⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠁⠀⠀⠀⠀⡀⠠⠄⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⡀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡈⠸⠀⠀⠄⠊⠠⢀⢐⠀⠃⠁⠀⠀⣀⠠⢄⠰⠂⢈⢱⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠡⠐⡒⠁⠲⠐⡈⢐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⢱⠀⠂⠂⡁⡢⠂⠀⡀⠄⠒⢈⢡⠐⠈⠤⠤⠌⢀⠈⠂⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⡀⠂⢂⢀⠀⠀⠀⠀⠁⠄⠄⡈⠂⠄⢀⠀⠤⠀⡐⠒⠊⠛⠙⢃⠠⠘⢲⠈⠠⠔⠀⠒⢁⠐⡨⠘⠁⠀⠀⠀⠀⠀⠁⠄⠨⢐⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠡⡠⠀⠤⠀⠀⠀⠀⠀⠈⠀⢌⡀⢂⠀⡂⢁⠠⠈⠠⠁⠡⢀⠑⢄⠚⠠⢁⢐⠀⠥⠀⡊⠀⢄⠀⡀⠀⠀⠀⠁⠊⢀⠡⠈⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠂⡀⢂⠨⢀⠡⠈⠠⠐⠠⢈⠈⡁⠠⠐⡀⠡⠀⠌⡀⠂⠄⠨⠐⢄⠀⠠⠂⠐⡀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠄⡀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢐⠀⢂⠠⠠⠐⢈⠠⠁⢂⠐⢀⠂⠄⢁⠊⠂⠥⡐⠀⠌⡀⠅⠨⠐⠑⢄⠂⠁⠄⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠐⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⢀⠐⠠⠈⠠⠐⠠⠈⠠⠐⠈⡀⢐⠀⢂⠈⠄⡈⠄⠁⠄⢑⡠⠐⢀⠂⡁⠂⠈⠕⢈⠀⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠌⠠⠐⠠⠀⠀⠀⠀⠀⠀⡠⠐⢀⠂⠄⠨⢀⠁⢂⠁⠄⠱⠀⡁⢂⠐⠠⠈⠠⢀⠑⡀⠄⠡⢈⠠⢀⠑⠰⡀⠠⠐⠀⠈⢂⢐⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠨⠐⢈⠠⠘⠀⠀⢀⠠⠐⠀⠌⠠⠐⢈⠀⡂⢈⠠⠐⢈⠆⠁⠄⠂⠠⠁⠌⠐⡀⢂⠐⡈⠄⠂⡀⠂⢈⠀⠈⠃⠠⡀⠈⠀⢂⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠔⢈⠠⢐⠂⠀⢀⠂⢀⠨⠀⠅⠂⡁⠄⢂⠐⡀⢂⢈⠨⠀⠌⠠⠁⠌⠐⡈⠄⠂⡐⠠⢁⠂⠁⠀⠈⠀⢀⠈⠀⠡⠀⡁⢂⢐⢂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠄⡑⡀⡂⢐⠀⡂⠆⠠⠀⢀⠀⠂⠁⠀⠂⠐⠀⠂⡐⠀⡇⠈⠄⠡⠈⠄⠁⡀⠂⠁⠀⡀⠘⡀⠀⢁⠀⠁⡀⠄⡈⠄⠅⡐⠠⠈⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠔⡈⠄⢂⠠⠐⡀⠢⢰⠀⠅⠨⠀⠜⠐⢐⠐⢐⢀⠡⠀⠄⠌⠇⠠⠐⡀⠄⠄⠂⠠⢀⠐⠠⠀⡂⠥⢐⠠⢈⠐⡀⢂⠐⢐⠘⠠⠈⠄⡘⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡨⠐⠠⠈⠄⢂⠁⡌⠂⡎⠠⠁⠌⠌⢂⢁⠂⠌⠠⢀⠂⣁⠊⠀⢃⠁⡂⡐⠠⠁⠌⡐⢐⠈⠄⠡⢠⠩⢡⠐⡀⡂⡐⠐⣬⠠⠈⡌⠨⢐⢀⢱⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠐⡐⠈⠄⠅⠌⠄⢢⠂⡁⢂⡡⢊⠠⢁⠢⠂⠄⠅⠌⠄⡂⠠⠀⠂⠘⡄⠂⠄⠡⠁⠅⡐⠊⡄⠅⡁⠚⠀⠀⠣⡀⠂⠄⢥⢳⠀⠅⡇⡁⢂⠔⡀⢇⠀⠀⠀
⠀⠀⠀⠀⢀⢁⢂⠡⠡⠨⠠⠁⡆⡐⢘⠈⠄⡂⡐⡐⠈⡎⠤⠁⢚⣾⣾⡍⠉⠉⠀⠐⠁⢅⠡⢁⠢⠌⠀⠠⢂⢂⣧⡀⠑⠒⠓⠁⠡⢺⡪⢂⢁⢂⢐⠐⡐⡐⠨⡀
⠀⠀⠀⠀⡨⠐⡀⡂⠅⠌⠄⠅⡇⡐⡐⠨⢐⢀⠂⡤⠋⠀⠀⠀⣸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠘⢤⠊⠀⣈⠔⢨⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢄⢂⢐⠠⠂⡂⠆⠅⢇⠀⠀
⠀⠀⠀⠠⠠⢁⢂⢐⠨⠠⢑⠐⡐⡀⡂⠅⡂⠔⠸⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡆⡐⠨⢐⠠⢡⢁⠚⡀⠀
⠀⠀⠀⢂⢑⢐⢐⢐⠨⢈⢐⠐⡔⢐⠠⢁⠂⠅⢢⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⢁⠂⢅⠢⠨⢸⢐⠈⢥⠀
⠀⠀⠘⣐⡐⠐⡆⡂⢌⠐⠄⠅⡇⠢⡈⡂⠅⠅⠅⡢⡀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⠔⠀⡈⠄⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⡠⢂⢂⢑⢐⠨⠨⡘⠢⡨⠨⡀
⠀⠀⠀⠀⠀⠀⡇⡂⠢⠡⠡⡑⢨⢐⠐⠄⠅⠅⢅⢢⠈⠢⢀⡀⠀⠹⡿⠁⠀⠀⠀⠀⡀⠄⠁⠠⠀⠠⠀⠐⠀⢻⠟⠀⠀⠀⣀⠀⠄⠂⢠⢂⢂⢂⠢⠡⡁⡂⠀⠈⠂⡅
⠀⠀⠀⠀⠀⠀⢐⠨⠨⡈⡢⢨⢜⡄⠅⢅⠕⠡⡂⣜⢔⢀⠀⠀⠉⠀⠐⠒⠒⠂⡈⠁⡀⠄⠈⡀⢐⠀⡈⠀⠂⠈⠀⡀⠀⠄⠀⠄⠀⢂⢦⢂⢂⠢⠡⡁⡂⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠘⡈⡂⣢⠊⡕⣗⢵⠡⠡⠨⢂⢢⡳⡕⡕⢄⠁⡀⢁⠀⠂⠠⠀⠡⢄⠀⠄⢠⠠⠂⠐⠠⠀⡁⠌⠀⠠⠐⠀⠂⢐⢜⡝⣜⠁⡢⠡⡑⡐⠌⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠣⠂⠁⠀⢓⡕⡧⣣⠑⢅⢑⢜⢮⡪⡳⡕⡧⣢⠀⠠⠈⠀⠐⠀⠄⠂⠠⠀⠠⠐⠀⠂⠁⠀⢀⠐⠀⢄⡢⡲⣝⢕⢽⡘⠀⠈⠐⡔⡨⢸⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢝⠚⠀⠹⡐⠴⠈⢕⡝⣎⢗⢝⢪⢝⢖⣆⣁⡈⠀⠄⠐⠀⠐⢀⠐⠀⠂⣀⣁⡠⣔⢎⠇⠸⡱⡣⣏⢮⠁⠀⠀⠀⠀⠐⠹⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠈⢎⢮⠓⠁⠀⢗⠵⡕⠎⠚⠉⠉⠃⠪⠂⠒⢘⠽⠉⡒⡵⣱⢣⡫⠁⠀⠉⢚⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠘⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠔⠀⠀⠀⠑⢈⠣⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠄⠂⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠂⠠⢈⠐⠈⠄⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠠⠁⠄⠂⡁⠌⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡀⠂⡁⠄⠁⠄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠁⢂⠀⡂⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢐⠠⢀⢁⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠊⣀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⡀⠄⠂⠠⠀⠄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢥⡱⡂⣄⠀⠀⠀⠀⡌⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⢔⠪⠃⠀⢀⠀⢈⠐⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⡀⠅⠂⡁⢐⠠⠐⠠⢀⡠⠂⠁⠠⠐⠀⡀⢀⠀⠂⡕⢄⠀⠌⠣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⢎⠪⠌⢀⠐⠀⠠⠀⠄⠂⠱⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠐⢀⠂⠄⠐⢈⠀⢒⠀⢀⠐⠀⠄⠂⠀⠠⠀⠂⠈⢪⡲⣹⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡢⡩⠀⠄⠠⠈⠀⠄⠠⠀⠄⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠂⠈⡈⠠⠈⡅⠐⠀⡀⠂⠠⠐⠈⠀⠐⠈⢀⠀⢹⣸⢶⢦⢤⢄⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⡤⣴⢾⢕⢯⢆⠂⠐⠀⠄⠂⠐⠀⠐⠀⡰⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡢⠂⠁⠀⡀⠂⢀⠐⠈⢀⠐⠀⠠⢸⢮⡳⡳⣝⢵⡽⡽⡽⣹⢽⢽⣻⡳⣕⣗⢵⢝⢮⣻⢝⡵⣳⣩⠢⡐⡀⠐⠀⡁⢄⡥⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠢⡂⠠⠀⠐⠀⠀⠂⠀⠄⠈⢀⡾⢵⢝⣝⢮⡳⣻⢝⣝⢮⡳⣝⣗⣝⢮⢮⡳⡽⣕⢗⣝⢮⡳⣳⠑⠣⠮⠮⠕⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡔⡄⢅⢀⡁⣈⢀⣐⠌⢳⣝⢵⡫⡮⡳⣝⢮⡳⣕⢗⣝⢮⡳⣕⢗⢧⡳⣝⢮⢳⢕⣗⣝⢮⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠓⠀⠉⠀⠀⠀⠈⠑⣫⣯⣝⡮⡳⣝⢮⡳⣕⢯⡺⡪⣏⢗⣝⢮⡳⣝⣵⠣⠳⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣟⣯⡞⠮⡺⠪�