⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣢⣦⢾⡶⣷⣻⡾⣿⢾⡿⣷⢾⣴⣤⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⣶⡿⣿⢯⣿⣫⣟⣿⣺⢿⡽⣯⡿⣽⣻⢷⣻⣟⣿⣦⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣵⣿⣻⢾⡯⣿⣳⣯⢿⡽⣾⡽⣟⣽⣟⣾⢯⣟⣿⣳⢿⣵⣟⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣵⣿⣻⡽⣷⣻⣯⢿⡽⣷⣻⣯⢿⣝⣿⡽⣷⢯⣻⣯⢿⡾⣽⣟⣾⡽⣯⢿⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾⣿⣻⣷⣻⣞⣿⢽⣗⣿⣫⣿⣽⣗⣿⣽⣻⢾⣯⣟⣯⢿⡾⣯⣟⣷⣻⢾⣯⣟⣿⡺⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣳⣟⣾⣳⢿⡾⣽⢯⡿⣞⡿⣾⢞⣷⣻⢾⡽⣷⢷⣯⢿⣽⣻⣗⣿⣳⣟⡷⣷⣻⣞⣿⣽⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣗⡿⣞⣷⣻⣯⢿⡽⣿⢽⣟⣽⣻⡯⣿⢽⣯⣿⣫⣿⣞⣿⣳⢿⣞⣷⣻⣞⣿⢽⡷⣻⣞⡷⣯⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡻⣵⡿⣽⢯⡿⣵⣿⡝⣿⣽⣫⣟⢽⣽⣯⣻⡻⣾⢞⣷⣻⢾⣳⣟⣯⠟⠾⣫⣾⢯⣟⡿⣽⢯⡿⣽⡷⣻⣿⣦⡐⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣡⣿⢽⣯⢿⣽⢷⣻⣧⢻⡌⢷⣻⣎⠳⢟⣮⠳⣝⣻⡽⣯⢿⣯⢾⣿⠐⡂⡄⢹⣯⢿⣽⣻⣽⣻⢷⣻⣟⣾⡽⣿⢷⣤⡀⠄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢣⢻⡇⣻⡾⣻⡷⣻⡷⣿⡌⢷⡈⠻⣮⠻⢦⡽⢛⡊⣠⠜⠉⢿⣞⠯⣿⠈⠀⡒⢀⣿⣻⣞⣯⡾⣯⢿⣯⣾⡽⣯⣟⣯⣟⡿⣿⢷⣷⣾⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢌⠑⠁⠹⡇⢾⣿⠊⣿⣯⢿⡽⣷⡈⢗⠀⣨⠵⣡⡲⠕⢋⡱⠀⠄⢨⣿⡅⣿⡆⠊⠀⣼⣯⢷⣟⣾⣻⣽⣟⣾⣞⣿⣳⣟⣾⡽⣯⡻⡻⡞⡷⣿
⠀⠀⠀⠀⠀⠀⠀⠄⡸⠁⠀⠀⢀⠀⡹⢈⢿⡇⠽⣾⣏⢻⣽⢷⣄⠈⡁⠐⠉⠱⠔⠃⠀⠀⠀⠀⣿⣖⢸⣿⠱⣾⣻⢾⡯⣿⣺⣯⣾⣳⣟⡾⣷⡽⣷⢯⡿⣽⣻⢿⣾⣬⣹
⠀⠀⠀⠀⠀⠀⢀⠪⠀⡀⠀⠂⠠⢀⡇⠱⡠⠙⡧⠈⠻⢦⡙⠿⢌⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢫⣿⡰⣿⣇⣿⢯⡿⣽⣻⣞⣷⡽⣷⢯⡿⣾⢯⣟⣯⣟⢯⣟⡿⣮⡿⣽
⠀⠀⠀⠀⠀⠀⠢⠈⠀⠄⡨⠢⡑⠍⢌⠘⡤⠑⠁⠀⠀⠠⠀⠄⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⣿⣽⣞⡿⣞⣿⣻⡽⣷⣻⢾⡯⣿⣽⣻⡽⣯⢿⣞⡿⣷⢾⣮⣝⡾⣿
⠀⠀⠀⠀⠀⢈⠈⠂⡔⡕⢱⠑⡈⠢⡁⢊⠄⢤⠌⣄⠁⠀⠠⠀⠑⠢⢅⠠⠀⡀⠀⠀⠀⠀⠀⠄⠀⣽⣾⣝⢯⣟⡷⣯⢿⡽⣯⣟⣿⣺⢷⣯⢿⣫⣿⡽⣯⣟⡿⣞⣯⣿⣻
⠀⠀⠀⠀⠀⠀⠳⡀⡇⠨⡂⢕⣨⣢⣑⠌⠢⡂⠢⡐⢥⠀⠂⠀⠀⠂⠈⠢⡒⠂⢈⠀⠀⠀⠂⣠⠞⡈⣷⢿⡌⢏⣿⣻⣮⣻⣟⡾⣷⣻⣟⣾⣻⣟⣾⣻⣗⣿⣽⡻⡷⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠢⠣⡱⣑⡾⣷⣝⣿⣏⠢⠘⢔⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⢠⣀⢨⡒⢅⠊⠀⠹⣟⣷⣄⢻⡷⣯⢿⣞⣿⡻⣾⣽⣞⣷⢟⣾⣳⢿⣞⣷⣟⣿⠳⠿
⠀⠀⠀⠀⠀⠀⠀⢸⢈⠆⢺⡿⣯⢾⠷⣡⣈⠉⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠡⣩⠂⠃⠀⠈⠸⢯⣟⣯⡿⣽⣇⠓⢚⠙⣷⣻⢾⣽⣟⣽⢯⡿⣞⡷⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⡤⡙⡦⡉⢅⠡⠢⠠⢌⠑⡒⢢⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠂⢪⢆⠁⠀⠀⠀⠈⠢⠽⣯⢿⣽⡿⣿⠿⠻⠙⠋⢚⣳⣟⣯⡟⠏⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢐⠒⡸⢂⠅⡑⠡⢂⢅⠞⠤⡃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢄⣻⠀⠀⠀⠀⠀⠀⢀⠈⠛⠾⣽⣻⢿⣶⣧⣦⢿⡯⣿⠎⢁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠨⡪⠐⠅⢪⠈⡢⣡⡉⠒⣄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠦⠁⠀⠀⠀⠀⠄⠠⢀⡀⠢⠐⠀⡉⠻⢞⡷⣯⢿⡽⠃⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠁⢌⠊⢌⡪⣢⠡⠼⢄⡈⢳⠈⡑⢢⡀⠄⠀⠀⠀⠀⠀⠁⢌⠀⠂⠜⣀⠂⢀⠀⡁⠐⠐⠉⢀⠀⠄⢐⠄⢁⠀⠀⡈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⣝⠠⡁⠪⠔⠁⡜⠰⠓⢤⠙⢈⠀⢂⠈⢧⢐⡀⢀⠀⠀⢂⠀⠆⠄⢅⠠⠀⠀⠀⠈⠂⠈⠀⠀⠀⠠⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠁
⠀⠀⠀⠀⠀⠀⡰⠑⢥⣾⡑⠀⡸⣡⡋⠳⡌⠃⠀⠂⠀⠂⠄⡈⠈⠒⠀⠁⠀⠀⠂⠠⠀⠈⠦⡁⠠⠀⡀⢀⠀⠐⡀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠠⢀⠈
⠀⠀⠀⠀⠀⠐⣇⠘⢄⠋⡕⠀⠼⠠⡈⠳⡘⠙⡄⢧⠈⡐⠀⠄⢈⠀⡁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠒⠐⠐⠄⠄⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠐⠀⠌
⠀⠀⠀⠀⠀⠀⡳⣈⡢⢁⡝⠀⠘⣤⠙⡎⠀⢰⠁⠀⠇⠠⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡈⢀⠁⠄⢈
⠀⠀⠀⠀⠀⡠⠣⡀⠈⢇⠪⢇⠀⠈⠲⠁⠀⠈⡀⠈⠦⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠠⢀
⠀⠀⠀⠀⠐⡁⠀⠱⡠⣀⢈⠈⢅⡐⡀⢄⠡⡀⠄⠀⠀⠱⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠀⠄⠠⠀⡈⢀⠂⠄
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⠈⠓⠬⣉⠎⡢⢑⠐⠄⠀⠀⠀⠱⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡈⠀⠄⢀⠐⢀⠂⠠⠀⠄⠠⢁
⠀⠀⠀⢠⠣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡁⢆⢁⠂⠀⠀⠠⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠐⠀⠄⢀⠐⢀⠂⡐⠀⠔⠀⠌⠠⠁⠢
⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⢆⠢⠁⠌⢀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠐⠀⢀⠈⡀⢂⠀⡂⠐⠄⢌⠨⡀⢅⠈⡂⠡⡑
⠀⠀⠘⠌⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⡨⠁⠀⠀⠀⠰⠀⡇⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⠀⠀⠂⢀⠁⡈⠠⠀⠄⡀⠢⠠⠨⡈⡢⡑⠔⠌⠤⢑⠌⡢⢘