Undergarment Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⣷⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡄⠀⠀⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⢿⣒⣶⣖⣺⡇⣼⣶⠀⠀⠀⠀⠀⠀⢿⣷⡄⢀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⡿⠿⠿⣿⣿⣿⣿⠀⠀⠀⢀⣤⣀⣼⣿⣧⣼⣿⣿⣠⣤⡄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⠀⠀⣠⡾⡉⡀⠀⠀⠀⠀⠀⣴⣿⣦⡀⠀⣠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣻⣇⣸⣥⣿⣷⣿⠆⠀⠀⠀⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠰⡀⢀⡀⠀⠠⠀⣀⠀⠤
⠀⠀⢀⠀⠀⠀⠀⠀⣼⢿⣿⣧⡀⠀⣠⣾⡿⣷⠀⠀⠀⠀⣷⣶⣾⡇⠀⠀⠀⣿⣦⣼⡆
⠀⠀⢸⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣴⣿⣿⣷⣯⡆⠀⠀⠀⢻⣿⣿⡷⠾⠽⠶⢿⣿⣿⠁
⠀⠀⢸
⠀⠀⢸
⣠⣤⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣌⡂⡄⢠⠎⣄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⢣⣼⣧⡍⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⣉⠄⠺⡟⠻⣿⠉⠒⠄⠐⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡋⠔⠈⠀⡐⡐⠀⠀⢹⡇⠀⠀⠀⠀⠀⠨⢑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡫⠊⠀⠀⠀⠰⠷⠁⠀⠀⠴⠟⠀⠀⠀⠀⠀⠀⠀⠑⢍⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡠⢃⣀⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠔⠋⠀⠘⠢⠤⠃⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣖⠉⠁⡿⠉⠒⢬⡣⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠠⣎⢱⠄⠀⠀⠀⢀⠤⡈⢦⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠒⠊⠁⢀⡤⢤⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣸⠈⢆⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠘⢆⠼⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⡸⠀⠀⢀⡖⠒⢦⠀⠀⠉⠀⠀⠈⢧⣀⣀⣰⠋⠀⠀⣞⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠈⠧⡄⠼⠀⠀⠀⠀⢀⡠⠿⠄⠤⢿⡰⠎⠆⠈⠒⠊⠀⠀⠀⡤⠒⠢⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠤⠤⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠠⠤⠄⠀⠨⠇⠠⠔⠋




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠤⠤⠤⠴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣀⣀⠈⠉⠉⠐⠒⢲⡦⠤⠤⢤⡤⠀⠐⠒⠒⠒⠒⠒⠈⠉⠉⣠⠤⣄⠀⠀⠀⢠⠞⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⣄⣀⠗⠀⠀⠀⠀⠈⠣⣀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢀⡇⠀⠀⠈⠒⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠢⣄⠀⠀⠀⠀⠉⠈⠉⢀⣀⠤⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠴⠋⠀⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⡠⣄⡀⠀⠀⠀⠀⠀⢠⡖⠒⢦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⣹⠀⠀⠀⠀⠀⠸⣄⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡚⠁⠀⣀⡀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢼⡁⢸⠆⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠁⢀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⠛⠻⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠛⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠐⠀⠄⠀⠌⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⢿⣆⠀⠀⠀⣠⣿⠏⠁⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⠂⠐⠠⠁⠀⠀⠀⠀⢻⡷⣄⢰⡟⠁⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠄⠀⠂⣼⣷⣿⣿⣧⡀⠂⠀⠀⠐⠀⠀⠀⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⣀⣄⣀⣀⣡⣶⣿⠿⠟⠛⠻⠿⣿⣦⣥⣀⠀⠀⠀⢀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠟



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣦⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⡶⠾⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⢂⠈⠈⠀⠀⠈⠀⠃⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠷⠾⠶⣤⣀⡀⠅⠒⠀⠐⠈⠀⠀⡀⠀⠈⣠⣴⠶⠛⠛⠛⢛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡁⠀⠂⠀⠉⠛⢦⡠⠀⠀⠀⢀⠐⠀⣠⠞⠁⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⢀⠂⠈⠀⠈⢻⡆⠂⠁⠄⠂⣼⠏⠠⠀⠄⡐⠈⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡀⣷⠀⠀⠀⢰⡟⢈⠠⢁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡠⠀⠂⢻⣄⢀⡠⣹⡇⡐⠀⢐⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⡾⣮⢖⠿⠚⡇⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠷⠷⠦⠽⠛⠁

⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣶⣶⣶⣶⣖⣒⣒⣒⣒⣚⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣒⣒⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣸⠷⠶⠦⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠴⠾⣇
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡿⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣆
⠀⠀⢰⣏⡙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢷⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⢉⣹⡆
⠀⠀⠈⢿⠻⣿⠶⣬⡙⠳⣦⡀⠀⠀⠀⠀⣾⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⣧⠀⠀⠀⠀⣠⠶⠋⢁⣤⠶⣿⠟⣽⠁
⠀⠀⠀⠈⢳⡜⢷⡀⠙⠳⣄⡙⠳⣄⠀⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⣇⢀⣴⠟⢁⣠⠾⠋⢁⡼⢃⡼⠁
⠀⠀⠀⠀⠀⠻⣦⠻⣦⡀⠈⠻⣦⡈⠻⠏⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠹⠟⢁⣴⠟⠁⠀⣠⠟⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠈⢳⣌⠳⣄⠀⠈⠻⣦⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣠⡞⠁⠀⢀⡾⢋⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠳⣄⠀⠈⢷⡄⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⣼⠋⠀⢀⡴⠋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣌⠻⣦⡀⠹⣆⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡾⠁⣠⡶⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣙⠷⣼⣧⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⣿⣥⠞⣩⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢮⣙⣧⠈⣷⠀⠀⠀⠀⠀⠀⢠⡟⢠⡿⣫⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠾⣧⣤⣤⣤⣤⣤⣼⠷⠾⠛⠁

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢸⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢹
⠀⠀⠀⢸⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢸⠀⠀⠀⠀⠀⠰⠂⠉⠉⠉⠓⢄⠀⠀⠀⡠⠒⠈⠉⠁⠰⡀⠀⠀⢠⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤
⠀⠀⠀⠀⣇⠀⢱⠀⠀⠀⠀⠀⠀⢰⠅⠀⡟⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠆⠀⡘⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⢸⣤⠃⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢀⠀⠀⠀⢀⡜⠉⢣⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢋⠀⡆⠀⠀⠊⣯⠑⠀⠀⡄⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠉⠒⡖⠉⠀⠀⠀⠉⢑⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠧⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢡⠀⠀⠀⠀⠠⢶⣶⣂⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣶⣶⡄
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠪⡀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠰⣤⣼⠀⠀⠀⠀⠀⠀⠀⢸⣤⡾⠆⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠄⢀⣀⠀⠤⠀⠂⠉
⠀⠀⠀⠀⠀⠉⠹⠿⠿⣶⣶⣶⣾⠿⠟⠉⠁
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢸⠀⠀⠀⠀⡏⠉⠛⠒⠒⠲⠶⠦⣴⠶⠶⠖⠖⠚⠛⠋⣧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠀⠘⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠀⠀⠀⠀⠢⡀⠀⠀⠀⢀⠄⠀⠀⠀⠘⠀⠀⠀⠰⡧⠠⣄⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⣠⡤⣾⠀⠀⠀⠀⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠘⢦⡀⠙⢄⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⡜⠁⠀⠀⠀⢀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸
⠀⠀⢠⠀⣠⣄⡤⢀⠀⢀⠗⠀⢀⠀⢤⣀⣄⠀⠂⠀⠀⠀⠀⠀⠱⣄⠀⠱⡀⠀⠀⠀⡴⠁⢀⡴⠋⠀⠀⠀⠀⠀⣸⡀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⡆
⠀⢀⡼⠋⠁⠀⠀⠀⠘⣧⣠⣾⠃⠁⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠈⠳⢄⢐⠀⠀⡜⣠⠔⠁⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠉⢯⡁
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠸⠛⢅⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠒⣄⣠⡔⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠚⠉⠀⠀⠙⠣⠤⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣴⣒⣠⣤⣮⣤⣤⣤⣔⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠛⠙⠛⠋⠉⠁
⠀⠀⠀⠀⣀⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠒⠆
⠀⠀⠀⠠⠋⠀⠑⡄⠀⠀⠀⠀⠀⠀⠔⠁⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⡀⠔⠃
⠀⠀⣠⠁⠀⠀⠀⠈⢆⠀⠀⠀⢀⠎⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⢇⠀⠀⠀⠀⠀⠈⣄⠀⢀⠆⠀⠀⠀⠀⠀⢈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣆⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠉⠂⠤⠀⠀⠴⠒⠒⠚⠦⠀⠀⠠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠋

⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⢠⠒⠄
⠀⠀⠀⢴⠈⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠐⠀⠀⠠⡀⠰⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈
⠀⠀⠀⠠⢄⡀⠀⠀⠀⢀⣨⠀⠀⠀⠀⢀⣞⡂⠄⡀⠉⢂⡄⣂⣹⣆⠀⠀⠀⢀⠂⡜⠁⠀⠀⠀⠀⠲
⠀⠀⣠⣾⣿⣿⣆⣀⣼⣷⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⡶⣿⣿⣿⣿⣿⠀⠀⠀⡦⠎⠑⠀⠀⠀⡠⠘⠂⠠⡀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠘⣿⣿⣿⣟⣁⣘⣿⣿⣿⡿⠀⠀⠀⣅⠀⠁⠀⠀⣶⡀⠀⠀⠁⡅
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣽⡿⠀⠀⠀⠀⠁⠀⠃⠈⡁⠉⠁⡆⠀⡆⠀⠀⠀⠈⠳⠶⠶⠚⠁⠙⠓⠞⢻⠁
⠀⠀⠀⠻⡟⠛⠏⠻⢉⠿⢿⡇⠀⠀⠀⠀⠀⣀⣸⡀⠐⠂⢠⡅⣸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠠⠀⠄⠎
⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣴⣒⣒⣒⣒⣒⣒⣒⣒⣢⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⢠⠄
⠀⠀⢀⣿⠦⠠⢀⣀⣀⢄⡶⣹⡗⠀⠀⠀⠫⣉⣶⣶⣶⣷⣶⣎⢉⡹⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠙⢿⠻⣿⣿⣿⣿⣩⠞⠁⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⠟⠉⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠉⢻⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠑⠂⠐⠄⡀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠂⠄⠀⠀⠤⠊⠠
⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠨⠁
⠀⠀⠈⠛⢦⡄⠀⣠⠖⠋
⠀⠀⠀⠀⠀⣿⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠏⠙⠒⣄⣠⠒⠏⠹⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⡄⠀⠀⠀⠘⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁
⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠈⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣴⠛⢑⡗⣺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢒⠙⢒
⠦⠷⠾⠿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣞⠀⠱⣴⣷⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠶⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⠿⠛⣿⠻⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⡀⠑⡴⠁⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠶⣶⣆⠰⣶⣶⣶⠖⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠃
⣀⡀⡀⢀⣠⡄⠀⠀⠀⠀⠀⠱⠰⠇⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⢀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠻⠉⢓⡤⡖⠛⢹⢓⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⣷⡇⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⠯⠴⠼⠶⠿⢟⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣧⣤⣤⣼⣷⡶
⠀⠀⠀⠀⠀⠀⠿⣿⣧⡄⡀⠈⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠶⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣒⣆⡀⠒⢲⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡎⠀⠰⠎⠀⠡⠀⠀⠀⠀⠀⠀⠙⠛⠿⠏⠛⠛⠀⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠶⠃⠻⠲⠧⠨⠄⠰⠂
⠀⢤⣤⣀⣀⣀⣀⣀⣀⣀⡤
⠀⠀⠉⢿⣦⣿⣿⣦⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠤⠤⢤⣤⠤⠤⢤⣤⡄
⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣷⣦⣄⣴⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣽⣿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠿⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀
⣿⣿⣿⣿⣟⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣿⣟⣟⣿

⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡴⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢆⠀⠀⠀⢀⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡄
⠀⠀⢠⢁⣀⣈⡓⢄⠀⠀⠀⠀⠀⢀⡤⢊⣁⣀⣸⠀⠀⠀⣸⣀⣀⣉⠢⣄⠀⠀⠀⠀⠀⢀⠔⣋⣀⣀⣣
⠀⠀⠸⠛⠛⠛⠿⣷⣵⣄⠀⠀⣠⣧⡾⠟⠛⠛⠻⠇⠀⠀⣿⣿⣿⣿⣿⣮⣧⡀⠀⢀⣴⣵⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣦⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣄⡀⠀⠀⣀⣴⣿⣶⣾⣷⣦⣀⠀⠀⢀⣠⠃⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠯⠉⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠩⠟⠀⠀⠀⠸⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠇

⠀⠀⢐⣒⣲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣒⡂⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀
⠀⠀⣠⣀⣀⣈⣉⣿⣿⣿⣿⣿⣿⣿⣿⣉⣁⣀⣀⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁⠀⠀⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋

⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡸⠥⣤⣀⣀⣉⡉⠉⠉⠉⠉⠉⠉⣉⣁⣀⣠⡤⠷
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢃
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢮⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠆
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠳⣝⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⠝⠁
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠈⠑⢌⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁
⠀⠀⠈⠙⠪⢄⣂⠠⢀⡀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⢀⠠⠄⣒⠬⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣆⠀⠀⠀⠀⠀⢠⡮⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠪⠵⣤⠊⠀⠀⠀⠀⢣⡴⠮⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢠⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⢤
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⠋⠛⠓⠒⠒⠒⠶⠾⠿⠭⠭⠵⠶⠖⠒⠒⠒⠛⠉⡇
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠑⣍⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢋⠜⠁
⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠈⠳⡀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⡴⠃
⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠈⢢⡈⢆⠀⠀⠀⠀⠀⢠⠋⡰⠋
⠀⠀⠀⠀⠉⠒⠭⢐⡂⠄⡀⠀⡠⠄⠀⠀⠤⡀⠀⣀⠤⣂⡨⠕⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡆⠀⠀⠀⢀⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠒⠁⠀⠀⠀⠀⠐⠚⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠷⣠⣞⠾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠠⠀⠟⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡠⠀⠀⡷⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡢⠆⠰⣀⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢠⠤⢈⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠂⠀⠀⡠⢌⢆⠀⠀⠀⠀⠠⠀⠈⠂⠊⢀⠤⡙⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠀⠀⢀⢄⠀⠣⠜⠈⢆⠀⠀⡀⠀⡠⠤⡀⠀⠈⠒⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡀⠀⠣⠀⠃⠀⠀⠀⠸⠉⠹⡠⠂⠀⠀⠁⠀⡠⠀⢄⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠓⠂⠉


⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠤⡈⠉⠀⠐⡆⠀⢲⠂⠀⠀⠀⠀⠀⠉⡤⠠⡄⠀⠰⡊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠧⣉⡀⠁⠀⠀⠀⠉⠂⠁⠀⠀⢠⢄⠀⠀⠀⠓⠐⠃⣀⡠⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⣠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠂⠒⡄⠀⠀⠀⢎⢀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡚⠁⣀⡀⠀⢈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠘⠤⠋⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣿
⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⡏
⠀⠀⠀⠈⢆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜
⠀⠀⠀⠀⠌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡄
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿
⠀⠀⠀⠀⠀⢿⢁⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣿
⠀⠀⠀⠀⠀⢸⡌⢳⠿⡴⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠛⠙⠂⠋⠛
⠀⠀⠀⠀⠀⠀⢷⠂⠀⠀⠀⠀⠈⡑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⢆⡀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠑⡀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⡀⠀⢢⣯⡀⠀⣄⠀⡄⣀⢀
⠀⠀⠀⠀⠀⠀⢠⡄⠐⠘⠂⠀⠈⠁⠀⢈⠉⢁⠉⡉⠀⢀⡀⠀⣀⢀⡉⠉⠀⠚⠐⠓⠓⢄
⠀⠀⠀⠀⠀⠰⠤⠤⠐⠀⠒⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠀⠈⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⣀⣠⡄⠤⣀⡀⠤⣠⡀⢤⠄⠤⠀⠄⠲⠀⠒⠒⠂⠒⠉⠀⠁⣠⠶⠚⠰⠆
⠀⠀⠀⠀⠀⠀⠖⠤⢤⢉⡉⠀⠂⠠⠀⠤⠄⠠⣀⠀⠀⠤⠄⠀⠒⠐⠀⣀⡩⠤⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠉⠓⠀⠲⠒⠰⠄⠤⠠⠤⠀⠖⠒⠪⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⢀⠂
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠
⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡀⠾⠄⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠒⣚
⠁⠛⠓⠢⠤⢀⡈⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠀⣁⣠⠠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⣀⠈⠲⢄⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⢀⠠⠔⠚⠁⢀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡁⠺⢆⡀⠀⠘⠂⠠⡜⠂⣀⡠⠠⠤⠊⠁

⠀⠀⠀⠀⢠⡄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡇⠑⠀⠀⠀⠠⠁⠁
⠀⠀⠀⠀⡏⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⢸⢹⡀⠀⠀⠀⢀⡞⡃⠀⠀⠀⢰⢹⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠛⠄⠀⡀⢀⠂⠀⠢
⠀⠀⠀⠿⠇⠀⠀⠀⢀⠘⠣⡀⠀⠀⠀⣼⢚⣓⢦⣤⡖⣚⢓⣷⠀⠀⠀⢨⠘⣇⠀⢠⠟⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠄⠐⢀⠁
⠀⠀⠀⠀⠀⠲⠀⠀⠘⠀⠀⠰⠀⠀⠀⢩⣍⣈⡼⣻⣆⣀⣨⡟⠀⠀⠀⠊⠀⠘⣆⡜⠀⠙⠀⠀⠀⠀⠀⠀⢀⠀⠡⠰⡇
⠀⠀⠀⠀⠀⠀⢡⡐⠀⠀⠀⢀⠀⠀⠀⠀⠈⠻⣬⠿⠛⠛⠉⠀⠀⠀⠀⠐⠀⠀⠾⠇⠀⠘⠀⠀⠀⠀⠀⢀⠎⠛⠒⠚⠓⠚⠶⡀
⠀⠀⠐⠂⠂⠀⢈⡃⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⢫⣖⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢠⠈⠈⠀⠒⠴⠀⠁⠠⠼⠀⠀⠀⡼⣄⠀⢸⣿⡇⠀⠀⠘⢢⠀⠁⡞⠀⢐⡆⠈⡓⠈⣅
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠠⠀⠒⠀⠀⢐⠁⠀⠉⢻⣿⡇⠀⠐⢀⠤⠂⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⢄⢠⣤⡄⢀⠀⣀⠀⡀⢠⠀⡄⠰
⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠘⠋⠃⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠀⠀⠈
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠐⠲⢦⠀⠀⠀⠀⠶⠊⠁⠀⠀⠠⠄⡀⠀⠀⠀⠀⠠⡄
⠀⢻⢀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠠⡀⠀⠄⠀⠀⠀⠀⢀⠂⢀⡀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⣰⢆⠀⠀⠀⠀⠀⣶
⠀⢸⣀⣆⠀⠀⠀⠀⠀⣀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠓⠘⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠘⠉⠀⠀⠀⠀⠀⠀⣿⢸⡄⡀⠀⠀⢸⣿
⠀⢸⣏⠁⠉⢢⠀⡐⠀⢀⠜⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢈⡆⡰⠉
⠀⠈⣌⠓⠢⢶⣖⠶⠚⠁⡰⠀⠀⠀⠀⠀⡇⢱⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⠤⠔⠋⠛⠤⠴⠁
⠀⠀⢈⣁⠸⠄⠽⠀⠼⢠⡀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠁⠸⠇⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢰⢃⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡆
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢄⠹⡀⠀⠀⠆⠀⠠⡄⠀⠀⠠⠀⠀⠈⢂⠀⠔⠀⠏⢸⠀⠀⠀⠀⠁
⠀⠀⣼⢄⣀⠀⠀⠀⣀⠠⠤⠀⠀⠀⢘⠀⠀⠑⠌⠒⠀⠀⠀⠀⡇⠀⠀⠀⣆⠀⠑⣾⠻⡗⠋⢀⣾⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰
⠀⠀⠻⢘⡉⠥⠒⠉⠀⢀⡠⠆⠀⠀⠀⠑⠒⠒⠊⠉⠉⠉⠓⠋⠀⠀⠀⠀⠍⠛⠟⠁⠀⠙⠻⠟⠅⠀⠀⠀⠀⠙⠢⣄⠀⠀⡴⠊
⠀⠀⠀⠀⠙⡄⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⡤⢀⡀⠀⠀⠀⣀
⠀⠀⠠⠠⠀⠀⢀⣀⣀⠀⠤⠀⠀⠀⠀⠀⠶⠒⠂⡆⠄⣀⡲⠒⠆⠀⠀⢀⠀⡈⠋⠭⠩⠍⠫⣀⠠⠀⠀⠀⡈⠁⠒⠬⢽⣒⠯⠔⣃
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢀⠀⠀⠀⠐⢦⣄⣸⣯⠀⢸⣣⡴⠚⠀⠀⠀⠈⠉⠢⡀⠀⣠⠊⠀⠀⠀⠀⠀⢀⠙⠂⠆⣀⠠⠔⠋⠁
⠀⠀⠑⠀⠤⣄⡠⣄⡠⠤⠖⠚⠀⠀⠀⠀⠀⠀⠙⠻⠄⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠄⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡍⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠑⠛⡇⠀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠀⠐⠶⣦⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠈⣿⡤⠔⠀⣈⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠠⢶⣾⣯⠙⠿⠿⠒⠢⢤⣤⣽⣿⣿⣾⣛⣿⡦⢀⣼⣿⡆⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣂⠈⢀⣩⣿⣿⣿⣿⣿⡍⠙⢿⣿⣿⣿⣇⡔⠀⢀⠠⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⡟⢻⣿⣿⣿⠏⠀⢀⣾⣿⣿⣿⣿⣿⡟⢛⠤⣀⣄⣈⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣅⣈⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⠻⠿⠟⠀⠀⠘⢿⣿⣿⣿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠪⠻⣧⠠⠀⠀⠀⠠⢆⣈⣿⣿⣿⣷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⠄⠀⡜⢢⠀⠀⠀⠙⠻⢩⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠄⠀⠘⠁⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⡃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠠⠾⣿⣿⣿⣿⠀⠁⠘⣷⣶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⣀⣠⣿⣿⡿⠇⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣿⣿⠿⠿⣿⣿⣿⣿⠁⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠙⡄⠀⠿⣷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⡏⡂⠀⠀⠈⠻⢇⡀⠘⢀⡿⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⡏⠢⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠑⢤⣼⠇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⡳⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢊⣿⣿⣿⠷⢶⣦⣀⠀⠀⠀⠀⠀⠹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢱⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⣾⠟⠋⠁⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⡙⢷⣄⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⢷⡄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⢾⡿⠿⢿⣿⣟⢿⣿⣿⣿⡟⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠳⢍⠙⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⡿⢿⣿⣿⡟⠿⣿⡿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠞⠁⠀⠀⠙⠻⣿⣷⣦⣍⣙⡒⠾⢭⣤⣈⡙⠲⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣝⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠋⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣬⣍⣙⠓⠲⢮⣭⣳⢮⣷⣵⡝⣿⣿⣿⣿⣿⣿⣿⣿⣦⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠭⢷⣒⣶⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⠓⠂⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣶⠈⠀⠹⣿⣿⣿⣿⣿⢧⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠰⠿⠋⢁⠀⣀⣀⣀⡀⢀⡠⠤⠒⠉⠁⠙⣿⣿⣿⣿⣧⡐⢀⣿⣿⣿⣿⣿⣷⡁⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢰⡶⠿⠗⠀⣠⠊⠉⠀⠉⠻⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠸⡟⠘⣿⡞⣿⣾⣿⣿⣿⣿⣿⣿⣷⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣷⠬⢄⣀⢠⣀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣀⡼⡟⢺⢋⠳⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⠀⢃⠀⡇⠀⠙⠢⢄⣀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣆⠀⠈⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⡇⡿⣸⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢠⠉⠉⠑⡄⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡇⠁⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣏⠄⠀⠀⠈⢆⢈⡤⠴⠒⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡟⣦⠀⠀⠀⠀⠡⡄⠐⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡏⢿⣿⣿⣿⣿⡟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢱⠘⣷⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⣠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠇⢸⠻⣿⣿⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡇⠈⢿⣄⠀⠀⠀⠀⠈⠂⠀⠒⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⣀⣤⣖⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣷⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣌⣿⣯⣟⣹⣬⣿⣷⣿⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⣿⣿⠿⠛⠻⠻⠿⠿⠿⠿⣿⣿⣿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡧⢶⣶⣦⣀⣀⣠⣶⣶⣖⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣈⣻⣿⣿⡏⣒⡀⣿⣿⣿⣿⣿⣿⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡹⠟⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠦⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡇⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⣅⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣍⣁⠲⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣆⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢰⣿⣿⣦⣀⠀⣰⣟⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇

⡇⠀⠀⠀⡼⢁⡀⢀⠀⠄⠈⠁⢀⣠⣶⣾⡿⠿⠿⢛⣛⣯⡽⠟⠁⠀⠀⠀⠀⠀⣬⡛⢯⡿⠶⠾⣿⠯⣽⣿
⡇⠀⢀⡼⠛⡞⠀⣀⡭⠅⣠⠌⢉⣉⣡⣤⣤⣶⣿⣿⣿⡏⠀⢀⣀⠀⠀⠀⠀⠀⢨⣙⡧⣾⣿⣿⢹⣿⣿⡟
⣇⡴⢁⡖⣉⣅⣉⢀⡀⠀⠤⠶⢟⣻⣭⢾⣿⣿⣿⣿⠻⠇⠀⠈⣭⣿⡋⠁⠀⠀⠈⠻⠟⠋⢹⣿⣾⣿⠋
⡷⠘⠋⠀⢁⣤⣤⣤⣤⠐⣶⣿⣿⣯⣾⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢿⡏
⣇⡀⠀⣀⣠⢟⡴⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⡏⠁⢴⣯⣿⣷⡞⡻⠋⣰⣿⣿⣿⣿⢿⠟⠋⣰⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠁
⡧⠚⠋⡹⣩⢋⠀⣠⣾⣿⡟⢻⠹⡟⠓⢀⣼⣿⣿⣿⣿⣿⠃⠀⠲⣄⠀⠀⠁⠀⠀⠀⢰⣿⠋
⡇⡠⢀⢞⣥⡵⠾⠟⢻⡟⠀⡌⢰⣇⣴⣿⣿⣿⣿⣿⡿⠃⠀⠱⡄⠀⠙⠂⠤⠤⠀⠊⢸⢇
⣏⢀⡴⡙⠋⠀⢁⠎⣼⠁⢰⢃⣿⠼⣿⣿⣿⣿⣿⣏⠤⠤⠤⠄⡈⢦⠀⠀⠀⠀⠀⠀⢸⣷⡱⡀
⣇⠚⡜⠀⢀⡴⠃⡼⠃⣠⣿⠋⠄⢠⣟⣬⣿⡿⠉⣀⠤⠄⠀⠠⢌⣻⡳⣖⠲⠄⠀⠀⢸⠀⢻⣷
⡏⢰⣄⡠⠊⣠⠴⡁⠚⣰⠃⡴⢚⣫⣵⢿⡿⠖⠉⠀⠀⠀⠀⠀⠀⠙⣷⡈⠳⣄⠱⡀⠸⠀⠀⠻⣆
⣇⣾⠋⣠⡞⢁⡞⢀⡟⠃⢀⡴⢚⡭⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣷⡀⠸⠷⣕⠀⡀⠀⠀⠙⣧
⡟⠡⢂⡼⠀⢋⡠⠋⢀⠔⣉⠔⣷⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣷⣄⡀⢘⣷⣇⠀⠀⠀⠘⢷⡀
⡇⠹⢉⣴⠞⠋⠀⡰⢃⢒⣵⡠⢚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡽⣧⢈⡙⠷⡿⣆⠀⠀⠀⠘⣿⣄
⣧⠴⠋⠀⠀⣠⠞⣠⣾⠟⣩⠃⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣆⡿⠂⢡⡌⡳⡀⠀⠀⢸⡟⣦
⡏⠀⢀⡴⣾⠟⣨⢛⡡⠞⡠⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⠷⡀⢸⣿⣟⢞⣆⠀⣸⣏⢻⡆
⣇⠔⣡⠾⠗⠈⢸⢁⡴⠊⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢋⣷⢄⣼⣾⠉⠹⣟⣾⣷⡏⠘⣮⠇
⡇⢠⠁⠀⣠⠟⣀⣿⣤⣡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢠⣿⢋⣥⣾⡏⠹⡽⣆⠀⣿⣾⣿⣇⠀⡽⠁
⡧⢡⠴⣽⠃⢭⡽⠋⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣼⣷⡄⢿⣿⣇⠈⢻⣽⣦⠘⣿⣿⡿⠚⠁
⡗⠀⠀⠀⠐⠣⠄⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢀⡼⢋⠻⣧⣼⣿⣿⠀⣿⣿⣿⣿⣿⡿⠃
⡇⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⡴⠟⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡁
⡇⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⢶⠶⠞⠛⠛⠶⣽⣄⣀⣀⣂⣬⣼⣿⣿⣿⣯⣷⡹⣧
⡇⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢻⣿⣿⣧⡠⠀⠆⠀⡀⠉⠻⣛⠛⠉⠭⠲⠶⠶⣉⣭⣝⢻
⡇⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢻⡿⣿⣿⡷⣶⠾⠀⠀⠀⠀⠈⠑⠂⢈⡑⢮⡿⠷⣿⢿⡆
⡇⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⣷⣾⣿⡖⣯⣾⢦⠀⢀⠀⠀⠀⠀⢺⠻⣦⠙⢷⡐⢾⡇
⡷⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣤⣎⢀⣀⠀⠀⠀⢐⣼⡇⠀⢹⣾⠁
⡇⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⠟⠉⣼⣷⣽⣇⣠⢤⡀⠀⠊⠹⣯⣦⡀⣿
⡇⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣽⣷⣶⣷⡿⣿⢋⠎⡔⠘⣀⢠⡀⠀⠑⢇⡿
⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣉⢻⡿⠟⣊⠵⣫⠞⣠⡬⠤⠀⠵⡴⠄⠨⡇
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⠈⠙⢿⣛⠛⠴⠎⠤⢚⣡⡤⠓⠤⢎⣻⣨⡄⣷
⡇⠳⡄⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠈⢿⡀⣀⠃⠖⣿⣷⣧⢃⠀⠀⢸⣿⡇⡿
⡇⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠈⢧⡶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⣯⡶⠋⢐⣿⠸⢸⠀⠀⢸⣿⣷⡇
⡇⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠈⠳⣷⠀⠀⠀⠀⠀⠀⢀⣠⡾⡿⢺⠗⠐⠤⢤⣉⣄⢸⠀⠀⣾⢿⣿⠃
⡇⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠙⢧⣤⣤⣤⡴⠞⣫⣽⣄⣷⡘⠀⠈⠔⣠⣿⣿⢶⡃⢠⡟⣽⡟
⡇⠀⠀⠀⠀⢀⣴⣉⣳⣦⡀⠀⠀⠀⠀⠀⠉⠳⢴⣄⣠⡙⠿⣏⠻⢻⣶⣶⣾⣿⣻⣼⣆⣱⡞⣼⣿⠁
⡇⠀⠀⠀⠀⣾⠉⠉⠙⢿⡿⣷⣄⡀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢶⣉⠋⣹⣿⣯⣿⣿⣿⣏⠰⢿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣦⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⢻⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⠓⠯⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣷⣤⡀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡷⠂⠀⢀⣀⣠⣤⣶⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⠟⠀⠀⠀⠀⢈⣹⠟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠹⣆⠀⠀⢀⡴⠋⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠈⢷⡴⠋⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡀⠀⠀⠀⠀⣠⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⣀⢠⣾⣿⣿⠿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠈⠙⢻⣿⣿⠏⠀⠀⠀⠈⠛⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡟⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣡⣦⣄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⡦⠀⢀⣠⡾⠛⠛⠒⠒⠒⠚⡛⣦⣄⠲⢄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⠒⠿⢶⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣿⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⢠⠚⠁⡄⠸⣇⠀⡀⠀⠀⠀⠀⠀⡠⠀⠐⠋⠀⠘⠁⠚⠿⢿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡄
⠀⠀⠀⠀⠀⠈⠄⡄⠰⠀⢛⣼⠀⠀⠀⠤⠂⠉⠀⢀⡀⢀⣀⣐⣄⣠⠀⠀⠈⠉⠻⠷⠀⠀⠀⢀⣀⣀⣀⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠾⣦⡳⡀⡟⠀⠀⠀⠀⢀⢴⠀⡀⠙⠿⢿⣿⣿⣽⣝⣢⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠙⠿
⠀⠀⠀⠀⠀⠀⢀⠠⠟⠿⣿⣿⠀⣴⠋⠀⡂⠀⠈⠉⠲⣶⣤⡝⠋⠃⠀⠹⠗⠒⠢⡀⠀⣀⣤⡶⢦⣤⣞
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠹⣇⢸⠇⣴⡹⠁⠄⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣇⣀⣉⡿⠛⠐⣼⣿⠛⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣠⠎⠉⠉⢿⠀⠀⠀⠉⣤⣷⠀⠈
⠀⠀⠀⠀⠀⠰⡀⢆⠀⠀⢀⠀⡋⢸⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣀⡴⠞⠡⠄⠀⠀⠠⠀⠀⠀⠀⣠⢏
⠀⠀⠀⠀⠀⠀⠙⢦⣥⣄⣈⢳⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠞⠁⢸⠂
⠀⠀⠀⠀⠀⠰⠁⠈⠀⠈⠉⢿⡏⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢿⡆⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠶⠐⠆⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⡟⠀⢑⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠑⢆⡀⠀⠀⡀⣿⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢴⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡦⣤⣰⡿⠀⠀⠀⡀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⡀⢁⠬⠁⠌⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⣁⢰⣧⡞⠂⢤⠀⠈⠀⠀⠀⠀⠐⠖⠆⢸⡷⠋⣠⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢯⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢲⣄⡈⠳⣏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢷⠀⠀⢰⡇⠀⠀⠀⠀⠀⣾⢉⢻⣶⠿⠀⣀⣷
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠲⢿⡇⡀⠘⡟⠀⠀⠀⠀⠀⢹⣧⣾⠁⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢋⡒⣾⣿⣀⣶⡇⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠻⠿⠋⡕⠀⠀⠀⠀⢀⠐⠉⡒⠖⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⢲⣆⠁⠑⣄⠀⠀⣾⡠⢠⣇⠨⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠋⠆⠀⢻⡆⠀⣿⢃⣾⠃⢸⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠰⡸⡄⠈⠟⠀⢹⣾⡏⢠⡏⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣀⢷⡁⠀⢸⠀⠈⣽⡁⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠸⠏⠿⡀⢸⠀⠀⠙⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣙⣛⣛⣹⣯⣩⣉⣿⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣽⣭⣽⣿⣯⣭⣭⣭⣭⣿⣭⣭⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡖⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢵⠉⢆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡰⠉⡾
⠀⠀⠀⠀⠀⠀⠀⡜⡘⠀⠈⡆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠁⠀⢃⢣
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⣧⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣼⠀⠀⢸⢘
⠀⠀⠀⠀⠀⠀⠀⢡⣦⠀⣌⠝⠛⠐⠯⣖⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣲⠽⠒⠛⠫⣡⠀⣴⡌
⠀⠀⠀⠀⠀⠀⠀⠈⢫⣾⠋⣴⣿⣿⣶⣄⠙⢾⡲⣄⠀⠀⠀⠀⠀⠀⣠⢞⡷⢋⣠⣶⣿⣿⣦⠙⣷⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⢠⣿⣿⣿⣿⣿⣿⣄⡙⢮⢢⠀⠀⠀⠀⡔⣵⢋⣠⣿⣿⣿⣿⣿⣿⡄⣏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⢸⣿⣿⣿⣿⣿⣿⣿⣆⠈⣷⠑⣠⣄⠊⣼⠁⣰⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣩⢸⣿⣿⣿⣿⣿⣿⣿⣿⣠⣏⣼⣵⣮⣧⣹⣄⣿⣿⣿⣿⣿⣿⣿⣿⡇⣍⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣿⣿⠿⠿⢿⡚⠛⠒⠒⠚⠛⠛⠋⠙⠛⠛⠓⠒⠒⠛⢓⡾⠿⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠗⠓⠀⠀⠀⠀⠒⠺⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠉⠈⠑⢤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⢹⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⠀⠀⠀⠙⠷⡠⣶⣶⠆⠼⠋⠀⠀⠀⠆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⢣⣀⣀⠠⠊⠀⣸⣏⠀⠑⠄⣀⣀⡜⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣈⣿⡤⠴⠖⠋⣹⣏⡙⠒⠦⠤⣿⣁⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣷⣿⡿⠟⠛⠟⠻⢿⣿⣾⢷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣁⣾⡿⠛⠀⠀⠀⠀⠀⠀⠛⢿⣷⣊⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠉⠉⠉⠙⢿⡿⠋⠉⠉⠉⠁⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⠤⠤⣼⣷⠤⠤⠴⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠳⠶⢦⣤⣄⣀⣀⣀⣿⣿⣀⣀⣀⣠⣤⡴⠶⠞⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣛⠷⠶⣤⣤⣀⣉⣉⠉⠉⠉⠉⣉⣉⣀⣠⣤⡶⠾⣛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣌⣉⡉⠉⠛⠛⠛⠛⠉⢉⣉⣡⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⠀⡤⢍⠙⡍⠙⠛⠛⠛⠛⠋⢩⠋⡩⢤⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠑⠺⡀⠀⠀⠀⠀⢀⠗⠊⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣶⣶⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠉⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠉⠙⢿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡄⠀⠈⢧⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡜⡄⠀⠀⠳⢍⡲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢀⢎⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡆⢀⣀⣀⡉⢢⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠾⣇⠀⠀⠀⣮⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠩⣄⠬⡉⢉⡓⠳⣄⠀⠀⠀⠀⠀⠀⡠⠞⣉⣠⠖⠊⠘⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠉⠓⢶⠈⢳⡀⠀⠀⢠⠎⣠⠔⠁⠀⠀⠀⠀⠘⡄⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠈⢹⡀⢹⡀⢀⡏⡶⠁⠀⠀⠀⠀⠀⠀⠀⣇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣧⣸⣈⠁⠀⠀⠀⠀⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⣀⣠⠤⠤⠾⣍⣏⡻⢬⠗⠒⠒⠒⠶⠤⠤⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡞⠉⢁⣀⣠⡤⠤⠤⠶⠖⠒⠲⠶⠶⠤⠤⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⢙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣈⠙⠲⠦⠤⢤⣤⣤⠤⠤⠴⠖⠒⠉⢠⣴⣻⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣓⣦⣄⡀⠀⣾⢲⣖⣶⢻⡀⢀⣴⡧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⡿⢦⠈⠉⠉⠉⠉⢁⣾⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣧⠀⠀⠀⢠⣏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣾⡄⠀⢀⣞⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡇⠀⡼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣴⠷⠋

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

⠀⠉⢠⡟⠂⠐⠖⠢⣄⠀⠈⠢⡙⢧⠂⠂⣇⢠⡾⠁⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣷⣴⠀⡎⠜⣱⠂⠀⡼⢡⡟⣠⠞⡙⠻⠖⠈⡙
⠀⠀⡶⠁⠀⠐⠀⠁⠀⠱⣄⠀⠁⡌⢣⡐⡌⠎⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣐⡄⣿⡄⢠⠇⢸⣣⠋⡐⠀⠀⠀⠀⢈
⠀⠰⠁⠀⢀⡑⡀⠀⠀⠐⠈⢧⠀⠘⢆⢻⡌⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣷⣾⣈⣤⡇⣴⠀⠀⠀⠀⠀⠈
⠀⣱⣎⢐⣉⣈⡀⠐⠄⠀⠄⠀⠱⠀⠀⡛⠀⠀⠀⠀⣸⣿⣿⣿⠟⠋⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣻⡛⢿⣧⣹⠦
⠀⠁⠐⠎⠀⠂⡈⠓⠀⡁⠢⣄⢀⠀⡛⠀⠀⠀⢀⣼⣿⣿⣿⠋⢀⣀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣄⣂⣿⢟⠿⡿⡶⢾⠙⠿⣦⡀⠂⡀
⠀⠀⢣⡄⠀⠈⠀⠀⠀⢘⠢⣙⡆⡶⠁⠀⠀⣠⣾⣿⣿⣿⠃⢴⣤⡄⠀⠀⣨⣭⣁⢻⣿⣿⣿⣿⡿⠽⠿⠇⢹⠗⠚⠓⠐⠛⠋⠘⠛⠂
⠀⣄⠀⠐⠀⢲⠀⠀⠀⠈⣀⠡⠞⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⣿⣿⣿⣿⣷⡄⠀⠀⠈
⠀⠀⠀⠁⠀⡀⢢⠐⠀⡎⠀⠠⠅⠀⠀⠒⣳⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣝⣶⠶⠒⠶⣶⣶⣤⣤⠤⠄
⠀⡟⠀⠺⢦⡈⠂⠇⠰⠈⠀⣂⣀⣀⣤⢶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⢀⠹⠂⠀⢠⣿⣿⣿⣿⣿⣿⣿⣽⣆⢂⠀⣸⠦⠋
⠀⣽⠁⠀⠀⢩⡐⢄⢀⠀⠚⠉⠉⠉⣰⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡾⠞⠁⠀⠀⠀⠀⠀⢀⣴
⠀⡀⠘⢭⠀⠂⢱⠶⠌⢳⢄⣤⣤⣾⣿⢿⣿⣿⣿⣿⣿⣿⡇⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠠⢀⠂⡀⠛⠃⣸⠞⣩⣤⣾⣭⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⢲⡎⢀⠄⡰⢊⠀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠐⠟⠿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠻⣽⠃⠨
⠀⢡⠈⢇⣠⠼⠄⠤⠊⡰⢋⣼⣿⣿⣿⣿⣿⣿⠳⣘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⠂⠟⡐⢀
⠀⣼⣠⢋⢁⠠⠤⠄⢀⠙⣻⣿⣿⣿⣿⣿⢳⡏⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⡄⠀⠀⠀⢀⣾⡿⣿⢋⡴⡀⣴⢁⣟
⠀⣄⡀⠈⠑⠳⢈⠓⢀⣼⣿⣿⣿⣿⠟⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡈⡇⠀⠀⢠⣿⣿⡿⣁⠌⠱⣰⢃⡾⢠
⠀⣦⠄⣁⠆⠀⠈⠱⠈⣿⣿⣿⣿⣇⣴⡿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣷⠀⣠⣿⠇⡌⠛⠄⠠⡴⢡⠊⠁⢸
⠀⠁⠀⠄⡀⠀⢀⣀⠈⢮⣿⣿⣿⣿⠏⠀⠈⢿⡙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣽⠏⠘⣿⣿⡿⢛⡤⠶⠶⠚⠃⠐⠁⠘⣰⠶
⠀⠀⢀⡀⢐⠈⡀⢉⡑⠾⠿⣿⣿⡿⠀⠀⠀⠀⢻⡄⠘⢷⡀⢠⡾⠀⠀⠀⠀⢀⡾⢃⡾⠁⠀⠀⢹⣿⡏⡶⠈⡿⠀⠠⠔⢀⠴⠁⠤⣨
⠀⠋⠀⠈⠏⠠⠠⠞⠉⢀⡴⢿⣿⡇⠀⠀⠀⠀⠀⢻⡄⠈⣷⣾⠃⠀⠀⠀⢠⡾⢡⡟⠀⠀⠀⠀⢸⣿⣇⡹⢀⣤⠊⢤⣤⠔⠒⠒⠶⢬
⠀⠰⠄⠘⣤⠋⠀⣠⠨⡘⣡⢞⡻⣿⡀⠀⠀⠀⠀⠀⢿⡀⣿⡏⠀⠀⢀⣴⣟⣴⠏⠀⠀⠀⠀⠀⣸⣿⠞⣡⠛⠩⠉⠒⠀⠠⠅⣀⣂⣀
⠀⡀⠈⢣⠀⠀⠐⠄⠐⢧⡁⠆⣸⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣷⣶⣶⣦⣶⣶⢟⣵⢆⡝⠉⠑⠒⠦⢤⣤⠀⣀⠀⢰
⠀⠂⠀⢸⡀⠒⠁⠄⠠⠀⢳⠘⣽⠟⣿⠋⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣿⣾⣿⢣⠇⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉
⠀⠄⢀⡆⢹⡀⠄⠐⠄⠃⢈⣧⠏⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠻⡤⠀⠀⠄⠀⠐⢀⡀⢀⠠
⠀⠤⠘⠁⡎⢰⡀⠀⠰⠀⢸⢉⠠⡏⣿⣇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠂⢄⡈⠓⢂⠒⠒⣈⠀⠁⠈⣡
⠀⢀⠈⠠⠁⡌⢧⠀⠀⡆⢹⣞⢀⡁⣿⣿⣷⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⣿⣿⣳⣌⣙⠒⠤⠭⠬⠍⠥⠩⣄⠰⣎
⠀⠌⢀⠃⡘⠀⠀⠾⢀⠂⡾⠛⢹⡇⢻⣿⣿⠂⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣄⠀⠈⠙⠋⠙⣂⣐⣮⡛⢌⡑⠌⠙
⠀⡶⠀⠀⡔⢀⠌⢠⣿⡞⢀⠎⠘⣇⢸⢹⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⢸⣿⣿⣵⠦⣌⡐⠂⠄⠈⣀⣁⠢⣍⡂⢠
⠀⠀⠀⠐⢠⠃⠐⠙⠇⢡⠎⢀⠝⢿⣸⣺⣿⠁⠀⠈⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠃⠀⠀⠐⠶⣿⣆⣒⠦⣭⣉⠛⢛⣛⣂⣉⣥⠤⢬⣉
⠀⠞⠀⠀⠀⠘⢰⡎⠀⠆⠀⠌⢠⠆⣹⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠉⠀⠿⢿⣄⠐⠶⢈⡙⠻⢤⡶⣴⠶⢾⠛⣹
⠀⡜⠀⠂⣠⠂⡟⠀⠌⠁⡜⢠⠟⢠⢿⣵⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢈⡳⠎⠑⠈⠖⢡⡾⠁
⠀⣁⠈⡄⠡⡘⢀⠁⠀⡘⢀⠋⣰⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠁⣜⢠⣧⠖⣩⠖⠜⡡⣾
⠀⠁⠘⠀⡄⠇⡈⠀⠀⠁⢆⡜⠋⢑⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⠟⠉⣠⠚⡉⠆⠰⡀⢹
⠀⡀⠄⠀⠱⡂⡁⠀⠠⢈⣞⣁⣠⣿⣯⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣬⣥⣶⣷⠈⡄⢱⠘⡀⢛
⠀⡇⠀⢈⠀⡇⢄⢨⠖⠚⡉⣉⠼⣿⣿⣿⣮⣻⣿⣿⣶⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⡿⠘⢧⡈⠈⡄⡇⢸⣅
⠀⠁⠀⠆⡀⢰⡎⠑⠊⢀⡶⢉⠜⢻⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⡄⠢⠘⠤⠟⢋⣉⠤
⠀⠾⡅⢖⢁⡞⠀⠀⡔⢀⢁⠁⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⡿⣄⠈⠲⣉⠋⠀⠰
⠀⠰⢀⡄⠾⠠⢀⡾⢠⠏⡄⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢹⣿⣷⣄⡙⣀⠞⠡
⣀⣢⣉⣙⣆⣀⣜⣴⣁⣀⣁⣀⣀⣀⣀⣀⣀⣈⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣿⣋⣀⣀⣀⣀⣀⣀⣘⣿⣧⣉⣉⣉⣚⣓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣤⣄⣰⣾⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣶⣄⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣴⠀⠙⣿⣿⣷⡀⠀⠀⠀⠈⠙⠛⠛⠿⠿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⡃⠀⠈⠻⢿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣦⡀⠀⠀⠈⠙⠳⠶⠦⢀⡀⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣄⣀⡀⠀⠀⠰⢛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣷⣤⣄⣬⣽⠿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡿⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠛⡿⠿⠿⠿⠻⣿⣿⣿⣍⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠲⠒⠀⠀⠀⠸⠏⠀⠃⠀⠀⠀⠀⠈⢿⣿⣿⡀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣦⣀⣀⠀⠁⠀⠀⠰⠿⠶⠿⠂⠀⠀⣴⠟⠀⠀⠀⠀⣴⠿⣿⣿⣿⣧⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⠱⡆⠀⠀⠈⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣶⣶⣶⣤⣤⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠞⠋⣹⣿⣿⣿⠏⢀⠀⠉⠙⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠟⠋⠁⠀⢠⣿⣿⣿⣿⣇⠉⠻⢦⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣿⣤⣤⡤⠦⠶⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⡶⠾⠿⠟⠋⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣯⣄⣀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠁⣀⣀⣀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣠⠀⠀⠀⠈⠁⣼⣿⣿⣿⣿⣿⡯⠀⠉⢳⠶⠂
⠀⠀⠀⠀⠀⠀⠀⣰⣵⣶⡿⠟⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠾⠿⠟⠋⢁⣠⣴⠞⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⣿⡇⠀⢠⡌
⠀⠀⠀⠀⠀⠀⢰⠟⢋⣁⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠁⠀⠀⣿⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠸⠛⠋⠉⠁⠀⠈⢉⠙⠋⠉⢿⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡈⠀⣼⠃
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢰⡟
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣀⣠⡴⠛⠁
⡄⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠒⠞⠋
⡇⠀⣀⣀⣤⣭⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣬⣥⣀⣀⠀⠀⠀⠀⢀⣾⠋
⠰⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠻⠛⠁⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠀⢠⣿⠃
⠀⠈⠻⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣤⣴⣿⣇⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣄⠀⢠⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠉⠉⠙⠛⠳⢦⡄⠀⣿⣷
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣾⣿⠟⠿⠿⣿⣿⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣽⣿⣿⡆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠀⠀⠚⣭⣶⣿⣷⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⣴⣾⣿⣷⡀⠈⣿⣿⣿⡇⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⣛⢉⣩⣭⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⢛⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣰⣿⣤⠀⢀⣠⣿⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠛⢛⣉⣉⣭⣬⣽⣿⣦⣀⣀⣀⣴⣿⣿⡿⣣⣾⣿⡿⢋⣿⣟⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣟⣵⣿⣿⡟⢠⣿⣿⣿⡏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣬⡙⢿⣿⣿⣿⣿⠿⠛⢛⣉⣉⣉⣩⣭⣥⣤⣶⣿⣿⣿⣿⣿⠟⢁⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣦⡙⢿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⢛⣉⣥⣴⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⢀⣶⣦⣬⡙⠿⣿⣿⣿⣿⣦⡹⣿⠿⠛⠻⠛⠛⠛⢛⣋⣥⣶⣿⣿⣿⣿⣿⣿⣷⣤⣬⣉⠁⣁⣴⣿⣿⣿⡆
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣌⠻⢿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⣛⣉⠽⣟⣫⣴⣾⣿⠋⣾⣿⣿⡇⣴⣄
⠀⠀⢀⣴⣦⣄⣈⡙⠻⢿⣿⣿⣿⣦⡙⢿⣿⣟⠻⠛⠻⠿⠿⠿⢛⣫⣴⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⢸⣿⣿⡟⢠⣿⣿
⠀⢠⣾⣿⣿⣿⣿⣿⣶⣤⡙⠿⣿⡿⢻⣾⣷⡾⢿⣿⣿⣿⣿⡿⢿⣿⣿⡿⠟⣛⣯⣥⣤⣴⣷⣿⣟⣋⣠⣴⣿⣿⣿⢁⣿⣿⡿
⢀⣙⣙⠛⠛⠛⢿⣿⣿⣿⢿⣷⣾⡆⣷⣿⣿⣴⣷⣿⠛⣿⣿⣿⣎⢻⣿⣿⣦⠀⠉⠉⠉⠉⠛⠛⠻⣿⣿⣿⣿⠟⢡⣾⣿⣿⠇
⣿⣿⡿⠿⠛⢛⣛⠟⣽⢃⣿⣿⣿⡇⢹⣿⣿⣷⣹⣿⡇⢹⣿⣿⣿⣆⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣙⡻⢧⣴⣿⣿⡿⠋
⠋⣡⣴⣾⣿⣿⢋⣾⡇⢸⣿⣿⣿⡇⢸⣿⡟⢿⣿⣿⣿⡈⢿⣿⣿⣿⣇⢹⣿⣿⡇⠀⠀⠀⠀⠠⡄⠘⠛⠁
⣿⣿⣿⣿⠟⣵⣿⣿⡇⢸⣿⣿⣿⠇⣾⣿⣇⠸⣿⣿⣿⣷⠈⣿⣿⠿⠟⠀⠉
⣿⣿⠟⢡⣾⣿⣿⡿⢀⣿⣿⡿⢣⣾⣿⣿⣿⠀⠿⠿⠟⠛⠁
⠟⢁⣴⣿⣿⡿⠏⣴⣿⣿⣿⠇⠘⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣉⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣡⣾⣿⣷⣿⣆⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠏⣼⠋⠉⠁⠀⠙⣯⡛⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡇⣴⣿⠀⠀⠀⠀⠀⠈⣿⣣⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢆⣾⣸⣿⣿⣆⠀⠀⠀⠀⢀⣿⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⠻⣿⣿⡦⣀⣀⠤⢻⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⡞⣾⡿⡰⣿⡿⠁⠀⠀⠀⠈⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⢣⢎⡼⣹⢃⡿⣫⠃⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⠈⡜⡰⢣⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠃⢡⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢸⠀⣼⣿⣿⣄⠀⠀⢠⠀⠀⢀⣴⣿⣿⣿⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠔⢻⣿⣿⣿⣷⣦⣎⣦⣴⣿⣿⣿⣿⣿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⣟⠛⠛⠉⠉⠉⠉⠙⠛⠛⠛⠛⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡎⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢰⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡌⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠙⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠹⣷⣤⣀⡀⠀⣀⣠⣶⡿⠃⠀⠀⢧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⣠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢠⢸⠀⠀⠀⠀⠀⠘⢻⠟⠁⠀⠀⠀⠀⠀⢠⠃⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣯⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣎⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁

⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⣀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠧⢤⣀⠤⠚⠊⠉⠉⠀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⡛⠿⠛⠹⣲⣶⣦⣴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⣠⣟⠳⠿⣶⣤⣄⡈⠈⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠣⣿⣶⣭⣕⢋⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠺⠿⠸⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣀⣠⣤⣶⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠰⡄⠀⠀⢀⠀⢀⣞⡁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠙⠀⢄⢀⠐⠂⠉⠛⠐
⠀⠀⠀⠀⡀⠠⡄⢸⠑⠺⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⢬⢻⠀⡀⠀⣠⣄⠄
⠀⠀⠀⠘⢿⠼⠤⠴⠉⠀⠁⠂⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣶⠋⠉⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⣀⢀⣠⣠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⡇⠀⠀⢀⡦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣳⠶⠀⠈⠀⠉⠀⠉⠙⣋⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠉⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢧⣄⠀⠀⠀⣀⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⣿⣏⢹⣟⠿⣿⣟⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⡐⠐⢶⠄⠦⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⠈⠡⡗⢤⢳⠄⣤⣱⠉⢨⠕⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⢹⣲⠇⠊⠀⠀⣼⣷⣦⣚⣙⡁⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⡖⣸⣤⡾⠎⠀⠀⠀⠙⢯⣟⡟⣷⣿⣜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠨⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢺⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠈⠉⠁⣶⠆⠀⠀⣶⡆⠁⢸⣼⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠠⢀⠄⠛⠳⣤⠴⡋⠗⢀⣼⡿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⠗⡄⠇⠀⠀⠘⠀⠀⠁⠀⠘⢿⣷⣜⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠁⡄⢸⠀⠀⠀⠀⠀⢰⠀⡄⣎⣿⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣏⠀⢄⠀⣸⠀⠀⠀⠀⠀⠸⡁⠀⠏⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠎⢈⠁⠀⠄⠀⠀⢀⠀⠉⠀⢲⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡔⠛⠃⠀⠂⠀⠀⠐⠀⠄⠀⠀⠘⣯⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠣⢀⣀⣀⣀⠀⠠⠤⠔⠓⠒⠒⠐⠒⠂⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠒⠂⠀⠉⠉⠉⠉⠉⠁⠀⠐⠂⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣶⠄⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣴⡆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢠⠒⠒⢄⠸⠀⠀⠀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠇⠀⠀⠀⠀⠀⣀⠤⡀⠀⠀⠸⢀⠔⠒⢢
⠀⠀⠀⠈⠢⣀⠀⠈⠣⣀⠘⡄⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠃⠀⠀⠀⢀⠔⠀⠀⡜⢀⡠⠋⠀⠀⡠⠊
⡈⠁⠉⠒⠤⢀⠑⠤⡀⠀⠑⠌⡒⢄⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢀⠆⠀⠀⡠⠔⠁⢀⠔⡈⠔⠁⠀⡠⠔⢁⠠⠐⠊⠁⠉
⠈⠀⠀⠀⡀⠀⠈⠁⠚⠶⠀⠀⠈⠉⠀⠀⠀⠈⢢⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡠⠔⠁⢀⠔⠁⠀⢠⠊⠀⠀⠀⠈⠉⠀⠀⠰⠞⠂⠉⠀⠀⡀⠄⠐⠊
⡎⠀⠀⠀⠐⠋⠶⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠙⠭⠀⣀⣁⣈⡁⠠⠄⠊⠁⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠎⠓⠀⠈⠉⠈
⠈⠁⠀⠒⠂⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠒⠂⠀⠉
⠀⠀⠀⢌⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⢉⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠒⠠⢄⡀⠀⠀⠀⠀⢀⡠⠃⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠣⣀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠀⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣤⣤⣤⣤⣤⣬⣭⣴⣶⣤⣤⣤⣤⣤⣴⣦⣬⣥⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠃⠀⣠⣿⣿⣿⣤⣶⠟⠟⠛⣿⠛⠋⠁⠀⣠⠀⢻⣿⣿⣿⣶⡶⠀⠀⣤⠖⢈⡄⠀⠀⠀⢸⣿⣿⠉⡛⠛⠁⠀⠀⠀⠀⠀⠘⠻⢿⡿⠟⠋⠁
⣋⣤⡾⢿⣿⣿⣿⣿⣿⣀⣀⡀⠉⠀⠀⠀⠠⣼⣳⡀⠀⠀⠀⠀⠀⢀⣸⣿⢰⣿⣇⠀⠀⠀⠺⠿⡃⠀⠀⠀⠀⠀⠀⠐⠂⢤⠄
⡿⠟⠀⠀⠀⢀⣽⣿⣿⣩⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠿⠟⠁⠠⠷⢶⣾⡿⠿⠀⠀⠀⠀⠀⢀⡀
⣠⣤⠀⠀⠐⠿⠻⢿⡏⠀⠰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠿⠇
⠉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠈⠙⠀⠀⠀⠀⠀⠀⠀⠠⣤⡠⢶⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠩⠄⠀⠀⠀⢸⣷⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠆⠠⠠⡄⠀⠀⠀⠶⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢤⠔⠀⠀⡤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛
⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠈⣿⣿⠆⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣿⣶⣦⣹⣿⣶⣄⣀⡀⠀⠀⠸⠿⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣟⣿⣿⣿⣧⣿⣿⣿⣾⣷⣶⣶⣬⡀⣀⣤⣠⣤⣄⣠⣤⣀⣀⣀⣀⣀⣀⣄⣀⡀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣦⣤⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢶⠒⠒⠒⠛⠉⠉⠉⠉⠙⠉⠙⠒⠒⠒⠒⠒⢤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣶⣤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠿⣿⣧⣏⡒⢄⠀⠀⠀⠀⢀⣴⣬⣦⢩⢿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠛⣿⠻⡸⣿⣿⣷⠀⢀⣴⣿⣿⠻⡸⣼⢻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣿⣿⠏⣵⣼⣿⡿⢟⢩⣬⣷⣷⠏⣹⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠌⠵⠇⠘⠁⠀⠀⠈⠙⠿⢯⡤⠯⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⢰⣲⡖⠶⣰⠄⢀⠲⣶⠖⠒⣖⣶⠒⢋⡏⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣟⢿⣄⣈⠛⢾⣽⣆⣢⢯⣻⠋⣧⣀⡙⠳⣯⣷⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣟⢦⡈⣏⣹⣞⣯⣟⡿⣇⣰⣔⢩⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢿⣮⠵⣭⢿⣿⣷⣾⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡖⠋⡺⠛⣽⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠂⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⢛⠵⢒⠒⠘⡻⣦⣀⣄⢈⣻⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⣿⣿⣿⠡⢃⢎⡴⢚⣻⣿⣮⣿⡿⣟⢿⡇⢻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠻⢿⣿⣿⣤⡈⠘⣷⠏⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠐⠯⣿⣿⣦⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣦⣴⣶⣾⣿⣿⠟⠜⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣭⠀⠀⠀⡤⠶⠖⠚⠛⠋⠀⠀⠀⠚⠛⠓⠲⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡏⢿⣿⣿⣶⠿⢟⠁⠀⠀⠈⢠⣶⡿⠲⠆⠀⡀⠀⡍⣴⡟⠛⣶⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⢤⣬⣿⡇⠈⣿⡷⠃⠀⠀⠀⠙⠓⠓⠂⠀⠀⠀⠀⠹⡉⠈⠈⠁⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣮⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⣵⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⠛⢛⣽⣶⣷⣪⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⣻⣿⣿⣿⣷⡟⠄⠀⠀⠀⠀⠀⠀⠀⡠⢊⣄⣂⣜⡄⠀⠐⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣷⠘⢦⡀⠀⠀⠀⠀⠀⠣⢤⣀⣀⡴⢙⠁⡀⢀⠐⠆⠉⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣿⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠁⠀⠃⡀⠀⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠓⠦⣀⠈⠙⢿⣶⣤⣄⣀⣀⣀⣤⣆⡅⠀⠀⠀⠀⡼⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⢀⣤⣵⣄⠀⠙⠻⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠘⣧⠀⣿⡿⢿⣿⡄⠀⠀⠀⠀⠈⠉⢻⣿⡀⠀⠀⠀⠛⢿⣿⣷⣤⣳⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⢾⡀⠀⠀⠉⠀⠸⣇⠀⠙⣷⣄⣀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠓⢄⡉⠉⠉⠁⠀⠉⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠻⠀⠀⠈⠛⠻⠿⣷⣄⠀⠀⠀⠀⡼⠛⢀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢰⠀⠀⠀⠀⠰⠶⢿⣿⣦⠀⠀⢠⠇⠀⠀⠢⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠇⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⠀⢸⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢸⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠋⠉⠉⠉⠉⠓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⣞⢻⡝⢦⡀⣤⡄⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⢠⡼⣦⡙⠈⠻⠀⡇⠀⡼⠁⠀⠀⠔⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⣤⡀⣠⢶⠄
⠀⠀⠀⠀⠀⠀⣱⠦⠀⠀⠀⠀⣿⢠⣇⣀⠀⣀⣈⡀⣀⡀⣀⣒⣂⣄⡀⠀⠀⠀⠀⠀⢷⠀⠀⡞⠐⠟⡵⠛⡹⠃
⠀⠀⠀⠀⠀⠘⠓⠒⠙⢢⡀⠀⠘⣿⡇⠀⠀⠈⠀⠈⠀⠀⠁⠉⠑⠉⠑⠀⠇⠉⠀⠂⢾⣀⡼⠁⠀⠀⠀⣛⣲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠘⡇⢀⡠⢤⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⡥⠊⠀⡼⠋⠀⠀⢀⣠⠤⠨⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠃⡀⡀⠀⠀⣀⡀⠀⡀⣀⠁⠀⠈⠃⡀⠀⡠⠊⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢚⠈⠈⠀⢁⠀⠀⠀⠀⠀⡸⠀⠓⠑⡡⠊⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠑⠠⢀⠀⠁⠀⠀⠒⠉⢀⡀⢤⠎⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠠⠤⠄⠀⢀⢀⣀⠀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠑⢀⢤⣾⣷⣦⣀⣀⣀⣀⣀⣀⣀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⣹⣿⠯⠃⠀⠀⠀⠀⠀⢡⢠⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⡄⢃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⠄⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠰⠘⡘⣇
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⡀⠀⣀⡴⠋⠁⠀⠙⣆⠀⠀⠀⠀⠀⠀⠃⢁⢻
⣖⠻⠻⠲⡴⠛⠁⠀⠀⠀⣀⡴⠋⠉⠛⠁⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⡀⠄⠘⢺⡛⠁
⠹⣆⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⣇⠀⠀⠀⠈⣧
⠀⠘⢦⡀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠸⡆
⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠈⠓⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣤⣤⣤⣤⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢔⣠⣾⣿⣿⣿⣿⣿⣧⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣁⣥⠖⠛⠛⠛⠿⣿⣿⣿⣿⣿⣷⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡗⣒⣂⡀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣧⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢋⣹⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄⠀⠀⠀⠀⣀⡀⠀⠀⠸⣿⣿⣿⣿⣆⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⢰⣯⣥⣾⠃⠀⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡿⣿⣿⣿⣿⣧⡀⠀⠈⠈⠉⠀⢠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠹⣿⣿⣿⣿⣿⣷⡶⠆⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣹⡄⣈⣛⣛⣛⣛⠻⢫⡄⠀⠼⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠟⠉⠙⣿⣿⣿⣿⣿⡄⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⣧⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⣿⣿⣿⣿⣿⣆⠈⣻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⣾⠟⠛⠟⠚⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠊⠁⠀⠀⠀⠀⠺⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠫⠛⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠉⠒⢤⣀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⢿⣻⣵⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣛⣯⣽⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⣛⡻⣿⣿⣿⣿⣿⡞⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⡠⠴⠊⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣱⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠿⠟⠛⠛⣻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⠛⠛⠛⠿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠂




⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣶⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡄⠀⠀⠀⠀⠀⣠⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡹⣿⡿⣦⠀⠀⢀⡼⢃⣻⡍⢳⡄⠀⠀⠀⢀⣃⠀⣠⣀⣰⠢⡀⠀⠠⠒⠉⣨⡀⢙⣠⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣽⡓⣻⠣⣿⣆⠀⣼⢷⠿⣇⣨⡴⣇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣧⣿⣿⣿⣿⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⡾⡷⣮⣿⣶⡿⠾⣶⠿⡶⠿⡿⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⠛⠿⣿⡷⠿⣿⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⠀⠀⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢾⣤⣀⡀⠀⠀⠀⠀⣀⣤⠷⠻⣄⠀⠀⠀⠀⠀⢠⣄⠑⣄⠀⠀⠀⠀⢀⠜⢁⣶⣤⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢮⣹⣆⣈⣿⣉⡙⣻⢶⡞⣋⣙⣿⣄⣀⣼⣽⡖⠀⠀⠀⢸⣿⣷⠈⢣⠀⠀⡠⠁⢰⣿⣿⠉⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣯⣽⣿⣿⣿⠿⠿⣿⣿⣯⣭⣿⣤⣀⡀⠀⠀⢁⠘⣿⣿⣆⣀⣵⣮⣄⣴⣾⣿⣿⢀⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⠛⠉⠉⠉⠉⠙⠛⠛⠻⠿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧⠠⠀⠀⠀⣖⠚⠆⠀⠀⠄⠀⠀⠀⠀⢰⡿⣿⣻⣦⠀⠀⠀⠀⠀⣴⠛⣹⠋⢷⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡗⢣⠀⢸⣈⡴⠀⠀⠀⠰⠀⠀⠀⢰⢧⠀⠸⡆⢿⠷⡀⠀⢀⡏⠀⢠⡇⡄⠓⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠚⢛⡀⡘⠒⠒⠀⠀⢀⣀⡀⠀⠀⣸⣥⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⠾⢧⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⣀⣀⣀⡀⠀⣀⣠⣔⢂⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣻⣿⣿⣿⣿⣿⣿⢿⣿⡤⢐⡀⣀⠀⠀⠀⠄⠈⢷⡀⠈⢮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⣼⣿⢸⣿⠶⠟⠛⠋⠀⠀⠀⠀⠀⠀⠙⢷⣾⣧⣠⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣚⣿⡿⣻⣿⣧⡈⢷⣿⣷⣔⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣽⣸⡿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣿⠿⠉⠛⢿⣿⣾⢿⠉⠋⠛⠿⠿⠤⠀⠀⠀⠀⠀⠑⢹⣿⣿⣿⣿⣿⡿⠛⠛⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢛⣹⣿⣿⣧⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⣠⣾⣷⡿⢟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢄⠀⠀⠀⠀⠀⠀⠀⠈⣉⣽⠿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⣀⣀⠀⠀⠈⠀⣈⡴⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⢿⡏⠛⠑⣲⠚⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢈⡘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄
⠀⠀⠁⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⣿⣼⡇⢸⣿⣧⠙⣿⠃⢸⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⢹⡇⢸⣯⣿⠀⣿⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠛⠘⠃⠙⠁⠛⠁⠋⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⣤⡴⠆⣶⢸⡆⢿⣟⠀⣿⢹⡇⢹⣿⡏⢸⡏⣿⢸⣷⣾⢰⡶⠆⠀⢀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣇⠠⠈⣿⠀⢹⣿⡇⢸⡏⠀⣿⢿⡇⠀⣿⠁⢸⣇⣿⢸⣿⣿⣸⠋⠀⠀⠛⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠨⡇⠀⠙⠶⠀⠻⠃⠘⠛⠃⠛⠈⠋⠀⠙⠀⠈⠙⠋⠘⠋⠃⠛⠃⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠦⠒⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠀⣖⠀⠀⠀⠀⣀⠀⠀⡀⢐⠀⠥⠒⠉⠀⠀⢀⡀⢠⠜⢲⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⡶⠖⢉⠛⠉⣹⠟⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⣠⠎⠀⡴⠁⠀⠀⠁⠀⠀⠀⣿⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⡸⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡟⢀⡟⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡀⠀⢀⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⣠⠏⢀⡞⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡇⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⢠⠞⢁⡴⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⢸⠀⠀⠀⢸⡇⠀⠀⠰⠁⣴⠏⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⡇⠀⢸⠀⣄⠀⢸⡇⠀⠀⢁⡞⠁⠀⢀⠐⠀⡀⢠⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⠀⢸⠀⣿⠀⢸⠀⠀⢠⡟⠀⢀⡴⠋⣀⠜⣠⠃⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠸⠀⠘⠀⡏⠀⠀⠀⠀⣸⠀⠠⢋⣤⠞⠁⡰⠃⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡀⠀⠀⢸⠅⠀⠀⠀⠀⠁⠠⢁⣿⠃⠀⣰⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠁⠀⢀⡎⠀⠀⠀⠀⠀⡴⠁⣸⠃⠀⣸⢇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠀⠠⠶⠋⢀⠀⠀⠀⠀⢀⠁⢀⡞⠀⢠⡟⠛⠁⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠋⢀⡀⠀⠀⠀⠀⠁⠰⠶⠋⢀⡼⠃⢀⢿⠀⠘⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⣠⠔⠛⠷⢦⣄⣀⠀⠀⠉⠀⣠⠋⢸⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⢀⡞⠁⠀⠀⠀⠀⠨⠋⠒⠒⠒⠛⠀⠀⠀⡤⢤⠤⣧⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⢄⠀⠀⠀⠆⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠃⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠞⢡⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢰⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢷⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠄⠀⠀⣧⠀⢀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⡆⠀⠀⡇⢾⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠓⢤⣿⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⣧⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠋⡀⠀⠀⣼⢣⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⢀⡠⠾⣀⣀⣀⣠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠄⢰⡇⠈⣆⠀⠀⠀⠀⠀⣿⠈⠉⠉⠁⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡾⠀⠀⢹⡄⠀⠀⠀⠀⢻⠀⢀⠔⠁⢀⡠⠊⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠀⠉⠃⠀⠀⠘⣇⠀⠀⠀⠀⢸⡄⠀⣠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⠀⠘⠷⡄⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠘⢣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠠⣤⣤⣤⣤⣄⢀⣃⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠤⣄⡀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢹⠀⠉⠉⠉⠉⠓⠚⠓⠤⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⣄⡸⡄⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀
⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠈⠂
⠀⠈⠀⠀⠀⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⢿⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠂⣀⣀⡀⠀⠀⠄⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⡏⠉⠉⠉⠉⠀⠀⠐⠂⠀⠀⠀⠀⠠⠁
⢀⣴⠟⠁⠈⠛⠁⠀⠀⠀⠈⠳⠤⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠇⢹⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠑⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢦⠀⠀⠀⠀⣀⠀⢠⡀
⠀⠠⠀⠀⠀⠉⠒⢤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠉⠀⠀⠀⠀⠀⢹⡦⢤⣤⣀⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠠⠤⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠀⠀⠀⠀⠀⠀⡠⠄⠘⠀⠘⣄⠴⠆
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⡀⠂⠀⠐⠉⠁⠀⠙⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠰⠶⠶⠤⢤⣤⣄⠀⠀⠀⠀⠀⢠⡤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⠀⠀⠀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⢤⢏⡌⠡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⢡⠹⣤⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡛⣿⣴⣟⡲⣾⠛⠈⢦⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡜⠑⢛⣷⢶⣻⣦⣿⣙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⣧⣿⣿⣶⣻⣏⣰⣃⣆⣸⠶⡀⠀⠀⠀⠀⠀⢀⠾⣇⣐⣙⣆⣿⣿⣶⣿⣿⣼⣿⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⣿⣻⣿⣿⢻⣿⣧⣽⣿⣿⣃⣽⡀⠀⠀⠀⢠⣋⣚⣿⣿⣯⣸⣿⡟⣿⣿⣏⣿⣁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⣿⣟⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⣿⣻⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣬⣹⣿⣿⣿⣿⣿⣿⣿⡿⠏⢸⣿⡇⠹⢿⣿⣿⣿⣿⣿⣿⣿⣏⣥⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣛⣻⣛⣿⣿⣿⣿⣿⣿⣴⣤⣿⣿⣷⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠿⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⡾⣿⢾⣾⣿⡿⣿⢻⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣜⠸⣿⣆⢿⡶⣄⣒⡀⣠⡇⠀⢸⠿⣦⣼⣿⠿⠃⠀⢿⣄⠠⣃⣴⢿⡷⣼⣿⢈⡵⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢸⣁⣍⣽⠾⢧⣹⠁⣐⠀⣧⠄⠊⢠⡀⠞⢠⡄⠑⠦⣾⠀⣀⡄⣯⠼⠿⢯⣠⣡⢆⣰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⠉⣉⢳⣀⣷⡼⣈⣿⢷⡟⠀⠐⡟⣀⡀⠈⢛⠂⠀⢻⡿⢯⣤⠳⣾⣄⣞⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠓⠛⢻⢡⣾⣉⣳⣼⠧⣠⠏⡉⠋⡇⠀⢀⠙⣄⠘⣧⣾⣩⣽⠄⢝⡉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣈⡌⢹⣏⡄⢈⣧⠟⠉⠙⢣⡟⢖⡇⠠⣽⣏⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⣮⣈⡗⢻⡍⣬⣧⠈⣰⣼⣇⣟⢻⡇⣼⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠙⣏⢉⣍⡍⣀⢻⣍⡍⢩⡏⢁⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠋⢀⡽⠄⡷⡟⢼⣁⢈⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣯⣘⣝⣟⣨⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣫⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⢿⣻⣽⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣽⣛⣭⣵⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣛⡛⠿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠈⠙⢦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣷⣮⣝⣻⢿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣴⣧⣤⣤⣶⣯⣥⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣭⣽⣭⣿⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣷⣷⣾⣿⣶⣿⣷⣿

⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢆⣀⣀
⣿⣿⣿⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠂
⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⡿⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡆⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡄⠘⠁
⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠃⠘⠢⡁⢸⠀⠀⡇⠀⢣⢰⠀⠀⠂⣿⠀⢸⢰⡀⠀⠁⠀⠀⠀⠀⠀⢀
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠈⠀⠈⠀⠁⠈⠁⠀⠉⠀⠁⠀⠀⠈⠀⠈⠀⠁⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠀⢀⡀⠀⡀⠀⠀⠀⡀⢉⠀⡁⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣨⠀⢈⡆⠀⣧⠸⡀⠀⠇⢸⠀⡇⢈⠑⡄⠀⠀⢀⣠⡀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⢿⣿
⠀⠀⠀⠀⢰⣬⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⢸⣿
⠀⢠⡀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠛⣉⡻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠛⠃⠀⠚⠃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣄⠰⠶⢂⣰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠏⠉⠻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⢤⣞⢹⣿⣦⣤⣾⣿⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣼⣥⠼⣶⣶⣬⣻⣿⣿⣿⣿⣿⠧⡀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⠀⠙⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⢿⡇⠀⠀⠀⠈⣿⣿⣟⣿⣿⣿⣧⠀⡼⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⣧⠀⠀⠀⢀⡸⢏⣾⣿⣿⣿⣿⣿⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢰⠋⠛⢿⣿⠿⠛⢹⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠗⠀⡇⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠒⠒⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠲⠶⠶⠤⠤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⠤⠴⠶⠶⠚⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠓⢲⠶⡶⠒⠒⠒⠲⠖⠒⠒⠒⠒⠲⡶⢶⠒⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⣼⣭⣍⣙⣛⣓⠲⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣶⣿⣿⣿⡄
⠀⠀⠀⠻⣄⠀⠀⠉⠉⠙⠛⠲⠦⣭⣙⡓⠦⣤⣀⠀⠀⠀⠀⠘⣦⠹⣄⣠⠶⠚⠲⢦⣀⡼⢣⡞⠀⠀⠀⠀⣀⣤⠴⣖⣻⡽⠿⠛⠋⠉⠉⠁⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢬⣝⡲⢤⣀⠀⠈⢳⡟⠁⠀⠀⠀⠀⠙⣷⠋⠀⢀⣤⢶⣻⡽⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣄⣀⠀⠀⠀⠀⠀⠈⠉⠓⢮⣟⣶⠟⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⡿⠞⠉⠁⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⠤⣤⣤⠬⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⢤⣤⣤⠤⠴⠶⠖⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡶⠃⠀⠀⠀⠉⠀⠾⠋⠃⠀⢠⡄⠀⠙⡛⠀⠉⠀⣼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠇⠀⢰⣦⡀⠀⠀⢀⣤⠆⠀⠀⠀⠐⠦⠁⠀⠀⣄⠈⠛⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃⠀⡀⠀⢻⡿⡦⠀⠉⠉⣴⠆⠀⠀⣰⣦⣧⣄⠀⠈⢳⡆⢐⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣇⠀⠈⢠⣤⣅⣄⠂⠀⣸⡟⠻⠿⠴⠟⠉⠉⠻⢿⣶⣴⡞⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡛⣷⠴⠛⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢾⡄⠀⠴⠟⠛⠷⢶⡄⠀⠀⠀⠀⠀⠈⣋⣥⣼⣿⣿⡇⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢼⡇⠀⠀⠀⣀⣠⣠⣀⠀⠀⠀⠀⡴⣻⣿⣿⡿⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣧⠀⠙⢻⣿⣿⡏⠙⡹⠀⠀⠸⡇⠙⠻⠿⠧⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡉⠀⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⢻⡀⠀⠀⠀⣿⣿⡞⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣇⠀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣤⡀⠀⢠⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠘⠀⠈⢄⠀⠀⠀⠀⠸⢿⡿⣿⣿⠟⠀⠘⣿⣿⣭⣽⣿⠛⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣴⠆⠀⠀⠣⣄⠀⠀⠀⠀⠙⠋⠉⠀⣴⣾⣿⣿⣿⣿⣿⡗⠸⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⢿⣦⣤⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⡏⢿⣻⣏⢿⣿⣷⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠷⣬⣯⣤⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀
⠀⠀⠀⠀⠀⣠⠔⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣦⣄⡀
⠀⠀⠀⠀⢠⠁⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⠛⣷⣄
⠀⠀⠀⠀⠃⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⣿⣆
⠀⠀⠀⢈⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢻⣿
⠀⠀⠀⡌⠀⠀⠀⢀⣿⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⡇⠀⢀⡴⣿⣿⡿⠋⠀⣠⣾⣿⣿⡟⠛⢶⣶⣶⣶⠶⠖⢶⣶⣶⣶⡤⠄⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠁⠀⡞⣼⣿⡟⠁⢀⣾⣿⣿⣿⡿⠀⠀⠈⣿⣿⣿⣷⡄⠀⢹⣿⣿⣿⣦⡈⠑⣤⡀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢐⣇⣿⡟⠀⣠⣿⣿⣿⣿⠟⠀⠀⣾⡆⠈⢿⣿⣿⣿⡄⠉⠙⣿⣿⣿⣷⠒⢿⣿⡄⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⡀⠀⣿⡿⠀⢰⣿⣿⣿⣿⠏⠀⠀⣼⣿⣿⡄⠈⢿⣿⣿⣷⡄⠀⠹⣿⣿⣿⡀⢸⣿⣿⡀⢀⣿⡟⠀⠀⠀⠀⠀⢸⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⢷⣿⣟⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⠍⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⠋⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡜⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣦⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠛⠿⢷⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣦⣀⠀⠀⠘⢷⡀⢀⣀⠀⠀⣠⡀⢀⡤⠋⠀⠀⢀⡀⣄⠀⣀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢾⣿⣹⣿⣿⣷⡔⣢⣀⢻⣼⣿⣷⣾⡿⣷⣾⣥⠄⡺⢇⠁⡷⢼⣴⡷⢬⣽
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣷⣿⡛⣿⡲⣿⣿⡟⠫⢐⣿⣿⣟⣿⡿⣿⣦⣻⡗⣿⡚⣟⣵⡿⢭⠟⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⢿⣿⣿⣷⣿⣧⣶⣶⣿⣻⣬⣿⣼⣱⣭⣿⣿⣼⣯⣟⣫⣿⣱⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣀⣬⠿⠟⠛⠛⠻⠿⢿⣛⡟⠛⠻⣻⠻⡟⣿⠿⠿⠿⢿⣿⣟⠛⠉⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢛⡿⣿⡷⢲⡶⠶⠶⣾⡿⣾⡿⣷⡾⠶⣶⣶⣶⣶⣶⣶⣶⣶⢶⢶⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠯⠼⠁⣄⣹⣤⠛⣳⡼⣻⢶⢧⡿⢀⡸⡧⣠⣯⠳⣿⡒⢛⣷⠝⣧⣏⣿⣍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⢶⣦⣄⣼⣾⡟⢭⢀⣇⡀⠈⡋⠹⡄⠸⢉⡧⢺⣏⠻⣯⡼⢆⣥⢷⣀⣼⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⡄⠁⢀⠼⣿⣼⠏⢿⢻⠒⠆⡠⠝⠣⠒⠒⢳⣏⢠⡿⠛⠛⢛⠻⢭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢧⣀⡼⢿⣖⠹⣄⣦⣌⠐⠒⠘⡄⣠⠆⢹⣟⠓⡖⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣏⣷⣤⠎⡁⡜⠛⠋⡩⢹⡥⣰⣿⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⢿⣶⣦⣧⣦⣵⣤⣤⣥⡿⠅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠁


⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⠤⢤⠀⠰⢤⣴⠰⢠⠀⠠⣤⣴⢠⢤⠀⢰⢤⣤⢰⢤⠄⢫
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⡤⡤⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠐⠒⢺
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢀⢡⠃⠀⠀⠀⠀⠀⠀⢡⢠⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⡄⡌⠀⠀⠀⠀⠀⠀⠀⠈⡀⡄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢠⢡⠈⡀⠀⠀⠀⠀⠀⠀⠀⢁⢡⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄⡘⡇⠁⠀⠀⠀⠀⠀⠀⠀⠘⠈⡀
⠀⠀⠀⠀⠀⢀⡄⠀⠄⠤⠄⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⢤⠤⢤⣤⣄⣀⠀⠁⠂⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠜⠀⠀⣀⣀⣀⣀⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠑⢌⠉⠀⠛⠿⣦⡀⠘⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠙⠋⡉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠐⠀⠈⠳⡀⠘⠀⠀⠀⠀⠀⠀⠀⠄⢀⠞⠁⢀⡀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡈⠀⠠⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⣠⠀⡀⠀⠀⠀⠀⠀⡄⠀⡁⠈⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠸⠀⡅⠀⠀⠀⠀⠀⢁⢰⠃⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠄⠄⠤⠤⠿⠼⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣦⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢕⡇⠀⠀⠀⢋⠆⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣝⡞⣷⠀⠀⠀⠀⢠⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⡀⠀⠀⢀⡾⢻⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣸⣧⣧⣻⡏⠒⠒⠁⣧⡑⢄⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡿⣿⡟⣿⠀⠀⠀⢰⠂⠙⠶⠶⠶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢇⢸⢇⣿⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡀⡈⣼⠮⢂⠀⠀⣰⣷⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⡞⠁⢀⡏⢀⣾⣿⣿⣷⠀⠀⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⣿⣤⡎⣡⣾⣿⣿⣿⣿⡗⣠⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠉⠉⠀⠀⡍⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⣘⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣈⣆⠇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇⠀⠀⠀⠀⠀⣠⡾⠛⠉⢻⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣷⡀⠀⠀⢀⣾⡟⠀⠀⠀⠀⢣⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⠘⣿⣤⣴⣿⡿⠀⠀⠀⠀⠀⠈⡆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⢹⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡼⡇⠀⠀⠙⠿⣧⠀⠀⠀⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⡇⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⡌⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡣⣧⣣⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡇⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣼⠐⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠀⠠⢤⣤⣄⠄⢀⣀⠀⠠⠄⠒⠈⠉⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡐⣼⣿⣯⣁⠂⢝⣆⣀⣀⣀⣀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡾⠿⢿⠳⠦⠉⠲⣿⣧⣤⡀⠀⡎⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢧⠀⠀⠀⠀⠀⠸⠟⣿⡎⣯⢸⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘⠋⡄⢻⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢹⣆⠀⠀⠀⡀⠀⣧⢇⢿⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠙⠷⠲⠊⠀⢀⠘⠑⢄⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠳⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢰⣿⣿⣿⣦⡀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⣼⣿⣿⣿⣿⣷⡄⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⣿⣿⣿⣿⣿⠿⠷⠼⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠉⠉⠉⠒⢤⡀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣲⣽⣾⣿⣏⠁⠀⠀⠀⠀⣉⠉⢙⡛⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⣪⡵⠛⠙⡛⠣⣿⠛⠀⠀⠀⠸⣿⣶⣿⣷⣿⣷⣶⣍⡻⣷⣦⡀
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⡟⠀⡐⢀⠻⣄⠻⢰⠀⠀⠀⠀⠈⠙⢿⣿⣟⠻⣿⡝⠻⣮⣿⣿⣦⡀
⠀⠀⠀⣠⢟⡽⠛⣩⣿⣿⣿⣿⣿⡁⠀⢃⢿⡀⠈⠳⢄⣧⡀⠀⠀⠀⠀⠀⠙⣿⣧⠘⢿⡄⠈⠻⣿⣿⣷⡄
⠀⠀⠀⠃⡾⢁⣾⠟⠙⠉⣩⣾⣿⣇⢀⠀⠘⢷⣤⣇⢠⠈⠙⠷⣦⡐⣄⢢⡀⠈⢿⣧⠀⠻⡀⠀⠙⣿⣿⣿⣆
⠀⠀⠀⠀⡇⢸⣧⢠⣴⠟⠁⠸⣟⣿⣦⣷⡀⠀⠙⠻⣿⣦⡘⣶⣄⠙⣾⣆⢳⡄⠘⣿⡇⠀⠀⠀⢀⠸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⠀⠈⠻⣏⠻⣿⣶⣄⡀⠀⠉⠻⢮⣿⣷⣌⢿⡎⣿⡀⢸⣿⠀⢧⡀⠸⣆⢹⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣛⣿⣿⣶⣦⣄⡈⠙⢿⣾⣿⣼⣧⠀⣿⡇⢸⣧⠀⣿⡌⣷⣿⣷⡙⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠶⢦⡄⠀⠀⠀⠠⠶⠛⣋⣉⣁⡀⠀⢉⠙⠿⣦⡀⠹⣿⣿⣿⠀⢸⣧⢸⣿⡄⢹⣧⣿⢸⣿⣷⡡
⠀⠀⠀⠀⠀⠀⠀⠰⠞⢳⣦⡀⠀⠀⠀⠀⠟⠹⣿⡟⡿⠟⠋⠀⠀⠘⣷⡄⢹⣿⣿⠆⢸⣿⣼⣿⡇⣸⢿⣿⢸⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢘⣿⣿⠀⢸⡟⣿⠿⣷⡿⠘⣿⣸⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⢸⣿⡟⠀⣼⠃⡿⠀⣿⠇⠀⠃⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣸⣿⣾⣿⠇⢰⡟⠀⠀⠀⠟⢸⠇⣼⣿⣿⣿⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣷⣿⢿⣿⡟⢀⣿⠃⡀⠀⡄⢀⣿⣰⢟⣿⣿⡏⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣼⠁⣼⠙⢏⣿⣿⠁⣼⡏⢠⠇⢰⡇⣼⣿⢋⣾⣿⣿⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠟⠃⠀⠀⢠⡏⠀⣿⠀⣼⣿⡇⣼⡿⢠⡟⢠⣿⣿⠟⣡⡾⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⢀⣿⡟⣸⣿⢣⣿⣱⣿⡿⢋⣴⠟⢡⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⢸⣿⣧⣿⣣⣿⣿⠟⠋⣴⠟⠁⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠰⢸⣿⣿⣿⣿⡿⠟⠁⠀⠈⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢧⣿⣿⠟⠋⠀⠀⠀⠀⡠⢂⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢿⡄⡜⣿⡿⠀⣠⠆⠀⣠⠞⣡⡾⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⡼⣷⡸⣽⣷⣾⣧⢎⡾⢃⣴⠏⣴⠋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢠⡇⠘⣷⣽⣿⣿⡿⠋⣴⠟⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡜⠻⣿⣿⣷⠞⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⢱⣠⢈⣿⣿⣦⡾⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣾⣿⣿⢠⣿⣿⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⣿⣷⣾⣿⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏⢿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟

⠀⠀⠀⠀⠀⠀⢠⠀⡄⢠⠀⠄⠀⠀⠄⠀⠠⠀⠀⠀⠀⠠⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣤⡄⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⠟⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣦⣾⣿⣿⣿⣟⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢰⣿⣿⣿⢿⡆
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣟⣿⣷⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⢀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⣠⣴⣿⠟⠁⣿⣿⣿⣿⣾⣿⠉⢣⣀⣀⣀⣿⣿⣿⣿⠛⣿⣿⣷⣦⠀⠀⢀⣴⣿⣿⡿⠋⠈⣦⣀
⢦⡴⠞⠛⠉⠀⠀⠀⣿⣿⣿⣿⣿⠃⡄⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⣾⡿⣿⣿⣆⣿⣶⣿⡟
⣩⣀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⣷⡿⠋⠀⢸⣿⣿⣿⣿⣟⡃
⠛⠁⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⣿⣿⣿⣎⠻⣿⣿⣿⣿⠻⣿⣿⣿⣿⠆⠀⣼⡇⠀⠀⠘⣿⣿⣿⣿⣥⡾
⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⢹⣿⣿⣿⢿⣿⣿⣌⠛⠛⣿⣿⣿⣿⣿⣿⣆⠀⠘⠿⠇⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣷⠄⢌⣿⣿⠛⠿⣿⣿⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣆⣿⣿⣿⣿⣿⣿⣿⠃⡀⢸⣿⣟⠀⠀⠈⣿⣿⡗⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⡀⠀⢭⠃⠀⠀⢸⣿⡿⠁⠀⠀⠀⢸⣿⣿⡟⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠇⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠘⣿⣿⠇⠹⣿⡇
⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢈⣿⣿⣷⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⡟⡇
⠀⠀⠀⠀⢸⡿⢻⠿⠈⣻⣿⣿⠏⠀⠀⠀⠀⠘⣿⣿⠁⠀⢀⣀⢨⣿⣧⢤⡄⢠⣶⣶⣿⣿⡗⣀⠸⣻⣷
⠈⠀⠀⠀⠀⠀⠘⠒⣶⣿⣿⣟⠓⢀⠀⣀⣼⣿⣿⡇⠀⠉⠙⠡⠴⠉⠀⠈⠢⠀⡨⠉⡈⠻⠟⠹⠿⢿⣿
⣠⣀⣠⣤⣤⣄⣀⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟⠉⣼⣿⣿⣿⣿⣶⣶⣤⣌⣀⣀⠀⠀⠀⠀⢲⡶⣿⣿⣿
⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣤⣀⣤⣄⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠻⡿⠉⠉⠉⢻⣛⡿⠛⠋⠉⠉⢉⣽⣿⣿⣿⣉⠉⣉⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢀⣶⣿⣆⠀⣀⠘⠻⣟⠀⢀⣀⠀⢀⣿⣿⡶⠛⠀⣨⣻⠀⢸⡇⠀⠀⠀⠀⠀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⣿⣿⣿⡰⣿⣿⣷⣬⣦⣄⣾⣿⣯⣼⣷⣿⡌⢙⣷⣬⣷⡶⣶⠀⠀⠀⢠⣾⣶⣄
⠀⠦⠄⠀⣴⡶⣤⣄⣀⣰⣿⣿⣦⣤⣀⣀⡀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⣹⣿⣾⣿⣿⣿⡶⠉⠀⡶⣶⣿⣿⣿⣯⡄
⣿⣋⠀⢀⣏⣷⣿⣿⣿⠿⠋⣹⠟⢻⠋⠉⠀⠀⠀⠉⣻⣿⣿⣿⠻⣿⣃⣠⣆⣂⡉⢹⢻⣿⠿⠟⣻⣿⣶⣿⣿⣿⣿⣿⣿⣾⡟⢀⠀⠀⠌⠥⣿⣿⣿⣿⣿
⢿⣿⠿⠋⠙⠿⠻⣿⣯⣰⣴⠷⠀⠼⣀⢀⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣯⣤⣦⣶⣶⣟⣿⣻⣿⣿
⣼⣧⣤⣤⠄⠀⢋⣛⣋⣉⢠⢤⡤⢀⣴⣿⣷⣤⣤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡿⠿⠿⠿⠭⠩⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⡿⠿⠿⣿⣿⣿⣿⣿⡿⣳⡾⠻⣿⣚⠉⡉⠛⠛⠺⣿⣿⣷⣿⣿⣿⣯⣭⢍⣭⣿⣿⣿⣿⣿⡿⠿⣿⣿⡿⣿⣿⢿⠿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣾⣯⣽⣿⣿⣿⣿⣿⣿⣷⣶⣶⣧⣀⣉⣀⣥⢶⣿⣿⣿⣿⣿⠿⠿⠶⢤⠿⠋⠛⠿⢿⣿⣶⣿⡿⡷⣶⣤⣶⣤⣄⣾⣧⣶⣿⣿⣿⠿⠿⡿⠿⣿⣽⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣵⣷⣾⣿⣿⣯⣭⣿⣵⣶⣾⣿⣿⣶⣖⣒⣚⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣤⣿⣿
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠠⣤⣾⣿⣿⣿⡄⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣡⣢⠊⠉⠙⠻⠿⡟⠹⠀⠔⠊⠀⣀⠤⠐⠖⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⠒⠂⠈⠉⠉⠙⠄⠀⠀⠀⠀⠀⢸⡆⠀⠔⠈⠀⡠⠊⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⣀⠔⠂⠺⣿⢷⢶⣄⡀⠀⠀⠀⠀⠳⣀⣀⠔⠈⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠉⠐⠠⠾⠄⢀⠻⢿⣦⠔⠀⠀⠀⠐⡀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⢻⡏⠿⢛⠀⠀⠀⠀⠀⠰⣄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢄⠀⠈⢀⣿⣧⡀⠀⠀⠀⠀⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣼⣿⣿⣿⣄⠀⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⠇⠰⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠙⠲⣦⣤⣤⣤⣤⣤⣤⣴⠞⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⢿⣿⣿⣿⡿⠟⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠻⡿⠋⠁⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣻⡿⣿⣿⣿⣿⣡⣴⣿⣭⣭⡟⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠿⠛⠉⠙⠿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⡹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⢶⠀⠀⠰⠾⠛⠁⠀⠀⠀⠈⣿⣿⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⡇⢰⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⣿⣿⣯⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣿⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢈⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣝⢻⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣿⣻⣿⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⠟⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣻⣿⣿⣿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢻⣿⣧⡉⠻⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠀⠀⠀⠀⠀⠀⣀⡶⣶⠄⠀⠆⠀⠀⠀⠀⠈⡎⣹⣿⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢽⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠻⣆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⣿⣆⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⢸⡀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣠⣀⣀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋

⡟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣾
⡿⢿⣿⣟⣿⢻⡿⣿⣿⠟⣿⡿⢿⡿⠿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿
⡇⠈⠙⠛⠛⠓⢒⣉⣀⣀⠀⠀⠀⠀⠀⠀⠓⠻⢛⣹⣃⡉⣳⣱⡈⠛⠿⣿⠶⠎⣉⢁⠻⣽⠿⠿⠿⠿⢿⣏⣿
⡇⠀⠀⠀⠀⠀⠈⠀⠉⠉⠁⠀⠉⠒⠀⠀⠀⠀⠐⠉⠻⢿⣿⣿⣽⣿⣏⣩⣴⢤⢤⣀⣈⡉⠉⠁⣐⠂⠠⣴⣿
⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠶⣭⣛⠻⠉⢿⣿⣿⣷⣶⣶⣮⣿⣟⣹⣿⣿⣤⣤⣼
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠈⠡⠀⠄⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠟⠙⢛⠛⣿⣿⣿⣿⣿⣿⣿⣟⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⢽⣛⠛⢻⣿⣿⣿⣿⣿⣿⣿
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣀⣁⣉⣛⣟⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⠔⠚⠉⠉⠉⠒⢄⠀⠀⠀⢀⡠⠐⠂⠀⠐⠒⠤⡀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠦⡀⠀⠀⠀⠀⣠⠖⠉⠉⠉⠓⠦⡀
⡐⠁⠀⠀⠀⠀⢀⡀⠀⢣⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢀⠞⠁⠀⠀⠀⠀⣀⠈⢦⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠘⢦
⡇⠀⠙⠀⠀⠀⠈⢁⠀⢸⠀⡇⠀⠀⠀⠆⠐⠇⠀⣠⠀⢰⠀⠸⠀⠘⠃⠀⠀⠈⠉⠀⠈⡇⠀⡇⠠⡄⢀⡀⢠⡄⢹⠀⢸
⢣⠈⠢⣀⠀⠀⣀⠞⠀⡜⠀⢣⠐⢇⠠⣀⣄⣠⠤⠜⠀⡄⠀⢸⡀⢲⣖⣖⣺⣹⣭⡟⢀⡇⠀⢣⠀⠳⣈⠁⠀⣠⠞⠀⡸
⠀⠳⢄⡀⠉⠉⢀⡠⠞⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠑⢄⡉⠛⠚⠒⢁⣠⠎⠀⠀⠀⠳⢄⣈⠉⢉⣁⡠⠞⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢀⡤⠒⠒⠛⠒⠢⣄⠀⠀⠀⠀⢀⡤⠖⠒⠒⠶⢄⡀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠦⣀⠀⠀⠀⠀⢀⡠⠖⠒⠚⠒⢦⡀
⢠⠋⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⠞⠁⠀⠀⢀⠀⠀⠈⢣⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⣆
⡇⡀⠀⠈⠁⠈⠁⠀⠠⠈⡆⢸⠁⠀⠶⠀⠀⠘⠓⠀⠀⣷⠀⡘⠀⠀⠀⠁⠀⠀⡤⡀⠀⡇⠀⡇⠀⣠⡀⠀⠀⢀⣀⠀⢸
⢇⢣⠀⠀⠀⠀⠀⠀⣰⢡⠁⢸⡀⢤⣔⣒⣒⣒⢦⠟⢀⡏⠀⢱⠀⢀⢔⣚⠉⠉⠉⠂⠀⡇⠀⢇⠀⠈⢡⠔⢤⠈⠁⠀⡜
⠈⠢⣙⠢⠤⠤⠤⣚⠵⠃⠀⠀⠳⣄⠙⠢⠤⠒⠁⣠⠞⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠈⠢⣀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⢀⡤⠒⠚⠓⠒⠦⣄⠀⠀⠀⠀⢀⡠⠖⠚⠛⠒⢤⡀⠀⠀⠀⢀⡤⠒⠚⠛⠒⠦⡀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠒⠦⣄
⢠⠋⠀⠀⠀⠀⢀⠀⠈⢳⡀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠙⡄⠀⢠⠋⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠱
⡇⠀⠨⠛⠀⠐⠻⠀⠀⠀⡇⠀⡇⠀⡤⠢⠈⠀⠔⠄⠀⣷⠀⣾⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠐⠀⢀⣀
⢧⠀⠐⠒⠒⠒⢶⡲⠂⢀⠇⠀⣇⠀⠀⠀⢰⡦⠀⠀⢀⡏⠀⠸⡀⠀⠀⠀⣀⡀⠀⠀⡸⠀⠀⢇⠀⠒⡖⢾⠉⠉⠉
⠈⠳⢄⡀⠀⠀⠀⣀⡴⠋⠀⠀⠈⢦⡀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠑⢄⡀⠋⠀⠈⣠⠔⠁⠀⠀⠈⠳⣄⡙⠋⠀⠀⣀⡤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠛⠉⠙⠛⠿⠟⢻⠅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⠁⡀⠀⠂⠀⠀⠐⠃⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢠⠢⢈⣶⣤⣤⣤⣦⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠚⠀⠀⢀⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀

⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⣼⡀⠀⣸⡇⠀⠀⢀⣄⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠸⣿⡿⠿⣿⡿⠃⠀⠀⢀⣀⠀⣀⣀⠀⠀⠀⠿⠿⠿⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠛⠛⠛⠿⠃⠀⠀⣀⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⡆⠀⠀⠸⣿⣿⣿⣿⡿⠁⠀⠀⣴⣶⣶⣶⣶⡄⠀⠀⠻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⢀⡄⠀⢰⠀⠀⠀⠀⠀⣰⠶⣆⠀⠀⠀⠀⠀⢀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⠀⠀⠀⠀⣸⠃⠀⢸⡇⠀⠀⠀⢠⡇⠀⢸⣆⠀⠀⠀⠀⣸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⢿⣷⣤⣾⣿⠀⠀⠀⣾⣧⢀⣼⣿⡄⠀⠀⣰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⢀⣀⣈⣉⣀⣀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠉⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠐⢦⣤⣤⣤⠤⠀⠀⠸⣿⣿⣿⣿⡿⠄⠀⠀⢿⣿⣿⣿⡿⠀⠀⠀⡾⢷⣶⣶⣶⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠈⠛⠟⠋⠀⠀⠀⠀⠁⠀⠉⠋⠁

⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡀⠀⠀⠀⢀⡆⠀⠀⣆⠀⠀⠀⣀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢠⣾⣇⣀⣀⠀⠀⠀⢀⣶⣄⢀⣨⣇⠀⠀⠀⣼⣷⣶⣶⣿⡄⠀⠻⢿⣿⣷⣾⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠇⠀⠀⠸⠿⠿⠻⠿⡿⠂⠀⠀⠻⠿⠿⠿⠿⠃⠀⠀⢠⣄⣁⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⣤⣤⣤⣤⣤⡀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⠿⠀⠀⠈⢿⣿⣿⣿⠟⠀⠀⠀⠻⣿⣿⣿⠟⠁⠀⠰⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣛⡻⣿⣭⣯⠻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡏⠁⠀⠀⠀⠈⠉⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣗⣿⠁⠀⠀⠀⠀⢀⡀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⣿⣿⣿⢿⣟⣒⣂⠀⠀⣫⣭⣥⣄⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⢻⣿⣿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡿⣿⡿⣹⣿⣿⠟⠀⠀⠙⠿⠿⠋⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣿⣰⣿⣟⣿⣧⡀⠀⠲⠶⠂⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⣾⣵⣟⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣿⢿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢳⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣶⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣾⢿⢻⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠸⡿⠓⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠐⠃⠀⠀⠈⠉⠉⠉⠉⠋⠉⠙⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⠿⢶⢤⣦⣤⣄⣄⣀⣀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠧⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠉⠢⡀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⠖⠲⢤⣄⡈⢆⠀⠀⡜⢁⣠⡤⠒⢲⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣂⡀⢙⢿⣺⡄⢰⣗⡗⡋⣀⣐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣵⣫⡥⢍⢳⣥⣽⣫⣿⣭⣨⢞⡭⣬⣽⣮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⢚⣯⣧⠿⢟⡻⢧⣿⣙⠐⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠛⠿⠿⠔⠴⠿⠿⠻⠿⠿⠿⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⣤⢤⡤⣤⣤⣤⠤⣤⣤⣤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢀⠛⡒⢀⡀⣠⠞⠁⠘⢦⡀⣀⡀⢲⢻⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠻⣶⣃⣽⣈⢵⣄⠀⠀⣴⡏⣝⣏⣲⡾⠛⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠪⢝⢆⢁⠤⣄⢂⠐⣢⠤⢉⣒⡫⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⢺⣱⡿⢿⣮⠗⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢡⡦⣴⡅⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠷⠿⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁