Amorousness Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡝⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠘⠛⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⡀⠀⠀⠈⠲⣄⠀⠀⠀⣴⣾⣿⠛⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠑⠤⣀⡏⠹⠏⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠈⠑⢆⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠔⠒⠣⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠤⠠⠔⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠴⠃⠀⢀⡤⠒⡉⠉⠀⠈⠉⠑⢦⠤⣠⣀⠈⡗⠒⠒⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⣠⢾⡁⠀⠸⠖⠚⠛⠢⣦⡀⠀⠀⠀⠙⠧⡀⠀⠀⠀⠀⠀⠀⠀⢱⡴⡖
⠀⠀⢰⠋⠀⠌⠁⠈⡇⠀⠀⡼⠁⠀⠃⠀⠀⠀⠀⡀⠰⠈⠃⠂⠰⡩⠛⠛⠓⢤⠀⠀⠀⠀⠀⠀⠘⡎
⠀⠀⠈⢦⡀⠀⢀⠞⠀⠀⠀⡇⠀⠙⣶⠤⠤⠴⠊⠀⠀⠀⣴⡁⠈⣇⠀⠀⠀⢨⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠲⠁⠀⠀⠀⠘⠷⡀⠀⢯⠉⠉⢣⠀⠀⠀⢸⢻⡿⠋⠀⢱⢀⣧⠎⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡑⠤⠤⠇⠀⣠⠔⠋⠀⠀⠀⠸⠋⢘
⠀⠀⠀⠀⠀⠀⣠⢒⡬⣇⢦⡀⠀⠀⠀⢀⡹⣆⢠⣏⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠰⡅⠣⢽⠙⠏⡹⠀⠀⡞⠉⠻⣀⣈⠽⣆⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠆⢀⠇⡠⢛⡤⣔⠀⠀⠀⠀⡀⠀⢈⡆⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡟⢉⠾⣀⣅⠀⠀⠉⠂⠤⠊⠀⣠⢾⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠦⠄⠩⠖⠢⣀⠀⢀⠞⠁⠘⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⠉⠁⢀⡀⠤⢎⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢀⣀⣀⣀⣀⡤⠒⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠖⢁⣜⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⣄⠀⠀⠀⢠⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢺⡙⡤⠖⠹⣞⠝⢲⠀⠀⠀⠀⢠⡏⡝⠀⠀⠀⠀⠈⠀⠀⢀⣬⠘⠳⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠓⠂⠒⠫⣀⠞⠀⣀⠤⠐⣋⠤⡇⠀⢱⣄⡀⠀⠐⡖⠉⠈⠙⠲⠤⣄⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⣀⠤⠃⠀⠀⠁⠈⠉⢀⠞⠀⠀⡼⠀⠉⠓⠒⠸⢃⣄⡀⠀⠀⠀⠉⠒⠤⠍⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⣁⠠⠤⢄⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠙⠂⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣄⡀⢀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⠿⠛⣋⣭⣿⣭⣿⣿⣿⣏⣛⠈⠉⢻⣿⣿⣿⣿⠇⠠⣭⢥⣽⣿⡿⠿⠿⠟⠉⢠⠀⢼⣿⣿⣿⣿⣿⣷⣶⣤
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣴⣾⣿⣿⣿⣿⣯⣭⡿⣿⠅⠀⠀⣿⣽⣿⣿⣾⣦⣘⠛⢀⣻⣿⣶⣦⡀⢀⣀⢋⡦⣸⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢁⣠⣶⣿⣿⣿⣿⣿⣿⣾⣶⣯⣽⡇⠀⠀⢀⣿⣿⣾⠏⢹⣿⣿⣷⣦⣿⡏⠈⠙⣷⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⡈⠏⠉⠁⠁⠀⠀⣾⣿⣿⣷⣤⠘⣿⣿⣿⣿⣿⣷⣴⣠⠈⠿⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠰⢾⣿⣿⣿⣿⣿⣿⣿⣟⠿⠛⠁⠀⠀⠀⠀⠀⠀⢠⣹⣿⣿⣀⣉⣼⣿⣿⣿⣿⣿⡿⠛⢻⡆⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⡿⢷⡀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣶⣆⣶⣶⣤⣄⠈⠁⠈⠙⠛⠛⠟⠛⠛⠉⢁⣾⡅⠀⠚⠀⠀⢀⢀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⣩⣿⣿⡟⠀⠈⢷⣄⠀⣨⣭⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠹⡆⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠠⣿⣿⠟⠀⢀⣠⣼⣿⣿⣽⣦⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⣃⣠⣤⡶⠂⣀⡤⠀⠀⠀⢿⣷⣤⣶⠆⢠⣤⣄⣀⣿⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⠏⢠⣴⣿⣿⡻⣽⣿⣿⣇⡁⠭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣭⡵⠖⠋⠁⠀⠀⠀⠀⢈⣿⠟⠁⠀⠀⠉⠛⠿⢿⣄⢳⠹⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣰⣿⣿⣿⣿⣯⣻⣫⣯⣽⣿⣷⣦⣍⡛⠿⠿⢿⣿⣿⣍⣙⣛⣛⣛⣛⣉⣀⣀⣀⣀⡠⠀⠒⠀⠀⣾⣧⣀⣀⡀⠀⠀⠀⠀⠘⣿⡆⠀⢻⣿⣟⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠙⠻⠿⠿⠿⠿⠿⠻⠛⠛⠉⠁⠀⣀⣤⣼⣿⠟⠉⢭⠅⠀⠀⠀⠀⠀⠹⣿⣾⣼⣿⣟⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⣶⣶⣾⣿⣿⣿⣿⣧⠘⠶⡤⠄⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣠⣴⡶⠊⠀⠀⢸⣿⠿⢻⣿⣿⣿⡇
⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⢿⣿⣿⣿⣿⣿⣿⠿⠿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢀⣿⣿⣿⡿⠛⠿⢿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣧⠐⢧⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣿⣿⣿⠏⠀⠀⠀⢀⣾⣿⡇⢸⣇⠀⠶⢿⡇
⠀⠀⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠉⣿⣿⣿⣿⣿⣿⣷⡌⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣾⣿⣿⡿⠀⠀⠀⣠⣿⢋⣬⣽⣿⣭⣴⣿⣿⡇
⠀⠀⣿⣶⣴⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣷⡄⠉⢻⣿⣿⣿⢿⣭⣾⣿⣿⣿⣿⣿⡟⢻⣿⣿⠁⢠⣴⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣷⡾⠛⢿⣿⣿⡿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠚⣿⠉⠐⢛⣻⣿⣿⣿⡿⠷⠻⣿⣾⣿⣿⠄⠈⣿⣿⢿⡿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⣀⣀⣼⣿⣿⣿⣿⣯⣭⢌⣹⣿⣧⡄⢀⣿⣿⣿⣿⣿⠿⠟⣿⣿⠃⢠⢿⣾⣿⣿⣿⣿⣿⣿⣷⣶⠟⣿⣇⢹⣿⠀⠀⣾⣿⣶⣶⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠼⠟⠞⠛⠛⢛⣛⣛⣳⡀⢻⣿⣿⡐⣾⢩⣤⣶⣿⣷⠿⢾⣿⣿⣷⡏⠀⣽⣿⣿⣿⣟⣉⣿⣿⣿⠂⢻⣿⣾⣿⡇⢠⣿⡟⣈⣙⣿⣿⣿⣿⣿⠿⠛⣿⠁
⠀⠀⢀⠈⣿⣿⣿⣿⣿⣿⡿⠇⠀⢹⣿⡇⡿⠨⣿⣿⡿⠷⠶⢾⣿⣿⣿⣷⡄⢸⣿⣿⣿⣿⣿⣿⣿⣛⠀⠘⣿⣿⣿⡇⢸⣿⣭⣭⣿⣿⡿⠿⠶⠟⣀⣤⣤
⠀⠀⠀⠀⠈⠩⣿⣿⣿⣿⣿⣿⣗⠀⣿⠃⣷⢰⣿⣿⣷⣶⣞⣛⡻⠿⣿⣿⣿⣦⢿⣿⣿⣿⣿⣿⠃⠚⠀⢀⣾⣿⣿⠇⢸⡿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠿⠛⣋⣴⣿⡾⠏⠰⠿⣿⣿⣿⣿⣿⣟⣛⣛⣛⢻⣿⣈⢿⣟⠻⣿⠶⠿⢀⣴⠟⢩⣿⠟⠀⠾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡏
⠀⠀⠀⠀⠘⠛⠛⠛⠂⠀⠀⠾⠿⠽⠿⠁⡬⣭⣿⣿⣿⣿⣿⣿⣿⡞⠛⠛⠺⠛⢿⣦⠭⣷⣤⠀⠘⠋⠷⡿⠟⡽⠶⣶⣠⣯⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠔⡰⠀⠀⡠⠂⠐⠂⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠠⠀⠠⠗⠒⢄⠀⠀⡘
⠰⠐⠀⠀⠀⠈⠑⢄⠀⠀⠀⣀⣀⣀⡎⠀⠀⠀⠀⠀⠀⢸⡀⠤⠶⠶⠂⣰⠪⢄⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢃⡐
⠀⠀⠀⠀⠀⠀⢀⣤⠓⠉⠛⠛⠋⢨⣿⣷⣄⠀⠀⠀⡔⠉⣴⣶⣦⠀⢀⣥⣴⣶⠑⡄⠀⠀⠀⠡⣀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢠⠿⣣⣾⣿⣷⡄⠀⠘⠛⠛⠉⣆⠀⣾⢀⡀⢿⣿⠏⣀⣘⠿⠟⠋⣠⣶⠀⠀⠀⢠⢢⠈⠐⠀⣀⠊
⠀⠀⠀⠀⠀⢈⡀⢈⠍⠈⠋⢄⠔⠈⣉⠑⣼⣿⢰⡟⠱⠂⢁⠔⠧⡀⠀⠱⣀⠀⣿⣿⠀⣀⡐⠁⡎
⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⠌⡏⢸⠁⠠⠀⠀⡀⠀⡀⠀⢰⣿⡇⢈⠇⢨⠀⢤⢀⢼
⠀⠀⠀⠀⠀⠀⠙⡏⠇⠀⠀⠀⠀⠀⢀⣸⣿⠀⠸⢦⠀⠀⢠⡄⠀⠀⢀⠞⢛⣠⠊⡜⠉⠑⡬⠤⠙
⠀⠀⠀⠀⠀⠀⠀⢛⣲⠒⢒⡄⠀⢨⠩⠙⠛⡤⠚⡘⠀⠀⠊⠁⠐⠀⢘⢀⡿⣁⠎⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠂⠀⠊⠀⡀⠀⢣⠀⢀⣀⣀⡀⡤⠀⠀⠀⠀⡟⠁⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢣⢀⠀⠀⡀⠒⠊⣀⡤⠴⢁⡑⢄⠢⡐⠒⠃⠀⠀⢀⠔⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠰⡀⡀⡄⠁⠙⢄⡠⠃⠀⠀⡇⠀⠱⠤⠄⡤⠒⠉⠁⢀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠉⠀⠀⠀⢀⠤⠾⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⡆⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⢔⡁⠤⠄⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⢪⡮⠒⠈⠉⢁⠒⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⡄⡀⠀⡁⠀⠀⠀⠁⠀⣠⠲⠋⠀⠁⠀⠀⢀⡅⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⡈⡇⠀⠀⠸⣗⢺⠃⠂⠐⠤⡀⠠⢐⡩⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠉⠑⠤⡀⣠⠃⠁⠳⠡⢤⡔⠈⣈⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠚⠒⢒⡃⠀⢠⠔⠃⠂⠀⢘⠉⠉⠁⠀⢀⡨
⠀⠀⠀⠀⠀⠀⠸⠄⣀⣀⠀⡠⠊⠀⠀⣇⣀⡀⠀⠠⠊⠁
⣶⣶⣶⣶⡶⣶⡶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⢶⣶⣶⠶⡶⣶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⢀⣀⡀⣀⠀⠀⣀⣠⠤⠶⠶⠶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠄⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣠⠶⠛⠉⠉⠉⠉⠉⠉⠉⠳⠦⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢀⠀⠀⡆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣤⣀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠈⠂⠊⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠸⣿⡿⠿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⡀⠀⢈⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⡁⠀⠘⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⣿⠿⠋⠁⠀⢀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠐⠄⠙⢿⣐⠒⠉⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⡿⠀⠀⠀⢰⡿⠛⠉⠀⠀⢀⣠⠾⠛⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢦⣄⠈⠢⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠈⠀⠀⣀⡴⠞⠋⠱⠖⠐⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠄⠈⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⠥⠀⠀⠀⠀⠀⠀⢄⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⠀⠀⠙⠛⠛⠛⠶⣤⡀⠀⠀⠀⡠⠚⠡⠒⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢎⠀⠈⠂⢀⠀⠀⢠⣴⡖⠁⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⠢⣇⠙⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠑
⠀⠀⠀⠠⠐⠢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠘⠠⡀⢀⠂⠀⢀⠠⠤⠀⠤⢀⠀⠀⠀⢷⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠈⠁⢀⣴⠁⠐⡤⠴⠤⣄⠁⠉⠑⠶⡀⠀⠀⠀⠀⠀⢇⡤
⢐⠁⠈⠀⡸⠀⢠⠃⢈⠀⠀⠀⡠⠐⠈⢑⢰⡝⠁⠓⠄⠀⠀⠀⠀⠸
⠈⠒⢄⠜⠀⠀⢸⡀⠀⡏⠄⡍⠀⠀⢰⣷⡄⠃⡄⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠘⠠⢐⠀⠨⠀⣀⠜⠃⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢔⡱⣿⡦⡄⠀⡀⢬⠇⢾⡀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠣⠉⡈⡠⢥⡀⠁⠈⠁⠈⠑⡄⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣊⡷⠰⢁⠈⠐⠠⠀⢀⠼⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠑⠒⠉⡰⠤⠔⠁⠀⡣⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢈⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣒⢂⡞⠁⠀⠀⠀⠀⣃⠀⠀⠀⡀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠲⣢⠄⢳⡟⢲⠀⠀⠀⣰⠙⠀⠀⠀⠁⠀⢀⣌⠳⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠁⠉⢣⠊⠀⠄⠬⠔⡟⠀⠱⠤⣀⣸⣅⠈⠙⠒⠤⣉⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⡠⠊⠀⠠⠃⠀⠀⠈⠉⠑⠆⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡉⠤⠠⢄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠔⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠢⠤⠦⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⢤⠀⠀⠀⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠢⠀⠀⠀⠀⠀⢀⠄⠀⠀⠂⠈⠀⠘⠤⢌⠢⠀⠀⠀⡇⢠⠀⠀⠠⠊⠀⠈⠃⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢱⠄⠀⣀⠔⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠱⠐⡄⠀⠆⠈⡄⠀⠘⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠸⠀⠀⠁⠀⠈⠑⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⣁⣀⠀⠈⠐⠄⢀⠔
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⣥⣄⠀⠁⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠰⡇⢀⡠⠠⡀⠁⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡄⣀⠀⠀⠠⢀
⠀⠀⠀⠀⠔⠊⠀⢣⠀⠀⠀⠀⠀⠀⡀⠀⡄⢀⣠⡄⠀⠀⣀⠀⠀⠀⡸⠟⠻⠏⡈⠀⠀⠀⠀⠀⡐⠉⠀⠀⠀⠀⠀⠙⢦
⠀⠀⢀⠎⠀⠀⠂⠹⠀⠀⠀⠀⠀⠘⠋⠑⠊⠁⠀⠋⠈⡋⠀⠀⠉⠁⠉⠑⢖⠒⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⢦
⠀⠀⡌⠀⠠⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵
⠀⠀⠁⠀⠂⠀⡐⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⡇⠀⠇⠀⡇⠐⠕⠤⡈⠐⠠⣀⠀⠀⠀⠀⠀⠀⡠⠋⠐⢄⡀⠀⠀⠀⣀⠀⠂⢠⠊⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⢰⠊
⠈⡁⠓⠀⠘⠀⠀⠤⠆⠀⢈⠓⠀⠐⠉⠁⠐⠒⠂⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠑⠢⠘⣀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⣠⠂
⠀⢨⠀⡀⠀⠑⠄⠀⠀⠔⠉⠀⠃⠐⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⠃⠀⠈⠈⠈⠂⠀⠀⢄⠀⠀⠀⣀⠎
⠀⠈⡄⠘⠄⡀⠀⠑⡊⠀⡡⠀⡂⡆⠀⢀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠑⣴⢡⡀⢘⠀⠀⠀⠉⠒⣨⠏⣩⠏
⠀⠀⠘⢄⠀⠈⢢⠀⠑⢲⠑⠁⠉⠋⠒⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠐⠃⠠⠆⠃⠀⢇⣀⣀⠤⢊⠕⠉
⠀⠀⠀⠈⠢⡀⠀⠑⢠⠌⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡘⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⠀⠠⠄⠀⠤⠒⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣂⣀⣰⣤⣤⣄⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣏⣝⣟⣩⣛⣻⢋⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠈⠈⠉⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⢠⠗⢃⠈⠈⠠⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⠣⠙⠦⢄
⠈⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠠
⠀⢣⡄⢠⡴⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠐⢿⠏⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠉⠓⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣄
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⠓⢄⡀⠀⠀⠀⠀⠀⠈⠀⠈⢀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⢈⠆⠀⠀⠉⡁⠉⠀⠀⠀⠀⡀⠀⠄⠀⠀⠐⢄
⠀⠀⠀⠀⠱⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠈⠀⠐⠄⠂⠀⡄⠀⠀⠀⠀⠀⠃⠀⠘⡄⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠑⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠈⠳⡆⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣄⠀⠀⠀⠒⠠⠤⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠆⡂⢠⠀⠀⠤⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠠⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠰⣄⡀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⡆⠀⠀⠀⠀⠀⠈⣄
⣐⣀⣀⣀⣀⣀⣀⣀⣠⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠶⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢴⠑⡤⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠋⠀⣀⣄⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⣇⠤⠴⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠉⠒⢉⡕⠀⢠⠋⠀⠀⠀⠀⣁⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠀⠘⠦⠤⠒⠒⠉⢹⡀⠀⠀⠀⠘⠒⢦⢀⣀⠤⠔⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠶⢋⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠖⠲⢤⠀⠈⠁⠀⠀⠀⠠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠊⠀⠀⠀⠀⠀⣀⣤⠤⠤⡄⠀⠀⠀⠀⠀⢀⠀⢀⡼⠶⡤⠖⠂⠀⠀⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋⠀⢶⣤⠷⠐⠒⠒⣶⣢⡼⠚⠁⠀⣀⠷⠀⠀⠀⠀⢀⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣳⣤⡎⠀⠀⠀⣀⡠⠽⠋⠄⠀⠀⠞⢋⡉⡆⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⠉⠉⠉⠉⠉⠉⠉⢉⡭⠄⠀⠉⠉⠉⠉⠁⠀⠀⡜⠀⠀⣀⣋⠭⠞⠑⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠀⠤⠒⠶⡖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⡜⠁⠀⠀⣠⠮⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⢀⠎⠀⠀⠀⠀⠀⠀⠘⠓⠋⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⣀⠤⠒⠉⠀⠀⢀⡮⣄⠀⢀⣠⠴⡀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⡞⠀⠈⠛⠉⠀⠀⠙⢦⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⠷⠤⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠲⠏⠀⠈⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠐⢄⡈⢢⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⡤⣴⣏⠰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠻⡣⠙⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡀⡀⢤⠤⠤⠚⡣⢝⢿⡉⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠇⢧⠎⡰⢡⠜⠁⣔⡳⣍⡁⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢮⢍⠌⡏⣰⡵⠃⠀⠀⠙⠴⣁⠥⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡿⢿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞⡡⢋⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠈⠓⠒⠚⠥⠀⠛⠑⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠀⠀⠀⡆⠘⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠌⠀⠘⢦⡀⠀⢃⠀⡀⠀⠀⡜⠁⠙⠀⠀⠇⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀⠑⠤⡀⠈⠈⠙⠢⡉⢦⡀⠀⡃⠀⠀⠹⡀⠀⡠⠊⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⢄⠀⠈⠀⠀⠀⠀⠈⢆⠙⣄⠀⠀⠀⠀⠈⠊⠀⠀⠀⠜⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠔⠂⠭⢅⣂⣤⣤⠖⠚⢇⠈⢖⠲⠄⠐⠤⡀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠘⡀⠈⢧⢀⢠⠀⣧⠀⠀⠀⡘⠁⢹⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂⠀⠀⠀⢸⠀⡠⢶⠁⠀⢠⠊⠃⢀⡸⠀⠙⠀⢸⠀⠀⠀⠇⠀⠉⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⡀⠀⠀⠈⢶⡄⣀⠑⠒⣒⠚⠓⠀⠀⠀⠀⠀⠀⠣⡀⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠉⠂⠀⠘⠄⠀⢰⢣⠞⠉⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢰⠁⠀⠸⣡⠶⠂⠀⣸⡟⡄⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠤⢀⠴⠦⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄⣀⢀⠂⠀⠀⢰⠛⠃⠀⠀⠿⠇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠴⠊⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡂⠂⠤⣀⠀⠀⠀⠸⡀⢨⠀⢸⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⢹⠀⠀⠀⢀⠴⠒⠐⠣⢼⣠⠮⠠⠤⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢀⡠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠑⠄⠀⡰⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⠔⠁⠀⠀⠀⡃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢠⣞⣉⣿⡐⡄⠀⢀⠎⠀⢠⠖⡹⣦⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢁⣠⡤⣄⠻⠿⠿⢿⣷⡔⢄⢸⠀⠀⠘⠤⠝⢁⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠒⠆⠈⠩⠤⠔⠊⢉⡠⠛⠉⠀⠀⠀⠀⠨⣿⣦⢸⠀⠀⠀⠀⡠⢫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠤⠔⠒⡆⠀⠀⢱⢻⡎⠣⠤⠴⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣄⡀⠀⠀⢞⠀⢀⡠⠃⢘⣷⣀⡠⠔⢊⡱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⢀⠖⠉⠀⠀⢸⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣀⢿⡁⠀⠀⠀⠀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⣿⡼⠧⠀⠀⡄⡀⢠⡱⠞⢘⣖⢪⣉⣩⣉⡍⠖⡚⠭⣔⡤⡀⡈⠛⠛⠁⡀⣀⣐⡄⣤⣤⣤⠀⡄⠀⠀⣄⣀⠀⠀⢰⢯⡈⠻
⠀⠀⣿⠁⠠⢴⡽⢋⡡⠷⠜⠛⠉⠉⢤⣄⠀⠀⠈⠉⠁⠛⠳⢬⡎⢌⣡⣀⣰⣊⢥⠶⠶⠲⠔⠨⠏⡀⠖⠀⠛⣻⢪⣅⠄⠁⠜⢻
⠀⠀⣿⢰⣞⠣⠖⢳⣔⠀⢰⢀⠀⢂⡴⠉⠁⠀⠀⠀⠀⠀⠀⠺⠀⣧⠄⣼⢙⠆⠀⠀⠀⠀⠀⠲⠄⡠⠆⠀⠀⠈⠛⢮⠷⠧⣄⣼
⠀⠀⣿⠾⠈⡟⠄⣥⢈⠀⠘⢣⠔⡉⠠⢽⡄⠂⠀⠀⢀⠠⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⢀⡠⠄⠒⠒⠀⠠⢀⡀⠀⠀⢸⣵⣦⣧⣼
⠀⠀⣿⢠⠺⡇⠀⠰⡆⠠⡰⠟⠐⠓⠆⠀⠘⠀⢀⠜⠈⠀⢴⠀⣤⣤⣤⣄⣀⡀⡠⠊⠁⣀⣆⠀⠀⣤⣾⠾⣍⢢⡀⠀⡄⠄⡤⣽
⠀⠀⣿⠀⣰⣇⣈⡀⠴⢠⠟⠂⠀⠀⢠⠒⠀⢀⠊⣠⠀⢀⠀⢀⣀⣀⠙⠑⠘⡿⠁⠀⣀⡿⣁⣀⣀⣫⣫⡔⡻⡏⠱⠀⣷⠐⢒⣿
⠀⠀⣿⠀⢚⡇⠘⡏⡉⣿⢀⣄⠈⣉⡍⠒⠂⢶⠀⠁⣴⠗⢛⣩⣥⢬⣕⡆⠀⡇⢠⡚⠁⣰⡅⠀⠀⢠⠙⢃⠉⣻⠀⣇⣰⢃⡌⣿
⠀⠀⣿⡘⠏⡇⠠⡿⢶⡾⣉⠠⢴⠤⢄⡀⠀⠀⠀⡼⢡⡞⠋⠁⠀⢠⡈⣇⣠⡆⠘⢧⣠⣿⣿⣿⣶⣾⠆⠙⣾⠁⢀⠃⣿⠀⠀⣇
⠀⠀⣿⢁⣾⡇⣀⠀⠘⡻⠁⠀⠀⠀⠉⣁⣀⡀⠰⣔⣆⣀⠛⠦⣾⣿⣿⠻⠟⠛⣿⣧⣿⣿⣟⣿⣻⣿⣟⣿⣍⢻⠞⠀⣆⢆⠈⣷
⠀⠀⣿⠘⣋⠇⠻⠗⠀⡇⢠⣿⣤⠀⢈⣈⣻⣥⣸⣿⠌⢹⣿⣧⡉⠉⠀⠠⢶⡀⣬⣳⣿⡈⠈⠙⠿⠋⠙⣻⠿⠋⢀⡀⣯⡈⠂⣿
⠀⠀⡏⠈⢏⠁⠀⠀⠀⡇⠀⠉⠁⢠⣿⡿⠻⠒⢸⡿⠛⢤⡽⢿⡿⢒⡄⠻⡄⠿⢀⣼⣶⣮⡁⠒⠂⠉⠉⠁⠀⠀⠛⠁⠉⠀⠀⣿
⠀⠀⣷⠢⣤⡇⠀⠀⠀⡇⠀⠀⢢⡿⡽⢁⡷⢻⡷⢈⠑⢄⠑⢾⣧⢿⣧⣀⣈⣐⣾⢷⣀⢀⣉⠀⣂⣠⠀⡠⣠⣂⡀⠀⢶⡀⠂⣷
⠀⠀⣿⢛⠼⣃⣿⣷⡄⡿⠒⣠⣯⡋⠠⢾⣇⡶⠀⣾⣷⣦⣈⢂⡽⠾⡏⢱⡼⣦⣿⣶⠟⣹⡷⠾⣿⠃⠀⠴⣶⣯⣡⡀⡤⡦⠤⣿
⠀⠀⣧⠀⢀⣿⣿⠿⠁⠉⣿⣻⣤⡿⢃⣴⣿⡇⢠⠼⢧⣤⣈⠁⠀⠸⢫⣿⣿⣽⠟⠁⢠⣇⠸⢰⣿⡇⠀⠀⠡⠊⠊⠀⠗⠈⠂⣿
⠀⠀⣯⡡⠻⠟⣁⡀⠀⢘⡀⠀⢀⡴⡟⠀⠿⣧⠄⡀⠀⡌⡝⠱⢶⣴⣷⣾⡿⠃⠀⠀⣸⠟⢿⣇⣸⣇⠀⠘⠄⠀⡄⠀⣿⡲⠒⣿
⠀⠠⣷⠀⢰⡆⢹⣧⠣⢈⡵⠾⣿⢁⡆⢳⠀⢺⣷⣴⢆⠜⣴⣿⢞⠿⢼⠿⠤⣤⣤⣬⠿⡶⠾⣶⣱⡍⠀⠰⢸⢠⠀⠀⣙⠻⡶⣿
⠀⢀⡷⢴⠰⡇⠈⠁⠀⡞⠀⠷⢻⡶⠬⠆⣑⠘⠻⣷⣧⡿⡟⢨⡳⢤⣉⢀⡀⠐⢂⣀⠀⢀⣀⠀⠈⠀⠀⢝⣿⠻⠄⠀⣿⡏⢀⣿
⠀⢰⡏⡄⢺⡅⠠⡄⡘⠀⠀⠒⠃⢺⡶⡶⠿⡷⣶⢟⣿⢻⡇⠀⠙⠀⠼⣿⠀⣸⢿⡇⠀⠈⠸⠆⠀⠀⠂⠀⠁⠉⠀⠤⠌⣴⡇⣿
⠀⢸⡏⠉⣚⡇⠈⠠⢃⠤⠀⢀⠀⢘⡇⠀⠰⣊⢡⢾⠃⢀⠀⠀⠀⠐⠀⢿⣎⣩⡟⠀⢀⡠⢔⣂⠉⠉⠉⣀⣢⢀⡀⠀⢘⠏⠄⣿
⠀⢸⡇⣆⢹⡇⠀⡼⠁⠀⠀⢘⠀⣈⡱⠆⠀⠀⣛⠋⢶⣷⣲⣦⣶⠒⢿⣟⠛⠃⠄⡠⠋⢰⣿⠍⠷⠀⢀⠈⢋⡇⠙⢆⠸⣗⡄⣿
⠀⢸⡇⠈⢣⡇⣀⣉⠐⠀⡬⢿⠀⣻⠀⠐⠀⢰⢛⠀⢹⠉⠉⠀⠈⢳⡈⢿⡗⠲⢦⠁⠀⡸⠇⠘⢿⢠⠿⡀⠶⠅⠀⠈⢷⣄⠁⣿
⠀⢸⡣⠥⣸⡇⠋⠘⢠⢎⣰⠖⣋⠉⣄⢻⡏⠉⠉⠉⠉⣰⡖⡆⠀⢹⢻⡌⢿⢦⡀⠀⠀⢀⠲⡄⢺⣡⣆⣀⡆⣶⠀⠀⠘⢁⡶⣿
⠀⢸⣇⠀⢰⡆⠀⠀⡸⣹⠃⠀⣀⣀⠐⣎⢶⡀⠉⡄⢀⢚⣷⠲⠆⠈⡆⣗⠢⢝⣿⡟⣿⣿⡗⣏⣬⡅⠀⣉⣴⣿⣯⣽⣿⠟⣁⣿
⠀⢸⠛⠃⠁⡃⠈⢠⢃⡟⠀⠘⠀⠋⠀⠈⠺⢷⣄⠀⠀⠙⠂⠀⢀⡀⠷⠘⣦⡀⠀⠻⣣⠭⠼⢿⠯⠛⢾⡉⣉⣻⣏⡔⠘⠢⠄⣿
⠀⢸⡀⠀⢐⡇⠀⢈⡃⢷⡀⠀⠠⠀⠈⠁⠀⢳⣍⠳⢶⠆⠀⢀⡄⡀⡀⠷⢯⣭⠟⠀⡈⠳⠶⣊⡙⠒⠻⣛⡧⠜⠋⠀⣆⠀⠀⣿
⠀⠸⠇⠈⠑⡇⠀⠈⠻⠅⠿⠓⠀⠀⣄⠀⠀⠀⠈⠉⠁⠀⠀⠈⠽⣿⠶⠀⠠⠀⠀⠄⣠⠀⠀⠀⠀⠀⠀⠠⣠⡀⡦⠀⢣⠤⠤⣦
⠀⢸⡄⢂⢱⠀⠘⠓⠀⠀⠄⠀⠀⠂⢻⡿⢿⡁⠀⠀⠐⠃⠀⠀⠈⠉⠐⠂⠀⠀⠀⠐⠃⠀⠀⠀⠈⠀⠀⠐⠬⣙⡒⠀⢠⠀⠀⣿
⠀⠘⠒⠚⠒⠒⠒⠒⠒⠲⠖⠒⠒⠒⠒⠒⠒⠒⠒⠲⠖⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠄⠰⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠯⣽⣷⣄⣀⣠⣤⣶⣾⣃⡀
⠀⠀⣠⣴⣶⣶⣶⣿⣫⡴⠟⠛⢉⡿⠋⠉⣉⡿⣯⠉⠙⠛⠓⠶⢦⣤⣀⡀
⠀⠘⡽⢵⡯⠭⣿⣿⠋⠀⠀⠀⢸⡟⠋⠻⠟⠀⣹⠆⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⠶⢶⣦⣄⡀
⠐⠚⢃⢎⣴⡿⣿⠃⠀⠀⠀⠀⠀⠻⣦⣤⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦
⠈⠉⠡⠞⠏⢸⡏⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⣼⠏⠀⣠⠤⢤⡀⢀⣤⣤⡀⠀⣀⡀⢻⣷
⠀⠀⠀⠀⠀⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡟⠀⡾⠁⠀⠀⠙⠋⠀⠀⢻⡄⢿⣿⠀⣿
⠀⠀⠀⢰⡿⠛⠛⠇⠀⢀⡞⢡⣴⡶⠶⣶⣤⡀⢠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⠀⢸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⡿⠀⣿
⠀⠀⠀⢻⣇⠀⢀⣀⠀⡾⢀⣿⠃⠀⣰⣧⣬⠻⣦⡀⠈⠉⢹⡟⠛⠛⠛⠻⠿⠿⠛⠛⠋⠁⠀⠀⠘⣇⡀⠘⢷⣀⠀⠀⠀⣠⡾⠃⠀⠚⠃⢀⣿
⠀⠀⠀⠀⠙⣿⠟⠋⠀⠃⢸⣿⠴⣾⡇⠻⣿⣧⣸⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⠴⠟⠁⠀⠀⠙⠳⠦⠞⠋⠀⠀⠘⠿⢀⣾⠏
⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⣿⡸⣿⣷⣾⡿⠋⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⢤⠴⢶⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠸⣧⡙⣻⠏⠀⠀⢠⣿⠁⠀⠈⣇⠀⠀⠀⠀⠀⠀⢠⣤⢤⣀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠿⣧⣤⣤⣴⠟⠁⣴⠀⠀⣿⠀⠀⠀⠀⠀⡶⠊⠁⠀⠙⣧⡀⠀⠀⠀⠀⠀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠲⢤⣉⣁⣀⣤⠞⠃⠀⢀⣿⡄⠀⢠⣤⣴⠷⣄⠀⠀⣠⣾⠃⠀⠀⢲⠟⠛⠋⠉⣀⣸⡆
⠀⠀⠀⠀⠀⢀⣿⠛⠷⠶⠶⠶⠶⠶⠶⠶⠿⠿⠶⠚⠛⢛⣯⣭⣷⣀⡼⣻⣽⣧⣼⠶⢾⠅⠉⠀⠀⠀⠈⣗⠒⠋⠉⠉⠀⢻
⠀⠀⠀⠀⠀⣼⡇⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⠃⠀⠀⠞⠏⡀⣨⣿⣿⠛⠗⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣀⣤⣼⡆
⠀⠀⠀⠀⢀⣿⠀⠀⠀⢿⡄⠀⠀⢀⣾⠋⠉⠉⣧⠉⣴⣿⣿⣷⡄⣈⡾⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⡇⠀⠀⠿⣿⠿⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢿⣦⣤⣼⣧⣤⣤⣶⣿⣤⣿⣿⣿⣿⠟⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⠈⠹⣿⠻⣯⠻⠏⠁⣠⣄⣤⠟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠢⠿⠆⠸⠇⠀⠀⣯⢀⣿
⠀⠀⠀⠀⠛⠻⠿⠟⠚⣿⠿⠿⠿⠿⠟⠛⠻⣦⣀⠀⠀⠀⢀⣰⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠒⠒⠒⠒⠦⣄⠀⠀⠀⣠⠿⠶⠖⠒⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠐⢧⣼⠓⠶⣤⣤⣤⠟⠀⠀⠰⣧⣼⠳⢦⣤⣤⣴⠟
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣰⣿⠿⣿⣷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⢿⣿⣆
⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⡿⢿⣟⣲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⣻⡿⢿⣿⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠙⣿⣟⣿⣿⠉⣹⣷⣾⡿⢏⣙⣿⡳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣿⣋⡹⢿⣷⣾⣏⠉⣻⣿⣻⣿⠋
⠀⠀⠀⠀⠈⠻⣿⡿⣿⡟⠉⠙⠛⠶⡟⠉⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠉⢻⠷⠛⠋⠉⢻⣿⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠖⠒⠢⠤⠷⣄⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⣠⠾⠤⠔⠒⠲⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣀⣀⣀⣀⣀⣈⣑⣦⣀⠀⠀⠙⢷⣄⠀⢀⡀⠀⣠⡾⠋⠀⠀⣀⣴⣊⣁⣀⣀⣀⣀⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⠙⢦⣾⣷⡴⠋⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠈⠙⣲⣿⣿⣖⠋⠁⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣽⡿⠿⠿⠛⠻⢿⣛⡻⡿⠿⠿⣿⣿⣿⣿⠻⠿⢿⣟⣛⡿⠟⠛⠿⠿⢿⣯⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⠤⠴⠒⢒⡲⠋⠀⠀⢀⡠⠚⣹⣿⣿⣏⠓⢤⡀⠀⠀⠙⢶⡒⠒⠦⠤⣽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢋⣀⠤⠔⢺⠏⠀⠀⣀⠔⠉⢀⠜⡸⣿⣿⢏⠣⡀⠉⠢⣀⠀⠀⠹⡗⠢⠤⣀⡙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠉⢀⡠⠖⢻⠖⠒⣹⠃⠀⣠⠋⡰⠡⣿⣿⠈⢆⠙⣄⠀⠘⣏⠒⠲⡟⠲⢄⡀⠉⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢶⡋⠀⢠⠋⠀⢰⠃⠀⡼⠁⣰⠃⠀⣿⣿⠀⠘⡆⠈⢧⠀⠘⡆⠀⠙⡄⠀⢙⡶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⢶⡇⠀⣠⠃⠀⡜⠀⢠⡇⠀⠀⢻⡟⠀⠀⢸⡄⠀⢣⡀⠘⣄⠀⢸⡶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⣿⣦⢾⣀⣠⠟⠀⠀⠀⠘⠃⠀⠀⠀⠻⣄⣀⡷⣴⣿⣽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⠿⢻⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⠀⠀⢸⠟⠻⡟⠙⢿⠻⣏⣵⣼⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠻⡉⢻⠙⣿⣦⣿⡇⠀⡇⠀⠎⢀⢿⣧⢽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡋⠓⢄⠀⣈⡃⠡⢸⣿⠁⠀⠁⢀⠶⠚⣻⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠉⠲⢦⣝⡇⠙⠦⣼⣿⠀⠀⡴⠃⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣉⠓⠦⣌⠻⣄⠀⢿⡇⠀⠘⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠈⠑⢈⣳⣼⣇⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠒⠶⢿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠂⣠⠔⣨⠞⢿⡏⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠤⠒⢋⣥⡞⠁⢀⣿⡇⠀⢠⡀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠴⠚⢋⢴⣀⡴⠋⢿⣇⠀⠀⠳⡀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⠚⢁⠌⢰⠀⡆⣼⣿⠀⠀⠀⠘⠲⣤⡝⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣮⣤⣯⣼⠿⠋⣿⠀⢸⠀⠀⡆⢰⣾⣛⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢿⣦⣾⣦⣴⣷⣾⡉⡏⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣷⣧⣦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⠇

⢼⣶⣀⡀⠀⠀⠀⣀⣲⣿⠀⢸⡿⣒⣄⠀⠀⠀⠀⣀⡲⢶⡆⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠐⣿⣷⠦⡀⠀⠀⢴⣿
⢸⣛⣭⣵⣄⣤⣮⣭⣍⣻⠀⠀⡿⠉⠛⢷⡀⢠⠞⠛⠛⣿⠀⠈⠳⢉⣾⣿⣶⣴⣶⣾⣿⡌⠼⠁⠀⠹⢿⣀⣱⣤⣀⣠⡿
⠀⢿⣿⣿⠟⡟⢿⣿⡿⠄⠀⠀⢈⠔⠂⠀⢹⡏⠀⠀⢠⠁⠀⠀⠀⠘⠙⠉⢸⡿⣿⠈⠋⠟⠀⠀⠀⠀⣞⠁⠀⠋⠀⠙⣷
⠀⠀⠻⠇⠀⠁⠈⡿⠃⠀⠀⠀⢸⡀⠀⠀⠈⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠂⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⠷⠆⠀⠳⠿⠁
⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉⠑⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⡾⠛⠁⢀⠀⠀⢀⠀⠉⠛⡇⠀⠹⣿⢶⣄⠀⠀⠀⠀⣠⡾⠿⠃⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢰⡿⠒⠀⠀⠀⠀⠀⠒⠻
⠁⢴⣿⡁⠀⡄⠀⣽⣷⠄⠀⠀⠀⢻⣆⣻⣷⣀⣠⣾⡏⠸⡎⠀⠀⠀⠀⣠⣾⣤⣶⣀⠀⠀⠀⠈⢀⣴⡖⠁⣀⠈⢑⣶⡄
⢰⠻⣿⣥⡴⡷⣄⣽⡿⠰⠀⠀⠀⠰⠋⣀⠘⠻⠻⢃⣄⠱⠀⠀⠀⠀⢼⡙⠟⠉⠿⢙⡆⠀⠀⠀⠾⣿⣏⣠⣿⣄⣹⣿⠇
⠀⠀⠈⠙⠁⠁⠙⠉⠀⠀⠀⠀⠀⠀⠣⡴⠟⠀⠐⠿⢆⠊⠀⠀⠀⠠⠜⠛⠀⠀⠘⠛⢥⠀⠀⠀⠁⠈⠛⠋⠈⠘⠟⠋
⣠⢤⣀⡀⠀⠀⠀⠀⢀⣄⡦⣄⠀⠄⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢤⣄⠀⠀⠀⠀⠀⠀⠀⢠
⣷⠛⠉⣀⡀⣀⣀⢀⣀⠉⠛⡎⠀⠀⢸⣧⡀⠀⠀⢀⣾⠄⠀⠀⢸⣿⠿⣢⠀⠀⣠⡺⢻⡿⠀⠸⣄⣾⣦⠀⠀⠀⣠⣶⢠
⠙⢒⣻⣟⣯⣿⣟⣟⣿⡟⡒⠃⠀⠹⣟⣙⣿⣤⣴⣿⣉⣿⠇⠀⠀⠙⡆⠛⣷⣶⠛⢠⠛⠁⠀⠀⠻⡎⢻⣷⣤⣼⡟⢩⡞
⠀⢿⡿⣿⡾⢙⠹⣿⣿⣧⡗⠀⠀⢀⡾⢿⡿⢻⡿⣿⡿⣦⠀⠀⠀⢸⣿⠸⠁⠉⢇⣼⠇⠀⠀⠀⠀⣾⣟⡉⢻⢉⣟⣶
⠀⠈⠙⠛⠁⠀⠀⠈⠋⠉⠀⠀⠀⠈⠻⠞⠁⠀⠀⠘⠻⠛⠀⠀⠀⠐⠙⠀⠀⠀⠀⠋⠃⠀⠀⠀⠀⠈⠋⠀⠀⠀⠛
⠐⠰⣠⡀⠀⠀⠀⣀⠠⠒⠀⠀⢠⣤⣄⡀⠀⠀⠀⣠⣴⡆⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⢰⣦⣤⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠈⠈⢢⣦⡚⠈⠀⠀⠀⠀⠀⣏⡉⠉⢀⡀⠈⠉⠙⡇⠀⠀⡂⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠘⣿⢰⠋⠀⣠⠀⠘⢷⢹
⠀⠀⠌⠃⠺⡟⠃⠯⡠⠀⠀⠀⠀⢉⡷⠾⢿⡿⠗⠶⣌⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢨⣾⠟⠉⢹⠉⠳⣾⡎
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠷⡴⠋⠙⢳⡴⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠧⣀⠀⠘⠀⣠⠼⠃

⠀⢀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀
⠀⢾⡿⢍⠳⠈⠛⠙⢶⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡶⠋⠟⠡⠞⡩⢿⡇
⠀⠸⣗⠄⡱⣚⠭⠭⠶⢷⣬⣟⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣛⣥⡾⠶⠭⠭⣓⣎⠠⣻⠃
⠀⠀⠹⡗⢽⣱⢊⣭⠤⠬⢜⣛⣿⢷⣽⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣯⣾⣽⣋⡡⠥⠬⣭⡑⣾⡇⣺⠏
⠀⠀⠀⠹⡆⢿⢺⠀⢲⡖⠐⠠⠭⡝⡿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢿⣿⢿⢫⠭⠄⠂⢲⡖⢀⡇⡿⢰⠏
⠀⠀⠀⠀⢹⡄⢦⠑⢤⣙⡢⢄⣀⠀⠁⠂⠝⣺⣻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⣝⡫⠒⠍⠀⣀⡠⢔⣃⡤⠋⡼⣠⠏
⠀⠀⠀⠀⠀⣷⠈⡷⣄⡤⠉⣑⠒⠡⠕⣴⣄⡈⠚⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⡝⠓⠁⣀⣦⠪⠘⠒⣈⠉⢄⣠⣾⠡⡾
⠀⠀⠀⠀⠀⢸⡅⢻⣦⣉⢉⠈⠳⠥⠤⠶⡉⡙⠷⣠⡛⣽⢟⣷⡀⠀⠀⠀⠀⠀⠀⢀⣞⡿⣫⢋⣤⠖⢋⢉⠖⠤⠭⠞⠡⡉⣉⣴⡟⢨⡇
⠀⠀⠀⠀⠀⠸⣇⢸⠟⣩⡥⠤⡤⠤⣀⡀⠙⠢⠄⢂⠙⠿⣿⡿⣿⡄⠀⢀⡀⠀⢠⣟⢯⣿⠟⠋⡐⠠⠔⠋⢀⣀⠤⠤⠤⢬⣍⠻⡇⣺⠇
⠀⠀⠀⠀⠀⠀⢿⡼⣜⢧⣠⣰⣰⣀⣸⣈⣹⡐⢠⢈⡳⢅⠨⠻⣼⣿⡔⣿⣿⢢⣿⣳⠟⠅⡸⢞⠀⡄⢒⣏⣀⣅⣀⣇⣆⣄⡼⣣⢧⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠮⣓⣤⣀⡉⠀⠀⢀⣀⠈⠉⠓⠺⠶⣄⡀⠘⢿⣷⣿⣿⣾⡿⠁⢀⣠⠶⠓⠊⠉⠁⣀⣀⠀⠀⢉⣀⣴⣛⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢲⣴⣶⢢⣤⣤⣩⣵⣄⣀⣩⣷⡾⣿⣧⣾⡿⢷⣾⣍⣀⣠⣮⣁⣥⣥⡔⣶⣦⡖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡉⢱⢁⡛⠀⡇⢻⠙⠉⢹⣿⣿⣮⡿⣿⣿⢿⣤⣿⣿⡏⠉⠋⡟⢹⠀⢛⡈⡌⣉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠠⢇⠀⡇⢼⡐⠂⠀⠸⠛⣿⡇⣿⣇⢸⣿⠛⠇⠀⠐⢂⡇⢸⠀⡼⠄⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⣠⢣⠈⣻⡀⠹⡀⢹⡀⠂⠀⠀⢹⠀⣿⣿⠈⡏⠀⠀⠒⢀⡏⢀⠏⢀⣟⠀⡜⣄⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⢀⡒⡀⠐⢣⡀⠙⣄⠉⠱⣄⣡⠇⠀⣿⣿⠀⠸⣌⣠⠊⠉⡠⠋⢀⡜⠢⢀⢒⡀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀⢘⡒⡄⠉⠷⣄⠀⠙⠲⢼⠏⠀⠀⢻⡗⠀⠀⠹⡦⠖⠋⢀⣠⠾⠉⡠⢐⡁⢀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠤⠦⣀⠁⠟⢲⡾⠁⠀⠀⠀⠈⠁⠀⠀⠀⠘⢷⡖⠻⠈⡠⠴⠥⠀⢸⠉
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⣜⢂⣭⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣭⡐⣣⠒⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⡌⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠚⠑⠀⠒⠋⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⠌⠛⠓⠀⠘⠓⠒⠲⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠘⠳⣄⡀⠛⠀⢻⣧⠀⡤⠐⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢀⣤⠀⠀⠀⠛⠀⢀⡼⠿⠖⠀⠸⠿⢷⠀⠀⠀⠀⣷⠀⠂⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡁⠀⠀⠉⠠⣀⣐⣶⣶⣾⣷⠶⢶⣶⠶⢶⣾⣶⣶⣶⣀⠠⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⡁⠠⣈⣵⠶⠟⠛⠉⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⡌⠙⠻⢷⣮⣁⠀⠉⠁⠈⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠀⡠⠂⣤⡾⠋⠁⠀⢀⠀⠀⢈⣀⣤⣦⣤⣤⣌⣉⣁⡀⠁⣀⠀⢀⠉⠻⣷⣤⠀⠀⠀⠿⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⣴⡾⠋⠀⢀⡠⢀⣹⣶⠾⠛⠛⠉⠉⠉⠉⠉⠛⠛⠿⣿⣦⣼⣃⡀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠊⠉⠁⠄⢠⡾⠋⠀⠀⠀⢀⣾⠟⠋⠁⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠙⠻⢿⣷⡆⢀⠀⠀⠙⣷⡀⠂⠘⠛⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢀⠊⣰⠟⠁⠠⠀⢻⣶⡟⠁⠀⠀⣀⣴⣾⠿⠿⠛⠛⠛⠿⢿⣿⣶⣦⡀⠀⠈⠻⣿⣦⠀⡄⠀⠈⢿⡄⠀⠀⢀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠒⠈⢁⠈⣠⡟⠀⠀⠀⣳⣿⠏⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣆⠀⠀⠈⢿⣷⡀⠀⠂⠈⣿⡆⠀⠚⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⠈⣠⣿⠀⠀⢠⣼⣿⠃⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠸⣿⡇⠀⠀⠀⢸⣇⠐⠀⠀⢀⡛
⠀⠀⠀⠀⠀⠀⢀⡴⠟⠂⠀⣿⡏⠀⠀⢀⣿⡟⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⡀⠀⢰⣿⠃⠀⠲⢏⡀
⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⡇⣿⡇⠀⠀⢈⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠤⠤⠤⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⣿⡄⠀⠀⠈⣳⠄
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⡇⣿⡇⠀⠘⠛⣿⣇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠰⢾⠃
⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠁⣿⣧⠀⠀⠉⢻⣿⠀⠀⠈⣿⣦⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⣼⣿⡀⠀⠀⢸⣿⠃⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠘⣿⡀⠀⠀⠘⣿⡆⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⠀⢰⣿⠟⠓⠀⠀⣼⡏⢀⠀⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⢠⢿⣧⡀⠈⢀⣸⣿⣦⡀⠀⠀⠛⢿⣷⣤⣤⣀⣀⣀⣀⣀⣤⣴⣾⠟⠋⠀⠀⢀⣴⡿⢧⠀⠋⠀⣸⡿⢀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⡈⠸⢻⣷⡄⠀⠁⣠⢿⣷⣦⡀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⣠⣿⣏⠑⠀⠀⠀⣰⡿⠁⠈⢠⣤⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟⠁⠀⠃⡘⢿⣦⡀⠀⠈⠉⠻⢿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠿⣥⠈⠓⠄⢀⣼⠟⢀⠆⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⣤⠘⢀⠙⠿⣦⣀⠀⠀⠀⠈⠙⠿⠛⣿⢷⣶⠶⣶⣿⠟⠛⢛⡿⠉⠤⠀⠀⢀⣠⣴⠟⢁⠔⠉⢰⡦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠐⠲⣌⠛⠿⣦⣤⣄⣠⡀⠀⠀⠀⠐⠀⠀⠉⠀⠀⠀⠀⣀⣠⣶⡾⠟⢋⡥⠲⣀⡀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⠀⠀⠉⠢⠀⢉⡙⠛⠿⠿⢶⣶⣶⣶⣶⡶⠶⠶⠾⠟⣛⣋⡡⠤⢈⠁⠀⠀⠹⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠓⠓⢾⡆⠀⠈⢁⡄⠒⠒⠂⠀⠀⠀⠀⠀⠒⠒⠀⠀⢉⡁⠀⠀⢸⠓⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠦⠄⠻⣇⠀⢀⡀⣶⡄⠀⠀⣸⣆⠀⠀⢀⡘⠛⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠁⢸⠟⠶⠂⢹⡏⠑⠺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⢀⣀⣀⠀⠠⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡌⢉⣉⠁⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠓⠶⠶⠶⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣤⡀⠀⠀⠀⠀⠠⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠉⠉⠉⠉⠁⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡉⠀⠙⠂⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡶⠦⠤⠤⠴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⣤⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⠀⢠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⠀⠸⣷⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢠⣿⡆⢰⡿⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⣧⡞⠙⣧⡼⠃⠙⢿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣦⣌⣉⣀⠀⠘⠃⠀⣀⣠⠷⠙⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⣤⣄⡈⠉⠉⠉⠀⠉⠉⠉⢁⣠⡴⠞⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠉⠉⠛⠛⠓⠒⠚⠛⠛⠉⠁⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣬⡛⠛⠛⠒⠀⠀⠀⠀⠀⠐⠛⠛⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠶⠶⠶⠶⠶⠶⠊⠉⠉⠁

⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢿⡟⠈⠳⣄⠀⠀⠀⠀⡤⠊⠁⣻⠗⠀⠀⠀⣿⡇⣶⣤⡀⠀⠀⠀⠀⠀⣠⣴⡆⣿⡇⠀⠀⠘⢻⡷⡦⣄⠀⠀⠀⠀⠀⣠⣴⡾⣿⠓
⠀⠀⠀⡉⠁⠀⠈⠁⣴⡄⠊⠀⠀⠈⣩⠀⠀⠀⠀⠸⣷⡙⠻⣤⣇⢀⣄⢀⣧⡼⠛⣱⡿⠀⠀⠀⠀⠫⢿⡹⡟⠀⣀⣸⣀⡠⠘⡟⣽⠯⠃
⠀⠀⠀⠛⣳⠖⠶⠖⣿⡗⠲⠖⠲⡛⠋⠀⠀⠀⠀⠀⠙⣷⡾⠉⠉⣹⣿⡉⠉⢹⣾⡟⠁⠀⠀⠀⠀⠀⠻⣿⣛⡈⣹⣿⣏⠁⣘⣿⡟⠁
⠀⠀⠀⠘⡀⢠⠔⠀⣿⡇⠑⢤⠀⣈⠀⠀⠀⠀⠀⠀⠸⣿⣷⠘⠻⠏⠏⠿⠛⢸⣿⡿⠀⠀⠀⠀⠀⢠⣿⣿⣿⡤⠁⢸⠈⢠⣟⣿⣿⣆
⠀⠀⠀⠀⢳⣀⣀⠂⠀⠀⠂⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡞⠂⠀⠛⣾⣿⡟⠃⠀⠀⠀⠀⠀⠸⣨⣻⣴⠇⠀⠀⠀⠘⣧⣝⣉⡇
⠀⠀⠀⠀⠞⠉⠉⠀⠀⠀⠀⠉⠉⠆⠀⠀⠀⠀⠀⠀⠀⠸⠁⠁⠀⠀⠀⠀⠀⠁⠹⠀⠀⠀⠀⠀⠀⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠱
⠀⢿⣿⣶⣦⢀⠀⠀⠀⠀⢀⠀⡠⣴⣾⣿⡇⠀⠰⣄⣀⣶⣶⢤⡀⠀⠀⠀⣠⢴⣶⣆⣀⡶
⠀⠈⣿⡯⠨⠈⠳⣄⠀⠀⡠⠋⠩⠁⣿⡏⠀⠀⠀⠘⢷⣮⡁⠀⠈⣶⣷⡎⠀⠀⣩⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⢸⣿⣀⣈⣀⣈⣶⣞⣁⣊⣀⣰⣿⠃⠀⠀⠀⠀⠈⢿⡿⠛⣋⣹⣿⣉⡛⠻⣿⠏⠀⠀⠀⠀⠳⣦⠀⠀⠀⠀⢺⣿⡗⠂⠀⠀⠀⡰⡾
⠀⠀⠀⣿⠁⠈⣉⠉⣿⣏⢉⡉⠀⠹⣯⠀⠀⠀⠀⠀⠀⠸⣇⠀⢉⡉⠃⣉⠁⢠⡿⠀⠀⠀⠀⠀⠀⠈⠀⡤⡖⠉⠘⣿⡇⠈⢐⣤⡄⠋
⠀⠀⠀⢿⣥⣀⡀⠘⠙⠙⡀⣁⣡⣼⡟⠀⠀⠀⠀⠀⠀⠀⣸⠲⠚⠁⠀⠙⠲⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠿⠄⠀⠀⠄
⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⠀⠀⠀⣾⣷⠶⣄⡀⠀⠀⠀⠀⠀⠀⣀⠴⣾⣿
⠀⠴⠀⠶⣤⢀⠀⠠⢴⡤⠀⢀⡀⡴⠖⠰⣄⠀⠀⡏⠁⠂⠀⠉⠢⣠⣦⡠⠚⠁⠀⠂⠉⡇⠀⠀⠹⡶⠻⠹⠊⠶⡄⠀⢀⡶⠂⠏⠝⠳⡏
⠀⠻⣷⡀⠀⢀⣠⠔⢻⡃⢤⣄⡀⠀⡐⡾⠋⠀⠀⠑⠒⣶⠶⠦⠤⣾⣿⡦⠤⠴⢶⠒⠒⠁⠀⠀⠀⠷⣀⣀⣀⣀⣸⣶⣏⣀⣀⣀⣀⠸⠃
⠀⠀⠈⢃⠈⠉⠀⠀⣼⡄⠀⠈⠉⠀⠛⠁⠀⠀⠀⠀⠘⠁⢀⠄⢀⡿⡿⣇⠀⢤⠀⠙⠀⠀⠀⠀⠀⢰⡏⠀⠈⡀⠘⣿⡧⠀⡉⠀⢘⣦
⠀⠀⠀⠈⠢⢀⠠⠜⠁⠈⠢⢀⠀⠘⠀⠀⠀⠀⠀⠀⣀⡦⠀⠠⠃⠀⠀⠈⠣⠀⠠⣆⡀⠀⠀⠀⠀⠸⣇⡀⠀⠀⠋⠹⠙⠀⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠈⠛⠛⠁
⢀⣤⠂⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⣄⠀⠀⢰⣿⣿⠳⢄⠀⠀⠀⠀⢀⠴⢻⣿⣷⠀⠀⠀⢾⣷⣶⣤⣀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣷
⠀⢿⠂⠀⠙⢦⡀⠀⠀⠀⠀⢠⠞⠁⠀⢘⠟⠀⠀⠘⣿⡏⠂⠀⠐⠀⠀⠀⠀⠀⠊⣿⡟⠀⠀⠀⠘⣿⣿⠷⣟⣳⣄⠀⢠⣖⣻⡾⣻⣿⡏
⠀⠀⡛⢐⠢⠀⠈⠀⢰⡄⠔⠁⠠⠐⠘⠋⠀⠀⠀⠀⠹⢿⡥⠄⠐⢶⣷⠖⠀⠤⣽⠟⠁⠀⠀⠀⠀⠘⣿⣷⡉⣿⣿⣿⣿⣿⣉⣴⣿⠏
⠀⠀⠈⠂⢠⡤⠀⢠⣾⣧⡄⠠⣤⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠾⡟⠆⠀⠂⠈⣷⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣼⣿⣧⡀⢸⣿⡅
⠀⠀⠀⢰⠏⠀⠀⠀⢿⡇⠀⠀⠀⢵⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⡾⠀⠁⠸⣦⣤⣼⠟⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣟⠉⢹⣿⣿⡿⠃
⠀⠀⠀⠈⠃⠼⠄⠀⠀⠀⠀⠤⠅⠂⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⢩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠁⠀⠀⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁

⢰⣶⡶⠖⠛⠛⠒⠒⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠒⠒⠛⠓⠶⣶⣶
⣿⡋⡉⠀⠀⠀⠀⠔⠂⠀⠀⠈⠙⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠒⠄⠀⠀⠀⠈⡉⣻
⢻⣀⡀⠠⠄⠐⠀⠂⠈⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠤⠀⢀⣸
⠀⠛⢦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⢠⣴⣤⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠞⠃
⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⣀⠠⠀⠀⠉⠐⠀⠤⢀⣀⣙⣷⣴⣿⣷⣴⣟⣁⣀⠠⠄⠀⠈⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⢶⡖⠒⠉⠁⠀⠀⠀⠀⢀⣠⡤⠖⠚⠛⠛⠻⣻⣿⣿⠛⠏⠛⠛⠒⠦⣤⣀⠀⠀⠀⠀⠀⠉⠑⠒⣶⠎
⠀⠀⠀⠀⠀⠀⠳⣦⣀⠀⠀⢀⣠⡶⠋⠁⡀⠄⠤⠀⠀⢀⡞⣿⡟⣆⠀⠀⠠⠄⠄⠀⠉⠳⣦⣀⠀⠀⢀⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡟⠉⠀⠀⠀⠀⠄⠀⠀⢀⣾⠐⣿⡇⢸⣆⠀⠀⠀⠄⠀⠀⠀⠀⠙⢿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠃⠀⠠⠂⠀⠀⠀⠄⢸⣿⠀⢿⠇⢸⣿⠁⠄⠀⠀⠀⠢⠀⠀⠱⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⡠⠁⢀⠀⠀⠀⠀⠀⢻⠀⠘⠀⢸⠇⠀⠀⠀⠀⠀⠀⠱⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⡐⠀⢀⠂⠀⠀⠀⠀⠀⠈⡇⠀⠀⡏⠀⠀⠀⠀⠀⠀⢣⠀⠐⡄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢀⠆⠀⠀⠀⡆⠀⢀⠀⣃⠀⢀⡇⠀⠀⠀⠆⠀⠀⠀⢂⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡜⠀⠰⠀⠀⠆⠀⢸⣠⡏⠀⠈⣧⣸⠀⠀⠂⠀⠘⡀⠈⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⡇⠀⢠⠀⢀⣼⠋⠀⠀⠀⠈⢻⣄⠀⢰⠀⠀⢇⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠿⢾⡿⠛⠛⠀⠀⠀⠀⠀⠘⠛⠻⣿⠾⢿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀⣀⠀⠀⣠⣶⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠀⠀⢹⣿⣿⣿⣿⣿⠋⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣿⣿⣟⠉⣹⣿⠋⠁⣰⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠏⣠⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠿⠿⣿⣿⡿⠟⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠀⡀⠀⠀⣀⣠⣤⣴⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣏⣠⣄⣀⣤⣿⣿⣿⣿⣿⣿⣛⢛⣿⣛⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⡡⠿⢿⡇⣸⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣿⣿⣿⣽⠛⠛⠋⠉⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀


⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠻⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢀⣜⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠔⣠⣤⠊⣠⠀⠀⣨⡕⢶⣤⠤⣤⠣⠁⢨⣄⡀⢀⣀⣀⣀⣀⡄⢀⣀
⠀⠀⠀⠀⠈⢢⡉⠠⠈⠻⣄⠀⠙⢆⠀⠐⡄⠈⠉⠓⠀⠀⠀⠈⢰⢧⠀⠈⠀⠐⠈⠒⠀⡀⠀⠈⠁⠉⢸
⠀⠀⠀⠀⠀⠂⢀⣀⠈⠉⠀⠀⠁⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠸⢶⡦
⠀⠀⠀⠀⠀⠀⢸⠐⠀⢀⡄⢃⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠁⠀⢀⡀⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠬⠥⠴⠤⠤⠴⠧⠤⠴⠾⠄⠀⣶⣾⡄⠀⠀⠤⠔⠒⡲⢀⠾⣆⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠅⠀⠀⠀⠔⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⢦⣼⢳⡀⠀⠀⠀⢀⡴⠋⠁⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡙⣸⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡴⠲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡇⣈⠵⠃
⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠈⢧⣡⠃⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠉⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣯⠤⠤⠤⠤⠤⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣺⣿
⠀⠀⠀⠀⠀⠀⢸⠛⢦⣄⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⢀⡀⢀⡀⢒⡒⣒⠂⠀⣀⡴⠚⠁⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠳⢦⡀⠀⠀⣤⣀⡄⠀⠀⠈⠯⠟⠀⠘⢭⢟⡤⠞⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠶⣄⠈⠳⠁⠀⠢⠄⠀⠀⢄⡠⠖⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⣠⠴⠶⢤⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡟⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣙⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣟⣛⣟⣛⣻⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⡭⡄⠀⢸⣿⣿⠇⠀⠁⠈⠁⡰⣤⣀⢈⢿⣿⣿⡄⠀⠀⠀⠀⠒⠦⣀⠀⣤⠃⠼⠄⠠⡀⠀⡋⠀⠀⠀⢠⠏⣸⠏⠀⡿⢁⠀⣹
⣿⣿⢰⠅⠠⣿⡈⣿⠀⠀⠀⠀⠀⢀⠨⡿⣧⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠑⢿⡞⠀⠀⠈⣿⢀⡧⠠⠀⠀⣾⡐⣼⠄⢣⡦⡴⠚⢯
⣿⣿⠷⠂⠀⣿⢡⡏⠀⠀⠀⠀⠀⡀⠁⠐⢬⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣼⠟⠒⢶⡾⣿⡾⣠⠠⠤⠄⣿⣿⣿⡄⢢⡵⡖⢀⡄
⣿⣿⣿⣿⣦⣿⣿⣷⠄⠒⠀⠀⠀⠙⢦⡀⣤⣾⠛⣷⣟⠻⣇⠀⠀⠀⠀⣰⡾⣏⠁⠀⠈⢁⣟⣟⣿⡟⠋⠉⢻⣿⣟⠙⡆⢱⡧⠙⠑
⣿⣯⡀⠀⠀⢸⠶⠀⠀⠀⠀⠀⠀⠀⠀⡙⠚⠃⠀⠘⣿⣄⣿⣿⣦⡀⠀⣠⣋⠀⠀⣀⠙⢧⣓⣮⣿⣷⣄⠤⣬⣯⣿⣶⡇⢀⠗⢠⣿
⣿⣿⣿⡥⠭⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣴⣄⣤⠙⠛⢿⣿⣿⣷⣄⠀⠹⣞⠀⠈⠀⠼⣵⢹⣯⣿⣍⣙⣿⠁⣰⣿⠡⢸⡆⢸⣿
⣿⣿⣾⡧⠀⢸⣶⣶⣶⣶⣶⡶⠶⠶⣶⣾⣿⣿⣿⣷⡄⢸⣿⣿⣿⣿⠿⠿⣶⣾⣿⠅⠰⣿⡟⢀⠛⠉⠉⠉⡁⠙⢾⣿⣏⢸⡄⠀⢸
⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⡷⠔⠚⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⠐⠶⢶⣿⣿⣇⣦⡿⢧⡟⢀⣔⠒⢦⣴⡷⠀⢻⣿⣴⣥⠀⠸
⣿⣿⣿⡇⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⣟⣛⢿⣿⣿⣿⣿⣭⣴⣶⠞⠛⡿⣿⣍⢿⡟⠀⣾⡛⣷⣆⠙⠀⠀⠘⣿⣻⣄
⣿⣿⣿⡏⠀⢸⣿⣿⣿⡄⡇⠀⠀⠀⠀⠆⠀⢸⡿⣿⣿⣯⣿⡿⢿⡟⠛⠛⠀⣤⠀⣮⣿⡌⠃⢹⡟⢹⣿⠉⠛⠆⠀⠀⠘⣿⣿⣄⣤
⣿⣿⣿⡇⠀⢘⣿⢿⠏⠣⠃⠀⠈⠁⢰⣾⣎⢸⡆⢸⣟⣿⠻⠇⡤⠖⡇⠀⠠⡧⠈⠳⠡⡇⢹⣸⢀⠘⠋⠀⠀⣀⠤⠙⣿⡿⣿⣷⣾
⣿⣿⣿⡷⠀⣈⣿⡷⠀⡇⡤⠀⢀⣄⣼⠍⠙⠛⢳⣶⣿⣿⡶⢰⢳⢼⣳⣴⡀⠐⡖⡀⠀⠳⣼⠟⠢⠶⠶⠞⢏⠙⣶⣤⢹⣿⣿⣼⡇
⣿⣿⣿⡏⠁⢸⠏⠄⠀⡷⡃⠀⠀⠈⣁⣠⢰⣾⡛⠉⠛⣿⣧⣄⣡⣼⣿⡿⣽⣶⣿⠛⠁⢠⣄⡀⠀⠐⠄⢧⠈⠰⠃⠀⣸⣿⣯⡰⠄
⣸⣿⣿⢃⠀⠀⣀⠀⣆⢸⠈⠑⠄⠌⠁⢹⡏⣿⣶⡦⣛⣛⣻⣿⢿⡿⠿⠿⠿⠿⢣⣠⣴⣾⠻⡇⠀⠀⠘⢿⣧⠀⠀⢼⣿⣿⣿⣿⡄
⣿⣿⣳⣻⠴⠊⠀⡀⠠⣾⣦⠀⡐⠀⠊⠿⡇⠸⣿⣿⣯⣍⣟⡁⠀⣈⠗⠀⣀⡴⢾⣿⠙⢿⠀⡇⢠⠀⠀⠀⠀⠂⣀⡌⣿⣿⣿⡿⣇
⣿⡏⠀⠀⠀⠀⣦⠁⠀⠈⠛⠀⡟⠻⡿⣿⣿⠀⢹⡟⢶⣟⣿⠟⣻⣷⣴⣟⣻⣷⢸⡟⠀⣿⠀⠹⡄⠀⠐⠀⣦⡞⡏⣸⣿⢸⣿⡿⠟
⠋⠀⢀⣠⣴⠶⠛⠀⢀⣸⣿⠀⣧⢀⡎⣿⣿⣇⠀⣿⣻⡟⠉⠀⠀⠀⠈⠹⣿⡇⢸⡇⠀⢛⠐⣯⣷⠀⢠⣴⠃⠀⠀⣿⣿⠨⣿⣄⣰
⣶⢋⠭⣿⢁⡄⠀⡀⠀⠈⢿⣄⢹⣿⣿⡇⣻⣿⡄⢹⣿⢅⡀⠀⠀⡀⢀⡀⣹⣾⣾⠀⠀⣸⡆⢸⣻⣧⠈⠁⠀⠀⢠⣿⣿⢠⣿⣿⣿
⠀⠀⠀⠰⠄⠃⡀⠀⠀⢳⣌⣉⣩⣁⠈⠹⠉⣿⡿⠘⡃⠈⠁⠀⢘⢹⣆⠛⣮⡯⡿⢀⣼⣿⣧⠀⠿⢿⣆⠀⠁⠀⣸⣯⣻⢲⣿⣿⣿
⣤⡇⣦⢀⣀⠀⠀⠀⣀⣾⠉⠛⢻⣿⣟⡃⢸⣾⢃⡄⡿⢠⡀⠀⡾⠀⠿⣿⣻⣿⣷⣿⣿⡟⣿⡄⠐⣿⣿⡄⠀⢠⣿⣿⣿⠘⣽⣿
⣿⠷⢤⣴⢡⠈⡛⠀⣀⣠⡀⠀⠀⢻⣏⠷⡮⠏⢻⣿⡃⢀⡿⠛⠳⠶⠶⠿⠿⢋⣋⣻⣿⡇⠿⡁⠀⣿⡿⣷⡀⠸⣿⣿⣿⣼⣿⣯
⠶⠚⠋⠉⢭⣉⠙⠓⠶⠀⡙⠂⠀⣈⢩⣿⣿⠀⠀⢹⡤⠛⠀⠀⠀⠀⠰⠀⠀⠀⠁⠉⣰⣷⠀⠀⣰⠏⠀⢻⡇⠀⣬⣾⣿⣵⣿⣧
⠷⠄⣀⣶⢶⠈⢧⡀⠀⠐⢸⣷⡶⠀⠀⢈⡿⣣⣤⣸⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠙⢻⣇⠰⡟⠀⠀⠀⣿⡈⢹⣿⣿⡏⣿⡏

⣰⠋⡻⠻⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠟⢟⠙
⣿⣷⡿⣖⣿⠿⣆⡙⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢋⣰⠿⣿⣲⢿⣾
⠻⣿⣍⣺⣿⣷⣾⠟⠛⠉⠃⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠘⠉⠛⠻⣷⣾⣿⣗⣩⣿
⠀⠹⣿⣿⣿⣿⠍⢑⣒⣤⣤⠾⣦⣟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣻⣴⠷⣤⣤⣖⡊⠩⣿⣿⣿⣿⡏
⠀⠀⢹⣿⡏⣙⣶⣶⣾⣿⣿⣶⣾⡿⠷⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠾⢿⣷⣶⣿⣿⣷⣶⣶⣋⢹⣿⡏
⠀⠀⠀⢻⣿⡍⣽⣿⠿⠛⠁⠀⠈⢧⡀⠈⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⡼⠁⠀⠈⠛⠿⣿⣯⢩⣿⡟
⠀⠀⠀⠸⣿⣶⡿⣇⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠹⣿⣧⡀⢀⡀⢀⣼⣿⠏⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣸⢿⣶⣿⠇
⠀⠀⠀⠀⠹⣄⣷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⣠⠏
⠀⠀⠀⠀⠐⣿⣿⣄⣀⣀⣀⣀⣀⣤⣤⣤⣀⣀⣀⣥⣤⠿⣿⣿⠿⣤⣬⣀⣀⣀⣤⣤⣤⣀⣀⣀⣀⣀⣠⣿⣿⠂
⠀⠀⠀⠀⠀⠈⠛⠛⣻⡿⢿⣟⡛⠛⠿⢛⡿⠟⠉⠁⠀⠀⣿⣿⠀⠀⠈⠉⠻⢿⡛⠿⠛⢛⣻⡿⢿⣟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡰⣿⠖⠊⠈⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠹⠦⠀⠀⠁⠑⠲⣿⢆
⠀⠀⠀⠀⠀⠀⠀⡟⢹⣡⠄⠀⠀⠀⠼⠓⠒⠂⠀⠀⠀⠀⣿⡯⠀⠀⠀⠀⠐⠒⠺⠧⠀⠀⠀⠠⣬⡏⢻
⠀⠀⠀⠀⠀⠀⠀⢻⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣼⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠙⠃⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠗⢦⡆⠀⠀⢠⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⡄⠀⠀⢰⡴⢺⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣟⣿⣖⣻⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣟⣲⣿⣻⡽⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣍⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⣹⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠆⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⢹⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣯⣤⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢰⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣤⣴⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⢈⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣷⡄⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠋⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⢸⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⡀⠀⣿⡏⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠛⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡀⠉⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢷⣦⣀⣠⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠻⣁⣉⣉⡠⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢹⣷⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⢀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡈⠳⣦⣴⣦⣄⠀⠀⠀⠉⠛⠛⠋⠉⢁⣴⣤⣀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠡⠝⢿⣿⣿⣿⣦⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠈⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡀⢠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⡿⠁⠐

⠀⢠⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡄
⠀⠀⢻⣧⣏⡗⠚⠷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡿⠟⠲⣏⣯⣿⠁
⠀⠀⠈⣽⢤⣿⣋⣈⣽⡇⢨⣿⣷⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣽⣯⠰⣿⣉⣉⣹⣥⣿⡇
⠀⠀⠀⠹⣯⣼⣯⣀⣨⣛⡧⠤⡿⠷⠮⢷⣄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⣠⣴⠯⠶⠿⡯⠴⣟⡯⣀⣨⣟⣼⡟
⠀⠀⠀⠀⠘⢿⣾⡉⡱⡗⡀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⢀⠀⠀⢀⣼⣿⡉⣳⣿⠏
⠀⠀⠀⠀⠀⠠⡿⣫⢽⣿⠛⠂⠀⠉⠒⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⠔⠋⠁⠀⠛⢾⣻⡽⣧⡟
⠀⠀⠀⠀⠀⠀⢹⣿⣤⢷⣀⠀⣀⣀⠤⠴⠦⠄⠀⠀⠈⠳⡀⢀⣀⢀⡴⠋⠀⠀⡀⠠⠶⠤⠄⢀⡀⢀⣸⢧⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠘⣾⣿⠎⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⣤⣶⣶⣶⣶⠒⠒⠤⣤⢤⣼⣿⣿⣷⣤⣤⠀⢀⣶⣶⣶⣶⣶⣤⣤⣴⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠀⠀⠀⠀⠉⠁⠀⠀⠀⣠⠞⢿⣿⡟⢄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡽⠓⠂⠀⠀⠉⠀⠀⣀⡤⠊⠁⠄⢸⣿⠀⡀⠙⠢⣀⠀⠀⠈⠀⠀⠀⠒⠻⣜⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣦⣻⣄⣀⠀⠰⢊⠠⠒⠋⠁⠀⠀⠜⠀⢹⣿⢠⠘⠄⠀⠀⠉⠒⠦⠘⠲⠀⣀⣀⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡅⢀⡔⠁⠀⠀⠀⠀⠀⠀⣤⠃⠸⡟⠈⢷⡀⢀⠀⠀⠀⠀⠀⠱⣄⠀⣵⣹⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡿⣟⣿⣄⣀⣠⡆⠀⡰⠁⣰⠃⠀⠀⠀⠀⠈⢧⠀⢣⡀⠈⣦⣀⣀⣼⣛⡟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣽⣻⣽⣿⣷⣺⣇⡴⠃⠀⠀⠀⠀⠀⠀⠈⠳⣤⣿⣲⣯⣿⣽⣿⠽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡀⠈⡂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠂⠀⠀⠂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠄⠛⠈⠐⢂⣀⣼⠯⣼⡀⠀⠌⠁⠀⣸⠆⠐⠴⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢧⠄⣤⠀⠐⠎⠀⠀⠳⠀⠈⡁⠀⢀⡴⠔⠀⠈⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣍⠀⠀⠛⠀⠀⢰⠂⢂⠩⡀⠀⠠⠀⠀⠰⠉⠂⢀⢓⠀⠀⠀⠀⠀⡁⠀⠀⠐⠘⠳⠦⠄⢀⠔⡀⠀⣠⠀⠀⢀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠰⠄⢊⠀⠂⠀⡇⠂⠁⣶⣀⠀⠠⠀⠀⠀⢠⠀⡀⠀⠐⠒⢦⣀⠀⠀⢀⡀⠀⢀⣀⡀⠼⢇⣀⡀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⣠⣄⠈⠀⠀⣰⣾⠉⠀⠐⠠⡄⠦⠀⠀⠀⡠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢸⡆⠈⡁⠁⠈⡠⠊⠁⠈⠑⠀⠀⢀⢁⡀⠋⠈⠩⠁⠀⠀⠥⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠂⠀⠀⠀⠅⡆⠀⠀⠀⠥⠀⠀⠄⠄⠀⠀⢤⡀⠠⠀⠀⠀⠀⠁⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⣠⢀⠀⣡⡈⠀⠀⠀⢀⠀⣀⠀⠃⢀⠀⠀⣰⣀⠀⢀⡀⠙⠆⠀⠀⠀⠀⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⠆⠈⠋⣃⠉⠀⡀⢰⡟⠀⠁⠀⢀⠍⡀⠊⠁⠀⠀⠚⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⠀⠀⠀⢭⡀⠰⠛⠸⠇⠀⠀⠀⠉⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡖⠡⣄⠠⠀⠙⡠⠀⠀⠈⠪⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⡄⠐⢶⠀⠱⠆⢸⠂⠀⠀⠉⢈⡁⠂⣴⠆⠀⠀⠐⠲⡈⠀⠐⡊⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠤⣁⣀⣀⣀⠀⠀⠀⠀⢧⡀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠉⠙⠛⡦⠴⠖⠒⠋⠀⠀⠀⢠⠀⠀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⡀⢉⠴⠡⠄⠀⡠⠀⠑⠄⠀⡀⠀⠈⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠄⠀⠀⠀⡤⠀⢀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠿⠀⠄⠀⡁⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠈⡄⠁⠀⠄⢦⡰⠀⠙⠂⠉⠀⠴⢠⡀⣀⠀⠚⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣈⠀⢢⡀⠀⠀⡀⣘⡄⠀⠀⠀⠀⠀⢆⡇⠀⡀⠀⡀⠸⠋⠶⠆⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⡐⠋⡉⠉⠐⠉⠑⢳⣦⣴⣤⣚⡉⠀⠀⢈⡁⢆⠀⠀⠀⡅⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠈⠋⠔⡲⠦⠄⢸⣇⢧⠙⡇⠐⠀⠉⠀⡈⡗⢀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠑⠲⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠦⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠚⠙⡆
⠀⢀⡤⢄⣾⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣤⠴⠊⠁
⠀⣏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠖⠒⠒⠒⠛⠳⢤⡀
⠀⠘⠦⠤⠤⣴⠇⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠖⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⢀⣠⡤⠤⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⡷
⠀⠀⠀⠀⠀⣀⣉⣳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠃
⠻⣏⠉⠉⠉⠉⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢸
⠀⠈⢳⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢦⣀⣀⡀
⠀⠀⠀⢳⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡠⠚⠉⢧
⠀⠀⠀⢸⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤⠊⠀⠀⠀⢸⡀
⠀⢀⣠⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠸⡇
⢰⠋⠁⠈⠁⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠸⣆⠁⠀⠀⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣄⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⠶⠛⠓⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠶⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣰⡥⠤⠖⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠁⡴⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣤⠙⠋⠋⣴⢾⡠⠼⠐⡀⠀⠀⢠⡄⠒⠁⡇⠀⢹⢨⡟⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠑⠰⠀⣀⠀⠀⢿⣷⣮⣻⣖⢶⣀⠂⣸⠁⢀⠰⠁⠀⣼⢾⡅⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢢⠖⠋⠁⠀⠀⠀⠉⠳⡝⢿⣷⣯⣶⣿⣤⣎⣤⣆⡤⢖⣘⣾⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣺⣿⠿⠿⠛⣉⠉⠉⠁⠀⠀⠀⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⡄⣄⡀⠀⢀⡠⠰⢸⣿⡟⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⣀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⡇⠁⠚⠣⣀⠜⠣⠀⣹⡿⠃⠀⠀⠀⠀⣡⠀⠀⠂⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⡆⣀⠀⠀⠀⢀⠸⡿⣷⡆⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠃⣟⠀⠀⠀⢸⠀⠗⣿⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡞⢘⣉⢈⠉⠉⢈⡁⠙⣿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⡇⠉⠁⡑⠊⠓⠉⢺⠉⠀⣿⣸⣧⠀⠀⢠⠀⠀⠀⠀⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠸⠁⠀⠀⡀⠀⠀⠀⢸⠀⠀⠸⣿⠈⠒⣄⠏⠀⠀⠀⠀⠀⠀⡸⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢃⠀⠀⠘⠂⠀⠸⣈⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⢀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠑⠒⠒⠄⠀⢀⣀⠀⠀⠀⣀⣀⡀⠀⠄⠒⠒⠉⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡋⠀⠀⠀⣀⡤⠤⠤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠈⠑⢦⡀⠀⠀⢀⡤⠚⠁⠀⠙⢦⠔⠋⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠳⠔⠋⠀⠀⠀⠀⣰⠃⢠⠴⠒⠲⢄⡤⠖⠒⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠳⡄⠀⠀⠀⠀⠀⠀⣠⠞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣴⢋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣣⠀⠀⠈⠢⣀⠀⠀⣠⠞⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄⠀⠀⠈⠳⠚⠁⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠤⠤⠴⠖⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⣄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⢿⡟⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟⢁⣀⡀⠀⠀⠀⠀⢀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⣼⣿⣿⣿⣿⡍⣡⣶⣿⣿⣿⡆⠀⠀⠔⠁⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⠇⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢻⣿⣿⣿⣿⣿⣿⡏⢉⣡⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠟⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢦⣬⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣏⠀⢹⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⡿⠛⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⢰⣦⠀⡟⢦⣠⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⣤⡭⠧⣧⡚⠁⠀⢹⠁⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠲⣦⣄⡀⢇⠀⢰⠍⠀⠀⠀⠙⢆⠀⠀⠀⠀⢨⡦⡴⢤⣀⣀⢀⣀⠤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠓⠦⣀⠀⠈⢇⠈⢻⣄⣸⣆⠀⠀⠀⠀⠈⠀⣄⣆⣠⠋⠀⠀⠀⠋⠘⢿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⣾⣧⡴⠚⠿⣿⣿⡿⠿⣄⣖⠙⠢⣄⠀⠉⠛⢿⣦⡀⣠⠖⠛⠛⠻⣿⠓⠒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⡿⠁⠀⣠⠿⣯⠄⠀⠘⠫⠿⢶⢤⣿⢿⣇⠀⣠⣄⣠⣾⣥⠴⠚⠋⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠛⢷⡀⠀⠀⠇⠀⠞⠁⠀⡼⠷⢖⣰⣋⠙⢻⡀⠁⠀⠈⣾⠃⠀⠈⠛⠖⣤⣴⡊⠉⣻⠟⠉
⠀⠀⠀⠀⠀⢀⣠⣤⠒⣾⠀⠀⠀⠀⠀⠁⠐⢓⡀⠀⣀⡤⠾⢧⠀⣜⠉⠉⠉⢢⠱⣄⠀⢀⡎⠀⠠⠄⠀⠈⠉⠛⣇⠴⠁
⠀⠀⠀⠀⠀⠈⠉⢉⣲⣿⣄⣀⡠⠲⡤⣥⣚⣛⣉⠵⠿⠀⠀⣰⠋⣹⣤⠶⣦⣸⣶⠿⣶⣿⣦⠀⠀⣀⣠⣤⣤⣴⣧⠤⣀⡀
⠀⠀⠀⢀⣤⣴⣞⣁⠬⠿⠛⠛⢣⡄⡠⠋⠁⠀⢀⠀⠀⢀⡾⠁⠀⠙⢦⣴⠄⠌⠃⠲⣶⣾⠻⡏⠉⠁⠀⠀⠹⣿⣏⣀⣀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠔⠋⠉⠓⠋⠀⠀⠀⠀⠀⠙⠶⢦⣖⡆⠄⣅⣸⡧⠀⢀⣀⠀⠀⠀⣽⣿⣿⣋⠉⠉⠉⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠾⠿⢤⡀⠀⢘⡇⠀⠀⣿⣿⣯⠛⠻⣕⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠧⠴⣾⣿⣿⣷⡀⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠌⢿⡌⠙⠒⠒⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⢘⡔⢁⠀⠀⡠⠃⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣾⣛⠀⠈⠢
⠀⠀⠀⠀⢀⡈⠀⡜⢀⠴⠡⣚⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠂⠄⠐⠖⠋⠀⠘⣿⠋
⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠏⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡗
⣶⣤⣾⠂⠀⠒⡄
⠉⢻⡿⠃
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠤⣉⠊⡆
⣀⣴⣄⣠⠔⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣁
⠀⣽⣿⡏⠁⠐⠄⠀⠀⡠⠊⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢀⣷⣶⡾⠆
⠴⠋⠀⠀⠀⢀⡁⢀⣌⠀⢐⠀⠀⠠⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠟
⠀⠀⠀⠀⡠⠜⢻⣿⡇⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⢀⣴⣶⣶⣦⣄⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋
⠀⠀⣴⣿⣿⣿⣿⣦
⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⡀
⠀⠀⠰⢻⣿⣿⣿⣿⣿⣿⠻⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣼⡿⢈⣿⣿⣿⣿⠀⠀⠙⠛⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⠈⠛⢻⣿⣿⣿⡇
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡄
⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⣽⣿⣷⡀⠀⠀⣼⣿⣿⣿⣿⣿⣷
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣄⣴⣿⣿⣿⣿⣿⣿⣿
⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⡏⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⡿⢁⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿
⠀⠀⠀⢻⣿⡇⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⠀⠀⠀⠀⣼⣿⣿⣿⡿
⠀⠀⠀⠈⣿⡇⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣠⣤⣤
⠀⠀⠀⢀⣿⡇⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣥⣤⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⢿⣿⡀⠘⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⡄⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢻⣿
⠀⠀⠀⠘⠈⠿⠷⠂⠀⠻⠿⠆⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠛⠻⠛⠃⠀⠀⠘⠿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠟⠛⠛⣍⠉⢹⡏⠉⣩⠟⠛⠻⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠉⠀⠀⠀⠀⠀⢈⣻⣾⣷⣞⣁⣀⣀⡀⠀⠀⠉⠻⢿⣷⣄
⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⣀⣴⣶⣮⣉⠉⣹⢿⣿⣿⡛⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣿⣆
⠀⠀⠀⢀⣾⡿⠋⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣦⣸⡇⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⢠⣿⡟⠁⠀⢀⣴⣿⣿⠋⠁⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⢠⣿⡟⠀⠀⣰⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣆⠀⠀⠈⠳⠄⢀⡀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⣼⣿⠁⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⠀⠀⠀⠀⠋⢹⣦⣿⣶⣄⠀⠀⠈⣿⣧
⢠⣿⣏⣴⡾⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⡇⢙⣿⣿⣷⡀⠀⢹⣿
⢸⣿⣿⠏⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣶⣦⡀⢸⣿⣾⣿⣿⣿⣷⠀⢸⣿
⢸⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠉⣿⣿⣿⣧⢻⣿⣿⣿⣿⠀⢸⣿
⠈⣿⣷⠀⠀⢸⣿⡇⠀⣠⡾⠛⠛⠛⠒⠒⠒⠶⣦⠀⠀⢸⣿⡇⠀⢀⣼⣿⣿⣿⡄⢀⣿⣿⣿⣿⠀⣼⣿
⠀⢻⣿⡄⠀⢸⣿⡇⠀⠹⣿⣿⣷⣶⣶⣶⣶⣿⡿⠀⠀⢸⣿⡇⠐⢿⡿⢻⣿⣿⡇⣾⣿⣿⣿⡏⢠⣿⡟
⠀⠈⢿⣷⡀⢸⣿⡇⠀⠀⠘⣿⠛⠛⠛⠛⢻⡟⠀⠀⠀⢸⣿⡇⠀⠈⣡⣿⣿⣿⣷⣿⣿⣿⣿⣧⣾⡿⠁
⠀⠀⠘⢿⣿⣾⣿⣿⣆⣀⣼⣛⣧⣀⣀⣀⣟⣻⣄⣀⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣶⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣛⣛⣛⣛⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠛⠿⠿⣿⣿⣿⡟⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣄⡈⠛⢿⣷⠀⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠉⠉⠉⠙⠻⠆⠀⠈⠀⠈⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⢷⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣧⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣹⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⢿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⡀⠈⠳⣦⡀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢰⣆⠀⠻⣦⡀⠙⣿⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⣦⠀⢻⣿⣄⢘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⣿⣧⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠙⢏⠛⠃⢤⠀⠀⠉⠉⠳⣄⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⠀⢄⣠⣄⡈⢦⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢺⠝⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢦⣀⣃⠀⠉⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣿⠞⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣨⠟⠒⢀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠘⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠗⠒⠛⠉⠀⠈⠲⢝⢢⡀⠀⡀⠀⠀⣀⠀⡼⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⡑⡾⠁⠀⠀⠈⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠂⠛⠐⠒⠢⢄⣀⠀⠀⠙⣜⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠘⣞⡄⠐⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏
⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⢀⠀⠀⠀⢳⡀⠘⢂⡔⠓⠂⠀⠀⠀⠀⠠⠶⠊⠁⠀⣷
⠀⠀⠈⠑⠤⣄⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⡀⢳⠀⣄⢹⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡇⢰⠸⠀⠀⣠⠚⠁⠀⡀⡠⠊⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢲⣄⣀⠠⣄⡰⠖⠛⠁⠈⠛⢦⡀⡆⢰⠃⠀⢀⢀⠟⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡤⠧⢤⣿⠀⠀⠈⠿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⠚⠛⠒⠦⣄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡦⣾⠤⠖⠛⠛⠛⠲⣄⠀⣀⣀⣀⡈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠃⠀⠀⢀⡀⠀⠀⠀⣼⢰⡖⣪⡅⠪⣽⢃⠀⢀⡤⠀⠀⠀⣤⣟
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣆⠀⣀⡇⠉⠓⣤⡾⢻⣌⠳⢭⣠⠞⠉⢧⠞⠁⠀⠀⠀⠀⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣨⠟⠋⠀⠀⣼⠋⠀⠀⠉⠳⣄⠟⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠤⠐⠒⠠⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠒⣒⢦⡄⠀⣠⡶⠶⠋
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⣦⡀⠀⠀⣀⣉⠤⠞⠁⣠⠎⠹⣄⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⡠⠞⢋⠟⠉⠀⠀⠀⣀⡾⠁⠀⠀⠀⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢤⣄⠀⠀⠀⠀⠰⢟⣁⢀⣠⢞⠟⠒⠒⠉⢩⠑⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣤⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⣉⡩⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠶⠚⠉⠂⠈⠙⠓⠛⠛⠲⠶⠴⠖⠚⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢠⠞⢻⣭⠭⢽⡓⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣚⡯⠭⣭⡟⠳⡄
⠀⠀⠀⠀⣿⠀⠻⠿⠁⣀⣈⠑⢄⠙⠻⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠿⠋⡠⠊⣁⣀⠀⠿⠟⠀⣿
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠻⠟⠁⠀⠑⣄⠈⠢⡑⢄⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠀⠀⡠⢊⠔⠁⣠⠊⠀⠈⠻⠟⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠹⡄⠀⣤⣤⠀⠀⠔⠋⠀⢱⣄⠈⢆⡑⣄⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣠⢊⡴⠁⣠⡎⠁⠉⠢⡀⠀⣤⣤⠀⢀⠎
⠀⠀⠀⠀⠀⠀⡇⠀⠛⠛⠀⡌⠀⣀⣠⠃⠈⠳⣜⠛⢌⠳⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⠜⡡⠛⣣⠞⢁⠘⣄⣀⠀⢡⠀⠙⠛⠀⢸
⠀⠀⠀⠀⠀⠀⢧⠀⠀⣴⣤⡅⠘⠿⠟⠀⠋⠀⠈⢳⣄⠱⣝⢄⠀⠀⠱⡀⠀⠀⠀⠀⢀⠎⠀⠀⡠⣋⠎⢠⡞⠁⠀⠙⠀⠹⠿⠃⢈⣤⣦⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠈⠢⣄⠙⠋⠁⢀⣤⣤⠀⠀⠀⡰⢅⣈⠳⣌⠁⢧⡀⠀⠱⠀⢀⡀⠀⠎⠀⢀⡼⠛⣩⠞⣁⡈⠆⠀⠀⠀⣠⣤⡀⠸⠙⠋⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⡘⡈⠛⠛⡀⠀⠀⡇⠀⠋⠙⠪⣑⠦⣙⢦⡀⣨⣿⣿⣅⢀⡴⣋⠴⣊⠔⠊⠙⠁⢸⠀⠀⢀⠛⠛⢃⢂⣀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠇⠀⡀⠀⠡⣾⣿⠧⣀⣉⠉⣁⡲⣵⣬⣓⢿⣿⢻⡟⣿⡿⣞⣥⣮⣴⣊⡉⣉⣀⠸⣿⣷⡌⠀⢀⠀⠸⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣄⡈⠄⠀⢣⣠⠤⠷⢒⠒⡉⠩⣿⣭⣍⣛⣿⣿⣿⣿⣛⣩⣭⡿⠍⢉⠐⡒⠾⠤⣤⡜⠀⠠⢃⣠⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡩⠟⠋⠁⣀⣠⣤⣈⡑⠾⢦⠶⢾⣿⣻⣿⣿⣿⣿⣿⣿⡷⢶⡴⠷⠊⣁⣤⣄⣀⠈⠙⠻⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠊⠍⣙⡽⢃⠠⢀⣶⢶⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⡶⢦⡀⠄⡘⢯⣋⠩⠙⢦⡀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⣺⢟⣪⠴⠛⠉⢀⣠⣶⣫⣽⠿⣻⡿⠃⣸⣿⣿⣇⠈⢿⣯⠻⣼⣝⡶⣄⡀⠉⠛⠦⣕⡻⣗⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠙⢉⠁⠀⠀⡀⠘⠋⢉⠔⠁⡼⢻⠁⣠⢿⣿⣿⡿⣆⠈⡛⢷⠈⠢⡉⠙⠓⢄⠀⠁⠈⡉⠋⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠔⠁⠀⢀⢸⣿⠀⣐⠁⢠⠟⠄⡄⢠⠇⢸⣽⣯⡇⠸⡄⢡⠠⠻⡄⠈⣂⠀⣿⡇⠀⠀⠈⠢⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⠁⢸⡏⣸⡿⠀⣟⡜⡐⢀⡎⠀⢸⢾⡷⡇⠀⠱⡀⢂⢳⣳⠀⢿⣧⢹⡇⠈⠄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣄⠂⠀⠈⢰⡿⠁⠀⠟⡰⠁⡸⠀⠀⠘⣿⣿⠇⠀⠀⢧⠈⢆⠻⠀⠈⢿⡆⠁⠀⠘⣀⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢦⡀⠀⠆⠀⠀⠀⡰⢁⣶⠇⠀⠀⠀⠛⠛⠀⠀⠀⠘⣶⡈⢆⠀⠀⠀⠐⠀⢀⡴⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠳⠴⡀⠀⣀⣠⡷⢋⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡙⢾⣄⣀⠀⢀⡦⠞⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠙⠊⠉⠉⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠉⠉⠑⠋⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⣠⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⡿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣰⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠙⠻⠿⠿⠿⣿⣿⠿⠿⠿⢿⣿⡿⠿⠻⠿⠟⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠢⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡠⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠀⢸⠀⡏⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢌⢇⠘⠀⠀⢀⣾⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡈⠓⠄⢀⠼⠚⠓⠒⠲⡟⠁⠠⠊⢀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡏⠀⢛⢀⠘⠃⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⢒⣒⠭⠛⠧⡀⠘⠚⠀⢀⡼⢍⠑⠒⠤⣄⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢀⠤⠀⠀⢨⣷⠒⠒⠁⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠴⠁⡰⠁⢀⣿⣿⠀⢰⠀⠳⡀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠻⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡿⠟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡸⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⠤⠒⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠳⢄⡀⢀⡰⠁⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡠⠊⢁⡠⠖⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠣⡀⠀⢀⡠⠴⠊⠁⠀⣠⠊⢀⠴⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⢀⠜⢁⡔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠣⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡠⠃⠴⠋⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠢⣄⠀⠳⡄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠁⠀⠈⠢⡀⠀⠀⠀⠘⡌⡄⠀⠀⠀⢠⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢦⡀⠀⠀⢃⢱⠀⠀⢠⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠢⣀⠸⡈⡆⢠⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡓⢧⠱⣎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢠⠂⢬⠁⡇⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⠀⢁⡠⠤⠆⠀⠀⠀⠩⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠌⠀⠀⢰⠊⢠⣲⢅⠀⠂⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⡏⠀⣯⣿⣧⡅⢀⣀⣄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⢣⡀⣛⢭⢏⢀⡮⠉⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡅⠀⡇⠀⢀⡔⡊⡙⢢⣠⠢⠀⡀⠀⠔⠡⠉⠑⣶⠛⠉⠳⡀⠀⣀⠂⣄⠀⡴⢐⡒⢦⡀⠀⠰⠀⣄⠐⠄
⠀⠀⠀⠀⠀⠀⡀⣨⢔⣻⠞⢸⠁⠀⣸⣿⣾⠁⠚⡃⠂⡂⠐⢀⢀⠀⠀⢾⣫⡶⢠⢱⠈⠆⠄⢨⡵⠀⠘⠂⢹⣾⣇⠀⠀⠆⢻⣃⡦⠑⣄
⠀⠀⠀⠀⠀⡈⣠⡾⣻⣿⡆⢀⡶⠛⠓⢏⡯⢠⡤⠃⢘⠀⠀⠀⠈⠡⢀⢀⣿⠠⠀⠐⠂⠀⠀⠀⡁⡃⣠⢄⠘⢿⠾⠞⢷⢃⢉⣌⡷⡈⠈⢢
⠀⠀⠀⠀⡴⢹⠷⣋⡼⣮⣿⡏⢀⡀⠀⢢⣀⠋⣇⠐⠍⣷⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⢡⡼⢁⡠⠤⡜⠀⢠⡄⢻⡼⣎⣛⡈⠰⠄⢆
⠀⠀⠀⢰⢠⡟⣟⣹⠟⡫⠟⠀⢸⡇⠀⠈⢋⣴⢾⡇⠀⣿⠀⠀⠀⠀⢀⡜⠘⣇⠀⠀⠀⠀⠀⣰⠶⠔⠋⣀⣜⠀⡄⢸⣷⢸⣿⠉⠳⣟⡓⠴⡈⡄
⠀⠀⢀⢣⠟⢚⠟⠁⠊⢠⣤⣦⡀⢿⢀⠀⡨⠝⠋⣠⡎⢿⠀⠀⠀⠀⣏⢸⠃⢸⡇⠀⠀⠀⠀⢁⣀⣠⣾⠿⠌⠓⣶⣿⠏⣘⣻⣝⢦⡈⠑⣦⠉⠹⡄
⠀⢀⢢⣮⠟⡡⣪⠜⣰⡿⠀⠘⢻⠷⡇⠀⠀⣠⢶⣻⣥⠞⠀⠀⠀⠀⠘⣄⣠⡟⠁⠀⠀⢀⣄⠛⢋⣉⣀⣤⣤⡇⡿⠛⠚⠋⢹⢹⡧⡛⣁⠈⠳⣧⠠
⠠⠗⠋⣁⢰⣄⢷⡼⡟⠀⠀⠰⡝⣴⢻⣸⣯⠟⠛⠉⣿⠀⠀⠀⠀⠀⠀⢾⣼⡇⠀⠀⠀⣠⣂⡤⠈⣻⡿⢸⡿⡎⡰⠁⠐⠁⠘⡄⣷⡹⣄⡟⠀⠈⠓⠦
⢠⢞⠃⠀⠂⠹⢸⣯⣧⠀⠀⠀⠨⣟⡸⣿⣏⠀⣀⠀⣿⡀⢀⡤⠀⡠⠄⠿⡾⠗⠀⠤⠄⠠⠤⢄⠠⡽⣷⡞⢨⡟⢄⠈⠀⠀⠀⡅⣿⣿⣽⠆⡏⠨⢀⢦
⢈⠉⠀⠀⡠⠀⠀⢸⡉⡚⠇⠀⠀⠈⢻⣜⢿⢰⢋⣜⣏⣀⡄⢀⣀⢲⡏⡂⠂⡀⢷⡊⠀⠀⠙⠐⡀⢦⡛⣰⠟⣠⠆⠀⠀⠀⠠⢣⣿⠟⣉⡼⠃⠃⠀⠘⡆
⣠⠛⠰⠊⣠⣶⣾⣷⣷⠛⡆⢀⡀⠀⠈⣿⡖⣢⢟⣽⣿⠴⣿⢚⣫⠥⠎⠀⠸⠃⠐⠭⣝⢶⣷⠐⠸⠘⠃⣩⡄⡿⠀⠀⠀⢀⢫⣿⠭⣾⣣⢃⡀⠀⡘⠣
⠻⠂⢠⣶⣿⣿⡟⡁⡞⠀⣝⣈⣏⢀⣀⢟⣼⠷⣻⠵⣿⣿⣧⣘⣃⡷⠟⠻⠿⠛⠻⢗⣙⣠⡿⠦⡙⠼⠧⠥⣤⡳⣤⣘⣁⡣⠂⠛⡀⡘⢿⣽⣿⣿⣇⠀⢂
⠀⠿⢾⣿⣿⠁⡔⡃⣇⣤⣬⣾⠟⢟⣡⣾⡒⠛⠛⠉⠈⠉⠀⠀⠈⠀⢠⣤⡀⣀⡀⠀⠉⠀⣀⡀⠀⠀⣀⣀⣄⡱⣿⡍⠻⣽⣿⣛⠘⢁⡌⡋⣮⣿⡟⢀
⠀⠀⠠⣜⣇⣴⣧⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⡷⣄⣠⣄⣿⣿⣟⣿⣿⣷⣠⣿⣽⣿⡷⢶⣟⣽⣿⣿⣿⢿⡾⣗⣶⣿⣷⣼⣠⡇⢸⡿⡢⠁
⠀⠀⠀⠈⠛⠶⠿⠼⠿⢿⢿⣻⣿⠿⠛⠙⢟⠻⢿⠿⣿⣿⡿⢿⣿⣿⣿⣷⣿⣿⢿⣿⣿⣷⣿⠫⠛⡹⠆⢉⣿⣯⠭⠙⠻⡟⠋⠁⠉⠈⠙⠗⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠁⠀⠂⠙⠋⠑⠂⠩⡏⠯⠋⠉⠀⠈⠛⠉⠀⠈⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢿⣿⣿⡷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣆⠶⣿⡿⣦
⠀⠀⠀⠀⠀⣿⣿⣿⡦⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⣦⣹⠆⢋⣾⣿⠞⠁
⠀⠀⠀⠀⠀⠈⢿⣿⡿⠲⣦⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⢿⣿⡏⠀⣰⣿⣯⠋
⠀⠀⠀⠀⠀⠀⣸⣿⣿⡦⣄⡀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⢿⣮⡂⠀⡀⣿⣏⡇
⠀⠀⠀⢠⣤⣻⣿⡿⢆⣷⣿⣿⡾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⣦⠝⠉⠛⠈⠘⢿⣷⢀⣴⣿⡗⡁
⠀⠀⢀⡦⣿⣿⣿⣀⠀⠀⠙⢿⣳⣼⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⣉⣣⣤⣴⣶⣦⣔⣠⣤⡞⠉⢹⡿⠝⢄
⠀⠀⠀⠲⠙⡿⢻⡏⢴⣿⣿⠿⡻⠛⠋⠆⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡯⠿⠿⠛⠛⠼⡻⠟⡿⡻⠆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣬⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣰⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⠢⣄⢠⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡈⢑⠀⠈⠀⠀⠀⣀⠀⢆⡀⠙⡿⠛⠁
⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠨⠄⣶⣿⠁⠀⠰⣤⠺⠝⠋⠀⠀⠀⠀⢀⣤⣾⣯⢹
⠀⠸⣿⣏⢛⣳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⣴⣛⡯⢰⡯⣽
⠀⠀⠘⣿⣿⠩⠭⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣿⣅⣿⣗⡏
⠀⠀⠀⠘⣷⠖⠂⣀⣀⣀⣀⣱⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣯⣷⣈⣷⠙⣧⣀
⠀⠀⠀⠀⣿⡛⠻⠋⠉⢉⣩⣿⢿⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦⡹⠿⠿⣿⣿⣧⡽⣳⠖⠛⠃
⠀⠀⠀⠀⠹⣧⣀⠘⢀⣾⡿⠃⢸⡇⠀⡈⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣄⠺⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠛⢻⡏⠠⠊⣸⡇⢀⢀⡤⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡛⠂⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣀⣘⣧⣼⣠⣧⣟⡻⣧⠀⠀⠀⠀⠀⠀⠀⠉⠩⠛⠛⢉⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠛⠻⢿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣶⡶⠤⠤⠤⢤⣄⣀⡀⠀⠀⣠⣤⡀⠀⠀⠀⣿⣿⣿⠛⠛⣛⣛⣛⣻⣿⠿⠿⣿⣿⡿⠛⠻⣿⣿
⠀⠀⠘⠻⢿⠀⠀⠀⠀⠀⠀⠈⡆⠀⣿⣿⣷⠀⠀⠀⣿⣿⣿⣤⣀⣿⣿⣿⣿⣿⣿⣿⡜⣿⡇⠀⠀⢿⣿
⠀⠀⠀⠀⡾⠀⢴⣶⣦⣶⣦⠀⡇⠀⢹⣿⣿⠀⠀⠀⣿⣿⣿⣿⡏⣿⡟⠉⠙⠋⠉⣿⡇⣿⣷⠀⠀⢸⣿
⠀⠀⠀⢠⡇⠀⠈⠻⣿⠿⠃⢰⠃⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⢳⣿⣿⣦⡀⢀⣴⣿⢸⣿⣿⠀⠀⣾⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⠇⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⣠⣿⣿
⠀⠀⠀⠸⡄⠀⢠⣤⣤⣤⣶⣾⣷⡆⠸⠃⠀⠀⠀⠀⣿⣿⣿⣿⡬⣷⣿⠿⠛⠛⠛⠋⠈⢙⣿⣴⣿⣿⣿
⠀⠀⠀⠀⠙⠤⠴⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣝⣛⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠛⠛⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠚⠉⠉⠉⠙⠓⠒⠤⢀⠀⠀⠀⠀⠀⣀⣰⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢡⠀⠀⢀⣾⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢿⠀⠀⢸⣿⠃⠈⢻⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣤⢊⡥⠖⠒⠀⠸⠀⠀⠀⢹⠀⠐⠒⠦⣝⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢰⣍⠀⠀⠠⠔⠒⠋⣡⡎⠀⠀⣀⣠⠄⢳⠂⠀⠀⠈⠀⢠⣀⡀⠀⠁⠘⣿⢿⢿⣿⣦⣄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⣏⠙⠲⠄⠀⠐⡋⠆⠁⠀⠀⠀⠀⠀⡴⠋⠀⠐⢧⡀⠀⠀⠀⠀⠀⠀⠁⡿⠄⣿⣿⣿⣷⣤⡉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⡎⢣⣀⠀⠀⣀⣑⠄⠀⠀⠀⠀⠀⢲⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⡿⣁⠈⠳⣄⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢹⠀⠈⠑⣶⡴⠛⠀⠐⠲⣤⣀⡀⣸⠀⠀⢀⣀⡜⢣⡀⠀⢀⣀⣤⠶⠒⠘⠿⣿⡿⠛⠓⠚⠓⢶⠾⠟⠲⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⣀⣀⣠⠞⠁⠀⠀⢀⡾⢥⣾⣿⣷⣦⣴⠋⠈⢣⡈⠙⣟⠉⠙⣮⠀⢀⡤⠞⠁⠀⠀⠀⣠⣴⠭⠽⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠀⠀⢠⠀⣸⣿⠀⠀⠀⠀⠀⠀⠞⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠙⠲⣌⡳⢀⠈⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⣸⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⢸⠋⠑⢺⠦⠤⠤⠤⠴⠒⠠⠤⠤⠤⠄⠒⠛⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⢠⣧⣾⣿⣿⡧⢤⣤⣀⣤⡤⠶⠊⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⢠⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠏⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⣾⣿⠿⠟⠋⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠤⠤⠖⠒⠚⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⡀⠀⠀⠀⠀⠀⠀⠔⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠈⠀⢀⠀⠶⣿⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠢⡀⠀⠀⠀⠃
⠀⠀⠀⡀⢤⠠⡀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠄⠀⢀⠀⠀⢀⠀⠀⣀⠀⠀⠀⢦
⠀⠀⠀⡀⠈⠁⢀⡞⠁⠤⠤⠤⡈⠀⠐⠦⠀⠀⠀⠀⠀⢃⡄
⢐⠀⠁⡨⠀⢰⠁⢌⠀⢀⠀⠐⢈⠂⡞⠁⠑⡄⠀⠀⠀⢸
⠀⠁⠒⠀⠀⠸⠄⠸⡀⠲⠀⠀⢾⠗⠉⢂⡤⠃
⠀⠀⠀⢀⣤⡀⠀⠁⠀⡄⢠⠂⠁⠀⠀⠁⡀
⠀⠀⠨⡙⠻⢋⡀⠰⠉⠃⠘⢇⠀⠀⠀⢀⠃⠀⠀⠀⠀⢠
⠀⠀⠀⡔⡲⣠⠉⠑⠀⠄⢀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⠐⠊⢁⠆⠐⠁⢈⡄⠀⠀⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⠃⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠅⣀⣀⣀⠔⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⢀⣠⢐⢡⣢⣄⠀⠀⠀⢴⠀⠀⠰⠀⠀⠰⣇⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠃⠘⢧⠜⢀⢀⡊⡴⠀⣄⣀⠠⣖⠙⠢⢬⡑⠤⡀
⠀⠀⠀⠀⠀⠀⠈⠒⠀⠒⠈⠀⠀⢀⠔⠁⠠⠁⠀⠉⠙⠒⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠅⠐⢄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⣠⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠀⠀⣸⣿⣿⣷⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠋⠉⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡄⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠸⣿⣿⣏⠘⢿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠉⠛⠿⣿⣿⣿⣿⣿⣿⣎⠻⣿⣿⣿⣿⣿⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠈⠻⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠙⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⡟⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠁⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣷⣄⠀⠙⢿⣄⢠⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠈⠻⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⡴⠿⠟⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣭⡉⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠻⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣼⣥⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣡⣏⣾⣥⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⢋⠏⢁⡼⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡯⠖⣦⠔⠉⡠⠞⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣿⠁⢰⣃⠔⠊⢀⣠⣾⠟⠁
⠀⠀⠐⣿⣿⣗⢶⣶⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡞⠀⢀⣴⣿⣟⣥⣴⣶⣤⣄
⠀⠀⠀⠈⠻⣦⣿⠧⣽⡿⠛⢿⣷⣦⣀⠢⠀⠀⠀⠀⠀⠀⢸⠀⡰⢀⣴⣿⡿⠛⣁⠤⠊⢉⡻⢿⣧
⠀⠀⠀⠀⠀⠈⠻⢿⣯⡉⠙⢻⣄⡉⠛⠻⢷⣤⣀⠀⢀⣀⣿⣠⣷⠿⡫⠾⠶⠭⠴⣂⣉⠠⠄⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣍⣀⠀⠭⠓⠒⠀⠤⠭⢿⣾⣿⣿⣯⣴⣫⡤⠤⠤⣤⡒⠉⠑⠢⢤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣤⣤⣴⣶⡿⠿⣋⠝⣿⣿⣿⣒⠤⢄⣀⠀⠈⠉⠒⢤⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠛⡻⢖⠉⠀⡌⢸⢻⢻⣖⠱⢦⣀⠉⠑⠒⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣀⠞⣄⣸⠀⠈⡆⢣⢿⣦⠀⠈⠉⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣍⠣⠱⡀⠡⠀⢿⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠾⣦⣶⡾

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣴⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣨⢿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠋⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠀⠈⠃⠀⠀⠀⢼⣿⣷⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⢿⠉⠉⠙⠛⠿⠿⣿⣿⠟⠛⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢚⣿⡿⠁⠀⠀⠀⠀⢸⣿⣿⡟⠛⢻⡆⠀⠉⠹⠛⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⢸⡏⠙⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠋⢡⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢐⣠⣼⣿⣿⣿⣶⣿⣅⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡎⠙⣿⡿⣿⣿⣿⣿⣿⣟⠒⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⢀⣤⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠹⠄⠀⠈⠙⠀⠉⢻⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⠴⠉⠑⡆⠀⣤⠀⠀⠠⠛⠉⢳⡄⠀⠀⠀⠀⠀⢸⣄⠀⡀⠉⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⣌⠑⠆⠠⠴⠃⠀⠈⢷⣄⡀⠀⠀⠈⡧⠀⠀⣾⣦⣴⣦⠈⠉⠈⢦⣀⡤⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣀⡰⡁⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⢸⡏⡼⠏⠘⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⣠⡄⠰⠚⠲⢤⡀⠀⠀⠈⣴⠀⢀⡴⠋⠛⣦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⢰⠃⠀⠀⡀⠀⠀⠙⠲⣄⠀⠘⠶⠉⠀⠀⢀⡟⠀⣰⠟⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⢄⡠⠂⠀⢸⡇⠀⡼⠀⠀⠀⠷⠀⠀⠀⠀⢈⣀⠀⠀⠀⠀⠀⠉⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⣷⠀⠐⠛⡄⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡽⢆⠀⠀⠀⠀⣾⠏⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⠀⠀⢠⡀⣠⠇⠀⠀⢀⡄⠀⣤⠀⠀⠀⠀⣼⠃⠸⣿⣇⠀⠀⠀⢹⡀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠀⠀⠉⠁⠂⢀⣀⡞⠀⠙⠀⠀⠀⠀⠘⣷⡐⠢⠿⠿⣿⠖⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠈⠀⠀⠈⠁⠁⠀⠀⢀⡀⠀⢠⣊⣀⣤⣬⣝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠚⠹⡄⠈⠉⠋⠁⠀⠀⠀⠀⢠⣄⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣭⣿⢻⡆⠀⠈⠓⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⢀⡏⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⡺⠈⣿⣤⠀⠀⠀⠀⠀⣠⡶⠀⠀⠈⠳⡄⠀⢠⡷⠤⠀⠀⣠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠈⢻⣆⠀⠀⠀⠀⠈⠉⠲⠦⡀⠀⠀⠀⠀⠀⠀⠒⢺⣿⠁⠀⠀⢀⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠃⠐⠒⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠏⠀⠀⠀⠀⠒⠚⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⣀
⠀⠀⠀⢀⠔⠰⠄⠀⠀⢠⠁⢓⠁⠌⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠀⠉⠉⠀⠐⠂⠠⢄
⠀⠀⠀⠈⠳⡀⠈⢆⠀⢀⢀⡣⢪⣾⣿⣿⣷⠄⠖⠂⠀⠶⠴⢊⣠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠈⢆⠈⢢⠈⡏⠀⠸⣿⣿⠋⠉⡬⠀⠀⠀⠀⠀⠰⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠉⣂⣀⣀
⠀⠀⠀⠀⠀⠀⢻⢣⠀⠳⡄⠀⠀⢀⡍⢀⣤⡄⠀⠀⣀⣀⠀⠀⢨⣷⣾⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠹⡘⠺⣆⠀⠳⠀⢰⣿⢇⣿⢿⣿⠀⣾⢿⣿⣧⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠢⢼⡆⠀⠃⣿⠇⣿⡇⢤⣿⢰⡏⣤⣹⡟⠀⠀⢰⣿⣿⣿⣿⣄⡀⠀⣠⣾⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡰⠿⠀⠿⠷⠶⠿⠿⣧⣬⣼⠇⠀⠀⠘⢻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠀⠀⡀⢴⣴⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢠⠐⠒⠢⢄⠀⠀⣇⣵⣞⠀⠀⠈⠁⠀⠀⠀⠀⡐⠁⣀⣄⡠⠒⠛⢳⣿⣿⣿⣿⣿⣿⣏⠱⠛⠟⠟⠛⠒⠀⠐⡄
⠀⠀⠰⠁⠀⠀⠀⠀⢑⣾⣿⣿⣿⣦⣤⣤⡀⠀⢀⣠⣾⣿⣿⣿⣿⣷⡀⠀⠀⠈⢿⣿⣿⣿⣿⠿⠠⢀⠀⢀⡀⡀⠀⢀⠆
⠀⠀⠐⡤⢀⡀⠤⡒⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣠⠀⡸⠿⠿⠋
⠀⢀⠌⠀⠀⡆⠀⠈⠢⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⠻⣿⡿⡁⠀⠀⠀⠈
⠠⡆⠀⠀⠤⠴⡀⠀⠀⠈⠒⢌⡉⠉⠁⠉⠉⠉⠉⠉⠣⢠⠀⠀⠀⣁⠜
⠀⠙⠦⡀⠀⠀⠑⠤⠄⢠⠄⠠⠌⠲⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠊
⠀⠀⠀⠈⠢⡀⠀⠀⠀⢸⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠈⠢⡀⠀⢸⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣴⣶⣶⣶⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢻⣿⣋⣌⣿⣿⣿⣿⣿⣿⣷⣾⣷⣮⣽⣦
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⡀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢰⣿⣿⣿⠟⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣭⣿⢿⣿⣿⡘⣿⡄
⠀⠀⠀⠀⣾⣿⣿⡿⠀⣸⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⠛⠉⠀⠀⠻⢿⣿⣿⣿⣷⣾⣿⣿⣷⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣋⣠⣼⣿⣿⣿⣿⣿⣿⣿⣯⣽⣯⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠿⠿⣿⡇
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⡿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡄⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣷⠉⣰⡿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣷⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣻⣿⣿⠀⣿⣟⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣤⢻⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡔⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣤⣿⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣿⣿⣿⣿⣿⢽⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⣿⣿⣿⣧⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠸⣿⣿⣿⣿⡌⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠻⣿⣷⢻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠹⢿⣿⣿⣷⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⢸⣏⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣷⣤⣭⣿⣿⣿⣜⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣸⣿⡏⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣽⣶⡀⠀⠀⠀⠀⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⢀⣼⢿⣿⣹⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⡌⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⠃⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⡆⠀⠀⣿⡏⠈⠀⢸⣿⡏⠛⣾⠦⠤⢤⣴⠤⠔⠋⠀⣿⠀⠁⣿⡟⠻⠛⣿⣿⣿⣿⣿⡿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⠸⢿⣇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢾⣿⡄⠀⣰⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⣿⣿⡅⠀⣾⣿⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣄⣿⣄⣾⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡃⣄⠹⠿⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠇⣰⣶⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⡿⢻⡟⣿⢸⣿⣷⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⣿⣿⣿⣿⣸⣿⣿⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠻⠿⢿⣿⣼⣿⣿⣿⡿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠈⠉⠋⠉⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⡀⠀⡀⠈⠃⠀⠀⠀⠀⠈⠙⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡄⠀⠀⠀⠀⠀⣰⠟⢀⣿⣯⣾⣽⡧⠀⠀⠀⢀⠈⠀⠀⠀⢋⠀⠁⠀⠀⠀⠀⠀⠀⣤⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⣿⠀⠀⢿⣯⢋⡼⠃⠀⠀⠀⠛⠃⠀⠴⠀⢸⡄⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢰⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢠⡄⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠂⠐⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠨⡆⠀⠀⠀⠀⠀⠛⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⠿⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣄⡀⠀⠀⠀⠀⣀⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⣤⣤⣤⢤⢀⣀
⠀⣿⣿⣻⣿⣶⣌⣙⣲⣤⣀
⠀⠙⣿⡟⢿⣿⡿⠿⣟⣿⣿⣷⣶⣄⡀
⠀⠀⢙⣿⣿⠿⣗⣀⣻⡏⠉⢉⣿⣿⣷⣄⢀⠀⠀⠀⠉⠂
⠀⠀⠀⢹⣿⣛⣿⡤⠜⠧⡀⢿⣿⠋⠉⣯⣳⣀
⠀⠀⠀⠈⢻⣟⣻⣷⡊⠈⣉⣒⣤⡀⢾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄
⠀⠀⠀⠀⠨⣫⣭⣽⣯⣀⠀⣀⣹⣿⣷⣮⢅⡀⢻⣷⡀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠘⣷⠶⢾⡤⠀⠀⠠⠿⠿⢷⣄⣉⣳⣿⣧⣠⡞⣿⡿⠂⠀⠀⢀⣀⣠⣤⣤⣤⣀⣴⡴⠤⠤⢀⣀
⠀⠀⠀⠀⣠⡿⢻⣋⣉⣽⠖⠒⠉⠉⠉⠁⠐⠚⣭⣿⣿⣿⣿⣿⣷⣶⣟⡛⠋⠛⠉⣹⣿⡟⠓⣾⣯⠭⣝⡛⠶⣬⣻⡦⣤⡀
⠀⠀⠀⠀⢺⡿⢻⣧⠀⢇⠀⠀⠀⠀⣀⡴⠒⠉⢉⣽⡿⢽⣿⣿⡛⠿⣍⠉⠛⢀⣶⣿⣟⡁⢰⣿⠃⠀⢈⣿⡙⢻⡿⣿⣭⣿⠦
⠀⠀⠀⠀⣿⡟⣛⣿⡔⠉⡙⠓⢒⡿⠃⠀⢀⡔⣡⣟⣿⡿⠛⢡⡀⠀⠠⡙⠲⣾⣿⣿⠛⠟⠙⠗⠒⢶⡛⢤⣽⠿⣟⠻⣯⣹⡷
⠀⠀⠀⠰⣿⣟⣫⣿⣷⣾⣇⣠⣏⡀⢀⣴⠟⣰⣿⣿⠟⣰⠁⠀⣇⠀⠀⠘⣷⢌⣻⣧⣵⣄⢀⣤⣴⣶⡿⣿⣻⣧⡽⠗⠛⠃
⠀⠀⠀⠀⣼⠿⠛⣍⣿⡿⣻⣿⣿⣿⣺⡎⢰⣿⡿⠋⡼⠃⠀⢀⡟⣦⡀⢀⡜⢦⣹⣾⡿⠿⣿⢽⣷⠾⡿⠛⠛
⠀⠀⠀⣸⡡⠀⠀⠫⠋⠙⣿⠈⠉⢭⠎⠀⠈⠁⠀⣼⣟⢷⡄⢸⠀⣘⠈⡀⣧⣼⢻⣿⠇
⠀⠀⡼⠟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡏⠘⣋⣾⣷⣾⣀⣿⣶⡿⢷⣙⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢻⣿⣿⣏⣿⢸⣯⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡿⣧⣿⣿⣿⣿⡿⠿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⡟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃

⠀⠀⠀⠀⢀⣀⠀⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⠶⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠆⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⡴⠀⠈⠐⣀⣧⢞⠁⢠⠀⠀⢀⠠⠀⠀⠄⠚⢢⡄⠊⠉⠹⡄
⠀⡲⠐⠄⡀⠀⢀⣼⣗⠇⠈⠀⠀⠈⠀⠀⢧⠀⢀⢸⠏⠜⢈⣶⡋⠈⠓⠀⠀⣠⠞⠀⠀⠀⡁⠀⠀⢠⢃⠄
⠀⠸⡂⠬⣟⣴⡯⠦⠯⠋⠀⠀⠀⠀⠀⠂⠈⠿⣿⣴⣮⠜⠉⠜⢁⠠⠀⠀⠀⠘⢄⡢⣀⠢⠁⣀⣀⡜⠁
⠀⠀⠄⠀⡐⡿⠗⠀⠀⠤⠀⠀⠀⠀⠀⢀⡴⠚⢹⠋⠟⣛⣿⢣⡀⠀⠀⠀⠀⠀⠀⣉⡻⠛⢻⠛⠓⡀
⠀⠀⠀⠈⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⡆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⣶⣶⠶⠶⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣶⠶⠶⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠉⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢰⣿⠁⠀⢀⣴⣶⡶⠾⠿⠿⣷⡀⠀⠀⠀⢠⣤⠀⠀⣿⡀⠀⢠⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠸⣿⡀⠀⠛⠃⢸⣿⠀⠀⠀⢹⣧⠀⠀⠀⠛⠻⠷⠾⠟⠿⠾⠟⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠙⠿⣦⣤⣶⣿⠷⠶⠾⠿⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⡄⠀⢀⣀⣀⣀⣀⣀⣴⠶⣦⣤⣤⣤⣴⠟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢷⣶⠟⠋⠿⠋⠉⠛⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢀⠀⢀⣤⣄⣀⣤⣄⣠⣤⡄⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠾⠋⠙⢿⣦⡾⠛⠻⠟⠁⠙⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢠⣿⣆⣴⣶⣄⣠⡤⠀⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠏⠻⠋⠀⠉⠉⠀⢰⡟⠻⣷⣠⡾⠛⣿⡇⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠹⠿⢁⣾⠏⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⡟⠁⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣀⣀⣠⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠐⢀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠁⠀⠀⠀⠂⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠠⡠⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠤⠀⠦⢴⠶⠄⣀⣀⠀⢀⠀⠀⠀⡀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠐⠞⣢⠶⠁⣰⠀⣒⣠⠠⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠠⠀⠀⠀⠐⠒⢛⡐⢸⠏⠀⠀⠀⡰⠃⣿⠀⣸⡇⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢣⠇⣸⠀⠀⠀⠈⠐⢣⡟⣨⢻⡆⠀⠀⠀⠀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢨⡇⠋⠀⡟⠀⠀⠀⠀⡒⢼⡇⣸⢺⡇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⡇⠀⠅⢀⣀⠀⡀⢀⣿⢠⡜⢸⡇⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣋⠀⠠⠀⢫⣽⠉⢛⡇⡤⠚⢾⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⡇⠀⠀⣀⣈⣀⣉⣁⣉⣻⠀⣋⡸⠃⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠈⡀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠒⠂⠐⡄⠀⠀⠀⠤⠲⠀⠀⠀⠀⠀⠀⠀⠐⠄⠁⠘⢠⠀⠀⠉⠀⢄⠁⠀⠀⡁⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢁⠠⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠠⡰⠀⠄⠘⠀⠀⠁⡉⠬⢠⠀⠀⠈⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠑⠈⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡀⠀⠀⠀⠀⠆⡀⡜⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⣇⠠⡀⠀⠀⠀⠀⠀⡄⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⡴⠀⠀⠀⠀⠀⢹⣯⠀⠀⠀⠘⠘⡸⡞⡀
⠀⠀⠀⠀⠀⠀⠀⠅⠏⠅⡀⠀⠀⠀⠀⠀⡴⠄⠘⠈⠉⠉⠉⠀⠀⠁⠙⠀⢚⠧⠋⠁⠀⠠⣀⠵⢼⠄⠀⡀⠈⠒⠰⢇⡵⣠
⠀⠀⠀⠀⠀⠀⣀⣂⠛⠖⠃⠀⠀⠀⠠⣄⣵⣤⠅⠀⠀⠀⠀⠀⠀⠀⠐⠦⢛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⠿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⡿⠿⠶⣄⡀⢀⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⡷⠀⠰⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠈⢹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⢦⣄⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡛⠛⢯⣏⡁⠉⣹⡏⠉⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣦⣙⣿⡿⢼⣿⣠⣽⣌⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣮⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣼⣿⣿⣿⣿⡛⡟⢋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⡟⠉⠉⠙⠻⣿⣿⣿⣿⡇
⠀⠀⣸⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣶⣿⣿⣿⣿⣿⣿⣆⠈⠻⣿⣷⣈⠛⢦⣄⡀⢈⣿⣿⣿⣿⣧⡀
⠀⢀⣿⡿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡄⠀⠈⠻⣍⡳⢤⣈⠙⣿⣿⣿⣿⣿⣿⣷
⠀⢸⣿⠁⠙⢺⠟⠛⠛⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡄⠀⠁⠀⠙⠀⠀⠀⠈⠙⠶⣝⠷⣾⣿⣿⣿⣿⣿⣿⡇
⠀⢸⡏⠀⠀⣿⠀⠀⠀⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠸⣇⠀⢀⢀⡀⡆⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠟⠋
⠀⠸⣧⣀⣀⣯⣀⣀⣠⣧⣄⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣴⣿⣶⣿⣿⣿⣷
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠿⠛⠛⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠈⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⣐⠄⠀⢀⠔⠂⠒⠀⠢
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠔⠊⠀⠀⠀⠀⠀⠄⠀⠼⠒⠠⡀⠀⢀
⠰⡲⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⣀⣀⡜⠀⠀⠀⠀⠀⠀⠀⣇⠤⣴⡶⠖⢒⠾⢅⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢰⢀⠂
⠀⠀⠀⠀⠀⠀⢀⣠⠳⠘⠛⠟⠛⢩⣾⣷⣄⠀⠀⠀⡠⠊⣡⣶⣶⠀⠀⣨⣤⣶⡜⢆⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⢠⡾⢣⣶⣿⣷⡆⠀⠘⠿⠟⠋⣣⠀⣼⠀⠀⠻⣿⡿⢀⣘⠿⠿⠟⢀⣼⡀⠀⠀⠀⡤⡌⠐⠢⣀⡐⠁
⠀⠀⠀⠀⠀⢈⠀⢈⠍⠛⠫⢁⡠⠂⠉⠐⢤⣿⠀⡿⠫⠆⠑⡤⠪⡄⠀⠙⣄⡀⢾⣿⠃⠀⢀⠎⢐⠁
⠀⠀⠀⠀⠀⠸⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⡜⡻⠀⠇⠀⠀⠀⠀⠀⠌⠀⠀⣿⣿⠈⡙⠰⡁⢡⠄⡠⠆
⠀⠀⠀⠀⠀⠀⠙⡏⢧⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⢤⡀⠀⠀⡄⠀⠀⠀⡸⠟⢃⡔⢡⠊⠑⢄⢧⠴⠃
⠀⠀⠀⠀⠀⠀⠀⢲⢲⠶⡖⣆⠀⢈⡩⠹⠿⢄⠞⢺⠓⠀⠚⠓⠐⠀⠀⡃⣼⠏⡰⠃⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠣⠤⠐⠀⢀⠀⢱⠀⢐⠀⠀⠀⣀⠠⠀⠀⠁⢹⠊⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠒⠖⠒⢈⣀⠠⣊⠑⢄⠣⠠⠄⡇⠀⠀⠀⣠⠊⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡈⠀⠆⠋⠱⡈⠁⠇⠀⠀⠑⠈⠢⡬⠈⣁⡴⠒⠉⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠁⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠑⠈⠁⠀⠀⣀⠤⠥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠆⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠤⠤⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣔⣜⠥⠒⠒⡒⠢⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⢸⠀⠀⠀⠂⠀⢀⣅⠞⠁⠂⠁⠀⠀⢱⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠈⠀⡇⠀⠀⠐⡖⢤⡟⡡⠀⢄⠀⢀⠠⠐⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⠋⠓⢄⡀⢀⠌⢰⠻⡶⠀⡠⠜⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠼⠤⠤⠦⠀⢀⡠⠟⠉⠀⢸⠒⠓⠂⠉⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢀⠔⠁⠀⡎⠀⠀⠀⢀⠸⠃⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⢀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⡘⢶⡿⢈⣷⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⣾⣆⠃⠋
⠀⠀⠀⢠⣴⣶⣶⣶⣾⣶⣦⣸⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠀⠀⠀⠀⠀⠀⠀⣀⠛⠶⠟⠀⠀⠀⠀⠀⠀⢲⣤⡀⠀⣥⣭⣿⡆⠺⠟⠁
⠀⠀⠀⢼⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠘⠻⢋⢻⣷⣶⠆
⠀⠀⠀⢾⣿⣸⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⢰⣿⣿⣿⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⡿⡾⠿⠛⠁
⠀⠀⢰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣆⠀⠘⢀⣐⣒⢢⣄⡀
⠀⠀⠀⠙⠺⣬⣿⣿⣿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⡟⢿⣿⣠⣴⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⡟⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⣏⡿⣿⣿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠃⣴⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⡜⢿⣿⠟
⠀⠀⠀⠀⠠⣤⡀⠀⢀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠀⢸⡿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠻⠿⢧⠄⠀⠀⠀⣾⣤⠀⠁⠀⠀⠀⠀⢀⣿⣿⠿⠃⢸⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⢤⣤⣶⡚⣠⣶⠀⣠⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣾⣿⣿⣿
⠀⠀⠀⢴⣦⡈⢻⣿⣿⡟⠁⠈⠁⢠⡜⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⣿⣿⡿⠀⠀⠀⢠⣼
⠀⠀⠀⠸⣿⣿⡄⠈⠛⢿⠟⠃⢤⣜⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡟⣿⣿⣿⣿⠇⢠⢀⣴⠿⡟
⠀⠀⠀⠈⢻⣿⣿⡏⢀⣰⣾⣆⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⣿⣿⣿⣿⠟⢀⣼⣿⡿⠞
⠀⠀⠀⣀⣼⣿⣿⣷⡀⠈⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⠹⣌⠛⢃⣴⣤⣤⣤
⠀⠀⠀⠉⠻⢿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠉⠁⠈⢉⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠂⢸⣿⣿⣿⣿⡿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠻⠛⠀⠠⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⣠⡶⠆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡉⠉⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⠘⢿⡇

⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠎⠀⠴⣾⣿⣿⡇⠈⠃⠉⠛⠂⠀⠀⠂⠀⠀⠀⠰⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠤⠒⠚⣸⣿⣿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⡇⠀⠀⢀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⠋⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼
⠀⣀⠔⠂⠀⠦⢄⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠠⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⢀⣾⡟
⠴⠁⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⠙⢂⣀⣀⣀⡀⢀⡠⠐⠒⠦⢄⠀⠀⠀⠑⠇⡀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣤⣾⡿⠋
⠆⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢡⠀⠀⡀⠀⠐⢄⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢤⣶⣶⣼⣷⣷⣤⣾⣿⣿⣿⠖⠈
⠀⠀⠀⠀⠠⠀⠀⠐⠠⠀⠄⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⢳⡎⠀⠁⠈⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠛⢛⣟⣿⣿⣿⣷⣶⣖
⣦⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠎⢳⠀⠀⠀⠀⣠⠤⠀⠠⠤⢺⣿⣿⠘⠻⠟⠿
⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠊⠁⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣆⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣻⣷⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡄
⠀⠀⠀⠈⠀⣦⢄⡀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠐
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠤⠱⠶⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⡇⠀⣾⡇⠀⢀⠔⠀⡠⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠢⣀⣀⣀⠀⠉⠙⢿⣿⣿⣇⢰⣿⣷⣾⣿⡵⠄⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠘⠛⠛⠛⢻⣿⣾⣿⡿⢿⡅⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠂⠸⡄⠀⠀⡼⠀⠀⣠⡄
⠀⣠⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠈⠁
⠀⠠⢀⣸⣆⠀⠀⠀⠀⠀⢉⡂⠀⡀⠀⠀⠀⠀⠀⠀⠠⠶⠒⣈⡽⠟⠉⠉⠻⣧⠀⠉⠀⠀⠀⠶⠤⢄⠀⢀⠌⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⠂
⠀⠀⠀⠀⠹⠞⠀⢀⠂⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠴⢒⢀⠝⠉⠡⠚⠐⠆⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⠀⠀⠀⠀⠀⠀⣠⣄⡀⢸⠀⠀⠀⠀⠀⢀
⠀⣀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠋⠉⠉⠻⢟⢿⣇⣠⡀⡀⠀⠀⠀⠀⠐
⠊⠁⠀⠀⠀⢀⠀⠀⢀⠈⠈⠲⠤⢶⡀⠀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⢀⠀⢠⠀⠀⢀⣴
⠀⠀⠀⠀⡤⣈⣀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢳⣇⠀⢸⣧⣤⣤⣶⡿⠛
⣀⠀⠀⠀⡎⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠹⣆⣧⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⠿⠿⠛⠂
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡄⠀⡌⠀⠀⠀⢀⣴⣾⡥⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢿⣍⠀⠀⠀⠐⠊
⠀⠀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠀⠀⠀⠤⣄⡀⠀⠀⠀⣹⣿⣿⣄⣄⡆⠀⣴⣾⣿⠯⢀⡄⠀⠂⠐⠀⠀⠲⠀⠠⠁⠈⠀⠈⠳⣍⡀⠒⠒
⠒⠠⢀⠀⠀⠀⠁⠀⠀⠢⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠙⢿⣿⡏⠃⢠⣿⣿⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣷⣶⡅⣤⠛⠲⣹⣿⣣⠄⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠆⠀⠀⠀⠐⠚⠋⠉⠁⠘⠞⠛⠙⠋⢻⡿⢿⠅⣀⣶⠋⣵⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣾⣿⣆⠀⠀⠀⠤⡘⠃⠃⡁⠄⠀⠀⠀⠄⡀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠉⣛⡛⠻⠁⠀⠀⠀⠉⠀⠄⡁⠄⢄⢠⢀⡂⠐⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⠀⠰⠦⠀⠠⠃⠀⠀⢹⠢⠋⠛⠃⠀⢀⣀⡄⠗⠀⠀⠠⠥⠂⠀⡽⠯⡃⠀⠐⢃⠠⣤
⠀⠀⡤⠛⢂⠔⢄⠐⢏⠴⡟⢠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡄⠀⣨⡟⠀⠀⠀⠀⠀⡠⢮⡰⡀⢳⡶⠈⠉⡴⠄
⠀⠀⢀⡣⠠⣀⡘⠀⠞⠀⡀⠀⠃⠡⠀⠀⠀⠀⠀⠀⠼⡆⠊⠶⠴⠁⠀⠀⠀⠀⢾⢷⢛⠛⠁⠁⢨⢃⡀⢈⠁
⠀⠀⠠⠀⠈⠃⡇⠀⠄⠀⠵⢁⣁⣠⣶⢂⠀⠀⠀⠀⠀⠀⠂⢠⡀⠀⠀⠀⡴⠻⠀⢓⠀⠀⠀⠀⢈⠠⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠀⢃⢀⡠⠼⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⡴⠀⠈⠀⠁
⠀⠠⢠⣄⢀⣀⢄⣈⠀⠀⠀⠀⠠⠤⠒⠶⠈⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⢈⣦⡥⠀⠀⠀⠠⠄⠀⠄⠠⠀⠀⠀⠘
⠀⠀⣉⣐⠅⠄⡽⠢⠋⠌⠐⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠀⠀⠀⡄⠀⠀⢀⠠⡞⠈⢀
⠀⠈⠀⡰⠂⠃⢨⣀⠀⠄⠀⠀⡀⠠⢠⠤⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⢿⠆⠀⡀⣀⠐⡤⠲⠘⠀⠘⡃⠈⠙
⠀⠀⠸⣴⡗⡧⡛⠀⠛⠉⠀⡘⠎⣾⡁⠂⣤⢶⠈⠐⣖⣡⡄⢿⣾⠏⣿⣹⣿⠞⠰⣇⣇⡰⠁⠀⠤⠃⢤⠁
⠀⢀⠀⠀⠀⠀⢁⡷⢠⠀⡖⠀⠀⢨⢠⠀⠯⠄⡱⠳⠚⠿⣋⢁⢱⠂⡝⡋⢁⢀⡀⠘⠀⠀⠰⠀⠀⠠⠨⢀⣀⣀
⠀⠠⠀⠀⠀⠁⢊⡀⠈⠈⠁⠀⣰⠋⢳⠖⠁⠀⢰⡞⢀⣠⣸⠈⢞⡟⠀⠀⡍⢄⠒⢤⠀⠀⠀⠀⠀⠀⠀⠏⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⢺⢁⣿⣄⡟⣁⣘⣶⠃⠀⠀⢀⠇⡨⠀⠂⠀⠀⠀⠀⠀⡀
⠀⣈⣄⣠⣆⣀⣁⣀⣠⣒⣀⣼⣿⣬⣿⣼⣡⣞⣼⣍⠋⠈⠉⠁⠠⠀⠀⢸⠅⠃⠀⠀⡆⠀⠀⠀⠀⠀⠂⠈
⡀⠋⠉⠿⣉⠙⠟⠉⠉⠉⠉⠋⡋⠙⢻⠛⣿⣿⡏⡙⠀⠠⠒⢃⡐⠦⠀⠀⠐⠒⢃⠀⠀⠀⠀⠁⠀⠈⠀⠀⠐
⠔⠓⠐⠂⠐⠊⠘⣞⣻⣟⣻⠛⢛⠚⡛⣛⣿⣿⣿⣇⠀⠀⠀⠈⣁⠀⠀⠀⠁
⠂⠓⠺⠟⠛⠛⠛⠙⠛⠛⠑⠑⠚⠛⠛⠛⠋⢿⡟⠙⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠋⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⣤⠾⠻⣶⡀⢀⣿⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⠁⠀⣠⣾⠃⣸⡇⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢸⡇⠀⠀⠉⠀⢀⣿⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢿⣄⠀⠀⠀⣸⡇⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠶⢶⣾⣧⣀⠀⠀⠀⠙⠻⠶⢶⣿⠟⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣼⡏⠛⢷⣄⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢀⣴⣟⠛⢷⡄⠀⣠⣶⠀⠀⢀⣶⣶⠀⢠⡶⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣾⢻⡿⠶⣴⣿⡾⠛⣿⠀⢀⣾⠋⣿⣠⡿⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠃⢸⣇⠀⢀⡿⠀⠀⣿⡄⣼⠏⠀⠈⣿⠟⠛⠁⢀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣸⠏⠀⠀⠻⣶⠿⠁⠀⠀⢸⣿⡏⠀⠀⠀⢿⡄⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⣀⣴⠟⠁⠀⣶⠀⠀⢀⡄⠀⠀⠀⠀⢿⡄⢰⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠸⣧⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⢿⡀⠀⣸⡇⠀⠀⠀⠀⠈⢷⡿⠁⠀⢺⡟⠛⢻⡆⠀⠀⣰⡶⣦⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⠻⣶⠟⠀⣶⢷⣆⠀⢶⡄⢀⡶⠀⢹⡷⢿⣟⠁⠀⣰⠏⠀⢸⡇⠀⢀⣠⡶⠟⠋⠉⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⣶⠟⠀⠈⠻⠟⠃⠀⠘⠃⠀⠛⠀⢰⡟⠀⠀⣾⣡⣴⠟⠉⠀⠀⠀⢰⡟⠃⠀⣿
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣠⡶⠶⠶⣦⠀⠀⢸⡇⠀⠀⣿⠇⠀⣼⡿⠋⠀⠀⠀⣀⣤⡀⠘⢷⣤⣼⠏
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣰⡟⠀⠀⠀⣀⣤⡀⠀⢀⣠⣄⢠⡾⠋⠀⠀⢰⡟⠀⢀⣀⠀⠀⢠⣿⠀⠘⠋⠀⠀⠀⣰⡟⠋⣹⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠚⠋⠀⢀⣴⡾⠛⣿⡄⣠⡿⠉⣿⣿⠁⠀⠀⢠⡿⠁⢀⣾⣿⠀⠀⣸⣿⠀⠀⠀⠀⠀⣸⡿⠿⠿⠋⠀⠀⢀⣤⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣀⣴⠟⠁⠀⠀⢹⣧⡿⠀⠀⣿⠇⠀⠀⢠⡿⠁⠀⣼⠿⣿⠀⢠⣿⣿⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⣠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡁⠀⠀⠀⠀⠸⣿⡇⠀⠀⠟⠀⠀⠀⣾⠃⠀⣰⡟⠀⢿⣤⡾⢹⣿⠀⠀⠀⢠⡿⠙⣿⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠉⢿⣄⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢰⡿⠀⢰⡟⠀⠀⠀⠉⠀⠈⣿⠀⠀⢠⣿⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⡿⠁⠀⠀⠀⠀⠀⠀⢻⣇⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⡶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣿⡿⠿⠿⠛⠛⠋⠉⠉⠁⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⡿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⣠⣶⣿⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⠟⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣾⡟⠉⠈⠙⢿⣿⠏⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⣴⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣿⣇⠀⠀⠀⠈⠁⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⠶⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠴⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⢿⣦⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⠿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠠⠶⠿⠿⠟⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⠶⠶⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⠶⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⡶⠾⠟⠃⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣾⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣴⣾⣿⡿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⢠⣾⢿⡆⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠛⠿⡿⠏⠸⠃⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣠⣤⣴⣶⣿⡿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣾⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠒⠒⠒⠤⠤⣄⣀
⠀⠀⠀⠀⠀⣤⢀⠴⠊⠉⠀⢀⠠⠔⠒⠉⠉⠉⠉⠉⠀⠈⠙⢦⡀⣀⠀⢀
⠀⠀⣔⠻⠁⠘⠁⣀⠠⠀⠀⠀⠀⢀⠠⠤⠒⠒⠋⠉⠀⣠⡤⠖⣋⢵⡴⠃
⠀⠀⢈⠗⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉⠁⠀⢉⡵⠞⣇⠀⠀⠀⠀⠀⠀⠀⣀⠤⢐⣲⠶⠓⠒⠒⠲⠤⣄⠀⠀⢀⣠⣤⣄⡀
⠀⢠⠏⠐⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠀⠀⢀⣀⣠⠴⠞⠉⠀⠰⣹⡀⠀⠀⣠⠔⡶⢊⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠒⠛⢦⣋⣤⠂⠈⠙⣆
⠀⡞⢠⠁⢠⠃⠀⠀⠀⢠⠀⠀⠒⠒⠚⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣿⠃⢠⡞⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠽⡝⣆⢉⡀⢸
⢠⡇⢸⠀⠈⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢹⢀⣿⢠⠀⠀⠀⠀⠀⠀⠐⠒⠢⠤⢄⣀⠀⠀⠀⠀⠀⢀⡽⣼⡀⠀⣸
⠈⡇⠘⠆⡴⢁⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⣸⢸⠺⢀⠧⣄⣀⣤⣀⡀⠈⠉⠒⠤⣀⠀⠉⠒⠦⣤⡴⢞⠅⣧⡤⠖⠁
⠀⣿⠀⢰⢡⡏⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⢀⠇⠈⢇⣸⠠⠀⢀⡄⠀⠉⠙⢦⡀⠀⠈⠳⡀⠀⠀⠀⠐⠋⡴⡇
⠀⠸⣇⠀⠈⠣⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⢇⡞⠀⠀⠀⢿⢀⠄⠀⠁⠀⠀⠀⠀⠓⢸⡍⢣⠁⠀⠀⢀⠀⡰⢣⡇
⠀⠀⠹⣆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣀⠴⣫⠋⠀⠀⠀⠀⢸⡘⠂⠀⠀⠀⠀⠀⠀⠀⠈⠶⠎⠀⠀⠀⣡⠞⢁⣾⠁
⠀⠀⠀⠈⠳⢬⣓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣡⠞⢁⡀⠀⠀⠀⠀⠀⢳⡤⠤⠞⠁⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⢀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠤⣄⡀⠀⣠⣤⠤⠶⠚⠉⠀⠀⠙⠃⠀⠀⠀⢀⣄⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠘⡎⠀⢒⣈⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢽⡀⠾⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠉⠀⠀⠀⠈⠉⠓⡆⠀⡠⠒⠲⠿⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠁⠀⠉⠢⡀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⠀⠀⠀⠀⡠⠂⠑⠒⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢄⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠈⡄⢇⡀⠀⠀⠀⠈⠛⠿⠃⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠒⠂⠀⠀⠀⠀⠀⢠⠀⠈⠑⠒⠤⢤⠖⢦⣤⠤⠖⠋⠹⠀⠀⠀⠀⠀⢸⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢣⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⢏⣢⠀⢀⡠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠈⠉⡩⡅⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡤⡆⠀⠀⣠⣴⢺⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⠊⠀⠰⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⣈⠙⠒⠛⠂⠛⠋⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡴⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠙⢓⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣊⣑⠀⠑⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠭⠃⡔⢚⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣦⠤⣄⣀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢀⠀⠀⠀⣀⠊⠀⠠⠀⠙⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⡀⠀⡿⠉⠑⣶⠟⠀⠀⠀⠀⠀⣅⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠁⠈⣾⠇⠀⠀⠀⠀⠀⠢⠐⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⡀⢀⠂⠀⠀⢠⡟⣇⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠐⠀⠀⠀⠠⣧⡄⠀⣺⣵⣷⠆⠀⠀⢀⢁⠎⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠙⢯⠁⠀⠀⠘⣿⡏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠺⠟⠯⡀⠀⠀⠈⠓⣄⣀⠀⠹⠧⢂⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠓⡄⢸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠉⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠤⠄⠀⠤⠀⠀⢀⡀⠀⢀⡀⢠⡔⠋⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⡀⡘⠁⠀⠼⣇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⢀⠀⠠⠋⠉⠦⠄⠀⠀⠀⠀⠀⢇⠀⠀⣀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢲⣾⣿⣿⡟⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠉⠀⠀⢀⠀⠀⠀⠀⢀⣰⡿⠁⢻⡏⠈⠉⢄⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠚⠁⢁⠀⠀⠀⠐⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡡⡩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢡⢄⡞⣆⢴⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡉⠂⠀⢰⠁⠀⠒⡃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢰⠀⠀⢸⠀⠀⢀⠇⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⢢⡀⠀⠸⡀⠀⠸⣄⣀⡈⣠⡀⣾⠀⠀⡸⠀⠀⢀⠆⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢧⣀⠀⢱⡀⢐⡋⠈⠁⠉⠘⣻⠃⣰⠁⡀⣰⠋⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠈⠢⣀⠐⢽⣷⡀⠷⠈⣷⡀⠤⠀⣄⡏⢰⠃⣰⣾⠟⢀⡴⠋⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠦⣤⣄⠈⠗⢸⡟⠙⡾⠴⣿⢠⣄⣤⢺⡷⠼⡔⠙⣿⠐⠏⡀⢀⡴⠚⠉⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠤⣀⡀⠀⠿⣿⡟⠓⠈⠈⣀⣸⣇⠜⠀⠘⠀⣿⣀⣈⠀⠐⠛⣿⡽⠟⠁⣀⡠⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡉⠙⠢⡤⢵⠋⠀⣾⢋⡥⠊⠉⠉⠉⠉⠢⣍⢻⡄⠈⢻⢇⡤⠒⠋⡁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡛⠇⠛⠓⡲⡿⠞⢙⢋⠊⠀⠔⠀⠀⠀⠀⠀⠈⢆⠛⠙⠞⠷⡶⠞⠃⠲⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠀⣛⠁⠀⠐⠀⢘⡏⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢁⣏⠀⠰⠀⠀⢘⡁⠰⠄⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡄⣠⣴⣦⣮⣽⣁⡘⡄⠀⠀⠀⠀⠀⠀⠄⠀⡜⣀⣨⣭⣦⣖⣤⡄⣨⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠀⢉⣉⢄⠦⣹⡀⠉⢿⣘⠦⣀⠀⠀⠀⠀⡠⠞⣸⠏⠁⣸⠡⢤⢌⣉⠉⠉⠉⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⣁⡤⣾⣅⡈⠀⠉⠙⢻⡆⢌⠉⢉⠁⣺⠛⠉⠀⠈⣀⣽⣦⠄⡉⠑⠲⠤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⢀⡤⠟⠝⠉⣌⢱⣏⣈⢦⢬⣿⡼⠖⠺⢸⣧⠤⢮⣀⣷⢉⣌⠙⠷⠶⣄⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⡠⠊⢀⣼⣼⠃⡸⠁⣟⠆⠤⠄⢺⡇⠋⡀⢻⣼⠂⠈⠢⡀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣸⠃⠀⣰⠁⢸⣋⠀⠀⢀⢈⡹⠀⢫⡀⠙⠳⡄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⡞⠁⠀⢠⠛⠲⢉⠉⠻⠃⠻⠋⢁⠱⠚⢣⠀⠀⠱⡄⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⡜⣡⠢⢀⡧⢪⣹⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠙⠃⢠⣇⠛⠁⠃⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠈⠛⣶⣤⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣠⣶⣶⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⢀⠀⡀⢀⡀⠀⠀⢀⢀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣶⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣯⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⢻⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠀⠞⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠍⠁⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⡀⢀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⣀⣀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣿⣻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⡿⠤⣴⣏⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣸⠏⣠⠶⠦⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡿⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢻⠀⡇⠀⠀⣧⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣧⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢱⠀⠀⣿⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⡆⠀⢸⣄⠀⣸⠁⠀⣀⣀⡀⣀⡀⠀⠀⣿⣸⡿⢘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢳⣀⣈⣁⣉⣉⣉⣉⠥⠬⣽⣧⣙⡖⡒⢫⣥⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠉⠉⢹⡇⠀⠀⠀⠀⠘⢦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠛⠛⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢠⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡘⠀⠀⢸⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡇⠀⠀⣸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢰⠃⠀⠀⢸⠃⠀⠀⡿⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⡟⢀⠀⠀⡾⠀⡀⠀⡇⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡈⠀⢠⡇⠀⠁⠀⡇⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢰⠀⠀⡸⠀⢸⠀⢸⠃⠀⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⠇⠀⢀⠇⠀⡇⠀⡸⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠎⢠⠀⡸⠀⢰⠇⠀⡇⠀⣼⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⠇⠀⡈⢀⡎⠀⠃⠀⣼⠀⠀⠀⠀⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⢀⡞⠀⠈⠀⢠⡇⠀⠀⠀⢰⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⣙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠀⠉⠉⠉⠉⠀⠀⠀⢀⡾⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⡇⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣤⣿⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠓⠲⠶⠶⠶⣦⣤⣀⣀⣀⣀⣾⣃⠒⠒⠐⣰⣿⣤⣤⣆⡴⠟⠓⠒⠒⠚⠃⠂⠀⠀⠀⠀⠀⠛⠶⠶⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣠⡶⣀⡀⠀⠀⠤⡤⠤⢀⣬⡅⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢨⣭⡀⠤⢤⠀⠀⠀⠀⣀⢲⣄
⠀⠀⠹⣿⣯⡗⠲⣄⠘⠦⣀⢠⣼⠁⣼⢙⡖⠀⠐⢤⡀⣿⢻⢀⡤⠂⠄⢲⡋⡧⠀⣯⡄⣀⠔⠃⣠⠖⢺⣽⣿⠏
⠀⠀⠀⠀⠉⢻⣧⡌⠢⡀⠈⠉⠉⠙⣏⣹⡏⠀⢐⣛⣿⣿⣼⣿⣟⡂⠀⢸⣏⣹⠏⠉⠉⠁⢀⠔⢡⣼⡟⠉
⠀⠀⠀⠀⠀⠀⠈⠿⣷⣭⡓⢤⠀⠀⠈⠉⠀⠀⠈⠈⣹⡇⢹⣏⠁⠁⠀⠀⢉⠁⠀⠀⡤⠚⣭⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⡽⢖⠋⠀⠀⠀⠀⢠⠇⡏⢹⠹⡀⠀⠀⠀⠀⠉⡲⢯⣶⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣙⣆⡀⠠⠀⠀⠀⠀⢸⠀⣷⣾⠀⡇⠀⠀⠀⠀⠄⢀⣰⡋⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢿⣷⣄⣰⠀⣴⢶⡌⠀⠘⠃⠀⢁⣶⣦⠀⡆⣠⣾⡿⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢀⠛⢿⣀⣏⢁⠔⠂⠀⠀⠐⠢⡉⣹⡀⡿⠛⡀⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠁⠈⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠇⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠀⢚⣦⠀⠀⠀⠀⠀⣴⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⣀⠀⣿⠛⠛⠛⠁⣰⠇⣸⢃⡴⣠⢀⡴
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⡇⠀⠀⢀⡟⡰⢃⡾⢉⠇⢠⡿⢠⡏⣾⢳⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢸⡇⠀⡈⠿⠉⠀⠸⠇⡾⢰⡏⠈⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠘⠃⠀⢸⣦⡄⠀⢀⣀⣀⣘⡁⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠐⠋⣴⠗⠋⠉⢉⡉⠉⠙⠋⣴⠗⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⢶⣄⠀⠀⡀⠀⡀⣽⠈⡆⡃⣼⠃⣸⠇⣰⠏⣀⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠹⣇⢸⡇⢸⡇⣼⠀⡷⠁⡿⢠⠟⣰⠏⢰⠏⣸⠤⣶⣆⡔⣤
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠙⢦⡀⢹⢸⡇⠸⡀⢸⡇⠳⠘⠁⠸⠃⠿⠀⣿⣰⠏⣰⠋⣿⡠⠛
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠈⢻⣼⠘⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⡧⠐⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠐⠢⢄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣦⡀⠀⠀⠐⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠠⠀⠀⠘⠻⣷⣄⠀⠀⠀⠉⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⣄⣠⣀⡴⠚⠳⠆⠀⠀⠠⠈⢽⣷⣄⠀⡠⠄⠀⠉⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⢻⡽⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡂⢤⣀⠀⠀⠀⠀⠀⠀⣤⣦⣘⣧⣹⣿⣿⣦⡀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠻⣿⠀⠀⠀⠀⠀⠀⠀⠙⣉⠙⠉⠃⠙⠻⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠘⠁⠀⠀⠈⠁⠉⣶⣉⢶⠄⠀⠀⠀⠀⠀⣿⣷⣜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⠀⢿⠂⡀⠀⢀⠌⠉⢲⣒⣛⠷⠀⠀⢰⠿⠟⢿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠈⣷⠀⣤⠁⡄⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠏⠀⠀⠄⣹⡄⠀⠸⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢷⡄⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠐⣿⣯⠀⠉⢑⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠠⠿⠂⠀⠀⠀⠀⠚⠀⠀⡀⠀⠀⣶⡆⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁⠀⠀⠹⠀⣯⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠠⡆⠟⠲⠚⡃
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⣮⠆
⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣀⣤⣴⣶⣿⣦⣤⣤⡄⠀⠀⠀⣀⠀⠀⠀⡌⠁
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠻⠿⠿⠟⠉⠀⠀⠒⠀⠀⠀⠀⠐⠚