⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡢⣆⢦⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣹⡪⣞⢮⡳⡝⡮⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⡦⣣⢗⡵⣓⢦⡢⣮⡳⣝⠮⡃⡃⡁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣢⡳⣜⢴⢜⢮⡫⣞⣵⣯⣞⡵⣫⢞⡵⣝⢾⢝⣗⣝⢮⡳⣕⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠎⡚⣞⣞⢵⣫⣫⣷⣿⣻⣗⣿⣺⣽⢮⡳⣝⢮⢯⡳⣵⣳⡳⣝⢮⡳⣕⣖⣦⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣝⢮⢮⡷⣷⣻⡾⣽⣽⣻⢾⣽⢾⣳⣝⢮⡳⡳⣽⡣⣗⢽⡺⣳⢝⡮⣺⣺⣗⣯⣷⣲⡲⣎⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⢞⢮⢯⡿⣽⣻⣞⣿⡽⣾⢯⣿⢽⣻⡯⡮⡳⣝⣝⢞⣮⡳⣝⢮⡳⣝⢮⡳⣽⡾⣗⣿⣾⢾⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢓⢦⢢⡰⣸⡪⣮⡫⡯⣟⣯⣿⣳⢿⣺⢿⣽⣻⣾⣻⣽⡯⡯⣫⢞⢮⡫⣞⣟⢮⡳⣝⢮⣳⣻⢽⣯⣟⣷⢿⡽⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣗⢽⡺⣺⡪⣞⣝⢞⢿⣺⣟⣿⣻⢿⢾⢷⣯⣷⡯⡿⣹⡪⣏⣗⢽⢺⡺⣳⣻⡺⣝⢞⢮⣫⣷⢿⣞⣿⡽⣿⢽⡠⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣕⣟⢞⢮⡳⣳⡽⡽⡾⡽⣺⣪⡳⣝⢵⡳⣕⢯⡫⣞⢮⡳⣕⢯⡳⣝⢮⣞⢮⡳⣝⢧⡳⣻⣽⡮⣓⣏⡋⠧⢫⢆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣝⢮⢮⣫⡳⣝⢵⡻⣻⣺⢽⡪⣞⢮⡳⣟⢮⣳⢳⢝⡮⡳⣝⢮⡳⣝⢮⢟⣗⢗⡽⣜⢕⠽⢕⠧⢏⢎⠞⡜⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡫⡮⣳⣳⡳⣝⢮⡳⣝⣞⢮⡳⣝⢮⡳⣽⢮⢻⢎⠟⣓⣿⣿⡭⡍⠭⢍⠫⡊⣓⣿⣾⠾⡽⡽⣵⡳⡳⡳⣝⢞⢮⢆⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣝⡮⣾⢺⡪⣟⢮⣳⢯⡳⣝⢮⡳⣝⡧⡑⢕⠔⢕⢸⠹⡫⡳⡑⡍⡪⢪⢘⢔⠻⢿⢟⠜⡌⡳⣏⢯⡫⡮⣫⣳⢽⠬⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡢⣕⢗⡽⣹⡺⣝⢮⢯⢯⣫⢷⣟⢮⡳⣝⢮⡇⢎⢆⡣⠣⡑⡑⠌⡂⡃⠪⢘⠐⠕⢔⢕⢅⠣⡑⡅⢽⢮⡳⣝⢽⢵⢝⡭⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢝⢮⡳⣝⢞⢮⡳⣝⢮⡳⣝⣯⢮⡳⣝⢮⣳⣫⢓⠻⢟⢷⢷⢷⣦⡂⠐⡈⠠⢈⠐⣰⣴⣶⣿⡮⡌⡾⣕⢯⡺⣕⡯⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⢞⡽⡳⣝⢮⣻⣺⣳⢽⣻⢾⣵⣫⢮⡺⡵⡈⠂⠅⢐⠀⡂⠄⠡⠐⢈⠠⠈⠩⢁⢃⠢⢪⣜⢗⣷⢽⣺⡳⣝⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⠷⣵⣳⡽⡯⣾⢽⢮⡯⣟⣍⡛⠓⠡⠈⠄⠂⠠⢈⠐⡈⠠⠰⠊⢐⠠⢀⣚⡷⣷⣻⢽⣳⢧⢯⣺⣪⡗⠇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠽⠳⠻⠚⢿⢽⢽⣳⢯⣻⣺⣽⣾⣤⣅⣂⢁⢁⠂⡔⡲⢙⡆⣁⢦⣖⣟⡾⣽⣺⢾⢽⣺⡽⣽⠺⠑⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⣻⣺⢽⣞⣗⠿⡷⣿⣽⣟⣿⡷⣿⡾⡪⡃⡫⡘⡔⣟⡮⣟⡾⡽⡝⠙⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡹⢙⢈⢊⠪⢌⢝⠳⣟⠷⢟⢟⠏⡮⡪⡨⡊⣆⣯⣯⣅⣁⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠪⠂⡐⠠⠠⠐⡨⠐⡁⠇⡺⡬⡲⡓⢝⡺⣜⡗⠩⢈⠨⢈⠫⡿⣯⡷⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⠡⠐⠐⡀⠅⠠⠁⡐⢐⢐⣨⢪⡺⠍⢂⢸⣷⣓⢵⠉⣄⠐⢸⣻⣽⢯⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢀⠐⡈⠄⢂⣨⡧⣶⢾⣽⣿⣥⣆⣂⠨⢈⠙⣾⣻⣽⣷⣔⣈⡤⣟⣾⣻⣽⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣠⡲⣇⣢⣢⢖⣯⢟⣷⣟⣯⣿⣻⣞⣯⣿⢽⣻⣦⣆⣹⣯⡿⣾⡽⣯⣟⣯⡿⣽⢾⡯⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢷⣻⢿⣽⣪⣪⣺⣽⢾⣳⣿⡽⣗⣯⡷⣟⣿⢽⣻⣞⣿⣞⣯⣷⣟⣿⣺⢷⣟⣯⢿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣾⣻⣗⣿⣺⣽⣾⢿⣻⣗⣿⣻⣽⣻⣽⡾⣿⣽⣾⢿⣞⣿⣞⡷⣯⢿⡽⣷⣻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀ ⠀⠀⠠⣱⢷⣻⣾⣽⡯⣿⢾⣻⣽⢾⡯⣿⣾⣻⣞⣯⡷⣷⢿⣯⡷⣟⣾⣟⣿⣻⣿⣻⣾⣻⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣯⢿⣳⣿⣞⣯⡿⣯⢿⣾⣻⣽⣟⣾⢷⣯⡷⣟⣯⡿⣷⣟⣯⡷⣿⢽⣻⣾⣳⣟⡾⣝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠑⢿⣽⡷⣯⡷⣟⣯⡿⣾⡽⣞⣷⣻⣟⣾⣻⣽⣷⡿⣿⣞⣷⣟⣯⣿⡽⣾⣳⠯⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀