Bohemian Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⣀⡠⠤⠲⢸⣷⡆⠐⠒⠃⣿⣦⢄⣀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⡠⠦⠹⠇⢧⠀⠄⣾⠁⢷⡀⣠⡬⠋⡏⠀⠈⢙⣴⣏⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⣶⣟⣄⠀⠐⣧⡈⢿⠘⠹⡄⢸⠀⢹⡇⢸⠇⡠⡶⠒⣹⠋⠙⢦⡀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⠋⠈⢸⡄⠙⠄⢃⣳⠼⠒⠋⠉⠉⠉⠙⠓⠳⢤⣸⠃⣴⠅⢀⡀⢀⣵⣟
⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⢶⣷⡦⢤⡌⢏⢲⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣁⢪⡝⠋⣩⠟⠉⢦⡀⢀⡤⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⢷⢤⣈⡰⠋⡀⠔⠒⠒⠲⣶⡀⠀⠀⣰⡶⠖⠒⠒⠄⡈⢫⠐⡉⢥⣀⡀⣨⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣧⣤⣄⣢⣑⣸⠃⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⠚⢉⣠⡽⠟⢻
⠀⠀⠀⠀⠀⠐⠒⠚⠉⢉⠋⢳⣤⣠⣬⡇⠀⢴⣄⡀⠀⣀⣠⠀⠀⠀⠀⠀⢠⣀⡀⠀⣠⣴⠄⠸⡍⢠⣤⣄⠀⠈⠇⢀⡤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠉⣀⣈⣧⠀⠈⠜⡝⢟⠿⠋⠁⠀⠀⠀⠀⠉⢏⢿⠟⠟⠂⠀⣰⣟⣭⣁⣉⣵⡿⡟⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣞⡋⢉⣁⣿⠑⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠈⢹⣄⣠⢍⠢⡀⢠⠇
⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⢬⠀⠙⠛⠂⣈⣧⠀⡀⠀⠀⠀⠀⡀⣠⣀⣠⡀⡀⠀⠀⠀⢀⠀⢰⠯⢍⣉⠙⢣⣠⣘⣂⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⠞⠉⣠⠞⢦⡀⠀⠀⠀⠀⠈⠀⠈⠁⠈⠀⠀⠀⠀⠀⣠⠯⣄⣢⠉⠉⠛⢻⠋⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢛⣓⣡⡴⢙⢦⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣤⠎⠑⢦⣄⠙⣇⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⠣⣀⡴⢋⣠⣋⠄⡘⢹⢺⠓⢦⠤⠴⠞⢓⡚⠡⡈⠳⣔⢠⠈⠑⢟⠷⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⡿⢉⠉⠀⡾⢀⣾⠌⡸⠈⣇⡆⣷⠈⢧⡇⠽⣆⣿⠀⣀⠔⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠉⠑⠺⣷⠍⡁⠈⣿⡰⠟⠀⠘⡆⣾⢀⢀⡨⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠀⠀⠀⠀⢿⠀⠉⠛⠻⣿⠰⠒⠒⠛⢿⡌⠁⠀⠀⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣍⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⠀⣿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⣩⣿⣿⠀⠹⢦⡉⠛⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⢤⣳⡿⢟⢿⡃⠀⣓⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢊⣴⠟⠉⠼⠫⢋⠀⠀⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣶⠏⠀⠀⣠⢦⣷⣿⡄⢸⡷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠁⠀⠀⢠⣽⣿⣿⢨⡀⠈⡁⢿⣿⣷⣤⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠘⠁⠉⢛⢿⡇⠀⣽⠛⠋⠻⣆⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠠⠞⠊⠳⡄⠁⠿⡀⠨⡤⠀⠘⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠈⠂⠠⠀⡏⠀⡄⠃⠀⢁⣤⡙⢈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠶⠞⠀⠀⠁⠀⣄⠀⠐⠋⢩⡔⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠍⠢⠝⠂⣿⠀⢼⡛⣾⡑⣿⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣄⣀⣀⠠⣛⡄⠈⣡⣤⡤⠤⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣴⣦⣭⣷⡟⣻⠀⠉⠰⣾⣶⣦⣈⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠀⣸⠟⠀⢺⣦⣭⣙⣋⣈⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣴⣯⣞⠀⠘⢟⡋⠉⠉⠛⢻⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⣖⣻⡅⠀⠌⢁⣴⠾⠛⠛⠳⣝⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠛⢉⣿⠃⠘⠂⣾⠁⣴⠳⠒⠀⢸⢻⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣴⣿⠀⠜⣃⠻⡀⢿⣹⠦⣤⢞⣿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡽⢀⡼⢹⣿⣌⡄⠙⠛⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢄⡀⠉⠀⢈⣷⣜⠻⢿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣴⣦⠤⠂⠁⠀⠻⣿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣩⣥⣤⣶⠟⣤⡀⢷⣄⠉⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⣩⠟⢂⠉⢻⣾⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢿⠀⣿⡀⢈⡇⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠿⣷⠈⣽⣟⣠⢿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠿⢶⡖⢻⣿⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠦⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂


⣿⣿⣿⡿⠿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢤⣀⠀⣼⡷⢀⡄⠀⣦⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⢤⣤⣄⡀⠲⠂⠀⢠⠛⠋⡄⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠘⢿⡆⠛⢳⡿⠃⠀⢻⣿⣦⣤⣤⣶⣿⠃⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⢈⣴⣯⡄⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⢠⠞⢉⠓⢦⠀⠀⠀⠙⠻⢿⠿⠟⠁⠀⠀⠀⠘⠁⢠⣿⣿⠇⠀⠀⢐⣿⣻⡂⠀⠠⣤⣤⣾⣿⣿⣿⣷⣦⣤⠄
⠀⠀⠀⢏⠰⢀⠰⢈⠇⠀⠀⠀⠄⠂⠀⠂⠀⠀⠀⠀⠀⠠⠾⡟⢫⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠈⠳⠤⠖⠋⠀⠀⠀⠀⡍⠀⠂⠘⠀⠀⢰⣷⠀⠀⠂⡂⡃⠀⠀⠘⠃⠀⠀⣿⠀⠀⠀⠀⠈⠹⠟⠟⠁
⠀⠀⠀⢴⡄⣀⡀⣤⡄⠀⠀⠀⠀⠀⠀⠨⠀⠀⠘⠋⠀⠀⠀⠃⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⢀⣤⣤⣀⡀⠀⢀⢠⣶⡄
⠀⠀⠀⠈⠀⠙⠃⠈⠀⠀⢤⣤⡄⠀⠀⠀⡀⡄⣄⢠⢄⠀⢠⡤⠀⠀⡄⣠⢤⣀⠀⡀⠀⢴⣿⣿⣿⡟⠃⠀⣸⣇⡉
⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠉⠀⠀⢳⣿⡇⢧⢻⢸⡮⠀⢸⣳⡀⢰⡇⡿⡇⡟⢸⠃⠀⠀⠀⠉⠁⠀⠀⠀⠙⠊⠉
⠀⠀⠀⢰⣿⡿⠿⢿⣿⣇⠀⣠⣦⡀⠈⠃⢁⡈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠀⠀⢀⡀⣀⠀⣀⠀⣀⠀⣀⠀⡀
⠀⠀⢀⣼⣿⡇⠀⠈⣿⣿⣀⠈⠛⠁⣠⣶⣿⣿⣿⣷⡄⠀⠀⠁⠀⠛⠀⠐⠇⠀⠼⠆⠀⠈⠀⠈⠀⠈⠀⠀⠀⠁⠀⠁
⠀⠰⢿⣿⣿⣿⠁⣼⣿⣿⣿⠄⠀⠀⠘⡛⠛⡛⢛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄⠀⣄⠀⢤⠀⢠⠀⢠⡀
⠀⠀⠠⠍⠠⠅⠀⠈⡅⡉⠄⠀⠀⠀⠀⢁⡅⠆⢀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠇⠀⣨⡄⠀⢄⢁⢄⢁⢄⢉⠤⠈⠤⡈
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠁⠠⠁⠀⠁⠤⠉⠀⠈⠄
⠀⠀⢘⣴⡀⣄⠀⣀⡀⣀⠀⡲⠧⠀⠀⣰⣿⣿⠿⢿⣿⣧⠀⠀⠀⠀⠐⣤⣥⡔⠀⠀⠀⠠⠂⠔⠀⠠⠂⠔⠀⠠⠂⠄
⠀⠀⠀⠻⢿⣈⣿⣿⣿⣏⠺⠿⠀⠀⢠⣿⣿⣧⡀⢀⣿⣿⣄⠀⠀⠀⢁⠻⠿⢃⠁⠀⠀⢠⡄⢠⡄⢠⡄⢠⡄⢤⠄⣤
⠀⠀⠀⠀⠈⠁⠙⠛⠋⠉⠃⠀⠀⠀⠈⠙⠛⠋⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠁

⠀⡀
⠀⣁⣈




⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢀⠔⣠⠆⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠢⡙⢦⠀⠀⠀⠀⠀⠀⡴⢋⠔⣡⠞⡡⠂⠀⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠳⣌⢢⡑⢄⠀⠀⣠⠊⡰⢁⠔⡡⠊⡠⢺⠀⠑⢌⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣄⠑⣌⢢⡑⢌⠓⠚⡡⢊⡴⢡⠎⣠⢊⠔⢩⢀⡑⢄⠑⢌⠢⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢮⡳⡈⠢⡙⢦⡁⢈⡴⢋⠔⢁⠞⡵⢋⠔⢹⢣⡙⠢⡑⢄⠑⢎⠢⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡗⢀⡑⢌⠢⡈⠢⡁⢈⠔⢁⠔⡡⠊⡠⢁⠔⢹⣷⠬⢦⡼⢦⡵⠤⡵⢬⡦⢼⠦⢤⡤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢢⡑⢄⠑⢌⢢⡀⢀⡴⣡⠊⡠⢊⡴⠃⠀⢸⣯⠖⡩⠔⢁⠔⡩⠔⡁⠔⣡⠖⠁⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠙⢦⡑⠌⠳⡍⢨⠞⠡⠊⡴⠋⠀⠀⠀⢸⣿⠋⡠⢚⡵⠊⡠⢊⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠑⢌⠣⡀⢀⠔⡡⠊⠀⠀⠀⠀⠀⢸⣿⣫⠖⢁⠔⣡⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣀⡀⠀⠀⣀⣀⣼⡇⠀⠀⠀⠀⠈⠳⣄⢠⠞⠁⠀⠀⠀⠀⠀⠀⢸⣿⡠⢊⡡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣀⠀⠀⢀⡀
⠀⠙⠉⠀⠉⠉⠙⠻⢇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣻⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡟⠉⠀⠀⠀⠰⠂
⠀⠖⠂⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣦
⠀⠶⠦⣄⣠⣀⡀⠀⠀⠀⢀⡀⠀⠀⠐⢱⡖⠀⡀⣄⢀⠀⠀⠀⣀⡄⢠⡄⣠⣀⣀⢄⠀⢀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣾⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠆⣀⠀⠀⠀⡀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⡄⠀⠈⠁⠀⢀⠀⠹⠛⠀⠀⠈⣄⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠁⠀⢦⠀⠀⠀⠘⠣
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⢠⣾⠀⠀⠀⠈⠁
⠀⠀⠀⣸⣆⠀⠦⠀⠀⠀⠀⠠⠀⠀⠀⠐⣦⠀⠀⢸⡄⠀⢻⣷⢠⡞⠀⢠⣶⠖
⠀⢀⣨⠐⠹⣶⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⢹⣿⡿⡋⣼⣗⣠⣾⡇
⠀⠈⠳⣄⡀⠀⠉⠛⠿⠀⠀⠀⠀⠀⠀⠐⣤⣤⣼⣿⣷⣿⣿⣽⣿⢯⡍⣹⡿⠓⠂
⠀⠀⠀⠀⠉⠓⠒⠶⠃⠀⠀⠀⠀⣄⣀⣠⣶⣯⣠⣿⣿⡏⣴⣾⣿⡿⣿⣿⣥⣤⡾⠶⣄⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⣸⣿⢿⡿⣷⣬⣿⡿⣿⣯⣥⣯⣍⠀⠀⠀⠀⠀⠀⠀⣘⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣴⣾⣿⡿⡷⣿⣟⠓⣿⣏⠀⠈⠃⠀⠀⠀⠀⠀⠀⠏⠛⢷⣤⡀⠀⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡏⠈⢹⡇⣾⡟⠛⢿⡝⠛⠻⣧⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠉⢷⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠋⠀⠙⣷⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣦⠀⡀⠠⡏⠀⠀⠀⠂⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⡀⠈⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⢏⢩⡔⢠⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡄⠀⠂⠀⠀⠛⠿⠄⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠒⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⣈⡧
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⠀⠱⡀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒⠒⠒⠒⠲⠤⣀⠀⠀⠀⢀⠎⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⢀⠴⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠛⢲⣄⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢁⡴⠋⢳⡀⠀⣸⠛⢧⡀⠀⢠⠏⠙⢦⡀⡞⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⣼⣎⣀⣀⣀⣳⣴⠃⠀⢀⣹⣤⣏⣀⡤⠤⠿⠓⠚⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⢠⠤⢤⣀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢤⠀⠀⠀⢀⡇⠀⠀⠀⣧⠀⠀⠀⢀⠄⠀⠀⠀⠀⡇⠀⠀⠀⣀⡠⠤⠖⠒⡆
⠀⠀⠀⢸⠀⠀⠀⠉⠑⠒⢤⠀⢸⠀⠀⠀⠀⠀⠈⢳⡤⣤⠾⠄⠀⠀⠀⠸⠦⡤⢴⠋⠀⠀⠀⠀⠀⡇⠀⢮⣉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣀⣀⡤⠤⠒⠊⠉⠀⠀⡇⠀⠀⠀⠀⢀⣀⣀⠈⠀⠀⡀⠀⡀⠀⠀⢁⣀⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠉⠑⠲⠤⢤⠇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢰⠋⠀⠈⡇⠀⠀⠈⠉⠁⠀⢰⠋⠀⠈⢣⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠈⠢⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⠃⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠙⠲⢤⣀⣀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢆⠈⠙⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢏⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣶⣾⣿⣿⣷⣶⣶⣦⣤⣀⠀⠀⠀⠀⣀⠀⠨⡿⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⣶⣾⣿⡿⠟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⠟⠻⠇⠀⢀⣀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢿⡟⠛⠉⢹⠀⢀⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠀⣴⣷⡄⠀⢸⠀⠸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡏⠀⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⣾⣷⡄⠘⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠨⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠄⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠃

⡿⢿⠿⠿⠿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⢿⠟⠛⠛⡟⠛⠛⢻⠿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⡉⢀⣨⣦⣴⣶⠷⢶⣶⣷⣤⣤⡟⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⡁⣠⡿⠟⠉⢡⡄⠠⠤⠄⢠⡄⠉⠙⠻⣷⣬⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⢠⡿⠋⠀⠖⠁⢠⠖⣴⣒⣤⢬⠁⡈⠓⢀⣌⠻⣷⡈⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⢲⠟⡠⠛⣃⢼⣡⣿⣿⣿⢿⣿⣿⣷⣬⡤⡄⠁⢄⠹⣿⡈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣠⣏⣤⠀⣠⣧⣾⣿⠋⡟⢆⡰⢹⠉⠻⣿⣷⣖⢢⠈⣤⢹⣧⠼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣼⢠⠀⠘⣤⣿⡿⢫⢑⣷⡾⠿⢿⣖⡩⠋⢿⣿⡟⢤⠠⡀⣿⡄⢻⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⡇⣤⠀⣏⣿⣿⠿⢐⣿⢯⣾⠿⣷⢻⣏⡩⠻⣿⣿⢮⡀⣁⢸⡗⢺⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⡇⠀⢰⢛⣿⣿⢖⣻⡟⣿⢳⣿⢻⡏⣿⣉⡱⣿⣿⣶⡁⢉⢸⣇⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢙⡇⣤⢨⢿⣿⣿⢖⣿⡇⣿⢸⣿⢸⣷⣿⣏⣐⢿⣿⣧⠃⣤⡘⣿⠰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⡇⠤⠘⣾⣿⣿⣖⣿⡇⣿⢸⣿⢸⣿⢸⣟⣒⣼⣿⣇⠕⢠⠀⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⡇⠶⠂⢅⣿⣿⣤⣻⡇⣿⢸⣿⢸⣿⢸⣏⠱⢾⣿⣏⡆⣶⠀⣿⠒⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠃⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠁⠀⢀⣀⡀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⡠⠔⠊⠉⠀⠀⠀⠉⠉⠒⢄⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡜⠀⠀⢀⡠⠔⠂⠉⠉⠉⠒⠢⡀⠀⠀⠈⢆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡜⠀⠀⡠⠋⠀⠀⣀⠀⢀⣀⠀⠀⠈⢢⠀⠀⠈⡆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢰⠀⠀⡰⠁⠀⡰⠋⠀⠀⠀⠀⠙⢢⡀⠀⢣⠀⠀⠰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⡇⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⡄⠀⠀⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⢰⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢰⠀⠀⢃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠤⠺⠤⣸⠀⣀⠼⠤⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⡅⠀⣸⡀⠀⡇
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠈⠋⠀⠀⠀⠈⠳⠤⠤⣀⠀⠀⠀⠀⠀⣠⠴⠃⠠⢧⠎⠁⠈⠳⠗⠊⢁⠉⠲⡀
⠀⠀⢀⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⡆⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠈⢀⡇
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢀⠜⠀⠀⠀⠀⡱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠄
⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠈⠉⠉⠙⡄⠀⠀⠀⠀⣰⢆⡀⠀⠀⢀⠜⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣸⠗⠂⠐⠊
⠀⠀⠀⠀⠀⠀⠀⢸⠒⠤⠤⡞⠁⠀⠉⡏⠉⠁⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠋⠈⠓⢤⠤⠚⠁⡇
⠀⠀⠀⠀⠀⠀⣠⠘⢤⠀⠀⡇⠀⡠⠴⠇⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢢⠀⠀⢸⠀⢀⠔⠲⡄
⠀⠀⠀⠀⠀⠀⠣⣀⠴⠀⠀⡇⠀⠱⡀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⠇⠀⢸⠀⠘⠤⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⣄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⢀⠖⠚⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢤⠠⡄⠀⠀⠀⠀⡠⠐⢦⣄⠀⠀⠀⠀⢀⣴⠠⠀⠀⠀⠀⠀⣠⣶⣄
⠀⠀⠀⣼⡏⠆⠀⠀⠀⢀⢸⢸⣿⠂⠀⠀⠀⢰⠀⢸⣿⣷⠀⠀⠀⢸⣿⡓⡀⠀⠀⠀⠀⣿⢹⣿⡇
⠀⠀⡸⣿⡇⡄⠀⠀⠀⢸⣿⣸⠃⣾⡀⠀⠀⢸⠀⢸⣿⣿⠇⠀⠀⠁⠹⣧⠗⡂⠀⠀⠀⢿⣼⣿⡇
⠀⢀⣿⣿⠗⠁⠀⠀⠀⢸⣿⣿⣾⣿⠇⠀⠀⢸⣦⣸⣿⠟⡆⠀⠀⣠⠐⡗⡁⡄⠀⠀⡀⠀⢹⣿⡇
⠀⣾⣿⣿⠀⠀⠀⠀⢈⡈⠙⣿⡿⠛⠀⠀⠀⣿⣿⣿⡡⢠⠇⠀⢀⣿⣿⣯⡴⡇⠀⢰⣷⣄⢸⣿⡇
⠀⣹⣿⣿⢀⡤⠄⠀⣮⣳⢴⡋⠀⠀⠀⠀⠀⠙⠻⣯⡴⠋⠀⠀⢸⣿⣿⣿⣻⡇⠀⢸⣻⣿⣿⣿⡇
⠀⠹⣿⡟⠆⠀⠀⠀⠳⣭⡿⠀⠀⠂⠀⠀⢀⣄⠄⡏⠀⣠⠀⠀⢨⠙⢿⡿⠏⠃⠀⠀⠙⣿⣿⣿⡇
⠀⠀⠙⠛⠈⠁⠀⠀⠀⠁⠇⠀⠀⠀⠀⠀⠸⣿⣿⠓⠊⠁⠀⠀⠘⢄⣿⢁⠄⠀⠀⠀⠀⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠂⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⢀⡤⢖⣢⣤⣠⡤⢤⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠧⠤⣤⣤⣤⣤⣀⡶
⠀⠀⠐⠀⢰⣿⣴⡫⢍⣛⣋⠽⣿⣵⣞⢋⣡⣶⣶⣦⣤⣀⡀⢸⣇⣒⣐⣞⣛⣛⣛⣛⣫⣟⣦⡀
⠀⠀⠀⠀⠀⠙⢿⣿⣤⣂⣀⠐⢄⠑⣄⠙⢿⣿⠿⠿⠛⠋⠁⠀⠙⠋⠉⠓⢬⣻⡿⠿⣿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠉⠉⠉⠈⠁⢠⣴⣶⣶⣶⣶⡒⠉⠉⠙⠻⣿⣿⣿⣿⣶⣀⣀
⠀⠀⠀⠠⠠⠄⠄⣤⣮⣤⣼⣟⣆⣦⡤⠀⢿⣿⣿⣿⣿⠿⠿⠖⠒⠒⠒⢊⣟⣿⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠛⠉⠉⠉⠀⠀⠀⠉⠙⠛⠃⠀⠀⠄⠀⠀⠤⠡⠏⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⢀⠃⡇⠀⠀⠀⡾⢷⠀⠀⠀⡔⣧⠁⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢆⠀⠀⠘⣇⠓⣄⠀⠀⡇⢸⠀⠀⢸⠁⢣⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⣄⠀⠀⠀⠘⡌⠳⡄⠀⢸⣁⢸⠀⢰⠁⠈⡆⢠⡟⢀⡞⠀⢀⠞⢡⠃⠀⠀⠀⣀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠲⢤⣀⠀⢹⠀⠈⢦⡸⡄⢈⠳⡜⠀⠀⢃⡎⠠⢸⠀⡴⠁⢀⡏⠀⢀⡔⢩⠖⠋
⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠈⠢⣶⠊⠦⣀⢣⠀⠀⠑⠗⠀⠀⠇⠀⠀⢘⠐⠀⠙⠋⠀⠀⡜⣠⠒⢉⢀⡜⠀⠀⠀⠀⢀⣠⠤
⠀⠀⠀⠀⠀⠈⠻⢟⡒⠤⣄⣀⠘⢦⡀⠄⠉⣃⠀⠀⠀⣂⣤⠤⠤⠤⠤⣦⣁⠀⠀⠀⠘⢛⠉⠃⡜⠁⣀⣠⠤⢒⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠈⠉⠒⠻⢌⡀⠰⣦⠔⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⢀⡀⠠⠼⠒⠉⠁⢀⠔⠋
⠀⠀⣠⣴⣂⠤⣠⡤⠤⣀⠈⠳⣄⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡁⠀⠀⠀⡠⢞⣁⣀⣀⡤⠒⠶⢤⡴⠃
⠀⠀⠀⠀⠈⠓⠦⠦⣤⠨⠉⠉⠩⢅⠂⢀⠎⠀⠀⠀⠀⡀⠀⠀⠂⠐⠀⠀⢀⠀⠀⠀⠀⠹⣄⡰⠊⠉⣁⣄⣌⣄⡤⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡉⡲⠤⠀⠁⡞⠀⠀⠀⠀⠘⠃⠀⠀⠂⠐⠀⠀⠘⠁⠀⠂⠀⠀⢻⠁⠠⠤⠴⣋⣀⡀
⠀⠰⠶⠶⢾⣏⣉⣉⡁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢈⣉⣉⣹⡷⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠖⠒⠶⡀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡼⠄⠐⠶⣦⣍⣉⠁
⠀⠀⠀⠀⣀⣀⡠⠚⣉⠙⠡⣈⣀⡒⠙⠉⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⠉⠂⠑⣀⣀⣈⡈⠙⠖⠒⢤⡀
⠀⠀⢠⠞⠒⠲⠶⠚⠉⠉⢉⡵⠊⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠁⠀⠀⠀⠙⢤⡀⠉⠛⠒⠙⠒⠯⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⢀⣀⠤⡖⠀⠓⢀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠛⠂⠄⠓⣦⠤⣀⡀⠈⠢⣄
⠀⠀⠀⠀⠀⢀⣴⣮⠥⠒⠋⠉⢀⡼⠀⢉⣤⡄⠀⠀⠀⠝⢿⡒⠒⠒⠒⠛⠃⡀⠀⠀⢱⣀⡤⢈⡳⡄⠉⠙⠒⠬⣵⣦⡀
⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⡜⠡⣠⠼⠋⡜⠀⠀⣠⠂⡄⠂⡄⠀⠀⢀⠐⠀⣹⢄⠀⠀⢣⠉⠲⡀⠙⠢⡀⠀⠀⠀⠈⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠁⠀⣸⠁⢀⠞⠀⡇⢚⡞⢡⠀⠀⡜⢧⡁⠌⡎⠳⡀⠈⣇⠀⠉⠒⠢⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠀⠀⠀⢠⢃⡴⠁⠀⡼⠁⣼⠃⠸⡀⢀⠇⠀⡇⠁⡇⠀⠈⢦⡘⡄⠀⠀⠀⠙⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠎⠀⠀⠀⢳⠀⡇⠀⠀⡇⢸⠀⠀⠑⢆⡘⡄⠀⠀⠱⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⢠⢯⠟⠀⠀⠀⢷⡾⠀⠀⠀⢸⣱⠃⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣤⣤⣤⣶⣤⠤⠤⣤⠤⠴⠶⣶⣶⣶⣶⣶⢶⡶⡶⢶⢶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⢿⣿⣿⣷⡀⢻⡀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢿⣿⣿⣦⠁⠀⢰⣿⣿⡿⠀⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠙⢿⣿⣷⡀⢸⣿⣿⡇⠀⠀⢠⣾⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠳⣦⡀⠀⠀⠻⣿⣷⢸⣿⣿⠇⠀⣴⣿⣿⡿⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠈⠻⣷⣄⠀⠘⢿⡞⣿⣿⠀⣼⣿⣿⠏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠈⠻⣿⣦⣌⠃⢻⡿⣸⣿⡿⠃⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈⠻⣿⣷⡘⢧⣿⡟⠁⢀⣤⣾⣿⠟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠘⢿⣷⢸⣿⢁⣴⣿⣿⣿⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠹⣇⠋⣾⣿⣿⣿⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠈⠀⣿⣿⣿⣿⠃⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⢿⣶⡘⣿⣿⣿⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⠶⠶⠶⠖⠀⠀⠀⠀⠀⠠⠤⣤⣤⣤⣁⠘⢿⣿⡄⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣄⠻⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⣀⣀⣈⡁⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠦⡽⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣶⠶⠶⣦⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⠀⠀⠙⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⢤⡄⣿⠀⢷⣄⡀⠀⢀⣼⠀⣿⢰⡟⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⢢⡶⡀⠀⢸⡇⣿⠀⠀⠙⠛⠛⠛⠉⠀⣿⢸⡇⠀⢸⠐⠈⠀⡄⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⢡⣯⣾⣃⠀⢸⡇⢉⣋⣉⣛⣉⣋⣙⣉⣉⣋⢸⡇⢸⠀⢸⢀⠈⡇⣿⢸⡇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⣶⢸⡇⠀⢠⣿⣿⠃⠘⠁⢸⡇⣿⠀⣠⠀⣀⣀⠀⠀⠀⣿⢸⡇⢸⣸⢸⢸⡀⡇⣿⢸⡇⣿⠈⠉⠈⣉⣉⣉⡉⠉⣿
⠀⠀⢸⠀⢶⣔⢦⡐⢦⠻⠂⣻⢸⡇⠀⠘⠛⠋⠀⠀⠀⢸⡇⣿⠀⡟⣾⣟⣠⣾⡆⠀⣿⢸⣧⣄⣄⣈⣠⣄⣅⣠⣼⡇⣿⠀⣠⣮⣿⣷⣿⣗⠀⣿
⠀⠀⢸⠀⢶⡝⠂⣙⢠⡔⠀⣾⠈⣙⣛⣛⣛⣛⣛⣛⣛⣛⡃⣿⠀⣿⣿⡛⡿⣿⣯⡄⣾⢠⡶⠶⠶⠶⠶⠶⠶⠶⢶⡄⣿⢰⣿⣿⣯⠷⠿⠿⠀⣿
⠀⠀⢸⠀⢦⡙⠆⣉⢠⠹⠆⣿⢸⡇⠀⠀⢀⣀⣀⣀⣀⢸⡇⣿⠀⠿⠿⠿⠿⠟⠿⠃⢿⢸⡇⠀⠀⣠⠴⢶⣴⡆⢸⡇⣿⠘⠟⠛⠁⠀⠀⠀⠀⣿
⠀⠀⢽⣀⣀⣁⣀⣉⣀⣀⣀⣽⢸⡇⠀⠀⠉⢉⣉⣉⣅⢸⡇⠻⠶⠶⠶⠶⠶⠶⠶⠶⠿⢸⡇⠠⠀⠀⠀⣼⣏⡇⢸⡇⠿⠶⠶⠶⠶⠶⠶⠶⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⢸⡇⢠⣤⣤⣤⣭⣭⡥⢘⡇⣿⠋⠉⠉⠉⠉⠉⣉⠉⣿⢸⡇⠀⠀⣠⣾⣿⣿⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠶⠶⠶⠶⠶⠦⢸⡇⣿⠀⠲⢦⣶⣶⣿⣿⠃⣿⢸⡇⠀⠘⠛⠛⠋⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠾⠇⣿⠀⠓⠶⠄⠀⠚⠋⠀⣿⠈⠉⠋⠉⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⡀⠀⠀⠰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⡀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠻⠿⠏⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠒⠛⠋⠀⠀⠀⠀⠀⠁⠒⠒⠛⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⠛⠀⠀⠀⠙⠉⠃⠀⠂⠀⠙⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⢀⡤⢶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡏⠘⢦⣀⠀⠀⠀⡞⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡏⠧⣤⣄⣀⠀⠀⢳⡀⠀⠈⢳⡀⠀⣧⠀⠀⠱⣄⠀⠀⠀⠀⡔⢻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⣇⠀⠀⠀⠀⠙⢦⡀⢳⠀⠀⠀⣧⠀⣸⠀⠀⠀⢈⠇⠀⠀⢰⠇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⣄⡀⠈⠓⢦⡀⠀⠀⠀⢣⡾⠀⠀⠀⣸⡤⠏⠀⢠⠞⠉⣀⣀⡤⠋⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠉⠢⡀⠀⢹⡄⠀⢀⣠⠥⠴⠖⠒⠚⠶⠤⢤⣘⠲⠏⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠹⣤⠞⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢀⣴⣛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣌⠁⠈⠙⠦⣀⠀⠀⣠⣤⠖⠋⠉⠉
⠀⠀⠀⢀⡀⠀⠀⠀⢀⡤⠶⠶⢤⡘⠓⠋⣡⠖⠚⠛⠒⢦⡀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⠤⠿⡄⠀⠀⠀⠈⠉⠉⠀⡾
⠀⠀⠀⠈⠳⣦⣴⣞⣁⠀⠀⠀⠀⠙⣆⡴⠁⠀⠀⠀⣀⣀⣹⠀⣠⠀⣤⡀⢸⣁⣀⣀⠀⠀⠀⠀⢻⠀⠀⠀⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣿⠇⠀⠀⠶⢯⣹⣥⠟⣢⠟⣆⡇⢣⣘⢯⣽⣩⠷⠦⠀⠀⢸⡇⠐⠾⢥⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠬⢷⡶⠄⣿⠀⠀⠀⠀⠀⠈⠉⠉⠀⣠⠇⣇⠀⠈⠉⠉⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠐⢷⣤⠬⠃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣤⣤⣄⠀⠀⠙⠚⠉⣉⡷⣤⡄
⠀⠀⢠⣄⣀⡴⠋⣀⡤⠦⣄⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣠⣞⣳⣲⣤⡄⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⢳⡙⠦⣄⡤⠞⠁
⠀⠀⠀⠈⠙⠒⠛⠁⠀⠀⠈⠓⠦⢤⡶⠘⢷⣄⠀⠀⠀⠀⠀⠀⢽⣯⡭⠋⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠷⣄⡀⠀⠀⢰⡋⠉⠀⠀⠀⠀⠀⣀⡴⠟⠀⠐⢧⡀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠼⠀⠀⠀⠀⠀⣀⠙⠻⠂⠀⠓⠲⠤⠤⠴⠒⠋⠁⠀⠀⠀⠀⡟⢹⡀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⣀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣇⠀⠉⠉⠉⠉⠙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⡴⠋⠉⠉⠓⠒⠋⠙⣆⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣸⠙⣆⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠛⠛⠙⠒⠋⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢀⣀⠜⠙⣆⠀⠀⠀⢸⡀⠈⠙⠛⠓⠲⣄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡏⠀⠀⠀⠈⠙⠒⠲⢄⢳⡀⠀⠀⠀⠀⠸⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⡇⠀⠀⠀⠀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⢒⣒⣒⣒⣒⢒⠂⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢂⣥⠔⠂⠙⢁⠁⠀⢸⡇⢀⠀⠉⠋⠒⠦⢍⣒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⠔⠚⠁⠀⢰⡀⠀⠀⡀⠀⣸⣷⠀⠀⠀⠀⣠⠂⠀⠀⠓⢢⡑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣡⢞⡁⠠⠀⠀⠀⠘⣧⠀⠀⠁⠀⣿⣻⣠⠎⠀⠰⠇⠀⠀⡀⠁⠀⢈⡳⣍⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠴⠁⠈⠻⡶⣄⠀⣄⠀⠀⠈⠀⠀⢰⣿⣿⣿⢀⠄⠀⠀⠀⠀⠁⣠⢴⡟⢁⠈⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⠞⠀⠂⢀⡀⠙⣮⣿⣾⣦⡘⣄⡠⢔⠚⣛⣛⡛⠺⢄⣀⠀⢀⣴⣿⣵⠋⠀⠀⠀⠀⠳⡘⠄
⠀⠀⠀⠀⠀⠀⠀⢠⢃⡏⣁⠀⠀⠀⠁⠀⠈⠻⣟⡿⡋⠔⠈⠀⠀⠀⠀⠀⠀⠁⠪⢑⢯⣿⣿⠗⠋⠀⠀⣀⣀⡀⠻⡘⡄
⠀⠀⠀⠀⠀⠀⠀⡎⣜⠀⠀⠙⠲⠎⠀⠀⠀⢠⢎⠄⢀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⡀⠱⡹⡥⠒⠀⠀⠋⠁⠅⠀⠈⣡⢡
⠀⠀⠀⠀⠀⠀⢰⠠⠁⠀⠀⠀⢀⡀⠀⠀⢤⠎⠂⠀⠀⠀⠈⠙⠓⠀⠀⠚⠋⠁⠀⠀⠀⠐⠱⡀⠀⠀⠂⠤⡄⠀⠂⠨⡄⡆
⠀⠀⠀⠀⠀⠀⢸⢘⣀⣈⣠⣤⣤⣽⣿⣿⣿⢰⠀⠀⠘⠒⠂⠒⠀⠀⠀⠀⠒⠂⠒⠃⠀⠀⡄⣷⣶⣦⣤⣤⣄⣀⣀⣀⡇⡃
⠀⠀⠀⠀⠀⠀⢸⢸⠉⠉⠉⠙⡛⠛⠻⠿⢿⠘⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠇⣿⣿⣿⣟⠛⠛⠋⡉⠉⡁⡇
⠀⠀⠀⠀⠀⠀⠸⠈⡂⠠⠀⠘⠂⠠⠀⠀⠉⣆⠄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠠⡰⠓⠂⠀⠈⠁⠀⠀⠀⢠⠁⠇
⠀⠀⠀⠀⠀⠀⠀⢇⢫⠀⠀⢐⣀⣀⠁⠀⠤⢚⣎⢆⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠐⡱⠃⠀⠀⢀⡰⠦⣄⠀⠀⡙⡸
⠀⠀⠀⠀⠀⠀⠀⠘⡌⣦⠈⠉⠉⠀⠀⣠⢴⣿⣿⣷⢄⡢⢀⠀⠀⠁⠈⠀⠀⡀⢔⣨⣾⣿⣦⡀⠀⠀⠀⠀⠀⠉⣸⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⢦⠀⠀⠀⢀⣠⢟⡿⠟⠁⠐⠉⠑⡦⣬⣭⣭⡥⠖⠊⠙⡌⠻⡿⣿⡳⣄⠈⠁⠠⠀⡴⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠢⡀⠁⣼⠗⠋⢀⠀⠀⡀⠀⠀⠀⣿⣿⣿⠇⠀⠀⡀⠀⠀⠙⠀⠙⠾⣦⡀⢀⡖⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣘⢮⡁⠀⢀⠈⠀⠀⢰⠇⠀⡰⠋⣯⣿⢀⢠⠁⠁⠸⡄⠀⠀⠀⠂⢀⠱⢣⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢍⠣⢤⡀⠀⠠⠋⠀⠀⠀⠀⢿⡏⠀⠉⠀⠀⠈⢇⠀⢀⡤⠔⡡⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠌⣒⠰⠤⣠⣀⠀⠁⢸⡇⠀⢀⡁⣄⠠⠔⢛⠥⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠭⠥⠬⠥⠭⠤⠄⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⣿⣷⣸⣿⣆⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡻⣿⣷⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⠿⠿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⡅⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠷⠀⠀⠀⠀⠀⢠⡀⠀⢠⣤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠈⠻⠆⠈⠋⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⢀⣷⣄⠀⣀⠈⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⣤⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⡀⢸⣿⣿⣿⣿⠂⠀⠈⠀⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠳⠀⠉⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣧⠀⠹⣿⠟⣁⣤⣴⣷⠀⠀⠉⢡⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣤⣦⡀⠀⠀⠀⠀⠀⢰⣤⣀⡀⠈⠠⡀⢃⣾⣿⣿⣿⠟⠀⠀⠀⡆⠀⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⢹⠃⠀⢉⣉⣡⡤⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⢀⣴⠁⠀⠀⠀⠀⠀⠀⡀⠙⠻⠿⠿⣌⣼⣿⣿⣿⢟⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢰⠀⣼⣧⣀⠀⠀⠀⠋⠉⠛⠛⢩⡟⡀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢰⠋⠀⠀⠀⠲⠐⠻⣿⣿⣿⣦⡀⠀⡀⠀⣦⣿⡼⠁⢠⡈⠁⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⢠⣶⠀⠰⡇⠀⠈⠛⠿⠛⠧⡄⢻⣶⣿⣭⣴⣶⣿⣿⡆⢠⣿⡟
⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⢠⣦⠀⣿⡟⠀⠀⢧⡀⠀⠀⠀⠀⠀⣇⣾⣿⣿⣿⡿⠋⠈⣿⣧⣿⡿⠁
⠀⡀⠀⠀⣀⠀⠈⠙⠶⣦⠘⢋⠀⠀⣾⡇⣀⡀⢻⣿⣶⣦⣄⠀⠋⠘⣍⠉⢹⡇⠀⡀⠸⣿⠟⠁
⠀⠻⡄⠐⠃⠀⣤⡀⠀⣠⡕⠃⣾⣧⠋⢸⠛⣿⠈⠻⣿⣿⣿⣆⣠⣶⣽⣧⣸⣷⠟⠁⠀⡇⠀⣀⣤⣶⡶⠶⠦
⠀⢙⡂⠀⠀⣠⠈⠛⠂⠙⠟⣠⣿⣿⡇⠀⣠⣿⡀⠀⡈⠉⠛⢿⣿⣿⠏⠸⣿⡏⠀⠀⠀⡶⠛⠛⠉⠁
⠀⠀⠉⠀⠐⡡⠀⠀⠰⣶⣬⠃⢻⡟⢀⠀⠙⠸⣿⣰⡗⠀⠀⢸⡿⡛⡄⢰⢿⡇⢀⣠⠎⠀⢀⣤⠆⢀
⠀⠀⢰⡄⠈⣰⠃⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⣾⠟⣿⣿⣦⡀⠘⠀⠙⢾⣾⣤⣿⣿⠏⠀⣰⣿⡷⠛⠉
⠀⠳⣄⡓⠀⠁⠀⠀⠀⠀⢰⣶⡄⣰⠏⢀⠆⠉⠀⢻⣿⣿⣷⠀⣠⣾⣿⣿⣿⡟⠉⢀⣾⡿⠋
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⢁⡥⠀⠀⠀⠀⠀⠀⠙⠻⢏⢀⣿⣿⣿⠿⠋⠀⠰⠟⠋⣤
⠀⠈⠛⠒⠘⢂⠀⠀⠀⠉⠙⠳⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠉⠉⠀⠀⠀⢢⠄⣠⣾⣷⡶⠶⠖⠂
⠀⠀⠓⠦⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣀⠀⠀⣀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡄⢸⣷⡄⢸⣿⣿⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣀⠹⣿⣿⣷⣸⣿⣿⣾⣿⡟⠀⣠⣿⠀⠀⣠⣾⡟⠉
⠀⠀⠀⠀⢤⣀⣀⣀⣀⣀⠀⠈⢻⣿⣷⣿⣿⣿⢿⣿⣿⣿⣿⣷⣿⣿⣏⣤⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣷⣦⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⡻⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣿⣵⣿⣛⣛⣻⣿⣿⣿⠿⣿⣿⣞⣿⣷⣤⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠶⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣶⣿⣶⣿⣿⣹⣿⣟⣋⡁⣀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠃
⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⢿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣤⡈⠉⠉⠉
⠀⠀⠀⠐⠚⠿⠿⠛⠉⠀⣴⣿⣿⡿⣿⣮⣻⢿⣿⣿⣿⣿⡿⣿⣵⣿⣿⡿⣿⡿⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡍⠻⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠋⣿⣿⡟⢹⣿⣿⠻⣿⣇⠀⠙⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⢹⡏⠀⢸⣿⣿⡀⠈⠙⠳⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠃⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⣼⡟⢸⡇⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠐⣿⣦⠀⠀⢰⡏⠀⢸⡇⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⣀⣠⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡻⣦⡀⠀⠀⠀⣿⠟⣷⡀⣿⠀⠁⢸⣇⠀⠀⣼⠟⣿⠀⠀⣀⣤⠶⠿⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠌⠻⣤⡀⠀⢹⣇⣬⣷⣿⡶⠾⠶⠿⣦⣾⣧⣠⡟⣠⠞⢋⠔⢡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠈⢷⡈⠢⢈⣻⡷⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣄⠂⣰⠏⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣦⣤⣈⣻⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡟⠀⣀⣀⣤⣶⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣄⢩⣿⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⣿⡋⢉⣿⠟⠁
⠀⠀⠀⠠⠤⢶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣋⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣍⠛⠛⠛⠾⠛⣿⠇⠀⠀⠀⠤⠐⠚⠿⠒⢄⠀⠀⠀⢀⠀⠀⡀⠒⠿⠒⠂⠤⠀⠀⠀⢹⣯⠠⠤⠀⠈⢉⣻⣿⣶⣶⣤⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣬⣁⣐⢐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠨⣿⣤⠶⠟⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣟⣫⣁⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠍⢛⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⡛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠯⢈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⡁⠄⢈⣹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⢦⣤⣉⠢⠹⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠷⠶⠶⠶⠾⠛⢡⡾⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠿⣧⠀⠀⠀⠉⠻⢦⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣥⡶⠛⠉⣿⠶⣶⣤⣄⣀⣀⣀⣀⣀⣤⣴⡾⢿⣟⠳⣤⣺⣧⠀⠀⠀⠀⠀⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠉⠀⠀⠀⣿⠀⠀⢸⡏⣿⠛⢻⡏⠈⢻⣄⠰⠀⣿⠀⠈⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⣸⠇⠘⣠⡟⠀⢸⣧⣿⠁⠀⠀⢹⡆⠃⣼⠀⠀⠀⠀⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣤⡾⠋⠀⠀⠈⣿⡟⠀⠀⠀⠀⢿⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣇⣾⡹⣿⡟⣼⣿⡜⣿⢟⣽⣧⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⡻⢿⠸⠿⠛⠊⠈⠋⠋⠛⠈⠛⠻⠿⢽⣟⣭⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⣿⠟⠁⠀⠀⠶⠀⠀⠘⠁⠀⠀⠋⠀⢀⣤⠙⠿⣿⣾⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣯⣭⡇⠀⠀⠛⢀⣤⣦⢶⣿⠿⠟⠺⠿⢶⣶⢠⣀⠀⢠⡄⠩⣷⣾⡿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⠏⠀⠃⢀⡰⣿⠟⢋⡠⠄⠒⠂⠀⠐⠂⠤⡙⠻⣷⢄⠀⠀⠘⠿⣽⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢯⣭⡧⢠⠄⢠⣿⡿⠀⠀⠉⣠⣴⣾⣿⡿⢷⣶⣄⠈⠳⠀⠻⣷⡌⠛⠀⢶⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣽⠃⠀⢀⣝⠿⢁⠆⢠⣾⣿⣯⣿⡿⠷⢾⣿⣿⣷⡄⠀⡄⠘⣫⡄⢠⡌⢟⣵⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣯⣿⠖⠂⢸⣿⠇⡎⢠⣿⠛⣿⡟⠁⠀⠀⠀⠙⣿⣿⣿⡄⠘⠀⢹⣿⠈⠁⢰⣯⣝⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣭⡟⠀⠀⣶⣭⠀⠀⣾⣿⣿⡿⠀⣠⣶⣶⣦⠀⢸⣿⠿⣷⠀⠁⠈⣵⡆⠴⠘⡿⣫⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣭⣶⠶⢦⣿⣿⢠⢠⣿⣻⣿⡇⢠⣿⣿⢿⣿⣇⠈⣿⣾⣿⡆⢠⠀⠻⠇⠀⠀⣼⣛⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣝⠁⢠⣿⡇⠀⢸⣿⣿⣿⠇⢸⣿⠇⡄⣿⣿⠀⣿⣟⣿⣇⠀⠀⢸⣿⠰⠆⣿⠿⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣾⡖⠺⣿⡇⠁⢸⣏⣹⣿⠀⣸⣿⠀⡇⢿⣿⠀⣿⣿⣿⣿⠀⠀⢘⣟⠀⠀⣐⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠁⣀⣾⡇⡀⢸⣿⣿⣿⠀⢹⣿⠀⡇⢸⣿⠀⣿⣏⣽⣿⠀⡀⣼⣟⠰⠂⡿⠿⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣻⠇⠉⢙⡏⠀⠘⣿⣾⣿⠀⢸⣿⡀⡇⢸⣿⠀⣿⣿⣿⡿⠀⠀⢹⡟⢀⠠⣜⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠈⠁⠀⠀⠈⠛⠋⠀⠀⠙⠁⠃⠘⠁⠀⠘⠛⠛⠁⠈⠀⠙⠃⠈⠀⠿⢿⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢀⣨⣤⣄⠀⠀⢀⣤⣤⣤⡙⡇⢦⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧⠤⣾⣿⣿⣿⡄⠁⠈⢻⠐⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠇⠀⢹⣿⣿⣿⡄⢨⡆⡈⠀⠙⣿⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠏⠀⠀⠀⠀⠈⠉⠁⠀⣼⠘⠁⢀⡰⢨⠇⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠸⢀⠀⠀⠀⠀⠀⠀⡄⡏⡞⠀⣾⠓⠈⠛
⠀⠀⠀⠀⠀⠀⠐⢯⠁⠂⡆⠈⠆⠀⠀⠀⠄⢀⣇⠃⡇⡇⢇⡅⢿⠀⠿
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠆⡌⠀⠐⠀⠀⠀⠘⣜⠄⢧⡈⠋⠃⣌⢠⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⣰⠺⢃⢢⠰⢰⡆⣔⠀⠃⠈⠉⠉⠻⣇⢈⠳⠄⠃⠁⣄⣼⡇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠃⡂⠀⡆⠂⢀⢸⠃⠀⠀⠀⠀⠀⠀⠘⣯⠋⠓⣦⠄⣬⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠀⠁⠀⠃⠠⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠰⠆⠐⢫⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠈⠀⠆⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⡎⢠⢰⡄
⠀⠀⠀⠐⠺⢈⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡄⣾⠶⡼⡀⡂
⠀⠀⠀⠈⠀⠈⠀⠃⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣏⡁⠹⣇⢰⡃⠇
⠀⠀⠀⠀⢀⣀⢀⢰⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡝⠹⠱⠀⠇⠈⣳
⠀⠀⠘⢾⠈⣹⠂⡔⠞⡅⠀⠀⠀⠀⠀⠀⠀⡔⣐⡇⠜⢠⣼⠄⢱⠶⡃

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠠⡀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⠀⢀⣾⡗⡀⠀⠀⠀⠀⠀⠊⣼⢠⣿⣦⠀⠀⠀⠀⠀⢸⠀⠀⢿⣿⣷⡀⠀⠀⠀⠀⢸⣿⣏⡀⢐⠀⠀⠀⠀⠀⢰⣿⢹⣿⣷⡀
⠀⠀⠀⢠⣿⣿⠁⡇⠀⠀⠀⠀⢀⡇⢣⢸⡿⠇⠁⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠼⣿⡗⠈⡄⠀⠀⠀⠀⠀⢸⣷⠛⣿⣿⠇
⠀⠀⢀⠘⣿⣿⠀⡕⠀⠀⠀⠀⢸⣿⣮⢸⡗⢀⣦⡄⠀⠀⠀⢸⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⡇⠈⢿⡴⠑⣩⠀⠀⠀⠀⠼⣿⠆⣿⣿
⠀⠀⢸⣷⣾⣯⠔⠁⠀⠀⠀⠀⢸⣷⣯⣻⡰⡻⣿⡇⠀⠀⠀⢸⣦⡄⢸⣿⣿⠯⡄⠀⠀⠀⡇⠀⢸⡥⠊⣠⠀⠀⠀⢀⠀⠈⢻⣿⠟⡆
⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠰⠠⡽⣿⣿⣾⣟⡯⠀⠀⠀⠀⣸⣿⣿⣿⠟⠁⣸⠇⠀⠀⢰⣾⣦⣸⡔⣡⣴⠀⠀⠀⣼⡀⠀⢸⣷⢞⡇
⠀⢸⣿⣿⣿⡃⠀⠀⡨⠀⠀⣴⣄⠀⢸⣿⡿⠋⠁⠐⠀⠀⠀⣿⣿⣿⣟⠴⢊⡼⠀⠀⠀⣸⣿⣿⣿⣾⢋⡽⠀⠀⢠⡻⣾⣦⣸⣿⣿⡇
⠀⢈⣿⣿⣿⠁⢀⡀⠠⠀⠀⣦⡙⠷⣼⠁⠀⠀⢀⠀⠀⠀⠀⡏⠛⢿⡧⣲⠞⢠⠀⠀⠀⢿⣿⣿⣿⡵⣉⡇⠀⠀⢨⡛⣯⡻⢿⣿⣿⡇
⠀⠘⣿⣿⣿⠄⠁⠀⠁⠀⠀⢤⣝⡛⣞⠁⠀⣠⠄⠀⠀⠀⠀⠄⠄⠘⡏⠁⠀⡰⠀⠀⠐⡉⠻⣿⣿⣾⡿⢣⠀⠀⠀⠻⣧⣻⣻⢷⣿⡇
⠀⠀⠘⢿⣿⠅⣠⠎⠀⠀⠀⠈⢻⣳⡏⠀⠀⠀⠀⠀⠀⠀⣰⣿⣏⣹⠀⣀⡴⠃⠀⠀⠀⡇⠀⢸⣿⢋⢀⠐⠀⠀⡀⠀⠈⢇⣵⣿⣿
⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠁⢁⠀⠁⠀⠀⠀⠘⢆⣾⠃⡠⠂⠀⠀⠀⠐⠀⠀⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡗⠂⠁⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢀⣃⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣀⣠⠞⡩⠔⣚⣋⣤⣐⣀⡩⣭⠖⢒⣶⡦⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠂⡐⠲⢦⣙⣉⣩⣽⣷⣦⡀⠀⡁
⠀⠀⠀⠒⠀⠈⢿⣷⣾⣟⠯⢭⢄⣓⣩⠽⣿⣵⡾⣟⣉⣀⣴⣿⣿⣿⣷⣶⣤⣄⡀⠘⡀⡰⠠⡒⣏⠟⠛⠿⠿⢿⣿⣷⣄⡀⡐⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠥⠅⠀⠐⢄⠘⢔⠄⠙⢽⣿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠐⠃⠓⠛⠓⠤⣠⠤⢴⣿⣿⣿⣿⣿⠟⠉⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠓⠶⠤⠑⠆⠺⠒⠀⠙⢛⣀⣀⣀⣀⣠⣤⣤⠴⠒⠢⠴⣶⣶⣶⣶⣿⣷⣦⣄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡢⠦⠶⠦⣤⠄⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⡿⠧⠐⠒⠂⠀⠐⠂
⠀⠀⠐⠀⠀⠀⠢⠀⠤⢠⣾⣇⣤⣥⣿⣓⣣⣤⣧⡴⠀⠘⢿⣿⣿⡿⣿⣿⡿⠛⠛⠛⠒⠚⠛⠋⠉⣡⢋⡿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⡙⢿⠿⠿⠾⠛⠛⠛⠉⠁⠀⠀⠀⠉⠛⠻⠛⠯⠄⣀⡀⣀⠀⢀⠠⣄⠘⣱⣟⠠⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⡏⢱⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⠀⠀⠀⣸⠘⡆⠀⢰⠁⠈⡆⠀⡠⢖⢡⠇⠀⠀⠀⢀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⢕⠤⡀⠀⡇⢢⠑⢄⡜⠀⠀⢣⡞⠰⢡⠃⠀⢀⠴⡺⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⢌⠑⠼⡤⢓⡩⠧⠴⠦⠼⢝⡓⢼⠤⠊⡡⢊⠜
⠀⠀⠀⠀⠀⠀⠶⢶⣒⠦⣀⠀⠑⡄⠑⡤⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⢤⠊⢠⣊⠤⠤⣄⣀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠢⠍⣩⢋⠎⢀⣠⣤⣤⡀⠀⠀⢀⣤⣤⣄⡀⠱⡙⡤⠂⡡⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⠶⠯⡎⠀⢀⣴⣶⣦⡙⡆⢰⢋⣴⣶⣦⡀⠀⢱⠽⠮⣀⣀⡀
⠀⠀⠀⠰⠶⠶⢎⣉⡁⠀⠀⠀⠰⡇⠀⠀⠙⠛⠃⠀⠇⠸⠀⠘⠛⠋⠀⠀⢸⠆⠀⠀⠀⢈⣉⡹⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡲⣖⢇⠀⠀⠀⠀⠀⠨⣄⣠⠇⠀⠀⠀⠀⠀⡸⣲⠶⣍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢊⠤⠚⣜⢆⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⡰⣡⣋⣐⠢⡱⡄
⠀⠀⠀⠀⠀⠀⠾⠛⠉⠙⠒⠒⡩⠃⡠⠓⢄⡀⠀⠀⠈⠁⠀⠀⢀⡠⠚⢄⠘⢄⠀⠉⠲⠬⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡡⢊⡠⠒⡗⢬⣕⡖⠲⠖⢲⣊⢥⠚⡖⢄⡑⢌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡨⠒⠁⠀⢠⢃⠆⡼⢣⠀⠀⡜⠑⢄⠡⢸⠀⠈⠒⢕⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠁⠀⠀⠀⢰⢃⠕⠊⠀⠸⡀⢀⠇⠀⠸⡄⡏⠀⠀⠀⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢇⡸⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⡄⠀⢿⠀⠀⠀⣾⣿⣿⡿⠁⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡘⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄⠀⢸⣿⣿⡟⠀⠀⠀⢠⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠙⢿⣿⡆⢸⣿⣿⡇⠀⠀⣴⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⢻⣿⠘⣿⣿⠃⢀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠹⡇⣿⣿⠀⣾⣿⡿⠁⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⢸⡟⣸⣿⡟⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⢿⣿⡄⢣⣿⡟⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⣿⡌⣿⢁⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠈⢷⠀⣿⣿⣿⣿⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀⢻⣿⣿⣿⡇⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⡄⢻⣿⣿⡇⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣒⠶⠒⠒⠀⠀⠀⠀⠀⠀⠀⠒⠶⢦⣤⣤⣀⠹⣿⡇⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢧⡘⠧⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡆⠀⠀⠠⠀⠀⠀⡄⠀⢀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⣔⣧⡀⠔⠀⠀⠁⠀⠀⠀⠀⠂⠀⡎⢧⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠔⠀⠀⠰⣟⣹⡯⠁⠠⢶⣀⡈⢽⣏⡿⠂⠀⠀⠀⠀⠸⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠛⢴⠀⠄⣄⠢⠉⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠞⠀⠳⡞⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠁⠀⠀⢢⠀⠀⠀⠀⠀⡀⠀⠦⠷⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠜⠒⠒⠀⠀⠐⠒⠓⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⠁⣾⣿⡟⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠻⣿⣿⡄⠙⢿⠃⢘⡿⠃⢸⣿⠟⣉⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢸⣧⡴⢧⣤⡞⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⡄⢉⢷⣸⡿⣴⠾⣥⣶⠞⡉⢠⣿⣿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⡟⠋⠀⠀⠀⠀⠉⢿⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣈⣉⠉⢹⣦⡟⠉⠀⠀⠀⠀⠉⢻⣞⡛⠉⣠⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠠⢤⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⠿⠳⢦⡟⠀⠀⠀⠀⠀⠀⠀⠀⢽⡛⠚⠿⠛⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣄⣈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠳⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣤⣶⣤⣬⣟⠀⠀⠀⠀⠀⠀⠀⢀⣼⠳⢦⣶⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡉⣉⣧⣄⠀⠀⠀⠀⣠⣼⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠋⣀⣬⡽⣦⣀⠀⠀⠀⠀⣀⣼⠻⣅⣀⣉⡉⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⢀⡼⠛⢳⠖⢻⡇⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣿⣿⠃⣈⡴⠿⢛⡷⠞⣾⡏⢷⣁⠘⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠏⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣉⣴⣿⣇⢠⣾⡅⢠⣿⣄⠘⣿⣿⣧⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣼⣿⡿⢀⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠀⠀⠀⣰⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡄⠀⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣦⡀⠀⢿⡄⠀⠆⠀⠀⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠊⢁⡀⠘⠿⠀⠀⠀⠘⠀⠸⠟⠀⠐⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠢⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⡄⠀⠴⠶⢦⣄⠀⠀⠀⢀⣤⠴⠦⠄⠀⣤⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⠀⠀⡈⠀⠀⢺⡉⠟⡉⠁⠀⠀⠉⠉⠏⣹⠗⠀⠈⠀⠠⠄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠒⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⢃⠀⠀⠀⠀⢀⣀⣈⣉⣀⠀⠀⠀⠀⠀⠛⠻⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡀⠈⠀⠀⠀⠀⠈⠛⠚⠚⠋⠀⠀⠀⠄⠀⠐⠄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⡀⠀⠠⠂⢀⣶⠀⠀⠀⠀⠀⢰⣆⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⣾⠃⠀⡀⠀⠀⠆⠀⢻⡆⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣸⠃⠀⢠⡏⠀⠀⠀⠀⠀⢳⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠁⠀⠀⠰⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣿⣛⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣴⠀⠀⠀⠀⠈⠀⠀⡆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⢸⣆⣀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣀⠀⠘⢦⣄⣄⠻⣿⣷⡌⣿⡄⡄⠀⢀⡄⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⣀⡙⢿⣷⣼⣿⣿⡿⠷⠥⠧⡊⠄⣠⠞⢁⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣽⣷⣿⣿⡿⣿⣤⢀⣀⣠⡀⠘⢯⣼⠟⠁⠀⣀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⣭⣻⣿⣿⣷⣿⣿⣿⣷⣾⣿⡇⢠⣶⣶⣦⡀⠀⠡⠖⣻⠭⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⣤⣽⣿⣿⣿⣿⢟⡇⠈⠛⢿⣾⡇⠀⠀⢸⠳⠒⠉⠉⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠓⠛⠛⠉⠉⢻⣿⣿⣛⢁⣾⣇⣀⠲⠤⠀⠀⠀⠀⢠⢴⣾⠷⠄⠤⠤⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⠤⠤⠖⠛⣿⣿⣿⣿⢿⣾⡿⠅⠀⠐⠖⠀⠀⠀⣩⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⢻⣿⣿⣿⡁⢹⡟⠛⠀⠀⠀⠀⣬⡈⠙⠛⠛⠛⠲⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠞⠛⠁⠀⢨⣿⣿⣿⣿⣷⣦⣿⣤⣤⣦⢶⣶⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⠶⠋⢰⡿⠋⠹⣿⡈⣿⣿⠸⣿⢧⠙⣷⠀⠈⠀⠃⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣠⠎⠀⠀⣸⠟⠀⠙⣻⠀⠘⣿⣆⠹⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⣺⠁⠀⠀⠀⢹⡆⠀⠘⡿⠀⠻⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠁⠀⠀⠀⠀⠈⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⣀⠤⠒⠉⠉⠉⠉⠉⠒⠦⣄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⡜⠁⠀⠀⢀⡠⠔⠒⠒⠒⠦⣄⠀⠀⠀⠈⢣⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⡎⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⢇⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡸⠀⠀⡰⠁⠀⢠⠔⠊⠉⠉⠑⠢⣄⠀⠈⢦⠀⠀⠘⡄⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠃⠀⢠⠃⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⢣⠀⠘⡆⠀⠀⢇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠸⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢸⠀⠀⠸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡌⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠒⢇⡠⠒⠛⠒⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢀⡤⠧⢄⢸⢀⡠⠤⢄⡀
⠀⠀⠀⠀⢀⣰⠁⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠙⠈⠑⢢⠀⠀⠀⠀⢀⡴⠊⠉⠉⠘⠎⠀⠀⠀⠙⠁⠀⠄⣀⠙⡄
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠃⢀⠇⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠀⢠⠃
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠊⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢆
⠀⠀⠙⢄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠈⠉⠉⢣⠀⠀⠀⠀⢀⠖⢄⡀⠀⠀⡠⠋⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⠾⠉⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠒⠒⢲⠋⠀⠀⢸⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⡏⠀⠉⠲⡤⠒⠉⢸
⠀⠀⠀⠀⠀⠀⢠⠔⠣⡄⠀⢸⠀⠀⡴⠺⠖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠳⡄⠀⠀⡇⠀⡠⠚⠢⡀
⠀⠀⠀⠀⠀⠀⠸⣄⣠⠇⠀⢸⠀⠀⢧⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⠀⡸⠀⠀⡇⠀⠳⢄⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢣⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠁⡠⠴⠧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⣀⡀⣀⣀⣀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡠⠒⠉⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡠⠊⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠑⢄⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡜⠀⠀⢀⠔⠊⠁⠀⠀⠀⠉⠒⢄⠀⠀⠈⠆⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡸⠀⠀⡰⠁⠀⣠⠤⠀⠀⠤⣀⠀⠈⢣⠀⠀⠘⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⠃⠀⢰⠁⠀⡜⠁⠀⠀⠀⠀⠈⠳⡀⠀⢇⠀⠀⢃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⠀⡇⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠹⡀⠘⡀⠀⠸⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢡⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⢠⠖⠉⠉⠑⢇⠔⠚⠁⠒⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⢀⠤⠃⢄⣸⣀⠤⠤⣀
⠀⠀⠀⢀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠉⠳⡀⠀⠀⠀⡠⠊⠁⠀⠉⠛⠀⠀⠀⠉⠀⠐⠠⠄⡇
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⢠⠃⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢀⢠⡃
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⣰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠈⠒⠤⠤⢆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣰⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢤⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠈⡦⢄⣀⣠⠜⠉⠑⡦⠀⠒⠁⠀⠀⠀⠀⠈⠲⠤⠤⢤⠒⠲⢄⣀⡠⠞⠿
⠀⠀⠀⠀⠀⠀⣀⢧⡀⠀⢸⠀⠀⣠⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀⠀⠀⡇⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⣅⢀⡕⠀⢸⠀⠘⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⢘⠄⠀⡇⠀⢇⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢆⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠋⢠⠤⠧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠰⠰⠆⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⢠⣿⡄⠀⠀⠐⢿⡄⠀⠀⣾⣇⠀⠀⣤⡄
⠀⠀⠀⢳⣥⡞⠀⣠⣘⢉⡀⠀⠲⢒⠥⠀⡐⣬⡖⡀⠀⣜⣛⣡⠀⠀⣼⡿⡇⠀⢀⣙⠋⠀⢰⡿⣋
⠀⠀⠀⢰⣿⠄⠀⣿⣿⣿⡟⠀⡈⡡⠴⠀⢳⣗⣷⡏⠀⢿⣿⠟⠀⣼⡟⠛⠃⢠⣌⣫⣤⠀⠈⠳⠞
⠀⠀⠀⠀⠙⠁⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠐⠾⠏⠀⠹⠷⠋⠀⠀⠙⠋⠁⠀⠀⠈
⠀⠀⠀⢠⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⣀⡀
⠀⠀⠀⣿⢁⠀⢰⡾⠻⡆⠀⢠⣄⢀⠀⠀⣼⡟⡆⠀⠀⠀⠁⠀⠀⣽⣳⠀⢀⣾⢟⣻⡇⠀⠀⠺
⠀⠀⢀⢸⣿⠆⠘⣳⣵⠋⠀⢨⠛⠟⡃⢀⠓⣷⣧⠀⡰⣄⢚⠁⠀⡻⢿⠆⠀⢻⣮⣽⡷⠀⣀⠳⠞
⠀⠀⠘⣯⠍⠀⠀⢺⣟⡃⠀⢻⣿⡿⠁⠈⣷⠉⠀⠸⣿⠏⠀⠀⠀⢹⣶⡇⠀⠈⣝⣩⠄⠘⠿⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠙⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⡴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⣼⣯⠳⣄⠀⠀⣀⡴⣳⡇⠀⠀⠀⠀⣠⣤⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣗⠶⣤⡀⢀⣿⣹⣤⣿⢿⣿⣻⡡⣹⣧⣀⣀⣤⣾⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡨⡟⢛⣫⠿⠓⠛⠛⠛⠛⠿⢿⣘⣿⣯⣽⣋⣸⠃
⠀⠀⠀⠀⠚⢶⡶⣄⣀⣀⣠⣿⡏⣷⣡⠝⠛⠛⠻⣶⣄⠀⠀⠀⠀⠀⠙⢻⣽⢻⣷⣤⣀⠀⣀⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠙⢪⡙⢛⢻⣿⢠⡞⠁⠀⠀⠀⠀⠈⠹⣇⠀⠀⠀⠀⠀⠀⠙⣿⣭⢟⠟⣫⣭⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠟⢩⡏⠀⣠⣴⣶⣶⢦⣄⠀⢹⣤⣶⣶⣖⣓⠶⣤⡙⣿⣷⢊⡞⠁
⠀⠀⠀⠀⠀⢀⣠⢤⣿⣯⡀⣾⠠⠏⠿⣟⣉⣿⡿⢿⡇⣾⡿⠿⢿⣿⣿⡿⣾⢷⣿⣷⣿⣇
⠀⠈⠑⠶⢿⣭⣿⣿⣿⠿⢶⣿⠀⠀⠀⠠⠶⠶⠒⣿⠃⣿⣏⠙⠒⠶⠄⠀⠀⠀⢸⣿⣿⣏⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠿⣿⡇⠀⠀⠀⠀⣠⣾⣿⣀⣙⣿⣿⢦⡀⠀⠀⠀⠀⢸⣿⢿⣷⣉⣭⠿⠛⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣳⡼⣿⡄⠀⢴⠟⠁⢋⣛⣿⣿⣿⠋⠀⠙⢦⠀⠀⢠⣿⣟⢿⡏
⠀⠀⠀⠀⠀⠀⢀⣴⡾⠎⣔⣻⡿⣿⣆⠀⡰⠒⢿⣛⣛⣛⣛⡿⠒⠒⠀⠀⣠⣾⣟⡫⠙⣇
⠀⠀⠀⠀⠀⡶⠿⠚⠉⠙⠛⢿⣿⣮⠻⢷⣤⣀⣀⣩⣿⡿⠁⠴⣦⣤⣴⣿⣯⣶⡶⠿⠿⢮⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣻⣿⣦⣬⣿⣿⣯⣥⣴⣶⣾⢿⣟⣻⡾⢏⣾⡁⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⠟⢹⣿⢹⣽⡿⣷⡽⡜⢸⣼⡿⠛⠷⣌⡻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠋⠀⠀⠸⣟⣞⡿⠁⠀⠙⣿⣸⠋⠀⠀⠀⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠁⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⡸⡇⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢆⠀⠀⠀⡇⡇⠀⠀⢠⠃⡇⠀⠀⢀⢜⠇
⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠘⡌⢢⠀⠀⡇⠸⠀⠀⡎⠠⠀⠀⣠⠋⡜⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢝⠢⡀⠀⢱⠀⠣⡀⡇⠀⠇⠸⠀⢸⠀⡔⠁⡰⠁⢀⡤⢊⠎
⠀⠀⠀⠀⠐⢦⢄⡀⠀⠈⢢⠈⠢⡀⡆⠀⠑⡇⣀⣸⣇⣀⢸⠎⠀⢠⣃⠔⠁⡠⠃⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠑⢬⡙⠢⢤⣑⡀⠈⣻⠴⠚⠉⠉⠀⠀⠉⠉⠙⠢⣜⠁⠀⡔⣁⠤⠒⢋⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⣩⠞⢁⣠⡤⣀⠀⠀⠀⠀⣀⣀⣀⠀⠙⢎⠁⠀⣠⠔⠁
⠀⠀⠀⠉⠛⠭⣉⠀⠂⠒⠚⣲⠃⠠⣾⣛⣽⣟⣷⠀⠀⣾⣿⣿⣿⢷⠀⠈⢧⠞⠐⠒⠒⢂⡠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⢒⣤⣀⡇⠀⠀⠘⢙⣿⣏⠙⠀⠀⠛⢹⣿⡏⠛⠀⠀⠘⣆⣠⠔⠊⠁
⠀⠀⠀⠐⠒⠭⢍⣉⣀⠀⢸⡇⠀⠀⠀⠀⢀⡀⣠⡀⠀⣤⣄⣀⠀⠀⠀⠀⠀⡇⠀⠉⠉⢐⣒⡤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠋⣇⠀⠀⠐⢿⡟⢦⣤⣍⣉⣠⡼⣿⣷⠀⠀⠀⢰⠓⠎⡉⠉
⠀⠀⠀⣀⣤⣖⠉⠠⠤⠤⢤⠼⣆⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠙⠃⠀⠀⢠⠏⣄⠀⣈⣑⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⢀⣨⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣃⠀⠈⠲⢄
⠀⠀⠀⠀⠀⢀⣔⡪⠄⠒⠉⡡⠃⠀⣨⡟⠲⢤⣤⣀⣀⣠⠤⠔⠚⣅⠀⠘⢌⠁⠒⠠⠕⢦⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠜⣁⠔⠊⡰⠀⢠⢚⠀⠀⢦⠀⠀⠃⡀⠘⡑⠢⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡧⠊⠀⠀⢠⠃⡰⠁⠀⠀⠸⠀⠆⠀⡀⠈⢄⢡⠀⠈⠒⢵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣎⠜⠀⠀⠀⠀⠇⠀⠘⡀⡇⠀⠀⠣⣇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⡜⠀⠀⠀⢣⡇⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡔⠀⠀⠀⠀⠀⢨⠀⢀⠀⢀⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡌⠘⠀⠀⠁⠀⠀⠀⠀⠀⠘⢀⠁⢀⠁⠀⠂⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⠀⠈⢃⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⡀⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠆⠀⠁⠀⠠⠀⢀⠀⠈⠀⠘⠀⠀⠀⠈⢠⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢡⡀⠀⠀⠀⠸⣶⣺⣶⣻⠽⣏⣿⣣⣯⠆⡄⠀⠀⠀⠈⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠗⠉⠋⠀⣸⡁⠀⠃⠙⢣⡦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠂⠈⠁⠀⠲⣴⣦⣣⣶⣴⠈⠈⠺⣷⣦⣦⣳⡽⠦⡔⠀⠀⠈⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠘⠀⠀⠀⣩⡿⢏⢱⡦⠙⠀⠘⠩⣷⠖⡐⢌⠛⠉⠉⠃⠂⠀⠒⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⡄⠀⠀⠇⠁⠀⠄⠇⠀⠀⠀⠀⠀⠀⢻⢄⠀⠀⢢
⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠁⠄⠀⡠⠂⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⢀⡤⠂⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣄⡴⠀⠀⡀⠀⠀⠈⠁⡀⠀⠠⢄
⠀⠀⠀⢀⠀⠀⠁⢀⠀⠀⠰⠏⠉⠀⢀⣦⠀⠀⠁⡀⠀⠀⢀⡤⠒⣋⢁⣀⠠⡾⠁⠀⠀⠀⠀⠀⠁⠠⢀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠁⠋⠑⡄⠀⢀⣀⢠⡏⠀⠊⠅⣹⠕⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⣘⢛⣻⠅⠀⠀⡆⠇⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⣾⠋⠁⠭⠆⠂⠀⠙⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠈⠀⠀⠘⠀⠀⠀⠀⠐⠒⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠰⡋⣡⣄⢤⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠁⠆⠤⠋⠀⠀⠀⠀⠀⠢⣥⣷⡢⠚⠃⠀⠀⠰⠀⣡⠀⠀⣉⡁
⠢⡀⠀⠀⠀⠀⢀⢤⠀⡇⠀⠀⣇⣈⠁⠖⢁⡇⠀⠀⠦⡀⠀⠀⠀⠸⠁⠀⠀⠀⢃⠆⠃⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠱⡧⠒⣡⠞⠀⡔⠉⠑
⢄⠉⠒⡄⠀⠀⠈⠀⠱⠈⠒⠘⣄⡀⣀⠖⠉⠀⠀⠀⡇⠘⢄⠀⢀⠞⠑⠚⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⡌⠑⠈⠀⠀⡘⠀⠚⢉
⡼⠃⠀⡅⠀⠀⢠⠀⠀⡆⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠰⡐⡌⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⡀⢸⡀⠻
⣀⢀⡘⠀⠀⠀⠈⠢⣧⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⡾⠀⢀⠠⠐⢐⠂⠀⠀⠀⠀⡠⢶⡀⡠⠄⠤⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠤⢍⠁
⠀⠀⠀⣀⠀⠀⠠⠖⠋⠀⠀⠀⠀⡰⠋⣉⢑⠒⠢⡀⠀⠀⠀⣦⣴⣗⠨⠔⠁⠀⠀⠀⠀⢰⠁⠸⠙⡑⠀⠐⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠉
⢏⠀⡜⢸⠀⠀⠀⠀⠀⠀⠀⠰⠉⢠⠀⣤⠀⠂⠀⠣⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠃⢀⠊⠀⠉⠁⠉⠐⠢⡀⠀⠀⠀⡔⠁⡤⠴⡄⢹
⠈⢷⡄⡜⠀⠀⠀⠀⠀⠀⠀⠐⢄⠈⠀⡀⠀⡇⢠⠔⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⢐⠁⣭⠐⡄⢠⠃⠀⠀⠀⠣⡄⠳⠼⢠⠃
⠀⠤⠹⣄⡀⠀⠀⠀⠀⠀⢀⡀⠠⡄⠀⠀⡵⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢂⠀⠈⠣⠤⠊⠀⡁⠀⡠⡀⠀⠀⣸⠒⠐⠁
⠒⠀⠒⠊⠙⠆⠀⠀⠀⠀⠈⠫⡁⠊⠥⡈⠇⠀⠀⠀⠀⠀⠀⡠⠤⠤⠠⢀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠈⠙⠤⠤⠊⠈⠀⠐⠃⢰⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠁⠒⠁⢰⠀⠀⠀⠀⠀⡔⠋⠄⠀⠀⠂⠑⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⡼⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠠⡀⠀⢠⠊⠁⢀⣑⠒⠒⢤⠀⠀⠀⢆⠀⠀⠀⠀⢸⠆⠨⣀⠀⠀⠀⡇⠀⠀⠀⢀⠠⢄⠀⣀⡀⠀⠀⠀⣀⠤⡞⠋⠠⢄⠀⠀⠀⣠⠌⡁⣄
⠀⠀⠀⠀⠈⠓⠾⠀⠸⣩⡌⢓⠆⠀⡇⠀⠀⠀⠂⠀⠀⠀⠈⠧⣤⣤⠀⢀⡀⢳⠀⠀⠀⠁⠀⣀⠉⠀⠈⡃⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⣹⡐⠀⡿
⠙⠦⡀⠀⠀⢨⠀⠀⠰⣈⢁⡆⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡠⠁⢸⡄⢶⡆⢳⠉⡆⠀⠀⠀⡀
⠀⠀⠈⠀⠀⠀⠁⠒⡄⠀⠉⠀⡘⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠢⡀⢇⠀⣈⠳⠖⠈⢉⠤⠃⠀⠀⠀⠳⣄⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠈⠀⠀⠀⠀⢠⠂⠀⠊⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡦⠀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠓⠦⣖⠱⣒
⠄⡴⠀⢀⠔⠐⢄⠀⠀⠀⠀⠀⢠⠋⠆⢀⠜⠀⡄⠸⠉⠆⠀⣸⠀⠀⠀⠀⠀⠰⡀⡆⢠⢀⡼⢓⠒⠒⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠅
⠀⠠⡉⢐⠉⠀⠘⢆⠀⠀⠀⠀⠇⡀⠘⠘⠄⡀⠉⣄⣁⠃⠀⠈⠆⠀⠀⠀⠀⠀⠑⢷⡶⠊⠙⠀⠒⠈⠀⣀⠄⡠⠒⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⡜⠀⢡⡐⠠⠂⣀⠜⠀⠀⠀⠀⡀⡇⢀⠀⢀⡦⣀⠠⢀⠴⠂⠉⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠀⣠⠅⠀⠄⠄⠀⠄⠉⡡⠀⠀⠀⠀⢀⠜⠀⠜
⠤⠔⠋⠈⠒⠊⠁⠀⠀⠀⠀⠀⠡⣇⣬⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⠀⠀⠀⠀⠈⢀⡨⠂⠀⠀⡠⢊⡠⢊⣠
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⢠⡄⡐⡩⠉⠲⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠒⠤⡀⠠⠔⠊⠀⠀⠀⠉⠙⠛⣝⢻⣭
⠢⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠔⢩⠀⠀⣡⢴⠀⠀⢠⠊⢡⠀⠈⢄⠑⠢⣀⣡⠔⢒⡖⠀⠀⠀⠀⠀⢄⠈⡀⠑
⠉⠒⠸⠂⡠⠒⠒⠒⢄⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠠⣡⠀⠀⠃⢠⢶⡕⠃⠀⢠⠁⠀⠘⠀⠀⠀⠑⠠⠻⠃⠒⠁⠀⠀⠀⠀⠀⠀⠈⠢⠇
⠀⠀⢀⡠⠇⢀⡠⢦⠨⣄⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠐⠒⠈⢅⡀⣠⠾⡀⡉⠓⢄⡸⡸⢠⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣠⠀⢄
⠀⠀⠈⠀⠀⠃⣠⡌⠠⡄⠘⡆⠀⠀⠀⠀⠈⢣⡔⠦⠀⡀⠀⠀⠀⠀⠀⠀⣑⠧⠤⠤⢙⡏⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡌⠀⠀⠄⡀⠀⠙⡄
⠀⠀⢈⡶⠀⠸⠬⠤⠋⠁⡠⠃⠀⠀⠀⠀⠀⠰⠈⢂⣄⠁⠀⠀⠀⠀⠰⢉⡀⠀⠐⣒⠺⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⡔⠁⠀⠐⢀⠸⡀
⠀⡴⠃⠀⠑⠤⠀⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠘⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠁⢀⠁⠄⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠳⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠆⠀⠀⠀⠀⠀⠀⠇⡙⠸⢃⡄⠀⠀⠀⠘⠣⠘⠦⢄⠁⠀⡔⠊

⠀⡶⠚⢲⠖⠒⠚⠲⡲⢶⡶⠒⠒⠲⠶⠖⠒⡶⠶⠖⠒⠒⠒⠒⠒⠲⠶⣶⠀⠀⢰⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲
⠀⣿⠄⠀⠐⠀⠀⢤⠙⣟⡄⠠⠂⢠⡀⠀⠘⠛⠛⠁⠀⠀⡀⢀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⣿⠀⠀⠥⠤⠀⠀⣾⢁⡀⠀⠀⣦⠀⢢⡀⠀⠀⠐⠀⠀⢠⡉⠀⠀⢰⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⡗⠀⠀⡴⠀⠀⢈⣏⠋⠁⠀⠀⠀⠘⣻⠟⠁⢀⠤⠀⠀⠈⠀⠀⣀⡀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⣿⠀⢳⣶⢆⡀⠟⢠⠆⠀⢦⠆⠀⢚⡅⠠⠀⠀⠀⠀⠙⠤⠀⠂⠀⠘⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⡧⢐⠻⣧⠮⢀⠀⠀⠀⠀⠃⠄⠈⢿⣯⠖⠀⠛⠄⠀⠘⠀⠀⠀⢤⠚⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠛⠉
⠀⣿⠈⣳⠾⠆⢹⠀⠀⡀⠀⠀⠀⠉⠉⠉⠈⡂⠀⠀⠀⠀⠹⠛⠛⠀⠀⣿⠀⠀⢸⡇
⠀⡗⠀⢠⠀⠀⡍⣤⢀⠁⠀⠀⠀⠘⠃⢄⠈⠀⢀⡠⠗⠀⢄⠀⠀⠀⢘⣿⠀⠀⢸⡇
⠀⡟⢀⠃⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢼⡀⠖⠫⢤⡂⠀⠚⠗⠉⠀⠈⣿⠀⠀⢸⡇
⠀⣷⠈⠀⠀⣦⠚⠐⠂⠖⠀⠀⠀⠀⠀⠛⠍⡁⢣⡀⠐⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡿⢰⣰⣢⠈⠓⠒⡒⠀⠀⠀⠀⣠⠀⠠⡆⠈⢂⠃⠀⠀⠀⠙⠒⠈⠀⣿⠀⠀⢸⡇
⠀⣿⠚⠊⠙⠁⡓⠈⠁⠀⣀⠠⠄⠈⢃⠀⠃⠀⠇⠀⠀⠀⠀⠀⣰⠴⠃⣿⠀⠀⢸⡇
⠀⣿⡄⠀⠀⡃⠈⠉⠉⠁⣤⠀⠀⢠⡉⢌⣣⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣿⠀⠀⢸⡇
⠀⣯⠐⠠⠊⠉⠀⠀⣀⣿⠿⠀⢀⡈⠄⠀⢹⣇⢐⠄⠀⠀⣀⣤⠩⠀⡏⣿⠀⠀⢸⡇
⠀⣿⡅⠀⠀⠀⠀⠠⠘⡿⡃⠐⠋⠱⠄⠀⠈⠉⠋⠀⠀⠈⠍⣉⠙⠀⠈⣿⠀⠀⢸⡇
⠀⣏⣀⡀⠀⠀⠀⠀⣽⢃⡄⠀⠀⠶⢀⢲⡀⠀⠀⠀⠀⠀⢲⠅⠀⠀⢰⣿⠀⠀⢸⡇
⠀⣿⣭⣤⣤⣤⣤⣤⣥⣭⣤⣤⣤⣤⣬⣽⣿⣥⣤⣦⣤⣤⣤⣤⣤⣤⣤⡿⠀⠀⢸⣧⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⢀⢀⠈⣰⢻⣦⠀⠁⠀⠂⠀⠠⢀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣷⣄⡀⠀⠀⠐⠀⠘⠀⡏⢸⢻⡆⠀⠀⠀⠀⠂⣤⢞⢻⡁⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢹⡎⢏⢿⣀⡀⠀⠆⠀⡶⠀⢸⣾⢿⠀⠀⠠⢀⡼⢋⡞⡼⠃⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⢂⠀⠀⠄⢠⡇⠘⡆⢟⣧⣁⠀⢰⠁⠀⢸⣿⣾⡇⠀⣠⠞⠀⡼⡼⣸⠆⠀⡀⢀⠊⡀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢛⢶⡶⣄⣢⡐⠈⠄⠠⣇⠀⠘⡌⠜⡹⣦⣎⣀⣀⣼⣾⣯⣿⣶⠋⠀⣰⢷⢣⢿⠆⠄⠀⢀⣈⠴⢋⣵⣟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⢦⠙⢮⡫⣙⢾⣦⣀⣹⠀⢀⣹⠾⠟⠛⠉⠁⠈⠁⠀⠀⠈⠉⠙⠒⠿⣦⣍⣿⣀⡴⠖⠉⢀⡴⣫⠟⡁⠀⠀⢂
⠀⠀⠀⠀⠀⠐⠀⠀⠢⠀⠂⢧⡀⠑⢮⡳⡱⣽⡿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣅⠀⢀⡴⡫⣺⠟⠁⠀⠠⠂⠀⠐
⠀⠀⠀⠀⠀⢀⠀⠀⠁⠢⢀⠂⢳⡄⠀⢉⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⢿⣮⣾⠟⠠⢐⠠⠂⠁⣂⣀⣐⣀⡀
⠀⠀⢠⣤⣥⣤⣤⣀⣤⣀⣠⣄⣀⣽⣴⠏⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠤⠖⠓⠚⠛⣉⡩⢝⣿⢋⠁
⠀⠀⠀⠌⠻⣟⡓⠮⢭⣲⣭⣝⣪⣿⠁⠀⠀⠐⠚⠛⠋⢛⣛⠙⢲⣄⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠓⠶⡶⠤⠀⠀⠀⠙⣄⣀⠤⢒⠭⣱⡾⠋⠅⠀⢀
⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠉⠛⢿⠃⠀⠀⠀⠀⡠⠖⣿⣿⣿⣿⢿⣌⣷⡀⠀⠀⣰⣟⡁⣶⣤⣴⡾⣷⠦⣄⠀⠀⠀⠂⠸⣯⣭⣶⠟⠉
⠀⠀⠒⠀⠀⠀⠀⠀⠊⡛⢦⣀⡏⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⢷⢶⡿⠿⡇⠀⠀⣿⠚⠁⠁⠈⠿⡿⡿⣄⡯⠁⠈⠀⠂⠂⢿⢿⣅⣀⡀⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠀⣀⣀⣤⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⢸⠀⠀⠈⠉⠛⠣⠤⢤⣀
⠀⣀⣥⣶⣖⣛⣛⣀⣒⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡶⠶⠶⠶⣶⣤⣼⠿⠛⠃
⠀⠉⠙⠒⠂⢤⣄⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⠛⢛⡁⠂
⠀⠀⠀⠠⠀⠀⠀⣀⠈⠉⢙⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠙⠤⣄⠀⡀⠒⠀⠤
⠀⠀⠀⠀⠈⠁⠀⠀⣒⡶⢿⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡤⢄⡀⠈⠲⢦⡀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⣤⠾⣓⡮⠗⠛⠉⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⠒⠤⠤⠤⠞⠛⠓⠦⠶⠐⠗⠒⠠⠀⣂⠶⡶⣼⣿⣾⣷⣾⣿⣶⣦⣽⣷⣖⠲
⠀⠀⠀⢈⣴⣿⣒⣋⣁⣠⠤⡤⠤⢶⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⣀⡀⠂⣠⠖⡟⠠⠂⠔⢀⠴⠗⠚⠞⠛⣿⢭⣛⠭⣛⠯⣝⡻⠿⣟⠛
⠀⠀⠀⠈⠉⠌⠉⠁⢀⠄⠀⠀⢂⣼⡫⢟⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠄⠈⠃⢀⡰⠓⠉⠒⠋⣐⣼⡿⡀⠀⠘⢇⠈⠑⠢⡉⠒⠉⡑⠂
⠀⠀⠀⠀⠀⠈⠄⢀⠠⠂⠀⠀⣼⢋⢔⡵⠋⠈⠛⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣀⡤⢾⡿⣕⣝⠮⡳⢄⠈⢣⡀⢅⠠⢀⠐
⠀⠀⠀⠀⠀⠀⠐⢁⠀⢀⠀⣸⢗⡵⠋⢀⡠⠶⠫⡿⡙⣿⡷⠦⠤⣄⣀⣀⣀⣉⣀⣀⣠⣭⣶⠞⠏⠀⠸⡇⢌⠛⠳⣿⣦⣑⢄⠹⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣾⡷⣋⡤⢚⠉⠔⠀⢠⡿⡱⡸⠁⠀⡰⠟⣿⣿⡏⡏⠀⠀⡿⠻⣾⡆⡜⡄⠀⣿⢄⠠⠑⠀⠈⠙⠛⠷⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢉⠀⠀⠀⠀⠀⠠⢰⣷⢣⠁⢀⡞⠃⠀⠻⡏⡇⡇⠀⢠⠃⠃⠀⢻⣼⡔⡀⢸⠂⠠⡀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢁⠌⠀⠀⣸⣇⠏⡠⠉⠠⠀⠀⠂⢿⡅⠇⠀⡞⠀⠀⠐⠀⠈⢷⡰⡘⢇⠀⠐⠄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢹⣏⠜⠁⠀⠀⠐⠰⠀⢸⡇⡀⢸⠀⠀⠆⢀⠀⠀⠀⠙⢿⡆⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠐⠰⠀⡆⠀⠰⠀⠹⣷⠉⡀⠀⢰⠀⠀⠂⠃⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡤⠤⠤⢤⡤⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣸⠁⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⢀⡟⠀⠀⣸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢧⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣼⡇⠀⢀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⣸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⣼⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢠⣿⠃⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⣾⠀⣸⣿⣿⣇⣼⣿⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣿⡀⣿⣿⣿⣿⣿⡇⢠⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⢹⣿⣿⢃⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡀⠀⠀⠸⣿⣿⡆⠀⠀⣾⣿⣿⣿⣿⢸⣿⣿⣿⠇⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢷⡀⠀⠀⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠏⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⠀⠘⣧⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⣷⡀⠸⣧⠀⢿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠿⠷⠤⠿⠦⠼⠿⠿⠦⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠘⠀⠀⠀⠀⠀⣴⣿⣿⡄⠀⠀⣀⠀⠀⣀⣤⡴⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠦⣄⣠⢘⢦⡀⣷⠙⠋⡝⣿⠙⡱⠀⣾⡏⢽⡁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⢀⡀⡀⢻⠀⢌⠲⡝⢷⣸⣾⠏⡜⢀⠇⣸⣴⣷⡾⠋⢂⢸⡇⠀⢀⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⡶⣤⣄⣠⢧⣹⢱⠈⠀⡇⣶⣿⣿⢠⣤⣎⣶⣾⣿⣿⡷⣖⢡⠟⣿⣟⡗⣡⠖⢿⢫⠍⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣝⠻⣿⡻⣿⣼⣮⣾⣿⣿⣿⠿⠿⠿⢿⣿⡿⣿⣿⣿⣿⣥⣾⣿⣿⠏⠁⠁⢸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⢠⣿⠁⡗⢡⣳⣿⣿⠿⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣇⣀⡎⡠⢡⡿⢀⠴⣀
⠀⠀⠀⠀⠀⠀⢠⣒⣀⣀⣄⣀⣬⣦⣿⣵⣷⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣵⣿⣧⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣋⠡⢌⣻⣏⢿⣿⣿⠟⣃⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⣉⣿⣿⣿⣿⣏⣋⠩⠈⡹⣡⠞⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠂⠨⣿⣷⣿⣿⠟⣫⣾⣿⠿⢿⣿⣿⣿⣶⣤⡀⠀⠀⠀⣤⣶⡿⠿⠛⠛⠛⠿⣿⣾⣿⣿⡟⡧⠴⢂⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡃⣹⣞⣿⣿⡏⣼⣟⣿⣆⣤⡶⢾⣶⣴⣿⣽⣷⡀⢀⣾⣿⣿⣦⡴⣶⣶⣽⣦⣼⣿⣿⣿⣯⣶⣶⠿⢷⢶
⠀⠀⠀⠀⠀⢠⠄⠀⢀⣠⣿⡻⣿⣟⣿⢱⡿⠿⡿⣿⣧⣹⣿⣏⡠⣽⣿⢿⡇⠰⣿⣿⣿⣍⣻⣿⣏⣼⣿⣿⠻⡟⣿⣿⣟⡛⠿⢿⣦⣤⣀
⠀⠀⠀⠀⠀⠙⢶⣾⣿⣷⠤⡈⢺⣾⣿⣀⠁⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⣁⡀⠀⡀⠈⠁⠀⠈⠀⠉⠀⠀⠉⠀⡀⣽⣿⡯⠖⢊⡡⢔⣩⠽⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣉⣹⣷⣾⣾⣿⣿⡼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠠⠸⠿⣿⣿⣟⣛⣛⣾⠛⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣫⣿⣯⣿⣿⣿⢀⠀⠀⡘⢶⣮⢶⡄⠀⠀⣷⣼⢻⡖⢻⡶⢇⠀⠀⢴⡦⢀⠠⢀⢀⣴⣿⣿⣿⣿⣿⠬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⡡⢐⠀⢽⣿⣧⡒⠀⠘⢶⡿⠴⠀⢀⠂⣀⣀⣈⣠⣄⣀⡈⠁⠀⠈⠶⠞⠂⠆⢾⣿⣿⡟⢿⢌⠛⡳⣄⡀
⠀⠀⠀⠀⠀⢠⣀⣴⣿⡷⢟⣨⣴⣬⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠈⢙⣿⠛⠛⠛⠛⢻⡋⠀⠀⠀⡀⢠⠄⣸⣾⣿⣝⣟⢀⠢⣌⡙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⢙⡡⢽⣿⡿⠛⢛⡿⣿⣶⣄⠀⠀⠀⠀⠀⠛⢻⡷⣶⣾⠯⠀⠀⠀⠀⣰⠃⣾⣿⠿⠟⠿⢿⣿⡟⠿⠛⠛⠉⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢻⡟⠈⡈⠭⢿⣿⣿⣿⣷⣤⣤⣀⡀⠀⠀⠁⠈⠀⡀⠠⣔⢤⣠⣼⣿⣿⣧⡌⠁⠀⠼⡴⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣮⠜⣟⣿⣷⣿⣿⠿⣿⣿⣿⣷⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⡿⣻⣿⣝⣟⠦⢄⣉⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣾⠞⠛⣽⢻⣿⡿⢛⠫⢛⣿⣿⣟⣿⣿⠻⣟⣿⣿⣿⡟⠟⡇⢿⣿⢟⢿⠛⠻⠿⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠁⠜⠈⣿⣠⡇⣦⣿⢾⡿⣇⣿⠻⣮⣿⣿⣿⣿⣿⣆⠱⡺⣟⠰⠐⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡟⠙⡼⣋⠻⣾⣧⣘⡟⠁⢳⠏⠭⠓⢵⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠋⠀⠀⠀⠁⠀⠈⠛⣿⠇⠀⠀⠀⠀⠀⠀⣨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢄⡀⠀⢀⡼⠀⠀⠈⠑⢄⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢠⠈⢣⠉⢦⠞⡀⡖⢰⢸⠰⡌⢣⠔⠁⡏⢣⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⠚⠉⠉⢀⠠⢍⡙⠲⣄⢠⡏⣰⢹⢁⣞⣸⠀⡇⢸⡆⢀⣀⣼⠤⠤⠖⠁⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢌⠢⡍⢢⡙⣢⠼⢞⣻⠩⠩⠩⠩⠭⢭⢓⠺⢴⣋⡤⢄⣢⠔⣡⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡇⢨⢧⢘⡶⣋⠕⠊⠁⠀⢀⣰⠴⢢⠄⠀⠀⠈⠒⠬⡳⢯⡴⢚⡡⢃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠯⠥⠀⢳⡘⣦⢋⠞⠁⠀⠀⠀⠀⠠⡇⠰⠖⢸⡃⠀⠀⠀⠀⠈⢦⡳⣏⣠⠋⣰⠇⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣤⠤⣤⣽⢡⠋⢀⣀⠤⠴⢤⣤⣄⠘⠳⠒⠋⣠⣤⡴⠦⠤⣀⡀⢳⡙⣤⠞⠁⠀⠔⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠡⠤⠶⢶⠇⠇⠀⠁⢀⣠⠤⣀⠀⠈⡁⠀⠀⢈⠁⢀⣠⠤⣄⡀⠈⠀⢣⢹⣉⡙⠲⢔⡁
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠐⣊⣉⣙⣻⢸⠀⠀⠛⢉⣻⣿⣟⡳⠄⢳⠀⢀⠇⠰⢟⣻⣿⢏⡙⠋⠀⠘⡍⡷⠮⢕⣀⠙⠒⠒⢺⠁
⠀⠀⠀⠀⠀⠀⣎⣠⣀⡀⠒⡦⢬⣽⢸⠀⠀⠀⠀⢀⣉⠀⠀⠀⠈⡆⠸⠀⠀⠀⠀⢁⣀⠀⠀⠀⢠⠅⣟⡓⠒⢒⡀⠀⣠⠃
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⢢⣌⡑⠺⡜⡄⠀⠀⣲⣵⣮⢳⠀⠀⢀⡃⢐⡀⠀⠀⣶⣷⣾⢳⠀⠀⢸⢸⣍⣩⡭⠅⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡀⠉⣩⢷⡹⡀⠀⠛⠿⠟⠃⠀⠀⠀⢀⠁⠀⠀⠀⠛⠷⠿⠋⠀⡠⢣⣿⠦⠤⠶⠾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠅⣀⡞⢁⡔⢳⡝⣄⠀⠀⠀⠀⠈⢻⡭⠴⠬⣿⠟⠁⠀⠀⠀⠀⡰⣡⢯⡈⢧⠀⠠⠤⢽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢋⠔⣋⡽⢦⡑⢦⣀⠀⠀⠀⠈⢭⡍⠀⠀⠀⠀⣀⠴⣩⢾⡍⢧⢃⢸⡍⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠊⠞⡡⢖⠫⠽⢓⣯⣍⣒⠆⠤⠌⠤⠤⠆⣒⣫⡕⢮⡙⢦⡙⢌⠣⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣨⠤⠤⠴⡖⠋⢰⡇⢸⠉⡟⣻⠋⡏⡿⢸⠇⠻⣄⡙⠲⠍⠂⠀⠀⢈⡳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡎⠀⠀⠀⠀⡇⡎⠈⣇⠸⡄⡇⡏⢸⠙⢡⢎⠀⢄⠀⡏⠉⠓⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⠁⠈⠢⡀⠑⠀⠁⣰⠃⠀⠑⠢⠷⠧
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡶⠶⠶⠶⠶⠶⣶⣶⣶⢶⣶⣶⣶⡶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⡀⠀⠀⠈⠻⣿⡈⣿⣿⣿⠇⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠘⢷⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⡟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠹⣿⡇⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⡙⢿⣿⣿⣿⣦⠀⠀⢹⡇⠀⠀⢰⣿⣿⡟⠁⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡈⠻⣿⣿⣷⡄⠀⣷⠀⠀⢸⣿⠏⠀⠀⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠹⣿⣿⣿⣿⣿⣿⣦⡀⠙⠿⣿⣆⢹⡄⠀⣼⠏⠀⠀⣠⣿⣿⣿⣿⠟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠙⢿⣇⠀⡟⠀⠀⣴⣿⣿⡿⠋⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢻⣴⠃⠀⣼⣿⡿⠋⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣷⣄⠀⢿⡀⣸⡿⠋⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣦⠘⣧⡟⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣿⡇⠀⠀⠀⠀⣼⣿⣿⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀⠈⠉⠙⢿⡀⠀⠀⠀⣿⣿⣿⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣮⣿⣄⠀⠀⣿⣿⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒⠶⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠈⠉⠻⣦⠀⢹⡟⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣙⢷⣼⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠿⠿⠿⢿⣧⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠶⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⢣⣿⠀⠀⠀⠀⠀⢰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⢀⣾⣿⢁⣾⡏⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠘⣷⡹⢷⣶⣤⢄⣸⣿⣇⣴⣿⣇⠀⣤⣴⣶⡿⢫⡿⠁⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢷⣄⣀⣘⣛⢀⣭⣼⣿⣿⣿⣿⣿⣿⣿⣿⣬⣏⢡⣶⢟⣁⣀⣠⢟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣍⣛⠛⡣⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡕⠛⠛⣛⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡛⣹⣧⢹⢿⢿⣿⣿⣿⣛⢻⣿⣿⡿⢿⢿⣁⣷⡈⣙⡛⠁
⠀⢠⣤⣀⡀⠀⠀⠀⢀⣴⣿⡿⠇⣿⣿⣾⣶⣶⠖⣾⣿⣿⡿⣷⡖⢶⣾⣾⣾⣿⡇⠿⢿⣷⣦⡀⠀⠀⠀⢀⣀⣤⡄
⠀⠀⠈⠻⢯⣓⠶⠶⠿⣛⣩⣴⣆⣿⣿⣿⣿⣿⣾⢿⣿⣿⣷⣿⢿⣿⣿⣿⣿⣿⡇⣦⣦⣌⣛⠿⠶⠶⣚⡽⠋⠁
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠁⢹⣿⣿⣿⣾⣯⣼⡿⣷⡿⣿⣸⣿⣿⣿⣿⡿⣥⠉⠉⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡹⣿⣿⣿⣿⣿⣭⣭⣍⣹⣿⣿⣿⣿⡿⠱⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⣣⡈⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⡺⣷⡙⢿⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠴⠶⠿⣿⣯⣽⠾⠛⠁⣿⡇⣄⢩⣭⡅⢀⣭⡅⢾⡇⣿⡇⠈⠓⠦⣬⣭⣿⠿⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⣱⡟⠈⢿⣿⡌⢿⣇⠘⢿⣜⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡯⠵⠟⠉⠀⠀⠈⠻⣿⡜⣿⠀⠀⠙⠳⠯⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠸⠀⠀⢠⠇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠣⡀⠈⢧⡀⠀⡀⠀⣿⠀⠀⢠⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠙⠶⣄⠀⢻⣄⣸⡇⢠⣷⣀⣾⠇⠀⡨⠀⢰⡏⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠈⠳⣦⣤⡽⠂⠀⠁⠀⠀⠀⠀⠈⠉⠀⠺⢧⣠⠿⠁⠀⠔⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣆⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣠⠆⢀⡴⠚⠉
⠀⠀⠀⠀⠀⠉⠑⠒⠶⢶⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⡛⠉⠁⡀⠔⠊⠁
⠀⠀⠀⠠⠤⢄⣠⡤⣦⣄⠁⠀⠐⠚⠉⠉⠉⠳⠄⠀⠀⢀⣶⠛⠻⠋⢻⣷⡄⠀⣷⠶⠉⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡌⠀⠀⠀⠴⠚⠿⠻⢳⠀⠀⠀⢸⡏⡀⠻⠿⠟⢻⡄⠀⠘⠛⠿⠋
⠀⠀⠀⠀⠈⠉⠉⢁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠋⠀⠀⠀⠀⠀⠀⢀⣄⠀⠄⠀⠒⠒
⠀⠀⠀⢀⠤⠤⠴⠋⢉⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⣤⠛⠂⠀⠀⠀⠀⠀⠀⠸⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⣮⣥⡀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠉⠀⠈⠉
⠀⠀⠀⠀⠐⠊⠀⠀⢀⣴⠟⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⠰⣏⠉⠀⠒⠀⠄⢀
⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⠀⡼⠓⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢦⡀⠙⠛⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠞⠁⢠⡞⠁⣿⠒⢂⡤⣤⠰⢻⡟⣿⡁⠀⢻⡀⠈⠒⢄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡸⠁⠀⡇⠀⣿⠀⢳⠀⠘⣧⡀⠹⣄⠀⠉⢣⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⠁⠀⢻⠀⠸⡆⠀⠈⠃⠀⠘⢆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡇⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⢠⠞⣷⠀⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡟⣆⠀⠀⠀⠀⡌⠀⠹⣧⠀⠀⠀⣰⢋⢿⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣅⠘⡌⣿⣄⠀⢰⠁⠀⡇⣿⡆⢀⡜⠁⣞⣬⡆⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⢛⢶⡶⣤⣄⠀⠀⢾⠀⠘⡔⣹⣷⣆⣠⡤⠷⠿⣿⣎⠀⡼⡿⣽⡆⠀⢀⡀⠔⣫⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡘⠪⡪⡻⡶⣾⣄⡤⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢾⡧⠚⠁⣠⢞⣽⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⡀⠈⢪⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣞⣵⡿⠃⠀⠀⠀⠀⠀⠄
⠀⠀⣀⣀⣤⣀⣠⣀⣀⣀⣹⣤⠎⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠧⠴⠀⠒⢒⡻⣽⡿⠃
⠀⠀⠈⠹⢿⡓⠿⢷⣭⣟⣿⠃⠀⠀⠒⠉⣋⣙⣉⣳⣤⡀⠀⠀⠀⣠⡶⠛⠓⢚⠲⠂⠀⠀⠈⢦⣠⡔⢊⣡⠞⠉
⠀⠀⠀⠀⠀⠙⠲⣄⠀⢉⠏⠀⠀⠀⠀⠋⢿⣿⣿⢛⣾⣷⠀⠀⣼⡯⠼⢶⣾⣿⡗⣢⠀⡀⡀⠈⣷⡾⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣹⡾⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠀⠀⣯⠀⠀⠀⠙⠉⠈⠋⠀⠁⠠⠀⢹⠉⠒⠲⠤⢄⣀⡀
⠀⣀⣤⣶⣖⣛⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⡿⠖
⠈⠉⠓⠂⠠⠤⠀⣀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⠀⢰⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠲⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⡴⣫⣿⠿⠳⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢀⣀⣀⡰⠦⣄⣀⠀⠄⠀⠀⠀⠀⢀⡾⣿⣴⣤⡠⠑⢦⡀
⠀⠀⠀⣠⡾⢟⣋⣁⣀⡤⢤⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⢂⡐⠄⠀⠀⠀⠠⠀⣠⠚⠏⠙⠉⠛⠛⠛⠞⠛⠂
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣰⡫⣻⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠤⠀⠀⠀⣠⣼⡧⡀⠈⢂⡀⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠁⠀⣰⢋⡤⠊⠀⣨⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⡟⢮⣯⣪⡂⠀⠣⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⡠⠖⠉⠁⣿⠀⡿⠉⢉⡶⣶⢶⡶⠒⠚⣿⣿⡉⠂⠀⢳⠀⠀⠙⠛⠶⢥⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠟⠃⠉⠀⠀⠀⠀⣯⠢⠁⡠⠋⠀⢿⠸⠀⠀⠸⠀⠉⢳⣧⠀⢸⠀⠀⠀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢨⣏⢂⡄⠁⠀⠀⠸⣧⠀⢀⠃⠀⠀⠀⠙⣆⡈⠃⠀⠢⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⠀⠀⠀⠀⠀⢻⣄⡸⠀⠀⠀⠀⠀⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣯⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⣟⣻⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠶⠤⣀⠀⠀⠀⣀⡠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⠳⡤⠊⠁⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠚⠃⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠶⠤⡀
⠀⠀⠀⠀⠀⠀⣰⠃⠀⣀⣀⠀⠀⣠⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠧⠼⠀⠀⠈⠚⠁⠀⠀⠀⡖⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣇⠀⢀⣀⠀⠀⠿⠀⠀⠀⢀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠸⡄⠘⠚⠀⢀⠤⡄⠀⠸⣍⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠘⠒⠃⠀⠀⠀⠀⠘⠃⠀⠀⠀⣀⣀⣠⠤⠤⠤⠴⠶⠶⠤⡤⠤⠤⣄⣀⣀⣀⡤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠒⠲⠦⠤⠤⢶⠖⠒⠒⠉⠉⠁⠀⢵⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡤⣬⠤⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠁⠈⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⢤⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢏⠀⠀⠀⣸⠀⠀⠀⣠⠊⠉⠲⡄⠀⠀⠣⡀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢹⠀⠀⠀⣸⡀⠀⠀⠀⠀⠈⠢⣀⡔⠁⠀⠀⠀⠳⣀⠀⣠⠇⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢤⠴⠃⠀⠀⡞⠉⠙⡆⠀⠀⠀⠀⠀⠭⠀⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠑⢤⠔⠁⠀⠀⢠⠖⠓⠛⠚⠲⡄⠀⠀⠀⡎⠉⢣⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠙⡆⠀⠀⢠⠚⠲⡀⠀⠀⠘⢆⠀⠀⢀⡴⠁⠀⠀⠀⠑⡖⠋⠀⠀⠀⣠⠋⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠒⠁⠀⠀⢘⡦⡞⢇⠀⠀⠀⠀⠑⠶⠋⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠙⣄⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⡟⡈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⡆⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠀⡇⢇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀⡀
⠀⠀⠀⠀⠈⠣⠀⠀⣀⠄⢀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢂⣀⢀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⠀⠀⠀⠀⠀⠀⠌⣠⠀⡀⠈⡄⠀⠀⣀⠀⠀⠀⢀⡀⠀⠁⠚⠐
⠀⢁⠀⠀⠀⢀⢁⠊⠀⠀⡀⠱⠀⠐⣀⣀⠀⠀⡈⠊⠂⠒⢁⠔⠉⠀⠈⠉⠒⢌⠢⣀⡈⠜⢀⠤⠀⠄⠄⡀⠄⠠⠠⠄⡰⠀⡴⠊⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠁⠀⠀⠀⢀⠀⠄⢀⠀⢀⠀⠀⢀⡀⠤⢤⠄⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠆⠀⢀⠔⠁⠀⠀⠀⠀⠈⠄⠁⠀⠉⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠔⠉⠉⠐⡄⠒⠤⠠⠄⠂⠀⠀⠀⣁⡀⠬⠚⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠈⠀⢀⡄⠀⠠⣂⠀⠨⠀⠀⠀⠀⡇⠀⠀⠊⠀⠀⠀⠈⠂⠀⠀⡇
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠤⣀⠀⠘⢄⠐⠄⠐⠀⠃⢀⠌⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⡜⡠⠔⠐⢄⢧⠀⠀⠀⠀⠤⠤⠄⠀⠀⡔
⠀⠈⠢⣀⣤⠤⢂⠔⠒⠉⠐⠀⠀⠀⠀⠊⠀⠀⠀⠀⢃⠀⠀⠑⠒⠀⠐⠊⠁⠔⠉⠉⠆⠠⠄⣀⡉⣀⠠⠊⠰⡃⠀⠀⠠⠈⠓⢄⡀⠀⠀⠀⠀⠤⠊
⠀⠀⠀⠀⠀⢠⠃⠐⠕⠀⠐⠀⠃⠀⠀⢄⠈⣀⡈⠀⠄⠀⠀⠀⠀⠀⢀⡀⠀⠁⠄⠤⠂⠀⠀⠀⡀⠀⠀⠀⠀⢈⣐⠊⢁⣀⠀⠀⠀⠀⠀⠀⠄⠒⠐⠒⠢
⠀⠀⠀⠀⠀⠘⡄⠀⠊⠉⠉⢀⠆⠀⠀⠈⠐⠀⠒⠈⢀⠀⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⡆⢀⠔⠉⠀⠀⠀⠀⠀⠉⠲⡄⠀⢀⠀⢤⠀⠀⡄⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠑⠒⠐⠂⠡⠐⠒⠒⠄⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠑⠒⠉⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢰⠀⠡⡤⢤⠄⢀⠆
⠀⠀⠯⡷⠈⠁⠈⠑⡀⠀⡄⠒⠒⡄⡀⡀⢡⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠢⠀⠀⢀⠁⠀⠁⠒⠬⠅⠒⠁
⠀⠀⢰⠀⠀⠀⠀⠀⡃⠀⠘⠒⠊⠀⠀⠁⡸⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠌⢠⠁⠀⠀⠔⠈⠁⠉⠢
⠀⠀⠀⠲⢆⣌⡠⠗⠁⠀⠐⠤⢀⣀⠠⠔⠁⠀⠘⡀⠀⠀⢀⢠⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢀⢀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡀⠀⠈⢰⣂⠀⢦⢠
⠀⠀⠀⠀⠀⢀⠤⠒⠈⠀⠈⠁⠲⡄⠀⠔⠉⠈⠱⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢆⠀⣁⠇⠀⠁⠒⠠⠤⠄⠂⠉⠀⠀⠀⣀⣀⡀⠈⠪⠴⠮⠑⠁
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⠆⢄⠀⠐⡰⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣄⠀⠀⢀⠠⠤⠤⢤⠀⠀⡞⢷⡏⠀⢀⠐⠈⠀⠀⠀⠁⠢⡀⠐⠍⢝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⢸⠀⠈⢁⣠⠤⢄⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠄⠀⡔⠁⠀⠀⠀⠀⠱⠀⠉⠉⠁⠠⠁⠀⠀⠀⠀⠀⠀⠀⠘⡈⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⠠⠠⡀⠀⡘⠀⠀⠈⠀⠀⢀⡀⠑⠄⠀⢀⠋⠀⠀⠀⠈⠢⡀⢐⠀⠊⠀⠀⠐⠀⠀⠆⠀⠀⠀⡂⠀⠊⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠚⠓⠒⠒⠁⡰⠁⠀⠀⢺⡇⠀⠸⠗⠀⠐⠀⠇⠀⠄⠀⠠⠐⠀⡇⠘⢄⠀⠪⠝⠀⠀⡰⢀⡠⠄⡀⢡⠀⠀⠠⠤⠀⠤⠂⠀⢀⠇
⠀⠀⠀⠀⠐⠠⣀⡀⠀⠀⢀⡠⠜⠀⠀⠀⢄⠀⠀⠈⠀⢀⠔⠁⠀⠣⡀⠂⠀⠒⠁⡠⠁⠀⠀⠑⠂⠤⠐⠊⠀⢋⡀⠀⠈⠀⠑⢄⣀⠀⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠀⠁⠀⠀⠀⠀⠈⠐⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⢁⠹⣦⣀⣀⣀⣀⡀⢹⡀⠀⠀⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠙⢓⣦⡤⡖⠚⠉⠂⢻⢻⡀⠀⠀⢃⢾⣿⠖⠦⢤⣀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢀⡠⠞⠂⠛⡇⢷⡀⠌⣸⡇⠈⣇⠔⡱⡴⠋⣾⠀⣀⠀⠉⣞⣿⡀⠀⠀⠀⠀⢀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠳⣶⢯⢦⡀⠀⠰⣧⠈⢿⠀⠃⢧⠀⡏⠘⢸⡇⢠⡷⢀⢔⡿⠋⣹⠃⠈⠳⣄⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⠀⢉⣇⠉⢳⠘⢌⣳⡬⠷⠚⠛⠁⠉⠘⠛⠓⠾⢄⣆⡾⠀⣴⠃⠀⠀⠀⣩⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⣤⣤⣴⣧⣄⣀⡀⢟⠦⣄⡱⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣼⠁⢔⣫⠟⠋⣹⠟⠳⡄⠀⠀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠍⠚⣦⡉⠙⠄⣱⠋⠀⣀⣤⣤⣤⣤⡀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠻⣞⠁⠴⠚⠁⠀⠀⠙⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠠⠙⠭⠛⡶⠁⡔⠉⠀⠀⠀⠀⠙⠳⠀⠀⠀⠞⠋⠀⠀⠀⠀⠉⢢⠘⣄⣊⡽⠟⠳⠮⣿⣿⠁
⠀⠀⠀⠀⠀⠄⣀⣀⣀⣤⣤⣾⢟⠒⠒⠾⠦⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡭⠤⠴⠾⡋⠁⠸⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠠⡬⢀⠿⣲⠴⠶⡞⠀⠐⡿⣦⣄⣀⣠⣴⣆⠀⠄⠀⠀⠀⣰⣦⣤⣀⣠⣴⢿⠂⠀⡧⣂⣤⣶⣤⠀⣀⣃⣠⡴⠟⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⠠⣉⣀⣿⡀⠀⠐⠃⠏⠿⠹⠉⠈⠀⠀⠀⠀⠀⠁⠋⠏⠿⠹⠉⠀⠀⣰⡿⣤⣤⣤⣤⣽⣛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣞⣋⠉⠉⣀⣽⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⣀⠤⣑⢀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠛⢉⡁⠈⠛⠫⠁⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠾⣉⡉⠙⠻⡇⢄⡚⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⢧⠀⢀⡼⠟⢋⡸⢧⡈⠀⠀⠀⠀⠀⠉⢉⠉⠛⠉⠙⠁⠀⠀⠀⠀⠈⢠⣾⣐⢄⠹⠛⠒⠾⢿⠟⠛⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡯⢤⣴⡚⠁⢀⡽⣤⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡴⢣⣄⠉⠳⣧⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢋⠥⡄⠀⠀⣰⠟⢡⡞⡙⢲⢤⣄⣄⡀⠀⠂⠀⣠⣄⣠⡴⠚⢯⣄⢆⢙⠳⢤⣣⣵⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠀⠁⠀⠙⢦⣞⡡⠴⠯⠊⣠⠂⢸⠏⠉⡉⢻⠛⠉⡉⠩⣆⢀⢳⡀⠙⣾⠀⠀⠀⣩⠞⠉⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⡿⢂⣅⠀⢩⡇⣠⡾⠚⣰⠂⢸⣳⠀⣿⠀⢹⡾⢈⠻⣄⡯⡀⢤⠞⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⠙⠺⣾⠇⡐⠀⢸⣇⠞⠃⠈⠘⡇⣞⠀⣀⣠⠼⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⣟⠈⠉⠙⠛⣿⡇⠲⠒⠒⠛⢻⣇⠉⠀⠀⠀⠀⠈⠙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦

⣶⣶⠶⣶⣤⣶⣶⣶⣴⣦⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣂⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠦⠠⠄⣠⠄⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠒⠦⢤⣀
⠀⠀⠀⠀⡴⠋⢀⣄⠀⣠⣤⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢶⠋⡀⡱⠆⠐⢮⠁⠠⡶⠆⠀⠀⠀⣴⠟⠁⣠⡀⠀⠀⣤⡄⠑⢤
⠀⠀⠀⢸⡇⠀⠺⠿⠀⠛⠛⠀⢸⡆⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠉⠲⣄⠀⠀⠀⢸⠋⢉⣁⣀⣀⣃⣉⠁⢻⠄⠀⠀⣼⠏⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⢱⡄
⠀⠀⠀⠀⠳⣄⠀⠀⠉⠀⣀⡤⠟⠀⠀⠀⢰⠋⠀⢰⡷⠀⢀⣶⠀⠀⠈⡄⠀⠀⠸⣄⠸⡀⠀⠀⢀⡟⢀⡾⠀⠀⠀⡿⠀⠀⣀⣬⡤⠤⠤⠤⠤⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣄⡾⠁⠀⠀⠀⠀⠀⡇⠀⡀⠀⠁⠀⠀⠉⢀⡀⠀⡇⠀⠀⠀⠘⠳⠬⣲⡖⠭⠔⠉⠀⠀⠀⠀⢷⠀⠀⣇⠀⠀⠀⠀⠀⣰⠏⠀⢠⡟
⠀⠀⠀⢴⢻⡆⠑⣙⣙⣁⣀⣤⡄⠀⠀⠀⢡⠀⠳⣄⠀⠀⠀⢠⠞⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠈⠦⣄⣀⣠⡼⠋⢀⣴⡟⠁
⠀⠀⠀⢮⢀⡭⠭⠭⠭⠭⢭⣙⡇⠀⠀⠀⠀⠳⢄⣀⠈⠀⠀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠛⠉⠉⠉⠓⠲⣤⡀⠀⠉⠛⠶⠤⠤⠴⠖⠛⠉
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣤⣤⣤⣤⣤⣄⣀⣀⣈⣻⣆
⠀⠀⠀⢀⡤⠞⠋⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⢦⡀⠀⠀⠀⠀⠀⢸⠛⢿⣿⣿⣿⡿⠋⢿⣿⣿⣿⣿⢿⡆
⠀⠀⢠⡞⠀⢀⣀⡀⠀⢀⣀⣀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⢹⡀⠀⠀⠀⠀⢸⠀⢠⡌⠉⠉⠀⠀⠈⠙⡛⠛⠁⣿⠇
⠀⠀⡞⠀⠉⠀⠙⠃⠀⠉⠀⠛⠃⠈⡇⠀⠀⠀⠀⢀⡄⠀⠀⠀⡠⠋⠈⣦⠀⠀⠀⠘⣆⠀⠻⢦⡀⠀⢀⣠⡼⠃⢀⣼⠟
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠇⠀⠀⠀⡀⠈⠳⠄⠀⠀⠀⢀⡴⠛⢦⠀⠀⠀⠈⠢⠀⠀⠀⠈⠉⢁⣀⡴⠟⠁
⠀⠀⠹⣆⠀⠀⠀⠀⠤⠶⠟⠁⣠⠞⠀⠀⠀⠰⡇⠀⠀⠀⠀⢀⠞⠁⠀⢀⡼⣀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠈⠛⠶⠤⣤⣤⣤⠶⠞⠁⠀⠀⠀⢀⠄⠙⠲⠆⠀⠀⠀⠀⡠⠔⠋⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⢰⠀⠀⠀⣾⡇⠀⠀⣶⠀⠀⠘⢆
⠀⠀⠀⠀⠀⣠⠖⠛⠉⠀⠉⠉⠙⠦⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⣠⣄⡀⠀⣀⣤⣶⠶⠶⢶⣤⣞⠉⣧⠀⠀⠀⡏⠀⠀⠀⠻⠃⠀⠀⠿⠀⠀⠀⠸
⠀⠀⠀⠀⣰⠃⠀⣴⡆⠀⢰⣶⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣹⠟⠋⠀⠀⠀⠀⠀⠀⠙⢿⣇⠀⠀⠀⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⢰⡇⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠀⣀⣀⡀⠀⢀⡴⠚⠂⠀⢻⡆⠀⠀⢷⡀⠈⠳⣄⠀⠀⢀⣠⡾⠁⠀⠠
⠀⠀⠀⢸⡇⠀⣶⠲⣄⢀⡖⠲⣄⣴⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠛⠉⠙⠿⠀⠘⠏⠀⠀⠀⢸⡇⠀⠀⠀⠳⣄⡀⠈⠉⠉⠉⠁⠀⢀⡔⠁
⠀⠀⠀⠀⠻⣄⠈⠀⠈⠉⠁⠀⠈⣁⡼⠁⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣾⠁⠀⠀⠀⠀⠀⠉⠛⠒⠲⠖⠒⠚⠉
⠀⠀⠀⠀⠀⠈⠓⠲⠤⠤⠤⠶⠞⠋⠀⠀⣴⠟⠋⠁⠀⠀⠈⣉⠲⢄⡀⠀⠱⣄⠀⠈⠓⠤⠤⠤⠞⢁⣴⠟⠁⠀⠀⠀⣠⠴⠒⠒⠒⠶⣤⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣾⠋⠀⢰⣿⠀⠀⢸⣿⠀⠀⠑⣄⠀⠈⠑⠂⠤⠤⠤⠐⠀⠉⠀⠀⠀⠀⢠⡞⠁⣰⡄⠀⢠⡄⠀⠙⡄
⠀⠀⠀⡰⠊⣠⠀⢀⡀⠙⠆⠀⠀⠀⢸⡏⠀⠀⠀⠛⠀⠀⠈⢋⣀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⠁⠀⠘⠁⠀⠀⠘
⠀⠀⢠⠀⠀⠉⠀⠘⠁⠀⠀⠀⠀⠀⢸⡇⠀⢸⠋⠉⠉⠉⠉⠉⣹⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠖⢺⠉⡏⢩⡍⢧⠀⠇
⠀⠀⠀⢀⠓⠄⠀⠀⠚⡀⠀⠀⠀⠀⠀⢷⡄⠈⢧⡀⠀⠀⢀⣴⠏⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠡⠴⠴⠤⠷⠶⠒⠃⡸
⠀⠀⠀⠈⠑⠲⠶⠖⠊⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣉⣛⣛⣉⣥⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⣀⣀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⣀⠀⣀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⡀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⠚⠉⠈⠈⡀⠈⣔⡀⠀⠀⡠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠎⢠⡆⠱⣶⣻⣠⡇⠀⢿⣷⣼⠟⠁⣼⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⣀⠀⠀⣼⣇⠀⠻⣿⣷⠁⠀⠸⣿⠇⠀⢸⣿⣿⡯⠐⢁⡾⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠻⢾⣿⣆⠀⠀⣙⣀⣤⠤⣋⣀⡀⣸⠟⠉⠀⢠⣿⣷⡃⣜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⠀⠀⠉⣻⣴⠿⠉⠙⠀⠈⠉⠻⠟⢳⣤⠀⣰⣿⡽⠿⠧⠜⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠲⠼⣿⣷⣄⡾⣃⣠⣤⣤⡀⠀⠀⣤⣴⣶⡦⡬⣻⡉⠀⠀⣠⣤⣶⣛⢬⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠈⢷⣄⠀⠀⢀⡞⢀⣵⣿⣿⣽⣿⠀⢸⣿⣿⡿⣿⣿⠮⣿⡠⠾⢿⣿⣯⣬⡀⣤
⠀⠀⠀⠀⠀⠀⠁⠀⠠⠧⠭⠿⠓⢺⡧⠈⠉⠛⠛⠉⠁⠀⢸⡏⠈⠛⠋⠀⣠⢿⡇⠀⠀⠀⠈⢉⣓⣠⡄
⠀⠀⠀⠀⠀⠐⠦⠄⣤⣀⣀⣀⠀⢻⡶⣀⠀⠀⠐⠀⢴⣦⡼⠟⠃⠠⠀⠀⠀⢺⠧⣶⣶⣿⠿⡍⠋⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⡹⠟⠉⣷⣠⠀⠀⠀⠤⣴⡶⠶⡷⠖⠀⠀⠀⢀⡞⠀⠈⠙⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⣀⣠⡞⢿⣶⣀⠀⠀⠀⠛⠛⠄⠀⢀⢠⣼⠟⠘⢿⣷⣶⣤⣌⡓⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠖⠊⠉⠉⠋⣽⡟⠀⠀⠙⡿⢦⣤⣤⣀⣤⣠⠤⠞⠋⢧⣄⡀⠀⠻⣿⠣⠌⢋⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠏⠀⣀⡴⣿⡇⠀⢀⣇⠈⠀⢸⣷⣄⠀⠹⣿⣿⠷⡄⡘⢶⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠻⢾⡿⠀⢠⣾⣿⡀⠀⣼⡿⡻⡄⠀⡟⠀⠀⠠⠠⠍⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⣸⠃⣴⠟⢻⠋⠃⠀⡟⠀⠑⠙⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠾⠁⠀⢀⡅⠸⢢⡇⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣜⢿⣿⢏⣿⡞⣿⡿⣫⡆⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣚⢿⣿⣼⡿⠿⠛⠛⠛⠛⠘⠻⠿⣿⣿⢿⣻⣵⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⣿⠟⠀⠀⠠⡦⠀⠀⠂⠀⠀⠛⠀⢀⣄⠻⢿⡏⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣯⣿⡇⠁⠐⠓⣀⣤⡴⣶⡿⠿⠸⠷⢶⣶⣀⡁⠀⣤⡙⢯⣷⣶⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣞⠟⠀⠃⢠⡺⡿⠟⣁⠤⠒⠒⠀⠀⠒⠤⣙⠻⣷⣈⠀⢀⡹⢯⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣿⣷⠰⠄⣰⣿⠟⠀⠈⢁⣴⣾⣿⣿⢻⣷⣦⡈⠳⠀⠻⣷⡘⠁⠐⣶⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠁⠀⢀⣭⡋⠰⠃⣴⣿⣿⣽⠿⠛⠻⣿⣿⢿⣄⠐⡀⠩⣷⡀⣤⠙⣵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢷⣾⡏⠁⠸⣿⠁⠃⣼⣏⣹⡿⠁⠀⣀⠀⠈⢻⣿⣿⡄⢰⠀⢹⠇⠀⠀⣷⣾⡭
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢾⣁⡀⣿⣷⠀⢠⣿⣿⣿⠃⢠⣾⣿⣷⡄⠘⣟⣹⣷⠀⠀⠰⣿⠘⠃⢛⣽⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⡏⠁⡿⠇⢰⢸⣿⣿⣿⠀⣿⣿⠛⣿⣿⠀⣿⣿⣿⡆⢃⠀⣯⡄⣀⢸⣾⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣟⣄⣀⣿⡇⠌⢸⡿⣿⣿⢠⣿⡇⠀⢸⣿⠀⣿⣷⣿⡇⠸⠀⣿⡏⠋⠘⣯⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣛⠇⠀⣿⡅⠀⢻⣷⣿⡇⢸⣿⡇⢀⢸⣿⠀⣿⡿⣿⣿⢀⠀⣶⠆⡄⣸⣮⣝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣭⣦⠒⢿⠇⢁⢸⣿⣽⡇⠈⣿⡇⢸⢸⣿⠀⣿⣿⣿⡿⢸⠀⣿⠇⠀⣩⢿⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡖⠀⠘⠃⠈⠈⠻⠿⠃⠀⠙⠇⠐⠸⠏⠀⠹⠷⠟⠁⠂⠀⠿⠀⠃⢿⡷⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⢻⣿⣿⡋⠉⠉⠉⢻⣿⣿⣿⣟⠉⢻⣿⣿⣿⡟⠉⠉⢙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢸⣿⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⡀⢸⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⣿⣿⣧⠀⠀⠀⣿⣿⣿⣿⡇⠈⣿⣿⣿⡇⠀⠀⣾⡿⠟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⣿⣿⣇⠀⠀⢿⣿⣿⣿⣷⠀⢻⣿⣿⡇⠀⡰⠋⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣿⣿⣿⣿⡄⠀⠸⣿⣿⣿⣿⠀⠈⣿⣿⢃⣾⣷⣿⣿⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣿⣿⣿⣿⡀⠀⢿⣿⣿⣿⠀⠀⢹⣿⠋⠁⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠘⣿⣿⣿⠀⢀⡼⠁⠀⣀⣠⣤⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠈⣿⣿⣿⣿⡆⠀⠹⣿⣿⣀⣾⣴⣶⣿⣿⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠘⣿⣿⣿⣷⠀⠀⢻⣿⠟⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀⠀⠀⠘⣿⣿⣿⡆⠀⣰⠏⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⣿⣿⣿⣿⡄⠀⠀⠈⢿⣿⡇⣰⣏⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢿⣿⣿⣿⣆⠀⠀⠈⢻⣿⠟⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠻⣿⣿⣿⡄⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⠀⠙⢿⣿⣿⡀⢠⣟⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣧⡀⠀⠙⢿⣷⡟⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢿⣿⣄⠀⢀⣿⣁⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠤⠴⠽⠿⠦⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠯⠤⠴⠶⠶⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣤⡀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣀⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡿⠿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⣶⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⣿⣷⡆⠀⠀⠀⠀⢀⣿⣄⠀⠀⠀⠹⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣿⣿⣿⠃⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠀⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠙⣿⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠐⣿⣿⡿⠀⠀⠀⠀⢀⣀⣿⣀⡀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠄⠀⠀⠀⢺⣿⡿⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠀⠀⠀⢾⣿⡷⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⢾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢠⣿⣿⣿⣆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣽⡀⠀⠀⠀⠀⠈⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢻⡹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⢸⠘⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠛⠘⠀⠛

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢰⡦⠀⠀⠀⠀⢀⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⣤⣤⡄⠀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣠⣄⠀⣠⣀⠀⢀⣴⡄⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣤⣀⠀⣤⣤
⠀⠀⠀⠀⠀⠄⠀⠠⠿⠷⠀⠘⠛⠃⠀⠀⠀⠀⢭⠍⠀⠙⢻⡦⠈⠻⣧⠀⣿⠿⡷⠸⠿⠶⠀⣿⣶⠀⠘⠀⠄⠀⠴⠏⠀⠛⢿⡿⠀⠿⠿⠂
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⣠⣶⠀⢠⡶⠀⠀⣠⣤⠀⢀⡀⠀⠀⣠⣄⠀⢀⣤⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠚⠁⠀⠻⠗⠀⠰⠆⠀⠐⠒⠓⠀⠶⠒⠀⠈⠒⠃⠀⠛⠃⠀⠘⠟⠀⠘⠛⠛⠁⠻⠛⠀⠀⠉⠛⠀⠈⠁⠀⠈⠉⠛⠀⠻⠗
⠀⠀⠀⠀⢠⣤⡀⠀⢀⣄⠀⢄⡀⠀⠺⠆⡀⢠⣴⣶⡄⢰⣤⣖⢠⣤⣀⠀⠀⢠⠀⢰⣆⠀⠀⣀⢀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠑⠆⠀⢀
⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠀⠈⠁⠀⠀⠈⠛⠀⠙⠉⠀⠈⠛⠋⠈⠉⠛⠀⠀⠉⠃⠀⠙⠛⠀⠙⠛⠁⠈⠛⠃⠀⠘⠁⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⣤⠀⢠⣶⣦⠀⣴⣶⡄⢰⣿⣷⠀⢿⣦⠀⣄⢀⣄⠀⣴⣤⠀⣀⣽⡇⢀⣼⡄⠐⢶⣄⡀⠀⢀⡴⠀⣈⣁⠀⢀⣤⠀⢰⡟⣻
⠀⠀⠀⠀⣀⣉⡁⠀⠀⠀⠀⠉⣉⠀⠈⣉⡁⠀⠀⠉⠀⠉⠈⠉⠀⠙⠋⠀⠉⠉⠁⠀⠉⠁⠀⠀⡈⠁⠀⣈⠁⠈⠛⠛⠁⠈⠉⠁⠈⣉⡉
⠀⠀⠀⠀⣿⣿⡇⢀⣤⠴⠀⢀⣄⠀⠀⠉⠁⢠⣄⣠⡄⠙⠿⠇⠀⢸⡇⠀⠻⣷⣄⠀⠀⠀⠀⠀⡗⠀⠀⣿⠆⠀⡀⠀⠀⣠⣿⣆⠀⣿⣟
⠀⠀⠀⠀⠠⠀⠀⠀⠁⣀⠀⣤⣭⠀⠀⠈⠀⠀⠉⡉⠀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠘⠗⠀⠘⢿⡿⠀⠸⠍⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠋


⠀⠀⠀⠀⠀⠠⠆⠀⣀⡀⠀⠶⠶⠀⠀⠤⠄⠀⣠⣄⠀⢀⣀⡀⠀⣠⣄⠀⢠⠀⠀⠀⠴⠇⠀⢠⣤⠀⠀⠀⠀⠀⢠⣦⠀⢠⠠⡄⠀⠴⠄
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠓⠚⠂⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⠉⠁⠀⠁⠀⠀⠐⠀⠛⠀⠘⠉⠀⠀⠀⠀⠀⠘⠋⠀⠈⠈⠃⠀⠙⠃
⠀⠀⠀⠀⠐⣶⠀⢴⣀⣶⠀⡘⠂⠀⠀⠓⠀⠐⡇⢰⠄⡴⠿⣦⢰⠟⢷⡄⢸⣿⠆⠐⠿⠗⠀⣴⢳⡄⠀⠿⠂⠠⠾⠳⠄⢾⢿⡗
⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠛⠛⠁⠀⠃⠀⠀⠁⠘⠀⠓⠚⠛⠙⠓⠺⠃⠀⠛⠀⠐⠒⠖⠀⠛⠙⠃⠐⠖⠓⠀⠚⠛⠀⠻⠿⠟⠀⠀⡀
⠀⠀⠀⠀⠘⠛⠀⠀⠰⠀⠀⣿⣿⡄⢸⣿⡇⠰⡶⠶⠆⢠⣾⠋⢰⣦⣤⡄⢰⣶⠀⠰⣿⡟⠀⢀⣀⠀⢻⣷⣿⠀⣀⣼⡇⠺⢣⣄⢸⣿⣿⠆
⠀⠀⠀⠀⣘⣉⡁⠀⣀⣀⠀⢁⡉⠁⠀⠉⠀⠀⢀⡀⠀⠈⠉⠀⠀⠀⡉⠀⠀⠀⡀⢀⡉⠀⠀⣀⣀⡀⠀⠉⠁⠀⠉⠉⠀⠀⠈⠉⠀⣉⣉
⠀⠀⠀⠰⣿⣿⡗⠰⣿⣿⠂⠈⠉⠀⣴⣿⣦⠀⣸⣇⠀⣿⢿⣷⢠⣦⣴⡄⣠⠞⠁⢴⣄⣴⠄⠉⠉⠁⠲⣾⠄⢠⣶⡆⠀⢀⣴⠆⢸⣿⣿⡆
⠀⠀⠀⠀⣬⣭⡄⠀⣩⡁⢀⣤⣤⡄⢠⣀⣀⠀⠉⠉⠀⣁⣠⡀⢨⣭⣭⠀⣅⠀⠀⠈⢉⡉⠀⣀⣤⡄⠀⠀⠀⢀⡍⢁⠀⠀⠀⣠⠀⠉⠁
⠀⠀⠀⠀⠻⠿⠇⠰⣿⡇⠘⠿⠿⠇⢺⣿⡿⠲⣿⣿⠖⠿⠾⠟⢸⣿⣷⠄⠙⢦⣄⠐⠿⣷⠘⣿⣿⠇⢠⣤⣤⠘⠷⣾⠀⠠⠞⠁
⠀⠀⠀⠀⣤⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠈⠓⠀⠈⠻⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠈⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣶⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠂⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⣄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉




⠀⠀⠀⠀⣆⠀⡆⣀⠀⢠⠀⡄⠀⡆⢠⠀⢠⠀⠀⠀⠀⡄⠀⡄⢰⡄⣰⠀⣄⠀⣤⢀⠀⠀⣄⠀⠀⠀⣤⠀⣠⠀⠀⠀⡄⣀⠀⣀
⠀⠀⠀⠀⠛⠛⠃⠙⠛⠋⠀⠛⠛⠁⠘⠛⠋⠀⠛⠛⠁⠙⠛⠃⠘⢛⡛⠀⠛⣛⠃⠘⠓⠛⠈⠛⠛⠀⠙⣛⠋⠈⢛⡚⠁⠙⠛⠋⠘⠛⠛⠁
⠀⠀⠀⠀⢠⡤⠀⠀⠰⠆⠀⠰⠶⠀⠀⢠⠀⠀⣶⣶⠀⠀⠀⠀⠀⣛⣓⠀⠈⠉⠀⠰⣮⡗⠀⣸⢷⠀⠈⢉⠀⠀⢈⡉⠀⣼⠛⡧⠀⠟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠈⠁⠀⠈⠁

⣝⡫⣺⣷⣿⡥⣟⣥⣾⡿⢦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣾⡶⢿⣿
⢟⠿⡿⣿⡿⡿⠟⢻⣣⡛⣾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⣶⡤⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⣙
⠀⠈⠃⠀⢀⠏⣄⠰⣬⠫⡛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠘⣭
⠺⣢⠸⠃⠈⢠⡜⠷⠁⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠳⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠳⡆⠀⠀⠀⠀⠀⠀⠀⠰⠃⠐
⠆⠉⡾⠇⠾⠎⠁⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢰⡀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢀⡀⠀⠀⠈⠄
⡇⠀⠐⢸⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈
⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢸⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠠⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡁
⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠾
⠀⠀⠀⠀⠀⠠⠀⢠⣦⠀⠀⡀⠀⠀⠈⠀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣄⠀⠀⠄⢀⡀⠀⠀⣀⡀⠤⠠⠀⣤⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣦⡀⠀⣸⣧⡧⢶⣿⡇⠀⢀⣌⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠁⠀⢿⠀⠀⠀⠀⠀⠄⠀⠀⡠⠀⠀⠈⢀⡃⠈⣦⢌⡛⠿⠿⠋⣡⡦⠌⢹⡈⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢠⠀⠀⠀⠄⠀⠀⠀⠀⢿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢡⠀⠀⠀⠚⢞⡜⠓⠈⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠱⠂⠀⠀⠘⠂⢀⠀⠀⢾⠗⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡄⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⢘⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠢⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡆⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⣂⣆⠀⡀
⠀⡀⠀⣀⡀⡀⢀⠀⢠⣀⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡄
⢀⠀⠚⢠⠂⡇⢀⣀⣴⣿⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣞⣿⣿⣇
⣏⣆⣀⣐⣀⣁⣎⣙⣉⣛⣃⣽⣀⣀⣀⣀⣒⣂⣀⣀⣀⣀⣾⣇⣀⣀⣾⣶⣀⣀⣀⣀⣄⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣿⣼⣿⣏
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡠⠒⠊⠉⠉⠑⠢⡀⠀⠀⢀⠄⠒⠀⠀⠐⠢⣄⠀⠀⠀⢀⡠⠒⠒⠒⠲⢄⠀⠀⠀⠀⡴⠊⠉⠉⠙⠲⢄
⡜⠀⣀⠀⠀⠀⡄⠀⢹⠀⠰⠁⠀⠀⠀⢀⠀⠀⠈⢂⠀⢠⠊⢀⠀⠀⠀⣀⡀⢳⡀⠀⡜⠀⠀⠀⠀⠀⠀⠈⢳
⡇⢀⠉⠀⠀⠀⠀⡄⢸⠀⡇⠀⡀⠘⠃⠙⠀⢤⠀⢸⠀⠈⠀⠉⠁⣀⣀⣀⣀⠀⡇⠀⡇⠰⡀⣤⠀⠶⢈⡇⠈
⠱⡀⠓⠤⠤⠤⠊⢀⠜⠀⠰⡈⠃⠂⠔⠢⠒⠊⠀⠌⠀⠸⡄⠻⣽⡯⣇⣳⠟⢀⠇⠀⠱⡀⠑⠤⣤⠤⠚⢀⡜
⠀⠈⠒⠤⠤⠤⠒⠉⠀⠀⠀⠈⠂⠄⠀⠀⠠⠔⠊⠀⠀⠀⠈⠲⠤⠌⠁⠤⠔⠁⠀⠀⠀⠈⠒⠤⠤⠤⠒⠉
⠀⠀⢀⣀⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢄⣀
⢀⠔⠁⠀⠀⠀⠀⠙⢆⠀⠀⣠⠊⠁⠀⠀⠀⠉⢢⡀⠀⠀⡴⠊⠀⠀⠀⠈⠑⢄⠀⠀⢀⠔⠉⠀⠀⠀⠈⠳⡄
⡎⠀⠀⠰⠀⠒⠀⠀⡈⡆⢰⠃⢀⡄⠀⠀⠶⠀⠀⢳⠀⡘⠀⠀⠐⠀⠀⢀⡀⠈⡆⠀⡎⠀⣀⠀⠀⠀⠀⠀⢸
⡇⢆⠀⠀⠀⠀⠀⢀⢇⠇⢸⠀⣤⠤⢤⣤⣒⡶⠀⡼⠀⢃⠀⢀⡤⡖⠒⠓⠣⠀⡇⠀⡇⠀⠛⣠⠤⡀⠛⠀⣸
⠘⢮⡒⠤⠀⠠⢔⡪⠊⠀⠀⠣⡈⠛⠥⠤⠚⢁⡴⠁⠀⠈⠢⡀⠈⠁⠀⠀⢀⠔⠁⠀⠘⢄⡀⠀⠀⠀⣀⠔⠁
⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉⠁⠀⠀⠀⠀⠀⠈⠑⠒⠒⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⢠⠔⠊⠋⠑⠒⢤⡀⠀⠀⠀⢀⠔⠚⠉⠑⠢⣄⠀⠀⠀⣠⠔⠒⠉⠙⠲⢄⠀⠀⠀⠀⣠⠔⠚⠉⠙⠒⠦⡀
⡰⠁⢀⡀⠀⢀⠀⠀⠙⡄⠀⡰⠁⠀⠀⡀⠀⠀⠈⢆⠀⢰⠁⢀⠀⠀⠀⠀⠀⢳⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠈⠁⠀⠉⠃⠀⠀⡇⠀⡇⠀⠊⠂⠀⠈⠑⠀⢸⠀⢇⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⡇⠀⠀⣀⣀⣁⡠⠤
⠱⡀⠈⠉⠉⠉⠛⠁⡠⠃⠀⢣⠀⠀⠀⠿⠀⠀⢀⠏⠀⠘⣄⠀⢀⠤⠠⠄⢀⡜⠀⠀⢣⡀⠘⣍⠇⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⠚⠊⠙⠋⠋⠈⢹⠁⡀⠉⠉⠁⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢉⠙⢄⢀⠀⠀⠀⡰⠄⠀⣷⣤⠂⣄⣂⣠⠋⢸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠉⠀⠀⢠⡄⠈⣧⢠⣼⠆⣿⠀⠀⢹⣿⣧⣾⠟⠁⢀⣿⠁⠀⠀⠄⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⣄⡀⠀⠀⢀⣾⡆⠀⠘⢾⣿⣷⡎⠀⠀⠘⣿⣿⠏⠀⠀⣾⣿⢿⣶⡸⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠳⢷⣤⣙⣷⡀⠀⠀⠉⢻⡅⠀⠀⠀⢻⠏⠀⠀⠀⣿⣿⡿⠟⠁⠀⣰⡟⠀⠆⠡⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠹⠄⠀⠀⠉⠻⢾⣷⠀⢀⣠⣤⣴⡶⠒⠒⢦⡴⢤⣤⣠⠟⠉⠀⠀⠀⣼⣿⣧⢇⠀⠘⠂⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣼⠗⠀⠀⠀⢀⣬⠾⠯⠓⠀⠀⠁⠀⠀⠉⠙⠻⠻⣛⣳⣄⠀⢀⣾⣿⡿⠿⠶⠶⠀⠠⣲⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢦⡀⠐⠳⢤⣟⣯⣿⣆⣰⠿⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣍⣚⠷⡞⠉⠀⠀⠀⢀⣠⣴⡞⠣⣌⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠐⢛⣦⡀⠀⠀⠀⠀⣸⠏⠀⢋⣭⣛⣛⢭⣟⣦⠀⢠⣾⣿⣿⣿⣿⣿⣏⠙⣻⡄⠀⣠⣾⣯⣷⠻⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠰⣋⣝⢦⣤⣀⢠⣟⠀⢰⠿⣏⡿⢾⡿⡯⠏⠀⠸⣿⡿⢿⣟⢟⣿⡿⠟⣭⣷⡈⠙⠛⠛⠻⠿⢟⣟⡂⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠿⠵⠒⠋⠉⠉⢹⡏⠁⠀⠀⠈⠉⠁⠀⠀⠃⠀⠠⣿⠇⠈⠉⠛⠁⠀⢀⡶⣿⡇⠀⠀⠀⠀⠀⠀⣀⣉⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠐⠶⠤⣤⣀⣀⣀⣀⠀⠀⠀⢹⡡⡅⠀⠀⠀⠠⠔⠀⣀⣤⡀⣀⣿⣷⠇⠀⡀⠐⢀⠈⠀⣿⢃⣤⣴⣶⡶⠛⠏⠉⠁⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⢛⣯⡿⠚⣧⠀⠐⠀⠀⠀⠀⠀⠈⣉⡙⢋⣁⡀⠀⠐⠁⠀⠈⠐⢁⣿⠀⠙⠫⣿⣅⣡⠖⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⣸⠖⠚⠀⠀⠀⠹⣗⠊⠀⠀⠀⠀⠓⠚⢏⡛⢛⡻⠋⠋⠀⠀⠀⠈⣤⣴⣃⡀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠈⠉⢀⣀⣀⣠⣤⢴⣾⠉⢿⣌⢠⡄⠀⠀⠀⠰⠛⠛⠢⠀⠀⠀⠒⣖⣻⡟⠁⠙⢿⣿⣿⣷⣦⣤⣈⣒⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠚⠋⠉⠉⠙⢡⠿⡿⠁⠀⠀⠙⠿⣷⣀⢀⣀⡀⠀⢀⠀⡀⣁⣬⡷⢟⣉⡀⠀⠀⠈⠻⣼⠉⠁⡀⠉⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠀⠀⠈⡟⠀⠀⠀⣠⡴⣿⠀⠉⠛⢻⠲⠾⠟⠛⣻⣋⠉⠀⠈⣿⣿⣷⣠⣀⠀⠙⢧⡀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠞⠀⣀⣴⠞⣉⣾⠟⠀⠀⢀⣿⡆⠀⠀⠀⣿⣿⣦⡀⠀⢸⡿⡿⡿⠋⠓⠀⢈⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣠⠞⠉⠀⠈⢱⠺⠃⠀⣠⣾⣿⣧⠀⠀⢰⣿⣾⢻⣧⠀⠈⣷⠋⠀⠀⠀⠀⠠⠄⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⡀⠀⠀⠰⠇⢀⡾⠿⠸⠿⠛⠆⠀⢸⠏⠙⠿⡝⢧⠀⢩⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠁⠀⠈⠆⠀⢢⡀⣼⠀⠀⠀⣠⡂⠑⠒⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠀⠀⠶⠀⠞⢡⡞⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⢠⠰⡆⡶⡀⣶⠀⡦⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⢤⣳⠀⢇⢻⠸⡆⢸⡆⢻⠀⢱⣾⠀⡇⠸⠀⣩⠀⡇⢰⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⣞⢧⣹⡀⢦⢣⠸⡌⡇⢻⠈⡇⡾⢠⠀⣯⢀⡇⣴⠀⡟⢸⢁⡟⡡⢸⡷⢃⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡘⢦⡙⢷⡈⢧⠹⡇⢸⡇⢣⢧⢻⣸⡄⣿⡇⢸⡆⢸⡸⡇⣿⢠⡇⡟⣸⢀⡇⢸⣧⠞⣴⠋⣠⠆⣀
⠀⠀⠀⠀⠀⠀⠀⣤⣑⢦⡑⢄⠻⢄⠙⢆⠱⡇⠈⣿⡈⢾⡎⠇⢧⢹⡅⢸⣇⠈⢿⠇⡏⢼⣠⠷⢃⣼⠃⣼⢃⡜⢁⡜⢁⡾⢃⣀⡀
⠀⠀⠀⠀⠀⡀⠲⣌⣦⡉⠛⠮⣧⡀⠱⣌⠡⣿⠀⢻⣷⣄⠑⠺⢈⣺⠀⢸⣿⠀⢸⣀⡷⠋⢁⣴⠟⣡⠞⣡⠏⢠⣏⣴⠿⠚⣩⠏⢀⡤
⠀⠀⠀⠀⠦⣉⠓⢤⡙⢷⡌⠲⣄⠉⠳⢬⣥⡈⠳⡀⠹⣿⣷⣦⠀⢿⠀⣸⣿⠀⠈⡿⠀⣴⡿⢁⡼⢁⡴⣁⡴⠛⢉⣠⢖⣼⡥⠞⢉⣤⠖
⠀⠀⢀⠛⠂⣌⠑⠦⢌⡓⠻⣦⠈⠻⣶⣄⠈⠙⠳⢽⣄⠈⢿⡟⢻⡏⢹⠇⠘⡟⠠⣧⣼⣿⡇⣼⢁⡾⠋⠁⣠⣴⠟⣡⠟⢁⡤⠞⢉⣤⠖⢋⣤
⠀⠀⣄⡉⠲⠦⣝⡲⠤⣉⠒⠬⣓⣄⠈⠻⣷⣶⣄⣤⡟⢿⢈⣣⠤⠶⠒⠒⠒⠶⠤⣅⡀⠻⠛⣿⣏⣀⣴⣾⠟⢁⣼⡿⠚⣫⡴⠚⣩⡴⠚⢉⣠⡄
⠀⣶⠶⠿⢗⡢⢤⡙⢲⣤⠽⠶⠬⣝⣦⡀⠈⢿⣯⠉⡡⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⠋⢉⣿⡿⠁⢠⣾⣭⣶⣯⣥⣶⣏⣡⠶⢞⢋⡼⠦
⠠⣍⢳⣜⠦⢙⣒⣛⣁⣤⡴⢶⣶⣤⣌⠙⣰⣷⢀⠞⠁⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠙⡄⠉⣿⣶⢋⣠⣤⣤⣤⣤⣤⣴⣶⠞⣫⠾⠶⠖⠂
⢐⡒⠶⠮⠷⣦⣭⣍⣠⣤⣤⣤⣀⠈⠛⢿⣷⢀⢩⠴⣚⡉⠭⠍⢩⡄⠀⠀⠀⢠⠗⣒⠭⢉⣓⠢⠘⡆⢀⣽⡿⠛⠋⢉⣉⣉⣭⡤⠶⠾⠷⠶⠒⢚⣋
⢨⣭⣉⣉⣑⣒⣒⠒⠒⠒⠀⠠⢿⣷⡄⣿⡄⣼⠀⣼⣗⢰⣿⣿⣀⡀⠀⠀⠀⠠⣤⣿⣿⠉⠲⠧⠀⢹⡄⣉⣷⠶⢯⣭⣭⣉⠴⢖⡒⠒⢋⣉⣉⣉⣥
⠠⠤⠤⡤⣤⣭⣭⡭⠿⠛⠛⢋⣉⣠⣤⡷⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⣿⣤⣤⣤⣤⣤⣈⣉⣉⡉⠛⠻⠿⢟⣒
⣈⡵⠛⠋⠉⠠⠴⠶⠶⠿⠿⠿⠿⠿⢿⡶⠆⢣⠀⠀⠀⠀⠀⠀⠘⠶⢆⢀⡶⠼⠃⠀⠀⠀⠀⠀⠀⠀⡁⠙⣿⠛⠛⠋⠋⢉⣉⣉⣩⣭⡭⠥⠶⢀⣊
⠨⠭⠥⠶⠶⠖⣒⣦⣤⣤⣤⣤⣀⣀⣀⢿⠖⠘⠀⠀⠀⠀⠀⠀⠀⠀⣨⡉⢀⣀⣀⠀⠀⠀⠀⠀⠀⣰⠈⠻⡇⠒⠺⢿⡭⠭⣭⣍⣉⣉⣛⠒⠒⠲⠶
⠐⠒⠛⢛⣉⣁⡤⠴⠖⢚⣋⣩⠟⢉⣡⣾⣶⡆⠐⢆⠀⠀⠀⠀⠈⠋⠉⣙⣋⡭⠁⠀⠀⠀⠀⠀⣠⠋⢻⡿⠿⣿⣦⣄⠉⠻⣖⡲⠤⢬⣍⣉⡙⠓⠂
⠀⠚⠋⠉⣀⡤⠤⠖⢚⣩⠏⢀⣴⠿⠛⣉⣬⠷⣶⠈⠓⢄⠀⠀⠀⠀⠴⠛⠓⠲⠄⠀⠀⠀⡀⣞⣁⣻⡟⢧⡀⢄⡉⠻⣷⣤⣌⡛⠿⠯⣄⣛⠋⠽⠇
⠀⠐⢛⣩⠥⢒⡾⠿⠋⣁⣴⠟⢁⡴⠋⣩⠋⢀⣼⣿⣇⣀⣌⠒⢤⢀⣀⣀⣀⣀⡀⡠⠔⢫⣤⣸⠟⢿⣷⡄⠹⣦⡈⠲⣄⣉⣙⣛⣛⣩⡵⠬⣷⡄
⠀⠀⠉⠔⡶⢋⠴⠚⢛⣉⣁⡴⠋⢀⡼⠃⣠⣿⡿⠋⠈⠙⢻⣶⣿⣤⣾⣄⢠⣦⣠⣿⠶⣼⣏⠀⢢⣄⠙⠿⣦⠘⢿⠢⣌⠙⢦⡉⠱⢦⣈⠒⠌
⠀⠀⠀⠙⣇⡤⣲⠞⢋⡽⠋⣠⢔⡿⢁⣾⠟⠋⣠⣴⠃⣴⣿⠟⢁⡆⠀⢻⣿⡇⠈⡗⣄⠙⣿⣆⠀⢿⡑⢤⡈⠳⣄⠳⣄⠑⠢⣈⠓⢤⡈⠑
⠀⠀⠀⠀⠁⠖⢁⠴⢋⡴⠟⣡⡞⢁⡟⢁⣤⠞⣱⣿⠀⣿⠃⢠⡎⣷⠀⢸⣿⠇⢰⢧⠸⡄⣿⣿⠀⣸⡙⢦⡙⢦⣈⠃⠙⣧⡲⢌⠓⠦
⠀⠀⠀⠀⠀⠀⠈⠰⠋⡠⢊⡟⠀⣩⡖⠉⡟⢀⠇⢸⡄⢿⡄⢸⢰⡟⣆⠀⣿⠀⣾⠈⣇⢃⠸⣿⠀⢯⠁⠌⠳⣆⠙⢧⠢⢜⡏⠲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⢴⡏⢹⠀⣼⠃⡼⢰⠃⡇⢸⡇⢸⠀⡃⣿⡄⢿⠀⡟⡆⠹⡈⠂⡙⠷⡌⠳⣌⠳⣌⠳⣌⠳⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠏⣸⠃⡴⢁⠇⡼⢡⠞⢠⡞⢘⠃⡇⢷⠈⢠⣇⢻⡀⢳⠀⢯⠓⣮⠂⢹⡳⣌⠻⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠃⡜⠀⡇⠏⣼⠁⡇⢸⠀⡇⢸⡄⣸⢹⠀⢧⠈⣇⠘⡆⠸⠉⠺⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⢸⠀⠇⢸⠀⡇⡌⢳⡟⡆⢇⠸⠆⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢈⡧
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠱⡀⠀⠀⠀⠀⣀⡤⠔⠒⠒⠛⠛⠒⠒⠢⠤⣀⠀⠀⠀⢀⠎⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⢀⠴⣋⣁⣀⣀⠀⢀⢀⣀⣀⣀⣠⡤⠴⠟⢲⣄⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢁⡴⠋⢳⡀⠀⣸⠛⢧⡀⠀⢠⠏⠙⢦⡀⡞⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⣼⣎⣀⣀⣀⣳⣴⠃⣀⣀⣹⣤⣏⣀⣤⠤⠿⠓⠚⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢠⠤⢄⣀⡀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢤⠀⠀⠀⢀⡇⠀⠀⠀⣷⠀⠀⠀⢀⠆⠀⠀⠀⠀⣇⠀⠀⢀⣀⡠⠤⠒⠒⡆
⠀⠀⠀⢸⠀⠀⠀⠉⠑⠒⢤⠀⢸⠀⠀⠀⠀⠀⠈⢳⡤⡤⠾⠀⠀⠀⠀⠸⠢⡤⢴⠋⠀⠀⠀⠀⠀⡿⠀⢮⣉⡀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⢀⣀⡠⠤⠒⠊⠉⠀⠈⡇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡀⠀⡀⠀⠀⢀⣈⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠉⠑⠒⠤⢤⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢰⠉⠀⠈⡇⠀⠀⠈⠉⠁⠀⢰⠋⠀⠈⢣⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠈⠢⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠓⠤⠴⠃⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠙⠲⢤⣀⣀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢆⠈⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣸⠀⠀⠀⠘⡇⠀⠀⠀⢸⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠹⣄⠀⠀⠹⣆⠀⠀⠀⢳⠀⠀⢠⡿⠀⠀⠀⠀⡏⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠘⣦⠀⠀⢹⣷⡄⠀⢰⡀⠀⠸⣿⡀⠀⠀⢸⠃⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠈⡛⣶⣤⡀⠀⠹⣷⣄⠀⣿⡇⢀⣼⣷⡀⣸⣿⠇⠀⠀⣸⠀⠀⢰⡏⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠈⠻⣦⣄⠀⠁⠸⣿⠀⠤⠙⠋⠉⠉⠉⠉⠉⠉⠙⠻⡣⠄⢤⣴⡏⠀⣠⣿⡇⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣀⣀⠀⠈⠻⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠟⠋⠀⠀⠰⠋⠀⠀⢀⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠝⢀⢤⣴⠇⢀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠲⢦⣤⣤⣄⣤⣠⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢿⡷⠶⠿⠛⠀⠀⠀⣠⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⠋⠀⠀⠀⣀⣀⣤⣠⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣤⣶⣶⣀⡀⠀⠀⢲⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠐⠒⠒⢤⣤⣤⣴⣶⣷⣤⡎⠀⠀⠀⠛⠉⠀⠀⠀⠀⠈⠙⢧⠀⠀⠀⠀⢠⣾⡏⠉⡹⠋⠁⢽⣿⣿⠀⠀⠀⣿⡿⠋⠀⣀⣤⣤⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠁⠀⠀⠀⢀⣠⠴⢾⣿⣿⣷⣶⠀⠀⠀⠀⠀⢸⡟⠁⡈⢶⣿⣿⣿⠻⡷⠀⠀⠀⢿⠶⣾⣿⠟⠉
⠀⠀⠀⠀⣀⣀⣠⠤⡤⠴⠶⠒⠿⣿⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠚⠀⠀⠀⠀⠀⢻⡇⣸⣃⠄⠈⠉⢀⠀⠿⠃⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡤⠄⠒⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⠿⠋⠛⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⢀⣀⠿⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⡀
⠀⠀⠀⠀⠀⠋⠁⠈⠁⠀⠀⣠⠾⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢈⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠙⠿⠿⠿⠛⠛⠓⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⢘⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠇⠰⠿⢿⣶⡦⠄⠲⠀⠀⠀⠀⠀⠀⢀⣆⡀
⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⣴⣿⠿⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠶⠿⠿⢄⠀⠀⠀⠀⠀⠀⠀⣠⣯⠉⠙⠛⠒⠲⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⢠⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣡⡊⠀⠹⣿⣦⣄⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠉⠀⠀⠀⣠⣿⠁⠀⢈⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡠⠂⠐⣮⠻⢿⣀⠀⠀⠈⠉⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⣴⡿⠃⢨⣿⣻⡒⢂⡦⣤⣬⠨⠿⢻⡟⢿⣿⡁⠀⠀⣿⣆⠀⠈⠙⢦⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⣿⠁⠀⢀⡏⠀⢠⣿⠁⢸⣇⠀⠀⢸⣧⠀⠙⢷⡄⠀⠈⠛⢷⡄⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣴⠋⠀⠀⢸⠇⠀⠸⣿⠀⠀⣿⡃⠀⠈⠻⣷⡀⠈⠻⣆⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⡼⠀⠀⠀⣿⠁⠀⢸⡇⠀⠀⠀⢸⡀⠀⠀⠙⢆⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡏⠀⠀⠀⡇⠀⠀⠀⠈⢧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣆⠀⠀⣾⠀⠀⢰⡇⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠸⣧⠀⠀⣿⡄⠀⣿⡆⠀⣿⡇⠀⣼⡇⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠀⠈⢿⣆⠀⠹⣷⡄⢹⣿⡀⢻⣷⠀⣿⡇⢰⣿⠁⢠⣿⠃⠀⣀⡔
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠹⣷⣤⡀⠻⣷⡄⢻⣷⠀⣿⡇⣿⡏⣸⣿⢁⣾⡟⢀⣾⡏⢀⣼⡟⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⡈⠻⣿⣆⢹⣿⠘⢛⣀⣭⡤⠤⢤⣬⣅⣘⠛⢰⣿⠏⣠⣾⠟⢀⣴⡿⠁
⠀⠀⠀⠀⠈⠲⣦⣤⣀⡈⠙⢿⣦⡘⠟⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⡘⠟⢁⣤⣾⠟⢀⣠⣶⠞⠁
⠀⠀⠀⢀⣀⠀⠀⠉⠛⠿⣷⣤⠙⢣⡾⠉⠀⠀⠀⠀⠀⢴⣾⣶⡄⠀⠀⠀⠀⠈⠹⣦⠙⢋⣤⣴⡿⠟⠁⣀⣀⣤
⠀⠀⠀⠀⠉⠛⠿⠿⣷⣶⣌⠛⣰⠏⠀⣀⣀⡀⠀⠀⠀⠹⠻⠻⠁⠀⠀⠀⣀⣀⣀⠈⢷⡘⢋⣩⣤⣶⡿⠟⠉
⠀⠀⠠⣤⣤⣤⣤⣤⣤⣌⡛⢡⡿⠞⠛⡉⣉⣙⠓⢦⡀⠀⠀⠀⠀⣠⠞⡋⣉⣉⡉⠛⠻⣧⠙⣛⣋⣁⣤⣤⣶⠶⠖
⠀⠀⠀⠀⠀⠉⣉⣉⣉⡛⠃⣼⠁⠀⠸⢄⣿⣿⠠⠂⠹⡄⠀⠀⡼⠁⠨⢐⣿⣿⠨⠂⠀⢸⡆⠛⠛⠛⠛⠉
⠀⠀⠒⠶⠿⠿⠟⠛⠛⠿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⠿⠿⠿⠿⠟⠛⠛⠂
⠀⠀⠀⢀⣀⣤⣶⣾⠿⠿⠇⢹⡄⠀⠀⠠⣷⣷⣧⠀⠀⢐⠀⠀⡀⠀⢀⣴⣶⣆⡀⠀⠀⣼⠁⢿⣶⣦⣤⣤⣤⣄⣀
⠀⠀⠈⠉⠉⠉⢁⣠⣴⣶⣶⠌⢧⠀⠀⠈⠛⠟⠏⠀⠀⠈⠉⠉⠀⠀⠐⠿⡿⠟⠀⠀⢠⠏⢴⣶⣬⣉⣉⠉⠉⠉⠉
⠀⠀⠀⢀⣤⡶⠿⠛⢉⣤⣶⣶⠌⢷⡀⠀⠀⠀⠀⠀⣠⡴⣂⢴⣦⣀⠀⠀⠀⠀⠀⣰⠏⢴⣦⣌⠛⠛⠛⠿⠶⠦
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢁⣴⣶⠆⠙⢦⣀⠀⠀⠀⠀⠉⠒⠚⠁⠀⠀⠀⢀⣠⠞⠁⢶⣦⡙⠻⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠚⠋⠉⢀⣴⡿⠋⣠⣾⠿⢀⣍⠓⠶⣤⣄⣀⣀⣀⣀⣤⡴⠖⢋⣅⠸⣷⡈⠻⢿⣦⣄⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⣸⡿⠁⣰⣿⠋⣴⡷⢀⣤⡍⣭⡍⣡⡄⢰⣷⠘⣿⣆⠙⢿⣦⡀⠈⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢰⣿⠃⢸⣿⠁⣾⡟⢰⣿⠃⣿⣧⠘⣿⣆⠘⢿⣦⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡿⠃⠀⣾⡏⠀⣿⡇⠘⣿⡇⠘⣿⡄⠘⢿⡆⠀⠙⢧⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⡟⠀⠀⣿⠃⠀⢻⡇⠀⠹⡇⠀⠈⢿⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠏⠀⠀⠸⠃⠀⠀⠃



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣛⣙⣻⣉⡟⣻⡟⣹⡟⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⢀⡏⣇⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢻⠱⡄⠀⠀⢸⢀⢸⠀⠀⠀⠀⡰⢻⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⠀⠀⠀⠀⡼⠀⢣⠀⠀⡟⢸⠘⡇⢀⠔⡫⢂⠞⠀⠀⠀⢀⡤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡔⢝⠢⣄⠀⣇⠸⡈⠳⣤⠃⢸⠀⢣⡏⢰⠁⡏⠀⢀⡤⢚⠕⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠑⢄⠙⠺⣶⢚⡩⠽⠶⠚⠲⠾⠭⣙⢲⡷⠚⢁⠔⢁⠞
⠀⠀⠀⠀⠀⠀⠐⠶⣶⡒⠦⣄⠀⠈⢣⡀⠑⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡔⠁⣠⣋⣤⣤⣀⣀⣀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠢⣉⡉⣹⢳⠞⠀⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠀⠘⢶⢻⣡⠔⢒⣒⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣤⣃⡎⠀⠈⢉⣭⣭⡙⢦⠀⢠⠞⣩⣥⣍⠉⠀⠈⣆⣣⡔⠁
⠀⠀⠀⠀⣀⣀⡤⠤⠒⠒⠚⠋⠉⣁⣸⠀⠀⠐⠫⣾⡿⠕⢘⠀⢈⠐⠹⣾⡯⠗⠀⠀⢹⣀⣉⠉⠛⠒⠒⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠉⠉⠉⠓⠒⠠⠤⠤⣀⣀⣸⡀⠀⠀⠀⠀⠀⠀⡜⠀⠘⡄⠀⠀⠀⠀⠀⠀⣸⣀⣀⡠⠤⠤⠐⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡏⢧⠀⠀⠀⠀⠀⠀⠑⠲⠒⠁⠀⠀⠀⠀⠀⢠⠏⣟⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣊⡉⠉⠄⠊⣹⡼⢧⡀⠀⠀⠀⠑⢒⣒⣒⠒⠁⠀⠀⠀⣠⠿⣼⡥⢤⣑⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠐⠛⠉⠁⠉⠉⠉⢉⠜⢀⡴⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠳⢄⠙⣄⠀⠈⠙⠲⠿⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠔⢋⡠⠚⡟⢮⣍⣳⠒⢲⠒⢲⣊⢭⡞⢻⠦⣀⠳⣌⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢁⡡⠖⠉⠀⢀⠇⡸⢠⠟⡆⢸⠀⡞⠓⣄⢱⠈⡇⠈⠑⠦⣁⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠉⠀⠀⠀⢠⠏⣊⠴⠋⠀⢧⢸⢠⠇⠀⠘⡄⢰⠃⠀⠀⠀⠈⠙⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠁⠀⠀⠀⢸⠈⢸⠀⠀⠀⠱⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣇⡇⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠹⣿⣿⣿⠀⠀⠀⢸⣇⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⢻⣿⣿⠀⠀⠀⠈⣿⠀⠀⠀⣿⣿⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⢿⣿⡇⠀⠀⠀⣿⡆⠀⠀⣿⡿⣠⡾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠘⣿⣧⠀⠀⠀⢸⣷⠀⠀⠟⠙⠉⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠸⣿⡆⠀⠀⢸⣿⡇⣠⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⠀⠀⠀⢻⣷⡀⠀⢸⡟⣱⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡄⠀⠀⠘⣿⣧⠀⠈⣀⣀⣠⣤⣤⣤⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⢻⣿⢇⣾⣿⣿⣿⡿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢻⣿⣿⣆⠀⢸⠏⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⢻⣿⣿⣦⠀⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⢻⣿⠏⣼⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠈⠟⣈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠈⢻⣿⡆⢰⠿⠟⠛⠉⠁⠀⠀⠀⠀⠈⠉⠉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⡆⢰⠀⡌⢀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠘⠎⠀⠀⠁⠘⡇⡜⢠⠁⡞⣠⠃⡰⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠠⠈⠀⣦⠀⢳⠄⢰⡇⠃⣾⠀⣡⠏⡼⠁⠊⣨⠋⡠⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡈⠧⡈⠢⣦⠱⠀⠏⠀⠉⠀⠸⠇⢰⡏⠀⠏⡼⠁⢁⡞⣡⠞⣁⠔⢁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠣⠙⠂⠽⣄⠀⠀⠃⠈⠀⠀⠀⠃⠀⠋⢀⡾⠘⢁⣴⠋⠀⣡⠞⣡⠔⢁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠺⣗⡂⠲⠀⠀⠀⠀⢀⠀⠐⠋⠀⠀⠐⠀⢘⠁⠐⠁⠠⠴⠎⠵⣻⡥⠚⣁⡤⢢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣐⠒⠄⠈⠒⠂⠱⢄⡀⠀⢀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⣠⡶⠻⠉⣀⣤⠤⠒⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡉⣐⠲⠖⠦⡀⠀⢀⠅⠀⣀⠀⠀⠆⠀⠀⠔⠀⠀⡀⢀⠙⠈⠁⢀⣠⣬⠥⠖⢒⣉⡁⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠀⣠⠄⠀⠀⡸⠀⠘⠻⠿⠃⠚⠀⠘⠱⠻⠿⠟⠃⠀⡄⠀⠉⠑⠚⠋⠩⠿⠤⠄⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣉⣭⣥⠤⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠲⠶⠞⠛⠒⠒⠒⢒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠀⢀⣀⣀⢠⠄⠀⠀⠈⡂⠀⠀⠀⠀⠀⣤⡤⠀⠀⠀⠀⠀⣐⠃⠒⠲⢤⣤⣠⠤⠶⠤⠄⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⠇⢀⠄⠊⠀⠈⠀⡂⠀⠀⠀⠒⠲⠀⠀⠀⢀⣔⠉⠑⠢⠤⠄⡱⢭⣍⣓⠒⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣈⠀⠰⠃⣠⡆⠀⡠⠊⠁⠔⠀⢀⠀⣠⠄⠤⠒⠁⠀⠓⢤⡄⣤⡄⠈⠲⡪⢍⣉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡡⢞⡁⠀⡼⠀⠠⠊⠀⢠⡇⠀⡄⠀⣶⠀⠈⢷⡠⡀⠉⢳⣍⠲⢍⡙⠢⢄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⢴⠟⢀⡴⠀⠀⢸⣃⠀⣷⠀⠘⣷⠀⢨⡱⣄⠀⠐⢦⡑⠤⡉⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⢁⠆⡚⠀⠀⠰⠇⣽⠀⣿⠰⠀⡙⣏⢤⠳⡌⢯⡐⢄⠙⠦⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠴⠀⠁⡟⠀⢹⠀⣀⢧⠸⡄⢧⠘⢄⠑⢄⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠇⠸⠈⡇⢩⠈⡄⠱⠀⠣⠈⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠇⠈⠄⠐

⠀⡀

⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⣠⣀⣀⣀⡀⠀⠀⠀⠂⠀⠰⠀⠀⠀⠀⢤⠀⢠
⠀⠀⠰⠦⠰⠆⠀⠀⠀⠀⠀⢀⠔⠄⠀⠀⠐⠆⠴⠆⠀⠀⠀⡱⠞⠉⠉⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁
⠀⣀⡀⣀⣀⢀⡀
⠀⠧⠴⠜⠳⠦⠟⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠂⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⣛⣫⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠐⡀⠐⠂⠀⢠⡄⠠⠄⠀⠄⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⢂⢀⣐
⠀⠀⠀⣄⣀⠀⠀⠘⠇⠀⡀⠀⠀⠸⠃⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠈⠀⠄⠁⢄⠀⠀⢴⠀⣤⠀⢣
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠂⡀
⠀⠀⠐⠀⠐⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠐⠂⠀⠀⠀⠀⠀⠒⠀⢂⠀⠀⠀⠘⠿⠐⠿⠂⠀⠀⠐⠓⠀⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠚⠛⠃⠀⠀⠀⠀⠈⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⡀⠀⠀⠠⠄⠒⠠⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⢠⣦⣤⣤⣤⣤⠀⠀⠈⠁⠀⠈⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠊⠀⠀⠀⠑
⠀⠀⠈⢀⡈⠁⠀⠀⠀⠘⢃⡈⠃⠀⠀⠀⠉⠉⡌⠉⠁⠀⠀⠀⠁⢀⠉⠀⠀⠀⠀⠂⣀⠐⠀⠀⠀⠀⣍⣀⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⢀⠄⠀⠠⡀
⠀⠀⣌⠀⢀⣡⠉⠀⠀⠀⠀⠀⠀⣨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠮⠁⠀⠀⠀⢁⡈⠋⢁⠈⠀⠀⠈⠂⠀⠐
⠀⠀⠫⠅⠩⠍⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢶⣶⣿⣶⠖⠀⠀⠣⡚⠀⢒⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⣿⡄⠀⠀⢀⣿⡇⠀⠀⣼⡇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⢸⣿⡄⠀⠀⣿⣷⠀⠀⢸⣏⡅⠀⢠⣿⣿⠀⠀⣸⣿⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠈⣿⣧⡀⠀⢸⡻⣦⡀⢸⡼⢷⡀⡾⢹⠏⠀⣠⣿⡟⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣄⡀⠀⠈⢿⣗⣤⡀⠈⢯⡋⠲⣈⡿⠈⠳⣼⠑⠁⠳⠃⢾⡠⠾⣋⡾⠀⣠⡾⣿⡟⠀⠀⣠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⣀⠀⠳⣯⡻⠒⣼⣈⣠⣼⣷⣶⣿⣿⣿⣿⣿⣶⣾⣧⣤⣸⡷⠈⠣⣴⠋⣀⣤⣶⣿⠏
⠀⠀⠀⠀⠀⠲⣤⣀⠀⠀⠀⠙⢾⣿⡻⠒⢿⣄⣴⣿⣿⣿⣿⣿⡿⢻⢻⢸⡿⠻⢿⣿⣿⣿⣿⣿⣶⣤⡟⠈⡡⣨⠞⠁⠀⠀⣠⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣤⡄⠀⡹⣎⣠⣾⣿⣿⠟⠛⠙⠉⠁⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⠙⠻⢿⣿⣿⣦⣼⡣⢄⠶⠖⣺⡿⠋
⠀⠀⠀⢤⣄⣀⡀⠀⠈⠛⠶⣦⠅⣠⣿⣿⣿⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠘⢻⣿⣿⣦⢀⣴⠚⠁⢀⡀⣠⣴⡶⠒
⠀⠀⠀⠀⠙⠻⣿⡶⢶⡖⠒⠚⣷⣿⣿⣏⠁⠀⠀⠴⠇⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠠⡣⠀⠀⠀⢙⣿⣿⣿⡉⢁⡛⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⣄⣰⣿⣿⠟⠃⠀⢠⠁⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢃⠀⠐⠻⣿⣿⣷⣴⠋⠁⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠛⠿⣿⣶⣶⣶⠒⢀⢈⣿⣿⠿⠁⠀⠀⢈⣴⡟⣛⣻⣿⣿⣿⣷⣄⠀⠀⣴⣾⣿⣯⣭⡉⠻⢷⡌⠀⠀⠀⠽⣿⣿⣇⠐⣄⣚⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠉⠀⠉⢛⢷⣼⣿⣿⠂⠀⠐⠀⠀⣿⣿⣿⢥⠶⣬⣽⣿⣿⠀⢸⡟⡿⢭⡴⡾⢿⣮⡀⠀⠀⠀⠀⠐⣻⣿⣿⠿⢅⡀⡀
⠀⣀⣠⣤⣤⣤⣶⣮⣂⠀⣿⣿⣗⠀⠀⢀⠀⠘⣿⡛⠿⠶⠿⠷⠞⣿⡿⠀⢸⠀⠑⠚⠛⠲⠟⠋⠀⠀⢀⠀⠀⠀⢐⣿⣿⣂⣲⣬⣽⣿⣿⣿⠷⠖⠂
⠀⠀⠈⠙⠛⠓⠈⠛⢙⠷⢿⣿⣯⠀⠀⠀⠀⠰⢿⣦⠀⠀⠀⠀⠀⣿⡟⠀⢸⣇⠀⠀⠀⠀⠀⠀⣶⡔⠀⠀⠀⠀⢸⣿⣿⠛⢣⢄⡀⠈
⠀⠀⠀⢀⣀⣀⣴⢞⣡⣄⢸⣿⣿⣅⡄⠀⠀⠀⠳⠿⡦⠀⠀⣠⣾⡏⢵⣄⣀⠉⣷⠀⠀⠀⠠⡬⠞⠁⠀⠀⠀⣔⣿⣿⣿⣤⡶⣵⣻⣶⣴⣶⣤⣀
⠀⠐⠛⠛⠛⠛⠛⠛⠉⡩⠛⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⡾⠋⠉⠙⢻⡟⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠑⡆⡀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣠⡾⣦⣦⣬⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣥⣶⡴⠾⠯⠿⡿⠷⠦⣤⣤⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠶⠷⢬⣿⣶⣤⣄⡀
⠀⠀⠀⡤⠾⠿⠿⠟⠋⠉⢠⠊⠈⢻⣿⣿⣶⡄⠀⠀⠀⠈⠉⠈⣻⣶⣤⣤⣤⣼⣿⠁⠉⢀⠀⠀⠀⢀⣴⣿⣿⣿⣋⡄⢐⣄⠀⠉⠉⠙⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⡿⠗⢻⠛⢿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⠟⠉⢍⠙⠶⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢀⣠⣾⠿⠟⠋⠀⢠⡧⣠⡤⠿⢿⣿⣦⣄⡀⠀⠀⠲⠚⠙⠉⠈⠉⠂⠀⠀⢀⣠⣰⣿⡿⠟⢟⠶⣇⢺⣆⠀⠈⠉⠛⠛⠷
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⣿⡷⠋⢀⣧⣀⣨⣿⠿⢿⣶⣤⡀⣀⢀⣀⠀⠀⣀⣤⣴⣿⠿⠛⡿⢤⡒⢈⡄⠈⠿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠋⠀⠀⣘⣧⡟⠁⢸⢠⣠⠾⠉⢉⣻⡟⠛⢛⡿⣏⡉⢉⠷⣤⢢⢧⠀⢻⣞⣷⡀⠀⠈⠙⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣼⣿⠏⠀⠀⡿⣿⠏⠀⣾⣼⡇⠸⣦⢻⡇⢸⡇⣶⠀⠈⣾⣾⠀⠀⠙⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⣰⣿⡎⠀⠀⢸⣯⡅⠀⢱⣯⡇⠈⣷⡟⠀⠀⠙⢿⣧⡀⠀⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⣾⡟⠀⠀⢸⣿⠁⠀⠹⣿⡄⠀⠀⠈⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠟⠀⠀⠀⢸⡇⠀⠀⠀⠙
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣈⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⢠⠀⠐⠀⡀⢀⢿⣷⡀⠀⠀⠘⠐⠠⢠⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡿⣧⣄⠀⠀⠄⠄⠀⠀⡿⢸⢿⣇⠀⠀⠀⠀⠀⣤⢾⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢧⠹⡜⣧⣀⠀⠀⢠⢸⠀⠀⣄⢹⡆⠀⠀⢠⡔⠃⡞⣾⠁⠀⠀⡀⢁⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⢄⠀⠀⠀⢾⠀⠱⡸⡻⣦⡀⠸⡇⠀⠀⡏⣸⣿⠀⣠⠞⠀⣼⣼⣽⠂⠀⠀⠀⢂⠀⡀⣁⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠌⡛⢿⡳⣤⣤⡀⠀⠀⢺⡄⠀⢣⡱⣩⣷⣾⣀⣤⣴⠷⡾⣿⣷⣃⠀⣰⢿⣓⣿⠆⠀⠀⣀⠠⠚⣡⣾⡏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⢮⡫⣙⢿⢦⣤⡇⢀⣠⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣿⣠⠖⠋⠁⣠⢞⣵⠋⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠂⠈⠂⠀⠐⢧⡀⠙⠮⡣⣳⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⣠⢞⢵⣿⠋⠄⠀⠐⡁
⠀⠀⠀⠀⡀⠠⠀⠀⢈⠂⠄⡀⢷⡀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣿⠋⠀⡀⢐⡁⣀⣤⣄⣡⣤
⠀⠀⠰⢶⣶⣶⡶⢴⡶⠤⣶⣦⣴⣷⡞⠀⠀⠀⣀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠹⣟⠐⠋⠀⠉⣁⠤⣒⣼⠟⠁
⠀⠀⠀⠀⠙⠯⣉⠓⠲⢽⣶⣭⣵⡏⠀⠀⠀⠉⠉⣉⣀⣉⣁⣹⡷⣄⠀⠀⠀⠀⢠⣴⢛⠉⠉⢉⡚⠛⠒⠀⠀⠀⠘⣦⡤⡂⢉⣒⣽⠏
⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⢉⡟⠈⠀⠀⠀⠀⠚⠹⣿⣯⣿⠟⣳⣽⣇⠀⠀⢰⣯⣤⣿⢶⣾⣿⣿⠓⣦⠄⠀⠀⠀⠸⣷⣼⠟⠋
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⣳⣾⠁⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠛⠛⠛⠿⠀⠀⣿⡉⠀⠀⠀⠙⠛⠓⠲⠛⠀⠀⠀⠁⠀⢻⠓⠤⠤⣄⣀⡀⠈
⠀⠀⠀⣀⣀⣠⡤⠶⢾⣿⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢸⡀⠀⢀⣀⣈⣉⣛⣓⣤⣀
⠀⠶⣾⣿⠯⠀⠀⠈⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣟⣛⣛⣿⡻⠿⠛⠉⠁
⠀⠀⠉⠁⠈⠑⠒⠀⠤⣄⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢄⢀⣸⡿⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠛⠋
⠀⠀⢀⠀⠠⠄⠀⠀⡀⣠⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠙⢦⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⣠⢞⡫⣿⡿⠿⡆⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣤⣄⡀⣀⠀⣀⠀⠀⠀⠀⢀⠀⢠⣿⡕⣦⣤⢀⡈⠳⢤⡀
⠀⠀⠀⠠⢀⣶⣻⠽⠒⠉⠁⠀⢀⣙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠉⠉⡠⠄⢉⠁⠀⠀⠀⠀⠀⠀⠀⣠⢿⡓⠿⠓⠾⡿⠶⢮⡭⠿⣦⡀
⠀⠀⠀⠰⠟⠻⠴⠒⠒⠛⠙⢋⠉⣽⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠶⠂⠀⡖⠁⠁⡀⠀⠀⠀⠀⢀⣜⠁⠀⢢⡀⠀⢀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠈⠄⠀⠀⢂⠀⠀⠀⣼⠗⠨⣪⠛⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⢀⣠⣶⠟⢯⡢⣀⠀⠱⡄⠂⠌⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⣴⡯⣢⠞⠁⢀⡤⢾⡿⣶⣥⣀⣀⡀⠀⠀⠀⠒⠀⠀⢀⠠⣄⣤⡖⠋⢹⠏⠻⡷⣭⣦⡑⢄⠙⠆⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⣯⠞⣁⠤⠞⠉⠀⢹⡇⡌⡟⠀⠉⣽⢿⣿⣾⢷⠓⠚⢻⣿⡿⡁⡑⡀⠀⡷⠐⠀⠈⠙⠛⢶⢵⣬⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠕⠛⠡⠀⠀⠀⠀⣹⡵⡘⠀⢀⠾⠀⢸⣏⢻⢈⠀⠀⡎⠀⠙⢷⣱⡑⠀⢹⠄⠀⠀⠀⠀⠀⠁⠌⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⣿⠣⠁⡰⠃⠀⠄⠀⢿⡸⢈⠀⢰⠀⠀⠀⠈⠛⣧⢠⠸⡄⠀⢠⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⣿⣄⠾⠀⠀⠀⠀⠀⠈⣿⢰⢀⡇⡆⠀⠀⠀⠀⠘⠷⣦⡄⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡿⠁⠂⠀⡀⠸⠀⠀⠀⠹⣾⡜⠁⢀⠈⠀⠀⠆⠅⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠁⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣄⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠤⣄⣤⣿⠀⡿⣠⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣉⢉⢉⡉⢹⢸⠀⠀⠉⠙⠛⠀⣗⡇⠀⡠⠀⠀⠀⢸
⠀⠀⣀⣀⣀⣀⣀⣀⡀⡇⠀⢹⠀⡘⠁⢸⢨⠭⠭⠭⠭⠭⠭⡅⡇⠰⣄⣀⣀⠀⢸⣠⠤⠤⠤⠤⠤⠤⡄
⠀⢸⠁⠀⠀⠀⠀⠀⡇⡇⠀⠘⠃⠃⠀⢸⢸⠀⠀⣰⢶⣾⡀⣿⣇⠀⣈⢛⣋⣀⣸⢼⠀⠀⠄⠤⡤⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⡇⣙⣚⣓⣒⣚⣛⣛⢸⠐⠓⣿⡟⠉⠀⣯⡽⠭⠿⠯⠭⠽⢭⣸⠀⢤⡦⠀⡆⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⡇⡇⢀⣀⢤⠄⠀⢸⢸⣀⣋⣀⣙⣊⣁⡯⡇⠀⠠⣐⡀⠀⢸⢿⣀⣈⣁⣀⣁⣀⡇
⠀⠈⠛⠛⠛⠛⠛⠛⠃⡇⢐⣚⣚⣒⡂⢸⣰⠒⠒⠒⠒⠓⠒⣦⡇⢈⠀⠁⠀⡁⢸⠂⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠁⠐⠒⠂⢸⢹⠀⠀⠙⣿⠃⠀⣏⣧⣈⣀⣀⣀⣁⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢸⠀⣤⣦⣿⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⣭⣭⣤⠟


⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣷⠄
⠀⠀⠠⠤⠄⠀⠀⠀⠠⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠦⠴⠏⠉⠉⠤⠤⠤⠤⠤⠂