Head cold Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⢀⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠘⠁⠀⠀⠀⣈⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣋⣡⣴⡾⠇⠀⢠⣌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣤⣶⣶⣿⣿⣿⣿⠋⣴⣶⣦⡀⠹⣆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⠇⠀⠁⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⣛⠛⠻⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⢠⣿⣿⣿⣦⠘⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠔⢀⠀⠆⠀⠀⡀⠀⢀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⡀⠀⢀⠀⠀⡐⡀⡆⠀⠀⠀⠀⠀⡄⠀⠘⢁⣁⠁⠀⢠⡄
⠀⠃⠀⠀⠐⡀⣀⠄⠀⢠⡀⢸⠊⡀⢠⠀⠀⠀⣀⣠⣔⡄⠀⢠⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠰⢀⡄⢢⠳⠊⣄⠰⠂
⠀⠘⠤⠀⢈⣁⠱⠚⠈⠓⠎⣉⠅⠠⠚⠀⠀⠀⠊⠉⢙⢸⣩⠒⠒⠦⠖⠠⢁⠀⠀⠀⠀⠀⠀⠀⠀⢋⣀⠛⠘⢅⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⠂⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠠⠤⠐⢠⡄⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠀⢈⡒⠀⠘
⠀⠀⠀⡀⠉⡀⠀⠄⠀⠠⠀⢀⠋⢐⠀⠀⠀⠀⠀⠀⠉⠠⠀⠁⠂⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠐⠀⠀⠑
⠀⠀⠀⠐⠈⠤⡂⠀⠀⠀⢐⢬⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⣂⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⡀⡀⠌
⠀⠀⠀⠀⠄⠰⡍⠉⠛⠸⠜⠐⠨⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡂⠀⠡⠀⠈
⠀⠀⠀⠀⠀⠀⠂⠑⠂⠠⠔⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣉⣤⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⣡⣤⣤⣄⠙⢿⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣧⡈⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⠘⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣄⠙⠻⢶⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⢋⣴⣿⣿⣿⣿⣿⣿⡿⠏⠐⠶⣦⡙⢷⣄⠹⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⣿⣿⣿⣿⣿⣿⣿⠋⣤⣶⣦⡄⠈⠿⠀⢻⡄⢹⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⣤⠈⢉⡉⠐⣿⣿⣿⡿⢀⣶⣶⣤⣤⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠹⣧⡈⢷⣄⡈⠉⣉⣠⣾⡿⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣦⡘⢷⣤⣉⠛⠠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⣌⠙⠻⠂⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢀⠀⠀⠀⠀⠺⣷⣶⣶⣤⣤⣀⠙⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⢘⣉⡀⠀⣒⠀⠄⢹⣿⡉⠛⠻⢿⣷⠘⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠤⠒⠂⠀⠉⠀⠀⢸⣿⣿⣿⣷⣦⠹⠇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⣧⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠁




⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⣠⠞⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠈⣀⡴⠚⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡏⣴⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⢀⡴⠒⠦⣄⡤⠦⣄
⠀⠀⠀⠀⠀⢰⠀⠹⣜⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠃⠀⠀⢀⡏⠀⠀⠀⠈⠀⠀⣸⠧⠤⣀
⠀⠀⠀⠀⠀⠀⠆⠀⡏⠳⣤⠖⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠀⠀⠒⠋⠁⢸⠆⠀⠀⠀⠀⠘⠁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡦⠀⠀⠙⠛⠛⢯⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠉⠙⠲⠬⣽⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣷⠀⣷⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠞⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣀⣀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠸⠁⠸⠁⠀⠀⠀⠀⢸




⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠇⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⢄⢠⠠⠤⠀⣄⠀⡤⡄⠀⠀⡤⠄⠀⡠⠀⢠⠄⠠⢤⢀⠄⢄⢠⢠⢠⠤
⠀⠀⠀⠀⠀⠀⢀⣉⣀⣀⡠⢹⠀⣀⣸⠘⣆⣏⡇⣀⣀⣏⣁⠀⣈⡶⢸⡀⣈⢱⡸⣀⡘⣸⢱⣘⣉⡀
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢀⠠⠐⠒⠒⢦⠖⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠄⠚⠠⢀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠁⡔⠁⣐⡎⠀⠀⢩⣒⠙⢄⠀⠄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⣦⣃⠠⠀⡖⠉⠉⢣⠄⠠⣘⣴⠀⣀⡀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⠂⠁⠀⠀⢀⣾⠃⠔⡟⣰⢻⠘⠆⢛⠀⠘⣧⠀⠀⠈⠐⡀
⠀⠀⠀⠀⠀⠀⢠⠒⠀⠐⠁⠀⠀⢀⣠⣎⠉⢠⠎⢠⠇⠈⠀⠘⡀⢦⠂⠙⣧⡀⠀⠀⠈⢂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⡴⠋⠁⠈⣱⠏⠀⡞⠀⠀⠀⠀⢃⠈⢖⠈⠉⠻⣦⡀⠀⠀⠡⡀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⣠⠎⠀⠀⠀⣰⠃⠀⢰⠁⠀⠀⠀⠀⠘⠀⠘⡄⠀⠀⠈⠻⣆⠀⠀⠐⡄⠀⠐⡄
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠐⠏⠀⠀⠀⠀⢧⣤⣄⡈⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠉⠻⣦⣀⣤⠞⠋⠉⠁⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠤⠀⠊




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⡀⢀⡀⣀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⣾⣬⣷⣦⣟⣧⣤⣾⣿⣿⣿⣾⣿⣶⣶⣷⣶⣿⣬⡷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣯⣿⣿⣿⣜⣿⣿⣯⣴⡻⣿⣿⣿⣻⣿⣿⣛⣯⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡇
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⠹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⡄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢾⡿⠓⣍⢻⣿⣿⣿⣿⡿⢿⡇⠀⠁⠀⠀⠠⣦⠀⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣤⣤⠃⠸⣿⣿⣿⠃⠀⢻⡁⠀⠀⣼⠈⠀⠀⢾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠿⠏⠀⠈⠺⣿⣿⠷⠀⠈⠥⠀⠀⠁⠀⢀⠀⠀⢻⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⠀⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⠄⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣭⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠉⠛⠛⢶⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣿⣿⣏⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⡆⠀⠀⠀⠀⠀⠛⢦⣄⠀⠀⠀⠀⠈⢻⣿⣿⠟⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⢟⢧⣿⣿⣿⣿⣿⣿⣯⣠⡂⠀⠋⡀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠐⣾⣿⣿⡄⠀⠀⢈⣿⣷⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⢰⡄⠀⠀⠸⣿⡀⢀⣠⡤⠄⠉⣿⣧⣤⣤⣌⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠁⠀⠀⠀⠀⠙⣿⠀⠀⢀⡶⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⡀⣶⣿⠀⠈⣿⠁⠀⠀⠀⠐⠀⠂⠒⠀⠒
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣽⣿⣉⢤⣤⣽⡆⠀⠀⠀⢀
⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣾⡿⠟⠋⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠖⠒⠲⣶⣾⠟⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⡠⣶⣿⣷⡀
⡶⠒⠒⠒⠒⠒⣶⣿⡵⡂⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣴⣿⣿⣿⣿⣷⠖⢲⡶⠲⣶⠖⢲⣶⠒⢲⣶⡶⠒⠒⣲
⡇⠀⠀⠀⠀⢸⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⣼⢿⣆⣿⣿⣿⣿⣿⣿⡆⢸⠇⢠⠏⢀⡿⠁⣠⡿⠋⠀⣠⣾⡿
⣧⠀⠀⠀⠀⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠙⠻⢿⣿⣿⣿⣿⣧⠈⠀⠜⢀⡞⢁⡼⠋⠀⣠⡾⠟⠉
⣿⣷⡀⠀⠀⢹⣿⣿⣯⣠⣤⣶⡶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣄⠀⠀⠈⠠⠋⢀⡴⠞⠉⢀⣠⣴⡿
⡇⠘⢿⣄⠀⠘⣿⣯⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⠀⠀⠀⠀⠐⠉⢀⡤⠞⠛⠉⢀⣀
⡿⣦⡀⠙⢦⡀⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⢄⠙⠐⠃⠴⢀⣠⠎⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠤⠶⠞⠛⠛⠉
⣧⡈⠙⠦⡀⠑⠀⡟⠻⣿⡀⠀⢀⣀⣀⣒⣂⣀⠀⠀⢶⣋⣥⠞⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠶⠶⠶
⡏⠙⠲⢤⡈⠀⠀⠙⣆⠙⠧⠀⠀⠀⢬⣭⣭⢿⣷⣀⣀⡉⠀⠀⠀⠀⠀⣿⠤⠤⢤⣤⣤⣄⣀⣀⠒⠶⢦⣤⣤
⡟⠛⠒⠤⠄⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⠀⠘⣟⡒⠤⣤⡤⠟⠛⢇⠀⠀⠀⠀⠀⠉⠉⠙⠒⠦⠬
⡷⠶⠶⠄⠀⠀⠀⠀⠀⠈⠉⢳⡺⣶⣤⣀⣠⣾⡟⣸⢷⡀⠻⢌⡙⠲⢭⣛⠲⠾⢦⣀⠀⠀⠀⣴⣋⣳⡄
⣧⠤⠔⠒⠀⠀⠀⠀⠀⠀⠀⣠⣷⣝⢿⠟⢋⡼⢱⠇⣼⣄⠸⣄⡈⠓⠢⢄⡉⠀⠀⢿⡷⣄⠀⠘⢿⣏⡹
⣧⡴⠖⠂⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠃⢰⡿⢃⡟⢰⠃⡏⠀⠘⣯⡉⠲⢦⣈⠁⠀⢸⣅⠈⢦⣤⣜⣥⣀
⡇⣀⡤⠖⢁⠀⠀⣠⠾⠛⠛⠻⣿⠏⠀⠈⠁⠞⢡⡏⢠⠇⣤⠀⠈⢿⣶⣄⠈⠁⠀⠸⢧⡀⣿⣤⣙⣤⣿
⡟⠋⣠⡶⠁⣠⠞⠁⠀⠀⠀⠀⢸⣦⣀⡀⠀⠀⠀⠀⡼⠀⢽⠀⣀⣀⣈⠛⣿⣦⠀⠀⠀⠈⣿⢸⢸⢸⣸
⣧⣾⠋⢀⡴⢋⠔⠁⠀⠀⠀⠀⣸⣿⣿⣿⢶⣶⠆⠀⢫⠀⢸⡏⠈⣯⠜⢁⣈⣻⣷⣄⣀⡾⠛⠛⣾⣾⣟⢳⡴
⡟⠁⣠⢏⡴⢃⡰⠀⠀⠀⠀⢠⠇⢹⣷⢻⠈⡿⠀⠀⢸⣀⡼⠁⠀⠙⠋⠉⠁⠉⠉⠛⢹⡇⠀⠘⢛⣿⣿⠀⡇
⣇⡼⣣⠞⣠⠏⣠⠂⠀⠀⠀⠘⠒⠾⣿⡇⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣄⣀⣈⣿⣿⣴⡇
⡟⣱⢋⡾⣡⠞⢁⠔⠀⢰⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣠⣤⠶⠋⢹⡇⢼⣭⣭⣭⣵⣦⣷
⣿⣣⠟⡴⢋⡴⢁⠄⢀⣸⣇⡀⠀⠀⣿⡏⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠⠄⢚⣉⣤⣤⣤⣼⣧⣘⢿⣿⣿⣿⣿⣿
⣿⢯⡞⣱⠏⡰⠋⡀⢾⠾⠧⡇⠀⢰⣿⣧⠀⠀⠀⢘⡁⣀⡄⠘⠋⠁⠀⣙⣋⣀⣀⣀⣤⣬⣿⡏⢿⣿⣿⣿⣿
⣿⣟⡼⢣⠞⣡⠞⢠⠾⢷⣴⡷⡄⢸⣿⣿⡄⢀⣠⢼⠋⠉⠀⠀⠀⠀⣀⣽⣏⣏⣹⣹⣏⣿⣽⣇⡀⠙⢿⠇⡟
⣿⣿⣵⢋⡼⢁⡔⢹⠀⠘⣿⡇⣧⣾⣿⡿⠟⠉⢠⠟⠚⢦⠀⠀⠀⣼⠋⠀⠀⠨⣭⣿⣿⣿⣿⡄⢹⣦⣴⠾⡧
⣿⣿⣧⠟⡴⢋⡴⢻⢳⣒⣚⡋⡿⠋⠁⠀⠀⠀⣸⠲⠶⢻⣀⠀⠀⣿⣄⡀⠀⠀⠈⢩⣽⣿⣿⣇⣼⣿⣠⡼⠃
⣿⣿⣿⣞⣡⠎⣠⣾⠈⢿⣿⣿⡇⠀⠀⠀⢠⣞⠙⣿⠿⣛⡛⡇⠀⣿⠈⠉⠉⠙⠛⠛⠛⠉⠉⠉⠀⣿
⣿⣿⣿⣿⣿⠾⠛⢹⠀⠀⠻⣿⡇⠀⠀⠀⢸⣿⡯⠀⠛⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡿⠟⠉⠀⠀⠀⢸⠲⠤⠤⣴⡇⠀⠀⠀⢸⣿⡇⠀⠀⠘⢷⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡏⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠛⠁⠀⠀⠀⢸⣿⡇⠀⠀⠀⢈⡇⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡤⠖⠋⠀⠀⢿⣍⡉⠛⠛⠛⠛⠛⢻⣿⣏⣨⡟
⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⢦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠉⠙⠛⠻⠶⢶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢹⣇⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣴⠟⠀⠀⠀⢸⣇
⠀⠀⠀⣠⡾⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⢰⡶⣦⣤⣤⣄⡀⠀⢿⣦⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢰⡏⠀⠀⠀⠀⠈⣿⠀⠀⠀⢤⣤⣴⣾⠇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠦⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⣀⣤⣄⠀⠀⠀⠸⣧
⠀⠀⢸⡇⢀⣶⣶⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡅⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⣀⣤⣴⢾⣿⣿⣿⠟⠀⠀⠀⠀⢿⡄
⠀⠀⢸⡇⢸⣿⣿⣿⠀⣿⠀⠀⠀⢀⠀⠠⠶⠟⠋⠁⠀⢀⣴⠾⠋⠀⠀⠀⠻⠛⠋⠁⠀⣸⡿⠟⠁⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⢸⡇⢸⣿⣿⣿⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⢸⡇⢸⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠂⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⢸⡇⢸⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⢸⡇⢸⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⢀⣼⠇⢸⣿⣿⣿⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣾⠃⠀⣼⣿⣿⣿⡄⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⣿⣿⣿⣿⡗⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠹⣧⡀⠈⠛⠛⠋⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠛⠷⢶⣤⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢐⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣄⠀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠗⠊⡟⠉⠉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠇⠀⠜⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠘⠀⢀⡀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠁⣀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⣿⠇⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠢⠄⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣀⠤⠤⡀⢀⣠⣀⢀⣀⣤⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠤⠤⠀⢀⠄⣀⣀⡠⠤⠤⢴⡁⡀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢀⢠⡀⠀⠀⠀⠀⣀⠀⠀⢩⠽⣓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡰⣄⠀⠉⠪⠁⠐⠄⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠶⢤⠇⡡⢠⣬⢡⠦⠤⠒⠊⠁⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢀⣚⣀⠆⠀⠁⠈⣀⢢⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⢸⠈⠉⠉⠀⠀⠈⠉⢸⡦⣤⠄⠤⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⠗⠘
⠀⠀⠀⠀⠀⠀⣴⠒⠲⠄⣠⢖⣁⡠⠤⠴⠒⢮⡉⠳⡄⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⡹⠦⢤⣀⠱⣄⠀⠀⠀⠀⢰⠤⠄⢦
⠀⢸⠒⡛⠀⡡⠛⠒⠈⢫⠉⠉⠁⠀⠀⠀⠀⠁⢹⠒⠃⠀⠀⠀⠀⠀⢸⣗⡫⠟⠉⠀⠉⠁⠈⠉⠛⠷⠉⠉⠁⡟⠀⠀⠨⠈⠉
⠀⣸⠦⡆⠀⣿⠗⠒⢒⢼⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠒⠃
⠀⠳⠤⠇⠀⣗⠤⠤⠄⠚⠀⠀⠀⠀⡀⠀⣠⠤⠈⠢⠤⣤⣤⣤⡤⠤⠜⠁⠀⠀⠀⠀⠄⠙⠀⠀⠤⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢉⣁⣤⡴⠶⠾⠛⠁⢀⠀⠠
⣤⣤⣤⣤⣤⣤⣧⣴⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣅⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠄⠒⠀⠀⠀⠀⠤⠒⠒⠒⢄⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢳⣑⣁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡴⢫⢀⠀⠀⠀⠀⡰⢄⠔⣩⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⠀⠀⠀⠀⠀⠸⣤⢇⠇⣠⢞⡁⠀⠀⠠⠞⣉⡄⠰⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠠⠃⠀⠀⠀⣀⣀⠠⠤⠼⣄⠙⠛⢡⢰⠀⠀⠀⠠⠐⠂⠀⠀⡸⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⠂
⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⡠⠒⠉⠀⠀⢀⣀⢀⡀⠙⠢⡀⠑⠂⠀⠀⠀⠀⠀⢰⠛⠀⠀⡗⢦⣀⠉⠐⠂⠤⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⠔⠊⠁⠀⠀⠀⠑⢢⡈⠢⡀⠀⠀⠀⠀⢀⣼⠁⠀⠀⡇⠀⠈⠙⠲⠤⢀⣀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠑⢄⣀⡤⠖⠋⠀⡄⠀⢈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠁⠱⡀⠀⠀⠉⠂⢹⣆⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠑⡄⠀⠃⠀⠀⠙⣦⡀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠘⣸⠀⠀⠀⠀⠈⡿⣦⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⡏⠀⠀⠀⠀⠀⠐⠘⢷⣄⠀⠀⠀⢀⡤⠞
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⢦⣴⠏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⡀⠔⠊
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⢿⡄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣄⣀⡀⡃
⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⡀⠀⠀⢀⣀⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⢠⠀⠄⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢠⣆⠂⠤⠀⠠⠒⣸⠀⡠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠎⠁⠉⠀⠶⠀⠀⠀⢉⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⢀⠃⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠃⠂⠠⠀⠂⠃⠀⠀⠀⡅⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠔⡊⡀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⢐⣉⣁⠀⠈⣉⡇⠀⠀⠁⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠈⡄⡠⠄
⠀⢀⠤⠒⠈⢁⠊⠈⠀⠁⠰⠀⠉⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠄⠈





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⢈⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠇
⠀⠀⠀⡀⠀⠁⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⣉⠔⠁⢘⠀⠀⣀
⠀⢰⠉⠁⠀⠀⠀⠀⠈⠉⡁⠀⠀⠀⠨⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡠⠔⠊⠀⠀⠀⠘⡀⢌⠁⢉⢂
⠀⢠⠀⡃⠁⠀⠀⢸⠘⠀⡇⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⡨⠈⠈⠁⢸⠗
⡎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⢆⠁⡆⣒⡂⠘⢢
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠀⠀⢀⠀⡠⠀⠀⠈
⡇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠃⠀⠀⡈⠀⠀⠀⠀⠜⠈⠑⡀
⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣬⣧⣯⣤⣤
⣿⣿⣁⣍⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠰⠗⠀⠀⠠⠄⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⢀⠄⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠿⣿⣷⣌
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿⢋⡁⠀⣩⠙⣿⣦⣤⡀
⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠗⠀⠀⠉⠒⡄⢀⣾⣿⣿⢿⣄⠐⡤⢀⡐⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⡀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠿⠻⠛⠇⠀⠀⠀⠀⠀⠀⠀⢻⠿⠏
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠊⠀⠂⠂⠀⠉⠠⠀⠀⠮⠂
⠀⢀⣤⠀⠀⠀⢸⣿⠿⠿⠿⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠂⠀⠄⡄⠀⠀⠀⠀⠀⠀⢾⠂⠀⠀⡀⠁⠀⠀⡈⠐⠑⠀⠀⠀⠃⡄
⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠄⠀⠀⠁⠀⠄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡀⢻⠀⠈⠿⠀⠀⠀⠀⠀⠀⢀⣸⡟⣳⡟⢿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⢀⣾⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢲⡿⣼⣿⣁⣿⣣⢡⠀⠀⡄⡄⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⣾⣿⣿⡀⠀⠀⠀⠀⠲⠿⠀⠀⢀⡼⣿⠀⢸⡧⢼⣿⣼⣿⣉⠂⠀⠈⠐⠀⠀⠈⠎⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡆
⠀⠀⡛⠿⣿⣿⣦⣄⡀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠁⠈⠉⠉⠉⠀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠁⠀⠀⠉⠉⣿⣷⣶⣦⣤⣌⣁⠀⣤⣶⡇⠀⠀⠸⢞⡊⠀⠀⠈⡅⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠸⣷⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠠⠒⠃⠀⢠⠀⠂⠀⠀⠀⠀⠤⠐⠂⠲⠂⢠⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠿⠻⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠠⡒⠇⠀⢤⣠⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⢹⣿⣿⠀⠀⠠⠀⠀⠄⢀⠀⠀⢠⠀⠤⠀⠄⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⣿⣿⣿⣿⡟⠈⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⡀⠀⠀⢘⣿⣿⠀⠀⠀⠰⢊⠆⠀⠀⠀⠀⠀⠀⠄⢨⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠀⠘⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠈⠁⠩⠃⠀⠀⠀⠀⠀⠀⠰⡐⠂⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⢸⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⡀⠀⠀⠀⣀⣀⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠇⠢⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠈⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢅⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⡁⢣⠏⡿⣄⢿⢸⡭⠇⢹⠉⣞⢙⡆⣷⡰⡇⠪⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁⠈⠀⠁⠈⠈⠈⠀⠀⠈⠀⠈⠋⠀⠁⠁⠁⠉⠋

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⣴⣿⣿⡿⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣽⣿⣿⣿⣦
⠀⠀⠀⠀⢀⡀⠀⣄⢸⣿⣿⡇⠀⠀⠀⠀⠠⠀⢸⣇⢀⠀⠀⣀⡈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⢠⡀⢸⡇
⠀⠀⠀⠀⠀⢆⠀⠶⠀⠛⣹⠁⠀⠀⠀⠀⠀⡀⠈⠓⠴⠄⠀⠶⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠂⠘⢋⡟⠀⠀⠀⠀⠀⠀⠀⢹⠀⠶⠀⠰⠄⠸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠁
⠀⠀⣤⡀⠀⢀⠚⡣⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⢛⢌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠀⢱⠄
⠀⠀⠠⠛⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠗⠤
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡄
⠀⠀⠀⢠⣿⢷⣿⣿⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⢻⡆
⠀⠀⠀⠈⡏⢀⡄⠀⢤⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⢘⡅⠀⣹⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣉⠀⢘⡃⣸⠁⠀⠀⠀⠀⠀⠀⠀⢿⠐⣏⠀⢐⠂⢸⠁
⠀⠀⠀⠀⠓⠘⠂⠄⠘⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠂⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠂⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⣀⠀⠀⠖
⠀⠀⠀⠀⡀⠁⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠄⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⠤⡤⡀
⠀⠀⠀⠀⠀⠇⢀⣀⣀⡸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠰⠦⢐⠀⠂⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠂⠀⠂
⠀⠀⠀⠂⠀⠲⠤⠽⠬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠞⠛⠛⠛⢻⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠠⠆⠀⠶⠜⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣉⠉⢉⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⣿⠐⡂⠀⢐⠂⣸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠐⡒⠀⢘⠂⢸⠃
⠀⠀⠀⠀⠀⠣⠀⠛⠀⢋⡽⠁⠀⠀⠀⠀⠀⠀⠀⠫⠈⠧⠀⠰⠃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠋⠄⠘⠁⠍⠀⠀⠀⠀⠀⠀⠀⠀⠸⡢⠛⠂⠘⠅⠯
⠀⠀⠀⠂⠀⢐⠒⣀⠠⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡀⣀⠘
⠀⠀⠀⠀⠁⠀⠔⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠔⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠊⣁

⣿⣿⣿⣿⣿⣟⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⡀⠀⣤⡴⠀⠠⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠠⢀⡀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣾⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠌⢀⠂⠙⠉⠀⣄⡀⠀⠈⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣸⣿⣿⣿⣿⣾⣷⣿⣿⣿⣟⣿⣿⣿⠀⠀⠀⢠⡇⢠⡈⠀⠜⠁⠋⡁⠀⠀⡴⠀⠀⢸⣿⣿⣿⣟⣾⣿⣿⣿⣻⣿⣶⣿⣿⣿⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⠿⣿⣿⣿⣿⣟⢺⣿⣿⣿⠀⠀⠀⠜⡇⢠⠉⠉⣡⡤⠭⡗⠠⢤⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⡟⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⢀⠘⠀⢀⣡⣦⠀⠀⠀⠈⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢿⣿⣯⣟⣇⣾⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠰⡬⣄⡚⠦⢿⠻⠁⠁⠀⠐⡰⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣛⣣⣧⣿⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⠆⠀⠠⠘⠓⠀⢠⠜⢰⡧⠀⠀⢸⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠊⠀⠀⠀⠀⡤⠤⠤⠤⠼⠇⠀⠀⢸⣿⣿⣿⣯⣵⣿⣿⣿⣿⣻⣿⣿⣿⣟⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⠄⠄⠄⠀⠀⠠⠈⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀⣀⣀⣤⣤⣴⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⢟⣯⣽⣶⣶⣶⣶⣾⣷⣶⣶⣯⣻⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣫⣾⣿⣿⣿⠟⠛⠛⢿⡟⠉⠉⠻⣿⠿⠿⢿⣿⣿⡝⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣵⣿⠋⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠿⠿⠿⡝⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠛⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣁⠀⠀⠀⠀⠀⢀⣶⣤⣀⣠⣾⣦⣀⣀⣴⣦⣄⣀⣠⠀⠀⠀⡼⠷⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠛⠀⢀⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⠈⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⠆⠐⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣟⡄⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⠀⠀⢿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠈⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⡿⢆⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⠖⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⢸⣿⣿⣷⣄⣠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣠⣤⣤⣼⡟⣿⣿⡄
⠀⠀⠀⢸⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠈⠙⢿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡇
⠀⠀⠀⣼⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡇
⠀⠀⠀⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⣿⣇
⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿
⠀⠀⠀⢸⣿⣹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣾⡏
⠀⠀⠀⠀⣿⣧⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠿⣿⣿⢷⣿⠁
⠀⠀⠀⠀⢿⣿⣾⠛⠛⠋⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⣿⡟⣾⡟
⠀⠀⠀⠀⢸⣿⣿⡄⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣻⣿⠁
⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣽⣿⡏
⠀⠀⠀⠀⠀⠘⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣏⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠈⣿⣿⢳⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢷⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣴⢯⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣮⢿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣼⢯⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⡻⣿⣶⣄⣤⡁⠀⢀⡀⠀⠀⡀⠀⢀⣀⣀⣤⣤⣴⣿⣿⣿⢯⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⢫⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣝⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣴⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣾⣟⡿⠿⡿⢿⡿⠿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠦⣄⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣆⡴⠻⡆⠀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡤⠴⠚⣉⠈⠉⠀⢀⡉⠉⠹⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⡀⠀⠈⠭⠀⣆⠀⠨⠍⠀⠀⣸⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡄⢹⡄⠀⠀⠀⠸⠃⠀⠀⠀⣼⠁⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢂⠀⢳⡀⠀⠰⠯⠷⠀⠀⣰⠃⢀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⠀⠀⠙⣶⠤⠤⠤⢴⡞⠁⠀⠈⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⢣⡀⠀⢀⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⣟⢮⡳⣌⠉⠉⠀⢻⠉⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠻⣷⠍⠀⠀⠀⠀⠈⡶⠓⢦⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣄⣀⣸⠁⠀⠀⠱⣄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠚



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡣⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢰⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢨⠘⣿⣿⠗⠀⢀⣀⣀⣠⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⣠⣴⣿⠛⠋⠙⠉⠉⠉⠉⠛⠛⠛⠓⠶⣶⣄⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡾⠋⠀⣿⠀⠀⠀⠀⠀⠀⢤⣤⣤⣄⡀⠀⢨⣿⠁⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠂⢀⡾⠋⠀⠀⠀⢿⣦⣀⣀⣀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⢸⡏⠻⣦⡀
⠀⠀⠀⠀⠀⢀⠤⢂⣴⠟⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠷⢦⣤⣀⠀⠙⢿⣿⡀⢸⡇⠀⢹⡃
⠀⠀⠀⠀⡀⠃⢠⡟⠁⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣾⣿⣧⡀⠻⣦⣼⣷
⠀⠀⠀⠀⠢⡀⢿⣇⣀⣀⣄⣀⣀⣙⣻⣶⣶⣶⠶⢶⣶⡶⠶⢶⣶⣤⣀⣠⣿⣿⣿⣿⣶⡄⠉⠃
⠀⠀⠀⠀⠀⠈⠐⢌⠉⣿⣵⡎⠉⠉⠉⠀⠉⠙⠛⠋⠛⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣜⡅⢻⡍⠁⢀⣠⣤⣶⣶⡶⣶⠶⠶⣶⣶⣶⣦⣤⣤⡀⠈⠀⠹⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣄⣸⣇⣼⣿⣿⣋⣽⣿⣿⣷⣶⣶⣶⣶⣦⣤⣴⣽⣛⢷⣦⡀⡘⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠷⢿⣿⣿⣿⡛⡽⣿⣶⣶⣶⣤⣍⡛⢦⣍⠛⢿⣿⣿⣿⣿⣷⣺⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠃⣬⢷⢹⣿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣷⡀⢻⣶⡌⠉⠙⢿⣿⠻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⣿⢹⡇⢻⡿⣷⣜⣿⣿⣿⣿⡿⡟⠇⣸⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠈⠀⠀⣮⣩⣿⣿⣿⣿⣣⣴⣂⣧⣤⣶⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡀⠀⠀⠙⢿⣻⣿⣿⣿⣿⣿⣿⣟⣭⡿⠃⠀⠀⠞⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⣧⡀⠀⠀⠀⠉⠁⡀⣀⣀⣀⣀⣉⢠⣤⣬⣦⣶⠋⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡈⠳⠦⠶⠶⠾⠛⠛⠋⠉⠉⠉⠉⠫⡙⠊⠹⡏⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⢀⡀⠀⠀⠀⠀⠄⠠⠤⡀⠀⠘⠦⠀⠠⡄⢻⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⢸⠇⠀⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣼⠀⠀⠀⠙⣿⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⢰⡇⠀⠀⠀⢰⡿⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣷⣤⣾⣷⣿⡇⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣿⣿⡿⢿⣿⣇⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠛⠆⠀⠙⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠿⢿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣥⣤⣷⣤⣧⣤⣶⣥⣾⣯⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⠀⠀⠈⣉⠁⡀⠀⠀⢉⣉⠀⠀⢸⡿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⡀⠀⠀⠈⠁⡇⠀⠀⠈⠀⠀⠀⢸⠃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢣⠀⠀⠀⠀⠛⠒⠀⠀⠀⠀⢀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢴⠿⠿⠷⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠀⠤⠐⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⡤⢶⣾⡄⠀⠀⠀⣤⣶⠦⡤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⢀⠸⡀⠘⠙⢇⣴⠆⢠⣟⡉⠀⢨⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠳⣾⠳⢌⠂⠄⣈⣇⠀⠀⠈⠣⡶⠻⡷⠊⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣀⣀⣀⣀⠀⠀⠀⠀⣀⠄⠂⠓⠦⠤⢀⡈⠈⠁⠐⠒⠂⠤⣈⡄
⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⣀⠔⠁⠀⠈⠢⢄⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢀⠠⠤⠒⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠁⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠐⠒⠒⠈⠉⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⣠⠃⠀⠰⡇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠁⢀⣀⣀⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⠐⠚⠫⢤⡈⠙⠫⡙⠲⣤⡀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠞⠁⠀⠀⠀⠀⠀⠈⢦⠀⠘⡄⠀⠙⢦⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣀⣠⠄⠀⢠⣤⣄⣀⠀⠀⠀⣠⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠁⠻⠟⠀⠀⠸⠟⠁⠀⠁⠀⢀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠶⠄⠔⠂⠀⠓⠒⠒⠂⠀⠀⢸⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣠⠤⠒⣸⠭⠍⠛⠛⠓⠲⠤⠀⠀⠀⠈⣳⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣯⠟⣋⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣮⠛⢃⠴⠊⠙⠲⢤⣀⣀⠀⠀⠀⢀⣀⡤⠖⠉⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣊⠷⢈⡤⠚⠁⠀⠀⠀⠀⠀⠀⠉⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠑⠈⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠳⠈⠳⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⢹⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠘⢦⡀⢆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⣾⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠠⠁⠈⡆⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⡇⠀⠀⠀⠙⢄⠀⠀⠀⢠⣴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡇⠀⠀⣴⠧⠤⠶⠒⠶⠾⠥⢾
⠀⠀⠀⠀⠀⠀⢠⣤⣀⣤⢎⡡⠤⠤⣴⣶⣦⣭⣭⣭⣭⣭⣭⡙⡏⠙⠻⢦⣠⣤⣤⣤⣤⣤⣤⣤⣤⡝⢦⡇⣠⢂⣼⡄⠀⠀⠀⠀⠀⠀⢺⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⣧⠀⠀⢀⡀⣤⠀⠀⠀⠀⠀⠀⠘⢦⣸⡇⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠰⣬⣀⡀⠩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠳⠖⠄⠲⡇⠀⠀⠀⠀⢠⡀⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠼⡾⣒⠒⣒⣚⠁⠈⠍⠀⠀⠀⠀⠀⡀⢤⢤⡀⠀⠀⢀⠴⢎⡸⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣥⠋⠉⠁⠘⢆⠀⠀⠀⠀⠀⠀⠸⣛⡓⠞⠁⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠀⠤⠰⠂
⠀⠀⠀⠀⠀⠀⠸⣍⣒⣒⣊⠠⢹⠤⠤⠤⠒⢻⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⢇⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠸⠀⠀⠀⠀⢘⠀⠑⢤⣏⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⣒⣒⣉⠍

⠀⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⡀
⠀⠀⠀⣼⡿⣿⡿⢿⣿⣷⡀⠀⠀⠀⠀⣾⣿⡿⣿⣿⢿⣧⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⢿⣿⡿⣿⡄
⠀⠀⠀⠙⣀⠀⡀⠸⠿⣿⠁⠀⠀⠀⠀⢻⢏⣭⠀⢤⠸⡏⠀⠀⠀⠀⠀⠀⠫⡌⠉⡀⢸⢿⡿⠀⠀⠀⠀⠀⠘⡇⢠⠈⢨⠀⡿
⠀⠀⠀⠀⠈⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠘⠀⠴⠀⠀⠁
⠀⠠⠄⢀⣀⠚⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠓
⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣿⡀
⠀⠀⠀⣿⠛⠋⠿⠛⢻⠇⠀⠀⠀⠀⠀⠸⡇⢠⠈⢩⠀⡿⠀⠀⠀⠀⠀⠀⢿⠁⡀⠉⣁⢸⠏⠀⠀⠀⠀⠀⠸⡏⢠⠈⢩⠀⡿⠁
⠀⠀⠀⠘⡀⠁⠀⠁⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠗
⠀⠀⠀⠀⣃⠤⡤⢜⠀⠀⠀⠀⠀⠀⠀⠀⡈⡂⣄⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⣈⣭⣍⣉⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⣬⣤⣀⡁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀
⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠘⡏⠈⠉⠉⣿⣿⡟⠀⠀⠀⠀⠀⠸⡟⢉⠙⠛⠉⡿⠁⠀⠀⠀⠀⠀⢿⠁⡄⠉⡅⢸⠇⠀⠀⠀⠀⠀⠸⣏⢠⠈⢩⠀⡿⠁
⠀⠀⠀⠉⢁⠈⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠑⡈⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡌⠀⠈⡈⠁
⠀⠀⠀⠀⠐⠒⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐
⠀⠀⠀⠀⡠⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠘⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣒⣭⣽⣿⣾⣿⣿⣿⣿⣿⣶⣄⠠⠚⠀⣀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠁⠀⠀⠹⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⡀⠠⠴⠒⠊⠒⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠈⠻⢿⠻⠆⠁⠈⢿⣿⣿⣧⠐⠢⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢉⠁⠀⠀⠀⢀⠀⠀⠀⢀⠀⢣⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠀⠀⠒⠈⣁⠀⠀⠀⠀⣉⠉⠓⢄⠀⢠⠼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⡀⠀⠉⠀⠀⡀⠀⠀⠈⠉⠀⠈⡇⠀⣀⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠁⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣹⣿⣦⡁⠀⠀⠖⠋⠓⠂⠀⠀⠀⢀⡼⡆⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⡶⡦⢤⢤⣀⣠⢤⡤⢶⠋⠀⢱⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡼⠁⠘⣄⠽⠲⢏⣠⠃⠀⠀⠀⢣⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⡖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣄⡀⢀⣠⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠉⠉⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⠛⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⣛⠋⠉⠉⠛⠛⠉⠉⠙⣛⠁⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠈⠑⠦⠤⣤⡤⠤⠶⠊⠁⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠤⠖⠚⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⣧⡀⣀⠀⠀⢠⡿⢴⠄⠀⠀⢀⣴⠟⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢻⣫⣽⡀⠀⠘⠙⡿⠀⠀⠰⢯⣻⡦⠀⠀⠘⣆
⠀⠀⠀⠤⠤⠶⠖⠒⠋⠁⠀⠀⠀⠈⠁⠘⠇⠀⣀⣀⣁⣀⠀⢠⠟⠁⠀⠀⠀⠀⠈⠙⠒⠲⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢊⣡⠤⠤⣌⡑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡞⣠⠚⠒⣄⢳⠈⡇
⠀⠀⠀⠐⠒⠒⠶⠶⠦⠤⠤⢤⣄⡀⠀⢸⡀⢧⠙⢤⠤⠋⡼⢀⡇⠀⢀⣠⡤⠤⠤⠴⠶⠶⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠱⢌⡙⠒⠒⢋⡡⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⡢⠀⠀⠀⠀⠀⠀⠀⢀⢴⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⣎⡆⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦
⠀⠀⠀⠃⠠⠂⠁⠢⠀⠀⠀⠀⠀⢰⠁⢾⢺⠀⣃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠗⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿
⠀⠀⠀⠀⠐⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⠸⡘⠥⠇⠀⠀⠀⠀⠀⢰⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⡄
⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢄⠤⣠
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⠴⠖⢶⡦⢄⠀⠀⠀⠀⠀⠈⠑⠒⢪⠶⡍
⠀⠀⠀⠀⣊⡠⠠⠃⠀⠀⠀⠀⠀⠀⢀⠄⠒⢀⠃⠀⠀⠀⠀⠀⠈⠐⠂⠐⠓⠪⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⡅
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠃


⠀⠀⠀⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠥⠐⡄⠀⠀⠀⠀⠀⠰⢶⡠⠧⣠⡲⠀⠀⠀⠀⠀⠀⠀⢰⠱⡀
⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⣧⡄⣇⠀⣰⠀⠀⠀⠀⠀⢤⢘⠄⠘⢊⠶⠆⠀⠀⠀⠀⠀⢀⡄⠀⣅
⠀⠀⠀⢀⣀⡸⢹⠁⠄⠀⠀⠀⠀⠀⠹⠗⠙⣿⡿⠀⠀⠀⠀⠀⡠⠻⡐⡔⠚⣷⠀⠀⠀⠀⠀⠀⠉⣁⢃⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣛⣭⣭⣿⣿⣷⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠹⢟⣴⢹⣿⠿⢿⣷⠀⠀⠀⢠⡄⣀⣙⣀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣛⣫⣭⠾⣷⣾⣿⣿⠶⣶⣿⡘⠇⠀⠀⢀⠀⣼⢟⣭⡻⣷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣛⣿⣟⢿⣗⣳⣿⣿⠟⠀⠀⠀⠈⠁⢻⣮⣻⣿⡿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣪⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⠃⠉⣉⠉⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣯⣛⣋⣽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣛⣛⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⡜⣿⣿⣿⡿⣱⣶⣶⣶⣆⣾⡁⡀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⠟⣵⣿⠏⠶⢭⣥⣾⣷⡱
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡨⣽⣿⣿⣿⡟⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣚⡻⠿⠟⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢸⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣷⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡄⡠⡄⢀⡤⢀⣠⣠⢀⣀⣤⡠⡤⣠⢄⢠⣀⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠏⠣⠛⠀⠝⠈⠇⠟⠛⠛⠋⠀⠇⠳⠜⠙⠛⠃⠽⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠖⣀⡤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⣰⡧⠊⠵⠚⠒⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡁⠀⠀⡞⠀⠀⠉⠀⠀⠀⠀⠀⠠⠤⢭⡵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡅⠀⠀⠀⠀⠀⢀⣀⣐⢶⣒⠒⠺⠆⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠶⡾⠍⣯⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⣉⠀⠀⠀⠀⠀⠰⠭⡀⠀⠘⣦⠒⠒⢤⡀⠀⣈⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢩⠞⠁⠀⠀⠀⠀⠀⠀⠐⢾⡆⠀⢻⠀⠀⠀⣹⣞⠁⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠈⢧⡠⠚⠉⠸⣍⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡇⣤⡀⠀⠀⠀⠀⢀⡴⣸⡁⠀⠀⠀⠙⠲⢤⡼⠁⢈⡉⠉⠉⠒⠒⠒⠒⠒⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠇⠹⣾⠦⠶⠒⠻⡤⠝⣇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⢶⠢⠄⣀⣀⣀⣀⡠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⡀⠀⢀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠇⡯⠜⢒⡹⠿⠛⢦⡀⠀⠀⠀⢀⡠⠤⠄⣲⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡄⠀⠷⠚⡇⠀⠀⠀⠀⠑⢤⡀⠀⠁⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⢱⡀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠉⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡄⡀⠀⠀⢣⠘⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢠⠊⡸⠋⠀⠀⠘⡀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⡰⠃⢠⡃⠀⠀⠀⣠⠓⣼⡛⣲⣀⡀⡰⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣀⡜⠁⠀⠈⠛⠒⡞⡩⠏⠘⠓⠢⠼⡁⠞⠁⡀⢹⠈⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠙⠲⢤⣀⡀⢠⠁⠗⡆⠀⠀⠀⣾⢸⢲⣴⠋⠁⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠤⠶⢚⡿⠍⢺⢨⣼⢳⣤⣦⡙⢧⡘⣸⠧⣤⡀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡩⢦⠀⡇⢠⢚⡀⠀⢸⠀⠀⡘⠒⠚⡜⡈⡄⠷⠼⠉⠉⣗⠇⢸⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠋⣗⣰⠀⠁⡟⠞⠁⠀⣠⠇⠀⢰⠁⡇⡇⡅⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⣠⠞⢼⣿⠦⣄⡈⠉⠉⠁⠀⠀⠀⠦⠴⢀⡇⡇
⣠⠴⣖⣴⡖⣆⠀⠀⠀⠀⢀⠀⠀⢀⠔⠁⠀⢉⠟⠁⠀⠀⠀⠀⠀⠉⠑⠒⠶⣤⡤⠤⠤⠬⠟⠘⠁
⣸⣈⣀⣋⣈⠷⠀⣞⡉⠋⢡⡯⡀⠫⠤⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠒⠢⠴⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣤⣜⣉⠁⠈⠉⠉⠙⠒⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣾⠫⠭⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣭⣭⡿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣽⣿⣿⣿⡿⠿⣿⣿⣿⣿⠏⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣿⣿⣿⣿⣿⣿⡀⠀⠘⣿⣿⠁⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠛⠉⠰⣶⣶⣶⡗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⣿⣿⣿⣿⣿⡿⠤⢄⠀⠀⠀⠀⠀⢉⣉⠛⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣶⣌⠻⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣛⡉⠂⠈⠛⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠤⠀⠀⣤⣶⣭⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⣿⣿⡿⠋⠁⣴⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⣿⠿⠛⣃⣸⣿⣿⣿⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⢰⣿⣿⣷⣿⣾⣿⣿⣿⣿⢻⠟⠙⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⡿⠻⠈⣠⣞⡈⢻⣿⡀⠀⢀
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⣠⣾⣿⣿⣅⡀⠘⢧⠀⣸⣷⡀
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠟⠻⠿⣿⣿⣿⣿⣿⣾⣿⣿⣏⠛⠓⠀⠀⢐⡀
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠉⠙⠛⠻⠟⣿⣿⣦⣀⡀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠉⠉⠉⠛⠋⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣼⣿⣽⣻⣤⣶⣶⣄⠈⠻⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⣿⣿⡀⠀⣠⡈⣲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡿⠀⠀⡆⠀⠀⠀⠤⠤⠤⠄⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣠⣤⠴⠒⠋⠉⢁⣀⠤⠄⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣄⣀⠀⠀⢀⣉⠉⠙⠳⢦⣄
⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⣶⡋⠁⠀⠀⠀⠀⠀⠀⢴⡄⠀⠠⠤⠀⠀⠀⠀⠀⠀⠈⠙⣶⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠶⠶⠦⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠁⢀⣀⣀⣤⣴⠞⠋⡀
⠀⢠⣭⠿⣿⠷⠦⠤⢤⣤⣤⣤⣄⣠⣤⣄⠀⢀⣀⣀⣀⣀⣠⣀⣤⠤⠤⠤⠴⠖⠒⠚⠋⠉⠻⠋⠁
⠀⠀⢀⡆⠉⠙⠓⠒⠒⠠⠴⠶⠶⡤⠄⠀⠀⠴⠲⠐⠀⠀⠀⠀⠤⠖⠠⠀⠀⠀⠉⠀⠀⣀⣀⣤⡤⠴⡖⠉
⠀⠀⠈⠇⠐⠒⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡴⠶⠶⠞⠛⠋⠉⠉⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠃⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⠀⢸⠇
⠀⠀⠀⠀⢿⡔⠀⠁⠀⠀⢀⠤⠤⠤⠄⠀⠀⠀⢀⣀⣤⡤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠁
⠀⠀⠀⠀⢹⡟⠦⢤⠀⢀⡎⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠉⠐⠈⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠉⠀⠀⠀⣐⠶⠖⠷⠦⣤⢄⣀⣀⣀⣠⣤⣤⣤⣤⣤⡔⠚⠒⠉⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠠⠒⠉⠀⠀⠀⠀⡀⠈⠀⢀⡴⠋⠁⠀⠀⠤⡄⠀⠉⠉⠲⢄⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⢻⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠊⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⣧⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠰⠁⠀⢸⡀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠈⠂⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠤⠐⠂⠉⣁⣤⠌⠒⠒⠲⠖⠛⠁⠀⠀⠀⠐⠀⠀⠄⡁⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣄⡀⠀⠘⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⣠⠟⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⣤⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠛⠁
⠀⠀⠀⠐⢧⣄⣀⣐⠂⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠳⠶⠶⠶⠖⠒⠒⠊⠋⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⣾⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢹⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣌⣼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⢟⣿⣷⠶⠤⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠘⣿⣿⣿⣿⡟⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢻⣿⣿⣿⣶⠀⠀⠈⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠛⢿⣿⣿⠙⠆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠿⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠟⠷⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⢀⣤⡶⣶⣤⡾⠛⢿⣆⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⣠⣾⠛⠿⣦⣴⢶⣦⣄⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢠⣿⠁⠀⠀⣉⣤⡶⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢶⣦⣌⡀⠀⠈⢻⣆⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⠿⠛⠋⣠⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡈⠛⠻⢶⣄⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⣈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡅
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⢸⡇⠀⠀⠐⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⣄⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠘⢿⣄⠀⠀⣿⠁⠀⠀⠀⠀⢀⡀⠀⢾⡃⠀⠀⠀⠀⠀⠀⠙⠁⢀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⢿⡇⠀⢀⣼⠟
⠀⠀⣸⡟⠀⢠⣿⠀⠀⠀⠀⢀⣾⠃⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⢻⡄⠀⠀⠀⠀⣸⡇⠀⠸⣧⡀
⢠⣾⠋⠀⠀⠈⣿⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢸⡇⠀⠀⠉⢻⣦
⠈⠻⣦⡀⠀⠀⣿⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⢻⡄⠀⢸⡇⠀⢸⡇⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣼⡇⠀⠀⣴⡾⠋
⠀⢀⣾⠃⠀⠀⣻⣧⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⣤⣤⡴⠞⠋⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠈⠻⢶⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⣀⣠⣿⡀⠀⠀⢻⣆
⣴⡟⠁⠀⠀⣼⠟⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠉⠙⣿⡄⠀⠀⠹⣷⡄
⠙⢷⣄⠀⠀⣿⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣼⡇⠀⠀⣴⠟⠁
⠀⣸⡟⠀⠀⢹⣆⠀⠀⠀⢻⡇⠀⣴⠶⠛⢉⣿⠇⠰⣶⣶⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣶⣶⡶⠀⢿⡟⠛⠻⢦⡀⢀⣿⠃⠀⠀⢠⡿⠀⠀⠘⣿⡀
⠀⣿⡁⠀⠀⠀⠻⣦⡀⠀⢸⣧⢸⣇⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢈⣿⢸⣿⠀⠀⣰⡿⠁⠀⠀⠀⣹⡇
⠀⠈⠻⢷⣦⡀⠀⠈⠻⠿⢾⣧⠀⠙⠷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠁⢸⣿⠾⠟⠋⠀⠀⣠⡶⠟⠋
⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠈⣿⠀⠀⠀⢠⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣦⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⢻⣧⡀⠀⣿⣀⣿⠀⠀⣠⡶⢦⣀⣀⣤⣄⡀⠀⠀⣀⣤⣄⣀⣴⠶⣦⡀⠀⢸⣇⣹⡇⠀⣠⡿⠁⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣦⣌⡉⠁⠀⠀⠉⠀⠀⠙⠉⠁⠈⠛⠛⠛⠋⠀⠉⠙⠁⠀⠈⠁⠀⠀⠉⣉⣤⡾⢿⣦⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠉⠛⣷⣶⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣶⡿⠋⠁⠀⠀⣹⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠙⢿⣦⠀⠀⢀⣿⠀⠀⠀⢀⣿⣿⡛⠿⠟⠉⢻⡟⠛⠛⠛⣷⣶⣿⠿⠟⠛⠛⢛⣿⠋⠉⠀⢛⣿⡆⠀⠀⠀⢿⡇⠀⠀⢠⡾⠛⠉
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢸⣇⠀⠀⠀⣿⣏⠉⠁⠀⢀⣴⡾⠿⠿⠾⠿⣿⣯⣄⣀⠀⠀⠀⣼⡇⠀⠀⠀⠸⣿⡀⠀⠀⠀⣸⡇⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠛⣷⠀⠘⣿⣉⠀⢠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢶⣿⣧⣀⣀⠀⣾⡟⠁⠀⠀⣼⠟⠁⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⠀⠀⣿⠀⠀⢾⣯⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠁⠀⠀⠀⣿⠆⠀⠀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠹⣧⠀⠀⠉⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢶⣦⡀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠁⠀⣠⣶⠾⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⠀⠀⠀⠀⠙⠻⠿⢶⣶⣦⣶⡶⠿⠛⠉⠛⠛⠿⠶⣶⣦⣤⣄⣀⣀⣀⣠⡿⠃⠀⠀⢰⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠐⠠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣁⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠯⢥⣉⠓⣄⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠒⠒⠀⠀⢄⠀⠀⢎⢡⢀⡀⠀⠀⠙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠉⢉⣳⣄⠀⠀⠀⠢⠀⠀⠏⠀⠀⠃⣢⠄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⠀⠘⢯⡟⠀⠁⠈⠁⠐⠎⠀⣰⣠⠎⠁
⠀⠀⠀⠀⠀⡠⢀⠄⠀⣀⡤⠴⠶⠒⠒⠚⠛⠛⠁⠈⠉⠉⠉⠙⠇⠀⠀⠤⠐⠈⠀⢠⣯⠟⠒⠒⠲⡄⢠
⠀⠀⠀⠀⠀⢀⠁⢠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⡶⠟⠉⢠⠇⢠⢁⠂
⠀⠀⠀⠀⠀⠀⠄⠸⣄⠀⠖⠢⠄⡀⠀⠀⠀⣀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠋⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠓⡒⣶⠒⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡼⢸⠀⠀⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡇⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠈⣀⠤⠒⠉⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠲⡇⠀⠀⠀⢸⠀⢸⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠴⣮⠄⠒⠒⠂⠀⠀⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠚⠀⠃⠀⠳⠶⠶⠒⠒⠒⢂⡀⠈⣩⠜⠀⠀⠀⡀⠀⠀⢂⡀
⠀⠀⠀⠀⠀⢀⡀⢀⠀⣀⠜⢲⠆⠤⠤⠄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⠔⠊⠀⠀⠀⡜⠀⠐⠀⠀⠀⢄⣠⡪⡙⠭
⠀⠀⠀⠀⢠⢋⡴⢋⠞⣁⠤⠞⠓⠒⠛⠒⠒⠒⠒⠂⠠⢤⣀⠴⠋⠀⠀⠀⠀⣠⠞⢹⡀⠃⠘⠀⠀⠈⢟⢍⠂
⠀⠀⠀⠀⠈⠈⠁⠫⢈⣡⣃⣠⡤⢤⠶⢶⡚⠋⠉⠉⡹⡟⠁⠀⠀⠀⢀⡴⠞⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⠁⠀⢸⠀⠀⠉⠒⠤⣸⠁⠃⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⣇⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⡏⡌⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣇⠸
⠀⠀⠀⠀⠀⠀⠀⣄⠘⢅⠓⠤⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠼⢀⣹⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠻⢻⡟⠛⠛⠛⠛⠛⣿⠿⠿⢿⠟⠉⠁⠀⠀⠀⠈⠁⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠓⠒⢿⠀⠀⠀⠀⠀⠀⡷⠤⠤⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣻⠎⠟⠀⢯⠀⠀⠀⢀⣶⡺⣭⡷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋⠁⠀⠀⠛⠒⠒⠒⠀⠤⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠐⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡎⠀⢀⡞⢁⡄⠀⣀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠠⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⠋⠀⡼⡱⢻⢡⢾⣿⠀⢀⠔⢹⢀⡴⡟⣠⣶⠀⣠⠀⠀⠀⠘⣆⠀⢧⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⣸⠀⢠⡟⠁⠸⠃⠀⢸⠔⠁⠀⢸⠋⠀⠟⠁⣧⠞⣽⣠⡄⠀⠀⢸⡄⢮⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢀⡵⠀⢸⠀⠀⢀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⣃⢼⠇⠈⡇⢸⡀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⣠⢼⠶⡶⠛⠋⠁⠉⢳⡀⠀⠀⣠⠼⠷⡦⣄⠀⠀⠀⠋⢸⠀⢀⡇⠈⠁⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠋⢸⠀⠀⠀⠀⣠⣄⡇⠀⣰⠁⠀⠀⠉⠲⢷⣀⡀⠀⢸⠀⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣰⠋⠀⠀⠘⢦⣀⠀⠀⣩⠞⠀⠀⡇⠀⠀⠀⢀⡶⢠⠃⠀⠀⣸⢏⠭⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢈⡇⠀⠀⠀⠀⠀⠉⠉⡽⠗⢺⡃⠀⠘⠲⢤⣀⣀⠴⠋⠀⠀⢠⡿⠟⡲⠂⢸
⠀⠀⠀⠀⠀⠀⠻⡒⠿⢀⡠⠼⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢆⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡼⠓⢤⡄⠀⠘⣆⣀⣤⡴⡴⠗⠳⠺⠖⠻⠴⡦⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⡴⠙⢧⣀⣀⣀⣀⡀⣤⠖⠂
⠀⠀⠀⠀⠀⠀⢰⡛⠛⠛⢒⠟⢀⡨⠟⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣠⡀⠀⢀⡤⠊⠀⠀⠀⠷⣖⠒⠒⠈⠁
⠀⠀⠀⠀⡀⠀⠠⣇⠀⠀⠛⠉⠉⠀⠀⠀⣌⠙⠲⠦⢤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠴⠞⠉⠀⠀⢤⣄⢻⠒⠪⠿⣷⢀⡆
⠀⠀⠀⠀⢳⠀⠀⢯⡀⠀⠀⠀⢴⣖⡒⠶⠮⠷⢤⣄⣀⣤⡀⢀⡀⠀⡴⠒⠉⠉⠓⣌⢆⡟⢢⣸⠀⠙⠳⠆⢸⠁⣹⠀⡟
⠀⠀⠲⡀⠘⢦⠀⠰⣇⣀⠀⠀⠀⠀⠉⠓⢦⡀⠀⢹⢸⠀⠉⠉⠓⠚⠧⠤⠶⠤⠤⠼⠎⠃⠀⠙⠃⠀⠀⢠⡞⢈⡇⡴⠁
⠀⠀⠀⢸⠀⠙⢤⠀⠀⠹⢦⣠⡤⢤⡀⠀⡀⠙⢦⣼⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠃⢯⠀⠟
⠀⠀⠀⠈⠣⠀⢧⠀⠀⠀⠀⡧⠀⠀⠉⠉⠙⠒⠉⠉⠉⠳⠤⢤⡤⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠴⠁
⠀⠀⠀⠀⣅⠀⠀⡇⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠙⠉⠛⠓⠒⠚⢹⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⣿⡏⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣼⣿⣧⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢛⣿⣿⣿⣿⣿⣿⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡿⠣⠞⣩⣷⣦⣽⡿⠁⡉⢻⡿⣿⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢠⡞⠁⣤⠾⠿⠿⣿⣿⣿⣦⣸⣌⢿⡼⣷⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⣷⡿⠃⢈⣵⢺⣿⣾⣿⣿⣿⣿⣿⣿⡄⠐⢭⣿⣦⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢀⡀⠋⠀⠀⠀⠛⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⣾⡇⠀⢀⡀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⣹⣛⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣛⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠨⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢠⡞⢀⣻⢃⡀
⠀⠀⠀⠠⣴⣶⣶⣴⠀⠀⣀⣴⣦⡆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣰⡟⠀⢢⡟⡾⠀⠂
⠀⠀⠀⣠⣿⡿⠿⠿⢿⣿⣿⣿⣿⠇⢀⡈⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⡟⠀⢀⡟⠂
⠀⠴⣾⣿⠇⠀⠀⠀⣤⣿⣿⣿⣿⡆⠘⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠞⠛⠀⣦⣿⣷⣤⣀⡤
⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⡆⠘⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠘⣿⣿⣶⠀⣼⠟⠛⠿⣿⣿⡇⠀⠀⢻⡄⢻⣿⣿⣿⠟⠁⠀⣠⡞⠀⣸⣽⣿⣿⡿⠿⠟⢿⣿⣿⣷⡀
⠀⠀⣦⠼⣿⣿⡄⠀⢀⠀⠀⠻⣿⣧⠀⠀⢻⣧⢸⣿⣿⠏⠀⡠⠾⣿⢀⡼⢻⣿⣿⣿⡧⠀⠀⣿⣿⣿⣿⡇⠐⠆
⠀⠀⠀⢠⠈⢿⣿⣷⠘⠦⠀⠀⢸⣿⠀⠀⠈⣿⢸⣿⠙⠂⠀⠁⠘⠋⣿⠃⣾⡿⠿⠛⣁⣼⣷⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠠⣽⣿⣇⣴⢠⠀⢸⣿⡀⠀⢀⣿⣿⡇⠀⣀⠀⡀⠀⣾⡏⣸⡿⠦⠴⢛⣉⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⣎⠻⣿⣤⣀⠀⠛⠃⣴⡿⣻⣿⣷⣤⡉⠠⠶⣦⠈⠈⠃⠄⢠⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡮⠻⣿⣧⣄⠀⣩⠷⢿⣿⣿⣿⣿⣷⣦⣤⣤⣾⣿⣶⣶⣿⠿⠋⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⣝⣻⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠀⢀⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢉⡏⢻⣿⣿⣿⣿⣿⣿⡿⢸⣏⢻⡇
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢺⡇⢸⣿⣿⣿⣿⣿⣿⡇⣼⡏⠈⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⣻⡇⣀
⠀⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠁⣤⠉
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⣴⣶⡶⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠻⢿⣷⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣟⠇⣰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠆⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠿⠛⢿⣿⣶⣦⣄⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⢀⣴⡾⠟⠉⠁⠀⠀⠀⠀⠉⠙⠿⢿⡿⢿⣿⣿⣦⣤⣀⠀⣶⣦⣤⡠⠆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠛⠀⠀⠀⠀⢀⣠⣶⣷⣦⡀⠀⠀⢠⣾⣿⣯⠙⣻⣿⣿⡇⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠉⠈⠹⣿⣿⣿⣣⡴⠟⠛⠻⣿⣾⣿⡟⢠⢻⣿⣧⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣁⣀⣀⡀⠀⠰⠿⢿⣿⣿⣿⣿⣦⣤⣿⣿⣿⣿⡌⠣⠙⢿⣿⡇⣰⣧⠄⡀
⠀⠀⠀⠀⠀⠀⡀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠸⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⠃⠿⠇⠀⠁⢀⠀⠄
⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⣭⣽⣟⠻⢿⣿⣿⣿⣿⣷⣶⣶⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⡄⣦⠘⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣻⣿⣰⠦⠽⠭⠟⠛⡛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣆⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡿⠟⣋⣥⣴⢶⣶⣷⣶⣶⣶⣶⣤⣤⣁⠉⠛⢿⣿⣿⣿⣿⢿⣿⣿⣿⡟⢻⠋⢷⣄⢱
⠀⠀⠀⠀⠀⠀⠀⡀⣼⢀⣤⣶⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣦⡀⠹⣿⣿⣿⡾⠻⢿⢹⡇⡿⠀⢸⠙⠎⡅
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡊⡙⢹⠋⠻⣷⠺⢶⡷⠾⢷⣶⡾⣿⣿⣿⣿⣿⠿⣿⣄⠘⣿⣿⣇⠀⠀⣿⡇⠁⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡚⢷⣶⣿⣶⣴⣧⣤⣤⣿⣀⣌⣿⣄⣽⣿⠃⣸⠿⠀⢹⣿⣿⠀⠀⢸⡇⠀⣀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⠿⠟⠛⠛⠛⠛⠛⠿⠛⠿⢿⡟⠁⠀⠓⢒⣾⢃⢻⣿⠃⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠠⣾⡀⢲⣷⡆⠀⠀⣀⡀⠛⢷⣄⣰⣿⣯⣼⣧⠹⣇⠀⠀⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣷⣞⣿⣇⣿⣿⡿⢨⣼⣿⣡⡖⡀⠰⣿⣿⡐⣿⣿⣇⢻⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣷⣄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⡀⢠⣽⣧⠹⣿⣿⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⡄⢧⢼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡀⠀⢻⣷⣿⣿⣧⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⢀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡈⣿⣿⣿⣿⣿⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠸⠻⠄⠘⢿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣍⡿⠛⣉⣶⣼⢋⠈⣿⣿⠀⠀⣿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣶⣾⣿⣿⣿⠿⠟⠛⠛⠻⠿⠯⠤⣤⡺⢍⠟⣱⣧⠀⣿⠻⠀⠀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠻⠿⠿⠿⠿⠶⠿⠿⠿⣀⠒⣿⣵⠏⣸⣿⣯⢰⣿⡀⠀⠀⢹⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣹⠋⢰⠛⣿⣿⢸⣿⡃⠀⠀⢸⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⡾⢠⣇⣶⣿⣿⡞⢿⣷⡀⠀⠘⣇⠨⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣰⣏⠸⠿⣿⣿⡅⠀⣿⣇⠀⠀⢹⠁⠀⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣫⣿⣿⣶⣶⠿⣿⣇⢰⢸⣿⠀⠀⠘⡆⠰⠀⠂⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡏⢹⣿⣷⠀⠀⢹⣏⠸⢼⣿⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣿⠀⠀⠸⣿⠈⢸⣿⠆⠀⠀⠈⠆⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⡄⢀⣴⣿⡄⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠈⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⡤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣁⠈⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⣁⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⡄⠰⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢤⣄⠈⠉⢳⡖⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠰⠆⠀⢦⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠄⠀⢠⡄⠀⠠⠄⠒⢢⣤⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⢃⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢀⣤⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡦⠀⠀⣠⠤⠘⠛⠋⠛⠓⠦⣀⠴⠤⠤⠄⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠸⠋⠀⠀⡄⠂⡐⠀⢀⠈⢠⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠰⠆⠀⠀⠀⢀⠀⣠⠶⠛⠻⣦⡇⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡆⠀⠀⠀⠠⠢⢌⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠟⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⣹⠦⠀⠀⡴⢋⠁⠀⢀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⣀⣀⣈⡠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠉⠉⠉⠁⠆⠰⡆⠸⡷⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠶⠀⡘⠋⠛⠀⠀⠀⠠⣀⠀⢻⡇⠀⠀⢉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣠⠀⣾⠂⢀⡆⠀⠀⠀⠀⠀⡙⠳⠀⢻⠀⢴⢣⣴⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠛⠀⠙⡀⠨⠿⠀⢰⠇⠀⠀⠙⢢⡄⠀⡐⠂⡸⠃⣄⠉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠀⠀⡀⠃⢀⠤⡀⠈⠰⡇⠀⠙⠦⠀⠀⠀⠴⠁⣴⠆⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢀⠄⠈⠀⠀⠆⠐⢧⠀⠀⠲⠀⠀⠒⢀⠐⣀⠔⢁⡄⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠶⠉⠀⠀⠀⠀⠰⠀⢺⣇⠀⠀⠀⠀⣠⡎⠀⢁⠴⠉⡠⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠌⠠⢄⡰⠊⢀⡠⠀⢁⠤⠈⣠⠆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠄⠁⠤⠒⠁⠄⠚⠀⠔⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡁⠒⠊⡁⠒⢉⡀⠒⠈⢀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠄⢈⡁⠄⢉⡀⠠⢄⡉⠢⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠄⠒⠤⠐⠂⠀⡤⠐⠀⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⣁⠰⠂⠁⠀⠀⢉⣀⠀⠈⠐⠂⣤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠛⠛⠛⠻⠛⠁⠀⠀⠀⡀⠀⠀⢴⠀⡼⠁⠀⡠⠛⣁⠤⠘⢣⠀⠁⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⠐⠀⠠⣤⡐⠂⣄⠐⡜⠁⢀⡾⠀⠉⠁⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⡄⠈⡀⠀⠀⠀⣠⣴⠎⠀⠐⠲⢄⡈⠳⢄⠲⣀⣰⠟⠓⠂⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣴⣾⠛⠻⡟⠀⠀⠘⠓⠤⣈⠢⠈⣠⠾⠷⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠃⠀⠀⠀⠐⠒⠒⠒⠛⠀⠀⠐⠆⠀⠀⣾⠛⠛⠛⠃⠀⠇⠀⠀⠈⠒⠀⠁⠀⢸⠟⠒⠒⠒⠂⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠰⠀⢈⡁⠐⠀⠀⣏⠉⠃⠀⠀⠀⠀⡄⠀⢈⣁⠀⠐⠀⢺⡦⠤⠀⠀⠉⠉⠉⢉⣁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⢘⠉⠁⡀⠀⠀⠀⠀⢁⠀⠀⠤⠀⢀⠀⠘⡦⠠⠀⠀⠀⠀⠀⠈⠛⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⡄⠀⢺⠉⠉⠁⠀⠀⠀⠀⠘⠀⠀⠒⠂⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⢺⣁⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣉⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠒⠀⠀⠰⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠈⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢀⣼⠓⠐⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠂⠀⠁⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠁⠀⠀⢀⡴⠋⠉⠉⡉⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⠂⢰⠶⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡠⠞⠉⠀⠸⠛⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡴⠟⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡦⠀⠀⠉⠉⠀⠀⠀⠀⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠤⠴⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣈⢀⡤⠤⠤⠤⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⡦⠤⠤⠴⠶⠆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠒⠒⠛⠛⠋⠉⣉⣁⡀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⢀⣀⣀⢀⠀⠀⠀⠃⠸⡇⠀⠒⠒⠒⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⢉⣉⣁⣀⣀⣈⣁⡹⠲⠶⠶⠿⠷⠶⠶⣤⣤⠆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠿⠿⠿⠛⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠀⠀⠉⠁⠉⠹⣿⣿⣿⣿⣦⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠄⠀⠸⣿⣿⠏⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⡄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⣠⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠒⠂⠀⠀⠀⠀⢀⣀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠈⠂⠀⠀⠤⠊⠀⠀⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠚⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢊⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠒⠙⡒⠠⠀⠀⠀⠀⠀⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠚⡇⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣀⡀⠀⠀⠀⠀⠀⠀⠀⢃⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⠂⣧⣤⡇⣿⣭⠐⢯⣃⠉⡟⠁⠀⣾⣹⢆⡾⡄⢸⢸⢧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃⠃⠀⠃⠓⠒⠈⠒⠋⠀⠃⠀⠀⠛⠀⠘⠉⠙⠘⠘⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠚⠦⡑⢦⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢗⠬⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⡤⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⢀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡠⠀⡔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⡇⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⣇⠀⠌⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⣆
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠠⠤⠤⢴⣶⠈⠃⠀⠀⠀⡰⠁⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠈⠻⢄⡀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠊⠀⠀⠀⠠⠘⠇⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠔⠒⠉⠀⠀⠀⠀⢰⢁⠊⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠱⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣵⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⡩⠁
⠀⠀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣬⣤⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉⠉⠙⣆⠈⠻⡒⠦⣀
⡶⠲⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢱⡀⠈⠑⢦⡀
⣧⠀⠀⠀⠀⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠉⠉⠙⠓⠒⠦⢤⣀⡀⠀⠀⠀⠀⠘⡆⠀⢳⠀⠀⠀⠹⡄
⢹⠀⠀⠀⢀⡀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣄⡀⠹⡄⠀⣇⠀⠀⠀⠹⡄
⢸⠀⠀⠠⡏⢻⠤⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣧⠀⠸⣄⠀⠀⠀⡇
⢸⠀⠀⠡⠇⢸⠬⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣠⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣯⠙⠲⣤⡇
⢸⡆⢀⣈⣰⢸⣀⣀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣤⣄⡀⠀⠉⠉⠒⠲⠤⢤⣀⡀⠀⠀⠀⡤⢄⣸⡇⢀⣸⣀⡀⠈⢳
⢸⡇⠀⠐⢸⢸⢐⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢸⡁⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠉⠙⣆⠀⠛⠀⠈⠓⠁⠉⠀⠼⠒⡞
⠀⡇⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⡇⡀⠀⠈⡇⡆⠀⠀⠀⠀⠀⠀⠮⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⡄
⠀⡇⠀⠖⢺⢸⠐⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢹⣟⣀⡴⠃⠇⠀⠀⠀⠀⠀⠀⢰⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡄
⠀⣇⠀⠀⠭⡾⡏⠅⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡀⠀⠀⢻⡠⢄⡀⠀⠀⠀⣀⠀⠀⠀⠀⣼⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⢿⠀⠀⠈⡇⡇⣀⠆⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡖⣏⡟⡇⠠⠶⣝⠂⠀⠀⠀⣰⣿⠏⠀⠀⠀⠉⣹⠁⡀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⢸⠀⠉⠭⡇⡇⠅⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡳⠙⠇⠀⣀⠀⠸⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⡇⢀⡄⠀⢠⡀⢸⠻⡄
⠀⢸⠀⠀⡃⣷⡇⢁⡐⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠬⣍⠙⠚⠃⠀⢀⣠⠤⠖⠒⠓⠲⣄⠀⠀⠀⠈⠋⠙⠶⠋⣸⠇⠀⡿⡄
⠀⢸⡄⠴⠗⣿⡇⠉⠀⠈⡇⠀⠀⠀⠀⠀⢀⣀⣀⣠⠴⠒⠲⠤⣄⠀⠀⠈⠳⠤⠖⠚⠙⢇⠀⠀⠀⢀⣠⣿⡆⠀⠀⠀⠀⠀⠀⡴⠃⠀⣸⠃⡟
⠀⠘⡇⠀⠈⢽⡇⢁⠀⠀⡇⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢩⡟⠉⣟⠉⠓⢦⣸⠁⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⣠⠞⠁⠀⣰⠃⣸⠃
⠀⠀⡇⠀⠀⢸⣇⣀⡌⠉⣷⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⢹⡄⠀⠉⠓⠤⣄⣀⣠⠟⠀⠀⠀⣠⠞⠁⠀⠀⡴⠃⣠⠏
⠀⠀⡇⠀⠋⢸⣿⠀⠀⠀⢹⠀⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⠚⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢀⡜⠁⡴⠃
⠀⠀⣧⠀⠀⡼⠉⢲⡀⠀⢸⠀⠀⠀⠈⠙⠲⡄⢀⣄⠀⡀⡞⣇⣤⡇⠀⠀⠀⢀⣼⣇⣀⣀⣀⣀⣠⡤⠖⠊⠁⠀⠀⠀⠀⠀⠈⢀⡞⠁
⠀⠀⢿⠀⠀⢷⣖⡿⠃⠀⢸⠆⠀⠀⠀⠀⠀⡇⡟⢹⠀⣷⠇⠸⠯⡇⠀⠀⠀⠸⣄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⢧⠃⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⣰⢯⡉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⢸⡀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠓⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠒⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠾⠟⠉⠀⢾⣿⣶⣄⣀⣉⠀⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⣦⡀⠀⠀⠀⠉⠙⠺⣉⠉⠉⠈⠛⠋⠳⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠛⣿⣠⡄⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠊⠀⠑⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⢹⣯⣁⠹⣿⣶⣶⣶⣿⣿⣶⣶⣤⣶⣶⣶⣦⠀⣠⠀⠀⠉⠄⠀⠀⠀⠊⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⣤⡀⠀⠀⠙⠻⣦⣽⣿⣿⣿⣭⣿⣿⣿⣟⡿⣿⣍⢻⠇⣹⣆⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠾⠋⠀⢃⠀⣤⣤⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⢃⡜⠋⠀⢀⣿⣿⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣿⣾⣿⣽⣋⣙⣿⣷⣂⡀⠀⢀⣀⣀⣤⣶⣿⣿⣷⢼⠆⣶⣦⠈⣿⣿⣇⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢮⣧⡘⣿⣷⠈⣿⣿⠀⡄⠀⠀⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣴⣤⣤⣄⡀⠈⠉⠻⣿⡅⢸⣿⠀⠀⢀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⡟⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣦⠀⠀⠙⣷⠸⣿⠀⠄⢠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣿⠿⣿⠏⣿⠇⠰⣶⠦⣴⣶⣠⣆⠈⣿⣿⣿⡂⢄⠀⠹⣧⢿⣷⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠹⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢿⣿⣿⡇⠈⣷⡀⢹⣿⣿⠀⠀⡀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⠀⠀⠄⣠⣆⠀⡄⢈⣤⢠⣴⣤⢀⠀⠀⠀⠈⠻⢇⣾⣿⣧⢈⣿⣿⡇⠀⠁⠀⠀⠀⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢧⡀⠺⣿⣿⣦⣹⠈⠙⠃⠈⢉⠘⠳⠮⠁⠀⠀⠈⣿⡟⡟⢹⡿⢩⠁⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠈⢻⣄⠹⣿⣿⣿⣷⣿⣿⢦⠄⣀⠀⠀⠈⠛⠀⠀⢹⣿⣷⠹⡇⣼⠀⠀⠀⠀⠀⠀⡏⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠿⢀⣿⣿⣿⣿⣿⠧⣾⡶⠆⠑⠀⠀⠀⣠⠀⢸⣿⣿⡄⣆⢹⠀⠀⠀⠲⠀⠀⡃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⣿⣿⣿⣿⡧⠚⢋⣰⣲⣼⠀⠀⠀⠀⢀⣾⡟⣿⡇⠃⣸⡄⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠋⢀⣾⣿⣿⡛⠛⠤⠀⠃⠀⢸⡿⣷⢸⣷⡇⢻⡇⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣛⡛⠛⠚⠠⠅⠐⢂⣀⣀⣶⣬⢠⡏⢸⣿⡇⢹⡇⠀⠀⠀⠀⠀⢈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢟⠉⠍⠉⠛⠓⠶⣶⣶⣦⣭⣟⡟⣤⠇⠀⣿⣿⣇⢼⣿⠀⠀⠀⠀⠀⠈⣇⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠑⠲⢄⣸⡟⠉⠘⢸⣿⠀⠀⢿⣿⣇⠼⣿⡀⠀⠠⠷⠦⠀⢧⠀⠆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣄⠀⣿⡟⠀⠀⢀⡻⣿⣿⣿⣧⠀⠀⡀⠀⠀⢸⡆⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⡏⠁⠀⠀⠈⡿⠙⢻⣿⣿⡄⠀⠀⠀⠀⠀⢻⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⣿⣏⣀⣠⡤⢾⠗⠀⢸⣿⣿⡇⠀⠀⣤⣈⠀⠘⣧⠶⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⢛⣉⣥⣤⡶⠶⠁⠀⠹⣿⣿⠀⠀⠀⠁⠀⣠⢽⡀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⣥⣤⣤⠴⠶⠁⠀⡄⢻⣿⡇⢀⠀⠀⠀⠈⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⣿⣄⠰⣇⠀⠀⠁⠘⣿⣿⡀⠠⢠⠶⡀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢻⣿⠟⠁⠀⠀⠐⠀⢻⣿⣇⠀⠀⡹⠧⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠹⣦⣄⣀⣤⣤⣆⠘⣿⣿⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣠⢸⣿⠏⠈⢈⣹⣄⠻⣿⣧⠈⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⣏⢿⣿⣇⠈⣉⣻⡆⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢹⡼⣿⣿⠏⠉⠙⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⣤⣴⣶⣦⣤⣄⠀⠀⠀⠀⢀⣴⠶⣤⣤⠆⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄
⠀⠀⠀⠻⢷⣶⣿⣷⡾⠟⠀⠀⠀⠀⠘⠷⢶⣾⣷⣶⡀⠀⠀⠀⠀⠀⢀⣽⣿⣅⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⢻⣿⡿⡆⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠇⠀⠀⠀⠀⠀⡏⣾⢿⡇⡇⠀⠀⠀⠀⠘⠋⣿⣿⢿⣿⡏⠓
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⠁
⠀⠀⠀⠀⠀⢿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠸⡿⠸⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢽⣧⣿⣿⣧
⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠸⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢹⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠙⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⢀⣴⣶⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⣹⡟
⠀⠀⠀⢠⣿⣏⣿⢀⡇⠀⠀⠀⠀⠀⢰⣶⣠⣿⣷⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄
⠀⠀⠀⠀⠈⣹⡿⠀⣿⠀⠀⠀⠀⠀⠈⠻⣽⣿⣿⣧⠀⠀⠀⠀⠀⠀⣹⡟⢙⣿⠂⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⡆
⠀⠀⠀⠀⠀⠛⠃⠐⠛⠀⠀⠀⠀⠀⠀⠐⠛⠛⠀⠛⠀⠀⠀⠀⠀⠀⠛⠁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣲⣤⢄⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠼⠴⣧⡼⣮⣉⣆⣹⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠈⠀⠣⠚⢹⢓⡧⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡜⣩⣻⣦⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠈⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠤⠭⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⡠⠒⠲⢄⠀⠀⠀⠀⠈⢢⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⡟⠟⠓⠋⡡⠤⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠲⡃⠀⠀⠠⠤⠴⠊⠀⠀⠀⠀⠀⢒⣚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠃⣰⠃⣀⠤⠒⣉⡦⠤⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠑⢄⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠊⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢷⠿⠓⢻⠉⠇⠇⠃⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣀⣀⠵⠴⠚⠁⠀⠀⡘⠀⠀⣀⠤⠚⠁⠘⣄⠀⠀⠀⠀⠀⠀⢀⡴⢎⡑⣠⠓⠓⠒⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣮⠤⢄⠀⠀⠀⠀⠀⠀⢀⡤⠒⠃⠀⣰⠧⣄⣀⣀⠀⠈⢢⣀⣀⣤⢴⡚⢧⡑⣢⢝⠁⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢄⡤⠊⢀⡠⠴⠒⠁⠀⠀⠀⠀⠈⠉⠁⠈⠑⠾⠦⠭⢶⠞⠋⠁⠀⢀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⡤⠖⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠊⠓⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⠖⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⢎⡀⣀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⡦⢋⡀⢀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠾⠛⠛⠛⠳⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⣴⠾⠛⠛⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⣠⡾⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣿⣏⣠⡾⠟⠀⠀⣿⣆⣀⣀⣀⣠⣤⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣠⣴⠾⠛⠁⠘⠟⠛⠉⠀⠀⠀⠀⠙⠛⠉⠉⠉⠉⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣠⡶⠛⠛⠋⠉⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠙⠻⣦⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠘⠋⠉⠙⠃⠀⢠⡀⠀⠀⠀⠛⠉⠉⠛⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠐⠟⠻⠆⠀⢹⣇⠀⠀⠀⠾⠛⠗⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⢀⣠⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⢾⣧⣴⣴⣿⠆⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⣻⡷⢶⣤⣤⣤⣤⣤⣴⠶⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢻⣆⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣷⡀⠻⣦⡀⠀⠀⠀⠀⠀⢸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣾⣧⠈⢿⣄⠈⠻⢶⣄⣀⣀⣴⠟⠁⠙⣷⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⣿⠙⢷⣄⠙⢷⣄⠀⠈⠉⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣄
⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠙⢷⣄⠙⠻⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡻⢦⣄⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡾⠋⠻⣦⡀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠘⣧
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⢻⡆
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⣦⣄⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠘⣷
⠀⣼⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⢿⡄
⢀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⢸⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣄⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⡄⠁⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣭⢷⠞⠓⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡹⠁⠀⡞⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠤⠒⠂⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠿⠏⠀⠀⠛⢃⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠉⠛⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⡠⠔⠒⢢⣀⡠⠤⣄⣠⠤⠶⡄⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠖⠲⠄⠤⡠⢀⡠⣤⠤⠤⠤⢾⣁⣀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠄⡀⣌⠀⠀⢠⠀⠀⣀⠀⠀⢨⡬⢏⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⢄⡀⠈⠑⡍⠀⠪⡉⠀⠀⠀⢘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣹⡉⢐⣁⣀⢡⣢⠤⠴⠒⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⡠⠇⢺⣐⣒⣒⣙⡠⢄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⡏⠛⠓⠒⠂⠒⠚⠛⢹⣆⢀⣀⢀⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⡧⠴⡇
⠀⠀⠀⠀⠀⠀⢀⠤⠤⣄⠀⢀⡤⠊⠁⣀⣀⣤⣴⠚⠳⣄⠀⠀⠀⠀⠀⠀⢸⡇⡝⠁⠀⢹⣄⣀⠀⠹⣄⠀⠀⠀⠀⠀⢨⣀⣀⣸
⠀⢠⣄⡴⠄⠀⡸⠯⠭⢼⡴⠓⠒⠚⠉⠁⠀⠀⠈⠳⣤⡸⠀⠀⠀⠀⠀⠀⢸⣇⡷⣤⠤⠚⠲⠎⠛⠒⠮⣷⡄⠖⠒⠆⣾⠈⠀⢸⠰⠲⠖
⠀⢸⣒⢲⠀⢸⣤⡤⠤⢄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡗⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣯⣀⡀⣼
⠀⣸⠉⣸⠀⣘⡋⠀⠀⠊⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⡄⠄⠀⡀⠀⠀⠀⠀⠠⠤⠤⠜
⠀⠈⠉⠉⠀⠙⠭⠭⠭⠄⠀⠀⣀⣀⣤⣄⠐⠚⠋⠁⠈⠉⠛⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⢀⠤⣤⠶⠺⠛⠋⠉⠁⠀⠰⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠰⢠⡀⡠⠄⠀⠀⢚⣙⠆⠤⡀⠀⠀⠐⠞⠀⠀⠤⡀⠀⠀⠀⠀⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠈⡡⢁⣀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢮⣨⠗⠐⠒⠠⠤⠀⣀⣸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡂⠀⢐⢀⠠⠔⠀⠒⠐⠀⠀⡀⡀⢉⠀⣠⡄⠀⠀⡠⢄⠀⠀⠍⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠦⠄⠑⢢⢀⡀⣐⢊⠀⣼⠉⠈⠀⠀⠀⠀⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢢⠀⢀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⢐⡂⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠈⢨⡧⢠⠞⠙⠉⠑⠶⠤⣄⣂⣀⡀⣀⡠⡤⠞⡹⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⠀⠀⠀⠀⠀⢸⠁⣰⠋⠀⠀⠠⠁⠀⠘⡆⢲⠀⠀⠐⡰
⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠠⢄⠀⢨⠅⠀⣀⢀⠠⠒⠐⠀⠾⠴⣧⣀⠀⢪⠅⠀⠀⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠊⠀⡞⠀⠚⠛⠷⣄⡀⠀⠀⠀⠀⣀⠉⠉⠳⠀⠀⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠤⠀⠀⠙⠂⢄⡀⠀⠢⠁⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠸⣇⠀⠀⠉⢀⣠⠲⠤⠶⡾⡗⠲⠤⣤⣀⣀⣠⡞⠀⠀⠀⢀⣄⠀⠀⠋⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢨⠇⠀⠩⡇⠀⠀⠀⠁⠀⠀⣇⠫⠀⠀⢸⡄⠀⠀⡠⠀⠀⠀⠠⡤⠃⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢳⡦⣤⣤⣤⡤⠤⠾⢼⠐⢒⠀⢳⣒⣚⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣽⣯⡝⣆⠁⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠸⠦⡄⠀⠀⢳⢠⡇⢹⡄⠀⣀⢠⡷⠀⠀⠀⡞⢳⠀⠀⠭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠒⣧⠀⠀⠈⣶⠀⠀⣷⣧⢹⣾⠛⠃⠀⠀⠙⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢹⠐⠆⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣾⠈⠀⠀⠈⠿⣀⠀⢄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡔⠀⡠⠋⠀⠀⢰⠛⣆⠀⠀⠀⠀⠈⠡⢸⡇⠁⠀⠀⠀⠀⠀⠀⢀⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠁⢾⠉⢠⠎⠀⠀⠀⢀⡏⠀⠘⡆⠀⠔⡆⠀⠀⢠⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⢀⡏⠀⠀⠀⠀⢸⠁⠀⠀⢻⡀⠈⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⠠⠤⣤⣄⣸⠀⠀⠀⠈⣇⣠⠤⠤⠤⢤⣬⡄⠀⢐⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠖⠒⠒⠒⠀⠈⢳⡀⠀⡜⠋⠈⠉⠀⠒⠦⣌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⠒⠒⠒⠚⠁⠀⠙⠛⠛⠒⠶⠶⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢀⣾⣿⠃⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⣾⣿⠃⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⡇⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿
⣄⣀⠀⠀⠀⢀⣄⣀⠀⠀⠀⢀⣀⡀⠀⠀⣼⣿⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣤⣄⣀⣤⣼⣿
⠹⣿⣧⠀⠀⠀⢻⣿⡆⠀⠀⠈⢿⣿⡄⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠁
⣰⣿⡿⠀⠀⠀⣼⣿⠇⠀⠀⢀⣼⣿⠃⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡆
⣿⣿⠀⠀⠀⠰⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠙⠻⠿⣿⣧⣤⣤⣄⡀
⠙⣿⣧⠀⠀⠀⢻⣿⡆⠀⠀⠈⢿⣿⡄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠛⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠛⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣆
⣰⣿⡟⠀⠀⠀⣼⣿⠇⠀⠀⢀⣾⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⡿⠗⠀⠀⠀⠀⠺⢿⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠋⠉⠀⠀⠀⠈⠋⠉⠀⠀⠀⠈⠙⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢶⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡶⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣿⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣦⣄⣀⣀⡀⠀⠀⠀⣀⣀⣤⣤⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠛⠻⠿⠿⠿⣿⡿⠿⠿⠿⠟⠛⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠂⣠⣾⣿⣯⣽⣻⢦⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣜⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠹⣿⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⢏⡼⠃⠘⣧⠀⢹⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏⣙⣿⣿⣿⣿⣿⣿⣿⠟⢃⡾⠁⢀⠀⣻⡇⠀⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⢾⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣯⣽⠇⢰⡞⠁⢠⣿⠀⠘⡇⠀⠂⠀⣿⣿⣿⣿⠱⣾⣿⣶⣦⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠉⠻⣿⣿⣿⣿⣯⣿⣿⣿⣧⣾⣿⣿⠀⢸⠁⠀⣿⡏⠀⠀⠀⠀⢀⠞⣿⣿⣿⢋⣿⣿⡀⠁⡙⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⠀⠀⣿⠇⠀⠀⠀⠀⠈⢠⣿⣿⣿⣿⣾⣿⣾⡿⠓⠻⡿⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠄⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⣿⠧⠀⠀⡠⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢰⣿⣄⢀⠔⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠹⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣾⣿⣿⣿⣆⠀⠀⠀⠘⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⢿⣿⣿⣿⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⡇⠉⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⡀⠛⢷⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣿⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⣿⣿⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡶⠤⣌⣹⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⢸⣿⣿⣿⣿⣮⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣦⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⠀⢸⣿⣿⣿⣿⣿⢏⣴⣭⣙⣛⣛⡛⢛⣫⣵⣾⣿⡿⠀⢸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢛⣛⢛⣫⣵⣿⣿⣿⣿⣿⣿⣿⣦⣝⡛⢛⣛⠁⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣤⠀⠀⠸⣿⣿⢋⣭⢉⣭⡍⣭⡝⢿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⢿⣄⠀⠀⠻⣿⣬⣉⣈⣉⣥⣙⣥⣿⡟⠀⠀⠀⢀⡀⠰⣦⣸⣿⣠⡶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⢿⣄⡀⠀⠙⠻⢿⣿⣿⣿⣿⠿⠋⠀⠀⢠⣤⡈⣿⡀⠈⢻⣿⠋⠀⢸⡏⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠇⠀⠉⠓⠀⠀⠀⣴⣶⣤⣴⣶⡀⠀⠀⠀⢀⣈⣽⣿⣧⣀⢨⣿⢀⣠⣿⣿⣍⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⠰⣾⣿⣿⣿⣿⣿⣷⡶⢀⡄⢉⡉⠀⠀⠀⣉⣿⣿⣟⡁⠀⠀⢈⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⠿⣫⣴⣿⣷⡬⢙⣻⣿⡟⠋⢸⣿⠈⠛⣿⣿⣛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣘⣛⢁⣿⠁⢀⣼⣿⣄⠀⢸⣇⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣅⠰⠟⢹⣿⠙⠷⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠈⠁⠐⠂⠀⠀⠀⠀⡠⠉⠉⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠠⠄⠠⢤⣀⠀⠢⡀⢀⡠⠖⠛⣇⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣀⡤⢄⣠⡀⠈⠑⢂⣸⠁⠀⠀⠀⠃⣤⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠉⠻⣷⠀⠀⠙⣦⠀⠀⠀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠿⠀⠀⠂⠉⢢⠀⠀⠈⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠠⠤⢤⠀⠀⠀⠀⠈⠋⠡⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠖⠒⠂⠀⠀⠀⠀⠀⢰⡦⠾⠁
⠀⠀⠀⠀⠀⠀⢰⠚⠉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠘⣦⣀⣰⣧⠟⢄⠒⠠⣄⡀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠘⡄⠀⠀⠘⠛⡀⠀⠀⡆⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠳⠒⠒⠒⠒⠒⠒⠺⣥⠀⢳⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣶⣈⡷⠀⢸⡇⠉⠙⠢⣄
⠀⢀⡤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠁⠀⠀⠘⣿⠄⠀⠀⠈⢢⡀
⢠⠛⠀⢰⠀⡤⢟⣳⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣀⣀⡀⣿⢾⠀⠀⠀⠀⢳
⡸⠀⠀⢰⡟⢰⡏⢉⡁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⡹⠉⠉⠉⢡⠞⢰⠀⠀⢰⠀⢣
⢿⠀⠀⢸⠁⠀⠙⠋⡄⠸⢧⣀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣷⠋⠀⠀⢀⠔⠉⠀⢸⠀⠀⠘⠀⠈⣇
⠸⣇⢠⡞⠀⠀⠀⠀⠃⠀⣰⠏⣀⡠⠀⠀⠀⣠⠞⠁⢰⠁⠀⠀⠀⠏⠀⠀⠀⢸⠀⠀⠀⡇⠀⢸⡄
⠀⠈⣙⠇⠀⠀⠀⠀⢠⣠⠓⢤⡌⡘⠛⠛⠋⠀⠀⠀⢸⠀⠀⠀⢀⡀⠀⠀⠀⡾⠀⠀⠀⡇⠀⠈⣇
⡄⠐⠋⠀⠀⣆⠀⠁⢰⠀⠀⠀⠣⢆⠀⠀⠀⠀⢀⡴⠛⣧⡴⠚⣯⠣⠀⠀⠀⡇⠀⠀⠀⢇⠀⠀⣿
⠀⠑⢤⡀⠀⠇⠀⡇⢸⠀⠀⠀⠀⠈⠳⠶⠶⠞⠉⠀⠇⢸⠃⠈⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⠀⠀⢿
⠀⠀⠀⠳⣸⣄⣀⡇⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠸⡆⠀⡘
⠀⠀⠀⠀⠙⠁⢨⠁⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣟⠀⣫
⠀⠀⠀⠀⠀⠀⠘⠓⠒⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⢏⠣⠄⣦⡀⠀⠀⠀⢀⣸⡆⠀⠀⠀⢠⡿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠒⠒⠀⠀⠀⠈⠐⠂⠛⣦⢠⠴⡏⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣀⣀⡀⠀⢀⣀⡀⠀⠐⠶⢶⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⠀⠉⠉⢁⡟⠀⠀⣠⣞⠤⠒⠈⡀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡼⠁⠀⠀⠀⢀⡼⠁⠀⡰⠁⠈⢆⠀⠀⠉⠲⣄⡔⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠐⣫⠀⠀⠀⠀⢀⠞⠀⠀⢠⠃⠀⠀⠘⣆⠀⠀⠀⠘⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠸⡇⠀⠀⠀⢾⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⠦⣤⣄⣠⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠈⢧⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⠀⠀⠀⠀⠹⣄⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠷⣤⣼⡏⢱⠀⠀⠀⠀⠀⢈⡇⠀⢸⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⡐⠙⠳⠶⠾⠆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⠃⠀⠀⠀⠀⠀⣇⠀⠀⢀⣧⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠖⠛⠉⠁⠀⠀⠀⠀⠠⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡤⠄⣀⣀⣀⣀⣀⣀⣀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠸⣧⡀⠀⠀⢀⣀⣠⣤⡴⠶⠶⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇⣀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠈⢉⡜⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠒⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢀⡀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠈⠠⠦⠀⠀⠀⠀⠀⢀⠀⢆⠈⡎⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠄⠀⢀⠘⠇⠀⠀⢀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⡀⠀⠀⠒⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⡀⠀⣻⣿⡟⠛⠛⢀⣴⣶⣶⣶⡆⢀⣴⣶⣾⣿⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⣿⣄⠀⢰⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⠿⢿⣿⣿⠿⠿⣿⣿⣿⡄⣿⣿⣧⡀⠀⣾⣿⣿⡿⠿⠃⢸⣿⣿⡟⠛⠁⠀⠀⢀⡴⠟⠛⠁
⠀⠀⠀⠀⠀⢠⡿⠟⠁⠸⣿⣿⣷⣶⣶⣤⡀⠀⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡿⠃⣿⣿⣿⠿⠀⢻⣿⣿⣆⠀⠀⠘⢿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠈⢹⣿⣿⣿⡆⢸⣿⣿⡆⢸⣿⣿⡿⣿⣿⡉⠀⠀⢻⣿⣿⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠈⠉⠙⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⣿⣿⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡇⢻⣿⣷⡀⠀⢸⣿⣿⣦⣤⣤⡄⠀⢻⣿⣿⣷⢠⣤⣤⣴⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣿⠿⠋⠀⠘⢿⣿⡇⢸⣿⣿⡇⠈⢿⣿⣿⣄⡘⠛⠛⠛⠛⠛⠁⣀⣼⣿⣿⣿⢸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⠉⠉⠁⠀⠈⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠃⠘⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠇
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠃⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣶⡖⠀⠀⠀⠰⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣾⣴⡄⠀⣀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣏⠀⠀⠀⠀⠀⠀⠁⠐⠠⠀⠀⠀⠀⠀⠀⢀⣿⡿⠇⠀⠀⠀⠀⠀⡀⠀⠀⠴⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠛⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠈⢆⡘⢣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠀⠀⢠⣄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⣿⣷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠙⡍⠁⠒⠒⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⠄⡄⢀⣀⣘⡄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢒⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣾⣿⡞⢶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠻⣿⣿⢀⡙⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⢟⣉⣸⡄⢱⣿⣇⠀⠀⠀⠀⠀⢢⠀⠀⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢘⣋⣤⣤⠈⠉⡉⢉⢉⠉⠉⡉⠲⣿⣿⡿⣶⡟⠁⠀⠀⣄⡀⠀⠈⣆⠠⣟⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠋⠊⢧⣀⠀⠀⠺⣦⣼⡐⣿⣷⡘⠻⠇⠸⢧⠀⠀⠀⠹⣇⠀⠀⢸⣧⠛⢤⠀⢘⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠙⠻⠿⣷⣦⣟⢰⣿⣿⣿⣦⠀⠀⠈⢆⠀⠀⠘⣥⠀⠀⢸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠹⣇⣀⡀⠀⠀⠀⠀⠀⠀⣀⣛⣿⡀⠻⢿⠟⠋⠘⠷⠄⡈⣤⣀⠀⠻⠀⠀⢸⣟⣀⣤⠄⠀⡄
⠀⠀⠀⠀⠀⡀⢰⣿⣿⡿⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠗⢦⡀⣀⠀⠀⠀⠔⠀⠀⠃⠐⠀⠀⠀⢸⣿⡏⠁⠀⠀⣷
⠀⠀⠀⢀⡞⢠⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣟⠁⠀⠈⣀⡈⣿⣦⠠⡄⠀⠀⢹⠀⠀⠀⠀⠸⣿⣿⣤⣀⣀⣸⣇⣀⣤⠆
⠀⠀⠀⢸⠀⢸⣿⣿⣷⡍⠆⠙⢿⣿⣶⣦⣤⣤⣤⣶⣿⣿⡿⠁⠈⢀⣿⣧⠈⢷⠀⣇⣤⠀⠀⢧⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⡿⠟⠁⠀⢀
⠀⠀⠀⠈⠀⠀⠹⠿⢿⣶⠀⠀⠀⠀⠙⠋⠙⠉⠉⠉⠉⠀⠀⠀⠀⣸⡇⠈⠇⢸⠀⢻⣿⠀⠀⢸⡀⠀⠀⢸⢣⣿⠟⠉⠀⠀⣀⡀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⣤⣶⣶⣷⣾⣾⣿⣿⣿⣷⡶⣬⣛⢾⣇⣨⠇⡏⠉⠀⠀⠘⡇⠀⠀⢸⢸⠋⠀⣀⡴⠞⠉⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⣤⢔⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⣧⠙⣿⡄⣷⡆⠀⠀⠀⠇⠀⠀⡇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣼⣵⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠿⣿⣿⣿⣿⣾⣇⠹⣧⣿⠀⠔⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡘⢿⣿⣿⣦⣶⣾⣿⣷⣶⣤⣄⠈⢻⣿⣿⣿⢹⣶⢇⠇⠀⠁⠀⣬⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠈⠿⣼⣿⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⠞⣿⡴⠀⠀⠀⠀⡇⠀⠀⠀⠄⠸⣷⠦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⢰⣿⠇⡇⠀⠀⠀⣷⠀⠀⠀⡇⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠰⠀⠈⠙⢻⣿⣿⣿⣿⣿⡏⠀⣿⣿⡏⣼⢣⣾⡇⠄⠀⢀⡿⠠⠤⠠⡧⠴⠿⠿⠤⢤⡄⠀⠤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢀⣻⣿⣿⣿⣿⣿⣿⣧⣴⠟⠟⠈⠁⣾⣿⠰⠀⠊⠸⣧⡄⠠⠄⢶⠠⣶⣶⣶⣎⣿⡀⠀⢹⡆⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠛⠿⠿⠿⠿⠿⠿⠻⠛⢻⣄⠀⠀⣴⠏⠻⡇⢞⡀⠀⠖⢷⠀⠲⠘⣦⠰⠶⠾⠿⠟⠃⠘⠈⠿⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢠⣿⡟⢠⣮⣿⡈⡀⠘⠀⠛⣧⠀⢀⠸⣆⠐⣾⣿⣧⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⣿⣷⠂⠉⠹⣇⢱⠀⠀⠀⠘⣆⠀⠀⠱⡀⠿⠿⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⠻⣿⣿⡇⠀⠀⣿⡌⢇⠀⠲⠄⠛⣆⠀⠀⠙⠂⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠹⣿⣧⡀⠀⢻⣷⡘⣄⡀⠀⠀⠈⢦⠀⠀⠙⠂⠀⢤⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢻⣿⣇⠀⢈⣿⣧⠈⠷⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⢻⣿⠀⠀⠨⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠘⠛⠁⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⠶⠶⠛⠻⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡾⠋⢀⣀⣤⣤⡄⠀⠀⠠⣤⣤⣀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢡⡀⠈⣁⣀⣠⡄⠀⠀⠠⣄⣀⣈⠁⠀⣌⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣸⡇⠰⢮⣭⡁⠀⠀⠀⠀⠈⣩⡥⠦⢠⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⢤⡀⢦⡴⢀⡤⠁⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣷⣤⣤⣾⠁⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⡸⠋⠀⠀⠹⢇⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⠙⠳⠦⠴⠞⠋⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠋⠉⠁⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠉⠙⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠶⠶⠶⣶⣶⡿⠛⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⣤⣾⣷⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣾⡟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢈⣵⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⡏⠀⠀⠀⠀⠀⢠⣿⣿⣾⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢨⣿⣿⣿⣿⣿⣿⣯⠀⢸⡏⠀⣼⠏⠀⣼⠟⠀⣠⣿⠟⠉⠀⢀⣼⣿
⡇⠀⠀⠀⠀⠀⣼⣿⣿⢟⡫⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢿⣜⣿⣿⣿⣿⣿⣿⣿⡄⢸⠃⢠⡟⠀⣼⠋⠀⣴⡿⠁⠀⣠⣴⣿⠟⢻
⣷⡀⠀⠀⠀⠀⣿⣿⣿⡯⢒⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⡇⠈⠀⠘⠁⡴⠃⢠⡾⠋⠀⣠⣾⠟⠋⠁⠀⣸
⣿⣿⣄⠀⠀⠀⢹⣿⣿⣿⢕⣠⣤⣶⣶⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⠁⠴⠋⠀⣠⠾⠋⠁⢀⣠⣴⣿⢿
⡇⠘⢿⣦⠀⠀⠘⣿⣿⣡⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⠀⠀⠀⠀⠀⠀⠋⠁⣀⣤⠶⠛⠋⠉⠀⣸
⣷⣄⠀⠙⢷⡀⠀⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⡀⢲⠀⡶⠀⣤⠀⢀⡤⠂⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠊⠁⣀⣠⣤⡶⠾⠿⢿
⡏⠙⠷⣄⠀⠹⣄⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣈⡙⠀⠀⠀⢈⣠⡴⠟⣁⠄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣀⣀⣸
⣷⣦⣀⠈⠓⢄⠀⠀⢻⡀⠹⣿⡀⠀⠐⠲⠶⢦⣤⣴⣶⣄⠀⠈⠻⠶⠞⠁⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠙⠛⠛⠛⢻
⣇⡀⠉⠓⠦⡀⠀⠀⠀⠹⡄⠙⠓⠀⠀⠀⠰⠶⣶⡶⠿⣿⣤⡤⣤⣄⡀⠀⠀⠀⠀⠀⡿⢒⡒⠶⠶⠶⠶⢤⣤⣤⣈⣛⠛⠷⢶⣶⣾
⡟⠙⠛⠒⠤⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⡀⠀⠸⣏⣛⠶⢤⣴⠶⠛⠋⢻⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠦⣼
⣷⠶⠦⠤⠄⠀⠀⠀⠀⠀⠀⠉⠛⢷⡰⣶⣤⣄⠀⢀⣴⣿⢁⡿⣤⠀⠻⢄⠙⠲⢬⣉⠓⠶⠼⣧⣀⠀⠀⠀⠀⣠⠞⠑⢦⡀⠀⠀⢸
⣧⣀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡝⢿⣿⠿⠛⢩⠃⣼⠃⣿⡀⢰⡀⠉⠲⠤⣀⠉⠑⠂⠀⢸⣷⣄⠀⠀⠈⢿⣿⡏⣷⠀⠀⢸
⡏⠉⣀⣠⠄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣮⠋⢠⣴⡏⢰⠇⣸⠋⡗⠀⠻⣶⠖⢤⣄⡉⠑⠂⠀⠀⣿⠈⠳⣄⠀⠀⣿⠛⠁⠀⠀⢸
⡿⠛⠉⠀⣀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠇⠀⢼⠟⢀⡟⢠⡏⢸⡇⢄⠀⠸⣦⣄⠈⠙⠳⢄⠀⠀⢻⠦⠀⣸⠟⢻⡜⠻⣦⠀⠀⢸
⣇⣠⡶⠋⢁⡴⠀⢀⣴⠟⠉⠉⠉⠙⢿⡏⠀⠀⠀⠀⠋⢀⠞⠀⣼⠁⣺⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⠘⠲⢄⣿⢷⡶⡶⠶⣿⠀⠀⢸
⡟⠁⢀⣴⠏⠀⣠⠞⢁⠀⠀⠀⠀⠀⠸⣷⣆⣀⡀⠀⠀⠀⠀⢠⡏⠀⢿⠀⣀⣀⣀⣀⠙⢿⣷⡄⠀⠀⠀⠀⣿⣸⠄⡇⠆⣿⠀⠀⢸
⣧⣴⡟⠁⢀⡾⢃⡔⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣾⡆⠀⠘⣇⠀⢸⡏⠀⢸⡥⠞⠀⢀⣻⣿⣦⡀⢀⡾⠋⠛⠷⣿⣿⡟⢻⣤⢾
⡿⠋⠀⣴⠏⣴⠋⢀⠄⠀⠀⠀⠀⢀⡟⠸⣿⡎⣿⠘⣿⠀⠀⠀⣿⠀⣰⠃⠀⠘⠓⠛⠋⠉⠉⠙⠻⠟⢻⡇⠀⠈⠿⠿⣿⣿⠀⣿⢸
⡇⢀⡾⢡⡞⢁⡴⠁⣀⠀⠀⠀⠀⢸⣧⣄⣿⣿⠘⠀⠏⠀⠀⠀⢸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠐⢛⣻⣿⣀⣿⢸
⣷⠟⣰⠏⣰⠏⣠⠞⠁⡀⠀⢀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢿⡏⠛⠶⠶⠾⠿⠟⠋⣿⢸
⣏⣼⢃⡾⢃⡼⢃⡠⠊⠀⠀⣿⡇⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠋⠁⠀⢼⡇⢸⣿⣶⣶⣶⣿⡇⣿⢸
⣿⣣⡟⣱⠟⣠⠞⢁⡔⠀⣠⣽⣧⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣠⡄⠀⠀⡀⣀⣤⡴⠟⣩⣤⣶⠶⠶⠶⠾⣧⣤⡻⣿⣿⣿⣿⡇⣿⢸
⣿⢏⣼⢋⡼⢃⡴⠋⢀⠐⣷⠾⠶⣿⠀⠀⣾⣿⡇⠀⠀⠀⠀⣬⢀⣠⣤⠀⠉⠀⠀⠀⣭⣤⣤⣄⣀⣀⣤⣤⣿⡧⠙⣿⣿⣿⡇⣿⢸
⣿⡿⣡⡞⣡⠞⢁⡔⠁⣠⢿⣥⣤⣿⡀⢀⣿⣿⣿⠀⠀⢀⣠⠿⠛⠁⠀⠀⠀⠀⠀⢀⣿⣇⣿⢹⡏⢹⠉⣯⣸⣇⡀⠈⠻⣿⡇⣿⢸
⣿⣷⠏⣴⢋⡴⠋⣠⠾⡇⠀⢾⣿⡇⢻⢸⣿⣿⣿⡷⠖⠋⣰⠯⢷⡀⠀⠀⠀⢀⣾⠛⠉⠉⢙⣛⣻⣿⣿⣿⣿⡍⠙⣦⣀⣈⣴⣿⢸
⣿⣿⡾⢡⡞⢡⡾⠁⣸⡧⣄⣀⣿⡷⣿⣽⠿⠋⠁⠀⠀⢰⣇⣀⣠⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⣷⠀⢹⡟⠉⢀⡿⢸
⣿⣿⣿⡏⣴⠏⣠⠞⢘⡇⢷⣾⣶⣶⣿⠁⠀⠀⠀⠀⢀⣼⣄⣉⣡⣿⣶⡄⠀⢸⡿⠶⣤⣤⣤⣤⣤⣼⣿⣿⣿⡷⠶⢿⡷⠾⠋⠀⢸
⣿⣿⣿⣿⣧⡾⢃⣴⣻⡇⠈⢿⣿⣿⣿⠀⠀⠀⠀⢰⣫⡀⣽⢟⣛⣭⣍⡿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⣿⣿⣿⣿⣿⡿⠟⠋⢹⡇⠀⠀⠻⡿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠈⠻⣿⣯⣟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⣿⣿⠿⠛⠉⠀⠀⠀⢘⡷⢤⣤⣤⡴⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠹⣷⡝⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⡟⠁⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⠴⠿⠋⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⣿⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⣀⣤⠾⠃⠀⢸⡟⠻⠶⠶⣦⣤⣤⣤⣤⣶⣶⣾⠟⣻⡇⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠈⠛⠶⢦⣤⣤⣤⣀⣀⣠⣼⣿⣿⠶⠛⠁⠀⠀⠀⢸
⣿⣀⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼

⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⣼⣇⣠⣦⠀⠀⠀⠀⠀⠀⠀⠔⡾⠁⠀⠀⠀⠀⠀⠀⣴⣀⣰⡄⠀⠀⠀⠀⠀⢠⣴⡶⢦⡀⠀⠀⠀⠀⠀⠀⠚⢃⡀
⠀⠀⠀⠀⢿⡿⠿⠉⠀⠀⠀⠀⠀⠰⠿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠧⣄⠍⠁⠀⠀⠀⠀⠀⠀⢸⣿⡦⠁⠀⠀⠀⠀⠀⠋⠼⠛⠓

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⣠⣥⡢⠀⠀⠀⠀⠀⢀⣴⡖⢲⣦⠀⠀⠀⠀⠀⠀⣴⣶⣴⡄⠀⠀⠀⠀⠀⢰⡆⣾⣇⡀⠀⠀⠀⠀⠀⢠⣶⣷⣶
⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⢸⡿⠇⠸⢿⠇⠀⠀⠀⠀⠀⢠⣴⡟⠁⠀⠀⠀⠀⠀⢸⣿⡟⠸⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀
⠀⠀⠀⠀⣇⣈⠻⣦⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠿⠏⠰⠏⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⡶

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣠⡦⠷⣦⡀⠀⠀⠀⠀⢸⠏⡛⡛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⣿⡇⠀⠀⠀⠀⠀⣸⣿⢆⣀⡀
⠀⠀⠀⠐⣿⠧⠸⢿⡇⠀⠀⠀⠀⢰⣬⣥⣥⣼⡇⠀⠀⠀⠀⠘⠐⡂⡳⠐⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠸⠻⠸⣭⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡊⡀⠈⡁⣀⣠⡤⠄⢀⠀⠠⠌⡡⠔⠊⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠄⢼⣇⣁⡤⠛⣉⣥⣤⣶⣶⣤⣀⠐⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠒⠀⠐⠒⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠋⠄⢿⡏⠐⣿⣅⣂⣸⣭⣿⣿⣿⣧⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⣿⣀⣷⡀⠛⠉⠉⠉⠉⠛⠛⢿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⡇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⠀⣼⣯⣼⠁⠀⠀⠀⠀⠀⠀⢀⣼⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣀⣀⣿⣏⡐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡠⠀⠀⡿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠞⠄⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠉⢹⣿⠙⣤⣄⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡇⡟⣿⡄⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣽⡊⠒⢦⡠
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡏⠐⢠⠀⡀⣷⡏⢿⡢⢰⣲⣾⡿⠟⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡶⠟⠿⠉⠉⠍⠐⢶⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣹⠀⣧⡿⠀⠈⢻⣌⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢓⠀⠀⠀⠁⠀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⣼⠀⣿⢇⠀⢈⣹⣿⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⢤⣤⣤⣴⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢰⣿⠀⠀⠀⢸⣇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣀⠀⠀⢀⣟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠉⣇⠈⣿⠀⠀⢩⡟⠟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⠉⠉⢉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⣿⠀⢹⠀⠀⣸⣗⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠦⠀⠐⠒⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣿⠀⣿⡄⠀⠙⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈⣁⢁⠈⣁⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠈⠛⠀⠿⠗⠈⠀⠯⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠈⠐⠀⠀⠘⠀⠁⠁⠀⠃⠃⠑⠘⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠃⠈⠘⠈⠛⠉⠘⠀⠁⠀⠀⠃⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠔⠂⠉⠉⠉⠁⠒⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⡀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢀⣤⢮⠁⠓⠀⠀⠀⠀⠘⠃⢉⠶⣤⡀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣸⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⢻⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠠⠖⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠆⠀⠀⠈⢧⠀⢠⠊⣡⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠽⠋⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠀⠀⠀⠀⠀⠰⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣿⠀⠀⠀⠀⠀⠀⡠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢿⠀⠀⠀⠀⠀⡼⢲⢶⣦⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣅⠀⠀⠀⠀⡟⢁⣎⡀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠻⣄⠀⠀⠈⠛⢋⣉⣉⡙⠛⠀⠀⠀⠀⠀⠀⠀⢠⢺⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣧⡀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣀⠀⠁⢸⠦⡀⠈⠀⡇⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⢩⠆⠀⠀⠀⠈⡿⠀⢠⣄⠠⠤⠔⠋⠁⠀⠀⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⢛⡁⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⠜⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢸⡞⢣⠀⠀⠀⠀⠀⡠⠔⣁⠔⠁⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢸⢣⡈⢆⠀⣀⠴⢊⡡⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⣴⠴⢇⠀⠀⠀⠀⠈⡄⠳⡈⠋⣠⠔⠋⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⢰⠁⠀⠘⣃⠀⠀⠀⠀⢱⡄⠙⠉⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠙⡀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⡤⠾⡁
⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠾⣄⡀⠈⠦⡀
⠀⣴⠁⠀⠀⠀⠀⠀⠀⢀⡼⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⣠⠞⠁⠙⣄⠀⠱⡀⠀⠀⠀⠀⠀⣰
⠰⠁⠀⠀⠀⠀⠀⠀⢀⡾⠀⡇⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⢶⠉⠀⠀⠀⠀⠘⢃⠀⢹⡀⠀⠀⠀⢠⠋
⡇⠀⠀⠀⠀⠀⠀⢀⠾⠃⢰⠀⣀⢴⠃⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡄⠀⣡⠀⠀⢠⡎
⡇⠀⠀⠀⠀⢀⡴⠟⠾⠦⠛⠁⠀⢸⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⣀⠞⢱
⠀⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣛⣠⠞⠁⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠉⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠈⠛⠛⠛⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠓⠛⠒⠲⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠖⠒⠒⠲⠶⠦⢤⣀⡀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠁⠀⠀⠀⢀⣄⠀⣾⡦⠉⠓⠶⣄⣀⠀⠀⠀⠀⠙⠲⡄⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣋⣹⠆⠀⢠⢦⣜⡿⠶⠿⣇⣠⣯⠇⠀⠉⠓⠦⣄⡀⠀⢰⢷⡀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢉⠀⠀⡀⠈⢙⡏⠀⠀⠀⠀⠈⢧⣀⣰⣶⠀⠀⠀⠙⢦⡟⢰⠛⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣩⠇⠘⢻⡶⠋⣲⡄⠀⠀⠀⣀⡄⠈⣏⣀⣠⡄⠀⠺⢿⡄⠈⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⢀⡴⣼⠇⠀⠁⠀⠀⠀⠀⠉⠉⠀⢀⡟⠉⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣠⡿⣏⠀⠚⠓⠚⠙⠲⠚⠳⠴⠂⠘⣇⡀⢀⣾⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣯⣉⣙⣛⣓⣚⣯⠽⠚⠹⣎⢳⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠸⡄⠀⠉⢩⡟⠛⠛⠳⠦⠄⢀⡟⢦⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢀⡶⢿⡶⣶⣚⠛⠲⠤⠤⠴⠶⢻⡇⠈⢧⡀⠘⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣇⢸⠃⣇⠀⣼⡄⢀⣆⠀⠀⠈⠀⠀⠀⣿⣤⡏⠀⠀⢈⡽⠛⠓⢦⣀⠀⠀⣇⠀⠈⢳⣠⠞⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⡾⠀⠸⣤⠏⣇⡼⢻⠀⠀⠀⢀⡔⠀⠿⢻⡄⠀⠀⢸⠁⠀⠀⠀⡉⠉⣻⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⢼⡇⠀⣾⡄⠁⠀⠉⢀⡟⠀⠀⠀⠘⠄⠀⣠⠞⣛⡛⠛⣻⣄⣀⣀⣠⡿⢯⣅⣠⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡴⠃⠹⡄⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⣉⡿⠿⠿⢧⣤⢭⠴⣯⣭⠟⢺⣷⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠃⠀⠀⢹⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠒⣿⣋⣓⣋⡽⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣦⣤⣤⢼⣶⡾⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⢀⠞⠇⠸⡘⡆⢙⣌⠳⣭⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡉⢷⠎⢠⡖⠒⣧⢹⡋⢉⡇⠹⠤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠉⠉⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠙⠛⠁⠀⠀⢳⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣋⣀⣰⣤⣴⣦⣄⣤⠄⠉⠉⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣶⣿⣿⣿⣿⣿⣿⣿⣏⣫⣜⣿⠶⣁⠀⣠⡶⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣏⣼⣿⠶⣿⣿⣿⣿⠟⢳⣟⣿⣶⣿⢾⡻⣿⡉⠚⢛⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⣾⣿⠿⣿⣿⣿⣿⣿⣷⣿⣙⣿⣍⣿⣿⣿⣿⣿⣿⣾⣿⣿⣧⣰⣿⣿⣿⣦⠀⠀⢀⠀⠀⠁⠀⠌⠀⠀⠀⠀⠘⣧
⠀⠀⡠⠂⣥⣿⡿⢿⠿⢿⡍⢉⠻⣿⠿⢷⣽⣻⣬⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⣸⡟⠁⢂
⠠⡎⠀⠰⣿⣿⣧⣾⠀⢸⣿⣷⣶⣌⢷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⠛⢿⡿⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⡀⠀⠈⣇
⠀⠳⠀⠀⠛⠻⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠃⢀⣿⣦⡈⠁⠂⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⢀⠀⠰⠋
⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠉⠙⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⠇⢰⣾⠟⠻⣿⡆⠀⠀⠀⠀⢈⣶⡄⠀⠀⠀⠀⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⢀⡠⠂⠀⠀⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⡏⣱⣿⡟⣤⠀⢹⣧⡀⠀⠀⠀⢺⡆⠀⠀⠀⠀⠀⣰⣏⣀⣀⡤⡀
⠀⠀⠀⠀⠀⠀⢠⡖⠀⣀⣤⣴⣶⣿⡿⢿⣷⣶⣶⣤⣈⠉⠛⢿⣏⣿⣿⣿⣧⣿⠀⠀⡿⠃⠇⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⠋⠉⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢰⠏⣠⣾⢿⣿⣿⣽⣛⡷⠾⣿⣿⡟⠛⠛⠿⢶⡄⠈⡿⠿⢿⣿⠻⠠⠀⠇⠐⠻⠂⠀⢸⡇⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⣸⠃
⠀⠀⠀⢀⣤⡀⡀⠙⠧⠴⣏⢠⡟⠉⠻⠛⠻⢿⡦⠀⣤⠀⣈⠀⢦⣀⢠⡄⢱⡄⠀⠀⠀⣀⠀⠀⠀⢠⠁⠀⠀⠀⢠⡿⣿⣿⣿⣷⣤⣀⣿⣀⢀⣀
⠀⠀⠀⠈⠹⣿⡇⠀⠀⠀⠀⠀⠘⠂⠙⠒⠤⠘⠄⠀⠀⠁⠈⠀⠘⠋⢀⡀⢸⠃⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⠀⢠⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⣸⠃⢀⡄⠀⠀⠀⠀⠀⠛⠀⠀⠀⡾⣡⡿⠟⠉⠉⠀⠉⣉⡉⣉⣉⣯⣧⠞⠁
⠀⠀⠀⠀⠀⢾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⣾⠇⠀⠀⠀⠐⠀⠀⠀⠀⡘⣴⡟⢁⣀⣠⣤⣴⠾⠟⢛⣭⡟⢉⠔
⠀⠀⠀⠀⢸⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢣⡙⠀⠀⠀⢀⡟⠀⠀⠀⢠⣽⠋⠀⠘⠋⠉⠀⠀⠀⠰⠋⡠⠊⠀⢀⡰⠄
⠀⠀⠀⢸⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢧⣼⡇⠀⠀⢠⣿⠁⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⢀⡴⠋
⠀⠀⠀⢈⢹⣇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢸⣿⠀⠀⠀⢸⠇⠀⠀⠀⣾⡇⢀⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡉⠰⠶⢿⣿⣶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⢻⣿⡇⣿⡅⠀⠀⢀⡿⠀⠀⠀⣼⠉⢸⣿⣀⣰⡆⠀⠀⠀⡌
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣶⣤⣀⠀⠀⠀⠙⠋⣱⣿⣿⣇⣿⠆⠀⠀⢸⠃⠀⠀⢰⡿⠀⣀⡛⠉⠉⠀⠀⠀⠀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣧⡄⠀⠀⢀⣴⣿⣿⣿⢩⡇⠀⠀⠀⣾⠀⠀⠀⣼⠃⣠⣿⡇⠀⠀⠀⠀⠀⡖⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣠⣶⠿⣿⣿⣿⣿⣿⡗⠀⠀⢰⡟⠀⠀⠀⣿⢠⣎⣙⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⠋⠁⠀⣴⢹⣿⡟⣿⠃⠀⠀⠈⡇⠀⠀⠀⣏⠰⣿⣿⡇⠀⠀⠀⠀⢸⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡿⠁⠀⠀⠋⢸⣿⣇⣿⠀⠀⠀⠀⣷⠀⠀⢠⣿⠀⠉⠛⠿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢿⣦⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠈⣿⠀⠀⠀⣟⠀⣶⣶⣶⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⣿⣄⡀⠀⠀⣿⣿⣿⣿⡗⠀⠀⠀⢹⡄⠀⠀⣿⠀⣼⣿⣿⣧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⢸⣿⠋⠀⠀⠀⣿⣿⠋⢿⣇⠀⠀⠀⠀⣥⠀⠀⢸⣆⠀⠙⠛⢿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣶⣿⣶⠶⠶⠀⣿⣿⣟⠀⣿⠆⠀⠀⠀⢿⡄⠀⠘⣿⠀⣀⣰⣤⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣾⣿⣿⣤⣤⣤⣼⢻⣿⡷⠀⣿⡃⠀⠀⠀⠠⣇⠀⠀⢻⡈⢿⣿⣿⣿⡀⠀⠀⣇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠈⠛⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃⠀⠀⠻⠀⠀⠅




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁⠈⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠈⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⣄⣀⠀⠀⠀⠈⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠈⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠁⠀⠀⠀⠀⠀⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠠⠂⠠⡲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠋⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⣄⣉⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡄
⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣄
⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆
⠀⠀⠀⣴⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⡀⣤⢀⠀⠀⢠⠞⠓⣦⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡆⣠⠀⠀⣰⠚⠳⡄
⠀⠀⠀⠀⢘⣱⣶⣎⡃⠀⢸⠀⠀⣿⢀⣀⡀⠀⠀⠀⠀⠠⠤⣏⢙⠦⠄⠀⣿⢀⡀⡇⢀⣀
⠀⠀⠀⠀⠈⠁⠛⠈⠁⠀⢸⠀⠀⣿⢀⣀⡀⠀⠀⠀⠀⠀⠚⠈⠅⠑⠀⠀⣿⢸⡇⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⡇⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡆⣿⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⡇⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣸⣧⡹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣼⣇⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢸⣿⣿⣷⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⣿⣿⡧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⣛⣛⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⣛⣛⣥⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠦⠤⣀⢈⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠾⠉⠉⠑⠢⢌⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠲⠤⣄⠀⠀⠈⢆⠀⠘⣾⠓⠒⣤⣀⢤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢰⣻⡧⠤⠖⠮⠥⣠⠋⢠⣎⡤⠚⠑⠃
⠀⠀⠀⠀⢀⠀⡀⠀⠀⣀⣀⣤⡤⠤⠤⠤⠤⠽⠗⠒⠒⢿⡆⠀⢀⣀⠴⠃⠀⣾⣿⠥⠤⣄
⠀⠀⠀⠀⢃⡎⢠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠉⠉⠉⠀⠀⠀⠸⠋⣤⡶⠒⢨⠇⢸⢸⠁
⠀⠀⠀⠀⠘⡄⢻⡀⠠⠤⣀⢀⡀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢓⠲⠶⡶⠶⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠞⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⠀⠀⠀⠀⣀⣠⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠘⠈⡠⠔⠊⠁⠀⢸⡷⠅
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⣺⠀⠀⠀⢸⣇⢸⠉⡱⢦⡀⠀⠀⠀⠀⠀⠀⢀⣿⢲⣮⠔⠂⣀⣀⠒⠲⠿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠉⠉⠉⠑⠒⠚⠛⠒⠓⠒⠛⠒⠚⠛⠉⣉⠭⠽⢼⠚⢁⠐⠢⣀⠀⢉⣶⣄
⠀⠀⠀⠀⢠⢄⡤⢀⡴⠋⣿⢓⣲⣒⠒⠲⠤⠤⠤⠤⠤⠤⡤⠖⠉⠀⠀⢀⣼⠀⡸⢸⠀⠈⣦⣾⡣⠙⠂
⠀⠀⠀⠠⢁⡎⣰⢫⢔⡈⠉⠉⠉⢀⣉⣉⣩⠭⢭⣴⠖⠋⠀⠀⠀⢀⡴⠋⠘⡇⠀⠀⠀⠀⠀⠚⠆
⠀⠀⠀⠀⠀⠑⠉⠒⠚⢶⠒⠛⡏⠙⠧⣄⡀⢀⠞⡇⠀⠀⢀⣠⠖⠋⠀⠀⠀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⢠⠀⡆⡟⠀⠀⡇⠀⠀⠀⠉⠚⢤⣥⠴⠞⠋⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡎⢰⠱⣇⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢿⠸⡄
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠉⣿⢷⣶⣶⠶⠦⠤⠤⠤⣤⣴⣶⣶⠶⠒⠚⠋⠉⠁⠒⠉⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠤⢴⡇⠀⠀⠀⠀⠀⣧⣀⣠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠽⠆⢸⡀⠀⠀⠀⣤⠶⣓⡦⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠚⠃⠀⠀⠛⠊⠩⠥⠤⠬⠆

⠀⠀⢢⠀⠀⠀⢠⣴⣤⡄⡔⠀⠀⠀⠀⢰⡀⠀⠀⣤⣤⣄⠀⡴⠀⠀⠀⠀⠰⡀⠀⢰⣿⣿⡆⠀⣰
⠒⠒⠚⠓⣶⣶⣾⣿⡿⠚⠓⠒⠒⠒⠒⠚⠓⠒⢒⣿⣿⠟⠚⠓⠒⠒⠒⠒⠒⠓⣒⣾⣿⡟⠓⠒⠛⠒⠒
⠀⠀⠀⢰⣿⣿⣿⣿⣶⣾⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣠⣄⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢷⣄⡀
⠀⠀⠀⠈⠙⣿⣿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠿⡿⣿⣿⣿⣷⡀⠉⠛
⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠉⢻⣿⡇
⠀⠀⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢿⡗⠀⠀⠀⠀⠀⣠⡾⠟⠋⠁⠀⠈⣿⡇
⣤⣶⣶⣶⣴⣿⣭⣿⣧⣄⣀⣠⣤⣄⣀⣤⣴⣏⣠⣤⣸⣧⣄⣀⣀⣤⣄⣿⣅⡀⣀⡀⢀⣀⣿⣧⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⡟⢻⢉⢫⡝⠏⢩⢻⠉⣿⡉⣝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⣇⣾⣜⣣⣆⣰⣙⣀⣋⣃⣫⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠩⢙⡯⠩⣽⡍⢭⡍⢹⡏⠍⣭⣭⡏⠩⣽⡍⢭⣿⣿⢩⡍⢻⠏⡹⣯⠹⢩⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣇⣰⣜⣏⣘⣹⣁⣿⣷⣘⣣⣦⣙⣠⣇⣘⣹⣁⣋⣿⣿⣘⣣⣞⣰⣆⣹⣃⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠒⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠁⠉⠉⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⡀⢀⠠⠠⠄⠀⠐⠂⠀⠖⠀⠉⠈⠀⠚⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⡆⠀⠀⠀⠀⠀⢀⠤⠤⢤⡀⢀⠤⠒⠢⢄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣀⣀⣠⠤⡤⡇⠀⠀⢀⣈⢫⠀⣀⡀⠀⢹⠉⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠤⡖⠂⠀⠀⢀⡀⠀⠀⠀⠘⢄⠀⠀⠁⢀⡧⣌⣀⣀⠜⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⢊⣶⠲⡁⢈⡒⢈⡦⠀⢬⠛⣁⠖⠉⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠠⠀⢀⠀⡎⠀⠀⠀⠀⠀⠈⠂⠒⠂⠄⠐⠥⠄⠠⠩⠴⠤⣀⣀⣀⣀⡠⠤⡔⡆
⠀⠀⠀⠌⠀⡸⢀⠎⠀⠀⣄⣀⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⡄⠀⢳⡸⢦⠃
⠀⠀⠀⢀⠀⠀⠈⡆⠀⠰⡋⠀⠁⠀⣇⣠⣄⢀⡀⠀⠀⠀⡆⠙⠋⠉⢱⢨⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⢅⠀⠀⠠⢤⡇⢀⠀⡈⢉⠉⣀⠣⠀⠀⠀⠘⡼
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠑⠰⠃⠀⠁⠀⠀⠀⠀⠀⠐⡶⠂⠀⠰⠋⢁⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠊⠑⠂⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠲⠒⠢⣤⡲⠤⠤⠤⢤⠤⠤⠒⠴⢶⣒⡒⠉⠉⠙⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⠀⠀⠁⠲⢄⡀⠀⠀⠉⠙⠦⢄⣀⠀⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⡄⠀⠀⢈⠂⠀⠀⠀⠀⢰⠉⠁⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠈⢀⡰⠁⠀⠀⠀⠀⢰⣅⣀⠀⡘⠀⠤⠐⠂⠐⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣁⠀⢠⠎⠤⠀⠀⠀⠀⠉⠁⣰⠀⠀⠀⠀⠀⢀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠥⣄⡀⣉⡥⠤⠤⠤⠤⠤⠤⠖⠚⠓⠚⠉⠉⠉⠉⠉