Cell Emojis & Symbols

⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⢀⡠⠋⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀⡠⠑
⠊⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠇⠀⠚⠛⠋⠀⠀⡠⠋⠀⠀⠐⠁⠀⠀⠂⠀⠀⠀⠀⠀⡀⠠⠛⠛⠁⠀⢀
⠀⠀⠀⡀⠘⢏⠀⠀⠀⠂⠀⠀⠀⠈⠩⠂⠠⠐⠀⠁⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠄⠀⠀⠀⠐⠱⠂⠠⠐⠀⠁
⠀⠀⠀⠐⡀⠘⢶⣼⡆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠙⣦⣼⠄
⢀⠀⠀⠀⠈⠢⣄⣀⣯⣢⠃⠀⡠⠐⠒⠁⠈⠀⠀⠀⠠⡀⠀⠀⠀⠐⠦⣄⣰⢷⡢⠀⠀⠠⠐⠐⠉⠁
⠀⠑⢄⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⣠⠤⠈⠐⠐⠀⢀⠈⠢⡀⠀⠀⠀⠀⠉⠈⠀⠀⠈⠀⠀⣤⠤⠀⠂⠂
⢍⢦⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠠⣕⠄⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⣶⠟⠂⢹⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠒⠶⢤⣴⣾⠿⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠉⠒⠦⣤
⣀⣤⡽⠋⠀⠀⡀⠄⠠⠤⣤⣀⠀⠁⠠⡀⠀⠀⠀⠀⢀⣀⣄⡷⠃⠀⠀⠀⠀⠄⠤⣤⣀
⠛⠉⠀⠀⠀⠈⠀⠀⠀⣤⡐⠸⡇⠀⠀⠀⠉⠐⠒⠒⠚⠋⠉⠀⢀⠔⠀⠀⠀⠠⣤⡀⢹⡆⠀⠀⠀⠉⠒⠒⠒
⠀⠀⠀⡈⠀⠀⠀⠀⢠⠿⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡰⠣⠀⢸⠇
⠀⠀⠀⠁⢀⣶⡦⠖⠁⠀⢠⠎⠀⠀⠠⠀⠀⠠⡀⠀⠀⠀⠀⠠⠀⣠⣷⡴⠂⠀⠀⡠⠋⠀⢀⠄⠀⠀⠄
⠀⠀⠀⢔⣤⠀⠀⡀⠠⠂⠁⠀⠀⠈⠀⢴⠀⠀⠈⢂⠀⠀⠀⠀⢄⡀⠁⢀⠀⠄⠊⠀⠀⠀⠂⠀⡴⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣆⣀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⣄⣀⡀⠈
⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠐⢄⡈⠛⢻⣀⡀⠀⠀⢀⠀⠤⠄⠀⠀⠀⢀⠀⠀⠀⠀⠑⢄⠈⠛⣇⣐
⠀⠐⠁⠀⢀⠀⠄⠄⠀⠀⠀⠢⡀⠀⠀⠀⠉⠛⠛⠓⠁⠀⠀⠁⠀⠀⠄⠄⠠⠀⠀⠑⢀⠀⠀⠀⠀⠙⠓⠛⠊
⠀⠆⠀⠐⢯⠀⠀⠀⠀⠈⠓⣄⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠁⠀⠀⠀⠈⠲⣄⠱⠄
⠀⠠⠀⠀⠈⠑⠤⣀⣀⣀⣈⣦⡇⠨⡆⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈⠓⢤⣀⣀⣀⣸⣬⡆⢸⡀
⠀⠀⠑⠄⠀⠀⠀⠀⠉⠉⠛⠉⡠⣲⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠛⠁⣠⡾⠀⠀⠀⠀⠀⢀
⠙⢶⠀⠀⠀⠐⠠⢄⣀⣰⡶⠷⠛⠁⠀⡠⠀⠀⠀⢀⡉⠹⣦⠀⠀⠁⠂⠤⢄⣀⣠⡾⠟⠋⠀⠀⠀⠂⠀⠀⣈
⣂⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣹⣓⣀⣹⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣻⠭⠥⠒⠒⠒⠒⠒⠠⠤⠤⣍⣙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢤⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⡫⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡙⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⢏⠞⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠒⠒⠦⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢯⠎⠀⠀⠀⠀⠀⡠⠊⠀⡠⠚⠉⠒⢄⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⣿⣿⣿⣿
⣿⣿⣿⣿⢯⠋⠀⠀⠀⠠⠀⡜⠀⠀⢸⠀⠀⠀⠀⢨⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡀⢠⢀⠀⠸⠸⣿⣿⣿
⣿⣿⣿⢏⡎⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠦⣀⣀⡤⠊⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡇⢻⣿⣿
⣿⣿⡟⡸⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⣿⣿
⣿⣿⠃⡇⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⠀⡄⠀⢹⢸⣿⣿
⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⢸⢸⣿⣿
⣿⣿⡇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⣿⣿
⣿⣿⣧⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⣿⣿
⣿⣿⣿⣆⢣⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠜⠑⢀⠀⡐⠀⠀⠀⠀⠀⠒⠀⠀⠀⢂⡎⢰⣿⣿⣿
⣿⣿⣿⣿⣆⢣⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣧⡳⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣎⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡰⠃⣠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡓⢄⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣍⣑⣒⠒⠂⠀⢀⣒⣒⣭⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⣭⣙⣫⠅⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡰⠁⠻⡅⠙⡷⠈⢣⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⣒⢢⣀⠀⠀⠀⠀⣇⡇⠀⠀⢘⣴⠁⠀⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⡈⢄⡁⡈⢃⠰⠠⠈⢱⠀⠀⠀⠱⣜⠢⣠⠼⢋⠣⣤⢊⡜
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⠀⠀⣳⠁⡢⠂⠉⠐⢄⣧⠀⠀⠀⠈⠢⠬⣀⣀⡤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⣰⣿⣠⠃⠀⠀⠀⣾⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣗⣿⢫⣃⢌⢀⣀⣄⡴⣳⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠵⣒⣚⣛⣯⠭⠚⠁⠀⣀⠄⠐⠋⢡⠀⠉⠉⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢲⡤⠤⠒⠒⢢⢤⠲⠈⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡔⠁⠀⠀⠀⠀⠠⣻⡟⢎⠐⠃⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡀⣇⠀⠀⠀⠀⡴⠊⣀⣼⣼⡎⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠁⡌⣉⠓⡈⢏⠔⠋⠀⠀⠈⣉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⣈⠒⠆⠎⠀⠀⠀⠀⢈⡟⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⡀⠂⢄⡀⠀⠀⣠⠞⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠭⠭⠭⠒⠋⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⠷⢦⣤⣤⠴⠶⠾⠟⠛⠻⠏⠉⠙⠛⠛⠳⠶⣦⣤⣤⡤⠖⠚⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⡃⠀⠀⢤⡀⠀⠀⠀⠙⠿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢹⡐⠀⢻⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡄⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣠⠎⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⣤⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡴⠋⠉⠉⠙⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠁⠀⠈⠈⠀⠁⠀⠉⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⡄⣀⣠⣀⣤⣥⣴⠴⠴⢥⣗⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠠⡀⣀⣁⣧⣤⡴⠴⠶⠒⠛⡛⡉⠉⢉⠐⠀⠂⠀⠀⢠⠈⢻⣔⣀⣀
⠀⠀⠀⠀⠀⠀⡀⢠⡵⠗⠚⢻⠉⠉⣁⣆⣄⠀⢀⠂⠁⠀⠨⠁⠂⠠⠄⡨⠀⠀⣀⠈⠠⠀⣯⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠐⢠⡟⣀⢓⠀⠀⠀⠀⠐⢢⠐⠋⡁⡁⠈⠅⠀⠁⠀⠀⠀⠁⠄⡀⠄⠀⢃⣼⠗⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠈⢹⡇⢀⠐⢀⠠⡄⠀⠳⠊⠋⠡⠰⢁⠐⠠⠡⠁⠀⣀⣀⣨⣤⡤⠶⠶⡻⠃⠁⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠩⢳⣄⠒⠀⠰⠄⢀⣀⣀⣀⣤⣤⢤⠶⠖⢚⠻⠙⠉⠉⠈⠀⠁⣡⡴⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠡⠊⠋⠛⠛⠋⠋⠏⠡⠁⠀⠈⠀⠀⠀⢄⡀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⡴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠰⠤⠴⠶⠦⣶⠷⠰⡁⢴⠸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣤⠴⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠁⠀⠀⠀⢀⣤⡀⠀⠀⣀⡤⢶⠖⣦⢤⣀⠀⠀⠀⠙⠿⢶⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⣀⠀⠀⠀⠀⣀⣰⠋⠀⢻⠀⠸⣇⢣⠘⣄⢻⡜⣃⠿⣆⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠈⢸⠩⡏⣹⠗⣤⡀⠙⣧⠀⠀⢸⠀⠀⠈⠙⠓⠿⠼⠧⠿⠴⠋⣀⠀⠀⠀⠘⢿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠕⠀⠀⠀⠈⢿⣁⡷⢸⣁⡿⣄⠙⢦⣤⡞⠀⢠⡞⠉⠙⣦⠀⠀⠀⣠⣼⠋⠀⠀⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢀⡀⠀⠀⠀⠀⠙⠲⣟⣸⠲⢿⠀⠀⠀⠀⠀⠈⠣⣤⡴⠋⠀⠀⢸⣿⡟⠀⠀⢠⠋⠀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢁⡞⠉⠉⣧⠀⠀⠶⣄⢀⣀⡉⠛⠋⠘⠷⠀⠀⠀⠀⠀⠀⠀⡤⠒⡄⠸⡉⠇⠀⠀⣌⣰⢦⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠳⢤⠴⠋⠀⡠⢤⣊⠽⢼⣃⠡⣟⣒⣳⡭⣀⣢⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⡐⡼⡹⡧⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢄⡀⠀⠀⠀⠘⡳⢼⡯⡿⠗⣁⣬⣥⣥⣬⣈⠢⢕⡆⠀⠀⠀⠀⠙⠛⠀⢀⣀⣮⣿⣾⣽⠃⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣔⡴⠡⠾⡀⠀⠀⠐⢶⢏⢌⡴⠋⠁⠀⠀⣀⣤⣌⠙⢦⡀⠀⠀⠀⠀⠀⠀⢠⡗⣲⡿⠟⡹⠁⢨⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⣿⣿⡗⠉⠀⠀⠀⠀⠑⠊⡾⠁⠀⠀⠀⢰⣏⠀⣸⡇⠈⣷⠀⠀⠀⠛⠇⠀⠙⠉⠑⠒⠈⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⡗⢃⡴⠲⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠛⠛⠉⠻⢿⡆⠀⠀⠀⠀⣠⠖⠲⡄⠀⠀⢤⡞⢁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⢸⣋⡒⣋⣧⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⣾⠀⠀⠀⢠⢾⣉⣣⣙⡇⠀⢠⡟⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠸⣭⡼⢉⡼⡇⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣩⡿⠁⠀⠀⢰⡛⠦⠰⢤⡽⠣⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⢳⡴⠂⡴⢻⠀⢀⠀⠀⠈⠛⠶⠶⠶⠖⠛⡙⢶⠄⠀⠀⣿⢹⣂⡳⢞⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⢠⡟⠻⠵⠏⠀⠀⠤⢊⡐⠠⠌⠀⠒⠬⢀⠀⣀⣀⠀⠀⠈⠉⠉⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⡀⠀⠀⠀⠀⠀⣤⠟⠟⠒⡀⠐⠩⠀⢠⡞⠉⠉⢳⠀⠀⠠⣲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⢯⣄⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⠶⠶⠶⠶⣾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠔⠒⠒⠒⠒⠒⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⣀⣤⣠⣀⡀⠀⠀⠀⠀⠠⠉⠓⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⣆⣥⣍⠹⢲⣀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠺⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡙⢆
⠀⠀⠀⠀⠀⢠⠞⡀⠀⠀⠀⠀⠀⠀⠔⠡⠤⠀⠂⣁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⡀
⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠡⠀⠐⢂⠐⠀⡁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠠⡀⠈⣤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⢇
⠀⠀⠀⡏⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡁⡐⢦⣡⢈⣵⣀⢋⢂⡀⢰⢎⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⣀⡜⠅⠉⡬⠹⠘⡀⢅⡈⢼⡀⠐⠄⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠐⡌⠙⠂⣊⣡⡟⢭⡟⡃⡪⣘⠡⠁⢁⣒⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣆⢔⣢⠉⡝⠈⢩⠈⡓⣰⢶⣅⠈⡎⠠⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⡇⠀⠀⠀⢬⠰⠠⡂⠀⠀⠀⠠⠋⠡⠚⢠⢫⠇⢾⢀⠣⢎⣒⣢⡔⠐⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸
⠀⠀⠀⢹⠈⠀⠀⢊⠰⠚⠀⠀⠀⠀⠀⠁⢈⠚⡠⡠⢅⣣⢎⠶⠈⢗⢀⠔⡃⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠄⣂⠊⢋⠀⠎⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠠⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣡⢃⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⢁⠘⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠗⢵⠎⠂⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⡨⢅⠏⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠄⠀⠀⠀⣞⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠿⠃⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠙⡏⡇⠗⡤⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠈⠥⣅⣸⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣊⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣈⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⣐⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⣀⡀⠂⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⣂⣈⡤⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉⠁

⣀⣀⣀⠀⡀⢀⢀⠀⠀⠀⡀⡀⢀⠀⡀⣀⣀⠀⣀⡀⢀⣀⡀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀
⣿⣿⣿⣿⣿⡏⠍⠋⢻⣿⠋⢻⠉⠿⢩⡉⣽⡟⢩⣙⠋⠭⡏⢻⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣧⣾⣧⣬⣥⣴⣬⣤⣦⣼⣤⣿⣷⣌⣥⣤⣭⣧⣬⣧⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣀⢀⠠⠤⠄⠐⠒⠒⠊⠉⠉⠁⠀⠀⠈⠉⢉⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡞⠉⠀⠀⢀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠈⠐⠀⠀⠀⢠⣢⠀⢰⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠠⣝⠆⠀⠀⢸⣿⣆⠀⠀⠀⣂⣬⡦⠈⠙⠤⠒⠂⠄⠀⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠆⠉⠀⠀⠀⠀⢻⠿⠀⠀⢈⣍⢋⠔⠁⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⢠⠖⠀⠀⠀⠋⠀⠀⠀⠀⡜⣁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡆⢞⠉⠀⠀⠀⠀⠀⠀⠀⠂⢔⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⢼⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡱⢸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⠄⠠⡆⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠒⢒⠁⡾⠋⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣧⠠⣚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠄⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠅⠀⣴⣜⡀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡞⠀⠀⠱⢘⠣⠁⠀⠀⠛⠋⠀⢊⣉⠆⠀⠀⠀⠀⠠⠹⠀⣸⣿⡇⠀⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠤⠶⠶⠶⠶⠒⠒⠶⠶⠶⠦⠤⠤⣤⣀⣀⠀⠙⠛⠀⠀⢠⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠉⠉⠙⣛⡛⠛⠉⣀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⣀
⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠱⣄
⠀⠀⠀⡜⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⡐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⢆
⠀⠀⢰⠃⠀⠄⠀⠀⠀⠒⠠⢽⣷⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⢸⡆⠀⠄⠀⠀⠀⠀⠀⠀⠈⠣⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣇
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠋⠀⡏
⠀⠀⠀⠈⢧⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠑⢄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣀⣀⡀⠀⠐⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣖⣒⣒⣶⣴⡴⠿⠭⣭⣕⠂
⠀⠀⠀⠀⢀⢴⣽⠞⠋⠉⠁⠠⠉⠁⢠⡄⢀⣤⢬⡀⠙⣦⠣⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⡛⠉⢙⣓⠒⡚⠛⠉⠉⠉⠛⠲⢤⣀
⠀⠀⠀⢀⢣⠟⠀⠀⣠⡷⠾⠿⢷⣶⣄⠀⡘⢯⡝⢳⡄⠘⡇⠄⠀⠀⠀⠀⠀⢀⡞⠉⠘⠀⠰⢀⣀⣠⢮⠤⣚⣁⡇⠀⡀⠛⡤⠤⣄⠉⠳⣄
⠀⠀⠀⠰⣮⣴⣲⣼⡏⠀⠀⠀⠀⠙⢻⡇⠀⠀⠻⣷⠇⢀⡇⠀⠀⠀⠀⠀⠀⡾⢠⠞⡿⡇⡀⠈⡋⣠⢶⣶⡉⣁⡴⠞⠁⠢⣞⣟⢾⢷⢀⡈⢣
⠀⠀⠀⣰⡇⡭⡻⣻⠅⠀⠀⠀⠀⢀⣼⠇⠄⠘⢀⠐⠀⡼⠀⠀⠀⠀⠀⠀⠀⡇⣜⡝⣌⠈⣻⠠⠯⣾⣿⣿⣷⣷⣿⣥⣆⠀⣾⣿⡈⢦⣇⠀⠈
⠀⠀⠀⣼⠸⠿⢆⡿⠀⠀⠀⢀⡴⠛⣡⡤⠄⣙⢞⡂⢰⢇⠃⠀⠀⠀⠀⠀⢠⡇⠿⠮⣻⣾⡇⠀⣨⢗⡿⢛⣿⣭⣍⡉⠹⣛⣬⣬⣟⣦⢻⡇⢀
⠀⠀⢸⡇⢠⠀⣼⠃⠀⠀⢀⣾⠁⣾⣡⡶⠞⢹⡗⡏⢸⢘⠀⠀⠀⠀⠀⠀⢸⡇⠺⡶⡿⣧⣷⡇⠀⠋⡼⠋⠛⢀⠉⢻⠀⢙⡻⠈⠋⢾⠔⡇⢠
⠀⠀⣾⣀⠀⢸⡇⠀⠀⢠⠞⠁⠺⡥⠋⣀⣈⣳⣯⣟⢸⢸⠀⠀⠀⠀⠀⠀⠈⣇⢸⡤⢡⣣⣶⣿⡿⣿⡷⠰⣿⠇⠐⢹⠣⡍⢁⠠⢄⣾⠋⠀⡾
⠀⢴⠇⠀⠀⣾⠀⠀⠀⢸⡄⠀⠀⣴⢿⠋⠉⢹⡉⢹⢸⡄⠀⠀⠀⠀⠀⠀⠀⠘⣎⠙⢬⡉⣾⣾⡽⢦⣷⣆⠀⢁⣨⢯⢾⣤⡯⠀⠆⠀⠘⢠⡇
⠀⣿⣰⣿⣆⢿⡆⠀⠀⣸⠇⢠⡀⣿⠄⠒⣀⢸⡥⠞⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠹⠃⡃⠯⢫⡏⠂⠉⢙⣹⢛⡝⢛⣾⠁⠂⠞⢳⡎⢸⠁
⠀⢻⠸⣿⢿⣆⠻⢿⡿⣋⡤⣾⡌⢿⡄⠙⠿⣏⣧⡆⣸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠃⠹⠆⠀⢳⣼⠞⣷⠛⡩⠔⠻⢫⠟⣸⠎⢐⡏⣼
⠀⡘⢦⠙⠿⠿⠃⠑⢲⣵⣿⣿⠷⠈⣻⣿⣿⣯⣾⣡⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠠⠈⠀⢀⡈⠛⣀⡥⡞⡶⠃⠸⣯⡀⠉⣰⠇
⠀⠘⠈⠳⣄⠀⠀⠁⠘⢳⣛⣁⣀⠐⠓⠾⣛⣫⢟⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠤⣬⣁⣀⠋⠘⠋⠀⠉⢀⣀⣩⠼⠃
⠀⠀⠀⠐⠬⠛⣛⡛⠿⠭⠭⠿⠭⠭⠉⠉⠭⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠋⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣄⣀⡀⢀⣀⣀⣀⣀⣀⣀⡤⠤⠄⣀⣀
⠀⠀⣠⠴⣶⡾⠾⢿⠿⠚⠚⠛⠛⠓⣛⠛⠛⠛⠛⣛⡉⠉⠛⠓⠒⠒⠾⠷⢴⡆⢀
⢠⣾⡔⣋⣄⣀⠐⢫⡤⠶⢦⣥⡀⢈⣥⣤⠚⣳⢦⣍⢁⣀⡀⢀⡤⣤⣬⣥⡀⠙⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣬⡞⢰⣿⣿⣿⣀⡿⣽⡒⣯⣿⣇⣯⠞⣿⡿⣷⣖⣿⣸⣭⣿⢸⣷⣯⣿⣾⠇⠀⣼⣭⣤⣤⡀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠈⢳⡀
⡏⣇⠈⡻⠿⣿⠿⠿⠉⠹⣿⣿⣶⣫⢾⣿⣷⡟⠙⠟⠟⢓⠛⠿⠿⠋⠁⢉⣁⡴⠃⡰⠀⠀⠸⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠘⣮⠣⣄⣀⡀⠘⠀⠀⠀⠈⠉⠉⠀⣀⣈⣤⣤⠤⠤⠤⠤⣔⣖⣒⡛⢫⠭⠥⠒⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣀⣀⣠
⠀⠀⠙⠲⠬⣭⣍⣉⣉⣭⣭⠭⠽⠿⠟⠛⠋⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⣀⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢠⠄⠀⠀⠀⠈⠉⠓⠒⠤⣀
⠀⠀⠀⡔⠁⠂⡀⠀⠀⠀⣀⡀⠀⠁⠀⡀⠁⠀⠘⠘⠀⢀⡀⠀⢠⢲⣲⢦⣤⡄⠉⠳⢄⡀
⠀⠀⢸⠁⡀⠀⠀⢠⢎⣯⣽⣽⣽⡗⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢟⡿⢺⣵⣻⡄⠀⠀⠙⢦⡀
⠀⠀⡇⠀⠀⠀⠀⠈⠺⠿⠇⠿⠿⠚⠁⠀⠄⠀⠀⠀⠀⠠⡔⢠⠀⠀⢑⠫⠿⠽⠎⠀⠀⠀⠀⠑⡄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡⠀⠀⣉⠌⠦⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⢇⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠄⠠⠀⣃⠤⠒⠒⠒⠤⣀⠀⠀⠀⠀⠈⢦
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡈⠀⠘⠀⡜⠁⠀⠀⠀⠀⠀⠈⢫⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠱⡄⠐⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠄⠘⣉⢸⠀⠀⠀⠀⠀⢀⣤⣄⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⢦⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣨⠂⠄⠀⢸⡀⠀⠀⠀⠀⠈⠛⠋⢁⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠑⢄⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣍⠁⠒⠀⠀⠳⡀⠀⠀⠀⠀⢀⣀⠞⠀⠀⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⠷⣞⡆⠀⠈⠀⠀⢠⡚⣯⣷⢿⡿⠐⠀⠁⠀⠀⠈⠑⠒⠤⠒⠊⠁⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⣗⢿⣻⡮⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣀⠤⠴⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠑⠒⠒⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⣚⣉⣉⣀⣀⣀⣀⣀⠈⠉⠙⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣭⠴⠒⠋⠉⠉⠁⠀⡀⠀⠀⠀⠉⠉⠉⠙⠓⠲⠦⢤⣈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣊⠵⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⡀⠁⠂⠐⠂⠀⠀⡀⠀⠰⠠⠀⠀⠀⠀⠀⠀⠙⣄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡰⢣⠏⠀⠀⠈⢀⡀⠀⠀⢀⡠⠖⠋⠁⣀⣀⡀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠈⢧⢹⡄
⠀⠀⠀⠀⠀⠀⡰⣣⠋⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⡴⠋⠁⠀⠉⠳⡄⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⣧⢳
⠀⠀⠀⠀⠀⣰⢱⠃⠀⠀⠁⠀⢀⠈⢀⡞⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⢈⡀⠀⠐⠸⡎⣇
⠀⠀⠀⠀⣰⢣⠇⠀⠀⠤⠀⠀⠀⠀⣸⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠉⣁⠀⠄⣧⠸⡄
⠀⠀⠀⢰⢃⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠓⠶⠶⠊⠁⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢰⠈⡄⠐⠁⢸⡀⣧
⠀⠀⢀⡏⢸⠁⠀⠈⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⡀⠀⠆⠠⠘⡇⢸
⠀⠀⢸⠃⣿⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠋⠀⢠⠀⠄⠈⡇⢸
⠀⠀⢸⠀⣇⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠈⠁⡌⢁⠈⠀⡇⢸
⠀⠀⢸⠀⣿⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⢄⣀⣀⣀⡤⠴⠚⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⢸⡄⢸⡀⠀⠀⠈⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣼
⠀⠀⠀⣇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠂⠀⠁⢐⠛⠂⠣⠀⠀⠀⢠⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠁⡞⢀⡇
⠀⠀⠀⠘⡄⢳⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠠⠬⠀⡀⠀⢀⠟⣂⠀⠈⠢⠄⠀⠀⠁⠀⠀⢀⢀⡀⠀⠀⠀⠀⡀⣸⠁⢸⠁
⠀⠀⠀⠀⠹⡌⢧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠛⠁⠀⠋⠀⠁⠘⠀⠢⠀⠀⠀⠀⠀⠐⠀⠂⠀⠈⠆⠀⠀⢠⠃⢀⠇
⠀⠀⠀⠀⠀⠹⡌⢧⠀⠀⠀⠀⠈⠀⡀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⣰⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠘⢦⠳⣄⠀⠀⠀⠐⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠤⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⢀⠄⠀⡰⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣄⠀⠀⠀⠰⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠔⠁⢀⡞⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠳⣄⠀⠈⠀⠁⠀⠀⠀⠐⢄⠢⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⠏⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠳⢤⡀⠀⠀⠀⠀⠀⠀⠲⡀⢂⡀⠀⠀⠀⠀⠰⠀⠀⡄⠀⠀⠀⠀⢀⡠⠞⠁⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡙⠳⢤⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣈⠙⠒⠶⠤⣤⣀⣀⣀⡀⢀⣀⣀⣠⠤⠶⢚⣉⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠦⠤⠤⠭⠭⠭⠴⠶⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠓⣀⠀⡖⠀⠌⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣁⠀⢠⣭⣃⣿⣀⡀⠀⡂⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠠⡿⣝⠀⠉⠻⡀⡄⣻⠍⡫⡄⠀⠚⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣧⣬⡙⠂⠛⣻⣻⠶⣍⠴⣿⢀⠚⠣⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢭⡀⣾⣫⠹⣅⢪⡍⠋⠆⠴⠂⠤⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠤⠐⢡⣲⠛⡏⣨⡹⠟⣬⡍⠀⠂⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠖⠙⠻⠄⠘⠻⢃⡐⠟⠁⠈⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⡀⠐⠀⠀⢈⠵⠐⣃⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⣤⣤⣀⣀⣠⣠⣤⠤⠴⠖⠒⠉
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠎⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠠⣄⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⠙⠓⠲⠶⠖⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠦⣤⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠐⠀⠀⢀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠐
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⠀⠀⠀⠀⣀⡴⠏⠀⠀⠀⠀⠀⠙⠷⣤⣀
⠀⣀⠀⠙⠲⠦⢤⣤⡤⠤⠤⠶⠚⠙⠓⠶⠶⠶⠶⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠤⠴⠊
⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠈⠙⠦⣤⣀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⣀⣴⠿⢿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠚⠁⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣏⣀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠳⠶⠶⠶⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠛⠀⠐⠁⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠀⠌⠀⡘⠀⢀⠇⠀⠀⠈⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠲⣄⡀⠉⠢⡀
⡄⠀⠀⢀⣤⣀⠀⠀⠀⢀⣤⡄⠈⡆⠀⠀⠀⠀⠐⠀⢸⣄⠔⠀⢠⠞⠀⠆⠀⡀⠠⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠑⠀⣀⣽⡆⠀⠘
⠀⠀⠀⠙⢝⢿⣷⠀⠀⣾⠉⡗⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⡄⠀⢀⠞⠀⢰⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠸⠴⠋⠀⡇⠀⠀⠀⠀⠀⢀⠒⡰⠋⠀⠰⠁⠀⡼⢀⣠⠎⢀⡞
⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠐⣅⡠⠀⠀⢀⡜⠁⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⠔⢡⠈⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⣉⠙⠒⠦⡀⠀⢀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣶⣶⣶⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠟⠛⠛⠻⠿⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⠙⠷⢶⣤⣤⣤⠶⠞⠁⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⠀⣀⣠⣤⣤⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠈⠙⠛⠿⠶⠶⠶⠶⠾⠟⠛⠉⢀⣤⡶⠟⠛⠉⠁⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠉⠉⠉⠛⠻⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠈⣿⡄
⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢀⣿⠁
⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢸⡇⠀⠀⠠⣄⣀⣀⣠⣤⣶⠟⠋⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⢸⣧⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠙⠿⣦⣄⣀⣀⣀⡀⠀⠀⢹⣇⠀⠙⢷⣦⣤⣀⣀⣀⣠⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⢸⡟⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⡀⠀⠀⠀⠀⠀⢀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⠶⠶⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⢸⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠴⠀⡀⠀⢆⠀⠈⢃⡎⠀⠀⠀⢰⠁⠀⠀⠸⠆⠀⠠⠠⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠸⠖⠀⠀⠂⠀⠀⠡⣤⡀⠀⠀⠀⠀⢈⠂⠀⠀⣸⠤⠀⠈⠤⠤⡞⠉⠀⠀⠀⠈⢂⣀⣀⡶⠃⡼
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠬⠷⠂⠀⠀⠀⠂⠒⠀⠗⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠸⣄⡀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠤⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡅⠀⠠⠛⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⠤⠴⠞⢨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⢀⣤⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠀⠀⠀⠀⠀⠀⠑⠒⠤⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀


⣿⣿⢿⠿⡿⠻⠿⡿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⣄⡄⠀⠀⣤⢠⣤⠀⠀⠀⢀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠔⠎⡛⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠃⠇⠘⠂⠉⠈⠀⠀⠀⠀⠀⠁⠈⠁⠀⢀⡀⢸⣳⡠⠔⠁⠀⠀⢀⡜⠓⢤⡀⢰⠋⠀⠀⠠⠀⠠⠀⡇⠀⣰⣦⡁
⠀⠀⠸⢇⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⣼⣠⠏⠀⠀⠀⠐⠀⠈⠀⠀⠈⠀⠸⡆⢄⠀⢅⠀⣐⠼⠁⠀⡇⡜⠃⠀⠀⠀⣀
⠀⠀⠀⠘⡆⡜⠀⠀⠀⠀⣤⠀⡰⠂⠀⠀⠼⢾⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⠒⠒⠟⣶⠛⠛⠉⠁⠀⠀⠘⡇⣇⠀⢀⡰⢶⣝
⠀⠀⠛⢄⢻⡇⠀⠀⣠⡞⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⢉⠦⠤⠤⠴⢠⡤⣧⡄⢼⠆⠀⠀⢰⠉⠂⠌⣐⠀⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢳⣈⠉⢉⣀⠜⠋
⠀⠀⠀⠈⢢⡇⠀⠀⢈⣧⡀⠀⣸⡄⠀⠀⡠⡄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠘⣆⠌⠐⠀⢁⠂⠈⠂⡼⠀⣠⣤⣄⠀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢳⠀⠀⠁⠘⠛⠙⡍⠀⠀⠀⠉⠁⠀⠀⠀⣄⠀⠀⠀⠀⠀⣴⡽⠀⣀⡠⢤⠤⡬⠷⣤⠤⠤⠤⠖⠋⢀⢮⠞⠁⠀⡴⢹⢀⣤
⠀⠀⠀⠀⠀⢹⢠⠶⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⡿⠋⠀⢸⠁⠠⠀⠀⠀⠄⠈⢦⠀⠀⠀⠀⢸⠘⠒⣦⡸⠁⠓⠊⡼⠀⣾⣿⡆
⠀⠀⠀⠰⢆⢸⡏⠀⠀⡠⠤⠤⣀⣀⠴⠊⠳⡀⠀⠀⠀⠀⣀⡠⣤⠄⣀⣀⣀⠘⣦⠈⡀⠀⠀⠄⠀⣘⠇⠀⠀⠀⠈⠓⠊⠡⡷⠒⡆⣸⠁⠀⠙⠛⠋
⠀⠀⠀⠀⠈⢾⡇⠀⠘⣅⠀⠀⡼⠉⠓⢦⣸⠁⠰⠀⠖⠉⠃⠀⠀⢀⠠⠐⠉⠉⠉⢷⠚⠛⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣵⠖⢲
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⢦⡞⠁⠺⠇⢠⡏⢢⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⣤⣤⠀⢠⣤⡄⠀⣠⣤⡀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠠⠂⢼⠙⠲⢄⣠⠋⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣠⡇⠀⢸⢹⠀⠀⠀⡿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡴⠞⠁⠉⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣾⠀⢸⣿⡇⠀⢸⢸⠀⠀⠀⣙⠇⢸
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡀⠠⠍⡇⠀⠀⢀⢏⣌⢧⠀⠀⢸⣿⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⣧⠄⢾
⠀⠀⠀⠀⠠⠀⡷⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠛⠀⡇⠀⠀⢸⢸⣸⢸⠀⠀⠘⠋⠀⠈⠛⠁⠀⠘⠛⠀⠀⠀⠿⠤⠼⠰⠆
⠀⠀⠀⠀⠀⠀⡧⡇⠀⠁⠀⠀⠀⠀⠀⠀⠐⡿⠷⢶⠂⠀⠀⢸⢰⡏⠙⢆⡇⠀⡆⠘⣆⣡⠞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⢹⡇⠀⢠⠀⠀⠀⠀⠀⠀⢀⠇⠂⠘⡆⠀⠀⠀⠶⠁⠀⠘⠃⠘⠃⠀⠀⠀⠀⠀⠀⢠⣾⡁⢀⣀⣠⠔⠛⠓⢤⣠⠎⠀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⡴⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⢹⡀⠀⠀⠀⢀⠀⢀⠀⠀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠈⠛⠿⡀⠀⠀⠀⠈⠋⠀⠤⠆⠀⠈⠁
⠀⠀⠀⠀⢀⡞⠤⣈⣆⠀⠀⠀⠀⠀⠀⣸⠃⠠⣕⠀⠀⢧⠀⠀⠀⠉⠀⣿⠀⠀⠀⢰⡆⠀⢇⣤⣤⣀⣀⣀⣀⡀⠘⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡞⠈⠘⠋⢹⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⠀⣤⡄⠀⢶⠀⠀⠀⠣⠤⠄⣀⣀⣀⣀⣤⠀⠀⢠⣤
⠀⠀⠀⠀⢣⣀⠀⠀⣸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⠐⣿⠀⠀⠀⢸⢿⠀⣿⡇⠀⣿⠀⠀⡤⡄⠀⠀⠀⣶⠀⠀⡷⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣠⡀⠙⠿⢿⡿⢿⡿⠿⠿⠿⠿⠛⠀⠀⠀⢿⠀⠀⠀⢸⢸⠀⡇⡇⠀⣿⠀⠀⡇⡇⠀⠀⠀⢸⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢖⣢⣀⣀⣹⣷⣤⡤⠼⢷⣦⠤⠤⠄⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⢸⢸⠀⡇⡇⠀⡿⡆⠀⡇⠇⠀⠀⢸⢸⠀⠀⣷⡇⠀⠐⢶
⠀⠀⠀⢀⣤⣤⠔⠊⠉⡿⠃⠀⠐⠖⠁⠀⠺⠊⠀⠀⠀⠀⠀⢤⡄⠤⠤⢾⠤⠶⠶⠾⠾⠶⠷⠷⠶⠻⠗⠦⠗⡖⠒⠒⣼⠛⠒⠲⠟⠓⠿⠆⠙
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠘⠈⠀⠈⠉⠁⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⠂⢄⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠨⠓⠁⠀⠉⢀⠀⠙⠴⡀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠂⢀⣂⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠸⠀⠀⠘⡖⡇⠀⠀⠀⢠⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠠⠀⢀⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⣀⡀⣀⣀⠄⢁⣀⣀⢀⠄⣀
⠀⠀⠀⠀⠀⢁⠄⠀⠀⠀⠐⠤⠀⢤⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⠀⠀⡄⠈⣁⠀⠢⠀⠾⠙⡮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠈⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢂⠴⠒⠄⠀⢰⡀⠂⠀⠄⠄⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⣀⠐⢀⠀⠀⠀⠀⠠⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⣸⡏⠀⠀⠉⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠛⢷⣦⣄⣀⡀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠁⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁⢀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣤⣤⣤⣀⣀⣠⣤⣤⣴⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠛⠛⠻⣧⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⠛⠷⠶⣶⣤⣴⡶⠶⠿⠛⠉⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⢿⣿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠈⠙⠛⠷⠶⣶⣤⣶⠶⠟⠋⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣀⣀⡀⠀⠀⢀⣀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠓⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠈⠙⣶
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠇⢐⠀⠀⡀⡀⠀⠀⠀⠀⢀⣴⠾⠿⣶⣆⡀⠀⠀⠠⠅⠠⠅⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠄⠀⠙⣿⡇⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠠⣒⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢁⡿⠋⠀⠀⠀⠐⠒⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⠀⣀⣀⠀⠀⠀⠀⠘⠓⠀⠶⠛⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠸⣶⣴⡟⠀⠀⠄⠡⠄⠀⠀⠀⠈⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢘⣵⡿⠛⢿⣧⠀⠀⠀⠀⠉⠁⠀⢀⡄⡀⠴⡀
⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠁⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⢃⠁⠈⠀⠠⢹⢃⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⢀⣿⣧⣀⣀⣠⠁⠀⠀⢻⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠁⠀⠈⠀⠴⠆⠤⢀⠁⠞⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠤⢀⡶⢾⣷⣦⠀⠀⠀⠀⠀⠀⢀⠀⠀⠓⠒⠀⠀⠈⠀⠀⠀⠀⡴⠖⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⡇⣄⠀⠀⠀⠒⠁⠀⠀⠀⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠂⢀⡀⠈⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⡗⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠘⠀⠉⠁⠐⠀⠀⢀⠀⠀⠀⠠⠁⣠⡄
⠀⠀⠀⠀⠀⠰⠁⠈⠀⠀⠀⠀⠀⠃⠠⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⢂⠈⠁⠉⠀⠀⠀⠀⠀⠐⣆⠂⠘⠀⠀⠀⠀⡀⣠⣶⣷⣾⣧⣄⢰⣿⣿⡇
⠀⠀⠀⠀⠂⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠨⠂⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠊⠀⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⢽⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠂⠀⠈⣿⣿⣿⣿⠀⠀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣿⣿⣿⣿⣿⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠛⠛⠋
⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣴⣖⡠⠀⠀⠀⠀⠀⠣⣐⠲
⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢀⠀⠈⠙⠋
⠀⠀⠀⠀⠠⡤⠅⠋⢰⠀⠴⠢⢀⠀⠰⣄⣠⠠⠴⠃⠀⠀⠀⠀⠀⠀⢄⣤⣤⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠣⠀⠀⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢆⠐⢚⣋⡁⠜⠀⠀⠀⠀⣀⡀⠀⠀⡀⠀⢀⣀⣤⠀⠀⡆
⠀⠀⠀⠀⣦⠀⠅⠀⠈⠈⠙⠠⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠻⠒⣤⠤⠄⠀⠴⠃
⠀⠀⠀⠀⠈⠻⠶⣶⡶⠖⠛⢋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⠖⠿⠃⣠⠀⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠐⠸⢂⡆⠀⢰⠀⠀⠠⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉⠓⠤⠶⠾⠟⠃⠀⠀⠒⠛⠒⠊⠁⠀⠀⠀⠀⠀⣀⡀⠀⡀
⠀⠀⠀⠀⢤⠀⠀⠐⠀⠛⠀⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠸⠋⠁⠉⠀⡀
⠀⠀⠀⠀⠀⠑⠂⣀⠄⣀⠀⠂⠀⠀⠀⠀⠀⢻⣄⠰⠦⠤⡀⠀⠀⠀⠀⠛⠦⠤⠤⠆⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⣈⠉⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠎⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⣶⣶⣤⠀⠀⠀⢀⣤⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣳⡀⠀⠀⢷⣹⡆⠀⠀⠀⢀⣤⣶⣄⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣯⢿⣿⣧⠀⠀⠂⠈⠁⠀⠀⠀⣾⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣧⢙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⣟⣿⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡄⣿⡇
⠀⠀⠀⣼⢯⣦⣿⠉⠀⠀⠤⠤⣀⠀⠀⠀⠀⣿⡆⠀⠨⣿⠀⣾⡇
⠀⠀⠀⠘⠛⠃⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠋⢠⣿⡇⠀⠀⠀⠀⡰⣖⠦⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡈⡼⡿⡿⠀⠀⠀⠀⢰⡁⣼⣤⠀⠀⠠⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⠟⠁⠀⠀⠀⠀⢸⣉⡁⠻⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⢈⠃⠉⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠔⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠟⠓⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⢠⡞⢻⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣸⣿⠀⡇⠀⠐⡹⠇⢘⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⣿⣶⠇⠄⠀⠣⣤⠞⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢉⠀⠀⠄⠙⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠛⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠉⠀⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣤⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣤⣀⣠⡴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡴⣆⠀⠀⢸⡇⣿⠀⠀⠀⣀⡴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠉⠁⣿⣿⣿⣷⡀⠀⠀⢀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣼⣃⣿⠤⠒⠉⠁⠀⣠⣤⡾⠿⣦⠀⠀⠀⢀⣤⡴⣤⡀⠀⠀⠀⠉⠲⢄⣸⣿⣿⣿⣿⣷⣄⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠸⣿⣇⠀⢀⢿⡄⠀⠀⢸⡏⠀⠈⡇⠀⠀⠀⠀⠀⢀⡿⠿⣿⡿⠿⡿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⡀⠸⠋⣿⣿⡄⣠⡄⢠⣄⡙⣶⣾⣇⣠⣶⠀⠀⢀⣾⣼⣿⣤⣾⣭⣽⣥⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠈⠉⠒⣾⣿⣿⣿⠀⠀⠀⠀⢙⣿⣿⣿⠀⠀⠀⠀⠸⠿⠿⠿⡿⠿⢿⠿⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇⠀⠀⠀⠀⠀⠾⠿⡿⡿⠿⠃⠀⠀⠀⠀⠻⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣜⣈⣧⣀⣀⣀⣀⣀⣀⣠⣇⣣⣄⣀⣀⣀⣀⣀⣤⣇⣧⣀⣀⣀⣀⣀⣀⣀⣤⣇⣀⣈⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣆⠀⠀⠀⠀⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⢿⣿⡷⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⣹⣧⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠻⠿⣀⠀⠀⠀⠿⠿⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀⠀⠀⠀⠘⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⢀⣴⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣴⣶⣗⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣀⣀⣶⣴⠚⢿⠿⠿⠟⠻⠁⠛⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⡦⢀⣠⣾⣿⣿⣿⣿⣿⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠃⠙⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠲⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠏⠉⡙⠢⣄
⠀⠀⠀⠀⠀⠀⡞⠡⠔⠊⠉⠉⠲⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣤⠗⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡈⠀⠂⢀⠂⢀⠛⡆
⠀⠀⠀⠀⠀⠘⡄⠀⠲⠄⠀⠤⠔⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠀⢸⣍⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⠀⠂⠈⠠⠀⢤⠇
⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡼⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡐⠀⠐⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⡄⡀⡀⡤⡄⡀⡀⣀⢀⡀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⡄⣀⢄⢀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢠⡀⢀⡀⢀⡠⣀⡀
⠀⠀⠀⠀⠀⠁⠁⠉⠀⠀⠁⠀⠉⠀⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠈⠀⠀⠁⠈⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠖⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡴⢧⠀⠀⢀⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡠⢧⠀⣀⡈⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠄⢸⠃⡴⠉⢹⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢐⡇⣏⣸⠁⠘⡄⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠳⣍⣀⡇⢀⢜⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⢦⣈⠁⣀⡴⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠬⠭⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠧⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠖⠆⠒⠲⠰⠢⠘⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠖⠲⠄⠦⠶⠓⠸


⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⡠⠊⡡⠤⠤⣄⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣊⣭⡭⠭⣕⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡋⠀⠀⠄⠉⠳⣄
⠀⠀⠀⠀⢠⠃⡞⠀⠀⠀⠀⠑⡄⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡞⠀⠀⣠⡀⠈⡇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⡊⠉⡳⠀⢀⡤⣄⠈⡆
⠀⠀⠀⠀⠘⡆⢇⠀⠀⠀⠀⣠⢇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠑⠖⠚⠙⠧⠖⡁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢧⣼⠗⠲⣼⡇⢘⢆⡇
⠀⠀⠀⠀⠀⠈⠪⣓⣢⣖⣊⠵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣈⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡙⠦⠬⠶⢟⡫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠊⠉
⠀⠀⠀⠀⡀⡤⠆⠆⡄⡴⢠⢠⢄⢠⠴⣢⠄⠀⠀⠀⠀⠀⠀⠀⠀⢦⡆⡄⡄⢤⢠⡄⢠⠴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡤⠤⣀⠖⠤⡀⣴⢴⢢⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠏⡆⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢦⡀⠀⠀⠀⠀⠀⠀⡜⡎⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⢆⠳⡙⢆⠀⠀⢀⡤⡞⡜⠀⠚⠂⣿⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⠀⢀⡱⠬⠖⠛⠉⠈⣻⣰⡥⠤⠖⠊⢫⢳⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠴⠗⠒⠉⠁⣀⡠⠤⠖⠒⠉⠉⠈⢷⡀⠀⠀⠀⢇⢇⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠉⢁⣀⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⢀⣀⡬⣧⢀⣀⠤⢾⣿⣠⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡳⣤⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⣖⡺⠯⠗⠓⡏⠀⠀⠀⢸⣼⣿⡿⠟⢡⡾⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⠼⢄⡀⠳⠗⠀⠀⠀⠀⠀⣀⡤⡴⣖⡹⠷⠛⠋⠁⠀⠀⠀⠀⣇⣀⣠⣴⣾⠟⠋⢀⡴⠋⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⣾⣷⡜⠌⠓⢖⣉⡹⠀⢀⣤⢴⣲⣹⠓⠛⠉⠀⠀⠀⠀⣀⡠⠤⠔⠒⠻⡍⠳⡜⢯⠁⣠⠴⠋⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠚⠉⠁⠀⢀⣀⡤⠴⠒⢎⠉⠀⠀⠀⠀⠀⠀⠈⢦⡈⢢⡙⣅⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⣽⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⠀⢀⠀⢀⣀⣀⡠⠤⠶⢶⣯⣭⣭⣭⣭⣭⡭⠭⠶⣒⡤⠄⣀⠀⠀⠀⠑⣄⠙⣌⣳⡀⠀⣀⣴⣿⣿⠟⣡⣾⡿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠁⠒⠫⠵⠶⠶⠮⠍⠚⢉⣤⣾⣿⡿⠋⣁⣾⠟⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠉⣠⡾⠋⠁⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠛⠉⣀⡴⠚⠁⠀⠀⠀⠀⢀⣴⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠋⠁⣠⠴⠊⠁⠀⠀⠀⠀⠀⢀⣤⠿⠋⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠋⠉⢀⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠛⠛⠉⠉⢁⣀⣠⡤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⣠⣶⡿⠋⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⣀⣴⡾⠛⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⢀⣠⣴⠾⠛⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⣤⣶⡶⠶⠞⠛⠋⠁⠀⠀⣀⣠⣴⡾⠟⠋⠁⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⢄⣀⣀⣀⣤⣴⣶⡾⠟⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠥⠄⣀⣀⣀⣀⠠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⢤⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠉⠙⡚⢯⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡀⢶⣶⡄⠇⠀⣱⣻⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠱⠃⠀⣘⣹⠀⠀⠀⡠⠚⢁⣀⢀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡱⠃⠀⠀⢠⡏⠀⠀⢰⠁⠀⡄⣠⡦⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠌⠀⠄⢰⣄⡀⢀⣀⡴⠋⠀⠀⠀⠀⣠⠏⠀⠀⠀⢸⠀⠀⠁⢿⡆⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣄⠻⣿⠿⠛⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⢇⠀⠀⠈⢷⢀⠀⠀⢆⠠⠒⠨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠠⣄⠉⠁⠈⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠈⢿⣿⣾⣿⡆⠀⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⣘⣓⣒⣀⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡉⣩⡶⢂⡦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣴⣂⣂⡑⣊⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠂⠀⠀⠀⠐⠢⢤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢀⣠⠠⠅⠴⠈⠉⡉⠘⠒⣦⢤⢀⡀⠐⢉⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣠⢂⣤⠤⠔⠀⠀⠀⠀⠀⠀⠀⢤⢀⣀⠙⢦⡀⠐⣘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠊⠼⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣹⣶⣾⡄⠰⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠁⠀⠀⠀⠈⠙⠲⠤⣀⣀⣀⡀⣠⣠⣤⣶⣿⣿⠟⠜⠀⡀⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠍⠁⠉⠁⠀⢀⣰⣄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢄⣀⠀⠀⠀⠀⠐⠂⠐⣊⣉⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠉





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

⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠐⠂⠐⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣀⠀⠀⡥⠀⠀⠀⠀⠤⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢠⡎⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢃⠀⠀⠊
⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠁⠠⠀⠄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠐⠂⠄
⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡜⠀⡃⠀⠀⠀⡤⣖⡧⠀⣂⣀⠔⢒⠄⢀⣠⠀⠀⠀⢠⠀⢀⢢⣄⣐⠄⠀⠀⡅⠀⠀⠀⠀⠀⠀⡐⠀⠀⠁
⠀⠀⡀⠀⠠⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠨⢰⡌⢇⠀⢀⣼⠀⢠⠔⠀⢠⡜⢸⢀⠇⢀⡠⠮⠂⠀⠀⡇⠀⠀⠙⠦⠀⠙⠁
⠀⠀⠃⠃⠉⠁⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠋⠁⠚⠁⠐⠚⠀⠙⠀⠋⠀⠀⠊⠁⠛⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⠆⠀⠀⠁⠶⠦⠠⠶⠦⠼⠿⠧⠷⠄⠴⠶⠴⠄⠠⠶⠶⠦⠿⠇⠀⠘⠀⠀⣖⠀⠤⠀⢐⡀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢀⣀⠀⠀⠈⢠⡄⠀⠀⠈
⠀⠀⠒⠀⠀⠀⠀⠄⠀⠀⠀⢠⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠇⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠣⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⠉⢀⣀⣀⡀⠊⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠢

⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠋⠛⠸⠀⠀⢸⡏⠉⣻⣿⠋⠻⣿⡿⠿⠛⢻⣿⠉⢿⠹⡏⢻⠨⠃⠀⠈⡏⠉⣹⣿⠋⠻⣿⡿⠿⠛⢻⡟⠉⠏
⠀⠈⠀⠇⠀⠈⡇⢠⡇⠙⠀⠀⣼⠁⠀⠀⠸⣟⠐⠀⢸⠁⠀⠀⠃⠀⠀⡇⢠⡇⠙⠀⠀⣾⠁⠀⠀⢸⣇⠸
⠀⠐⠀⠀⠀⢠⠇⠸⠇⠀⠀⢰⡇⠀⠀⠀⠰⡏⠀⠀⠈⠀⠀⠀⠀⠃⢀⡇⠸⠇⠀⠀⢰⡇⠀⠀⠀⢸⠋
⠀⠀⠃⠀⠀⠿⠀⠀⠀⢀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠓⠀⠀⠺⠀⠀⠀⢀⠀⠸⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠉⢀⠐⠂⠒⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠠⠟⣆⠀⠬⡁⠄⢉⡟⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠁⢈⡃⠠⠆⠀⠀⠀⠀⠀⠀⣡⠀⠀⡀⡙⠇⡐⠀⢨⠄⠈⠵⠊⠀⠀⠘⠛⠀⠠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣿⠀⠀⠀⠀⠀⣸⣿⣄⠀⠆⠀⠀⠈⠈⠉⢰⠆⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⢄⠀⠀⠈⠀⠀⠀⠀⠈⠈⠻⠛⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣿⡀⢀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣁⡆⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⣤⡀⢸⠿⡷⢦⣻⢲⢲⡖⣾⡗⡦⡏⡷⡟⡟⢹⣶⣠⡇⠈⠉⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠄⠀⠀⠁⠀⡀⠀⠉⠉⠈⠈⠀⠉⠀⠀⠁⢉⠁⠁⠀⠈⠀⢉⡇⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⢒⠇⠀⠙⠛⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⢀⡁⠐⠀⣠⠀⢤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⡖⠀⠀⠀⠂⢀⡀⣙⠕⠀⠀⠘⠦⠤⠀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⡟⠚⠃⠀⡆⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣉⠑⡋⠐⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣄⣀⠀⡀⠀⠈⠛⣷⣶⡄⣰⠇⠀⠰⣄⠀⠀⠀⠀⠀⠁⢀⠨⢁⠊⠁⣄⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⣼⡟⣻⡟⠀⠀⠀⠘⠛⠛⠥⠂⠀⠀⠹⣷⡆⠀⠀⠀⠀⠀⠀⠋⣀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠉⠀⠀⠀⠐⠀⠉⠐⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡀⠉⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠰⠇⠀⠈⠁⠀⠀⠖

⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢉⡉⠉⠛⠛⠿⠟⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣾⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢻⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣦⣄⠀⠀⠀⠙⠻⠿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣄
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢀⣀⣤⣦⣜⣶⠟⠉⠀⠀⠀⠀⠀⠀⠈⠲⠇⡩⠛⠓⠁⠣⠘⠕⢊⡡⣨⠜⡟⣶⠃⠄⡈⢄⡔⠈⡀⠀⠁⢀⡰⠀⠉⠀⡜⣸⠂⠀⠀⠀⠀⠂⡄
⠀⠘⣯⣿⢽⣟⣿⠃⠀⠀⠀⠀⠀⠀⢀⢀⠤⠀⠌⠂⠑⠙⠊⠔⠡⢊⠄⡁⠠⢑⢸⠇⠀⠈⣀⡭⡀⠈⢀⣼⣷⡞⠁⠀⠠⢰⢁⡟⠀⠐⠀⠀⢊⠀⠀⠐⠃
⠀⠀⢋⡡⢞⡾⠁⠀⠀⠀⠀⠀⠔⠉⠈⢀⠀⠤⠰⠂⠐⠀⠂⠒⠀⠄⡈⠊⣤⣴⣴⣌⣠⠁⠎⠀⠀⣡⣾⣿⣿⡟⠀⢀⢠⡇⡘⠀⠀⠀⠀⠀⠄⠒⣂
⠀⠀⢟⣅⣮⡆⠀⢀⡠⠂⠁⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢼⡏⣿⡧⣻⡎⠉⠆⢂⣼⣿⣿⣿⠟⠀⢀⢀⠊⠐⠀⠀⡀⠤⠔⣢⠫⠐⠀⠂⠄
⠀⢀⣄⣿⢿⠀⡐⠂⠀⡠⠘⠀⠀⠀⠀⢀⣠⣴⣶⣞⣛⢿⣿⢿⣶⣶⣤⣈⠻⠿⠿⠟⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⢀⠎⡴⠀⠀⢨⣴⡮⣜⣖⣫⠜⡄⣇
⠀⢘⣯⠷⣿⠂⠀⡠⠊⠀⠀⠀⣀⣴⣾⣿⣿⠿⠋⠁⠉⠀⠉⠙⣶⣛⣿⣿⣷⡀⠀⠀⠠⣼⣿⣿⣿⡟⠀⠀⠀⠀⢊⠜⡡⢪⡴⣟⡇⣯⢵⡸⠆⡫⠩⠈
⠀⢘⣾⡾⠁⠀⠌⠀⠀⠀⣠⣾⣿⢟⣻⠟⠋⠀⡀⢨⣔⣧⡄⢔⠘⣿⣿⣴⣿⣿⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣴⣻⣃⡇⢻⠜⢃⠐⠂⠀⠆
⠀⠀⡟⠀⠀⠂⠀⠀⣠⣾⣿⠟⡝⡡⠱⠀⣀⢨⡾⣿⣿⣿⣻⠐⡆⢸⣿⣭⣿⣿⡇⢸⣿⣿⣿⡟⠀⠀⠀⠁⠄⠀⢈⣾⡋⢓⠃⠜⠠⠀⠀⠀⢀⣀⣤⢤⠄
⠀⠀⠀⠀⠂⠀⢀⣾⣿⣿⣧⢞⣤⢄⡆⡐⣠⣛⢏⠀⢆⠋⢿⠀⡇⢸⣿⣿⣿⣿⠁⣿⣼⣿⣿⡁⠀⠠⠀⠀⠠⢀⣿⣛⣉⠃⡘⠀⠁⢀⡴⠛⠉
⠀⠀⠀⠌⠀⣰⣿⣿⣫⢻⢿⠻⠍⠿⠀⢤⣯⣦⣷⣥⣂⣳⣻⣿⠇⣼⣿⢟⡧⠜⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⡠⠚⠋⠉⠉⢀⠀⠀⠐⠋⢀⠀⠤⠀⣠⣤⡀
⠀⠀⠘⠀⣰⡿⣟⣃⠄⠀⠀⠐⠀⡌⢸⠋⠣⠊⢿⣿⣄⣞⣧⠂⣰⣿⢣⡔⠄⠀⠠⠹⣾⣿⣿⣿⣿⣶⡦⡄⠖⠘⠙⡉⠉⠉⠩⠛⠫⠱⡄⡀⠲⢿⡏
⠀⠀⠃⢰⣿⡷⡷⣧⠀⠀⠀⠈⡂⢱⠹⡀⠀⢀⣼⣶⠟⠃⢁⣼⣟⣿⡿⠃⢀⠀⠀⠑⣝⠿⣿⣿⠟⠋⠁⡀⡄⢠⢢⠂⠄⠆⠦⠤⣀⠀⠈⠂⠑⠆⡉⠠
⠀⠀⢠⣿⠯⡜⠉⠍⢐⣁⣀⠀⣑⠇⡈⠀⠀⣾⣿⣏⣠⣶⢿⠷⠷⣋⣀⠀⠰⠀⡆⢤⠀⢠⠈⠃⣀⠔⠈⠀⢀⣀⣤⣤⣤⣤⣤⣀⠀⠉⠂⢄⠀⠀⠘
⠀⠀⣸⡟⠀⢄⠀⠀⣾⣿⠗⡨⡨⠁⠅⠀⢸⣿⣿⣿⠽⢒⣍⣿⣼⣿⣿⣷⣦⡀⠁⠈⠠⠈⢀⠆⠁⢀⣰⣾⠿⠿⠿⠿⠿⠿⠿⠿⣿⣶⣄⠀⠙⢄⡀⠈⠄
⠀⠀⣿⢮⢰⠆⡑⢾⠁⠘⢁⠠⠆⠧⡁⠀⣿⣿⡿⢁⣜⣷⠟⠉⠀⠈⢿⣿⣿⣿⡄⠀⢀⠔⠁⠀⡴⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠙⢷⡄⠈⠳⡀⡁
⠀⠀⣟⠀⢘⠃⢌⡄⠀⣴⢶⢾⣷⡠⠀⢸⣻⣿⠁⣾⣽⠃⠀⠀⣦⣄⣀⠀⠉⠙⡎⡖⠚⠀⢠⠞⠤⠤⠞⢳⡄⠀⠀⠀⠀⠀⠀⢯⠀⠉⠳⣌⢻⡆⠀⠙⡅
⠀⠀⠘⣤⣶⣶⠄⠑⡀⢸⢹⢹⠞⣯⠀⢸⣿⣿⠀⣱⣿⡀⠀⠀⠘⣿⣿⠇⠀⠀⢇⢈⠀⠀⡎⢠⠖⠲⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠁
⠀⠀⡌⠜⠿⣻⣔⡤⠢⡐⠀⠠⡂⢸⡄⠘⣿⣿⡄⠹⢞⣷⣄⠀⢰⣿⡿⠆⠠⠀⣠⠖⠀⢰⠓⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⢺⠂⠀⠁
⠀⠀⣦⠘⠢⡏⠽⢧⠀⢀⠀⡀⢧⣇⣻⡆⠘⢿⣿⣦⣈⠁⠋⠿⠋⠉⡀⣠⠴⠋⡡⠀⠀⠸⣠⣆⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆
⠀⠠⣏⣆⡁⠀⡌⠌⣴⣖⣦⡘⠀⢙⡯⣿⠀⠀⠈⠛⡡⢍⢁⣀⣐⣉⠍⠥⠚⡡⠐⢀⠀⠀⢻⣿⣄⠀⡏⠀⠙⢆⢠⡀⠀⠀⠀⠀⠀⠀⣴⠋⢳⠆⠀⠀⡇
⠀⠀⢿⡿⠈⠑⢄⡐⢿⢟⣞⠛⢆⠀⢻⣾⣷⡀⠀⠀⠀⠀⢄⠀⠀⢀⣀⡄⠂⠁⣡⣤⣖⠀⠈⠻⣿⣆⠙⠢⠤⢄⡀⠈⠓⢦⢣⠀⠀⠀⠈⣲⠏⠀⢀⡼⡇
⠀⠐⢸⢲⣷⣶⣦⡁⠈⠸⢹⠀⡀⠃⠌⠹⢿⣿⣷⣤⣀⠀⠚⠈⠋⠘⠀⣁⣤⣾⣿⡿⡀⢀⠂⠀⠙⠿⣷⣄⠀⠀⠳⡄⢸⡜⢰⡀⢀⣠⠞⠁⠀⡄⠪⠯⡗
⠀⠀⠈⠨⠨⢩⢿⡿⣔⠐⠀⢀⠁⠣⣖⣀⠀⠙⠟⢿⣿⣿⣿⣷⣷⣿⣿⣿⡿⡿⢋⠀⢀⠀⠠⠁⢀⠀⠈⠙⠳⠤⠄⢌⢼⠡⠤⠖⠉⠀⡀⠔⢉⡒⣝⣝⡃
⠀⠀⣄⠰⣄⢆⡆⠘⠻⠧⠈⠈⢢⡀⢀⠁⠠⣖⡄⡠⠄⠍⠩⢙⢙⢋⡉⠡⡔⠈⠈⠀⠀⠀⠐⢆⣠⣿⣦⠂⠀⠀⠀⠀⠀⠀⠀⠄⠖⠈⢀⠠⢎⢪⡵⢣⠁
⠀⠀⠠⡠⠉⢻⣿⣿⣆⢰⠀⠢⠄⡵⡯⣅⡲⡀⠁⠀⠐⠀⢁⠉⠊⠁⢀⠀⣠⠄⠖⢲⠂⢀⡄⢬⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢎⠝⣪⠟⡨⠃
⠀⠀⢁⢋⠊⠐⠎⢿⣷⣿⣦⣤⡁⠀⡁⠊⣔⢎⡉⠺⢄⡤⣠⢄⡂⠱⠵⣯⣇⣤⣧⡦⣆⣴⣿⣿⣿⣯⡟⠁⠀⠀⠀⠀⠀⠀⢀⡠⠿⣚⡱⠞⣡⠞
⠀⠀⠀⡀⠈⠐⠲⣦⣿⣮⣿⣿⣿⣲⣶⡀⢠⠋⢄⠐⡀⠖⢈⡁⠂⠀⣘⠻⡿⢿⣟⡫⠿⠛⠻⣿⠿⠉⠄⠀⠀⠀⠀⠀⠀⣠⡤⠶⠚⣩⡴⠞⠁
⠀⠀⠐⠊⠵⠒⠦⠄⡛⠿⠿⢿⢟⠹⡛⠛⠻⠶⢆⣻⢷⡨⠀⠹⡗⡶⠿⠻⠏⠹⠈⠁⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠠⡾⠾⠔⠃⠉⠁⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣾⣶⣾⣷⣶⣶⣿⣶⣶⣶⣶⣷⣾⣶⣷⣶⣷⣶⣶⣶⣶

⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⢀⠄
⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣇⣀⣠⣤⣬⣄⣤⡤
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠙⠁⠀⠀⠀⠀⠀⢲⣶⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣀⡀
⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢹⣿⣿⣿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡏⠁
⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⡟⠸⠿⠟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⠟⢻⡿⢿⣿⣿⣟⡿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⣿⢸⣿⡇⠀⢸⣿⣿⣿⠏⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⠀⠀⢸⠁⠿⠀⠀⢹⣧
⠀⠀⠀⠀⠀⢻⣾⣿⠇⠀⢸⣿⣿⡟⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⠁⠀⢸⡆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⢸⡇⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⠾⠀⠀⠀⠀⠈⠁⠀⠀⠀⠻⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠷⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠄⠀⠤⢠⣀⠀⠀⠈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠢⣄⡀⡀⡀⢠⣷⣧⣠⣿⣇⠂⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⢈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣋⡀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠄⠀⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢞⠂⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠏⠁⠄
⠀⠀⠈⠉⠉⠉⢩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⠶⠿⠛⠛⠛⠛⠛⠋⠉⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⠠⢹⠏⡟⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠟⠀⠀⠸⠐⠀⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠓⣤⠤⠤⠤⠤⠶⢍⡀⠀⠀⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠃⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠓⢄⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⡤⠃⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢚⠦⣀⢀⠓⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢸⠀⠀⠁⠀⢱⠀⠑⠂⠔⠂⠤⠤⠤⠴⢂⠀⡖⠊⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣂⣀⡀⢀⡤⠐⠒⢷⡦
⠀⠀⠀⠀⠀⠀⠘⠴⡀⢠⢧⡄⢆⠰⠤⣀⠀⡷⠶⡏⠁⢠⠀⣹⠹⣄⠀⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠳⡳⠈⠉⠀⠀⠘⠦⠤⠔⠣⢀⠌⠁⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⡀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠂⠽⣝⠉⣝⢉⡎⡃⠀⢤⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠛⢆⠈⣷⡱⣤⠜⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⢄⠀⠀⠀⠀⠉⠀⠀⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⡦⠤⣀⠀⠀⣀⣰⠖⠉⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠤⠇⠀⠀⠀⠀⠘⡇⠀⠀⠀⣸⡑
⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠜⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⢸⠀⠀⣀⣀⠤⠄⠃⠀⠰⡤⠤⡴
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⡩⢷⠈⠉⠉⣹⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⠀⠀⢀⠔⠊⢆⡀⠑⠒⢶⡁⠉⣉⠽⣄⠀⠀⠀⢀⠄⠀⢇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠡⠀⠈⠀⠀⠀⢋⠉⠉⣷⠀⠙⠤⠤⠋⠳⡀⠐⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⠑⠊⠁⢇⠀⠀⠀⠀⡰⠁⠃⠀⠀⠀⠀⢀⠇⠐
⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠤⠊⠀⠑⢄⡀⠀⠀⠀⡠⠊⠒⠒⠒⠉⠀⠀⠈⠢⠄⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠋⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠿⠛⠉⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡟⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠛⠉⠛⠁⠀⠀⠈⠀⠀⢠⣿⣿⠏⠀⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⢀⣀⣤⣄⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠈⠛⠳
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡈⣿⠘⣿⣿⣿⣷⡀⠀⣇⠀⠀⠀⠀⣄⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⡇⢠⣿⣄⠀⠈⠃⠀⠈⠁⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⡄⠄⠀⠀⢠⣾⣿⣿⠇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠻⣿⡿⢻⣿⣿⣿⣿⡄⠈⠙⠛⣑⠦⠤⠞⣂⠉⠉⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣧⡀⠀⢠⣿⣷⣶⣾⣿⡆⠀⠀⠀⢸⣿⣿⠀⣀⡀
⠀⠀⠀⠀⢰⣿⡿⢿⣿⣿⣿⣿⢿⣿⡿⠋⠀⠀⠉⡛⠿⢾⣿⢿⣿⣿⣿⠉⣿⡀⣠⣶⣿⡟⢿⣿⣭⠁
⠀⠀⠀⠀⡀⢈⠁⠀⢸⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⣿⣿⠀⠘⡿⠛⠛⠁⠀⠈⠻⢿⣷
⠀⠀⠀⠀⠉⠁⠀⠀⢸⣿⣿⣿⠀⡀⠚⠀⠀⠁⠀⠛⠀⠈⠀⣴⣿⣿⣿⣆⠀⠁⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⣿⣦⣄⣀⣀⣠⡄⠀⠀⣸⣿⣿⣿⣿⣿⣧⠀⠀⠙⠀⠀⠘⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣦⣠⣿⣿⣿⣿⣿⣿⣿⣆⢠⣾⣶⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⠟⢿⣿⣿⣿⣿⡜⢋⠝⠛⠛⠛⠛⠛⠫⠤⠀⣿⣿⣿⣿⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⡇⠘⣷⣿⣿⣻⡟⣛⠁⠀⠀⠀⢀⡀⢀⣀⠀⠀⠈⢛⡋⢙⣛
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠃⠀⠛⠙⠿⠟⠁⠿⠦⠀⠐⠛⠉⠀⠀⠉⠁⠀⠀⠚⠛⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⠉⠉⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⠳⠶⣤⣤⣤⡴⠞⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⡀⠀⠀⠀⠀⣀⣠⡾⠋⠀⣀⣠⣤⣶⠶⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⡈⠉⠛⠛⠛⠛⠛⠋⠉⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠀⠈⠉⠛⢷⣤⡀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⢼⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⣸⠇⠀⠀⡀⠀⠀⣀⣠⣴⠟⠁⠀⣰⠏
⠀⠀⠀⠀⠸⣧⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠘⣷⡀⢻⡆⠀⠀⠉⠛⠛⠋⠉⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠙⠷⣦⣄⣀⣀⡀⠀⢸⣧⠀⠻⢶⣤⣤⣀⣤⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠉⠉⠁⠀⠀⢸⡟⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⣄⣀⣀⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⠻⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⣿⠈⠻⣦⣈⣻⣶⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠀⣷⣄⠙⢷⠌⠉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠻⣯⡙⠃⣿⡏⠻⠿⠛⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣄⠰⣶⣾⡄⢻⣤⣤⣤⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠻⣦⡈⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢶⣄⠙⢷⣌⠻⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡈⠻⣦⣨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⢷⣄⠙⠷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣤⣶⣤⣸⣿⢠⣌⣻⣦⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠛⠛⣧⠘⡿⠿⠆⠙⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠿⣯⡉⠻⣦⡀⣿⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣀⡐⢷⣄⠙⢿⠀⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡙⢷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠉⣭⠙⠒⠒⠒⠒⢉⠙⠒⠒⠒⢻⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⠉⠉⠙⢢⣄⣀⣤⠬⣁⣀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢃⡼⠁⡏⠀⠀⡀⠈⢻⢶⣧⠀⠀⢀⣠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠋⠩⡀⠀⣸⣤⣀⣉⡰⠃⠸⣄⡤⠖⣉⡽⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⠀⠀⠀⠀⠑⠦⣝⡋⠀⠀⠀⠀⠀⠀⢀⣀⣤⢚⡡⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣞⠣⣄⠀⠀⠀⠀⢀⡎⠉⣿⣒⢒⣺⠋⡹⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡆⠗⠀⠏⠑⠠⡔⠉⡇⢀⠂⠀⢀⠃⣞⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠳⠞⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⡘⢠⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠒⠤⢀⡀⢴⢰⠃⠀⠀⠀⡇⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡠⠴⠋⠀⠀⠀⠀⡆⡃⠀⠀⠀⠀⢠⠔⠋⠂⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡕⠀⠀⠀⣀⠴⠥⡀⠀⠀⠀⡠⠃⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⡶⢄⣈⡷⠀⠀⠈⣢⠤⠒⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠀⠀⠈⢳⠧⠶⢊⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠃⠀⠀⠀⣴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⡟⠀⢦⣀⣀⠀⠀⠀⣀⣀⠠⠜⢄⠀⠐⡠⠀⠀⠀⢢⠉⠉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⢧⣀⠀⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠈⠉⠒⠒⢲⠁⠀⠀⠀⠠⠀⢀⠀⠀⡇⠀⠀⢸⠀⠈⡇⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡞⠒⠦⢄⣀⠀⠀⠀⠀⠀⢱⠀⡠⠊⠀⠀⠑⢄⠀⠀⠌⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⠀⢀⠂⠀⠐⠒⠩⡉⠉⠉⠉⢣⡀⠀⠀⠀⠈⠉⠉⠙⣳⠃⠰⡀⠀⠀⠀⠀⠀⡆⢠⠂⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠸⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠊⠉⠁⠀⠈⢆⠀⠀⠀⠀⡰⢁⠆⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⢨⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠂
⢹⣿⣿⣷⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠏
⠘⣿⣿⣿⣷⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠛⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⡿⠃
⠀⠹⣿⣿⣿⣷⣄⠀⠙⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠹⣿⣿⣿⣿⣷⣤⣀⠈⠉⢁⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣶⣭⣭⣙⣛⣛⣛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⢡⣶⣬⡻⣿⠀⠀⣴⣿⣿⣿⣷⡄⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⠀⣿⣿⣿⣿⣷⠘⣿⣿⣷⡜⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⣦⣄⠙⠿⣿⡿⠟⠁⣠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣦⡹⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇⣿⣿⠟⣋⡙⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣶⣤⣄⡀⠘⠿⣿⣿⣿⠟⠁⣫⣴⣿⣿⠇⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⡿⣫⣾⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡙⠿⣿⣿⣿⣿⣿⣿⣟⢡⣾⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠙⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡟⠀⢸⣿⣿⣿⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇⠀⢸⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⢹⣿⣿⡏⣿⣿⡏⠀⣿⣿⣿⠇⠀⠀⠿⣿⠇⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⢿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠔⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣍⠀⠁⠀⢂⠲⢀⡀⠀⠀⠐⡡⢂⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢊⠈⠁⠀⠀⢠⡀⣋⠀⠈⠉⠚⠠⠅⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠐⠐⠀⠄⢀
⠀⠀⠀⠀⠀⡀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⣼⢤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⡀⠉⠀⠂⠀⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⣀⣸⡀⠀⠀⠀⠈⠈⠐⠘⠀⢰⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡐⠈
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢇⠀⠀⠀⣠⡿⢛⡿⠈⠐⡠⢄⣠⠀⠀⠀⠀⠀⠈⠉⠚⠃⢇⡖⡤⣀⠀⠀⠀⠀⠀⠑⠀⣀⠔⢢⠜⢠⡀
⠀⠀⠀⠀⡠⠸⣽⣿⠿⠗⢛⣴⣵⢄⢘⡶⣶⡇⢀⠄⡀⠀⠀⠉⠀⠀⠒⠠⠀⡀⠀⠀⠀⠀⠁⠙⠽⢃⢂⢤⠀⠀⠄⠠⠃⠀⠂⣴⣿⣦⡀
⠀⠀⠀⢠⠽⡺⢁⡰⣢⣿⡶⢣⡇⠘⡻⢀⣍⡗⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⢄⣀⡀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡆
⠀⠀⠀⡾⠄⢡⣳⠉⠹⡿⠂⢴⡟⢲⠁⠀⢖⠅⢀⡐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠠⠀⠀⠀⠀⠠⠿⠿⠿⠛⠁
⠀⠀⢰⣄⠀⣘⠉⢠⡔⠂⢹⣿⠰⠈⠓⠊⠀⠁⠀⠀⠐⠀⠀⠀⠉
⠀⠀⢸⠀⠩⡇⢐⡺⠋⠸⠃⢐⣤⡟⡄⠀⠀⠈
⠀⠀⠀⠀⠴⠇⢈⡟⢶⣤⠶⢦⣟⣴⠇
⠀⠀⠀⠈⠠⢺⡞⢛⠻⠮⡶⡾⢞⣁⠈
⠀⠀⠀⠀⠘⠿⠗⠀⠀⠒⢃⣍⣄⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠘⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠄⠀⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣡⡀⢠⠴⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠄⠞⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠺⡶⠿⠿⡷⠀⢠⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢠⠄⠤⣀⡤⠔⠢⡀⠀⠀⠘⡷⠤⠒⠉⠀⡀
⠀⠀⠀⠈⠁⡀⠀⡿⠀⠀⡇⢀⡀⠀⠀⠀⢀⡎⠉⠑⡍⠀⠀⠀⢀⣧⢠⡼⠵⢤⣀⣱⠀⠄⠊⠀⠀⢘⠉⢵⠈
⠀⠀⠀⠀⠀⠁⡰⠥⠀⠀⢣⠀⠁⡠⢤⣴⠟⠻⠀⢀⣼⠀⠀⠐⣅⠈⢲⡀⠀⣤⠑⣾⠉⠛⢆⠀⢠⢬⠉⢁⣤⡀
⠀⠀⠀⠀⠀⢠⠃⣠⣀⣠⣌⣆⠀⠣⡴⡅⠀⠀⠙⡿⠿⠀⠀⠀⢈⡉⢶⠬⠦⣄⣠⡏⣦⠤⠚⠀⡈⢊⡴⣋⠜⠁
⠀⠀⠀⠘⢁⣿⣿⣿⣿⣿⣿⣿⣆⠀⣷⠹⡄⠀⣐⣒⣒⡄⠀⠀⢀⣸⠈⢆⣀⡤⠛⠒⠊⠀⠀⠀⡤⣋⠜⢁⣀⡀
⠀⠀⠀⠀⢎⢻⣿⣿⣿⣿⣿⡿⢈⠀⢸⠖⠚⠚⠒⠒⠒⠚⣆⠀⠀⠙⠀⠀⠀⠀⣠⣴⠶⢒⡚⠿⣿⠁⠀⠀⢸⣏
⠀⠀⠀⠀⣈⠑⠒⠂⡒⠒⠒⠈⠁⠉⠘⠒⠦⠤⠤⠦⠆⠒⠃⠀⠀⣤⡆⠀⠀⡾⢋⠖⡒⠒⡌⢢⠙⡄⠀⠀⢸⣍⠁
⠀⠀⠀⠈⠚⠁⠀⠀⠻⠯⠀⠀⠀⠀⠀⠄⠀⠀⠀⠔⠒⠄⠴⠀⠀⠈⠁⠀⠀⡇⠸⣀⡛⠘⡜⡸⠀⡇⠀⡆⢸⠮
⠀⠀⠀⠀⢴⢀⡶⠚⢲⡀⠀⠓⠺⠰⠀⠀⠀⠀⠀⠊⠒⠤⠴⠂⠀⢠⣤⠀⣄⠹⣤⠈⢉⠉⠐⣡⠞⠀⠀⠧⠿⠁
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠁⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠀⠀⢠⠴⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⣠⠞⠛⠳⡄⠀⢰⡆⠉⠂⣴⠶⠶⠄⠈⢀⡀⠀⢀⡄⠀⣠⠤⢤⡒⠀⢸⣂⣀⣀
⠀⠀⠀⠀⠐⣆⠀⠀⢀⡇⢰⡇⠀⠀⠀⠉⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⣾⢧⠀⢸⡇⢰⡇⠀⠈⠁⠀⢸⠉
⠀⠀⠀⠀⠀⡌⠛⠒⠋⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⣟⠛⠛⠂⠀⡟⠈⣇⢹⠃⢺⢁⠀⠀⢀⡀⠸⠶⠖⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢧⣀⣀⣠⠄⠀⣸⡇⠈⠁⣿⣤⣤⡄⠸⠃⠀⠘⠟⠀⠈⠦⠤⠖⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⢒⣤⣤⡄⢀⡴⣊⠥⢄⠉⢆⠲
⠀⠀⠀⠀⢠⣄⠈⠉⠀⠀⠠⡆⠀⠀⠘⠀⠈⢫⡂⢠⠛⠒⠠⢼⡀⠛⠑⠙⠀⠀⣀⠁⡀⡄⢠⣾⡔⠁⠀⢀⠃⣼⢠⣤
⠀⠀⠀⠀⠀⠁⢰⠭⡆⢰⠯⡆⠀⡟⡇⠀⠀⠀⡸⠡⠀⠀⠠⠀⠈⠟⢠⣄⣤⠀⠳⣄⢀⡗⠋⠀⠀⠀⢴⣧⡴⠃⢸⠏
⠀⠀⠀⠀⢶⣾⣯⡿⡽⠽⠯⡿⠿⠿⣿⠟⠀⠘⢇⠀⠀⠀⠀⠀⠀⡆⠈⠻⠃⠀⢂⡬⠛⠬⡀⠀⠀⠀⠔⠈⠀⢻⣮⡄
⠀⠀⠀⠀⢸⣿⠀⣶⡇⠀⣿⡇⢠⣿⣿⠀⠀⢀⡠⠣⡀⠀⠐⣂⣀⡇⠀⠀⠀⡰⠉⠀⠀⠀⠉⢦⠀⠀⠀⠀⣦⣠⠟⠁
⠀⠀⠀⠀⢸⣿⠀⣿⡇⠘⠿⠇⠀⠿⣿⡆⠀⡉⣀⢀⠙⠉⠀⠀⠀⠀⠐⠀⠙⠁⠀⠀⠀⠀⠀⠸⠟⠄⠀⠀⠸⠛⠋
⠀⠀⠀⠰⠿⠿⠿⠯⠿⠿⠯⠿⠽⠽⠽⠿⠃⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠎⠉⠉⠁⢈⣉⠡⣬⣩⠲⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣶⣾⣿⠳⠓⠉⠀⠀⠀⣠⣾⡿⠇⢫⣯⡀⠘⢛⣻⣟⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡴⢋⣶⣽⡯⠊⠀⡄⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⣮⠠⠖⠋⠁⠉⢻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⡎⣰⢿⣿⠏⠑⠒⢶⢧⣀⠀⣠⣿⡿⠁⠀⠀⣠⠞⢹⡆⠀⠀⠀⠀⣼⠉⠻⣼⣇⡀
⠀⠀⠀⠀⠀⠐⣿⣟⡛⠀⠄⠀⡠⠎⠀⡸⢋⣿⣿⠁⠀⣠⠞⠁⠔⠘⠁⠀⠀⠀⢠⡏⠀⠀⠬⢿⣿⡀
⠀⠀⠀⠀⠀⢸⡏⣟⠀⠠⠼⠊⠀⢠⣟⣀⣼⣿⣅⣠⣼⣁⣠⣎⣀⣾⢀⣀⣀⠀⣎⡠⠊⠀⠀⠸⠁⢷⡄
⠀⠀⠀⠀⢠⣿⠖⣦⣶⣶⡿⢿⣷⠛⠛⢻⣿⣿⡿⠋⢉⠕⠛⠋⠉⠈⠙⠛⢛⡿⠛⠿⠿⣶⣦⣦⡄⠘⣧
⠀⠀⢀⡴⣾⣿⠾⠋⠁⠀⣠⠋⠀⠢⠀⢸⣿⣿⢀⣴⠂⠲⣶⡄⠁⠐⢀⣴⡞⠀⠀⠀⠀⠀⡜⠉⠻⣴⣾⡇⠠
⠀⠀⣿⣷⣿⣿⣇⠀⠀⡼⠁⠀⠀⠀⢐⣿⣿⡯⠊⢠⠀⠀⣸⣿⢠⣀⣈⠿⡀⠀⠀⠀⢀⡼⠀⠀⠀⣿⢻⠗⠒⢂
⠀⠀⠘⣞⠃⠛⢿⣦⣾⣀⡀⠀⠀⢀⣞⣷⣿⣧⠀⠀⠉⠚⠋⠁⢨⠉⠁⠀⠀⠄⡀⠀⡘⢁⣀⣴⣾⠟⠁⠀⣠⠊
⠀⠀⠀⢿⠿⣶⣤⡈⢉⡛⠛⠷⠶⢈⢘⡛⢛⣧⠀⢀⡀⠠⢴⣧⠀⢀⣀⠀⣤⡐⠩⠭⠁⠉⢉⣩⣤⣴⣶⣿⡇
⠀⠀⠀⠈⣦⣹⣿⣿⡤⠚⠋⢀⡖⠀⢲⣶⣤⡄⠀⠀⢀⢤⠤⣤⡄⣤⣴⠂⠀⡲⠂⠀⠊⣀⠴⣶⣽⣿⠉⠋
⠀⠀⠀⠀⠸⡇⠹⣶⣷⡀⠀⣸⠁⠀⣼⣿⣿⡇⢀⡠⠖⠁⠈⠉⠉⠋⣿⣤⡎⠀⠀⣠⠞⢣⣴⣿⣿⠋⡄
⠀⠀⠀⠀⠀⠘⠄⠙⢳⣅⣤⡇⠀⠀⢿⣿⣿⡛⠹⠁⢠⡶⠃⠀⠀⢰⣿⠁⠉⠐⠲⠅⠠⢿⣿⡟⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠢⡈⠻⣿⣿⣶⡖⢻⣿⣿⡇⠁⣠⡟⠁⢀⠄⠚⣼⡇⠀⠀⣀⣼⣶⣷⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣌⠛⢿⣿⣟⣿⣛⣃⣰⡏⠁⠀⠀⠀⠀⣿⣧⣶⣾⣿⡿⣿⣯⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣯⡉⠛⠻⠾⣷⣤⣤⣬⣬⠁⣠⠿⠛⠛⠉⢁⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠲⠶⠶⢞⣿⣷⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢼⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠁⠢⠄⢠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡠⠄⠒⠃⠀⠀⠀⡤⠠⠄⠀⠀⠄⢀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⢀⠍⠊⣁⠤⠀⠚⠠⠤⠄⠐⠁⢀⠠⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣊⠥⠂⠉⠀⠀⢀⣀⡀⠤⠐⠂⠈⣁⠠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡾⠋⠀⠙⣭⠀⠀⠐⠂⠈⠁⢀⣀⠠⠤⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠱⠟⠰⠖⠊⠀⡼⣁⠄⠋⠉⠀⠀⠀⠀⠀⠀⡀⠤⠤⢄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⠀⠈
⠀⠀⠀⠈⡉⡲⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⣀⡠⠠
⠀⠀⠀⠀⢯⣮⢂⠆⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⣾⠃⠀⠀⢀⠤⠆⡴⡋⠁⠌⡈
⠀⠀⠀⠀⠀⠳⡁⠰⡉⠀⠐⢎⡂⢄⡀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⢀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠐⠇⠈⠠⠀⡫⢊⠎
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠨⠠⢀⡑⢦⡈⠰⠄⠐⡈⠉⡏⠣⣀⠀⠑⠀⠀⠁⠒⠂⠚⠉⠀⠀⠀⠀⣄⠆⢀⡠⢊⢔⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢊⠢⡀⠠⢂⠜⠆⠄⡀⠀⠐⡼⠀⠐⠢⣵⡠⢄⡀⡀⠀⠀⠀⠀⠀⠀⢀⢞⣈⠖⣊⢔⣁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠒⠓⡀⠀⠀⠘⢇⠀⠈⠂⢴⠂⢝⠳⣌⠲⣎⠓⠀⠀⠀⠀⠀⠠⠠⠔⠊⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⣠⡀⢀⠲⡂⠀⠀⠈⢧⡀⠧⠄⣤⡞⠁⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠸⠄⠀⠀⠀⠙⠃⠐⠰⠌⠶⠆⠌⣂⡀⠀⠀⠀⠄⣀⡀⠀⡲⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠂⠠⣄⠀⠀⠀⠀⠐⠂⢰⠆⣀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠒⢁⠴⡄⠉⢢⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠂⠀⠀⠀⠀⠑⠕⠀⠀⠀⠀⢬⠆⠀⠙⢀⢈⡁⠀⠀⠀⠀⠀⠀⢤⣀⠉⠠⡤⠣⣤⠀⢰⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⢈⠑⠢⠀⠀⠀⠕⣀⡸⠑⠠⠈⠒⢆⠀⠀⠀⠒⢅⣸⡄⠀⠗⠪⠀⠘⢣⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⠀⠈⠡⠒⠂⠉⠁⠈⠀⡨⠀⠀⠀⢤⢡⡈⠂⠄⠀⠀⠡⠀⠀⠀⠈⢀⣀⢆⡀⠀⠦⠔⠈
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠉⠀⢺⡆⠒⠒⠀⢲⡈⠀⠀⠀⠀⠀⠐⠴⢄⠀⠀⣠⠞⠃⠀⠀⠀⠀⡁⢄⠉⠐⠂⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠧⠄⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢦⠀⠀⠀⠑⢦⣁⠚⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⠀⠘⡐⠓⡀⠀⠀⠀⢻⠻⣄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠂⠀⠂⠀⠋⠀⠀⠀⠀⠀⢿⠏⠳⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡔⠈⠂⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠠⠀⠀⠀⠁⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢰⣄⠀⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⣱⡀⠜⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠓⢀⠀⠀⠈⠉⠰⠆
⠀⠀⠀⠀⠀⠀⢉⠀⢄⠀⠀⠀⠀⠀⠀⠉⢀⡀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⢸⡀⢈⡆⠀⠀⠀⠀⠠⡄⠀⢛⣆⠀⠸⣅⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠇⠈⠁⠀⠀⠀⠀⠈⢠⠄⠘⢀⠱⡀⠀⢰⠄⠀⠀⠀⠀⢐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠙⣷⠀⠀⠐⠁⠀⠁⠈⠀⠀⠄⠀⠀⠲⣄⠀⠀⢐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠳⣮⣾⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠙⣆⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣴⠏⠳⣄⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣈⣁⣀⣀⣀⣀⣠⣚⣁⣀⣀⣌⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠠⡠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡛⠒⠂⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣶⡐⠠⢀
⠀⠀⠀⠀⢀⡞⠁⠰⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣶⣦⣤⣀⣠⣄⣽⣿⣶⡄⠀⠀⠀⠀⣠⠀⠀⠀⠀⣴⣴⣿⠟⠀⣸⡇
⠀⠀⠀⠀⢸⣇⠀⠈⣿⣿⣿⣿⠏⠛⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⣿⣿⣿⡿⠃⠀⠀⠀⣴⡇⢠⣤⣴⠾⠟⠋⠁⢀⣠⣿⠇
⠀⠀⠀⠀⠀⢿⣧⣀⠘⢿⣿⣿⣧⠀⠀⢣⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣦⣠⣀⣠⣤⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠘⢿⣿⣶⣼⣿⣿⣿⣧⣄⣿⠀⠀⠀⢠⣾⣷⣾⣷⣶⣦⣦⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⣶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣂⣵⠞⠛⠉⠁⠀⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣴⣿⣿⡛⢉⣀⣀⣀⣤⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⣀⠈⠙⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡠⠀⢀⣴⣶⣟⣿⣶⣄⡀⠀⢆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢠⣴⠿⠇⠀⠀⠀⠠⡀
⠀⠀⠀⠀⡼⠀⠀⠠⠁⠉⠁⠉⠙⠿⡅⠀⢸⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣸⣿⠀⠀⠀⠀⡔⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢸⣇⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⠰⣧⡀⠀⠓⢦⣀⣀⣀⣀⠀⠀⠀⢨⡇⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⣀⣠⣤⣾⡿⠋⠀⠀⠀⠀⠀⠹⣿⣷⣄⡀⠀⠀⠀⠀⠀⣀⣠⣾⠃⠀⠀⠀⠀⠻⢿⣿⣦⣤⣄⣤⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠦⣤⡴⠛⠛⠛⠶⣤⡀
⠀⠀⣠⡴⠛⠉⠉⠉⠛⠶⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢰⡏⠀⠀⠀⠀⣀⣤⠶⢦⡀⠀⠀⠀⠀⠀⢹
⣸⠇⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡇⢀⣠⠶⠛⠉⠀⠀⠀⠙⠳⣤⣀⡀
⣿⠀⠀⠀⠀⠀⣀⡤⠞⠋⠈⠳⣦⣀⡀⠀⠀⠀⢸⡇⠀⠀⢸⣧⡟⠁⠀⠤⠄⠀⠀⠀⠀⠠⠤⠍⠙⠛⣧
⣿⠀⠀⣾⠛⠉⠑⠲⠀⠀⠀⠀⠖⠋⠙⢛⣿⣀⠈⣇⣀⣀⢨⣟⣡⣀⠀⠰⠆⠀⠀⡀⠀⠀⠶⠀⠀⠀⢙
⣿⠀⠀⢹⡄⠀⠘⠃⠀⠀⡄⠀⠘⠃⣰⠏⣠⡿⠛⠉⠉⠉⠙⠻⣧⠉⢷⠀⠀⢀⠀⠋⢀⠀⠀⠀⢀⡾⠛
⢻⡀⠀⠀⢻⣄⠀⠀⠠⢈⡀⠄⠀⠀⢻⣶⠏⣀⣤⠖⠛⢦⣄⡀⠈⣷⣾⡁⠀⠈⠉⠉⠁⠀⠀⣠⠟⠁
⠘⣧⡀⠀⠀⠙⢦⣄⠀⠀⠀⠀⣀⡴⠋⣿⡟⠭⡆⠀⠀⠀⠈⡿⠛⣿⡋⠛⢶⣄⣀⣀⣠⡴⠟⠁
⠀⠈⠛⠶⠦⣤⣤⣽⣿⠲⠖⣿⢯⣤⣤⣯⣀⠀⠁⠀⠰⠀⠀⠁⢀⣹⡇⠀⢠⣿⣉⣉⣹⣇
⠀⠀⠀⠀⠀⠀⣀⡼⣿⣿⣿⣿⣧⣀⠀⠈⠹⣦⡀⠀⠉⠁⠀⣠⡞⠉⣠⡴⠟⠻⠿⠿⠿⠛⢶⣄⡀
⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠈⠙⣷⣶⣶⡛⠁⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣴⠞⢉⠀⠈⠛⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣇⣀⣿⣛⣻⣇⣀⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⡉⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢀⠰⠀⠒⠄⠀⠀⠀⠀⠀⠠⠰⠀⠀⠄⠆⢀
⠀⠀⠀⠀⠀⠀⠸⠤⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⢃⢀⠞⡍⠌⣂⠄⡀⠀⠀⠀⠀⠐⢯⢨⠭⠅⠭⠝⠹⡆
⠀⠀⠀⠀⠀⠓⠖⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⡤⢁⡀⣤⠀⠀⡖⠒⠀⠀⠀⠀⠀⢀⠉⠲⡠⠔⣇⡄⠴⠀⠀⠀⠀⠀⠇⡀⠀⢆⠂⠀⠠⠇
⠀⠀⠀⠀⠀⠰⠉⠈⠉⠡⠀⠀⠀⠀⠀⠀⠀⠨⠀⠨⠤⠄⠅⠀⠑⠀⠀⠀⠀⠀⠀⠈⠚⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠱

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠀⠀⠈⠀⠠⢀⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡉⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠄⠀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠊⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠰⣤⠚⡖⣰⠒⠀⠀⠀⠀⠀⠀⢠⣾⠒⢠⡘⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠠⡢⠀⣠⠀⡔⠄⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠀⠀⠈⢬⡱⡀⠰⠹⠀⠀⠀⠀⡔⠁⠀⠈⣤⡀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣈⠉⠀⡀⠡⠂⠀⠀⠀⠀⡄⠀⣠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⢀⠁⠐⢄⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢦⠀⢤⠂⠀⠀⠀⠀⠀⠀⢀⡔⠓⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠦⠔⠒⠒⠢⠴⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⣠⢤⣔⣒⣒⣈⣉⣭⣭⣭⣀⣒⣒⡢⠤⣄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠛⠉⠙⢻⣿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣻⣿⣿⡀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⠘⢿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣀⣥⣭⣭⣅⢉⣾⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⣷⣄⠙⠻⢿⣿⣷⣶⣤⣤⣄⣀⣀⣠⣤⣤⣶⣿⣿⣷⡧⠄⣠⣴⠬⠶⣾⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣄⣀⣿⣿⣷⡄⣀⡉⠙⠿⢛⣻⣿⣿⣿⣿⣛⡛⠿⠛⢉⣀⣴⣾⣿⣿⣀⣠⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠿⡿⣿⣿⣿⣿⣗⡏⠉⠑⠒⠲⠶⠤⠤⠤⠤⠶⠒⠒⠊⠉⢹⣿⣿⣿⣿⣿⢿⠿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⢻⡆⠇⢱⠀⠀⠙⠿⢿⣴⣒⡠⠀⠀⠀⠤⠄⠀⠀⠀⢄⣒⣾⡿⠿⠋⠀⠀⡎⠸⢰⡟⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢸⡇⢰⠈⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⢠⠀⡇⢸⡇⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⢸⠃⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠸⠀⠀⠀⡰⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⢦⠀⠀⠀⠇⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⣠⠄⠈⡁⠀⠘⠀⠾⠂⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣦⡠
⠀⠀⠀⠀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣷⡀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣝⠷⠾⢓⣆⠀⠂
⠀⠀⠀⢠⡤⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⣦⠖⣾⣯⡉⠀⠀⡀⠀⣴⣷⣿⣼⣿⠗⠀⠀⡀⠀⠀⠀⠀⠀⠟⣷⢿⡏⠉⠂
⠀⠀⠀⢀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋⠐⣴⣷⣏⣩⡅⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⣄⠀⠀⠀⠀⠀⢤⠿⠙⠍⠋⠈⠀⠀⠀⠀⣠⣤⢰⣦⣄⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⡀⢰⡤⠀⡀⠀⠀⠀⠀⢺⡃⢰⡆⠀⠈⡀⠀⠀⠀⠻⠀⠀⡅⢻⠋
⠀⠀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠁⠰⠆⠀⠀⠈⠁⠀⠙⣦⡀⣿⠀⠀⠀⠘⢶⡰⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠿⠿⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠁⡀⠀⢠
⠀⠀⠀⠰⠂⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢶⣴⣇⣀⠀⠀⠀⡀
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⣾⣭⣻⢻⡷⡔⠚⠁
⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠈⠀⠀⠐⣟⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⣼⣩⡄⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢿⡿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠁⣼
⠀⣀⠀⠀⠀⠰⣄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠐⢀⣠⣞⠁⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠘⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠿⠀⠀⠀⠀⠄⠉⠙⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢈⠀⠀⠀⠀⠰⠦⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⠈⢘⠀⠀⠐⠀⠀⠀⠘
⠀⠀⠀⠀⠁⡄⠃⡀⠀⠀⠀⠀⠀⠀⠀⠂⠐⣲⣾⣿⣞⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡚⢲⡄⠀⠀⠘⠟⢻⣿⠛⠃⠀⠀⠂⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠎⣰⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠉⠳⡖⠀⠀⢠⠆⠀⠈⠀⠂⠀⠀⠀⡀⠠⢥⣮⣿⣿⣷⡀⠀⠠⠄⠀⣇⠀⠉⢸⠀⠈
⠀⠀⠀⠀⠂⠀⠀⠀⢠⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠛⠻⠻⠟⠑⠦⠀⠀⠀⠉⠀⠀⡀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣴⠆⠁⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠒⠀⠀⠠⣤⠉⠀⠀⠄⠀⢨⠀⠀⡀⠀⢼⡴⠁⠀⣀⣀
⠀⠀⠀⠀⠁⠀⠀⡼⠀⠀⠀⠀⠀⣀⡴⠒⣫⡟⠁⠀⠀⠀⠀⢰⡀⣠⡔⠀⠀⠀⠛⠀⠀⠀⠀⠀⠗⠀⠞⠀⠀⠈⠁⠐⣴⠛⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠞⣁⡴⠚⠁⠀⠀⠀⠀⠀⠐⠘⢿⣿⣆⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡾⠁⠀⠀⢄⣸⣿⣯⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣸⣧⠖⠚⠙⠋⠉⠁⠀⠀⠀⠀⠀⠙⠲⠦⠤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢴⣶⡄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠉⠉⠀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢀⡀⠠⠤⠤⠬⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⢀⠃⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⠸⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠾⠰⡦⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣧⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡜⠀⠑⢧⠈⠉⠒⠒⠀⠒⢄⠀⡇⠀⢀⢼⠀⠀⠀⣴⠄⠀⠀⠀⠀⣰⡆⠀⠘⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢰⡷⠀⠀⠘⡀⠉⢑⢈⠸⠀⠀⠀⠉⠀⢰⠀⠀⠀⠉⠀⠀⠀⠑⠁⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡏⠀⠀⣷⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢡⡥⠊⠈⠒⠇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆⢰⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⡀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢧⢂⠀⠀⠀⠀⠘⢄⡀⠄⠀⠀⢀⠎⠈⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⢀⢦⡠⡄
⠀⠀⠀⠀⣤⢄⣀⣄⠀⠀⠀⠙⢷⠢⢀⣀⠀⠀⣀⡀⠤⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⣤⠤⠤⠐⣆⠁⠀⠀⠸⠙⠂⠃⠖⠱
⠀⠀⠬⣓⠀⠀⠁⡎⠀⠀⠀⠀⠀⠁⠀⢀⣠⢠⠇⠀⠀⠘⠂⠠⡤⢀⡀⠀⠀⠀⠀⡀⠴⠙⠦⡀⠀⡀⠟⠢⢄⡠⡒⠃⠀⠀⢀⠇
⠀⠀⠀⠉⠂⠢⡀⠈⠢⡀⠀⠀⠀⠀⢰⣫⠶⡁⠀⠀⠀⠀⠀⡠⠁⠀⠈⠑⠢⢔⠉⠀⠀⢀⠀⢉⡞⠀⠀⠀⠀⠈⠢⡉⣖⠂⠉⣆
⠀⠀⠀⠀⠀⠀⠑⡄⠀⣈⠦⠀⠒⠚⠣⢄⡀⠑⠢⠀⠠⠄⠊⠀⠀⠀⣀⡀⠀⡠⠓⠢⡀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢀⠜⠉⣆⡠⠄⠀⠒⠒⠠⣈⠢⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⡲⠅⠀⠒⠘⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢀⠊⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢧⠵⢌⠑⡆⠀⠀⠀⠀⣷⣊⡀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⣴⢤⡀⠨⠇⠀⠀⣴
⠀⠀⠀⠀⠀⠘⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢀⣈⢣⠀⠀⠀⣽⠁⠀⠑⠂⠿⠲⠆⠀⠀⠀⠀⠇⠀⠀⠀⣿⠀⠙⠳⠦⣤⣤⠟
⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠇⢏⠈⡎⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠁⢸⠀⠀⠀⡇⠀⠀⠀⠙
⠀⠀⠀⠀⢠⠚⢪⡖⠀⠀⠀⠀⠤⠀⠁⠀⠀⠀⡼⢄⡞⠀⡦⢤⠤⠒⡀⠈⠁⠀⠀⠀⠖⠀⠀⢄⠄⣀⠃
⠀⠀⠀⠀⠸⡀⢇⢀⡠⡀⠀⠀⠀⠐⠈⣀⠤⣎⡠⠚⠣⡜⠀⠓⡄⢇⠑⢄⠀⠙⠀⠂⠀⠀⠀⠀⣠⡮⢀
⠀⠀⠀⠀⠀⢄⠀⠀⢀⠌⠡⠒⠀⢂⡽⠀⠠⠦⠄⠀⣠⠃⠀⠀⠱⡈⢳⠆⠷⢰⠂⠠⣤⠴⡖⠋⡩⠄⠚⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠐⠂⢻⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⣀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁⠉⡀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠟⠀⠘⠉⠉⠁⠉⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢰⡖⠤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡶⠋⠀⢀⢸⠀⡠⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⡴⠿⠛⠛⢛⠛⠛⠋⠈⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠖⠉⢰⡴⠚⠁⠀⠀⠀⠘⠾⠊⠀⡠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠐⡀⣰⠁⠀⠀⡠⡀
⠀⠀⠀⠀⡀⠄⢠⣤⠔⠉⢀⡤⠞⠁⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠳⠿⠥⠀⠀⠶⠃⠸⠀⠀⠈⠣⣀⠀⠀⢀⣨⠔⠁⠀⠀⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⠋⠀⠀⠀⠀⠶⠂⠀⢠⠏⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⡤⡸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⠀⠸⢦⣾⣥⢤⣄⠀⠀⢆⠠⢴⢀⢀⠎⠀⠉⠒⠢⢄⣀⠀⢀⣀⠀⠀⠀⠀⠙⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠋⠀⢰⣖⡒⢤⡀⠀⣄⢠⠀⠀⠀⣾⢁⣀⣠⣟⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⠟⠢⠀⣠⠀⡮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡮⠉⢹⠗⠋⡀⠐⠂⠂⠀⠀⠉⠛⢡⠏⢀⣽⠂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⢀⠤⠐⠋⠁⢀⠏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⡿⠋⠉⢳⡄⠀⠀⠈⠢⡐⠀⠀⠠⢀⡸⠀⠀⠀⠀⢀⠀⣹⠧⢒⠆
⠀⠀⠀⢰⣿⣶⣤⣄⡀⠘⢄⠀⠀⡆⢀⠀⠀⢠⡄⠀⠀⠀⠸⣅⠀⠀⠚⠀⠀⠀⠀⠀⠘⡀⠒⠊⠉⠀⠀⠀⠀⠀⢺⡷⠛⠊⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣷⣦⡼⠀⠀⡏⠙⠀⠓⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡞⠙⡆⠀⠼⠁
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢄⣀⠀⢠⣤⡌⢀⡄⠀⠀⠀⠸⡿⠀⠀⠀⢀⣼⠙⡣
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⠖⠛⠃⡀⣀⠀⠀⠀⣞⣽⣿⣿⣿⢶⠚⠁⠘⠀⠉⠀⠁⠀⠀⠀⠀⢿⠀⢀⣴⠟⠊⢹⡅
⠀⠀⠀⠀⠄⠸⣔⡽⠀⠀⠠⢀⣘⣧⣤⠤⠴⠞⠁⠀⠀⠀⠀⢹⢹⠁⣀⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠭⡟⠁⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢳⣏⠉⠐⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⡞⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠉⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠹⣧⠀⠀⠂⢉⣻⣦⣤⡼⠀⠀⣀⡀⠀⠀⠀⠀⢃⢿⣿⠀⣿⠀⠀⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠸⣍⣀⠈⡧⠀⠀⠘⠉⠀⠃⠀⠀⠄⠸⡸⣿⡇⢻⡇⠘⠋⠉⠻⡁⢘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠧⡡⠃
⠀⠀⠀⠀⣀⡀⠀⠀⢈⡴⠒⠛⠻⣏⠃⠀⠀⠀⣀⡠⢠⡀⠀⠠⠀⡇⢿⣿⣘⣿⠀⠀⠀⠀⢸⢎⠀⠀⠀⠀⠑⠁⠈
⠀⠀⠀⠀⠛⠁⠀⠀⠾⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠘⢼⣟⡩⠋⠀⠀⠀⠰⠃⠀⠳⣄⠀⠀⣠⣐⣒⣒⢒⡄
⠀⠀⠀⠀⢞⣹⠀⢴⡦⠀⠀⠐⠋⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢠⠆⠀⢀⡠⠖⠉⠀⠀⠈⡟⠛⠛⡟
⠀⠀⠀⠀⠀⣀⣶⠶⠦⢤⣤⡤⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠙⠙⠈⠂⠾⠃⠀⠀⢰⣹⠆⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⣀⣤⣤⣤⣤⣖⠚⠉⠉⠉⠉⠉⣷⠴⠶⠶⠶⠶⠚⠃⢀⠀⠀⢠⢊⣁⣀⣨⣌⣲
⠀⠀⠀⠀⡀⠸⣄⣀⣀⣠⠇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠸⢿⣿⣿⣿⣿⠟⠰⠄
⠀⠀⠀⠰⠀⠀⠀⠀⠘⠒⠒⠒⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣤⣤⣤⣄⡀
⠀⠉⠉⠛⢿⣿⠟⢻⡿⢿⡿⢿⣿⣿⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻
⠀⠀⠀⠀⢸⡇⠀⠈⠁⢸⡇⠀⢹⠁⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⡀
⠀⠀⠀⠀⠺⠇⠀⠀⠀⠘⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀⡀
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠉
⢀⣿⣿⡇⠀⣠⣿⣤⣶⣶⣶⣧⣦⣤⣤⣭⣤⣤⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⢈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠈⣿⣿⣿⠀⠉⠙⠛⠟⠛⠛⠛⠛⠿⡟⠛⠻⠋⠉⡉⠉⠁⠐⠂⢀⣤⣾⣷⣿⣷⠀⢰⣿⣷⣇⠀⠀⠀⠲⠂⣀⣤
⠀⣽⣿⣿⠀⠀⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⣻⠿⣋⡅⠀⠈⠻⠿⠋⠀⠰⠂⠀⠐⡿⠿⠁
⠀⣽⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⠁⣠⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⣋⣄
⠀⢻⣿⣿⠘⠀⠀⣠⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠛⠉⢀⣀⣤⣿⣿⣶⣦⡄⠀⠀⠸⠿⠁⠈⠛
⠀⢿⣿⣿⡆⠀⠐⣎⣹⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠈
⠀⡴⠌⣯⡇⠀⠘⠻⡋⢿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠈⠻⣿⡿⠿⠛⠉⠀⠰⡦
⠀⢰⡆⣿⠇⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠈⠀⢠⠀⣀⣀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⣠
⠀⠘⢣⡏⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⢠⡀⢠⠜⣿⣿⠃⠀⠀⠀⢁⠀⠈⠋⠀⢠⣌⠀⠉
⠀⠀⠼⠃⠀⠀⠀⣀⣠⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠐⠃⠀⠄⠐⠀⢀⡀
⠀⠀⠐⠀⠀⠀⠀⢻⣿⡿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄⠀⠠⠖⠮⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋
⠀⢀⡀⠀⠀⠀⠀⠀⠉⢵⣮⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣟⡢⠀⡀⠀⠀⠀⠀⠈⢩⣿⠋⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⣀⣤⡀⢰⡶⠀⠀⠀⠘⠛⠈⠙⠛⠒⠌⣢⡤⢤⣀⠀⠈⠉⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⣤⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⢉⣉⡁⡀⠀⠀⠀⣤⣤⠀⠀⠀⠂⠣⠀⠉⠛⠷⠿⠷⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠟⠋⠁
⠀⠀⠀⠀⠀⠘⠿⠇⠁⠤⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢀⠤⢄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡘⠦⠜⠀⠀⢳⠀⠀⠀⠀⠀⠀⢀⡤⠖⡒⠒⠦⣄
⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠸⠆⠀⠀⠀⠀⠀⣞⠂⠜⢩⡄⠒⢸⡇⠀⠀⠀⠀⠀⣀⡤⠤⠦⢤⡀
⠘⢦⡉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⣀⠀⠀⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⢳⡀⡀⠀⠀⢀⡹⠃⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⢸⠆
⠀⠀⠉⠓⢤⣈⠳⣄⠀⠀⠀⠀⠠⣮⠂⠉⠁⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠓⠛⠉⠀⠀⠀⠀⢀⠇⠀⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠉⠘⢦⠀⠀⠀⠀⣟⢄⡀⣀⡮⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⣄⠀⠀⠀⠀⠀⠀⠀⣴⣃⠤⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠠⣖⠢⠀⣸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠩⠍⠓⢦
⠀⠀⠀⠀⠀⠀⢸⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡏⠀⠀⠀⠈⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⣤⡀⠀⠀⠀⢷⠎⠘⠁⠀⢘
⠀⠀⠀⠀⢠⠞⠁⠀⠈⢳⡄⠀⠀⢀⡠⠤⠤⠖⠋⠵⠃⠀⠀⠀⠈⠃⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠫⡖⠉⠁⠙⣿⠀⠀⠀⠈⢎⠀⣀⡥⠞
⠀⠀⠀⠀⠸⣄⠀⠀⢀⡼⠀⠀⣰⠋⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⢒⡳⣤⡀⡚⡋⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠈⠙⠚⠉⠀⠀⠀⠘⢆⡀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠊⠉⠳⣄⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠈⢺⡯⠒⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⣀⣀⣀⣀⡸⠃⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢷⣤⣀⣪⡞⢇⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⢰⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠉⠉⠀⢀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠷⠶⠟⠋
⠀⠀⠀⠀⠀⢀⠇⢀⠞⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⠓⠒⠚⠉
⠀⠀⢰⡀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⢦⡀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠛⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⣧⠄⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⠤⠚⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠶⠀⠀⠈⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣠⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⣄
⣴⡗⠀⠀⠈⠃⠀⡠⡖⠀⠀⡏⠀⠀⠰⢦⠎⢢⠀⠀⠈⣁⠀⠀⠀⡑⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡿⠇⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠠⠂⠁⢶⠀⠀⠀⣲⣿⠀⡤⠀⠁⠀⠀⠀⠀⠀⠀⡀
⠉⠀⠀⠀⠀⠀⢀⢴⡜⠃⢰⡂⠀⠀⠀⠀⠈⠀⠀⢰⣿⠃⠐⠇⠀⡠⠠⠀⠀⠀⠀⡀⠀⠄
⣆⠀⠀⠀⠀⠈⠛⠶⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠠⠄⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡌⠀⣧⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⢀⠠⢀⡀⠀⠀⠀⠊⠀⠀⠀⠀⢀
⣘⣣⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⠠⠀⢐
⠸⣾⠍⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠐⢀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠁⠀⠀⠨⠀⠈
⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⣠⠀⠀⠀⠂⠀⠀⠀⢀⡀⠀⠈⢠⡀⠀⠨⣀⠀⠒⠀⠀⠀⠀⠀⠀⢀⠀⡀
⡄⠃⠚⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠉⠀⡇⠀⢱⣄⡀⠈⠐⢸⢀⠰⠀⠠⠀⠀⠿⠟⠠⠠
⠐⠘⠀⢀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠇⠀⠈⣡⡠⢀⡀⠀⠈⢄⠸⠠⠙⠢⡀⡼⠀⠀⠀⠀⠄⠰⣅⠀⠀⠈⠃
⠀⠀⠢⣿⣀⣀⡁⠉⠀⢀⠀⠀⠀⠀⠀⠂⠜⠉⠈⠀⠀⠠⠀⠁⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃
⠀⠀⠀⡇⠀⠀⠉⡆⠀⠈⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠠⠄⣀⠀⠀⠀⠓⡀⠀⠐
⠀⡎⠀⠈⠲⠤⣀⣇⣀⣀⠀⠀⠀⢢⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠄⠀⠀⠀⢠⠠⠴⡖
⠘⠀⠀⠀⠀⠀⢸⡁⠀⠈⡄⠀⠀⠀⢤⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢹⠊⠁
⢀⣀⣄⣨⣤⡄⠀⠙⠒⡾⠦⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠜⠃⠀⠈⠛⠧⠂⠀⠘⢇⡀⢸⠀⠀⠀⢀⠐⠀⠀⢀⡄⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢺⡇
⡠⡀⠀⣀⡝⠀⠀⢄⠀⠀⠈⢹⡒⡄⠀⠀⢀⣤⣄⣀⡀⠀⠀⠀⠠⠄⠀⠄⢀⣀⠀⠀⠀⡀⠀⠂⠀⠀⠈
⠀⠀⠂⠈⠀⠀⠀⠘⠉⠀⠀⢀⠙⠧⠄⠀⠀⠉⠉⠙⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢈⢐⡂⠀⠀⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠂⠠⢄⣀⣀⠠⠤⠶⢀⠀⠤⡄⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠟⠒⠻⢤⣀⠀⣀⣐⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠰⢠⠏⢿⠁⠀⠉⠙⣶⣦⡀⠀⠀⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠊⢅⠀⢘⢦⣀⣁⡼⠁⢇⣠⠔⣉⢽⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡰⠀⠀⠀⠈⠑⠬⣙⡁⠀⠀⠀⢀⣀⢠⠤⠎⠡⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡾⣝⠢⣀⠀⠀⢀⡼⠀⡝⠭⠭⡟⣁⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⢈⢀⠃⠈⠐⠁⢰⠀⠃⠀⠰⢀⠇⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢱⡀⠀⠀⠀⡘⡠⠀⠀⢠⡆⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡜⠀⠈⠁⢉⠁⠁⠀⠀⠀⠃⡠⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢈⠄⠀⠀⢀⣠⡊⠀⠀⠀⢀⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⢄⡀⡯⠀⠀⠑⣄⠠⠊⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠏⠀⠈⢳⠤⠖⡊⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡄⠀⠣⠄⠀⠀⠀⢠⠄⠒⠁⠤⠀⠤⠀⠀⠀⠳⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠤⢀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⢸⢄⡀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⡀⠀⠀⡈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠠⠀⠀⠉⠒⠒⠠⠐⢦⠀⠈⠉⠒⠒⠂⠤⢤⡄⠐⠁⠀⠀⠀⠈⠀⢀⠄⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠒⠉⠀⠨⡀⠀⠀⠀⢀⠂⠂⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠁⢠⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣟⣟⣿⣻⣛⣟⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣟⣻⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠉⠉⠉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣖⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠦⠀⠀⢃⡝⠀⣸⡀⠀⠀⡄⠄⠀⠀⠀⠀⠸⣷⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢶⠚⠒⠢⢰⡆⠠⢳⡇⠀⠘⣶⡀⠰⠤⢄⡀⠀⠀⠉⠙⠛⠓⠲⢦⡀
⠀⠀⠀⠀⠀⠀⡌⠠⠐⣐⢟⣁⣐⣒⢪⣕⣢⣚⣑⠒⡒⠋⠳⠄⠀⢄⠙⠄⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⡇⠅⡐⠐⢉⡔⡠⠞⢙⠯⠋⡉⢠⠠⢄⣀⣉⠉⠓⣮⡃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⡀⠄⠄⡄⠑⠨⠠⠒⠁⠤⠂⠁⠁⠔⠀⠐⠉⡑⢤⢹⡇⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⡟⢉⣉⠙⠛⠋⠊⠈⠛⠓⠒⠒⠒⠂⢼⢪⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠠⠄⠄⠄⠀⠀⠀⠀⢈⡍⠄⠀⢸⢼⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠂⠀⠄⠈⡇⢠⢠⢠⠤⠤⠀⠀⠀⠀⠀⢠⠠⠀⢸⢾⠀⠀⠀⠀⠀⠘⢷⣄⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⣅⡁⢀⣤⡇⠀⠀⠀⠀⠀⠀⠠⠦⠈⠤⡄⠭⢤⢼⡀⡇⠂⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠆⠁⢁⡇⠘⠀⠘⠘⠚⠀⠀⠀⡒⠒⠒⠒⠂⢸⠞⡇
⠀⠀⠀⠀⠀⠀⡄⡆⡂⡾⡇⢀⡀⡀⣀⢀⠀⠀⠀⠉⠀⠙⠙⠀⢸⣠⡇⡄
⠀⠀⠀⠀⠀⠀⠆⠁⡇⠫⠣⠌⡁⡁⣁⣈⣀⣀⣀⣀⣀⣒⣀⣀⣸⣃⡇⡇
⠀⠀⠀⠀⠀⠀⠆⢃⢇⠘⠒⠢⣄⠠⠉⠀⠀⠂⠀⠉⠀⡤⠉⢁⠀⣸⡇⡇
⠀⠀⠀⠀⠀⠀⠰⣌⠛⠷⠯⠷⠥⠠⠤⠤⠤⠤⠤⠤⣬⣐⣠⠽⠦⢃⡔
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⢰⡿⠛⠶⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣄⡀⠀⠀⠀⢀⣀⣀⠀⠤⠤⣤⡀⠀⢠⠖⠁⣀⡉⠒⠀⠀⠀⢠⣴⣶⣄
⠀⠀⠀⠀⢈⣤⠀⣴⠀⢠⠄⢙⠀⠒⠐⠄⠀⠂⠀⠀⠀⣠⣾⣯⣾⣿⣿⣿⣦⣀⡀⠀⣿⣿⠀⢤⣤⠿⡟⠀⠄⢈⣡⣍⣡⣄⠀⠡⡀⢸⣿⡈⠹⣷
⠀⠀⠀⠀⠈⢁⣀⡝⠀⣤⡠⢇⢤⡈⠂⠐⠇⠈⠀⠀⠀⣿⣿⣾⡿⣿⣿⣿⣴⣿⠿⣶⣄⠈⠠⢤⣀⣷⡀⠈⢰⠙⣣⡄⢿⠟⣀⠀⠐⠀⢹⣷⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⣸⣌⡷⣚⠦⡆⠰⠀⠀⠀⠙⢿⣿⣿⢛⣿⠏⣡⣬⡛⢿⣿⣿⣦⡀⠉⠉⠁⠀⠈⣿⡀⠀⠀⣠⡛⣶⡇⠀⠀⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢣⣾⡤⣖⣂⣝⡛⠒⠀⠀⢠⣆⠀⠀⠙⠻⠿⠿⣿⣷⣶⣿⡷⣿⣿⣿⣿⠀⠀⠀⠀⣤⣼⣶⠆⣀⣟⠈⣿⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣋⣉⠓⣱⠧⢿⡛⠽⡇⢰⠆⠀⢿⡿⢿⡏⠐⢒⡤⣤⡈⠙⠿⣿⣿⣿⠿⠃⠀⠀⠀⠃⣿⠉⠀⢀⡅⢉⡉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠲⣿⣤⣤⣿⡕⠃⠉⠀⠀⠈⠳⠦⣤⣐⣂⣰⠖⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⣾⠁⠈⠉⣿⡇⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢦⡄⠀⠐⢦⠈⠛⠙⢈⠻⠊⠀⠀⠀⠀⠀⣤⣤⣍⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣯⡄⣴⣿⣧⠙⣷⠀⠀⠀⢰⣿⢿⡇
⠀⠀⠀⠀⠳⠤⣀⠑⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣴⡾⢧⠿⣿⢻⣿⡄⠀⠀⠀⠀⠀⢀⠀⢰⡄⠀⠀⠀⠀⠀⢡⡘⣄⣿⣿⣿⣴⣿⡇⠀⠀⣾⣯⣿⠇
⠀⠀⠀⠀⣑⡂⠬⡷⢬⠳⡄⠰⣷⣦⣀⠀⠀⠀⢸⣏⢀⣬⣤⢰⣄⣽⡇⠀⠀⠀⠀⠂⢸⡀⢼⠁⠀⠀⠀⠀⠀⠀⠐⡌⣼⣿⣿⡟⣿⣧⠀⠀⢿⣿⠏
⠀⠀⠀⠀⠈⠑⠿⣾⡻⣧⣌⠒⣀⡉⠁⠀⠀⠀⠀⢿⣯⣵⠿⠈⢛⣉⠀⠀⠀⠀⠀⠀⠈⣇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢿⣿⣿⣿⣿⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⣈⣙⡻⣍⣻⠷⢶⣦⡽⣖⡲⠄⠀⢈⣥⣾⣷⢸⣟⣿⣷⡄⠀⠀⠀⠀⡸⢦⡅⢀⠈⡀⠀⠀⠀⠀⠀⠈⠢⡙⠿⠿⢃
⠀⠀⠀⠀⢰⣿⠉⡸⠓⣻⣷⢞⣩⡍⢿⣿⣽⣯⡶⠀⠘⣿⡕⢋⣄⣹⣽⣿⡄⠀⠀⠄⠑⠂⠀⢷⠀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠁⠸⢃⣾⠉⣉⡛⠳⣜⣧⢹⣽⣧⣽⡇⠀⠙⠿⠶⠟⠈⠛⠉⠀⠀⠀⠀⠀⠈⠀⣼⢀⣿⣶⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⡀⠈⢁⡞⢀⠙⣤⣿⣽⡸⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠙⢀⠀⠀⠀⠀⠀⡀⣠⣾⢿⢾⣽⡇⠀⠄
⠀⠀⠀⠀⣼⣿⢿⡆⠘⣷⠛⡿⡟⠋⣸⠛⠀⠈⠉⠁⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⢀⣇⠇⠀⠆⠀⠀⠀⠀⠀⠰⣽⣷⢟⣻⡿⣿⠃⠈
⠀⠀⠀⠀⢸⣧⢸⡏⠀⣴⢾⠆⡷⠀⠉⠀⢀⠈⢡⣀⡀⠔⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠠⠐⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⢻⣏⣼⣿⣿⡇⠠
⠀⠀⠀⠀⠀⣿⣼⡇⠀⣿⠆⠀⠀⠀⢠⡦⠉⠙⢀⡄⢸⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⣶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠉⡽⢿⡗⣄
⠀⠀⠀⠀⠀⠿⠛⠀⠀⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣠⠚⠁⢠⠙⢨⠃⡅⠀⠀⠀⠀⠀⠀⢀⡙⣤⠂⣌⡀⠁
⠀⠀⠀⠀⠀⠀⣠⠶⠟⠓⢦⡀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠰⠷⡈⠁⠀⡄⠀⠀⣤⠔⠁⠀⠀⠀⠸⠀⠀⣷⠄⠀⠀⠀⠀⢀⠈⠠⠐⢊⣈⡄⠁
⠀⠀⠀⠀⠀⢰⠏⠀⠠⠤⠀⠉⢢⡀⠈⠃⠀⠀⠀⠀⠀⠰⡖⠓⡆⠀⢱⠀⠀⠀⠀⠁⠀⠀⠋⡀⠀⠈⠉⡄⠀⢀⣾⡿⢛⡷⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡟⠐⠀⠀⢀⡀⠀⠀⡇⠀⣀⢤⣤⣤⣤⢦⣤⡈⠆⣀⣤⣸⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⡿⢂⠀⠀⠋⠀⠑⢲⠇⢸⣉⠩⠉⠛⠀⠀⠛⣵⠀⠀⠈⠉⠳⡄⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⡼⢃⡄⢠⣤
⠀⠀⠀⠀⠀⡆⢸⠀⡀⠰⡦⢀⠎⠀⠸⣭⣆⠿⣿⣿⢶⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠶⠊⠛⢶⣄⠀⠙⠛⠶⣾⡿⠵⠦⣖⡛⢛⣋⠉
⠀⠀⠀⠀⠀⣷⠘⠀⡇⠠⠄⣸⠀⠀⠀⠀⠀⠈⠉⠉⠈⠀⠀⠀⣀⣀⢀⣤⣄⢀⣤⣶⡆⢀⣀⠀⠀⠙⠛⠒⠛⠛⢉⣠⡶⠚⠡⣿⣻⣽⣦
⠀⠀⠀⠀⠀⣿⠀⢠⠃⠐⢠⡇⠀⣾⢻⣧⣞⢻⣆⣤⡀⠀⢀⣤⣿⡿⢣⡏⣽⠏⢉⣿⠰⣿⠛⢿⡄⠀⠀⠀⣠⠴⠋⠁⣠⣄⠀⣤⡛⠁⠉⣰⣾⡆
⠀⠀⠀⠀⠀⣾⠀⢘⠀⢀⠞⠀⠀⣽⣿⣍⣙⠋⢿⣉⣧⠀⢻⡏⠻⣶⡟⣤⣿⣿⣿⠿⣇⣿⣤⣾⠃⠀⢀⡴⠁⢀⡴⠞⠉⢡⡄⠉⠁⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠳⠾⠖⠋⠀⠀⠀⠛⣽⣟⠙⢷⣶⠙⣏⠀⠨⣷⣴⠟⠹⠷⣾⣏⡹⢦⣿⠎⠉⠁⠀⢠⠞⠁⠀⢻⡀⠀⠀⠀⢡⣤⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡷⣦⠁⠀⠀⠀⠠⠾⠷⠚⠛⠒⠚⠛⠀⠀⠀⠀⠀⠘⢧⠘⠇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡶⡄⠀⠀⠀⠀⠀⠰⢋⣉⢉⣤⣤⡀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⢃⡐⠶⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣾⣿⠎⠙⢿⣧⣴⡀⠀⢰⣶⡤⢻⣟⢼⣇⣸⣯⣤⠀⠀⠙⢢⣄⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠸⣿⣙⣤⢢⣄⠛⠋⠀⠀⠀⠉⢰⡟⠻⣯⣹⣿⣿⡟⠃⠀⠉⠀⠈⣙⠶⠦⣬⣦⣼⣤⡜⠀⠀⠀⣦⠀⠀⢀⠜⠀⣠⠄
⠀⠀⠀⠀⠀⠉⢭⣤⣯⡿⠀⣴⣿⡦⠀⠀⠈⠿⣶⡿⣿⣷⡾⠏⠀⠀⠀⠀⡎⢀⣤⡤⢴⢾⡿⠟⠓⠒⢐⡾⠧⠴⣮⡁⠐⢶⣥⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⣾⠙⣯⣵⣶⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠤⠊⠀⠀⠀⠁⠀⠀⢀⡠⠒⠂⠀⠀⠀⢀⠈⠐⠤⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⢂⣴⣾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣴⣾⡟⣼⡇⣾⣿⣿⣿⡇⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠚⢿⠟⠂⠀⠀⠀⠀⠀⣀⡤⣷⣿⣿⣿⣇⣿⣇⢿⣿⣿⣿⣿⣦⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢖⣵⣿⣷⣾⣿⣿⣿⣿⣿⡸⣿⡌⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⣠⣶⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣦⡙⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣝⠿⠶⣤⠄
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢿⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣌⠻⣿⣿⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣾⣿⠀⠀⢀⣰⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⠀⣶⣿⣿⣿⣿⣿⣿⡿⠟⢩⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⡀⠙⠟⠉⠁⠈⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⣼⣿⡟⠀⠈⠙⠉⠸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣰⣦⣀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣧⠹⣿⣆⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠹⣿⡆⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⢻⣦⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣶⣶⣾⣿⣶⡶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠀⠘⠀⠘⠃⠀⠛⠀⠀⠀⠸⠀⡂⠀⠀⢀⠀⠁⠘⢀⠀⠠⠀⠀⠐⠂⢐⠀⠀⠀⠐⠟⠀⠘⠃⠀⠃⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣬⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣿⣽⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢶⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⢇⠈⢧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣧⠈⢇⠈⢆⠈⢇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠠⣦⠘⢇⠘⣇⠘⣇⠙⢧⠈⢆⠈⢆⠈⢆⠈⣦⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣈⣿⣿⣿⣿⣿⣿⡙⡇⠘⣄⠈⣆⠘⣆⠈⣆⠸⡧⠸⠇⠸⠇⠸⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠛⠋⠈⠛⠛⠿⣿⣿⣿⣿⣧⠘⡄⠈⡄⠘⡄⠈⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢋⣬⠐⡄⠘⣄⠸⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⡷⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡠⠞⠋⠁⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⣀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⢶⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠁




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣶⣤⡀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠿⣿⠿⠿⠆⠀⠀⣾⡿⠉⠉⠉⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠀⠰⠀⠰⠆⠀⠀⠀⠉⠁⠀⠘⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣶⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠛⠛⠛⠀⠀⠈⠁⠀⣤⠀⠀⣠⠀⠀⠀⠀⠀⣴⣷⣀⣭⣄⣾⣦⠀⠀⠀⠀⣠⣤⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠖⠀⠘⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡆⠀⠀⠀⣸⠟⣿⣿⣿⠟⢿⠻⣇⠀⠀⣾⡟⠻⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⣤⣄⡀⠐⠒⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⣿⠀⣿⣿⣿⣷⣡⠀⣿⠀⠀⣍⠀⠘⠃⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣶⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠐⠋⠀⣿⣿⣿⣿⣿⠀⠘⠀⠀⠾⠆⡀⠉⠁⠚⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⠉⣿⣿⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣿⣿⠇⠀⠀⠀⠀⠸⠅
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⣤⣤⣄⠀⠀⠀⠀⢠⣀⡀⢠⣀⡀⠀⠀⠀⢀⣴⣿⣿⣀⣿⣿⣦⡀⠀⠀⢀⣤⣤⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠉⠉⠉⠀⠉⠉⠉⠀⠀⠀⠀⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢀⢐⡥⠈⠂⠒⠐⠑⠂⠒⠂⠐⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠀⠄⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠊⠀⠀⠀⠀⢂⡁⠤⠀⠀⢀⠀⠈⣀⠦⠁⡀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣠⢄⡀⠀⠉⠈⠁⠀⠀⠀⠀⠐⠆⠲⠋⠀⠀⠀⠀⠄⠀⡀⠅⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠠⠀⠈⠈⠳⣕⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠘⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠳⡄⠀⠠⡢⡢⡁⠀⢁⠽⠟⠀⠀⢀⠀⠂⣠⠀⠀⠀⠄⠪⡄⠀⠀⠈⠁⡀⡄⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠸⣗⠄⢀⡌⠔⠊⠁⠀⠀⠀⠀⠀⠜⠠⠄⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠐⠂⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠘⣆⠀⡀⢀⡠⠖⠊⠁⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠖⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠘⢶⠟⠓⠊⠁⠀⠀⠀⡀⠀⠂⠀⠀⢤⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⣀⢐⣈⠵⠼⠛⢕⣉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠸⠈⠰⠀⠀⠂⠀⠀⠀⠀⠀⠦⠄⠀⠀⠄⠀⢘⣀⢔⠄⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠳⠾⠉⠀⠀⠀⠀⣈⣩⡶⢦⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⢤⠄⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡄⡀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠄⠢⡎⠀⠀⠈⠀⠤⠠⠴⠈⠀⠀⠈⠀⠀⢀⣠⣎⢊⢌⠎⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡤⠒⠩⠢⠁⠂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⢸⠀⢡⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠃⠐⡐⠂⠒⠐⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣟⠀⠀⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠂⢰⠻⠾⠀⠀⠲⣠
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣸⠀⠀⠀⠀⠀⢉
⠀⠀⠀⢀⣀⠀⣄⣦⠀⠔⠛⠐⠀⠠⠀⢠⣠⡌⣤⢀⠀⠀⣀⠄⠠⠀⠄⠠⠀⠄⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠚⠉⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⡀⠄⠒⠀⠐⠂⠄⠀⠀⠈⠖⠀⠀⠀⠀⠀⠂⠀⠀⡆
⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠠⠒⠀⢀⠑⠄⢀⡏⠀⠁⠀⠀⠀⠀⠀⠐⠇⠠⠈⢀⠤⠀⠀⠀⠀⠀⠀⠄⠀⡅⠠⢤⠀⠠⠈⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠂⢁⠐⠘⠆⠀⠀⠀⡂⠀⠀⠀⠀⠄⠀⠀⠁⠀⠑⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⠁
⠀⠀⠀⠀⠀⢀⠀⢀⠔⠡⠀⠀⠈⠀⠈⡤⠀⠀⠄⠀⠀⠀⠀⠀⢀⠌⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⣆⠀⠇⠀⠀⠀⠠⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠒⠙⡀⠀⠀⣀⡅⠀⠀⠆⠀⠀⡀⠒⢤⠤⠤⡰⠋⠄⠀⠀⠀⠀⠀⢁⣀⠀⠀⠠⠌⠐⠀⠈⢐⢦⠀⠀⡤⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣢⠤⠤⡠⠐⣇⠀⠀⠌⠀⡄⠀⠀⠀⢀⡴⠒⠒⠂⢄⠑⠤⣄⡁⡤⠴⡇⠈⡆⠀⠀⠠⠀⠀⠁⠀⡀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠑⠀⠘⠊⢀⠀⠀⠀⢀⡁⠀⠀⢀⠃⠔⣮⠉⠨⠄⣏⠄⠠⠜⢀⠀⠘⠄⡅⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⠀⠀⠀⢲⠩⠘⠂⠠⣠⠔⠀⠐⠀⡀⠀⢡⠆⠀⢀⠈⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣃⢀⢘⠀⢂⠰⠂⠒⡒⠊⡅⠀⠰⠀⠐⢀⡄⢴⠐⡡⠂⡅⠀⠀⠀⠀⢐⠋⠁⠀⠀⠀⢋⢊
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠭⠉⠈⠉⠀⠈⡚⢸⢴⡀⠁⣀⠀⢠⠅⠸⠁⠀⠈⠉⢧⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁⠀⠀⠀⠈⠀⠀⠈⠝⡂⠀⢀⠑⣀⢡⡅⠀⠀⢸⢄⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⠀⠀⠀⢾⡇⠀⢰⣷⠋⠇⡏⠉⡉⠙⢸⠂⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠨⣄⣀⣠⣭⡄⡄⠰⠰⠗⠸⠀⠁⠒⠰⠐⠆⠈⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢘⣇⠡⠀⠀⠀⠀⣾⠀⢹⡞⠀⠊⠀⣬⣨⣄⠀⠀⢰⠴⠤⠆⠀⠀⠀⠀⠀⠀⢨⢃⡓⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣦⢄⠈⠠⠟⣹⠯⠫⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠃⠠⠿⠀⠉⠠⠀⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⡀⠈⠃⢈⠀⠆⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⠈
⠀⠀⠀⠀⠀⠀⡆⠡⠐⠀⢴⠆⠀⠁⠀⢦⡀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⣀⠢⡀⠊⢦⡍⠆⠀⠐
⠀⠀⠀⠀⠀⠀⢻⠐⡀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠘⢆⠀⠀⠨⡠⠈⣳⣙⠆⠀⢊⡀⢂⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠹⠆⠈⠿⡆⡀⠱⣊⣴⢛⣦⠁⠒⠀⡁
⠀⠀⠀⠀⠀⢀⠀⠀⠠⠈⠠⠀⠀⠁⠀⢄⡀⠀⠀⠀⠀⠂⠑⠌⠰⠄⠀⢌⠉⣣⣈⠈⣀⡀⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣁⠀⠀⠀⠀⠀⠙⣁⡔⠀⡀⠀⠐⡀⠃⣈⠁⢀⡛⠀⠂⠲⠖⠐⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠉⠄⠀⠀⠈⠒⠲⠀⡰⠡⠳⠆⠰⠄⢤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠠⠀⠃⠀⠂⠠⠀⠉⠀⠀⠐⠄⠈⠀⠁⠀⠀⠀⠄⢠⠂⠀⠀⠀⠆⠄
⠀⠀⠀⠀⠂⠀⠅⠀⢴⠀⠄⠀⠀⠀⠀⠀⢀⠀⠴⠀⠀⠀⠀⠈⠑⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠰⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⠀⢈⠠⠀⠀⠘⢁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠤⠄⠀⠂⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠁⠀⠆⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠁⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠠⠈⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠠⠠⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠩⡈⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢡⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣆

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⢀⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⡄⠀⣄⠀⠀⠀⠀⠀⢸⣍⠋⠀⠀⠀⠱⠀⠀⠀⠀⠀⢀⡴⠋⢊⠔⡡
⠀⠀⠀⠘⡀⠣⢀⣀⣀⠀⠀⠀⠀⠀⠈⠟⠀⣔⠉⣦⠀⠀⠀⠀⠀⠘⡓⠠⢀⡀⢠⠓⡄⠀⠀⣰⣍⡊⠐⡡⢀⠔
⠀⠀⠀⠀⠈⠒⠒⠒⠁⠀⠀⠀⠀⢼⡣⠀⠀⠈⡶⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠊⠉⠉⠁⠀⢸⡌⠈⠉⢀⡴⠃
⠀⠀⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⡀⠀⠀⠀⠀⠀⠀⠑⠚⠊
⠀⠀⠀⠰⣊⠾⣸⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⢤⢀⠔⡭⠁⠀⠀⠀⠀⠙⡗⠯⠳⡀
⠀⠀⠀⠀⠙⡆⠙⢆⠀⠉⠀⠀⠀⠀⠀⣔⢎⡹⢥⣮⡀⠀⠀⠀⠀⠀⠈⢢⡰⣉⢎
⠀⠀⠀⠀⠀⠈⢆⠀⠑⢄⠀⠀⠀⠀⠀⢠⠧⣪⡲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⢀⠄⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⡴⠢⣄⡀⠀⠀⠀⠀⠀⡃⢊⠥⣰⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡼⡓⠤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠽⠲⠶⠯⣅⠀⠀⠀⠀⠀⠐⠤⠭⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⢷⡦⠀⠀⠀⠀⠀⣇⣰⠶⠲⣆⢨⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠴⡦⢼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠘⠋⣡⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢈⣤⡀⠈⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⠟⠿⣿⡀⠈⠑
⠀⠀⠀⠀⠀⠀⡛⠁⠀⠀⢠⣤⣤⡌⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠁⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣈⣥⡦
⠀⠀⠀⠀⢀⡀⡁⠀⠀⠠⠿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣄⠋⠁⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⡂⠀⠀⠀⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⢰⣶



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣋⣍⣹⣟⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢰⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠠⠁⠠⠬⠐⢂⠀⠀⠀⠀⡦⠀⡐⠉⢂⢀⡀⠀⠀⠀⣀⣧⡵⠖
⠀⠀⠀⠀⠀⣥⠜⠁⢢⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠸⡀⠁⠐⠀⠂⢉⡜⠀⠀⠀⠈⢢⡠⣯⡟⣍⣈⠏⠀⠀⢼⠁⢈⡇
⠀⠀⢀⠀⡀⠀⠀⠰⢀⠀⠀⠘⡁⣤⠁⠠⡀⢀⠀⠀⠑⠤⡄⠀⡠⠎⠀⠀⠀⠀⠐⣁⣸⠨⢵⡋⠑⡀⢀⡴⠚⢯⠉
⠀⠀⠈⡉⠅⠀⠠⢂⠌⠀⠀⠀⠐⢄⠀⠀⠐⣄⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⠈⠉⠀⠀⠈⠉⠚
⠀⠀⠀⢰⣶⣿⣷⣷⣀⠀⠀⠀⠀⠈⠰⢤⡾⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠤⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠀⠑⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⡈⢀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠃⠀⠐⠃⠀⠀⠀⠠⠘⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⡠⢊⠆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠐⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠉⡇⠠⠀⠀⠀⠀⠀⢻⣷⠄⠀⠠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡦⢄⠤⠖⠁⠀⠀⠀⠀⠀⠠⣠⣶⣶⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠢⠄⠄⠊
⠀⠀⠀⠀⠀⠠⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡡⠀⠀⠀⠀⠈⢤⡐⢀⠆
⠀⠀⠀⠀⢠⠁⠏⠀⠀⠀⠄⢀⠴⠀⠀⠀⢠⠿⢿⠿⣿⣿⡿⠷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠂⠐⠔⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠒⢀⠸⠈⠂⠀⠀⠀⠀⠀⠀⢂⡀⠀⡰⢀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣈⠁⢈⣤⣤⡗⠀⠀⠀⠀⠀⢠⠊⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡍⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂
⠀⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠀⡆⠀⠄⣀⠀⠀⠀⠀⠀⠀⡠⢐⣤⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠠⠐⠁⠀⠑⢩⠁⠀⠀⠀⠀⠀⠀⠈⢿⡇⣀⣀⠀⡄⠀⠀⠀⢰⡸⠏⠟⢀⢰⠀⠀⠀⠈⡀⣬⢢⠀⠀⡠⡂⠥⠀⠠⠁
⠀⠀⠀⡁⡄⠀⠐⠒⠄⠅⠀⠀⠀⠀⠀⣠⣴⡾⠛⠻⢿⠃⠀⠀⠀⠀⠀⠃⠈⠄⢉⡀⠂⠀⠀⠀⠐⠠⢙⢠⣌⠜⢁⠠
⠀⠀⠀⠑⠈⠀⠒⠨⠈⠀⠀⠀⠀⠀⠠⡈⠛⠃⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⣀⣰⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠓⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠂⠠⠈⠈⡄⠀⢤⠀⠀⢀⠀⡀⠀⠀⠐⠀⠀⡈⠀⠀⠀⠀⢀⠨⠶⠥⠄⠈⠂⠑⠀⡀⢀⠄⢈⠀⠴
⠀⠀⢈⠁⠃⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠈⠀⠁⠢⠀⠁⠁⠁⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠐⣀⢀⠂⠂⠰⢑⡁⢐⡴
⠀⠀⠈⠀⠁⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡒⠀⡀⠀⠀⠐⠐⠐⠀⠸⠀⣀⠀⠀⠀⠀⠀⠂⠐⢂⢀⠒⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠆⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠤⢄⡀⠀⠂⠀⠂⠢⠀⠲⢦⡀⠐⡥⢄
⠀⠈⠀⡄⢀⠀⡀⠀⠀⠘⢃⢂⠀⠈⢀⣄⠀⠀⣀⠀⠀⠀⢀⣁⠓⠈⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢥⡤⡄⠀⢰⠄⢀⠀⠀⠛⠀⠑⠊
⠀⠀⡀⠓⠀⠄⠠⠼⠀⠀⠈⠀⠀⢀⠐⠀⠁⢨⠙⡞⡛⠀⠈⠈⠀⡂⢠⠁⠂⠀⠊⠁⠀⠛⡏⠏⠭⠉⠁⠀⠁⠀⠀⠐⠐⠀⢀
⠀⠀⠀⠠⠖⠐⠦⠀⠀⠀⠀⠀⠀⢐⠴⠀⢠⠀⠑⠊⠁⠤⠁⢊⡀⠀⡸⠀⠀⠐⠀⠀⠠⠀⠀⠀⢀⣀⢠⡀⠀⠀⠀⠒⠀⠠⠄⡀
⠀⡎⠉⢣⠀⠇⠀⠀⠀⠥⠀⠀⡀⠈⠶⠀⠪⡨⠀⠂⠤⠐⠁⠃⢀⣈⠀⠀⠀⠀⠁⠈⠀⠀⢼⡥⣞⠛⡘⠁⠂⠈⠂⠴⡆⠠⠀⠂
⠀⢄⠂⡴⠀⠀⠀⢀⠀⠀⠀⠀⠘⠃⠀⠆⠀⠀⠀⠀⣤⠀⠀⠀⠠⠌⠀⠸⠂⠀⣤⠀⠂⢴⡀⡠⠔⠊⠤⠄⠀⠐⠍⠀⠐⠄⠀⡀⢀
⠀⠀⣀⠀⠀⠀⠐⠆⠀⠀⠀⠉⠁⠀⠀⠀⠤⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⠀⠂⢀⠈⢀⠈⠀⡀⡀⠀⠀⠀⢀⡐⢿⠂⣀⠀⠀⠉⢙
⠀⠀⣏⡂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠄⠀⠀⠈⠀⠀⠠⠔⠀⠘⠀⠀⢃⠀⠠⠀⠀⢀⠐⠐⠀⠀⠀⠀⢹⠀⠀⠀⠁⠀⠑⠁
⠀⢀⠀⠀⠐⠀⠀⠠⠀⠀⡄⣤⢠⠀⡀⠂⠀⠀⠀⡀⠄⢶⣲⣤⠂⠀⠀⠀⠀⠄⠀⠙⡀⠀⠀⠈⠈⠀⡰⠁⠀⠈⠀⠀⠀⠠
⠐⠪⢬⠐⠲⠀⠀⠀⠀⠀⠀⡆⠀⠐⠠⠂⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠴⣂⠀⠀⡀⢠⣒⡆⢀⠀⠀⠁⠀⠀⠠⠂⠄
⠀⠤⡀⢃⠂⠀⠀⠀⠀⠀⠀⠀⠁⠓⠒⠀⠀⠄⠀⢰⣖⠀⠀⠀⠀⠀⠀⠸⠁⠐⠈⠁⠀⠀⡁⡀⠁⡒⢀⠁⠂⠘⠄⠀⠃⠀⠰⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠀⠂⠀⠐⠈⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⢼⠀⡶⠆⠀⠀⠀⠀⠀⠀⣀⡈⢁⣈⢁⠈⠁⣀⠀⠀⠀⠀⠀⡇⠠⠶⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉
⠀⠀⢸⢠⣀⣠⠀⠈⠀⠀⠀⠀⣿⠉⠉⠉⠈⠈⠀⣿⠀⠀⠀⠀⠀⡇⣤⣀⡄⠀⠀⠀⠀⠀⠀⢠⠂⠀⠈⢢
⠀⠀⣼⣀⣉⣁⢰⣶⡆⠀⠀⠀⠿⠒⠂⠒⠂⠐⠂⠿⠀⠀⠀⠀⢀⣇⣈⣉⡀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠟


⠀⠀⠀⠀⢠⠶⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠲⠀⠖⡆⠀⠀⠀⠀⠀⠀⡠⠖⡒⠤⡀
⠀⠀⠀⠀⢸⣄⣀⡼⠀⠀⠀⠀⡖⠉⠉⡩⡉⠉⢱⡄⠀⠀⠀⠀⢰⣴⠀⣀⡇⠀⠀⠀⠀⠀⡜⠴⠜⠃⢶⢸
⠀⠀⣠⡶⠋⠀⠁⠀⠀⠀⠀⠀⠈⠂⠈⠀⠀⠀⠊⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠰⣔⠇⠘⢓⠜
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⣶⡎⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢂⣠⣤⢂
⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣀⡽⠀⠀⠀⠀⠀⠀⠈⠀⢻⢿⠃⠘
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠉⠘⠂


⠀⠀⠀⣤⠈⠛⢫⣄⠀⠀⠀⠀⠀⢀⡨⠄⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢙⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡐⠠⠄⠃⠎⠀⠀⠀⠀⠀⠀⢀⣂⣈⡄⠀⠀⠀⠀⠀⠀⠀⣄⣿⡀
⠀⠀⠷⣶⠾⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠋⡒⠉⠁⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣠⣿⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠈⠉⠉⠉⠐⠒⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⢋⣷⢄⡀
⠀⠀⠀⠀⢀⣀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠛⢁⣴⠟⠁⠀⠉⠢⡀
⠀⠀⠀⠻⣥⠄⠒⠒⣉⣱⢤⡀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠉⢀⡴⠋⠀⠀⠀⠀⠀⣠⣿⣄
⠀⠀⠀⠀⠀⠑⢯⣁⡩⠶⣔⠛⢢⣀⠀⠀⠀⠀⡜⠀⠀⠀⣀⣠⣤⣶⣿⣿⡿⠟⠋⠁⠀⡠⠞⠁⠀⠀⠀⠀⠀⣠⣾⣿⡿⣩⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣖⠊⣛⣥⡨⠷⢄⠀⣼⣶⣶⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⢉⣼⡿⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡤⠝⠶⣊⠙⣻⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⢁⣴⠟⠁⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠉⠙⠿⣖⣒⡭⢿⠤⠤⢐⣒⡺⡍⠑⡔⢢⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠋⣠⠶⠋⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠙⡹⣆⠈⠆⢿⠿⣿⠛⢸⠀⢱⠴⠁⢀⣠⣤⣶⣿⣿⠿⠛⠉⣀⠴⠊⠁⠀⠀⠀⠀⢀⣼⠟⢉
⠀⠀⠀⠀⠀⠀⢈⠀⣀⣀⠤⢄⠀⠀⢡⢻⠀⢸⠘⢃⣉⣶⣼⣤⣾⣶⣿⣿⡿⠟⠛⠉⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⣠⡶⠋⢁⣴⡿
⠀⠀⡠⢤⣤⡲⢯⠭⠔⠒⠊⠉⠀⠀⢸⡿⠄⢺⣿⣿⠿⣉⡩⠷⢤⡚⢫⣄⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⣴⡿⢛⠇
⠀⠀⠑⠉⠀⠀⠀⠣⡀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢣⣤⣤⣼⠶⣖⠊⢙⣧⣤⠵⢤⡀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⣠⣶⠟⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠖⠁⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠙⢤⡤⠜⠳⣎⣉⡦⣤⠔⠚⠉⠀⢀⣠⣴⠟⠋⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣤⣤⣤⡼⠶⣉⣉⠹⠤⠘⠓⢤⣴⠾⠛⠉⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢍⣀⣀⣀⣬⣷⣤⠴⠖⠚⠉⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠫⠥⠀⣀⣀⣀⠠⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢠⣶⣷⣶⣄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠸⣿⣿⣿⡟⠀⠀⠀⠀⡀⣰⣀⡀⠀⠀⠀⠸⣿⣿⣿⡇⠀⢦⣴⣿⣿⣿⣿⣆⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷
⠀⠀⠈⠉⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⡗⠀⠀⠀⠈⠉⠁⠀⠀⠾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢀⣠⣦⣄⠀⠀⠀⠸⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣦⡀⢀⠿⢿⣿⢿⣿⠋⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⠀⠀⠀⠀⠉⠉⠁
⠀⠀⢼⣿⣿⡄⠀⠀⠀⢻⠿⣿⡿⠿⠏⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠋⠀⠱⠀⠀⠀⠀⠀⠀⠙⢿⡿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⡀
⠀⠀⠘⠻⠛⠁⠀⠀⠀⠀⠀⠸⠅⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⢀⣴⣤⣤⡀
⠀⠀⠀⠀⢀⣴⣶⣶⣤⠀⠀⠀⠀⣾⣿⣿⡄⠀⠻⠿⡿⠿⠁⠀⠀⠀⠀⣼⣿⣿⣿⣶⡄⠀⠀⠸⣿⡿⠃⠀⠀⢀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠂⠀⠀⠀⠙⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠻⣿⣿⣿⡿⠃⠁
⠀⠀⠀⠀⠠⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠙⢻⣿⣿⣿⣿⠟⠀⠀⠀⠀⣴⣶⣦⡀⠀⠰⠋⠈⠁⠙
⠀⠀⣠⣤⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⣴⣀⣰⠀⠀⠀⠀⠿⣿⠇⠀⠀⠀⠀⠈⠘⠏⠀⠀⠀⠀⠀⠀⢿⣿⡿⠇⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⣿⣿⣿⠆⠀⠀⠂⠀⠀⠀⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⢰⣿⣿⣿⡷⠀⠀⠀⢀⣠⣤⣦⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉
⠀⠀⠀⢠⣶⣶⣶⣄⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⣼⣿⣿⣿⣿⣧⢄⠀⠀⢀⣶⣿⣿⣷⣦
⠀⠀⠀⢸⣿⣿⣿⣿⠀⢀⠈⠁⠘⣿⠿⡿⠉⠛⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠓
⠀⠀⠀⠼⠛⠛⠛⠃⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⠀⠘⠻⠿⣿⠟⠻⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣷⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⢰⠆⠚⢠⡄⠀⠀⢀⣀⣠⣶⠶⠲⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣰⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢙⠁
⠀⠘⢠⠓⠀⠂⠀⠀⠿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⡤⠈⠀⠀⠰⠋⠀⠀⢀⣄⢀⡀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠘⠁⢠⡀⣤⠀⠒⠁⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣴⡖⠀⠀⡰⠆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⡀⠀⣿⡄⠀⠁⠀⠀⠀⠀⠠⣠⣦⡀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠙⠋⠀⠀⠰⢂⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠡⠙⠀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠁⠀⠀⠈⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠈⠀⠀⠀⠲⠄⡀⠀⠀⠐⠀⠟⠀⠐⠠⠠⠀⠀⠀⠀⠀⠀⠈⢽⠿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠑⠀⠂⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠚⠁⠀⢀⡀⠀⣐⠁⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠄⠈⠀⣀⠀⠈⠀⠀⠀⡀⠶⣳⡋⠀⣀⠞⠁⠀⠀⠀⠂⠀⠀⠂⠁⡀⠉⠬⠋⠀⠃⣀⠤⠀⠀⠀⠀⣠⡾⠀⢟⡷⠧
⠀⠀⠀⡰⠃⠀⠀⢀⣀⠤⠤⠤⠀⠀⠀⠀⢀⣸⠄⠀⠀⠀⠊⣙⣧⠝⠓⠊⠡⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣻⣧⢰⡌⠷⢶
⠀⠀⡐⠃⠀⢀⠀⠈⠄⠘⠀⠀⠀⣠⠀⠀⠈⠁⠀⠀⠀⢠⠈⠁⢠⡤⠂⠠⢐⡏⠀⡀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⢄⠀⠈⠉⢠⣸⡗⣶⠒⠠
⠀⠀⡗⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⢨⠀⢀⠀⠀⠰⠀⠀⠈⠆⣠⠃⠀⢰⡂⠀⠀⢈⠀⠀⠀⡼⠥⠋⠑⠀⠀⢀⡀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠋⢿⢿⠟⡄⠃
⠀⠘⡟⠀⠀⠄⠀⠀⠀⠀⠴⠆⠀⠀⠀⠁⠀⠀⠂⠀⡀⠴⠄⡁⠀⢰⡇⢱⣄⠀⠋⠀⠀⢀⠃⠀⡢⠀⠀⠀⠙⠃⠠⠀⠀⢀⠔⠀⠀⠀⠀⠀⠘⠉⠃⠁
⠀⢠⣣⠀⠀⠀⠀⠒⠒⠂⠀⠀⢀⣀⠀⡰⠀⠀⢻⢩⠘⣴⣋⠴⢅⣿⡁⢀⣹⠀⠀⠀⠀⠈⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⢰⡄⠰⠄⡀
⠀⢨⡇⠀⢈⣿⠆⠀⠀⠀⠀⠠⠌⠁⠀⠙⠖⠀⠸⡄⠾⠀⡁⢠⡾⠻⣄⡺⡌⠀⠀⠀⠀⠀⣄⠀⠘⢹⣇⠀⠀⠀⠀⠀⠆⠀⢡⠀⠄⠀⠈⠀⢡⡀⢀⠃
⠀⠀⠁⠀⣸⡉⠀⠄⠀⠀⠀⠀⢆⠀⢈⠀⡀⠄⠀⠹⣄⠀⢱⠏⠐⡃⠙⡼⠀⠀⣖⡐⣲⣾⣴⠁⠀⠘⡇⠀⡄⠂⠀⠈⢤⣀⠀⠀⢀⠀⢠⡀⠈⠁⠙⠈
⠀⢢⣶⡆⠋⢾⠶⣄⡀⠀⠀⠀⠚⠴⠁⠀⠀⠀⣀⠀⠙⢶⡟⠀⣤⠰⠂⠀⠀⠀⠀⠜⢹⢛⡤⠗⠀⠀⠀⠀⠉⡀⣀⠀⠀⠉⠉⢉⣁⣀⣹⡦⠖
⠀⠀⠈⠁⠀⠰⠟⠋⠁⠀⠀⠀⠀⣐⡒⠤⡖⠀⠀⠀⠀⠀⠙⠢⠄⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢎⡵⡻⠁⠀⢀⡞⠋⠉⠈⠹⡇
⡀⠀⠀⠀⠀⣤⠘⠀⢀⡠⠔⠚⢒⣨⠥⠲⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠋⠀⠀⠀⠀⢸⡃⠀⠀⣀⡼⠁⠠⣄⡀
⠸⠗⡄⢐⡟⠁⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠑⠀⠀⠀⡀⠀⠀⠠⠀⠀⢠⠀⠀⠀⠀⣮⣀⠉⠀⠰⣾⠁⠀⠀⠀⢠⣴⡦⢼⣧⠶⠛⠉⠀⠀⡒⠸⡆
⢰⣾⣵⡊⣀⡔⠀⠀⠀⠒⢈⡀⠀⠄⠀⠈⡄⠀⠀⠀⠀⠈⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠐⣶⠃⠀⠀⠀⠀⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠐⠟
⠈⠾⣬⡷⠋⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠤⣀⣀⠴⠛⠢⠀⠀⠀⠀⠀⠀⢄⡙⠀⠀⠀⣀⣤⣿⡤⠶⠞⠁⠀⢀⣈⣲⠀⡀
⠀⠀⠈⡇⠀⠀⠀⠀⡌⠐⢀⡈⠈⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⡀⠠⡄⠀⠀⢀⠠⠄⠀⡆⣠⠘⠃⠀⠀⠀⢸⡏⠁⠈⣿⠀⠀⠀⡀⢹⣿⡿⠚⠏⢹⣦⡀
⠀⠀⢀⡅⠀⢀⠀⠀⠆⠠⢠⠀⠌⠁⠁⠀⠀⠀⠀⠂⠀⠀⠰⠄⠵⠀⢀⠀⢈⡀⠀⠀⠠⡀⠤⠈⠀⠀⠀⢸⣇⣠⠴⠋⠀⠀⠀⠉⠊⣵⠀⠀⠀⠀⠉⠛
⠀⠀⢸⡟⠀⠘⢆⠀⠀⠙⠀⢀⣀⠀⠀⠀⣀⣠⡤⠤⠤⠄⠶⠶⠀⠀⠈⠀⠀⠉⣁⣀⣀⡈⠀⠁⠀⢠⡞⢻⡏⠀⠀⠀⡈⣷⠀⠀⠀⢷⣠⡄⠀⣠⢾
⠀⠀⠘⠃⠀⠀⠈⢦⡀⠀⠀⠘⠼⠀⠈⠉⠈⠁⠀⠀⠠⠀⠁⠀⠀⠀⣠⡤⠴⢿⡿⠿⠟⠋⠀⠀⢀⣼⠷⠋⠀⠀⠀⠈⠉⠳⡀⠀⠀⠀⠉⠳⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣾⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⠶⠶⢶⣶⡶⢶⡶⠶⠶⠶⣶

⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀⠀⠀⡠⠋⠊⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⣀⠀⠀⡠⠋
⠜⠁⠀⠀⠔⢂⠀⠐⠢⠀⠀⠀⠀⠀⠀⡇⠀⡴⢿⠶⠋⠀⠀⢀⡼⠁⠀⠀⠔⢂⠀⠐⠢⠄⠀⠀⠀⠀⠀⡇⠀⡰⢿⠷⠋
⠀⠀⠀⢸⠀⢲⡖⠀⠀⠈⢢⠀⠀⠀⠀⠰⣰⣤⠀⠀⣀⠠⠐⠁⠀⠀⠀⢸⠀⢲⡖⠀⠀⠈⢢⠀⠀⠀⠀⠰⣖⣤⡀⠀⣀⠠⠐
⠀⠀⠀⠘⡀⠀⢳⣀⢀⡄⠀⢁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢳⣀⢀⡄⠀⢁⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠐⢄⠀⠛⢿⣿⠄⣊⠀⠀⠀⠀⢀⣀⠤⠠⠤⢀⡀⠀⠀⠀⠀⠀⠐⢄⠀⠛⢿⣟⠀⣊⠀⠀⠀⠀⢀⣀⠤⠠⠤⢀⡀
⠢⡀⠀⠀⠀⠀⠑⠶⣦⣿⡯⠟⠀⢀⠔⠈⠁⠁⢀⡀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠑⠶⣤⣿⡮⠟⠀⢀⠔⠉⠁⠀⢀⢀
⣀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣢⡶⠀⠉⠁⠉⠀⢠⣀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⣢⡶⠀⠉⠁⠉
⣬⠳⡀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⢧⡀⠀⠀⠀⠀⠠⢬⠳⡈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⢧⡀
⣾⣷⠗⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠈⠓⠦⣤⣤⣤⣾⡷⠇⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠈⠓⠦⣤⣤
⠉⢁⡰⣿⠃⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠉⠉⢁⡴⣿⠃⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠑⢄
⡿⠷⠚⠁⠀⢀⠔⠊⠀⠀⢀⡉⡙⢷⡄⠀⠀⠉⠒⠤⠤⣐⣒⣒⡾⠿⠛⠁⠀⢀⠔⠈⠀⠀⢀⡉⡙⢷⡄⠀⠀⠉⠒⠤⢤⣐⣂
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠈⢹⡗⠂⢘⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠈⢹⡗⠆⢘⡿
⠀⠀⠀⢰⠀⠀⡀⠀⠀⡠⠋⠊⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠀⣠⠋⠊⠀⣸⠃
⠀⠀⠀⡇⠀⡴⢿⠷⠋⠀⠀⢀⠔⠁⠀⢀⠔⠂⠀⠐⠢⢄⠀⠀⠀⠀⠀⡇⠀⡴⢿⠷⠋⠀⠀⢀⡼⠁⠀⢀⠔⠂⠀⠀⠢⢀
⠀⠀⠀⠰⡶⣤⡀⢀⣀⠠⠐⠁⠀⠀⠀⢸⠀⢲⡖⠀⠀⠈⢢⠀⠀⠀⠀⠱⣵⣤⡀⠀⣀⠠⠔⠁⠀⠀⠀⢘⠀⢰⡖⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢳⣀⡀⡄⠀⢃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢳⣀⠠⡄
⠀⠀⠀⠀⢀⣀⡀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠘⢄⡀⠛⠿⣿⠀⣈⠀⠀⠀⠀⢀⣀⡀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠐⢄⠀⠛⠿⣿
⠀⢀⠔⠉⠁⠁⢀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠑⠶⣮⣿⡪⠞⠀⢀⠔⠈⠁⠀⢀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠑⠦⣦⣿⡯
⠀⡘⠀⠀⣢⡶⠀⠁⠉⠉⠀⢤⣀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣢⡶⠀⠈⠉⠉⠀⢤⡀⠈⢢⡀
⠀⠁⠀⠀⠈⠧⡀⠀⠀⠀⠀⠠⣬⠲⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⠧⡀⠀⠀⠀⠀⠀⣌⠲⡀⠱⡀
⠀⠘⢄⠀⠀⠀⠈⠓⠦⣤⣤⣤⣾⡷⡗⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠈⠓⠦⣤⣤⣤⣼⣷⡇⠰⡃
⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠉⠉⢁⡴⣻⠃⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠉⠉⢁⡴⣳⠃⠀⠀⠀⢀⣀⣀
⡙⢷⡄⠀⠀⠉⠒⠤⢤⣐⣒⣒⡿⠽⠛⠁⠀⢀⠔⠈⠀⠀⢀⡉⡙⢷⡄⠀⠀⠉⠒⠤⢤⣐⣐⣲⡿⠿⠛⠁⠀⢀⠔⠈⠀⠀⢀
⠇⢘⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⢹⡟⠆⢘⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄⡀⠸⢀⣄⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⣀⣴⣾⡯⠤⠾⠛⠁⠀⠀⠀⢠⣿⣥⣿⣿⡷⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠛⠋⠂⠈⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣇⠄⠈⣾⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⣸⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠏⠈⡄⠀⡠⠔⢏⠉⢹⣇⣇⣦⠖⠁⠀⣴⣿⣿⣿⡆⠀⠀⠀⠤⣿⡀⠃⣼⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⣢⣇⣴⠿⣦⣤⣼⣿⠾⣷⣆⠀⠹⣿⣿⡿⠃⠀⠀⠀⡖⣿⣏⣹⣿⢸
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠄⠄⠀⠀⠈⠁⢻⣄⣽⣿⡿⠧⠤⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⢸⣿⣹
⠀⠀⡤⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⢰⠀⠀⢻⣥⣽⣷⠀⣠⣴⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⡷⡿⡇⢸⢿⢾
⠀⠀⣤⣀⣴⡇⠀⠀⠈⠉⠛⠶⢶⣾⣄⣤⡀⠀⠈⢿⡏⠀⣿⡓⣶⣵⡀⠀⠀⠀⠀⠀⠀⣿⠃⡇⢸⡎⢿⡀
⠀⢾⣿⠋⠙⠳⣄⠀⠙⠓⠶⠟⠋⠀⠈⠉⠉⠀⠀⠈⢿⣤⣴⡝⢿⣟⠁⠀⠀⠀⠀⢀⣾⣹⡿⢹⣾⢹⣿⣷⡄
⠀⢠⣿⣦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠁⠘⣉⠀⠀⠀⠀⠀⠀⠁⣟⣷⢿⣿⢾⣘⠆⠁
⠀⠉⠀⠀⠉⠳⢴⣄⣀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠻⢥⡑⣄⠀⠀⠀⠀⠘⠁⠈⠁⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢂⠀⠀⠉⠁⠀⠀⠀⠀⣠⡖⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠈⠙⠲⢕⢄⠀⠀⠀⠀⠀⠀⠀⢹⣤⡆⠀⠀⣰⠃⠀⠘⠋⠉⠠⠂⡀⠐⠀⣆
⠀⠀⠀⠠⠐⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⢀⢈⠀⠀⠀⣄⠁⠰⠆⠀⡀⣐⣰⣿
⠀⠀⠀⠀⠘⢠⣤⡀⠀⣠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⣤⠶⠷⠛⣨⠀⠀⠈⣶⣴⣰⣾⣥⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠛⠏⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡤⠚⠋⠀⣀⣴⣾⣯⣄⣀⠀⠈⠙⠛⠛⠛⠛⠁⠀⡀
⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠠⠀⠀⠀⠀⣤⠚⠋⠀⢀⣤⡼⢟⣋⠥⠭⠯⠥⠭⠵⠖⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠈⠁⠀⠀⢀⠐⠁⠀⣠⢾⠀⣀⣴⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠔⠂⠄⠀⠄⠒⠐⠁⠀⠰⠿⠷⠛⠋⠉⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣶⣀⡀⠰⠄⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠱⣆⠴⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣰⣾⡿⠀⠀⠀⠀⢨⡆⣄⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣤⣶⣦⣠⡤⠾⣧⣿⣟⠉⠻⢧⠀⠀⠀⣶⢸⣷⣿⠁⣸⣿⣅⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠙⠿⠈⠫⣄⠀⠙⣿⣿⡀⢀⡀⠀⢴⡆⣻⣿⠛⣽⢢⡿⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠘⣿⣷⣸⡏⠀⢸⠃⢀⣏⠀⡢⠿⣷⢶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⢰⡄⠈⠁⠀⢀⠀⠀⣤⠼⣿⣿⡇⠀⢹⡗⠈⠉⠈⠀⢸⣿⡿⠉⠉⡉⢉⣷⡷⣦
⠀⠀⠀⡠⠖⠛⠛⠛⠉⠀⠓⠀⠀⠘⣿⣼⣿⣇⠀⢸⠇⠀⠀⠀⠀⡜⠁⠀⠀⠀⢀⡾⠃⠀⠈⠓
⠀⠀⠀⠶⣤⡤⠄⠄⡀⠀⠀⢀⠈⠰⣼⣿⣿⣿⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠈⠉⠶⣝⣻⣿⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡌⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠳⠶⣥⣌⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡇⠄⠀⠀⠀⠀⠠⠚⡀
⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣙⣿⣿⣶⣶⡄⠀⠰⢷⣿⣿⣩⡖⠛⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⣷⣶⠶⠚⠋⠉⠁⠀⠈⠻⢿⣿⣶⣤⣀⠈⢹⠉⠙⢷
⠀⠀⠀⠀⠀⠸⣇⠠⠈⠉⠉⢉⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⣞⣻⡿⢿⠆⢺⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠙⠷⠶⣦⣤⣀⣀⣀⡀⠀⠀⢐⠊⣿⡶⠉⣘⣿⣤⡷⠾⠈⣿⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⣿⠛⠃⠀⢴⠛⠋⠁⣠⠞⠀⡟⢧⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠁⠀⠀⢠⠋⠀⢀⡌⠁⠀⢸⠃⠈⡆⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⣾⠀⠀⠀⠘⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠃


⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠅
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠤⠐⠃⠀⠀⠀⢀⠂
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⢄⠀⠀⠆
⠀⢸⠀⠀⢀⢀⡀⠄⠀⠀⠀⠀⠐⠂⠐⠄⠠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠰⣿⡷
⠀⢸⠀⠀⣠⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢸⠀⠀⠀⠈⠞⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⢀⠀⠀⠀⠀⠀⠀⢂
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⢄⠀⠐⠈⠀⠀⠀⠀⠀⠀⢂
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠀⠤⠄⠀⠀⠀⠀⠁⣀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⠤⠀⢠⢄⡈⠉⢅⡢⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠃⢤⠀⡠⠀⠐⠓⠁⡈⠠⠀⠡⠀⠀⠀⠀⠁⠂⠀⠀⠤⠤⠄⠔⠐⠈
⠀⢸⠀⠀⠀⠀⠀⠀⠌⠠⠊⠉⠈⠣⣝⠀⢀⡀⠀⠀⠻⢢⢅
⠀⢸⠀⠀⠀⠀⠀⠀⠢⠂⢀⢄⠀⡸⠀⠀⠀⠁⠀⠀⠀⢇⢀⡱
⠀⢸⠀⠀⠀⠀⠀⢀⢠⠀⠈⠈⠢⠙⢆⠀⠀⠦⡀⠀⠀⠀⠈⠂⠀⢀⠠⢄⠐⡄
⠀⢸⠀⠀⡠⠐⠈⠀⠸⠀⠀⠀⠀⠀⠢⢉⡒⠐⠊⢁⠡⠐⠓⠈⡁⠀⠀⠀⡱⠁⠀⢀
⠀⢸⠀⠄⠱⠀⠀⢀⡀⠧⠰⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡁⠠⠐⢈⠔⠁⠀⠀⠀⠈⠀⢄
⠀⢸⠀⠄⠀⠱⠂⠙⠀⣠⠁⠀⠀⢀⣀⠌⠠⠀⠀⠠⠂⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⢸⠀⠀⠀⠀⠀⢂⡐⠀⠀⠀⠀⠀⠀⠁⠐⠠⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠓⠤⡀
⠀⢸⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠄⠒⠈⠀⠀⢀⠠⢤⠊⠁
⠀⢸⠀⠀⠀⣠⠁⠀⠀⠀⡠⢊⠌⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⢀⠠⠄⠂⣁⣤⢐⠊
⠀⢸⠀⠀⠠⡀⠈⡢⠔⠈⡠⢀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠤⠂⣉⠠⠔⠀⠃⠈
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠜⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⡐⠀⢀⠀⠀⠀⠀⠁⠀⠵⠆⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⡀⢀⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠢⡈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠤⠂⠀⢄⢅⡀⠀⠂⠁⠂⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠄⡀⢀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠒⠒⠀⠨⣀⠼⠀⠀⠀⠈⠈⢀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠔⠀⠂⠀⠀⠀⠀⡆⠀⠒⠀⠢⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠂⢀⣀⠒⠁⠀⠀⠀⠠⠤⠤⠾⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⢀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠐⢂⠴⠀⠠⡀⠀⠚⡛⣛⠛⠉⡉⠉⠀⠀⠄⢀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠘⣤⠀⠈⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠲⢇⢠⣦⡄
⠀⠀⠀⠀⢄⢠⠀⠀⠀⠀⢲⡄⣀⣁⠀⢀⣀⡀⠀⣀⣀⡀⠀⣀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⡀⠀⣀⠀⢀⡀⠀⠀⢦⡘⢁⡤
⠀⠀⠀⠈⠙⢚⠀⠐⠀⠀⠀⠀⣿⣹⠇⠈⣿⠀⣸⠏⠈⣷⠀⣿⠀⠀⢠⡏⠀⢻⡄⣼⠁⢈⡁⠀⠘⣆⣼⠃⠀⠀⡀⠉⠁⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢙⡇⢀⣿⠀⢻⣄⣀⡿⠀⣿⣀⣀⠸⣇⣀⣼⠃⢿⣀⣈⡿⠁⠀⣸⠇⠀⠀⠠⠁⠠⡁⠀⠤⠆
⠀⠀⠀⠀⢀⠀⠠⠀⠄⠀⠄⠀⠉⠉⠀⠈⠉⠁⠀⠉⠉⠀⠀⠉⠉⠉⠀⠉⠉⠁⠀⠈⠉⠉⠁⠀⠀⠉⢀⠀⡐⠀⠐⢄⢀⡂⠂
⠀⠀⠀⠀⠀⠀⠓⠀⠈⠀⠀⠀⡁⢈⠆⠀⡰⠢⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠆
⠀⠀⠀⠀⠀⠠⠒⠤⠒⠀⠀⠈⠂⠁⠀⠈⠐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⠀⠁⠒⠈⢀⠀⡘⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⡀⠀⣖⠁⡐⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠑⠄⠀⠀⠀⠈⠁⠀⠀⠠⠰⠀⠀⡐⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠰⠂⠛⠂⠀⠀⠘⠃⣑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣗⠀⠃⠀⠃⠀⠆⠐⡵⠀⡔⠀⠠⠀⠀⢲⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢀⠀⠀⠄⠀⠁⠐⢀⠐⠈⠁⠀⠀⠀⠆⠑⠁⠐⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⡄⠀⠀⠠⠟⠂⠐⠢⠄⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢴⠂⢮⡉⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠁⠉⠁⠀⠀⠀⠀⠀⣤⠟⠻⠀⠀⠈⠀⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠦⠤⣤⠀⠀⣀⣀⡀⠀⠀⠀⢀⠀⣤⣠⡄⠀⠀⠀⢶⠄⠈⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⣶⡝⢿⣷⣷⣿⣷⣿⣿⢿⣿⡄⢀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠛
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢧⡀⠀⠀⠀⠀⢸⡤⠛⠛⢻⣿⣦⠛⣽⣿⣿⣿⡁⠀⣩⣀⠀⠀⠀⠀⠀⣠⢤⣤⡀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢺⣿⠃⡈⣳⣤⡀⠀⠀⠀⠀⠋⠘⢛⣿⣿⣷⡹⡿⣿⣿⣧⣴⣿⣿⣶⣤⠀⠀⠀⡇⠀⠄⠇⠀⠘⣿⡇⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣀⣿⣿⣿⠀⢀⢀⣆⠀⣰⡟⣿⣿⣿⣿⠖⠉⡙⣿⣿⣿⠋⣳⠙⣦⠀⠀⢡⠀⠀⡄⠀⠀⡿⢿⣾⠋⠻⡆⢠⣯⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠇⠁⡘⣿⣿⣿⣷⠌⢹⣿⠿⡦⢞⢆⣿⣿⡟⣰⢏⡇⢃⠰⠀⠀⠀⠀⠁⠀⠀⣿⣿⣇⠀⢠⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⢰⣀⠉⠉⠉⡏⠀⢾⣧⢐⠄⠀⡺⣿⣿⣷⠿⢸⠃⢸⠀⠀⠀⠀⢠⠄⢀⠀⣿⣺⣿⣶⣌⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⡘⠀⠀⠀⠀⠟⠷⢻⣿⠏⠀⠄⠿⢿⣿⡇⡤⢸⠆⠘⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠛⠃
⠀⠀⢐⠳⠀⠀⢠⣾⣿⣿⣯⣭⣉⡁⠀⠁⠀⠀⠀⠀⡀⣄⢈⡧⠤⠀⠀⠀⣾⣿⠀⣤⢻⡄⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣯⣭⡏⡀
⠀⠀⠀⠀⢰⣷⠾⣿⣿⣿⣿⠡⠀⠀⢀⠀⠀⣠⣴⢶⠀⡟⠁⠛⣷⠎⠀⠀⣿⣿⣤⡀⢰⣷⣦⣀⠀⠀⠀⠀⠀⠀⣶⡉⢉⣿⠿⠏⠰⠇
⠀⠀⠀⠀⠀⠨⣦⠘⠻⣿⣧⠀⠈⣸⣺⡤⠎⣡⡜⢉⠀⡏⠀⠹⠇⢀⡆⠸⡿⠂⠼⣏⣠⠁⠈⠸⠿⠷⡾⠿⠛⠀⢧⡀⢀⡽⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⣷⣄⣩⣽⣇⢸⣿⢷⣥⠞⠁⠀⣠⡿⠋⠀⢿⡇⣼⣇⣠⣶⡀⣸⣿⣷⣜⠃⠠⠀⠀⠀⠀⠀⣰⡿⠁⠘⣧⣰⣷⣾⣆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⢯⡽⠹⠻⡻⠽⣄⠀⠀⡇⣿⠁⠀⠀⠘⣿⣿⣿⠟⠻⢿⠿⣿⣿⣿⡀⡀⠒⠀⠀⠀⠀⡿⠁⠀⠀⠸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡟⢦⣴⠃⠀⢩⣴⣿⡆⠀⡇⣿⠀⠀⠀⠀⢹⣿⣧⣤⣤⣀⡉⠩⢽⣿⢹⡵⢦⣤⡛⠆⢰⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣏⣿⡭⠀⠀⠀⢿⣸⠇⠠⡅⣿⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⣷⢸⣿⣸⡏⠘⢈⣿⡆⢸⣤⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣤⣤⣤⣞⠁⡀⢀⢀⣇⣀⣀⣀⣀⣸⣿⡿⠠⠤⠄⣀⣍⣸⣿⣏⢷⣀⢸⡷⡇⢰⠉⠉⠛⠉⠙⣿⣿⣯⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣠⣠⡤⣌⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⠀⠈⢻⣿⣿⣾⠳⡾⢿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⠳⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⣌⠀⠀⠈⢻⣿⣿⣶⣤⣾⡇⠰⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⡀⣙⢿⡷⠶⠶⣤⡾⠙⡏⠀⠀⠀⠀⢀⠀⠸⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⢀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠀⢸⣿⠀⣶⣾⠀⠀⠀⢠⣶⣟⢿⡟⠀⣬⠀⡼⠯⠈⠀⣠⡄⠀⠀⠀⢸⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⡈⣿⡼⠳⠃⢰⣶⠀⣥⡌⠀⠀⠀⡀⠘⢳⡶⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠢⣹⠈⢿⣈⡇⠰⠚⠀⠉⠃⠀⠀⣶⠀⠀⠀⠀⠀⠀⢰⣶⡄⢼⣶⠀⠀⠠⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠸⠷⠋⠀⠘⣧⡼⠁⠀⠀⠀⠀⠀⠑⠊⠃⠀⠂⠀⢀⡀⠉⠻⠷⠿⠋⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠔

⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠐⠀⠠⢀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄
⠀⠐⠐⠀⠘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠗⠀⢠⠇
⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⢦
⠀⠀⠀⠁⠀⠄⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠈⠁⢀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡎⢱⠀⠀⢀⣠⡄⠀⣀⡠⢀⡀⣀⠀⠀⢀⠀⠀⣀⣀⡀⠘⡂
⠀⠀⠀⢲⠀⠘⠀⠀⠈⠀⠀⠌⠀⠀⠀⠈⣠⠼⢁⣀⡇⠀⠕⠀⣄⢸⠠⢁⡀⢟⠀⠈⡇⠀⠀⠦⠀⠃⠁
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠁⠀⣀⣈⢀⣁⣀⣈⣀⣁⣀⣀⣈⢀⣉⣁⣀⡇⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠉⠈⠉⠈⠉⠉⠉⠈⠉⠉⢉⠉⠉⠉⠁⠀⠀⡀⠃⠈⠀⠘⠀⠈⠈
⠀⠐⠀⠀⠀⠠⠀⠀⠀⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠰⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠄⠡⠤⠤⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠄⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⢶⣶⡶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣤⣤⣬⣥⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⣉⣩⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣉⣩⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠋⠛⠛⠋⣛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⡀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠠⠴⣶⠖⠢⠄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠂⣠⠶⠛⠒⠆⡀⠐⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠒⠒⠁⡄⢠⡄⢠⠈⠒⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠌⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⠀⢀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⠆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠁⡌⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⢰⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡄⠠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠈⠢⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⠔⠁⠀⠀⡠⠂⠀⠀⠋⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠀⠘⢤⡀⠀⠈⠢⡈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀⠔⠁⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠢⡄⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠂⠁⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠈⠐⢄⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠊⠁⡠⠊⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠧⠤⠄⠠⠤⠤⠠⠤⠤⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄⠈⠑⢦⡀
⠀⠀⠀⣰⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠀⠠⠠⡔⢢⠤⠄⠀⠸⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠢⡀⠙⣆
⠀⠀⠀⢹⠀⠃⠒⠀⠒⠒⠒⠀⠒⠂⠀⠒⠒⠒⠒⠒⠒⠂⠖⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠲⠐⠀⠒⠒⠒⠒⠀⠐⠂⠒⠂⠒⠒⠒⠂⠒⠘⠀⡏
⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠚⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠃⢤⣀⣀⡀⣴⢿⡿⢠⣀⣀⣀⠠⡞⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣀⣭⣹⣯⣨⣅⣀⣈⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡿⠋⣍⣀⣀⣀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠸⠿⠟⠀⠀⢰⣶⠀⣰⠚⣿⣭⡁⢳⣿⠋⠀⠘⠇⠀⠀⠈⠙⠛⠉⠀⠙⠛⠋⠀⠙⠓⠋⠁⠈⠛⠚⠉⠈⠑⠒⠀⡀
⠀⣴⣤⣴⣶⡦⢤⠀⣿⣤⣿⣤⣥⣬⣭⣤⠀⠀⠀⠀⠀⠎⠈⢳⡞⠁⢱⡞⠁⠹⣶⠁⠈⣶⠁⠈⣷⠏⠀⢳⠋⠀⢹⡀⠀⠀⠀⠀⠀⠀⠺
⢀⡈⢵⠌⡋⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠈⢧⠀⢸⢧⠀⢠⢿⡀⢀⢿⡄⠀⡾⣇⠀⡸⣇⠀⡸⢷⡀⣠⠀⣶⣦⣀⣶
⠛⠟⠀⠺⠿⠀⠀⠻⡼⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠈⢁⡁⠀⠀⠱⠄⠉⡌⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣁⡀⠀⢻⣿⣿⣿
⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠐⣦⣷⣷⣾⣷⠲⣞⠀⠀⠀⠀⢀⣀⡀⠀⠁⠠⠀⠀⠋⡙⠛⠉
⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣧⠀⠀⠀⠀⠚⠢⠀⢀⠀⠈⢑⣾⣿⣟⠋⣋⣿⣿⣿⣿⡥⠖⠀⣾⣿⣷⠀⢸⠀⠶⢐⣇⣇⣐⠈
⣛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠽⠿⠽⠀⠀⣀⣠⣤⠖⠀⢠⣄⠰⢼⣿⣿⣿⡴⣿⣿⣇⣨⣿⣧⣀⡀⠈⠉⠁⠀⠈⠀⠀⢉⡙⠉⢛⣀
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢐⡀⡊⠉⠁⠀⢙⠀⠐⠊⢀⠽⢷⣿⣿⣤⣽⣿⣾⣿⢻⣇⠀⠀⠀⠀⠀⠀⠠⠀⠀⣸⣇⠀⠈⠩
⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⣸⠃⢈⣶⡶⢾⣉⠀⠀⠄⠂⠐⠊⢻⣧⣼⣿⣿⣾⠿⡃⠈⠀⠀⠄⢀⣀⣤⣤⣤⠀⠐⠦⠄⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠙⢧⠀⣉⠀⠀⠀⣈⠀⠀⠉⠄⠀⠐⠃⠀⠈⠏⠃⠘⠄⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣸⠁⡨⠷⠚⠛⢥⠀⠀⠀⣠⡄⠈⠀⠠⠀⣤⣦⡄⠀⠀⠀⣴⡶⣾⣿⠿⠿⠛⠋⠁⠸⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠲⣧⠀⠲⣀⣀⣠⡴⠀⠀⠀⠨⠀⠘⠀⠀⠀⠻⠿⠃⠀⠀⠊⠉⣈⣉⣁⡀⠀⢀⡀⠘⠳⣶⠤⣿⣿
⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⠇⠴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣴⡄⠀⠀⠀⠀⠉⠙⠋⠁⠉⠁⠀⠰⠶⠀⠶⠈⣁
⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⣀⡉⠀⠰⠤⠀⠀⠀⣘⣃⢀⠀⠤⠀⠠⣶⡶⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⣀⡄⣄⠠⠀⠀⠂⠂⠘⢿
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠈⠛⠘⠿⠿⠺⠗⢠⣴⣄⠈⣿⣿⡿⣧⣶⠆⠀⡉⠉⠀⣿⣿⣿⣿⣷⠀⠀⠀⠰⡀⠂⠀⡃
⠀⠀⠒⠒⠀⡍⢳⣧⣻⢽⠀⠀⠀⠀⣀⣄⠻⠃⠈⢹⣷⡄⠛⠛⠋⠉⠿⠀⠘⠃⢠⠀⠻⣿⣷⡿⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠦⡈
⠀⣾⠀⠀⠀⠀⡾⠃⡵⡞⠀⠠⠀⣀⡿⡿⠃⣠⡰⣾⣿⣵⣶⣦⠄⠀⠠⢴⣲⡀⠀⠠⡦⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠃⢀
⠂⠹⠂⠀⠀⢰⡱⣿⣻⠋⡄⠠⢀⣛⠁⣾⠆⣽⣧⡋⠘⢿⠓⢮⠑⠀⠀⠋⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢠
⠄⠀⠀⠀⠀⠀⠓⠛⠁⠀⠀⠀⠈⠁⠈⠛⠀⢸⣿⣧⠀⢀⡀⣀⣠⡤⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇⠈
⠀⡀⠀⠾⢶⡄⠀⠀⠀⠀⢠⡴⣶⣿⣽⣦⡄⡀⢻⣿⣷⣽⡯⠽⠋⠀⠀⣀⠀⠀⠀⠠⠀⠄⠀⠁⠀⢠⣶⡄⠀⠀⠄⣁⣨⣀⡀⠄⠈⠁
⠠⡇⠰⠶⠀⣄⠰⠆⠀⢠⣿⣿⠏⢻⣿⣿⣧⡀⠈⢻⣿⢿⣶⡾⣶⠀⠈⠙⠀⠀⠀⣀⠀⢠⣦⡄⠀⠈⠙⠁⠀⣰⡿⣿⣿⣿⢻⣶⡀
⢠⡇⢴⡤⠀⢿⠀⣦⠀⠀⠻⣿⣿⣶⣾⣽⡇⠀⢀⣿⣿⣾⣿⣷⠿⣷⣦⠀⠈⠀⠀⠈⠁⠈⡟⠁⠀⠿⠂⠈⢠⣿⣷⣿⣿⣿⣿⣿⣇⠈
⠈⠃⣀⣀⡀⢀⣀⣘⠀⠀⠀⠈⠉⠻⢋⠁⠀⣀⠀⢉⡛⠻⠿⣿⣶⡆⢰⣾⣶⣄⠛⣶⡦⢾⣿⡧⠰⣿⡆⠐⠂⢿⣿⣻⣿⣿⣯⣿⠇⠒
⣴⣿⢶⣆⣙⣻⢶⡨⡕⡲⡦⠀⠀⠄⠀⢠⡀⠈⠃⠈⡉⠛⠀⠀⠈⠀⠀⠙⠋⠀⠘⠃⠀⣠⣯⠀⠰⠆⠰⡶⠀⠈⡙⠿⢿⠿⠟⡉⠀⢀⣤
⣪⣿⣼⣟⡡⠪⠿⠄⢅⣧⣤⡽⠀⣀⣀⡿⠁⢰⠄⣴⠏⠐⠞⠛⢲⣤⠖⠛⠓⢦⡶⠛⠓⣎⡵⠚⠓⠲⣤⠖⠓⠂⠀⠁⠘⠀⠁⠀⠀⣿⢿
⢠⣧⣄⠀⠠⠴⠶⠀⠀⠀⢀⣄⣼⣿⣿⣿⠀⢠⡄⢠⡴⠀⠀⠀⢠⣟⠀⠀⢀⣿⡀⠀⠀⣸⡃⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉