⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢰⠃⠑⠁⠑⠉⠉⠊⠉⠉⠉⠉⠉⠊⠉⠉⠉⠉⠉⠉⠉⠉⡉⠉⡉⠍⠉⠩⠉⠉⡉⠉⠉⠑⠉⠉⠉⠊⠑⠉⠈⠊⠁⠉⠊⠉⠈⡁⠁⠁⠑⠈⠊⠐⠑⠐⠁⠉⠊⠈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠊⠉⠊⠉⠊⠁⠉⠊⠊⠉⠉⠉⠊⠉⠊⠈⠑⣆⣀⠀
⢸⠠⠀⠀⢂⣀⣐⣀⠠⡈⠀⠈⠀⣠⠈⣀⣄⣂⣀⠐⣀⣅⣀⡀⢀⡀⠀⠠⣀⠄⢀⢠⣐⣀⠀⠀⣄⡀⠀⠠⣄⠀⢄⣁⣀⣄⣀⠀⠀⠁⠀⢠⡀⠀⡀⡠⠀⠠⡀⠀⠀⢄⠈⠀⠐⢀⣁⣀⠁⠀⢠⠀⢀⠠⠀⠀⠁⡀⢀⠠⣀⣄⢀⠀⠠⠀⠁⣿⣿⠅
⢸⠠⠀⠀⣽⠛⡙⣉⠀⢷⡀⠀⣡⠇⠀⣟⣉⣉⣉⠀⣧⣀⡭⠇⠈⠳⣄⣴⠏⢠⡞⠉⠉⠙⣶⠀⣿⢷⡄⢐⣗⠀⢸⣋⣉⣉⣁⠀⠐⠀⠁⢸⣧⡴⠊⠁⠀⢸⣷⣄⠁⢸⠀⢀⡶⠋⠉⠉⢷⡄⠸⣇⠀⣰⣧⠈⣰⠃⠀⣾⡉⠉⠃⠀⡀⠄⠈⣿⣿⠂
⢸⠠⠀⠀⢾⢛⣉⣉⠀⠘⣷⣰⡟⠀⠀⣟⣉⣉⣁⠀⡏⠙⠷⡄⠀⢀⡽⠉⠀⠘⣆⣀⣀⣠⠿⠀⣿⠈⠹⢶⣗⠀⢸⣋⣉⣉⣁⠀⠀⠄⠀⢸⡏⠙⠶⣀⠀⢸⡆⠙⢶⣽⠀⠈⢧⣀⠠⢀⡼⠇⠀⢻⣤⠇⠻⣶⡏⠀⣠⡉⢙⣶⠀⠁⠀⠀⠠⣿⣿⡁
⢸⠂⠀⡀⠙⠋⠉⠉⠀⠄⠉⠋⠀⠀⠄⠙⠉⠉⠁⠀⠁⢀⠀⠉⠀⠘⠁⡀⠠⠀⠈⠉⠉⠉⠀⠀⠛⠀⠀⠈⠃⠀⠈⠙⠉⠉⠁⠀⠀⠀⠄⠘⠃⠀⠀⠉⠀⠘⠁⠀⠀⠛⠀⠀⠈⠙⠙⠋⠁⠀⠠⠈⠋⠀⠀⠋⠀⠀⠈⠛⠛⠉⠀⠀⠠⠀⠐⣿⣽⠂
⢸⣦⣤⣤⣤⣦⣴⣤⣦⣤⣤⣴⣤⣦⣤⣤⣤⣴⣤⣴⣤⣤⣤⣦⣴⣤⣥⣤⣤⣤⣦⣤⣦⣴⣤⣦⣴⣤⣦⣴⣤⣦⣴⣤⣤⣤⣦⣴⣤⣥⣤⣤⣤⣤⣤⣤⣦⣤⣴⣤⣦⣤⣴⣬⣤⣴⣤⣤⣴⣤⣦⣤⣤⣤⣦⣴⣤⣥⣤⣤⣤⣤⣦⣴⣤⣦⣬⣿⣿⠀
⠸⠯⠟⠽⠻⠞⠿⠺⠯⠿⡽⡟⡾⢯⠿⠽⠯⠻⠞⠿⡺⠯⠿⣞⡿⣾⣿⣽⣯⣿⣾⢿⣾⣿⢾⣿⡾⣿⣾⣿⣾⢿⣾⣿⣽⣿⣾⡿⣾⣻⣽⣯⣿⣽⣯⣿⣾⣿⣾⣷⣿⣽⢾⠯⠟⠷⠻⠽⠻⠞⠯⠻⠝⠯⠯⠷⠻⠯⠻⠻⠻⠫⠟⠯⢟⠯⠿⠳⠿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣟⣿⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣯⣟⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡫⣿⣿⣿⣟⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣷⣿⣿⣯⣿⣿⣿⣿⣿⣽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⡿⣿⣿⣯⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣽⡿⣿⣿⣿⢫⡻⡽⣻⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣻⣻⢿⣷⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣾⢿⣿⡿⣗⠱⡱⢱⠱⡸⡨⢣⢓⢝⠪⡋⡏⡏⡞⡝⡎⡗⣕⢧⢫⢳⢕⢧⢳⣻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠹⣟⣿⣟⣿⣯⠨⡊⡢⠱⡰⡘⢌⢢⢑⢅⠕⢔⢑⢌⢪⠸⡸⢰⠱⡱⡱⡱⡱⡳⣽⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠿⢞⢿⣾⡷⠐⢌⠜⢌⠢⡊⡢⡑⢔⠢⣑⢑⢌⢢⢡⠱⡘⡔⡱⡑⡅⡇⡇⡯⣺⣿⣿⣿⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢾⡯⢈⠢⠣⡑⢅⠣⢪⢘⢬⢪⡢⡣⡕⡕⡅⡇⡕⢜⢌⢪⠸⡨⡪⣪⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢎⢿⡃⠂⢕⣿⣾⡿⣿⡿⣿⣾⢳⢝⢮⡺⣝⣽⣷⣿⣾⣾⣾⣾⣾⡸⣸⡪⣿⣿⢽⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡕⣽⠨⠨⣺⣳⢷⢟⡷⣫⣗⣗⡯⡯⣳⢽⣺⣺⣺⣺⣵⣻⣺⣞⣞⢿⢜⢮⣿⡯⣯⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠪⠅⡱⣽⠃⢵⣙⣷⡕⢳⣳⢯⡻⣮⡳⣗⣿⣺⢵⢷⣷⡆⠨⢯⢯⢯⢳⣿⢝⣗⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠄⠀⠄⠀⠄⢍⠂⠜⡜⣆⡉⠿⠽⠨⡸⡺⡽⡝⡜⢮⣻⣺⣕⠻⡿⣝⠕⢠⢹⢝⢎⢗⡽⣝⠎⠄⠄⠄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠁⡀⠂⠀⠄⠠⠐⠀⠂⠄⡁⡂⠔⠠⢑⢌⠢⢫⠫⡟⡽⡹⣪⢫⠊⢌⠪⣺⢵⢳⢝⢞⠮⢎⢏⢎⠪⡪⣪⢟⠎⠌⠌⠨⠐⠠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢕⢐⠠⠐⠈⢀⠂⢐⠠⠈⠄⡡⢐⠠⠡⠨⢐⢢⠣⡑⡕⡱⡱⡱⡑⢅⢑⠠⢑⠵⣝⢜⢜⢌⢎⠪⡢⢡⢣⢣⣇⡑⡌⡐⡡⢁⠊⠄⠅⣪⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢐⢰⠀⡂⡨⡀⣂⣂⡢⣑⣔⣔⡦⣷⢽⣳⡁⢇⢇⢇⢇⢇⢣⠣⡊⡐⠠⠨⢀⠣⡣⡳⣑⢕⢸⢨⢪⢪⡪⣗⣯⡿⣽⣻⣞⣷⣫⢯⣗⣷⣣⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢷⣱⡳⣝⢮⢮⢯⣗⣷⣫⢿⣽⢯⣷⢧⢣⢫⢮⣪⢪⡳⣜⣦⢧⣣⢇⣧⣣⣗⣮⢾⢸⡸⣸⣪⣳⢽⣽⡷⣟⣯⣿⣺⣗⣯⣻⣺⣗⣗⣕⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢽⢮⢮⣗⡯⣟⡽⣞⡾⣽⣻⣞⣯⡿⡯⣃⡳⣕⢧⡳⣝⠽⡝⡿⣞⣟⣾⣳⢯⡫⣎⢧⢯⣺⡺⡮⣿⢾⣟⡿⣗⣿⣺⣞⣾⣺⢷⣯⢷⣳⠅⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠄⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣞⣿⢽⣳⣳⢯⣷⣻⢽⠯⢟⢎⢷⢳⢱⡱⣽⡎⢎⢷⢝⣎⣞⣮⡟⣞⣗⢷⣳⣵⣝⡮⡯⣟⣮⢯⢿⣟⣯⣿⣺⣳⣳⣕⢧⢯⣎⢟⢽⢻⣺⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⢯⢿⠺⡙⡣⡃⡆⣕⣕⡵⣕⡯⣟⣞⣾⣿⠕⠈⢪⢳⢝⣟⢟⡿⣻⢟⡿⣟⢿⣻⢽⢯⣗⡯⠣⠸⣿⣯⣿⣗⣿⣺⢾⡽⣽⢮⢯⢯⣗⣗⣕⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡨⢐⢄⢕⡼⣜⣮⣻⣺⣺⣽⣳⡯⣿⣯⣷⣿⠕⢈⠀⠕⣝⢮⣗⣟⡾⣽⣺⡽⡽⣞⡯⣗⠗⠡⠈⢌⢿⣾⢿⣷⣻⣞⣯⢯⣗⡯⡯⣗⣗⡷⣳⢽⣺⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠢⡹⣜⢮⣗⡯⣗⣷⣻⣞⣷⣻⣞⣯⣿⣽⣷⣿⡇⡂⢂⠁⡂⡓⢗⣗⢯⣗⣗⡯⣟⣗⠟⡐⠈⠄⠅⢌⢿⣟⣿⣿⣺⣽⣞⣿⣺⡽⡯⣗⡷⡯⣗⡯⡾⡽⣽⢬⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⢌⢎⢞⣮⣻⣺⢽⣳⢷⣳⣟⣾⣳⣟⣾⣳⣿⢷⣿⣿⡔⢐⠐⡀⠂⠅⠌⢣⣷⣯⣿⣗⡁⡂⢂⠁⠅⠨⢸⡿⣿⣻⣿⣺⣗⣿⣺⣗⣟⣯⢯⡿⣽⣳⢯⣟⣽⢽⡽⣵⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢕⢰⢱⢸⣳⣳⡳⣯⣟⡾⡯⣗⣿⣺⣗⣿⣺⣽⣾⡿⣿⣽⣯⢐⠐⠠⠁⠅⢌⣾⣿⣿⣿⣿⣷⡀⡂⠈⠄⢡⣟⣿⣻⣿⣻⡾⣽⣞⡷⣯⢷⡯⣯⣟⡷⣯⣟⡾⣽⡽⣽⣺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡕⣕⢵⣫⣞⡾⣝⣗⡯⣿⢽⣳⣟⣾⣳⣯⢷⣟⣾⣿⢿⡷⣿⢵⠈⠄⠡⢨⢪⢻⣿⣟⣯⣿⡗⡙⢢⠡⣈⣾⣳⢿⣽⣯⣿⢯⡷⣯⢿⡽⣯⣟⡷⣯⣟⡷⣯⣟⣷⣻⣽⣞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢅⢇⣗⢽⡺⡮⣯⣟⡾⣽⢯⣟⣷⣻⣞⣷⣻⣯⢿⣽⣾⡿⣿⡽⣿⡌⠨⢘⠘⠜⠜⠽⣿⣿⡏⠢⠈⠄⠂⡼⡾⣞⣯⣿⣾⢿⡯⣿⢽⡯⣟⡷⣯⣟⡷⡯⣟⣷⣟⣷⣻⣞⣷⢯⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⢅⢧⣳⣫⢯⡯⣷⣻⡮⡳⣕⡔⣜⢗⡿⣞⡷⣟⣯⣷⢿⣻⣯⣟⡷⣯⡀⢂⠨⠈⠌⠌⣿⡿⡇⢁⠐⠀⣂⡿⣽⣻⣞⣷⢿⡿⣯⢿⡽⣯⢿⡽⣗⣯⣟⣯⢿⣞⣷⣯⡷⡿⣞⣿⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠐⡀⢕⢜⢔⢷⢽⢽⢽⣳⣯⣯⡺⣜⢮⡳⣽⣻⣽⣻⣽⢷⣻⣿⣻⣷⣻⡽⣯⣷⡠⠀⠡⢈⢰⡿⣿⣗⡀⠄⢁⢾⢽⡳⣗⣷⣻⣟⣿⡯⣿⡽⣯⢿⣽⣻⣞⡷⣯⣿⣽⢷⣿⣽⢿⣻⣽⡷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⢂⢇⣗⡽⡽⡽⣯⢯⣷⣻⡮⣯⣗⡯⣯⢷⣻⣞⣷⣻⡯⣿⣽⣯⣷⢿⡽⡷⣯⣧⡌⠀⠂⣾⢿⣿⣻⡅⠠⡸⡯⡯⡯⣗⣷⣻⣽⣿⣽⣳⢿⣽⣻⣞⣷⢯⣟⡷⣿⢾⡿⣷⢿⣟⣿⢷⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢂⢪⡺⣪⢯⢯⡟⡽⣻⢞⢯⡿⠵⡗⡯⢿⠽⣓⠫⠚⢷⣻⢽⢷⢿⡽⡯⡟⣯⠷⢷⢻⡀⡁⠿⢿⣻⢿⢯⠠⠯⡯⢯⢻⢵⢳⣟⢷⢿⡾⡽⡻⣞⢷⢻⣞⢯⢿⠽⣟⡿⡿⢿⢿⡻⣟⡿⠷⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠌⢠⡣⡯⡯⣯⢿⠄⠀⢠⠀⠀⠇⠀⠀⠀⠀⣀⡃⠀⢀⠀⢙⠀⠀⠀⡃⠀⠀⠄⠀⢀⣀⠇⠀⢠⠀⠉⣀⠀⠀⢀⡀⠀⢀⠀⠀⠻⠀⢸⣟⡇⠀⠘⠀⠀⡏⠀⢠⠀⠀⡇⠀⠘⠀⠀⠋⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠂⡢⢯⢯⢿⣽⣻⠅⠀⢸⠀⠀⠀⠀⡆⠀⠀⠉⡆⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⡂⠀⢸⣤⣤⣟⠅⠀⢸⡅⠀⠀⠀⠀⠀⠀⢸⣯⡇⠀⠀⠀⠀⠁⠀⢸⠀⠀⡇⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢐⢜⡽⣽⣻⢾⣯⡃⠀⢸⡆⠀⠀⠀⡇⠀⠀⠾⠃⠀⢐⠀⠐⠀⠀⠄⠀⡄⠀⡁⠀⠰⠶⡂⠀⠸⠀⠀⣞⠀⠀⢸⠆⠀⠈⠀⠀⡀⠀⢸⡿⣮⠀⠀⠀⡀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⡉⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢰⢱⡯⣗⣟⣯⣷⣥⣤⣼⣧⣤⣤⣤⢧⣤⣤⡤⣦⣤⡴⣤⣬⣤⣤⣥⣤⣧⢤⣤⣤⣤⢤⣧⢤⡤⡤⡦⣯⢦⢤⣜⡧⣤⡤⣦⢤⣧⣤⣼⣿⣻⣤⣤⣤⣧⣤⣤⣿⣤⣤⣧⣤⣼⣤⣤⣷⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡱⡽⣽⢯⡿⣽⣯⣿⣻⣯⣿⡿⣗⣿⢽⣞⡷⣟⡿⣞⣿⣽⢾⣳⣿⣻⣽⣯⢿⣺⣗⡯⣟⢾⢽⢝⡯⣟⣗⡯⣟⡮⣟⡾⣽⢯⢿⣽⣯⣷⣿⢷⣻⡾⣷⣟⣷⣿⣳⣟⣾⣿⣟⣿⣟⣿⣟⣯⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀