⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠄⡀⠂⠌⡹⣝⢯⢿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠐⠠⠉⡄⠑⡌⣛⠾⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠄⠡⢀⠣⠰⢌⣣⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠡⠀⠄⠃⠆⢆⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⣒⣲⢦⣴⣭⣶⣶⣶⣷⣿⣷⣾⡷⢖⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⠐⡈⠐⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⣴⣶⣾⣿⣭⢟⣫⣭⣽⣮⣿⣿⣿⣽⣿⣷⢿⣶⣷⣿⢿⣿⣿⣿⣽⣯⣭⣧⣖⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠡⢀⠁⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⣢⣖⣿⣿⣿⣿⣾⣽⣿⡻⡿⣟⣽⣾⣟⣿⣟⣻⣳⣿⣿⡾⠯⢟⠛⢋⣙⡮⠯⢵⣫⡽⣛⣷⣛⠿⣿⣿⣾⡷⣆⡀⠀⠀⠀⠀⠀⠐⠀⠂⠈⢄⠡
⠀⠀⠀⠀⢀⠀⢀⣠⢤⡟⢡⣾⣟⣿⣿⣿⣿⣿⣿⠿⣫⣾⣷⡿⣿⢾⡿⢟⡕⣿⡹⠙⠂⢈⠔⠁⡐⠉⡠⠔⠁⠂⠀⠀⢀⠠⠈⠈⠍⢩⢛⣿⣷⢽⡲⣄⠀⠀⠀⠀⠂⠁⡈⠄⠂
⠀⠀⠀⠀⡜⠠⣃⢄⣟⣼⣿⡿⣸⣧⣿⡿⣿⡟⡇⣸⣿⣿⣿⣿⣿⠟⡠⡃⠠⠃⠀⡠⠀⢀⠄⡀⠀⠘⠀⠀⠀⠀⡀⠃⠀⠄⠀⠛⠘⢃⣿⣤⣿⣿⣿⣤⢀⠀⠀⠀⠀⠄⡀⠘⡀
⠀⠀⠀⠖⣂⡟⢰⣿⣿⡷⣿⣞⣿⢻⣵⡷⡫⢫⣾⣿⢿⡟⡱⢡⠗⢡⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠒⠉⣉⡀⠍⢻⢷⣵⡄⢀⠠⠀⢀⠀⠁⠄
⠀⠀⢨⡇⣞⣇⣾⣿⣾⠧⡟⣿⠩⣹⣿⠇⣗⢇⡿⡾⠉⢎⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢤⢈⣫⠙⣷⣝⣀⡀⢈⠠⠀
⠀⠰⡘⣵⡛⣿⡛⡯⣏⣇⢃⢯⠰⢻⢸⡳⢏⠞⣀⠇⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⢧⣫⢦⠹⡆⠠⠀
⠀⠐⢥⠳⣏⠀⢻⠘⡵⡎⠇⡎⠠⠘⡰⠁⡉⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠙⢒⠛⢦⠀⡁⠠⠀
⠐⢄⠈⢣⠁⠣⡐⠑⠀⠈⠁⠀⠀⠀⠁⢠⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠄⠉⠋⢁⠂⠄
⠀⠀⠱⡀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣁⣀⣖⣠⣴⣵⣏⣶⣮⣆⣓⣎⡀⡂⠤⢂⠄⡁⡐⠨⢀⠡⠂⠄⢂⠁⡀⠠⠈⢀⠂⠠⠈⠄⠡⢂⠀⠄⡀
⠀⠀⢀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⣵⣾⣿⢟⣫⣭⣽⡿⢷⠶⠷⣿⠷⣶⣷⣵⣭⣜⢢⡜⡠⠑⢢⢁⢂⠡⡈⢄⠢⢀⠅⡂⢄⠨⠄⡉⠄⠃⡄⢃⡐⠀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣲⣿⢿⣿⢿⣻⣯⣋⡃⣽⣛⣫⢿⣟⣷⢻⣽⣫⣍⢛⣹⢫⡿⣿⣿⣝⡦⡌⢦⠱⡌⢢⠘⡄⠢⢐⠠⡈⢄⠡⠌⢂⠰⢀⠂⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡾⢿⡴⢋⣾⣿⣷⣿⡿⡗⢏⣥⣰⣫⣫⣿⣻⣶⣿⣷⣿⣾⣳⣳⢳⣞⡾⡭⣝⢪⢵⡘⡥⢊⠴⣁⠎⡐⠤⢈⡐⠈⠄⢂⠂⠡⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢅⣻⢟⣴⠿⣼⣯⣟⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣾⡽⣦⣻⢼⡭⢶⠱⣊⠔⢂⠡⢀⠡⠈⠠⢈⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡱⣪⢃⣾⣻⣻⣾⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣏⣗⢫⢓⡌⣎⠤⡑⠀⡄⠀⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠐⣣⣟⣾⣿⣿⣿⢿⠛⠉⠀⠀⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠻⣬⡣⠖⡜⡈⠖⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣴⣿⣿⣿⡿⣿⢋⠀⠀⠀⠈⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣿⣿⣿⣿⣿⣿⣿⡿⢯⠻⢃⠁⢂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⡿⣽⣽⣿⣷⣤⠄⣀⢀⢀⡀⢈⠻⢿⣿⣿⣿⣿⠿⣟⣫⣷⣿⣯⣿⣿⣿⣿⡿⢏⡏⠋⠈⠡⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⢉⠁⠙⠑⡈⣦⣵⣂⢷⡯⣶⣞⣼⣾⣪⣿⡿⣿⣷⡲⣻⣿⣟⣻⣷⣿⣟⣿⣿⠺⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠓⣿⣿⢻⣵⣿⣿⣿⢻⣽⡟⣿⡟⢻⣽⣿⣿⢫⠋⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢍⠉⡛⢛⠿⠿⣟⣺⢛⠓⠛⠫⡙⠊⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠌⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡰⣈⠁⢆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀