⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢮⣖⢦⡢⣄⣄⣀⢀⢀⠀⠀⢀⢀⡀⡄⡄⢄⢀⣀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣠⣠⣀⣰⣹⣕⡵⣕⣝⣔⣵⣸⣸⣡⣣⣫⣸⣪⣪⣪⣪⣱⣣⣣⡳⣕⡃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⡿⣞⣯⣟⣷⣟⡿⣯⢿⣽⣞⣿⣺⢷⣻⣞⣷⣻⢾⣺⣗⣟⣾⣺⢽⣳⢽⢀⢀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡽⣷⣻⣗⣿⡽⣿⢽⣞⣷⣻⢾⣻⡽⣞⣷⣻⣽⣳⣗⣟⣾⣺⢽⣺⢯⠟⠙⠉⠋⠙⠺⣢⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢿⣽⣻⣾⣳⣟⣯⣿⡽⣾⣻⣽⢷⣻⡽⣗⣿⣺⣗⣟⡾⣵⣫⢗⠏⠋⠀⠀⠀⠀⠀⠀⢸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⢾⣗⣿⣞⣿⣺⢷⣻⣽⢷⡯⣿⢽⡯⣿⣺⣗⣟⡾⣽⣳⢽⠝⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢯⣿⣺⣷⣻⣽⢿⣽⢾⣻⣽⢿⡽⣯⢿⣺⣽⣞⣯⣗⡯⣞⠁⠀⠀⠀⠀⠀⢀⢤⠞⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣷⣻⣾⣻⣾⣻⣽⢿⣽⢾⣻⣽⡯⣿⡽⣞⣷⣳⣗⣟⣆⣀⣀⣄⡤⠴⠚⠊⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣞⣿⣺⣷⣻⢷⣻⡿⣞⣿⣽⢾⣻⢷⣻⣟⣾⣳⣳⣳⠯⠑⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡷⣿⣞⣿⣻⣽⣟⣯⣷⣟⣿⡽⣿⢽⣞⣷⣳⡳⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣳⣟⣷⣟⣷⢿⣽⢾⣻⣾⣻⣯⢿⡽⡾⣕⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣂⡒⢄⡷⣿⢽⣾⣯⢿⣻⣽⣟⣿⢾⣻⡾⣟⡿⣯⣃⠂⠄⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡻⡽⣻⢽⣻⡾⣽⣟⣯⡷⡿⣞⣿⣳⢿⢽⣻⢞⡾⠛⠝⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡳⡝⡮⡫⡳⡹⡱⡫⡳⣛⢻⢹⢺⢹⢹⢹⢪⢫⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡺⡸⡪⡪⡪⡪⡪⡪⡪⡪⡪⢪⠸⡐⡕⡸⡨⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣕⢧⡳⣕⢵⡱⡕⡵⡱⣪⢪⡪⡪⡪⡢⡱⡸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣽⣿⣾⣾⣾⣾⣾⣾⢷⣷⣷⢷⣟⣾⢾⣳⣿⣾⡾⣶⢷⣷⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢾⠿⡾⡿⡿⣾⣷⣿⢿⡷⣿⢿⣳⢿⣻⣽⢾⡻⠟⠿⢟⢿⢽⡾⣿⣻⣿⣦⡄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⡫⣞⢽⣪⡲⣲⢪⡣⣇⢧⠳⣌⣿⣜⢜⡳⡂⠀⠀⠀⠀⠀⠀⠈⠹⢯⣷⣿⣦⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡸⣪⢗⢵⢝⢎⣗⢽⢜⢮⡫⣺⣿⡇⡗⡵⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣿⣻⣇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡫⣮⢯⡳⣝⢵⡳⣝⢮⡳⡹⣼⣯⡗⣝⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣾⢝⣽⡳⣝⡮⣳⢝⡮⣺⡪⣏⢾⣷⡯⣚⢮⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣾⢽⣯⢿⢵⣫⣞⡵⣯⣞⣞⢮⣻⣷⡯⡮⣳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⣟⣯⣿⣯⢯⣗⣗⣟⡷⣗⡿⣽⣽⣯⢷⣻⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣿⠃
⢀⣠⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢿⣺⣿⣳⣟⡮⡷⣯⢿⢽⢯⢷⣻⣽⣻⢾⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣿⠗⠀
⠑⠋⠛⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣄⣤⣤⣤⣤⣤⣤⣠⣀⣀⡀⠀⢸⣻⣗⣿⡿⣾⣽⣯⡿⣽⢯⡿⣽⣟⡾⣽⣟⣮⡀⠀⠀⠀⠀⠀⢀⣠⣶⡿⣟⣯⠗⠀⠀
⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⡾⣟⣿⢿⣾⢷⣿⣻⣯⣟⣷⣯⢯⡷⣟⢾⣯⣧⣶⣶⣾⣾⣿⢿⣷⠿⠏⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣿⣽⣿⣟⣿⣟⣿⣻⣟⣿⣻⣯⣿⣽⣿⣾⣿⣺⣿⣿⣿⣿⣿⣿⣽⣽⣷⣯⣯⡿⣽⣳⣻⣻⡿⣿⢿⡽⡾⠷⠻⠋⠁⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣺⣻⣝⢟⢟⡻⣻⢿⣿⣿⢿⣿⣯⣗⣿⣟⡗⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣾⣿⣿⣽⣿⣟⣿⣿⣽⣾⣿⣿⡿⣟⣿⣿⣿⣿⣿⣮⡯⣿⣿⣿⣿⣗⡿⣾⡻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣷⣿⣿⡿⣟⣿⣿⢿⣿⣿⣿⢿⣿⢿⣿⣾⣿⡿⢿⡻⠹⠙⠙⠹⣿⠛⠛⠷⣿⣗⡿⡽⡿⡻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣟⣿⣻⣯⠀⠀⠀⠀⢈⡿⠀⠀⠀⠹⣻⢽⡯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⡿⣿⣾⣿⣿⣻⣯⣿⣿⣿⢿⣟⣯⣿⣿⣟⣯⢷⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠈⠻⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⡟⠙⠐⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⣀⠀⢼⣿⣯⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣽⣾⣿⣿⣯⣷⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣻⣿⣿⡿⣿⣿⣿⣾⣿⣯⣿⣿⣿⣿⣯⣿⣿⢿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⡿⣿⣽⣿⡿⣿⣿⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣯⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⣿⢿⣷⣿⣿⡿⣟⣿⣿⣿⢿⣯⣿⣿⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠽⠾⠿⡿⡿⡿⣿⢿⡿⡿⡿⠿⠻⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀