Reports Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣼⣿⣿⣇⠀⠀⣸⣿⣿⣧⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠻⠿⠿⠿⠿⠿⠿⠟⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠾⠋⠀⠀⠰⠷⠶⠶⠶⠶⠶⠶⠾⠆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡀⠀⣠⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠞⠉⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠟⠁⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣀⠀⣠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠛⠁⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠻⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠈⠁⠈⠀⠀⠀⠒⢠⠀⡄⡀⠀⠈⠉⠀⠒⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣈⠐⠂⡀⠀⠂⠢⠆⠄⠠⠀⠀⠀⠀⠀⠉⠐⠂⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⠄⠠⠄⠀⢀⠀⠠⢘⠁⠆⡩⠀⡔⠀⠺⠁⠀⢀⠀⣁⣤⠀⠀⣸⠒⢠⠤⢄⣀⠀⠀⠀⠈⠀⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠊⠠⠤⠤⢀⣀⠀⠀⠀⠄⠁⠀⠂⠨⡀⠚⠀⠅⠠⠃⠀⡆⢘⠏⠀⣰⡴⠁⠨⡈⠢⡀⠁⠀⠀⠠⠀⠂⢀⣶⣽⡀
⠀⠀⠀⠀⠀⡐⠁⠀⠉⠍⡒⠲⠬⠈⠁⠀⠀⠀⠀⠤⠀⠁⠊⠦⠃⠐⣄⠠⠅⢀⠀⠃⠈⠠⠬⡠⠠⢀⠄⠀⠀⠀⠀⠀⣠⡿⡟⣔⡇
⠀⠀⠀⠀⡐⠀⠀⠁⠀⠠⠂⠀⠡⠀⠀⠐⠀⠀⢀⠄⠀⠀⠀⠁⠀⠀⠀⠉⡐⠀⠀⠤⢈⡁⠒⠀⠉⢠⠄⣀⠀⠀⠀⣰⠯⡞⣾⣫⡾⡆
⠀⠀⠀⢀⠁⢀⡁⠈⠀⢀⠀⠀⠀⠀⠌⠀⠀⠀⠄⠂⠀⠀⠀⣒⠀⠀⠀⠀⡡⠋⠰⢤⠄⣄⣀⠀⡁⠀⠒⠓⠈⣠⡞⠕⢨⣾⢏⡡⣶⣷
⠀⠀⠀⠈⡄⠀⠈⠁⡒⠀⠀⠀⠀⠘⠠⠄⡀⠀⠀⠀⠀⠀⠈⡁⠀⢀⡀⡐⢀⡘⠒⠂⠰⠤⠌⡀⠉⠀⠀⣀⡴⡡⠪⣾⣿⢃⣾⣾⡻⠃
⠀⠀⠀⠀⠈⠂⢀⣤⡀⠈⠀⠀⠀⠉⠒⠰⢤⠁⡀⠀⠠⠀⢄⠠⠂⠄⡐⠀⠐⠂⠠⠀⠒⠠⠄⠀⠀⢀⢔⢝⡥⠠⡺⡿⢫⢮⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢀⠀⠠⣀⡁⠀⠀⠭⠉⠀⠒⠄⠀⠁⠀⠀⠈⠤⠀⢐⢀⠀⠀⠀⠀⣀⠽⣡⣷⣮⠞⡝⠕⢡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠡⠀⢄⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠈⠁⠀⢰⢣⢾⣿⡟⠣⠊⡂⣮⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠠⠄⡀⠂⠀⠀⠀⠄⠈⠀⠐⠀⠀⠀⡯⡏⠀⠙⠴⣁⣦⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚⠒⠦⢴⣤⣁⡀⠀⢧⠀⠀⠀⢀⡠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠃⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠀⠀⠀⠀⡀⠐⡀⠀⠀⠀⠀⠂⠄⠀⠀⠄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠀⠀⠀⠿⢦
⠀⠤⠀⠀⠠⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠐⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⣀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠆⠀⢠⡄⠁⠀⣶⠀⠰⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠣⠀⠀⠠⠁⠀⢄⡀⢱
⡇⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠃⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸
⡇⠀⣼⡏⣿⢀⣠⣧⣀⢐⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠅⠀⣿⡟⣿⡇⠈⠏⠀⠀⠉⠂⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠃⠀⠀⣀⠄⠁⠀⠀⠀⢸
⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡄⣰⠀⠀⠀⠀⢠⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠼⡄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠘⠷⣄⠀⠀⢸⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡇
⠀⠈⠒⠠⠄⡀⣀⡀⠀⠀⠀⢠⠀⠀⠈⠛⠶⣼⡦⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢤⠀⠀⠠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⣼⠀⠀⠀⣧⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠛⠛⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⡸⢅⠀⠀⠀⠀⢀⠽⣀⠀⠀⣏⠀⠀⢈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡢⢄⡤⠊⠀⠀⠉⠢⣤⠔⠁⠀⠀⠉⢒⠏⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣤⣤⣤⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠀⠀⢀⡤⠤⠖⠒⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⢀⡠⣤⢠⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⠀⡇⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⡴⠋⠀⢸⢸⠀⣨⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡇⢸⢸⠀⡇⠀⠀⠀⡼⠁⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⢳⠀⠀⢰⠃⠀⠀⢸⠈⣫⠴⠊⢹
⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⠀⡇⡇⢸⢸⠀⡇⠀⠀⢠⢣⡄⠀⠀⡿⡄⢶⣄⠈⢇⠀⠀⡀⡇⠀⠘⣆⣤⡀⠘⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⠀⡇⡇⢸⢸⠀⡇⠀⠀⢸⢸⠁⠀⣸⠓⠟⢾⡎⠓⠾⡄⠀⡇⢸⠀⠀⢿⠀⣗⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠴⠧⠾⠼⠴⠷⠧⠼⠾⠤⠧⠄⠀⢸⢸⡀⢀⠇⠘⠀⠀⠀⠘⠀⣇⠀⡇⢸⠀⣴⣚⣈⣓⡍⠉⠁⠀⠀⠀⠀⢰⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢧⣸⣆⠀⢀⣐⣀⠀⢀⠿⡀⣇⡞⠀⠀⠀⠀⠀⠀⠀⣰⢦⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠙⡷⣄⠀⠀⠀⠀⠘⢮⡗⣨⣳⡤⣉⡡⡴⢋⣤⣱⡟⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠓⣤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣛⣀⡉⠙⡇⠀⠀⠀⠀⠀⠈⢁⣽⠇⠀⠀⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠰⠯⠇⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⡠⠖⠒⠦⣼⠀⠶⠶⠆⠀⡇⠀⢀⡤⠴⠖⣺⠋⡏⠀⠀⠰⠊⡆⠹⡖⠶⢤⡀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⣀⣈⣉⣁⣀⡇⠀⡜⠀⠀⠀⣹⠀⡇⣀⣀⡀⣤⡇⢴⡃⠀⠀⠸⠀⠀⠀⢸⣉⣉⣉⣉⣉⣉⣿⣏⣹
⠀⠀⠀⠀⢀⡹⠀⠀⠀⡆⠀⠀⠈⣇⠀⠀⠀⠀⡇⠀⠀⠐⡇⠀⡇⠀⠀⠀⢀⡇⠀⡟⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡴⠦⡄⠀⠀⢸
⠀⠀⠀⠀⡎⠀⠀⠈⠳⡷⠊⠀⠀⠈⢧⠀⠀⢰⠀⠀⠀⠀⢧⠀⣇⠀⠀⠀⢸⠀⣸⠀⢠⠀⠀⢱⠀⠀⢸⠀⠀⠀⠳⠴⠧⠀⠀⢸
⠀⠀⠀⠀⠱⢄⣀⣀⣀⣀⣀⣀⣀⡠⠏⠀⠀⡞⠀⠀⠀⠀⠘⡆⢸⡀⠀⠀⡟⢰⠃⠀⢸⠀⠀⢸⠀⠀⢸⠀⣒⣒⣒⣒⡒⠒⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢰⠀⠀⠹⡀⡇⠀⢰⢁⠏⠀⠀⣼⠀⠀⠸⡄⠀⠸⣤⣤⣤⣤⣤⣤⣤⣤⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⠀⠀⠀⢷⢸⠀⡏⡞⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣯⠭⢌⣣⣀⡞⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣼⡝⢶⣄
⠀⠀⠀⠀⠀⠀⠈⠉⣽⣷⡾⠋⠀⠀⠀⡵⣝⣯⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠧⠤⠤⠴⠊⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣷⣄⣠⣾⣿⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⠶⠖⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠲⠶⠶⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠆⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢦⠞⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢦⡞⠁⠀⠘⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠲⣤⡴⠋⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠰⣄⡴⠋⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⣄⣠⠞⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⣤⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⠋
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠿⢟⡁
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⡇
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠇
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⣾⡿⠿⠿⠿⠿⠿⠿⢿⡇⢠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠺⠿⠿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣇⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⣺⣿⡇⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠅⣠⣦⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⡟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢻⡇⡿⣫⣾⣿⢟⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⢣⣾⣿⠟⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⡏⣭⣭⣭⣭⣭⣭⣭⡍⢻⣿⣿⢟⣵⡎⢛⣵⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⢟⣵⡿⣫⡄⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⡏⠭⠭⠭⠭⠭⠭⠭⠍⢱⡿⠫⠤⢹⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣨⣥⣾⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣇⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣸⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢇⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⢶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠉⠉⠛⠻⠷⣶⣦⣄⣀⠀⣼⡷⣶⣦⣤⣿⣆⡀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣴⣶⣤⣄⣀⡀⠈⠉⠛⢿⡿⠀⠀⠀⠉⠉⠛⠛⠿⢷⣶⣶⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢠⣿⠁⠉⠉⠛⠻⠿⣶⣦⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⣶⣦⣤⣀⣀⠀⠀⠀⠀⢰⣿⠉⠉⠙⠛⠻⠿⢷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⢰⣿⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⣶⣦⣾⠿⠿⠷⣶⣤⣤⣀⡀⠀⠈⠉⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢀⣿⠃⠀⠀⠀⠀⠘⠛⠿⢶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠙⢻⣿⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢰⡿⠀⠀⠀⠀⠰⠶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣶⣤⡄⠀⠀⠀⠀⣸⡟⠀⠀⠉⠙⠛⢿⣷⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⡟⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⢰⣿⠁⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢠⣿⠁⠀⠀⠀⠀⠙⠛⠻⠿⢷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣸⡏⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢶⣦⣤⣄⣀⠀⠀⠈⠉⠛⠻⢷⣾⠏⠀⠀⠀⠀⢀⣿⠃⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⢸⡿⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢀⣿⠃⠀⠀⠀⠀⠈⠛⠛⠻⠷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠆⠀⠀⠀⢠⣿⠃⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠷⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⣼⡏⠀⠀⠀⠀⠘⠿⢷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠗⠀⠀⠀⠀⣾⡏⠀⠀⢀⣿
⠀⠀⠀⠀⢀⣿⠇⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣼⡇
⠀⠀⠀⠀⣸⡟⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢰⣿
⠀⠀⠀⠀⣿⠃⠀⠀⢰⣿⠁⠀⠀⠀⠀⢴⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢶⣶⣤⠀⠀⠀⠀⢸⡿⠀⠀⠀⣾⡇
⠀⠀⠀⢸⡿⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⢰⣿
⠀⠀⠀⣿⠃⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢀⣿⡇
⠀⠀⣸⡟⠀⠀⢀⣿⠇⠀⠀⠀⠀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠷⣶⣤⡄⠀⠀⠀⠀⢰⣿⠀⠀⠀⣸⡿
⠀⢠⣿⠁⠀⠀⣼⡏⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣿⠃⠀⠀⢀⣿⠃
⠀⣼⡇⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⣾⡟
⢰⡿⠀⠀⠀⠘⠛⠿⢶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⣶⣤⣤⣀⡀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣰⡿
⢾⣷⣤⣤⣀⡀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠀⠀⣼⡇⠀⠀⢠⣿⠃
⠀⠀⠉⠉⠛⠻⠿⢶⣦⣤⣄⡀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠉⠙⠛⠻⠷⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢷⣶⣦⣤⣴⡿⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⢶⣶⣤⣤⣀⡀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢷⣾⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠤⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⠉⠁⠐⠒⠂⠠⠬⠩⠉⣉⣐⠒⠒⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⡻⢶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣶⣶⣦⣤⣼⣿⣿⡿⢟⣛⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⠿⣿⣿⣿⣿⠿⠿⠷⣾⣩⣿⣶⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠺⠿⠿⠟⢋⡁⠀⢹⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠤⠁⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣉⣒⣒⣚⡣⢼⢠⣶⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠥⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡒⠒⠤⠤⠶⣿⣽⠀⠁⢸⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣤⣍⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢭⣉⣉⣭⡏⣺⡰⡿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡏⠉⠉⣿⣿⣿⣿⣶⡶⠤⣄⣀⡀⠀⠀⠈⣁⣒⠲⠖⢒⣯⣽⣁⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢤⣼⣿⣿⣿⣿⣿⣿⣷⡆⠶⢿⣿⣒⡶⢦⣬⣉⣩⣭⣷⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⢿⣿⡀⠀⣼⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢥⣍⡛⢻⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠋⠉⠉⠉⠁⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣗⣦
⠀⠀⢀⡠⠶⠾⠧⠤⠤⠤⠤⠤⠤⠤⠀⠠⠤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠂⠀⠉
⢀⠔⠁⢀⡤⢴⠖⣰⢶⠖⠀⠂⠒⠒⠒⠒⠒⠀⠀⠠⠠⠤⠤⠤⠄⢀⢀⣀⣀⣀⣀⡀⡀⣀⡀⣀⠀⡀⢈⢉⠉⡉⠈⠁⣀⣀⣀⠒⠒⠢⢄⠄⠌
⠈⠀⡰⡝⠀⡏⢠⢵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⢹⠄
⡀⠀⣛⠃⠀⡇⠠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡟⠆⠈⡅
⠀⠀⠴⠀⠀⢋⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡇⢰
⠀⠀⠘⡄⠀⢸⠸⣺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠇⠘
⡇⠀⣉⡅⠀⢐⠆⣖⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⠁
⡇⠀⠠⡄⠀⢸⠁⠈⠛⠻⠶⠶⠶⣦⠶⣶⣴⢶⣶⣔⣄⣴⣤⢴⠠⢤⣤⢤⢤⠤⡤⡤⢤⡤⠀⣀⣀⣀⣀⣀⣀⣀⢀⡀⣀⣠⣴⡵⠁⡆⡀⠀⡆
⡇⠀⠰⠁⠀⡎⠀⠃⠀⠤⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠐⠒⠒⠒⠚⠒⠛⠛⠃⠛⠋⠁⠀⠀⡇⠃⠀⡇
⡇⠀⠰⠀⢸⠃⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠂⠤⠄⠀⠀⠆⠃⡇⠀⡇
⠁⠀⠰⠀⡞⠀⠀⢠⠈⠀⠀⠘⠷⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⡀⡇⠀⡇
⠀⠀⠰⢰⠃⠀⠀⡉⠤⠀⢀⠞⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠛⢦⠀⠀⢰⡇⡇
⠀⠀⠔⣯⢠⠀⠀⠙⠶⠤⡤⠤⠚⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠠⠔⠁⡸⠀⠀⠀⢤⡇⢸⠃⠀⠀⠀⣰
⠀⠀⡄⠘⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠒⠒⠀⢰⠀⠂⠒⡶⠒⠒⠒⢲⡆⠀⢠⡤⠀⠀⠠⡄⠀⠀⠀⠀⠀⠉⠒⠤⠄⠊⠁⠀⠀⠀⢸⡇⢠⠀⠀⠀⣴⠃
⠀⢆⠐⠂⡘⠑⠲⠀⠭⠽⠣⠄⡀⢄⠹⢐⠲⣶⣷⣤⣄⣴⣇⣀⣠⣤⡾⠄⡀⣼⣧⣄⣠⠄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠘⡇⢸⠀⣀⡷⡋
⠀⠀⠑⢤⣀⢉⣑⠚⠐⠃⠂⠀⠄⠬⠤⠦⠤⠁⠤⠀⢌⣀⡍⣨⣉⣉⣉⢉⠋⣐⠂⡒⠐⠂⠑⢐⠖⡒⠰⠠⠠⠟⡯⠭⠽⡭⢬⢤⠀⡴⠁⣸⡵⢋⣷
⠀⠀⠀⠀⠘⣹⠉⠈⣽⠉⠀⠐⠒⠒⠒⠂⠰⠀⠤⠤⢀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⠒⠒⠀⠀⠠⠬⠤⠥⠤⠤⠥⠃⠐⢊⣠⠶⠋⠙⠛⠁
⠀⠀⠀⠀⠀⠙⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠒⢋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡂⡔⡛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⠶⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡜⣧⠀⠀⠀⠀⠀⢀⡴⣊⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⢿⡝⣿⠿⠿⢶⢖⡭⢾⣿⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣩⣿⣿⣷
⠀⠀⠀⣴⡾⠿⠿⠿⠿⠿⠶⠶⢶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠿⠟⠛⠛⠋⠉⠁⢀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⠀⢠⡟⠉⣉⣉⣉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠸⣟⠀⢸⡇⢸⠁⡄⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡀⠀⠀⠀⢀⡀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⣯⠀⢸⡇⡏⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⠶⢶⣄⠀⣿⠂⠀⢸⡇⣧⣴⣿⢸⡇⣿⡇⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢸⡇⢧⠈⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣧⣀⣨⣿⠀⣿⠀⠀⢸⡇⣿⣿⣿⢸⡿⠿⠇⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢸⡇⠘⠓⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⠋⠁⢸⣿⠀⠀⠸⡇⠿⠙⠉⠀⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡶⠿⣶⡄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⡀⠀⣸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⡏⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠻⠿⠋⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣶⢶⣦⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⢸⡇⠀⠀⠻⢶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⠀⢠⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⠦⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡼⠁⡇⠀⢸⣿⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇
⠀⠀⠘⠛⠳⠶⢶⣶⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠁⠀⠙⠛⠛⠃⣼⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠶⠶⣶⣶⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⣤⣶⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣶⣿⠷⠾⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠚⠉⠉⠉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⡉⠛⢯⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⢀⣀⣹⠕⠬⠵⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠀⠀⣠⠔⡏⠁⠠⠀⠀⠀⠀⢀⡤⠔⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠀⠜⢣⠀⠙⢴⠒⠒⠦⣄⢰⠃⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢘⠀⠀⢻⠀⠀⠀⠀⠙⢴⠀⠀⡰⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⢣⡀⣀⠀⠀⠈⢷⠾⠔⠒⠉⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⡔⣩⣭⡓⠀⠀⠀⠈⠲⠤⣀⣠⠞⠀⠀⠈⠒⠒⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢶⠀⠀⠀⠀⠸⡀⢀⣀⠔⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⡼
⠀⠀⠀⠀⢀⡠⠔⠊⠁⠈⢆⠘⢄⣀⣀⡄⠠⣈⠁⠀⢀⡠⠊⢀⡿⢿⡁⠐⠒⠒⠶⠶⠶⠒⠚⠉
⠀⣀⠄⠊⠁⠀⠀⢀⡀⣔⡮⢣⠀⠈⠉⠱⢄⣘⡟⠒⠋⠰⣍⠢⠧⠾⢼⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠈⠀⢄
⠹⣄⠀⠀⠀⢀⠔⡡⠃⠀⠀⠀⡷⠲⠀⠀⠀⠀⢤⠒⠂⠉⠀⠀⠀⠀⠀⢀⠉⠒⠒⡦⠀⠀⠀⡴⠢⡴⠋⠀⠀⠀⣀⠤⡀⠈⠢⡀
⠀⠈⢦⠀⠐⡇⠸⠔⣂⠀⠀⠺⡠⡇⠂⠒⣒⣒⣉⣵⠀⣠⠊⠀⠀⠀⠀⠘⣦⡀⣾⣓⠂⠤⠼⢠⣀⠇⢀⠠⢐⢨⠄⡰⠁⠀⠀⢀⣱
⠀⠀⠀⠳⡀⠘⡄⢺⡀⠀⠀⠀⠀⠸⡏⠁⠀⠀⠀⠘⠋⡸⠀⠀⠀⠀⠀⠀⢱⡙⠁⠀⠉⠉⡖⠁⠁⠲⠔⠂⡈⠉⡜⠀⣠⠔⠊⠉
⠀⠀⠀⠀⢱⠀⠘⣄⠱⠀⠀⠀⠀⣀⠵⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠑⢤⡀⠀⠀⠰⣁⠜⢠⠊
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⡝⠐⠒⠒⠒⠒⠂⠈⠉⠈⢆⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⣰⠃
⠀⠀⠀⠀⠀⢱⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⣄⣀⣀⣀⣀⣠⠠⠤⠄⠃⠀⠀⠀⠀⠀⠀⠀⠑⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠭⢭⡭⠭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠓⢌⣀⣀⣅⣈⣛⡷⠒⢂⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⡿⠿⠛⠛⠛⠛⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠈⠁⠀⣠⣤⠀⢀⣄⣴⣿⣽⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣻⡿⢭⣽⣿⣿⡄
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣠⣯⣶⣿⣿⣿⣿⣿⠿⠿⡿⠛⠂⠉⠉⠙⠉⠈⠀⠀⠀⠀⢸⣿⣿⢻⡇
⠀⠀⠀⠀⢠⡏⠄⠀⠀⠀⠀⠀⣸⣿⣿⡿⠉⠉⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⡇
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣆⣀⣠⣤⣤⣶⣾⢿⣿⡿⣾⡇
⠀⠀⠀⠀⢸⠀⠀⠀⢰⠀⢀⣿⣿⣿⣿⣧⣤⣤⣤⣤⣶⣶⣾⡶⣶⡶⢿⡿⠛⠛⠛⠉⣰⣎⠉⢹⣿⣧⣿⡇
⠀⠀⠀⠀⢸⣾⣭⣀⣧⡀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠈⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣶⣿⣿⣿⣿⣟⠀⣿⣿⣦⠀⠀⠀⠀⠀⠀⣀⠠⡄⠠⠔⠠⣀⣈⣭⣥⣴⠿⢟⡿⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠙⠛⢁⣀⣀⣠⣂⠈⣀⣀⡄⠤⠤⣒⣒⣋⣭⣥⣴⣶⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⡶⠿⣛⣋⣉⣩⣭⣽⣿⣶⣾⡿⠿⠿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⡿⠿⠿⠟⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⣤⡄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣶⣶⣶⣶⣶⣦⣤⣤⣶⣦⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⣤⣶⣿⣷⣶⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣸⣿⠻⣿⡇⠀⠈⠉⣿⡟⠉⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⢠⣿⣿⣶⣿⣿⠀⠀⠀⠈⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢿⠏⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⠻⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠟⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠙⠻⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠟⠋⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠈⠙⠻⣿⣷⣤⣤⣾⡿⠟⠋⠁⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁



⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⢶⣶⠶⠶⠶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠴⠶⠒⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠤⠶⠖⠚⠛⠋⠉⠉⣁⣀⣀⡠⠤⠤⠐⠂⢻
⠀⠀⠀⠠⡶⠖⠒⠛⠋⠉⠉⢁⣀⣀⡠⠤⠤⠒⠒⠒⠊⠉⠉⢀⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣧⠀⠐⢒⣊⣉⡉⠉⠀⠀⢀⣀⣄⠀⣤⢶⣄⣶⠟⠙⣿⡁⠀⠀⠀⠀⠀⠀⣧
⠠⣶⣿⣿⣿⠀⠀⢻⣯⣽⡃⣾⡭⣷⢸⣧⣽⠇⠻⠾⠋⠘⠃⠀⣉⣁⣀⠤⠤⠄⠒⠂⢻⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶
⠀⢿⣿⣿⣿⡇⠀⠘⠗⠈⠛⠈⢉⣉⣈⠿⠤⠤⠖⠒⠒⠉⣉⣁⣁⣤⣤⠤⠶⠶⠒⠒⠸⡇⠀⢀⡴⣺⣿⣿⣿⣿⠟
⠀⠸⣿⣿⣿⣧⠀⠒⠒⠉⠉⣉⠁⢀⣀⣤⠴⠖⠚⠛⠋⠉⠿⢭⣄⡀⠤⠄⠒⠒⠊⠋⠀⢁⡴⣫⣾⣿⣿⣿⡿⠃
⠀⠀⢿⣿⣿⣿⠀⠐⠊⠉⠉⣠⡶⠋⠥⠴⠖⠒⠒⠉⢉⣁⣀⡠⠌⠻⣦⡔⠒⠒⠉⣠⠞⣩⣾⣿⣿⣿⡿⠋
⠀⠀⠸⣿⣿⣿⡇⠀⠉⠁⣾⠏⡤⠤⠤⠒⠒⠊⠉⢁⡌⢀⣠⠤⠤⠔⠈⠻⠂⣠⣞⣵⣿⣿⣿⣿⡿⠋
⠀⠀⠀⢿⣿⣿⣷⠀⢉⢸⣿⠠⠤⠔⠒⠒⠚⠉⢁⢎⣇⠀⣀⣤⠀⠐⠒⠀⢘⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠸⣿⣿⣿⡀⢀⡀⢿⣆⠰⠒⠒⠊⠉⢰⠛⣉⢹⡀⠄⠉⠐⠒⢀⣴⣿⣿⣿⣿⡿⠋⢠
⠀⠀⠀⠀⣿⣿⣿⡇⠀⠤⠄⠻⢷⣄⠛⠉⣁⣛⣀⣤⠤⠁⠐⠂⢀⡴⠛⠻⣿⣿⠟⠉⠠⠂⠸⡇
⠀⠀⠀⠀⢸⣿⣿⣷⠀⠠⠲⠒⠂⠙⠳⢦⣅⡀⠠⠤⠴⠆⢀⡴⠋⠀⠀⣠⠟⠁⠠⠴⠖⠒⠀⣷
⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠒⠒⠉⠉⢁⣀⣈⠭⠙⠓⢀⡴⠋⠀⢀⣠⠞⠁⡤⠤⠤⠒⠒⠊⠁⢸
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠒⠋⠁⠘⠀⠀⣀⣤⣤⡴⠋⠀⢀⡴⠋⢁⣀⠤⠤⠔⠒⣲⣿⣎⣁⣾⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠈⠉⠀⠀⠀⠀⠀⣠⠟⠀⣠⠶⠋⣀⣀⡠⠤⠤⠴⠒⣶⣿⣿⣿⡿⠋⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠈⡉⠀⠠⠂⢠⡾⣇⡴⠞⠁⢀⣀⣀⣄⠤⠤⠔⣲⣾⣿⡿⠟⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⣰⣯⠞⠋⠀⠀⠉⠁⠀⠀⠀⢀⣠⣾⣿⠟⣉⣀⣠⣤⣤⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣽⣯⣽⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣭⣭⣭⣭

⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⡇⣶⣶⣶⡆⡅⠀⠀⠀⠀⠀⡀⢰⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⣧⣿⣧⣤⣇⡇⠀⠀⠀⠀⢸⣿⣡⡮⡃⠀⠀⠀⠀⠀⣿⢀⡀⠀⠸⠿⠀⠀⠀⠀⢸⠟⢸⡀⠘⡆
⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠒⠁

⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢈⣿⣏⠀⠀⠀⠀⠀⢰⠒⡊⠉⠁⠒⠂⠀⠀⠀⠀⢀⣀⣠⣤⣶⡆⠀⠀⠀⠀⠀⠀⠋⠛
⠀⠀⠀⠀⠈⢻⠛⠀⠀⠀⠀⠀⢸⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠻⢿⡏⠀⠀⠀⠀⠀⢠⣇⣸⡄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠿⡇

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠶
⠀⠀⠀⣠⠬⢠⢤⢧⡄⠀⠀⠀⢀⠛⠒⣒⠒⡖⠒⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⡗⡆
⠀⠀⠀⠈⠈⠋⠉⠛⠁⠀⠀⠀⠸⠛⠛⠰⠺⠟⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠻⢧⡀⠀⠀⠀⠀⢰⣶⣶⠀⠈⠃⠀⠀⠀⠀⠀⠀⣠⣶⠟⠿⠻⡆⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠈⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠼⡿⠿⠿⠿⠻⠇⠀⠀⠀⠀⠀⣏⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠚⠉⢉⡉⠉⢉⠉⣉⣉⠁⠂⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠊⠛⠷⠡⠀⠼⢧⠢⠧⠧⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣒⢀⣀⢀⠀⣐⣒⠂⠐⠂⠈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠸⠀⠀⡀⡀
⠀⠀⠀⠀⠀⢰⠀⠠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⣀⣀⣤⠄
⠀⠀⠀⠀⠀⠘⠀⠄⠀⢀⡦⢤⡿⢄⠀⠀⠀⠀⠀⠂⠼⠩⢷⡆⠀⠀⠀⠀⠀⠀⢠⠒⡒⣒⢏⠖⠃⠈⠉⠛⠛⠒⠛⢴⡹⡯⣿⡄
⠀⠀⠀⠀⠀⠀⣤⡀⠊⠀⠘⣄⠀⢺⠪⣄⠀⠀⢀⢶⣌⣻⡯⡄⠀⠀⠀⠀⠀⠀⡄⢠⡇⣿⣾⠨⣭⣷⡦⢀⠀⠔⠄⡀⠕⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠱⠂⠀⠰⠋⢉⠂⠉⠃⡀⠀⠀⠀⠀⠀⠈⠁⠛⠉⠁⠑⠀⠠⣤⣤⣤⣤⣤⠆⠠⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠒⢠⢸⠁⠉⢱⢲⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢙⣻⡝⠣⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⡆⢰⡈⢐⡧⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠘⠻⣟⠿⠆⠀⠉⠉⠲⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠋⠋⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⢤⣧⣤⣭⣇⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠈⡄⠈⣿⢻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢠⡆⠀⠀⠀⠀⠤⠷⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⠀⠒⠒⣾⢲⠒⠒⠶⠶⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣀⢄⣈⡠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠁⠈⠊⠁⠂⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢰⣁⣸⠀⠀⠀⣀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⢄⣠⠌⡄⣐⡃⠀⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡤⠤⠤⠤⠶⠒⠒⠂⠀⠉⠁⠀⠀⠆⢱⣅⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢌⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣄⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⢀⠀⢀⠀⢻
⠀⠀⠀⠀⠀⠀⣆⠈⠃⠀⠉⠀⠈⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠘⠶⣄⡤⠤⠤⠰⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠃⠀⠀⠀⠀⠓⠒⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣄⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠉⣀⣄⣀⣠⠈⢣
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⣼⠆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢖⢻⢆⡶⠍⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠉⣉⣠⠴⠊
⠀⠀⠀⠀⢠⡤⠀⠀⠂⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⢢⢱⠸⡀⠀⠀⠀⣀⡀⠀⠀⠀⠈⡭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠨⠁⠐⠂⠀⠀⠀⢀⠀⠴⠀⠀⠀⢱⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠰⡀
⠀⠀⠀⠀⠀⠀⠒⠂⠉⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣀⢰⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣃⠀⠀⠠⠉⠋⠀⠀⠀⠀⡇⠀⠀⢠⡤⢤⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠐⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠃⠈⠀⠘⠉⢸⡄⠀⠀⠀⠠⠀⠒⠀⢀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡖⠒⢳⡶⣋⠛⢫⢻⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠼⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠦⡤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢳⠀⣀⡇⠈⠀⠀⣘⠃⠀⠀⠀⠀⠀⠀⠀⢀⡄⡄⢠⣤⡀⣤⣤⠀⠀⣃⠀⠀⢐⠃⠀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⡇⢸⣸⡇⢸⡇⠀⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠄⢃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⡀⣤⠰⡀⠀⠦
⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠒⠛⠛⠛⠛
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠁⠀⠀⠀⠀⠀⢠⡤⠶⠤⠤⠤⠤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠂
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⢸⠀⠀⢀⡶⠲⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡶⡄⠀⠀⠈⢀⡄⠈⠑⠀⠀⢸⡇⠀⠘⠃⠀⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⢅⠀⠀⠀⣭⠀⠀⢀⡇⠀⠘⠓⠀⠶⣴⡦⠤⠤⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷
⠀⠐⠒⠀⠀⠀⠀⠒⠂⠀⠀⠙⠙⠋⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠊⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠙⢦⣀⣠⡏⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣀⡿⠽⠷⠶⠞⠳⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⣠⠖⠦⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⣀⣀⡠⠔⠊⠿⠶⠚⠋⠁⠀⠀⠀⠱⣄⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡏⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⣧⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⡿⢿⢀⣠⢲⢶⣄⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠸⣾⠀⠉⠉⠀⠀⠀⠀⠓⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢸⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢹⠀⢀⢿⡄⠀⠀⢴⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣀⣻⣄⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⣸⣃⣀⣼⠀⣸⠁⢹⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡭⡿⠦⣄⡀⠈⠀⠀⠀⠀⠀⣠⠾⣭⠝⢋⣹⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⡀⠀⡈⠉⠓⠒⠐⠒⠒⠉⢠⣀⣼⠊⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢉⡋⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄⢸⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠐⠚⠃⠀⠀⠀⠀⠀⠀⢰⠖⠒⠒⠒⠓⠚⠒⠒⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢀⠄⠒⠂⠀⠓⠲⢥⡠⠴⠒⠒⠐⠒⠠⢄⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢸⣁⠀⠀⠄⢀⡀⠀⠀⠀⠀⢀⣀⠀⠄⠀⣀⣧⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠸⣷⡶⠶⠶⠶⠶⣷⣶⣶⣾⡷⠶⠶⠶⢶⣶⡯⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⡖⠒⠐⠒⠒⠐⠒⠚⠓⠐⠒⠓⠒⠒⠒⠒⠒⠒⠂⠒⠚⠒⠒⠂⠒⠒⠒⡲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣤⡤⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠵⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠠⠀⠐⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣥⣬⠻⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣩⣭⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠌⠻⠟⣰⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠻⠿⢣⣿⣿⣿⣿⣿⣿⣿⡿⢫⣾⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢡⣿⣷⣝⢿⣿⣿⣿⣿⢋⣴⣿⡿⠟⣛⣋⣉⣙⡛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣷⡍⣩⣍⢱⣿⡿⣫⣴⠟⣛⣩⣭⣍⡛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⢸⣿⣿⣿⣿⣿⡌⠿⠿⣠⡟⣰⡟⣡⣾⣿⣿⣿⣿⣿⣷⡜⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣶⣿⡎⣿⣿⣿⣿⣿⣿⣷⣾⣿⢁⣿⢰⣿⣿⣿⣿⣿⠟⣡⣿⣿⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣬⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⠸⣿⣿⣦⣙⣫⣾⣿⣿⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣧⡙⢿⣿⣿⣿⣿⣿⡿⢣⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⠻⣶⣭⣉⣛⣩⣥⡾⠟⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣷⣦⣭⣭⣩⣭⣥⣶⠂⠀⣼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⡤⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣲⢰⣖⢰⣶⣆⡔⢲⣰⣖⡆⢲⠆⣶⢰⣄⡆⡴⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠸⠷⠘⠋⠈⠳⠞⠙⠋⠃⠘⠀⠛⠘⠘⠇⠓⠾⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠒⠂⠒⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠒⠀⠐⠀⠐⠒⠂⠒⠂⠢⡀
⠀⠀⠀⠀⠀⠀⠈⠀⣀⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⡶⢶⡶⣶⣶⣶⠶⠶⠶⠶⠶⣶⣶⣶⠶⠶⠾⢻⡖⣿⠻⠿⠿⠿⡩⠿⡿⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠈⠁⠁⠌⡀⠀⠀⠨⣀⡅⠈⠁⠀⠀⠌⢈⡑⢳⣍⡇⠊⠉⠉⠉⠙⠝⡅⠇⠀⠉⠉⠉⢈⡗⠀⠊⠈⠀⠀⢱⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠄⠀⠀⡀⢈⠇⠄⠀⠀⠀⠀⠀⠀⠈⠢⡑⠃⠀⠀⠀⠀⠀⠈⠆⢸⠀⠀⠀⠀⢘⠃⠄⠀⠀⠀⠀⣹⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠀⣿⠀⠀⠀⠀⡢⢀⠀⢀⠇⠄⠀⠀⠀⡠⠤⠱⠀⠀⠀⠱⠀⠀⠀⠀⠀⡀⠀⠈⣌⠀⠀⠀⠀⠘⠑⠀⠀⠀⠁⠀⠸⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⢑⠀⡂⠑⠠⠋⢰⠀⠀⠐⠰⡋⣐⡃⡃⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠘⠀⠀⠀⠀⠘⠂⠐⠁⠀⡀⠁⠺⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡀⠀⢀⠨⡂⠑⠆⠸⠀⠀⠐⡀⠸⠵⢂⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠃⢰⠐⠠⠀⠀⠠⠀⢼⢘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠔⠀⠀⠠⠀⢅⠲⡀⢃⠀⠀⠈⠒⠒⠁⠀⠀⢀⠂⡒⠀⠀⠀⠀⢀⡈⠄⠀⠀⠀⠀⡄⠠⠈⠀⠠⢀⠠⠀⣹⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢁⠠⡀⠁⠀⣀⣄⡰⠄⠑⢄⡀⠀⠀⢀⡀⠔⠡⢞⢽⢀⣀⣀⣀⡈⡿⡰⣀⣀⣀⣀⡄⠠⠀⠴⠀⠀⡀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠤⡊⠥⠈⠡⠤⠄⠀⠄⠉⡉⠒⢂⠡⠀⠀⠈⠂⠄⠀⢠⠄⠀⠙⠂⠂⠀⠀⠒⠬⡀⠂⠁⢀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⠐⠁⣠⠄⠄⠀⠀⠀⠀⠈⡀⢑⠗⢸⠄⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠈⡌⡆⠤⠈⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⡀⣤⡃⠀⠀⠀⣀⢃⠀⠀⠐⠈⠶⡸⠂⡆⠀⠀⠀⢠⢠⠈⠀⠀⠀⢢⣘⡿⠜⠀⠀⠀⡄⡸⠂⠀⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢄⠖⡆⠀⠂⠀⠀⠐⠀⠀⠀⠀⠂⢙⢂⡂⠁⠀⠀⠀⢀⢰⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢠⠓⠀⠄⠀⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠡⠀⠻⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠘⠌⡏⣇⠀⠀⠀⠀⠀⠨⠌⡄⠀⠀⠘⠏⢷⡀⠀⢻⣄⠯⣿⣳⣿⣾⢿⡷⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠡⠔⡘⠁⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠹⡉⠀⠀⠀⠀⠀⠀⢄⡁⠀⠀⢠⣄⠠⢤⠀⠈⢿⢷⡙⠿⢿⣿⣿⡯⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢤⠂⠀⠀⠀⠀⢸⢀⢺⠘⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⢡⠀⢀⣴⡄⣶⣼⢧⢀⣀⠰⣴⡤⢦⢤⣤⠦⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⣦⠘⣄⣒⠒⢰⣤⢫⠊⠜⡦⠡⡤⠖⠊⠈⠄⡠⣍⡩⠓⡭⣗⢛⢊⢸⣭⠩⠰⠁⠂⠇⠈⠈⠤⠀⠱⠀⠻⢆⠄
⠀⠀⠀⠀⠀⠀⠀⢱⡄⣀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⢀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠤⠭⣍⣉⣩⣉⣉⣍⣍⣩⣉⣈⣍⣍⣉⡉⣩⣋⢘⡐⡓⣂⢒⢒⡒⡂⣂⢚⢐⠐⡁⡋⡉⢉⢉⢉⡉⡉⣉⡉⣡⡭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠁⠁⠉⠉⠈⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠈⠄⠈⠀⠁⠀⠀⠀⠀⠀⠈⠀⠈⠄⠈⠀⠀⠄⠂⠁⠂⠅⠂⠄⠂⠁⠈⠄⠊⠀⠂⠁⠀⠀⠀⠐⠀⠂⠁⠀⠀⠈⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢀⣤⠄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠻⠟⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢤⣄⣤⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠴⠟⠷⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠳⠾⠋⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢷⡾⠋⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢶⡾⠋⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣯⣽⣿⣷⣯⣿⡲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⢛⣵⠟⠋⠙⡛⢛⠋⠙⠻⣮⡛⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⣱⡀⣀⣤⣴⣶⣶⣿⣾⣇⡀⣀⢴⣾⣷⡦⣀⢀⣸⣷⣿⣶⣶⣶⣤⣀⢀⣌⡳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣉⡁⠰⠀⡄⡈⠙⣟⡣⠋⡱⢞⠙⢜⣻⠋⢁⢠⠀⠆⢀⣉⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠸⣵⣿⡆⠃⠃⡇⣞⡷⣁⣤⣼⣧⣤⣈⢾⡷⠨⠘⠘⢰⣿⣮⠇⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠐⠂⠬⠍⢛⡴⣞⣛⣺⣿⢋⣤⠤⠤⣤⡉⣻⣗⣛⣳⢦⣛⠩⠥⠐⠂⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢒⣻⣿⡀⠩⠍⠤⣾⢸⢱⣶⢿⣰⣫⡶⢋⡙⢲⡝⣆⡿⣶⡎⡏⣷⠤⠩⠍⢀⣿⣟⡒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⣵⠟⠉⠹⣷⢤⢮⡵⣾⡾⢛⣳⠁⠀⡐⠺⢁⡈⠗⢂⠂⡙⣞⡛⡵⣷⢾⡷⡤⣾⠏⠉⠻⣮⡻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣏⣿⣏⠀⢀⣜⡭⠊⠁⣼⠋⣴⢫⡶⣸⡈⣮⣿⡦⢴⣿⣷⢁⣆⠶⡝⣦⠙⣧⠈⠑⢭⣣⡀⠀⣹⣿⣹⡄
⠀⠀⠀⠀⠾⢶⠶⢾⢾⣿⣿⢈⢾⢾⡱⢎⠶⣿⠀⡇⣏⠰⡁⠰⢈⣏⢾⡷⣹⡁⠆⢈⠆⣹⢸⠀⣿⠶⡱⢎⡷⡷⡁⣿⣿⡷⡿⠶⡷⠶
⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣏⠀⠈⢫⣓⢄⡀⢻⣄⠻⣜⢶⠹⢁⢽⣿⠗⠺⣿⡿⡈⡏⠶⣣⠟⢠⡟⢀⡠⣞⡝⠁⠈⣹⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⡻⣦⣀⣰⡿⠊⢾⣷⢿⢾⣬⡽⡌⠠⠥⢴⡈⢁⡦⠬⠀⢀⢯⣥⡷⡿⢞⡷⠓⢿⣆⣀⣴⢟⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢼⣽⣿⠁⣐⣂⠒⢿⢸⡸⠿⣾⠹⡔⠧⣌⣡⠼⣣⠏⣧⠿⢇⡇⡿⠂⣒⣂⠈⣿⣯⡥⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠠⠄⢒⣒⣬⢳⢯⣭⢽⣯⣌⠛⠒⠒⠛⣁⣽⡯⣭⡭⠞⣥⣒⡒⠠⠄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⢰⡻⣿⠇⡇⡄⡃⢮⡷⡉⠛⢻⡟⠛⢉⢾⡷⢨⢠⢰⠸⣿⢟⡆⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣉⠁⠰⠀⠃⢁⣠⣯⡕⣄⡱⢎⣠⢪⣵⣄⡈⠘⠀⠆⠈⣉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⡹⠁⠉⠛⠻⠿⠿⣿⢿⡏⠁⠉⠺⢽⡿⠗⠉⠈⢹⡿⣿⠿⠿⠟⠛⠉⠈⢋⡵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠙⣮⡻⣦⣄⣠⣥⣬⣆⣠⣴⢟⣩⠋⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⣟⣻⢿⣿⣟⣿⠵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣄
⠀⠀⠀⣼⣿⣿⣿⡟⠁⠀⠀⠀⣠⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣄⠀⠀⠀⠈⢻⣿⣿⣿⣦
⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⢀⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⡀⠀⠀⠀⠻⣿⣿⣿⣧
⠀⣸⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠈⢿⣿⣿⡄⠀⠀⠀⢹⣿⣿⣿⣇
⢠⣿⣿⣿⡟⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⢻⣿⣿⣿
⣸⣿⣿⣿⠃⠀⠀⠀⣾⣿⡿⠀⠀⠀⢠⣿⠏⠀⠀⢀⣴⠄⠀⠀⠀⠀⠠⣦⡀⠀⠀⢻⣿⡄⠀⠀⠀⢿⣿⣷⠀⠀⠀⠸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⣾⡟⠀⠀⢠⡟⠁⠀⠀⢀⡀⠀⠀⠈⢿⡀⠀⠀⢿⣷⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⢸⣿⣿⡆⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⣿⣧⠀⠀⠘⣧⠀⠀⠈⢹⡏⠀⠀⠀⣼⠃⠀⠀⣼⡿⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⣿⣿⣿
⢻⣿⣿⣿⡀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠸⣿⣆⠀⠀⠘⠳⠄⠀⣸⡇⠀⠠⠞⠁⠀⠀⣰⣿⠃⠀⠀⠀⣾⣿⡿⠀⠀⠀⢠⣿⣿⣿
⠸⣿⣿⣿⣧⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⣀⣼⡿⠃⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⣼⣿⣿⣿
⠀⢻⣿⣿⣿⣆⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠙⠿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠈⠿⠋⠀⠀⠀⢀⣼⣿⣿⠃⠀⠀⠀⣰⣿⣿⣿⡏
⠀⠀⢿⣿⣿⣿⣆⠀⠀⠀⠘⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠁⠀⠀⠀⣰⣿⣿⣿⡿
⠀⠀⠈⢿⣿⣿⣿⣦⡀⠀⠀⠀⠙⢿⣿⣿⠂⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠐⣿⣿⡿⠋⠀⠀⠀⢀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⣀⣠⣤⣤⣴⣶⣾⣿⣿⣷⣿⣿⣿⣷⣶⣶⣶⣶⣦⣶⣆⣤⡀
⠀⠀⠀⠀⠀⠀⣤⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⢀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣦⣤⡀
⣰⣿⣿⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣿⣿⡿
⠙⠛⠛⠓⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠉
⠀⢀⡠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠣⡄
⣰⣾⣇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⢻⢀⣴⣿⣦
⠿⠿⠿⠋⠁⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠀⠈⠻⠿⠟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠰⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠰⠳⠆⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⣄⢼⡍⢹⡍⡿⢠⡤⣤⣤⣤⣤⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢸⠁⠀⠙⠚⠖⠛⠀⠀⠀⠀⠀⠀⠀⢸⣈⡻
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⡀⠀⠀⢾⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⠀⢸⣷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣆⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⢀⣾⠃⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠙⠢⠤⣤⣀⣀⣀⣠⠤⠴⠚⠁⠀⣠⡿⠁⠀⠀⢀⣴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⢯⠶⣤⠶⠛⠋⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠋⠀⠙⠟⠛⠿⠶⠶⣶⣶⣶⡶⠶⠶⠟⠋⠉⢻⡄⠀⠀⠘⣦⣴⠶⠛⢻
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢻⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣀⡦⠟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢴⠋⠀⢀⣠⠴⠿⣤⢤⣶⠋⠀⠀⠀⠀⣠⠖⠓⠦⣤⠞⠛⠛⢢⡀⠀⢸⣿⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⢷⡀⠀⠀⣸⡏⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣿⡄⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠹⣄⢀⣿⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⠴⠛⠁⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡜⢻⡄⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣴⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⢻⡿⠷⠶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠶⣶⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡆⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⠀⢀⣀⣀⡀⣼⡀⠀⣀⣀⡀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣭⣭⣭⣭⣥⣽⣥⣭⣭⣬⣥⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⢉⣿⠉⠉⠉⡏⠉⠉⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢸⡟⠀⠀⠀⣯⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⡇⠀⠀⠀⣿⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣉⣹⡇⠀⠀⠀⣿⣉⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⠀⠀⢀⣿⣿⣿⣿⣿⣷⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⢀⡤
⠀⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⠛⠋⠀⠂⠀⠀⠀⠀⠀⢤⣴⣶⡄⠀⠀⢀⣴⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢿⠏⠀⢠⣸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡿⠏⠉⠛⠉⠙⢿⡇⠀⠀⠀⠀⡀⠀⠒⠡⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠄⠰⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠀⠐⠀⠀⠘⣦⣤⣤⣤⣄⡀⠀⠀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⢰⣿⣿⢟⣀⢰⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠈⠙⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠀⠘⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⠿⠿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣸⡟⠃⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⠈⣻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⣼⣿⣿⣿⣿⢦⠀⢀⣀⢼⣯⣴⣦⣄⠀⠀⠙⠋⠉⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⡼⣿⣿⣿⣿⣿⠀⠙⠛⠁⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠋⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⠇⠿⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣀⡤⠶⠶⠦⣄⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⣈⠻⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⣠⡾⠋⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⠾⠋⠀⠀⢠⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣠⣤⠶⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠶⠒⠒⠒⢚⠉⢩⢡⢄⢳
⠀⠀⠀⠀⠀⠀⢠⣤⣭⣉⣛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⣉⣉⢉⣀⡀⠠⣤⠀⡠⠴⠶⣄⠀⠀⠀⠀⠀⠈⠈⣆
⠀⠀⠀⠀⢀⣴⣞⡄⡟⠀⠀⠀⠀⠀⠀⠀⠲⣞⢢⡀⢋⠀⢹⠹⡉⣙⠙⢞⢆⠱⣽⡄⢸⠀⠻⢶⣤⣦⣄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⣤⢻⣻⣷⡳⣹⡀⠀⠀⠀⠀⠀⠀⠀⢏⢣⡱⣜⡄⠈⣇⢻⠻⠂⠀⠳⣱⠏⢮⣾⠀⠀⠲⡦⠤⠿⠂⠀⠀⣀⣀⡠⠀⠹⡄
⠀⠀⠈⢏⢯⢻⣿⣧⢳⣳⡀⠀⠀⠀⠀⠀⠀⠸⠦⠑⠮⠇⠀⠸⠾⠦⠼⠃⢀⣈⣀⣀⣡⡤⠤⠤⠴⠒⠒⠚⠉⠉⠁⠐⢒⣂⡄⠹⡄
⠀⠀⠀⠈⢾⡎⢿⣿⡷⣳⣳⡀⠘⢓⡒⢒⣛⠓⣒⣒⠚⠉⠉⠉⠉⠉⠉⠉⠩⠔⣒⣂⣒⣒⣛⣀⣠⠤⠤⠔⠒⠒⠚⢉⠁⣀⡄⠀⠙⡄
⠀⠀⠀⠀⠈⣿⡜⡿⣷⡽⣷⡱⡀⠐⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⢒⣋⣉⠩⠭⠭⠥⠤⠴⠶⠖⢢⡀⠐⠶⠚⠘⣚⣉⡠⠥⠴⠂⠀⠘⣄
⠀⠀⠀⠀⠀⠸⣷⢹⣿⣣⡻⡵⡱⡄⠀⠒⠃⠲⠶⠐⠒⠂⠒⠀⠀⠈⢣⠀⠀⡠⡠⡠⢠⠀⠀⠀⢠⢳⡀⠈⠭⠥⢄⣒⣒⣠⠠⠤⠄⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢻⢣⢳⢿⡳⡹⡝⡜⢄⠈⢉⣉⣈⣉⣉⣁⣀⣀⡀⠀⠀⢧⠀⠐⠱⠱⢱⠁⢠⢃⢇⠏⣷⠀⠈⣉⠁⠤⠤⠐⠒⠒⠋⠁⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠈⡞⡏⣏⣷⠹⣿⡜⢎⢦⠀⠐⣒⣒⣀⣀⣀⣈⢭⡅⠀⠈⣆⠄⣠⢧⢣⢣⢃⣆⣎⣼⣼⠇⠀⠀⢒⣈⣉⣭⠥⠤⠴⠆⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢸⢹⠸⡼⣏⠽⡟⣎⢧⢣⡀⠰⠒⠒⣒⣒⣛⣋⣉⡀⠀⠘⠚⠛⠛⠛⡉⣉⣉⠀⣄⣠⠄⠀⠀⠀⠒⢒⣒⣂⣛⣛⣀⣀⡀⠀⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡎⣇⢣⢻⡖⠽⡘⡌⢧⢱⡀⠉⠭⠤⠴⠖⠒⠒⠒⠀⠀⠙⠚⠛⠀⠛⢉⣁⣀⡄⠀⠀⠀⠀⠀⣈⣩⠤⠤⠤⠤⠤⠤⢄⢀⡂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⡜⣎⣿⡭⢧⣣⠸⡆⢃⠀⢒⣚⡉⠉⠉⠉⠉⠀⠀⠘⢋⣉⣉⣁⣨⠁⠉⠠⢐⠒⡒⢃⣀⡷⣠⢄⡤⠀⣠⣶⢄⡿⣟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣳⡘⣎⢿⣿⣿⢀⣇⢸⠀⠠⠤⢤⠤⡖⢲⣒⡆⢠⠈⡽⢴⡗⣲⣳⣟⣻⣋⣽⣭⣋⣭⣭⣡⣍⣩⣃⣨⣷⣿⣾⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡳⣌⠳⠬⣿⣿⣫⡟⡄⢰⣖⣟⣩⣯⣏⡼⠤⠟⠒⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢍⣓⣿⣿⣿⠴⠴⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⡠⢀⠄⡠⢖⡤⢊⠴⢂⠔⢊⠴⢂⠴⢊⠤⢊⡴⢂⠔⢃⠔⢂⠔⣃⠔⣀⠔⡠⢂⡤⠂⡠⢂⡴⢂⠔⢀⠔⢁⠔⣠⠞⡠⢂⠔⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠊⠰⠋⠔⠁⠔⠁⠔⢡⠖⢁⠔⠁⠴⠫⠖⢡⠔⢡⠔⢡⠞⢡⠞⠥⠊⡴⠋⡴⢉⡔⢁⠔⢁⠔⠁⠔⠡⠞⠡⠊⠐⠃⠚⠐⠁⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⠋⠁⠾⢛⣿⠿⠃⢸⠇⠀⠈⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⡀⢸⣿⠃⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⢺⣿⠆⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠆⢠⣿⣿⡿⠃⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢸⠙⠚⠛⠁⠉⠉⠉⠗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣾⡿⠟⠁⠀⠀⠘⣧⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠰⠀⢾⠀⠀⠀⠀⢀⣆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠓⠁⠀⠰⠀⠀⠀⠀⠀⠀⡘⠀⣸⠀⠀⡀⠀⡾⠘⣇⠀⠀⠃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⡤⠔⡃⠀⠁⠀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡇⠀⠈⠛⠀⠀⠀⠀⠀⠈⠁⠀⣷⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠓⡖⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠼⠇⠈⠁⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠤⠼⠯⢍⡉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⠀⠀⠀⡇⠀⠀⣀⣀⣀⣀⡀⡄⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠨⣧⣴⠶⠶⠶⢶⣷⣦⣧⣶⡷⠶⠶⠶⣶⣜⡇⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⠒⠒⠒⠒⠚⠒⠚⠒⠒⠚⠒⠒⠒⠒⠓⠓⠒⠒⠒⠚⠓⠀⠐⠒⠒⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠀⠠⠤⠄⠠⠤⠄⠠⠤⠤⠀⠤⠤⠀⠤⠤⠤⢤⠄⠤⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠆⣤⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⡀⠻⣟⡉⣽⣋⠙⠙⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠀⠀⠀⣽⣎⣽⠳⣦⡘⠻⣧⠀⠀⠀⠀⠀⠀⠀⢠⡄⠚⢛⠒⡒⢒⠋⡏⠛⢶⡶⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣾⣆⢻⣷⣸⠀⠀⠀⠀⠀⣠⡼⢶⡗⠤⡠⠈⠀⠡⠘⠠⠘⠨⣑⠈⠉⠙⠛⠲⣄⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣷⣇⣶⡒⢤⡀⠞⠁⠀⠀⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠐⢛⣏⣀⣀⣀⣴⣲⢪⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⢸⠉⠀⠠⠉⠓⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⣆⠀⠀⠀⢸⠺⠭⠭⠽⠀⢹⠏⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⢳⠏⠀⠀⠀⠀⢤⡀⠀⢀⡄⢾⣅⠹⠆⠈⠁⠀⠀⠀⠉⠉⠿⣝⣻⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢦⡀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢀⠲⣝⢆⢦⡙⠒⡌⠋⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣴⣇⣶⣆⡼⠀⠀⠀⠀⢴⡛⢦⡣⣙⠷⠙⠉⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡠⣼⡀⠻⡌⠓⣀⣤⡀⠀⠀⣀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠷⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⡄⠹⣎⠳⠄⣠⡾⠋⢾⣥⠶⠛⠉⠀⠀⠀⣾⣩⡿⠿⡍⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⣅⠀⢰⠋⠳⡝⠦⠝⣠⠞⠁⡠⠊⠓⠀⠀⠀⠀⠀⠀⠀⢯⠹⠮⠽⠟⣼⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠻⣌⠷⣤⠏⣠⡞⠁⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠿⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡀⠀⠀⠀⠈⠉⠉⠉⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⣥⣀⡀⣀⣀⣠⣤⠞⠋

⣶⣾⣿⣿⣿⡿⣾⣿⣷⠀⡂⠀⠤⠤⠤⠤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢱⡆⠀⠀⠁⠀⡆
⣿⣿⣿⣭⣤⣤⣬⣻⡿⠀⠅⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠲⠆⠰⠯⠍⢸⣧⠠⣤⡄⠤⠷⠤⡦
⠛⠹⠃⠹⣼⠟⢻⣿⣿⣄⡁⠀⠠⠄⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣶⣤⠀⡀⢈⣧⣤⣤⣧⣤⣤⣤⣷
⢦⠀⣤⡀⠀⠐⠻⢺⡿⠋⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣾⡥⠅⠀⠈
⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⢘⣯⣼⡇⠸⡋⠐⣿⡉⢿⠏⢹⠏⢹⡟⣻⡟⢻⡟⢻⡟⠛⢛⣠⡀⢀⣤⣤⣶⣤⣤
⠀⠀⠀⠀⠀⣀⣀⣀⠀⣤⣤⣤⣸⣿⣿⣇⣨⣅⣨⣿⠷⠱⠖⢻⠦⠸⣿⣿⣷⣾⣷⣾⡟⣧⢈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠿⢿⣿⠰⣶⣆⠀⠀⣶⠂⠀⠀⡀⠀⠀⢀⢢⣄⣐⡦⠀⢸⢻⡿⢿⠟⠛⣿⡿⠈⢹⣽⣿⢿⣿⣿⠛⢿
⠀⠀⠀⠀⣠⠀⣨⡟⠰⡿⡏⡁⣧⣇⣈⣀⣀⡃⠀⠀⢸⣷⢮⣭⡖⠀⢸⠀⢀⢈⣀⣘⣛⠁⠀⠘⢉⢻⡤⠭⠤⠾⠟
⠀⣀⡀⠀⣀⠀⣿⡇⠀⠟⠿⠇⠮⠌⠉⠉⠿⠇⠀⠀⣈⣛⡚⢋⣃⠀⠸⣷⣈⣙⡉⠉⠉⠀⠀⠰⠾⠾⠏⠤⠀⠤⠃
⠀⡿⠇⣀⣠⣾⣿⣷⠰⠶⠆⠐⠶⠀⠀⠀⠀⢀⡀⠚⠛⣿⡅⣶⣏⠀⠀⢸⣿⣿⢫⣄⡀⠀⠀⠀⠀⠊⠀⣤⡀
⠀⣤⡆⣿⣐⣈⣸⣿⠰⠦⠀⠀⠒⠀⠀⢀⡀⢸⣧⡴⠶⠿⠦⣿⠇⠐⣶⠂⣿⣧⡼⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⣛⣿
⠀⠛⠇⠀⠀⠀⠀⣿⢠⠰⠄⠀⠀⠀⠀⠀⠀⢸⣿⡟⢖⠀⠀⠉⠀⢰⠖⠀⢹⣿⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⡄⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣧⡸⠷⠶⠶⠄⠤⢄⠀⢈⡿⠦⣙⠠⠞⠀⠀⠀⠀⣵⣶⣼⣿⣿
⡲⣶⣷⣶⣶⣶⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢨⣦⣼⣷⠂⠀⠀⠀⠀⠀⠀⠀⠁⣿⢻⡿⠿
⣿⣿⣷⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡀⠀⣾⠁⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣈⠩⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⢘⠉⢰⡏⣃⠀⣇⣀⡀⠀⡀⠀⢀⣠⠀⢰⣾⡟⠋⠉
⠿⢓⣿⠃⢀⢈⣷⣿⣿⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⡇⢸⣿⣿⣡⣧⣤⣿⣏⡁⢺⣣⡴⢉⣛⣴⡙⢋⣷⢒⠶
⠒⠛⠛⠙⠁⠚⠛⠿⣿⣿⣿⣏⣀⠈⠇⢰⣿⡆⠀⠀⠀⠀⣄⣈⣘⣻⡟⠛⠛⠛⠛⠳⠛⠛⠉⠙⠟⠗⠖⠺⠾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣷⣶⣄⢉⠀⠀⢀⣴⣶⣿⣿⣿⣿⡇⢀⢰⣶⡄⡆⢰⣶⣶⣶⣷⣷⠀⠀⠠⣿⣿
⠀⠸⠋⠁⠛⠀⠀⠀⠀⣿⣿⣿⣾⣿⣯⣛⡋⠀⢠⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣼⡟⡛⣿⡟⠀⠀⠀⠀⠈
⠀⠀⠈⠁⠉⠀⠀⠀⠀⡙⠛⠛⠛⣿⣟⣿⣦⡴⣾⣿⣿⡦⠛⢿⣿⣿⡟⠿⠿⠿⠿⠀⠀⠛⠓⠄⠈⠉
⠀⠐⠖⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣼⣿⣿⣷⣿⣿⣿⠆⠀⠸⠿⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢶⣶⣶⣶⣶⣴⣴⣤⠄⡠⠀⠀⠘⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢛⣉⣑⣶⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⡵⠊⠁⠀⠀⠉⠂⠁⠢⠀⠀⠩⢭⣭⣭⠍⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢸⠁⠀⡷⡀⢀⣇⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢸⡀⠸⠒⠃⠀⠁⠀⡀⠀⠀⠀⢩⣭⣩⣭⡉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡑⢄⣀⠀⢀⣀⠔⡡⠄⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⡦⣉⡭⠀⠔⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠤⠤⣤⣤⣤⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠭⠭⠭⠽⠭⠿⠿⠯⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡶⠶⠶⠴⠶⠦⠀⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⣿⢹⡏⣭⣭⠹⠏⡿⢹⢡⣌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡌⢸⡇⣭⣽⡄⢀⠀⣼⢦⡍⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣷⣼⣧⣭⣽⣧⣼⣤⣿⣬⣡⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⣛⣛⣛⣛⣛⣛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠩⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⠀⠀⠀⢀⡴⡊⠉⠉⠉⠑⠢⡀⠀⠀⠀⢀⠔⠊⡇⡏⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠉⣿⠀⡇⠀⠀⠀⡾⠊⠁⡄⢀⠞⠀⠀⠘⡄⠀⠀⡎⠀⠀⡇⢓⡩⠒⡆
⠀⠀⠀⠀⠀⢀⣄⣸⠈⢹⠀⣿⠀⡇⠀⠀⡸⠁⢀⣼⠵⠋⠀⠑⣄⠀⢣⠀⠀⢧⠤⡀⠛⠁⠀⢀⠇
⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⣿⠀⡇⠀⢠⡧⠴⠋⢉⡁⠀⠀⣙⠈⢦⡀⢣⠀⢸⢠⡧⣤⡤⠴⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⡇⠘⣦⠀⠀⢠⠀⠀⢠⢾⠋⣸⠘⠓⠒⠚⠂⠀⡀⠀⠀⢈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⣦⡀⠀⠀⠀⠹⡄⣏⠳⣄⡛⣃⡴⠃⡞⢰⠃⠀⠀⠀⠀⠀⡼⠚⠦⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⣂⣉⠙⡆⠀⠀⠀⠈⠋⢳⡇⠈⠀⣧⡾⠗⠁⠀⠀⠀⠀⠰⠽⠀⠀⠀⠓⠁
⠀⠀⠀⠀⢀⠔⠂⠒⢾⠀⣒⣒⠀⡇⠀⡠⠲⡊⠉⢣⠙⢲⡊⣸⠉⠑⠲⠤⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢎⠀⠀⢀⠘⠀⢤⠀⠐⠃⠰⠀⢨⡇⠀⢘⠑⣻⠛⠙⠀⠀⡏⠀⢹⠀⠀⡷⠒⠒⢒⡒⠚⠛⠺
⠀⠀⠀⡰⠋⠀⠢⣼⡴⠀⠘⢢⠀⠀⡆⠀⢸⠇⠀⠘⠊⠑⡏⠚⠀⠀⢷⠀⠀⡄⠀⡇⠀⠀⢇⣸⡀
⠀⠀⠀⠱⣄⣀⣀⣀⣀⣀⣀⡼⠀⠀⠇⠀⢸⠀⠀⠀⠀⠈⢧⠀⠀⠀⢸⠀⠀⡇⠀⡇⢐⣒⣒⣒⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⣼⠀⠀⠀⠀⢀⡜⠀⠀⠀⢸⣀⠤⢱⡀⠣⠤⠤⠤⠤⠤⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣏⡉⣺⡄⠀⠀⠀⡀⡧⡀⠀⠀⣾⠒⣉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣸
⠀⠀⠀⠀⠀⢀⣀⣀⢶⣶⠯⠭⣷⣔⣁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣾⣗⢦⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠸⠷⡟⠁⠀⢀⠜⠮⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠉⣉⣉⡀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢃⣴⣶⣿⠆⠀⢀⠀⠙⠿⠿⠷⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠁⠘⠋⠉⠀⠀⠀⡉⢆⠀⠀⢰⣷⠀⠀⣸⡉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠩⣸⡀⠀⠀⠀⢸⣿⠀⢀⡇⠸⣆⠀⠀⠉⠀⠀⡏⣁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡈⣃⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⢠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢀⣄⣀⣤⣴⣶⠀⠀⠀⡎⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠈⢿⡿⠿⠿⡿⠀⢀⡜⠀⣠⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠈⠑⠒⠊⠀⣠⠊⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠠⠤⠤⠤⠊⢁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠻⡿⣋⠘⡄⠠⣿⠏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠸⡌⢛⡕⢫⠀⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⠟⠓⠢⡑⠮⣀⡰⠖⠛⠛⠂⣀
⠀⠀⠀⠀⠀⡀⠠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠉⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠈⢦⠌⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⡼⠮⠑⠃⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡤⠀⣠⠤⢤⡀⣽⣿
⠀⠀⠀⠀⣏⠁⠀⠀⠀⢰⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡜⡇⠀⠘⣿⠿⠃⠇
⠀⠀⠀⠀⠓⠤⠤⠤⠤⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⡠⠇⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⣀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠒⠂⠀⠈⠉⠁

⠀⠀⠀⠀⠀⢀⡠⢴⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⢠⢯⢾⣽⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣚⣻⣿⣿⣛⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠘⠚⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠭⢽⣿⣿⠭⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⣄⣿⣿⣿⣿⣿⣇⣦⡀⠀⠀⠀⣷⢾⣿⣿⣿⣿⡷⡷
⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⠀⡀⠀⠀⠀⢸⡀⣿⣿⣿⣿⣿⠇⡇⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⠂⡇
⠀⠀⠀⠀⠀⢇⠻⣿⣿⣿⡿⢠⠇⠀⠀⠀⠀⢳⡘⠿⠿⠿⢋⡴⠃⠀⠀⠀⠀⠸⣄⠻⢿⡿⠛⣰⠃
⠀⠀⠀⠀⠀⠈⠳⠬⣽⡭⠔⠋⠀⠀⠀⠀⠀⠀⠈⠛⢲⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡟⠋
⠀⠀⠀⠀⠀⠀⠀⣤⣼⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣧⣤

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣗⡂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣲⣖⣿⣿⣿⣿⣗⣶⠂
⠀⠀⠀⠀⢸⢹⣿⣿⣿⣿⡏⢻⠀⠀⠀⠀⢰⢸⣿⣿⣿⣿⣿⢠⡆⠀⠀⠀⠀⢸⡁⣿⣿⣿⣿⡇⡿
⠀⠀⠀⠀⠈⢦⡙⠿⠿⢟⣡⠋⠀⠀⠀⠀⠘⣆⠻⣿⣿⡿⢋⡼⠁⠀⠀⠀⠀⠈⢧⡙⠿⡿⠟⣠⠃
⠀⠀⠀⠀⠀⠀⠉⠙⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⡷⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡗⠋⠁
⠀⠀⠀⠀⠀⠠⠤⠴⠧⠤⠤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠷⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠧⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠀⠀⠀⡴⠞⠛⠛⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇⠀⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠖⠛⠋⠁⠀⡀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⢀⣠⠀⠂⠀⠔⣴⠀⠽⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⣦⣲⠐⡱⠊⠎⠀⠀⠈⠉⠀⠀⢠⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠛⠣⠐⠒⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢾⣆⣀⣀⣤⠴⠖⠛⠋⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠿⣦⣄⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠋⠳⠈⠁⠀⠀⠈⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢰⡆⠀⠀⣀⣤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠰⡀⠀⣀⠀⠀⠉⠻⢿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠈⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠉⠂⠉⠀⠀⠀⠀⠀⠈⠙⠛⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⣵⢄⠐⠆⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⡄⠀⠀⠘⢷⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠛⢿⣷⣄⠀⠀⠀⠸⣧⣠⡶⠟⠁⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠈⠁⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣀⠀⠀⢰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⠈⠉⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣀⡀⠀⣀⣸⠇⠀⠀⠐⠒⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁⠀⠀⠲⢦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡸⢂⣀⣁⠀⠀⢸⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣭⠥⠄⠀⣀⠾⢿⡟⠿⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠖⣩⣹⣦⣤⣾⠋⠀⡸⢛⠀⠙⡂⣀⣀⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣶⣿⣧⠀⠹⣏⣿⢻⠀⠀⣸⢁⠁⣀⠀⣡⠍⠛⠲⣤⡀⠉⠒⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠟⠛⠻⢧⡀⣿⣽⡾⠀⠀⠙⣿⣶⣤⡾⠋⠀⠐⣿⣿⣷⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⣄⣿⣿⠏⣴⣾⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⣠⣿⣿⠂⠀⠀⠀⠀⠀⢀⣖⣭⡼⠾⣿⢣⣾⡯⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠁⠀⠀⠀⠀⠀⠀⣼⣿⠛⠛⠙⠀⠀⠀⠀⠀⠈⠁⠀⠀⣼⣣⣿⣟⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣇⠀⠀⠀⠀⠠⠠⡀⠀⠀⠆⣰⣱⢏⣿⣿⣧⡰⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢢⠀⠄⡀⠀⠀⠃⣫⣿⣿⣿⣧⠀⠈⢀⡀⡐⣁⠀⡟⢰⣧⠟⢰⣿⣿⣿⡁⠀⠀⠀⠀⡠⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠲⣦⣿⣿⣾⣿⣷⡡⢀⣴⣿⣿⣿⣿⣿⢿⡶⠊⠓⠘⠋⠊⣾⡿⣾⠀⠸⣹⣿⡵⠿⠚⠛⠛⠊⢋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡀⢢⡙⠻⠿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⡤⡠⣤⣭⣷⣯⣐⣮⠟⠉⠀⠀⠀⠀⠄⣂⡭⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⢌⣳⣳⣤⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⢿⣿⣿⣷⣼⣿⣿⡿⠛⠉⠀⠀⠀⠀⡀⢊⠥⢊⡴⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠻⣿⣷⡀⢀⡀⠀⠀⠀⠀⢀⠄⠀⠙⢿⣿⣾⣿⣿⣿⠿⣿⣆⠀⠀⢀⠠⠀⡊⡤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠘⣿⣿⣷⣱⢀⢀⢀⡂⠀⠪⣪⢊⣫⣿⣿⡿⠋⠁⠀⣸⣿⣇⠠⣴⣢⢩⣸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠠⢾⡛⢿⣿⣿⣾⣿⣿⣅⡟⠉⠛⢿⡿⠋⠀⠀⠀⢰⣿⣯⣿⣽⣿⣾⡿⠿⠿⣯⣵⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡆⢀⠀⠀⠀⠘⣧⠤⠍⢿⣿⣿⣿⣟⡀⠀⠀⡼⢁⡄⡠⢀⢠⣿⣿⣿⣿⣿⣏⣅⠀⠀⠀⠈⢻⣞⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⡂⠀⠐⠀⠀⣿⣆⠚⣿⣻⡿⠿⣿⣿⣶⣼⢁⡟⣰⣣⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿⡯⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠒⢐⣻⣧⣱⣀⣀⣀⣾⣷⠶⠻⠶⢦⡞⣾⠯⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⣾⠤⠶⡤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡃⠈⠉⠀⣀⣀⢹⣿⣷⡒⣠⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣦⡠⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠿⢿⣿⣿⣿⠏⣿⡟⣃⣶⡿⡶⢢⠈⡻⣷⣼⠑⢉⡂⢍⡩⢯⣽⣿⣿⣿⠁⠀⠀⠀⠀⣾⣟⠋⢐⠃⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⣻⢿⣿⣵⢽⡿⣯⣿⠷⢶⣻⠺⡓⡛⣻⣚⠿⢗⡮⠳⡞⠿⠿⣿⣟⠀⠀⠀⠀⢰⣿⣧⠞⠛⠮⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣰⣁⠀⡸⠷⠵⠋⢻⣤⣢⣸⣿⣶⠨⣆⡁⢤⣠⣿⠯⠄⠐⠋⣁⣿⠇⠀⠀⠀⠀⣼⡿⣴⣆⣐⢟⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠦⢤⢉⠀⣠⡴⢒⢘⣍⣭⡿⠿⡿⡿⠻⠛⣛⡛⠛⡛⠛⠛⠛⠛⡿⠀⠀⠀⠀⢠⡟⠓⠋⠙⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠘⠛⠀⠴⠀⠘⠩⡴⣎⣶⠟⠉⠀⠀⠀⠁⠐⠟⢿⢮⣠⡀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣁⡀⠀⠠⠀⡔⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣿⢟⠋⠀⠀⡾⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣵⣴⢀⣼⣻⣋⣭⡉⠓⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⣷⢯⣿⣿⣿⠀⠈⠠⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡖⠒⡴⢲⣰⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠉⠈⠋⠈⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡾⢉⣤⣤⡉⢷⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠒⠺⣿⠀⠀⠀⠘⠷⠾⠃⠀⠀⠀⣿⠗⠒⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣾⠋⢠⣄⡈⢻⡄⢈⣛⣛⣛⣛⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⡀⠈⠉⠁⣸⠇⢈⣉⣉⣉⣉⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠈⠛⠶⠶⠞⠋⠀⠈⠉⠉⠉⠉⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⡀⢰⡆⢸⡇⠀⠘⣛⣛⣛⡛⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣀⣿⣸⣇⣸⣇⡀⢈⣉⣉⣉⣉⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢰⡇⠀⠀⣴⠶⠤⣤⣤⣤⡤⠶⠶⠛⠉⢹⡇⠀⠀⢸⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠠⠶⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠙⠲⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⢷⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢿⣿⣿⣿⢇⣤⣶⣶⣶⣦⣄⠀⠀⠘⣆⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⠀⠀⠳⣄⠀⠀⠀⣏⠀⠀⣹⠆⠀⠀⠀⠀⢀⣴⠏⠀⠀⠈⠁⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⣬⣷⣤⣤⡼⠟⠛⠉⠀⠀⢀⣠⡴⠛⠁⠀⠀⢀⠀⠀⣬⣭⣭⣽⣿⣿⣯⣭⣭⡅⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠙⠛⠓⠒⠛⠛⠉⣿⣀⣀⠀⠀⠀⢸⡇⠀⣩⣭⣭⣽⣿⣿⣯⣭⣭⡅⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⣿⣿⣿⣄⡀⠀⠀⠀⢀⣰⣿⣿⣿⠙⠂⠀⢸⡇⠀⣩⣭⣭⣽⣿⣿⣯⣭⣭⡁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⣿⣿⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⢉⣉⣉⣽⣿⣿⣯⣉⣉⡁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠙⠿⠿⠏⠀⠀⠀⠈⢧⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠉⠛⠻⠛⠛⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣤⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉



⣿⠿⠿⠿⡿⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣻⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣤⡀⠀⠀⢠⡀⠀⣤⣤⣤⣤⡄⣠⡀⠀⢀⣄⠀⠀⣠⡀⢀⣠⣤⣄⡀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣷⡄⠀⢿⡇⠀⣿⠉⠉⠉⠁⢸⣇⠀⣼⣿⡄⢀⣿⠁⣾⡏⠉⠉⠃⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⠈⢿⣄⢻⡇⠀⣿⠶⠶⠶⠀⠀⣿⢀⡿⠘⣇⢸⡇⠀⠘⠻⠶⢶⣄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⠀⠀⢻⣿⡇⠀⣿⣀⣀⣀⡀⠀⢸⣿⠇⠀⢻⣿⠁⠀⣤⣀⣀⣠⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠉⠀⠀⠀⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣠⣀⣀⣀⣀⣀⣀⣄⡀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠘⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠟⠃⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠻⡄⠀⣠⠴⠂⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠴⣾⠁⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⡿⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣼⢃⡴⠚⠙⠲⣤⡀⣾⡟⠀⠀⠀⠀⠀⠹⣆⠀⠚⠉⡉⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠈⠙⢯⡿⢦⡀⠀⠀⠀⠀⠙⢧⡀⡿⠙⡟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣿⡄⠀⠀⠀⢀⣀⣭⡇⢀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠦⢤⣽⣷⣿⠤⠖⠚⠉⠉⠀⠙⠛⠓⠞⠷⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠴⠶⠶⠶⢦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢦⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠄⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠻⠶⢾⣷⣤⣬⣵⣤⣤⣵⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⣙⢦⡀⠀⣠⡟⠀⠀⠶⠖⣻⣿⠟⠓⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠁⣿⠂⢙⡾⠋⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡶⠟⠁⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾⠀⠀⠸⣧⠶⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣄⣤⠶⡿⢲⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣽⣷⣭⡙⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⣈⡝⢿⡀⠙⢿⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⣇⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣠⣶⣶⠾⠷⠿⠿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡯⠀⣿⠷⢼⡇⢰⠜⠃⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣄⢹⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠠⣾⣥⣤⣤⣤⡴⠛⣻⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢳⠀⠸⡇⠈⡇⢸⠀⢸⡿⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢷⣜⣧⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⢷⠀⣿⢸⠇⢸⣇⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⣄⡈⠻⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠸⡇⢸⡿⠀⠈⣿⢹⣆⣿⣤⢴⣶⡤⠶⣶⣆⢹⣷⠤⢀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠛⠋⠛⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢿⡀⠀⠸⡇⠀⣿⣸⣧⣤⠶⠛⠉⠉⠀⠀⢰⣿⠀⠀⣿⡽⠦⠿⠄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠛⠶⠖⠲⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠃⠀⠀⢿⡤⠼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣀⢹⡗⣶⠶⢆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣒⣚⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠈⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⡚⣣⡇⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠈⠙⠓⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠸⣿⡃⠀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⡟⠉⠉⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⡏⠙⣿⠉⢻⣏⣉⣠⡤⠶⠚⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⢿⣆⣠⣿⠀⠘⣏⠉⠀⠀⠀⠀⣿⣤⠤⠤⠴⠶⠶⠶⠖⠚⠛⠛⠛
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠁⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠂⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣧⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣤⠤⣤⠼⠟
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠛⢛⣻⣿⣿⣿⠿⠧⠤⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠒⠒⠂⣉⡿⠋⠀⣮⠈⠛⠓⠶⠶⢠⡿
⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠁
⢀⣀⣀⣤⣤⣤⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⣤⠴⠶⢿⡷⠻⠛⠒⠒⢠⡄
⠻⣤⣀⣀⣈⠹⣦⣹⣯⣭⡿⠿⠛⠛⠋⠉⠁⠀⠀⠉⠉⠐⠶⠾⠶⠧⠶⠶⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⣿⣿⡿⠛⣿⣿⣿⡿⠋⢘⠇⠀⠉⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠐⣿⣿⠁⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠆⣿⡇⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡎⠑⠓⠋⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠶⣴⣿⠿⠋⠀⠀⠀⣻⡄⡀⠀⠰⠒⠂⠀⠀⠀⠀⢰⠀⣯⠀⠀⡁⠀⡰⡄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣙⠂⠀⠘⡀⠀⠀⠀⠀⢀⠎⢀⡿⠀⢀⡇⢐⠃⢻⡉⠀⠈⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢤⣀⠀⠀⠀⠀⠀⢀⡴⢯⡤⠔⣊⡆⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣎⠀⢀⠉⠉⠒⠒⠒⠋⠁⢀⣠⡗⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣽⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⣀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠓⠛⠂⠀⠀⠀⠀⠀⠐⠒⠚⠒⠐⠓⠛⠒⠒⠛⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠔⠂⠉⠉⠉⠁⠲⡴⠂⠉⠉⠉⠁⠒⠄⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣧⠐⣒⣀⣒⣠⣄⡀⡇⣀⣤⣐⣐⣀⡒⣰⠆⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠙⠛⠻⠿⠛⠋⠉⠉⠉⠛⠛⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣤⡄⢠⣤⠀⢀⣤⠀⢠⣤⠀⣤⣤⠀⣤⡄⠀⣠⣀⣤⠀⢠⣤⡄⢠⣤⡬⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁⡍⢨⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠊⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣫⣭⣭⣭⣭⣝⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣾⣿⠿⠿⠿⠿⣿⣷⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⠟⣵⣾⣿⣿⣷⣮⠻⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢈⣼⣿⣿⣿⣿⣿⣿⣧⡁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡋⢉⣛⣛⣛⡛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣭⣍⣩⣭⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣘⠿⢿⡿⠿⣃⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣆⣰⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣛⣛⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠛⡇⠀⣀⢀⣀⣤⡄⡀⠀⢀⣀⣄⡀⠈⡗⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⢀⡇⠀⡟⢾⣻⡗⠀⢳⣦⡏⣙⢲⡅⠀⡇⢸⠀⡏⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠈⠉⠀⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢀⡤⠤⠤⠤⢤⡀⠠⠤⠤⠤⠤⠤⠤⠤⠄⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⡇⠰⠶⠶⠶⠶⠶⠶⠶⠀⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠘⠒⠒⠒⠒⠚⠁⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠀⠀⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠀⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠀⠀⣉⣉⣉⣉⣉⣉⡁⢈⣉⣉⣉⣉⣉⣉⡁⠀⡇⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⣇⠸⠤⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠶⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠶⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢮⣥⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⢸⣆
⠀⠀⠀⣀⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢹⡆
⠀⠶⣯⣥⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠦⢤⣀
⠀⠀⠀⠀⣤⣾⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠛⣻⠖⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠛⠛⠛⠋⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⣠⠾⠋⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣴⠏⠀⣠⠄⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⢀⡾⠁⠀⣛⣁⣀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⢀⡾⢡⡾⠛⠉⠉⠉⠛⠳⣿⠀⠀⠀⠀⠀⢀⣴⢠⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣤⡤⠾⢯⣄
⠀⠀⢀⡾⠁⡟⠀⠀⠀⠀⠀⣿⣽⣿⠀⠀⠀⠀⠀⠀⠛⢻⡇⠀⠀⠀⠀⠀⠀⠈⢷⣤⠞⠋⠁⠀⠀⠀⠀⠈⡷
⠀⠀⣾⠃⠀⢿⣀⠀⠀⠀⠀⠀⣠⣿⡀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⢾⣿⠿⠶⡤⠀⣼⠃
⠀⢰⡟⠀⠀⠀⠉⠛⠲⠶⠖⠛⠁⠈⢿⣦⣤⣤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠀⣀⣤⣠⣤⣄⣀⣤⡴⣆
⠀⢸⡇⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⢤⣤⣶⣶⣿⣯⢤⡀⠀⠀⠀⠀⣾⠛⠻⡄⠹⣆⡿⠁⠀⠀⢹⡄
⠀⢸⣇⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠛⠛⠉⠉⠁⠀⠀⣿⣀⣷⣀⣠⣤⣤⣧⣦⣀⡿⠼⠋⠀⠀⠀⠀⠀⢿
⠀⠀⢻⡄⠈⠛⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠰⡞⠋⠉⠀⠀⠁⠀⣀⣤⣦⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠙⢦⡀⠈⠉⠉⠉⠙⠛⠒⠶⠤⠄⠀⣀⣠⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⣶⠚⣷⠀⠀⠀⠀⣴⣾⣿⣿⠿⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣤⣤⣤⣴⠶⢾⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⡶⠶⠤⣤⣷⡇⠀⢸⡄⠀⠀⠀⢹⣿⣯⣤⣶⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠸⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡟⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠸⡇⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇⠀⠀⠀⠀⣿⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠛⠉⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣀⣠⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠻⣤⣤⣤⠴⢶⠖⢿⠋⠉⠁⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠘⣧⣠⡴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠰⡄⣿⠀⠀⠀⢸⡆⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠣⣽⠀⠀⠲⠲⣷⣶⣷⣖⣚⡋⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⣐⣊⣉⣀⣸⣇⣈⠉⠉⢀⣀⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⣈⣷⣿⠬⠽⢛⣫⠟⠻⢭⣛⠋⠭⣽⣩⣏⣉⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠶⠒⠒⠋⠁⠀⠀⠀⠀⠀⠉⠑⠒⠲⠤⠤⠼⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠚⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣴⣞⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢻⠀⠀⠰⣄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⢀⡛⣓⣒⣋⣀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠀⠉⢉⠁⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⠓⣄⠁⢢⠀⠇
⠀⠀⠠⠄⠀⠀⠑⢤⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠈⠑⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢠⣤⡤⢀⡀⠀⠆⠀⠀⠑⣼⣿⡆
⠀⠀⠀⠀⢴⣾⡦⢀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠜⠀⠘⠏⠀⠀⠀⠀⠀⢉⠛⣅
⠀⠤⠦⣄⠀⠀⠇⠌⠀⠃⠘⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣷⣦⠖⠋⠉⠀⢸
⠀⠀⠀⠈⠑⠄⠈⠀⠀⡤⠀⠹⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠈⠀⠀⠀⠂⠀⠉⠋⢄⢀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠁⡘⠆⣀⢠⠆⢰⠀⠀⠀⠀⠀⠀⠀⠠⠈⠂⠀⠀⠀⠀⠠⣾⡦⢠⠀⠀⢦⡄⢀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠝⠘⠠⠄⠠⠀⠀⠀⠀⠐⠀⠀⠁⢀⡠⠑⠉⠂⠄⠀⠀⠨⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⠘⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠶⠒⠖⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣿⠶⠲⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠚⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣹⠀⠀⠈⡉⠉⠉⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠛⢻⠀⡇⠀⣿⠀⠀⠈⠉⠉⠉⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⣿⠀⠀⠀⣿⠀⣾⠉⢹⠂⣿⠀⢸⠀⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⣿⠀⢸⠀⣿⠀⠸⠀⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⣿⠀⢸⡃⣿⠀⢸⠀⡇⠀⣾⠀⢰⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠟⠀⠀⣀⣀⣀⣙⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⣿⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⡴⠋⠁⠀⠀⠉⠳⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡀⠀⣠⠞⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠤⠤⠤⠤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⡆⠀⠙⠛⠁⠀⠀⣸⠧⠤⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡄⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣤⠞⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤





⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠲⠶⠶⠶⠶⠶⠶⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⢀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢀⣀⣰⡟⣾⢸⡇⠀⠚⠿⠿⠻⠟⠛⠿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⣷⢺⢽⡇⣿⢸⡇⠀⢀⣀⣀⣀⣀⣀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠙⠛⠛⠛⠛⠛⠛⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠛⠛⠛⠛⠛⠀⢀⠔⠁⠀⢸⣿⣶⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⣀⣀⣀⠀⠄⠀⠀⠀⢸⣿⣿⣿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠉⠉⠀⣶⣶⣶⡖⣼⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣤⣤⣤⣤⣤⣤⠀⠘⢿⡟⣼⣿⣿⡿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣈⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⢻⡀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡞⠛⢧⠈⠹⠉⠀⠀⠀⠘⠷⠶⠶⠶⠶⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⡟⠛⠁⠈⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣠⡞⠁⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠈⢳⡄⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⣄⣠⡶⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⣸⠃⠀⣾⣿⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣠⡟⢹⣆⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠘⠋⠉⠻⠃⠘⡿⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢢⡄⣀⡀⢀⣀⡀⢀⣀⠀⢀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⡰⣗⠛⢸⣀⡷⢮⣀⡇⡏⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⣠⣾⠟⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀
⠀⠀⣠⣾⠟⠁⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣰⣿
⢠⣾⣿⣥⣤⣤⣤⣴⠿⠃⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢹
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢸
⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⣾⠃⠀⢠⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡟⠀⠀⣼⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠃⠀⢀⡿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢀⣿⠀⠀⣾⠇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⢸⡏⠀⣼⡏
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⢿⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⢀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⡇⠀⠀⣼⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⠛⠛⠛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⠀⠀⠀⠀⢸⣧⣀⣴⡟⢸⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣃⣀⣀⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⠀⢀⣴⠟⠉⠙⣿⠀⢸⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠸⣧⠀⠀⢀⣿⡇⢸⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣤⡶⣶⠄⠀⠀⢹⣇⣴⠟⠋⠀⢸⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣀⣠⡾⠟⢡⣼⢋⣠⡤⠀⢸⡿⠋⠀⠀⠀⢸⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠛⠛⠋⠀⠀⠘⠁⠀⠀⠀⠀⢸⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠋⠉⠁⣠⣤⣤⣀⡀⠈⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢟⣽⠿⠿⠀⠀⠀⡀⠈⠉⢉⣉⣉⠀⠀⡿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠻⡋⠀⠋⢀⢀⣀⠀⠀⠀⡇⠀⡎⣿⣯⠗⠀⢰⠿⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠳⢇⠀⠀⠿⣿⠯⠃⠀⠀⠑⢄⠀⠀⠀⠀⠀⣸⣤⣾⣿⣿⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡄⠀⠀⠀⣿⣿⡿⣿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⡼⠛⢋⣵⣾⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡌⣿⣿⣿⣷⣄⠀⠀⠀⠈⠿⣉⣉⠝⠁⢀⡜⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠙⠦⢄⡀⠀⠀⠀⢀⡤⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠛⠋⠉⡆⠀⠀⠀⠀⢻⣿⣿⡟⢟⠛⢻⡿⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡞⠀⠀⠀⠀⠀⢻⡉⠢⡄⠀⠀⡽⢻⣦⢸⣄⣹⣇⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡧⠚⠀⠀⠀⠀⠀⠀⠀⠑⢢⡦⣀⡀⠘⣄⡽⠁⢡⢀⣀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢎⣙⣗⠤⠤⢠⣤⣤⡤⠤⢔⣊⢥⠟⠁⠈⠑⢌⠒⡴⠋⠀⠀⠀⠙⢦⣄
⠀⠀⠀⢠⠤⣸⡉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠹⡉⠙⠒⠂⠒⠀⠉⠉⠀⡜⠀⠀⠀⠀⠈⡟⡀⠀⠀⠀⠀⠀⠀⡄⢆
⠀⠀⠀⣏⠓⠀⠀⠀⢱⡀⢀⡀⣀⣀⣀⣀⣀⣀⣠⠤⠔⠋⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠳⣁⡃⣠⠒⠉⢳⢤⠇⡼⡀
⠀⠀⢸⠆⠀⠀⠀⠀⡌⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⡆⠀⠈⡖⠊⠀⠇
⠀⠀⠀⠉⠙⠓⠒⠚⠢⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠌⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠘⠢⠔⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢀⣀⣀⡀⠀⠀⣀⣀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣿⠤⠤⢤⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠶⠂⢤⢺⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⢸⠀⠀⠀⠀⠀⠀⡟⠢⢄⣀⣀⣀⡀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣓⣼⣿⣿⣧⣸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿
⠀⠀⢀⣿⡟⣰⡦⠀⠀⠀⠀⠀⠀⠀⢴⣆⢻⣿⡀⠀⠀⣿⣿⡿⠁⣰⠏⢙⣿⣿⣿⣿⣿⣿⣿⡋⠹⣆⠈⢿⣿
⠀⠀⢸⣿⢠⣿⠃⣿⠃⣢⣾⣶⡘⣿⠘⣿⡄⣿⡇⠀⠀⣿⣿⡇⠀⡿⠀⣾⠁⣼⠟⠉⠉⣧⠈⣷⠀⢿⠀⢸⣿
⠀⠀⢸⣿⠘⣿⡄⢿⠄⠻⣿⠟⠡⡿⢠⣿⠃⣿⡇⠀⠀⣿⣿⡇⠀⣿⠀⢿⡀⣻⣄⣀⣠⣟⢀⡿⠀⣿⠀⢸⣿
⠀⠀⠈⢿⣧⡙⠟⠀⠀⢠⣿⡄⠀⠀⠻⠋⣼⡿⠁⠀⠀⣿⣿⣿⡀⠘⣧⣨⣿⣿⣿⠀⢿⣿⣿⣥⣼⠃⢀⣿⣿
⠀⠀⠀⠈⠋⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠙⠁⠀⠀⠀⣿⣿⣿⣷⣴⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣦⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟
⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀
⠀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧
⠀⢿⣿⣿⣿⣿⣿⣧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿
⠀⠈⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠃⠀⠀⠈⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣀⣤⣤⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢸
⢠⣾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⢠⡄⣶⢠⡆⣠⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢹
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⢻⡌⣧⠹⠈⠃⠛⠘⠃⠛⠸⢃⡞⣠⠇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣶⡀⠀⠀⢸
⣿⡅⠀⠀⠀⠀⠀⠀⢴⡈⢷⠙⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠾⢡⡞⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸
⣿⡇⠀⠀⠀⠀⣤⡙⢦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⡴⠂⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸
⣿⡇⠀⠀⠀⠰⢦⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠶⣋⣡⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⠿⠟⠋⠀⠀⠀⢸
⣿⡇⠀⠀⣈⡛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⠀⠀⠈⢩⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠠⣬⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠐⠖⣛⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠰⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⡅⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⠀⠀⠀⢸
⣿⡇⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠟⠀⠀⠀⠀⢸
⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣀⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⣿⣟⣿⣿⡻⣿⣟⣿⣿⣻⣿⣿⣿⡇⣿⣿⣭⣭⣭⣭⣿⣯⣽⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⢿⠿⡿⠿⣿⢿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⡇⣿⣷⣶⣶⣾⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣙⣛⣛⣛⣛⣃⣿⣯⣭⣭⣽⣿⣿⣿⣧⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢟⣭⣿⣯⣝⢿⣿⣿⣿⣿⣿⣿⡏⣽⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣾⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⡇⣿⡟⡛⣫⣭⣵⡆⣿⣭⣽⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⡃⢿⣿⣿⣿⣿⡇⣛⢿⣿⣿⣿⣿⡇⣿⣧⢃⠻⠿⢿⡇⣿⠶⢶⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⢲⣒⣒⣚⣒⠂⣿⢸⣿⣿⣿⣿⡇⣿⣿⡘⣸⣿⣷⣶⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡜⣷⠹⣿⣿⣿⣿⢱⡟⣼⣿⣿⣿⣿⣧⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠘⠳⣮⣭⣭⡶⠟⠈⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠩
⠀⠀⠀⡗⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠄⠒⠈⣡⣤⡰⠦⠑⡄
⠀⠀⠀⣇⠀⠀⠉⠁⢀⠈⢀⠉⣀⣤⢤⣤⡀⢠⣼⣯⠻⠶⠶⢶⡄⠘⣄
⠀⠀⢀⣸⡄⠠⠭⠄⠹⣷⣜⣷⡽⣦⡴⠀⠙⢾⣿⠛⠧⠿⢒⡚⠡⠀⠈⠢⡀
⠀⠀⣺⡄⢳⡄⠐⢄⣢⠙⣯⡙⠿⠮⠿⠖⣚⡥⠄⠀⠐⣋⣩⢤⣒⣪⠭⠃⠈⠢⣀
⠀⠀⠻⣿⣄⠹⣄⠀⢉⣤⡬⠥⠖⠒⢋⣭⠥⠶⠚⠲⡈⠓⣫⣭⣶⠶⢊⣠⣤⣤⣤⠍⢒⡒⠒⠦⢄
⠀⠀⠀⠙⢿⣦⠘⢦⡀⠲⢶⣿⠿⠃⠈⠲⣄⠀⠀⠀⠀⠦⣄⠘⠿⣛⡭⠵⠖⠋⠁⠀⠀⠀⠀⠑⡄⣣
⠀⠀⠀⠀⠈⢻⣷⡀⠳⣄⠀⠀⣈⣿⣧⠦⠈⠳⡀⠀⠀⣀⡨⠽⠒⣀⠑⣄⠀⠀⠀⠀⢀⢤⣤⣾⡷⠃
⠀⠀⠀⠀⠀⠀⠻⣿⡄⠘⢧⡀⠀⠰⢶⡾⣷⣶⠬⢓⣉⣠⣤⠶⣛⡁⠀⢸⣆⣠⡶⠷⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⢙⠲⢤⡈⠩⠽⠯⣝⣫⠭⣶⣿⣿⣻⠿⠓⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠎⡝⢧⠀⢐⣭⣶⠿⠗⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⣀⣸⡧⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣿⡿⠛⠛⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿

⠀⠀⠈⠠⠀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠂⠄⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠀⡄⠀⠀⠀⠀⠀⢀⡀⠠⠤⣀
⠄⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡡⠡⢂⢊⢀⠔⠢⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠁⠱⠀⠀⠀⠀⠀⠀⠀⠀⠜⠔⣀⠀⠀⢡⡂⠐⢠⢃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠊⣜⠑⠀⠀⠀⠈⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠰⢹
⠀⠀⢀⡫⡡⡀⠀⠀⠀⠀⠀⠀⠐⠉⠐⢄⡁⠀⠊⢂⠄⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⠀⠀⢐⠀⠀⠘
⡠⠔⠁⠠⡑⠜⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⣀⠀⠸⠀⠀⠰
⠖⢢⠣⡱⡑⡌⢊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⡀⠆⠑⡐⠈⠀⢀⠱⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠁⠒⠋⠀⠀⠀⠕⠳⡘⢌⢆⠀⠀⠀⠀⠀⠤⡀⠀⠠⠄⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⣠⠀⠀⠀⠀⠀⠀⠀⠑
⠑⡀⠂⠠⠰⡀⠀⠈⠀⠘⠣⠃⠀⠀⠀⠀⠀⠀⡖⠂⠀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠈⢆⠑⠑⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠂⠁⠀⡔⠂⠐⢄⠡
⠀⠀⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠈⠁⠀⠀⠀⠀⠈⠀⠂⠁⠀⠂⠐⡀⠌⠀⡚⠀⡄
⠀⡪⠐⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡉⠀⠀⠀⠀⠨⠁⣐⠀⠐
⢈⠴⠋⠢⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⡠⠥⠂⠀⠀⠅⠐⡀⠤⠐⠁⠀⠀⢀⠄⡪
⠀⠁⠁⢀⠄⡀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠣⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⠂⠈⠄⠒⠈⠀⠀⠀⠀⠀⠀⡠⢀⠅⣋⢄
⠀⢀⠀⠀⠈⠂⡡⠊⠀⠀⠀⠀⠀⠀⠀⠃⢀⡀⠉⡐⠈⠔⡀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⢆⠁⠊⠀⠡
⠆⢐⠂⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢆⠤⢀⠀⠅⠀⠀⠒⠦⢅⠀⡬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢡⠊
⠊⡡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⢄⠰⠀⡱⠀⠀⠈⠠⠃⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠉⡉⠀⠀⠀⠈⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠐⠂⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠅⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠉⢰
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⢀⢁⡐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀
⣀⣰⣃⣰⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣊⣮⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣑⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀
⣿⣿⡋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿






⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉












⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠠⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡴⠶⠶⢦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠛⠈⠛⠛⠛⠛⠀⠛⠒⠒⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣤⠴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⡏⠀⠀⠀⣬⢻⡄⠠⣤⣤⣤⣤⣤⣤⣤⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣇⠻⣦⡾⠁⣸⠇⠠⡤⠤⠤⠤⠤⠤⣤⠄⢀⣿⠴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠶⢤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠐⠒⠶⢦⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠞⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣀⣀⣀⣀⢸⡇⠐⠶⠶⠶⠶⠶⠶⠶⠞⢧⣀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢧⡀⠀⠀⢀⡼⠁⠐⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠁⠒⠒⠛⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠁⠀⠀⠈⢳⡄⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠛⠛⠛⠛⢸⠇⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠦⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠧⣶⡄⣶⣠⣶⢄⣶⢰⠆⡖⠂⡗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠀⠉⠀⠉⠈⠉⠈⠀⠉⠁⠁⠁

⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠩⠩⠉⠉⠉⠉⠉⡉⠭⠭⢉⠉⠍⡃⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡠⠐⠁⠀⠐⠉⠂⠙⠢⢇⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠐⠉⠀⠀⠀⠀⣈⠒⠡⠀⠀⢤⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⢀⠠⠐⡂⠩⠄⠒⠂⠈⠉⠉⢉⡱⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠂⢀⠀⠀⠁⢤⣴⡨⠔⠚⢁⢐⠆⠀⠀⠀⢛⣽⡉⠀⠀⡄⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠠⠤⠤⠌⢀⣀⠀⢰⠌⠄⡄⢠⠀⡯⣿⡆⠀⠀⢹⢺⣇⠰⣄⠂⠀⠀⣠⣀⡟
⠀⠀⠀⠀⠐⠰⢀⠅⠄⠀⠀⠈⢸⠀⡔⢹⣌⡀⠐⠬⠃⠀⠀⢂⡑⠃⠀⡏⠀⠀⢠⠏⢰⡇⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⠁⠀⡀⠂⠀⠀⢠⠁⠸⣆⠘⠬⠀⡁⠀⠀⢐⠉⠉⢫⠇⢀⠜⢃⠀⠀⢰⣮⢿⡇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡊⠀⠀⠀⢀⠁⠁⡌⠂⠠⣏⠠⠆⢀⣌⠰⠦⠤⣄⡤⠴⠒⠩⢓⡆⠉⠂⠀⠚⠉⡇
⠀⠀⣀⠑⠒⠒⠤⣀⡀⢰⢄⠈⠒⠒⡚⠛⠉⠀⠀⠈⠄⠪⡨⠓⠀⣀⠀⠀⡉⠆⠀⡛⠓⠓⠒⠀⠀⠄⢀⡀
⠀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⢄⠀⠱⡡⡀⢀⢐⠁⠘⠀⠀⢸⡉⠘⡤⢠⡽⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⣓
⠀⠉⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠲⠓⠊⠹⠈⠂⠊⠀⠀⠈⡄⠁⢉⠁⠈⠉⠉⠉⠉⣿⠉⠉⠉⡏⠉⠩⠇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢃⠀⠀⠀⣀⣑⣊⠀⠤⢔⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⡀
⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⡟⠀⠀⠀⠤⠀⠐⠐⠒⠘⠉⠙⡉⠉⠉⠀⢻⠀⠰⠀⡇⠆⠆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠸⡇⠀⠀⠀⠀⢀⠀⠂⠀⠀⡄⠐⠜⠄⠀⠁⢸⠀⠘⠀⡇⠀⡃
⠉⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠩⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠏⠉⠉⠙⠍⠉⢹⠀⠘⠉⡇⠀⡛⢍
⡠⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠏⠀⠀⠁⠀⡙⠁⠈⡿⠀⠀⠀⠀⠀⠀⠀⠈⠄⢸⠀⠀⠀⡇⠀⡅
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡐⠱⠀⠀⢰⠀⢀⠁⠀⠀⠁⠅⠀⠀⠀⠀⠀⠀⠐⠈⠄⠑⠂⠀⡇⠀⡇
⠀⠈⠇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠖⠁⠋⠀⢀⠏⠀⣘⠁⠀⠀⢙⢙⠀⠘⣁⠀⠀⠀⠀⠀⠉⢇⠀⠈⠣⡄⠏
⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⢠⠚⠂⠈⠉⠀⡉⡸⠀⠀⠰⠘⢸⢈⠁⠈⠉⠀⠒⡢⠀⠀⠈⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠯⠈⢻⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⢸⣿⣿⢀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠸⣿⠑⠀⢀⡠⢤⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠶⠆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣑⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣯⡿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠟⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣭⣉⣿⣉⣏⣛⣉⣙⣍⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢿⣿⡿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡦⣇⣽⣼⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣫⢖⡯⠭⣗⢮⡻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣷⣷⡞⠛⢶⣷⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡅⣹⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡃⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡋⣻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣭⢫⣵⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣾⡇⢻⣷⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣏⣀⣀⣈⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⡢⡀⠐⠶⣤⣈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠤⠀⠀⡀
⠀⠀⡄⠀⠀⠁⠚⣦⠀⢑⡌⠳⢀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⢀⠀⠓
⠀⠀⠐⡄⠀⠀⠀⠻⠄⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠁⠀⠀⢸⠇⠀⠒⠀⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠀⠧⢀⠀⠀⠀⠁⠴⠒⠀⢄⣳⠀⠈⠀⠀⠀⡀⠀⠀⠸⡆⠀⠁⡀⢺⣧⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⡈
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠠⠃⠀⢱⠀⠑⣆⠀⠀⠈⣿⡗⣄⠀⢃⡰⠄⠂⠀⠄⡄
⠀⠀⠀⠀⠀⠶⠀⠄⠀⠀⠀⠀⠀⠀⠑⠈⢛⠂⠈⠀⠀⠀⢷⣤⠀⠀⠀⠉⠋⠉⠉⠉⠁⠀⡤⠆⡴⣾⠁
⠀⠀⠀⠀⠀⠀⠈⡀⣀⡀⠀⠀⠔⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠙⢭⠿⣁⠄⡀⢤⡖⢒⢪⢂⠔⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠘⠂⠈⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⢀⣀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢠⠄⣠⠴⠀⠀⣼⠃⢀⡤⢦⠀⡤⠀⡤⠈⡽⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣰⣇⠞⠓⣏⣴⣂⠜⣏⡠⡾⠀⣏⢼⣃⢼⣃⢾⣃⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⠀⠀⠀⠁⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡼⠃
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢶⣾⡿⠃⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢄⡀⣤⠏⢠⣷⣿⣿⣷⣶⣦⣠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢨⢻⣴⠩⣴⣾⣿⠟⠛⣉⣽⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⢬⡟⣝⡻⣧⣶⣴⠿⠏⠀⠂⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡗⠓⢰⣿⠛⢻⠟⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡠⣿⣿⣿⠂⠘⣿⣿⣾⠦⠂⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣸⣷⣿⢋⡝⣳⣾⣿⣿⣿⠁⠀⠀⠀⠉⠧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⡟⣯⣱⣿⣸⣿⣧⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⣻⣾⣽⣿⡏⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠏⠀⣿⣿⣿⡿⢃⡾⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣾⡿⠁⠀⠀⣿⣟⣣⣴⣿⣼⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⡀⠠⣦⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠀⠀⢀⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣤⠈⠉⣤⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣿⣥⠶⠖⠲⠻⣿⡟⢻⣿⢿⣯⣿⣿⣟⠛⠿⣿⣿⣿⠀⠘⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠀⠛⣁⣴⣤⣄⣀⣃⠛⠛⢷⣿⣿⣿⣿⣿⡀⠀⠈⠉⠁⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠘⠛⠛⠛⠻⣿⣿⣿⣿⣶⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢻⣿⣿⡿⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⡀⠀⠀⡠⠄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⠿⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⢻⣿⣿⡿⣿⣿⡤⠂⢀⡀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⣤⣤⣶⣾⣿⣿⣿⣿⣩⣤⢶⡶⢶
⠀⠀⠀⠀⢀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⣿⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡅⠀⠀⠀⣿⣿⣿⣿⡿⠿⠛⢉⣭⣤⣴⣯⢓⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⣿⡁⠀⣤⣈⠀⣴⣮⡴⡶⠒⠸⠆⡀⠀⠀⣿⣿⣿⣿⣶⣾⣻⣿⣟⠋⢁⣴⡊
⠀⠀⠀⠀⠀⠀⠐⠒⠖⠒⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⢠⣠⡭⠁⠀⠀⠉⠀⠀⠀⠀⠴⣷⠶⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣀⣠⠴
⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⢠⣿⣿⡇⢄⡶⢲⡶⠶⢇⠀⠀⢀⣀⣤⠁⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⡠⠬⠯⢽⣿⣿⣿⡄⢀⠀⠘⢳⣞⣾⣿⣿⣿⣤⣄⣀⣀⣀⠀⠀⠈⠉⠙⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢰⣶⡀⣀⣀⣀⣀⣀⣀⣸⣿⡟⠁⠀⡠⣿⣿⣿⣿⣿⣿⣧⣄⡀⠈⠛⠉⠻⣿⣿⣿⣧⡟⣿⣿⡛⡛⣹⠿⣿⣿⣿⣿⣿⣿
⣯⣤⣶⠶⠶⠛⠛⠿⠋⠛⠛⠋⠁⠛⠛⠋⠀⠁⠉⠉⠀⠈⠉⠁⢸⣿⣤⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣾⣿⣿⣿⣿⣼⣿⡟⣿⢷⢻⢞⢃⣩⡿⢿⣿⣿
⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢆⣿⣿⣶⠟⢋⠁⠈⠉⠉⠀⠈⠙⠉⠉⠛⠛⠛⠿⠻⠭⠿⠧⠉⠀⠀⠛⠀⠉⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠸⢿⣿⡛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠄⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⠖⠂⠀⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⣦⣘⣿⣏⣀⣠⠀⠀⡀⢀⣶⠀⢀⣾⡇⠀⣀⣸⣷⠤⠐⠦⠁⠈
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠰⠈⠉⠀⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣧⣿⣿⡟⢸⣿⣿⣍⣤⡶⠔⠂⠀⠁⠀⠀⢀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡀⠀⠀⣤⡄⣀⠀⠀⠀⠀⢀⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠀⠀⠀⠀⠀⠀⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣄⣼⣿⡿⠿⠯⠀⠀⠀⠈⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢿⣿⣿⢂⣀⠀⠀⠀⠀⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⢿⡇⢀⣤⣾⣿⣿⣿⣦⠀⠀⠀⠀⢀⣀⠔⠉⠀⠀⢀⡞⢁⣿⣿⣷⡄⠀⠀⠈⠁⠳⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⢀⣴⣿⠋⠀⠀⠀⠀⢸⠁⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⣿⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⢈⠛⠉⢷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡈⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢃⣆⣤⡦⣮⡄⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣴⣿⡟⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣾⣭⣀⣠⣿⠻⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠰⠋⠀⠀⣿⣿⣿⣿⣿⡿⠟⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣧⣄⣇⠀⠀⠀⠀⠈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢻⡛⡛⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⡇⢸⡄⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⣿⠈⣿⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⠂⢹⡇⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢃⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠻⣿⡆⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⡇⠀⠨⡄⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⠛⠛⠃⠀⠿⠇⢀⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⢍
⠀⡖⠊⠉⠉⠉⠙⠒⠒⠦⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠈⢠⠊⠀⠀⠀⠀⢡
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⡇⠀⠀⢰⡆⢸⠀⠀⢰⡆⠀⢸⠖⠦⠤⣀
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠄⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠱⡀⠀⠀⠁⡰⡀⠀⠈⠁⠀⡜⠀⠀⠀⠀⠙⢦
⡆⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠰⡄⠀⠀⠈⠂⠀⠈⠀⠈⠂⠀⠐⠊⠀⠀⠀⠹⠲⠀⢸
⡇⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢸
⡇⠀⠀⣼⣿⢹⣿⠀⠀⢰⣇⠀⠀⡰⠒⠲⠧⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠂⠠⡄⠠⠄⠀⠀⠀⠀⠒⡹⠁⠀⠀⠀⢸
⡇⠀⠀⣿⣿⣾⣿⡇⠛⢻⣿⠛⠃⠘⠦⣤⠄⠘⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢸
⢱⠀⠀⣿⡏⠉⣿⡧⠀⠈⠋⠀⠀⠀⠀⠸⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠟⠣⠄⣁⡈⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⢸
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠄⢰⠇⠀⠀⠀⠀⠀⢀⡼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⣄
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠹⣷⣄⠀⠀⠀⠀⢸⠶⠖⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠲⠶⠶⢏
⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠙⢷⣤⡀⠀⢨⠗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢏
⠀⠀⠀⠉⠒⠲⠤⠤⢀⣀⣀⡀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⠛⠷⣮⣷⡖⠒⠒⠒⠒⠒⠒⣶⠒⠒⠒⠒⠒⠒⠒⠒⢲⡖⠒⠒⠒⠒⣖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣇⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠾⠿⣝⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠙⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠿⢤⡀⠀⠀⠀⠀⠀⢀⠼⠧⣀⠀⠀⠀⣇⠀⠀⠀⢀⠇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⢄⡀⠀⡠⠚⠁⠀⠀⠉⠲⢄⡀⣠⠖⠁⠀⠀⠀⠙⠲⢤⡏⠓⠰⠤⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⡤⠶⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠑⠢⢍⣛⠲⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⠑⠦⣌⣉⠒⠤⡉⠛⠷⢶⣤⡀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⣑⠢⢭⣑⠢⢍⡙⠒⠤⣀⠀⠀⠈⠙⢷⣄⠐⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠓⠮⣍⣒⠦⢍⣓⠢⠄⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⣭⣷⣦⣌⣁⠐⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣯⣭⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣤⣄⣠⣤⣴⣟⣁⣤⣤⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢻⣿⢿⣿⠻⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⢛⣛⣛⣻⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⡜⠘⣯⠉⣿⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠖⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠁⠀⠘⢧⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⢶⡟⣀⣠⣤⣴⣦⣬⣿⣿⣿⣿⣿⣿⣏⣉⣉⣉⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⢀⣤⡶⠟⠛⠉⠁⠀⠀⣼⣿⠏⠉⠀⠀⠀⠀⠂⠠⠤⠤⠤⠤⢤⣭⣉⣉⣉⣉⣉⣉⣛⣛⣛⠛⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⢶⡾⠿
⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠒⠒⠒⠒⠤⠬⠭⠭⠭⠭⠭⠍⠉⠁⠀⠀⠀⢠⡟⠳⢤⣀
⣼⠃⠀⠀⠀⠉⠑⠢⠄⠀⠀⢿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣒⡒⠒⠒⠢⠤⠤⠤⠤⢬⣭⣉⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠛⠶⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠄⢀⣀⣀⣀⣀⣈⣉⣉⣉⠉⠉⠉⠉⠉⠉⣛⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣀⣀⣤⣽⡷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣦⣤⣭⣭⣭⣭⣽⣿⣿⣿⣿⣷⣶⣶⣶⣶⡦⠄⠤⠄⠀⠀⠀⠉⠁⣀⣀⠀⠀⠙⣟⠛⠋⠉⠉⡟
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢤⣀⠀⠀⠀⠀⠉⠉⠉⠉⣉⣉⣩⣭⡿⠿⠟⠛⠛⠛⠛⠛⠒⠶⠶⠾⠿⠛⠀⠀⠀⢸⠃
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣤⠞⠛⠉⠁⢀⡀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⡀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠴⠶⠶⠒⣛⣉⣉⣤⡤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣈⡳⣭⣑⠒⠀⠀⠀⠀⠒⠚⢉⣉⣉⣩⣭⣤⡶⠶⠖⠛⠛⠛⠉⠉⠁

⢀⡀
⠸⣷⠶⠟⠻⡿⠻⡗⢶⣆
⠀⠙⠳⢦⡛⠾⣦⡀⠀⢻⣧⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣛⣛⣉⣩⠭⡽⢧⣄⣀⡀
⠀⠀⠀⢶⡷⡶⢆⠝⢳⣄⣍⠙⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⠁⢌⠠⢁⠡⢈⠬⣀⢇⠩⡹⡿⡃⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠉⠻⣷⣯⣆⠹⣿⢧⡇⠀⠀⠀⠀⠀⢀⣤⠾⠿⣯⠍⠆⡔⠀⠂⠀⠂⠐⠀⡂⢃⣂⣹⠀⠀⠈⠉⠙⠳⣦⢀⠄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⢻⣼⢡⣶⡒⢤⡀⠰⠟⠁⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠻⣿⣀⣀⣀⣀⣰⣒⠶⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡇⣾⠛⠁⠀⠙⠳⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⣆⠀⠀⠀⠀⢸⢩⣟⣛⣛⣻⠉⠉⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠏⠀⠂⠀⠀⠀⢠⣞⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡟⠙⣷⡀⠹⠆⠀⠀⠀⠘⠳⠮⠭⡭⠥⢤⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠘⢦⡀⠀⠰⣅⣈⠻⣦⡌⠃⠀⠀⠀⠀⢀⣤⠀⠀⠀⠉⠒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡼⠛⢦⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡀⠰⡙⢷⣝⢮⡳⣄⠉⡳⠈⠁⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣴⣾⣥⣞⡇⣰⠃⠀⠀⠀⠀⠠⣾⠻⣦⡱⣌⣻⠳⠙⠈⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⣤⡇⠘⢷⡈⠻⠀⣁⣀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢷⡝⢿⣄⠈⠁⣠⡾⢋⡿⢀⣠⣴⠾⠋⠁⠀⠀⢠⠖⣒⣒⣒⣒⢲⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠇⠀⢀⣤⣤⣙⢦⡀⢹⡄⢉⣴⠾⠋⢀⢬⣛⠋⠉⠀⠀⠀⠀⠀⠀⢸⠀⣟⡭⠴⣿⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠻⢦⣄⢸⣆⠀⠹⡎⠙⢛⣴⠟⠁⠀⡴⠁⠀⠉⡑⠀⠀⠀⠀⠀⠀⠀⠸⣆⣙⣓⣚⣛⣸⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠙⠷⠶⠃⣠⡾⠁⠀⠀⠈⣀⠀⠠⠚⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣹⠃⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢭⣝⡁⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠳⠶⠶⠶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⣆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣿⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣁⣀⣀⣀⣀⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢴⣄⣾⠋⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢻⡏⠀⠀⠀⠀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣽⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⣠⡶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣷⡟⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⠿⠶⠶⠶⠶⠶⠶⢶⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⢸⣿⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣿⠀⢸⣿⣿⣏⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⣶⣿⣿⡇⢸⣿⠀⢲⣤⠀⢻⣷⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁⠀⠀⣿⠀⢸⣏⠙⢿⣧⠈⢻⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡆⠈⣿⡄⠈⣿⣿⣿
⠀⠀⠀⠀⠙⢻⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡇⢸⣿⠀⣠⡿⠃⢰⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣷⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡿⠁⣰⡿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⡾⠃⠀⠉⠛⢿⡇⢸⣿⠀⠉⠀⣠⡿⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠛⣋⣴⣿⣷⣦⣄⢀⣿⠀⣸⣷⣿⡿⣁⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣷⣷⣾⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠃⠔⢯⠀⠀⠀⡀⠀⢀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠅⠀⠀⡄⡜⠀⣸⣇⣀⣈⣧⢀⡎⠀⠀⡀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⠀⢹⣀⣤⣶⠾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠻⠿⢾⣧⣤⣄⣀⡜⢈⠃⡐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣆⣀⣶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣾⣼⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⡴⠟⠉⠀⠀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⣄⠀⠀⠀⠀⠀⠉⠛⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⢠⠾⠋⠁⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⣤⣀⠙⠳⠄⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣏⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠈⠙⣷⡄⠀⠀⠀⠀⣴⠟⠁⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⠶⣦
⠀⠀⠀⠀⠀⢰⡟⠀⠈⠙⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣀⡈⣿⡀⠀⠀⢸⡟⣠⣤⡀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⢿⣿⠇⠀⠀⢸⡧⠻⣿⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠛⠂⡿⠀⠀⠀⠈⢷⠙⠛⠁⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⡇
⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠸⣷⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣠⣤⣴⡶⢾
⠘⠻⠷⠶⢶⣦⣴⣦⣤⣤⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣿⠿⠿⠛⠛⠋⠉⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠿⠿⠿⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⢀⣠⣤⣶⣾⣿⡿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⢶⠾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠤⡀⣤⠶⢲⣖⣦⡞⠛⢦⠘⣏⣦⠾⢻⠛⠇⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣸⣟⠁⣿⣆⠀⣏⠈⢧⣤⡾⢠⣿⣤⠀⣸⡆⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠋⠉⠉⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡶⠶⠖⢈⣿⡄⠀⢹⣩⠦⢸⡏⢳⡆⠀⣀⠼⠏⣀⡀
⠀⢀⡤⠚⠀⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠳⢀⡄⢸⡏⠿⠄⠾⠽⠦⠸⠧⠞⠃⢮⡀⠀⠀⠀⣹
⠀⠈⢳⡤⠀⠀⠰⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢰⠋⠁⠀⠈⢹⡄
⠀⠀⠸⣤⣀⠤⠤⠤⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⢠⡎⠁⠀⠉⣶⠋
⠀⠀⠀⠀⠧⣄⣀⣠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣤⡤⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⡿⢻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⠿⣿⣿⠿⢿⠿⠿⠿⢿

⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠒⠚⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣭⣥⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⢰⠂⡆⡃⠢⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣯⣽⣿⣽
⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠹⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣗⣪⠄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⣤⠀⠀⢀⣠⣥⣀⡝⣎⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⢣⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢀⣿⣷⠐⠚⢿⡗⢉⣉⣭⠵⠒⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠣⠅⠀⠀⠈⠀⠀⠀⢢⣀⡀⢀
⠀⠀⠀⢸⣿⣿⡇⢦⢞⡃⢐⣿⠶⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢙⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂
⠀⠀⠀⢸⡿⠿⣷⢿⣿⣯⡜⠿⡿⢿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⡀⠀⠀⠀⠀⢀⡀⢰⠀⠀⠀⠐⣂
⠀⠀⠀⠈⢻⡀⠛⢛⣿⣈⠻⣾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣌⠐⠄⠀⠀⠀⠰⠸⠦⡄⠀⠴⢀
⠀⠀⠀⢀⣼⡷⣦⣀⣙⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⢤⣀⠀⠀⢀⣴⣾⣠⣿⢀⡌⣸
⠀⠀⢠⣾⣿⡝⠳⢽⣋⣏⢰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠐⠒⢿⣿⣿⣏⢁⠼⠋⠥⠂
⠀⠀⢸⣿⣿⣿⣦⠸⢿⣿⢸⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠶⠒⢀⠀⠉⢀⣀⣬⠔
⠀⠀⢸⣿⣿⣿⣿⣦⣸⣿⣄⣿⣿⣿⣿⣿⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣾⣍⡙⠛⢿⡿⠢⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠦⠦⠠⠙⠻⠿⠿⠛⠒⠂⠀⠀⠀⠹
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠟⠃

⠀⠰⠛⠂⠀⠀⠀⢰⠂⠀⠀⢰⠟⠂⠀⠀⠰⠱⠀⠀⠀⠀⠀⠞⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠂⠀⠀⠀⢀⡖⠂⠀⠀⠐⠲⡆⠀⢠⡖
⠀⠀⡆⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⡀⠀⢰⣷⣾⣷⠀⠀⠀⠀⣖⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠲⠀⠀⠀⠀⠆
⠀⠀⠬⠏⠀⠀⠀⠀⠀⠀⠀⠀⣬⡅⠀⢿⡿⠹⣿⠆⠀⣀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠐⠓⢾⡗⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⢰⢿⠁⠀⠀⠛⠁⠀⠀⠀⠀⠁⠀⠘⠁⡀⠀⠀⠀⠀⠀⠀⠘⠙⠀⠀⠀⢀⡈⣀⡀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠘⣷
⠀⢠⣼⠇⣼⡇⠐⠏⠸⠀⠀⠀⠠⠀⠀⠀⠆⠀⠆⠀⠀⠀⠛⠀⠀⠀⠀⣴⡾⠇⢰⠿⣦⠀⢸⣧⠀⠻⠖⠀⠀⢀⠴⣄⠀⠀⠀⠼⠇⢹⡇
⢠⣤⠉⠀⠛⠁⠀⠀⠀⢠⠖⠀⠀⠀⣰⠶⠀⠀⠀⡤⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⠀⠀⠘⣃⡛⠀⡞⠀⠀⠀⠉⠉
⠘⢀⠀⠀⠀⠀⠀⠀⣠⡄⣤⢤⡄⠀⢛⣀⠀⠀⢀⡅⠘⣥⡀⢠⣦⠀⣦⠀⠀⠰⠋⢀⣀⣰⡆⠀⠀⠈⠀⠀⠀⣠⣉⡁⠀⠀⢰
⠀⠈⠃⠀⠀⠀⠀⠀⣿⣷⠙⠾⠃⠀⠘⠛⠀⠀⠿⠧⠴⣿⠇⠈⢿⡏⠙⠟⠀⣶⣆⡏⠻⠿⠃⠀⠀⠀⠀⡀⠀⠉⠛⠃⠀⠀⣀⣾⡟⣿

⡚⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⢠⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⡔⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠐⢂⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⡂
⠀⢀⡀⠀⠀⠀⠀⠉⠘⠀⠀⠀⠀⠀⠀⠀⢘⡀⢀⠀⠀⠀⠁⠛⠀⠀⠀⠀⠀⠀⠀⣁⠀⡀⠀⠀⠈⠀⠃
⠀⠠⠰⠝⠂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠄⠰⠑⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠮⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡠⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠⡂
⠀⠀⠀⠀⠨⠄⡠⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠥⢂⠄⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠅⢀⠄⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠒⠸⠉⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠂⠊⠁⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠐⠀⠋
⠤⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢂⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⡠⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠁⠀⠀⠀⠀⠀⠀⠨⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠅⢀⡄
⠀⠀⠄⠀⠀⠀⠀⠐⠰⠑⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠮⠁⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠐⠀⠏⠂
⠀⢈⢁⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⡠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠥⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⢀⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠸⠁⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠂⠮⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⠐⠏
⠀⠀⠀⠀⠀⢠⠪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣔⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡚⠄
⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠊⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠒⠘⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠓⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐
⠀⢀⡀⠀⠀⠀⠐⠈⢡⡪⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⡁⣔⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢂⠉⡐⠅
⠀⡀⠂⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢁⠄
⠀⠈⠘⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠓⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠈⢠⠪⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠅⣔⠅⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠢⢨⡚⠄
⠀⠀⠀⠠⠐⢂⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡠⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢂⠤
⠀⡀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⢐⡠⢀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⣂⠄⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣦⣍⠻⡷⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢛⣛⣛⣛⣛⣛⣛⣛⣛⣿⣛⣛⣛⣛⣛⣻⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠥⠭⠭⠭⠭⠥⠽⠯⠤⠬⠭⠭⠭⠤⣽⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡐⢒⣒⣒⣒⣒⣒⣒⣶⣒⣒⣒⣒⣒⣻⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠘⣿⣿⣿⠿⢛⣥⡆⣿⣿⡇⢹⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⣛⣛⣃⣶⣭⣥⣶⣾⣿⣿⡇⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⡿⠿⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣭⣭⡥⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡘⣿⣦⡙⢫⣴⣿⢂⣬⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⡜⠋⠸⠆⠙⣡⣿⣿⣿⣷⣦⣝⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⡇⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠠⠄⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⡿⢘⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⣦⢠⡆⣴⠂⠰⡶⢠⠖⢦⢰⣆⣰⡆⣶⡂⢰⢲⡄⢰⡖⡄⣴⡒⢰⢶⣄⡴⢶⡀⡶⣦⠰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠄⠻⠼⠃⠬⠟⠰⠇⠸⠦⠞⠸⠹⠏⠇⠿⠥⠼⠹⠆⠸⠏⠧⠿⠭⠸⠉⠈⠧⠼⠃⠏⠷⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠈⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢲⣶⣶⣶⣶⣶⣦⣴⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡀⣶⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⡉⠛⠿⣿⡟⢉⣉⣉⣉⣉⣩⣹⣿⢏⣉⣉⣉⣉⣩⣭⣍⠋⠻⣯⣭⣿⠶⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣲⣶⣿⣏⣙⣛⣿⣯⡽⠌⡟⠯⣯⣽⣿⣿⣟⣋⣹⠍⣿⣿⣿⡟⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⢚⣿⠿⠿⣾⣟⢿⣿⣿⣿⣿⣿⣷⣿⡿⢟⣻⣿⡿⠿⠿⣟⣾⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣷⣿⣷⣸⣿⣷⣽⣻⣿⣿⡿⠿⢿⣖⣿⣿⣿⣡⣿⣷⣾⠿⠓⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣯⣭⣛⣻⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢽⣾⣿⣿⣿⣟⣛⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣛⣛⣿⣿⠷⠖⢲
⠀⠀⠀⠀⠀⠀⠀⢠⢷⣿⣟⣋⣿⣛⣹⣿⣋⣿⣛⣹⣿⣋⣽⣿⣩⣿⣋⣹⣟⣩⣿⣋⣽⣿⣩⣿⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠛⠟⣹⣿⢿⣹⡿⢏⢝⡟⡩⢿⠟⣭⡿⢫⣽⠻⢋⣽⢿⡩⡿⢿⡹⢿⣿⢻⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣯⠵⠇⠈⢟⣭⢈⣟⣥⠐⣋⣥⠘⣫⡄⢛⣡⡄⣚⣥⠚⣋⡼⢹⣫⣌⠀⢈⣶⠟⣿⡀
⠀⠀⠀⠀⠀⠀⢿⣛⠶⢞⡛⠶⠲⠶⣶⠶⠶⠶⠾⠶⠶⠶⠶⠲⠶⠶⠖⠒⠒⠲⣶⡖⠒⠛⢃⣈⣓⣮⡝⣡
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡷⣶⣿⣿⡿⠿⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣯⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁


⣿⣿⣿⣿⣛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣻⣿⣻⣛⣛⣻⣻

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠂⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⣀⠀⠀⠂
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠉⠀⠀⠈⣠⣾⡇
⣿⣿⣿⣿⡏⠛⠁⠈⠻⣿⣿⣿⣿⡟⠛⠻⣿⣿⠉⠛⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠒⠀⠀⣠⣴⣶⠀⢀⣡⣴⡆⠈⠀⣀⠀⣡⣾⣿⣿⣧⠀⠀⠁
⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠻⣿⣿⡇⢀⡀⠈⢿⠀⣠⡀⠘⣿⣿⣿⣿⣿⠀⠀⠀⢀⠀⣼⣿⣿⣿⣡⣾⣿⣿⡇⠀⠀⣡⣾⣿⣿⣿⣿⣏⠀⠀⡀
⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣷⣾⣿⣦⡈⣿⣿⣿⣦⣹⣿⣿⣿⣿⠀⠀⠀⠀⢰⡟⠻⣿⣿⣿⠟⢿⣿⣇⣠⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠉⠉⠉⠉⢿⣿⣿⣿⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠠⠀⠀⠀⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣠⡞⣡⡈⢶⡄⠀⠀⠀⣿⣿⣿⠀⠀⣰⣿⣿⣿⡿⡛⣩⣍⠛⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⡟⣿⡟⢿⣿⣿⣿⣷⣄⠀⠀⢹⣧⣙⣃⠞⠁⠀⢰⡄⣿⣿⣿⠀⠀⢸⣿⣿⣿⣷⣧⣙⣋⣤⣼⣿⣿⡿⠋⠀⠀⠀⢠⣴⢀⣤⠀⠀⠂
⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠙⠿⣿⣷⣄⠀⠉⠙⠀⠀⠀⠀⣸⣇⣿⣿⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣶⣶⣾⣿⣶⣾⣶⣿⡷⠀⠀⡀
⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣀⣀⡈⢻⣿⣷⡀⢀⣀⣀⣠⣴⡿⢸⣿⣿⣿⣀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠏⠺⠿⠿⠿⠿⠿⠿⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⢿⣿⣿⣷⣌⠉⠉⠁⢠⣿⣿⣿⣻⠭⡤⠀⠈⢻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⡆⢀⣴⣿⣿⣿⠿⣯⣓⣐⡒⡀⠀⠙⣿⡿⠀⡀⠀⠐⠀⠀⠀⠀⣀⠀⠁⡠⢇⣈⠷⠀⠀⠁
⣿⣿⣿⡇⠹⢿⣿⡏⢹⣿⣿⣿⣇⡀⢀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣛⣷⣛⡬⠩⣤⠄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢰⣷⣄⣀⣴⣷
⣿⣿⣿⣷⣄⢼⢾⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⡿⣭⣽⣛⢛⣀⠒⠀⠀⢠⣧⠀⡠⠀⠀⠠⠀⠀⠀⠄⠀⣼⣿⣿⣿⠿⠇
⣿⣿⣿⣿⣿⢻⢾⡇⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠘⣿⣿⣿⣿⣛⣻⠭⡬⠍⠀⠀⠀⣾⣿⣷⣄⣀⠀⠀⠀⠠⣀⣠⣾⣿⣿⡟⠄
⣿⣿⣿⣿⣿⢻⣻⣧⡀⠀⠉⠈⠉⠉⠉⠉⠁⠀⠀⢀⣴⢿⣿⣿⣯⡿⢯⣓⣐⡒⠀⠀⢠⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⡿⠋⠀⠀⠀⠒
⣿⣿⣿⣿⣿⣻⣹⣿⡿⢷⣤⣄⣀⠀⠀⢀⣠⣤⣶⡿⠋⠈⣿⣿⣿⣷⣟⡬⠩⡤⠀⠀⣿⣿⡇⠙⢻⣿⢿⣿⣿⡿⣿⡟⠋⠀⠀⠀⠐⠀⠀⠀⠂
⣿⣿⣿⣿⣿⣽⣽⣿⠃⠀⢻⣿⣿⣷⣼⣿⣿⣿⣿⡗⠀⠀⢹⣿⣿⣭⡽⣘⢛⡀⠀⣸⣿⣿⡇⠀⢸⣿⠀⠙⠋⠀⣿⡃⠀⠃⠀⠀⠀⠐
⣿⣿⣿⣿⣿⡟⣿⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⣟⣻⠥⡤⠁⢀⣿⣿⣿⡇⠀⠈⢿⣆⡀⢀⣸⡿⠁⠠⠤⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠸⣿⡿⢯⢛⣀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠛⠛⠛⢋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⢿⣿⣟⡬⠉⢠⣿⣿⣿⣿⣇⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀
⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⣙⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡿⠀⠀⠀⢻⣿⠥⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠰⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣀⡘⣛⣛⣛⣛⣛⣋⣴⣿⠀⠀⠈⣿⢛⣿⣿⣿⡏⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠛⠆⠀⠀⠀⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠘⣾⣿⣿⣿⣷⡖⠀⠀⠀⠂⠀⠀⢀⣴⡿⠋⠃⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠖⣏⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⡿⠋
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠛⠿⠿⠆⠀⠙⢿⣿⣿⣿⣿⣿⣿⠏⠀⣿⣧⠁⠀⠀⠈⠁⠀⢀⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣬⣤⣀⠀⠉⠀⡀
⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡏⠀⠀⣿⣿⣇⠀⠀⢀⠀⠘⠛⡛⠛⠛⢛⠛⠛⠛⠋⠉⢹⣿⣿⣿⣧⠀⠀⠁
⣿⣿⣿⣿⣟⣇⣟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⡀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠽⠯⠿⠥⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⣿⣿⣿⡇⠀⢠⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢤⠬⣞⣻⣿⡧⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢂⣚⣽⣿⣿⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠄
⣿⣿⣿⣿⡇⠙⠈⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢥⣿⣿⡏⠐⠂⠀⠀⠒⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⣴⣿⣾⣷⣿⠀⠀⠂
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡐⠺⢿⡆
⠀⠀⢀⡤⠖⠛⠉⠙⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠀⢀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠠⡈⠊⠀⠀⢀
⠀⢀⠎⠀⣠⠖⢻⠋⡼⡿⠋⠁⠉⠉⠈⠉⠉⠉⠉⠉⠙⠐⠒⠒⠒⠒⠒⠒⠖⠴⠠⠤⠀⠀⠀⠀⠀⠒⠲⠒⠒⠒⠐⠒⠐⠂⠤⠤⠤⣀⠉⠒⢕⠐⠀⠀⠀⠀⢸
⠀⠈⠀⡘⡎⠀⣿⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⡄⠀⠇⠀⠀⠀⠀⢸
⠀⠂⠀⡧⠅⠀⣵⢸⢾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠹⠀⠦⠀⠀⠀⠀⢸
⠀⠀⠀⣛⡀⠀⢸⠀⣽⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⣽⠀⠆⠀⠀⠀⠀⠀⢸
⠀⡀⠀⡅⡇⠀⢸⡇⡭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⢠⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠄⡇⠀⢸⡇⢻⣙⣤⡀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢻⢸⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠖⡇⠀⢸⠃⠀⠉⠙⠛⠛⠛⠗⠟⠛⠁⠁⠦⠦⠿⠿⠿⠻⠗⠿⠶⠦⠆⡖⡖⣶⣲⣖⡒⡂⡀⡤⡤⠤⠤⣤⢤⣤⢤⣴⣾⠷⠃⢸⢸⠀⢸⠀⠀⠀⠀⢸
⠀⡇⢀⠚⠁⠀⡟⠀⡁⠀⠐⠉⠁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠃⠈⢸⠀⢸⠀⠀⠀⠀⢸
⠀⡇⢸⠸⠁⣸⠃⠀⠁⢀⠤⠤⢄⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸
⠀⠁⢸⢸⠀⡟⠀⠁⢰⠀⠀⠀⢈⡟⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠠⠀⣄⡀⠀⠀⠀⡄⢸⠀⢸⠀⠀⠀⠀⢸
⠀⠀⢸⠘⣼⠃⠀⠀⣤⠓⠀⠠⠊⢀⡟⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡌⠀⠀⠀⢨⢹⡄⠀⠘⣇⢸⠀⠴⠀⠀⠀⠀⡄
⠀⢠⠀⠂⢻⠐⡀⠀⠈⠛⠲⠶⠖⠋⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡆⠀⠀⠀⠀⢧⡂⠠⠒⠁⡰⠁⠀⠀⠸⣾⠀⡆⠀⠀⠀⢠⠏
⠀⠘⠀⢧⢈⢆⡀⠐⣄⣀⣀⣀⣀⢀⣀⣠⠋⠉⠉⢹⠊⠉⠉⡿⠉⠉⠉⢱⡏⠀⠐⡶⠒⠒⠒⢢⡀⠀⠀⠀⠀⠀⠉⠓⠒⠉⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⣰⠎
⠀⠀⠡⡀⠓⢴⣉⡙⢀⠖⠹⢃⠒⡖⠠⠼⠯⠽⢷⠿⡶⢶⡾⣿⣤⣤⣴⣾⢧⣀⡸⠷⠶⠶⢆⣺⠇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡠⠖⠀⡧⠀⡇⢠⣾⠉
⠀⠀⠀⠈⠲⢤⣤⣄⢉⣈⡁⠛⠚⠒⠚⠒⠓⠂⠐⠐⠣⠴⠤⠆⠤⠤⠦⠎⠠⠏⣠⣉⣌⣁⠋⡝⠉⠋⡑⠰⢲⠒⠞⢺⢛⠛⠱⢠⠤⡆⡴⠃⣰⣷⢋⣿⠆
⠀⠀⠀⠀⠀⠀⣟⡀⠀⣼⠁⠈⠉⠉⠉⠉⠁⠐⠒⠒⠂⠠⠤⠤⠤⠄⢀⣀⣀⣀⡀⣀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠒⠒⠒⠚⠒⠐⠚⠂⣉⠠⠞⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⠛⢹⢉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⣸⡸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠮⠵⢀
⠀⠀⠀⠀⠀⠁⠀⢀⡌⠀⠀⠁
⠀⢀⠄⡀⠙⠂⠀⢶⠀⠂⠀⠀⠀⠈⠁⠐⠀⠀⠈⠃
⡀⠂⠂⠄⡓⠀⠘⠀⠊⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠃⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠈⠃⠄⢀⠤⠀⠀⠀⠀⠀⠀⠀⣤⣠⡤⢚⠁⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⠐⠀⠀⠀⠀⠑⠄⣈⠛⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠐⢃⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠀⠲⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠀⡀⠀⠀⠀⠐⠀⠂⠀⠀⡀⠠⠐⠁⠁⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶