Bird of prey Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠾⠃⣀⠀⢀⡀⢀⣀⠀⠀⣀⣀⡀⠀⠀⠄⢠⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⡀⢀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⡞⠀⠀⠀⡀
⠀⠀⠀⠀⡠⣻⡐⠄⠠⠀⠈⠉⠉⠉⠉⠁⠀⢌⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡈⠁⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠞⠙⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠈⣄⠀⢀⣤⡄⠀⠀⠀⠀⣃⠀⠀⡴⠀⣄⠉⠑⠄⠶⠃⠀⠀⠀⠀⠀⢰⣟⠀⠀⢀⠠⠊
⠀⠀⠀⠈⠀⠃⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠑⠒⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⣉⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠠⣄⠄⠀⠀⢀⣐⣊⣀⠐⡄
⠀⠀⠠⡄⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠓⢒⡀⠀⠀⠀⠀⠀⠀⠀⠛⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠠⠎⠉⠉⠛⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠠⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠞⡉⠤⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⣢⣰⠿⢾⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠰⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⣠⠔⢠⡄⠀⠀⠀⢀⣄⠒⣄
⠀⠀⠀⠀⠀⠀⠈⠀⣀⠤⠔⠀⠂⣾⠀⠐⠀⠀⠀⠀⢀⠈⠀⣠⠀⠀⠀⠀⠀⠀⠀⢛⢠⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠸⡄⠤⠀⢘⠀⠀⡄⣟⠀⠰⠀⠗
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣀⠄⠂⠀⠀⠀⢉⠁⠀⠃⠠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠻⠃⡞⠻⡀⠀⠀⠿⢻⢘⡿
⠀⠀⡽⠯⠠⠤⣴⣶⢴⣞⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠀⠀⣃⢀⠀⠀⠀⠀⠀⢀⠠⠀⠙⠿⠁⠾⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⢳⣦⡞⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠍⠠⠀⠀⠀⠀⠀⠀⠈⠓⠠⢀⠀⠈⣉⣉⣀⣀⣀⠀⠀⠀⠀⠀⠀⢶⠀⢸⣟⠃⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠱⠀⣀⠀⣀⠀⣠⠤⢤⣖⣶⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠠⢄⠀⠀⠰⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠶⠤⢤⣤⣤⡄
⠀⠀⠀⠠⠀⢈⡠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢺⡃⠀⠀⠀⠀⠀⠀⡈⠒⠤⠀⠀⠀⠖⠈⠂⠀⠀⠰⠄
⠀⠀⠀⠀⠐⠋⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡴⠀⢦⡀⠀⢀⡟⠁⡈⣤⠻⡀⠀⠀⠀⠀⠀⠀⠙⠅⠢⣄⠙⡂⣀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⡀⠒⠈⠐⠀⠀⠈⢀⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⡉⠉⠃⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠙⡙⠃⠀⠀⠛⠀⠊⠁
⠀⠀⠀⠐⠀⠒⠀⠠⠔⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠿⠈⠹⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣄⡀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠘⣎⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢿⡁⠐⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⢠⡂⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣠⡞⠁⢠⢎⠹
⠀⠀⠀⠀⠀⠀⠀⠀⣌⢳⣀⢰⣾⣷⠀⠀⠠⠀⠀⠀⠀⠀⡞⠀⣸⣟⠒⡀⠊⢆⢄⠀⡀
⠀⠀⠀⠀⠀⠀⣠⡘⠚⡘⣶⣿⠟⠁⠀⠀⠠⠀⠀⠀⠀⠀⡿⢠⣿⡷⠭⡂⠐⣨⠊⡘⢰⠆
⠀⠀⠀⠀⠀⡰⠉⡹⢛⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⣾⡿⠍⠭⣒⠁⠴⢎⡴⠃⣴⠆
⠀⠀⠀⠀⢰⣷⣾⣷⣿⣻⠟⣀⡀⠀⠀⠀⠀⠀⡂⢠⡴⣿⣶⣿⣿⣧⣶⣶⡅⠀⣤⡴⠿⠋⠁
⠀⠀⠀⢀⣿⣿⣿⡟⠛⠀⢹⣛⣀⡀⠀⠀⠀⠀⠀⣼⣷⣿⣤⠁⠀⠀⠀⠈⣁⡀
⠀⠀⠀⣾⣿⢟⠝⢳⠀⠰⠎⣿⣟⣁⣀⢀⡄⠀⢸⣿⣿⠅⠙⠀⠀⠀⠀⠀⠈⠁⠀⡀⡀
⠀⠀⠼⠋⣴⠃⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⢿⡎⠈⢻⡿⣦⠀⠀⠀⠀⠀⠀⡆⢀⠈⢳⠇
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⡶⠀⢀⣤⣄⣿⣷⣶⣶⣧
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠃⠐⠿⣿⡿⠉⢈⣿⣿⣿⠀⠀⠐⢀⡀⢀⣀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⢘⣿⣷⣤⣤⣴⠆⠀⠀⣸⣿⣶⣼⣿⠿⠿⠀⠀⠀⣀⡵⠇⠉⡘⣠⣄⣀⣴⣋⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢿⠿⠛⢀⠀⣿⣿⣿⣿⣿⣡⣤⣴⡇⢀⠈⠋⠀⢀⣼⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣮⣉⠿⢿⣿⣿⣿⣿⣿⣧⣄⣠⣴⣾⣿⣿⡿⠋⠁⠀⠐⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣴⣤⡄⣠⣾⣿⣿⣄⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⣮⣿⣿⡿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠴⢯⢩⠇⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢘⣿⣻⡽⠛⡻⠛⣽⣿⣿⣿⣿⣿⣶⣆⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⣿⣿⠇⠈⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠂⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠉⠀⠠⠀⢲⣒⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠚⡀⠎⠀⠄⠀⠨⡉⠁⠁⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠠⠂⠀⠀⠀⠄⢉⠕⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢾⢑⡲⣴⠆⣤⡀⢄⣴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⢢⡢⡀⡤⠀⠀⣉⠓⢀⠄⠁⠙⡁⠂⢙⣓⠷⠦⣀⠐⠢⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡐⢪⠑⢈⠨⠓⢈⠇⡀⠀⢀⠠⣀⠑⠄⠙⡀⣄⠁⠷⠈⠈⠊⡁⡅⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠰⠀⠁⠎⠀⡜⠀⡰⠉⡦⠘⠢⠀⠐⠄⠀⠰⣄⠄⠦⡀⠀⠀⠀⣠⡕⠎⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠘⠀⠀⢳⠈⢰⠉⠢⣅⠒⣄⠀⠰⠄⠀⠂⡀⠐⠄⠂⠀⠀⠁⣠⡌⠄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⢄⠐⢄⠀⠀⣀⠐⢄⠱⣄⠀⢀⠀⠈⠀⡄⠁⢠⡀⠤⢠⠈⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣌⠣⠀⣙⠂⠈⣷⢌⡷⢼⣶⢤⡀⠘⡦⠼⢦⠄⢹⣼⣤⡄⠇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⢦⡙⣦⣌⠢⡀⠹⣷⡀⢀⠛⢕⣄⠈⢧⡀⠜⣆⠀⣇⢹⡇⠅⣽⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⢻⣿⣿⣟⡶⣶⣌⡪⡒⢅⢈⡑⢽⣮⡈⠣⢌⡃⠌⢹⣇⠇⣿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣯⡙⣾⣳⣧⣨⡍⠐⢦⣤⣀⡈⠀⠙⢿⢦⣧⢱⣿⢹⣻⠷⢹⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠹⣷⡙⣷⣆⠈⢃⣙⣻⣛⠷⡶⣴⣤⣎⡤⣾⡿⡇⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠳⣿⡙⠮⢻⣗⢤⠩⣿⣌⣩⠹⠶⢲⢦⡖⣿⠁⣷⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠓⠃⠑⠝⢧⡀⢻⣵⣿⠳⢶⣿⡶⣶⣿⣧⣿⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠂⠀⡐⠽⣶⡃⠀⠀⠠⣙⢶⣿⢻⣷⣾⣿⣿⣶⣿⡟⣿⠀⠀⠄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠸⠉⠛⣆⠀⠀⢌⠓⢝⠇⣿⠻⠋⠿⢥⠸⠱⠸⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠉⠳⣄⠳⣄⠢⡙⠆⠈⠀⠀⢸⡀⠀⠳⠀⠳⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠀⠀⠀⠐⢀⡀⡀⠀⠉⠳⡈⠳⡌⠲⣄⣃⠀⠀⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⢀⠄⠒⠈⠀⠀⠀⠠⠋⠀⠀⠀⠀⠈⠉⠢⡈⢢⡘⣄⠀⠑⠄⠀⠀⠀⠀⠀⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠔⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣦⠑⢌⡁⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⢐⡀⡄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠁⠀⠈⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠲⠦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⢷⣶⣦⣄⣀⠀⠀⠀⢠⣤⠀⠀⠀⠀⣾⢿⢷⠦⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣯⣿⡶
⠀⠀⠀⠀⠀⠀⢒⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣨⡒⣿⣷⠆⣷⣤⣀⡂⠀⢀⣆⠀⢐⣾⡉⢿⣿⣷⣿⠽⠇⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣷⡀⠀⠿⢷⣿⣿⡿⠏⠛⠋⠀⠀⣾⠑⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠉⣀⣠⣄⡀⠀⠐⠯⠶⠔⠛⠻⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠀⠀⠀⣙⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣀⠀⣳⠤⡦⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡶⠆⠚⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣢⣤⣴⣤⡀
⠀⠀⣼⣿⣿⣿⣿⣼⣶⡀⠀⠀⠀⠉⠽⠛⠛⠉⠁⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢿⣿⣿⣾⢟⣿⣿⣿⢿⣿⣿⠃
⠀⢰⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠐⠂⢐⣽⣶⣶⡶⢤⣤⠀⠀⠀⠀⠀⠀⠀⣿⢻⣻⣟⣵⡿⠟⣾⣿⣧⢹⡿⡄
⠀⢸⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣥⣿⣿⣿⣷⡄⠉⠀⠳⢄⠀⠀⠀⠀⣿⡿⢹⣧⠀⢙⣿⣿⣿⣿⢠⣿⡅
⠀⢺⣿⡿⢿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⣿⣭⡙⢻⣿⣷⣶⢀⣾⡄⠀⠀⠀⣿⣷⠻⣿⠛⢫⡍⢛⣿⣯⣻⡧⢸
⠀⠀⠉⠇⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⣿⠿⣿⣿⠿⠿⢿⣵⠄⠀⢀⣔⢿⣭⣅⠛⣿⡿⠿⢊⣻⣿⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⣰⠏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢸⡇⢠⢏⡼⢋⡠⢚⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣼⣠⣯⣾⣵⣿⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⢟⣋⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢠⣐⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣒⣈⣽⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢈⣩⣽⣿⠶⠤⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⢿⣟⣛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠛⠩⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡦⠀⠀⠀⠀⣸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⡂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠑⠒⠀⠠⣈⣿⣿⣯⣬⣵⣔⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣬⡀⠬⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⢭⣻⣿⣿⡿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⠋⠨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⠀⣤⣶⣿⣶⡟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠻⣿⡿⠉⠀⠈⣵⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠉⢡⣿⣿⣿⣛⡿⠃⠀⠀⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠈⠛⠛⠉⠻⣿⠆⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⢀⡄⠀⣷⠀⠹⠟⠁⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⡀⢀⣴⣿⢠⣾⣿⣄⣤⣄⣠⣀⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⣴⠟⢿⡿⠋⢨⡿⠛⢿⡏⠹⠿⠟⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⠀⠀⠘⠃⠀⠈⠀⠀⠂⢹⡆⠀⣾⡿⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⡈⠻⣿⣿⣶⣶⣄⣄⠀⠀⠀⣿⡇⠀⢻⣇⠀⢈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡆⢸⣿⠀⠀⠀⠉⠀⠀⠒⠛⠁⠀⣾⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢫⣿⡿⠛⣻⣿⣿⣧⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠃⠰⡿⠟⠁⠸⣿⣿⣿⣿⣧⠀⢸⣿⠀⠀⠀⠀⣰⡇⠀⠀⠀⠛⠀⠀⣹⣿⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢯⣄⠀⢠⣤⡀⢿⣿⣿⢿⣽⢿⣿⠀⠈⠛⠀⠀⠀⠀⣿⣿⠄⣠⣾⠃⠀⢾⣿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠓⠘⠿⠃⢸⡿⠁⠘⢟⠁⣼⣿⡾⠟⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠛⢁⡀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⢰⣶⣾⡿⠈⠛⣋⣱⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⣠⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠠⠈⠁⢀⣴⠶⣿⡿⠛⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⢸⣟⠛⣿⣀⣿⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⣀⡴⠀⠀⣠⣶⡟⢻⣤⡿⠓⠀⢀⣀⣀⣉⠀⠀⠀⠀⠀⢠⡼⢋⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢀⣀⠀⢸⣿⠃⠀⣼⡿⢋⣴⣿⣿⣿⠀⠐⡿⠛⠛⠁⠀⣤⠀⠀⠀⠈⠁⣸⣿⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣿⠛⣤⡈⢙⡀⠘⠛⢡⣾⣟⣿⣿⣿⠂⢸⣧⣤⡄⠀⠀⠛⠀⠀⠀⠀⢼⣿⠋⢰⣿⣿⣿⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢉⣰⡿⠃⠿⠏⢀⣴⣿⣿⣿⣿⣿⣿⠀⠘⠛⠉⠀⠀⠀⣶⡶⠆⠀⠐⠀⠀⠀⠀⠹⢿⣿⣿⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⣿⠁⣠⣷⣄⣾⣿⣿⣿⣿⣿⡿⠋⢰⣶⣤⡄⠀⠀⠀⠀⠃⠀⠀⣴⡄⢀⠀⠀⢀⣤⣿⠟⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⡁⠀⠻⢋⣿⣿⣿⣿⣿⣿⣿⠇⢴⣦⣤⣤⡀⠀⠀⠀⠀⣤⢠⡀⠈⠀⠿⠃⠀⠀⣹⠇⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄⠀⣿⡟⠀⣦⣿⣿⢿⣿⣿⣿⣿⡿⠀⣼⣿⣿⡉⠁⣀⣾⠂⠀⠁⠘⣿⠀⢰⠀⠀⣿⢀⣆⠸⠃⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣿⠀⠟⠀⠀⢿⣿⣯⣾⣿⣿⣿⣿⣧⣾⡟⢿⠟⠛⠀⠉⠙⠀⠀⠀⠀⢁⣀⠀⠀⠀⢀⣿⠃⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⣦⠈⢻⣿⣿⣿⣿⣾⣿⣿⣤⣿⣀⡀⠀⠈⠀⠀⠀⠀⢀⡼⢿⠇⠀⢰⣾⠋⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⢠⣾⠀⠀⠈⠀⣾⣿⢻⣿⡿⣿⣿⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢀⣾⠉⣤⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⢹⣿⡇⣶⡄⣰⣿⣿⡾⢋⣴⣿⣿⣿⣧⡤⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠟⠀⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣦⠈⣿⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣼⣿⡏⠛⠀⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠻⠆⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠻⠏⢛⡀⠀⠀⠈⣳⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠄⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠸⣿⡏⢠⣶⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⡠⠒⠂⠀⣆⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⣰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠻⢇⠀⢻⣿⣼⣿⡄⣿⣿⣿⣿⣿⣿⣿⡟⠀⠉⠀⠀⠀⠀⠋⢠⠄⠀⢠⡾⠁⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃⠀⣿⡇⠀⣤⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠈⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⣤⣀⢴⡀⠀⢺⣦⡌⠀⣰⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠈⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⣀⣿⣿⣎⡀⠀⢸⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⢶⣶⣾⠃⠀⠀⠀⠀⠀⠠⠎⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⣼⣇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢰⣼⣿⣿⡿⠀⠀⢀⣶⡀⠿⠁
⠀⠀⠀⠀⠀⣿⣿⣿⠟⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣁⣰⣷⣄⣠⣿⢻⣿⣯⡄⢀⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠿⠛⠁⠀⠁⢠⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⡿⠛⢋⣿⣿⠃⠰⣿⣿⣶⣿⡿⠛⢿⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠃⣿⣿⣿⣿⣿⣿⣿⣿⢧⣿⠏⠁⠀⣼⣿⣿⡄⠀⠈⠻⢿⣯⣀⣤⣿⣄⠀⠈⠁⠐⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣼⣿⣿⣿⣿⣿⣿⣿⢿⡿⠁⢠⣤⣀⡿⠟⠉⣀⣴⣦⣤⣤⣤⣼⣿⣿⣿⠿⠶⣤⣤⣤⣤⣤⣤⣤⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⡇⠘⣇⠰⣼⠋⢠⣷⣤⣾⣿⠏⠀⠀⢀⡀⠀⡀⠈⣷⠀⠀⢸⡇⢀⡀⠀⣤⡄⢠⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁⢀⡿⠀⠙⠀⠘⠋⢠⣿⣿⠀⠀⠀⢸⡇⠀⣿⡆⣿⠀⠓⠘⠁⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣠⡾⢁⣴⡄⠀⠀⠀⠸⣿⣿⡀⢠⠀⢀⡇⠀⣿⣧⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠉⣿⣿⣥⣾⣿⠇⠀⠀⠀⠀⢿⣿⡇⠘⣇⢸⡇⠀⢸⣿⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⠀⠀⠀⠆⠀⢸⣿⡇⠀⠿⣿⡇⠀⢸⣿⠀⢀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⢀⣾⣿⣿⣿⣿⣿⠀⠀⠀⣀⠀⠈⣿⡆⠀⠀⣿⡇⠀⢸⣿⣇⡿⠀⠀⠀⠀⣿⠀⠀⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁⠀⠀⢿⣇⠀⠀⣿⠃⠀⢸⣿⣿⡇⠀⠀⠀⠀⢻⠀⢠⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⣼⣿⡟⠁⣸⣿⣿⠁⠀⠀⢀⣷⡄⠀⢸⣿⠀⠈⠇⠀⠀⣸⡏⣿⠁⠀⠀⠀⠀⠘⠀⠈⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⣴⣿⡿⠀⠈⣿⢿⠃⠀⠀⢀⣾⠏⠀⢀⣿⣿⠀⠀⠀⠀⠀⣿⠁⣿⠀⠀⠀⠀⠀⠐⠀⢰⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⢠⣿⣿⠇⠀⣶⠙⠀⠀⠀⠀⣾⣿⠀⢀⣼⠿⣿⠄⠀⠀⠀⠀⢻⠀⠏⠀⡆⠀⠀⠀⠀⠀⠸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⡏⠀⣾⠛⠆⠀⠀⠀⠸⣿⣿⣦⣸⣿⠀⣿⠀⠀⡇⠀⠀⠆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⣿⡿⠀⣼⠻⡆⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⣿⠁⣠⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢆⣿⡇⣠⠉⠀⠀⠀⠀⠀⠸⠀⠀⢠⣼⡇⠀⠄⠘⠿⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⢳⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠠⠞⡰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠀⡶⢀⡶⠀⠖⠀⠀⠄⠀⠀⠀⠀⠁⠉⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡶⠃⢸⠏⠀⢹⠁⠀⠃⢠⢄⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⢉⡀⢠⡤⠀⠴⠀⠀⠒⠀⠁⠀⠀⠀⢰⣆⣶⡀⣄⠠⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠐⠋⣀⡈⢠⣤⠀⠰⠖⠀⠠⠀⠀⠀⠀⠀⠀⠘⠮⢻⡽⡾⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⠀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡅⢠⠸⠣⠔⡄⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣄⡀⠠⠀⠀⠀⠄⢀⡰⡜⡆⢹⠷⠾⠖⠰⠀⢠⠀⠀⠀⠀⠀⠉⡉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠀⣁⣤⠞⠊⠑⠇⠲⠰⠰⠀⠄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠰⠋⠁⠀⠀⠀⠐⠀⡄⠀⠄⠀⠀⠀⠈⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣡⠡⡊⠀⠆⢀⠰⠀⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⡀⠀⠂⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣾⣿⠛⡔⠤⠀⢀⠂⠀⠀⠀⠀⡄⠀⠂⠀⠄⢀⢀⣤⣄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢾⣿⣿⠿⠛⠑⠉⠀⠀⠀⠁⠀⠀⠀⠀⢚⠀⠤⠀⠀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⠁⠀⠀⠀⠀⢀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣄⡽⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡂⠂⠈⠀⠀⠀⠚⠁⢀⠲⠙⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣿⠟⠀⠀⠀⠀⠈⠀⠀⡴⣣⣦⣼⣷⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣡⢉⡄⠀⠀⠀⠈⠀⠀⢀⣴⣿⢿⣇⡿⢿⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⠁⣿⠁⣠⠆⠀⠀⠀⠀⠚⠛⣃⣼⡋⢁⣤⠇⠀⠂⠁⠀⡀⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣴⡇⣀⠈⣰⡏⣀⠀⠀⣠⠀⣠⣿⣿⢋⡟⠀⣠⡄⠀⠀⠀⠀⠉⠀⠀⠈⠃⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠹⢿⣃⡖⠈⠃⠙⡠⠞⠿⠀⠋⢹⣥⣿⡧⠈⣽⡇⠀⠀⠀⠀⠀⠀⢀⣶⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣠⢟⣴⠀⠀⢨⠁⠀⣀⣤⠎⠀⣀⠀⣠⣾⣿⢟⣀⡤⢛⣵⡷⠀⠀⠀⠀⠀⠶⠺⠁⠀⠀⠀⠀⠐⠛⠂⠆⠀⠐
⠀⠀⠀⠀⠀⠈⠁⠿⠋⠀⢀⢻⡤⠜⠛⠻⣤⢸⣿⠋⠉⠈⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢿⠀⣀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⡿⡃⢰⣾⠇⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈⠘⠁
⠀⠠
⠀⠐

⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⣀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢾⡈⠣⠆⠉⠀⠃⠀⠀⠀⠀⠈⢀⣄⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠪⠼⠇
⠀⠀⠀⠀⢀⠰⠚⠅⠀⣀⠀⠴⡃⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠑⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡀⠠⠐⠁⠀⢀⣛⣒⣒
⠀⠀⠀⢠⡋⠈⠡⡀⠀⠀⡙⠀⢃⠀⠀⠀⡀⠤⠂⠀⠐⠒⠒⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠈⠇⠀⠀⠈⠀⠀⠄⣀⠉⠉⠠
⠀⠀⢀⠜⠃⢀⡀⠴⡟⠀⠄⣤⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠅⠂⠀⠠⡀⠂⠀⠀⠀⠀⡂⠀⠀⠉⠁⢠
⠀⠀⠈⢀⠀⢀⡤⢿⡀⠀⠀⡀⠄⠀⠀⠀⠀⢂⠐⠂⠂⠈⠁⠀⠀⠀⠈⠠⡈⠀⠀⠄⢀⠀⠀⠠⠤⡀⠀⠀⠀⠠⠁⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠆⠠⠂⠄⠀⣬⡉⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠪⠤⠤⡄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢈⡀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠃⢇⡠⢾⠆⡸⣆⣡⢾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣮⡁⠀⠀⠀⠀⠀⠀⠘⠀⢤⣀⢰⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠤⡀⠁
⠘⠷⠶⠆⠋⣰⠇⠁⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⢀⠀⡀⠠⢁⡀⠉
⠀⠀⠀⠀⠈⠓⠰⡄⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢂⣅⠀⠀⠀⠀⠧⡀⠀⡘⠀⠐⡀⢁⠀⠀⢀⠑⠄⠁
⠀⠀⠀⠸⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠄⠀⠂⠀⠀⠁⠀⠀⢀⡐⠂⠀⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⠤⠶⠂⠀⢈
⠀⠀⠀⠀⠸⡆⢀⠀⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠘⠗⠂⠠⠌⠦⠉⠳⢬⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣁
⢀⡠⠤⠴⠶⣤⠤⠄⠀⠀⠤⣌⠳⣤⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⠉⡛⠲⠖⠤⠀⠀⢠⠄⣠⡆⠃⢸⡀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠐⠛⣡⢤⠴⠆⠀⠐⠒⠒⢣⣄⠀⠸⣇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠾⠃
⠤⠬⢤⡼⠐⠂⠁⠀⠀⠀⠈⠻⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠀⢒⣐⣚⣓⠚
⠀⠰⠇⠠⠂⠈⠠⡀⠀⠔⢀⠀⠈⡷⠆⠈⢳⡄⠘⣄⠀⣠⠀⠐⠂⠀⠀⠀⠀⢀⣀⠤⠀⠀⢩⠀⠀⠀⠿⣤⡁⠉⠉
⠀⠀⠀⠉⢋⡄⢀⠐⠃⠀⠁⠀⠀⠘⡖⣠⠀⠈⠀⢸⢳⡏⠀⣠⡴⠤⠂⢤⡖⠋⠁⢀⠀⠀⠋⠐⣌⠼⢷⡀⠘⠓⠀⠀⠀⠀⠀⣴⠲⡴⠤⣠
⠀⠀⠀⠀⠀⠙⢯⣀⠀⠁⠀⠀⠀⠀⠀⠓⠂⠀⣧⠈⠀⡄⠀⣈⢰⠆⠴⡆⠈⢱⠛⠉⠀⠀⠐⣄⡈⠆⠀⠛⠀⠀⠀⠀⠀⠀⡐⢾⣨⣿⣷⢠⡖⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⠀⠀⣀⠀⢀⡄⠀⠀⠿⣦⠄⠁⠀⡉⠀⠚⠀⢠⣀⡀⠀⢠⡄⠙⠄⢨⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡡⣄⣛⣁⣰⠷⢣⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡤⠎⢀⠎⢈⡳⠔⣶⡇⢰⢀⠀⠀⢌⢛⠀⠆⠀⠀⠘⠴⠞⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠄⢉⡅⠠⡟⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⠏⠃⠀⠈⢉⠹⠷⣌⢠⣾⠀⢳⠈⡂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣴⣌⠰⢀⠶⢌⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⢲⣉⡠⠀⠀⠀⡀⠀⠙⠂⠐⢆⡀⣱⡏⣰⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⣃⣭⣥⣨⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⠟⠀⠠⠀⠈⠀⡆⠀⠀⠀⠀⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢠⡔⢢⡍⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣺⠁⠀⠀⠀⢰⠇⠀⠀⠀⡀⠼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢲⡁⡾⠂⠀⣀⠀⠀⣤⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣘⣉⣐⣘⣷⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠠⠆⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣖⡰⢆⣰⠀⠖⠀⠀⠀⠀⠀⠈⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠰⠀⢘⠁⠸⡁⠨⠃⠈⠁⡴⢰
⠀⠀⠀⠀⠀⠀⠠⠆⠐⠂⡘⠃⣈⠁⠀⠁⠀⠀⠀⠀⢳⣾⢣⡧⠔⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣐⡂⠀⠘⠤⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⢀⡠⣖⡙⠨⠛⠛⠒⠂⠀⠀⠀⠀⠀⠉⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⣠⠶⠋⠁⠈⠁⠁⠉⠈⠁⠐⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣄⣃⠄⢀⠀⠀⠄⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣈⡈⠁⠀⠀⠠⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡋⠔⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⡤⢤⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⡿⠟⠛⠁⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡀⠈⠀⠀⠐⠁⠠⠀⡉⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠼⠋⠀⠀⠀⠀⠀⢀⣯⣿⢾⣿⠀⠀⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣱⢃⡞⠀⡀⠀⠀⠀⠰⠿⢏⡿⠟⢙⡁⠀⠐⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⢀⡔⣡⡏⡈⢁⣾⢁⠀⢀⡀⢀⣶⡿⢻⠂⣉⡄⠀⠀⠠⠓⠀⠉⠢⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡉⠀⠙⢿⡡⠂⠉⠙⠰⠺⠇⠉⣩⣵⢿⠀⣽⡇⠀⠀⠀⠀⠀⣰⠆⠀⠀⠈
⠀⠀⠀⠀⠀⠞⣵⡄⠀⣠⠀⣠⣶⠃⢠⣤⢤⠾⣿⣧⣴⠨⠿⠃⠀⠀⠀⠀⠋⠋⠀⠀⢀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠐⣼⠋⠁⠈⠋⠙⠃⢀⡤⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⢰⡗⢀⣴⡆⠰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈⠁⠋


⣿⠿⠿⠿⠿⢟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⣠⣾⡿⢾⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢀⣤⣾⣿⣼⣯⣧⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣟⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣹⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⢿⣷⣤⡀⠀⡀
⠀⡀⠀⠀⠀⠰⠀⠀⡆⠀⣀⠀⢀⠀⢀⠀⢀⠘⣿⣯⠹⣿⣿⢁⢀⡄⠀⠀⠀⢀⢀⣠⢸⡇⠀⣠⠀⠌
⠀⢸⣀⠰⡆⢠⣇⣸⢀⣼⣯⡷⣾⣶⣼⣇⣠⣾⣿⣿⣧⣼⣿⣮⣼⣷⣾⣿⣠⣾⣿⣿⣼⣿⣿⣿⣀⣀⣼⣧
⠀⢸⡇⢳⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣤⣄⣰⠄
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡸⢿⡿⢛⡛⣛⠿⢿⣿⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⡇⠀⢿⣿⣿⢿⡖⢢⣉⠛⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣾⣷⣤⣼⣿⣿⣶⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠉⢻⠛⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡟⠛⠓⠠⢬⣽⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣿⠟⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢶⡈⠻⢿⣿⡟⠀⠀⠀⠀⠀⠸⣿⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⢠⡌⠛⠀⢈⣿⣿⣶⣄⣴⡦⢠⣾⡧⠀⠀⢀⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢿⣿⢷⡄⢻⠟⢿⣿⣿⣋⠁⠀⠉⠁⠀⠀⠈⢿⣿⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠇⠀⢻⠀⠁⠀⠀⠀⠙⠉⠙⠷⣿⣦⡄⠀⠀⠀⠀⠹⣷⡌⠻⣦⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠿⣏⠀⠀⢀⡀⠀⠈⠛⠀⠈⠻⢄⠀⠀⠀⢠⠀⢀⠃
⠀⠀⠀⠀⠀⠠⠗⢀⡀⠀⠀⠀⠀⢸⡇⠀⢀⣶⣄⠀⠀⢸⣧⠀⠀⠘⣧⠀⡀⠈⠀⠀⠘⣿⠀⣀⠀⠀⠀⠀⠀⠐⡄⣰⡟⠰⠇⠀⠁
⠀⠀⠀⠀⠀⣀⠀⠈⠉⠀⠀⠀⠀⢸⣷⠀⡼⣿⣿⠀⠀⠀⠛⢷⠀⠀⠈⠀⣷⠀⠀⠀⠀⠈⠀⢻⠀⢠⠀⢠⡄⠀⢺⣿⣷⠤⠔⠂
⠀⠀⠀⠀⠀⣛⠁⠘⠃⠀⣀⡀⠀⢸⣿⣿⠁⠈⠁⠀⠀⢀⡀⠀⠀⢠⡄⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠈⡟⠀⣼⣿⣇⣠⡆⠀⡀
⠀⠀⠀⠀⠘⡋⠀⡖⠂⠈⠛⣡⣤⣾⣿⠃⠀⠀⠀⣷⡀⠘⣷⡀⠀⠘⣿⠘⡇⠀⠀⣶⣄⠀⠀⠀⠀⠸⣿⠀⠀⠰⣿⣿⣿⠀⠘⠒⢿⡆
⠀⠀⠀⠀⠀⣀⣴⠃⢰⣤⡼⠋⠉⢹⡟⠀⢠⠀⠀⠈⠻⣿⣿⣿⡄⠀⠘⡇⣿⡀⠀⢸⣿⠀⠀⣰⣾⡇⠀⠀⠀⠀⣿⣿⣿⣾⠀⢠⡄
⠀⠀⠀⠀⠀⡀⠁⢠⣶⡀⢡⣄⣠⣼⣷⠀⠘⢧⣴⡄⠀⠹⡿⣿⣿⡀⠀⢹⣿⠁⠀⢸⠃⠀⢰⡿⠁⠀⠀⠀⠀⢀⣹⣿⣿⠉⠁⠘⠋⠘
⠀⠀⠀⠀⠀⢀⣤⡈⠁⠛⠀⠉⢹⣿⣯⠀⠀⢺⣿⠇⠀⡀⠀⠻⣿⣣⠀⢻⣿⠀⠀⠈⠀⣠⠈⠀⠀⠀⠀⠀⠀⣸⣹⣿⣿⡆⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠖⠀⣨⣿⣷⠀⣤⠈⢿⠀⠀⢻⣤⠀⠙⠃⠀⠛⠃⠀⡀⢰⡄⢻⡀⠀⢠⣦⣤⠀⢸⣿⣿⣿⣿⣿⣆⣠⣄
⠀⠀⠀⠀⠀⠀⣾⢶⣶⡶⢤⣴⣿⣿⣿⠀⠸⣷⡀⠁⠀⠀⣿⣿⣆⠀⣀⠀⠀⢰⣿⡟⠀⢸⣿⡿⢿⣿⣿⠁⢸⣿⡿⣿⣿⣿⠉⠉⠛⠂
⠀⠀⠀⠀⠀⠀⣧⠀⢀⣠⣤⣀⣽⣯⣿⡄⠀⠘⠿⣦⣀⠀⠈⠹⣿⢹⣿⣧⠀⣿⠻⠇⢰⣦⣿⠁⣼⣿⡏⠀⠘⠁⠀⣿⣿⣿⣆⠠⣦⣀
⠀⠀⠀⠀⠀⠀⠃⣰⠿⠏⠈⣿⠉⢻⣿⣇⢠⣆⠀⠈⢻⣿⡄⠀⠀⠀⠘⣿⡔⠁⠀⠀⣾⣿⡇⣸⠟⢉⣴⡄⠀⠀⠀⠙⢻⣿⣿⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠻⠻⠆⣀⠄⠘⠷⣾⣿⣿⠺⣿⣄⠀⠀⠟⠈⠀⠈⣶⡀⣦⠀⠀⠀⢸⣿⠏⢠⠏⢀⣾⣿⣥⠀⠀⠄⠀⢸⣿⠉⠻⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⣠⣴⣄⡀⣿⣿⣆⠘⢿⣷⡀⠀⠀⢀⢀⣼⣷⣹⣇⣰⣶⣿⠋⠀⠀⠀⠘⣻⣿⠃⢸⡇⠀⢸⣿⣿⣧⣤⠼⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠙⣿⣿⣿⣿⣧⣄⠈⠁⠀⠀⠀⠻⣇⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠰⠟⠁⢀⣾⣿⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠿⢿⣿⣿⡿⢿⣷⣄⠀⠀⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣾⣿⠃⠀⢸⣿⣿⣷⣀⣘⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈⢿⣿⣇⠀⠉⠛⠓⠦⡄⠀⠀⣤⡿⣿⣿⣿⣿⣥⡿⠀⠀⠀⠀⠈⣿⠇⠀⠀⣸⣿⣿⡟⠙⠿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⣿⣿⠉⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠁⠀⣾⣀⣿⣿⣿⣧⣄⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⡍⠻⣿⣶⣄⠀⢻⣿⣿⡏⢻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡙⠋⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢸⣿⣿⡄⠈⢿⡟⠀⢠⣿⣿⣇⠈⢿⣿⣿⣿⣧⡄⣤⠀⠀⠀⢘⣿⣿⣿⣿⡿⢷⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡀⠁⠀⣿⣿⣿⣾⠀⣼⣿⣿⣿⣿⣿⠃⠀⠀⣠⣾⡿⠿⣿⣿⠃⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀⠀⣼⣿⣽⣿⣧⣿⣿⣿⣿⣿⡟⠀⣠⣾⠋⢿⣡⣶⣼⣿⡃⠾⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢰⡿⢃⢤⣞⠉⠙⠋⠙⢳⣤⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢹⣿⣿⡿⠀⠙⣿⣿⠟⠁⠀⠈⣧⣀⣤⡟⠀⠀⠀⠀⠂⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢿⡀⠙⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣶⣬⣦⣶⠀⢀⣤⣴⣶⣷⠀⠀⢠⣶⣿⣶⣶⣶⣤⣤⣤⣤⣤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠿⠿⠿⠃⠿⠷⠟⠉⠀⣀⡅⠀⢠⣿⠟⠀⠉⠉⠉⠉⠙⠛⠛⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣸⡷⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⣠⣤⣤⡄
⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡿⠀⠀⣹⡟⠀⠀⠀⣀⣀⣶⣿⣿⣿⣴⡟⠛⠉
⠀⠈⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠂⠀⠀⠀⠀⠀⣦⡀⣽⡿⠀⠀⠀⠙⠉⢀⣌⠿⢃⡄⢰⣦
⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣄⣉⣁⣉⣙⣿⣋⠁⠀⠠⠴⣿⠷⠿⠓⠀⠂⠀⠀⠀⠸⣿⣶⡟⠃⠸⣧⣦
⠀⠀⠿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⣀⣤⣶⣷⣾⣿⣾⡇⠈⣿⣿⢀⣼⠋⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢸⣿⡿⠿⣿⠛⠛⠻⠛⠋⠙⠛⠸⠃⠀⣿⠃⠈⠻⠧⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⣀⣤⣤⣤⣴⡄⠀⠀⠀⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣿⣿⣿⣿⠿⠿⠿⠿⠛⠿⠟⠿⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣤⣤⣤⣾⣷⣶⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠈
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⢀⣀⡀⠀⣤⣴⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⡿⠿⠿⠟⠸⠿⠛⠋⠀⠛⠛
⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⢀⡀⠀⣆⢀⣤⣄⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠃⠀⣿⣯⠉⠉⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢩⠷⣤

⠀⠀⠀⠀⠀⠀⠀⠀⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠐⠶⠈⠳⠘⡄
⠀⠀⠀⠀⠀⢠⣙⠂⠀⡰⠋⢉⠓⠀⠀⠀⠀⠀⠠⠀⡤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆⠰
⠀⠀⠀⠀⠀⠡⢊⣀⠐⣇⠀⠂⠩⠑⠒⢂⡠⡦⣆⣀⠀⠘⡄⠐⠖⢰⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢈⡩⠔⠀⠈⢦⡀⠀⠀⢀⠀⣴⡕⠁⠀⠀⣠⠀⠀⡀⠁⠀⠀⠀⠀⠀⢸⢃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈⢏⡅⢀⡄⠀⠉⠲⠰⢾⣾⣽⠃⠀⠐⠰⢌⣦⡀⠁⠀⢀⠀⠀⠀⠀⠀⠈⡄⠀⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠱⡝⡄⠀⠀⠀⠀⠀⠹⠛⣶⠶⠘⠠⢆⣍⡓⠐⠄⠃⠀⠀⠀⠀⠀⠀⠈⠂⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠦⠄⠤⠁⠙⠁⠸⠃⡀⠌⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠫⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⢁⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠎⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠔⠂⠀⠠⠀⠀⠀⠀⢀⠊⠁
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⢔⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠤⠆⠀⠁⠠⠂⠀⠁⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠁⠀⡀⠀⠀⠀⠀⠀⠀⠈⢄⠄⠀⠀⠀⠺⡀⢄⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠒⠀⠀⠒⠀⠀⠁⠀⠀⠀⢠⠔⠲⠆⠀⠀⠈⠨⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⡜⢠⢄⠀⢀⠳
⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣁⣀⣀⣰⣀⣀⣐⣀⣑⣁⣀⣁⣀⣀⣀⣐⣂⣙⣀⣊⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠱⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣉⡙⠓⢄⠀⠀⠀⠀⠀⢀⣀⣲⣶⣤⣤⣤⣤⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⢀⣀⣀⡀⠀⠀⠀⠈⢉
⠷⠄⠐⠂⠉⠙⠛⠻⢿⣿⣿⣿⠋⢁⠀⢉⣻⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠿⠟⠛⠻⢿⢠⣤⣈⠓⠀⠀⠀⠀⠈⠙
⠓⠒⠒⠒⠒⠀⠀⠀⠀⢹⡇⢿⠀⠹⣿⣿⣿⡿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⣿⣿⣦⣤⠀⢸⢸⣿⠿⢿⣶⣶⣶⠿⠀⢀
⡤⠚⠛⠁⠀⠀⠀⠀⠀⠈⠑⠈⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⢀⡾⠟⠀⠀⠀⠔⢚⣷⣿⣿
⣭⠭⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⣻⡗⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡾
⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶
⣠⡦⠶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⠀⣪
⢁⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢮
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠠⠤⠀⠀⠀⠀⠀⠁⡾⠂⠀⠀⠀⠀⠀⠀⠀⠐
⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣄⠠⢄
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡙⢶⣄
⣠⣶⢖⣤⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡂⠀⠓⠀⠀⠀⠀⠀⠀⠀⠑⢆⠠⠄⠀⠀⠀⠀⠀⢴⣤⠀⢷⡿⣿⣷⣝
⡿⢱⣿⣿⠟⠀⣠⠿⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠀⠀⢸⣶⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠢⢿⡻⣟⣿
⣷⣿⢿⣋⣴⣪⡥⠀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⣶⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣶⣄⠻⣿⣽
⢾⣿⣿⡿⢫⡿⠷⠛⠁⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢿⡏⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢦⡀⠀⠀⠀⠠⠤⣄⠈⢽⠠⣘⣿
⢟⣽⡿⣡⣷⣷⢆⠀⡠⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠙⢶⣜⣿
⣼⡟⣾⣛⣭⡷⠋⠊⠀⡀⠀⢀⠂⢀⣴⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣹⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡀
⠀⠀⠀⢄⡀⠀⠀⠀⠹⣦⣀⠀⠀⢷⡄
⠀⠀⡀⠀⠙⢶⣤⣀⡀⠈⠻⢷⣄⡈⢻⣦⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈⠳⣤⣀⠉⠛⠿⣿⣶⣤⣙⠻⣦⣝⢷⣄⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡆
⠀⠀⠀⠀⢨⣙⣻⣷⣶⣦⣭⣛⣿⣿⣶⣽⣷⣿⣷⣌⢦⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣼⡇⣸⡇⠀⡄
⠀⠀⢀⠠⣄⣈⡙⢿⣿⣿⣿⣿⣿⣟⡿⢿⣿⣿⣿⣿⣷⣶⣼⣗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⢣⣿⢠⣿⠁⡼⠁
⠀⠀⠀⠈⠙⠻⠯⢭⣿⣿⣿⣿⡓⠒⠠⠦⣀⡉⠙⠛⣻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⣿⣴⣿⣾⣿⣿⣏⣾⠃
⠀⠀⠀⠀⠀⠐⠾⠿⢿⠿⣿⣿⣷⣶⣶⣶⣾⠃⡀⢿⣿⣿⣿⣿⣿⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡠⣿⣦⡀⣿⢿⣿⣿⣿⣿⡟⣡⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢿⣿⣿⣿⣿⣿⣿⣙⣿⣿⣿⠁⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣛⣳⣿⣿⣿⣿⠊⣨⣼⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠄⠰⢿⣿⣿⣿⡿⢛⣩⣥⣽⣿⣿⡓⠘⣿⣿⣦⡀⠀⠀⠀⠀⣀⣰⣶⣿⣯⡽⠿⠿⠿⠟⣍⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣯⣟⣡⡴⢿⣿⣷⡄⠈⢿⣿⣷⣄⠀⢶⣿⣯⣭⣴⣂⣴⣶⣾⣷⣿⣿⣿⠿⠟⠻⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣶⣾⣿⣿⣿⣧⣤⠙⠿⠛⠿⠿⠿⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢭⣟⠛⣶⠈⠉⠀⠀⢀⠐⠶⢤⣄⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢽⣿⡿⣿⣿⣿⣿⣿⣿⣿⢦⣴⠗⠭⣉⣭⠍⡳⠂⠀⠸⣦⣤⣶⡍⢩⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⠟⢛⠋⠬⢭⣀⠘⡻⣻⣿⣿⣿⣶⢞⡿⠋⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⣀⣴⠂⠀⠀⠀⠀⠘⣯⣾⣼⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣐⠲⢦⣤⣤⣤⣤⣤⣤⣤⣴⡶⢟⣿⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⠀⢠⣤⣴⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⣿⢿⣿⣿⣿⣿⣡⣿⡿⠋⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠏⠹⣿⣿⣿⠁⢀⣤⣶⡾⠋⢀⣿⣿⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠈⠤⡇⠙⠋⠀⠀⢸⣿⢋⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⣤⣄⠀⠈⢿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⡾⠀⠀⠀⠈⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠞⠘⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡷⠃

⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⣈⣑⣶⣾⣿⣶⣶⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣿⣿⣶⣒⣋⡥
⠀⠐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣠⣤⣤⣄⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡓⠂
⠀⠀⠀⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣴⣾⣿⣿⣿⣿⣷⣬⡉⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠛⠋⠉
⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⣽⣾⣮⣧⡀
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣙⠿⠿⣿⣿⣿⣿⣶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⡘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⣿⣿⣷⡊

⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠰⣄⡙⠦⣌⣢⣤⣀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⠠⠊⣀⠔
⠀⠀⠀⠀⠀⠀⠐⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣦⣤⣺⣧⣠⣶⣀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⡟⣻⣿⣾⣿⡶⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢟⣽⣿⣿⣿⣿⣿⣿⣿⣛⣿⣟⡛⠿⠛⢛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⣄⣼⣿⣿⡟⢻⣿⣿⣼⡿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⡽⣾⣿⣿⣧⣿⣿⣻⣿⣿⣍⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢫⡿⣿⣿⣿⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠛⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠉⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣴⣦⣄⡀⠀⠠⠏⡠⢤⢴⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⠂⣿⣿⠟⠁⠀⣠⠋⠀⠘⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠄⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⣰⠀⠀⠀⠀⠀⢰⠀⠉⢂⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠈⠑⠒⠺⠌⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠁⠂⠐⠒⢠⢆⡀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡤⠛⠐⢇⡴⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠋⠑⠗⢢⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⠋⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠰⠄⠰⢎⡀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⢠⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠒⠒⠃⠀⠤⠔⠀⠀⠀⠀⠠⣀⡠⠄⣀⠤⠀⢀⡞⠀⠀⠨⣄⠀⡄⠠⠃⠈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡼⢁⣠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣥⠤⠄⣄⢂⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢀⠔⠀⠀⠀⠀⠀⠀⡤⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣀⠄⠊⠀⠀⠀⠁⣀⡻⣆⣀⡶⠁⢀⠠⣀⢌⣂⠘⠧⠴⠒⠁⠀⠀⠀⠀⣀⡠⠀⡴⠁
⠀⠀⠀⠀⣀⠤⠒⣉⡥⠂⠹⠬⠉⢉⣀⠠⠭⢉⣐⠠⠍⢘⠡⠟⠓⠉⠉⠁⠀⠀⠀⠀⠀⣄⠀⠌⠀⢘⢟⡤⠊
⠀⠀⣔⠫⠀⣤⡊⠣⠀⠁⠀⢀⣀⡀⠀⢩⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠎⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠩⠋⠉⠁⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣳⠾⠓⠈⣙⣷⢦
⠀⠀⠀⠀⠀⠀⢀⠔⢀⡤⣠⢔⣿⠧⠴⢶⢉⡀⠀⠀⠀⣠⣢⢔⡠⣂⡠⡔⠙⠉⢁⠃⢋⡽⠖⢀⠮⢅⡈⠙⠣⡀
⠀⠀⠀⠀⠀⣠⠵⣋⠕⢈⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠚⠛⠛⠉⠁⠁⠀⠱⠦⡄⠞⠚⠁⠀⢇⠘⣲⠟⡩⠓⢄⢈⣿⢻
⠀⠀⢀⠤⢊⠕⢊⡠⠂⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠈⠓⠋
⢀⢔⡥⢊⣡⠖⠁⢀⠴⠊⠁
⠳⠅⣴⠫⣀⠥⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠙⠻⢿⣷⣿⣿⣿⣿⣧⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠙⠿⣿⠟⠉⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠐⠐⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡌⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣟⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⡁⠠⢀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⣀⣤⣤⣤⣄⡉⠉⠙⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢦⡀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡙⠻⢿⣿⣯⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠁⠶⠦⣄⢀⣀⡀⡀⠀⠈⣉⣽⠿⠿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠉⠛⠻⢿⣿⣿⣿⣿⣷⣦⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠷⣄⠀⠠⠈⠫⠀⠀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠄⣠⣤⣾⣿⣏⠀⠀⠙⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⠛⠉⠻⠛⠦⠀⠀⠀⠀⠀⢠⡿⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠏⠀⠀⠀⢳⣄⠀⠀⠀⠀⢀⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠈⢀⣀⣀⣾⣁⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡶⣿⣿⡿⠟⠛⣿⣄⠀⠀⠀⠀⠀⠀⢠⠐⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⣿⡏⠀⠀⠊⠹⣿⣷⣄⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⣀⠀⠀⠘⠀⣠⠾⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢿⡿⠀⢢⡄⠰⣔⠻⢿⣷⡀⠀⣶⣿⣷⡗⣤⡀⢸⡟⠁⠀⠀⠀⠉⠀⣄⣂⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣷⣤⠀⠀⣀⡈⣃⠀⠀⠁⢰⢆⠈⢿⣿⣿⡄⠀⢾⡟⠛⠛⠕⣻⠷⠶⠶⢀⠀⠒⠒⠛⠉⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠈⠻⣿⣿⣀⣴⣿⣻⣭⣿⣿⣧⡀⢫⡀⢎⠻⣿⣷⡀⠀⠀⠉⠀⠉⠀⠀⠤⣄⠼⠇⠐⠶⠶⠤⡤⠤⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠙⣼⣿⣿⣿⣿⣯⣿⣿⣿⠎⡇⠈⡀⠙⢿⣷⡄⠀⠀⠀⣤⣤⣤⡤⣀⣴⣾⡟⠛⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠹⠀⠘⢿⣿⠀⠀⠾⠋⠁⠠⠾⠿⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠿⠿⠛⠛⠛⠉⢁⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠂⠀⠀⢀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠊⡉⠀⡤⠀⠀⠀⣲⣦⣴⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠤⢁⠀⠈⢠⣬⠀⠈⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡶⠖⠄⣀⡐⠁⠀⠁⡖⢡⣎⢄⡰⠃⠀⠂⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣡⡶⣷⡞⠁⠉⠡⣮⢔⡫⣕⡠⣻⠟⠁⠀⠀⠈⠀⠂⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠙⢋⣀⣤⣴⣿⣷⣾⣭⡵⠋⠀⠀⠀⠀⠀⠀⢄⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡡⠶⠀⠳⠬⣿⣿⣯⣿⣿⣿⡏⠉⡀⠀⠀⠀⠀⠀⢐⣵⠛⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⢿⣶⣶⠿⠶⠾⡶⠒⠒⠈⠉⠉⠀⠀⠀⠀⠀⢀⠐⡶⠛⠀⠀⡠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠁⠤⠞⠍⠿⠠⣷⠾⡄⡀⠀⠠⠀⠀⠀⠀⠀⠀⡠⠞⠛⠛⠛⣦⡴⢊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡀⠀⠀⠀⠀⠀⠀⣼⣯⣵⣿⠃
⠀⠀⠀⠀⠀⠀⣀⠔⠮⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⢀⠠⣪⠾⠋⠉⠃
⠀⠀⠀⣀⢔⡪⢖⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣦⣶⠋⠁
⢀⣴⣿⢟⠟⣨⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣶⠿⣿⠿⠟⠉
⠈⠻⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣯⡙⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⣾⠟⠉⠉⠀⠸⢿⣿⣿⣦⠀⣀⠀⠀⠀⠀⠄⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠻⣆⠀⠀⠀⠂⣸⠟⣩⣿⣿⡇⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣹⣿⣿⠿⣿⠷⠆⠁⣿⣿⣷⣈⡃⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠃⢀⣹⡟⠃⢀⢰⣏⣿⣿⡿⠇⠀⠀⢸⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡹⠀⠂⣠⢻⣴⡟⠀⠀⠈⢱⣿⣿⠇⠆⠀⢀⣾⡎⢩⡉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣿⣿⢧⣾⣽⣾⣿⠇⠀⢀⠀⢘⣿⣿⣦⣤⣦⠙⠊⠀⡀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⣾⣧⣽⠿⠁⠀⠀⠁⣼⣿⣿⠟⢻⣾⠇⠀⠀⠲⠋⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢿⠿⣿⣽⠏⠁⠀⠀⠀⢀⣼⡟⠁⠉⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤
⠀⠀⠀⠀⠀⠀⢀⣰⣵⣿⣭⣷⣿⡾⠋⠁⠀⠀⢀⣶⡾⠛⠁⠀⠀⠠⣂⠀⠞⡋⢃⠘⠁⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣏⠥⠂⠀⣀⣠⡿⠛⠉⠀⠀⢀⣴⢶⠛⠋⠀⠀⠀⠀⢑⠂
⠀⠀⠀⣠⣾⣿⠿⢛⣯⣽⣿⠕⢢⣾⣿⡿⠁⠀⠐⠀⢀⡴⠟⠁⠘⣀
⠀⢠⡾⠟⠋⢡⡴⠟⠁⠰⣿⣴⠀⡙⣟⣷⣄⣀⣠⠞⠃⠀⠀⠀⠀⠹⡄⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⣟⣿⠀⠙⠉⠙⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡗⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠃⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⢀⣠⣶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣩⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠠⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠆⠀⠀⠐⠀⠀⢀⣯⣿⣿⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢹⣿⡿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢸⣯⣷⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣷⢛⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣀⠘⠁⢀⠀⠀⠀⡀
⠀⠀⠀⠀⢨⡦⠀⠉⠀⠀⣄⡆⠬⠔⣁⣀⡀
⠀⠀⠀⠀⡜⠀⠐⡷⠞⠼⢫⣤⣤⣶⣮⢾⣴⠶⣤⡀
⠀⠀⠀⠀⢣⡀⠆⠀⠀⠀⠀⠻⠟⠹⠽⣶⣿⡾⣾⣿⢦
⠀⠀⠀⠀⠀⠙⠓⣶⣦⡆⠀⠀⠀⢀⣿⣿⣻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠀⠀⠀⠈⢿⣜⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠸⣿⣮⣽⣿⣿⣿⡿⣷⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠘⣿⣿⣿⠟⠍⠀⠘⡮⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠠⠀⠘⠿⣉⠀⠀⣀⣠⠽⣮⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⡄⠰⡄⠀⠈⠛⢻⣅⢀⡀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠛⠐⠂⠐⠀⠙⢿⠮⢅⣂⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠆⠀⠀⠀⠈⢲⣶⡄⠢⡘⣿⣷⣖⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣥⣄⢀⣀⣼⣿⡿⠀⠀⠈⠃⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⢻⣿⣿⣿⣟⣻⣭⣥⠤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⣠⣤⣴⠾⠋
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠟⠉⠙⣆⣦⡙⠋⠁⠀⠀⠀⠀⣀⢀⣠⠴⠂
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠟⠈⠉⠛⠷⣶⣤⣤⣤⣤⣴⣶⣶⣿⠿⠛⠉⠀⣠⣴⠞⠋⠀⠘⣆⠀⣠⣤⣶⣿⠿⠋⠁
⠀⢀⣾⣿⣿⣿⠿⣿⠿⠋⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠀⢀⣤⡶⠟⠉⠀⠀⠀⠀⣀⣤⣾⣿⠿⠛⠁
⠀⠸⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⣠⣴⠾⠛⠁⠀⠀⠀⠀⣀⣴⣾⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣤⣤⠟⠋⠁⠀⠀⠀⢀⣠⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡇⢌⠀⠀⠀⠀⢀⣠⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⠀⠀⠈⢆⢀⣤⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠁⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠉⠀⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⢠⡄⠀⠀⠀⠈⠐⠢⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠈⢆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⡄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠐⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠉⠀⠀⠀⠀⢄⠁⠂⠀⠤⠄⠄⠠⠀⠁⠢⠀⠀⠀⠀⠐⠀⠈⢉⡹⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠈⠓⠤⢀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠄⠀⠀⠀⠦⣄⡑⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠉⠒⠠⠤⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠐⠄⠀⠄⠠⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠢⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠪⠁
⠀⠀⢀⠤⠤⠤⠤⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠋⡉⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠔
⡠⠤⠄⠐⠀⠤⠤⢄⣀⣀⡀⠈⠉⠉⠘⠃⣀⣈⡉⢉⣉⡁⠀⢀⣠⠴⠊⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡁⠔⠁
⢑⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠋⠉⠀⠀⠀⠀⢠⠔⠚⠉⠀⢀⣴⢿⠢⠤⠀⠀⠂⠀⠐⠆⠄⠖⠁⠄⠠⠔⠈
⠣⣀⠀⠈⠉⠁⠀⠀⠒⠒⠒⠀⠀⠀⠈⠉⠁⠀⠀⠀⠣⢤⡤⠴⠞⢋⣀⠨
⠀⠈⠑⠂⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⠀⠀⠀⠀⣀⣀⠤⠖⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠓⠒⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠶⠶⠖⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠢⠠⣀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠯⠉⠙⠙⠛⠙⠻⠟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣷⠆⠀⠀⣹⣿⠿⠛⣿⣿⣿⡿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠚⠋⠀⠀⠀⢠⠇⠈⠀⠀⠿⠿⠿⠆⠀⠈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢴⣤⣤⣄⡀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠈⠻⢿⣿⣿⡖⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⠟⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢲⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣦⠀⠀⠀⠀⠀⠀⠀⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡙⣶⣾⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣠⣶⣆⠀⢀⡀⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣡⣶⣿⣿⡟⢹⢻⡔⣎⢷⢽⣄⠱⣤⣀⣀⠹⠋⠁⠹⣿⣿⣀⠀⠀⠀⠀⠀⠀⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣿⢨⣷⣼⣿⣾⣾⣿⠛⠋⠀⠀⠀⠀⠀⠹⣿⣿⣿⣤⣀⡀⠀⣴⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⠞⠋⠈⠁⠈⠛⢷⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⣸⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⢠⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⣾⣿⠟⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢀⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣴⣿⣵⣿⣥⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢛⡿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣼⣻⣄⠁⢀⡩⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠋⠁⠀⢈⣿⠁⠰⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢀⠬⠁⡙⣭⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡰⠷⠆⣰⣞⣷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠂⠀⠀⣀⣼⠳⣶⡻⢮⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢠⠀⠸⣿⣿⢄⠛⣬⣙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⡐⠃⠀⢸⣿⢠⡈⢻⣬⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⣀⣠⡞⠃⠀⢠⣿⣿⣷⣼⣻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⣥⣀⣀⡀⣀⡘⣯⡄⢉⣻⡗⠠⣾⣿⣿⣝⢷⡼⠋⠀⠀⠀⠀⠀⠈⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢿⣿⢉⠘⠛⠛⠻⣿⣿⣿⣿⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣦⣠⣿⣿⡠⡀⡄⠀⠀⠈⠻⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠆⠛⠉⠉⠀⠉⠛⠻⠉⠻⣥⣡⡀⠀⠀⠀⠀⢤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⣀⡀⠤⠒⠉⣀⣤⣔⣾⣾⣷⣦⡌⠡⣄⢘⠅⠀⡄⣥⡁⠑⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠁⠈⠀⢰⣦⣴⡒⢘⡷⣺⣿⡿⠛⠉⠀⠀⠘⣿⣿⣿⣠⣼⡧⢀⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠘⡿⣇⢀⣀⣴⣤⡶⢿⢿⣿⣿⣧⣿⠞⠋⠀⠀⠀⠀⠀⠀⠨⣷⠙⠛⣳⠼⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣷⣠⣸⢾⡇⡛⡽⣿⣿⣾⣝⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣉⢿⣾⣾⣽⣾⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡷⣾⠟⠉⢣⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⢾⡿⣯⣏⣀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⢀⠈⡏⠸⠟⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⢀⣄⣴⡏⣾⣷⣴⢤⣤⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⠀⠟⠀⠋⠈⢱⡈⣱⡏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣌⢻⡷⠞⠢⠄⢀⣸⠇⢹⢇⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⣿⠛⣿⣿⡈⠃⠀⠀⠀⠀⠀⠀⠈⢈⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠛⣡⣞⣿⢿⣷⠀⢿⠀⠀⢀⣦⠀⢀⠋⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⡘⠷⢸⡟⠸⠟⢹⣿⠿⠀⠀⠀⠀⢸⡟⠘⠛⡀⢘⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠼⠛⢇⣉⣥⠆⠀⠀⠀⠀⠈⠀⢡⣾⡇⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡤⠀⣠⡴⢯⡼⠇⢀⣀⣙⠀⠀⠀⢀⡼⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⣤⠀⠿⠁⣼⢟⣾⣿⣿⠀⣜⠉⠁⢰⡆⠀⠀⣠⡿⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⣡⡟⠸⠇⣠⣾⣿⣿⣿⠀⠟⠋⠀⠠⡤⠄⠀⠉⠁⠘⢿⣿⡇⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢠⣾⣴⣿⣿⣿⣿⠃⣸⠖⠂⠀⠀⠀⠀⠰⢀⡄⠀⢶⡟⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢸⡷⢠⣼⡿⣿⣿⣿⡏⣸⣟⡛⠁⣰⠀⠈⢹⡄⠄⠀⢆⣈⠳⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠘⠁⢈⠻⣷⣿⣿⣿⣿⡟⡟⠉⠀⠈⠀⠀⠀⣤⡄⢀⣸⠇⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⣦⠀⠈⢱⡞⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠈⠉⠀⣼⡁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣏⠘⡿⣾⣤⣿⣿⣭⣾⣿⡿⠖⠀⠂⠀⠀⠀⠀⠀⠀⣰⠛⠁⠀⡜
⠀⠀⠀⠀⠀⠀⢠⣿⣹⡶⠃⠙⣿⣿⣿⣿⣿⣿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠐⠣⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣾⡏⢩⣥⢀⡀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠟⠧⠘⣿⣼⡇⣿⣿⣿⣿⣿⠉⠋⠀⠀⠛⠠⠀⣠⠋⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⢠⠀⣀⡸⠃⣰⣿⣿⣿⣿⠃⠀⠀⠀⠀⠁⢀⡄⠁⠀⠀⣰⠏
⠀⠀⠀⠀⣸⣿⣍⠀⣹⠇⣼⣿⣿⣿⣿⣿⠀⠀⠀⢀⣤⣴⠋⠀⠀⠀⠰⠋
⠀⠀⠀⢰⣿⣿⠃⠀⣿⣾⣿⣿⣿⣿⣿⡿⢀⡀⢀⣶⣿⠟⠀⣠⣷⠘
⠀⠀⠀⠸⠛⠁⠀⡄⣿⣿⣿⣿⣿⣿⣏⡿⠟⣻⣿⠁⣿⣯⣾⠟⡿⠀⢀
⠀⠀⠀⠀⠀⠀⠰⢁⣿⣿⣿⣿⣿⣳⠏⡀⢠⣿⠷⢀⣈⠛⠓⣶⣷⣤⣀⣀⠁⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⢍⢤⡛⢹⡤⣶⡏⠉⢉⠉⢉⢙⡁⠈⡏⢉⠉⠍⠉⠛
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢀⡜⣀⠀⠉⢰⣿⡇⠀⠀⠁⢸⡎⠁⠁⠁⢹⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣹⣿⣼⣿⠀⠀⠀⢿⡇⢰⣀⡇⢸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⢀⣿⣿⣿⡿⠀⠀⠀⢸⡇⠈⢿⠃⠘⣿⢠⠀⠀⢸⡀⢀
⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⣿⣿⢿⣿⠇⠀⠈⠀⠘⣧⠀⣸⠁⢰⣿⡇⠀⠀⠀⠇⢸
⠀⠀⠀⠀⠀⠀⣾⡏⠀⣼⡟⠡⣬⡟⠀⠀⣸⠇⢀⣿⠀⠁⠀⣸⢹⡇⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⢀⡿⠀⣸⣿⠃⣴⠃⠁⠀⣸⡏⠀⡼⢿⠀⠀⠀⠹⠀⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢸⡇⠀⣿⡏⡸⣏⠀⠀⠀⢻⣿⣶⣧⣼⠀⠰⠀⠀⠀⠃
⠀⠀⠀⠀⠀⣼⠀⣰⡟⢠⠃⠈⠀⠀⠀⠈⣿⠁⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠘⠀⠾⣧⢆⠀⠀⠀⠀⠀⠐⣿⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠁⠀⠀⠀⠀⣀⣉⣤⠈⢁⠴⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⣙⠀⠘⠿⢿⣿⡿⡇⠀⠈⢨⣹⣻⡟⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⡾⠋⡀⠀⠀⠈⢿⣦⡇⠀⠀⠀⠉⠳⡄⠀⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠯⠈⣠⣼⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠶⠀⠠⢀⠻⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⢸⢋⡄⠀⠀⠀⢀⢠⡆⠀⠰⣄⡀⢐⠿⠂⠀⠈⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⣀⣴⣀⢀⡃⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠉⠘⠛⠉⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠂⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠰⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠎⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣰⡧⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠟⠃⠀⢀⣀⣤⣤⣾⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⣦⣤⣤⣵⣶⣤⣄⡀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠾⠟⢿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⣄⠠⢤⣑⠢⡄⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣲⣾⡿⠿⠿⡻⢟⢿⣆⣀⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣁⣩⡽⢷⡍⠂⠐⣄⣠⣄⠛⠿⠿⡑⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣬⣕⣒⣻⣣⠔⣂⣤⣭⠶⠦⠴⠄⠠⠀⠈⠿⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⠉⠉⠀⡴⠉⠀⠀⠀⠀⠀⠀⢀⡂⠰⡀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣭⡍⣩⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡧⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡤⢾⣿⢾⠁⠀⠀⠀⠀⠀⠀⢦⡾⠷⠤⠀⠀⠀⠀⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡟⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠐⡤⠀⠈⢳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢸⠃⠀⠀⠂⠀⡀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⢻⡏⠀⢀⣇⣀⢀⠙⠒⣋⠉
⠀⠀⠀⠀⠀⣠⢔⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⠀⠀⠀⠈⢧⠐⠋⠈⠉⠀⠀⢠⡾
⠀⠀⠀⠀⣼⠁⣵⣠⣿⡇⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⡀⠡⢀⣦⠀⠑⢊⡞
⠀⠀⠀⠀⢿⣿⣷⣿⣿⡟⠀⠸⣷⣤⣤⣤⣤⣄⣀⠀⣀⣿⡿⠋⠉⠲⢬⠀⡈⢁⠒⠳⠤⠁
⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⢸⡧⣘
⠀⠀⠀⠀⠸⠋⣿⠉⣿⣿⡄⢀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⢀⡞⠙⠐⠉⢻⣿⣮⣧⠀⠉⠉⢿⣿⣿⡇⠀⠀⣠⣆⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠸⣀⠀⠀⡴⠁⣻⡟⠻⣿⠀⠀⢸⡟⢿⡇⠀⠀⠉⠙⠓⠃⠙⠊
⠀⠀⠀⠀⠀⠻⠂⣤⣧⣰⣿⣇⠀⠀⠀⠀⢸⡷⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡿⢿⣿⠛⠋⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⡀⠀⠀⠀⠀⠈⠳⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠀⠀⠀⢸⣦⣄⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣷⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣯⡹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣯⡻⣿⣿⣾⢷⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣿⣟⠟⣿
⠀⠀⠀⠀⠀⢀⣴⣖⢿⣬⣿⣿⣿⣿⣿⢿⣻⣾⢷⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣬⡻⢧⡙⠳⠙⣧
⠀⠀⠀⠀⣠⣾⣿⣽⣿⣿⣻⣯⣻⣯⣿⣿⣿⣻⣼⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡳⢮⠻⣌⠃⣈⠀⣏⣆
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠷⣿⣷⣿⣟⣾⣿⣿⣷⢿⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣾⣿⢦⣧⣬⣶⣶⣷⢿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⢿⣿⣿⣿⣽⣿⡿⣛⣿⣦⣀⡀⠀⠀⠀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠮⡷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣰⣿⣯⣥⣍⣿⣿⣿⣶⣾⣿⣿⣿⣿⠿⠏⠛⣿⣿⣿⣯⣇⡀⢁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣹⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣿⣯⢏⡝⠉⡄⠨⡎
⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣯⣭⣴⢶⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣶⣿⣶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣶⣶⣦⣴⣶⢶⣶⣍⣙⠻⣿⡟⢫⣅⢬⣭⣬⠩⡛⢿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣂⡀⣀⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⡿⢭⣛⣿⠟⢻⣿⣙⢿⡧⠄⠳⠝⢿⣿⣿⣾⣿⢿⣿⠇⠈⠌⠙⠿⠛⣻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣿⣿⡆⣴⣿⣿⠿⠷⣦⠆⠀⠘⢿⠟⠛⡋⢅⣀⡤⠰⠄⠂⠒⠘⠛⠓⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠠⢉⣟⢣⡻⣅⡤⠀⠀⣸⡅⣀⣙⣟⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠓⠛⠃⠀⠞⠙⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠐⠿⣿⣿⣿⣿⣿⡿⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⣉⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣀⣬⣤⣤⣰⣿⣿⣿⡠⠀⠀⠀⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠙⠡⠶⠭⠛⠻⠛⠿⣯⣻⣟⠀⠀⡀⣹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡀⠀⠀⠀⠻⢿⣿⣿⣯⣾⡿⠿⠛⠻⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢈⣿⢊⠀⠈⣿⢃⣬⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⣀⠸⠇⡴⣚⡿⠑⠞⣉⣏⣿⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⠞⠛⠈⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡛⠢⠜⣱⠏⠁⢚⡛⠟⢉⡃⠈⠋⡀
⠀⠀⢀⠀⠀⠈⣷⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠉⡀⠰⢤⠄⠐⡚⣁⢠⣿⣷⠶⢸⠁
⠀⠀⠈⠀⠀⠀⣿⢧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠁⠈⠀⠿⣤⡀⠀⢀⠑⢀⠐⠀⡽⢀⣾
⠀⠀⠀⠀⠀⢰⡇⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠱⢿⡆⢤⡞⣷⡴⣳⠛⠉⠛
⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⢡⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠸⣏⢉⣀⣬⣾⠟⠁⠀⠁
⠀⠀⠀⠀⣠⣶⡟⢛⠉⠉⠐⠶⡟⠀⢀⣀⣀⠶⠖⠀⠀⢀⡖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⠟⠛⠟⠛⢁⣠⡴⢆⡅⢀
⠀⠀⠀⣰⠟⠀⢀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠀⠀⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢉⣀⣀⣰⣿⣿⣿⡓⠀⠐⠀⡀
⠀⠀⣴⠿⣟⣦⢽⡗⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠞⠁⠛⠋⡉⣿⢻⣿⣼⣿⣿⡃⢑⣀⡇⢠⡁
⠀⢰⣟⠚⢁⡤⢞⣿⠈⠀⠀⠀⣂⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⣸⠁⢀⠀⡀⠓⠈⠙⠋⣼⡿⣿⣈⣤⡟⣣⣄⠍
⠀⠸⣿⠟⡁⢳⣾⣿⣦⡀⣠⡼⠟⠋⠃⣤⢀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⣭⣁⠀⠉⠰⠋⠀⠈⠉⠛⠙⠻⠷⢶⡀
⠀⠀⣿⣶⣾⡿⣿⣉⣿⣿⠏⠀⠀⠀⠀⡀⠀⣷⢈⡁⠳⢦⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⣀⠀⠙⢛⣿⠏⠀⣤⡄⢀⣤⠀⠀⢃⠀⠀⣷⡀
⠀⠀⣿⣿⣟⣵⣿⣿⣿⣿⠄⠀⠀⠀⠀⠿⣘⣥⡄⢢⡄⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⣠⣴⠟⠡⢀⠀⠉⣥⡼⠁⠀⠀⠎⠀⢢⣼⡁
⠀⢠⣯⣿⣿⣿⣿⠿⢋⣤⣾⣷⣤⣴⣯⡾⣿⡟⠃⠰⣿⣷⡞⠛⠳⠶⠖⠀⠀⠆⠉⠑⡆⢸⣿⡏⠀⣴⡿⢦⣴⢻⣦⠂⠀⣴⠀⠀⠀⠗
⠀⠀⢹⣿⣿⢟⣵⣾⡿⡿⢿⢿⣿⣿⣿⣷⠋⠛⢯⠉⢃⣛⣠⡿⣤⣤⠅⠀⠀⠀⠀⠀⠁⠘⣿⣿⣾⣿⡇⠈⣧⢈⡉⠀⠀⢀⣀⠘⡴
⠀⠀⢿⣿⣵⡿⠟⠁⠀⠀⠀⠀⠉⠙⢻⣿⡖⢂⣀⣀⣄⠈⣧⠐⠒⠂⠖⠀⣀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣇⠀⠈⣷⣤⣹⣶⡾⠃⡘⠁
⠀⠀⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣎⡩⢶⠧⠀⠛⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⠀⠸⣿⣿⠛⠁⡞⠁⠀⠀⢴
⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡗⠉⣀⡤⠴⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⠀⠹⣿⣿⣖⡁⣄⠹⡿⠓⠐⠀⣀⣠⣖⠃
⠀⠀⠙⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⢋⣠⠄⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⠠⠈⣂⠓⠦⠈⢿⣿⣇⠀⠛⢳⠶⣶⠴⡟⡈⠁
⠀⠀⡀⠂⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢢⠀⠈⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⢼⢁⣅⡀⢀⠀⢈⠻⣿⣧⣴⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⠦⠉⣠⢹⣿⠲⠗⠶⢦⡴⢤⠀⠀⠙⣶⠖⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⢘⡗⢢⢴⣄⣷⡈⣿⣿⣿⡁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡈⠀⠀⠠⠄⠀⠀⢀⠀⠘⣦⣤⠀⠀⠀⠀⠀⠀⠈⠀⢚⡧⠠⠄⢿⡈⣹⠻⣿⣿⣶⣟⣿⣷⣄
⠀⠀⢀⠀⠀⠀⠀⠀⠳⠆⠀⠀⣁⢀⡀⠀⠀⠀⠀⡃⠀⠀⠉⠳⢿⣿⣀⣠⣴⣿⣩⣭⣿⡆⠠⣾⣿⠉⣻⡛⣳⣾⣿⣿⣿⣿⣿⣿⡗
⠀⠃⡂⠐⣐⠀⠀⠀⢤⡬⢙⡿⠋⠘⠛⠛⠓⡀⢰⠇⠀⠀⠀⠀⠀⠀⠈⠙⠙⠛⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣭⣽⣿⡿⠛⠍⠁
⠀⠀⠁⠀⠉⠀⠀⠈⠁⠀⣰⡶⠶⠆⠐⣆⣘⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢉⣀⣀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠐⠃⢮⠻⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠶⣾⣿⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠉⠀⠀⠀⠀⠀⠀⠲⠦⠴⠖⠂⠀⠀⠈⠀⠀⠀⠀⢸⡟⠋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢆⠀⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠈⡶⠀⡄⢀⠀⠀⠀⠠⠀⠀⠀⠀⠐⣾⣿⣾⠀⠉⠃
⠀⠀⠀⠘⣦⠀⢁⡜⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⣇⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠸⡄⠈⣀⡾⠁⠀⠀⠀⠀⠀⠛⠳⠀⠀⠠⠈⣿⣿⠞⠀⠀⠀⠀⠀⠁⠘⠂⠻
⠀⠀⠀⢄⣠⢹⣦⢹⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠂
⠀⠀⠀⠠⠇⠌⠸⣆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣹⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⡀⠀⠀⢀⣀⡤⠤⠤⠖⠚⠁⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⢿⠏⠀⠀⠀⠀⠀⠈⢁⠀⠁⠉⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠈
⠀⠀⢀⠀⡝⠲⠀⠒⠋⠹⠈⣤⠀⠀⢀⣤⠖⡀⠀⠂⠀⠚⠠⠎⠀⠀⠀⠇⡔⠙⠂⠀⠁
⠀⠀⣆⡜⡇⡀⠀⠀⠀⠀⢧⡟⠀⠠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⢎
⠀⠀⠛⠃⠉⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠁⠀⠀⠈⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⢀⣠⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣠⣶⣦⡌⠛⠁⣘⠛⡿⠀⠀⣴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠸⣿⣿⡿⠃⠀⢘⡏⠀⠠⠴⢾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠉⠉⠀⢀⠔⠈⠁⠉⠠⠶⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣀⣴⠀⣤⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠁⠀⠀⠀⠀⠀⠴⠖⠂⠀⠀⠂⢨⣏⠀⠛⢱⣿⣿⣷⡀⠙⣆
⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⠆⠀⠾⢛⠃⠀⠀⠈⠻⢿⡿⠃⠀⣿⡄
⠀⠀⠀⠀⠀⢀⡿⠰⠀⠀⠀⠀⠀⣀⣄⣞⣿⢷⢯⡿⡄⢀⣻⣄⠀⠀⠀⢀⠀⠀⣀⣾⣿⣿⣶⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢋⣤⢿⣽⣩⡿⣿⡷⢸⣿⣸⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢀⣟⣹⣿⣡⡿⣟⣿⡏⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢹⣿⣿⡄
⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⣸⠛⣭⣼⠿⠷⣿⣾⡇⡼⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣧
⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⢻⠿⣷⣯⡟⣰⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⣷⡇⠀⠀⠀⠀⠀⣾⡿⣿⣩⣿⡾⣩⣾⣱⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⡿⠷⠿⢟⣩⣶⣏⣿⡇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢻⣿⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣤⣤⣶⢞⣷⠞⣩⢟⣯⢤⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢹⣿⡆
⠀⠀⠀⠀⣿⣷⠀⠀⢠⣦⣈⣬⠣⣶⣧⣠⡾⢛⣾⡃⠀⡉⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢻⡇
⠀⠀⠀⠀⢹⣿⡆⣀⢛⣇⡨⠷⠖⣠⣿⣿⡶⠋⠛⠋⠁⢠⡟⡡⠚⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⢀⣀
⠀⠀⠀⠀⢸⣿⡇⠛⠣⠊⡰⢀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠐⠧⢤⡆⠀⠉⠛⠻⠿⠿⡉⣩⠜⠿⠟⠟⠉⠁⠀⠙⠁
⠀⠀⠀⠀⣸⡏⣠⡔⢁⢖⣵⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠐⢦⣠⣆⣣⣀⣀⣉⣄⣵⣾⡆
⠀⠀⠀⣀⡩⢊⡠⢊⣵⣿⣿⡟⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⢿⣿⣿⡿⣿⣿⣿⡇
⠀⠀⣠⣉⠔⡋⢴⣿⣿⡟⠎⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣽⣿⡇⢹⢿⣿⡇
⠀⢀⡉⠔⢻⡇⣿⣿⡟⡐⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣮⢿⣾⡈⣿⣷
⠀⠀⠀⠀⠸⢳⣿⠃⢰⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠹⡇⢹⡟
⠀⠀⠀⠀⢦⠞⠟⢀⢃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⣸⣧
⠀⠀⠀⠀⠀⢰⣴⡏⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⣶⠃⡿⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠃⣼⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⢰⠞⢇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠂⡜⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⢸⠟⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠊⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⢀⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⣿⠇
⠰⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣟⣽⣿⣿⣿⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣿⡻⢿⣿⣿⣿⡟⢃⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠃⠠⡌⢗⢀⣿⡭⢁⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠿⢓⢲⡖⢶⠛⣾⡟⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠌⠄⠹⠊⣷⡈⡧⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⠛⣀⠬⠁⠄⠥⠨⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠄⢒⣀⣁⠥⠭⠤⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⣚⣩⣤⣶⠶⡟⡟⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣋⣁⢀⡉⣀⣤⣨⣾⠃⣀⣀⣀⠀⠀⢀⣀⣤⡤⠖⠒⠂⢉⣤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡖⠁⢀⠁⠙⠛⠓⠑⠶⡴⢦⢌⠀⠌⠉⠩⠻⠝⠉⡏⢶⣄⣀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠀⠀⠚⠄⠀⢀⠃⠘⠇⠿⡌⣻⣷⡞⠸⠞⠀⠷⠟⠛⠉⠀⠙⠛⠦⠤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠃⠈⠶⡾⠶⠾⠿⠷⠿⠛⣿⠓⠴⢞⣻⣾⣿⣿⠿⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣿⡿⠋⠉⠀⠀⠀⠀⣤⣖⣂⣨⡅⠁⠨⠀⠅⠈⠀⢿⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣋⡤⠖⠊⠉⠉⠉⠉⠙⢿⣟⡀⠧⠠⠠⡇⢼⠀⢰⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣨⡭⢠⣅⣼⣬⢨⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣛⣚⣓⣻⣶⣤⣭⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢿⣿⣿⣿⣿⣗⢰⠋⡲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣷⣿⣽⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢰⡇⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠉⠅⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⡷⣴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⣀⣿⣷⠃⠸⢦⠀⠀⠀⠐⠀⠀⠀⠀⢠⡿⣀⠡⡏⠻⣷⣷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⢖⣿⠒⡒⠶⠍⠉⠀⢰⡇⠀⠀⠀⠀⢀⢰⡆⠀⠑⠺⠛⠿⠃⣰⣄⡟⠗⢿
⠀⠀⠀⠀⠀⠀⠛⠱⣄⣀⠺⠋⠂⠠⣽⡦⠸⢦⣩⠄⠀⡉⠁⠀⠀⠀⠠⡟⠘⠀⠀⠰⣿⢛⠛⠇⣽⠟⠒⠀⢸
⠀⠀⠀⠀⠀⠀⢰⠏⣹⣄⠀⣃⣠⠰⠈⠂⠓⢠⢸⠃⠰⠀⠀⠀⡤⡄⠀⠀⠀⢀⣀⠀⠈⢐⡆⣰⣯⣿⠯⠀⢸
⡄⠀⡀⣀⠀⣠⠄⢸⡏⠀⠸⣿⣿⡍⠀⠀⠀⠈⠈⠀⠀⠀⢠⠈⢀⣇⣰⣤⣠⣤⣼⡤⠈⠋⣠⠃⣘⣻⣿⡷⢸
⠁⠀⠈⠻⡧⠁⢀⣼⣰⣶⣐⣿⠏⠉⠀⡇⠘⢆⠀⠀⠀⠀⣤⠀⣸⣧⣽⣿⣿⣏⠀⡿⢷⣠⠋⢀⡛⢿⣿⣿⢿
⢀⡀⠀⠀⠁⣴⠟⢸⠃⠹⢿⡏⠂⠰⢤⡇⠀⠸⡆⢠⡄⠀⠂⠀⢋⣿⣿⣿⣹⣿⣷⡀⢀⣷⣄⠻⠁⢙⣏⠙⢸
⠃⠑⣂⠀⠰⢿⡀⡋⠀⢀⣫⣅⣀⣴⣾⡇⠀⣶⡶⠛⢳⡇⢀⣡⣾⣿⣿⣿⠉⡿⡟⣧⣾⣿⠁⢘⠃⢨⣷⡇⢸
⠀⠘⡟⡲⢤⣼⡏⣼⡖⢉⣿⠙⠻⣿⣿⠅⠀⠀⣿⡖⢸⣧⡈⠙⡷⡩⠿⠙⣿⠃⢀⣾⣽⣿⣷⣿⣾⣿⣉⣿⣸
⠀⠀⠈⠀⠀⣿⡍⢿⡇⣸⡏⠀⠀⢡⢸⠁⠀⠀⢰⠎⡉⠹⠃⠀⠀⠀⠀⠀⠋⠀⢨⡿⢿⣿⣿⣿⣿⣿⣿⡯⢽
⠀⠀⠀⠈⠚⠉⠑⠈⠓⠙⠃⠐⠀⠘⠙⠂⠒⠠⠒⠜⠳⠦⠴⠰⠆⠰⠤⠄⠠⠴⠟⠤⠎⠥⠼⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠙⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢼⡋⠿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣿⡽⣿⡟⠛⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣗⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟⢸⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣾⠿⠋⠀⠀⠀⢀⡀⣀⣾⠇⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠿⠿⢿⣿⣛⣻⣿⣦⣤⣤⣦⡾⠶⠛⠋⠁⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣿⣶⣿⣿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⠒⠲⠟⢛⣿⡿⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢰⣾⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣷⣦⣄⡀⠠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣿⣿⣿⣶⣦⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⡙⠿⠿⣿⣿⣿⡿⠿⠋⠙⠻⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠈⠓⠂⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣍⣭⣿⠁⠀⠀⣀⣀⣀⣉⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣄⣠⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠈⠻⣿⠻⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠈⢳⡤⣀⠀⠀⡞⢹⣷⣶⣄⣀⠀⠀⠀⠀⠈⢳⣄⠀⣩⣽⣿⣿⣿⣿⣿⣟⣋⣉⣤⣤⣴⣦⡀⠸⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡑⡼⠃⠀⠛⢿⣿⣿⣶⣄⢀⣰⣶⣶⣽⣿⣿⣟⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⢻⣷⣤⣬⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡆⠀⢀⣠⣬⠿⠿⠿⠻⠟⠉⠛⣿⣿⣿⣿⣿⣿⣿⡿⢻⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢷⣮⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⡟⠻⠛⠋⠀⠀⠀⢠⣤⣴⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣋⡁⣀⣠⣤⣶⣾⠶⠿⢿⡯⣁⣀⣀⡁⡀⢀⣠⣤⡄⣀⣀⣠⣆⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⣸⣿⡿⠟⠉⠁⠀⠀⠀⠀⠙⠋⠉⣹⢛⣟⠛⠣⡈⢑⣄⠀⠈⢮⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠷⠁⢠⡿⣾⣨⣦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⠿⠅⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣄⠀⠀⣠⠆
⠸⢻⣭⣤⣴⣶⣦⣄⣨⣴⣦⣤⣀⣤⣶⣿⣿⣿⣿⡷⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⠋
⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠉⠀⠀⠰⠆⠀⠀⠀⣀⣤⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠓⢤⠈⠰⠃
⠀⠸⣿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠐⠲⡄⠈⠛⠻⠹⣿⣿⠿⠋⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⢨⣿⡟⢿⠿⣿⠇⠀⠀⠀⠀⢀⡠⠤⠚⡛⣟⢭⣁⠈⠁⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠐⠟⠀⠈⠀⢹⠁⠀⠀⠀⢀⠀⠀⠀⠀⠛⠙⠋⠉⠠⢠
⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠭⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣷⡄
⠀⠀⠀⠀⣄⠠⣦⣤⣀⡀⣠⡀⢠⡄⡀⠀⠀⠀⠀⢠⣾⣿⣦⣀⢀⣴⣤⣀⣤⣤⡀⣤⠀⠻⣿⣾⡄
⠀⠀⠀⠀⠈⠠⡎⠀⠏⠑⢺⠿⣿⠿⠿⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣘⣿⣷⣶⡐
⠀⠀⠀⠀⠀⠀⣼⡂⢀⣠⣦⣿⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣇⠀⠀⢀⣤⢚⣿⣿
⠀⠀⠀⠀⠀⠴⠾⠿⢿⠟⠉⠉⠀⡀⢀⣶⠀⢠⣶⣸⣿⣿⣿⣿⣿⡿⠻⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣧⣴⠾⡷⣉⣤⣿
⠀⠀⠀⡀⠀⠑⢲⣬⣶⣤⣤⣤⣼⣷⣿⣿⣷⣿⣿⣿⣿⣋⣿⡿⣿⣦⣴⠀⢠⡾⢯⢿⣿⡗⠳⠗⠻⠏⠙⠙⠉⠘⠁⠀⠈⣰⡇
⠀⠀⠘⠻⣿⣿⣯⡷⢏⡿⠻⡿⠿⠿⠿⢿⡿⢿⢿⠿⢿⣟⣻⣿⡍⠻⣿⣷⠛⣹⣿⡟⠁⠀⠂⠀⠀⠀⠀⠀⠀⢀⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠙⠻⢿⣿⣦⣴⣠⣄⣀⡀⠋⠛⠉⠉⠉⠉⠊⣷⡜⠛⠙⠚⠡⡿⠇⠻⢿⡀⠀⢠⠀⢀⣀⣠⣤⡴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⢿⣿⣿⣿⣷⣷⣶⣦⣶⣶⡿⠋⠧⢀⡀⠃⠈⣁⠀⠀⠀⠀⠈⢉⣹⣿⠟⠻⠋⠀⢀⠀⣀
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠙⠉⠉⠙⢛⣿⣿⣿⣿⣀⠀⢀⠈⣹⠠⠄⠉⣇⡀⠀⠀⢤⣼⣿⣿⡆⡶⠆⠠⢐⣿⡟⢀
⠀⠀⠴⣾⣉⡶⡀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢸⣿⠟⠻⢿⣷⣆⣀⢃⠁⠀⠀⢀⢨⣥⣴⣶⣿⡿⠿⠿⠇⠀⣠⣴⣟⡉⠙⢿⣀
⠀⠀⣘⡋⠉⠁⢙⣷⣄⣀⢤⣀⠀⠀⠀⠀⠀⠋⠀⠀⠸⡿⠛⠻⣿⣻⣟⠛⠛⠋⠋⠉⠀⠀⢀⠐⠀⢁⣼⣟⣥⣤⠴⠀⠀⢹⡆
⠀⢀⣿⣿⣄⠠⠈⠛⣯⣿⣿⣾⣦⢀⡐⠀⠀⡀⠀⣀⠘⣧⠀⠀⣿⠻⡐⢥⠀⠠⠀⠀⠈⠉⠆⠁⢰⡞⣯⡿⠃⠏⠀⠀⠀⣾⡇
⠀⢰⣿⣿⣿⣿⣾⣬⣻⣿⣿⣿⢛⡭⠋⠀⠀⠀⠈⣿⠃⡎⠰⠠⠛⢷⣤⣀⠀⢀⠰⣺⣷⣿⡄⢖⣿⢿⣧⠄⠊⠀⠀⠀⢀⣾⣃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⡟⠻⡿⢁⣄⡴⣤⠀⢰⡿⣇⠰⡆⠀⢿⠷⣽⣿⣷⣄⠦⠿⣻⣿⡅⣸⡇⠀⠈⠉⠀⠀⠀⣠⣿⡿⠁
⡀⢸⣿⣿⣿⣿⣯⣾⣋⢨⣱⡆⠈⢠⠀⠁⠄⣀⡴⢛⠽⠀⠀⠀⣼⣷⡦⣛⣿⣿⣦⣴⣿⡏⣿⡉⠀⢄⡪⢽⣧⠼⣓⣛⠛⢛⠂
⢒⡸⢟⣿⣿⣿⢶⢮⣽⣿⣿⢷⡄⠸⡱⣦⣾⠉⠶⠉⡀⠆⠀⢁⡏⠀⠀⢁⠁⠆⣤⠤⣽⡷⢿⣀⣾⣣⢠⠽⠃⠋⠁
⣧⣴⣿⣿⣿⣿⣾⣿⣝⣿⠿⠈⠽⣿⠊⢈⣧⢘⡀⢉⠀⣀⡀⠘⠶⠄⠠⠀⠈⠀⠦⠀⠈⠙⣺⠏⠉
⣿⣾⣿⣿⣿⣿⣿⣿⣵⣿⣶⣴⣿⣿⣶⣿⣿⠿⢿⡝⠋⠃⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠫⠚⡁
⠀⠀⠀⠲⣶⣶⣛⣛⣛⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣟⣯⣾⠿⠛⠉
⠀⠀⠀⠀⠘⢟⣝⠿⣿⣿⣾⢿⣧⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣴⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠘⣿⣮⣍⣭⢾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⢳⣍⣅⣊⢻⡿⣿⣿⣿⣿⣟⣿⣿⣿⢿⣿⣷⢿⡋⠁⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠨⠍⠀⢪⡢⣻⣿⣿⣿⣟⣾⣿⣿⣫⠿⠻⣽⢏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠘⠒⠚⢙⣿⡿⡿⡿⣿⣻⠬⠚⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠖⠒⠀⠈⠈⠉⠉⠉⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⡆⢀⣀⣀⡈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣯⣭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠸⢯⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⡞⠀⣠⡄⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⡠⠄⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢹⡇⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠈⢳⣴⣿⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⠀⠀⠀⢹⣇⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣵⡀⠀⠀⡐⠂⠀⠤⠤⠴⠞⠻⣍⠛⠳⠶⢾⣿⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠙⠓⠶⠤⣤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠀⣒⠉⢀⠄⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢄⠀⠀⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⣶⣶⣶⣶⣴⣶⣷⣾⣷⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣻⣿⣮⡽⣯⣽⣟⣯⣿⣿⣿⣿⠿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⡿⣿⠿⡘⢝⣿⡏⣛⢻⡿⣿⣿⣯⣯⣾⣿⣖⠎⠿⣣⠵⣟⢻⣷⣶⣦⣀
⠀⠀⠀⠀⠀⣠⣾⣿⢿⣿⣿⣿⣟⣷⣹⣟⠾⣏⣉⣤⡀⢠⠈⣽⠿⡿⣾⣇⠉⠔⢊⡀⠀⢆⢻⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢀⣼⣿⣿⣷⣯⣿⣿⣿⣿⣿⣿⣧⣽⣶⣿⣿⣿⣷⣽⣿⣷⣈⠁⠀⣾⣧⠀⠰⣴⣰⢚⣿⣿⣿⣿⣿⣿
⠀⠀⣠⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣗⣿⠋⠈⣼⣿⣿⣏⡏⠛⠛⠛⠛⠛
⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣹⣿⣶⣶⣿⣿⣿⣿⡇
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢇⡿⣿
⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢹⣿⣿⡏⠨⣷⣷⠁
⢠⣿⣿⡿⠟⠋⢡⡟⣿⣻⣿⣿⣿⣿⣿⣿⣾⣾⣿⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⡇⠘⠿⠋
⠻⠛⠁⠀⠀⢀⠞⠜⢸⣿⣿⣿⣿⣿⣿⣿⣿⡋⢻⡇⠈⠛⠿⢿⡿⠏⠀⠀⠀⠀⣿⣿⠿⠃⢀⣴⢣
⠀⠀⠀⠀⠀⣠⠋⣰⣿⣿⣿⣿⣿⣿⡿⠟⣻⡧⣾⣙⢿⣿⣷⣦⣿⣽⣄⡀⣀⣐⣋⣁⣀⣠⣾⣿⣎⢠⣀
⠀⣀⣷⡞⠔⣁⡜⣱⢻⢿⣿⣿⠙⠛⢀⣸⣿⣽⣿⠿⣯⠿⠹⣿⣽⢿⣿⢿⣮⣿⢿⠿⢿⠿⣛⣻⣿⣦⣬⣿⠯
⠐⠛⠛⣿⣶⣥⣚⡱⠋⠞⡸⠁⣠⣴⣿⣽⠛⠻⠇⣠⣿⡟⠀⡻⣇⠀⢹⣼⢐⣏⣀⣀⣠⡞⢫⣿⣿⣿⠉⠉⠁
⠀⣀⣉⣽⡿⣿⠿⣿⣿⣿⣷⣿⣿⣛⣻⣤⣾⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠫⠄⠖
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠽⠿⠿⠿⠿⠿⠿⠿⠭⠉⠛⠛⠛⢛⡛⠛⠍⣹⣟⠉⠝⠛⠋⠉⠛⠀⠂⠀⠀⠁
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠌⢽⡟⢉⡓⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢰⠀⠃⠉⠀⠆⡠⡸⢻⣦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⣿⡆⠀⠀⠀⠀⣣⢊⢘⣿⡄
⠀⠀⠀⠀⠀⠀⢨⣿⢲⡄⠀⠈⠉⠀⠀⣠⢠⠀⣝⡴⡋⡿⢿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⣤⣬⣄⠈⣯⡦⢘⣇⠴⢻⣧⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠘⠛⠛⠺⢶⡿⣿⡶⠷⠛⣁⡽⢋⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠙⣇⡀⢰⢀⡀⢚⣯⢯⣇⢹⠂⠐⠊⣤⣴⣟⡿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⠶⠀⠀⣲⣼⣟⣡⣴⠆⠀⠀⣀⣌⣉⣺⠿⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢹⣷⣰⣻⣞⠋⣾⡁⠙⠊⠟⡁⠻⣇⣈⢻⣞⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡸⣿⣿⣿⣿⡇⣭⡁⠙⠛⠘⢿⡷⠈⠡⡴⢩⣭⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⢸⣿⣏⢉⡿⣾⣟⡈⠻⡷⢂⣀⠘⠃⠠⣤⣤⡿⣴⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡼⣿⣿⡜⢣⠈⢡⣾⣿⣦⣬⢿⡟⠰⡤⠈⢉⣥⣌⣥⡾⣦⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⣻⣧⡈⣁⣠⠴⣓⣐⢿⣿⣿⣌⡛⠃⣇⣛⣿⣧⣤⣤⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⡘⢯⢾⡻⠾⣮⣟⣋⣥⢹⣻⣾⡟⣿⠟⣿⡿⢹⣿⣿⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⣷⡐⢶⣉⣐⠛⠿⡍⢹⠇⡉⣩⣵⡟⠿⡿⠻⡟⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣷⣺⣟⣿⣿⣮⣝⠻⣿⣦⣶⣿⣾⣿⣿⡿⣝⠳⢤⣤⣽⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⢿⣿⣻⣿⣿⣿⣶⣝⡮⣻⠝⠷⠍⣡⠹⠿⣦⣤⣉⣹⣿⣧
⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⡿⠿⠛⠶⣿⣿⣯⣿⣿⣿⣿⣾⣥⣀⣈⣥⣜⡳⣄⣉⠛⠻⢿⣿⣄
⣰⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠈⠉⠻⠿⢿⣿⣿⡻⢿⣿⣿⣿⣿⣿⣷⣮⣭⣭⣼⣿⡿
⠻⢿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣽⣿⢼⢇⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣧⣳⣿⢮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿
⣿⣿⠟⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠀⠀⢀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠺⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡐⠀⠀⠀⠈⠀⠀⢠⣴⠿⡿⣿⣶⣀⣀⠔⠀⡆⠀⠀⡈⢀⡇⠀⠄⠀⠀⣀⣄⡤⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠑⠀⣢⣀⢀⠀⠈⣀⣴⠉⠀⠤⠂⢢⠀⠀⢈⡇⠀⠠⠶⠛⠃⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠹⢀⡀⠀⠠⠀⠂⠠⠘⠪⠼⠁⠂⠈⠀⠐⠁⡇⠠⠀⢘⠃⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠁⣰⠟⠳⠀⠀⠈⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡧⠀⠔⠉⠈⠋⠙⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠈⢸⡇⢠⣄⡀⠀⠀⠀⠂⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢴⣿⣿⣧⣷⡄⠀⠀⠀⠀⠀⠄⠂⠄⠈⠉⠙⠶⠤⠄⢄⠀⠀⢴⠂⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⢷⠀⣀⠀⠉⠁⠀⠀⠀⢀⣿⣿⣿⣷⡖⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠆⠊⠈⠐⠤⠢⠈⠦⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⡀⠀⠀⠨⠾⠇⠤⢳⣍⠂⠄⠀⠀⠁⠀⠉⠉⠁⠁⠀⠀⠠⠄⠈⠀⠀⢢⣾⣾⣮⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⢁⠀⠀⠀⠀⠴⠚⠟⠁⠀⠄⠀⠀⠀⠀⠠⠀⣀⠀⠀⢀⣰⠎⠀⠀⠀⣽⢿⣿⣿⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⢀⢸⡄⠀⠀⠄⠀⠀⠀⠊⠳⣄⠀⠁⠀⠄⡄⠈⠉⠘⠀⠀⠈⠠⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣡⡤⣤⠆⠊⠃⠀⠐⠦⡄⠀⠀⠀⠀⠈⠱⠦⠐⠀⠈⠀⠀⣤⣤⣄⡀⠀⠈⢢⡄⠂⠐⠂⡐⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠻⠦⣀⠀⠀⠀⠀⠱⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠘⠀⠀⣴⠷⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠂⠀⢸⡇⠀⠀⠀⠀⠈⠀⠀⠀⠘⠻⡿⠃⠀⠀⠈⡄⠠⠁⠀⣤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⣉⣀⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠁⠀⠀⠄⠁⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠆⠈⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣴⣿⣿⢤⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣁⣀⢀⣤⣶⣿⣿⣿⠇⢸⠳⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⠟⣛⣥⣶⠟⣿⡟⢩⡿⣫⢀⠘⣇⠀⠀⠀⠑⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢢⡤⠖⣪⡅⣰⡿⠟⠫⢀⣠⢾⡿⠐⠛⢱⣿⠘⠄⠻⠣⡄⠀⠀⢀⣙⢮⠁⠐⠦⣄⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠘⠋⢠⠖⠀⢀⣠⢴⣿⠏⡄⠀⣀⣴⣿⣿⡆⠰⣀⡀⠀⣷⢄⡀⠹⠄⢀⡀⠀⢈⣙⠓⣿⣼
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡵⣆⣥⡶⠞⢁⣴⣿⠟⣾⠁⡼⠿⠋⢀⡝⠻⣄⠀⠁⣠⣀⠓⠈⡀⢀⡆⠙⢰⣴⡠⣃⣏⣻⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⠋⠸⢋⣥⡶⠾⣿⡿⠋⠰⢃⠾⠁⢷⠀⢿⠁⠑⡹⠀⠀⠀⠹⠀⠸⡇⠸⡇⡀⣿⣿⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠁⢀⣨⠉⢁⣤⡆⠉⠀⠀⣴⡟⠀⢰⡈⠀⣼⢠⠄⠃⡈⠣⠀⡇⢀⣆⣧⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠶⠾⠛⠛⠛⠓⠻⠛⠛⢿⣄⢠⡀⠁⠘⢠⣾⡷⢐⣿⣸⣠⡀⣇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢳⢸⡇⣰⢀⣨⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣇⠀⣶⠀⣤⣼⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢿⣇⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠛⠻⣿⣿⣿⣿⣾⣿⣿⣿⡷⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣿⣿⣿⣿⣿⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠹⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⠿⠿⠿⠛⠉⠉⠁⠀⠀⢻⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠁⠀⠀⣠⣶⣶⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋

⠀⣠⠴⠒⠉⠉⠉⠑⠒⠦⣄⡀
⡰⠁⠀⠀⠀⠀⠀⠀⠀⠃⠘⠉⠳⣄
⣇⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠚⢣⣀
⢋⡉⢢⠀⠘⠿⠀⢀⣄⠀⠀⠀⠀⠀⠠⢋⣆
⣇⠈⡟⠀⢀⣄⣀⠉⠀⠀⠅⠀⠂⣀⠚⠑⡺⢆
⠘⢶⠀⠀⠈⠉⠉⠁⠀⠀⠀⣀⠤⠴⢚⠩⠐⠨⡳⣤
⠀⠘⡄⠀⢀⡘⠃⠀⠀⠄⡩⠑⠂⠀⣀⠐⠈⠅⠈⠊⠑⣄
⠀⠀⢸⡄⠈⠁⠀⢀⣴⠀⠀⠄⠐⠀⠀⠀⢈⡳⠮⠳⠶⣬⡷⣄
⠀⠀⠀⡁⢠⠴⠚⠀⠘⡖⠪⠥⢀⠴⠂⠔⠉⢒⡀⢩⢁⡈⣠⡿⣧
⠀⠀⠀⠏⠁⠀⠀⠀⠀⢿⡤⢌⡐⢋⣠⡆⠀⡡⢋⣐⠗⣎⠩⠾⢃⣿⡄
⠀⠀⠀⠘⡀⠀⠀⠀⠀⠘⣷⡉⠠⠍⠱⠢⠐⡿⢛⢳⣚⡡⣿⣿⢧⣍⢷⣦⡀
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠘⢿⣄⠒⠂⣠⠎⠴⠏⠾⣿⠗⣷⣦⡘⢿⣧⡹⣿⣦⡀
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠈⠻⣷⣍⠉⠴⢛⣴⣦⣛⢿⣮⡙⢿⣦⡛⠷⢨⣿⣟⢦⡄
⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡙⠻⠿⠿⢃⣬⣝⣓⢾⣥⣟⡿⠿⣿⣛⠿⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣼⣆⠀⠀⠀⠙⣿⣿⣷⢶⣦⡉⣛⠻⠿⠿⣭⣭⣍⡀⢤⣤⣙⣶⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣧⠀⣤⡀⣄⠙⢿⣿⣾⣥⣈⡛⠿⠷⠶⣭⠛⠿⠷⠶⠝⢛⣷⣬⣹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣬⢿⣿⣯⣷⣽⣶⣿⣿⠛⠻⠿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡽⣿⣿⣿⣿⣿⣿⣫⣓⠶⣤⣙⣿⣿⣄⠈⠉⠛⠿⢿⣷⡀⠀⠙⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⡦⣈⢻⣿⣾⣿⣿⣿⣿⣷⡀⠁⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠈⢱⣤⣬⣿⢿⣿⣿⡿⠿⠿⠿⡽⣿⣄⢄⢂⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠰⠿⠾⠶⢬⣵⡢⠜⣪⠿⢧⣼⣿⣿⢥⠀⣀⣀⣹⣌⠛⣦⠉⢦⢢⡀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠙⣮⡇⠀⠀⠙⣿⣧⠀⠀⠀⠀⢀⡈⢦⡈⢳⠈⢧⡱⡜⢄⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠿⠶⠦⠴⠤⠬⠥⢤⣴⣓⡧⠤⠤⢤⣿⡿⠤⠾⠿⠥⠬⠟⠰⢿⣦⠠⡀⢳⡝⣎⢧⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠙⣎⢯⣳⡼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡘⣧⡳⣽⣜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡝⢾⣯⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣮⡻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣜⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣝⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣎⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⢿⣆

⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⢿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣯⣤⡶⠀⠂⠀⣄⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣶⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣨⡷⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠛⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣤⡀⠀⠀⢹⢡⠙⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣄⠄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⡙⢂⡆⠀⠄⠀⠀⠂⠈⠠⠀⠀⠀⠀⠀⠀⡤⠞⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣷⣾⠃⠄⠀⠀⠀⠂⠁⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣳⣀⣹⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⣵⢧⠋⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣛⠛⠛⡂⠀⣀⣀⣤⣾⣿⣿⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⢀⣀⠛⠛⠉⠀⠀⠿⡿⣿⡏⢍⡊⠼⢽⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢝⠧⠻⣷⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣄⠰⡱⡑⣕⢮⡝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⣶⠔⠔⣱⢲⢿⢸⢹⠈⢢⣤⢀⡆⠈⠁⠓⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠴⢪⢦⢃⣇⣎⣄⢤⠖⠉⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡈⠀⢄⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠑⠋⠊⠉

⠀⠀⠀⠀⠀⠀⠤⠴⠶⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⢀⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣙⢿⣿⣦⣴⣾⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⡷⢒⣙⣼⣿⣿⠏⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣟⣪⣾⣿⠛⠉⠉⠀⠸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢛⣿⢿⠟⠉⠁⠀⠀⠀⠠⢀⠀⠉⣹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⠽⠴⠖⠀⠀⠀⠀⢀⣄⡀⢀⣴⣿⡟
⠀⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣶⡶⠖⠛⠛⠿⣿⣿⣿⣥⣦⣤⣦⡤⢠⣤⣤⣛⣭⣶⣿⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣶⠤⠊⣵⣷⣿⠿⠛⠉⢩⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢃⣠⠾⠛⠉⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠋⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⣴⠾⠛⠁

⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠚⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⢸⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡇⠀⠈⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠈⠻⣿⣿⡆⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⠁⠀⠀⠹⢿⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠋⠹⣏⠀⠀⠀⢀⣾⡿⠿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠙⠋⠻⣆⠀⠀⠀⠰⠾⣿⣿⠟⢿⣿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠉⠉⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠘⠃⠀⠈⠃

⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⣴⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⡇⠀⠀⠀⠀⠐⣿⣿⡆
⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠈⠛⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⢼⡿⠛⠂⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣦⣀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠉⠁⠀⠸⣿⣿⠋⠉⠿⢀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⢀⣤⣤
⠀⠀⠀⢸⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⠀⠀⠀⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠴⠂
⠀⠀⠀⠈⢿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠉⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠙⠙⢿⡇⠀⠀⠀⠀⠀⢀⣿⡿⠏⠀⠀⠀⠀⠀⠈⠻⡿⣿⣿⣄⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠹⠏⠉⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡋⠂⡀⣠
⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠠⠀⠀⢀⠀⠀⠰⢿⡀⢈⡝⠛⠁⠀⡀
⠀⠀⠀⠛⡀⠀⠀⠤⣀⠀⠀⠀⢀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⠀⠈⠁⠀⠈⠀⠀⠀⠀⠑⠊⠁⢀⠀⣰⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠃⢀⡠⠂⢀⠀⠀⠸⡇⣀⠀⠀⠀⢴⠋⠀⣄⣀⠀⠀⠠⢀⣠⢠⡀⣐⣦⠀⠀⠀⠰⠚⠀⠉⠉⠙⡆⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠈⠃⠈⢁⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⡁⠀⢂⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠄⠀⠚⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⢀⣀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢁⠈⢀⡀⠀⠀⠀⠈⠉⠑⠀⢹⣠⡤⣤⢴⣶⠲⢶⢲⡶⢄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠑⢦⣀⠀⠈⣷⣠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠱⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠁
⠀⠀⠀⢀⡀⠀⢦⠀⢠⡀⠀⠈⡿⢷⠰⡆⠀⠙⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠰⢳⡄⠈⠧⠚⢣⠀⠀⠀⠈⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠒⠚⠻⢧⠀⠄⠀⠀⠀⠀⠘⢶⢸⣠⡍
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢨⡄⠀⠁
⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠻⠁

⠀⠀⠀⢰⡷⣦⣄⣀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠈⣿⣮⣻⣿⣷⣄⣼⣿⢟⣵⣿⡟⠁
⠀⠀⠀⠀⠘⣿⣿⣾⢻⣿⢽⣯⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠙⠿⠏⢫⣼⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢀⣠⣴⣦⢶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⣴⣾⣿⣿⠀⠀⠀⣠⣶⣿⣿⣿⣿⣮⡻⣿⠿⢿⣷⡀
⠀⠀⠀⠀⢀⣀⣀⣿⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣩⣟⡛⠛⣠⡀
⠀⠀⠀⠀⠀⠉⠛⢿⡆⢀⡤⠀⠈⠉⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⢩⣼⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣼⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢠⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣺⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣧⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⣦⣤⡀⢀⣤
⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣿⡿⠛⠉⠉⠙⣿⣿⣿⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁⢻⡆⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢟⣟⢿⣽⣟⣷⣿⣿
⠀⠀⠀⠀⢀⣾⡟⠉⠀⣾⠋⠀⠀⠀⠀⠀⢻⣿⡿⠀⣿⡄⠙⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⡈⠛⠛⢀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣯⡿⣿⡿⣾⣿⣿⣿
⠀⠀⠀⢠⣿⠏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⣠⣼⣿⣧⠠⣌⠻⣿⣿⣿⣿⣿⢸⠀⣿⣿⣷⣶⣿⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣷⣿⣾⣧⠙⠿⠟⠃
⠀⠀⠀⣼⣿⣷⠀⠀⠀⢻⠀⠀⢀⣀⣀⡀⠀⠈⢻⣿⣿⣿⠀⣿⣦⠈⠻⣿⣿⣿⠘⣇⠘⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⠟⠀⣦
⠀⠀⠀⣿⡏⢿⣇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡄⠀⠹⣿⠿⠀⣼⣿⣧⠀⡘⢿⣿⠀⢻⣦⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⠄
⠀⠀⠀⢹⣷⠀⠙⢧⡀⠀⠀⠈⠛⠛⠛⠛⠛⠃⠀⠀⠀⢀⣴⣿⣿⣿⡇⢹⣆⠹⠀⠈⠻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⢠⢿⣆⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⠁⣼⣿⡆⢠⠸⣦⣈⢉⣀⣶⣄⠀⠀⠀⣀⣀⣤⠖⠀⠀⠀⠀⠀⣠⡾⠃
⠀⢀⡴⠋⠀⠻⣦⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣄⣀⡀⠀⠀⠀⠀⠀⠠⢾⣿⣿⡗⢸⣧⡙⠿⠿⠉⠻⢿⣿⣾⣽⣟⣋⣁⣀⣀⣀⣤⣴⡾⠟⠁
⠀⠉⠀⠀⠀⠀⠘⠳⣄⡀⠀⣿⣿⣿⡿⣿⣯⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⢫⣥⣾⣿⣿⣶⣶⡄⠀⠀⠈⠉⠛⠛⠻⠿⠿⢿⣛⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢘⡛⢿⣶⣿⣿⣮⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡽⣿⣻⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢿⡏⣿⣯⡻⣟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠾⠿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢹⣿⣷⡘⢯⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⠈⣿⣿⣿⡌⢳⡝⣷⣄⠀⠀⠀⠀⠀⠀⣼⣿⣿⡌⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢹⡆⠸⣿⣿⣷⡀⢻⣎⢿⣷⣄⠀⠀⠀⠀⢿⣯⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡘⣷⡀⠹⣿⣿⣧⠀⢿⣆⠻⣿⢷⣄⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡇⢻⣧⡀⠙⢿⣿⠀⠘⣿⡆⠹⣧⡙⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢻⡄⠹⣷⣄⠀⠁⠀⠀⣿⣿⠀⢹⣷⠈⢿⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠻⠀⠙⠶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠘⠃⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠈⠉⠉⠉⠉⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⡰⠁⠂⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠰⠁⠰⠶⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⢀⡠⠤⠓⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢶⣥⠤⠤⠤⠤⠤⠤⣬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠉⠁⠒⠒⢆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠉⠉⠐⠢⢄⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠑⠢⢄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠉⠉⠛⣍⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠌⠉⠉⠧⠀⠀⢀⡢⣈⠪⢕⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠙⠳⣦⢌⡙⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⠽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⡖⡆⠀⢴⠴⠐⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠁⠣⢄⡸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⠒⠾⣥⣞⡉⠤⠄⠑⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⡻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣻⣿⣿⣾⣿⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣿⣟⠛⣿
⠀⠀⠀⠀⠀⢀⣴⣶⢿⣬⣿⣿⣿⣿⣿⣿⣻⣾⢷⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣬⡻⢧⡙⡳⢙⣧
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣻⣯⣻⣯⣿⣿⣿⣿⣼⡗⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡳⢮⠻⣌⠃⣘⠐⣟⣆
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣛⣾⣿⣿⣷⢿⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣾⣿⢦⣧⣬⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⢿⣿⣿⣿⣽⣿⡿⣟⣿⣦⣀⡀⠀⠀⠀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠯⡷⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣱⣿⣯⣥⣯⣿⣿⣿⣶⣾⣿⣿⣿⣿⠿⠏⠛⣿⣿⣿⣯⣇⡀⢁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣹⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣿⣯⣟⡝⢉⣤⢨⡎
⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣻⣯⣭⣴⢾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣶⣿⣶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣶⣦⣴⣶⣶⣶⣏⣙⠻⣿⣟⢫⣅⢮⣭⣬⠩⡟⢿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣃⡀⣀⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⡿⣯⣟⣿⠟⢻⣿⣙⢿⡗⠄⠳⡝⢿⣿⣿⣾⣿⢿⣿⠟⠈⠍⠹⠿⠛⣻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣿⣿⡆⣴⣿⣿⠿⠿⣦⠟⠀⠘⢿⠟⢻⡋⢅⣀⡤⠴⠔⠂⠒⠘⠛⠓⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠠⢉⣟⢣⡻⣅⡤⠀⠀⣸⡇⣀⣙⣟⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠓⠛⠃⠀⠞⠙⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⣀⠈⠀⠀⠦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⢀⠆⠀⡀⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠈⠁⠀⠂⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠁⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⢰⠀⠀⠀⠆⠈⢿⠃⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣿⣏⡿⢵⣞⡣⢄⣈⣀⠓⠀⢄⡀⠐⢀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢀⠔⠂⠛⣆⠀⠀⢀⡴⢏⠻⣾⣿⢻⠟⡟⢿⣾⣧⣯⣽⣒⣬⢆⡀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⠻⠆⣵⠃⠀⠀⠀⠐⠀⠀⡼⢮⣝⣉⣷⡿⣦⣖⣠⣶⣝⡻⡿⣿⣿⣿⡾⠾⢳⣮⣯⡃⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠆⡶⠃⠀⠀⠀⠀⠀⠀⠀⣋⡄⣠⠨⡬⣡⣴⣎⡁⣟⠁⢱⢧⢬⡿⣿⢥⣄⣀⣷⣾⣷⣾⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸⣛⡶⣃⣴⠶⠏⠀⣿⠉⣿⣱⡿⠠⠾⢷⣄⢺⣿⣻⣿⣿⡿⢿⣿⣾⣿⡆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣾⡤⠀⠀⠀⠀⠀⠀⠸⡖⠷⠛⠁⢠⡄⠟⡇⠘⣯⠉⢠⣤⣦⣼⠇⣷⡜⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡳⡞⠂⢴⠇⢠⡇⢸⡇⠀⢀⡉⠃⠱⣦⣛⣁⠻⣿⣿⣿⣿⣶⣿⣦⣽⣿⣿⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⣿⠀⢾⠀⠈⣇⣸⡃⠀⢸⠃⢠⣾⢌⣽⣏⣹⠿⣷⣽⡛⣿⣅⣿⣿⣿⡿⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣎⡀⢰⠀⣠⡟⢿⠙⠀⣾⡄⠀⡉⢈⡟⠻⢿⡧⣦⡉⢻⣿⣿⣿⣿⣿⣿⣀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠠⠀⢿⡋⢠⣄⣀⡉⠙⠀⢘⠛⠓⣶⠂⠀⡻⠷⣾⡟⢿⣿⣿⣿⣿⣿⣿⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢠⣾⡃⠠⣧⠀⠸⣷⠂⠨⣷⣦⠻⠀⠿⣧⣄⣤⡿⠘⣿⣿⣿⣿⣿⣿⣿⣷⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣾⡿⣅⢤⣶⣀⣼⣯⣿⣷⣶⣦⣸⣿⠘⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣅⣁⣝⠊⢦⡵⣽⢮⣋⠈⠙⠻⠀⢹⡇⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣻⡿⢿⣿⣷⣶⣿⣮⣣⣝⡳⣄⠀⠀⣸⣗⠀⣿⣿⣼⣿⣿⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⡀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠹⢻⡏⠀⠀⠀⢿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣿⣿⢿⣯⠻⣧⠹⣿⡀⠨⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡾⠁⠀⠙⢧⣀⢀⢳⡀⣀⡀⣤⢀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠹⣿⣿⢿⣯⣿⣿⣿⣿⣿⣿⣿⢻⡆⢻⣇⢹⣷⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠫⡀⠀⠀⠀⠸⣿⠟⠛⣿⣧⡄⠳⠇⠀⠶⠒⠿⠛⠳⠀⠀⠀⠀⢹⡟⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣌⢻⣦⠙⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⡠⣴⠋⠀⠀⠈⠀⠀⠀⠀⠀⣀⣴⣹⣟⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣝⣿⣿⢿⣟⢿⣿⣿⣿⣿⣦⠙⣷⣌⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⠋⠀⠘⢄⠀⠀⠀⡀⠈⠀⣠⣾⠏⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⣰⠏⠀⠀⢠⡏⡏⠙⢶⣬⣷⣿⣿⣿⣿⣿⣷⣮⣿⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠆⠀⣠⣾⠟⠁⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠶⣦⣀⣤⠋⠀⠀⠀⠀⡇⡇⠀⠈⠀⠈⢽⣿⠋⠿⣿⣿⣿⣿⣿⣿⡇⣻
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠐⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⢌⣉⣻⢛⣿⢷⣴⢶⡇⠀⢈⣿⣿⣤⣤⠀⠀⠸⠇⣷⢀⠀⡀⠀⣾⣿⠀⠀⠙⢷⣌⠛⡿⢻⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⢼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⢠⡀⢱⠆⣿⣾⣾⣿⣶⣿⣿⠀⠀⠀⠀⠙⣦⡀⠀⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠙⠛⠛⠉⠙⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣼⣿⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⣿⡄⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣷⣶⡲⢤⣀⡈⠉⠀⠁⠀⠈⣹⣿⣿⣿⣿⣿⢸⡇⢸⣿⣿⡿⠋⣿⣿⠀⠀⠀⠀⠀⠀⠸⣷⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠉⠉⠁⠀⠀⠉⠛⠻⣿⣿⣿⣿⣼⡇⠀⢹⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⡇⠀⠈⣻⣷⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣟⣧⠀⢠⠃⠻⠤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠠⡰⣔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣷⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢾⣿⣆⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⢀⣷⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣿⣿⣿⣿⡿⠿⠛⠛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣩⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡵⣡⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷⣛⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣳⣿⣷⣋⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⡻⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⡿⣣⠾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⣷⢽⣄⠤⠐⠘⣿⡵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣼⢿⡏⡿⣵⢋⡞⣡⢄⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣻⡯⣟⠡⣿⢟⣴⡏⠚⠁
⠀⠀⠀⠀⠀⠀⢠⡛⢿⣿⣫⣯⢞⡟⣴⠻⢾⡷⣒⢤⡀⢀⣠⣞⡿⠿⠟⠃⠠⢦⢋⣾⢟⢅⡀
⠀⠀⠀⠀⠀⠀⢸⡡⢻⣟⣿⣟⢏⣾⡟⡰⢋⠞⣁⣪⣿⣿⣿⡿⢽⠃⠂⠀⢠⡿⠟⢙⡤⠆
⠀⠀⠀⠀⠀⠀⣾⢰⡻⣻⣟⣥⣯⣯⣾⣷⣿⠿⠏⡽⠿⡻⠾⠶⠨⡁⢸⡹⠮⣠⣲⠟
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣽⣷⣷⠾⣋⡭⣿⠿⠟⠛⠛⠀⠀⠀⠀⣤⣿⣯⣴⢃⣱⠒⣱⣦⢀
⠀⠀⠀⠀⠀⠀⠈⠻⠟⢋⣿⢟⣥⡾⠌⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣾⣿⣽⣿⣿⣶⣷⣮⢢⠁⢂⡴⠃
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣷⡜⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿⣟⢟⣿⣷⠻⣿⣿⣿⣿⢷⡞⠽⠇⠌
⠀⠀⠀⠀⣠⣾⠛⠛⠛⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⡟⣿⣧⣎⣿⣿⣷⠹⡿⣋⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⢰⣿⣿⣀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⢽⣿⣪⣿⡇⣙⡵⣿⡮⢿⣌⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣠⣿⣀⠉⠛⠋⠀⠀⠀⣀⣀⠀⠀⢀⣺⣿⣿⢍⣻⣿⣭⣷⠿⣿⣿⣫⣽⣫⠵⣪⡴⣫⠿⣿⠰⡤⢠⣤⠀⢠⣤⠠⢶⣤⢶⣿⣻⡿⠟⠁
⠀⠘⠛⠋⠉⠉⠀⠲⣶⣶⣾⣿⣿⣿⣶⣽⣿⣿⢿⣿⣾⣥⣾⣽⢿⣿⣟⣿⣽⣷⣰⣶⣼⣴⣿⠵⣾⠴⣋⡤⣖⣫⢵⣺⣭⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣽⣿⣿⣿⣿⣷⣿⣾⣷⣿⣿⡟⣯⣿⣟⣾⣿⣶⠯⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣯⣽⡟⣛⣟⠛⡛⠛⠛⠛⠛⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⢯⣿⣑⢆⣭⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣷⣶⣶⣦⣤⣢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⢯⡷⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢹⡃⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣦⣠⣄⣼⡇⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⠀⠀⠀⣙⠛⠻⠃⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢃⡀⢀⣻⣿⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠋⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢼⠁⠀⠀⠐⠁⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢾⣲⣿⢿⣿⡟⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠰⠀⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣦⡀⠀⠀⠀⠀⠀⠘⣿⡆⣸⣿⣾⠏⠀⠀⠀⠀⡀⢉⠿⢶⡇⠀⠀⠀⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣐⡄⢀⠂⠈⢙⣭⣟⣿⣦⡀⠀⠀⠀⣼⡯⢿⣿⠟⠁⠀⠀⠀⠀⠠⣤⠗⠐⠊⠲⣖⣠⣤⣼⢃⠴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠉⢳⣮⣁⠘⠧⢠⡙⠿⣿⣷⣄⠀⠀⠻⣇⣾⠃⡄⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⣀⠙⡒⠢⠀⠀⠈
⠀⠀⠈⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣷⣤⡀⡀⠀⠀⠀⣘⠛⢦⣀⡊⠛⢒⠿⢏⣽⣆⠀⠀⡩⠁⠀⠡⡀⠀⠀⠀⠘⠁⠀⢀⣀⠖⠊⣠⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⠃⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⢿⣔⣣⣞⡷⡻⣿⣟⣽⣿⣇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠀⢠⣀⣈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠀⢻⡄⠀⠈⠉⠉⠀⠀⡺⠗⣬⣵⣝⣋⣿⣾⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠈⢿⣙⡀⠄⢀⠀⠀⣬⣿⣤⣭⢽⣽⣏⢷⣿⣿⣿⣷⡀⠀⠀⠀⠀⠰⡀⠀⢉⠀⠀⠀⠀⠈⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠈⢿⣷⣴⣄⣥⣾⡏⠉⢿⣝⡿⢿⣿⣦⣝⣿⣿⣿⣷⡀⠀⠀⠀⠀⣁⠀⠠⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠠⣄⠙⣿⣿⣿⣿⣇⠠⢄⣿⣷⣿⣿⣿⡿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠸⠀⠀⠀⡄⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠛⠻⣿⣿⣶⣾⣍⣿⣿⣿⣿⢿⡮⣛⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⠀⣹⠀⠀⠀⠸
⠀⠀⠠⠄⠐⠂⠠⠀⠀⠀⠀⠀⠤⡀⡀⠀⢸⡟⠃⢀⠔⣠⢤⣭⣿⢿⣿⡟⠛⠻⣧⡁⠀⢾⣿⣿⣿⣿⣆⠀⠀⠀⠠⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⠸⠃⢀⣞⠞⠁⠀⠈⠙⣿⡛⣿⣷⣼⣿⣿⣴⣗⣹⡻⣿⣿⣿⣦⠀⠀⠀⢷⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣨⠇⠀⠀⠀⠀⠨⣁⠈⠻⣿⣿⣿⣧⣿⣷⣿⣿⣿⣿⣿⣧⠀⠘⢻⡇⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⡟⣦⠀⠀⠀⠀⠰⢾⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⡀⠈⢿⡀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⡴⢋⠸⠀⠀⠀⠀⠀⢰⣿⣤⢠⠀⢻⡝⢏⢹⣿⣿⠻⣿⣯⠻⢿⣿⣦⡀⠃⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⢸⠀⠀⠀⠀⠀⠀⢹⣧⠀⡠⠈⢳⣾⡆⣿⣿⣧⠘⠻⡗⠠⣀⠙⠃⠈⡀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡥⠀⠀⠀⠀⠀⢀⣼⣿⠗⣨⡱⠆⡔⢿⡈⡌⠛⣿⣇⢰⣮⠳⣹⠏⠀⠂⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠁⠀⠀⠀⠀⠀⢘⣿⡿⠀⠉⠀⢰⡷⡌⢧⣘⡀⣿⣿⡘⠁⢀⣄⠀⠀⠀⠀⠀⠀⠙⠷⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠀⠀⠀⠀⠀⠀⣠⣺⣿⡟⠀⠀⠀⠄⠉⠁⠀⠘⣿⣷⡈⠿⣧⠀⠉⠁⠘⠚⠖⠀⠀⠂⠀⠀⠙⠒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⡀⠠⠖⢹⣄⡷⣸⣿⡆⠀⠀⠀⠀⠀⠠⡄⠀⠐⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠻⣇⢿⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡈⠑⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣷⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠘⠀⠀⠀⠀⠀⢨⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡜⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠘⢷
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⣉⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠤⡤⣤⣶⣿⣿⡍⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠈⠋⠛⠆⠶⠿⠏⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⡀⠀⠉⠀⠄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⡀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣶⣤⣤⣶⣤⣀⠀⢽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠙⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣿⣿⡿⠻⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠂⠘⠉⠀⣀⣠⠶⠋⠙⠻⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠉⠀⠀⠀⢀⡴⠖⠋⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⢀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠄⠀⣀⠀⠀⠀⠀⠀⡯⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⡞⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⠀⢀⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠙⣞⠙⠿⣧⣀⡀⠀⠀⠀⠀⢀⡴⠖⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣻⣟⣗⢀⠀⠀⠀⠀⠀⢰⢤⠀⠀⠀⠀⠀⠀⢠⠆⣸⠀⠀⠀⠀⢀⠀⢤⠜⠿⣾⣟⣻⡁⠁⠀⠀⠀⠹⢫⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣠⣿⣿⣿⣿⣾⣟⣷⡀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣀⠜⠚⠃⠀⠀⠠⠼⢿⡧⡄⠀⠀⠀⣸⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠉⢾⣿⣿⣿⣿⣿⠣⡟⢻⡟⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⢺⠁⠙⢃⠀⠀⠸⠇⢠⢸⡗⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⢏⣈⣩⣀⣞⢀⣷⡇⣿⣿⠃⠠⠀⠀⠀⠀⠰⡆⠀⢸⡇⠀⠙⡀⠀⠀⣀⠀⡁⠀⣸⡀⠷⠀⣴⠛⣢⡰
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠀⣰⣻⣴⢯⡮⠬⡌⡇⣸⣏⣿⢷⡿⠐⣴⠂⠀⠀⠀⢀⡦⠈⠄⢻⡟⠠⣗⠀⠀⠙⡌⠁⡔⣸⣥⣷⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣠⣤⣭⣾⣟⣩⠭⢲⣵⣷⡿⢾⣫⠿⠁⣰⡯⣧⠀⠀⠀⡾⣷⡄⠐⣌⣡⠀⠉⣇⠀⠉⠑⠀⠻⡍⢝⣇⡀⠀⢈⣤⡴⠶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⢹⣿⣿⣿⣿⡀⣽⣿⠟⢛⠀⣸⠻⠃⠰⣿⠃⠸⠄⠀⢀⠏⣽⠁⠀⡻⣿⣿⣆⠈⠛⡛⠋⣡⣿⠛⢸⡟⠛⠷⡌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠏⠘⠁⠀⠸⠘⠏⠀⡀⣦⠛⠀⢠⣇⠀⢘⠀⠛⡄⠀⠃⠀⠀⠛⠿⡟⠻⢷⠿⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠃⠀⠀⣭⣤⡿⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⣄⠀⣀⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⣰⠄⠹⣼⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡤⣱⢀⣀⠡⡡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⣒⠄⠬⠩⠛⢫⣃⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⡜⣬⢟⢦⡭⡫⠵⠔⠦⢌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⡾⠝⠥⡳⢄⢤⣽⠇⣑⠴⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣞⡛⠴⠳⢪⣃⠤⡞⠰⡍⠰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⣦⣩⣉⣒⣉⡡⡭⠚⡠⠧⣶⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢱⣿⡾⣯⣿⣟⡿⠋⢡⡾⣟⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣴⣿⣿⣭⣿⣿⣟⢱⣥⣼⡿⠿⠁⡀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⣺⣽⡿⠋⢉⡀⡚⡦⠿⠖⣰⣰⣫⣥⣔⣺⣦⣶⡿⠾⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⢄⡤⢂⣵⢾⢫⢗⣩⣶⡟⣙⣋⣽⣼⣶⡶⠿⠿⠾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣾⠷⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣀⢖⡵⠋
⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠘⠅⢀⡠⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⢄⡀⠀⠀⠀⠀⢺⡀⠄⢣⠘⡁⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢤⡌⢢⡀⠀⠀⢰⠯⠔⠧⠋⠁⠫⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⡰⠽⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣠⠔⢛⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⢸⡘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢀⠔⠁⠀⠀⠀⠣
⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁⠀⡨⢳⠦⢼⡬⢼⡅⠀⠀⠀⠀⠀⠀⠀⠈⢆⣠⠖⢁⠂⠀⠀⠀⠀⠀⡐⡀
⠀⢀⣤⡒⠊⠁⡀⢀⢔⣵⠎⡴⠃⠀⠈⣗⣊⠹⣆⡀⢠⡀⠀⠀⠀⣀⡰⡀⠑⠘⠦⠀⢀⠀⠀⠀⠰⡘⢦⡀
⣠⣞⣁⠤⢒⡡⢚⣵⠟⡡⠊⠀⠀⠀⠀⡯⠔⢂⡹⣦⠄⠉⠒⠖⢒⡫⢖⢷⠀⠀⠀⠀⠀⠀⠀⠑⠠⠱⢄⣫⡢⢄
⠉⠙⠖⠾⢽⣞⣛⡥⠊⠀⠀⠀⠀⠀⠀⡷⠊⡡⠒⢉⢳⣞⡅⠀⠀⠀⠪⠔⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠓⠊⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⢀⠜⠁⠀⠉⠷⣇⡼⡀⠀⠀⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠁⡠⠊⡐⠁⠀⠨⠓⢅⣰⢠⢰⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡞⢁⠜⢀⠎⢀⠁⢠⠀⠉⠉⠉⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠎⢀⠎⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢎⠀⠰⠀⢠⠁⢀⠀⠈⠀⢠⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠇⠀⠸⠀⢈⠀⠀⡇⠀⡧⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣧⠀⢸⠀⠀⣇⠀⠸⠀⢄⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣄⢸⢁⠀⣸⢦⡀⢧⠀⠑⠚

⠀⠀⣤⣶⣦⣄⡀
⠀⠘⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣥⣶⣶⣬⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠿⣿⣿⣷⣶⣾⣿⣿⣾⣶⣶⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⢀⣼⣟⢹⣇⡬⢋⡿⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⡴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⡿⠇⠈⢩⡴⠿⢣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⢿⣿⣿⣶⣽⣿⣿⣿⣿⣿⠿⡝⣿⣻⣟⣿⡗⠀⠀⣀⡀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢻⠿⢿⣿⣿⣏⣿⣿⣿⣿⣿⢻⣏⢷⣴⠋⠀⠀⢱⠰⡂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠉⠛⡻⣿⣿⣿⣿⣿⣗⣶⡽⠃⠈⠠⢠⠌⠆⣂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂⠦⠉⠉⠝⠋⠩⠅⡐⠀⠁⠈⢖⡈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⢹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣲⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠈⡅⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢳⡄
⠀⠀⠀⢰⡛⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀⣾⠀⠀⠀⠹⡄⠀⠀⠀⢀⣀⣀⣤⡤⠤⣤⣄⡀
⠀⠀⠰⡏⠉⠉⠒⠲⠭⣝⡲⠤⣄⣀⡀⢀⡞⠃⠀⠙⢥⣀⣀⣠⡤⠽⠖⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠈⠉⠒⠪⡟⢙⠇⠀⠀⠀⡀⠙⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⣗⣲⠶
⠀⠀⠀⠀⠀⠀⣞⠋⠉⠉⠉⠉⠉⢹⠃⢸⡀⠀⠀⠀⠀⠀⠀⡀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠷⠆
⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠚⠛⠋⢹⠀⠀⠳⡀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠴⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠈⠉⠚⠒⠒⠒⠉⠀⠀⡀⠈⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⣀⡤⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⠦⣄⣘⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢽⣷⣦⣭⣽⡟⢲⣖⠒⠒⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣏⣿⣯⢻⡀⠀⠀⠀⠀⣠⠴⠚⠋⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢶⣤⣶⠋⣁⣀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡻⣽⠁⠈⣳⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢳⡬⠷⡞⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⣦⣠⣷⠒⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢠⡇⠀⠘⣧⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠋⠀⠀⠀⠈⠓⠖⠚

⠀⠀⠀⠀⠀⠀⠀⠓⠶⠦⣄⡀
⢀⡈⠙⣯⡙⠀⣀⣄⠀⠀⡜⢿⡄
⠈⠙⢇⠀⠀⠀⠙⠋⠀⠀⠹⣿⣿⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⢠⢂⠀⠀⠀⠀⠀⠀⠆⢨⡟⣿⣿⡀
⠀⠘⢏⠀⠀⠀⠀⠀⠖⠀⠀⠁⠀⠈⠃⠀⢀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠘⠄⠀⠀⠀⠀⠀⢹⣦⠀⠀⠘⠀⢀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠛⣤⡀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡀⢶⠏
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠷⣦⠀⠀⠀⠐⠦⣈⠀⠄⡀⠀⠀⠀⠈⠳⣄⠀⢀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠨⠞⡯⠃⣰⣤⡓⠂⠄⣙⠂⠀⠁⠂⠀⠀⠀⠈⢓⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠐⠨⠙⢮⢽⣻⣷⣦⡤⣉⡐⠂⠤⠄⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠒⠠⠀⠩⣿⡿⠿⣿⠿⠿⠿⣶⣶⣤⣤⣤⣄⣀⣀⡀⠐⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢰⡟⠀⣠⣾⡿⠋⠈⠳⡝⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠀⢠⣟⣠⣾⡿⠋⠀⠀⠀⠀⢉⢦⠈⠻⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠏⢀⣼⣬⣿⣿⠊⠀⠀⠀⠀⠀⠀⠀⠘⠱⣄⠈⠢⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠷⣦⣝⢾⣿⠿⠛⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣠⣤⡄⣰⠿⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣉⣤⣴⡿⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠀⠀⡠⠼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⡧⠠⠫⠀⢀⡀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣲⣄
⠀⠀⠀⠀⠀⠀⢠⢾⢕⡓⡃⠀⢀⠔⡈⡳⠻⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢰⣦⢨⠛⣦
⠀⠀⠀⠀⠀⣴⡽⣞⠷⠊⡃⠈⠅⠙⢋⣴⠀⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⣹⠀⠰⣬⠛⣓⡬⠧
⠀⠀⡤⠊⠀⠀⠘⠩⠯⠇⠀⠀⠉⣠⣆⠀⠀⠀⠀⠀⠀⡰⠋⣤⠄⢀⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⡄⡁⢀⡀⣉⢀⣁⠱⣞⢧
⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⢀⠀⠈⢹⡼⣚⡟⠅⠠⠘⠀⠀⠀⠀⠀⠀⢸⣾⣿⣮⣥⣤⣿⣿⡟⠀⣣⡷⣄
⠀⡾⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣐⣿⠀⠀⠀⠀⠀⢸⣴⣃⠰⠆⢹⠁⡾⡃⠸⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠤⡹⡆
⠀⣯⠄⠀⠀⠀⣻⡁⠀⠀⠀⠀⣐⢺⣽⡇⠀⠀⠀⠀⠘⣏⡽⠟⠠⣝⣤⣤⣕⣸⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠪⢗
⠀⣵⠮⠄⠀⠀⣻⡇⠀⠀⠀⣒⣚⣶⣽⠃⠀⠀⠀⠀⠀⡗⠖⠍⡌⠱⠸⣿⣿⣿⣆⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣍⢺
⠀⢾⣼⣬⣦⣄⣿⣇⣤⢴⣶⣴⣯⢿⡟⠀⠀⠀⠀⠀⢸⣛⢛⠉⡦⠼⣂⡘⣿⣿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢷⠀⠱⣾⡇
⠀⠸⣿⣯⣿⣍⣿⣿⢘⣿⣴⡿⢰⣿⠁⠀⠀⠀⠀⠀⣾⢰⣄⠀⢐⣓⡒⣄⡻⣿⣿⣿⠀⠀⠀⠀⠀⢟⠻⣿⣿⡿⠋⠁⠀⠀⢸⡧⡽⣾
⠀⠀⣯⣋⡛⠺⣿⣿⣁⣋⣮⣴⣵⣯⣶⣀⣀⠀⠀⢠⣯⡿⣥⢊⣃⣂⣉⣤⢶⣻⣿⣿⡇⠀⠀⠀⠀⠈⡆⢀⠉⠀⠀⠀⣠⠀⠀⣿⠿⣿
⠀⠀⠸⣟⡛⣹⣿⣿⠙⣋⣻⢿⣿⣇⡤⠀⠀⠈⠀⢸⣯⡯⡷⢿⣽⣽⡯⣟⡯⣇⣿⣿⠁⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠐⠁⠀⠀⣿⣧⡇
⠀⠀⠀⢫⠹⢿⣿⡿⠟⢹⣿⡟⠈⠁⠀⠀⠀⠀⠀⢽⣿⣟⡿⡚⢶⣶⣛⡧⢾⣥⣿⠏⠀⢀⠀⠄⠐⠂⠀⠀⠈⠂⠒⡒⠀⠚⠒⠻⠿
⠀⠀⠀⠈⣆⠞⢻⣋⢃⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣟⡢⠿⢿⣾⢷⢟⣻⣿⡟⠀⠈⢁⡤⠀⠀⠀⠈⠓⢻⣿⡌⠉⠛⠏⣽⣿⡖
⠀⠀⠀⠀⢸⡜⡞⢫⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⢹⡿⢫⠿⠳⢞⣱⣿⠟⠀⢀⣀⣀⣉⣉⣉⣁⡀⠀⠀⣿⣷⣄⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⣿⢿⡟⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠶⣤⣴⣴⡾⣻⣿⣿⣯⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢹⣷⣿⡏⣿⡇
⠀⠀⠀⠀⠀⢸⡆⠘⣻⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣤⢦⣴⣟⣵⣿⣿⣿⡿⠊⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⢻⣿⢣⣿
⠀⠀⠀⠀⠀⠸⣷⣦⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠄⢀⠾⣿⡿⠋⠉⠈⠛⠃⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⣿⠁⣿⠀⠀⠀⠀⠀⠀⠀⣾⡧⢠⡔⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣶⣿⠀⠀⠀⠀⠀⠀⣼⣣⡶⣫⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠼⣿⣠⡾⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢇⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣠⢞⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣰⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠸⠟⢠⣞⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⢠⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢔⣠⠴⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣖⣛⣭⠄⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⢀⣤⣾⣿⣿⣿⣿⣟⢫⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⣿⣿⣿⣄⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣷⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⠟⠋⠉
⠀⠀⠀⢒⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠠⠟⢻⣿⣿⣿⣿⣿⣧
⠀⢈⣩⠽⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠠⠞⠩⠿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠟⠋⠋⠁⠀⠘⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⢀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠐⣶⣿⣾⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏⡇
⠀⠀⠀⢾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡄⠄
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡤⠄⠀⠀⠀⠈⠙⢛⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢙⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⡒⠀⢠⣾⣿⣿⣿⣿⡄
⠀⠀⢀⣤⣀⠀⠀⠻⣿⣿⣿⣿⡏⠙⠛⠿⠿⠿⣿⣿⡿⠿⣿⣿⠿⠮⠉⠃⠀⠀⠀⠉⡿⠁⢹⡿⠻
⠀⣉⣉⣻⣿⣿⣶⣦⣄⠋⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣶⣶⣶⣶⣚⣉
⠀⠀⠀⠚⠿⠿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣐⣠⣤⣀⣰⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠒
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣩⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣶⡶⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⢠⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⡀⠀⠉⠉⠙⠛⠿⠿⠿⠻⠿⠻⠿⠋⣽⣿⣿⣿⣿⡉⠉
⠀⠀⠀⠀⠿⠛⢿⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⢻⡎⠳
⠀⠀⠀⠀⠀⠠⠟⠁⠁⣼⠏⠀⠀⠀⠘⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠗⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠈⠙⠁