀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠋⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠻⢍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠶⠛⣹⡇⠀⠀⣀⣄⡀⢠⡎⠛⠀⠀⡟⠀⠀⠑⣄⡀⠀⠀⢰⡟⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡇⠀⠀⠛⢈⣷⠈⠻⠶⠶⠚⠀⠀⠀⠀⠈⢻⡶⢤⣀⣁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣲⠄⠀⠀⡴⠃⠀⠘⠢⠤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⢶⠀⠀⣷⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⡴⠃⠀⣯⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠁⢸⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⡆⠘⠦⢬⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣷⣶⣦⣤⣀⠀⠀⠀⠀⢀⣴⠟⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣤⣾⣿⠿⠛⠉⠉⠀⠀⠀⠈⠉⠙⠻⢿⣿⣦⣄⣴⠟⣁⣰⣿⠃
⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠟⢡⡾⠛⠻⠟
⠀⠸⢿⣿⣶⣤⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⣿⣶⣦
⠀⠀⠀⠈⠙⠋⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⣋⣀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⡛⠛⠛⠿⠿
⣠⣤⣤⣤⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠛⢿⣿⣿⡗⢀⣠⣄
⠙⠛⠛⠛⠛⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⣿⣷⣶⣿⣿⣁⣴⣿⠟⠁⢀⣤⣦
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⠋⢿⣷⣀⢻⣿⡛⢿⣿⣿⣷⣶⣿⣿⣁⣀⣴⣶⣄
⠀⠀⠀⢀⣠⣄⠀⢿⣿⡀⠀⠀⠀⣠⠞⢁⣴⢻⣿⣴⣿⣿⡿⢿⣿⣷⠈⣿⣧⠀⢹⣿⡟⠛⣿⣿⣙⣿⣷
⠀⢰⣿⡿⠿⠛⠀⠈⢿⣿⣄⣀⠞⢁⡴⢿⣿⠘⣿⡿⢁⣿⣿⣦⣬⣁⠀⠘⠛⠀⠀⠻⠃⠀⠘⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⢁⣴⣿⠀⢸⣿⠀⠀⣰⣿⠟⣿⡏⠛⠻⠟
⠀⠀⠀⠀⠀⠀⢀⣴⡿⢁⣴⠋⢹⣿⠀⠀⠉⠀⠈⠛⠃⢀⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⢠⡿⢁⣤⣾⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⠟⣿⣏⠛⠛
⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣅⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠘⠃⠀⠄⠅⠀⠉⠠⠀⠠⠀⠃⡐⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠸⠀⠸⣀⠀⢀⡤⠒⢲⠒⠀⠉⠁⠐⠢⡄⠃⡈⡉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡲⠃⠀⠀⢨⠀⠀⠀⠀⠀⠀⠘⠀⠉⠀⠈⠀⠀⠀⠙⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠇⡠⠂⢄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⢀⠘⠀⢡⢠⠀⠀⣠⠳⠁⠀⠀⠀⠀⡂⠀⠐⠤⠂⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢈⠃⠀⡟⠀⠀⠀⠁⠀⠀⠋⠀⠀⠉⠀⠉⠀⠁⠈⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡵⡀⠃⠃⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢔⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣐⡾⢁⡴⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠊⡢
⠀⠀⠀⠀⠀⠀⠀⡈⠔⠒⡄⠘⢅⣾⣁⠸⠶⣕⣒⠒⠒⡀⠀⠀⢈⡀⠤⣀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⡂⠻⣄⠙⣆⠀⢹⣦⢉⡒⢠⠏⠀⣯⠘⡆⠈⠑⡦⡤⣤⠤⢴⠎⠁⠰⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢹⣆⠜⠋⠁⠀⠀⠀⠸⠀⠀⣿⡀⠄⠀⠀⠁⢠⠀⠀⠀⢸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠌⠀⡒⠀⠂⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡑⡈⢂⠀⣤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⠀⠀⠀⠀⣠⠀⠀⠀⠐⡘⠄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠁⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠱⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠤⠤⡌⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇⢃⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠀⠀⠀⠁⠒⠒⠒⠒⠂⠁⠀⢀⣀⣠⠤⢇⠈⠆⠐⠊⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠱⠶⠶⠶⠶⠶⠚⠛⠛⠛⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠂⠤⠤⠄⠀⠐⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠢⠐⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⢀⠄⠀⠴⡮⢀⡀⠀⠀⠀⠀⠈⠁
⠠⢴⣆⠀⠀⠀⠐⠀⠀⠀⠀⢰⡄⠉⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠂⢀⡀⠀⠀⠀⡄⠀⠀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⣸⠀⠀⠀⠀⠸⠁⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⡎
⠀⠀⠠⠀⠁⠀⠉⠋⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠐⠀⠁
⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠀⠀⠀⢀⠠⡀⠀⠀⠐⠀⠂
⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣥⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢦⡞⣏⠰⣶⣶⣮⣴⢞⣤⣤⠰⣦⣀⠐⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣿⣾⣿⠖⢤⢀⣀⡨⣀⣀⣦⣗⢰
⢐⡻⡄⠋⢙⠿⣟⣡⠈⢨⢸⡗⣫⠩⡟⠊⡩⡼⢀⣀⣀⢀⣄⠀⣀⣠⣠⠟⣤⢝⡎⢼⠨⣏⣮⠹⣿⠫⡐⢓⢻⠤⢳⠾⣇
⠈⠇⢑⠦⡋⠏⢹⡼⢎⣽⢨⠶⣨⡷⠚⣶⢑⣯⣘⣿⠹⣏⠋⣿⣿⢿⣟⣿⢍⣜⣄⡼⡔⠿⢆⠓⢿⡧⠙⢠⠙⡏⣸⢳⡟
⠐⠀⠾⡖⣩⠓⢞⠋⠂⠶⢊⢌⡽⠋⢔⣠⠋⡐⣬⡱⢏⢀⣀⠈⠗⠞⠣⠊⢄⣿⡸⢆⡲⢌⣆⠛⠶⢦⢘⠀⢅⠱⢗⠊⢍
⠀⠀⣰⠀⡌⡙⠂⣀⠀⠀⠀⠀⠀⠋⠀⠓⠀⠙⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠀⣆⠁⢲⠌⢺⡷⠄⠈⠀⠁⠙⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠄⠀⠀⠀⢀⣤⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡀⠀⣻⣤⡲⢀⣤⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠀⠀⠠⡼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣿⡀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣌⡆⠀⠀⢠⠟⠞⢃⡾⣹⢧⡹⡏⡜⠛⠁⠀⠀⠀⠀⣀⡤⠒⠋⠘⠁⠠⣶⡀⠀⢸⡞⢻⣀⡀⠀⠀⢠⣷⡄⠀⠀⠀⡩⡀⣥⣾⡀⣠⣿⣄⣾
⢀⣄⢠⣿⠀⣰⡰⡚⠛⣸⣦⣴⣃⣶⣤⣴⣏⠀⠙⠿⠀⠘⠁⢀⡠⢔⣾⡏⠀⠀⠀⠀⠀⢴⣿⣆⠀⠼⣇⣾⡍⠁⠀⠀⣦⢷⣰⢀⠀⢰⡟⢻⣠⡏⢳⣿⣸⣇⢿
⢸⣿⣫⣭⣷⣿⢉⣿⢉⣻⡿⢩⠇⢸⣧⠁⣾⡆⣸⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⣎⣻⢸⡤⣀⣽⣏⠹⠶⠀⣠⣤⣿⡉⢼⠁⠼⣷⣚⣡⣇⣸⣧⣹⠙⡛
⢸⣿⣽⡉⣿⠟⠁⣈⢈⡉⢹⣿⣠⣾⣧⡖⢁⣿⣏⠈⡝⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣷⢿⣌⣥⠈⠾⡟⠳⠇⠀⢈⣡⣯⡛⠆⣉⠰⣾⡿⢋⣍⠁⢹⡻⡦⣿
⢸⣯⣴⣿⡝⠛⠋⠈⣿⡛⠛⣿⣿⣿⡿⣥⣿⢿⣿⣆⣻⣿⣾⣧⣦⣷⣶⣴⣄⣤⣄⡤⠚⣇⣼⣮⣙⣥⡀⡇⠀⣀⣴⡌⢀⣧⢥⠔⢣⡴⢤⣿⣯⣿⣷⣼⣿⣁⣿
⢸⣿⣿⣿⠧⠾⣀⣼⠞⢷⣶⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣟⣯⣛⣧⣿⣿⠖⣉⣿⠈⣧⡙⢯⢻⣿⣟⡣⠾⣙⣿⠅⠞⡔⠢⠗⢤⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⡄⣠⠏⠈⣄⠀⢳⣀⣻⡇⠿⠛⢿⠈⠟⠻⡿⣿⠟⠙⠛⠙⠿⠿⣿⣿⡿⢛⣡⣻⠆⠁⣹⡚⠻⡿⢓⠦⠓⠆⣹⢇⡠⠃⡤⠧⡘⠾⢿⡿⡿⡻⠻⠻⣿
⠈⠋⠻⢯⣀⡴⠦⠤⠊⠓⠛⢿⣄⣀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⣡⠀⣥⠀⢈⣍⠠⡬⠃⠈⠋⠸⡀⠤⣹⠇⡴⢳⠀⡉⠙⡈⢃⠁⠀⠀⠀⠁
⠀⠀⠦⠴⠁⠀⠦⠀⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠐⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⠀⠀⠀⢀⠀⢐⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⣝⡀⠀⢀⠁⡊⠀⠀⠈⢻⢦⠀⠴⢂⡿⡄⠀⠀⠹⡦⡐⠐
⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠈⠓⢍⠓⢤⣁⠀⠀⠀⠀⡟⢀⠀⢀⠞⠀⡟⠀⠀⢠⠀⠹⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⢧⠀⠈⠳⡀⠐⣦⡙⢮⠢⢸⠁⢠⠃⠀⡠⠋⠀⠈⡇⠀⢀
⠀⠀⠀⠀⠀⠀⢖⠈⣀⣀⡀⠀⠀⢸⡤⠀⠀⢧⢠⣇⠹⡌⢳⠘⣆⡿⢀⡞⠁⠀⠀⢀⠆⢀⠎⢣⠀⡀
⠀⠀⠀⠀⠀⠈⠀⣈⠉⢳⡎⢢⡀⢸⠀⠶⡀⠈⠻⠂⠀⢱⠀⠀⡿⢽⣼⠀⢠⠀⢀⡌⢀⠠⡀⡏⠐⠁
⠀⠀⠀⠀⠀⠀⠈⢹⠙⠸⡇⢀⠱⣾⡄⢁⠈⠢⡀⢠⡄⡞⠀⢀⠇⡄⠋⡠⢁⢣⠏⠀⢸⢣⢰⠇⢀⢦
⠀⠀⠀⠀⠀⢀⣆⢺⡊⠀⠹⣇⠰⠠⢄⠀⠂⠀⠰⠂⢠⠃⢀⡞⡐⢸⠐⠀⢠⠃⠀⡰⠃⣸⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠘⡏⢦⣀⠀⠀⠈⠳⣝⢤⡈⠲⠀⠀⡄⢸⢠⠟⠐⠀⢾⠀⠂⡇⣠⠏⠀⣀⡎⠀⣠⢀⡸
⠀⠀⠀⠀⠀⠀⠱⣄⠈⠑⠢⢤⡀⠘⡄⠱⠀⠀⠀⠆⠈⠿⠀⠀⠀⠀⠀⡆⢹⠇⡄⢠⠎⢀⡰⣻
⠀⠀⠀⠀⠈⢻⠆⠈⠲⢄⠀⣄⡙⠶⣗⠀⠁⠀⠀⠘⢄⠈⠂⠰⠀⠀⠀⠐⡤⠊⢄⣮⠖⢋⡜⠁⣦⡀
⠀⠀⠀⠀⠀⠘⠒⠠⠱⠀⠱⡀⠂⠑⠄⠀⡄⠀⠀⠀⠀⠉⠂⣤⠀⠀⠀⠀⠄⠐⠎⠥⢈⠏⡀⠀⠁⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠢⡀⡇⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢐⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠙⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠈⠨⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠓⢤⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠂⢀⠁⠀⠀⡀⠁⠀⠔⠀⡀⠀⡀⠀⠀⢐⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠐⠀⠀⠈⠈⠀⠀⠁⠀⠈⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⢸⣿⡄⣸⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠃⠞⠁⠟⠹⡆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⠀⠀⠂⠀⠀⠀⠀⠃⠉⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣶⢦⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠶⡡⢿⡞⡏⠀⠀⠀⠀⢰⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡂⢀⠟⡉⠀⠁⣘⢳⢿⡄⢸⠀⠀⢨⠻⠻⣤⣇⢀⣖⠦
⠀⠀⠀⠀⠀⠀⣴⣥⠄⡀⠀⣹⡗⣡⣾⣿⠃⣰⡇⡸⠀⣿⠿⣄⣠⣿⢣⡀⠈⠹⣾⠁⠀⠈
⠀⠀⠀⠀⠀⠠⠋⠉⠛⠳⢾⣯⣾⣻⢳⠃⢠⣿⠇⡇⠀⢿⣾⡘⡈⢻⣇⢻⣤⠬⢶⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⠉⠭⠙⠒⠋⠛⢻⣿⡄⠚⠛⠓⠓⠘⠻⡏⢁⡤⠔⠈⣷⣀⠄
⠀⠀⠀⠀⠀⠢⡀⠀⠀⡀⠘⠇⠀⠀⠀⠀⠀⠀⢸⢸⡇⠖⠂⠀⠀⠀⠀⣹⣀⠀⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠈⠑⠤⠷⠞⢀⣀⠀⠀⠀⠀⠀⡄⢸⠃⠀⠀⠀⠀⢀⠌⠀⢠⠋⠱⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠈⠉⠀⠉⠉⢉⡉⠉⢩⡀⠴⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠛⠁

⠀⠀⣤⡀⢻⢀⡤⠄⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⢀⣩⢝⠯⣤⡀⠀⠀⠀⢀⠐⠳⠄⠀⠀⠀⠁⠀⠀⠀⠀⠙⠇⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠉⠁⢼⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠂⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⡍⠀⠀⡀⣀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠀⠀⠀⠀⠀⡇⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠐⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠀⣃⣁⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡈⠃⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠃⠓⠀⠀⠀⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⡤⢤⢤⡀⠀⠀⠁⠐⠃⠀⠁⠀⠀⠸⡿⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠒⢂⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⢀⡡⣆⣾⣶⠬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢈⠊⠀⠀⠀⠀⣠⢀⡄⠀⠀⠀⠰⢂⠚⢛⢂⡄⠀⠀⠀⡘⡈⠀⠐⠀⣀⠀⠃
⠀⠀⠝⠽⣿⡛⡏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⡛⠤⠀⡃⠆⠀⠀⠘⠀⠀⠀⠠⠂⢀⠀⠂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣹⣗⠄
⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄⢊⡵⠋⠀⠀⠀⠀⣀⢠⡖⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡴⣎⡤⣀⡀
⢀⡆⣸⣿⢀⠀⠀⠀⢠⡴⠂⡀⠀⠀⠁⢀⠀⠂⠀⣀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⣇⠀⠀⢨⣿⡰⠷⢻⣇⣀⣀⡀⡀
⣿⣿⡿⣿⣾⢁⡀⡀⠄⠖⢀⠐⠀⠀⠀⢻⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠖⠀⠀⠀⠀⣤⠀⠀⢸⣳⡀⢀⠿⢟⠳⢟⠀⠀⢿⠉⢡⣿
⢹⣿⠉⢿⣿⣿⣿⡵⠞⠀⠎⠈⠱⡁⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠤⠀⢏⡳⣤⢨⠔⢉⣌⢻⡞⠙⢌⠻⣦⣽⣿
⠙⣳⢾⢀⣿⣿⠟⢐⠃⠀⠋⠐⠥⠈⢿⣿⣦⣶⣤⣰⢠⡀⡀⠀⠀⣀⣀⣴⣀⣠⣫⣊⡀⡷⠋⡁⢜⡹⠒⠂⠀⠻⡧⠀⠈⢀⠈⠙⣿
⢨⠅⠰⠐⣏⣤⠴⠇⡤⠞⠀⠀⠀⠒⢼⣿⣿⡿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠁⠁⠙⠻⣡⠘⣏⡴⠒⣡⣄⢦⣍⠄⠀⠈⠳⢶⣿
⠸⠧⠈⣼⠛⠁⠤⢠⠶⠀⢠⢾⡍⠢⣄⣻⣿⢓⣿⣯⡿⠛⣉⠀⠀⠀⠈⠙⢿⡡⠔⠉⠛⠷⣎⣤⡿⠔⠊⠀⠹⠄⣙⠀⠀⠠⡀⠀⢈
⠀⠀⠀⣹⠋⡀⢤⠂⢀⠌⠛⢄⢻⡂⠈⠉⠉⠉⠻⣠⢔⣻⠼⣿⢦⣤⣤⣤⠆⠔⠀⠚⢀⠒⠼⣿⣷⣣⢶⠀⠈⣦⡀⠙⢤⠀⡀⠐⢬
⠀⠀⡴⠋⠀⠀⣛⡖⠋⠀⠀⢤⡁⢹⡰⠦⡂⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠂⠚⠉⠀⠊⠑⠲⠟⠉⠀⠘⠂⡖⢛⠁⠐⠢⣷⠃⠀⠘
⠀⠈⠀⠁⠒⠛⠉⠙⢦⡤⠄⢀⠷⠇⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⢠⡶⠄⠀⠀⢃⠐⣀⠙⢷⣾⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠈⠀⠈⠉⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⢀⠀⢘⡅⠀⢀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣆⢸⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡆
⠀⠞⡞⢻⡇⠰⣾⣷⢖⡶⡖⣀⣤⣤⡌⢓⣂⣄⢘⠇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡀⢠⣿⠯⢿⠓⠒⣦⣠⣤⡔⢂⣤⣼⣰⣇⢰⡆
⠀⣛⣥⠘⠉⢛⠿⡟⢙⡄⠁⡩⢨⣗⢩⡍⢹⡟⢊⣩⣬⠇⣀⠀⠀⢀⣤⠀⠀⣀⣀⣀⠾⢠⡮⢳⠈⣼⠨⣯⣳⡍⢿⣿⠯⠀⠛⡚⢇⠤⢳⠼⣧
⠀⡯⠘⢶⣸⢓⡖⢹⣅⣷⠚⡳⢘⣵⠈⣓⡭⠶⣤⠒⣶⡂⢿⣿⢻⡏⠛⢿⣿⣯⣿⡿⣺⣙⢈⢫⠀⣶⡠⢿⢧⠱⠶⣿⣆⠩⢠⡈⢻⠒⠸⢱⡼
⠀⠈⢀⣯⡜⠸⣆⡠⠿⣠⠋⣡⡬⡀⢛⡿⢖⡱⢉⣞⠉⡍⠺⣻⡤⠁⠡⠈⢿⢁⡹⡂⡩⠌⢹⡏⣹⠁⢩⡘⢄⠱⠦⣬⡁⠲⡆⠓⠘⣥⡋⡜⠳
⠀⠀⠀⠙⠀⠺⠤⡘⠀⡀⠀⠉⠐⠉⠛⢡⣄⢱⣏⠀⢢⣸⠓⠈⠁⠐⠒⠀⠈⠀⠀⠈⠁⠉⢚⡙⢉⠑⠏⠉⠒⢇⣠⡁⠉⠠⠆⠨⠂⡌⠔⢠⡉
⠀⠀⠀⠃⠐⠁⠄⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡇⢈⠧⠀⠘⡟⠠⠩⠉⠁


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

⠀⣴⣶⣶⣶⣶⣶⣾⣶⣿⣶⣾⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣽⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣟⣿⣽⣻⣽⣿⣿⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣽⣿
⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣽⣿⣷⢿⣷⣽⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣶⣿⢯⣿⢏⣼⣿⡿⣿⣷⣭⣹⣿⣿⡿⠟⡩⠿⣿⣿⣿⣽⣿⣾⣿⣾⣿⣿⣾⣿⣿⣷⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠠⠀⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿⣿⣿⡻⣷⣿⣿⣽⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣯⡛⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⠤⠀⠂⠯⢩⡛⠂⠭⢥⣿⣉⣟⠛⠻⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣷⣿⣼⣿⣽⠩⠭⠤⠘⠛⠠⠈⣉⠂⠐⠀⠀⣤⢄⠒⠄⡀⠠⢀⠀⢙⡋⠠⠖⠦⠤⣼⣻⣿⣧⣾⡟⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⡿⣥⣾⣿⣧⠀⠀⠀⡠⠊⠀⣾⡇⠰⠀⠃⠀⠶⠘⠆⠀⠄⠐⠄⡀⠘⠃⠰⠇⢾⢠⡆⠿⣿⣿⣿⡇⢀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⡿⠋⡀⠸⣿⣿⡻⣄⣈⣠⠀⠀⠈⠀⡄⠀⠀⠀⠀⢄⠀⡀⠸⠀⢀⠄⠈⠀⠢⡄⠀⠀⠀⠈⠀⠉⠻⣿⢃⣼⣿⡄⣾⣿⣿⣿⣻⣿⣿⣿⣿
⢸⣿⣿⣿⣿⡏⠉⢁⣼⣟⡃⠈⢁⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠱⠀⠀⠰⢶⢶⠶⠶⢷⠆⠩⣿⣧⡶⢤⣦⣤⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣷⣿⣿⡇⠀⠜⢻⣿⣟⠂⠀⠐⡎⣀⡀⡥⢇⠀⠀⠀⠂⠀⠀⠆⠀⠀⠀⠈⠃⠀⣸⡇⠀⠀⢶⠄⠈⡟⣸⡗⠈⢹⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⡇⠀⠠⣿⣿⡿⠰⡀⠈⠃⠿⠷⠄⠄⠀⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⠀⠀⢫⡇⠀⣷⣸⡧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠙⢹⡟⡟⠃⢀⠢⣿⣿⡑⠒⠧⠀⠀⡀⡀⢀⠂⢀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠃⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠑⠀⡈⠑⠃⢡⠐⠀⡆⠐⠀⠀⡀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠈⠉⠉⠙⠛⠛⠋⠛⠛⠛⣧⣴⣾⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠡⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⣿⣿⣿⢿⣿⣿⣿⣿⣷⡬⠙⠿⠛⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⣙⣿⣿⣿⣿⣿⣟⠻⣿⡻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠙⠛⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣶⡿⢯⡿⢿⣿⠿⠛⠋
⡃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣁⠀⠞
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢄⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠀⠀⠈⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⡀⠀⢁⠆⠀⠀⠁⠆⠀⠀⠀⠁⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠤⠀⠀⠀⠀⠀⠀⠀⠀⢑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠑⡠
⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⠀⠀⠀⡆⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠌⠢⡀⠀⠈⠃⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⡏⠀⠀⡠⠂⠀⠀⠀⠈⢆⠀⢀⡀⡨⠓⠀⠀⢀⣀⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡆⠀⡀⠀⠀⢠⣮⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⣰⠀⠀⠀⠀⠈⠸⡁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠈⠉⠀⠸⠀⠠⠀⠀⡀⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠀⢀⡀⠀⠍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠀⠀⠠⠀⣀⠀⠀⠰⠀⡠⠀⣀⡠⠜⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠑⢄⠀⢱⡁⠉⠀⢀⣀⡀⠄⠁⠀⣠⠂⠀⠀⡀⢠⠀⠀⡈⢁⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠠⠆⠀⠀⠀⡀⠀⢀⠀⢅⠊⠓⠤⣄⣀⣀⡠⠤⢚⠀⠠⠒⠉⠁⣈⢀⠀⠘⢇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠁⠈⠐⠐⠀⠉⢄⡰⠁⠀⠀⠑⠂⠤⠀⠤⠤⠵⠒⣉⠒⠤⠤⠤⠚⢉⠀⠄⠀⠀⠉⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢁⠠⠄⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡣⣀⠐⠁⠎⠐⢀⠂⠈⠀⠀⢠⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠊⡑⠒⠒⠒⠚⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠎⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⠧⢄⡠⠤⠤⢀⣰⡦⢄⣰⡖⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠔⠁⠀⠀⠠⠤⠤⠀⠢⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠈⠉⠉⠁⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠏⠢⠄⣀⣀⣀⢤⢤⣳⣄⠀⠀⠀⠀⠀⠀⠀⡙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⡀⡇⠀⠀⠌⠀⢙⠒⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠤⠀⣬⣯⡍⣉⠉⢢⢸⠄⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠻⠊⠉⠙⠒⠒⠚⠁⠸⡁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢎⡔⡦⡙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⡰⠋⠁⠀⢰⢈⣆⡠⠀⠀⠀⠀⠀⡘⠒⠑⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠀⠀⠀⢻⠈⡄⠀⠀⠀⠀⢠⠗⡂⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠀⢀⠀⠀⠈⠀⠱⡀⠀⢀⣰⣏⡰⠙⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢣⠈⢆⠀⠀⠀⢀⡱⠒⡿⢻⠏⢀⠜⢄⠀⠀⠀⠑⠤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣼⠀⣾⡀⠀⠑⢄⠀⣍⡠⣬⠞⠁⡠⠂⠀⢸⠄⠒⠄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠜⠈⠑⠣⣠⠼⠆⠸⡇⠁⡠⠊⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠁⠀⠀⡾⡖⠁⠀⡼⡄⡙⠀⠈⠀⠇⠀⠀⡆⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠀⣀⠎⠀⢁⣎⢰⠀⡸⠁⠀⠰⢿⡀⠀⠀⡇⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠔⢁⠌⠸⠈⠀⠀⢀⠃⠀⢻⠄⢰⢀⠠⣺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠓⣤⡠⠊⢣⡀⠀⠀⣸⠅⠊⠀⠉⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⠬⢽⠉⡏⠬⠤⠝⠦⠤⠃⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠌⡝⠃⠀⠀⡀⠠⠀⠤⠄⠂⢀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆⡇⡇⠀⠀⠃⠀⠀⠀⠀⠀⡄⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠁⠃⠀⠀⠑⠠⠤⠤⠐⠊⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠙⡃⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠚⢳⣤⠤⢤⣤⣤⠤⠤⠤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠀⠒⠒⠂⢠⠨⠛⣆⠀⠹⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣀⣀⡤⠒⠂⠉⠉⠁⠉⠉⠻⣆⠉⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠠⠘⠒⠒⠒⠚⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⠄⢀⠠⡵⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣠⠴⠇⠀⠀⠀⢟⠎⠙⢆⣀⠔⠒⠒⠒⠲⢮⣀⢀⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⢈⡲⣄⠀⠀⠀⢉⡹⢇⠀⠀⠠⠤⠤⠀⠤⠋⠀⠀⢀⠠⢀⠶⠒⠠⠔⣉⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⠞⠁⠀⣈⠧⢤⡀⠚⢓⡶⠋⠀⠤⠪⠅⠐⠀⢶⡀⢀⡄⠀⠀⠀⠘⡀⠀⠰⣈⣓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡷⠈⠀⠀⠀⠙⣯⡤⣶⠉⠀⠀⠤⢤⣄⣠⠀⠀⢠⡙⣺⢦⣀⣀⡠⢼⡂⠀⠠⠈⠳⢤⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⢘⣯⠶⢃⣀⣀⢀⠂⠀⡠⠀⠊⠁⠐⠀⠤⠈⠄⢠⠄⠀⠀⠽⠿⡤⠤⠆⢶⠣⠴⠄⠨⠙⠷⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠊⠀⡀⠂⠀⢀⣵⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠦
⠀⠀⠀⠀⠈⢁⠀⠀⠈⢇⠀⠀⠀⠀⠄⠐⠐⠇⢀⡈⠃⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄⠀⠀⠀⣿⠸⢀⢀⠆
⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠳⠂⠐⠒⠋⠉⠉⠀⠀⠁⠀⢀⠭⠐⠠⣄⠀⠀⠀⠀⡰⠆⠗⣇⠀⡀⡀⠀⠩⢅
⠀⠀⠀⡐⠀⢀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⢄⠨⣀⢘⠀⠀⡰⠃⠀⠀⠉⠂⠀⢄⠀⠀⠀⠡
⠀⠀⠀⠃⠐⠈⠰⠚⠂⡀⠀⣀⠀⠀⠀⠀⠀⠈⡅⠢⠎⢰⠤⠔⠂⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⣀⣸⣀⣠⠴⠒⠊⠁⠀⠀⡀⠀⠰⠄⡀⠀⠐⣤⠀⠀⠺⠀⡀⠀⠀⠂
⠀⠀⢐⡒⠀⠀⠠⠤⠨⠓⠈⠁⠉⠀⠀⠀⠟⠦⠓⠂⠉⠀⠑⠊⠈⠀⠀⠀⠀⠀⠀⠒⠒⠒⠠⠸⠱⠈⠑⠒⠒⠂

⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠘⣇⠈⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣸⠀⠀⠀⠀⠀⠀⠀⠰⠋⡇⣦⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠰⡆⣨⢛⢥⡶⠀⠀⠀⠀⠀⠀⢸⢸⣄⣧⣠⠀⠀⠀⠀⠀⢀⠀⣲⠛⡏⠀⠀⠀⠀⠀⠀⡾⡿⠜⡄
⠀⠀⠀⣷⣼⢀⡾⣤⠦⡼⠀⠀⠀⠀⢸⣾⠈⡟⠙⢻⠀⠀⠀⠀⠀⠘⢟⢫⣿⣀⡀⠀⠀⠀⢀⠠⢇⠠⣤⢷⡄
⠀⠀⠀⠘⠦⠾⠇⠘⠃⠁⠀⠀⠀⠀⠀⠁⠰⠛⠖⠁⠀⠀⠀⠀⠀⠀⠀⠱⠐⢯⠃⠀⠀⠀⠘⣷⠛⠘⢆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠜⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀
⠀⠀⠀⠀⠘⡿⡏⡅⠀⠀⠀⠀⠀⢀⢀⠀⣿⢣⣀⡀⠀⠀⠀⠀⠀⠀⠎⣼⣿⢰⡀⠀⠀⠀⠀⠀⡄⣾⡼⣃⡋
⠀⠀⠀⠀⠸⣧⠛⣇⠀⠀⠀⠀⠀⠈⠉⢷⠾⡌⠛⢛⡁⠀⠀⠀⠀⢰⣜⣇⢹⡾⣧⡀⠀⠀⠀⠀⡁⢏⢁⢹⠠⠇
⠀⠀⠀⠸⣆⣿⢶⣼⠀⠀⠀⠀⠀⠀⣻⡛⠀⡿⣆⠛⠇⠀⠀⠀⠀⠸⣿⠹⢰⡀⠻⠀⠀⠀⠀⠠⣼⡾⢿⡘⢾
⠀⠀⠀⠀⠀⠰⠏⠋⠀⠀⠀⠀⠀⠀⠈⠃⠻⠅⠙⠋⠀⠀⠀⠀⠀⠀⠙⠘⠋⠃⠀⠀⠀⠀⠀⠀⠈⠃⡎⠻⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠘⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⢺⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠻⣿⡄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⡀⣰⣿⡿⠃⠀⢠⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡼⣿⠁⠀⠀⠸⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠘⣿⣿⣷⠈⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣦⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⢤⡀⠀⢀⣿⣷⡀⠀⠀⣸⣿⣿⡟⠀⢤⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⢋⣼⣿⣿⣿⣿⡇⠀⠀⠀⢱
⠀⠀⠀⢤⡀⠀⣴⡿⠀⡀⠙⣿⣿⣿⣿⡆⠀⠀⢸⣷⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⠁⠀⠈⣿⣷⡄⠀⠀⠙⠻⣿⣿⡿⠁⢸⣿⣿⣿⣿⣿⣷⣤⣀⣤⣿
⠀⠀⠀⠈⣿⣶⠟⠀⠀⣿⣦⣾⣿⣿⣿⠇⢀⣴⣿⠟⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⣿⣿⡀⠀⠀⠀⠀⠉⣰⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⣇⠀⢀⣿⣿⠀⡀⠀⠘⢿⣿⣿⣿⢟⣴⣿⣿⠋⠀⢠⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⠀⣿⣷⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟
⢿⣿⣿⣿⣿⣇⣀⣽⣷⡀⠈⠙⠿⢋⢸⣿⣿⣿⣷⣤⣾⡇⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⣿⣌⣿⣿⣿⣿⣿⡿⢡⣾⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣠⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⢠⠀⠙⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⡟⠀⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣷⣶⣿⣿⣿⣿⡟⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣘⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⡟⠛⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⢀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⣥⣴⠂⠀⠀⣤⣦⣄⡀⠀⠀⠀⠀⠀⠀⢈⣦
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠐⠒⠂⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠈⠀⠀⠀⠀⢤⣤⣤⠖⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⣀⣿⣿⣷⣶⣦⡤⠀⠀⠀⣀⡉⢀⣀⣀⣀⣠⣴⠟⠛⠛⠛⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⣛⣫⣼⣿⣷⣴⣦⣤⣄⠽⠋⠀⠈⠉⠛⠿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠤⢤⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠶⠀⠀⠀⠀⠀⠀⢀⣾⢷⣶⣾⣿⣿⣿⣿⠿⠿⣿⣿⣥⡀
⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠹⠿⡿⠏⠉⢠⡤⠒⠛⣛⣿⣿⡿⠆⠄
⠀⠀⠀⠀⢠⣄⣀⠀⠀⠬⠉⠀⠀⠈⢉⠛⠓⠶⠶⢆⠀⠐⠀⠀⠀⠈⠀⢀⣿⣿⣿⠿⠷⠶⠶⣶⠶⠚⠟⢶⣄
⣿⣧⣲⠴⢤⣄⠉⠉⠉⠀⠀⠀⠀⡀⠈⠷⢠⠄⠄⠀⠀⠈⣉⠀⠀⠐⡖⠀⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⠯⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣌⠙⠛⠛⠂⠀⠀⠀⠀⢠⣶⠶⠀⠀⠀⠀⠀⠀⠀⠂⢠⣾⣿⣿⣶⣤⠀⠀⠀⠀⢀⣤⣦⣄⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⣄
⠤⠀⠰⠆⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠄⠃⠀⠤⢀⣶⣾⣿⣿⣿⣿⣇⣠⣴⠆⠊⣳⣿⣿⣿⣿⣶⣦⣝⠓⣤⣄⣀⠉⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀
⠀⠀⣀⣀⡀⣭⣍⠛⠉⠁⠀⠀⠀⠒⡀⠀⢤⣴⣤⣮⣽⣿⣷⣶⣶⣾⡾⠛⣼⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣮⣍⠁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴
⠲⠦⠍⠉⠁⠛⠉⢁⡀⢶⣶⣮⣷⣾⣿⣿⣶⣾⣿⣿⣿⣭⣟⣟⣿⣿⣿⠛⣡⣽⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⡀⣀⣠⡴⠄⠺⠿⠿⠆⠀⠐⡛⠛⠛⠛⠛⠛⠻⣿⠛⠟⠛⠛⠏⠙⣿⣾⣶⣶⣶⣶⡤⢶⣶⣶⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⣿⡿⠿⠶
⠘⠉⢁⣀⣀⢠⣀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣿⠀⢹⣦⣠⣦⣶⠶⠂⣀⠀⠀⣬⣿⣿⣦⣤⣄⠉⢉⠛⠿⢿⣿⣿⣿⣭⣉⣉⣉⠉⠉⠁⠀⠀⠈⠙⠓

⠀⠀⣸
⡇⣠⠿⠀⡀⣤⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢹⣿⡉⠀⠺⣳⠀⢸⡟⠀⡄⣸⠀⠀⠀⢸⡄⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐
⣾⣥⣌⠉⣠⡼⠂⠸⢁⣘⢻⣯⢀⣼⣴⡯⠤⣾⣠⠀⠀⡀⠁⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡿⢿⡶⠂⠠⠤⠀⢀⣬⡍⠼⣥⠀⣿⣿⠛⠂⢹⣿⣄⣼⢁⡀⢀⠁⡀⠀⣰⠀⡀⠀⢠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⢀⡀⣀⠀⢠
⡷⠨⣙⣟⡳⠒⠀⢸⡟⣀⣘⠟⡯⣿⠛⡘⣋⡀⣽⣿⠟⣛⣦⣿⣼⡇⢰⡿⢻⣧⣶⡾⠇⣴⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣠⣶⣿⣡⡏⣾⣿⣵
⠀⠀⠀⠀⠈⠈⠀⠘⠛⣷⠍⠂⣼⡻⠂⠠⠤⠀⢐⣿⣧⠆⠰⣿⡗⠭⢿⡥⠄⢿⣿⡷⠼⣿⡿⢲⣾⣆⡀⢀⣄⣤⣿⣿⣿⣿⣿⣿⡿⠥⠽⠿⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⢯⡃⢦⣤⠄⠀⠉⢻⣇⠐⠈⢻⡿⢠⢽⣏⣀⣋⣿⡥⠀⢹⣇⣈⣿⡿⣿⣿⠿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠶⠾⠿⣯⣘⡛⣷⣀⠊⢿⡧⢴⢺⣄⣐⠈⣷⡆⢸⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠚⠛⠶⠿⠿⠿⠒⠛⠉⠉⠁





⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣿⣵⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢇⣤⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⢠⠟⠙⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⡀⢠⡆⠀⠀⠀⠈⣧⠄⠐⠸⠀⡤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠀⡰⠉⠷⠀⠐⠑⠀⠃⠀⣤⡠⠞⡷⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠸⢤⡇⠀⠄⠀⠀⠀⠀⠀⠀⠉⠀⠐⡁⠘⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠈⠁⡐⠂⡰⠁⠀⠀⠀⠤⠂⡳⠀⠇⠆⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣦⡱⡄⠲⠇⠘⠀⠀⠀⠀⠀⠀⢀⠀⠞⠀⠔⢁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠌⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠊⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠈⡤⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢥⡀⠀⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠮⠴⠖⠖⢤⣼⢶⡿⣷⣾⣷⠋⢻⣿⣷⡖⠦⢤⡄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣤⣤⣴⠞⠋⠀⣸⠁⢀⣿⢻⣆⠀⠙⢯⣟⠉⣷⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢹⣁⣀⣠⣶⡇⠀⠸⣿⠄⣻⡆⠀⠈⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠈⣿⠷⠏⠛⠉⠘⠶⠶⠞

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡔⠊⡉⠽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠀⠃⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠷⢤⣤⣀⠀⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠻⠶⢦⣄⡀⠉⢻⣌⣻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⡴⠞⠂⠀⠀⠀⠀⠈⠙⠳⢦⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠻⣌⣿⠓⠶⣤⣀⡀⠀⢀⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠉⣙⢷⣄⠀⠀⠀⠀⠈⢷⡀⠀⠀⣸⠏⠀⠀⣤⠟⠙⢳⣤⠹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⢸⣿⢛⣿⡦⠶⠶⠒⠲⠶⠦⣄⣰⠏⠀⠀⣾⣄⣠⡾⠛⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⣄⣀⣀⢀⣹⣿⣥⡀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⣸⠓⣾⠏
⠀⣤⠄⣀⡄⠀⠀⢀⣀⣤⣤⡴⠶⠶⠶⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⢿⣗⠀⠀⣀⣀⣤⠶⠟⠁⠀⠀⢰⢏⣽⠷⠒⠚⠓⠳⣦⡀⠀⣠⠀⢀⡀
⡼⠁⣾⠋⢀⣤⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⣴⠄⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣾⠋⠁⣠⢴⣶⠦⡤⢸⡇⠀⢸⠀⢸⠃
⠃⢸⡇⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠾⠓⠋⠀⠀⢠⡾⠃⢀⡾⢀⡟
⠀⠘⣇⠀⢻⣆⠀⠀⣤⠶⢂⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠈⠀⠚
⠀⠀⠈⠀⠀⠙⠳⢶⣤⣀⣉⣁⣀⣉⣁⣀⣠⣤⡶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡬⠉⡉⢹⡿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢰⡇⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠾⠀⠀⠀⢀⣀⡤⠖⠛⠙⣿⣓⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⠀⠀⠀⠀⠀⠶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⢻⡭⠭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣹⠟⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣄⡤⠚⣁⡠⠤⠤⠤⠤⠤⠤⣄⣸⣿⣛⣂⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣉⠀⢤⣿⣀⣀⠀⠀⠀⠒⠻⢤⣄⣿⠀⢠⡟⠙⢷⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣼⣧⠀⣿⠟⠒⠒⠒⠒⠶⢤⣤⣀⠀⠈⠉⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠑⠀⠉⠉⠉⠛⠛⠒⠒⠲⠞⠛⠖⠾⠷⠶⠶⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣩⣥⣤⣤⣾⠞⠉⠉⠑⠲⢤⣀⠀⠀⠀⠈⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠁⠀⠀⠈⣿⠀⠀⢷⠀⣄⠀⠙⢷⡀⠀⣠⣟⠻⣟⡳⠦
⠀⣠⠆⢀⣴⠀⢀⡴⠋⠀⢸⣏⠉⣉⣹⣟⣛⠛⠛⠛⠛⠛⠓⠒⠒⠶⠶⠶⠶⠖⣲⡶⠛⠁⠀⠀⠀⠀⠀⢀⣼⣯⠀⢀⡿⠀⣿⠀⠀⠈⢷⣾⡟⢌⡳⡄⠙⠓
⣼⠁⣴⠟⢁⣴⠋⣀⡤⠶⠟⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⠒⠶⠦⣤⣀⣴⠞⠉⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢻⡄⠈⠀⠀⠁⠀⠀⠀⠀⠳⡸⣆⠙⠂
⠇⢸⡏⠀⣿⠁⠞⠉⣴⠃⠀⠀⠀⢀⣀⣀⣤⣤⡴⠶⠶⠖⠛⠛⢛⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈
⠀⠀⠙⠂⠉⠛⠲⠾⠷⢶⡶⠿⠛⢻⡏⠉⠛⢶⣄⡀⠀⠀⠀⣰⠏⣽⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⣷⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⠁⠀⠀⢸⡇⠀⠀⠀⠈⠛⠷⣤⣀⡟⠀⢹⡄⠀⠀⣀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣿⠀⢠⡀
⠀⠀⠀⠀⢠⡗⠀⣿⢰⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡷⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⢸⡁
⠀⠀⠀⠀⣾⠁⣸⠇⣼⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆⠸⡇
⠀⠀⠀⢸⣇⠀⢻⡄⠛⠶⣤⣄⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡏⠀⢸⣧⠀⣿
⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠉⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⡶⠖⠛⠛⠛⠉⠉⠁⠀⠶⠚⠉⠁⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠉⠉⢙⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⢉⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠶⠶⢾⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣤⣤⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠳⢶⡀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣀⣠⡼⠷⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡞⡓⠦⠟⠑⠛⠀⠀⢹⡆⠀⠀⠀⠀⢀⣼⢯⣍⣙⡶⠶⠟⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠛⠲⠶⠤⠤⠤⢤⣤⣤⣼⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣠⡇⠀⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠋⠀⢀⣾⠟⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⢀⣴⣿⡏⠀⠀⣠⣴⡏⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⢳⣿⢀⠠⠾⢻⣿⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⣿⡿⣱⣿⡿⠃⠀⠀⣼⣿⣼⠠⠄⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣽⣿⣿⣿⣿⣆⣠⣾⣿⢿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢟⣿⡿⣿⣿⣿⣿⣿⢣⣼⣿⡎⢷⣇⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⢏⣾⡟⣸⣿⣿⣿⣿⡇⡾⣿⣿⢻⡌⠻⣆⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢀⢠⠁⣿⠟⣿⡿⣰⣿⣿⣿⣿⣿⡘⣇⢸⣿⣿⣿⠀⠙⣷⣄⠠⡱⣄
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣿⣾⣿⡇⣿⢸⣿⣿⣿⡿⠧⠀⣼⡈⣿⣿⡇⢢⠘⣿⣧⠙⣿⣦
⠀⠀⠀⠀⠀⠀⢸⡀⣼⢿⣿⣿⠀⣿⣼⠳⠏⢠⣷⢀⣼⣿⠃⣸⢹⡇⣸⡄⠸⣿⡇⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣧⢹⡜⣿⣿⡄⢹⣿⠀⠀⢸⣾⡈⠻⣿⢀⡟⣹⠇⣿⣥⠀⣿⣷⠀⢻⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠹⣿⣿⣿⣦⡿⣷⣄⡈⠻⠏⠀⡹⣾⣱⠏⣼⣿⣧⠀⣿⣿⠀⢸⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⣻⣯⣛⣿⣿⡝⢷⣂⠠⣼⣷⣿⢏⠼⠁⢸⡇⢰⣿⡟⠀⣸⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠀⠈⠋⣉⣿⣿⣿⣷⡙⠿⠿⠯⠁⠀⢀⠴⠋⣴⣿⠟⠁⣰⢃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⣿⣿⣿⣿⣷⣦⣤⣤⣤⣷⣼⣿⣿⠗⢀⡴⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣹⣿⣿⣿⣿⣿⣿⣿⠿⢛⣡⠴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣧⣴⣿⡭⠤⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠿⠟⢻⠟⢿⠿⢻⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣰⡆⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣤⣿⣿⣤⣶⣶⣶⣯⣴⣷⣭⣿⡆⠀⠛⢲⡀⢛⣒⣶
⠀⠀⢀⢸⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⡏⢿⣿⣿⠀⢀⢀⡐⢾⢈⣦⣠⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⣰⣶⠟⣤⢿⡏⢈⡷
⠀⢀⣾⣿⣿⣧⣷⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠋⠀⢀⡌⣦⣠⢷⣏⣿⣯⣯⣧⡐⠀⠀⠉⢈⣍⣉⣩⣍⣭⣥⣤⡄⠀⣜⣿⣶⣼⠟⢹⣏⣸⡅
⠀⣼⣿⠏⠀⢿⣿⣇⠀⠀⠸⣿⣿⢿⢿⣿⣿⠃⠀⠀⢀⣜⣾⣷⣿⣿⣿⣛⣿⣿⣼⣿⣿⣷⣦⡰⣼⣿⣍⠓⢺⣟⡋⠉⠉⠀⠘⠿⠞⢿⣶⣿⠿⠛⠁
⠀⠻⢧⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⣁⣸⣿⣿⡟⠛⠈⠑⠀⠀⠀⠀⡀⠀⠘⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⡀⣴⢠⡀⠀⠀⠀⠀⠀⠀⠆⡐⠂⡆⠐⠀⠀⠀⠀⢰⠀⠀⠀⣿⣿⣿⣿⠇⢀⠀⠀⠀⠀⣠⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⣸⢼⣠⢧⣟⣳⢼⡰⣅⣷⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠉⠁⡄⣑⣘⣄⠀⣀⣙⣦⡴⣃⠀⠀⠀⢠⡀
⠀⠀⠀⢸⡆⠀⠀⣯⣻⣾⣾⣿⣿⣿⣾⣷⣿⣷⣿⡄⠀⠀⠀⠀⠀⢀⣼⣧⣶⡤⣄⠀⠀⠀⠀⢀⠙⠐⠀⠑⠿⠟⠀⢠⣞⣿⡻⣞⡽⠁⠀⢠⣤⣽⠆
⠀⣰⢠⡿⢁⡀⠀⠈⠛⠛⢉⠉⠉⠉⠙⠛⠛⠛⠋⠀⠀⠀⠀⢠⣼⣽⣿⣿⣿⣿⣿⣤⠀⠀⣀⡴⠛⢦⡞⠀⠀⠀⠀⠀⣹⣟⣿⣏⠀⠀⢰⣛⣦⣹⣦
⠀⢿⡈⢷⣼⣄⠀⠀⠀⣠⢿⣆⣤⠀⠀⠀⠀⣦⣷⡄⢣⣀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣞⠀⡄⡟⢷⡜⣮⣷⣴⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⢈⣽⣿⠁
⠀⢸⡿⢸⣿⣿⠇⠀⣰⣯⢾⣿⣏⣦⠀⠀⠲⣬⡟⣷⣦⡙⣷⠀⠁⢸⣿⣿⣿⣿⡟⠃⠸⣧⣿⣿⣵⣿⣟⣿⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⣿⡟⠋
⠀⣾⣷⣾⣿⣿⠀⠀⣿⡇⠀⠋⢹⡿⠀⠈⣧⣿⣷⣿⣼⣷⣡⣧⠀⠻⣿⡟⢽⣿⡅⠀⠀⣽⣯⣿⣿⣿⣿⣻⡇⠀⠀⣀⣤⣴⣆⡀⠀⠀⠀⠉⢻⣿⣦
⠀⠈⠛⠛⠛⠋⠀⠀⠈⠛⠀⠐⠋⠁⠀⠀⠙⠛⠉⠋⠋⠙⠛⠋⠀⠀⠘⠀⠘⠋⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠃⠀⠾⠿⠿⠿⠿⠟⠛⠗⠀⠚⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⡽⠛⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠒⡿⠒⠂
⠀⠀⠀⣠⣼⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠖⠁⠴⡏⠀⠀⠀⠀⢀⠀⠔⠠⠏⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡀⠀⠀⠀⢸⣧⠀⡴⢏⣤⠒⢦⠄
⠀⣴⢠⣿⡿⢀⡀⠀⠀⠀⣠⡶⠂⣀⠀⠀⠀⢁⣀⠀⡐⠀⠀⣠⢤⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⣸⣷⡄⢿⠛⢿⣣⣀⣀⡄⢀⡀
⣾⣿⣿⣿⣇⣾⠁⡀⢀⢠⠠⠶⢂⠈⠀⠀⠀⠁⢸⠇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⢠⡀⠀⠀⠸⢻⡄⠀⢸⣿⠟⠷⣦⠒⠀⠈⣯⠉⢉⣸⡇
⢺⣿⠛⢨⣿⣿⣼⣷⣟⣠⠰⠀⡼⠁⠲⡴⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⡀⢀⠞⠿⣁⠀⢙⡀⠚⡀⠲⣧⠔⢦⡘⢶⣤⡙⢻⣴
⢸⢿⣃⣤⢹⣿⣿⣿⠟⣡⡄⠀⡤⢀⡠⠀⢲⣶⣴⡄⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡬⢓⠇⢠⣠⡼⠆⠙⡧⣎⣀⠚⠙⢢⣻⠆⠀⠑⠂⠘⠿⣿⣿
⠠⡾⠋⢃⢸⣿⠟⠋⢁⠇⠈⣀⠀⠀⠀⠓⡀⣻⣿⣿⣿⣷⣶⣼⣴⣶⣠⣄⢀⣠⣴⣶⣼⣷⣶⣾⡶⠣⠤⣘⣇⡠⢢⣈⡶⠉⢀⡄⠀⠀⢻⡃⠀⠀⠠⣄⠀⢈⣿
⢀⣆⠀⠋⢘⣥⡴⠚⠛⠠⠖⠃⠀⠀⣀⠀⠉⢺⣿⢿⣿⢻⣟⣿⣻⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⡿⠂⢀⡀⠈⠡⣴⠀⢹⡴⠞⢉⡴⣦⠜⢶⡤⠀⠀⠀⠘⠛⠿⣿
⠸⠧⠆⢁⡞⠋⠀⠰⠂⡴⠦⠀⠀⡾⣯⠉⠢⣄⡛⣏⠍⣋⣽⣥⡿⠟⢁⡀⠀⠀⠀⠀⠈⠛⣿⡀⠄⠈⠉⠻⢷⣎⢢⡼⠷⠒⠉⠀⠘⠦⢄⡙⠀⠀⠐⣆⠀⠀⢈
⠀⠀⠀⠈⡽⠋⡀⣀⡔⠀⢀⠌⠛⢇⠘⡶⠀⠈⠉⠉⠉⠉⠏⢀⠤⢾⣧⣽⣷⣤⣀⣀⣠⣤⠌⡀⠀⠐⠂⡐⠦⣽⣿⣯⠔⡠⣆⠀⠀⢦⡄⠈⠛⣤⠀⠈⠁⠢⣌
⠀⠀⢀⡼⠓⠀⠁⠰⢤⡤⠟⠀⠀⢀⡓⠙⡇⢠⣄⢨⠀⠀⠀⠙⠺⠿⠛⠒⠓⠚⠛⠛⠚⠡⠀⠤⠐⠃⠠⠟⢦⣤⡹⠟⠊⠁⡼⡀⢠⠿⠟⠀⢠⢈⣦⣾⠀⠀⢻
⠀⠠⠞⠡⠀⣀⣤⠖⠾⡄⠀⠀⠀⠈⢳⣠⠝⢦⡉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡀⡀⠙⠈⢢⠀⢄⠀⠻⣆⡀⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠚⠃⠐⠊⠉⠂⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⣤⠄⠞⠉⠣⠠⡀⠀⢦⡨⠦⠀⢨⣿⢯⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠰
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣽⣭⣿⣭⣯⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣾⣶⣾⣿⣿⣶⣾⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⣀⡀⢀⣀⡀⣀⣀⣠⠀⣀⣀⣤⣄⣀⣀⣀⣀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠪⠐⠁⠐⠫⠀⠀⢜⣻⠒⠋⡙⢣⣌⠻⡛⣿⣿⢦⡽⣿⣿⣿⣿⣿⣯⣿⢯⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⣤⡶⠒⡒⠲⡯⣌⣉⠘⣥⡘⢠⣝⣧⣹⣿⣼⣾⣇⣷⣼⣟⣿⣿⣿⣏⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⢶⠟⠿⡛⠟⣀⣀⣬⣤⣶⣭⡈⣹⢷⡍⢈⠘⡿⣿⣧⣹⣿⣱⣿⣿⣿⡟⣷⡿⣷⣾⡏⡇
⠀⠀⠀⠀⠀⠀⣴⠿⠿⣾⡄⠀⢀⡴⠁⡞⠋⠁⠀⣫⡧⢄⣉⠻⢷⣤⢈⣳⡘⣿⣏⡜⢷⣾⣿⣯⣿⣷⣿⣿⣿⣿⣟⡇
⠀⠀⠀⢠⢈⣳⣾⢧⢛⡋⣁⠐⢛⡶⠚⠁⠀⠀⣰⣿⡷⢊⣭⠌⣁⡹⣧⣈⠙⠺⣯⡳⣾⣿⣯⣽⣻⣿⣿⣯⣳⣿⣿⡇
⠀⠀⠀⠀⠙⣿⠀⠀⠸⠁⠀⠀⣨⠀⣀⣠⡄⣉⡡⠶⢗⡾⠡⢤⠶⢷⢶⡽⣷⣧⣹⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣀⢵⠁⠀⢦⣤⠄⠚⠁⠖⠁⠀⢰⠋⠻⢓⠞⣷⠓⠢⣀⣭⣥⣌⡙⢿⣍⣿⡦⣿⣿⣿⣿⣿⣟⣿⡿⠿⠿⡇
⠀⠀⠀⠀⡉⢾⣖⠃⠐⠅⠀⠀⡀⠀⠀⠀⢿⡂⣤⡾⣏⣽⣄⠀⠀⠀⢥⣬⠟⡛⢿⡿⣧⣿⣿⣿⣿⣿⣿⣿⣽⢹⡿⠅
⠀⠀⠀⠀⡜⣽⣿⣰⣿⠂⡖⠄⠀⠀⠀⠰⣤⣿⠵⡴⠞⣐⣧⡤⣰⣯⡐⠯⢶⡌⢻⣿⣿⣿⣼⣿⣿⣿⣟⣿⣷⣟⢡⡖
⠀⠀⠀⠈⣡⠀⡧⢿⡿⡆⠀⠀⢄⠀⣄⠀⣨⡽⠿⢻⡟⡽⣿⠡⠑⠀⣹⡟⢦⣉⠛⢻⣿⣿⣿⣿⣿⣽⣿⣿⣿⡤⠟
⠀⠀⠀⠀⢸⡄⠁⣐⠲⠀⡀⠀⢀⣤⣭⣾⠍⣷⠶⠎⢑⢷⠻⣄⠀⠀⠉⠀⠀⠈⠳⣌⠻⣿⣼⣿⣿⣯⣾⣿⡟⠁
⠀⠀⢸⠀⣸⣧⠀⠩⠄⣰⡷⠰⡟⡏⢻⡿⣿⣧⠀⠀⠋⠈⠀⠛⠀⠀⠒⣀⣀⡀⡀⢻⣦⣽⣿⣿⣿⣯⣻⣿⠇
⠀⠀⠠⠀⣿⣿⡇⠀⠀⢘⢰⣼⣷⣄⠳⣦⡻⣥⣧⠀⠀⠀⠀⣦⣶⣪⣿⣿⣿⣿⣿⣆⡽⣿⣿⣿⣧⣿⣿⣿⠂
⠀⠀⢠⢸⣿⣿⣷⡂⡀⢘⣴⣟⣿⡾⡽⣸⣿⣧⡿⢈⡘⠀⣼⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡙⣿⣿⣿⣿⣿⣟
⠀⠀⠸⣼⣿⣿⣾⣷⣴⣝⣷⣾⣷⣿⣶⢙⠿⢿⣿⡏⢑⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣟⣿⣿⣿⡇
⠀⠀⢰⣿⣿⣿⣿⣿⡿⣿⡓⢸⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠛⢚⣿⣿⣿⣯⡁
⠀⠀⣸⣿⣿⣿⣾⣿⠹⣿⣿⣿⣿⣿⣿⣏⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⣵⣿⣿⣿⣿⣅
⠀⠀⠘⢻⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣯⣿⡏⢻⣿⣿⠛⣉⠛⠛⢿⣿⣿⡷⢷⣿⢻⣟⡼⡷⣿⣿⣿⣿⣧⠐⠀⠀⢀
⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠈⠛⠛⠿⢿⣿⣿⠹⠋⠡⠞⠉⠑⠀⠀⢹⡏⠀⢹⣹⡊⣿⠇⠶⣿⣿⣿⣿⡿⠃⣀⣤⣿
⠀⠀⠀⠘⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠤⡄⠀⠀⠚⣛⠋⢸⣇⠀⠀⠈⠉⠉⣳⣮⣹⣿⣿⣿⣿⡛⠇⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⣶⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠘⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠀⠀⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣥⣼⡿⠟⢉⡰⢿⠻⠟⢆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠊⠉⠀⠀⠀⠀⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠀⣄⣠⠄⠘⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⡀⠀⡀⠃⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣧⣠⣀⣤⣀⢀⣀⠀⠀⢀⣀⡀⢠⣀⣀⣀⡀
⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⢸⣿⢯⣿⣿⣿⣿⣿⢿⣻⣷⣿⢟⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠉⠉⠉
⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠸⠿⠾⠿⠿⠿⡟⢵⠿⠿⠿⢦⡿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠷⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠞⠛⠛⠛⠛⠶⣦⣤⡾⠋⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣤⣤⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡶⠶⠶⠶⢶⣤⡀
⠀⠀⠀⢀⣀⣴⠟⠋⠉⠻⠋⠀⠀⠈⢻⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⣴⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⡀
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⡀
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠙⠛⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⠟⠉⠉⠉⠉⠻⡶⠛⠛⠛⠷⣦⣰⡟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠷⣦⡀
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⣽⣿⢭⡇⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢤⡆⠀⠀⠀⠀⠀⠀⠀⠈⠾⠽⠶⠁⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⣺⣗⢺⠂⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣻⣿⣻⡟⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠃⠀⠀⠀⠀⠀⢾⡷⣽⠆⠀⠀⠀⠀⣴⡖⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠹⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠸⣗⡿⣗⡟⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⢲⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⠙⠀⠀⠀⠈⠩⠣⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⣺⢿⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⡩⡄⠒⡖⠂⠬⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁⠈⢌⢠⢐⡠⠐⠝⢂⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠐⠒⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠃⠭⠀⠀⣈⠁⠈⢚⠠⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣤⡄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣆⣤⣤⠴⠷⠋⣷⡙⠄⠀⠨⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠏⠈⣪⠖⡡⠂
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⠡⠐⠈
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣀⡀⠄⢀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠄⠂⠈⠫⠑⠂⠀⠒⠊⠁⠀⠀⠀⠀⠙⢸
⠀⠀⠀⠀⠀⠀⠀⢰⢰⠀⠀⠀⠀⠠⠀⢁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈
⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⡈⡍
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠁⠀⠀⡀⠔⠀⠀⠀⠒⠒⠂⠈⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠌⠀⡀⠁⠙⠦⠀⣀⠀⠈⠀⠀⠈⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢈⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠨⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠢⣙⠒⠶⠤⠤⠄⠰⠂⢒⡩⠔⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠉⠉⠐⠒⠒⠈⠉⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠒⠒⠊⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⢸⣿⣿⣹⣿⣿⣛⣿⣍⠉⣿⡏⣹⣿⣿⣿⣟⠉⢉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⠉⠙⣿⣿⣿⣍⠉⢛⡛⠛⠉⠉⠉⣛⣿⣹⣿⡇
⢸⣿⣿⣿⣿⡟⢸⡇⢸⡀⣿⣗⠸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣿⣿⣿⣿⢰⡿⡇⢀⠀⢿⣀⣿⣿⢿⣿⡇
⢸⣿⣿⣿⣿⣷⣿⣧⣾⣷⣾⣿⠀⢹⡟⠉⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠈⣿⠁⢸⠀⡇⢾⣤⢸⣿⣿⣿⢸⣿⡇
⢸⣿⣿⢿⣿⣿⣿⣿⡿⠻⡟⠉⠀⠈⠁⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⣷⠇⢿⣿⣿⣿⣿⢸⣿⡇
⢸⣿⣿⠸⢿⣿⣿⣿⡇⠀⠇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⢻⢸⣿⡇
⢸⣿⣿⢸⣿⣏⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠐⠏⢸⣿⡇
⢸⣿⣿⢸⠛⣿⡏⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠈⣿⣿⠈⠐⠟⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣿⣿⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢸⣿⡇
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣿⣿⠀⠀⠂⠀⠀⣤⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⢇⠀⡄⠀⠀⠀⢀⡀⠀⠀⠀⢷⠄⠰⠀⠀⠀⢸⣿⡇
⠀⣿⣿⠀⠃⠀⠀⠀⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣸⠀⠃⠀⠀⠀⠀⡏⠀⠀⣸⣟⡆⠁⠀⠀⠴⠈⣿⡇
⢠⣿⣿⠀⠀⠀⠀⠀⣸⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⠀⡶⠀⠀⠀⠀⡇⠀⠀⢸⡿⠀⠀⠠⠈⠀⢠⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣿⣰⡁⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⡇
⠸⣿⡿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡇⠀⠀⠀⠀⡇⠀⠀⣿⠇⠀⠀⠀⠀⠀⢸⣿⡇
⢰⣿⡇⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠇⠀⠀⠀⢀⡇⠀⣀⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠈⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢻⠀⠀⠀⢀⠈⡇⠀⣾⠋⠀⠀⠀⠀⠀⠀⠸⣿⡇
⢰⣿⡇⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠘⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣿⣿⡄⠀⢀⢠⣿⣾⣿⣠⠀⡄⠀⣤⠀⠀⢀⣿⡇
⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⠿⠟⠛⠚⠛⠛⠛⠛⠛⠛⠒⠛⠛⠛⠓⠚⠸⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⢀⣼⣇
⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣤⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⢰⣷⣿⠇⠀⢻⣷⡆⠀⠀⢰⣷⣿⠿⣿⣿⣆⡄⠀⠀⠀⠀⢻⣿⣿⣿⣶⣇⠀⠀⠀⠀⢸⣿⣿⠻⣿⣶⠀⠀⠀⠀⡄⣿⣿⠏⢿⣿⡆
⠀⠀⠸⣿⠁⣰⣦⠈⣿⡇⠀⠀⢸⣿⠿⠀⠘⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠐⣶⡿⠁⡀⠛⣿⠀⠀⠀⠀⣿⡿⠁⠀⢸⢿⡿
⠀⠀⠀⠙⠦⡙⢛⠴⠋⠀⠀⠀⠀⠻⠄⠀⠀⠠⠟⠀⠀⠀⠀⠈⠻⢿⣿⡿⠟⠁⠀⠀⠀⠀⠹⠄⢾⡿⠀⠋⠀⠀⠀⠀⠙⠧⠀⠀⠀⠜⠁

⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣿⣷⠀⠀⠀⠀⠀⠀⣆⢸⣿⡄⡄⠀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀
⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀⠀⠀⠀⢠⡄⣾⣿⢿⣿⣀⡄⠀⠀⠀⠀⣶⣻⣿⣷⢀⡄
⠀⠀⠀⢸⣿⣿⣿⣇⣷⠀⠀⠀⢈⣿⣿⠏⠻⣿⣿⣶⠀⠀⠀⣼⣿⣿⠋⢻⣿⣧⠀⠀⠀⣿⣿⣿⠟⠈⣿⣿⣿⠀⠀⢰⣦⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣷⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⡇⢀⣦⠀⣿⠇⠀⠀⠀⢻⣿⠁⠀⣼⢿⡟⠀⠀⠀⠸⣿⠉⢀⣆⠉⣿⠏⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⠛⠻⠟⠛⠁⠀⠀⠀⠀⠈⠁⠘⠟⠁⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠛⠟⠀⠉⠀⠀⠀⠀⠈⠙⠚⠛⠛⠋

⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⢀⢀⣼⣿⣷⡄⠀⠀⠀⠀⡀⠈⣿⣧⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⠀⡀
⠀⠀⢀⣿⣿⣿⢿⣿⣿⡄⠀⠀⢀⣿⣄⣽⣿⡟⢀⡀⠀⠀⠀⡀⢻⣿⠛⢿⣿⣆⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⡀⣿⣿⣿⣾⣿
⠀⠀⠸⣿⡟⠉⠀⠹⣿⡇⠀⠀⢸⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⣿⣾⠋⢠⡀⢻⣿⠀⠀⠀⢠⣸⣿⡏⠿⣷⣾⡆⠀⠀⢸⣿⣿⣿⠛⢿⣿⣿
⠀⠀⠀⠻⣧⡀⠀⣰⠟⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⢿⣄⣿⣧⡼⠃⠀⠀⠀⠀⠻⡏⠁⠐⢻⡟⠀⠀⠀⠈⢿⡿⠷⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠆⣀⡀
⠀⠀⠀⣠⣄⠀⠀⠀⠀⠠⢤⢀⢬⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡸⠄⡀
⠀⣠⡾⠟⠁⠀⠀⢀⠀⠐⣊⣠⢤⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣠⠀⡾⠃⠠⣄⠒⠆
⠠⠿⠿⠿⠄⠀⠀⠀⠀⡀⠈⣤⠀⢀⠰⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠛⠂⣮⠅⣈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁⢨⠀⡠⠀⠺⠇⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠵⠂⠨⡡
⡀⠀⠀⠀⠀⠀⠀⠀⡘⠂⠉⠙⠒⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠆⠀⠰⠀⠀⠀⢠⣅⣋⣭⡟⠒
⠈⠀⠀⠀⠀⠀⠀⢐⡤⠾⠀⠼⠀⢳⡤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⢹⡿⠄⠿⢶⣶⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢂⣉⣉⣉⣉⡐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢁⠴⡞⢇⠀⢀⡜⡇⠈⠑⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡴⠃⠀⠘⢎⠳⣘⠁⠇⡠⠖⡡⠛⢆⠱
⠀⠀⠀⠀⣀⣀⡀⠤⠤⠤⠤⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠠⡆⠰⠇⠤⠤⠤⠬⠧⠙⠀⢋⣀⠊⠀⠀⠈⡄
⣠⡖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠉⠙⠐⠒⣶⠄⣠⡄⢠⣤⡍⡉⢉⢐⡃
⡇⠉⠛⠶⠤⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⣘⣄⢀⣤⠾⠷⠟⠃⣿⢄⠳⡀⠀⠀⢠⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢍⠉⢊⡁⠊⠀⢠⡄⣿⠈⠳⡙⢄⡠⠋⡰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠑⠤⣀⡀⣇⣿⠀⣠⠼⠋⣡⠞
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠀⢈⣿⠀⠤⠒⠋
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⢸⡀⢀⣄⣀⠀⠀⠀⣀⣀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢄⣀⠀⠀⢸⠇
⠀⡇⡏⠀⠈⠉⠉⠉⠀⠀⣀⠤⠒⠛⣗⠢⢤⣀⣀⣀⣠⠤⠒⠋⠁⠀⠀⠀⠀⠈⠳⡄⣸
⠀⡇⢹⠀⠀⠀⠀⢠⠖⠉⠀⠀⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿
⠀⢳⢸⠀⠀⠀⠀⡆⠀⠤⠒⠉⠙⡄⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡇
⠀⢸⠈⡇⠀⠀⠰⡀⠀⠀⠀⠀⠀⠈⠂⢠⠸⣀⣀⣀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣸⢸⡇
⠀⠘⡆⡇⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣠⢳⠁⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⡿⣸
⠀⠀⡇⢻⠀⠀⠀⠀⠱⡌⢦⢀⣠⠴⠚⠁⢸⠀⠀⢸⠃⠉⠁⠀⠀⢹⠀⠀⠀⢠⡇⣿
⠀⠀⢣⢸⠀⠀⠀⠀⠀⠈⠓⠉⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢣⡇
⠀⠀⢸⠈⡇⠀⠀⠀⠀⠀⠀⢠⠞⠉⠒⠤⣈⣆⠆⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⣾⢸⠃
⠀⠀⠈⡆⣇⠀⠀⠀⠀⠀⢠⠏⠀⠀⣆⠀⠀⠉⢳⠦⠀⠀⠀⠀⠨⠜⠀⠀⠀⡇⣾
⠀⠀⠀⣇⢸⠀⠀⠀⠀⠰⠃⠀⢀⠎⠀⠉⠒⠄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡏
⠀⠀⠀⢸⠘⡆⠀⠀⠀⢠⡗⠀⠈⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠃
⠀⠀⠀⠘⡄⣇⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾
⠀⠀⠀⠀⣇⢸⡀⠀⠀⠀⠀⠈⠑⠢⢄⡰⢊⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡇
⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠁
⠀⠀⠀⠀⠸⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡟
⠀⠀⠀⠀⠀⣷⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⡇
⠀⠀⠀⠀⠀⢸⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼
⠀⠀⠀⠀⠀⠀⢷⠘⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠘⡷⢦⣈⠛⠲⠶⠤⣤⣤⣤⣤⡤⠴⠶⠚⠋⣁⡤⢞⡏
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠉⠛⠲⠦⠤⢤⣤⣤⣤⠤⠴⠶⠚⠋⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣶⣶⣶⣶⣶⣶⣾⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⠀⠈⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⢀⠄⠀⡆⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠰⠳⣄⡎⠀⠀⠁⠠⠀⡜⢆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠈⠆⠀⠀⢀⠀⠰⠁⢸⠈⠰⠃⢰⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡏⠲⢦⠀⠀⠀⠀⠀⠈⢄⠁⠀⢸⠀⠀⠀⡌⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⡘⠘⡐⢚⡇⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⡀⠀⠳⠉⠀⠌⡰⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢤⣄⡀⠀⠀⠀⠀⠂⠄⠐⠂⠀⠀⠁⠄⠂⠀⠀⠤⠊⠉⠈
⠀⠀⠀⠀⠀⣼⠹⣭⢍⣠⢴⠚⢠⣀⣤⡤⡖⠢⣤⢼⣐⠒⣦⠀⢲⢢⡀⠘⠛⣻⣻⡄⢠
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠻⠼⠛⠛⠊⠳⣾⡴⠃⠰⣌⣠⡼⠷⣤⣸⡖⠲⠮⠯⠟⠛⠛⠃

⣽⣷⣶⡇⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣉⣿⣶⣿⣿⠿⠛⠃⠉⠙⠐⠉⠀⠀⠈⢻⣿⡷⠀⢠⠀⠀⠀⢻⣿⣿⣿⣷⣿⣿⣿⡿⢾⣏⣹⣿⣿⣿⣯⣨⣿⣤
⣿⢿⣿⣆⣸⣶⡆⢹⣧⣿⣿⡟⢹⡟⢤⡿⣽⣿⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⡗⣄⢷⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠹⣿⠟⢻⣿
⣿⣶⣶⣿⡿⠻⣿⣿⣿⣿⣿⣿⣶⠀⢺⣅⢠⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣼⡀⢁⠀⠀⠈⢿⣿⣟⠁⠘⠿⢛⢃⣀⣀⠀⠀⠀⠈⣳⣿⢻
⣽⣟⠉⠛⠛⠀⠉⠁⠀⠈⠉⠀⢤⣷⣾⣇⢸⣿⠃⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠈⠇⠘⡦⠀⠀⠸⣿⣾⣄⠀⢀⣤⣛⠉⢻⣿⣷⣶⣺⣿⣿⣿
⣿⡿⠷⠀⠀⠁⠁⠀⠀⠀⠀⠀⡾⢿⡏⣸⣿⣿⢠⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤⢤⣇⣰⡀⣿⠀⠀⠈⠛⡝⢻⢿⣷⣤⣿⠳⠾⠷⠾⣿⣿⣿⣧⣾
⠿⣧⣴⣶⣦⣤⣤⣤⣀⠀⠀⣘⣧⣬⣿⣿⣿⡷⠀⠀⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣙⣿⣟⣹⣿⡄⠀⠀⢹⣿⣿⣿⣿⠋⠀⠀⠀⠀⣾⣟⣿⣿⣿
⣿⣟⣿⣿⣿⣿⠉⣿⣿⣿⠠⣗⡟⢿⣤⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⠿⣷⣿⠀⢻⣻⡟⠃⠀⢰⣏⠻⠟⣹⠟⢣⣤⣤⡀⠀⠻⣿⣿⣿⠿
⣿⣿⡉⠟⠛⠿⣿⣿⣿⣿⣤⡿⣄⣼⣿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⣻⣧⣀⢸⡟⡿⡀⠀⣴⣾⣻⣿⣿⡇⠀⠀⠙⢿⣧⣀
⠿⣿⡃⠀⠀⠀⣯⣬⣿⢟⡿⠻⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣧⡄⢿⣿⣿⠸⣷⣿⡷⢀⣿⣿⣿⣿⠿⢇⠀⠀⢴⣿⣿⡏
⣀⣼⣧⣀⠀⠀⠐⠂⡸⣹⣿⣿⣿⣿⡿⢳⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢤⡇⢸⣿⡿⠀⣿⢿⠅⠙⣹⣿⣅⣽⠀⠀⠀⠠⣾⣿⣿⣿
⣟⣻⣧⣜⣷⣶⣿⡟⢣⣼⡇⠸⣿⣿⣦⣿⣼⠀⠀⠀⠒⠒⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⣿⣿⢸⡇⠸⠇⣿⡆⢸⣇⠀⢄⣹⣟⠿⠿⠀⠀⢀⣤⣿⣿⣿⣿
⡟⠉⠻⣿⣿⣷⣛⣿⣏⡿⣷⠰⣿⣏⠒⣧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠃⡝⠛⢿⠃⡆⠀⣴⣿⣾⢿⡀⣿⣟⣿⣦⣤⣦⣴⣾⣿⣹⣿⣿⣏
⠀⠀⠀⠀⠀⠉⠉⢘⣿⠓⣿⠀⢋⣁⠠⣟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣶⠆⠀⡇⡀⣿⣿⠙⣸⣿⣿⣤⣿⣿⣿⣽⠿⣿⣿⣉⣿⣿⣿
⠀⠀⠀⠀⠀⢲⠂⢸⡇⢸⡇⢈⣬⡏⣼⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠀⢸⣿⣿⠀⢸⡏⣇⢹⡟⠀⠙⣿⣿⠿⣿⣿⠿⠿⠷⢿⢿⣅⣿⡟⠙
⣶⣤⣴⢶⣴⣯⣤⣼⠀⣸⠁⢿⣿⣿⣿⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⢸⣿⡟⠀⣼⠁⢻⣿⡇⠀⠦⣼⣿⣾⣿⣿⣿⠀⠀⠀⠀⠟⢹⣷⣾
⣿⣿⠿⣿⣿⣽⣿⣻⡆⢿⣄⣈⡾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠔⢺⣯⡿⠀⠀⠘⠻⠁⠐⠛⢠⣸⣿⠠⠀⠨⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⢀⠻⣿⣿
⣿⣿⣦⣾⣿⣷⣿⣿⣄⡀⢹⣿⣿⣿⣿⣿⡏⠸⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⡛⠀⢀⡀⠆⡀⠀⢿⠀⣾⠘⣇⣠⣦⢠⣿⠻⣿⣋⣿⡿⣿⠧⣠⣴⣿⣿⣿⣻
⡞⠁⢹⢻⡟⠛⠉⠘⠘⣇⠙⠍⡉⣸⢿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⣦⡜⠡⠀⠀⠀⡀⢒⣶⣷⡎⣸⣿⡷⠈⠉⠙⠿⣿⣿⣿⣏⣿⣿⣿⣷⣿⣿⣿⣽⣿⡿
⣿⣶⣾⠈⠀⠀⠀⠀⠀⠸⣿⣰⣷⣿⣾⠛⡇⠀⡀⠀⠀⠀⠐⢯⣹⣿⣦⠠⠀⠠⠌⣠⣾⣿⣏⣹⣿⢿⡇⠀⠀⠀⠀⠈⠛⠉⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿
⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠉⣿⡄⠀⠀⡇⠀⠀⠀⠀⠀⢻⡿⠋⢀⠀⠀⣴⣤⡿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣽⣿
⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠔⠛⠁⠀⠀⠁⠀⠀⠀⣶⣶⠞⠁⡀⠁⠀⣼⣍⠟⠀⠘⢽⣿⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣉⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠲⢦⣄⠀⠀⠀⢀⣠⡤⠶⠶⠒⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠻⣆⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠛⠛⠧⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢀⣀⣀⣀
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠉⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠈⣻⡶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣀⣀⣀⡴⢾⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠿⢫⡟⠛⠛⠻⣤⣞⡃
⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠙⢦⣤⣀⣀⣠⣤⡞⠉⢳⣦⣤⣀⣀⡀⠀⢀⣀⣤⣏⡿⣠⣭⡗⠀⠀⠀⣿⠉⢿⠄⢀⣄
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣯⠹⣤⣤⣤⣽⡜⢯⣩⣟⢻⣍⣧⠙⣴⣧⣴⠟⣦⣤⡾⠡⣿⡭⠤⠟⠼⠧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣶⢂⡶⢶⣲⡶⣠⠶⠶⣦⡴⢶⡶⣀⠰⡖⢶⣔⡶⠒⣆⢲⡒⠒⠓⢶⣘⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⢾⡉⢰⡇⠀⠀⢸⣗⠛⢰⠋⣷⠙⣻⡏⠀⠀⠘⣧⠛⢶⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⢧⣤⣤⣤⠟⡴⠞⠛⠳⢷⡙⣦⣀⣀⡴⣷⣦⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡴⠒⢲⢆⡘⣇⠀⠀⠀⣰⢃⡄⢶⠶⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡞⣷⠙⢲⣶⠚⢡⡏⠹⣎⢧⣍⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⢁⢺⡄⠙⠶⠛⠹⠷⠖⠋⠀⣸⠂⢰⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠟⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠷⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠀⠀⢿⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠙⢦⣌⢷⣄⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠙⣆⠙⠷⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢛⡓⠦⡀⠈⢷⡀⠀⠀⠀⣿⠀⠀⠘⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⡄⠀⠻⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⣰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡆⠀⣠⠟⠀⠀⠀⠹⣄⡀⢻⠀⠸⣇⠀⠀⠀⠀⠀⢰⡇⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⠟⢰⡏⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠘⢷⣄⠀⠀⠀⠀⠻⣄⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⡉⠀⠀⠻⢦⣄⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠉⠳⢦⣄⠀⠀⠈⠳⡄⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠀⠀⠈⠉⢻⠓⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠃⠀⠀⠀⠀⢠⣸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⢠⡞⠀⠀⠀⢀⣤⠟⣿⠃⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠀⠀⢹⡆⢀⡤⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢁⡖⠉⠀⣀⡴⠞⠋⠀⣰⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠘⡇⢸⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠘⢧⣠⠞⠁⠀⠀⠀⠘⠁⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢰⠇⢸⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠁⠀⠀⠀⠀⠀⢠⢀⠞⠁⠀⣰⠋⣠⡾⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠃⢀⣠⠄⣀⣴⠆⢀⠀⠀⠀⠞⠀⠀⠀⠀⠟⠉⠁⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠰⣶⣤⣛⠀⠈⢁⣾⣿⣷⣿⣿⠃⠀⠀⣀⠀⠖⢀⣀⡀⠀⠀⠀⠀⠀⣰⣯⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⣍⣻⣦⣄⡀⠀⣀⠐⣀⣈⠛⠻⣟⣻⣿⣿⡿⠟⣋⠀⠀⣠⣤⣤⣤⡶⠿⠛⠁⠀⠀⠀⣀⡼⠯⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠴⠶⠛⠻⢶⣤⣄⣈⣁⣰⣿⡿⣯⣥⣤⣀⡛⣿⣿⣶⣬⣍⠀⠻⠷⠶⢶⣆⣤⣴⣾⣿⣶⣶⣶⠤⠴⠶⠖⣶
⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⣤⡤⠤⠶⠶⠚⠋⠉⠉⠉⠉⠁⣀⡤⣿⣿⣿⣿⣿⣿⣿⡿⠿⢶⠲⠶⣿⣿⣭⢤⣭⣍⣛⡛⠛⠶⠦⢤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠋⢑⣻⡿⠟⠋⢁⣀⡈⠉⠉⠉⠋⠉⠛⠒⠾⠭⣅⣀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⣯⣉⣀⠤⠖⠚⠉⠉⠀⠠⠖⠛⢉⣀⡙⠚⠙⠛⠶⠶⢦⣄⣠⣤⡴⠶⠯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠐⠀⠀⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠈⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠀⠄⠀⠀⠀⠀⠚⠀⠀⠀⠃⠈⠀⠀⢒⡔⠫⢣⠀⠀⠀⠈⣡⠎⢱⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠥⢌⣧⠀⠐⠀⠀⠀⠀⣠⠞⢳⠀⠐⠂⣠⣞⠉⠙⢺⡁⠀⠀⣰⠏⠉⠑⢿
⠀⠀⠀⠀⠀⠀⡀⠁⢾⡥⢄⣀⣹⡄⢀⠀⢀⠁⢠⣎⣉⡙⣇⠈⣠⣔⠉⠙⡓⢿⠃⠀⠀⡽⢋⠙⡒⢞⠂⠀⠄
⠀⠀⠀⠀⠀⣀⠀⠠⣎⣄⣀⣀⢱⡀⠀⠄⠀⠀⣼⣁⣤⣘⣏⢀⣔⠉⠙⢓⠲⡼⠆⠀⢈⠗⠒⡒⠓⣼⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣔⣁⡂⠀⠈⠹⡁⠀⠄⡀⠰⠯⣤⣆⣈⣫⠁⠉⢹⠒⡶⠤⠵⠐⠀⠓⠒⡶⢤⠤⠬⠆⠰
⠀⠀⠀⠀⣀⣀⣈⣀⣀⣏⣹⣉⣉⣁⣀⣀⣁⣀⣀⣸⣼⣤⡤⠤⠤⠾⠴⠦⠤⢤⣤⣀⣀⣀⣇⣼⡤⠶⠤⣄⡀
⠀⠀⠀⠀⣿⠉⢉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⡠⠤⠄⠠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⠉⣷
⠀⠀⠀⠀⣿⠀⠀⠉⠑⠒⠒⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠒⠤⣀⣀⠤⠤⠤⠒⠂⠀⠀⠀⠉⠉⠙⠒⠒⠒⠋⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⠤⠤⠒⠢⠄⣀⠀⠀⠀⠀⠐⠉⠉⠑⠢⢄⣀⣀⠤⠤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠈⠐⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⢤⡀⠀⢀⠔
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠸⠠⠀⠀⢠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠐⡒⠀⠀⠀⠀⠀⠀⢄⠠⡜⢣⡜⣤⠀⢠⡄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠐⠆⠰⠀⠀⠀⠀⠀⠀⠰⠄⠄⣤⠢⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣄⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⠀⠀⣀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠁⠒⠀⡠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠛⡄
⠀⠀⠀⠀⢀⠔⣁⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⢠⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠈⠢
⠀⠀⠀⠠⡁⠐⠉⠀⡀⠀⠀⠀⠈⢀⠄⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⢀⡀⢤
⠀⠀⠀⢀⠐⢀⣤⠐⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⡀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠠⡃⠀⠂⠀⠁⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⢈⠶⠀⣄⠈⠫⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⡀⠀⠀⠀⠀⠀⢀⠠⠈
⠀⠀⠀⠁⠀⠈⡅⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠣⣸⣎⠐
⠀⠀⠑⠰⠀⢀⣀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠊⢀⠀⠀⠀⠀⠀⠠⢄⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠆⠀⠀⠀⠀⢐⣀⣠⡊⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠙⠉⠀⠁⠘
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠒⠄⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣶⣿⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠘⠠⠄⢀⣀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⠁⠠⣀⠁⠀⠀⣀⣦⡀⠀⠀⠀⠀⠘⠤⠃⠀⡄⢀
⠀⠀⠀⠈⡄⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⢀⠆⠀⠀⠈⠀⠀⡆⠀⠀⠀⠀⠀⠔⠁⡃⠘⠂⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠻⠩⠦⢀⣀
⠀⠀⠀⠀⠘⠉⠉⠁⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠐⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⡀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀⠗⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠤⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠁⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⣀⣀⠃
⠀⠀⠀⠀⠀⠢⣤⠀⠀⢀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⡀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⠟⠁⠀⠀⠀⠀⠀⢀⡀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠬⢼⠯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣠⠆⠡⡞⠁⠀⠀⠀⡀⠠⠄⠎⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⠀⠀⠀⢸⡇⠠⠖⣃⡆⢤⠄
⢀⣴⡠⣦⢀⡄⠀⠀⠀⠴⠞⢠⡀⠀⠈⣁⣀⠀⠁⢀⠠⢴⡟⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⣼⣷⠘⠟⣻⢧⣄⣠⡄⣀
⣾⡯⡐⠁⣌⣷⣀⣠⠴⠐⠋⣄⠑⠀⠀⠀⢿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠞⠁⠀⠀⠀⢰⡄⠀⠀⣏⣷⠀⣸⠿⡏⠛⢟⠁⣀⠻⡈⠑⣿⡀
⣿⠞⣫⢹⠮⣻⣿⣷⠆⠁⠘⡃⠉⠱⡀⠈⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⠄⠘⢉⠳⡄⠸⠊⣱⣌⡙⣿⠟⢆⠙⠳⣬⣿
⠿⢒⣬⡏⣳⣬⡿⠇⠘⠶⠊⠙⠻⢤⡱⣦⣤⣴⣀⣦⣀⠠⣿⢄⣀⣰⣄⣤⣴⣿⣶⣞⢝⣁⢸⠞⣩⡀⣏⠔⠈⠻⣀⠻⣍⣶⡤⣄⡀⠙⣿
⢴⡇⠘⡟⢿⣥⡤⢾⣤⡴⠛⣆⣷⠀⠛⢿⣿⣿⣿⣿⣿⡉⢿⠓⣿⣿⣿⠏⢹⣿⣿⡧⠚⡈⠙⢋⡅⡛⣁⠄⠂⣷⣈⠦⣿⣿⣄⠈⠳⠶⣿
⠚⢱⣼⣿⠟⣉⡤⠤⣤⠇⠀⡡⠻⠓⢦⣐⢿⣯⣯⣻⡯⡤⠆⠠⡹⣿⠿⠞⣨⣿⣏⡀⠚⠛⠓⢦⣿⣤⠇⠒⠊⠉⢻⣄⡘⢇⠀⠰⡄⠀⠘
⣠⠟⣧⣿⠟⢁⣠⠖⠁⣠⠃⠀⠀⣄⠀⠉⠻⣿⣮⠟⠈⢠⢚⠞⠁⠺⣊⠈⣀⡼⢁⠀⢰⠛⡓⠦⣽⣿⡄⣠⡆⠀⠸⣦⠀⠹⣤⠀⠈⠐⣤
⠻⠲⣿⠟⠀⠀⠐⢲⠾⠃⠀⠱⣄⠈⣧⣴⣄⡏⢻⡂⢀⠴⣴⣄⡀⠈⠣⠀⠉⠢⠐⠒⠇⠰⠛⢢⡶⢟⠋⠀⠿⡄⡴⠃⠙⠲⣬⣷⠇⠀⠘
⠀⠐⠁⠀⠠⠔⠋⠁⠄⣀⠀⠀⠘⢶⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⢠⣆⡀⠀⠀⠀⠠⠀⠙⢧⣼⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠁⠀⠈⠊⠉⠀⠈⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣊⠿⠋⠀⠀⠀⠀⠀⠀⣀⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠐⢋⡽⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⡠⠚⠀⠘⠏⠀⠀⠀⠀⣀⠄⠚⢁⠛⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⠀⡘⣩⣶⡍⠐⠖
⢀⣼⣠⢶⢇⢰⡄⠀⠀⠀⠰⢞⢋⢠⠀⠀⠀⠀⢤⡄⠠⠀⠀⠤⠒⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⣴⣿⣦⢛⠋⣹⠿⣤⢤⣤⠀⡄
⣾⣿⠔⢊⢠⣹⣧⣀⣠⡠⠐⠛⠋⡄⠓⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠀⠀⢰⡆⠀⠀⢸⣉⢗⠀⢈⠿⢋⠙⠻⡏⠀⣀⠹⡎⠁⠺⣿
⣿⡥⢟⠘⣻⡝⢿⣿⣿⡤⠈⠁⠲⠁⠉⠱⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡄⠀⠎⠛⢧⡀⢸⠖⠋⣤⠉⢻⣾⡿⢆⠙⠳⣦⣹⣿
⣿⡤⡎⣳⠻⢄⠻⣿⣏⠠⢆⣠⠜⠢⢶⣀⠳⣠⢀⡀⣀⠀⣄⠀⢀⡼⣷⠀⠀⢀⡀⣀⣄⣴⣦⣤⡴⡫⠗⢰⣦⠖⠚⢄⡓⢠⠔⠺⡏⠙⣆⠻⣤⡈⡁⠀⠘⢿⣿
⢀⡴⠋⣿⢼⡿⠋⢉⣠⠂⠀⣠⣆⡄⣄⠉⣷⣼⣿⣿⣿⣾⣿⣾⡤⢾⡧⣵⣿⣿⣿⣿⢻⣿⣿⣿⣗⡱⠳⢦⣥⠔⠛⣦⠟⠁⠀⢤⡘⢦⠈⣿⣿⡏⠙⢷⣄⣀⣿
⢹⣧⠀⣽⣄⣿⠿⠋⠉⠛⣿⠁⠈⠗⣿⣄⡀⠘⢿⣿⣿⢿⣿⣿⠆⢃⠓⠸⣿⣿⣿⢁⡏⣿⣿⡿⠉⢀⣮⣠⣀⢸⡥⠓⡖⠁⣀⠼⣷⡞⠙⢿⡛⠗⢄⠀⠉⠙⢿
⠉⣀⣿⢿⣿⣡⠄⠚⠁⡷⠛⢀⠀⠏⠙⡈⠙⢦⣭⣿⣾⢱⣼⢗⡳⠛⡀⢰⠘⣿⢛⠇⡔⠚⣿⠤⠂⠀⢁⣀⠉⠻⣳⣞⠉⠉⠁⠀⠈⡟⠢⠤⣡⠀⠀⠳⡀⠀⢸
⣰⠟⢱⣾⡿⠃⣠⣤⠟⠀⢠⠞⠀⠀⠀⣶⡀⠀⠙⢻⣿⡤⠋⠁⠠⡎⢹⠡⠈⠙⣥⠆⢀⡰⠇⡔⠁⣺⠉⢫⠑⠲⢿⣷⣆⡴⣿⠀⠀⢹⢦⡀⠘⢷⡆⢀⠀⠙⣶
⠛⠒⢿⡿⠋⠀⠀⠘⢲⠶⠏⠀⠀⢳⣄⠀⢻⣰⣶⣘⡍⢻⡃⢠⠤⠦⣾⣀⠀⠈⠫⠀⠈⠑⠂⠚⠋⠇⠰⠋⠙⣶⡾⢋⠋⠀⠸⢧⢠⠞⠀⠉⠳⢦⣵⠾⠀⠀⠸
⠀⠀⠊⠀⠀⠤⠴⠋⠁⢀⣀⠀⠀⠀⠸⣦⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⢀⡆⠀⠈⠀⠀⠀⠀⡀⠙⢷⣤⣴⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠈⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠛⠉⠙⠖⠀⠀⠑⠖⠒⠂⠀⠋⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠓⢓⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠈⠀⠠⠀⣶⡶⠀⠀⠀⠠⠀⠀⠀⠀⠐⠖⠓⠶⠀⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣭⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠚⠓⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠑⠛⠙⠚⠓⠓⡁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠀⢠⠖⠀⠉⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠚⠐⠳⠖⠤⠂
⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠌⣠⠖⠈⠈⣧⣽⠋⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡐⠧⠢⠔⠦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡯⠀⠈⠐⠉⠈⠒⠚⠀⠙⢆⢀⡠⢀⢀⠀⢠⡄⠤⠒⢲⣴⣧⣶⡜⢚⣒⠘⢀⠀⡀
⠀⠀⠀⠀⠀⡠⠐⡐⠒⠊⠉⠁⢡⣹⣷⢰⠀⣰⣶⡆⠀⡄⣯⡴⣄⢅⠠⡌⡛⣿⣏⣴⠀⠌⠁⢉⡁⡀⣀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⣌⠄⠈⠉⢅⣐⣋⠏⡁⢹⠀⢀⡇⠀⣠⠤⠼⠂⠂⡁⠀⠈⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⢀⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠋⠉⠁⠉⠐⠄⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠀⠀⠀⠀⠂⠁⢈⢑⠄⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢠⠔⠀⠀⠀⠀⢀⠀⡀⠷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢶⢡⢣⠂⡠⠀⠒⠈⢉⠖⠛⢠
⠀⠀⠀⠀⠀⠀⢀⠠⠃⠀⠀⠀⢀⢀⢀⣨⠙⠷⠎⣨⠁⠀⠀⠀⠬⠉⠀⢑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠
⠀⠀⠀⠀⠀⠊⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠉⠢⡀⠁⠀⠀⠀⠀⠀⠀⡲⠁⠀⠶⣌⠀⠂⠀⠤⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠠⠂⠁⠀⠈⠈⠂⡈⠢⡀⠀⠀⠀⠀⣠⠃⠀⢀⡇⠀⠉⠒⠤⠄⣀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⢀⣤⠔⠋⠀⣆⠀⣈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⢰⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠱⡀⠀⣀⠀⠸⣆⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⣾⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠑⡀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣱⠀⠀⠀⠀⠈⡙⢦⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⣠⠞⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣀⣀⠀⠀⢀⣀⣀⠄⠂⠁
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⢿⡄
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠶⠶⠶⠶⠶⠦⠇
⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⠶⠟⠋⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⡞⢀⡠⢤⡤⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠋⢠⣯⠞⡩⠞⠓⠒⢤⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢈⠀⠀⣰⠁⠀⠉⠀⠀⠀⠀⠀⠠⠤⠬⡭⠦⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣀⣀⡲⣖⡒⠚⠷⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠰⣾⠯⢹⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣉⠀⠀⠀⠀⠀⠐⠯⣄⠀⠈⢣⡖⠒⠦⣄⠀⢀⣁⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡴⠁⠀⠀⠀⠀⠀⠀⠀⠲⡷⠀⠸⡇⠀⠀⢈⣷⡋⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣹⠀⠀⠹⣄⠔⠉⠁⣯⣉⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⢠⣀⠀⠀⠀⠀⠀⣰⢆⡇⠀⠀⠀⠈⠓⠦⣤⠇⠀⡉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠙⢾⠢⠶⠖⠚⢧⠽⢻⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠳⡖⠤⢀⣀⣀⣀⣀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⠀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⣾⠼⠓⣉⠿⠟⠳⡀⠀⠀⠀⠀⣀⠤⠤⢐⡶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣇⠀⢰⠒⢻⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠘⡆⠸⡆⢸⠀⠀⠀⠀⠀⠀⠈⠙⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡀⠀⠀⠀⢹⡀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢀⠜⢡⠋⠁⠀⠀⡇⢁⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⠏⠀⣇⠀⠀⠀⢀⠞⣼⣟⢳⣆⡀⢠⠖⢦⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣀⡰⠃⠀⠀⠙⠒⢲⢩⠿⠀⠓⠒⠶⣏⡼⠋⢀⢘⡆⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠓⠦⣄⡀⠀⡇⡟⣶⠀⠀⠀⢰⡅⡗⣤⠏⠉⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡇⠀⢠⠤⠴⠖⣿⠯⢻⠄⣧⡟⣤⢴⡌⠫⣀⢣⡿⢤⣄⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣫⢶⠀⢃⠠⡜⡂⠀⢸⡇⠀⠈⠓⠚⢱⢳⢸⢸⠤⠋⠁⢹⡾⠈⣇⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⠊⢵⢸⠀⠉⢎⠓⠁⠀⣀⡧⠀⢀⠇⡌⠸⢠⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⢀⠜⢮⣿⡶⢤⣈⠉⠉⠉⠀⠀⠀⠸⠤⠇⡟⠾
⣠⠴⣖⣴⡖⣆⠀⠀⠀⠀⢀⠀⠀⢀⡴⠁⠀⢉⡹⠁⠀⠀⠀⠀⠀⠈⠉⠒⠲⣦⣤⠤⠤⠬⠽⠓⠃
⣸⣊⣀⣋⣈⡷⠀⣾⡉⠋⢁⡾⣀⠸⠤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢜⡝⠁
⠈⠁⠀⠀⠀⠀⠀⠛⠒⠢⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣥⣴⣋⠉⠀⠉⠉⠉⠒⠢⡄

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠦⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠄⠀⠀⠀⠁⠀⠐⠀⠀⢀⠀⠀⠂⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠤⢆⡑⠀⡴⠜⢃⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⢰⠸⣿⠒⠒⣶⠂⣄⣤⡖⡇⠀⡀⣄⠀⠓⢴⡚⢚⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠒⠚⠓⠛⠀⠀⠁⠒⠕⠊⠇⠀⠠⡅⠹⡤⠀⠀⡠⠞⠒⠓
⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠀⠈⠟⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠴⠖⠒⢲⠀⠀⠀⢀⠆⠀⠈⠙⠒⡐⠈⠱⢲⡷⠄⠀⢢⣄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⡀⢀⡀⠀⠠⢜⡁⠀⠀⠁⠀⠀⣀⠀⠀⣗⠀⠀⠁⠀⠀⠀⠓⠛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠶⠀⠈⢄⠀⠄⡀⡀⠀⠀⠀⡰⠁⠉⠀⠀⣀⡠⠴⠒⢉⢄⡂⢼⠤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠀⠀⠀⠀⠉⠒⠢⠅⠀⠀⠀⣇⣄⣠⠶⣻⢶⠦⢖⠋⠁⠀⣤⡁⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠸⡍⠉⠒⠶⣄⠁⠂⣬⡇⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇⠀⠀⠈⠢⠄⠀⠀⠙⠢⡈⠑⢤⣀⠀⠀⠙⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣉⣠⣤⡠⡼⠶⠂⠀⠈⠓⠦⣈⣉⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠠⠔⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣠⠤⡤⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠁⠀⠀⠀⣀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢀⠐⢛⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢚⣊⡤
⠀⠀⠀⣈⣙⣬⣁⠄⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⣤⠖⠂
⠀⠀⠠⡌⣁⠼⣤⠤⠀⣠⠀⠀⠰⠀⢀⠀⠀⠀⠰⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⢀⡤⣵⣶⢦⣞⠳⢚⣃⠀⢀⠆
⠀⡀⣀⣉⣼⣧⣠⠴⠂⠀⡆⠀⠀⠀⠈⢠⠀⠀⢀⠀⢀⠀⠠⠀⠃⠀⠀⠘⠀⢈⣻⣭⢿⣟⣽⣟⣉⡁⠁⠈
⠈⠡⣬⣉⣠⠷⣦⣀⡴⠖⠉⠐⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣞⣷⣻⡶⣾⣧⣽⡆
⠀⣠⣬⣭⣡⣴⣇⣉⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣟⣽⠶⠒⢛⣿⠶⣤⠶
⠀⣄⣀⡩⢀⣠⣧⣉⠑⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠛⠛⠛⠳⠲⠖⠃
⠐⣧⠘⠓⠋⢠⣏⠉⢩⡧
⠀⠀⠈⠿⠲⠃⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⡆⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⣈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠟⠛⢉⠉⠉⢉⠻⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠀⠀⠘⠛⠀⣀⠘⣃⣠⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠰⢂⡀⠙⠛⠡⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡤⣤⣀⣤⣤⠀⣅⣁⣤⣦⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠓⠦⢤⣀⠀⠀⠀⣸⣿⡿⠿⠿⠿⠿⢷⣿⣾⣿⣦⠀⣀⣤⠶⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣐⠉⠀⠀⠀⠀⠀⠀⠀⢉⣻⣿⢻⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡉⠻⠒⠀⠀⠀⠀⠀⠀⠈⠁⢹⣧⢻⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠒⠊⠀⠀⠀⠀⠀⠀⠀⠶⠈⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠄⣀⣠⡀⢀⠀⠀⠀⠀⠛⠀⠈⡟⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠭⠛⠋⠉⠉⠉⠉⠈⠋⠙⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠠⣴⣶⣿⠶⣾⣮⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣒⡈⠠⣺⣿⣿⣿⡶⣬⣄⣂⣀⡀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠶⠞⠉⠐⠋


⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⠢⠀⣀⠀⠀⠀⠀⠀⢱⣄⣦⣀⣄⡈⡄⠀⠀⢰⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢀⠼⠋⠀⠀⠀⠀⠀⠀⠉⠓⠢⠄⠀⠀⠈⠙⣏⢡⡄⠐⣀⠀⠠⠈⠉⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡜⠁⢤⣄⡀⢿⣳⠄⠀⠈⣢⣄
⠀⠀⠀⠀⠀⠀⠀⢈⠌⠀⢉⣭⣋⢀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠲⢴⡶⠿⠓⠦⢌⠙⣿⣏⣥⢤⠀⣒⠀⢈⠁
⠀⠀⠀⠀⣀⡤⠊⠁⠈⠈⢉⡀⣠⠛⠀⠀⣊⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⠙⠀⢀⣠⣼⣉⡁⠀⠀⢹⣥⣤⣑⢧
⢀⣠⡴⠟⢋⣡⢴⣂⡀⢈⣿⣝⠢⠙⢶⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣆⠀⠏⠙⠿⣅⠘⢧⡉⠉⠹⠯⢆⣄
⠀⠀⣤⣶⠏⠁⠀⠈⠙⠀⣬⣤⣴⣶⣌⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⠂⠒⠺⢥⣌⠋⠷⠶⠶⢆⠋⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⣤⣶⣾⠶⠖⠛⠛⠓⠛⠻⠿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢼⣲⣼⣿⣷⣂⡀⠀⣀⣀⣤⣀⡀⠁⠢⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠒⠚⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⠟⠿⢿⣿⣿⣿⣯⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⣀⣀⡀⠈⠊⣛⣿⠶⠶⠘⣛⡒⠉⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⠿⣯⣓⡀
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠛⣻⣿⣿⡿⠟⡿⠉⡀⠀⡀⠚⢢⡷⠀⠀⠀⠐⠀⢰⠳⠘⠀⠈⠀⠀⢠⠍⢉⡿⢻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿
⣿⣿⣿⣯⣹⣿⣿⣿⣿⣿⣿⡿⠟⠉⠑⢤⣗⢐⡧⠊⡄⡄⢀⡿⠃⠀⠀⢀⠀⠹⡀⠀⢶⠃⢀⣴⡍⠀⣀⠀⠀⠈⠻⣿⣿⣿⣷⣾⣿⣾⣿⡿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢀⠀⠀⢹⡎⠀⡀⣀⠀⢸⡗⠀⠀⢠⠋⠀⣠⣧⡤⢸⣦⣿⠟⠋⠀⠀⠀⠀⠀⠀⣬⠻⣿⣿⣿⣿⣿⣿⣧⣾
⣿⣦⣿⣿⣿⣿⣿⣿⣿⣃⠶⢀⡄⢈⡖⠀⠈⠃⢀⡟⠁⠀⢨⣿⠀⡆⠉⠁⠀⠀⣹⠁⢠⡏⠁⠀⠀⠀⠀⣰⠃⠀⢠⠇⠐⣙⣿⣯⣽⣿⣿⣿⣿
⣿⣿⣿⣿⡿⢷⣿⣿⠃⠀⠀⡤⠽⢤⣤⡀⠀⢈⡀⠀⠀⢀⣾⠟⣠⠁⠀⠀⠀⠀⡏⣰⡿⠀⠀⠀⠃⠀⡈⠁⠀⠀⣼⠚⠉⠁⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣦⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠃⢼⡧⠀⠉⢁⣿⣶⡟⠀⠀⠀⠀⢀⡿⣿⡇⠠⠀⠀⠀⣸⠃⠂⣠⠞⣁⠀⠀⡴⠀⣿⣿⣿⡟⣼⣿
⣷⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠘⣟⡻⢓⠀⠀⠀⠀⣼⣿⡟⣧⠂⠀⠀⢠⣿⣤⠾⠷⠞⠿⠶⠸⣷⡛⢸⣿⣿⣿⡿⣿
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢒⡢⡄⠀⠀⠀⢻⣄⠀⢿⣿⣿⣇⠀⠀⣼⣭⣿⠁⣻⣀⣀⡴⠟⠋⠁⠀⠀⠀⢀⠰⠀⠈⠣⣼⣿⣿⣿⣿⣿
⣿⣯⣽⣿⣿⣿⠀⢀⢀⠐⣄⠀⠀⠀⣀⣨⠶⠶⠚⠛⠛⠳⣼⣿⣿⣽⣦⣼⣿⣿⡁⢀⡿⣿⣷⡆⢠⣖⠀⠀⠀⢠⠏⠠⠄⠀⣀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣛⣿⣿⣆⡀⢿⣀⡼⠶⢛⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⣿⣿⣟⢙⣿⣿⠋⠀⠀⠀⠀⠀⠀⢶⠲⠾⠤⠀⣀⠀⠈⢩⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡏⠁⠀⠁⣀⠴⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢀⠀⠀⠀⠀⣸⣿⣻⣿⣿⣿
⣿⣿⣿⣿⣟⢻⣧⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠉⠓⠂⢠⣿⣿⣿⡿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⣿⣯⣿⣿⣿⢿⣻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣽⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢻⣿⣿⣿⣿
⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⣿⣿⣿⣿⣻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣿⣿⣿⣿⡿⣶⣤⣄⣀⢠⣾⣿⣿⣿⣿⣻⣿⣯⣿⣿⣷⡀⢀⣀⣠⣴⣺⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠛⠉⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠋⠀⠸⣿⠆⠀⠀⢾⡿⠀⠈⢻⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣀⣤⣤⡴⣄⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⡇⠀⠈⠉⠛⠛⠷⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⡟⠀⠀⠀⠀⢧⠀⠀⠐⠦⠤⣤⡤⠤⠖⠂⠀⢠⠇⠀⠀⠀⠘⡇⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠿⣧⡀⠀⠀⢰⡾⡷⢶⣦⣤⣤⣤⣤⣤⣶⣶⠶⠿⣶⠀⠀⠀⣠⡿⠷⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢸⣯⣭⣭⣭⣇⣀⣀⣀⣀⣮⣭⣭⣭⣿⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠋⣿⣰⣽⡇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣿⢹⣼⡇⠀⢠⡶⣦⠀⠀⠀⠀⠀⠈⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿⣾⣿⡇⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣻⣻⡇⠀⢰⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠉⠉⠀⠀⠀⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠸⣧⡿⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠦⣄⣀⠀⠀⠀⠀⠀⢀⣀⡤⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠛⠷⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠈⠹⣧
⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⣠⡶⠶⣶⣾⠟⠛⣷⣄⣠⣤⣤⣴⣶⣶⡶⢶⣶⢶⣶⣤⣤⣀⣠⣾⠛⠻⣶⣶⠶⣶⣄⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⢀⣸⡏⠀⠀⣈⣥⡶⠾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢶⣤⣀⠀⠀⢻⣆⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣤⡾⠛⠉⣁⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣈⠉⠛⢷⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣅⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⢀⣀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢷⡄⠀⠘⠷⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣀⡀
⠀⠀⢻⡆⠀⢰⣿⠀⠀⠀⠀⠀⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⢠⡄⠀⠀⣄⠀⠀⣄⠈⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⣄⠀⠀⠀⠀⠀⣿⡆⠀⢰⡟⠁
⠀⣠⣾⠇⠀⢸⣯⠀⠀⠀⠀⣰⡏⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠹⣆⠀⠀⠀⠀⣿⡅⠀⠸⣧⣄
⢸⣿⡁⠀⠀⢸⣏⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⡇⠀⠀⡿⠀⠀⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣿⡁⠀⠀⢀⣿
⠀⠉⣿⡆⠀⠘⣿⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢸⣇⠀⠀⣿⠀⢀⡟⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⠃⠀⢰⣿⠁
⢀⣴⠟⠀⠀⣠⣿⠷⢶⣦⣿⠀⠀⠀⠀⠀⠀⠐⠓⠛⠋⠁⠀⠀⠘⣿⠀⠀⣟⠀⢸⡇⠀⠀⠈⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⣿⣤⡶⠾⣿⣄⠀⠀⠻⣦
⢿⣇⠀⠀⠀⣿⠁⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠋⠀⠀⠁⠀⠈⠃⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠈⣿⠀⠀⠀⣸
⠀⢹⡇⠀⠀⣿⡀⠀⠀⠘⣿⠀⠀⢀⣤⣴⣤⣤⠀⣀⣀⣙⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⣁⣀⠀⣤⣴⣦⣤⡀⠀⠀⣾⠃⠀⠀⢠⣿⠀⠀⢸⡏
⠀⣾⠃⠀⠀⠸⣷⡀⠀⠀⢹⡆⣴⠏⠁⠀⣾⠋⠀⠉⠋⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠛⠉⠀⠘⣷⠀⠈⠹⣆⢸⣿⠀⠀⢀⣾⠃⠀⠀⠘⣷
⠀⠻⣶⣄⡀⠀⠘⠻⣦⣤⣼⡇⠹⢧⣄⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⡼⠏⢸⣯⣤⣴⠿⠁⠀⢀⣠⣴⠟
⠀⠀⠀⠉⢿⡄⠀⠀⠀⠈⢹⣧⠀⠀⠉⢁⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡈⠉⠀⠀⣸⡏⠉⠀⠀⠀⢠⣿⠉
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠘⣿⡀⠀⢰⡟⢹⡇⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢸⡏⢻⡆⠀⢀⣿⠃⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣦⣈⠛⠛⠁⠀⠘⠋⠉⠻⠶⠛⠛⠷⣶⡶⠾⠛⠛⠷⠟⠉⠙⠃⠀⠈⠛⠛⣁⣴⣿⡁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠈⠙⠻⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠋⠁⠈⣿⡆
⠀⠀⠀⠀⠘⠷⣦⡄⠀⠀⢸⣿⠀⠀⠀⢠⣾⡿⢿⣿⠟⠿⣿⠶⢶⠶⣶⣤⣤⣴⣶⠶⠶⠾⣿⠟⠛⠉⠿⣷⡄⠀⠀⠀⣿⡁⠀⠀⢠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣼⡇⠀⠀⢰⣿⡛⠛⠀⠀⢀⣤⣿⣶⣶⣶⣾⣟⣉⠀⠀⠀⠀⢰⡿⠀⠀⠀⢰⣿⠁⠀⠀⠀⢸⡇⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠘⢿⣆⠀⢸⣟⠃⠀⣠⣶⠟⠁⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣦⣾⣧⣀⠀⠀⣴⡿⠃⠀⠀⣰⡿⠁⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⢸⣿⠀⠈⣿⣃⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠷⠿⠃⠀⠀⠀⣿⠀⠀⠀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠈⢿⡄⠀⠉⠛⢻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⢻⣆⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⣦⣄⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠉⠻⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⣴⠶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠉⠛⠿⢶⣦⣤⣤⣶⡶⠟⠋⠙⠛⠷⠶⣦⣤⣤⣄⣀⣀⣀⣠⡿⠋⠀⠀⢰⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡶⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⡟⠄
⠀⠀⢀⣤⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⢴⠋⠀⠀⠀⠀⠠⠠⠎⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⣼⠀⢴⢎⡠⢤⡀
⢠⣆⣼⡗⣠⠀⠀⠀⠴⠂⠠⠀⠀⠈⢀⡀⠀⠀⠠⠠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⢀⣼⡆⠛⠛⠿⣤⣠⡄⣀
⣿⣿⡟⣷⣿⣠⣠⠔⢐⠁⡄⣀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠐⣆⠀⠀⣯⠃⠀⡜⢛⠉⣻⡁⣀⠹⣀⠠⣿⣀
⣸⡿⢀⢻⣿⣿⣿⢞⠁⠈⡁⠀⠡⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠕⢀⢘⣤⠓⣄⡔⠀⠶⢌⣯⠈⠣⠈⢳⣮⣿
⢨⡿⢛⢰⣿⠟⠃⠌⠁⡀⠀⠀⢂⢘⣿⣷⣿⣶⣼⣤⣄⣀⡀⣠⣤⣤⣷⣤⣼⠴⠅⣘⣋⠀⣀⡥⠋⣁⠀⠈⢿⠀⠀⠠⡀⠈⣿
⢠⡀⠊⢈⣥⠖⠋⠀⠚⠀⠀⣄⡀⠑⢿⣿⣟⣹⣽⣽⣿⡿⠟⠟⠿⠿⣿⣿⠟⠀⡀⣈⡑⡆⢈⡶⢋⡠⢦⠜⢦⡂⠀⠀⠉⠙⢿
⠘⠋⠀⢯⡥⠀⠁⠚⠁⢤⡜⢳⡀⠱⠦⢷⣀⢼⠷⠛⣡⠤⣄⠀⠀⠀⠈⠻⠀⠀⣀⢉⠻⣄⣾⢉⢁⠀⠀⡁⠂⢄⡀⠀⠢⢀⠠
⠀⠀⢠⡎⠀⠂⣏⡀⠔⠀⠈⠤⠷⠀⣀⢀⠀⠈⠰⣭⡼⠏⢿⣽⠶⠾⠟⠁⣨⠈⢁⣁⣉⣻⣿⡷⢛⣧⣀⣿⣷⢄⣻⣇⣧⠈⢿
⠀⠠⠋⠀⣀⡤⠶⣅⠀⠀⠈⢳⡠⠲⡍⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠀⢩⠉⠛⠙⢦⠠⠉⠻⣏⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠠⠀⠊⠁⠐⠄⠀⠠⠳⠄⠸⠿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠀⢸⠇⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠿⠀⠀⠀⠀⡆⢸⠇⠀⠀⢠⡀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠘⠀⠀⠀⢰⠃⠉⠀⡼⠀⠀⣇⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠁⠀⢠⡏⠀⠙⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇⣶⠀⠀⢰⡀⠈⠁⢠⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢀⡟⠀⠀⠈⣷⠀⠀⢸⡄⠀⠇⠀⠀⣀⢠⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠸⠇⢄⠀⡄⠀⠀⠀⠈⠁⠀⠀⢠⠀⠛⠈⠀⣧⣰⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⠀⠀⢀⠈⠀⠁⡀⠀⠀⠚⠀⠀⣆⠛⠀⠀⡇⢸⢿⠋⠀⡀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⠋⢸⠃⠀⠀⠃⠀⢰⣆⠑⡆⣼⠘⠀⠀⠙⠀⠀⠄⢠⠀⠈⠀⠠⡇⠸⠁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠈⣷⢈⡁⡀⠀⠀⢀⠘⠛⠀⠁⠻⠀⠀⢠⡀⠀⠀⣄⠀⠰⠀⠀⠀⠃⡿⠀⣆⠸⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠃⠀⠐⡋⢻⠃⣷⠀⠀⠚⡀⠀⠀⠀⠀⠀⢰⡌⡅⠏⢠⠟⡄⠀⠄⡀⠀⠀⣀⠠⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢸⡸⠇⠑⠴⡄⢡⣲⠐⣄⡇⠀⢀⠀⡄⢈⠀⡀⠿⠀⡾⠠⣠⡇⠸⠇⢀⡜⠃⠀⠀⢰⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢸⠂⠓⡆⣠⡄⠀⢧⣴⣾⡿⣶⡔⡌⠇⠀⠘⢠⡇⠀⠀⠀⠀⠋⡠⡖⣠⣸⣧⡘⠁⠀⠈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡦⠘⣄⠀⠈⠛⠈⠃⠙⢟⠃⣿⠿⣦⠀⠶⠀⠈⠓⣦⠀⠀⠐⠈⣷⠏⣻⢁⣼⡿⠋⢀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣤⣌⠀⡮⡄⢈⣗⣦⡼⢆⢠⡄⠻⠃⠀⠀⠊⡰⣿⠄⠈⠀⠰⢷⠠⠉⠈⠈⣴⢒⡊⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢈⣛⣠⠤⣿⣄⡧⢴⣾⣿⡹⠇⢸⣧⢱⣄⡶⠀⠆⠁⠁⠁⢠⣄⠀⢁⣈⡀⢻⣧⢀⣿⡌⠃⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠞⠁⠀⠀⠀⢹⣿⣆⡈⠯⠀⣀⣀⠀⠈⠛⢡⣿⣧⠤⠓⠠⢾⣿⣦⡈⠿⠃⢘⠡⢜⣿⡛⠁⠀⢉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⣀⣀⣠⣾⡾⠁⠀⠀⠀⣨⡿⠂⠒⠒⢿⠏⠀⠀⠀⠀⠀⠙⢿⡟⠈⠁⠙⣄⠀⠈⠙⢿⣯⡭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⡽⠯⠽⠿⣵⣠⢤⣄⣾⡿⣇⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⣠⣿⣷⣤⣠⢔⣿⡆⠀⣠⡴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠐⠾⠶⠛⠿⠧⢯⡥⠿⣿⣓⠠⠄⠂⠾⠛⣋⠿⠷⠿⠿⠳⢿⠶⠾⠾⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠉⠉⠉⠉⠓⠲⠤⣄⠀⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣥⠤⠤⠤⣄⣀⡀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⣠⡀⣾⡍⢓⡢⢄⡀⠀⠀⠉⠲⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠯⠟⠀⠰⣶⡽⠓⠛⠦⣾⠃⢀⡉⠑⠦⡀⢰⢳⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢶⠄⢴⣤⠜⣀⠀⠀⠀⢈⠲⡟⠁⡀⠀⣬⡏⠸⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢠⣤⡇⠀⠉⠀⠀⠈⠛⠀⢹⠛⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠼⠻⡀⠉⠙⠉⠓⠋⠒⠂⠘⣄⣠⠿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠦⠤⢤⣤⣤⣴⡾⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡁⠉⠛⡿⠛⠻⠥⠀⢸⢎⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣿⡇⠀⡀⠀⠀⠀⢠⠋⠀⢀⡔⡻⠶⢾⣛⣒⠶⠖⠒⣿⠈⢧⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢣⡇⢰⢀⣷⢀⣧⠀⠀⠀⡀⢸⣷⠃⠀⢠⠎⠈⠙⠦⢤⣼⠀⠀⠳⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢦⣼⡌⢡⡄⠋⠘⠋⡜⠀⠀⠐⠁⢈⡼⠧⠴⢼⣄⠀⢀⣶⣖⢉⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⠎⠙⡄⠀⣼⠀⠀⠀⠀⠀⠈⣉⠿⠷⢿⣭⠭⣿⡽⠛⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣞⠀⠀⢸⣀⡼⠀⠀⠀⠀⠀⠀⠙⠚⠉⠻⠯⠿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⡽⠛⡻⡙⢟⢿⣲⣄
⠀⠀⠀⠀⠀⠀⠀⢏⢳⡎⢰⠒⣧⢳⣋⡷⠹⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠿⠛⠛⠻⠿⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠶⠦⢤⣼⡿⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠟⠛⠛⠛⠛⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⣠⣴⠞⠋⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿
⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣿
⠀⠀⣿⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣾⡟
⠀⠀⠹⣷⣁⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠄⠀⢈⣴⡟
⠀⠀⠀⠙⢿⣦⣁⡂⠀⠀⡀⠄⠀⢀⣀⣠⡾⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣉⡀⠂⠥⣑⢂⣡⣶⠿⠋
⠀⠀⠀⠀⠀⠈⠛⠻⠿⢶⣶⣶⠶⠿⠛⠯⣅⣂⢐⠠⠀⠄⠀⠀⠀⡀⡀⠰⠐⢜⣿⣯⣐⡀⡀⠀⠀⠀⠀⠀⢀⢀⢠⣼⠟⠁⠉⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠾⠶⣷⣷⣷⠶⠶⠟⠛⠉⠀⠉⠛⠷⢾⣦⣵⣤⣬⣬⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠂⢀⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠄⠀⠀⢢⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠀⡀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠄⢠⠠⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠐⢀⢀⠄⠈
⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⡀⡀⠂⠀⣀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢐⠀⠂⠁⢁⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢀⠐⠁⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠃⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠌⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⡀⠀⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠌⠀⢁⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡠⠁⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠅⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠠⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢠⡀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠀⢀⣀⡀⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠘⣼⡇⠠⠀⡀⢸⣿⢆⣣⠀⠀⠀⡄⡧⠋⢀⠆⠀⠠⠀⠀⠀⣄⡇⢸⣧⠀⢸⢡⠇⢀⡇⢸⠀⢾⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡄⠡⡄⣿⠀⠂⠀⢸⣿⠈⠉⢀⡴⠔⠃⡧⠐⠁⠀⢀⠆⠀⠀⠀⠈⡆⢸⣏⣤⡏⠠⠓⠀⠇⢸⡇⠸⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡿⢸⡞⠁⠀⠀⣿⣆⠀⠆⠀⠀⢠⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠐⢼⡟⠻⠰⠁⠠⠀⠐⢾⡇⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⡄⠀⢷⠀⠀⠀⣿⡿⠐⠀⡀⠀⠸⠴⠶⠚⡋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⠀⣂⠀⠀⢘⡿⠂⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢾⡄⠀⠀⣿⡟⠤⠉⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⣿⠻⠀⠀⠀⠀⠐⡈⢋⣁⣤⠅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⢰⣿⣇⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣿⠀⡆⢀⢀⠀⠀⠁⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⣿⣯⣄⢀⣀⣥⠤⠤⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⢹⠂⣇⣎⣁⡀⠀⠀⡀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣭⡿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣼⣄⡟⠻⠿⣷⡄⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣿⠃⠀⠀⠀⠀⠀⣀⣴⡾⠛⢿⣿⣿⣿⣿⣿⣿⡜⢱⣄⡀⠈⠀⠠⣿⠘⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀⠀⢸⣿⣿⠂⠀⠒⠀⣀⣴⠾⠛⠁⠀⠀⠀⠈⠻⢿⣿⣿⣿⡏⢸⣿⣿⣦⡀⠀⣿⠸⢃⡀⣴⣄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡏⠹⣷⣾⣿⣿⢀⣠⡴⠟⢋⠥⠘⠀⠀⡀⠄⠀⢸⢠⢀⠉⠻⣿⡷⠇⣿⣿⣿⣿⣷⣿⣸⠀⠹⣧⣿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡃⠐⠀⢸⣿⣿⣼⠋⠀⠄⢼⣤⢤⣤⣴⣃⡴⠇⠈⠘⢽⡌⣆⣸⡷⢀⣿⣿⣿⣿⣿⣿⣿⣦⣄⠡⠜⢉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠆⠈⣀⡴⠞⠉⣿⡇⢰⡇⢀⣧⡞⠁⠠⠤⡀⡄⠀⠀⠀⠀⠸⠀⣇⣿⣿⢷⠉⠻⣿⣿⣿⣿⣏⣋⣥⣶⡟⡋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⠶⠛⠁⠀⠀⠇⣿⢃⡀⣥⡾⠁⠑⠎⠀⠀⡁⡇⠀⠀⠀⠀⠀⠀⢹⣿⣗⡈⠀⢀⣼⣿⣧⣼⡏⣻⣵⣾⣷⢹⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣴⠾⢛⠁⠀⠀⢲⡀⡆⠀⣿⣿⠙⡍⠁⠀⠀⠀⠀⠀⠃⠇⠐⠀⠀⠀⠀⠀⠘⣿⣟⠃⣰⣿⣿⣿⡿⣿⠃⢿⣿⡏⢻⣈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⠀⠐⠂⠺⠛⠉⢦⣿⣟⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢻⡏⢰⣿⣿⣿⣿⡇⣿⡄⢺⡻⡇⠈⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⢸⠀⠀⠀⣿⣿⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⣼⣿⢸⣿⣿⣿⣿⡇⣿⡆⣿⡇⣧⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠈⠀⠀⠀⠀⠀⣻⡏⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⢀⣾⣿⡇⡄⣿⡿⣿⣿⡇⣿⡇⢿⡇⡿⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⡀⢤⣄⣸⠃⠀⠀⢰⠀⠀⠀⣿⣧⡔⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⣰⣿⣿⣿⣟⣥⢛⣿⣿⣿⣷⣿⠂⠸⡃⣷⡇⣿⠃
⠀⠀⣰⣦⣔⣃⣹⣷⣾⣿⣿⠀⠀⠀⠈⠀⡄⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣾⣿⣿⣿⣿⡏⠅⣸⣿⣿⣿⢿⣿⠀⡆⠁⣿⠇⡿⡇
⠀⢸⣿⣿⣿⣿⢿⣿⣻⣿⣿⠀⠀⠀⠀⠀⠃⠀⣿⣿⢿⡅⠀⠀⠀⠀⠀⠀⣄⠀⠀⣴⣿⣿⣿⣿⣿⣿⣏⠀⣿⣿⣿⣿⢸⣿⠚⣿⠀⣿⠀⣿⡇
⠀⢸⣿⣿⣿⣓⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃⠏⠀⠀⠀⠀⠀⠀⠃⠠⠾⠿⣿⢿⠿⠿⠿⣿⣯⠴⢻⠿⣿⣿⠑⣿⣆⡿⡿⣯⣿⣿⠁
⠀⠀⠉⠉⠉⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠧⢰⠀⠐⢠⢿⠿⠇⠁⠀⡃⠠⢀⡿⠿⠁⡀⠀⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⢠⠀⠇⠀⠘⠇⠀⠀⠃⠀⢸⣽
⠀⠠⠠⢠⣄⣄⣤⠤⠄⠤⠤⠤⠴⢤⠴⠶⠖⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠰⠀⠀⠐⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡶⠀⠀⠀⠀⠀⠉⡙⠐⠋⠀⠀⠀⠀⠠⠈⢄⡜⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠄⠐⢛⠘⠟⣾⠀⠀⠀⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠘⠻⡉⠀⠀⢀⠔⡀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⢰⠋⢣⠀⠠⠎⠔⠂⡠⠆⠁⠄⠀⠀⠀⠀⠀⣁⣩⡁
⠀⠀⠀⠀⣠⣀⣠⠀⣤⠤⣤⠄⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠛⠚⠒⢥⣤⣠⣀⡀
⠀⠀⠀⠀⡮⠤⢬⡦⠶⠢⠤⠤⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠉⢣⣤⠠⣤⡀
⠀⠀⠀⠠⠴⠤⠴⡖⠰⠴⠄⠲⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⣈⣉⢁⡉⠑⢢⡐⠀⠈
⠀⠀⠀⠐⠒⠒⠶⠦⣄⢤⡠⢴⡔⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⡀⠁⠉⢐⡇
⠀⠀⠀⠈⢐⡇⠂⠂⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠈⠃⠀⠀⢐⠂
⠀⠀⠀⠀⢈⡁⠀⠀⠠⠜⠆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠋⠉⠉⠁⠀⠀⠀⠀⠈⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠐⡀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠠⠊⠱⡁⠂⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠤⠶⠦⠚⠒⠲⠶⠦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠢⠀⠀⣁⡤⠤⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠴⠚⠉⠀⠀⠀⠀⣀⣀⡤⠤⠤⠖⠒⠒⠚⠉⠉⠉⠻⡒⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⢀⣠⠤⠒⠊⠩⠥⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢱⡀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣤⠚⠉⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⢰⣖⠄⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠁⠸⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡓⠦⠤⣀⣀⣀⡀⣀⣀⣀⡠⠤⠖⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⠳⢤⣀⣀⡉⠉⡁⢀⣀⣀⠤⠲⡅⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡼⢩⣏⡏⠉⠉⠁⠀⠀⠀⠘⡄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢠⠀⡼⠀⡾⠰⡇⠠⠈⠀⠀⠀⠀⠀⢹⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⢠⡇⠀⣧⣀⣱⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠤⠋⢸⣠⣀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠉⠈⠙⠢⡀⠀⠀⢀⣀⠎⠁⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣄⣀⣀⡤⠏⠉⠉⠉⠙⠦⣤⣀⣠⠤⠟

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠟⣋⣥⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣤⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⡟⣿⡿⣹⡇
⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣇⣟⣼⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠋⣾⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣧⠸⣿⣿⣿⣿⣿⣿⣿⡇⢸⠏⣼⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⠀⣿⣿⣿⣿⣿⣿⣿⡇⠛⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿⣿⣿⣿⣿⣿⠃⢠⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⢛⣾⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⡇⠘⣿⡿⢋⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣇⠀⠟⣴⣿⣿⣦⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣅⣤⣤⣤⣤⠀⣾⣿⣿⣿⣿⣷⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿⣿⣿⡇⠀⣿⠿⠿⠟⠛⠛⠁
⠀⠀⠀⠉⠉⠉⠉⠛⠛⠿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠈⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠂⠉⠁

⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠛⠛⠳⣦⡀
⠀⠀⠀⠀⢀⡾⠃⠀⠀⢀⣀⣀⣸⣷⣤⡀⠀⣀⣀⣀⣀⣠⣀⣀⣀⠀⠀⣠⣾⣏⣀⣄⡀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢸⡇⠀⠀⣠⡟⠁⢉⣩⡴⠾⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⢦⣌⡉⠈⢻⣆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠻⣶⠟⢉⣥⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣬⡉⠛⣶⠟⠁
⠀⠀⠀⠀⠀⠀⢹⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡟
⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⢰⠇⠀⠀⠘⣇⠀⠀⣷⠀⠀⠀⠀⠀⢀⡇⠀⠀⠸⡆⠀⠀⠀⠀⠀⢻⡄
⠀⢶⡄⠀⣼⠇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠘⠷⠀⠈⠁⠀⠀⠀⠠⠞⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧⠀⢀⡶
⠀⣠⡟⠀⣿⠀⠀⠀⢠⡏⠀⠘⠂⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⣾⠀⠀⠀⠀⠀⠀⠇⠀⠹⡄⠀⠀⠀⣿⠀⢻⣄⡀
⠸⣏⠀⠀⣿⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⡀⠀⡇⠀⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⣸⠗
⠀⣸⠇⠀⢹⣇⣀⣰⡇⠀⠀⠀⠀⣀⣠⡴⠋⠀⢸⡇⠀⡇⠀⣿⠀⠙⠦⣤⣀⠀⠀⠀⠀⢸⡇⣀⣰⣏⠀⠸⣏
⣾⡋⠀⢰⡟⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠘⠃⠀⠃⠀⠛⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⢹⡆⠀⢈⣷
⠈⣿⠀⠸⣇⠀⠀⢻⡆⢀⣤⠶⢦⡄⢠⣈⣻⣦⠀⠀⠀⠀⠀⠀⣴⣟⣁⣄⢠⡴⠶⣤⡀⢠⡟⠀⠀⣸⠇⠀⣾⠁
⠰⣏⡀⠀⠹⣦⡀⠸⡇⢾⡁⢀⡟⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿⢸⡇⢀⣴⠏⠀⠀⣹⡇
⠀⠉⠛⣷⠀⠈⠙⠻⣇⠈⠙⠛⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡛⠛⠁⢸⡟⠋⠁⠀⣴⠟⠉
⠀⠀⠀⡿⠀⠀⠀⠀⢿⡀⠀⣞⢹⡇⠀⣠⣄⠀⣀⣀⠀⠀⢀⣀⠀⣠⣄⡀⠀⣏⣻⠀⢀⡿⠀⠀⠀⠀⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢶⣭⣉⠀⠈⠁⠈⠉⠉⠉⠛⠛⠉⠉⠙⠁⠈⠁⠀⢉⣭⡴⠿⣦
⠀⠀⠀⠰⣤⡄⠀⠀⣿⠀⠀⠈⣽⢿⣷⡶⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⡴⠶⢻⣿⡁⠀⠀⣾⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⣿⠀⢰⡏⠀⠀⣾⡛⠃⠀⣠⡼⠷⠶⢶⣿⣉⡀⠀⠀⣾⠁⠀⠀⢿⡁⠀⠀⣹⠆⠀⢿⡁
⠀⠀⠀⠀⢸⣇⠀⠀⠙⡇⠈⣯⡅⢠⡞⠉⠀⠀⠀⠀⠀⠉⠙⠛⠳⠿⣦⣤⣰⡟⠁⠀⢰⡋⠀⠀⣸⡗
⠀⠀⠀⠀⠀⠉⣿⠄⠘⣧⠀⠙⠶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⠁⠀⣾⠉
⠀⠀⠀⠀⠀⢰⣏⠀⠀⠘⣷⡀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠈⠙⠛⢶⡀⠈⠛⢦⣄⣀⠀⠀⢉⣻⣶⢦⣤⣀⡀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣴⠛⠛⠁