Happy 21st birthday Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⢰⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠿⣿⣿⣿⡆⢸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⡿⣀⠴⣺⡄⢠⡾⡴⢲⡆⣰⣏⠖⣲⢄⣴⢀⣴⠀⠘⠛⠛⠃⠀⣿⣿⣿⡇⠈⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢠⠇⠙⠋⠉⠉⢸⡿⠓⠋⠉⣸⡞⠚⠋⠁⠈⣡⠋⠁⠀⠀⠀⢀⣼⣿⣿⡿⠁⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⣾⣿⣿⠟⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣶⣶⡆⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡇⠀⠘⠋⠁⠀⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀⣀⣠⢴⡄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠉⠉⣿⡏⠉⢰⣏⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⣄⠀⣰⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠞⠃⢰⣿⠀⣀⣿⡟⠁⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣴⢶⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⢿⡿⠛⢛⣿⠀⣴⠟⣿⠀⢸⣷⠟⣷⣾⣷⠟⣷⣤⣶⢀⣿⠟⠟⠀⠀⢀⣴⣿⡿⠛⠉⠀⣿⣿⠷⣤⣙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣿⠇⠀⠈⣿⣼⡟⢸⣿⠁⣾⣟⠀⣿⣿⣟⠀⣿⣿⡏⢸⡟⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⣿⣿⢷⣤⠟⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣴⣾⣯⣤⣀⠀⢻⣿⣧⡼⢿⡾⣿⣯⣼⠏⣿⣯⣼⠟⠹⣷⣿⣷⢶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠋⠉⠁⣠⡿⠟⣿⣆⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿⠇⠀⠀⠀⣴⢿⣷⣿⣿⠋⣀⣀⣀⠀⡀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⣸⣿⠁⠀⣸⣿⠀⠀⠀⠀⢀⣿⠀⠀⢠⣿⡄⠀⣤⣼⣇⣼⠿⢿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠰⣿⡿⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⠀⣀⣀⣿⡿⣤⣴⡟⠁⢰⣾⠇⣴⠿⡇⠀⣀⣸⣿⣇⣸⣿⡏⠉⠁⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⢰⣿⠃⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢩⣿⡟⠉⠉⢿⣧⣼⣿⠈⣿⣼⣷⣶⡍⣿⣿⠉⣹⣿⣠⣶⣦⠀⣠⣶⣶⣿⡟⢀⣤⣶⣦⡄⢸⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⣸⣿⠁⠀⠀⢸⣿⣿⡏⢠⣾⠋⣿⡿⣠⣿⡇⢠⣿⣿⠋⣿⡟⢰⣿⠏⢰⣿⡇⣼⣿⠁⣼⣷⢼⣿⠀⣾⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⢀⣴⣿⣧⣄⣀⣴⡿⠛⣿⣷⠟⠁⠀⢿⣷⡿⢿⣧⡿⣿⡏⠐⣿⣷⠿⣿⣤⣿⣿⣶⠿⣿⣴⣿⣿⣼⠿⣿⣿⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠈⢣⡾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠎⣿⡤⢹⡇⠀⠀⠘⣧⠀⠈⠁⠀⡜⠀⠀⠀⣼⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠇⢸⡇⠀⠀⠀⠈⠣⡀⡠⠊⠀⠀⠀⢀⢿⡇⢸⣿⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣽⣾⣆⣀⠠⠔⢋⣦⣴⣶⣿⣶⣧⣌⡉⢽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⢀⣿⣿⡟⠋⠉⠙⢿⣿⣷⡜⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⡿⠟⣿⣿⡇⠀⢀⠘⠛⠛⠀⣴⠀⠀⣸⣿⣿⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣽⣽⣷⠊⣿⢀⡀⢀⡔⣿⣀⣾⣿⡿⢿⡆⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⢚⠈⡟⣯⣿⠀⣿⡃⣿⢸⣇⢻⡟⡻⠍⠓⠂⢙⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠸⣄⣿⣿⡛⠣⠙⠃⣨⣾⣿⡿⠋⠀⠀⣀⣀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⠃⢀⣾⣿⣿⣿⣦⠀⣼⣿⣿⣿⣶⣶⣶⣾⢹⣿⢲⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⢸⢸⠂

⠀⠀⠀⠀⠀⠀⣿⠀⢰⡟⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⠀⠀⢀⣄⢀⢠
⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡇⠀⢹⡇⢸⠘
⠀⠀⠀⠀⢠⠴⣿⠒⢻⡏⠀⣴⠟⣹⡇⢰⣧⠊⣷⣰⣦⠊⣦⣼⡆⢸⡇⠀⠀⢠⡿⠁⠀⢸⡇
⠀⠀⠀⠀⠘⠴⠟⠀⠘⠷⠴⠿⡴⠻⠧⢻⣿⡹⠏⢸⣗⡽⠇⠸⡧⢻⡇⠀⣰⡟⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢀⣀⢀⠖⢹⡇⠈⣛⠓⠒⠒⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⣻⠀⠀⠀⠄⡜⠃⢠⣇⡘⢳⡟⢸⠑⠄⠼⠁⣼⠁⡇
⠀⠀⠀⠀⠀⠀⠐⠱⣿⠈⢻⡄⣽⠀⢑⣦⡀⢠⡇⠀⢸⡧⣪⣄⠀⣠⡤⣿⠊⠀⣠⠤⡄⠀⡄⠀⡄
⠀⠀⠀⠀⠀⠀⢠⢶⠟⠀⣼⠃⣿⢀⡞⢸⡇⡸⣇⠀⢼⡷⠁⣿⡠⣿⢀⣿⠀⢰⡏⡰⣇⡰⣇⣰⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊⠁⠀⠈⠉⠀⠀⠉⠀⠈⠁⠈⠁⠀⠉⠁⠈⠉⠈⠉⠀⠉⠀⠉⠀⢉⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣰⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄⡈⡇⠀⣸⢼⢡⣤⢠⣤⣤⡄⣤⠄⢰⠇⣠⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⡁⠀⠀⠀⠈⠈⠉⠛⠁⠋⠀⠛⠀⠀⠈⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡎⢹⡆⠀⠐⠋⠙⡿⢴⡯⠔⠀⠀⣼⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠄⡠⢸⡗⢫⣄⡏⢰⠋⢾⡇⢸⡗⣶⢰⡖⣿⢠⡞⣲⠀⡇⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠸⣃⡼⠋⠓⠙⠀⠀⠓⠛⠃⠻⠘⠒⠙⠘⠓⠙⠊⠳⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⢀⣤⡄⣠⡄⣶⡆⠀⠀⠀⠀⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠦⠄⠀⠀⢀⡰⠃⣸⠁⠀⠀⠀⢠⡷⠼⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢶⡶⠆⣉⣩⠅⢉⣠⠀⣴⠇⠨⣵⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠘⣿⡛⠠⣤⣛⠁⠲⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⣡⣼⣧⣼⣇⣴⠞⣳⠀⣷⠊⣿⠀⣷⠊⣿⠀⣶⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣿⢰⣏⠰⣿⢼⣇⣠⡿⢺⣇⣠⡿⢺⣇⢾⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠰⡇⠀⠀⠀⠀⣼⠁⠀⠀⣼⠁⠀⠀⢀⠔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⣶⠿⣷⡿⢀⣴⣷⣿⣀⠀⢠⣆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⡟⢻⣿⣿⡀⠸⠋⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣾⠟⢀⣿⡏⢌⣉⡇⢸⣇⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⣴⣾⣟⣡⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⡶⠀⣽⣧⣀⡙⣉⣿⣛⣿⠉⠛⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣸⡇⢶⣋⡇⠙⡶⢠⡏⢠⡗⣽⢁⡞⢹⠇⣰⠏⣥⢠⡇⣼
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡿⠁⣰⡿⠇⠀⠗⠙⠗⠙⠀⠓⠻⠆⠻⠖⠻⠈⠛⠹⢢⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⡆⣿⠀⠀⡰⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⡀⠀⠀⠘⠀⣿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠴⠴⣂⣛⠀⠲⣄⠀⠠⡄⣶⢰⠖⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⡄⣎⠁⠀⢷⡀⢸⠄⠀⠧⣿⢺⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠙⣷⢈⡇⡇⡌⡇⡇⢯⠀⠀⢸⡇⢸⠄⣚⡀⠛⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠸⡆⢷⠀⣿⠙⡇⢇⠰⠃⠇⠡⠄⠀⡀⡱⣆⠀⢂⡖⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠉⠀⢁⡄⠄⠀⣶⠰⠸⡇⠀⡇⠘⣧⡸⡇⢸⠀⣶⡏⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣸⡆⡧⠀⠀⢿⣴⣴⡇⠰⢁⠄⠈⠧⠁⣈⠀⣿⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠈⡇⠸⠃⠇⠀⠀⠀⠈⢀⠀⣩⠡⣄⢰⣆⢀⣿⠀⣿⡏⢙⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣀⠀⡤⠀⡆⣶⠈⠻⣤⣰⡇⡄⣿⠀⢹⡆⣿⠀⣿⡇⠘⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⡆⢻⡄⣿⠀⠀⣿⠒⢠⡈⠻⡇⠁⡿⢐⠈⠇⠿⠀⠛⠓⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠇⢺⣷⠘⣷⠹⡇⠀⣿⠀⠚⣀⠀⠙⠊⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢻⣆⠋⠀⠁⠀⠀⠈⠉⢁⢓⡀⣷⡔⡠⠊⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠠⣠⠀⠁⠒⠲⠛⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣲⡆⣶⢶⣤⢤⣀⠀⠀⠀⠀⠀⣐⡒⣴⣦⢶⣲⢶⡖⣶⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⡍⣯⢹⣿⣿⢯⣯⡾⣿⣏⣿⠿⢦⠀⠀⠀⢸⢿⢿⣹⢾⢿⣿⣥⡿⡾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⢻⣽⣿⣼⡟⢹⢛⢸⡧⣥⠝⣻⣼⠫⣿⡄⠀⣘⢛⣛⣛⡚⣹⣟⣿⣇⡇⣟⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣷⣸⣾⣹⣖⠿⣿⠈⠉⠈⢹⠝⣿⢿⢽⣷⡯⣇⠀⠛⠛⠋⢋⡃⣿⠟⡛⢻⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠛⠛⠂⠚⠓⠀⠀⠀⣘⣃⣛⢘⣚⣜⣉⡃⠀⠀⠀⠀⢸⢽⣿⣿⢽⣾⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣭⣷⡴⡾⣥⣬⡭⠁⠀⠀⠀⠀⢨⣞⡹⡿⣱⢞⡛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⡿⣿⢿⣇⢗⠿⠃⠀⠀⠀⠀⠀⠀⢸⡏⣿⢰⣿⡌⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⡆⣿⢸⡾⠘⠋⠀⣀⣀⡀⣀⠀⠀⠀⠀⢸⡇⣿⣼⣿⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡿⢿⣿⢿⠻⣇⣿⣡⢧⢯⡇⣿⠧⣾⠃⠀⠀⠀⠘⣻⣭⣏⣠⣛⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⢹⢡⣯⣧⣯⡌⣼⣭⣷⡼⡼⣌⣽⣇⣭⣯⠄⠀⠀⠀⢰⢏⣽⣿⠉⣿⠹⣏
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠿⠟⠿⠯⠏⠼⠁⠿⠿⠿⠇⠇⠿⠿⠿⠿⠾⠀⠀⠀⠀⠸⠾⠿⠿⠆⠿⠩⠼⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠀⠀⢰⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡏⠀⠀⠀⠀⠈⣧⠀⠀⣠⣄⠀⠀⢀⡏⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢰⣿⣆⠀⣻⡿⠀⢀⣼⡇⠀⠀⠀⠀⣽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⣀⣾⣿⡿⣳⡿⣿⡰⣿⣿⣷⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣷⣭⣯⣭⡾⠋⠀⠘⢷⣜⢿⣿⣷⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠛⠛⠓⠀⠀⠀⠈⠁⠀⠉⢉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣶⣶⣦⣤⣼⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠄⣴⣿⠀⠀⠀⠀⣀⣤⡶⠖⠒⠶⣶⣦⣄⠀⠀⠀⢈⣉⣭⣶⡆⠀⠀⡀⣿⣦⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠳⠏⠐⠂⢀⠀⢰⣿⣿⠁⠀⠀⠀⢸⣿⣿⣇⠀⠈⠉⢹⣿⣿⡇⠀⡈⠐⠂⡻⢞
⠀⠀⠀⢀⠀⣀⡘⠐⠀⣁⣀⠀⠀⠈⠛⠛⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⢸⣿⣿⡇⠀⠀⣀⣀⠐⠂⢃⣀⠀⣀
⢠⠖⠒⠺⢶⡕⠾⠂⣞⢥⣤⠹⡄⠀⠀⠀⠀⠀⠀⣀⣾⣿⠟⠁⠀⠀⠀⢸⣿⣿⡇⢠⡏⣠⣬⣳⠐⠿⢪⡶⠗⠒⠲
⢳⡀⢟⡦⠐⠞⢳⣦⣌⣛⣡⠼⠃⠀⢀⣠⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠈⠧⣌⣙⣡⣴⡞⠳⠆⢰⡻⢂
⠀⠉⠉⠀⠀⠀⠑⡚⠀⠀⠀⠀⠀⣰⠋⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣛⠊⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠘⢈⡠⠐⠂⠠⠀⣧⣾⣿⣿⣿⣿⣷⣶⣤⣤⣶⡟⠀⠀⢸⣿⣿⡇⠀⠔⠀⠂⢀⠁⠃
⠀⠀⠀⠀⠀⠀⢀⡋⣷⣌⠩⠀⠀⠿⠋⠉⠉⠙⠛⠿⠿⠿⠿⠟⠁⠠⡤⠾⠿⠿⠷⠦⠄⠈⣥⣾⢙⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠻⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⣦⣄⠀⣶⠀⣴⡄⠠⣤⡀⠀⠀⠀⠀⠟⠉⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣼⣉⣿⡄⠀⡏⢻⣦⣿⠀⣿⡇⠀⣹⡇⠀⠀⢀⣀⣄⣀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠁⠙⣷⠀⠰⠃⠀⠸⠷⠀⠇⠀⠙⠟⠀⠿⠇⠴⠛⠁⢀⣴⠟⠁⣠⣿⠋⠀⠘⠋
⠀⠀⠀⠀⠀⢀⣾⡟⠀⣀⣠⣾⡟⠀⣀⡤⢤⡄⢀⣠⠴⣶⣶⠀⢀⣤⠴⢦⣄⢠⣾⠋⠀⣰⡿⠁⣠⢦⡄⠀⣠⡦
⠀⠀⠀⣤⠖⣿⡿⠛⠛⠛⠋⠁⠀⣼⣏⣤⠾⣣⡿⠁⣼⡿⢀⣼⡟⠁⣴⣾⢋⣾⠇⠀⢰⡿⠁⠀⢁⣾⠇⣼⡿⠁
⠀⠀⠀⠛⣰⠟⠁⠀⢀⣤⢦⣄⢸⣿⠋⠀⣀⣿⡇⣰⣿⡇⢸⣿⠀⣸⣿⠏⣼⡏⠀⠀⣿⠃⠀⠀⣾⠏⣼⣿⠃
⠀⠀⠘⠛⠷⣦⣀⠀⠸⠷⠀⣿⠀⠙⠳⠚⣁⠘⠛⣻⡿⠁⠈⠻⠞⠁⠛⠚⠻⠷⠃⠀⠻⠞⢁⡀⠻⠞⣽⠏
⠐⢻⣶⣿⠿⢮⡻⢷⣶⣤⠾⣫⡿⠿⣿⣮⡻⠶⢾⣫⣾⠇⠐⢿⣿⠗⠐⢿⣿⣶⡀⠘⣿⣷⠻⠧⠴⣞⣵⠒⠀⣾⣿
⠀⢸⣿⣿⠀⢸⣿⣷⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⢹⣿⠂⠀⢸⣿⣿⣷⠀⣿⣿⠀⠀⢸⣿⣿⠀⣴⡿⠃
⠀⣸⣿⣿⡀⣸⣿⣿⡆⢀⣿⣿⣇⣠⣿⡿⠁⠀⣸⣿⣿⡀⢀⣾⣿⣄⠀⣼⣿⣿⣿⣇⣿⣿⡀⠀⣸⣿⣿⣾⣿⣅
⠀⢹⣿⣿⠁⢹⣿⣿⠋⠈⣿⣿⡟⠛⣿⣿⡆⠀⢹⣿⣿⠁⠀⢻⣿⠃⠀⢻⣿⡟⣿⣿⣿⣿⠁⠀⢹⣿⣿⣿⣿⣿
⠀⢸⣿⣿⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⢸⣿⠀⠀⢸⣿⡇⢹⣿⣿⣿⠀⠀⢸⣿⣿⠘⣿⣿⡆
⢀⣸⣿⣿⣀⣼⣿⠇⠀⢀⣿⣿⣇⠀⣿⣿⣧⡀⠈⣿⣿⣦⣠⣿⡿⠀⢀⣸⣿⣇⠀⢿⣿⣿⡀⢀⣸⣿⣿⡀⢹⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣆⠸⠇⣴⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠈⠋⠐⠀⠋⠁⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⡇⠀⠋⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠛⠀⢸⣿⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⢿⣿⠘⠁⠀⠀⠀⠐⠋⠉⢻⣿⡆⠈⢻⣿⡆⠀⠀⠀⠀⠀⠈⠃⣿⡿⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⡈⠛⠁⢈⡋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢙⡁⠈⠛⢁⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠳⡌⠿⣿⣿⣶⣬⡙⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢋⣵⣶⣿⣿⠿⢡⠞
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣌⡉⠛⠳⠟⠂⠀⠀⠀⠀⢀⣼⣷⣶⣶⡇⠀⣸⣿⡇⠀⣀⡄⠀⠀⠀⠐⠻⠞⠋⢉⣡⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠀⠀⠲⠤⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠤⠖⠀⠀⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⣶⣿⣿⣿⣖⡢⠀⠀⠀⠀⠐⢶⣤⡀⢸⡇⢀⣴⡶⠂⠀⠀⠀⠀⢔⣒⣿⣿⣿⣶⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⢈⠁⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣄⡐⢂⣤⣤⣤⣤⣄⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⣀⠀⢀⣀⢀⡀⠈⢙⣛⣋⢉⡁⠀⣀⠈⢉⣙⠛⢉⣁⢀⡀⠀⠀⠀⡀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣨⡿⠀⢸⣿⠀⢸⣿⢀⣿⠂⠈⠀⣿⠀⠁⠀⣿⣀⢸⣿⠀⢸⣿⠀⢻⡆⠀⡰⣷⡀⠈⢻⣆⠠⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣇⢈⣿⠂⢸⣿⠀⢸⣿⠙⣷⡀⠀⢀⣿⠀⠀⠀⣿⠀⢸⣿⠀⢸⣿⢀⣼⠇⣰⠂⠻⣷⡀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⠁⠀⠀⢁⣠⣤⣭⣀⣀⠀⠀⣀⣀⣥⣤⣤⡁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠙⠛⠛⠛⠁⠂⠙⠛⠛⠛⠉⠙

⠐⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠒⠂⡐⠂⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠒⠂⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢠⠀⠀⠀⡞⣉⠅⠀⠁⠀⠀⡜⠃⠀⠀⠀⢀⠀⠠⠀⠀⠀⠉⠁⠈⠠⠠⡌⡀⠀⠀⠀⠀⠀⠠
⠠⠀⠀⠀⣬⣄⠀⠀⠀⠠⢠⡳⣧⠀⡀⢰⣡⠀⠀⠀⣔⠁⠀⠀⡃⠀⣠⠓⡧⢀⡄⠀⠀⠀⠀⠀⠐
⢰⠀⡀⠸⠏⢻⠆⠀⠰⣶⡨⠄⢘⣈⡄⠾⠉⠇⠀⢠⡛⢧⡔⡜⢾⣏⡽⠀⣒⢸⣦⠀⢀⠀⠀⠀⢠
⠘⢰⣦⠆⡃⠌⠑⢍⡏⠈⣿⡂⢼⠏⠹⢖⠀⣵⢁⠐⠄⣸⠿⣧⢀⣻⣨⠀⢼⠇⠈⣇⣠⡀⣠⢀⣖⡆
⠸⡛⠀⡃⣬⡐⠀⢻⠇⠀⡇⢁⠘⠃⠈⠡⠀⣾⠛⡎⠓⢊⠀⢿⠰⡇⢙⣇⡻⣬⡀⣿⠉⢿⣁⣼⠀⢺⠄
⠰⠩⠀⣆⡏⠠⠀⠠⡅⢰⠉⢱⠀⢈⠀⠐⠀⣝⠀⡇⠀⣙⠀⠹⢈⣅⠐⡗⡇⠨⡥⣟⠀⢼⠉⣻⠁⢰
⠘⠒⡬⢹⠛⠃⠠⠀⣷⢘⠂⢴⠀⡂⠀⠀⠀⢫⠀⣇⠓⢬⡀⠺⠠⠇⢨⡇⠆⢈⡗⣿⢄⣺⠀⢾⠢⢼
⠀⢸⡝⠻⠄⢽⢐⠠⡏⠸⢀⣺⠁⠉⠘⠅⢀⠀⠤⡏⠀⠐⠈⡟⠀⠀⠰⠇⠑⠠⡏⠘⡂⢼⠀⣽⡅⢺
⠀⢨⡂⢬⡁⢺⠈⠐⠃⢰⡂⢼⡴⠛⢢⠀⠈⠄⠕⠃⠀⠀⠀⠀⠀⠀⠈⢀⡮⢐⡇⠀⠡⢚⡀⠎⠀⠉
⢸⠀⡅⢰⠆⣹⠀⠀⠀⠐⠃⠂⠈⠉⠉⠀⠀⠀⠀⠀⠐⠈⠀⠒⠁⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠉
⠈⠐⠂⢨⠥⠒⠀⠀⠀⢰⠀⠆⠀⠤⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰⠉⡠
⠀⠀⠉⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⡈⠀⠘⠈⠁
⢀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠘⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣽⣿⣯⡿⣷⡄⠀⠀⣠⣾⣻⣿⡇⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⠃⠀⢸⣿⣽⡇⠀⣾⣻⣾⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⣠⣾⣳⡿⠁⠀⠈⠉⠀⣿⢾⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣾⠟⠀⠀⠀⠄⠀⠀⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣾⣿⣁⣀⣀⡀⠀⠀⠀⢀⣿⣽⡇
⠀⠀⠀⠠⠀⢀⣀⣀⠀⠀⢀⣠⣤⡀⢺⣯⣿⣽⣯⣿⣽⣯⡿⠀⠀⠀⠈⢿⣾⠇
⠀⠐⣶⡖⣶⣯⡈⣿⣧⢠⠏⠀⣽⣿⠀⠀⠀⠀⠀⣀⣀⣴⣄⣠⣦⣤⠞⠀⠀⡀⠀⣴⡆⠀⠀⠀⣀
⠀⠘⠞⠁⡇⣿⣷⢸⣿⠏⠀⣠⣿⠏⠠⡦⣾⡇⠀⠁⢸⣿⠁⣽⡿⢀⡀⠀⠀⣀⢀⣿⠇⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠙⠿⠋⣿⣿⠀⠙⢻⣷⣤⣴⡷⣹⠛⣿⡇⣼⡟⠀⣿⣿⢹⣿⠀⣾⡏⢻⣿⠀⣼⡟⢳⣿⢃⣾⡇⣼⣷⢰⠆
⠀⠀⠀⠀⠁⠀⠀⠿⢇⡴⠋⠉⣿⣷⣿⣧⠇⢰⣿⣷⣿⡇⣸⠿⠇⢸⣿⡸⢿⡷⢿⣟⣰⠿⠷⢻⣿⡼⣿⡞⣻⣿⠏⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣬⣴⠿⠃⠙⠋⠀⠈⠛⠁⣿⣧⠇⠀⠀⠘⠛⠁⠠⠀⠘⠛⠁⠀⠀⠘⠋⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠈⢸⡇⠀⢀⡄⣤⠀⣄⠠⡄⣄⠠⡄⣤⢀⡄⠀⠂⠀⠀⠘⣧⠀⠈⢻⣧⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡟⠀⢸⡇⠀⠻⡄⢿⡀⣿⠴⠏⣿⠴⠟⢿⠘⣷⠈⠢⠀⠀⢠⡟⠀⠀⢸⡿⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠞⠁⠀⠀⠓⠀⠀⠀⠀⢈⣿⠀⠀⣿⠀⠀⠀⢸⡿⠀⠀⠀⢀⡾⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⠇⠀⣸⠏⠀⠀⠀⣼⠃⠀⠀⣠⡞⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠈⠁⠀⢀⠀⠉⠁⠀⢀⡴⢋⣀⣀⡀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠇⠀⠀⠀⠀⢀⠀⢀⡾⠁⡁⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢤⣐⠁⢀⡀⠀⠀⠀⣾⠆⣾⡇⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢳⣴⠀⠿⣀⠀⣿⠀⢿⠃⢰⣄⡤⢼⡏⠀⣠⠄⣤⠀⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣼⠿⡄⠠⡏⠀⢿⠀⠹⠇⣾⠹⣇⠸⡇⠀⢻⡀⣿⠰⣏⢸⡧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠈⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⢻⡆⠚⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢡⠀⣿⠀⠀⣀⡀⠀⣀⡀⢠⡶⠲⣶⠀⢰⣦⡤⢶⣄⣀⡀⣀⣀⠀⢴⡆⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷⣦⡆⢠⣿⡄⠁⢸⡇⠀⢸⡇⢹⣧⡸⣿⣦⢰⣿⣀⣾⡇⣸⡟⠈⣿⠀⣼⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠐⢶⣄⠀⠐⣿⡗⠒⣿⡇⢰⣦⠐⢈⣁⣸⣿⠈⣍⡀⣿⡟⢺⡋⣠⣼⣟⣁⣸⠇⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⠷⠀⣿⡂⣼⠟⠘⡃⠿⠇⠁⠿⠇⢸⣿⡤⠀⣿⡏⣉⡀⠋⠀⠻⠃⣿⠁⠛⣉⣙⣋⠁⣤⡄⢀⣤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣴⡿⢰⣿⠉⢻⣦⢸⣿⠀⠻⠧⠴⣶⢸⣿⣱⢘⣿⠋⢸⣿⠀⣼⡏⢙⣿⡀⢸⡇⢨⣿⡀⣿⡇⠸⣿⡄
⠀⠀⠀⠀⠀⢀⠀⢿⣄⠘⠛⢀⣼⡿⠹⠿⠠⣶⡎⠸⣿⡌⠙⠉⠘⠛⠀⠘⠟⠀⠻⡧⠜⢿⡿⠘⠛⠊⠛⣯⣽⣓⣁⣸⠇
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢠⣼⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠀⠀⡀⠀⠀⠀⠀⢀⠟⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠒
⠀⠀⢀⠈⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠐⠇⡀⠀⠀⢠⠀⠀⠀⠀⢀⡀⠀⠀⠀⢲⠈⠉⠐⠃⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠃⡀
⠀⠀⠈⠀⠉⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠸⠃⠀⠀⠀⠀⠀⢀⠀⠀⣠⣄⣀⠀⠀⠖⠀⠀⠀⠀⠠⠀⠀⠀⠤⠀⠀⠐⠈
⠀⠠⡀⠄⠀⠀⣠⡀⠛⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢁⡀⠀⠀⠀⠀⠈⠹⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⣤⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠋⠀⠉⠀⠀⠞⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠷⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉
⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠤⠀⠀⠀⠀⠉⠂⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡀
⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⣠⣶⣾⣷⣶⣄⠀⠀⠀⣠⢾⣿⣶⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠀⠀⠙⢿⣧⠀⠀⠁⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⢿⣤⡀⠀⠀⢠⣾⣿⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣴⣿⣿⠏⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣅⣀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⢷⣶⠀⠀⠐⠚⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢀
⠀⠰⡶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠃⠀⠁
⠀⢠⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣾⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣤
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠟⠀⠁⠀⠀⠀⠐⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⢤⡀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠁⠀⠉⡅⠈⠀⠀⠀⠀⠈⠀⠀⠀⠶⠀⠀⠀⡀⠀⠀⣦⠀⡄⠀⠀⠽⠆⠀⠀⠀⠀⠀⠀⡀
⠀⣼⠁⠀⢴⡄⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣍⣀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢀
⠀⢹⡤⠀⠤⠐⠂⠀⣀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⠘⣀⠀⠀⠀⠀⢀⡈⠋⠀⠈⠁⡀⠀⢠⡴⠀⠀⠀⠀⠀⠀⠀⢠⡋⠀⠀⠐⠢⡄
⠀⣼⠀⢠⣀⠀⠀⠸⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠲⡶⠘⠆⠀⠀⠀⡈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠛⢠⡤⠘⠇⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠌⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⣠⠰⠹⢿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⢀⡴⠲⣶⣶⢶⣄⡀⢀⣿⡶⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⣠⢤⡀⠂⠈⠹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⢏⡈⡔⢡⣁⠊⠑⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠄⣸⣥⣤⣤⣤⣭⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣷⣾⣿⣿⡿⢃⠀⠁⢹⠛⡁⢁⣤⡄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣠⡾⠁⢈⡼⢣⡈⠀⠁⠈⣌⠁⠀⠀⢿⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢢⢄⡄⠀⠀⢸⠁⠉⡀⠉⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⡠⠈⠹⠧⠖⠋⠀⢀⠑⠀⠀⡔⠙⠀⠀⣤⠈⣿
⠀⠀⠀⠀⠀⠀⠐⠀⠸⣿⣿⣿⡏⢡⢇⡃⠃⠀⠀⠈⠀⠘⣧⠀⣀⣠⠤⠚⠁⠀⠀⠀⠠⣎⡀⢀⣀⠠⠤⠔⡚⢿⢃⡤⠊⠀⢀⠃⠀⠰⠀⡉
⠀⠀⠀⢄⡶⡾⣆⠀⠀⠹⣿⡟⡅⣵⡎⠙⢓⢠⠄⠀⠀⡀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡸⠀⠀⠀⠀⡇
⠀⠀⠀⠾⣿⢾⣧⠀⠀⠀⠙⢷⣴⡹⡇⠂⠈⠁⠀⠀⠀⣿⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢰⠁
⠀⠀⠀⠰⣿⣷⠁⠀⠀⠀⠀⠀⣿⣇⠈⣺⠀⠀⠀⠀⠀⠈⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠁⠀⢸⡆⠇⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⡌
⠀⠀⢀⣄⠘⣻⡅⠀⠀⠀⢀⣉⠁⠹⢆⣿⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡑⢌⣣⠸⢸⠀⠀⢰⠀⠐⠃⠀⠀⢀⡠⠖⣹⣷⡄⠀⠀⠐⣉
⠀⠀⠀⠏⠳⠹⣗⠀⠀⠈⠙⠛⠋⠀⡜⠛⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠑⡄⣏⠦⡘⠀⠀⠸⣄⢸⣤⡠⠖⠁⢀⣼⣿⣿⣷⠀⠀⠀⠈⠐
⠀⢠⣶⡆⠀⠀⢋⣄⠀⠀⠀⠀⢰⣾⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣠⠀⠠⢸⠄⠃⠀⠀⢠⣿⣿⣯⠀⠀⢠⣾⣿⣿⣿⣿⡄⠀⠀⢠⠂⠠
⠀⠀⠻⠇⠀⠶⡀⠉⠀⢀⡠⠴⢏⡈⠙⠳⢷⢶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⢀⡧⠔⢻⡄⠀⠘⢃⣿⠻⠀⣰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⢸⠁⠀⣿⠙⠓⣿⠌⠉⣲⣤⣀⣀⣀⣹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡧⠶⠀⠸⣶⣾⣿⣷⡀⠀⣸⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⣸⡀⢼⡷⠟⠀⣈⢻⢫⣮⢿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⢻⣿⣿⣿⣷⣠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢠⠟⣡⣤⠀⢰⣯⢓⠾⡀⠡⠊⢀⣀⣀⠀⠀⠀⢀⡴⠴⠦⠼⢄⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢠⡄⢀⡎⠀⠀⠀⡎⢸⠯⠜⡲⠆⠞⠁⠀⣀⠤⣊⠖⠲⣭⠻⢆⠀⠈⣞⣙⣋⠱⠂⠀⡝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣠⢿⠀⡜⠀⠀⠀⣸⢁⣤⣦⠞⠀⣴⣶⣶⣿⣶⣶⣶⣾⣷⣿⣶⣾⣶⣶⣿⣷⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣰⣷⡾⣷⠁⠀⠀⠀⠧⣛⣃⣍⠀⠀⠈⠉⠹⠋⠉⠉⣉⣉⣭⣭⠭⢭⣿⡉⠉⠉⠉⠉⢩⣿⣿⠉⠉⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠉⢉⣿⣸⣄⣀⣀⣀⣀⣀⢠⡭⣍⡁⠚⠻⢞⣁⢾⡏⢩⣟⠷⠀⢠⣞⠛⣿⡀⠀⠄⠀⢸⡇⡀⠀⠀⠀⣠⠊⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠛⠛⠛
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡠⢀⢽⢏⡀⣤⠀⢀⣐⡄⠀⠀⢹⣯⠆⢤⣴⣾⣿⡇⠀⠀⣀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠓⢾⡎⣧⡻⠃⠀⠺⠙⠁⠀⣾⠎⢹⣯⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣦⣸⡆⢹⢁⣶⡿⠂⢠⣶⡇⠀⣴⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣇⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣾⣿⣷⣇⡈⢉⣤⠤⠽⡿⠒⠋⠙⣇⠁⠀⠉⢻⣿⣿⣿⣿⣿⣏⠙⣷⢀⠤⠀⠰⠄⠂⢀⣩⣽⡮⠁⠈⠈⠁⠚
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⠉⠀⠀⡰⠉⢦⠀⠀⠹⣆⠀⠀⣾⠉⢿⣿⣿⣿⣿⣦⢹⡄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⠛⠉⠀⡆⠀⢠⡅⠀⠀⠳⡀⠀⠹⣦⣀⣿⠀⠀⠙⣿⣿⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⣇⣀⣀⡈⢻⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢠⣁⠀⠘⣆⠀⠀⠀⠙⣿⣿⢿⣿⣿⣶⣤⣤⣀⢀⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡞⠀⠀⠀⢸⣿⠻⢿⣿⣶⣄⣀⠀⠈⣧⠀⣼⠀⠉⠛⣻⣿⣿⡿⠋⠈⠋⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠓⠚⠁⠀⠀⠀⠀⠃⠀⠀⠈⠙⠛⠿⠿⠗⠛⠚⠋⠠⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⠚⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⣶⣿⣶⣯⣭⣿⣧⣿⣿⡿⣿⣿⡧⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⠟⠹⠿⠟⢻⡟⢻⣿⡶⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡷⡌⣿⣿⡆⠀⠀⠀⢈⠉⠉⠈⢳⡦⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⠀⠀⢺⣿⣿⣿⠀⣸⣇⡇⠀⠀⠀⠀⠀⣶⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢨⣿⣿⣿⣿⡟⠀⠀⠀⠉⠉⠛⢀⣿⠿⠁⠀⠀⠀⠀⠀⠹⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠗⠀⢤⣾⣿⣻⢋⣿⣿⣶⣤⣀⡀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⢰⣿⡽⣱⣿⣼⣿⡇⠙⠋⠉⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠙⢶⣤⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⢁⠀⢀⣠⣤⣴⣶⠿⣿⡿⣇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠘⡟⢜⢿⣿⣿⣷⡻⣧⠄⠀⠀⣾⣧⣭⣿⣿⣿⣿⡟⣼⣿⢟⣿⣿⣿⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⣰⡿⢸⠘⠿⣿⣿⢿⣟⠛⢦⣠⠏⠀⢺⣿⣿⣿⠟⣱⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣍⠀⠀⠀⢠⣿⢣⣯⣴⡇⠨⣿⣿⡿⠃⠀⠘⠂⢀⠀⢹⡿⢻⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢀⣠⣾⣿⣿⣿⣿⣿⠀⠚⠁⠀⠀⠀⠀⠁⠘⡀⠀⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣇⣀⣩⣿⠟⠉⢻⣿⣿⡏⢇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢿⣿⣇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢣⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣭⡤⠀⠀⠀⠰⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠶⠿⡥⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡰⠳⣄⠀⠀⠀⠀⠀⣠⠞⠑⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠖⠉⠙⠦⣀⠀⠀⠀⢸⠁⠀⠈⠓⠦⣄⡀⢰⠃⠀⠀⠈⢧⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠙⠲⠤⣬⡦⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡰⠓⢶⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣠⠶⠚⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠘⠦⣀⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⢢⠀⣀⠀⠀⠈⠁⠀⡜⠉⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⢹⡹⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡿
⠀⣴⣤⣶⢀⠀⠈⢇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃⠀⡀⣶⣤⣦
⣻⣽⠞⠻⢯⠇⠀⠘⢎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠃⠀⠸⡽⠟⠳⣿
⠙⢯⡀⠀⠸⢤⡀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⠶⢦⠀⠀⠀⠀⠀⠀⠀⢀⡜⡱⠃⠀⢀⡤⠃⠀⢀⡽
⠀⠀⠈⠳⡀⠀⠙⠦⣀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠀⠹⡍⠉⠉⠉⠉⢉⠏⠀⠀⠀⠀⠀⢀⣠⣯⠞⠀⣀⠴⠋⠀⢀⠞⠁
⠀⠀⠀⠀⠙⢦⠀⠀⠈⠳⢄⡀⠙⠻⣦⣄⠀⠀⠀⠀⠈⠓⠒⠒⠚⠁⠀⠀⠀⠀⣠⣴⠿⠋⢀⡠⠞⠁⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠳⢤⡀⠉⠛⠷⢦⣤⣤⣀⣀⣀⣀⣤⣤⡴⠾⠛⠋⢁⡤⠞⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠉⠳⡶⣄⠀⠀⢘⡇⠀⠀⢸⡃⠀⠀⣠⢶⠞⠉⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⡰⠃⠀⠙⠒⠋⠱⢤⡤⠞⠙⠒⠋⠀⠈⢆⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢦⡀⠀⠀⠀⠀⠀⠀⡟⢆⠀⠀⠀⠀⠀⠀⢰⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⢦⣷⠀⠀⠀⠀⠀⣰⣷⣬⡇⠀⠀⠀⠀⢀⣼⣶⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣸⣿⠀⠀⠀⠀⢸⣿⢀⣼⡏⠀⠀⠀⠀⢾⣏⣀⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣫⠿⣇⠀⠀⠀⠀⢀⣏⡹⠿⡄⠀⠀⠀⠀⢸⣛⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡠⠖⡏⠀⠀⠀⠀⢸⣁⡤⢖⡇⠀⠀⠀⠀⢸⣁⠤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠖⢋⣿⠀⠀⠀⠀⢸⡧⠒⢉⡇⠀⠀⠀⠀⢸⠵⠚⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢚⣩⣧⠀⠀⠀⠀⢸⡶⢚⣩⡇⠀⠀⠀⠀⢸⠔⣋⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢉⡤⡟⠀⠀⠀⠀⢸⠋⣉⡤⡇⠀⠀⠀⠀⢸⢋⣡⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡥⠒⣧⠀⠀⠀⠀⢸⣏⠥⠚⡇⠀⠀⠀⠀⢸⣩⠴⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⢚⡏⠀⠀⠀⠀⢸⡥⠴⢚⡇⠀⠀⠀⠀⢸⡤⠖⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠷⠾⠥⠿⠤⠖⠒⠒⠚⠓⠚⠛⠓⠒⠒⠒⠒⠺⠶⠯⠽⢄⣀⣀
⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠢⢤⣀
⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⢀⡤⠒⠢⣄⠀⠀⠀⣀⠀⠀⡞⠉⢧⠀⠀⢰⠋⠀⠉⢇⠀⡞⠁⠹⡄⠀⢀⡞⠉⠉⢳⡀⠀⣠⢤⡀⠀⢀⠀⠀⣷
⠀⠀⠀⠘⡇⠀⠀⡾⠀⠀⠀⢸⠀⠠⡏⠉⡇⠀⡇⠀⢸⠀⠀⡅⠀⠀⠀⣀⠀⢻⠀⠀⡇⠀⢸⠀⢀⠀⠀⡇⠀⡇⢰⡇⠀⢸⢳⣤⠏
⠀⠀⠀⠀⠳⣀⣠⠃⠂⠀⠀⢸⠀⠀⡇⠀⠣⠤⠃⠀⡞⠀⠀⢹⠀⠀⠀⡇⠀⠸⣤⠀⡇⠀⠸⡀⠀⠀⢸⠁⠀⡇⠀⠓⠲⠋⡼
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡀⠀⣹⠠⠀⠀⠀⠘⡇⠀⠀⠘⡆⠀⠰⡇⠀⠀⡇⢰⠇⠀⠀⢧⢀⠀⠘⢤⡤⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠈⠀⠀⠀⠙⡚⠁⠀⠀⠀⠀⠸⡅⠀⠀⢀⡇⠀⠒⠙⠒⠚⠁⢸⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠤⠴⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⢀⣀⡤⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢤⣀⡀
⢀⡠⠖⠋⠉⠀⢀⣠⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣄⡀⠀⠈⠙⠲⢤
⡎⠀⠀⠀⠀⢼⠉⠀⠘⣆⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠉⣷
⠙⠦⣀⠀⠀⠈⠓⠦⢤⣀⣉⠓⠒⠲⠤⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠴⠒⠒⣋⣁⡤⠴⠚⠁⠀⠀⣀⡴
⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡈⠉⠉⠙⠒⠒⠒⠒⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠒⠋⠉⠉⢁⣀⣀⡤⠤⠖⠚⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣙⣓⣒⣒⣶⣲⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣖⣶⣒⣒⣒⣋⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠋⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⣀⣴⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠟⠛⣿⣿⠀⠀⠀⢠⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠉⠀⠀⢀⣴⣿⡏⠀⢀⣴⡿⢋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⢀⣴⣾⣿⠟⠀⠠⠚⠁⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢀⣴⣿⣿⣿⠏⠀⠀⠀⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣴⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣤⣤⣤⣤⣄⡀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠁⢀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠋⠉⣇⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣍⠀⠀⠀⣿⡈⠀⠀⡀⠀⠀⠀⠀⠀⡀⠆⠀⠔⠈⠀⡇⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⡏⠐⢤⣴⣿⣿⡆⠀⠀⠀⣀⠀⠀⠀⠀⠘⣦⠀⠀⢀⠃⠠⠀⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠋⠉⠁⠀⠈⠙⢻⣿⣿⠆⠀⠰⠂⠈⠀⠄⡀⠀⠁⠐⠠⠊⠀⠔⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⡀⠀⠀⠀⢹⠀⠘⣿⡷⠀⠀⠀⠀⠄⠀⠀⠁⠄⠀⠀⠀⠙⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠠⠁⠀⠀⠀⡆⠀⠀⠡⠀⠀⠀⠁⠂⠀⡔⠀⠀⠀⠀⠀⠀⠀⢡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢜⠀⠀⠀⣰⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠏⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⣼⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠖⠀⠈⠒⠇⠀⠀⠀⢠⣤⣤⣀⣀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣿⠿⠛⠉⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢉⠙⠉⠀⠈⠁⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⢀⠆⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠰⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

⡿⢿⠿⠿⠿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡄⠤⠤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡎⠀⠀⢰⠃
⠀⠀⠀⠀⠲⠬⠥⡤⣬⣭⣬⣭⡵⡇⠀⠀⠀⠀⠀⢀⠠⢼⠤⠀⢰⡧⠤⢤⡧⣀⠀⠀⠀⠀⠀⠀⠀⢀⣁
⠀⠀⣠⡒⠓⠶⠶⠤⠶⠶⠶⠾⠟⢓⣠⠀⠀⠀⠀⠀⠒⠶⠶⠦⠤⠤⠴⠶⠶⠖⠇
⠀⠀⢨⠉⠙⠛⠛⠳⠚⠻⠛⠛⠙⠋⢽⠀⠀⠀⠀⡏⠁⠚⢀⠆⠀⠀⠜⠀⠒⠀⡆
⠀⠀⠈⡄⠈⠀⠀⠀⠐⠚⠀⠃⠃⠀⣼⠀⠀⠀⠀⢷⠠⠤⠄⡀⢀⡠⢂⠤⠤⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠙⠛⠒⠒⠶⠖⠶⠖⠓⠒⠋⠉⠀⠀⠀⠀⠀⠉⠛⠓⠒⠒⠒⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉

⠀⠀⠀⠀⠀⢶⣈⣹⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⢸⠀⠀⠀⡆⠀⡄
⠀⠀⠀⠐⣛⣉⡥⢴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠔⠺⠖⠺⠛⠈⠿⠉⠏⠉⠏⠉⠟⠐⣢
⠀⠀⠀⠈⠁⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆⣙⠒⠒⠒⠒⠒⠒⡒⠒⢒⣒⡋⡉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠠⠀⠀⠀⠀⠀⠀⢁⠌⠁⠖⠁⡐⠃⠈⠀⠛⠀⠉⢙⣐⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⠉⠉⠒⠚⠛⠒⠚⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢨⠀⠀⠀⠖⠄⢀⠒⡒
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠤⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠶⠦⠶⠶⢾⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⡀
⠀⠀⠀⣴⡪⡩⣍⣉⡆⠀⠀⠀⠀⣼⠉⠈⠒⠒⠶⠶⠶⠘⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠠⠟⡉⠟⠫⠍⠟⢓⡀
⠀⠀⠀⡇⠐⠊⠁⠂⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⢘⠛⠛⠛⠛⠛⠛⣛⣛⡀
⠀⠀⠀⡗⢒⠶⠒⠚⡇⠀⠀⠀⠀⣿⣿⠠⠤⠴⠂⠖⠀⣦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢘⣉⣉⣉⣙⣛⣛⣋⡉
⠀⠀⠀⢠⣀⣀⣠⣤⠄⠀⠀⠀⠀⢸⣌⡊⠀⠀⠀⠀⠀⠠⣼⠀⠀⠀⠐⠂⠀⠖⡆⠀⠀⠀⢘⠿⣷⣭⣮⣾⣿⣿⡷
⠀⠀⠀⠨⠌⣩⡥⠠⠀⠀⠀⠀⠀⠸⢦⣄⣉⣙⣓⣒⣒⣂⡏⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠀⠈⢿⣿⣾⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠁⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠴⢚⡿⠀⠀⠀⢀⣶⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⢳⡄⠀⠀⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠺⡏⠙⠹⠓⠗⠛⠉⡷⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠇⠀⠀⢀⣠⡴
⠀⣀⡀⠀⠀⠀⠀⠈⢳⡝⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡄⠀⣧⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣾⠤⠶⣾⢕⡠⠶⠚
⢀⣬⡿⣷⣶⡖⠶⠒⠉⠀⠀⠙⡆⠀⠀⠀⠀⢀⡾⠀⡾⠀⠀⢸⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠠⡿⠟⡒⠒
⠀⢬⣭⣻⣯⠶⣠⡤⠦⢤⡀⠀⣷⠀⠀⠀⠀⣼⠃⢸⠇⠀⠀⠈⡷⠶⠶⢶⡞⠀⠀⠀⢹⠀⠀⢻⡄⠀⠀⠀⠀⠀⣸⠃⠀⡴⠛⠉⠉⠉⠉⠙⠛
⠀⠀⠀⠾⠖⠋⠁⠀⠀⠀⣧⠀⢸⡄⠀⠀⢰⡏⠀⢾⠀⠀⠀⠀⢷⠀⠀⣸⠁⠀⠀⠀⣿⠀⠀⠈⣷⠀⠀⠀⠀⢠⡏⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣇⠀⠀⣿⢄⡀⠘⣆⠀⠀⠀⢸⡄⣰⠃⠀⠀⠀⢀⣿⠀⠀⠀⠸⡆⠀⠀⠀⡞⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢹⠀⢰⠇⠀⠉⣱⠘⢧⣤⠤⢬⣧⣏⣀⣀⡀⢀⡾⠙⣧⣤⠴⠞⣧⠀⠀⣸⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⡏⠀⠀⣰⠏⠀⣎⣵⣤⡴⠷⠾⢤⣀⡿⡾⠁⠀⠘⣇⠀⠀⠸⡆⢠⠏⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡿⠀⠀⢠⡏⠀⠀⠀⢈⣆⣀⣀⣀⠀⠈⣹⠇⠀⠀⠀⠘⣆⠀⠀⠙⠞⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠀⠀⣠⡴⠋⠁⠀⠉⠉⠉⠉⠛⢦⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⡞⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⢦⡀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡾⠁⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠤⠚⠓⡀⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⣀⠀⠀⠀⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠾⠆⠀⢀⠟⡆⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢠⠎⠀⠱⠀⠀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⣃⠄⠀⡋⠀⢠⠀⡇⠀⡴⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠳⢄⡣⠜⡀⢘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢱⠀⠇⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠈⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠉⡉⠉⠉⠙⠛⠉⠉⠉⠉⠉⠉⠉⠙⠃⡅⠀⠀⡇⠀⠀⠀⠀⠤⠤⠀⠠⠀⠒⠒⠒⢂⣤⡔⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠡⣌⠉⢛⠀⠀⢠⠤⣀⣄⡀⡀⠀⣀⡀⠀⠀⠀⠲⠂⡀⠱⠁⠀⠀⠀⠀⠀⡀⠀⣠⡀⠦⠤⣇⣀⠄⠐⠁⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢣⠀⠀⠉⠁⠒⠛⠓⠈⠁⠎⠤⠥⠤⠥⠤⠤⡤⠤⠤⠀⠠⠀⡀⠒⠒⢚⣂⠀⠀⠈⠉⠁⠀⠓⠀⠈⠀⠀⠀⠘⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣠⣤⣠⡤⣄⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠁⠈⠁⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣀⠀⢤⣼⢶⣄⣠⢏⢧⡨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠳⠙⠡⠋⠐⠩⣻⡷⠄⠀⢰⠹⡇⠠⣤⡾⣫⢛⡽⣲⠖⡲⢋⠔⣡⢜⣿⡂⠀⣰⢟⢟⡽⡫⠣⠢⠪⠃⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡗⢈⣳⣒⠤⣰⣆⣀⡈⠛⣦⠠⠆⠈⠛⠹⠋⠈⠀⠉⠀⠀⠈⠀⠀⠀⠀⠉⠉⠘⡙⠁⢁⣀⡀⢠⡄⡀⠤⠰⠆⢺
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠎⠱⠦⠴⡤⢬⣸⣗⣈⣲⠠⠿⠔⣂⡐⠤⣶⠴⠂⣲⠠⡿⠤⠒⣂⡒⠚⠓⢊⣡⣤⣍⣉⣉⡴⢲⡳⠖⢻
⠀⠀⠀⠀⠀⠀⠀⣠⡗⣦⢆⠄⡀⢀⠀⠉⠀⠈⠁⠽⠉⠙⠛⠋⢽⠉⠛⠋⠛⠉⣤⠈⠓⠚⠋⣭⠉⠛⠛⠉⠀⠛⠀⢀⢀⢀⡼⣥⣤⢨⣀
⠀⠀⠀⠀⠀⠀⡼⠀⠛⢶⣥⣊⣔⡡⡢⢚⠔⡠⣠⢆⡤⡠⢀⠴⡤⢂⠄⡀⣀⢀⡀⡠⢠⢀⣄⣀⡰⣢⢂⡖⣴⡿⠿⠛⠛⠛⠚⠘⠉⠀⢻⠳⡀
⠀⠀⠀⠀⠀⢰⢥⡶⢤⣤⠀⡀⢀⠉⠙⠓⠾⠾⠵⠯⠞⢛⣷⣛⡒⠓⠮⠶⠵⣫⡾⠾⠓⠋⠉⠉⠉⠉⠉⠙⠋⠀⠀⢀⠀⠀⠀⣴⣲⣄⠀⠀⡛
⠀⠀⠀⠀⠀⠀⠟⢧⡓⣷⡺⡫⣻⡀⠐⣧⣦⠀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣀⠀⠀⠀⢀⣤⣶⡲⣤⣴⢿⣧⣀⡠⠛⠛⠛⠗⡞⠁
⠀⠀⠀⠀⠀⣶⠠⠤⠀⠀⠈⠈⠪⠿⠴⠾⣻⣤⣿⣟⣽⠉⠀⣾⡻⣧⣤⣶⣛⣿⣦⣶⣣⣮⣧⡀⣤⣼⠟⠓⠙⠘⠋⠁⠀⡀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⣿⡀⢠⠄⣀⣀⠀⣀⠀⣀⡀⠀⠀⠀⠈⠛⠳⠜⠛⠛⠛⠉⠈⢈⠁⠉⠀⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⢀⣲⡇
⠀⠀⠀⠀⠀⣿⣌⣙⣃⣬⣩⣝⠋⠋⣉⣉⠣⣐⣶⢀⣤⣀⣰⣄⢠⠄⠠⣀⣤⣀⣀⣄⡀⢠⡄⠀⠀⢠⡦⣀⠤⠤⠾⠷⠊⣈⣻⣉⣀⣤⡥⡇
⠀⠀⠀⠀⠀⣠⠈⠉⠉⠁⠳⠈⠙⠋⢩⡍⠱⠿⠿⠦⡶⠦⢤⣭⡤⠖⣓⠫⣭⣁⣠⠤⢬⣟⣋⣡⠬⠽⣯⣅⣤⠶⣶⠲⠛⠡⠮⠉⠉⠀⠁⡖
⠀⠀⠀⠐⣒⣻⡷⢹⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠸⠂⠀⠀⠀⠛⠀⠀⠀⠀⠀⣉⢀⠀⠄⠄⠂⠀⠀⣠⡴⠄⠀⣀
⠀⠀⣉⠤⠔⢂⡱⠦⢄⡉⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢄⡠⡠⢂⠄⡰⣠⢢⢂⠜⠜⠁⠀⠁⠀⠀⢀⣠⠄⢒⠽⠋⢀⡤⠊⠁⣀
⢀⡠⠄⠒⢉⡠⠔⠒⢉⡡⠝⠓⠒⣠⠤⠄⢀⣀⠀⠀⠁⠐⠁⠈⠊⠐⠋⠚⠱⠋⠊⠐⠁⠊⠈⠀⠀⢀⣀⡀⠤⣤⠖⠂⡩⠛⣁⠔⠁⡠⠖⠁⣀⠔⠋⣠⠖
⢠⠔⠂⠉⣀⡤⠖⠈⣁⡤⠒⠋⠁⣀⡤⠖⠉⠉⢀⣉⠍⠉⠀⡂⠴⣒⠴⠖⢒⡴⠖⠀⣉⠝⠋⢉⡴⠋⢁⠴⠋⢀⠔⢋⡤⠊⣀⠔⠋⢀⡴⠊⢁⡴⢊⡠⠖
⠀⠖⠊⢉⣀⠤⠖⠋⠁⢀⡤⠖⠋⠀⣀⡤⠖⠈⠁⣀⡤⢖⣩⠔⠊⢀⡤⠖⠁⣠⠔⠋⢀⣠⠞⢁⡠⠞⠁⡠⠞⣁⠔⠁⣠⠞⠁⡠⠚⠁⡠⠒⠁⠀⠋⠀⠈
⠀⠀⠊⠉⢀⣀⠤⠒⠊⢁⣀⠤⠐⠉⠁⢀⡠⠔⢋⣡⠖⠉⢀⡠⠖⠁⢀⡤⠊⠁⣠⠔⠉⢀⡴⠋⢀⠔⠊⡠⠞⢁⠴⠊⢠⠔⠉⠀⠀⠈
⠀⠀⠀⠀⠉⠀⠀⠤⠒⠉⠀⢀⡠⠔⠊⠁⡠⠔⠋⠀⣠⠔⠉⢀⡠⠒⠁⢀⡴⠋⠀⣠⠞⠁⢠⠖⠁⠀⠊⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢰⣿⡆⢠⣿⣧⠀⠀⣾⣷⠀⢰⣿⡆⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⢸⣿⡇⠈⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠁⠀⣿⣿⠀⢰⣿⡟⠀⠈⣿⣷⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣴⣿⡟⠀⠀⠹⣿⣧⣾⣿⠃⠀⠀⢻⣿⣦⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⢠⣿⡿⣿⣷⡀⠀⠀⢠⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⣿⣿⠀⠀⣿⣿⠁⢸⣿⡇⠀⠀⣿⣿⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢹⣿⡆⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡏⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⢸⣿⡇⢸⣿⣿⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠈⠛⠁⠀⠙⠁⠀⠀⠙⠋⠀⠈⠛⠁⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⡀⠀⠀⣠⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠉⢉⣿⣿⡇⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣥⣤⣤⣤⡀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠻⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠰⣖⣲⡄⠀⠀⣯⣹⠆⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡥⠆⠀⠀⠿⠤⠇⠀⠀⠃⠘⠂⠀⠰⠛⢾⡄⠀⠈⡱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⣠⠆
⠀⠀⠀⠀⠀⢀⡴⠀⠀⠈⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣠⣶⠀⡀
⠀⠀⠀⠀⠀⠙⢾⣡⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⠞⠁
⠀⠀⢠⠶⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⠻⣶⣤⡀⠀⠀⠀⠀⣰⣶⣶⠀⢀⡀⢰⡆⠀⠀⠀⠀⠀⠀⢠⠴⢦⡀
⠀⠀⠘⢦⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⣿⣿⣷⠀⠀⠔⢺⣿⣿⣿⠀⢀⢳⢸⡇⠀⠀⠀⠀⠀⠀⠹⣴⠆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⣀⡀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠿⠁⠀⣿⣿⣿⠁⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠐⢲⣿⣷⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⣿⡖
⠀⠘⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⡀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⣀⣀⣀⣀⣀⣀⣸⠃⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⣸⣿⣿⣿⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠙⠻⣿⣿⣿⣿⡿⠋⣀⣤⣶⣾⣿⠿⣿⡛⣿⡏⣿⡏⣿⡏⣿⡏⣿⣿⡏⣿⡟⣿⡿⢿⣿⣷⣦⣄⡀⠻⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⢀⣿⣿⣿⣵⣿⣿⣿⣿⣟⣿⣧⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣧⣿⣯⣿⣿⣿⣿⣷⣼⣿⣿⣯⡁
⠀⠀⠴⠿⠿⠛⠻⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠟⠁
⣿⣿⠛⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⠀⠀⠀⢀⠀⠀⢣⠆⠀⠀⢠⠀⠀⠀⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠀⠀⠠⣋⡆⠀⡆⡆⠀⢠⣀⠀⠀⠀⡙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆⠀⠀⢠⢉⠀⠀⡇⢡⠀⢠⢃⠀⠀⢰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⠤⠄⢺⢸⠒⠒⣇⡜⠒⢺⠸⠒⠒⢺⠀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠉⠣⠇⠀⠀⢸⢸⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠒⠁⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠢⢄⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣀⠤⠔⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡆⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⢀⡀⠀⠀⣀⣀⠞⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣧⠏⠀⠀⠑⠒⠒⠒⠉⠙⢆⡴⠁⠀⠱⠀⡁⠈⠢⠴⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠃⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠛⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠻⡿⠰⢀⠐⠸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⣤⣤⣀⣾⠋⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⠻⢷⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆⣀⣀⣀⣀⣀⣠⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣰⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣎⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢡⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠉⠉⢍⠉⠈⠙⢿⣿⡿⠟⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣤⣴⡄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣸⣿⣿⡇⠀⠀⢸⣿⣠⣴⣾⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣧⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢰⣦⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡄⠀⠀⣏⢸⠀⠀⠀⣼⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡼⠁⠀⠀⣹⣯⠀⠀⠀⠻⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡇⠀⠀⣧⢾⠀⠀⠀⣾⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡇⠀⠀⣧⢾⠀⠀⠀⢿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⢾⢹⡗⠒⠒⢿⡼⠓⠒⠒⢿⢹⡗⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⢤⣄⣀⣈⣉⣡⣈⣀⣀⣈⣀⣀⣀⣈⣉⣡⣤⡭⠴⠞⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⡆⠀⢀⡭⢯⡉⠉⠉⣭⡍⣩⣍⠉⠀⢠⢤⡄⢸⡿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣧⢀⡞⠀⠄⢧⣀⡼⠣⠙⡁⢹⡀⠀⡟⠐⠙⠚⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣈⣉⣀⣁⣐⣀⣉⣀⣐⣀⣀⣂⣛⣚⣁⣈⣀⣂⣐⣀⣟⣀⡀
⠀⠀⠀⠀⠀⢀⣞⠉⠉⠉⣩⡭⢭⣉⠉⠉⠉⠉⠉⣩⡭⢭⣉⠉⠉⠉⠉⣩⠽⠿⢭⡉⠉⢳
⠀⠀⠀⠀⠀⢸⠛⠲⠶⠛⡁⠐⠀⠈⠳⠤⡤⠴⠚⠁⡒⠀⠉⠓⠶⠶⠛⠁⠀⡁⠀⠉⡙⢹⡇
⠀⠀⠀⠀⠀⢸⣄⠀⢀⣠⣥⣄⡉⢀⣂⣠⣈⡀⠠⠀⠁⢀⣨⣤⣤⣀⡀⠁⠀⣀⡤⠖⠒⢾⡇
⠀⠀⠀⠀⠀⢸⣏⣉⣉⣤⢤⣌⣉⣉⣡⠤⣤⣉⠛⠒⣛⣉⡤⠴⠦⢬⣉⣉⣉⣩⡴⠖⠚⢺⠇
⠀⠀⠀⠀⠀⠀⣧⠉⢉⢀⡀⣤⢉⡉⢀⢠⡀⢩⠉⡉⠉⠁⡀⢰⠀⡀⣈⠉⣉⠀⣀⠀⡄⢸
⠀⠀⠀⠀⠠⠤⠿⠶⠾⠾⠷⠿⠴⠧⢼⣼⣧⣼⣤⣷⠼⠦⠷⠾⠶⠷⠿⠶⠿⠶⠿⠴⠷⠟⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠰⣖⣶⠀⠀⢸⣭⠇⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⢂⠀⠀⠷⠞⠁⠀⠙⠈⠃⠀⠰⠛⢻⠀⠀⢸⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⢠⣊⠄⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡴⡟⣠
⠀⠀⠀⠀⠀⠀⠀⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣀
⠀⠀⠀⠰⡏⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢶⣦⣤⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠠⣄⠀⡄⠀⠀⠀⡞⣉⠷
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⢻⣿⣿⣆⠀⠀⠐⠛⢿⣿⣿⣿⠀⠀⠈⠀⠁⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⢰⣿⣿⣿
⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶
⣿⡏⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⢀⡄⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣥⣤⣤⣤⣤⣴⣾⡇⠀⠀⠀⢈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠈⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣤⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡏
⠀⠘⣿⡿⠟⠽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣙⠹⢻⡿
⠀⠀⠹⣷⡿⣫⡙⣷⣄⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⣤⠠⠀⠀⠀⣤⠀⠀⠀⣤⢠⡀⠀⠀⠀⠀⠀⢀⣴⣿⡑⢨⣽⡿⠁
⠀⠀⠀⠙⣷⣭⠕⣫⢿⣦⡀⠀⠀⠀⠀⠹⠀⠀⠀⠿⠀⠀⠀⠀⠘⠇⠀⠀⠿⠹⠄⠀⠀⠀⠀⣠⣾⢏⡚⠟⣷⡟⠁
⠀⠀⠀⠀⠈⠻⣮⣵⠞⠭⢻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡙⢤⣽⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣼⣿⡏⣙⡿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⣛⣯⡢⢝⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣾⣫⡔⣾⢫⣝⣿⢿⣿⢶⢶⣶⡶⠶⣾⡿⣛⣻⡫⣶⢱⣔⣫⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣮⣴⣇⣻⣟⣾⣨⢁⡇⠭⢿⣧⣙⣢⣷⣶⠾⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣿⣿⣿⣿⣿⣿⣿⣟⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢷⠀⠀⠀⠀⠀⠀⡟⠈⣷⠀⠀⠀⠀⠀⠀⢰⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡼⠀⠀⠀⠀⠀⠘⣇⣀⡿⠀⠀⠀⠀⠀⠀⢸⣄⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⡀⠀⠀⠈⣿⠀⠀⠀⠀⣀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⣿⡄⠀⠀⣿⠀⠀⠀⣼⣿⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⠶⠿⠶⠶⠾⠿⠿⠷⠶⠶⠿⠶⠶⠶⠿⠿⠿⠶⠶⠾⠷⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⣴⠶⢦⡀⠀⣴⠶⣦⠀⠀⢠⡟⠉⠻⣆⠀⣼⠏⢿⡀⠀⡿⠙⣧⠀⢰⡿⣆⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠃⠀⢸⡇⢀⡿⠀⢸⡇⠀⣼⠁⠀⠀⠙⠶⠏⠀⢸⡇⢸⡇⠀⠘⠷⠾⠃⠙⢧⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⠞⠁⠀⠀⣷⢀⡿⠀⠀⠀⠀⠀⠀⠀⠈⠷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡀⣀⢀⣀⢀⣀⣀⢀⡀⢠⣀⢰⢀⣀⣄⣠⣀⢀⣠⢀⣀⢀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠓⠟⢹⠾⢻⠿⠋⡿⠁⠘⠞⠛⠛⠁⠳⠛⠘⠙⠾⠒⠟⢸⠏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠄⣀⢠⡀⢀⢀⠀⡀⡀⢀⡀⡄⢀⢠⡄⣀⡀⡤⣀⣀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠀⠀⠘⠁⠀⠁⠁⠈⠁⠁⠀⠈⠀⠉⠀⠁⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠂⠀⠒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠀⠐⠒⠤⡀
⠀⠀⠀⢀⠐⠉⠀⠀⠉⠰⠁⠀⠀⠀⠀⠀⠀⠈⠳⡊⠉⠈⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠉⡖⠁⠀⠀⠀⠀⠀⠀⠈⢖⠈⠉⠈⠁⠢⣄
⠀⠀⡐⠁⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⢂
⠀⢠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄
⠀⢸⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡄⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠰
⠀⠀⠐⡄⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⡱⢒⠤⠤⠐⢎⡁⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⢀⠕⠢⠤⠤⠐⣌⠀⠀⠀⠀⠀⡠⠚
⠀⠀⠀⠀⠀⠈⠁⠐⠖⠀⡩⠀⠈⠀⠉⠈⠩⠉⣀⣐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠈⠁⢰⠂⢈⠡⠀⠀⠀⠁⠈⠏⠡⣀⡒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢃⡠⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠀⠀⣇⠘⠉⠣⠀⠀⠀⠀⠀⡆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡱⠦⠀⠀⠀⠀⠀⡂⠘⠃⢀⠀⠀⠐⠋⠧⡀⠹⠍⠛⠀⠀⠠⡀⠚⠃⡄⠀⠀⠀⠀⢀⠴⢎⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⠗⠂⢀⠀⠀⠀⠀⢃⠠⢲⣤⠄⠰⠁⢰⠄⢸⠀⢴⠄⠰⠀⢤⡇⠄⠶⠀⠀⠀⠀⠀⡀⠐⠢⠂⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠈⠛⠶⣤⠖⡏⠀⠸⠀⠀⣷⣄⡀⠀⠈⠀⠀⣀⣠⡇⠀⡅⠀⢸⠓⢦⣤⠶⠛⠁⠀⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠃⠀⠀⠈⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⠁⠀⠀⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣀⣄⡀⠈⢻⡇⠀⠙⠉⠈⣿⠏⠀⠀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠎⠀⠀⢱⠀⠀⠈⠀⡖⠐⢮⠀⠀⠀⡸⠀⠹⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠀⠀⠀⢠⡆⠀⠀⢀⡇⠀⠘⡆⠀⢠⠇⠀⠀⠙⠒⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⠃⠀⠀⠀⠀⠳⣄⣠⠼⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⢲⡶⠒⠲⣶⠒⠒⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⢸⡇⠀⠀⣿⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢿⠿⠿⠿⠇⠀⠀⠿⢿⠿⠿⠿⠿⠦

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡄⣿⡀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣬⡻⠟⠋⠁⡱⠒⡑⠊⣁⣠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⡠⠚⣡⣾⣿⣿⡿⢃⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠰⢯⣟⡣⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⣠⡅⣰⠟⠋⠀⢀⣤⠀⠈⣹⠶⡀⠀⠀⠀⠀⢈⡎⠱⡄⠉⡿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢡⡴⠀⠀⠋⢀⠀⠀⢄⢰⠜⢆⠀⠀⠀⢀⣷⢠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡘⠇⠀⠀⠀⠙⠁⠀⣤⣉⠆⠈⣇⠀⢀⣾⣷⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⠀⡄⡰⢊⡁⠀⣀⡠⠤⠀⣿⣠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠁⠁⠸⣶⣶⣶⣾⠏⢀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⡐⢄⠈⠛⠛⢉⢄⣾⣿⣿⣿⣿⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣦⣥⡒⢂⣵⣿⣿⣿⣿⣿⢵⣾⢫⣯⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⡇⠈⠑⠼⡏⠙⠉⠙⢟⠛⠛⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠁⢃⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠛⠿⠿⣿⣿⣿⣶⠶⢤⣤⣀⠀⠀⠀⠒⠒⠰⠀⠓⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⠇⠀⠈⠸⣩⢻⢦⡀⠀⣠⡼⢤⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣼⡀⢀⢃⢇⠀⠄⢋⢞⠃⢀⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⢩⠇⡸⠌⠈⡄⣴⠟⡌⣄⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡏⠈⠊⡤⢇⣄⣠⠟⢁⣤⡾⡿⡱⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⢰⢰⠀⢡⣠⣟⡥⢶⣏⠁⠁⠁⠃⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⡆⠆⢾⣠⡾⠛⠁⠀⠀⠙⣶⡰⡆⠀⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣶⡞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⣥⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⡄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⢹⠀⠀⠀⠀⠀⠀⡞⠁⠀⢸⡇⠀⠀⠀⠀⠀⢀⡾⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⡞⠀⠀⠀⠀⠀⠀⠹⣄⣀⡼⠃⠀⠀⠀⠀⠀⠈⣧⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡭⣭⣀⠀⠀⠀⠀⠀⢠⣶⣛⣩⣤⡀⠀⠀⠀⠀⠀⢠⡤⠷⢋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣍⡁⠛⢻⠀⠀⠀⠀⠀⠠⣧⣉⠉⢸⡇⠀⠀⠀⠀⠀⢸⡟⠛⠋⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠙⠛⢻⠀⠀⠀⠀⠀⠀⡇⠀⠙⢻⡇⠀⠀⠀⠀⠀⢸⡋⠛⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠶⣤⣼⠁⠀⠀⠀⠀⠀⡏⠑⠦⣼⡇⠀⠀⠀⠀⠀⢸⠗⣤⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⢸⡅⠀⠀⠀⠀⢠⣇⠀⠀⠐⡇⠀⠀⠀⠀⠀⢸⠁⠀⠉⢹
⠀⠀⠀⠀⠀⡴⠋⠙⠲⣄⠀⢀⡤⠤⠤⠾⣿⡲⢤⣼⠧⠒⠒⠒⠦⣤⡏⣱⠖⠚⠛⠒⠲⣄⠀⣠⠼⠷⢤⣄⣼⣄⡤⠴⠒⠦⣤⡀⣠⠤⠤⢤⡀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈⠻⠏⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠉⣿⠏⠀⠀⠀⠀⠈⡿⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢸⡏⠙⠲⠶⠖⠲⢤⣄⡀⢀⣀⣤⠶⣄⡀⠀⠀⠀⢀⣠⠴⣦⡀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⢀⠾⣤⡀⠀⢀⡼⠉⠳⠶⠶⠖⢿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠑⠋⠉⠀⠀⠈⠑⠲⠶⠶⠞⠉⠀⠉⠓⠲⠚⠉⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⣀⣀⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⡸⠋⠉⠁⠙⠻⣇⣰⠟⠉⠀⠀⠀⠉⠙⣆⣀⡴⠞⠛⠛⠉⠙⠛⢦⡀⢀⡴⠛⠋⠉⠈⠉⠛⢦⣀⡴⠖⠛⠛⠓⠶⣄⣀⡴⠚⠛⠛⠛⠳⢦⣸⡶⠞⠛⠲
⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠟
⠙⢦⣄⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⣠
⠀⠀⡏⠉⠉⠐⠋⠉⠓⠶⢤⣤⣤⣤⡤⠖⠚⠢⣤⣀⣀⣀⣀⣀⣤⠶⠚⢶⣀⠀⠀⠀⠀⢀⣠⠾⢧⣀⡀⠀⢀⣠⡞⠛⢦⣤⣀⣠⡴⠃⠈⠙⠒⠒⠋⢹
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠑⠓⠒⠒⠉⠁⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡇
⠀⠀⡇
⢀⣠⢧⡤⣄⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀
⣞⠁⠀⠀⠈⠙⢦⣠⠞⠋⠉⠉⠉⠙⠓⣦⡴⠚⠉⠉⠁⠀⠈⠉⠳⣄⣠⠞⠉⠀⠀⠀⠀⠉⠳⣤⡜⠋⠉⠉⠙⠳⣄⢠⡞⠋⠉⠉⠉⠙⢳⣤⠞⠉⠉⠉
⣏⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠉
⠈⠳⠶⣤⣄⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⡴⢶⣄⠀⢀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠲⠶⠶⠶⠖⠉⠉⠛⠶⠦⣤⣤⣤⡴⠶⠋⠉⠻⣦⡀⠀⠀⢀⣤⠴⠋⠳⣤⣄⣀⣤⡴⠋⠉⠳⠶⠦⠶⠞⠁⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⢀⠀⠀⣀⠀⠀⡀⠀⠀⢠⠐⠈⡁⠁⠉⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠
⠀⠀⠀⠀⠀⢫⠀⠀⠀⠠⠌⠀⠈⠀⠀⠀⠀⢂⠐⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⢏⠀⠀⠀⠀⡀⠀⠓⠳⠢⡀⠀⠀⠀⠐⠆⠀⠀⠀⠄⠁⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⠄⠀⠀⠀⢀⠀⡐⠃⠉⠂⠀⠽⡃⠀⠀⠀⠀⠀⡀⠠⢆⠰⠀⠈⠀⠀⠐⠁⡘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣎⠦⣁⡇⠘⣠⣀⠀⡌⣅⠀⡀⠘⠀⠇⢀⠀⢠⠂⠘⠀⠀⠈⠀⡇⠈⠐⡎⠀⠀⠀⢊⠀⡀⠂
⠀⠀⠀⠈⠀⠀⠀⢀⢀⣁⠉⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠁⠀⠀⠀⠉⣐⠴⠂⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡀⠊⠁⠤⠨⠀⠀⠀⠀⠀⠂⠀⠀⠀⠅⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠊⠉⠁⣲⢄⡀
⠀⠀⠠⠀⠃⠑⠂⢠⡠⠚⠗⠂⢐⢲⠀⠀⠐⠒⠀⠐⠂⠐⠂⠀⠀⠀⠀⠀⡤⢠⠀⠤⣶⣶⠄⠀⠰⠁⢁⡨⠂⠀⡁⢡
⠀⢀⡀⠄⠀⠀⠚⠆⡐⠁⠀⠀⠈⣚⠄⠀⠀⠀⠀⢀⡆⠙⢲⠀⠀⠀⠀⡎⠀⠎⠐⢠⡀⠀⠀⠀⠀⠈⠀⠀⡤⠈⢱⡈
⠀⠀⠀⠠⣠⠌⠼⡀⠳⡄⠄⠀⡜⠁⠀⠀⠤⢠⡚⢁⠛⢆⡀⠑⠊⠑⠊⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠰⠂⠃⠀⠀⠀⠀⠰⡊⠀⡰⠆⢚⡙⠁⠐⠀⠈⠀⠀⠠⠄

⠀⢠⠄⠀⢠⠄⠀⠀⠀⡀⠀⠀⠀⠠⡂⠀⢠⠄⠀⠀⡀⠤⡄⠠⠄⡀⠀⠀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⢠⠀⣀⣀⢀⡀⣀⡀
⠀⡔⠃⢀⠌⡑⠄⠀⡈⠆⡀⢠⢲⠐⠃⠀⢘⣃⡄⠰⠈⠁⣾⠈⣰⠃⢄⠀⣿⢁⡽⣈⠇⢨⡇⣯⡋⣿⡇⡠⠂⠐⢄⡀⠤⢌⡀⠥⠄⣀⠄⠀⠠
⠀⢐⠀⠀⢈⢌⡄⠈⡥⠔⢀⠸⠐⠚⠀⠘⠌⢤⠠⢀⠬⡲⠍⠹⣤⡀⠤⠁⠻⠿⠿⣿⣿⡿⠿⠿⠿⠛⠀⠃⠰⢤⠸⢁⣰⡀⠑⣠⣀⠙⠀⣦⡄
⠀⠢⠸⢀⠂⠈⢥⠢⣒⢐⠈⠀⢘⠌⠊⠐⠤⠄⠊⠀⢨⡄⠃⢀⠘⠁⠀⡰⢄⢀⠀⠀⠛⠁⡘⡂⠀⢠⠄⠡⢈⣁⠠⠈⠒⢁⡌⠓⠉⡰⢀⣉⡁
⠀⠀⢀⠶⠀⡀⠤⢀⠀⠠⡢⠠⢊⢍⠢⠀⠀⢐⡄⠀⠉⡀⢀⠃⢀⠔⠈⠁⠁⠀⢰⠀⠀⢠⢜⠦⠀⢠⣜⠀⠀⠁⠀⠈⠒⠁⠈⠑⠋⠀⠈⠁
⠀⠔⠱⢆⠈⠁⠆⡐⡠⢒⣓⢔⠀⢡⠄⡇⡠⢚⠢⡀⠀⠀⠀⠀⣻⡈⠀⠤⠈⠀⢈⠓⢠⢸⠀⢸⠀⡧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢄⠃⠂⠄⠐⡰⠀⠅⣉⣈⢈⠘⠀⠊⠀⠰⠀⡰⠀⠰⠒⡂⠪⣈⠒⠒⢐⣀⠩⠄⠐⠁⢸⠀⢸⠀⢃⢈⡇⠠⠰⢆⠀⠸⠝⠀⠀⢉⠟⢄
⠀⠀⠈⠁⠀⠀⠀⢀⡂⠥⠥⠮⠀⠉⡝⠀⠂⠤⠄⠁⠀⠱⠁⠀⠀⠀⠀⣀⢸⡆⠄⠀⠀⠘⠂⠊⠀⣆⠠⠅⠀⠀⠁⠠⠀⠠⠀⠀⡖⡀⠈⠄
⢀⠄⠀⠠⡄⡀⠊⠀⣈⡢⠐⢙⠗⢖⠒⢒⠶⠒⠒⠒⠶⠒⠐⡲⠫⡀⠀⡄⠈⠁⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⡄⠀⢠⠀⠀⠸⠁⢁⠂⡈⠄
⢘⣠⣤⣤⢸⠙⡀⣾⣿⣿⠁⠀⠀⠈⢧⠎⡀⠐⢄⠌⠀⠙⠴⠃⠀⠀⠲⠀⠀⡂⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⡐⠈⠑⠈⢱⠀⡇⣀⢀⠁⠀⠸
⠈⣿⣿⣿⠏⠀⠈⠚⠛⠫⣇⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⢀⠠⠤⢐⠶⠒⠶⠤⢤⣀⠐⡀⠀⠀⠀⡸⠉⠀⠂⠱⣑⠀⡀⡄⠀⠀⠀⠀⢀
⠀⠈⢩⠉⠀⠀⠀⢀⠄⠀⢈⠷⡠⢀⣀⣸⢄⣀⠀⠀⠀⢠⡇⠀⠀⠀⠐⢠⠀⡄⢈⠀⢗⡈⠲⠒⠊⠀⠀⠀⠀⠀⠳⡀⠐⠁⠀⠄⠀⠘⠋
⠀⡀⢼⠤⡀⠀⠠⠃⠀⠁⡌⠈⠀⡁⠀⠈⠀⠀⢘⠀⠀⡖⠀⠰⠒⠐⠔⠠⠂⡇⢨⠀⠁⠂⠀⠀⡄⠀⠀⢀⠜⠀⠀⠑⠀⠀⠀⢄⠸⠄⠀⡠
⠀⠀⠈⠀⡀⡀⢨⠀⠀⠀⣨⠀⡎⠐⡠⡀⠀⠀⠔⢱⠀⠳⠀⢂⠀⢁⢀⡀⠀⠀⠀⠀⢀⡄⠀⠀⠘⠋⠈⢀⠒⠐⢂⠀⠀⠀⠂⠐⢄⠈⢄⡀
⠠⠤⠀⢰⠻⡀⠐⡄⠀⠀⡖⢀⣃⠋⠀⠂⠐⠂⢈⢘⠀⠈⠖⡔⠀⠀⠉⠈⠁⠀⠉⣄⠖⠀⢀⠀⠀⣀⣀⣘⠁⠀⠉⣄⣀⣀⠂⠀⢀⣀⡀⠁⡀
⠀⠆⢀⠋⠀⠱⠀⠡⢀⢀⠃⢎⠣⢌⡀⠀⠀⣀⠠⢜⡁⠀⠀⠈⠀⠓⠐⠒⠒⣉⣀⠀⠀⠀⠟⠂⠘⢿⡟⠁⠀⠀⠀⠈⢿⡿⠃⠀⠀⠀⠐
⠀⠀⡞⠀⠀⠀⢣⠀⠉⠀⣀⣠⣁⠀⡔⠀⠀⠒⠐⢉⡀⢤⡔⢀⠀⠀⢀⠐⠨⠂⢀⠄⣉⢀⣠⡀⠀⡊⡅⠐⠀⠀⠐⠀⢸⠢⠀⠔⠒⠉⠉⣉⠢
⠀⢸⡀⡀⢀⡀⡀⡇⠀⣸⠀⠈⠁⠈⠋⢁⠀⠀⣠⠊⠄⠐⢀⢀⡀⡀⠐⡮⠥⣐⡥⠊⠀⠈⠚⠡⠀⠀⠁⣀⠀⠀⢀⣠⠁⠀⢠⡀⠙⠁⡂⠡⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡄⠂⡂⡂⢨⠀⡰⠃⠀⠀⠂⣈⠜⠃⠀⠀⠡⠀⠀⢃⠀⢀⠄⠀⢖⠦⠀⣨⣀⣽⣼⣧⣠⡇⠀⠀⠁⠂⠀⠐⠀⠁
⠀⠀⠀⠘⠻⢃⡀⠀⠐⠦⠘⠐⡀⠂⠀⡈⠸⣥⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠑⠂⠬⠔⠁⠰⡷⢀⣀⣀⣈⣉⢙⣛⡋⠉⠁⢀⣀⡀⠀⣀⠠⢿⠂
⠀⣴⠋⢲⡶⣃⡱⢀⠀⠀⢀⢀⠀⠀⠀⢀⡀⠀⠀⠀⡸⠖⠉⠉⣣⡴⠒⠒⠀⠀⠀⠠⠀⠀⠁⢀⠇⠀⠞⠈⠀⠀⡅⡏⢉⠉⠈
⠀⡖⠝⠋⠉⣑⢡⠨⠁⢂⠀⡀⠀⠀⠰⡈⠈⠐⠀⢨⣑⠢⠤⢀⣀⠇⡀⠀⢠⠀⠀⠀⠘⠄⠀⠘⠋⠑⠇⠰⠐⠠⠃⠷⠛⠦⠀⠖⠀⠒
⠀⣀⠀⠨⠁⠀⠠⢨⠀⣠⣀⠀⠀⢀⠀⢓⢵⠃⠀⠀⢀⡀⠐⠀⠞⡰⠙⡄⠈⢱⠀⠀⡀⠀⢤⠀⢤⠀⡤⠀⠀⢀⢀⠀⢀⢀⢠⢀⡤⠀⢀
⠀⠈⠢⢴⠀⠤⠒⠊⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⢣⡈⠁⡜⠀⠘⡐⠓⠒⡺⠀⠀⡀⠀⢈⠃⢸⠐⠐⠂⠀⢨⠅⠀⣨⠌⠀⠂⠀⠀⡀
⠀⠔⠒⠂⠀⠀⠐⠂⠠⢠⠣⠀⠀⠀⠀⠀⠃⠇⠀⠣⠀⠀⠐⠀⠀⠀⢙⠀⠀⡄⠀⠀⠀⢀⠈⠀⠀⠀⡀⠈⡀⣉⢀⡀⢁⠀⠈⠁⣀⡀⢀⡀⣀
⠀⠀⠀⠸⠉⠇⠀⡠⢀⠀⠀⠁⠀⠀⡄⠀⢀⣀⢀⣀⣀⣀⣀⣀⢬⠼⢆⢢⡰⠀⠀⢨⠀⡀⠀⢀⡃⡇⠀⠘⠁⢘⠂⢰⠀⠈⠂⠠⠡⠄⠀⡂⠁
⠀⠓⠀⠀⠀⠀⠀⠋⠀⠈⠒⠀⠈⠉⠀⠀⠘⠀⠈⠒⠒⠒⠒⠒⠒⠐⠀⠀⠀⠀⠉⠀⠁⠘⠁⠙⠀⠃⠀⠃⠀⠈⠀⠈⠈⠈⠁⠀⠀⠈⠀⠓

⠀⠀⠀⢀
⠀⠀⠔⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢃⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠂⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈⠀⠀⠘⠆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡁⠀⠀⠈⠀⠀⠈⠀⠀⠀⠈⠀⠀⠡⣀⠀⠀⠇⠀⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠉⠉⡀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠰⡀⡔⠑⡄⠀⢀⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠀⢀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⠀⠀⠀⠀⠀⢸⠀⠀⠌⣓⣃⠀⢀⠀⢄⣂⠀⠀⠠⠀⢀⠀⠀⠀⠌⠀⠈⠂⠊⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠄⣀⠀⢸⠀⠀⡋⢩⠇⠀⠈⠀⢿⣽⡗⠀⠀⠰⣯⣿⠄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠅⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠰⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠐⣌⠲⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⠦⢈⣑⡦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠈⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣶⣶⣶⢶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣴⣶⣦⣶⣦⣾⣧⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⣭⣷⣶⣿⣛⣟⡻⣿⣷⣾⣭⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣾⣿⣿⣿⣿⢿⣿⣿⣧⡘⢿⣿⡏⠹⣷⣾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠻⠿⠿⣿⣯⣴⣿⣿⣿⣿⣿⣶⣮⣍⣀⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⡿⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⠿⠿⠧⠀⠀⠀⠀⠤⠬⠭⡙⢿⣿⣇⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⡿⢁⣤⣤⡀⢶⡀⠀⠀⢀⣤⣀⡀⠀⢿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢁⣹⣿⣿⡇⠘⠋⠀⠀⠀⠁⠀⠀⠉⠀⠈⠉⠀⢸⣿⣿⣄⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⢿⠀⠀⠀⠀⠀⢾⣇⠰⠄⢀⠀⠀⠀⠀⣾⣿⣿⣿⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⡇⠀⠀⢀⣤⠤⠀⠶⠦⣌⢳⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠈⠻⢿⠿⣿⣿⣷⠀⠀⠘⣿⣿⣿⣿⣿⡾⠋⠀⠀⣸⣿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣴⣞⠀⠀⣠⣿⣿⣿⣆⠀⠀⠈⣉⣉⡋⠍⠀⠀⠀⢠⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠉⠁⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⠾⠟⠈⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣷⣤⣴⣤⣤⣤⣿⣿⣿⡟⠻⣿⠋⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣿⡿⣿⣿⣿⣿⣿⣿⣿⠃⠃⠀⠛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⢈⢷⣹⣿⣿⡏⠘⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠙⢿⣿⣿⣿⠟⠛⠁⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢀⣦⡸⡞⣇⢻⣿⣦⡀⠑⠺⢻⣆⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⢸⣿⢿⠟⠀⠀⣀⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣾⣿⣿⣿⣹⡜⣿⣿⣿⣷⣦⣄⡙⠳⡄⠀⠀⢀⣼⠟⠁⢀⣠⣶⣿⡿⠀⠀⠀⣼⣿⣵⠏⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣻⣿⣧⢻⡏⢻⣿⣿⢿⣿⣿⣷⣌⠢⣠⣞⣡⣶⣾⣿⠛⡉⡁⠀⠀⣇⣼⣿⣿⡏⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠃⠈⠋⠛⠁⠂⢻⠛⣿⣷⣽⣏⠋⠉⠁⠀⠈⠀⡐⠀⣆⣿⣿⣿⡾⠀⣤⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠈⠀⠀⠐⠀⠈⠃⠈⠃⠘⠛⠀⠀⠀⠀⠀⠘⠁⠀⠛⠛⠛⠛⠁⠘⠙⠋⠐⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⣀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠣⠌⠙⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣠⠆⠞⢤⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠙⢂⢀⡖⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢑⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠡⠀⠁⠀⠀⠀⢀⣀⡀⢄⠠⠤⡠⣤⠒⠲⡖⠂⣼⡃⢠⢤⠆⠤⣤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡁⢮⡽⣽⠵⣈⡓⢡⢶⢑⢇⡈⠊⠱⢎⢒⣃⠮⢜⣛⡍⠌⣓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠸⠀⠀⢡⠀⣴⣡⠇⣈⡗⢡⠲⠑⠮⢟⡃⡰⡗⢿⡺⢈⡛⠧⠊⢰⢢⠓⡧⡀⢎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠆⠀⠀⠀⠡⣇⡤⡘⠒⡡⡑⠃⡐⡲⢧⡫⡈⡗⢄⠤⠘⣬⢲⢢⠈⣚⠀⣉⠚⡠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢱⡌⢱⠉⡉⡚⢠⢻⢡⢴⢭⠋⢫⡁⣚⠦⢝⡡⡯⠮⠼⢦⢠⣄⡭⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⢀⠂⠀⠀⠄⠙⢹⠀⣶⢥⡇⢸⡼⡁⢸⠀⠈⠑⠁⠀⠘⣬⠃⡶⢫⣸⢸⢸⣱⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡠⠁⠀⠀⡐⠀⠀⢸⡄⠈⠉⠀⢸⠋⢯⣿⠀⠀⠀⠀⠀⠀⠉⠒⠁⢸⡐⢸⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⡠⠈⠀⠀⠀⡐⠀⠀⠀⠀⢣⡀⠀⠀⢸⠸⣵⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠀⢀⠈⠀⠀⠀⠀⡀⠈⢣⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⡠⠀⠁⠀⠀⡀⠔⠈⠀⠀⠈⠶⠂⠤⠤⠤⠄⢀⣀⣀⣀⣀⣀⣀⠠⠤⠤⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠠⠄⠂⢁⡀⠀⠐⠀⠁⠀⠀⠀⠀⢀⠠⠘⢷⠐⠲⠒⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⢂⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣖⡐⠂⠀⠉⠀⠀⠀⠀⢀⡀⠠⠀⠂⠈⠀⢀⠠⠘⠀⢈⠋⠉⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠠⠀⠀⠀⡀⠀⠀⢀⠄⠦⠴⠊⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠲⣿⣿⣿⡷⠂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡗⠀⠀⠀⡿⠛⠻⡇⠀⠀⠐⣾⣿⠂
⠀⠀⠀⠀⠀⢀⣴⠋⠀⡀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⡀⠈⢳⣄
⠀⠀⠀⠀⡄⣾⢃⣤⡾⠁⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀⠀⠀⢀⣠⣿⣿⡆⠀⠀⠀⠀⠹⣦⣄⢻⡆⡄
⠀⠀⢀⣾⢻⣿⣿⡋⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⠙⢿⣿⣿⡄⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⣻⢿⣿⢻⣆
⠀⠀⣸⣿⢞⣵⡟⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠀⠀⢨⣿⣿⡇⠈⠉⢹⣿⣿⡇⢠⠶⠆⠰⣶⠆⠘⣷⣝⢾⣿⡄
⠀⡄⣿⣧⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⢸⣿⣿⡇⠈⠳⡄⠀⣿⠀⠀⠈⢻⣦⣿⡇⡄
⢰⣇⢹⡿⣱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⢸⣿⣿⡇⠘⠶⠏⠀⠟⠀⠀⠀⣷⡹⣿⢀⣷
⢸⣿⡘⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡜⣸⣿
⡘⣿⣿⣿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣿⣿⡟
⣷⡙⣿⡏⣾⡇⠀⠀⠀⠀⠀⣀⠀⠀⢀⣾⣿⣿⣃⣀⣀⣀⣀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣀⠀⠀⠀⠀⠀⣿⡎⣿⡟⣱
⢹⣿⣌⠃⣿⡇⡀⠀⠀⠀⠟⠉⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡇⣀⡈⠉⠁⠀⠀⠀⡀⣿⡇⢋⣼⡿
⠀⡙⢿⣷⣿⡇⢿⣆⠀⠀⠀⠐⡿⠂⢈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⢉⡀⠙⠻⠂⠀⠀⢀⣾⠇⣿⣷⣿⠟⡁
⠀⠸⣦⣈⠻⢿⢸⣿⡄⠀⠀⠀⠀⠰⡿⠃⢠⠀⢀⠀⠀⠀⠀⢀⡀⠀⢴⡀⠈⠿⠀⠀⠀⠀⠀⣼⣿⢸⠿⢋⣡⡾⠁
⠀⠀⠘⠻⣷⣦⣄⣻⣿⣻⣦⡀⠀⠀⠀⠀⠋⠀⢸⠏⠀⡯⠀⠈⡿⠀⠀⠛⠀⠀⠀⠀⣠⣾⣻⣿⣃⣤⣶⡿⠛
⠀⠀⠀⠀⠈⣉⠛⠻⠿⠷⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠱⠿⠟⠛⢋⡉
⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣶⣶⡿⠿⠟⠒⣀⣤⣤⠤⣤⣤⣤⡤⢤⣤⣄⡐⠚⠿⠿⣷⣶⣶⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣤⣤⣶⣾⡿⠟⣫⣵⠟⠉⠀⠈⠙⢷⣭⡛⠿⣿⣶⣦⣤⣤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢾⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⠆⠀⠈⠉⠁
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⢀⣿⠹⣇⠀⠀⢠⡟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠘⣧⣤⡟⠀⠀⢸⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡞⠁⠀⠀⠀⣬⣿⣤⠀⠀⣀⣹⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣻⣿⠀⠀⢠⣿⡴⣿⠀⠀⣿⣩⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⣡⣿⠀⠀⢸⣏⡴⣿⠀⠀⣿⡡⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢠⣿⠀⠀⢸⡏⡰⣿⠀⠀⣿⢁⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⢉⣿⠀⠀⣸⣟⣀⣿⣀⣀⣿⣁⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⣦⣤⣤⣤⣤⢦⣤⣀⣀⡀⠀⠂⠀⠀⣀⣀⣨⣤⡶⢦⣤⣤⠶⠶⠾⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠒⠒⠉⠳⠤⠤⠔⠋⠻⣍⣛⣛⡿⠛⠻⢍⣉⣀⡤⢊⡁⠑⠢⠔⠊⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⣤⣤⣬⣤⣤⣌⣀⣬⣥⣤⣤⣄⣐⣀⣀⣒⣀⣀⣀⣀⣀⣤⣥⣤⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣿⣤⣧⣀⣍⡀⣹⠉⢹⠁⢹⠀⢹⠉⢹⠉⢹⠉⢹⠉⢹⠉⡏⢹⢀⣇⣰⣼⡷⠾⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⢠⡄⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠳⠿⠶⠾⠶⠾⠶⠾⠶⠾⠶⠿⠾⠛⠛⣛⠉⠉⠀⠀⠀⠀⠛⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣠⣤⠀⢴⠄⣀⡀⠁⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⢀⢀⡀⠉⠠⠀⢀⣿⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠻⣤⣤⡴⠟⠛⠛⢷⣤⣀⣀⣠⣴⠿⢷⣴⣦⡈⠈⠋⠀⣀⣴⠟⠛⠛⠶⠶⠶⠟⠁⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠄⠀⠀⠀⠀⠀⠃⠀⠀⠉⠉⠁⠀⣀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠠⠄⠀⠀⠀⠀⠀⠈⣁⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡶⣤⣤⣄⣀⣀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⢀⣀⣀⣠⣤⣤⢶⣶⠟⠛⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣷⣀⣠⡿⢿⡉⠛⠛⣿⣿⠲⠶⠶⣶⡶⠶⢶⣶⡶⠞⠛⢻⣿⠛⠉⣹⠿⣧⣀⣸⣟⣷⣾⠿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⠛⠛⠳⠾⠿⢶⣾⣯⣽⣦⣤⣾⣋⣷⣤⣾⣍⣿⣦⣴⣿⣽⠷⠾⠿⠟⠛⠛⠉⣉⣠⣤⣶⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣽⢿⡟⠶⣶⣶⣤⣤⣤⣤⣤⣈⣉⣉⣉⣉⣉⣉⣀⣀⣠⣤⣤⣤⣤⣴⣶⠶⢻⡟⢿⣁⣿⠙⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⡈⣷⡾⠃⢸⣇⣴⠏⠹⣇⢠⡿⠙⣯⢉⣽⠋⢻⡍⣼⠏⠹⣇⣼⠃⠘⣧⣾⠁⠈⣿⣇⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⢦⣤⣿⣃⣀⡀⣿⡟⠀⠀⢻⣾⠃⠀⠈⣿⡏⠀⢀⣻⣿⣀⣤⣼⣿⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢿⣶⡍⢿⠇⣱⣾⢏⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⠷⠙⠟⠀⠘⠟⠁⠛⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠖⣠⡦⢈⣵⡇⡄⠀⠀⠀⠈⠁⠀⠀⠀⢠⢸⣮⡁⢰⣄⠲⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⣿⠠⢸⣿⠇⠻⠋⡠⢂⠄⠀⠀⠀⠀⠀⠀⠀⡐⢄⠙⠟⠘⣿⣇⠄⣿⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠁⣦⠹⠟⠜⡠⢊⠔⠁⠴⠛⢿⣧⠀⢾⣿⠀⠈⠢⡑⢄⡢⠹⠋⣰⠈⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣉⠈⠃⠿⢀⡴⢊⠔⠁⠀⠀⠀⠀⣸⠏⠀⢸⣿⠀⠀⠀⠈⠢⡙⢦⡀⠿⠘⠁⣉⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢌⠻⣿⣿⣷⢦⡘⢄⠣⡀⠀⠀⠀⠀⣰⠋⢀⠀⢸⣿⠀⠀⠀⠀⢀⠜⡠⢃⡴⣾⣿⣿⠟⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣤⣬⣭⣓⠉⢀⠑⢌⠢⡀⠀⠾⠿⠿⠟⠀⠾⠿⠄⠐⢀⠔⡡⠊⡀⠈⣚⣭⣥⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⢙⣋⣀⣈⣛⣀⠑⢌⠢⡀⣀⡀⢀⡄⢀⣀⢀⠔⡡⠊⣀⣛⣉⣀⣙⡛⠉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⡿⠿⠿⠁⠀⠁⠀⠈⠻⠌⠡⠟⠁⠀⠈⠀⠈⠿⠿⢿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠐⠂⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠈⠛⠿⠟⠛⠛⠁⠀⠛⠛⠻⠿⠛⠉⠀⠀⠀⠀⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⢹⡇⣹⡆⠀⣿⠃⠀⣿⠁⢹⡆⠘⠉⣿⠁⠃⠈⣿⠀⢸⡏⠀⢸⡇⠙⣶⡀⠀⣼⡄⠀⠻⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⡦⠀⣿⠁⠀⣿⠙⢿⡀⠀⠀⣿⠀⠀⠀⣿⠉⢹⡇⠀⢸⡇⢀⣿⠃⡰⠈⢿⡄⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠁⣉⣁⣀⣉⣁⣀⡉⠁⠈⠉⣀⣀⣉⣀⠀⠀⠉⠀⣈⣁⣀⣈⠉⠉⠀⢈⣁⣀⣈⣁⣀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠁⠂⠛⠛⠛⠛⠛⠓

⠰⢶⡤⠀⣤⡄⣤⠀⠐⣠⣀⠤⣄⣀⠤⠀⠄⠀⠤⠤⡤⢤⠠⠤⠄⣶⣄⣢⣴⢄⣰⡖⠶⠶⠒⠒⠒⢒⣲⣶⣶⡶⢶⣶⠶
⠈⡈⡡⠀⠠⠁⠢⠄⠜⣠⠴⢫⠥⡀⠉⠺⣩⢈⢳⣃⠜⡀⠀⠀⠈⠋⠁⠐⠚⠐⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⢰⠟⠻⠋⣴
⠀⡤⣠⠒⠉⠉⠉⠓⢤⠁⠈⠉⠁⢣⡀⠈⠐⠀⡰⠁⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠍⣩
⠀⢧⠃⠀⠀⠀⠀⠀⢘⡦⡀⠀⠀⢀⠉⠒⠒⠎⠑⠁⠀⠀⠀⢨⢧⢻⢀⡈⢰⢴⡧⠇⠑⡠⠀⠀⠀⠀⠀⠀⠀⠀⢋⡏⣿
⠀⡿⠀⠀⠀⠀⠀⠀⡼⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡇⢠⣶⢰⠉⡎⠀⢸⠁⠀⠀⠀⢀⣀⡀⠀⢀⡘⠒⢹
⠈⡇⠂⠀⠀⠀⠀⡼⠁⡰⢳⠀⠀⡸⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣁⣁⡈⠉⠘⣀⣁⠀⠈⠀⠀⠀⠀⢎⠀⠈⢲⠋⢹⠀⢸
⠀⠞⠆⣄⣠⠴⠊⠀⡰⠃⡜⢀⠰⠁⠀⠀⠡⢦⠈⣷⣤⠀⡐⣶⠁⢹⣄⡀⠉⢃⢦⡑⡄⠀⠀⣠⠀⠬⢦⡀⠀⢀⠎⡀⠼
⠀⠀⡅⢺⠓⡤⠀⠔⠁⢠⢯⠏⠀⠀⠀⠀⠀⠋⢖⠡⡌⠀⡽⠊⢰⢸⡔⢽⠀⠸⢀⠃⡇⡹⠆⢸⣀⢀⠀⢙⠶⠊⢟⡬⠂
⠀⠈⡢⠸⣄⠀⠀⠀⢠⠓⠋⠀⠀⠀⠀⠀⣄⠗⡨⢠⡇⠰⠇⠀⠼⠸⠗⠸⣀⡗⣋⠞⠸⢧⣾⣼⣾⢸⡀⠘⠄⣀⡠⢣⢷
⢸⡄⡠⠸⠈⠃⡶⠊⠆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⣴⣏⠁⣠⣯⠎⠸⠆⠀⠀⠀⠀⢺
⠈⣿⠡⠈⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠚⢁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠰⠚⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡳⠀⠠⠓⠀⠀⠀⠀⢸
⠀⣇⠂⡞⠛⠚⠶⣄⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠙⡇⠀⡔⠀⠀⠀⠀⢰
⠰⣹⢰⡇⠀⠀⠀⠈⢳⡥⠦⣄⣠⠤⠖⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡈⣿⠀⠘⠇⢠⢈⡆⠀⠘
⠀⠛⡉⠣⣎⣉⠭⢽⣾⠀⠀⣏⣀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣀⠈⣉⠀⢀⠠⢄⠀⢀⠤⡬⢀⠋
⠀⠈⠐⠀⡇⣏⠉⠉⠛⠓⣾⠓⠛⢷⠢⠬⠋⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠂⠀⠘⠲⠃⢰⠤⣬⠂
⠀⠠⠐⠚⡛⣿⠀⠀⢀⠀⣿⠀⠀⠼⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⢅⡂
⠀⠀⠀⠀⠁⡿⠀⠀⢸⠀⡏⠀⠀⠇⠀⣎⣩⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠛⠛⠋⠭⠭⠭⠭⠭⠉⠛⠃⢈
⠀⠀⡆⠀⢠⡇⠀⠀⢈⢀⡇⠀⢸⠠⡄⠀⠀⣀⡈⠉⠛⠛⠒⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠒⣡
⠀⡀⢘⢾⠋⣇⠀⠀⠀⢸⠁⠀⣈⢀⠃⢰⢰⠃⠙⠉⠓⠄⠒⠑⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⡀⠀⢸⡁
⠀⠀⠜⠒⠢⠭⠿⢿⣿⠛⠒⣒⠉⠸⠤⠬⠥⠤⠤⠀⣀⣀⣀⣀⡧⣀⠀⣀⠀⠀⠀⠀⢜⣀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢣
⠠⠀⠚⢿⠥⡤⠤⡤⠶⠶⣶⢖⡟⠉⣓⠠⠼⠿⠷⠶⠶⠶⠶⢒⡠⡿⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠁⠊⠙⠈⠀⠈⠃⠃⠊⠘⠘⠲⠊⠘⠀⠀⠒⠋⠟⠙⠂⠓⠈⠀⠓⠖⠀⠀⠲⠒⠒⠒⠒⠈⠉⠉⠉⠉⠁⠒⠒⠂⠒⠋
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⠀⠀⠀⠀⠀⢸⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡏⠀⠀⠀⠀⠀⡿⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢀⣀⠀⢰⠛⢦⠀⠀⣀⡤⠖⠚⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠓⠶⣤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠇⢀⡴⠛⠁
⠀⠀⠀⠀⠀⡏⠉⠳⣾⠀⢸⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡴⠶⠦⣄⡀⠀⠀⠸⠶⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⣀⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠞⠈⣇
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡴⣄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⡴⠋⠀⠀⢹
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⢰⣄⣀⡸⠁⠈⠒⠀⠚⠁⠈⠑⠦⠴⠈⠳⢄⣀⣀⣀⣠⡶⠋⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢘
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⢀⡴⠒⠒⠦⡀⠀⠀⠀⠀⠘⡆⢀⡤⠚⠛⢤⡘⡇⠀⠀⠀⠀⠸⠧⠶⠶⠒⠛⠋⠉
⡏⠀⠀⠦⠼⠂⠀⠀⠀⠢⡄⠀⠐⣦⣤⠜⠀⠀⡎⠀⣠⣤⡀⠹⠀⠀⠀⠀⠀⢹⡏⠀⣶⣶⠀⢷⡇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢁⠀⠀⠀⠀⢧⠀⠙⠿⠃⢠⠇⠀⠀⠀⠀⠈⣧⡀⠙⠋⢀⡿⡇
⢻⡀⠀⢀⠀⠀⢠⠞⠉⠀⠈⠦⠤⠎⠀⠀⠀⠀⠈⠳⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⣧⠉⠒⠒⠉⠀⢹⠀⠀⠀⠀⠠⣶⠒⠒⠲⠶⠤⢤
⠈⣧⠀⠀⠁⠀⡇⠀⢀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣸
⠀⠈⢧⡀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠳⡄⠀⢰⠇
⠀⠀⠀⠙⢦⣀⠀⠙⠓⢲⡆⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠹⣦⡏
⠀⠀⠀⠀⠀⠉⠛⢦⡄⢸⠇⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠶⣯⡞⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⠙⠒⠤⢤⣄⣀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⠀⢀⡾⠃
⠀⠀⠀⠀⢀⣤⠤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢑⡞⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⣴⠋⠀⢰⠋⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡾⢄⠀⠀⠀⢠⠞⠁
⠀⠀⢰⡇⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⠜⠀⠀⢠⠏
⠀⠀⠀⢷⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠘⡇⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢠⡞⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠑⢄⡀⠀⢀⡴⣏⠀⠀⠈⠙⠳⢤⣄⣀⣠⡴⠟
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⣢⠞⠀⠘⡆
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣠⣾⣁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣧⢀⣠⠞⠁⠀⠈⠙⠶⣄⣸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠠⠲⡦⠀⠀⠂⠤⠠⣄⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠄⠀⠀⠀⠀⢦⣤⡶⠆
⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠒⠁⠀⡀⢢⡢⠀⠈⠀⢠⡀⠀⠾⠓⠆⠀⢰⠸⠁⠀⠀⡀⠀⠀⠀⢀⠰⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣅⠀⠀⠀⠀⢁⡼⠀⠁⣼⠁⠀⠀⠀⠄⢷⣄⠀⠀⣀⣴⡟⠀⠀⢠⠜⠁⠀⠀⠀⠀⠊⠀⠀⠀⡄⡘⠀⠀⠀⢀⠠⠘
⠀⢀⠠⠆⠀⠀⠀⢀⡷⡤⢠⣈⡀⠒⠖⠒⠛⠀⠀⢠⠇⢠⠀⠀⠆⠘⠢⡉⠙⠛⠛⠉⠀⠀⠀⠀⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣧⡇⠀⢀⢀
⠀⡁⠀⠀⠈⠄⢀⣤⣠⣴⣾⣿⡿⠒⠀⠀⠀⠀⠀⡾⠀⠈⠀⠀⠀⠀⠦⠀⠱⡖⡲⠲⣶⠂⡀⠀⠀⠀⠀⠀⡙⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⢡⠀⠣⠄
⠀⠁⠤⠤⠞⠀⠸⣿⠿⠿⠛⢉⠑⠁⠀⠠⠆⢠⡄⢠⡄⠀⠀⠀⠀⠈⠀⠀⠀⠋⠀⢀⠉⠀⠲⡀⢠⡀⠀⠀⣓⡀⠀⠁⠀⠀⠀⠀⠀⠸⠀⠘⣧⠀⢻
⠀⢀⠀⠀⠀⠀⠀⠀⠒⠒⠊⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠊⠆⠀⠀⠀⠀⠦⡀⠀⢐⠎⠁⠀⡈⠀⠀⠀⠀⠠⠬⠐⠀⠀⢠⡀⣤⡀⠀⠀⠀⣄⡁
⠢⠈⠥⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⣄⣀⠀⠃⠀⠀⠀⡀⠀⠰⠞⠒⣆⠳⡄⠀⠀⠀⡠⠄⣤⡀⠀⠀⠀⡛⠀⠀⣞⠸⡃⠕⢸
⠀⠃⠀⡼⠻⣦⠀⠀⠀⠀⢂⠲⠙⠶⠠⠁⠉⠀⡀⠀⠠⠀⠀⢂⠀⠀⠀⠈⢧⠙⠆⢠⠊⠀⠀⣿⠇⠀⣤⣤⠀⠀⠀⢰⠀⠁⠈⢨⠀⢀⠄⠀⠀⠃
⠀⠀⠰⠛⡁⠘⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡘⠠⠄⠀⠀⠈⠓⠒⠃⠀⠀⢠⡿⠀⠀⣉⣉⡇⠀⠀⠐⠄⡣⣠⠞⠀⠘⠀⠠
⠀⠀⠀⠀⠤⠄⢧⠀⠀⠀⠀⠃⠀⠀⠀⠠⠀⠂⠀⠀⠀⢤⢨⠄⢲⢚⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣤⣤⠄⠀⠄⠀⠀⠀⠀⠄⢁⣲⡀
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠘⠀⠀⠀⠀⠂⠀⠂⡀⠀⢸⢐⠀⡠⠴⡕⢾⡄⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋
⠀⠀⠐⠀⠀⠀⠀⠙⠓⠛⠀⠀⢂⠀⠀⠀⠀⠀⠘⠧⠀⠘⢨⠈⠳⠊⢀⣸⠃⢀⠆⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠐
⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⠀⠀⢰⠀⠀⣟⠋⠁⠀⠂⠀⢀⡿⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠰⣶⣶⣴⣶⡦
⠀⢠⡄⠘⠃⠀⠀⠰⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣽⠀⢠⡃⠀⣠⡾⠁⠀⠀⠀⢀⡀⠁⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⡆⠀⢀⡀
⠀⠈⣗⣄⠀⠀⠀⠀⢫⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠀⠀⣠⣄⠓⠒⠋⠀⠀⠁⠚⠛⠁⠀⠀⠀⠀⠈⠀⠢⠀⠀⠀⠀⠀⠘⠒⠒⠒⠚⠀⠟⢻⣿⣿⣦
⠀⠀⢋⡈⣳⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣤⡎⠙⠏⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⠤⠄⠀⠀⠋⠈⠉⡀
⠀⠀⠈⠉⠁⢀⠀⠀⠀⠀⢰⣲⠆⢠⠀⠐⢶⣯⠈⠛⠁⢀⣀⠄⠀⠤⠚⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⢃⠀⠀⠀⠠⠂⠁⠢⠀⠀⠀⠠⠄⠐⠁
⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠈⠙⠀⠘⠉⡀⠈⠀⠀⠀⠀⠠⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣛⣫⡇⠀⠃⠀⠀⣠⠀⠀⠀⠀⢠
⠀⠀⠀⠀⣀⢀⣀⡀⠀⣶⣖⠄⠐⠒⠂⠈⣁⠠⠼⠀⠀⠄⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠂⠉⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠠⠂⡀⠀⠀⠀⠀⠀⡠⠆⠁⠀⠀⠀⠀⠀⠀⠂⠀⠠
⠀⣠⡀⠀⠀⢀⠄⠒⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⠂⠄⠀⠐⠚⠙⠈⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠘⣀⡀⠀⠀⠈⢀
⠀⠉⢁⠰⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠠⠂⠀⣸⣷⠟⠁⠀⡤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣷⣿⠀⠀⠸⣭⣡⠴⣼⠁
⠀⠉⠀⠊⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠠⠤⣤⠄⣄⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢹⡏
⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡀⠆⢰⠀⠀⢠⣋⡴⠃⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⣠⢲⢓⢀⡀⠀⠀⠴
⠈⢠⣴⣦⡀⠀⠀⠀⠀⠂⢀⡀⠒⠂⠉⢁⡀⠀⠀⠀⠈⠀⠀⢢⡄⠀⠀⠄⢀⠀⠘⡄⠀⠀⢠⠃⢰⠆⢀⠔⠲⠀⠀⠀⢹⢀⢂⠀⠃⠀⠀⠀⠀⠡
⠀⠘⣏⣼⠇⠀⠐⠠⠄⠂⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠘⢢⠀⠉⠀⠄⠁⠈⠀⠘⠇⠀⠀⠁⠀⣤⡶⢾⡿⠿⢿⠀⠀⠰⠒⠣⠼⠃⠠⠶⠤⣤⣤⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⠤⠤⠚⠁⢴⡒⢪⣹⡇⠀⠀⠀⠀⠀⠠⠜⠀⠲⢿⡟⠀⠀⠀⠘⡆⠀⠸⠀⠀⠀⠀⠀⠀⠀⢘⠖⠲⠘⠳⠚
⠀⢀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠠⠀⠀⠀⠀⠘⠛⠉⠉⠀⠀⠀⢁⠀⠂⠐⠁⣠⠦⠀⠐⠀⠀⠀⠀⣧⠀⠀⠇⢀⣟⢟⠣⠀⠀⠀⠁⢀⠀⠐⡂
⠀⠈⠈⠁⠀⠠⠂⠚⠉⠀⠀⠀⠀⠀⠀⠬⠱⠤⢀⠀⠀⡠⣦⠀⠀⠘⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⢀⣨⡤⡄⠒⠀⢀⡀
⠀⠀⠀⠀⢀⣤⠤⠀⠐⢠⠀⠀⠀⡄⠀⠀⠏⠊⣹⢇⢼⠉⢉⡇⠀⠀⠉⠁⢠⣶⣶⠀⠀⠩⣤⡤⣧⠀⠀⠠⠀⠀⠈⠑⠠⠆⡰⠔⠃⠀⠻⣿⣾⣿
⠀⠀⠀⠀⠈⢿⡀⠀⡠⠋⠀⠀⠀⠀⠈⠀⠀⠀⠍⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⢄⣁⠤⠀⠀⠀⠙⠛⠁
⣀⣀⣀⣀⣀⣀⣁⣴⣁⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠌⠀⢈⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠓⢲⡖⠒⢦⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠃⠀⠘⣦⠤⢤⣀⠤⠤⠼⣄⡤⠦⠤⢄⡠⠦⡀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⡄⠀⢰⡇⢠⠀⢻⠀⠐⠃⢈⡇⠀⠒⠀⡷⣄⠉⢁⡤⠇
⠀⠀⠀⠀⠀⠀⠀⡔⠓⠒⠿⣧⣀⣸⡀⡴⡆⠘⡆⠀⣖⠋⣇⠀⢰⢚⢁⣸⠀⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠠⣏⢩⡏⠩⠙⣏⠉⣹⠉⠏⢻⠉⡙⡿⢉⢹⡌⢋⡹⠂
⠀⠀⠀⠀⠀⠀⠀⣇⣀⣘⣠⣏⣨⣇⣀⣸⠜⣆⣿⣰⣆⣼⣀⣡⢇⡴⣄⡿⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡽⠃⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⠈⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⢻⡀⠀⡇⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢁⣀⠈⠳⡀⢣⠈⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠁⠀⠀⠀⠀⠀⠀⠉⠙⠢⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⣠⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠳⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢉⣿⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠛⠒⢦⡄⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⡙⣎⣧
⠀⠀⠀⠀⠀⠀⢠⢏⡄⣤⣀⡀⢸⣿⠉⠉⠁⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⣧⢹⣎⢧
⠀⠀⠀⠀⠀⠀⡾⡼⢸⣿⣿⡟⡼⢸⢠⡠⣶⣶⣦⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⣮⠳⣄
⠀⠀⠀⠀⠀⢸⢳⢃⣿⣿⣿⢱⡇⠸⡇⡇⢿⣿⣿⡜⣇⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣿⣷⣌⠳⣄
⠀⠀⠀⠀⠀⣿⢸⢸⣿⣿⡟⡾⠀⠀⢷⢹⢸⣿⣿⣧⢻⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⣿⣿⣿⣷⡌⢷⡀
⠀⠀⠀⠀⠀⠘⢮⣛⢟⣋⡼⠁⠀⠀⠸⡜⣆⢿⣿⡿⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣿⣿⣿⣿⣆⢻⡄
⠀⠀⠀⠀⠀⠀⢨⡿⠉⠁⠀⠀⠀⠀⠀⠙⠮⣽⣭⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⣿⣿⣿⣿⡿⠶⢿⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡤⠾⠛⠉⠁⠀⠀⠈⢷
⠀⠀⠀⠐⠶⠾⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠷⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⣀⣤⣴⣾⣿⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠚⠉⢿⣿⣿⣿⣿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠾⠟⠋⠁

⣿⣿⠻⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣣⠟⣤⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡞⢋⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠛⠛⠉⠀⢀⣀⠀⣀⣀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠉⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣷⣾⣄⣀⡀⠀⠈⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠙⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡿⠋⠶⣴⣴⣶⣿⣿⣿⣿⣏⠉⢿⣿⣿⣿⣿⡏⠈⣿⣿⡿⢿⡿⣿⣿⣿⣿⣿⣧⣤⣤⠀⠘⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡏⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⢉⣿⣇⣀⣀⠁⣿⠇⢸⡇⠸⣿⣿⣿⣿⣿⣥⡀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⢸⣿⢂⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⡆⢶⡿⠋⣻⣿⣿⡇⢻⣤⣾⣿⣾⣿⣿⣿⠛⢿⠀⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣧⡟⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⠗⠛⠳⣶⣿⡿⠛⠁⢸⣿⣯⣽⣿⣿⢿⣿⡄⠈⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠘⠃⠀⣾⢫⣿⣿⢿⣿⣿⢿⠟⠛⢻⣀⣀⣤⡄⠁⢠⣤⣤⣤⡛⠿⠿⠋⠁⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠴⠋⣿⣿⡄⢀⣼⢿⣿⣿⣇⢀⠀⣈⠻⠿⣿⣷⣦⣴⣾⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣯⣴⣶⣾⣿⣿⣿⣛⠛⠿⣾⣿⣿⣡⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧⣥⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⣐
⠀⠀⠀⠀⠀⢸⣧⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⢿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⣿⣿⣿⠟⠿⢻⣿⣿⠟⠋⣿⣿⡿⣿⣿⣿⠟⣹⢻⡏⣭⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣼⣿⢁⠀⢀⢿⣿⡇⣿⣀⣉⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠤⢴⣦⣴⡶⠄⣿⣿⣿⣿⣿⣦⢦⣾⣿⣿⣿⣿⣤⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣤⡤⠤⢤⣤⣤

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢸⣭⠇⠀⠀⠀⠀⠀⠀⠀⠘⢍⡹⠃⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠧
⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣺⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢒⠢⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⢠⠾⠥⠭⣭⣉⡆⠀⠀⠀⠀⠚⠶⠆⠀⠀⠀⠀⠓⢤⣀⣧⡇⠀⡀⠀⠀⠀⠀⣠⠖⠒⠒⠢⣄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢯⡉⢉⠐⠒⠒⠢⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢰⠁⠀⠀⣠⠤⠞⢧
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠒⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠜⠉⠁⢀⠤⠼⡆
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢠⡆⠀⠀⠀⠀⠀⢸⡏⠉⠉⢢⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠳⡤⠊⠉⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣸⡇⠀⠀⢸⠁⠀⣰⣷⡀⠀⠸⣷⣀⣠⠼⠁⣀⡀⠀⠀⢻⣆⠀⠀⡜⠀⠀⠀⠀⠈⠲⢤⣀⣠⠏
⠀⠀⠀⢼⠉⠑⠖⠁⠈⡇⠀⠀⠀⣿⠋⠉⠉⡿⠀⢠⡿⠀⣧⠀⠀⣿⠉⠀⠀⣿⠉⠉⢳⡀⠀⢿⣄⡜⠁⠀⠀⠀⠀⠀⠀⠀⠸⣽⡆
⠀⠀⠀⠈⢦⡀⠀⠀⣠⠃⠀⠀⢰⣿⠀⠀⠀⡇⢀⣿⠋⠉⠹⡄⠀⢿⡄⠀⠀⣿⠀⠀⣠⠇⠀⢈⡿⠀⠀⠀⠀⢏⠓⣹⠂⠀⠀⠈⡆
⠀⠀⣀⠀⠀⠙⠢⠔⠁⠀⠀⠀⠀⠉⠀⠀⠈⠁⠸⠏⠀⠀⠀⢧⠀⠈⠁⠀⠀⣿⠛⠋⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠘⠟⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣴⡏⠙⠳⡄⠀⠀⢸⡇⠀⢰⣾⠙⠳⡄⠈⠉⢹⡏⠁⢀⣀⠀⠀⡀⢸⡇⠀⠙⡄⠀⠀⠀⠀⢶⡄⠀⠀⢰⠀⠀⡸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠇⠀⠀⣾⠃⠀⢸⣟⣀⠴⠃⠀⠀⢸⡇⠀⢸⡇⠀⠀⡇⠸⣷⠀⠀⢹⠀⠀⣀⠀⠘⣷⡀⢠⠇⠀⠀⡇
⠀⠀⠤⣀⡄⠀⢸⣿⠉⠉⠲⡄⠀⣿⠀⠀⢸⡟⠳⣄⠀⠀⠀⣼⡇⠀⣼⡷⠦⢼⠇⠀⣿⠀⢀⡞⠀⣼⠟⡄⠀⠘⣧⡞⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠇⠈⠉⠁⠀⢸⡇⠀⠈⠀⠀⠀⠈⠀⠀⣿⠀⠀⢸⠀⠀⠿⠖⠋⠀⣼⠯⠤⡇⠀⠀⢹⡇⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠻⠦⠖⠁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠻⠀⣠⡀⠀⠀⢴⠏⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣦
⠀⠀⠀⠀⠀⣠⡀⠀⣀⡁⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⡞⠳⠊⡆⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠀⠀⠀⡇⠙⡖⠦⣄⡀⠋⠀⠀⠖⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣸⣄⣧⠤⠔⢲⠀⠀⠘⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡀⡇⣨⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⡭⠤⠔⠲⣚⣉⠽⡆⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠰⡶⠀⠀⠀⠀⢡⠀⢣⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠖⠒⠉⠉⠀⠀⣀⣱⡀
⠀⠀⣀⠀⠀⠀⠀⠀⢸⡤⠊⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠲⠶⠀⠀⠀⠀⡧⣤⢤⣖⣲⡭⠿⠽⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠴⠊⠀⠙⠦⣀⣀⡠⠋⠀⠀⠀⠀⠀⠀⠀⢟⠏⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⠿⣇⣤⣴⣾⣿⣄⠀⠀⣸⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⣲⣿⣿⣿⣿⣿⣧⡀⣻⣿⣿⣟⣋⡉⠀⢀⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡈⢀⣉⣛⣿⣿⡏
⣠⣶⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠀⢀
⣿⣿⣧⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿
⢿⣿⣯⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣼⣿
⣤⡛⠛⠀⢸⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡟⠉⢁⣠
⣿⣧⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⣿
⣿⣿⡧⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠘
⠘⠋⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠋⠉⠈⠉⢻⣿⣿⣿⣿⣷⡄⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿
⣤⡄⣀⡀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⢤
⣿⣿⡿⠃⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠀⠘⠛⠋⠉⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠰⣶⣾⣿
⣿⠏⠀⠀⠀⠀⠀⠈⠛⠛⠿⠂⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡆
⢀⣴⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠛⠃
⣿⣿⣿⣇⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢴
⣿⡟⢿⣿⡆⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢘
⠉⠀⠀⠙⠁⠀⠀⠀⠈⠿⠿⢻⣥⣴⣾⣿⣷⡄⠀⠀⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣦⣠⣤⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢻⡿⠟⠋⠉⠀⠀⠸⣿⣿⣿⡿⣿⣆⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢄⠀⠀⠀⠀⠀⠀⣠⠎⠱⡄⠀⠀⠀⠀⠀⠀⠀⡠⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠈⡆⠀⠀⠀⠀⠸⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡋⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠜⠁⠀⠀⠀⠀⢀⣑⣒⡊⠀⠀⠀⠀⠀⠀⠀⠱⠄⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠯⠥⢶⡄⠀⠀⠀⠀⣿⡉⠒⠚⡇⠀⠀⠀⠀⠀⢸⠳⠮⠽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠢⢼⡇⠀⠀⠀⠀⣿⠈⠉⠓⡇⠀⠀⠀⠀⠀⢸⠙⠒⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿⣀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⠤⢤⡇⠀⠀⠀⠀⣿⠈⠙⠒⡇⠀⠀⠀⠀⠀⢸⠉⠲⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠧⠤⠔⠒⠓⠒⠒⠒⠒⠋⠀⠀⠀⠉⠈⠀⠐⠒⠒⠛⠒⠒⠂⠾⠤⠤⣀⣀
⠀⠀⢀⣠⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠤⣄
⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⢰⠁
⡇⠀⠀⠀⢠⢻⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠑⢦⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠦⡀
⢧⣀⢀⡠⠎⠈⢧⡀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠑⠤⣀⣀⢀⣀⣀⡠⠔⠋⠀⠀⠀⠀⠁⠀⠈⠒⠤⠤⠤⠔⠉⠀⠀⠈⠓⠤⠤
⠀⠀⢹⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⣀⠤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠶⠶⠦⠤⢄⡀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠲⠤⣀⠀⠀⠀⠀⢸
⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠈⠙⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠈⠙⠒⠒⠚⢹
⠀⠀⢸⣀⡠⠤⠖⠒⠒⠒⢄⡀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠉⠉⠀⠀⠈⠑⠦⢄⣀⣀⠤⠤⠴⠒⠒⠒⠒⠒⠢⢄⣀⣠⠤⢚
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⢀⡸⠆⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⠒⠚⠢
⠀⠸⡀⠀⣰⠋⠁⠉⠦⠒⠒⠒⡤⠖⠒⢤⠤⠒⠢⡤⠔⠒⢤⡠⠔⠒⣤⠴⠒⢤⠤⠤⢤⡠⠔⠢⢤⠒⠒⠒⣎⠁⠀⠈⣆⣀⠴
⠀⠀⠈⠉⠉⠢⠤⠔⠳⣄⣀⡠⢇⣀⢀⡸⣄⠀⣀⣧⣀⢀⣠⢆⡀⢀⣧⣀⠀⣧⡀⠀⢀⣇⠀⢀⡸⢇⣀⣠⠞⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠙⠿⠛⠋⠁⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠔⣿⣋⣛⣛⣛⡙⣋⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⠀⢾⡇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡿⠀⣿⡇⠀⠀⠀⡆⠃⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢀⡀⠀⠀⠀⠀⡸⠃⠀⣿⡇⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⢸⡇⠀⠀⠀⡰⠁⢀⠀⣻⡇⠀⠀⠀⠀⠀⣀⡀⠤⠀⢀⣒⢨⣁⠀⡔⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⠷⠟⢛⣀⣭⡅⠄⠘⢻⣦⣥⠐⡆⢰⡆⢸⡇⠀⣿⠘⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⣒⠀⣭⠩⠅⠀⠸⣷⠀⡜⠐⣿⢀⠀⡌⣧⢸⡷⢼⡇⠀⠀⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠄⠒⣈⡅⢁⡔⠐⣦⠈⣿⠀⣿⠠⠀⠀⠀⢹⣧⠁⠀⣿⠙⠐⣀⢙⢸⡇⠈⢻⣄⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⣒⡈⠡⣤⠀⢸⢻⡏⠀⢸⡇⣾⢀⡄⣿⡆⣿⠀⣿⠀⡄⠀⠀⠀⣿⠀⢘⣿⠀⠘⠃⡠⠘⣋⠠⠤⠬⢑⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠸⣧⠀⣿⡄⡎⢸⡷⢲⣾⡇⣿⠈⠃⣿⠁⣿⠀⣞⠁⠀⠀⠀⢐⣛⣀⠨⠭⠉⣉⠋⣡⡀⢠⠀⠀⠐⠎⡈⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣴⢸⣿⠁⢸⡇⠀⢸⡇⠹⣆⠠⠋⠐⢛⠠⠴⠞⢐⣂⢠⡄⣶⠙⣷⡄⠀⡇⢰⡇⠛⠋⠀⠀⠀⠐⠁⠒⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠟⠀⠘⠓⣀⡉⠥⠐⠂⣈⠉⠁⢰⡆⢳⡆⢸⡇⠀⠁⣿⠀⡏⢿⡄⠃⣿⡇⠀⣾⠀⠰⣶⣾⠋⠉⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠈⢈⡥⠐⣆⠀⣿⠀⠈⠀⠀⢸⡇⣸⠃⢸⡇⠺⠀⢿⠀⡇⠈⢿⡇⠸⣇⠀⠿⠀⠀⣿⣿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠐⠀⠀⣼⠁⡀⢻⡆⣿⠖⠂⠀⢀⠼⣷⡌⣷⢸⡇⠀⢴⠅⢱⠇⠀⢈⡻⣄⡀⠄⢀⠈⠀⣿⣿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⣿⠈⠃⣼⠃⣿⠀⠀⠀⢨⡀⠋⡰⠋⢘⣃⠰⠶⠖⡁⠊⠉⣤⣤⠀⠀⣼⣿⠀⠀⣿⣿⠖⢿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⠹⣆⠠⠏⠀⠟⢀⣀⠠⠤⠀⣒⠂⡉⠠⡄⢀⡶⠁⠙⣷⡀⣾⣿⡆⢠⢿⣿⠀⠀⣿⣿⠀⢀⣤⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢀⠠⠄⠐⠒⣈⡉⠀⡶⢲⣿⠉⠙⢸⣧⡀⠀⣼⡇⢀⡀⣿⡇⣿⠸⣿⡎⢸⣿⠀⠀⣿⣿⠀⠸⠿⠆⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢿⣿⠀⢸⣿⠀⢸⣧⠃⢠⠇⢸⣿⠀⡄⠀⠙⢻⡆⣿⡇⢻⠁⣿⡇⣿⠀⢿⣇⢸⣿⠀⠀⣿⣿⣀⣀⠤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⣿⡇⠀⣿⡇⢘⣿⡄⡸⠀⢸⣿⠘⠇⠰⢄⡘⠁⢿⡇⠀⠀⡿⠁⣿⠀⠶⠃⠸⠿⡀⠊⠩⠍⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⣿⣷⠀⢸⣷⣸⢹⣷⡇⠀⢸⣿⠀⠀⣴⡋⠉⡆⠈⠷⠄⠜⡁⠀⠩⠀⠒⠂⠉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢹⣿⡆⠀⣿⡇⠀⣿⠀⠀⢸⠿⣀⣀⣈⣠⡼⠁⠒⠀⣡⠄⢄⢀⠠⡄⣠⠖⠁⣤⢸⢆
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⢿⣿⡀⠘⢀⡀⠩⠄⠂⠐⠀⠀⠀⡉⠁⠈⢹⠆⠀⣘⣃⡼⠀⣠⠜⠁⠀⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠠⠞⢂⡀⠤⠄⠊⠻⠿⠶⠀⠀⠀⡠⠒⠒⠢⠤⣁⣈⠂⠓⢶⣾⠃⠙⠿⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⢹⡆⠀⠀⠀⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢠⠀⠐⠂⠀⠔⠀⠀⠀⠀⢈⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠄⠈⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠂⢄⡄⠀⠉⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠴⠀⠄⠈⠶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠑⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀⣉⢨⢁⣀⡀⠀⠘⠄⠀⠀⠀⠌⠀⠀⡃⡆⠀⠠⠀⠐⠀⢒
⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⣀⠀⠀⠂⣂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢁⠈⢔⢤⠀⠐
⠀⠀⠀⠀⠀⢁⡹⠈⠉⠁⠠⣹⠀⠀⠉⠀⠈⡀⣄⡀⠀⠀⢀⡨⢃⠇⡈⠛⠛⠀⠀⠁⠐⠊⠀⠠⡀⠆
⠀⠀⠈⠀⠀⢒⠀⣄⠀⠀⠀⠚⠀⠀⠀⠠⢚⠧⠀⠣⠤⠦⠔⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⠅⠀⠁
⠀⠀⠀⠈⠀⠀⠡⠌⠄⠂⠁⠀⠀⠀⠄⠁⢠⡤⠟⠒⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢠⡈⠳⣄⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠱⠂⣠⡞⠁
⠀⠀⠀⠀⠱⣄⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡞⠉
⠀⠀⠀⠀⠀⠈⢡⠊⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⠷⢦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢿⣦⠀⠀⠀⠀⠀⢸⠃⡀⠀⢀⡀⢿⡄⠀⠀⠀⠀⣠⡾⠋⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠹⣷⡀⠀⠀⠠⢾⠀⠁⠂⠈⠁⣸⢧⠀⠀⣴⡾⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠈⠻⢦⡀⠀⢻⡄⠺⣯⡗⢀⡿⠃⢀⣾⠟⠀⠀⣠⠞⣠⣴⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠙⢦⡀⢻⠦⣤⡴⣿⣠⣴⡿⠋⠀⣠⣾⢫⡞⡀⠀⣀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⠀⠈⢷⡈⢰⣦⣴⠛⠛⠁⠀⢀⣾⠛⠁⢸⠀⣉⣁⡀⠀⢙⠆⠀⢤⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠁⠈⠀⠉⠀⠀⠀⢀⡾⠁⠀⠀⠸⣆⠻⠾⢁⢠⡋⠀⠀⡂⢿⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠈⢓⡖⠉⠸⣀⠀⠀⢹⡄⠀⡼⠁
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣆⣤⣶⣋⢳⣤⡾⠉⠳⣶⠟⠀⡸⠁
⠀⠘⡕⢦⣄⣏⢳⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣻⣿⢋⠀⠀⠀⠙⢿⣿⠀⠤⠤⠄⠒⠒⣲⡾⠁⠀⣼⠃
⠀⠀⠙⣦⡈⠁⠘⣇⡀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣼⣿⡇⢠⡄⡄⢠⡦⠸⣿⣿⡆⢒⣛⣛⣻⠏⠀⢀⡾⠃
⠀⠀⠀⠈⠙⠶⣤⡈⠛⠷⣤⣄⡀⠀⠀⢿⣇⠀⠀⠀⣿⣿⡇⠈⠁⠀⠀⠀⠀⢻⣿⡏⣉⣁⣼⠏⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠉⠛⠷⣦⣸⣿⠲⢤⣀⣙⢿⣿⡀⢿⣥⣽⠃⣰⣿⠟⢀⣠⡿⠃⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠙⠿⣧⣤⣀⣀⠀⠉⠳⣄⣉⣡⢴⡏⠀⣴⣿⡋⠙⢦⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⡿⠙⢻⣿⣷⣦⣼⠉⠁⢘⣧⣾⣿⣿⣿⣦⡾⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢧⣠⣿⣿⣿⣿⡏⠛⣶⠛⠀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠉⠻⣿⣿⣿⣷⣾⣿⣷⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡠⠀⠀⡀⠀⢀⠀⠀⢀⢀⠀⢀⣶⠆⠀⠀⠀⠀⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣆⠀⠰⢡⢣⢂⠎⠀⡘⢸⡄⠀⢸⢸⠀⣼⠏⢀⠀⡄⡄⠀⣏⡏⡀⠀⣀
⠀⠀⠀⠀⠀⠠⡀⢀⡀⠀⠀⠀⠀⠘⡜⠀⠀⠀⠏⠎⠀⢀⠃⢻⠀⠀⠘⠀⠰⠏⣰⠃⠀⣧⡇⠸⡻⡰⢁⢞⡁
⠀⠀⠀⠀⠀⠀⣈⠢⡙⢦⠀⠀⢀⠀⠘⠀⣿⡀⠀⠀⠀⢸⠀⠋⠀⠀⢀⣴⠀⠀⠁⠀⠀⠃⠀⢀⡁⠰⡣⠋⠀⠀⠀⢀⡤⣚⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢌⠳⠅⢀⣿⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣮⡟⠀⠀⠀⡄⠀⠀⠀⣞⡇⠈⠁⠀⠀⢀⠐⠉⢈⣁⡠⠤⠄
⠀⠀⠀⣀⣐⠢⠤⣀⡀⠙⠂⢘⣼⠀⠀⠀⡞⣷⠀⠀⠠⢳⡀⠀⠀⢰⢻⡆⠀⠀⢰⢹⡀⠀⢠⣾⡆⠀⠀⠀⢰⣻⠀⠈⠐⠒⠒⠘⠃
⠀⠀⠀⠀⠈⠉⠓⠒⠪⠄⠀⣴⣷⡆⠀⠀⡇⣿⠀⠀⠀⢻⡇⠀⠀⢸⢀⡇⠀⠀⠈⣾⡀⠀⢸⠀⡇⠀⠀⢠⣾⡇⠀⠐⠒⠲⠶⠭⠭⢥⣄⡀
⠀⠀⠀⠀⣒⡒⠫⠭⠙⡁⠀⢻⠀⡇⠀⠀⡇⣿⠀⠀⠀⡓⣿⠀⠀⢸⢸⡇⠀⠀⢸⠋⣿⠀⢸⠀⡇⠀⠀⡸⢰⡇⠀⠲⠭⣒⡢⠄
⠀⠀⠀⢀⣀⠤⠖⠊⠉⠀⠀⢸⠀⣷⠀⠀⡇⢿⣀⣠⠬⠄⣿⠤⠤⢾⢘⡧⠤⠤⢼⡆⣿⣀⣸⠄⡇⠀⠀⡇⣸⠀⠀⠀⠀⠀⠉⠓⠲⠤
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡼⡀⢿⠉⠹⣳⣞⡦⠀⠀⡇⣿⠀⠠⡞⣼⢷⡄⠀⠘⡇⣿⠀⡸⠀⣏⡒⢲⠁⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣠⣇⣸⣄⠀⠈⠉⠀⠀⠀⡇⢸⠀⠀⠉⠉⠉⠀⠀⠀⡇⢹⠀⠑⠿⠷⠃⣸⢰⡇⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠘⠞⠗⠋⠀⠀⠀⠀⠀⣖⠣⢾⣦⠀⠀⠀⠀⠀⠀⢮⣳⣞⡷⠀⠀⠀⠸⢜⣺⠽⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣃⡤⠞⡗⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣠⡀⠀⣀⣤⣀⡀⠀⠀⢠⠖⢦⣀⡟⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠑⠛⠉⠉⠑⢄⣀⡤⠞⠋⠉⠑⢄⣀⣠⡾⠁⠉⠁⠉⠛⠉⠀⠀⠉⠳⢤⡼⠀⠀⠀⠀⣻⣢
⠀⠀⠀⠀⠀⠀⠀⠘⣟⠣⣀⠀⠀⠀⠀⢀⣀⢶⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⢦⣠⠀⠀⢀⣀⣤⣾⢿⠁⡠⠤⠤⢤⣀⣀
⠀⠀⠀⣠⠴⠒⠋⠉⣸⠉⠻⢝⣓⡒⢖⣼⠂⠈⣽⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠮⡁⠀⢯⡽⠿⠻⠟⠉⠁⢸⡧⡄⠀⠀⠀⠀⠈⠙⢦
⠀⠀⣞⣁⣀⠀⠀⠀⢝⣄⡀⠀⠀⠈⠉⠉⠚⠑⠻⠦⠯⣽⣒⣒⣒⠶⡶⣖⢒⡒⣏⡭⢽⠴⠲⠉⠉⠁⠀⠀⠀⠀⢀⡼⡣⠇⠀⠀⠀⠀⣀⣀⡼
⡠⠊⠉⠉⠉⠀⠀⠀⠀⠙⠼⣶⡤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠙⠛⠋⠉⠛⠁⠀⠀⠀⠀⠀⢀⣀⠤⣶⣮⠽⠋⠁⠀⠀⠀⠀⠈⠉⢙⡇
⠙⠦⠤⠤⡤⠶⠃⠀⠀⠀⠀⠈⠉⠓⠽⠿⣗⢦⣤⣀⣤⠤⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⣤⣴⣒⣺⣭⠽⠛⠛⠁⠀⠀⠀⠀⠀⢠⣤⣤⡤⠶⠋
⠀⠀⠀⠀⠳⣄⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠙⠓⠚⠛⠓⠋⠙⠛⠋⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠙⢯⣁⣀⣀⠀⣀⣠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢽⣷⠤⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠳⠦⣤⣤⣤⣀⣤⣤⡾⠗⠀⠀⠀⠀⣠⣾⠃⠀⠀⢀⣀⣠⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠿⠿⠿⠻⢿⡀⣿⣷⣶⣦⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡅⢠⣤⣴⡆⠸⠁⢸⣿⣿⣿⡟⠀⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣉⣉⣀⣄⠀⢸⣿⡿⠋⠀⠀⠰⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⢇⣴⣿⣿⠁⠀⠀⠀⠀⠻⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣄⣀⡀⠀⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠟⠛⣋⣉⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣌⣉⣉⠙⠛⠶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠟⣛⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣭⣓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠛⠉⠉⠉⠉⠉⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⠹⣿⣿⣿⡿⠿⠟⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢠⡿⢋⣁⣄⣀⣀⣀⡀⠀⠶⣶⣶⠀⣤⣶⣾⣿⣿⣛⠻⠟⠆⠀⠀⢸⣿⠷⢋⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣰⣿⡿⢿⣿⣿⣿⣿⣶⣶⡎⠁⠀⣡⣾⣿⠟⢿⣷⣿⣆⠀⠀⠀⠸⣿⢠⡟⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⡿⢡⣶⠿⠛⢻⣿⢿⣿⣿⡇⠀⠀⣿⠙⣧⣐⣂⣉⠄⠁⠀⠀⠀⠀⠈⢼⣷⣌⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡏⢠⡐⣿⣿⣦⣤⣤⣶⠆⣽⣿⡇⠀⠀⠈⠁⠈⠉⣉⠁⠀⠀⠀⠀⠀⠀⣤⠀⡼⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⢸⣷⡙⠏⠉⠉⠀⠉⣼⢿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀⢠⣿⠘⠑⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠘⣿⣿⡀⠀⠀⠀⠘⣡⡞⠛⠁⠀⠀⠈⢛⠃⠰⣄⠀⠀⠀⠀⠀⢀⣾⣿⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣇⠹⣿⣧⣀⣴⠆⣴⣿⣦⣴⣶⣤⣶⠞⠻⠋⠀⠈⠳⠀⠀⠀⠀⣾⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢽⣆⢻⣿⣿⣿⢠⠟⢻⣿⣿⡿⠁⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣼⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣿⣿⣿⢀⣀⣈⢁⣤⣶⣶⣆⣤⣤⣶⣶⣬⣷⡄⠀⠀⠀⠀⢠⡟⢸⣇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠀⠀⠹⣿⡇⢺⣿⣿⣿⣿⣿⣿⡋⣛⠋⠁⢀⠉⠙⠀⠀⠀⠀⠀⠟⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢃⣾⠁⠀⠀⠹⣿⡀⢿⣿⣌⠛⣿⣿⡿⠿⠶⠿⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⣿⡿⠀⣰⣶⡰⣎⠻⡌⣿⣿⣦⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠂⢀⡶⠀⣸⠃⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⣿⣧⣾⣿⣿⠇⢹⣷⣤⡘⢿⣿⣦⣄⣀⣀⣀⣀⣠⣴⣾⡿⠋⣀⣴⡿⠃⣰⡏⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⣿⣿⣿⠋⣴⠄⢿⣿⣿⣦⣙⡛⠿⠿⠿⠿⠿⠟⠛⠋⣠⣾⣿⡿⠁⣰⡿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⢠⡏⣼⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣿⣿⣿⣿⣧⡾⠋⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⢸⡇⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⢰⡎⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣸⣿⠘⢧⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣛⡉⠁⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠈⠻⣮⠻⠿⠛⣛⣉⣭⣤⣶⣾⡿⠋⠀⣠⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⣠⡶⣾⣿⣿⣿⣿⣿⡿⠋⠀⢠⡾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⡄⠊⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠘⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣖⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢻⣵⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠈⢻⣿⣭⣍⣻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢃⠀⠙⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣆⠀⠈⠻⣿⣿⣿⣿⣿⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡄⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡝⣦⡀⠀⠈⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡳⣄⠀⠀⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣀⡈⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠚⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠿⠛⠛⠛⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⢀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⣴⣿⣿⡿⠿⣿⣿⣷⡀⠀⣴⣾⣿⣿⣿⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠛⠛⠋⠀⠀⣹⣿⣿⡇⠀⠛⠛⣿⣿⣿⠀⠀⠀⣀⣿⣿⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠟⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠉⠀⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣶⣶⡖⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠛⠛⠋⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣠⣤⣤⣶⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠙⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡠⡠⢀⢀⢀⠀⡀⡀⢀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠠⠂⠈⠀⠀⠀⡀⢈⠀⠀⠁⡀⠊⠄⠀⢡⣁⠂⠊⠊⠐⡁⢀⠀⢂⠀⢀⡀⠢⠊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡨⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀⢂⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠄⡠⠂
⠀⠀⠀⠀⠀⠀⠀⡈⠐⠈⠀⠀⠀⠆⠱⡀⠀⠀⠀⡈⠱⡀⠀⠀⠀⠜⢣⠀⠀⠀⡌⠱⡀⠀⢘⠈⣆⠀⠀⠀⠀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠄⢀⠇⠀⠀⠀⠄⠇⠅⠀⠀⠸⡀⠈⠇⠀⠀⠇⠀⠑⠀⠀⢐⠘⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⡡⢂⠀⠁⡴⠂⠐⢠⠀⠀⠀⠀⠓⠀⠀⡀⠀⠌⠚⠢⠀⢀⠐⠛⢠⠁⠐⠉⠹⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠊⠀⢸⠀⢰⣤⠡⡊⠀⠀⡲⢽⠠⡪⠐⠴⠀⠠⠀⠈⡄⢠⢸⡆⣳⢀⡀⡄⢀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠧⠻⠈⠂⠀⠦⢰⢸⠀⠐⠄⠨⠠⠀⠀⠕⡇⠜⡇⠂⠐⠸⢣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢹⠀⠀⠀⠀⠠⢸⠀⠀⠀⠀⢨⢤⠀⠀⡇⠸⡇⠀⠘⠨⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⡆⠀⢸⡀⢀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠈⢈⠁⠀⣀⢈⢉⠉⡁⡈⡩⠈⠉⡉⢀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠻⡀⢒⣈⡀⠀⠀⠠⠂⠔⣁⣊⣊⣊⣀⠁⠐⠔⠡⠢⠀⣂⣊⣈⡐⠐⠀⠌⠄⠄⢂⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠈⠉⠂⠈⠓⠒⡰⠊⠉⠁⠑⠀⠀⡉⠉⠓⠒⠒⠒⠓⠁⠀⠲⠋⠉⠑⠂⡮⠉⠉⢉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠶⠷⠶⠶⡖⣖⡶⠿⢶⣶⣶⢶⣶⢶⣶⡖⣴⡶⣖⢢⠯⠯⠽⠿⠭⡟⠿⠽⠽⠿⠏⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠈⠀⢠⡄⠈⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢂⠀⠀⡀⠄⢐⠆⠂⣀⠦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣥⠤⠠⢤⢤⣄⢠⣭⠤⡤⡬⠥⠤⡤⢠⣤⣐⣂⢀⢤⢤⢤⢽⠉⣉⡉⢹⣭⡉⠙⣿⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡉⠉⠉⢩⠉⠉⠉⠩⠈⠉⠉⠉⠩⠁⠉⠩⠉⠉⠙⠛⣉⠉⢙⣩⣉⣉⣛⣝⣉⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠐⠒⠀⠐⠀⠈⠉⠀⠀⠀⠀⡉⠁⡈⠉⠁⢉⠀⡁⢈⢉⠉⢀⢉⡩⠄⠄⡉⡈⣉⢉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⠧⣀⡹⠦⣄⠠⢑⡴⠦⠞⠉⠳⢌⣠⠷⢢⡐⠐⡡⠖⣄⠁⢠⠞⠙⠳⣬⢐⠝⡇⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⡘⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⢧⣋⠞⠁⠀⢀⠀⣀⣀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣀⡀⠀⣀⣀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠑⠀⠑⠁⠊⠓⣐⣓⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢢⡆⢠⢒⣔⢢⠀⠜⣩⡍⠍⢉⠤⡤⢌⠀⠩⢭⠍⡍⢩⢭⡬⢍⠉⡕⡬⢢⠀⠔⣦⠒⠰⣞
⠀⠀⠀⠀⠀⢀⡀⣀⣘⢋⡀⠈⠊⠩⠃⠀⠈⠟⠿⠃⠀⢪⡱⠞⠀⠐⠒⣴⠃⠈⢟⣲⠏⠀⠁⠉⠞⠀⠩⠙⠂⠀⠹⠀⡀⣠⡀⡀⡠⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠐⠉⠛⠉⠛⠉⠛⠙⠛⠰⠒⠒⠒⠲⠒⠶⠒⠶⠲⠶⠰⠶⠴⠖⠶⠖⠶⠒⠲⠂⠜⠫⠋⠑⠉⠊⠈⠁⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣧⠀⡆⠀⠀⠀⣀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡆⣷⣄⠀⡞⠁⠀⠀⠀⠀⣀⡤⠿⢒⡒⣒⡶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⡏⠙⢻⡇⠀⠀⢐⣠⢞⡥⠚⢉⡩⠥⠬⠽⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣾⠗⠊⠉⣻⢉⣠⠤⠴⠶⠒⣩⣿⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣿⡇⠀⠘⡏⠉⣻⠋⠀⡤⠾⣷⡉⠀⢀⣠⠴⠚⠁⣿⣆⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢿⡇⠀⢀⡇⠀⣿⠀⠀⠀⠀⠀⠙⠳⠯⠤⣤⣴⢚⡟⢻⢣⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⢸⡇⠀⣾⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⠃⡼⠀⣸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⢸⡇⠀⢻⡇⢸⡇⠀⠀⠀⢀⠀⠠⡀⠀⢿⢸⠀⡇⢠⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣸⡇⣸⠁⠀⢸⠃⣸⢧⡄⠀⠀⠺⠃⠀⠛⣀⣼⡟⢸⣧⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣇⣿⠀⠀⢸⡇⣿⠀⠻⠶⠚⠚⠛⠛⠉⠁⢾⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣿⡇⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣸⣃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣦⣤⣍⣯⡏⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠒⠒⠲⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢀⣤⠀⣠⣤⣄⡀⠀⣠⣤⣤⡀⠀⢀⣤⠀⣀⠀⣀⠀⣀⢀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠏⣾⡟⠉⢹⣷⣼⡟⢉⣭⣤⠀⣾⣿⡇⠶⣶⣄⠀⣴⣦⠀⢶⣆⣰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠙⢿⣶⡾⠋⠘⢿⣶⣿⡿⢸⡿⠛⣾⠀⣸⣿⣸⣿⣿⣆⠈⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠛⠋⠙⠛⠀⠘⠃

⠀⠀⢀⡀⠀⡀⠄
⠀⠀⣀⠀⠀⠂⢠⡔⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠀⡀⢤
⠀⠈⢹⠀⠸⡇⠘⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢲⡘⠀⠞
⠀⠐⠀⠀⠀⠤⠄⣠⡁⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢴⡄⠈⠋⠠⠖⠐⢀⠖
⠀⠀⢣⠀⠀⠀⠀⠀⠁⡰⠶⠾⠀⠀⠀⠀⠀⠰⣶⠀⣀⠀⠐⣀⠀⠶⠄⠠⢆⠀⠀⠀⠀⠈⠓⢦⠀⢥⠀⠀⠀⢀⠻⢦⡀
⠀⠀⠀⠓⣤⡀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⠀⠀⠁⠀⠀⠞⠁⠈⣃⠀⠀⠀⠀⠀⢸⡀⠀⠙⠀⠀⡜⠁⠒⠛
⠀⠀⠀⠀⠀⠹⠂⠸⡄⠀⠀⠀⠀⠀⠀⢀⠰⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣐⠆⠀⠀⠀⢱⠄⢀⡠⠴⠁
⠀⠀⠀⠀⠀⠀⣧⠀⢿⡀⠀⠀⠀⠀⠀⣘⡷⠄⠚⠓⠂⠘⠛⠒⠤⣀⠀⠀⠀⠀⢞⠁⠀⠀⠀⡰⠃⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠙⡄⠘⣧⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠳⢄⠀⠀⠈⢓⠀⢀⡞⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢃⡀⢘⣄⣠⠏⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠈⠆⠀⠀⣸⣄⠎⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⢰⡞⠉⠉⢹⠇⠀⠀⠀⠀⠀⠊⠁⠙⠀⠀⣀⡤⠚⡁⠀⠀⠸⠀⠀⣿⠏⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡎⠀⠀⠀⣤⣤⠤⠤⠤⠔⠒⠋⠁⣠⣼⡇⠀⠀⠐⠓⢶⣟⡀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡇⠀⠀⠀⢻⣿⣄⣀⣤⣤⠤⠚⠛⢻⡟⠀⠀⠀⢦⡤⠎⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢰⠀⠀⠀⠈⢿⣿⡿⠿⠇⠀⠀⣸⠋⠀⠀⠀⢠⠟⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢳⡀⠀⠀⠀⠙⠓⠀⠴⠖⠊⠀⠀⠀⢀⣴⠋⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠛⠒⠒⠢⠤⠤⣄⣀⣀⣀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠰⠒⠒⠂⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠉⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢣⡀⠀⢀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⠤⠒⣇⣀⣀⠀⠀⠀⠀⠀⠀⢀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⡄⠀⠀⠉⠉⢹⠉⠉⠚⠟⠓⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠀⠀⠀⠀⠀⠀⣧⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⣳⠀⠀⠀⠀⣠⠜⠁⠀⢠⡄
⠀⠀⠀⣀⠀⠀⢀⣤⠔⠊⠁⠀⠀⠀⢻⣀⠠⠔⠊⠁⠀⠀⠀⢸
⠀⠀⢘⠁⠄⢀⣈⣃⣀⣀⠀⢀⡠⠛⠊⠉⠣⠀⠀⠀⠀⡀⠁⢰
⠀⠀⠈⠁⠀⠤⠤⠤⠤⠤⠀⠿⣈⠀⠄⠀⠀⠰⠀⠤⠄⣛⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡼⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠛⠛⠛⠛⠛⠓⠲⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⣴⠶⣄⠀⣿⠀⢹⡆⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⣿⠀⠸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡴⠛⠉⠛⠲⣤⡀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢀⣠⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⢠⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢹⡄
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⣷
⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢦⣀⠀⠀⣠⠏⠳⣄⠀⠀⢀⠿⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢸
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡧⣤⣤⠎⠀⠀⠈⠈⠉⠁⠀⠀⠈⠙⠓⢋⠀⠈⠳⠤⣤⣤⣤⠴⣞⠁⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣀⣀⣸
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⠴⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠘⡆⠀⡤⠒⠋⠉⢶⡀⣿⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠛⠛⠉⠉⠁
⣿⠀⠀⠰⣄⡸⠂⠀⠀⠀⠐⢆⣠⠀⠀⢰⣄⣀⡴⠁⠀⢀⡇⠀⢀⣤⣀⠀⠱⠀⠀⠀⠀⠀⠀⢹⡼⠀⣠⣶⣦⠀⢻⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠸⡇⠀⠺⣿⡿⠀⢀⡇⠀⠀⠀⠀⠀⠈⣧⠀⠙⠿⠛⠀⣸⣿
⢿⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠲⣆⠀⢀⡇⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠀⠀⢹⠳⢤⣀⡀⠴⠋⢹⡄⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣀⡀⣀
⠸⣇⠀⠀⠢⠀⠀⢰⠋⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠹⣧⠀⠀⠈⠉⠉⠉⢹
⠀⢹⡄⠀⠀⠀⠀⢸⠀⠀⠠⠽⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⡾
⠀⠀⠹⣦⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣼⠁
⠀⠀⠀⠈⠳⣤⡀⠀⠈⠛⠲⠤⡤⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠚⠁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣰⠇
⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡇⣼⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⣲⠶⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣀⣤⠶⢦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢉⡟⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⢀⡾⠋⠀⢀⡾⠁⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⢀⡾⢄⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⢸⠃⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⣸⠇⠀⠀⢠⡞⠁
⠀⠀⠀⢸⡆⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⢠⡟
⠀⠀⠀⠀⢷⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣠⡟⠉⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡼⣯⠀⠀⠀⠀⠉⠛⠶⣤⣤⣤⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠉⠳⣴⠏⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠟⠉⠛⠶⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣆⣴⠟⠁⠀⠀⠀⠀⠀⠉⠛⠶⠟

⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⢀⠄
⠀⠀⠀⠀⠀⠀⢁⡼⣞⠀⡼⡃⠀⠀⠀⠀⠀⠾⠛⣷⡟⢻⣿⠛⣷⢾⡇⠀⠀⠀⣀⣤⡀⣀⣤⡀⢀⣤⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠛⠒⣨⡏⠚⡇⠀⠀⠀⠀⠀⣠⣾⣻⣇⣸⣿⣾⣃⣸⡇⠀⠀⠀⢉⡼⠳⣇⠀⣹⢏⣴⠃⠙⡇
⠀⠀⠀⠀⠀⠀⢀⠔⣡⠏⠀⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠰⠋⠀⠙⠃⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⢀⡀⢀⠀⢀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⢀⣀⠀⣠⡤⠀⠀⠀⠀⢠⡖⣠⠁⣶⣶
⠀⠀⠀⠀⠀⢋⣽⡏⢹⣏⡿⢿⠀⠀⠀⠀⠀⢉⣴⠇⢰⣿⡆⠈⣻⡟⢻⡇⠀⠀⠀⠀⠀⠀⠉⣤⡌⢁⡤
⠀⠀⠀⠀⠀⠾⠿⠷⠾⠿⠶⠾⠀⠀⠀⠀⠐⠛⠛⠓⠀⠉⣴⣾⣯⡄⠸⠇⠀⠀⠀⠀⠠⠞⣠⣷⣤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣀⣠⡀
⠀⠀⠀⠀⠀⢠⣠⠄⣄⢠⡀⡄⠀⠀⠀⠀⠀⠉⣩⡿⣿⠉⣿⣩⡿⠛⣿⠀⠀⠀⠀⠀⠀⣡⠟⣇⣥⣀⣤
⠀⠀⠀⠀⣀⣚⣿⠀⣀⣚⡁⡇⠀⠀⠀⠀⠀⠘⣛⣒⣛⣚⣛⣛⣒⣚⡛⠀⠀⠀⠀⠀⠀⠷⠖⢛⡼⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠓⠂⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠛⠛⣉⠉⢉⣉⠉⠙⠛⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠀⢰⣖⡃⢸⣽⡇⣿⣽⠀⣾⡇⠐⡶⠄⣈⠙⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⠆⠘⣆⠀⢻⡥⠜⠗⠃⠛⠙⠘⠛⠇⣸⠃⣰⢏⣴⢆⡈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⣠⠦⠙⢿⣡⠜⠋⣠⢤⣶⡾⠿⠿⠿⠿⣶⣶⢤⡀⠁⠞⣿⢏⡴⢶⡌⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠈⠳⣼⠆⣡⣴⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠪⣗⢤⡀⢿⡾⠋⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡀⠀⠀⠀⡴⠟⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡱⣄⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡠⡼⠰⠀⠀⡞⠀⠀⠀⠀⠀⣴⣿⠿⣿⣷⡀⣶⣿⣿⡇⠀⠀⠀⠀⠘⡆⠀⠰⡘⡤⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣷⣧⣧⡇⣼⠀⠀⠀⠀⠀⠀⠿⠟⠀⣼⣿⠃⠁⢸⣿⡇⠀⠀⠀⠀⠀⢸⡄⣧⣧⣷⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠛⠛⠛⠛⠛⠓⠤⣴⡄⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⢸⣿⡇⠀⠀⠀⣶⡴⠘⠛⠛⠛⠛⠛⠛⢧⣤
⠀⠀⠀⠀⠒⠒⠒⢶⣾⡗⠀⠠⣴⣤⠄⠀⠀⢬⣽⡇⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣭⠄⠀⠀⢤⣴⣤⠀⠐⣿⣦⠶⠒⢰
⠀⠀⠀⠀⢸⠀⠀⠰⣿⣇⠀⠀⠋⠛⠀⠀⠀⢼⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⢸⣿⡇⠀⠀⠀⣿⡷⠄⠀⠀⠘⠋⠃⠀⢀⣿⡷⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⢺⣿⣟⣲⣶⣶⣶⣶⣶⠶⢾⡛⠃⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠛⣻⠶⠲⣶⣶⣶⣶⣶⣚⣿⣿⡀⠀⢸
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢻⡁⠁⠃⠁⠀⠀⠳⣄⠀⠀⠈⣿⠁⣿⡁⣾⣇⢸⣿⢸⣿⠀⠀⢀⡴⠃⡀⠀⠁⠁⠁⣱⠋⠉⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⣴⣾⠟⢠⡈⠳⢦⣤⣄⣀⠀⠀⠀⠀⠀⢀⣀⢤⣠⠴⢋⣀⡲⠿⣆⡀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣆⠘⢃⡴⣿⡶⣣⡄⡉⠛⠻⠿⠿⠽⠿⠿⠿⠛⠋⢡⣤⠘⢯⣷⣄⠈⢡⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡘⠋⣰⢿⡿⢱⠏⣶⣰⢲⡶⠆⣶⣦⢸⣟⡂⢻⡷⣄⠛⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣘⠁⠟⠀⣿⠏⢸⣟⡀⢿⢿⡀⠾⠟⢈⣡⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⣤⣤⣬⣭⣥⣬⣤⣴⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣇⠀⠀⠀⠀⠀⠀⠀⢠⡷⣄⠀⠀⠀⠀⠀⠀⠀⣸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⠘⡆⠀⠀⠀⠀⠀⢠⡟⠀⠘⡆⠀⠀⠀⠀⠀⣰⠃⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⠸⠃⠀⠀⠀⠀⠀⠈⠳⣽⠴⠃⠀⠀⠀⠀⠀⠘⠆⡧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠳⡄⠀⠀⠀⠀⠀⠀⣴⠛⣦⠀⠀⠀⠀⠀⠀⢠⠞⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⡄⡗⠀⠀⠀⠀⠀⠀⡇⣠⣿⠀⠀⠀⠀⠀⠀⣿⢠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⢻⠀⠀⠀⠀⠀⢰⡇⣠⡟⠀⠀⠀⠀⠀⢀⡟⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢸⡀⠀⠀⠀⠀⢸⡇⣁⡿⠀⠀⠀⠀⠀⢀⡇⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣿⠶⠾⠶⠶⠶⠶⠶⠾⠷⠿⠿⠶⠶⢶⣶⣶⣼⣧⣭⣿⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⣿⡁⣄⠀⣰⡟⠁⠀⠉⠻⣦⡐⠦⠤⠤⠖⣡⣴⠟⠉⠉⠉⠻⣦⣄⠐⠤⣤⣤⡤⠔⢀⣴⠟⠉⠉⠻⣦⡀⢠⢆⣿⠁
⠀⠀⠀⠀⠀⠀⠘⠷⣶⡶⠟⠀⠀⠀⠀⠀⠈⠛⠻⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⠶⠾⠛⠁⠀⠀⠀⠀⠈⠻⠷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠛⠛⠂⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⠶⠶⠈⣿
⠀⠀⠀⠀⠀⠀⣠⣴⠿⢷⣤⣤⣤⣤⣤⣤⡶⠾⠛⠛⠛⠛⠛⠛⠶⣶⣤⣤⣤⣤⣴⠶⠾⠛⠛⠻⠶⢦⣄⣀⢀⣀⣠⣤⣤⣴⣿⣀
⠀⠀⠀⠀⠀⠰⣿⠀⣀⣀⡀⠀⢉⣉⣁⣀⠀⠀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣉⣉⡀⠀⣀⣀⣀⣀⠀⠀⢀⣉⣛⠛⠋⠉⣀⣀⣀⢈⣿
⠀⠀⠀⠀⠀⠀⠙⠷⣦⣭⡶⠾⠛⠉⠉⠙⠻⠶⣦⣭⣥⡶⠶⠟⠛⠋⠉⠉⠉⠛⠷⠶⣮⣭⡶⠶⠟⠛⠉⠙⠛⠷⠶⢾⣿⠶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠐⠶⠶⠶⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠶⠶⠶⠖⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠖⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠟⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠰⡀⠀⠀⠀⢠⠀⠀⠐⢀⠐⢠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠈⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠤⣓⣂⠘⠠⡀⠀⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⢰⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠀⠀⠀⠁⠀⠀⠀⢠⠀⠁⠀⠀⠀⠇⠀⡆⢸⠀⡁⢀⠀⠄⢠
⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⡇⢸⠄⡇⢸⠀⠦⣜
⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠤⠀⠀⡁⠈⡀⢁⠈⡀⠡⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢐⠒⡦⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠐⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠂⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠀⢀⠀⢠⠆⠀⠀⠀⢠⡞
⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⣿⠀⡄⡀⠠⣿⠄⣿⣀⠀⣠⣤⡟⢀⣤⠀⠀⣀⠀⡀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸⡿⣷⣴⡞⢷⡶⣇⢠⡏⢼⠠⣏⢸⠥⣟⢠⡆⠀⡏⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⣀⣼⣧⣿⠴⠓⠻⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⣿⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢀⠀⡄⠀⠐⡒⠂⠁⠀⠀⠈⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡁⠀⠉⠙⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⡀⠉⠀⠄⠀⠠⠀⢁⠀⠸⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠤⠀⠀⠄⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⡈⠂⢀⠀⠀⠀⠤⠤⠐⠂⠰⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢠⣈⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢈⡉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠈⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⣿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿