Frost Emojis & Symbols

⠀⠆⠀⣤⡊⢀⠀⢻⠀⠀⠐⠠⠊⠁⠀⠒⠒⠲⠤⠤⣀⠀⠀⠀⠀⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠄
⣡⢈⣹⣿⡑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢠⡀⠀⠀⠀⠀⢀⣴⠚⠁
⢥⠘⣩⣋⣋⡷⢶⠶⠊⠀⢠⠀⠀⠀⠀⠀⣠⣰⡭⠖⢋⠴⠃⠀⠀⠀⠀⠈⠱⠀⢀⠤⡘⠋
⠀⢘⣻⣿⣿⠿⢁⠔⠤⠂⠀⠀⢀⠠⠴⢀⡤⢯⠄⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠄
⢞⣞⢹⣿⣿⣫⠕⠦⠄⠀⠀⠀⠈⠀⠀⠀⠀⠙⣸⢷⠂⠀⠤⠌⠤⠀⠀⠀⠀⠀⠀⢀⠹⡍⠿⠛⠋⠀⠉⠂
⠲⣴⣯⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⣯⡛⠿⢧⣄⠀⠣⢀⡀
⠸⠛⠉⠁⠀⠀⠀⠀⡀⠈⣛⡁⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡾⢆⠧⠽⣥⠀⠈⠁⡠⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⣖⠀⠁⠀⠰⣆⠀⠀⠀⠀⠀⢐⡘⠂⠒⣙⣷⠀⣠⣤⣶⠖⠉
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠐⠀⠀⠀⠑⣶⠟⡅⠆⠀⠀⠀⢨⠄⠀⢼⠀⠀⠀⠀⠀⠀⠀⠚⢹⣦⡀⠈⠣⠴⣢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠴⠉⠀⠀⠀⠀⡿⠀⠀⠐⠠⠁⣤⡀⠀⠀⠀⠃⢸⡷⠁⠁⠀⠠⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣬⠯⡶⣰⠀⠀⠚⠁⠀⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡾⣫⡾⢙⣥⢰⠀⠀⠀⢀⠀⠠⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠩⢭⠄⡟⢡⣿⠀⢂⡀⠉⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠍⠓⠀⡸⢁⣿⡏⣐⣿⡶⠀⠀⠈⠱⢦⠀⠦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⣿⠏⣠⡽⡶⠯⠥⠀⢰⠀⠀⠠⠄⠄
⠀⠀⠀⠀⠀⠈⠉⠀⢀⠤⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠋⡠⣏⣛⡨⠈⠀⠀⠀⠀⢀⡄⠈⢣⠂⢁
⠀⠀⠀⠀⠀⠒⠶⠀⠉⠁⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠰⠃⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠙⠠⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡆⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠺⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢰⠄⢀⣴⡞⠋⠁⠀⠙⠀⠀⠐⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠐⠠⠄⠀⠀⠀⠀⣠⣤⣀⠀⠀⡀

⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠕⠊⢀⣈⠍⠉⡉⡩⣰⡼⠿⣀⣀⠀⠀⣄⣠⣀⠀⠀⠀⢁⠤⢀
⠀⠀⠀⠐⠃⢹⣷⣀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣤⡐⣲⣿⣚⣎⣘⣟⣳⣪⣿⣿⣿⣿⣿⣽⣶⣿⣿⣞⣻⣿⣿⣶⠦⣠⣝⡤
⢢⠀⢀⠀⠂⠰⣿⡹⣄⡀⢄⠀⠀⡨⣧⡤⡾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠘⢇⠀⠀⠀⠀⠘⣿⣎⢷⣾⣧⣚⡰⠿⣿⠓⠹⣟⠉⢻⡏⢻⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠳⡀⠀⠀⠔⢈⠿⣿⣿⣿⣿⣄⠀⠘⢧⣄⣻⣆⠀⣻⣞⣿⣿⣹⣿⣿⣻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢤⡀⠙⢶⣦⣄⡄⡸⠏⠛⢯⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣽⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⠿⣿⣾⣽⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠠⡀⠀⠙⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣾⡛⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠑⠷⢯⢄⣀⣙⣿⣿⣿⣿⣿⣿⠏⢀⣼⣿⡿⠉⠰⠛⠛⠛⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠦⠥⠤⢴⠶⢛⣻⣿⣿⣿⠋⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠺⠋⣹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠰⠒⠤⠤⠤⠶⠛⣻⣿⣿⠛⠀⠀⢸⡟⠃⠃⠀⠀⠀⠀⠀⣀⣀⣈⣛⢿⣿⣿⣿⣿⡟⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣷⣸⣿⣝⠿⠋⢿⡟⠉⣿⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⡀⢠⡀⠀⠀⠀⢄⠀⠆⣸⣿⣿⣿⣿⣿⣿⣷⣿⣏⠀⠀⠀⠀⢀⣾⣿⠇⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠃⢸⣷⡄⠀⠀⠀⠈⠇⢸⣿⣿⣿⣿⣿⣿⣯⣿⣿⠳⠀⠀⠀⣻⡿⠉⠈⠋⢁⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠁⠀⠀⣴⣶⣌⣻⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠸⡟⠓⠀⠀⠀⠀⠀⠀⠘⠝⠛⣟⣻⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠃⠴⠄⠀⠀⠀⠀⠀⠀⠀⠞⠻⣿⣿⠛⢁⠀⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟⠈
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⣶⣦⣦⣤⣀⠀⠀⠀⠀⠀⠀⠟⠟⠀⠉⠀⠀⠀⠀⠀⠤⢾⣿⣿⣿⡟⠁⢀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁⠻⠏⠙⣸⡢⢼
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠿⣿⣇⠀⠀⠀⠀⠹⣿⣟⣿⣿⣿⣯⣉⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢀⣥⢃⢳⢾
⠀⠀⠀⠀⠀⠂⠿⡿⠿⢛⣁⡀⢺⣍⠀⠀⠀⠀⠀⠙⢿⣿⣯⣿⡿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⠋⣀⣀⢤⡾⣿⣿⡨⢥
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡞⠛⣿⣿⣷⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⡆⣴⣶⣾⡿⢀⣠⠖⠀⣁⣜⣳⣮⣿⣿⣿⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠋⣿⣿⣦⡐⠠⠀⠀⢀⠀⠈⠛⠻⠿⡿⣷⣿⠿⠟⣃⣈⣢⣴⣿⣿⣯⣿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠃⠘⢿⣦⣄⡀⠀⠉⠀⠀⠠⡐⣤⣢⡓⣾⣯⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠛⠻⣿⣶⣬⢶⡤⠰⠇⠄⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠂⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠿⢧⣀⣀⣀⣀⣴⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠙⠚⢁⡠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣡⡌⠱⣄⡏⠀⣀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢹⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠉⠢⣡⢠⠎⠀⡀⢹⡆⢀⣤⠶⢤⡀⠀⠀⠈⠓⠀⡆⢠⡶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣶⠀⠀⢹⡟⠀⠀⠙⠊⠁⢻⣤⠄⠀⢱⠀⠀⢀⣠⣮⣤⣀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⠀⠀⠀⢠⠶⠶⢆⠈⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠴⠚⠉⠁⠀⡀⠈⣇
⠀⠀⠀⠀⢠⣤⡄⢸⡀⠀⠛⠀⠀⠀⢏⠀⠐⠊⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⣀⣤⠾⠿⠤⠤⡀⠀⠀⠈⠷⠾⠃
⠀⠀⠀⠀⠹⠦⠇⢀⣙⣶⣤⣤⣀⣀⡈⠳⣄⡀⠀⠀⠀⠀⠀⡬⠓⠶⣤⠴⠋⠁⠀⠀⠀⣀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢰⢉⣄⠀⠀⠀⠀⡠⠖⠛⠉⠉⠛⠒⠒⠒⢺⡇⠀⠀⢸⠂⠀⠀⠀⠀⠀⠱⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⠟⠀⠀⠀⢘⠀⠀⣤⡄⠀⠀⠀⠀⠀⠈⢻⠶⠖⠛⠢⣄⠀⠀⠀⠀⣴⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡴⠇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⢳⣄⠀⠀⠙⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠷⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⢹⡿⣶⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣀⡤⠞⣿⠀⠀⢀⡀⠀⣠⠲⠀⠀⡇⠈⠳⣄⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢺⠀⠀⠋⢹⡆⠹⠶⠤⠞⠀⠀⠀⠈⢳⡦⣄⣉⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢲⡆⠀⢀⡞⠀⠀⠓⠤⠴⠛⠀⠀⠀⠀⠀⠀⢀⡴⠄⠀⡷⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣀⣀⡴⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠖⠁⠸⢧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⡀⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠴⠇⠀⠉⠉⠁

⠙⣧⣀⡄⠂⢙⣳⡾⠈⢷⣬⠾⠏⠿⣷⣩⣠⠟⣵⣿⠛⠙⡄⢀⠀⠀⠀⠀⠀⠀⢿⣷⣾⢃⡴⠂
⣐⣛⠘⢻⠀⠀⠉⠀⢐⡾⢻⣞⣠⠅⠾⢿⣧⣼⣿⣿⡇⣸⣥⣿⣤⡀⠀⠀⠀⢀⠘⢽⣻⣿⣿⡶⠂⣤⣠
⠿⠻⢇⡘⢠⠴⢀⠀⠁⠒⡟⣿⠁⡀⠀⠤⢸⣿⡿⢇⣰⣿⣿⣿⠣⣦⡀⣠⠀⢼⣿⣨⡿⠿⢉⣿⣧⣜⣿
⠀⣠⠊⢀⠹⢧⣼⣰⠔⠂⠀⠫⠀⠁⢀⣠⠄⠘⣇⠀⣿⠻⣿⣥⣴⢻⣿⡯⠀⡨⢿⣿⣷⡀⣛⣻⣿⣺⣿
⣀⣀⣁⠢⠀⢀⡈⠁⡤⠀⠀⣀⠀⠀⠀⢀⠀⣈⡷⠞⠀⠉⠠⢟⡙⢿⣿⣿⠿⠃⣾⣿⣈⣁⡉⣿⣽⠾⢿
⢠⠬⢋⡁⢘⠋⠀⠀⡘⣧⣞⡡⠄⡀⠀⠁⠀⢸⣇⣄⠂⠀⠀⠉⠀⡠⠞⣿⡾⠾⢿⢻⡟⣿⠛⢹⡧⠃⢼
⠚⠛⢛⣋⣿⡳⠤⢀⡀⠞⠻⣯⣉⡠⠀⠀⠀⢠⠁⠁⠁⠀⠀⠀⠉⢐⡲⢼⣧⡤⠸⠋⠃⠧⠀⠈⢀⡞⠋
⢲⠖⢋⡀⠸⡄⢀⡾⠧⢄⠠⠋⢿⣴⢒⠆⡀⠘⠀⠀⠀⠀⢀⡀⠀⠷⢧⣽⣷⢒⠀⣞⣠⠀⠀⠀⠾⠃⠁
⠀⠀⠒⠒⢓⠛⣝⡻⠆⠀⢀⠀⡨⣻⣧⡈⠀⣰⠆⢀⡶⠀⠀⢱⣄⠀⢶⢐⠏⣽⣿⣟⣁⡀
⠀⠀⠋⠉⠁⢀⡟⠻⠶⣖⢋⠤⠀⠀⠋⠱⣤⢁⡴⠟⡀⠀⠄⢔⣉⣡⡜⠋⢃⠉⢻⡉⠻⣷⠀⠀⣲⣄⣾
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠸⡃⢀⡀⠀⠀⢊⢹⣿⡔⣊⣁⣄⠀⠒⠢⢌⣷⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛⠛
⠞⠀⠀⠀⠀⠁⠀⠀⠠⣄⣽⣎⢀⣴⠀⠀⢞⡉⢿⡿⠭⡅⠀⠴⢛⡽⢷⣆⠈⢀⣀
⡷⢀⠀⢠⡆⠀⡆⠀⠀⠉⣩⣷⣿⠁⠀⠀⢤⣔⠛⣷⠊⠁⡄⢰⠈⠐⠁⣸⣄⣩⣥⠄⠀⢀⡠⣸⡃⡀
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠿⢿⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⡠⠤⠐⠒⠒⠚⠁⠀⣀⣠⣄⠀⠀⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⠴⠋⢀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠻⣿⣷⣦⡀⠈⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠎⠀⠀⠀⠀⢠⣞⣡⣤⣤⣤⢀⣀⠀⠈⢿⣎⠻⣦⡀⠹⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣳⣾⣶⡰⠇⠀⢰⣿⣾⣿⣿⠏⠁⠈⠉⠻⣶⣄⣿⣆⠙⣷⡀⢺⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡉⠿⢻⡟⠀⢰⠿⣿⣿⠿⠏⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⡘⣇⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠀⢛⣠⣶⣤⡁⠊⠀⣍⣀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣿⠄⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡀⢠⣿⣏⠣⡄⢀⡾⣷⠀⠒⣾⡀⠉⠀⣀⡄⠘⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⣿⡇⢸⣿⣿⠀⠈⠀⠇⠏⠰⣾⣽⠁⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢸⣿⣿⣧⡀⠀⠀⠀⢠⣿⣿⠠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⡇⣸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⠸⢆⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡏⣽⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣶⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⡄⠈⢻⣿⡟⠙⠋⣹⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠘⠀⣸⢿⢿⣷⡌⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣵⣿⣿⣷⣴⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣿⣿⣿⣿⡿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠟⣻⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣫⣿⡟⣡⣾⣮⣅⣵⣿⣿⣿⣿⣿⣿⠻⠿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠛⣿⣿⣿⣿⣿⣿⢿⠿⠟⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⢀⣀⣤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⡤⠤⣤⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⣾⣿⣿⡏⠀⠀⠀⠀⣀⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠰⡀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⠙⢶⣿⣿⣿⣿⣿⣇⢀⣠⢴⡟⠁⠀⢀⣤⠤⠤⠤⣄⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠈⢧⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⡾⠋⠉⠉⠉⠀⠈⡛⠀⢾⠁⠀⠀⠘⠦⠴⠶⠶⠞⠀⠀⠀⠸⢧⡤⠤⢬⡽⠇⠀⠀⠀⠀⢳⡘⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⠀⣤⣦⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢳⡇⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢦⣈⣇⣀⣀⣀⣠⠤⠤⠄⠤⢤⡴⠚⠲⣄⣀⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⡸⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠻⢄⠈⠄⠀⠠⠀⠈⠀⡀⠀⠹⢦⣤⡜⠀⠆⠀⢈⡀⠈⠙⠒⠶⢤⣴⣇⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠑⢦⣀⠂⠀⠄⠀⠀⡤⠀⣀⣀⠀⠀⠄⠀⠀⡀⠀⢀⠁⢀⣠⠖⠁⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠈⠛⠲⠦⠤⠴⠖⠛⠁⡈⠙⠲⣤⣄⣀⣠⡤⠴⠚⠉⢀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢠⠀⠀⠀⠀⠒⠀⠃⢠⠀⠀⣤⠀⠀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠆⣀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠦⠀⠐⠆⠀⠣⠀⠈⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⢶⡈⠁⠀⠀⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢧⠀⡄⠀⠁⠀⠀⠀⠃⠀⠀⠀⠆⠀⠀⠀⡞⠀⠀⠇⠀⠀⠜⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠆⠀⡶⠀⣰⡇⠈⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⣆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⠀⠀⠆⠀⠘⢁⡤⠞⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢀⣀⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣄⠀⠀⠀⣿⣿⠀⠀⠀⣠⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠙⢿⣿⣷⣄⠀⣿⣿⠀⣠⣾⣿⡿⠋⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣄⡀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣠⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⣦⣀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⣀⣤⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠿⠛⠋⠉⠀⠀⠉⠛⢿⣿⣷⣦⣄⣀⣿⣿⣀⣠⣴⣿⣿⡿⠟⠉⠀⠀⠉⠙⠛⠿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣶⣦⣤⣀⡀⠀⠀⣀⣤⣶⣿⣿⠿⠋⠉⣿⣿⠉⠙⠿⣿⣿⣶⣤⣀⠀⠀⢀⣀⣤⣴⣶⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣾⣿⡿⠟⠋⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠙⠻⢿⣿⣷⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠟⠋⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠙⠻⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠛⠉⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠉⠛⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⣠⣾⣿⡿⠛⠀⣿⣿⠈⠙⢿⣿⣷⣄⠀⠀⠀⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠀⠀⠀⣿⣿⠀⠀⠀⠙⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛



⣿⣶⣶⣾⣶⣷⢾⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⠿⣷⠾⠶⠶⠿⣿⡿⠿⠿⠶⠿⠷⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠖⠒⠒⠲⠒⠚⠓⠒⠛⠛⠛⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠤⠤⠤⢤⠤⠴⠞⠋⠀⠘⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⠞⠋⠁⠞⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⡇⠀⠀⠀⠀⣿⣇⢳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡴⠶⠶⠶⠶⠖⠛⠁⠀⢀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⣿⣷⡀⠀⠀⢰⣇⠀⠀⠀⢀⣿⣿⡌⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣟⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⣼⡇⠀⣼⠀⣦⠀⠀⣿⠀⢰⣧⠀⠀⣰⠇⢹⣧⠀⢀⡾⣿⠀⠀⠀⢸⢹⣿⣇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡅⠀⣰⡆⠀⢀⣿⣇⠀⣰⢻⣿⠀⣯⠀⣿⡄⢠⣿⣆⢸⣿⣇⠀⢣⠀⢸⣿⡆⢸⠀⣿⡆⠀⠀⣸⠸⣿⣿⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⣿⣄⢸⢿⣯⠀⡿⠈⣿⣴⣿⣼⣿⡇⣿⣿⣿⣾⣿⣿⠀⢸⠀⢸⢿⡇⣸⠀⢿⡇⠀⠀⡏⠈⡇⢹⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⣿⡄⢸⠈⣿⣳⡇⠀⢹⣿⡿⠟⢻⣿⡧⠈⢻⠁⣿⣿⡆⡜⠀⢸⠁⢿⠏⠀⢸⡇⠀⠀⡇⠀⠁⢸⡇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣧⣿⠀⢻⣿⡇⠀⠀⡿⡷⠀⠈⢿⠀⠀⢸⡀⣿⢹⣧⡇⠀⢸⠁⠀⠀⠀⣸⡇⠀⢸⡇⠀⠀⢸⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡏⢿⠿⢻⡄⠀⠉⠁⠀⢠⠃⣿⠀⠀⢸⡄⠀⢸⠄⢿⢠⠽⡇⠀⠘⠀⠀⠰⢠⣿⡇⠀⢸⡇⠀⠀⣸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇⠸⡇⠈⣧⠀⠀⠀⠀⣿⠀⢿⠀⢠⠘⡇⠀⣸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⢿⣿⠀⢸⡇⠀⠀⡇⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡄⠁⢸⣿⡆⠀⠀⠤⣿⡄⠘⠀⠈⢀⣷⠀⣿⠀⠀⠀⢸⣷⠀⠀⠀⠀⣸⢿⢸⣿⠀⣾⡇⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣷⠀⢸⡿⣿⡀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⣿⠀⠀⠀⢾⣿⡆⠀⠀⠀⡏⢸⠘⣿⠀⡏⡿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡇⢿⡇⠀⢨⣿⣿⠀⠀⠠⢸⣿⣧⣿⡄⠀⠀⢸⣿⡇⠀⠀⢸⠇⢸⠀⣿⢠⠇⢷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣧⢸⡇⠀⠈⡏⣿⢤⠀⠀⢸⠁⣾⣿⡇⠀⠀⣿⢸⡇⠀⠀⣼⠀⢸⠀⣿⣼⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⢸⣿⢿⠀⢸⣿⢿⡆⠀⠀⣾⠀⡇⢸⡇⠀⠀⡯⢸⡇⠀⠀⡏⠀⢸⣰⡛⠃⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠁⢸⡀⢸⣿⢸⡇⠀⠈⣿⣰⣧⢸⡇⠀⢠⡇⢸⡇⠀⢠⡇⠀⢸⣿⠀⠀⠀⢸⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣸⢸⣾⡇⠀⠀⡿⣿⠙⠈⡇⠀⢸⡇⢸⣧⠀⢸⠀⠀⠈⣿⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠸⣿⡇⠀⢰⡇⠋⠀⠃⡗⠀⢸⡇⢸⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⣷⠀⢸⡇⡾⢻⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠘⢻⠀⢸⡇⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡿⠀⠀⠀⠀⢸⡄⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢠⠇⠀⠀⠀⠀⠀⠿⠁⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠀⠉⠁⠀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⣀⡠⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠐⠋⠛⢧⣈⡰⠳⢶⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠉⠉⠀⠲⠟⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⢈⠧⡀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣁⡀⠈⠂⠙⣀⠈⠛⣤⣶⢆⡔⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠉⠐⠀⠈⠷⣶⡿⣿⡦⠄⠒⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠈⠧⡀⠀⠀⠀⠠⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠎⠀⢀⠀⠀⠀⠈⡦⡀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⣀⡤⠞⠁⠀⠀⢠⠀⢀⡄⡸⠀⠘⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣠⡴⠈⠉⣇⠀⠀⠀⠀⠈⠀⠀⡴⠁⠀⣷⠀⠀⣶⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡶⠋⡿⠁⠀⠀⠘⠀⠀⢀⠀⠀⠀⡼⠁⠐⠂⢿⠀⠀⣿⠏⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢀⣇⠀⠀⠀⡇⢻⠀⠀⠀⣠⠞⠠⠊⠀⢀⣼⠆⢀⡟⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢀⡈⢋⠑⠒⢻⠀⠸⠆⣠⣾⣁⡔⠁⠀⠄⠚⢩⣷⠎⠀⢃⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠘⠀⠀⠂⣀⣾⢛⣿⣟⠿⠀⠀⠀⠐⢉⠘⣿⠂⠀⠘⡄⠹⡆
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢀⠀⠀⢘⡀⠀⢸⠛⠛⠉⠁⠙⢀⠀⠀⠀⠀⠀⠀⠙⢧⡐⡀⠳⠀⠀⢆
⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠈⠀⠀⠘⠁⣀⢶⡀⠀⠀⠐⠂⠀⠑⣄⠀⠀⠀⢀⡀⡼⣱⣯⠘⢶⠀⠈⢆
⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠚⣏⠑⢄⠀⠠⠀⠀⣰⠟⠳⡀⠀⠀⢸⠛⠛⣾⡴⣤⠃⠀⠈⠂
⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⠀⠠⣞⡆⠀⠙⢄⣀⠀⢹⡄⠀⠙⢆⡸⠃⠰⠄⠈⠚⡄⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠐⠂⢿⠁⠀⠀⠀⠈⢠⠇⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⢃
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢶⠀⢂⠀⡾
⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡏⠀⢠⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢂
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⡀⠀⠁⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡄⠀⠀⠀⠇⠀⠀⠀⠀⣈⠀⠀⠀⡃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠘⣌⠁⠂⡠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠗⠀⠠⠀⠀⢰⠀⠀⠀⠀⠀⡿⠀⠀⠀⣧⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢀⠇⠀⠀⢠⡅⠁⢀⣴⠿⡿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠁⠀⠀⢂⠀⡏⠀⠀⠀⠈⠃⠀⠘⠋⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠐⢉⡠⠁⠀⠀⠀⠀⢰⠀⢀⡠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠈⢡⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠈⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢰⢀⡀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠲⠤⡀⠀⠈⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡤⠦⠀⡆⠀⠀⠀⠀⠳⣼⠄⠀⠀⢀⠀⠀⡄⠀⠏⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠢⣤⡀⠀⠀⣨⠀⠀⠀⣸⡤⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⢲⠉⠀⠈⠲⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣀⠀⢀⡸⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠤⢻⠃⠀⠈⢑⠉⠀⠀⠹⠓⢤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡒⠀⠏⠀⠈⠀⠀⠀⢾⠦⡀⠀⠀⠀⠈⠇⠀⠖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀
⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡞⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤
⠀⠈⠂⠱⣄⣱⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⣐⣠⠴⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠀⠀⡀⠀⡀⠀⠈⠢⣧⠞⠁⠀⠀⠀⢰⠀⠸⠥⠞⠁
⠀⠀⠀⠀⠐⠛⠃⢠⣄⠀⠀⠀⠇⠀⠀⢠⣄⠐⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠓⠄⣀⠀⠀⠀⣄⠀⠀⢀⣧⠴⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠉⠈⢑⠊⠁⠀⠁⡖⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠉⣶⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡨⣄⡀⠀⣀⢧⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢄⡀⠀⣀⡿⠀⢀⣀
⠀⠀⠀⠀⠤⣤⡄⠘⠋⠀⠀⠙⡏⠀⠀⠘⠋⠠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠞⠿⠀⠀⠈⠋⠁⠀⠈⡟⠦⣤⠄
⠀⢀⠐⠰⠋⡙⠀⠀⠀⠀⠀⠔⠓⠀⠀⠀⠀⠀⠹⠙⠲⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢲⠀⠐⠁⠀⠁⠀⢀⠔⡟⢦⡀⠀⠀⠀⠸⠀⢠⡒⢦⡀
⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠉⠀⠀⠀⠀⠀⠀⠁⢀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠙⡆⣴⠟⣁⣀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⣤⡄⠀⠀⠈⢧⡈⠛⠷⠸⠟⢉⡿⠀⠀⠀⢀⡞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠹⡄⠀⠀⠀⠉⢳⡄⣠⠞⠉⠀⠀⠀⢠⡞⢠⡞⠁
⢀⣠⣄⠀⠀⠀⢀⠀⠀⠀⠀⠘⢧⠀⠹⣄⠀⣾⠳⢼⡇⣾⡴⠒⣦⠀⢠⡟⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤
⣿⠉⠉⠛⣶⠀⣿⠳⣄⠀⠀⠀⠈⢧⠀⠙⣆⠙⠳⣄⠀⠉⣠⠴⠃⢠⠏⢠⠏⠀⠀⠀⢠⡾⢧⡀⣴⠋⠁⢈⣿
⠈⠳⣤⣤⣬⣛⠻⠀⢷⠀⣶⠓⢤⠈⢧⠀⠘⣆⠀⠈⡇⢸⠁⠀⢠⡏⣠⠏⢰⠞⢳⡄⢸⠀⠸⠟⣩⠴⠶⠛⠁
⠀⠀⠀⣴⠚⠋⢁⣠⣘⡳⠾⠀⢸⡄⠈⠳⢤⣷⡀⠀⣷⢸⠀⢀⣼⡴⠋⠀⣾⠀⢼⠶⣋⣤⣀⠛⠳⣦
⠀⠀⠀⠉⠑⠚⠉⢁⡬⠿⠂⠀⣀⠳⣤⡀⠀⢳⡳⡄⣿⢸⢀⣞⡽⠁⣀⡴⠏⠀⠀⠾⢭⡄⠈⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠿⢤⡴⠖⠛⠉⠙⠶⣍⡳⢦⣷⡹⣿⢸⣾⣿⠴⣛⡥⠖⠋⠙⠓⠲⠤⠗
⠀⣴⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⢤⣤⣤⣭⡷⠬⠙⠋⠈⠉⠴⣾⣥⣴⣶⣶⠖⠒⠚⠒⠛⠛⠛⠛⠛⠛⠛⣳
⠀⠈⠛⠒⠒⠒⠒⠒⠚⠓⠒⠒⠛⠉⠉⠉⣉⡽⣛⣥⠀⢀⣀⡛⢯⣍⡉⠉⠉⠙⢛⣛⣛⣛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⣟⠛⠳⢦⣀⣀⣤⢶⣫⠵⠊⢡⢫⣿⢸⣞⣏⠓⢮⣝⠓⠶⠖⠋⠁⣀⣿⠀⢀⡀
⠀⠀⠀⢸⡛⠳⢦⣌⣙⣦⠀⠉⣩⠖⠋⠀⠀⣰⣳⠋⣿⢸⠘⣎⢧⠀⠈⢳⡀⢀⣲⣯⣥⡴⠞⠋⢉⡿
⠀⠀⠀⠀⠉⣳⡄⠈⢩⡴⣶⠀⡟⠀⠀⢀⡖⢳⠃⠀⣿⢸⠀⠘⣮⢧⡄⠈⣇⠘⡏⢻⡄⢀⣰⣾⣋⣀
⢀⡶⠛⠛⢋⡥⣦⠀⣿⠀⠻⣄⡿⠀⢠⡞⢀⡞⢀⣠⠟⠸⣄⠀⠻⡄⠹⡄⠻⣄⡵⠀⣧⠈⣿⠻⣯⠉⠙⢶⡄
⣼⣇⣀⣠⡿⠀⢿⣀⡿⠀⠀⠈⠀⢠⠏⢠⢏⡴⠋⢁⡀⢀⡈⠓⢦⡹⣄⠹⣆⠈⠀⠀⠹⢤⡟⠀⠈⠳⢦⣴⠿
⠀⠈⠉⠁⠀⠀⠀⠙⠁⠀⠀⠀⣠⠏⣠⠏⠾⣤⡴⢋⡿⢸⡙⢦⣬⣧⠙⣆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣰⠏⠀⠀⣀⡴⠋⠀⠀⠙⢦⡀⠀⠀⠘⣆⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣸⣇⣀⡴⣧⢻⠳⣄⣙⣧⠀⠀⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢸⠋⠘⣦⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣠⣿
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠢⣄⠘⠃⣠⠴⠀⠀⠀⠢⢀⢠⠀⠀⠀⠀⠀⠀⢴⣷⣙⣋⣾⡦⠀⠀⠀⠸⣿⣿⣷⣾⠇
⠀⠀⠀⠀⢠⡄⢹⠇⣄⠀⠀⠀⠈⢀⠀⠙⠃⠀⠀⠀⠀⠀⢴⣶⢛⡛⣶⣦⠀⠀⠀⠺⡿⣿⡿⢿⡆
⠀⠀⠀⠈⠁⠀⢸⡆⠀⠉⠀⠀⠀⠀⠂⠐⠀⠄⠀⠀⠀⠀⠀⠁⠻⠟⠈⠁⠀⠀⠀⠀⠀⠙⠃


⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠂⢠⣴⣦⡄⠒⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠙⠃⢠⡀⠸⠃⠀⠀⠀⠘⠃⢀⡆⠀⠛
⠀⠀⠀⠀⡀⠸⢿⡿⠂⣀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠴⠆⢈⡁⠰⠆⠀⠀⠀⠰⠆⠀⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃


⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⡀⠲⡆⢠⡀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⢀⣀⠸⡷⣤⣤
⠀⠀⠀⠀⠀⠀⢶⡖⠀⠀⠀⠀⠀⡚⠁⡰⠧⣈⣣⠀⠀⠀⠀⣹⣿⣿⣯⠀⠀⠀⠀⠀⣿⠸⡷⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⣀⣄⠘⠃⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠈⠉⢱⡷⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠉
⠀⠀⠀⠳⢤⡄⣚⢠⡴⠛⠀⠀⠀⠀⠀⣀⢄⠀⠀⠀⠀⠈⠁⠀⢀⡀⠠⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠈⠁
⠀⠀⠀⢀⣠⠌⠧⠡⣄⣀⠀⠀⠀⠀⠀⠳⠼⠀⠀⠀⠀⠀⠀⠀⠘⡛⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠋⠁⠀⣹⠀⠀⠉⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⡀⠀⢿⡏⠀⣠⡀⠀⠀⠀⠱⡀⠀⡜⠀⠀⠀⢀⣤⠀⢈⣿⠀⢀⣄⠀⠀⣀⠀⠀⠇
⠀⠀⠀⠘⠐⡖⢩⡉⣿⠁⠁⠀⣀⣀⡀⣼⢲⡄⠀⡀⠀⠀⠉⠛⡂⠋⠞⠋⠉⠀⠀⠈⠁⣶⣷⡄⠈
⠀⠀⠀⢠⣤⠓⢬⡥⠛⢔⡆⠀⠀⠀⢀⠌⠙⢄⠀⠀⠀⠀⣶⠟⠡⡦⠙⠳⡖⠀⠀⠔⠈⠉⠋⠁⠢⠄
⠀⠀⠀⠀⠉⠀⠶⡷⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⠆⠀⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠆⠠⠤⣀⣀⣀⡤⠖⠚⠉⠑⠢⠤⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣉⠴⠒⠓⠂⠀⠠⢤⡤⠤⠒⢊⣉⡁⠒⠠⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣡⠴⠚⠉⠉⠐⠂⠤⠤⠖⠊⠉⠁⠉⠙⠒⠲⠔⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢴⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠁⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠊⠀⠀⠈⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢻⠙⠢⢤⣀⣀⣠⠤⠶⣦⣄⣀⡈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣦⣠⣤⣧⠴⠒⠋⠙⠲⢄⡀⠀⠉⢻⣿⣿⣷⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⣀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⣿⣧⠀⠈⠳⣄⠀⢀⣀⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⣾⢱⣿⡿⠃⠈⢧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⡆⠀⣀⠼⢻⡉⠀⠉⠳⣄⠀⢀
⠀⠀⠀⠈⠁⠀⠀⠀⣼⣿⠀⠁⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⢋⣹⣿⠉⠀⠀⢸⡇⠀⠀⠀⠈⠳⣼⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡆⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢷⣀⡀⠄⢿⠀⠀⣿⠀⠀⠀⢸⣇⠀⠀⢾⣷⠀⢨⠗⢄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⣇⠀⣠⠄⠀⠀⣀⡾⠓⠁⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢹⠀⠀⠀⠀⠙⠳⣦⣈⣋⣠⠏⠀⠀⠑⢄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠙⠋⠀⣀⠴⠚⠁⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⠀⠉⠀⠈⣇⠀⣀⢀⢀⡀⡈⢻⣿⣟⠀⠀⠀⠀⠀⠱
⠤⠀⠀⠀⠀⣺⡟⠛⣿⠀⣀⡶⠛⠉⠀⠈⠛⠒⠢⣤⡞⠋⠁⠀⠀⠀⠈⠙⢆⠀⠀⠈⢠⡤⠤⠤⣤⣤⠬⣿⣿⣉⣁⡀⠀⠀⢰
⠀⠀⠀⠀⢀⣿⠀⢀⣿⡟⠉⠉⠙⠒⠦⣄⠀⠀⢀⣿⠀⠀⠀⠰⠖⠲⡄⠀⠈⡆⠀⠀⣾⠀⢰⣾⡿⢿⣿⣿⡏⠀⠀⢹⠀⢠⠏
⠀⠀⠀⢠⠞⣧⠖⠉⢸⡇⠀⠀⠀⠀⠀⠈⠳⣄⠈⢿⡀⠀⠀⠀⠀⢀⡇⠀⢀⡇⠀⢰⡇⠀⠘⣏⠀⠀⢻⣿⠃⠀⠀⣸⠀⡞
⠀⠀⢀⣸⣿⡇⢺⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠓⠦⠤⠔⠋⠀⢀⡾⠀⠀⣼⠁⠀⠀⢻⡀⠀⢸⡟⠀⠀⠀⡇⡜
⢀⡴⠋⠀⠹⣷⣈⠛⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠲⢤⣀⣀⣀⡤⠶⠋⢀⠀⠀⠙⠢⡀⠀⠈⠁⠀⣾⠁⠀⠀⢀⡿
⢸⠀⠀⠀⠀⢹⣿⣿⣷⣿⠀⠒⣀⣈⣀⡀⠤⠤⠶⠶⠒⠒⠒⠓⠂⠈⠉⠉⢙⡒⠦⣵⡀⢹⠀⠀⠀⡼⠃⠀⠀⠀⠘⠁
⢸⠀⠘⠉⣁⣠⣿⣿⣿⡟⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣖⣒⣞⠙⠀⠀⠀⠙⠾⠀⢠⠟⠁
⢸⠀⢀⡟⢡⣾⡿⢿⣿⡇⠀⠙⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠉⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⣀⡀
⢸⠀⡼⠀⠘⣿⠀⠀⣿⠃⠀⠀⠀⠀⠉⠉⠉⠛⠛⠒⠂⠀⠤⢤⣤⣤⣤⣤⡤⠤⠤⠒⠋⠟⠁
⠘⣸⠇⠀⠀⢻⡆⠀⣿⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟
⠀⣿⠀⠀⠀⠈⢷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿
⠀⠟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⣋⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠠⠀⣀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⡄⡀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠠⣰⠏⠀⠀⠀⠀⠀⠀⠀⢶⣄⡠⠀⣄⡑⢾⠂⣁⢀⡀⢠⡄⠀⠀⠀⢠⡀⠀⠀⠀⢸⣟⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣤⠒⢤⢍⠀⡀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠂⣼⣈⣿⣊⣹⠼⠿⠉⠁⠀⠀⠀⠀⠉⠿⣶⣐⠶⠧⢶⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠰⠦⠶⡦⢶⠰⣿⡧⢺⠰⡗⠗⠗⠒⠀⠀⠀⠀⠀⠶⣍⢸⡏⢾⢘⡋⢑⠶⠀⠀⠀⠀⠀⠀⠀⠈⣤⣓⢽⣾⣝⣣⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠖⢤⠜⣏⡁⠈⠀⠀⠀⠀⠀⠀⠀⢒⣦⠜⢋⢉⣾⡉⠙⢣⣄⡂⠀⠀⠀⠀⠀⠀⣬⣾⢵⣩⢩⣻⢷⣮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠂⠀⠀⠎⢷⡀⠀⠀⠀⠀⠀⠀⠺⠋⠁⠀⠉⡡⣶⢌⠁⠈⠈⠙⠃⠀⠀⠀⠰⠋⠈⠈⠀⢔⣴⠂⠀⠈⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⢀⠀⢀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠑⠶⠔⠁⠀⡀⢀⢠⠀⠀⠀⠀⠀⠀⠀⢆⠈⡆⣿⢰⠃⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣆⠆⠀⡐⡼
⠀⠀⠀⠀⠀⠀⠀⠘⡣⠧⢀⡀⢲⠂⢀⡄⠷⡙⠀⠀⠀⠀⠀⠀⠘⠳⢦⣕⡶⠼⢴⡊⠴⠞⣠⡀⣀⠀⠀⠀⠀⡐⠂⠙⠀⠄⠝⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢘⠀⢈⡁⢀⠀⠀⠀⠀⠀⠀⣤⠒⠉⠉⡁⢾⠀⠀⠀⣽⡣⠤⠤⠖⠋⠀⠀⢠⣴⠵⡦⠈⢰⣿⢈⠀⢦⠶⠶
⠀⠀⠀⠀⠀⠀⠀⣠⠑⡆⠈⠀⢙⡁⠈⠃⡶⣊⠀⠀⠀⠀⠀⠈⠉⣥⡖⣪⢷⠲⠺⠓⠝⡓⠂⠀⠀⠀⠀⠀⠀⠈⡀⢨⡄⠆⢠⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡰⠄⡀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⡐⠁⡘⢸⡇⠡⣄⠱⡀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠐⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠚⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡥⠒⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠐⠢⣿⠔⠂⣤⠀⣠⡄⠀⠀⠀⠀⠀⠠⠽⣎⠀⣤⠀⣱⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢁⠦⣀⠀⢘⡃⢀⣂⣥⠧⠀⠀⠀⠀⠀⠈⠙⣷⣷⣀⠀⢿⠀⣀⣿⡞⠉⠀⠀⠀⠀⠀⠀⢠⣄⠙⣀⠍⣠⠋⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢱⣿⣿⡎⠓⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⢉⣃⣾⣞⡋⠀⠀⠁⠀⠀⠀⠀⠤⣦⢤⠤⢥⣽⣿⣯⡍⢡⣄⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢑⡲⡦⠙⠃⣹⡉⠈⠋⡶⣉⡁⠀⠀⠀⠀⠀⣁⣶⡖⠛⠁⣾⠈⠙⣶⣖⡀⠀⠀⠀⠀⠀⠀⣤⠖⠃⡼⡛⢅⠒⣤⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠃⢀⡠⢴⣌⠀⠀⠁⠈⠈⠀⠀⠀⠀⠘⠉⠀⠇⠀⡠⣽⣄⠀⠛⠈⠙⠂⠀⠀⠀⠀⠀⠀⢀⠾⠀⡇⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠉⠀⠉⠃

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠤⠬⢆⠀⠀⠀⠀⠀⠑⣀⡂⠁⠀⠀⠀⠐⠀⠆⠐⠂⠀⠀⢰⠑⠆⠖⣿⡇⠀⠀⠀⡊⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⡸⠧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢸⠀⡀⡀⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠠⠤⠄⣤⣥⣀⡠⠀⠀⠠⡤⠄⠡⢄⠀⠀⠀⢀⡜⣆⣞⡅⠀⠀⢘⣀⡁⢢⡇⣛⠀⠀⠀⣇⢂⣲⣦⡄
⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣤⡀⠀⣠⡀
⠀⠀⠀⠀⠄⠒⠢⣠⠔⢢⠀⠀⠀⢐⠀⠂⠁⠀⠀⢠⠌⢐⣀⠀⠢⡀⠀⠀⠂⠀⢪⣦⣀⡄⠀⠀⠑⢊⣈⡛⡀
⠀⠀⠀⠀⠈⠡⠁⡘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠈⠀⠚⠁⠀⠀⢀⠁⠸⡍⠙⠃⠀⠀⠀⠈⠂
⠀⠀⠀⢀⡤⠄⢠⣇⠀⠀⠀⠀⢠⡦⣒⢶⡦⠀⠀⠀⣴⣴⣰⣤⡄⠀⠀⣄⡤⠀⣄⡅⠀⠀⠀⠀⣀⡄⠀⢬⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠀⣦⣠⡆⣤⠶⣦⢠⣶⡆⠲⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠽⠃⠟⠹⠇⠻⠦⠟⠰⠯⠟⠀⠿

⠀⠀⠀⠀⢀⠂⠀⠀⠂⠀⠀⠀⠀⢈⠀⡑⠄⠀⠀⠀⠀⢠⢀⡕⠄⠀⠀⠀⠐⣄⢀⡵⡀⠀⠀⠀⠐⣳⣴⠢
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⡄⠧⠴⠃
⠀⠀⠀⠀⠠⢤⣄⣤⠤⠀⠀⠀⢘⢭⣆⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣠⡄⠀⠀⠀⡅⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠈⠉⠈⠁⠂⠀⠀⠀⠉⠋⠉⠉⠃⠀⠀⠀⠉⠉⠉⠈⠁⠀⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢰⠊⠀⠀⠂⠀⠀⠠⠀⠀⠈⠂⠀⠀⠀⣷⡄⠄⢠⡾⠀⠀⡐⠁⠀⢱⠀⠀⠀⠀⠀⡐⠁⢀⠁
⠀⠀⠀⠀⣰⢰⢄⠈⠀⠀⠀⠀⣁⠇⠀⠰⣎⠆⠀⠀⠈⠀⣀⠀⠁⠀⠀⠳⠈⠡⡌⡄⠠⡀⠀⠀⡆⡙⠚⣁
⠀⠀⠀⠀⠱⣩⠠⡀⠐⡀⠀⠀⠰⢂⣀⡀⡈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡞⠀⢛⠣⡿⠭⠅⠀⠀⡊⠁⠀⠉
⠀⠀⠀⠀⠀⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠁⠉⠁⠁⠀⠀⠉⠈⠀⠉⠁

⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣟⣟⣻

⣰⣷⡟⠆⠀⡼⠛⣧⣠⣿⣿⠆⠀⠈⣸⢃⠛⡄⠛⠃⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡴⠋⠠⠆⠀⠀⠈⠀⠜⠋⠁⢀⣶⣀⣥⣤⡶⢿⣿
⠿⣿⠠⣀⠀⠏⠀⢹⣿⡿⠃⢀⠀⢀⣿⡿⠀⠀⠀⠀⡼⡋⠀⣠⠘⠘⠃⠀⠀⠀⠀⠄⠀⠠⢠⠮⠃⠈⠏⠇⠀⠀⠀⠀⠀⠘⡇⠀⠘⢿⠿⠿⣳⣶⠿
⠀⢿⣿⣿⢷⣦⣤⣾⡟⠁⠀⠀⠀⢸⣿⡇⠀⠀⠀⡸⠁⠉⠃⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠁⠀⠀⠀⠈⠓⡀⠀⠀⠀⢁⣠⣤⡀⠀⢀⣾⣿⡦⢰
⣀⠈⢿⣿⣷⣤⣭⣿⡇⠀⠀⡀⢴⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠐⠀⠀⠀⠀⢀⡄⠘⠂⠀⠀⢀⠀⠀⠈⢠⣗⣄⣰⡴⢾⣿⣿⣧⠀⠘⠟⢿⣷⣿
⣿⣿⣥⣿⡿⡿⣿⣿⠧⣤⡾⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠈⠀⠤⠄⠀⠀⠠⣿⠿⠿⠿⠿⣿⣇⠀⠘⢦⡙⣝
⣯⣿⣿⣿⣿⣿⣾⣄⡀⠻⠳⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣩⡀⣦⠀⠀⠀⠉⠀⠀⠀⠙⢮
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⠧⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⢿⢿⣿⡀
⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⢏⣅⠀⠈⠻⢎⠉⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠉⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠟⡉⡞⠹⣿⣯⡛⠛⣿⠃⠀⠀⠁⠀⣴⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠐
⢈⠛⢀⣶⣬⣿⡟⢦⣍⠣⣔⠆⠀⠀⠾⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⠁⠀⠀⠀⠀⠀⠀⢠⡆
⢸⡟⣾⣿⣿⣿⣿⣾⣝⣆⠈⣓⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣴⡇⠀⠀⢀⡠⠋⠁⠀⠀⠙⠦
⣻⣸⣿⣿⣿⡿⠛⠛⠻⠿⢻⣿⡃⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡮⣄⠀⠀⠀⠀⠀⡀⢀⣄
⣿⣿⠛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿
⣿⡟⠑⡷⣬⡀⣿⣿⣿⣿⢿⡈⢤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⢰⠀⠀⠀⣀⣬⡿⠀⠀⢀⣠⠐⢀⠀⡀⠈⠉
⢸⣤⣠⣻⣾⣿⣹⡿⢿⣛⠛⠉⣁⠾⠉⠉⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣰⠀⢀⠆⡀⡉⠉⠻⣿⠃⠀⢀⡜⡧⢡⡈⢇⠀⣤⣄
⠹⣷⣿⣿⠻⣿⣟⠧⡄⠹⠷⣷⢿⣦⣤⡀⣀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⢀⣿⡯⠀⡘⢠⣸⣇⠀⠀⡟⠀⢀⣴⣿⡇⢻⣷⡈⢻⡄⣛
⡟⣿⣿⣿⢢⡄⢻⡄⣌⠐⠿⣿⣿⣭⣉⡉⣉⠀⠀⡀⠀⢦⡄⠁⠀⠀⠈⠁⠀⠙⡈⠀⠡⠃⠠⣦⣶⣿⠋⠈⠹⣿⣿⠁⢀⣰⡈⠛⢜⢨⣿⡇⣺⣿⠏
⣿⡀⠻⢿⣿⣿⡾⣧⣈⣵⡾⠿⣿⣿⣏⣯⡍⠙⠓⠀⠀⢨⠀⠀⠀⠀⠀⣠⠂⠁⠀⠀⠀⠀⢀⣹⣿⠏⢀⠀⠀⢹⣿⡠⢠⣎⢻⣿⣮⡉⢿⣿⠏⢘⡴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠁⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠄⢐⠂⠁⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠠⠠⠸⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠄⣀⠠⠀⠀⠀⠀⠀⡠⠂⠐
⠀⠀⠀⠀⠀⠁⢀⡐⠀⠀⢄⢤⡀⠀⡆⡌⠄⠐⠈⠠⠀⢀⠀⠀⠀⢀⠀⠀⠀⠠⠀⢈⠊⠀⠀⢄⠀⠐⠀⠂⠈⠄⣀⠠⢀⠌⠁⠂
⠀⠀⠀⠀⢀⢀⠀⠁⡀⠀⢈⡓⠈⠈⠁⢂⡀⠊⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⢐⠀⠀⠀⢀⠆⠂⢆⠀⠈⠄⠀⠀⠐⠀⡂⠈
⠀⠀⠀⠀⠀⠀⠆⢀⡂⠀⠂⠄⠀⠀⠈⡀⠎⠁⠂⠑⠈⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢈⠔⠁⠀⠠⠀⠀⠰⠀⠀⠀⠠⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢈⡥⠖⠆⠍⠢⠀⠀⠁⠄⡀⠠⡀⠀⠀⠀⠒⠁⢈⠄⠀⠀⡀⠀⠀⠁⠈⠀⠀⠂⠀⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠂⢀⠉⡄⠀⠀⠄⠀⠀⠀⠄⠂⠃⠀⠈⢀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠀⠀⠀⠀⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠠⠈⠀⠄⢀⠀⠑⡠⠂⡄⠤⠀⠀⠀⠀⠄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⡀⠀⠀⠀⠂⠢⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⡀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⢀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠈⠠⢀⠈⠀⠰⠐⠴⠈⠆⠎⡄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⢀⡠⠀⡀⠀⠈⢀⠘⠁⠀⠀⠀⠀⠘⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⠀⠀⠈⠀⠂⠀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠾⠂⠀⠀⠀⠀⠸⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⡀⠀⡈⠀⠃⠀⠀⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠹⢠⠸⡳⠁⠀⠄⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠠⠀⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠻⠿⠟⢷⡆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡾⠃⠀⢰
⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠛⠋⠀⠀⣠⡟
⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣤⣴⠞⠋
⠀⠀⠀⠀⣿⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠁
⠀⠀⠀⠀⢻⣆⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠻⣆⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠙⠷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁
⠀⠀⠀⠉⠙⠛⠷⠶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡶⠶⠟⠛⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡖⠋⠛⢶⡀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⢻⡄⣰⠏⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⢠⣤⣤⡤⠶⠄⣤⣤⠀⠀⠀⢠⣄⢠⣤⡄⠀⠀⣤⡄
⠀⠀⠀⠀⢠⣤⣤⣤⣽⣷⣦⣄⣿⣇⣤⣤⣶⣞⣛⡉⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⡀⣿⣿⠀⠀⢀⣿⣿⢸⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣟⠉⠉⠉⢉⣽⠛⣿⢿⡛⠛⠛⠛⠛⠛⣿⡃⠀⠀⠀⠀⢸⣿⠋⠉⠉⠁⣿⣟⠀⠀⣸⢻⣿⢸⣿⡇⠀⠀⣿⣟
⠀⠀⠀⠀⣿⣿⣉⣀⣴⠟⠁⢸⣿⠀⠹⣦⡀⠀⠀⠀⣿⠁⡾⠋⠙⣷⢸⣿⡇⠀⠀⠀⣿⣿⠀⢰⡏⢸⣿⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⣿⡏⠉⠉⠀⠀⠀⢸⣿⠀⠀⠈⠻⣦⣀⠀⣿⠀⠛⢂⣰⠟⢸⣿⡇⠀⠀⠀⢿⣿⣠⡟⠀⢸⣿⠈⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠉⠛⣿⠖⠛⠋⠁⠀⠘⠟⠃⠀⠀⠀⠈⠛⠋⠀⠀⠘⠛⠁⠛⠛⠛⠛⠙⠛⠛⠛⠃
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⢢⣹⠉⣙
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣤⡿⠉⢳⡀⠀⠀⠀⠀⠀⢰⣿⠀⣇⠀⠀⣸⢿⠉⠛⠀⠀⠀⠀⣤⡀
⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⢰⣿⢃⠀⠘⡇⠀⠀⠀⠀⠀⢸⣿⠀⠈⢙⣋⣁⠋⠁⠐⣿⡆⠀⠀⣿⣿⡄⠀⣠
⠀⠀⢀⡾⡋⠙⢷⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡯⣿⠀⣸⠁⠀⠀⢠⡖⢢⣸⣿⡀⢠⣿⡇⢸⣴⣿⠀⢿⣧⠀⢠⠇⢹⣿⡾⠁
⠀⠀⣾⢷⡇⠀⠀⡇⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⣿⡇⡟⡰⣃⡀⠀⠀⣿⠀⠀⡿⣿⡇⢸⣿⡇⠘⢸⣿⡀⠀⡟⠛⠋⠀⣠⢿⣷⡀
⠀⠀⣿⢸⠃⠀⢀⠇⠀⠀⣿⡇⠀⡠⠶⠦⡀⠀⣿⡷⠊⠀⢩⡅⠀⢀⣿⡇⠚⢱⢿⣷⠀⣿⣇⠀⣼⢿⣧⢠⡇⠀⠀⢰⠋⠈⢿⣧
⠀⠘⣿⡿⣶⡦⡼⠤⣤⣤⣿⠧⠄⢧⡀⣀⡹⣆⢹⣿⡀⠀⢻⣿⢠⡏⣿⣧⠀⣼⠈⣿⡄⢸⡿⡶⠏⠈⠿⠾⠁⠀⠀⣿⠀⠀⢸⣿
⠀⠀⣿⣇⠀⡼⣱⢻⣷⢉⡏⣿⣇⠁⠙⠶⠷⣿⠛⣿⣷⡀⠘⢿⡿⠀⠘⢿⣶⡟⠀⠈⢿⣄⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠸⣿⡞⢰⠏⠸⣿⢸⣿⠻⣿⡄⠀⠀⢠⡿⠀⣌⠿⣷⡶⠢⢄⠠⡦⢀⡤⠶⣤⠐⣶⡞⢲⡦⢀⡤⠶⢦⠒⣶⠖⠸⢷⣶⠾⠷
⠀⠀⣷⢻⣿⣼⠀⢸⣿⡼⣿⢸⡟⠿⣿⣿⠿⠃⠀⣿⣿⣿⠻⠿⠾⢯⣿⡟⠀⢀⣿⠇⣿⡇⢸⣿⡟⠀⠀⣿⠇⣿⡇⠀⢸⣿
⠀⢀⣿⠇⢻⣿⡇⠀⣿⣧⠇⢸⡇⠀⠀⠀⠀⠀⠀⣿⡟⣿⣧⠀⠀⠘⣿⢁⣠⡾⠋⠀⣿⡇⢸⣿⠁⣀⡾⠋⠀⣿⡇⠀⢸⣿
⠀⢸⣿⣤⣤⡙⠿⠀⡘⢿⣦⣼⠇⠀⠀⠀⠀⠀⠀⣿⠇⡈⠻⣷⠀⠀⣿⠉⠁⠀⠀⠀⣿⡇⠸⣿⠿⣅⣀⠀⠀⣿⡇⠀⠸⣿
⠀⠈⠉⠉⠉⠙⠛⠛⠛⠓⠉⠹⠾⠿⠿⠿⢷⣶⣶⣿⠀⠳⠶⠟⠀⠀⠿⠀⠀⠀⠀⠠⠿⠧⠘⠿⠀⠀⠈⠁⠠⠿⠧⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡜⠁⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠂⠀⠀⠴⠀⠀⠀⠘⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠁⠀⢀⡸⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⡘⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⠈⠀⠀⡜⠁⠀⠀⠀⠀⡄⠀⠀⠀⠇⠀⠀⠀⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⢰⠇⠀⠀⠠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣧⠀⠀⠈⠀⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡟⠀⢸⡄⠀⠀⠇⠀⠸⠀⠀⠀⠀⠆⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣿⠀⢸⡃⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠀⢠⠀⠁⡀⠘⠁⠀⠀⢸⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⢇⠀⢺⠀⠀⠸⠇⢰⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠆⠈⢣⠀⠷⡀⠀⢀⡈⠀⠀⣠⠇⠀⠀⠋⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠢⣀⠐⠀⠉⠠⠞⠁⢀⡴⠃⠀⣤⠦⣀⠀⠀⠈⠉⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⢶⠀⠀⠒⠉⠀⠀⠀⠀⠰⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⡀⠀⠁⠀⢰⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡀⠖⠀⢀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠉⠁⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠛⠓⢠⡀⠶⠀⠀⠰⢀⡆⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠶⠁⠀⠀⠀⠀⠁⢀⡇⠸⠀⠘⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣾⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠰⡀⠰⠄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠖⣀⣶⡤⠞⠁⠀⠀⠀⠀⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⠲⠻⢶⣤⣭⣉⣁⠀⠈⠉⠉⠉⠉⠉⠀⠈⢉⠙⣆⡰⠃⣠⠶⠛⠓⠐⠒⠈⣉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠂⠀⠱⠖⠲⠾⠋⢉⣍⣀⣤⣤⣄⣀⣤⣤⡤⣤⡄⠈⣤⢈⣷⠌⢡⣤⠄⠚⠛⠓⢠⡀⠃⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠉⠁⣀⣀⣠⡤⠀⠀⠀⠀⠰⠶⡄⠐⠂⠈⢉⠤⠟⠛⠛⠛⠁⠀⠀⠁⠀⠸⣞⠡⠞⠃⠄⡀⠀⠀⠀⠀⠈⠶⠀⢰⡄
⠀⠀⠀⠀⠀⠀⡰⠊⠀⣠⠖⠊⣉⠛⠉⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⠀⠴⣆⠀⠀⢀⣄⡀⠁⠀⠀⣀⠀⠀⠀⣇⣸⠁
⠀⠀⠀⠀⢀⠞⠀⣠⠞⠁⣠⠛⠀⠀⠀⠀⠀⠀⢀⠈⢠⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢀⡿⣦⡀⠘⠏⠰⠾⢦⣄⠈⠀⠀⣶⠉⣿
⠀⠀⠀⢀⠉⠀⣘⠃⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⡇⠀⠀⠀⠀⠈⠙⠃⣠⡞⣀⠇⡸⠃
⠀⠀⠀⠘⠀⠀⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢴⠀⠈⠓⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⢀⣤⠾⢯⠚⠁⡸⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣤⣄⣀⡀⠛⢣⣶⣶⣶⡄⠀⣶⣴⠾⠟⠁⠀⠘⠋⡰⠛⠂⠀⠀⠀⠠⠾⠁
⠀⠀⠀⢀⠀⠀⠈⠉⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠛⠳⠶⠶⠀⠀⠉⠁⠀⠀⠀⠀⠰⠎⠀⠀⠰⠶⠖⠈⠀⠀⡀
⠀⠀⡄⠈⢀⠀⠀⡆⠀⠀⠘⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠒⠀⣀⠃
⠀⠀⠁⠀⠀⠀⠀⠙⠦⣀⣀⡤⠃⢀⡠⠦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢤⣤⠿⠏⠁⠀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣴⠎⠀⠀⠈⠉⠉⠁⠀⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠀⢀⣉⣉⠲⠂⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠓⣤⠀⠀⠀⠀⠀⠀⠛⠛⠓⠀⠀⠤⣤⣤⣤⣤⣤⣤⣤⠤⣤⣤⣤⣤⣤⠤⠒⠒⠛⠋⠀⠀⢀⣀⣠⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠶⠆⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠤⠶⠶⠆⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⣠⡴⠂⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢰⣿⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠘⣿⣄⠀⠀⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠘⢿⣦⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣿⡆⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢀⡿⠁⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⢒⣚⣓⣋⣉⣉⣉⣉⣉⣉⣓⣚⡒⠒⠒⠲⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠻⣷⣶⣄⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠁⢸⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠓⢆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡼⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡤⠊⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣁⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁
⠀⠀⠀⠀⣠⠤⠖⠒⢛⣉⣩⣭⣬⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣭⣭⣍⣉⡛⠒⠲⠤⣄
⠀⠀⠀⠈⠣⣄⣀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣾⣿⣿⣿⣿⣿⠿⠟⠀⣀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠦⢬⣭⠭⣉⣉⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣉⣩⠭⢭⡤⠴⠒⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠦⠤⠤⠤⠤⣤⣤⣥⣤⡤⠤⠤⠤⠤⠖⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢿⠶⣳⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠏⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣞⣿⡖⢀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠉⢳⢷⣖⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣋⠹⣿⣿⡟⣬⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣴⡏⢹⡟⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢫⠟⠺⢯⢃⡔⠘⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⣽⡠⣿⣧⠈⢛⠒⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡔⣣⠶⡴⠃⠈⠱⠒⣤⡀⢀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠶⢻⣏⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⢈⡍⠈⢨⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟⠋⢠⣿⡿⢎⠨⠛⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠠⡿⠗⠀⣉⣴⠁⣀⡀⣠⠈⣆⡀⠀⠀⣀⢠⠖⠋⠉⣵⡾⢏⡽⢿⣴⣗
⠀⠀⠀⠀⣀⠠⠀⠐⢡⣮⣭⣭⡥⣶⣤⣤⣍⢄⣬⣿⣦⣼⣿⣿⠇⠐⣋⣥⣴⠂⠀⣾⣿⢞⣶⣷⣢⣄⡤⠄
⠀⠀⠀⠀⠙⠛⣶⣾⠋⣼⣿⣿⢁⣨⣿⣿⣿⣧⡀⠙⠛⠿⠟⠁⢀⣾⣿⣿⣿⣷⠀⣿⣿⠈⠱⣿⣏⠁
⠀⠀⣀⣤⢶⣿⣿⡏⠀⣿⣼⡟⠘⠛⠛⢿⣿⣿⣇⢀⣠⠤⣀⠄⢻⣿⣿⠿⠿⠛⠁⢻⣿⢷⡦⣿⣿⣷⣦⣤⣤
⠀⠀⠙⠻⢦⣽⡷⣧⠀⢹⣯⣷⠀⠀⠀⠈⠉⢁⣿⢸⡯⠀⠘⣷⠸⠃⠀⠀⠀⠀⠀⣾⣿⣿⡀⢸⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠉⠿⣿⣦⣾⣄⢿⡀⠀⠀⠀⢰⣿⢏⡼⠣⣄⠀⠻⣧⢄⡀⠀⠀⠀⠀⣿⣿⣿⢡⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣬⣧⣀⣀⣠⣾⣿⣿⣿⣽⣤⣾⡛⠻⣯⡓⢤⡀⢀⣘⣷⣿⣵⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⢿⣿⣅⣈⣁⣤⣼⣷⣟⣠⣼⣿⡶⣦⣬⣭⣷⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⡀⠘⢿⠙⢿⣿⣿⣶⣿⣻⠿⠉⠿⢠⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⣿⣼⡟⣿⣿⢿⣿⣿⣤⡀⠀⠀⢨⣽⣿⡿⠀⠀⠀⢀⣡⣼⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⠏⣼⣿⣼⣇⣿⣿⣿⣿⠙⢿⣧⡀⢺⣿⣉⣴⠀⣀⣴⣿⣿⣿⣿⣿⢻⣿⣿⡇⢰⡄
⠀⠀⠀⠀⣴⠋⢐⣿⣿⣿⣿⣿⣿⣿⣿⠀⢾⣿⡛⠚⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠇⢠⣿⣿⣿⠀⢹⡀
⠀⠀⠀⣼⢃⣼⠟⣈⣿⣿⣿⣿⣿⣿⣿⣷⣸⠗⠇⣤⣤⣴⣿⠟⣿⣿⣿⣿⣿⣿⠀⢀⣿⣿⢿⣷⣦⡁
⠀⠀⢀⣼⣿⣵⣿⠛⠒⠻⠉⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⡏⣿⡌⠫⣿⢻⣧⣄
⠀⣰⣿⢟⣋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⡁⠉⠙⠛⠛⠓⠛⢋⡽⠛⠻⣿⣯⣠⠏⠙⢿⣠⡄⠠⣀⠉⠉⠐
⠻⠇⠀⠀⠉⠀⠀⠠⠄⣀⠀⢀⠀⠀⠀⠀⠀⠁⣤⡀⠀⠀⢠⠼⠋⠀⠀⠀⠈⠹⣿⡄⣀⡈⠻⠛⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠲⣶⠖⠀⠀⠀⠀⠀⠀⠀⠈⠡⠖⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⢦⣠⠀⢤⠀⢸⠆⡘⢀⡴⡇⠀⣤⠀⣀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣵⡀⣀⣤⠀⠐⢷⡀⢁⡿⠂⠀⣄⡀⢀⣎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡒⢤⡑⡀⣄⠓⠞⢠⢀⠊⡥⠐⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⢻⠀⢽⡀⢤⠈⢇⡸⢀⡄⢀⡯⠀⠏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⡛⠞⢌⡒⠿⢠⡇⠿⠗⡩⢳⢚⡰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠳⡄⣝⠺⢟⢸⡕⡿⢗⡯⣠⠞⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠲⢖⠀⠬⣿⡆⢻⣳⠿⡶⣦⡛⢰⣷⠥⠀⣒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⣄⡀⠒⣾⣆⣛⣤⠺⠖⣤⢛⣰⣳⠂⢀⣠⣦⠂
⠀⠀⠀⠀⠲⠖⡿⠤⢈⣹⠇⠐⠶⢞⡗⣻⡷⠸⣋⠶⠂⠐⣏⣁⠤⢾⠲⠖⠀⠀⠀⠰⣉⠶⠶⠄⢈⣉⡡⢚⠶⢸⠂⣾⣟⠰⡇⠶⡰⣍⣉⡁⠤⠶⠶⡹
⠀⠀⠀⠀⠀⠀⠀⠴⠎⠀⢀⢿⠇⣼⠿⢖⡷⠿⣧⡘⡟⢁⠀⠨⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⠋⠀⠤⠿⢎⣤⢻⢼⠦⡛⣬⠹⡿⠤⠀⠙⠏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠊⡥⢦⣊⠴⣶⠘⡇⣶⣦⢑⡬⢦⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠔⠋⣯⢶⣾⢸⡣⣷⡦⣏⠙⢦⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⠄⠚⡱⠀⠂⡶⢧⠈⠈⠂⡑⠀⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣤⠄⣺⠀⠙⢀⡏⢹⠀⠃⠈⣗⠢⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠎⠙⠀⠚⠀⢸⠆⠡⠈⠱⡇⠀⠙⠀⠈⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠹⠀⠉⠙⠀⠠⣾⠅⡌⡷⠄⠀⠋⠁⠈⢏⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢘⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣠⠀⠘⢼⡗⠁⠠⣀⠛
⠀⠀⠀⠀⠀⠀⠀⠰⣢⠀⠀⠀⠻⡀⠀⢜⣇⠀⢀⡟⠀⠀⠀⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣆⡆⠀⢍⣫⡜⢠⢺⡑⡀⢧⣟⡡⠀⣀⡸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢆⡄⢈⣻⡀⢸⡇⢀⡸⡁⢠⡠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡿⢗⡢⠌⣧⠎⣼⣧⢱⣬⠡⢔⡺⡟⠓⠂
⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⢉⠹⣦⢨⠷⢮⡷⠾⡏⣴⠞⡉⢀⡀⣤⠄⠀⠀⠀⠀⠀⠀⠸⠓⢸⣄⣣⠘⡈⣳⣶⢌⣦⢋⡱⣼⡡⣖⡏⢱⢁⣎⣴⡄⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠺⢦⡟⠻⡦⢽⢣⢷⢟⠳⡴⠧⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠒⣉⠱⠟⠤⣧⣰⣺⠮⠷⣷⣆⣮⠔⠿⢌⡙⠒⠈⣀
⠀⠀⠀⠀⢶⠦⠰⠶⠱⠶⠶⢎⡷⠿⣓⣎⣹⣎⠾⢾⡱⠶⠶⢮⠴⠆⠴⡶⠄⠀⠀⠀⡶⢾⡶⣏⠰⠶⣿⣎⠱⣿⡇⣏⡱⢸⡳⢌⣱⣷⠶⠆⣽⢾⠷⢖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠾⡧⣽⢗⣪⡝⡺⣮⢴⠳⡶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠤⣉⣐⣦⠐⡟⠹⢽⣖⡶⡯⠏⢟⠢⣴⣊⣉⠤⠀⠉
⠀⠀⠀⠀⠀⠀⠐⠟⠉⠁⣈⣰⠟⢚⡦⢞⡷⢶⡇⠻⢆⣁⠈⠁⠻⠂⠀⠀⠀⠀⠀⠀⢠⣤⢹⠟⡹⢡⢃⡸⠷⠂⠟⣌⡱⢻⠐⡿⣇⡰⡈⢏⠺⠯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠎⠃⢈⣽⠁⢸⡇⠘⢫⡁⠘⠱⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⣶⡎⠥⢒⡟⠆⢻⡟⡸⢓⠒⠬⡱⣧⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢰⠝⠀⠀⢀⣬⠋⠀⢪⡗⠀⠘⣧⡀⠀⠀⠻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠎⠏⠀⢊⣝⢣⠘⣸⣡⠁⡜⣏⡒⠀⠉⠳⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⠙⠀⢠⢾⡷⡀⠐⢙⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠊⠃⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⢄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠃⠀⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡰⠳⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡇⠀⠧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⢴⡆⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡒⢲⣞⣛⣿⠟⠿⠿⣿⠿⠿⠿⣿⣛⣳⡖⢒⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢸⡇⣠⠖⠉⠀⠀⠀⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣄⢈⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠈⠙⢿⡄⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢿⡄⣠⠀⣠⠾⠛⢩⠭⣉⠉⣱⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣙⣿⡞⣿⡞⠁⢀⡤⣀⢳⡈⠑⢎⠀⠙⢄⠀⠀⢀⣤⠖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⣠⠞⠋⠙⠉⢻⣷⣿⣿⡿⣶⣽⡄⠙⣇⠀⠈⢆⠀⢸⡠⠖⠉⠀⢠⠃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⢼⡇⠁⠀⠀⠀⠈⢳⠀⠹⡀⠀⠸⡀⢸⠀⠀⠀⠀⢸⠤⠒⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢷⣄⠙⢦⡀⠙⣾⠀⣀⡠⠖⠁⢀⣼⠖⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠤⠜⠙⡄⠀⠀⠀⠀⠀⠀⡸⠀⠉⠓⢚⣿⣽⠉⠉⠉⠑⠒⠊⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠦⣤⠤⠴⠁⠀⣤⠶⠋⠁⠘⢟⠢⣄⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢊⠥⠂⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠠⠤⢀⣈⠀⠀⠀⠀⠀⠄⣀⣀⠤⠀⠀⠄⣊⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡜⠁⠀⠀⠉⠉⠒⠒⠒⠒⠒⠒⢄⠀⠀⠈⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡇⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠃⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠤⠤⠤⠤⠀⠀⡀⠀⠠⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠉⢉⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠨⠄⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠐⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⡰⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⠐⠊⠁⠀⠀⢀⡠⠴⠚⣀⡤⠶⢦⡘⠄⠘⢦⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠞⠉⠀⠀⠀⠀⣠⠴⠊⠁⢀⡠⢄⢳⡀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⡇⠀⠘⡄⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡀⠀⠀⠘
⠀⠀⠀⠀⢸⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠉⠁⠀⠀⠀⢳⡀⠀⢰⠘⡄⠀⠀⠀⠀⠀⡇⠀⡀⠇⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠰⡀⠀⠀⡄
⠀⠀⠀⠀⢸⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⢰⠁⠀⠀⠀⠀⠀⠃⠀⠅⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⡇
⠀⠀⠀⠀⠈⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠸⠀⠀⠀⠀⠀⠀⠙⢄⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⡃
⠀⠀⠀⠀⠀⠈⠣⡀⠘⠆⠀⠀⠀⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⡀⠈⠙⠃⠀⠀⢀⣀⣠⠤⠤⠴⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠘⠚⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣊⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡷⠆⠀⠀⠀⠀⠀⠀⠰⣶⣶⠆⠀⠀⠀⠀⠀⣀⣀⠀⣤⠀⠀⣤⢀⣀⡀⠀⠀⠀⠀⢠⠦⣀⣀
⠀⢠⣶⣟⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⢈⠉⢉⠁⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠘⠻⢿⡞⠉⠙⣷⡿⠟⠁⠀⠀⠀⠀⠈⡿⠋⠉⣹⣧
⠀⢿⠋⢻⣿⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠘⠀⢤⡄⠘⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⠖⠋⣼
⠀⠈⠳⠾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠊⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠃⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣏⠃
⠀⠀⢀⣤⠀⣮⠷⠉⠁⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢠⣾⣿⡎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⣠⣶⠆⠀⠀⠀⠀⠠⠚⠿⣿⣿⣻
⢤⡜⣯⠻⠀⠉⠂⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⣿⣟⠕⠁⠀⠀⠀⠀⠀⠸⠿⡿⠿⣿⠿⡿⠟⠀⠀⠀⠀⠀⠐⠄⣀⠀⠈⠉
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉


⠀⠀⠀⣚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡖⠂⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⢀⣜⣛⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⣄⠀⠀⠀⠀⠀⠀⡴⠻⠓⠒⠺⠗⣄⠀⠀⠀⠀⠀⠀⠠⣿⣿⣯⠀⠀⠀⠀⠀⠀⠐⢛⣩⣥⣶
⠀⢈⣍⣋⣩⡄⠀⠀⠀⠀⠀⠀⠀⣸⣟⡻⢝⣻⡇⠀⠀⠀⠀⠀⣷⣶⡖⣶⢲⣶⣿⠀⠀⠀⠀⠀⠀⣆⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆
⠀⠈⠁⣉⡁⢘⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣧⡿⠃⠀⠀⠀⠀⠀⣿⣿⣡⣿⣢⣿⡟⠀⠀⠀⠀⠀⠀⠸⣦⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡧
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠋⠀⠀⡼⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⡟⠀⠀⢀⡞⠁⠀⢀⡜⠁⠀⡠⠚⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⡀⠀⢀⡎⠀⠀⢀⡏⠀⢀⡞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⢳⡀⢸⡀⠀⠀⠈⣇⠀⢸⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⡇⠈⢧⠀⠀⠀⠈⡆⠘⣆⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠋⠀⠀⠈⠓⠦⠤⠞⠁⠀⠈⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣠⣤⣀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠈⣇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣿⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣀⠀⠀⢀⠏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣈⠉⠛⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⡿⡿⢿⢷⡿⠿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣾⡿⢿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⠀⠀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⣿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠸⣇⠀⠀⠀⢀⣠⡾⠋⢰⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⠙⣿⡶⠺⣿⠏⢀⣴⠿⠛⠋⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⢸⣷⡟⠀⠀⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠾⢋⣀⣠⣤⣬⣥⣴⣦⣤⣤⣄⡀⠀⠀⠀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⣀⣤⠶⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⢻⣷⣶⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠛⠻⢧⣄⣀⣀⣀⡀⠀⣀⣀⣀⣠⣤⡴⠾⠛⠋⠁⠀⠀⣿⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⠈⠉⠉⠉⠉⠉⢩⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠟⠋⠉⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⣃⣴⠶⢶⡄⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣤⡤⠶⠚⠛⠛⢦⣤⣤⡴⠛⠁⠀⠀⠀⢸⣿⠏⠀⠀⠀⣿⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠷⣄⣀⣠⠟⠁⠀⠀⠀⠀⠀⠀⣿⣏⠀⢀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⣀⣠⣴⠟⠻⠿⠛⢿⣦⣀⠀⣾⡇⠀⠀⠀⠀⡿⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⡀⠀⠀⠀⢰⡟⠉⠉⠁⠀⠀⠀⠀⢀⣹⣿⠇⣿⡇⠀⠀⠀⣼⠇⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠖⠛⠛⢷⣄⠀⠀⠻⢷⣶⣶⡶⠶⠶⠶⠟⠛⠉⠁⢸⣿⣄⢀⣠⡼⠋⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡿⠟⠳⣾⠟⠛⠛⠳⠶⠄⠀⠀⠀⠀⠀⣾⡟⠛⠋⠉⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢠⡾⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠶⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⣦⣤⣤⣄⣀⣠⣤⣤⣤⣶⠶⠾⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣀⣀
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣻⢩⣿⣟⢿⣿⣿⣻⣿⣿⣿⢿⣟⣽⣿⣟⣿⣿⣿⣠⣶⣶⣶⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠸⣿⣿⣿⣿⠿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣟⣷⣿⡿⠟⠛⠛⠛⢿⣿⣿⣿⣷
⠀⠀⠀⠹⣿⣿⡿⡿⣿⣿⢿⢸⣿⣿⢿⣿⡿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣷⠛⠉⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢩⣽⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢈⣿⢿⣿⣿⠿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡟⣾⣿⣿⣿⣿⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣸⣿⣿⣿⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⣧⣽⣿⣿⣿⣿⡝⣤⣤⣾⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠻⠟⠻⢿⣿⣿⡏
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣷⡆⠀⠀⠀⠈⠉
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠴⣀⠀⡗⢀⡤⠆⠀⠀⠀⠠⠀⢀⠀⢀⠀⠀⠀⠀⢠⣶⡞⣟⢳⣶⡄⠀⠀⠀⠾⣷⣿⣿⣶⡦
⠀⠀⠀⠀⠀⣀⠑⡷⢀⡀⠀⠀⠀⠐⠀⠀⠻⠢⠀⠀⠀⠀⠀⢠⣭⡜⠿⢧⣽⡄⠀⠀⠀⣴⡿⣿⣿⣿⣦
⠀⠀⠀⠀⠛⠁⠀⡧⠀⠑⠂⠀⠀⠀⠈⠀⠠⠀⡀⠁⠀⠀⠀⠈⠙⠱⣿⠎⠙⠁⠀⠀⠀⠀⠀⠻⠟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠠⠀⠀⣅⠀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⡦⠀⠉⠀⣤⡄⠀⠀⠀⢠⠄⠀⡅⠀⢤
⠀⠀⠀⠀⠀⢘⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⢀⣀⠀⠗⠀⣀⡀⠀⠀⠀⢀⡀⠐⠇
⠀⠀⠀⠀⠐⠈⠉⠛⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣦⠀⠉⠁⠀⠀⠀⠈⠁⠀⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⣰⡀⠙⠏⢰⡄⠀⠀⠀⠀⢠⣤⣿⣤⡄⠀⠀⠀⠀⠰⣶⢘⡛⣴⣶
⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⣉⡀⠜⠓⢤⣼⠀⠀⠀⠀⠠⠿⣿⡿⠇⡀⠀⠀⠀⢠⣷⡘⡟⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢤⡦⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠸⡿⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠻⠦⡄⢘⡃⣤⠞⠇⠀⠀⠀⠀⠀⣀⢄⠀⠀⠀⠀⠀⠉⠀⠀⢈⠀⠄⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⢀⢀⡄⠳⠜⣤⣀⡀⠀⠀⠀⠀⠀⠣⠼⠃⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠚⠊⠀⢸⡇⠀⠘⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⢰⡷⠂⠀⡀⠀⠀⠀⠀⢦⠀⠀⡰⠁⠀⠀⠀⢀⠀⠀⢻⡇⠀⢀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠰⢫⢤⠖⠛⢦⡞⠹⠂⠀⠀⠀⠀⣱⢤⠁⠀⠀⠀⠀⠛⠳⢦⠹⢣⡴⠟⠃⠀⠀⠑⠂⣤⣦⡀⠰⠂
⠀⠀⠀⠀⣀⣸⣄⢛⣸⢇⢤⡀⠈⠉⠉⠉⡳⠺⡁⠉⠉⠁⠀⣀⣤⠶⢠⡐⠶⣤⡀⠀⠀⣀⠀⠛⡿⠓⢀⡀
⠀⠀⠀⠈⠓⠁⢈⣽⡀⠀⠋⠀⠀⠀⠀⡜⠀⠀⠱⡄⠀⠀⠀⠈⠁⠀⣼⡅⠀⠈⠁⠀⠀⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣌⣛⣛⣡⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠁⠀⠀⠀⠀⠈⠛⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⢀⣀⣀⡀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣿⠉⠟⣻⢛⡏⢩⣿⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡉⠉⢿⣶⣾⣿⣿⣷⣾⡿⠉⢉⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠙⠿⠿⠿⠿⠋⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣄⣀⣀⣠⣴⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⣾⡟⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠸⣿⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠻⣧⡀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⢉⣉⣀⣠⣤⣤⣤⣤⣤⣍⣉⡙⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⢀⣴⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⠙⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠈⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠷⠞⠋⠀⣿⣿⣀⡈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠈⢻⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣤⠟⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣉⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠉⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠉⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠙⠿⠷⠶⠶⠶⠶⠶⠶⠶⠿⠛⠁⢀⡀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⢶⣶⣦⣤⣤⣤⣤⣶⣶⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣶⡶⠶⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠁⡄
⠀⠀⠀⠀⠀⡀⠀⠘⠄⠰⠁⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠖⠚⢃⠐
⠀⠀⠀⠀⠀⠉⠐⠂⣰⣦⢘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠽⣻⠮⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠠⠂⠉⠠⠀⠄⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡄⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠌⠁⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠉⠀⠀⠀⠀⠀⠀⠰⣶⠀⡀⠀⠀⠀⡄⣿⠇⠀⠀⠀⠀⠀⠀⠀⠆⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠓⠁⠀⠰⠆⠀⠀⠀⠀⠐⠢⢿⡁⢠⣦⠀⣿⠗⠂
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠿⢠⡾⠋⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣤⠤⠽⡏⠈⡿⠥⢤⡄
⠀⠀⠀⠀⠀⢠⢴⠶⢆⡻⡱⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⣄⣷⣤⣬⣧⣤⣽⣾⣥⣤⣯⣤⣴⣭⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⠀⣴⣃⣠⡞⠙⣦⣀⣳⡄⠈⠃⠀⠀⠀⠀⠀⠐⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⢳⣴⠉⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢀⣿⣿⣿⠃⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣩⡟⡇⠀⠀⠀⠾⢻⣍⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⣦⣽⣿⣧⢶⣰⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣷⣧⣼⢿⡷⣾⣿⣏⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⢾⣙⣿⣽⣿⣿⣿⣷⣾⣧⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⣿⣟⣿⣽⣏⣻⡙⢿⣿⣾⠂⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⠀⡀⢁⢠⠃⢀⡀⠦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠛⣻⣿⣿⣩⢿⣿⣭⡁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢺⠟⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇⠃⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⢀⠃⠈⡀⠀⠉⠪⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠘⠻⠛⠋⠀⠀⠘⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡪⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣬⠐⠂⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠟⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁


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

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀
⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⡦⣠⣿⣿⣿⣿⣧⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⣿⠋⠉⠉⠹⣿⡆
⡇⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢸⡇⠀⠀⠀⠸⠿⠷⠛⠿⠿⠿⠿⠿⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⡀⠀⠀⢠⣿⡇
⡇⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⢸⡇⠀⠀⠀⠘⠁⠀⢿⠇⠀⠃⠀⢾⠇⠀⠀⠀⢸⡇⠀⠀⠀⢼⣿⣻⣿⡄⢠⣿⣏⣠⣄
⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠻⠿⠟⠀⠀⠙⠿⠟⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⣠⣶⠓⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⡇⠀⠀⠀⠀⠀⢰⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣽⣭⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣴⣶⣷⣆⠰⠛⠃
⡇⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣴⣿⣿⣿⠿⣫⣦
⡇⠀⠀⠀⠠⠶⠦⢠⣬⣽⡍⠴⠶⠄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢿⣿⣿⣶⣿⣿⠟
⡇⠀⠀⠀⠀⠀⠈⠙⠿⠿⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠻⠿⢿⣿⠿⠟⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⢀⡀⠀⠰⡆⠀⢀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣶⣄⠀⠀⣠⣶⡆
⡇⠀⠀⠀⠀⠈⠙⣠⣤⣤⣄⠊⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢙⣿⣶⣶⣿⡃
⡇⠀⠀⠀⠰⠖⠀⣿⣹⣏⣿⠀⠲⠆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠻⠿⠿⠿⠿⢿⠿⠟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣿⠟⠉⠉⠹⣿⡆
⡇⠀⠀⠀⠀⢀⡠⠈⠛⠛⠉⢄⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣶⡇⣰⣿⣁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣿⣆⠀⠀⣠⣿⡇
⡇⠀⠀⠀⠀⠈⠁⠀⠰⠇⠀⠈⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠀⠉⠟⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠿⢿⡿⠿⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠒⠒⠤⠭⠍⠉⠁⠂⠀⠈⡇⢸⠁⠀⠘⠃⠈⠉⠩⠤⢒⠂⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⠀⠰⢍⣀⡀⠀⠀⠀⠀⠀⠐⠶⠧⠴⠶⠀⠀⠀⠀⠀⠀⢀⣈⡱⠆⠈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠈⠉⠒⠢⠤⣤⡉⠉⢉⣉⣁⣀⣀⣐⣀⣀⣀⣀⡈⡁⠀⢬⡥⠄⠒⠂⣉⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⡐⠂⠀⠠⠤⠀⣀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠀⠘⣋⣡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠉⠐⠒⠒⠂⠀⠀⠀⠠⠤⠤⠄⠀⠀⠐⠒⠒⠂⠈⠉⠉⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡆⢠⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣤⠒⠈⠉⠉⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠈⠉⠉⠐⣦⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠒⠀⠸⠀⠖⠠⠄⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠐⠊⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣿⣿⣿⣿⣿⣟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⢿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡷⣿⣿⣿⡞⣿⣿⣿⡏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣷⣿⣶⣭⣟⡿⣷⣭⣽⣷⣾⣿⣿⣿⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⣿⣿⣿⡷⣿⣿⣿⣿⢧⣿⠿⣛⣿⣻⣿⣿⣿⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿⣿⣿⣷⣿⣭⣭⣽⣾⣿⣻⣿⣿⣷⣿⣿⡟⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢸⣿⣿⣿⣿⣿⢿⣛⣭⣻⣿⣿⣝⣿⣯⣿⣿⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⣿⣿⢿⣽⣿⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⠃⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣹⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢹⣿⣿⣽⡿⣿⡿⣛⣯⣾⣿⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⠤⢀⣈⠉⠉⠉⠉⠉⠉⠉⠉⣀⡀⠤⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠠⠤⠤⠤⠤⠤⠤⠄⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡟⣻⠀⠀⣤⢀⣠⣤⣄⠀⠀⠀⠈⠛⢻⡟⢛⣃⣤⣤⣄⢸⡇⠀⠀⣿⠆⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣀⣤⣿⢸⣏⣠⣄⠀⠀⠀⠀⠀⢸⡇⢸⣁⣤⣤⡿⢸⡇⠀⢀⣿⠀⣿⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠉⠉⣿⢸⡟⠉⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟⢿⣤⡀⠸⣿⣀⣼⣿⡀⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠀⠀⠉⠈⠛⠛⠙⠀⠀⠀⠀⠀⠈⠃⠘⠁⠀⠙⠁⠀⠙⠛⠋⠘⠃⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⢰⡞⠛⢲⡄⣶⢠⡶⠚⠻⣶⢐⡃⣿⠀⠀⠀⠀⠀⠀⣰⠶⠛⢷⡄⢸⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠙⢷⢸⣷⡶⠟⠃⣿⢸⣶⣶⠟⠋⢸⡇⢸⠀⠀⠀⠀⠀⢸⡏⠀⠀⢸⡟⣸⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⠟⢸⡇⠀⠀⠀⣿⢸⡇⠙⢷⠆⢸⡇⢸⠀⠀⠀⠀⠀⠘⢷⣤⣴⠿⠁⢿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠷⢀⡀⠀⢀⡀⣴⠾⠛⠿⣦⢀⠀⠀⣘⡛⢻⡿⣿⠀⠀⣀⠀⠀⢠⣿⣧⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢸⡁⣀⣼⡇⣿⣤⣶⡾⠋⣴⠀⣿⡉⠁⢸⡇⣿⣷⣼⣿⠀⠀⡾⢹⣿⣇⠀⣾⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣶⢸⡟⠋⣿⡇⣿⡛⢿⣄⠀⣿⢠⡌⠉⢳⢸⡇⣿⠻⠟⣿⡀⣸⠛⠛⢿⣿⢠⡌⠉⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠘⠃⠀⠙⠃⠙⠃⠈⠛⠁⠙⠈⠛⠛⠋⠘⠃⠋⠀⠀⠛⠃⠃⠀⠀⠘⠛⠊⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⡀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢀⡴⠦⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡴⠶⢶⣄⢿⡆⠀⢀⡾⢡⡶⠶⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠦⣤⡀⠀⠀⠀⠀⣿⠀⠀⢠⡟⠁⠀⢈⣿⠸⣿⣀⣾⠃⢸⣧⡴⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣄⣠⠇⠀⠀⠀⠀⣿⣤⣦⡌⢷⣄⣠⣾⠃⠀⢻⣿⠇⠀⢸⣇⣀⣀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠸⢦⠥⡴⠁⢠⡆⢀
⠀⠀⠀⡀⠀⠀⠀⠠⠾⣖⣠⣐⡾⠆⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⣡⠚⠤⣄⠌⡎⢅⡀⠜⢢⣁⣠
⠀⠀⠈⠂⠲⢢⣖⡐⢗⣁⣽⣄⣱⢗⣐⣤⠶⠀⠊⠀⠀⠀⣄⡈⠀⢥⠱⠄⢆⣓⠆⡴⢫⢅⠀⢀⡀
⠀⠀⠀⠀⠀⠈⢡⡧⢴⣟⠩⢙⣷⢤⡧⠁⠀⠀⠀⠀⠀⠠⠌⡗⡪⠥⢬⢱⠡⣷⢹⠸⡤⡭⡒⡏⠤⠄
⠀⠀⠀⡀⢀⡨⣗⢓⣼⠿⢶⠿⢯⣔⢶⡧⣀⠀⡀⠀⠀⠀⠔⠋⠁⣱⢈⡥⡁⡆⡋⣬⡐⡹⠈⠑⠤
⠀⠀⠐⠐⠁⠀⠈⠠⣍⣖⠲⢒⣎⡅⠈⠐⠀⠑⠈⠀⠀⠀⠰⠶⠲⣁⠯⣖⡄⡇⣔⡨⢅⣱⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠠⠎⠉⠀⠈⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣸⠀⢀⡼⠬⢆⠀⢸⡆⠑
⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣤⣀⠀⣦⠀⢹⠁⣠⡆⢀⣠⠄⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⠄⠀⡆⠀⢄⠀⢀⣤
⠀⠀⠀⠀⠀⠘⠱⣦⢰⠓⣈⠀⢳⢠⠶⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⢠⡸⠿⣸⢈⡰⠎
⠀⠀⠀⠀⠈⠻⢴⡒⡵⣯⠻⢻⠷⡔⡲⠜⠋⠀⠀⠀⠀⠀⠀⠐⠉⠠⣭⠔⡡⠨⡉⢨⡥⠄⠕
⠀⠀⠀⠂⠗⠒⢊⠘⡟⣂⠪⢂⡺⡧⡠⠒⠒⠗⠂⠀⠀⠐⠂⠒⠒⠿⣀⠨⡀⠒⡨⢠⡘⠟⠐⠂⠒
⠀⠀⠀⠀⠠⠾⠙⣒⢝⠧⣾⡮⢟⢕⡚⠱⠦⠀⠀⠀⠀⠀⠀⠐⠅⠊⣥⢐⡮⠈⣦⢠⣭⠃⠕
⠀⠀⠀⠀⠀⣰⣞⠋⣘⠎⢲⠈⢞⡈⢻⣶⡀⠀⠀⠀⠀⠀⠀⠀⡄⠟⢀⠊⠘⡟⠸⢠⠘⢳⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠋⠀⢼⠄⠈⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⡃⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢘⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣿⡇⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣨⣇⠀⠀⢀⡄
⠀⠀⣠⢀⠐⡦⣄⡉⣠⢴⠂⠀⣀⠀⠀⠀⠀⣶⣴⡀⠀⠙⡏⠀⢠⣾⣴⠀⠀⠀⠀⠀⠘⠻⣦⣞⣾⣆⣷⠿⠟⠀⠀⠀⠀⠀⠀⠀⠙⣶⡆⢹⡏⢰⣷⠃
⠀⠨⢟⠿⠤⣹⣾⠷⣵⣞⠠⠷⡛⠇⠀⠀⠀⠉⠋⠉⢲⠚⠳⠖⠉⠙⠉⠀⠀⠀⠀⣤⣤⣾⣭⣿⣿⣿⣯⣽⣦⣤⡄⠀⠀⠀⣀⣄⣤⣀⢙⣾⣷⢋⣀⣴⣀⣠
⠀⠀⠀⢠⣞⣹⣿⣶⣿⣯⡱⠦⠀⠀⠀⠀⠀⢤⣶⡴⠊⠳⡎⠑⢦⣴⡤⠀⠀⠀⠀⠉⠉⠺⣽⣿⣿⡿⣿⡭⠊⠉⠁⠀⠀⠀⠀⠀⢉⣀⡴⢙⡏⢦⣀⡉
⠀⠀⠀⢀⣠⡔⠙⣭⠇⢳⣄⢄⠀⠀⠀⠀⠈⠋⠙⠁⠀⣼⣧⠀⠈⠻⠙⠀⠀⠀⠀⠀⢰⣾⠃⠓⣽⡕⠋⢿⡶⠀⠀⠀⠀⠀⠀⢀⡼⠛⠃⢼⡧⠘⠛⢦⡀
⠀⠀⠀⠀⠛⠜⠀⠉⠀⠂⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇

⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠏⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆
⠀⠀⠀⠀⠀⣀⠑⢿⠒⣀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢠⢰⣬⣧⡆⡄⢀⣀⠀⠀⠀⠀⠀⠛⠆⠀⠀⣀⡠⠀⠀⠛⠃⠀⠀⠀⠀⠀⠈⢶⣆⣴⣼⣧⣦⣰⡾⠁
⠀⢀⣿⣮⣆⢹⣷⣶⣾⡟⢠⡵⣿⡆⠀⠀⠀⠈⣩⡾⣯⣹⣏⣿⣷⣍⠁⠀⠀⠀⢀⣀⠀⠠⢰⢋⣯⣟⢷⡠⠀⠀⢀⠀⠀⠀⠀⣀⢰⣿⠿⣽⣯⢿⣿⡆⣀
⠀⠀⠁⠊⣠⣿⣿⣿⣿⣿⣤⡉⠂⠀⠀⠀⠀⠀⠲⣿⡶⢻⡟⣶⣿⠖⠀⠀⠀⠀⠘⠛⠀⠀⢻⡜⡿⡟⣼⠃⠀⠘⠓⠁⠀⠀⠘⠛⢻⣿⣯⢿⡿⣽⣿⡟⠛⠁
⠀⠀⠀⠀⡉⣭⠛⣿⠟⣥⡀⠀⠀⠀⠀⠀⠀⠞⠋⠹⢹⣿⣿⡏⠏⠙⠷⠀⠀⠀⠀⠀⢀⡀⠀⠉⠓⠋⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⡟⢿⣿⣿⡿⢻⣤
⠀⠀⠀⠘⡿⡋⠄⠉⠀⢋⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⡠⡄⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⢾⡧⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠸⣿⡄⠀⠰⣽⠿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣸⣇⡄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣼⣿⣿⣤⡀⣀⡀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣸⣇⠀⣀⣤
⠀⠀⠀⠀⠀⠈⢣⣀⡰⠉⠀⢀⠀⠀⠀⠀⠀⠙⢷⣿⣿⢿⡿⣷⣧⡶⠋⠀⠀⠀⠀⠸⢿⣿⠋⢻⠀⡸⠋⢻⣿⠿⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣿⣾⡿⠁
⠀⠠⠿⢾⠆⠀⢎⡀⡱⠆⠰⢷⢿⠆⠀⠀⠀⠀⣭⣿⣯⢿⣿⣽⣿⣏⠀⠀⠀⠀⠀⠀⢸⡷⠶⠒⣁⡱⠲⠶⣿⠀⠀⠀⠀⠀⠰⠶⠾⣿⣿⣿⣿⣿⣿⡷⠶⠆
⠀⠀⠀⠀⠀⢀⡞⠁⠱⣀⠀⠀⠀⠀⠀⠀⠀⡴⠾⣿⢿⣾⣷⡿⡿⠷⣆⠀⠀⠀⠀⢨⣿⣿⣦⣾⠁⢹⣦⣼⣿⣮⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⢿⣷⡀
⠀⠀⠀⠀⢬⣿⠃⠀⠐⣿⣥⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢹⡏⠃⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⢻⣿⣿⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⠋⠁⠀⢹⡏⠀⠈⠙
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⡿⢆⣀⣠⢾⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣨⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣄⡀⠐⣿⡇⠀⢀⣀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠸⡇⠀⠀⣤⡄
⠀⠠⣤⡤⣰⣾⣀⣿⣇⣼⣦⣤⣤⠄⠀⠀⠀⠀⠘⢶⡄⠘⡇⢠⣷⠃⠀⠀⠀⠀⠀⠀⠘⢿⣦⡏⢾⠎⣧⣿⠏⠀⠀⠀⠀⠀⠀⠀⠉⣶⣆⣿⣯⣦⣶⠋
⠀⠀⠀⣹⣦⣼⣿⣿⣿⣯⣤⣎⠁⠀⠀⠀⣀⣠⣤⣄⣈⣶⣷⣁⣠⣴⣄⣀⠀⠀⢀⣤⣦⣝⣬⣹⣾⣾⣩⣔⣥⣶⣄⠀⠀⢠⣤⣄⣤⣯⣿⣿⣿⣿⣵⣤⣠⣤⡄
⠀⢀⣤⣞⣁⣽⡿⣿⡿⣯⣉⣓⣄⡀⠀⠀⠀⠀⢉⣠⡔⠉⠟⢢⣀⡁⠀⠀⠀⠀⠀⠉⠉⠾⣤⡞⢻⡛⣦⣜⠂⠉⠀⠀⠀⠀⠀⠀⠈⣮⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠈⣏⡰⠿⢆⣹⠁⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⢸⡧⠀⠛⢢⡀⠀⠀⠀⠀⠀⢸⡿⠇⠒⣽⡕⠋⢿⡿⠀⠀⠀⠀⠀⠀⢠⡶⠉⠉⠻⡗⠁⠉⣶⡆
⠀⠀⠀⠀⠀⠟⠁⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⢀⡤⢷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣠⠖⠛⠉⠀⢸⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠀⣴⣤⠀⠀⡼⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡉⣰⣿⠃⢠⣾⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠉⠉⠉⠉⠉⠉⠁⣿⡇⠰⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠻⣦⣌⠳⢦⣄⡀⠀⠸⣿⠁⠀⠀⠀⠉⠛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢙⡻⠶⣤⣭⣉⣛⣓⡒⠒⠒⠒⠒⠒⣂⣈⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠛⠳⣦⡉⠉⠉⠛⠛⣛⣛⡛⠛⠛⠛⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⢸⣇⡀⣀⣴⠟⠋⠙⢷⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠈⠛⠷⣶⣤⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣴⠾⠟⠛⠁⠀⠀⠀⠀⠈⣿⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⠶⠟⠃
⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣩⣿⣟⡉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠛⠛⠶⠶⣶⣶⣤⣤⣤⣶⡀⠈⠉⠉⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠾⠟⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⣶⡆⠀⠀⢰⣶⡀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡆⠀⠀⢠⡿⡆⠀⠀⢰⣿⡄⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⠀⢀⠁⢸⠀⠀⢸⠀⢃⠀⠀⠸⠀⡇⠀⠀⡆⠈⡄⠀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣸⣿⣿⣄⢀⣼⣿⣿⡀⢀⣿⣿⣧⡀⢀⣿⣿⣇⢀⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢨⣭⢩⢭⣭⣭⠩⡭⣩⣭⣭⠭⡭⣭⣭⣍⢭⠍⣭⣭⡭⠭⣭⡅⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣛⣻⣿⣿⣛⡛⣿⣿⣿⣛⣛⣿⣿⣿⢻⣛⣿⣿⣟⣛⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣾⣾⣿⣿⣶⣷⣾⣿⣿⣾⣷⣿⣿⣿⣾⣶⣿⣿⣷⣶⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢻⡇⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠹⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢸⡇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣹⣿⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣿⠀⠀⠀⣷⠀⠀⣦⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⣶⠀⠀⢀⣿⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣨⣿⣿⡀⠀⢠⣿⠀⠀⠈⠳⡄⠀⡇⠀⢀⣰⠟⠀⠀⣿⡇⠀⠀⣿⠀⢀⣼⠿⠷⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠻⣄⢸⡇⠀⠀⠀⠀⠙⣦⣧⡴⠋⠀⠀⠀⠀⢻⡇⠀⠀⣿⣴⣟⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠤⠴⠶⠾⢿⣇⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⣾⠃⣀⡴⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⣾⣯⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢸⡇⠀⠀⠀⢀⣠⠞⠁⠀⠀⠉⠉⠉⢙⣧
⠀⠀⠀⠀⠀⠲⣦⡀⠀⠳⣄⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣸⡇⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⣠⣴⠖⠀⠀⠀⣠⡴⠆
⠀⠀⢀⣤⣠⣤⣼⣿⣦⣤⣼⣷⣤⣤⣤⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣥⣶⣏⣥⣤⣄⣀⣀⣠⣤⣾⣟⣁⣀⣠⣴⣿⣁⣀⣀⣠⣞⣉⣀⣀⣀⣀⡀
⠀⠀⠈⠉⠉⢉⣼⠟⠁⠀⣠⡾⠃⠀⣠⡿⠋⠀⠈⠉⠉⠉⠉⠉⢉⣥⢿⡟⢧⡀⠀⠀⠀⠀⠈⠉⠙⠻⣍⠉⠀⠙⠿⣍⠉⠉⠉⠻⣯⡉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠠⡾⠋⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢸⠁⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠈⠻⣦⡀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⡀⠀⣀⣀⣀⣀⡴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣉⠿⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠙⣶⡶⠶⠶⠦⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⣶⣾⠋⠀⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢉⡏⠀⢀⡇⠀⠀⠀⠀⣀⣼⣧⡀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣷⡒⠲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢿⣿⡿⠁⠀⢸⠇⠀⠘⠗⠀⠀⣠⡾⠛⢹⡇⠙⠶⣄⡀⠀⠀⢻⠀⠀⢀⡏⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢹⡇⠀⠀⢸⠀⠀⠀⠀⠀⠸⠏⠀⢀⣼⣧⡀⠀⠀⠙⠃⠀⠀⠀⠀⢸⡇⠀⠘⣿⣟⠛⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⡟⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣏⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣿⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣿⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣞⠭⢈⡭⠭⠅⠢⠤⣄⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠦⠍⠉⠭⠕⠒⠀⠀⡀⠀⡎⠀⠰⢆⠀⠉⠻⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢩⠀⠞⠁⠀⠈⢈⠅⠐⠂⠀⠀⢀⠀⠀⠤⠀⢈⠀⠀⠀⠈⢅⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡠⠂⢖⠀⠰⠂⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠃⠀⠘⠀⢨⠂⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡏⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡚⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢃⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡀⠀⠀⢀⣴⠄⢠⣿⡄⠀⢀⡀⠀⠀⣹⣿⣄⢸⣿⡄⠀⢰⡄⠀⠀⣾⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡁⠀⠀⣸⣿⣧⣼⣿⣷⠄⣿⣷⡀⢀⣿⣿⣯⢿⣿⣟⡀⣾⣧⡀⣰⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢝⢦⡠⢽⣿⣿⣿⣿⣿⣦⣿⣿⣆⢴⣿⣿⣿⣽⣿⡿⠄⣿⣿⣟⡹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⢿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣭⣩⣴⣧⣤⠼⠿⣿⣻⠿⠿⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠘⢿⣿⣷⣦⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠈⠙⣿⣿⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⣠⣿⡿⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠀⠀⢀⣾⣿⠟⠁⢀⣴⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢸⡟⠁⠀⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣾⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠟⠛⠋⠉⣉⣁⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡈⠉⠉⠛⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠗⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣕⡳⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠰⡇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⠀⣀⣤⡤⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠠⣿⢊⣉⡤⠤⠤⠤⣄⡀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⢹⡀⢀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⢀⡟⠉⠀⠀⠀⠀⠀⠀⠹⣆⠀⠘
⠀⠀⠀⠀⠀⠀⠈⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢹⡌⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣿⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⣰⠏
⠀⠀⠀⠀⠀⠀⢀⠀⢷⣹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣛⣻⣷⣤⡴⠶⠞⢛⣡⡴⠛⠁
⠀⠀⠀⢀⣤⡾⠁⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⣋⣩⣥⠴⠶⠶⣿⡉
⠀⠀⣴⣿⠋⠀⢀⠀⠀⠈⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠉⠀⢄⡀⠀⠀⠈⠻⣦⡀
⢀⣾⣿⠃⠀⣠⢣⠂⠀⠀⠀⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠙⢆⠀⠀⠑⣮⣷⡀
⣸⣿⡏⠀⢠⣽⡏⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠘⣿⣿⣧
⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿
⣿⣿⣇⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⣸⣿⣿
⠸⣿⣿⡄⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣏⠉⠛⠒⠒⠒⠒⠒⠒⠉⠉⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⢠⣿⣿⡏
⠀⠹⣿⣿⣄⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⢶⣤⣤⣴⡶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⢠⣾⣿⠟
⠀⠀⠙⢿⣿⣦⡀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠁⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠙⢿⣿⣦⣄⠀⠈⠛⠷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠟⠋⠁⠀⠀⠀⢀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣄⡀⠀⠈⠉⠛⠛⠿⠷⠶⢶⣶⣶⣤⣤⣤⣤⣶⣶⠶⠶⠾⠟⠛⠛⠉⠉⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢶⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⡶⠿⠛⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣙⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠉⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣬⡙⢓⢄⡀
⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠈⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠘⣿⠈⠀⠙⠦⣀
⠀⠀⠀⢠⡋⠁⠀⠀⣄⡀⠀⠀⠀⠀⠀⣸⠧⠓⡀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠈⡇⠗⠂⠢⡘⣿⠗⣫⠴⠞⠉⠀⠀⠀⡅⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠⢀⠁⠊⠉⠀⢄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠀⠀⠀⠈⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⢔⡨⠐⠁⠀⠀⠀⠀⠀⠑⢠⠀⠀⡇
⠀⠀⠀⠀⠘⡐⠀⠀⠀⠢⣼⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣧⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡤⠖⠁⠀⠀⠀⠀⠈⠳⢦⡰⢄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠀⠀⣀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣥⣀⡀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠚⠁

⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡾⠠⠀⠟⠓⠀⠉⠉⠙⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠋⠉⠉⠁⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⢸⠃⠀⢡⠄⢀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠰⡄
⠀⠀⠀⠀⠀⡞⠀⠀⢎⡞⠐⠀⠒⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠄⠙⣦
⠀⠀⠀⠀⣸⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢸⡀⢠⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⢣⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⣠⠎
⠀⠀⠀⠀⠀⢳⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠁⠀⡄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠶⠴⠤⣄⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢀⣀⣀⣀⡀⣰⠞⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠄⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⠀⠀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠀⡀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⢀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀⣤⡶⠂⠀⠀⠀⠀⢠⠖⠀⠀⠺⡿⠚⣡⣀
⠀⣤⣶⠇⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡄⠀⡉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠠⠞⠠⢀⡀⠀⠀⠀⠘⣠⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠉⠛⠂⠀⠀⠀⠠⡄⠂⠜⠀⠀⠀⢰⡇⠀⠈⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⣉⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢀⡗
⠤⠠⠤⠄⠀⠤⠀⠄⢰⣇⡃⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠤⠤⠄⠀⠀⠀⠈⠀⠉⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠑⠀⠀⠁⠈⠀⠁
⠀⠀⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠆⠒⠂
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠊⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⡀⠀⠀⠘⠓⠐⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠤⠒⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣻⣿⣛⡟⠿⠿⠿⠿⢿⢷⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠔⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣸⠀⠀⡀⣾⡖⢛⣛⡓⣚⣒⢿⣿⣿⣿⣿⣿⣿⣿⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠸⠁⠀⠀⠘⢹⡇⢸⣿⣿⣿⣿⣿⡏⠀⠈⢉⡋⢹⣿⢐
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣈⠃⠀⠱⡀⠀⠀⠀⠀⠀⠆⠀⣼⡇⢸⣿⣿⣿⣿⣿⣿⡶⠿⠾⣷⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⣴⠒⣄⠀⠀⢸⣧⣱⡀⠀⡇⠀⠀⠀⢠⠀⢀⠀⢿⡇⢨⣿⣿⣯⣽⣿⡏⠀⡀⠀⠈⠉⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠸⣿⣿⠇⠀⡇⠀⠀⠰⡟⠀⠈⠁⣿⡇⢸⣿⣿⣿⣿⣿⣖⣀⡀⡀⢠⠀⣼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠰⠀⡈⠉⠀⢀⠃⠀⠀⢈⡇⠸⠀⠀⣽⡇⣸⣯⣿⣯⣽⠹⣿⡿⢿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠄⡀⠀⠀⠀⠀⠀⠸⣶⣿⠇⠀⠰⠁⣔⠀⠀⠘⡆⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣏⡏⠀⡀⢙⠁⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠈⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡌⠙⠀⠘⠱⠠⡀⠆⣿⡇⢸⣿⡿⣿⣿⢓⡧⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠘⢠⣀⣀⠘⠯⣄⠀⠀⠈⠐⠒⠒⠂⠁⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⢸⠆⠶⠓⠶⣶⣶⣲⣒⣚⣓⣒⣠⣼⣭⣥⣭⣭⣄⣀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡨⣯⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉
⠀⠤⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠑⠤⠴⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠂⠠⠀⠀⠀⠠⠤⠾⠂⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁⠂⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠂⡰⠁⢠⣀⡀⠀⢠⣂⡰⠀⠈
⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢿⣿⠃⠀⠸⣿⠇⡂
⠀⠠⢤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠉⠀⣀⣴⣀⣀
⠀⠀⠀⠀⠓⠢⠤⠔⢦⣄⣤⣤⣄⡀⢀⣀⣀⣀⢀⣀⡠⠤⠤⠤⠀⠓⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⣳⣤⣀⣀⣠⣼⡩⢤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣶⣤⣀⣀⣀⣀⣀⣀⣤⣴⣶⣁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠭⢽⣿⣛⣿⣿⣿⣿⠷⠶⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠛⠛⠛⠛⠛⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠿⠉⣀⡸⠇⠀⠀⠈⠹⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠛⠃⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠿⠿⠿⠿⠿⣄⡀⢀⣠⠿⠿⠿⠿⠿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⠉⠀⠀⠀⠀⠀⠉⢳⡞⠉⠀⠀⠀⠀⠀⠉⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣤⠿⠃⠀⠀⠀⣤⡼⠃⠀⣤⡟⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⡿⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⣶⠶⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⣶⡎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⠁⣿⡏⢸⣿⠁⢹⡏⠈⣿⡇⢹⣿⠈⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⡟⢳⣾⠛⣶⡞⢳⣶⠛⣷⡞⢻⣶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣇⡼⢿⣠⠿⢇⡸⠿⣄⡿⢧⣸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⣸⣇⠀⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣶⠉⢱⡎⠉⣶⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣤⡜⢣⣤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢇⡸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠⡢⠀⠀⢀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢐⠂⠠⠂⠀⠀⠠⠂⠂⠀⢀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠒⠀⠐⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠈⠒⠠⡀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⢠⠒⢢⠉⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠇⠀⢘⠄⠜⠁⠀⠀⠀⠀⠀⠀⠐⠁⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠢⠀⠐⢴⠟⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⠰⣆⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⡡⠀⡐⠀⠀⢂⢿⠁⠀⠀⠀⠀⠀⠂⠀⠀⢀⠈⠀⠣⠔⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⠀⠀⠀⠈⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣦⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⢠⠔⣋⣉⡀⣉⣏⣀⣀⡠⠐⡇⠐⠒⠂⢄⡠⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠡⢤⠠⠄⠒⠉⠁⠈⡍⢹⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣁⠀⡄⠄⠀⢢⠀⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⡂⠀⢈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⢙⢃⠄⢞⣉⣀⣣⠁⢃⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡛⢨⠀⡀⣸⣠⣄⣤⡄⠈⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠟⠤⡤⠬⠉⠙⠀⠂⠀⠓⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠐⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠘⢥⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠚⢿⡟⠃
⠀⠀⠀⢠⠀⠀⢠⢖⡄⠀⡇⢸⠀⠀⠀⡅⠀⢀⢰⠀⠀⠀⣄⠀⠀⣴⣷⡄⠀⣿⡇⠀⠀⢸⠀⠀⢀⣼⡀
⠀⠀⡐⠉⢧⠆⠠⠀⠆⠀⠐⠌⠀⠀⠂⠀⠂⠈⠀⠈⠀⡸⠿⡤⠀⠿⡿⠇⠀⠻⠇⠀⠐⠀⠐⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠄⠀⠀⢐⣉⡆⠀⠀⣌⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠀⠀⢀⡇⠀⠀⡇
⠀⠀⠀⣄⠀⠀⠈⠉⡇⠀⠰⢤⠀⠀⡉⠁⠀⡀⠀⠀⠀⠰⡀⠀⠀⢸⣿⡇⠀⢿⣿⠀⢸⣿⠀⢠
⠀⠐⠚⠃⠸⠀⢴⠀⢃⡄⢰⠀⡀⠀⡇⠀⢐⠙⡂⠀⢰⣾⣶⣶⠀⡼⡿⢣⠄⢸⣿⠀⢸⣿⡄⣿⣿
⠀⠘⠂⠀⠚⠀⠀⠀⠀⠀⠈⠀⠁⠀⠉⠀⠀⠈⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠈⠋⠀⠈⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢠⠎⠁⠀⣰⣣⠀⡷⣄⡇⡌⢉⡇⡆⠀⢸⡯⠁⠀⠀⠀⡏⢉⡆⡞⠳⡄⡔⢦⡄⡖⢢⠀⠀⠀⢲⣒
⠀⠈⠳⠶⠲⠃⠈⠗⠃⠘⠇⠷⠚⠑⠓⠶⠘⠶⠂⠀⠀⠰⠟⠉⠀⠓⠚⠁⠳⠚⢡⠷⠋⠸⠦⠦⠸⠤
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠤⠀⠀⠀⠐⠓⠠⠄⠀⠀⠀⠀⠶⠈⠐⠴⠂⠀⠅⠒⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢸⣿⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⣴⡄⠀⠀⠈⢻⣷⣼⣿⣰⣿⠋⠀⠀⠀⣴⡄⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⢹⣿⠀⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⠀⢸⣿⢃⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣙⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣾⣿⣿⣟⣡⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⢿⣷⣄⠀⠀⠀⢸⣿⠀⠀⠀⢀⣴⣿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⣸⣿⡀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣶⣄⡀⠀⠀⠀⠀⠀⠙⣿⣿⠟⠛⢿⣿⡟⠁⠀⠀⠀⠀⠀⣀⣴⣾⠟⠃
⠀⠀⠀⠀⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⠁⠀⠀⠀⢹⣿⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠀⠀⠀⠀⠀⢈⣿⣷⣤⣀⣴⣿⣏⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⢻⣿⠋⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣿⠟⠁⠀⠀⢸⣿⠀⠀⠀⠙⢿⣷⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠟⢛⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣟⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢹⣿⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⢸⣿⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⠿⠇⠀⠀⠀⣰⣿⢻⣿⢿⣷⡀⠀⠀⠈⠿⠇⠀⠙⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⢸⣿⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣠⠖⠒⠒⠦⣴⠋⢳⣀⣀⣤
⠀⠀⠀⠀⠀⡏⠉⠉⣿⡄⠀⠘⢦⡀⣴⠛⠓⢦⢸⠁⠀⣤⠉⢻⠀⠘⠋⠀⡽⢧⠀⠉⣀⡬⠇
⠀⠀⠀⠀⠀⢻⠀⠀⠈⠁⠀⠀⠈⡇⡏⠀⠆⠈⢿⡄⠀⢀⣠⠾⡇⠀⢰⠊⠁⣼⠀⡞⠁
⠀⠀⠀⠀⠀⠈⣇⠀⠀⣤⡄⠀⠀⢷⡇⢠⠶⡄⢘⣧⣤⣼⣤⣀⡙⢚⡭⠤⢤⣨⠟⢧⡀⣀⣤⣄⡤⠤⢤⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢹⡀⠀⢹⠻⣤⣤⣼⣉⠉⢠⠼⠟⠚⢻⡇⠀⣄⠙⣾⠀⢀⠀⢿⢦⡀⠙⠁⣠⣿⡀⠀⠤⣞⠀⠀⠀⢠⡟⠉⠙⣇
⠀⠀⠀⢀⡤⢦⡀⠓⢚⡯⠤⣄⣿⠉⠙⡇⠘⢲⡄⢰⢿⡇⠀⠿⠃⣿⠀⣈⡀⠈⣇⣷⠀⡏⠁⣏⠉⠃⠀⢈⡇⣤⠶⣼⡇⠀⠀⣿⣠⣄⡀
⣾⠛⠉⣿⡀⠀⠛⣶⠏⠀⣤⡄⢻⡀⠀⠷⠶⡞⠃⠘⢻⣧⣤⡤⠞⠳⠤⠏⠹⠴⠞⠛⠚⠃⠀⠙⠲⠤⠴⠚⢸⡇⠀⠈⠃⠀⠀⠟⠁⠀⣿
⢹⡄⠀⠈⠁⠀⠀⢸⣆⠀⠹⠃⢸⣇⣀⣤⠴⠟⠛⠋⠉⠀⠀⠀⣴⠛⠶⣄⠀⣠⠖⠛⣦⠀⠀⠀⢠⡴⠶⠶⠶⠛⠒⡂⠈⠀⠈⠉⠒⢛⠛⠛⠛
⠀⢿⡀⠀⢶⣆⠀⠈⣿⠲⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣏⣀⣀⡭⠉⢅⣀⣀⣻⣤⣤⡀⢸⡇⠀⠂⠀⠀⠊⢆⠀⠀⠀⠀⠄⠁⠀⠀⡀
⠀⠈⣧⣀⣸⠟⠖⠒⢋⣤⣀⡴⣆⢀⣄⡀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣧⠈⡗⠀⠀⠀⠀⠠⠚
⠀⠀⠈⠉⠁⠀⡴⠦⣾⠀⣿⠁⣿⠋⣸⣃⠀⠀⠀⠀⢺⠥⠭⠬⠤⠤⢬⠀⣄⠀⠇⠒⠒⠒⠒⣿⠀⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⢠⣷⣄⠘⣇⢸⢠⠏⡰⢋⣽⣃⣀⣠⣤⠸⡇⠀⠀⠀⠀⢲⠀⠉⢀⢸⣄⠀⠀⠀⣿⠀⣿⠀⠄⠁⠀⠀⠈⡄⠀⠀⢸
⠀⠀⠀⢀⣤⣬⡷⠾⠷⠾⠛⠈⠉⠋⠉⠉⠉⠁⠀⢸⠀⣧⠀⠀⠀⠀⠁⠀⠀⠀⠙⠃⠀⠀⠀⣿⠀⢻⠂⠀⡀⠈⠀⠀⠄⠀⠀⠰⠂⠀⢀
⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣿⠀⢸⡄⠂⠀⡠⠀⠀⠀⠀⠀⠸⢀⠐⠁
⠀⠀⠀⠸⡗⠂⠈⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇⠀⠀⠀⠀⡀⢒⠀⠀⠘⠁⠀⠠
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠈⠀⢸⠀⠀⢀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⡀⠀⡀⠀⠀⡀⠀⠀⣿⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⡇⠀⠈⣇⣀⣊⣀⣀⠀⢀⡀⢀⣈⢀⣀⡐⠁⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣤⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠈⠛⠁⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⠀⠈⠻⣷⣤⡀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠿⣷⡿⠟⠻⢿⣦⡀⠈⠻⢿⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⣿⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⣠⣾⡿⠋⠀⠀⣀⣀⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⠟⠉⠀⠀⣴⡿⠟⠛⠿⣷⡿⠟⠛⠿⣷⡄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣀⣀⣀⣀⣈⣀⣀⣀⣀⣹⣿⣀⣀⣀⣀⣀
⠀⠀⠀⣰⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣦
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⣿⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠋⠉⠉⠉⠉⠛⢿⣦
⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢻⣧⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⢸⣿⡀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⢿⣇⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣤⣴⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠈⠙⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢳⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⡇⢸⣇⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠛⠛⠛⠋⠉⢉⣭⣭⣭⣽⡇⢸⣯⣭⣭⣉⡉⠉⠙⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⢸⣯⣀⡀⠀⠀⠀⠀⠀⠀⠘⠳⠶⣶⣾⣧⣼⣷⣶⣤⣭⣽⠇⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⣿⠀⢉⣭⡛⠛⠲⣶⢤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⠴⠞⡿
⠀⠀⠀⠀⠀⠈⠛⣷⠦⣿⣴⣏⣉⣳⣀⣠⠞⠫⡉⠉⡩⠿⣛⠛⢛⡩⣍⠉⠉⣩⠭⡄⢠⣞⣹⡾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠚⠷⠶⠾⠶⠿⠶⠶⠷⠾⠶⠶⠛⠛⢹⠇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣠⠶⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⢒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠤⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣀⠀⠀⠀⠀⠀⠻⣤⣀⡀⠀⠀⠀⢀⣀⡴⠗⠀⠀⠀⠀⣠⢾⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠛⠿⢶⣶⣤⢄⣀⠀⠉⠉⠛⠛⠛⠋⠉⢀⣀⣤⢴⣶⠿⠗⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠭⠭⠭⠭⠭⠽⠟⠒⠊⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⡔⠉⠀⠈⠁⠀⠀⡞⣀⡴⠗⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⠒⠒⠀⠛⠛⠚⠉⠴⠛⠋⠙⠛⠓⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠛⠲⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⡴⠶⢺⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⣠⣤⣴⣤⣶⣶⣶⣾⣿⢿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣏⣹⣿⣛⣿⣿⣿⣯⣽⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⣿⣿⢿⣿⣽⣿⣿⣟⣿⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠚⠛⢯⣽⣿⣿⣿⣿⣿⣿⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠲⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣻⣿⣿⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡤⢰⣟⣿⣿⣿⣧⣼⣿⣼⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢁⡰⠠⢐⠀⢸⣿⡋⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠰⠀⢸⠋⡀⠀⢀⠹⠿⣿⣾⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣾⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠂⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿
⠀⠀⠀⡀⠄⠀⠘⠂⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠄⠒⠀⠠⢄⢀⡆⠇⠀⡀⢘⠄
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠃⠠⠘⣦⠤⣴⠄⠐⠈⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠠⠀⠀⠀⠀⢷⠄⠔⣀⡙⠀⠀⠀⡀
⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠂⠀⠀⠀⠀⠈⢂⠐⠁⢱⠀⠀⠃
⠀⠀⠀⠈⠀⠀⡄⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣄⣀⡄⣀⡀⠀⠧⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣌⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠉⠰⠂⡀⢠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠈⠂⠀⠀⠀⠀⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠸⠁⠘⠓⠒⠒⠒⠚⠛⠓⠀⠒⠒⠒⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠰⠂⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢉⡟⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⣲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠑⢆⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣠⡀⡜⠀⠀⠀⠀⠀⠀⠈⠦⣰⢂⠜⠀⠀⠀⠀⠀⠀⠀⢣⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠢⠤⢩⣯⠤⠤⠤⡄⠀⣠⠖⠒⠒⠿⣠⠤⣄⠀⠀⡰⠒⠒⠒⠴⠧⠤⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢰⠋⠀⡿⠤⠒⡆⠀⠠⠦⣾⠀⠀⢸⡄⢠⠖⣢⠀⣀⡀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢰⠀⣿⠀⠀⢸⠈⠁⠀⡇⠸⠀⡇⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠈⠀⠁⠀⣿⠀⠀⢸⠀⠀⠀⠉⠊⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠀⠀⠈⢢⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠣⠤⠄⣀⣀⡀⠤⠋⢲⠀⠣⣄⣀⠀⠀⢀⡠⠇⠀⢠⡾⢄⡀⠀⠀⠀⢀⡸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠐⣒⠂⠀⠀⠀⠀⢀⡼⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠙⠦⠤⣄⣉⣉⡭⠤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠋⠉⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠢⡀⠀⠀⠀⠀⠀⠀⢠⠋⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠂⡅⠀⠀⠀⠀⠀⠀⠘⡄⢠⡠⠃⠀⠀⠀⠀⠀⠀⠀⢸⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠤⢼⣭⣤⠤⠤⢤⠀⠀⡠⠤⠬⣿⠥⠤⢄⡀⠀⠀⢀⠤⠄⠀⠛⠥⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡀⢠⠖⢆⣠⠲⢤⡞⠁⢸⠓⠒⠐⡄⠀⡔⠒⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢇⣀⠇⠀⡇⠀⠀⢸⢱⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠂⠠⡀⠀⠀⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⡇⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢣⠀⡉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⠀⠀⠀⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⣉⡐⠒⠒⠒⠒⠒⠊⠈⣃⠈⠒⠠⠄⠀⠠⠤⠚⠁⠀⣴⠋⠢⠤⠄⠀⠀⠤⠐⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡑⠒⠒⠀⠀⠈⢀⣀⣀⣈⠉⠬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠐⠒⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠸⡄⣠⠜⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣟⣁⡤⠔⠋⠉⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡴⠋⠉⠉⠀⠀⢑⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠉⡠⠄⠒⠒⠊⠁⠈⡳⣄⠀⠀⠀⠀⠀⠀⢀⠿⠒⠒⠒⠒⠒⠊⠁⠀⢈⡦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠉⠉⠉⠉⠉⠁⢀⡠⠞⠁⠘⡆⠀⠀⢀⣤⠴⠃⠀⠠⠤⠤⠤⠤⠤⠤⠒⢉⡤⠋⢢⣄
⠀⠀⠀⠀⠀⠀⠀⢟⡓⠢⠤⣀⣀⣀⣀⣒⣒⣉⣀⡤⠤⠔⣛⡇⠀⠈⢿⠒⠤⠤⠤⢀⣀⣀⡀⠠⠤⠶⠥⠒⠒⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⠒⠒⠒⠤⠤⠤⠤⠤⠐⠒⠂⠉⢉⠃⠀⠀⠘⡤⠤⢄⠤⣄⡠⢤⣀⠤⢄⡠⠤⢄⣼
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡇⣀⠤⠔⠒⠒⢒⣒⠒⠒⠀⠒⠒⠊⡙⠀⠀⠀⢀⠃⠀⠀⢠⠎⢁⠔⡲⡍⢆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⣜⡹⢨⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡀⠀⠀⠸⡀⢾⣋⠴⠁⡘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠓⠐⠋⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢣⠀⠀⠀⠙⠢⠤⠤⠊⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠤⠒⠒⠒⠒⠒⠤⠤⠤⠤⢼⠁⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠖⣶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠺⠀⠹⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠠⠤⠐⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠋⠁⠉⠁⠀⠙⠛⠷⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣤⡾⠛⠋⠉⠉⠀⠉⠙⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⡀⠀⠀⢸⡇⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⢸⡇⠀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣸⣇⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢦⣤⣀⣀⠀⠀⡀⠀⠀⣸⣇⠀⡀⠀⠀⠀⣀⣀⣠⡴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠉⠉⠛⠛⠉⠙⠋⠉⠛⠛⠛⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠿⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣶⣤⣀
⠀⠀⠀⠀⠀⢀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⣀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄
⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⢀⡴⠚⠻⣿⣆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢀⡠⠶⠂⠈⠉⢹⣿⡿⠀⣰⠏⠀⠀⠀⣿⣿⠀⢀⠴⠋⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⣰⠟⠀⠀⠀⣀⡀⠀⣀⣀⠀⠀⣿⣿⣄⠸⠇
⠀⠈⠁⠀⠀⠀⠀⢸⣿⡇⣴⠃⠀⠀⠀⠀⣿⣿⣠⠋⠀⠀⠀⣸⣿⡇⠀⠀⢀⣤⣤⣸⣧⣸⡃⢀⣠⣤⡿⢷⣾⠒⢾⣿⡇⠀⣿⣿⡀⣠⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠁⠀⠀⠀⠀⢰⣿⣿⠏⠀⠀⠀⠀⣿⣿⠀⠀⠀⣾⣿⠟⠀⢀⡿⠛⠀⣿⣿⠀⢸⠁⠀⣼⣿⠇⣼⠛⠿⠿⠋⠀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⢸⣿⡿⠀⠀⢀⣿⣿⣇⢀⡜⠁⠀⢀⣿⡏⢠⡏⠀⠀⢿⣿⣦⠋⠀⠀⠀⢀⡟⠀⣿⣿⡇
⠀⢀⣤⣤⣄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⡜⠉⠛⠛⠋⠀⠀⠀⠸⣿⣿⠏⠀⠀⠀⠘⠛⠉⢀⣴⡆⠀⣸⠀⢀⣿⡿⠁
⣴⣿⣿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠶⠋⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠈⠳⠟⠛⠁
⣿⣿⠃⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⣤⠾⠋⠁
⣿⣧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠛⠛⢿⣿⣿⠛⠛⠛⠛⠛⠋⠁
⡿⠁⠀⠀⠀⠀⠀⠀⠀⣴⡶⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣠⣤⡀
⠁⢀⣴⠞⠛⣿⣷⣄⠀⣿⡇⣸⠇⠀⠀⠀⠀⠀⠀⠀⣴⣦⢠⣿⣿⡏⢹⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣀⡾⠋⣿⣿⣿⡄
⣶⠋⠀⠀⣰⣿⣿⡟⠀⠈⣿⠿⠤⠶⣶⣶⣶⡆⠀⠀⡈⠁⠘⣿⣿⣷⣾⠁⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣾⣿⣷⣼⡆⢹⣿⣿⣿⡇⠀⢻⣿⣿⡇
⠀⠀⢀⣼⣿⣿⠟⠁⣠⡾⠋⠀⠀⣴⣿⣿⠏⠀⢠⣾⣿⣷⠀⠘⣿⣿⣿⣤⡀⠀⠀⢠⣾⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠇⠈⣿⣿⣿⠀⠀⢸⣿⣿⣷
⠀⠀⣼⣿⣿⣏⣠⡾⠋⠀⠀⠀⣼⣿⣿⠏⢀⣰⠿⣿⣿⣿⠀⢀⣿⠋⠻⢿⣿⣿⣿⣇⢸⣿⣿⣧⠀⠀⠀⣰⡿⣿⣿⠟⠀⠀⠛⠋⠁⠀⠀⠀⠛⠛⠋
⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⢸⣿⣿⣯⣤⠾⠁⠐⣿⣿⣿⢀⣾⠇⠀⣠⡾⠛⢿⣿⣿⣾⣿⣿⣿⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⠻⣿⣿⠿⠃⠀⠰⣿⠀⣠⣾⣿⣿⠃⢻⣿⣿⡆⢀⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠈⢿⣿⣿⡿⠃
⠀⠀⠠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡄
⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣷⣶⣦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠠⠄⠠⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠈⠐⠂⠤⠤⢀⣀⣁⣀⡉⠀⠨⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣉⣀⣀⡀⠀⠀⠀⣀⡸⠐⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢂⣀⠬⠉⠀⠀⠀⠀⢀⣁⣠⣬⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠑⠤⠤⠄⠒⠒⠚⠉⠉⠁⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⢀⠆⠀⠀⢠⠱⡀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡄⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠄⢈⡥⠀⠀⠀⠐⠂⣡⠔⠊⠑⣄⠂⠈⢦⠀⠀⠀⠀⠀⠀⠁⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠁⠀⠀⢀⡠⠖⠉⠀⠠⠒⠐⠘⢆⠀⠀⡑⡀⠀⠀⠀⠀⢰⠀⠀⠆⠀⠀⠀⠀⢘⠄⠀⠀⢠⠀⠀⢸
⠀⠀⠀⢀⡠⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠇⡄⠀⠀⠀⠀⢸⠀⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢰
⠀⠀⠀⠀⢆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠘⢄⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠒⠊
⠀⠀⠀⠀⠀⠣⡀⠑⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⢣⠄⠀⠀⠀⠀⠀⠀⠑⢼⠀⠙⠒⠀⢀⣀⣀⠤⠤⠴⠖⠚
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠠⣀⠀⠀⠀⡠⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠉⠛⠲⢤⣀⠀⠀⠀⠀⣀⡴⠒⠛⠉⠉⠓⠲⢄⡀
⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⣼⠀⠀⢰⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠹⠀⠀⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠊⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣀⣀⣀⣀⡀
⣠⠴⠒⠚⠋⠉⢉⣉⣉⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣭⣭⣭⣭⣝⣛⣒⣒⠤⣄
⣿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⣿⠀⠀⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠁⠀⣿
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡤⠤⠴⠦⠤⣄⡀
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣀⠀⠀⠀⠙⢢⡀
⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠴⠊⠉⠉⠉⠳⣄⠀⠀⠹
⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠓⠒⠒⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣀⡀⠀⢠⣤⠀⣶⡆⣼⡏⣽⢇⣶⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢠
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⢀⣾⡏⢻⡆⣸⡏⣰⣿⣴⣿⢛⣡⡾⠁⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⢸⡏⢙⣿⠛⢿⣴⡟⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⣰⡯⠤⠴⠶⠒⠚⠉⠁⠀⢀⣀⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⣾⣿⣀⣤⣴⡞⠷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢀⣀⣀⡤⠴⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠓⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⢀⣠⡼⠃

⢸⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣷⠛⣩⣍⣻⣿⣿⠇⢺⡿⠿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⠿⡄⠙⠋⢸⣿⠋⠀⣬⠰⣾⣟⠀⠘⣿⣿⡟⠉⣿⣿⣿⣿⣿⣿⢿⣿⠛⠉⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘
⢈⣠⠀⠀⠀⠀⠈⠉⠁⠀⠉⠸⠿⠟⠀⡜⢻⣿⠐⠃⢸⣿⣿⣿⣿⣿⣾⣧⠀⢀⠹⠿⠿⠿⢿⣿⣿⣿⣿⡟⡋⠉
⣿⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠷⠐⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣦⡄⠀⠘⣿⣿⣿⣽⣿⣶⣶
⣿⣼⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠁⣀⣴⣿⣿⡿⠿⢿⣿⣿
⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢡⠂⠙⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣀⣼⣿⣿
⣿⣾⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⢀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⡖⠀⠀⠀⠀⠀⠀⡇⢙⠛⣿⣿⣿⠏⢀⢤⡀⠀⠤⡄⠀⠀⠀⠀⠈⢿⡿⠋⠉⠛⢿⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠗⠁⠀⠈⠻⠏⠀⢸⡿⠁⠀⣿⡿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢿⣿
⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿
⣿⣻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⡇⠀⠀⠐⠈⠀⠀⠀⠀⠀⠒⠐⠠⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉

⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⣠⡒⠒⣄⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⠀⠀⠀⠀⠀⠀⢀⡠⢄⣄⡀⣀⠀⠀⠀⠀⢠⣴⠆⢠⣄⣀
⠀⠀⣐⢷⣄⢠⠞⢁⡔⠒⠛⠄⢀⠄⠒⢤⡀⣐⣿⣯⣭⡇⠀⠀⠀⠀⠀⣞⣷⣿⢻⠻⣧⠁⠀⠀⢰⡇⠘⡧⠀⣹⠻⢷⡄
⠀⣶⡛⣯⣿⡌⢦⡼⡄⠀⠀⣰⠸⠀⠀⠴⡇⣿⣯⣿⣿⡋⣀⠀⠀⠐⡄⢿⣇⣹⣀⢃⣹⣶⠀⠀⣾⣿⣆⠈⠓⢻⣼⣾⡇
⠀⢹⣿⣾⣿⠀⠙⢷⣴⣧⣽⠃⠀⣤⣴⡶⠁⠸⣶⣷⡿⠹⡙⠂⢀⣀⡇⠸⣯⣿⣙⣏⣽⠏⠀⠀⢸⣿⣿⣿⡦⣽⣻⣿⠁
⠀⠘⣿⣿⡏⠀⠀⢸⡿⢟⡃⠀⠀⢹⡿⠁⠀⠀⣿⣻⡇⠀⢹⣏⣩⡿⠀⠀⢀⣹⡟⣏⠁⠀⠀⠀⠈⣟⠉⠉⠉⠉⢻⡏
⠀⠀⢻⣽⠀⠀⠀⠀⣟⣻⠀⠀⠀⠘⡡⠀⠀⠀⢸⣿⠂⠀⢸⣿⢽⡇⠀⠀⠘⠛⠍⠙⠋⣠⡤⠄⠀⢿⡀⠀⠀⠀⣼⡇
⠀⠀⠈⠃⠀⠀⠀⠀⠘⠇⢰⣶⣒⡆⢀⡶⢶⠀⠈⢿⡀⠀⠀⠻⠻⠁⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠠⡀⠉⠓⠚⠛⠁
⠀⠀⠐⠂⠆⠀⣼⣿⣗⣆⢸⣿⣽⡧⠰⠃⢸⡇⣴⡟⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⣁⢀⣿⣷⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⢰⠀⣿⣿⣿⠹⢸⣟⢿⣧⠀⠀⢸⡇⣿⡇⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⢾⡆⢈⣓⡛⠛⠀⠀⠀⠀⠀⠉⠉⠋
⢀⣦⠀⢠⣼⡆⣿⣿⣿⠰⢸⣿⣻⡇⠈⡇⢸⡇⣿⣇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣶⣇⠀⣠⠀⠀⠀⠀⡀
⠈⠧⠤⢼⠿⠋⣿⣿⣿⡘⠸⣿⣿⡏⠀⣇⣾⡇⠈⠛⠻⠟⢃⣴⠖⢲⣯⣿⢦⢀⠀⡼⠋⠛⢻⣿⠉⡁⢹⡂⣀⣊⣊⣝
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠉⠉⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⠃⠑⢦⣷⣴⣦⣶⠟⠀⢻⡬⣿⢟⣿⡿⡏⡆
⠀⢀⠤⠤⣄⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣀⠀⢠⡘⢛⣿⡿⢿⣏⡀⠀⠀⡿⠉⠛⡏⠀⠀⢸⣿⣯⣦⣿⣤⣿⣵
⠀⡅⠀⠀⢸⡆⢰⠋⠀⠉⢲⠀⡎⠀⠈⢱⢰⠋⠉⠉⢳⠘⣿⠿⠿⠿⣿⣿⠀⠀⠀⣠⡄⣾⠁⠀⠀⠀⠻⣿⣏⣿⣻⡿⠏
⠀⠿⣂⣤⣾⠇⠹⢄⣀⣀⠟⠐⢧⣀⣤⠾⠈⠷⢀⣴⠿⠀⠿⣀⣀⣠⡿⠇⠀⠀⠀⠩⠁⠿⠃⠀⠀⠀⠀⠿⣿⣉⡯⠗

⠀⠀⠀⠀⠀⠀⠀⣀⣶⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀
⠀⠀⠀⠀⣶⣾⣤⢼⣿⣿⢄⣸⣷⣆⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⡀⠀⠐⣷⣿⡁⠀⠀⠀⠀⠀⠀⠠⣼⡆⣀⣿⣿⣁⢰⣇⡄
⠀⠀⠀⣀⠉⡿⠿⣦⣻⣡⡾⢿⣏⣉⡀⠀⠀⠀⠀⠀⢀⠀⠾⢿⡇⢸⣷⣯⡍⠀⠀⠀⠀⠀⠀⣈⠉⣻⠿⣥⣼⣧⣨⠿⡟⠛⡀
⠀⠀⠰⣿⣿⡷⠶⣾⣿⣿⡟⠻⡿⢿⡟⠀⠀⠀⢠⣿⣾⣷⣤⣤⣿⣾⣧⣴⣿⣴⣿⠤⠀⠀⠀⢾⡿⡿⠶⣿⣏⣿⣿⠤⣿⣾⠥
⠀⠀⠀⠉⣤⣿⣿⣋⣼⣏⢻⣾⣧⣄⠀⠀⠀⠀⠀⠋⠘⠋⠶⣶⡏⢹⣷⣦⠝⠀⠛⠀⠀⠀⠀⢉⣀⣹⣶⠛⢻⠟⢳⣴⣃⣈⠃
⠀⠀⠀⠀⠛⠻⠉⢉⣽⣯⡉⠹⠟⠃⠀⠀⠀⠀⠀⠀⠠⢶⣾⡋⠃⠘⢫⣿⡶⠆⠀⠀⠀⠀⠀⠀⠚⣻⠉⢙⣿⣟⠋⢹⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠘⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠙⠃

⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠙⠻⡄⢠⡄⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⠁⣀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣃⠀⠀⣸⡿⠇
⠀⠀⠀⠀⠀⢰⣦⣹⣾⣧⣏⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠋⣷⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣸⠟
⠀⠀⢿⡿⠶⠶⣶⣿⣿⣿⣿⡶⠶⠶⣾⣷⠀⠀⢠⣶⣿⣦⣼⣧⣼⣿⡧⢴⣧⢤⣾⣵⡆⠀⠀⣾⡷⢶⡶⠶⣿⣿⠶⢶⡦⢾⣷
⠀⠀⠀⠀⠀⠘⠋⣱⢿⡟⣯⠹⠗⠀⠀⠀⠀⠀⠀⠀⠋⠀⠾⢥⣼⠉⣷⣤⠿⠆⠈⠁⠀⠀⠀⠀⠀⠀⢀⣼⠋⢻⣦
⠀⠀⠀⠀⠀⢀⣴⠃⠘⠋⠈⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⡙⠿⠋⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⡏⠀⠀⢹⣦⡄
⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠁⠀⠀⠛⠿⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣄⡤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣤⡄⠀⢀
⠀⠀⠀⢀⣤⣧⣈⣿⡷⢾⢯⣄⣿⣆⠀⠀⠀⠀⠀⠀⠀⣻⣦⡿⠀⠀⠀⢻⣶⣟⡀⠀⠀⠀⠀⠀⢠⣤⡼⣻⣭⣽⣏⣳⢿⡃
⠀⠀⠀⣺⣉⡿⢷⣾⣿⣿⣶⣟⣳⣴⣧⠀⠀⠀⠀⢠⡌⠉⠛⢷⡄⣴⣠⡾⠋⠉⢉⠀⠀⠀⠀⣀⣨⢿⡿⣭⣷⣿⣾⣛⣧⣽⡿⠂
⠀⠀⠠⣟⢻⡟⣿⣾⣏⣹⣿⣿⣩⣏⣽⠅⠀⠀⠠⣤⣿⣦⣤⣿⡿⠛⣿⣿⣥⣴⣿⣥⡀⠀⠀⠙⢿⢿⣛⣳⣿⣹⣿⣿⣽⣇⣧⡄
⠀⠀⠀⢿⡟⢯⣹⢾⣽⣻⡽⢯⣼⣛⡿⠁⠀⠀⠀⠠⠟⠁⠀⢻⡿⣶⢿⣿⠃⠈⠻⠌⠀⠀⠀⢰⡾⣟⢏⣵⢿⡿⣟⢳⣾⣱⠋⠁
⠀⠀⠀⠈⢺⣿⠟⢶⣿⠼⣷⡛⢻⠿⠁⠀⠀⠀⠀⠀⠰⢶⣶⡟⠀⠉⠀⣹⣿⡷⠆⠀⠀⠀⠀⠀⠀⣼⡿⣝⣻⣗⣻⣥⠞⠻⠇
⠀⠀⠀⠀⠀⠈⠐⠻⠻⠼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠷⠀⠀⠀⠉⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⠇⠼⠡⠤⠤⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣟⣛⣿⣼⣧⣿⣟⣻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡴⠛⢻⣿⣿⣶⣶⣶⣶⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢋⣤⡄⠀⠉⠉⠉⠉⠉⠉⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡜⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠲⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣍⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⡚⠉⠻⣿⣿⣿⣿⡿⠉⠑⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⣽⣶⣄⠀⠀⠀⠀⣠⣶⣯⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠳⠶⢶⣶⣶⣶⠶⠶⠋⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣍⣉⣙⠛⠛⣛⣉⣩⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⡴⠖⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣕
⢸⠀⠿⣷⡀⣀⠀⠀⠀⠀⠀⠂⠀⢀⠀⡀⠀⠠⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢿⣦⡀⠀⠀⠀⠀⢰⣶⡆⢰⣶⢰⣶⡖
⠀⢀⣔⡛⣿⠋⠀⠀⠀⠀⠀⠱⠀⠠⡀⠀⠠⠀⠀⠀⠀⠀⠈⣛⣿⣿⣯⡄⠀⠀⠀⠀⠀⠸⣿⣧⣾⣿⠇⠀⠀⠀⠀⠀⠉⠉⣰⣶⠉⠉⠁
⠀⠙⠛⠛⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆
⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣦⠄⠀⠀⠀⠀⢀⣠⠖⠦⣄⣀
⠀⣾⠿⠿⠿⢷⡄⠀⠀⠀⠀⠀⣀⡤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡟⣗⠀⠀⠀⠀⠈⡍⠀⠀⠀⠀⢩⡍
⠴⢿⠐⠀⠐⢸⠷⠀⠀⠀⠀⠈⠳⣀⠀⠈⣳⣤⡤⠀⠀⠀⠘⠛⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⣰⡏⣸⡇⣿⠀⠀⠀⠀⢰⢧⣴⣶⣦⣶⡜⣇
⠀⠈⠳⠅⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠛⠀⠿⠀⠀⠀⠀⠈⠈⠛⠛⠛⠛⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠀⠀⡀
⠀⠀⢰⡆⠀⠀⠃⠀⠀⠀⠀⠀⣻⣛⣭⣥⠀⠀⠀⠀⠀⠀⣰⣿⣿⣹⣿⣆⠀⠀⠀⠀⠀⠀⠶⠄⠀⣻⡆⠀⠉⠁⠀⠀⢀⡴⢞⣻⣷⣤⡄
⠀⢀⣼⣷⣴⡖⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⠃⠀⠀⠀⠀⠀⢻⣿⠿⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠐⠦⠀⠀⠠⣫⢾⣛⣻⡿⢿
⠀⢀⣍⣭⣡⣀⠀⠀⠀⠀⠀⠀⢹⡟⠿⣿⡆⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⢸⡇⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠙⠻⣿⣸⣇⣿⠟⠋⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣀⣀⠀⠹⣿⣿⠏⠀⣀⣀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡄⠘⣿⣿⡆⠀⠈⣻⣿⣿⣿⡟⢠⠀⣿⣿⠀⡄⢻⣿⣿⣿⣟⠁⠀⢰⣿⣿⠃⢀⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠿⢦⣌⢻⣧⡀⠘⠛⠿⣿⣿⣷⣿⠀⣻⡟⠀⣿⣾⣿⣿⠿⠛⠁⢀⣼⡟⣡⡴⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⣬⣿⣿⣷⣦⡘⠻⢿⣿⣿⣿⡆⢸⡇⢠⣿⣿⣿⡿⠟⢁⣤⣾⣿⣿⣥⣴⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠈⠉⡙⠻⠿⣶⣄⡉⢻⣿⣷⣼⣧⣾⣿⡟⢉⣠⣶⠿⠟⢋⠉⠁⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣿⣦⣘⣷⣦⣈⡙⠻⣷⣿⣿⣿⣿⣿⣿⣾⠟⢋⣁⣴⣾⣃⣴⣿⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⣿⠿⢫⡿⠟⠋⣉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⠙⠻⢿⡝⠿⣿⠋⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⢀⣉⣤⣶⡿⠛⢉⣼⣿⣿⢿⡿⣿⣿⣧⡉⠛⢿⣶⣤⣉⡀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⢿⣿⣿⣿⠟⢁⣤⣶⣿⣿⣿⠇⢸⡇⠸⣿⣿⣿⣶⣤⡉⠻⣿⣿⣿⡿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⠾⢋⣽⡟⠁⢀⣤⣴⣿⣿⣿⣿⠀⣼⣿⠀⣿⣿⣿⣿⣦⣤⡀⠈⢿⣯⡙⠷⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⢠⣾⣿⡇⠀⠈⣻⣿⣿⣿⣧⠙⠀⣿⣿⠀⠋⣼⣿⣿⣿⣟⠁⠀⢸⣿⣷⡄⠘⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠛⠛⠀⢰⣿⣿⡆⠀⠛⠛⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣀⣴⣿⢻⡟⣿⣦⣀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠃⢸⡇⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋


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

⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠙⣛⣸⣦⡀⠀⠀⠀⠀⠘⣿⡀⠀⠀⣀⢼⡧⣀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢀⣴⣇⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢄⡱⡄⠀⠀⠀⠀⢸⢳⠀⠀⠘⢿⡿⠃⠀⠀⡜⡇⠀⠀⠀⠀⢠⢎⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⢖⡦⣀⣀⣀⣇⢣⣀⠀⢸⡇⠀⣀⡜⣸⣀⣀⣀⢴⡲⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⢻⡲⣤⠔⠛⡌⢯⡑⢼⡧⣊⡽⢡⠛⠢⣤⢖⡟⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⣀⠀⢸⢀⠟⢺⣶⡒⠙⡜⡌⢻⡟⢡⢣⠋⢒⣶⡗⠻⡀⡇⠀⣀⣀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠭⣙⡛⠤⣜⠀⠑⣤⡘⣿⢾⡷⣿⢃⣤⠊⠁⣣⠤⢛⣋⠭⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠘⢎⠳⡒⠭⣶⣬⣿⡻⣏⣹⢟⣿⣥⣶⠭⢒⠞⡱⠃⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠶⡷⣿⡷⠶⠶⠶⡷⣿⠶⢾⡷⣏⣹⣏⣹⣏⣹⢺⡷⠶⣿⢾⠶⠶⠶⢾⣿⢾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⢠⣞⡴⠥⣒⠿⢛⣽⣵⡯⣹⣮⣿⡛⠿⣒⠬⢧⣱⡄⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣒⣩⣵⠒⢫⠀⡠⠛⢡⣿⢾⡷⣿⣌⠛⢄⡀⡝⠒⣬⣍⣒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠉⠉⠀⢸⠈⣦⢼⠿⠥⣠⢣⢃⣼⣧⡘⡜⣄⠬⠿⡧⣴⠁⡇⠀⠉⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣼⠵⠛⠢⣤⢃⢞⡡⢺⡗⢭⡳⡘⣤⠔⠛⠮⣧⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠮⠗⠉⠉⠉⡏⡜⠉⠀⢸⡇⠀⠉⢣⢹⠉⠉⠉⠺⠵⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⡱⠃⠀⠀⠀⠀⢸⡼⠀⠀⢠⣾⣷⡄⠀⠀⢧⡇⠀⠀⠀⠀⠘⢎⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣭⢹⠟⠁⠀⠀⠀⠀⢠⣾⠀⠀⠀⠉⢺⡗⠉⠀⠀⠀⣿⡄⠀⠀⠀⠀⠈⠻⡏⣭⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠉⠀⠀⠄⠤⠔⠑⠁⡀⠀⠂⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠀⠠⠀⠀⠀⢀⠀⠐⠒⠂⠀⢸⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⠋⠵⠖⢀⠀⡀⠀⠄⠠⠬⠥⠤⠀⣈⠐⠀⠁⠈⠠⣦⡭⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⠿⠿⣿⠶⡶⠶⠤⠤⣬⡭⣭⣭⣤⣤⣥⣴⣷⣶⣾⡷⢿⠖⢉⡠⠔⠚⠉⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠉⣙⣓⣿⣿⠿⡓⠸⠃⢸⠆⣠⠔⠚⠓⢢⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⡠⠶⣒⣶⡿⠿⢃⠈⢔⢸⠞⠁⠀⠀⠀⠀⢳⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠄⠀⠒⣒⣚⡯⠅⢀⠅⢸⠀⠀⠀⠀⠀⢀⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣁⠀⣩⣤⠭⠥⢶⣒⠁⠔⠂⣿⠀⠀⠀⢀⣠⠞⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⣐⣚⣭⠭⠵⢂⠉⠀⣷⠀⡐⠊⠉⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢒⣒⣚⣫⠭⠀⡂⢠⠏⣠⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠄⠭⢵⣒⣚⡋⡠⢂⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡃⢀⠀⠀⠀⠀⠀⠀⠀⠤⠈⢥⣼⣿⣿⡋⠐⣠⣋⡀
⠀⠀⠀⣀⡤⠤⠖⠒⠶⠞⠉⢉⣉⡉⠭⢿⣦⣄⡀⠀⠀⠙⠈⢛⠈⣿⣿⣿⣟⣣⣴⣿⢿⣿⣿⣟⣛⣛⠛⠶⠶⠦⣤⣄⡀
⠀⠀⢾⣡⣀⣄⠀⠀⠆⠀⠀⠀⠀⠰⠀⠈⠙⠤⢀⣀⣀⣀⣀⣉⣍⣱⣾⣷⣦⡦⠟⢉⣠⠄⠀⠀⠀⠉⣉⡳⠀⣠⢈⡹⣹⠆
⠀⠀⠀⠙⠺⢭⣛⡒⠲⠶⠤⠤⠥⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣉⣀⣭⣤⣀⣠⣤⣶⣶⣶⡾⠤⠴⠶⠖⢒⣛⣭⠵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣐⣂⣀⣠⣤⣤⣤⣤⣤⠤⠴⠶⠖⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠃⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⢾⣄
⠀⠀⠀⠀⠀⠀⠀⣰⢻⠻⡐⠀⠀⠀⢠⡟⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠰⣟⣲⠿⣋⣩⡝
⠀⠀⠀⠀⠀⠀⠀⠉⣥⣀⣤⡀⣤⣤⣠⠁⠀⣠⡮⣭⡁⠀⠘⡆⠀⢸⣿⣿⣫⣭⣽⡿⣻⡆
⠀⠀⡠⠒⠚⠓⠶⣄⠉⣿⣏⠉⣉⣿⠟⢠⠞⠉⠀⣠⠼⠶⠴⢧⡀⠐⣿⣷⣶⣶⣶⣿⣉⠀⠀⣸⢷⣤⡀
⠀⢸⡁⠀⠀⠀⠀⣽⡆⢻⣿⠿⣿⣿⠀⣮⠁⠀⣴⠃⠀⠀⠀⠀⠨⡀⢻⣞⣽⣟⣩⣿⠇⠀⢸⡼⣊⡿⢽⣄
⠀⠀⠇⠀⠀⠀⠖⢾⡀⢸⣿⡟⣿⡟⠀⠀⡏⣶⠻⡄⠀⠀⠀⠀⢸⡇⠀⣿⣯⣭⣿⡏⠀⢲⡟⣓⢻⡆⣽⣷⡆
⠀⠀⢰⣠⢀⣶⣤⡞⠁⠈⣷⡟⣿⠇⠀⠈⠳⣿⣤⢈⣖⠒⢘⣿⡿⠀⠀⢸⣟⠟⣿⡇⠀⠙⣷⣷⣦⣭⣽⣶⠃
⠀⠀⠘⣭⢻⣽⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠋⠀⠀⠀⠘⣯⡿⣿⠀⠀⠀⢻⣿⡿⣷⠺⡟
⠀⠀⠀⢆⣹⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣉⡏⠀⠀⠀⠀⠀⢿⠿⣿⠀⠀⠀⠸⣾⡷⢿⢿⠇
⠀⠀⠀⢸⠷⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣶⢛⠃⠀⠀⠀⠀⠀⢸⣶⡏⠀⠀⠀⠀⢹⣿⢾⡟
⠀⠀⠀⠀⢂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿