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