Zodiak Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⢤⠤⡤⣤⢤⣤⣀⣀⡀
⠀⠀⠀⢴⣟⣑⣠⣴⣾⣧⣤⠄⣤⣤⠭⠴⣶⣤⣬⣿⣷⠖⠛⠉⠁⢀⣴⡚⠏⢫⠈⠀⠋⠓⢦⡈⠉⠓⠶⣄⡀
⠀⠀⠀⠈⠙⠛⠉⠁⠀⠉⠏⢸⢻⣀⣀⡠⣠⡿⠟⠩⠁⠁⠀⠀⠀⠘⢦⠱⠘⢘⠀⠀⠀⠠⢋⠵⣦⠀⠁⠢⢍⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠓⠛⠛⠉⡝⢰⠻⣷⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⡬⣀⡀⠄⠀⢨⠞⡣⠀⠀⠀⠀⠁⠠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⢛⡀⠰⣦⠐⠁⠈⢻⣷⡀⠀⠀⠀⠀⠀⢀⠎⠎⠐⡉⢷⠅⠈⠔⣈⡗⠀⠀⠀⠀⠀⠀⠈⠆⠳⣄
⠀⠀⠀⠀⠀⠀⢠⣮⠄⢲⡀⠨⠄⠀⢹⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⡠⣣⣞⠥⠊⢐⣭⡇⠐⡈⢤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⡘⢦
⠀⠀⠀⠀⠀⠀⠈⠙⠤⠞⠛⣧⡀⢀⠞⠀⠀⠀⠀⠀⢻⣧⡀⣄⡨⠞⠯⣿⣀⠀⢀⠞⠍⢁⣠⠏⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣠⡠⡏⠀⠀⠀⠀⣀⡠⣚⡟⢞⡖⠋⠉⢴⠋⠀⡠⢟⣐⣒⡶⡷⠾⠿⠿⠛⠛⠛⠛⠿⢷⣦⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠀⡇⠀⢀⣄⡾⠗⡉⠔⠊⢁⡤⠰⠋⢀⠤⠛⠁⣤⠖⠊⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣉⠒⢤⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠂⢰⠃⠰⣪⢋⡵⠊⢀⡠⠖⠉⢀⠤⠖⠀⣀⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⢳⠀⠀⡎⠉⢳⢦⡙⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⠀⢵⣯⢋⢀⡤⠁⢀⣤⠔⠈⣀⡴⠚⠁⠀⠀⠀⠀⣠⠤⢀⠀⠀⠀⠀⣠⠇⢸⠀⢀⡇⠀⡸⣆⢹⡎⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢷⠀⣾⠫⡇⡽⠀⡸⠃⠀⣠⠞⠉⠑⠲⠤⠤⠤⠖⠋⠀⠀⠘⢀⡀⠀⠸⠤⠴⠋⢀⠜⠀⣰⠁⢹⡄⢸⡌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠃⠈⣧⣸⠯⠯⣇⠾⢁⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠋⠉⠛⣆⠀⠀⣠⠔⠁⢀⠔⠁⠀⢸⡇⢸⠻⡘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠐⠀⣼⠁⢀⠊⡈⣶⢾⢻⠉⠉⠙⡟⠳⡒⠒⠚⠛⠋⠉⠀⠀⠀⠀⣄⡦⠤⢀⣧⠴⢶⡁⠀⡄⠦⡀⣳⣏⠀⠱⡄⡀⠀⡠⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢰⠇⠀⡏⠀⣠⠇⠈⠳⣕⢄⠀⢸⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡧⢀⢯⠀⠀⠀⠉⠒⠻⢦⣿⣉⠢⣹⠀⠙⠮⣀⢷⣽⢏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢀⡼⣋⠔⠃⠀⠀⠀⠈⠓⠓⢬⡆⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡟⠹⢟⠞⠀⠀⠀⠀⠀⠀⠌⡼⠻⡛⠏⡄⠀⠀⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⢋⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢯⣹⠶⠄⠀⠀⠀⠀⢀⡜⡠⡾⠃⠀⠀⠀⠀⠀⡠⢂⠞⠁⠀⠱⡈⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢴⡶⠞⠁⢺⣿⢻⠙⢦⡂⠀⠀⠀⠀⠀⠀⣀⠒⢎⢀⠵⡟⣊⢀⣀⡀⢀⠜⢠⠷⣇⣀⣀⠀⠀⡠⢊⡴⠋⢀⡀⡀⢀⡅⠀⠙⣄⣀⣀
⠀⠀⠀⠀⠀⠠⣼⢿⣴⣮⣴⡿⠉⠋⠀⠀⠙⢦⣑⠄⡀⠀⠸⠽⠶⠽⠿⠞⠫⠿⢷⣴⣾⣏⣶⣽⢶⡵⠤⠾⠗⣊⡴⠋⠀⡼⢾⣷⣶⣭⡶⣴⠾⠴⠤⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣅⡂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⢒⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠬⢥⣤⣤⣤⣬⠭⠤⠶⠒⠋⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⠶⢀⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⢸⣋⠘⣇⣀⡼



⣶⣶⣶⣶⣶⡾⣷⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢾⡿⠿⠷⠶⣶⡾⠷⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⣖⣻⣿⣶⣄⠀⠀⠀⠀⢀⠘⠈⢲⣦⠥⠀⠀⠀⠀⢀⣠⡶⠰⣆⠀⠠⣄⠀⠀⡴⡶⠃⠀⠀⢳⣦
⠀⠀⠀⠘⠱⠘⣻⣼⣌⢿⡆⠀⠀⢀⡬⠂⣛⡏⢻⡆⠀⠀⠀⢠⣾⠹⠂⠀⠉⣿⠳⡼⠀⠀⢙⠋⢲⠀⡓⣏⠋
⠀⠀⠡⢴⣶⣿⠛⣡⠟⢈⠇⠀⠀⣾⡇⣀⣑⠃⣿⠁⠀⠀⠀⣾⣯⣼⠆⠀⠀⣨⣿⡳⠀⠀⠈⢤⡿⡆⡷⣼⠂
⠀⠀⠀⢀⣿⣉⣥⡤⠞⠀⠀⠀⠀⢿⡇⡇⠀⠐⣿⣀⣀⠀⠀⠘⠮⠻⠷⣤⢠⡾⠛⠀⠀⠀⠀⠋⠁⣸⠁⢉
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢀⠀⣀⣶⣀⠀⡀⠀⠀⠀⢤⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⡶⣀⡀⠀⠀⠀⣰⡇⣴⢸⣆
⠀⠀⠀⠀⠏⠀⣿⠀⠸⠁⠀⠀⢡⠴⣋⣼⠀⠀⠙⢉⣑⡄⠀⠀⠀⠉⠈⠉⢻⣿⠁⠀⠀⠀⠀⢿⣀⣿⢀⡿
⠀⠀⠀⣄⡤⡤⣿⢠⣤⢴⡄⠀⠘⣟⣡⡀⣤⣾⣷⡀⠉⠀⠀⠀⢀⠂⢠⣤⣤⣾⣲⠀⠀⠀⠐⢷⣿⣿⣾⡾
⠀⠀⠀⠈⠉⠁⣿⠂⠉⠉⠀⠀⠀⢸⡟⣳⠀⢙⡇⢙⡄⠀⠀⠀⠋⢠⠋⢟⠀⠘⠁⠀⠀⠀⠀⠙⢙⣿⡏⠋⠁
⠀⠀⠀⠀⠀⠛⠀⠈⠀⠀⠀⠀⠀⠋⠘⠋⠀⠛⠁⠋⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃

⠀⠀⠀⠀⠀⢀⣤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⢢⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣆⠀⠀⢰⣄⠀⡶⡄⣀⣴
⠀⠀⠀⠀⠀⠠⠤⠘⠁⠀⠀⠀⠀⠀⢀⣠⣤⡦⠦⣀⡾⠀⠀⠀⢀⡀⠀⠀⠙⠒⡙⠃⠀⠀⠘⠿⡿⡆⢾⡿⠃
⠀⠀⠀⠐⣿⣥⣤⣀⠀⣅⡄⠀⠴⠛⠙⢦⣤⣶⣿⣿⢗⠂⠀⠀⠙⣠⣷⠶⣆⣿⠃⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⡾⠁⠙⡿⠙⠻⡆⠀⠀⠀⠀⢀⣀⣸⠏⢹⠘⠀⠁⠀⠀⡞⠙⣇⠐⢾⡛⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠈⠁⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠁⠘⠃

⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⢷⠀⠀⠀⠀⠀⣠⣶⠀⠉⠛⠁⢾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⡠⣄⡴⠟⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠂⠄⠀⠄⠈⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠀⠀⠈⠁⣿⠁⠀⠀⠀⢹⣶
⠀⠀⠀⠀⠠⣤⣄⣠⠄⠠⣤⣠⣤⡄⠀⠀⠀⠘⣿⣇⡀⡀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⣄⣀⠀⠠⣦⠄⠀⠤⡾⠀⠀⠀⠀⠀⡀⠉⣀⣄⠀⡀⠙⠋
⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠙⠿⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠘⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠒⠁⢠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠙⠏⠛⠛⠙⡃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠛⡧⠀⠀⠀⠀⠀⠀⢀⣤⠋⠀⠀⠀⠉
⠀⠀⠀⠀⢘⣀⠀⠐⠂⠐⠂⠁⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣴⠀⡄⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⢿⣤⣤⠆⠀⢤⣄⣾⠄⠀⠀⠀⠀⢠⣀⣀⡤⠄⠠⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠈⠛⠀⠀⠁⠀⠈⠋⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠄⠉⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠀⠐⠀⠐⠀⠀⣀⡤⠀⠀⠀⠠⣄⢀⠀⠐⠀⠈⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠈⢦⣤⠄⠀⠀⠀⣤⣶⠃⠀⠀⠀⠀⠘⢿⣦⡄⠀⠀⠠⣼⣿⠃⠀⠀⠀⠀⠻⣿⡿⠟⠂⠘⠿⣿⡏⠀⠀⠀⠀⢻⣦⣤⣤⠀⠀⣠⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠀⠀⠀⡴⠀⠀⠠⠁
⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⢠⡆⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣯⣤⢿⡆⠠⣀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠆
⠀⢀⠀⠠⠀⠔⣤⢸⣇⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⠀⠈⠃⢞⠀⠁⠀⠀⢰⠤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠉⠐⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠠⠄⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠆⠀⠀⠰
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠆⠐⠀⠀⠀⠀⠂⠀⠀⢤⠤⠄⠀⠀⠘
⠀⠊⢳⡎⠁⢈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣴⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⢰⣠⡀⠀⠂⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠂⠀⠀⠀⠀⠀⠀⢸⣀⣸⠇⠀⠀⠀⠐⠂⠒⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠂
⠀⠀⠀⡀⠄⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠈⢺⡆
⠀⠀⠀⠀⠀⠀⣫⡔⣮⡂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠸⠃⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⢜
⠀⠀⠀⠀⠂⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠈
⠀⠀⠀⠀⠀⠀⠓⣊⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠆⠠⠂
⠀⠄⠀⢈⠃⠀⠀⠡⠀⠀⢄⠀⢀⣀⡀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠖⢆⠀⠀⠀⠀⠀⣀⣄⣤⡀⢀
⠀⠀⠠⠈⠀⢀⠀⠀⠀⠀⣀⠀⠿⠽⣉⡀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠸⢆⠇⠀⠀⠀⠀⠀⠹⠘⢈⡁
⠀⠀⠀⠤⠐⠉⠀⠀⠀⠀⠀⠀⠑⠚⠒⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠰⠀⠀⠀⠠⠀⠂⠰⠤⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡊⠇⠀⠀⢠⠂⠀⠀⠀⠚⡰⠄⠣⠔⠂⠜⣂⠀⢀
⡄⠀⠀⠀⠀⠀⡆⠀⠈⡀⢀⠀⠀⠀⠀⠀⠀⠒⠀⢣⠀⠀⠀⠀⠀⣐⡶⠃⡄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⢉
⠁⠉⠀⠀⠀⠀⠁⡀⣀⠀⠉⠀⠰⠀⠀⠁⠀⠈⠄⠀⠀⠤⢀⠀⠀⠈⠋⠰⠟⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠃⠀⠀⠀⠀⠁⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⢀⡘⠀⢀⣀⣄⣠⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⡀⠀⡀⠀⢠⡘
⠀⠀⠀⠀⢠⠛⢣⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠥⠀⠀⡅⠀⠀⡏⠈⠀⢃⡀⠀⠀⠀⠀⢀⣀⠶⠀⠀⢀⠁⠈⡁⠂⠁
⠀⠀⠀⠀⠛⠒⠚⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⠀⠀⠀⠀⠀⠘⠾⠀⠀⠀⠀⠀⠚⠁⠀⠠⠤⠉⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢀⢄⠖⠓⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢈⠆⠒⡃⡐⠁⠀⠈⠉⠃⠀⠀⠀⠀⠰⣉⢭⣓⡃⠀⠀⠀⠀⠀⠛⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⡀⠀⠀⠀⣀⡋⠉⠂
⠀⠀⠘⣡⣄⣵⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠉⠀⠀⠀⠀⠀⠐⠀⣰⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡞⠁⠡⠀⠀⡠⠀⠙⡆
⠀⠀⠀⠀⢈⠅⢀⠀⠀⠀⠈⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠘⡀⠀⠂⠀⠀⠀⠀⠰⠀⠀⠶⠧⠀⠀⠀⠀⠠⠶⢆⠀⠰
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠘⠂⠀⠀⠐⠀⢀⠀⠆⢀⠀⠀⠀⠀⡀⠏
⠢⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⡀⠀⠀⠀⡀⢠⠀⢀⠀⠀⡀⠀⠀⠠
⠀⠃⢸⣗⡄⠘⠂⡄⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⢻⠃⠀⠁
⣀⣀⣀⣀⣀⣐⣂⣉⣀⣀⣀⣐⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣈⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⣤⣶⡍⣛⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡖⢷⣄⣀⡀⢀
⠀⠀⠀⣠⣾⡾⢋⣉⢻⡞⠷⡼⠃⠀⠀⠀⠀⠀⠀⢀⣿⠠⢟⡛⢿⣯⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢡⡾⠃⠈⢹⣦⢷
⠀⠀⣰⠿⣿⣧⡈⠓⠛⠁⣠⡇⠀⠀⠀⠀⠀⠀⠀⠘⢉⣳⡄⠛⠚⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡾⠀⠀⢀⣼⣼⣿⡿⡆
⠀⠀⢈⣀⣿⣿⣿⣿⣿⡿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣨⣷⡇⢀⣀⣼⣶⣾⡶⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⠀⢀⣤⠘⣵⣿⣿⣷⡏
⠀⠸⡏⣽⠛⠛⠛⠁⣿⣇⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠃⣱⢆⣞⣙⢻⡟⠁⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠷⠦⣿⡇⣾⣿⡿⠟⠋
⠀⠀⠳⠌⠙⠢⠆⠀⠙⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠛⠁⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣤⣾⡿⠟⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣶⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡷⠀⣀⣗⡀⠀⣿⣿⣦⡀
⠀⢿⣿⣇⣻⣷⣿⣥⡀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠈⠹⣿⢿⣿⡿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢰⣿⣿⣿⣄⣠⢣⣔⣧⣀⣼⣿⣿⣿
⠀⠈⠛⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣽⣿⠿⣷⠛⢻⣿⣿⡿⠇⠀⠀⠀⠀⠀⢨⣿⡏⢻⡟⠖⠁⠀⠈⠛⣻⡏⣿⣿⡀
⠀⠀⠀⠀⠀⠉⣼⡿⠟⠛⢻⡻⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠏⠁⡤⢚⣫⠿⢁⠆⠀⠀⠀⠀⠀⠸⣿⠃⣾⣿⣦⣄⢀⣤⣾⣿⡇⢻⡿
⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣸⢁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⢀⣤⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣶⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀⢴⣿⣭⣿⣇⡴⠋
⠀⠀⠀⠀⣺⡿⠋⠀⠀⢀⡄⡄⠀⠀⠀⠀⠀⠀⠰⡞⠿⣿⣿⣿⡏⠞⢛⠤⠒⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠹⣀⣤⡄⠀⢺⣾⠃⠀⠀⠀⠀⠀⠀⠀⣇⠸⣿⣿⣿⣟⣬⣥⣴⣶⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣼⣿⣿⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣏⣻⣿⡿⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⣿⠏⠻⢷
⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡉⠀⠀⠀⣹⠋⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⢹⡏⠓⠢
⠀⠀⠀⠀⢻⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃⠀⠰⠶⠃⠐⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⠀⠌
⠀⠀⠀⠀⠀⠀⢲⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠞⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣻⣇⣄⣀⣉⣲⡄
⠀⠀⢯⣁⣆⣀⣼⣧⣄⣀⣆⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣰⣿⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⠋⠉⠉⣩⠏
⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋⠉⠹⣿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣿⣿⣷⣿⣿⣿⡆⢐⠉
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠁⠀⠀⠀⠀⣨⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⣿⢻⡟⠛⠛⠛⣻⡦
⠀⠀⠳⢶⣶⠶⠃⠀⠱⢶⣶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠙⡈⠣⡀⠀⠋⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⢀⠈⠀⠀⠠⠖⠉⠑⠣⠐⠀⠂⢁⠀⠠⠀⠠⠐⠀⠀⠐⠠⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⡁⠸⠃⠀⠀⠀⢣⠀⡇
⠀⠀⠀⠀⠨⡇⠀⠀⡁⢤⠀⢀⠄⡠⠀⠀⣮⠀⠀⠀⢁⠲⡄⠀⠱⢂⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠙⠂⠀⠈⠢⡄⠠⠌⠀⠀⠞⠁⠀⠀⠀⠀⠂⣄⢀⡄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠤⡀⠀⠀⠀⡠⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠘⠁⠀⠀⠀⠁⠱⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠆⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⡰⠶⡄⠀⠀⣂⣀
⠀⠀⠀⠀⠀⠀⢐⠀⢀⠂⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠢⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡊⡉⣡⠋⠀⡐⠣⡌⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⣀⠒⠀⠀⠃⠀⠀⠀⠀⠀⡖⠀⠀⠈⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡇⠀⠀⠉⠑⡅
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⠠⠀⠊⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠀⠇⠀⠀⠀⠂⠀⠀⠂⢀⠀⡀⠂⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⠀⠀⠍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠐⠒⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠉⠢⠐⠀⠀⠀⠕⠉⠉⠀⠀⠀⠀⠀⣠⣒⠄⠀⠀⠀⠀⠐⠒⠃⠀⡠⠀⠐⠂⠚⠂⡄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⠠⠀⢰⡀⢀⡀⠀⢀⠀⢀⠙⠀⠀⠈⡀⠀⣀⠀⠀⠀⠀⢼⠋⠀⠀⠀⢱⡀⠀⠀⠀⠺⠊⢏⠀⠀⠀⣶⣀⡀
⠀⠀⠀⠀⠀⠘⠛⠁⠁⠀⠂⠀⠁⢸⠁⠀⠉⠀⠀⠉⠁⠐⠂⠀⡒⠀⠉⠁⠀⠀⠀⠀⢰⠀⠘⠄⡠⠊⠂⠀⠀⠀⠓⢀⡤⠤⠀⠠⠉⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⣶⣒⣒⣒⣒⡲⠶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢞⣻⣭⣿⣷⣿⣿⣿⣿⣿⣿⡿⢿⣿⢛⣿⣿⣾⢯⣭⣝⡳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⣫⣷⣿⣿⣷⣿⣯⣯⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣟⣾⣿⣿⣿⣿⣶⣍⡳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣫⣾⣿⣿⣿⣿⣿⣿⣯⣿⡞⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⡳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣻⣮⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⢟⣵⣿⡿⣟⣿⣯⣿⣯⣿⣿⣟⣻⣿⣿⣿⣷⣾⣿⣿⠛⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣮⡻⣄
⠀⠀⠀⠀⢀⡼⣣⣾⣿⣿⣿⣿⣳⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⡿⠿⠿⠿⠿⢿⣿⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡵⣫⣾⣿⣿⣿⣷⣜⢧⡀
⠀⠀⠀⢀⡾⣱⣿⣿⣿⣿⡿⢿⣿⣯⡳⣝⢿⣿⣿⣬⣿⠿⡛⠉⠁⠀⠀⠸⢷⡿⠆⠀⠊⠈⢉⡙⠻⣿⣿⣿⣿⣿⡷⣫⣾⣿⣿⣿⣿⣿⠻⣿⣎⢳⡄
⠀⠀⢀⡞⣽⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣗⠽⠿⠋⠐⢦⣿⡆⠀⠀⣀⣀⣘⣀⣀⡀⡀⠀⢴⢳⡖⠂⠙⠿⠫⣪⣾⣿⣿⣿⣿⣿⣟⣿⣿⡿⠿⣧⢻⡄
⠀⠀⣾⣱⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠈⢃⣤⣴⣿⣿⣿⣿⣿⣛⣿⣷⣶⣬⣉⠀⠀⠀⠀⠘⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣆⢷
⠀⣸⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⢆⡄⠀⢤⣶⢿⣿⣧⣿⣿⣋⣍⡉⠙⠛⢷⣾⡿⢻⣿⡄⠀⣀⣴⣄⢻⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣇
⢀⡿⣽⣿⣿⣿⡭⢿⣿⢿⡿⣿⣿⣿⡟⠀⠏⡼⠁⣴⡿⢿⣿⣯⣶⣌⡉⠃⡈⢿⡀⠀⠀⠙⢿⣿⣟⣹⣦⠹⠞⠉⠀⢻⣿⣿⣿⡿⢿⠿⣟⣓⣯⣽⣿⣧⢻
⢸⡇⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⣝⣉⠁⠀⠀⢀⣰⣿⣿⢟⡛⠻⠟⠋⣻⠋⠀⠴⡇⠠⠄⠀⠈⣿⡟⣿⣿⣧⠀⠀⡐⠈⣡⣤⣶⣶⣿⣿⣿⣻⣿⣿⣿⣿⠸
⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣸⡟⠀⠀⠀⢐⣿⡿⣿⣿⣿⣄⣀⣈⡇⠀⠙⠃⡏⠈⠂⠀⠀⣹⣿⣿⠟⣿⡎⣁⣀⡀⢸⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⢸⣷⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⡇⣤⡞⠣⢨⡿⣷⣿⣿⠹⠿⠉⠉⢣⠀⠀⠀⢿⠁⠀⠀⠀⣿⠛⣿⣿⣿⡇⣧⣝⣳⢸⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣿⡇
⢸⣼⣿⣿⣿⣿⣿⣻⣿⣿⣏⣻⣿⣧⠀⠀⢀⣈⣿⣿⠿⢿⡷⣶⣦⣀⣠⣿⡢⠄⡼⠀⠀⢀⣼⣿⣿⣿⣿⣿⠇⠈⠁⠁⣸⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⡇⣿⣿⣿⣿⡿⣿⣿⡿⣿⣿⣟⣻⠖⠉⠀⠀⢹⣿⣶⣿⣿⣿⣟⣫⣴⣾⣍⣀⣀⣠⣴⣿⣿⣿⣯⣬⣿⡟⠀⠀⠀⢀⣫⣟⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⢰
⠈⣷⣻⣿⣿⣿⣿⣟⣾⣿⣷⣿⣾⣿⣷⠠⡾⣴⣄⠹⣿⣻⣿⣼⣿⣿⣿⠟⢿⣿⣟⣿⣿⣿⣿⣽⣿⣿⠟⣤⣾⣆⠀⣼⣿⣿⣿⣿⣷⣿⣷⣺⣯⣽⣿⣟⣼
⠀⢹⡟⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣧⡀⠈⠀⠀⠸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣠⣿⣿⠣⠀⠀⠉⠋⣼⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⡿⣿⢉⡏
⠀⠀⢷⣹⣿⣿⣿⣿⣯⣀⣿⣿⣿⣿⣿⠿⣷⡤⠀⠀⠀⠀⣬⠙⠻⢷⣾⣿⣯⣽⣿⣷⡿⠟⢛⡉⣀⠀⠀⠀⢠⣾⡟⣿⣿⣷⣿⣿⣿⣿⣯⣿⣿⣿⠏⡾
⠀⠀⠈⣧⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣫⣠⣶⣄⡀⢠⣿⡶⠀⠈⠈⠨⣭⢭⡅⠉⠁⠀⠸⣿⡿⢆⣠⣶⣄⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡟⣼⠁
⠀⠀⠀⠘⢧⡹⣿⣿⣿⣿⣿⣿⣿⡿⣯⢟⣽⣿⣽⣯⣿⣶⣬⣅⡀⠀⠀⠀⠯⠼⠃⠀⢀⢀⣀⣤⣶⠿⣿⣿⣿⣿⣝⢿⡻⣿⣿⣿⣿⣿⣿⣿⠏⡼⠃
⠀⠀⠀⠀⠈⢳⡝⢿⣿⣿⣿⡿⣫⣾⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⠃⣿⣿⣷⣶⣶⣶⣿⣿⡙⣿⣿⣿⣦⣾⣿⣿⡛⣿⣷⣝⢾⡻⣿⣿⣿⡿⢋⡞⠁
⠀⠀⠀⠀⠀⠀⠙⣮⡻⣿⣯⣿⣷⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⣏⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠟⣵⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣙⣿⣯⣿⣿⣿⣿⣿⢏⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣙⢿⣿⣿⣿⣿⣿⣇⣸⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢮⣙⠿⣿⣽⣿⣿⡿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⠿⣋⡥⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢬⣝⣛⣳⡿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣤⣿⣿⢿⣿⣛⣫⡥⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠭⠭⠭⡭⠭⠭⠶⠶⠖⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠚⠉⠉⠉⣉⠶⣊⠥⠊⠕⣳
⠀⠀⠀⢲⠠⠴⠒⠉⠑⣄⣘⠣⡀⢀⣀⢀⡤⢚⠉⠀⠀⠀⠀⣀⠤⡲⡯⠐⢻⡔⠀⠀⢐⢿⠢⣄
⠀⠀⠀⠀⠉⠉⠉⢱⣔⣉⠙⠢⣌⡇⠀⠉⠙⠲⣄⠀⠀⡠⡪⢗⡝⣎⠀⢀⡽⠀⠀⠀⢤⡏⠈⠪⡳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣴⡷⠓⠠⢀⠀⠀⠀⠀⠀⢈⣷⢞⣪⠖⢋⠰⢃⠤⠞⠒⠀⣀⠤⠋⠀⠀⠀⠈⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢈⠀⠀⠀⡠⠞⣋⠔⠈⡠⠄⢊⡠⠚⣁⠤⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣷⠀⢰⣂⠤⢺⠀⠀⣾⣯⠎⢁⠔⠉⡠⠔⢉⣠⠞⠓⠒⠚⠋⠉⠉⠑⠒⠒⠒⠒⠒⠦⣀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡏⠀⢸⠀⠀⣿⣯⣐⠁⣰⠉⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡈⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀⠀⢸⣿⢺⣴⣡⡞⠁⢀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢃⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣇⠀⠘⠷⠟⠀⠁⠹⡄⠉⠈⠀⠀⠀⠀⠈⠉⠳⣄⠀⠀⠀⠀⠀⣰⠃⢸⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⠳⣤⣄⣘⣲⣤⠀⠻⡏⠉⠉⠛⠓⠓⠒⡖⠋⠉⢦⣄⡀⠀⠀⢧⠀⠐⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⠤⠚⠁⢀⣰⠁⠈⢧⡀⠸⡄⠀⠀⠀⠀⠀⠸⡀⢠⡏⠀⠉⠓⠦⡀⠙⢻⣯⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠃⢠⠞⠉⠉⠀⠀⠀⠀⠙⢆⠙⢦⡀⠀⠀⠀⢀⠇⣨⠇⠀⠀⠀⠀⠈⢳⡘⠏⢦⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⡗⠀⠀⢀⠞⡸⠁⠀⠀⠀⠀⠀⠐⡴⢳⢸⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡰⡚⠉⣠⠟⠈⠢⡀⠀⠀⠀⠀⠀⠀⣠⠚⠁⣳⢀⠖⠉⢰⠋⠀⠀⠀⠀⠀⡠⠎⣠⡾⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠋⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠉⠓⠒⠃⠙⠒⠀⠉⠀⠀⠀⣀⠵⠊⠀⠘⠛⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⣤⣀⣀⣀⣀⣀⣠⡤⠤⠒⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠂⢠⡀⢠⠠⡄⣤⢠⠄⡄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠋⠓⠘⠜⠃⠛⠊⠢⠃⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡆⠰⡄⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡄⠀⠓⠀⠁⠀⠀⠀⢀⡌⠿⡷⢘⣦⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠄⠋⠀⠀⠀⢀⡀⢠⡄⢀⡁⠘⠂⠿⠈⡛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⣠⠋⡰⢋⣠⢾⣾⢞⠁⠐⠆⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢉⣴⠟⠛⠂⠀⣐⢠⠂⣿
⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠰⡺⠟⠁⠀⠀⠢⠀⠈⡝⣇⢹⡄⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⢈⡁⠀⠀⢀⠎⠀⣰⠁⠀⠀⠀⡴⠁⢋⡀⠀⠑⣄⠙⠛⠀⠀⢇⢻⣆⢻⣄⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠈⠛⠀⢀⡟⠀⢠⣿⠀⠀⠀⠀⢧⠀⠀⠘⠀⠀⠈⠛⠁⠀⠀⠀⠑⢿⣄⠙⠂⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⢿⣆⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢱⡄⠉⠓⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⠻⢦⡀⠀⠀⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⡄⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⣄⠀⠀⠀⠉⠓⠄⠐⠀⠁⢰⢶⠄⠤⠀⢀⡔⡂⠀⠀⢿⡀⠀⢲⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⡻⠾⢿⣴⡾⢣⡻⡆⠀⠸⣇⠀⢸⡇⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠟⣿⣏⡡⢘⣿⠠⡾⠁⠀⠈⢧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⣄⡰⠀⠀⠀⢿⠏⠋⠠⠃⠀⠁⣰⠋⠰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⢀⡁⠱⠘⣆⠀⠀⠀⠀⢨⠀⠀⠼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣿⡧⠀⢀⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠖⣄⡄⢰⢲⡆
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢤⠢⢀⠀⠀⠀⠀⠨⣷⢂⢈⡖⠷⠂⣠⢾⡇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠢⣁⢲⡁⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⡏⣀⠇⠈⠠⣨⠘⠆⠀⠀⠀⠀⠀⣴⡆⠘⠁⠀⠂
⠀⠀⠀⠈⠃⠀⠁⠈⠃⠀⠀⠀⠀⠀⠐⠛⡃⠀⠀⠀⠀⠀⠀⠀⠷⣿⡘⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠇⠀⡈
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⢚⠋⠁⠀⢰⠀⠀⠀⠀⠀⠀⡄⣄⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⣀⠀⠼⠋⣏⠄
⠀⠀⣼⣶⣴⣄⠬⠀⠂⠀⠀⠀⠀⠙⠷⣿⡆⡆⡑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⠇⠀⠁⠓⠀⠤
⠀⠀⢿⣿⣿⣿⡄⠶⢤⠀⢠⠀⠀⠀⠀⡄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠠⠀⠀⠳
⠀⠀⠘⠙⠘⠙⠀⠀⠀⠀⠈⠀⠀⠀⠀⠺⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⠂⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠉⢆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⠤⠀⠢⡢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠆⠩⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⣛⣓⠒⠆⠀⠀⠀⠀⡸⠘⡆⠀⢈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠣⠼⠟⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡟⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⢻⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣾⣿⣿⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠁⠀⠀⣀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠠
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠟⠀⣀⣀⡉⠻⠂⠀⠛⢉⣁⣀⡀⢳⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⡠⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⡄⠀⠀⢿⣿⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠋⢠⠎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠀⠙⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⡿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⢦⣤⣄⣀⣀⠀⠀⠀⢀⣀⣀⣠⣤⡴⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⠉⠛⢛⡛⠛⠉⠉⠉⠀⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣏⢞⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣧⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⣾⣛⣯⢟⣛⢲⡀⠀⠀⠀⠀⠀⡀⠀⢳⣶⣿⣿⣿⠃⠀⢠⣿⣿⣷⣾⣻⣷⡶⣤⡴⣶⢾⣿⡟⣶⣦⣄⠀⠈⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣷⣿⣾⣸⣬⣒⣿⡇⠀⡤⠖⠶⠴⠟⠀⣹⣿⣿⠿⠿⣧⠀⠘⠿⠟⠻⣯⣿⣿⣷⢹⡷⣻⢿⣿⠇⣽⣿⣿⣆⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⣠⠤⠤⣐⡋⠉⠁⠋⠉⣉⣽⠟⠈⠦⣤⣶⡶⣶⢞⣿⠟⠿⣿⣛⣽⠀⠀⠀⠀⠀⣻⢿⣿⣷⣿⣷⣿⣿⣟⢿⠿⣿⠿⡛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢛⣛⠛⠉⠁⠀⠀⢰⣿⣿⣸⠱⣿⡇⢎⠵⣹⣿⣿⠀⠀⠀⠀⣾⡿⠛⣿⡇⠀⠀⠀⠀⢿⣿⠀⠈⠳⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⡀⢰⣏⡽⠀⠀⠀⠀⠀⣼⠿⠛⠛⣿⡿⣿⣟⣿⣿⣿⠃⠀⠀⠀⠀⠙⠓⢤⣿⠃⠀⠀⠀⣀⣸⠟⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣿⣾⣿⡄⠀⠀⠀⠀⢰⣯⠀⠀⠀⣿⡀⣿⡉⠉⢈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣨⣁⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⠗⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⣀⣘⣿⣿⣿⣿⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣘⣿⣷⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⢿⣿⢿⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⢀⡴⡏⢕⡪⢿⣿⡿⠿⠟⠛⠉⠋⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⠀⠀⢠⢳⣿⡿⣿⡿⣷⣄⠀⠸⢯⣵⣿⣻⣼⢔⡢⣻⣾⣟⣿⠂
⠀⠀⠀⠀⠀⢸⠟⢻⣶⡷⠋⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣬⢿⢻⣿⣿⠀⠹⠿⠀⠀⠿⠾⣿⢀⣼⠇⢸⣿⠀⠀⠈⢻⣿⣽⠛⠓⠶⠿⣷⣾⣇
⠀⠀⠀⠀⢀⣾⣷⣾⣿⠀⠀⣤⣶⣷⠾⠿⢻⣿⢱⠫⣿⡷⡟⣮⡷⣝⣿⠀⠀⠀⠀⢀⡤⣾⣿⣿⣿⣶⡿⠏⠀⠀⠀⠸⣿⡏⠀⠀⠀⠀⢈⡿⢹⡇
⠀⠀⠀⠀⠈⣵⣿⠉⠁⠀⠀⠀⠁⠁⠀⢀⣼⣿⠷⣾⣿⠿⠶⠿⠻⢿⡟⠛⣿⠂⠀⢸⣿⠉⢠⣞⣷⡶⢶⡄⠀⠀⠀⠾⣿⠃⠀⠀⠀⠠⠞⠁⠼⠃
⠀⠀⠀⠀⠀⢿⡝⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠁⠀⢹⣇⠀⠀⠀⠀⠈⢻⣾⠇⠀⠀⠈⠻⣦⣤⡿⠟⠀⠀⠀⠀⠀⣰⣶⠦⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣰⣦⡀⠀⠀⠸⣷⠀⠀⠀⠀⠹⣶⡄⠀⠀⠀⠀⠻⣶⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⠀⠀⣴⣿⡷⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⠿⠖⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣠⣤⣶⣤⣤⣀⣠⣴⣿⣦⣤⣾⠀⠀⠀⠀⣿⢞⣼⣧⣸⣿⣿⣿⣿⣧⠛⣿
⠀⠀⠀⠀⢄⠀⠀⠀⡀⠀⣾⡟⠓⢒⠃⢀⡀⠀⠀⠀⠀⠀⠘⠟⣿⢋⠝⣿⡋⣿⡏⣿⡝⣿⣿⣿⠿⠃⠀⠀⠀⠀⠹⣿⠱⣒⣽⢻⣿⣟⡿⣿⠀⠈
⠀⠀⠀⠀⢿⠟⠛⢿⣿⢀⣽⣿⣦⡿⠛⢿⠏⠀⠀⠀⠀⠀⠀⠀⠹⣶⣵⣟⣧⠻⣧⣙⣿⣦⠟⠁⠀⣀⣄⡀⠀⠀⠀⠈⠻⣾⡿⠞⠇⠋⠐⠁
⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⠿⠟⣁⣢⣀⣀⣄⣀⣤⣤⠀⠀⠀⠀⠸⣿⠉⠉⠙⠉⠹⣿⠋⠀⢀⠞⠁⠈⢻⡄⠀⠀⠀⡸⠽
⠀⠀⠀⠀⢸⣿⠿⡏⠀⠀⠀⠀⢀⣠⡝⣯⣻⣾⣟⣩⡤⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠛⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⢠⣶⢶⣄
⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⠀⢚⣻⣾⠟⠙⢁⣨⣤⡀⠀⠀⢀⣴⢞⢛⣶⣀⣴⣶⣾⣷⣄⠀⠙⠋⠀⠸⣇⢠⠔⣟⡫⣟⢿⡻⣿⡇⠂
⠀⠀⠀⠀⠀⠹⣯⣻⣿⣿⣷⣿⣾⡏⣿⣿⣠⣴⣿⠟⠓⠀⠀⠀⣿⢿⣬⡶⣿⣯⡿⣿⣿⠿⣿⣷⠀⠀⠀⠀⠙⢿⣼⣷⣿⣦⣯⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣼⣿⣿⣿⣿⢿⡿⠋⠀⠀⠀⠀⠀⢹⡸⣿⣷⣿⣫⠟⣦⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⡆⠀⢿⡿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡬⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣿⣷⣾⣿⣷⣼⠇⠀⠀⠀⠀⠀⠀⠀⡸⠋⣼⠁⠀⠈⣧⠀⠈⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠙⠛⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠃

⠀⠀⣴⣿⣿⣦⡀⠀⣴⣿⣿⣦⠀⠀⠀⢸⡆⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⣾⣟⠁⠀⠀⣻⣷⡄
⠀⠸⡟⣉⠛⣿⣿⣿⣿⠛⡉⣻⠇⠀⠀⠘⣷⣶⣶⣶⣶⣞⠃⠀⠀⠀⠉⠀⠀⠀⢧⣀⣀⣤⣄⣀⣩⠁
⠀⠀⠙⠋⠀⢻⣿⣿⣿⡈⠉⠉⠀⠀⠀⠙⠉⣿⣿⣿⣿⠙⠓⠀⠀⠀⠀⠀⠀⠈⢒⣿⣿⣿⣿⣿⣗⠊
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣟⠀⠀⣴⣶⣶⣶⣀⣀⠀⠀⠋⣟⠙⢙⡏⠂
⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⣀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠈⠀⠉
⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠉⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆⠀⠀⣠⠀⠀⣤⣄
⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⣿⣿⣿⣿⡆⣀⣀⠀⠀⠺⣿⣿⣿⣿⣿⣿⡁⠀⠘⠣⠂⠀⣐⡠
⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠈⠟⠛⠻⠿⠟⠁⠀⠀⠀⠉⣿⣿⡀
⠀⠈⠙⣿⣿⣿⣿⣿⠋⠁⢀⣦⣴⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠹⣿⡇
⠀⠀⠀⠈⠛⠟⠉⠁⠀⠀⠸⢿⡿⢿⣿⣿⠿⠿⠿⠃⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⠀⣶⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⢹⠀⠀⢙⡁⠀⠀⠀⠀⠀⠐⠚⠁
⠀⠀⠀⠉⠀⠀⠀⠐⠟⠀⠀⢀⣀⣀⣦⣤⣴⣀⡀⠀⠀⠈⠃⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠐⠛⠓⠀⠀⠀⠀⠀⢠⣤⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣿⣿⡏⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⢀⣴⣷⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⣿⣿⣧⠀⠀⠀⣿⣿⣿⣿⣿⣧⣤⠀⠀⢾⣿⡿⠀⢰⠆⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⡀⠀⠀⠙⠿⠿⠿⠟⠛⠛⡷⠀⠀⠘⡇⠀⠀⠀⠸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡎⠉⠙⠀⠀⠀⠀⢀⣴⣶⣶⠚⠁⠀⠀⠀⠘⣶⡶⠒⠂
⠀⠀⠀⠀⠀⠅⠀⠀⠀⠰⠄⠀⠀⠈⠙⠇⠀⠀⠀⠀⠸⣧⡼⡿⠿⠿⠒⠀⠀⠀⠀⠀⠈⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡤⠐⠒⠒⠒⠒⠒⠒⠀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠱⠙⡄⢠⠀⠀⠀⠀⠀⠀⣀⡹⠂⠈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠊⠂⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⢀⡴⠋⡠⠆⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡑⢄
⠀⠀⠀⠀⡠⠊⠀⠈⢠⠔⠁⠐⠀⠀⠀⠀⣀⣀⡀⣀⡀⠀⠀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠔⡕⡩⠊⠣⡀
⠀⠀⠀⡴⠁⠀⠀⠀⢀⠀⠀⠀⠀⢀⠔⠉⣀⠴⠞⡉⠀⠀⠀⠀⣉⠲⢤⡀⠙⢢⡀⠀⠀⠀⠀⠸⠀⠀⠀⠙⡄
⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠛⠳⠾⠕⣠⠟⠁⠀⠀⠀⠀⠈⠻⣄⠪⠶⠞⠛⠒⣄⠀⠀⠀⡀⠀⠀⠀⠘⡄
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣰⣗⢄⣀⣠⠶⠋⠁⠀⠀⡀⠀⠀⢀⠀⠀⠈⠑⠶⣄⣀⠄⣺⡆⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠇⠀⠼⡀⠀⠀⠀⠀⣾⢯⠛⢟⠉⠀⠀⠀⠀⠀⠀⣠⡂⢈⣄⠀⠀⠀⠀⠀⠀⠉⡛⠋⡽⣷⠀⠀⠀⠀⡌⡋⡂⠃
⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⢦⣀⡧⢄⠀⢀⣤⠤⠴⠿⠶⠷⠿⠤⠤⣤⡀⠀⡤⢬⣀⡼⢱⡏⠀⠀⠀⠀⠀⠁⠀⢸
⢐⠀⠀⠀⠀⡀⠀⠀⠀⠈⠳⣦⣈⣢⠤⢷⠃⠀⢠⠊⠉⠀⠈⠉⠱⠀⠀⢘⡾⠤⣚⣀⡴⠟⠀⠀⠀⠀⡆⠀⠀⠀⢀
⠸⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⣀⣭⣭⣙⣻⡆⠀⢆⠀⠀⠀⠀⠀⢀⡄⠀⢸⣟⣋⡭⣭⣀⠀⠀⠀⠀⠀⠇
⠸⡠⠄⠀⠀⠀⠀⠀⢠⠤⠴⠋⢀⣤⣭⣍⣹⣞⠠⠙⠢⡀⢠⠖⠋⠔⣲⣏⣩⢥⣤⡀⠙⠦⢤⡀⠀⠐⠀⢀⠀⠀⢸
⠀⡏⡒⠖⠒⠀⠄⠀⠾⠀⠀⢀⡿⠀⠀⠀⣤⣬⣧⣅⣀⣸⣅⣀⣬⣾⣥⣤⡀⠀⠀⢷⡀⠀⠈⠇⠀⠄⢠⣐⡈⠽⡆
⠀⠸⠂⠂⠀⠀⠐⠀⠀⠀⠀⠯⣤⡀⠀⢲⡂⡀⡀⠉⠉⠉⠉⠉⠉⣀⢀⢸⡆⠀⢀⣤⠽⠀⠀⠀⡈⠀⠀⠁⠈⢡⠁
⠀⠀⢣⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⢀⢀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠑⣄⢀⣠⡉⢉⠄⠣⢀⠀⠀⠀⠀⠣⠀⠒⠂⠀⠂⠃⠘⠀⠂⠀⠀⠀⠀⠠⠂⠔⢄⠐⠢⠀⡔⠁
⠀⠀⠀⠀⠀⠈⠣⡀⡍⠉⠀⠀⠀⠐⠒⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣆⠀⠀⠀⠑⡲⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⢢⠄⡀⠈⠉⠁⢲⣶⣾⣿⢻⣟⣻⣿⣿⣿⣦⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⢀⡆⡰⠀⠀⠀⠀⢸⣿⣿⡿⠿⠟⢿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠤⠤⠤⠄⠼⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠐⠒⠒⠒⠒⠒⠒⠀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠱⡎⣦⠠⠀⠀⠀⠀⠀⠀⢀⡨⠲⠈⠑⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⢀⠎⠈⡀⠀⠄⣀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⡠⠊⢠⠔⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⢀⠀⠀⠀⠀⢀⣝⢄
⠀⠀⠀⠀⢠⠎⠀⠘⠁⠔⢁⠀⠀⠀⠀⠀⠧⢠⠀⠎⠘⠙⠭⠸⠸⠩⠸⠀⠀⠀⠀⠀⠢⠀⠠⡣⢪⠋⠑⣄
⠀⠀⠀⡰⠁⠀⠀⠀⢀⣐⣁⣀⠀⡠⠐⠂⠒⠲⢤⣀⠀⣀⣀⡀⠀⣠⠔⠀⠀⠒⠠⣀⣀⠈⣤⡐⡅⠀⠀⠈⢆
⠀⠀⡰⠁⠀⠀⣖⠈⠁⠀⠀⢀⣀⣙⡳⢦⣀⠀⠀⣹⡶⠩⠉⣩⢽⡁⠀⢀⣀⠶⣟⣁⣀⡀⠀⣂⠁⢢⠀⠀⠀⢇
⠀⢠⠁⡀⠀⠀⠘⣆⠐⠒⢿⡉⠉⠁⠈⣠⣿⣫⣥⡞⠀⠀⠀⠀⠸⣦⣭⣻⣧⡈⠠⠈⠉⣹⠟⠋⣠⠇⠀⠀⠀⠈⡆
⠀⡆⠀⠸⢅⠁⠄⠀⠑⢄⠀⠈⠒⣬⢤⣽⠟⠏⢏⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣥⣌⣤⠞⠁⢀⡴⠁⠀⠀⢃⢉⣱⢸
⢀⠁⠀⠀⠀⢀⠀⠀⠀⠈⠻⣆⣸⠈⣿⠉⠀⠠⡄⠙⠿⣦⣀⣾⠿⢏⠠⠱⡜⠻⠉⠞⣆⣠⠊⠀⠀⠀⠀⠈⠈
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⢸⠀⡠⠊⢀⣠⠚⠉⠹⡅⠈⢂⠘⢶⡀⠀⢠⠠⠹⠁⠀⠀⠀⠀⢃
⢸⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣀⡀⢠⣸⡎⠀⠀⣯⣧⢀⡀⠀⡇⢀⣾⡆⣎⠻⣦⡾⣧⣰⠀⠀⠀⠀⠀⡄
⠈⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⡾⠋⠁⠀⠠⡟⢿⣦⠉⠀⠹⣾⡿⠁⡾⠀⠙⠹⣷⣿⠀⠀⠀⠀⠀⠃⠀⡀
⠀⠏⣁⠖⠒⠀⠂⠀⠀⠀⠀⢘⣟⡀⠀⠀⠀⡅⣈⢨⡟⠀⠀⢀⢻⠰⠀⢠⠀⠀⢦⢬⣿⠀⠀⠀⠀⠰⠠⣔⣒⢬⣽
⠀⠘⡗⠂⠀⠀⠈⠆⠀⠀⠀⢾⣿⠀⠀⠀⠰⡌⠀⡜⠀⠈⣦⢀⢼⣆⠐⢋⡴⠀⠊⣻⠇⠀⠀⠀⢠⠁⠀⠈⠈⢁⠇
⠀⠀⠱⡀⠀⠀⠀⠘⠂⠀⠀⠈⢻⡧⠼⠦⠖⠻⢲⣿⠀⠀⢰⠀⢀⣿⠦⠚⠒⢐⣾⠁⠀⠀⠀⡰⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠱⡄⠀⠀⠀⠀⠑⠄⠀⢀⣿⣦⣛⣛⡛⠭⢿⣆⠀⢸⣤⠾⠿⠿⠛⠩⣾⣿⣧⡀⢀⠌⡀⡀⠀⠀⢀⠎
⠀⠀⠀⠀⠈⢆⣀⡠⡏⠈⡇⠑⠻⣿⡿⣂⠀⠀⠀⠠⠛⣆⠈⣇⠀⠀⠀⣐⡾⢟⡟⠿⠋⠁⠢⡐⠓⠄⡠⠃
⠀⠀⠀⠀⠀⠀⠑⢄⡸⠉⠀⠀⠀⢀⣷⣬⣙⣋⠁⡇⠀⢯⠆⢘⠀⠀⣋⠩⠄⠊⠁⠀⠀⠀⠀⠈⡲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⢀⣾⣿⣿⡯⣻⢻⣿⣷⣾⣶⠉⠀⠀⠐⠂⠐⠂⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠻⢿⣿⣿⡳⡿⣾⣿⣿⣿⣿⠀⠀⠀⠠⠤⠴⢈⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠚⠿⠿⠿⠿⠤⠤⠐⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⣠⠤⠖⠶⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⠶⠔⡀
⠀⢾⣤⠀⡠⢒⠠⢹⣆⠀⠀⠀⠀⠀⠀⢨⣠⠈⠚⢽⣍⣀⢢⣈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢠⣆⢈⣑⡣⢀⢀⣬⣿⠖⠀⠀⠀⠀⠀⠀⡌⠘⣄⣤⢠⢹⠁⣛⣼⡄⠀⠀⠀⠀⢀⡶⠆⠀⢀⡀
⠈⠛⠁⠀⢀⢴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠺⣡⠇⣈⡿⡠⡿⠀⠀⠀⠀⠀⠀⠻⠶⠶⠛⢀⠀⠄⠴
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⢴⣌⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠘⠷⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣶⢶⠶⣄
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠆⠒⠛⠒⠳⢤⣄⠄⠀⠀⠀⠀⠀⠀⠮⠍⣘⢁⣀⣃⡉⡽
⠀⠂⠀⠀⠐⣗⠀⠀⠪⠁⠘⠀⠀⠀⠀⠑⢀⠐⠂⠀⢀⠀⠋⢹⣥⡀⠀⠀⠀⠀⠀⡆⣎⡉⠈⡁⠌⣱⡐
⠀⠀⠀⠀⠀⠙⢦⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠫⠁⠀⠒⡼⠟⠃⠀⠀⠀⠀⠀⠉⠱⣛⠖⠓⢞⠯⠉
⠀⠀⠀⠀⠀⠐⠁⠈⠉⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⡀
⠀⡐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⡐⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⡷⠃⠀⠀⠰⠹
⠀⢳⢿⠂⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠖⠐⢒⠳⣄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡂⣠⢃⣟⣀⢒
⠀⠀⠁⡠⠂⡉⢉⣼⡒⢭⠲⢤⡀⠀⠀⠀⣄⠐⠀⢀⠀⢙⠁⢀⠀⣀⠨⠃⠀⠀⠀⠀⠀⢒⡾⣄⠤⣹⢾
⠀⠀⠀⠉⠃⣀⡘⠀⠶⢐⡻⠶⢦⠀⠀⠀⠹⢶⣀⣌⠓⠦⣜⣛⣛⣌⠁⠀⠀⠀⠀⠀⠀⠘⠁⢛⠑⡄⠈
⠀⠀⠾⠅⠈⠀⡤⠒⠋⠉⢨⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠌
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⡁⠀⠀⠀⡃⠼⠃⠀⠀⠀⠀⠀⢠⠶⣶⢖⢦⣀
⠀⠀⠀⠀⠀⠃⣡⡔⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠸⣏⣃⢂⣋⡽⠂⡂⠀⠀⠀⠀⠀⢸⡟⠀⠳⣼⣿⡀
⠀⠀⠀⠀⠘⢻⣴⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢬⠞⠋⣳⣴⣋⠙⠪⠅⠀⠀⠀⠀⠀⢹⡿⠦⠞⢛⠙
⠀⠀⠀⠀⠀⠘⡤⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⣲⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠦⡤⣾
⠀⠀⠀⠀⠀⠀⠐⠈⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠼⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⡁⠚
⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣈⣠⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠁⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣶⣶⣶⣤⡀⠀⠙⠢⣄⠀⠀⠀⠀⢀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣛⠻⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⠓⢦⣤⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢃⣻⡿⢯⣶⣶⣮⣡⣤⡀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣣⡏⣴⣶⢟⡿⢫⠗⣴⣖⢤⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣘⠋⠀⠈⢹⣿⢏⣼⣇⠹⣦⣛⣛⣛⣛⣻⣿⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡟⡞⣭⣍⣷⣶⣾⣿⢟⣿⣯⣋⣋⣛⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣛⡃⣀⠀⠀⣿⠃⣷⣬⣭⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠙⠀⣤⣛⢰⣤⠍⠺⣿⡿⣱⣿⣿⣿⣿⣿⣮⡹⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡎⣇⠸⠷⢟⡼⣻⣿⣿⢹⡿⣙⢛⣋⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠈⠓⠾⣛⣵⡿⣻⢯⢈⠳⣝⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡚⡛⣂⠀⠀⠀⠾⣶⣾⠇⡇⡏⡾⣰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠋⠛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⣥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⣯⢷⣶⣦

⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣀⡲⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣴⣿⣿⣮⢻⣇⠀⠀⠀⣀⣴⣾⣿⣿⣶⣦⡀⠀⠀⠀⠀⣠⣶⠶⢶⣦⡀⠀⠀⣠⣶⠾⢶⣦
⠀⠀⠀⠀⣴⠓⢀⣠⣾⣿⣿⣿⣿⡧⠟⠀⠀⢰⣿⣿⡿⣿⡛⠛⢻⣿⣇⣀⡀⠀⣿⠁⠀⣨⢻⣿⣄⣼⣿⢫⡀⠀⢹
⠀⠀⠀⠀⢿⣶⣿⣿⣿⡿⠋⣿⠉⠁⠀⠀⠀⢸⡟⠉⠀⠀⠀⠀⠀⢹⣿⠛⠁⠀⠙⠛⠞⢣⣾⣿⣿⣿⣿⣮⠛⠺⠛
⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⣠⣤⣄⠀⠀⠠⣴⣿⣧⡄⠀⠀⣀⡀⣠⣾⣧⠀⠀⠀⠀⠀⠀⢸⣟⣻⣿⣿⣋⡿
⠀⣾⢯⣿⣿⣿⣿⢻⣿⣦⣱⣼⡿⠀⠀⠀⠀⠨⢿⣿⣶⣶⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡟⠁
⠀⠙⠿⠟⠉⢿⣭⣿⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡉⢀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢠⠂⢐⢀⣴⣿⣿⣿⣷⣄⢰⠀⢢⠀⢀⣀⣻⣶⣼⣾⣷⣿⣾⣾⣧⣾⣇⣀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠘⢾⣿⡾⣿⣿⣿⣿⣿⣿⣿⣽⠞⠀⠀⣙⣿⣿⣯⣿⣿⣿⣿⣯⣿⣿⣟⡉⠀⠘⠿⣿⣿⣶⣿⣿⣿⣷⣿⣿⡿⠋
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣉⣿⣿⣿⡿⣿⣿⡿⢿⣿⣿⣏⣁⠀⠀⣴⡾⣿⣿⣿⣿⣿⣿⣿⡿⣦⡄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣹⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣉⠀⠘⠛⠀⢿⣛⣿⣿⣿⣟⣻⠇⠙⠛
⠀⠀⠠⠀⢸⣿⠿⣿⡿⢿⣿⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇
⠀⠈⠑⠒⠛⡏⠁⡽⡇⠉⠟⠓⠒⠉⠀⠀⠀⠀⠛⠛⠛⠙⠛⠉⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠙⡿⠿⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁
⠀⢀⣴⠀⠀⠀⠀⡀⢀⠀⠀⠀⠐⣤⠀⠀⠀⠀⠀⠀⢀⣠⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣴⣦⡀
⠀⠸⣿⢠⡇⠀⠠⠀⠀⠆⠀⢺⡀⣿⠇⠀⠀⣀⡴⠚⠉⠁⡇⠈⠙⠳⢤⡀⠀⠀⠀⣼⡿⠋⠁⠈⢹⣶
⠀⠀⠘⣇⠀⠀⣶⣦⣴⣶⠀⠀⣺⠂⠀⠀⠐⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠀⠀⢸⣿⡗⠀⠀⠀⠰⠋
⠀⠀⠀⢻⣤⣴⣿⣿⣿⣷⣄⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⠀⠀⠀⠈⣾⣿⣷⣦⣤⣄⣀⣀⡀
⠀⠀⠀⠀⡨⠝⡿⣿⣿⢟⠫⢅⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⣿⠀⠀⢠⣤⣤⡄⠀⠀⣸⠿⣿⣿⣿⡿⠉⠉⠍⢳
⠀⠀⠀⠊⠐⢈⠔⠈⠑⠠⠑⠄⠑⠀⠀⠀⠈⠉⠁⠀⢀⣼⣿⣦⠀⠀⠉⠉⠀⠀⠀⠁⠘⠁⠟⠉⢷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠾⠃
⠀⠀⠀⠀⠀⢀⡤⡶⢲⣶⣤⡀⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⢀⡰⠂⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⢰⣿⡷⠀⠀⠰⣿⣿⣦⡀⠀⠀⠀⠈⣷⣦⣀⣀⣀⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⡄⠰⡾⠁
⠀⠀⠀⠀⣿⣿⣷⣧⣼⡶⠿⣿⣿⣷⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣶⣄
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣤⣿⠋⢻⠀⠀⣰⣿⢼⣿⢻⣿⡿⣻⡧⣽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⣀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⠋⠀⢹⣿⣿⣿⡏⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢸⡁⠈⠘⠟⠛⢻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠙⠶⢶⣾⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡶⠿⣿⣿⠛⢻⣿⡿⠷⢶⣶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠃⢻⡇⠀⠀⠰⠀⣄⠐⢡⠄⡁⠀⠀⠀⣾⠇⠙⠻⢟⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⡟⠃⠀⠀⠀⠈⣿⠀⠀⠀⠉⠉⠀⡌⠉⠀⠀⠀⢰⡟⠀⠀⠀⠸⡹⡈⠛⢟⡶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠙⠠⡑⠇⠈⡱⠀⠀⠀⠸⣄⣀⣀⣀⣀⣶⣇⣄⣀⣀⣀⣼⠁⠀⠀⢠⢊⠤⠈⠠⠠⠋⠛⢟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⡋⠀⠀⠀⠘⠤⠘⠁⢂⣶⢤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣜⣌⣒⡁⠂⠀⠀⠀⠀⣡⣿⢶⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠃⠙⢿⣆⠀⠀⠀⢀⡾⠖⠋⠀⠀⢀⠁⠀⠀⠀⠓⠀⠀⠠⡤⠀⠛⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⢀⣴⠟⠁⠙⢝⢦⡀
⠀⠀⠀⠀⢠⣾⣏⡀⠀⠀⠀⠉⠳⣤⡴⠋⠀⠀⣀⡀⠀⠀⠄⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠂⠀⠙⢦⣴⠋⠁⠀⠀⠀⡨⢿⣷⡄
⠀⠀⠀⣰⡿⢋⠃⡎⠡⢠⡄⢀⡴⠋⢀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠘⢦⣀⣄⡀⠔⣃⡐⠙⣽⣄
⠀⠀⣰⣿⠡⠊⡌⢱⢁⠔⢠⡞⠁⠀⠈⠁⠀⠀⣄⠠⠆⠀⠀⠒⠀⠀⠀⢨⡄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠐⠗⠀⠀⢀⠀⠈⠻⡦⡀⠎⠂⠡⢌⠾⡽⡄
⠀⢠⣿⠃⠀⠲⠀⣅⠎⢠⡏⠀⠀⠀⠁⠂⠯⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣤⠀⠀⠀⠀⠘⠃⠀⠀⠘⣞⣐⠁⠀⠀⠀⠱⣻⡄
⠀⣾⣋⠀⠀⠀⠀⠈⢡⡟⠀⠀⠒⠀⠀⠖⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠤⢀⠀⠀⢠⡄⠄⠀⠀⠀⠀⠘⠁⠀⠀⣴⠢⠀⢺⡄⠀⠀⠀⢀⣀⣥⣷
⢸⣯⠛⠛⠷⠶⠤⢤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡄⢔⠘⠟⠙⣧⠀⠀⠙⢦⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢻⠶⠒⠛⠉⠉⠉⣾
⣼⡏⠀⠀⠀⠀⠀⢸⠇⠀⠰⠂⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠠⢁⠀⢹⠖⠀⢠⡇⠀⠀⠀⢣⠀⠉⠀⠀⠀⠀⠐⠂⠀⠐⠀⠀⠛⠀⠘⡇⠀⠀⠀⠀⠀⢿
⣿⠇⠀⢀⠤⢄⠀⢸⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢀⠀⠀⠘⠷⠡⠀⠀⣚⠀⠘⠃⡇⠙⠀⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⣧⡌⢹⠩⡌⠢⣸
⣿⣤⠒⣘⡈⠄⣂⣾⠀⠀⠀⠃⠀⠀⠀⢠⠀⠀⠘⠁⠀⠀⢐⠇⠂⠀⠙⡄⠀⠀⢿⠀⠀⠀⢰⠀⠰⠷⠀⠀⠀⠀⠀⠀⠙⠁⠀⠙⠃⠀⣿⡄⠤⠵⠘⠆⣿
⣿⡾⠔⠀⠁⠀⠈⢹⠀⠠⠄⠀⠚⠀⠀⠄⡀⠀⠀⢠⣄⠀⠀⠁⣐⠢⡄⠻⣲⠀⡜⠀⠀⡰⠁⠀⠀⠀⢀⢄⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⡇⠐⢌⡃⠼⠠⠋
⢿⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠙⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠓⠀⠀⠙⠊⠀⠀⠀⠀⢀⠀⠀⡀⠀⢠⡇⠀⠀⠀⠀⠀⣾
⢸⣷⢀⣀⣤⣤⡶⠾⣷⠀⡈⠀⠀⠀⠻⠞⠀⣀⠀⠂⠀⠀⠀⡀⠴⠆⡄⠀⢠⠀⠀⠀⠲⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠰⠀⠀⠀⣤⠀⣼⠓⠶⠦⣤⣤⣀⣳
⠀⣿⣛⠋⠉⠀⠀⠀⠘⣧⠁⠀⠠⠐⠁⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠀⠀⠀⣠⡀⠀⠀⠔⠷⠂⠀⠀⠀⠀⠀⣠⠆⢠⡄⠀⣼⠇⠀⠀⠀⠀⠈⣿⣿
⠀⠘⣿⡄⢀⠔⣱⠀⡤⠘⣆⠀⠂⠄⠀⠰⠷⠀⠈⠀⠻⠃⠈⣀⠀⢠⠀⠀⠠⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⠘⠟⠀⠀⠀⢸⠏⡴⣒⣦⠀⠀⣰⣿⠇
⠀⠀⠹⣿⡰⢀⡄⢌⢀⡓⡘⢧⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠊⠉⠀⠀⠀⠀⣴⠀⢀⡾⠟⣉⠳⣭⢌⠳⢠⢻⠏
⠀⠀⠀⠹⣿⡍⠀⠈⠂⠀⠀⠈⠳⣄⠀⠸⠗⠀⠀⠀⠀⠀⡰⡄⠀⠀⠀⠀⠀⠀⠀⠠⠷⠂⠀⠀⠁⠀⠀⠘⠗⠀⠀⠀⣠⠟⠁⠈⠁⠑⠢⠉⣼⣫⠏
⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠀⢀⣠⠞⠳⣄⡀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⡀⠀⠈⢀⠀⠀⠀⢀⣠⠞⠻⣦⡀⠀⠀⠀⠀⡴⡵⠃
⠀⠀⠀⠀⠀⠈⢻⣷⡄⢀⣴⡾⠃⠀⠀⣨⣽⣶⣄⡀⠈⠰⠀⠀⠀⠀⠛⠀⠀⡀⠀⠀⠀⠘⠓⠀⠀⠈⢁⣠⡶⠋⠁⠀⠀⠈⠻⣦⡄⢠⣮⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣋⠀⠀⠀⠀⠀⡘⢠⠀⠉⠛⠶⢤⣄⣀⢀⠀⠀⠈⠁⠀⠀⠀⣀⣀⡤⠶⠛⢩⣏⠣⡔⣎⠂⠀⠀⠀⣨⢻⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠰⢠⠏⡞⡙⠀⠀⢀⡏⠉⠉⢩⠙⠿⠛⠫⢭⠉⠁⢹⡄⠀⠀⠀⠍⢧⡙⣎⢧⣱⡤⣪⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣖⣶⣯⣸⢱⠁⠀⠀⣼⠁⠀⠀⢸⡉⠻⢸⠋⡻⠀⠀⠘⣷⠀⠀⠀⠈⠐⣉⡾⣻⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣤⡄⢰⡟⠀⠀⠀⠀⠣⠃⠔⡡⠁⠀⠀⠀⢹⣧⢠⢤⣶⡭⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⠿⠶⣶⣶⣶⣬⣿⣰⣶⣶⠾⠼⠾⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠰⠒⠒⠒⠒⠒⠒⠒⠐⠆⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠑⣄⠱⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠶⡈⠑⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠉⠀⡷⠀⠁⠀⠀⠀⠀⠀⠀⠀⡽⠏⠀⠁⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⣀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠒⠀⠀⠈⠀⠐⠀⠠⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠞⠁⡤⠚⢀⡤⠀⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢲⣶⣶⣾⣷⡄⠀⠈⠀⠢⠀⠀⠀⠀⢀⠴⣢⡱⣄
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠔⠊⠠⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣉⣺⣿⠋⠀⠐⠶⡔⠒⠤⡀⠀⠀⠁⠀⠀⠋⠞⡵⠁⠈⢢⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣱⡏⡶⣻⠀⠀⣠⣦⣴⣶⣿⠁⠀⠀⠀⠀⠀⢀⠪⡀⠀⠀⠀⠱⡄
⠀⠀⢀⠎⠀⠀⠀⠀⢀⠐⠀⠀⠀⡰⠀⠀⡠⠄⡀⣄⠀⢀⣯⠀⢹⣿⠾⣷⣄⠀⢰⡿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠘⡄
⠀⠀⡜⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠣⠀⠐⠇⠘⠆⠉⠀⢸⣧⠀⡨⢿⣶⣽⣯⣀⡀⠻⠲⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠱
⠀⡸⠀⠑⠶⠄⠄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣦⣍⣈⣹⡛⠻⣿⣶⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⢠⠤⣄⢣
⠀⠇⠈⠈⠉⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠈⠙⠿⢿⣯⣟⣿⣍⣺⣿⣌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⡄⡦⠊⠘
⢸⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠈⢧⠀⠀⠲⢍⡉⢻⣿⣿⣿⣟⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⢸⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⡀⠀⡀⠀⠀⠀⠑⠀⢈⣻⣿⡿⢹⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠠⢝⠀⠀⠀⠀⠀⢡⡿⡿⣡⡿⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠸
⢸⡄⣀⠀⠀⠀⠘⠀⣀⣤⡶⢶⢖⠀⠀⡰⠀⠐⢄⢀⣀⣀⣀⣀⠀⠑⣄⠀⠃⠂⠻⢿⡿⠿⣣⣈⣲⣦⠀⠑⠀⠀⠀⠀⠀⡇
⠀⡞⠤⢀⣠⡤⢴⡿⠋⠀⡀⠀⠀⠀⡀⢀⣤⣤⣀⣉⡉⠉⢉⣁⣠⣶⣿⠂⢆⡉⠀⠈⢣⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠐⠀⠀⣁⡒⢦⢸
⠀⠰⡴⠋⠀⢠⣿⠎⠐⠈⠐⠀⠀⢈⡵⢿⢿⠿⠛⠛⠛⠛⠟⠉⠉⠀⠀⠙⢦⡉⠁⡀⠀⠣⡀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠁⠈⠳⡐⣮⠝⡝
⠀⠀⢣⠁⠀⠘⣿⣄⣠⣶⣶⣶⣾⡏⠀⠙⢌⡀⠀⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠈⡄⠀⠳⢦⣤⠇⠀⠀⣠⡀⠀⠀⠀⠀⡸
⠀⠀⠈⢆⠀⠀⡃⢍⢉⠻⣆⠉⠿⠁⠀⠀⠀⠙⢦⣀⡐⢈⣻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣦⣄⣠⠃
⠀⠀⠀⠈⢦⠈⠑⠛⠚⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣄⡀⠀⠈⣉⣰⣾⡻⡏⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠐⠉⠑⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢚⣻⣿⣧⣶⡝⣿⡶⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠘⢮⡉⠉⡆⠚⠀⠀⠈⠀⢢⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠐⣿⣿⣿⣿⣿⣿⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⠀⢀⣀⣀⡀⠀⠤⠔⠂⠉⠈⠀⠀⠀⠈⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⢸⠁⡒⠂⠀⠀⠀⠀⠀⠀⠀⠐⢂⠈⠁⡂⠀⠀⠀⢈⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠠⠎⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⢀⡠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠐⠒⠠⠤⠤⠤⠤⠤⠤⠤⠒⠒⠚⠉⠁
⡿⠿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿

⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⠀⠀⠀⢤⣤⣄
⠀⠀⢰⣿⡿⣿⣿⡿⣆⣀⣠⣿⣾⣿⢿⣿⡄⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠿⠷⠂⠀⠀⠀⠒⠿⢿⠇
⠀⠀⢸⡏⢀⠙⠛⣿⣿⣿⣿⣿⠛⢋⠀⣹⡇⠀⠀⠀⠀⢛⣛⣾⣿⣿⣿⣷⣿⡛⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣦⣀⣠⣤⣤⣤⣄⣀⡴⠁
⠀⠀⠈⠛⠚⠁⠀⣿⣻⣿⣿⣻⠀⠙⠚⠋⠀⠀⠀⠀⠘⠛⠋⣿⣿⣿⣿⣿⡍⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠬⢿⣿⣿⣿⣿⣿⠯⡥⠚
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠁⢘⡯⢻⢿⣿⠿⣟⢭⡃⠈
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⢺⣿⣿⣿⣿⣷⣤⣤⣄⠀⠈⠀⠻⣄⠀⣠⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣿⣿⣿⣿⣿⡿⠁⠀⠀⣾⠏⡀⠀⠉⣉⡛
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣸⣿⢿⣿⢿⣿⣧⢠⣦⣤⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⢈⣁⢀⣀⡠⠖⠁
⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣞⣻⡷⠂⠀⠀⠀⠀⠹⠟⠋⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⣄⣀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠹⣿⣿⣿⡿⣿⠏⠀⠀⠀⢲⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠋⢺⠙⠁⠀⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠘⠋⠏⠛⠿⠿⠿⠿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡁⠀⠀⠀⢩⠀⠀⠀⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢴⡷⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠻⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠺⠛⣳⣿⣿⣿⣿⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⣟⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⢶⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠇⠀⠀⠀⠀⢀⣤⣶⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢰⢻⣿⣿⡃⠀⢤⡄⠀⢀⣿⣧⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⣷⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠈⠈⣿⡟⠀⠀⠙⠉⠀⣾⢿⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠙⠿⠿⠿⠛⠉⠉⠉⣹⠇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢻⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠉⠉⠛⠂⠀⠀⠀⠄⠀⠀⣤⣶⣶⣶⠖⠁⠀⠀⠀⠀⠀⠘⣶⡶⠶⠶⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⢠⣶⠀⣸⣿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠈⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠛⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣷⢀⣤⡀⢠⣄⢀⡁⣆⢀⡇⣠⣀⢠⣄⢈⢡⠀⡀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠑⢤⣨⠤⣷⣏⢨⢸⡏⡏⠹⣯⠥⢸⢸⠀⢸⢸⡀⣇⡱⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠥⣈⣹⠀⠁⠁⠀⠁⠉⠉⠀⠀⠈⠀⠉⠁⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣦⣀⡁
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣟⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⠛⠋⠀⠀⠀⠠⠄⣉⠉⠀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⢿⣿⣆⣀⠁⠀⠀⠀⠸⠿⣣⡾⠋⢹⠄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠟⠀⠀⠀⠀⠀⣴⠟⣁⠀⠻⢄⡉⠀⠀⠀⠀⠀⠀⠻⢿⡆
⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⣻⣿⢀⠀⠀⢹⠄⣀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠁⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⡿⠃⠀⠀⠘⠋⠸⣿⠂⣾⠟
⠀⠀⠀⠀⠀⠁⠀⢻⣇⠘⠿⡿⢇⣄⠐⠃⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⣀⠀⠙⠀⠁⢰⣿⣴⠆
⠀⠀⠀⠀⠀⠸⢿⣦⡀⠀⠀⠀⠸⠃⢴⡄⣠⣤⣴⣶⣶⣶⣶⣶⣷⠀⢸⣦⣧⣈⠀⠀⠀⠀⢈⣙⡋
⠀⠀⠀⠀⠀⠀⢠⣠⣀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⠿⠿⠟⠿⡟⢛⣣⡀⠻⣿⣿⣶⠀⠀⠀⢙⣉⠁⠴
⠀⠀⠀⠀⠀⠀⢺⣿⡯⠀⠀⠄⠀⠠⣆⠁⢸⠀⠀⠀⠀⠀⢤⡙⠈⠛⡵⣢⡙⢿⣿⣇⢀⡠⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠈⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠙⠈⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⢠⡀⡀⠀⢸⣿⣿⣿⣷⡄⠀⠀⠀⠀⠃⠀⠈⢻⣿⣿⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⢀⣿⡟⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣼⡟⢀⣿⡿⣀⠀⠀⠀⠀⣀⣀⡀⣀⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡿⢁⣾⣿⠛⢋⣵⣶⣿⣿⣿⣷⠁⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠁⠀⠀⠀⠀⠉

⠀⠀⠀⡔⣒⢶⠋⠙⡆⠀⠀⠰⣏⣳⡀⣠⠚⣓⡆⠀⠀⠐⠛⢦⡤⢴⠚⠙⠀⠀⠀⢀⡴⣖⠒⠲⣄
⠀⠀⠀⠳⠼⢸⠘⠒⠃⠀⠀⠀⢠⠏⠉⠙⢦⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠘⠒⠋⠀⡨⣟
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢧⣀⣠⠞⠀⠀⠀⠀⠀⣄⡼⠤⢼⣀⣤⠀⠀⠀⠀⠻⢧⣈⣧⠼

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⣄⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡞⠉⠙⡆⠀⠀⠀⠉⢸⠁⢹⢉⡇⠀⠀⠀⠀⠀⢀⡔⠒⢦⠀⠀⠀⠀⠀⠘⢻⡏⢹⢹
⠀⠀⣖⠻⡇⢀⡜⠁⠀⠀⠀⠀⢸⠀⡿⢸⣤⡄⠀⠀⠀⠤⠤⠷⠀⢯⡤⠶⠀⠀⠀⠀⢸⠁⣼⢸⡀
⠀⠀⠉⠉⠁⠘⠶⠂⠀⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠀⠁⠀⠿

⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⣩⠛⡇⠀⠀⠀⠀⠚⣿⠉⢹⠀⠀⠀⠀⠀⢀⡔⠳⠞⠶⠳⠆⠀⠀⠀⠀⠀⢳⠀⢰⠃
⠀⠀⠀⢲⣴⠁⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⣤⣄⠀⠀⠀⠀⡴⢲⠞⣶⢻⠄⠀⠀⠀⠀⠛⡟⠉⢻⠛
⠀⠀⠀⠎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠼⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠈⠧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣿⣄⠀⠀⠀⢠⡄⢠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣡⣴⣾⣿⣧⡙⣇⡟⡀⠀⠀⠀⠙⠘⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣯⣹⣿⣿⡉⠟⠡⡇⠠⢠⡀⠀⡐⢻⣿⡿⢿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⠙⠲⡶⢀⠀⡿⠙⣯⠃⡞⠈⠁⠉⠻⣿⣿⣿⣿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣢⢴⠎⡼⢡⠌⠉⠀⢈⠀⠀⠠⡀⠀⢿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⡀⠀⠀⠈⢦⢾⣀⢴⣄⣐⡤⠀⠁⣀⣠⡤⢤⡾⢿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠛⠀⠀⠀⠀⠉⠓⣈⣭⡽⠛⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠂⣸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⢠⡄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢐⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠁⠀⢴⣿⣿⣿⣿⣿⣿⣿⣽⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⢿⣿⡿⠛⠃⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⡀⠀⢀⣠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠻⣿⣿⡿⢫⣽⡇⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣬⠻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡀⠉⠁⠀⠀⢠⣄⠈⠳⣤⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣦⣶⢿⡿⠃⠀⢠⣼⡿⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⢿⣿⣶⣤⣄⣤⣿⣿⣷⣀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⣀⢀⠀⡀⢀⡀⢀⢀⡀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⠁⡇⢸⠀⡇⠘⠁⣘⢸⡇⣯⣇⠘⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠈⠀⠁⠀⠁⠁⠈⠁⠁⠈⠁⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠤⢽⣋⣛⣛⣛⣙⣋⣛⣋⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣠⣴⣖⣆⣦⣶⣵⣵⣰⣲⣢⡄⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣰⡼⠼⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠳⢏⣖⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣲⠾⠋⠁⠀⠀⠀⠀⡀⣴⡀⠀⠀⢰⡆⠀⢀⢀⢰⣦⠄⠀⠀⠀⠈⠙⠷⣆⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⠾⠋⠀⠀⠐⣄⠀⠀⠀⠍⢅⣼⡇⠀⠀⢀⡄⠀⠀⠉⠙⢿⡄⠀⠀⠀⣠⠃⠀⠀⠙⠷⣓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⠟⠁⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⡜⠁⠳⠀⠀⢠⡄⠀⠀⣠⢼⡆⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠈⠻⣕⣄
⠀⠀⠀⠀⠀⠀⢀⣜⡾⠃⠀⠀⠈⠃⣠⡀⠃⠀⠈⢤⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⡤⠁⢠⣿⣇⣈⡀⠀⠀⠀⠘⢷⣣⡀
⠀⠀⠀⠀⠀⠠⣺⠞⠀⠀⠀⠀⠀⠐⢿⠃⠀⠀⠀⠈⢣⠀⠀⠀⠀⢀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠜⠁⠀⠀⣿⣾⣿⠇⠀⠀⠀⠀⠀⢳⣗⠄
⠀⠀⠀⠀⠠⣰⠏⠀⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⣦⠦⠷⠞⠚⠚⠒⠶⢭⣖⡤⣄⠀⠀⠀⠀⠁⠈⠉⠒⠀⠀⢀⡤⠒⠀⠹⣞⡄
⠀⠀⠀⢀⣳⡏⠀⠀⠀⠀⠀⠐⠂⣄⡀⠀⠀⠀⢀⣔⣯⠟⠋⠀⠀⡒⡄⠀⠀⠰⣦⠀⠀⠙⠳⣯⢤⡀⠀⠀⠀⢀⣠⠐⠂⠁⠀⠀⠀⠀⢹⣖⡀
⠀⠀⠀⣜⡾⠀⠀⢀⠀⣰⣦⣀⠀⠀⠉⠒⠀⡰⣾⠞⠁⡤⠀⠀⠀⢁⡠⡠⡄⣀⡁⠀⠀⠤⠠⠈⠻⡼⣀⠀⠒⠉⠀⠀⡤⠀⠠⢄⠀⠀⠀⢷⣣
⠀⠀⢀⢾⠇⠀⠀⢻⡿⣿⣿⠉⠀⠀⠀⠀⠰⣱⠋⠀⠀⠀⢀⡴⣮⠶⠛⢋⡙⠓⠫⣕⡦⡀⠀⠀⠀⠙⣷⣆⠀⠀⠀⠀⠻⠿⠿⠿⠀⠀⠀⠸⣾⡀
⠀⠀⢸⢽⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⣱⡏⠠⢀⠀⢠⣺⡞⣁⣸⣷⣾⠋⠚⢇⣈⢻⡵⡀⠀⣴⠆⢸⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠅
⠀⠀⠨⣿⠀⠀⢀⣀⠀⣀⠀⣀⡀⠀⠀⢘⢺⠁⠀⠀⠀⡐⡿⢀⣽⣿⣛⣧⠘⠃⡈⢋⡀⢿⠅⠀⠀⠀⠐⡯⠇⠀⠀⢀⣀⠀⣀⠀⣀⡀⠀⠀⣯⠁
⠀⠀⢐⣿⠀⠀⠈⠉⠀⠉⠀⠉⠉⠀⠀⢠⢼⡄⠀⡀⠀⠱⣷⠈⣹⣿⣿⡇⠀⠀⠁⣜⠁⣾⡢⢀⠀⠀⠀⣗⡆⠀⠀⠈⢩⡀⢉⡀⠉⠁⠀⠀⣿⡀
⠀⠀⢸⣺⠀⠀⠀⢸⣿⣶⡀⠀⠀⠀⠀⠈⡭⣇⠘⠀⠀⠘⢽⣧⡉⢹⡿⣿⡦⣴⡎⢉⣼⡳⠃⠀⠀⠁⣸⢳⠀⠀⠀⠐⢯⡅⣸⡿⠀⠀⠀⠀⣗⡇
⠀⠀⠈⠿⡆⠀⠀⠠⢼⣿⣿⣾⠃⠀⠀⠀⠰⡹⣄⠀⠀⢀⠈⠻⢟⠶⣤⣌⣡⡤⠔⡫⠗⢁⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⠸⢛⡿⢇⠀⠀⠀⢰⣷⠃
⠀⠀⠀⢫⢷⠀⠀⠈⠉⠁⠘⠛⠀⠀⣀⠄⠀⠐⡽⢦⡐⠂⠀⠀⠀⠈⠑⠑⠓⠉⡁⠀⠀⠀⠀⢀⣴⠫⠏⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡝
⠀⠀⠀⠈⠽⣇⠀⠀⠀⠀⠀⠠⠄⠋⠁⠀⠀⠀⠀⠫⡟⣦⣄⠀⠀⠘⠀⠀⠀⠰⠒⠀⠀⣠⡴⡿⠇⠁⠀⠀⠀⠈⠙⠠⠄⠀⠀⠀⠀⠀⣸⠯⠁
⠀⠀⠀⠀⠘⡹⣆⠀⠤⠚⠁⠀⠀⢰⣾⣦⠀⠀⠀⠀⠀⠑⠫⠟⢖⡶⢤⢤⢤⠤⠶⣒⠯⠗⠋⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠈⠓⠤⠀⣰⠯⠃
⠀⠀⠀⠀⠀⠐⠽⢦⠀⠀⠀⠀⠀⢠⠾⡇⠀⠀⠀⢀⡔⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⢢⡀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⡴⡯⠂
⠀⠀⠀⠀⠀⠀⠈⢯⢷⡄⠀⠀⠠⠋⠀⠘⠀⠀⢀⠚⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠀⣀⣄⠠⠀⠀⠃⡀⠀⠠⠇⠚⠁⠀⠀⠀⢠⡾⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢜⣦⡀⠀⠁⠀⠀⠀⢀⠌⠀⢰⣿⡁⠶⣶⣤⠀⠘⠃⠠⣾⣿⣿⣿⣦⠀⠀⠡⡀⠀⠀⠀⠀⠀⢀⣴⡣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢭⢶⣄⠀⠀⢠⠋⠀⠀⠈⠙⢻⡤⠼⠋⠀⠘⠃⠀⠈⠻⠉⢉⠃⠀⠀⠀⠙⡄⠀⠀⣠⡶⡭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠹⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⡏⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠽⢳⢶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⡴⡎⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠚⠚⠩⠯⠯⠟⠟⠻⠻⠽⠹⠭⠓⠓⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣶⣶⣤⣿⣧⣿⣧⣴⣶⣦⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⠿⣿⡟⢯⣿⣿⢈⣏⢸⣿⢩⣿⠛⡿⢿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣿⣿⡘⣟⠻⣷⠸⠿⣿⣸⣿⣸⡏⣸⡏⣼⢡⣿⠋⣽⣿⣿⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣧⠘⢋⣿⣿⣿⣷⣘⣧⣿⠾⠛⢛⣛⡋⣛⣛⣛⣛⢷⣾⣃⣾⣿⣿⣿⣏⠻⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡀⣨⢟⢥⢿⠞⣩⣶⣟⣉⣗⣃⡀⠀⡱⠉⠀⠀⡟⠙⢳⡏⣽⣷⣮⡑⢾⢧⣝⢯⡀⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⡿⣵⢍⢷⡇⣠⢓⣷⣿⡿⣱⡮⣳⡀⠃⠁⢀⡀⢀⠈⠁⢹⣿⣷⣒⢣⠀⣷⢧⢵⡽⡷⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⢇⢁⠠⣎⢡⣽⣵⣀⣇⠦⠟⢃⣸⡀⡀⠀⢸⡧⠀⠀⠀⠈⣻⣇⣔⣽⣤⢩⡦⠀⢁⣧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⣼⣼⣒⣣⡟⣿⣾⣿⣿⡇⣧⡛⣻⡿⠿⠓⠛⠣⡷⠀⠀⠀⠰⡟⢟⣿⣟⣾⡟⣇⣓⣺⣽⣤⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⡀⠘⣛⡄⣿⡿⠘⠁⢠⣼⢀⣉⠽⠿⠆⠀⠀⠀⠀⠁⠀⢸⠳⢾⣷⣸⣥⠀⠙⠸⣿⡇⢙⡙⠀⡀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⣴⢳⣰⢿⣆⠀⣀⢀⡿⡕⣿⠟⠀⠀⠀⠸⡆⠀⠀⣦⠈⠘⡀⢙⡿⣇⢀⡀⢀⣾⢗⣽⢷⡄⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠿⣣⢋⣿⢳⢭⡛⢻⣬⣽⣷⣿⡆⠀⠀⠀⢸⠇⠀⠋⣾⢃⠀⢹⡿⣿⣿⣼⠻⣻⢽⢻⣿⢢⡻⢿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠾⣟⣫⣴⠋⣡⣟⣿⠋⢉⣸⡿⣇⣠⡄⠀⠈⠀⡰⠀⡏⢸⣿⣿⣧⣈⠉⣿⣟⣧⡈⠳⣍⣛⡿⢦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠆⢰⣌⡷⢻⢿⣞⢾⣥⣿⣟⣀⣾⣏⢵⠀⠀⡰⠁⡆⠇⣿⣿⣎⣘⣼⣷⣿⢞⣾⢿⠳⣏⣴⠀⠎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢠⣿⣶⢛⣦⡉⠈⣿⢇⣲⢮⣻⡧⠀⠚⠂⠀⠘⢃⢸⠏⠙⣫⠶⣂⠿⡟⠈⣡⣎⢳⣿⣷⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⢁⠘⠉⠛⢻⣿⢍⣉⢉⣼⣿⢤⣀⣀⠀⠀⠈⢻⡆⢰⡌⢉⣉⢽⣿⠛⠃⠙⠠⢙⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣆⣡⣗⡿⣼⠼⡿⣷⣿⣡⡞⣻⠇⠀⠁⣠⡟⣇⣘⣷⡿⣟⢼⡼⣇⣷⣇⣾⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡏⢿⣷⣿⡗⡙⠋⠉⣿⠟⣷⡏⣰⡟⢾⡏⠉⠙⡓⣿⣷⣿⠏⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⢉⡁⢂⡶⠈⠀⡽⢵⠫⡻⠿⢳⢽⡅⠈⠰⣆⠆⡉⠁⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠜⠳⠆⠀⡙⣺⣼⡆⣾⣾⠊⠁⠀⠷⠟⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠁⠑⠁⠙⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣥⣶⣿⣿⣯⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣽⣿⣿⣶⣬⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠛⠉⢁⡀⠀⣀⠀⠀⠀⠀⢀⠀⢀⠈⠉⠛⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠁⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⢠⠓⠃⠀⠀⠀⠀⠈⠙⢿⣿⣿⣦⡐
⠀⠀⠀⠀⠀⠀⠀⠐⣠⣿⣿⡿⠋⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠁⠁⠀⠀⠀⠀⣀⣀⠀⠙⢷⣟⣿⣄
⠀⠀⠀⠀⠀⠀⠈⣼⣿⣿⠏⠀⠀⠐⠒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠠⢀⠀⠀⠀⢀⣀⠀⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠈⠀⠈⠀⡤⠈⠀⠀⢠⠄⠀⠀⠀⠀⠠⠄⠀⠀⠀⢦⠀⠁⠀⠁⠀⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⡿⠀⠔⠤⠂⠄⠀⠀⠀⠀⣠⠀⠀⠀⣀⠰⣄⣧⣼⣠⠆⣀⠀⠀⠀⢀⠀⠀⠀⠀⠜⡃⠀⠁⠀⢿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⡇⠀⠁⠉⠈⠁⠀⠀⠀⠀⠁⠀⠀⠢⢬⣿⣿⣿⣿⣿⣿⡥⠔⠀⠀⠈⠀⠀⠀⠀⠀⠂⠘⠁⠀⢸⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⢒⣻⣿⣿⣿⣿⣿⣿⣟⡒⠀⠀⠀⠀⠀⠼⠆⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠘⣿⢿⡇⠀⠀⠤⢠⠀⠀⠀⠀⠀⡀⠀⠈⢉⡽⢿⣿⣿⣿⣿⡿⢯⡉⠁⠀⢀⠀⠀⠀⠀⠀⠔⠄⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣯⣷⠀⠀⠀⢀⡀⠀⠀⠀⠀⠑⠀⠀⠀⠰⠋⡼⢹⡏⢧⠙⠆⠀⠀⠀⠊⠀⠀⠀⠀⠾⠠⠁⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠸⣿⢿⡆⠀⠀⠈⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣷⣿⣄⠀⠀⠀⠀⠀⢀⡀⠓⠀⠀⠀⠑⠂⠀⠀⠀⠀⠐⠊⠀⠀⠀⠞⢀⣀⣀⠀⠀⠀⠀⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⢀⡠⠊⠁⠀⠀⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠘⠀⠀⢸⢸⢀⡀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣷⣄⠈⠁⠀⠀⠀⠀⠠⡤⡤⠀⠀⠈⠁⠀⢀⠤⠀⠀⠀⠀⠈⠈⠸⠇⣠⣾⣿⣿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠃⠁⣀⠀⠀⠀⠀⠼⠀⠸⠄⠀⠀⠀⢀⣠⣾⣯⣿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣻⣶⣶⣶⣶⣶⣶⣶⣶⣻⣻⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠿⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠖⠂⠉⠛⠓⠂⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠀⠀⠀⢀⡄⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⢿⣿⣷⣦⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢠⠆⠀⠉⠀⠀⡀⡀⠀⠀⠀⡠⠐⠀⠂⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣴⣿⠃⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠒⠄⠴⠀⠀⠇⠀⢀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌
⠀⠀⠀⠀⠀⠀⠀⠐⢠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⢄⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⡈⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣄⡀⠀⠀⠀⠀⠈⠉⠉⠛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⡇⠀⠈⠀⢀⠀⠀⠀⢣⡀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠻⠀⠢⠀⠀⢄⠀⠦⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⡇⠙⠀⠀⠀⠃⠀⠀⠀⣿⠯⠏⠠⠀⠀⠀⠐⠸⠭⠟⠀⠀⠀⠀⡀⠀⠀⠑⠀⠀⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⠀⠅⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⠀⠀⠀⠀⣴⣾⠟⠻⣿⣟⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⠀⢠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⡆⠀⣿⠿⣿⣿
⠀⠀⠀⠀⠀⠸⡿⢿⣿⡿⠿⣿⣿⡀⠘⢤⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠁⠰⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠋⢀⡀⠀⢻⣷⡀⢸⣆⠈⠣⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣷⠀⣸⣿⡇⠈⠉⢷⠁⡚⠐⢩⡒⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⡀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠂⠛⠀⠘⢿⣿⣿⡿⠃⠀⠀⠀⠂⠀⠀⠈⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠶⢦⠀⠀⡄⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠐⣄⠀⠈⠀⠀⠀⠀⠀⡠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠒⠀⢸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠉⠀⠀⠀⠀⣤⠶⠢⠄⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⢠⣀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠷⠶⠶⠶⠶⠶⠿⠛⠛⠉⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⠀⢀⢀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⡇⣿⢸⣟⣸⡽⢿⡇⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠁⠈⠁⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠸⢢⢀⠀⠞⠀⡰⡄⠂⠀⠀⠀⠀⢐⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣶⡾⠓⣙⣛⡳⢦⣀⡔⠁⣦⡶⠶⠂⠐⠄⠀⠒⠀⣀⠀⠁
⠀⠀⣤⠐⣄⣊⠂⢀⠀⠈⡁⠉⢠⠀⠔⠓⠰⢶⠋⠁⣴⠋⣙⠀⠓⢦⠹⠗⠀⠥⣀⢄⡰⠀⠠⠇⡤⠀⠘⢇
⠀⠀⢨⣽⡏⠁⠀⣈⡁⠀⠐⠀⠘⢠⡀⠁⡠⠃⠀⠆⠀⢙⡅⠰⠃⡜⠟⡀⠃⠀⠀⠀⠿⠻⣖⠄⠢⠀⠀⠀⢈
⠀⢀⢙⣺⡥⠐⠚⢧⣹⡀⠃⢀⠀⢀⠀⠐⡆⠀⢄⡁⠀⢈⡁⠂⠸⠈⠀⡁⠈⢽⠾⣤⣠⠇⢹⢠⠀⠘⢇⡤⠎
⠀⠤⣬⣭⣤⠠⠄⠈⠙⢗⢀⠈⠀⢀⡠⠈⡑⠀⠠⣇⠀⢈⡠⠆⠀⡀⠀⠄⠀⠀⠱⠛⡸⢀⠋⡌⣣⠀⠊⠁
⠀⠀⢙⡞⢋⡀⠀⠦⡀⠴⠀⠘⡃⠊⠉⠑⢇⠀⠀⠘⢦⣤⣀⣤⠤⢬⡗⠠⣬⣆⣠⠿⠯⠁⢈⢁⡀⠐⠀⡈⠁
⠀⠀⠘⠶⣃⣀⣀⣬⡗⠀⣻⡖⠛⠀⢰⡀⠸⣦⣀⡀⢀⣾⠽⢋⣈⣹⣯⣨⡙⠻⣧⣀⢈⣤⡿⠀⠀⢀⡄⠀⠢
⠀⡏⠛⠻⠇⠀⠈⠉⠃⡄⡁⠁⡾⣴⢉⠁⠄⢿⣗⠒⢉⢩⡷⠞⠺⠧⠄⠇⠲⠾⣄⣁⢚⣿⠇⣠⢤⠆⠈⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠀⡔⠂⠈⠸⣱⠀⠀⠙⢛⠒⠂⣾⠂⢄⠈⠁⡀⠐⢻⠖⠂⠛⠁⠀⠈⠘⢢⠀⠐⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠠⡄⠆⠁⣆⢀⠆⠁⠀⠀⠀⠀⠙⠀⠉⠓⠒⠀⠀⠐⠚⠀⠀⠈⠀⠀⠀⡀⣰⠀⢰⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⢀⠀⠜⡁⢀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠠⠀⣆⠠⠆
⠀⠀⠀⠀⠠⡖⢆⠀⠀⠀⠹⠃⢔⠃⠀⢠⡀⠂⠀⠀⠂⠀⢨⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⡘⢢⠘⢏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⡷⠂⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢲⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠰
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠑⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠲⠤⠤⠤⠤⠤⠬⠤⠤⢤⣤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⡄⠀⠀⠀⠀⢀⡀⠀⠀⠐⢢⡀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣀⣿⠀⠀⠀⣠⣞⣱⡟⢚⣷⣞⠁⠀⠀⠀⠀⣾⡷⢂⣤⣤⣐⢾⣷
⠀⠀⠀⠀⠀⠀⠀⣶⡛⠛⣽⡿⣷⠄⠀⠀⠈⠉⣿⣱⡋⠙⣿⡀⠀⠀⠀⠀⠉⢳⣿⣿⣿⣿⡟⠋⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⣀⣡⡆⠀⠀⠀⢠⠤⠿⣿⣉⡾⠛⢛⡿⠀⠀⠀⠈⠶⠛⠛⠛⠛⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠂⠐⠉⠉⠉⠁⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣀⠀⠀⠀⠀⠀⠲⢶⣿⡄⠤⠠⠀⠂⠀⠰⡁⠀⠘⠄⢀⣴⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣾⣿⣷⡄⠀⠀⠀⠀⢨⣿⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢿⣿⣿⠆⠀⠀⠨⠙⠻⠿⣿⣿⠙⠷⠀⠀⢠⡿⣿⠋⠙⣿⠉⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠃⠀⠀⠀⠘⠓⠈⠉⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣆⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⢠⣾⣿⡏⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⡀⠂⠀⠀⠀⢺⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡄⠘⢿⣏⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⠻⡇⠀⠀⠀⢠⣾⣿⣿⡟⣁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⠀⠙⠷⠂⠀⠘⣷⠛⣿⠏⣽⠏⢀⡇⠀⠀⠀⢠⡿⢹⢹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⢿⠀⠀⠀⠀⠶⠒⠒⠒⡟⠒⠀⠂⠀⠀⣠⣾⣷⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣖⠀⠀⠀⣀⣀⣀⠀⢰⣿⠀⠰⠶⠀⠠⣿⢟⣿⠯⠉⣿⡷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠉⠁⠈⠟⠀⠀⠀⠙⠛⠉⠀⣾⣿⡆⠀⠀⠀⠀⠀⠈⣡⣄⣠⡿⠷⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠚⠋⠉⠁

⠀⠀⠀⠀⢀⣤⠶⣤⡀⠀⠀⢀⣤⠦⣄⠀⠀⠀⠀⠀⠀⠀⠰⢦⡀⠀⠀⠀⠀⠀⢀⡴⠆⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⣀⣀⣀⣤⣤⠤
⠀⠀⠀⠀⢸⡇⠀⠈⢷⡀⢠⡟⠁⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠃⠀⠀⠘⡇⣼⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠚⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠉⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠚⠶⠰⠲⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠒⠆⠛⠷⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠲⠆⠲⠐⠆⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣴⡄⢠⣴⡄
⠀⠀⠀⠀⣠⠶⠿⣏⠉⠉⠉⠉⠙⠛⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⠘⣷⠏⠘⡗⣠⣄
⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⣠⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠋⢻⡄
⠀⠀⠀⠀⠙⠲⠖⠋⠀⠀⢸⡇⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⢠⠞⠉⠙⢷⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⣸⠇
⠀⠀⠀⠐⠶⣤⣄⣀⣀⣀⣀⣹⣶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⡼⠁⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⡟⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣿⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⣤⣴⣿⠁
⠀⠀⠀⠀⠀⠠⠂⣠⢠⣄⠄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⢠⡀⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠋⢉⣁⣀⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡴⢻⣆⡴⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⣿⣿
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⡿⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⣠⣾⠟⠁⣿⡀
⠀⠀⠀⣀⣀⣀⣿⣦⠀⠀⢀⣼⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣾⡟⠁⠀⠀⠛⠃
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠻⣶
⠀⠀⠀⢴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⣷⠆
⠀⠀⠀⠀⠀⠀⠤⠠⠸⠔⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠶⠀⠆⠆⠶⠰⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠐⠆⠦⠶⠒⠆⠆⠲
⠀⠀⠀⠀⣀⣀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⢹⡆⢠⡞⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡀⢀⣠⣦⠀⢀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⣿⡾⠁⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠁⠘⠷⠛⠁⠘⠷⠛⠁⠈⠧⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢹⣇⡴⠖⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣿⠷⠶⢾⡷⠶⠶
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢘⣿⡁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⡴⢿⡀⣠⡴⢷⡀⣠⡴⢿⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠙⠓⠞⠋⠀⠀⠀⠀⠀⠐⠟⠉⠀⠈⠛⠉⠀⠈⠟⠉⠀⠈⠓⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⡀⣤⣤⡤⠄⢤⠠⣤⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⢤⡠⣤⢤⠠⡄⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠠⡤⠀⡤⢤⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⡰⢻⣿⣿⣶⡀⠀⠀⠀⠀⠀⣠⣤⠴⠧⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢰
⠀⠀⠀⢣⠀⣽⣿⠉⠁⠀⠀⠀⠀⢸⣿⡀⠀⠿⢯⣷⣤⠀⠀⠀⠀⠀⢹⣿⣷⣤⣤⡠⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣮⣄
⠀⠀⠀⢈⡼⠋⠘⢇⠀⠀⠀⠀⠀⠈⠻⠻⠧⠀⣈⣿⠏⠀⠀⠀⠀⠀⠈⢿⡿⠿⢿⠃⠀⠀⠀⠀⠀⠈⠹⣟⣿⡿⡿⣿⣿⠂
⠀⠀⠠⠞⠇⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠈⠻⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⠇⠜⠜⠀⠀⠀⠀⠀⠀⠀⠀⠨⠽⠄⡠⠃⠃
⠀⢰⣶⠀⠠⠀⠀⠠⠄⠀⠀⠀⠀⠀⡤⡆⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⢲⠂⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢰⡆⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢺⣶⣴
⠀⠀⠸⡯⣦⠉⢹⣿⠷⠀⠀⠀⠀⠠⣐⠋⣁⣀⡉⢙⡢⠀⠀⠀⠀⠀⠴⣿⣿⣤⣌⡁⠀⠀⠀⠀⠀⠀⠈⠟⠒⣿⣿⣿
⠀⠀⠈⢀⡅⢶⡏⠃⠆⠀⠀⠀⠀⠀⠉⠻⢿⣿⡿⠯⠁⠀⠂⠀⠀⠀⠀⣫⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠹
⠀⠀⠀⡜⠃⠞⢆⠱⡄⠀⠀⠀⠀⠀⢀⡆⠀⠁⠀⢠⠈⠀⠀⠀⠀⠀⠀⠁⠙⠛⠻⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠲
⠀⠀⠠⠤⠈⠀⠈⣀⡀⠀⠀⠀⠀⢠⢤⠄⠀⢀⠀⠈⢀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠤⠄
⠀⠀⠀⣰⣒⠒⣿⣦⠀⠀⠀⠀⠀⠀⠀⠁⢀⣄⠀⠋⠀⠀⠀⠀⠀⠀⣀⠀⣠⣿⣷⢀⣐⠀⠀⠀⠀⠈⡿⣿⣿⣆⠀⡀
⠀⠀⠠⠄⠠⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠀⠀⠀⠀⠀⠀⠀⠼⢻⣿⣷⣾⣿⣏⠸⠀⠀⠀⠀⣀⣉⣿⣿⣿⣧⣼⠖⠂
⠀⠀⠀⠀⢠⠏⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⡀⠀⠀⠀⠀⠀⠀⢐⠋⠋⠙⠻⢿⢦⠀⠀⠀⠀⠀⢨⣨⢟⡙⠛⣿⣿⡇
⠀⠀⠀⣀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢀⢠⠄⠀⠀⠀⠈⠉
⠀⠀⠘⠒⠃⠀⠈⠁⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣬⣥⣭⣤⣤⣀⣀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⢿⠿⠛⠉⠛⠛⠻⣿⡿⠿⢿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠠⠀⠤⠤⢭⣀⡀⠈⠿⢿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠐⢀⣴⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢽⣄⠘⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣹⣿⠃⡸⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠈⠓⢌⠙⢿⣿⣿⣿⣿⣿⣷⡄⠄
⠀⠀⠀⠄⣠⣿⣿⣿⣿⣿⣿⣿⠇⠀⢡⠀⠀⢸⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⡙⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⡏⢠⠀⢋⠀⠰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢹⡦⡙⠻⣿⣹⣿⣿⣦
⠀⠈⢰⣿⣿⣿⣿⣿⣿⠟⠁⠀⠜⠀⣼⠀⠢⠘⡆⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡑⠀⠀⡀⠉⠻⣿⣿⣆
⠀⠀⣿⣿⣿⣿⣿⠟⠁⣀⡴⠡⠊⠈⠱⠀⠀⠀⠈⠄⠈⢆⢣⣀⣤⣤⣀⠀⠀⠀⠰⠶⠓⠶⠀⠈⣆⠀⠀⠀⠄⠈⢻⣿⡄
⠀⢸⣿⣿⡿⠋⠀⠰⠊⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⢦⡀⢘⢿⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⢿⣇
⡄⣼⣿⡿⠁⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⢸⣿
⡁⣿⣿⠃⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⠀⠀⠀⠀⠇⣸⣿
⠁⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⢠⠀⢠⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡘⠀⠀⠀⠀⢱⣿⣿
⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠈⠳⡄⠄⢊⣏⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠜⠁⠀⠀⠀⣠⣿⣿⡇
⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣈⡙⠂⠀⠀⠀⢀⣫⣿⡿⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢀⣠⣾⣿⣿⣿⠁
⠀⠡⠘⣿⣿⣷⣄⡑⠲⠦⠤⠀⠀⠤⠀⢀⣒⣒⣷⠶⣿⣻⢷⡄⠀⠀⠀⠀⡍⠀⠀⠐⠁⢀⣲⣯⣵⣾⣿⣿⣿⣿⣿⠏⠄
⠀⠀⠀⠘⣿⣿⣿⣿⣶⣦⣤⣤⣴⣶⣿⣿⣿⣿⣷⣶⣦⡙⢦⣳⣄⠀⢀⣂⡀⠆⣰⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠘⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢧⣟⣸⡄⠈⣿⣯⣿⢧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣷⣤⡹⣦⡀⢿⣿⣿⠻⠟⠻⣿⣿⣿⣿⣿⣿⠟⠡
⠀⠀⠀⠀⠀⠀⠀⠌⠻⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠋⡿⠾⣻⣿⣦⡌⠀⣼⣿⣿⣿⠟⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠘⣿⣿⣤⣾⣿⠿⢛⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠄⣆⢿⠿⠻⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠋⠉⠁⠴⠚⠆⠂⠀⠌


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣶⢰⢰⡴⠰⡆⡖⡄

⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠄⠀⠀⢠⠀⠀⠀⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠐⢀⣀⣀⡀⣸⣀⢀⣀⣐⠀⠀⠀⠉⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢫⡀⢀⡀⠤⠐⠂⢉⣀⡀⠀⠀⢸⠀⠀⢀⣀⡈⠑⠒⠠⠄⠀⠀⣨⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⢀⠤⠻⠁⠀⠀⠰⠿⣿⠿⠿⠆⠀⠸⠀⣼⣯⠽⣷⡾⠷⣦⠀⠀⠹⠣⢄⠀⠀⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡠⠀⠀⠀⠀⠐⡀⠀⣾⠀⠀⣀⣀⣾⠀⢰⠀⢾⠿⠀⣿⣮⣶⡿⠀⡐⠁⢀⣀⠀⢠⡀⢀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠈⣠⣤⣤⣄⡄⠀⠹⣄⠘⠐⢿⡟⠛⠁⠀⢸⠀⠀⠀⠐⠛⠈⠙⢀⡔⠀⠀⣾⣉⠁⠀⠙⢢⡀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣊⠀⢀⠔⠀⠀⣿⣿⣿⣿⣇⠀⠀⠈⣄⣀⡤⠶⠒⠚⠉⢹⠁⠘⠒⠂⢤⣄⣀⡞⠀⠈⢩⣼⣿⣷⡀⢰⡀⠀⢀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⢰⠏⡈⢳⠮⣄⡀⠀⠼⢟⢿⣿⡯⠀⣀⡴⠛⢧⠀⠈⡶⠇⠀⢸⠀⠈⠸⠁⠆⢠⠟⠳⣄⡀⣿⡿⠿⣽⠛⠋⠀⣀⡬⢶⠋⠈⢷
⠀⠀⠀⠀⠀⢠⠏⠄⢁⠎⠀⠀⠙⠲⢤⡈⠉⠉⢁⡞⢁⠔⢀⠈⢧⣠⡥⠶⠶⠛⠒⠶⠦⣤⣠⠏⠀⢀⡂⠙⣆⠀⠀⠈⣡⠴⠊⠁⠀⠈⢆⠀⠈⢧
⠀⠀⠀⠀⠀⡾⠀⠀⡌⢀⣤⡶⠷⠶⣄⠉⠓⣶⣏⠐⢁⠖⢡⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠠⠋⠀⢈⣷⡖⠋⠀⣠⣶⣶⣤⣄⠈⡄⢠⠸⡆
⠀⠀⠀⠀⢸⠃⠈⠀⠀⢻⣿⣷⢰⠋⢹⡆⢰⠇⠈⠙⠲⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⠚⠉⠀⢷⡀⠸⣻⠻⠏⠛⠿⡄⠰⠐⡀⢛
⠀⠀⠀⠀⠸⠠⠄⠂⠀⢸⣧⠉⢀⠙⡻⠃⡟⢀⠐⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠂⠨⠈⡇⠐⢶⣾⣷⣾⡞⠀⠀⠂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠉⠏⠉⠉⠉⠉⠁⠉⠉⠙⠉⠉⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⣉⠈⠉⠉⠉⠉⠉⠈⠈⠏⠁⢸⡇
⠀⠀⠀⠀⢷⠀⠀⡀⠀⠀⠀⣠⣤⣔⠀⠀⡆⠲⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⢀⠀⠆⡟⠀⠀⣀⣠⢤⣦⡀⠀⡁⠀⢸⠇
⠀⠀⠀⠀⢸⡄⠁⢡⠀⠉⣿⣏⣻⣿⠀⡀⢻⡀⠀⢀⣨⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⣀⠁⠀⣼⠁⠀⣸⣿⣿⠏⠉⠁⢠⠀⡂⣼
⠀⠀⠀⠀⠈⣧⠀⠀⠆⠀⠉⢀⡟⠛⠃⠀⣀⣷⡞⠉⠀⠆⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠈⠙⣶⣇⡀⠀⠘⠟⢿⣧⠖⠀⠇⠈⢠⡏
⠀⠀⠀⠀⠀⠸⡄⠀⠘⡄⠀⠈⢀⣠⠴⠋⠁⠀⠙⣦⠢⣀⠀⠀⡽⠳⠦⣤⣤⣀⣠⣤⡤⠶⠻⡅⠐⣆⡠⢀⡞⠁⠀⠉⠲⢤⣀⠀⠀⠀⡜⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠹⡀⠀⠐⡤⠖⠋⠀⣠⠞⢛⣿⡆⠈⠓⢦⣀⡼⠁⢠⠀⡄⠀⢸⠀⠒⠠⠀⠀⠹⣄⣠⠞⠋⣠⣶⣿⣷⢦⠀⠈⠑⠀⣔⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠻⡎⠀⠘⢆⠀⠰⣿⠟⢻⣿⡗⠃⠀⠀⠟⠓⠶⣤⣁⡀⠀⢸⠀⣈⣀⡁⠵⠖⠛⣇⠀⠀⡟⢿⣿⣫⢸⡦⠀⢀⠖⠀⠉⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⢀⠻⠄⣰⣿⡆⠀⢠⠞⢀⡀⠲⣶⣤⡄⠉⢹⠁⠉⣁⠀⣤⣄⡀⠘⢆⠀⠻⣿⣿⠀⠾⢃⡴⠋⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⠀⠀⠀⠀⢠⠋⠀⠀⠀⢠⡟⠀⠶⠄⠸⠀⢸⣿⣣⣿⢿⣿⡄⠀⢂⠀⠀⠀⠀⠀⠋⠄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠐⠤⣀⠀⠀⠀⠀⠀⣸⠁⢀⠀⠀⢰⠀⠸⣿⡅⣿⣷⣽⠃⠀⠀⢡⡠⠔⠉⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡰⠃⠀⠢⠤⣀⡰⠿⠀⠀⠀⠀⢸⠀⠀⠈⠛⠀⠈⣁⠠⠤⠒⠉⠳⣄⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠐⠀⢤⠉⠑⠒⠒⠂⢺⠒⠒⠒⠂⠉⠁⠄⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⠤⠀⠀⠸⠀⠀⠠⠤⠴⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣴⣾⠿⠿⠛⠿⢿⠿⠋⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣾⡟⠉⠀⠀⢀⡄⠈⠁⠀⠀⠀⠀⠀⠀⠩⡙⠛⠉⠉⠉⠙⠻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠺⠉⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⠀⠀⠀⢀⢇⠀⠀⠐⠀⠀⠀⠀⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠂⣼⣿⣿⠃⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠰⠀⠀⠀⢁⠀⠠⠤⢀⠀⠀⠀⠀⠀⢻⣿⣿⣟⣿⣿⣿⣧⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⡀⠠⢀⠙⠻⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡶⣤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠈⠀⠀⠈⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⠀⠀⡀⣠⠶⠶⡄⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⠈
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠈⠲⢬⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣟⢹⣿⣿⠟⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡛⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣇⠀⠀⠀⠀⠀⠑⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⡤⠐⠂⠀⠂⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣄⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⡀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠈⠉⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣶⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠟⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠛⢁⠄⠊⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠀⢀⠀⠀⠀⢠⣿⣿⡻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⢿⣿⣿⠁⠀⠀⣀⣀⣥⣴⣾⣿⡷⣶⣶⣾⣿⣿⡀⠀⢡⡀⠀⠀⠛⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠧⣀⢸⣿⣿⣿⣿⣿⣿⣶⣿⣿⠀⠙⠛⠓⠀⣼⣿⣶⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⠿⠿⢿⣿⣿⣿⣿⣧⠀⠀⠀⠚⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⢸⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠒⠂⢓⣚⣓⡚⠂⠳⠶⠴⣄⡀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡖⡠⠔⠙⠉⠀⠀⠀⠀⠀⠸⡟⠉⠉⠛⠋⠛⠛⠶⢮⣌⡒⠆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠐⢉⡀⠀⠀⠀⠀⢰⣶⣦⡀⠐⣗⠀⠀⡀⣠⠀⠀⠀⢠⡟⠀⠀⠈⠚⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠛⠋⠙⢧⣀⠀⠀⠻⣟⠋⠀⠀⠩⠀⠘⠋⠀⠘⠀⠠⠶⠀⢀⡐⠀⠀⠀⠙⢧⡄
⠀⠀⠀⠀⠀⠀⣀⠟⣡⣿⠀⣠⢦⠀⠀⠹⢦⡀⢀⣠⣤⣴⣶⣾⣶⣦⣦⣂⣠⢄⠂⠀⠀⠋⢉⣷⠘⠀⠀⢀⡙⡄⠣
⠀⠀⠀⠀⣰⢶⠒⣤⡟⠁⢾⠁⠀⡀⠀⢀⡜⣯⠴⣿⡉⠀⠀⠀⢸⠄⠀⠀⠉⡿⠾⣷⣄⠀⠀⠀⢀⣠⡶⠟⠷⣄⢠
⠀⠀⠀⠀⣀⠇⠘⠾⢦⡄⠀⠀⠀⢀⣔⡵⠋⠀⠀⠘⣷⡀⠀⢀⣸⣄⡀⠀⡼⠁⠀⠈⠛⣿⡤⣴⠍⠉⢀⣀⠈⢻⡘⠳
⠀⠀⠀⢀⡍⠀⠀⠀⠀⠈⠳⢦⣠⣶⡾⠷⣄⡀⠀⣤⠶⣚⠩⠥⠔⠇⠭⣝⠢⣀⠀⣠⠶⠋⠱⡽⡄⣸⣿⣿⣷⠈⢿⡀⠇
⠀⠀⠀⢴⠁⠀⠆⠈⠅⠀⠀⢀⣾⡟⠀⠀⠈⢩⡾⡡⠞⠂⠀⠀⠀⡀⠀⠈⠙⢪⡣⡁⠀⠂⠀⠹⡜⡀⠉⠷⠁⠀⠀⡧⢨⡄
⠀⠀⠀⡻⠀⠀⠀⢄⠀⠄⠀⣺⣿⠁⠀⠀⢀⡟⡸⠁⠀⠀⠠⣴⣴⣆⡀⠀⠀⠈⢻⣹⣤⠤⠤⠴⣷⡃⣀⣀⣀⣠⣴⣶⠘⠠⡄
⠀⠀⠈⡇⠀⠀⠀⠉⠀⠀⠀⠾⣿⣤⠶⠶⢸⠁⡅⠀⠀⠀⣺⡏⠁⠙⣿⠤⠀⠀⠐⡃⡗⠀⠀⡄⣼⣯⠉⠉⠀⠀⠀⠀⢈⡖⠁
⠀⠀⣄⡖⢰⣶⠶⠶⠒⠂⠛⡆⣧⠀⠄⠀⢿⡆⣆⠀⠀⠀⢺⣷⣤⣾⠟⠉⠀⠀⢰⢃⡇⡈⠐⠀⣿⡯⢀⡶⠶⠀⠀⠀⢀⢂
⠀⠀⠀⢰⠀⢒⠀⠀⡀⠀⠀⢷⢹⡀⠀⠀⢘⣿⡘⣀⠀⠀⠀⠟⠏⠈⠀⠀⠀⣠⢍⡾⠀⠛⠶⣄⣿⠇⠘⡇⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠘⡄⢸⡄⠀⣋⡄⠀⠈⠄⢷⣠⡶⠋⠈⠻⣌⡓⠄⣀⢀⠀⡀⣀⡤⢚⣵⡷⠃⡀⠀⢀⡼⣱⡀⠀⠈⠃⠀⠀⢀⣧
⠀⠀⠀⠀⢹⠈⢷⠀⠀⠀⠀⠀⣀⢿⣷⣆⠀⠀⠀⠈⢛⣶⣾⣭⣵⢩⡤⠖⠓⠋⠙⣦⡀⢠⠞⡕⠀⠈⠓⠶⣄⣀⠀⡨⠒
⠀⠀⠀⠀⠀⠡⡌⢱⣀⣀⡴⠞⠋⠈⠹⢿⣷⣤⡀⣠⡿⠀⡀⠉⠉⢰⡇⠐⠀⠄⢀⣼⣿⠣⠞⣀⠏⠁⡀⠀⢀⠋⠴⠻⠆
⠀⠀⠀⠀⠀⠠⡓⣀⠋⠁⠀⠀⣴⣤⣦⠀⠘⠿⠛⡿⠧⣤⣁⣀⣀⣘⣛⣤⣴⣾⡿⠟⡋⠀⠘⠋⢀⠐⠀⣰⢋⡤
⠀⠀⠀⠀⠀⠀⠉⠸⠧⡀⠀⠀⠛⠛⠛⠀⠀⣰⠋⠉⠓⠢⠴⠨⠍⣭⠑⠛⠙⠈⠁⠀⠘⢆⠀⠀⢀⡠⠞⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡧⣄⠀⠀⠀⠀⣰⠃⠀⢠⠆⠀⡀⠀⠀⢹⠀⠠⠀⣶⡄⠀⠀⠀⠻⠶⢉⠥⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠤⣀⡸⢧⡄⢀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠊⠉⠃⠀⠀⣀⡀⢼⣤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠗⠐⠤⣍⢙⣓⡒⠶⠶⠖⠾⠀⣀⡀⠀⡰⠴⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠘⠌⠃⠽⠆⠉

⠀⠀⠀⣀⣤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢤⣄⡀
⠀⣠⠞⣡⠀⢠⣶⣄⣉⢳⡄⠀⠀⠀⢠⡞⠉⠀⢀⠀⢀⠈⠳⣄⠀⠀⠀⢀⡾⠫⣄⠀⠀⠀⣬⠙⣦⠀⠀⠀⠀⡴⠋⣡⣤⣀⣤⣌⠙⢦⡀
⢠⡏⠀⠘⡇⡾⢸⡧⠞⠀⢻⠀⠀⠀⡟⠀⠹⡴⠛⢦⠏⢳⠀⢹⡆⠀⠀⣾⠀⠀⢹⣄⠀⣸⡃⠀⠸⡇⠀⠀⣸⠁⠸⡇⠀⣿⠀⢨⠇⠈⣧
⠘⡇⠀⠀⢻⠇⢸⠇⠀⠀⣸⠀⠀⠀⣧⠀⢲⣠⠶⣄⡴⣆⠀⢸⠇⠀⠀⢿⠀⠈⢹⠏⠉⢻⡉⠀⢠⡇⠀⠀⢻⡀⠀⠀⠀⣿⠀⠀⠀⢀⡟
⠀⠹⣆⡀⠈⢀⡟⠀⢀⣴⠋⠀⠀⠀⠘⢦⡀⠉⠀⠉⠀⠈⣠⠟⠀⠀⠀⠘⢧⡐⠋⠀⠀⠈⠓⣠⠞⠀⠀⠀⠈⢷⣄⠀⠀⠛⠀⠀⣠⡾⠁
⠀⠀⠈⠙⠲⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠛⠉

⠀⠀⣠⠶⠛⠛⠛⠓⢦⣄⠀⠀⠀⠀⠀⣠⡴⠚⠛⠛⠛⠶⣄⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠲⣤⡀⠀⠀⠀⠀⢀⡴⠞⣛⣉⣛⠳⢦⡀
⠀⡾⠁⠘⢧⣤⣤⡾⠁⠘⣧⠀⠀⠀⣼⠋⠀⢲⠶⠶⣶⠃⠈⢷⠀⠀⠀⣰⠏⢠⠴⠒⢛⡶⢦⡈⢳⡀⠀⠀⢠⡟⠀⣾⠁⠀⠈⢻⠀⠹⡄
⢸⡇⠀⢰⠋⠀⠀⠹⡆⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⢠⠴⢤⠘⠦⠼⠃⠈⡇⠀⠀⣾⠀⣠⠿⡄⠀⢠⡟⠀⠀⣷
⠈⢧⠀⠘⣦⣀⣀⡼⠃⢀⡾⠀⠀⠀⢷⡀⢀⣸⣤⣤⣿⡀⠀⣼⠁⠀⠀⢻⡄⠸⢦⣼⣁⣠⡴⠃⣸⠃⠀⠀⠸⣆⠘⠛⠁⢠⡏⠀⠀⣠⠇
⠀⠈⠳⣤⣀⣉⣉⣀⡴⠞⠁⠀⠀⠀⠈⠻⢮⣁⣀⣀⣀⣥⠞⠁⠀⠀⠀⠀⠙⢦⣄⣈⣉⣀⣠⠾⠃⠀⠀⠀⠀⠙⠦⣄⣀⣈⣓⣾⡶⠋
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⢠⡞⣏⣀⣀⣄⠈⠙⢷⡀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠀⠉⠳⣄⠀⠀⠀⢀⡴⢻⣁⣀⣠⡀⠉⠳⣄⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⣈⡛⣦
⢠⡏⠀⡏⠈⡇⢸⡧⡄⠀⢻⠀⠀⠀⡿⠀⣀⣰⠋⠉⣧⣀⠀⢹⡄⠀⠀⣾⠁⢸⠀⣿⠀⢹⠀⠀⠸⡆⠀⠀⣸⠃⠀⠀⠀⢀⡼⠿⠁⠘⣇
⠸⡇⠀⡇⠀⡇⢸⡇⡿⠀⣸⠁⠀⠀⣇⠀⠬⠤⠤⠤⠬⠥⠀⢸⡇⠀⠀⣿⠀⢸⠀⣿⢀⡞⢀⠀⢠⡇⠀⠀⢻⠀⠀⢐⡶⣯⠀⠀⠀⠀⡟
⠀⠻⣄⠁⠀⣁⡼⢿⣁⣰⠏⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠘⣧⡈⠀⠉⢺⣀⠟⣃⡾⠁⠀⠀⠈⢧⡐⠋⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠈⠛⠶⠤⠴⠶⠛⠁⠀⠀⠀⠀⠀⠈⠛⠶⠦⠴⠶⠚⠁⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠶⠞⠋⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠌⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⡀⠄⠀⠀⠀⠨⠄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠬⠉⠀⠀⠀⠈⠨⠄
⠀⠀⠀⢀⣀⠀⠀⣀⡀⠀⠀⠐⠀⠀⠈⢀⡀⠀⢐⣀⠈⠀⠀⠀⠀⠀⠀⠶⠆⠀⠰⠆⠀⠀⠀⠀⠀⢀⣀⡂⠀⠀⣂⣀⡄
⠀⢀⠈⠀⠉⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡈⠉⠀⠈⠉⡡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠄⠀⠀⠄⡀⠀⠀⠀⠐⠂⠒⠁⠀⠀⠌⠐⠀⠂⠀⠀⠀⡀⣠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⠁⠀⣀⡀⠀⠀⠀⠀⠡⠌⠐⠃⠀⠐⠓⠈⠀⠁⠀⠀⢀⠀⢀⠀⠀⢀⠀⠐⠀⠀⠀⠀⡀⠠⠰⠶⠀⠐⠶⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠈⠀⠀⠀⠀⢁⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀


⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠨⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠀⢀⠀⡂⠀⢀⣀⢀
⠀⢀⢀⣀⠀⠀⢂⠀⠀⠠⠀⠀⠀⠀⢀⠐⠒⠀⠐⠒⠀⠀⠀⠀⠀⡀⢃⡀⠀⢂⡀⡅⠀⠀⠀⠀⠀⠈⠈⠉⠀⠈⠉
⠀⠈⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⣄⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢀⠆
⠀⠄⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠙⠉

⠀⠀⠀⢂⡀⠀⠂⠂⠐⠐⠀⠀⠀⠀⠀⢂⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠐⠐⠐⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠟⢿⡋⠉⢁⣠⣀⠀⠀⣀⣄⡈⠉⢹⡟⠛⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⢉⣀⠀⠀⠘⣧⠈⠀⣿⠻⣷⣾⡿⣿⡃⠀⣿⠁⢀⣦⠀⠈⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⣀⠀⢸⣿⣠⡄⠄⢹⡄⠀⠈⠀⢻⠿⠀⠁⠀⢸⠇⠀⢀⣿⣥⣄⠀⠂⢈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣏⠀⢀⠉⡛⢻⣾⣿⣿⣀⡀⠈⡇⠀⠂⠀⠸⠃⠀⠄⠀⡟⠀⢰⣿⠟⠛⢿⣿⠿⠟⠀⢀⣽⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠈⠳⣄⠀⠀⠟⢹⣧⠈⡍⠉⠀⣸⣄⣠⣤⣤⣤⣤⣄⣴⣁⠀⠜⢿⣦⣠⣾⡿⠐⢀⡴⠛⠁⠀⠙⢿⣆
⠀⠀⠀⠀⠀⢀⣼⡟⠁⢰⡶⢂⣂⡈⠳⣄⠈⠀⠃⢀⣤⡶⠟⣛⣩⣭⣥⣤⣤⣬⣭⣍⣛⠻⢶⣤⡙⠁⠋⠀⡠⠋⠀⡀⠰⢶⣶⡈⢻⣧⡀
⠀⠀⠀⠀⢀⣾⡟⠀⢸⡿⢇⣸⣟⠁⠀⠈⠳⣤⣾⢟⣩⣶⣟⣿⣿⣷⣿⣿⣉⣿⣯⣽⡿⢿⣶⣬⡛⢷⣤⠊⠠⣤⣄⣴⡾⠛⢹⣷⠀⢻⣷⡀
⠀⠀⠀⠀⣼⡟⠀⠘⣿⢣⣼⡏⠁⡀⠀⢀⣾⢟⣵⠛⣿⡿⢿⣟⣻⣷⡿⠿⠿⠿⠿⢿⣿⣟⣿⣯⣿⣦⡙⣷⡀⠀⢻⣏⣠⡾⠟⠻⢦⠄⢻⣧
⠀⠀⠀⢰⣿⣁⠀⠄⠈⠈⡿⠀⠀⠀⢠⡿⢣⡾⢿⣿⢿⣿⠿⠿⠿⢏⣠⡤⢤⡀⠀⠀⠊⢝⢿⣏⣹⣿⣿⡌⢿⡄⠀⠛⣧⡀⠀⠀⠀⣀⣠⣿⡆
⠀⠀⠀⣿⡏⠙⠛⠳⠶⠤⣌⣀⡀⢠⣿⢣⣿⣷⣾⣿⣿⡇⠀⣀⠀⠀⠁⠀⡀⢻⡄⠀⠀⠀⠑⡙⣿⣿⢻⣿⡌⢿⡄⣀⢠⠤⠒⠚⠛⠉⠁⢹⣷
⠀⠀⢠⣿⠁⠀⠀⠀⠀⣀⠂⠀⠉⣿⡇⣾⣧⣽⣿⠟⠋⠉⠒⠛⠀⠐⡄⢀⠠⢤⡇⠠⡤⡀⠀⠈⡾⣿⣿⣿⣿⢸⣿⠀⠀⢀⡤⣨⣶⣦⡠⠀⣿⡀
⠀⠀⢸⣿⠀⠘⠀⢰⣟⣹⣆⡂⠀⣿⡇⣿⣯⣹⣇⠀⠄⠀⠀⠀⠀⢬⠀⠀⠖⠦⡍⠘⠧⠆⠁⠀⠁⣿⣏⣿⣿⡆⣿⠀⠀⣾⡇⠻⢷⠾⣿⡆⣿⡇
⠀⠀⢸⣿⠀⢸⣿⣟⣛⢋⠛⠿⠀⣿⡇⣿⠻⣿⣿⡷⠚⠁⠀⠀⠀⢾⠀⠀⠀⠐⢡⡴⠀⠀⠀⠀⠀⣿⣿⡉⣻⠇⣿⠀⠀⣿⣷⠶⣦⡀⣿⡇⣿⡇
⠀⠀⠘⣿⡀⠿⢋⠋⠉⠀⠀⠀⠀⣻⡇⢿⣶⣿⣾⣇⠀⠀⠀⠀⠀⠀⢧⡀⠀⠒⢸⡔⠂⠀⠀⢀⢲⣿⣿⣿⣿⢸⣟⠀⠀⠑⠻⠶⠟⠐⠋⠀⣿⠁
⠀⠀⠀⣿⣇⠀⣀⣀⣤⡤⠶⠒⠋⠙⣿⡘⣿⣽⣿⡟⣷⣶⠖⢀⠀⠀⠘⠀⡀⠀⡸⠂⠀⠀⠀⣣⣿⣧⣼⣿⢇⣾⠋⠉⠙⠒⠶⢦⣤⣄⣀⣸⣿
⠀⠀⠀⢸⣿⠛⠉⠉⠀⠐⡀⠀⡀⠀⠹⣷⡜⣿⣿⠿⣿⣿⣄⠀⠀⣠⣤⡳⠅⠀⠀⠀⣀⣤⣾⣿⣿⣽⣿⢏⣾⠏⠀⢀⣴⣶⡄⠄⠀⠈⠉⣿⡇
⠀⠀⠀⠀⢻⣧⠀⣤⣴⣾⣥⣤⣿⣄⠀⠘⢿⣎⠻⣾⣿⣿⣿⣿⣿⣧⣿⣿⣟⣿⣿⡟⢻⣿⣏⣹⣿⠟⣡⡿⠃⣴⣁⣼⢷⣾⣯⡤⠀⠀⣸⡟
⠀⠀⠀⠀⠈⢿⣆⠈⠙⢿⡈⢹⡟⠙⠇⠀⣠⠿⢷⣬⡛⢿⣿⣼⣿⣿⢛⣿⡿⣿⣿⣿⣿⣿⡿⢟⣥⡾⠛⢄⠀⠟⠉⠹⣦⣀⠈⣷⡆⣰⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⢀⠀⠄⠀⣠⠞⣡⣄⡀⠙⠻⢷⣮⣍⣛⡛⠿⠿⠿⠿⢟⣛⣩⣵⡾⠟⠋⡀⠀⠀⠱⣤⠁⠀⠁⠛⠛⠋⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⣠⡾⠁⠀⠈⣸⡗⠀⠀⠀⠈⢙⠛⠛⠛⠛⠛⠛⠛⠛⡏⢁⣾⣤⠀⠀⠀⢦⡀⠌⠻⣦⡀⠘⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡾⠋⠀⠈⢠⣾⢋⡴⠀⢀⠀⠀⡼⢀⣴⡆⠢⣄⣀⣠⡤⢷⠈⢻⣜⣇⠸⣦⣈⣷⣦⣀⠈⢻⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠸⢷⣾⣇⣠⠟⠀⢰⡇⠨⢾⣷⠏⡙⣿⣭⡖⠸⣇⠀⡉⠻⢦⣿⠟⠛⠈⣡⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⡈⠛⢿⡀⠀⣿⠀⠀⢸⣿⣀⣀⣿⠁⠀⠀⢻⡀⠀⠀⠀⠀⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣦⣤⣼⣇⠀⠂⠀⠙⠛⠛⠂⠀⠀⠀⣸⣧⣤⣴⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠷⠶⠶⠶⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣿⣿⣿⠿⢿⣶⣶⣒⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠛⠉⠛⣿⣿⣷⣿⣾⣿⣷⣾⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢶⣶⠞⠫⠀⢿⠟⠃⠀⢿⢹⡏⣿⣿⣻⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡿⣦⣿⣿⣽⣿⣇⣿⣿⢹⣷⢹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠲⢶⣾⣦⣄⠀⠀⠀⠀⠀⣃⣴⣿⢣⣿⣿⠏⣿⣿⣿⣾⡟⢸⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⣿⣿⣷⠀⠀⠀⢰⣿⣿⣷⣿⣿⣥⣸⡿⣿⣽⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣏⣀⡞⠀⣠⣠⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣻⣿⣧⢏⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⢻⡟⠁⢀⣶⣿⠋⣿⣿⣿⣿⡿⢝⠋⢻⡯⠻⣷⣹⣼⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣥⣿⣷⣿⣿⣿⣿⢿⣿⣿⣟⠿⠀⣤⣀⢻⣄⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣙⣿⣿⣿⣿⣿⢻⣿⣯⣙⠛⢦⣾⣿⠘⣿⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢰⡊⢉⡉⣿⣿⣿⣿⣿⣿⣿⣙⠁⠀⢀⣹⣟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠋⢁⠀⣷⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⠏⣼⣼⣿⣿⣿⣿⣿⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠉⠀⠀⠙⠻⣿⣿⣿⣿⣿⡜⢻⣿⣿⣼⣿⣿⣿⣿⣿⣿⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢠⡾⢫⣿⣿⣿⣿⠈⠙⣿⣿⣿⣿⣿⣿⣿⢻⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠘⠀⣸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⣤⣤⣴⣾⣿⣿⣿⣿⠇⠀⠀⣾⣿⣿⣿⣿⣿⣆⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⠿⣿⠛⣒⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣧⡤⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣄⡀⠀⠀⠀⡠⢒⡜⡉⠠⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠐⠶⢶⡒⠒⠬⠌⠃⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⡺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠔⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢦
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢠⠎⢉⡉⠓⢄⠀⠀⠀⠈⠀⠀⠀⠊⡠⠒⡁⠠⠐⠂⠀⢠⠊⠉⠉⢢⠀⠀⠀⠱⡀
⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠜⠀⠀⠈⢆⠈⠂⠀⠀⠀⠀⠀⠀⠜⢀⣀⢀⡀⣀⠀⠀⠃⢀⠆⠀⠀⡇⠀⠀⠀⠱⡀
⠀⠀⠀⠈⡎⠀⠀⠀⠀⠀⢰⠰⡀⠀⠈⠆⠀⠐⠒⠐⠂⠄⠀⠔⠁⠀⠀⠀⠈⠑⠰⠀⠸⠀⢰⢀⠇⠀⠀⠀⠀⢣
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠔⠉⠀⠀⠀⠀⠈⠀⠀⠀⠤⠃⠌⠀⠀⠀⠀⠀⠀⣡
⠀⠀⢠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⣮⡆⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠒⠉⠁⠀⠀⣀⡩⠄⠀⡀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⡄⢀⠎⠀⠉⠉⠉⠫⡝⠉⠀⠀⠀⠘⢆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠘⡤⠃⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠰⠃⠀⠠⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠑⠤⢤⠀⠐⠪⠿⠒⠐⡀⠒⠉⠀⠀⠀⠀⠀⠀⢀⠔⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠐⠂⠐⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣴⢖⣒⣢⣄⡀⠀⠀⠀⠀⠀⠀⠪⠑⢦⣄⣐⡀⠀⠀⠀⠀⠀⠀⣠⡆⢀⡆⠀⠀⣀⠀⠀⠀⢀⣤⡶⠀⠀⠐⢦⣤⡀
⠀⠀⠀⠀⡰⣋⠱⢿⢏⡍⢻⣷⡀⠀⠀⠀⠀⠷⡀⣤⡶⢿⣯⠁⠀⠀⠀⠀⣀⡾⣛⠁⠈⠻⢶⣴⡌⢷⠀⠀⠹⡾⠆⣤⠁⡄⣼⢿⠃
⠀⠀⢠⠀⠀⣉⣤⣾⡿⣿⡆⢿⡇⠀⠀⢀⣾⠇⠀⠔⣻⢸⡿⠀⠀⠀⠀⢸⣿⡇⣁⠀⠀⠀⠙⢧⣽⢸⠀⠀⠀⣁⠀⣷⠀⣆⢹⡜
⠀⠀⠀⠑⡿⣿⣿⣍⠴⠟⣹⠸⠃⠀⠀⢸⣿⠀⡤⠼⠅⢸⡆⠀⠀⠀⠀⢸⡿⣶⣯⠃⠀⠀⠠⣾⡿⠞⠀⠀⠀⠘⣾⠛⡀⡟⣾⡇
⠀⠀⠀⠀⢰⣿⣤⣤⠴⠞⠃⠀⠀⠀⠀⠈⡿⢀⡇⠀⠀⣽⣇⡤⣄⠀⠀⠀⠻⠬⠛⠻⡦⠀⣾⠛⠁⠀⠀⠀⠀⠈⡈⢀⣿⡀⢁⠇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⡠⠀⣀⢐⣄
⠀⠀⠀⠀⠲⡤⠒⢾⠗⠒⡴⠂⠀⠀⠀⠈⣹⡇⣰⠀⠀⢤⡤⠤⣀⠀⠀⠀⠀⠐⠗⠶⠶⢸⢻⡶⠛⠁⠀⠀⠀⢸⡟⠁⡏⠘⣿
⠀⠀⠀⠀⡐⠁⠀⣼⠀⠠⠉⠄⠀⠀⢨⣟⣡⠶⠃⠀⢀⣀⠀⢠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⢈⠣⣴⡇⢠⢟⡀
⠀⠀⠀⢰⣤⠶⡧⣿⡤⣧⡶⢾⠄⠀⠀⠻⣥⣶⡀⡶⣿⣿⣧⡀⠀⠀⠀⠀⢀⡍⠁⣶⣦⡶⠾⢯⡻⠀⠀⠀⠀⢑⣿⣿⣏⣿⣟⡊
⠀⠀⠀⠀⠉⠉⠀⣿⢖⠈⠉⠁⠀⠀⠀⠀⣿⠋⣷⠀⠈⣧⠀⢱⠀⠀⠀⠀⠈⠁⢴⠉⢿⠀⠀⠋⠀⠀⠀⠀⠀⠉⠊⢸⣿⡃⠱⠉
⠀⠀⠀⠀⠀⠀⠛⠈⠈⠓⠀⠀⠀⠀⠀⠚⠃⠚⠃⠀⠚⠋⠐⠛⠀⠀⠀⠀⠀⠀⠘⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⢀⡀⠀⢀⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠴⠾⠓⡢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣄⠋⣩⣹⡇⠀⠀⢸⢷⣦⣘⠛⣃⣴⣿⠆
⠀⠀⠀⠀⢀⣀⣐⠛⠋⠀⢀⡀⠀⠀⠀⢀⣀⡤⠶⠷⠟⢛⣩⣵⡇⠀⠀⠀⢀⡤⠀⡀⠀⠀⢉⢠⠇⠀⠀⠀⠀⠙⠻⠿⡇⠻⠟⠋
⠀⠀⠀⠀⢸⣟⣶⣴⣤⡀⢰⢌⡆⠀⠰⠋⠉⠙⠧⣴⣾⣿⣿⡿⣍⠂⠀⠀⠀⢅⣼⣿⠶⡇⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⣟⡧
⠀⠀⠀⢠⡏⠀⠙⣿⠋⠻⢿⡀⠀⠀⠀⠀⠀⠀⢀⣘⡯⠉⡇⠳⠈⠀⠀⠀⢰⠋⠹⣇⠐⠶⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠈⠈⠀⠀⠒⠃⠀⠀⠚⠁⠀⠀⠀⠀⠈⠭⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠙⠂⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠾⣿⠃⠀⠀⠀⠀⢻⡿⢶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠋⢉⣀⣠⡄⢻⣄⠀⡄⡄⠀⣼⠃⣤⣀⣈⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⠴⣚⠉⠁⣸⠁⠀⠙⠛⠁⠙⠛⠁⠀⢹⡀⠉⢙⡲⢄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣼⢟⡳⠄⢿⠀⠀⠀⢠⡄⣤⠀⠀⠀⢸⡇⠴⣛⢿⡄⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣸⡏⠇⠙⡦⢸⡄⠐⠋⢹⣧⣿⠉⠓⠀⣼⠠⡞⠁⠏⣿⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠁⣀⡄⠘⢧⠀⠈⠰⠋⠀⠀⠀⣸⡿⣿⡀⠀⠀⠈⠳⠈⠀⢠⠟⠀⣄⡀⠉⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⠟⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠙⢷⣌⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⣴⠀⠁⠀⠀⢰⣮⢠⣀⣀⡀⠀⠘⡇⡟⠀⠀⣀⣀⣠⢨⣦⠀⠀⠀⠁⣰⣄⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣾⠇⡼⠉⠳⣄⡀⠀⢸⡏⠢⣈⡫⠙⠦⠀⠀⠀⠠⠞⠩⣋⡠⠊⣿⠀⠀⣀⡴⠋⠹⡄⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⣀⠀⠉⠀⠘⣷⣰⠁⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⢱⣰⡟⠀⠈⠁⢀⣀⣀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣷⠏⠁⢀⠀⠀⠀⠀⠈⠻⡄⢀⡀⠼⠀⠀⠀⠹⠄⣀⠀⡾⠋⠀⠀⠀⠀⢁⠀⠉⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⣾⡆⠀⠀⠀⢠⠀⢀⣞⡧⣄⣀⣀⣀⣀⡤⣟⣆⠀⢠⠀⠀⠀⠀⣾⣆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⡀⠀⠀⣸⠀⢸⣿⠀⠻⢦⡀⡠⠶⠃⢸⣿⠀⢸⡀⠀⠀⣰⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⠛⠛⢁⣴⡄⠙⠶⣤⠤⠀⠠⣤⡴⠞⠁⣴⣄⠙⠛⠛⠁⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠉⠒⠒⠋⠁⠙⠦⣄⠙⠶⠤⠴⠞⢁⡤⠞⠁⠉⠓⠒⠊⠁⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⢆⠀⠀⣀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢀⡀⠀⢀⢆⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡑⠦⣌⡙⠓⠒⠾⠷⠶⠤⠴⠶⠿⠖⠒⠛⣉⡤⠖⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣉⡛⠶⢤⣀⠀⡀⡀⢀⣠⠴⠞⣋⣡⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⡈⠳⡇⡷⢋⣠⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠠⠊


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⡀⣔⣀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠯⣭⣵⣿⠛⣁⠤⠬⣙⠻⢗⢆⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠚⣚⣃⠢⣼⡇⡌⠀⠈⡇⠘⢣⠄⡢⢶⣿⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡔⠁⠈⢹⣧⣬⣥⣥⠬⠉⠙⢳⣔⣡⠖⣐⡪⢽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢇⠐⠦⠛⢉⣭⢬⠉⠑⢸⣄⣠⣿⡗⠶⣦⣌⠳⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢠⡴⠟⣢⢷⣾⣰⢻⣷⣧⣵⣿⣿⣿⡿⠿⠧⣥⠧⡈⢂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣧⠀⠈⠙⠶⢏⠐⡉⢀⡩⣊⣹⠾⠛⣩⣿⣿⡿⢿⣯⡓⠲⣴⡤⠃⢌⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠈⣄⠀⠀⢀⣆⢀⣴⡿⢦⡞⠁⠠⢨⠋⡝⠉⡇⠘⢬⠻⢖⠢⡕⢢⣤⣊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠄⠈⠲⢨⣾⠟⠁⣸⡄⠀⠀⢁⠀⠇⠀⠀⠀⠀⡇⠀⠁⠈⠀⣿⡉⠉⠙⠩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣈⠀⠰⠚⡼⡏⠁⣸⡿⡀⡠⠀⢮⡈⠔⡔⣴⢀⡴⠉⠁⠛⠶⢾⣿⠏⠀⠀⡰⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠤⠦⠋⣨⠆⣿⡄⠑⠐⣷⠀⠑⢀⡀⢀⠙⠆⠸⣏⡼⠂⠀⠀⠀⢈⣡⣼⠁⠀⢠⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⢿⡌⢈⠲⢍⠛⠀⠀⠨⠑⣛⡵⠾⡿⣿⣿⣛⠛⠛⢛⡩⣿⠀⠀⠃⣀⣀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠐⠀⠀⠻⢿⠊⠿⢲⠖⢶⣰⡟⠉⠀⠀⠈⠺⣻⣖⡞⠛⢻⣴⠃⠀⢰⡈⠠⠄⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠘⢓⠊⢀⠲⣯⠻⢄⡀⠀⠀⣀⠀⠱⣷⣝⠦⢼⡏⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠒⢾⡏⡾⠶⠌⠀⠀⠡⠶⠖⠃⢹⢒⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸
⠀⠀⠠⠄⣶⠀⠀⠀⠀⠀⣤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥
⠀⠀⠀⢸⡏⢠⡞⢶⣴⢿⡇⢰⡆⣼⢣⡆⣾⠓⡀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠋⠙⠁⠙⠀⠈⠉⠋⠈⠉⠉⠉⠋⠙⠊⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠓⠒⠒⠒⠒⠒⠶⠶⠶⠤⠤⠤⠤⠤⠶⠶⠒⠒⠒⠛⠋⠁

⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈⠀⡖⠀⠀⠀⠀⠐⡂⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠴⠢⠐⠀⠀⠀⠰⡶⡈
⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⠉⠀⣇⡀⠀⠀⠐⡄⠁⠀⠀⠀⠀⠈⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⡀⠀⠀⢀⠀⠈⠉
⠠⠂⠀⢀⠄⠁⠀⠀⠀⠑⠄⡀⠀⠨⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠄⠀⠀⠀⠀⠀⠀⠦⠄⠁⠁⠀⠈⠀⡤⠆
⠀⢀⡀⠀⠀⠁⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⢀⠀⠄⠈⠀⠀⠀⠀⠁⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡈⠁⠀⠀⠀⠁⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠴⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠠⠆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠤⠂⠐⠀⠀⠒⠤⠀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠄⠀⠀⠀⠀⠄⠀⣀⡀⠀⠂⠸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠐⠄⠠⠂⠀⠀⠀⣄⢀⡹⠀⠀⠀⠀⢀⠀⢀
⢂⠀⠀⠄⠀⠁⠀⠀⠈⠀⠠⠀⠀⡘⠀⠀⠘⡁⠅⠀⠀⠀⠀⠀⠈⠸⡀⠃⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠅⢀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠁⠀⠰
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠤⠀⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⠐⠐⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠀⠀⠀⠀⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠄⠀⠇⠀⠀⢀
⠀⠀⢁⠤⠀⠂⠀⠀⠐⠀⡀⠂⠓⠀⠀⠀⠀⠀⠃⠠⠀⠀⠀⠀⠄⠘⠀⠀⠀⠀⠀⠀⠄⢈⠀⠀⠀⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠄⠘
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠰⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⣥⡄⠀⠀⢠⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀
⣀⣀⣀⣉⣠⣁⣁⣀⣉⣀⣈⣈⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣉⣈⣉⣉⣁⣀⣉⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢷⡀
⠀⠀⢀⠔⠋⠁⣉⠙⢦⡀⠀⠀⠀⠀⠀⠀⡀⠀⢔⣩⣴⠃⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠙⠡⣤
⠀⠀⣘⠛⠆⣜⠐⠊⢸⣷⡀⠀⠀⠀⠀⠀⠓⠏⡄⠈⠻⡲⡆⢷⣜⡆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠙
⠀⠀⠿⠖⠒⠪⣐⣶⡿⠋⠁⠀⠀⠀⠀⠀⠘⠂⢈⣾⢣⠇⣧⡔⢺⡇⠀⠀⠀⠀⠀⣮⣁⣠⡴⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢴⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⣀⠋⠣⠟⠀⠀⠀⠀⠀⠀⠈⠍⠭⠢⠰⣤⣌⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋⠀⠤⠤⠀⠀⠀⠀⠀⢰⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠤⠶⠶⠦⣄⣄⣀⠀⠀⠀⠀⠀⠀⢴⡚⠱⠃⠈⠆⢓⣢
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢒⠃⠰⠀⠀⠀⠈⠧⢉⠴⠀⠐⠀⠜⠛⣣⡀⠀⠀⠀⠀⠀⡴⣁⠋⠙⠋⡉⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠚⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠗⠁⠐⢨⠿⠇⠀⠀⠀⠀⠀⠙⠻⠷⠔⠣⢜⠯⠋
⠀⠀⠀⠀⠀⠀⠐⠂⠉⠑⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀
⠀⠀⡄⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠔⣂⢍⠀⠀⠀⠀⠀⠀⠀⡈⡷⠂⠀⠐⢢⢷
⠀⠀⠹⡾⠆⠀⢀⣦⠀⡀⡀⠀⠀⠀⠀⠀⠰⠻⠒⠚⡙⢧⣀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⡕⣰⣈⣧⣜⣚
⠀⠀⠀⠀⢰⠃⠨⢁⡾⡟⠮⣓⠦⠀⠀⠀⢰⡆⠒⠀⣀⠈⠁⢀⣀⡀⡘⠃⠀⠀⠀⠀⠀⢸⡿⣌⠨⡽⢿
⠀⠀⠀⢀⡈⢁⡀⡜⠐⢓⣪⠝⠓⢢⠀⠀⠀⠛⠥⠤⠜⠛⠶⣉⣩⣷⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⢀⠄
⠀⠀⠀⠘⠓⠀⠀⠖⠈⠉⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠀⠀⠀⢠⠄⡄
⠀⠀⠀⠀⠀⠀⡌⡀⢀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠙⣠⢤⠀⢠⢬⡜⠁⠀⠀⠀⠀⠀⢰⣫⠋⡍⢳⣄
⠀⠀⠀⠀⠀⢀⣅⢼⢣⣠⡇⠀⠀⠀⠀⠀⠀⠘⢄⣹⢶⡊⣲⢾⣅⠜⠀⠀⠀⠀⠀⣸⣯⡁⢉⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⠟⢻⡋⠄⠀⠀⠀⠀⠀⠀⠈⠋⠁⣐⣿⣟⡀⠉⠋⠀⠀⠀⠀⠀⣼⣿⡉⢇⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠫⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡄⣼⡗⠀⠀⠀⠀⠀⠀⠀⠈⠈⣙⠟⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⢄⣠⣂⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢮⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⣟⠢⠀⢀⡠⢶⣶⣶⣦⣤⣾⣿⡓⢦⢄⠀⠀⠆⣿⣶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣭⡿⣶⡿⣂⣴⣿⣯⣽⣿⣧⡛⢾⣶⣥⣻⣶⢿⢧⠹⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⠷⣿⣷⣿⣿⣿⣥⠞⢹⠇⠹⣿⣶⡌⢻⣿⡍⠛⠩⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣧⣺⢜⡿⠋⣿⣥⣔⠊⠀⠀⣘⣟⣿⡀⠋⠁⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⣩⣯⡤⠺⠇⠴⠦⠉⠀⠉⠴⠎⠙⢧⣢⣤⣬⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢉⡰⠭⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢂⡈⠋
⠀⠀⠀⠋⣿⠀⣀⡀⣀⣼⠀⡂⣠⣄⢀⣠⡀⠀⠀⠉⠁⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠰⠧⠜⠧⠙⠧⠿⠘⠷⠿⠾⠼⠧⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣄⣄⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡴⢿⣅⣥⣍⡽⡿⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁

⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢽⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣦⣘⠀⠀⠀⠀⠀⠀⣀⣞⡡⠤⠤⣧⣄⣀⠀⠀⠀⠀⠀⢜⡁⣠⣄⢈⡣
⠀⠀⠸⣿⣿⣿⣿⣿⣿⡖⠁⠀⠀⠀⠓⠻⣿⣻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠇⠙⢁⣄⡜⠿⠹⡀⠀⠀⠀⠀⠀⠈⠹⢿⣿⠏⠁
⠀⠀⠀⠘⣿⠋⢁⡗⡹⠀⠀⠀⠀⠀⠀⠀⠹⡹⠁⢀⢋⠏⠀⠀⠀⠀⠀⢠⠏⢀⠾⡅⠀⢤⡀⠀⠀⠀⠀⠀⠀⠄⠈⠀⢠⠈
⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡾⣻⡿⣄⣀⣐⠋⠀⠀⠀⠀⠀⠠⠆⠓⠛⣿⣷⡆⠀⠀⠀⠀⠀⢀⡀⠀⣬⣿⣷⣀⣱⠀⠀⠀⠀⣠⣶⠚⠋⣭⣄⣄
⠀⠀⠌⢀⣻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠐⠂⠐⠂⣼⠿⣿⠃⠀⠀⠀⠀⠠⠟⣿⣿⣶⣾⣿⡯⠨⠀⠀⠀⠀⠿⣿⣦⡤⠉⠙⣿⡿
⠀⠀⠀⠸⠁⠙⠛⠛⢧⡀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠘⢦⠀⠀⠀⠀⠀⡎⢘⠉⠉⣛⢿⡲⠀⠀⠀⠀⠀⠀⠀⠈⢭⠴⠚⠛⠁

⠀⠀⠀⠀⣀⣀⣤⣤⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⢠⣠⠀⠀⠀⠀⢀⡀⠀⠀⣀⣠⣀⡄
⠀⠀⠀⠞⠹⠿⣿⣿⠿⠄⠀⠀⠀⠀⠈⣿⣿⣿⣦⠀⣀⠀⠀⠀⠀⠀⠀⠙⠀⢀⣤⡄⠀⠘⠉⠀⠀⠀⠀⠺⠗⠤⢖⣿⣿⣽⡇
⠀⠀⠀⠀⣠⡾⠛⢿⡄⠀⠀⠀⠀⠀⣠⣈⣼⣿⣿⣷⣼⠗⠂⠀⠀⠀⠀⠀⠀⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠳
⠀⠀⢀⡾⠋⠀⠀⠀⠙⢦⠀⠀⠀⠀⠺⠤⠏⠍⠉⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠄⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣄⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⡶⢶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠋⠀⠀⠀⠉⠉⠀⠀⠀⠀⠘⡟⠿⢿⠿⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣶⠟⠛⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠂⠀⠀⠱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣾⡟⠃⠀⠀⡀⠀⠂⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣌
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣛⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣽⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡀⠁⠀⠀⡀⣠⠤⢄⠀⠀⠀⠴⠒⠲⠄⠀⠀⠀⠀⠀⢢⠀⢀⣀⠀⠁⠀⢀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⠆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠿⠛⠛⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣹⣿⡟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⡉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣤⣀⡀⠀⠃⠀⡀⢀⣀⣀⣠⣤⡴⠾⠛⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠀⠀⠚⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⢸⣹⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⡆⢰⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣏⣿⣽⣿⠿⢿⣿⣟⡇⢙⣿⣿⠿⠿⣿⣯⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⢿⣿⢿⣛⣯⣽⣶⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣶⣯⣽⣻⠿⣿⡿⣠⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⢟⣄⢰⣿⡟⠋⠁⠙⠛⠉⠛⢿⣿⡇⢸⣿⠁⠈⠉⠉⠉⠙⠻⣿⣿⡆⣠⡻⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣿⡿⣫⣾⣿⣿⣧⢻⣿⣶⣄⠀⡀⠀⠀⠘⢿⡇⣸⣟⣀⣄⠀⠀⠀⠀⠀⣿⠟⣼⣿⠿⢷⣝⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⣵⣿⣿⠿⠿⠓⢾⣷⡹⣿⣶⣶⣷⣶⣿⡿⢣⡇⢸⣷⣾⣿⣿⠀⣼⣷⣶⢎⣾⣿⠅⢿⣿⣿⣿⣮⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣫⣾⣿⣯⠁⠀⠀⠀⢈⡹⣷⡙⣿⣿⣿⣿⣶⣾⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⢃⡾⣫⠅⠀⠀⠙⠿⠿⠟⣷⣝⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣤⣝⠛⢵⣿⣯⣝⠁⠀⠀⠀⣀⢶⣽⣾⣿⡜⠿⢛⣫⣭⣵⣶⡆⠀⠀⠈⠉⠙⠛⠿⢣⣿⣿⣿⣿⣿⠄⠀⢀⣠⢄⣻⣿⡮⠛⣫⣤⡀
⠀⠀⠀⠀⢀⣾⣿⣟⣴⣶⣭⣛⢯⣾⣶⣤⡜⢿⣧⣿⡿⠋⠁⠈⢿⣿⣿⠿⠿⡇⢀⠀⠀⠀⠀⠀⣠⣮⣙⢿⣿⣿⣯⣦⣾⣿⣿⠿⣛⣭⣶⣦⢻⡿⣷⡀
⠀⠀⠀⠀⣼⢻⡿⣼⣿⣿⣿⣿⣷⣮⣝⠻⣿⣾⡿⠋⠀⠀⠀⠀⣈⠱⠖⣛⡉⠥⠬⣍⣛⡳⠆⣀⠿⣿⣿⣷⡝⢿⣿⣿⠟⣫⣵⣾⣿⣿⠿⣿⣯⢿⣿⣧
⠀⠀⠀⢰⣿⣿⢳⣿⣿⣼⣿⡏⣤⣉⣽⢻⣶⡍⡁⠀⠀⠀⣴⢟⡥⠚⠉⠀⠀⠀⠀⠀⠉⢻⣷⣬⠳⣌⠻⣿⡿⠊⢩⣶⣿⣿⠟⠉⠁⢘⣾⣿⣿⡞⣿⣿⡄
⠀⠀⠀⣾⣿⡟⣿⣿⣿⣿⠟⠀⣿⣿⣿⣿⡟⣰⣿⣷⢄⡘⣡⠋⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⣿⣿⣿⣌⢃⡀⠀⠀⠀⢿⡯⣭⠀⠀⠐⠃⠉⠉⠛⣷⢻⣿⣷
⠀⠀⢀⣿⣿⣧⣿⣿⣟⠀⣴⡈⣿⣿⣿⣿⢇⣿⣿⡟⣼⢱⠃⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢀⣿⣿⣿⣿⡌⣧⠀⠀⠀⠸⣿⣿⣿⣧⣶⣶⣶⡔⣤⣿⣼⣿⣿
⠀⠀⠸⠿⠿⠹⠿⠿⠿⠶⠿⠧⠿⠿⠿⠿⠸⠿⠿⠧⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣧⠸⠄⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠇
⠀⠀⢰⣶⣶⣰⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠐⡆⢃⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⢲⣶⣾⡆⣶⣶⣶⠶⣒⣒⠶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⣿⣿⡟⣿⣿⡟⠙⠿⠟⠛⣠⠼⣿⡆⠀⠀⠀⢿⡘⡄⠀⠀⠀⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⢃⡟⣼⣿⣿⢱⣿⠿⠃⠘⠛⢿⣿⣎⢿⣿⢻⣿⣿
⠀⠀⠀⢿⣿⣧⣿⣿⠄⢀⣀⠀⠀⠀⠀⣻⣷⠀⠀⠀⠈⢡⡙⣄⠀⠀⣿⣿⣿⣄⣠⣿⣿⣿⣿⣿⡿⢋⡌⠱⢿⣿⠏⣾⡏⠲⠀⠀⠀⠀⠛⢻⣾⡿⣼⣷⡿
⠀⠀⠀⠸⡿⣿⡼⣯⣤⣸⣿⣧⣤⣤⣤⣿⠿⣃⡠⣾⣿⣧⡙⢮⡓⢤⣘⢿⣿⣿⣿⣿⣿⣿⡿⢋⡴⠋⠀⠀⠀⢀⣘⠿⣷⣿⣤⣈⣤⡄⠀⣀⣿⢧⣟⣿⠇
⠀⠀⠀⠀⢻⡾⣷⣻⣿⣿⣿⣿⡿⢟⣫⣴⢿⣿⣷⣜⢿⣿⣿⣶⠉⠰⠦⣭⣉⡛⢛⣉⣭⠴⢆⡉⠀⠀⠀⠀⣠⣾⠛⠻⣦⣝⡻⢿⣿⣿⣾⣿⣟⣾⣿⡟
⠀⠀⠀⠀⠈⢿⣿⣯⠻⠿⣛⣥⣾⡿⡿⠉⠚⢿⢿⣿⣷⣍⡻⠋⠀⠀⠀⠀⠀⠁⢰⣶⣶⣿⣿⣷⠀⢀⣠⡾⢿⣿⣿⠆⠙⠻⢿⣶⣭⣛⠿⢟⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠛⣫⣤⡺⣿⣿⣭⣕⢇⠸⣳⠀⣿⣿⣿⡿⢣⣶⣤⣄⣀⡀⠀⠀⠻⠿⢟⣛⣫⣥⣶⡜⣿⣷⣦⠛⠀⠀⢀⣴⣿⣿⣿⣿⢗⣤⣝⠛⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣝⢿⣿⣿⣞⠆⣹⣯⣔⣹⡿⣡⣿⡿⠿⠋⠉⠛⢿⡇⢸⣿⣿⣿⠻⢿⣿⣿⣌⢿⣿⣆⠀⢠⡿⣿⣿⣿⡿⣫⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡻⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⡀⠀⠀⠈⠃⢠⡇⢸⣿⣿⣿⠃⠈⢹⣿⣿⣎⢻⣿⡧⠀⣱⣿⡿⢫⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢷⣝⢿⣿⣿⡟⣼⣿⣿⡿⠃⠀⠀⠀⣤⣼⡇⢼⣟⣋⠁⠀⣴⣿⣿⣿⣿⣧⢻⣾⣿⡿⣫⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣮⠋⠸⣿⣿⣏⣁⠀⣀⣤⣷⣿⣿⡇⢸⣿⣿⡗⠀⢸⣿⣿⣿⣿⣿⠇⠙⣵⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⣾⣿⣶⣯⣟⣻⠿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣷⣿⠿⣟⣻⣽⣶⣿⣷⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣶⣶⣿⣿⡅⢨⣿⣿⣶⣶⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠇⠸⠟⠛⠛⠛⠋⠉⠁

⠀⢠⠴⣄⠀⣤⢦⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠠
⠀⠀⠀⠸⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡎⠀⠀⠀⠀⠀⠀⠀⠀⢊⣀⣥⡤⢤
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠸⠥⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣢⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠁⠈


⠀⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠐⠳⡖⠳⡖⢳⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠦⡄⠀⠀⠀⠀⠀⠀⠚⢶⠚⣶⠛⡆
⠀⣠⠖⠲⣄⡼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠸⡏⡷⠀⠀⠀⠀⠀⢸⡅⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠀⡇
⠀⠻⣄⣀⠟⢇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⢁⠀⠀⠀⠀⠀⣉⣥⣤⣤⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠑⠑⠂


⠀⠀⠀⠈⠒⣲⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠀⠀⠀⠀⠀⠀⠀⠴⠒⢦⣀⣠⠖⠲⠄⠀⠀⠀⠀⠀⠀⢠⡀⢠⠂
⠀⠀⠀⢤⡞⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠐⢤⠀⠀⠀⠀⠀⠀⣠⢤⣀⠉⢁⣤⢤⡀⠀⠀⠀⠀⠀⠀⢺⡷⢿⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠂

⠀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠉⠉⠀⠉⠉⠉⠁⠀⠀⠀⠐⠀⠁⠁⠁⠉⠉⠉⠀⠀⠀⠀⠀⠉⠋⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⡁⠉⠀⠼⠛⠛⢼⣿⣯⡻⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢿⣷⠈⢿⣿⣿⣿⣷⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣯⣽⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠛⠻⠿⢶⣦⠀⠀⢀⣤⡶⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠚⠓⠂⠀⠀⠀⠀⠠⡔⢦⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿
⠀⠀⠀⠀⠀⠠⠈⣿⣿⣷⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣾⣿⣿⣿⣿⡟⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠟⠛⠛⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣤⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⡀⣀⢀⠀⡀⢀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⡏⢸⡇⣯⠻⢸⠀⡏⢸⡗⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠀⠉⠈⠀⠁⠈⠈⠁⠀⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠤⢽⣋⣛⣛⣛⣙⣋⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⡀⢀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡖⠒⠒⠲⠿⣦⠀⡀⢸⡇⢀⠀⣴⠿⠖⠒⠒⢲⣶⡤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⠾⢿⣷⣄⠀⢀⣀⣤⣭⣾⣷⣭⣤⣀⡀⠀⣠⣾⡿⠷⣴⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣏⡟⢁⣈⣻⣿⣿⣸⡏⠈⠉⢹⠞⠳⡏⠉⠁⢹⣇⣿⣿⣟⣁⡈⢻⣹⡄
⠀⠀⠀⠀⠀⠀⢸⡞⣧⠀⢙⡿⠙⢀⡙⡸⠀⠀⢫⡀⢀⡝⠀⠐⢇⢋⡀⠋⣿⡉⠀⣼⢳⡇
⠀⠀⠀⠀⠀⣼⠷⢿⣮⡻⠏⠀⡰⡋⠀⠀⠀⠀⠈⠅⠨⠁⠀⠀⠀⠀⢙⢆⠀⠹⢟⣵⡿⠾⣧
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣸⡽⠀⢤⣤⣤⡀⠀⠀⠀⠀⢀⣤⣤⡤⠀⢯⣇⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⣿⢁⠀⠀⠀⠀⠀⢹⡇⠀⠀⠳⠀⠇⡆⠀⠀⢰⠸⠀⠞⠀⠀⢸⡏⠀⠀⠀⠀⠀⡈⣿
⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⢳⣤⡀⠀⣀⡞⠀⠀⠀⠀⢳⣀⠀⢀⣤⡞⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⣿⢸⡀⠀⠀⠀⠀⠀⠀⢿⣓⣼⣼⡧⠄⠀⠀⠠⢼⣧⣧⣚⡿⠀⠀⠀⠀⠀⠀⢀⡇⣿
⠀⠀⠀⠀⢻⡎⢛⣫⣭⣛⢦⡀⠀⠘⣿⡭⠁⠱⠛⠦⠴⠛⠏⠈⢭⣿⠃⠀⢀⡴⣛⣭⣝⡛⢰⡟
⠀⠀⠀⠀⠀⠻⡾⠁⠀⠈⡇⣷⠀⠀⢹⣽⠖⢂⠀⡠⢄⠀⡐⢲⣯⡏⠀⠀⣾⢸⠁⠀⠈⢷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣱⠏⠀⠀⠀⢹⣯⡶⠞⠁⠈⠳⢶⣽⡏⠀⠀⠀⢹⣎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⢸⣀⡄⠀⠀⠀⠀⢠⣀⡇⠀⠀⠀⠀⢉⡿⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢷⡀⠀⠀⠀⠀⠙⢿⢿⣸⣇⡿⡿⠋⠀⠀⠀⠀⢀⡾⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⢦⡀⣠⠔⣒⡂⠀⠙⠋⠀⢐⣒⠢⣄⢀⡴⠋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣙⠁⡟⠉⠙⡆⠀⠀⢰⠋⠉⢻⠈⣋⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠄⣾⠁⢰⡆⠈⣷⠠⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣿⣿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣴⣢⣤⣤⣤⣄⡀⠀⠀⠀⢰⣄⣀⣀⡦⠀⠀⠀⠀⠀⠀⠀⢲⣴⣶⣴⠂⠀⠀⠀⠀⠀⠀⠀⣠⢂⣾
⠀⠀⠀⠀⡀⠀⠀⠀⣸⡿⠛⡿⣿⣿⣿⣿⣿⣆⠀⠀⢹⣿⣿⣿⡁⠀⠀⠀⠤⠀⠀⠀⠻⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠿⢻⠟
⠀⠀⠀⠀⢀⠀⠀⠀⠋⢀⡜⢁⠇⢹⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⠀⠀⠠⣴⣾⣿⣦⣴⣶⣿⣶⣄
⠀⠀⠀⠄⠀⠐⠂⠀⠀⠀⡠⡀⠀⠚⣽⠋⢩⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⣹⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡁⠀⠀⢀⡠⠏⠀⠀⠀⠀⢸⡆⣿⢩⡟⣿⠀⠀⠀⠀⠀⠰⠿⢠⡏⣻⣿⣿⠿⠁⠀⠀⠠⢞⡽⢁⡤⠟⣻⡿⠉⠋
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣦⠺⣇⠀⠀⠀⠀⠀⣀⣀⠀⢀⣴⡾⠻⣶⡄⠀⠀⣰⣶⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⡿⠿⠍⡂
⠀⠀⠀⠀⠛⠀⠀⠀⠀⣿⣿⣁⡈⢿⣷⣿⣿⣶⡄⠀⣸⡟⢻⡇⢹⡿⢃⡀⠸⣿⠀⢸⣿⣿⣿⣄⣀⣀⣄⣀⣤⣴⠀⠀⠘⢹⣿⣷⣄⣀⣀⣀⡤
⠀⠀⠀⠀⠒⠀⠀⠀⣰⣿⣿⡟⠹⠆⠀⣴⠞⠛⣿⠀⢸⡆⢸⡇⠀⠀⠧⣹⢀⣿⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⠉⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠠⡀⠀⠁⠀⠀⠀⢸⣦⠿⣿⠀⢰⣧⠀⠀⣴⠏⠀⠈⢿⣌⡻⠦⠤⢖⣣⡾⠃⠀⠸⣇⢸⡟⠛⠛⠻⣿⢿⡄⠀⠀⠀⢠⠞⢻⠟⠛⠻⢿⣿⠁
⠀⠰⠁⠀⢀⡀⠀⠀⠀⠀⠀⠘⠃⠀⠛⠓⠚⠉⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠿⠸⠁⠀⠀⠀⠛⠀⢸⠀⠀⠀⡏⠛
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠑⠀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠈⠂
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⡀⣀⣤⣀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠘⣇⢿⣿⠏⠀⢀⡀⠀⠀⢠⣾⡟⠁⠀⣀⣤⣤⣄⠀⠀⢠⡟⢻⠆⠀⠀⠀⣿⣿⡇⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠆⠀⠀⠀⢀⡤⠤⡌⢱⣿⣞⠛⠉⠁⠀⠀⣿⣿⣷⣄⣰⣿⣿⣿⣿⡆⠀⠸⢿⢶⣿⣿⣶⣾⣿⣿⠃⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠄⠀⠀⠀⣏⢰⣃⡼⠘⣿⣿⡷⢶⡄⠀⠀⠹⣿⣿⣿⣿⡿⢿⢻⣿⠃⠀⠀⠀⣸⣿⡿⠿⢿⡿⣏⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠏⠈
⠀⠀⠠⠀⠀⠀⠀⠀⠙⠦⢤⠤⠖⢹⣏⣀⠺⠇⠀⠀⠀⢈⠛⢛⠁⠀⠀⠈⠃⠀⠀⠀⠀⢻⡘⣧⠀⠀⢷⡘⠦⠀⠀⠀⠛⠼⠁⠀⠼⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⡤⠠⠄⠠⠀⠀⠀⠀⠀⠑⠀⠀⡐⠤⠄⠀⠀⠀⠀⠀⠀⡀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠆⠀⠀⠀⠂⠀⠀⠀⠀⠐⠐⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠿⠿⠿⠛⠛⠿⠿⠿⢷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠀⠀⠀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⢺⠷⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣴⡄⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢀⠀⠠⠦⠀⢀⠀⠀⠀⠘⠀⠐⠂⢀⠀⠀⠈⣴⡆⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠁⠂⠀⠀⠀⡀⠀⠀⢉⣴⣶⠿⠿⠿⢷⣶⣄⠘⢆⠀⠘⠇⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠰⠆⠀⠀⠀⢀⡀⠁⠀⢠⣿⠏⠀⣀⣀⣀⠀⠙⢿⣆⠀⣶⡄⠀⠰⠀⣴⣶⡄⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢠⣄⠀⠀⣴⣤⠀⢀⣡⣴⣾⣿⠀⢰⣿⣿⣿⡇⠀⢸⣿⡀⢉⡀⠀⠀⠀⠈⠉⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠉⠀⠀⠉⠁⣰⡿⠋⠁⠀⠀⠀⠀⢿⣿⣿⠇⠀⣼⡿⠀⠻⠃⠀⠀⠀⡀⢀⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣴⣤⠀⠀⢰⣿⠃⠀⣴⣿⣿⣦⠀⠘⣿⠏⠀⣰⣿⠃⠐⠂⠀⠀⠀⢈⡁⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠁⠀⠀⠘⣿⡄⠀⠻⣿⣿⠟⠀⢠⠏⠀⣰⣿⠃⠀⠀⠀⣀⠈⠁⠈⢡⣄⢀⡀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⣄⠀⠀⠀⠀⢹⣿⣦⣀⡀⢀⣀⣴⡏⠀⢰⣿⣷⣶⣶⣦⡌⠛⠀⠀⠀⠈⠁⣬⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠙⠁⠀⠶⠀⠀⠀⠀⠙⠛⠻⠟⠛⣿⡇⠀⢾⣿⣿⠀⠀⣿⡇⠀⠀⡀⠀⣤⠀⠉⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠈⠀⢰⡦⠀⠆⠠⣶⠄⠀⠂⢻⣧⠀⠈⠛⠋⠀⣠⣿⠃⠀⠈⠉⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠃⠀⢰⣶⠀⠀⠀⠀⠤⠀⠰⠀⠻⣷⣦⣤⣤⣾⡿⠁⠀⠀⣀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠈⠉⠉⠉⠐⠂⠀⠀⠀⠉⠁⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⢀⠸⠆⠒⠀⠉⠀⡙⠡⠄⠀⢰⣶⡆⠀⠀⢀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠀⠶⠀⠀⠁⠀⠀⠀⠀⢉⣠⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢷⣶⣶⣶⣶⣶⣶⡾⠿⠟⠛⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⣀⣤⣶⡿⠟⠋⢉⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢠⣾⡿⠋⠁⢀⣴⠾⠛⠛⠛⠙⠛⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣿⡏⠀⠀⡴⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⡇⠀⠘⠁⠀⣠⠔⠚⠋⠉⠙⠛⠷⣦⣄⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠛⠦⠀⠀⠈⠀⠤⠴⣶⣦⣤⣀⠀⠈⠻⣷⣄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠂⠀⠀⠀⠀⠀⠀⢤⣤⣈⡙⢿⣷⣄⠀⠘⣿⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⠘⢿⣧⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⠀⠀⠀⠀⠀⢀⡄⠀⠀⡀⠘⡇⠀⢻⡄⠀⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠀⠀⢠⡇⠀⠀⠀⠈⠁⠀⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠖⠀⠀⡠⠀⠀⡀⠀⡾⠁⠀⠀⣠⡿⠀⢠⠂⠀⠀⠀⢰⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣼⡁⠀⠀⠃⠘⡁⠀⣠⣾⠏⢀⣴⠏⠀⠀⠀⢀⣾⠃⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡼⢠⣾⠟⠁⣠⡾⠃⠀⠀⠀⢀⡾⠃⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣠⣾⠟⠀⠀⠀⢀⡴⠟⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢰⣿⠃⠀⠀⢀⠴⠋⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠀⠀⠀⠔⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠙⢿⠀⢀⣴⣿⠁⠀⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⡀⠀⣿⣿⡧⢼⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣴⡇⠀⣿⣿⡇⠈⠀⣰⡇⠘⣿⣿⠀⠀⣸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠐⠛⠛⠃⠐⠛⠛⠃⠀⠚⠛⠃⠀⠈⠙⠃⠐⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⠀⠤⠦⣤⣄⠀⠀⣠⣤⠶⠤⢠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⢀⡴⢠⠄⠉⠀⠀⠉⠠⣴⠦⠀⠃⠀⠀⢸⣿⣟⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠟⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠙⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣽⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠙⣏⢼⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠉

⣷⣟⣛⡿⠿⠿⠿⠿⠷⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣄⣀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠛⠉⠉⠉⠙⠋⠉⠉⠀⠀⠈⢻⣻⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣴⠿⠛⠛⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡠⠒⠀⠍⢲⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⣾⡿⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡨⠀⠀⠀⠀⠀⡄⠀⠀⠀⣿⣷⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣾⣿⡏⠀⠀⠀⠘⠁⠀⠀⠂⡀⠀⠀⠈⠁⢠⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠅⠐⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠂
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇⠀⢀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⢻⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣾⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠄⠀⠠⠞⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠂⣼⣿⣿⣿⣿⡏⠰⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠄⠀⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⣀⡀⣀⡁⠚⠉⠑⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠗⣠⠶⠚⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢿⣿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣾⣿⣿⣿⣦⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠛⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⠟⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⡀⠀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢛⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠻⠶⣦⣤⣀⡀⠀⠃⠀⢁⣀⣀⣀⣀⣤⣴⠶⠿⠋⠅⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠀⠚⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣸⡵⣿⡇⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠃⠃⠘

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



⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⢀⡤⠖⠒⠂⠀⠀⠀⠀⠀⠀⠄⠀⢠⡀⠠⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢿⣷⠀⠀⠀⢠⡆⠀⠚⢻⠹⠀⠀⠀⠀⠀⠀⣾⡿⠋⠉⣙⣿⣧
⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⢶⣄⡀⠀⠀⠀⠀⠀⣠⡴⢦⡀⣿⠇⠀⠀⠀⢰⣿⣷⣶⣷⣤⢤⠀⠀⠀⠀⣿⣧⠠⡤⢁⣾⣿
⠀⠀⠀⠀⠈⠁⠙⠻⠟⠟⠀⠉⠀⠀⠀⠀⠹⣄⣿⣀⣀⡿⠏⠀⠀⠀⠀⠉⢍⠉⠉⣙⠕⠃⠀⠀⠀⠀⠘⢿⣧⣤⣼⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⣴⣿⣿⡏⠐⢠⠀⠀⠀⠀⠀⠰⢾⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠰⣾⣦⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠐⠄
⠀⠀⠀⠀⢸⠋⠛⠉⣿⣶⣶⡄⠀⠀⠀⢠⣾⠉⠁⠈⠈⠟⠓⠀⠀⠀⠀⡄⠀⣻⣿⡐⠀⠀⠀⠀⠀⠈⡚⠀⠀⣀⠀⠐⡃
⠀⠀⠀⠀⠘⢧⣤⣷⣤⠟⠋⠀⠀⠀⠀⠀⢉⠀⠐⠓⠢⠤⠆⠀⠀⠀⠀⠓⠄⠉⠿⠁⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡿⠿⠄
⠀⠀⠀⠀⠀⠈⠙⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠈⠀⠛⠺⠿⠖⠋

⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⣀⣤⣄⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⣴⣍⣁⣨⣭⠲⠀⠀⠀⠀⣸⠏⣝⠛⠉⣋⠉⢷⠀⠀⠀⠀⢸⠋⠉⡏⠉⠉⠁⠀⠀⠀⠀⣠⠿⠄⠀⠤⠧⡄
⠀⠀⠀⠀⠸⢳⣿⣗⣺⣿⡲⡁⠀⠀⠀⢹⢾⠿⡅⠈⡹⢿⣾⠀⠀⠀⠀⣀⠀⢠⣷⢀⣀⠀⠀⠀⠀⠈⢿⣇⣀⣀⡀⣸⡍
⠀⠀⠀⠀⠀⠋⠟⠹⢝⡟⠅⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠉⡉⠈⠉⠈⡉⠁⠀⠀⠀⠀⠘⢽⠋⠉⠙⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣭⣿⣯⣉⠉⠛⠳⢶⣤⣀⠀⢀⣠⣴⠶⠛⠋⢉⣩⣿⣯⣅⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣭⣀⠉⠉⠙⠛⠷⢦⣌⠉⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠈⢉⣥⠶⠟⠛⠉⠉⢉⣨⣭⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠉⠛⢦⣀⣠⠖⠛⢉⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣤⣀⣈⠙⠳⢦⣄⣠⠞⠋⠁⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⣠⡾⠋⠐⢚⣿⣿⣯⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣭⣿⣿⣟⠒⠈⠻⣦⡀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢠⡾⠋⠀⣠⡾⠋⠁⠀⠈⠙⠿⣦⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠉⠀⠀⠉⠻⣦⡀⠈⠻⣦⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠐⠿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠗⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣸⣿⠛⢻⣷⣦⣄⡀⠀⠀⠀⠀⢿⡀⣸⠇⠀⠀⠀⠀⣀⣤⣶⣿⠛⠻⣿⡀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢠⣿⠃⠀⠘⣿⠀⠙⣿⣷⡶⣤⣀⠘⣧⡟⢀⣠⡴⣾⣿⡟⠁⢸⡟⠀⠀⢻⣧⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢰⡏⢀⣠⡤⠀⠀⣼⣿⣆⠀⠀⠙⠳⠦⠼⣿⣧⠀⠉⠛⠿⠟⠋⠁⢠⣿⡿⠤⠶⠛⠁⠀⢀⣾⣿⡄⠀⠠⣤⣀⠈⣷
⠀⠀⠀⠀⠀⠀⣾⠟⢋⡿⠁⠀⠀⣿⠇⠻⣧⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⢠⡿⠃⢿⡇⠀⠀⠹⣏⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠐⢿⡄⠀⠹⣧⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢠⡿⠁⠀⣼⠗⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠻⣦⠀⢻⡆⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣾⠃⢠⡾⠋⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠘⢷⣘⣿⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⢸⡟⣰⠟⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⣠⡿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣧⡀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠘⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢴⡄⠀⠀⠀⠀⣿⡋⠀⠀⠉⠛⠻⣿⣷⣶⣶⣿⡿⠛⠋⠁⠀⠈⣻⡇⠀⠀⠀⢀⣴⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⠙⣦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢹⣿⣿⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡞⢹⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⠻⠿⠿⠿⣷⣄⡀⠀⠀⢀⣤⡿⠋⠻⣧⣄⠀⠀⠀⣀⣴⡿⠿⠿⠿⠋⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡟⠋⠁⠀⠀⠀⠀⠉⠛⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⣶⣶⣶⡶⠾⠛


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⢰⡿⠻⣦⢀⣴⠟⣷⡄⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠙⢻⡿⠻⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠿⠿⠷⢶⠄
⠀⠈⠛⠀⢻⣿⡇⠐⠟⠀⠀⠀⠀⠀⠀⠀⢈⣿⡶⢾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⣻⡇⢠⡾⠿⣶
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⡛⠋⠀⣘⣷⣶⡿
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠟⠛⠛⠻⠷⣦⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁


⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⢠⣶⢶⡶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣶⢶⡶⣶⡶⣦
⠀⠀⠀⣾⡏⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣿⢷⡄⠀⠀⠀⠀⠀⠀⣠⣍⣉⡛⠛⣋⣉⣥⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⣿
⠀⠀⣴⢿⣧⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⣿⣸⡏⠀⠀⠀⠀⠀⠀⠈⢹⡿⠁⠀⠹⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⣿
⠀⠀⠿⣶⡿⣾⡇⣠⡤⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⠁⣿⣤⣄
⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠁

⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣤⡀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣠⡄
⠀⠀⠀⡀⠈⢉⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⢿⣾⠋⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣦⡿⣦⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⢸⡟
⠀⠀⠈⢻⣶⡟⠁⠘⠗⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⡿⠛⣷⡆⠀⠀⠀⠀⠀⠉⢉⣤⣉⣤⣈⣡⣌⠁⠀⠀⠀⠀⠀⠀⠐⠾⢿⡿⢿⡿⠿⠂
⠀⠀⣴⠟⠙⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣼⠟⠿⠛⠁⠀⠀⠀⠀⠀⠶⠾⠛⠿⠋⠻⠟⠻⠆⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠠⠤⠴⠄⠂⠒⠲⠀⠠⠶⠤⠄⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠣⡰⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⠀⠀⠀⠀⠀⠀⠓⠈⣷⠉⠅⠀⠀⠀⠀⠀⠀⠀⠐⣤⠊⠘⠃⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡐⢣⣴⠶⠛⣛⠛⠒⢤⡠⠄⣈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢫⣽⣿⣿⣿⣷⡀⠀⠐⠂⠀⢰⡛⣿⡘⢣⠉⡆⠀⡀⠱⠀⠀⠀⠀⠂⠄⠀⡀⠀⠀⠀⠀⠀⠀⢈⠳⢄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢩⣼⡟⣃⣾⡿⠟⠅⠈⠀⠀⠀⠀⣼⡟⣻⣷⣾⠟⠓⠪⠁⢾⡃⠀⠀⠀⠀⠀⠁⠪⢀⢄⠀⠀⢀⡴⢋⠟⡮⠳⡄
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣧⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡔⣿⠃⠒⢀⣄⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢡⠎⠀⠀⠘⢆
⠀⠀⠀⠀⡴⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⡀⣀⣠⢼⡟⠻⡰⠀⡈⠙⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⠄⠀⠀⠀⠈⢧
⠀⠀⠀⡼⠀⠀⠀⠉⠛⢿⠟⠅⠀⠀⠀⠀⣀⣄⠚⢋⢥⣌⣿⡘⣿⠀⠣⢔⣓⣤⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠀⠀⠀⠀⠀⠀⢣
⠀⠀⡸⠁⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣀⣜⣯⠐⣀⣠⢌⣣⣿⣿⣻⣷⡄⠀⠐⣸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠀⠀⠀⠀⠀⠀⢧
⠀⢠⠇⠘⠢⣄⣀⠀⠠⠀⠀⠀⠀⢰⣿⣅⡤⠦⣤⣹⣿⡿⣻⣫⣾⠫⣽⣿⡿⣛⣽⣿⢿⣿⡕⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⣀⣀⠘⡆
⠀⡼⠀⠒⠺⠢⡀⢠⠃⠀⠀⠀⢠⣾⡿⠉⠀⠀⠀⠈⠿⡽⣱⡟⢦⣿⣿⢏⣴⣿⡿⢍⣶⠿⠉⢢⢴⡖⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡇⢁⣀⠇⢛
⢀⡇⠀⠀⠀⠀⠀⡌⠀⠀⠀⢀⡟⣽⠀⠀⠀⠀⠀⠀⢸⣇⡟⢬⣿⡿⣱⣿⣿⠯⣹⠞⡗⡀⣪⣾⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠘⠀⠀⢸
⠈⠃⠀⠀⠀⠀⠀⢁⠀⠀⠀⣸⡇⣿⡀⠀⠀⠀⠀⠀⢸⠟⣿⣷⣿⢳⣿⡿⢍⣾⣫⠁⢄⢾⣽⢿⠅⣸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠰⠄⠄⠀⠀⠀⠠⠐⠀⠀⢀⣿⢿⣿⣧⠠⢄⠀⠀⠀⢸⣿⣿⣿⢇⣿⣿⣝⣲⣯⠅⢣⣟⣿⣃⢨⡴⠇⡼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐
⠈⡀⠀⠀⠀⠀⠀⡈⠀⢠⣾⠙⣸⡟⣿⠀⠀⠀⠀⠀⠸⡇⢿⡾⡿⣿⣟⢢⡟⡔⡈⣞⣾⠋⢃⡾⡄⡟⠭⣰⠏⠁⡗⠂⣴⡄⠀⠀⠀⠀⡀
⠘⡇⡠⠄⠀⠀⠀⢃⠀⠀⣿⣦⣿⠃⡻⡆⠀⢀⠀⠀⠀⣷⣤⡝⢾⣺⣷⣾⣱⠣⢹⢿⡟⢎⢸⣧⣽⣖⣹⠃⠀⡘⠀⣾⣿⠃⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢰
⠀⢼⠒⠤⢤⡤⠄⠘⡀⠀⠀⢸⣿⡀⢳⣷⠀⠀⠳⠀⠀⠸⠦⠿⠷⠽⠿⠿⠟⠛⠛⠉⠁⠀⠀⢤⡤⠋⣱⣄⠀⡧⢰⣿⠏⠀⠀⠀⠀⠀⠆⡀⠬⢍⣓⣶⡘
⠀⠸⣠⠊⠁⠀⠀⠀⠰⠀⠀⢠⢹⣿⣅⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣴⣿⣿⣗⣁⣾⡄⠀⠀⠀⠀⠀⡌⠈⠙⢖⢲⡬⢧⠇
⠀⠀⢫⠉⠁⠀⠀⠀⠀⠑⠄⠘⠻⣿⣿⣸⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠍⠁⠿⣿⣿⡏⠀⠁⠰⠄⠀⢀⢎⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠪⠀⠀⠙⠛⠿⢿⣿⣦⣤⣴⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣀⢄⣪⠟⠏⠐⠀⠐⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠻⠿⠟⠋⠉⣙⠛⠻⠿⠿⠟⠛⠛⠉⠉⠉⠉⡉⠉⠈⠁⣀⡀⢀⠀⠒⡀⠄⠀⠊⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⠤⠤⡙⠄⠀⠀⠀⠀⠀⠀⡀⣻⠀⠀⡦⢠⢠⠄⠀⠤⢠⠄⢦⠠⠄⡄⢀⡀⠀⠈⠐⠘⡛⠀⠀⡄⢐⠄⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠘⢦⠠⠤⠼⡀⢀⡄⠀⠁⠐⡄⡀⢀⠀⠀⠃⠘⢒⠃⠚⠚⠘⠒⠃⠈⠀⠂⠘⠒⠃⠂⠁⢀⡠⡀⠊⠀⠊⠑⢄⠈⠉⢆⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡍⠁⠀⠀⠀⠀⠀⠐⠐⠤⣀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠑⣲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⠀⠒⠀⠀⠴⠔⠲⠤⠄⠉⠓⠀⣩⡤⠤⠄⡀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⢀⠇⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠃⠠⠀⡄⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡠⠾⢀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚⣋⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠒⠛⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⡶⠶⡶⢶⣶⣶⠶⣶

⠀⠀⣤⣤⡀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄
⠀⠀⠁⠈⢷⢸⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⠿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱⢀⣀⡀
⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⡏⠀⣹
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠴⠛⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠉⠁⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠊⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠉⠉⠁


⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⣀⣠⣄
⠀⠀⠀⠀⠸⠉⢻⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⠁⢸⡇⠘⣇⣀⡀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠉⢷⡀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⢹⠁⢹
⠀⢠⠞⠉⠻⣆⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⡇⢸⠇⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⢸⠀⢸⡄
⠀⠘⢦⣀⡠⠏⢧⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡋⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣇⢀
⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠄⠤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡫⠡⠤⠄⡄⠄⠄



⠀⠀⠀⠀⠙⠒⣲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠲⣤⣀⣠⡴⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⡔
⠀⠀⠀⠀⣠⠞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠉⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣧⣼⡧⠂
⠀⠀⠀⠰⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣸⠇⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⠦⠶⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠠⠒⣀⣀⡀⡤⣀⣀⢀⠀⠀⠀⠀⠀⠀⣠⢀⠀⣀⡀⠀⣀⡀⡀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⣀⢀⡀⡀⠀⠀⠀⠀⠀⠀⢠⠄
⠈⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠂⠈⠈⠀⠉⠁⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣄⢠⣠⣦⢰⢞⣄⣠⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠆⠙⠀⠤⢈⡃⠱⣻⣿⡌⢚⠡⠤⠀⠋⠠⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⠁⢀⠤⠄⠒⠂⠀⠲⡄⠀⠉⣼⠁⣰⣂⠤⣴⣒⠢⠤⣀⠨⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠔⠚⠁⣀⡆⠉⠤⢕⣶⣿⣝⠀⠃⣨⣕⣁⣀⠀⠈⢡⣀⠀⠓⠲⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣁⠤⠖⡠⠋⠀⢀⡶⠟⠛⠛⠻⠟⠹⠏⠉⠉⠛⢳⡀⠱⡙⢄⠒⠦⢈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⠔⠀⠐⡊⠀⣠⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠑⢤⣌⠂⢉⠂⠀⠡⣑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠻⠃⡔⠊⠀⣰⠏⡀⠐⢲⠿⠤⠀⡤⣤⣤⢤⠀⠄⠿⡖⠁⢨⣿⣏⠀⠑⢢⠘⢟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡷⠁⠐⠀⢀⣴⡏⡤⠁⢠⠁⠀⠀⠀⠙⢦⠴⠋⠀⠀⠀⠈⡄⠈⢤⢹⣖⠀⠀⠁⠈⢷⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠃⡀⠀⡄⠀⣼⠛⠁⠀⠀⠀⠀⠀⢀⣠⡼⢧⣄⡀⠀⠀⠀⠀⠀⠈⠂⣿⠀⢩⠀⠀⠘⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖⠉⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⢀⢰⡃⠃⣰⣆⠘⢨⡇⡀⠀⠀⠀⠀⢠⣿⠁⠄⠀⠀⠈⠲⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⠀⠀⢀⢷⡼⣯⡾⣦⠀⠀⠀⡸⠏⠀⢠⣿⣿⡄⠀⠙⢇⠀⠀⠀⣠⢸⣿⠃⠞⡀⠀⠀⠳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠛⡅⢠⡄⢸⠀⠁⢸⣧⢻⣧⣤⣾⡀⠀⠀⠈⠉⠉⠁⠀⠀⢀⣷⣤⡼⢱⡿⠁⠈⠀⠇⢠⡄⢨⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⠇⠀⡆⠀⡄⢹⡏⢡⣿⣿⣿⠄⡀⠀⢀⡀⠀⢀⡢⣿⣿⣿⡄⢸⠃⠀⠀⢰⠀⠘⣇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢹⠐⢰⠃⡀⠀⠀⠸⡼⣿⡄⠻⠏⠂⠒⠁⠈⠒⠐⠛⠟⢠⣿⢧⠇⠀⠀⢠⠘⠆⠁⡏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡀⢡⠇⠀⠃⠀⡃⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢸⠀⠘⠀⠸⡌⢠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣸⣷⢀⢀⠸⡁⡈⠷⠲⣶⣶⣿⠿⠿⣿⣶⣶⢖⡇⢁⢈⠇⡄⡀⣿⢇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣻⣽⡮⣆⠳⣙⠺⠃⡀⣙⣿⡇⢾⣷⣸⣿⡧⢮⠜⠗⣫⠎⣰⢵⣯⣟⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⡿⣦⣲⢷⣅⣆⠸⣿⡿⠜⣗⣿⣿⢂⣷⣠⡾⣆⣴⢿⣿⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣿⣿⠾⢹⣿⣷⣿⣷⣾⣿⣿⣤⣿⣿⡯⠗⣿⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⢉⠛⢿⡶⢾⡿⠛⡉⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣤⣴⣿⣷⣾⣿⣦⣤⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡛⢙⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠖⠒⠒⠒⠒⠒⠒⠲⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠑⣄⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠉⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠘⠀⢹⡇⡸⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡊⠈⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠤⠀⣐⠒⠒⠒⠂⣒⡀⠤⠈⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⠜⠀⡀⠀⠀⠀⠤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣥⣀⠁⠠⢀⠄⣀⠂⠀⠀⠀⠀⠀⢠⠑⢄
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠞⢁⣠⠞⠁⠠⡢⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣫⣭⣿⠿⠿⠛⠻⣂⣀⠀⠀⠀⠐⠀⡀⠀⢀⡴⣩⠞⡶⠱⣄
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠈⢁⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢅⣤⠾⢿⠃⠒⠈⠙⠳⠄⠉⣻⡄⠀⠀⠀⠀⠢⡈⠐⣡⠊⠀⠀⠈⢦
⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢡⣾⡥⠟⣼⠃⠀⠀⢼⣷⣏⠿⡿⠁⠀⠀⠀⠀⠀⠈⠢⠱⠄⠀⠀⠀⠀⢷⡀
⠀⠀⠀⡜⠀⠀⠀⠀⠐⢀⠄⠂⠀⠀⢠⠇⠀⢀⡐⠆⡀⢲⠀⠀⢸⠇⠈⣿⣿⠿⣿⣦⠀⠀⣾⠿⢯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠡⡀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⡸⠀⠀⠀⠀⠀⠀⢊⠂⠀⠀⠀⠾⠤⠄⠘⠇⠈⠧⠈⠀⠀⣿⣆⠀⡸⣿⣶⣮⣧⣄⣀⠘⠛⠒⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢣
⠀⢠⠁⠈⠢⠦⠤⠄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⣿⣦⣙⣉⣙⠝⠻⣿⣷⣶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢠⡀⣆⢤⡈⡆
⠀⡞⠀⠈⠙⠲⡀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠛⠻⢿⣿⣯⣻⣿⣎⢺⣿⣯⡉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡇⢠⠤⠇⢸
⢠⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠳⡄⠀⠀⢬⡛⠻⠿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠈⠀⠀⠘
⢸⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠉⠀⠀⠀⠈⠣⡀⠁⢯⣿⣿⡇⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠆
⢸⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠑⠀⠐⢆⠀⠀⠀⠀⡀⠠⢲⣾⣿⠇⣼⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⣀⣀⡀⠄⠖⠋⠀⢀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡀⠁⢠⣿⣷⣧⣿⣋⢀⡀⠈⠂⠄⡀⠀⠀⠀⠀⠀⡄⠃
⠘⣇⠤⠀⠀⠀⠀⢃⣀⣴⠾⠛⢛⢛⠄⠀⠠⠟⠀⠈⠳⣄⢶⣤⣦⣤⠄⠀⠈⣶⠀⠁⢈⡈⠙⢿⠛⠛⠒⠦⠼⣿⡆⠀⢘⠀⠀⠀⠀⠀⢡⠀⢀⠀⠀⠀⢰
⠀⢯⠒⠲⡦⠴⠖⣾⢟⠁⣀⡰⠁⠀⠀⢀⠀⣢⣶⣶⣤⣤⣄⣀⣀⣠⣤⣼⡿⢿⡇⠱⣄⡀⠀⠀⢣⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠤⠭⣙⣒⣺
⠀⠘⣴⡋⡀⠀⢸⡟⠄⠈⠀⠀⠀⠀⠀⣡⡿⣿⢿⠟⠛⠋⠙⠛⡹⠋⠉⠀⠀⠀⠈⠳⢄⡉⢁⠀⠀⠣⡀⠀⠀⠮⢀⠀⢠⠀⠀⠀⢀⠀⠈⠙⢭⣲⡆⢣⠇
⠀⠀⢱⡀⠀⠀⢸⣿⣀⣠⣶⣶⣶⣾⣿⠁⠀⠻⢄⣀⠀⠪⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡕⠀⠀⠈⠆⠀⠙⢧⣬⡸⠀⠀⠀⣢⡁⠀⠀⠀⠀⠀⡜
⠀⠀⠀⢣⠀⠀⢸⠉⡉⢉⠫⣅⠉⢻⠏⠀⠀⠀⠀⠙⢦⡀⠀⠢⣙⣕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡈⠀⠈⠢⠀⠀⠀⠀⠀⡀⣱⣿⣿⣷⣦⣄⠀⡼⠁
⠀⠀⠀⠀⢣⡀⠈⠲⠷⠼⠣⠞⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⣀⠀⠀⠑⠔⠂⣢⣾⢫⣝⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⢠⠔⠢⡕⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠬⡷⣾⣿⣫⣝⢾⣥⣭⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠈⢦⡠⠄⠺⣀⣀⠇⠀⠁⠃⢆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣾⣿⣿⣿⣿⣷⣝⣫⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⡇⠀⠀⠀⠀⠀⠀⠙⠊⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠑⠈⠀⠙⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣈⡁⠒⠂⠈⠉⠩⠍⠐⠒⠈⢉⣀⣄⣀⡀⠀⠀⠀⠈⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⡈⠀⠀⠀⠀⢠⠃⢠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⣉⠇⢀⠀⠅⠀⠀⠀⣀⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢄⣐⠚⠤⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⠄⢒⡡⠤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠦⠤⠤⠤⠦⠴⠶⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣤⠟⣹⣿⣿⣿⣿⡷⠶⢾⣿⣿⣶⣾⣛⡋⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡿⢿⣿⣿⣿⣿⣿⣿⣋⣷⣶⠈⣿⣿⣟⢛⣉⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣿⣯⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣾⡟⠉⠀⢹⣿⣏⠻⠦⠀⠀⣾⣿⢹⡿⠉⢿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢠⣾⣛⣥⡉⠙⠻⣿⣧⠀⠈⠻⡁⠘⠛⠁⠀⠀⠀⠀⠘⢇⣼⡷⠘⢿⣿⣿⡗⣾⣿⣿⡏⢿⣯⣿⡿⣿⣷⡿⣿⣾⡻⣦
⠀⠀⠀⠀⠀⡝⣿⠋⠉⠀⠀⠀⢹⣷⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢈⡿⢷⣤⣾⡿⣿⣥⣿⣿⣿⣧⢸⣿⣿⡇⠸⣿⣷⡘⣿⣷⢻⣷⣄
⠀⠀⠀⠀⠀⠳⣦⣀⣀⡀⣀⣀⣴⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡁⣠⣿⡿⠡⢿⣿⣿⣿⢹⣿⣸⣿⣿⡇⢸⣿⣿⠀⣿⣿⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠉⢿⡉⠻⢻⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠘⢁⣴⣿⣿⡿⢃⣴⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⡀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠹⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣠⡾⣻⣿⠟⡁⠰⣾⣿⣿⣿⢩⣿⢿⣿⡇⣾⣿⣿⣧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣮⣿⠿⠟⣿⡇⠀⠀⠀⠀⢀⣿⣿⢸⣿⣿⣿⣿⣿⣧⣾⣿⣾⣿⣿⢸⣿⣿⣿⣿⣿⢁⣿⣿⣿⢿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣿⣿⠃⠀⢠⡿⠀⠀⢸⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡇⢸⣿⣿⣿⣿⣟⢸⡿⣿⣧⣼⠇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣻⣷⣿⠶⠛⠀⢀⣤⠸⣿⡾⢟⣿⣿⣿⣿⣿⣿⣿⣏⢻⣿⢿⣿⣮⡿⢮⠻⣯⡛⢿⡉⠟⣧⢸⣿⢇⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣤⢩⣿⡇⠀⠀⠀⣸⣿⣾⡏⡐⣈⣿⣿⣿⣿⣿⣿⣿⠻⢦⡙⢢⡙⢷⡝⢦⠹⣿⣷⡈⣿⢠⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡛⢀⢻⣷⣶⣼⣿⣿⣿⣿⢹⣷⡟⣿⣿⣿⣿⣿⢿⣿⣦⣀⠈⢠⠙⡌⢿⣆⠀⢹⣿⣷⣿⢸⣿⣿⣦⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢙⡿⢾⣿⣿⣿⣿⣿⣿⣿⣼⣿⣇⡙⣿⣿⢻⣿⣦⣈⠑⠀⡙⢿⣿⡆⢸⣿⣧⠈⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡎⠹⢻⣿⣿⣿⣿⣿⣿⣿⠯⢳⣿⣿⣧⣉⠙⠛⠻⣦⡹⣮⣿⡇⠈⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠿⣷⣿⣿⣿⣿⣿⣿⣷⣿⣦⣾⣧⠻⣿⣟⠛⠒⠄⢈⠹⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⡶⠐⢿⡇⠀⢲⣄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡑⠀⠀⣄⠀⣱⣸⡿⣡⡾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣶⠞⠻⠃⠀⠀⢻⣄⠠⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣭⡟⣿⣿⣿⣷⣶⣌⣻⣿⠟⢰⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⡏⠀⠀⢻⡄⢀⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠿⣿⣿⡻⡏⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠈⣷
⠀⠀⠀⠀⢰⢀⣤⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦⣰⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡟
⠀⠀⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠈⣿⣿⣿⣿⣿⡟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣾
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣞⠁⠸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⢀⡾⠃
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣻⣿⣿⣿⣿⣿⣿⠈⠙⢶⡟⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣥⣴⡟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⣰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢿⡟⣿⡟
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠐⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣲⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⣤⣦⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⢿⡿⠋⠙⢿⠀⣀⣾⣤⢾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣸⣇⣠⣤⣼⣿⣷⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠠⠈⠀⠐⠁⠀⠀⠈⡛⣶⣦⣄⡀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣠⠖⢁⡴⠚⠁⡄⡀⠀⡤⠴⠀⢸⣿⣿⣿⣶⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⣴⡏⡐⠀⠀⢀⠎⠀⠈⠈⠀⢃⠀⡀⠻⣿⣿⣿⣿⣦⡀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠰⠀⠀⠀⣾⣿⢠⠀⠀⡄⠈⣧⠤⢤⡄⠀⡤⠷⢦⡄⠢⠄⢈⠙⢿⣷⠀⠀⠀⠆⠀⢀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢸⣿⣿⠀⠈⠀⠀⠀⣹⠟⠁⠀⠀⠘⠙⡀⠀⠀⠈⠂⠀⣼⣿⣇⠀⠀⠠⡀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⢀⠀⠀⠩⠀⠀⠀⣿⣿⣿⡆⠀⢀⢐⠾⠃⠀⠀⠀⠀⠀⠀⠂⠐⠂⠀⣰⣞⢻⣿⣿⠀⠀⠀⠇⠀⠀⡀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⣿⡿⢿⣄⢺⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠐⠛⠟⠘⢹⣿⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⢸⡏⣶⡀⣿⡜⢷⣷⠶⣄⠀⠀⠀⡴⠋⡀⢀⠤⠁⠀⠀⠀⢨⠅⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠐⡀⠀⠀⠁⡘⠻⠛⠙⠀⡁⠀⠁⠄⠀⠀⠁⡀⠀⠠⠻⡆⠨⠂⠀⠀⠃⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠙⠉⠁⠀⠈⢀⡀⠀⠀⠀⠀⣄⠁⠀⠀⠘⠀⠐⢆⡠⢂⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⢤⡤⠀⢄⠀⠝⠂⠀⣀⡴⠦⠀⠤⠊⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠈⠙⠂⠲⠶⠤⢤⠤⠤⠴⠖⠋⠉⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠸⠜⠼⠣⠈

⣿⢿⠿⡿⣿⢟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣛⣛⣻⣿⣟⣟⣻⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣞⡤⠀⠠⣜⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⣀⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣯⣽⣆⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠘⢿⡀
⠀⠀⠀⠀⣀⣀⢴⢘⢻⡅⡿⠅⠀⠀⠀⠀⢀⡻⢶⣶⣶⣶⣞⣀⠀⠻⠟⠃⠀⢀⡞⠉⢹⣿⣿⣿⣿⠿⠿⠋⢁⡄⠀⠀⠀⠀⠹⣦⣠⣿⣿⣿⣧⣠⠞⠁⠠
⠀⠀⠀⠀⠿⠿⠛⠃⠈⠁⠀⠀⠀⠀⠀⠀⡡⠉⠋⠛⠿⣿⣿⠋⠙⠂⠀⠀⠀⠈⠀⠀⠘⣿⣿⣿⠛⠳⠀⠀⣿⣇⣀⡀⠐⣙⠢⠤⠙⣿⣿⣿⡟⠡⠤⢚⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⣄⣈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⣤⡴⠛⠛⠋⠀⠀⠈⠈⢉⠢⠈⠉⠉⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⢀⡀⠀⢀⣀⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣤⡶⠖⠒⠠⢤⡀
⠀⠀⠀⠈⠁⠒⠉⢹⣿⠉⠓⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⣆⠀⠀⠀⣴⡿⠋⣀⣀⣠⣤⣶⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣶⡗⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⢼⣿⣿⣿⣿⣿⣿⠀⠀⢰⣿⠁⣾⣿⣿⣿⡇⢘
⠀⠀⠀⣷⣾⠾⣿⢸⡇⣶⣶⣦⣴⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠉⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢿⠇⠸⡿⣿⣿⣿⣿⡟⠀⠀⢸⣿⡀⠘⠿⠿⠿⠿⣼⣷
⠀⠀⠀⠙⠻⠞⠋⣼⣇⣻⣶⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⣀⡀⢀⣀⣼⣿⡿⠋⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⣠⣾⡿
⠀⠀⠀⠀⠀⠰⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠘⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠋

⠀⠀⠀⢀⣴⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⠟⠀⠀⠁⢆⠀⠀⠀⠀⢠⡾⠛⠉⠩⣭⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣷⣍⠉⠉⠁
⠀⠀⠀⣬⡿⣿⣿⣿⠆⠀⠀⠀⢘⡄⠀⠀⢀⡿⠁⠀⣠⣿⣿⣿⣛⣿⠀⠀⠀⠀⠀⠀⠀⡀⢀⢠⣤⢴⢲⣴⠀⠀⠀⠀⠀⠀⣴⣿⣯⣿⣿⣯⣽⣇⡀
⠀⠀⠀⢻⣿⣿⣽⠋⠀⠀⠀⠀⣰⡇⠀⠀⠸⡇⠀⢰⣿⣿⡿⣿⣿⣿⣷⡆⠀⠀⠀⠀⢸⡇⡼⠿⠻⠘⠈⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠘⣿⣿⣿⢻⣿⣆⣠⣾⠟⠀⠀⠀⠀⠑⠀⠘⣿⣿⣿⡟⠀⣝⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣇⠈⠁
⠀⠀⠀⠀⠀⠻⠿⠿⣿⠻⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡀⢀⠔⠉⠉⠙⢷⡄⢀⣠⠄⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⣄⡀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠙⠀⠀⢀⠀⠀⠐⣷⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣿⣶⡄⠀⠀⠀⠋⢽⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡆
⠀⠀⠀⣀⠀⠄⠊⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⣿⣏⣀⠙⠁⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⣿⢸⡟⠿⠇
⠀⠀⠛⠋⠀⠀⠀⣠⣤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠙⠯⣿⠿⣿⣄⡀⠐⠋⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠨⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠛⠻⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠻⡏⠉⠠⣤⣤⡀⢀⣠⣤⡀⠀⣹⠛⠛⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠁⣠⠀⠀⠀⢿⠀⠂⢿⠛⣿⣿⢟⡿⠁⢠⡟⠀⢰⣆⠀⠈⠙⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢁⣀⣀⡀⣿⣤⡆⠂⠸⡇⠀⠀⠀⢹⡋⠀⠀⠀⣼⠁⠀⣨⣿⣶⣄⣈⣀⡈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠻⣦⠀⠄⠨⢉⣿⣿⠻⣧⣤⠀⢹⠀⠁⠀⠘⠁⠐⠂⢠⠇⠀⣾⣿⠉⠉⣿⡟⠛⠁⢀⡴⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⣀⢈⠳⣄⠀⠘⡀⣿⠀⠀⠀⣀⣬⣶⡶⠶⠶⠶⠶⢶⣾⣤⣀⠈⠿⣶⠾⡿⠃⢁⡴⠋⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⣰⡿⠃⢀⣹⡏⣤⣤⠈⠳⢄⠀⢉⣠⡶⠟⣫⣥⣶⣶⣿⣿⢿⣿⡶⣶⣬⣝⡻⢶⣄⡀⢀⡴⠋⠀⠐⢈⣹⣿⣧⠈⢿⣆
⠀⠀⠀⠀⣰⡿⠁⢠⣿⠉⣶⠾⠂⠀⠀⢈⣵⡿⣫⣴⣿⣯⣿⣿⡟⣿⣿⣶⣿⡿⣿⣯⣿⣿⢶⣍⠻⣯⡀⠛⢷⣾⠟⢁⣤⣿⣆⠈⢿⣆
⠀⠀⠀⢠⣿⠃⠀⠘⠇⢻⡟⠂⠐⠂⢠⣾⢋⣾⣯⣿⣿⣽⣿⣼⡿⠛⠉⠉⠉⠙⠛⠿⣥⣿⡿⢿⣷⡜⢷⡄⠀⢻⣶⠛⠉⠈⠙⠃⠈⣿⡄
⠀⠀⠀⣾⡿⢦⣤⣀⡀⠘⠃⠀⠀⢠⡿⢡⣿⣉⣿⣯⡿⠋⠉⠉⠑⠾⠋⠹⣦⠀⠀⠈⠐⠝⣷⣾⣿⣿⣎⢿⡄⠀⠘⠒⢀⣀⣤⠴⠶⢻⣧
⠀⠀⢀⣿⠁⠀⠀⠈⠉⠉⠑⠒⢤⣿⢃⣿⢻⣿⣿⡿⠷⢔⣴⠄⠀⡀⠀⠃⢘⡆⠀⠀⠀⠈⢎⣿⣷⣿⣿⡌⣿⠖⠊⠉⠉⡀⣀⣀⠀⠀⣿⡀
⠀⠀⢸⣿⠀⠄⠀⢠⡶⢦⠀⠁⢸⣿⢸⣿⠿⣿⢃⠀⠀⠀⠀⠀⣀⠃⠐⣩⡝⡧⢨⡍⡀⠀⠀⢸⣿⢿⣿⣧⢻⡇⠀⢠⣟⣸⣿⣹⣷⡄⣿⡇
⠀⠀⢸⣿⠀⣤⣶⡾⠷⠿⢷⡄⢸⣿⢸⣿⣿⣿⣦⡤⠖⠀⠀⠀⣸⠀⠀⠉⢨⣃⣀⠉⠀⠀⠀⢸⣿⡿⠿⣿⢸⡇⠀⢸⣧⣤⣍⡉⢻⡇⣿⡇
⠀⠀⢸⣿⠀⣾⠿⠟⠃⠈⠀⠀⠸⣿⢸⣦⣼⣟⣿⠀⠀⠀⠀⠀⠈⣆⠀⠀⠤⢾⡡⠄⠀⠀⠀⣸⣿⣿⣾⡏⣼⠇⠀⠈⢽⣯⣿⢇⡟⠃⣿⡇
⠀⠀⠈⣿⡄⠀⠀⠀⣀⣀⡤⠤⠚⢿⡆⢿⡟⣿⡿⣶⣤⣤⠀⠀⠀⠘⠀⠀⠀⢺⠀⠀⠀⠠⣢⣿⣉⣿⣿⢡⡿⠓⠒⠤⢤⣄⣀⡀⠀⢀⣿
⠀⠀⠀⢻⣷⠖⠛⠋⠉⠄⠀⠀⠀⠘⢿⡜⢿⣿⡷⣾⣿⡇⠈⠀⢀⣀⠢⠁⠀⠁⠀⢀⣪⣾⣿⣏⣿⣿⢣⣿⠃⠀⣀⣤⣄⢀⠉⠉⠛⣿⡟
⠀⠀⠀⠈⣿⡄⢀⣀⣰⣞⣀⣸⣆⠀⠈⢿⣮⡻⣷⣿⣿⣿⣷⣾⣏⣿⣿⣖⣶⣶⠾⢿⣿⡟⣿⣿⠟⣱⡿⢁⡄⢠⢿⣤⣿⣀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠘⣿⡄⠙⠻⣏⠉⣿⠟⠻⠀⠀⡹⢷⣮⡻⢿⣿⣭⣿⣿⠛⣿⣿⣿⣿⣷⣾⣿⡿⢟⣡⡾⠯⡀⠸⠋⠙⢷⣄⠉⢻⡤⢰⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠈⠀⠉⠀⠀⡠⢋⣀⡀⠉⠻⢷⣮⣝⣛⡿⠿⠿⠷⠿⠿⣛⣫⣥⡶⠟⠉⠀⠀⠙⢦⡈⠀⠈⠙⠿⠟⢡⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⢀⣴⠋⠀⠈⢉⣿⠀⠐⠀⠈⠉⡟⠛⠛⠿⠟⠛⠛⠛⡋⢡⣶⡄⠀⡀⠰⣄⠀⠙⢶⣄⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡟⠁⠀⠁⣴⠏⣡⠆⢂⠀⠀⣸⢀⣴⡆⢢⣄⣀⣠⡔⣥⠉⢷⣹⡆⢳⣄⣸⣷⣄⡀⠙⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠻⢶⣯⣠⠾⠀⢠⡇⠨⢾⣷⠏⡩⣿⣿⠆⢸⡆⠈⠉⠿⣶⠿⠛⠁⢉⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⠉⠻⡇⠀⣼⠁⠀⠸⣿⣄⣠⠿⠀⠀⠀⢿⠀⠀⠀⠀⣀⣤⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣦⣴⣧⣀⣀⡀⠈⠙⠋⠀⢀⣀⣀⣸⣧⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣾⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠠⢀⡀⠀⠠⢀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⡠⠐⠊⠀⠀⠀⠀⠒⠀⡀⠀⠀⠄⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢐⠂⠀⠀⠈⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠉⠉⢀⡀⢀⠀⠀⢀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠅⠀⠀⠀⠀⠈⠌⠀⠀⠀⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠂⠠⡀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠙⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⣧⠀⠀⢠⣇

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠂⠀⠀⠀⠒⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢏⠡⠀⣀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠐⠆⢂⠱⢸⡸⠀⢀⡠⠖⠋⠁⠀⠀⢀⡤⠤⠤⢄⡀⠈⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣶⣉⣠⠔⠉⠀⠀⠀⠀⠀⢰⣅⡠⠀⠄⠀⣻⡄⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⢸⣼⢽⡿⡹⡕⠀⠀⠀⠀⠀⠀⠀⠘⠛⢀⡶⢿⣟⣿⡇⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠈⠻⢤⣭⠔⠁⠀⡠⠤⠠⠤⣀⠀⣿⣾⣿⡇⣿⣿⣿⡷⣆⠤⠔⡒⠤⣀⠀⠹⡄
⠀⠀⠀⠀⠀⣶⢀⢻⠞⢔⠢⣮⠂⠁⠀⡁⠀⢄⣿⣿⣯⢠⣿⣿⣿⣿⣷⠉⠁⣀⠉⢪⢢⠀⢹⡀
⠀⠀⠀⠀⠀⠘⢨⣍⠈⠸⠄⠸⠈⠡⠀⢡⡞⠯⣿⣿⣷⡌⢿⠹⢋⣿⠉⠑⢦⠁⢀⠄⠈⡄⠀⡇
⠀⠀⠀⠀⠀⠀⣼⢀⡨⢒⣦⣶⢧⠀⠀⣯⠹⣄⠻⣿⢟⡴⢢⡞⠢⠈⠀⠀⠀⣧⠀⠀⠘⢜⢦⣹
⠀⠀⠀⠀⠀⣔⣭⠔⠀⠁⢸⠀⠈⢦⣌⡈⠱⡹⣄⠉⢪⡁⠘⠁⠀⠐⡄⠀⠀⣾⠀⠀⠀⠀⠑⣮⠤⡢⡀
⠀⠀⠀⠀⠀⡀⢸⠀⠀⡴⠈⡇⣦⠈⣯⢥⢄⡟⡜⢆⠀⠑⢄⠀⠌⡀⢸⠀⠂⡘⡦⣀⠀⡂⢄⡇⠀⠁⡇
⠀⠀⠀⡀⢄⡿⠀⢧⣄⠀⠀⢳⡉⠁⠃⣠⠋⣥⣘⠮⢧⡀⠈⠙⣧⡀⣼⢦⣀⠈⡗⠎⠇⠘⣢⠀⡀⢶⡄⣀
⠀⠀⠀⡔⡋⠀⠀⡈⢎⠀⢀⠤⠳⠤⠚⢹⠀⠈⠉⠀⠀⠙⣿⠉⠁⠀⠐⠚⠀⠀⡇⠸⡰⢴⠃⠁⠀⠈⢑⢄
⠀⠀⡂⠀⠂⠀⠂⠂⠀⠳⣿⡀⠀⠀⠀⠀⠁⢰⠶⠶⠶⠺⣟⢳⢶⣶⡶⡖⠒⠉⠁⢀⠜⠁⠉⠻⡀⠀⠨⠀⣑
⠀⠀⠀⠀⠀⠂⠄⠀⡀⣠⡼⠢⣄⠀⠀⠀⠀⡼⠉⠙⠛⠛⠛⠻⣠⣠⣤⣇⠀⣀⠔⠁⠀⠀⠀⢐⠓⡍
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢘⠀⠀⠈⠑⠢⢄⣐⡇⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦⢎⣙⢾⣀⢀⠶⡰⢗⠀⢇⣸⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠶⠶⠶⠶⠶⠦⣄⣠⠴⠶⠶⠶⠶⠦⢤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⠁⠀⠀⠀⠀⣀⠀⠠⡄⠀⣿⣿⠀⢠⠄⠀⣀⠀⠀⠀⠀⠈⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣾⠋⢀⣀⠀⣰⣋⣀⣀⣀⠀⠀⣿⣿⠀⠀⣀⣀⣀⣙⣆⠀⣀⡀⠙⣷⣽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡿⠛⠁⠀⣸⠏⠻⣦⠀⠀⠀⠈⠙⠲⣿⣿⠖⠋⠁⠀⠀⠀⣴⠛⠻⣇⠀⠈⠻⢿⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣶⡾⠋⢠⡏⠀⠈⠀⠀⠀⣠⣤⠤⠴⣾⣷⠦⠤⣤⣄⠀⠀⠈⠁⠀⢹⡆⠙⢷⣶⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢫⡿⠋⠀⠀⠀⠳⠀⢀⡴⠶⠾⠋⠀⠀⢀⡿⢿⡀⠀⠀⠘⠷⠶⢦⡀⠀⠎⠀⠀⠀⠙⢿⡝⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⠃⣰⡟⡡⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠹⡀⢀⠏⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢌⢻⣆⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡸⠁⠀⣀⡀⠀⢰⡏⠀⠀⣀⡀⠀⠀⠀⠀⢁⡈⠀⠀⠀⠀⢀⣀⠀⠀⢹⡆⠀⢀⣀⠀⠈⢇⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣇⡴⠋⠉⠇⠀⢸⡇⠀⠀⢹⣙⢻⡶⠀⠀⠀⠀⠀⠀⢶⡟⣋⡍⠀⠀⢸⡇⠀⠸⠉⠙⢦⣸⡘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠏⠀⡀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠘⢁⠀⠀⠀⠀⠀⢀⡈⠃⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⠀⠹⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣠⡾⠁⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⢷⣄⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⡇⠀⠀⠀⠀⠀⠙⠲⣦⣀⠀⠀⣿⡀⠀⠀⠀⠀⢀⣿⠀⠀⣀⡴⠖⠋⠀⠀⠀⠀⠀⢸⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⡆⠀⢀⣀⠀⠀⠀⠹⣦⡄⠘⢯⣙⢉⡉⣉⡽⠃⢠⣴⠏⠀⠀⠀⣀⡀⠀⢰⠀⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢿⡄⢱⢀⡏⢿⠀⠀⠀⠀⢸⡀⠀⠀⠈⠛⠛⠁⠀⠀⢀⡇⠀⠀⠀⠀⡿⢹⡀⡎⢠⡿⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡈⢿⠁⠈⢳⡀⠀⠀⢸⣆⠀⢀⣠⡴⢦⣄⡀⠀⣰⡇⠀⠀⢀⡞⠁⠈⡿⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣠⠀⠀⠱⡀⠀⠀⠙⡛⠋⠁⠀⠀⠈⠙⢛⠋⠀⠀⢀⠎⠀⠀⣄⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣴⠟⢷⡀⠀⠓⠀⠀⠈⢳⣦⡀⠀⠀⢀⣴⡞⠁⠀⠀⠚⠀⢀⡾⠻⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠁⠀⠈⣷⠀⠘⠀⠀⠀⠀⠀⠻⢶⡶⠟⠀⠀⠀⠀⠀⠁⠀⣾⠁⠀⠈⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣄⢣⡀⠀⠀⠀⠀⠀⠀⢀⡜⣠⠀⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠤⠶⠋⠙⣆⠱⣄⠀⡄⢠⠀⣠⠞⣰⠋⠙⠶⢤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣌⠳⣧⣸⠞⣡⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣸⣇⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠋⠉⠉⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⣀⡀⠀⠀⢀⣴⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣩⣤⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⢋⣠⣤⣤⣬⣙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⢠⡇⢸⣿⠿⠟⢡⣿⣿⣿⣿⣿⣿⣷⣤⠶⠶⠶⡤
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢁⣾⠇⠘⣵⣦⣤⣾⡟⣫⣽⡿⠛⣩⡤⢀⣠⣤⡄⠀⠀⠀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⣡⣿⣿⢀⣾⣿⣿⣿⠏⣼⣿⠏⢠⣾⠋⢰⣿⣿⣿⣄⢻⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣤⠸⣿⣿⠟⠁⠀⠙⠻⠇⣾⣿⣿⣿⠏⣠⣿⣿⠀⢸⣿⣆⠘⢿⣿⣿⣿⣷⣶⣶⣶⣿⣯⣭⣙⠻⣿⣷⣦⣄
⠀⠀⠀⢿⣦⣤⡄⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⣼⣿⣿⣿⣆⠈⠿⣿⣶⣦⣬⣭⣭⣭⣭⣭⣭⣭⣛⢿⣿⣷⢹⣿⣿⣿⣆
⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣸⡿⢛⣛⡉⠻⣷⣶⣤⣉⣉⣉⣉⣩⣭⣭⣿⣶⣝⢿⡏⢿⠿⢸⣿⣿⡿⠃
⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁⣿⠀⣿⣿⣿⣆⣿⣿⣿⠿⠿⣿⣿⣿⠁⣴⣶⣿⣿⣶⣴⣶⣶⣦⣬⣉⠁
⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⢰⣿⣄⠙⠛⠛⢛⣩⣵⣾⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⣠⣴⡄⣭⣤⠀⣤⡀⠀⠀⠀⢸⣿⣿⠁⠀⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠻⣿⣿⣿⣿⣿⡇
⠻⠿⠛⢿⣿⠇⠻⡿⠀⠀⢀⠈⢿⡿⠀⣄⡀⠉⠉⣉⣥⣶⣿⣿⣿⣿⠟⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣋⠉⢩⣍⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⠀⣿⣿⡟⠋⣉⡉⠉⢿⣿⡿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⢻⣿⠀⢸⣿⣿⣤⣾⡟⣡⣿⠟⣿⣿⣿⣿⡟⢻⣿⣿⡿⠿⣿⣿⣿⣿⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠈⢿⡀⠈⠛⠛⠛⢉⣴⣿⢃⣾⣿⣿⣿⣿⣧⠘⣿⣿⢡⣶⡆⢀⡤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠈⠻⣶⣶⣶⣾⣿⠟⢁⣾⣿⣿⣿⢟⣿⢋⡀⠻⣿⡌⠿⠿⠿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠛⠀⠀⠀⠀⠀⠉⠉⣉⣠⣤⣾⣿⠿⢋⣱⡟⣡⠏⣠⡄⠙⠻⢷⣶⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢈⣉⣉⣤⣤⣶⣿⡟⢰⡿⢠⣿⢡⣾⡶⢀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⣡⣤⡈⣥⣤⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⠁⢾⣿⢸⡇⢸⣿⠃⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏⠻⣿⠇⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠙⠿⣿⡿⠋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢡⣶⣶⣭⠉⣷⣦⣤⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠇⢿⣿⣿⠿⢿