Marching band Emojis & Symbols

⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡦⠀⡄⠀⠀⣦⠀⠀⡶⠀⠀⠀⠀⡗⠀⠀⠀⠀⢲⠀⠀⠀⠀⠘⣆⠀⠀⢀
⠀⢸⣿⠂⣀⣀⣀⠀⠀⢰⣶⡇⠀⠀⠀⣿⡇⣶⣷⣄⣶⡿⢀⣿⠇⠀⠀⠈⣿⡇⣶⡦⠀⢰⣿⡇⠀⠀⠀⢘⣿⣀⣴⡛⢠
⠀⣿⣿⡎⠀⠀⠙⣧⠀⣾⣿⠀⠀⠀⢸⣿⣴⣿⣿⣿⣿⣿⢾⣿⣦⣦⠀⢸⣿⣾⣿⡖⠀⣼⣿⣷⣤⠆⠀⢸⣿⡿⠻
⠀⣿⣿⡷⣄⣀⡴⠏⠐⣿⣿⡦⠀⠀⢸⣿⣟⣿⡿⢹⣿⡗⢸⣿⡏⠁⠐⢸⣿⣟⠟⠀⠀⢸⣿⣿⠉⢂⠀⢸⣿⣇
⠀⣽⣿⡆⠀⠀⠀⠀⠀⢻⣿⡗⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣸⣿⣿⡀⠀⢸⣿⣧⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⢻⣿
⢰⡿⠹⣷⠀⠀⠀⠀⠀⢀⣿⡇⠙⢛⣿⠋⣿⡟⠙⣿⠇⢹⣿⠇⣿⠁⢠⣿⠋⣿⡆⠀⠀⣿⠃⢻⡇⠀⠀⠀⣼⣿⠃
⣿⡇⠀⢻⣆⡀⠀⠀⠰⡟⣿⡇⠀⣾⣏⠐⠿⠗⠂⣿⠀⢾⣿⠄⠛⠂⣿⣇⠀⠘⠟⠃⠀⣿⡀⠈⣷⡀⠀⠀⢿⡇
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣴⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁
⠀⠀⠀⠀⠀⢀⣾⣶⠂⠀⠘⠀⠀⠀⠁⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣧⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠟⠀⠀⠀⢀⠐⠛⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⣿⡯⠹⠶⠦
⠀⠀⠀⠀⠀⠈⣿⠛⠀⠀⠀⠀⠋⢠⢀⡟⢐⠢⡀⠀⠀⠀⠀⠀⢤⣶⣤⢆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠨⡟⠀⠀⢠⡀⠀⡆⡾⠁⣃⡤⠁⠀⠀⠀⡰⢠⡞⠈⠀⢸⠀⠈⠀⠀⠀⠂⢠⠄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣓⠸⠟⣗⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⡷⠀⠁⠈⠉⠃
⠀⢐⠀⠠⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠀⠀⠠⣲⠀⢠⡀⠀⠀⠔
⠀⠘⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⢸⣯⣤⣽⣹⡆⠁
⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢁⣨⣅⠀⠀⠉⠻⠲⠿⣀⠶⠃⠀⠀⠀⠀⠀⣶⡾⠿⠞⠳⢄
⠀⢄⡀⠈⠂⠀⠈⠁⠀⠀⠀⠀⠀⠙⠊⠆⠈⠻⣿⣧⣄⡠⠀⠀⠀⠀⢂⢀⡄⠀⠀⠀⠀⠾⠇⠀⠀⠜
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠘⠒⠀⠀⠀⣨⠿⠟⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡦
⠀⠀⠠⠀⠀⠡⡀⠰⠀⠠⠄⢀⣢⠀⠀⡈⣠⣴⣶⠂⣀⣱⣤⣤⡄⢠⣴⣭⣧⢠⣤⣤⣄⣿⣁⣀⣀⡀⣨⣿⡀
⠀⠉⠀⠀⠀⠀⠀⠶⠀⠠⠦⠘⠛⠂⠐⠓⠉⠀⠁⠀⠀⠉⠀⠉⠁⠈⠉⠉⠉⠀⠉⠉⠁⠀⠉⠉⠉⠀⠀⠈

⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠯⣭⣽⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⡤⠞⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡀⠀⠀⠀⢳⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠱⣁⡂⠒⢈⣉⢀⣐⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⢠⠀⠉⢹⠀⠀⠒⠐⠒⠀⢸⠞⠁⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢡⣈⠠⠀⢀⠀⠀⠀⠣⠩⠀⠂⠙⡄⠨⠤⠄⠠⠀⡌⠑⡖⢦⡄⡀⠀⠀⠀⠀⠃⠀⣲⡡
⠀⠀⠀⠀⢀⠴⠍⠁⠀⢤⢀⡀⢀⠀⢀⠌⠀⠀⠀⢈⡀⠥⠭⠬⢀⡅⠎⢀⠞⠉⠐⢄⢀⠀⢀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⡈⠁⢺⡖⠔⢉⠀⠀⠀⠁⡸⠔⠀⠀⠀⠅⢠⠊⠉⠐⠀⠠⠈⠈⠒⠃⠀⠀⠀⠀⠣⠠
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⡇⠀⠀⢹⡇⡀⠘⠀⠀⠤⠀⠆⠀⠀⣀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠙⠀⢄⠤⠀⠃⠀⠑⠀⡇⠀⠀⢀⡁⠀⡈⠀⠀⠀⠀⠀⡤⢄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠐⠀⢳⠀⠀⡈⣿⠫⠇⠂⠀⠀⠀⠐⠀⠈⠀⠂⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣄⡀⠇⣿⡄⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⡆⠠⢘⠀⠀⠰⠀⠀⠀⠀⡔⠀⠀⠈⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⢀⠀⠀⠀⠀⠇⠀⠀⠀⣤⡄⢻⣆⠀⠀⠀⢸⠀⠀⠀⠈⢀⠀⠀⡔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⣿⠄⠈⠉⠀⠀⠀⠨⠠⠀⠀⢀⠀⠠⠀⠀⣀
⠀⠀⠀⠀⠀⡀⠀⡀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠲⠷⠲⠄⠼⠃⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢦⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠄⠀⠀⠀⠀⠀⣠⠞⠁⠀⣈⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⢀⠀⢀⡜⠁⢀⠔⠋⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⢀⠞⢀⡴⠁⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠋⣠⠊⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⠞⠁⠀⠀⠀⠀⠀⠀⢀⣠⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⡔⠁⠀⠀⠀⠀⠀⡀⠔⠊⠉⢀⡤⠧⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⡴⠋⠀⠀⠀⢀⡠⣐⡉⠀⣀⠴⠊⠁⠀⠀⠀⠀⠈⣑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡤⠞⠁⠀⠀⠀⡠⠒⠁⠀⣅⡽⡮⠥⠤⠤⠤⠤⠄⣲⡊⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠴⠋⠀⠀⣠⠶⢍⢣⡙⣄⠀⠀⠀⢸⡧⠙⠈⠋⢁⠜⠀⠀⠀⡠⢤⢀⠠⠤⠒⠉⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣆⠀⣠⠔⠋⠀⠀⠸⡀⠑⡌⢢⡀⠀⠈⠉⠀⠀⠀⠉⡾⢏⠢⣼⠁⡎⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⢀⡠⠤⣠⡇⠀⠈⢦⡙⢦⠀⠀⠀⠀⡀⠄⢳⡌⠃⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡰⠁⠴⠲⢂⠀⠀⠀⠀⠑⣄⡱⠆⠂⠁⠤⡶⠃⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⢸⡅⠀⠙⠒⣶⠾⣉⡗⣦⡤⠐⢀⠡⠔⢦⡉⠢⢤⣟⢙⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠆⠘⠒⠃⠮⣑⠢⣾⣍⣰⠚⡫⢂⣨⡟⠛⡿⠿⣍⣉⡑⠤⠋⠈⢟⠚⠛⠉⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⢤⣄⠤⠐⢋⠠⢴⡂⠋⢰⣧⡀⢹⠖⠆⠠⣍⠢⠹⡛⠊⡀⠘⠳⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡤⠖⠈⠀⠀⠀⠈⡱⢶⠃⠈⠺⡚⠀⡄⠈⠣⡘⠋⡑⢌⢇⠀⠙⠒⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠀⠈⠂⢁⠀⠀⠘⢦⠑⠀⠑⢵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠂⠤⣄⣸⠀⠀⠀⠀⠳⡘⠀⠈⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡖⠒⠒⠒⠒⠓⠒⠒⠁⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⣤⠒⢎⠁⠀⠈⠒⢄⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⢦⡀⠀⠀⠸⡀⠀⠱⡀⠀⠀⠀⠉⠲⡴⠊⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⠁⢸⠀⠈⠂⠀⠀⢣⠀⠀⠈⢦⠀⠀⣠⢊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠀⡏⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠉⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠀⡀⠀⠀⠠⠀⠀⠀⠀⢠⠀⠀⠁⠐⠲⠀⠀⠀⠈
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠢⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢰⠀⠆⡆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⡴⠀⡸⠀⠀⠀⠀⡄⠢⠀⠃⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⡆⠀⠀⠀⠰⠀⠀⠀⠂⠰⠀⠀⢰⠀⠀⠃⠀⠁⠒⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠸⠀⠀⠀⢀⠇⠀⠀⡀⠀⠃⠀⠀⠈
⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠈⠀⠀⢠⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣦⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⣀⣀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠰⢺⣷⠀⠀⠀⠀⠀⠀⠀⡜⠁⣰⡟⠡⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⡠⣿⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠻⠆⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡞⠃⠁⡃⢠⠖⠋⠉⠙⠳⡀⠑⢦⡄⠀⣖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⢧⡴⠚⡧⡇⠀⣄⡰⠞⠷⢿⣢⡼⠟⠻⢿⡿⢟⡛⠛⢛⣛⡓⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣘⣿⡟
⠀⠀⠀⠀⠀⠀⠸⡆⠀⣠⣺⠠⢿⡍⠤⢶⡾⠁⠀⠀⠀⠀⠈⠢⡁⠀⡨⠛⠛⠉⠉⢫⠢⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠁⠹⢶⣶⣟⣲⡟⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣄⠀⠀⢀⡀⠀⠑⡐⠀⠀⢠⣤⣶⣠⣿⣩⣯⣽⣭⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢦⣭⡑⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡈⠁⠹⠿⠓⠚⠻⠛⠀⠀⠀⢙⣿⡏⠁⠈⡉⣹⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡗⡎⠀⠀⠀⠀⢈⡤⠀⠀⠀⠀⠀⣄⡴⠊⠀⠀⠀⠀⠀⡄⡇⠀⡀⢿⢿⣗⢠⡾⠋⠹⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⣰⣾⣿⣷⠀⠀⠀⠀⠛⠲⠤⣀⣀⣀⠀⠀⡀⣡⣾⣿⠈⣿⣿⣿⣷⣾⣥⡿⢾⣧⣴⣀⣠⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣮⡇⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⠀⡠⠽⠿⠀⢈⣿⠛⣛⣿⣤⢨⣿⣿⡿⠟⢡⣦⣴⡷⢦⣽⠿⣿⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣇⠀⠀⠀⠉⠻⣿⠇⠀⠀⠀⠀⣴⢲⠁⠀⠀⠀⣠⡿⠃⠀⠀⢿⣿⣟⢛⣥⣴⣴⣿⡿⠋⠀⢸⠿⠷⠄⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠋⡟⢸⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡏⠉⠐⠠⣤⣤⡟⠣⠀⠀⢀⣸⣿⣿⣿⣿⠇⠀⠉⠀⠀⠀⠈⠢⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠧⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⡄⠀⠚⠋⢿⡅⠀⢰⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⢳⣆⣸⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⢿⣯⣤⣴⣶⣿⣿⡇⠀⠈⢻⣿⣿⣿⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠐⢫⡤⠖⠋⠉⠉⠁⠈⠳⣦⡀⠀⠀⢀⣴⣿⣿⠿⠿⠛⠋⣉⣽⣿⡇⠀⠀⣿⣿⣿⣿⣿⣄⣿⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢹⡗⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠁⠈⠑⢄⣼⣼⡿⢃⣼⣿⣄⡾⠃⠀⠈⠉⢻⣯⣳⣦⣭⠽⠇
⠀⠀⠀⠀⠀⠀⠈⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠙⢿⡻⡝⠷⣤⣀⣀⣠⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⡿⠆⠉⠉




⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣄⠒⢀⠁⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡄⠘⣀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠂⠈⠀⠌⣑⢲⣶⣶⣉⡝⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⠊⣋⣚⣙⡁⣸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⣸⡠⠀⢂⡀⠈⠀⠀⠒⠒⠒⠒⠒⠠⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠛⠉⡰⢂⢴⢦⠒⠂⢀⢶⣦⠑⣲⠁⠀⠀⠀⠀⠘⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡵⣼⠀⠠⣙⠜⠀⠀⡠⠙⠜⠀⣏⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣀⢦⡁⠀⠀⢖⠲⠶⢶⠆⠀⢠⠋⡁⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⠽⠦⠻⢶⠤⢌⣁⣀⠡⠤⠶⠷⣚⡿⠋⠈⠇⠀⢺⣷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠉⠹⠉⠉⠀⠐⠚⢳⠚⡀⠘⡏⠁⠀⠀⠀⠣⠘⠊⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠪⡠⢀⡨⠙⠂⡤⠤⠜⣻⠙⡍⣁⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢰⣤⠊⠀⠀⠑⠋⠉⢻⣉⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠈⠉⢹⣅⡀⠤⠐⠋⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠀⠈⠉⠀⡉⠽⠂⠂⠀⠀⠀⠀⠪⢐⢄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡅⠀⠀⠀⡌⠀⠀⠀⠁⠂⢄⡀⠀⠀⠀⠑⠱⡖⠈⠀⢠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠐⠠⣀⡀⠜⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠁⠁⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⢀⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⢀⣼⣷⡀⠀⠀⢠⣦
⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⢘⣿⣶⡄⠀⠀⠀⢸⣿⣿⡁⠀⠀⠀⠀⠀⠈⢻⣿⡇⢀⣶⣎⡍⣤⡄
⠀⠀⠀⣴⣿⣯⣠⣶⣿⣿⣿⣶⣄⠀⠀⢀⣿⣿⣅⣼⣿⡗⡆⠀⠀⢀⣾⣿⣿⣿⣄⣀⣤⠀⠀⢰⣿⣿⣿⣿⣿⡏⠀⠈
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⠉⠉
⠀⠀⠀⢻⣿⣿⡿⣿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣛⡟⠁⠀⠀⠀⠀⢸⣿⣿⣿⠁⠈⢣⡀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⢹⣿⣿⡇⠈⠉⠛⠉⠁⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠁⠀⠀⢻⣿⣿⡇
⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⣻⣿⣧
⠀⠀⣰⣿⠏⢻⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠉⣿⣿⠀⠀⠀⠀⠀⢰⣿⡏⠈⢿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⢀⣿⡟⠀⠈⢿⣧⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠘⣿⣧⣤⠀⠀⠀⢸⣿⠁⠀⠸⣿⡆⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠘⠛⠓⠀⠀⠈⣿⡷⠶⠀⠀⠀⢸⣿⣿⡄⠀⠀⠘⠋⠀⠀⠀⠀⠘⠻⠆⠀⠀⢻⣿⣤⠀⠀⠀⠘⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⣀⡀⠀⠆⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠆⠀⠀⠘⢆⣀⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠱⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠠⢢⠗⠢⠴⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⢃⣄⠁
⠀⠀⠀⠀⠀⠀⠀⡀⢄⠂⠀⠒⠀⢤⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⢠⢀⣥⠒⠲⡄⠀⠀⠀⠀⠀⢠⠀⢴⡀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⢀⡐⠂⢠⠀⠀⠀⢠⡀⣸⡄⠀⠀⢠⡿⠁⠀⠀⠤⠌⠀⢠⠈⣹⢗⣀⠀⠀⠀⠀⠀⠀⠛⠂⢀⣾⡷⠶⠀⠀⠉⠁
⠀⠀⠀⠀⠈⢷⣀⡙⣆⣂⣠⣬⣿⣿⣶⠀⢀⢺⠇⠺⣖⡀⠀⠀⠀⢰⡟⠁⠉⢉⠉⠳⣄⠀⠀⠀⠀⣠⣾⢅⡀
⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣿⣿⣧⣿⡟⠃⢸⣾⣠⡧⣠⡭⠋⠀⠀⢿⠀⠀⠀⠈⢙⣷⣿⡄⠀⣠⣼⠟⡡⡺
⠀⠀⠀⠀⠀⠀⠻⢿⡷⣿⠿⢛⠏⠁⠁⠀⠐⢷⣴⣾⡿⠁⠀⠀⠀⠸⠆⠀⢀⡔⣯⣿⡿⠁⢠⡹⢻⡿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⢳⣤⡿⠿⠟⠛⠁⢀⠜⢲⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡇⠀⠀⠀⠀⢀⡜⣧⠀⠀⠀⠀⠀⠀⢀⡿⣇⠀⠀⠀⠀⠉⢀⡟⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢹⡄⠀⠀⠀⡞⠀⠸⠄⢀⠀⠀⠀⠀⡋⠀⠿⠄⠀⠀⠀⠀⢼⠃⠀⠀⠛⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠂⠉⠀⠀⠈⠂⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠘⠂⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⣾⣷⡄⠀⡾⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠞⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠿⣛⣀⣀⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢋⣵⣾⠿⡿⠿⢽⣿⣯⣭⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣀⣿⡏⠀⠀⠀⠤⣄⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⣿⣷⠀⠀⠀⠀⠀⠀⠈⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠈⢻⣦⡀⠀⠀⠀⠀⢰⣏
⢀⣠⣤⠴⣶⣶⣦⣤⣤⣤⣀⡀⠀⠘⣿⣷⣀⠙⠷⣤⡀⠀⠙⣯⣫⡷⣄
⣏⢠⣶⣶⣶⣤⣭⣙⠻⢿⣿⣿⡛⠲⠧⣶⣾⣯⣝⢺⣙⣶⣶⡾⠛⢾⣟⣷⣤⣄
⠸⣌⢿⣿⣿⠋⠉⠉⠛⠶⣌⡛⢷⡀⠀⠀⠙⣿⡿⢷⢧⣶⣾⠀⠀⠀⠹⣿⣿⣾⣽⣆
⠀⠈⠦⣻⡿⠀⠀⠀⠀⠀⠈⠙⢦⣷⠀⠀⠀⠸⣦⣤⣀⡀⠉⠙⠶⣤⣀⠈⠻⢿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢻⡟⢻⠛⠻⠿⢷⣦⠈⣿⣶⡜⢎⢻⣿⣧⣿⣿⣦⣤⣀⣀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⣿⣿⣶⣦⣤⣤⣴⠀⢹⠻⠀⢻⣿⣿⣿⡟⣩⢿⣿⣿⣿⣳⣀
⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢿⣉⣇⣈⣉⣹⣽⣤⣼⣤⣦⣼⣿⣿⣿⣞⣵⡿⣫⣭⣿⣿⡿⣧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⡴⠿⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣾⣿⣿⡿⠿⣿⠛⢋⣦
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠦⣴⠞⠋⢉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠟⠙⢿⣿⣿⣷⣔⡿⢦⡀
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⢈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⣿⠦⡀
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠟⢁⣴⣿⣷⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⢸⣿⣿⣷⣀⣀⣀⣠⣤⣤⣶⠶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⣿⣿⣿⣷⣯⣝⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠶⠶⠶⠿⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢤⠃⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡄⠀⠀⡀⠠⢄⠀⠀⠀⠀⢰⣀⣈⡑⠆
⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⢄⠀⣀⡀⣰⡀⢀⡀⠁⠀⠀⠄⠂⠰⠀⠀⠀⠀⢞⠉⠙⢻⠄
⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⢢⡌⠀⠠⢁⣠⠄⠀⠀⠀⣠⠀⢴⠄⠀⠃⡀⠈⠒⢛⣎
⠀⠀⠀⠀⠀⠈⡁⠀⠲⠀⠀⠉⡦⠄⣀⠈⠈⠉⠓⢁⣤⢠⡤⢀⣀⠜⢰⣄⡰⠊⠉⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠨⠭⠥⠈⠷⠦⠀⢀⠀⡖⠈⣅⠀⠀⠀⠭⠤⠽⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠥⠭⠭⠥⠤⢤⡀⠀⠵⠛⠐⠚⠄⠐⠉⠀⠀⠁⠀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠤⢤⠃⠤⠤⠐⠖⠀⠀⣠⠀⠀⠀⠀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡘⢀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡄⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢰⠀⠀⢀⡇⠘⠀⠀⠀⠀⢀⠀⡇⠀⠀⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⡀⠀⢸⠀⡇⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⠀⣸⠀⠀⠀⠀⡇⠀⠘⠀⡇⠀⠀⠘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⣿⠀⠀⣋⣀⠏⠀⠁⠀⡇⠀⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠘⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⢒⠒⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠆⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠛⠛⠛⠛⠛⢻⡇
⠀⠀⢀⣤⣤⣤⣶⣶⣿⣿⣿⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⣀⢀⣠⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣼
⠀⠀⣿⣈⣉⣹⣿⣿⣷⣿⠀⠀⠀⣞⣿⡷⠀⠄⠀⠀⠀⠀⢻⣿⡇⢼⣿⡟⠉⠁⠀⠀⣿⣿⣤⡴⠖⣇⠀⠀⠀⠀⠀⠀⣾⣿⣧⡴⡏
⠀⠀⠈⠙⠋⠙⣿⣿⣿⠟⠀⠀⠀⢻⣿⣿⣄⣸⡀⠀⠀⠀⢸⣿⣿⡿⠟⠁⠀⠀⠀⠀⣿⣿⣾⣷⣼⠇⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣟⠀⠀⠀⠀⠘⣿⣿⣏⢻⣿⠃⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣴
⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⣿⣿⡟⠻⠿⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣼⡟⢿⡇
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⣴⣿⠿⣿⡇⠀⠀⠀⠀⣸⣿⠃⢻⣿⡀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠈⢿⣿⣏⠀⠀⠀⢾⣿⡋⠀⣿⣷⡀⠀⠀⠾⣿⡃⠀⢸⣿⣇⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⢴⣏⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⢙⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⢺⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣧⣶⣦⣤⣄⣀⠀⠀⢠⣽⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⣿⣿⡟⠢⠀⢀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⣾⣿⣿⣷⠀⣾⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣄
⠀⣸⣿⣿⣿⣿⣶⣿⣰⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠸⣿⣿⣿⣿⣿⣿⠟⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢹⣿⣿⣿⣷⣤⣤⡤⠶⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣁⣤⣤⣤⣤⣤⡀⠀⠀⠀⣿⣿⣿⣿⡿⠗⠒
⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠾⢿⣿⣿⣿⣿⣿⡁⠀⠀⠀⢿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⢻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⡸⠏⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠙⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁
⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠈⢿⣿⣧⣀⡤⠀⠀⠀⠀⠀⠀⢼⣿⣿⡇
⠀⠀⠸⣿⣿⣭⠛⠀⠀⠀⠀⠈⠿⠿⣿⣧⡄⠀⠀⠀⠀⠸⠿⣿⣧⠀⠀⠀⠀⠹⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀

⣠⣼⣷⣾⣷⡄
⣿⠉⠀⠀⢸⡟
⠀⠲⡄⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⣤
⠀⠀⠀⣄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠙⣄
⠀⠀⠀⠏⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⣿⡄⠀⠚⠛⠓⠒
⠀⠀⠀⠀⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⢈⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠈⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⡴⣶⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠸⡆⢸⠦
⠀⠀⠀⠀⠀⠀⢸⠆⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣀⣀⡀⠀⠐⠶⠶⠶⢶⣶⠃
⠀⠀⠀⠀⠀⢰⡄⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣯⡈⠻⡿⠄⠸⠇⠀⢸⣿
⠀⠀⠀⣀⣀⣈⢀⣠⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠘⠛⠙⠋⠛⠋⣿⡁⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⢀⣀⡀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⢀⠙⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢀⠀⣀⠶
⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠃⠀⠸⣇⣉⠀⠻⣄⠀⡀⠀⠀⠀⠀⡀
⠀⠀⢀⡄⠀⠀⠘⣿⠀⠀⢸⡇⠀⠀⢀⣴⣦⣤⣤⠀⠀⠈⠉⠈⢀⣤⠟⠃⠘⠛⠓⠚⠛⠛⠛⠟⢤⣤⣠⢿⡆
⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⣿⣿⣦⡀⣾⣟⢉⣿⣇⠀⣠⠄⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠁⠀⠀⠀⠈⣓⠈⠛⣿⣿⠁⠁⠙⠏⠰⠏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠦
⠀⠀⡀⠀⠀⠀⠠⠈⠉⠀⠀⠿⠉⠀⠀⠀⠀⠀⠘⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠙⠄
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠉⠉⠉⠿⣷⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠀⠘⠓⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠀⠀⠩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠉⠻⣷⡄⠀⠄⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⠀⠠⡆⠀⠀⠀⠀⠀⣶⠀⠒⠀⠰⠟⠛⢿⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⠀⠸⠇⠀⡤⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠰⠂⠐⠂⠀⠀⠀⠀⠀⠀⠴⠶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⠻⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣸⡀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠿⠀⠙⠛⠛⠋⠙⠛⠉⠉⠁⠀⢰⡦⠀⠈⠛⠛⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣧⠀⠀⠀⠀⣀⠀⠤⠶⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⢆⡀⠀⠀⠀⠀⠀⢀⣴⠟⠃⠠⡇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠁⢶⣆⠀⠠⠶⠟⠁⠀⠀⠀⠃⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⣌⣿⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠘⢿⡽⠁⠀⠀⠀⠀⠀⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀⠀⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠻⠿⠦⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⡈⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⣀⠋⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣠⠉⠀⠀⠀⠸⣿⡿⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠈⠻⢶⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣤⡤⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠀⢀⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠿⠿⠿⣶⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣌⣭⣶⣶⣶⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⢖⡊⠉⢍⠢⡀
⠀⣤⣞⢹⣿⣖⣒⣋⡥⠒⠒⣄⡏⢷
⠀⠀⠈⠛⠉⠉⠉⠀⠀⠀⠀⠿⢖⣊⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢃⣦⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⠉⡀⠷⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣔⠃⢩⡀⠀⠀⠀⠀⠀⠀⢸⣶⣴⣄⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠀⠀⢱⠄⠀⠀⠀⠀⠀⠀⢿⠛⢻⡄⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠗⠁⡀⣨⣆⠀⠀⠀⠀⣀⣸⣤⠤⠠⠄⠀⠀⠄⢸⠿⣿⣷⡀⠀⠀⢸⠁⠀⣤⡬⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⡀⣐⡾⣿⡦⠒⠈⠁⣿⣿⡿⠀⠀⠀⠀⠀⠀⢼⠀⠈⢿⣿⠏⠒⠸⣇⠀⣿⣿⣿⣄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡅⠘⢩⠷⡤⢤⡀⠤⠐⠛⠀⠉⠉⠉⣩⣿⣷⡟⠐⠒⠠⣿⣀⠀⠀⣸⢧⡈⢿⣿⣿⣷⡀⠙⡄
⠀⠀⠀⠀⠀⠀⣴⠒⡄⠘⣿⡿⠿⣇⢻⡁⠈⡗⢻⢣⠂⣴⠚⠉⡈⠙⠒⢦⣄⠘⠿⢟⣁⣀⣀⣰⠁⠀⠉⡶⠃⠀⣙⣦⡈⠛⠛⣵⠇⢸
⠀⠀⠀⠀⠀⠀⠈⠛⣇⡠⠚⠣⣄⡠⠚⠇⠀⢹⠀⠙⡗⢇⢰⣿⣿⣿⣷⣦⠈⢧⡀⠀⠸⢿⣿⠟⠒⢠⠞⠀⢀⡴⣫⢵⠛⠛⠿⢥⡤⠞
⠀⠀⠀⠀⠀⠀⠀⡠⠚⡄⣀⠔⣿⣷⢀⡼⡀⠘⣦⠅⠱⣸⣎⢿⣿⣿⣿⣷⣀⡈⡷⠠⠄⢄⣀⢀⡴⠋⠀⢔⡡⢎⠁⠺⢗⣄⠀⠀⠱⡀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠎⠀⣠⡞⣅⠀⡹⠛⠁⠀⢣⢀⣇⠀⠀⢿⠀⣷⠨⢙⣛⣛⣛⡴⠃⠀⠀⠀⣠⡟⠀⣀⣔⠉⠀⠘⠛⢏⠉⣤⣵⡀⢀⡵⣶⣿⠃
⠀⠀⠀⠀⡔⢁⢀⠼⢧⣞⡸⢏⠀⠀⢀⠄⢿⠀⣼⠁⠡⢸⡆⡇⠀⠀⢨⡏⠀⠀⠀⠀⢀⡞⠃⣠⡾⠟⢋⣉⣙⠳⢦⡀⠱⡿⠿⠿⡋⠀⢸⡿⠂⣀⡤⠒⣩
⠀⢲⣶⣾⠀⣸⢿⡄⡠⠻⡄⠀⣷⠎⠁⠀⢸⠀⢘⠳⠚⢮⣇⠗⠀⠀⢸⡇⠀⠀⢀⡴⠃⢠⡖⠟⣡⣾⠷⠂⡩⠋⣤⡅⠀⠘⡄⠘⠙⣀⣠⢋⠥⠀⡶⠋⠁
⠀⢀⠟⣼⡖⠁⢀⠼⣄⠀⣹⡎⠀⢱⣦⠀⠀⣆⢨⢧⢤⣜⢻⣝⡑⠀⠀⢷⣀⣴⣿⡴⣿⠞⣡⣾⠝⢁⡊⢀⣴⠟⠋⠑⡄⠀⢈⠤⠖⠉⡠⠶⡖⠉
⢀⡌⠀⡞⢢⣠⢮⡀⢽⡿⠁⠘⣄⠀⠻⣣⡀⠈⣾⢼⠁⠀⢹⣎⡀⠀⢰⣳⢮⣏⡉⠛⡡⢾⠟⣡⠔⣡⠔⠉⠁⢀⡤⢒⡙⠚⣃⡤⢚⣉⡇⠀⠇
⣾⣄⠼⣇⢿⣗⣤⢹⠋⠀⠀⠀⠘⢆⠀⠑⣕⡀⢼⣿⡻⠚⠣⣽⣱⢹⠞⣿⣳⠞⡡⢾⣗⣲⣜⣡⣚⣻⢂⠤⣾⠉⠊⣡⠴⢾⠉⠀⡿⠼⢷⠒⢻
⠈⠻⣷⣽⠧⠬⣼⢏⠀⢀⣀⣀⣀⣈⣷⣄⢈⠃⢈⣉⢣⣀⣀⣿⡡⣏⣰⠞⣡⢮⡽⠛⠁⠏⣽⣴⣾⣋⣷⠻⣩⠶⠋⠁⠀⠀⡇⠀⢰⣀⡼⠖⠛
⠀⠀⠈⠻⣦⣠⠿⢋⡰⡫⠋⢀⠈⢠⠋⢉⠝⠉⡙⠉⡹⠋⠻⠛⢻⠛⢓⠞⠛⡿⠒⡿⠷⠾⠿⢿⠿⢿⡯⢿⣤⣤⣤⣤⣀⣠⣴⣚⣉⡠⠴⠚⠋
⠀⠀⠀⠀⠈⣹⠎⣭⣬⣓⡶⠋⡠⠋⠀⠋⢐⣎⢀⡴⠤⡴⢁⣴⠁⢠⡊⢀⡌⠀⡔⠀⡰⠁⢠⠃⢀⠎⠀⡔⠁⡠⠂⢀⠎⢀⡌⠀⣼⡖⣶
⠀⠀⠀⠀⣼⣵⠳⠖⠺⣏⣀⡄⢀⡐⠁⢠⣾⣶⠏⢀⠜⠀⣠⣟⣶⠋⢀⠍⠀⠉⠁⢸⠃⣰⠇⠤⡧⠠⣜⠀⠀⠁⢠⠆⢀⠞⠀⣼⡇⠐⠸
⠀⠀⠀⠀⣧⣸⡧⢤⣄⣀⠉⠙⠛⠓⠞⠓⢶⣃⣠⠂⢀⡴⢿⡿⠁⣠⠃⢀⠌⠀⣰⣯⣲⠋⢀⠞⠀⣠⣧⣤⠏⠉⡹⠉⢻⠆⣼⠍⠀⡰⠁
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠙⠒⠢⠤⢄⣀⣀⠉⠉⠛⠉⠻⣀⣴⡁⣠⠊⢀⣼⠿⡿⠃⢀⠎⠀⣰⣿⣷⠏⠀⡔⠀⢠⣟⡚⢋⠓⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠠⠤⢀⣈⡉⠙⠒⠚⠉⢹⣁⣠⠋⠀⣴⠟⢿⠏⠀⡜⠀⢠⣿⣿⢃⠂⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠦⠤⢌⣉⡋⠙⠁⠈⣇⣀⡜⠀⣀⠟⠒⢁⠎⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠦⠤⣍⣙⡃⢀⣅⠎⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠈⣰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⡇⢀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⢁⡀⡀⠀⢀⣀⡤⠬⠧⠀⠐⠂⠁⠙⠂⠀⠉⠀⠐⣈⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⣅⠬⠄⠡⠀⠀⠀⣤⡄⠀⢸⠀⠀⠀⠤⠴⠋⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠶⠀⠀⠈⠀⠀⠀⠈⠀⠂⠀⠂⠘⠓⠢⠐⠒⠁⡇⠀⠀⠐⠒⠓⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⡤⠀⡀⠀⠀⠠⠀⢩⠁⠀⢇⠙⠐⡸⠁⠀⣀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠠⠁⠀⠀⣰⠀⠄⠀⠀⠉⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠂⣤⠀⠀⢀⣀⣀⠐⠀⠀⠀⠀⣀⠔⠁⣀⠀⠀⠀⠀⠀⢀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⢰⠙⡄⠀⢠⠟⣅⠀⠀⠀⠉⠉⠀⠀⠰⠉⠆⠀⠀⠀⠀⠠⠞⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣄⣰⣿⣀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠞⠋⠉⠀⠀⠉⠙⠳⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠃⠀⠀⠀⣿⣿⠀⠀⠀⠘⣧⣴⠄⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡇⠀⠀⢀⣤⣼⣧⣤⡀⠀⠀⢸⣇⡀⢀⣤⣀⣿⠷⠶⠿⠶⣾⣟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣸⠋⣡⣀⡽⠋⠀⠀⣠⣤⡀⠀⠈⠻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⣆⠀⠸⠿⠿⠿⠿⠇⠀⣰⠟⠿⠉⡿⠁⠀⠀⠀⠻⡿⠃⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢻⣦⣤⣤⣠⣤⣴⠛⠿⠆⠘⠻⡇⠀⠀⢠⣾⣿⣿⣿⡆⠀⠀⢈⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠙⢳⣦⠿⣣⡈⠛⠀⠀⠀⠀⣶⣿⡀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡴⠟⠛⠛⠛⠛⠿⣇⣠⣄⠀⠀⠀⠀⣠⣿⣤⡈⠋⠛⠛⠛⠃⣠⣾⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⠋⠀⠀⢠⣶⣦⠀⠀⠈⢻⣅⣀⠀⠀⠀⠈⠀⢸⡿⠛⣶⡶⠺⣿⡁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⣈⣛⣋⡀⠀⠀⠀⢻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⣸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢷⡀⠀⢸⣿⣿⣿⣿⡇⠀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢦⣈⡈⠉⠉⣈⣠⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠙⣿⠋⠹⠷⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠿⠿⠿⠷⠖⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣺⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣿⠿⣿⢿⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠀⠀⠀⢀⣤⣤⣶⣿⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣿⠟⠁⣠⣧⡀⠙⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣭⠶⠷⢮⣷⣄⡀⢨⣿⣿⠟⠉⢉⡟⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⣷⣶⣶⠸⢿⣿⡟⠛⣿⣿⡟⠛⣻⣿⠿⠁⣤⣀⣤⡄⠀⠘⣏⣟⠙⢦⣴⠋⢹⣼⢡⣾⣿⡏⠙⢲⣿⣄⣀⣿⣿⡿⠇
⠀⠀⠀⠀⠀⢀⣰⣿⠯⣤⣞⠹⣿⣦⢀⣿⣷⣦⣀⣇⣠⣶⣿⣇⢠⣾⡿⡛⣻⣿⣶⢄⣦⢷⣴⠋⠉⢢⣼⣹⡏⠹⣿⣿⣄⣼⠁⢀⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠙⣿⣦⣜⣉⣽⣿⠃⠈⣿⠋⠛⣿⣿⡟⠋⢻⡏⢸⣿⣤⠛⢣⣼⣿⠀⣤⠙⣭⡛⠟⣭⡟⣡⡄⠀⢾⡿⣿⣿⣿⣿⡿⠟⠟
⠀⠀⠀⠀⠀⠀⢶⡟⠛⢿⡿⠛⢹⡷⠀⠸⣇⠀⣨⣭⡀⢀⡾⠀⢶⡉⢿⠿⠿⠟⢀⡖⠹⣇⠈⣤⣤⡀⢀⡿⠀⠀⠐⣆⠀⠈⠛⠋⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠈⢧⢰⣿⣿⡄⡼⠁⠀⠀⠹⣆⠿⠿⢇⡾⠁⠀⠈⣧⢰⣿⣿⠄⡾⠁⠀⢹⡌⠿⠿⢃⡾⠁⠀⠀⠀⠹⣆⢼⣿⣷⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⠀⢻⣷⣶⣿⠃⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⢻⣷⣶⣿⠁⠀⠀⠀⠀⠀⢹⣬⣭⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠀⠸⣿⢿⡏⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⢁⡴⠖⠲⣄⠀⠖⠒⢦⡀⠐⣶⠒⢶⡖⠲⣆⠰⣖⠀⢲⡖⠀⣲⠆⣠⠖⠲⢦⡀⢲⡶⠖⠀⡇⣰⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣟⠚⠒⠛⢰⡏⠉⣻⡇⠀⡇⠀⢸⠇⠀⡿⠀⠹⣤⡟⢹⣰⠏⠀⣧⠀⠀⣸⠇⢸⡇⠀⠀⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠉⠛⠛⠁⠈⠙⠛⠛⠃⠘⠛⠀⠛⠃⠘⠛⠀⠀⠙⠀⠀⠋⠀⠀⠈⠛⠛⠉⠀⠛⠃⠀⠘⠛⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⢹⡏⠉⠉⢻⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⣦⠀⠀⠀⢠⡞⢷⡄⠀⠀⠀⣴⠻⣦⠀⠀⠀⢠⡞⢳⡄⠀⠀⠀⠀⢰⡟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠋⠀⠀⠀⠈⢳⠟⠁⠀⠀⠀⠙⣾⠋⠀⠀⠀⠈⢳⡞⠁⠀⠀⠀⠀⠈⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠶⣄⠀⢀⡴⠞⠳⣦⡀⠀⣠⠶⠛⠶⣄⠀⢀⡴⠞⠳⢦⡀⠀⠀⠀⠀⣴⠟⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠶⠶⠆⣿⠀⣿⠰⠶⠶⢸⡇⠰⡇⠶⠶⠆⣿⠀⢾⡁⠶⠶⢈⡇⠀⠀⠀⠀⣿⠐⠛⠋⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠾⠋⠀⠈⠳⠦⠴⠟⠀⠀⠙⠶⠤⠾⠋⠀⠈⠳⠦⠴⠟⠁⠀⠀⠀⠀⠈⠛⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠞⡇⣎⣳⢸⢻⠛⡦⣿⡇⡟⢳⢸⡁⣇⡗⣾⢻⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣤⣶⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣿⣿⣿⣯⡀⠀⠀⠀⢻⡗⠀⠀⠀⡀⠈⣻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣟⠀⠀⢾⡷⢾⣿⣶⣿⠚⠿⠚⢿⣿⣿⠟⠀⠀⠀⠀⣤⣶⣶⣄
⠀⠀⠀⣁⠀⠀⠀⣠⣴⣶⣤⣿⣿⡿⣿⠟⠻⠿⠃⠀⠀⠀⢴⣿⠿⣿⠆⠀⠐⢾⡟⠡⣿⣦⡀⠀⢸⠏⠉⠛⠻⠄
⠀⠀⠀⠙⣷⡀⢸⡿⠋⠉⠉⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠈⢷⣦⡻⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣴⣦⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣧⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣤⡄
⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⣤⡉⠁
⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣥⣤⣤⣤⣤⣼⣿⣧⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣬⣿⣿⣧
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡔⠒⢶⢲⠒⡖⢲⠒⡖⢲⠒⡖⢲⠒⡖⢲⠒⠒⠀⠀⠀⠀⠀⡴⠒⢲⣶⠒⢂⣔⠒⢂⣰⠒⢒⣴⠒⠒⣔⠒⠒
⠀⠀⠀⠀⡇⠀⠈⡏⠀⠈⡏⠀⠈⡏⠀⠈⡏⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡍⠀⠈⡍⠀⠈⡍⠀⠀⡍⠀⠀⢉
⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠸
⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣄
⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡃⠀⠀⡇⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡃⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⣡⣀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡇⢠⠊⠻⣶⠊⠻⣶⠊⠻⣶⠊⠻⣆⠀⠀⢸⠀⠉⡇⠀⠀⠀⡇⠀⣠⣅⠀⣠⣅⠀⣠⣅⠀⣠⣅⠀⠀⠀⠀⣶⣶⡀
⠀⠀⠀⠀⡇⠈⠢⠤⠚⠢⠤⠚⠢⠤⠛⠢⠤⠃⠀⠀⠀⠉⠈⠀⠀⠀⠀⡇⠀⠻⠿⠃⠻⠿⠃⠻⠿⠃⠻⠿⠃⠀⠀⠀⠉⠛⠁
⠀⠀⢠⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠆⠀⠀⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠨⣀⣀⣀⣀⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣆⣀⣀⣀⡄⠀⠀⣿⣿⣿⡿⢹⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⢽⣿⣿⣿⡇

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣾⣦⣠⣾⣦⣠⣶⣦⣠⣶⣦⣠⣾⣦⡀⠀⠀⠀⢸⡇⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠿⠟⠙⠿⠿⠙⠿⠿⠙⠿⠿⠙⠿⠿⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣸⡇⢠⡤
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠿⣯⠉⢉⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣨⢷⡋⠁⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣷⣦⡴⢿⣿⣁⣀⣻⣾⣿⢡⣤⣼⣷⣴⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⣡⡜⢻⣿⡎⠉⠛⠻⠋⠋⢠⣶⡿⣫⣽⣭⡹⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⡄⣾⣿⣿⡠⢻⠃⣴⣿⠇⠀⠀⠀⠀⠀⢠⣿⡇⢿⣯⣽⡿⣹⣿⡁⣠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⣱⡿⢋⣿⣿⣿⡿⡟⣯⣴⣦⣴⣀⡀⠀⠀⢹⣿⣦⣭⣯⣷⣿⣿⣇⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⡿⠋⠀⠀⣀⣿⢻⣧⡶⣾⣿⣷⡆⠀⠀⠀⠛⠉⠻⠻⣿⣬⣿⣷⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢼⡇⢾⣏⠶⣩⣷⣻⣟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣆⠀⠀⠀⠀⠀⠐⢿⣾⡛⠻⣛⣷⣿⠉⢀⡀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠈⣿⠃⠀⣀⣀⣶⣤⣼⡍⠛⠻⠏⣛⣸⣷⣾⠿⢿⣿⣶⡆⠀⠀⢰⡿⠋⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⢰⡿⢠⣤⣿⣿⣿⣿⣿⣿⡟⠀⣀⣿⡿⠻⣷⡀⠀⣨⣿⣿⣿⠀⢸⣿⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⣶⣿⣶⣿⣤⡄⢠⣿⣿⣿⠁⠀⣿⣿⣿⠟⣹⣿⠁⠀⣈⣷⡾⠋⠁⢹⣿⠗⠈⠛⣠⣀⣌⢛
⠀⠀⠀⠀⠀⠀⠀⠠⣴⡿⠋⠀⠀⠀⠈⢻⣷⡆⣹⣿⣿⣷⣾⣿⣿⠿⠃⢙⣿⣶⡿⠋⠁⢻⣦⣀⣾⡿⠦⣤⡿⠛⠛⠛⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⡀⠀⠀⠀⠀⠀⣿⡷⠀⠈⠿⠛⣿⠛⠛⠀⠀⠈⠙⣿⣷⣦⣤⣤⣿⣿⠟⠇⢺⣿⠀⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⣴⣿⣿⣦⢿⣿⣇⠀⠀⠀⣀⣼⡿⠇⢻⣧⢀⣴⣶⣄⠀⣿⠇⠀⠀⠁⠈⠛⠙⣿⡏⢩⣴⣦⣝⣿⣄⢀⣄⠀⣠⣿⡋
⠀⠀⠀⠀⠻⢿⡿⢋⣼⣿⢻⣿⣿⡿⣿⠛⠋⠀⢸⣿⠸⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⡿⠈⢿⣿⠿⡿⠛⠃
⠀⠀⠀⠀⢠⣶⣶⣾⣿⣥⣾⡿⠋⠀⠀⠀⠀⠀⠈⠻⣷⣬⣭⣵⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣭⣭⣤⣶⡿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿
⠀⠀⠀⣼⡿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠙⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠘⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢐⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣉⠟⣉⢷⡄⠀⠀⠀⢀⣀⡀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣴⣾⣷⣷⣦⠀⢠⣿⡗⠿⠿⠟⢻⣿⡆⠀⣴⠿⣿⣿⣷⡄⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠋⠃⠉⣿⣛⡇⠀⠀⠀⢸⣿⣿⣍⣿⣿⡶⢻⣿⣿⡀⠴⢄⣿⣿⣷⡀⣿⣦⠚⠛⣿⣿⣦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣤⣉⣴⣿⣽⣧⣀⡀⠀⢿⣿⠋⢁⠙⣿⣿⣀⠙⢿⠛⠶⣿⣿⠿⢿⣿⣿⣿⡐⢀⣼⣿⣿⣯⣅⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣇⠉⠉⠉⣼⣿⣿⣿⣟⠻⠻⣿⣧⣀⣼⣿⣿⣿⠀⠀⠀⠀⠶⠀⢰⣾⣿⠿⣿⣿⡿⠉⢻⣿⠷⡏⢉⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⡶⣶⣿⠁⢹⣟⡿⢤⡠⠿⠿⠉⠹⠿⣟⣼⡁⠲⠶⠚⡟⢾⣿⣿⡏⠃⠘⣿⡇⣰⣿⢀⣹⣤⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠈⠁⢸⡟⡤⣿⣸⠀⢺⣗⠀⠀⠀⠀⠀⢨⣿⠀⠠⠀⢀⣷⢦⣿⣿⣇⣤⡀⣿⣷⡿⢡⣿⠾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠀⢀⣿⣷⠂⢀⠀⠀⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⡧⠸⡇⢿⣿⠇⣉⠀⠙⣿⡿⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣭⡹⠿⣿⣿⣿⡏⠀⢸⣿⣿⣗⠀⠀⠐⣿⣿⡆⠀⢿⣿⣿⣿⣿⣿⠃⠘⠧⠈⣿⣦⠀⠶⠿⣷⢀⡇⢾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣧⣥⣼⡟⠻⠁⠀⠇⣿⣿⣿⣷⣄⣼⣿⣿⠇⠀⣼⣿⢿⣿⣿⡏⣀⣴⣾⣯⠀⠋⠀⣤⣴⣿⣿⣷⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⠃⡐⣾⡶⣾⣯⣿⣿⣿⣿⣿⣿⣀⣼⣿⡇⠀⠉⢻⣟⣻⣿⣿⣿⠃⠀⠀⣀⢀⠸⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡏⣿⢿⣿⣿⣿⡿⠿⣧⢈⣿⣿⣿⠀⡀⠀⣸⣿⣿⣿⣿⣹⣆⠀⠘⢶⠀⢹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣿⣭⣽⣥⣿⣿⣿⣿⣾⣿⡿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡦⣤⣧⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠁⠀⣿⣿⣿⣿⢻⣿⣿⣿⣿⣁⣾⣇⠀⠀⠸⣿⣿⣿⣿⣯⣿⣿⠃⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⡇⠈⣿⣿⣿⣿⡏⢾⣿⡄⠁⡄⠉⠻⣿⣿⣿⣿⡿⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⠁⠀⣿⣿⣿⣿⡷⣿⣿⣿⣾⡇⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠻⢿⣿⠀⠀⣿⣿⡿⣿⣗⣿⣿⣿⣿⠄⠀⠀⢸⣿⠛⠛⠃⠀⠀⢹⡍⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠀⠀⣿⠀⢽⠀⠀⣼⠃⠰⣿⣟⣿⣿⣿⡿⠀⠀⠀⢸⣿⠃⠸⡇⠀⠀⠀⣧⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢹⣠⣿⠀⠀⣿⣷⠾⠘⣯⢹⣿⣿⠃⠀⠀⠀⢸⣿⠀⠀⠁⠀⠀⠀⢿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢻⣾⣿⡿⠀⠀⠀⠀⠀⣿⠀⢈⡀⠀⠀⠀⠸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣿⣿⠀⠀⠀⠀⣿⣿⡃⠀⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢿⡇⣩⡀⠀⠀⠀⠀⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⣿⣿⣇⠀⠀⠀⢸⣿⡇⢰⣿⣿⠀⠀⢺⣿⣻⣷⣄⠀⠀⠀⠀⣾⡇⠙⡇⠀⠀⠀⠀⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⢹⡇⠰⠏⢹⣿⠀⠀⢀⣾⣿⡇⢸⣿⣿⣧⣤⢸⣏⣉⣻⣯⡷⢤⠀⠠⣿⣿⠂⡇⠀⠀⠀⢰⠀⣼⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣸⠀⠀⡄⠈⣿⠂⠀⠀⣥⣼⡇⢺⣿⠶⠶⡟⠘⢇⣿⣸⣿⠇⠘⠦⠶⢿⣿⠖⠃⠀⠀⠀⢨⠀⢹⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠨⣽⣷⠷⠶⢄⣀⣀⠀⠀⠀⠄⠀⠁⠀⠈⠉⢹⣶⣶⣾⠿⠶⡦⠄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠄⠀⠀⢀⡀⠀⣈⡉⠁⢀⡁⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠃⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡀⠀⣀⠀⢀⡀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⠟⠀⣿⡇⠼⢿⣤⡴⢿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣷⡀⣿⡇⢀⣾⡏⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠹⠗⠻⠇⠸⠟⠀⠸⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠿⣿⣶⣷⠄⠀⣠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠛⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⣡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠆⠀⠀⠊
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢻⣿⣿⣧⠀⢀⠘⢋⣿⣿⣿⣿⣿⠃⠀⢻⣧⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠈⡃
⠀⠀⠀⠀⠀⣸⣿⠿⣿⣿⣿⢸⣿⣿⣿⣾⡿⡷⠟⠛⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠸⠄⠀⢠⡃⠀⠠
⠀⠀⠀⠀⠀⣿⣿⣤⣿⣯⣤⠈⣿⣿⠙⠋⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠙⣋⣿⣿⣿⠀⡿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡯⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⡗⠿⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠸⡷⠾⠿⠿⢿⡇⠙⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠛⠛⣿⣷⣦⣤⣀⣈⠄⠇⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣤⣴⣶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⢿⣿⡿⠁⠈⠈⣗⣀⣤⣴⣾⣧⣤⣤⠤⠤⠀⠐⢲⠒⠂⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⠟⠁⣼⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡥⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣿⣿⠙⠿⣿⣻⡿⠟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡃⠀⠀⣿⡉⠀⠀⠀⠀⠿⠿⠿⠿⢿⡟⠋⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⡆⠀⡸⣿⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⣿⣴⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠇⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠙⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡟⢛⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⡿⠙⠘⣛⠻⣿⠟⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣗⠀⠚⠉⢀⠍⣠⣿⣿⣿⣿⣿⣿⣿⢿⡿⢲⣯⡅⠶⠾⠙⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠂⢀⠀⢀⣼⡿⡿⠻⢋⣿⠟⠓⠛⠵⠿⠋⠫⠀⠀⠀⢠⠀⠀⠙⠙⠋⣻⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿
⣿⡟⠓⠀⠀⠁⣾⡏⠁⠀⠀⢊⣹⡤⠴⢂⣴⠒⣀⣶⠤⠶⠾⣶⣶⣦⣤⠀⠀⠀⢮⣡⡼⠯⠹⡍⠩⠗⠈⠉⠉⠛⠛
⠀⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠚⠃⢀⡴⠛⠃⣄⣘⣫⣿⣿⣷⠤⣄⠈⠉⠻⠄⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡌⠀⠘⣻⣿⣿⣿⣿⣿⡿⠈⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠉⠁⠀⠀⠈⠿⣿⣿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠠⠆⠤⠘⠉⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣴⡟⠁⠀⠊⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢛⣁⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠠⢀⡀
⠀⠀⠀⠲⣂⢀⣀⣀⢤⣐⣪⠥⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠆⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⣠⡄
⠶⠾⣿⣿⣿⠿⠿⠟⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠃⠀⠈⢀⣀⣀⣀⠀⠄⠀⣠⣴⣿⠿⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⢙⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⡃⠀⠀⠀⣶⣄⠀⢠⣤⡜⣿⣦⣸⣤⣠⠆⠀⠀⣀⡤⢤⡀
⠀⠀⠀⠀⠀⣠⣦⣦⣀⠀⠀⢘⠁⠺⠃⠀⠀⠶⣿⣿⣷⣾⣿⡧⣾⣿⣿⠏⠀⠀⠀⢘⢿⡃⢼⣿⠆
⠀⠀⠀⠀⡰⢿⣇⣻⡿⠀⠀⢸⣶⣾⡆⠀⠀⠀⠀⠙⠻⢷⣿⠄⢼⡏⠀⠀⠀⠀⣴⠛⣷⡋⣡⠞⢷⡄
⠀⠀⠀⠀⢸⣾⡿⡿⠀⠀⠀⢠⣿⢻⣷⡀⠀⠀⠀⠀⠀⠘⣹⣶⣾⡃⠀⠀⠀⠀⠈⠁⠘⠿⢿⠀⠚⠁
⠀⠀⠀⠀⣼⣿⣧⣶⣄⠀⠀⣻⡇⠀⣿⡇⠀⠀⠀⠀⢠⣾⡿⠟⠿⣷⡄⠀⠀⠀⠀⠀⣯⣴⣽⡀
⠀⠀⠀⠀⣿⡟⠉⠛⢻⣷⠀⠿⠃⢀⣟⠀⠀⠀⠀⠀⠘⠿⣦⠀⢠⡿⠃⠀⠀⠀⠀⠚⣿⠉⠛⡇
⠀⠀⠀⠀⣿⡇⠀⢀⣾⠋⠀⠋⠃⠀⠁⠀⠀⠀⠀⠀⠠⠀⣸⡷⢹⣇⠀⠀⠀⠀⠀⠀⡿⠀⢻⡇
⠀⠀⠀⠀⢹⡇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⡗⠀⢸⡇
⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠠⠀⠟⠛⠛⣤⠄⠀⠀⠀⠀⠀⠾⠿⠿⠧⠄⠀⠀⠀⠀⣇⠀⢸⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠒⠂⠀⠰⠀⡄⠀⡄⠒⠀⠀⠀⠀⠀⠀⠁⡛⠀⠐⠀⠀⠉⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠂
⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡀⠂⠀⠁⠖⠒⠛⠛⠛⠟⠁
⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠐⠻⢦⠀⠀⠀⠀⠀⠀⠑⠂⠄
⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡤⠐⢤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⡻⠀⢀⡞⠀⡠⣶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢾⡄⡔⠀⢻⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡿⠃⠀⠀⠈⣧⢁⠀⠀⢻⡞⡆⠀⠀⣠⣤⣶⣦⣤⡤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢦⠀⠀⢀⡤⡞⠋⠀⠀⠀⠀⠀⠀⢸⡆⡆⠀⠀⣷⣷⣶⣾⠏⠙⢩⣿⠟⠟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠸⣶⡺⠟⠉⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣶⡶⣿⣿⡟⠿⠻⣷⠀⠈⠳⣌⢂⠀⠉⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⢸⣁⣀⢀⣀⠀⠀⠀⠀⣰⡿⠿⠊⠋⢻⣿⣿⢸⣿⢂⢀⢡⡀⠀⠀⠈⠈⢢⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠺⠛⠛⠉⠁⠀⢀⣤⣼⣯⡀⠀⠐⠀⡀⢻⣟⣾⠻⣧⠠⠀⠡⠡⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⠀⣆⠀⠀⠀⠀⢀⣼⣿⡿⣿⣷⣄⠀⠀⠀⠈⠉⢉⠣⡘⣧⠠⠠⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⠤⢤⣌⢳⠀⠀⠀⣾⣿⠉⠀⠉⢿⣝⣆⠀⠀⠀⠀⠀⠀⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⢠⢟⡆⠀⢰⣏⡏⠀⢀⣴⣿⡉⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⢀⡟⣸⠀⢠⡟⣸⢀⣶⡿⡟⡩⢷⡱
⠀⠀⠀⠀⠀⠀⣸⢣⠃⠀⠈⣹⣷⣾⣿⡄⠘⡓⡀⠧
⠀⠀⠀⠀⢀⣤⣿⣶⣶⣶⣿⣿⡿⠛⣿⠙⡄
⠀⠀⠀⢀⡾⣿⣿⣿⢂⠙⢝⣿⠣⠨⡿⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⣿⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡶⠿⢦⡀⠀⢀⡴⠿⢶⡄⠀⠀⢀⡴⠿⢶⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣄⠀⣸⡇⠀⢸⣇⠀⣠⡟⠀⠀⢸⣇⠀⣠⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠛⠉⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠃

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⢀⣀⣀⠀⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⣀⡻⣿⣿⠟⣁⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣤⣬⣛⣛⣥⣤⠀⠀⠀⣠⣬⣛⣟⣫⣤⠰⣿⣿⣿⡆⣠⣬⣻⣟⣫⣄
⠀⠀⠀⠀⣾⣿⣿⣶⣶⣿⣿⣷⠀⠀⠀⣰⣶⣭⣿⣵⣶⡽⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⢿⣵⣽⣛⣯⣶⣝⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⠿⠛⠁⠀⠀⠻⠿⣿⣿⠿⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠛⠉⠀⠀⠀⠙⠛⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⠀⠀⠀⣴⣿⣶⠀⠀⠀⢰⣾⣷
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣴⣽⣿⣵⡄⠀⢠⣮⣿⣯⣦⠀⢠⣬⣿⣯⣦⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⢠⣼⣿⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣺⣿⡛⠛⣻⣿⡛⠛⣻⣿⡀⠀⠀⠀⠀⠉⠉⣵⣾⣽⣷⣮⠉⢹⣾⣽⣷⣾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡗⠸⣿⣿⡿⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠀⠸⣿⣿⣿⡿

⠀⠀⠈⠀⠀⠈⢢⠀⡀⠀⠀⣈⠀⠀⠀⠀⠈⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⣂
⠀⠀⢀⠈⠁⠀⠈⠳⣼⣄⡐⠁⠀⠀⢀⡴⠖⠒⠶⢦⡀⠁⠂⠀⠈⠀⠀⠀⠆⠀⢀⠠⠶⠀⠀⠀⠀⠀⢂⠀⠀⠈⠀⠀⡀⠤⠀⠐⠂⡅⠄⠉
⠀⠔⠈⠀⠀⠀⠔⠀⠀⠃⠐⠀⠀⢠⡏⠀⠀⠀⠀⠈⠞⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠐⠀⠀⠀⠀⠀⠰⠔⠈⠁⠀⠐⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠸⣁⠀⠐⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠂⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠙⢦⣄⣀⣠⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢾⠃⠀⠆
⠀⢰⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠉⠤⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⡀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡀⠀⡀⠀⠀⠀⠘⢀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⣀⣤⣀
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠛⢧⣤⣴⣋⡷⢭⠗⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠹⠆
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡼⠃⠀⠰⣤⣠⡦⠤⠈⢿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢦⡄⠀⠁
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢋⡷⣼⢱⢣⡖⢸⡱⡞⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⣈⡼⠃⠛⠛⠛⠛⠙⠛⠁⠀⠀⠀⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠟⠃⢁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⣴⠖⢂⣶⠂⣠⣤⠆⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠊⠁
⠀⠀⠀⠀⠺⠶⣤⡤⠂⠈⠁⠀⠀⠀⠀⠈⠳⡶⠾⣇⠀⠀⠃⢠⣾⣣⢎⡟⣡⢠⠀⡤⣄⡃⣠⣄
⠀⠀⠀⠀⢀⡀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⢑⡞⠋⢠⣄⣴⠏⢿⡇⣾⢰⣧⣿⣬⡧⣿⣴⣧⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠇⠄⠈⠁⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠙⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢶⠒⡺
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⡄
⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡇
⠀⠚⠸⠅⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁
⠀⠀⠁⠀⠀⠀⠀⠀⡀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉
⠀⠀⣄⠨⡄⠀⡐⢴⣤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴
⠀⠀⡤⢘⣇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢰⢓⠄⠺⡀⠂⠀⠀⠀⠠⡄⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠐⠂⠁⠀⠀⠀⠀⠀⠠⠀⠀⠈
⠀⣉⠀⣓⠋⠁⠉⠀⠀⠀⠠⣦⡴⠃⡠⣤⣤⡀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠔⠎⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠈⠀⠀⠋
⠀⣿⠀⠈⠂⠀⢀⠀⠀⠀⠐⣷⠁⠸⡃⢣⣸⣷⣶⣥⠖⠚⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡈⡀⠀⠀⠀⠀⠀⠆⠀⠀⢰⠂⠄⢀⠈⠀⡀⣀⡀
⠀⡿⢀⠀⠀⠀⠀⠀⠉⢠⣿⠏⠀⠀⠛⠈⠸⡿⢿⠗⠀⢰⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠐⠇⠈⠀⣉

⠀⠀⠀⢴⣶⡄⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤
⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆
⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣀⣀⡀⠀⠀⢈⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⣹⣿⡿⠤⡀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣗
⠀⠀⣼⣿⣿⣿⠀⢰⣾⠇⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣸⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣧⡀
⠀⢰⣿⣿⣿⣿⣤⣾⢇⣄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡷
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠛⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢻⣿⣿⣿⣿⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣇
⠀⠀⣹⣿⣿⣿⣿⡿⠁⠀⠀⢸⣿⣿⣿⣿⡏⠉⠉⠉⠉⠀⠀⠀⠈⣿⣿⣿⣿⠿⠟⠁⣀⣀⣀⣀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡤⠤
⠀⠀⢻⣿⣿⣿⣿⠁⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⡗⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠸⣿⣿⣿⣿⣧⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⣿⣧
⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠸⡿⠿⠿⠿⠿⠿⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠻⣿⣿⣇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠋
⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁
⠀⠀⢸⣿⣿⡿⢧⡄⠀⠀⠀⠀⢾⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠈⢿⣿⡷⠟⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⡴⡊⠍⠉⠒⡀
⠀⠴⣯⢹⣿⣓⣊⡥⠒⠉⢢⡏⢱⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⡏⡍⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣶⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⡤⣉⠄⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠀⠸⠆⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠘⠡⡀⠀⠀⠀⠀⠈⠇⠀⠃⠀⠀⠀⠀⢠⣿⣷⡄⠀⠀⠀⣰⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢙⠀⢠⣾⣷⣀⠠⠤⢐⣶⣿⡍⠀⠀⠀⠀⠁⢸⠉⠻⣿⣦⡄⢀⣳⠀⢨⣿⣵⡌⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠅⢰⠿⣍⣃⡀⢀⣈⠻⠏⠠⠤⠤⢀⣤⣤⡾⠀⠀⣉⡟⠀⠀⠈⣧⡘⢿⣿⣿⣦⠈⢣⡀
⠀⠀⠀⠀⠀⠀⡀⢀⠀⣤⣤⣶⡔⢺⠓⠠⡮⣶⣁⠝⣡⠄⠒⠒⠤⣄⡀⠘⣿⣿⠇⠀⠀⢸⠋⠐⠢⡴⠁⠑⢄⠙⠿⠿⢇⡆⢱
⠀⠀⠀⠀⠀⠀⠳⢾⢀⡼⠯⡀⢉⠴⡇⠀⢳⠀⢻⡬⡇⣴⣶⣶⣶⣄⡙⢦⠀⠈⢹⣿⣿⡿⠤⣠⠞⠀⢀⡼⣫⡽⠶⢶⣞⣀⡼
⠀⠀⠀⠀⠀⠀⢀⡴⢇⠀⡠⣾⣷⠄⣰⠀⠙⣤⠌⢧⣱⡹⣿⣿⣿⣿⡄⡀⢧⠠⠀⣈⠁⠀⡴⠃⢀⢔⣽⠋⠰⠷⢄⠀⠈⠣⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡠⠊⢀⡰⢏⠀⢙⠟⠉⠀⢣⠀⣇⢈⠘⣿⢸⡎⢛⡻⠿⢟⣡⠏⠀⠀⠀⣨⡟⠁⢠⡔⠁⠀⠛⢾⡋⣩⣱⡀⢀⣱⣴⣿⠁
⠀⠀⠀⢀⠜⣀⡠⠿⣴⣂⢿⡁⠀⢀⠄⢾⠀⣿⠁⠀⢻⢸⠁⠀⠀⣿⠁⠀⠀⠀⢀⡜⠋⣤⡿⠛⣉⣉⡓⠶⡄⠑⢿⠿⠿⡏⠀⢸⡿⢂⣠⠴⢚
⠀⢲⡶⣎⢀⡿⢧⢀⠜⢇⠀⣱⡖⠁⠀⢰⠀⢙⠖⠒⢼⡜⠃⠀⠀⣯⠀⠀⢀⡴⠃⢠⡾⢋⣤⡿⠒⢈⠝⣡⣜⠀⠈⢆⠘⠙⣀⣴⠋⠤⢀⠖⠉
⠀⢠⠛⣾⠊⠀⡠⢳⡄⣨⢏⠀⠸⣦⡀⠀⢄⠼⢳⠠⢎⡻⣭⠀⠀⢸⢄⣴⣿⣶⡻⢋⣴⡯⠊⡰⠁⣠⠞⠉⠉⢂⠀⣈⠶⠞⣁⠤⠓⠈⠁
⢠⠃⢰⡏⣦⡖⢤⣐⠿⠃⠀⢣⡀⠙⡵⡀⠈⣷⢺⡀⢀⠿⣤⢀⠠⣭⣾⢯⡌⢉⡔⠻⢋⡴⢋⡔⠁⠀⢀⡤⣊⠍⠋⣡⡴⣪⣭⡆⠀⡃⢀
⢿⣦⡻⣇⠿⠵⣆⡏⠀⠀⠀⠀⠳⣄⠈⢮⡄⢻⣫⡝⠉⠰⣧⢧⠟⠻⡿⢋⣔⣻⡯⣽⣮⠾⣥⣿⢔⣺⡇⢈⡤⠖⠛⡇⠀⠝⠚⠛⢁⣧
⠀⠙⠷⣞⠓⢲⣏⡄⢀⢴⠤⢤⠤⢿⠧⣤⢤⣬⣤⣷⣤⣶⣧⣺⣴⣋⣴⣿⡞⣩⣄⣃⣶⣾⠯⣼⣭⡥⠚⠁⠀⠀⠀⢐⣀⡸⠶⠚⣉⡥
⠀⠀⠀⠈⠳⣎⡩⠔⣥⣁⣠⠅⣴⠁⣴⠁⡠⠀⠠⠂⠠⠋⣠⠋⢀⠋⢉⠍⢉⠍⢉⠟⠉⡟⠉⡝⠋⡻⠒⢲⠒⢲⠖⢺⠦⢴⣒⡉⠁
⠀⠀⠀⠀⡠⢋⣜⣻⣷⠞⠁⠄⠀⠔⠁⣰⣃⡴⠋⡙⠃⢲⣇⣰⠇⢠⠆⠠⠎⢠⡎⢀⡞⢀⡜⠀⡔⠁⠠⠁⢠⠂⢠⠋⢀⡟⠃⢻
⠀⠀⠀⠀⡟⣿⣉⠀⠙⠲⠿⢤⣦⣠⣾⣛⠟⠀⠔⠀⣰⣿⡾⠁⡠⠃⢀⠀⢀⡼⢤⡾⠁⡙⠁⢹⠆⢠⠆⠰⡧⢤⡇⢀⠾⠃⢀⠆
⠀⠀⠀⠀⠓⠛⠉⠉⠓⠒⠢⠤⢀⣀⠀⠈⠓⠻⠤⠾⠣⡿⠀⡔⠁⡠⠁⢠⣾⣿⠟⠀⡜⠀⣠⣿⣭⠏⠀⠌⠀⣰⢥⠾⡥⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠄⣀⡈⠉⠓⠺⠧⠴⠟⢹⡏⢀⠞⠀⣰⠿⣿⠏⠀⠌⠀⣰⣿⡖⡐⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠤⠤⣈⣉⠛⠒⠒⠋⢰⣃⣀⠎⠀⡰⠯⠝⡰⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠤⢍⣉⣛⠛⠁⣀⡰⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡏⢁⠎

⣿⣿⣿⣿⣷⣝⢿⣿⣻⠿⡿⣿⣷⡈⢟⣍⡀⠀⠀⢷⠀⠈⠇⡇⠀⠅⠀⡏⠘⢡⣿⢫⣿⣿⣟⡿⣫⣾⡿⠛⣽⣿⣿⣿⣯
⣿⣿⣿⣿⣮⠣⣅⢍⠻⣧⠀⢀⠈⠀⠀⠈⠓⠀⠀⠈⠀⠀⠂⠀⠀⠀⢰⠀⠁⠘⢠⠏⡿⡹⠟⠐⡵⠋⠠⣈⣿⣿⣿⣿⣿
⣿⣿⡿⣿⣿⠷⢌⠓⢕⠌⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠞⡐⠁⠀⠂⠀⢠⣴⠿⠀⠀⢠⣽⡿
⣿⣿⣿⡶⢌⠛⠦⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⠁⣀⣴⠢⣟⣿⣿
⣿⣯⣯⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠉⢀⣾⣿⣿⣿
⠿⠿⣿⣷⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠟⣛
⣛⣷⣶⠤⠈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄⣺⣿⣿
⣿⣛⡉⢁⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⠛⠛⠙⣿
⣦⣤⣄⣤⣤⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣐⣚⣛⣿
⣿⣅⣬⣩⣬⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⡿⣿
⡿⣿⣿⣛⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣤⣤⣀⣀⣒⣤
⣿⣿⠷⠶⠒⠊⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣻⣿
⣟⣡⣤⢴⢒⠊⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠢⠭⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⠟⠋⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⣀⠈⠽⣿⣮⣽
⣿⢿⣻⣽⣾⠟⣫⠄⣨⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⡙⢿⣿⣿⣽⣿
⠗⣯⣽⣿⡿⣛⡵⠊⠀⡠⠂⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡢⡙⠾⣶⣴⣿⣿⣿
⣿⣿⣟⣋⣘⡉⠄⡤⠊⡠⠂⠞⠀⡴⠁⢠⠇⢀⠀⠀⢀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣄⢢⡀⠀⣀⣙⣿⣦⣶⣿⣿⣿⣿⣿

⠀⠂⠩⢤⣂⡠⠤⢀⣀⠀⠀⠀⠀⠀⠀⠰⣖⠒⠒⠠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⢀⣐⡤⠍⠒
⠀⣠⠊⠀⠀⢣⠀⠀⠀⠈⠉⢠⠄⠒⣄⠠⠌⢳⣌⠓⠲⠄⠀⡉⢣⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠠⢄⣀⣀⣀⠀⠠⠤⣄⠐⠲⡄⠈⠀⠀⠀⠀⡜⠀⠀⠑⢄
⠁⠘⠀⠀⠀⠀⠱⡀⢀⡠⠊⠁⠀⠀⠀⠄⠈⠯⣉⠀⠀⠈⠢⡇⠀⢛⡉⠉⠉⡰⢎⠁⠈⠉⠉⠉⡹⠉⠣⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠑⢄⠀⢀⣌⣀⣀⣀⡀⣀
⠀⢰⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠈⢆⢀⣨⢛⣶⠌⠀⢣⡄⠈⣇⢀⡜⠁⠈⢢⠀⠀⠀⡔⠁⠀⠀⠈⠑⢄⡀⡰⠀⠀⠀⠀⠀⠀⠀⠉⠊⡇⠀⢀⣀⣀⣀
⠀⢸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡤⠄⠒⠈⠉⠁⣉⠬⠷⠦⣤⠶⢴⠇⢉⠝⠑⡄⠀⠑⡤⠊⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⠉⠉⠉
⠀⢸⢠⡤⠤⠤⠤⡄⠀⢤⠔⠊⠉⠀⠀⠀⠀⠀⠀⠈⡖⢒⡐⣏⠪⢊⡯⣭⠗⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢀⠊
⠀⢸⢸⡇⠀⠀⠀⡇⠀⣿⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⣓⣈⣉⡁⣉⣉⣀⣰⠂⢀⠊⠉⠑⠒⠂⠀⠐⠠⡄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠈
⠀⢸⢸⡇⠀⠠⠀⡇⠀⠼⡀⠀⠈⡄⠀⣀⠤⠔⠒⢉⣉⡈⠤⠤⠤⠤⠤⠤⠤⠼⢆⡉⠀⠀⠀⠀⠀⠒⠒⠠⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉
⠀⢸⢸⡇⠠⠁⠀⡇⠀⡄⢣⣀⡔⣗⢯⡤⠒⠚⠭⠀⠀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠠⠚⣳⠀⠀⠀⠀⠀⡎⠉⡝⣏⠉⠙⠉⠉⠙⠉⠉⢹⠉⠉⢹⠉⠉⠉⠉⠙
⠀⢸⢸⡇⠁⡀⠀⡇⠀⡃⠸⡩⠷⠳⠋⠀⢈⡝⢉⠄⡶⣞⠛⠃⠀⢀⠶⡖⣶⠚⣞⡿⠉⠉⠀⠀⠀⠀⡇⠀⢣⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠰⡀
⠀⢸⢸⠇⠘⠀⠀⡇⠀⠃⠀⡗⢉⡙⡇⠰⠋⠀⡈⢸⡟⡹⠀⠀⠀⡆⢸⡟⢹⠀⢸⡇⣀⢀⡀⣀⣀⡀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⡤
⠀⢸⢸⡆⠀⠀⠀⡇⠀⠃⠀⠃⠀⢼⡙⢾⣄⠀⠘⠢⠔⠁⠀⠀⠐⠌⠒⠔⠁⠀⢸⢆⠀⡅⢁⡈⠀⠁⣧⣀⣼⣧⣀⣤⣀⣀⣤⣀⡀⠸⠀⠀⠸⢄⣀⣤⣀⣨⠄
⠀⢸⠸⣆⣀⣀⣠⠇⠀⣄⣀⣀⣳⡦⢡⣆⠀⠀⠀⠀⢰⠦⠶⡶⢶⡶⡾⠀⠀⠀⡾⠛⠠⠆⠀⠁⡈⠀⠀⢹⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⡀
⠀⢸⠀⢠⠄⠤⠤⠄⠤⣤⠠⠄⣤⠷⠍⠐⠳⣄⠀⠀⠀⠳⢄⣀⣠⠝⠁⠀⢀⡜⣁⣩⡶⣶⠤⠤⡗⠒⠲⣁⣤⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⢸⠀⠀⡎⠉⠁⠉⠉⡏⣿⣍⠝⡿⠛⠟⠛⠞⢻⡖⠲⢤⡠⢤⡤⠶⠒⠛⠛⠛⠉⠁⠔⠋⠉⠁⢹⠀⠀⢸⣿⡉⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠀⠇
⠀⢸⠀⠠⡇⠀⠀⠀⢐⣧⡗⡩⠋⠙⠚⢲⠞⠉⠉⠀⠀⣤⣄⣛⡶⠾⠃⠀⠐⡎⣁⣀⡀⣀⠤⠤⢼⣀⣾⣾⠽⣿⠭⠤⡬⠤⡬⠤⡤⠤⡥⠤⡤⠤⢬⠥⢽⠀⡂
⠀⢸⠀⠇⡇⠀⢀⠀⢸⠏⡌⠀⠠⠤⢴⣩⠀⠀⠀⠀⠀⢠⣄⣀⣧⢤⡶⠀⠀⢹⣠⠊⠐⠚⠢⠤⠜⠂⠤⣼⠀⢸⠀⠀⡀⠀⡇⠀⡇⠀⡇⠀⡆⠀⠀⠀⠀⠀⡆
⠀⢸⠀⠀⡇⠀⠂⠀⠘⢳⣜⠢⣀⣀⠴⠊⠉⢓⡤⠔⡒⠈⣻⠁⢹⠀⢀⡀⠀⠀⢫⠀⠀⠀⠀⠰⠀⠄⢀⢹⠀⢸⠀⠀⡇⠀⡇⠀⡇⠀⡇⠀⡇⠀⠀⠀⠀⠀⡇
⠀⢸⠀⣂⠧⣀⠠⠠⢰⣧⡧⢙⡾⠁⣠⢀⡴⠋⠀⠀⠨⢥⠾⠦⠬⣽⠛⠁⠀⠀⠀⢳⡁⠉⠀⠀⠀⠀⠀⣸⠤⠼⠤⠤⠦⠤⠧⠤⠧⠤⠧⠤⠧⠤⠴⠤⢤⢀⡃
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⠛⢻⠁⠂⠐⠂⠈⠀⢀⣀⣀⣰⣤⡶⠀⠀⢀⡠⢏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⣉⣀⢸⡥⠐⣒⡉⠥⠒⢏⠢⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡭⣉⣉⡉⠩⠭⠤⠤⠒⠖⡋⢉⡁⠀⠀⠀⠀⠑⢌⠳⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠃⠀⠀⠀⠀⠀⣠⠴⠿⠍⡁⠀⠀⠀⠀⠀⠀⠀⠙⠢⡙⢤⡄⠀⠠⢄⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠈⠳⢝⣦⠔⠁⠀⠀⡠⣻⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⡇⡌⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⠀⢀⡝⠀⠀⢀⢎⡕⠁⠀⠀⢸
⠀⠀⠀⠀⠈⠉⠉⠀⠘⠁⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠙⠢⢄⢀⡠⠊⠀⢀⠔⡵⠋⠀⠀⠀⠀⢸⣀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢰⠀⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⣀⠔⡣⠊⠀⠀⠀⠰⠤⠆⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣀⣀⣀⣠⡤⠾⠤⢀⣉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠴⢲⣯⡃⠀⠀⠀⠀⠀⠀⢀⡠⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⣀⠤⠠⣄⣠⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠛⠊⠉⠀⠻⡍⣅⣀⣀⣀⣀⣀⣐⡶⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢘⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⢳⠀⠀⡠⠄⢿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠸⡆⠉⣹⠀⡜⠀⠀⠈⡦⠊⠈⠱⡀⠀⢠⠊⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠻⠖⠁⠀⠈⢢⠀⢰⠀⠀⡀⠀⡇⠀⣊⠑⣠⣎
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢢⠀⡀⠀⠀⠀⣄⠀⠳⡏⢀⠀⡇⢠⠃⡔⠞⠉⠀⣁⠀⢀⠐⠂⠲⡄
⠀⠀⠀⠀⡠⠤⢤⣸⠤⡴⠁⠇⠁⢄⠀⠀⡏⡧⡀⠱⡟⢠⠇⡼⠀⡧⠂⢀⠀⡏⠀⢘⢄⣀⡀⡇
⠀⠀⠀⢸⠀⠀⢠⢻⢢⢨⠏⠀⣠⣀⠑⠲⢇⡇⠱⣀⡅⡘⢀⡧⠊⣣⢔⣋⣰⠥⠆⠀⣉⠈⣨⠁
⠀⠀⡔⠉⢂⣤⠎⣤⣀⡉⠀⠠⣇⠀⠑⠢⠴⣑⣤⣶⡿⠟⢍⠉⠭⠤⠳⠤⢤⠤⠄⣼⡀⠀⠈⡆
⠀⢸⠁⠀⠀⠀⠀⢀⡨⠐⠒⠛⠊⠀⠀⠈⢉⠽⠫⡀⠕⢶⡰⣉⠒⠧⢀⣀⡎⠦⣀⣡⡭⠦⢼⠃⠈⢆
⠀⠈⢇⠀⠀⠀⢀⡎⠀⠀⣼⡤⠤⣔⡲⠚⣣⠊⢠⢃⠘⡆⠉⠊⠑⢦⡀⠈⠉⠉⣵⠋⠐⢄⠀⠀⠀⡸
⠀⠀⠈⢆⠀⢰⢡⢣⠖⢉⠭⠔⠈⠁⣠⡞⡡⢤⠃⠈⡄⠘⡄⠀⠘⣄⣙⣯⣐⠒⣿⡄⢀⣸⡄⡀⣰⠁
⠀⠀⠀⠘⡗⢶⢸⡇⠀⠀⠀⠀⣠⢊⠎⡜⠀⠀⡇⠀⢱⠀⠘⢄⢰⠋⠀⠈⡆⠀⠀⠀⠀⣇⡇⣧⠇
⠀⠀⠀⠀⡇⠈⠄⢷⣄⠀⠀⠀⣧⡇⠀⢧⣀⣠⠇⠀⠈⡆⠀⠈⠛⣦⣀⡴⡃⠀⠀⠀⠀⠉⢀⡏
⠀⠀⠀⠀⡇⠀⠰⡆⢹⠓⠂⠉⢸⡇⠀⠀⠉⠒⠈⠐⢄⢸⡀⠀⠀⠈⠉⠀⠑⢦⠀⠀⠀⢹⠋⠇
⠀⠀⠀⠀⠃⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠸


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠉⠩⠓⢄⠀⠀⠀⠰⠚⡀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠉⢲⣾⣿⡂⠈⠈⠀⠀⠀⠈⢉⡀⠀⠀⠀⢀⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡟⢈⠠⣀⣤⣶⣶⣆⠀⠀⣤⠀⢴⣿⣿⣧⡀⢀⡀⠴⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣤⣤⣿⣿⣿⣿⠏⠀⠀⣿⡄⠘⠻⣿⣿⠃⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⡧⢸⡇⠀⢠⣿⣿⣷⡐⠠⢄⣠⣿⣧⠀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠇⣴⣿⣿⡁⣸⣧⠀⠀⢿⣿⣿⣿⣷⣾⡿⠀⣹⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⠋⢰⣿⣿⡿⢻⣫⣿⣄⡀⠈⢿⣿⣿⣿⣿⣾⣿⠟⢻⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⠀⠈⢿⣿⣿⣿⣷⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠘⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⢀⣴⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⢠⣿⠋⢿⡇⠈⠩⠁⠀⠀⣠⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⢿⣿⣷⠀⠙⠻⠆⠸⠿⠦⠀⠀⢀⣶⡿⠟⠁⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠀⠘⠛⠛⠳⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⡀⢠⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⣈⡰⠃⠀⠀⡀⠀⠀⠠⡂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠱⠀⠀⠀⢣⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠗⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⣰⡄⠀⠀⠈⠆⠊⠚⠲⡄⠀⠀⠀⠀⠀⢀⣤⣶⣲⣦⡈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⡀⠀⣤⡂⠀⣄⣀⠀⠸⣆⠀⠀⡀⠀⠁⠀⠀⠀⠦⡀⠈⠐⠆⠀⠀⠀⠀⠀⢻⠁⠂⠈⠙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⠀⠀⠀⠐⡄⠸⠇⢈⠉⡉⠀⢀⣀⠀⠀⠀⠀⠀⢃⠀⡄⢠⠀⡈⠉⠽⠀⠀⠸⠷⠶⠴⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠒⠉⡀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠴⠤⢯⢀⡐⢤⠀⠤⠊⠕⠐⠙⠋⠂⢰⠠⠰⠂⠀⢐⣤⠠⡛⢃⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠛⠉⠉⠀⠠⡽⠶⠤⣀⠀⠀⠀⠀⠀⠉⠀⣱⣒⣒⣲⡾⢄⣀⡤⠋⢀⡷⠤⣰⠋⠀⠀⠁⠋⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡇⢀⣟⣋⣙⣊⢹⣷⣤⡀⠀⠀⠀⢠⡶⠀⢉⠀⠀⠀⠀⠀⣚⣉⣉⣭⡶⣤⢄⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡅⢨⡭⠥⠤⠤⠶⠄⠀⠀⠀⡰⠀⠸⠇⣄⡙⢃⣠⠀⠀⢡⢠⠤⠄⠠⠴⠆⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠐⠒⠒⠒⠒⠒⠒⠒⠒⢶⡀⠠⠤⠛⠀⢠⡀⠀⠛⠤⠀⠀⠀⠀⠉⠀⠉⠉⠑⠲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⠀⠤⠤⠤⠤⠧⠆⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⢠⡀⠀⠀⢸⠀⠀⠀⡇⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠈⡇⠀⠀⠸⠀⠀⢀⠃⢈⡄⠀⠀⠀⢸⡄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠀⠀⠀⢸⠇⠀⠀⡇⠀⠀⡄⠀⠀⢸⠀⠀⡇⠀⠀⠀⠈⡅⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⡇⠀⠀⢠⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣾⠀⠀⠀⢰⠴⠄⡷⠤⠤⡆⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣿⡄⠀⠀⠣⠠⠤⠃⠄⠀⠈⠀⠀⡇⠀⠀⠀⠀⡁⠄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡔⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣸⣀⣄⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠠⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⢌⠀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣀⣆⣀⣀⣹

⣿⣿⣿⣿⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠈⠀⠀⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠛⠛⠛⠛⣻⠛⠛⢻⡟⠛⠛⢻⡟⠛⠛⢻⠛⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⡇⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣤⣼⡁⢀⣠⣼⡀⠀⣠⣼⣇⠀⣠⡼⣧⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣯⠀⢀⡿⢾⡁⠀⣹⢸⡇⠀⢹⠖⣇⠀⢈⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣈⣛⣋⣁⣈⣛⣛⣋⣀⣙⣛⣋⣀⣈⣛⣋⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠣⠤⠔⢑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠒⠁⠀⢀⡀⠀⠈⠢⠤⣄⣀⠖⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡔⠁⠀⠱⠀⠀⡴⢪⡇⢸⡱⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⡀⠀⠘⠤⠤⠊⠀⠀⠳⣄⣘⠖⢤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⢈⡆⠀⣀⡀⠀⡴⢊⡶⠝⠀⠁⠉⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠕⠈⠉⠁⠘⠵⢮⠀⠈⠆⠂⢀⠅⠀⡎⢉⠆⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⡐⠉⠑⠀⠀⡎⠀⠀⠀⠀⠈⠑⡄⣈⡁⡠⠬⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠤⠀⢠⠃⠀⠐⠤⠔⠁⢀⢉⠆⠀⠀⠀⢀⡀⣍⠀⠑⠁⠀⠀⠀⢠⠤⠋⠑⠊⡆
⠀⠀⠀⠀⠀⡌⠜⡠⡾⡍⢑⠀⡤⠤⣀⡅⢔⢄⠀⠀⠀⠘⣃⡬⠀⠀⠀⠀⠀⠀⡴⠀⢎⢉⠆⢨⠃
⠀⠀⠀⠀⠀⢣⠑⢴⠁⠟⠀⠉⠀⠀⠀⠀⠈⠪⡒⠀⠐⠂⢸⡆⠒⠤⣀⣀⢔⡲⠀⢇⠤⣀⡔⠚⢠⢖⣄
⠀⠀⠀⠀⠀⠀⠱⡀⢸⢈⡑⠠⢐⡩⠃⠀⠀⠀⠀⠀⠘⠀⢸⠁⠘⠐⠒⠒⠁⠀⠀⠀⢔⠤⡀⠀⠳⣰⡃
⠀⠀⠀⠀⠀⠀⠀⢃⡼⢠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠆⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⠭⠥⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡳⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢐⠠⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⣉⡡⠂
⠀⠀⠀⠀⠀⠀⠀⡠⢃⠄⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⡄⢈⢹⠰⡄
⠀⠀⠀⠀⠀⠀⣎⠔⠁⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠘⡈⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠃⠀⠈⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣴⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡶⠟⠓⠂⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣷⣄⠀⠀⡈⠀⠀⠀⠀⢠⣴⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠄⠀⠀⣀⠀⠀⠀⠈⢿⠉⠉
⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠈⠹⠟⠉⠉⠁⠀⠀⠀⠙⠉⠓⢶⣦⣄⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠾⣻⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⠁⠀⠀⡾⠛⠋⠀⠀⠀⠀⠀⢼⣷⠾⠋
⠀⠀⠀⠀⠀⠀⠿⠃⢠⣶⣄⠀⠀⣀⠀⠀⠀⠀⢀⢄⣐⣔⣢⣄⠀⠀⠀⠀⠘⡄⠒⠀⢷⠐⠀⠀⠀⢀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⣸⣿⣿⣿⡟⠀⠀⠀⠀⠀⣰⠀⠀⡠⠤⢤⣴⠟⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢘⠿⠃⠀⢈⠀⠀⠀⠁⠀⠈⢰⣿⣷⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠂⠭⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠉⠀⢐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠘⠀⣼⣄⠀⠀⠀⠀⠀⠀⣠⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢶⡄⠀⠀⠺⠟⠉⠁⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢁⣠⣿⡀⠀⠀⠀⠀⣀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⢀⠀⣤⡐⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⣻⣧⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⢿⣿⢛⣿⣿⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠊⠻⠿⢿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⡷⠀⠛⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣄⣈⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈
⠀⠀⠀⠀⢤⣾⣔⡀⠀⠈⠀⠀⢀⡤⠒⠀⠀⠀⠀⠀⠀⠀⡰⠶⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⢀⠄⠀⢲⣿⣿⠁⠀⠀⢀⡄⠉⢳⠆⠀⠀⠀⠀⠀⠀⠀⢁⠀⠹⠿⠈⠛⠛
⠀⠀⠀⠀⠀⠻⣉⠀⠀⠀⠀⢰⢆⡟⡸⠑⠄⠀⠀⠀⢀⠔⣽⠶⠛⡄⠀⠀⠂⠀⠀⠀⠠
⠀⠀⠀⠀⡀⠸⠃⠀⠀⢷⡀⣾⣾⡀⠥⠒⠂⠀⠀⢠⠏⠺⠉⠁⢀⠅⠀⠈⠀⠀⡀⠻⠘⢷⣤⡀
⠀⡀⠀⠀⡀⠀⠀⠀⠀⠶⠧⠭⠽⡀⡤⠵⠀⢀⣰⡎⢀⡂⠊⢄⡀⠀⠀⠀⠀⠈⠋⢀⡈⠉
⠀⡖⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢻⣿⠀⡟⣜⠲⠋⠁⠀⠀⠀⠀⠀⠀⡀⠁
⠀⡀⠘⡀⠀⠀⠈⠓⠠⠄⠈⡛⡅⠀⠄⠐⡖⠈⠛⠻⣗⣻⡆⢀⡦⠀⠀⠀⠀⠀⣾⣶⣤⡤⡄
⠀⠀⠐⠄⠀⠐⠂⠀⠀⠀⠀⠘⠴⡄⠘⢿⣷⣄⠀⠀⠀⠀⠈⠁⠁⡀⠀⠀⠀⠀⢙⠁⠀⠀⡤
⠘⡇⠀⠐⠀⠀⠠⠀⠀⡂⠀⠀⠆⢀⠀⠀⢙⣿⡟⠀⠀⠀⠀⠀⢳⡇⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀
⠀⢰⡀⠀⠣⠀⠀⠘⠀⢈⣀⣐⠀⠘⢣⣤⣾⠜⣀⣦⣤⣤⣤⣤⣜⣃⣀⣄⣄⣼⣿⣃⣀⣀⣈⣿⡇
⠘⠓⠀⠐⠠⠀⠤⠈⣤⡬⠟⠃⠀⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡛⠛⠛⢻⡟⠛⢻⡟⠛⠛⠛⣿⠛⠛⢻⡟⠛⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣾⠃⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢠⡿⠀⠀⢸⣇⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⢸⣧⠀⠀⠀⣯⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⣿⣷⠀⢠⣾⣿⣦⢠⣾⣿⣦⢰⣿⣿⡆⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠙⠛⠋⠀⠈⠻⠿⠋⠈⠻⠿⠋⠈⠻⠟⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠤⠄⠦⠀⠀⠠⠴



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿

⠀⠁⠀⠀⠀⠀⠈⣄⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣀⢀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡓⢀⠀⣀⠀⠈⠐⠄⠀⠀⠈⠁⠠⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⡦⠉⠶⠀⠠⠆⠀⠐⠠⣀⣀⡄⣀⠀⠀⠂⠀⠀⢠⣶⠀⠀⠀⠀⠉⠑⢀⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⢰⠆⡀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠈⢹⣿⠃⠀⠀⠀⠀⠀⠀⠈⠂⢄⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⢀⣼⠆⠀⠀⠀⠀⠁
⠀⠀⢀⡀⠰⠀⠀⠀⠈⠲⣄⣀⣄⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠘⠧⠄⠀⠀⣌⠁⠀⠀⠀⡀⠀⠀⠊⠳⡄⡀⠀⠀⠀⠒⠐⠒⠄⠀⠀⣿⡿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠙⠒⠂⠄⠀⠀⠴⠆⠀⠀⡀⠀⠁⠘⠀⢀⡀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⢀⡈
⠲⡄⢀⠀⠀⠀⠀⠢⢄⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠋⠂⠀⠀⠀⡸⠁⠀⠀⠀⠀⠒⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡈
⠂⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠠⢲⡄⠀⠀⠀⠀⠀⠀⠀⢁⣠⣄⠀⠄⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⠀⠰⠠⢀⡀
⠀⠀⢨⠅⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⠖⠒⠈⠀⠈⠃⢉⠀⡆⠀⠀⠀⠀⣴⡁⡄⠀⠉⠀⢤⠀⠠⢤⠀⡀⠀⠀⠀⠀⡀⣠⠠⡌⠆⠀⠀⠀⠀⠘⠋⠲⠆⠤⢀
⠉⠐⢒⠀⠀⡀⠀⠘⠠⠄⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠂⠈⠀⣀⡤⣖⠀⠠⠼⣿⡇⠀⠀⠀⠀⠀⠳⢠⡄⣀⠉⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⠀⠄⠔⠼⠆⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠰⠖⠐⠦
⠀⠤⠀⠀⠀⠀⠀⠀⠀⡀⠌⠀⢀⡄⠶⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠂⢈⠅⠀⠀⠀⠀⠀⠀⠀⠈⠘⠳⣾⣿⠁⠀⣧⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠖⢀⠠⠤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠄⣀⠄⠰⠐⠈⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⢰⠟⠃⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠀⢀⡀
⠀⠀⠀⢀⠘⠣⡄⠀⠐⠀⠀⠀⡆⠀⠀⠀⠈⠛⠃⠀⠀⣡⡄⠀⡠⠒⠈⠉⠢⠀⠀⠀⠀⠘⠠⠀⠀⠉⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠄⠀⠁⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠷⠀⠀⣄⠠⡀⠀⠀⠀⠠⠀⠘⠀⠀⠙⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠈⠑⠂⠤⣄⡀⠀⠀⠀⠆⠀⢐⠇⠀⠀⠀⠀⢀⠠
⠀⠀⢀⣀⠀⠀⠀⠀⠈⠢⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⠐⡖⡀⠀⠀⠀⠀⠀⠀⠉⠓⠄⢀⡀⠀⠸⠀⠀⠘⠠⠓⠈⠇
⠘⢶⠿⡿⠃⠀⠀⠠⡄⠀⢈⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡀⢀⡰⢣⠁⠀⢀⡠⠋⠀⠈⠄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⢂
⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⢻⡉⠀⠀⢀⡤⠂⠠⠀⠀⠀⠀⢀⡼⠀⠀⠀⠖⠈⠀⠀⠙⠀⠉⠁⠀⠀⠀⠀⢰⠇⠀⢀⠀⠀⢄⠀⠀⠀⠀⠜⠆⠘⢾⣿⡿⠶⠃⢸
⠀⠀⣷⠏⠀⠀⠀⠘⡇⠀⠀⠀⠀⣠⠋⢠⡀⠀⣠⢀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠁⠀⠀⢀⡰⠗⡀⠀⠀⠀⠀⡄⠀⠀⠀⢹⡄
⠀⠀⠹⢈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠄⠀⠙⢦⡉⢅⠀⡀⠀⡠⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢌⠁⢄⠑⠀⠀⢀⣤⡇⠀⠀⠄
⢀⠀⠨⡆⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡃⠓⠉⠀⠀⠀⢂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠐⠀⠀⠀⠀⣈⠀⠁⠀⡄⠀⢀
⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠀⠀⠀⠀⠀⠴⢶⣄⣰⠃⠀⠀⠀⠀⡄⠀⠠⣶⣧⣴⡀⠀⠂⠀⠸⠄
⠀⠀⠀⢠⠀⠀⠀⠀⢰⠄⢤⠂⠀⠴⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⢠⠀⢀⠀⡀⠀⠀⠈⠻⢿⡇⠀⢀⢴⠆⠀⠀⠀⠉⠁⠀⠀⠀⠁⠃⠸⠂⠐
⠀⠀⠀⢸⠀⠀⠀⠀⠆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠔⠁⠀⠾⢶⣾⡄⠀⠀⠀⠀⠀⢀⠕⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠂
⢴⡀⠀⢨⡀⠀⠀⠐⡟⠀⠀⠀⠀⠀⠀⠀⠦⠀⣀⠀⠀⠒⠁⠀⠀⢠⠞⠀⠀⢀⡀⠀⠛⠉⠁⠀⠀⠀⢔⠔⠁⠀⠀⠠⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡠⠠
⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠋⠀⢀⠀⠀⢀⠔⠀⠀⠀⢀⡞⠝⠁⠀⠀⠀⠀⢀⠖⠈⠀⠀⠀⠈⢠⡞⢨⡄⠀⠀⠀⠀⠤
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡔⠁⠀⠀⠀⡀⡉⠄⠀⠀⠀⢀⠚⣶⠋⠀⠀⠀⠀⠀⠘⠆⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⢇⡀⠀⠀⠀⠀⠀⠉⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠰⠂⠀⢠⠊⠊⠀⠀⠀⣾⣾⣃⠀⠀⠀⡀⠀⢀⠄⠀⢀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠚⡰
⠀⠙⠦⡀⣀⠀⠀⠀⠀⠀⠀⠁⢰⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡐⡁⠀⠀⠀⠀⠀⠘⠉⠟⠀⠐⠉⢷⣶⣯⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦
⠄⠀⠀⠀⠁⠀⢀⠀⠀⠀⢠⠞⠀⠳⠰⣀⠀⠀⠤⠘⠀⠀⠀⡼⠋⠀⢀⣤⡀⠀⠲⠀⠠⣀⠀⠀⠀⠈⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠈
⠰⠂⣀⡀⠀⣄⣈⠈⠀⠀⡁⠀⠀⠁⠀⠈⠃⠀⠀⠀⠀⠀⠚⠀⠀⣀⣹⡟⠳⣀⠀⠀⠀⠈⠉⠉⣒⣂⣄⣀⣀⠀⡒⠂⠀⠀⠀⣀⠀⡀⣈⠐⠀⡀⠀⠀⠂
⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⡇⠀⠀⢠⠁
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣷⠀⠀⠂
⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⣿⠀⡌
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣿⣰⠁⠀⠀⠀⠀⠀⠀⣀⣄⣠⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠘⡆⠀⢸⠇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⣀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⡀⣵⠀⡟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏
⡟⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣀⣶⠀⠰⣿⣿⣿⣿⣿⡆⠀⠠⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡆
⡇⠈⢃⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣇⣰⣇⠀⠀⣾⣿⣿⣿⣿⣿⣇⠀⠈⢿⣿⣿⣿⣿⣾⠃⠀⠀⠀⠀⠙⠻⠿⣿⡿⠿⠋
⣷⣤⣾⣆⣠⣆⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡇⠀⠘⢿⣿⣿⣿⣿⣿⣿⠋⠀⣿⣿⣿⣿⣿⣀⣤
⣿⣿⣿⣿⣿⣿⡆⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣶⠇⠀⢻⣿⣿⣿⣿⣿⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣄⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣷⣤⡤⠀⠹⣿⣿⣿⣿⣿⣿⣿⡀⠀⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡿⠛⠁⠉⠉⠉⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⠛⠛⠛⠛⠛⠉

⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⣬⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠩⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⢐⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠠⠐⠈⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠠⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠰⠁⠀⠐⠊⠀⠈⠀⠈⠁⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠠⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⡠⠊
⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣽⣽⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣷⣾⣿⣷⣶⣶⣶⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣹⠛⠂⠀⣤⣄⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠛
⠀⠈⠀⠘⠋⠎⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠰⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢠⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⡁
⠠⠤⠄⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⡄⠐⣂⠀⠀⠀⠀⡀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠂⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢠⠒⣶⠀⠀⠀⠀⢰⢺⠀⠛⠀⠀⠀⠀⠊⠉⡀⢨⠀⠀⠀⠈⢻⠈⡃⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡗⠀⠀⣤⣤⣀
⠀⠘⣄⠨⠀⠀⠀⠀⢈⡤⣶⢷⠤⠀⠀⡀⢴⡺⢿⠈⠓⠄⠀⢰⡞⡷⠳⠦⠀⡤⢚⡞⠓⠄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃⠀⢰⣿⣿⣿⠀⠀⢰⣶⣦
⠀⡎⠡⠁⢉⠀⠀⢠⢀⠀⠀⠀⠀⠀⠈⡂⠀⡅⠁⠀⡀⠀⠀⠸⠘⠡⠀⠁⡆⠱⠈⠁⡆⡘⠀⠀⠀⠀⠀⠀⣰⣤⣤⡀⠀⠀⠀⠀⠉⠁⠀⠀⠾⠿⣿
⠀⠇⠀⢀⠀⢰⠀⠀⠜⢀⢀⠴⠀⡆⠀⢀⠀⣀⡤⡀⠋⡔⠠⢰⣶⣠⠀⣰⠃⠸⣼⠂⣰⠁⠀⠀⡀⠀⠀⠠⣿⣿⣿⠃⠀⢠⣿⣶⣶⠀⠀⢀⣀
⠀⠸⠯⡏⢀⡎⠀⠀⠄⠻⠾⠁⣼⠁⠀⠀⠉⠉⡝⢁⡜⠀⠀⠀⢠⠊⡰⠃⠀⢀⢦⣞⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠺⠿⢿⠇⠀⠀⣾⣿⣿⠇
⠀⠂⣴⣠⠞⢀⠀⠀⢀⠰⢇⡼⣁⡀⠀⠀⠀⣈⠤⠎⣀⡀⠀⡀⠓⣩⠁⣂⢠⠀⢰⠀⠡⠀⠀⠀⠀⠀⢠⣿⣶⣦⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⠛⠛
⠀⡆⠲⠉⠉⠉⠀⠀⢸⠉⢺⡀⠘⠀⠀⠀⠀⠱⢰⡄⠙⠆⠀⡇⣀⠻⡄⢜⡜⠀⠆⠀⣘⠦⠀⠀⠀⠀⠸⠿⢿⡏⠀⠀⣾⣿⣿⡇⠀⠀⣤⣄⣀
⢠⠇⢠⢣⠘⣦⠀⠀⡿⠀⢀⢣⠳⣄⠀⠀⣆⢸⡄⠘⠀⠢⣀⡀⠈⠃⢳⠀⠱⡀⠸⠀⠳⡄⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠛⠛⠟⠀⠀⢰⣿⣿⡿
⢠⣄⢸⠀⢀⠚⠉⡦⣇⠘⡏⠀⢄⠈⢱⢀⠋⠈⡢⠀⠠⡀⠡⠑⠈⠁⠀⠑⡐⢰⠀⠀⠀⢘⡃⠀⠀⠀⠀⠀⠲⣤⣾⣿⣿⣷⣶⡄⠀⠈⠉⠙⠃
⠈⠃⡘⠀⠀⠀⠆⠀⢠⢠⠀⢠⠄⢱⣀⠀⠉⠐⠁⠀⢰⠀⡄⣄⢒⡀⠀⠀⣥⠛⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⠶⠄
⠀⣘⡃⠀⠀⠀⣤⡆⣰⣸⡀⠀⠀⢸⡿⢈⣄⣛⠀⠀⠀⠀⣶⠏⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣭⣭⣭⣤⣤⣤⣬⣼⣯⣭⣤⣤⣤⣬⣥⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣤⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⢃⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣀⣼⡯⣽⢟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡚⠈
⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⢆⣀⣠⠄⠀⠀⣀⢤⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡃⠀⠀⠈⠟⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠈⣿⣫⣵⠊⠉⠈⠀⢀⡠⠰⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠘⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠙⢿⣿⡇⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠚⠉⠻⣯⠃⠀⠀⠉⠀⠃⠀⢄⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⣄⠈⡁⠀⠀⡀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢽⡁⠀⠀⢠⡟⠃
⠀⠀⠀⠀⠀⠀⢀⠊⠉⠒⠄⣀⠀⠀⠀⢀⠊⠀⡾⠀⠀⠀⠀⠀⢰⣿⡉⢹⣄⠀⠸⣿⢠⣶⣽⢿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⣰⣤⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠑⣾⠋⢐⡼⠁⠂⠀⠀⠀⠀⡼⡟⣿⠘⣿⡇⢸⡟⣤⣿⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⢀⢿⠃⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠐⠀⢀⢀⠀⠈⢀⡞⠀⠀⠀⠀⠀⠀⢰⠃⢱⣿⠀⣷⢹⣼⠁⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⢰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢘⢐⣾⣏⠀⣿⣸⡷⣸⡙⢿⡿⣿⣇⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⢀⣯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠐⠙⣷⣿⣿⢋⣴⣀⣼⣿⡶⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡌⠉⠽⢃⣵⣮⡹⣿⠀⠀⠀⠀⠀⠀⠀⡀⠈⠃⠀⠀⢀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⣼⡿⣷⣀⢠⡔⠻⠿⠃⣾⠀⠀⠀⠀⠀⠀⣰⣼⣦⠀⠀⣱⣿⣿⠁⠀⠀⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⣿⡿⡈⠃⢠⣶⣿⣿⠀⠀⠀⠀⠀⣠⡯⠀⠀⡄⠋⡀⠈⡄⢠⡄⣤⣤
⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⢠⠇⠀⠐⠀⠀⢀⠄⠈
⠀⣿⣷⣦⣴⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠘⠁⠈⠀⠈⠘⠋⠀⡏
⠀⣿⣿⣟⣟⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⣁⠀⠀⠀⠰⠂
⠀⣿⣽⡟⣏⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡟⠃

⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡈⠉⠃
⠀⠀⠀⠀⠀⢀⢀⠁⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠸⠿⠿⣿⣼⠛⠿⠃⢸⣿⠟⠦⠾⠿⠃⠀⠀⢀⣿⣿⣧⠀⠀⠀⢸⣿⣿⡀⠀⣸⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣤⣤⣤⣤⠀⠀⠀⣾⡿⠘⣿⣆⠀⠀⢸⣿⢻⣧⢀⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⡆⠀⣡⣿⠀⠀⠀⢸⣿⠉⠉⠉⠉⠀⠀⣼⣿⣷⣤⣿⣿⡄⠀⢸⣿⠘⣿⣼⡿⢸⣿⣿⣏⠘⠃
⠀⠀⠀⠀⠸⡿⠀⣿⣿⢀⡤⠂⢸⣿⣶⣶⡶⣶⡆⣰⣿⢏⢱⣤⠬⢻⣿⡀⢹⣿⠀⢿⣭⡇⢸⣿⡿⠇
⠀⠀⡀⠀⠀⠀⠀⠀⢠⠞⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⢠⣦⠀⠀⢠⣤⠄⠀⠀⣠⣤⠀⠀⣠⣴⣶⣦⣤⡀⠀⠀⣤⣤⣤⣤⣤⣄⠀⠀⣠⠈⣇⡀⢀⣤⣤
⠀⠀⠀⠸⣿⡇⠀⡾⠁⢀⡀⢀⣿⡏⢀⣾⡿⠉⠁⠉⠋⠹⡄⢀⣿⡏⣩⣭⠙⣿⡇⠀⣿⣶⢉⣴⣿⠟⠁
⠀⠀⠀⠀⢿⣷⢸⣶⠀⢿⣧⢸⣿⠁⢸⣿⡇⠀⠀⠑⢄⣾⡇⠀⣿⣷⣾⣿⣿⠟⠃⠀⣿⣿⣿⢿⣷⡀
⠀⠀⠀⠀⠸⣿⣿⡏⠀⠸⣿⣿⡏⠀⠈⢿⣷⣀⣀⣀⣼⣿⠃⠀⣿⡇⠀⠙⢿⣷⡀⠀⣿⣿⠁⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠛⠛⡡⠄⠀⠛⠛⠁⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠛⠃⠀⠀⠈⠛⠛⠀⠛⠛⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⠀⠐⠉⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠃⢀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣤⢚⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⣥⣾⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⠟⠁⣯⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣀⣞⣱⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣱⠾⢿⣟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠁⢘⢈⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣠⠜⣯⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣜⣯⠽⠟⠿⣽⣷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⢹⣿⠀⢀⣿⣿⠀⠘⠛⠿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡯⠟⠉⠀⠀⢸⣏⢐⡜⣿⣿⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡯⠟⠉⠀⠀⠀⠀⠀⢸⡇⠨⡄⣾⣿⠀⠀⠀⣀⣀⣀⠤⠄⠀⠀⠛⠛⢛⣟⣿⡇
⠀⠀⠀⠀⠀⠀⣀⡤⢺⠿⠛⠁⠀⠀⣀⣀⡀⣤⣤⡤⠜⠳⠔⠚⠋⠉⠉⠁⣂⣈⣀⡤⠤⠴⠴⠒⠛⢻⣿⡟⠁
⠀⠀⠀⢰⡖⠚⠛⠛⠁⠀⣈⣉⣁⣀⣤⣤⣴⡤⠴⠶⢶⣶⣶⣶⣾⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠓⢾⣿⡟⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⢈⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⣾⣿⠀⠀⠀⢀⣀⣀⣀⣠⡤⠤⠤⢲⣶⢾⢿⣿⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣿⠶⣶⠾⠿⠛⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⢰⡤⣖⣿⢿⣿
⠀⣠⣤⡄⠐⠒⠲⠖⠲⠞⠛⠛⠟⠛⠋⠛⠋⠉⠃⠀⠀⠤⠤⢠⡀⠐⠂⠠⠄⣀⠀⠀⢀⣤⣠⣶⣼⣿⣷⣿⣿⣷⣿
⣸⠿⠂⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠤⡀⠒⠂⠤⠄⠉⠓⣒⣢⣬⣩⣷⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⣿⣟⣓⠲⢚⡤⠀⢤⡀⠀⠀⠠⢀⠀⢀⠀⠀⣠⣬⣭⣿⣶⣤⣿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉⠉⠁
⠻⣿⣿⣷⣻⣿⣾⣧⣵⣶⠶⠿⠶⠿⠛⠛⠛⠋⠉⠉⣿⠋⢛⣿⣿⣇⡀
⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⠛⠉⠉⠉⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣀⣀⣈⣴⣶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⡀⠀⠀⣠⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⡄⠀⣤⣿⣧⣤⣤⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⢿⣿⣦⡾⠋
⠀⠀⠀⠀⠀⠀⠠⠶⣤⣤⣶⣴⣿⣿⣿⣿⡄⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⠋⣁⣤⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡇⣾⣿⣿⣿⣿⣿⣿⣷⣬⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⣡⣶⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⢘⣇⣿⣿⣿⣿⣿⣿⡏⢿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⡴⠻⣿⢿⢿⣿⣿⣿⣿⡧⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⠏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠻⠿⠿⠛⠁⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣠⣄
⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆
⠀⣠⣶⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣄⡼
⠀⣿⣿⣿⣿⣦⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠁
⠀⢐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⢹⣿
⠀⠀⠀⣹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⢿⡇
⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠃⠀⠀⠀⠘⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠄⠐⠖⠓⠆⠶⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢃⣀⣀⣀⠀⢠⣤⠀⠀⠀⠻⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⢾⠛⣹⡛⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⣿⣿⢰⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡠⣤⠉⡟⢡⢤⢠⡤⢤
⠀⠀⠒⢳⡄⢴⠚⠆⠙⠉⠙⢣⡞⢳⠄⣴⠛⠂⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡛⣸⣿⣿⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠠⠄⣟⠠⠀⢀⡤
⠀⠀⢑⣺⣟⣻⠛⡀⠀⠀⢀⣩⡟⢹⣛⣿⡖⠁⠀⠀⠀⠀⠀⢸⡏⢹⣿⣿⡇⠉⠀⣿⣿⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⢿⣰⡖⢈⣿⠐⢶⣸⢷⣖⡀
⠀⠀⠾⠉⣹⣿⣷⣟⠛⣧⣿⣿⣷⣿⣿⡉⠹⠆⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣦⣦⣤⣿⣿⣷⣿⣤⡀⠀⠀⠀⠀⠀⠘⠉⢹⣷⠏⣳⣾⢿⣷⡏⢷⣿⠉⠙
⠀⠀⠀⠈⠉⠁⠲⠞⠁⠠⠚⠙⠷⠂⠉⠉⠀⠀⠀⠀⠀⠀⠻⣿⡟⠻⠛⠟⠛⠛⣛⣛⣛⣛⣻⣿⡻⠤⠀⠀⠀⠀⠀⠈⠛⡛⣿⣿⣿⣼⣿⣿⡟⡛⠋⠁
⠀⠀⠀⠘⠛⠛⠛⠚⣚⠃⠘⢛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⠿⠋⠀⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠗⠆⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢢⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⡀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⢰⣿⣷⣖⠀⢀⡀⢀⣿⠁⡶⢎⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣤⣾⣃⠀⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣧⣄⡀
⠀⠀⠀⡛⡿⠃⠹⢷⡌⠙⡿⢻⡼⠃⠘⢿⠻⡀⠀⠀⠀⠀⠀⠀⠀⡉⣉⣠⡾⠧⣿⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡀⢸⡇⠈⢻⡄
⠀⠀⠀⣦⣴⣤⡆⣦⠙⢷⣴⠋⣰⢰⡦⣦⠶⠁⠀⠀⠀⠀⠀⠀⠀⡇⣯⣟⠀⢠⡇⠀⠘⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣈⣷⣾⣧⢤⡀⡇
⠀⠀⠀⡀⠀⠀⢁⣠⣶⡟⠙⢷⣌⡰⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⠃⠛⢿⡄⣿⣇⢀⣠⣁⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⡿⢩⡀⢸⣿⣷⣶⣤⣀
⠀⠀⠀⠡⣴⣾⣿⣿⡿⠁⠀⠨⢻⣿⣾⣅⠐⠀⠀⠀⠀⡀⢰⠦⠶⠴⠦⠐⣹⣿⣿⣏⠰⠶⠶⠲⣀⡀⠀⠀⠀⠈⠛⢿⣯⣥⠆⠙⢶⣷⣦⢦⣭⣿⠿⠋
⠀⠀⠀⠀⠑⢄⡘⢿⠁⢀⡉⡀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠄⠙⠃⠁⠀⠀⠀⣿⣿⣿⠈⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⢰⣆⣼⣤⡖⣦⡆⢹⡶⣶⢸⣶⢶⠆
⠀⠀⠀⠀⠀⠀⠈⠚⠠⠤⠤⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡤⣷⣮⣵⣾⣦⣽⣶⣵⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠠⡦⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡄⠰⠶⣶⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣀⡺⠇⠸⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⣿⡇⢸⣿⡇⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⢀⣄⣀⡀⡠⢺⡗⢄⢀⣀⣰⡀
⠀⠀⠀⢠⣶⣿⣿⣿⣧⠜⡟⣿⣿⣿⣿⣷⡤⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣿⠇⠸⣿⡇⠀⠀⢀⡴⠃⠀⠀⠀⠀⣾⡿⣿⣿⣧⣷⣸⣇⣘⢼⣿⣿⠿⡷
⠀⠀⠀⠀⢿⣼⣿⣿⣿⡀⡃⣸⣿⣿⣿⣾⠇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠁⠰⡾⠛⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣹⡟⡛⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣿⢻⣿⣿⣷⢷⣿⣿⡿⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣳⠄⠀⠇⠀⢴⣋⣀⣀⡀⠀⠀⠀⠀⠀⠈⣯⣹⣿⣿⠻⣤⣧⣌⣿⣿⣏⣽⠁
⠀⠀⠀⠀⣤⣻⣶⡿⢿⣷⠈⡟⢿⠷⣾⢥⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠻⠚⠃⢘⡚⠛⠿⢿⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣽⣲⣮⢹⠭⣯⣿⢿⣿⡄
⠀⠀⠀⠀⠛⣏⣋⡙⠚⠛⠀⢃⠙⡛⠋⢘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈⠛⠛⠈⠙⠋⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠻⠏⠐⠠⠀⠀⠀⠀⠀⠁⢸⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⣹⠀⠰⠄⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣀⠀⠀⢸⣧⣄⡀⢐⡤⠀⠀⣴⡀⠈⢁⣀⠀⠀⣠⠉⠰⠆⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⡄⠈⡃⠀⡀⠀⠀⢀⠀⢻⡻⠀⠀⠋⠀⠀⣿⣦⡀⠠⠶⠀⠀⢸⣅⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠙⢄⡀⣀⠰⡀⠀⠫⣡⠘⠀⠀⡀⠄⢲⠀⠹⠁⠀⡀⠀⢳⡄⢘⠋⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠁⠀⠐⠁⢠⠳⣄⢳⠐⠉⠑⠀⣤⡑⢦⡀⠐⣆⠀⠨⡑⢧⠠⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⠻⡈⢢⠀⠀⠈⠂⡸⣎⠢⡀⢀⠀⡇⢹⡌⠀⠀⠀⢰⠀⠀⡈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠄⣼⠇⢿⡄⢣⠐⣦⠀⡿⠛⣦⡁⡈⢳⡇⣼⠰⣦⣀⠱⣞⠀⣴⠔⣆⡠⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠐⠃⠂⠈⠉⠀⠠⠘⠏⠉⢀⣉⠀⢀⠂⠷⠙⠀⣙⠁⢀⠘⠂⢙⣀⣉⣄⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢶⢺⣏⢠⡄⠤⣀⢀⡄⠻⠀⠦⠰⢼⠀⠛⢫⡀⣌⡰⡆⢰⡏⢤⠠⠀⢇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⠈⠀⠅⠀⢰⠀⡇⠀⠀⠀⠀⠘⡄⣇⠀⠁⠀⠀⢣⢸⠀⠀⠒⠀⠀⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⡴⣧⠃⠀⢀⡀⠀⢰⢷⠃⠀⢤⣄⠀⡰⠋⠀⠀⢦⠀⣘⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⢃⣠⠄⠸⡀⠀⠠⢈⣄⠁⠈⡆⠀⠀⢡⣤⡀⠹⡀⠀⢀⣼⠁⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢄⠀⠀⢣⠀⠀⢺⡙⠳⠀⠘⠀⠀⢿⡶⣄⠀⠡⣠⠃⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠃⠀⠀⠀⠙⣧⠁⠀⠀⠠⠀⠀⠙⣦⠀⠀⠱⠀⠀⠹⣮⡁⠀⠀⠀⠀⠈⣿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⣠⠀⠀⠀⢘⣷⣄⠀⠀⠀⠀⣰⢿⣧⡀⠀⠀⢀⣴⣿⡛⠆⠀⠀⠀⠀⠈⢮⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡰⠁⠀⠀⢀⣿⣿⣯⠃⠀⠀⢰⠃⢸⣿⡑⠀⢀⣘⢸⣿⠇⠀⠀⢀⠀⠀⠀⠈⠃⠘⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣴⠺⢿⡟⠈⡇⠀⣰⡗⠀⣸⠀⢿⠀⠏⠘⣼⠇⠀⠀⠀⡋⡀⠀⠀⠀⠀⢿⣿⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⢴⠆⠀⠀⠀⡟⣻⠟⠀⠀⠘⠀⠁⠀⡔⠃⠀⠈⢁⣄⢶⣟⠀⠀⠀⠲⡌⢧⠀⣀⡀⠀⠘⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⡄⠀⠀⠀⢀⡜⡇⡏⠀⠀⠀⠀⠲⣿⠹⡄⠀⠀⠀⠀⢹⣆⢻⡶⠂⠁⠀⣽⣦⠏⠁⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣸⠁⠀⠀⠀⠈⠛⢧⡿⠄⠀⠀⠀⠀⠘⣿⡿⡶⠚⠁⠀⠀⠛⢿⠁⠀⠀⠀⠘⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡘⠆⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⣿⠻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠾⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢸⠅⣸⡀⣷⢼⠅⣿
⠀⠀⠀⠀⠀⠀⡏⠀⠙⠁⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⢉⡁⠈⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠙⣆⡀⢾⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡰⠟⠋
⠀⠀⠀⠀⢀⡠⣀⠀⠙⣀⡁⢀⡠
⠀⠀⠀⠀⢸⠀⡿⡁⣧⠏⣇⢸⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣄⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⠆⠙⠚⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⢹⠛⣽⡏⣿⢿⡟⣟⡿⢿⢟⢿⠉⣿⡿⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⣜⣛⣴⣜⣮⣤⣧⣤⣼⣮⣾⣰⣽⣦⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡰⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠙⠛⠛⠛⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠀⣿⡈⠉⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⢟⣳⠶⣞⣻⣿⣿⡿⠂
⠀⠀⠀⡄⣠⢀⣀⢩⡅⡄⣀⢩⡁⣀⡄⢠⠔⠖⢀⣄⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠻⣿⣼⣿⣿⣿⣷⣿⠟
⠀⠀⠀⠃⠉⠈⠛⠂⠙⠃⠋⠐⠙⠉⠃⠊⠃⠃⠋⠈⠁⠙⠀⠀⠀⠀⠀⣀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢉⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⡴⢰⡀⡆⢠⡶⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣏⣹⡉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠁⠈⠁⠁⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⡀⢄⠠⢄⣀⢠⡄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠰⠁⠏⠘⠉⠚⠸⠣⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠈⠻⠟⠑⢦⣠⡶⠂⠠⣦⣶⣶⢆⠀⠀⠀⠀⠀⢠⣶⣀⡀⣸⣀⡀
⠀⠀⠀⠀⠀⢶⣾⠛⣶⠀⠀⠀⠀⠀⠀⢣⠴⣿⠉⠀⠀⠀⢈⡹⣗⠈⠀⠀⠀⠀⠛⣿⣿⣿⣿⡿⠹⢧
⠀⠀⠀⠀⠀⢀⣧⣤⣄⠀⠀⠀⠀⣠⣤⣧⠂⡿⠗⠀⠀⠘⢿⣥⠂⠒⠿⠀⠀⠀⠀⠘⣿⣩⣿⠗⠆⢰
⠀⠀⠀⠀⠸⣿⣿⡘⡿⣁⢀⡤⢺⣿⣿⣿⣿⣿⢰⠟⢀⣴⡆⢰⣿⣿⣿⣿⣿⡷⣾⠛⣼⣿⠏⣴⠚⠁
⠀⠀⢠⣾⣿⣵⠆⣬⠁⠛⢻⠷⡿⣿⣴⠋⢙⣦⠎⣸⣛⠹⢇⠰⢾⣿⣟⣿⣿⢷⣧⣴⣿⣿⣾⡇
⣀⡱⠉⠀⣿⡟⠳⣤⠄⠀⣼⣻⣁⡾⢾⠁⠉⣏⠀⠈⠉⢰⡈⠺⢿⣿⣿⣿⣿⣾⣼⣿⣿⣧⡙⠇
⠀⠀⠀⢸⠿⣏⠀⠀⠀⠀⠛⢻⣿⠃⠀⠀⠀⠈⠀⠀⠀⠸⡇⢘⣻⣻⣿⣿⣿⣯⣿⡿⠙⡿⠃⣀
⠀⠀⠀⢸⢀⠸⠀⠀⠀⠀⠀⠘⠁⢰⡄⠀⠀⢀⠀⠀⠀⢸⣧⣼⣿⠟⠻⣿⡟⠙⠿⣁⠀⢀⣸⡿
⠀⠀⠀⠈⣸⡀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⠀⠈⠀⠀⠀⣼⠿⠃⠾⡀⠀⡿⢡⣄⠀⣵⢡⣿⠏
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢨⠀⣾⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡇⢸⠁⠈⠈⠘⠄⠈⠀⠠
⠀⠀⠀⠀⢸⢡⠀⠀⠀⠀⠀⢸⣇⣿⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⢸⣀⣤⠀⣠⣶⠀⣾⣾
⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠘⠙⡏⠀⠈⡇⠀⠀⠀⠈⠀⠀⠀⢸⠀⠜⣹⢋⣴⠿⣫⣾⣿⡿
⠀⠀⢰⣀⡀⠀⠀⠀⠀⠀⠀⢰⠀⢷⠀⠀⣇⠀⠀⠀⠀⣄⠀⠀⣾⣆⡠⢯⣬⣴⠾⣿⡿⢿⠇
⠀⠘⠋⠉⠀⠀⠶⡶⠀⠀⠠⠾⠻⠃⠀⠀⠙⢿⡀⠀⣾⢻⠀⠀⠹⠿⠵⠄
⣤⣤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣧⣤⣤⣷⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⢀⣴⣿⡿⠿⣿⣦⡀⣾⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣻⣻⣟⣿⢸⣿⣿⡶⢃⣿⣿⡇⣿⣻⣟⣿⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⣿⣿⣿⣿⠘⢿⣿⣿⣽⣿⡿⠃⣿⣿⣿⣿⣿⣿⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⠀⠋⠉⠙⠛⠉⠁⠀⠀⠉⢻⡟⠉⠀⠀⠈⠙⠛⠋⠉⠙⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣿⣿⣿⣿⣿⡟⠻⣿⣷⣄⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⣠⣾⣿⠟⢹⣿⣿⣿⣿⣿⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⣿⣿⣿⣿⣿⡇⠀⠈⠻⠿⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠛⠛⠁⠀⢸⣿⣿⣿⣿⣿⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣴⣿⡿⠋⢹⣿⣿⣿⣿⡏⠙⢿⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⡇⠀⠀⠀⠙⠋⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠙⠋⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠸⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠘⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠂⠀⠀⢔⠉⠀⠀⠢⠄⠀⠀⢀⠄⠀⠀⢀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣉⠔⠀⠀⡄⠀⢀⠖⢊⠀⠉⢀⠀⠀⠀⣀⠄⢁⠤⠀⠀⠀⠀⢁⡀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠀⢸⡥⠂⠀⢀⡼⠋⠀⣞⢡⠀⠀⠈⣱⠦⠐⢤⠆⠁⠀⠐⢠⣀⠠⠘⣥⠂⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠈⠀⠀⠀⡟⠀⠀⠀⡼⠃⠀⠀⣾⠊⠀⠀⢠⠟⠀⠀⢸⡄⠂⠀⠀⣰⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⡁⠀⠀⠀⢠⠀⠀⠀⠘⠁⠀⠀⠀⡀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠐⠀⠠⡅⠀⢀⠀⠀⠄⠀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠙⠀⠀⠀⠀⠁⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢤⣀⣀⠀⠀⠉⠀⠀⡀⠀⠀⠁⠀⠔⠁⠀⠁⠀⠂⠚⠀⢀⠔⠀⠐⠀⠀⠀⠴⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⠀⠀⠈⠉⠉⠐⠀⠀⠀⠀⠀⠲⠤⠀⠄⠀⠐⠀⠀⢤⣀⣀⣀⣀⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⠀⠀⠀⢠⠀⠀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠦⣀⣀⣨⡀⠀⠁⢅⡀⠀⠀⡁⠀⠀⠗⠚⠀⠀⠀⡆⠀⠀⢤⢤⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⣧⠀⠑⠊⠉⠁⠈⠹⡀⠀⠢⡸⠲⠤⠠⢦⡀⠀⢠⠐⣤⣀⠀⣈⡀⠀⠀⠀⠀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠆⠀⡀⠀⠀⠀⢸⢣⠀⠀⠀⠀⠀⠀⢨⣆⠀⠀⠙⠀⠈⠀⠉⢿⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠘⡀⡁⠀⠀⢀⠇⠀⢃⠀⠂⠀⠀⠀⡼⠈⡆⠀⠀⠀⠀⠀⢀⡟⢆⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⠀⠀⠀⢸⠃⠀⠀⣾⠀⠀⠈⢄⠆⠀⠀⢰⠃⠀⠐⡀⢠⠀⠀⠀⡸⠀⠈⡄⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢰⠇⠀⠀⢀⡜⠀⠀⣸⢩⠀⠀⠀⢸⠆⠀⢠⡇⠀⠀⠀⢡⡔⠀⠀⢰⠃⠀⠀⠰⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠘⠀⠀⢀⡏⠀⣇⠀⠰⠁⠀⠀⡾⢹⡀⠀⠀⡸⠃⠀⢠⣿⠀⠀⠀⠀⢇⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢰⠇⠀⠀⠀⣸⠀⠀⣸⠁⠀⠘⡀⢰⠀⠀⢰⠇⠈⡇⠀⠘⠁⠀⠀⣼⠈⡇⠀⠀⠀⠘⡄⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡟⠀⠀⠀⠀⡇⠀⢠⡏⢇⠀⠀⣇⡜⠀⠀⣾⡀⠀⢸⠀⢰⠀⠀⢠⡇⠀⢰⠀⠀⠀⠀⢡⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢸⠁⠀⠀⠀⡰⠀⠀⡼⠀⠈⡆⠀⢸⠃⠀⢸⠃⢣⠀⠀⢇⡸⠀⠀⡼⢇⠀⠀⡇⠀⠀⠀⠀⢦⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⣜⠀⠀⡆⠀⠀⠀⢀⡃⠀⠀⠃⠀⠀⢃⢀⡖⠀⠀⡏⠀⠸⠄⠀⢈⠇⠀⢰⠃⢸⡀⠀⠸⡄⠀⠀⠀⠈⡆⠀⠘⡆
⠀⠀⢤⣤⣒⠂⠉⠀⠈⢹⣃⣀⠤⠄⠚⠒⠢⣼⠀⠀⢀⣼⣹⠇⣀⣰⠀⠀⢀⡴⠚⡛⠀⠀⠘⠀⠀⢁⣠⠶⢿⡀⠀⠀⠀⠻⠀⠀⣙⡄
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒⠒⠒⠒⠓⠶⣮⣉⣀⣀⣀⣀⡿⣀⣴⣋⠠⠚⠓⠂⢤⡆⢀⡴⠋⣀⡴⠋⠀⠀⠀⠀⢀⡶⠚⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠶⠶⠶⠞⠀⠛⠒⠋⠁⠀⠀⠀⠀⢠⣴⣯⣤⠾⡏⣤⢤⣄⢠⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⠀⠀⠀⠠⢤⠄⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠩⠉⠀⠀⠀⢰⠶⡶⠆⠀⠀⢀⣿⣫⠇⠀⠀⢀⣿⣻⡧⠏⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣥⢠⣤⡀⠀⢀⣠⡄⢭⣀⠀⠀⠀⢀⢠⣀⠀⠀⣾⣯⢈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⢸⣿⣷⠀⣾⣿⡇⢸⣿⣇⠀⣼⣿⠀⣿⣷⠀⣿⣿⡈⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢾⣿⣿⣦⣿⣿⣿⢻⣿⣿⠀⣿⣿⠄⣿⣿⡇⣿⣿⡇⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⢿⡯⡤⣿⠿⠈⢿⡿⣿⣼⣿⣿⡆⣿⣿⣄⣿⣿⡇⠿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠀⠀⠘⢊⣯⠉⣈⠃⣿⠋⢐⡨⠿⢇⠸⠋⠑⠋⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠉⠀⠘⠀⢸⣇⡈⣰⠂⠃⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠆⠀⠀⢠⠘⠿⠠⠀⠀⠀⠀⠀⠈⠉⠁⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⢰⣆⡀⠀⡀⠀⠿⠁⡀⠀⠀⢰⣶⢰⣶⡀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠈⠉⠉⠈⠉⠉⠀⠉⠛⠂⠀⠉⠉⠀⠉⠁⠚⠋⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⣦⣶⣶⢶⣴⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢟⢟⠛⡿⣟⡹⣻⢗⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣇⣿⣻⣍⣿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠉⠁⠉⠁⠁⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⣶⡶⠶⢶⣶⠶⠶⣶⡶⠶⣶⣶⠶⠶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⡇⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⡇⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠁⠀⢸⣿⠀⠀⣿⡇⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣿⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿⡀⠀⣸⣿⡀⢀⣿⣇⢀⣼⣧⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣏⣹⣿⢸⣿⣉⣿⣿⣉⣿⣿⣏⣹⣷⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠙⠛⠁⠀⠙⠛⠉⠉⠛⠋⠈⠛⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠶⠶⠶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⠁⠈⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀
⠀⠀⣀⠀⣶⣂⠄⠀⠀⠀⣠⠀⠀⠩⣖⣫⠄⠀⠀⠀⠀⡀⢐⣫⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣮⣍⡒⠁
⠀⠀⢹⡎⠳⣷⣤⠴⠂⣰⡏⠀⠀⠀⢿⢻⡈⠄⠀⠀⠀⠀⠀⢩⣷⠚⠁⠀⠀⠐⠀⠀⠀⠘⣿⣯⣀
⠀⠀⠈⢿⣦⣸⢶⣤⣴⡟⠀⢀⠔⢀⣐⣶⣶⣤⡄⠀⠀⠀⠀⣾⣿⣽⣦⣾⢳⡆⠀⣴⡿⣿⣿⣂⠈
⠀⠀⠀⠀⢻⡷⣽⣻⣫⠀⠀⡆⠰⣷⣿⣿⣿⡿⠈⣑⢀⣤⢸⡿⣿⣿⣿⠛⠿⠃⣾⣿⣱⡏⠳⢹⠊⠰⢄
⠀⠀⠀⠀⠼⣿⣭⣟⣷⠄⠀⢧⡀⠙⢿⣿⣿⢷⠜⡁⠛⠃⢠⡇⣿⣓⣿⠀⠀⢀⠉⣻⣯⡅⠀⣀⡄⠀⠚⠁
⠀⠀⠀⠀⣿⣿⡷⢾⣿⡀⠀⠠⠙⣦⣤⣭⣤⣴⣾⡇⠀⠐⣾⢻⣿⣮⣷⠀⠀⠀⠀⡿⣿⣷⣴⣿⡆
⠀⠀⠀⠀⠁⠙⣝⣿⠛⠁⠀⠀⠘⢻⣟⣿⣿⣿⠟⠃⠀⠀⠈⢮⣿⡏⠁⠀⠀⠀⠀⠀⠙⣿⣿⡀
⠀⠀⠀⠀⠀⢠⡟⢾⡇⠀⠀⠀⠀⠀⠹⣿⣿⠃⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⢰⡟⠙⣷⡀
⠀⠀⠀⠀⢠⡟⠀⠀⢿⡀⠀⠀⠀⠀⢀⣼⣽⡄⠀⠀⠀⠀⢀⡼⠹⣇⠀⠀⠀⠀⠀⢠⡿⠀⠀⠘⣧⡀
⠀⠀⠀⢠⡿⠀⠀⠀⠈⣷⡄⠀⠀⠠⡿⠁⢸⣿⡀⠀⠀⢀⡞⠀⠀⢻⡄⠀⠀⠀⠀⣼⡇⠀⠀⠀⠘⠋⡀⠖
⠀⠀⠀⠐⠓⠒⠂⠀⠀⠀⠂⠉⠁⠉⠲⠀⠀⣋⣁⣀⠀⠘⠡⡄⠀⠈⠠⠒⠋⠀⠀⠐⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠚⢻⡟⠓⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣀⣤⠴⠶⠚⠓⠲⠦⣤⣀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⠏⢀⡴⠒⢦⣤⠤⢤⡤⠤⣤⣀⠀⠀⠀⠙⣄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢰⠃⠀⣸⣇⣀⡼⠃⠀⢸⡇⠀⠀⠈⠙⠲⣄⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡟⠀⢰⠃⠀⠀⠀⢠⠞⠉⠉⠳⡄⠀⠀⠀⠘⡆⠀⢻⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢰⠚⢻⡇⠀⡟⠛⢻⠀⠀⠀⠀⢻⡀⠀⠀⢀⡿⠀⠀⠀⠀⡟⠛⢻⠀⢸⡟⠓⡆
⠀⠀⠀⠀⠀⠀⢸⡄⠈⡇⠀⢻⠀⠈⢧⡀⠀⠀⠀⠓⢶⡶⠛⠁⠀⠀⢀⡼⠁⠀⡟⠀⢸⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⢹⡄⠈⢧⠀⠀⠙⠳⠤⣄⣀⣸⣇⣀⣠⠤⠞⠋⠀⠀⡼⠁⢠⡏⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠹⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢾⠁⣠⠏⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⢦⣀⠈⠓⠦⣄⣀⠀⠀⠀⠀⣀⣠⠴⠛⠶⣋⡴⠃⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠳⢦⣄⡀⠉⠉⢹⡏⠉⠉⢀⣠⡴⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⡀⠉⠙⠛⠛⠛⠛⠋⠉⢀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⡶⠲⠖⢶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠤⠤⠤⢤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣖⣺⠭⠽⠒⠒⣛⣛⣉⣩⣭⣭⣭⣭⣭⣭⣙⣛⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⣴⣒⣒⣶⡿⢛⣩⣥⣴⣶⣶⣿⣭⣭⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢼⣿
⠀⠀⠀⠀⠀⣠⢔⣚⠭⢖⣒⡭⠭⠽⠖⣿⢾⡞⠉⠉⠉⠉⠉⠙⠛⠛⠯⠽⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡿
⠀⠀⢀⡴⣫⢖⣩⠴⠋⠉⠀⠀⠀⠀⠀⣿⡼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⠀⠀⢀⣠⠴⣚⠽⢋⡴⠃
⠀⠀⡼⣸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡝⠮⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡿⢖⣊⡭⠖⢋⣡⠖⠋
⠀⠀⣧⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⣉⠛⠻⠿⠿⣿⣿⣿⠭⠭⠟⠒⠋⣉⡤⠴⠒⠉
⠀⠀⢹⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠶⠶⠶⠒⡶⣲⣿⠋⠁
⠀⠀⠘⣷⡙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣊⠴⣳⠃
⠀⠀⠀⠈⢿⡲⢭⣓⠦⢤⣀⣀⡀⠀⠀⣀⣀⣀⣠⡤⠤⢖⣒⣫⠭⣒⣫⠵⠊
⠀⠀⠀⠀⠀⠉⠓⠮⢭⣓⣒⣛⣛⣛⣛⣛⣓⣒⣒⣺⠭⠽⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠀⠂⠂⠀⠀⠉⠉⠁⠁⠈⠁⠁⠁⠉⠀⠈⠈⠉⠈⠉⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠀⠠⠤⠠⠀⠠⠀⠠⠠⠀⠀⠂⠀⠀⠂⠀⠀⠈⠉⠈⠁⠁⠁⠈⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⠐⠀⠀⡀⠈⠁⡄⢀⠀⠀⠀⠀⠉⠨⣆⠀⠀⠀⠀⠀⡃⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠤⠠⠀⠀⠂⠀⡀⠀⠀⢀⣤⣿⡏⢇⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠰⠒⠂⠐⠀⠉⠄⠀⠐⠀⢀⠠⡤⠈⠁⠂⠀⣀⣰⠂⡘⡦⠍⠘⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣤⣤⣀⣀⢈⠁⠀⠀⠚⠁⠀⠂⠀⠐⢀⠴⢰⢢⠀⠃⡏⠀⠂⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠛⠛⠀⠁⠁⠉⠀⠈⠉⣖⠀⢤⣀⠀⠀⠋⠘⠉⠉⠀⣦⠀⠀⠀⠀⣠⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡌⡇⡀⢀⠀⠢⢅⠆⠀⠽⠃⠀⠀⣸⢭⣀⠄⠸⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢽⣇⣧⠀⠐⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠋⠃⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠠⠀⠀⠀⠠⠤⠤⠤⠄⠤⠤⠤⠀⠀⠒⡄⢀⠀⡔⢀⠀⠒⠐⠢⡄⠀⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⡄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⡡⢀⣄⣠⠘⣁⣄⢄⡤⡅⢀⣆⣤⡤⡤⢤⠄
⠀⠀⠀⠀⠀⠤⠀⠒⠖⠒⠂⠰⢾⠱⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠈⠉⠉⠀⠨⣇⣭⣴⣱⢀⡀⠀⡅⢶⡥
⠀⠀⠀⠀⠀⠀⢰⠆⣀⠀⠀⠘⠉⠉⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘