⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⡀⠀⠀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣢⣤⢠⡲⡕⣝⡱⢸⠨⠨⠨⠐⡀⡂⠐⡀⠄⠐⠈⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣢⡦⢮⡾⢗⢫⢝⢪⣻⣿⡻⡪⡨⠨⠂⠅⡂⠄⠡⢀⠐⠠⢡⠨⡰⡲⡵⣬⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣄⢖⡼⡽⢷⡵⣌⣧⢵⣵⣷⣟⡷⣕⢧⣷⡑⠅⠅⡂⠌⡐⢐⠈⠌⡪⠪⡨⡂⢅⠕⣝⢷⣌⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣾⣮⣗⢯⡻⣿⡫⣙⢕⠕⡱⡱⣿⡫⣎⢣⢑⠄⠅⡂⠡⠠⠁⠌⠄⡩⠏⠐⢱⡱⢭⢻⢧⠅⡂⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡲⣝⢵⡫⣟⣿⣾⣿⡾⣜⢽⢖⣆⣕⣢⣪⣨⡺⣪⣾⣷⣌⠢⡈⡂⠅⠨⠐⢀⠂⠡⠑⡐⠡⢁⢂⢐⠀⡙⠷⡦⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢮⡳⡵⣝⣮⣳⢽⣕⢷⢝⣟⣿⣿⣯⣗⢽⢿⠾⠽⠿⡻⡻⡿⡫⡣⡣⡊⢄⠅⠅⠌⠠⠀⠅⠐⠠⢈⢐⢔⡀⡂⠂⠄⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⢝⣗⣯⢯⡯⣷⣳⢯⢷⢽⣝⣗⡯⣞⣽⣻⣿⣷⢯⡻⣕⢕⢐⠌⡌⢕⢵⣵⣷⣕⠌⡌⠌⢌⠨⠠⠡⠈⠄⡐⡱⢝⡖⢕⠥⡐⡀⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣞⣵⢯⣟⡷⡯⣟⣞⡷⣽⢽⣝⡷⣝⢾⢽⣳⣳⣳⡻⣿⣿⣻⣺⣻⣷⣳⢾⣞⣿⢿⡻⡹⡜⡔⠅⡅⢌⠂⠅⡅⢅⢂⢪⢪⠪⡂⠀⠈⠂⢅⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣔⣟⣾⢾⣯⢷⣟⣯⢿⣺⣽⣺⣽⣺⢽⢾⡽⣽⣺⣺⣺⢽⣝⣿⣿⣿⣷⣿⠩⠫⡳⢻⢜⢌⢧⣫⣾⡕⡌⡢⡑⢅⠕⢕⢕⡕⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣜⣷⢿⣽⡿⣽⣯⢿⣞⣿⣳⣗⣷⣳⢯⣻⢽⡽⣗⣷⢯⣞⣷⣳⡳⡽⣿⣿⣿⡝⡌⡌⡂⣿⢆⣿⣻⡻⡝⣎⢎⢜⢔⢱⢱⢱⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⢾⣽⣟⣿⣯⣿⢿⣾⣻⣽⣞⣷⣻⣺⣞⣟⣾⣫⡿⣽⣽⣟⣾⣞⡾⣽⢽⣺⣻⣿⣯⣶⣆⣦⣺⡭⡻⡧⡣⢝⣾⣯⡺⣜⢼⠸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣽⡾⣯⣿⣿⢾⡿⣞⣷⢷⣻⣞⣾⣳⣳⢗⣟⣾⢽⣗⣿⣿⡷⣿⡽⣯⣻⣞⣞⣿⣿⣿⢾⡽⣗⣟⣧⣿⣼⣑⡿⡕⣿⣞⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣸⣳⣯⡿⣯⣿⣿⢯⡿⣯⡿⣽⣗⣿⣺⣞⢾⢯⡿⣞⣿⣳⣿⣿⣿⢿⣽⢷⣳⣗⡷⡽⣿⣪⠱⠩⠩⣺⣞⢿⣾⡙⢿⡕⢽⡎⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⣗⣿⢷⡿⣟⣷⣿⣟⡿⣯⣟⣷⣻⣞⡷⣽⢽⢯⣻⣽⣯⡷⣿⣿⣿⣻⣽⣟⣷⣳⢯⡯⣿⣕⠅⡃⠇⣳⣯⣻⡺⣯⣖⣿⡳⣯⣪⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢸⣳⣟⣿⣻⣿⢿⣿⣯⣿⣻⣞⣷⣟⡾⣯⢯⣟⣯⣷⣻⣾⣻⣿⣿⣯⣿⣻⣞⣷⣫⢿⢽⣳⣧⣥⣱⣱⣸⢿⣷⣯⢯⣟⣿⣎⡮⡧⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣞⣯⣿⣽⣾⣿⡿⣷⣻⣽⣻⣞⣷⣟⣯⣟⣞⣷⣻⣺⢷⣿⣿⣯⣿⣽⣟⣾⢷⣻⡽⣯⢯⣿⣿⢾⣯⣯⢯⢿⣿⣿⣺⣪⢿⢽⢯⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡷⣯⣟⣷⣿⣽⣾⣿⡿⣽⣯⡷⣿⣞⣿⣺⣞⣗⣿⣺⡽⣻⣿⣷⢿⣽⣾⣯⣟⣯⢷⢿⡽⣯⡿⡹⢩⠱⡹⣽⣻⢮⢿⣿⣷⡯⣗⣿⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣯⢷⣿⣻⣾⣯⣿⣟⣿⣽⡾⣟⣷⣟⣾⣳⣟⣾⣺⡽⣯⣿⣿⣾⡿⣟⣷⢿⣺⣯⣟⣯⣟⣷⣟⢜⢰⢑⢹⣺⣳⣻⢽⣺⣻⡿⣎⢿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣯⣟⣷⣟⣷⡿⣷⣿⣿⣽⢿⣻⡷⣿⡽⣞⣷⣳⢯⣟⣷⣿⣾⣷⣿⢿⣻⣟⣿⣺⣽⣞⣷⣻⣷⣷⣲⣷⣯⣞⡷⡽⡯⣞⡷⡽⣪⣻⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣷⣻⣾⣻⣽⣿⣻⣷⣿⢾⣿⣻⣽⣯⣿⣻⢾⡽⡽⣾⣺⣿⣾⣷⢿⣿⣻⣽⣻⢾⣳⣟⣾⣻⡿⡿⡽⡷⣟⡾⣽⢽⢽⣳⢽⢽⡣⣻⣗⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣷⣻⡾⣿⣽⣾⣿⣻⣾⣿⡽⣯⣷⢿⣞⣯⢿⡽⣯⢷⡿⣿⡾⣟⣿⣯⣿⡽⣯⣿⣻⢾⡽⣿⣯⡳⡸⡸⡰⣯⢷⣻⢽⣺⢯⢯⡇⣗⣷⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡷⣯⡿⣯⣷⣿⣽⣿⣻⣾⣻⢿⡾⣿⡽⣯⣯⡯⣯⣷⡿⣿⣻⣿⣻⣾⢷⣿⣻⡾⣽⢯⣿⣿⣗⡇⡎⡪⢸⣽⣫⡯⣟⡾⣝⣗⡏⣗⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣟⡷⣟⣿⣽⣾⢿⣽⡿⣯⣿⣻⡿⣽⣻⣽⣞⣯⢷⣻⣿⣻⣿⣽⣯⣿⣻⡾⣷⣟⣯⡿⣷⣿⣿⣾⣗⣿⢷⣳⢷⣻⣗⡯⣗⣟⡎⣞⡷⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢺⡽⣯⡿⣷⢿⣿⣻⣿⣟⣾⣯⡿⣿⣽⢾⣳⡯⣿⢵⡿⣯⣷⡿⣷⢿⣽⣟⣷⣻⢷⣻⣿⣿⡿⡯⡻⡻⡫⣿⢽⣳⣗⢿⡽⣺⢪⣻⣯⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀