Recreation Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⣿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠙⠿⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⡿⠟⠁⠀⠀⠀⠀⠈⠻⣷⣟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣱⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣎⡳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣵⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⣷⣤⡀
⠀⠀⠀⠀⣠⣴⣿⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⣽⢦⣄
⠀⠀⠀⠀⠉⠹⠋⠁⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠈⠻⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⢀⣀⡀⢶⡶⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⢀⣀⣻⠿⠟⠃⠀⠈⣛⠁⣴⣦⣀⠀⠘⠿⠿⣟⣀⡀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⣶⣤⣴⣾⡿⢻⡏⠻⢿⣷⣤⣶⣿⣿⣿⣷⡄⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⡟⢿⣿⣟⠛⠁⠀⣼⣧⠀⠀⠙⢛⣿⣿⢻⣿⣿⣿⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢰⣿⣿⣿⡇⠀⢹⣿⣀⣀⣀⣿⣿⣀⣀⣀⣸⡿⠁⢸⣿⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠈⣿⣿⣿⣇⣀⠀⠀⠻⣦⡀⠀⠀⢀⣼⠟⠀⠀⢀⣸⣿⣿⣿⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠙⠿⠿⢿⣿⣿⣿⠆⠘⢿⣄⣰⡿⠁⠰⣿⣿⣿⡿⠿⠿⠋⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠽⠿⠿⢿⡯⢀⣿⡟⠀⠀⢠⡿⣯⡀⠀⠀⣿⣿⠈⢿⡿⠿⠿⠯⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⢸⣷⣾⣿⠃⢀⣴⠟⠁⠈⢻⣦⠀⢸⣿⡇⣿⡇⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣼⣯⣽⣭⣤⣾⣯⣤⣤⣤⣤⣽⣷⣤⣽⣥⣿⣧⣤⣤⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⢻⡷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⠝⡡⠃⡇⢸⠈⢎⠻⡢⣄
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡟⡿⡟⠻⢻⡟⠛⡟⢻⠟⠛⠛⡇
⠀⠀⠀⠀⠀⢸⡏⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⢠⡱⠀⡇⣌⠇⢀⠃⢸⡄⠀⢰⠁
⠀⠀⠀⠀⠀⠀⢻⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⢀⢧⠃⡸⢰⡟⠀⡸⠀⣆⠇⠀⡎
⠀⠀⠀⠀⠀⡠⡿⠛⡹⡳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡜⠀⣎⠇⢠⠃⣾⠁⢠⠃⢰⢹⠀⢠⠃
⠀⠀⠀⠀⡰⢱⢱⠀⠇⠘⣌⢪⡢⣄⠀⠀⠀⠀⠀⡰⠁⡜⠀⣜⠎⢠⠃⣼⠇⢀⠎⢀⠆⡆⠀⡜
⠀⠀⠀⡜⢀⠇⢸⠀⠸⠀⠈⢆⠙⢦⠑⢄⡀⢀⠜⢀⠎⢀⢎⠎⢠⠃⡼⡏⢀⠎⢀⠎⡘⠀⢰⠁
⠀⠀⠚⢟⠻⡿⣯⠻⡿⡥⠶⣼⠷⠶⡷⠶⣝⢯⡴⠁⡠⣣⠊⡰⠁⡜⡜⣠⠋⢠⠊⡜⠀⡠⠃
⠀⠀⠀⠀⠙⠿⣎⢢⡑⣌⢦⡈⠢⡀⠈⠢⡈⠢⡙⢮⣴⣡⠞⣠⣾⣯⠞⢁⡴⡡⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠿⣮⡳⣝⠢⣈⠢⢄⡈⠲⢌⡓⢮⣓⠿⢿⣿⣥⣶⣿⣫⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠮⠽⢶⣭⣶⣤⣍⣓⣮⣽⣳⡶⠭⠝⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠈⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠀⠈⠑⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂
⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⡀⠀⠀⢠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣶⢰⣗⣿⣴⣸⣧⣿⣸⡧⣆⣾⣾⣷⣼⣧⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⡇⡀
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⠿⠿⠟⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠃⠛⠀⠀⠉⠀⠀⠀⠀⠀⣤⡀⢹⣏⡤
⠈⠉⠉⠙⠛⠿⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣈⣩⡷⠒
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠓⠒⢂⡓⠒
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠛⠦⠭⣽⣿⣒
⠀⣀⣶⡄⣀⣘⣃⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⣽⠆⠸⢾⣉⢻⣟⣡
⠘⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢍⠴⣾⣛⣁⠀⣉⠙⣟⣡
⠀⣿⣿⣿⣿⣿⡿⣧⣦⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⠀⠀⠭⣽⣿⠥⠶⠀⢈⣿⣿⣥
⠀⠻⠿⢿⣿⡿⠟⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠷⡿⠾⠅⠐⠚⠻⠃⠈⠐⠫⠟⠻⠯

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠄⠀⠴⠶⠶⠶⠶⠄⢶⠀⠘⠒⠚⠋⢿⠙⠙⠛⠛⠛⢻⠛⠃⠐⠃⠁⢰⠄⠤⠴⠶⠶⠦⠤⠤⠤⠤⠀⠀⡀
⠀⠀⠀⢠⣖⢉⠉⣛⠁⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⢚⠀⠀⠀⠀⠀⣒⠀⠐⠒⠒⠒⢘⠂⠀⠀⠒⠒⢺⠀⠂⠀⠐⠂⣒⠐⠒⠀⠀⠂⢘⠛⠋⣰
⠀⠀⠀⠀⠘⠉⠑⠛⠢⠤⠤⠤⢤⣴⣤⡤⠤⠀⠀⢼⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢼⠀⠄⠀⠀⠀⠤⢤⠤⠤⠤⠄⠚⠒⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣠⠏⠀⠀⠀⠀⠀⠀⠸⣇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠁⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣒⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣂⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠐⠒⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠄⠂⠀⠁⠘
⠀⠀⠐⢀⡀⠤⠤⢀⠀⠰⢄⠈⠀⡀⣀⠀⢂
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠈⢤⠈⠅⠠⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠱
⡇⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡂⠈⠐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁
⠁⠈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠈⠀⠈⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣤⣴⠆⠀⠀⠀⠀⠀⠀⣀⣘⣧⣀⡀⠀⠀⠀⢠⣤⣿⡀⠀⠀⠀⠀⠀⠰⣦⡾⠙⣿⣿⣿⡇
⠀⢀⣀⣿⣿⣉⢳⣀⣀⠀⠀⠀⠀⠀⠙⣯⠉⠉⠁⠀⠀⠀⠘⣿⣿⠙⠃⠀⠀⠀⠈⠛⢻⣷⣀⠘⠿⣿⡇
⠀⢾⠀⢈⣧⡟⣏⠀⢈⡆⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢠⡏⢻⡄⠀⠀⠀⠀⠀⠀⠀⢷⠘⠇⠀⠙⠇
⠀⠈⠉⠉⠈⠀⠈⠉⠉⠀⠀⠀⠘⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠘⢦⣿⡴⠃⠀⠀⠀⠀⠀⢀⣠⣤⣠⠀⠀⠀⠀⠀⢰⡆⢤⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣷⣦⣄⡀
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠈⠓⣀⣤⣀⠀⠀⠀⠀⠀⠙⠉⠛⠉⠛⠉⠛⠁
⠀⠀⠀⣸⠀⣟⡀⠀⠀⠀⠀⠀⠙⡿⠿⡿⠃⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡄⠀⠀⠰⣝⢦⣄⢀⣀⡀
⠀⠀⣀⣽⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠾⠿⢿⡿⠁⠙⠿⠄⠀⠀⠈⠳⠽⡯⠭⠽⡤


⠀⠀⠀⠀⠐⣿⣿⠂⠀⠀⠀⠀⠀⢠⢼⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢤⡛⣤⡀⠁⣀⠀⠀⠀⠀⠀⣠⣿⢸⣷⠀⠀⠀⠀⢰⣦⣠⣾⢿⣿⣦⡀⠀⠀⠀⠀⢠⣾⣿⣿⣷⢐⡀
⠀⠀⠉⠪⣻⣛⣉⣛⣂⠀⠀⠰⣾⣿⣾⣿⣶⠂⠀⠀⢽⣷⣿⠋⢸⣿⣿⣏⠀⠀⠀⣶⡿⠿⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠀⠀⠀⠀⠁⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣶⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢠⠶⣦⣠⣤⠀⠀⠀⠀⠀⢸⣁⣀⣏⣀⣸⣿⡟⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁
⠀⠀⡀⢀⣶⣿⡿⠟⠓⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢰⣤⣤⣾⣯⣭⣭⣤⡆
⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠙⠚⠉⠉⠉⠉⠑⠊⠀⠀⠀⠀⠀⠀⠉⠙⠟⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⠀⠀⣀⡀⠀⠀⠀⠀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠂⡞⠀⠐⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⠀⠀⣠⠊⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣐⠦⠔⠃⠘⡠⠃⠀⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢄⠈⢄
⠀⠀⠀⠀⠀⠀⠐⠁⠀⡐⠁⠀⠀⢀⡝⠶⠒⠉⡸⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠁⡹⠋⠀⠀⠆⡰⣡⣤⣿⣦⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠿⠟⠛⠛
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠄⠚⠀⠒⢒⡒⠘⠛⠛⠛⠛⠛⠓⠴⠦⠤⠼⠦⣘⣻⠂⠀⢓⡀⠄⣢⡆⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠦⠄⠀⢀⢷⠏⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢈⠔⠀⡸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠴⢄⠠⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡴⠺⣮⠄⣞⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⣀⣑⡛⠋⠺⡡⠃⠸⠂⠒⠒⠂⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠀⠀⠁⠂⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠀⢀⣄⣠⣤⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿
⠀⠀⠀⠀⣨⣶⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣤⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⢚⣿⣿⣿⣿⣿⡿⢿⣿⣶⣶⣄⠀⠀⢤⢰⣎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢄⣤⡄⠸⣿⣿⣿⣿
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣶⣿⣳⠛⢿⣿⣧⣄⠈⢹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠼⠶⠂⠀⠟⠛⢻⣿
⠀⠀⠘⡻⣿⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿
⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢸⡆⠀⣀⣀⣤⣄⣤⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠀⠀⠀⠈⠻
⠉⠛⣻⣿⣿⠏⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣠⣶⣶⣆⠀⢸⣿⣿⣦⣄
⠀⠀⠈⠉⢣⣾⣿⣿⣿⣿⣿⣿⠛⣿⣿⠿⠛⠂⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣿⣿⣷⣼⠟⣿⣿⣿⡀⠀⢈
⠀⢀⠀⠀⠸⠟⠙⠉⣿⣿⠛⣛⠂⠀⠀⠀⠀⠀⢸⣇⠽⢛⣟⠻⣿⡿⠿⠫⡏⣸⣿⠝⢛⣿⣿⠿⣿⢿⣆⣽⣿⣿⣇⠀⢨
⠀⣿⣇⣠⣀⠀⠀⠀⢸⣟⢠⣿⣶⡄⢰⣿⡄⠀⢸⣿⡗⢺⣿⠖⣿⠀⢃⡀⠁⠙⣿⡆⠀⢸⡇⠀⠉⣾⣿⡿⢿⣿⠃⠀⠠
⠀⣿⣯⡸⣿⣂⡀⠀⢸⣿⣄⢻⣿⣧⣿⣿⠆⢴⣿⣿⣿⡞⡿⠇⠛⣰⣿⣿⣧⠀⡟⢇⠀⠀⠁⠀⠀⣹⡉⢽⣖⣿⣀⡀
⠀⢹⣿⣼⣿⣿⣷⠀⢸⣿⣿⣸⠿⣿⣿⣿⣆⡏⢹⣿⣯⠹⡄⠀⠀⣽⣿⣿⣿⡐⠗⠘⠃⠀⠀⡔⠉⢹⠈⠛⣯⣿⣅⠈⡆
⠀⠸⡿⠏⣿⣿⣿⡇⢀⣿⣿⣿⠃⠈⣿⢿⡟⠃⢸⣿⣿⠀⠁⠀⠀⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⣆⢀⡜⠀⠀⠾⠿⠤⠜
⠀⠀⠁⠀⣿⣿⡟⠃⠉⣿⣿⡟⡇⠀⣿⠈⣧⠀⠀⣿⡏⠀⠀⠀⢸⠇⣿⣿⣇⢻⡄
⠀⠀⠀⠀⢿⣟⠀⠀⠀⣿⣿⡇⠁⠀⢻⠀⠹⠄⠀⣿⡇⠀⠀⠀⡟⢸⣿⣿⣿⠈⣧
⠀⠀⠀⠀⠸⣿⠀⡀⠀⢹⣿⣧⡀⠀⠈⠀⠀⠀⠀⠛⠃⠀⠀⢠⣇⠈⣿⣿⡏⠀⠉⠀⠀⢠⡀⠀⢀⡀⠀⢠⠀⠀⡀⠀⡀
⡇⢸⠀⣿⢴⡿⠧⣷⠀⣿⣿⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⢸⣿⠀⠀⠀⠀⠀⢸⡗⠒⢺⡗⠒⢺⡗⠚⣗⠒⣿
⠇⢸⠀⣿⠐⡗⠐⣿⠒⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢸⣿⣀⣼⣿⣤⣿⣿⣿⣿⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠘⣇⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣬⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣾⢿⣿⣷⠻⡖⠄⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⡀⠀⠿⣟⠁⣙⣿⠇⠀⢀⣾⣴⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣾⣿⣷⠀⠀⠑⡀⠈⠁⠀⠙⣌⢻⡟⢦⣸⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⠈⠀⣼⡟⢻⣟⣍⠳⢹⡁⠀⠀⠙⢆⠀⠀⠀⠀⠛⠻⢻⣿⣽⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣾⡄⠀⠫⠟⠄⠘⠋⠑⠘⢿⠄⠀⠀⠈⠁⠀⠀⠀⢤⡀⠀⠙⠀⠘⠟⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢛⣿⣛⠯⡳⠄⠀⠀⠀⠀⠀⠀⠠⣈⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠂⠀⠀⠀⠀⠀⠈⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⢯⣻⣟⠣⠈⠀⠀⢀⠀⠀⠀⠀⠀⠈⠑⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠰⢄⡀⢤⡀⠠⣠⡀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⡠⢊⡀⠓⠀⠀⠈⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢄⠀⠈⠀⠀⢀⣴⣶⡠⣀⣀⠀⠁⠠⣄⠀⠈⢀⡀⠀⠀⠀⠙⣦⣄⡀
⠀⠀⠀⠀⠔⠁⠀⠃⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡠⢄⡤⢴⣾⣿⠛⢣⡴⢌⠀⠴⠞⠻⣿⣿⣨⠉⢉⣱⡦⣀⣀⡠⠈⣉⣩⣵⣦⣄⡀⠀⠈⠙⠲
⠀⠀⠠⠖⠶⠶⠶⠒⠒⠛⠉⠙⠻⢿⣿⣶⣆⠤⠊⠈⠁⠀⠀⣈⠥⢠⣤⠎⠁⠀⠀⠀⠀⠀⠉⠉⣉⡽⠚⡿⠟⠁⠀⠀⠉⠙⠋⠯⠭⠭⠷
⠀⠀⠀⠀⠐⠒⠒⠒⠶⠦⢤⣶⣶⣯⣭⣵⣆⣒⣒⣒⣒⣒⣊⠒⠲⠾⢁⣐⣛⣛⣛⣛⣛⡁⠀⠀⠉⠀⠀⠠⠤⠄⠀⣤⣤⠤⠤⠤⠤⠤⣌⣉⣀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠉⠁⠩⠤⠵⠖⠒⠒⠂⠃⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠁

⠀⠀⡄⠀⠀⢀⡀
⢠⡄⣇⡆⢠⢸⡧⠀⢰⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈
⠸⣿⣿⡧⢸⣾⡷⢸⣿⡇⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡀
⠀⠈⠉⢻⡾⠿⣿⣸⣿⣇⣿⢸⡇⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠙⠉⠉⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⢿⣾⢠⡀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡀⠰⡆⠸⡇⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣩⢠⢵⠀⡁⢠
⠀⠰⠃⢄⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢂⣤⢌⢠⡯⠷⠠
⢨⡇⢾⣍⣈⣰⣖⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢨⡈⣰⢦⣄⣫⡷⣾⣙⠈⠄

⠀⠀⠀⠀⢀⡔⣰⡖⢶⡄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣀
⠀⠀⠀⢠⡟⢰⣿⡇⢸⣿⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡅⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠸⡇⢸⣿⡇⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⢢⠀⠀⠀⠀⣤⢾⡼⢦⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠘⠿⣷⣶⣤⣄⣀⣠⣤⠤⣤⣤⣄
⠀⠀⠀⠀⠻⡈⣿⡇⣸⡿⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⠟⢻⡇⠀⣿⣶⣶⣾⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⣿⣿⣿⣏⣹⣧⣬⣿⣤⣿⡿⠟
⠀⠀⠀⠀⠀⠈⣎⣓⣛⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣼⣿⣄⣼⣿⣿⡇⠀⣿⠟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⢻⣿⡿⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣸⣀⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣉⣿⠿⣤⣿⡇⠀⣿⣄⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣾⣿⠇⠀⠙⠿⠻⠟
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⢿⣿⠟⣿⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⢠⣄⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠿⠿⢿⡦⣿⣿⡟⠛⠂⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⣿⣿⠂⠙⠉⠀⣠⣤⣤⣤⣶⣶⣶⣿⣷⡀⠀⠀⠀⠀⠀⢤⣤⣀⣤⡤⠀⢠⣿⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⡇⠀⠀⣰⣿⣿⣿⣿⡿⠿⠛⢻⣿⣧⠀⠀⠀⠀⠰⣿⡿⢿⡛⠆⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠘⠋⠀⠀⣰⡿⣽⣿⣿⣿⣇⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠉⠀⠈⣧⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⣛⣿⣯⣭⣭⣴⣶⣶⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠰⢿⠇⣿⣿⣿⣿⣿⣶⣶⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣼⣛⣛⣋⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣒⣒⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⣿⢟⣭⣽⡻⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠆⣉⢐⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣆⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢺⣿⣿⢇⣿⠀⠀⢸⠋⠉⠛⡗⠀⠀⠀⠂⠭⠀⠀⠀⡝⠀⠈
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆⠀⢀⣆⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣮⣵⣾⣿⠀⠀⣿⣀⠀⢀⡇⠀⠀⠀⠀⠀⣀⣀⣀⣧⣀⡀
⠀⠀⠀⠀⠀⢠⣿⠛⡛⠻⣿⣿⣆⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣉⠉⠙⠛⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣇⠀⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⣿
⠀⠀⠀⠀⠾⠿⠿⠷⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⢶⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋⠉⢂
⠀⠀⢰⣶⣶⣿⣿⣶⣶⡆⠀⠀⠀⠀⡍⠉⠉⢹⠀⡤⢤⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠐⠲⡶⠓⠁⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣦⣴⠏
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⠇⠀⠀⠘⡆⠀⠀⠄⠀⠀⠀⢻⡿⡿⠿⠿⡿⢿⠇⠀⠀⠀⠀⠀⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠸⠿⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣦⣤
⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢘⠀⣿⡴⢸⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠿⡿
⠀⠀⠀⠀⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠐⠀⠟⠑⠸⠛⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠁⠋⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⠌⣡⠀⠀⠀⠀⠀⠀⠀⠈⣿⢀⣴⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⣿⠟
⠀⠀⠀⣸⣿⡟⢻⣿⣇⠀⠀⠀⠀⠀⠀⠘⣴⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣼⣿⡄⠀⠀⠀⠀⠀⠀⢾⣿⠞⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⡶⠗⡴⠷⢦⠀⠀⠀⠀⠀⠀⣿⢈⣽⣿⣿⠉⡇⠀⠀⠀⢠⣾⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄
⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣤
⠀⠀⠀⣸⣿⣿⣧⣶⣆⠀⠀⠀⠀⠀⠿⠟⠻⠿⠿⠛⠟⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠞⠿⠿⠛⠁
⠀⠀⠰⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇

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

⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠂⠀⠀⠈⠀⠨⠆⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠉⠉⠀⠀⡇⠙⠀⠀⠀⠀⡀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⡁⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠁⠭⠀⠀⠀⠀⠀⠀⠀⠠⠘⠄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠀⠀⠀⠀⠀⡜⢹
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡘⠀⠸
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠁⢠⠃⠀⡏⠀⠠⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠚⠀⠠⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡤⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠉⠉⠁⠈⠹⣁⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠒⢠⣧⠔⠀⠀⠀⠀⠁⢀⠓⠀⠀⠀⣁⠀⠀⠖⠦⠏⠁⠈⠉⠀⠀⠀⠀⠀⠀⢀⡉⠁
⠀⠀⠀⠐⠁⠔⠀⠈⠛⠀⠒⠠⠀⠀⠔⠁⠀⠀⠀⠀⢨⠂⠀⡄⠤⡤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡀
⠀⣀⡠⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⠀⠀⠀⠀⠅⡂⠀⠀⠀⠀⠐⠍⠩⠑⢄⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠈⠁⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⠀⠀⠀⠀⠀⠀⠃⠉⠀⠌⠀⠀⠀⢠⠀⠀⡇⠀⢌⠆
⠀⠀⠲⠬⠃⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠞
⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⡀⠀⠀⠀⠀⢺⢂⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠈⠉⣽⡀⠀⠀⠀⡌⠀⡇⠀⡀⠀⠀⡀⠀⠀⠙⡛⠓⠛⠦⢀⣀⢀⠀⡄
⠀⠀⠀⠨⠇⠀⠀⠀⠀⠇⠀⠀⠘⠀⢧⡀⢀⠋⠁⠀⠀⡐⠀⠠⣡⠀⠿⣀⠈⣡⢈⡇⢰⡃⢀⡄⣴⣀⠝⡿⢠⣁
⠀⠀⠀⠀⠐⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠂⠀⠙⠂⠀⠉⠉⠉⠉⠁⠈⠉⠀⠁⠈⠁⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠉⠑⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠐⠹⣀⡀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⢂⡀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣖⠁⠐⠀⠈⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠳⣄⣀⠠⠂⡄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⢻⡱⠮⣿⢻⢄⠀⢀⠌⣉⠿⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠄⠘⠒⠒⠊⢸⡞⠛⣡⣎⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠳⠀⠀⠀⠀⠈⣋⠉⣁⣀⣁⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⠀⠀⠀⠀⠀⡠⠤⠐⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣾⠀⠀⠀⠀⠀⡏⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠀⠀⠀⠀⠀⡗⠠⣄⠀⠀⠀⢣
⠀⠀⠀⠀⠀⡴⠚⡛⠓⠧⡌⡄⠀⠀⠀⣀⣀⡃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠇⠀⠈⢓⠀⠀⡆
⠀⠀⠀⠀⢸⠁⡀⠀⠀⠀⠘⢢⠀⡠⠊⠀⠀⠀⡌⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠙⡏⢣⠈⠳⣀⠤⠼⠦⠮⣄
⠀⠀⠀⠀⠘⣎⠀⠘⠄⠀⢀⣺⣸⠀⠀⠀⠀⡸⠁⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⡔⠁⠀⣷⣠⠁⠀⠀⢀⣀⣈⣆⣀⡀
⠀⠀⠀⠀⠀⠈⡱⠦⠒⠒⠻⣴⠉⡄⠀⠀⢖⡃⠀⠀⢰⠁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⠀⡀⠀⣉⠆⣠⡛⠶⢤⡴⡐⠉⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡷⠜⡱⢲⣤⣧⣤⣖⠋⢰⠃⠀⠀⣩⣽⣲⠆⠀⠀⠀⢰⠒⠉⠙⡌⢠⠖⠒⠼⢎⠀⠉⡲⠊⠠⠁⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⡠⠖⠒⠋⠉⠳⡒⠚⢤⠖⡡⠋⠀⠀⠀⠀⠙⠻⡇⠰⢀⡿⣡⠞⠀⠀⠀⠀⡼⣤⠀⠀⡸⠁⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⢧⠀⠀⡷⠗⠒⠒⠒⠒⢤⣀⢀⡽⠿⠋⣰⠃⠀⠀⠀⠀⠀⢧⠙⢦⡔⡧⢄⡰⢀⣀⣀⣀⣇⣀⣀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⠄⠀⡇⠀⠀⠀⠀⠀⠸⡎⢻⡐⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⣀⠜⢱⠉⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢷⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠒⠀⢀⠒⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⢀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠢⠈⠀⡀⠀⠀⠀⠀⢈⠀⠀⢁⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠑⠂⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠈⠀⠀⠁⠄⠄⢀⠀⠄⠀⠀⠀⢁
⠀⠀⠀⠀⢀⠀⠒⠤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠈⠖⠀⠀⠁⠀⢀⠂
⠀⠀⠀⠀⡅⠀⠀⠀⠁⠂
⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⡀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠁
⠀⠀⠘⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡈⠀⠀⠀⠄⠀⠀⠀⠀⡄⠀⡄⢡⠀⠂⠀⠀⠀⠀⠰⠄⠠
⠀⠀⠀⡀⡀⠰⠀⠀⠀⡀⠀⢀⠠⠀⠀⠀⡀⠀⡀⠀⠀⢂⡀⠀⠀⢈⠀⠂⠀⠀⢢⠡⠀⠂⠀⠔⠀⠀⠀⠀⠤⠀⢇⠀⣁⢀⠀⠀⠀⠀⠀⠁⠒
⠀⠀⠀⠀⠤⠀⠈⠀⠀⡀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠈⠉⡀⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠀⢂⠃⠙⠆⠀⠀⠀⡄⣀⡀
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⠉⠀⠀⠄⠂⠀⠀⠀⠀⠀⠠⠏⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⢀⠠⠆⠀⠤⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣴⣻⣏⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⢀⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⢠⡄⠀⠉⠀⠀⠠⠄⢘⠀⠀⠀⠒⠘
⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠂⠁⠀⠐⠀⠀⠀⠃⠠⠤⠀⠀⠀⠀⠸⠁⢜⠀⠀⠀⠈⡤⠊⠀⠀⠁⠀⠀⡈⠀⠈
⠀⠀⠀⠀⠠⠂⠀⠢⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠈⠀⠀⠀⠀⠁⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠘⠀⠀⢀⡆⠦⡀⠀⠀⠰⡄⠰⣤⠀⠀⠀⠀⠀⢀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⣀⡀⢛⡛⡛⠆⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣻⣷⣤⡀
⠀⠀⠀⠀⣿⣽⣿⣿⣿⣿⣷⣷⠶⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⠿⠿⢧⡀
⠀⠀⠀⠀⠈⠉⠙⠙⠛⠯⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢄⣠⣾⣤⣠⡼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⡶⢶
⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠻⢿⢷⣦
⠀⠀⠀⠀⠀⡄⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠿⠿⣿⣿⣿⣶⣄⣾⣿⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠀⣈⣉⣀⠰⡛⠃
⠀⠀⠀⠀⠀⠣⣄⠰⡧⢀⡠⠃⠀⠀⠀⠀⠀⠀⢀⣀⠛⠻⣿⠟⠿⣿⠟⢿⡿⠛⢿⡿⠛⣿⡿⠋⣀⡀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⣿⣿⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠔⠚⠂⠤⠤⠤⠤⠐⠓⠂⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠘⠛⠃⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣴⣶⣶⣬⣯⣭⣕⣀⣀⣼⠁⡄
⠀⠀⠀⢠⣬⣧⣤⣤⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠙⠛⠿⢿⡟⠛⠋⠀⠈⠁⡇
⠀⠀⠀⠘⢿⣿⣿⠋⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣾⠉⠻⣆⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣴⡏⣸⣷⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠚⠀⠘⠓⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠱⠦⣀⠀⠀⠀⠀⠀⠃⠀⠰⠛⢆⠀⢁⠼⠶⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠈⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠐⠿⠍⠐⢂⠈⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠲⣦⣤⣀
⠀⠀⠀⠀⠀⢠⣿⣴⢿⣿⣧⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠁⠀⠀⠀⠀⠀⠰⢤⣀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣻⣦⡀
⠀⠀⣠⣴⣶⣦⣶⣴⣶⣶⣤⣭⣩⣀⣀⣠⣶⠋⠀⠀⢀⣾⣷⡟⠀⠀⠀⠀⠀⢿⣷⣾⣿⣷⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣅
⠀⠀⠘⠿⠛⢿⣿⣿⣿⡿⠿⠿⣿⠋⠄⠙⠳⠀⠀⠀⡾⠟⠋⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⠻⠿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠏⠙⠛⣲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶

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

⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠐⢿⠉⠉⢙⡟⠂⠀⠀⠀⠀⢸⣇⣿⡆⢸⡆⠀⠀⠀⠀⣠⣼⡓⣲⣀⣀⡄⠀⠀⠀⢠⣦⣼⣭⣉⣉⣣
⠀⠀⠀⠘⢿⣿⠿⠁⠀⠀⠀⠀⠀⣻⣏⣿⡅⣸⡇⠀⠀⠀⠀⠙⠿⣿⠿⡿⠟⠃⠀⠀⠀⠛⠛⠻⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉

⠀⠀⠀⢀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⣻⣿⣿⣿⡗⠀⠀⠀⠀⢀⣀⣀⣾⣓⣄⣀⠀⠀⠀⠀⢀⣾⣷⣾⣿⡆⠀⠀⠀⠀⠀⢩⡏⣩⢹⡏
⠀⠀⠀⠿⣿⣿⣿⠟⠀⠀⠀⠀⠘⢟⠟⠛⠛⢟⡿⠀⠀⠀⠀⠸⣍⡟⠹⣍⡿⠀⠀⠀⠀⠀⢸⣷⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠰⣿⠗⠖⠀⠀⠀⠀⠀⠀⣀⣀⢸⣴⣳⠂⠀⠀⠀⠀⠀⢶⡿⣿⣷⠆⠀⠀⠀⠀⠀⣠⣾⡆⣿⣦⡀
⠀⠀⠀⢠⠏⡏⣠⣴⠀⠀⠀⠀⠀⣿⣛⣟⣟⣿⠃⠀⠀⠀⠀⠀⢸⡇⣛⣻⠀⠀⠀⠀⠀⠚⢛⣿⣧⣿⣟⠓
⠀⠀⠀⠶⠿⠿⠿⠿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠓⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⡴⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⣼⠦⣰⣦⠀⠀⠀⠀⢀⣠⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⢎⡀
⠀⠀⠈⡇⢠⣤⡀⢹⠁⠀⠀⠀⠀⢿⣿⠿⢿⡟⠀⠀⠀⠀⠀⣿⣿⢿⣶⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡯
⠀⠀⠐⠓⠚⠚⠓⠚⠀⠀⠀⠀⠀⠏⠣⠀⠸⠝⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠈⠀⠀⠀⠀⠀⠠⠟⠛⠛⠇

⠑⠦⣴⡞⣳⠄
⠀⠀⢿⠽⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣤⣀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⡀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⡿⠿⢿⣿⣿⣦⣤⣾⣿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣀⣈⠙⠛⠻⢿⣿⣿⣿⢶⣶⣶⡲⣤⡢⣤⣀⠀⠈⠻⣿⢹⣿⡟⠻⢌⣛⣻⣿
⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣭⣗⣄⡀⠀⠻⠿⣿⣕⡮⠿⣿⣿⣿⣿⣿⣷⣦⣴⠿⣾⡟⠋⠀⠀⠛⣿⡇
⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣳⣦⣤⣀⠙⠿⣷⣗⡻⡛⠻⢿⣿⣏⠁⠀⠈⠉⠓⠦⡴⠞⠉
⠀⠀⠀⠀⠀⠈⠙⠷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣷⣝⣷⣦⣄⣹⣿⣿⣷⣤⣉⠙⠻⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⡿⠿⠭⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣾⣿⢿⣿⣻⣿⣿⠟⠁⠀⠈⠙⠻⢦⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣄⡀⠀⠀⠀⢀⣤⢖⣫⣙⠛⢿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣶⣿⣷⣾⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡉⠛⠷⣦⣤⡤⠤⠄
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣾⣵⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣡⣤⣼⣿⣶⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢤⣤⡶⠯⠷⠆⠈⠁
⠀⠀⠀⠀⠀⣠⡴⢞⣫⡥⠀⠀⣀⣀⡈⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⢀⠏⣤⡤⠤⣵⠄
⠀⠀⠀⣠⢾⣿⠞⢉⡿⠗⠋⢉⣽⣋⣀⣀⣀⣀⠈⠲⣄⠀⠀⠀⠀⣎⣀⣬⣽⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠁⠀⢀⣴⠯⠭⣄⣀⠈⠳⡀⠀⢰⠀⢉⡻⠂⢀⣠⠴⠒⠉⠁⠀⠀⣀⡀⠈⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢯⣍⡀⠙⡄⠘⣠⠎⣠⠖⠉⣀⣀⡐⠿⣖⠢⢄⣙⣍⠓⢦⣽⡷⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⢤⣌⡓⢌⣾⣣⠞⣁⣤⢭⣿⣬⠉⠉⠻⠆⠀⠈⠛⠀⠀⠈⠇⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⢀⡄⢀⣠⡶⢂⣀⣿⣶⣇⣨⣟⢿⠛⢒⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⣾⣠⠋⣧⠎⠶⠚⠉⢰⠉⢹⢨⠀⢿⠳⣄⢸⡇⢰⡆⢈⡓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣴⣼⠈⠟⠀⠋⠀⠀⠀⠀⠈⣹⠅⣾⠓⠋⠀⠘⠟⢻⡞⣿⢸⡇⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⣿⠀⠀⠀⠀⠀⠈⠀⢽⡜⢷⣿⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⢿⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡿⣷⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠤⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣠⠴⠻⡄⠀⠀⣀⢀⡾⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡄⢹⡀⠀⠀⠀⣀⠙⣆⢰⣿⡎⣧⠊⠔⢲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢦⣣⠀⢷⠒⠚⠋⠁⠀⠈⢷⣇⠁⠹⠸⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠙⠀⣸⠦⠀⠀⠀⠀⢀⡤⠬⠿⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠁

⣏⠉⣹⣋⣛⣛⣛⣛⣏⣛⣛⣛⣟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡐⢦⡀⠀⣸⡇⢀⣤⣤⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠿⢿⣿⣿⣄⣿⣷⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣿⣿⣿⣿⣿⣾⣿⣾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⢻⣿⣿⣿⣿⣿⣍⠉⠛⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⢠⣾⠃⢀⣀⣀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⣾⣿⠟⠿⠁⢻⣿⡄⠀⠀⠉
⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣿⣷⣿⣷⣿⣟⠻⢿⣷⡄⠀⠀⠀⠀⣸⠏⠀⢀⣼⣿⡗⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠁⠀⠀⢀⣠⣴⡿⢿⣿⣿⣟⠙⠛⢷⡄⠈⠁⠀⠀⠀⠀⠟⠀⢠⡾⢹⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⡿⠉⠀⢠⡿⣿⣿⣿⣆⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠈
⠀⠀⠀⠀⠀⠻⠀⠀⠀⣿⡇⠀⢻⡝⢿⡄⠀⠸⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠻⣎⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⣿⣿⠀⣿⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣾⣿⣿⣾⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠻⠿⠿⠟⠛⠛⠃
⣿⣿⣿⣿⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣛⣿⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣤⣦⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣄⠀⠹⣿⣿⣇⠀⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⣐⣺⣿⣿⢀⣿⡿⢂⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣅⣘⠿⣿⣷⣼⣿⣿⣾⣏⣴⣿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣫⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⣠⣶⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⡿⣿⣿⣗⠙⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣯⣾⡿⠛⣵⣿⡟⠀⣿⣿⢻⣿⣿⣿⣿⠋
⠀⠀⠀⠀⢀⣀⡀⢀⢱⡇⣰⣴⣶⠦⠀⠀⢠⠟⢁⣾⡿⠛⣡⣾⣿⡿⠀⠀⣿⣿⠈⠘⢻⣿⣿⡄
⠀⠀⠠⢄⣛⣿⣿⣿⣆⣿⣿⣿⠁⠀⠀⢀⠀⠀⠸⠉⠠⣾⣿⣿⡿⠁⠀⠠⣿⣿⠀⠀⠀⣿⡿
⠀⠀⠀⠀⢈⣹⣾⣿⣿⣿⣿⡗⠀⣠⣿⣿⣿⣷⠄⠀⠀⠉⠉⣿⠃⠀⠀⠚⣿⣿⠀⠀⠀⠉
⠀⣠⣤⣴⣾⣿⣻⣿⣿⣿⣿⢇⣼⡿⠟⠉⠉⠉⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠁
⠀⠈⠉⣉⣭⣭⣿⣿⣿⣿⣿⣾⣿⣷⣾⣿⣿⡿⠇⠀⠀⢰⡟
⠀⠀⠺⠟⢋⣻⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣷⣧⡀⠀⢀⣿⠁
⠀⠀⢀⣴⠿⠻⢫⣿⣿⣿⣿⠈⣿⠿⣿⣦⡉⠻⠻⡀⣼⡇
⠀⢀⡾⠁⢀⣴⡿⣹⣿⢻⣿⣇⠘⣇⠘⣿⡏⠀⠀⢰⡿
⠀⠈⠀⠀⢸⡿⠱⣿⡿⠈⣿⣿⠄⠘⣆⠉⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠈⠉⠀⠀⢹⡄⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣿⣾⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⣴⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣼⣿⣿⣿⣿⣿⣿⣿⣯⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡺⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⣿⠑⠰⢭⣽⣷⣴⣶⣤⣤⣄⢀⣀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣮⠳⠀⠚⠋⠉⠉⠉⠉⠉⠁⠀⡿⠿⣿⠂
⠀⠹⠟⠦⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠟⣩⠀⣿⣿⡄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠉⠑⠀⢤⣄⣀⠀⠀⠀⣠⣄⠀⠀⢸⠛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠿⠿⠂⠀⠈⠑⠯⠄⠀⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⣀⡭⠵⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣤⣶⣶⡤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣼⣿⡿⣿⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠾⠃⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠤⢖⡂⠠⣿⣯⡿⠐⢯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣟⡳⣿⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣤⣤⣄⣀⠐⠀⠤⠤⣀⣠⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣷⣦⣤⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉



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

⠀⠀⠀⢠⣼⣦⣠⣾⣥⡄⠀⠀⠀⠀⢀⣭⣿⣦⣴⣶⣂⠀⠀⣈⣿⣧⣾⣯⡄⠀⢤⣶⣿⡆⣠⣴⣤⡀
⠀⠀⠀⢀⣼⣿⠛⣿⡿⠗⠂⢀⣦⣄⣾⣾⠉⡟⠉⣿⢹⠀⠈⢩⡿⢻⣿⠏⠋⠀⠿⣿⠁⡆⢿⡍⠻
⠀⠀⠀⠀⢰⠃⠀⠈⠛⠀⠀⠛⣟⠙⡏⠃⣼⠁⠀⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⡇⠀⠛
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣀⠀⠀⠀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⠐⣦⣼⣀⡤⠀⠀⡼⣿⣷⣴⣿⣿⠂⠀⢐⣶⣿⣷⣴⣿⣏⠀⠀⢀⣠⣤⣾⣿⣯
⠀⠀⢀⣰⣦⡿⠿⢍⣿⡿⢿⣗⠀⣾⢹⡏⢩⠉⠙⠃⠀⠈⢹⡟⣹⠻⣿⠿⠁⠀⠽⢿⠛⢋⠙⠣
⠀⠀⠠⠟⣿⠙⢦⠏⠀⡟⠀⠈⠀⠈⠀⠁⠘⡆⠀⠀⠀⠀⠈⢁⡇⠀⠉⠀⠀⠀⠀⠈⠀⠘⡄
⠀⠀⠀⠀⠈⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢸⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡟⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣯⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠇⠀⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⣰⢏⣤⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠀⡀⠀⢀⣼⡿⠛⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣽⣿⠇⡴⢁⡴⠛⠉⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣾⣷⠟⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣿⠋⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣾⠇⢸⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠡⢪⠏⢠⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢠⠏⠀⠁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢠⠏⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⢠⠟⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⢠⠏⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢠⠏⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⢫⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠛⠉⣠⠴⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣆⡼⠃⠀⠀⠀⠀⠀⢂⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⣠⣾⣯⡶⠞⠁⣀⣀⣤⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⢘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⢀⣠⣾⠿⠿⠿⠛⠛⠉⠉⢁⡴⠛⠛⠛⠛⠛⠒⠒⢒⠲⣿⣭⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⡿⠁
⠀⠀⠀⠀⠈⠉⠉⠙⠓⠶⢶⣿⣿⣿⣿⠯⠭⠍⠹⠍⢩⣇⠛⠦⠀⠀⠀⠀⠀⠈⠉⠉⣭⡛⠛⢿⡛⠻⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠁⠀⣭⡽⠿⢿⣶⣶⣦⣤⣭⣙⣛⣓⣒⣒⣒⡒⢒⣒⣂⣈⣠⡤⠞⠁⢀⡄⠀⢀⣠⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣠⠄⠀⠀⠙⠻⣷⣄⡉⠉⠉⡿⠛⠛⠿⣿⣿⣿⣷⣶⣶⣶⠾⠋⠀⣊⣥⡴⠾⠛⠋⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⠀⠀⠀⠈⠻⣿⣶⢦⣤⠴⠀⠀⢸⡟⠻⠿⣷⣶⣶⣶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⣶⣦⣀⣠⣤⣤⡀⠀⢀⣠⣤⣄⣀⣤⣶⣤⡀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠛⠛⢻⠿⠛⠛⢿⣿⣿⠟⠛⠻⢿⠟⠛⠛⢿⣿⣿⣿
⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣧⡄⠀⢀⣴⣿⣿⣿⡿⣷⣦⡄⠀⠀⠀⢸⣿⣿⣿⣿⠛⠉⢁⠀⠀⠀⠘⠻⣿⣿⠛⠁⠀⠀⢀⡈⠙⠻⣿⣿⣿
⠀⠀⠀⠛⠉⠀⣾⠁⢻⣷⠈⠛⠀⠛⠁⣼⡟⠈⢷⡀⠉⠛⠀⠀⠀⢸⣿⣿⣿⣧⣶⣿⠏⣼⣧⠈⣿⣦⣽⣧⣶⡏⢠⣿⡄⢻⣷⣦⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠋⠀⠀⠀⠀⠀⠙⠀⠀⢸⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⢰⣿⣿⣧⣿⣿⣿⣿⣿⣧⣾⣿⣷⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⣠⡶⠢⠄⢀⣀⣀⠠⠐⠶⣄⠀⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢹⡿⢋⣽⣿⣿⣿⣿⣿⣿⣍⠻⢿⠁⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⡆⠈⠁⠢⠠⡄⣤⠠⠤⠔⠊⠁⠀⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢸⣿⣾⣿⣿⣿⣿⣿⣛⣿⣽⣾⣿⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡄⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡜⣧⠀⠘⣿⣿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⢹⡇⠀⠸⡇⠀⠉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡇⠀⢀⣼⣿⠀⠀⠀⡀⠂⠀⠈⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⡇⠀⢈⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠖⠐⠂⠀⣀⠀⣀⣠⡤⠖⠐⠛⠛⠀⠀⠀⠀⢸⣿⢀⣺⣿⡿⣇⠀⠀⠀⢠⠀⠀⠀⠀⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠟⠉⠀⢹⣷⡀⠀⠈⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⢸⣿⡳⡄⠀⠀⣀⣲⡄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠘⢿⡇⣿⠀⠀⠸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⢀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠁⠀⣼⣂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⢻⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠖⠶⠾⡿⢭⠭⠭⢽⣟⣛⣛⣛⠶⠦⠤⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⡴⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠻⠶⠶⢶⣶⣿⡇⠀⠀⣠⣾⣿⣷⣷⣿⣤⣀⡀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠊⠉⠛⠻⣿⣿⣟⣓⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣦⢀⣤⣆⢠⣄⡀⠀⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣤⣴⣶⣶⣾⣿⠿⠛⠿⠇⠘⠓⠛⠸⠛⠓⠘⠓⢛⣳⠒⣶⣶⠶⠒⢀⣠⣤⣤⣴⡶⠶⣛⣛⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣤⣬⣭⣽⣿⣿⣿⡏⠉⠉⠉⠀⠀⠀⠀⣀⣀⣀⣀⣛⣛⣣⣬⡭⢭⣾⡿⣭⡿⠿⠛⠋⠉⠭⠀⠀⠊⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣶⣾⣾⣿⣿⣆⣡⣄⣀⣀⣀⣀⠀⠁⠈⠁⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣯⣭⣉⣭⣿⠍⠉⢉⣙⡋⠉⠉⠉⠀⠓⠂⠒⠂⠂⠀⠐⠀⠀⠀⠀⠀⢛⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡐⢒⣊⣛⣛⣛⣻⡿⠇⠚⠉⠉⠉⠙⣁⣤⣤⣼⠿⠟⠋⠉⠉⠀⠛⠷⢶⣾⡿⠛⠉⠁⠀⢀⣉⣷⠞⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⠉⣉⣀⣀⣀⣀⣀⣀⡀⠉⠉⣉⣉⣀⣀⡀⡠⠄⠀⠀⠀⠐⢉⣠⢤⣤⢤⣬⠉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠿⠛⠛⠒⠒⠒⠒⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠊⠁⠈⠀⠀⠀⠉⠉⠛⠻⠷⠶⠶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠯⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⣀⠀⠉⢳⠢⠄⠄⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⠋⠙⢷⣌⣀⣀⣀⣤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡒⡪⢊⠍⢠⠋⠀⠀⠈⠛⠲⣒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡖⢳⡀⠘⢄⠀⠀⠀⠀⠀⡀⠑⢍⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⢄⡋⠒⠄⣉⠉⠉⠁⠐⠢⡄⠈⠆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠙⠒⠤⢄⡉⢴⢤⠔⠒⠓⠒⡼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡄⠀⠀⠘⢣⣼⠞⠴⠐⢺⣝⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠁⣠⠞⠀⠀⡰⠡⠊⡌⠀⠀⡻⡏⠉⢹⡤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣙⠏⠁⠀⠀⠀⡰⠱⠁⢰⠀⡠⠊⠀⠈⠁⠊⠀⠀⠀⠉⠉⠉⠙⠛⠒⠈⠋⠓⠲⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠰⠁⡇⢀⠆⠀⠣⡀⠀⠀⠀⠀⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡇⠀⠐⣎⠀⠀⠀⠈⠂⣄⠤⠊⣁⠔⠊⠀⠀⢀⡠⠔⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠃⠀⡘⠀⠑⢢⣤⠔⢉⡠⢓⣍⠀⠀⣀⠴⠊⣁⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢸⠀⠀⣃⡠⠒⣁⠤⠛⠓⣶⠋⣨⠖⢉⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⢠⢾⣶⠾⣁⠔⠊⠁⠀⢀⡮⢓⣽⠡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢀⡰⠟⡫⠕⠈⠀⠀⠀⡠⠒⣁⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⡡⠔⠉⠀⠀⠀⡠⠒⣩⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⣁⠔⠉⠀⠀⢀⡠⠐⡩⠔⠉
⠀⠀⠀⠀⠀⢠⠒⠐⠂⣁⠔⠊⠀⠀⢀⡠⠒⢁⠔⠊
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠨⣁⣈⡡⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣾⣿⣿⣿⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣷⣿⣯⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⣿⣿⣿⣿⣿⣍⡁
⠀⠀⠀⠀⠀⠉⠉⣛⣿⣿⣿⣿⣿⣿⡛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⣾⠿⢛⡽⣿⣿⣿⣿⠻⣿⣦
⠀⠀⠀⠀⠀⢠⣾⡿⣿⣿⣿⣿⣿⡻⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣴⠋⣸⣿⠏⢿⣿⡄⠀⠙⠇
⠀⠀⠀⠀⠀⠟⠁⠀⣿⣿⠹⣿⣏⠻⣄⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡴⠟⠁⠀⢘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠈⠛⠦⠘⢦⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⣿⣿⣴⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⡠⢀⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣧⡅⠢⡀⠐⣿⣦⡀⠀⠠⢊⣶⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣻⣧⠬⠢⡙⣿⡰⠠⢁⣾⠛⢁⢄⣤⡶⠦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⠿⡷⣶⣤⣐⠺⣗⡑⣘⣧⡇⣽⣭⣯⣴⠟⠛⢉⣁⠠⣄⢤⡨⣤⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣈⠍⣉⠁⡛⠻⠿⣵⠢⡘⣯⣅⠟⡵⢋⣰⣦⡙⠧⠙⠳⠙⠃⠙⠈⠑⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣵⢟⡾⣻⠇⡾⢃⡾⣱⡾⡄⠨⣙⢺⣼⣿⠏⠔⢃⣨⣍⣭⣭⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠀⠀⠀⢠⣂⣥⡦⣶⡶⠖⠐⣾⣡⠛⡏⢙⡭⠥⠬⣍⢩⡉⠛⠫⢛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢛⠝⠃⢡⣠⢞⣩⠿⣀⣸⢈⡿⠫⡐⢦⣦⣼⣳⡗⢰⡄⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⡄⠿⢈⣧⣾⢃⡈⠀⡇⡁⢨⡌⢦⡄⠩⠙⠋⠇⠘⢱⣷⡲⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣴⣆⠻⠀⠘⠇⡇⢬⡏⠀⠁⠁⠐⣯⡈⡼⡂⠀⠀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠸⠉⠀⠀⠀⠀⠡⡸⠍⢀⡄⣀⠠⠞⠀⠘⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠦⠘⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠒⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣴⠏⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⡄⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠈⢿⣆⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣏⣀⣀⣠⣏⣀⣀⣀⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣒⣩⣭⣥⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣭⣭⣉⡒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠴⠶⠾⠛⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⢢⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠞⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢊⣁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉
⠀⠀⠀⡠⠔⠒⢊⣉⣭⣭⣼⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣭⣭⣉⡑⠒⠢⢄
⠀⠀⠀⠙⠢⠤⣈⣙⡛⠿⠿⠿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⠿⠿⠿⢛⣋⣁⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⢖⣒⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⣒⡲⠽⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣤⣄⣀⡀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠦⣤⣬⣍⣉⣁⣈⣛⠛⠛⠛⠛⠀⠀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡘⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣄⠀⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠙⠒⠶⠤⣄⣤⣤⣶⣶⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣭⣭⣛⡻⠟⠛⠋⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⡿⠋⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠨⣽⣛⣲⠦⣤⣄⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡒⠤⢭⣙⣛⠻⠟⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠾⢿⣿⣛⡶⠦⣤⣀⣀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣍⣀⠐⠤⢌⣙⡛⠿⠿⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠾⢭⡽⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣭⣍⣒⣒⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⣦⡀⢀⣴⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠻⠿⠿⣷⣬⣿⣾⣷⣶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠿⣿⣿⣿⣿⣿⣿⣷⣿⠿⠿⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠶⣤⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣾⣟⣻⣿⣿⣿⣿⠏⣿⣿⣿⣿⡿⣿⣷⣌⠛⢿⣦⣀⣀⡀⠀⠀⠀⠀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⣾⢩⡾⢋⣿⣉⣸⣿⠇⢰⣿⣿⣿⣿⣧⡈⠙⠻⣆⠀⠙⢿⣍⡙⠻⣶⣤⣼⣯⣶⠶⠺⠲⢤
⠀⠀⠀⠀⢲⣴⣿⣿⣿⣟⣁⣤⣼⣧⣄⣸⣿⡟⠉⣿⣿⣷⠀⠀⣠⠶⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⡛⠓⠦⡀
⠀⣠⠖⠟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢿⡇⠀⠀⠙⣿⡆⠐⠋⠀⠀⢀⣤⣶⣿⣿⣿⣿⣹⣿⣽⣷⣤⡀
⠀⠀⣴⡾⣻⣿⣿⣿⣿⣿⣿⠿⣿⣦⡈⠀⠘⠃⠀⠠⠀⠈⠃⠀⠀⣠⣴⠿⣫⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⣆
⠀⣼⠏⡼⢻⣿⣿⣿⣿⣿⣿⣷⣄⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣴⠟⠛⣿⣿⣿⣿⡇⠸⣿⠈⠻⠀⠉
⠀⠇⠀⠀⣸⡏⣼⡟⢸⣿⡇⠙⢿⣇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⣿⡇⠀⠸
⠀⠀⠀⠀⣿⠁⠹⠀⠀⢿⡇⠀⠀⠉⠣⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠿⠁
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠨⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣇⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⢀⣷⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠧⠿⠛⠾⠿⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣠⣀⠀⠐⠋⣀⣀⣀⣀⣀⣠⣤⣤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣿⣾⣿⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠋⣷⡖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠓⠚⠉⠁⠀⠀⠀⡄⠂⣈⠽⠟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣟⣿⣿⣿⣿⣻⣿⣿⣿⠋⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡉⡀⠠⠀⠤⣶⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⢿⣿⣿⣿⣿⡿⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⢠⠉⠾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣨⢉⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⣤⠈⢒
⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⢯⠄⢀⠀⠀⠀⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠼⠟⠄⠒⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⠹⠄⠀⠀⠀⢀⣤⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⠋⠀⠐⡢
⠙⠙⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⣾⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠬⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣾⣿⣿⣿⣿⣿⡿⠻⠛⠛⠃⠀⠀⠐⠁⠀⠀⠀⠀⠘⠉⠠⡾⠊⠀⢀⣴⠡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⠟⢠⡄⠀⣰⣶⣆⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⢀⡀⠀⠀⠀⠀⠀⠴⠃⠈⠉⠀⠈⢁⢩⠛⢟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠁⢩⠿⣿⣿⣿⢿⣷⠀⠈⠝⠲⠆⣠⠀⠣⢄⣥⣤⣾⣴⣿⣟⡷⣮⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠃⠀⠈⢻⣿⡾⣿⣔⣂⠀⠰⠂⣀⣘⣧⣬⣽⣶⣿⣿⣿⣿⣿⣿⣋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⡀⠀⠀⠀⢀⣹⣷⣽⣿⣿⣷⣿⠿⠟⠛⠛⢻⢿⠋⣿⣿⣻⡿⣿⡛⣷⣿
⠀⠀⠀⠀⠀⢐⣀⠀⠀⠀⠀⠀⠀⢖⡞⣨⣤⣶⠶⠿⣛⣿⣿⢿⡛⠛⠉⠁⠨⢧⠀⠚⠺⠤⣧⠰⣿⣞⣽⣄⣽⣯⣿
⠀⠀⠀⠀⣀⡻⢿⣤⣤⠃⣀⣀⣀⡮⢸⣙⣯⠿⠿⠛⠋⠁⠀⠈⠃⠀⠀⠄⠀⠀⠀⠀⡀⠤⢤⣄⠙⣭⢿⣿⣿⠏⢁
⠀⠀⠀⠀⠉⠁⠂⣀⡠⠔⢛⡠⠁⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠐⠚⠂⠤⣬⡯⠟⣻⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠿⡄⢀⠀⠀⠀⠀⠀⣴⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣾⣿⠟⣿⣇⠀⠈⠲⠈⠁⠀⠴⣾⣿⢻⣿⣦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠋⣿⣿⢸⣿⣿⣆⠀⢀⠀⠀⠀⠀⠀⠘⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠈⠃⠀⠀⢺⣷⠀⡏⢹⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⣴⣿⠟⠇⠀⠀⠀⠀⠁⠀⠀⢾⢟⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠹⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣿⣿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡄⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠚⠏⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣯⣿⣿⣿⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠣⢿⣿⣿⣿⠋⠛⠿⣿⣿⣿⣿⣷⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣟⠀⠀⠀⢿⣇⡄⠀⠀⠀⠈⠉⠉⠛⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣽⣲⣶⣦⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣵⣿⣿⣟⣽⣯⣿⣄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠻⡿⣿⣆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠻⣿⣿⣿⣷⣽⣯⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠈⠻⣿⣿⣻⣯⣟⣿⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣦⣄⣀⠀⠀⣿⣿⣿⣷⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣶⣿⣿⣿⣿⣎⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣯⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢽⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣷⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣾⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠒⠀⠉⣙⣻⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣴⣧⣾⣷⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡹⢿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⡩⠿⣿⣿⣄⠀⠀⣠⣾⣿⣟⠻⠿⠟⢛⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠑⣞⣿⣿⣦⣴⣿⡿⠯⠝⠋⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣟⣻⣷⡶⣶⣶⣾⣿⣟⢿⡷⡶⣖⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⡿⣿⢿⣟⡿⣿⣿⢒⣿⣿⣟⣽⣿⣿⣿⣿⣯⣿⡷⣿⡉⠛⠟⠚⠓⠁⠙⠈⠎⢿⠛⡏⢻⢶⣄
⠀⠀⠀⣀⣴⣿⢿⣿⡿⢫⢞⠞⡡⠋⠊⠐⠋⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠙⡷
⠀⢀⣼⣿⣿⢫⡾⠋⠐⠁⠈⠈⠀⠀⠀⢀⣾⣿⣿⣿⡿⢱⠏⢻⣿⠻⠟⢻⣿⣿⣿⣿⣆⡀
⢰⣿⡟⣹⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⠙⠀⠀⠀⣸⣿⠀⠀⠠⠘⢸⠘⢿⣿⣿⣦
⠨⡟⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠀⠘⣿⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡝⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⠁⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣫⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢻⡿⣄⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⠈⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣚⣩⣵⣶⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠋⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣊⣥⣶⡿⠿⡿⠟⣛⣉⣭⣤⣴⣶⣶⣶⠾⠿⠿⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠉⠀⣰⣿⣿⣿⡿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⣀⠴⠛⠉⠀⠀⠴⠞⠛⠉⠁⠈
⠀⠀⠀⠀⠐⠈

⠀⠀⠀⠀⠀⡄⡀⣀⡀
⠀⠀⠀⠀⢀⠀⠣⠍⠙⢙⠒⠦⣄⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠑⠚⠋⠉⠉⠀⠀⠉⠉⠙⠻⠦⠄
⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡴⠀⠐
⠀⠀⠀⠀⠀⠉⠉⣹⢯⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⡛⠛⠛⠻⠓⠒⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠈⠀⠀⠀⠀⡀⠚⡁
⠀⠀⠀⠀⠀⠀⠘⠉⠻⢿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠁⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⠁⠀⢐⡢⠆
⠀⠀⠀⠀⠀⢈⠀⠀⣀⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⣸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⢿⣿⣿⣿⡿⠋⠁
⠛⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣤⣶⣶⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠩⠿⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋

⠀⠀⠀⢠⡴⠖⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠲⢶⡄
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⢻
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼
⠀⠀⠀⣿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡴⠞⠁⠀⣼
⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣴⠞⠁
⠀⠀⠀⢹⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠁
⠀⠀⠀⠘⣧⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠘⢷⣄⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋
⢀⣤⣤⣤⣤⣤⣤⣤⣬⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣯⣥⣤⣤⣤⣤⣤⣤⣤⣄
⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟
⠀⠀⠀⠉⠙⠛⠲⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠚⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠒⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣠⣶⣾⠟⠉
⠀⠀⠀⠀⠀⠉⠻⣿⣷⡄⠀⢀⣴⣿⡿⠋⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣆⠀⢠⣿⣿⠋
⠠⠴⣶⣶⣶⣤⣤⣘⣿⣿⣀⣿⣿⠟⢀⣀⣀⡀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⢀⣠⣤⣤⣤⣘⣿⣿⣦⣿⣿⣥⣶⣾⣿⡿⠿⠓⠂
⠀⠀⠀⠉⠉⠛⢻⣿⣿⡿⣽⣿⣿⣾⣿⣿⠿⢿⣷⣦⣄⠀⠀⠀⠀⠀⠰⠾⠿⠟⠛⠻⢿⣿⡿⢿⡷⠺⣿⣿⣯⣉
⠀⠀⠀⢠⡴⠿⠟⠛⠛⠁⢸⠁⠈⢿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠁⠀⣇⠀⠉⠉⠉⠛⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠈⠛⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠛⠉⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⣷⣾⣿⣷⣻⡇⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⣴⠆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣥⡀⠀⢀⣴⣿⣿⠾⠶⠾⠗⠁⠀⠀⠀⠀⢿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⣤⠀⠀⠀⠈⠙⠿⣟⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣤⣄
⠀⠀⠀⠀⠤⠴⣶⣶⣿⣿⣿⣿⣿⣶⣶⣿⣶⣶⣆⣀⣀⣀⣀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠶⠶⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡼⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⡀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⠁⠀⠀⠀⠀⠀⢼⠀⠀⠀⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣤⣤⣤⣄⣀⣀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣀⣠⣴⣾⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿
⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⠴⠾⠿⠿⠿⠿⠿⠃⠀⠀⠀⣇⣀⣀⣀⣀⡤⢤⣤⠶⠶⠶⠾⠿⠿⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠛⠿⠷⠶⠾⠿⠿⠿⣿⣿
⠀⣦⡤⠀⣀⣀⣀⡀⠀⢰⣶⡦⣤⣤⣄⣀⣀⣀⣀⣀⡛⠋⠉⢙⣇⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⠋⠛⣿⣿⣿⡿⠀⠀⠀⠀⠀⢤⠰⡶⢠⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣲⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣶⣾⣿⡿⠓⠠⠾⠂⢺⣿
⠀⣿⡟⢻⣟⣿⣿⣿⣿⣿⡿⠧⠀⣀⣠⣤⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣥⣄⠀⠀⢠⣤⣾⣿
⠀⣿⣥⣼⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠠⣼
⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠿⠟⠉⠁⣘⣛⠋⠛⢻⡿⠛⠛⠉⠉⠉⠻⠟⠀⠀⠉⠀⠀⠈⠁⠀⠀⠁⠀⠻⠂⠀⠈⠁⠈⠉⠉⠉⠁⡀⠀⠀⠀⠀⢸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⣥⡀⡉⠀⠀⠀⡶⠀⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣬⣉⢐⣷⣼⣿⣾⣿⣾⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⠁⠀⠀⠀⠀⠀⠐⠀⢻⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣈⣻⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⠀⠀⠀⠀⠂⠐⢀⣾⣿⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠓⠀⠀⢶⡶⠶⣿⣧⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠛⢿⣿⣿⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⣉⡉⡁⠀⠠⣴⣷⣦⣼⣯⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⡆⣰⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣷⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⡉⠉⠉⠉⠙⠋⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣷⣤⣭⣿⣩⣿⣿⣿⣟⡛⠛⠻⠆⢠⣤⣤⣶⣶⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣶⣦⣀⢀⠀⠀⡀⠀⢀⠀⠀⣠⣾⣿⡏⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣿⢿⣿⣿⡟⢁⡤⣄⣦⣁⢨⣷⢠⣾⣿⣿⠅⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠉⠀⠛⠛⠋⢠⣿⣵⣶⣯⢽⣷⣿⣾⣻⣿⠟⠀⠀⠀⢘⠂⢈⠃⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⣿⣿⠟⣻⢇⣠⣂⠀⠀⠈⠀⠐⠅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢠⡴⣿⣿⣿⣿⣷⣶⣿⣿⣿⣟⣿⠟⣡⣾⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣈⣀⣉⣉⣻⣿⣿⣿⣿⣾⣿⢟⣀⡀⡠⣄⣤⡤⠄⢀⣀⡄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⢿⣿⢸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠘⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⡿⡯⠊⣿⢶⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠃⠁⣀⡾⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⡨⢿⣯⡙⢿⣿⣯⢿⣿⣆⠀⠢⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⣠⣿⠉⠀⠀⢀⣼⡿⢈⣼⣿⢿⡷⠙⢰⣿⡇⢿⣻⣿⣷⣾⣧⣀⠻⣿⣿⡟⣿⣅
⠀⠀⠀⠀⠀⢰⡿⠃⠀⠀⢀⣾⠟⠀⣾⠏⠟⠏⠁⠀⢸⣿⠇⠀⠻⡟⣿⣿⣿⢯⡀⠉⠹⠻⠹⢿⣦
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣸⠃⠀⠈⣿⠊⠀⠀⠀⠀⢸⣿⠀⠀⠀⠁⠊⣷⣯⣟⠆⠀⠀⠀⠀⠸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠃⠀⠀⠀⠀⠀⠸⡽⠀⠀⠀⠀⠀⠉⢹⣾⡇⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⢠⣾⠋⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠘⠻⣦⣄⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠠⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠙⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠿⠟⠛⠛⠋⢹⣷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢶⣄⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣤⣄⠀⠀⣀⣀⠀⢀⣤⣄⡀⣤⣄⣀⣤⣤⣤⣴⠆⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢰⡟⠁⠀⠀⢼⡟⢻⡇⠈⣿⣭⣄⣿⣉⣸⣿⡤⢸⡷⠆⣼⣧⡾⠟⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣇⠈⢿⣦⡄⠀⠘⠷⠟⠁⠀⡿⠉⢠⣿⠉⠹⣷⡄⢸⣷⢠⣿⣵⣶⣟⠋
⠀⠀⠀⢀⣤⣴⠶⠟⠛⠛⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠉⠉⠙⠛⢶⣤⡀
⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣤⣤⣤⣴⣶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣃⣴⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣾⣿⣫⣿⣿⣟⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣟⠉⠉⠉⠛⢻⣿⣿⣿⣿⣿⣿⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡻⣿⣀
⣿⣷⣶⣦⣤⣀⣀⣉⣀⣀⣀⣈⣙⣷⣶⣦⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡿⢿⡾⣿
⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⠿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠦⠠⢤⣴⣶⣶⣶⡦⠀⠀⠀⠀⠀⢠
⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢀⣴⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⠇⠀⠀⠀⠀⠀⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠀⠀⠀⣠⡿⢋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡘⢷⣤⣤⣤⣾⠋⢀⣾⡿⠶⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⢳⣼⡏⠀⣿⣥⠶⠋⠁⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⡿⠏⢀⣀⣘⣏⣤⣤⣄⣀⡀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣠⡴⠖⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⢳⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠿⢧⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠟⠋⠉⠀⢸⡇⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠉⠉⠉⠉⠉⢉⡍⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⠾⠋⠉⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢷⡀⠀⠀⠀⣠⣟⠁⠀⠀⠀⣠⡀⠀⠀⠀⣸⣯⣴⠶⠶⣄⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡀⠀⠀⠈⣷⡶⠖⠛⠉⠙⠳⣦⣴⠞⠋⠀⠀⠀⠀⣿⡟⠁⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠛⢦⣤⡞⠁⠀⠀⠀⠀⠀⠀⣼⣧⣀⣀⣀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⠀⠀⠀⠀⢀⣤⢴⡾⠋⠉⠋⠉⠻⣷⣄⢸⣿⠀⠀⠀⢀⡟⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣴⣿⡄⠀⠀⢠⣿⣁⣀⣀⣀⣀⣀⣠⣤⡾⠟⣸⡟⠀⠀⢀⣼⠃⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠀⠀⢹⣦⣤⣀⣭⣿⣿⣏⣉⠉⠉⠉⠀⠀⠀⣿⣷⣤⠶⠛⠁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⡼⠋⠀⢹⡏⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣿⣦⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠚⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⡴⠶⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠉⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠤⠤⠤⠤⠀⡀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠔⠙⣇⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠢⠀⠀⢀⠃⠀⠀⠈⠙⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⢴⠚⠛⡏⠉⠉⠙⠒⠒⠦⢤⣀⠀⠀⠈⠑⠄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢴⡇⢈⣀⠀⢈⢦⣀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢿⡖⡢⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢾⡇⠃⡸⡿⠓⠢⢄⣀⡀⠀⠀⠀⠱⡀⡠⠃⢀⠤⠒⠈⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠻⣾⠗⣒⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠳⡀⠁⠀⠀⢀⠤⠹⠆⠀⠀⠈⠑⢦⠀⠙⠤⢄⣁⣀⡀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠚⠛⠛⠒⠚⠟⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⢦⡀⠒⠴⠾⠇⠂⠀⠀⢀⣴⠋⠡⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠫⢁⠀⠀⠀⠀⠀⢀⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣈⡑⡖⠢⠤⠤⠶⠞⠋⠀⠀⠀⢙⡦⡀⠀⠀⠀⣀⠀⠀⣀⣀⡀⠀⠉⠐⠂⠄⠊⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢀⡠⠤⠒⠋⠁⠀⠀⠑⠢⢤⡄⢀⣀⣀⠀⠈⠉⠒⠠⠤⢠⡤⢞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠘⠋⠉⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠳⣄⠤⢀⠑⠄⣀⡠⠄⠀⠀⣁⡠⡃⡧⠀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣳⡀⠀⠀⢹⡳⣖⠨⠥⠐⢖⡉⠈⠀⠉⠲⠖⠁⢈⢤⡁⢊⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⠁⣡⠄⠀⡤⠓⠂⢀⠰⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⣨⠧⠬⢤⠤⠴⠦⢒⠒⠂⠀⠀⡠⠀⠙⡈⠀⢜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡒⠈⠙⠊⡉⠷⠚⠒⢊⠇⠀⢣⢢⣄⣀⠀⣀⣀⡠⠴⠒⠋⠁⠀⠀⠈⢆⠀⠀⠀⠈⠀⠒⠈⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⡥⠤⡖⠒⠈⠁⠀⣀⣤⡁⠈⠡⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠙⠒⠖⠒⢋⠁⢀⣡⠶⠚⠉⠉⠉⠉⠉⠙⠒⠒⠲⠦⡤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⡉⣡⣖⠐⠋⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⢀⠀⢷⢄⣀⡠⠎⠀⠀⠤⡔⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠀⠀⠑⢠⡀⠀⠀⠀⢀⣈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠀⢄⣀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⣀⠀⠀⠑⠢⡉⠁⠀⠀⠈⠲⣀⢀⠠⠄⠀⠀⣁⠠⠄⠒⠀⠁⠀⢀⣀⠬⠗
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠛⢠⠀⠄⠄⠒⠁⠀⠀⠀⠈⢦⠄⠂⠉⠁⣀⣀⡤⠤⠒⠚⠉⠁
⠻⣍⠒⠢⠤⠤⠤⠄⠀⠀⠀⠠⠤⠄⠐⠒⢒⣘⡦⠤⣤⠊⠀⠐⠂⢈⣁⣠⡤⠤⣔⣢⡥⠴⠒⠚⠉⠉
⠀⣌⠛⠒⠶⠶⠶⠶⠶⠒⠒⠒⣚⣋⠩⠭⠔⠒⠒⠀⣉⣉⣉⠭⠭⠷⠒⠚⠋⠉⠁
⠀⠙⠦⣍⡉⠐⠒⠈⢉⣉⣉⣉⣠⡤⠤⠴⠒⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⣶⣾⣿⣶⣆⣀⠈⠻⢿⣟⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⣲⣦⣄⡐⠦⢿⣧⡨⠻⡿⢟⣿⣧⡴⣶⡟⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠟⣭⠱⣤⢤⣤⡤⢷⠶⠴⠶⡟⣛⡋⣽⣿⣿⠁⣿⣿⡇⠱⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⠁⢺⣿⣶⡄⣿⣿⣿⠇⢛⣽⣿⣤⣸⠋
⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠠⢀⣙⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠤⣤⢤⣀⠙⣻⣿⣿⡏⠀⠸⠿⢻⣃⡙⠃⠀⠀⢽⣿⣿⣿⣿⡇⠀⠀⢠
⠀⠀⠀⠀⢀⡰⠋⠀⠀⠁⠒⠂⠒⠒⠚⠹⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠊⠉⠙⠙⠳⠤⣄⡀⠀⣰⣿⣿⣿⠆⠀⠀⠈⠙⠁⠈⠉⠀⢀⡴⠃
⠶⠶⠶⠶⠿⠶⠶⠶⠿⣿⡶⠒⠒⠒⠲⠶⠶⢬⣝⣦⣄⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢭⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣤⡄⠀⠀⣀⣀⣀⡤⠤⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣩⡭⣿⡿⡿⢻⢫⠉⠉
⣠⡤⠴⠶⠶⠶⠖⣶⣾⣿⣶⠔⠋⠉⠉⠀⣀⣀⣤⡤⠴⠖⠒⠛⠛⠛⠒⠲⠤⠤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⡫⢕⡴⢋⠞⠁⡇⢰
⣀⡀⠀⠀⠀⠀⠐⣾⣭⣿⣥⠤⠶⠖⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣛⡶⢖⡺⢟⣻⠗⣛⣗⣛⣿⢋⣉⣿⣏⣽⠉⠉
⠉⠉⠉⠉⠛⠛⠙⢿⣷⣾⣳⣤⣄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣩⣾⣡⡟⢛⢳⣾⣿⢾⣿⣾⡶⠿⠟⠛⢛⣭⠵⠂
⠲⢤⣀⠀⠀⢶⣤⣭⡿⣿⡏⡙⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⠏⠀⠉⠁⢀⣿⣿⠁⠀⣠⡿⠀⠀⣠⣴⡿⠁
⠀⠀⠈⠙⠓⠾⣭⣟⣣⣽⢺⢿⣄⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢰⡃⠤⢠⢴⡶⢿⡿⢧⣀⣾⡿⢃⡤⢚⣵⡟⠁
⠀⠀⠀⠀⢵⣤⣼⣯⣛⣿⢿⣷⣿⣿⣿⣗⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠤⠾⠡⠶⣈⣩⠃⣠⡾⠋⠀⣩⠄⠀⠹⣿⠟⢋⣴⡿⠋
⠀⠀⠀⠀⠀⠹⣿⡿⣛⣿⣶⣮⣙⠛⠿⢶⣯⣭⣛⡛⠛⠿⠿⠷⣶⣾⣶⣶⣶⣤⣤⣶⣶⣶⣶⣿⣦⢼⣿⠡⠀⠚⠉⢳⣤⠞⣡⣴⣿⠟⠁
⠀⠀⠀⠀⠀⣦⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣦⣤⣤⣉⣉⡉⠙⠛⠛⣛⠿⠋⣩⣤⣤⠤⠤⠴⣒⣩⣴⣾⠿⠋
⠀⠀⠀⠀⢀⣶⣾⣯⣿⣿⣟⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣀⣾⣿⣶⣶⣶⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠤⠾⠿⠿⠟⠛⠛⠛
⠀⠀⠀⠀⠉⠙⢿⣿⣯⣻⣿⣿⣷⠀⠀⠀⠀⢀⡀⣴⣧⣰⣶⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠿⢭⣍⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠠⣬⣿⣿⣿⣿⣿⣿⣯⣤⡶⣷⣶⣿⣿⠟⢹⣿⣿⣁⣰⣶⣦⣄⣂⣻⣶⣀⣀⣀⣠⣴⣶⣶⣯⣥⣴⣬⣤⣤⠶⠾⠟⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢲⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⣺⣿⡿⠿⠿⠿⠟⠛⠛⠉⠉⠁
⣤⡤⠶⠶⠟⠛⠛⠙⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⢼⣤⣤⣮⣤⡾⠕⠶⠐⠒⡖⡴⠒
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠿⠇⠀⠀⠉⠉⠉⠈⠁⠀⠀⡤⠴⠶⠿⠟⠛⠉⠑⠦⠄
⠀⠀⠀⠀⠀⠀⣤⣠⣤⣬⣍⣉⣿⣶⣒⣂⣠⣤⣀⣀⣤⡤⠤⠤⣤⣤⣤⣤⢤⣤⣤⡄⣤⣤⣤⣤⡄⢐⡲⠶⢶⣤⡴⣶⣦⣶⣶⣶⠖⠁
⠀⠀⠀⠀⠀⣠⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠓⠒⠚⠛⠛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣤⠞⣛⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠟⠋
⠀⠀⠀⠀⠹⡏⢹⣮⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋

⠀⠀⢀⡀
⠀⠀⢹⣿⡄
⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⠀⣿⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⣿⠸⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⡏⡌⡜⡿⣿⣮⢻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣻⡿⡟⣆
⠀⠀⠀⢠⣧⣧⢳⣹⡽⣝⣷⣝⢷⣽⡻⡿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠉⠹⡿⢿⠿⢿⣿⣿⣿⣷⣿⣾⣢⣝⣰⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⠷⠷⣳⠾⡺⡿⡷⣷⠆⡦⣤⣸⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢊⠉⠘⠛⢯⠤⡌⠻⢾⣛⢿⡟⠿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠛⠱⣂⢂⠀⡉⠙⠙⠛⠖⠫⠛⢛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⠀⠀⠀⠀⠤⡀⠈⠑⠢⠂⠀⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⢀⠠⠴⠂⠀⠀⠀⠀⠀⠀⠁⠁⠘⠀⠃⠊⠐⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢄⠄⠀⠀⠐⠠⢀⠀⠀⠂⠤⡀⡀⠉⠙⠛⠲⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⣀⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠈⠑⠀⠀⢀⠀⠁⠀⠀⠀⣀⡉⠉⠙⠛⠛⠿⠶⠶⠶⠶⠶⠶⠾⠛⠛⣭⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢄⡀⠀⠀⢀⠀⠙⠒⠲⠤⣄⢈⡉⠁⠐⠒⠀⠀⠠⠠⠐⠐⢂⣡⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠆⣀⣀⡀⣀⠀⠀⠈⠈⠀⠀⣠⣄⣦⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠙⠚⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⢶⣶⣿⠿⠿⠷⠶⠤⠀⠀⢰⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢸⣿⣟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠶⠛⠛⠻⣷⠶⠿⠶⠤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣸⡀⠉⠉⢉⣙⡛⠛⠛⠛⠻⠟⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣤⣦⣴⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣤⣄⣠⣍⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣄⣠⣤⣄⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠶⠲⠒⠒⠚⠓⠒⢲⣯⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⡟⠛⠓⠖⠶⠶⠶⠶⠒⠒⢶⣶⠞⠛⠒⠒⠒⠲⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢩⣭⣭⣭⣭⣭⣉⣉⡹⠿⣿⣿⣿⣿⣿⣯⣉⣩⣭⣭⣤⣬⣭⣭⣭⣭⣭⣭⡍⢉⢩⠉⠉⠉⠉
⣀⣄⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣴⣴⣶⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠶⠀⣦⠀⠀⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⢀⡀⠀⠀⢰⡬⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠃
⠀⠀⠀⠀⠈⢀⣔⠈⠁⠁⠀⠐⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠫⠷⠷⣥⣿⠟⡆
⠀⠀⠀⠀⠀⢠⠉⠉⠉⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⢀⣅⠉⠉⠉⠉⠉⠀⡇⣠⠀⠀⠀⠀⠀⡀⢁⠀⠀⠀⣷⠉
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⡰⠁⢀⠀⠀⠀⠀⠀⠀⢿⠀⠁⣶⠀⠀⣤⠰⢷⡿⠃⠀⠀⠀⠀⠠⣼⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢠⣥⣠⣷⣦⣤⣶⣿⣤⡤⠂⠀⠠⡄⠤⣤⠄⠿⡀⢸⡏⠀⠈⣷⠀⢸⡇⡀⠀⠀⠀⠀⣠⣼⣦⣀⣀⣀⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣇⣀⣸⡧⠴⠞⠛⠛⠚⠒⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢲⠆⠀⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠎⠀⠀⠁
⠀⠀⠀⠀⢀⢀⡛⠛⠛⠛⠛⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⢴⣁⠀⠀⠀⢉⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠶⠷⠾⠿⡆
⠀⠀⠀⢠⡅⠘⠉⠁⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⣰⢀⠀⠉⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡟⠈
⠀⠀⠀⠀⢷⣄⠀⠀⠨⣶⡥⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⡔⢀⠀⠂⡂⠰⡆⢈⠄⠀⠀⠀⠀⠀⠀⡃⠑⢤⡀⠀⠀⠀⣧⣻⡀⠲
⠀⠀⠀⠀⠀⠉⣦⣀⠀⠁⠁⣀⡄⠀⠀⠀⠀⠀⠀⠠⠐⠂⠳⣄⠀⠀⢀⠼⠛⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠉⠉⠉⠉⢠⠃⢁⠖
⠀⠀⠀⠀⠀⠀⢼⣦⣭⡉⠉⠔⠁⠘⠃⠀⠀⠀⠀⠶⠶⣤⣄⣉⣛⠚⠉⠠⠤⠔⠂⠀⠀⠀⠀⠴⠷⣤⣑⡦⢤⡀⢀⣤⡴⠤⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠒⠤⢄⣀⠀⠀⢰⡒⢲⣿⠉⠛⠟⠛⢃⢻⡇⠀⠀⠀⡠⢠⡀⠀⢀
⠐⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⣿⠛⠿⢤⡃⠉⢰⡦⡀⠄⠊⠀⡀⡀⡰⡡⡑⡡⡡⢁
⠀⠀⠈⠃⠉⠢⣀⠀⠐⢤⣆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣀⠶⢉⠍⡸⠀⠀⠀⠀⠜⡐⠑⡔⢉⠜⠔⢠⢃
⠀⠀⠀⠲⢄⠀⠈⣈⠂⠈⠀⠉⠒⠤⣀⣸⣿⣿⣿⣿⣇⢨⣟⡳⠈⠀⠀⠀⢀⠎⠈⢈⢎⢀⢊⠎⡠⠁⠢
⠀⠀⠀⠰⢄⡈⠂⠁⠑⠀⠢⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣏⡇⠀⠀⠀⠀⠠⠂⠂⠀⣁⣀⡀⠔⡠⡑⡡
⠀⠀⠀⠀⠀⠈⠓⠆⢄⠐⠀⢄⢠⣠⣴⣾⣿⣿⠏⠙⢿⣿⣿⣷⡀⢀⣀⠤⠤⠤⠖⠋⠁⠀⠉⠒⠥⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠝⠛⠛⠋⡉⢀⣼⣿⣿⠿⢿⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠉⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡉⠲⢤⣀⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣛⠶⢥⣄⣀⡀⠀⠀⠀⠠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠠⡈⠉⠠⣄⣀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠲⢄⠉⠉⢍⡁⠀⠘⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠈⠲⢄

⢀⡞⠀⠀⠀⠀⠀⣰⢠⠆⢀⣠⣄⠂⠀⠀⠀⠀⣛⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠄
⢋⣿⣿⡰⠴⠓⢋⡵⢋⡯⠻⣾⣥⣤⣀⣀⠀⡀⢼⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣻⣿⠀⠀⣠⠴⢚⣷
⠀⣴⡇⠀⢀⡴⣫⠖⠁⠀⠐⠟⠹⠟⠳⠙⠢⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣥⠀⢀⣿⣣⠐⣷⣡⣀⡬
⠀⢹⣷⡰⢋⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠜⠛⠉⢿⣠⡏⡁⡁
⠀⢸⡏⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⡸
⠀⢸⣧⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⠃
⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸
⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⠀⠇⠀⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⢸⡿⢸⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⢸⡄⠈⠆⠀⠐⠀⢓⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⢸⣧⣀⣄⡀⠀⠀⠠⣋⡁⠀⣀⠀⠠⢦⠀⠀⠸⣆⠁⣀⡤⠤⢤⡜⠠⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⢸⡇⢹⣷⣿⢆⠀⣼⡿⠁⠘⠋⡀⠀⢐⡠⡀⠀⠙⠲⠧⢤⠴⠊⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣼
⠀⣸⡇⢸⠘⡜⢆⡳⠿⣿⡇⠀⠀⠀⠀⠀⣁⠴⡀⠀⠀⠀⢸⣤⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢸
⠀⣿⠃⢸⢀⡽⠋⠀⠀⣞⠉⠀⠀⠉⠉⠉⠀⠠⠗⠊⠙⡀⠩⣋⠳⠀⠀⠀⠀⠀⢀⣀⡀⢀⣠⢄⠀⠀⠀⢀⣴⡿⣿⢸
⠀⣿⠀⢸⠈⣷⡀⠀⠀⠙⠓⠲⣄⠈⠒⠒⠒⠿⣶⡶⣻⠷⡀⢸⠉⠀⠀⢀⡴⠞⠉⠀⠈⣏⠀⠀⢇⡠⠔⠿⡿⠁⣿⢸
⠀⣿⠀⢸⠀⠘⣷⡀⠀⠀⣠⠎⠀⢢⣉⣒⠢⢠⠏⠀⠀⢔⣅⠸⢂⢤⠖⠉⠀⢠⠋⠀⠀⢸⡠⠖⠉⠀⠀⡰⠃⠀⡟⢸
⠀⣿⠀⢸⠀⠀⠈⢻⣦⣠⡏⠀⠀⠀⡄⠈⢒⠞⠀⢀⣰⣿⣿⡔⡱⢃⣀⣀⣤⠃⠀⠀⠀⢸⠀⠀⠀⢀⡜⠁⠀⠀⡇⢸
⠘⣿⠀⠀⡄⠀⠀⠀⠣⣼⠁⣠⠔⠒⢳⠔⠁⠀⣴⠋⢷⣿⠿⠋⠉⠀⠀⠀⢸⣇⠀⠀⠀⡄⠠⠀⣠⠎⠀⠀⠀⢰⡇⢸
⠀⣿⠀⠀⡇⠀⠀⠀⠀⢹⣾⣄⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣿⢀⡤⢊⠇⢀⡔⠁⢀⣠⡄⠀⢸⡇⢸
⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠉⠙⢿⣷⡾⠟⠋⠀⠀⠁⠀⠠⠀⠀⠄⠘⠀⠀⠀⣿⣡⠖⣡⠔⣯⣾⡿⠿⠟⠁⠀⢸⡇⢸
⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⠶⢨⣀⣽⣿⠿⠿⠃⠀⠀⠀⠀⢸⡇⢸
⠀⢸⡀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠉⠉⠉⠁⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿
⠀⢸⡇⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠃⠀⠀⠉⠻⠷⠶⣶⣶⣿⣶⣿⣿⣿⣿⣾⣿⣷⣶⣶⣴⣶⣶⣷⣦⣴⣦⣤⣤⣤⣦⣴⣷⣤⣤⣄⡀⠤⣀⡿⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢻⠁⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡬⠿⢿⣿⣶⣦⡀⠛⣸⡇⢀⠖⣁⠀⢈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠙⠿⣷⠀⣼⣣⠏⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣭⣭⣶⣦⣬⣵⣟⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣡⣤⣜⣿⣿⣾⣶⡭⢙⣷⣶⡶⠦⣝⣣⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠿⢿⣿⣿⣯⣾⣟⡟⢩⣿⣿⣦⡘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠸⣿⠋⣿⢛⣿⣿⢸⠈⠉⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢻⠋⡰⠈⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣤⣏⠀⠀⠀⠀⢀⠀⣀⡀⡀
⠀⠀⠀⠀⣠⣤⡤⣤⣤⣤⣤⣤⡉⣭⣭⣽⡿⠿⠩⠈⣿⠤⢬⣕⣛⠚⢻⠉⠉⠩⠁
⠀⠀⢠⣤⡴⠦⣴⣿⣿⣿⣽⠻⠷⠛⠉⠁⠀⠀⠀⢀⣹⣧⠀⠉⠙⠻⢛⣛⣛⠓⢲⣖⣀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠠⢶⡶⢶⣶⣦⣭⣙⢂⣀⡀⡀⠀⠀⠀⠀⠈⠉⠳⣄⠐⠀⠀⠙⠻⢿⣷⠍⠉⡉⠈⠉⠉⠛⠛
⠀⠀⠀⠀⣤⣤⣶⣿⣶⣬⣭⣾⣶⣿⣿⣿⣧⣈⠓⠒⠓⠈⠀⠈⠳⠄⠀⠀⠀⠴⠾⠒⠿⠿⠶⠆⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠤⠒⠶⠶⣾⣯⣤⣀⡈⠛⠋⠀⠲⠾⠿⠿⠿⠻⠒⠒⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡉⠉⠉⠀⠒⠀⠂⢉⣭⠁⢀⠒⢒⣠⡈⠅⠀⠀⢒⠀⠀⠠⠀⠒⠂⠠⠬⠁⢐⠒⠠⠬⠁⢒⡀⠬⢉⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠈⠓⠘⠉⠛⠛⠒⠰⠾⠿⢶⠴⡖⠲⠑⢲⠁⠲⠀⠹⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠚⠁
⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢶⣄⠀⠀⠄⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣥⣤⣤⣤⣋⣶⢀⠀⠀⢠⢴⣟⠿⠿⢿⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⡽⠟⠋⠉⡁⠺⣀⣀⣠⣂⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢰⠯⠿⠋⠈⢁⡴⠋⠀⠀⣀⠘⣷⠈⣷⡄⠻⢟⠟⣤⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠈⡐⡘⠲⣿⣷⠀⠈⠰⢼⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠑⣿⠀⢹⣿⡆⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠸⠀⠀⣿⠀⠘⣿⡇
⠛⠛⠛⠛⠛⠛⠛⠿⠟⡏⠀⣠⣿⠿⠋⡙⠂⠿⣿⣤⣽⠉⣿⡇⠀⠙⠛⠻⠛⠻⠿⠿⠿⢿⣿⡟⠻⠛⠻⠿⠿⠯⠟⠛⠛⠛⠚⠛⠛⠻⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠉⠉⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠋⡀⠥⠠⠄⠀⢀⣀⣀⣀⡀⠀⠁⠀⠀⠀⠀⠒⠶⠦⣠⣄⡋⠁⠁
⠁⠉⠉⠉⠐⠛⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠤⠤⠤⢶⡤⢄⡈⠛⠓⠀⠀⠐⠒⠲⠶⠆⠀⠀⠀⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢴⡆⠀⠀⡀
⠀⠀⠀⠀⢄
⠀⠀⠈⠁⢈⡤⣬⠤⠤⠴⠀⠦

⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠘⡄⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⣦⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢹⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗
⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢰⠶⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠤⠆⠙⠶⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⣀⣀⣀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⣾⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠙⢥⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡇⣠⠀⠛⣗⡄
⠀⠀⠀⠌⠈⢁⣄⡨⢄⢾⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠞⠁⠂⠙⠛⠪⠷
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣝⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⡿⠈⠻⣿⣿⣦⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡃⠀⠀⠈⠉⠛⠟⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⣼⣿⣿⡿⣻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⣾⣿⣿⡏⣼⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⣿⣿⣿⡟⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠿⠿⣿⠛⠛⠛⠛⠃⠠⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣆⠀⠀⠀⣠⢠⣼⣾⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣀⣄⣀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡆⢀⣾⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⣿⣿⣿⣶⣤⣆⠀⠀⢿⣿⣿⣿⣷⣾⣿⣟⣯⣿⣇⣰⣶⣴⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣿⣿⣿⣷⣦⣼⣿⣽⣿⣿⣿⣿⣿⣡⣾⣷⣿⣯⡝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠈⠙⠿⢿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣥⣤⣴⣴⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⠿⣟⣿⠟⢿⠿⢿⣶⣾⣿⣝⣿⣿⣿⣿⣿⡿⠿⣿⣿⡿⠻⢿⣯⢻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⣀⠐⠋⣽⡿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈⠁⠈⠛⡙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡏⠀⢈⣵⣿⡿⢿⡟⣿⣿⣿⣿⢿⢿⣿⡟⢿⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣷⣦⡀⠀⠀⠀⣰⣿⣿⠋⢀⣴⣿⣿⡿⡝⣿⠀⣿⣿⣿⣿⣎⠟⠹⠋⡌⡿⢻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⢀⣼⣿⣿⠀⢠⣿⣿⢻⡝⠣⠉⠀⠀⣿⣿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠘⠉⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢿⣿⣷⣾⣿⣿⣋⣰⣿⣿⣹⣧⣵⣶⣶⣶⣤⣿⣏⢸⣟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢣
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⢿⡿⣿⣿⣷⣶⣀⡀⢨⣿⣿⣿⣿⡟⢉⣿⣿⣿⣿⣿⠟⠋⠉⠈⠉⣿⣿⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠁⠀⢲⣿⣿⣿⣾⣿⣿⣿⣿⣵⣾⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⣿⣟⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣸⣿⣿⣯⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣶⣶⣤⡀⠀⠀⣿⡯⠀⠀⠁⠃
⠀⠀⠀⠀⠀⣴⣿⢿⠛⠉⠈⠁⠀⠛⠛⢛⣩⣿⣿⣿⣿⣿⣿⣿⣏⠉⠁⠀⠀⠀⠀⠉⠛⠀⢠⣿⡇
⠀⠀⠀⠀⠰⡟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢹⣿⡟⣿⡿⣿⡿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⣿⣿⠀⣿⡇⣿⡇⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⢹⡇⠀⢻⣇⢹⣧⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠈⠁⠀⢸⣿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⢰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⢀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆

⠀⠀⣟⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣿⠀⢩⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢺⡁
⠀⠀⢻⠀⠀⠙⣟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣺⠟⠛⠀⢸⡇
⠀⠀⣼⠀⠀⠀⠈⠢⠢⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⢿⢦⣤⣖⣝⣅⣄⡀⠀⠠⡇
⠀⠀⣿⠀⠀⠀⠀⣀⣘⣄⣀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠂⠈⠛⠳⠦⠮⢿⠿⠋⠀⠀⠠⡇
⠀⠀⣿⠀⠀⠀⠀⠉⠋⡉⠉⠁⠀⠄⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠠⠂⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠠⡄⠀⠀⠀⠠⢀⠀⠀⠀⠀⠠⠆⠉⠀⠁⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⡇⠀⠀⠀⠀⠈⠙⠓⠅⣠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠭⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⡅
⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣘⠊⡱⠦⠲⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⠄⠀⠀⠀⠀⠀⠀⠀⢰⣿⡅⠀⠀⣆⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣷⣦⡀⠀⠀⢰⣽⡇⠀⠀⠀⠀⠀⣤⣾⡿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⣀⣀⡸⢿⣿⣷⣄⠘⣿⣧⠀⠀⢠⣠⣾⡿⢃⣤⣾⣿⣶⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠞⠛⠛⢉⠛⠉⣉⣩⣿⣿⣿⣷⣿⣝⢷⣿⣏⢀⣼⣿⣿⣟⣡⠿⢻⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⢛⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡟⠉⣀⣀⣀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠿⢯⣉⠹⣿⣄⣀⠙⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠈⠁⠀⢉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡍⠻⠿⣿⣶⣄⠀⠉⢻⣿⡙⠂⢀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣳⣿⠆⠀⠀⠙⠓⣠⡀⠀⠈⠙⠀⠸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠻⠁⠁⣼⣿⣿⣿⢿⣇⠉⢻⣟⣻⢦⠀⠀⠀⠀⠸⡿⣶⡆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣰⣿⣿⠂⣯⢻⡅⠀⢸⣯⢹⣷⠀⠀⠀⠀⠀⠈⢏⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⣇⣾⠁⠀⠀⢿⡗⣿⠀⠀⠀⠀⠀⠀⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠁⠀⠀⢻⡇⠀⠀⠀⠸⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣀⣠⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠛⠛⠶⣄
⠙⢿⣍⠁⠀⠀⠉⠉⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣠⡼⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠛⠋⠀⠀⠀⠀⣀⣤⡴⠶⠶⣤⣄⣸⡇
⠺⠛⠛⠛⠛⠛⠛⠲⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠉⠀⠀⠀⠀⠀⣨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣤⣤⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡿⢀⣴⠞⠛⠉⠉⠉⠉⠉⠛⠛⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢸⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡴⠶⠖⠛⠋⢩
⠀⠀⠀⠀⠀⠀⢸⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣷⣦⣿⣀⣠⣤⣤⣤⣤⣤⡤⠶⠶⠶⠚⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠁
⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣀⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠤⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣤⡀⢀⣴⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣀⡈⠻⣶⣄⢹⣿⣿⣏⣴⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⣴⡀⡠⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣠⠏⣀⣀⣀⣉⠳⣿⣿⣾⣿⣿⣿⣿⣿⣿⡥⠶⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣾⣿⣽⣿⣿⣿⣿⣷⣿⣥⣂⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⣰⠃⣠⣵⣶⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⢿⣶⠀⠀⠀⠀⠀⠀⠀⠤⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣆⣄⡄⠀⠘⣆
⠀⠀⠀⠀⢰⠃⣀⣽⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠿⢷⣦⠀⠀⠀⠀⠀⢒⣺⣿⣿⣿⣿⣿⣿⣴⡷⣿⣿⣿⣿⣿⣿⡾⠀⠀⠘⡆
⠀⠀⠀⢀⡏⠀⠉⢉⡤⠴⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠲⡄⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⡥⠄⠀⢹⡀
⠀⠀⠀⣼⠀⠀⠀⠁⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣮⡁⠀⠀⣧
⠀⠀⠀⡇⠀⠀⠀⢀⣾⡿⠋⠉⠉⣿⣿⢿⣿⣿⠟⠀⠀⠀⠀⠈⠙⣿⣷⠀⠀⠀⠀⠀⠀⠊⣼⣿⣿⣿⡀⠋⠈⣿⣟⠻⢿⣿⠿⢿⣿⣿⣿⠷⠂⠀⢹
⠀⠀⢰⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⣿⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠐⠛⢿⠿⠟⠏⠀⠀⣿⡇⠀⠀⠁⠀⠸⡿⣿⢿⠂⠀⠀⢸⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⠀⠀⠀⢀⡀⢀⣄⣀⠀⠀⠀⢸
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⠀⠀⢲⣶⣿⣿⣿⡷⠦⠄⠀⡟
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⡴⠚⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠚⠁⠀⠀⠀⣿⡇⠀⣠⡿⣻⠛⢿⣏⠀⠀⣸⠁
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠈⢧⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠁⠀⠀⠀⠀⢿⣷⠀⠉⠀⠁⠀⠈⢏⠀⢠⠇
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠳⣄⠀⠀⠀⠀⠑⠒⠀⢀⡀⠂⠉⢀⠔⠁⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠈⣦⠏
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠙⠲⠦⠄⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠄⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣏⣉⣉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠶⠶⠦⠤⠄⠀⢀⣀⣤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠲⠶⠶⠶⠶⠒⠚⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⡇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡿⠀⠀⡰⠋⠀⢀⠞⠀⣠⠖⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣇⠀⢰⠃⠀⠀⡏⠀⢰⠁⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⡆⢸⡀⠀⠀⠳⡀⢻⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠃⠀⠳⣄⣀⣠⠃⠈⠢⠤⠋

⠀⠀⠀⠀⠀⠀⠀⠠⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠖⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠉⠲⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡇⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⡀⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣈⣉⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢸⡟⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢸⡇⢨⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⢸⡟⠋⢔⣩⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣼⠇⠀⠀⢸⡇⠀⠀⠠⠐⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢸⣿⠀⠀⠀⢸⣷⣠⣤⢶⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⣿⠀⠀⠀⢸⣏⢽⠚⣙⠥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⢼⣿⠀⠀⠀⢸⡇⠈⠍⠁⢀⣠⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣿⣿⠀⠀⠀⢸⣏⠤⠐⣿⣿⣅⡀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢿⣿⡀⠀⠀⢸⣗⠂⠀⠘⣏⣁⠠⠀⣀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⢸⡧⠀⠀⠀⢹⣶⣒⠅⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠘⣧⢻⣆⠀⢸⣿⡶⠟⠁⠈⢫⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢚⠀⠀⠰⠀⠀⠀⠀⢿⣼⣿⡆⢸⣿⣾⣯⠆⠀⠈⠅⠀⠀⣀⣠⣤⢶⡷⠤⠴⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣿⣿⣾⣿⣿⣿⣂⣀⣤⡾⠿⠟⠋⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣿⡟⠀⢀⣥⣄⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠩⣽⣭⣰⣆⣾⣿⣿⡿⣟⠛⣉⠉⠁⠀⢀⣁⣀⣠⣤⣀⣼⣿⣀⣀⣀⣀⣠⣤⣲⣶⣒⣖⢒⠀⢐⣂⣀⣂⠀⠀⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠿⠿⢿⣿⣿⣻⣭⣙⣛⣭⣉⣹⣉⣛⣯⣙⣛⣛⠙⠋⣉⣫⣝⠋⠙⢉⣩⣍⠋⣛⣝⣛⣛⣛⣟⣛⣿⡉
⠀⠐⠂⠀⠀⠐⠒⠒⠒⠶⠷⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠀⠈⠈⠁⠛⠛⠉⠉⣼⣷⣶⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠿⠷⣶⣶⣶⡶⢶⣦⣴⣤⣴⣶⣶⣶⣶⡶⠤⣦⣤⣴⣶⣶⡶⣶⣶⣶⠶⠶⣶⣶⣒⣛⣛⡛⠻⠛⠛⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣙⣓⣒⣀⣉⣙⣋⣭⣿⣿⣭⣌⣉⣹⣷⣶⣦⣤⣈⣉⡵⢶⣦⣶⣶⣶⣞⣛⠛⠉⠉⠉⠉⠉⠙⠋⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣀⣉⣭⣤⣤⣤⣤⣴⠶⠦⠶⠶⠶⠶⠶⠶⣶⣦⣬⡍⠙⠛⠿⠿⠭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠒⠚⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣤⣶⣶⣶⣶⣶⣿⣿⣤⣤⣤⢤⠀⠤⠀⠤⠠⠀⠀⠐⠒⠲⠶⠀⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠀⠀⠐⠒⠐⠶⠦⣤⠀⠉⠉⠉⠻⠿⠿⣿⣿⣿⣿⣷⣶⣧⡌⠉⠉⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠴⣶⠚⢋⡥⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠊⠉⠁⣄⢀⣀⣀⡉⢁⣀⣉⠀⠀⠀⠀⠀⠒⠂⢠⠄

⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢈⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢋⣿⣿
⢰⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣿
⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣯⣽⣅⣨
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡌⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⢿⡟⠛⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡇⣶⣶⣶⠰⣶⡖⠀⣶⣶⣶⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡇
⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣭⠀⣿⠁⠀⠉⠙⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⡃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠂⣶⢀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡇⣿⣿⣶⣀⣶⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠙⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⣩⣍⣉⣀⣉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⡇⣿⣿⣿⣿⣿⣄⠰⠦⠴⠄⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣾⣶⡄⢰⡇⣿⣿⣍⠁⠉⠁⠀⢀⣤⣴⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣠⣿⡇⠟⠛⠛⠓⠀⠀⠒⠚⠻⠿⠀⢹⣧⣿⣿⣿⣿⣿⣿⣯⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣷⣤⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣴⣦⣤⣤⣴⣿⣛⡀⣙⣿⣿⣿⣿⣿⠀⢘
⠀⠀⠀⠀⡀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠰⠾⠿⢷⣿⣿⣿⣾⣟⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠿⠛⠿⠟⠿⠿⢿⠽⠿⣿⠿⠿⠿⣿⣿⣿⣿⠀⠈
⠀⠀⢀⣼⠃⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⣷⠀⣷⠀⠀⠀⠀⠀⠀⢰⠇⣴⢀⡶⢠⣿⣿⣿
⠀⢀⣌⠀⡀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠘⣦⠹⣿⡟⢹⡿⠟⣻⣿⣿⠀⣿⠀⠻⣄⡉⣀⣀⠀⠀⠀⠀⡀⢰⡇⣼⠇⢸⣿⣿⣿
⢠⣎⣿⣾⠿⠥⠀⠸⣿⣿⣿⠀⠀⠀⡦⠀⠀⠀⠀⠠⠹⣷⠈⣷⠹⡿⢺⡟⢺⣿⣿⣿⠠⣿⣦⡀⢘⣿⡟⢋⣽⠀⠀⠀⣇⢸⠀⣿⠀⣿⣿⣿⣿⡇⢀
⢸⡿⠛⠁⠀⠀⠀⠀⠈⢿⣿⠀⠀⠰⣿⣿⡟⠒⠀⠸⣷⢿⡶⠉⠀⢱⠀⠁⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠈⠀⠉⠀⣿⣿⣿⣿⡇⠈
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⣻⠄⠀⢠⡼⠙⠧⠤⠤⠤⢴⠀⠁⠀⠀⢸⠀⠀⠀⣸⣷⢶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⢀⣿⣿⣿⣿⡷
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣂⢀⣀⣀⣼⣧⡄⠀⠀⣸⡇⣴⣶⣿⣿⣾⣟⠱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢸⣷⣶⣿⣦⣤⣤⣿⣀⣀⣆⣀⣤⣍⣉⣉⣉⠉⣍⢻⣿⣿⣿⣿⣭⣿⣿⡟⢧⣬⠾⣍⡻⠯⢙⡿⠋⣽⣿⠛⢻⣿⣛⣿⠉⢿⠿⢿⡏⢹⣷
⠀⠀⠀⠈⣿⣿⠋⠉⠙⠉⢻⣏⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣽⣿⣛⢻⣟⠛⠲⣬⠓⠀⣼⠋⢠⣜⢫⣄⡙⠦⣉⡛⢠⣿⣤⣿⣷⡾⠿⠆
⠀⠀⠀⠀⡏⣿⣧⠀⠀⠀⠐⠟⠁⠐⠛⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣇⣿⣯⣭⣽⣗⣈⠩⣅⠚⠒⣮⡍⠀⣦⡭⠑⢤⣝⠣⢤⣛⢢⡄⠛⠋⠁
⠀⠀⠀⠀⡇⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣿⣿⣿⣯⡀⠻⣆⣉⠩⠶⠈⠩⣭⡙⠒⢲⣭⠓⠒⣚⣭⠤⢤⣛⡣⡄⠀⠀⠀⡄
⠀⠀⠀⠀⡇⠀⢹⣇⠀⠀⢀⣀⣀⣀⣤⣤⣤⣶⠶⢿⣿⣿⣯⠻⣯⣁⡤⠽⢇⡈⢭⣔⠒⠙⣳⣤⠙⢚⣋⠭⠴⢦⣍⡳⠦⠴⠛⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢴⠦⠾⠛⠛⠛⠋⠉⠉⠉⠉⠀⣀⡒⢲⣾⣧⣼⡿⣇⠈⣽⡞⠛⠉⢻⣦⠙⠛⣛⡡⠤⠤⣌⡙⠷⡤⠶⢛⣃⣤⠀⠃⠀⠀⠀⠀⠀⢿⠤
⢰⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⣤⣶⠾⡛⠻⣧⡈⠻⠗⣛⣩⡤⣤⣌⠙⣧⠀⣤⠾⠛⣀⡀⠿⣏⡁⠀⣤⠀⠀⣀⠀⢀⠀⠐⢶⣈⡃
⢸⣿⣿⡏⢠⡀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⡶⢿⣶⡞⠿⡿⠿⢛⣉⣡⣀⠘⢿⣄⣀⣠⣿⠷⠀⠀⣶⡞⠛⠉⠛⣧⣌⠙⠛⣋⣿⡿⠿⢷⣶⠷⢶⣶⣟⠁
⠸⣿⣿⠀⠈⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠈⠁⠀⠙⠛⠂⠒⠟⠋⠀⠈⠻⣦⡈⠙⢋⣉⣴⠶⠰⣦⡈⠻⢶⡴⠾⠉⣉⣠⣈⠛⣷⣄⣀⣀⣽⡿⠆⢸⣿
⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⠛⠒⠀⠰⠶⠾⠋⠉⢹⣷⣯⡉⠛⢋⣁⣴⡶⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠖⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠤⢀⠢⠈⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⡑⣠⣴⣿⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿⣿⠏⠁⣠⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠟⠋⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢡⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠀⢀⣠⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣾⣿⣿⣿⡿⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⠟⠋⣠⣾⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠛⠉⠁⢠⣼⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠛⠁⠀⢀⣤⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠛⠁⠀⢀⣤⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠠⣴⣾⣿⣿⣿⠿⠛⠁⠀⣀⣤⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⢀⣠⣴⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠓⠛⠒⠛⠒⠛⠚⠓⠑⠛⠃⠙⠒⠛⠂⠚⠻⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⠤⠔⠒⠒⠀⠀⠀⠉⠁⠀⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠁⠀⠀⣀⡠⠤⠄⠒⠒⢚⣋⣩⣄⣠⣤⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⢀⠠⠤⠒⢊⠉⣀⣤⣤⣶⣷⣿⣿⣿⣿⠿⠛⣩⠟⠁
⠀⠀⠀⠀⠀⠀⣠⠞⡁⠆⠒⢉⣀⣤⣶⣶⣿⣿⣽⣿⣿⣿⠿⠟⢋⣉⠠⠒⠈
⠀⠀⠀⠀⠀⣼⠿⣟⣠⣶⣾⣿⣿⣿⣿⡿⠿⠟⢳⢋⡁⠠⠒⠂⠁
⠀⠀⠀⠀⠀⢧⠚⠛⠛⠛⢛⣉⣉⣭⠤⠂⠀⠊⠉⡆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤
⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡠⠠⣀
⠀⠀⠀⠀⣏⣲⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠁⠀⠀⠀⢡⣄
⠀⠀⠀⠀⢻⡇⠠⢇⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⡀⠀⠀⢀⡠⣀⡀⠀⠀⠀⢠⠀⢛⡔⠈⠰⡄⠀⠩
⠀⠀⠀⢐⠩⡴⡄⢊⢰⡈⠐⠀⠄⠒⠾⠿⢿⣿⠿⢿⣤⣤⣄⢩⣁⣸⣿⣦⣄⠒⣰⡄⠊⠀⠀⠀⠟⠊
⠰⣾⣿⢀⠟⢀⡀⢢⣄⠩⠁⣤⡤⠀⣀⣀⣼⣿⣋⣉⡿⠶⠖⠚⠻⠿⠿⣿⣿⠟⠛⠒⢠⡦⠨⢧⣤⠆
⠈⢹⡇⠀⣴⣄⣀⣉⠹⠐⠂⠨⣉⠛⠻⠿⠿⣿⣿⣿⣿⡀⣀⢀⠀⠀⢤⡀⠁⠀⠀⣰⡎⠀⠀⣼⡿
⠀⠸⣵⣾⣿⠓⡤⢞⠋⠉⢀⡀⠈⣇⠀⠀⠀⠀⠀⢹⣿⡇⠀⡌⠀⢠⡍⠘⠫⣽⣿⣷⣤⡶⢟⡫⠄
⠀⠀⢸⣿⣿⡀⠀⠈⡄⠀⡂⠀⡀⢸⡀⠀⣀⢠⡴⣾⣿⣇⣼⡠⢀⢟⠁⠀⠀⠐⢢⣿⠅⠈⠁
⠀⠀⠈⠹⠿⢿⣷⣶⣧⣠⡇⠀⢀⠘⡇⣸⡶⡤⠄⢞⡫⠩⢍⣀⡇⢸⠘⡀⠀⢸⢥⣇⡀⠀⠀⠀⡠⠄
⠀⠀⠀⠉⠉⠰⠦⢭⣭⠔⠙⡀⢸⢂⢛⡊⠠⣇⠀⠂⠐⠀⣸⠉⠩⠛⠉⠣⡄⠸⡀⡈⠈⡙⠶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠙⡳⢲⠀⠾⣀⢁⣀⠀⠉⠉⠋⠭⠀⠀⠀⠒⠀⠔⠫⠴⠋⠉⠋⠠
⠀⠒⠒⠤⢀⣄⢀⠤⡟⠡⠀⠀⡄⠃⠘⠀⠉⠀⠐⠂⠀⣀⠀⠈⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠐⠛⢌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢀⠀⠀⡤⠊⠉⠉⠉⠒⢤⡀⠀⢀⡴⠂⠉⠉⠉⠒⢄
⠀⠀⠀⠀⢠⠃⠀⠎⠀⠀⠀⠀⠀⠀⠀⠉⢲⡏⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠸⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⢠⠇⠘⢦
⠀⠀⠀⠀⠀⠀⠘⠀⠣⡀⠀⠀⠀⠀⠀⢸⡀⠀⣸⠇⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣠⠤⠒⠒⣚⣉⣉⣉⣉⣭⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣭⣭⣭⣭⣭⣵⣶⣒⣤⣤⣀
⣟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣿
⢿⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⣇⣀⡀⠀⣀⡀
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⢸⠇⢀⣀⣀⡀⠈⠳⡄
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⢤⣤⠤⠤⠄⠤⠀⠀⠀⠀⠀⠀⣾⠞⠁⠀⠀⠈⢆⠀⠘
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣠⣤⡀⣰⠆⣾⣰⣯⢟⡴⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢀⡴⠃⠀⡰
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⣸⡏⠀⢰⡿⢾⡷⣿⣼⠋⠙⠶⠛⠁⠀⠀⠀⠀⣴⣃⣀⠤⠤⠒⠊⠁⣀⠤⠊
⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⣿⣡⣤⡾⠳⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣀⣀⡠⠤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣄⣀⣀⣀⣀⣤⣤⠤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⡙⠻⣶⣌⣷⣶⣿⣶⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⠀⠻⣿⣷⣄⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣶⣽⣿⣿⣦⢻⣿⣿⣿⣿⣿⣿⠛⠟⠋⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣴⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⣿⡿⡿⠿⢸⣿⣽⣿⣿⠇⣻⣿⡛⠹⢾⣿⣿⣿⣶⣶⣤⣴⣤
⠀⠀⠀⠀⠀⠠⣄⠻⣷⣼⡇⣠⡄⠀⠀⠀⣠⣶⣿⠿⠟⠋⢁⣾⣿⣿⣧⡀⠀⢾⣿⢻⣿⣿⣿⣿⣇⢀⣀⣀
⠀⠀⠀⣰⣿⡿⣿⣷⣿⣿⣿⡿⢀⣀⡀⣸⠿⠋⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⡿⢿⣿⣬⢻⣿⣿⣿⡿⣟⣻⣿⠟
⠀⠀⠘⠛⠛⡛⣿⣍⠻⣿⣿⣿⣋⡉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⢻⡿⡷⢀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣿⣯⣭⣭⣿⡟⠛⠛⣯⠙⠲⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⣿⠀⠀⠀⠿⠛⢿⣿⣯⡿⢻⣿⣿⣯⣝⣛⣿
⠀⠀⠀⣸⡟⠉⠁⠿⠿⠁⠹⣶⣝⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⢞⣵⣿⣿⣇⢨⣿⣭⡉⠛⢿⣿⡄
⠀⠀⠀⠉⠀⠀⠀⠀⢀⣶⠀⠀⠙⣷⣄⠀⢹⣇⣽⣿⣰⡏⠀⠀⠀⣿⠀⠀⢰⣿⠈⣿⣿⡇⢿⣾⣿⣿⢿⣷⣶⣯⣃
⠀⠀⢸⣷⣶⣦⣴⢟⣿⣷⣄⠀⣀⣈⡙⢿⣄⢿⣿⣿⡿⣢⣤⣤⠀⣿⠀⠀⠸⣏⠀⠛⢹⡇⠈⠘⣿⣿⠀⢹⣇⠉⠉
⠀⢐⣶⣾⣿⣿⣿⣾⣿⣿⣿⡶⠦⣬⣙⢿⣿⣾⡿⢿⣿⣿⣙⣛⠇⣿⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠘⠿⠀⠀⠃
⠀⣿⣛⣉⡙⠛⣿⠁⠴⣿⣷⣦⣴⣾⣿⣿⠿⣿⣧⠸⣏⣉⡉⠛⠓⣿⠀⠀⠀⠀⠀⠀⣿
⠀⣼⠿⠛⣿⣿⢻⣧⡶⢾⡇⠹⠇⠉⠁⢠⣾⣿⠛⢻⣿⡙⠻⢿⠆⣿⠀⠀⠀⠀⠀⢀⣿
⠈⠁⠀⠀⣿⣿⢾⠙⢧⢸⣷⡀⠀⠀⠞⠋⢀⡏⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠛⠁⢸⡄⠈⢹⡈⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠁⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠂⠘⠛⠛⢻⠷⠶⠶⠶⠶⣶⣶⣾⣿⣶⣦⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣼⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄
⣀⣀⣠⣀⣀⣀⣀⣀⣀⣉⣉⣻⣿⣿⣉⣉⣉⣉⣈⣉⣉⣙⣻⣿⣿⣿⣿⣋⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⡍⢽⡟⠛⠛⠛⠛⠛⢛⠻⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠠
⠀⠀⠀⠀⠀⠐⠂⠀⠠⠁⠀⠀⠀⠀⠐⠀⠁⠀⠀⠰⡆⠀⢀⡆⠀⡀⠀⠁⠈⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣨⠀⠀⢤
⠖⡂⠄⡐⠀⠂⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠺⠈⠀⠀⣰⣿⡇
⢀⡨⠁⣿⡌⣙⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⡀⠀⠀⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⡳⠆
⠀⢃⠀⠈⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡈⠁⢘⠎⠠⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠰⠀⠀⠈⡀
⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠂⣼⡁⢀⠂⠶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠠⠸⠉⠇⢸⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⣿⣿⣿⠿⠂⠈⢠⣶⡷⠿⠿⣿⣶⣶⣶⣿⣷⣶⣾⣷⣷⢶⣤⣖⡂⡁⣠⢲⣶⣶⣿⣶⣶⣶⣤⣶⣶⢶⣦⡶⢇⠠⠴⣖⡒⢒⣲
⠙⠻⠉⠀⠠⠁⡄⠘⠳⠄⠀⠁⠐⠓⣀⠙⠛⠻⠈⣩⡁⠚⠿⢿⡽⠁⠀⢉⣽⣿⣿⣿⣿⣶⡿⠿⠷⠾⠂⢒⠠⠀⠩⠍⠑⠿⠃
⠀⠀⠀⠀⢀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡈⠀⠈⠀⠀⠉⠉⠀⡶⠁⠀⠭⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠠⠤⠀⠈
⢀⡀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠀⠀⠀⢃⠀⠀⠉
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠠⣀
⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⠀⠉⢛⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⠉⠉⠉⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡇⠈⣯⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡇⠀⢸⡎⢿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡇⠀⠀⣷⠈⢯⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡇⠀⠀⢹⡀⠘⡆⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡇⠀⠀⢸⡇⠀⢹⡈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⡇⠀⠀⠸⡇⠀⠈⣇⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⣿⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⡇⠀⠀⢸⠇⠀⠀⢸⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢰⡇⠀⠀⣾⠀⠀⠀⢸⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡏⠀⠀⠀⠸⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣼⡇⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⡀⠀⠀⢀⣿⣷⣧⣤⣤⣤⣄⣀⠀⠀⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗⣚⣯⠭⣴⣶⣾⣿⠭⠽⡿⠶⠒⠒⠒⠛⠛⠛⢛⣶⣿⣿⣷⣶⡶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⠵⠶⠚⠋⠉⠁⠀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⠿⠟⠛⠉⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣷⣶⣶⣶⣶⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⢀⣀⣀⣀⣤⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠈⠉⠉⠉⠉⢉⣀⣀⣀⣀⣀⣤⣤⡤⠤⠴⠶⠶⠒⠚⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢴⠀⠰⡘⢂⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⡘⠃⠀⠀⠀⠀⠀⠀⠀⡤⠛⠀⠀⠀⢖⠀⠀⠠⡀⠀⠾⡧⠛⣦⡀⠀⠀⠀⠀⠀⢄⡠⣀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⢺⠀⡰⠀⠀⠀⠃⠀⠀⠀⢈⢳⣀⡀⠬⡟⠓⠀⠀⣀⠈⣣⠤⠨⠁⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠫⠶⢤⠄⠀⠀⠰⠚⣀⣍⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢒⢠⡤⠄⢎⡁⡐⠚⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣀⣰⣤⠀⠰⠀⠈⠀⠀⠀⠀⠖⠀⠀⠀⡀⠀⠀⠀⡀⠀⢤⠨⣿⠁⠰⠆⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡟⡏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⢸⡀⡀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢻⡇⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠛⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡈⢉⠝⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⠀⠀⠲⠟⢁⡿⠿⠍⠈⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠩⠙⠉⠛⠛⠋⠈⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠰⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠃⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡇⠀⠸⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣔⠀⢀⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠒⠛⠀⠛⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣄⠀⠀⠀⢹⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣹⣯⡀⠒⠒⢺⠀⠀⠀⠀⢀⣠⣤⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣻⣦⣤⣼⣶⣶⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⡙⠂⢀⣀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣛⣿⣻⣿⡆⠀⠀⡀⠀⡀⠐⢢⣄⠀⠠⠤
⠀⠀⠂⣀⣀⣈⣩⣭⣿⣾⠷⣾⣯⣽⣿⣷⣿⡟⢻⣿⣿⡻⣿⣿⣯⣭⣴⣶⣶⣿⣿⣿⣿⣿⣟⣷⣀⠀⠐⠒⠒⠒⠒⠂
⢀⣀⣤⣧⣤⣄⣀⣙⣿⣶⣷⣶⡾⣿⣿⣿⣛⣓⣟⣿⣯⣤⣤⣿⣿⠿⠿⣿⣿⣿⣞⣻⣿⣿⣿⣶⠶⠶⠦⠀⠉⠁
⠀⠁⢭⣉⣙⣋⡉⠉⠛⠦⣤⣀⣬⡭⠉⠙⢿⣿⣍⡁⣾⣿⣿⡿⠉⠛⠉⠀⠀⠐⢺⣖⣛⣛⣿⣿
⠀⠀⠀⠀⠉⠉⠀⠀⠛⠓⠶⠤⣤⡤⠄⠀⠀⠈⠁⠒⠒⠺⣿⡗⠀⠈⢉⡉⠉⠉⣉⣤⠴⠖⠀⣀⠀⠈⠉⢉⣉⠙⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠈⠛⠁⠀⠀⠉⠉⠉⠁⠒⠀⠀⠀⠈⠉⠉⠛⠛⠋⠁

⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣷⣶⣦⣄⣀⣴⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⣯⠉⠻⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠉⣿⡿⠋⠀⢸⣿⠀⠙⣿⣿⣇⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠸⣿⠃⠀⠀⢸⣿⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣁⣀⣠⣾⣿⣿⣷⣦⣸⣿⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⣿⠿⣿⣿⡙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⣿⠏⠀⢹⣧⠈⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠿⠀⠀⠀⣿⡆⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠸⣿⣧⡀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠙⠻⣿⣷⣄⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⢀⣾⣿⠏⠀⢀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢀⣿⠟⠁⠀⢰⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⡇⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠟⠛⠛⠉⣉⣉⣁⣀⣠⣤⣤⣤⣤⣄⣀⣀⣈⣉⠉⠉⠛⠛⠷⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⢁⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣑⡶⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢹⣿⢀⡠⠤⠒⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠐⣿⣡⠤⠖⠒⠒⠢⣄⡀⠉⢧
⠀⠀⠀⠀⠀⠀⢧⢀⡄⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈
⠀⠀⠀⠀⠀⠀⢸⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢀⡟⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢷⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣿⣄⡀⠀⠀⣀⣠⡴⠋⣠⠾⠁
⠀⠀⠀⠀⢀⡤⠂⠘⣧⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡶⠿⠛⣛⣋⣩⡥⠖⠋⠁
⠀⠀⣠⣾⠟⠁⠀⠀⠘⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡥⠶⠛⠋⠉⠁⠙⢷⣄
⠀⣴⣿⠋⠀⢠⠂⠀⠀⠈⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠉⠀⠀⠀⠐⢦⡀⠀⠀⣝⣧⡀
⢸⣿⡏⠀⢠⣣⠃⠀⠀⠀⠀⠙⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⢲⣷⠀⠀⢾⣿⣧
⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⡄
⣿⣿⡇⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢸⣿⣿⠃
⠸⣿⣷⡀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠛⠛⠲⠶⠶⠶⠶⠖⠚⠋⣩⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⣼⣿⡟
⠀⠻⣿⣷⡀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⠦⣤⣤⣴⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣼⣿⡿⠁
⠀⠀⠙⢿⣿⣦⡀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠉⠻⠶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠾⠛⠉⠁⠀⠀⢀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⣀⠀⠈⠉⠙⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠛⠉⠉⠀⠀⠀⢀⣀⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠾⠟⠛⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣠⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠂⠐⠊⠊⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡖⠀⠀⠀⠀⠀⠂⢀⠉⠁⠀⠃⠀⠠⠂⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣤⣬⣉⣴⣀⡀⠀⠤⠿⠟⠓⠰⢂⣠⣀⣠⣔⣿⡫⠇⢀⣠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⠉⠉⠈⠉⠉⠉⠙⠉⠙⠛⠛⠛⢿⣿⣿⣿⡏⢽⠀⣚⠁⣀⠤⠤⣈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡬⢭⣭⣿⡷⠇⢐⠄⣧⠊⠁⠀⠀⠘⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠬⢵⣒⣺⠍⢀⠁⡇⠀⠀⠀⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠥⠬⠽⣖⣒⠁⠐⢠⠇⠀⠀⢀⡴⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⣗⣒⠠⠀⢸⡆⠂⠉⢁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠨⠭⢭⣷⣒⠀⠁⣼⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠨⣈⣭⣿⡷⢂⢊⡜⠁
⠀⠀⠀⠀⣀⣀⡠⣤⠤⠀⠒⠀⠚⣦⣀⠀⠀⠀⠠⠀⣦⠡⣶⣿⣿⢟⣁⣴⣿⣶⣶⠶⠦⠤⣤⣄⣀⡀
⠀⠀⡞⡉⢁⠀⠀⡄⠀⠀⠈⠉⡀⠈⠻⢉⣀⡀⠀⠀⠚⡒⣿⣿⣍⣩⡽⠋⣉⠉⠉⠉⠛⠒⣢⠀⢈⠙⣟⡆
⠀⠀⠙⠻⢿⣓⡲⠤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣈⣉⣉⣉⣉⣉⣭⣉⣁⣀⣴⣶⣶⣦⡤⠤⠤⠖⣚⣻⠽⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠶⠤⠤⢤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣥⣤⣤⡤⠤⠤⠶⠖⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⡤⣀⣄⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢲⣄⡀⠀⣿⠀⠀⢻⣦⢀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⡇⢇⠈⠲⣿⠀⠀⢸⠋⠀⠑⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢲⣿⠀⠀⠘⠀⠀⠀⠈⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠆⠀⠇⠀⠀⠀⠈⠸⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢰⠀⠀⠀⠀⠀⠀⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠥⠏⠀⠀⠀⠀⠀⠀⡯⡞
⠀⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠠⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⡆⠀⠀⠀⠀⠀⠀⡠⠄⢄⠠⠹
⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⠀⠀⢐⢺⣿⣇⣀⠠⠀⠀⠀⠀⠀⣾⣧⢰⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⣠⡿⠁⡠⠀⠀⠈⠉⢻⣿⣥⣺⣟⣛⢋⡀⠀⠀⢻⣿⣜⠀⠐⣉⣤⣾⠏
⠀⠀⠀⠀⠠⠯⠹⠏⠀⠀⠀⠒⠛⠚⣿⣿⣿⡇⠀⠀⠀⠀⣀⣽⣋⣭⣶⣿⣿⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣏⣈⣿⣇⣀⣤⣶⣿⡿⣿⡿⣛⠉⠉⠛⢁⣜
⣶⣶⣦⣄⡀⢀⠀⠀⢀⠀⠀⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠶⣞⣋⣥⣼⡇⣸⣿⡀⣶⡄
⠿⠿⠿⡉⠀⠀⠀⣅⡘⠛⠶⣬⣽⣛⣻⣿⣿⣿⣋⣿⣿⣿⣧⣤⣶⣿⣿⣿⣿⡇⢻⠉⠀⣿⣿⡷⣶⣶⣮⣔
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡉⠛⠛⠛⢯⣽⣟⠛⠉⠀⠉⠉⠛⠂⠉⠉⠉⠀⠀⠀⠀⢿⣿⣗⠀⢩⣍⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠆⠀⠉⠁