Hold up Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢉⠄⢄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠀⠈⠀⠁⠀⢃⡀
⠀⠀⠀⠀⠀⠀⠐⡐⠠⠑⢄⠠⡐⠅⠢⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡀⢀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⢠⣠⠐⠁⠀⠀⠀⢉⠛⠀⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⡀⠐
⠀⠀⠀⠀⠁⢽⠀⠂⠀⠁⠁⢀⠄⠀⠀⢫⡀⠀⠀⠀⠀⠀⠠⠀⠺⠶⠶⠟⠃⠐⠠
⠀⠀⠀⠀⠰⡆⠀⡀⠠⠄⢠⠄⠀⠀⠀⠀⢳⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣀⡀⠀⢀
⠀⠀⠀⠀⢴⣀⡴⢤⣄⠀⠀⠀⠀⡰⡤⢤⡈⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠈⠉⠉⠃⠉⠁⠀⠘
⠀⠀⠀⠀⠀⢘⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠓⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠀⠀⠀⠀⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣨⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⣛⢋⢀
⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠒⠒⠀⠐⠂⠀⠀⠀⠉⠀⠀⡀⡀⠀⠀⣀⠗⠲⣒⣑⠄⠀⢠⡄⠀⡤⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣧⡴⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⠄⠐⠄⠀⠂⠀⠀⠀⠀⠀⠀⠃⠁⠉⠈⠀⠀⠀⠀⠀⠈⠉⠀⣢⠜⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢹⢠⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣘⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣀⠀⠀⢠⢴⡀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢸⣿⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⡇⠀⢻⣿⣏⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⣄⠂⠐⡟⠄⡀⠐⠣
⠀⠀⠀⠀⠀⠀⠀⢨⣛⣺⠀⠘⣌⣩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣓⣸⠀⢅⣖⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠦⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠻⣄⣀⠤⠞⠁⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠙⢄⡈⢉⠝⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⢀⣀⠠⠤⠤⢤⣀⣀⠀⠀⠀⠀⢘⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢤⠀⢸⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⢲⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⢩⡄⣬⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠰⠇⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠘⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠄⠀⠀⠠⠠⠀⣀⣀⣀⠀⠤⠄⠈⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡇⠀⠀⡀⠀⠀⠀⠃⠀⠘⠀⠀⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠀⠈⠀⠀⠀⠈⠈⠉⣉⠁⠐⡁⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⣀⣀⣀⣀⠤⠐⠊⠁⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⡠⠒⠒⠒⠢⠤⣤⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠮⠴⠚⠉⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠈⠑⠲⠦⠤⣄⣀⣀⣀⣀⣀⡤⠔⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢳⣠⠤⣄⣠⣄⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⠀⠈⠁⠀⠉⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⡶⠯⡍⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠉⠛⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⠀⠀⢀⡠⠖⠋⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⠚⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢠⢫⠖⡢⠀⠀⠀⠀⠀⠀⢔⠲⡕⡄⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢑⡇⠎⡸⣩⢍⠀⠀⠀⠀⡫⡍⠇⠃⢰⠊⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⡄⠁⠀⠀⢡⠊⠀⠀⠀⠀⠑⡌⠀⠀⠈⢰⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠛⠒⠋⠈⠀⠀⠀⢀⡆⠀⣀⡀⢀⡀⠀⢡⡀⠀⠀⠀⠁⠛⠒⠓⠦⢤⣀
⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⢀⡠⠔⠚⠛⠉⢠⠊⠁⠈⠀⠈⠓⡄⠉⠛⠓⠢⢄⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠔⠚⠉⠓⠦⣄⠀⢀⡞⠀⢋⡓⢊⡛⠀⢳⡀⠀⣠⠔⠚⠉⠓⠢⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠘⠒⠺⣇⠘⢄⡉⢉⡠⠀⣸⠗⠒⠁⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢦⠀⠿⡀⠊⠜⠣⡑⢈⡟⠀⡴⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⠀⢸⠀⢸⡄⠁⠀⠀⢈⢠⠃⠀⡇⠀⣤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⡇⠀⠛⡄⠐⠂⢠⠚⠀⣸⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠦⣄⣐⢀⡀⣂⣠⠴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⢀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⠀⠈⢃⠀⠀⡀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠋⠉⠿⠛⠻⣄⠈⢄⠀⠀⠻⢷⣤⠴⠖⠒⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⡠⠊⠀⠈⠑⢾⢦⠆⠈⠁⠀⡀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⢀⠁⠀⠀⡀⠀⠈⡇⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢨⣿⠀⠀⡇⢼⡇⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠋⠉⣱
⠀⠀⠀⣼⠀⠀⠀⠀⣀⠀⠀⠀⠘⢄⠈⠉⠀⡐⠁⠄⠀⠀⠔⠁⠀⠀⠀⡜⠀⠀⠀⠀⡧⢤⣤⣀⣀⣀⣠⣇⠀⣴⠍⠀⢼⠃
⢀⡴⠞⣷⠀⠀⠀⠁⠙⢄⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠑⡟⢠⠤⠊
⠸⣆⠀⢸⡄⠀⠀⠀⠀⠀⢳⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠹⣆⠀⢧⠀⠀⠀⠀⠀⠘⡄⢀⡋⠐⠒⡒⠐⠒⡂⠉⠈⣧⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⣸⠃⠺⢆⠀⠀⠀⠀⠀⠙⡈⠉⠉⠉⠉⢉⡋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠
⡤⢯⠁⠀⠀⠈⢣⠀⠀⠀⠀⠀⠐⠤⣄⠈⠙⠃⣈⣢⠄⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠑⢘⠄⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠖⢹⡇⠀⣿⠀⠀⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⢖⡒⠖⠒⠸⢇⢀⠿⠐⠒⠒⡒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠄⢀⡠⠔⠋⠀⠒⠤⣀⠀⠀⠀⠀⠘⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠴⠶⠒⠒⠛⠛⠋⠉⠉⠉⠉⠙⠛⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇⠀⠀⠀⠀⠀⠘⢯⡁⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠤⡷⠀⠀⠀⠀⠀⠀⠀⠈⠉⡡⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠔⡲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⢏⠉⠢⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠡⠄⠙⣀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠄⠀⢈⣀⣰⡶⠬⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⡧⠀⢦⠀⠀⠀⢀⡴⠊⣠⠐⠉⠁⠀⠀⠹⠀⠀⢀⡉⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⠃⠈⢆⢣⣀⡴⠁⠀⡜⠁⠀⠀⠀⣾⣀⢀⠀⠀⠀⢱⡀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠲⠞⠀⠀⠈⢢⢏⣠⣄⡀⣇⠀⠀⠀⠀⡸⠉⠀⠂⠈⡇⠀⠃⠀⠀⠹⡒⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢟⠀⠀⠣⠈⠲⠤⠀⠤⠇⠀⠀⠀⣀⠃⠀⠀⠀⠀⡜⠙⠢⠀⠀⠀⠀⠀⠉⠁⢒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠓⠆⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡟⠑⠀⠘⠐⠒⠒⠦⣄⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠁⠒⠢⠤⢤⠤⠀⠄⠀⠁⣠⠏⢀⠠⡀⠀⠘⠮⡳⢄⠈⣳⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⡀⠀⠀⠠⠄⠊⡡⠂⠀⠀⠈⠓⠢⠤⣤⣤⠴⠋⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⣠⢎⠁⠂⠀⠀⠀⢠⠂⠀⠀⡄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠸⣁⠄⠃⠊⠐⠒⠄⣤⠀⠀⠀⠀⠀⣁⡒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠾⡝⣓⡲⠤⠤⢊⠄⢰⠂⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠉⡶⠶⠞⠀⠀⠀⠉⢧⣬⠳⢍⡙⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⠦⠀⢤⠀⠀⣀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠒⡤⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠏⢛⠉⠙⠛⢲⠶⡞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡒⠻⣍⠙⠣⡀⠀⡨⣣⠀⢀⡜⠀⠙⢓⡶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⣀⡵⢤⣸⠟⠊⠁⠀⠈⠺⣗⣴⡁⣠⠞⠁⣰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠭⢷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠒⠀⠁⠈⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣾⡟⠃⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠋⣡⣤⣶⣶⣦⣴⠟⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣉⡉⢁⣤⣦⣤⡄⣤⠃⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠉⣉⣩⣤⣢⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢺⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡏⠀⢀⣀⣹⣿⣿⠃⠀⠀⢀⣀⣀⣀⣠⣤⣤⣴⣆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡟⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠻⢀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⡘⢷⡾⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢾⠃⢠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⣠⣴⠇⢠⣼⣿⣿⡟⠀⠀⠀⠀⠀⣀⣀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠰⠟⢁⣴⣿⢃⣿⣿⡇⠀⠀⣠⣴⣿⠟⢻⡿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⠀⠀⠀⣤⠟⢹⠏⣸⣿⣿⠃⠀⣼⡿⠋⣴⠆⠈⠁⠛⠙⠻⣿⡿⣿⣦
⠀⠀⠀⠀⠀⠀⢸⡏⠀⣼⣆⠀⠘⠁⠀⠀⢀⣿⣿⣿⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⢻⣿⠀⢀⣀⠀⠀⠀⠀⢀⣀⣹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⠙⣇⠀⠀⠀⠀⠀⢀⡔⣿⣿⣿⣿⠀⣴⣿⠂⠰⣿⣿⠇⠀⣿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣿⡀⠘⠆⠀⢀⣀⠀⠋⢀⣿⡏⣿⣿⡄⢿⣻⠀⠀⠈⠀⠀⠀⣿⠀⠈⠁⢸⣿⣍
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠂⠚⠛⢁⣴⡟⢸⣿⢇⣿⣿⡇⠈⢿⡇⠀⠀⠀⠐⠠⢾⠇⠀⣤⣾⣿⠟
⠀⠀⠀⠀⠀⢸⣿⣷⣄⠀⠀⣤⣿⡿⢁⣿⡟⢸⣿⣿⡇⠀⠘⣇⠀⠀⠀⢂⣀⣀⠀⠀⣧⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣦⣼⣿⣿⢧⣽⣿⠇⣼⣿⣿⣷⣤⡶⣿⠀⢀⠀⠉⠉⠙⠃⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⠀⣿⢹⣿⣿⣿⡅⠈⠃⠈⠻⣶⣴⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡽⣿⣿⣿⣿⡅⣿⡇⢰⣿⣾⣿⣿⡟⣿⡀⠀⠀⠀⢄⡈⢻⣿⠿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⣿⣿⡗⢹⣿⡿⠇⢹⡿⠀⢸⡿⣿⠇⣿⡇⢘⣷⡀⠀⢀⡆⢿⠀⠁⠀⠸⣿⡿⢿⣦
⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠿⠑⢢⣿⠃⠀⣸⡇⡟⠀⣿⡇⠘⢹⣧⠀⢸⡿⢸⣧⣠⣶⢠⣿⢡⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⡌⠺⠃⣼⠃⣿⠃⢀⠀⣿⡇⠀⠙⣿⡄⣿⡁⠀⠈⠹⣧⣸⡿⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠈⡄⠀⠀⢠⣿⠀⠉⠀⡘⠀⣿⡇⠰⠶⣿⣷⡈⣿⡄⢦⡀⢿⣿⣇⢠⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠹⡄⠀⣾⠏⠀⠀⣠⡟⠀⣿⠇⠀⠀⢀⡉⠛⠻⢿⣌⢻⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠙⠢⠁⠀⠀⣰⡿⠀⢸⣿⠀⠀⠀⠀⢿⣆⠀⠀⠉⠙⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠠⡝⠁⠀⣾⡇⠀⠀⠀⠀⢸⣿⡄⠀⢰⣿⣿⣿⣠⣿⣿⠵⣿⣿⣻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠁⠀⢰⣿⠃⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⣿⣉⣩⣿⠀⠀⢸⡉⠻⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢻⡆⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⡾⣿⣿⢃⣾⣿⣿⡆⠉⢹⣿⠀⡰⣾⣿⣆⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠛⠄⠀⠀⠀⠀⠿⠿⠀⢠⣧⠀⠀⡇⢸⣿⠈⡏⣿⣿⣦⣤⡴⣿⠀⣿⣿⣿⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⡏⠀⠀⠸⡟⢫⠀⣷⣿⣿⣿⣿⣡⣿⣰⣿⣿⣇⣼⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢿⡏⠉⠋⠉⣿⠹⣿⣿⢻⣿⣿⣿⣿⣿⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠉⠀⠈⠁⠈⠉⠻⠟⠟⠛⠈

⠀⠀⠀⢠⡇
⠀⠀⢀⠚⠻
⠀⠀⡆⠒⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄
⠀⢰⠁⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡔⡠⠂
⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠰⡐⠀⣾⣦⣤⣀⣀
⢀⠼⢄⡤⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠉⠉⠉⠚⠯⢁⠀⣔⡠⠞⠁⠈⠁⠀⠈⠙⠢⡀
⡜⠂⠴⠀⠀⣸⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡍⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠢
⢏⡀⣘⠀⣧⠿⣤⣀⣀⣀⣀⣀⡞⠀⠀⠀⠀⠀⠀⡇⠀⠀⣶⡀⠀⠀⣾⡆⠀⡘⠀⠀⠀⠀⠀⠀⢡
⢀⡇⠀⢠⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⡀⠀⠀⠘⠄⠀⠈⠔⠃⠀⠉⠁⡰⠁⠀⠀⣀⠀⠀⠀⢸⡀
⢺⠁⠑⠘⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⢸⠙⠦
⢸⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠑⢠⢀⣀⠀⠀⠀⠀⢀⣀⠔⢺⠀⠀⠀⠀⠀⡼⠀⣰
⠈⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠧⠤⠤⢯⡀⠹⠀⠀⠥⢠⠆⠀⠀⠀⠀⣰⣃⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⢀⠤⠃⡤⠤⡀⠠⠎⠀⠀⠀⠀⣰⠋⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠉⠀⠀⠀⠀⢀⠞⠣⡀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠈⠑⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠀⢸⠇⢸⡓⠢⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⠀⢸⡇⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠙⠍⠁⠈⢉⡆⢼⡉⠁⠈⠙⠉⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⠬⠔⠊⠁⠀⠀⠈⠑⠢⠥⣀⣈⠤⠃

⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⣠⡄⠀⠀⠠⡀⠀⣠⣤⠀⠀⡔
⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⡚⠁⠀⠀⠀⠀⠙⢦⣁⢀⣠⠞⠀⠀⠀⢀⣐⣀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠸⣿⢻⡇⠀⠀⠀⠀⠸⣿⡿
⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⣿⣸⡇⠀⠀⠀⠀⣰⣿⣿⡀
⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⠻⠟⠂⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢸⢣⠀⠀⡆⠀⠀⡞⡇⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣄⠀⠀⠀⠀⢸⠈⡆⠠⢻⠀⣸⠃⡇⠀⠀⠀⠀⣠⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⢳⠘⢦⠀⠀⠀⢸⠀⢹⣸⣾⢀⡏⠀⡇⠀⠀⠀⡴⠃⡞⠀⠀⠀⠀⠀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣗⢦⣀⠀⠀⠀⠈⣇⠀⠳⡄⠀⢸⠀⠀⢿⣿⡾⠀⠀⡇⠀⢠⠞⠀⣸⠁⠀⠀⠀⣀⡴⡺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡈⠳⢤⡀⠀⠸⡄⠀⠘⢆⣸⠀⠀⣸⣿⠃⠀⠀⣇⡰⠃⠀⢀⠇⠀⢀⡤⠚⢁⡞⠁
⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠙⠲⣄⣷⠀⠀⠈⠻⠀⠀⡿⣿⡇⠀⠀⠟⠁⠀⠀⣜⣠⠞⠉⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠈⠙⢯⣒⠶⢤⣄⣀⠀⠈⢧⡀⠀⠀⠙⠂⠀⠀⠀⢀⣀⡝⣿⡇⠀⠀⠀⠀⠀⠘⠋⠀⠀⢀⡜⠁⠀⣀⣠⡤⠖⢒⡽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠉⠙⠓⠢⠵⡄⠀⠀⠀⠀⠀⠀⣼⠶⢆⣿⣇⣀⣀⡀⠀⠀⠀⠀⠀⢠⠯⠔⠚⠋⠉⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠾⠋⠁⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠲⠶⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠦⠀⠀⠀⠀⠀⠀⠀⢰⣟⣁⠀⠀⣰⣇⡀⠀⢸⠀⠀⠀⠀⠀⠀⠴⠿⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⠶⠖
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⣠⠴⠾⢿⣿⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡱⠶⠀⠀⠀⠀⠀⠀⠀⠀⣶⡾⠃⠀⢀⠀⠀⠈⠉⠙⠳⣄⠀⠀⠀⠀⠀⠶⢎⣉⡁
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⣠⠾⠦⠤⠤⠤⠤⣤⣼⣆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀
⠀⠀⠀⠰⠒⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⣶⠖⠀⠀⠀⠀⠀⢰⣿⡅⢀⡴⠁⠀⠀⣀⠀⠀⠀⠀⠀⠉⢧⠀⠀⠲⣖⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢸⠈⢧⡞⠀⢀⡤⠶⠟⠖⠚⠛⠋⠉⠙⠻⡄⠀⠀⠈⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⣀⣠⡤⠴⡲⠋⠀⠀⠀⢸⡄⢸⡀⠀⡼⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⣖⠢⢤⣄⣀⠈⠑⢦⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⠥⠖⠚⠉⠉⠀⢀⡞⠁⠀⠀⣠⠆⠀⣧⠀⢷⣾⡃⠀⠸⣷⣴⣿⡆⠀⠀⠀⣸⠃⠈⢣⡀⠀⠉⠙⠓⠲⠬⣷⣤⡀
⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⣠⠋⠀⣠⠴⠋⡽⠀⠀⢸⡆⠀⠈⢳⣤⣀⣹⣿⠉⠛⠲⠄⢠⠟⠦⣀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⡴⠊⠁⠀⢸⠃⠀⣠⠎⢱⡀⠀⠸⣿⡿⢿⣿⣦⠀⠀⠀⡼⡆⠀⠈⠓⢦⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠞⠉⠀⠀⠀⢀⡏⠀⡴⠃⠀⢸⢧⠀⠀⣿⠀⠀⠉⠋⠀⠀⠀⡇⢹⡀⠀⠀⠀⠉⠳⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⡼⢠⠞⠀⠀⠀⢸⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⡄⢧⠀⠀⠀⠀⠀⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠋⠀⠀⠀⠀⢸⢠⠏⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⢸⡜⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⡶⣶⡶⣶⡶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠮⣥⣌⡉⢽⣶⣦⣄
⠀⠀⠀⡟⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠯⠭⠭⠭⠭⠭⠭⢉⣉⣉⣉⣉⣉⣑⣄⠀⠀⢀⣼⣶⠒⠁⠉⢷⣮⣿⣿⣿⠟⣆
⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣾⣿⠏⠀⠀⠀⠀⠉⠉⣭⣿⣿⣿
⠀⡴⢁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣿⠋⢖⣲⣤⠀⠀⠀⠀⠉⢛⣿⣿
⣸⠤⠘⠃⠉⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⡜⠀⠀⠉⠙⠀⡰⢻⣖⠦⢸⣿⠇
⡧⠀⠀⢁⣀⣥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢀⡄⠀⡀⢠⡇⠀⠈⢠⡿⠏
⢳⠒⢈⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠐⠆⠤⣉⡠⢀⣴⡯⠞
⢸⠃⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⡳⡀⠐⠲⠄⠈⠀⣺⠋⠁
⠀⠑⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠱⡙⠲⣤⣤⣴⠾⠃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠇⢀⣤⣿⡆⠉⠁⠀⡘⢣
⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣠⣿⣿⢃⠀⢀⠌⠀⠈⠣
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣴⣿⣿⡷⢈⠔⠁⠀⢠⠤⠀⠀⠑⠠⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢻⣿⣿⣿⠟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢢
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠹⠟⠁⠀⣀⣠⡄⠀⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡀⠱⡾⠛⠉⠁⠀⠀⠀⠀⢠⡤⠀⠀⡴
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠉⠢⣥⡀⠦⠤⣀⡀⠀⠀⣼⣧⢀⡼⠁
⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠤⠖⠈⠐⠢⠵⠦⠀⠀⠉⠑⣶⢻⡟⠋⠀⠀⢠
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡛⠁⣴⣊⣹⠦⠀⠺⡆⠐⠒⠰⠃⣾⠁⡄⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⣾⠜⢀⣤⣤⡤⠀⠸⡀⠀⢀⣼⢥⣈⣀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⣿⡏⠙⡇⠀⠀⠘⢿⢌⠁⠂⠠⣀⠵⠀⢸
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠠⣼⡀⠀⠀⠘⠈⡆⠀⠀⠈⠓⢀⣼
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⡀⠑⠦⠤⢴⠀⠀⠀⠀⢀⢀⡾⠃
⠀⠧⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠘⠦⣤⣴⣯⣥⡇⠀⣄⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠓⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡇⠣⡀⠀⠙⣿⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠖⠒⠚⠛⠉⠉⠉⠉⠉⢉⣭⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣖⠒⠒⠒⠒⠒⠛⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣆⠀⢒⡒⠒⠲⡒⠒⠒⠒⠒⠒⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠄⠒⠒⠒⠛⠋⠉⠉⠀⠀⠀⣈⣉⠓⠦⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡝⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠔⠋⠉⠉⠛⢦⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣄⡀⠀⠀⠀⡶⠟⠀⠀⠀⠀⠀⠴⠊⠉⠉⠀⢠⠃⠀⠀⠀⠀⠀⠈⢳⡀⠘⡆⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣙⡄⠀⢠⠇⠀⠀⠀⠀⠀⠀⡠⠚⠉⠒⢤⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢻⠀⠀⠀⠀⠸⠉⠁⠀⠎⡷
⠀⠀⠀⠀⠀⠀⡰⠋⢁⣠⣤⣄⠈⠙⠲⠏⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣱⠀⠞⣷⠀⠀⠀⠀⠀⡿⠀⠸⡆⠀⣤⠶⠤⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠉⠑⠒⠚⠉⠂⠀⠀⢀⣀⠀⠀⠀⠤⢾⣀⠀⠀⠀⠈⢸⠇⠻⠟⠀⠀⠀⠀⢠⠇⠀⠀⡇⠀⠈⠓⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠒⢤⣄⣀⣀⣀⡤⢤⠀⠈⠘⣄⠀⠀⠀⠀⠀⠉⠢⡀⡠⠊⠳⣄⡀⠀⠀⣀⣴⣃⡀⠀⠀⡇⣀⠤⠂⢀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠙⠉⢡⠊⠀⢹⠀⢰⣏⣠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠍⠀⠀⠤⠠⠞⠀⡜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠌⠦⡀⠀⠀⠉⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⢀⡴⢤⡀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⣀⠀⠀⠉⠒⠤⠤⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠐⠧⣈⣙⡂⠑⠄⢒⣋⡭⣇⠀⢀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠤⠤⠤⠤⠴⣶⢶⠊⠁
⠀⠰⣏⣁⡤⠀⠀⠋⠁⠀⠀⠘⠋⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⠒⠒⠒⠒⠺⠏
⠀⠀⠀⠀⠀⠀⣆⣠⠄⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⣹⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⠀⠀⠀⠀⠸⢻⠟⠋⠁⠀⢸⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⢰⡚⣽⣿⡖⢲⣦⣤⣾⣤⡤⠒⢲⣻⡻⢯⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠭⠭⠭⠭⠉⠙⠛⠉⠉⠉⠛⠻⠽⠽⢷⣿⡭⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠂⠃⠘⠐⠂

⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣷⣦⣄
⠀⠀⢰⡏⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠫⠭⠉⠏⠩⠭⠭⠭⠥⠤⠈⣑⣦⡀⠀⠀⣰⣿⠉⠉⠻⠿⣿⣟⣋⣷
⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⣛⡓⠲⣀⣀⣀⡀⢹⣿
⢀⠎⣰⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⣿⠀⠰⠟⢣⠀⣭⠥⡉⢺⠃
⡞⣊⡡⠴⠺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⠿⠠⠤⢴⠂⠀⠘⠃⠁⣞⡆
⣯⣡⠤⢒⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠳⣒⢦⣍⠃⠠⣞⠙
⢸⡤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠙⠒⠂⠈⣀⡼⠁
⠈⠢⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢋⠲⣤⣤⣤⣶⡏
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢀⣤⣶⠘⠛⠉⢠⢣
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⣨⣿⡿⠇⠀⡠⠀⠀⠣
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣼⣿⣿⡇⡰⠊⠀⢠⠤⠀⠀⠐⠄⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠹⡋⢀⣠⣤⠶
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣦⣹⡛⠉⠀⠀⠀⠀⠀⢠⡇⠀⢠⠇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢉⠠⣙⡳⠈⠉⠓⠦⢄⣀⣿⣷⡵⠋
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⡛⢉⡤⢌⡉⣙⣇⠀⢀⣀⡽⣸⠇⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣴⣻⠋⣉⣁⡀⠙⡄⠀⠈⢡⣛⠐⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⡿⢿⡅⠀⠙⢶⣖⠛⢍⢉⠑⡄
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡧⢄⣧⠀⠀⠈⡎⠆⠀⠀⠙⠆⢀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠉⠲⢤⢤⡇⢈⠀⠀⢀⢠⡿
⠀⠧⠤⠄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⢳⠤⢤⣮⣥⡞⠀⣄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡄⢦⡀⠈⣻⠷⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠋⠉⠉⠑⡢⡀⢀⠄⠂⠠⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣬⠁⠀⠀⠀⠈⢣⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠀⠀⢀⡀⠀⠀⠘⠀⣄⠀⠀⠀⢸⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⡠⣴⣶⡖⡒⢸⡀⠀⢯⣿⠀⠀⢸⠐⢾⠃⠀⠀⡸⠇⠷⠂⠐⠢⢄
⠀⠀⠀⠀⣰⠋⠀⠈⠉⠀⠁⠀⠱⣄⠀⠁⢀⡰⠁⠑⠄⣀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢰⠇⠀⠀⡠⢤⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⢸⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢸⠑⠛⠈⠋⠁⠓⡖⠒⡖⠒⠒⠒⠒⠒⠒⠲⠖⠒⠒⠒⠒⠚
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⢀⠉⠑⠢⠤⠤⠤⡤⠤⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⠔⠂⠉⢲⠧⠤⡄
⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠘⢤⠀⠀⠒⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢀⠀⡀⠀⢸⡀⠴⡃
⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠑⠰⠆⠉⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⠤⠐⠈⡀⢰⢤⣔⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢊⣁⡀⠤⠒⣉⡠⠔⠊⠁⠉⢻⢯⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠒⠂⠤⠤⠤⠤⠤⠤⠐⠒⠺⡅⠒⣈⠡⠄⠒⠈⠉⠀⠀⠀⠀⠀⠀⡸⠀⠘
⠀⠀⠀⠀⠀⠀⢀⠤⢂⠡⣶⡰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⢰⠉⠴⡊⠁⢰⣿⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠁⠀⢈⡁⢺⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⡇
⠀⠀⠀⠀⠀⢰⠊⠐⠜⠉⢟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠤⠃
⠀⠀⠀⠀⠀⡠⠛⠀⠀⠀⡸⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠑⣶⠀⢀⠜⠁⠀⠀⠉⠒⢠⣤⠤⢤⡤⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⢸⠀⡇⠀⢀⣀⡀⠠⠤⠤⠀⣀
⢀⣠⡤⠄⢀⡈⠉⠉⠉⠁⠉⠉⠁⠚⠀⢇⠀⠀⡠⠀⠓⠉⠀⠀⠀⠀⠀⠀⠠⢄⠛⠲⢦
⣎⠁⣠⡎⠀⠀⣀⡀⢠⣤⣤⠤⢤⡤⣤⡤⠇⠸⣀⣀⣤⠤⠤⠤⠤⣤⣤⣀⣀⣀⡵⣄⣠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠠⠈⠀⠀⠉⢀⡤⠊⠀⠈⠉⠐⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⢀⢠⣢⣴⣶⣿⣿⣖⣊⣁⣀⣈⣉⡣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡥⢖⣢⣽⣿⡶⠿⠿⠿⡿⠿⠿⠿⣿⣿⣿⣿⣯⠉⠛⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⣿⡿⠉⠠⠖⠒⠀⠀⠀⠈⠙⠒⠝⣿⠋⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠁⠀⠔⠉⠙⢢⡀⠀⡔⡉⠉⢢⢸⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠉⠙⠆⠀⠀⣿⣤⣠⠃⣀⢻⣷⠆⠀⢸⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠋⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠄⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⢀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⣀⡀⠀⠀⠙⠷⠶⡾⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠁⠒⠠⠄⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⢠⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠛⡀⠀⠀⠀⠀⠀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡇⠀⠀⣀⡀⠀⢸⢀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡃⠀⡈⠀⠑⡀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⣈⠀⠀⠀⠔⠠⠒⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⡄⢰⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠤⠐⠁⠀⠉⠙⠒⠂⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⣛⢻⣏⡛⠛⠛⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡿⢛⡽⠂⣩⣿⠟⢿⡳⠯⢷⣦⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⣠⠞⠉⠀⠀⠀⠙⢧⣀⡈⠙⢛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠇⠰⢻⣄⠀⢀⣤⣤⣄⠈⢻⣿⣷⠀⣿⣏⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢿⣿⣿⡏⠀⡀⣿⠉⠀⠀⠠⠤⣄⠀⢸⣿⠿⢷⣽⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⠇⢼⡇⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠀⣼⣿⣷⣶⣦⡀⠀⠀⠀⢀⣿⣿⣏⠀⠀⡿⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣧⣼⣿⡇⠛⠟⠛⠁⠀⠀⢀⣾⠟⣸⣿⡄⠀⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⢻⣿⣿⣿⣦⣤⣤⣤⠶⢾⡿⢡⣾⣹⣿⠃⡜⣰⢠⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢠⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣵⣥⣾⣿⣿⣿⣿⣄⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻⣿⣿⢻⣿⣿⠿⠏⠀⣀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠋⠀⠀⠀⣾⣟⣵⣶⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣽⡷⢀⣀⣰⢠⣿⣿⣿⣿⣿⠛⣿⣿⣶⣿⣿⣿⣿⢿⣧⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣠⣉⣽⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡻⣿⡟⣸⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣖⣾⣿⡿⢡⡯⣹⣧⠿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠀⣼⡴⣥⡿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡗⣿
⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⡟⠀⠀⣿⣷⢛⣿⣤⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⢿⣼⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⠟⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⣭⣿⣿⣿⣷⣿⠆
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣁⣘⠿⣋⠹⠆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣻⣵⣿⣿⣷⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠉⠉⠐⠒⠢⠄
⠀⠀⠀⠀⠀⠀⣀⢀⡀⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠐⠀⠀⠠⢀⠀⠀⡀⠈⠀⡠⠤⠤⠒⠒⠢⢄⠀⠈⠤⡀
⠀⠀⠀⡠⠂⢁⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠈
⠀⠀⠈⢀⠔⠈⠉⢢⠀⠀⠀⠀⠀⠀⠀⢣⠀⡐⢁⠀⣀⣀⣠⣀⣀⡀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠰
⠀⠠⠔⠁⠀⠀⠀⠀⠣⠠⠤⢀⡀⠀⠀⢸⢐⠀⡈⠉⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡠
⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠄⠀⠈⠑⢤⡜⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢰⢓⠅
⠀⠀⠈⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠠⠀⠘⡀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢪⠦⣀⠀⠀⠄⢐⠈⠀⣀⠔⠁
⠘⠣⢄⣀⡑⠄⣈⡐⠒⠒⣁⡠⠔⠁⡠⠶⠆⠢⠠⠼⠤⡀⠀⠁⠘⢢⣤⣀⣀⣀⡠⠊⠀⠈⠙⢒⠶⠒⠶⣋⠁
⠀⠀⠀⠠⢠⠜⠁⠉⠉⠘⠉⢦⡈⡉⠀⢀⡦⠤⠤⡄⠀⠋⠉⠉⠋⠟⠐⢤⠤⠄⠒⡣⠤⠎⠀⠁⠐⠄⠀⠈⢆⡑⠒⠢⢤
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⡒⠁⠀⠘⠃⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⢀⡠⠬⢁⡀⠂⠠⣤⠤⠤⠄⠤⠆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢇⡀⠀⠀⠄⠠⠤⠤⠤⢼⣀⡀
⠀⢀⣳⡀⠀⠀⠉⠈⠈⢂⡔⠐⠒⠂⠀⠀⠀⠚⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⢀⡢⠀⠀⠀⢰⣀⣀⣀⣀⣦⠤⠔⠊⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡖⠒⠊⠉⠈⠳⣄⡴⠀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡀⠀⠀⠀⠀⠈⠁⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠈⢉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠛⠫⡉⢢⡉⠢⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡄⢄⠀⡄⠲⡈⠣⠑⢲⠤⡄⠀⣼⠖⠀⠀⠀⡠⢺⠋⢍⠒⠀⠀⠁⠀⠁⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣸⢸⡀⠣⠘⠀⠁⠀⠄⠈⣏⣥⠾⠃⠀⠀⠀⠀⡔⡏⢀⡀⠀⠀⠀⠀⡀⠀⠀⢸⡱
⠀⠀⠀⠀⠀⠀⠀⢠⠿⡀⠀⠂⠀⠀⠀⢀⡄⠀⡏⢀⣀⠀⠀⠀⠀⠀⠈⢇⠀⠁⡀⠀⠀⢀⡁⠀⠀⡬
⠀⠀⠀⠀⠘⣷⣀⣈⡽⠳⡀⠀⠒⠒⠒⠉⢀⡴⠡⡋⠈⠃⢀⠐⠠⡆⠀⠈⠢⡀⠙⠂⠐⠊⠀⢀⠔⠁⢠⡤⢤
⠀⠀⠀⠀⠀⣈⠉⠁⠀⠀⠈⢓⢶⡴⠦⣚⣉⣠⠔⠉⠁⠀⠈⠓⠚⠳⢄⣠⢤⡌⠓⡴⢤⣤⠖⠥⣤⠴⠚⢦⡼
⠀⠀⠀⠀⠸⣉⡹⠒⠒⠾⡉⠉⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⢤⠀⠀⠀⠀⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣀⣀⣀⠠⢧
⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⠀⠀⠀⠀⠀⢀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⡀⠀⢀⡱⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⠒⠒⠒⠲⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠒⠃⠑⠘⡀⣠⣻⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⣠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠦⠠⠃
⠀⠀⠀⠀⠀⠀⠰⢏⣈⡹⠃⠀⠰⣛⣃⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠁





⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣴⠋⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⡄
⠀⠀⠀⠀⢿⣄⣀⣼⠀⠀⠀⣀⠀⠀⠀⢀⣼⣿⡿⠟⠉⠛⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣹⣧⣀⣠⠇
⠀⠀⠀⠀⠀⠹⣟⠉⠀⣀⠠⠀⠄⣀⠀⢺⣿⡿⠄⠀⠀⠀⠠⠙⢻⣿⣿⡀⠀⠀⠀⠠⣴⠏⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣆⢂⢁⠀⠐⢀⡀⠐⢿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⡀⠀⠐⠀⠋⠀⡀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿⣿⣿⣿⣦⡼⣿⡆⠀⠳⣶⡾⠀⠀⣼⣿⡿⠋⠁⣠⣴⡦⠀⣀⡰⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠸⣿⠿⠟⠛⠙⠋⠼⠗⠄⠀⠀⠠⢾⣏⠉⠀⠐⠿⠿⠟⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢀⣠⡤⢄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠴⣿⣾⣷⣤⣤⣤⣄
⣾⡇⠀⠀⣡⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⢈⣿⣿⣿⣿⣿⣿
⠹⢿⣶⣾⡷⠀⠀⠀⠀⠿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢸⣿⣿⣷⣶⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠻⣏⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣆⠀⠑⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⢰⡏⠀⢰⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⡆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡾⠀⢠⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⠀⢰⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢀⡇
⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⡿⠿⡿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣴⠏⠀⣿⡞⠋⠀⠀⠀⠀⠀⠈⠙⠯⣍⣧⡀⠀⢀⣤⡄⠙⢿⣿⡄⢹⡿⠛⢿⣿⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⡿⢻⡏⢀⡼⠃⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠙⠳⣄⣀⢉⣷⣶⣾⣿⣷⡼⠁⠀⠀⠙⢿⡄⢻⣿⣿⣿
⣿⣿⣿⣿⠿⢿⣿⣿⣼⢧⣸⡿⠁⠘⣧⣜⡁⠀⡴⠃⠀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⢷⣿⠉⣿
⠉⠀⠀⢈⠛⠶⣼⣿⡧⣴⣿⡅⠀⠀⢹⡄⢠⡾⠁⠤⠴⠖⠛⠓⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⢹⡾⠁
⠒⠀⡈⠁⠀⠀⠀⢻⣿⡟⠉⠛⠶⢤⣈⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠂⠀⠀⢸⡇
⠈⠈⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠈⣿⣦⠤⠴⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⣀⣀⣀⣀⣸⣿⣧⠤⠤⠄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⡇⠀⠀⠀⠸⣷
⠀⠀⠒⠚⠛⠉⠉⠉⠀⢿⣷⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏⠈⣿⣿⣿⣿⣽⣿⣿⣿⠿⣿⡏⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡏⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣠⣄⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⣿⡏⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡁⠀⠀⠀⢻⣧⣿⣦⡀⠀⢸
⠀⠀⠀⠀⠰⣤⣁⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠨⣿⡇⠀⠀⠘⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⣤⡀⠘⣿⣾⣿⣿⣦⢸
⠀⠀⠀⠀⠰⣿⣿⣿⠛⠁⠀⠸⣿⣇⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣃⣀⣿⣿⣷⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⢿⣷⡄⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⣀⠀⢠⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣻⣿⣿⣿
⠀⠠⡄⠀⠀⠘⡇⠀⠈⣻⣦⣀⣸⣿⣦⣶⣶⣶⡄⠀⠻⣿⠀⠀⠀⢿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⠿⡿⠟⠛⠛⠉⠘⣿⣿⣿⣿⣿⣿⣿
⣴⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠙⠀⠀⠀⠀⠀⠰⠛⣹⣿⣏⠉⠁⠀⠀⠀⠰⠟⠛⠃⠀⠀⠀⠠⣦⣜⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⣶⣖⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⣘⡿⠿⠟⠿⠿⣿⣿⣻⡏⠁
⠀⠀⢠⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⠞⡡⢋⣄⡠⠙⢄⠈⢣⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡜⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡀⠡⠆⣐⡀⠀⠀⠣⡏⠀⢑⡿⠋⢤⣿⠟⠀⠀⡧⢄⠀⠀⠀⠀⠀⠀⠐⡀⠂
⠀⢰⠃⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⣀⠀⠀⠀⠀⠠⡇⢠⡈⠙⠁⣀⣼⠒⠀⢀⡗⠋⠀⠀⠐⠒⢈⠒⠄⢠⢂
⠀⡏⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠄⠀⠀⠀⢇⠀⠙⣉⠉⣵⠃⠀⠀⡼⠁⠀⠀⠀⠀⣀⠼⡤⠔⠃
⢸⠁⠀⣀⣤⠀⡶⠶⠖⠒⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡂⠄⠈⢦⣀⠈⠭⠄⢀⣠⠞⢁⠀⠠⢄⡂⠭⠚⠃
⠿⠒⠋⠁⣿⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠬⠙⡹⠋⠉⠡⠀⠀⠒⠈⠁
⠀⠀⠀⠀⢸⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡀⡇
⠀⠀⠀⠀⠘⡇⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠐⠀⡇⠀⠀⠸
⠀⠀⠀⠀⠀⢿⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢠⠁⢠⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡇⠈⠦⠊⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠐⡉⡬⠁⠀⠠⠤⠴⣄⡉⠒⠤⠤⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠸⠠⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠄⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠙⣦⣄⠀⠀⠀⢀⡇⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⡇⠀⠀⠀⠀⡠⠔⢻⠤⢄⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠙⢍⣙⣀⣈⡨⢥⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠹⣄⠠⠐⢉⣠⠖⠋⠀⠀⠑⡄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⢤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⣤⣤⣀⣀⡾⠋

⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠿⠿⠋⠽⠿⢿⣿⣿⣿⣿⡿⢃⣾⡿⢡⣿⣇⢹⣆⢻⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠟⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⡟⢡⣾⡿⢁⣿⣿⣿⡄⢻⡆⢿⣿
⠀⢘⣯⣿⣿⣿⡿⠋⢁⣤⣶⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣤⣿⡿⢁⣾⣿⣿⣿⣷⡀⢿⣾⣿
⠀⠘⣻⣿⣿⣿⡀⠀⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡀⣰⡄⠀⠀⠀⢿⢿⣿⣿⡟⢁⣾⣿⣿⣿⣿⣿⣧⠈⣿⣿
⠀⣾⣿⣿⣿⣿⣿⡖⣞⣿⢿⣿⣿⠇⠀⠀⠀⠀⢀⣤⠤⡽⢿⡻⡄⠀⠀⠀⢸⣿⣿⡀⠺⣟⠸⣿⠏⢹⣻⣿⡆⢸⣿
⠀⣴⣿⣿⣿⣟⡁⢻⡟⣿⢴⣿⢿⠃⠀⠀⠀⠐⠉⠀⠰⣦⣾⡉⣷⠀⠀⠀⠸⣿⣿⣿⣤⡟⠓⣿⡖⣾⡄⠋⣠⣾⣿
⠀⣿⣿⡟⠫⢾⣿⡿⣸⡟⠈⢡⣾⠃⠀⠀⠀⠀⠀⢀⣼⡶⣿⡗⠓⠀⠀⠀⠀⠻⣄⠙⢿⡄⣾⣿⠇⣿⡇⠀⠻⢿⣿
⠀⠈⠻⣧⣾⣿⡾⣷⣿⣷⣶⣾⡟⠀⠀⠀⠀⠈⠻⠿⠿⣽⢿⡇⠀⠀⢴⣦⠆⠀⠟⢳⣄⢹⡟⢹⠀⣿⡇⣠⣀⣼⡿
⠀⠀⠀⠘⠻⣿⣿⣉⡿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣿⣤⣄⠀⠀⢀⡄⠀⠈⠛⠻⢿⣄⣸⡀⣧⢻⣿⣿⠿⠛
⠀⠀⠀⠀⠈⠀⠉⠹⡌⠁⠀⠀⠀⠀⢀⣀⣤⠀⠀⣸⣿⣿⣿⣿⡏⠶⠂⠉⠀⠀⠀⠀⠀⠙⠛⠛⣷⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⣤⣶⣾⣿⣿⣿⣿⣿⣿⣧⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⠀⢐⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠓⡞⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣃⠀⠐⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠉⠉⣿⣿⣿⠿⠿⠟⠛⠉⠉⣀⣠⠉⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⡀⢀⣀⣰⣤⣤⣴⣶⣾⣿⣿⠏⣿⠀⠀⣾⠘⣸⣿⡄
⠀⠀⠀⠀⠀⠀⣧⠀⣠⣇⠀⠦⣇⡀⠁⠰⠤⠄⠒⠦⠝⣿⠿⠟⠛⠛⠿⡋⡁⠐⠿⢄⣡⣿⡄⠞⠀⠈
⠀⠀⠀⠀⠀⢸⡿⠰⠋⠉⡠⠆⢁⠉⠁⠈⠀⠰⠆⠒⠀⢀⠀⠂⠀⣧⠀⠈⠈⠐⢲⡦⡀⠈⠱⠔
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣼⡰⠀⠁⠀⠀⠀⡆⠀⡀⠀⣸⣴⣿⡀⠠⠄⠀⠀⠀⠀⠀⢰
⠀⠸⠀⠄⠀⠀⠀⠀⠀⠰⣶⣶⣶⠀⠀⠋⠀⠲⠸⣿⠘⢋⠄⠉⠉
⠀⠀⠀⠀⠀⠂⠀⠀⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠔⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⣄⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⢻⣿⣿⣿⣿⢿⣾⠟⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠚⠯⣎⣿⠈⠟⢛⣛⣹⡇⠀⢸⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣫⣾⣿⣟⢻⣿⣿⣷⣤⣄⠉⠉⢀⣰⣿⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡆⠈⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠻⠗⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⣿⣿⣦⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣟⠾⣄⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⡏⠻⢿⣿⣷⡀⠀⠈⢙⠛⠿⠻⣻⡏⣿⣷⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⡇⠁⢀⠸⡿⢿⣿⣿⣿⣿⣧⣼⣿⣿⣇⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣃⣿⣻⣟⣿⣿⠿⣷⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠇⢸⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠀⠀⠸⠉⢉⣽⠟⠉⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⡔⢻⣿⣦⣀⣙⣿⣇⠈⠀⠀⠀⠀⠀⠁⠺⢷⣴⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠀⠀⠂⠀⠀⠀⠀⠀⢰⠁⣼⣿⣿⣍⠻⠿⣿⣤⣳⣶⣦⣄⠀⠀⠀⠘⢿⠇
⠀⠀⠀⠀⠀⠀⠀⡝⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣫⣷⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⢫⠟⠟⢻⠟⠡⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⠿⢿⣿⠿⠟⠛⠋⠁⠀⠀⢈⠈⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣰⣾⡼⡋⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⣟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡛⠯⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⡿⠁⣀⣠⢉⠐⠀⠀⠂⠀⠀⡀⠀⣹⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠛⠛⠁⠈⠛⢻⡶⠂⠀⠈⣙⢿⣶⣿⣷⡀⢂⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⣬⣧⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣄⡀⠈⠓⢘⣆⣀⣿⣾⣿⣿⣿⣷⡈⣄⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣷⣄⠀⠀⢠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⢷⣌⠳⠀⠀⠀⠀⣠⣾⣿⣿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠄⠐⠟⠛⠋⠁

⠀⠀⠀⠀⠀⣠⡄⣶⢦⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣦⣀⢄⡄
⠀⠀⠀⠀⢈⡟⠓⠙⠛⢷⣷⠁⠀⡰⠋⡹⠟⠟⠛⠳⢽⠯⡁⢱⠀⢀⡤⠞⣫⢶⣤⡘⠙⣶⢶⣶⢿⢿⣿⢿⣧⣤⣀
⠀⠀⠀⢀⡎⠀⠀⠈⠁⠀⢸⡀⠀⡇⣼⠁⡆⠀⠀⠀⡀⣇⢇⠇⠘⡏⢠⠐⠡⠀⠉⢻⢺⣿⡗⢁⠀⠀⠋⡀⢿⣿⣿
⠀⠀⠀⠙⢇⠘⢄⣀⡤⠂⡼⠋⢠⡷⠉⢆⠈⠒⠒⠊⣠⢿⠘⡄⠀⢣⡘⢤⠤⠂⢀⡼⠂⠛⣇⠘⠤⣀⠴⠁⡎⠹⡍
⠀⠀⠀⠀⢀⣱⠦⠤⢴⠊⠀⠉⠉⠀⠀⣀⣵⠒⠒⢎⠁⠀⠣⠝⢀⣠⡼⠒⢲⣒⡋⠀⢰⠷⠈⣲⠤⠤⢔⠊
⠀⠀⣀⢴⠉⡜⢦⡔⢺⢱⡄⠀⠰⣶⠊⢡⠟⢼⡀⢿⠣⡠⡦⠐⣇⡴⠁⠀⢸⣀⡟⠊⣁⣀⠼⡧⢖⠦⡜⢄⡀
⠘⠛⠀⢘⡟⠀⠀⠀⠈⡏⠈⢢⡄⠀⢠⠋⠠⠀⠀⠸⡀⠹⠖⠠⡎⠀⠀⠀⠸⠀⠁⠀⠟⣠⠊⠀⠀⠀⢰⡀⠸⠓
⠀⠀⠀⠀⣱⠲⠄⠤⢤⠃⠀⠈⠀⠀⠀⡽⠒⡶⡄⢴⠓⠀⠀⠀⡜⠉⠀⠀⢸⠁⠀⠀⠀⠙⠿⢖⠶⣲⡿⠵
⠀⠀⠀⣠⠃⠀⠔⡄⠘⡄⠀⠀⠀⠀⢠⠃⡸⠀⢧⠘⡄⠀⠀⠘⢦⠴⠓⠤⡜⠂⠀⠀⠀⠀⡤⣇⠀⠀⣣⢄
⠀⣀⠼⠤⣀⡞⠀⣟⣀⣸⡀⠀⠠⣊⣉⡻⠁⠀⠸⡟⠓⣄⣠⣒⣳⠀⠀⠀⣱⠒⣢⠀⢤⣚⣙⠇⠀⠀⠙⣉⣲⠄

⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣤⣄
⠀⠀⠀⠀⠀⢸⣿⠛⠉⠛⠿⢿⡟⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢰⣿⠻⠿⠿⣿⢿⣿
⠀⠀⠀⠀⠀⠸⡏⠒⠀⠀⠐⢈⡇⠀⠀⠀⠀⠀⠀⣿⠋⠀⠉⠁⠈⣿⡇⠀⠀⠀⠀⠀⢸⣯⠐⠀⠀⠐⠀⣿
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢻⡀⠤⣤⠤⣰⠇
⠀⠀⠀⠀⠀⠀⠈⠳⠶⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⡾⠟
⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠆⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠘⠛⠛⣓⠚⠛⠛⠓⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⢰⣿⣿⢿⠟⢿⣿⣇⠀⠀⠀⠀⠀⠀⣾⡏⠈⠛⠛⠛⣿⠆⠀⠀⠀⠀⠀⢀⣿⢿⢿⣿⢿⣿⣷
⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⢹⠊⠁⠀⠀⠁⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⠀⠈⣿⣷
⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠄⠀⢀⣼⣿⡄
⠀⠀⠀⠀⠘⣿⣿⣷⡄⠠⣾⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢘⠛⠋⠀⠀⠉⠛⡃⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣤⣤⣼⣯⡍
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⣠⣶⣄⣤⣤⣤⣤⣤⣦⡄⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⣿⣿⠛⠛⠉⠙⣿⣧⠀⠀⠀⠀⠀⠙⣷⣿⠿⢿⡿⢻⣿⣾⠃⠀⠀⠀⠀⣾⣿⠟⠛⢿⠻⢿⣿⡄
⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠻⡿⠀⠀⠀⢀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣄⠀⠀⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠷⠄⠠⠾⠟⠛⠁
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⣳⢟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢖⡀⡙⠉⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠈⠁⠹⠏⠒⡦⠔⣼⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠤⣹⡧⠐⠀⠀⠐⠓⡖⠂⣵⠂⣟⠁⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡆⠀⢸⡧⢍⡧⣼⠗⠂⠀⣼⠋⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⣨⠀⠀⠀⠙⠀⠀⡀⠏⡀⢤⣰⣿⣿⣽⡔⢷⣶⡄⠈⢁⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡇⢌⣛⢾⣷⣿⣿⣿⣿⣿⢧⣄⡬⠳⢟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠗⠀⠀⠀⠀⠀⠀⠄⢸⠀⠀⠐⠰⣁⡠⣬⡿⠋⣿⣿⣯⠿⣷⣿⣽⣷⣢⠞⣂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡴⠢⠆⢠⡟⣦⢼⡇⠀⠈⣿⣳⣦⣿⡿⣿⣿⡧⡿⣭⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⢀⣦⣍⡁⡦⢠⡛⣽⣻⣿⡇⠀⠈⢻⣿⢾⣿⣷⣿⣿⣛⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⠌⠁⠈⠃⠰⠀⢀⠘⠻⣏⢰⣲⣾⣷⣿⣾⠋⠈⠀⠀⡀⠹⢿⣿⣿⣿⣿⣿⠿⠁⠠⠆⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠈⡆⠀⢀⢀⡈⠄⠛⡿⠕⢿⣾⣿⣿⡿⠟⠋⠀⠀⡠⠎⠰⢠⠘⣿⣷⣿⣿⣿⣾⣼⡿⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠂⠀⠰⢱⡆⣠⣚⡟⣷⡄⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠂⠀⠂⠰⡄⢸⣿⣿⣿⣿⣩⢷⡀⠙⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⣠⢫⣿⠪⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠐⠀⠀⠤⣷⣂⠀⣿⣿⣿⣿⣿⣿⣹⡡
⠀⠀⠀⠀⠀⠀⠀⢠⣄⡌⡉⣿⣿⠿⢗⣮⣹⣳⠏⠀⠀⠀⠰⡀⠤⠤⠄⠀⠐⠘⢂⡘⠀⠀⠐⢼⣏⣀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⣼⣴⣼⣿⣿⣿⡿⢿⣿⡟⠀⠀⠄⠀⠈⡉⠀⠌⡀⠄⠀⠀⠂⠁⠀⣀⠀⢻⡿⢾⠿⢿⡿⠿⣿⣭⡷
⠀⠀⠀⠀⠀⠀⡀⠐⣾⣿⣿⣿⣧⣽⣟⣶⠟⠀⠀⣠⣦⡀⢃⡕⢀⠀⠠⠈⠐⡀⢀⢠⠈⠅⢀⢹⣷⣿⣷⣿⠷⣦⣈⢿⣿⠄
⠀⠀⠀⠀⠀⠀⠙⢀⣻⣿⣿⣿⣿⣿⡏⠁⠀⣠⣾⣿⣯⣽⡀⠚⢠⠌⠀⠁⠀⢂⠉⠀⠀⠖⣻⡶⢁⣿⣿⣶⣼⣿⣟⣧⣟
⠀⠀⠀⠀⠀⠀⢤⡏⡿⢻⣿⣏⣻⣁⣤⣤⣤⣿⣿⣿⣯⣽⣷⣤⣄⢈⠱⠐⢠⠀⢠⠑⣦⣷⣿⣧⣾⣿⣿⣿⢿⣿⡟⣿⣻⣗⢠⣀
⠀⠀⠀⠀⠀⠂⠈⠁⠙⠛⠛⠙⠻⣿⣿⡟⠻⣿⠙⣿⣓⣾⡷⠿⣿⣦⣤⣤⣶⣦⣶⣾⣿⣿⣿⣿⣿⣿⠿⣿⠄⢹⡖⡟⠉⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠰⠲⡟⠘⠋⣿⣿⣿⡛⢿⣿⣿⣿⠛⠻⠿⠿⠛⠉⠀⠙⠀⠈⠙⣾⡷⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⢹⡋⠓⢸⣿⡧⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣶⣶⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠴⠶⠶⠶⠤⢤⣤⣤⣤⣤⡤⠶⠶⠚⠛⠛⠛⠛⠉⠛⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⣸⠇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⡏⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⣇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣷⠀⠀⢀⠀⠈⢧⠀⠀⠀⠀⢀⣤⣶⣿⣗⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⡈⠙⠛⠷⣦⠀⠀⣠⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠉⡀⠀⠘⠀⠀⠈⠳⠤⠴⠲⠛⠉⠉⣉⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣉⣛⡒⠢⠤⣄⣀⠹⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠙⠛⢶⣄⠀⠀⠀⠀⠀⣠⠾⠛⠉⠉⠉⠉⠉⠛⢦⡀⠁⠘⠓⠂⠀⠲⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⡤⣀⡇⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⣾⠃⠀⠀⠀⠀⢀⣀⣀⡀⠀⢻⡆⠀⠀⠀⠀⠀⢸⡷⠚⢶⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣇⠀⠈⢻⡀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣴⣶⣶⣦⡄⢹⡆⠀⣸⠇⠀⠀⠀⢠⣾⣿⣿⣿⡿⠷⣄⣿⠀⠀⠀⠀⢀⣿⠀⠀⣼⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⣦⣀⠈⣣⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⡿⠏⠀⣹⣾⠇⠀⢿⡀⠀⠀⢠⣿⣿⣿⣿⢯⡀⠀⣽⣿⠀⠀⠀⠀⣼⠷⠶⠛⠁⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠛⣆⠀⠀⠘⣧⡀⠀⠸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠸⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⣆⠀⠀⠈⠛⢶⣤⣌⣿⣿⡿⠟⠉⠀⠀⠀⠀⠙⢷⣄⡀⠈⠻⣿⣿⣿⣿⡿⠃⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⠋⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⢢⣄⣀⣀⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⠀⠀⠀⠀⠠⣀⣀⣠⠆⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠙⢷⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣦⣤⣄⣈⣉⣉⣀⣀⣤⣤⡴⠶⠛⠋⠉⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢓⣾
⠀⠀⠀⠉⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠉⠉⢹⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⢤⣤⣤⣤⡤⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⡸⠀⠙⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠶⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⢣⡀⣀⡜⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢘⠃⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⡛⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⡴⠋⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠒⠀⡀⠀⠁⠀⣉⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠉⠁⢠⡄⠂⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠓⠦⠀⠀⡖⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠓⠄⠀⣀⠴⢃⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠎⠀⠀⠀⠀⠀⠠⠃⠤⠤⠤⠤⠤⠤⠐⠚⠙⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠂⠰⠆⠘⠁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢣⠈⠇⠀⢀⠉⠹⠉⠁⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢰⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠤⠤⠀⠒⠁⠸⡇⠀⠐⠀⠀⠀⣤⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠸⣄⣀⡀⠀⢤⣤⠿⣄⡀⠀⣀⠴⠃⠀⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠀⡀⠠⠤⠌⠓⠃⠀⠀⠈⠙⠻⢤⣀⣀⣀⡠⠞⠁
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠋⢀⠞⠀⠀⠺⣿⡷⠿⠿⠃⠀⠀⠀⠾⡁⠀⠀⢀⣀⣇⠀⠀⢿⠿⠇⠀⠀⠀⠶⡁⠀⠀⣠
⣶⡏⣀⠓⠄⡀⢉⠩⠀⠀⠀⠀⢠⡀⠈⢉⠀⢸⣿⣏⣽⣷⡮⠀⠀⠀⠀⢠⡀⠈⢉⠁⠀⠉⠁⠀⣴⡆
⣷⣿⡂⠤⣍⠱⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢁⣿⣿⣿⣿⣿⡷⢀⣄⠰⠆⠉⠀⠐⠢⠄⠀⠐⠀⠀⠀⠀⣀
⠋⠀⠐⠿⢩⣤⣴⣾⣿⣷⣶⣄⠀⠠⠀⠀⠀⢹⣿⣿⣿⣟⠀⠟⠻⠷⢿⣶⣦⣄⠀⠀⠈⠀⠀⠄⠀⠀⠙⠈
⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄⣸⡞⢡⢃⠈⣿⣿⣿⣿⣷⣦⣤⣀⠀⠈⠉⠙⠛⠦⢠⠄⠈⣤⣤⠂⠀⢠⣶⣶⣄
⠃⡄⢰⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣿⠄⢀⣿⠿⣿⣯⣈⣉⡙⠟⠷⢶⣦⣤⣀⡀⣰⠀⠀⢛⣫⣄⠀⠻⢿⣿⣿
⠐⠈⠀⢈⣛⣿⣿⣿⠛⡟⠛⠿⠿⠿⠿⢃⣷⣿⣿⡿⠀⠠⠈⠙⣆⡀⠀⠀⡁⠈⣿⡿⠙⠒⠈⢮⡹⣧⣀⣿⡯⠁⢀
⡀⠓⠠⣿⠟⠉⠻⠿⠀⠙⢀⡵⢶⣾⣿⣿⣿⣿⣿⣿⣦⣄⡀⠈⠉⠉⠀⠀⠉⠛⠛⢠⣿⠈⢐⣾⣿⣾⣯⣷⠆⠀⠠
⠏⠘⠂⠀⠀⢀⠀⣠⠶⠒⢺⢣⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⣶⣿⣡⣾⡿⢟⡝⡿⠹⣿⣎⠁⣠
⣀⢀⡀⠀⢃⣸⣿⣿⣄⣀⢆⠚⣁⠈⠰⠈⠹⢿⣿⠉⠋⠙⠛⠿⣿⡿⢿⡿⣷⣾⣿⣿⡬⡆⢠⠉⣴⣷⣦⠙⢿⡆⢻
⠛⠀⢁⣆⠃⠘⢿⡿⣩⣴⡶⠆⠉⠀⠀⠀⠀⣾⣿⣷⣦⣤⣤⣀⠀⠀⣿⣷⣾⣿⣿⣿⣧⡀⠀⡾⠙⢟⣏⣶⠒⣤⣼
⠀⠀⠈⣿⠀⠈⠐⠹⣿⠛⠁⣀⠸⠁⡠⠓⢴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢂⠎⢀⡈⠂⠻⠿⠉⠋⠉
⣶⣿⣿⣿⣿⣶⣾⣾⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣄⣵⢟⣿⠟⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⣀⣴⣿⣿⣫⡾⠋⠀⠀⠈⠙⣷⡄⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠛⠛⠁⣠⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠞⠋⠁⢦⠀⣉⣉⢀⡀⠀⠀⠀⠀⠈⠙⠲⣦⡼⠋⢀⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠰⠞⠁⠀⠀⠀⣠⡾⠟⠋⠉⢿⣄⣀⠀⠀⠠⠾⢓⡷⠋⠀⢀⡴⢀⠈⠃⠀⠀⣼⣷⠀⠀⠀⠀⠀⢠⡔⣉⠀⠀⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢈⠻⣆⠈⠐⠆⠹⠴⠚⠠⠀⠀⠀⣼⣧⠀⢰⣿⣿⡇⠀⠀⡤⢶⡿⠋⣁⠀⢀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠐⠈⠻⣦⣀⣠⠔⠊⢀⡴⠂⠀⢿⣿⣀⣸⣿⣿⣷⠔⠋⠐⠉⠀⠀⢁⡰⠏⣈⣽⣿⡆
⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢠⠞⢀⡔⠁⠐⠀⠀⠀⠀⠀⠀⠘⣶⣿⠟⠁⠀⠐⠋⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠠⠖⢁⣤⡾⠟⠛⠁
⠀⠀⠀⠀⠀⠻⢿⣶⣦⣤⠁⡴⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠏⠀⠀⠀⢠⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⡀⠠⠚⠛⠋⣠⣄⣠⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣶⣾⡿⣶⢦⣠⠀⠀⠀⠀⣤⣾⡷⠃⠀⢀⡄⠈⠁⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⡇⢰⣿⣦⣤⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡍⠙⠋⠀⠀⠀⠚⢻⣄⣦⣼⣿⠟⠀⠀⠀⣾⣿⣄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⡿⠟⠛⠉⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢦⣤⡀⠀⠀⠈⠉⠹⠿⠧⠴⣦⣴⣷⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠏⠀⠀⢐⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠈⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⢀⣀⣠⣠⠖⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⢰⣦⡀⠀⠀⠀⠉⠳⢦⣤⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠐⠻⠿⠋⠀⠀⠀⠀⠀⢀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠙⠲⣀⣤⠀⣀⠉⢷⣤⣰⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢟⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠛⢿⣿⣧⣶⣾⣿⣿⣿⣿⣿⣿⡿⠛⠟⠉⠻⠉⠉⠀⠀⠀⢀⣴⡀⣀⡤⠀⠀⠀⢀⡴⠟⠉⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⠃⣨⣿⣿⣿⣿⡿⠿⠗⠀⠀⠀⠀⠀⡠⠀⠀⣠⣶⣿⣿⠟⠋⠀⠀⠀⠀⠁⠀⠀⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣾⣿⣧⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠠⣿⣿⣿⡆⠀⠀⢀⣤⣌⠛⢦⣬⣿⣿⣥⡤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⢀⣠⠦
⠀⠀⠲⣄⣙⣿⣿⠏⠀⠀⠺⣿⡿⠃⠀⢨⣿⣿⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠑⠲⢦⣭⣿⣿⡶⠋
⠀⠀⠀⠀⠉⢻⣿⣿⠆⢀⣾⣿⣿⣄⠀⣸⣿⣿⡇⠀⠀⠀⠀⣹⣿⣯⣤⠦⠀⠀⠀⢨⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⠀⣾⣿⣿⡏⠻⠆⣹⡿⣿⡇⠀⠀⠀⣰⡿⣿⣷⡀⠀⠀⠀⢀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢰⣿⠉⣿⠰⣿⠉⣿⡇⠀⠀⣿⣇⣾⡇⠀⠀⠀⠩⣾⠿⠿⢿⡆⠀⠀⠀⠈⡿⢸⡇
⠀⠀⠀⠐⣿⣿⢀⣿⠀⠀⠀⣽⡁⠀⠀⠿⠇⣿⡷⠀⠀⠀⠀⠙⠀⠀⠘⠁⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⣼⡇⠀⣿⠀⠀⠀⢸⠁⠀⠀⢠⡄⢸⡦⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡉⠘⡇
⠀⠀⠀⠀⠈⣿⣠⣿⠀⣯⣤⣿⡇⢀⠀⢿⡇⣹⡇⠀⠀⠀⢠⡡⡶⣶⠞⠁⠀⠀⠀⣀⣷⣸⣄⡀
⠀⠀⠀⠀⢀⢹⣿⣷⠄⠘⠿⣿⡷⠛⠀⢨⣻⣿⠇⠀⠀⠀⠀⢛⣿⡆⠠⠄⠀⠀⠀⠘⣿⣿⠋
⠀⠀⠀⠀⣡⣾⣿⡆⠀⠀⠰⠿⡷⠀⣀⣈⣿⣿⣀⣀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⢭⣿⣿⠓⠢⢀
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠐⠊⠀⠰⢾⠿⡂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⡀⠀⠀⢀⡜⠁⠀⠑⢤⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣸⠁⠀⢠⡞⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣤⠈⣇⠀⣧⠀⠀⠀⣀⣀⣀⣀⣠⣤⣀⠀⠀⠀⠀⢰⠇⢠⠏⠀⢠⠏⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠀⠈⣧⢹⠀⠸⡄⡾⠛⠫⢅⡀⠀⠀⠀⠈⠳⡴⠶⢦⡏⢀⡎⠀⢠⠏⠐⠀⣠⠖⠦⣏
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠸⡆⣇⠀⢿⡇⠀⠀⠀⠉⠓⣦⠒⠂⠀⢘⡆⡜⠀⡼⠓⢶⡏⠀⢀⡔⠁⠰⡆⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢻⣹⠀⢸⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠸⡸⠀⡰⠀⠀⠀⢹⣠⠏⠀⠀⠀⢣⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⣿⣇⣀⡇⡠⠖⠒⠦⣄⡟⠀⠀⢀⡷⠁⢰⠁⠀⠀⢀⡘⠓⢄⡀⠀⠀⠈⠳⣄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢀⠔⠁⠀⠈⠻⠀⠀⠀⠀⠀⢱⡤⠤⠼⣅⢀⣧⡖⠋⠉⢉⡿⡄⠀⢷⠀⠀⠀⠀⢀⣀⣾⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠤⢧⣀⣀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠹⣜⡉⠉⠲⠋⠀⢸⣄⡼⠴⠖⠚⠉⠉⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠉⠉⠉⠓⠒⠒⠦⠤⢤⣄⣀⣠⣁⣀⡀⠀⠀⢀⣀⣸⡧⠤⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣩⠟⡏⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳
⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⡇⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛
⠙⠛⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣾⠋⠁
⠀⠀⠀⠀⠀⠈⢻⡓⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⣀⠤⠊⠉⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⠉⠓⠶⢤⣀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠣⠖⠋⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⢣⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠰⡖⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣲⣦⣼⣗⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠋⠲⠀⠀⠀⢀⢄⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠕⠀⠀⢀⡃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡾⠃⠀⠀⠐⠈⠛⠲⠶⠂⢺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠹⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠂⠄⢦⠀⠀⠀⠀⢀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠂⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⡀⠉⠁⠂⠀⠐⠃⠀⠀⠁⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠡⠴⠤⠌⡠⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠢⠀⠠⠂⠀⢢⠔⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⢀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣆⠀⢠⠛⠇⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠄⠀⡀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢩⣭⢽⡾⠀⠰⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁
⠀⠀⠀⠦⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢄⣁⣀⢈⣀⠀⠈
⠀⢀⠴⡻⠛⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠣⠀⡁⢈⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠀⠂
⠀⢈⠘⠁⠀⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⡦⡈⠁⠀⠀⠀⠀⠀⢀⠔⠁⠨⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⠀⠀⡆⡄
⠀⠊⢀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡹⠀⠀⠀⠀⠀⠀⠢⡈⠢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠀⠀⣿⡇
⠀⢤⠤⢃⡈⢟⣘⢆⡠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠡⠒⠀⢀⣎⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣶⣶⣷⣿
⠀⡦⠀⠢⠄⠀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⣿⣿
⠀⢳⢠⠄⠀⠈⠀⠀⣼⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇
⠀⠘⡀⡀⠀⠀⠀⢠⡿⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠘⢷⣦⣤⣀⣼⣿⣦⣹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠇
⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⠯⠍⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠠⠛⠓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⡀⠱⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡆⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⡀⠀⢸⠃⠀⡄⠀⢐
⠀⠀⠀⠀⣀⡀⠐⠂⢀⣼⣿⡆⠈⠀⢠⠏⣀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⣥⠾⢛⠁⣠⢀⡬⠈⠁⠖
⠀⠀⠀⠈⠀⠀⣠⡿⢷⣶⣿⣝⣋⣉⣀⠀⠚⠁⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⢋⣄
⠀⠀⠸⠀⠀⠀⠀⠀⠸⢻⡿⠷⠀⠉⠁⠈⠉⢷⢄⡆⠙⢉⣭⣭⡭⠽⠿⢓⣒⣠⣤⡄⣶⠶⣦⠉
⠀⠀⠀⠈⠲⣶⠖⠂⠈⠉⠀⠀⠀⠀⠀⠀⢈⠃⢋⣀⢠⡍⣴⡶⠖⢻⡆⣸⡏⣿⣀⡀⣿⣠⡿
⠀⠀⠀⠀⠀⠈⠁⢀⡼⣿⣅⣀⣀⣠⠤⠐⠚⠀⢸⣿⣼⡇⢸⣧⡆⠸⣷⣿⠁⣿⣉⡅⣿⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⣀⣀⣰⠀⢾⡏⢿⡇⢸⣧⢶⣀⣻⣯⠀⢻⡏⢱⣾⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠒⠀⠉⠈⠀⠀⢿⠀⢸⡷⣮⣵⢾⡇⢸⠟⣯⣁⠀⢸⡇⢸⣿⣷⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⣸⡄⣿⠰⣶⣿⠈⣿⣿⠀⣿⣭⡄⠘⠷⢟⣘⣧⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠑⠒⠒⠒⠚⠋⢹⡇⠻⡶⠟⠻⣂⣛⣫⠤⠅⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠲⠤⠤⠤⠄⠒⠊⢳⠶⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇

⣿⣿⣦⣀⣈⡧⢳⣟⠓⢧⠆⠉⣾⠻⣷⣿⣿⣿⣷⣽⣯⣯⠉⠀⠸⠏⠰⠀⠻⠈⠛⠈⠋⢿⣶⠼⠃⠀⡟⠀⢸
⣿⣿⣿⣿⣿⣶⠮⢍⡀⠈⠙⠆⠸⠛⠻⣚⣻⣿⣟⣝⡶⢿⣳⠉⠁⣯⣤⣀⠒⠀⠀⠀⠀⠈⡏⠀⠘⠠⡄⠔⢋
⣿⣿⠟⠛⣳⢾⣿⢸⢟⠖⠂⠀⠁⢸⡟⠯⠉⠛⠉⠉⠛⠃⠀⠰⠏⠛⡿⡻⡡⢀⠂⠄⠀⠢⠠⠀⢁⣠⡷⠖⠛
⣿⣿⣿⣿⣧⣼⡿⣛⣋⣴⠶⢀⣘⡓⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⡞⠀⠋⠁⠰⠆⣠⠀⠑⢰⡄⠠⠀⠸⠁⠀⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢤⠟⠘⠁⠀⠙⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡅⣿⢸⠃⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣁⣼⣴⡶⠀⠀⠀⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⣶⠞⡿⣵⣿⢿⡇⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣴⣷⠀⠀⠀⠈⣅⡀⠀⠀⠀⠄⠈⠁⢄⠒⠋⠸⡆⠉⡟⠁⢀⠀⠀⢀⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⢀⣄⢤⠋⠁⠃⠀⠀⠀⠀⠀⡽⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡴⣢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠴⠠⠄⠀⠀⣀⠀⠀⠚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⣾⠁⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠀⠈⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣧⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⣰⣿⣅⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠏⣡
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣬
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡶⡟⠃⣄⡀⠀⠀⠀⡶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠗⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⠿⠶⠀⣷⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣿⠏⣹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣰⢉⣶⣴⠀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠰⢤⣼⣿⣭⣿⡗
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠀⠀⠀⠀⠀⠀⣨⠚⢻⣣⣾⠿⠅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⡠⠬⠍⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠐⠀⢢⠊⠑⡄⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠈⠃⣄
⠀⠀⠀⠀⡄⠒⠁⠀⠀⠀⠀⠀⠧⢤⢤⡙⠢⢄⠀⠀⠀⡠⠊⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡰⠃⢀⠏⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠇
⠀⠀⡸⠉⢹⡀⠒⠂⠀⠀⠂⠐⠂⠇⠀⠈⠁⡀⠀⠀⠙⠤⡤⠄⠀⠄⠒⠂⠊⠉⡦⡀⠀⠀⡆
⠀⢘⠂⠀⠘⠄⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⢀⠋⠑⢄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠈⣦⡜⢄
⠀⠈⠀⠽⠛⡀⠀⠀⠠⠀⠀⠀⠀⠒⢒⠶⠂⠀⠀⠈⡂⠐⢄⠀⠤⠄⠀⠀⠀⠀⠑⣉⠘⡄⠀⠱⡀
⠀⠀⠀⠀⠀⠈⢒⣤⣀⣀⣀⣤⡄⠚⠡⢠⠶⠀⡠⡘⠀⠀⢀⠡⣶⣤⢤⡤⢠⡶⢥⡒⢄⠘⠦⠴⠁
⠀⠀⠀⠀⠀⢠⠃⠀⡿⠀⡠⠒⢌⠫⡁⠐⢶⠋⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠸⡀⠀⢱⢸⠦⢠⡄
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⡆⠀⠀⠄⠈⢂⠀⢂⠀⠀⠀⠀⠠⣰⡇⠀⠀⠀⠀⠙⠎⠉⡈⢆⣤⠁
⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠈⠢⢀⣀⠠⠞⠀⠀⠂⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠠⠃⢀⠁⠀⠈
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⣠⢇⣀⠐⠁⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⣄⣀⠄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠈⢁⡖⠒⠒⡔⢲⠛⡫⢴⡁⠂⠀⠀⠀⠀⠀⠀⠐⡴⠒⢄⢀⠝⠣⠤⡀
⠀⠀⠀⠀⠀⢀⣤⠪⣀⠀⣰⣁⠎⡰⠄⠀⠈⢆⠀⠔⠀⣀⣀⣀⢎⠀⠀⡘⢣⡔⠁⠀⠈⢱
⠀⠀⠀⢠⠊⠩⣀⡩⠂⠈⠁⠈⠧⠤⠊⠀⠀⡸⠀⠀⠐⠁⠀⠀⠁⠁⢸⠁⠈⠁⠐⠀⠀⡸⠀⢄⡀
⠀⠀⠀⠸⡀⠀⠀⠀⠀⡰⠀⠀⠈⠦⠄⠤⠐⠁⠀⠀⠰⢄⠀⠀⠀⠀⡌⠀⠀⠢⢀⡠⠞⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠈⠈⠁⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀⣀
⠀⠀⠚⠀⢔⡀⠧⠬⢆⠐⡀⠀⢀⠌⡲⠀⠀⠐⡂⠀⡐⡆⠠⠄⠉⠡⠀⠡⠄⠀⢀⣞⣋⠺⡇⠼⠋⡵⢆
⠀⠀⠠⠢⠊⢠⢰⣞⣦⠙⡈⡂⠈⠂⠀⠠⠄⠀⣯⡇⠀⡁⠀⣾⢰⡣⠀⠀⡂⠀⣬⡝⠋⠢⢠⣀⡈⠠⡴
⠀⠀⢰⠚⡀⠢⡘⠻⠫⢀⢅⠐⠀⣀⠀⠒⠂⠀⢀⠂⢀⠀⢀⠀⠀⢀⠀⠀⠁⠀⢁⣄⣀⣰⡇⠈⠶⠚
⠀⠀⠀⠸⣑⠤⢈⠁⡄⢢⠔⠀⠠⡘⠇⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⡠⠀⠀⠉⠁⠈⠀⠈⠀⠀⠞
⠀⠀⠀⠰⣦⡑⢘⠂⣊⠁⠠⠀⠀⠀⠀⢠⣄⠰⡍⠀⠀⠀⣄⠀⠀⠀⠀⡁⢒⠁⣘⠂⡀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⢰⠺⣋⡀⠀⡜⠃⠀⢀⢊⠑⡀⠀⠀⠉⠐⠂⠀⠀⠀⠀⠁⠀⠀⢂⡁⠏⠑⠩⠃⡁⠢⠀⠂⠈⣤
⠀⠀⠀⠀⠙⠉⠐⠀⠀⠀⢨⡆⢞⠕⠀⠀⠀⣀⡈⡀⠀⠀⢀⡠⣄⠀⠀⡛⢄⡐⢀⡔⠁⠀⠀⠰⠐⠤⣀⠶
⠀⠀⠀⠀⠀⠐⠁⠀⠃⠀⢄⣨⠌⣤⡅⠀⠀⠇⠻⠅⠀⠀⠀⠛⠉⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⢀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⣀⡄⡔⢄⠀⠀⠙⠀⠈⠀⢠⣄⠀⠀⠂⠀⢀⢀⡀⠀⠀⠂⠈⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⣐⡮⢧⢇⣂⡌⡑⠀⠀⠠⠂⠀⡀⠀⠀⠀⠠⠊⢬⡄⣬⢓⣀⠀⠀⠀⠐⠀⠀⣀⡍⠉⢢⠀⠀⠒
⠀⠀⠀⠙⢾⠞⢹⣾⣙⡊⠀⠀⡀⠀⢀⠰⠀⣀⠀⠨⡀⠘⡋⠛⢸⡸⠃⠀⠀⠀⠀⠘⠜⠃⠀⠠⠇
⠀⠀⠘⠀⠂⢠⠁⠀⢀⢀⢀⠀⠁⡌⠽⠿⠇⠀⠀⠀⠅⠀⠂⠀⠢⠀⢀⠀⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠉
⠀⠀⠀⠘⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠐⠂⠠⠀⠀⠀⠓⠛⠛⠛⠃⠀⠃⠒⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠉⣀⣀⠀⠉⠉⠁⠀⠀⠀⠀⡉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠒⢤⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣠⠊⠀⠀⡠⠐⠀⠀⠀⠀⠀⠢⢀⠀⠈⠉⠒⠢⢄⡈⠢⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡰⠁⠀⡴⠉⠀⠀⠀⠀⠀⣀⠀⠂⢄⡈⠐⠢⠄⣀⣀⠈⠢⠈⠈⠆
⠀⠀⠀⠀⣀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠰⠁⠀⠊⢀⠄⢀⠔⠁⡠⠊⠀⠑⠢⣀⠈⠐⠢⠄⣀⠀⠈⠁⠈⠂⠸
⠀⠀⢠⠤⣣⠹⡄⢂⡐⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠤⠐⢊⡡⠖⢁⡠⠊⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⣀⡉⠉⠉⠁⠀⢀
⠀⠀⢀⡵⡌⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⣊⡩⠄⠒⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠉⡋⡙⠏⡀
⠀⠀⠀⠑⠤⣀⠀⣀⠔⠉⠑⢄⠀⠀⠀⠀⠰⠹⡨⢁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠴⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡧⠊⢀⡇
⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⣄⠈⠚⢆⠀⠀⡔⠒⡄⠀⠀⠐⠤⣠⠆⠀⠀⠀⡆⢐⠀⠀⠀⡌⢀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠑⠢⡜⠀⠐⣠⡨⠢⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠞⠂⡅⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠑⠲⢅⣀⠀⠈⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠚⠁⠀⠀⠘⢮⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄⠀⠈⠁⠒⠒⠀⠀⠀⠒⠂⠉⠀⢠⠑⢄⠀⠀⠀⠸⡀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣬⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⢀⣀⠠⢤⠤⣀⣀⣀⠠⠔⠁⠀⠀⠑⢄⠀⠀⢈⡖⣈⣽⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢉⠏⡿⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⡟⡏⢇⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠑⠤⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠣⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⡣⠀⠀⡀⠔⠊⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢠⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡝⣠⡏⡀⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠤⠃

⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠾⠿⠿⠄⠀⠀⠀⠀⠀⣠⣴⣶⣦⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⢰⣿⣻⣿⣿⡗⠀⠀⠀⠶⢿⠉⠁⠈⢻⠿⠂⠀⠀⠀⡿⠉⠀⢹⡀⠀⠀⢠⣿⣏⠉⢉⣿⡆⠀⠀⠀⣬⠔⠿⢣⡄
⠀⠀⠀⠀⠀⠀⠑⠈⠀⠁⠃⠀⠀⠀⠀⠀⢀⠚⠂⠀⠀⠀⠀⠀⠀⢂⠀⠀⣀⡁⠀⠀⠘⣿⡇⠀⠀⣿⡇⠀⠀⠀⠳⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣈⣀⣁⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠰⣛⣳⣮⣟⡗⠀⠀⠀⠈⠛⠀⠘⠋⠀⠀⠀⠀⠀⡐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠃⠀⠃⢲⠀⠂⠀⠃⠀⠀⠀⠀⠈⣦⢹⡀⠀⠀⠀⠔⠧⡀⢠⠗⠠⠀⠀⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢰⠰⠀⠐⠁⠀⠠⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠿⢼⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡄⠀⠀⠀⠀⢀⣤⠤⣤⡀⠀⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⠀⠀⣴⠿⠿⣿⡆⠀⠀⠀⠀⢠⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠸⠿⠟⠋⢿⠀⠀⠀⠀⠸⢋⠀⠙⠇⠀⠀⠀⠀⠟⠉⠉⢹⡄⠀⠀⠀⠀⢛⠀⠀⠈⡇⠀⠀⠀⠀⢸⠛⠉⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⢀⣶⡀⠀⠀⠀⠰⣤⣦⡼⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡈⠁⠉⡀⠀⠀⠀⠀⢀⡈⠀⠹⣿⠃⠀⠀⠀⣀⠌⠉⠁⣀⠀⠀⠀⠀⠤⠆⣤⢐⠦⠀⠀⠀⠀⢰⠊⠀⠈⠴⡄
⠀⠀⠀⠀⠀⣴⣾⣿⣾⣾⣿⣶⡄⠀⠀⠀⠇⠤⠴⠀⠙⠁⠀⠀⡙⠄⠀⠴⠇⠀⠀⠀⠀⠘⠦⠠⠤⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣤⠀⠀⠀⣤⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⠀⠀⠀⠀⣴⣿⡿⢿⣦⠀⠀⠀⠀⠠⠾⠛⠽⠦⠀⠀⠀⠀⠠⣶⣶⢾⡆⠀⠀⠀⠀⢀⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠘⠤⠀⠆⠙⠀⠀⠀⠀⡟⠛⠀⠚⢻⠄⠀⠀⠀⢰⠖⠛⠛⢷⠀⠀⠀⠀⢹⣿⣿⣯⣿⠀⠀⠀⠀⣿⣿⠉⣻⢿⡄
⠀⠀⠀⠀⠀⠀⠈⢀⠀⡀⠈⠀⠀⠀⠀⣿⣄⠀⢠⣿⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠘⣧⣴⣤⡏⠀⠀⠀⠀⣿⡇⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠄⢀⠐⠄⠀⠀⠀⠀⣿⠋⠀⠈⠿⡀⠀⠀⠀⢀⠉⠒⠊⠁⠀⠀⠀⠀⠀⢙⠛⢛⠀⢀⠀⠀⠀⠙⡻⠀⠘⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⣿⣦⣄⡀⢀⣠⣾⡄⠀⡀⠀⢀⣄⣠⡀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⠓⢺⣿⣿⣇⠀⠀⠀⠀⠸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠁

⠰⣿⣷⣶⣴⣶⣤⣀⡀
⢴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣤⣄
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣼⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠛⠛⠿⠿⠟⣠⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠚⠛⠛⠒⢼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣾⣿⣿⣿⠇⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣦⣤⡶⠦⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠟⠛⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠸⠶⠞⠙⠀⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠁⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⢀⣾⣿⠟⠛⣛⣻⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⣧⠀⠀⠀⠸⠟⠓⠻⠿⣿⣿⡿⠟⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠟⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠂⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⢢⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡠⠂⠁⢀⠔⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⠀⠤⠀⠈⠀⠀⡰⡄⠀⠀⠀⢀⠊⠀⣀⠖⣠⡄⠐⠈⠤⢂⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡄⠁⠀⠁⡹⠏⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠉⠀⠀⠘⢿⠁⠀⠢⠔⠈⠀⠀⠀⠀⠠⠡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⢠⣄⠀⠀⠀⢰⣦⠀⡀⠀⠀⠘⠀⡠⠋⠀⢀⠀⠀⢀⣅⠀⠀⠀⢾⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠛⠀⠀⠐⡌⠁⠀⠁⠀⠀⠀⠀⡇⠀⡆⠄⠀⠀⠀⠛⠁⠀⠂⡌⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⢀⣠⠄⠁⢀⠀⠀⠀⠀⠀⢡⢣⠀⣸⡄⠀⠀⠀⠀⠀⠄⠄⠀⢀⢜⠎⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⡀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠡⢳⣇⡈⠀⢄⡀⠀⢀⣀⡠⠔⠃⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⢂⢸⣟⠦⠤⠵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠆⠈⠀⢀⠉⠀⣠⡄⡄⠈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢌⠢⡑⠜⠘⠚⠒⠂⠀⠀⠶⠂⠠⠀⠲⠄⠀⠀⠀⠒⠒⢑⠁⠀⡀⠀⠀⠀⠀⠀⠀⠉⠈⠢⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠢⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡒⠒⠲⠒⠲⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠀⠀⢺⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠀⠺⠂⣠⡉⠰⠦⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡁⠐⠗⢀⡉⠀⡦⠀⠘⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠤⠜⢁⠤⠄⠸⠟⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡐⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠀⠀⡆⠀⠀⠀⠀⠸⠀⠀⢀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡐⠁⠀⠀⠀⠀⠀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣅⡈⠉⠡⠃⠀⠀⠀⠀⠀⠏⠉⠁⢈⡤⠆⠀⠀⠀⠀⠀⠀⢌⡁⠀⠀⠀⠀⠀⠀⠀⢈⣁⣀⠥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠚⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢣⠀⠀⡎⠀⣹⠟⢧⣌⣧⣠⠏⠀⣼⣷⡀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡂
⠀⠀⠀⠀⠀⣾⢙⣯⣸⡆⠀⠀⠁⠀⠀⠁⠀⠁⠃⠀⠁⠀⠀⠁⠀⠀⠑⢼⣷⠋⠀⠀⢀⣨⠟⠁
⠀⠀⠀⠀⠀⠹⢿⣤⡿⠃⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⡄⣦⡔⠂⠀⠈⠉⠉⠉⠉⠀⠀⠀⢠⣴⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠄⠀⠀⡀⢧⣙⠂⠁⠠⣄⠀⠀⠀⠀⠨⠃⠀⠀⠀⠙⠀⠉⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠄⠀⠀⠀⠀⠠⣀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⢀⡤⠄⢀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢀⡡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠂⠀⠀⠐⠲⠦⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠂⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡤⠂⠰⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⠐⠀⠀⠻⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠐⢿⠇⠀⠀⠀⠀⢀⠀⠀⡎⠀⠄⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡀
⠀⠀⠴⠀⠀⠀⠀⠀⠀⠺⠿⠀⣿⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⢀⠄⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠐⡄⠀⠀⣀⣀⣀
⠀⠀⠀⠀⣠⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠀⠀⡄⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡈⠁⠀⠀⠀⠀⢠⣿⣿⣿⠁
⠀⠐⢰⠆⠀⢀⡀⠀⠀⠈⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⠐⢸⣿⣿⣿
⠠⠀⠀⠀⠾⢠⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⢀⣠⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠻⠟⠀⠀⠀⣀⠌⢿⣿⣿⣆
⠀⠄⠀⠙⠀⣐⠂⡉⠀⠀⡾⠃⠀⠀⠀⠀⠁⠄⠁⠄⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⡾⢶⡀⠀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⠤⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⢠⠀⡀⡀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠠⣶⡿⠗⠀⠀⠀⠀⠙⠁⠀⠀⠀⢸⡟⣯⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡈⠈⠀⠈⠐⢀⠵⠀⠅⠉⠈⠀⠀⠀⠠⠀⠀⠀⢁⣠⡄⠀⠀⠀⠀⠀⠀⢤⣤⡄⠈⠻⣷⡿⠀⠀⠀⠃⠂⠀⠀⠀⠁
⠈⠀⠂⠐⠀⠀⢠⣦⡄⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⢾⣿⠿⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠠⠔⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠘⠁⠀⠀⢠⣤⡴⠀⠀⠻⠿⠃⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⢀⡴⢿⠗⣦⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⠸⣿⡗
⠀⠀⠲⣀⠀⢀⣾⣛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⡀⠀⠀⠀⢠⣤⣤⡄⠀⠰⣾⣿⠆⠀⠘⣧⡞⢹⣿⠀⠀⠁⡆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠲⢁⠀⢸⣯⡿⣼⠃⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠃⠀⠀⠉⠁⠀⠀⠀⠈⠉⠋⠁
⠀⠀⠤⢁⠀⠀⠹⠿⠋⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢃
⠀⠀⠀⠁⠀⢀⡀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⢀⠀⠀⠀⠀⠸⠷⠅⠀⠠⢀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠀⠚⠛⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠈⠀⢠⡄⠀⠀⡶⠀⠀⣤⠀⠈⠀⠀⠃⠀⠀⢀⡤⠔⣶⡖⠒⠤⣄⠀⠀⠀⠀⠀⠀⣴⠀⢰⡦⠀⠀⠀⠀⠀⠠
⠀⣀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⠀⢀⡔⠁⠀⢰⠃⢳⡀⠀⠈⠳⡄⠀⠀⠀⠀⠐⠃⠀⠀⠁⠜⣃⠀⠌⢀
⠴⢁⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣖⠒⠒⡿⠒⠚⢿⡒⢛⡻⢿⠀⠀⠀⣘⠂⠀⡘⠀⡐⠀⠁⠀⠀⢡⡉⠡
⡂⢺⡒⠘⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠙⢶⣧⡀⠀⢈⣷⡋⠀⢸⠀⠀⠀⠉⠃⠀⠀⠀⠀⣠⣂⣜⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⢶⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⢿⣫⣷⣾⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢤⡶⢉⡠⠒⠉⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⣀⣤⡶⢶⣾⢟⣶⣿⣿⣟⣵⣿⣿⣿⣿⣿⣯⣭⣟⢿⣿⡀⠀⠀⠀⠀⣀⠴⠚⠉⣥⠀⣴⣏⠄⢂⠴⠋⠀⠀⠐⠀⠁⠙⠋⠓⢄⠀⠁
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣶⡿⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠁⢀⠔⢁⠀⠊⠇⠀⠀⠀⠀⠀⠠⠁⠊⡄
⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢻⡀⠀⠀⠀⡀⠀⠒⠁⠀⠀⠀⣀⣸⠋⡠⢴⡄⠀⠀⠀⠐⠃
⠀⠀⠀⢿⣿⡿⣿⣿⣿⣿⣿⣟⣻⣿⣷⣬⣽⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⢄⠀⠠⡄⠀⠂⠀⠀⠔⢶⠀⠀⢻⠋⠀⡂⠀⡀⠐
⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⢁⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠈⣈⣁⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠒⣶⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⢿⢿⣿⣿⣿⣷⣇⣶⣿⣷⣦⣤⣀⣤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡡⡤⡤⢄⡀⠀⠘⠸⠋⠁⠉⠑⢶⢺⣿⣿⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⣾⣝⣻⣿⣿⣿⣿⣿⣯⣥⣤⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠘⠀⠹⠦⢀⠀⠀⠀⠈⠙⠋⠙⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣯⣿⡿⡿⣿⣿⣿⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠒⢠⠤⣀⣀⣀⡀⡄⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣾⡶⠷⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⠁⠂⡄⠀⠀⠀⡀⠀⣈⣀⡠⠈⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢿⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⢰⠀⠀⡂⣠⠂⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⣿⣿⣿⢿⣿⣿⣿⡀⡄⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⠀⣶⠀⠀⠈⠐⠀⠀⣄⠻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⢹⣿⢾⣿⣿⣿⣿⠗⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⢧⣷⡀⢷⡀⣴⠀⠀⠀⢀⣠⣥⠶⠢⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠙⠛⢿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠐⠬⣽⣾⡿⢿⣾⣄⣀⠟⠁⠀⢀⡆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠓⠋⠉⠀⠈⠙⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡜⢦⣀⣐⣠⣤⡼⣃⡤⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡸⠁⠀⠀⠀⠀⠀⠀⠉⢳⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠃⠀⢿⠇⠀⠀⠰⡿⠀⠈⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠵⠛⣶⢤⢄⢀⡤⡔⠾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡷⠋⢠⢀⠋⠑⢶⡴⠒⢣⢸⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⠗⢄⡼⣘⠀⠀⠈⠀⠆⠻⣼⡦⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢀⢾⣧⠇⠀⠀⠀⠀⠂⠚⢿⣇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣾⢄⡿⠀⠀⠀⠀⠀⠀⠀⢸⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡨⢠⣧⣀⠀⠀⠀⠀⠀⣀⣤⡞⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⠈⠉⠉⣉⡋⠉⠉⠊⠀⠩⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢰⠿⣇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⠀⡿⣚⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠉⢑⣤⣷⠋⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⢄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⢗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠉⠈⠘⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠰⠦⣀⢨⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠈⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡄⠛⠒⠣⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⣠⡀⡾⠀⠀⠀⠀⠀⠀⠊⠁⠀⣒⠀⠀⠠⠖⠀⣠⠯⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢸⡗⢒⡦⠭⠛⠛⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⢋⡀⠤⠢⡀⠀⠅⠠⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠥⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠡⠿⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢀⡠⠐⠂⠉⠀⠀⠁⡇⠀⠠⠀⠀⠰⠂⢤⠧⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣲⣀⣿⣴⣍⣹⠂⢄⠀⠀⠀⠀⠀⢀⣠⠔⢃⠜⠁⠀⠈⡐⠰⢀⣀⡀⠤⠀⠛⠉⠉⠀⠒⠲⠦⠤⢤⡤⣤⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠋⠁⡇⠉⢿⠳⠤⢀⡁⣀⠤⠒⠈⠀⠁⠀⠀⠆⢄⠀⠀⠀⠱⠀⠀⡠⠐⠀⠉⠁⠉⠀⠀⠀⠠⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀⠀⡇⠀⣼⣷⠀⠤⠀⣀⡀⡀⡉⠀⠀⡠⡆⠀⠀⢑⡄⠀⡘⠠⠊⠀⢀⠠⠐⠈⠉⠀⠈⡀⠂⠀⠈⠰⠇⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⣠⠄⠒⠹⠶⠦⠤⠶⠟⠁⣀⠤⠐⢙⠀⠀⠁⠀⠀⠃⠃⠀⠀⡦⡆⠀⢀⠃⠀⣰⠡⠤⠠⠤⢀⣀⠀⠀⢀⡤⠠⠔⢁⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠂⠤⠌⢁⡐⠀⠠⠔⠂⡁⠄⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠀⡐⠀⠈⠉⠀⠒⠂⠒⠈⠭⠅⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢷⠐⡈⠄⠛⡞⠛⠁⠒⠀⠤⢼⣀⡄⣀⡰⠄⠨⠇⢠⠀⠀⡄⠈⠄⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠠⠐⣈⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠠⠊⡱⠊⠀⡸⠣⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⣀⠔⠈
⠀⣋⠠⠐⠉⠀⠈⠉⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠒⠂⠠⠤⠀⠀⢀⣄⠔⠊
⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠰⠶⠀⠉⠉⠉⠉⠉⠉⠉⠑⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠐⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⡀⠠⡀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠸⠃⢀⠀⢀⣀⠤⠄⣻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠀⠤⠤⢈⣻⠀⠀⠐⠄⠡⠀⠀⠀⢃
⠀⠀⠀⠀⠰⠆⠀⣾⠖⠋⠀⠀⠀⢁⡤⠀⠚⠓⠠⢄⠀⠀⠀⠀⠀⢀⡔⢊⣀⠚⠣⣄⠘⠓⠤⣀⠀⠀⠀⠀⢸
⠀⠀⢀⣤⠷⢄⣸⠇⠀⠀⠀⠀⣴⠋⢠⣾⣟⣡⣤⠈⣦⠀⠀⠀⠀⣏⣰⣿⣿⣴⣇⣈⣣⠀⠀⠈⢻⡆⠀⠀⢸⡀
⠀⠀⡟⠀⠀⢤⣽⠀⠀⠀⠀⠰⡶⠖⠛⠛⠉⠉⠉⠙⠛⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠸⣧⠀⠀⢸⡄
⠀⠀⠻⣤⣀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢈⠇
⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⢦⣄⡀
⠀⠀⠀⠘⣆⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣀⠀⠀⠀⠀⠀⢀⣀⡤⡶⠁⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠹⠇
⠀⠀⠀⠀⠹⣆⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠷⣍⠑⠒⠚⠒⠒⠈⠉⣰⠇⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤
⠀⠀⠀⠀⠀⠈⠳⣤⣀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⣤⣤⠴⠛⠁⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡒⠀⠲⡄⠀⠀⡴⠊⠀⣠⠞⣠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠻⠷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢘⣀⡊⠀⠀⠛⠥⠚⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠉⠉⠳⠄⠀⠀⣤⠒⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠲⣶⠒⠋⠀⠀⠀⠉⠓⣶⠖⠖⠒⠢⢤⣤⣄⣀⡀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠳⢦⣤⣤⣤⡤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡉⠉⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠳⠶⠶⠲⠞⠉⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠶⠦⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⣠⣤⣿⠶⠶⠶⠶⠶⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡌⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠑⣄⢾⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣟
⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠈⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⢀⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢈⣤⠀⠸⠀⢠⠀⠀⢿⣀⢙⡄⠀⠀⠀⠀⢠⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠉⠛⠷⣄⡼⣆⡠⠞⠉⠋⠁⠀⠀⠀⠀⠺⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢄⣀⡠⠀⠀⠀⢸⡇⠀⠀⠀⠹⡄⠀⠀⢀⣀⣀⣠⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢀⣀⣀⢸⡇⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠀⠀⠀⠀⠀⠀⢻⣀⣀⡀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⣠⠴⠷⠮⢬⣁⠀⠀⠉⠉⠉⠉⠉⠉⢩⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⢈⣁⣠⣤⣤⠄⠀⢹⣄
⢠⡞⠁⠀⠀⠀⠀⠉⠓⡄⠀⠀⠀⠀⠀⠀⠻⠤⣄⣹⡆⠀⠀⠀⠀⠀⠀⣧⠴⠚⠉⠉⠀⠀⠀⠀⠀⠈⠉⠲⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠂⠀⠀⠀⠤⢤⡴⠋⠉⣇⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠛⠒⠒⠲⠶⠶⠶⠶⠶⠶⠷⠤⠤⠤⠤⠤⠴⢤⣤⠼⠋⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣾⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠏⠉⠁⠀⠠⣆⡀⠈⠘⠏⢝⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣯⠔⠊⠁⠀⠀⠈⠁⠀⠀⠀⠁⠒⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠟⣾⠷⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣠⣷⣾⣿⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠘⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⡄⣸⣾⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟⣱⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⣽⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢀⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣯⣿⣵⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣠⢴⣿⣿⠿⢛⢋⣽⣭⣴⣿⣿⣿⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⣤⣃⣿⢩⠀⠀⡟⣼⣿⢻⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣠⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠟⠀⠀⠀⠀⢨⣿⣿⣷⣿⣟⣿⣿⣷⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢻⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣜⢿⣿⣿⣿⠿⣿⡿⣿⣿⣤⡄⠀⠀⠀⢰⣿⣟⠿⣿⣿⣿⣿⣿⣷⣿⣿⣟
⠀⠀⠀⠀⠀⠈⡽⣿⣿⣿⣮⣻⣿⣿⣷⣿⣿⡿⠛⠩⢤⣄⣀⠀⣶⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡨⠻⣿⣿⣿⣿⣿⣿⣿⣯⣌⡙⢀⡒⠈⣧⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣴⣤⣭⣦⣄⣛⠂⡙⣻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠙⢿⡿⠿⠿⢿⣿⣿⣿⣟⣟⠛⠛⠛⠃⡂⢈⣽⣿⣿⣿⣻⣿⣿⣾⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⢾⣿⣿⣿⣿⣿⡌⠉⠁⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠒⠒⠒⠂⠸⠿⣍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠄⠀⣠⠦⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣔⠤⢊⣉⠤⠚⠁⠀⠑⢦⣘⣳⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡍⠒⠉⠉⠂⠀⠀⠀⠐⣋⡉⠙⠀⠍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⣰⠀⠈⣴⣶⣷⠀⠀⠀⢸⣷⣦⡉⠀⢰⡶⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢐⡇⠀⠀⠻⠿⠏⢀⠀⠀⠘⠿⠿⠃⠀⠀⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⡤⠈⣁⠀⠀⠀⠀⣤⡠⠤⠤⢠⡆⠀⠀⠀⢀⡍⠠⠴⠥⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡄⠀⠀⠀⠙⢾⡓⣛⠮⠁⠀⠀⠀⡌⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⡤⠄⠤⢤⠐⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡖⠒⠒⠒⠒⠂⠈⠉⠁⠢⠔⠈⠉⠁⠐⠒⠒⠒⠒⢲⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠀⠀⠈⠉⠀⠀⠀⠀⠚⠛⠚⠛⢻⠒⠂⠀⠀⠈⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡰⢀⠀⠀⠀⠀⠀⠀⠀⠠⣛⡓⠛⣻⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⡯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠻⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⠃
⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠀⠀⠠⠤⠤⠴⣈⢃⣊⡸⠤⠤⠄⠀⠀⠐⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀
⡖⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓
⡇
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠤⢤⡤⣤⠤⢤⡤⣤⢠⣤⣀⣀⣠⡤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠤⠤⠤⠤⠤⢤⣄⣀⣀⡤⢠⡤⣤⠤⢠⡤⣤⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢰⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠠⠤⢤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢢⡤⠤⠄⢲⠊⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣦⠾⣇⠀⠀⠀⠀⠀⢸⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⣶⡌⣌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢇⢱⣶⡈⠳⣄⠀⠀⠀⢸⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠓⠀⠀⢀⠞⠉⠐⢤⠟⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠹⡄⠂⠉⠳⠀⠀⠘⠉⢱⠀⠸⠀⠂
⠀⠀⠀⠀⢀⠦⢦⠀⠰⠀⠄⠀⠀⠸⠃⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⢀⠇⠀⠠⠄⠼⠀⠰⠦⢆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⡆⠀⢘
⣤⣤⣤⣤⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡜⠦⠼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢱⣄⣀⣀⣀⣀⡀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠀⣒⣒⠒⡒⢒⣒⠒⣒⣂⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠈⠉⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢏⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⢂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡⠈⡎⡀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠾⢶⣦⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡠⠭⡀⢣⠀⠀⠀⠀⢠⠞⠁⡀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢀⠀⠀⠀⣠⠐⢀⠁⠀⢤⣽⣒⠶⣶⡄⠀⡦⠀⠀⢘⢸⠁⠀⢃⢸⠀⠀⠀⠀⠃⠀⠀⠠⠀⠀⠄⠀⠀⠀⠈⠇
⠀⠀⢠⡁⠀⢾⡟⠐⠊⠁⠀⠀⠀⢀⠀⠀⠘⢄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣄⠀⠀⠀⡄⢂⠀⠀⠀⡀⠀⠀⠀⢀⡀⡐⠠
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢇⠀⠀⠑⢌⠑⠒⠉⠀⠀⠀⠀⠈⡡⠃
⠀⠀⠀⠀⠀⠛⢆⠀⠀⠀⡀⠀⠀⠰⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣀⠀⠀⠀⠁⣢⠄⣀⣀⡤⠔⠊⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠈⠐⠊⠀⡋⢄⠁⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠙⣦⣤⠴⢊⠛⠜⠁⢠⡃⠑⢆⠀⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⠘⠧⠛⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⢸⡁⠀⠀⣀⣁⡠⠞⠐
⠀⠀⠀⠀⠀⠀⢀⠞⠁⡧⡸⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡉⠁⢸⠁⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠘⠢⠀⡷⠃⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢣⡰⠀⡀⡄⠀⠀⠀⠘⠤⠛⢤⡼⠀⠀⠀⠀⠀⠠⢀⡏⢀⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢬⡀⡀⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠪⡨⢛⢾⠇⠸⠃⡇⠀⠰⠒⠼⠀⣠⢤⡧⠀⢀⠀⠀⠀⠀⠀⢑⢀⡀⠎⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⠁⠀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠹⠤⢅⠼⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠵⠄⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡆⢀⣾⣿⣟⣿⡿⠁⠀⠈⣿⣿⣿⣿⡻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⢻⣿⡏⡇⣿⣇⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠐⠦⣄⠸⣾⠘⣷⣿⢹⣿⡿⠟⠟⠁
⠀⠀⠀⠀⠀⠀⣰⣸⣿⣿⣿⢯⡇⠘⠒⣶⡲⣄⠁⠘⠧⠸⣟⣹⣿⣡⠔
⠀⠀⠀⢀⣞⣼⣿⣿⣿⣿⡟⣾⡇⠀⠀⠀⠀⠈⠀⠀⠀⠺⠋⢸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠄
⠀⠀⠀⠀⠈⢿⠻⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠐⠀⠀⣰⣿⣿⡿⠘⡁⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⣿⣿⣿⠱⣄⠀⠀⠀⠀⠀⣀⣼⣿⣿⡟⠁⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣧⡈⠳⢤⣀⣠⡾⣿⠻⡿⣿⠃⠀⠟⠁
⠀⠀⠀⠀⠀⠂⠠⠄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠙⡡⠿⠀⠀⠁⢀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢧⡄⠀⡴⣀⣦⣴⠶⠀⠈⠣⣖⣆⠸⣏⢹⠂⠙⣏⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⠘⣿⣿⢻⣯⡛⣀⣀⣀⣷⡓⣧⢹⡟⠳⡄⠿
⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠓⠛⠈⠋⠙⣿⣿⣿⡟⠿⠙⠏⠑⣤⡀⠀⠀⠀⠐⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⡤⠀⠀⡂⠀⠀⠀⠀⢠⠒⢒⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠁⠀⠀⠀⠁⠀⢰⠈⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠁⠠
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡆⠀⠀⠀⣨⠔⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣀⣤⣤⠆⠀⠀⠀⠻⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠰⠞⠋⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠇⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢠⠀⢀⠀⠀⠾⡇⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⢻⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣀⣀⡀⠇⣠⠤⣾⣾⡐⢒⣲⠖⠀⢀⣰⠦⠤⣤⡧⠤⣤⠌⠀⢀⣀⠠⡘⡀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣤⡶⠞⠀⡀⠐⠓⠛⢺⠆⠈⠉⠉⠃⠉⢛⠉⠁⠈⠀⠈⠉⠉⠉⠩⢉⠉⠁⠀⠘⠶⠯⠹⢿⣭⣓⠠⠀⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠴⠀⠄⠀⠀⠀⠀⢀⠃⠀⠀⠀⡰⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⡈⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀⡠⠒⠀⠰⢮⣒⢔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠰⢭⡗⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠸⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠸⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠂⠀⠀⠀⠰⠇⠀⠀⠀⠀⢠⣴⣢⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢄⠀⠀⠀⠀⠀⠀⠈⣠⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠉⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⡠⠀⠀⠀⠈⠢⣉⠒⠀⠠⠤⡴⠚⠁⠀⢤⡢⢀⣀⠐⠒⢁⡀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⠭⠭⠿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠑⠚⠛⠛⠉⠀⠀⠣⣓⣤⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⡀⠄⢄⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠐⠐⠀⠀⠂⠔⠀⠢⠂⠔⠠⠠⠂⠀⠠⠀⠀⢀⠀⡀⡀⢀

⠀⠀⠰⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⢹⠈⠉⠈⡇⡞⠀⠈⠁⢻⠀⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⠈⠁⠂
⠀⠀⢰⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢹⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⢸⠘⣷⠛⠓⠒⠒⠒⠒⠒⠲⣿⢸⠛⠛⠉⠉⠛⠀⠀⠐⢰⠄⡎
⠀⠀⢸⠀⢠⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⡆
⠀⠀⢸⠀⠸⡆⠀⠀⠀⠄⢰⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⣿⠀⠄⠀⢀⠀⠀⠀⢸⡇⢸⠈⠰⠀⠀⠀⠀⠰⠁⡎⢰⠇
⠀⠀⢸⠀⠀⡆⠀⠀⠀⡤⢄⠀⠀⠀⠈⠁⠁⠀⠀⣾⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠠⠄⢰⡇⣈⠀⢀⠀⠀⠀⠀⢀⠀⠁⠸
⠀⠀⢸⠀⠀⠛⠀⠀⠀⡇⡄⠀⠉⠶⢂⡁⡃⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠈⠀⠐⠤⠤⠄⠊⢰⠁⣶
⠀⠀⢸⠀⠀⢸⡄⠀⠠⠃⠉⠀⠉⠉⠈⠀⠀⠐⢄⣹⠀⠀⠀⡆⠀⠀⡇⢸⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣁⡽⠤⢣⠀⠀⠀⠀⠂⠀⠀⡘⠀⡍
⠀⠀⢸⠀⠀⠀⣧⠌⢤⣠⣤⡤⢔⠒⠒⠂⠉⣋⠉⡥⠄⠀⠰⠇⠤⠄⣇⣸⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠈⠡⣴⠚⡇⠀⠀⠀⠀⠀⠀⠀⡇⢀⠁
⠀⠀⢸⠀⠀⠀⠀⠀⡞⠁⠠⠒⠋⢓⣤⠔⠋⢥⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⠀⣼⡾⠛⡛⠲⡄⠀⠀⠀⠀⣼⠀⠇⠀⠀⠀⠀⠀⠀⢠⠁⢸⠃
⠀⠀⢸⠀⠀⠀⠀⢸⠀⢀⠔⠊⠉⡝⠁⡄⠀⠐⠉⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⢠⣀⠀⠀⠸⠀⠀⠀⠀⣿⠀⡆⠐⠀⠀⠀⠀⠀⢸⠀⠘
⠀⠀⢸⠀⠀⠀⠀⠸⠀⡎⠀⠀⡜⠀⡸⠀⠀⣶⡀⠂⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠙⡀⣿⠱⡀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠄⡆⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠷⠀⠀⢰⠀⠀⠃⠀⢸⠀⠙⢤⡀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠀⠀⣿⣿⠀⢃⡸⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡌⠀⠀⠀⢠⠃⠀⠀⠀⠈⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠃⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠑⠀⢸⠀⢰
⠀⠀⢸⠀⠀⠀⠀⡴⠊⠉⠈⡇⠀⢀⡴⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⢢⡀⢸⠉⠉⠓⣤⣠⣀⣤⣖⡇⢸⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⢸⠀⠀⠀⠈⠃⠀⡶⠰⣇⢠⠋⠀⠀⠔⣴⡀⠀⠀⠀⠀⠀⠀⢀⢠⣶⠀⠀⠀⢁⣸⡀⠀⠀⣏⠀⢀⠀⠀⠁⠈⢈⣉⣉⣁⠙⠚⠃⠀⡄
⠀⠀⢸⠀⠀⠀⠀⢡⠀⢱⡞⠉⢿⠀⠀⠀⠘⠋⠁⠀⠀⠐⠂⠀⠀⠀⠙⠋⠀⠀⠀⠙⠉⢓⠀⠀⡇⠀⠀⢀⡤⠬⠭⠭⣍⠉⠉⠉⠉⠉⠉⠁
⠀⠀⢸⠀⠀⠀⠀⠘⠀⠈⠳⠄⢄⠁⠀⠀⠀⠀⠀⠀⡒⠀⠒⡂⠀⠀⠀⠀⠀⠀⠀⡠⢤⠊⠀⠀⠇⠀⠀⠏⠑⠒⠒⠊⠙
⠀⠀⢸⠀⠀⠀⠀⠀⡆⠁⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⢁⡀⣀⠂⠀⠀⠀⠀⠀⣀⠞⠁⠀⡇⠀⢠⠀⠀⡸⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠀⠀⠀⠀⠀⣷⠛⢢⣀⣀⠀⠈⠑⠦⢄⣀⠀⠀⠈⠀⠀⠀⣀⡠⠴⠚⠉⢀⣀⣠⠟⠲⣼⠀⡤⢏⠐⠂⠠⠤⠀⠲⡧⣄
⠀⠀⠸⠀⠀⠀⠀⠀⢹⢶⠶⡏⠈⠉⠙⠐⠒⠖⠾⢿⣏⣉⢉⣻⠟⠏⠒⠊⠉⠉⠙⠁⢹⠶⡖⢹⠀⣇⠀⠹⣶⣴⣴⡖⠉⠀⠈⢦
⠀⠀⢈⠀⠀⠀⠀⠀⢀⡼⠀⠷⢂⣄⣀⣠⣄⢰⡆⠀⠀⠉⠁⠀⠀⢰⣤⣤⣤⣤⣤⣤⠎⠀⣣⡘⢸⢸⠂⠠⠘⠛⠋⡠⠄⠤⠤⠤⠷
⠀⠀⢠⣄⣀⣀⣀⣀⣠⣿⠊⠉⠉⠉⠉⠉⠉⣿⣀⠤⠶⠶⠖⣀⡀⡀⡇⠀⠀⠀⠀⠀⠀⠉⢡⡕⢺⡈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⡄
⠀⠀⢠⠀⠀⠀⠀⠀⢀⣇⣀⠠⠤⠤⠤⠤⠔⢉⡀⠤⠤⠤⠤⢀⣈⠉⠛⠤⠤⠤⠤⠆⠀⠄⠌⠿⡟⠛⡌⠉⠉⢸⣿⣷⠀⠒⢳⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠀⠀⣘⣀⣀⣤⢤⡤⠤⠤⠔⠁⠀⢠⠀⢤⠒⢀⣀⡉⠐⠲⠤⠤⠤⡦⠤⠤⠤⠀⣿⠆⠃⠀⠀⠀⠚⢀⣀⣀⠸⡟⠉⠉⠉⢿⡀⠀⠀⠀⠀⣿
⠀⠀⢸⠀⠀⠀⠀⢠⠉⡀⡈⠁⠀⠃⣠⠶⡂⠠⠀⠤⣋⣙⣌⡱⣄⠊⠀⠀⡀⢀⡔⠿⢶⡷⢤⠀⠉⢶⡦⣬⠭⠤⠭⠭⠽⠽⠿⠥⠤⠤⠤⠼⡧⠀⠀⠀⠀⠿
⠀⠀⢸⠀⠀⠀⢀⡏⡟⢀⠜⠒⢆⢁⡌⡶⢶⠀⠀⠀⠀⡀⠀⡉⠘⠀⣴⠶⠶⣆⢘⡂⢸⡅⠾⡐⣂⣈⢇⠀⠀⠀⢀⡀⣤⠤⠤⣄⡀⠀⠀⠀⠰⠄
⠀⠀⢸⠀⠀⠀⡾⠁⠼⣀⣘⣏⠱⡎⠀⠀⡞⠲⠶⠦⣠⣥⣤⣥⠤⠔⠛⣄⠀⠈⠷⠤⢤⣉⠤⠋⡙⠳⠛⣆⠀⠐⢯⡀⢀⣄⣀⠴⠃⠀⠀⠀⠀⢳
⠀⠀⢸⠀⠀⣘⠓⠛⠀⢉⣁⣀⣀⡱⡴⠞⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠆⠤⠠⠀⢀⣈⣉⣉⣉⡉⣦⠀⢀⡔⠋⠙⠀⠈⠐⢢⠀⠀⠀⠈⢇
⠀⠀⠸⢰⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢊⠁⠉⠉⠆⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠑⠒⠒⠓⠒⠊⠁⠀⠀⠀⠀⠸⡄
⠀⠀⢠⠀⣄⠀⠠⠒⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠐⠂⠀⠀⡾⠠⠄⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⢶⠀⠁
⠀⠀⢸⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⢸⠀⠘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⢈⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡏
⠀⠀⢈⡀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣧⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⠃⠀⠀⠀⠀⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠐⠺⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠍⠃⠀⠀⠴⣟⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢰⣿⣿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣺⣿⣷⣄⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣷⡆
⠀⠀⠀⢸⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢰⣿⣿⡏⢻⣿⣿⡆⠀⠀⠀⠀⠀⠋⢹⡏⣹⡿⠋⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠃
⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃


⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⣀
⠀⠀⠀⠀⠀⢠⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣧⣄⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⡿⠿⠿⠆⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠄⣬⢀⣬⡀⣀
⠀⠀⠀⢰⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⢸⣟⡉⢹⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡗
⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠉


⠀⠀⠀⠀⡀⢀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣠⣄
⠀⠀⠀⢰⣷⣯⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣄⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⣤⣤⢀⡀⠀⠀⠀⠀⠰⠁⠰⠾⢷⣶⣆⠀⠀⠀⠀⠀⠀⢙⣳⣿⣿⣆
⠀⠀⠀⠸⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠿⠿⢿⣿⣷⠟⠁⠀⠀⠀⠀⠀⢷⣆⠀⠢⡙⠋⠀⠀⠀⠀⠀⠀⣾⣉⣹⣿⣿⡆
⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋


⠀⠀⠀⠀⣀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤
⠀⠀⠀⠀⣿⣄⢘⣿⡇⠀⠀⠀⠀⠀⠀⢹⣧⣾⣷⣆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣧⣄⠀⢤⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣼⠆⠀⠀⠀⠀⠀⠀⠀⣀⣷⣿⣿⡇
⠀⠀⠀⢀⣽⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠶⠆⠀⠀⠀⠀⠠⣬⣿⣿⣿⡀⠎⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠁


⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⣸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⡅⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠸⠇⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣧⡾⠂
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣤⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠈⢷⣭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣅⣾⡟⠀⠘⠙⠿⣿⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠠⠠⢼⣦⣿⣿⣿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⢹⣿⡿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⡿⠃⠈⠋⣹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠴⠞⠛⠇⠀⠛⠟⠛⢿⡿⠻⣿⠇⢠⠀⠀⠀⢀⠆⠀⠀⠀⣠⣦⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⣬⠀⠀⠀⠀⠀⣸⠀⢀⡀⠀⠀⢘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡼⢳⠋⠀⡀⠀⠀⠀⣹⣿⣿⠁⠀⢀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣿⣿⡀⠐⠀⠀⡀⢀⡀⠀⠀⢠⡄⣡⣤⣦⣾⣷⠆⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⢻⣿⣟⡁⠀⠀⣀⡓⠸⡿⠄⢥⣾⣧⣿⣿⣿⣿⠿⡆⠀⠀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⠀⠉⠉⣳⠷⣶⣶⣾⣿⣿⣿⣿⣿⠀⠠⡆⠀⣿⡇⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⡿⠉⢀⠀⠰⠒⠲⠉⠘⠛⢉⠈⡉⢿⣿⣿⣿⠀⠀⠀⠀⣿⠁⠀⡀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⠷⣗⠀⠾⠀⠀⢠⠀⠀⠉⠀⠈⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⠀⠀⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⡄⠀⠀⢸⡆⢀⣿⠀⠀⠀⠀⠀⢀⣀⣾⠁⠯⣉⣧⠀⠀⠀⠀⠀⠀⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣦⣼⡿⢸⡟⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⣿⡇⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃⠼⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠱⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⣸⣆⣰⣿⣿⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣀⣼⣿⣿⢿⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠘⠛⠛⠉⠉⠘⠁⠷⢦⣈⡇⢀⡄⠀⢀⣠⣞⣿⠄⠀⠀⠖⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣾⡟⠉⠉⠉⠉⠛⠛⠒⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⠆⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⢀⡤⠖⠋⠁⠾⠥⠤⠤⠔⠚⠉⢡⣞⣁⣠⠤⠖⢫⡇⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠋⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢀⣤⡀⣠⡀⠀⠀⠀⠀⠀⢻⠀⠀⢸⠃⠀⠈⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⣤⡀⠀⢷⣀⡀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡀⣠⣄
⠀⣤⣼⡏⣿⠉⣇⣤⡄⠀⠀⠀⠈⡷⢦⣼⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠿⡇⠀⠀⠀⠀⣸⠇⣤⣤⡏⢻⡏⣿⣴
⣴⡽⣮⠓⠘⠀⡟⢸⡇⠀⠀⠀⢸⡇⢢⣟⠀⠀⠀⠀⢸⣷⣿⠀⠀⣠⠃⠀⠀⠀⠀⣾⣽⡆⠀⠀⠀⠀⠹⣀⡴⠢⣠⡟⠀⣿⠘⡇⠈⠐⢫⡾
⠈⠛⢶⡀⠀⠜⠁⣸⠇⠀⠀⠀⠀⣧⠀⡟⠀⠀⠀⠀⠀⠉⠁⠀⢞⠉⠁⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠛⡴⢫⡏⠀⠀⢾⡀⠹⠆⠀⣠⠞
⠀⠀⠈⠳⣦⠀⠀⢻⡄⠀⠀⠀⠀⠘⢦⣭⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣼⠃⠀⢠⡴⠋
⠀⠀⠀⠀⠹⡆⠀⠀⢻⡄⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠈⡟⠒⠂⠤⠤⠤⠤⢤⡶⠀⠀⠀⠀⠀⠀⠀⢀⣧⡼⠃⠀⠀⡼⠃⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⣠⡞⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠙⢦⣀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠈⠓⠶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⢀⣠⠞⠃⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢀⣽⣷⣶⣤⣄⣈⣳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣡⣴⣶⣿⣇⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠲⠤⣀⣀⣀⣀⡀⠤⣴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⢧⡀⠀⠀⠀⠀⢀⡰⠋⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠓⠴⠶⠒⠋⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠫⠭⣉⡉⠛⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠛⢉⣉⠭⠝⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠉⠓⢦⣙⠻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⣋⡴⠚⠉⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠞⠁⠀⠀⠀⣀⣤⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣓⣶⣤⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣦⣤⣶⣒⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⢀⣄⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣇⣾⣿⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⣿⣇⣼⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣟⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡷⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣯⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⣼⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠘⠌⠁⠀⠀⠀⠀⠈⠃⠐⠂⠁⠀⠀⠤⠄⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠦⡀⠀⠀⠀⠀⣆⡙⢄⠐⠀⡀
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠈⠡⠚⣁⠀⠀⠀⠀⠄⣢⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⣶⠢⢹⡆⡴⢋⡩⠉⠻⣆⠴⢋⣽⡄⠠
⠀⠀⠀⠀⡅⠀⠈⡀⠀⠠⠀⠀⢠⣿⡄⢀⡀⠀⢻⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢧⡸⢟⣀⡼⢻⡇⣾⡿⠀⠀⣿⣤⣉⠈⠀⠀⠂
⠀⠀⠀⠀⠠⠀⠑⠚⣢⠅⠀⠀⠨⠟⠁⠀⠉⡷⠀⠀⢀⠀⠀⠀⢐⡀⠀⠀⠀⠐⠀⠀⠀⠙⠙⠃⠀⠀⠳⢬⣀⣀⠴⠟⠀⢹⠆⣀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢅⠈⠥⡀⠀⠀⠀⠀⠦⢠⡦⠀⠀⠀⠌⠀⠀⠀⠘⣭⡀⠀⠀⠐⠠⠒⠃⠤⠀⠀⠘⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠘⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣜⡤⢢⠴⠀⠀⠀⠀⠈⠂⠀⠀⡀⠀⠀⠀⠤⡀⣀⣀⡀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⢯⣤⣬⠦⠄⠠⠠⠀⠰⠀⠀⠀⠘⠭⠅⠀⠀⠀⢠⣀⡀⠀⢀⣤⢠⣅⠀⠡⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢰⢀⠋⡍⢸⠈⣿⡄⠀⠠⠆⣼⢃⠕⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠘⣷⡋⠀⠤⠀⠀⠛⢧⢊⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠀⠴⠀⠀⠋⢀⢨⡇⡀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠢⠤⠤⠤⠤⠤⠤⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡭⠥⠤⠤⠤⡯⢤⠌⠰⠇⠐⠆
⠀⠀⠀⠀⠀⠀⢀⠈⢀⢄⡰⢀⣀⣻⣤⣤⡀⠀⠀⠶⢒⣲⣤⠴⢶⠒⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⢥⡐⠅⠀⠁⠀⠉⠛⠛⠛⠁⠀⠀⠀⣀⣀⣀⠐⠨⠋⠉⠉⠙⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠩⠴⡮⢄⢀⡀⣂
⠀⠀⠀⢀⡄⢉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠈⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠓⠐⠒⠀⠀⠀⠒⠀⠂⠀⡁⠈⠑⠈⡋⣂
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸
⠀⠀⠀⢸
⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠠
⠀⢀

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

⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⢰⣿⣿⠛⢿⣿⡿⠷⡄⠀⠀⠀⠀⠀⠀⣼⣿⠟⠿⠻⢿⣧⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣤⣤⡤⣤⣄
⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠙⠛⣿⣷⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠈⣿⠄⠀⠀⠀⠀⠀⠀⣿⠁⠉⠀⠁⠀⢹⣿
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠛⣿⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠁⠘⣿⡄
⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣀⠈⢀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⣿⣿⡿⠇⠀⠹⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣤⣬⣈⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣾⣿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣿⡿⠿⣧
⠀⠀⠀⠀⠀⢸⣿⠡⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢠⣿⠟⠃⠀⠈⠻⢿⡇⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠂⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠂⠀⠀⠤⢶⡄⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⠁⢨⡗⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⠄⢋⣥⣤⣍⣙⠣⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⣾⣸⡟⠉⠉⠻⢿⣦⣷⠀⠀⠀⠀⠀⠀⢸⡟⡉⠁⠀⢀⢻⡇⠀⠀⠀⠀⠀⣹⣿⡟⠋⠉⠀⠈⣿⣿
⠀⠀⠀⠀⠸⣿⣯⠀⠀⠀⠠⠀⣿⣿⠇⠀⠀⠀⠀⠀⠸⣿⣿⠇⠸⣿⣿⠃⠀⠀⠀⠀⠀⢹⡿⠀⠄⠀⠀⠄⠸⣿⠇
⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣴⣶⣤⣤⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠐⠂⠀⠠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠂⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠋⠀⠀⠀⠀⠀⠀⠀⠹⣶⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢀⣰⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣷⣆⠈⢿⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⣿⣿⣿⣿⣿⣿⠀⣰⣿⣧⢃⠸⣿⣿⣿⣿⣿⢿⡿⠀⠿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠐⠈⠛⠛⠛⠛⠛⡁⠀⣼⣿⣿⡏⠀⢉⠛⠛⠛⠛⠋⠀⠀⣠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⣤⢀⣶⡟⠁⢰⣿⡿⣿⡿⡆⠘⣷⣶⣄⡀⠠⠤⢪⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠐⢢⡄⠀⠀⠀⠀⢹⣿⣿⡄⡘⠉⠀⠁⠀⠈⠀⠀⠀⠁⠀⠈⢷⣠⣠⣤⣾⠏⠀⠀⠀⠀⢠⡴⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣥⠀⠀⣴⡟⠀⠀⠀⠀⠀⣿⠹⡇⢹⣶⠀⠀⢀⠀⠀⠀⠰⠀⠀⠀⣾⠃⣸⣿⡿⠀⠀⠀⠀⠀⢻⣧⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣹⡏⢹⣿⣧⠀⠀⠀⠀⠀⣿⠀⢻⣿⣽⣄⣄⣸⣀⣠⣀⣀⣠⣄⣤⣿⣴⡟⢹⡇⠀⠀⠀⠀⠀⣸⣿⡏⢹⣏⡀
⠀⠀⠀⠀⠀⠀⠰⠉⠙⠓⠀⠛⠋⠁⣰⠀⠀⠀⣿⡀⢸⣿⣈⡏⢻⠉⠋⠙⠉⢹⠉⠋⣻⣿⡟⠀⢸⡇⠀⠀⠀⣆⠈⠛⠛⠀⠘⠉⠉⠆
⠀⠀⠀⠀⠀⠀⠘⣄⡈⠳⠶⠶⠞⠻⢿⣆⠀⠀⠘⢧⡀⢿⡟⠁⠘⠠⠔⠺⠂⠘⠴⠃⠙⠿⢁⣠⠎⠀⠀⠀⣠⣿⠟⠳⠶⠶⠚⢁⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠠⠤⣤⣄⣀⠙⢷⣦⡀⠀⠙⢦⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠞⠁⠀⢀⣴⡾⠋⣀⣠⡴⠬⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣔⡚⠑⡦⣄⠑⠄⣘⣇⠀⠀⠤⠀⠚⠩⠽⢃⡠⣴⠊⢙⣢⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⠉⠂⢤⣀⠀⠀⠀⠀⣀⡤⠐⠉⣀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⣤⣈⡉⠲⠶⣉⣁⣤⡶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠚⣿⣿⣶⣄⣀⡉⠓⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⡁⢠⣴⣿⠿⠛⠉⠉⠛⠿⣷⣦⡄⠈⠑⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠶⠶⠾⠛⠓⠀⢉⣴⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣆⡉⠐⠘⠛⠷⠶⢶⠀⢠⣄⣤⣄
⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠉⢴⠶⠍⠓⠂⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠚⠩⠶⡦⠉⠀⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠘⢿⣄⣐⣶⡦⠔⠀⠀⣀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀⠀⠀⠰⢶⣶⣂⣤⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠀⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢲⡆⠘⠃⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢀⡤⠴⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⣠⡟⠀⠄⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡴⠏⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣽⣿⡏⠀⠀⠀⣰⣾⣿⡏⠁⠁⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣤⣄⣀⡿⠑⠫⣴⡄⠀⠀⣰⠊⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢠⣤⣈⠁⢀⣴⣏⢹⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠉⠉⠉⠁⠈⠀⣼⣿⣿⣿⡿⠁⠻⣿⣿⣿⣿⣿⣼⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣟⠲⢌⠛⢿⣿⣿⣿⡅⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠺⢶⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠈⣼⣿⣿⠋⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⡀⣀⣤⣾⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢠⣿⡟⠓⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⢠⣾⣿⣌⠛⢿⣿⣿⣿⣿⣿⣿⣟⣻⣨⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡌⢽⣿⣿⣿⣿⣿⣿⣿⠸⣿⣄⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⡿⣻⠋⢀⡟
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠻⡿⣿⣦⣉⠛⠿⠿⠿⢟⡿⠃⢀⡞
⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⠀⠈⠁⠉⠛⠒⠒⠒⢋⣠⠔⠋
⠀⠀⠀⠀⠐⠁⠀⠀⠀⢤⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠨⠍⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠖⠃⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁

⣿⣹⣿⣿⠇⢈⢿⡏⠉⠳⣵⡞⢠⢸⢷⣿⡹⣿⡈⠁⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⣿⢏⡻⣿⣘⠻⡿⢷⡄⣠⣍⢻⡉⠨⠈⠻⢿⣼⠋⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠛⠛⠛⠛⠫⡴⠒
⢿⣮⣄⣾⢿⣿⡇⣀⡀⣫⣻⣦⠙⢸⡀⠀⢀⢻⡇⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⢀⣴⡖⠛⠁
⣾⠿⠹⣿⣎⢹⠃⡫⠝⢺⡛⢿⡂⠀⠱⡄⣧⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣶⠃⠀⠀⠀⠀⢀⡤⠐⠐⠤⢀
⣿⡶⠸⡸⣿⣿⠀⣉⣳⠀⡘⣂⠻⠘⣆⢧⠹⣿⢦⠀⠀⠠⠀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢉⡠⢶⣮⡙⢒⠤⣁⢴⣀
⣿⣿⣿⣆⠈⠻⣦⣹⡇⣾⣾⡃⢠⣶⣈⡛⢦⡈⠋⠃⡀⠀⢢⠀⠀⡘⢆⠀⠀⠀⠀⠀⠀⠀⣴⢿⣰⣿⣧⡀⠙⢿⣟⣷⡌⢲⣿⣆
⣛⣧⡶⣹⣿⣶⣎⢏⣷⣿⠀⠀⠀⢙⣵⣾⣬⠓⣄⡀⢻⡶⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⣻⣿⢿⣷⣄⠈⠋⢿⣿⣆⠻⡟⠂
⣿⣿⣿⡄⣼⣿⣿⣿⣾⣟⠧⡳⠄⠘⣟⡁⠀⠁⠛⠻⡀⡄⠹⣶⠁⠁⠄⠰⣠⠀⠀⠀⣸⡏⠀⠀⣍⣿⣷⣭⣿⣿⣦⣤⣾⣿⠃⠘⠳
⡙⠻⣍⣭⣿⣿⣿⣿⣿⣿⢷⣦⣄⣈⠻⠻⢦⡄⣤⡄⢻⣿⣦⡌⠀⢸⣿⠀⢧⡄⠀⠀⣿⠁⠀⠀⠀⢻⣿⣿⠟⠛⠛⠛⠛⠉⠀⠀⠐⠶⣄
⣿⡀⣹⣮⡉⢿⣿⣿⡏⠈⠤⡈⣶⣭⣿⣷⣦⣅⡘⠇⡾⠛⢿⣿⣶⡡⣥⣴⠿⡄⣀⣸⣯⠀⠀⠀⠀⠀⣿⡇
⣿⠗⢡⣿⣿⣿⣿⣿⣷⣀⣿⣷⣻⣿⠿⢿⣿⣉⡙⠲⢤⠡⢄⣿⣿⡟⠉⠈⢳⣤⠹⣿⣿⡃⠀⠀⠀⠀⠀⢿⡄
⣀⣠⣾⣿⠟⢁⣀⠈⠻⣟⠟⣛⣿⣧⢀⣶⡻⠛⢛⠛⠀⠀⠉⣛⣿⣳⡤⣿⣼⡿⣎⣼⣿⣷⠠⡀⠀⢠⠀⠈⠻⢄
⣿⣿⣿⡟⠀⢁⣤⣥⠀⣿⣷⣈⠏⠙⢣⡻⣧⡀⣚⣡⢶⠚⠋⠉⢀⢀⠁⢐⠭⣿⣾⣏⢿⣿⣿⣯⡋
⣿⠿⠟⢇⠀⠈⣛⡙⠀⣿⣿⢿⣿⣄⣀⣉⡔⠛⠛⠻⣧⡔⠺⡄⠘⠋⣲⣄⣿⣿⣿⣿⣾⣿⣿⡦⠕
⣹⠤⠀⢀⣷⢤⣀⣤⣾⣷⡶⠛⠛⣿⣿⣾⣇⠀⢰⣿⡎⣿⣷⣼⣷⡾⠿⢸⣿⣝⡿⠿⣿⣿⣿⡟⠧⠤⢄⠀⠀⠠⠀⡀⠀⠄⠀⠈⠉⠀⡀
⣿⣶⣶⣿⣁⣸⣿⣿⣿⣿⣳⣾⣶⡼⣿⣿⣿⣷⣌⣭⣴⣿⣿⣟⡛⠀⣦⣸⣧⣾⠁⣶⣹⣿⣿⢧⡤⠤⢎⠉⠀⠀⠂⢀⡀⠀⠐⠂⠀⠀⠑⠀⡀
⣿⣿⣿⣿⣿⠿⢿⣿⠿⣿⣿⣿⣿⡱⣾⣿⡟⢋⣽⡿⢿⣿⣿⣟⣷⣄⣭⣽⣿⢿⣶⣾⣿⣿⣿⣗⠳⢤⣴⣉⡌⠛⠋⢄⠀⠁⡀⠒⠒⠒⢱⡖⠁
⣷⣾⣷⣿⣿⣿⡿⠁⣀⣀⠙⣾⣿⣷⣼⣟⣼⡟⣯⣹⣷⣿⣿⣿⣿⣿⣿⣿⠃⢁⣅⣈⣿⢟⣿⣿⣿⣁⠀⠀⣱⠦⡤⠎⡉⠉⣹⠶⠤⠄⠊⡀⢀⣼
⣿⡿⣿⣿⣿⣿⡇⠀⣷⣦⡀⠘⣿⣿⣯⣭⣭⣴⣮⣭⣿⣿⣿⢿⢫⡿⣝⣿⡄⠸⡿⢫⣯⣘⣿⣿⠙⠻⠿⠟⠛⣦⣀⣠⠜⠛⠅⠀⠀⠀⠘⠋⠈⢁⣰⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢰⠋⠀⠰⣶⠞⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⣤⣞⣵⣤⠟⠀⠀⠀⠙⢷⣼⡇⠀⠀⢳⠂
⠀⠀⠀⠀⣀⡠⠚⢷⡀⠀⠀⠀⠉⠛⢿⢦⡀⠀⠀⠀⠀⣽⣇⠀⠀⠘⢧⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⢤⣾⣿⠀⠀⠀⠀⠲⢦⣆⠀⠀⠀⠉⠓⠀⠀⠀⠀⢸⢻⢡⣀⠀⠀⠀⠀⣤⠀⠛⠂⢀⡾⠃
⠀⠀⠀⠀⠙⢧⣄⡀⠀⠀⠀⠙⠛⠶⡄⠀⠀⠀⠀⢰⠆⠀⠈⡽⢻⣳⠄⠀⠀⠈⠛⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠙⠳⠆⠀⠀⠀⠀⠀⠀⠀⠠⣤⡴⠋⠀⠀⢀⠀⠼⢃⣸⡆⠀⠐⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈⠁⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⣤⠀⠀⢠⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⣤⠀⠀⢀⡆⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡺⠋⠠⠴⠏⠀⠀⠀⠀⠀⠼⠿⣇⣀⠀⣿⡆⠀⢸⣿⠀⢀⡀⠀⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢇⣤⢾⣿⠀⢸⣇⢠⣿
⠀⠀⠀⠀⣀⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢄⣾⡿⣺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣻⠀⡦⠀⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⠁⠀⠀⠀⠀⠤⠤⣄⡀⠀⣿⣿⣿⡇⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠀⠀⠀⠀⠀⣀⣤⣀⠀⠈⠿⠿⠿⠷⠶⠦⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠃⠀⠀⠴⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠖⠒⠂⠘⠋⠒⠐⠒

⠀⠀⠀⠀⢰⡄⠀⠀⠀⡄⠀⣶⣄⠀⠀⠀⢴⣶⢶⠶⠶⡄⠀⣀⢻⣄⠀⠀⠀⠀⣀⡀⠀⣤⣤⡀
⠀⠀⠀⢀⣼⡇⠀⠘⠻⣋⡩⣍⡙⠀⠀⢠⡦⢀⣬⣤⣸⠇⠀⣿⣿⣿⠀⠀⠘⠿⠿⣿⠀⢿⣿⣿⠇
⠀⢀⠐⠀⠀⠉⠱⣦⣀⠘⠎⠳⠂⠀⠀⠀⠳⢍⡈⠛⠧⠀⠀⣿⣿⣿⠀⠠⡐⠂⣉⡥⠀⠚⠻⢿⣼⠄
⠀⣾⠥⠀⠒⠒⠒⠚⣻⡆⠀⢀⣤⡀⠲⠒⠂⠀⢀⠀⠀⠀⠀⣿⣿⣿⠀⡀⣭⣁⣩⡄⢀⣴⠁⣴⣿⣷⣦
⠀⢙⣃⣀⣠⠤⠄⠴⢿⡇⠐⠛⠛⠛⠂⢀⣤⡄⠀⠉⠀⠀⠀⠈⠉⠁⢄⢿⡧⢴⣿⣿⢇⢛⣆⢫⢉⡋⠉
⠀⠈⠉⣉⠅⢁⠀⢘⡆⡇⠀⠀⠄⠀⠀⠀⢹⡯⢆⡀⠀⠀⠀⡌⠐⡬⠶⠀⠩⠜⠛⡙⠎⢁⢁⡽⢹⣮⡆
⠀⠀⠘⠋⣠⡤⢖⣿⡇⡇⠀⠀⠀⠀⢈⡧⠀⣾⣿⣧⡇⠀⠀⠀⠀⢴⠟⠀⠀⠀⠀⢝⡡⠄⣼⠃⣿⣿⡇
⠀⠀⣤⠬⣭⣒⠋⠀⣰⡇⠀⠀⠀⠀⢨⡀⢤⣤⣶⠁⠀⣠⠀⣶⡄⠢⠤⣾⣖⠒⢖⡀⢀⠾⠃⢨⣿⣿⠋
⠀⢐⡟⠀⠀⠈⠉⣻⣿⠀⣠⣤⣠⠈⠀⢳⢿⣍⣉⠀⠙⢫⣵⣮⡅⠀⠀⠈⢿⣧⣼⠇⣼⣷⣦⣽⣿⠟
⠀⠈⠁⠀⠀⠀⠐⠾⠟⠀⠙⠉⠋⠀⠀⠈⠸⠿⠟⠀⠀⠲⠿⠿⠖⠀⠀⠀⠙⠛⠻⠠⣿⣿⣿⣿⠟


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠸⣷⡴⠶⠶⠟⠛⠛⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠶⠾⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣰⡟⢷⣄⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣠⡾⠋⠀⠀⠙⢷⣄⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣴⠟⠀⠀⠀⠀⠀⠀⠙⢷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢠⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠸⣿⠹⣧⡸⣷⠈⣿⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡛⢷⣄⠹⣧⠘⢿⣿⡇⠸⣧⣼⡟⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣾⣷⠀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣆⠙⠻⠿⠃⠈⠻⠃⠀⢿⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢿⡿⣦⣿⣿⣷⣿⣿⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⠛⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠰⣿⠷⣮⣷⡝⢿⠏⠻⠃⣿⣾⠋⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠙⠷⣮⡉⠁⠀⠀⠀⠀⠉⢁⡼⠃⠀⢀⣠⡀⢠⣶⣄⢀⣤⣄⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⡏⢿⣾⡇⣿⣼⡏⣿⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⠀⢀⣤⡀⢸⡿⠀⠀⣤⣬⣿⡌⠻⠇⠸⣿⠃⣿⣴⠶⣦⠀⣿⠀⠀⠀⢀⣀⠀⢻⣧⡀⠀⢀⡀
⠀⠀⠀⠀⢰⡟⣷⣸⡏⢹⣇⣿⢋⣿⢸⡇⠀⠀⢻⣮⡙⠋⠀⠀⠀⠀⠀⠙⠋⣴⠋⠀⣿⠀⠀⠀⣿⠹⣧⣼⣿⡇⢠⡿⣿⠀⢀⡀
⠀⠀⣾⡛⣿⣷⢸⣿⡇⢸⣿⡇⢸⡇⣾⡇⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⡼⠃⠀⢸⣿⣤⣄⡀⣿⡀⣿⣿⢸⡇⣾⠃⣿⣴⢿⡿
⠀⠀⠘⢷⡘⢿⡇⠛⠃⠀⠛⠁⣿⣴⡿⢷⡄⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣾⠁⠀⠀⢸⣿⣏⠉⢿⣿⡇⠙⠋⠸⣿⠏⠀⡿⢃⣾⠃
⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠈⢁⣴⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⠀⠀⠀⢸⣿⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢀⡴⠋⢸⡇⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢠⣿⠁⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡄⠀⠀⠙⠻⢶⡄⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢸⣿⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⢹⣧
⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⢿⡆
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠘⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠛⠂⠂⠄⡀
⠀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠁⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠔⠕⠀⠠⠊⠀⠀⠀⠀⠀⣴⡀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢀⣀⣀⣠⡤⠀⢀⠀⠀⠀⣀
⠈⠈⠀⠀⠀⠀⠀⠀⢀⠀⡴⠀⢓⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠘⠂⠁⠃⠄⠀⠈⠀⠡⡀
⡆⠀⠀⠀⠀⠀⡠⢀⠡⠚⠀⠀⠀⠁⡀⢦⠆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⢃⠀⣀⠀⠠⠁⠂⠀⠀⠀⠀⢠⠀⠀⠀⢹⢉⠀⣀⢀⠀⠀⢴⡃⠀⠀⡤⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠂
⠀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⣀⠓⢄⠇⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡘⠑
⠀⢽⢣⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⢀⢴⣡⠖⠁⢀⠂⠀⠀⠀⠀⠁⢀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠔
⠀⠀⠀⡗⠤⡀⠀⠀⠀⠀⠀⠄⡒⡟⢃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⡖⠂⣀⠲⣆
⠀⠀⠀⠃⠀⠁⠀⠀⢠⢧⣀⠘⠂⠇⠀⠓⠁⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠈⠂⠛⠊⠀⠁⠢⡀
⠀⠀⠀⠁⠀⡀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⢁⡀⡆⠀⠀⠀⠀⠀⠀⠧⡠⠀⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠠⢆⣀⣀⠀⠄⠀⠀⠘⠠⢤⠀⠀⠐⠢⠀⢈⡁⡗⠐⠲⡀⠠⢰⣇⣣⡤⠒⠠⢤⣀⡀
⠀⠀⠀⠀⠀⠉⠻⢦⠤⠠⠄⠀⡁⠀⠀⠤⠤⢖⣀⣓⣀⠀⠀⠀⢀⣁⡒⣒⡒⣒⣚⣋⣭⣭⣥⣤⣤⢤⣶⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠖⠶⠶⠶⠒⠒⠀⠉⠙⢌⠉⠉⠉⠀⠨⡁⠉⠉⠹⣔⠈⠉⠉⠀⢤⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢖⡀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣤⠀⠀⢠⣀⣐⡄⠀⠀⡤⠄⠸⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠠⠆⠶⡄⠀⢀⠠⠄⠀⡖⠀⡙⠀⠀⠉⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢄⠄⡆⠀⠀⡁⠨⠁⠀⠀⠀⠀⠀⠀⠂⣀⠠⠀⠈⠒⠐⠇⠐⠀⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠪⡂⠄⠀⠀⠈⠢⢴⠄⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢠⡾⢴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⣰⠟⠀⢾⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⡀⣤⣤⣤⣀⡄⠀⠀⠀⠀⡴⠋⠀⢀⠟⣣⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢄⣀⣠⣤⣬⠶⢿⣏⣹⣿⢲⣿⣾⠁⣷⠀⠀⢀⣼⣧⠟⡁⠀⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠐⠒⣲⠟⠁⠀⠀⠀⣠⣭⣭⣤⣿⢿⣮⣧⣿⣶⣶⢿⣏⣡⡞⠀⠀⢡⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢠⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠛⠷⣄⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠤⠤⢤⣀⣠⠾⠟⠉⠁⠀⠀⡤⢰⡇⢛⣟⠛⠓⠈⠋⠉⢙⡿⠉⠀⠀⢀⡀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠈⠈⠁⠀⠀⠀⠀⠸⣇⠀⠀⣠⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⡇⢈⡇⠀⠀⠀⠀⠀⠀⢀⣠⣿⠶⠟⠛⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢸⡇⣠⡄⠀⢀⣤⠆⠋⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣷⣘⣷⣴⣿⣧⠀⠀⠀⠄⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢰⠃⢸⡇⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠋⠁⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⣰⣿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⡟⣿⣧⣄⣀⣀⣃⣠⣤⣤⠴⠴⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠃⢿⣿⣄⣀⣠⣞⣀⣠⣤⣄⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣄⣠⣷⣶⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⠴⠁⠀⣀⣽⣭⣽⣿⣿⣿⡿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠛⠙⠓⠲⠶⠲⠷⠾⠿⠟⠛⠛⠁




⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣤⣄⢲⣦⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣠⡿⣠⣶⠆
⠶⣮⣿⣷⡿⠓⠤⠞⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠦⠚⠻⢻⣿⡷⠟
⠰⠶⢿⣏⡁⠀⠀⣠⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⡀⠀⣀⣠⠿⠾⠛
⠀⠀⠀⠀⠉⠉⠛⢇⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢀⡞⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠉⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠜⠙⢦⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠳⡄⠀⠀⠀⠀⠀⢸⣠⠞⠡⣤⠀⠀⠈⡷⠦⢤⣰⡃⠀⠀⠀⠀⠀⣰⠋⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠘⢦⠀⠀⠀⢰⡙⣇⠀⠠⠄⠀⠀⠀⠤⠀⠀⡟⣹⠀⠀⠀⢀⡜⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢳⣤⡀⠈⣣⣽⠀⠀⠀⠀⠀⣤⠀⠀⢠⡧⠃⠀⠀⣠⡎⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡴⠋⠀⠙⢶⠃⠈⣇⠀⠀⡶⠦⢴⠄⠀⡼⠹⡀⣠⠞⠁⠈⢢⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⣀⣟⠀⠀⠀⠀⠈⢣⡀⠈⠳⣄⠙⠛⢃⣠⠞⠀⣠⡿⠁⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣛⣷⡀⠀⠀⠀⠀⠙⠦⣀⡏⠙⠛⠉⢹⣤⠞⠁⡇⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠭⣥⠄⠙⣄⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⡼⠁⠀⠀⡇⠀⠀⢀⡞⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠀⠀⠀⢹⠀⣠⢯⡉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠇⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡀⠀⢴⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⣷⣮⣽⣛⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢦⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⡿⠿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠿⣿⣶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠿⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⡇⠀⠐⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠉⢦⡀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠢⡀⠀⠀⠀⢰⡟⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠈⠀⠘⠛⢫⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠀⠈⢘⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣄⣀⠀⠈⡏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢦⣀⡀⠸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠭⠥⢆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠃⢀⡜⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠰⠀⠀⣶⣶⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⠀⡄⠀⠀⠠⡇⠀⠈⠀⠀⢰⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣬⣇⠈⣿⣿⠿⣿⡆⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠁⠀⠀⠸⠁⢀⡀⠀⠀⠈⠻⠿⠿⣧⠀⠀⠀⠀⠀⠸⡇⢿⣿⣿⡿⠿⡟⠛⠛⠈⠀⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣂⣃⢀⣀⡀⠀⡻⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⢱⣼⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠿⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣄⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢲⣿⣿⣿⣷⣾⣿⣿⣿⡆⠸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⣿⣿⡷⠀⠿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⣬⠉⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⢀⣤⣤⣤⣴⣿⣶⣿⣧⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡆⠀⠀⠀⠀⢙⣛⣥⣾⠀⠀⠀⠀⠈⠿⠿⢷⣦⣄⡀⠀⢈⣀⡀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠠⠀⠀⠰⠚⠀⠀⢀⠀⣹⣿⣿⣿⣿⠀⠀⠀⠀⠠⠀⢀⠀⠉⢻⣿⣄⣿⡇⣇⢸⣿⣿⣿⣿⠉⣿⣿⣿⣿⠀⢸⡎⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣤⢀⣼⣤⣿⣿⣿⣿⣿⣤⣶⣶⣶⣶⣿⣿⣷⢀⣾⣿⣾⣿⡇⢸⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠃⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⠀⣿⣿⣿⣿⡿⠿⠛⠛⠋⢀⣀⣤⠤⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠋⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⣄⠈⠁⣶⣶⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠙⠓⠀⠉⠁⠈⠻⠿⠿⠛⠉⠉⠙⠛⠋⠹⠋⠉⠈⠋⠛⠙⠻⡿⠟⠋⠀⠀⠀⠀⢀⣠⡀⠀⠀⢤⡾⣿⣿⣿⣿⡿⣿⡟⠀⢀⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠓⢀⣤⣶⣮⢻⣳⡄⠀⢉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⣥⠀⢏⠾⠋⠀⠀⡀⢚⠕⠁
⠠⠶⠿⠿⢿⣿⣿⣿⣣⣾⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣽⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣭⣾⣿⠿⠊⠀⠀⢀⠤⡪⠚⠁
⢤⣾⡁⢠⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣼⣿⣿⠟⠁⠀⣀⠤⡺⠕⠉⠀⠀⠀⠀⠀⠀⣀⣴⣶
⢩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠉⠀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿
⡿⢟⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠂⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡟⢡
⣶⣿⣿⣿⣿⣿⣿⣿⣇⡿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠏⣴⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣟⣼⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣿⣿⣿⣿⠃⢠
⣿⣿⣿⣿⣿⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢀⣴⣿⣿⣿⣿⠃⢠⡟
⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⢀⣴⣿⣿⣿⣿⣿⡇⣠⡟⠁
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢹⣿⣿⣿⠟⠁⠀⠀⣰⣿⣿⣾⣿⣿⣿⣿⣿⠟⠁
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⡿⠋⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢹⡟⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣀⠀⠈⠛⠛⣿⠿⠃⠁
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉