Weak Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠉⠉⠉⠉⠉⠁⠐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢤⣄⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⣬⣻⣶⣒⡂⠳⠆⠈⡀⢝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⠒⠒⠤⠭⠷⠌⠁⠀⠀⠀⡇⠀⢇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣠⡷⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⢀⠀⣀⣠⣄⡀⠀⢠⠟⡄⠀⣠⠌⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⣰⢣⡞⢉⣀⢈⠙⢦⣿⠚⠙⣾⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢉⣾⠰⠋⠀⡀⠂⠈⢲⠶⠞⣹⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠉⡘⠛⠲⠿⠁⡔⠀⠖⠳⣚⠉⢳⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⢀⣤⣮⠵⠊⠁⠀⢈⠒⠒⠊⢁⡌⠀⠀⠈⠉⠉⢉⠴⠤⣄⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣠⡀⠀⠀⡼⠁⢠⠄⠀⠀⠀⠀⠀⠓⠲⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠲⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢦⠀⠀⠙⠛⠀⠀⡇⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢦⡀⠙⠲⠄⠀⠀⠀⠹⢄⣈⡙⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⢾⠉⠁⠉⢱⡆⠀⡇⠀⠀⠀⠀⠀⠀⠒⠂⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠸⠤⠚⠀⠀⠠⠜⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠘⠉⠳⣄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠞⠉⠀⢀⡠⠐⠐⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⢸⠀⠀⠀⢰⣆⠙⠢⠀⠀⠀⡰⠀⠀⠀⡼⠃⠀⠀⡠⠊⠀⠀⠠⠞⠃⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢸⣄⠀⠀⠀⣹⠀⠀⠀⠀⢠⠁⠀⠀⠸⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⢸⠀⠀⠀⠀⢮⠓⠒⠀⢠⡄⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⢸⠀⠀⠀⠀⠈⠃⠀⠀⣴⢱⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⣀⠴⠋⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢦⠀⠘⠀⠀⢀⣀⡤⠀⠀⣰⢿⡌⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠉⠊⠉⠁⠀⠠⠚⠁⠀⠑⢦⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡜⢀⠂⠀⠀⡀⠀⠀⠀⠀⠈⢧⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢼⠀⣜⠀⠀⠀⢱⡄⠀⠀⠀⠀⢸⠀⠀⠘⣆⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠳⡌⢦⡀⠀⢠⠇⠀⠀⠀⠀⡎⠀⠀⠀⠈⠙⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠹⣄⢧⣠⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠠⠤⠒⠒⠒⠒⠒⠒⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠠⠋⠈⠻⡀⠀⠒⠈⠉⠉⠉⠙⠲⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢦⡀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⠈⣀⡤⠀⣀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣠⠀⡐⡛⡟⢻⢻⢻⠻⠛⢫⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠚⠥⠟⠁⠒⠀⠠⠤⠔⠒⠒⠛⠛⠓⠛⠁⠙⠛⠓⠓⠓⠛⠚⠛⠛⠓⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠛⠛⠋⠉⠉⠉⠛⠛⠓⠲⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣍⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⡉⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⠟⠘⢛⣓⣒⣒⠓⠲⣖⣀⣀⠀⠀⢀⣀⣀⡤⠔⡾⠀⠀⠀⠀⠈⠻⢯⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⠖⠉⠀⠀⠈⢳⡄⠀⢀⠀⠈⠉⠀⠀⠀⠀⠀⡧⡶⠀⠀⠀⠀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠞⠁⠀⠀⠀⠀⠀⠀⢳⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠓⢦⡁⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⡇⠀⠀⠀⠀⠀⢀⡀⡼⠁⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢿⠀⢀⣏⣤⠤⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠹⣄⠀⠀⠀⢠⡏⢁⣇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⠓⠉⣡⠤⢦⣄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡴⠚⢿⠒⠲⠖⠛⠉⢹⡻⠤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠳⠶⠋⠈⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢧⣀⣈⠃⣤⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⢦⣄⠀⠀⠀⠒⠦⠀⠀⠀⠀⢀⣀⡴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠈⣣⠀⠀⠀⠈⠙⠓⠒⢶⠶⠶⠶⠒⠛⠉⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢛⡖⠚⠻⡄⠀⠀⠀⡎⢠⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⢀⡤⢤⣀⠠⣴⠊⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⠟⣷⠀⢅⠀⠀⡸⣥⡇⠀⠀⠀⠀⠀⠀⠰⡬⡇⠀⠀⢸⠀⡶⡞⢶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡽⠶⢤⡑⠞⣡⠏⡇⠀⠀⠀⠀⠀⠀⠀⣷⠸⠤⠚⣡⡜⣇⡸⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣾⠀⠙⠶⠃⢘⡇⠀⠀⠀⠀⠀⠀⠀⢾⠣⠔⠉⢘⣏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡮⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣋⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⡇⠀⠀⠀⠀⠻⡤⣤⣀⣀⣀⣀⣀⣀⡼⠀⠀⠀⠀⣚⡉⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣇⣀⣀⣀⣀⣀⣸⠀⠀⢸⣅⣏⠀⠀⠀⣇⠀⠀⠀⠀⠘⣟⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⢼⠿⠼⣿⡼⠿⣿⣤⣤⣍⣽⣉⣁⣀⣀⣼⣿⣉⣿⣑⣿⡋⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢠⠛⡷⣾⠏⠻⣿⡭⠻⡁⠀⠈⢳⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠉⠑⠋⠁⠀⠀⠈⠓⠾⠳⠦⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡒⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠃⠠⠊⠐⠗⣪⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠲⢤⡈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⠁⠀⠀⠀⢀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠄⠀⠀⠀⣠⣭⡈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡸⠁⠀⢺⣌⢱⠀⢴⣁⠇⠀⢣⢠⣄⠘⠛⠛⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⠐⢀⠞⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠒⠁⠀⠇⠘⠳⠎⢸⡀⠀⢄⣀⣀⠀⢸⠀⠑⠃⠐⠶⠆⠀⠀⠀⠀⡠⠊⠀⠀⠘⠈⢠⠏⡀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢺⠺⠘⠀⠀⠉⢿⡊⠁⠀⠈⢳⢸⠀⠀⠀⠀⣀⣀⠄⠀⣠⠊⠀⢀⠰⠀⠀⠔⠁⢊⡠⠀⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠔⣡⠋⠉⠑⠎⠇⠀⠀⠠⠀⠤⢂⡤⠊⠁⢠⢸⠘⣠⠖⠀⠀⠀⠉⢀⡌⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠉⠉⠀⠀⠀⠀⡎⠈⠁⠀⠀⠀⠀⠐⠂⠉⠐⠒⠀⠁⢀⡀⠸⠸⢀⠜⠁⠀⠀⠀⠀⠀⠈⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⢒⣋⣤⠤⠒⡹⢀⠂⡰⠃⡔⢀⠔⢀⠔⢠⠆⠀⢠⢂⠧⠇⣁⠔⠁
⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⢄⠖⡺⢉⢝⡵⡣⢂⠜⡰⢃⠞⣠⠊⡠⢋⡴⢋⡴⠁⡠⠂⡀⠨⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠠⣙⠲⠤⣁⣊⡔⠁⠊⠊⠐⠃⠞⡰⢃⠞⣡⠞⡠⠊⡠⠋⡠⠊⡠⢰⠁
⠀⠀⠀⠀⠀⠀⠲⠬⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⢈⠔⣡⠜⣡⠎⣠⠊⡠⠊⡠⠊⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢈⠔⡡⠞⣡⠞⡡⠊⡠⣊⠔⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡡⠚⡡⢊⡵⢋⠔⡫⢚⡁⢠⠏⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢈⡴⢊⠔⢁⢔⡵⢋⠔⢋⠀⡇⠀⢸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣩⠔⣡⠔⠁⢬⠐⡡⠒⡡⠄⢸⡀⠸⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠎⠀⠀⢚⡡⢚⠵⠂⡇⢸⠈⠰⣊⠔⢊⠀⢧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠄⢠⠀⠀⠀⡨⠒⣁⠔⠀⡇⡈⠀⡨⠔⣊⠡⠂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⡄⢸⠀⠀⠀⠴⢊⡥⠒⠀⡿⠀⠀⢒⡩⠔⢊⡁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠁⠸⡆⠀⠀⠚⣡⠔⣃⠄⢱⠀⢈⡥⠖⣊⠥⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢧⠀⠀⠈⠴⣊⠔⡂⠈⢇⠀⢒⣉⠤⣒⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠊⠔⠪⠶⠀⠘⠀⠀⠀⠈⢀⡰⠳⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡀⠤⠒⡀⠤⢳⠀⠀⠀⠀⠀⠀⠀⠙⠻⡭⢛⠡⢂⠤⠒⡩⠔⢊⡡⠔⣂⠠⢒⡢⠔⡠⠤⣠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠔⠊⠥⢒⠶⢦⡤⠤⢤⣤⠴⢐⡣⠐⣉⠤⠒⠉⠔⢊⡡⠖⠊⠀⠊⠁⠀⠉⠀⠈⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⣤⣄⡀
⢀⣀⣤⣤⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⢸⣿⡇⠀⣀⣀⣀⡀⠀⠀⠘⠛⠿⢷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⡿⠿⠿⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠋⠀⠀⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢠⣿⡁⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠶⠚⠻⡇⠈⠙⠛⠻⣶⠀⢠⣿⠋⡔⣀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⠃⢀⣿⠃⠀⠉⠁⠀⠀⡇⡇⣿⢸⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⣼⠏⠀⠠⠤⠴⠃⢀⡇⡇⣿⢸⠀⠦⣀⣀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠞⠁⠀⠀⠀⣦⠀⣰⠆⠀⠀⠉⠈⢠⠀⠀⡀⠀⡀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠛⠋⠉⠁⣼⠟⣷⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠘⠷⠶⠃⠸⣥⠆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢰⡟⢸⡇⠀⠀⠀⠀⠀⣠⠶⠚⠛⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣿⠁⠀⠀⠀⠀⠀⠻⠶⠶⠶⢦⣼⠇⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢰⣾⠛⠛⠻⣦⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⢻⡆⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠈⣿⢀⣠⣼⠏⠙⠻⣶⣄⣠⣤⠶⠟⢋⣡⡴⠟⣱⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣸⣿⡿⠉⠀⠀⠀⠀⠈⣿⣭⣤⡶⠾⠛⠉⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣴⠟⠘⣷⡀⠀⠀⢀⣠⡶⠿⠁⠀⠀⠀⢀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⣀⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠟⠉⠛⠻⢶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠒⠂⠉⠉⠉⠉⠉⠉⠉⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⣠⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡰⠃⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⡄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠆⠀⠀⢳⣠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢰⠀⠀⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⡿⠉⠀⠀⠀⠄⠀⠀⢴⡷⢀⠆⠠⠼⣟⡂⠄⠀⠀⠀⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠰⠤⠀⠂⠁⠀⠀⠀⠀⠙⢦⠀⠀⣰⠃⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠘⠻⠦⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⢹⡀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⡀⠀⣀⡴⠃⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠀⠀⢸⡧⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⢴⠋⡉⠍⠁⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡰⠁⠈⢦⠰⠁⠀⣰⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠘⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠙⠛⠦⠤⡤⠋⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢳⡄⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠚⠉⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣏⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠞⠉⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡖⠒⠒⠀⠉⠁⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣠⠴⢋⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⠢⠤⠜⠁
⠀⠀⠀⠀⠀⡼⠁⠈⢱⠀⠀⠀⢰⠃⠀⠀⣠⠾⣅⣀⡀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⢠⡶⢤⡟⣿⠃⠀⢰⠇⠀⣀⣀⡟⠀⠀⡼⠃⠀⠈⢉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⣾⣀⣾⢰⡏⠀⠀⣾⠲⡎⠀⠀⠁⠀⠀⠷⠠⡄⢰⠋⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠇
⠀⠸⢧⣾⠁⠀⢸⣧⡔⠁⠀⠀⠀⠀⠀⡠⠤⣼⡎⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡏
⠀⠀⢰⡇⠀⢀⡟⠐⣄⣠⠄⠀⠀⠀⢸⣀⠀⣸⠁⠀⢰⠏⣿⢤⣀⣀⢀⣀⣀⣤⠤⠤⠤⠐⠒⠊⠉⠀⠀⣸⠃
⠀⠀⠈⠳⠤⠞⠁⠀⠀⠙⠦⠀⠴⢀⣀⠝⢹⡇⠀⢀⡿⣸⡇⢰⡇⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡸⠓⠿⢉⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⡴⠃⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⠤⣄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢦⣤⣠⣁⣄⣀⣀⣀⣤⡤⠖⠋⠉⠏⠀⠀⣉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠉⠻⠋⠀⠀⠀⠀⠀⡤⢒⡛⠀⢠⡝⢓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢻⣅⠀⢸⡒⢲⡄⠀⠀⠀⡈⢦⡈⠉⠀⠀⣎⠐⠒⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢹⡂⠈⠀⠈⣱⢏⣀⠀⢠⡎⠀⠈⠙⠓⠚⠋⠁⠀⠀⠈⠀⠅⢈⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠑⠀⠀⠈⠉⠒⠒⠒⠉⠀⠈⢧⠘⣧⡀⠀⠀⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠐⠂⠀⠀⠀⠀⠀⠀⢀⢀⡤⠀⠀⢀⢾⠀⠀⠙⠷⣦⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠧⡚⠅⠀⠀⠀⠀⠀⠀⠈⠉⠔⠂⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣲⣐⣶⣤⣌⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⢿⣿⡿⠿⣿⣿⡿⣿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣶⣬⣦⣶⣦⣦⣷⣤⣾⣯⣼⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣾⢿⣟⣤⡿⣿⢿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⠀⣾⡟⠉⠉⠛⠿⣿⣿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣠⠿⠃⠀⠀⠀⠀⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣿⠃⠀⠀⠰⣦⣤⣤⣬⣛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇⠘⠀⠀⠀⠀⠙⡟⠃⢾⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⡆⠀⠀⣨⢀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⣲⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠃⠀⠈⠂⣄⠀⠀⠐⠒⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣇⠀⠀⠃⣠⢨⣿⣶⣶⢿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣼⣿⡏⣿⡄⠀⠀⠈⠻⣿⠋⠀⢸⣷⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣰⣿⣿⡀⠀⢠⣿⠟⣷⡄⢸⡇⣿⣞⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⣿⡿⣿⣿⣧⠀⠋⠙⣿⠟⠉⠸⠃⢸⣿⣿⣿⣿⣿⣄⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⡄⠀⠀⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⡄⢠⣿⣿⣧⠀⢠⢸⣿⣿⣟⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡸⣿⣿⣿⣿⣿⣟⢿⣿⣿⢿⡆⠀⢸⣿⣿⣿⣧⢹⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣎⠻⣿⣿⣿⣽⡌⢻⣿⣿⣷⡇⣿⣿⣿⣿⡏⣿⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣷⡜⢿⣿⣇⣷⠀⠻⣿⡿⠀⣿⣿⣿⣿⣿⣿⡿⣾⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣦⠹⣿⠸⣇⠀⢹⠃⢸⣿⣿⣿⣿⣿⣿⡇⢷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢿⢿⡿⠟⢿⣿⣷⡌⠁⢿⣤⠇⠀⢸⣿⣿⣿⣿⣿⠟⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠀⠀⠈⠉⠉⠉⠁⠀⠈⠀⠀⠀⠀⠈⠉⠉⠀⠀⠙⠀⠀⠙⠙⠛⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠀⢀⠈⣂⣄⠲⡀⠀⡰⢊⢅⣀⠉⠀⠘⠃⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠿⠀⠐⠬⢦⡥⠁⠀⠁⠡⠼⠃⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⢒⣀⢒⣀⣀⣖⠽⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡰⢚⠈⡤⣀
⠀⠀⠀⠀⠀⢠⠊⠔⢊⠍⢒⠄⠀⠀⠐⠖⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡑⠣⢶⠛⣐⠲⠤⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠶⢚⠪⠑⠌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠋


⣶⣶⠶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠩⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢄⣀⡀⠀⠀⠀⠠⠇⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⢠⣤⣽⡬⢳⠀⠀⠀⢀⣠⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⢶⡒⠑⠊⠃⠀⠈⠓⢾⣏⠹⡄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡟⡄⠂⠘⠃⠀⠆⠀⢳⡇⣹⡀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢱⡀⠰⠆⠀⠋⣠⣏⣛⠀⠀⣀⣀⣠⣄⡀⠀⠈⢧
⣀⣀⣀⣠⡔⠪⠭⢅⣀⣠⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡓⢲⣲⣖⠿⣙⡁⠈⠉⠈⡇⠀⠀⠀⠀⠀⠀⣸⣀⣀
⠀⠀⠀⠉⠉⠁⠈⠛⠛⠒⠒⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠜⠾⠒⠟⢳⠶⠓⠒⠒⠚⠳⠶⠦⠤⠤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⡶⠶⡴⠶⡶⣶⣶⣶⣶⣦⣬⣥⠤⠒⠋⠁

⠀⠀⠀⠀⢀⣤⢤⡄
⠀⠀⠀⠀⠘⠷⢿⡇⠀⠀⠀⠀⢀⢶⣠⡄
⠀⠀⠀⠀⠀⠀⠈⠛⠀⣠⣤⡔⠁⡠⢬⠷⠦⢴⡶⠤⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠙⢲⡗⠃⠒⠀⠂⠙⢢⡀⠀⠀⠀⠉⢳⡄⠀⠀⠀⢠⠇⠉⠉⢉⠁⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢻⡄⠀⡤⠞⠉⠱⣷⡄⠀⠀⠀⢰⡇⠀⣀⡤⠋⠀⡠⡔⠋⠤⠤⠤⡀⢉⠟
⠀⠀⠀⠀⠀⠀⠰⡏⠐⡇⠀⠀⠀⠈⢷⠀⠳⣄⣀⡼⢺⠀⠀⠀⠀⣸⠃⠀⠹⡄⠀⢈⡧⠊⠰⡘⠃⢖⠄⢸
⠀⠀⠀⠀⠀⠀⠀⢯⠀⣇⠀⠀⠀⠀⠘⡇⠀⠀⢤⠴⠃⠀⠀⠀⣠⠇⠀⠀⠀⡇⠀⠉⠠⣦⠄⣉⠛⠁⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣽⡀⠀⠀⠀⠀⣿⣄⡀⢺⢦⠀⠀⢀⡴⠃⠀⠀⠀⠈⠛⠒⢦⡀⠀⠀⠉⠀⢠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⠀⠀⠀⠀⢹⠴⠍⠛⠺⠄⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡴⠒⠛⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡤⠤⡤⠤⣤⣤⡤⣤⣤⣤⡤⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠰⡏⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⣀⣀⣀⣀⣧⣀⣸⣇⣀⣀⣀⣀⣠⣤⡤⠤⠤⠤⠤⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣀⣀⣀⣀⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣻⣷⣄⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠟⣋⡵⠶⢿⣷⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⡶⢛⣉⣤⣤⠶⣬⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣴⡟⢉⣀⣀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡀
⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣬⣿⡛⢷⡄⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀
⠀⠀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣷⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷
⠀⣸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠳⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇
⢠⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡉⠛⠛⠿⠟⠛⠛⠉⠉⠉⠉⠛⠛⠳⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠈⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⢹⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟
⠀⠀⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⢴⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿
⠀⠀⠈⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⣀⠀⠀⠈⠙⠛⠶⣦⣤⣄⣀⣀⣀⣀⡀⠀⢿⡄⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠁
⠀⠀⠀⠈⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣾⣫⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠈⠿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣄⠀⠀⠀⣾⣿⠋⠛⠛⠻⠶⠶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠳⠶⠶⠶⣤⣤⣤⣤⣤⣤⡶⠟⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠈⠉⠉⠉⠉⠉⠉⠐⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⠃⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠒⠲⢶⣄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⡇⠀⠀⠀⠈⠀⠀⠀⠀⠒⠠⠀⠠⠄⠒⠀⢄⠈⢷⠀⠀⠘⣇⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠂⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⡀⠀⠀⢠⣶⠀⡠⠀⢀⣾⡆⠀⠀⠀⠀⢀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⠛⠿⠁⠀⠀⠀⠀⠐⢀⣀⠈⠛⠀⠀⠀⠀⠉⠙⢦⡀⠀⠀⣼⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠁⣠⠆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⢻⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠄⠚⠁⢠⠀⠀⠀⠀⠀⠈⠓⠦⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡇⠀⣤⡶⠒⠒⠶⠒⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠐⠂⠠⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⡸⠀⢮⡃⣠⠊⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⠳⣼⣄⠀⠋⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠄⠈⠀⠤⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠈⠉⢘⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣠⠄⠂⢤⡀⠀⠉⠐⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠠⠟⢈⡿⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣎⡄⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣀⠀⠀⠀⢀⡠⠞⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠞⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠉⠁⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣠⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠳⡄⠀⠀⢀⠏
⠀⠀⠀⠀⠀⢀⡴⠶⣄⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⠶⠋⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⠓⠚⠋
⠀⠀⠀⢀⡀⡸⠀⠀⢸⠇⠀⠀⢀⡟⠀⠀⢠⡞⠣⢄⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⢠⠟⢲⡏⣽⠇⠀⢀⡟⠀⡀⣤⢼⠀⠀⢠⡏⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠾⣤⣾⢠⡟⠀⠀⣸⠟⣾⠁⠀⠀⠀⠀⠘⠒⠶⡄⣼⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠧⣾⠁⠀⢠⠿⡔⠁⠀⠀⠀⠀⠀⢀⠴⠤⣷⠃⠀⢀⣎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⣿⠁
⠀⠀⢠⡏⠀⠀⡜⠀⢧⣄⡄⠀⠀⠀⠀⢈⣀⠀⡎⠀⠀⣼⢃⣯⣤⣤⣤⢠⣤⣤⣤⣤⠤⠴⠖⠒⠉⠉⠀⠀⢀⡟
⠀⠀⠈⠳⠦⠶⠁⠀⠀⠈⠳⠤⠴⠯⣀⡨⠏⢹⠁⠀⢰⡏⣼⠁⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡾⠳⠏⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⡼⠃⠀⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠐⢀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣦⣤⣤⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⣤⣄⣀⣁⡄⠀⢀⣀⣀⡠⠤⠒⠉⢻⠁⠀⠘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠛⠿⠋⠁⠀⠀⠀⠀⣀⠄⢲⠀⠀⣹⠓⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢾⡃⠀⢳⠴⢄⡀⠀⠀⠀⠉⢦⡉⠛⠂⠉⢐⠃⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠷⡈⠑⠒⠊⢩⣎⠁⠀⠀⡴⠈⠐⠓⠠⠤⠐⠊⢁⠀⠀⠂⠩⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠆⠀⠀⠘⠃⠠⠤⠔⠛⠀⠙⢆⢀⣇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡼⠈⠎⡳⢦⣁⠈⡡⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢍⠃⣄⡵⣊⠔⠀⠀⠀⠁⠊⠩⠖⣠⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠟⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡣⠤⠀⠠⠤⠤⠤⡀⠒⠠⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣠⣶⠟⠉⢐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣵⣿⣿⣧⣀⠆⠠⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠂⡠⢂⢏⡠⠈⣀⡠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠋⠛⢤⢾⣷⠀⠬⠄⢲⡛⠒⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⠿⠀⠓⠤⠊⠀⠉⠋⠀⠀⠀⠀⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠉⡀⠀⠄⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠆⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡅⠀⠀⠠⠀⠐⠤⡀⠀⠀⠀⠠⠀⠀⠀⢠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠀⠀⢀⠀⠀⠈⠑⠢⠄⣀⣀⣀⣀⣤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⢠⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢁⠀⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠑⠢⡀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠈⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⡤⣀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠰⢀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠷⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⣀⡄⢀⡞⣧⠀⠀⠀⠀⠀⠀⠀⣤⠊⠥⠈⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⡶⡊⠑⠃⡀⠘⡀⠀⠙⠉⠀⡀⢊⠈⠑⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠧⣤⢦⡀⡀⠀⢰⠠⠂⠀⢰⠀⠈⠫⠬⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⡼⠀⠈⠙⢼⢃⣠⢼⡖⢶⣲⣺⠿⠲⡆⠐⢢⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠃⠀⠚⠿⠿⠿⣯⢴⣯⣾⣿⣿⣿⣿⣷⣿⣷⣿⣿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣂⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⣻⡛⣛⡟⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠀⠀⠀⠈⠲⣆⠀⠀⠀⢀⣐⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⢀⡤⢤⡀⣜⠉⠀⠀⠀⠀⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⢿⣌⣠⡿⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⡂⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠤⠆⠀⠀⠰⠤⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⠉⠀⠈⠢⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢄⡠⠒⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⢀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠡⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠡
⠀⣴⠥⢤⣤⣤⣤⠤⠤⠤⠤⢤⣾⡏⠀⠀⠀⠀⠀⠑⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢐⠂
⠀⣿⣭⣭⡭⠭⠭⠭⠭⠽⠿⠿⢟⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠁⠀⠀⡐
⠀⣾⣭⣭⣭⣭⣭⣽⣿⡿⠿⢶⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠀⠀⠒⠒⠒⠒⠒⠄
⠐⣾⠯⠴⠶⠶⠶⠶⠾⠿⠭⢭⣿⡟⢲⢖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⣿⣤⣤⣤⣤⣤⣤⣤⣷⠒
⠀⡻⠭⠭⠭⠭⠭⠭⠭⢭⣭⣭⡿⠁⠀⢪⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣿⣿⢾⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⢾⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⣀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢠⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠧⣈⣳⡀⢠⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⠿⠻⣿⣿⣿⣧
⠀⠀⠀⠀⠐⢏⡩⠽⠃⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢛⠻⢿⣿⡏⠀⠀⠀⠀⣶⠖⡤⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⣿⡿⠁⠶⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢠⣿⠃⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡴⠁⢘⢷⡀⠈⢦⣉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡸⠃⠀⠀⠀⠀⠀⣤⡼⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⡴⠾⣿⣿⠏⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠚⢷⣄⠀⠀⠀⠀⠀⠉⠀⠒⠙⠀⠀⠀⠀⠀⢠⡾⠳⣶⣦⣀
⠀⠀⠀⣀⡴⠖⠲⠞⠉⠀⢀⣀⣀⡾⠉⠛⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡿⠠⣄⡼⠃⢙⢧⡀
⠀⠀⢸⡋⠀⠀⠀⠀⠠⡶⠋⠁⠉⣹⠖⠂⠀⠙⠿⢭⣿⣷⣶⠖⠒⠻⣏⠁⢀⡿⠀⠈⠓⢶⡁⣰⠇
⠀⠀⠀⢿⣀⡀⠀⠀⢀⡟⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⡜⠷⠋⠀⠀⣴⠦⠴⠛⠁
⠀⠀⠀⠀⠙⢧⡴⠷⠋⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⠶⠞⠁
⠀⠀⠀⠀⢀⣤⢤⡀⠀⠀⢸⡦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⣠⠞⡱⠋⢹⣆⡀⠘⣇⠀⠈⠉⠙⠛⠶⠦⣤⣄⣀⠀⢀⡟
⠀⣠⠞⡡⢊⡤⣴⠏⠈⠙⢦⣾⡳⣤⣀⠀⠀⠀⠀⠀⢠⠟⠻⣿
⣼⣣⠎⢱⣭⡵⠛⢦⡀⠀⠀⠈⠙⢲⡽⠛⠓⠶⠦⢤⡏⠀⠀⢈⡇
⠈⠛⠛⠋⠁⠀⠀⠀⠙⠶⣤⣤⠴⠋⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢧⣄⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠠⣀⠸⢭⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣅⠢⢍⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣍⡾
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠉⠀⠀⠀⠈⠉⠉⠓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠦⣤⠜⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠐⣦⡀⠠⡬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠁⠀⠀⠉⠀⠀⠀⠀⠈⠢⢀⡿⣄⣨⠞⠉⠑⡦⠃⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣩⠔⠀⠀⠀⠀⠀⠀⠀⠐⢲⠖⠚⠋⣏⣀⣀⣸⡠⠀⠀⣹
⢀⡠⠤⢒⠒⣒⢒⡒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⣴⠉⠉⠀⠈⠉⣿⠦⠔⠓⠂⠀⠀⠀⠠⠤⠤⠤
⣏⠀⠀⠟⠸⠃⠟⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠀⠈⠑⠒⠒⠲⠞⠋⠀⠈⡷⠠⠤⢤⣤⢠⣤⠤⠴
⠈⠓⠾⠥⠭⢌⠠⣔⠒⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⢦⠁⢀⡆⠀⠀⣀⠈⠻⠶⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠑⠾⠅⠀⠀⠀⣠⠞⠉⠇⢝⢦⠀⠀⠈⠓⠚⢳⣀⢀⡿⣄⣀⣀⣀⡇⠀⠀⠀⠀⢀⣀⣀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠾⣝⡄⠀⠀⠘⠘⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠖⢤⠇⣠⡴⢚⣭⢍⢀⡜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠑⡜⢄⠀⠀⢀⢶⠂⠤⠤⣤⡤⣏⠻⡄⠗⠋⠁⠀⡰⠋⠐⠀⠉⠉⢩⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠦⠐⠚⢺⠀⠀⠀⠀⣀⣘⡄⢈⡲⡄⢠⠾⢷⠤⠤⠤⣀⣀⣟⠒⠂⠭⢖⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠀⠙⠋⠉⢹⣽⠴⠒⠋⠉⠀⠈⡛⠋⠀⠘⠃⠀⠈⢣⠀⠀⠀⠉⠁⠀⢀⠀⠀⡟⠒⠚⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⢀⣴⡿⠀⣠⠸⣤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⡆⣠⠞⢡⣿⡇⠀⠈⠑⠺⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡠⠤⢚⣥⠞⠁⢠⢎⢿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠒⠂⠉⠀⠀⠀⠘⠁⠀⠀⡞⠺⣌⡆⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣄⠀⠙⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠈⠣⣄⠚⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠈⠑⠚⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⣠⣤⣖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⢟⣓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢪⣅⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣉⣉⣣⣠⣤⡴⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠄⠀⢠⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⢄⠀⠀⢀⡈⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠒⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⢀⣀⣀⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⢨⠀⡠⠄⡸⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠈⠁⠁⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣀⣤⠶⢄⣰⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⣦⠀⠀⢀⣀⣴⠯⢾⡏⣱⣾⢾⣑⣻⢻⡙⢨⣿⡥⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣯⣴⣶⠾⠋⢀⡰⢷⣏⣩⡿⢷⣨⣞⠷⣢⣾⡿⠿⣘⣹⢫⡝⢠⣯⡍⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⠀⢠⡞⣹⢷⣈⡻⢖⣴⣿⠟⢇⣀⣛⢶⠆⣠⣿⣿⡕⢛⡟⠓⣙⣋⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠐⣷⠹⡧⣠⣿⢿⣈⣚⢵⡆⣠⣿⡿⣷⠈⠱⢯⣄⠁⣲⣞⠃⡼⡷⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣄⠀⠀⢀⢟⣩⣿⠈⠽⣥⢀⣽⣿⣶⠘⢱⢟⣤⠀⣹⣿⣯⡉⡴⣻⠟⠣⣘⣴⣸⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⡷⠀⢈⣿⠻⡁⣹⣷⡅⢋⡿⣁⠈⢻⣷⣪⡋⡴⢻⡿⠉⠣⣤⣍⠾⢠⣿⣷⠹⣁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠻⣿⣧⠀⠘⣅⣿⡝⡿⠋⠑⣶⣜⠛⣴⢿⡿⠛⢇⣀⣋⢴⡷⣠⣾⣿⡇⠉⢯⡄⣿⡝⡃
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣠⣾⠟⠀⢸⡟⢏⠱⣤⢟⡼⣿⠿⣇⣀⡛⣶⡆⢀⣽⣿⣶⠘⢫⢟⣅⢈⣿⣯⠘⣏⣹⠏
⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⣰⠟⠁⠀⠀⠀⢮⣯⠼⡿⢧⣘⣣⣦⢀⣿⣿⣦⠰⢛⡿⣋⠈⢲⣶⣮⠫⢾⢟⠉⣶⡝⣯⠁
⠀⠀⠀⠀⠀⠰⢯⡀⠀⠀⠸⢿⣤⠀⠀⠀⠀⠈⢩⢰⣽⢆⣿⣷⡗⠋⡿⢃⠉⢲⣾⣜⡛⡵⢿⠟⠙⢦⣦⡛⢾⣋⣿⠁
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠢⣄⠀⠈⠋⣟⡃⣜⣡⡄⣲⣶⣟⠧⢾⡿⠛⠧⣤⣍⠟⠴⣿⠋⣳⣎⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠙⢃⢻⣟⠜⣹⢋⡙⢲⣬⣼⠳⢾⢟⡙⣲⣦⣍⣛⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡈⠛⠐⣿⣟⢈⣽⣫⠍⣴⡶⢏⣩⣿⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠈⠉⠀⠚⠋⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠈⠉⠉⠉⠉⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⢷⣤⣤⣴⠞⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡄⠀⠠⣶⣶⡆⠀⢀⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⢹⣿⠁⠀⣼⣿⣿⣿⣿⠟⣩⣴⠶⠿⠿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣸⣿⡀⢠⣿⣿⣿⣿⠋⣼⠏⢀⣄⠀⠀⣠⣄⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⡇⣾⣿⣿⣿⡏⢸⡏⠀⠈⢻⣷⣾⡿⠋⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣇⠀⣠⣾⡿⠻⣷⣄⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠂⠻⣦⡈⠉⠀⠀⠈⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢀⡀⢀⣀⢀⣀⠀⡀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠳⡇⡾⢧⢪⣨⠀⡇⢸⠹⡞⢸⣒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠿⠿⠿⠿⢶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣰⡟⠉⠀⢙⣷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣴⠾⠃⣀⣤⣤⣀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠈⢻⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣧⣀⠀⠀⠀⠀⠀⠘⡇⠀⢀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⣋⣭⣿⣿⣿⠉⠉⠛⠿⢷⣶⣶⣶⡾⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⢁⣠⣾⠿⠋⢹⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⣠⣾⠟⠁⠀⠀⣼⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⢀⣾⡿⠁⠀⠀⠀⠀⣿⡇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⢠⣿⠏⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢿⡟⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀
⠀⠀⠀⣰⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣤⣶⣿⣿⣿⣿⣶⣤⣄⢀⣀⣠⣤⣤⣤⣶⣾⣿⣿⣿⣯⣭⣭⣥⣦⣴⣶⡶⠶⠒⠒
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⣛⣿⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣯⣥⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⠿⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠠⠻⠟⠛⠛⠛⢋⣽⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣤⣤⠄

⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⣿⣆⢸⡆⠀⣼⣷⠀⢀⡾⠛⠓⠀⠛⣿⠛⠀⣿⠈⣷⠀⣾⠀⣿⣛⠛⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⣿⠘⢿⠇⣸⠟⠻⣧⠈⢧⣤⡤⠀⠀⣿⠀⠀⣿⠀⠸⣿⠃⠀⣿⣭⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⠇⠀⠀⣀⠤⠒⠋⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⡀⠀⠀⢀⣔⣡⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⢳⡦⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⡏⠀⠀⠀⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⠘⣄⡤⠔⠒⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⡶⠶⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣁⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠉⠀⢠⡾⢻⣷⠀⣀⣤⣤⣿⡏⠀⠀⠀⠀⢠⣿⠃⠀⣰⡆⠀⠀⣼⡿⣶⡄⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⣿⡿⠟⠋⢸⣟⣁⣸⣿⣁⣀⣀⡀⠀⠸⣯⣀⣴⣿⣇⣀⣴⣿⣁⣸⡷⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⠛⠉⠉⠛⠋⠉⠁⠀⠀⠉⠉⠉⠀⠉⠉⣿⡏⠉⠉⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣶⣶⠶⠶⠾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉


⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⣤⣄⠀⠀⠀⢀⡀⠐⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠿⣇⢀⣠⣜⣷⣤⣾⣯⣤⣤⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡥⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠜⣿⣟⡿⢻⣿⢟⣽⣏⠛⠿⣿⣿⢉⠛⢻⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡜⠻⠟⠿⢾⣧⣾⡏⠃⢀⣴⡄⠻⢾⣶⣶⡴⠀⢀⡤⢶⣶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠘⢀⣀⣤⣨⣿⠓⠶⠛⢙⣿⣷⡬⠭⢄⡛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣤⣛⠁⢀⣹⣿⣿⣿⣿⣳⡐⠞⠛⢻⣧⣄⠀⠰⠼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣷⣬⣿⣿⡿⢏⣿⣿⣿⣿⣦⣞⣽⣿⣿⣿⣾⣿⣧⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⣭⣽⣷⣦⣩⡁⠘⠿⣿⣿⣿⣿⣿⣿⣿⢏⠀⠈⠉⠛⠛⠛⠹⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⣴⣿⡿⠿⠿⢿⡿⠿⠉⠛⢋⣻⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠤⠒⠀⠠⠂⠀⡚⣲⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣯⠉
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠼⢯⣿⣿⢻⣿⣿⡟⣿⣿⡿⠻⠛⠿⠋⠻⣟⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⢟⢸⣽⡽⠳⠿⠟⠀⠀⠀⠀⠐⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⢉⣵⡆
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⠀⢀⠀⠘⢂⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣰⠀⠀⠀⠀⠀⡤⠖⠃⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄
⠀⠀⠀⡼⠛⠛⠛⠛⠙⢦⠀⠀⣀⣠⣿⡿⢿⣆⣿⠀⠀⢀⢤⣤⣷⣤⣄⠀⠀⠀⢀⣽⣿⣿⣿⣾⣿
⠀⠀⠀⡇⠀⠔⡀⠢⠀⡸⠀⠀⣿⠀⠀⡾⢻⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⢲⠀⠀⠀⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠘⠦⣀⣉⡠⠔⠁⠀⠀⣏⠀⠸⠃⣠⠚⢂⠀⠀⠈⠻⣿⣿⠿⠋⠁⠀⠀⠀⠈⠉⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⢦⡀⠀⠀⠀⠈⣧⣿⣧⡀
⠀⠀⢠⠀⠀⣰⣷⡀⠀⠀⠀⠀⡯⠐⡖⢀⢠⡄⢹⡀⠀⠀⢰⢿⣿⡏⣧⠀⠀⠀⠀⢰⠉⠉⠉⣿⡂
⠀⠀⢸⣠⣼⣹⣿⣽⣦⣄⡄⠘⢧⣈⠁⠚⠈⠁⡴⠁⠀⠀⠀⢸⡏⣷⠀⠀⠀⠀⡄⣈⠩⢉⡉⢉⢀⡄
⠀⠀⠸⠉⠉⠹⠩⠏⠉⠉⠇⠀⠜⠋⠙⠋⠉⠉⠀⠀⠀⠀⠀⠸⠇⠿⠀⠀⠀⠀⠁⠉⠀⠀⠉⠈⠉⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠈⠓⠑⠀⠀⠀⠀⠀⠀⢀⡾⣿⣷⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣉⣋⣻⣷
⠀⠀⠀⣿⣿⢻⢻⣹⡟⢦⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⢰⠀⠠⠏⣰⠪⠇⠀⠀⠀⠐⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠘⢧⣽⣿⡿⠉⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣶⣶⣶⣶⣶⣶⣶⣷⠀⠀⠻⠿⣿⣿⡏
⠀⠀⠀⠐⠒⠚⠓⠒⠂⠀⠀⠀⠀⠀⠈⠋⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢙⠛⠉
⠀⠀⠀⠀⠀⣠⣦⠀⠀⣤⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⢀⣿⣗⡀
⠀⠀⠀⢀⣨⣿⡿⣷⣄⣿⠀⠀⠀⠀⢾⡿⣿⠆⠀⠀⢀⣼⣿⣿⠿⠋⠻⢷⡄⠀⠀⢠⡟⣿⣿⢧
⠀⠀⠀⣿⠀⢀⡾⣻⣿⣿⠀⠀⠀⣠⣤⣿⣷⣤⠀⠀⢸⣏⠀⠤⢀⠠⠄⢸⣿⠀⠀⠈⢠⡟⣿⠈
⠀⠀⠀⣇⠀⠼⠁⡠⠓⣄⠀⠀⢰⡟⣿⣿⣿⣿⠇⠀⠸⠿⠦⣄⣙⣀⡤⠿⠿⠀⠀⠀⠸⡇⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠒⠚⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⢳⠤⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠉⠙⢶⡄⠀⢠⡀⣀⠀⠀⠀⡂⠀⠀⠀⡠⠐⠁⢀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⠋⡠⠀⠼⠠⠄⠁⢀⣰⣄⡀⠀⠀⢂⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠈⣷⡴⡁⢀⣀⢼⡿⠷⡈⢿⣋⢈⡄⠀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣥⠀⠙⡿⠁⠀⠀⠀⣤⣸⣿⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠷⣽⣶⣲⣦⣤⣶⣽⡻⠆⣸⡟⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠢⠁⢊⠳⠒⡱⠿⣯⠻⣿⣿⣷⣔⠋⠈⡧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠉⡠⠘⠣⠀⠉⣹⣿⠛⢳⡨⠶⠎⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡤⠂⠄⠀⠉⠙⠀⠑⠄⠀⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠐⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⠀⢰⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠄⠤⢤⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢰⠃⠀⠀⠈⡇⠑⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⣄⠐⠈⢀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠸⡄⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⢿⣿⡆⠤⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠘⢄⡐⣄⢀⢀⡠⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠠⠄⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠈⠉⠹⡏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠠⠀⠠⠄⠂⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠄⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠐⠠⠀⠀⠀⢀⠀⣃⣤⠀⠀⠈⠀⠀⣿⠀⢸⡇⠀⠀⠘⣇⠰⡀⡀⠀⠀⣸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠚⢿⡟⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠨⡷⣬⡿⢶⠛⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣠⠀⠸⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢠⣄⡴⠀⢱⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠐⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠶⠶⠄⠙⠛⠛⠋⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⠙⣷⡀⠀⠀⠀⢻⣆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠉⠉⠉⠉⠀⠈⢷⡄⠀⠀⠀⢻⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣬⣿⡄⠀⠀⠀⠹⣆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⡏⠉⠉⠀⠀⠀⠀⢀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⣶⠶⠶⠶⠾⠿⠷⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠛




⡶⣶⢶⢶⡶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⡶⠶⢶⡶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣾⣿⣶⣶⢶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡌⡻⣿⣿⡄⠙⢿⣧⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⣹⢿⣿⡙⠈⢻⣷⡀⠀⠻⣴⣿⣃⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠿⣻⣿⣿⣿⣷⡀⠈⢿⣷⡀⠀⠙⣧⠀⠀⠙⣿⣿⡟⠀⠀⡀⠂⢸
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠋⠁⠀⢰⣿⣿⣿⡿⣿⣧⠀⠀⠙⢷⠀⠀⠈⠃⠀⠀⠊⢿⣿⣶⠈⠀⠀⡏
⠀⠀⠀⠀⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣧⠈⠻⣧⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⡇
⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⢀⣶⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠁⢀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⠁
⠀⠀⠀⠀⠐⠘⣿⣏⠀⠀⠀⢰⡾⣿⣷⣟⡧⡀⠀⠀⠀⠀⠀⢀⣺⠟⠱⣦⣄⣀⣀⣤⠤⣤⣍⠄⠂
⠀⠀⠀⠀⠀⠀⠸⡋⠀⠀⢴⣭⣭⣥⣤⣬⣥⣶⠀⠀⠀⠀⠰⣯⣥⣤⣤⣭⣥⣶⠆⠀⠐⣯
⠀⠀⠀⠀⠠⣤⣤⠇⠀⠀⠀⡾⣭⡉⢋⣭⡅⡆⠀⠀⠀⠀⠀⢶⣭⣭⣟⣭⡥⡶⠀⠀⠀⡧⣤⡤
⠀⠀⠀⠀⠀⠐⣽⠃⠀⠀⠀⠈⢫⣱⢾⡍⠛⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣿⡟⠃⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⢀⣾⡆⠀⠀⠀⠀⠀⠀⠁⠀⠐⣦⣀⣀⣀⣀⣠⡼⠀⠀⠉⠁⠀⠀⠀⠀⠀⠿⣆⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⣛⣻⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠿⠿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠢





⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄⣠⣶⣦⣤⣴⣶⣦⣤⡀⠀⠀⠀⢠⣄⠀⣶⣦⠀⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣉⡉⢠⣿⣏⣈⡁⢸⣿⡟⢿⣿⠀⠀⠀⣾⣿⠀⢿⣿⠀⣿⣿⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⢸⣿⡟⠛⠃⢸⣿⣇⣼⣿⠀⠀⠀⣿⣿⣀⣼⣿⠀⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠸⠿⠿⢿⠇⣼⣿⣿⠿⠃⠀⠀⠀⠻⠿⠿⠿⠋⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣠⡾⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠻⢶⡄
⠀⠀⠀⣿⠀⠀⢠⣤⠀⢠⣤⡄⠀⢠⣤⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⢀⣴⣦⣄⡀⠀⣤⣤⣤⣤⣤⠀⠀⣤⠀⠠⣤⠀⠀⢀⣤⠄⢠⣤⣤⣤⣤⡄⠀⢸⣿
⠀⠀⠀⣿⠀⠀⢸⣯⠀⢸⣿⢿⣦⠐⣿⠀⠀⢀⣾⠟⣿⡄⠀⠀⣴⡟⠁⠀⠉⠀⠀⠀⠈⣿⠀⠀⠀⠀⣿⠀⠀⢻⣧⠀⣼⡟⠀⢰⣿⣤⣄⠀⠀⠀⢸⣏
⠀⠀⠀⣿⠀⠀⢸⣿⠀⢸⣿⠀⠻⣿⣿⠀⠀⣼⡿⠶⢿⣿⡀⠀⢻⣧⠀⠀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠈⣿⣶⡿⠀⠀⢸⣿⠉⠉⠀⠀⠀⢸⣧
⠀⠀⠀⣿⠀⠀⠘⠛⠀⠘⠛⠀⠀⠙⠛⠀⠘⠛⠀⠀⠀⠙⠓⠀⠀⠉⠻⠟⠋⠁⠀⠀⠀⠛⠀⠀⠀⠀⠛⠀⠀⠀⠘⠛⠃⠀⠀⠘⠛⠛⠛⠛⠃⠀⢸⡟
⠀⠀⠀⠻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠃
⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠁⠀⠀⠉⠀⠈⠈⠀⠀⠁⠀⠈⠈⠉⠀⠈⠉⠁⠉⠁⠉⠁⠈⠈⠁⠉⠁⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠁⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⢅⡅⣿⢼⡷⣄⠀⢀⣠⣴⣾⣿⣿⣿⠿⢟⣿⣿⠉⣿⣿⣿⣿⣮⣧⡘⠀⠀⠀⠀⡮⣟⢨⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⢸⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡙⡇⢠⣿⣿⠃⣿⣿⣿⠋⠂⠀⢠⢆⣗⡿
⠀⠀⢸⢴⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⣿⠟⠛⠉⠉⠉⠁⠁⠈⠻⢃⣴⣿⣿⣿⠄⠀⣄⣿⠔⠃⠀⠀⠀⠀⢀⡄⣮
⠀⠀⢹⢸⡗⡗⣻⡗⡂⣤⣀⡀⠀⣸⣿⣿⣿⣿⣯⣭⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⢀⣶⣏⠚⠀⠀⠀⢀⣄⣎⣳⡃⣓
⠀⠀⠺⠰⡷⣓⣺⣖⡇⣿⢐⢗⢖⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⠴⠦⣤⡀⠀⠀⠀⣠⠴⢼⡉⠠⢺⠊⠀⠀⠀⣤⡶⢸⣇⡿⠒⠃⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠐⠇⠽⣿⣿⣿⣿⣿⡿⢳⡀⠀⠁⣠⣶⠦⠀⠀⠀⢈⠤⡶⣬⣷⠋⠀⠀⣤⣾⠪⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣫⣔⣣⡇⠀⠈⠙⠛⠁⠀⠀⠤⠀⢠⠀⠁⢹⡤⠼⠋⠉⠀⠀⠀⢀⢀⡀⡤⢤⡆⣶
⠀⠀⢀⢀⣀⣀⣀⣀⡀⣀⡀⠐⣶⡾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠾⠆⠀⠀⣧⠠⠄⠦⠶⠀⠏⠿⠴⠇⠯⠽⠧⠿
⠀⠀⢱⠘⣷⡃⣸⣔⡏⣻⣶⣷⣯⠀⠹⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⢶⣦⠀⢠⣷⣄⣀⣀⣀
⠀⠀⣨⢸⣧⡧⡘⠖⠇⠛⢼⣿⣿⣿⣿⣿⣿⠛⣻⣆⣀⡀⠀⠀⠀⠀⢸⣷⣶⣶⣾⡇⠀⣼⣋⡕⠢⣤⠀⢻⣣⡷⢰⡆⣤⢤⡀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣾⣿⣿⠻⣧⣄⡀⠀⠀⢸⡏⠉⠉⣿⠀⡰⠫⢀⣈⠂⢌⠢⡘⡀⠈⠉⠋⠧⣨⡇⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠃⠀⡟⠐⣄⠈⢿⣄⣀⣿⡤⢡⠄⠀⠀⠃⠈⠃⡼⣠⣀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣀⣠⣄⣦⠾⠋⠁⢹⣿⣿⣿⣿⣿⣿⠀⠘⠀⠀⠇⠀⡘⠙⡆⠉⠉⠁⣱⠏⠀⠀⠀⠀⠀⣠⠞⣏⣟⢐⣦⣀
⠀⠀⢰⢸⡫⣧⡾⠛⠉⠀⠀⠀⣿⣘⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⢳⠖⢶⣾⠃⠀⠀⠀⠀⡠⢾⣧⠀⠀⠁⢐⡗⣗⣘⡆⣄
⠀⠀⣸⠸⣳⡏⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠏⠻⣿⠛⣹⠓⠶⠦⠤⠀⣠⠎⠀⠀⢁⠀⠀⠀⠠⡆⠀⠀⠙⢷⡀⠀⠀⠁⢿⣒⡇⢿
⠀⠀⣸⡞⠉⠀⠀⣠⣤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠻⣦⠇⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⣆⠀⠀⠀⢹⠀⠀⠀⠀⢳⡄⠀⠀⠀⠘⠇⡟
⠀⠀⠹⣇⠀⢠⡞⠁⠀⠀⠀⠀⠈⠉⣦⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⢧⠀⠀⠀⢠⡷⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠘⢦⠈⣧⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡜⠀⠀⠀⠀⠀⡜⠈⢆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢣⡀⠀⣾⣗⣰⡄
⠀⠀⠀⠀⠈⠳⣽⡄⠀⠀⠀⠀⠀⠀⠀⢧⠀⡜⠁⠀⠀⠀⠀⣰⠁⠀⠈⢣⡀⠀⠀⢸⢷⠀⠀⠀⠀⠀⢳⣰⠁⢿⢘⡇⣆
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢸⠞⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢳⡄⢀⡞⢸⡄⠀⠀⠀⠀⠀⠹⣄⠸⡇⡇⣿⢵⡀
⠀⠀⠀⠀⠀⠀⠀⣹⡆⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⡏⠀⠀⣀⣠⣶⠀⢻⡞⠠⠼⡇⠀⠀⠀⠀⠀⠀⠘⢦⠩⡇⣭⠽⠕⡆
⠀⠀⠀⠀⠀⢀⢼⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠁⠀⣿⠀⢸⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⢹⡍⣿
⠀⠀⠀⢀⡶⣗⢸⣇⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡘⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢃⡿⢹⡇⣿
⠀⠀⢠⡹⣇⡧⢲⡗⡏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠐⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢳⣰⡇⡟
⠀⠀⠸⠺⠆⠯⠲⠯⠇⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⣠⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠛⡃⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠄⠀⠀⠀⠀⠀⠀⠀⢀⡨⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⠀⠀⠀⠀⠀⠀⠀⠈⠅⠂⢼⠆⠂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⠀⠀⠀⠀⠀⠀⠀⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠅⠃⠀⠀⠀⠠
⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄
⠠⣮⣤⣤⣤⣤⣤⣀⣀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄
⠀⡿⠿⠿⣿⣿⣿⣛⣙⣳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⡀
⠠⣿⣩⣭⣭⣭⣭⣽⣛⣿⡧⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠤⠤⠴⣿⣀⣀⣀⣀⣀⣠⡤
⢈⡿⣿⣿⣿⣿⡿⠭⠭⡿⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⡿⠿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⣲⣿⣿⠀⠁⠈⠉⠀⠀⠀⠀⠀⠂
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣿⡟⠀⢄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢞⠟⠁⠀⠐⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠
⢠⠀⠁⠀⠀⡀⠀⠀⠀⣤⡒⠂⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⢁⡄⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊⠀⠀⠈⢂⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⣠⣤⣴⣤⣄⣴⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢳⠀⠁⠁⠀⠀⠀⢀⠀⠀⠀⠈⠉⠙⠛⢿⣿⣿⣷⣶
⠓⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠄⠂⠀⠀⢹⡿⠃⠙⠉
⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⢄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠘⡁⠀⠂⠀⡐⢤⣄⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠀⠈⠉⣺⣿⣿⣠⣿⣶⣾⠆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣻⣿⠛⠛⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⣰⠁⠀⠶
⠀⠁⢰⡀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠤⠀⣲⠞⠛⠻⣇⠐⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⢠⠋⠁⠊⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠶⢦⣤⡄
⠀⠁⠀⠀⡀⠀⠀⠀⠠⠤⣀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠉⢘⡄⠀⢀⠀⠀⠀⣠
⠀⠀⠀⠆⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢹⣄⣸⣦⣶⣾⠋
⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠋⠀⠀⣠⣖⠀⠀⠀⠀⠀⠀⢈⣿⣿⡛⠉
⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⢄⡀⠀⠸⣿⣷⣤⡆⢠⣾⡿⠋⠄⠀⠀⠀⠐⠀⠈⠁⠀⠙⠦⣀⡒
⠁⠀⠀⠀⠀⠀⠀⢠⠀⠠⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣌⣙⠇⠀⡸⣿⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠔⠒⠊⠩⠛⠛⠛⡿⡿⢉⣤⢚⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⡾⠛⢿⣿⡿⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣦⠀⠉⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠋⢀⣤⡀⠙⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣿⣿⣿⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣦⣰⢀⣶⡀⡖⠲⠐⣶⠂⡆⣆⡰⢰⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠛⠘⠉⠃⠓⠚⠀⠛⠀⠃⠘⠃⠘⠒


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣾⣷⣦⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⡿⠿⠛⠛⠛⠉⠉⠛⠛⠛⠻⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣾⣿⡀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⣴⣿⡶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⣾⣿⣿⠟⠀⠀⠹⣿⣿⣷⣶⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⠟⠛⠉⠉⠀⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⢯⡀⠀⠀⠀⠀⠀⢸⠿⣿⡿⣿⣿⣿⣿⣿⣿⣆⣠⣿⣿⣿⣿⣿⣿⢿⣿⠿⣆⠀⠀⠀⠀⠀⢀⡼⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠇⠀⠳⣄⠀⠀⠀⠀⡇⠀⠈⣄⠈⠛⢻⣿⣿⣿⢻⡟⣿⣿⣿⡟⠋⠁⣠⠃⠀⠘⡄⠀⠀⠀⣠⠏⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⢦⣀⣴⣾⡇⣦⠈⠈⢱⣶⣾⣿⠟⠁⣾⣷⠈⠻⣿⣷⣶⡇⠀⢉⡖⢨⣿⣶⣤⡴⠋⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⣿⣿⣿⣷⠘⣷⣄⣸⣿⠟⡇⠀⢰⣿⣿⡆⠀⠈⡟⢿⣇⣴⣿⢁⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⡜⢹⣿⣿⣿⣿⣧⠹⣿⡿⠁⢀⡇⠀⣼⣿⣿⣧⠀⢠⣇⠀⢻⣿⠇⣾⣿⣿⣿⣿⡏⢠⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⡰⠁⢸⣿⣿⣿⣿⣿⡆⢻⡇⠴⠛⣇⢠⣿⣿⣿⣿⡆⢠⡝⠷⢼⡟⣼⣿⣿⣿⣿⣿⡇⠈⢣⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⡰⠁⠀⣼⣿⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢧⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠉⢉⣩⣅⣈⠉⠉⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⣦⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠐⣆⣤⣴⣶⣿⣿⡿⠿⣿⣿⣷⣶⣤⣤⡋⠈⢿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⡿⠸⣷⡄⠀⣿⣿⣿⣿⠏⠀⠀⠠⣿⣿⣿⣿⣿⠟⠀⠀⠘⢿⣿⣿⣿⣿⡇⠀⠀⠹⣿⣿⣿⣿⠀⢀⣾⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣷⢻⡄⣿⣿⠟⠁⠀⠀⠀⢰⣿⣿⣿⣿⣅⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⠀⠀⠀⠈⢻⣿⣿⢠⣾⢹⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡬⢻⣿⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠙⣿⡾⣡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣹⣦⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⡶⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⡷⣿⢻⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢠⡿⣟⣾⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣯⢧⣹⣧⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⡟⣸⠟⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣜⡏⢻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢻⣯⣹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢹⠀⠉⠈⠢⣀⠀⣀⡤⠊⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠸⡄⠀⠀⠀⡎⠉⠀⠀⠀⠠⣄⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢠⠧⣀⣀⠜⠀⠀⠀⠀⠀⠀⠈⢦⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠋⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠤⢎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠂⠀⠈⠉⠒⡖⠒⠒⠒⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢆⠀⠀⠀⣀⣧⣀⣀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣶⣾⣿⣿⣷⣦⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣼⣿⣿⣼⣿⣷⠀⠀⠀⠀⠀⠀⣾⣷⣭⣿⣿⣭⣾⣷⠀⠀⠀⠀⠀⠀⣾⣿⣧⣽⣿⣥⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⠛⠛⠛⠛⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⠛⠛⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣾⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣶⣿⣿⣶⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⣿⣛⣿⣿⠏⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣻⣿⡿⠀⠀⠀⠀⠀⠀⠹⣿⠏⣛⣛⣯⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢻⣿⣿⢻⣿⣧⠀⠀⠀⠀⠀⠀⣾⣿⠛⣿⣿⠛⣿⣷⠀⠀⠀⠀⠀⠀⣴⣿⡟⣿⣿⡟⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣼⣿⣿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣧⣤⣶⣤⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣹⣿⣿⣹⣿⣷⠀⠀⠀⠀⠀⢀⣿⡍⠉⢉⡉⠉⢩⣿⡀⠀⠀⠀⠀⠀⣾⣿⣍⣽⣿⣏⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⡿⣿⣿⡿⠀⠀⠀⠀⠀⠘⣿⣿⣶⣿⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⢿⣿⣿⠿⢿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣷⣿⣿⣾⡿⠃⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⢰⠄⠀⠀⢠⠄⠤⠤⠤⠤⢼⠰⡰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠁⠙⠁⠙⢸⣄⡀⣀⡀⠰⡆⡶⠀⢲⢸⢠⣢⢸⡄⠀⠀⠀⠀⣀⠀⠀⠆⢠⣀⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠤⠁⠐⠲⠀⠲⢸⠁⠄⠀⠀⠘⠃⠛⠀⠛⢸⢠⢤⢸⠀⠖⠰⠂⠖⢸⠀⠀⡁⢰⠀⡆⢀⠀⢤⠀⡀
⠀⠀⠀⠀⡲⠒⠒⠒⠒⠚⠒⢶⠀⠀⠒⠂⠒⢸⠀⢾⡶⣄⠈⠁⠙⠀⠉⢸⢀⣉⢸⠐⡰⠀⠒⢀⡙⢀⠘⡃⠰⠀⠆⢀⡀⠉⠀⢉⡍⡇
⠀⠀⠀⠀⠂⢰⠀⢰⠀⠐⠀⠸⠐⢃⣀⢀⣀⣘⠃⠏⠀⠂⠚⠐⠇⢸⠀⢸⠈⠊⢸⠄⢀⠀⡄⠀⠀⠀⠀⡍⢩⠉⠍⠩⠉⡁⢀⡘⠇⣇
⠀⠀⠀⠀⠀⠘⠁⠘⠁⠘⠁⢰⠀⢸⣙⢋⣏⣸⠀⠀⠺⠲⠷⠐⠿⠸⡶⠾⠶⠶⣾⠀⣬⠆⠈⠉⠁⠉⠁⠁⡶⢶⢶⠶⠶⠋⢈⣍⣈⣉⣉⡄⠉
⠀⠀⠀⠀⠀⠀⠅⠅⠀⡭⠀⢐⠀⢸⢤⢬⡥⢬⠀⠀⠲⠲⠶⠶⠶⢈⡿⠀⡇⢸⣿⠀⠦⠆⠐⠘⠂⠚⠂⡀⠗⠶⠶⠶⠶⠀⠀⡥⢬⠥⣤⡅⠀⡃
⠀⠀⢰⠿⠽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠷⠿⠿⠿⠿⠟⠯⠯⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠽⠿⠽⣷
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⢤⡀⡠⢺⠢⡈⣀⠘⡓⢄⢀⡄⡟⠠⡈⡠⢺⠃⢄⣀⠔⡗⢄⡈⡠⢺⠢⣀⣠⠜⡗⢄⢁⠔⢻⠢⡀⡠⠚⠳⣄⢁⠔⡛⢄⠀⡠⢺⠢⡀⣡⠞
⠀⠀⠀⠀⠙⠀⠘⠀⠈⠁⠀⡇⠀⠉⠀⡅⠀⠈⠀⢸⠀⠀⠁⠀⡇⠀⠉⠀⢸⠀⠈⠀⠀⡇⠀⠁⠀⢸⠀⠈⠀⠀⠀⠈⠁⠀⡇⠀⠉⠀⠸⠀⠘⠁
⠀⠀⠀⠀⢰⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢠
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⢀⡇⠀⠀⠀⣅⠀⠀⠀⢸⡀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⠀⢀⡇⠀⠀⠀⣸⠀⠀⠀⢀⡀⠀⠀⠀⣇⠀⠀⠀⢸⡀⢸
⠀⠀⢀⣀⣸⠔⣡⡙⢦⡴⢋⣌⠢⡠⠊⣤⡑⢄⠔⣡⣝⠢⡤⢊⢌⠳⢤⠔⡠⡐⢄⠴⢂⢌⠢⡠⠊⣤⡑⢄⠔⣡⣌⠢⡤⢊⣌⠳⣤⠞⣡⡑⢼⣀⣀
⠀⠀⢸⠤⠤⠊⣀⡈⠢⠖⠁⠀⠑⠴⠋⣀⠉⠢⠚⠁⠈⠣⠔⢁⣀⠑⠤⠊⠀⠈⠢⠔⢁⡀⠑⠴⠋⠀⠙⠢⠚⢁⡈⠳⠔⠁⠀⠑⠤⠊⣀⡈⠢⠤⢼
⠀⠀⢸⠀⠀⠀⠧⠝⠀⠀⠀⠀⠀⠀⠈⠤⠇⠀⠀⠀⠀⠀⠀⠠⠤⠂⠀⠀⠀⠀⠀⠀⠣⠼⠀⠀⠀⠀⠀⠀⠀⠣⠌⠀⠀⠀⠀⠀⠀⠘⠦⠇⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠈⡉⠀⠀⠀⣀⠀⠀⠀⢉⡁⠀⠀⠀⡀⠀⠀⠀⢁⠁⠀⠀⠀⡀⠀⠀⠀⣉⠁⠀⠀⢀⡀⠀⠀⠈⡉⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⢸⠣⣄⠞⠁⠑⢄⡴⠋⠈⠢⣠⠞⠈⠑⢄⠔⠁⠙⠦⡠⠊⠈⠳⣤⠔⠁⠑⢄⡴⠋⠈⠢⣠⠞⠁⠑⢄⠔⠋⠙⠦⡠⠊⠈⠳⣄⠔⠁⠑⢤⡴⢻
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠙⠊⠓⠁⠀⠀⠀⠀⠀⠑⠉⠊⠀⠀⠀⠀⠀⠈⠚⠙⠊⠀⠀⠀⠀⠀⠈⠚⠙⠂⠀⠀⠀⠀⠀⠘⠊⠑⠁⠀⠀⠀⠀⠀⠑⠋⠚⠁⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⢀⡠⣀⡄⠀⠀⠀⠀⠀⢄⡠⣀⠀⠀⠀⠀⠀⠀⣀⡄⣀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⢸⠳⣄⠔⣡⡑⢄⡠⢊⣍⠢⣤⠞⣩⡑⢄⠔⣩⡙⠦⡤⢊⣍⠳⣤⠔⣡⡑⢤⡴⢋⣍⠢⡠⠚⣩⡑⢄⠔⣩⡙⠦⡤⢊⣍⠲⣤⠔⣡⡑⢄⡴⢻
⠀⠀⢸⠑⠦⠊⠀⠈⠢⠔⠁⠀⠑⠤⠊⠀⠙⠦⠞⠁⠈⠳⠔⠁⠀⠑⠤⠊⠀⠈⠢⠔⠁⠈⠑⠴⠋⠀⠑⠢⠚⠁⠈⠢⠔⠁⠀⠑⠤⠊⠀⠈⠢⠞⢹
⠀⠀⢸⠀⠀⠀⣖⡢⠀⠀⠀⠀⠀⠀⢰⣒⡀⠀⠀⠀⠀⠀⠀⢰⣲⠀⠀⠀⠀⠀⠀⠀⢐⣲⠀⠀⠀⠀⠀⠀⠀⣒⣦⠀⠀⠀⠀⠀⠀⠀⣖⡆⠀⠀⢸
⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⢸⣹⣩⣹⣉⣹⣉⣹⣉⣩⣉⣉⣉⣏⣉⣏⣉⣏⣍⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣍⣏⣉⣏⣉⣏⣉⣉⣩⣉⣹⣉⣹⣉⣹⣹⣹⣹

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠉⠉⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⠈⠀⠀⠀⠀⡤⠀⠠⠔⠒⠐⠙⠄
⠀⠀⠀⠠⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠊⠀⣈⢵⣂⠢⠀⠐⠈
⠀⠀⠀⠄⢈⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠇⠀⢠⣈⢡⠇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠁⠀⠈⠐⠃⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠢⠤⡀⠀⠀⠀⠐⠂⠑⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠤⠐⡀⠀⠠⠐⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⣀⡀⠀⠀⠀⣀⣀⠀⡈
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠠⠠⠄⡐⠁⡢⠥⠈⡅⢰⢠⡄⢐⠁⠊⠁⠉⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠂⠁⠘⠒⠀⠅⠃⠀⠀⠀⠀⢼⠐⠁⠀⠀⠀⠀⠀⠀⠀⠂⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠡⠀⢘⠚⠀⠠⢀⠔⠉⠉⠑⠄⠀⠸⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢈⠄⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⠉⠉⠚⠒⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠪⢀⣠⠀⣀⣀⣀⡀⠀⠠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠢⢄⣀⠆⠈⠀⢠⢾⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠂⠄⠀⠀⠀⠈⠲⢇⣀⣄⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠶⣄⠀⠀⢹⡿⡇⠀⡗⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⡒⠈⠀⠀⠀⠀⠀⠐⠓⠙⠂⢀⣣⡰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣄⠙⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⠘⢃⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⣾⣿⣆⢻⣿⣿⣿⣿⣿⣿⢃⣾⣿⣶⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⣍⡛⠛⠛⣋⣥⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣤⣤⣤⣬⡹⣿⡟⣩⣥⣤⣤⣼⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⣾⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣾⣷⣿⣿⣿⣿⣿⣾⣿⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠉⠉⠉⠉⠉⠙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⢿⠀⢠⣤⡀⠀⠀⠀⣀⡀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠈⢠⣿⣿⠃⠀⠀⣼⣿⡇⠀⠘⠈⢿⣿⣷⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⠿⠟⠀⠀⠸⣿⡿⠁⠀⣠⣶⡌⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣼⣿⣿⠁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠈⣿⣦⡀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠻⢼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠈⢿⣿⣶⣤⣴⣾⣿⡿⠟⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⡀⠙⠻⠿⠟⠋⠁⠀⠀⣀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣶⣦⣤⣴⣶⣶⣿⣿⡿⠛⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠛⠻⠿⠿⣿⡿⠻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣯⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⣤⣤⣤⢤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠉⠩⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠉⠀⠀⣤⡀⠀⠀⠠⣤⣾⣦⡤⠀⠀⠀⣠⡄⠀⠐⢕⢦⡀
⠀⠀⠀⠀⠀⠀⢠⢞⠁⠀⠀⠘⢾⣿⣿⠓⠀⠀⢸⠟⢻⠀⠀⠐⢿⣿⣿⠟⠀⠀⠀⢙⢦
⠀⠀⠀⠀⠀⡰⠓⠁⠀⣠⠄⠀⠚⠉⠛⠀⢀⠀⠀⠀⠄⠀⢀⠀⠘⠋⠙⠂⠀⣴⡀⠀⠁⠱⡀
⠀⠀⠀⠀⣸⠁⠀⠈⢻⣿⣶⠟⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠘⢿⣿⣿⠋⠀⠀⢹⡄
⠀⠀⠀⢰⡁⠀⠀⠀⠘⠉⠙⠃⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠛⠉⠙⠀⠀⠀⠐⣷
⠀⡞⠉⣉⠉⠉⠉⠉⢉⡉⠉⠉⣉⠉⠉⠉⠉⣉⣉⠉⠉⣉⣉⣉⡉⢉⡉⢉⣉⠉⠉⣉⠉⣉⣉⣉⡉⠙⡆
⠀⡇⠀⣿⠀⢸⡿⣆⢸⡇⠀⠰⠿⣇⠀⠀⣬⠉⠙⠃⠀⠉⣽⠉⠁⠀⡟⠀⢹⡄⣰⡏⠀⣿⣭⠉⠁⠀⡇
⠀⡇⠀⣿⠀⢸⡇⠸⣿⡇⢠⡼⠳⢿⣆⠀⢻⣄⣠⡄⠀⠀⣿⠀⠀⢰⡇⠀⠈⣷⡍⠀⠀⣿⣀⣀⡀⠀⡇
⠀⢧⣀⣉⣀⣈⣁⣀⣈⣁⣈⣁⣀⣀⣉⣀⣀⣉⣉⣀⣀⣀⣉⣀⣀⣈⣁⣀⣀⣈⣁⣀⣀⣉⣉⣉⣁⣠⠇
⠀⠀⠀⠸⡦⠀⠀⠀⢠⣀⣠⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⣤⣀⣠⠀⠀⠀⠠⡾
⠀⠀⠀⠀⠹⡠⠀⢀⣼⢿⣿⣦⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⢠⣾⣿⣿⣄⠀⠀⡹⠁
⠀⠀⠀⠀⠀⠹⡅⡀⠀⠀⠁⠀⢤⣄⣤⠀⠀⠁⠐⠂⠐⠀⠁⠀⢠⣄⣤⡄⠀⠻⠁⠀⠀⡽⠁
⠀⠀⠀⠀⠀⠀⠘⢮⡢⡀⠀⢠⣾⣼⣿⠆⠀⠀⢹⣶⣾⠁⠀⠠⣾⣿⡿⠦⠀⠀⡠⣢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣆⠀⠀⠛⠁⠀⠀⠐⠛⢿⠟⠓⠀⠀⠀⠘⠃⢀⢀⡮⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢤⣠⡀⠀⡀⠀⠈⠀⠀⡀⢀⣀⣠⠬⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠈⠋⠙⠶⠿⣄⣤⠀⠀⠀⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⣀⡤⠖⠒⢦⣀⠀⠀⠉⣼⠀⠀⠐⠂⡀⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣄⠀⠀⣰⠛⠉⠀⣠⠤⠄⠉⢉⣳⣞⠁⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⡌⢻⡄⢷⠀⠀⠤⢖⡛⠁⢠⡾⣍⣿⡆⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢷⠈⠷⠞⠀⠀⣚⡛⠀⣠⢾⠻⡌⠈⣿⠀⠀⠀⠰⠆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⢠⡿⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠆⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣿⣄⠀⠀⠴⢒⠒⠀⢀⡼⠷⣤⡤⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⢤⣀⠉⠳⢦⣤⣬⣤⠶⣿⡀⠀⠈⢧⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠙⠶⠞⠻⣎⣻⠛⠁⠙⠶⣤⣸⠇⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⣆⠀⠹⡆⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠈⡇⢸⡄⠀⠹⡖⠒⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡃⢀⣀⣀⣠⣤⣄⠀⡇⠀⣧⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣷⠀⠀⢸⠋⠉⠉⠀⠀⢰⢹⡄⡇⠀⢸⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢹⠀⠀⢸⡆⠀⠀⠀⠀⠸⡏⣧⠻⣄⡼⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡇⠀⠀⣧⠀⠀⠀⠀⠀⢿⢸⠀⠈⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢹⡖⠋⠙⢳⣀⢀⣀⣠⡼⢸⣟⣛⣳⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⣼⡇⠀⠀⠀⢻⣯⠥⠖⠒⠋⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠞⠳⣶⠶⠞⠉⠁⠀⠀⠀⣴⠇⠀⢀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠘⣇⠀⠀⠀⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢠⡏⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣠⡤⠶⣾⣇⡀⢀⣀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢛⣻⣶⣄⣁⠛⠙⠻⣤⣈⣙⣋⣛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠙⠛⠟⠟⠳⠶⠞⠛⠛⠛⠛⠓⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢤⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠑⡄⢀⠀⢀⣾⣆⢠⣤⢀⠎⢰⣦⡄⣀⠀⢸⠀⢸⡇⠀⠒⣰⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠂⢠⡀⠀⠀⡼⣁⡀⠈⠳⣤⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⢛⣸⣿⣷⣾⣷⠀⠀⢻⠀⠀⡸⠃⡀⡄⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⠂⢀⣿⣄⡀⢸⣿⣿⠙⡖⠻⠶⠚⠛⠙⣿⣿⡆⠹⣿⣿⣠⣿⣿⣿⣿⡇⠀⢀⡀⠀⠀⡇⠘⠀⠀⠀⣇⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡎⠃⣼⣿⣿⠀⠀⠀⣠⠄⠀⠀⠸⠛⢁⠀⢹⣿⣿⡞⠿⣿⠟⠁⠘⢚⡅⠀⠀⢳⡶⣆⠀⣼⣿⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⢸⣿⣟⣹⣂⣀⠀⣿⣷⡀⢠⢠⣴⣿⠁⣸⣟⠣⣀⣶⡀⠠⣤⣆⣼⣿⣆⠀⣸⠀⢹⣧⣿⣿⣶⣀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣾⣟⣻⡋⣿⣿⣾⣿⣿⣿⡿⣿⠛⡣⣶⣿⡛⣿⣏⣉⠀⠀⠀⢿⣿⣿⣿⣿⣇⣰⣿⣿⣿⣿⣇⣿⡄
⠀⠀⠀⠀⠀⢀⠃⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣧⣶⡖⠃⠈⣽⡿⣋⡻⢿⣧⣄⡰⣄⠻⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⡿⣿⣿⣿⣃⡐⣳⢋⡾⣩⣾⣿⣬⣭⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠂⠀⣸⠛⣽⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣷⡜⢿⣿⣿⢷⡿⠻⠙⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣿⠟⠉⠈⠻⢿⠧⣿⣿⠿⠿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣆⠀⠠⠶⠶⠚⢻⣿⣿⣿⣿⣿⠟⣿⢟⣛⣉⣿⣿⣿⣿⣿⣷⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⠀⠀⠀⠘⠀⠈⠃⢹⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⣿⣿⡟⢹⣿⣿⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⡎⠀⠀⠚⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⡧⢸⣿⡇⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⠦⠄⠀⠀⠈⠋⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⣻⣥⣾⣿⣥⣼⣿⣧⣠⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠄⠐⠢⣄⣸⠛⠛⠛⠛⠿⠿⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠐⠂⠀⠂⠂⠁⠀⠂⠒⠐⠀⠀⠘⠐⠀⠐⠀⠈⠁⠓⠀⠀⠂⠀⠀⢐⠐⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠂⠀⠐⠒⠒⠂⠲⠀⠒⠒⠒⠂⠐⠒⠀⠂⠂⠒⠒⠂⠀⠴⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⣦⢠⠴⠴⢤⠈⠢⠤⠤⠤⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⢸⠱⠶⠆⠁⠀⠀⢸⡂⣀⣛⢸⣀⣀⣺⠀⠀⠀⢀⠶⡀⠀⠤⢼⣍⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⠀⣚⡊⢉⠊⠙⣻⠤⠤⣾⢸⠤⠤⢼⠀⠀⠀⠘⣦⠃⠘⠒⠻⢈⢸⠁⣀⣞⠃⠜⠓⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡖⣶⢶⢶⡇⠠⡅⠀⠀⠜⠀⠋⠁⠈⢡⠌⢹⡂⠂⣿⢸⣒⣂⣺⠒⣶⣶⠒⣿⣀⣨⣋⣹⢈⢸⣤⣬⣤⣦⣤⣭⣤
⠀⠀⠀⠀⠀⠘⠀⠘⠀⠦⠶⠶⠇⢩⢉⠉⡍⠉⠘⠃⠀⠃⢠⠸⠇⣸⠄⠤⠿⠸⢾⠹⠍⠏⠹⠉⡣⠟⣘⣛⣋⢚⡈⢸⠠⠤⠀⠦⠠⠦⠈
⠀⠀⠀⠀⢰⠀⣀⣈⣈⣁⡁⠘⡧⠤⡤⠤⣤⠄⠘⠂⠀⠃⠀⢸⡃⢸⣖⠒⡛⢛⣿⠶⠶⠶⠶⠆⠀⠀⣍⣍⣩⣻⡁⢸⠀⠒⠀⠗⠐⠒⢠
⠀⠀⠀⠀⢸⠀⢡⣥⣉⣭⣇⠀⡷⠶⠶⠶⠶⠆⠐⠂⠀⠓⠀⠘⠁⢘⣿⠀⡇⢸⣿⠶⠶⠖⠶⠶⡁⠀⣬⣌⣡⣽⡀⢸⠀⢀⡁⠀⣏⠀⢘
⠀⠀⠀⣼⠯⠿⠽⠭⠯⠿⠽⠭⠯⠯⠽⠽⠯⠯⠽⠹⠭⠯⠿⠽⠭⠯⠯⠽⠭⠯⠿⠽⠽⠯⠯⠽⠽⠯⠯⠭⠽⠭⠯⠿⠽⠽⠯⠯⠽⠭⠏⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠒⢄⡠⠊⠑⢌⣡⠞⠑⢄⡠⠞⠑⠤⣉⠔⠙⠢⣀⠴⠋⠢⣉⡴⠈⠢⢄⡠⠊⠳⢌⡡⠊⠑⢄⡠⠊⠑⢎⣁⠔⠙⢦⣀⠔⠉⠢⣉⠴


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡤⠤⠤⠤⠤⢤⠀⠰⣉⣑⠄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣈⣡⠀⣀⡏⣻⣿⣏⣁⣀⢀⢀⣄⠘⠛⠛⡓⢠⡀⣀⠔⢤⡀⠀⣎⣀⣀⣉⡄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠁⢀⡀⠁⠀⠧⡤⠄⠤⠄⠤⠿⠀⠤⡤⠠⠄⠤⠤⢤⠾⣥⢤⣤⠼⢶⠉⠠⡄⠈⣁⡴⠛⠦⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣄⡀⢠⣁⣀⣀⣘⡀⠀⣀⣏⣛⣛⣋⣀⣀⢀⣀⣅⣈⣃⣀⣁⣈⣳⠛⢛⡛⠛⣞⣒⣒⣒⣒⣛⠖⠒⠒⢻⡀⠣⠀⠃
⠀⠀⠀⠀⡤⠮⠈⠤⠡⣾⢠⠄⢠⠄⣶⡐⢣⡄⣀⠀⡀⠠⠟⢘⡛⠘⠃⢫⠴⢄⠋⡸⠋⠀⠀⠙⢮⠀⠤⠴⠀⡅⢠⣷⣄⢸⣧⡔⢣⡤⢤
⠀⠀⠀⠀⠣⠄⡀⢀⡄⢿⢸⢆⡰⡂⣽⠿⠭⡀⢻⢈⡃⠰⠄⢨⠉⡚⢶⠇⠤⢤⡷⠆⠀⠀⠀⠀⠼⠀⡂⢴⠀⠷⠤⠀⠆⢦⡷⠈⠁⠰⡼
⠀⠀⠀⠀⠀⠀⡥⠈⡅⠚⢨⠄⢠⡅⢻⣿⣼⠃⠶⠰⠆⠰⠖⢊⠀⡇⣨⠈⡯⢠⡄⠘⠛⠛⠛⠫⠒⠀⠆⠲⠀⠀⡿⠿⢻⢠⠄⡯⢈
⠀⠀⠀⣰⠿⢿⢹⠼⠧⡿⠼⠥⡯⠯⠿⠭⠿⠷⢶⠰⠶⠿⢷⠹⠭⡿⠯⠽⠽⠯⠿⠷⠰⡦⠶⠶⢹⠷⠦⠶⠼⠯⡯⠿⢽⠼⠯⠿⠿⢹⠯⢾⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠢⢄⡠⠊⠳⢌⣡⠔⠑⢄⣀⠔⠑⢄⣉⠔⠑⢦⣀⠴⠊⠢⣉⡠⠈⠢⣀⡠⠊⠣⣈⡡⠖⠑⢄⣠⠔⠑⢌⣁⠔⠑⢤⣀⠔⠉⠢⣉⠰⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀⠀⠀⠀⠀⠀⠠⠄⠤⢤⠀⠀⢀⡴⠒⢢⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⢄⠑⠒⠁⡀⠀⠀⠀⠀⡄⠀⡆⢸⠀⠀⠯⠀⠄⢠⠔⠀⠀⠀⠀⠀⠖⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠒⠚⠐⠒⠀⠖⢸⠀⠀⢈⠀⡄⢠⡄⢤⠀⠀⡄⢸⣀⣀⣀⣀⣀⡀⠀⠐⠂⣀⣀⠚⠂⢤⢈
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠭⠀⠰⠴⠀⠖⢘⢠⡁⠈⠀⡃⢘⠃⠛⠀⠀⡅⢸⠰⠆⠴⠒⠀⠀⠀⠀⠠⠅⢡⠀⡁⢭⠐⡀
⠀⠀⠀⠀⢰⠤⡤⠤⠤⠝⡢⢶⠀⠰⠤⠀⠦⠸⠢⣴⢠⠀⠁⠈⢁⠉⠀⠀⡅⢸⠰⠆⠀⠊⡉⠉⢈⢃⡐⠂⠚⢀⣀⠉⠠⠁⣌⠁
⠀⠀⠀⠀⢸⠐⠖⠀⠆⠰⠄⢸⠐⣃⣐⣀⣀⡛⠒⠁⠚⠐⠃⠟⢸⠀⠀⠀⠃⢸⠄⠄⠰⠀⡁⠀⣈⠈⠉⠥⠉⠸⠉⡇⢐⠀⠚⡀
⠀⠀⠀⠀⢸⠈⠃⡈⠃⡐⠂⢸⠀⣭⣭⣭⣭⡅⠀⠐⠛⠛⠛⠛⢻⣋⠙⡛⢋⣿⠀⣼⠀⠈⡁⠈⠉⠀⠿⠿⠯⠷⠕⡏⢉⣭⣭⣭⣭⡉⢹
⠀⠀⠀⢀⣸⣀⢀⣃⣀⣃⢀⣘⡀⣢⣰⣄⣤⣂⣀⣘⠚⠛⢓⠓⣻⣾⣀⣇⣸⣟⣀⣐⣀⣈⣁⣈⢉⣀⡒⡓⢚⠛⣒⣇⣀⣆⣖⣰⣴⣀⣨⣀
⠀⠀⠀⠽⠯⠿⠽⠽⠯⠿⠹⠭⠯⠿⠿⠹⠯⠯⠿⠽⠭⠏⠿⠽⠭⠯⠯⠽⠭⠏⠿⠿⠽⠯⠿⠿⠽⠏⠯⠿⠽⠯⠯⠿⠽⠽⠯⠿⠿⠽⠯⠿⠅
⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⢀⡀⠀⠁⠀⡀⠀⠀⠀⣀⠀⠉⠀⣀⠀⠀⠀⢀⠀⠈⠁⢀⡀⢀⠀⢀⡀⠈⠁⢀⡀⠀⠀⠀⣀⠀⠁⠀⣀
⠀⠀⠀⠙⢦⠔⠁⠈⢢⡔⠉⠉⠢⡤⠋⠈⠳⡤⠚⠉⠓⠤⠊⠈⠑⢤⠜⠁⠙⢦⠔⠉⠁⢠⠔⠁⠙⠢⡴⠊⠈⠣⡤⠊⠈⠲⡤⠊⠈⠑⢤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⠒⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣠⠖⠉⣠⠔⠁⣼⡡⠖⠚⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠘⠋⠀⢀⡞⠁⢠⠞⠉⣀⡤⠤⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⣠⣯⠴⠖⠛⠚⠊⠁⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⡠⠠⢄⡀⠀⠀⢀⣀⣼⠁⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠣⠤⠤⠼⠆⠀⣴⠟⠙⣦⠟⠁⠀⢀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠾⠤⠤⠴⠃
⠀⠀⠀⠀⠀⢠⠤⠤⣤⠀⠀⢰⡇⠀⡜⠁⠀⠀⠀⠸⠟⣋⡉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⢻⡀⠀⠀⢠⡤⠤⢤
⠀⠀⠀⠀⠀⠘⠶⠚⠁⠀⠀⢺⠏⣸⡇⠀⠀⠀⢠⠖⠋⠉⠙⠳⣄⠀⠀⠀⠀⠀⠀⢀⣀⣉⠀⠈⣇⠀⠀⠀⠙⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⣇⠀⠀⢠⡏⠀⠀⠀⠀⣀⣹⡆⠀⠀⠆⠀⠀⣿⣁⣈⣳⠀⢿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠖⢻⡀⠀⠈⣷⠖⠚⠋⠛⠋⣼⠀⠀⠀⣦⠀⠀⠣⡌⡩⠋⠀⢸⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣆⠀⢧⠀⠀⠘⠦⣤⣀⡤⠞⠁⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣁⣠⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⡴⠋⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⡼⣿⣿⣷⣼⠆⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠰⠦⣌⣻⣿⡟⠀⠀⠀⢀⡴⠋⠀⠀⢀⣸⡁⢸⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣤⣄⡀⠀⢸⡆⠀⠀⠀⠀⠀⠈⠙⠁⠀⢀⡴⠋⠀⠀⠀⠀⡏⠀⢈⡇⢀⠎⢙⡆
⠀⠀⠀⢀⣠⠤⠤⠴⠊⣉⠴⠚⠉⠉⠉⣿⣉⠉⢻⡄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢳⠀⠙⠉⠙⡶⠊
⠀⠀⡾⠋⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠈⠉⣻⠇⢛⡗⠶⠦⠤⠴⠚⣏⠙⣦⠀⠀⠀⠀⢀⡰⠋⣠⠖⠀⠁
⠀⠀⠳⡞⠀⠀⠀⡀⢀⠇⠀⠀⠀⠀⠀⠀⢠⡟⠁⠘⠀⠀⠀⠀⠀⢨⢻⡷⠛⢶⣦⣤⣶⡯⠴⠋⠁
⠀⠀⠀⠙⢾⣁⣸⠋⠉⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡀⠀⣸⠇⢰⣦⢀⠘⡇
⠀⠀⠀⠀⠀⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣀⡈⠃⠐⠾⠃⠐⠀⡇
⠀⠀⠀⠀⠀⠀⣰⠟⡹⣧⠀⠀⠀⠀⢰⡿⠶⣤⣤⣉⣉⠛⠛⠚⠚⠃⢠⡇
⠀⠀⠀⠀⢠⡾⢣⠞⢀⡼⡆⠀⠀⠀⣸⣯⠀⠀⠀⠈⠉⠙⠛⠳⠶⠦⣼⠃
⠀⠀⠀⣴⠟⡴⢁⣶⣿⡔⢻⡄⠀⣾⠉⠙⢦⣀⡀⠀⠀⠀⠰⢤⣤⣴⡋
⠀⣠⡾⣡⢮⠀⢠⡿⠁⠹⡄⠙⣛⡿⠳⡤⠞⠉⠛⠷⢦⣤⣀⣤⣶⣿⣧⣤⣀
⣴⢏⡔⢁⣠⡷⠋⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⢳⡦⠉⡷
⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠯⣴⣿⣶⡶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠢⠤⢤⡬⣉⣀⣀⣰⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠺⠶⠶⠦⣤⣭⣭⣉⣗⣻⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠤⠤⠤⠴⠤⠤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠰⠞⠛⠉⣉⡁⣠⣤⢠⣦⠄⣴⡄⢠⣦⢀⣤⠀⣈⡉⠉⠛⠶⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⡁⢠⣦⠄⠻⠆⠛⣃⣈⣉⡀⠉⠀⠡⠥⠬⠉⣈⣉⠀⠛⠃⠾⠗⢤⣦⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⢠⣦⠘⠻⢀⡠⠄⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⠬⣁⠸⠓⢠⣦⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⠰⠿⠊⡩⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⡉⠲⢷⢀⡈⠳⣄
⠀⠀⠀⠀⠀⣠⠟⢀⣈⠛⢃⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠛⢋⣀⠘⢧⡀
⠀⠀⠀⠀⣰⠋⣠⣈⠋⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠛⣡⡀⢳⡀
⠀⠀⠀⢰⠇⣀⣙⠃⡜⠀⠀⠀⠀⠀⠐⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠐⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠘⡄⠛⣁⡀⢷
⠀⠀⠀⡿⠀⣙⠋⢸⠀⣶⡄⢰⣦⡀⠀⣴⡆⠀⠀⣴⣶⡀⠀⠀⢠⣴⣶⣶⣄⢠⣶⣆⣴⣶⡆⢰⡆⢰⣶⡀⠀⢠⣶⠀⣶⣶⣶⣦⡆⢸⠀⢛⡁⠸⡇
⠀⠀⠀⡇⠈⠟⠁⡆⠀⣿⡇⢸⡿⢿⣆⣿⡇⠀⣰⣿⢻⣷⡀⠀⣿⡇⠀⠈⠁⠀⠀⢸⡿⠀⠀⢸⡇⠀⢻⣧⢠⣿⠇⠀⣿⣶⣶⠀⠀⠀⡆⠛⠇⠀⡇
⠀⠀⠀⣧⠐⠿⠂⢠⠀⣿⡁⢸⣿⠈⠻⣿⡇⣰⣿⠛⠛⢻⣷⠀⢻⣧⣤⣴⡦⠀⠀⢸⣷⠀⠀⢸⡇⠀⠀⢿⣽⡏⠀⠠⣿⣤⣤⣄⡀⢠⠁⠺⠗⢠⡇
⠀⠀⠀⢻⡀⢾⡖⠘⡄⠉⠀⠈⠁⠀⠀⠈⠁⠉⠁⠀⠀⠀⠉⠀⠀⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠀⠉⠉⠉⠉⠁⡜⠐⣾⠆⣸⠃
⠀⠀⠀⠈⢧⠀⢶⡖⠐⢄⠀⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⠜⠀⣴⡆⢠⠏
⠀⠀⠀⠀⠈⢧⡀⢲⡷⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢰⣷⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠳⣄⠐⠿⠆⣈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢋⡀⠾⠗⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡘⠛⢡⣦⠀⠐⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠈⢠⣦⠙⠛⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣌⡉⠸⠿⢠⣤⠄⣉⠐⠒⠦⠠⠤⠄⢀⣀⣀⣀⣀⠀⠤⠤⠤⠖⠒⢈⠁⣤⣀⠸⠿⠈⣉⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣄⣉⠈⠛⠃⢾⡖⢴⣦⢠⣦⠄⣤⡄⣴⠦⢠⣦⠔⣾⠂⠛⠃⣈⣁⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠦⣤⣬⣄⣉⣀⣉⣁⣈⣀⣬⣤⡤⠶⠶⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠴⡠⠔⢄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠉⠀⠀⠈⠛⠀⠀⠀⠠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⢀⡏⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣮⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡀⠀⢀⠀⠀⠀⠀⡀⠄⢒⠊⠁⠀⠀⠘⢐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡜⠀⠀⠀⢣⠈⢂⡜⢀⠎⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢦⠀⠀⠀⢣⡜⠀⠎⠀⠀⠀⡆⢀⡸⢂
⠀⠀⠂⠉⠉⠉⠉⠁⠈⠉⢉⣧⠀⠀⠀⠀⠀⠀⠃⠀⡹⡀⠀⢀⡞⠀⠌⠀⠀⠀⢸⠀⠁
⢐⣗⣶⣶⣶⣶⣤⣤⣤⣤⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡞⠀⠌⠀⠀⠀⠀⠆⠀⠀⠈⠆⠀⠈
⠀⡟⠛⠛⢿⣭⣭⢭⣭⣭⣵⡿⠀⠀⠀⠀⠀⠀⠂⠀⠀⢨⡞⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠀⠤⠤⠤⠄
⢤⣿⣭⣥⣭⣭⣭⣭⣽⣿⣷⣷⢤⡤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠦⠤⠤⠴⠶⠮⠴⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣿⣄⣀⣀⣀⣀⣀⣸⡤
⠘⡷⠿⠿⠿⠿⣛⣛⣿⣿⣿⠃⠈⢔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷
⣬⣿⣿⣿⣿⣿⣿⣧⣶⣶⣧⣤⣤⣬⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣥⣤⣤⣤⣥
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⢢⠴⢦⣄
⠀⠀⠀⣲⢢⢶⡶⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⣀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢿⣿⡻⢽⢚⣺⣿⠏
⠀⠀⢶⣁⣳⠺⣨⠬⣝⣫⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠁⣠⡶⢶⣆⠀⠈⠛⡄⠀⠀⠀⠀⠀⢀⢀⢄⣦⣟⣷⣷⢺⡝⢏⠋⡧⢏⠟⠁
⠀⠀⠀⠙⢾⡑⢳⣾⢳⢣⢿⣖⡿⣽⡟⣶⡴⡄⣠⢠⠀⡀⡀⠘⣆⠀⢾⣗⣼⣧⣙⡗⠀⢠⡇⣀⣠⢀⣢⣽⢓⣟⣿⣟⣏⢡⡌⠦⢡⢚⡸⠟⠁
⠀⠀⠀⠀⠀⠈⠛⠌⣍⢏⢟⣛⡞⡳⠯⣮⣔⡫⣹⣫⢞⣖⡇⡧⢏⣳⣨⢿⠿⠾⣿⣧⣶⣿⢷⡛⣫⢇⢞⢾⢺⣛⢛⢛⣘⠐⣼⣬⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢾⣟⣯⢇⠧⡳⠿⣿⡽⠜⡬⣀⡸⢚⣲⢿⣿⣷⣶⣿⡟⢮⡤⡁⡥⣰⢪⣻⡓⣣⢹⡞⢻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠦⣟⣁⢢⣈⡣⢜⡺⢖⡃⣏⢸⣿⣅⣠⣽⣇⣀⡟⠍⣏⠦⣉⣆⣩⠬⠞⠉
⠀⠀⠀⢠⢤⣠⣒⡆⡆⡆⢄⠠⡀⠀⢄⢆⠰⡙⡏⢷⢲⣾⢼⣾⠴⡿⣻⢿⡃⢹⣿⣟⠽⢛⠼⠼⢓⣻⠫⣀⣤⣀⣤⣀⠀⢠⣤⣀⣀⣀⣀⣀⢀⣀
⠀⢠⢓⣂⡸⠜⣌⢲⠃⡷⠛⡾⡴⠴⡈⡀⢗⡗⡎⢇⢫⢤⡽⢰⡉⢧⢹⠸⣧⣸⣿⣼⣓⡀⡦⢈⣙⣚⢃⠃⡧⠷⣵⢻⣼⣬⣿⣶⣷⣺⢻⢿⣩⣩⣟⣦⡀
⠀⠀⠙⠲⠶⣭⣔⢆⠹⡴⢷⡒⣏⢥⢹⢘⢣⢓⣩⡽⣋⢖⡥⢡⢍⣓⡸⠂⣿⢼⡿⣃⢷⠱⡚⡄⠢⠛⢾⢾⡯⣙⡹⣞⠻⠳⢺⢼⠹⢻⠫⣕⣮⢩⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠘⠒⠒⠚⠑⠚⠓⠚⠒⠚⠛⠋⠉⠈⠁⠀⠀⣿⣼⡇⠈⠉⠑⠳⠨⠵⠌⠦⠻⠿⣽⣟⣭⣯⠍⠾⠶⠌⠚⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣻⣻⣩⣙⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣭⣿⣯⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⠿⠟⠛⢋⡋⣙⣛⣟⢛⠿⠿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⠟⡻⠋⣉⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣩⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡃⠀⠀⠀⠀⠀⠀⢀⡀⢄⡀
⠀⠀⠀⠀⠀⠀⢜⠁⠀⠉⠳⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⣠⠔⠋⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠃⠀⣾⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⢀⣴⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⡿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⡀⠀⢿⡄⠀⠀⠀⢀⣤⠴⠶⠦⣄
⠀⠀⠀⠀⠀⠺⣄⣠⠶⠋⠁⠀⠀⠀⠀⢸⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⣠⡶⠞⠛⠳⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠘⣧⠀⠀⠀⠀⠙⠲⢤⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⣦⡀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⣀⣠⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⣥⣤⣄⣈⣷⠀⢸⣷⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⢻⡶⠶⠶⠛⠋⠛⠿⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢉⡽⠃⠀⠸⣧⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡆⠀⠀⠈⢷⣄⠀⠀⠀⠀⣀⡼⠋⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠙⠋⠀⠀⠀⠀⣿⣤⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⣿⣿⣿⠄⠀⠀⠀⠉⠛⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠐⠀⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣐⠒⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠾⠛⠋⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⣍⣻⡆⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢠⡟⠙⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⡾⠛⢻⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠘⠛⠶⠶⢿⣿⣿⠃⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⠶⣧⣀⣼⠋⠉⢻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⢈⣿⠀⢰⠏⠈⣷
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡶⠟⢋⣡⡴⠾⠛⠛⠛⢿⡿⠛⠗⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⢻⠛⠾⣯⣤⡼⠃
⠀⠀⠀⢀⡴⠞⠛⠉⠉⠉⠁⢀⡴⠛⠁⠀⠀⠀⠀⠀⠐⠷⣶⣤⣤⣼⢷⣤⣀⠀⠀⠀⠀⠀⣀⣤⠾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⣀⣤⠄⠞⠁
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠄⠀⠈⢛⠿⠿⠿⠿⠟⠋⠱⣦⣼⣿⣦⠀⠀⠄⠠⠖⣀⣴⠞⣁⣴⡾⠋
⠀⠀⠈⢳⣾⠃⠀⠀⠀⠀⠠⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣷⠿⣿⣷⣶⣶⣶⣿⣿⣥⠾⠟⠁
⠀⠀⠀⠀⠻⣦⣴⠃⠀⣾⠗⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⡧⠄⠀⠈⠉⠉⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠻⢦⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣻⣷⡀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡿⢁⡾⠁⢘⣷⠀⠀⠀⠀⠀⠀⣿⠗⠀⠈⠉⠙⠛⠛⠷⠶⣦⣤⣄⣀⣀⠀⣸⡇
⠀⠀⠀⠀⠀⣠⣾⠏⣰⠏⠀⣤⡾⢹⣇⠀⠀⠀⣀⣴⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢻⡟
⠀⠀⠀⢀⣴⡟⢁⡼⠁⣸⣿⣿⣷⠋⠹⣆⠀⢸⣿⡅⠀⠙⢷⣤⣀⡀⠀⠀⠀⠀⠀⠲⠶⣤⣴⣟
⠀⠀⣠⡾⠋⣴⡏⠀⠀⣹⡿⠃⠘⣧⡀⠻⠿⢻⣿⠿⣦⣠⠾⠋⠙⠻⢶⣦⣄⡀⠀⢀⣠⣴⣿⣿⣄⡀
⢀⣴⡟⣡⠞⠁⢱⣆⣼⠟⠀⠀⠀⠘⣷⣤⡶⠛⠁⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠛⠷⢿⣍⡙⠻⢶⣄
⣾⣯⣴⣏⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⣼⠏
⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣍⣩⣿⣵⣤⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠈⠉⠻⠿⠿⠃⠈⠉⢉⡿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣍⣙⣛⠒⠲⠖⠲⠤⣤⣤⣤⣯⣀⣀⣀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠶⣶⣶⣦⣼⣤⣭⣉⣙⣉⣻⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⡀⠀⠠⠤⠤⠄⠀⠐⠒⠒⠒⠀⠀⠀⠀⠉⠉⠑
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⡀⠀⠠⠤⠤⠄⠀⠀⠒⠂⠀⠒⠂⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⡔⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠻⣿⠟⠉⠀⣐⠶⠂⠦⣤⣤⡤⢀⣀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠛⠤⠄⠠⠜⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠄⠀⠠⠄⠒⠈⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠒⠒⠒⠀⠀⠈⠀⠉
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡀⠀⠀⠤⠤⠤⠀⠀⠒⠒⠒⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠐⢄⠀⠀⢀
⠀⠀⠈⠑⠂⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣠⠀⠀⠀⠀⠀⡤⠖⠃⠀⠀⠀⠀⣠⣾⣿⣿⢷⣄
⠀⠀⠀⡼⠛⠛⠛⠛⠙⢦⠀⠀⣀⣠⣿⡿⢿⣆⣿⠀⠀⢀⢤⣤⣷⣤⣄⠀⠀⠀⢀⣿⣷⣿⣿⣾⣿
⠀⠀⠀⡇⠠⠔⡀⠢⠀⡸⠀⠀⣿⠀⠀⡾⢻⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⢲⠀⠀⠀⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠘⠦⣀⣉⡠⠔⠁⠀⠀⣏⠀⠸⠃⣠⠚⢂⠀⠀⠈⠻⣿⣿⠿⠋⠁⠀⠀⠀⠈⠉⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⢦⡀⠀⠀⠀⠈⣧⣿⣧⡀
⠀⠀⢠⠀⠀⣰⣷⡀⠀⠀⠀⠀⡯⠐⡖⣀⢠⡄⢹⡀⠀⠀⢰⢿⣿⡏⣧⠀⠀⠀⠀⢰⠉⠉⠉⣿⡆
⠀⠀⢸⣠⣼⣹⣿⣽⣦⣄⡄⠘⢧⣈⠉⠚⠈⠁⡴⠁⠀⠀⠀⢸⡏⣷⠀⠀⠀⠀⡄⣈⠩⢉⡉⢉⢀⡄
⠀⠀⠸⠉⠉⠹⠩⠏⠉⠉⠇⠀⠜⠋⠙⠋⠉⠉⠀⠀⠀⠀⠀⠸⠇⠟⠀⠀⠀⠀⠁⠉⠀⠀⠉⠈⠉⠁
⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠈⠗⠑⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣉⣋⣻⣷
⠀⠀⠀⣿⣿⢿⢻⣹⡟⢦⠀⠀⠀⠀⢻⡿⡇⠀⠀⠀⢰⠀⠠⠏⣰⠪⠇⠀⠀⠀⠐⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠘⢧⣽⣿⠿⠉⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣶⣶⣶⣶⣶⣶⣶⣷⠀⠀⠻⠿⣿⣿⡏
⠀⠀⠀⠐⠒⠚⠓⠒⠂⠀⠀⠀⠀⠀⠈⠋⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢛⠛⠉
⠀⠀⠀⠀⠀⣠⣦⠀⠀⣤⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⢀⣿⣗⡀
⠀⠀⠀⣀⣨⣿⡿⣷⣄⣿⠀⠀⠀⠀⢾⡿⣿⠆⠀⠀⢀⣼⣿⣿⠿⠋⠻⢷⡄⠀⠀⢠⡟⣿⣿⢧
⠀⠀⠀⣿⠀⢀⡶⣻⣿⣿⠀⠀⠀⣠⣴⣿⣷⣤⠀⠀⢸⣏⠀⠤⢀⠠⠄⢸⣿⠀⠀⠈⢠⡟⣿⠈
⠀⠀⠀⣇⠀⠼⠁⡀⠓⣄⠀⠀⢰⡟⣿⣿⣿⣿⠇⠀⠸⠿⠦⣄⣙⣀⡤⠿⠿⠀⠀⠀⠸⡇⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⠒⠿⠛⠻⣷⣤⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠻⣿⠿⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢻⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠀⠀⠀⢀⡅⢀⡴⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣷⠀⠀⠁⠀⠀⠀⢸⡇⣾⣇⡴⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⡵⠀⠀⣀⣤⣴⣶⣤⠀⠀⠀⢴⣿⣿⠅⡏⣌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣆⡀⠀⠀⠀⣼⢿⣿⡏⠃⠀⠀⠱⢋⠉⠉⠇⠀⠀⠀⠋⠀⣦⠳⡇⣝⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⢻⡝⠙⢻⡄⣶⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢀⢃⡿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠈⠿⠙⠁⠀⢸⡷⠀⠀⠀⠀⣀⠀⠄⢸⢼⢘⡇⣿⠃⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢁⡠⢹⢺⣾⢘⣿⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠠⠁⠐⠒⠀⢸⣄⡜⢸⣿⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧⡐⠀⢠⣤⡄⠀⠀⠀⠀⠸⠈⢃⢸⣿⣾⣷⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠐⠂⠤⣤⡇⣸⢸⣿⠏⡇⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠦⣀⣀⣀⣼⡃⡹⣻⡬⣿⡀⣷⣹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠳⠋⣨⡟⣹⣧⢷⣿⢷⡟⣷⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠈⢰⡟⣡⣟⣿⣦⣿⠀⢷⠸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠁⠀⢰⣷⣿⣦⣿⣿⢻⡇⡸⠃⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣼⣿⢉⡿⠀⣿⢼⡏⡟⣴⣯⡛⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⡿⡇⣸⡇⣸⣧⣾⡿⣱⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠃⠿⣟⢠⡟⢻⣿⣿⣻⠛⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣼⡌⠀⣿⣼⡇⣿⡿⣧⣿⡤⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣮⣿⡇⢼⢿⡟⢻⣿⢁⠏⣽⣷⠇⣸⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠘⣿⡦⠃⢸⣿⣾⣿⢁⢋⠰⣏⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣇⢹⣧⠀⠘⢿⣷⢹⢯⡘⠠⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠸⣬⠹⠸⡾⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⣀⠀⠀⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡷⠀⠀⠀⠀⠰⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⣉⣹⣏⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣦⡀⠀⠩⣭⣭⣿⠁⢀⣴⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡟⠙⣦⣄⣀⣀⣠⣴⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠛⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠒⠒⠲⠦⠤⢤⣀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣶⣦⠤⠤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢨⣿⣿⡿⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠈⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠋⠀⠀⠀⠀⢠⠿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡃⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⡴⠃⠀⠀⠈⠉⠉⠉⠉⠛⠒⠒⠒⠒⠛⠉⠉⠉⠉⠉⠉⠁⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠿⠃
⠀⠀⠀⠀⠀⠈⠛⠻⣄⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣍⠉⠻⣶⣖⠒⠶⠶⠦⠤⠤⠤⠤⠤⠤⢤⣤⡴⠒⠒⣲⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣠⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠈⠻⣷⣄⠀⠀⣠⣾⠟⠁⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠈⠻⣷⣾⠟⠁⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣾⠟⠻⣷⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢿⣿⣷⣴⣿⡿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣿⣿⣿⣿⣄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠘⢿⣿⣃⣀⣀⡀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣈⣹⡋⣿⣿⡟⠉
⠀⠀⠀⠀⠀⣀⡤⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠙⠢⢄⡀
⠀⠀⠀⠠⣞⣁⣀⡀⠀⠀⠀⢀⢠⣤⣤⡤⢤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣽⡆
⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⡟⣛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⣴⣾⣷⣀⣀⣀⣀⡄⠀⠀⠀⠀⢠⠸⠻⠿⢟⢿⠟⠆⡈⢣⠀⠀⠀⣀⢰⡾⢿⡷⢖⣴⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠛⢋⣷⠤⠀⠀⠀⡌⠀⠸⠀⡌⠰⡀⠀⢠⣸⡀⠀⠀⠗⠇⠛⢶⢶⣾⠟⠋
⠀⠀⠀⠀⠀⠴⢶⣶⣦⣽⣿⣿⣿⣶⣦⣤⣺⣤⣴⣶⣤⠶⣴⣷⣤⣤⣟⣧⣤⣤⣤⣤⣾⣴⣿⡿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⡒⡛⣂⢀⡞⠛⣂⠀⠂⠀⠀⠠⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⡀⠀⠈⠡⠴⠆⠀⠀⠀⢨⡅⠤⣭⠠⡅⠄⣬⠀⠀⠀⢀⠶⠆⢀⠀⣶⣰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⡇⠀⠈⠀⣏⡉⢉⡉⢉⣹⠦⠀⢶⠐⠆⠀⠾⠀⠀⠀⠈⢶⠎⢸⡛⡗⢀⢸⢉⣥⣈⠃⢞⠑⠢
⠀⠀⠀⠀⠀⠀⡀⢀⠀⡖⣶⡶⠶⡇⢠⡀⡇⠀⠚⠀⠋⠁⠈⠀⡆⢙⡒⠐⢺⠐⣃⣀⣺⠓⣶⣶⠒⢺⡄⣸⡋⣧⠈⢸⣤⣤⣤⡧⣤⡤⣤
⠀⠀⠀⠀⠀⠐⠀⠚⠀⠦⠴⠶⡇⠹⠩⠉⠫⠉⡉⠉⠈⠁⢠⠰⠇⣬⠅⠀⢬⠀⣿⠸⠏⠧⠸⠅⡗⠚⠓⠛⠓⠓⠘⢺⠀⡄⠀⡴⠠⡆⢸
⠀⠀⠀⠀⢰⠂⢀⣀⣈⣀⣀⠐⡗⠲⠒⠖⠶⠀⠋⠋⠈⠃⠀⢐⡂⠰⠶⠶⡶⠶⣿⠒⠖⠶⠲⠂⡇⠀⣛⣛⣛⣛⡁⢸⠀⠂⡀⠘⣀⠃⢸
⠀⠀⠀⠀⢸⠀⣩⣭⣭⣭⣅⠀⣟⠛⠛⠻⠚⢀⡛⠓⠀⠃⢀⢸⡅⢀⣿⢀⡇⢸⣟⠊⠛⠛⠑⠁⡇⠀⣭⣭⣭⣭⡅⢸⠀⢀⡄⠀⢠⠀⢘
⠀⠀⠀⡷⠼⠶⠷⠾⠴⠷⠶⠴⠧⠶⠴⠧⠿⠼⠧⠶⠴⠧⠿⠴⠦⠿⠴⠦⠶⠴⠏⠶⠸⠦⠶⠼⠯⠶⠴⠦⠷⠴⠦⠿⠴⠤⠷⠴⠦⠶⠼⠾⡆
⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⡀⠀⠁⠀⡀⠀⠀⠀⡀⠀⠉⠀⢀⠀⠀⠀⢀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⡀⠀⠉⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠑⢄⡴⠈⠑⢄⡠⠊⠑⢄⡠⠊⠉⠢⣠⠆⠉⠢⣀⠔⠉⠳⣄⠐⠉⠑⢄⠴⠋⠑⢄⡠⠊⠑⢦⡠⠊⠉⠢⡀⠖⠉⠢⣀⠜⠉⠂⣀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢆⡀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⣀⡀⡀⠠⠀⠘⠤⠩⠂⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣀⡑⠀⡰⠅⠽⠿⠏⠡⣀⠀⠠⠄⠘⠛⠛⠃⠠⠆⣠⠄⠠⠀⠠⣉⣀⣀⣘⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡠⠐⠠⡄⠀⠀⠲⠖⠖⠦⠒⠶⠚⠒⠲⠖⠒⠖⠒⠖⠲⠞⣶⢲⣶⡒⢻⡀⠰⠄⠀⣠⠔⠉⠂⣀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⣠⣦⡀⠨⠀⠀⠀⠰⠆⠀⠴⠆⠛⠚⠛⠶⢄⢠⡤⢦⠀⡂⠀⠣⡀⢽⠉⣉⣉⡉⢮⠀⠀⠀⠀⡹⠂⠐⠀⠉⠀⠓⠘
⠀⠀⠀⠀⠐⠚⠐⠃⠘⠷⢘⠁⢘⠀⢶⣤⣴⠒⡤⠐⣆⠲⠊⢸⣍⢈⡁⠩⠒⠠⠁⡜⠁⠀⠀⠈⢲⠀⠗⢈⠂⣀⢰⣿⡆⠰⣖⢎⡱⠆⠲
⠀⠀⠀⠀⢙⠒⡖⢰⡆⠃⢸⠣⠼⠁⢼⣻⣿⠀⡅⠠⠠⠐⠒⠊⠀⢬⠛⣁⡂⣠⠛⠗⣀⣀⢀⣀⠘⠀⠷⠰⠂⠙⣉⣉⣉⠁⠟⡀⣁⠸⡆
⠀⠀⠀⠀⢘⡀⣆⢰⡆⣂⢰⡀⢰⣀⣸⣇⣿⡀⠛⠐⠛⣘⠛⣋⡀⣰⣰⣆⣀⣲⣀⡈⠋⠉⠙⠉⢘⡀⠛⠐⢂⣀⢛⣘⣙⢀⡂⣜⣫⣀⡇
⠀⠀⠀⡿⠼⠽⠯⠿⠬⠯⠼⠬⠧⠿⠽⠧⠿⠭⠧⠽⠽⠏⠿⠽⠧⠿⠼⠧⠿⠼⠧⠿⠼⠯⠿⠽⠯⠯⠼⠧⠿⠽⠯⠧⠼⠬⠷⠼⠦⠯⠽⠭⡇
⠀⠀⠀⡇⠈⠁⢀⠀⠀⠀⠀⡀⠀⠁⠀⡀⠀⡀⠀⡀⠀⠉⠀⣀⠀⠀⠀⢀⠀⠈⠁⢀⠀⠀⠀⠀⡀⠈⠁⠀⡀⠀⡀⠀⡀⠀⠁⠀⢀⠀⠀⠀⡆
⠀⠀⠀⠙⢄⠔⠉⠑⢆⠴⠊⠉⠢⡤⠊⠉⠢⡠⠞⠉⠢⣠⠞⠉⠲⣤⠔⠉⠑⢄⠔⠉⠑⢦⡴⠋⠙⢦⡴⠊⠉⠢⡤⠊⠉⠂⣤⠖⠉⠐⢄⠜⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡄⠀⠀⠀⠀⠀⢰⠀⠀⠂⠀⠀⣠⠞⠈⠱⡄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠒⡀⠒⡀⡂⠈⠀⠀⠤⠤⠤⠤⠤⢶⠐⠆⠀⠀⠀⠣⠤⠤⠤⠂⠀⠀⠀⠀⢀⡂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠈⠈⣣⠤⠄⢤⠀⡆⢘⡆⢀⣆⠰⠄⠀⠂⠀⠀⠀⠀⡄⠀⠈⠀⠴⠅⠈⠅⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠈⠉⠈⠉⡃⠀⠒⠀⠀⡅⢠⡀⣀⡘⠰⡂⠀⠈⠁⠉⠈⠁⡅⠀⡄⠐⡆⠲⠀⠀⣶⠀⡄⢀⡀
⠀⠀⠀⠀⠸⠂⡁⠘⠐⠋⠘⢻⢀⡈⠃⠈⠈⣃⢙⡟⢙⠀⠀⣀⢠⠀⢸⢠⡂⠀⠐⠂⠀⠀⡄⠠⠀⣥⠬⡥⡭⢤⢦⡄⠈⠀⢠⡇
⠀⠀⠀⠀⢘⠀⡇⢈⡇⠐⡇⢸⠁⣥⣤⣬⣤⡍⠈⡇⣉⣀⣁⣉⣸⡀⠸⢀⡀⣘⠃⣒⠚⠠⣤⢠⡤⡄⢈⣁⣈⣈⣀⣇⠰⠄⠀⠇⠀⠀⣀
⠀⠀⠀⠀⢸⠀⠰⠄⢀⠆⠀⢸⠀⠰⠤⠦⠤⠆⠀⣇⣉⣉⣉⣩⣘⣿⠉⡏⢹⡏⠀⢮⠀⠀⣄⢀⣀⠄⢨⣩⣩⣭⣈⡇⠠⠶⠴⠦⠴⠀⢸
⠀⠀⠀⣀⢼⠄⡜⢣⡤⣖⢀⠼⣄⣶⢶⡖⣶⠂⡤⢧⢀⡄⣡⢁⡴⣿⢠⡗⣼⢷⡄⣚⢠⡄⣤⢠⡄⡶⢀⡁⣅⢠⡠⡦⢰⠦⣖⢲⠾⣄⢼⢀⡀
⠀⠀⠀⡏⠩⠉⠉⠉⠉⠉⠉⠉⠉⠍⠉⠉⠉⠉⠉⠉⠉⠍⠉⠉⠉⠉⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⠉⠉⠍⠉⠉⠉⠉⠉⠉⠉⠍⠉⠉⠉⠉⠉⡇
⠀⠀⠀⢇⠀⢀⡠⢄⠀⠀⡠⢄⠀⠀⣠⠤⡀⠀⢀⠤⡀⠀⢀⠤⡀⠀⢀⠤⠀⠀⠀⡠⢄⠀⠀⡠⢄⠀⠀⡠⢄⡀⠀⢀⠤⡀⠀⢀⠤⡀⠀⢀⠇
⠀⠀⠀⠀⠑⠋⠀⠀⠑⠊⠀⠀⠁⠊⠁⠀⠈⠚⠁⠀⠈⠒⠁⠀⠈⠒⠁⠀⠀⠐⠊⠀⠀⠑⠊⠀⠀⠑⠊⠀⠀⠉⠂⠁⠀⠈⠒⠁⠀⠈⠒⠁

⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠿⠿⢿⠿⣿⣿⠿⠿⠿⡿⠿⠿⠿
⣿⣶⣶⣶⣷⣶⣶⣷⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣷⣾⣿⣿⣿⣷⣿⣶⣾⣿⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣾⣷⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⣿⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢟⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢰⡋⠉⠳⣄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠙⠓⠲⠾⠇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢾⣉⣀⣨⡽⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠿⠿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠙⣿⣿⣿⣿⣿⡟⠀⢀⣠⠤⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠐⠲⢦⠈⢿⡿⠃⠀⠀⠀⠀⠀⠀⣿⣧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢀⣿⣿⣿⣿⡿⠁⠀⠙⠷⢤⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠦⣤⠀⢸⣿⣿⣿⣿⠃⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⠟⠀⠺⠿⣿⡿⠁⠀⠀⠹⣍⠙⠳⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠉⠁⠀⠀⡈⢐⣿⠀⠀⠀⠀⠀⠙⠳⠴⠋
⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⠁
⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠋⠁⣀⣠⣈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣀⣉⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠉⠉⠉⢹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠙⢯⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣠⡴⠟⠛⠿⠛⠉⠀⠀⠀⣀⣀⢠⡿⠀⠉⠛⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢿⡟⠀⢦⣠⡼⠃⠀⣸⣿⡄
⠀⠀⠀⢰⡟⠉⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠙⠛⢷⣶⢤⠀⠈⠻⢿⣟⣛⠷⠶⢶⣦⣤⣦⡶⢶⣾⠛⠉⠀⢸⣧⠀⠀⠻⢧⣠⡞⠋⢀⡿
⠀⠀⠀⠸⣷⡆⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁⠀⡀⠀⠙⣧⡀⣠⡾⠁⠀⠀⠀⠀⠈⢳⣴⠟⠁
⠀⠀⠀⠀⠹⣧⣀⡄⠀⢀⡄⢀⣾⠃⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⣴⠟⠛⠁⠀⠀⠀⣰⡟⠛⠟⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣦⣄⣼⠷⠟⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠁⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⣤⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣴⢶⣄⠀⠀⠀⠀⣿⡶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢀⣴⠟⢁⠖⠙⢷⣄⠀⠀⢸⣇⠀⠀⠉⠙⠛⠛⠶⣦⣤⣄⣀⡀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⣠⡾⢁⡔⠁⠀⢠⡾⠛⢷⣄⡀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⢷⣶⣦⡾⠃
⠀⣠⡾⢋⣴⢋⣠⡏⢲⡟⠀⠀⠀⠉⠻⠿⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⣴⠋⡠⠋⠉⣞⣀⡹⠊⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠙⠛⠶⢶⣤⣤⣄⠀⠀⠀⠀⢸⡇
⠙⠿⢷⡴⠾⠛⠉⠀⠀⠀⠈⠻⣦⣀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠷⣦⣄⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠣⣄⡀⠀⣾⣉⡽⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣌⡑⠢⢄⡙⢲⠾⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣙⠳⢄⡀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣌⣿⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⠒⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣠⠖⠉⣠⠔⠁⣼⡡⠖⠚⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠘⠋⠀⢀⡞⠁⢠⠞⠉⣀⡤⠤⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⣠⣯⠴⠖⠛⠚⠊⠁⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⡠⠠⢄⡀⠀⠀⢀⣀⣼⠁⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠳⠤⠤⠼⠆⠀⣴⠟⠙⣦⠟⠁⠀⢀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠾⠤⠤⠴⠃
⠀⠀⠀⠀⠀⢠⠤⠴⣤⠀⠀⢰⡇⠀⡜⠁⠀⠀⠀⠸⠟⣋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⢻⡀⠀⠀⢠⡤⠤⢦
⠀⠀⠀⠀⠀⠘⠶⠚⠁⠀⠀⢸⠏⣸⡇⠀⠀⠀⢠⠖⠋⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⢀⣀⣉⠀⠈⣧⠀⠀⠀⠙⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠉⣇⠀⠀⢠⡏⠀⠀⠀⠀⣀⣹⡆⠀⠀⠆⠀⠀⣾⣁⣈⣳⠀⢿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠖⢻⡀⠀⠈⣷⠖⠚⠋⠛⠋⣼⠀⠀⠀⢦⠀⠀⠣⣌⡩⠋⠀⢸⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣆⠀⢧⠀⠀⠘⠦⣤⣀⣤⠞⠁⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⢸⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣠⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⡴⠋⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⡼⣿⣿⣶⣼⠇⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠰⠦⣌⣻⣿⡟⠀⠀⠀⢀⡴⠋⠀⠀⢀⣸⡁⣸⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣤⣄⡀⠀⢸⡆⠀⠀⠀⠀⠀⠈⠙⠁⠀⢀⡴⠋⠀⠀⠀⠀⡏⠀⢈⡇⢀⠎⢉⡆
⠀⠀⠀⢀⣠⠤⠤⠴⠚⣉⠴⠚⠉⠉⠉⣿⣉⠙⢻⡄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢳⠀⠙⠉⢙⡶⠊
⠀⠀⡾⠉⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠈⠉⣻⠋⠉⠓⠶⠦⠤⠴⠚⢋⠙⣦⠀⠀⠀⠀⢀⡴⠋⣠⠖⠀⠁
⠀⠀⠳⡞⠀⠀⠀⡀⢀⠇⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠻⢶⣦⣤⣶⡯⠴⠋⠁
⠀⠀⠀⠙⢾⣁⣸⠋⠉⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠉⠁⢀⣀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣰⠟⡹⣇⠀⠀⠀⠀⢰⡿⠦⣤⣤⣀⣀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢠⡾⢣⠞⢀⡼⡆⠀⠀⠀⣸⣯⠀⠀⠀⠉⠉⠙⠛⠳⠶⠦⣼⠃
⠀⠀⠀⣴⠟⡴⢁⣶⣿⡔⢻⡄⠀⣾⠉⠙⢧⣀⡀⠀⠀⠀⠠⢤⣤⣴⠋
⠀⣠⡾⣡⢮⠀⢠⡿⠁⠹⡄⠙⣛⡿⠳⡤⠞⠉⠛⠷⢦⣤⣀⣤⣶⣿⣧⣤⣀
⣴⢏⡔⢁⣠⡷⠋⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⢳⡆⠉⣷
⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠯⣴⣿⣶⡶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠢⠤⢤⡬⣉⣁⣀⣰⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠺⠶⠶⠦⣤⣭⣭⣉⣗⣻⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣤⣀⠙⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠟⢁⣠⣴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⣠⡶⠟⠛⢻⣄⠀⠀⠀⠀⠀⢀⣼⠛⠛⠷⣦⡀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠙⢷⣤⣤⣤⣴⠟⠁⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣆⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣼⠇⢾⣦⣤⣤⡀⠀⢀⣠⣤⣤⣾⠆⠻⣆⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠘⠿⠶⠶⠶⢶⣌⠹⣶⡟⣩⣴⠶⠶⠶⠶⠿⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⣿⡀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⣤⣤⣤⣤⡄⠀⣄⠀⠀⠀⠀⠀⢠⡀⢀⣤⣤⣤⣤⣤⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛




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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⢤⣤⣖⣒⣶⣮⣭⣭⣿⣿⣶⣶⣿⣶⣶⣿⣯⣭⣭⣴⣶⣒⣲⣤⠤⢤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢰⢸⣯⣭⣿⣷⣶⣾⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣶⣾⣿⣭⠽⡗⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢲⣤⠀⡆⠀⣰⣦⠀⢀⡔⠒⠆⠐⢶⠒⠂⣶⠰⡄⢀⡖⢰⣒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢸⠈⠻⡇⢰⠛⠚⢧⠘⢧⣤⠄⠀⢸⠀⠀⡿⠀⠻⡾⠁⢸⣭⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⣐⣲⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⣖⣂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠛⠻⠭⠽⠿⣛⣛⣻⢿⣿⠿⢿⣿⣿⣿⡿⠿⣿⡿⣛⣛⣛⠿⠭⠭⠙⠓⠒⠒⠊⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠒⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣿⣿⣿⣿⣿⣿⠈⣿⣿⣆⠀⡘⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⣿⣇⠹⡿⢿⣦⡁⠈⠻⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣆⠐⡤⠙⠛⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣄⠁⠸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢹⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⣀⣀⡀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣷⢦⣙⡻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣽⣿⣿⣿⣿⢽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢿⡸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠈⠛⠛⠿⠿⠟⠋⠀⢈⣶⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣾⡿⢿⣿⡿⠟⠛⠁⣴⡆⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⡇⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣰⢢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠿⠀⠀⠀⠐⣼⣿⣿⢻⠇⢸⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣀⡀⠀⠑⣽⠇⠀⠀⣸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣉⠁⠀⣼⡏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠰⡆⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠀⢠⣾⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣴⣶⣼⣿⣿⣿⠀⠀⡄⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠛⠋⠛⠚⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠋⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾
⠀⠀⠀⠀⠀⢸⣦⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠸⠿⠿⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠼⠿


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣦⣶⣤⣤⣤⣤⣤⣤
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⣄⠀⠀⢀⣠⣴⣿⣿⣷⡀⠀⠀⣀⣠⣤⠤⠦⠤⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣶⣽⣿⣿⣿⢿⣿⢣⣴⡾⡉⠨⠀⢀⡖⠰⠆⠇⢀⣦⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿
⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⡟⢀⣤⠆⣾⣯⡅⢉⡅⠺⠛⠛⠿⠿⢷⣾⣿⠷⠋⣤⣶⣆⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣳⣺⢷⣤⣯⣟⣰⣿⣿⢠⣄⣀⣀⣠⣠⣤⣤⠀⢈⣀⣴⠳⣿⣺⣯⣥⣷⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⢃⣻⣿⣿⣿⣏⣨⣿⣿⣯⢶⣾⣿⣿⣆⣀⡙⠛⢻⣦⢻⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⡿⠋⢞⣵⣾⣿⣿⣿⡿⣿⣿⣿⡿⠿⢡⣾⣿⣿⣿⣿⣿⣿⡄⢛⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⡵⢠⡿⣻⡿⠋⢉⣉⣡⠿⠿⠛⠛⣀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⡿⡿⠟⠁⠀⠀⠀⠛⠀⠀⠀⠈⠀⠀⠀⠀⠈⠛⣩⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠚⣻⣿⡿⣿⢟⣋⣩⣿⣿⠿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠈⠁⠉⠉⠀⠀⠛⠆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠛⡛⢛⣿⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⡞⠉⣧⣤⠤⠤⠤⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣇⡀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⢸⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⡀⠀⠀⣸⡀⠀⣸⡀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢳⢤⠗⠒⢚⡡⠝⠒⠒⠓⠞⠓⠚⠓⠛⠿⣇⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣻⠒⣦⠀⠀⠸⠀⠀⠛⠁⠀⠘⠃⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠓⠚⠳⡤⠜⠀⠀⠠⡂⢀⠀⢀⡀⠄⠑⠄⠀⠀⠀⢀⡤⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⡀⠀⠈⠉⠀⠀⢀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⠔⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠺⠤⠼⠶⠃⠀⠀⠀⢀⡠⠞⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣴⠃⠀⠀⠀⡀⡸⠧⣀⡀⠠⠤⠀⣀⣀⣀⣤⠤⠤⠖⠚⠉⢰⡦⣤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠋⠘⠧⣤⡤⠞⡿⠁⠀⠀⠉⠙⠓⠒⠒⡧⣤⣤⡤⠶⠶⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣠⠴⠒⠒⡆⡖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢱⡀⠀⠀⣧⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢣⠀⠀⢸⡄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢣⠀⠀⢇⠀⠧⢤⡀⠀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⡠⠄⠒⠳⣄⣉⣹⣏⣉⣙⣻⢩⡏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣾⣥⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⠁⣸⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠉⠓⠦⠚⠁⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠼⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠘⢛⠗⠀⠀⡴⠁⠀⠀⠛⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⠓⠂⠀⠀⠀⠀⢠⠖⠉⠉⠉⠳⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠙⠛⠓⠀⠀⠀⠀⠀⠀⢀⠴⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⡀⠀⢀⠴⠊⠁⠀⠑⢤⣀⣀⡤⠚⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⢀⠎⠀⠀⠀⠉⠀⠉⡶⠁⠀⡤⠀⠀⠀⠀⠘⡆⠀⠀⠰⡄⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠞⠀⠀⠀⠀⠀⢀⠞⠀⠀⡜⠁⠀⢀⡴⠀⠀⡇⠀⠀⠀⢳⡀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⢦⣀⡀⠀⠀⠀⡞⠀⢀⡜⠉⠒⢤⠎⠀⠀⢠⠃⠀⠀⣀⡼⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⠀⡇⠀⠀⢀⡎⠀⠰⠚⠁⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⡇⠀⠀⠈⢧⠀⠀⠸⡄⠀⡜⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢰⠃⠀⠀⠀⠈⢣⠀⠀⢱⣰⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣣⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠰⣅⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⣄⣀⡆⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⡏⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⣧⣴⡆⠀⠀⠀⢀⡭⠀⠀⠀⠀⠀⠐⠒⠒⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⡀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⢃⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⣄⣠⠴⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢇⣀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⢀⡀
⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠈⢻⠁⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⡤⠂⠀⠈⢿⣿⣿⣿⠋⠁⠈⠉⢻⡿⠿⠟⠃⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠈⠀⠀⠀⠀⠈⠛⠿⡇⠀⠀⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠉⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠀⠀⠀⠀⠀⠔⠈⢻⡇⠀⠀⣸⡏⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣇⠀⠀⣿⠀⠀⡄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢠⠀⠀⣿⠀⢰⡟⠀⠀⡃⠀⠈⠄
⠀⡀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢀⠀⢠⢸⠀⠀⢹⡇⣼⠇⠀⠀⠑⡠⠀⠈⢀
⠸⣿⣿⣿⣿⠿⠷⠾⢿⠂⠀⠀⠀⠳⠞⠈⠀⠀⠈⣿⡿⠀⠀⠀⠀⠐⠐⠒
⢐⣷⣒⣛⣛⣛⣿⣿⣿⣀⣀⣀⣀⣀⣀⣠⣗⣲⣀⣹⣃⣀⣠⣭⣀⣀⣀⣀⣀⣀⣀⣀⣾⡀⠈⠉⠉⠉⣀
⢘⣿⣿⣿⣿⣿⣿⣿⡟⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡆⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣭⣽⣿⣿⡤⢿⣯⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⡄⣠⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠠⢄⢀⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⡿⠿⡯⣿⣭⣿⣿⣿⡿⢻⣯⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⣿⣿⢿⡃⠀⠙⠁⠀⠀⢣⡍⢹⣿⠉⣃⣻⣻⣉⣉⣽⢿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠸⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣁⣽⠒⠁⠀⠀⠀⠄⠀⠀⠉⠀⣸⣿⣿⣿⣿⣿⣿⣿⣽⣧⣿⣽⣭⣿⣽⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⢿⣯⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢘⣿⡿⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⢲⣿⣦⡿⠿⣿⡿⠿⠟⠛⣡⠞⢡⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⢀⣠⣶⣾⡄⠀⠀⠙⠻⠿⠿⠿⠿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠊⢸⡛⡿⠧⠄⠀⣠⣤⣤⣎⣁⢠⠼⠁⢘⠋⣉⣩⣥⣤⡤⠔⠂⠁⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣟⣛⣿⣟⣿⣼⣿⣿⣿⣿⣿⣟⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣾⣿⣿⣿⣛⣿⡟⡙⣹⣷⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣧⣵⣼⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⡀⠀⠑⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠐⢄⢰⢄⡀⠀⠘⡄⠀⠈⡄⠀⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⢂⣀⠀⠉⡡⠌⠒⠲⢧⡠⢤⣰⠈⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠰⡄⠀⠐⠆⠀⠀⠈⠑⣠⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀⠀⠲⠀⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠮⡁⠀⠇⠬⠂⠀⠀⠀⣹⠋⠒⠤⣐⠢⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⡀⠀⠘⡝⠉⡹⠉⡇⠀⣰⠃⠀⠀⠀⢈⠳⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⢿⠿⡋⠀⠀⠀⠀⢨⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⠂⠋⠪⠧⠀⡇⠀⠀⢸⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⢠⠄⣀⡠⠅⠀⡇⠀⠀⢸⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠇⢠⠀⠀⠀⠹⢀⠀⠁⠀⠀⠈⠀⠀⠀⡀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠄⢃⠌⠙⡃⠀⠀⢘⠉⠣⡀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣶⠁⠈⡉⠩⠀⠀⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠸⣦⠤⠓⠀⠆⠀⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠈⢦⠀⠀⠈⡄⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡉⠉⢙⡓⠒⠒⠋⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠈⢹⠉⠉⠐⠚⠋⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠛⠐⠀⠄⠄⢀⠀⠀⡀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠌⠀⠀⠀⢀⡄⠀⠀⠀⠓⠄⠩⠄⣴⡏⢙⣦⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠈⣢⣿⣿⣿⣷⣤⠤⠄⠲⡆⠀⠙⠿⠚⠧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⡧⠤⠀⢰⣿⣿⣟⣿⣿⡿⠀⠀⠀⣈⠀⠂
⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⡀⠀⢁⡄⣼⢧⡀⠀⠈⣻⢿⣿⣿⠿⠛⠢⣄⢰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠈⠠⢦⠈⠁⠀⠀⠸⡀⠀⠠⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢆⢀⢀⡰⠐⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢲⡜⠁
⠀⠀⠀⠀⣠⡤⠖⠒⠲⠦⣄⠀⠀⣀⡤⠶⠒⠲⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠶⢤⣀⠘⠀⣠⠴⠖⠒⠲⢦⣄
⠀⠀⢀⡞⠁⠀⣀⣀⠀⠀⠈⠳⠞⠁⠀⠀⣀⣀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠄⠀⠀⠈⣳⠞⠁⠀⠀⢠⣀⠀⠈⢳⡀
⠀⠀⣾⠀⠀⠈⣡⠴⢤⡀⠀⠀⠀⠀⠀⡤⠦⢤⠉⠀⠀⢹⠀⠀⠀⠀⠀⠀⡏⠀⠀⠘⡡⢦⣤⠀⢴⡁⠀⠀⢀⣤⡦⣌⠂⠀⠀⣷
⠀⠀⣿⠀⠀⣻⡇⠀⠀⠹⠀⠀⠀⠀⠞⠀⠀⠈⣿⡃⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢘⣿⠁⣿⣿⡷⠀⠙⠆⠀⣿⣿⡇⢸⣟⠀⠀⣿
⠀⠀⠸⣆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡟⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣰⠇
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣀⠀⠀⣀⡴⠂⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣰⣊⣉⡦⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠈⠉⠉⠁⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠉⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣤⣤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣤⠴⠚⠋




⠀⠀⠀⣀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⠀⣀⣠⣤⣄⣀
⠀⡴⠋⠁⠀⠀⢀⠀⠉⢳⡄⠀⠀⠀⠀⠀⢀⡤⠴⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠉⠓⣄⡴⠋⠁⠀⠀⠀⠉⠓⣦
⢰⠁⠀⠀⠐⠒⠋⠀⢠⡏⠀⠀⠀⠀⣠⠞⠁⠀⢀⡀⠀⠀⠉⢷⡀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢀⣠⢠⠒⠠⣞⡀⠀⠀⠀⣄⡀⠀⠀⠈
⢿⠀⠀⠀⠓⠒⢶⣤⠀⠈⣳⠀⠀⠘⢧⡀⠀⠀⠀⠙⠒⠀⠀⠀⣷⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠀⠈⠙⠺⢥⣟⠑⢦⠀⠀⠉⠁
⢘⣦⣤⡄⠐⣚⠉⠀⠀⠰⡃⠀⠀⢀⡴⠃⠀⢶⣖⠒⠛⠃⠀⠀⣿⠀⠀⠀⠀⠀⢷⠀⠀⠠⣦⠤⠖⠀⠀⡴⠉⠉⠁⠢⠤⡴⠄⠀⢀
⢏⡠⢾⡀⣎⡟⠀⠀⢀⠖⢉⣉⣳⡜⢦⣀⠀⠀⠈⣛⡂⢠⣄⣰⠇⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⣀⣠⠤⠽⠶⠶⠶⠶⠶⠤⠤⣤⡞
⠀⠀⠀⠙⠦⣄⡀⠀⠘⠋⠉⣳⠀⢷⡦⢬⡇⠀⠀⢻⣱⣀⡷⣌⣳⠀⠀⠀⠀⠀⠀⠀⠙⢶⣋⠉⢁⣠⣤⣤⣶⣖⣒⡛⠛⣻⡿⠋
⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣤⡏⠀⣠⠇⠀⢀⣀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧⣤⣤⠤⢤⣤⡤⠾⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣭⣥⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠛⠉⠉⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡤⠶⠒⠒⠲⠶⣄⡈⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢰⠏⣠⣤⡶⣟⡳⢦⡈⢳⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣼⠀⣧⠘⢧⣄⣻⡼⣷⢸⡇⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠉⠳⣆⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠏⠀⠈⠓⠶⣤⡼⠷⠃⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣤⡤⠤⢤⣤⣘⡇⠀⠀⠀⠀⠀⢷⠀⠀⠙⣶⠀⠀⠀⠈⠉⠀⠀⢠⠏⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠈⢧⡀⠀⠸⣤⣤⡀⠀⠀⠀⠀⢾⠀⠀⢠⠟⠳⣄
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⢳⡀⠀⠀⠈⠳⣄⡀⠀⠀⠛⠀⠀⠀⠀⠘⣀⡴⠋⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⣿⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠙⠓⠦⠤⢤⠤⠤⠶⠛⠉⠀⠀⠀⠀⠀⠀⣘⣦⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⢸⠇⠀⠀⠀⣀⡴⠞⠁⣀⡴⠋⠀⠳⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⣀⣀⡀⠈⠙⠳⣄
⠀⠀⠀⠀⢸⡆⣠⠴⢛⣩⡤⠖⠛⠁⠀⠀⠀⠀⠈⠳⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⡴⠞⠉⠉⠉⠉⠙⠶⡄⠈⠳⡄
⠀⠀⠀⠀⠈⣷⡷⠚⣩⠴⠶⠶⠄⠀⠀⠴⠶⠶⢦⡀⠈⢿⡃⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣼⠁⠀⢠⠶⢶⡄⠀⠀⢹⡄⠀⢹⡄
⠀⠀⠀⠀⣼⠛⠀⠀⠡⠶⣾⠶⠀⠀⠀⠠⢶⣷⠦⠁⠀⡈⢻⡄⠀⠀⠀⢀⣀⣀⣸⣇⣀⣀⡀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢠⡾⠃⠀⠀⢸⡇⠀⠀⣷
⠀⠀⠀⠀⠸⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⡁⠀⠀⠀⢈⡀⠀⠀⠀⣼⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠙⢻⡀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⣸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⣿⠀⠀⠀⠈⠀⠀⠀⣸⠃⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠙⠶⠶⡆⠀⠀⠀⠀⢿⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢦⣀⠀⠁⠀⠀⠁⢀⣠⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠉⠀⠀⠀⠀⢈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠉⠳⢤⣀⣠⠴⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠉⠉⠉⠛⠲⢦⣄⠀⠀⠀⠀⠀⠈⢹⡷⠲⠶⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡼⠁⠀⣠⡴⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⠘⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⡞⠁⠀⠀⠀⢀⣙⣛⣂⣤⣤⣤⣴⠆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣆⣲⣾⣛⣋⣩⣭⣾⡿⠟⢋⡿⠁⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠉⠉⠹⡇⢶⣏⠁⠀⣴⠋⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠹⡶⠋⠙⠚⠁⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣄⣀⣠⣤⠴⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢄⠂⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⢐⠥⢢⠴⠻⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠐⠃⠲⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢰⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡀⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡈⠀⠀⠈⠀⠀⣰⣾⣿⣿⠛⠀⠩⢐⠀⠀⣾⠀⠀⢀⠄⠀⠙⢷⣧⡀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢿⠿⠛⠛
⠀⠀⠀⠀⢻⣆⡀⠀⢀⣾⣿⠟⠁⠀⠀⠀⡀⠤⠀⠈⢿⠀⠀⠀⢁⣵⠎⠘⣷⡉⠂⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠈⠻⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⣠⣴⠟⠁⠀⠀⠈⢻⣷⣤⣀⣀⣀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⡯⠟⠛⠉

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢
⠀⠀⠀⠹⣦⡀⠀⢶⣤⣀⣤⣦⣤⣴⣶⣤⣀⢀⢸⣦
⣤⣄⠀⠀⠘⢿⣶⣮⠿⠿⠟⠛⠛⠛⠛⠛⠛⠲⠿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣒
⠀⠀⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣷⣦⡀⢀⣀⣀⡠⠐⠋⠉⠉⠻⣧⣢⢄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡦⡄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⢄⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⢀⠰⠁⠀⠄⠀⠀⠑⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡄⠀⠀⠀⠐⢄⠈⠂⣄⠀⠀⠈⠀⠄⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠈⡀⠀⠀⠀⠀⠁⠀⠈⠆⠀⠀⠀⢂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢔⡈⠀⠀⠀⠀⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠
⠀⠀⠀⠀⢀⡠⢂⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠐⠀⠀⢀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⡀
⢖⠃⠀⠀⡄⠙⠀⠢⠀⠀⢀⡤⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⡀⠘⠀⠀⠑⠄
⠀⠁⠀⢀⡀⠀⢀⣠⠀⢐⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠀⠁⠡⠀⠀⠠⠀⠀⠀⡄
⠀⠀⠀⡴⢣⠚⠁⠀⠈⠀⠠⠂⠀⠀⠰⠀⠀⠀⠠⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠡⠀⠒⢀⡀⠀⠚⠀⠀⠋⠀⢀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣃⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣦⣤⣤⣤⣤⣴⣤⣤⣤⣤⣄⣁⣤⣤⣀⣀⣀⣀⣀⣀⣈⣠⣤⣤⣤⣤⣄⣁⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠷⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢰⡆⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠐⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠠⠾⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣼⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣾⠋⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣤⡴⠞⠋⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⣰⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡌⠛⠶⠞⠋⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣄⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⢻⣆⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⠹⣆⠀⠀⠀⠹⣆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣍⠉⠉⠉⠉⠉⢉⡍⠉⠉⠉⠉⠁⠀⠹⣧⠀⠀⠀⠹⣧⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡤⠾⠧⠀⠀⠀⠙⣧⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⣀⣀⣴⣟⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢰⠆⠀⠉⠛⠋⠛⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⣶⣶⣶⣾⠇⠀⠀⠀⠀⠀⠻⢶⠾⠃⠀⠀⠀⠀⠀⠸⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠢⡼⠁⠸⠖⠒⠛⠛⠓⠒⠒⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡆⠀⠀⢀⡀⠀⢠⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡇⠀⠀⢨⡇⠀⢸⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠰⣄⡳⠤⠔⣉⠶⠦⠶⠯⠒⠦⠚⠽⠶⡎⠑⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡳⢲⣄⠀⠀⡇⠀⠛⠀⠀⠐⠂⠀⠀⠀⠀⠒⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⠒⠚⢢⠤⠃⠀⠠⡠⢀⠀⢀⠠⠈⠀⠄⣀⠀⢀⡤⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢀⢤⡤⣄⠀⠀⠁⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡠⠂⠀⠙⠦⣀⠀⠀⠀⠀⠀⠓⠒⠛⠋⠀⠀⢀⣠⠴⠋⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣠⢾⠀⠀⠀⣠⠀⢀⡭⠒⠒⠒⠶⢤⡤⠤⣄⠒⠚⠉⢁⣸⠓⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠈⠓⠒⠚⠉⣹⢣⢖⡩⠛⠊⠉⢉⡹⠽⠮⢻⡝⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⠤⠒⠊⠉⠒⡤⢤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⣾⠁⠀⠘⢶⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡏⠀⠀⠀⠈⣇⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠀⠀⠀⠀⠀⠘⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠙⣌⠉⣷⣀⣰⣒⣀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣸⡉⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⡤⠟⠛⠉⠁⠀⠀⣹⣀⣠⠴⠋⢀⣤⠶⠋⠁⠀⠀⢤⣄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⡅⠀⠀⠀⠀⢀⣠⠞⠉⠁⠀⠀⣠⣞⡵⠚⠒⠦⣼⢧⡀⣻⠙⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⢉⡙⢷⣒⡲⠖⠒⠋⠉⠀⠀⠀⠀⠀⣼⡷⠟⣛⠛⠲⡀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠈⠉⣽⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢉⡴⠾⠝⠳⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⢠⡀⠀⢸⣿⢶⡄⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣴⣿⠃⡄⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⡏
⢠⡏⡼⢆⡼⠀⢸⡇⠀⠀⠘⢦⡀⠀⠀⢀⠀⠔⠘⣀⡴⠛⢁⡏⠀⠛⠺⡋⢉⡏⠀⢸⠁⠉⡇⠀⢸⡇
⠈⠙⠷⣟⠁⠀⢸⡇⠀⠀⠀⠀⠉⠓⠒⠶⠒⠚⠋⠁⠀⠀⣼⠁⠠⠒⢻⠙⡾⠧⣀⣼⡀⢰⠇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣦⣄⡀
⠀⠀⠀⠈⠳⣄⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣿⠁⠀⣰⠋⠈⡟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⡹⠆⣹⡄
⠀⠀⠀⠀⠀⠈⣷⣷⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠸⡄⡰⠃⠀⡼⠁⠀⠀⢸⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢻⠻⠖⢫⠟
⠀⠀⠀⠀⡖⠋⠉⣻⠗⠉⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢿⠁⢀⡼⠁⠀⠀⠀⢸⠋⣉⡉⢉⡽⠃⠀⠀⠀⠀⢸⠀⢰⠃
⠀⠀⠀⡔⠋⠒⢲⣇⣠⠇⣼⣠⢴⠄⠀⠀⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠸⡦⠞⠀⠀⠀⠀⣀⡼⠖⠋⠵⡋⠀⠀⠀⠀⠀⢠⠴⣤⠼⠦⡀
⠀⠀⠀⠳⡤⠤⠖⡯⠀⣤⣈⡿⠙⡆⠀⠀⠀⣴⠿⣄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠋⠁⠀⣠⠴⠊⠁⠀⢀⣀⠀⢠⠇⠀⢹⣀⡴⠓⣆
⠀⠀⠀⠀⠑⣴⣾⠟⣫⠋⣋⢀⡴⢣⣀⣀⡤⠥⠒⢻⡇⠀⠀⠀⣀⣤⡤⠖⠚⠋⠉⠀⠀⢀⡠⠖⠛⠧⡀⠀⠀⢠⡞⢉⢧⣸⠀⠧⡞⠁⣠⢴⠃
⠀⢀⣀⡤⠞⠕⡸⠀⠙⠋⠛⠘⣶⣻⣧⠴⢒⡾⠋⣇⣿⣀⣴⠾⣓⣪⠭⠷⣄⠀⢀⡤⠖⠉⠀⠀⠘⣦⡈⣶⣄⣯⠀⢻⢊⣡⢦⠀⣹⡋⣁⠜
⠘⠛⣧⢤⣀⣰⠃⠀⠀⠀⠀⠀⠈⠁⢀⡰⠋⠀⡞⠀⣀⡴⠛⠉⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⡷⢬⠷⣼⣘⡄⠈⠋⣧⠀⠙⠁⡿⡿⡄
⠀⢸⡏⠋⣼⠃⠀⠀⠀⠀⠀⣠⠴⠚⠉⢢⣀⡼⡷⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢦⠀⠀⠘⠒⠓⠃⠀⠀⢠⠇⠀⠙⢦
⠀⠀⠳⠞⠁⠀⠀⠀⠀⠀⠀⡿⠲⠤⣤⡴⠋⠘⠁⡿⠿⢍⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⡶⠋⠀⠀⠀⠀⠀⠀⠀⣠⣞⡴⢻⡭⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⠚⠁⠀⠀⠀⠀⣷⠤⣤⣤⣀⣀⣀⣀⣀⣠⡤⠤⠴⠖⠚⠋⠉⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣆⠀⠀⠀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠸⠒⣛⠏⠉⣿⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣾⡁⠀⠐⠚⠻⣤⣀⡾⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟
⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⠵⠿⠽⠗⠒⠒⠒⠛⠓⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⣖⣲⣲⡟⠛⠒⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠚⠛⠛⠛⠛⠓⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣠⠖⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⢦⡀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡞⠉⣿⣷⡀⠀⠀⠀⠀⠀⠀⢠⠞⢹⣿⣦⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⣹⣿⡇⠀⠀⠀⣦⠀⠀⢺⠀⢈⣻⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣸⣇⣀⣿⣿⣇⠀⢠⠖⠋⠀⠀⣸⣀⣸⣿⣿⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠘⠲⠶⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⣿⡇⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠘⠧⠤⠤⠤⠤⠤⠼⠇⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈






⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿