Upper Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⠖⠀⠀⡀⠀⠀⠤⢤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⠗⠐⠚⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠀⠀⠀⠀⠀⠀⠀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢿⡀⠳⣄⠀⢀⠀⠀⣠⠏⠉
⠀⠀⠀⠀⠀⠀⢀⡔⠊⠉⠙⠒⡾⣧⠀⠈⠳⣌⢓⡶⡁
⠀⠀⠀⠀⠀⡰⠉⠈⠢⡀⠀⠐⢇⡘⡷⠲⠉⢸⠋⣿⠿⣦⢀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠱⠀⠀⠀⠙⢿⡀⠱⣸⢸⣿⠀⠈⢦⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡸⢯⣦⣹⣿⠸⡀⠀⠈⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠑⠽⠃⠹⡟⠀⠙⠓⠐⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢱⢇⠀⠀⠀⠀⢀⡠⠔⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⡇⠀⢀⣀⣀⠼⣈⡒⣤⡤⠖⡉⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡗⠦⣤⣀⠀⠀⠀⠀⠀⠀⢨⣷⠈⠀⠀⠠⣤⡤⠟⢩⢺⠫⢙⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠙⠲⠦⣄⡀⢠⣧⡧⠔⢚⡩⠞⣉⣤⣖⢻⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠁⠀⠉⠀⠉⠀⢙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠁⠀⠀⢀⠀⠄⠀⡀⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠂⠈⡀⠀⠈⡀⠠⠀⠀⠁⠀⠀⠁⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⢁⠀⠀⠀⠀⠄
⠄⠐⡈⠀⠐⠀⢁⠠⠀⢂⠀⠄⠂⠈⡄⠀⠈⡀⠠⠀⢀⠠⠀⠂⠀⠀⠀⠈⡀⠀⠐⡀⠠⠀⠀⠁
⠐⡀⠡⠐⠂⠡⠀⠂⠀⡀⠄⠐⡀⠠⠐⠀⢁⠀⠀⢁⠀⠄⠂⠈⠄⠐⠈⠀⠀⠀⢀⠠⠀⢂⠀⠄⠂⠈⡀
⢁⠠⠄⢂⠈⠄⠒⠈⡀⠠⠈⣀⠠⠀⢂⠡⠀⠂⠉⡀⠄⠘⡀⠠⠐⠀⠡⠐⠂⢁⠀⠄⠀⡀⠄⠐⠀⠀⠐
⠀⠆⠈⡀⠄⠐⡈⠡⠐⠀⢁⠠⠄⢃⠀⠄⠒⡈⠄⠐⠈⣀⠠⠀⢂⠠⠄⢂⠈⡄⠀⠈⡀⠠⠐⡀⠠⠀⠂
⠒⠈⡀⠰⠘⣀⠡⠀⢂⠡⠄⠂⠈⡀⠤⠘⡀⠠⠐⠀⢡⠀⠂⢁⡀⠄⠐⡈⠄⠐⡈⠁⠰⠀⢁⠠⠄⢂⠈
⡈⢡⠰⠀⢁⠠⠄⢃⡈⠄⠒⡈⡄⠰⠈⣁⠠⠀⢂⠡⠄⠂⠈⡄⠀⠘⡀⠠⠐⣀⠡⠀⠂⢁⠀⠆⠘⡀⠄
⠡⠐⢂⢩⡄⠆⠘⡀⠤⠘⡀⠡⠐⠂⢡⠀⠆⢃⡈⠄⠒⡈⠄⠐⠈⢁⠰⠀⢃⠠⠄⢂⠈⡄⠒⠈⡀⠰⠐
⠄⢒⡈⡄⠒⠈⣅⠰⠘⣃⠡⠄⢂⠩⡄⠂⠈⡄⠤⠘⣀⠡⠐⢂⢡⠀⠂⢉⡀⠆⠘⡈⠤⠐⡈⢡⠰⠂⢁
⠘⡈⠤⠐⡈⢡⠰⠂⢣⠠⠆⢓⡈⠄⠒⡈⣥⠰⠈⣁⠠⠄⢃⠩⠄⢒⠈⡄⠲⠘⣀⠠⠐⣂⠡⠀⢂⢉⡄
⣁⠱⠐⣃⠡⠄⢂⢩⡄⠆⠘⡀⠤⠘⣀⢡⠐⢂⢡⠀⠆⢋⡈⠤⠘⡈⠥⠐⡀⢡⠰⠆⢃⠨⠄⢒⡈⡄⠒
⠰⠆⢃⡨⠆⢓⡈⡤⠲⡈⣥⠰⠈⣃⠡⠄⢃⠩⡄⠒⠈⡄⠴⠘⣁⠡⠐⢂⢩⠄⠆⢉⡀⠦⠘⡈⠥⠐⡂
⢖⢉⡄⠶⠘⣈⠥⠘⣂⢡⠐⢆⢩⡀⠆⢛⡈⠤⠒⡈⣥⠰⠊⢣⠰⠆⢃⡨⡄⢒⡈⡄⠲⠈⣁⠰⠐⣃⠡
⡈⡤⠲⡘⣥⠱⠔⣃⠩⠄⢓⢩⡄⠶⠘⣄⠵⠘⣃⢡⠔⢂⢩⡄⠆⢛⡌⠦⠘⡈⢥⠐⣊⢡⠰⠆⢋⡈⠆
⢵⠐⣊⢡⠰⠆⢫⡨⠆⢛⡈⡥⠲⡈⣥⠰⠎⢃⠩⠄⢓⡩⡄⢒⡈⣅⠰⠘⣃⠡⠔⢃⢩⡄⢆⢉⡄⠶⠘
⠜⢃⢩⡄⢖⢙⡄⠶⠘⣈⠵⠘⣃⢡⠔⢆⢩⡄⠆⢛⡈⠤⠘⣊⣥⠰⡂⢣⡰⠆⢛⡈⡄⢒⡈⣥⠰⡈⣣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣤⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⠛⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡯⠔⠐⠀⠀⠀⠀⠀⠀⠠⠀⠤⡈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣵⠋⠉⣁⡀⠀⠀⠀⠀⠀⣀⡉⠉⢳⡤⢽⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⣿⢿⠀⠀⣯⡿⠀⠀⠀⠀⠀⣯⡷⠀⠠⠇⢸⠇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⣿⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⢀⡾⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠳⣄⡀⠀⠀⠈⠉⠁⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠉⢛⢶⠶⠒⢶⢶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⢠⣴⣯⢘⡦⠤⣼⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡈⢿⣳⢠⢏⡏⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣼⣿⣿⣿⣿⣏⠉⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠰⣿⣿⣿⣿⣿⠃⠀⢸⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄⡀⠀⣀⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣠⠊⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢱⠤⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠘⣄⡠⠄⠒⠂⡀⠉⠉⠀⠒⠒⠤⣌⡀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⠔⠊⠙⡄⠀⠀⡀⡠⢢⠀⡀⠀⠀⡘⢀⠈⢑⠢⡀⠀⠀⠉⠑⠢⡀
⠀⠀⠀⠀⢀⡴⠊⠉⠀⣠⠊⠁⢠⡄⠀⠘⡄⠈⠁⠇⠀⡄⠀⠀⢠⠁⠀⠀⣈⡁⠈⠳⡀⠀⠀⠀⢡
⠀⠀⠀⠀⡎⠀⠀⠀⡔⠁⠙⢠⠤⢄⡀⠀⠘⡀⠀⢸⠀⠀⠀⠀⠆⠈⢠⠊⢠⠃⠀⠀⠈⢆⠀⡀⠘⢆
⠀⠀⠀⠀⡧⠄⣀⡜⠀⠛⠀⠀⠑⢄⡈⠢⡀⠐⡀⠀⢚⡀⠀⡈⠀⡴⢁⠔⠁⠀⠀⡀⠤⠚⢫⠀⠀⠀⠇
⠀⠀⠀⢀⡵⠀⡸⠉⠑⠂⠠⢀⡀⠀⠈⠑⠚⠀⠐⡴⠁⡇⠰⢀⡤⡏⠁⡀⠄⠂⠁⠀⠀⣤⠈⡆⠀⠀⡆
⠀⠀⢀⠊⠀⠀⡇⢀⠀⡀⠀⠀⠀⠉⠐⠂⠤⡒⠴⠁⠀⠘⠋⠁⢠⠋⠁⢀⣀⣀⡀⠈⣀⢀⠀⢰⠀⠀⠱⡀
⠀⠀⠸⡀⠀⠀⠁⠈⠀⣖⠂⠀⠒⢒⣒⠄⠀⠱⡀⠀⠀⠀⠀⠀⠣⢤⠀⠓⠂⠤⠤⠤⠴⠃⠀⢸⠀⠀⠀⠇
⠀⠀⠀⡸⠀⠀⡆⠀⠀⠰⠉⠉⠉⠀⢀⠠⠐⢈⠩⢳⣀⠀⠠⠄⠒⠉⠒⠀⠄⣀⠀⠀⠀⠀⠀⢸⠀⠀⠎
⠀⠀⠀⡇⠀⠀⢃⠀⠂⠀⣀⠠⠐⠈⠀⣨⠿⡅⠀⡌⠀⠉⠑⡃⠀⡴⢄⡀⠀⠀⠈⠁⠒⠠⢄⡌⠀⠀⡘
⠀⠀⠀⠱⡀⠀⣈⡦⠒⠉⠀⠀⠀⡠⠊⢡⠎⠀⠰⠀⢠⠲⡀⠘⡀⠈⢦⡈⠢⡀⠀⠈⠀⠀⡰⠁⠉⡞⠁
⠀⠀⠀⠀⢨⠃⠀⠈⢆⠀⠀⠛⠰⣁⠔⠁⠀⢀⠃⠀⢸⠀⡇⠀⠐⡄⠀⠙⠢⠼⠀⠰⠀⡰⠁⠀⠀⡇
⠀⠀⠀⠀⠸⡀⠀⠀⠈⠣⡀⠁⢀⡄⠀⠆⠀⠌⠀⠀⠸⠀⡇⠀⠀⠐⡀⠁⠐⠀⠀⣠⠞⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠈⠒⠒⡆⠀⠈⠲⢄⠀⠀⠀⡰⠀⠀⢠⠘⠚⠀⠠⠀⠀⠐⡀⢀⡠⠚⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⡀⠉⠒⢤⢅⣀⠀⠀⠀⠘⠃⢀⣀⡠⠄⠚⡉⡰⠐⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⢄⠀⠆⠀⢀⡀⠉⠉⠉⠉⣡⣀⠀⢀⣀⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠈⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⡐⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠋⠘⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠙⠿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠀⠀⠀⠀⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠛⠛⠃
⠀⠀⢀⣠⡄⠀⣠⣄⠀⠀⠀⠀⢠⣤⠀⠀⠀⢀⣤⣀⢀⣠⡀
⠀⠀⣿⣿⣿⣾⣿⣿⣷⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡿⠀⠀⠀⢸⡿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢠⡾⠛⢳⡄⠀⠀⠀⠀⠚⠉⠓⠀⠀⠀⠀⠿⠿⠟
⠀⠀⠀⠘⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢠⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⢸⣿⠀⠀⠀⢸⣿⠁⠀⠀⢰⣿⡄⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⠸⣿⠀⠀⠀⢸⣿⡇⠀⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⣼⠁⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⣴⡟⠀⢰⠈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣀⣤⠔⠀⠁⠠⠔⣛⠁⠀⠈⠷⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠘⣿⣔⠀⠀⠀⠀⢻⣿⠀⠀⠀⢠⠘⣧⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣊⣁⠀⠤⣀⠙⠿⣾⡖⠂⣻⣿⡄⠀⠀⢈⣳⡹⣷⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣷⣦⡈⢷⡀⠙⠻⣿⣏⣝⣿⡄⢠⣠⠀⠑⣦⠀⣰⠖⢮⡥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⢿⣿⣿⠿⠟⠻⢄⣶⣿⠇⣼⣿⠄⠀⢿⣇⠀⠈⠳⣇⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠙⣆⠀⠀⠈⣿⠁⢀⣾⠿⠀⡈⠀⠘⢷⣀⢳⢷⡀⠘⢃⣞⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡜⣦⢀⣸⣿⣷⣿⠇⢙⠀⠁⠀⠀⠀⠙⠻⢯⡓⢐⣩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣬⣿⡿⠈⢃⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⢀⢿⡗⠀⠀⠀⣀⡤⢄⡉⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠁⠀⠀⠀⣺⣿⡗⠀⠀⠀⡠⢔⣉⡀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⡐⢿⠛⠀⠀⠀⢠⠶⢿⣤⣀⠑⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢀⣾⣿⡇⠀⠀⠐⠀⠀⠔⠉⠡⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⡏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣹⣿⣿⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⢰⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠁⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠃⡠⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠠⠀⡜⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⣂⣀⠉⡉⠁⣆⣉⣀⠀⡀⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣀⠄⡆⣿⠀⠀⠀⠀⣿⠁⠀⠀⠀⡘⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢠⣤⣿⡛⣿⡇⢻⣆⡀⠀⣸⣿⡄⠀⠀⣰⡇⠀⠀⠀⠸⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣮⣿⣿⣿⣮⣵⣿⣿⣷⣾⣿⣿⣿⣷⣾⠟⣷⣀⣠⡎⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⡔⠉⣡⣌⠻⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠈⠉⠙⠁⠀⠀⠀⣼⠇⣀
⠀⠀⠀⠀⠀⠀⡀⣾⣿⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣷⡄⠀⠀⠀⠠⡴⠋⣀⠀⠈⣧
⠀⠀⠀⠀⠀⠂⣭⣿⡓⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣯⡠⠀⠀⠀⡀⠀⢘⣷⣀⡽⢄⡀
⠀⠀⠀⠀⠀⢸⢯⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⢿⣿⣿⣦⣿⣿⡆⠀⠀⠈⠓⠟⠋⢵⠀⠀⣿⡄
⠀⠀⢀⠐⠈⣛⠛⠦⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⣻⣿⣿⣿⣿⣿⡶⠆⠀⠀⠘⠀⠀⢸⣂⣠⣿⣁
⠀⠀⢀⠀⣤⣭⠇⠀⢈⢿⣿⣿⣿⣿⣿⣛⢿⡇⡿⢋⡉⢿⠈⡉⢋⠉⠏⢹⠋⢱⢺⡁⠀⠀⢑⠖⠋⠉⠀⠀⠘⣦
⠀⠀⠀⠀⣄⣡⡀⠀⠀⣼⣿⣿⣿⣿⡟⣿⣿⡇⣿⢛⡁⢸⠀⠃⢸⠀⠁⠸⠀⢸⣳⣧⠀⠀⠀⠀⠐⠾⡤⠀⠀⢻⡀
⠀⢀⠼⡤⠿⣏⠀⢀⣴⣿⣿⣿⣿⣿⣦⣿⣽⣇⣧⣈⣁⣸⠀⠀⢸⣀⣅⣀⠀⠈⣼⣿⡄⠀⠘⠀⠀⢶⣆⠀⢀⡼⠁
⢰⠁⢈⣇⣴⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣏⡉⠀⢀⠀⢻⣟⣧⠀⠀⠑⢤⠼⣛⠊⠉⠉⢢
⠘⠀⠒⢿⠏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣛⣻⣶⣾⣿⠟⢿⣿⠀⠀⢀⠀⠀⠿⣥⡄⠀⢸
⠀⡣⢼⣿⠒⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢾⣿⣦⣽⣿⣿⣿⠄⡈⢧⢱⠀⠈⠆⠀⠲⣦⡀⠀⡸
⡌⠀⠀⣄⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣾⣿⣿⣿⣿⣷⡤⣾⢷⡸⠆⠂⠀⢐⡤⠴⠬⠄⠚⠣
⣧⠀⠙⡏⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣾⡶⣿⠍⣭⣇⠀⠀⠈⠀⢠⣾⣆⡀
⠙⠤⣬⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢬⣤⣄⠀⠘⢿⡇⠀⠱⡀⠘⡟⠀⠀⡰
⠀⠀⠙⢷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠙⢿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠈⠀⠛⠀⠀⠀⢀⣈⣿⠿⠏⠉
⠀⠀⠀⠀⠈⠉⠉⠁⡀⢉⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢀⣀⣄⠉⠻⢧⣀
⠀⠀⠀⠀⠀⠀⢰⣠⠔⠛⠛⠏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠠⢭⠏⠛⠦⠀⠙
⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡷⠛⣹⠿⣿⣿⣿⡟⠈⣿⢿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⣠⡞⣡⡾⢋⡿⠋⠀⠀⠹⣿⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢣⣶⣾⣿⢿⠟⢠⡟⠄⠀⠀⠀⠀⠘⢷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠡⠇⠀⡟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠋⣴⣶⣶⡄⠀⠀⠀⠀⠀⣾⣷⡤⠄⠀⢸⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣩⣿⣷⠀⠀⠙⠿⠁⠀⠀⠀⠀⠀⢻⣿⠇⠀⠀⠈⢏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠃⣴⣿⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⣽⢋⠀⢠⣾⡿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠀⢚⠟⢱⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏⠀⠀⠋⢀⣰⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⢀⣠⠼⢾⣿⣿⣿⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡀⠀⠃⠀⠈⢿⣿⣼⣿⣿⣿⣿⣿⣿⣿⠙⢿⠏⠁⠀⠀⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠾⠃⣿⣿⣿⣿⣿⣿⠋⠹⠀⠀⠀⠀⠀⣠⡿⢏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠙⠿⠛⠉⢀⠉⠀⣁⡀⠀⠀⠀⡼⠋⠀⠈⢣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣠⡾⠉⢁⡄⠀⢰⣷⡄⠀⠀⠈⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡄⠀⢠⡶⠋⠀⠀⠈⠇⣠⠀⠀⢹⣦⡀⢰⡀⠀⢨⣗⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡀⠀⠀⠰⠶⡄⠀⠰⠁⠀⢰⠄⠈⠻⡧⠀⣠⣀⣿⠐⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠠⠃⠀⠀⠈⠀⣈⡷⠀⠀⠀⠀⢿⣩⡍⠀⣤⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⡆⠐⠂⠀⠀⠀⢹⡇⠀⣿⣿⣿⣿⣇⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡆⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⢼⡇⠀⣠⡄⠀⣴⡟⠳⣶⣿⣿⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠀⠀⣿⠀⠀⠰⠀⡀⠀⠸⡈⢳⠶⠁⣰⣾⣏⣹⣿⣿⣿⣿⣧⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠛⠀⠀⠀⠀⣠⠊⠀⠘⣴⠃⠀⠀⠉⠀⣴⠿⣄⣠⣿⢯⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠉⠀⢠⡏⠀⡀⠀⠉⣦⡄⠁⢳⡄⠈⠀⢀⡼⢣⣤⣿⢿⣷⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⠀⢈⣧⠀⠀⠀⣦⠈⢻⣦⡌⢻⡀⠀⣼⣷⣾⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠰⢸⣯⠀⣶⡈⢰⣶⡞⢉⣿⣾⣷⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⢇⣸⣿⡇⣠⣿⢿⣌⣻⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⠀⠙⣿⣿⣿⡟⠛⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣤⡜⠻⣿⣿⠿⣿⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣿⡇⠀⠀⠙⢿⣿⠇⠀⠉⠉⠉⢨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠛⣷⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣾⣤⡤⠀⠀⣠⡤⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠀⢀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣂⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠉⠻⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⠿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠁⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⢤⣄⣀⠀⠈⠊⠀⣀⣀⣤⡒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢱⠈⠙⠤⠤⠊⠁⡎⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠀⡞⠀⠀⠀⠀⠀⠀⢳⠀⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠁⠀⠀⠀⠀⠀⠀⠈⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡠⠤⠰⣿⣿⡮⡉⠛⠛⠻⠓⠛⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⠋⠀⠀⠀⢿⣽⡷⣝⢬⡠⢄⣐⣀⣤⢤⣈⠻⣶⣤⣄⣀⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡼⠁⠙⠁⠀⢀⡴⠈⠙⣿⣿⣬⣟⡷⡶⢦⣤⠤⢽⣿⣦⡀⡏⣉⣉⣻⣿⠊⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠳⡄⢀⠞⠉⠉⠁⠀⠀⠈⠛⠉⠙⠻⠿⠿⠃⠀⠀⠀⠱⡼⢛⣿⣿⣿⣉⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠋⠀⠀⠀⢀⡜⢀⣀⢀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⡽⠀⠉⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣄⢀⡀⠀⣠⠖⠋⠈⣃⠹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠰⠶⢤⣄⠀⠀⠀⠉⠋⠉⠑⣤⣤⣀⣭⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⠛⢁⠀⠀⠀⠸⣆⠹⣿⣿⣿⡷⠀⠀⢀⠄⠀⠀⠀⠀⠘⢳⢄⣀⡀⠀⠀⡸⠋⠙⠋⠙⠟⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠿⠟⣛⣁⠤⠴⠂⠁⠀⠀⠀⠀⠈⠓⣄⣈⡥⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠂⠀⠐⠠⡉⠿⣷⣦⡀
⠀⠀⠀⣀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⢸⠀⠈⣙⣿⣦⣀
⠀⠀⠀⢻⡍⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⡿⢸⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠈⢧⣄⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣈⡉⢁⠎⠠⢄⡀⠘⠻⢏
⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⣤⣤⣄⠀⠀⠀⢀⡀⠀⠀⢹⡄⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠉⢧⠀⠎
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢧⣤⣤⣴⣿⡇⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠉⣁⠘⠻⣿⣿⣿⣿⣿⡿⠃⢀⡞⠀⠀⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠁⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠘⢿⣿⣿⡿⢃⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢻⣿⡿⢋⣁⡤⠔⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⢦⡀⠀⠀⠀⢠⠎⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣌⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣶⣶⣿⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠢⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⢻⣿⠁⠀⠀⠁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⡈⢿⣧⡀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠔⠀⠘⠶⣤⣀⡀⠤⠀⠠⣀⠀⠀⣠⡴⠿⠻⠷⠦⣄⡀⠀⠀⠀⠿⣿⣿⡿⣡⠿⠶⠞⠳⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠂⠤⠬⠘⠁





⣶⣶⣶⣾⣶⡷⢿⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣾⡷⢿⠶⠾⠿⠶⢷⣶⠿⠶⠶⠿⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢰⠀⠀⠀⠀⣠⠂⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⢸⠃⣀⣀⢸⣿⣧⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⠘⠻⣿⡇⠻⠟⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣻⠷⣠⠥⣤⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣳⣤⠴⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣄⢀⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡤⢞⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢀⣴⣶⣮⣯⡹⠋⠉⠋⣽⡷⣶⣶⣄⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠙⢿⣿⡿⣯⣿⢿⠧⠦⣄⠀⢀⠒⣿⣟⣹⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⡐⠞⡿⢷⣿⣛⣿⡦⠤⠐⠀⢈⡙⢛⣿⣋⣽⣿⠛⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠓⢮⣽⡛⠿⠶⣒⡚⠀⠀⣞⢛⠛⣛⣽⡅⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣮⣹⢿⡿⠋⠉⡉⠀⠀⠉⠛⢿⡟⢛⣡⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷⠄⠿⡶⠃⠉⠀⠁⠀⠀⠀⠘⠛⠛⠦⣩⠄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠛⣄⣠⡿⠞⠁⠀⠚⢷⡸⠃⠀⠈⠻⢆⡈⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⡏⣁⠀⠀⠠⠚⠙⢛⡇⢘⡛⠓⠦⢄⣀⡹⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⢹⢏⣀⠴⠖⠋⢉⣉⡅⢈⣍⠛⠲⢤⡽⠷⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠃⠀⠀⠀⠞⢁⡤⠖⠊⠁⠈⡗⠈⠉⠙⠢⢄⠙⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⢼⠀⠀⠀⠈⠈⠀⠀⠀⠀⠸⡇⠘⠇⠀⠀⠀⠀⠀⠀⢠⣇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠇⠙⠇⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠙⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠃⠀⠀⠀⠀⢀⣤⡴⠤⢤⡀⠀⠀⢿⠉⠰⠒⢀⣤⣤⣤⣀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣮⣅⠀⠀⢠⠿⠶⢿⡿⠒⢰⣿⢏⠉⢻⣿⣿⡇⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡙⣆⠀⠀⣤⣾⣿⣟⠐⣾⣟⣃⠀⣸⣿⢋⠄⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣆⠀⠍⠉⢹⣯⢥⡟⠉⣻⣿⠿⠁⠁⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⢀⣠⣔⣿⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣟⠛⢷⣌⡁⠰⠷⠞⠀⣋⣥⠊⢈⣤⣀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⢠⠀⠀⠖⠾⣫⡙⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠈⠉⠋⠻⠧⣤⡽⠷⠿⣧⣄⡿⠛⢋⡀⠘⡀⠀⠀⠀⠀⠀⣤⣿⣀
⠀⠠⠀⠀⠊⢰⣁⣺⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⠁⠀⠀⠀⠀⠀⠟⡛⣿⣧
⠀⢘⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⢻⣈⡇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠎⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀

⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠿⣿⠁⣶⡎⢻⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣄⡀⣾⠉⢿⡄⠲⢤⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⣺⣕⡦⠴⠛⠻⣿⣶⣦⡝⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠮⢽⠯⢟⣫⣤⣔⠊⠙⠛⢦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣽⡞⠀⢾⣷⠄⣸⠿⣟⡠⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⢩⣿⡥⠌⣿⣇⣀⡤⣿⡻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣖⣠⣤⣖⣮⣟⣳⣽⣳⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⢈⡹⠿⡛⠯⢟⠮⠿⡷⠽
⣿⣿⣿⣿⣿⣿⣿⡿⢁⣤⣾⣿⠞⢠⣤⠤⠘⠳⣾⣧⣄⡹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⠉⣭⡿⢛⣻⣿⣍⠉⠙⠻⣧
⣿⣿⣿⣿⣿⣿⣿⠃⠛⢿⣿⣿⣄⠐⠿⠟⠀⣠⣿⣿⠟⠃⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣥⡀⠀⠈⢿⣯⣕⣤⣿⠟⠀⠀⣠⣼⡆
⣿⣿⣿⣿⣿⣿⣿⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢚⣿⣟⣛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣋⣭⠤⢶⡺⠉⢠⣤⠈⠳⣌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣛⡩⠔⠚⠉⠀⠀⠀⣈⣇⠀⠈⣽⡄⠀⢿⠜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⠕⠊⠁⣀⠀⠄⠀⠀⢀⠀⠈⠘⣧⣦⣄⡐⠼⠋⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡥⠔⠉⠀⣀⠀⠀⠉⠀⠢⡄⠁⠰⠀⠄⠀⠀⠉⠁⠀⠀⠀⢃⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⢉⠰⠂⠁⠀⠀⠀⠀⢠⠀⠰⣠⠀⠎⠸⡀⠊⡀⠈⠀⠀⠀⠀⠀⠀⠀⣘⣧⠘⣧
⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⠶⠖⠛⡩⠄⠒⠈⠀⠀⠠⠐⠀⢀⠀⠀⠀⡇⠂⠀⢣⠐⡀⢣⠈⠘⡄⢀⠀⠀⠀⠀⠀⣠⠽⢿⡄⠹⣆
⠀⠀⢸⡇⢡⡥⠼⠒⠶⠦⣄⡈⠁⠀⢀⠀⠀⠀⢀⠀⢀⡁⠘⡘⡆⠄⠹⠘⡀⠈⣇⢡⠘⠶⠃⢉⡡⠀⠀⠀⠀⠀⡇⢸⣦⠹⡆⢹⡄
⠀⠀⠀⣷⠀⡇⠀⠲⣶⡆⠘⡇⠠⠀⠠⠀⠀⡄⠰⡀⣋⡀⠀⡇⢡⡌⠄⢃⣉⠕⠘⠋⢠⣀⣠⣌⠀⣀⣀⠤⠔⢒⣙⣲⣴⣞⣁⣈⣧⡀
⠀⠀⠀⠘⣇⠙⢄⠀⠼⠧⣠⣷⠂⠀⠘⡄⠠⠃⠐⣷⠀⠄⢀⡇⠀⠘⠶⠀⠀⠤⣀⠤⠛⢉⡠⢈⣈⣥⣴⠶⠿⣻⣛⣛⠛⠛⣯⠉⠁
⠀⠀⠀⢀⣘⣦⠀⠙⠒⠚⠟⠉⠀⠀⠀⢳⠀⠶⡀⢹⣇⢉⡠⠷⠊⠀⢀⡠⠔⠉⣀⠤⣊⣥⠶⠿⠿⠁⢲⣲⠟⢉⣭⡉⠳⡄⢹
⠀⠀⣠⢺⡇⢹⡎⠠⢱⠄⠀⠀⠀⠀⠀⠀⣇⡀⠱⠖⠁⠀⠀⢀⡠⠒⠁⡀⢔⣪⣷⡟⠫⠀⠀⠀⠀⠀⣈⡁⠀⢘⣻⡀⠀⣿⢸⡀
⢀⡔⠻⢸⡇⢨⢿⡀⠀⢾⡄⣀⣀⡀⠈⠀⠈⠠⠀⠀⢰⠦⠂⠁⠠⢐⣯⡶⠛⡡⠀⠐⠂⡤⠀⠀⠖⠄⠐⢷⣄⡀⡉⣄⡴⠋⠘⡇
⣾⢐⠀⠈⣧⠸⣌⣧⠀⡼⠉⠠⣤⠙⣦⠀⠀⠀⠀⣀⠓⢀⣰⣴⢾⠋⠀⢃⠰⠐⢲⠐⠊⢱⠀⣰⠃⠁⠀⠈⠙⠉⠉⠁⠀⢇⠀⣧
⢻⠘⡇⠀⢹⡀⠈⠛⣶⠀⠳⣄⠙⢂⡾⢠⠀⣂⣭⣶⠞⠋⠁⣼⠘⠂⡄⢸⠀⠀⢸⡆⡀⢸⠀⡴⡄⢰⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿
⢸⠀⡇⠀⠘⣇⠀⢘⡜⠶⠶⠶⠿⠿⠖⡟⠛⣏⠈⣷⠘⣁⣀⡇⠀⠃⠐⢸⠀⠂⡀⡇⠃⠸⠆⢃⣃⠈⠀⢀⣀⠀⠀⠀⢒⣿⡇⢺⡇
⢸⠀⢹⠀⠀⣿⠀⠀⣶⠀⠀⠀⠀⠆⠀⡧⡤⢸⢠⠟⠠⠤⠼⠧⠆⠈⠓⠈⠉⠉⠁⠙⢉⡴⠋⠁⠀⠀⠀⠀⠀⠉⠒⢎⣭⠙⢧⠘⡇
⢸⠀⢸⠀⡀⣿⡀⢀⡝⢉⣉⠑⣄⠀⠀⠀⠀⠀⠀⣄⣠⣀⣄⠀⡀⠠⠤⠤⠤⠤⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡬⠭⠥⣝⣰⠷
⢸⡀⠀⡀⠁⢸⣇⠈⣇⠀⢿⠀⣿⠀⠀⠀⢀⣀⠫⠉⠁⠀⠀⠀⠤⠤⠤⠤⢤⡄⢰⠁⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠸⡇⢣⠃⠸⠀⣷⣀⣈⣱⣰⣾⣥⣯⣤⠤⠴⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠉⠃⡆⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣧⠘⠈⠀⡶⠭⠭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠸⡄⣣⠿⢥⣄⡤⠤⠤⠶⠖⠒⠂⠀⠉⠉⠉⠉⠀⢀⣒⣒⣒⣀⣀⣀⣈⣥⠥⠽⣦⠀⠀⠀⢸⠁⠀⠀⠇⡄⠀⠀⠴⠀⠀⠘⠂⠘⠀⠄⠀⠸
⠀⠀⠱⠏⠠⠤⠤⠴⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⢸⠀⠄⠘⠀⡇⠀⣐⠀⠀⡀⠀⢰⠀⠀⡂⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡆⠀⠁⠈⠈⠁⠙⠀⠉⠈⠉⠈⠀⠁⠙⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣶⣶⣯⣥⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⣀⠀⠀⣀⣀⣀⣀⡤⣤⣠⢤⣄⣀⣀⣠⣤⣤⣤⠀⡔⡖⣿⣱⣽⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡼⠿⠘⠁⠀⠓⠈⠐⢛⠛⣫⠏⡇⢟⡕⠙⠟⠛⠉⠉⠉⠉⠉⠁⠉⠩⠤⠀⠉⣼⣯⣿⠅
⠀⠀⠀⠀⠀⠀⢀⢔⡕⠁⠀⢀⢤⡠⡀⠀⠀⡰⣡⣾⡟⡼⢱⢿⡇⠀⡆⠀⠀⢠⣠⣤⣤⡄⠀⠀⠀⠀⡇⢩⣏⣿⡄
⠀⠀⠀⠀⠀⡠⡣⠋⠀⠂⠀⠀⠂⠑⠁⠀⠀⣿⣿⣧⢊⡆⣿⣿⡇⢠⠃⠀⠀⠊⠛⠛⠛⠀⠀⢀⠀⡜⡀⣾⣿⣷⠃
⠀⠀⠀⠀⡔⣵⣅⣤⣤⢠⢂⡀⡴⠴⢂⣀⣀⣰⡿⢃⢪⡇⣿⣫⣄⣁⣀⣀⠃⣀⡀⢃⣐⣛⣫⠭⡜⣸⡞⣼⣿⠃
⠀⠀⠀⢬⣬⣭⣿⠿⣶⠶⣶⣮⣭⣥⣭⣍⣙⣻⣣⣇⣬⣇⣿⣭⣿⣿⣿⣿⣿⣿⣿⣯⡭⠼⠶⠶⡾⣻⣿⣿⠁
⠀⠀⠀⢸⢻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣗⣒⣶⠿⣿⣄⣀⣈⣭⣙⣶⣖⣶⣿⣿⠇
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⢯⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⠀⠠⠂⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠻⢿⡿⠛⠀⠀⠂⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠡⠉⠀⠰⢦⡰⠃⡀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢀⡀⠀⢁⠀⠴⡂⢰⠃⡀⠀⠔⠀⠄⠀⠀⠰⠀⠀⠠⡄⡂⠀⠠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⠀⠀⠈⠁⠀⠙⠢⡑⠊⣿⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠈⢳⣿⠀⠀⠀⢠⠄⠀⣀⠄⡀⠀⠷⠟⠀⠀⢂⡀⠀⠘⠑⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠂⠀⣴⡆⠀⠀⠈⠀⠀⠀⠘⠈⢿⡄⠀⠄⢹⣤⢚⣁⡀⢁⠀⠢⢰⡀⠀⢢⣰⣀⠀⠈⠉⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠀⠋⠉⠰⢉⠳⠀⠱⠦⢄⣀⣈⣧⠠⢁⡾⠁⠀⠙⠁⠜⢁⠀⠀⢀⣤⡔⠊⠁⣀⡀⠈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⣄⠀⠀⠀⠄⠀⠀⠸⣆⣾⠀⠀⠀⠰⠀⢠⠞⢻⣴⡏⠀⠀⠀⢸⠭⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⢢⠰⣄⣀⠇⠀⠀⠀⠀⡀⠀⠹⣇⠀⡀⠀⠙⠁⢨⠨⠸⢾⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠑⠷⠤⠴⠶⢶⡶⠶⠿⣶⣤⣻⣦⠁⠐⠀⠀⠩⢛⠓⢾⡇⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠔⠀⢀⠞⠁⢄⠀⠀⠀⡬⠙⢿⣷⡀⠀⠀⣐⡙⠁⢈⡇⢠⠀⠵⢀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⡀⠰⣠⠇⠈⢉⡠⠃⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠕⠀⠄⢀⠂⠀⢁⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⣧⢋⢁⠁⢠⠿⢶⠶⠛⠁⠑⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠐⠃⢀⡀⠒⡀⠤⢢⣀⣀⣠⣄⣀⡛⢸⣿⣆⣡⡾⠃⢀⠄⠠⣎⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠤⢐⠚⠉⠉⠉⠀⠈⠙⠻⢿⣿⡿⠋⠀⠐⠊⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣠⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠏⠑⢀⠰⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢀⣄⡰⠾⠂⠈⢠⠰⡴⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠁⠀⣤⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠶⠶⠿⠿⠿⠿⠶⠶⠒⠒⠂⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣿⣦⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⢀⣤⣄⣀⣀⡴⢦⣀⣠⣿⡿⠛⠉⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⣠⠞⠁⠈⠉⢁⣠⣶⡿⠟⠉⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⡾⠁⠀⠀⣠⣴⣿⠟⠉⢀⣠⠶⣤⣀⣀⣠⡄⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⡴⠋⠀⣀⣴⣿⠟⠋⢀⣠⠶⠋⠁⠀⠀⠉⠉⠁⠹⣦⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⣀⣴⣾⠿⠋⠀⢠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣤⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢀⣤⣾⡿⠛⠁⣀⠀⠀⢻⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⣠⣶⡿⠛⠁⢀⣴⠞⠉⠀⠀⣼⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡿⠟⠉⢀⣤⠞⠋⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⢠⣾⣿⣿⠟⠉⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢀⣠⠶⠋⠁⣠⣴⡿⠟⠉
⠀⠀⠀⣰⣿⠋⢸⣿⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⣤⣾⡟⠀⠀⠀⠛⠁⣠⣴⣿⠟⠋
⠀⠀⠀⢿⣧⡀⠀⠹⣿⣄⠀⠠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠈⠻⢿⣦⣄⠘⢿⣧⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠀⢀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣌⠻⣷⣄⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠙⠛⠉⠙⠳⣦⠞⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣆⣀⣠⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⢀⣀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠏⣠⡗⠻⣍⡱⠀⡀⠀⠀⠀⢠⣧⣆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠
⠀⠀⠀⠀⠘⢈⡶⣦⣾⣗⠟⢉⡝⠀⠀⠎⢻⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠌⠀⢀⣿
⠀⠀⠀⠀⠀⠹⢆⣽⠉⠀⢦⣠⠀⠀⠀⠀⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠄⠁⢀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠈⠻⢶⠀⠈⠻⡄⠠⠁⠆⠀⠀⢸⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠊⠀⢸⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠚⣧⠀⢀⣷⣠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⣼⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠉⡇⠙⠻⠜⠉⠻⡄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⠀⣀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⣀⠀⢀⠀⣀⡀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠄⠀⠀⠀⣠⠲⠀⠀⠀⠄⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣿⠀⡀
⠀⣴⣔⡱⣄⠘⣄⡹⢿⡛⠓⠓⠛⠛⡇⠀⢘⡀⢀⢤⣰⠞⠛⢧⠀⠠⠀⠀⡠⠀⢚⠀⠀⡀⠠⠻⠀⠀⠀⢘⠀⠀⠀⣸⡏⠀⡀
⠀⣿⣿⡟⢸⡆⣼⠙⠲⣝⠢⡀⠀⠀⢹⣾⢋⠐⠁⠆⢷⢠⣴⣶⣶⣦⡀⠀⣗⠀⠀⢡⠈⠀⠀⠘⣶⠁⠀⠀⢱⠠⣿⣿⠙⠑⢳
⠀⣿⣿⣧⣌⣀⡏⠀⠀⠈⡷⠇⠀⠀⠘⡿⣧⠀⠀⢸⠈⢺⣿⣿⣿⣿⡇⠀⡿⠀⠀⠀⣶⠻⡀⠀⡿⡋⠀⠀⡙⢸⣻⣿⡦⠄⠀⡇
⠀⢸⡿⠃⡀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⣁⣠⣞⠀⠀⠙⠿⢿⣛⢀⡔⢻⡀⠐⢇⠀⢥⡀⠀⣴⠘⠒⠀⢠⠀⠀⠛⠛⢦⣼
⠀⠀⡧⣤⣭⠿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡭⠭⠭⣭⣽⡇⣼⡁⠀⠀⠀⠈⢧⠀⠉⠀⠈⡇⠀⠀⠈⠙⠀⡀⠀⠉⢰⡆
⠀⠐⠀⠀⢸⡄⢇⠀⠀⠀⠀⠀⠀⠀⢸⢯⣽⣿⡯⣛⣿⣿⠀⠀⠀⠘⣄⡴⠀⠀⠀⠀⣇⠀⠀⠀⠀⠄⠄⠀⠀⢞⡤⢀⠀⠀⠀⡄
⠀⠀⡀⠀⠈⠁⠰⠦⣤⣀⣀⣀⣠⠤⠚⣼⠀⢸⣿⣷⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠄⡠⠠⠀⣠⣀⣀⣴⠎⣀⠀⢀⠠⣤
⠀⠸⠤⣤⣤⡤⠂⠀⢨⡉⠀⠋⡅⠀⢠⣿⣦⣼⣿⣿⡿⠉⠏⠉⠉⠁⢿⡼⠦⣤⡴⡏⠀⠀⠃⠀⠀⠉⠈⠁⠀⠈⠉
⠀⠀⠀⣻⠀⣇⢀⡴⢾⣧⢰⣤⣇⠀⠈⠙⡟⡟⢻⠋⠁⠀⢁⠀⠇⠀⢰⢀⣀⢠⡤⠃⢀⠀⠀⢀⠀⠀⣀⣀⠀⣀⠀⠀⠀⠁
⠀⠀⠈⠿⠟⠚⠓⠀⠾⢿⣿⣟⠿⠿⠦⠀⣿⣿⣽⣀⠀⠀⢠⠀⡄⠀⢤⣄⡄⠸⢤⡀⢋⣿⣷⣬⣑⣢⠜⠳⠃⠻⠛⠘⠀⠪⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⢿⢊⡛⠚⡳⠄⠛⠖⢿⡤⠚⠉⠀⠀⠉
⠀⢀⠔⠁
⠔⠁
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣶⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠿⣷⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣄⣀⣤⣤⣴⠶⠛⠋⣀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⣁⣠⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣾⠿⠟⠛⠻⠿⣶⣤⡐⢶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡉⠙⠛⠻⠿⠿⠿⠏⣰⡿⢋⣤⣶⡿⠿⢿⣶⣄⠙⢿⡆⠹⠿⠿⠟⠛⠋⢁⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣤⡄⢸⡿⢡⣾⣿⡏⢰⣶⣦⣽⣿⣷⡀⣴⡄⢠⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⣿⡇⢸⣿⣿⣷⣬⣈⡙⢿⣿⣿⡇⠘⠃⢸⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣤⣄⢸⣷⡈⢿⣿⣏⠻⠿⠟⣸⣿⡿⢁⣿⠇⣀⣀⣤⣤⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠻⣷⣌⠛⠿⣷⣤⣾⠿⠋⣠⣾⠏⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣧⡌⠛⠿⣶⣦⣤⣴⣶⠿⠛⠡⠾⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⣤⠀⣥⣴⠀⢷⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠙⣀⠀⠪⠠⡁⠄⢽⠗⣶⠊⠑⣟⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠉⠄⢬⠀⠀⠀⢤⡋⢘⡤⣿⡀⣀⣿⡜⠃⠠⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠞⠀⠰⠂⢠⡄⠆⣈⠀⠝⣿⡤⠘⣌⠓⢼⡀⠀⣶⣾⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠂⠉⠀⢖⠀⠀⠀⢬⡀⢼⣷⣰⠧⢿⢸⡆⢻⣀⠉⣸⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠜⡳⢀⡀⠁⣀⠉⠁⠉⠻⣾⣿⠫⣄⠛⠔⢣⠸⢾⢤⠀⢸⡇⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢰⠻⠆⢋⡦⢡⠈⠉⠠⠜⠀⠀⢹⣿⡅⠘⣦⡄⣽⠀⠚⠀⠓⠈⠁⠋⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣀⢴⢛⠆⠀⠈⠁⠼⠪⠄⠒⢠⠸⣶⣷⠎⠹⠇⣾⠀⢁⡼⠉⢧⠹⡄⡁⢹⣞
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡐⡃⢠⣠⡈⠙⢰⡌⠘⠂⢃⡀⠙⢀⢸⣿⡿⠀⣠⢆⠀⣲⡞⣀⣀⠸⡆⡿⡗⠃⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠄⡈⠛⡁⠰⠈⣤⡁⢀⡒⠀⠀⣀⢸⣿⣷⣿⣷⢻⡀⣹⠀⢸⣇⣽⣓⠲⠧⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠺⢦⠚⡰⠚⠁⠒⡈⡀⠩⠠⠀⠙⢿⣿⣿⣯⠀⢷⣛⣿⠠⣸⠇⠳⢤⣖⣲⡇⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠩⠀⠙⠃⠄⠗⠀⠀⠸⣿⣿⡄⠀⠐⠜⠻⠴⠛⣾⣉⡉⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠖⠃⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⡶⣿⣿⣿⣧⣿⣿⣿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠛⠿⣿⡿⠛⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡛⠿⠿⠿⠟⢋⣥⣾⣿⡿⠃⠈⣿⠉⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⢻⣿⣷⣶⣾⣿⣿⡿⠟⠁⠀⠀⣼⠏⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⢻⡿⠋⠀⣀⣿⡿⠿⠟⠛⠁⠀⠀⠀⢀⡼⢋⣠⣤⣤⣈⡛⠙⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⣡⢚⡿⢁⣤⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⣰⢟⣴⡿⠋⠉⠉⠙⢿⣷⡄⠀⠉⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣹⡿⢡⡞⠁⣸⡷⠛⣿⣿⣿⣿⡿⠻⣦⠀⠀⣠⡞⢡⣿⠋⠀⠀⢀⡄⠀⠀⢹⣿⠀⠀⠀⠀⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⣼⡟⠀⢈⣷⠀⣿⠀⢀⣿⣿⣿⣏⠀⠀⠘⢷⣴⠋⣠⠗⠀⠀⠀⠀⣼⠁⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⣼⡿⠁⢀⣾⠃⠀⣿⠀⣼⣿⣿⣿⣿⠀⠀⢠⡾⠁⣴⠋⠀⠘⣿⡟⠛⠋⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣠⣤⣤⣀⠀⢻⣇⠀⠀⡇⢰⣿⣿⣿⣿⣿⠀⣰⠟⢀⡾⠁⠀⢀⣴⡿⠃⠀⠀⢀⡜⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢰⣿⣿⣿⣿⡄⠀⢻⣇⠰⡇⣸⣿⣿⣿⣿⠃⣼⠏⢀⡾⠁⠀⣠⡿⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠘⢿⣿⣿⠟⠀⠀⠈⣿⢰⡇⣿⣿⣿⣿⠏⣼⠏⢠⡞⠀⣠⡾⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⣿⡇⣿⣿⣿⡟⣰⡟⢀⡾⠀⣴⠏⠀⠀⣀⡴⠋⠀⣠⡟⠀⠀⠀⣀⣤⠖⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣿⣿⡿⢰⣿⠀⣸⡇⠜⠁⢀⣠⠾⠋⠀⠀⣰⣿⣀⣠⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⢇⣿⠛⢧⣻⣆⠤⠖⠋⠁⠀⠀⠀⠴⠿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣾⠇⠀⠀⠀⠙⠳⠿⢂⣤⣴⠶⠶⢟⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠟⢻⣿⠏
⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠀⠀⢀⣠⣴⣶⣿⣻⣛⣛⣛⣋⣛⠂⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⣼⡟
⠀⢀⣾⣿⠁⠀⠀⢀⡀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣧⣤⣤⣴⣿⣿⣿⣿⣧⣶⡆⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⢠⣿⡟⠁⠀⠀⠀⠈⠙⠳⢶⣾⣿⣿⣿⣿⣿⣿⠟⠁⢀⣀⣤⣤⡤⠤⠬⠭⠭⠍⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⣾⡿⠀⠀⠀⠀⠀⠀⢀⡀⢤⡽⠿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⣿⣧⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇
⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⢹⣿
⠀⠀⠈⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠐⠶⠶⢶⣶⡶⠶⠶⣶⣶⢿⣿⡿⠋⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠿⠿⠛⠛⠿⢷⣶⣦⣤⣤⣤⣀⣀⣤⣤⣤⠶⠞⠋⢠⣾⠟⠁⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣶⣤⡀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠻⢿⠏⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠛⠛⠛⠛⠛⠋⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⠿⢦⣀⣀⣀⠀⠠⠴⠶⢶⣄⠀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⠛⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠛⣀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠘⠋⠀⠀⠘⠛⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⣠⣦⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠈⠀⠀⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢿⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⣤⣶⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠰⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢹⣦⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠓⢶⡿⠛⠟⠃⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣾⡇⠀⠀⠀⢀⣀⣀⣾⠗⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⢾⣿⠃⢹⡷⠶⠶⠆⠈⠉⠉⣿⠀⠀⠈⢹⡷⢶⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠈⣿⡟⠉⠱⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣘⠋⠀⠀⠀⣿⣿⠀⠘⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠇⠀⣿⡇⠀⠀⠀⢻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡆⠀⣿⡇⠀⠀⣀⠟⠀⠈⣶⡄
⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⡇⠀⠀⡿⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢰⡇⠈⠁⠀⠀⢀⣼⣿⣧⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⣿⠃⠀⣄⠀⠀⠀⠀⠀⠈⢃⡀
⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⣶⣿⠟⠛⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣼⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⡸⠃⠀⠀⠀⢸⣿⡷⠾⣿⠶⢿⣿⣿⡀⠀⠀⠀⠀⣿⠀⠀⠀⣘⡁⣀⣿⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⢠⡇⠀⣿⣿⠀⠀⣿⠀⢸⡏⣿⣇⠀⠀⠀⠀⣿⠀⠀⣴⠟⠿⠟⢻⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠘⠀⠀⢸⡇⢠⣿⠇⠀⠀⣿⠀⢸⡀⢹⣿⠀⠀⠀⠀⣿⠀⢸⡄⠈⠀⠀⠁⢽⡇⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⣼⡄⠀⠀⢸⡇⢸⣿⠀⠀⠀⠿⠀⢸⡇⠈⣿⡇⠀⠀⠀⣿⠀⢸⣥⣾⠆⠸⣧⡌⠁⠀⠀⠙⣤⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠉⠀⠀⠀⢸⣿⢾⣿⠀⠀⠀⠀⠀⣿⡄⠀⢹⣿⠀⠀⠀⣿⠀⣾⠉⠀⠐⠂⠈⠁⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⣾⠀⠀⠀⠀⠸⡇⠸⣿⣄⣀⠀⠀⢠⣿⣀⣠⣿⠏⠀⠀⠀⣿⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠙⠀⠀⠀⠀⠀⡄⠀⠈⠛⣿⣷⠀⣸⣿⡿⠟⠃⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⡿
⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠏⢀
⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠛⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇
⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⡀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠉⠉⠙⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠀⠀⠀⠀⠀⠀⠠⠿
⠀⠀⠀⠀⢱⡄⠀⠀⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠈⢻⡆⢀⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣄⠀⠀⠀⢀⡼⠏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⡀⠉⢳⣶⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⡟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⡀⠀⣠⠀⠀⠀⠐⠇⠀⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠉⠁⠀⠛⠀⠀⠀⠀⠀⠀⡇⠀⣿⣀⣀⣀⣀⣀⣀⠀⢀⣄⠀⠠⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⢤⣤⣤⢀⣤⣤⣾⡇⠀⠀⠀⠀⠙⣷⣦⣤⠀⠀⠀⠀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢈⡃⠀⠀⠀⣤⠟⠁⠀⠀⠀⠛⠛⠛⠁⠘⣿
⠀⠀⠀⠀⠀⢀⡼⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠘⠃⣀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠟
⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠁
⠀⠀⠀⠀⠀⠈⠳⠖⠀⠄⠰⠶⠖⠋⠁⠀⠀⠀⠀⠐⠦⠀⠀⣀⣀⡀⠀⠴⠞⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠄⣤⣠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⣛⣉⣠⣶⣷⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣤⣿⣿⣿⣿⣿⣿⠟⣩⣶⣿⣿⣿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠏⢰⣿⣿⣿⣿⠿⠏⣡⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠉⠁⠀⠀⠀⠘⣿⣿⣟⣩⣶⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⢻⣿⠿⠛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣦⠀⠐⢻⡿⣯⣍⠀⠀⠀⠀⠁⠐⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⣼⠃⢰⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠂⣶⣦⡶⠶⠛⠋⠁⠀⠀⢀⡾⠃⠀⣿⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠚⠛⠋⠉⠀⢰⡟⠀⠄⠀⠀⠀⠀⠀⢠⡾⠁⠀⢰⡇⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣴⣿⣾⡀⠀⠀⣠⠟⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⣽⡟⠁⠀⠀⣴⠏⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⡀⢠⡟⢀⣾⣿⡷⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⣼⢁⣾⣿⣿⠃⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢠⠁⠀⠀⠀⠀⠀⢠⢃⣾⣿⣿⡟⢠⠟⠀⠀⢿⡖⠀⠂⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣣⠏⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⣿⣿⡿⢋⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡼⠰⣿⣷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣠⣴⡾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠈⠉⠠⠀⠀⠀⢀⣤⣼⣿⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⡴⣫⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⣀⠀⢄⣁⣀⡀⠀⠀⠈⣿⡿⠁⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣱⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠓⠚⠛⠛⠀⡀⠀⠀⠉⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⢠⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠀⠀⠀⣥⠀⠀⠀⠺⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠸⠟⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠴⠤⠀⠀⢘⡏⠀⠀⠀⠀⠀⠀⠉⢏⠛⠛⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⢀⣿⡟⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠄⠒⠂⠼⠃⠀⠀⠀⠀⠀⠀⠤⢸⡄⠀⠀⠀⠀⠀⠉⠙⠓⠲⠤⣤⣈⣿⣁⣰⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠗⠒⠠⠄⠀⠀⠀⠀⠀⣀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢿⣷⡦⣤⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⣦⢿⣝⢢⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⢳⣿⢦⣿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠉⠀⠙⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠶⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠋⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⣠⣴⣿⣿⡋⠀⠀⢀⣠⣴⣶⣤⣄⠀⢀⢀⣀⣀⣀⡀⠀⠀⠐⠷⣤⡶⠀⠀⠀⢀⣤⣄⡀⠀⣠⣶⣿⣿⣷⣦⡀⠀⢀⠔⡀
⠀⠀⠀⠀⠈⠙⢿⡿⣋⣀⡀⢠⣿⣿⣿⣿⣯⣿⣆⠸⡸⠟⣋⣀⣀⠀⠸⣦⡞⢿⣇⠀⠀⠀⠸⣿⣿⠇⢰⣿⣿⣿⣿⣿⣿⣷⠀⠊⠺⠿⢦⡀
⠀⢠⣾⢻⣟⡛⣮⠑⣿⣿⣿⠸⣿⠻⠿⣿⠯⢻⡿⠀⣰⣾⡟⢉⡻⣿⣦⠙⠓⠟⠛⠀⣀⣤⣤⣤⣄⡀⠘⣿⣿⣙⣉⣿⣿⡏⠀⣎⣤⢒⠀⠉
⠀⣿⣿⣼⡟⣽⣿⡇⢉⣉⣁⡀⠙⠿⣿⣿⡿⠟⠁⠀⣿⣿⣾⣿⣷⣿⣿⡆⢸⠷⢠⣾⣿⠿⣿⡿⢿⣿⣆⠈⠛⠿⠿⢛⡋⠀⠢⠯⣢⣏⠳⢀⣠
⠀⠘⢿⣿⣿⡿⢟⣴⡿⠿⠿⢿⣦⠀⠀⠐⢶⣠⡦⠀⠻⣋⣀⣉⣀⣉⡿⠀⣠⠄⣾⣿⠀⢀⠀⣀⠀⢸⣿⡆⠀⣰⡿⠿⠿⣷⡄⠚⠁⠀⠀⠸⣿
⠀⣠⣴⣦⣄⡀⢸⣷⡄⠀⠀⣰⣾⡇⠸⣦⡞⣿⡁⠀⠀⠈⠛⠛⠛⠉⠀⣿⣿⡇⢹⣿⠀⠈⠀⠈⠀⢸⣿⠁⠀⢿⣆⠀⠀⢸⡟⠀⣠⣶⠛⠓⣦
⣾⡟⠷⠶⢻⣷⠈⢿⣧⣿⣿⣿⡟⠁⠀⠙⠓⠛⠃⠀⠀⣠⣾⡿⣿⡿⣷⣮⡉⠀⠀⠙⠿⣿⣿⣿⡿⠟⣡⣴⠶⢾⣟⠿⠿⢛⣠⠀⣿⣸⠀⠀⣿
⢻⣇⣠⣄⣸⡿⠀⣀⡈⠉⠉⠁⣠⣾⣿⣿⡿⢦⡀⠀⣼⣿⡿⣷⠈⢰⣿⣿⣿⡄⠀⠀⣀⣀⣀⡀⠀⢰⣿⣤⣶⣦⣽⡇⢰⣿⣿⠇⠈⠛⠻⠛⠁
⠀⠉⠛⠛⠉⠀⣶⣿⣿⣶⡀⢰⣿⣿⡛⠉⣰⣿⣷⠀⣿⣿⣷⣌⠃⢈⣻⣿⣿⡇⢠⣾⣿⣿⣿⣿⣦⠈⢿⣁⣉⣈⡹⠃⠋⠛⠯
⠀⠀⠀⠀⠀⠀⠈⠛⢿⠟⠀⠈⢿⣧⣝⣾⣿⣿⠏⡀⠹⣿⣿⠛⠈⠛⣿⣿⡟⠀⢾⡿⣻⣿⣿⣻⣿⠀⢀⣬⣭⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⡛⠁⢼⡿⠂⠈⠙⠻⠿⠿⢛⠉⢀⡀⠈⠷⣶⣶⣶⡶⠏⠀⣿⡋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣧⠀⣠⣶⣾⣷⣶⣄⠀⢄⢸⢞⢣⢤⡀⢠⣤⣍⠁⢀⣰⡆⠙⠳⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⣸⡟⠉⠉⠉⠉⢿⡆⠀⢍⡷⢛⠆⠁⠻⣿⡿⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⣤⣄⣠⣤⣸⠃⣠⠀⡠⢀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⡄⠀⢸⣿⣎⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣧⠀⠈⠙⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠻⠿⣿⣿⣿⣿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⢸⣿⣿⣶⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣼⣄⣈⣉⣋⣉⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⠶⠟⠙⠶⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡀⢉⡉⠀⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⢄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢸⢸⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣒⣚⠚⢒⣪⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣿⣿⣿⡇⡔⠉⠉⠉⠉⠉⠉⠉⠉⠉⡭⠛⠉⠛⠭⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠤⢀⣀⣀⣈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢓⠁⠀⠀⠀⠉⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡃⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⡀⠀⢀⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠘⠒⣶⠒⡒⢲⠒⡖⢲⠒⣖⢲⡇⡇⠀⠀⠀⠀⠀⠀⢀⠜⠁⢣⢨⣛⡁⡆⠈⠢⠀⠀⠀⠀⠀⣔⣖⡤⡾⣸
⠀⠀⠀⢱⠆⠲⠐⠆⠰⠀⠆⠰⡀⠇⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢩⠉⡏⠀⠀⠀⡇⠀⣴⡎⠉⠹⠞⠉⠫⠏⠉⢳
⠀⠀⠀⠀⢯⡌⢥⠩⡌⢥⠨⡄⢡⡄⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣘⠀⡃⠀⠀⠀⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠘⡖⠐⠂⠒⠐⠂⠒⠐⡇⠣⣀⣀⣀⣀⣀⣀⣸⣤⣤⣤⣬⣶⣥⣤⣤⣤⣇⣠⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⢤⢤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⡤⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢠⡔⡲⡢⡤⠤⠤⠤⠤⠤⡤⣚⡸⡀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡇⠀⠀⣜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠮⣒⡣⠋⠉⠉⠉⠉⠉⢇⣣⣋⠇⠀⠀⠀⠰⣍⣉⣉⣉⣉⣉⣉⣉⠇⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣾⣿⣁⡀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣾⣶⡿⠿⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⡆⠀⠀⠀⠈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠙⠻⣟⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⠀⢀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⢩⣍⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣰⣿⣏⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⣿⣯⣿⣿⡄⠤⠀⠀⣀⡤⠤⣄⠀⠠⠤⠤⠄⣾⣿⣿⣿⠛⠿⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠛⣿⣿⡅⠀⣶⣦⠀⠀⠀⣠⣴⣾⣷⣦⡀⣿⡟⠙⢯⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠛⠒⣿⣿⠧⣼⣿⣧⣤⣤⣾⣿⣯⣍⣩⣿⣿⣿⣷⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠇⠈⢿⣿⣿⣿⠿⠁⠉⣉⣽⣿⠏⢽⣿⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣀⣀⣀⣉⣉⣡⣤⣤⣀⠈⣛⣁⣀⣿⣿⡄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠉⠁⠀⠀⠀⢸⣇⠀⠉⠛⢤⣀⠀⣹⢹⣧⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⡀⠀⣰⠞⠶⣀⢿⣄⠀⠀⠀⠈⠉⠻⢀⣽⣇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠴⠶⢾⣿⠇⠀⣿⣦⠀⠈⠃⠙⣷⣦⣀⠀⠀⠀⠀⠙⣿⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⣄⠀⠀⠈⠆⠘⠏⠳⢤⣀⣀⡄⠘⣷⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢷⣤⣤⣤⣿⣶⠀⠀⠀⠐⢶⣿⣾⣿⣻⣿⣦⠈⠇⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠁⠙⢏⡀⠠⣧⡀⠀⠀⠀⠙⢿⣿⣿⡙⢧⣄⠈⠀⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠁⠺⣿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣄⣽⣧⠀⢠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⢀⡀⢀⠈⠉⠛⢿⣷⡄⠐⠀⠀⠀⠀⠋⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢠⣾⣿⣧⣄⠀⠀⠀⠉⠑⠲⣆⡀⠀⠀⠀⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣾⠿⣷⠻⣿⢿⡄⠀⠀⠀⠀⢻⣅⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠀⢠⢜⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢲⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠿⢿⣿⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠠⠴⢬⣭⠷⢶⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠳⢦⣶⣦⣀⠂⠠⠍⢸⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠙⠓⠚⠂⠿⠿⠿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣄⣀⣀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣛⡿⢿⣿⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⢛⠹⠁⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠖⣢⡽⣠⣤⡀⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣫⠞⠁⢾⡿⣿⣿⣿⣿⣷⡄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣯⣾⠒⠋⢸⣿⣿⣿⣿⣿⣿⣡⡞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠁⠀⠀⠈⠉⠙⠛⠋⠛⠁⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠈⣿⡇⠀⠠⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠃⠉⠀⠀⠀⠴⢶⢯⠙⠂⠀⣴⣶⡶⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠛⠛⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⠀⠀⠀⠀⡠⠀⠀⡇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠏⡇⠀⠀⠀⠀⠀⠑⠦⡤⠧⠀⠀⡜
⠀⢀⣀⣀⠤⠤⠔⠒⠒⠚⠋⣹⣿⢿⡏⢹⠀⢳⡀⠀⠀⠙⠒⠲⠶⠶⠄⢀⠌
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⠁⢸⡇⠈⢧⡀⠙⠲⣄⠀⠀⠘⠓⠀⣠⣾⣤⣀
⢶⠀⠀⠀⢀⡞⣀⡀⠀⠀⢸⢿⠀⠀⠙⠲⢄⣷⠀⠀⠈⠁⠒⠒⠖⢪⠏⢹⣧⠈⠉⠓⠲⠤⣄⣀
⠈⣧⠀⢀⡾⠋⣽⠁⠀⠀⡼⢈⣦⣀⡀⠀⠀⢿⣷⡄⠀⠀⠀⢀⡴⢻⠀⠘⣿⠀⠀⠀⣀⣄⠀⠉⢻⡦⡄
⠀⢸⡆⠀⠀⣸⠃⠀⢠⣾⠎⠁⠈⠻⣿⡄⠀⠈⠻⢽⣆⠀⣠⣾⠐⠃⠀⠀⣿⡆⠰⠚⢁⡟⠀⠀⢸⡇⢳
⠀⠘⡇⠀⢰⠇⠀⠀⠘⣿⡄⠀⠀⠀⢻⣧⠀⠀⠀⠀⠙⣷⠏⣾⠀⠀⠀⠀⣿⡇⠀⠀⣸⠃⠀⠀⢸⠇⠘⡆
⠀⠀⢻⡀⠿⠤⢤⡄⠀⠘⣷⡀⠀⠀⠘⣿⠀⠀⠀⠀⢀⠏⠀⡿⠀⠀⠀⠀⢧⡇⠀⠀⠹⡀⠀⠀⣼⠀⠀⢻⡀
⠀⠀⠈⣷⠀⠀⢀⡇⠀⠀⠸⣇⠀⠀⠀⣿⠀⠀⠀⠀⣼⡜⠀⡇⠀⠀⠀⢀⣿⠀⠀⠀⠀⠈⢧⠀⡏⠀⣠⠜⡇
⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⠀⢻⡄⠀⢠⡟⠀⠀⠀⢠⡟⠀⠀⡇⠀⠀⢠⢾⡿⢤⠀⠀⠀⠀⢸⣴⢃⠞⠁⠀⢳⡀
⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠻⣄⣸⠃⠀⣤⡄⣼⠱⡄⠀⡇⠀⠀⢺⣼⣆⣸⠀⠀⠀⠀⠀⣿⢋⡤⠊⠀⠀⢧
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⣶⠦⠤⠞⠉⠉⠉⠉⢠⡇⠀⢸⠀⡇⠀⠀⡬⠷⠶⣋⠤⠄⠒⠒⢺⡷⣫⠶⠋⠀⠀⠈⠣⣀
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡸⠀⣧⠀⠀⣷⢀⣤⣤⣤⣤⠀⢀⣿⠞⠁⠀⠀⡤⠐⢒⠆⠈⢧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠟⢦⡀⠀⠀⠀⠀⢸⠐⠫⢤⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⠀⠀⢀⠔⣁⣄⠀⢀⡙⣄
⢀⡀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢀⣰⢌⣶⣄⠀⠀⢸⠀⠉⠁⠀⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⢸⠉⣿⣦⡀⠀⠟⢊⡴⠁⡼⠡⠉⠛⠳⣄⡀
⠈⠉⠙⠶⣿⣤⣀⣀⣀⠴⠚⠉⠀⠀⠀⠈⠙⠲⢼⡘⠓⠺⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡿⠀⠈⣷⣄⠀⠀⡾⠂⡸⠁⣠⠛⠶⠀⠀⠉⠳
⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡈⠓⠦⣤⣀⡰⠙⢿⣳⢼⠉⠉⠛⠒⠲⠶⠶⠶⠶⠴⠧⢤⣤⣸⣏⣃⠀⣠⣺⠇⡴⠁⠀⢠⠎⠈⡷
⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⣄⣀⠀⣨⠙⢲⣤⠳⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠇⡜⢳⣄⡰⠃⢀⡜⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣈⣍⣓⣖⣋⣀⣭⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⡽⠁⢻⠎⡲⠛⠊
⠀⠀⠀⠈⠛⠲⠦⢤⣀⣀⣀⣀⣉⣉⣡⣴⣏⡉⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣷⣾⣷⣤⣷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣄⡀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣀⣈⣩⣷⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠾⢷⣄⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠤⠶⠖⠒⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⢿⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⣄⠀⣤⣀⠀⠀⠀⢰⣿⣿⣿⣿⣻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣇⢿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⠋⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣦⣴⣤⡼⠿⠟⠛⠛⠀⠈⠛⠀⠀⠀⠀⠀⠐⠐⠀⠠⠄⢠⠀⠀⡠⠀⢀
⠀⠀⠀⠀⣾⣿⣸⣿⣉⠘⣿⡟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠈⠛⣹⣿⣿⣀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⢀⣰⣦⣤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠙⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣷⣤⣀⠀⠀⠀⡙⠉⠉⠛⠻⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃⠒⢦⠀⣀⣠⣿⣀⣀⣀⡀⠀⢀⠤⠤⠀⢴⣶⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⢡⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠻⢿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡁⠀⢼⣏⡀⠀⠀⠸⠻⠿⠿⢿⠋⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣷⠺⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⣌⠙⣶⣄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣞⡛⢿⣿⣯⣍⣉⠁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣾⣛⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡴⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠆⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣉⣉⣀⣀⣤⣴⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⠸⣿⠟⠁⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⢀⣠⣤⠀⠀⠀⠉⠙⠉⠿⠿⡟⠹⣿⡿⢿⠯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠣⣿⣷⡀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢃⢿⠟⠀⠀⠀⠀⠀⢀⠀⡀⠀⢹⣿⠇⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣷⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣗⠙⢻⣆⡀⠀⠀⠀⠀⠀⢒⡤⠤⠼⠿⢇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠙⢦⠈⢿⣿⣦⡀⠀⠀⠀⠀⠻⠷⠖⢀⠎
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣷⡄⠀⠑⠄⠘⠋⠻⣄⡀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠉⠛⣛⠛⠛⠋⠁
⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣤⡈⢳⣤⣀⣀
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣾⣿⣿⡷⢿⣿⣿⣷⡄
⠀⠀⠀⠈⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠙⢻⣿⡅⠈⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢸⣿⣿⡄⠸⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⣿⣿⣿⡀⣿⣿⣿⣿⡎⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢹⣿⣿⣷⣸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⡿⠛⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⣀⡀
⣿⣿⣿⣿⣶⣶⣤⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣷⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣰⣿⣿⠿⠟⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠋⠉⠀⠀⠈⠻⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⡏⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣾⡧⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣆⠈⠁⣀⣠⣤⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡕⣃⢹⢰⣿⠷⠲⣦⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⣿⣅⠀⢿⣟⠀⢶⡏⣸⢛⡿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠌⣻⣶⠋⠉⢛⣾⡷⣷⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⢳⣦⡘⠦⢸⣇⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢋⣤⣠⣾⣟⠛⢿⣿⠉⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣴⣶⡤⣤⣞⡃⠀⠀⠀⠙⣿⡧⠾⡛⠛⠛⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣻⣿⣿⣿⣿⣿⡇⢿⣿⡃⠀⠀⢀⣼⠟⠳⣶⣿⣶⣤⣤⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⡇⢿⡋⢡⣄⡀⣀⠏⠀⣧⣉⣰⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠈⠉⣿⡃⢀⣾⢿⣿⡛⢻⠉⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣿⣿⣿⣿⣿⣿⣿⣇⢸⣇⣀⣴⣿⠂⠀⣼⡿⠈⣷⠈⢲⣶⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⢿⣿⣵⠃⠀⢰⣿⠃⠀⠉⠄⢡⢻⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⢃⠎⠀⡼⠂⠀⠀⠀⠀⠈⢆⢻⡍⢌⢷⣀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣻⣿⣿⣿⡿⢡⠏⠀⣼⣤⡶⢾⡷⠶⣤⣄⠈⠊⢿⡌⢣⡻⣷⣄⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠛⢰⠋⠀⣴⣿⣿⡀⢸⡇⠀⠈⢻⠀⠀⠈⢻⡄⠀⢹⣾⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⠃⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢻⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣹⡇⢀⣤⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡌⢿⣕⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣿⠀⣞⠇⡐⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣷⢀⠀⡄⠀⠀⠈⣿⣿⢞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⢸⣿⣿⡃⠀⢠⠀⣄⠀⠀⠸⣿⠀⢻⣿⡟⠈⢀⣡⢀⠁⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣾⣿⣿⣧⠀⣞⡀⢿⣷⣤⣠⣿⣤⡿⠋⠀⠀⢿⡟⢸⣼⣈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢠⣿⡇⠀⠀⠀⠉⠛⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⣿⣿⢳⣼⣇⢠⢰⢠⣰⢠⣧⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣧⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣤⡎⠀⢐⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⠀⠘⠂⠸⠿⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⡛⠉⠁⢀⣿⣿⣿⡿⢂⣀⣤⣤⣤⣶⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠷⣥⡀⠙⢘⣿⣽⣷⣿⣿⠿⠿⠟⠛⢻⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠀⣠⣾⣿⣿⢟⣋⡁⠀⣀⣀⠤⠀⠘⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡐⠿⠛⠁⠀⠸⠯⠝⢿⠃⠛⠚⠛⠀⠙⠏⠒⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢻⡀⣼⠄⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⣀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠹⣜⠧⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⢀⣀⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣀⠀⢀⣠⠏⣿⢧⡀
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡾⠀⢀⡤⠶⠶⢤⣤⡶⠛⡿⠙⠟⠿⡿⠟⢁⠔⢻⠀⠳⡦⣄⣀
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⣣⠞⠁⠀⠀⠀⠀⣴⢣⣔⣾⠀⠀⢀⣷⡎⠁⠀⣾⠀⠀⡆⠀⠈⠙⠓⠦⠤⢤⡀
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⢹⠀⠀⠸⣿⡗⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠹⣄
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⠁⢸⠀⠀⣸⣿⡆⠀⠀⡇⠙⢎⠁⠀⠀⠀⢀⡟⠀⠀⠀⢿⡄
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣾⠇⠠⡀⠀⠀⠀⢸⡀⠀⣿⣿⣧⠀⢠⡇⠀⠈⢣⠀⠀⠀⣸⠁⠀⠀⠀⠸⡇
⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⡰⣃⠀⣀⢀⣾⠏⠀⠀⢣⠀⠀⠀⠸⠃⢠⣿⣿⣿⠀⢸⠀⠀⠀⡎⠀⠀⠀⢹⡅⠀⠀⠀⠀⣷
⠀⠀⠀⠘⢧⡀⠀⠀⢀⠾⣪⢕⣋⣵⡋⠀⠀⠀⠀⠀⢣⠀⠀⠀⡇⢸⣿⣿⣿⡀⡿⠀⠀⡘⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠙⠲⠚⠓⠚⠋⠉⠀⠀⣧⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⣾⣿⣿⣿⣇⡇⠀⠠⠁⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣿⣿⣿⣿⣿⠁⠀⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⡌⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠈⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠒⢐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡠⠗⠁⠀⠀⠀⠀⠀⠀⠄⠀⠸⠘⡜⡆
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠦⢸⣇
⠀⠀⠀⠀⠀⢼⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⠀⠂⢀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢢⢈⣽⡿
⠀⠀⠀⠀⢠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠤⠇⠀⡤⡆⠁⠄⠠⠀⠀⠠⠐⠀⠀⠀⠐⠀⠀⠀⠐⡂⢸⠈⡈⣯⡇
⠀⠀⠀⠀⠈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⣚⣉⣍⣭⡅⠀⠀⠀⠡⠀⠠⠀⠀⠠⠀⡀⠀⠀⠀⠀⠂⢸⡏⡿⣾⡂
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠨⠀⠀⠀⠀⢥⡤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⣆⢷⢘⣿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⡃⣨⢸⡏
⠀⠀⠀⠀⢸⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠺⡅⣧⡇⠁
⠀⠀⠀⠀⢸⠘⢣⣖⣒⣒⣢⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⢄⣄⣔⡈⡇⣷⠃
⠀⠀⠀⠀⠀⢇⣉⠉⠉⠉⠉⠙⠉⠛⠛⠻⠿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⠿⠿⠿⠛⣛⡃⡛⠉⠘⠋⠘⢑⣱⣿
⠀⠀⠀⠀⠀⠀⢀⣈⠙⠩⠮⠁⠀⠀⠀⠀⠀⠀⠒⣊⡹⠿⡿⣶⣶⣾⢟⡿⢝⡱⠶⠀⠒⠪⠍⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡟⡀
⠀⠀⠀⠀⠀⠐⠛⠒⠉⠲⠴⠶⠦⠤⠤⠤⠶⠖⠒⠲⠾⠷⢴⣿⣹⣽⡾⠿⠛⠒⠛⠓⠒⠂⠂⠄⠰⠰⠦⠶⠾⠛⠛⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣄⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⠀⡨⠋⢵⣶⣶⣝⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣾⣶⣭⣵⡌⢙⣆⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠞⠛⣿⣿⣿⣿⣿⡁⢰⣮⡅⠀⠛⠁⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠿⠙⠂⠀⠀⠀⠀⠐⢆⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⣶⣦⣄⡀⠀⠀⠀⠀⠈⠉⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠛⠛⠛⠋⠤⠀⢒⣀⡉⠀⠈⠀⠀⠈⣀⡒⠂⠤⠄⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡩⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣁⠒⠀⠠
⠀⠀⠀⠀⠀⠀⣠⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢒⡀⠉⠀⠀⠀⠀⠀⠀⠈⡅⣒⠐⠌⠤
⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠩⠤⠐⠂⡀⠀⠀⠀⠀⠀⡐⢒⣰⣸⣀⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⣒⣉⠭⠄⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠠⠄⠒⣒⠈⠉⠀⠀⠀⠀⠁⠉⣐⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⣀⣀⣀⠐⠈⠉⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⢹⣿⣿⣿⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠟⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠛⠛⠛⠻⠿⠶⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠾⠿⠛⠛⠛⠋⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⣶⠀⢰⠏⠁⠉⣧⠀⢰⡖⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢹⡇⢸⡄⠀⠀⣼⠀⣾⠇⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠈⣿⠛⠙⠒⠚⠉⢻⡿⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢹⣇⠀⠀⠀⠀⣾⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣈⣿⣀⣀⣀⣰⣟⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣛⣛⣛⣛⣛⣛⣛⣛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⢶⡶⢶⠶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⣿⡟⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠛⠛⠻⠿⠿⠶⠶⢶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠶⠶⠶⠿⠿⠛⠛⢻⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣤⣀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⠾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠿⠶⣶⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠀⣿⡇⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⡶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠶⢶⣦⣤⣤⣄⣀⡈⠻⣷⣿⣷⡿⠋⣀⣀⣤⣤⣤⡶⠶⠾⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠟⠛⠛⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣏⡌⠀⠀⣠⠂⠀⠀⠀⠉⢛⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢿⠇⠈⠚⠁⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠋⠁⢿⡇⠀⠙⢦⡀⠀⢤⡀⠀⠀⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠙⠓⠶⣬⣿⡀⠀⠀⠙⢦⡈⠙⠶⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠲⢄⠀⠀⠀⠀⢠⡇⠘⣷⣀⣠⡤⠔⠓⣾⢻⣷⡷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⢳⡀⠀⠀⠈⠳⣄⣹⡏⠈⢦⠀⢸⠃⢸⡿⠉⠛⣧⠓⢄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⠙⢿⡀⠈⢧⢸⡇⣿⣧⠀⠀⠈⢧⠀⣇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣟⣦⡈⢿⣧⡟⢸⠀⠀⠀⠈⢇⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⡏⢌⢿⡝⣎⣿⡇⠘⣧⣀⡀⠀⠘⣾
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠙⠒⠋⠀⠈⢿⠁⠀⠀⠉⠁⠘⠓⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠈⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠘⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣧⢯⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠲⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⢧⣀⠀⠀⢀⣠⠖⠋⠀⠀⠀⣠⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣼⡀⢀⡠⠤⠶⠶⠖⣛⠲⢦⣨⣻⠟⣋⣠⣄⢀⠶⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡿⠁⠀⠀⠀⠰⠶⣦⣬⠿⠋⢰⡛⡇⢧⠈⡿⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠛⠳⢦⣤⣀⠀⠀⠀⢀⡞⢰⣇⣀⡤⠖⠋⣩⠿⠋⠁⣠⠴⢮⣁⣨⠤⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⣸⠉⢻⠃⠀⢠⡴⣋⣁⡤⠖⢻⠯⣦⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣶⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠉⠉⠉⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡄⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣶⣆⠀⠀⠉⠉⠉⠉⠉⠀⠀⣰⣶⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠉⢹⣿⣿⣷⣶⣶⣶⣾⣿⣿⡏⠉⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠶⠶⠶⠶⠶⠶⠶⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣉⣉⣛⣛⣛⣉⣉⣠⣤⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣤⣾⣿⡀⠀⠀⠀⣠⠀⠀⠀⢠⣿⣷⣤⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⠿⣿⣿⣿⣿⠀⠈⠻⢿⣿⣿⣿⣿
⠀⠀⠀⢀⣘⣿⣿⣿⣿⡇⠀⠀⢾⣿⡷⠀⠀⢸⣿⣿⣿⣿⣃⡀⠀⠀⠀⣿⣿⣿⡿⠯⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⡇⠀⠀⠀⠀⠽⢿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⡇⠀⠀⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⢹⣿⣿⠃⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡀⠀⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⠁⠀⠈⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⠹⣿⠟⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣦⠀⣴⣿⡇⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢸⣿⣿⣿⡇⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠘⣿⣿⣿⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⢤⡙⠈⠶⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠄⠀⢀⠹⡆⠀⠛⠈⡥⠲⠖⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⢄⣀⡜⢠⡟⠲⠦⠤⠄⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣇⣠⡴⠟⠼⣷⣄⣀⣀⣀⡼⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠓⠶⣄⠀⢠⣞⠄⠠⡀⡤⠒⢢⡘⡏⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠈⢷⣟⠠⣸⡿⢻⠀⣿⣯⡟⠀⠀⠀⠀⣻⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠈⠱⠟⠋⠉⠁⠀⠀⠀⠁⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡞⠃⠀⠀⠀⠀⠀⢤⣄⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣄⣀⣀⣀⣤⠞⠀⠀⠀⠃⠀⣿⠀⠀⠀⠀⠀⠀⠸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡿⠋⢁⡀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⠻⢦⣤⣤⡴⠟⢁⠀⠀⠀⠀⠀⠀⠀⣿
⣀⣀⣀⣤⠤⠖⠒⠚⠉⠉⢉⡉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢺⡀⠀⠀⠀⠀⠀⠀⠠⠶⠋⠀⠀⠀⠀⠀⠀⣰⡇
⢻⠁⠀⠀⠀⠀⠀⠀⠒⠂⢤⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠙⠛⣶⡄
⢸⡀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠙⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⡰⡏⠀⠀⠀⣆⠀⢙⡷⣦⣤⣀⣀⣀⣀⣀⣤⣴⠾⠋⠁⠀⠀⠀⣬⠹⣦
⠀⡇⠀⠀⠀⢔⠉⡠⢀⢈⠀⡁⡀⢠⢸⡇⠀⠀⠀⠀⢀⠞⢱⠁⠀⢠⠖⠋⠀⢸⡗⠉⡀⣩⠟⢯⡏⠉⠡⠿⢤⣤⡄⠀⠀⠀⣷⡄⠈⢷⣄
⠀⣇⠀⠀⠀⢄⡉⠀⠈⠆⠉⡰⠁⢸⠘⡇⠀⠀⠀⡰⠋⣰⡏⠀⠀⠀⠳⠀⠀⢸⡇⠀⣿⠁⠀⢸⡇⠀⠀⢀⣴⠞⠀⠀⠀⠀⣿⠙⢆⠀⠙⢷⣄
⠀⢸⠀⠀⠀⠀⢨⣀⣀⣀⣤⠄⠀⠀⠀⣷⠀⠀⡼⠁⡴⢣⠁⠀⠀⠀⠀⢡⡀⣸⡇⢰⠉⡇⠀⢸⡇⠀⣠⠞⠁⠀⠀⠀⠀⠀⣿⠀⠈⠳⢤⡀⠙⠳⢦⣀
⠀⠸⡄⠀⠀⠀⠀⠨⠤⢤⠤⠃⠀⠀⠀⢻⡤⠚⠃⡼⠁⡌⠀⠀⠀⠀⠀⠀⠁⣿⢀⡏⠀⣿⠀⢸⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⠽⠓⠒⠂⣈
⠀⠀⡇⠀⠀⠀⢀⣰⠏⢻⡏⢆⡤⡄⠀⢸⡆⠤⠊⠁⢀⠁⠀⠀⠀⠀⠀⠀⠈⣿⢸⠃⠀⢹⡀⢸⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡾⠋⠀⣠⡴⠚⠉⠁
⠀⠀⢸⠀⠀⠀⣏⢸⡄⠀⡇⢸⠀⣡⠈⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠘⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢦⡴⠋
⠀⠀⠈⠛⢿⢁⠹⢆⠷⠴⢦⡼⠶⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢰⡆⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟⣇⠀⠀⠀⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣸⠅⢂⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢦⣀⡼⠧⢾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣄⡀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠒⠲⣶⠶⠾⣀⠋⠙⠛⢻⡇⠸⣥⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡏⠀⠀⠀⢸⠀⢀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⢸⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⡇⠘⣧⠀⠀⠀⠈⣇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠉⠓⠶⠤⣀⣀⣀⣀⣷⣀⣹⡆⠀⣀⣀⣼⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠤⣞⠉⠛⠙⠶⠶⠶⠶⠶⠛⠛⠉⠁⠀⢀⡄⠉⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣀⣀⣀⣀⣀⣤⣤⠴⠞⠉⠷⠤⠴⠇
⣾⣾⣶⡾⣷⣷⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷⣾⣶⣶⣶⣿⣷⣶⣿⣿⣿⣷⣾⣷⣶⣶⣶⣶⣶⣶⣶⣷
⣿⣿⣿⣀⣀⣤⣤⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣮⣉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⢣⠈⠙⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠈⢧⠀⠀⠘⢿⣿⡄⠀⠀⢀⣀⠤⠒⠒⠉⠉⢱⡄⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⣠⡔⠒⠒⠒⠒⠒⠒⢺⣿⣷⢶⠊⠉⠀⠀⠀⠀⠀⠀⢸⡽⣤⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣿⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠙⢷⡄⠀⣿⡇⠀⠀⠀⠀⠀⣠⣿⡿⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡠⠓⢄⠀⠀⠙⢶⣿⣇⣀⣀⣀⣤⣾⣿⠟⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⢄⠀⠀⣹⡿⠿⠿⠿⠟⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣸
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀⠜⠓⠊⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡷⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⡎⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⣸⡿⣿⡇
⠀⠀⠈⠢⡀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣤⠶⠯⠓⠒⠒⠚⠒⣿⡇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠶⡖⠒⢲⠶⣶⣾⣿⣷⠶⠶⠶⠶⠶⠶⠶⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠾⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣻⣿⡿⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠙⠛⠛⠁⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⡤⢤⣄⡀⢀⣤⠤⣄⠈⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠘⠒⠖⢻⡇⠘⠛⠛⠉⠀⣇⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠀⠔⠉⠓⠒⠋⠐⠄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠠⠤⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⡰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠻⡾⠛⠛⠊⠉⠀⠀⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡿⢧⠀⠀⠀⣀⣀⡀⠀⠀⠀⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⡍⢸⠇⠈⠄⠀⢰⣿⣿⡿⡆⠀⡀⢠⡇⠀⠀⠀⠑⠂⠄⣀⣀⡀
⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⢸⡄⠀⠘⢖⠁⣽⣿⣇⠙⡞⠀⡾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢢⣿⣿⣿⡰⢁⡞⠋⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⡀⠀⢻⣿⡿⠁⢸⡇⠀⠀⠘⡇⠀⠀⡆
⠀⠀⠀⡇⠀⠀⠀⠐⢄⠀⠀⠀⠀⣴⡾⠃⠙⠄⠀⡟⠁⠀⠘⠃⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠰⠁⠀⠀⠀⠀⠈⡇⠀⠀⠀⢻⣄⣠⡤⠀⠀⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤
⠀⠀⠰⡀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡼
⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⢄⠀⢠⡶⢾⢧⡇
⠀⠀⠀⢇⠻⠀⠀⠀⠀⠀⠀⣖⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡡⢂⠔⠄
⠀⠀⠀⠘⡄⠁⠀⠀⠀⠀⠀⢹⠉⠙⠒⠒⠢⣤⡀⠀⠀⠀⠀⠀⢀⠀⢀⠁⠈⠠⢀⠎
⠀⠀⠀⠀⢳⠀⠀⠀⠠⠤⠤⠼⠤⣀⡀⠀⢠⠃⢀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠒⠛⠛⠋⠛⠃⠀⢸
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⢀⣀⠀⠀⠠⠄⠀⠒⠀⠁⠈⠉⠉⠀⠒⠂⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠹⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⢂⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠶⠀⣻⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠉⢉⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⠆⠚⠀⠉⢻⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠲⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⣿⠀⠀⠰⠂⢸⣧⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⣤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣐⣄⣿⣦⣄⠀⢀⣀⣠⣴⣿⣿⢠⡏⠀⠀⠀⠀⣿⣼⣇⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⢀⣴⠜⢻⢿⡆⠀⠀⠀⠀⠀⠘⢻⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⠁⠀⢀⠾⠃⣹⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢡⣾⣿⣴⣃⣸⡇⢀⡄⠀⠀⠀⡶⠃⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⢠⣿⠀⠀⠁⠀⠀⢾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⠛⠉⠉⠈⣿⢧⠏⠄⢀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⣿⣿⡿⢿⣿⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠻⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠹⠿⢿⣼⠯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣠⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢿⠿⣷⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⢰⣶⣀⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠟⠛⠀⠸⠿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⣀⠀⣀⣀⠁⠀⠀⠤⣤⡤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⣀⣀⣀⣀⡀⠀⢰⣃⣀⣀⣘⣦⣀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣀⣀⣀⣀⠀⣠⣀⣠⣤⣤⣤⣤⣤⣴⣾⡇⠀⣷⣶⣾⣷⣼⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠛⠛⠻⠿⠿⠀⢻⠿⣿⣿⣿⣿⡿⠿⣿⣅⣠⣶⣧⣬⣿⣿⠿⠿⠿⠿⠿⣇⢀⣿⡿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠀⠰⠾⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⣉⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢀⡤⢠⣀⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⣷⡀⠀⠀⠀⠀⠀⡨⠟⠓⢤⣰⣶⠀⢰⠁⠀⠀⠈⢃⠀⠀⠀⢀⠄⠠⡀⢈⠀⠁⠀⠀⠠⠀⠀⠢⡑⡀⠀⠈⢂⢀⠐⡄
⠀⠀⢿⠹⣆⠀⠀⠀⡜⢀⡀⡈⠀⢹⠉⠀⠘⣄⠀⠀⢀⡆⠀⠀⠀⣵⣶⣶⣶⠃⠀⠐⠀⡀⠀⠐⠀⠀⠑⡘⢄⠀⠨⠈⠀⠐⠄
⠀⠀⢸⡄⠘⢿⠀⠀⣹⣄⠀⠀⢀⡜⠀⠀⠀⠈⠉⠉⠁⠹⣷⣄⠀⠙⡿⢿⣯⣿⣿⣿⣾⠀⠘⠀⠀⠀⠀⠈⢄⠒⠁⡄⠀⠀⢈
⠀⠀⠸⠇⠀⠀⠰⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⡄⢠⡶⠀⠀⠈⠛⠃⠀⠀⠀⠂⠉⠉⠉⢣⣄⠀⠀⠀⠀⠀⠀⠀⠢⡀⠡⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⢀⣀⣼⣥⣤⠀⠀⢀⣴⣶⣶⣦⣤⠄⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⢀⣀⠀⠀⠀⠀⠀⣼⠋⠙⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠰⢿⣿⣷⣿⣿⡁⠀⡀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠘⢾⣿⣿⣶⣲⠀⠘⠳⢾⡷⠾⢶⣄⠀⠈⠻⠿⠿⠋⠀⠀⢰⣿⣿⣿⠿⠗⠀⠧⠰⡷⠀⠀⠰⠆⠀⠀⠠⠊⠀⠀⠀⠀⠑
⠀⠀⠀⡽⠉⠉⠀⠀⠀⠀⠸⣇⡀⣠⡟⠀⠀⢀⠀⠖⡦⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠁
⠀⢠⣴⣧⠀⠀⣤⠴⢶⡀⠀⠉⠛⠉⠀⢠⣾⣦⣤⡀⠀⠐⢳⡀⢀⠠⢀⣸⠀⣆⡀⠤⡀⠀⠀⠀⠠⢢⠀⢃⠀⠀⠀⠂⠢⢀⠆
⠀⠀⠀⠉⠀⠀⠿⣤⡼⢋⡀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢠⣴⡀⡇⠀⠑⢤⣼⣿⣧⣤⠊⠀⠀⠠⠀⠀⠀⠀⠀⠁⠂⠄⠠⠐⠁
⠀⢤⣦⡀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠈⢄⠙⢿⡄⠀⡿⡱⠁⠀⣊⠀⠼⠓⡧⣤⣑⠄⠀⠁⠀⠸⠁⠀⠀⠀⢀⣴⣤⡀
⠀⠈⠻⡟⠀⠀⠀⢲⣿⣿⡇⠀⠀⠀⢀⠀⠀⠉⠦⢦⡔⠊⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠂⠀⡠⠤⠀⠀⠀⠀⠉⠛⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠶⠷⠒⠀⠀⠶⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠂⠂⠁⠀⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣆⠀⢀⡖⣶⠆⣦⡀⠀⠀⠀⠀⠀⠀⣤⠀⣀⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠀⠀⠉⠛
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠯⠽⠀⠒⠀⠀⠀⠀⠀⠰⢿⡟⢩⠧⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⡞⠛⠛⠛⠛⠛⠛⠉⠛⣷
⠀⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡀⠠⠀⢁⡀⠠⡄⠀⠀⠀⠀⠀⠦⣤⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣼⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠘⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣛⣻⣟⣛⠛⠛⠋
⠀⠀⠀⠉⠉⠙⠓⠒⠦⠤⢄⣼⠁⠀⠀⡒⠉⠙⣶⣄⠀⣤⠘⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣌⠉⠉⢉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣼⣿⡖⠁⠙⠟⢣⠀⠀⠀⣴⠟⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢀⡴⠋⣹⠆
⠀⠀⠀⠀⡠⠄⠀⢀⡀⢰⠡⠀⠀⢇⡘⠉⠑⣤⣤⡄⡘⠀⢰⠟⠀⠀⠀⠀⠀⠶⠆⠀⠀⣾⣿⡧⠀⠀⠄⠐⣏⣠⠞⡡⣪⣶⣄
⠀⠀⠀⠀⠠⣰⡀⠘⠀⠀⠀⡀⠂⠈⠻⣦⠀⠛⡿⠋⠀⠀⠀⠀⠀⠠⡶⢠⡜⠛⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠁⢻⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠂⠀⠈⠀⠠⠔⠊⠁
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢵⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⡄⠀⠀⡆⠰⠀⢀⣀⠄⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢁⡀⠀⠧⣈⡀⠬⠃⠀⠂⠓⠘⠂⠑⠐⠠⣙⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠘⣄⡠⠀⠀⠀⠀⠀⡀⢆⠐⠀⠀⠀⡀⠀⡄⠀⢠⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠸⡀⠀⢸⠠⡄⢼⡀⠰⠤⣇⠀⡸⢠⠉⠁⠤⠀⠀⡄⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⡀⠀⢧⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⢦⣀⠀⠀⢸
⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠐⠤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠣⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⣰⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠴⢿⣿⡟⠋⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠀⢀⣽⣶⡶⠆⠀⠀⠀⠀⠐⠞⣿⣿⠿⠂
⠀⠀⠀⢨⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣄⡀
⠀⣼⣿⣿⣿⣷⡆⠀⠀⠀⢠⣶⣾⣿⣷⣶⡄⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣦⠀⠀⠀⢰⣿⣿⣿⣿
⢠⡟⢿⣿⣿⣿⡇⠀⠀⠀⣾⠛⣿⣿⣿⠙⣧⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⠙⢷⡄⠀⠈⢿⣿⣿⣿
⢸⡇⠘⣿⣿⣿⡇⠀⠀⠀⣿⠀⣻⣿⢛⡀⣿⠀⠀⠀⣸⡇⢹⣟⣛⠁⣠⡴⠛⠁⠀⢠⣾⣿⡿⢿⡄
⠀⢻⡀⢼⣿⣿⣧⠀⠀⠀⠘⠷⢀⠉⠐⠄⠇⠀⠀⢰⡟⠀⠀⠈⠉⠘⠉⠁⠀⠀⠀⣸⣿⣿⣷⠈⣷⡀
⠀⢸⡇⢨⣿⣿⣿⠀⠀⠀⠠⠂⠀⢄⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣷⡈⢠⣶⠀⢀⣿⣿⣿⣿⡄⠙⣇
⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠑⡆⠀⠀⠁⠀⠀⠘⠇⠀⠀⠀⠀⠠⣿⣷⣸⠃⠀⣼⣿⣿⣿⣿⣿⠀⠙⠁
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⢻⣿⣿⣿⠿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿
⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⣸⠃⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣿⢹⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⣴⠇⣼⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄
⢸⠟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡰⣿⣿⣦⡀
⠀⠀⠀⢀⣴⣿⣷⣟⣿⣿⣿⣦⡀⣠⣤⣤
⠀⠀⠀⣾⣿⣿⣿⣿⣷⣏⠿⣿⣿⡿⠿⠋
⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠁⠀⠀⢀⡿⠁⢸⣿⣽⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁⠀⠀⠀⣾⣧⡀⠀⢸⣿⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠰⣿⣿⣿⠁⠀⠀⠀⠀⠈⠛⠁⠀⠸⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡀⠙⠁⠀⠀⠀⠀⢿⣷⠆⠀⠀⠀⠉⣿⡿⠏
⠀⠀⠀⠀⠀⣠⠷⠀⠂⠀⠀⠀⠀⠰⠧⢀⣀⠀⠀⠀⠈
⠀⠀⠀⢰⣿⣧⣴⣶⡇⠀⠀⠀⠀⠀⢀⣤⣍⠀⣀⣠⣴⣿⣄⡀
⠀⠀⠀⣾⣿⣿⣿⣿⠀⢀⣤⣶⣶⣶⣿⠟⣿⠛⢉⣿⠏⣹⠿⠻⣷⣶⣄⡀
⠀⠀⢀⣿⣿⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣷⣿⣶⠚⣿⡶⠃⠀⠀⠀⣀⡿⠛⢳⣄
⠀⠀⢸⣿⣿⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⠿⠀⠀⠀⠀⢠⣿⡇⠀⠘⣿⡄
⠀⠀⢸⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⢸⣿⣇⠀⠈⢿⣧
⠀⠀⠀⠉⠙⠛⠛⠉⠉⠀⠀⣿⣿⣿⣿⣿⠿⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣤⣀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⠀⢹⣿⣿⣿⣿⡿⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⡇⣄⡀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠘⣿⣷⡬⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣟⣶⣶⡎⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⡆⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⠓⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⣿⣿⣿⣿⣿⡏⢠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣿⣿⣿⣿⣿⢁⣾⣿⡏⠀⠀⠀⢀⣤⠶⠒⢾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣫⣾⠉⢁⡉⣿⣿⣿⡿⣿⣿⡟⠀⢀⣤⠾⠋⠁⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠇⢘⣷⣿⡿⠿⢱⣾⣿⣧⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠃⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣻⡇⢠⣄⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣶⣻⣿⣷⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠺⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠲⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡟⢠⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢫⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⡿⢿⣻⣛⣓⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⠿⢿⣿⡿⠿⠿⢟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠠⠶⢤⡀⠀⢀⣀⣀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠀⠓⠂⢻⠀⠤⠤⠀⢁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⠀⢀⠰⠆⠠⠀⠀⠀⡌⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠒⡂⠤⠈⡠⢊⠵⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠻⣷⡄⠀⠉⠀⢰⣡⢋⠊⣄⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⣿⣿⢻⠀⢩⡛⠓⠒⢂⡿⢿⠾⢻⠟⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⡿⠟⠻⠿⠛⠋⢉⡆⢸⡏⢸⡀⠀⠑⣄⡠⠼⣷⡀⢠⠂⠀⣰⢿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠀⢧⠀⠀⠀⠀⠀⡼⠀⠈⠁⠀⡇⢀⡴⣿⣿⡄⢻⣿⡀⠀⠀⠈⢆⠀⠈⠉⠙⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣦⠀⢸⡀⠀⠀⠀⠸⠁⠀⠀⠀⠀⢳⠈⣴⣿⣷⡀⢸⣿⣷⠀⠤⢂⡌⠱⡀⠀⠀⡔⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠧⡐⠦⣄⠀⡇⠀⣌⠉⠉⠉⠛⠃⠀⠀⢸⣴⣿⣿⣿⣇⣿⣿⣟⡃⢀⣾⠀⠀⠑⣄⠘⢀⣴⡇
⠀⠀⠀⠀⠀⠀⣴⠛⠧⣀⡀⠀⠈⢳⣷⡀⢿⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣷⠀⠀⠘⡆⠀⠀⠈⢲⡟⣹⣇
⠀⠀⠀⠀⠀⢰⡟⢆⠀⠀⠻⢯⡛⣯⣽⠇⠈⢻⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠁⠘⢿⣿⣷⠀⠀⠑⠀⠄⠀⠀⢻⡁⠈
⠀⠀⠀⠀⢀⣿⠇⠈⠑⠢⠤⠤⣙⣰⣿⠀⠀⣸⡿⡄⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠘⢿⣿⣧⣀⠀⠀⠀⠀⠀⠀⡳⢬⡳⡀
⠀⠀⠀⡠⣫⠵⣚⣛⡛⠛⠟⠞⣿⣿⣿⠀⠀⣿⠁⠹⡄⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠈⢿⣿⣿⢧⡀⠀⠀⠀⠀⢱⠈⠇⣷
⠀⠀⢸⣿⡿⠏⠁⠘⠙⠳⣄⣴⣿⣷⡛⡀⢰⡏⠀⠀⠱⡄⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢈⣿⣘⣧⡙⠂⠀⠀⠀⠀⢇⢀⡏⣠
⠀⠀⢸⠉⡜⠀⢀⡤⠆⠀⠈⡙⢿⡿⠛⢥⢾⢧⡀⠀⠀⠹⡀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢀⣾⣿⣟⠉⢿⣤⡀⠀⠀⠀⠈⠚⣴⠃
⠿⠿⠿⣦⣀⣴⠋⠀⡠⠔⣫⣾⡏⠐⠦⡀⠓⢟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⢾⣯⣛⣷⣄⢀⣠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠙⠿⠷⣶⣦⣼⣄⣏⣻⡅⡙⠶⢁⠐⠠⠜⣿⠿⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣈⡹⠿⠆⠉⠳⣝⣃⡀⠈⠗⠒⠙⠇⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⡀⣀⡤⣤⣀⠠⠴⠶⠖⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⠶⠖⠁⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⢀⢀⣀⠤⠴⠞⠓⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁

⠀⠀⣄⣺⠗⠁⠀⠀⠠⠀⠉⠁⠉⠉⠀⠷⡿⠀⠁⠀⠁⠀⠀⠉
⢰⠿⠉⠁⠀⣀⠀⠀⠂⠠⠀⠀⠐⠋⠀⠋⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠺⠇⠀⣤
⠀⠀⠀⠀⢈⠻⠈⠀⠀⢀⡄⢰⡄⠐⠂⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠁⠰⡇⢰⡾
⠀⠀⠀⠀⡎⠀⡄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠂⠀⠀⠐⣃⠠⡆
⠀⠀⠀⠀⠩⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⡨⠖⠺⠛⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡠⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⣀⡄⠐⠋⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡿⠿⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠐⠃⢀⣠
⡀⠀⡄⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⢿⣯⡀⢠⣟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠄⠐⠋⢀
⠀⠀⠄⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠃⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡙⡁⠀⠀⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠒⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⡷⠰⠂⣇⣸⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠢⠀⠀⠀⠀⠤⠆⠐⠃⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⠀⠀⢘⣿⣿⣿⣿⡇⠀⣳⠿⣿⣿⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤⠀⢠⣶⣶⠀⠀⠀⢀⣖⣂⠸⠯⠌⡒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢜⣭⣽⣙⠃⠠⠘⠀⡿⠻⠄⠀⠀⠀⠀⠐⠙⠀⠀⠀⠀⠈⠀⠈⣸⣿⣠⠴⠉⠀⣿⣇⠐⢃⡘⠉⣀⣀
⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠈⠘⠙⣬⠂⡶⠀⣿⡻⠃⠀⠀⠀⠀⠀⠐⠁⠀⠀⡀⠀⠀⠀⠀⢸⣿⣴⡖⢻⡟⣯⡍⢨⣯⢨⣿⡇⠉
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⠛⣧⡿⠺⣧⡤⢾⠳⠄⢀⡆⠀⠀⠐⡆⠀⠀⢤⡾⢶⡿⠿⡿⡆⠇⢹⠇⠘⡇⠸⠟⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠊⠣⠀⠻⠇⠀⣸⡗⣾⡧⠒⠀⠐⠀⠀⠐⠐⠒⠀⠈⠀⢈⣡⣤⠐⠢⡄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⠫⡹⣆⠔⡀⠳⡄⠀⠀⠊⠸⠃⠱⠠⡤⣤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣃⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣽⡔⢱⡷⠀⠙⢜⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⠁⣽⣇⠴⠆⠊⡹⠅⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⣃⠞⣹⠾⠆⠀⠀⢡⣡⣜
⠀⠀⠀⠀⠀⠀⠀⢰⠇⣦⢻⢃⠘⠃⠀⠀⠀⠀⠀⠣⣿⣣⣎⡀
⠀⠀⠀⠀⠀⠀⢠⡿⢚⣀⢃⣾⠄⠀⠀⠀⠀⠀⠀⢰⡀⣳⣵⠰⡄
⠀⠀⠀⠀⠀⠀⠀⡼⢡⡟⣨⣿⠀⠀⠀⠀⠀⠀⠀⠀⣱⢎⡁⢤⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠳⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡙⢼⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡴⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣣⠔⠊⠉⡴⠟⠛⠉⠉⠙⠓⣒⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⡞⡽⠛⠛⠳⠶⠶⠶⠤⠤⠤⠤⠤⠤⢤⣤⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢸⠏⠀⠀⣀⣤⡀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠙⢦⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣮⣤⠴⠿⠯⣭⣅⡀⠀⠀⢀⣨⣭⠽⠿⠶⣤⣌⣇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠐⠂⠀⠀⣿⠶⠶⣾⠀⠀⠐⠂⠀⠀⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠱⣿⡄⠀⠠⣿⣧⢄⡟⠀⠀⢹⡀⣾⣯⡄⠀⢀⡏⡴⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢏⠙⠶⠶⠿⠷⠟⠀⠀⠀⠀⠙⠾⠿⠶⠶⠛⢀⠇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠼⡄⠀⠀⠁⠈⠀⠠⣀⣀⡠⠀⠀⠀⠀⠀⠀⣸⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡶⠲⢤⡤⠤⢤⠴⠖⢶⢫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡹⢇⡀⣄⢠⠫⠀⡰⢠⢼⢏⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠇⠐⠀⠘⡁⠄⡩⠁⣨⠆⠈⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢠⣤⠎⠀⠀⠀⠁⡀⠡⠊⢀⣁⠀⣄⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠠⣬⠟⠀⠀⠀⠀⠀⡏⠀⠀⡏⠉⢹⠘⢦⡤⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠔⢙⠃⠀⠀⠀⠀⠀⡏⠀⠀⠙⠛⠁⠀⢛⠠⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠜⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠸⡀⠈⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡌⠀⠀⢠⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠱⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠉⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⣄⣀⡀⠀⠀⠀⠀⣸⢸⠀⠀⠀⠀⠀⢀⣀⣠⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢍⢙⣒⡲⡿⠚⢳⡚⣛⡉⠩⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠈⠀⣀⡧⠀⠸⣀⠀⠀⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠖⠒⠉⠁⠀⠀⠀⠀⠉⠓⠲⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠉⠉⠉⠀⠒⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠑⡀
⠀⠀⠀⡤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠐⠁⠀⠀⠉⠁⠐⠂⠉⠁⠀⠀⠈⡄⠀⠀⠀⠆
⠀⠎⠱⡁⢰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠄⠀⣴⣥⣤⣤⣀⡀⠀⠀⠀⠀⣀⣠⣤⣤⣴⡄⠀⢄⡀
⢃⢻⠀⡇⢸⡆⡇⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⡰⠉⠀⠀⠶⣛⡻⠝⠦⢄⠠⠾⢟⣛⠲⠀⠀⠈⠱⣀⣿
⠈⡈⠃⠀⠀⠀⠑⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣦⠇⠀⠀⠀⠸⣿⡇⠀⣼⠃⣹⣄⢸⣿⠇⠀⠀⠀⠀⣿⡎⢃
⠀⢣⠀⣀⡀⢀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⠿⣤⣀⡀⠀⣉⣤⠞⠁⠀⠀⠹⣧⣅⣀⠀⣀⣠⡼⡿⠀⡘
⠀⢰⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⡄⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠰⠃⠀⠉⠉⠉⠁⠠⠓⠊
⠀⠈⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠘⡓⠠⠤⠤⠤⠄⢚⠂⠀⠀⠀⡰⠁
⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠈⠒⠤⠤⠤⠒⠁⠀⢀⡠⠊
⠀⠀⠸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠒⣢⠤⠤⠀⠤⠤⢴⣒⠡⢄
⠀⠀⠀⡄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢡⡴⠛⠋⢄⠀⠀⠀⢀⠎⢋⠳⣦⠉⠐⠠⢀
⠀⠀⠀⢂⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢠⠿⢄⢀⠀⠀⠈⠀⠉⠀⠀⢸⢀⠼⢧⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠈⠆⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⢀⡟⠀⠎⠀⡀⠀⠀⠀⠀⠀⠀⡄⠘⡄⢸⡆⠀⡀⠀⠀⠑⠄
⠀⠀⠀⠀⠈⢄⠀⠀⠀⠑⠤⣀⠀⠀⢀⣀⠤⠐⠉⠀⠀⠀⠀⢸⠁⠈⢢⠀⠱⠀⠀⠀⠀⠀⡘⠀⢠⠃⠘⡇⠀⢹⢢⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⠴⣆⡤⠿⢦⡀⠀⠱⡀⠱⡀⠀⠀⡰⠁⡰⠁⠀⢀⡇⠀⠀⡄⢃⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢀⣀⣀⣀⣀⠠⠄⠒⠈⠀⢠⡏⠀⠀⠀⠹⡆⠀⠑⡄⠑⢄⠜⢀⡄⠀⠀⠀⢀⡇⠀⠀⡆⠈⡄⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⢿⠀⠀⠈⢆⠀⢀⠎⠀⠀⢀⣠⣧⢵⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠲⠃⠀⡴⠀⠚⠛⠛⠛⠓⡆⢸⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠁⢰⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⢠⡀⡄⢴⠃⠀⠀⠀⠀⠈⠀⠀⠈⠦⠀⠀⠀⠀⠜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠁⠁⠀⠀⠀⠀⠀⠀⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠤⠤⣤⠀⢠⡤⠤⠤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠻⡄⠀⠹⠿⠆⣀⡀⠾⠿⠁⠀⡼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⣽⠖⠒⠋⠉⠀⠈⠉⠓⠒⣾⡍⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⡀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⡴⠚⠛⠁⠀⠀⠀⠀⠀⠀⠹⠶⡖⠤⠤⣀⣀
⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠁⠀⠀⢸⡀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡸⠀⢹⠀⠀⠀⠀⠉⠑⠒⠦⣀
⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢠⡟⠋⢻⡄⠱⡤⠤⠤⠤⠤⡴⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢣⠀⢣⠀⠀⠀⢠⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠸⡀⠀⢣⠀⢠⠃⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠰⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⡀⠀⡇⠀⠀⢣⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠱
⠀⢠⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠙⣄⡇⠀⢀⣤⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⢀⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣄⣀⠼⠓⠒⠉⠀⠀⠀⠀⡴⠚⠳⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢆
⠬⣀⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣀⡸

⠀⠀⠀⠀⠀⢰⡏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⢹⡆
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢰⣱⠀⠀⢀⠤⠒⠉⢹⠉⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢸⣹⠀⡔⠁⠀⠀⠀⢸⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢸⣹⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠸⠼⢿⠀⠀⠀⢀⡰⡞⠀⠀⠀⠀⠀⢸⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠘⣄⡠⠖⠁⡰⠁⠀⠀⠀⠀⢀⡜⠀⡔⠒⠚⡦⡀⠀⠀⣀⠴⠦⡀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠈⠢⣀⢠⠃⠀⠀⠀⢀⡠⠊⠀⠀⣧⠀⠠⡇⣈⣖⣉⡖⠒⠒⡎⠲⣄⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠋⠁⠀⠀⠀⠀⣧⠤⢤⡇⡄⠀⢼⡗⠒⠒⡇⠤⠬⢤⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⡇⠖⠒⢺⣟⠉⠉⡇⣗⣒⣺⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡇⡋⣉⣹⣏⢉⢉⡇⡒⡒⣺⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠁⠈⠉⠉⠀⠉⠉⠁⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠸⣇⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡯⣀⠀⠀⠀⠀⠀⠀⡼⠷⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⢷⣦⡀⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⣿⣿⣿⣾⣿⣇⠘⣤⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠴⣻⢿⢿⠿⡟⡏⠁⠉⠛⠛⠛⠛⠻⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⣯⡟⡼⠈⡀⠐⠂⠀⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠠⠤⠐⠈⠀⠀⠀⠀⠀⠀⢘⡟⠠⠁⢸⣧⣀⡰⠼⠟⠓⠓⠓⠛⠃
⠀⠀⠀⣠⣦⣾⣷⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠀⠀⢀⡉⠉⠀⠀⣸⠀⠀⠀⠀⣠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⢀⣤⣿⣷⣤⣥⠶⠛⠛⠛⠛⠛⢣
⠀⠀⠸⠿⠁⠋⠛⡉⣿⢹⠀⢸⠛⠁⠀⠀⢠⣴⣿⠏⠀⠀⠀⢼⣿⢹⡥⡍⠀⠀⣹⣀⣀⣀⣀⣄⡇
⠀⠀⣤⠀⣦⣄⣦⣷⣿⢸⠀⠈⣿⡿⣷⣚⣻⣿⡏⠀⠀⠀⠀⠘⢿⣾⣇⣿⠷⠾⠟⠛⠛⠛⠛⠛⢇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢸⠀⠈⣟⠀⠀⢸⠇⠀⠀⠀⠀⠀⣤⣤⢸⡇⢹⠉⣿⡆⠀⠀⠀⠀⢀
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡆⣇⡀⠘⠀⠀⢸⣳⣄⠀⠀⠀⠀⠘⢿⣾⣧⣼⣶⣿⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠘⡿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⣀⣿⣿⣷⣦⡄⠀⠀⠀⠹⠏⠻⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠙⠉⠉⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠁⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⢤⣤⡄⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢈⡁⠀⠈⠀⢰⣶⡦⠀⠀⠀⠀⢠⡶⣶⡄⠀⠀⠀⢈⡅⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⠻⠿⠗⠀⠀⠀⠀⠀⠿⠿⠇⠐⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠘⣦⣤⣄⣠⣤⠟⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⣿⠁⠘⠀⠤⠀⠠⠄⠘⠋⢹⡇
⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠸⠏⡀⠀⠀⠀⠀⠀⠀⠀⠁⡘⡇⠀⠀⠰⠆
⣧⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠸⣥⡀⠀⠀⠀⠀⠐⢀⢤⠙⠁⠀⠀⠁⠀⠁
⠿⠀⠲⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⢈⡁⠲⠀⠶⠞⠹⡆⠀⠳
⠘⢧⡰⠀⠐⣃⠄⠈⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⢻⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠃⠀⠐⠛⡔⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠟
⠀⠀⠀⠀⠀⠀⠘⣦⡂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣴⣠⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠄⠀⠀⡀⠀⠁⠀⠀⠰⠀⠀⠀⠀⣀⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠘⠃⠐⠡⢀⣤⣀⣤⣤⣤⠤⠶⠋⠁⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢰⣄⡀⠀⠀⠀⢠⠀⠀⣀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣀⣀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡏⠀⠀⢃⠐⠂⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠠⡆⠈⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠰⡆⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⠂⠂⠀⠒⠀⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⣶⣿⣷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠈⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠉⠑⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⢹⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠼⠁⠰⠊⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡝⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢨⠉⠀⠀⠀⠀⠀⢁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⡴⠒⠂⢀⡜⠛⣦⡀⠀⠀⠀⠰⠂⠅⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⣆⠇⢴⡧⠎⠰⣿⣿⠉⠀⣀⣤⠶⠖⠺⠟⡛⠛⣦⠀⠈⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⣟⣲⡆⠿⠁⠤⠝⠉⠗⠋⠁⠀⠀⠀⠀⠀⠽⢂⡿⠀⠐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠓⠂⠀⠀⠚⢻⡟⠲⠄⠀⠀⠲⣶⢶⡏⠀⠀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢀⡾⠀⠀⠀⠀⠀⢀⡟⠈⡅⡀⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣀⣠⣴⠾⣿⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⡾⠑⠊⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣻⡀⠀⠛⠚⠻⣤⠴⠊⠀⠀⠀⢠⡞⠋⠛⡖⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢹⢷⡀⠀⠀⡀⠀⠀⠀⠀⢀⣴⠏⠀⠀⣸⠁⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⢸⡆⠙⢦⣬⣄⣀⣀⣤⠶⢻⠏⠀⠀⠈⠛⢛⡶⠃⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠉⡽⢛⠣⡀⡼⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢷⡟⠀⠀⠮⠀⢸⠀⢹⠃⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡇⠀⢰⠺⠘⠀⠀⢠⣏⡀⠀⠀⠀⡀⠈⠄⡏⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⣨⢸⢀⠀⠀⠀⠈⠙⠶⣄⠀⠡⠀⣼⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⠀⠈⠉⠉⠉⠉⠙⡄⠀⠈⠳⣄⢃⡇⣰⠃⠀⠀⠀⠀⢀⣠⠶⠛⠳⣸⡃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠈⠉⠉⠉⠁⢰⡇⠀⠀⠀⠘⣎⣿⣥⡄⠀⢀⣠⠶⠋⠁⠀⠀⢀⠀⠑⠄⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⢷⡤⠴⣄⣀⣤⡤⠾⠶⠒⠚⠛⢹⠋⠀⢰⠋⠉⠁⠀⠀⠀⠀⠀⢠⣄⠉⠲⢄⠐⢆⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⠁⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠀⠀⠀⠀⠀⣀⡴⠚⠉⠉⠳⢤⣀⠙⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣧⣀⣀⣹⠂⠒⠒⢚⣿⠁⢀⡄⠠⡀⠠⡄⠙⡏⠉⡟
⠀⠤⠤⢤⣤⣤⢤⣏⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠶⠾⠛⠋⠩⠠⠤⠤⠬⠤⠤⡄⠈⢀⣼⠀⢘⡇⢰⡇⠀⡇⢀⡷⠚⠲⠒⠶⠐⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠈⠓⠚⠓⠚⠂⠞⠓⠚⠁⠀⠀⠀⢠⡤⡔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢼⣤⣌⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠰⣆⡇⠀⠀⠈⢧⠀⠉⠙⠓⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠡⡄⠤⢀⠀⠀⠀⠀⠀⠀⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠸⡓⠊⠁⠀⠀⢱⡄⠀⠀⠀⠀⠐⡌⢧⠀⠀⠀⠀⠀⢀⡠⠄⠙⢧⡀⠀⠀⠀⢠⡎⠉⠉⠁⠂⢄⡀⠙⢷⣄⠀⠀⠀⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⣀⠀⠥⠐⠁⠀⠀⠀⠀⠀⠘⡌⢧⠀⠀⠀⣊⡥⠶⠛⠁⠀⠙⢦⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠲⣄⡉⠛⠛⠋⠉⠉⠙⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠰⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢣⠀⠈⠁⢀⣀⣠⡤⠶⠶⠟⠃⠀⠀⠻⣤⣬⡴⣈⣑⡄⠀⠈⠻⢶⣄⣀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠲⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠠⡀⣹⢱⡀⠀⠀⠀⠈⠉⠛⠛⠿⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣂⣤⣤⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠄⠒⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠐⠒⠢⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠂⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠶⠿⠿⠿⠿⡟⠁
⠀⠀⠀⠀⠀⠀⠱⠦⣄⣀⠀⠀⠀⠀⠀⠀⠉⠉⠰⠦⣤⣴⣶⣶⡶⠿⠿⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⣿⡟⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⠶⣶⣖⠛⠂
⠀⠀⠀⠀⠀⠀⡠⣟⡂⢤⣀⠀⠀⠈⠙⠓⠦⢤⣀⠀⢹⣯⣄⣀⣀⣀⣠⡤⠤⠶⣒⣚⣩⣭⣽⡿⠷⠶⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⢐⠀⠀⠉⠂⠈⠑⠢⢄⡀⠀⠀⠀⠈⠙⠚⠿⢿⣿⣿⠷⠶⠞⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⠂⣘⣇⡀
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠠⣌⠑⠢⢄⡀⠀⠀⠀⢀⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⢶⣒⣛⣯⣽⠅
⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣈⠑⠢⢀⣸⣿⣖⣀⣀⣀⣤⣤⣶⣖⣻⣽⣿⡶⠾⠛⠛⠉⠉⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠦⡄⢈⣙⣻⣭⡽⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠂⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡊⣟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⣿⣧⠄⠀⠀⠀⢀⣀⣀⡤⠴⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡷⠶⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣦⢀⡀⣀⣀⡄⠀⠀⣠⣤⣤⡄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠿⠛⠚⠊⠀⠑⣛⣿⣷⣾⣿⡻⠏⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢩⣭⣽⣏⣽⣯⡌⠀⠀⣀⣤⣶⣿⠿⣿⣿⡿⣿⣯⠟⠁
⠀⠀⠀⠀⠀⠀⣀⠀⣀⠒⠂⠔⢁⣾⡗⠈⣭⣟⣻⣛⣛⠋⣉⣩⠶⠆⠀⠻⡿⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠤⡉⢹⣷⡀⡟⠉⠁⠀⢸⡿⠟⠛⠛⡆⠉⠀⠉⠋⠁⠀⠁
⠀⠀⠀⠀⢀⣴⣦⣤⡈⢅⣿⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⣽⠉⠉⠇⠚⢁⠄⠀⠀⠀⠀⠀⢹⡆⠀⠈⠒⠚⠁⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠠⣾⣤⣠⠠⠆⠁⠀⠀⠀⠀⠀⠀⠘⡃⠀⠀⢘⣛⠩⠉⠀⠀⢀⣿⡀
⠀⠀⠀⣴⠀⠘⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢠⡴⠋⢸⢱⣦⣀⣀
⠀⠀⣰⣿⣧⡀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡞⢿⣶⣤⣤⣶⠏⠀⠀⠃⢸⣬⣤⠉⡙⠓⠲⢤⣄
⠀⢠⡟⢹⣿⣷⣦⡀⠀⠀⠀⠀⠀⣤⡤⢞⣫⣿⣿⣇⠈⠛⢿⡟⠁⡠⠐⠀⠀⢸⣿⣿⡿⣿⣿⣦⣄
⠀⣾⢁⣾⣿⣿⣿⣷⣶⣄⣤⣀⣤⣤⣴⣿⣿⣿⣿⣿⣆⠀⢀⣽⣼⡦⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⠂
⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣄⣾⣿⡟⠁⠠⠀⢀⣿⣿⣿⣿⢸⣿⣿⣿⣿⡇
⠀⠀⠀⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁⢸⣿⣧⠀⠀⠀⢸⣿⣿⣹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠛⠋⢿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⡟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣧⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⡿⣹⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠿⠙⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡀⢀⣿⣿⣿⡿⣱⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠿⣿⣿⡇⠀⣿⣿⣿⠃⠘⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠸⠟⠻⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠛⠛⠛⠛⢿⣷⠀⠀⠀⠀⠀⠐⠔⣤⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠓⠞⠋⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀⠀⢀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣅⣈⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⢉⣉⣉⣉⡁⢀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠤⠤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⠟⠛⠿⠿⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣣⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⣀⣤⣤⣾⠿⠛⠉⢀⣤⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠂⠀⠀⠀⠀⠀⢀⣠⣶⣶⣸⣰⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠻⣿⣿⡉⢹⣿⣇⣀⡀
⠀⠀⠀⠀⣠⣴⡄⡆⣰⣦⡀⠀⠀⠀⠀⣀⣤⣶⢨⡍⣁⠀⠀⠀⠀⠀⢸⣿⠟⢻⣿⣿⣿⣀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣌⣻⠿⠿⠿⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣾⡿⢻⣿⣾⣾⣿⣷⣀⠀⠀⠀⣼⡿⠀⢸⣿⣿⣿⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠰⣿⣭⣼⣿⣦⣴⣿⠂
⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⠀⠀⣸⣿⠁⣾⣿⣿⡿⠈⠛⠿⠿⠗⠀⠈⠁⠀⢸⣿⣿⠁⠀⢸⣿⢠⣤⠀⠀⠀⠀⢴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⣾⡟⣼⣿⣿⣿⡇⣿⡇⠀⠈⠁⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠸⣷⣶⣿⣿⣷⣶⣿⢂⣀⣿⣿⣿⣿⠛⠉⠻⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠈⠀⣿⣿⢿⣿⡇⠈⠀⠀⠀⠀⢸⣿⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⣀⣰⣿⣿⠋⠉⠻⣿⣿⣼⡿⣿⣿⣿⣷⣀⠀⣠⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⣸⣿⡏⠈⢿⣿⡄⠀⠀⠀⠀⠀⠰⣿⠟⠁⠀⠉⠻⣿⣷⣤⠀⢴⡿⡿⠋⢵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⣿⣿⠀⣿⣷⠀⠀⠀⠀⢠⣿⡟⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⣿⣿⡿⠿⣷⠆⠀⠀⢹⣿⡿⣿⣿⣿⢿⣿⡁⠈⠁
⣶⣶⣶⣶⣶⣿⣷⣶⣿⣿⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣿⣶⣾⣿⣶⣶⣿⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢠⠀⠀⠀⢀⣤⣾⡇
⠀⠀⠣⠀⠀⠻⠋⢀⠀⢀⠀⠀⠀⠀⠀⣤⣾⣷⣤⡀⠀⠀⠀⣌⠀⠀⠀⠸⠟⠃⢰⠀⠀⠀⡀
⠀⢀⠠⣑⠦⠥⠯⢔⡨⠀⠐⠀⠀⠀⢠⣴⣶⣴⣶⡄⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠁⠀⠀⢠⣿⣿⣿⣶⣧⣆
⠀⠀⠑⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠟⠍⠩⠯⠩⠻⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⢾⡭⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣗⣛⡯
⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠂⠀⠀⠀⠀⠘⠋⠁⠈⠁⣿⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣾⣿⣦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣵⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⣷⠦
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣯⣟⣿⣿⣿⣳⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠿⠿⠿⢀⣃
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤⠧⢼⠤⢤⢤⠧⢼⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣷⣿⣛⣿
⠀⠀⠀⢺⣿⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⣚⢞⡒⣖⢞⡲⠃⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⡿⠿
⣤⣤⣤⣬⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣭⣥⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⡙⢛⣩⡿⣟⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡈⠱⣶⣶⠆⠈⠫⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠈⠹⠅⠀⢉⡀⠀⠀⡜⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⢯⣭⠿⠁⣠⢾⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡖⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢐⡎⠁⣟⠛⠋⠁
⠀⠀⠀⠀⣴⠖⠈⠍⠉⠉⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⣽⣿⠃⢀⡾⡇⠒⠠⢄
⠀⠀⠀⠀⠀⠉⠐⠒⠒⠀⢟⠁⡨⠢⡀⠀⠀⠀⠀⡠⠊⡠⠂⡞⣼⣧⣴⠟⢸⠃⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠑⢄⡠⠊⠀⡔⠀⣼⠃⣿⣿⢏⡴⠻⡄⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠁⠀⢀⠥⢠⣯⢸⣿⠏⡞⠁⠀⡏⠀⣳⡀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠄⠊⠀⠘⠿⠟⢸⡏⢸⣅⢀⡼⠁⡰⠁⠑⡄⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠁⠀⠁⠉⢁⠜⠀⠀⢀⠜⠊⠁⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠄⠀⠀⢠⠊⢀⡤⠊⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢠⠀⠀⠀⠉⠖⠣⣈⣢⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣤⠀⠀⠀⠀⡄⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣿⠀⠀⠀⠀⠘⠑⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣶⣤⣤⣿⣇⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠠⢄⠀⠀⠌⠛
⠀⠀⠀⠀⠀⠀⠺⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠴⠟⠀⠀⢲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣶⠖⠒⠒⠒⠈⠙⢿⡃⠐⠒⠒⠒⠀⣤⡀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡏⠀⠀⣀⠔⠉⢲⠈⠃⠀⠀⠀⠀⠀⣠⡿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⠿⠟⠛⠰⣿⡛⠂⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠂⠀⢀⡘⠛⠛⠓⠀⠀⠀⠀⢀⠀⠀⢺⣶⡾⠋
⠀⠀⠀⠀⠀⠀⣶⡟⠛⠛⠻⣶⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠉⠁⠀⠀⠀⠀⣨⡿⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⣀⣠⣴⡾⠛⠁⠀⠀⠀⠀⢀⡠⠞⠃⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣧⠄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠛⢻⣧⠀⠀⠀⠀⠠⠤⠚⠉⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠘⣷⠄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠐⠒⠛⠃⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠒⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠴⠘⣿⣀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠠⠄⢺⡦⠀⠀⣸⠋⠐⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣼⣄⠀⣴⣦⠤⠐⠿⠖⢰⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠘⠚⠓⠖⠃⠀⢀⣀⣤⣶⣿⢷⣤⡀
⠀⠀⠈⡁⠈⠿⣯⣥⣤⣤⠤⣶⠖⠉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣄⣀⣀⣀⣤⣤⣤⡶⠾⠟⠋⠀⣠⠟⠀⠈⠛⠷⣦⣄⡀
⠀⣀⣀⠙⢶⡄⠀⠹⣭⠁⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠹⣍⠉⠉⠉⠉⠉⠀⠀⠀⠀⣀⡴⠋⠀⠀⢀⡀⠀⠀⠉⠛⠷⢶⣄⣀⡀
⣾⡋⠁⠉⠚⠛⢷⣄⢹⡇⢸⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠑⠂⠠⠤⣤⠤⠤⠐⠚⠉⠀⠀⠀⠀⠈⠻⣷⣦⣤⣀⡀⠀⠈⠉⠙⠛⠳⠦⣤⣤
⠈⠉⠓⠾⠷⠦⣤⡈⢨⡇⢸⠀⡇⠀⢠⡄⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡈⠉⠙⠲⠶⢦⣤⣤⣤⣤⠀⢰
⠀⠀⠀⠀⠀⠀⠈⢷⠀⡇⠸⠀⡇⠀⠈⣧⣤⡀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⣠⡾⠁⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⡇⠀⢠⡟⠉⢷⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⣰⠟⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣬⠛⠛⠛⢻⢠⣾⡁⠀⠸⣇⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢀⡾⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠈⠐⠻⣷⠉⢳⡄⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠟⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⣠⡿⠀⠀⢻⣄⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠙⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⣧⠀⠀⣠⠶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⡁⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢷⡼⠋⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣶⣶⣤⣤⣄⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠋⠉⢉⡿⠛⠛⠛⠛⠛⠛⢿⡿⠛⠻⠿⣿⠟⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣞⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠉⠷⡄⠀⠀⠀⠀⠈⢷⡄⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢹⡆⠀⠀⠀⠀⠀⠛⣄⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣧⢀⣀⣼⡀⠀⠀⠀⠀⠀⠀⣷⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠉⠉⠉⠙⠿⠓⠒⠚⠒⠙⡛⣩⡄⠀⣈⠉⠀⢻⡄⠀⠀⣠⡴⠞⠁⠀⢀⡀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣀⣌⣀⠀⠀⠀⠈⢁⣀⣤⡶⠛⣧⣀⣼⡷⠔⠛⢁⠀⠈⠁⣠⣾⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠻⣿⡛⠛⠛⠓⠊⠉⠉⠈⢣⠀⠀⠈⢀⣠⡾⠋⠀⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⣀⣀⣀⣤⣤⣼⣦⡶⠒⠋⠁
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠿⣿⣿⠁⠀⠀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⢀⣀⠀⠀⠀⢸⣿⣆⣀⣀⡴⠲⢤⣀⠀⠉⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠉⠁⢀⣀⣀⣤⡤⠤⠶⠖⠚⠛⠋⠉⠙⢦⣀⡀⠀⣿⣟⠉⠁⠀⠀⠀⠈⠙⠓⠦⣄⣀⠉⠙⠻⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⡀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢲⠋⠀⠀⠀⠀⠀⢠⡞⠛⢷⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡆⠀⠀⠀⢠⡞⠉⢳⡄⠀⠀⠀⠀⠸⣇⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡆⠀⠀⠀⠀⠀⠘⠧⡤⠞⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣇⠀⠀⠀⠈⠳⠤⠞⠁⠀⠀⠀⠀⠀⢹⡇⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⠀⢻⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣄⡀⠀⠀⢸⡄⠘⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⡞⠀⣸⣿⣧⣀
⠀⠀⠀⢀⣠⣶⡿⠟⠉⢹⣿⡀⠸⡇⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⠀⠈⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡾⠁⣰⣿⠋⠙⠿⣷⣦⣀
⠀⣠⣶⡿⠟⠉⠀⠀⠀⠈⣿⡇⠀⣧⠀⠀⣾⣿⡿⠋⣿⣿⠁⢀⣀⠈⢿⣿⣇⠀⢿⠀⢹⣿⠋⣀⡀⠙⣿⣿⡇⠀⠀⣼⠁⢠⣿⠏⠀⠀⠀⠈⠙⠿⣷⣦⡀
⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⠀⢸⣿⡿⠁⠰⢟⡋⠀⣭⣭⠀⢰⣿⣿⠀⢸⡄⠘⣿⣷⣭⡃⠀⣿⣿⡇⠀⣸⠃⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡀⠸⡇⠈⣿⣿⣆⠈⠉⠁⢠⣿⣇⣠⣾⣿⡟⠀⠈⣇⠀⣿⣿⣀⣠⣾⣾⡿⠀⣰⠃⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿
⣿⡏⠙⠻⢿⣶⣄⡀⠀⠀⠈⣿⡇⠀⣧⠀⠘⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⢻⠀⢸⣿⣿⣿⡿⠋⠀⣰⠏⢀⣾⡟⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⣿
⣿⡇⠀⠀⠀⠉⠻⢿⣶⣄⡀⢻⣿⠀⢹⡀⠀⠀⠉⠻⠿⢿⣿⠿⠿⠛⠁⠀⠀⠀⠀⢸⡆⠘⣿⡏⠁⠀⠀⢠⠏⠀⣾⡿⠁⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣷⠀⠀⢠⡏⠀⣼⡿⠁⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⢠⡴⠲⣤⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿⡀⢀⡞⠀⣼⣿⣷⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⡀⠀⠀⠀⠀⠀⠸⣄⣀⡼⠀⠀⠀⠀⠀⠀⣸⡆⠈⣿⣇⡾⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢿⣿⠁⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠛⢦⡀⣀⣀⣠⣤⠤⠴⠶⠒⠚⠛⠉⠁⠀⢀⣀⣾⡿⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠈⠉⠉⢀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⡿⠛⠋⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣶⣶⣶⣿⠿⠿⠿⠛⠛⠛⠉⠙⠿⠟⠉⠙⠻⢿⣷⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠙⢿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟⠁

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⣾⣿⣿⣶⠤⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠐⣛⣉⣤⡀
⠀⠀⠀⠀⠋⢻⣿⡿⠀⠀⠀⠀⠀⣾⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⠋⠁
⠀⠀⠀⢀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣻⣯⣀⣀⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⣴⣾⣿⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡆⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣾⡿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠛⠿⠿⠰⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠙⢿⣦
⠀⢀⣿⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡜⢿⡇⠀⠀⠀⠀⠀⣠⣿⣃⣈⣿⣿⣿⡿⠁⣿⡀⠀⣸⡏⢹⣿⣿⣿⠁⠀⢀⣴⡿
⠀⠈⢿⡄⠈⢛⣿⣯⣿⠄⠀⠀⢀⡴⢀⠘⠄⣸⡇⠀⠀⠀⠀⠐⠛⠛⠉⠙⣿⣿⡿⠁⢀⣼⠇⢠⣿⠁⠸⠛⠷⣖⣤⡶⠛⠁
⠀⠀⠈⣷⠀⢾⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⣿⠀⠀⠐⠤⣀⠀⠀⠀⣠⣿⣿⣿⣧⡾⠛⠁⢀⡿⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣿⡄⢀⣿⣿⣿⣿⠀⠀⠘⠘⠀⠀⠀⣿⡀⠄⠀⠀⠈⠙⣷⢰⣿⣿⣿⣿⣿⠇⠀⢀⣾⠁
⠀⠀⠀⠈⠁⠘⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⠀⠈⠙⠀⠈⠢⣄⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠈⠃⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⡀⢠⣤⣿⡀⠀⣀⡤⠂⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⢠⣿⡟⠀⢤⣴⣤⠄
⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⢰⣿⣏⠙⠀⠉⠁⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⠀⠻⠅⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣸⠏⣿⠇⠀⠀⠀⠀⢸⠏⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡿
⠀⠀⠀⣠⡏⣠⣿⠀⠀⠀⠀⢀⣿⠀⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿
⠀⠀⣰⡿⠠⢻⡿⠀⠀⠀⠀⢸⡏⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⡄
⠀⠘⠟⠁⠀⠸⣇⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⣩⡑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠟⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣤⣬⣤⣤⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢀⡀⠴⠆⣀⠠⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠐⠐⠀⠐⠀⠂⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⡁⠀⢀⣀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠤⠤⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠈⠁⠀⠈⠁⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡶⢺⡓⣢⡒⢪⠗⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣄⠠⣤⡄⢀⠎
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢪⠟⢦⢣⠈⣏⣹⡉⡏⢠⠖⢸⠉⠒⢄⠀⠀⠀⠀⠀⠀⢾⣿⣿⠗⠤⠖⡇
⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⠀⠸⠈⠣⡘⣿⡿⣠⠊⠰⣿⡷⠀⠈⡄⠀⠀⠀⠀⣠⡼⣿⣿⣦⣀⣤⣾⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣇⣀⣒⣒⣑⣀⣀⣙⣿⣗⣁⣀⣀⣈⣀⣀⣀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣥⣤⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⠀⠀⠀⠀⠀⠀⢧⡤⠶⠶⠶⠶⢲⠖⠛⠛⠛⠛⠛⠛⠁⠀⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⣄⠀⠀⠀⣰⠛⣆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠙⠦⡴⠁⠀⠈⢦⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠛⠁⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠈⠙⠛⠛⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠉⠁

⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣮⣭⣉⣉⡙⠛⠛⠛⠛⢋⣉⣉⣭⣽⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣠⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠉⠛⢶⡶⢶⡶⠛⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⣸⠃⠀⢷⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠚⠁⠀⠀⠀⠙⠦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⣶⣾⣿⠿⠛⠛⠿⣿⣷⣶⡶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣤⣄⣠⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠿⠛⠛⠿⢿⡿

⠀⠀⠠⡀⠀⠀⢤⡀⠀⠂⢦⠀⠀⠚⠀⠀⠀⠀⢹⡆⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠄
⠀⠀⠀⠈⠢⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇⠀⠒⠰⠻⡧⢀⠀⠰⠖⠓⠀⢠⣀⠁⠈⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠒⠄⠀⠈⠁⢈⠴⠚⠁⠀⠀⣀⠁⠀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠰⠁⠀⠀⠀⠉⠠⠐⠊⢀⠄⠈⠩⠵⠂⢀⣄⠄
⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⢸⣶⣆⠀⢆⠀⠀⠈⠀⠉⢁⣀⡤⠀⠒⢉⠜⡠⢀⠃⠐⠂
⠀⠐⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⢸⣿⡏⠀⠀⠁⠀⠀⠀⠀⠠⡶⠿⠷⠗⠚⢊⣴⣧⡀
⠀⠀⠀⠀⡁⠀⠀⡁⢘⠀⠀⠀⠀⣿⠀⠀⠀⠀⢺⣿⡇⠐⠀⠀⠀⢠⡆⠀⠀⠈⠓⠶⠠⠶⠻⠀⠙⠛⣠⠄
⠀⠀⠀⠀⡄⠀⠀⠁⢈⠀⠀⠀⠀⠸⠀⠀⣀⠀⠼⠛⠛⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠹⣿⣄⣀
⠀⠀⠀⠀⡀⠀⣀⠆⠀⢀⠤⠀⠄⠉⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⢸⡇⢈⢀⡤⠐⠂⠀⠀⠁⠀⠈⢘⣿⠟⠋
⠀⠂⠄⠲⠀⠀⠀⠀⠐⠭⠿⠿⡖⢂⡀⠀⠀⠀⢠⣄⡀⠅⠀⠀⢨⣿⡏⢠⡠⠧⠖⠀⠀⠰⠏⡀⠀⠼⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠄⠩⢈⣀⣋⣉⣁⡄⡀⠀⠀⠈⣿⠧⠴⠓⣺⣉⠲⣄⠉⢳⠀⠀⠀⠤⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠟⣻⣥⢬⣴⣿⣽⣷⠂⠀⠀⢘⣿⣿⣅⠀⠀⠀⣨⠟⢀⣀⣤⡴⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠡⠖⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣅⠀⡀⠀⣰⠍⠩⠃


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣦⣶⣶⣴⣦⣴⣿⣴⣷⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠉⠉⠉⠉⠉⠉⠁⠐⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡌⠉⠢⣀⣀⣀⣀⡀⠠⠤⠤⠀⠐⢂⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠇⣠⠶⠶⠄⠀⠀⠰⠶⢦⣀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠚⠁⠐⢦⠀⠀⢀⡄⠒⠲⠦⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣾⣷⠓⢲⡏⠀⣾⣷⠀⠘⣤⢄⣀⠀⠀⠀⡇⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠘⣁⠆⠈⢧⠀⠘⠁⠀⣠⠃⠀⠀⠉⠛⠰⡶⣤⢞⣉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠑⠒⠺⡋⠀⠀⠀⠑⠒⠐⠚⠁⠀⠀⠀⠀⠀⠀⢳⠀⠈⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠀⠀⠀⢀⡄⠠⠤⠤⠄⠚⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⢹⠀⠀⠀⠀⠀⢀⡀⠀⠄⠛⠀⠀⠈⠒⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠊⡉⡆⠀⢸⠉⠉⠈⠉⠉⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠀⠙⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⡇⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠉⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⢰⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⡰⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⢸⢸⠀⠀⠀⠀⠰⠁⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠘⠀⠀⠀⠀⠀⢸⡇⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⡴⡉⠲⠟⠳⠞⡙⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡇⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⢀⠃⠀⠀⡇⠀⠀⠊⡘⣀⣀⣇⣀⣀⣸⣇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⢡⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⢀⡈⠀⠀⢸⡇⠀⠸⠀⠟⠉⠉⢩⠉⠉⠙⠛⠸⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢀⣇⣀⣀⣜⡈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⡜⠁⠀⠀⢸⡇⠀⡠⣶⠀⠀⠀⢸⠀⠀⢸⠀⠷⠖⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⠊⢁⡤⢤⡈⠓⡄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⢀⠃⠀⠀⠀⠠⡇⠀⠇⠈⠀⠀⠀⠘⠀⠀⢸⠀⠸⠀⡆⠀⠀⠀⠀⠀⡇⠀⢠⠁⢀⠁⠀⠀⠈⠀⢸⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡘⠀⠀⠀⠀⠀⡇⢰⠠⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⡆⢠⠀⠀⠀⠀⠀⠇⠀⠀⣄⠈⢦⡀⢀⡴⠁⡌⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢀⠃⠀⠀⠀⠀⠀⠃⠘⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡎⡄⠀⠀⠀⠀⠀⠀⢀⠋⠒⠤⠀⠀⠤⠚⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⠀⠀⠀⠀⢰⠀⠀⢀⣀⣤⠗⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢿⣿⣦⠀⢸⢈⡷⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠀⠈⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⡆⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣷⣶⣶⣶⣶⣶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣷⣶⣾⣷⣿⣿⣿⣿⣿⣿⣶⣶⣾⣷⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠂⠀⡄⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⣠⣤⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠂⠀⠀⠀⠹⠉⠛⠋⢁⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡞⠻⣤⣶⣿⣁⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⣤⠀⠀⡄⣄⡝⢃⡼⠷⠃⠈⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣧⠄⠠⠀⠹⣶⠁⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠘⠡⣀⠀⢠⣿⣟⡀⠠⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣷⠂⠀⠀⢸⣏⢷⠈⢢⣜⣳⢆⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⡀⠀⢰⣿⡇⠀⠀⠈⣿⢸⡇⡄⠙⢣⣴⠋⠑⢆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⣾⠿⠃⠀⠀⠀⠈⢸⠇⡇⠀⢈⠁⢀⣉⡋⠓⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠙⢷⣼⣥⠀⠀⠀⠀⠀⠀⣸⠀⡇⢀⣼⠟⢩⣤⣿⠓⠀⠀⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠌⢻⡀⠀⠀⠀⠀⢀⡇⠀⠁⢹⠿⠉⠉⠛⠋⢦⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡘⠀⡀⢠⠌⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⣤⠠⠉⢹⣰⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⣀⣀⣀⠴⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⠆⠀⠄⠀⡤⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⡁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠀⠸⢇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠆⠹⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⡜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢸⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡀⣀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂⠁⠠⠁⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠴⠀⠺⢠⡄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⡡⠀⠄⠀⣔⣭⣄⡁⡀⠀⠀⠀⠘⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣳⡀⢀⠀⠠⠄⠀⠀⠀⠀⢀⠀⢠⠉⠛⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⠈⠠⠼⠀⠀⠓⠲⡖⣦⠀⠊⣴⠆⢀⠀⡔⡸⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⠂⠀⠀⠀⠀⢀⠂⠀⠀⠀⠠⣀⢉⠗⢦⠀⠁⠀⡠⠋⠀⠆⠀⢀⡀⠀⡀⠀⠄⠀⠀⠚⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠐⠁⠀⠀⠀⠀⠧⠆⠀⡀⠀⠀⢈⠙⠒⠊⣐⣢⡞⣒⢂⣀⢈⡤⢦⡍⠀⠈⠁⠀⢰⠖⢺⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⡠⠀⠀⢀⠀⠀⣤⠀⢀⠰⣾⡷⠄⢐⠉⠙⣇⣭⣒⣩⡀⠉⠁⠁⠀⠀⡔⠂⠘⠢⠜⠳⣆⢱⡦⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠟⠑⢄⠀⠀⠛⠀⠃⢀⡀⠀⠠⢻⡍⠀⠀⣿⠂⠀⠀⠀⠀⠐⡇⠀⠢⢰⣦⣄⡀⢠⠀⠈⠀⠀⡂⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⢈⡉⠐⠀⣠⢁⠄⠤⠀⠀⠙⢆⢀⣏⡴⠶⢗⠀⠄⠀⠀⠀⠆⠈⠛⠋⠁⢀⡖⡚⠂⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⣀⠈⢉⡀⣲⢤⣾⡀⢘⣿⡋⠀⢀⣨⠟⠂⢀⡀⠀⠈⠊⡰⠀⡄⠜⠀⠀⠠⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠰⠥⠤⣀⠴⠛⠃⠣⠀⠀⢹⡟⠁⠉⠛⣠⣥⣄⠀⢀⣉⣀⡀⠀⠀⠀⠏⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢦⠀⢀⣠⠖⠚⣧⡀⠀⠀⠀⠖⢮⡇⢠⠤⠤⡻⠿⠟⢀⡀⣸⠋⠙⣀⣠⡞⠀⠀⠀⠀⠀⠸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⡀⠈⢰⠊⠉⠓⢤⣾⡇⠀⢀⡇⢈⠂⠀⡧⠀⠀⠣⢸⢹⡞⠉⠁⠀⠈⡆⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⠿⣿⡇⠀⠀⠀⠀⠘⠛⠛⠛⣶⣾⣇⠹⠶⣶⣷⠶⠶⣤⠾⠋⠙⠦⠆⠁⠐⠁⠈⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠋⠀⠀⠀⠀⠀⣠⠆⠄⠠⠈⢿⣿⣦⣼⠇⠀⠀⠀⠘⠓⠀⠀⠀⠀⣒⣦⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⢼⣷⣦⣌⡙⠂⠄⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠉⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⣐⠀⠈⠭⠍⠁⠀⠀⠀⠐⠀⠀⠀⠠⢄⠀⣀⣀⢀⡠⠼⠛⠛⠿⣧⣤⠀⠄⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠐⠂⠀⠈⢉⣀⡠⢔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠀⠉⢉⣉⣁⣀⣀⣀⣀⣐⣒⣀⣒⣀⣤⣤⣤⣿⣗⣋⣐⣿⡿⠦⠁⠀⠀⠀⠉⠩⠭⠭⠭⠭⠥⠄⠙⠛⠛⠓⣒⠨⠇
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠋⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⢸⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠄⠠⢄⠀⢀⠤⠀⠠⢄⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠊⠉⢨⠀⠀⠀⠀⢱⠃⠀⠀⠀⢀⠍⠉⠲⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⢼⢆⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⡴⠋⠈⠢⢄⡀⠀⠀⠀⡀⢸⠀⡄⠀⠀⠀⢠⠠⠔⠈⠑⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⢰⠀⠀⡄⠀⠀⠁⠀⠀⠀⠃⢸⠀⠁⠀⠀⠀⠊⠀⠀⡀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠸⡀⠀⣈⢀⡀⠀⠴⠂⠀⠀⢸⠀⠀⠀⠐⠤⠀⢰⢀⡁⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠑⡌⠁⠀⠁⠀⠀⠀⠎⠠⢼⡦⠄⠢⠀⠀⠀⠉⠀⠈⡴⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣄⠠⡀⠀⢀⡀⠀⣠⠋⠳⡀⠀⢀⠀⠀⣠⠄⡰⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠑⢢⣤⣤⠕⠊⠁⠀⠀⠈⠒⠤⠤⠤⠔⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⢸⠇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣀⣀⣀⣀⣀⣀⣼⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⠤⣒⣉⠡⠤⠄⠒⠒⠢⢤⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⡇⣠⡴⠖⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⡖⠒⠒⠒⠒⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣄⣀⠀⢀⣠⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣾⠀⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠙⠀⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡴⣛⣭⣭⣮⣭⣭⡓⢦⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⢫⣾⣿⡿⠛⠈⠻⣿⣿⣦⡹⡎⢻⣿⡿⠿⣿⣿⠿⣿⣟⣯⣵⣿⣿⣿⠁⠙⣿⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⡅⠺⢸⣧⣼⣿⣿⣷⢹⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣧⣀⠈⠀⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡿⣷⢠⡉⢻⣿⣿⣿⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⢷⡹⣿⣿⣧⣈⠈⣁⣼⣿⣿⢣⡟⣴⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠳⣬⡻⠿⣿⣾⡿⠿⣛⡵⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠛⠉⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣁⠴⠚⠉⡥⠟⠒⠛⠛⠓⠒⢒⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢠⠞⡶⠲⠶⠶⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠞⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠙⢎⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣏⣀⣤⣾⣛⣛⡃⠀⠀⠀⠀⢙⣛⣛⣷⣦⣀⡈⢣⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⡟⠉⠈⠩⠍⠉⠙⣷⣠⣀⣼⠋⠉⠉⠄⠈⠉⢻⣿⣦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⣹⣷⠀⠀⢠⣶⣆⠀⡿⠀⠀⢻⠀⣴⣖⡀⠀⠀⣸⢉⡽⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠈⡟⢦⣀⣼⣿⣿⡼⠃⠀⠀⠈⢧⣽⣿⣃⣀⡴⠏⡘⠁⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⣣⠐⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣆⠀⠀⠀⠀⠀⠈⠒⠒⠊⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⡀⠀⠀⠀⢀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⠏⠈⠉⠋⡭⠙⢉⠁⠹⣼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠱⠈⡑⠀⠳⠜⠈⠤⡎⠐⣊⠃⠇⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠈⠢⠀⠆⠀⡌⢀⠜⡇⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢾⡿⠀⠀⠀⠀⠀⡆⠐⠁⣸⠦⢤⡶⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠐⣆⠁⠀⠀⠀⠀⠀⣧⠀⠀⢧⣀⡸⠁⢩⡖⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠐⠀⠬⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠌⠢⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣠⠋⠀⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⡄⠐⠈⢣⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠀⠀⠀⢰⠀⠀⠀⢰⠀⠀⠀⠀⠀⠁⠀⠀⠀⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣄⣀⣀⠀⠀⠀⡼⢸⠀⠀⠀⢀⣀⣀⣠⡤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠭⢉⡉⠱⡟⠉⢻⠍⢉⡉⠥⠻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⣀⠇⠀⠸⣄⡀⠈⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣷⣶⣯⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣶⣶⡆⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣖⠀⢻⣿⠀⢰⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣆⠘⡇⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢿⣿⣿⣿⣧⣰⣿⣿⣿⣿⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠈⢿⣿⣿⣟⣹⣿⣿⣿⠃⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣄⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⡇⠀⠀⠀⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢈⣽⣶⣀⣴⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣻⠿⣻⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢻⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠹⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⠀⠀⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠀⠻⣿⡿⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠀⠀⣠⢴⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠶⢒⣚⡓⠲⣄⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡞⠀⡞⠁⣠⡿⠋⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣤⠴⠋⠉⠙⠷⣄⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠤⡀⢨⣠⠼⣁⡤⠬⣏⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠱⠤⣄
⠀⢠⠞⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⢣⡀⠀⠀⠈⢻⣦⠀⠀⠀⢷⡄⠈⢦⠀⣼⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠈⢳
⣰⢋⡶⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠱⡄⠀⠀⠀⠙⣷⡄⠀⠀⠙⢦⣠⠇⠘⠶⠶⠶⠋⢀⡀⠀⠀⠀⠸⡇
⢻⡸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠈⢯⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣌⢢⠀⠀⠀⢷⠀⠐⠀⠀⠀⠀⡄
⠀⠻⢮⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎⡀⣼⠃⠀⠀⠀⠀⠀⠑⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡸⡆⠀⠀⠘⡄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠙⠲⣌⡙⠲⢤⣄⣀⡀⠀⠀⣀⣠⠴⢏⠀⠀⣟⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠢⣄⠀⠀⠀⠀⠀⠘⡌⡇⢧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣈⣉⣉⣉⣭⡴⠚⠉⠉⢣⠈⣆⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⢦⡀⠀⠀⠀⠀⢣⢻⡘⡆⠀⠀⢳⡀⠐⠀⠒⠊⢀
⠀⣀⡀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠳⡀⠀⠀⠀⠸⡎⣇⢱⠀⠀⠀⠙⠓⠒⡄⡞⠁
⠀⡏⠉⢉⠁⠀⠀⠈⠉⠓⠦⣤⡀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠹⣄⠀⠀⠀⢻⣹⡈⡆⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⢸⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⠀⠀⠈⡎⣧⢳⠀⠀⠀⠀⠀⢸⢳
⠀⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⢤⠀⠀⠀⠳⣀⠙⠲⠤⣤⣤⣤⣤⣤⠄⣀⡀⠀⠘⢆⠀⠀⠀⠣⣠⣀⠹⣼⣮⡆⠀⠀⠀⠀⢸⣾
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⢳⣾⠁⠀⠀⠀⠈⠉⠃⠒⠒⠒⠒⠒⠒⠒⠦⣌⠙⢦⡈⢧⠀⠀⠀⠀⣸⠇⠈⣿⣆⠀⠀⠀⢠⡿⠃
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣟⣿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠈⢧⠀⣧⢸⣀⣀⡤⠊⠱⠤⢄⠘⠛⠀⢀⣴⡟⠁
⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⠿⠀⠈⠑⠲⣄⠀⠀⠀⠀⢀⡜⡵⢺⡇⠀⠀⠀⡼⢀⡇⠀⠉⠀⡟⠀⠀⣀⣸⠀⠀⣠⡾⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡂⠀⠀⠀⠀⠈⠙⠒⠒⠒⢺⠚⢀⣾⠇⣠⠶⠚⠀⠜⠲⢦⠀⠘⢦⡤⠞⠉⠀⠀⢰⢹⠁
⠉⠒⠢⠤⢄⣀⣀⣀⣀⣀⣀⡀⣿⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠿⠃⢸⡃⠀⠀⠀⠀⠀⢈⠇⠀⠀⢀⣀⣀⡀⠀⠈⢯⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠉⠛⠲⠶⠒⠚⠉⠀⠀⢰⣫⠒⠚⠯⢽⣖⡯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠐⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⢠⣀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢀⢈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠥⠂⠈⠁⠐⠰⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⣀⡀⠀⠀⢀⣤⡔⠂⢸⠞
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢠⠀⠈⣿⣷⠆⠀⠈⠻⠟⠁⠘⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣦⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣄⠀⠄⣤⠄⠠⣀⠀⠀⠀⠀⢀⠀⡼⡄⠀⠀⠀⠀⠀⢰⠒⣲⡄⠀⠀⠀⠀⠀⡟⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⡀⠑⢄⡀⠨⠤⠀⠀⠀⠈⢀⠀⠘⠀⣰⣆⠀⠀⢹⠛⢛⡆⠀⠀⠀⠀⠀⡇⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠫⣁⣄⣠⡀⠀⠀⡀⠀⠀⠀⡰⠧⠼⡄⠀⠀⠉⠉⠀⠀⣀⣀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⣷⣬⣦⣝⠩⠒⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⣴⠋⠁⠀⠀⢰⡇⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠥⣀⣨⠉⠐⠂⢈⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠤⠄⢸⡇⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠴⠐⠂⢀⠀⠐⠂⠀⣁⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣖⡀⣀⠀⠒⠀⣈⢀⣐⣢⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠃
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⢈⣉⠋⠋⠙⠙⠉⢉⠁⠀⢀⢈⡉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠁⣧
⠀⠀⠂⠀⠁⠀⠃⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣡⠀⠀⠀⠻⠀⠁⠖⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄

⠀⣀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢾⡶⠄
⠀⠉⠹⠉⢁⣀⡀⠀⠀⠀⣟⣻⣿⣛⡆⠀⠀⠀⠀⠰⣿⣿⣿⠄⠘⠀⠀⠐⠂⠂⠀⠐⡷⣄⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⡇⠀⠀⠀⠀⣠⣤⣶⣶⣤
⠀⠀⠀⢠⡿⠟⢿⡀⠀⠀⣯⣽⣿⣭⡇⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⣯⢭⠉⠭⣹
⠀⠀⠀⣾⣧⢀⣼⡇⠀⠀⡷⢺⣿⢲⡇⠀⠀⠀⠀⠘⠛⣛⠛⠂⠀⠀⠠⠀⠀⠀⡄⡠⢤⠀⡸⠀⠀⠐⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠋⠀⠉⠀⠹⠀⠀⣏⣹⣿⣩⡇⠀⠀⠀⠀⢀⣀⣛⣁⡀⠀⠀⠀⠑⢤⠔⠀⠳⢤⠗⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⣶⠐
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠷⠾⠿⠶⠇⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠈⠢⠤⠤⠊⢀⣀⣇⣀⠈⠓⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠉⣿⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠐⣶⣿⣶⠀⠀⢸⣿⠀⡆⢸⠋⡈⢿⠁⣌⣥⡄⣤⠏⣀⢹⠁⡌⣻⣿⡇⢠⣸⠁⣿⠁⡏⢠⢙⡏⠈⣿⠁⡄⢻⡄⢠⣼⣿
⠀⠀⠀⠀⠉⠉⠉⠀⠀⢸⣿⠀⠃⣼⠀⡇⣠⠀⡿⣿⡇⣿⠀⣿⢸⠀⠀⢿⣿⡀⢸⢿⠀⣿⠀⡇⢀⣾⠁⠀⢹⡀⡀⢿⡇⢸⣿⣿⠀⢰⣶⢲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣼⣿⣦⣤⣿⣤⣴⣿⣇⣽⣧⣤⣾⣄⣷⣼⣿⣧⣀⣼⣤⣡⣤⣧⣸⣿⣼⣧⣼⣇⣿⣼⣧⣸⣿⡿⠀⠀⣿⢸⣿
⠀⢰⡄⠀⢀⣀⡀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣿⢸⡏
⠉⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠪⢀⡦⡆⠴⠀⠸⠄⠂⠔⡢⠀⠜⡌⠒⠤⠐⠄⠀⠀⠰⠠⠢⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏
⠀⢀⣴⣶⣶⣠⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣶⡆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣤⡄⠀⠀⢀⡀⡠⠤⠤⠤⢤⢀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣿⣿⣿⡿⠿⢿⣿⣿⣿
⠀⠈⢿⣿⣿⣿⣯⣿⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⢀⢀⣀⣀⣀⣀⣀⢠⣤⣤⠀⠀⠀⠀⣿⣿⣿⠰⣿⠆⣿⣿⣿
⠀⠀⠀⠙⠿⣿⠿⠋⠀⠀⠤⣾⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠜⠓⠚⠒⠒⠚⠛⠛⠛⠸⠿⠟⠀⠀⠀⠀⣿⣿⣿⣷⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣦⡶⠞⠋⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⣿⣿⡏⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⢨⢤⣀⡤⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⢿⡇⠀⢠⡇⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢂⡺⡿⠚⠉⠓⠤⠞⠙⠒⠥⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⣪⣃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠆⠀⠀⠀⠸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢤⠀⠀⠀⠀⢣⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠘⡄⠀⠩⡕⠲⠤⢀⣀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠤⢝⠳⣄⠀⠀⢱⠀⠀⢠⠀⠀⠀⠀⠀⢀⠄⢤⠂⡠⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢪⠂⠼⣀⠈⠓⠶⢤⣇⣐⠙⠀⠀⠀⠀⠀⢠⠀⢆⠂⢖⣊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢠⠄⠀⠉⠀⠒⢺⣾⣧⣦⣀⠀⠀⣠⠆⢀⣸⡷⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠙⡪⣟⢫⣑⡴⣁⣽⣿⣩⠔⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⠀⠀⠑⠤⢕⠣⣼⡁⢨⡉⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠈⡀⠀⠀⠀⠈⠀⠀⢁⡀⠙⡦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠉⠑⠤⡀⠀⠀⠀⠈⠗⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⣠⠶⠂⠀⠀⠀⠀⡄⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⢊⠀⠀⠀⠀⠀⠀⠀⢸⣄⢀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⠤⠤⠤⠤⠤⠀⣤⠤⠘⣁⠝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠠⠉⠙⢳⠤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢺⠛⠙⠠⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠭⠵⠖⠦⠤⠀⠀⠀⠀⠸⠤⠤⠴⠶⠪⠥⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠉⠛⠻⢍⣙⣟⣛⠻⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣻⠧⠰⠿⢋⣤⣬⣥⠀⠈⢿⠟⠋⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣷⡇⠠⢅⣿⡾⠟⠀⠀⠈⡎⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠸⢤⠀⠀⠀⣀⠀⠀⠀⢰⠞⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡘⢷⣖⠚⢛⣻⠟⠀⠀⣠⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣙⠛⠛⠁⢀⣠⠞⢹⣿⣿⣿⡷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢊⣿⠀⠀⢿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⣠⣴⡒⠉⡩⠉⠝⠋⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⢉⣽⣿⡿⠀⠀⣼⣿⠑⠢⣄⣀
⠀⠀⠀⠀⠀⠀⠑⠦⠤⣈⣀⣀⣀⡀⢤⣄⠜⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⠆⣠⠟⣼⡏⠀⣠⣾⠃⣿⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⢀⡴⠋⠑⢦⡀⠀⠀⠀⠀⢀⡴⠋⠀⡰⢃⢨⡟⢸⣿⣷⣾⡿⠃⢰⡏⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣄⠀⠀⠀⠈⠳⣄⢀⠴⠋⠀⢠⠞⠀⢠⣿⠄⣾⣿⣿⡿⢡⡴⠿⣏⠁⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢹⠄⠀⠀⠈⡂⠀⢸⡇⠀⣿⣿⡿⣡⠟⠄⠀⢽⠆⠀⠳⡄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⢀⡠⠞⢃⢰⣿⣧⠀⣿⡿⢁⡿⠀⠀⠀⣾⠀⠀⣰⠛⡄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡠⠒⠉⠀⠀⠸⠿⠿⠛⠀⣿⠁⢼⢧⠀⢀⣼⠃⠀⡴⠃⠀⠘⣆⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢰⡇⠀⠈⠋⠘⠛⠃⢀⡼⠁⠀⠀⠀⢈⠦⠒⠒⠂⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢰⢇⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠘⠐⠀⠀⠀⠀⣰⠃⠀⣠⠶⠋⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠀⠀⠀⠀⠐⠉⡷⠚⡅⠈⠢⣀⡤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⡇⠀⠈⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢧⢰⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠃⠱⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣦⣤⣤⣿⣷⡀⠀⠀⠀⠀⠀⢀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣟⣿⠛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⢦⡛⠘⢇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠀⠈⢢⢹⡄⠈⠁⠸⠒⠒⢒⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣈⠢⠴⠋⣼⢋⠓⠒⠒⠶⢲⠦⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢛⠟⠷⠶⠚⠙⠛⣷⡤⠤⠤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠉⠙⠳⣄⣰⠟⢩⣵⣤⠊⣩⢵⡼⠂⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠘⠳⣬⡼⠋⠱⣜⣿⡾⠅⠀⠀⠀⠼⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣤⠀⠀⠀⠀⠀⢀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⣠⠀⠀⠀⠟⠹⣧⠀⠀⠀⠀⠀⠸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⠾⠛⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠐⢷⣄⣀⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⢸⠇
⠀⢀⣀⣀⣤⠤⠴⠖⠒⠛⠛⠋⢉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡇⠀⠀⠀⠈⠉⠉⣡⡴⠟⠀⠀⠀⠀⠀⢀⡿
⢿⠉⠉⠁⠀⢀⠤⠄⠤⢄⡀⠀⢨⢸⡇⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠶⢦⡀
⢸⡀⠀⠀⠀⠁⠀⠀⠀⢀⡈⢆⠀⢸⡇⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⢸⠈⠙⡷⣦⣄⣀⣀⣀⣀⣀⣠⣴⠞⠋⠀⠀⠀⢸⠹⣆
⠈⡇⠀⠀⠀⣎⢁⠄⡸⡁⣰⢸⠀⢈⡇⠀⠀⠀⠀⣠⠎⡞⠀⠠⡔⠋⠀⢀⡿⠩⡈⣩⠟⣯⠉⠉⠸⠧⢤⣤⠀⠀⠀⢸⡄⠈⢷⡄
⠀⣧⠀⠀⠀⢄⠀⠀⠑⠀⠠⠃⠀⠀⡇⠀⠀⢀⡜⢁⣼⠁⠀⠀⠘⣆⠀⢸⡇⠀⣿⠁⠀⢸⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⠙⢦⡀⠙⢦⣄
⠀⢻⠀⠀⠀⠀⢧⣤⣤⡤⠆⠀⠀⠀⣿⠀⢀⡞⢀⡞⡌⠀⠀⠀⠀⠘⡄⢸⡇⣸⠘⡇⠀⣾⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸⠀⠀⠙⢦⡀⠙⠳⢦⣄
⠀⢸⡆⠀⠀⠀⠀⢀⣤⣖⠂⠀⠀⠀⢹⠖⠋⢁⠞⢠⠀⠀⠀⠀⠀⠀⠰⣼⢡⡏⠀⣿⠀⣿⢠⠟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠴⠚⠛⢀⣀⡬
⠀⠀⠇⠀⠀⠀⡤⡿⠈⡯⠱⡔⢦⠀⢸⡆⠒⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⣸⠁⠀⢹⡀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣏⠁⢀⡴⠛⠁
⠀⠀⢸⣀⣀⣸⡄⢿⠀⡇⠀⡇⣸⠀⠘⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠳⠟
⠀⠀⠀⠈⠣⠐⠈⠈⠛⠉⠚⠉⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⢁⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢰⡧⢷⡄⠀⣠⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠞⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠶⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠦⠤⢤⣤⣤⣼⠀⠝⠳⠶⢴⠄⢼⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡼⠀⠀⠀⣼⠀⢨⡏⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠰⡇⠀⠀⠀⢻⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢏⠀⢧⠀⠀⠀⠘⡆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠉⠓⣦⣄⣀⡀⡀⣸⣀⣸⣆⢀⣀⣀⣿⣀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠴⣏⠉⠙⠛⠓⠒⠒⠚⠛⠉⠉⠁⠀⣠⡍⠉⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠤⣤⣤⣤⣤⣤⡤⠴⠖⠋⠉⠳⠶⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠭⠴⠛⠉⠉⡩⢲⠎⢩⡏⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⣉⣀⡀⠀⠀⠀⠀⢛⠈⣢⣉⣴⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠺⠑⠒⠀⠈⠉⠒⠚⠳⠦⣀⡀⣠⣾⣿⣫⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⢟⠒⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡿⠧⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣒⠭⠒⠁⠀⠈⢄⡀⠀⠐⠭⠩⢛⣶⡦⣄⢀⢆⢆⣿⣿⣿⡿⠏⣏⣠⣷⣿
⠀⠀⠀⠀⠀⠀⣀⠠⠔⣊⣥⢖⠙⠓⡀⠀⠀⠀⠀⠀⠉⠒⠒⠚⢈⡡⣪⣾⣿⠀⢫⣾⣿⡿⠟⢁⣶⢿⣿⢻⢫⡇
⠀⠀⠀⠀⠀⢀⣿⢴⣢⣉⡁⢦⣀⠑⠒⡀⠀⠀⠀⠀⠀⢀⡠⣒⢵⣾⣾⣿⣿⠁⣿⣿⡟⠟⠚⠻⣿⣕⡕⣵⠋
⠀⠀⠀⠀⠀⠀⡏⠑⢶⡬⣉⡓⠶⣭⣕⣲⠊⣩⡽⠟⡩⢵⣺⣵⣿⣿⣿⣿⣯⣖⠛⠈⠀⠀⣀⡄⣿⣮⣾⠋⠱⠶⢤⣀
⠀⠀⠀⠀⠀⠠⡏⠀⠀⠙⡑⠪⢝⣒⢿⣾⣽⣓⣞⣿⣾⢿⣿⣿⣿⠿⠿⠏⠓⠒⣦⣤⣄⢦⡛⣄⣿⡿⣙⣑⣦⠀⠩⠃⣩⡕⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢭⣛⡲⠬⣅⡂⠉⢘⠛⣿⣶⣿⣧⣟⠛⠈⠤⠤⠔⡪⣻⣯⡟⢺⠁⣹⠟⠋⠉⡀⠺⢕⡶⢄⣠⣶⣷⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠳⠦⢌⣙⡋⣼⡿⠟⠋⠬⠍⠀⠀⠀⢀⢞⣵⢿⣿⡏⢃⡽⠓⠀⢀⠔⠁⠀⠀⠈⣉⡚⠻⢿⠷⢇⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⢉⠴⠉⢰⡁⠒⠂⢬⣙⣋⠉⠁⠀⠀⣀⣀⣀⢔⣕⣟⡷⣿⣿⣦⠯⠄⠀⠒⠁⠀⠀⠀⠀⠤⠀⠀⠀⡜⠀⢄⢨⠉⣱⡆
⠀⠀⠀⠀⠀⠀⢐⡯⠬⠥⠉⣀⣀⡒⠊⠭⠭⠶⣢⡤⠤⠊⢁⠖⣃⢴⡿⡯⣾⣾⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⠂⣸⢄⣠⠴⢊⠥⢻⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠤⢂⣐⡒⠒⠀⠭⠭⠤⠐⢀⣀⣀⡨⡭⠍⠐⣫⣺⣿⣿⣏⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⡚⠭⢒⡩⣴⣫⣵⣞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢅⣒⣒⣒⡰⠯⠭⠭⠥⠔⢛⣒⣐⡤⠭⡍⡭⣿⣿⠿⡟⠙⠿⣿⢤⣘⠓⢬⠑⠲⣔⡢⠭⣐⣻⣵⣶⣿⢿⣿⠿⠛⡋⢬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢕⠬⡛⢌⡙⠩⢋⢟⡹⡋⢞⠑⡋⠛⠝⡒⢛⢷⣕⡢⠀⠀⠀⠹⢦⢽⣒⣺⣽⣕⣛⣭⣶⣿⡯⠟⢛⠩⢔⣊⣥⠔⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠉⠢⢤⣄⡀⢀⣁⣈⣀⡁⠈⣈⣰⣥⡿⣻⣄⠀⠀⠀⠈⠃⢰⣿⣿⠿⢟⢚⣉⣥⡒⠥⢒⠮⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠨⣛⢥⡀⠴⠚⣋⣽⢿⣿⣿⡿⣿⣟⠝⠒⢄⡀⠀⠐⣗⣖⣯⣽⠿⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠸⡩⡉⠹⢿⣻⣷⣟⣯⡿⠟⠊⠀⠀⠀⠀⠈⠁⠊⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⡀⠈⡆⢐⣿⡿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢐⠀⠂⠐⡠⠂⠔⡠⠂⡀⠀⠄⡠⢀⠀⠀⠀⠠⢀⠄⠀⠀⡠⢂⠐⠀⠀⠠⢀⠄⠀⢀⠔⡠⡠⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠊⠔⠥⢂⠔⢈⢔⡠⢊⠔⢊⠔⡠⢂⠑⡨⠐⠡⢊⠐⡡⢂⠀⡠⡠⢊⠄⠁⢒⡑⡠⢂⡔⢃⠔⡡⢪⠊⠔⠑⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠘⠁⠊⠁⠊⠐⠁⠊⠐⠡⠂⡠⢊⠔⠡⠪⠊⠀⠁⠊⠀⠋⠀⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠐⠀⠨⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣠⣀⣤
⠀⠀⠀⠲⠀⠁⠀⠀⠀⠈⡔⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣼⠃⠀⠀⣠⣤⡄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢣⠐⠠⠤⠄⠠⠰⡿⡿⢿⠿⡿⠿⠿⢿⡟⠀⠀⠀⠸⠳⣿⡒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡂⠀⠀⠀⠁⠁⠉⡉⢉⢉⠩⠉⠉⢹⣇⠀⠀⠀⠀⠁⠁⠀⠀⠀⢤⣄⡀
⠀⠀⠀⠀⠠⠄⠤⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠀⠀⠀⠘⡧⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢖⠑⠛⣿⠀⠀⠀⣸⠀⠀⠀⣀⠀⢰⡏
⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣐⣾⠀⠀⠀⢸⡆⠀⠚⠺⢯⡿
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠠⠐⠃⠀⠉⠀⠀⠀⠀⠀⠉⠈⠉⠁⠁⠀⠀⠀⠀⠀⢚⡇
⠀⠀⠀⠀⠀⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠉⠉⠉⠁⠐⠂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡜⠉⠂⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⣤⡄⠀⠀⣤⣄⡀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠛⢒⣄⠀⢀⠄⡂⠥⡀⠀⠀⠀⠐⠄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠻⢿⠉⡇⠸⠿⠀⠹⠒⠠⠄⣀⡀⢀⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠄⢴⠋⠀⠘⠢⠤⠔⠁⠀⠀⠀⠀⢭⠑⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡠⠉⠂⡀
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢰⡏⠀⠀⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⡄⠀⠀⠰⠁⠀⠀⠈⠄
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠠⠁⠀⠀⠀⡠⠂⠹⠋⠉⢚⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢡⠶⠶⠗⠲⢾⡎⡀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⢀
⠀⠀⠀⡘⠀⠀⢠⠀⢸⡄⠀⠀⠀⠀⢸⠰⢦⠀⠀⠀⠀⢸⠀⠀⠔⢁⠀⣉⠣⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢠⠁⠀⠀⠀⠀⡆⠁⠀⠀⠀⠀⠀⠀⠂⠆⠀⠀⠀⠘⠀⠈⠀⡁⠀⠀⡂⡁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠆⠀⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⡄⠀⠘⡐⠀⠀⠀⠀⠀⢘⠆⢈⠂⢊⠀⠁⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠐⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠰⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣄⡀⠀⠀⣇⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠇⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⡆⠀⢀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣸⡀⠀⢀⠀⠀⠀⠀⣸⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢁⡀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣬⣿
⠀⠀⢰⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣿⡿⠉⡀
⠀⠀⢀⠱⣀⣠⣄⣤⠔⣂⠥⠤⢀⠀⠀⠀⠀⠘⢛⡛⢛⡛⠓⠀⠀⠀⠀⠸⠄⠀⠀⢀⡈⠁⠀⠠⠃⠀⠀⠀⣶⣤⣤⣀⣠⡄⡀
⠀⠈⢄⠣⣒⣒⣐⣒⣫⠔⠊⠁⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢾⣓⢀⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠒⠓⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡻⠴⢻⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣟⣛⣯⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⡄⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠸⠛⠓⠈⠛⠁⣿⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⣀⣤⣶⣄⡀⣿⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠧⢸⢽⣽⢽⠤⠿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣋⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠞⠻⠻⠛⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣿⣿⡷⢶
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣛⣿⣛⣛⣿⣛⣻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠘
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣉⣏⣹⣉⣉⣏⣩⣉⣉⠇⠀⠀⠀⠀⠀⠀⠀⢸⣾⢷⣶⣶⣦⣶⡾⠿⡇
⢀⡀⠀⣀⣾⣿⣀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣡⣨⣅⣨⣅⣼⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣷⣾⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠵⣷⣼⠮⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠟⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠱⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⣛⡳⣄⠀⠀⠀⠙⢦⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢡⠉⢈⡇⢸⡶⢤⣤⡀⠀⠁⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠈⠲⠊⢀⣞⠀⠀⠀⠈⠉⠲⠦⣤⡀⠈⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠳⠶⠖⠋⠹⡓⠶⠤⣀⡀⠀⠀⠀⠙⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠆⣩⣶⣤⠞⣹⡿⣿⣆⠀⠀⢨⡟⠶⣤⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠚⠻⢦⣈⣫⡇⠀⠻⠿⣿⠋⠀⠀⡞⠀⠀⠈⡇
⠀⠀⠀⡀⠀⠀⢀⣀⣀⡠⠤⠶⠶⠶⠶⠶⠶⣦⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠿⢶⣦⡜⠁⠀⠀⣸⠃⣠⠖⠚⠳⣄
⠀⠀⠀⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠲⠃⠀⣙⡆⢸⡆
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣉⡠⠟
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢫⠉⠙⠋⠉⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠶⣤⢿⡀⠈⠉⠳⢄⠀⠀⠀⣼⠁
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠠⢟⡀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠑⢄⠐⢯⣀
⠀⠀⠀⠀⣠⡦⠶⠲⠖⠲⣄⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⡰⠁⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠂⠀⠀⠁⢾⡁⠙⠶⣤⡀
⠀⠀⠀⢮⠄⠠⠤⠴⠞⠛⠻⣄⠀⠀⠀⠀⠀⣿⠸⡀⠀⠀⠀⡼⠁⠀⠀⠈⢲⠖⢶⣏⣀⣀⣀⣈⡷⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠱⡀
⠀⠀⠀⢧⣀⣀⣀⣠⡤⠴⢾⡏⠀⠀⠀⠀⠀⣿⠀⠙⣤⢀⣼⠁⠀⠀⠀⠀⢸⠀⢸⡠⠉⢻⡉⣽⠇⠀⠀⠀⠀⠀⠀⢹⠀⠀⡄⠀⢷
⠀⠀⠀⠀⢻⣤⣄⣀⣀⣠⣴⠏⠀⠀⠀⠀⠀⣿⠴⠊⠀⣾⠃⠀⠀⠀⠀⠀⢸⠀⢻⠀⣴⠊⢸⢿⠄⠀⠀⠀⠀⠀⠀⣼⠀⠀⣷⠀⢸⡇
⠀⠀⠀⠀⢿⡀⠉⢻⡏⠁⠀⠀⠀⠀⠀⠀⣀⣿⠀⢀⠗⡜⠀⠀⠀⠀⠀⢀⡴⠂⢸⡤⢇⠀⡾⠈⢧⠘⢦⡀⠀⠀⣠⠟⢳⠀⠸⡀⠈⣧
⠀⠀⠀⠀⠀⠙⣿⣋⡤⢶⡴⠒⠛⠛⠿⣍⠉⠁⣠⠏⢰⠇⠀⠀⠀⠀⠀⠻⡄⠀⠘⡇⢸⣄⡇⠀⠈⣻⠞⠉⠛⣿⠋⠀⠀⢷⠘⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠈⠹⠾⡁⠀⡿⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢳⠀⣿⠀⢀⡼⠁⠀⠀⠐⢷⡀⠀⠀⠘⣇⢱⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀⣤⢾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⣤⠋⢀⡞⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠙⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡿⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠘⢦⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠋⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠞⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⡀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠶⠦⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢰⡀⠀⠈⠉⠉⠉⠉⠛⠒⠶⠒⠲⡖⠚⠛⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢻⡄⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠛⢶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠾⠿⠛⠛⠛⠛⢧⠀⠀⠀⠀⠀⠻⣦⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣍⣉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠀⠀⠘⣇⣀⣠⡤⠖⠉⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⣶⡤⣤⣤⡤⠶⠶⠚⣛⣉⣠⡿⠤⠖⠛⠋⠉⠀⠀⠀⠀⢀⣼⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣟⣛⡛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⠋⠈⠛⠛⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣶⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠈⠻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣤⡄⠀⠀⠀⣤⣄⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⣠⣦⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠙⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⢀⣴⡆⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠻⢷⣤⣀⣀⠀⠀⢀⣀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠈⠙⠛⢻⣶⠊⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡼⠛⠁⢸⣿⣧⣤⣤⣀⠀⠀⠀⠀⠀⣸⡿⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⣿⣿⡟⠀⠀⠀⢸⣿⠋⠈⠉⠉⠀⠠⣶⣶⣾⣿⠄⠈⠻⣿⣿⡷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠀⣴⡆⠀⠀⠀⣿⣿⡇⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⣿⣿⡇⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⠀⠀⠀⠈⢃⣄⠀⠀⣿⣿⡇⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⣿⣿⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣿⣿⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⣿⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣠⣴⣿⣿⣷⣦⡀⠀⠹⠃
⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⢿⣿⠙⠻⣿⣦⡀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⠀⠹⣿⣿⣄⣀⡀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣤⣾⣿⣦⣤⣾⣿⡇⢸
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⣼⣟⠛⠛⠉⢱⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠉⠛⠛⠉⠙⠻⣿⣿⣿
⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⣄⠈⠻⠿⠀⠘⠃⣨⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠘⣿⣀⣴⡄⠀⠀⠀⠻⣿⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠇⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠿⣏⣀⣴⣦⣄⣼⣿⡇⠀⠀⠀⠀⠀⣰⣿⡿⠇⠀⠀⣾⡀⠀⠀⠀⠀⢠⣿⣿
⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⡟⠈⢻⡇⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⣿⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢩⣿
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣶⡇⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣿⡇⠀⠀⠀⠀⠀⢿⣿⣦⣀⡀⣿⣿⡄⠀⠀⢸⣆⣠⣿⣿
⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠀⠀⣴⣾⣿⡿⠛⠃
⣿⡃⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡇⠀⠀⠙⣿⡿
⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡇
⠿⠁⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⢸⠃
⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠿⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸
⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡄
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠙⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⢤⣾⣿⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣆⣀⣴⣶⡎⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⣴⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣶⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢠⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠶⠂⠀⠒⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠓⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠸⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⡸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⣸⣿⠀⠈⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢠⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣸⡆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣄⠀⠀⠀⠀⢻⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠈⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢻⣄⠀⠀⠀⢸⣷⣤⡀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠙⢷⣄⠀⠸⠟⠉⠀⣶⣶⡄⠀⠀⢙⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠂⣀⣀⠀⠀⠈⢻⣷⣶⡾⠟⠁⢀⣴⣶⣶⡾⠟⠋⠘⠿⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣤⣴⠛⠛⢀⣤⣤⣴⣿⠟⠁⣤⡄⠈⠉⠉⠉⠁⠀⣤⣤⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠿⠟⠛⠛⠀⠀⠀⠈⣱⡄⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠻⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣀⣀⣀⣀⣀⣀⣠⣴⣤⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡠⠚⣵⣿⣯⢙⣦⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢀⠃⢰⣿⠀⠀⠀⠐⡀⠀⠄⢱⡄⢿⣿⣎⣳⠄⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠈⠀⠘⣿⡀⠀⠀⠀⢡⡀⠀⠀⢹⣾⣿⢶⡋⠊⠀⠀⠀⢠⣿⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⣠⣤⣿⣦⣆⠀⠀⠀⠳⡀⠀⣾⣿⡟⡾⠁⠀⠀⠀⠀⠘⣿⠇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⣼⡟⠉⠉⠉⢻⣷⠀⠀⠀⠱⣸⣯⣿⣾⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⢣
⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⢧⣾⠃⠀⠀⠀⠠⠂⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡆⠀⠀⠀⠀⢣
⠀⠀⠀⠹⠇⠀⠀⠀⠀⠙⠁⠠⣀⠀⠀⢀⠃⠀⢀⠔⠈⠀⠀⣀⣀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠁⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠔⠁⠀⠀⠀⠈⠹⣯⣙⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢣⣰⣯⠍⢀⠟⠀⣀⠀⢲⣶⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠊⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠄⡌⠀⠀⠀⡀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠐⠁⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠁⠀⠀⣠⣤⢀⡀
⠀⠀⠀⢠⣦⡀⣀⡻⠦⠀⢴⡦⠀⠀⡇⠀⢀⡀⠀⠀⠀⠀⡀⢠⠂⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠵⣬⣤⡀
⠀⠀⠀⠈⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⠇⠀⠀⠀⠀⠑⠀⠀⠀⠘⠘⠀⠀⠰⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣂⣀⡀⠀⠀⠀⣇⠀⠈⠀⠀⠊⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣌⣩⣿⡭⣉⣭⠍⣀⠀⠀⠀⠀⠀⡇⠛⠤⠄⠶⠄⣶⡜⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢀⠀⢤⡀⠀⠀⠀⠀⠀⡠⣴⣿⣿⣿⣿⣿⣿⡽⠚⠉⠀⠀⠀⠀⠀⢰⠀⡄⠐⢄⠰⣾⡟⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⢀⡀⠀⠈⠙⢦⠀⠘⠛⢋⠙⣉⣙⣋⣁⣄⣡⡠⡤⠤⠤⠀⠀⠀⠀⠀⠈⠆⠀⡠⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠄⠐⣂⠙⠃⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣙⣻⣉⡟⣻⣟⣻⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠿⠿⠟⠻⠿⠿⠿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⢁⣀⣤⠤⠶⠶⠶⠶⠤⣤⣀⡈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿
⠀⠀⠀⠀⠀⠀⣿⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡈⠉⠛⠒⠶⠶⠶⠶⠒⠛⠉⢁⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣶⣦⣤⣤⣶⣶⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⠁⠲⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠁⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣷⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠿⡿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⢿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣮⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣋⣱⣾⣿⡿⠏⠙⠛⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣾⡿⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣾⣟⣿⡟⠙⣻⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣭⢠⣿⣿⣿⣟⡏
⠀⠀⠀⠀⢠⣾⣿⣾⣿⢱⣿⣿⣿⣿⣿⣿⠟⠉⣁⣿⣿⣷⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⠏⣿⠉⠉
⠀⠀⠀⠀⠈⠻⣿⣿⣿⡜⣿⣿⣿⣿⣿⣴⡶⣾⡿⣡⣿⣿⡿⠿⢿⡿⠸⣿⣹⣿⢍⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⢸⠿⣿⢷⣼⣿⡟⠻⢷⣿⣄⣿⣿⠿⠛⣁⣀⣀⣉⣀⣤⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠻⠄⠀⢠⣿⣿⡿⠶⣾⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⠀⠙⢉⣿⠛⢿⠿⢿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠿⢿⣿⣦⠴⠟⠁⠀⠀⠀⠈