⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣻⡯⣿⣻⢿⣿⢿⣻⣽⢯⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢸⣳⢽⣝⢾⢝⣗⡯⡇⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣿⢸⣳⣫⢾⢽⣝⢾⢽⡣⢿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢇⢼⣾⢾⣽⢵⣯⢿⣯⢧⢹⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⣟⣞⡿⣽⢯⣟⣿⣺⢯⢪⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡾⣽⣻⡽⣻⣞⢾⢱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣯⢷⡯⣟⡷⡯⣟⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣺⣽⣻⢾⢽⣞⣟⣮⡏⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣽⣿⣺⢮⣻⣽⣟⣾⡳⣯⣺⢽⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣟⣯⣿⣷⣿⢿⣽⣞⣗⣗⣗⣿⡺⣝⢮⣞⣿⣿⣻⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⣿⣻⣯⣷⣿⡿⣟⣷⣿⣿⣗⣟⣞⡾⣾⣿⣿⣷⣫⢾⣽⣿⣻⣿⣾⡿⣯⣿⡿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣷⣿⣟⣿⣽⣟⣿⣿⢿⣯⣿⣷⣳⣳⡻⣽⣿⣿⣳⢽⡽⣾⣿⣻⣿⣾⣿⣿⣟⣿⡿⣷⣿⠅⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⣾⣿⣟⣯⣿⣯⣿⣿⣻⣷⣿⣺⡺⣺⢽⣿⣟⣯⣗⡯⣿⣯⣿⢿⣾⢿⣾⣟⣯⣿⡿⣯⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣺⣿⣿⣾⣿⢿⣻⣽⣷⣿⣿⣻⣽⡾⣝⣞⣿⣿⣟⣿⢮⢯⣿⣯⣿⣿⣻⣿⣯⣿⣻⣟⣿⣟⣗⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣯⣷⣿⣿⣿⣿⣻⣷⣿⢿⡿⣿⢵⣳⣿⣿⣟⣿⡽⣽⣷⢿⣷⣿⣻⣷⣿⣽⣟⣿⣽⣟⣷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⡿⣷⣿⡿⣯⣿⣿⢿⣿⣟⣼⣿⣿⣻⣟⣯⣾⣟⣿⢷⣟⣯⣷⡿⣾⢿⣯⣷⣿⢿⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣿⣿⣟⣿⣽⣿⣿⣿⢿⣿⣿⣽⣿⣿⣽⡾⣿⣻⣿⣟⣷⣿⣽⣟⣿⣻⣽⣯⣿⢿⣿⣻⣾⣟⣿⡂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢐⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣾⡿⣷⣿⣯⣿⣿⣿⣿⣽⣿⢾⡷⣟⣯⣿⣯⣷⣿⢿⣟⣿⢾⣻⣽⡆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢰⣿⣿⡿⣿⣟⣿⣿⣿⣷⣿⣷⣿⣿⡷⣿⡷⣿⣿⣽⣯⣿⣻⣿⣻⣯⣷⡿⣾⣻⣿⣿⣻⣿⣻⡯⣯⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣼⣿⣿⡿⣿⣻⣿⣿⣿⣽⣿⣾⣿⣾⣿⢿⣻⣯⣿⣯⣿⣽⢿⡾⣯⣷⢿⣻⣟⣿⣿⣿⣽⣾⣟⣿⣳⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⢿⣯⣷⣿⣿⢿⣻⣽⣿⣯⣿⣻⣟⣯⣿⣟⣿⣽⣿⠛⢯⣷⣿⢾⣻⣽⡆⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣾⣿⣿⣿⣻⣽⣿⣿⣻⣿⢾⣯⣿⡽⣿⢾⣯⣿⡾⣿⠀⣸⣿⣯⡿⣟⣯⡷⠀⠀⠀⠀
⠀⠀⠀⠀⣿⣿⣿⣾⣿⣯⣿⣿⣿⣻⣿⣻⣽⣾⣿⡿⣿⣾⢿⣻⣟⣷⣯⣿⣻⣟⣷⣟⣿⣿⠀⣿⣿⢾⣟⣿⣻⣟⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢞⡯⣿⢿⣻⢗⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠗⠗⠯⠋⠑⠁⢁⠹⣸⣣⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠐⢀⠀⠄⠀⠄⠠⢈⢺⣾⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⣀⣄⠀⡂⣠⣨⢔⠨⣺⢃⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡸⡸⡼⢵⠘⠵⡕⢇⠁⣻⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠜⠨⠡⢈⠢⡈⢀⠢⡪⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⢄⡃⠣⡩⢀⢅⢪⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⢳⢡⢓⢢⢃⠢⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢆⠡⡁⠅⡂⡪⠨⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⡅⢑⠡⡁⠕⠡⢈⢰⢝⡮⣇⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢔⢦⡫⡯⣺⢵⡃⠀⣨⡀⢀⠡⢠⡺⡳⣝⢞⢮⡺⣳⢲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽⢕⢏⢮⡺⡝⣎⢷⡑⢠⠳⢧⠠⢈⢰⡫⡫⣎⢯⡣⣏⢮⢳⢳⢕⡽⣂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢮⢳⡹⡪⡮⡫⣎⢗⣇⢣⡙⡢⠣⡢⡓⡝⣕⢗⣕⢧⢳⡣⣏⢗⣝⢮⢯⠄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⣳⡣⡯⣝⣝⠮⡎⣗⢇⢣⢊⠢⡑⡐⠌⢌⢊⡷⣕⢗⢧⡳⡕⣗⢝⡵⣫⢟⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢮⡺⣕⣗⢵⣫⢞⡜⣇⡧⡢⡡⢂⢂⠅⣕⡽⣺⢝⣗⢧⡳⣝⢮⢳⡹⣪⢯⠇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣗⡽⣺⡪⣗⢵⡳⡹⣪⡯⣗⡮⡢⣢⢯⡳⣝⣗⢯⢾⢽⢝⣞⣎⢗⣝⢮⢏⣟⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡮⡯⣳⢝⡮⡳⣝⢽⢜⣽⣳⡫⡝⣗⡯⣞⣗⣗⢯⢯⡫⣟⣞⢮⢧⡳⣝⢵⣳⣳⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡺⣹⡺⣝⢮⡫⣞⣝⣗⢽⡺⣪⢯⡺⣪⡻⣺⡮⣯⣳⣫⢗⡽⣝⣗⡯⡮⣳⡳⣵⣃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣳⣝⢮⣻⡪⣗⣝⢮⡺⡮⣗⢽⣾⢵⢝⡎⣞⢮⢝⡾⣞⡾⣝⣮⡳⣵⡫⣯⡳⣝⣞⣞⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡵⣳⡳⡽⣺⡪⣗⢽⣕⢯⢧⢳⡹⣕⢝⣞⢼⢕⡯⡳⡯⣗⣗⢽⣺⡪⣗⡽⣺⣜⢞⣆⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣝⢮⢾⢽⢕⡯⣺⢕⣗⢽⡳⣝⣜⢮⡳⣕⢯⡳⡽⣹⣝⢷⢽⣳⡳⡽⣕⡯⣞⢮⣻⡺⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢽⡺⣝⢽⢵⣻⡪⣗⢽⡪⡷⣝⢮⢮⡳⡽⣜⣞⢮⢯⡺⡮⡯⡇⠙⠺⣝⡮⣞⡵⣻⣪⢞⠆⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⣝⢮⢯⣗⣗⢽⡪⡷⡽⣝⡮⣏⢷⢽⢕⣗⣗⢽⣕⢯⢯⢯⠃⠀⠀⠈⠛⢾⣺⢵⣳⡻⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⢕⡯⣺⣺⢮⣳⡫⣗⡽⣺⡺⣟⢽⣪⢟⡼⡮⣗⣗⢯⢯⣻⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡺⣝⢮⣻⢮⣳⡳⣝⡮⡯⣞⡽⣪⢗⡗⡯⡯⣻⡺⣪⢯⣻⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⢮⡳⣝⡷⣕⡯⣞⢮⢯⣺⢽⢕⡯⡾⡽⣝⡮⡯⣫⢗⡷⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣳⡫⣞⣗⢯⢞⣝⢮⣫⢗⣗⢯⢯⢞⡽⡮⣳⢽⣺⢽⢝⡾⣵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣗⢽⡺⡮⡯⡯⣺⢕⡷⣝⢮⢯⣳⢯⡺⣝⡞⣗⢗⡽⣝⢾⢵⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣗⢯⢞⡽⣝⣞⢵⢯⣺⡺⡽⡽⣝⢗⢯⡳⣝⡮⣻⣪⢗⡯⣗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⡗⠽⢝⣺⢵⡳⡽⣕⣗⢽⡺⣝⢮⢯⡳⡽⣪⢯⡳⣳⢽⢝⡮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⡊⡗⣝⢷⢝⡽⣪⢞⡵⡯⡯⣫⢗⡽⣝⢾⢕⣟⢮⢯⣫⢯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢷⡣⡯⣯⡳⡽⣕⢯⡳⡽⣝⣗⢯⢞⡵⡯⡯⣺⢵⡻⣮⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡳⣝⢮⣳⡳⡯⣯⣺⣳⢽⣝⣗⢗⡯⡯⣞⡽⣝⢾⣝⣞⣗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⡵⣝⢼⢕⣗⡽⣝⣞⣞⢮⢗⡷⡽⣽⡽⣝⡮⡯⣺⡳⣕⣗⣗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣔⢯⢞⣕⢯⣳⡳⣝⣗⡵⣳⢽⢝⡾⣝⡷⢸⡳⣝⣞⢵⡫⣞⢮⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⢪⡣⡫⣞⡵⣳⢽⢵⡳⡽⣺⢝⡽⣺⣳⡋⢸⢽⡺⡼⣕⢯⡺⣵⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⣕⢝⢼⢕⡯⣞⢽⣕⡯⣟⢮⢯⢞⣗⣗⡇⢸⣝⢮⡳⣕⢯⡺⣕⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡇⣏⢎⢷⢝⣞⡵⣳⠁⣷⡫⣗⢯⢞⣮⢷⠸⣺⢵⡻⣪⢗⡽⣳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡇⡧⡳⣹⡳⣳⢝⠃⠀⢳⣫⣗⢯⣻⣺⡽⡎⣯⡳⣝⢮⣻⡺⣝⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢧⢳⢕⢗⡽⡮⡋⠀⠀⢸⣳⢽⢝⣞⡮⣟⡇⢵⣫⢯⣳⣳⢽⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢕⢇⢗⢽⣕⠏⠀⠀⠀⢸⡺⡽⣽⣺⢽⢯⡇⢸⣞⣗⣗⣗⣟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⢸⡱⡳⡓⠁⠀⠀⠀⠀⡯⣟⣞⢾⡽⣯⡇⢸⣺⡺⡮⣞⣾⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⢜⡜⡞⠀⠀⠀⠀⠀⠀⣟⡷⡽⡽⣞⡷⣯⢸⣳⣫⢯⣗⡷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡯⣯⢿⢽⢯⡿⢸⣞⢾⢽⢮⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀