Infatuate Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠤⠤⢤⣴⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠠⠤⠤⠤⣡⣘⣿⣿⣿⣿⣟⠉⠉⠻⣿⣿⣎⡑⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠋⠁⡀
⠀⠀⠀⠀⠀⠀⠒⠐⠒⡶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢖⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠉⠀⠀⠀⠀⠐⠂⠀⠀⠈⠛⣍⢥⠀⠀⠠
⠀⠀⠤⠄⠠⠄⠀⠀⢸⣿⣿⠿⡽⢫⣄⣀⣼⠿⠿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠘⠂
⠀⠀⠀⠠⠤⠤⠄⠀⠈⣯⠼⠦⠭⠭⢽⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⣠⣶⣶⡶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣍⣙⣿⣿⠟⠁
⠀⠀⠀⠀⠠⠤⠀⠠⠤⠤⢤⣤⡤⢬⣴⣶⣦⣀
⠀⠀⢀⣀⡀⢀⣠⣦⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠠⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠁
⢀⣀⣠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⣒⣀⣀
⣲⠖⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠤⠍⢉⠉⠉⠉⠉
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉
⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡐⠆⠒⠂
⠀⠉⠉⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⠖⢒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠘⢠⠃⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡤⠖⠒⠒⠢⣄⠀⠀⠀⢣⡉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⠏⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⢸⠁⠀⢈⣿⡳⡀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠈⠙⢶⠊⠁⢙⡟⠋⠢⢄⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠰⡏⢻⠀⠀⠉⠻⠋⠉⠒⠒⣸⠇
⠀⠀⠀⢀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠤⠤⣀⠀⣀⡤⢶⣿⡗⢸⡧⠜⣆⠀⠀⠀⠀⠀⠀⠈⠀⠉⠑⠢⣄
⡜⢉⡕⣧⠒⠪⡄⠀⠀⠀⠀⣠⠖⢩⣃⡴⠖⠶⠖⠚⠉⠁⣀⡸⠟⣧⠘⣇⡤⠞⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠉⡏⢹⠉⢳⣖⠃⠀⠀⢀⡞⣁⠴⠚⢁⣠⠤⠒⠈⠙⠲⣿⣹⠀⠀⠸⡄⢳⠀⢀⡠⠗⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠉⠉⠀⠉⠀⠳⡀⢠⡿⠚⣁⡤⢶⠋⠀⠀⠀⠀⠀⠀⠘⡜⢄⠀⠀⢳⡈⢷⠉⠀⠀⣀⠼⠓⠲⢤⣄⣀⣀⣀⣀⣀⡤⠎
⠀⠀⠀⠀⠀⢀⡠⠟⣁⠴⠊⢹⠃⡟⠤⠤⠤⠤⠤⠤⠖⠚⠣⡈⢣⡀⠀⢣⡈⢧⡴⠊⠁⠀⣀⠴⠊⠁⠀⣩⠝⠉
⠀⠀⠀⠀⠀⠈⢦⣀⠇⠀⠀⡟⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⠗⠀⠀⠳⡄⠳⣄⠴⠊⠁⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⣾⠄⣀⣀⣀⣀⣀⣀⣀⠤⠴⢏⠀⠀⠀⠀⠀⠙⢦⡈⠢⢄⡤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠸⡆⢹⡒⠤⠤⠤⠀⠀⠀⠤⠤⠔⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢣⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⡤⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⣙⣒⣒⢒⣒⣒⣋⣩⣴⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠿⢯⡁⠀⠉⢳⡌⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⣠⠇⠀⡠⠋⣰⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠐⠚⠓⠒⠾⣄⡀⠀⠀⠈⢍⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⠀⢢⡐⢄⣙⠦⠀⠰⠦⠓⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣝⣦
⠀⠀⠀⠀⢀⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣰⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢻⣿⣿⡿⣧
⠀⠀⢠⣿⣿⡇⢰⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⢿⡿⠟⠋⠀⠻⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠈⣿⣿⣷⣿⣿⡟⢫⣶⣿⣿⣿⣿⣿⡿⠞⠛⠉⠀⠀⠀⠀⠀⠀⠙⣿⣿⡋⠛⢻⡄
⠀⠀⠀⠈⠙⠻⣿⣿⣷⢋⣽⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣶⠈⠃
⠀⠀⠀⠀⠀⠸⣿⣿⣿⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⢪⣯
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⣼⡇
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣟⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⣆
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⣹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⡏
⠀⠀⠀⠀⠀⠈⣿⡟⣿⣿⣷⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣹⣇
⠀⠀⠀⠀⠀⠀⢸⣧⢻⣿⣿⡎⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⢹⣧⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣬⣽⣿⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣮⣿⠉⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⣿⣿⣿⡄⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠛⣿⣿⣿⢿⣷⣄⠀⠀⠀⠀⠀⡠⢾⢹⣿⣿⣯⣿⣿⣿⣧⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣀⣼⡇⠀⢸⣿⠉⠛⠒⠒⠒⠈⠁⣿⢀⣿⠋⢉⣿⣿⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣿⡋⢨⣷⣶⣼⣿⠀⠀⠀⠀⠀⠀⠀⢷⣌⣿⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣥⠌⣃⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣏⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢌⣷⣿⣿⣿⣿⢹
⠀⠀⠀⠀⠀⣸⠟⠛⠛⠻⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠿⠾⠂
⠀⠀⠀⠀⢐⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⣠⢞⣁⢀⡀⠂⠀⠀⠈⠑⢀⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⢰⠃⢸⣿⢟⡿⠀⠀⣤⠀⡀⠀⡀⠀⠀⠀⠀⢀⠛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠄⠀⢹⠀⠀⠉⠈⠐⠀⠀⠀⠠⡄⠀⠓⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠀⠀⠀⠀⠱⣄⠈⠀⠀⠀⠀⠀⠐⣀⠟⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠑⠤⢄⣀⠀⠴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣦⣨⣤⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠿⣿⢿⡧⠹⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⢒⣤⣶⣶⡄⠙⣿⣿⣿⣿⣿⣿⣧⠀⢰⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⢦⣼⡟⠻⣿⣿⣷⡀⣻⢿⣿⣿⣿⣿⡟⠛⠻⡿⢿⣿⣯⠈⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣾⣿⣾⣿⠿⠛⠀⠁⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⠟⠙⢰⣿⣷⣦⣀⡀⠈⠙⠻⠟⠀⠈⠀⠀⠠⣾⠧⠤⠤⠴⠂⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⠃⠀⠄⣨⣿⣿⣿⣿⣷⣤⡀⠀⠀⡀⠐⠀⡀⠀⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⡏⠀⠀⠀⠈⠙⣿⣿⣿⠿⡿⢷⡄⠀⢄⣄⡀⠈⠃⠀⠀⠀⠀⣠⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣿⣶⣿⡆⢨⣥⠁⠀⠁⠸⣿⣿⣤⠀⠀⣿⣷⣬⣭⣥⣤⣤⣤⣤⣶⣾⣿⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⡟⠛⢿⣆⡼⠡⠤⠠⠾⣿⣿⣿⣷⣀⣪⣿⣿⣿⠟⠛⡛⣿⠋⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⡿⢿⣿⢿⣿⢿⡿⡿⠿⣿⡿⡿⣿⣿⣿⢿⣿⣿⣿⣿⠿⡿⣿⢿⣿⣿⡿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⣿⡿⣿⠿⣿⢿⡿⢿⣿⡿⣿⣿⢿⣿⡿⣿⢿⣿⣿⠿⣿⣿⣿⢿⡿⣿⢿⡿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢟⣡⣾⡇⠀⠀⠀⣠⡾⠿⢷⣶⡾⠿⢷⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⣴⣶⣿⣿⠇⠁⠀⠀⠀⢿⣅⠀⠀⠙⠀⠀⣸⡿⠀⠀⠀⢀⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣡⣤⣿⣿⣿⣉⡁⢾⡷⠀⠀⠀⠈⠻⢷⣤⣀⣤⡾⠟⠁⠀⠀⠀⢸⠟⣷⣦⣭⣉⣉⡙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣌⣩⣿⣿⣿⣿⣿⡆⠀⠀⠺⠷⠶⠶⠾⠟⠛⠻⠷⠶⠶⠾⠗⠀⢈⣤⡙⠿⣿⣿⣿⡇⣛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣡⣶⣾⣿⣿⣿⣿⠇⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⣿⣿⣿⣿⣿⠇⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢛⡁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣬⣭⣤⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣧⣙⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⢛⣛⡛⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣦⣭⣭⣭⣴⣾⣿⣦⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢋⣾⣿⣿⣿⣷⡹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢋⣭⣭⡙⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢡⣿⣿⣿⣿⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠘⣿⣿⣿⣿⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠁⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠈⠛⠛⠛⠛⠀⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠢⠖⠲⠶⡆⠀⠀⠀⠀⠀⠀⠄⢠⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢋⣀⣉⣁⣁⣉⠸⠀⠂⠒⠀⠀⡂⠘⠛⣛⡛⢀⣀⡀⠄⢰⠀⠤⠥⠤⠄
⠀⠀⠀⠀⠀⠀⠒⢒⠐⠀⠘⠀⢀⣀⣀⡀⠀⠐⠒⠖⠂⠒⠀⡒⠀⠀⠉⠉⢈⢁⠁⡀⠚⠀⠁⠤⠄⠆⠠⠢⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡉⡀⡉⣘⢻⣿⠛⠈⠒⠃⠒⠆⠐⠀⠃⠘⠓⠷⠻⠹⠛⠀⠟⠻⠀⠁⠆⠀⠠
⠀⠀⠀⠀⠆⠴⣠⢤⠀⠄⡤⢁⠁⢸⣿⠀⢸⠀⣰⡿⢿⣦⠸⣿⡟⠸⣿⡟⣰⡿⢻⣦⠂⠓⠒⠐⠂⣀⠀⠀⠐
⠀⠀⢀⣀⣁⡉⣉⢈⢉⢁⠁⢸⠀⢸⣿⠀⣬⡄⣿⡇⢸⣿⠀⢻⣇⢰⣿⠀⣿⣧⣼⣿⠘⡃⢘⡃⢩⠀⠛⠀⢀⢀
⠀⠀⢀⠀⡀⠀⠀⠈⠈⠈⠁⠐⠀⢸⣿⠀⣿⡇⣿⡇⢸⣿⠀⠘⣿⣾⡇⡀⣿⡏⠉⣉⢘⡇⠀⡂⠸⠾⠹⢻⠿⠹⠸⠨
⠀⠀⢸⡗⡖⣖⣖⡶⢶⡂⡂⠈⠰⠿⠿⠿⠿⠇⠘⢿⡾⠏⢸⠀⠻⡿⠀⡇⠸⠿⠾⠏⠸⠦⣄⣃⣐⣒⣒⡒⠒⠂
⠀⠀⠈⠉⠁⠅⠁⠬⠨⠀⠃⠀⠦⠮⠀⠒⠒⠂⠒⠂⠀⠀⠰⠀⠰⠂⠒⠶⠆⠀⣒⢠⡔⣃⣛⠛⢛⢟⢇⢗⡀
⠀⠀⠀⠀⠀⢀⢀⡐⢒⡒⠂⢑⠒⠒⠀⡿⠼⠽⠭⠧⠿⠿⠿⠀⠚⠐⠀⠐⠲⠂⠀⠀⡇⣤⢀⡀⠈⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠁⠈⠈⢸⠈⠋⠀⣀⢁⠀⡒⣀⠀⠀⠀⠀⠀⠀⠐⠀⢩⣀⠁⣩⠀⠇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠀⠠⠀⠒⠒⠒⠂⠈⠁⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣀⢀⡤⢄⡀⠀⠀⠀⠀⡠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠀⠀⠈⠆⠀⠀⠀⠀⠀⢼⠁⠀⠀⠉⠀⠀⢹⠀⠀⠀⠘⠀⠀⠀⠛⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣆⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠁⠀⠀⠀⠀⠸⠻⠷⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⣠⠔⠛⠉⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠦⢤⠤⠤⠶⣶⣾⣄⣀⣀⣀⣀⡀⢀⠀⠀⡀⠀⢀⣀⣀⣀⣀⣀⣿⣶⡶⠦⠤⡤⠤⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠈⣧⠀⠀⠀⠀⠀⠈⠀⢀⠃⠀⠈⠁⠀⠀⠀⢠⡇⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⢀⠰⠁⠀⠡⣀⡀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢅⠀⠀⠚⣄⠀⠀⠀⠀⠈⠙⠛⠉⢁⡔⠒⠒⢄⡈⠛⠛⠋⠁⠀⠀⠀⢀⡜⠀⠀⡔⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠱⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⣷⣦⡶⣿⡇⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢸⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⢠⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠙⠘⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⢡
⠀⠀⠀⠀⣼⠀⠀⢡⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣿⠀⠀⠈⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢹⡄⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣧
⠀⠀⠀⠀⠀⢻⣤⣤⣼⡇⠀⠀⠀⢰⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⣠⠃⠀⠀⣿⣤⣤⣀
⠀⠀⠀⠀⢰⠏⠁⠀⠸⡇⠀⠀⠀⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⡿⠀⠀⠉⢷
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⣀⠤⠾⠟⠀⠀⠀⡇⠀⠀⠀⠃⠀⠀⠀⠀⠠⠤⠀⠀⠰⠀⠀⠐⠒⠂⠀⠐⠒⠀⠀⠶⠤⠤⠤⠀⠀⠀⠀⠀⣇⠀⠀⠸⠇⠀⠀⠐⠿⠶⣄⡀
⠀⢠⣾⣋⣀⣀⣀⣀⣀⣀⣀⣦⠀⠀⢰⡄⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣿⠀⠀⢠⣄⣀⣀⣀⣀⣀⣀⣈⣻⣦
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁⠀⠀⠀⢹⡷⠦⠤⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣠⣤⣤⠴⠶⣿⠁⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠖⠛⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠛⠛⠛⠛⠻⠿⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇⢀⣀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠁⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⡦⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠋⠐⠀⠁⠀⢴⠀⠀⠙⠻⠿⠟⠋⠁⠠⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣾⣧⣀⣀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣧⣤⣤⣿⡿⠿⠟⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠋⠈⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣠⣿⣿⣿⠃⠀⠀⡟⠛⠛⠿⠿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⣾⣧⠀⠀⠀⠀⠀⠨⣤⣤⣤⡀⢀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⡄⢸⣿⡿⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣸⣿⡇⢀⣠⣤⣤⣴⣿⣿⣿⣿⣿⣿⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡒⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣄⠀⠀⠀⠀⠀⠀⢀⠊⠀⠈⣇⠀⠀⠀⠠⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⢿⣶⡤⣀⡀⠀⠁⣀⣀⣼⣿⣷⢲⠖⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠔⣦⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠱⠛⠒⣢⠽⠿⠤⠤⠤⣤⡀
⠀⢠⡏⠀⡘⡠⢀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠽⣆⢀⠀⠀⠀⠀⠀⠚⠛⠛⠒⣒⣦⣄
⠀⠈⣕⢂⡜⠀⠜⡰⡕⢉⠑⠄⢀⠀⠀⠤⢄⠀⠀⠀⠀⠀⢠⣞⡁⠀⠀⢸⠆⠀⠀⠀⠀⣀⠀⠀⠀⠈⣙⣲⣤
⠀⠀⠀⠀⢠⢊⠎⠀⡏⠀⡹⢻⠃⠀⠑⢢⢀⠇⢐⡭⠀⢄⠸⡀⠈⠒⣤⠊⠉⢣⠁⠀⢀⣟⡞⠀⠀⣾⠕⠒⢄
⠀⠀⠀⠀⢇⠘⠤⠔⡰⠀⠁⠉⠀⠀⢠⡱⣁⡠⡞⠸⠞⡳⡎⢈⠒⠊⢻⡋⡡⠿⠀⢀⡼⠋⠀⠀⢀⠐⣋⡡⠊
⠀⠀⠀⠀⠀⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⠉⠚⠚⠀⠀⠀⡇⠈⠀⣀⣀⣀⣉⣀⣀⠴⠊⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⠈⠉⠉⠀⣀⣀⡠⠤⠒⠁⢈⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣹⠀⢄⠀⣀⠀⣈⡇⠀⠈⢱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠉⢣⡈⡟⣇⣀⣤⠊⠁⠀⠉⠒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⡄⢣⡧⠓⠃⡉⢺⡉⡿⠫⣡⠤⣾⡧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠓⠛⠠⠤⢴⡺⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠉⠀⠈⠁⠒⠘⠓⠛⠒⠒⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⢿⣣⠀⢸⠁⠀⣞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠎⢀⡉⢷⡇⠀⠀⢰⡟⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠠⣄⠉⢶⡌⣠⠛⠀⠀⠃⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⣀⣼⠟⠉⠀⢀⠎⠀⠈⣁⣴⣽⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⢀⣀⡰⢇⡀⣈⡸⣇⠈⡷⠞⠁⠀⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠯⢽⢷⠠⡆⢝⡉⠷⠛⠁⠀⠀⠀⠀⢤⣏⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣿⠁⠠⡟⠋⣠⣤⡞⠠⡀⠀⠀⠀⠀⠀⠀⣤⠂⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣥⣄⣴⣾⣿⣿⣯⡹⣆⡀⠢⠀⠀⠀⠀⣸⡽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣷⢿⡟⠀⣀⠢⠀⠀⠙⠚⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣶⣿⣛⣁⠸⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⢿⣿⣿⡟⠛⣿⣿⣿⣯⣛⣵⣿⠤⡗⠃⠂⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣻⣟⢿⣿⣧⢃⣼⣿⣿⣿⣿⣧⣿⣿⣿⣒⣋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣿⡋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⡿⢛⣿⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⢴⣿⣿⣿⣿⡏⣻⣿⣿⣿⣿⣿⣿⣿⠟⣹⣿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠇⠁⠛⠻⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠋⡹⢿⡟⠉⠙⢻⣿⣿⠿⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠁⠀⠈⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⠑⠃⠙⠁⠐⠂⠂⠀⠀⠐⠢⠤⠴⠉⠀⠀⠀⡄⠧⠀⢀⡀⢀⢀⠀⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣶⣦⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⣿⣿⠀⠛⠀⣠⣤⣀⣤⣄⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠈⠁⠀⠀⢀⡴⢛⣿⡟⠀⠀⠀⣼⣿⢩⣿⡟⣹⣿⠃⠀⠀⠀⣠⠖⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠂⠀⠀⠀⠸⣧⣾⠟⠁⠀⠂⠀⣿⡿⢸⡿⠇⢿⡿⠆⠀⠀⠀⠀⢀⡉⠉⠁
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣦⣤⠀⠀⣠⣤⡀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⢀⣴⣤⣤⠀⠀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠉⠉⣿⡇⢀⣀⢀⣁⠀⣸⣿⣞⠁⣀⣀⣀⠀⣰⣾⣆⢀⣀⠀⣀⡙⠛⣋⣀⣀⠀⣰⣾⣆⠀⣀⣀⠀⠀⣀⣀⣿⡏
⠀⠀⠀⠀⠀⣤⢾⣿⠇⣽⣿⢻⣿⠂⢹⣿⠉⣾⡟⢹⣿⠁⣽⡿⠉⣹⡿⢸⣿⠃⣾⡿⢹⣿⠁⣹⣿⠉⣾⣏⣿⠗⣾⡿⢹⣿⠁
⠀⠀⠀⠀⠸⣧⣾⠏⠀⣿⠇⢿⡿⠄⣿⡿⠀⢿⡷⢿⡿⠄⢿⡷⠄⢿⡧⢾⡿⠄⢿⡷⢾⡿⠄⢿⡿⠌⠿⣿⠴⠂⢿⡷⢾⡿⠆
⠀⠒⠲⠤⠀⠀⠀⡀⠀⠀⠀⡀⠀⠘⠛⠃⠀⠀⠠⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠐⠀⣀⡄⠀⠀⣀⡤⠄
⠒⠚⠋⢀⠀⠀⢀⣤⣤⡀⢀⣤⡄⢴⣶⡆⣠⣌⠁⠀⠀⠛⣠⣤⠀⠀⠀⢀⡤⣶⣦⡀⢀⣴⣶⡄⠀⣬⠿⠏⠀⠱⢿⣥⠀⠀⠀⠤⠤
⠀⠐⠋⠁⠀⠰⣏⢸⣿⡇⢸⣿⡇⠈⣿⢇⣙⡋⣀⣾⣗⡀⣿⣟⣀⡀⠀⢿⣄⣾⣿⠃⣾⣿⡾⠁⣀⣀⣀⠀⣂⣀⠀⣀⡀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠉⢻⣿⡇⢸⣿⡇⣸⠏⢨⣿⠇⢹⣿⠏⢱⣿⢋⣿⡇⢀⠀⢹⣿⡏⢰⣿⡟⠁⣸⣿⠛⢿⡷⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⠷⠻⢿⠿⠃⠀⠸⡿⠆⠸⡿⠆⢾⠿⠸⡿⠷⠀⠀⠘⢿⣟⣿⣿⣅⡀⠻⣿⣦⠞⠐⣿⡷⢾⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣽⡿⠃⠈⠁
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠔⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠆⠀⢢⠀⠀⠀⢀⣩⣄⡀⠀⠀⠀⠀⠉⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⡘⠀⢠⠎⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⣄⠀⠘⡄⠀⠀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠊⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⢇⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⡧⠤⠤⠤⠄⠀⠀⠀⠀⢤⠤⠤⠤⢼⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠹⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠰⡀⠀⠀⠈⠓⠒⠂⠈⠀⠀⠀⠓⠶⠚⠉⠀⠀⢀⠆⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⢀⡧⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⢀⡤⠶⢾⣃⠤⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⢸⡷⠶⢤⡀
⠀⠀⠀⠘⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠇
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠹⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠟
⠀⢀⡤⠐⠈⠉⠀⠀⢈⠀⠀⢀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠀⠈⠁⠀⠉⠁⠐⠀⠀⢀⠀⡧⠀⠀⡁⠀⠀⠉⠙⠂⢤⡀
⠀⠙⠛⠛⠛⠛⠛⢛⡿⠀⠀⠸⣾⢤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⣷⠇⠀⠀⢿⡛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠚⠙⠛⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠘⠛⠋⠓⠋⠛⠛

⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠙⣦⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡄⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡤
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠷⠶⠶⣤⣶⣿⡀⠀⠘⠁⠀⠀⢀⡴⠚⠉
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣼⣿⣿⣿⡿⠋⠀⠀⠀⠐⠁⠀⠀⠈⠑⢦⡀
⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡥⢄⡀⡇⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠈⢱
⠀⠘⢷⣄⠀⠀⠀⠀⢀⡠⠚⠀⠒⠁⠀⠀⠀⠰⢧⡀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠁⠀⠠⣶⠋⠀⠠⠤⠴
⠀⠀⠀⠉⢳⡖⠀⡀⢁⠀⠀⠀⠔⢩⣷⠀⠀⠀⠀⠇⢸⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠈⢧⠀⠀⠸⠇⢀⠀⠀⠀⠉⠀⠀⣠⡞⠀⠘⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣀⢴⡇⠀⠀⢀⣀
⠀⠀⠀⠀⡀⠘⡆⠀⠀⠀⠾⠀⢀⣠⠆⠀⠀⡾⠀⠀⠀⠹⣿⡿⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠉⠁⢸⡇⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⢇⠠⡈⢆⠀⠶⣶⠶⢋⣿⠀⠀⡼⠃⠀⠀⠀⠀⠹⠀⠋⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠤⠤
⠀⠀⠀⠀⡾⡰⠙⠆⡑⢄⡈⠛⠛⢁⣠⣾⣿⣿⣶⡄⠀⠀⠀⠑⠄⡈⠃⣀⠄⠀⠀⠀⠀⠀⢠⡦⣄⠀⣰⡿⢻⡇
⠀⠀⠀⣸⠁⡇⠀⠀⢀⠘⣯⣽⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠁⠠⣀⡀⠀⠀⢀⣨⡇⠀⢻⣿⡇⢸⡇
⠀⠀⠀⡿⠀⠉⠀⠀⣩⡄⠈⠻⠿⠟⠉⢹⡌⢿⠁⠀⢠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣩⣿⡀⢸⡟⠁⠿⢷⣾⣿⣶⡷⠶⢶⣶⣶⣶⣤⡀
⠀⠀⠀⠙⠓⠂⣰⣾⣿⣿⣄⠀⣀⡠⠖⠋⣹⠘⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣧⣼⣿⣆⠀⠀⠻⣿⠿⣀⢀⠀⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣼⣿⣿⡿⠟⠛⠋⠉⠉⣿⣿⣿⣿⣧⠀⠀⢻⣧⠘⠒⠞⠛⢿⣿⡿⠟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣇⠀⢸⣿⣧⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠿⠧⠔⠒⢣⢀⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⡄⣸⣿⠈⠙⠛⠿⣿⣿⣿⡟⠀⢸⠁⠉
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⢀⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⡿⢿⠿⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⡇⠘⠁⠀⠀⠀⠈⠹⡀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢷⠐⡇⠀⠀⠀⠀⠀⢷⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⡀⣸⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠘⣆⢷⠀⠀⠀⠀⠀⠈⢧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣾⡆⠀⠀⠀⠀⠀⠈⢧⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠻⠿⠿⠿⠿⠟⠀⠙⢿⣿⣿⣿⣦⣄⣠⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⡠⠦⠀⡇⡄⢠⠢⡀⠰⣦⠀⡄⢰⠂⣠⠢⠀⠇⠀⠀⡴⠄⢠⠴⡄⢠⠤⠴⠄⢀⠀⠘⡇⠀⠚⠀⢰⠉⡆⠀⠇⠀⠯⢅⠀⠋⡗⠀⢛
⠀⠀⠑⠚⠀⠁⠁⠘⠚⠀⠑⠛⠀⠃⠈⠀⠘⠚⠀⠁⠀⠀⠓⠂⠈⠓⠁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠈⠒⠁⠀⠁⠀⠐⠊⠀⠓⠃⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣓⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣌⣙⡃⢉⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠿⠟⠻⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠐⢆⣼⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣿⣿⣿⣿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠢⠤⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠈⣧⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⠹⡄⠀⠀⠘⢆⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢻⠀⠱⡀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠸⡆⠀⠱⠀⠀⠠⣀⠀⠀⠀⠑⠀⠐⠄⡀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠹⡀⠀⠃⠀⠀⠈⠻⣄⠀⠀⢠⠎⠀⠈⠉⠒⠦⢤⣧⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠱⡀⠘⠀⣀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠶⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠱⡀⠀⠈⠳⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠡⠀⠀⠀⠈⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇⠀⠀⠀⠀⠀⣠⠴⠋⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⡼⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠤⠴⠋
⠀⠀⠀⠀⠀⠀⣇⠀⠀⣇⠙⠻⢧⣀
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠘⠧⡀⠀⠈⠛⠒⠶⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⣶⣢⣤⣤⣀⣀⣀⡀⠀⠀⠈⠉⠓⢦⣀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠻⢷⣦⠄⣀⡀⠀⠈⠑⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⡰⣶⡄⣴⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣴⠛⠛⠙⢭⡿⠉⠉⠛⠛⠿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⢠⣀⣀⣀⡀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢰⠃⠀⠠⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡗⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠛⣶⠋⠉⠙⣲⠖⢦⡤⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⡀⠀⠀⠀⠀⡞⠉⠛⠂⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⠠⠗⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣝⠦⣀⠀⡞⠙⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠛⠛⠛⠛⠒⠲⠿⣮⣕⣷⣄⠀⠀⠀⢠⡎⣀⡀⠀⠉⠉⠐⠒⠂⠤⢄⣳⡋
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡛⣿⡶⣤⢎⣀⠀⠀⠉⠁⠒⠒⠤⣀⣀⣀⣈⣿⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠚⠀⠀⠀⠻⡅⠀⠀⠉⠐⠩⠍⣉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠦⢤⣀
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢤⠔⠊⠉⠒⢦⣀⠤⠤⠤⣈⠓⠂⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢺⡃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠱⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡍⠉⠉⠛⠛⠛⠛⠒⠒⠒⠒⠚⠛⠋⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢻⠒⠀⠀⠀⠀⠉⣻⠉⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠁⠙⠉⠐⠒⠒⠒⠒⡟⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠤⠤⠤⠤⠔⠊⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠈⠑⠢⠤⠔⣐⣀⣐⡒⣄⣀⣀⣀⣠⣤⣶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡿⠛⡿⠛⠀⠀⠀⠀⠀⢀⡠⠖⠋
⠀⠀⠀⠀⢀⡤⢤⠀⠀⢠⠎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠐⡚⠁⠈⠀⠀⣇⠀⣀⣠⠴⠖⠋⠙⡄
⠀⠀⣞⠉⠺⣧⣀⣧⣴⡃⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣿⠉⠀⠀⠀⠀⠀⠘⣄⢀⡞⢻⡤⠢⣄
⠀⠀⠈⠒⠤⣄⠀⠀⠀⠻⠔⠲⡀⠀⠈⠙⠦⣄⡀⠀⠀⢇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣀⣀⡯⠛⠒⠊⣀⣠⠞
⠀⠀⠀⡴⠋⠁⣠⠖⠲⠤⢤⠼⠁⠀⠀⠀⠀⠀⠉⠓⠶⠾⠳⠤⠤⠴⠚⠁⠀⠀⠀⠀⣞⠀⢀⣠⠤⠤⣈⠉⠉⠓⣆
⠀⠀⠀⠳⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠙⠒⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⡸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡇⢸⣥⣴⣾⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠇






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⠠⠔⠋⠉⠉⠁⠘⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣵⣶⠁⠉⠀⠐⠒⢀⢃⣀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠿⠛⠿⠿⣿⣶⣶⣿⡈⠛⠟⠻⢦⣤⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⡶⢶⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⡄⠀⠾⣷⣆⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢻⣿⣾⣿⣟⠓⢶⣤⣤⣤⣤⣤⣀⡀⣀⣤⡀⠀⠀⠀⣠⣦⠀⠹⣦⡴⠿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣟⣿⣿⣿⣾⣿⠉⠉⠛⣿⣩⣍⣿⣿⠛⠛⠻⠛⢿⣯⣿⠃⢲⣽⣿⣶⣏⣰⣄⠃
⠀⠀⠀⠀⠐⠻⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⢰⣿⣿⣿⣤⠀⠀⠀⠀⠀⠘⢿⣿⣄⣿⡟⠋⠉⣿⣈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣟⢿⣧⣀⢀⣹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢿⣇⣠⠴⠚⠛⣻⣿⣿⣿⣿⡿⠔⠒⠲⣶⣶⣿⣿⣾⣿⣿⣿⣿⡟⢉⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡷⣦⡀⠀⣾⣿⣿⢿⣿⡇⠀⠀⠀⠈⣿⣿⣿⢻⣿⣿⣿⢿⣧⡈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⢳⣸⣷⣼⣿⣿⢏⣀⣿⡄⠀⠀⠀⠀⣿⢹⣿⣿⣿⣿⣿⣾⣿⣇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⢹⣇⣿⢿⣧⣼⣿⣿⣷⡀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣇⠀⠀⠀⢿⣿⡟⣿⡏⢉⣿⣿⠀⠀⠀⠀⣿⣾⡇⢿⣿⣿⣿⣿⣿⣿⣹⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠈⢻⣇⣿⡆⢈⣿⠙⣷⣤⡴⣦⣽⣿⣧⠀⠛⣿⣿⣍⢿⣿⣿⡟⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣤⡀⣼⣿⣿⣇⢸⣧⡾⠋⠙⢿⣿⣿⣿⣿⡇⢰⣿⣿⣏⠸⣿⣿⣏⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⡄⠈⢻⣿⣿⣿⡿⣎⣻⣿⡿⣦⣿⣿⣿⣆⣠⡿⠃
⠀⠀⠀⠀⠀⢀⣠⣀⣀⠀⠀⣸⣇⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣬⣿⣿⣶⣼⣿⣿⣿⣷⣽⡿⣿⣷⣈⣿⣿⣿⣿⣯⡀⠁
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣧⣶⡿⣿⡟⠻⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⡿⣿⣷⣿⣿⣿⣿⣿⢿⣿⡿⠿⡀⠀⢂
⢢⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣴⣿⣷⣴⡿⠋⣼⣿⣿⣛⣿⣿⣿⡿⣿⣿⢿⣿⣷⣶⡿⠿⣿⣿⣿⣿⣿⣄⠘⠟⠃⠀⠰⠀⠄⢀⣀⣀⣀⣀
⣜⡋⠉⠉⠙⠉⠙⠉⠉⠉⠉⠉⠉⠉⢉⡀⠀⠉⠻⡿⠿⠛⠛⠻⣿⣮⣉⠀⠈⠉⠙⠛⢷⣾⣿⣿⣿⣿⣿⣿⣷⣶⣌⣣⣶⣿⣿⡿⣿⣿⠿
⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣤⣤⣀⣀⡀⠀⠈⡉⠉⠙⠻⠿⢿⣿⣿⡿⣿⣍⣢⣾⢟⠁⣰
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⠹⣿⣿⣿⣶⣦⣤⣈⡇⢿⣎⣿⣶⣶⣼⣿⣿⣷⢾⢿⣿
⣿⣿⣿⣦⣀⣄⡀⢀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⢔⣲⣤⣌⣠⡾⣍⢹⣿⣿⣿⣿⣬⣭⣙⣻⣿⣿⣛⣛⡋⢠
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡀⢠⣼⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣷⣎⣵⢈⣛⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣷⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣤⡆⠀⠀⠀⠀⢀⣤⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⡌⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠚⠑⠀⠒⠋⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⠌⠛⠓⠀⠘⠓⠒⠲⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠘⠳⣄⡀⠛⠀⢻⣧⠀⡤⠐⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢀⣤⠀⠀⠀⠛⠀⢀⡼⠿⠖⠀⠸⠿⢷⠀⠀⠀⠀⣷⠀⠂⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡁⠀⠀⠉⠠⣀⣐⣶⣶⣾⣷⠶⢶⣶⠶⢶⣾⣶⣶⣶⣀⠠⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⡁⠠⣈⣵⠶⠟⠛⠉⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⡌⠙⠻⢷⣮⣁⠀⠉⠁⠈⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠀⡠⠂⣤⡾⠋⠁⠀⢀⠀⠀⢈⣀⣤⣦⣤⣤⣌⣉⣁⡀⠁⣀⠀⢀⠉⠻⣷⣤⠀⠀⠀⠿⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⣴⡾⠋⠀⢀⡠⢀⣹⣶⠾⠛⠛⠉⠉⠉⠉⠉⠛⠛⠿⣿⣦⣼⣃⡀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠊⠉⠁⠄⢠⡾⠋⠀⠀⠀⢀⣾⠟⠋⠁⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠙⠻⢿⣷⡆⢀⠀⠀⠙⣷⡀⠂⠘⠛⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢀⠊⣰⠟⠁⠠⠀⢻⣶⡟⠁⠀⠀⣀⣴⣾⠿⠿⠛⠛⠛⠿⢿⣿⣶⣦⡀⠀⠈⠻⣿⣦⠀⡄⠀⠈⢿⡄⠀⠀⢀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠒⠈⢁⠈⣠⡟⠀⠀⠀⣳⣿⠏⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣆⠀⠀⠈⢿⣷⡀⠀⠂⠈⣿⡆⠀⠚⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⠈⣠⣿⠀⠀⢠⣼⣿⠃⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠸⣿⡇⠀⠀⠀⢸⣇⠐⠀⠀⢀⡛
⠀⠀⠀⠀⠀⠀⢀⡴⠟⠂⠀⣿⡏⠀⠀⢀⣿⡟⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⡀⠀⢰⣿⠃⠀⠲⢏⡀
⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⡇⣿⡇⠀⠀⢈⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠤⠤⠤⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⣿⡄⠀⠀⠈⣳⠄
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⡇⣿⡇⠀⠘⠛⣿⣇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠰⢾⠃
⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠁⣿⣧⠀⠀⠉⢻⣿⠀⠀⠈⣿⣦⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⣼⣿⡀⠀⠀⢸⣿⠃⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠘⣿⡀⠀⠀⠘⣿⡆⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⠀⢰⣿⠟⠓⠀⠀⣼⡏⢀⠀⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⢠⢿⣧⡀⠈⢀⣸⣿⣦⡀⠀⠀⠛⢿⣷⣤⣤⣀⣀⣀⣀⣀⣤⣴⣾⠟⠋⠀⠀⢀⣴⡿⢧⠀⠋⠀⣸⡿⢀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⡈⠸⢻⣷⡄⠀⠁⣠⢿⣷⣦⡀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⣠⣿⣏⠑⠀⠀⠀⣰⡿⠁⠈⢠⣤⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟⠁⠀⠃⡘⢿⣦⡀⠀⠈⠉⠻⢿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠿⣥⠈⠓⠄⢀⣼⠟⢀⠆⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⣤⠘⢀⠙⠿⣦⣀⠀⠀⠀⠈⠙⠿⠛⣿⢷⣶⠶⣶⣿⠟⠛⢛⡿⠉⠤⠀⠀⢀⣠⣴⠟⢁⠔⠉⢰⡦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠐⠲⣌⠛⠿⣦⣤⣄⣠⡀⠀⠀⠀⠐⠀⠀⠉⠀⠀⠀⠀⣀⣠⣶⡾⠟⢋⡥⠲⣀⡀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⠀⠀⠉⠢⠀⢉⡙⠛⠿⠿⢶⣶⣶⣶⣶⡶⠶⠶⠾⠟⣛⣋⡡⠤⢈⠁⠀⠀⠹⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠓⠓⢾⡆⠀⠈⢁⡄⠒⠒⠂⠀⠀⠀⠀⠀⠒⠒⠀⠀⢉⡁⠀⠀⢸⠓⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠦⠄⠻⣇⠀⢀⡀⣶⡄⠀⠀⣸⣆⠀⠀⢀⡘⠛⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠁⢸⠟⠶⠂⢹⡏⠑⠺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⢀⣀⣀⠀⠠⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡌⢉⣉⠁⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠓⠶⠶⠶⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣤⡀⠀⠀⠀⠀⠠⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠉⠉⠉⠉⠁⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡉⠀⠙⠂⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡶⠦⠤⠤⠴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⣤⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⠀⢠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⠀⠸⣷⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢠⣿⡆⢰⡿⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⣧⡞⠙⣧⡼⠃⠙⢿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣦⣌⣉⣀⠀⠘⠃⠀⣀⣠⠷⠙⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⣤⣄⡈⠉⠉⠉⠀⠉⠉⠉⢁⣠⡴⠞⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠉⠉⠛⠛⠓⠒⠚⠛⠛⠉⠁⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣬⡛⠛⠛⠒⠀⠀⠀⠀⠀⠐⠛⠛⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠶⠶⠶⠶⠶⠶⠊⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠴⠂⠀⠀⠀⠀⠐⠶⠶⠶⠶⠾⣿⣭⣽⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠄⠂⠉⠀⢀⡤⠶⠶⢤⣄⣀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠩⣄⠀⣀⠠⠴⢾⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢸⡏⠀⠐⠈⠁⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠰⠃⢠⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣅⡀⠀⠠⢀⠑⢦⣀⠀⠀⠀⠀⠀⢀⣠⣿⡀⠀⠀⠀⣠⠇⠐⠋⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣄⠀⢸⠀⢹⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠁⠈⠉⠉⠛⠒⠰⠿⢿⣷⣝⢿⣿⡟⠛⠛⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠳⡍⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠰⠞⢛⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣠⠽⢦⣄⡀⠀⠀⠀⠈⠀⠀⠀⠂⢹⡦⡀⢀⢊⣾⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠻⠶⠶⠤⠤⣀⣠⠀⣼⣇⠘⠁⠊⢹⣿⡄⠀⠀⠀⠀⠐⢋⣼⠟⡝⣹⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠋⠁⠀⠀⠀⣸⣿⡇⠀⠀⠀⠐⣠⡾⣫⢎⡴⢁⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⡱⠶⠶⠶⠤⣤⣀⣑⠤⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣴⠞⣫⠞⣡⢎⣰⡿⠋⠛⠿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠐⣳⠟⠡⠊⢱⡿⠋⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠹⣶⣤⣤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡶⢋⣁⢄⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⣫⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣔⣳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠨⣿⣿⣿⣶⡂⠄⠀⠀⠀⠀⢸⣿⣿⣿⡇⢠⠏⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠶⡾⠿⠛⢿⡙⢿⣿⣧⣼⣠⡇⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣈⡓⠀⠹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢍⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⢭⣿⡗⠦⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣰⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧
⣿⠿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣿⣛⣿⣟⣿⣿⣟⣻

⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⣠⣶⣾⣶⣦⡀⠀⠀⠀⣠⣴⣾⣶⣦⣄⠀⠀⠀⢀⡴⣶⣷⡶⣄⠀⠀⠀⢀⣴⢶⣶⡶⣤
⠀⠀⠀⠀⣼⣿⣽⣿⣯⣿⣧⠀⠀⢸⣿⣭⣿⣿⣽⣿⠀⠀⢠⣿⣯⣿⣿⣽⣿⡄⠀⠀⣿⣧⣾⣿⣤⣿⡇⠀⠀⣼⣷⣽⣿⣧⣾⡧
⠀⠀⠀⠀⠸⣿⣟⣿⣻⣿⠇⠀⠀⠘⢿⣫⣷⣮⣿⠟⠀⠀⠀⢿⣜⡿⢿⣣⡿⠀⠀⠀⠻⣿⣾⣿⣾⡿⠃⠀⠀⠸⣧⣻⠿⣟⣼⠇
⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢠⣞⢿⣿⡿⣳⡄⠀⠀⢀⣾⢿⣿⣿⢿⣆⠀⠀⠀⣴⡿⣿⣿⢿⣦⠀⠀⠀⣰⡿⣿⣿⠿⣷⡀⠀⠀⢠⡾⢿⣿⡿⢷⡄
⠀⠀⠀⠀⢺⣿⣿⣛⡻⣿⡿⠀⠀⢸⣿⡟⠉⠙⣿⣿⠀⠀⠸⣿⣯⢿⡿⣿⣿⠂⠀⠀⣿⣿⢿⣿⢿⣿⡇⠀⠀⢿⣿⡾⣿⢷⣿⡗
⠀⠀⠀⠀⠈⠻⣾⣿⣷⠟⠁⠀⠀⠀⠻⢧⣄⣤⠿⠃⠀⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⠀⠘⠿⣧⣢⡿⠟⠀⠀⠀⠈⠻⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⣠⣤⣶⣤⣄⠀⠀⠀⠀⢀⣤⣴⣦⣄⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⣠⣤⣶⣤⣄
⠀⠀⠀⠀⣸⣏⣉⣭⣉⣹⣇⠀⠀⢰⣏⠿⣿⡯⢏⣷⠀⠀⢀⣿⣫⣿⣿⣝⣷⡀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠀⠀⣰⡹⠦⣶⠰⢏⡆
⠀⠀⠀⠀⠹⣟⢿⣿⡿⣻⡏⠀⠀⠸⣯⢿⣿⡿⢯⡿⠀⠀⠈⣿⡻⣿⣿⢟⣿⠁⠀⠀⢿⣿⡟⠻⣿⣿⠇⠀⠀⢹⣿⡿⠿⢿⣿⡏
⠀⠀⠀⠀⠀⠙⠛⠷⠛⠋⠀⠀⠀⠀⠈⠛⠷⠟⠋⠀⠀⠀⠀⠈⠛⠳⠞⠛⠁⠀⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠀⠀⠙⠛⠿⠛⠋

⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀⠀⠀⠀⣴⣾⣿⣿⣶⡄⠀⠀⠀⣠⡾⣿⣿⢷⣄⠀⠀⠀⢠⣶⣿⣿⣷⣦⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢼⣿⣾⣿⣷⣿⣷⠀⠀⢸⣿⣶⣿⣷⣾⣿⠀⠀⢰⣿⣷⣿⣿⣾⣿⠆⠀⠀⣿⣷⣿⣿⣶⣿⡇⠀⠀⢾⣿⣾⣿⣷⣿⡧
⠀⠀⠀⠀⠘⢿⣯⣿⣭⡿⠃⠀⠀⠈⢿⣿⣶⣿⣿⠏⠀⠀⠀⠻⣶⣲⣞⣶⠟⠀⠀⠀⠹⣿⣀⣀⣸⡿⠁⠀⠀⠘⢿⣿⣿⣯⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⢰⣿⢻⣿⡟⣿⡆⠀⠀⢠⣾⠻⣿⣿⢻⣦⠀⠀⠀⣾⡛⣿⣿⢛⣷⠀⠀⠀⣴⡟⣻⣿⡛⣷⡄⠀⠀⢰⣿⢻⣿⡟⣿⡆
⠀⠀⠀⠀⢻⣿⡟⠉⢻⣿⡟⠀⠀⢸⣿⢟⣿⣟⢿⣿⠀⠀⠘⣿⣿⣿⣿⢿⣿⠃⠀⠀⣿⣿⣭⣭⣽⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠈⠻⠷⣤⠾⠟⠁⠀⠀⠀⠙⠶⢶⠶⠞⠁⠀⠀⠀⠙⠿⢿⡿⠿⠋⠀⠀⠀⠈⠳⠶⡶⠶⠋⠀⠀⠀⠈⠻⠿⣿⠿⠟⠁

⣿⣿⡭⡄⠀⢸⣿⣿⠇⠀⠁⠈⠁⡰⣤⣀⢈⢿⣿⣿⡄⠀⠀⠀⠀⠒⠦⣀⠀⣤⠃⠼⠄⠠⡀⠀⡋⠀⠀⠀⢠⠏⣸⠏⠀⡿⢁⠀⣹
⣿⣿⢰⠅⠠⣿⡈⣿⠀⠀⠀⠀⠀⢀⠨⡿⣧⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠑⢿⡞⠀⠀⠈⣿⢀⡧⠠⠀⠀⣾⡐⣼⠄⢣⡦⡴⠚⢯
⣿⣿⠷⠂⠀⣿⢡⡏⠀⠀⠀⠀⠀⡀⠁⠐⢬⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣼⠟⠒⢶⡾⣿⡾⣠⠠⠤⠄⣿⣿⣿⡄⢢⡵⡖⢀⡄
⣿⣿⣿⣿⣦⣿⣿⣷⠄⠒⠀⠀⠀⠙⢦⡀⣤⣾⠛⣷⣟⠻⣇⠀⠀⠀⠀⣰⡾⣏⠁⠀⠈⢁⣟⣟⣿⡟⠋⠉⢻⣿⣟⠙⡆⢱⡧⠙⠑
⣿⣯⡀⠀⠀⢸⠶⠀⠀⠀⠀⠀⠀⠀⠀⡙⠚⠃⠀⠘⣿⣄⣿⣿⣦⡀⠀⣠⣋⠀⠀⣀⠙⢧⣓⣮⣿⣷⣄⠤⣬⣯⣿⣶⡇⢀⠗⢠⣿
⣿⣿⣿⡥⠭⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣴⣄⣤⠙⠛⢿⣿⣿⣷⣄⠀⠹⣞⠀⠈⠀⠼⣵⢹⣯⣿⣍⣙⣿⠁⣰⣿⠡⢸⡆⢸⣿
⣿⣿⣾⡧⠀⢸⣶⣶⣶⣶⣶⡶⠶⠶⣶⣾⣿⣿⣿⣷⡄⢸⣿⣿⣿⣿⠿⠿⣶⣾⣿⠅⠰⣿⡟⢀⠛⠉⠉⠉⡁⠙⢾⣿⣏⢸⡄⠀⢸
⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⡷⠔⠚⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⠐⠶⢶⣿⣿⣇⣦⡿⢧⡟⢀⣔⠒⢦⣴⡷⠀⢻⣿⣴⣥⠀⠸
⣿⣿⣿⡇⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⣟⣛⢿⣿⣿⣿⣿⣭⣴⣶⠞⠛⡿⣿⣍⢿⡟⠀⣾⡛⣷⣆⠙⠀⠀⠘⣿⣻⣄
⣿⣿⣿⡏⠀⢸⣿⣿⣿⡄⡇⠀⠀⠀⠀⠆⠀⢸⡿⣿⣿⣯⣿⡿⢿⡟⠛⠛⠀⣤⠀⣮⣿⡌⠃⢹⡟⢹⣿⠉⠛⠆⠀⠀⠘⣿⣿⣄⣤
⣿⣿⣿⡇⠀⢘⣿⢿⠏⠣⠃⠀⠈⠁⢰⣾⣎⢸⡆⢸⣟⣿⠻⠇⡤⠖⡇⠀⠠⡧⠈⠳⠡⡇⢹⣸⢀⠘⠋⠀⠀⣀⠤⠙⣿⡿⣿⣷⣾
⣿⣿⣿⡷⠀⣈⣿⡷⠀⡇⡤⠀⢀⣄⣼⠍⠙⠛⢳⣶⣿⣿⡶⢰⢳⢼⣳⣴⡀⠐⡖⡀⠀⠳⣼⠟⠢⠶⠶⠞⢏⠙⣶⣤⢹⣿⣿⣼⡇
⣿⣿⣿⡏⠁⢸⠏⠄⠀⡷⡃⠀⠀⠈⣁⣠⢰⣾⡛⠉⠛⣿⣧⣄⣡⣼⣿⡿⣽⣶⣿⠛⠁⢠⣄⡀⠀⠐⠄⢧⠈⠰⠃⠀⣸⣿⣯⡰⠄
⣸⣿⣿⢃⠀⠀⣀⠀⣆⢸⠈⠑⠄⠌⠁⢹⡏⣿⣶⡦⣛⣛⣻⣿⢿⡿⠿⠿⠿⠿⢣⣠⣴⣾⠻⡇⠀⠀⠘⢿⣧⠀⠀⢼⣿⣿⣿⣿⡄
⣿⣿⣳⣻⠴⠊⠀⡀⠠⣾⣦⠀⡐⠀⠊⠿⡇⠸⣿⣿⣯⣍⣟⡁⠀⣈⠗⠀⣀⡴⢾⣿⠙⢿⠀⡇⢠⠀⠀⠀⠀⠂⣀⡌⣿⣿⣿⡿⣇
⣿⡏⠀⠀⠀⠀⣦⠁⠀⠈⠛⠀⡟⠻⡿⣿⣿⠀⢹⡟⢶⣟⣿⠟⣻⣷⣴⣟⣻⣷⢸⡟⠀⣿⠀⠹⡄⠀⠐⠀⣦⡞⡏⣸⣿⢸⣿⡿⠟
⠋⠀⢀⣠⣴⠶⠛⠀⢀⣸⣿⠀⣧⢀⡎⣿⣿⣇⠀⣿⣻⡟⠉⠀⠀⠀⠈⠹⣿⡇⢸⡇⠀⢛⠐⣯⣷⠀⢠⣴⠃⠀⠀⣿⣿⠨⣿⣄⣰
⣶⢋⠭⣿⢁⡄⠀⡀⠀⠈⢿⣄⢹⣿⣿⡇⣻⣿⡄⢹⣿⢅⡀⠀⠀⡀⢀⡀⣹⣾⣾⠀⠀⣸⡆⢸⣻⣧⠈⠁⠀⠀⢠⣿⣿⢠⣿⣿⣿
⠀⠀⠀⠰⠄⠃⡀⠀⠀⢳⣌⣉⣩⣁⠈⠹⠉⣿⡿⠘⡃⠈⠁⠀⢘⢹⣆⠛⣮⡯⡿⢀⣼⣿⣧⠀⠿⢿⣆⠀⠁⠀⣸⣯⣻⢲⣿⣿⣿
⣤⡇⣦⢀⣀⠀⠀⠀⣀⣾⠉⠛⢻⣿⣟⡃⢸⣾⢃⡄⡿⢠⡀⠀⡾⠀⠿⣿⣻⣿⣷⣿⣿⡟⣿⡄⠐⣿⣿⡄⠀⢠⣿⣿⣿⠘⣽⣿
⣿⠷⢤⣴⢡⠈⡛⠀⣀⣠⡀⠀⠀⢻⣏⠷⡮⠏⢻⣿⡃⢀⡿⠛⠳⠶⠶⠿⠿⢋⣋⣻⣿⡇⠿⡁⠀⣿⡿⣷⡀⠸⣿⣿⣿⣼⣿⣯
⠶⠚⠋⠉⢭⣉⠙⠓⠶⠀⡙⠂⠀⣈⢩⣿⣿⠀⠀⢹⡤⠛⠀⠀⠀⠀⠰⠀⠀⠀⠁⠉⣰⣷⠀⠀⣰⠏⠀⢻⡇⠀⣬⣾⣿⣵⣿⣧
⠷⠄⣀⣶⢶⠈⢧⡀⠀⠐⢸⣷⡶⠀⠀⢈⡿⣣⣤⣸⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠙⢻⣇⠰⡟⠀⠀⠀⣿⡈⢹⣿⣿⡏⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠟⠛⠛⠻⠿⠿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⠤⢤⣀⡀⠀⠹⣿⡇⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⣤⣄⣀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡘⠃⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠴⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠦⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠘⠣⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⢤⣀⠀⠀⣸⠇⠀⠘⠲⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠍⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠃⠀⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⢰⡿⠟⢿⣏⠛⠛⠳⢤⠀⠀⠀⠀⠉⠓⠶⠤⠤⠤⠤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⠤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣸⠟⠛⠷⠶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠝⠋⢻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢠⡏⠀⠀⠀⠀⠀⠀⠈⡯⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⡤⠤⠴⠾⠷⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢻⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣧⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⣀⣤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⠷⠶⠶⠤⠤⠶⠶⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠐⠂⠈⠉⠀⠉⠉⠉⠀⠐⠒⠒⠂⠠⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢀⠴⠒⠒⢤⡀⠀⡠⠖⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⢄⡀⠀⡠⠖⠒⠒⠦⡀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⡌⠀⢐⠄⠀⠈⠎⠀⠤⡊⠔⢀⠈⡆⠀⠀⠀⠀⠠⠁⠀⡂⠄⡀⠑⣜⠀⢔⠡⢊⠀⢹⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠰⡀⠀⠄⢊⠔⡀⠔⣊⠔⡊⠅⢠⠁⠀⠀⠀⠀⢸⠀⠨⢔⠩⢂⠀⡠⢐⠡⢂⠕⠀⠐⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠄⢊⠔⡪⢔⠨⠐⡠⠁⠀⠀⠀⠀⠀⠀⢆⠀⠠⢊⠄⡨⢐⠡⢊⠅⠊⡠⠁⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠩⢂⠥⣢⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠁⡨⢂⠥⢊⠕⢊⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠡⠊⢔⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠃⠌
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⡀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⠀⡀⠆⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡠⠀⠠⠁⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⣁⣈⠀⠂⠐⠂⠀⠁⠀⠊⠐⠁⠊⠀⢁⣠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠐⠊⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⡀⠄⠀⡀⠄⠂⢀⠠⠀⣀⠤⠀⡀⠄⢀⠠⠀⣀⠄⠀⢀⠄⠀⡀⠄⢂⠠⠒⡀⠄⢂⠀⠀⡀⠄⢈⠡⠀⡁⠀⡠⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠡⠔⠊⢁⠄⠒⢈⠤⠒⣁⠠⠒⡁⠄⠊⡁⠔⢊⡠⠔⢈⠠⠐⡉⠔⢈⠠⠐⢉⠤⠂⡡⠐⠉⠠⠂⠁⠔⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠉⠀⠂⠁⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⡀⢠⠏⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣸⠁⡾⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡇⣸⠃⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⢡⠇⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣼⠏⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠋⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠣⣀⢀⠀⠀⠀⠈⠙⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠚⡇⠀⠀⢻⡌⠉⠑⠒⢤⣀⢠⣴⡿⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣭⢤⣀⡷⢶⡻⣇⠤⣄⠀⠉⢿⠳⡞⡋⡋⢛⠢⣤⠾⠟⡉⣽⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠝⢫⡀⠋⠡⠬⠝⠖⠚⠎⠵⠚⣀⠈⠙⡖⢐⡂⡀⡄⣀⠺⢠⡸⢫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡜⢣⠿⢶⣒⠆⠥⠌⣴⣆⣂⣐⣈⣑⣟⣶⡞⠤⠵⠎⣐⡂⠉⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢾⣉⠂⠼⣄⡈⡉⢙⠛⠁⠂⠐⠒⠒⠒⠂⢙⠋⠉⣀⠠⠶⠒⣈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠐⡦⠀⠠⢉⡉⢈⡉⠀⣉⡉⠀⢉⣈⣉⠠⠀⡴⠂⠉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠋⠁⠀⠀⠘⠀⠀⠀⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠂⠦⣄⡀⣸⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠼⠐⢂⣟⣸⡖⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣑⢲⠂⠤⢌⣈⣁⣀⣉⣁⡉⣉⡁⠤⠀⣶⠞⣉⠋⣿⢀⠐⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣤⣀⣀⣨⣀⣠⣤⡧⠴⠖⢧⢭⡬⡭⠼⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠛⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡀⠉⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢷⣦⣀⣠⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠻⣁⣉⣉⡠⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢹⣷⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⢀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡈⠳⣦⣴⣦⣄⠀⠀⠀⠉⠛⠛⠋⠉⢁⣴⣤⣀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠡⠝⢿⣿⣿⣿⣦⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠈⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡀⢠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⡿⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣠⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡛⠛⠿⠟⠛⠛⡉⠩⠍⣛⡻⠯⢝⣫⢿⣿⣿⣟⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠪⠓⠢⢍⠉⠉⠩⠤⠔⠂⠀⠥⢒⠈⠁⠈⠁⠊⠙⠙⢚⢗⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⡬⠷⡶⠖⡷⠿⢷⠥⠀⠈⠰⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠒⠀⠉⠛⠻⠿⣿⡿⢿⣿⡟⠛
⠀⠀⠀⠀⠈⢿⣿⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣫⣤⣦⣶⣶⣼⣿⣧⣦⣶⣡⡂⠘⠙⠰⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠐⠁⢀
⠀⠀⠀⠀⠀⡩⠳⠚⠐⠻⢻⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣶⣤⣈⠙⠻⢶⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣤⣶⣖⢠⣠⡖⢶⣦⡀⠉⠛⠻⢿⣿⣿⣷⣀⣉⠛⢿⣴⣤⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠿⠿⣿⣿⣿⣿⠅⠀⠀⠀⠀⢸⣿⢯⢙⣼⣤⣀⣽⣿⡇⠀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣦⡉⠻⢿⣮⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠋⢀⣾⠟⠛⡩⣷⠀⠀⠀⠀⠀⠀⠻⠉⠛⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠐⢹⣿⣿⣿⣿⡆⠄⠙⠻⣯⣢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠈⠻⠒⠒⠚⠻⠧⣀⠀⠀⠀⠀⠀⠑⢿⣶⣶⣾⠟⠀⠀⠀⠀⠀⠀⠁⠡⠈⠀⠁⠞⠉⠻⠿⡷⠂⠀⠀⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⣲⠦⣄⣈⠃⠶⡠⠄⢀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣦⣤⣀⣁⣀⣀⢀⡀⣀⣀⣀⣠⣤⡴⡾⢯⠣⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠜⠉⢉⡛⠛⠛⠛⠿⠛⠋⠙⠙⠩⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠙⠳⢦⡀⠀⠀⣀⡤⠶⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠹⣦⠞⠁⠀⠀⠀⢀⣀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣰⡟⢻⣿⣶⣿⣿⣿⣷⠀⢸⣤⡞⣹⣿⣾⣿⣿⣷⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⠇⢀⡿⣿⣾⣿⣿⣿⣿⣿⠁⠀⢸⠇⠀⠀⠀⢀⡀
⠀⢀⣤⣤⣤⡀⠀⠀⢳⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠋⠀⣸⠃⠈⠻⢿⣿⡿⠛⠁⠀⢀⡾⠀⠀⠀⠛⠉⡟⠷⡄
⠀⠋⠉⣧⠀⠀⠀⠀⠈⢳⡀⠀⠀⠈⠙⠻⠛⠉⠀⢀⣴⠋⠳⣄⡀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⢰⠃
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠉⠳⢤⣄⣀⣀⣀⣤⠶⠋⠁⠀⠀⠈⠙⠓⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⠤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣤⡤⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠚⠛⠛⠛⠛⠓⠒⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢠⠴⠒⠚⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠖⠲⠶⢤⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣴⣿⣿⣷⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⢿⣿⠟⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣿⣿⡟⠒⠲⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠶⣾⣿⡗⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣤⣤⣀⣀⣉⣉⣉⣉⣉⣉⣡⣤⣤⣶⣶⣿⣿⡿⠁⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⢤⣤⣤⣤⣤⡤⠤⠶⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠿⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣤⣀⠀⢀⣿⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⢹⡟⠉⡉⠙⣷⢸⡇⠀⢸⣷⡟⠻⣆⠀⣾⡆
⠀⠀⠀⠀⠀⢠⡀⠀⣾⡇⡻⣇⠀⣿⠀⢃⣼⠃⣾⠁⠀⢈⣯⡻⣄⠹⠇⣿⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⣆⢻⡇⠀⠛⢠⣿⡶⠿⣤⣄⣹⡆⠀⢸⣿⡇⠈⠳⣄⣼⣤⠤⢼⡶⢦⡶
⠀⠀⠀⠀⠀⠀⠈⠋⠘⣧⠀⠀⣸⠇⢻⡄⠀⠈⠋⠻⠶⢿⠀⠻⣦⣀⣈⣿⡆⠀⢸⡇⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠏⠠⠶⠷⠖⠛⠛⠛⠒⠚⠶⢤⣼⣯⣏⠀⠃⠀⠘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠶⠄


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

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⡇
⠀⢸⡇⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⡇
⠀⢸⡇⠀⠛⠛⠛⠻⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡟⠛⠀⠀⡇
⠀⢸⡇⠀⣀⣀⣀⣰⣿⣿⣆⡀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⣿⣿⣧⠤⠀⠀⡇
⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⢸⠿⠿⠿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⣿⡟⠉⠀⠀⡇
⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⠐⠿⠐⢀⠈⠶⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⠀⠀⡇
⠀⢸⡇⠀⣿⣿⣿⢀⣿⣿⣿⡇⢸⠀⠀⠀⠐⠀⠀⠈⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⡆⢀⣀⣀⣿⣿⣧⢸⠀⠀⡇
⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⢠⣶⡄⢀⣆⠴⠀⠀⠀⣿⣿⣿⡿⠛⠛⢿⣿⡇⢸⣿⣿⣿⣿⣿⢸⠀⠀⡇
⠀⢸⡇⠀⡿⣷⣿⢿⣿⣿⣿⣧⣼⡟⣿⣿⣤⣷⣾⣴⣤⡄⣿⣿⣿⣗⠀⢀⣸⣿⣇⣜⣿⣟⣿⣿⣿⣸⠀⠀⡇
⠀⢸⡇⠀⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣸⣿⣿⣿⣽⣤⣝⣦⣝⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⢸⡇⠀⣿⣷⣷⣾⣿⣿⣿⣏⣻⣿⣿⣿⣿⣿⡿⣿⣿⣦⣭⣿⣿⣿⣿⣿⣇⣉⡛⣿⡿⠛⣿⣿⣿⣿⠀⠀⡇
⠀⢸⡇⠀⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡗⡿⠿⠿⣿⣿⣿⢿⠀⠀⡇
⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠿⠿⠿⠟⢣⡤⠌⠁⣿⣿⣿⠼⠀⠀⡇
⠀⢸⡇⠀⣿⣽⣭⣽⣿⣿⣿⣿⣭⣯⣡⣤⣿⣯⣿⣭⣭⣭⣭⣭⣭⣿⣿⣿⣷⣤⣬⣥⣬⣭⣿⣿⣯⣥⠀⠀⡇
⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⢸⡇⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠻⣿⠿⠿⠿⠿⠿⠿⠀⠀⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠟⠀⠀⠟⠁⠀⠀⠀⠀⠴⠾⠇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣷⣶⣄⡀⠀⠈⢻⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠉⠛⠉⠀⢀⣤⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣏⠻⠟⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣛⣛⠛⠛⠉⠁⠀⣴⣶⣦⣤⣤⣄⣀⣠⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣄⠀⠀⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⡙⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢛⣥⣶⣦⡙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⢿⣿⣿⣿⣶⣝⠻⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⢇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣝⠿⣿⣿⣿⣷⣄⡀⠀⢀⣴⣾⣿⣿⣿⠿⣋⣴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡙⢿⣿⣿⣿⠟⠋⣴⣿⣿⠿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠉⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠀⢉⣉⠐⠒⠒⠀⠀⢀⣤⡭⠭⠙⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢠⣼⣛⠛⠿⠀⠀⠀⢀⣉⠁⠀⠐⣚⣹⠒⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠖⠒⠒⠒⠠⠤⠤⣄⠈⠛⠂⠀⠀⠩⡭⢁⠀⣴⣧⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⡄⠈⠘⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⢴⡟⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⢤⣅⠀⠀⠀⠘⠉⢉⣀⣀⠀⠀⠀⠀⢀⡴⠃⢸⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠉⠁⠀⠀⠀⠀⠂⠹⠏⠝⠃⠀⠀⠀⢨⠟⣀⠸⣼⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣡⠀⠿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡀⠃⡀⠀⠀⠑⠌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢃⠠⣶⡶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⣿⢏⣀⣀⡀⡄⡍⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡥⡎⣇⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⣠⠔⠙⢀⡄⢠⠾⠋⠋⠀⠻⠧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠶⠼⡀⠀⠀⠀⢀⣀⣤⣤⣶⠟⠁⠀⢸⠀⢀⡀⠀⠖⣠⡠⠀⣤⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣶⡎⠙⢿⣿⣿⣿⠿⠋⠁⠀⠀⠀⡜⠀⠛⠃⠐⠦⠈⠀⣸⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣶⢯⠉⠁⠀⠀⠀⠀⠀⠀⠀⣇⡐⣚⡄⠩⢶⣢⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠒⡋⠑⠨⠐⠀⠄⠀⠀⣀⣀⣀⣀⠈⠩⣅⣀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠑⠒⠀⠒⠂⠨⠩⠍⠉⠉⠁⠀⠀⠀⠈⠙⠓⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⡤⠤⠤⠤⠌⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⡒⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⣄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⢿⡟⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟⢁⣀⡀⠀⠀⠀⠀⢀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⣼⣿⣿⣿⣿⡍⣡⣶⣿⣿⣿⡆⠀⠀⠔⠁⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⠇⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢻⣿⣿⣿⣿⣿⣿⡏⢉⣡⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠟⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢦⣬⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣏⠀⢹⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⡿⠛⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠈⠁⠉⠄⠀⣠⣶⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⣴⠋⠘⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡰⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣆⠀⠰⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢸⣠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠼⠣⠂⠀⠠⠷⢄⠀⠀⢀⣴⠒⠒⠁⠀⢠⡟⢃⠈⠟⠘⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢠⠀⠓⠦⠘⡆⠀⢸⠿⠇⠰⠋⠹⠛⠃⠀⡆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡁⠀⡀⢀⣎⠠⠀⠀⠀⠁⠚⠋⠁⠀⠀⠀⢀⣠⠆⠀⢹⠀⢸⠀⢀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⠀⣸⣿⡄⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠁⠀⢀⣬⠀⡾⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠞⣧⠾⠋⠁⠀⠀⠠⠀⠉⠉⠁⠀⠀⠀⠀⢀⣰⢿⡟⡰⠃⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⡄⠀⠀⠀⢠⠀⠠⢀⣀⣴⣶⡿⣿⣨⠟⠁⡠⠃⠐⠟⠀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⡄⠀⠀⠈⠻⣿⣇⡀⠀⠀⠀⠒⣿⠟⢻⠛⠉⣷⠟⢁⠔⢋⢸⠀⠀⠀⠀⠀⠀⠀⠠⠴⠆⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⠁⠀⡀⠀⠀⠀⠈⠫⡧⣀⣀⣠⡼⠁⠀⠀⠉⠀⠉⠀⣷⡟⠠⣿⠇⠀⢨⡆⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠉⠁⢲⠀⠀⠀⠀⠀⠘⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⢀⡞⠀⠀⢿⣟⡄⠀⠀⠀⠀⢨⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣠⢀⠀⠀⠀⢠⠀⣀⢤⣄⡉⠃⠀⠀⠀⠀⠀⢈⡴⠚⢀⡞⠀⠀⠀⠀⠋⠙⠀⠀⠀⠀⡈⢁
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠆⣸⠃⠈⠳⣄⠀⠈⠁⠄⢠⠃⠀⠁⠀⠀⠀⠀⣠⠏⠀⣠⠏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠂⠀⠑⠄⠀⣰⣲⡀⠀⠀⠀⠀⠀⠀⡏⠀⡰⠃⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⣳⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⡞⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠉⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⡰⠣⠤⢠⣠⣤⣀⣀⣀⣤⠀⣼⠁⠄⠁⠀⢀⡀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⣇⠈⠀⠀⠀⠀⠀⠀⠀⢠⡷⠖⠲⠦⠤⠤⠤⠤⠀⠀⠀⠛⣤⠀⠀⠀⣈⠁⠀⠀⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌

⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⢣⠀⠀⠀⠌⠙⠃⡄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠚⢒⠀⠀⡀⠀⠐⠉⢃
⠀⠀⠀⢠⣀⡀⠀⡄⠓⠂⠋⠈⠈⠈⠈⠒⠒⠄⡀⠀⢀⡤⡀⠀⠀⢀⠱⠆⠊⠁⠁⠁⠉⠑⠐⠢⢀⠀⢀⡀⡀
⠀⠀⠀⡄⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣀⢀⠂⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢀⢀
⠀⠀⠀⣐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⡀⠠⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠠⠦
⠱⣀⢴⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠
⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⠀⠱⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠔⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⢻⡆⠀⠀⠀⠀⠀⠀⢰⠟⠛⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡜⠒⡀
⠀⠡⠠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⡀⠃⠀⠀⠀⠀⠀⠀⠸⣄⠀⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⠄⠄⠆
⠀⠀⠀⠠⢥⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡀
⠀⠀⠠⠁⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡋⠀⡐
⠀⠀⠀⠈⠈⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠁
⠀⠀⠀⠀⠀⠠⠈⠙⢢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⠉⢃
⠀⠀⠀⠀⠀⠀⠁⠂⠃⠈⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠑⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠈⠑⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠋⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠨⠓⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠓⡀⠈⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠄⠁⡱⠦⢤⠀⣀⠤⠶⠌⠐⠠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠌⠉⠐⢀⠠⠂
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠟⠛⠛⣍⠉⢹⡏⠉⣩⠟⠛⠻⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠉⠀⠀⠀⠀⠀⢈⣻⣾⣷⣞⣁⣀⣀⡀⠀⠀⠉⠻⢿⣷⣄
⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⣀⣴⣶⣮⣉⠉⣹⢿⣿⣿⡛⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣿⣆
⠀⠀⠀⢀⣾⡿⠋⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣦⣸⡇⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⢠⣿⡟⠁⠀⢀⣴⣿⣿⠋⠁⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⢠⣿⡟⠀⠀⣰⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣆⠀⠀⠈⠳⠄⢀⡀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⣼⣿⠁⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⠀⠀⠀⠀⠋⢹⣦⣿⣶⣄⠀⠀⠈⣿⣧
⢠⣿⣏⣴⡾⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⡇⢙⣿⣿⣷⡀⠀⢹⣿
⢸⣿⣿⠏⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣶⣦⡀⢸⣿⣾⣿⣿⣿⣷⠀⢸⣿
⢸⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠉⣿⣿⣿⣧⢻⣿⣿⣿⣿⠀⢸⣿
⠈⣿⣷⠀⠀⢸⣿⡇⠀⣠⡾⠛⠛⠛⠒⠒⠒⠶⣦⠀⠀⢸⣿⡇⠀⢀⣼⣿⣿⣿⡄⢀⣿⣿⣿⣿⠀⣼⣿
⠀⢻⣿⡄⠀⢸⣿⡇⠀⠹⣿⣿⣷⣶⣶⣶⣶⣿⡿⠀⠀⢸⣿⡇⠐⢿⡿⢻⣿⣿⡇⣾⣿⣿⣿⡏⢠⣿⡟
⠀⠈⢿⣷⡀⢸⣿⡇⠀⠀⠘⣿⠛⠛⠛⠛⢻⡟⠀⠀⠀⢸⣿⡇⠀⠈⣡⣿⣿⣿⣷⣿⣿⣿⣿⣧⣾⡿⠁
⠀⠀⠘⢿⣿⣾⣿⣿⣆⣀⣼⣛⣧⣀⣀⣀⣟⣻⣄⣀⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣶⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣛⣛⣛⣛⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠾⠿⠛⠛⠛⠻⠿⠶⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠟⠋⣁⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣈⠛⠿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⢉⣠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣿⣿⣿⣿⣿⣷⣦⣈⠙⠿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⡟⠿⣿⠟⢁⣀⠹⣿⣷⡄⣠⣤⣤⣤⣤⣤⣤⡄⣴⣿⡿⠁⣀⡙⢿⡿⠟⢿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤
⠉⠉⠉⠉⠉⠉⠉⠉⢿⣿⣿⡿⠛⣉⠓⢀⣤⡀⠻⣿⣆⠘⣿⣿⣌⠉⠉⠉⠉⠉⢉⣼⣿⡟⢀⣾⡿⠃⣠⣄⠐⢋⡙⠻⣿⣿⣿⠏⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⣄⠘⣿⣧⡈⢿⣿⣄⠹⣿⣧⠈⠻⠿⠂⠀⠀⠀⠀⠺⠿⠋⢠⣿⡿⠁⣼⣿⠏⣠⣿⡟⢀⣤⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠘⢿⣷⡀⠻⣿⡆⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⣾⣿⠃⣰⣿⠟⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠈⠻⠃⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠛⠃⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣶⣶⣶⣶⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣶⣶⣶⣶⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠄⠠⠄⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠁⠀⠀⠀⠂⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡐⠁⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⢠⢜⡂⠈⠁⠆⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⡖⠠⡆⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⣄⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠐⠢⣠⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣰⠋⠁⠀⠀⢴⡤⠖⠓⡦⠄⠠⠀⠘⠀⠚⢃⡤⣀⣀⠀⠀⠀⣟⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠤⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⣀⠒⡐⢆⠀⠀⠀⠈⡆⠀⠀⠀⠘⡶⠁⠀⠀⢀⠀⣄
⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠂⠀⠸⣇⠀⠀⠀⢸⡇⠀⡄⠀⠐⠀⢁⠈⠀⠀⠀⠀⠀⠉⣇⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠠⠀⠀⠐⠂⣨⡀⠀⠀⡄⣿⠀⠀⠉⠘⣷⡐⠁⠆⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡏⠰⠀⢀⠀⢸⡁⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡂⠐⢼⡇⠀⠀⠐⠚⣧⠀⠀⠀⢀⠀⠀⠀⠠⠀⠂⠘⡾⠀⠀⠀⠀⢬⢸⡗⠁⠰⠉⢐⠁
⠀⠀⠀⠀⠀⠂⡀⠈⠹⣤⠀⠀⠉⢀⢸⡏⠀⠀⠀⠀⠘⠳⣅⠀⣸⠀⠀⠀⠀⠆⣠⠞⠁⠀⠀⠀⠀⠀⣻⣠⠀⠀⡷⠈
⠀⠀⠀⠀⠐⠀⠀⠂⠠⠷⢄⣀⡀⠈⢹⡷⠀⠀⠀⢈⠀⠄⣌⠿⢧⣄⡀⢤⣰⠟⠁⠀⠀⠀⠂⠠⠂⠈⣿⠀⣠⡾⢁⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠉⢳⣼⡇⠀⠀⠀⠀⠁⠀⠁⢀⢀⣩⡿⠿⣧⣞⣀⠀⠀⠼⠀⠁⠀⣸⣧⡾⠋⠀⠈⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢹⡇⠀⡀⠃⠀⠀⢦⣠⡾⠛⠁⠄⠀⠀⠉⠙⠿⢿⣦⣀⡄⠀⣿⣿⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢙⢀⣴⠟⠛⠲⠀⠂⠰⠁⠀⠀⠀⢤⡀⡬⠻⢿⣦⣿⣿⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠘⠀⠀⢻⣆⠐⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⠀⠈⠀⠀⠹⣿⣿⡇⠀⢀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠁⢀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡅⠀⠀⠰⠀⢠⣿⣿⡇⠀⠀⠀⡵⠚
⠀⠀⠀⠀⠀⠀⠉⠑⣮⡀⠀⠀⣜⠁⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⣿⣿⠋⠠⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠹⣆⡀⣾⠁⡄⠀⠀⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠾⣯⠈⠀⠈⡄⠂
⠀⠀⠀⠀⠀⠀⢀⡰⠆⠈⠉⠛⠛⠻⢷⣤⣿⣿⡇⠀⠀⠀⠀⠀⣠⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⣼⣿⣧⠈⣿⣿⡆⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⢸⣿⡿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠛⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢻⣿⡟⠁⢹⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡀⠀⠀⠀⣿⣿⡇⠀⠈⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⡇⠀⠀⢀⣿⢸⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⡆
⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠶⠾⠿⠿⠷⠶⠶⠶⠾⠿⠿⠟⠛⠛⠿⠿⠛⠛⠳⠶⠶⠤⣤⣄⣀
⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠟⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠋⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠉⠻⣦⠀⠀⠀⠀⠀⠈⠳⢦⣴⣿⠟⠁⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⡀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠿⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⣾⣿⡿⠁⠀⠈⠁⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⠁⠀⠈⣿⣿⣷
⢸⣿⣿⣷⣄⡀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⣠⣿⣿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠉⠛⠻⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠟⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠤⣠⠤⢤⣤⠄⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠼⠾⠦⠸⠎⠦⠮⠞⢸⠂⡀⠀⠀⠀⠠⠌⠱⠦⡶⠶⠆⠀⠀⠀⡇⢈⠀⣈⠙⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⠀⠤⠄⠓⠂⠓⣂⢒⡒⠈⡁⣀⡠⢀⣈⠰⡇⠀⠀⠰⠲⠖⠗⠒⠂⠃⣸⠀⠅⠀⠈⢁⠀⣀⡀
⠀⠀⠀⡀⠠⢀⢀⠀⠐⠀⠤⠂⠴⠆⣴⣶⣶⡆⠠⢀⢠⢁⣀⠀⣀⢰⠆⢰⡆⡦⣧⠆⡦⠖⠄⣦⡤⠀⡆⠁⠋⠘⠀⠉⠉
⠀⠀⠀⠀⠀⠀⡀⠒⠐⠂⠠⠤⠤⢠⠄⣿⣿⠀⢀⠉⠈⣈⣬⣀⠁⣈⣁⣈⢁⣉⣉⡁⢉⣭⣉⠉⠉⡀⠁⡈⠉⠈⣁⠀⡀
⠀⠀⠀⠀⢚⢰⣣⡶⣲⣐⡒⣶⠀⡄⠀⣿⣿⠀⢰⠀⢰⣿⠛⣿⡆⠙⣿⡏⠈⣿⡟⢡⣿⠋⢻⣧⢁⠉⠉⡁⢉⠠⠄⣀
⠀⠀⠀⡄⡄⢠⠁⣄⢨⢀⢁⢈⠀⡇⠀⣿⣿⠀⣶⡆⢸⣿⠀⣿⣿⠀⢻⣧⢰⣿⠁⢸⣿⣤⣾⣿⢘⡃⠨⠡⠉⡍⡉⣉⣀⣠⢀⢀⢀⡀
⠀⠀⠀⡀⢀⠀⠀⠄⠉⠤⢨⠬⠀⠆⠀⣿⣿⠀⣿⡇⢸⣿⠀⣿⡿⠀⠘⣿⣾⡏⢀⢸⣿⠀⢀⡀⢸⡇⠀⣖⠀⠷⠟⠿⢻⠾⠛⠸⠹
⠀⠀⠀⣷⢲⢴⡲⣆⡶⢴⡆⢐⠀⠃⣶⣿⣿⣶⣿⡇⠘⢿⣶⡿⠃⢸⠀⢻⣿⠁⢹⠈⢿⣷⣾⠏⢸⠏⡄⠈⠐⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠉⠉⢉⡉⣉⣉⢈⡁⠸⠀⣲⣰⢒⠀⠤⠴⠄⠤⠤⠀⠀⠀⢨⠀⡤⠄⠤⠤⣤⠀⠀⡄⢀⡘⠁⠿⠟⠿⠻⠹⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠠⠤⠤⠤⢰⣲⣰⡶⡆⣖⢲⣶⣴⡆⠀⣤⢠⠀⠠⣤⡤⠀⡇⠈⠇⡀⠀⠀⠐⠒⠂⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠒⠖⠒⠒⠒⢠⡄⠢⠔⠠⠘⠃⡉⢉⠁⠉⡉⠨⠬⠍⠀⡂⢐⡂⠀⠄⠀⠤⠤⢐⡇⣛⡁⣃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠰⠖⠰⠄⠦⠒⠆⡀⢀⣀⣀⣀⡁⠰⡄⠈⠶⠂⠰⠷⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠛⠿⠿⣿⣿⣿⡟⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣄⡈⠛⢿⣷⠀⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠉⠉⠉⠙⠻⠆⠀⠈⠀⠈⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⢷⣶⡄⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣧⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣹⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⢿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⡀⠈⠳⣦⡀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢰⣆⠀⠻⣦⡀⠙⣿⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⣦⠀⢻⣿⣄⢘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⣿⣧⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡰⠁⠀⠀⠈⡇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠃⠀⠀⠀⠀⠸⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡰⡄⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡰⠁⠹⡄⠀⠀⠀⠀⠈⡇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠚⠀⠀⡰⠁⠀⠀⠱⡀⠀⠀⠀⠀⠋⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢠⠃⢰⠛⢄⠀⠱⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⡇⠀⠀⠳⡀⢳⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⡆⢇⠀⠀⠀⢳⠀⡇⣀⣀⣀⣸⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡤⠒⠉⠉⠙⠒⢤⡀⢈⡷⠋⠉⠀⠈⠉⠓⣌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⠀⡴⢦⠀⠀⠀⠀⠙⠞⠀⠀⠀⠠⣎⣳⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠉⠁⠀⡀⢠⣀⣇⣰⣁⡆⡄⠀⠈⠀⠀⡜⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢄⠀⠀⠀⢸⡊⠉⣞⣳⠀⢉⡽⠀⠀⢀⠞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⠳⣄⠀⠀⠏⡲⢪⠥⡖⠻⠈⢀⡴⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⡑⢦⡀⠀⠈⠀⠀⠀⡠⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⡓⢄⡀⣠⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢠⠂⢬⠁⡇⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⠀⢁⡠⠤⠆⠀⠀⠀⠩⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠌⠀⠀⢰⠊⢠⣲⢅⠀⠂⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⡏⠀⣯⣿⣧⡅⢀⣀⣄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⢣⡀⣛⢭⢏⢀⡮⠉⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡅⠀⡇⠀⢀⡔⡊⡙⢢⣠⠢⠀⡀⠀⠔⠡⠉⠑⣶⠛⠉⠳⡀⠀⣀⠂⣄⠀⡴⢐⡒⢦⡀⠀⠰⠀⣄⠐⠄
⠀⠀⠀⠀⠀⠀⡀⣨⢔⣻⠞⢸⠁⠀⣸⣿⣾⠁⠚⡃⠂⡂⠐⢀⢀⠀⠀⢾⣫⡶⢠⢱⠈⠆⠄⢨⡵⠀⠘⠂⢹⣾⣇⠀⠀⠆⢻⣃⡦⠑⣄
⠀⠀⠀⠀⠀⡈⣠⡾⣻⣿⡆⢀⡶⠛⠓⢏⡯⢠⡤⠃⢘⠀⠀⠀⠈⠡⢀⢀⣿⠠⠀⠐⠂⠀⠀⠀⡁⡃⣠⢄⠘⢿⠾⠞⢷⢃⢉⣌⡷⡈⠈⢢
⠀⠀⠀⠀⡴⢹⠷⣋⡼⣮⣿⡏⢀⡀⠀⢢⣀⠋⣇⠐⠍⣷⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⢡⡼⢁⡠⠤⡜⠀⢠⡄⢻⡼⣎⣛⡈⠰⠄⢆
⠀⠀⠀⢰⢠⡟⣟⣹⠟⡫⠟⠀⢸⡇⠀⠈⢋⣴⢾⡇⠀⣿⠀⠀⠀⠀⢀⡜⠘⣇⠀⠀⠀⠀⠀⣰⠶⠔⠋⣀⣜⠀⡄⢸⣷⢸⣿⠉⠳⣟⡓⠴⡈⡄
⠀⠀⢀⢣⠟⢚⠟⠁⠊⢠⣤⣦⡀⢿⢀⠀⡨⠝⠋⣠⡎⢿⠀⠀⠀⠀⣏⢸⠃⢸⡇⠀⠀⠀⠀⢁⣀⣠⣾⠿⠌⠓⣶⣿⠏⣘⣻⣝⢦⡈⠑⣦⠉⠹⡄
⠀⢀⢢⣮⠟⡡⣪⠜⣰⡿⠀⠘⢻⠷⡇⠀⠀⣠⢶⣻⣥⠞⠀⠀⠀⠀⠘⣄⣠⡟⠁⠀⠀⢀⣄⠛⢋⣉⣀⣤⣤⡇⡿⠛⠚⠋⢹⢹⡧⡛⣁⠈⠳⣧⠠
⠠⠗⠋⣁⢰⣄⢷⡼⡟⠀⠀⠰⡝⣴⢻⣸⣯⠟⠛⠉⣿⠀⠀⠀⠀⠀⠀⢾⣼⡇⠀⠀⠀⣠⣂⡤⠈⣻⡿⢸⡿⡎⡰⠁⠐⠁⠘⡄⣷⡹⣄⡟⠀⠈⠓⠦
⢠⢞⠃⠀⠂⠹⢸⣯⣧⠀⠀⠀⠨⣟⡸⣿⣏⠀⣀⠀⣿⡀⢀⡤⠀⡠⠄⠿⡾⠗⠀⠤⠄⠠⠤⢄⠠⡽⣷⡞⢨⡟⢄⠈⠀⠀⠀⡅⣿⣿⣽⠆⡏⠨⢀⢦
⢈⠉⠀⠀⡠⠀⠀⢸⡉⡚⠇⠀⠀⠈⢻⣜⢿⢰⢋⣜⣏⣀⡄⢀⣀⢲⡏⡂⠂⡀⢷⡊⠀⠀⠙⠐⡀⢦⡛⣰⠟⣠⠆⠀⠀⠀⠠⢣⣿⠟⣉⡼⠃⠃⠀⠘⡆
⣠⠛⠰⠊⣠⣶⣾⣷⣷⠛⡆⢀⡀⠀⠈⣿⡖⣢⢟⣽⣿⠴⣿⢚⣫⠥⠎⠀⠸⠃⠐⠭⣝⢶⣷⠐⠸⠘⠃⣩⡄⡿⠀⠀⠀⢀⢫⣿⠭⣾⣣⢃⡀⠀⡘⠣
⠻⠂⢠⣶⣿⣿⡟⡁⡞⠀⣝⣈⣏⢀⣀⢟⣼⠷⣻⠵⣿⣿⣧⣘⣃⡷⠟⠻⠿⠛⠻⢗⣙⣠⡿⠦⡙⠼⠧⠥⣤⡳⣤⣘⣁⡣⠂⠛⡀⡘⢿⣽⣿⣿⣇⠀⢂
⠀⠿⢾⣿⣿⠁⡔⡃⣇⣤⣬⣾⠟⢟⣡⣾⡒⠛⠛⠉⠈⠉⠀⠀⠈⠀⢠⣤⡀⣀⡀⠀⠉⠀⣀⡀⠀⠀⣀⣀⣄⡱⣿⡍⠻⣽⣿⣛⠘⢁⡌⡋⣮⣿⡟⢀
⠀⠀⠠⣜⣇⣴⣧⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⡷⣄⣠⣄⣿⣿⣟⣿⣿⣷⣠⣿⣽⣿⡷⢶⣟⣽⣿⣿⣿⢿⡾⣗⣶⣿⣷⣼⣠⡇⢸⡿⡢⠁
⠀⠀⠀⠈⠛⠶⠿⠼⠿⢿⢿⣻⣿⠿⠛⠙⢟⠻⢿⠿⣿⣿⡿⢿⣿⣿⣿⣷⣿⣿⢿⣿⣿⣷⣿⠫⠛⡹⠆⢉⣿⣯⠭⠙⠻⡟⠋⠁⠉⠈⠙⠗⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠁⠀⠂⠙⠋⠑⠂⠩⡏⠯⠋⠉⠀⠈⠛⠉⠀⠈⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠚⠉⠉⠉⠉⠓⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠉⠉⠉⠉⠑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠘⢆⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠉⠓⢴⡁⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠈⠙⠢⣄⠈⡆⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡼⠁⠀⠀⠀⠠⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘
⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢣⡜⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⣌⡉⠛⠻⠂
⡞⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣴⡟⠻⣿
⡇⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣴⣿⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣿⣧⣴⣿
⣇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⡀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠻⢷⣄⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠘⣷⡀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄⣞⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠘⢷⣄⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠙⠻⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠶⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠒⠉⠀⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⢀⢔⣿⠇
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⣾⣵⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢗⣦⣀⠀⠀⠀⠀⠀⠀⣀⠤⢺⣧⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⣿⠀⠉⠲⢤⣤⠒⠉⠀⠀⠀⢿⡄⠀⠀⢸⡟⠢⠤⠖⠊⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠁⢸⡿⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⢸⣷⣄⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠼⣿⡿⠷⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠰⠿⣿⡿⠄⢸⡇⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠷⡄⠀⠈⠻⣦⡖⠤⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠸⣀⠀⠈⠲⢄⠀⠀⠀⡠⠼⠧⡄⠀⠀⠀⢀⡤⠿⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⠃⠀⠀⠀⠀⠀⠉⠢⡄⠀⢱⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⠃⠀⠀⢣⠀⠀⠇⠀⠀⠀⠀⠸⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⡤⢃⡤⠒⠒⠋⠀⢸⠀⠀⠀⡠⠔⠒⠁⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠉⠉⠛⠉⠀⠈⠉⠉⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⡄⠀⡴⢦⣀⠜⠳⣤⠔⠁⠘⢦⠞⠉⠙⢦⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢠⠏⢘⣊⡤⠀⠉⠀⠀⠀⢀⠴⠂⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⡏⠘⠞⠁⠀⠀⠀⠀⠀⠰⡣⢂⡠⠖⠀⠐⠒⢦⣌⣇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣸⠀⠀⠀⠀⣰⣶⡄⠀⠀⠸⣷⣭⠤⠄⠀⠀⠀⠀⠹⢹⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠖⣾⠀⠀⣿⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢉⣿⠶⡄⠀⠸⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠇⠀⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⡝⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣸⠀⢀⣷⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⠞⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⢸⠹⡇⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⢀⣠⠤⠤⠿⢤⣸⠀⣇⠀⠀⢸⡛⢦⡀⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡇
⠀⢠⡋⠀⠀⠀⠀⣸⠻⡆⣿⠀⠀⢸⡇⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢹⡆⠀⠀⠀⢸⠇
⠀⠀⣟⠀⠉⠁⠀⣼⠁⣿⣿⠀⠀⢸⠁⠀⠀⠀⠀⠉⠛⠓⠶⠶⠦⠤⠶⠶⠒⠋⠁⠀⠀⠘⡇⠀⠀⠀⢸
⠀⠀⢸⣁⠈⠁⠀⢸⡤⠃⡏⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸
⠀⠀⠀⠻⠤⠤⠔⠊⠀⢰⠇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⢎⡀⠀⢀⣄⠀⠀⢀⣤⡀⠀⠀⠀⡠⣄⠀⠀⠀⣀⢤⠀⠀⠀⢷⡀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⡠⠋⠀⠳⠔⠁⠘⠦⠤⠋⠀⠙⠲⠤⠊⠀⠈⠓⠦⠞⠁⠈⠳⣄⣠⠎⠳⣄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠛⠛⠻⠷⣶⣤⡀⠀⠀⠀⢀⣤⣶⡾⠿⠟⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠀⠀⠀⠀⡘⣄⠳⣀⣀⠀⠀⠐⠘⣟⣿⣿⠋⠐⠲⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣌⠑⢦⣤⣤⡼⢾⣿⣽⣷⣭⣶⣾⣿⣿⣿⣿⣷⣶⣤⣧⣀⠈⠙⣷⣆⡀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⢤⣬⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡀⠙⢦⡀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢒⣲⣿⣿⣻⣿⣿⠋⢯⣹⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⠀⠙⠦⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠹⣿⣿⡇⠀⠘⢻⣿⣿⠟⠺⢷⣿⣿⣿⣿⣿⡟⣿⣿⡿⠻⣿⣮⣳⡀⠈⠂⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠁⢻⣻⣿⠀⠀⠈⣿⣂⠖⠈⠙⢿⣿⣿⣿⡿⠫⢿⡟⠀⠀⢿⢻⣷⡝⠄⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢰⢿⢿⣧⣄⠀⠘⢿⡀⠀⠀⠀⠏⠉⣉⣋⢠⠟⠀⠀⠀⠈⠀⢙⣿⡄⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠁⠀⠹⣿⣿⣷⣄⠀⠙⠦⣴⡊⡈⠸⡁⢘⠁⠀⠀⠀⠀⠀⢀⣄⢉⡿⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⢹⡗⣿⣿⣿⣷⣶⣤⡀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠊⠉⣿⠛⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢸⣿⠇⠙⣿⢶⡟⠉⠙⣏⡉⠉⠉⠉⠀⠠⠤⠐⠀⠀⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⣿⠀⠀⠘⣿⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⢀⣤⡾⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢿⣦⣿⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⢻⢷⣦⣀⣀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠂⠁⢠⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠋⠀⠀⠀⠀⠀⠿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡤⠤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⢀⢀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣠⣶⠿⢟⡃⣠⣴⡶⠀⠉⢉⡻⣷⣄
⠀⠀⠀⠀⠀⢠⡟⠀⣄⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣾⣿⣿⣿⠟⠀⠀⠀⠘⣿⢿⣿⡆
⠀⠀⠀⠀⠀⢾⠀⠀⡜⠶⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠘⢿⡟⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠘⠋⢹⣷
⠀⠀⠀⠀⠀⢸⡄⠸⡷⠤⠀⠀⠀⠋⠙⠀⠀⣯⡄⠀⠀⠀⠀⠀⠀⣇⠀⠀⠽⣿⣿⠀⣠⣄⡀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠈⡇⠀⡴⠀⠀⠀⠀⢤⡼⠇⠀⠙⡅⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠈⠹⠂⣼⡌⠁⠲⣦⢺⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡇⠀⢳⠀⠀⠀⠀⠀⠀⠀⣠⣼⠁⠀⠀⠀⠀⠀⠀⠈⢫⠷⠂⠀⠀⠀⣼⡇⠀⠀⠈⠁⠀⠘⠇⣀⡀
⠀⠀⠀⠀⢀⣾⠃⠀⢺⣶⡾⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠔⣫⣶⣿⣿⣦⣀
⠀⠀⡀⠐⠛⠁⠀⠀⠀⠈⠀⠀⠀⠙⠲⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣧⡷⠄⠀⠀⠀⠀⠀⣰⣯⣾⣿⣿⣿⣯⡄⠁
⠀⣸⡇⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣀⣴⢷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠷⠀⠀⠀⢀⣽⡆⠀⠀⠀⠀⠀⣼⠁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠟⡟
⠀⠘⠂⠐⠶⠛⠋⠀⠀⠀⠀⠀⠀⠿⠈⠉⠉⠈⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠋⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⢀⡠⠶⣂⡞⢡⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⡴⠂⠀⢈⡀⠀⠀⠀⢀⣶⡯⡾⡻⢹⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣟⣻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⠛⠰⢋⡀⠈⠇⢀⡴⠚⠉⠀⣀⡬⠒⠈⠀⢠⣼⣿⣿⣿⣿⣿⣟⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠛⠁⢀⣠⠷⡄⠙⠛⠻⡅⣀⡀⡆⠐⣿⣿⣿⣿⣿⣿⣿⣽⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢫⣏⠀⠀⠀⡰⢁⡿⠋⠀⡇⠀⢸⣿⣿⣿⣿⣿⡟⠃⠀⣛⣿⣿⣂⠤
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣠⠚⠁⢀⡞⠀⠀⢀⠞⠁⣸⠀⠀⣸⠃⠀⢸⣿⣿⣵⣟⢋⡔⠁⠀⢻⣯⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⢠⡞⠁⠀⢀⡞⠀⣀⡴⠃⠀⣰⠏⠀⠀⠇⠀⢠⣿⣿⣿⣿⣇⠎⢀⡀⠀⠀⠙⣿⡿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⢠⠏⠀⠞⠁⠀⠀⣼⠇⠀⠀⠀⢀⣴⣿⣿⣿⡿⢽⠏⠀⠀⠙⢦⡀⠀⠈⢻⣞⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣾⣿⣿⣟⡟⣣⠏⠀⠀⠀⠀⠈⠻⣄⠀⠀⠹⣭⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣾⣿⣿⣻⡞⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣿⣷⣿⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠛⠛⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠚⠉⠉⠉⠙⠓⠒⠤⢀⠀⠀⠀⠀⠀⣀⣰⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢡⠀⠀⢀⣾⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢿⠀⠀⢸⣿⠃⠈⢻⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣤⢊⡥⠖⠒⠀⠸⠀⠀⠀⢹⠀⠐⠒⠦⣝⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢰⣍⠀⠀⠠⠔⠒⠋⣡⡎⠀⠀⣀⣠⠄⢳⠂⠀⠀⠈⠀⢠⣀⡀⠀⠁⠘⣿⢿⢿⣿⣦⣄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⣏⠙⠲⠄⠀⠐⡋⠆⠁⠀⠀⠀⠀⠀⡴⠋⠀⠐⢧⡀⠀⠀⠀⠀⠀⠀⠁⡿⠄⣿⣿⣿⣷⣤⡉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⡎⢣⣀⠀⠀⣀⣑⠄⠀⠀⠀⠀⠀⢲⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⡿⣁⠈⠳⣄⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢹⠀⠈⠑⣶⡴⠛⠀⠐⠲⣤⣀⡀⣸⠀⠀⢀⣀⡜⢣⡀⠀⢀⣀⣤⠶⠒⠘⠿⣿⡿⠛⠓⠚⠓⢶⠾⠟⠲⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⣀⣀⣠⠞⠁⠀⠀⢀⡾⢥⣾⣿⣷⣦⣴⠋⠈⢣⡈⠙⣟⠉⠙⣮⠀⢀⡤⠞⠁⠀⠀⠀⣠⣴⠭⠽⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠀⠀⢠⠀⣸⣿⠀⠀⠀⠀⠀⠀⠞⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠙⠲⣌⡳⢀⠈⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⣸⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⢸⠋⠑⢺⠦⠤⠤⠤⠴⠒⠠⠤⠤⠤⠄⠒⠛⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⢠⣧⣾⣿⣿⡧⢤⣤⣀⣤⡤⠶⠊⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⢠⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠏⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⣾⣿⠿⠟⠋⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠤⠤⠖⠒⠚⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⣀⠀⠀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⡀⣸⠀⣆⠀⡇⠀⢰⡇⡒⢰⡧⣴⡇⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⣇⢠⡘⡦⣇⣻⣇⣿⣄⣷⣷⣿⠃⣇⣾⢣⣿⣼⣿⡿⣄⡴⣴⡆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⢌⣿⣧⣿⣿⣼⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⢟⣿⣿⣾⢋⣿⣯⣾⣣⡞⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠦⣀⣻⣖⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣡⡾⣃⡴⠋⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣨⣿⣿⣿⣿⣿⣿⡿⠿⠛⢋⠉⠉⣍⣉⡟⢩⢋⠻⡿⣿⣿⠿⣿⣿⣿⣿⣿⣻⣵⣿⣿⣾⣯⡴⢟⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠉⢹⣿⠁⠀⠀⠀⣶⡿⠻⠟⠛⣷⠎⠀⠘⣿⣿⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣫⣴⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣿⣿⡿⠋⠀⠀⠀⠀⠸⣿⡛⠀⠀⢘⣿⣧⡀⠀⣠⣿⣖⡁⠀⢿⡿⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣭⣥⣤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⢿⣧⡠⢔⡩⣻⠿⠷⠾⠿⡃⠀⠀⢀⣾⠁⢀⣠⣤⢤⣶⣴⣾⣿⠿⡿⣍⡛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢚⣋⡙⠻⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⠻⣿⣧⠞⡅⠀⠀⠀⠀⢳⡌⣠⠞⠁⣀⣢⣤⣶⣿⣿⣯⠛⢫⢔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢩⠟⢿⣶⣶⣿⣯⣭⣤⣌⣙⣿⣶⣦⣤⣤⣤⣶⣿⣷⣾⣿⣿⡿⢿⠛⠓⠀⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀⠈⠈⠁⠋⠉⠋⠙⠛⢛⠛⠟⠛⢿⠛⣿⠛⠏⠙⡟⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠁









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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⠤⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⢹⡀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⣿⣿⣿⣷⣦⡀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⣿⠀⠀⠀⠀⢰⡏⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠹⡿⠀⠀⢹⡇⠀⠀⠀⢸⣷⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⣾⣿⠛⠛⠛⠛⠛⠻⠷⠶⣦⣤⣈⠳⣄⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⢀⣀⣤⣤⣤⣤⣍⡻⢶⣄⠀⠀⠀⠀⠀⠀⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠋⠁⠀⠀⠀⠀⠈⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡴⠶⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⠴⣄⠀⠀⠀⠀⢀⣤⠴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣠⠏⠀⠀⠀⣰⠋⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢀⡾⠀⠀⢠⠶⠒⢦⠖⠲⢦⡀⠈⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠶⠖⠋⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠈⠷⣄⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠛⠶⣄⠀⠈⠳⣄⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣤⠀⢹⡆⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣠⡴⠚⠛⠛⠛⠛⠛⠛⠛⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣳⣼⡟⠓⠒⠶⠶⢦⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢦⣤⡟⠁⠈⠳⢦⣄⣀⣀⡼⠃⠀⠀⠀⠀⡇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠳⢤⣀⠀⠀⠈⠉⠙⣦⠀⠀⠀⠀⠀⡿⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹⡟⠓⠶⠦⠶⠃⠀⠀⠀⠀⠀⡇⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣧⣄⣠⣤⣤⣠⣤⣀⣠⡇⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⡗⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⢠⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⢸⡇⠀⣀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠙⣟⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠛⢾⡷⠛⠉⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⠶⠿⠶⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠾




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠄⠀⢠⠀⢀⠀⡠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⡀⠂⠒⠀⡆⠒⠀⠂⠀⠸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⡤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤
⠧⠤⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⢀⣀⣀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⢀⣀⡀⢀⣀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠐⠁⠈⠃⢸⠀⢸⠀⢰⢻⡀⢸⠤⡇⢸⠀⠀⡇⠀⠀⠀⠧⡀⠀⢸⠀⢸⣀⠇⡜⠀⠈⡆⣧⡜⠀⡧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣥⣤⣈⣤⣤⣃⣸⣀⣸⣀⣏⣈⣁⣸⣤⣏⣸⣤⣄⣧⣤⣀⣀⣤⣜⣀⣸⣀⣸⣀⣇⣘⣤⣔⣁⣇⣈⣆⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠦⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠙⢦⡀⠀⠀⢀⣴⣋⢀⡴⠚⠛⠲⣄⠀⣠⠖⠛⠛⢶⡄⣈⣷⡀
⠀⠀⠀⢠⡏⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠛⠻⢿⣷⡄⠈⢷⠀⡼⠋⠁⠉⠛⠀⠀⠀⠀⢸⠆⡏⠀⠀⠀⠀⠛⠉⠉⠉⢳⡀
⠀⠀⠀⡿⠀⢰⣿⠟⣋⣉⣀⠀⠀⠀⠀⢠⣾⣷⡆⠙⠟⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠈⠁⣾⠿⠿⠿⣷⠀⠀⠀⠸⣿⣿⡷⠤⢤⡀⢀⣿⠿⢶⣤⣀⠀⠀⠀⢀⡞⠀⠀⠀⢻⡄⠀⠀⠀⢀⣠⠶⢿
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠹⠏⠀⠀⠀⠘⡏⠓⢦⣤⡞⠀⠀⠀⠀⠀⠻⣄⡴⠞⠉⠀⠀⣿
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠚⢙⠆⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠠⣦⣄⣀⡀⠀⠀⣀⣠⣴⡄⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠤⠬⠃⠀⠈⠳⣄⠀⠀⠀⠀⣠⠞⢧⡀⠀⠀⠙⢦⣉⠉⠉⠉⣉⡴⠏⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠷⣄⣠⠞⠁⠀⠀⠛⢦⣀⠀⠀⠈⠉⠛⠋⠉⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⡤⠤⠴⠖⠛⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⣤⠤⠴⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠦⢤⡴⠁⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⢀⣤⠴⡾⠢⡀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⡴⠋⠀⠀⣧⠀⠀⠀⠀⢀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣤⣤⢤⡞⠀⠀⠀⠀⠈⠓⠦⠤⠴⠃⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣸⠀⠀⠀⠀⠀⠐⠉⠀⣡⠖⠋⠑⢦⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠠⠀⠘⣇⠀⣠⡤⠤⣄⠀⢠⡏⠀⠀⠀⠀⢳⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣿⣰⠃⠀⠢⠈⠳⣼⠁⠀⠀⠀⠀⠘⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠁⣿⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠘⠹⣄⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠉⠁⢹⠀⠀⠀⠀⠀⠀⢀⡼⠓⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣶⣦⣤⣤⣶⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠛⠛⣷⠚⠛⠛⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⡰⠋⠈⠀⠂⠀⠠⢄⡀⡀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢗⣩⡞⠠⠶⢦⣤⡄⠀⠀⠀⢀⣀⡈⣧⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠊⣹⠇⡧⠀⣠⣤⡀⠀⠀⠀⠛⠋⠀⠀⢹⡄⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡾⠃⢀⣿⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⠀⢸⣧⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⢀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⣆⡿⠛⠌⡆⠀⠀⢀⣤⣤⣤⠤⠀⠀⢀⣾⣷⠇⠌⢀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡿⠇⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⡵⠃⡠⠋⢈⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡇⢀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠄⠊⣻⢯⠋⢀⢊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣰⣄⠀⡰⠀⠀⠀⠀⠀⠀⠀⡍⢀⠀⣸⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣀⣭⠁⠀⠀⠀⠀⠀⠀⠀⠷⢫⣍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣦⣤⣀⣠⣀⣠⣤⣴⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⢀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡡⠤⠐⢂⣶⡤⠄⠈⠻⣿⢦⠀⠀⠈⠙⠳⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠠⠁⠋⠀⠀⢀⠄⣀⡈⠳⡆⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠠⠁⠀⠀⠀⢀⡖⢫⠏⢀⠀⠀⠀⠉⠉⠉⠉⠀⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣄⣈⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣶⣷⢿⠀⠀⢠⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣶⠀⠂⠀⠀⠀⠀⣄⣿⡏⡎⠀⡰⢃⡞⠀⢀⠀⣤⠴⠒⠀⠀⣬⣭⠟⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢠⡇⢰⠀⠀⠀⣤⣾⣿⣿⠃⡇⡰⢡⠎⠀⠀⠔⠀⠛⠀⠀⠀⠀⠈⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠂⢀⡇⠘⠀⠀⠸⢿⣿⣿⣿⠀⢱⣵⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡈⠀⠘⠃⠀⠀⠀⠀⠀⠸⢿⣧⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣻⣾⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢠⠃⣆⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⢿⣿⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⢰⡿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⣿⡇⠀⠀⠘⡇⠀⠀⡀⠀⢀⠃⠠⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋
⠀⠀⠀⠀⠀⠘⠀⠀⠰⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠳⠀⢃⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⣤⡄⢴⠃⠜
⠀⠀⠀⠀⠀⡆⢠⠁⣃⢾⣿⠇⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠾⠟⣡⣎⣜⣀
⠀⠀⠀⠀⠀⣀⢆⡼⢁⣾⣿⠁⠀⡆⠀⡇⠀⠀⠀⠀⠀⢳⠀⠀⠘⣀⠀⠈⠀⠁⠀⠀⠠⠀⠀⠀⠚⠁⠸⠁
⠀⠀⠀⢀⣴⠃⠞⢠⡟⣻⠇⠀⢰⠀⠀⣇⠀⠀⡄⠀⠀⠈⠗⠀⠀⠘⣠⠎⠀⢀⣴⠁⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠣⠤⠒⢉⠄⣠⠔⠊⠉⠁⠀⠀⡎⡌⠀⢸⠀⠀⡐⠀⠀⠀⢇⠀⠆⠀⡆⡆⠀⠀⠁⠀⠀⠀⠈⠀⣰⢃⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠁⡰⠃⠀⠀⠀⠀⠀⣼⡔⠀⠀⠀⠀⠀⡇⠀⡦⠀⢸⢰⡄⠀⠿⠃⠀⠀⠀⠀⢀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⢰⣿⠃⢹
⠀⠀⠀⢰⠁⠀⠀⠀⠀⣠⢞⡑⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠈⣸⠇⠀⠀⠀⠀⠤⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡣
⠀⠀⢀⠃⠀⠀⡤⠖⠋⠁⡞⠁⠀⠀⠀⠄⢀⠆⠁⢰⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠇
⠀⠀⡜⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡆⡰⠀⠌⡘⢠⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠇⡘⢠⢁⣾⠁⠀⠀⢸⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡠⢁⢃⢪⠃⡔⢁⠄⡾⠃
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡗⢁⡾⢡⢃⠌⢀⠎⠜⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⠾⣠⠟⣠⡗⠀⣠⠋⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠐⠁⠳⢋⡴⠋⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⡩⠴⠛⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⠛⠛⠛⠛⠛⠛⠿⣷⠄⠀⠀⠀⠀⠀⠀⢠⣶⡾⠿⠿⠛⠛⠛⠻⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣆⠀⣿⡇⠀⣾⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣆⡈⣿⣦⣾⣷⣶⣿⣁⣼⡟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣌⣿⡿⣿⡏⠿⠿⣿⡟⢻⣿⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣄⡘⠿⢷⡾⠟⢀⣠⣽⡿⠀⠀⠀⠀⠀⢀⣾⣦⣀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠷⠶⠿⠿⠛⠉⠀⢀⣀⠀⠀⠀⠿⠋⢉⣿⠿⢿⣿⠿⢿⣿⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠋⠀⢸⡿⠀⠘⠿⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⣠⣴⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠦⣄⣀⣀⣀⣀⣀⣀⣀⣠⠶⠒⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⢀⣀⣀⣀⡀
⠀⠀⢀⡴⠛⠉⠁⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠈⠉⠓⢦⡀
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⣽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏
⠀⠀⢰⠇⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⢹⡀
⠀⠀⣾⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠶⠦⠤⢤⣤⣤⣀⣀⣀⣀⣠⣤⣤⡤⠤⠶⠖⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⣀⣀⣀⣤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⡀⠀⢀⣀⣀⣀⡤⠤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣠⣤⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣷⣦⣾⣿⣿⣠⡶⠶⠒⠂⠉⠙⠲⣄⡠⠋⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⠛⠛⠛⢿⡿⠋⠛⠻⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣿⠀⠀⠀⢸⡇⠀⠀⠀⣹⣿⣿⣿⣿⡿⠟⠂⠄⠀⠀⠀⠀⠀⠀⠛⢿⣦⣀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠸⡆⠀⠀⢸⠀⠀⢀⣴⣿⣿⣿⠟⠉⠀⠀⠀⢀⣱⣦⡀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀⠀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⠀⠀⠀⠀⠹⡄⠀⠈⠀⣠⣾⣿⣿⣿⠃⠀⠀⠀⢠⡶⠟⠛⠛⠛⠓⠄⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⣀⣀⣀⣀⣠⣬⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠛⠳⠦⠤⣀⡀⠀⣀⢀⣴⣿⣿⡿⠛⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣄⠀⣀⣠⣤⣶⣿⣿⠿⠿⠿⠛⠛⢛⠿⠯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⡀⠼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢄⣀⣀⣠⣴⠖⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠋⠉⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⣀⡤⠖⠁⢠⣾⣿⣿⡿⢿⣦⡀⠀⠀⠀⠀⠀⠉⠻⡋⠀⠀⠀⠀⣀⣴⣿⣿⠟⠉⠉⠀⠀⠀⠀⣀⣀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠏⠁⠀⠀⢠⣿⣿⣿⠟⠀⠀⠉⠻⣷⣤⡀⠀⠀⠀⠀⠘⣶⣄⣠⣾⣿⠟⠋⠀⢀⡀⣀⣀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⢠⣿⣿⣿⣏⡀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⢈⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣾⣿⣿⣿⣿⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠁⠀⠀⠀⠀⠀⠀⢔⡁⢀
⠀⠀⠀⠀⠀⢤⡀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⠟⠄⠀⠀⠀⣱⣦⡀⠀⠀⠀⠀⣠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠁
⠀⠀⠀⠀⠀⠀⠙⣓⣤⣽⣿⠟⠻⣿⣿⣿⡿⠀⠀⠀⠀⠐⣿⣿⣿⣷⣶⣶⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⠀⠀⠈⠙⢿⡇⠀⠀⠀⠀⠀⠘⠲⣌⣙⣻⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠙⠲⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⣷⣄⠀⠀⠀⠀⠀⠈⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠊⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠑⢦⡀⢠⣿⠀⡠⠀⠀⠀⢹⠄⠀⠒⣤⠀⢸⡛⠂
⠀⠀⠀⠀⠀⠀⠠⠴⠚⠙⣷⠀⠀⠀⠀⠀⠙⢿⣿⣿⠏⠀⠀⠀⠈⢿⣿⠌⠀⠀⠀⠀⠘⠒⠀⠀⠦⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠸⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠙⢿⣿⣶⣤⣤⣀⣤⣴⣿⡿⠃⠀⠀⠀⠀⠠⠀⠀⢳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⠿⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣴⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣨⢿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠋⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠀⠈⠃⠀⠀⠀⢼⣿⣷⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⢿⠉⠉⠙⠛⠿⠿⣿⣿⠟⠛⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢚⣿⡿⠁⠀⠀⠀⠀⢸⣿⣿⡟⠛⢻⡆⠀⠉⠹⠛⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⢸⡏⠙⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠋⢡⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢐⣠⣼⣿⣿⣿⣶⣿⣅⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡎⠙⣿⡿⣿⣿⣿⣿⣿⣟⠒⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠒⢤⡤⠤⠴⠤⣤⠔⠚⠉⠉⠙⠲⣄
⠀⠀⢀⣠⠴⠒⠒⠦⣤⠋⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⢣⠤⠖⠒⠲⢤⣀
⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⣸⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢹⡀
⠀⢀⡇⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣀⣠⠤⠒⠉⠀⠀⠀⣇
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠉⠉⠙⠛⠓⠒⠒⠒⠛⠛⠉⠉⠉⠀⠈⣧⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡘
⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠑⠄⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⡀⠀⠈⢆⠀⡔⠁⠀⠀⠀⠀⢀⠠⠒⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠻⡀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢤⡀⠀⠀⠀⠑⢄⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⢳⣄⠀⠀⠀⠈⣇⠐⠄⣠⠤⣄⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠐⡂⢄⠀⠀⠀⠈⢆⠀⠀⠀⢫⠢⡄⠀⠁⠘⡄⠀⠁⠀⡄⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠘⡄⠣⡀⠀⠀⠈⢆⠀⠀⠀⢣⠻⣄⠀⠀⠰⠀⡇⢠⠁⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠘⡀⠐⡄⠀⠀⣸⡄⠀⠐⠉⡄⠈⢂⢀⣀⡇⠈⠪⢄⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⡇⠀⡇⠀⠀⠸⠱⠀⠀⠀⣱⢖⢜⠀⢀⡦⣀⠀⠀⠑⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠘⠁⠰⠁⠀⢀⠃⠀⠃⠀⠀⢸⡌⠛⣼⣡⠃⢸⠑⠤⣀⡀⣈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡌⠀⠀⡆⠀⠀⢀⠘⢄⠀⠀⠑⢄⢆⡰⠁⠉⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠘⠀⠀⠀⠇⠤⠤⡎⢢⣈⠑⢤⡤⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠀⠀⢣⣣⡸⠑⠮⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣶⣿⣿⣿⣿⠿⠻⢿⣷⣄⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣐⣚⣻⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡿⠉⠀⠀⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠟⠋⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠃⠀⠀⠀⠀⠐⣥⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⠿⠛⠿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⠁⠀⢀⠈⠳⣜
⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⣀⡀⣈⣿⣻⣗⣿⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣉⣉⣭⣭⣭⣭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣀⣈⣛⣻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠒⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⢄⡀
⠀⠀⠀⠀⠀⠀⣶⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡛⠲⠲⠶⠦⠤⠄⠠⠤
⠀⠀⠀⠀⠀⠀⠅⢝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠺⢯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣒⡒⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⢀⡠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠔⠂⠠⢄⡀
⠀⠀⢀⠃⠀⠀⠀⠱⡀⡠⠔⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⡍⢩⣭⡖⠢⣝⢆
⠀⠀⠘⡆⠀⠀⠀⠀⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠈⠢⢉⠠⠊⠀⣹⠂
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠒⠒⠂⠈⠉
⠀⠀⠀⠀⠈⠢⡄⢀⡀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠠⠤⠐⠀⠀⠉⠉⠉⠉⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢁⡀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⢗⡒⠂⢐⣂⣉⡀⠀⠤⠔⠙⣄⡀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠠⡤⠒⠒⠐⠈⠐⠒⢄⠀⡼
⠀⠀⠀⠑⡶⢇⠀⠀⢀⡠⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠉⠁⠀⠀⢇⡔⠒⠒⠒⠤⠤⠤⠞⢧
⠀⠀⠀⠀⡇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠂⠈⠉⠁⠒⠠⣡
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⠀⢀⠀⠠⠐⠀⠀⠡
⠀⠀⠀⠀⠀⠁⠒⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢢⠀⠀⡠⠄⠂⠄⡀
⠀⠀⠀⠀⠀⢠⡶⢳⠒⢒⣀⠤⣉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡠⢣⠊⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠙⢤⠀⠣⣈⠋⣀⠜⠑⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⡔⡠⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠄⠀⢙⠂⠤⠭⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⢇⠀⣀⡤⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠐⢀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠁⠀⠀⠀⠂⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠠⡠⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠤⠀⠦⢴⠶⠄⣀⣀⠀⢀⠀⠀⠀⡀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠐⠞⣢⠶⠁⣰⠀⣒⣠⠠⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠠⠀⠀⠀⠐⠒⢛⡐⢸⠏⠀⠀⠀⡰⠃⣿⠀⣸⡇⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢣⠇⣸⠀⠀⠀⠈⠐⢣⡟⣨⢻⡆⠀⠀⠀⠀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢨⡇⠋⠀⡟⠀⠀⠀⠀⡒⢼⡇⣸⢺⡇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⡇⠀⠅⢀⣀⠀⡀⢀⣿⢠⡜⢸⡇⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣋⠀⠠⠀⢫⣽⠉⢛⡇⡤⠚⢾⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⡇⠀⠀⣀⣈⣀⣉⣁⣉⣻⠀⣋⡸⠃⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠈⡀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠒⠂⠐⡄⠀⠀⠀⠤⠲⠀⠀⠀⠀⠀⠀⠀⠐⠄⠁⠘⢠⠀⠀⠉⠀⢄⠁⠀⠀⡁⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢁⠠⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠠⡰⠀⠄⠘⠀⠀⠁⡉⠬⢠⠀⠀⠈⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠑⠈⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡀⠀⠀⠀⠀⠆⡀⡜⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⣇⠠⡀⠀⠀⠀⠀⠀⡄⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⡴⠀⠀⠀⠀⠀⢹⣯⠀⠀⠀⠘⠘⡸⡞⡀
⠀⠀⠀⠀⠀⠀⠀⠅⠏⠅⡀⠀⠀⠀⠀⠀⡴⠄⠘⠈⠉⠉⠉⠀⠀⠁⠙⠀⢚⠧⠋⠁⠀⠠⣀⠵⢼⠄⠀⡀⠈⠒⠰⢇⡵⣠
⠀⠀⠀⠀⠀⠀⣀⣂⠛⠖⠃⠀⠀⠀⠠⣄⣵⣤⠅⠀⠀⠀⠀⠀⠀⠀⠐⠦⢛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠁⠀⠈⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣆⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢿⠀⠀⠀⣀⣠⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠑⠠⠁⠰⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣑⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣏⠀⠀⣼⣿⠟⠉⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣦⣴⣿⣿⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⢻⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠇⡀⠀⠀⠀⠻⣿⣿⣿⣿⠁⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡀⠀⠀⠀⠀⠈⠿⠿⠛⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠸⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠐⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣶⣶⣾⣭⣭⣭⣭⣽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢩⠞⣿⣿⣿⣿⣿⣿⡿⠟⠛⢋⣯⣽⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⣧⣿⣿⣿⣿⣿⡽⢁⣴⣶⣿⡿⠛⠉⠙⠉⡉⠙⢳⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⠿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣶⣶⣶⣼⣾⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠸⣝⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⡆⠀⠀⠘⠉⠁⢀⠀⠀⠀⠀⢻⣝⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠘⠻⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠉⠉⠈⢹⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣦⣴⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣟⣛⣉⣭⡟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠘⠯⣭⠿⠋⠀⠀⠀⠀⢀⠐⠰⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⠛⠛⠛⠛⢿⠋⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣦⣴⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠲⡒⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡐⠄⠨⠁⠑⠒⠢
⠀⠀⠀⠀⠀⠀⢀⠎⠡⠀⠀⠼⠁⠀⠡⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⠀⠀⡠⠃⡀⢀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡰⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠨⠑⠐⠂⠂⠉⠑⠤⠀⠀⢀⠀⢀⠠⠔⠉⠈⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠐⠈⠈⠀⠀⠀⠂⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣟⣵⠀⠀⢀⡴⢦⣤⠶⣤⠀⠀⠀⣴⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡭⢿⣶⣾⠿⣃⠀⠀⠸⣆⡀⠁⢀⡼⠃⠀⢠⣝⡻⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣜⣻⣿⣿⣶⡉⠀⠤⣤⣬⠿⠾⠯⣤⡤⠄⠀⣘⠿⣿⣿⡘⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢩⣶⣿⣿⠿⣃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣛⣛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣻⣥⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣼⣛⣯⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡛⠿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣛⣭⣝⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⠯⢽⣾⣿⣮⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣵⣿⣷⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠿⠿⠿⠇⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⠻⠿⠿⠸⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠇⣠⣤⡀⠀⠀⠀⠀⢠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡀⠀⠀⠀⠀⢀⣀⣼⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢻⠘⠇⠀⠀⠀⣀⣠⣽⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣦⡀⠀⢼⣂⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢀⠀⠀⠀⠀⠈⠈⠛⡀⠀⡐⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣾⡤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠙⠲⠅⠙⠅⠜⠇⠜
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠒⣂⡐⠒⡶⠂
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠁⠀⠀⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⡿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠉⠉⠉⠉⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠒⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠄⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠔⠊⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡶⠟⢋⣉⠁⠀⠀⠀⠀⠀⠀⠈⣉⡙⠻⢶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠻⣶⡾⠛⠛⠻⢷⡄⠀⠀⢠⡾⠟⠛⠛⢷⣶⠟⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⣿⡄⢠⣿⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠻⣷⣄⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣠⣾⠟⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⠻⢷⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⡾⠟⠁⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣾⠛⠲⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠶⠖⠛⣷⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢿⣦⣤⣤⣄⣀⣀⣀⣀⡀⠀⠈⢀⣀⣀⣀⣀⣠⣤⣤⣶⡿⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠛⠋⠉⠉⠙⠛⢿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠴⠆⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣄⣠⣤⡤⠶⠚⠉⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣾⣿⣆⠀⠀⠀⠤⡘⠃⠃⡁⠄⠀⠀⠀⠄⡀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠉⣛⡛⠻⠁⠀⠀⠀⠉⠀⠄⡁⠄⢄⢠⢀⡂⠐⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⠀⠰⠦⠀⠠⠃⠀⠀⢹⠢⠋⠛⠃⠀⢀⣀⡄⠗⠀⠀⠠⠥⠂⠀⡽⠯⡃⠀⠐⢃⠠⣤
⠀⠀⡤⠛⢂⠔⢄⠐⢏⠴⡟⢠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡄⠀⣨⡟⠀⠀⠀⠀⠀⡠⢮⡰⡀⢳⡶⠈⠉⡴⠄
⠀⠀⢀⡣⠠⣀⡘⠀⠞⠀⡀⠀⠃⠡⠀⠀⠀⠀⠀⠀⠼⡆⠊⠶⠴⠁⠀⠀⠀⠀⢾⢷⢛⠛⠁⠁⢨⢃⡀⢈⠁
⠀⠀⠠⠀⠈⠃⡇⠀⠄⠀⠵⢁⣁⣠⣶⢂⠀⠀⠀⠀⠀⠀⠂⢠⡀⠀⠀⠀⡴⠻⠀⢓⠀⠀⠀⠀⢈⠠⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠀⢃⢀⡠⠼⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⡴⠀⠈⠀⠁
⠀⠠⢠⣄⢀⣀⢄⣈⠀⠀⠀⠀⠠⠤⠒⠶⠈⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⢈⣦⡥⠀⠀⠀⠠⠄⠀⠄⠠⠀⠀⠀⠘
⠀⠀⣉⣐⠅⠄⡽⠢⠋⠌⠐⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠀⠀⠀⡄⠀⠀⢀⠠⡞⠈⢀
⠀⠈⠀⡰⠂⠃⢨⣀⠀⠄⠀⠀⡀⠠⢠⠤⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⢿⠆⠀⡀⣀⠐⡤⠲⠘⠀⠘⡃⠈⠙
⠀⠀⠸⣴⡗⡧⡛⠀⠛⠉⠀⡘⠎⣾⡁⠂⣤⢶⠈⠐⣖⣡⡄⢿⣾⠏⣿⣹⣿⠞⠰⣇⣇⡰⠁⠀⠤⠃⢤⠁
⠀⢀⠀⠀⠀⠀⢁⡷⢠⠀⡖⠀⠀⢨⢠⠀⠯⠄⡱⠳⠚⠿⣋⢁⢱⠂⡝⡋⢁⢀⡀⠘⠀⠀⠰⠀⠀⠠⠨⢀⣀⣀
⠀⠠⠀⠀⠀⠁⢊⡀⠈⠈⠁⠀⣰⠋⢳⠖⠁⠀⢰⡞⢀⣠⣸⠈⢞⡟⠀⠀⡍⢄⠒⢤⠀⠀⠀⠀⠀⠀⠀⠏⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⡇⣿⠀⢺⢁⣿⣄⡟⣁⣘⣶⠃⠀⠀⢀⠇⡨⠀⠂⠀⠀⠀⠀⠀⡀
⠀⣈⣄⣠⣆⣀⣁⣀⣠⣒⣀⣼⣿⣬⣿⣼⣡⣞⣼⣍⠋⠈⠉⠁⠠⠀⠀⢸⠅⠃⠀⠀⡆⠀⠀⠀⠀⠀⠂⠈
⡀⠋⠉⠿⣉⠙⠟⠉⠉⠉⠉⠋⡋⠙⢻⠛⣿⣿⡏⡙⠀⠠⠒⢃⡐⠦⠀⠀⠐⠒⢃⠀⠀⠀⠀⠁⠀⠈⠀⠀⠐
⠔⠓⠐⠂⠐⠊⠘⣞⣻⣟⣻⠛⢛⠚⡛⣛⣿⣿⣿⣇⠀⠀⠀⠈⣁⠀⠀⠀⠁
⠂⠓⠺⠟⠛⠛⠛⠙⠛⠛⠑⠑⠚⠛⠛⠛⠋⢿⡟⠙⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢰⡆⠀⠀⠂⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠀⠈⠋⠓⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡀⠀⡀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠾⠟⠈⠛⠛⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠀⠈⠟⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⣤⣤⣄⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢹⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠈⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⣠⡶⠁⠀⢸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠿⢿⠟⠛⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⠿⠆⢠⣿⣿⣿⣿⣿⣷⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢳⣤⠀⡀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠙⠛⠁⠀⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠸⠇⠂⡀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⣸⣿⣿⣿⣿⣿⠛⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣤⣤⣄⣀⣤⡀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⠤⠶⠂⢡⣄⠐⠶⢀⣀⡉⠘⠄⢄⡀
⠀⠀⠈⣹⠀⠀⠉⠋⠉⠂⢀⣀⠀⠘⠂⠀⠀⠀⠐⠀⠀⠀⢸⠆⠀⠀⠰⣄⣠⣽⢻⣿⣿⠃⠀⠀⣠⠞⠁⠀⠀⢤⣀⠉⠙⠶⢦⠉⠁⠀⠀⠀⠑⠄
⣶⣴⣦⣿⠁⠄⠀⠀⢠⡀⠈⢡⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠻⢿⣿⡿⠋⠀⠀⣰⠋⠀⠀⠂⠀⠈⢻⣶⣶⠀⠀⠁⠀⠀⠀⠀⠄
⣿⡏⣿⣟⣦⡴⣳⣶⡾⠁⢠⣦⣤⣼⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡏⠀⠀⠀⠀⠀⣀⠀⠀⠘⢿⣶⠃⠀⠀⠀⠀⠀⠀⢰⡄
⣿⣧⢻⣿⣿⡷⢉⣻⡄⣀⣰⣿⡄⣼⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⡇⢰⡀⠀⠀⠘⠻⣄⡀
⠻⣿⠿⣿⣷⣿⣿⣿⣿⣿⣿⣧⣇⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠰⡇⣸⡇⠀⠀⠀⠀⠀⠱⣄⣀⠀⠀⠀⠀⠠
⢶⣴⡆⢹⣿⣿⣿⣿⣿⡟⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣼⣿⡏⣆⣀⢠⡀⠀⠀⠀⠉⠁
⠀⠀⠁⠀⢻⣯⣉⣉⠙⠛⠻⠿⠿⣿⣧⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠙⠉⠉⠈⠛⢺⠗⠂⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠻⠏⠛
⠀⡀⠀⠀⠀⠀⠸⣶⣶⣶⣶⣤⠶⠶⠦⠈⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⣩⣄⠀⣴⡶⠀⠀⢠⣿⣶⡆⠈⠀⠀⠀⠀⠀⣀⠈⠀⠈⠀⢀⣾⣿⣿⣿⣿⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠙⣃⢀⣠⡤⠀⠀⣿⣿⣿⡧⠀⣷⣼⣿⠀⠀⠙⡇⠀⠠⣠⣿⡿⠋⣼⡿⣙⠓⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠊⢻⠀⢨⣇⠀⢰⡟⣿⣿⡇⢸⣿⠸⣿⡇⠀⠀⢁⣴⡶⠛⠋⠀⣾⣿⣿⡧⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣸⣷⣴⠟⢰⡟⠁⢸⣿⡇⣾⣿⡇⣿⣿⡄⠠⠛⠁⠀⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⣩⠿⣆⣰⣿⠀⠀⠈⢿⣷⣿⣿⡇⢻⣿⠇⠀⠀⠀⠀⣴⠟⠘⣿⠇⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣄⣤⣴⡿⠟⠃⠀⠀⠀⠈⢿⣾⣿⣫⣾⡿⠀⠀⠀⠀⡜⠁⠀⠀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡄⣰⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⢸⡆⠀⠈⣷⡛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⠀⠀⠘⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢉⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣶⣶⣦⣤⣀⠀⠘⠿⠿⠋
⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⡆⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⢰⣿⣿⣶⣶⡀
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣩⣙⣿⣿⣿⣿⣏⣍⡛⢿⣿⣿⣿⣿⣿⡿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠛⠛⠟⠉⠀⠉⢻⣿⣿⡟⠉⠀⠉⠻⠛⠛⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⣿⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⠿⣛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠱⣀⣀⠤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⠀⠀⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⣀⡄⠀⢀⠀⠀⠀⠀⠱⠊⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠚⣥⣄⡷⠤⢄⣁⠀⠀⠀⠀⠀⠀⠉⠉⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⡠⢺⣽⠚⠒⢤⣘⠈⢲⠻⡀⠄⠀⠀⠀⠀⠀⠸⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠉⠉⠁⠒⠒⠾⠤⣮⠃⠀⠀⢀⠼⠤⠮⣀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⣎⠉⠐⠢⡜⠦⣀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⡨⡕⠀⠀⠀⠀⠉⢫⡤⠀⠀⠀⠀⠀⠀⠰⣤⠤⠅
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠲⣜⠤⣀⢉⡶⠦⢤⣄⣀⣀⡀⠀⠀⠀⠈⠒⠠⠤⢄⡤⠔⠁⠀⠀⠀⠀⠀⠀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⠟⢉⡽⣯⣸⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⢠⠃⡠⢯⡔⠋⠁⠀⠀⣀⠼⣲⠀⠀⠀⣠⡠⠦⡨⢆⠀⠀⠀⠀⠀⠀⡤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⡰⠉⣇⢀⠇⡰⠓⠠⣉⠖⠒⠚⠛⣾⡾⠁⢀⠔⠋⡀⠉⡾⡁⠀⢀⢠⠦⠄⠤⠤⠊
⠀⠀⠀⠀⠀⠀⢰⠁⣼⣭⣾⢀⡇⢅⣷⠏⠀⠀⢢⣼⡿⢁⡴⠃⠀⢏⠈⠺⡄⠸⡦⢮
⠀⠀⠀⠀⠀⣀⣼⡾⢻⠁⠈⠳⠷⢿⡏⠀⠀⠀⠸⠜⡡⠊⠀⠠⠛⠒⣓⠆⢱⠀⢳⣀⡆
⠀⠀⠀⠀⡜⢝⠁⢣⠀⢣⠀⠀⠀⢸⠀⠀⠀⠀⢀⠜⠀⠀⣠⠒⠊⠁⣀⠤⠅⠒⠊
⠀⠀⠀⣤⡿⠒⠁⠈⢆⠀⢇⠠⡚⣇⣀⣀⣀⠤⣯⠶⠀⠀⠘⡄⡞⠁
⠀⠀⣞⢏⣀⣀⠀⠀⠈⢢⠀⠣⠧⢄⣀⣀⡀⢏⠀⠀⠀⠀⢴⠣⠜⡑
⠀⠀⠀⣨⡷⠋⠀⠀⠀⠀⠳⣠⠃⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⣇⠀⢡
⠀⠀⠀⠻⠢⣴⣶⠶⠀⠀⢠⡏⠀⠀⠀⠀⠀⢘⠑⠤⡈⠑⠤⣸⠀⠸
⠀⠀⠀⠀⢰⠱⣁⣠⣄⡄⠀⠉⡗⠀⠀⠀⠀⢸⡀⠀⠈⠒⢄⡁⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠈⠉⡟⠥⢔⣲⡇⠀⢀⠀⠀⠀⢦⠀⠀⠀⠀⠙⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠑⠵⠺⡻⡁⢀⠠⣒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠉⠉⠕⠠

⠀⠀⠀⠀⣠⣴⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣄
⠀⠀⠀⡼⠋⠉⠉⠉⠛⠛⠿⢿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠿⠿⠛⠋⠉⠉⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣖⣶⣶⣤⡀
⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠘⠿⣿⣿⣭⣄⣹⣿⣶⣾⣇⣤⠌⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠋⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠿⠿⣿⠿⠿⢿⢿⡿⢿⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣾⣿⣿⠀⠀⠀⠀⣿⣿⣷⣤⣿⣿⣿⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡄⠀⠘⠛⠻⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠛⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠉⠛⠿⠶⢶⣶⣶⡶⠶⠿⠛⠉⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣄⣀⣀⣤⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈