Festive Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡘⢡
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣛⡹⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣬⠑⠀⠀⣠⣿⣼⣇
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⢠⠿⠿⠁⠀⠀⡀⠀⠀⠀⢀⡡⠤⠀⠀⠀⠀⠐⣤⠇⠸⢿⠀⠂⠔⣿⣟⡠⠞⣷
⠀⠀⠀⠀⠀⠀⢸⡰⡆⠀⠀⠀⠀⢀⣶⡶⠀⠀⠀⠀⣳⡜⠀⠀⢘⡁⢠⡦⠀⡀⠀⠸⡆⡾⠀⠣⠈⠠⣼⣿⠟⠁⠀⣠⡇
⠀⠀⠀⠀⠀⣴⡾⠀⠀⠀⡀⠀⠀⠚⡉⠀⠀⠤⡄⠀⡶⢰⡖⣠⡜⠃⠀⠁⠘⢇⣠⡴⠇⠽⢤⣀⠀⡇⠿⢃⣤⠶⠋⠁⣿⠀⢸⡀
⠀⠀⠠⠁⣿⣿⠟⢉⡁⢈⣯⡠⠰⡿⠇⠀⢀⣶⡇⢠⣧⣼⡅⠘⠀⣤⣼⡟⣴⣿⣁⣀⣀⣀⣤⡬⠷⠁⢴⠛⣵⠀⣀⠴⠀⢀⣎⢸⡄
⠀⠀⣇⢰⣏⡩⣶⢁⠀⡂⠀⡱⣤⠁⠸⠿⠤⠙⠁⡞⠁⠈⣧⠀⠀⠏⠙⠃⣾⣅⡀⣀⡀⣀⡄⡠⣾⣷⠈⢆⣿⠊⢉⠄⠀⢘⡁⢀⣙
⠀⠀⠸⡢⣿⣧⡜⢂⣉⠔⣤⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡀⠀⠀⠀⢻⣯⣽⣯⣼⣧⠯⠻⢿⠇⠀⠀⠡⣔⠉⠀⠀⣫⡀⢀⣈⣇
⠀⠀⠀⠘⠻⢿⡿⠧⠁⠈⠔⠁⠀⠀⢀⣼⣤⣀⣀⠀⠀⠀⠀⡟⢃⠀⠀⢀⠀⠙⠶⣶⠶⠖⢒⡾⠋⠀⠀⠀⠀⠀⠀⣠⣾⣯⣥⣤⣤⣭⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣏⠉⢄⠀⠀⢰⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠂⠘⣧⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠌⡌⠐⠀⠆⠱⠈⠆⡆⠀⢘⠀⠠⠋⠰⠈⠀⠘⢠⠁⠚⠛⢠⢰⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠘⠛⠛⠛⣧⠀⢰⡆
⠀⠀⢀⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⢠⠄⣀⠀⡀⠀⡄⢠⠀⠲⠈⠁⢀⣿⡇⠀⢠⣤⣄⠈
⠀⠀⠈⣡⡤⢠⠄⠀⣴⡇⠀⠀⠀⠀⠀⢠⡄⠀⡤⡀⢄⠀⠁⠀⠁⠀⠀⠀⠀⠀⣸⣿⣿⠀⠹⠿⠟⠀⠀⢀⣄
⠀⠀⠀⠀⠰⠈⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠰⡿⣄⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠉⠁⠈⠉⠀⠁⠡⠶⢿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠐⡅⠀⢠⠃⠀⠀⠃⠈⢀⣴⣷⡀⠛⠛⢁⣀⣀⡀⠀⢀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠑⢸⠘⠀⠀⠐⠰⠾⣿⡿⠆⠀⠀⠀⠀⠈⠀⠈⠁⠀⠐⢛⡉⣉⢳⡀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣀⠀⠀⢠⣈⣠⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠖⢳⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠀⠀⠀⠙⠀⠀⠘⠖⠃⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⢀⣠⣶⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⢄⡒⠀⠀⠸⠀⠀⠀⠚⠁⠀⠛
⠀⠀⠀⠛⠛⠿⠇⠀⠛⠋⣤⡶⠀⡀⠀⠀⠈⠥⠀⠘⠂⠀⠀⢄⡐⠀⠀⠀⠸⡀⠀⠀⠀⡀⠀⠀⠛⠀⢀⢀⣀
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⡀⠀⠀⠁⢀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⠻⣿⣿⣿⣄⠀⠀⣶⡄⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠸⠇⡌
⠀⠀⠀⠻⠿⢿⣿⡷⠾⠛⠀⠀⢨⣯⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⡎⠀⠀⠀⣌⠀⠤⠀⠀⢠⡄⣬⠀⠀⢠⡆⠅
⠀⠀⠀⠈⠁⠈⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉

⣀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠿⠿⣿⠿⠗⠀⠀⠀⠀⠉⠀⠀⢀⣴⣄⠀⠹⠇⢀⡀⠀⢰⣿⣿⣷
⣿⡀⠀⠀⣿⣿⣿⠄⠀⠆⠀⣴⡁⠀⠀⠀⣾⣿⣷⠄⠦⠀⣾⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⠀⠉
⣿⣷⡀⢴⣿⣿⣿⣧⠀⠀⣼⣿⣷⡀⠀⣠⣿⣿⣿⡄⠀⠰⣿⣿⣿⡷⠀⢀⣄⣀⠺⠿⢿⣿⠿⠟⠂
⠻⠋⠁⣼⡿⣿⣿⣷⠀⢀⣼⣿⣷⡀⠀⣹⣿⣿⣿⡏⠀⠼⠿⣿⠿⠿⠄⣹⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⢈
⠀⠠⠄⠀⠀⠙⢠⡄⠀⢉⣿⣿⣿⡉⠀⠛⠛⣿⠛⠛⠂⣄⢠⠘⠀⠀⠰⣿⣿⣿⠧⠀⠉⣀⠀⠐⢻⣤⡞
⣤⣤⣤⠄⠀⠀⣾⣿⡀⠚⠛⣿⠛⠛⠂⠀⠀⠉⠀⣸⡄⣏⢹⠀⣼⡆⠼⠿⣿⠟⠷⢀⣾⣿⣷⡀⠿⠛⣿
⠿⢿⢿⠀⠀⣼⣿⣿⣧⠀⠀⠉⠀⢦⡼⣆⠀⠃⢠⣿⣧⠈⠉⣸⣿⣿⡀⠀⡉⠀⠀⣸⣿⣿⣿⡇⠀⠀⣿
⠀⠸⢸⠀⣰⣿⣿⣿⣿⣇⠀⠋⠀⣾⡿⠉⣀⠀⠛⢿⠿⠓⢰⣿⣿⣿⣷⠀⠉⠀⠰⢿⡿⣷⣿⡿⠆⠀⠏
⠀⠀⣰⡀⠛⠛⢻⡿⠛⠋⠀⠀⠀⠙⠶⠿⢫⣾⣿⣄⠀⢀⣿⣿⣿⣿⣿⣇⢰⡆⢰⣿⣿⣿⣿⣿⡆
⠀⣰⣿⣷⡀⠀⡀⡀⠀⠀⣴⣶⣶⡀⠀⠀⢿⣿⣿⣿⠆⠀⠀⠉⠻⠉⠁⠀⣾⣷⡀⠀⠀⠟⠀⠀⠀⠀⠈
⠀⣹⣿⣿⡉⢰⡍⣵⠀⢈⣿⣿⣿⡅⠀⣰⣿⣿⣿⣿⢧⠠⢶⣀⡴⠃⠀⠾⢿⡿⠷⠀⢀⡄⠀⣼⣶⣶⡀
⠴⣿⣿⣿⡿⠆⠉⠁⠀⢾⣿⣿⣿⡿⠄⣹⣿⣽⣿⣽⣯⠀⣾⢿⣷⣀⣁⠀⢀⡀⠀⢠⠀⠀⢠⣿⣿⣿⡄
⣰⣿⣿⣿⣷⣄⠀⠂⢀⣾⣿⣿⣿⣷⡄⠛⠛⢻⡟⠛⠛⠃⡄⠀⣿⣿⣿⣿⣿⠀⢠⣿⣧⡀⢻⣿⣿⣿⡟
⠀⠀⢻⠁⠀⠀⠀⠀⠈⠁⠀⢿⠀⠀⣄⠀⢄⠈⠁⢀⠀⢀⣿⡀⡟⡇⠀⢸⢸⠀⣠⣿⣿⡀⠟⠛⣿⠛⠛
⠀⣧⠀⠀⠂⣰⣶⣶⡀⠀⠃⠀⠀⠀⣸⣶⣏⠁⠀⣀⢀⣾⣿⣧⠀⠁⠀⠈⠈⠘⣻⣿⣿⡛⠀⠀⣉⠀⠤
⣼⣿⣆⠀⠀⣽⣿⣿⡏⠀⣶⣶⣶⣶⣿⠋⠙⠁⢠⣿⣎⠙⠟⠉⠁⢠⣾⣦⡀⠘⠛⢻⠛⠛⠂⣴⣿⣷⡀
⠻⡿⠟⠂⢾⣿⣿⣿⡿⠄⡟⡟⠛⢻⣿⠀⠀⠐⢻⣿⣟⠂⠀⠀⢰⣯⣿⣿⣷⡀⠀⡈⠀⠀⢠⣿⣿⣿⡅
⢠⡄⣀⠀⣼⣿⣿⣿⣿⣄⠧⠃⣤⠘⠟⠀⢀⣠⣿⣿⣿⣦⡀⠀⣴⢿⣷⢿⣿⣄⢰⡟⣳⠀⣹⣿⣿⣿⣏
⢾⣽⡅⠈⠁⠀⢹⠁⠀⠀⠀⣸⣿⡆⠀⠤⠀⣰⣿⣿⣿⣆⠀⠸⣿⡿⣿⣿⢿⡿⠀⠓⠃⡄⠉⠉⢿⠉⠉
⣤⣙⡟⠀⠐⠂⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠘⠋⠉⣿⠉⠉⠃⣰⣿⣿⣿⣿⣿⣿⡆⠀⣸⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣠⣿⡇⠀⠀⠀⠀⠀⡇⠰⠖⠘⡸⢙⡤⢤⢐
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠙⠋⠀⠀⠀⠀⠘⠐⠻⠈⠒⠀⠱⢎⣀⢒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⠀⠀⠀⠀⣰⣀⡀⢠⡴⡴⢤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠖
⠀⠀⠀⠀⠀⠀⠀⣿⣾⢿⣽⠀⠀⠀⠀⠩⠸⠿⣿⡖⣲⡀⠿⡗⠳⠗
⠀⠀⠀⠀⠀⠀⣴⣯⣿⠿⣷⠆⠀⠀⠀⢴⣶⢶⡂⣟⣸⣶⣶⡶⠲⠶⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠐⣿⣯⣿⢿⣿⣾⠄⠀⠀⢸⡿⣢⣴⣷⣶⣶⣶⡔⠤⠖⠀⠀⠀⠀⡼⠀⠀⠀⠔⠀⡰
⠀⠀⠀⠀⠀⣼⣹⣹⣿⣿⢻⣿⣦⠀⠀⠐⣪⣼⣿⣿⠟⢻⣿⡇⠄⡀⠀⠤⠀⠈⠑⠀⢀⣴⠃⢀
⠀⠀⠀⠀⢠⣟⣻⣿⣿⢿⣿⢿⡾⢀⠀⠈⠁⠐⣿⠁⢀⡈⢻⣇⠈⡇⢀⡰⢄⣠⣦⣾⢟⠋⢀⠄
⠀⠀⠀⠀⡷⣿⣿⣿⡏⠈⣹⣗⣿⠟⠢⣤⡀⠈⡿⡬⠿⠧⣿⡇⠀⠃⠀⠀⠈⠀⠁⢸⠁⣆⠊⣀⡀
⠀⠀⠀⠀⢛⠻⠁⢘⠟⢱⡏⠛⠀⡟⢰⡄⢀⠠⡴⣦⣾⡧⠾⠭⣦⠀⠀⠀⠀⠀⠀⠈⡇⡃⠀⠈⣤⠰⡃
⠀⠀⠀⠀⢷⡤⠀⣰⠀⠈⠀⢀⣿⡟⣾⠁⣸⡇⠠⠾⠿⠷⠶⠶⠶⠶⠺⠁⠀⠂⡤⠈⠁⠀⣿⠀⢻⠞⣧
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠉⠀⠉⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⡀
⠀⠀⠀⠋⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠐⠄⠃⠔
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠭⢹⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠄⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⠒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⢰⠄⠀⢀⠀⠀⠠⢠⠀⠀⠀⢀⠀⢸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠐⠀⠀⠀⠀⠠⠉⠀⡀⠀⡄⠀⡀⠘⠀⠀⠘⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡈⡴⢶⡎⣀⠀⠠⠄⠀⠔⠁⠑⠄⠀⠁⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡙⡚⠣⡀⠀⠀⠀⠀⠢⠀⠠⠃⠀⠀⠀⠀⠁⠀⠀⠲⠀⣨⠡⡠⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢴⣟⣳⠡⠠⠤⠀⠀⠀⠀⠀⠁
⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⢀⠀⠀⣠⠐⠫⠁⠃⡄⠀⢠⠀⢀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠠⢔⡀⠄⠀⠀⠤⠀⠌⢢⣀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠐⢨⢇⠜⡕⠂⠀⠀⣀⠀⠀⠈⠑⠀⠀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⠄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠀⡱⠒⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⣀⣤⠶⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠤⠤⢀⢀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡔
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠄⡀⠂⠀⠄⠀⢀⠀⢘⡀⠀⠀⠀⡠⠀⠀⠨⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠊⢁⡀⡋⠁⠀⠈⠂⠀⠀⢠⠃⠀⠀⠉⠀⢤⠈⠀⠀⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢄⠀⠀⠀⢐⢯⠀⠀⠀⠀⠖⠀⠀⣀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣶⣾⣧⣜⢦⠀⠀⣞⠀⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠀⠀⠀⠈⠙⢦⣠⠻⣧⣾⠇⠀⠀⠀⠀⡀⠀⠠⠂⢀⠀⠀⠀⠺
⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠇⠀⠀⠀⢰⣧⣼⡦⢿⣦⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⠀⠀⠀⠀⠁
⠀⠹⢄⠀⠀⠀⡄⠀⢳⠀⠀⠀⠀⣠⡆⠀⣨⣍⣭⠖⣩⡿⠉⠀⠀⠀⠀⢀⠀⠀⠀⣀⠶⠁⠀⠀⠐⠁
⠀⠀⠀⠛⠢⡈⢧⠀⠘⠀⠀⠀⠀⠉⢃⠀⠙⠛⢿⣶⣿⡇⠀⠀⠀⢀⠔⣙⣀⡚⣀⢠⠀⠀⢀⣠⡦⣉⣠⠊
⠠⡄⠀⢀⠻⣤⠿⠗⠀⠘⠄⠀⠀⠀⠀⣠⠋⠀⠀⠀⢸⡷⠀⣠⡴⠟⠛⠛⠙⡁⠉⣠⣦⣴⣿⢿⠔⣠⠃
⠀⠀⠀⠀⠄⠻⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⡄⣦⡀⠸⢿⠟⠢⢀⠀⠐⠗⠀⠻⠛⠋⣹⣿⡿⠀⢀⡿⠤⠄
⠀⠀⠀⠄⠈⠂⢒⣇⠀⠀⠀⠠⠀⣼⣷⠄⠘⠿⠟⢹⣶⣿⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠸⡟⢐⠀⠈⠁
⠀⠀⠀⠀⠐⠌⣻⣿⣦⡀⣠⢤⡾⠋⠀⠀⠀⣶⣤⣷⡟⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠹⣍⡻⠦⠈⠁⠐⠒⠀⠀⠈⠉⠉⣶⣤⣿⠀⠀⠀⣀⠀⠀⠠⠐⢀⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⣶⡿⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠛⢠⣤⣀⠀⠀⠙⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠁⠀⢸⡟⡤⠄⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⠣⣀⣀⡈⢒⠀⠐⠀⠀⠀⡀⠀⠊
⠠⠒⠂⠀⣰⠏⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂⡾⠀⢀⣄⣶⣤⣀⣦⢀⡾⠁⡌⠸⠛⠉⠠⠘⠋⠀⠀⠄⠘
⠀⠀⠀⠐⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢈⣻⡿⠍⢻⣿⡊⠀⠀⠀⠊⠁⠠⢄⠀⡠⠒⠀⠉⠛
⠀⠀⠀⢇⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡁⣨⡴⠚⠍⠉⢐⠦⣄⠈⠑⢦⡀⠀⠤⠒⠉⠉⠑⠒
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠍⠀⢾⣦⡷⠻⣾⡿⠀⠀⠀⠁⠀⠢⣤⣞⡁
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠷⠲⢤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢠⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢬⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠄

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀
⠀⠀⠿⠯⢠⣾⣧⡀⠐⠀⠀⣾⠁⢨⡷⠀⣴⢒⣦⠀⢉⡀⢰⢿⠂⣰⡿⣛⠻⣇⠀⡞⠀⠈⢳⠋⠀⠙⡆
⠀⠀⠀⣰⣿⡟⣿⣷⡀⠀⠀⠹⢤⣀⡀⠀⢻⣌⣉⢰⠫⣽⠂⠀⠀⠹⣷⣭⣾⡏⠀⠳⡀⠀⢺⠀⠀⡸⠃
⠀⠀⣴⣿⣷⡅⣥⣿⣿⣄⠀⣶⠆⢨⠟⣉⠉⠉⡉⠙⢳⡀⠀⠃⠀⠐⢶⠭⠶⠆⠀⠀⠙⢦⣸⣠⠞⠁
⠀⠈⢉⣽⡿⠇⡿⣿⣏⠉⠀⡀⠀⠘⠦⠿⠀⠀⣉⣀⣰⣧⣄⣀⣀⡘⠻⠖⠿⠃⢀⣤⡀⠀⠙⣡⡶⣤⡀
⠀⠠⠾⢿⣿⡆⣾⣿⡿⠧⠀⡁⢀⠀⠀⣀⣶⣀⠙⣿⣿⣿⣿⣿⡿⠟⠀⠄⠀⠀⠈⠛⠃⠀⠸⣕⡿⣫⡇
⠀⠀⣠⣿⣯⡃⢋⣿⣷⡀⢸⣎⣹⡁⠐⣺⣿⣗⠂⢸⣿⣿⡟⠁⠀⢠⡄⢶⢦⣤⠀⠸⣿⠟⠀⢉⣉⠉
⠀⠐⢛⠛⠛⣷⡟⠛⠛⠛⠈⠑⣋⡁⠀⠀⠛⠀⠀⢸⣿⣿⣧⠀⢸⡗⢠⣶⣦⠐⣷⠀⠀⠀⣰⢿⣿⡷
⠀⠀⣉⠠⡄⢉⠁⠀⣠⡀⠠⣾⣿⣿⣆⠀⠀⠙⠀⢸⡿⡟⡿⣧⠘⢫⡄⣩⣡⣬⠋⠀⢼⣿⣇⣠⡿⠃
⠀⢠⣭⡷⢳⣯⣄⠘⠿⠏⠀⠘⢿⣄⣸⣿⣯⠄⢀⣿⡁⠀⠀⣽⣧⠀⠀⠉⠁⠀⠀⠄⠀⠀⠉⣉⠀⢠
⠀⠀⠴⢋⡏⠳⠀⠀⢀⠀⠀⢀⣀⣉⣉⣉⣤⣶⣿⡿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣦⠀⠀⢀⣼⣿⣦
⠀⠀⠀⢠⣤⡄⠘⠁⠘⢷⣾⣿⣿⣿⠿⠻⣻⣿⡃⠀⢐⣻⣿⣿⡿⠛⠉⢀⣿⠏⣿⡆⢠⣾⣯⣩⣿⣷⡀
⠀⢰⣿⡟⣙⢻⣷⠄⠀⢸⣿⣿⣿⣧⣀⢀⢸⣿⣷⣦⣷⣿⡿⠋⠀⠘⠃⢸⡿⠀⣿⠇⢀⣼⣿⢙⣿⣧⡀
⠀⠀⢽⣧⣭⣾⡏⠀⠶⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣁⡤⢤⣀⠀⣰⡞⠁⣴⠟⠀⠀⣼⣟⠛⣿⣇
⠀⢰⣶⣭⢀⣭⣶⡆⠀⢰⣿⣿⣿⣿⠋⠉⠀⣠⣀⠀⡼⢃⢮⣷⡂⢧⠈⡀⠀⢡⡀⠀⠈⠉⠉⡿⠉⠉⠁
⠀⠈⠻⠿⠿⠿⠛⠀⢠⣿⢣⣿⡟⠁⠖⠀⠀⠛⠋⠀⠸⣆⡛⢟⣂⠏⠈⠁⣰⣿⣿⡄⢀⡄⠀⠠⡞⢳⠄
⠀⠀⠠⠀⠀⠒⣠⣴⡿⠃⣼⡟⠀⣰⠚⣶⣴⣾⣶⡀⠀⠀⣍⢉⠀⠀⠀⠐⣿⣧⣾⡟⠂⠀⢀⡀⢉⠉
⠀⢠⡶⣲⢶⣰⡿⠋⣠⣾⠟⢁⠀⠹⣾⣿⣿⣿⡿⠀⠐⢶⡿⢻⡷⠂⠀⠰⣿⣧⣾⡿⠂⠐⠶⢟⡻⠶⠂
⠀⠸⣞⣿⣳⠇⠀⠀⡿⠋⠐⣏⣱⠀⠘⢿⡿⠋⠀⠆⠐⠛⡿⢿⠛⠃⠀⠰⠛⢧⡾⠿⠂⠐⢋⡶⢾⠙
⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠻⢀⠘⡀⠃⠀⠶⠧⠊⠃⢡⡴⠀⠶
⠀⠡⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣼⠁⠀⠠⣤⢠⣆⣃⠦⢯⢡⠑⣄⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⠉⠓⠁⠈⠀⠀⠘⠑⠀⡄⠉⠉⢠⠀⠊⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠩⠁⠀⠒⠐⠊
⠀⠀⢠⡀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡀⠡⠀⠀⠀⠂⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣦⢤⡶⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠤⡨⠂⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢰⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡤⠔⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠠⢀⠀⠀⠀⠀⡀⠀⢀⠀⠐⢠⡄⠀⠀⠀⠠⣸⣀⠤⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢄⣸⣠⠄⠀⠀⠀⢀⡄⢂⠀⡀⠀⢀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠱⠘⠀⠀⠀⠠⠄⣻⠄⠑⠆⣲⢒⡐⠂⠀⠐⠛⡝⠒⠀⠈⠒⣀⢀⠰⢀⠄⡠⠎⠀⠐⠊⡍⠓⠀⠀⠀⢂⡚⢖⠀⠎⠡⣞⠠⠄⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠂⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠲⢈⡁⠆⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠐⠀⠀⠈⠁⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⠀⠀⠀⠘⢘⡠⢣⢜⡳⠜⠀⠁⠁⠈⠀⠀⠄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⠀⠀⠄⡹⠃⡭⠀⠀⠀⢀⠐⠑⠠⠍⠂⠑⢦⡀⠀⠀⠨⢐⠟⡠⠀⠀⠛⠀⢀⠀⠀⢠⠀⠀⠀⠀⡀⠠⠄
⠀⢀⠀⢿⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠂⠀⠀⠐⠄⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠀⠀⠀⠀⠀⠂⣼⠆
⠀⠀⠂⠠⡄⠀⠀⡀⠀⠀⠠⡇⠀⠆⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠆⢨⠅⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⠤⠢⠀⢠⠀⠀⠐⠀⢰⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢠⡆⠀⠀⠀⠈⡠⠀⠔⠤⠤⠄⠀⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠖⠂⠒⠀⣷⠀⠀⠀⠑⣄⠀⣀⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⣀⠌⠀⠀⠀⣴⠀⠔⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠒⣀⠋⠰⠀⠀⠙⠛⠀⢣⠆⠀⠀⠀⡄⠀⠀⠐⡞⠂⠛⠊⠀⠀⠒⠘⣀⠐⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⡄
⠀⠀⠀⠀⢤⠘⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠌⠀⠀⠀⠀⠀⠘⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⣠⠁
⠀⠀⠀⠀⠀⡀⠊⢠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠤⠑⢂⠄
⠀⠀⠀⠀⠀⢀⠐⠆⢀⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠘⠢⢜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⡰⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠔⠛
⠀⠀⢠⡄⠀⠀⠀⠉⡈⠂⡑⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠐⢃⡈⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠤⠄
⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠄⠄⠡⠄⠤⠀⠀⠠⠀⠀⠂⠀⠀⠀⡈⢀⣀⠀⠄⢂⠀⠀⠀⠀⣀⡰⠤⡀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠃⠀⠀⠀⠀⠂⠀⠄⠀⠀⠤⠠⠤⠀⠠
⠀⠀⠀⠐⠈⢀⠂⠀⠀⠀⠀⠘⢣⣽⡀⣁⢒⢡⡤⠭⠒⢠⠊⠉⠀⠀⠀⢉⣇⣀⠟⠀⠀⠀⠘⡙⡄⠐⠭⠤⡌⠐⣀⢀⣼⣔⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠐⠋⣕⠉⠉⠀⡃⠠⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⠘⡉⢢⠙⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠔⢲⢺⠀⡉⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡗⢒⠢⠔⠂⠀⠀⠀⠠⡄
⠀⠀⠀⢈⣰⠀⠀⠀⠀⠂⠒⢋⠁⡈⠐⠀⠀⠀⠀⠀⠀⠀⠗⠔⠃⠠⠄⠀⡀⠀⠀⠃⠘⠀⠃⠌⠀⠀⠀⠀⠀⠀⠈⠐⢉⠀⡉⠒⠀⠂⠀⠀⠁⣦⠀⠆
⠀⠀⠀⠈⡃⡄⠀⠀⠀⠈⠄⢨⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⡁⠠⠁⠀⠀⠀⢁⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠃⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠠⠀⠀⠐⠀⠀⠀⠄⢰⡈⠣⢀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠓⡄⠀⠀⠀⠀⠐⠂⢲⠀⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠁⠀⠀⢀⠞⠁⠃⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡯⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣙⣳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⡙⠿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⣤⣴⣿⣷⣆⠈⠁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠻⠟⢻⣿⠿⣿⣿⣿⣿⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣠⠴⠛⠃⣤⣾⣿⣿⣿⣬⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⣧⠄⢀⣴⣿⣅⢀⣿⣿⣍⡻⡟⣳⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⢆⢀⡷⣼⣿⣿⡿⠟⢁⡰⠎⣩⣿⣿⠏⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⢿⣿⢿⣿⣿⡟⠉⣃⢲⡛⠛⠛⠻⢆⣈⣤⣤⠴⠋⣹⠇⠀⠀⢀⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⢀⢉⡼⠋⠁⢠⣾⣿⣯⠝⢶⡤⡴⠾⠿⠟⢽⢳⣾⣥⣤⣤⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢶⠞⠛⣿⡀⢀⡀⣀⣾⣿⣿⣿⣶⣤⣤⣤⣷⣶⣶⡘⣟⣿⢿⣿⠿⢿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣁⢀⣤⠂⠀⢀⣨⣷⣨⣿⣿⣿⣽⣿⣿⣿⡇⠀⢹⣿⡟⠿⡿⢧⣿⣶⣤⣴⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣒⠉⠙⠓⠲⠧⢬⣁⠛⠉⣫⣼⣄⡠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡿⠛⢹⢿⣻⣽⠟⠋⢱⣾⠿⢿⣦⡽⠟⠛⠿⣿⡷⣶⡄⢠⡼⢋⢀⡀⠀⣠⣶⢶⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡂⣀⠠⣦⠰⠟⠉⠀⣀⣠⣾⣿⣄⣼⣿⣖⠶⢤⣤⡬⢿⣶⠊⠁⠙⠿⣿⣿⣶⣟⣁⣾⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⡆⣉⢩⣅⠀⠀⠘⣧⡀⣠⣾⠟⠛⠋⣠⣦⠀⠛⢿⣦⣆⡀⠀⠀⠙⠿⣷⣠⣤⣀⠈⠉⣉⠉⣭⣁⣴⣶⣾⠇
⠀⠀⠀⠀⠀⡸⢸⣿⣿⣦⡆⠀⠙⠣⣾⠟⣿⡷⢀⣷⣷⡟⠾⠁⣀⣴⣿⠧⣿⣷⡀⠀⠙⣿⣿⠂⠀⠀⡖⠉⢻⡟⣿⣿⣶⣿⣶⡏⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠐⣷⣾⡿⢣⡟⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠁⠀⠀⢻⣿⣥⣿⠒⢿⣿⡇⠀⠀⠈⢿⡿⣿⣇⠰⢲⠞⠀⠀⠙⢿⣇⡁⠀⠀⠀⠀⠀⣠⡄⢰⣞
⠀⠀⠀⠀⢀⡿⠙⢱⡟⠀⣴⣶⣶⣶⠀⣠⣴⠿⠋⢀⡰⣷⣾⣿⣿⠿⢿⠦⢾⣿⣷⣶⠀⢀⠈⠙⠻⢿⣿⣛⢿⡷⢦⣄⣠⣈⢉⠓⢦⣤⣤⣾⢣⢿⣼⣿⡇
⠀⠀⠀⠀⢸⠃⠀⡾⢸⣿⡘⣿⣿⣿⡟⢉⠀⠀⣰⣿⣷⡸⣿⣷⣿⣤⣴⣄⣴⣿⣟⣻⣾⣿⡆⢰⣶⣾⣯⠉⠳⣄⠸⠟⠛⠛⢾⣿⣤⣦⣤⣿⣿⡜⠛⡿⠤⠴⠂
⠀⠀⠀⠀⢸⡆⢸⡇⠀⣿⣧⠹⣿⣿⣿⡘⢷⣿⠟⢛⣉⢷⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠙⣿⠇⣾⣿⣿⡟⢀⡼⠋⣀⣀⠀⠀⠀⣽⢻⡽⠿⣿⡍⠀⢸⠷⠿⠂
⠀⠀⠀⠀⢸⣇⠈⢷⡖⠛⡿⠃⠉⣀⣀⣰⣞⢳⣼⣿⣿⠿⠿⢿⣿⣿⡇⠀⢀⣤⣤⣄⢸⠟⠀⠿⢿⣿⣧⣾⣵⣿⠟⠻⣿⡆⠀⣷⢰⣇⣴⡟⣠⢆⡇
⠀⠀⢀⡴⠾⣷⡀⠈⢿⡾⣧⣴⣿⣿⠋⠉⣿⣿⣿⣿⡇⠀⠀⠸⡏⢿⡇⣰⠿⠋⠉⢻⡟⠶⠦⠶⣦⣿⣿⣿⣿⠀⠀⠀⢹⣿⢠⡷⠖⢛⣫⣾⣡⣾⣀⡀
⠀⠀⠈⢰⣿⣿⠧⡀⢨⢷⣿⣿⠷⠛⠀⠀⣸⣿⣿⡏⠤⠀⠀⠀⣿⡘⣹⣿⠀⠀⠀⣼⠀⠁⠀⠀⣜⢹⣿⠟⣿⠄⠰⠆⣼⣾⣯⠶⢒⣻⣿⣟⣥⣤⢀⡉⠁⣀
⠀⠀⠀⠀⠉⠃⠀⢻⣿⣼⣟⢣⣀⠐⣠⢤⣿⣿⣿⣿⠔⣄⠶⢰⣿⠇⢻⣿⠣⡜⠃⠻⡇⠶⠀⠀⠉⣻⣟⢦⡀⠶⣤⠴⠂⣠⢿⣾⡟⠋⢠⡅⠈⡿⠏⠛⠃
⠀⠀⠀⠀⠀⣼⢻⡏⠙⢯⠙⠿⢿⣯⣤⡞⠁⣉⣿⡀⠀⠀⠑⠉⠉⠀⣸⡈⣳⣶⣶⡾⠿⢆⠰⡶⠚⢿⣿⣿⣿⣶⣦⣀⠶⠋⣀⠬⠛⢁⡾⠶
⠀⠀⠀⠀⢰⡇⡀⠃⣀⢈⣷⣀⣀⠈⢩⡙⠚⠻⠿⣷⣤⣀⣀⣀⣄⣰⣿⣿⣯⢿⣿⣇⢀⡜⠉⢱⣄⣼⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⣸⣧⣄⠀⠈⡄
⠀⠀⠀⠀⣸⢟⣡⠋⣹⣿⣿⣿⡿⢣⢸⡏⡷⣶⠀⠀⠉⠉⠻⠿⠿⢿⣿⣿⡿⠿⠿⠿⠛⠛⠀⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡟⠀⠀⡧
⠀⠀⠀⢀⡾⢹⡇⠀⢿⢻⠝⡏⠀⠀⢸⡇⡇⣿⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⣤⢸⠆⢶⠀⠀⠀⠀⠀⠀⠀⣤⢀⣀⣠⡄⣲⠴⢷⠀⣸⡿⢥⣠⠔⠋
⠀⠀⠀⢘⣷⠟⢳⡀⢀⡟⣰⠀⠀⠀⠘⠃⠗⠿⢸⡏⢸⡇⢸⡇⢰⡆⡆⠀⣿⠀⠀⢼⠐⡆⣷⠁⣷⠛⣤⣿⢸⣿⣿⠷⠟⠃⠈⢠⡏⠀⢠⡗
⠀⠀⠀⢸⡏⠀⢸⣿⣼⠁⠈⠉⠛⠲⠦⠤⣤⣀⣀⡀⠈⠁⠈⠃⠈⠃⡇⠀⠙⠀⠀⠚⠚⠃⠛⠀⠉⠀⠉⠀⠀⠀⣀⣀⠤⠤⠒⠉⢷⣠⠟
⠀⠀⠀⠈⢿⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣄⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⢴⡦⠶⠶⠖⠙⠃⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢰⠖⠟⠳⢲⡆⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣰⣿⣿⣿⣿⣿⣦⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⢰⣶⣶⡆⢱⣶⣶⡆⠀⠀⠀⠀⠀⠇⠀⠀⡀⠀⠾⢽⢯⠷⢀⢀⠆
⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⡷⠀⠀⢸⣿⣿⡇⢸⣿⣿⠁⠀⠀⢾⣦⡀⠀⠀⠀⠈⠀⢰⣿⢾⣿⣄
⠀⠹⣿⣿⣿⣿⣿⠟⢀⣴⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⡇⢸⣿⣿⠀⠀⠀⣸⣿⣧⣤⣄⠀⠀⠐⢿⠿⠀⠿⠿⠁
⠀⠀⠀⠉⠉⠉⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠸⠿⠿⠇⠸⠿⠿⠀⠀⠿⠟⠛⠉⠛⠛⠃⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣉⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⡁
⠀⠀⠀⠀⣸⣇⠀⠀⠀⠈⣚⣷⡴⠀⠀⢦⣴⣗⠚⠀⠋⣤⠀⠀⠀⠀⠀⢈⡉⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⢀⡻⢟⡀⠀⠀⠈⠁⢖⣿⣶⣶⣾⣳⠋⠁⠀⠈⢃⡄⠀⠀⠀⠀⣈⡠⠄⠀⠸⠿⠆⠀⣀⠂⠀⡀⠀⠺⠿⠂
⠀⢻⣿⣸⣿⣿⣇⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣤⠄⠀⢀⣈⣁⡤⠖⠀⠀⠀⠀⠀⠈⠉⠹⣷
⠀⠈⠿⠋⠀⠀⠙⠻⠋⠀⠀⠈⣿⠛⠛⢻⠏⠀⢠⣦⠀⠀⢡⡆⠀⠈⣡⣀⡠⠵⠚⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣰⣾⣶⣶⣧⣄⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠀⠎⠸⠂⠀⠈⠈⠁⠀⠂⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⣠⣶⣿⠓⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⣾⣿⣿⠟⠃⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠐⠻⠿⠿⠀⠀⣤⣄⣻⣿⣿⣟⣠⣠⠀⠀⠀⢀⣴⣄⡘⠇
⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠂⠀⠀⠙⢿⣿⣷⣆⣹⡿⠋⠀⠀⣴⣿⣤⣬⣿⡆
⠀⠰⠿⠿⠿⠿⢿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣄⠀⠀⠀⣿⣿⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠋⠈⠿⢿⠷⠀⠀⠿⠿⠀⠸⠿⠇
⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⢀⣬⠁⠱⠆⠀⠀⠀⠀⠀⠀⠀⣔⣰⣶⣠⣾⣧⢀⣤⠀⠀⣴⡟⠛⠉⠉⠙⢳⣆⠀⠀⢀⣴⣾⣦⡀
⠀⠀⠘⢆⡛⣡⡿⡿⠳⠇⢀⠀⠠⠀⠀⢀⣹⣷⣽⣿⣿⣿⣿⠃⠀⢘⣿⠀⠀⠀⠀⠀⠀⣯⠀⢠⣿⡿⠟⢻⣿⡆
⠀⠘⠛⣤⠢⣤⠷⢆⠈⣘⠋⠀⢇⠀⠀⠸⢿⣿⣿⠿⠿⠟⠃⠀⠀⠀⢿⣤⣀⡀⢀⡀⣰⡟⠀⢺⣿⣭⢀⠰⣿⣿
⠀⠀⠀⠉⠀⠀⠀⠘⠁⠙⠁⠀⠀⠑⠂⠤⠤⠤⠤⠤⠄⠂⠀⠀⠀⠀⠀⠉⠙⡿⡿⠛⠉⠀⠀⠈⠻⢿⣿⣿⠟⠃

⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢡⢠
⠀⠀⠘⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠆⠰⠈
⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠛⡘⠛⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠰⡇⠀⠀⠠⠤⠖⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠠⠄⠀⠀⠠⠠⠄⡄⠀⠀⠀⠀⡂⠆⠅⠀⠀⠀⠰⠰⠀⠀⠀⠀⡀⠀⠀⣠⢭⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠰⠠⠀⣃⣀⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡀⠀⠀⡀⢀⢀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⢸⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⡀⢇⣃⠸⡤⡀⣧⣅⠈⣆⠀⠀⠀⠁⠀⠀⠸⡿⠇⠀⠀⠀⠀⠀⡀
⠀⠀⣩⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⢀⡄⡄⠀⣄⠀⢠⠀⡀⡀⢠⡀⠀⡄⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⢀⣿⡇⠀⠈⠫⣶⡽⠃
⠀⠀⡡⢖⠀⠀⠀⠀⠀⠀⢀⠀⠀⠣⡀⠐⠇⠧⠀⠷⠄⠸⠀⠬⠄⠀⠧⠀⠧⠉⠆⠮⠐⠦⠰⠥⠀⠀⠠⠻⠛⠃⠀⠀⠀⣿
⠀⠀⠖⠉⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠁⠈⠁⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠁⠀⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⢴⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢄⠐⠀⣠⠆⠀⠀⠀⡆⠀⡆⠀⢀⢀⠀⠀⠀⢀⣀⡀⠀⠀⣄⠀⠀⠀⢰⡿⠁⠀⢈⠀⠄⠀⠀⠁⢈⠄
⠀⠀⠀⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠆

⠀⠀⣤⣤⣀⠀⣀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⢀⣀⣀⡠⠀⢤⣤⣤
⠀⠀⢹⣿⠋⠀⠹⣿⡏⠀⠹⣿⡯⠉⢽⣿⡷⠀⠲⣿⡶⠂⠰⣾⡶⠂⠰⣟⡶⠃⠈⢻⡿⠏⠈⢻⡿⠇⠀⠛⣿⠇⠀⠐⢿⠏
⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⠛⠀⠀⠈⠛⠀⠀⠀⠻⠃⠀⠀⠸⠃⠀⠀⠸⠇⠀⠀⠈⠏⠀⠀⠀⠛⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⡀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣀⡴⠄
⠀⠀⠀⠀⢛⣼⡺⠀⠀⠀⠀⠀⣻⣿⣿⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣿⣿⡟⠁⠀⠀⠀⠀⢠⣆⣀⠀⢿⣿⡘
⠀⠀⠀⠀⠀⠉⠁⠐⣶⣾⣇⠀⠈⠀⠉⠀⣾⣿⣿⠀⠀⠀⠀⠀⢴⣿⣿⡃⠈⠁⠈⠁⠀⠀⣀⠀⠘⢻⣿⣿
⠀⠀⠰⠖⠀⠀⠀⠰⠿⢿⡏⠁⠀⠀⠀⠀⠈⠉⠉⠀⢠⣾⣷⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⡞⡹⠇⠀⠈⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠘⠿⠾⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠒⠀⠀⠀⠀⠀⢀⠀⠁⠁
⠀⠀⠀⢀⡾⠿⠞⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣦⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠖⠄⠉⠃
⠀⠀⠀⣿⣧⠀⠀⢈⣿⡇⠀⠀⣀⣀⡀⠀⠀⡀⠀⡀⠀⣀⡀⢘⢋⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⡀⢀⡀⣀
⠀⠀⠀⠻⣿⠀⠔⢺⡿⠃⠀⠨⢛⣧⣧⠀⠀⡟⠶⢻⠀⡧⠄⢸⢴⠃⣯⢼⠁⢱⠎⠀⠀⠈⠁⠁⠀⠀⢛⠋⣉⣡⣟⡃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠈⢀⠃⠃⠈⠀⡁⠁⠈⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠊⠀⠀⠀⠀⠙⠁⠘⠃⠑⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠈⣠⣤⡀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣛⣛⣂⠀⠈
⠀⠀⠀⠰⣾⣷⣾⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡄⠀⠀⠐⠽⠿⠯⠛⠄
⠀⠀⠀⢀⣝⣛⡿⠿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⡖⠀⠀⠠⣄⣀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠉⠹⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠠⡾⠃⠀⠀⠀⡞⠛⠆⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣠⠤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⣀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠤⠄⠀⠀⠠⠤⠀⠄⠀⠀⠀⠀⠿⠿⠶⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣀⡀⢮⣷⡄⠀⠀⠀⠀⢀⡴⣆⣠⣄⣤⣀⡄⠀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢮⠅⠘⠛⠀⠀⠀⠈⠀⠠⢤⣀⠀⢠⣀⣈⡀⠀⠁⠠⠄⢀⣠⡼⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⣿⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠈⠀⠀⠀⠁
⠀⠐⠀⠀⠄⠤⠀⠀⠀⢀⣠⡝⠁⡈⣿⠇⠀⠀⠀⠀⠀⠀⢠⣦⣴⣦⣴⣦⣤⠀⠀⠀⠀⢿⡏⠋⠁⠀⠀⠀⠀⠒⠒⠲⠆⠐
⠀⠀⠀⠀⡀⠀⠙⠿⠛⠛⣿⣇⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠒⠒⠒⠘⠋⠙⠀⠀⠀⠀⠀⠐⠋⠋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⣭⣽⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠉⢱⣿⣿⣿⡿⣄⣴⣶⡖⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣴⠾⣿⢻⣟⢛⣿⣾⠟⢻⣷⣟⣩⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⡀⣀⠑⠾⣍⠈⣶⣾⣿⣾⣿⡟⣵⣢⣶⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⣢⡅⠐⣿⣣⣤⠞⣟⠷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣥⣾⣖⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣄⠀⠀⠘⣿⠁⢸⣿⡤⡴⡆⠈⡇⣿⣿⣿⣿⡿⣿⣶⢮⠙⠡⠷⠈
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⠀⠀⣸⣿⣧⣿⡟⣀⣠⣯⣾⠿⣿⣿⣿⣿⣧⡜⠋⡀⣰⠆⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠐⢻⡿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠟⠁⠸⡦⠚⣋⢤⣿⢿⣿⣿⣿⣿⣟⡿⢛⣿⣿⣿⡛⣟⠒
⠀⠀⠀⢀⠀⠀⠀⠠⠇⢙⢿⣿⣿⣿⣿⡿⠋⢀⣰⠓⣲⢶⣶⣇⣈⢀⣾⣿⣿⣿⣯⡾⢛⡋⢒⡀⠀⠈⠛⠁
⠀⣤⣶⣿⣿⣆⡀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⠴⠛⡷⣤⣌⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠿⠛⠁
⠀⢿⣿⣿⣿⣿⣷⠻⣧⠀⣤⡜⣿⣿⣿⣿⡤⢬⡛⠻⠟⣿⡟⣿⣿⣿⣿⣿⣿⣿⠉⡜⠉⢩⠓⠂⣠⣤⣤⣀⠀⢀⣀
⠀⠈⠻⢿⣿⣿⣷⣄⣿⣾⡏⣰⣿⣿⣿⣿⣗⠛⢨⡿⠷⣬⣥⣤⣿⣿⣿⣿⣿⣿⣶⣿⡶⣾⡂⣼⣿⣿⣿⣿⣶⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⡇⡋⢀⣻⣿⣟⣻⣿⣿⡿⠛⣻⠿⠻⣿⣿⣯⡥⡀⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⠉⠀⢿⣿⣿⣿⣿⣿⡾⠋⣟⣼⣟⣻⣿⣿⡇⠀⣿⠀⢀⣿⣿⣿⣫⠥⢈⠀⠀⡉⣉⣀⣾⣿⡿⠿⢿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⣾⣿⣿⣿⣿⣿⣷⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⡶⠢⠁⢠⠚⠙⠛⢛⣿⣿⠁⠀⢨⣿⣿⣿⣿⣏
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⢠⣿⣿⣿⣿⣿⣿⡟⠛⣤⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣘⡇⠀⠠⠐⠴⠿⠾⠛⠛⠁⠀⠀⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠈⢻⣿⣿⣷⠀⢸⣿⣿⣿⣿⣿⡿⣷⢾⣏⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣋⣷⣤⡀⣷⣶⣂⣠⣾⣧⠀⠀⢠⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⢘⣿⡿⣿⣿⠁⠠⢼⣧⣿⣿⢟⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⡽⠿⠻⡅⣟⢀⠀⢸⣿⣿⣿⣿⡅
⠀⠀⠀⠀⢰⣿⡏⣿⡟⠀⢸⣿⠁⠸⣿⣰⡶⠛⠙⠁⠀⣮⠀⠀⢸⠟⠇⠸⣀⠟⠛⢿⣛⣿⠀⠀⢰⡇⠁⠀⡇⢸⠀⠀⢸⣿⣇⢸⣿⣷
⠀⠀⠀⠀⠈⣿⣹⣿⠇⠀⢾⣿⡀⢴⣿⣏⠁⠀⠀⠀⠀⣿⠰⠄⣻⠶⠀⠄⣿⠀⠀⠀⠿⠀⠀⠀⢸⡇⠀⠀⡇⠈⠀⠀⠸⣿⡟⠸⠿⣿⠇
⠀⠀⠀⠀⠈⠛⠻⣿⣷⣄⠈⢿⠄⠀⠸⠿⠀⠀⠀⠀⠀⠿⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢾⠿⠃⢤⣾⡿⠇
⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⠤⣤⠤⣤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢦⡴⣟⠢⣶⣤⣌⣤⠞⣠⣄⡰⢦⢀⠀⡞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⣶⣿⠦⢤⡄⣀⣀⢅⣀⣰⣸⡀⡄
⠐⣻⡅⠋⠑⠿⡟⢫⠈⢨⢈⣟⢫⠉⣿⠓⣹⣩⢇⡀⡀⠀⣤⠀⠀⣀⣀⡼⢀⠮⠂⢪⡇⣿⣣⡉⣿⡿⢁⠺⡘⣯⠼⡦⢿
⠈⢇⢓⡦⡟⡾⢹⡼⢗⣙⢈⡷⣈⡻⠖⢢⠒⣶⡐⣿⡟⣿⡙⢿⣿⢽⡿⣺⣍⣌⣇⣰⣆⠺⠧⡑⠾⣷⠈⠠⡌⢓⢂⡟⡶
⠀⠀⠼⡆⢉⠗⢜⠓⠎⠴⢖⣄⡹⠟⢵⣡⠟⡀⢌⡙⢶⠉⣐⠈⠷⠠⠃⠄⡂⣹⣧⢧⢸⠌⣣⡘⢲⢦⠈⠇⢉⠘⠾⠑⢹
⠀⠀⢠⠀⡌⠁⠄⡐⠀⠀⠈⠀⠁⠚⠂⠛⠀⠸⠾⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠄⢄⠉⠠⡌⠳⠶⠌⠀⠁⠈⠈⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠁⠀⠑⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢠⡖⠀⠯⠛⠧⢽⡷⣀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣈⡉⠀⡀⠀⣠⣄⠀⠈⠐⠒⠓⣊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⡌⢳⡤⣴⢜⡛⢋⡅⠻⣖⡖⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡁⠀⠛⠋⢉⣀⠀⠻⣿⣺⡏⠑⢪⣠⡯⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⡀⣠⠶⠉⠘⠀⠀⠀⠀⠉⠀⠄⢀⠀⠉⠈⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣏⣹⡿⠖⠉⠠⠤⠠⠠⢶⣈⡓⠢⡤⠀⠤⠑⢀⠁⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠉⠉⠀⠈⠁⠀⠀⠃⠂⠙⢟⢷⡀⢀⡠⠠⣈⠉⠀⠈⠛⠉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⢠⠤⠠⡈⠂⠈⠳⣶⣆⣒⢾⠃
⠀⠀⠀⠀⠀⠨⡀⢠⡼⠠⠄⣠⢦⠂⠀⠊⠀⠀⠀⠠⠀⠁⠀⠀⠘⡆⢃⡈⠀⠏⢦⠀⢇⠈⠉⠉⣉⠇
⠀⠀⠀⠀⢠⠒⣿⠋⠇⠀⠀⢱⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢼⡿⠆⠒⣾⡦⠈⠛⠉⠋⢹⠀⠆
⠀⠀⠀⢠⠤⠹⢿⡴⠺⠆⠀⠐⣧⡀⠠⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⢳⣆⡀⡠⠶⠝⠦⠄⠋⠌⣿⣸⣆⣿
⠀⠀⠀⠀⠀⠀⢨⠆⠀⠀⠀⠀⢡⠘⢆⠈⠦⠀⠀⠄⠀⠀⡇⠀⠀⠘⠟⠇⢠⠀⠠⠄⠆⠰⠈⢙⢻⡏⡟⠑⡖⣒
⠀⠀⠀⠀⢐⡄⢘⡆⢀⠰⡂⡈⠪⠁⠈⣷⠄⠀⢤⣄⣀⡤⠰⠀⠈⢀⢠⡄⣠⠀⣠⢂⣠⣰⣦⣰⣤⠇⣥⢾⣶⡇
⠀⠀⠀⠀⠉⠉⠈⠉⠉⠁⠉⠉⠁⠈⠉⠉⠉⠉⠁⠈⠉⠈⠁⠀⠀⠉⠀⠉⠀⠉⠁⠉⠉⠘⠛⠙⠉⠉⠈⠉⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠈⠉⠂⠠⠠⠀⠀⠀⠀⠀⠀⡄⠊⠀⠈⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠛⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠠
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⠴⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⣧⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⢀⠀⠐⣌⣧⠄⠀⠀⢀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠿⡦⢴⡋⡀⠀⢸⡀⠀⠀⠀⠀⠠⠾⠁⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠠⠀⠈⠑⠃⠀⠈⢱⢾⡀⣀⣸⣁⣤⣄⣀⡀⠻⣦⠐⢦⠀⢄⠀⠈⢢⡀⠀⠑⢚⢟⡽⢃⡞⡀⠀⠀⠀⠀⠀⠈⣋⡰⢀⣧⡄⠀⠀⡄
⠀⢀⡤⠴⠒⠚⠛⠋⣶⡟⢻⠋⣩⡟⠁⠀⢐⡜⣄⠈⠙⢿⠁⣺⠁⢻⢀⡦⠈⠁⠆⠀⠶⠾⠷⠉⣅⠀⠀⠀⠉⠁⠉⠉⣉⢀⣤⠃⠀⠂⠠⠄
⠀⠀⠀⠀⠢⠄⠖⠂⡨⠊⠁⢻⡟⠑⢀⣠⢯⠊⢇⡀⠀⠈⠓⠋⠛⠛⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⡈⣠⢀⢈⡝⠻⠿⠛⠧⣈
⠀⠀⠀⠀⠀⠀⢀⡔⠁⢠⣺⠿⠂⢠⠋⠀⠀⠀⡨⡑⣆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⡃⢹⠉⠀⠨⢡⡀⠐
⠀⠀⠀⠀⠄⢀⡼⠖⠓⣂⢬⢀⣀⡠⠄⠀⠀⠀⠀⠁⠘⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠺⣥⠇⠰⠐⢦⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠐⠉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠄⠒⠁⣹⣁⣰⠂⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠿⠩⠖⠻⡤⠿⠥⣴⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡈⠁⣰⠶⡤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⠇⠀⠛⠀⠈⠂⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⡄⠀⠰⡀⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢣⠱⣤⠟⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠃⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠄

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠒⠠⡀⠀⠀⠀⠀⢀⡤⠒⠠⠀⠀⠀⣀⠀⣤⠀⠀⠀⠀⠀⢀⣀⠄⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⡀⠠⠀⠈⢀⠀⠈⠆⠀⠠⠀⡞⠠⠀⠈⠀⠀⢉⡉⣠⣅⢠⡄⠀⠀⡔⢎⠀⢸⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠅⣾⠀⠀⠀⠀⠀⠀⠍⢠⡌⢡⡄⠀⠀⠀⠻⣬⠑⡯⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠅⠀⢠⠀⠀⠡⡈⠀⠘⠀⠄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢁⠉⠀⠀⠀⠈⠀⠀⠀⠒⠊⠀⠀⠀⠀⢀⠐⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠚⡆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⢀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠐⠀⠀⠀⠁⢸⣏⠀⠀⠀⡀⢀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠆⠐⠤⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⢰⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠀⠐⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠀⢀⡁⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠈⠃⠀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠠⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠱⣀⢀⣀⣤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⠔⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⠂⠉⣩⠉⠑⠠⡀⠀⠀⠀⠰⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⢄⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⣐⣀⠔⠁⠀⠀⠠⢄⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠔⠉⡑⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣃⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⠑⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠃⠂⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠊⡀⠀⣀⠈⠳⣄⠀⠰⠌⡤⠂⡐⡄⠀⠀⠛⠀⢠⡷⢾⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠉⠁⠢⠀⡇⠈⠋⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠇⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠁
⠀⠀⠀⢠⡀⠢⡈⠀⠀⠰⠂⠀⠀⠀⠀⠐⢁⡄⠀⠑⡈⠀⡠⠁⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠈⠒⠴⣦⣿⣔⡾⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡷⠈⠀⡌⡆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⢀⠄⠀⠴⡮⢀⡀⠀⠀⠀⠀⠈⠁
⠠⢴⣆⠀⠀⠀⠐⠀⠀⠀⠀⢰⡄⠉⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠠⠂⢀⡀⠀⠀⠀⡄⠀⠀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⣸⠀⠀⠀⠀⠸⠁⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⡎
⠀⠀⠠⠀⠁⠀⠉⠋⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠐⠀⠁
⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠀⠀⠀⢀⠠⡀⠀⠀⠐⠀⠂
⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣥⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣤⠖⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⣤
⠀⠀⠀⠀⢠⣿⡇⠐⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠃⠀⠀⠀⠀⢀⢠⡀⡄⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠙⠶⠞⠃
⠀⠀⠀⠀⣾⣿⠓⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣷
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢀⠀⠀⠀⠀⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣸⢀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⢀⣘⣛⣂⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢁⡿⠄⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣹⢸⡟⣇⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡍⠰⠀⠀⠀⠀⠀⠴⣠⠄⡧⣠⠀⠀⠀⠀⣰⢻⡇⣿⢸⡇⢹⡄⠀⠀⠀⠐⠀⠀⠀⠆⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣬⠭⠵⣫⡴⣿⡂⠀⠀⣠⠇⡾⢸⣿⢸⣿⠈⢷⡀⠀⠀⠐⠀⠁⠀⠖⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠥⠞⠑⠃⠀⠀⠉⠈⠁⠉⠉⠈⠉⠀⠈⠁
⠀⠀⠀⢰
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠸
⠀⠀⠀⢸
⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⠃⠐⢩⢔⠀⠀⠀⠀⠀⠚⠥⠀⠀⠀⠀⠀⠀⢠⢋⡂⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⡔⠃⢒⡀⠀⠀⠀⠀⡈⠉⠭⠂⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠐⠠⠠
⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠃⠀⡈⠀⠰⠣⠀⠀⡀⠭⠭⠭⠙⠀⠀⠀⠀⢸⠗⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠄⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠋⠉⠌⠄⡠⠀⠀⢑⣒⣒⢒⣒⣲⠄⠀⠀⠍⠀⠀⠀⠀⡄⠀⠀⡄⠀⡀
⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠌⠌⠀⠀⠀⠀⠀⠀⠸⠡⠤⠬⡭⠥⠤⠄⠀⠐⠦⠀⢀⠀⠀⠀⠀⠀⠀⠁⣀⡐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣴⠋⠀⠀⠀⣀⠀⠈⢲⡀⠈⣧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢸⠀⠀⡈⠉⠛⠻⡿⣶⣿⣌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣸⡄⠀⠳⣄⠀⣸⠇⢸⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣦⡼⣿⣯⣴⣟⣡⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡐⡆⢀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣉⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣛⣿⣿⣿⣾⣦⣶⣶⣾⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⠟⠋⠉⢿⣿⣿⣿⠃⠀⠀⠀⠀⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⣿⣿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠸⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣷⡆⠀⠈⣿⡿⠻⢿⣿⠟⠛⠛⠛⢻⣿⡟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠉⠛⠛⠉⢹⣿⡏⣠⣾⡿⠃⠀⠀⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⢻⣿⠀⠻⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣇⠀⠀⠀⠀⢸⣿⡿⠋⠁⠀⠀⠀⠀⣿⠀⠀⠘⣿⠀⠀⠀⠀⢀⡾⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⢿⣄⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣿⠀⠀⠀⢀⡾⠁⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠘⠿⠆⠀⠀⠰⣿⡀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢠⣿⠀⠀⢠⣿⠃⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣨⣿⠋⣥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣿⣿⣿⣷⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡈⣻⣿⡟⠛⠫⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡾⢿⡏⢙⣧⣀⣴⠛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡀⢀⣿⠟⢿⡟⠛⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⡿⢿⣿⣾⣿⡿⠿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣏⠀⠀⣿⣄⣼⣇⠀⣀⡟⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⢹⣿⡿⣿⡿⠛⠻⣿⣿⠿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣦⣤⣼⣷⣴⣿⣇⠀⣀⣿⣧⣤⣾⣭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⠛⣿⣿⠛⠛⢻⣿⠟⠿⣿⣧⣽⣿⡏⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣴⣿⣿⣄⣀⣸⣿⣤⣤⣿⡟⠉⠹⣿⡿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⠛⢿⣿⡿⢻⣿⣿⣿⣧⣤⣶⠿⠿⢿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣼⣷⣤⣼⣿⣿⣿⣿⣧⣀⣿⣿⣏⡀⠀⣼⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⠋⢻⣿⢉⣻⡏⠀⠈⢻⣿⣿⠋⢹⣿⣿⣾⣿⠟⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣾⣿⣿⣿⣿⣿⣤⣾⣿⣦⣶⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣍⣀⣸⣿⠁⠀⢹⣿⠛⠛⣿⣿⣥⣿⡟⠁⠉⢿⣏⠁⠹⣟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⣿⣿⣿⣿⣦⣴⡿⢻⣶⣾⣿⠋⢹⣿⣧⣄⣠⣿⣿⣿⣿⣿⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⣛⣿⣿⠿⢿⣶⣾⠿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠉⠁⠀⠈⠉⠉⠉⠉⠁⠉⢫⠝⠛⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠀⠀⠀⠄⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⠀⠤⣄⠠⠀⠀⠉⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁

⣿⢿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡃
⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⠀⣇⠄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⡀⠀⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣒⠄⢠⡀⠄⠀⠀⢀⠙⠀⡋⡀⠀⠀⠀⢀⡀⠠⣚⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠅⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣉⠻⢀⠂⡀⠀⠀⡇⠀⠐⠀⢚⣩⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⡀⠀⠉⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠉⠠⠀⠀⠀⣠⠒⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢁⠀⡀⠀⠀⡀⠇
⠀⣀⠀⠀⠀⠉⠁⠠⠔⠁⠀⠀⠄⠀⠀⠈⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠖⣀⠠⠄⠀⢠⠀⠁⠓⠀⠄⡐⠶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠃⡀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⡄⠱⠞⠋⠀⠁⠀⠀⠀⠅⠖⠉⠀⠹⠒⠦⣀⡆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⢾⠀⡷⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⢰⠭⠶⠀⠠⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠂⢈⡄⠈⠁⠀⠈⠉⢠⡔⠀⠂⠀⢀⡀
⠀⠀⠀⠬⠳⣤⠀⠀⠀⠀⠠⡄⣧⠀⠀⠀⠀⢠⣀⡨⠆⠉⠀⠀⠀⠀⠀⠀⠆⠀⠶⢭⡚⠒⠃⠀⠀⠀⢼⠀⠀⠀⠈⠀⡈⠰⢗⡊⠵⠂⠠
⠀⠀⠀⠀⠀⠁⠀⢃⡶⢄⡀⠄⢃⡀⠰⣂⠐⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡭⠀⢄⡀⠀⠐⢿⠀⣰⡶⠞⠉⠀⠀⢤⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡌⠁⠈⢀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⠄⠀⠙⢷⠤⠌⠁⢫⣀⣄⣀⡀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠄⡰⢖⠁⠐⠄⠰⠀⢀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣠⡞⠘⣧⠀⠉⠳⠄⠀⢠
⠀⠀⠀⣀⡠⢷⠈⠁⠈⠀⢠⣦⢠⠀⠈⠉⠉⢠⣦⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢆⡀⠠⠒⠋⠉⠀⠀⡟⠀⠀⠀⠀⠀⡳⢆⡀⠀⢀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⠒⠄⠀⠀⠀⠀⠀⠀⠀⠈⠡⠞⠉⠈⠃⢠⡆⠀⠀⠀⠇⠀⢠⡄⠜⠈⠛⠰⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁⠀⠚⣠⢰⡖⠒⢀⠂⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠁⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢦⡞⣏⠰⣶⣶⣮⣴⢞⣤⣤⠰⣦⣀⠐⠯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣿⣾⣿⠖⢤⢀⣀⡨⣀⣀⣦⣗⢰
⢐⡻⡄⠋⢙⠿⣟⣡⠈⢨⢸⡗⣫⠩⡟⠊⡩⡼⢀⣀⣀⢀⣄⠀⣀⣠⣠⠟⣤⢝⡎⢼⠨⣏⣮⠹⣿⠫⡐⢓⢻⠤⢳⠾⣇
⠈⠇⢑⠦⡋⠏⢹⡼⢎⣽⢨⠶⣨⡷⠚⣶⢑⣯⣘⣿⠹⣏⠋⣿⣿⢿⣟⣿⢍⣜⣄⡼⡔⠿⢆⠓⢿⡧⠙⢠⠙⡏⣸⢳⡟
⠐⠀⠾⡖⣩⠓⢞⠋⠂⠶⢊⢌⡽⠋⢔⣠⠋⡐⣬⡱⢏⢀⣀⠈⠗⠞⠣⠊⢄⣿⡸⢆⡲⢌⣆⠛⠶⢦⢘⠀⢅⠱⢗⠊⢍
⠀⠀⣰⠀⡌⡙⠂⣀⠀⠀⠀⠀⠀⠋⠀⠓⠀⠙⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠀⣆⠁⢲⠌⢺⡷⠄⠈⠀⠁⠙⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠁⠈⠀⠈

⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠿⠿⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣯⣽⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠉


⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⣿⣷⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣿⣿⣿⣶⡷⢿⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⡾⠿⠿⠶⡿⠷⣶⡿⠷⢾⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠄⠀⠀⠀⢀⣤⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡀⠀⣻⣤⡲⢀⣤⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠀⠀⠠⡼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣿⡀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣌⡆⠀⠀⢠⠟⠞⢃⡾⣹⢧⡹⡏⡜⠛⠁⠀⠀⠀⠀⣀⡤⠒⠋⠘⠁⠠⣶⡀⠀⢸⡞⢻⣀⡀⠀⠀⢠⣷⡄⠀⠀⠀⡩⡀⣥⣾⡀⣠⣿⣄⣾
⢀⣄⢠⣿⠀⣰⡰⡚⠛⣸⣦⣴⣃⣶⣤⣴⣏⠀⠙⠿⠀⠘⠁⢀⡠⢔⣾⡏⠀⠀⠀⠀⠀⢴⣿⣆⠀⠼⣇⣾⡍⠁⠀⠀⣦⢷⣰⢀⠀⢰⡟⢻⣠⡏⢳⣿⣸⣇⢿
⢸⣿⣫⣭⣷⣿⢉⣿⢉⣻⡿⢩⠇⢸⣧⠁⣾⡆⣸⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⣎⣻⢸⡤⣀⣽⣏⠹⠶⠀⣠⣤⣿⡉⢼⠁⠼⣷⣚⣡⣇⣸⣧⣹⠙⡛
⢸⣿⣽⡉⣿⠟⠁⣈⢈⡉⢹⣿⣠⣾⣧⡖⢁⣿⣏⠈⡝⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣷⢿⣌⣥⠈⠾⡟⠳⠇⠀⢈⣡⣯⡛⠆⣉⠰⣾⡿⢋⣍⠁⢹⡻⡦⣿
⢸⣯⣴⣿⡝⠛⠋⠈⣿⡛⠛⣿⣿⣿⡿⣥⣿⢿⣿⣆⣻⣿⣾⣧⣦⣷⣶⣴⣄⣤⣄⡤⠚⣇⣼⣮⣙⣥⡀⡇⠀⣀⣴⡌⢀⣧⢥⠔⢣⡴⢤⣿⣯⣿⣷⣼⣿⣁⣿
⢸⣿⣿⣿⠧⠾⣀⣼⠞⢷⣶⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣟⣯⣛⣧⣿⣿⠖⣉⣿⠈⣧⡙⢯⢻⣿⣟⡣⠾⣙⣿⠅⠞⡔⠢⠗⢤⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⡄⣠⠏⠈⣄⠀⢳⣀⣻⡇⠿⠛⢿⠈⠟⠻⡿⣿⠟⠙⠛⠙⠿⠿⣿⣿⡿⢛⣡⣻⠆⠁⣹⡚⠻⡿⢓⠦⠓⠆⣹⢇⡠⠃⡤⠧⡘⠾⢿⡿⡿⡻⠻⠻⣿
⠈⠋⠻⢯⣀⡴⠦⠤⠊⠓⠛⢿⣄⣀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⣡⠀⣥⠀⢈⣍⠠⡬⠃⠈⠋⠸⡀⠤⣹⠇⡴⢳⠀⡉⠙⡈⢃⠁⠀⠀⠀⠁
⠀⠀⠦⠴⠁⠀⠦⠀⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠐⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⢒⠺⠓⠢⠒⠘⠏⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⣀⡀⠀⠀⡰⣴⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⡤⠤⠀⠤⡴⠀⠀⢠⠂⠁⠀⢀⠔⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡛⠚⠒⠐⠢⠐⠚⠁⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠠⠇⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠀⠀⠷⠀⠀⠀⢀⣠⠖⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠂⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢳⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠠⡴⠿⠁⠙⠉⠁⠀⠀⠀⠩⠆⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠙⠉⠃
⠀⠀⠀⠀⠀⠀⢐⠀⢓⠀⠀⠀⠀⠀⠘⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠜⡁⡐⠲⠂⠀⠀⠀⠈⠇⢀⣄⠀⠀⠀⠀⠀⠀⢰⡶⣶
⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⠀⠀⠀⠀⡀⣈⠙⠀⠀⢐⣖⣶⡞⠆⠀⠀⠀⠀⢤⠚⣁⡚⣠⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠌⣄⡀⠀⠈⠉⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠯⣣⠅⠀⠀⠀⠀⠀⠀⠀⣩⠃
⠀⠀⠀⠀⠀⠈⠛⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠄⢺⠠⠄⠀⠀⠀⠁⢀⣦⢤⣤⡤⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣠⠰⠿⠒⠛⠋⡉⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢤⡘⡛⡀⠜⠁
⠀⠀⠀⠀⠤⠴⠦⣐⡀⠁⣠⠦⠤⢀⣀⡸⠂⠘⠀⠙⠤⠖⠀⠀⠀⠛⠋⠙⠁⠀⠀⠀⢀⡀⠀⣴⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢐⡀⠀⠀⠆⣄⠈⠀⣠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⡀⠀⠀⠀⠀⠙⣃⠅⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠑⡐⠋⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠄⠀⠀⠀⠀⣤⢘⠃⠀⡀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⠀⠀⠀⠀⠠⣶⢈⠀⣠⢫⠄⠀⠀⠀⠀⠒⠤⢄⣀⠠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⡌⠰⠀⠀⠀⠀⠀⠲⢿⡆⣀⡌⠀⠀⠀⠀⠀⠀⠐⠘⠛⡀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⣁⢴⡂⠀⠀⠀⠀⠀⢭⣿⠋⠁⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⡀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠄⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣘⠉⠀⣚⠀⠛⣄⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⣋⣴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣾⣿⣿⣿⡿⠋⣁⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢟⣩⣶⣿⠟⣉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣫⣾⣿⠞⣫⠿⣫⣴⡿⢛⣡⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⣉⣴⣿⣿⡿⢟⣭⣶⡿⠋⣀⣴⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⢉⣴⣾⣿⣿⣿⡿⢞⡿⠟⣩⣴⣿⠿⠋⣀⣴⣶⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠾⢛⣽⣾⠿⢛⣥⢞⣫⣶⣿⠿⢋⣡⣶⡿⠿⠿⠿⡿⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣤⣾⣿⡿⣿⣿⣿⣵⣾⠿⣋⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⣋⣵⣾⠟⣫⣿⣿⣵⡾⠛⠉⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⢉⣴⡾⠟⣩⣴⣿⣿⠿⠛⠁⣀⡴⠞⢋⣡⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⢁⣤⡾⠛⣉⣴⣾⣿⠿⠋⢁⣠⡶⠟⣉⣤⣾⡿⠟⣁⣤⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⢁⣴⣾⣿⠿⠋⣁⣴⠾⢛⣡⡴⢛⣽⢞⣫⣶⣿⣿⣿⣿⢛⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⣤⣾⣿⠿⢋⣡⣶⠟⣋⣥⠾⣋⣵⣾⣿⣿⣿⣿⣿⣿⣿⡟⠋⡛⠛⠃⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣋⣥⡾⠟⣫⡴⠞⣉⣴⣾⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣵⡾⢛⣭⠶⠛⣡⣴⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⣟⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣛⣵⠞⢋⣠⣶⣿⣿⣿⣿⣿⢟⣫⣵⣾⣿⣿⣿⣿⣿⢟⣫⣵⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⢉⣠⣶⣿⣿⣿⣿⣿⢟⣭⣶⣿⣿⣿⣿⣿⡿⢛⣩⣶⣿⣿⣿⡿⣿⣿⣿⣿⠿⣿⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⠟⣋⣵⣾⣿⣿⣿⠿⣛⣭⣾⣿⠿⢛⣥⢞⣭⠶⣿⣅
⠀⠀⠀⠀⠀⠀⠸⠟⠉⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣾⣿⡿⢛⣩⣴⣿⣿⣿⣿⠟⣫⣵⣾⣿⠟⢋⣡⣾⣿⣿⣿⣿⣿⡿⠛⠿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⣯⣷⣿⣿⠿⣋⣥⣶⣿⣿⣿⡿⢟⣫⣶⣿⡿⠟⢋⣤⣾⣿⣿⣿⣿⣿⣿⠿⠋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣫⣴⣾⣿⣿⣿⡿⣛⣽⣾⣿⡿⠛⣉⣴⣾⣿⣿⣿⣿⣿⣿⡿⣛⣥⣾⣿⣿⣿⣯⡀
⠀⠀⠀⢀⣠⡶⣛⣿⣿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⣿⣿⣿⡿⣻⣵⣾⣿⠿⢋⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⢋⣤⠀⣠⡄
⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⡿⠟⢋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣾⣯⣵⡾⣋⣤⡶
⣸⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⠿⠛⣁⣤⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣵⣾⣿⣿⠟⣫⣿⣿⣿⣿⣻⡷⠻⣆⣠
⠉⠛⠉⠛⠁⠉⠁⠀⠀⠘⠋⢩⣽⣿⣿⣿⡿⠟⢋⣡⣴⣿⣿⣿⣿⣿⣿⠿⢛⣩⣾⣿⣿⣿⣿⣿⣿⣿⠻⣿⣛⣠⣾⠟⣉⣴⣾⣿⣿⣿⢛⡿⠛⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟⠉⠀⠘⢋⡽⢿⣿⡿⠿⠛⢋⣡⣶⣿⣿⣿⠟⠉⠉⠋⠙⠛⠉⠃⠺⠏⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⣿⣿⠿⣿⠟⠁⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣡⣞⣥⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢋⡵⢋⣤⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠞⠋⣼⣷⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣻⣿⣿⡿⣿⣿⢻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠄⠈⠀⢀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⡀⠀⠀⠐⡀⢀⣠⠀⠀⠀⠰⡷⡄⠀⡀⢦⡀⠆⠠⠆⢈⣧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⡄⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠁⠀⠀⠘⠃⢤⠍⢧⣿⣧⠉⢻⡄⠀⠶⣿⡿⠈⢣⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠒⠲⢛⡃⠒⠠⣄⠀⣺⡿⠂⠘⢧⣼⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠘⢯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣳⣄⠈⠙⣿⣧⠱⣄⠈⣿⣀⠀⢠
⠀⠀⠀⠀⢀⣤⣾⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⢈⠁⠀⣀⡀⠈⠂⠂⠖⠈⠉⠑⢻⣿⣄⣿⡀⢹⡶⢠⡯
⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣪⣌⣑⣈⣃⡀⠀⠀⠀⠀⢀⣀⣴⣿⣿⡙⢷⣈⣿⣼⣧⠖
⠀⠀⠀⢹⣿⡿⠟⠁⠐⠿⢷⣮⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠀⢀⡽⠛⣶⡶⠾⣿⣿⣄⣿⣿⠁⢀⣿⣿⣿⠗⠂
⠀⠀⠀⢸⡟⠀⢀⠀⠀⠀⢠⣿⣷⡀⢸⣽⡿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡟⡄⡩⠴⡞⠋⡀⠀⡀⠀⠈⢹⣿⣧⠈⣿⣿⡟
⠀⠀⠀⠿⠶⠒⣿⣧⣾⣿⣿⣿⠿⠇⠜⠻⡄⠀⠿⣿⠟⢿⡿⠿⣿⠿⢛⠛⠻⠋⠊⠟⠀⠂⠒⢿⡐⠛⣺⡟⢿⢀⡏⠑
⠀⠀⠀⠀⠀⠀⢀⣿⣶⣦⣬⢀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠛⠚⠛⠛⠛⠉⠋⠉⠁⠀⠀⠀⢀⣸⣿⣿⣿⡧⠀⢻⠶
⠀⠀⠀⠀⠀⣀⣄⣤⠛⢉⣈⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣠⣄⣀⣶⣿⣿⣿⠿⠟⣿⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠛⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠈⠉⣉⢉⢛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⡀⣿⣿⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⡿⣿⣿⣿⣟⡳⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⣯⣿⣿⣿⣿⣋⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣶⣶⣿⣿⣿⣿⣿⠿⠛⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠛⠻⣿⡿⠻⢿⣿⣅⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡿⠿⡀⣴⣿⣿⣿⣿⡟⢿⣿⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣆⣡⣾⣿⣿⣿⣿⣿⣿⡟⠦⠙⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣿⣿⣿⣿⣿⣿⣧⣽⣿⡛⢿⡿⣷⣶⣶⣦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢩⣍⡉⠉⢠⣷⣬⣾⣿⣾⣿⣤⣦⣿⣿⣿⣯⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣽⢿⢷⣈⣤⣽⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣾⣿⣿⢿⠿⡟⢷⣾⢿⣿⣿⣿⢿⣿⣷⣛⠿⢿⣾⣅⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⢟⡀⢀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣜⠛⠉⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡿⡏⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣝⣿⣿⣶⣿⣯⣽⣿⣿⣿⣿⠟⢿⣿⣏⣽⣿⣬⣛⣻⣿⡿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣟⣻⣿⠏⠛⠻⠇⠙⠃
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⠿⠻⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⡟⠫⠈⠁⣿⣿⣿⣿⣶⣿⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⣿⣷⣤⣼⣾⣿⣿⡿⠋⣻⣿⣿⣿⣿⣇⣰⠀⢀⣿⣿⣿⣽⠙⠿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⣿⡻⣿⣿⠿⢿⣿⠏⠹⣿⣿⣿⣏⠛⠻⣿⡷⠉⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠙⠋⠀⠀⠉⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⡀⠀⡀⠀⢀⠀⠀⠀⢠⠀⡄⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠁⠀⠀⠀⠁⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣀⡀⢠⠄⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠈⡀⠀⠀⢀⡀⢹⢦⣘⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠄⠄⠀⠀⡁⠤⠘⣁⠠⠐⠂⢢⣁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠀⠰⠤⠐⠞⢀⣀⡀⠀⠀⠠⠉⠱⠒
⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠶⠵⢆⡠⠀⠠⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⡠⠐⠀⠀⠀⠀⠒⢄⠀⠀⠀⠐⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠⠴⠀⠔⢄⡀⠀⡠⢁⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡀⠘⠀⢰⡆⠀⠰⡦⢠⠁⢘⠅⠕⣨⣀⠀⢀⡄⠀⡇
⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⡀⠀⢨⠀⠂⠀⣤⠀⠀⣀⠁⠁⠀⢀⡔⠁⠈⠑⢀
⠀⠀⠀⣠⠀⠀⠪⡜⠒⠒⡄⠀⠀⠀⠀⠈⠆⠀⠀⠀⠤⠈⠃⠀⡄⡠⢚⠀⠀⠀⡀⣶⢓
⠀⠀⠀⠁⠀⠀⠁⠉⣉⣩⢇⡀⠀⠀⠀⠀⠀⡀⠆⠤⠌⢠⠄⠊⢰⠀⢐⠦⢈⠀⠈⠃
⠀⠀⠀⠀⠀⠄⡀⠤⢤⣤⠺⠂⠀⠀⠀⠀⡀⡘⠉⠁⠐⠉⢄⣧⢀⠓⠂⠉⠁⠒⢘⠀⠀⡄⡀⡌
⠀⠀⠠⡏⠈⠁⢀⠀⠮⠘⠈⢀⠀⠀⠀⠀⠳⡃⠠⠬⠉⠐⠀⢙⠃⠀⠨⠀⣍⡍⢩⠑⠉⠉⠁
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⣴⠊⠁⠀⠀⠀⠀⠑⡒⢒⠒⠊⠀⢠⠀⠀⠀⠐⠈⣣⣼
⠀⠀⠀⠀⠀⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⣃⢇⠀⠀⠢⢀⠀⠀⠀⠉⡩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡢⢤⡤⣒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⢀⠀⠀⡠⣀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠨⣘⡯⠀⠀⠀⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⢇⡀⠁⣀⠔⠛⠭⠀⠀⠀⠀⠀⠀⠀⡲⢤⣄⡄⠀⡀⠇⢠⠀⣸⣴⠾⡓⢄⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠓⣄⣾⠕⣫⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠊⠔⠋⢑⣚⣿⣷⣊⣀⠀⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠐⢻⠟⠉⠉⠩⣭⢛⠟⣍⠓⠬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠔⢹⠙⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠁⡜⠀⠘⠁⠀⠀⠉⠲⣶⡄⠀⠀⠀⠀⠀⠀⠤⣐⣒⠾⠻⠀⠈⠀⢋⢷⣒⡒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠄⠀⠀⠀⠀⠈⡌⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣙⣇⠄⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⡀⠡⠀⣤⠀⣂⠗⠠⠀⠀⠀⠀⠀⠀⢴⡰⡀⠠⡀⡀⠀⣇⢘⠀⠀⢠⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⠽⠓⠤⠬⣙⣬⣠⡧⠊⣩⡅⠀⠠⠤⠀⠀⠀⠀⠩⠭⠛⠢⢔⡳⣄⡿⢁⠤⠒⠛⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠓⣂⢟⠿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⡻⡿⣓⠒⠣⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⠠⠎⠰⠈⠖⠈⢹⠳⡶⠀⠀⠀⠀⠀⠀⠀⠉⠀⢁⡔⢡⢇⠈⠑⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠙⠀⠀⡀⠀⠀⣃⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠿⠀⠀⠄⠀⠀⠸⢱⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠐⡹⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣾⡀⣀⡖⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢿⡏⠓⢿⣀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡁⣾⠛⠻⠿⣟⢆⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣁⡴⠶⠦⢭⣿⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠈⢦⣿⡿⡄⠉⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡆⠀⠀⢯⣷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡇⠀⠀⢰⡏⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⠀⠀⠀⠀⠀⠀⠀⢈⣾⣧⡤⣠⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠻⢿⣦⣀⡀⠀⠀⢠⣺⡾⢋⣩⡾⠟⠋⠉⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣵⣶⣌⣉⣙⣓⣪⡴⣫⣶⠟⠋⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣀⣤⣤⣤⣴⣶⣶⠶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡐⠐⠟⢨⣿⣿⣿⣧⣿⢗⣤⣀⢀⢀⣄⣠⣤⣶⣿⣿⣿⡿⠿⢛⣻⣿⣿⣿⣿⠏⠁⠂⠐⠲⠝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠛⣟⢛⣏⣸⣿⣈⠉⠛⠛⠿⠻⢿⡟⣿⠟⠙⠉⠘⡃⠸⣿⣿⣿⣿⣿⣷⡄⠀⠀⢰⣌⢸⡿⢂⣀
⠀⠀⠀⢠⡄⡀⢀⠀⠀⠀⠀⠀⠐⠒⠒⠒⡻⡻⠻⠟⢻⣿⣿⣿⣿⣶⣤⣤⠇⠀⠀⠀⣧⢀⠀⠀⣥⣾⣿⣿⣿⣿⣿⣿⣷⣔⢧⣾⣿⣿⡷⠿⠿⠿
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣀⡀⣠⣴⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣿⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⠿⠛⠉⠋⣧⡤⠆⠀⠰⣦⠀⠋⣹⡿⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠁⠀⠀⠀⠀⢶⣤⠿⠟⠛⠛⠋⢻⣿⣿⡿⣿⣿⠟⠛⠛⠛⠛⣿⣿⡆⠀⢻⣿⣇
⠀⠀⠀⠀⠀⢠⠀⠀⠴⠦⠤⣤⣤⣤⣤⣄⠀⠈⠂⣠⠸⡧⠤⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡏⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⠀⠉⠀⢹⠿⣦⣤⡶⣾⡑⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⡇⠀⢀⠀⠄⠀⠀⢸⣿⠃⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠰⣄⣷⡄⠀⢀⣀⠀⠀⢀⡏⠀⣀⣼⠑⠿⣇⠀⠀⠀⠀⠀⢃⡠⠠⣤⣬⣳⠟⢿⣿⡇⢸⡃⣼⣴⣷⣶⠧⠂⢸⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠠⢸⠟⠉⠙⠛⠺⢲⣿⣿⡿⣿⡿⠛⠳⣦⣨⠀⠀⠀⠛⠞⣷⣥⣾⣿⣿⣿⣤⣸⣿⢠⣼⠛⠉⠀⠀⠀⠀⠀⢻⠁⠀⠀⢀⣟⠠⠄⠀⢠⠄
⠀⠀⠀⠀⢠⠆⡇⠀⠀⠀⠀⠀⠌⠀⣿⡇⠈⠁⠀⠈⠀⠀⠄⣐⣥⣟⣆⣻⣿⠿⠛⠋⠉⠉⠈⣉⠀⢸⠀⠀⠀⢀⣀⣀⠀⣼⣤⣶⣶⠜⣿⢶⣶⣶⡆
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠘⠇⠀⠸⡇⢲⠐⣄⡤⣀⣀⣤⠘⠫⠙⠉⠁⠀⠀⠀⠀⠀⠀⣀⢻⠀⠀⣄⣀⣤⡾⣿⣥⣍⣵⣿⣿⣯⣍⣹⡿⠟⠋
⠀⠀⠀⢀⡃⡀⠀⣀⣤⣤⣄⠀⠰⢤⣤⠳⣼⣿⠿⠷⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠶⠿⠛⠓⠲⢯⣏⣉⣁⣀⣩⠟⠂⠉⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠉⠁⠈⠈⠀⠐⠂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⡔⠀⠀⢀⠀⠰⠀⠀⠐⠻⠿⠟⠋⠋⠉⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠱⣿⣄⣤⣀⡀⡀⠀⠖⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠚⠛⠛⠻⠿⣿⣿⣦⣦⣤⣠⣶⣶⣄⣀⣠⣄⣀⡛⢮⣀⡀⣟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⡀⠀⠀⠀⠉⠉⠛⠻⣿⣿⡿⠯⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠁⠈⠋⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠐⠂⠙⠂
⠀⠀⠀⠀⠀⠀⠒⠒⠻⠼⡁
⠀⠀⠀⠀⠀⠘⡂⢀⡄⠈⠉⠁
⠀⠀⠀⠀⠀⠘⠛⠻⠤⠄⢶⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠠⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⡨⠆
⠀⠀⠀⡀⠀⣠⠂⠀⠶⢷⡝⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢀⣀⠴⠶⢤
⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣵⣄⣤⣶
⠀⠀⠀⠶⠠⣰⠆⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡁⠀⢄
⠀⠀⠀⠀⠔⠁⠀⠒⣀⠀⠶⣀⡀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠁⢀
⠀⠐⠧⠄⡤⠀⠀⢀⠙⣀⡀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣘⢀⠤⠚⠐⠾
⠀⠀⠀⠀⠸⠇⠂⠉⡀⢀⡉⠹⠄⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠐⠪⣭⣋⠠⣤⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣤⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣶⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢄⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠀⠀⠈⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⡀⠀⢁⠆⠀⠀⠁⠆⠀⠀⠀⠁⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠤⠀⠀⠀⠀⠀⠀⠀⠀⢑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠑⡠
⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⠀⠀⠀⡆⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠌⠢⡀⠀⠈⠃⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⡏⠀⠀⡠⠂⠀⠀⠀⠈⢆⠀⢀⡀⡨⠓⠀⠀⢀⣀⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡆⠀⡀⠀⠀⢠⣮⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⣰⠀⠀⠀⠀⠈⠸⡁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠈⠉⠀⠸⠀⠠⠀⠀⡀⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠀⢀⡀⠀⠍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠀⠀⠠⠀⣀⠀⠀⠰⠀⡠⠀⣀⡠⠜⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠑⢄⠀⢱⡁⠉⠀⢀⣀⡀⠄⠁⠀⣠⠂⠀⠀⡀⢠⠀⠀⡈⢁⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠠⠆⠀⠀⠀⡀⠀⢀⠀⢅⠊⠓⠤⣄⣀⣀⡠⠤⢚⠀⠠⠒⠉⠁⣈⢀⠀⠘⢇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠁⠈⠐⠐⠀⠉⢄⡰⠁⠀⠀⠑⠂⠤⠀⠤⠤⠵⠒⣉⠒⠤⠤⠤⠚⢉⠀⠄⠀⠀⠉⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢁⠠⠄⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡣⣀⠐⠁⠎⠐⢀⠂⠈⠀⠀⢠⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠊⡑⠒⠒⠒⠚⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠎⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⠧⢄⡠⠤⠤⢀⣰⡦⢄⣰⡖⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠔⠁⠀⠀⠠⠤⠤⠀⠢⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⣱⠎⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢤⣿⣷⡶⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠞⠛⢁⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣶⣾⢹⡅⣠⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⡿⣧⣾⣷⣤⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⡋⣀⡿⠿⠛⢠⣼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢉⣁⣅⣠⠄⢾⠿⠾⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠏⠈⢻⣄⣘⣷⠈⠑⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠠⣤⠟⠁⡈⣠⡴⠋⠉⡾⠆⡷⠂⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢂⣾⣿⠋⡪⣴⣀⢱⡤⠶⢂⣤⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣄⠴⠲⠾⠿⢃⢠⣄⣼⠛⠋⡀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⢠⡶⢠⣀⣈⣉⠀⠴⠶⡅⠀⣀⡤⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠘⡻⠊⣸⡿⠃⠀⠘⠋⠀⠀⡈⠁⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀⣤⡂⠀⠈⠀⢸⡿⢁⠠⣀⠖⠀⠀⠀⠈⠓⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠐⣿⣾⠋⠀⠉⠀⢠⠀⠀⠈⠰⠄⣠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣄⠀⢀⠀⠄⢻⣿⣶⡀⠀⡰⢿⠀⢀⢀⠄⠂⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠀⠈⠁⠀⠀⡰⠃⠀⠀⢀⣧⡀⠀⠀⠦⠈⡀⠰⠊⠀⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠉⠒⠥⠰⠚⠃⠐⠒⠀⠚⠛⠟⢉⣠⠤⠊⡑⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠂⠐⠂⠀⠂⠠⣤⣦⠔⠀⠀⠹⠷⡰⠚⠓⠋⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⣤⣄⣀⠆⠚⢸⣿⠏⠀⠀⠀⣶⡆⠀⢈⡀⠀⠀⠒⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠄⣨⡼⠏⠀⠀⡸⢡⠀⣀⣠⠁⠀⠀⠀⠀⠘⢆⠀⢀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠁⣿⣿⣿⡇⠀⢠⡈⠀⢀⠀⠈⠡⠀⠈⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⠝⣿⠁⠐⠈⠁⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡄⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡙⢴⡧⢊⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⣸⣇⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣤⢸⡇⣤⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠙⡇⣬⡀⠆⠸⠀⣷⠀⠀⠓⢦⢣⣄⠰⣿⣺⣷⣿⠆⣠⡜⡴⠞⠀⠀⣾⠀⠗⠰⢀⣭⠸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠝⠿⣶⣦⡁⣤⣤⣄⢸⡯⣟⣷⡖⣡⣌⢲⣾⣻⢽⡇⢀⣤⣄⢈⣤⣴⠿⠫⠅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢁⡤⠉⣛⣿⣾⠯⠼⠻⡟⠉⡅⢿⣿⢸⠉⢻⠿⠧⢽⢷⣷⣛⠉⢤⡀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠙⠾⠛⡆⢿⣷⣮⣓⣼⡼⢧⣷⣒⣵⣾⡷⢠⠻⠷⠋⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⠴⠾⣟⣿⠂⠪⢭⣿⠉⢄⡠⠉⣿⡭⠕⠐⣿⣻⠷⠦⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠙⠻⢿⡿⢂⣚⣭⣿⣄⡈⢉⣠⣿⣭⣓⡐⢿⡿⠟⠋⠙⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⣴⣿⣶⠃⠿⠿⢋⠔⡻⢻⡟⡟⠢⡙⠿⠿⠘⣴⡿⣦⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠄⣉⣤⣵⢿⣿⣯⢱⡾⣧⣠⠇⣿⣿⢸⣀⣼⢷⡎⣽⣾⡿⣮⣤⡉⠡⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣈⣱⣾⠛⠋⣄⠙⠋⠁⢼⢟⣯⡿⠳⣌⣡⠞⢿⣽⡻⡇⠈⠙⠋⣠⠛⠛⣶⣆⡁⢄
⠀⠀⠀⠀⠀⠀⢀⡴⠦⠩⠁⠂⠼⠀⠟⠀⠀⠶⠋⡎⠁⠰⣿⢿⡿⣿⠆⠈⢱⠗⠦⠀⠀⠿⠀⡦⠐⠨⢍⠰⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣈⡉⣹⡏⢉⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⡁⢹⠏⢈⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⢹⡏⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢙⡣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠰⢁⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡀⢰⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠂⣀
⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠁⠉⠉
⠁⠒⠂⠁⠀⠃⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⣂⠥⣌⠂⠀⠤
⠀⠀⠀⠐⠀⠀⠊⠐⠄⣁⢀⠀⠀⠀⠒⠁⠀⠀⠀⢀⠔⠂⢀⠀⠀⣀⠤⠄⣀⡀⠤⠔⠊⠀⠀⠈⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢢⠆⠀⠀⠈⠈⠣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢇⠀⠀⡤⠀⡀⠀⠀⣆⠀⠉⠉⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⡇⠁⠘⠳⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⣀⠀⠀⣄⠀⠸⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡶⠁⠀⠀⡌⠀⠀⠀⠀⠀⠘⡳⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠁⠸⠞⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⢠⡇⠀⠀⠀⠀⠀⣀⢀⡁⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡄⠣⡀⠀⢠⠔⠋⠉⠀⠉⠑⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢃⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⡀⠀⢀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠢⣀⠀⠀⢠⠁⠀⣸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠈⢳⠒⢁⠀⠀⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⡄⠀⠀⢠⠀⠇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢁⠀⠀⢸⠀⢸⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⢸⡄⠀⠈⠀⢸⡄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡇⠀⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡧⠀⠀⠆⣀⠆⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠃⠀⠁⠘⣅⣰⠀⠈⠁⢸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡎⢇⠀⠀⢒⠄
⠀⠀⠀⡠⠔⠒⠒⢢⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠡⡀⠀⠀⢀⠜⠀⠈⢆⠀⠀⠀⣀⠤⣀
⠀⠀⠀⡱⠀⠀⠀⠘⢲⠶⡏⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠂⠀⠀⢊⠀⠐⢳⠂⠁⠐⠲⡛⠀⠰⠁⠀⠈
⠀⠀⠈⢳⠀⠀⠀⠀⠀⡆⢣⣴⠃⠀⡄⠀⡠⠀⠀⠀⠀⠀⠀⠀⢐⠇⠀⠀⠀⠐⣷⣖⡁⠀⠀⠀⠀⣑⣤⠀⠁⠒⠈
⠀⢠⠤⠶⠀⠈⠁⠈⠄⠉⠁⠈⡆⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⡡⠤⠀⠄⣀⣀⡤⠋⠀⠀⠀⠀⠀⠈⠢⢄⡀
⠀⢇⠀⢀⡁⠀⠀⠀⠐⠀⠀⢐⡍⡂⠀⠐⠀⣊⠀⠀⠀⠀⠀⢊⠤⠀⠀⠀⠀⢈⠽⠋⠙⠀⠂⠠⠐⠀⠛⢍
⠀⢰⠁⠀⠀⠐⠀⠁⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠒⠀⠀⠀⠒⠁⠀⠀⠠⠀⢌⢑⡶⠀⠀⠀⠀⠀⠀⠀⠴⣖⠉
⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠄⠀⠠⠤⠤⢤⠤⠀⠀⠤⠵⠂
⢹⠀⠀⠒⠒⠒⠁⠀⠈⠉⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠤⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠚⠀⠀⠀⡇⠀⡀⢀
⠀⠀⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠃⠀⠀⠀⢀⠀⠀⣀⣀⠤⠀⢀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠁
⠀⢧⣀⠄⠀⠀⢣⣀⠄⠀⠀⠀⠳⠤⠂⠘⠦⠄⠀⠘⠦⠔⠀⠈⠳⠤⠤⠂⠈⠳⠤⠤⠔⠊⠀⠳⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⢷⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣟⣘⡾⠾⣼⣼⣰⢢⢄⠀⠀⠀⠀⠀⠈⡈⠁⠀⠄⠀⠠⠀⠁⠀⠐⢀
⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⢀⠈⠻⡇⢠⣴⠀⠈⠉⠛⠺⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠶⠷⠶⢤⣄⣑⡆⠙⠂⠕⠚⣣⡰⠟⣠⠋⠀⠀⠀⠀⠰⠁⠀⠄⠀⠀⡀⡐⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠉⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠐⢠⡄⠀⠀⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠐⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⢸⠃⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠜⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠘⠆
⠀⠀⠀⠀⠀⢈⠀⠀⠀⡄⠓⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡈
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⡐⠒⠅⠀⠀⡒⠲⡁⠈⠀⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠴⠦⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠜⠒⠶⠶⠶⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠶⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⢫⣧⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⣠⡇⠐⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⣴⣾⠋⠀⠰⠷⠈⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣋⣭⣵⣾⣯⡐⠷⠀⠀⡀⠀⣿⣿⣿⣿⣦⡀⠀⠘⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠛⠉⣿⣦⣄⠈⠁⠀⠀⠉⠛⠿⣿⣿⣶⣶⢟⣿⠇⠀⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠅⠈⠀⠃⡸⠋⡣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣏⣀⣠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣫⣿⠃⢀⠀⠐⠜⠂⠆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣽⣿⡿⢛⡛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠠⠀⠤⠜⠀⢀⡠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢹⡿⣷⣌⢻⣿⣿⣿⠿⠿⢿⣦⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⡀⠀⠀⠐⠀⠀⠠⠄⠅⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣾⣿⡟⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠵⡀⢄⣴⣶⣿⣿⠥⢀⠀⠄⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⣿⡿⣏⣿⡇⠀⠀⣀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣴⢬⣿⣿⣝⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⢰⣿⣿⠀⠻⣾⣿⣦⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⠋⠈⢿⣝⡻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⣿⣟⢇⠀⠀⠉⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠉⠛⣿⣷⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠿⠟⢻⣮⡓⠦⡤⣶⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⣴⡿⠟⠿⣶⡀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣾⣿⠃⣴⢿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠹⠦⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⢠⣾⡿⠙⣷⠀⣀⣤⣤⡴⠶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⣴⣿⠟⠀⢀⣿⣿⣿⠟⠁⢀⣤⣾⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣝⡲⠶⢶⣶⣶⠿⠟⠋⠀⢀⣠⣾⡿⠋⠀⣠⣾⣿⠟⠁⣠⣶⣿⣿⡿⠛⠁⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠳⣝⡶⣤⣤⣤⣤⣤⣴⣶⡿⠟⠋⣠⣴⣿⠿⠋⢁⣴⣾⣿⣿⠟⠋⠀⢀⣴⠶⠶⢤⡈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣛⠶⣯⣭⣍⣉⣩⣥⣤⡶⠿⠛⠉⢁⣤⣾⣿⡿⠟⠉⠀⠀⠀⠀⣾⡇⢰⣆⣼⡇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣍⣉⣉⣁⣀⣤⣤⣶⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣉⣉⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡿⠏⠋⠻⣏⡀⠀⠀⠀⠀⠀⠀⢘⠿⢙⡙⢿⣁⠀⠀⠀⠀⠀⠀⠲⣾⣶⣿⣿⣮⠴⠀⠀⠀⠀⠀⠀⠑⠄⣒⠤⠁
⠀⠀⠀⠀⠀⠀⠙⣿⡄⡉⢠⡟⠉⠀⠀⠀⠀⠀⠈⢻⣧⢙⠃⣼⠋⠁⠀⠀⠀⠀⠀⠤⢽⣿⣿⣿⡿⡤⠀⠀⠀⠀⠀⠉⠀⡡⢛⠪⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠋⠝⠟⠈⠙⠀⠀⠀⠀⠀⠀⠀⠘⠩⠻⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠎⠋⠮⠑⠀⠀⠀⠀⠀⠀⠀⠚⠀⠰⠀⠈


⠀⠀⠀⠀⠀⠀⢀⡄⢐⢂⣔⡄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⡴⡴⡠⠀⠀⠀⠀⠀⠀⠀⣄⠦⣶⢖⡄
⠀⠀⠀⠀⠀⠀⣄⠐⣾⣿⣷⠢⠆⠀⠀⠀⠀⠀⢀⡀⠐⢬⡌⠄⠤⠀⠀⠀⠀⠀⠀⠈⣟⢲⣿⣷⢺⢒⠀⠀⠀⠀⠀⢘⡊⢡⣷⡎⠬⠅
⠀⠀⠀⠀⠀⠀⠀⣬⠹⠛⠏⡅⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠘⡹⣤⣉⣧⢟⠂⠀⠀⠀⠀⠀⠐⢢⢄⢸⢴⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠖⠞⢶⣀⠀⠀⠀⠀⠀⠀⠀⡰⢈⠬⢤⡔⠄⠀⠀⠀⠀⠀⠀⠀⠢⣴⠛⠶⠶⠂⠀⠀⠀⠀⠀⠐⢶⣶⣿⣶⣗⡄
⠀⠀⠀⠀⠀⠀⠂⣿⡸⠽⢀⡧⡂⠀⠀⠀⠀⠀⠐⠂⣗⡺⢇⡿⡘⠀⠀⠀⠀⠀⠀⠰⠈⣿⠶⡤⠤⠠⠀⠀⠀⠀⠀⠾⢿⠿⢿⠿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠎⠳⡓⠋⠒⠀⠀⠀⠀⠀⠀⠀⠠⠊⠍⡍⠈⠂⠀⠀⠀⠀⠀⠀⠀⠰⠋⢢⡖⠲⠀⠀⠀⠀⠀⠀⠀⠴⠰⢰⠦⠟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⣤⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⢪⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⠙

⠀⠀⣸
⡇⣠⠿⠀⡀⣤⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢹⣿⡉⠀⠺⣳⠀⢸⡟⠀⡄⣸⠀⠀⠀⢸⡄⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐
⣾⣥⣌⠉⣠⡼⠂⠸⢁⣘⢻⣯⢀⣼⣴⡯⠤⣾⣠⠀⠀⡀⠁⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡿⢿⡶⠂⠠⠤⠀⢀⣬⡍⠼⣥⠀⣿⣿⠛⠂⢹⣿⣄⣼⢁⡀⢀⠁⡀⠀⣰⠀⡀⠀⢠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⢀⡀⣀⠀⢠
⡷⠨⣙⣟⡳⠒⠀⢸⡟⣀⣘⠟⡯⣿⠛⡘⣋⡀⣽⣿⠟⣛⣦⣿⣼⡇⢰⡿⢻⣧⣶⡾⠇⣴⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣠⣶⣿⣡⡏⣾⣿⣵
⠀⠀⠀⠀⠈⠈⠀⠘⠛⣷⠍⠂⣼⡻⠂⠠⠤⠀⢐⣿⣧⠆⠰⣿⡗⠭⢿⡥⠄⢿⣿⡷⠼⣿⡿⢲⣾⣆⡀⢀⣄⣤⣿⣿⣿⣿⣿⣿⡿⠥⠽⠿⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⢯⡃⢦⣤⠄⠀⠉⢻⣇⠐⠈⢻⡿⢠⢽⣏⣀⣋⣿⡥⠀⢹⣇⣈⣿⡿⣿⣿⠿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠶⠾⠿⣯⣘⡛⣷⣀⠊⢿⡧⢴⢺⣄⣐⠈⣷⡆⢸⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠚⠛⠶⠿⠿⠿⠒⠛⠉⠉⠁





⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣿⣵⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⡼⠃⠤⣤⣙⠉⠀⡴⠆⠀⠀⠀⠀⢠⡀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡴⠛⣀⠀⠀⠨⣥⡀⣆⠛⠀⠀⠀⢶⣦⢸⣇⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠀⡷⢛⡁⠀⠀⠁⠲⣄⣧⢚⣥⠄⠀⢢⡝⠞⣥⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣝⣷⠋⢐⣷⠘⣶⣤⠘⡇⣚⠅⢀⠸⣦⠙⣏⢉⣥⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡈⠉⠙⠳⠾⣇⡠⣤⣹⣴⡟⣡⠀⣾⠀⡀⢠⠷⠿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣧⡘⠖⠈⢛⣶⣭⣽⣦⣁⠀⢿⡄⢹⡿⠖⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡄⢩⣟⠛⣩⢅⡟⠁⢶⣭⣝⣿⣷⣾⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣼⠿⠃⠾⠋⠸⠁⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣤⣶⠿⡿⢀⡿⢛⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣴⢾⢟⣿⡛⢿⣿⡿⣿⢻⣶⣶⢶⠿⣻⢿⠇⣾⡽⠿⣾⣷⠿⠶⢿⣿⡟
⠀⠀⣴⣾⣿⣿⣟⣓⣛⠻⢭⡿⢿⡦⣧⣿⠧⠾⢛⣡⠾⣶⠏⡼⢡⢖⣛⢲⢬⣙⣛⣛⣿⠃
⠀⠸⠋⠉⢸⡿⢉⡵⣛⡓⢦⡹⣯⣿⠾⢷⣔⡛⠛⠒⣋⣵⣚⣷⡐⢜⡽⢢⡇⡌⣭⣿⠃
⠀⠀⠀⠀⢸⡿⡏⠸⡩⠿⢢⠇⣿⣿⠻⣳⣴⣶⡦⣤⣤⣔⠿⣿⡷⣔⢶⡟⣇⣿⣼⡇
⠀⠀⠀⠀⠘⣇⡇⢸⣬⢍⣭⣾⣿⣿⣿⣿⣼⣿⣿⡌⣧⡙⣷⣬⣿⢾⣇⣦⣿⣿⣿⣅
⠀⠀⠀⠀⢰⣿⢃⣇⣹⣶⣿⣿⣿⠃⠉⠙⠻⠿⠿⠿⠿⠿⠿⠛⢿⣿⣼⣿⣿⢿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠈⠉⠛⠛⢿⣿⡇
⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⡿⠃⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⢸⡟⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⣿⠃⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣶⡿⠃
⢸⣿⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠉
⠸⣿⡆⠀⠀⠀⠀⠀⠹⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣠⢀⣸⣇⡀⣄⣤⠆⠀⠀⠠⣼⡆⢹⡟⢁⣿⡤⠀⠀⠰⣤⣆⢶⣼⣇⡶⢠⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⠋⠙⣿⡍⠀⠀⠀⠀⣛⡋⣙⣿⣿⣟⢉⣛⡀⠀⠀⠨⣭⣿⣿⣿⣿⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⠻⠷⡾⠟⡶⣤⡀⠀⠀⠸⢿⡏⢹⣇⠉⣿⠏⠀⠀⢀⣴⡶⣻⡿⡿⣟⢲⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠏⠓⠈⠀⠀⠀⠀⠀⠀⠉⢼⡷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⣀⠦⣽⠴⣀⣴⡶⠀⠀⠲⣤⣀⠀⣹⡁⠀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣝⣳⣾⣞⣛⠌⠀⠀⠀⠀⠀⠉⣛⣾⣶⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠛⡇⣿⢬⡛⣶⣦⠀⠀⣠⡶⠞⠉⢺⠒⠉⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⠀⠘⡏⠀⣠⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣶⣷⡚⠉⠁
⠀⠚⠒⠂⠒⠂⠒⠒⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠉⡇⠉⢻⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⣀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣀⣼⡖⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡿⣯⣏⣽⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⡘⢑⡙⠎⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡤⠠⡤⠄⠀⠀⢀⡀⠀⣀⣀⣀⣀⣠⢀⣀⣤⣭⡍⠛⠊⠻⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⢹⠄⠉⡀⢉⡖⠶⡈⠈⣃⢰⣿⠂⢹⠠⣿⣺⣇⣽⣿⣷⣤⣤⣴⣿⣿
⠀⠀⠀⠀⣾⢛⡁⠈⠀⠀⢂⠀⢆⠓⢠⡀⠠⠰⠀⢙⡯⢼⡛⠉⣗⡑⠘⠿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣯⠘⣷⠀⠄⢀⠀⠀⠈⠃⠐⠃⠘⠃⢀⠉⠀⡉⠉⠀⠀⠃⢷⡀⠈⢻⣟⣿⡟
⠀⠀⠀⢸⡏⢠⠈⢷⡄⠀⠀⠀⠂⣸⣟⠂⠰⣯⣤⣍⣻⣧⣾⣶⣸⣷⣴⣷⡔⣻⣿⡿
⠀⠀⠀⠈⠓⠉⠀⠁⠈⠆⠀⢠⡿⠛⣿⣿⣾⡛⣷⡞⠓⢽⣿⣯⣷⠛⢳⣏⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⡴⢃⠀⢠⢒⠎⠀⠀⠉⠉⠛⠛⠓⠒⠒⠒⠛⠛⠙⣦⢻⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠂⡀⣠⠀⠐⢫⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠿⡆⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⣈⡼⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣧⢳⠀⠀⠀⠠⠖⠂⠀⠀⣰⠖
⠀⠀⠀⠐⢠⠃⠀⠀⠀⠀⠓⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠃⠈⠁⠀⠉⠀⠀⠀⠀⣅⡈
⠀⠀⠀⠘⠎⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡖⠐⠐⠋⠀⠉
⠀⠀⠠⠖⠀⠀⠀⠀⠠⠄⠀⠈⠁⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠿⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠇⠀⠜⠃⠀⠀⠀⠀⠀⠘⠲⠓⠉⠁⠀⠀⠁
⣶⣶⠶⠶⢶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⠀⢈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡋⣳⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠰⡦⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣟⠛⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆⣤⡄⠀⡀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡏⢹⣧⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⡷⠁⠤⠀⠀⠀⢰⣿⠎⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⠀⠀⣿⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⣿⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣤⡈⢻⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢶⣄⡀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣿⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⣸⣇⠀⣶⠀⠀⠀⠀⠀⣤⣘⣿⣁⣤⠀⠀⠀⠀⠀⣴⠆⣰⡟⢀⣠⣴⠾⠛
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣾⠿⢿⣦⣿⡀⢴⡀⢠⣤⡈⢻⡿⠋⣡⣄⠀⣠⡀⢰⡟⣰⣿⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⠶⠿⠛⢷⣾⣇⠘⠻⣇⢸⡇⢀⡿⠛⢀⣿⣤⡿⣿⣉⡉⠛⠛⠳⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣉⠙⠳⣦⣽⣾⣷⣟⣠⡶⠟⠛⠿⢦⠄⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠗⠲⢶⣿⣿⣿⣿⠷⠶⠶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠐⠻⣶⣤⣴⠞⢋⣽⢿⡿⣟⠻⢦⣄⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠷⠶⢦⣤⣤⣈⣉⣿⣾⠛⣿⠁⣴⣾⠁⢸⡇⢹⣦⡄⢹⡿⢷⣤⣶⠶⠾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⣿⠏⣼⠇⠈⠋⠀⠉⢉⣠⣾⣧⡈⠛⠃⠸⠷⠨⣿⠻⣷⣶⣿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⣤⡶⠟⠋⠁⣼⠏⠰⠟⠀⠀⠀⠀⠀⠛⢉⣿⡍⠛⠀⠀⠀⠀⠀⠶⠀⢻⡏⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⣿⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠈⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣧⡈⠛⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⣿⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣿⠀⠀⠻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡶⣤⣤⣴⣤⡾⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠀⠀⠀⠀⣼⣋⣻⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⡤⠉⣗⡆⠀⠀⠈⠉⠉⠀⠀⢤⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⡶⢠⡾⠿⣷⣀⡀⠀⠀⠀⠀⢞⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠛⠛⠉⠛⢷⡤⡄⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⢦⣴⣄⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠏⠀⣽⠿⢿⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⡧⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡤⢸⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣖⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⣀⡀⣀⡤⣤⣴⣲⠖⠻⣾⠿⢯⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠭⠗⣾⣽⣶⣿⠽⠛⠓⠋⠉⢹⣯⣿⡆⠀⠀⠙⠛⠋⠀⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣎⣥⣆⣴⢞⣁⣻⣭⣽⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣀⣈⠳⣤⣶⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⣩⡿⣅⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠉⠉⠛⣽⣶⣖⣦⣠⢤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⢿⠇⠀⠀⠀⠘⣿⣭⡏⠀⢈⣿⣻⡷⠛⠻⣽⣶⣟⠦⢴⣲⠤⣖⠶⠾⡿⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢲⣾⣵⣧⡶⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠃⠀⠘⣯⣭⠇⠀⠀⠀⠀⠙⣦⣴⣾⣷⠚⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡤⣀⣴⣺⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢟⣶⣶⣖⣦⣤⢲⣤⣴⣢⣤⣖⣦⠴⣲⠶⠾⣽⡿⣷⠛⠉⠛⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠗⢲⠿⣿⣪⣿⣾⠇⠉⠀⠨⣿⣙⡧⠀⠀⠀⠀⠀⠻⣶⠿⠀⠀⠀⠀⠐⢿⡾⢿⡖⠽⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣯⣿⢿⣲⣤⣠⢤⡙⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠉⠀⠋⠀⣽⡯⣿⠊⠉⠓⠛⠛⢯⡿⣾⡽⠶⢾⡷⠶⢏⣶⣶⡭⠖⠺⠽⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣊⢥⡶⠃⠀⠀⢀⡴⣤⢈⡟⣻⠀⣠⡄⢠⠿⣸⣯⣿⡁⣠⣄⢰⡄⣴⣯⣬⣧⡀⠀⠀⠙⢶⠬⣳⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠰⠞⠒⢏⡤⠞⠹⠋⠱⠊⠷⠛⠋⠁⠈⡇⠀⠀⠀⢸⠋⠁⠈⠛⠙⠞⠻⠏⠻⠏⠲⢄⡟⠚⠳⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄⠠⠀⠀⠀⠀⡠⠐⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⣀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⣀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⠊⠀⠀⠀⠒⠲⣄⠀⠀⠀⠀⠀⠀⡀⠀⠓⠠⠀⠨⠄⠂
⠀⠀⠀⠀⠑⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⡠⠒⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡀⠤⠀⠀⠈⠐⠀⠀⠈⠉⠉⠀⣀⡀
⠀⠀⠀⠀⠂⠀⠁⠉⠉⠀⡀⠠⠔⣈⡀⠀⡀⠀⠠⠄⢂⢠⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⠤⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⢾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠨⠤⢄⣤⣬⣥⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠠⢔⣽⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠄⣀⠀⠉⠐⠒⠀⠄⠠⠤⠀⠀⠀⠀⠤⠄⠐⠷⢤⣤⣀⣀⣀⣀⣐⣐⣨⡧⢀⠠⣊⠔⠒⠉⠉⠉⠙⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⠒⠠⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢁⠀⡀⠀⡴⠃⢀⠠⠀⠀⠄⠠⢌⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⢧⡔⡥⠚⠉⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣈⠀⠀⠀⠀⠠⠀⠐⡀⠔⡯⠥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣟⠒⠤⠤⣤⠭⢤⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣌⡀⠉⠉⠙⠛⠛⠛⠛⠛⢑⣂⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠶⠖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠄⠠⠈⠠⠀⠀⠠⠂⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⡤⠀⣀⣀⠀⡀⣤⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢻⡀⣀⠄⢻⡟⡼⠀⠀⢠⢸⠁⠘⠁⢀⠀⠀⠐⠌⠀⣷⠀⡏⢆⠰⡅⣿⠘⠇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢣⢈⡎⣧⠇⠀⢸⣧⠀⡐⠋⠀⡘⠀⠀⢠⠎⠀⠀⠀⠘⣀⣷⡟⡁⠁⠀⢠⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠀⢹⠀⠀⢸⣿⠀⠁⠀⠀⣧⡤⠰⠋⠀⠀⠀⠀⠀⠀⣷⠰⠀⠀⠀⠀⢹⠆⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣆⠀⢸⣿⠠⠈⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣿⠻⠀⠀⠀⠀⠘⣋⣠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢽⠀⠀⡀⠀⠀⠀⢸⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⡷⣴⠒⠛⠁⠀⠀⠀⠀⢀⣠⣶⣶⣄⣸⣠⡾⣶⣦⡀⢰⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⠇⠀⠀⠀⠀⢀⣠⡶⢿⣿⣿⣿⣿⣿⡛⣅⠉⠙⠃⢸⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⡇⠀⠂⢀⣠⡶⠟⠉⠀⠀⠙⠻⣿⣿⣿⡇⢸⣿⣦⡀⠘⡇⢏⣀⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠉⠷⣿⣿⣇⣤⠞⠋⠅⠈⠀⢀⠀⠀⢠⠠⡄⡙⢿⡟⢸⣿⣿⣿⣷⣧⡇⠸⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠌⢀⣠⢼⣿⠛⡇⠀⢀⡶⠒⠲⠓⠐⠁⠀⠀⡗⣱⢾⣏⡼⣿⣿⣿⣿⣿⣿⠦⣀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢉⣠⠖⠋⠡⢼⡿⠀⢃⡴⠻⣤⠀⠀⢠⠀⠀⠀⠀⠀⢹⣾⡇⠃⠀⢙⣿⡟⣿⢶⣟⣿⡄⠆
⠀⠀⠀⠀⠀⠀⣤⡴⢞⠋⠀⠠⡄⡀⢸⣷⠓⡟⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⣿⡿⢀⣴⣿⣿⣿⡟⢸⣿⡟⡃⡄
⠀⠀⠀⠀⠀⠀⠈⠀⢻⠀⠂⠐⠛⠙⢾⣿⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡇⢾⣿⣿⣿⡏⡧⢸⡽⡆⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⢸⣿⠇⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⡟⢹⣿⣿⣿⡇⣷⢸⠇⣣⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠆⠀⠀⠀⠀⢠⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⣠⣾⣿⣷⡺⢿⣿⣿⣧⡏⠘⡇⣏⢸
⠀⠀⣤⣄⠴⣶⣼⣿⣾⠀⠀⠀⠂⡀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠐⢀⣾⣿⣿⣿⡟⡁⣾⣿⣿⢿⡇⣠⠃⣿⢸⣦
⠀⢸⣿⣿⡿⣿⣟⣿⣿⠀⠀⠀⠀⠁⢸⣿⣿⣅⠀⠀⠀⠀⠀⡄⠀⣰⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⢸⡗⣿⠀⡏⢸⡷
⠀⢸⣿⠿⠷⡯⠿⡿⠿⠀⠀⠀⠀⠀⢸⣿⣇⠋⠀⠀⠀⠀⠀⠁⠚⠻⡿⡻⠛⠟⢿⣧⠊⠟⠿⠿⢸⣷⡿⠟⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⡅⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠀⠀⠘⠛⠁⠀⠀⠀⢸⠋⠁⠀⠀⢰⡇
⠀⢀⠀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣄⡤⠼⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠚⡇
⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠞⠛⠛⠛⠛⠶⣦⣤⡾⠋⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣤⣤⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡶⠶⠶⠶⢶⣤⡀
⠀⠀⠀⢀⣀⣴⠟⠋⠉⠻⠋⠀⠀⠈⢻⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⣴⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⡀
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⡀
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠙⠛⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⠟⠉⠉⠉⠉⠻⡶⠛⠛⠛⠷⣦⣰⡟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠷⣦⡀
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⣽⣿⢭⡇⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢤⡆⠀⠀⠀⠀⠀⠀⠀⠈⠾⠽⠶⠁⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⣺⣗⢺⠂⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣻⣿⣻⡟⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠃⠀⠀⠀⠀⠀⢾⡷⣽⠆⠀⠀⠀⠀⣴⡖⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠹⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠸⣗⡿⣗⡟⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⢲⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⠙⠀⠀⠀⠈⠩⠣⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⣺⢿⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢠⠃⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⡞⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠷⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣷⠀⠇⡁⠀⠈⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢰⠱⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⣴⢃⠀⠸⡃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠈⠓⡀⠀⠀⠀⠀⠀⠀⠀⡠⢘⠀⠀⠜⣶⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⢄⠘⡿⡄⠀⠙⢆⢦⡀⠀⠀⠀⠀⠀⠜⡄⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢀⡼⡰⠃⢀⣾⣶⠃⡰
⠀⠀⠀⠀⠀⠠⠄⠀⠠⠀⠀⠤⠀⠑⢤⡌⠰⠄⠀⢠⡙⠆⠀⠀⠀⠀⠄⢧⠀⠀⠀⠀⠀⠀⠰⠘⠀⠀⢀⠀⠴⢋⠔⠀⣴⢟⣿⣿⡎
⠀⠀⠀⣀⠀⠀⡀⠀⢀⠀⠀⣀⠀⠀⡀⠙⢢⡴⢒⣒⠛⠓⠦⠌⠐⣤⣸⠀⠂⠀⠀⠀⠀⠠⠀⣃⣤⠂⢡⠴⠛⠛⢂⣲⣾⣿⣯⣾⣿⣄
⠀⠀⠀⠁⠀⠀⠁⠀⠈⠀⠀⠉⠀⠀⠁⠀⠈⠀⠀⠈⣌⣂⣀⡀⠢⣄⠈⢑⣦⠀⠀⠀⢀⣆⡈⢁⣠⠔⢀⣀⣂⣡⢰⣻⣿⣿⣿⣿⣯⣿⣆
⠀⠀⠀⠓⠀⠀⠃⠀⠘⠀⠀⠚⠀⠀⠃⠀⠐⠂⠀⠒⠠⠀⠀⠉⠳⠠⡙⡀⡀⢡⠀⣀⡄⠀⢀⠉⠠⠞⠉⠀⢀⡿⣹⣷⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢁⡜⢅⠀⠼⢠⡅⠦⢀⡀⢃⠌⠀⠀⢀⡾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢑⡉⠀⠁⠀⠀⢰⠀⠁⣊⣦⠐⢀⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠳⣐⠆⠚⠓⠰⣀⠞⢠⠂⢀⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⢃⠀⠀⠌⠹⠀⡏⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠆⠈⠛⠛⠂⠸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡆⠘⠃⠀⠀⠶⠃⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠜⢂⠠⠴⠦⠄⠺⠣⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠑⠐⠂⢊⠀⡠⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠭⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠛⠠⣀⣀⠄⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⣒⣒⣒⣒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠠⠬⠭⠭⠭⠭⠭⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠸⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡄⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢠⠉⣤
⠀⠀⠀⠀⠐⠆⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⠀⢀⠿
⠀⠀⠀⠀⠇⢀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠪⠆⠒⢸⡀⠀⡀⠤⢄
⠀⠀⠀⠀⠐⢄⡡⢀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠡⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣁⡠⠆
⠀⠀⠀⠄⣀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡐⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡡⠞⠁⠀⠉⠉
⠀⠀⠦⠀⠤⠄⠁⠈⠢⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣋⡀⠓⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠈⢀⠶⡈⠑⠆⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⣉⡤⠚⠁⠈⠶⣄⠀⢀⡙
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠁⠀⠈⠉⠒⠒⠀⠈⠉⠉⠀⠀⠤⣄⠀⠀⠀⠀⠄⠀⢀⠐⢄⠀⠀⡤⠒⠀⠒⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠞⠁⠀⠀⠀⠀⠈⠲⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⢀⠀⠀⡠⠄⠀⠀⠀⠀⢀⡀⠒⠒⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣤⡀⠁⠀⣾⠆⠀⠀⠈⠥⠄⠀⠀⠀⢀⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠇⠀⠀⠀⢠⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢠⠀⠀⠀⢆⠀⠀⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠑⠀⠀⠀⠉⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠠⠴⡿⠦⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣻⢢⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⣀⡤⠄⠠⠦⡄⡀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠈⠣⡀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⡄⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⢑⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⠏⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⣄⡀⠀⠀⢠⠁⠀⠀⣸⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠛⠦⠒⢛⠀⠀⠀⠻⡄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢰⠀⢸⠀⠀⢰⠀⢡⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⠀⢸⠀⠀⢸⠃⢸⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢢⠸⠀⠀⠈⠀⠈⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⣄⠀⠀⠀⠀⠀⣦⠀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠘⠀⠀⢀⡇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠂⠀⡿⠀⣤⡄⢠⡠⠚⢁⠖⠀⠤⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⡇⠀⠀⠀⠀⢣⣀⡎⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠘⢷⣄⣀⣀⣀⣼⠟⠙⠢⠤⠤⠴⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⣙⣿⠛⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⡀⠀⣠⣤⣤⣄⡀⠙⠏⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⢀⣸⣧⡀⠙⠛⠛⠛⠁⠈⠋⠀⣀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠈⢻⡟⠉⠀⠀⠀⠰⠶⠀⠀⠀⠙⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠇⠀⠃⠀⠈⠁⢀⣤⣶⣷⣦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⡀⠀⠀⠉⠉⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⡀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠈⠋⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⠁⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠉⠁⠉⠀⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣤⠀⠈⠋⠀⢰⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⣿⣿⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⡿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡀⣀⣤⣤⣄⣤⠤⠀⠀⠀⣀⠄⠘⠛⠉⠹⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡾⠟⠋⠀⠀⠀⠀⠀⠈⠛⠟⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣹⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠄⠀⢀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⣻⣤⡀⠀⡀⢀⢺⣗⠀⠀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠄⡄⠢⡝⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢝⡳⣦⢰⠟⠎⣀⣲⠵⣛⠫⠁⠀⠀⠀⠀⡀⡀⠠⣦⡀⢫⢛⠌⣠⡞⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣰⢒⢿⠐⣏⠂⠁⠀⠀⠀⠀⠀⠀⢐⠭⡻⠓⠂⠨⣷⣿⡯⠤⠄⢶⢶⠆
⠀⠀⠀⠀⠀⠀⢀⣠⡵⢿⠛⠸⠹⡾⠅⠋⠛⢥⡢⠄⠀⠀⠀⠀⠀⠀⠀⠐⠛⣡⢂⣝⡌⠱⠶⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⢭⠏⡂⠀⠀⠀⠸⠱⠆⠀⠀⠀⠀⠀⠠⢐⣸⡆⠀⠃⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⣟⡠⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠹⡅⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⢤⠀⠀⠁⢀⠀⣨⠾⠡
⠀⠀⠀⠀⠀⠀⠠⣁⡀⠀⠀⠙⡇⢀⠀⣆⢠⠶⡀⠀⠀⠀⠀⠈⠟⡺⠤⢀⡀⠀⠣⣸⣀⡪⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠊⠔⠩⠛⠒⢴⣶⣅⣊⠉⠑⠉⠂⠁⠀⠀⠀⠐⠋⠀⠀⢀⣀⠉⢶⡉⢹⢐⠊⠛⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢉⠔⢹⠱⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠔⡱⢻⠙⢆⠈⠹⡳⡣
⠀⠀⠀⠀⠀⠀⠀⠤⢄⡶⠉⠀⠀⠐⡙⣦⠴⠤⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠘⢁⡌⠀⠀⠃⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⠁⠀⠀⠀⠱⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢱⢯⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠚⠉⠉⠉⠉⠉⠓⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣫⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣫⢔⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣡⢞⠕⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡙⣆
⠀⠀⠀⠀⠀⠀⠀⣠⢋⡞⡵⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⢌⢳⡀
⠀⠀⠀⠀⠀⠀⣰⢣⢏⠞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⢆⢳⡀
⠀⠀⠀⠀⠀⢰⢃⠏⡎⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠆⢳
⠀⠀⠀⠀⠀⡏⡼⠸⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡈
⠀⠀⠀⠀⢸⠁⡇⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠇
⠀⠀⠀⠀⢸⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠘⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡄⡇⠇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠇
⠀⠀⠀⠀⠀⢧⢹⡸⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠰⢠
⠀⠀⠀⠀⠀⠘⣆⢧⢣⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢡⢃⡎
⠀⠀⠀⠀⠀⠀⠘⣎⢧⡳⡙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⢂⡞
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡱⣝⢎⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠣⡙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⣆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡜⣎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡘⡜⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢣⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢱⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠷⠶⠶⠶⠿⠟⠛⠛⠛⠛⠋⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠿⢿⣿⣶⣾⣿⣿⣿⣿⣿⣿⠿⠛⠚⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⠷⠶⠶⠶⠶⠶⠷⠒⠛⣿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣾⣿⣿⠿⠶⠤⠤⠤⠴⠶⠶⠶⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢶⣿⣿⣿⣶⣦⣤⣤⣤⣤⡤⠤⣶⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⣿⣿⣿⠳⠤⠤⠔⠒⠒⠒⢋⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣶⣶⣶⣾⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠘⠀⠁⠀⠀⠈⠀⠃⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣽⣷⣿⣾⣷⣿⣵⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡈⠂⠀⣠⡶⠟⠛⠛⠛⢶⣄⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣮⣿⡿⢋⣡⣤⣤⣤⣍⠻⣿⣯⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡉⠀⣼⠋⠀⠀⠀⠀⠀⠀⠙⣧⠀⢉⡠⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢷⣿⡏⣰⣿⣿⣿⣿⣿⣿⣷⡌⢿⣷⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠀⢰⡇⠀⠀⢤⡀⠀⡤⠀⠀⢸⡇⠀⠤⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣻⣿⠀⣿⣿⣟⢿⣿⢟⣻⣿⡷⢸⣿⣛⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠂⠀⢿⡀⠀⠀⡀⠀⠀⠀⢀⡾⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣽⣿⣆⢻⣿⣿⣿⣿⣽⣿⣿⠃⣼⣿⣭⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠋⡀⠈⢻⡄⠀⡇⢸⠀⢠⡞⠁⢀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣵⡿⣿⣆⠹⣿⡿⣿⣿⣿⢃⣾⣿⢷⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⠀⢿⠀⣵⣾⠀⣿⠀⠲⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⢾⢿⣿⡆⢻⡗⠉⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡿⠿⢿⡿⠷⠘⠃⠀⣛⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡲⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢰⠏⠂
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠂⠀⠀⠠⠒⠀⠚⠀⠀⢀⡠⠀⠂⠃⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⢸⡆⣰⣧⣆⠒
⣴⣧⣿⣇⡆⠀⠀⠘⣯⠐⠂⠀⠀⢲⡄⠀⠀⠂⠻⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢰⢷⠀⢠⣾⢷⣭⠌⠛⢶⠒⢠⡆
⣻⡟⢫⣿⣧⣷⣏⡠⠀⠔⠐⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡔⢯⡀⢘⡁⢂⠪⡦⠴⡘⢦⣝⢳
⠚⣯⡶⢹⣿⣿⠋⠂⠀⠂⠐⡈⢺⣶⣆⣤⣀⣤⢀⠀⠀⠀⢀⡀⣤⢀⣠⡊⠃⣴⠞⠃⢜⡺⠤⠉⠱⣿⠀⠈⠀⠙⢿
⢈⠁⠖⢸⣉⡤⠎⣠⠆⠀⠀⠐⠲⣿⣿⡿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠈⠈⠺⣂⠘⣯⡤⠒⣀⡠⣜⡀⠀⠈⠶⢾
⠸⠄⢠⣟⡉⠠⠠⠶⢀⣰⢿⠑⢠⣙⣯⢓⣹⣭⠟⣋⡁⠀⠀⠀⠙⢿⠀⠈⠙⠳⣎⢢⡟⠒⠉⠈⠣⠌⡂⠀⠠⡀
⠀⠀⢠⡏⠠⢰⠂⢠⠊⠙⢄⣯⡀⠀⠉⠁⠀⢄⣪⣷⣻⣓⣦⣴⡶⠂⠀⡀⠁⣌⠚⢿⣗⠄⣇⠀⢰⡆⠈⠲⡀⡀⠑
⠀⡰⠋⠀⢀⡬⣟⠀⠀⠀⢣⣀⠧⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠉⡋⠀⠀⠉⠐⠉⠄⠀⠁⢿⡁⣀
⠀⠀⠀⠈⠀⠀⠀⠙⠊⠀⠊⠁⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⠘⠉⠀⠄⠐⠤⠦⠀⠽⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠙⢿⣿⣶⣤⡀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡄⠀⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⣰⣿⣿⠏⠀⢀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⢰⡿⠛⠁⠀⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⠿⢶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡛⠋⠉⢀⣀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⡀⠬⣉⣛⣻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣫⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠸⣿⣀⣤⠶⠚⠛⠉⠙⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠐⣿⣡⠴⠶⠚⠛⠛⠲⢦⡀⠀⠻
⠀⠀⠀⠀⠀⠀⠈⣇⢀⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢻⡈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣿⡀⠀⠀⠀⠀⠀⢀⣠⠞⠁⣠⠟
⠀⠀⠀⠀⠀⠀⣀⠄⠻⣎⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣋⣽⣿⣶⠶⠶⠛⣋⣥⠴⠛⠁
⠀⠀⠀⢀⣴⡾⠁⠀⠀⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢛⣉⣥⡴⠶⠶⠻⢿⣍
⠀⠀⣴⣿⠏⠀⢀⠀⠀⠀⠹⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠛⠉⠀⠄⡀⠀⠀⠀⠙⢷⡀
⢀⣾⣿⠏⠀⢠⢣⠀⠀⠀⠀⠈⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠘⢦⡀⠀⠈⢮⣿⡆
⣸⣿⡿⠀⠀⡎⣾⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⢿⣿⣿⡄
⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⢸⣿⣿⡇
⣿⣿⣷⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢸⣿⣿⡇
⠸⣿⣿⡄⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠛⠓⠶⠶⠶⠶⠶⠶⠒⠚⠋⣹⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⣾⣿⣿⠁
⠀⢻⣿⣿⣄⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢦⣤⣤⣤⣤⣤⣤⣤⣤⡴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠏⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠙⣿⣿⣦⡀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠉⠻⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠋⠁⠀⠀⠀⣀⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⣀⠀⠈⠉⠛⠻⠷⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⠶⠿⠛⠋⠉⠀⠀⠀⠀⢀⣀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣤⣤⣀⣀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠷⠶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⡶⠶⠿⠟⠛⠛⠉⠉⠁
⣿⣿⣿⡿⢿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣠⣤⡄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣠⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡄⠀⡜⠁⡏⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠸⡈⠳⡀⢀⠎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠲⠁⠸⡇⠀⡜⠁⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠁⠳⡀⠀⡇⠀⠳⠋⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠞⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠓⠚⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠓⢄⡀⠀⠹⡦⢄⣀⣰⣄⡄⠀⠀⢀⡴⠃⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠓⢤⡀⠀⠙⣄⠤⠷⣄⠀⠈⠉⡿⠤⣤⡏⠀⢀⠴⠊⠁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⠋⠉⠁⠀⡀⣉⠑⠢⣌⡶⠋⠀⠀⠀⠀⠉⠒⠊⠁⠀⠀⠉⢶⣡⠔⠊⡉⣀⠀⠈⠉⠙⠒⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠁⠀⠉⠙⠂⡞⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢳⡐⠛⠉⠀⠈⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣠⠴⠚⢹⠁⠔⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠣⠀⡏⠓⠦⣄⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⣿⣛⣷⡀⠀⠀⠀⠀⠀⣾⣿⣹⣧⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⡻⠟⡢⠁⠀⣀⡀⠀⠀⢿⠿⢟⡜⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡈⠉⠀⣠⣤⣶⣶⢦⣄⠀⠉⢁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣨⣇⡀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠦⢤⣀⣑⣊⣀⡠⠴⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢹⠀⠀⠀⠀⠀⠀⠀⠀⢫⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠏⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⢦⡇⠀⠀⡆⠀⠀⠀⠀⠀⢸⣧⠎⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⡇⠀⠀⣇⣀⣀⣸⠀⠀⢸⣷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡿⠃⠀⠀⠈⠿⣶⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⡀⠋⠶⠴⠴⣄⡄⠀⠀⠧⢀⠀⡀⠑⢖⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠒⠂⢠⢀⣤⠀⠀⣰⠀⠘⣧⡄⠐⠛⠁⠛⠄⠀⣅⠂⠬⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠠⡔⡖⢻⢀⠠⠒⣄⢼⠝⠏⠋⠀⠀⠶⠋⠑⠒⠚⠷⠂⢀⣀⠀⠂⠰⣌⣆⠀⡀⠈⢹⣄⡄
⠀⠀⠀⠀⠀⠀⢠⠹⣦⣼⣧⣶⡿⠟⠉⠀⢓⡳⠂⠡⣄⠀⠀⠀⠀⠘⣧⠀⠀⢉⡧⢀⠀⢹⡦⣀⠀⠈⢿⡇⠀⡀
⠀⠀⠀⡀⢸⠀⠎⢀⣸⣿⢳⣏⣤⠤⠚⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠲⢇⣆⣷⢀⠁⠘⡧⢸⣦⡀⠎⢷⡴⢻⡆
⠀⠀⠀⣧⠀⡆⠀⡸⢋⡿⣾⣉⣆⢠⢠⡠⣴⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢿⠳⠷⣧⡄⣣⡑⡄⢼⣷⣾⠁
⠀⠀⢀⠸⡔⡉⢶⡟⣹⣱⡟⠓⠁⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠵⠂⠳⣄⣏⠘⠻⢷⣿⣀⣿⡃⣨
⠀⢳⠈⡦⢻⣦⡟⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢸⣿⡝⣿⡷⡴⠂
⠀⠈⢷⡄⠈⢿⣾⢃⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠳⣿⡇⢻⣇⢽⠄
⣄⠀⣸⢿⣆⣼⣟⣱⣿⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢻⡷⣼⣿⣷
⠘⣧⡽⣎⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⣸⣿⣼⣿⠏
⠀⠘⣧⣿⢸⣿⣿⢀⣄⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⡏
⠀⢀⢹⣾⢾⣿⣿⣨⠋⠡⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣀⠀⡓⢾⣿⣿⢳
⠐⢾⡤⣿⣤⣿⣿⡄⠀⢸⣿⣫⡀⠀⢠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⡻⡟⣾⣿⣿⢸⠇
⠀⠸⣿⢏⢿⣿⣿⣧⠀⢒⢸⣟⡆⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⡇⢈⣿⣿⡇⢸⠀⠠
⠀⠀⢹⡥⡂⢻⣿⣿⣧⠀⢖⣿⢩⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣿⠥⣼⣿⣿⣇⡏⠀⠏
⠀⠀⠀⠘⢷⡈⣿⣿⣿⣷⡨⣯⣾⣿⢘⣠⣀⡆⠀⠀⠀⠀⠀⠀⠀⢶⣷⣆⡀⠀⠈⢀⣤⡸⣿⣏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⣿⣛⠛⣿⣻⣿⣶⣄⢀⠀⢀⠀⠀⠘⢝⢯⣇⢆⣇⡞⣿⠩⢿⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠛⠻⣿⣷⣾⠼⠿⠿⠿⠯⣄⣠⣤⡄⢠⠴⠓⢉⣩⣍⠙⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠓⠦⠤⣼⣿⣥⠤⠄⣨⡤⠐⠀⠀⢈⠗⣅⠀⣲⣊⣹⣥⣾⣿⣿⣿⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡏⠉⢻⣿⣟⣩⣹⣷⣶⣊⣠⠶⢻⣯⠟⠉⠀⠀⢤⣤⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠿⠏⠀⠊⠁⠐⢤⠞⠂⠀⠐⠋⠁

⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⣰⣣⡄⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠘⣣⠒⠀⠀⠀⠀⢾⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⠀⣴⣶
⠀⠀⠀⢠⡯⠏⠁⡀⠀⠀⠸⢿⡿⠇⠀⠀⠀⠠⠰⠀⠀⠀⠀⠴⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⡂⠀⠀⢀⣾⣿⣿⣧⡀⠀⠀⠀⣷⣿⣆
⠀⠀⠀⠚⠉⠙⠛⠛⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠘⠉⠉⠉⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⢀⣼⣿⣧⠀⠀⠀⢀⠡⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⢾⣿⣿⡂⠀⠀⠀⢀⣽⣿⣄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣴⣿⣿⡄
⠀⠀⠀⣠⣾⣿⣷⣄⠀⠀⠈⠐⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠰⣿⣿⣿⡦⠀⠀⠀⣠⣾⣿⣧⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⣿⣿⣿⡷
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠙⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠦⢄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⢶⣀⠐⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠈⠀⠀⠀⠀⠀⠀⠐⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢋⠃⠀⠀⠀⠀⠀⢠⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⢀⢔⡗⣉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣦⡤⠆⣀⠀⠀⠀⢀⣲⠿⠛⠁⠀⣀⡠⢤⢎⣼⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⣀⣴⠀⠉⠙⠛⠓⠛⠛⠛⢿⣷⣄⢀⣷⣐⠥⢆⢀⡾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣛⢿⣿⣿⣿⣋⢵⣿⡟
⠀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠦⣽⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⡆⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠘⢃⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡶⠿⠿⣿⣰⡿⢿⠃
⠀⠀⠀⠀⠀⣾⣿⣡⣴⠀⠤⠄⠐⠰⠤⡁⠀⠰⠶⢦⣀⣀⠀⠀⠀⠀⠀⠋⠹⠉⠁⠀⠀⠈⠛⠙⠋
⠀⠀⠀⢀⢸⣿⣿⠟⡽⣶⣮⣰⠷⠶⠶⠾⠀⠀⣰⣋⣩⣿⣆
⠀⠀⠀⠊⣼⡿⢁⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⣽⡏⠁⠀⠀⠉⠢⠄⠀⠀⠀⢀
⠀⠀⠀⡟⡟⢰⣋⣿⡿⠋⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⡂
⠀⠀⠀⢹⡇⠀⢶⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢺⡄
⠀⠀⠀⢺⡇⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠼⣧
⠀⠀⠀⠸⣇⢼⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠐⣿
⠀⠀⠀⢶⣿⠿⠀⢸⡻⠀⠀⠀⢰⡇⠀⠀⣸⣧⠀⠀⠀⠀⣄⣠⠀⠀⢸⡆
⠀⠀⠐⠿⠟⠀⠀⠸⣿⣷⡀⠀⠈⠉⠂⠰⢿⣿⠜⠓⠒⠂⠁⠀⠀⠀⣾⣿⢆⡀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠶⠿⠗⠀⠀⠀⠘⠻⡷⠶⠂⠀⠀⠀⢠⣐⠠⣶⣩⣤⡥⠀⢈⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠲⢦⣄⠀⠀⠀⢀⣠⡤⠶⠶⠒⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠻⣆⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠛⠛⠧⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢀⣀⣀⣀
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠉⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠈⣻⡶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣀⣀⣀⡴⢾⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠿⢫⡟⠛⠛⠻⣤⣞⡃
⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠙⢦⣤⣀⣀⣠⣤⡞⠉⢳⣦⣤⣀⣀⡀⠀⢀⣀⣤⣏⡿⣠⣭⡗⠀⠀⠀⣿⠉⢿⠄⢀⣄
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣯⠹⣤⣤⣤⣽⡜⢯⣩⣟⢻⣍⣧⠙⣴⣧⣴⠟⣦⣤⡾⠡⣿⡭⠤⠟⠼⠧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣶⢂⡶⢶⣲⡶⣠⠶⠶⣦⡴⢶⡶⣀⠰⡖⢶⣔⡶⠒⣆⢲⡒⠒⠓⢶⣘⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⢾⡉⢰⡇⠀⠀⢸⣗⠛⢰⠋⣷⠙⣻⡏⠀⠀⠘⣧⠛⢶⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⢧⣤⣤⣤⠟⡴⠞⠛⠳⢷⡙⣦⣀⣀⡴⣷⣦⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡴⠒⢲⢆⡘⣇⠀⠀⠀⣰⢃⡄⢶⠶⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡞⣷⠙⢲⣶⠚⢡⡏⠹⣎⢧⣍⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⢁⢺⡄⠙⠶⠛⠹⠷⠖⠋⠀⣸⠂⢰⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠟⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠷⠋

⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⢀⣨⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣲⣫⠄
⠀⠀⠀⠀⢠⣄⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣍⣩⢧⠶⠂
⠀⠀⠀⠀⣄⣈⣱⣆⣠⠤⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣤⣩⣿⣦⣶⠛
⠀⠀⠀⢠⡌⠉⢡⣎⣁⣀⡄⠀⣠⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣉⣤⣾⣥⡼⠳⠶⠾⠃⢀⠀⣠⠄
⠀⠀⠀⠀⠓⠚⣯⠉⢉⢁⣠⠀⠋⡀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⠀⡞⠀⠀⠀⢰⠃⠁⢺⣤⠿⢶⣧⡾⢿⣶⠶⠾⠋⠊⠀⠟
⠀⠦⠐⠶⡴⠾⢛⡓⠟⠉⠁⠀⠀⣧⠀⡀⢀⡀⠀⠈⢧⡄⠠⢠⠆⠀⠐⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⣟⣫⡿⠶⢟⡿⠶⠶⠂
⠀⠀⠶⣦⣀⣴⠞⠛⣶⣄⣼⠻⠇⠘⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣳⣫⣿⣭⣿⣟⣻⡷⢶⡾⠇
⠀⢀⣤⣬⣡⣂⣴⣾⣇⠈⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣋⣽⠶⠖⠺⢛⣿⣷⣤⣤⣶⠄
⠀⠉⠉⠉⠙⡉⠉⣹⡋⠛⠋⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠛⠶⠼⠿⠿⣿⣧⣤⣤⡤⠆
⢀⡝⠛⢿⣍⣠⠾⠻⡿⢷⠆⣦⠆
⠈⣿⠏⣉⣈⠈⣀⢾⣧⣀⠀⡾⠟
⠀⠀⠀⠀⠿⠋⠏⠈⠙⠛⠋⠁
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣧⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⡿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢯⣄⠈⢷⣜⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢫⣿⣾⠁⢳⣄⠻⡷⣝⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣱⣿⣿⡟⢲⡀⠻⣧⡈⢮⡳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢃⣼⣻⣿⡟⣇⠀⠹⣦⡈⢯⡢⣙⢦⡑⢽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⡽⢋⣴⡟⣱⣿⠟⣴⡏⢷⣄⠈⠻⣦⡙⢮⡓⢭⣢⣌⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠖⢋⣴⣿⢋⣼⣿⢋⣾⠳⡇⠀⠹⣷⢄⠙⢿⡳⣽⣦⡈⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠵⠚⣩⠞⣡⣾⠟⣡⣿⠃⢠⡟⢦⡀⠈⠳⡙⢦⣙⢦⣉⠻⢶⣄⡀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⢾⡿⣫⡾⡿⠃⣠⢿⣦⡀⠻⣦⡀⠙⢦⡈⠓⢬⡳⢤⡈⠛⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠿⠛⣱⣟⡴⣫⠞⢁⣼⠟⢸⡏⠻⣦⡈⠻⣷⣄⡙⠦⣀⠈⠑⠻⢷⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢋⡴⠃⣴⣿⠋⣰⢻⠷⣄⠙⢿⣷⣌⠻⣝⠲⣬⡳⢤⡀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢁⡴⠋⣠⣾⠟⢡⣾⠏⣸⠧⣜⢷⣄⡙⠻⣟⢮⣝⣦⣍⠓⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⡴⣋⡴⣚⡿⢁⡴⣫⢏⡼⢻⠀⠈⠳⣝⢿⡦⣌⠙⠮⣙⠻⢿⣶⣤⣉⡙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⣡⠖⢋⡴⢋⡜⣡⠞⠁⢸⠀⠀⠀⠈⠓⢿⣮⣝⠲⠮⣽⣶⣄⡉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣡⠖⢋⡴⢋⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡈⠉⠙⠛⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⠿⠚⠉⢀⡴⢋⡴⠋⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣫⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣌⢿⡿⣡⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣀⠀⠙⢿⡿⠋⠀⣀⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢻⣿⣿⡄⣿⢸⡇⣿⢠⣿⣿⡟⠀⠀⠀⠀⢠⡄
⣀⣀⡀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⣬⣭⣿⣿⣸⣇⣿⣿⣭⣥⠀⠀⠀⠀⣴⣿⣇⠀⠀⠀⠀⠀⢀⣀
⠈⢿⣿⣿⣦⢹⡇⠀⠹⣿⣿⢠⡦⢸⣷⣄⡀⠀⠈⢻⡟⠁⠀⢀⣠⣶⡇⢴⡀⣿⣿⠏⠀⢸⡇⣴⣿⣿⡿
⠀⠀⠙⣛⣛⣿⣿⣜⡻⢾⣿⡿⠁⠘⣿⣿⣇⢠⡆⢸⡇⢰⡄⣸⣿⣿⠃⠈⢿⣿⡷⢟⣣⣿⣿⣛⣛⠉
⠀⠀⠀⠉⠉⠀⢶⣭⣛⢶⣽⣇⠀⠠⣬⣝⣿⣿⠁⢸⡇⠈⣿⣿⣫⣥⠄⠀⣸⣯⡶⣛⣭⠶⠀⠉⠉
⠀⠀⠀⣠⣾⣿⣶⡾⣿⠟⠛⠙⠻⣦⣄⡈⠉⢿⡄⢸⡇⢠⡿⠉⢉⣠⣴⠞⠋⠙⠻⣿⢷⣶⣿⣷⣄
⠀⠀⠉⠛⠛⠛⠋⢼⠏⢀⣀⠐⢶⣄⠉⠛⢶⣤⣻⣾⣷⣟⣤⡶⠛⠉⣠⡶⠂⣀⡀⠹⡧⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⡷⣿⡷⠶⠶⢾⣿⣿⣿⣿⡷⠶⠶⢾⣿⢾⣿⣿⣿⡷⠆
⠀⠀⣀⣤⣤⣤⣄⢺⣆⠈⠉⠡⠾⠋⣀⣤⡶⠛⣿⢿⡿⣯⠛⢶⣤⣀⠙⢷⠌⠉⠁⣰⡗⣠⣤⣤⣄⣀
⠀⠀⠈⠙⢿⣿⡿⢷⣿⣦⣄⣠⣴⠞⠋⢁⣀⣾⠃⢸⡇⠘⣷⣀⡈⠙⠳⣦⣄⣠⣴⣿⡾⢿⣿⡿⠋
⠀⠀⠀⣀⣀⠀⠶⢛⣭⡾⣿⡏⠀⠐⠛⣫⣿⣿⡀⢸⡇⢀⣿⣿⣝⠛⠂⠀⢹⣿⢷⣭⡛⠶⠀⣀⣀
⠀⠀⣀⣭⣽⣿⣿⢋⣵⢾⣿⣷⡀⢰⣿⣿⡟⠸⠇⢸⡇⠸⠇⢻⣿⣿⡄⢀⣾⣿⡷⣮⡙⣿⣿⣯⣭⣀
⢀⣼⣿⣿⠟⣹⡇⠀⣠⣿⣿⠘⠗⢸⡿⠋⠁⠀⢀⣼⣧⡀⠀⠈⠙⢿⡇⠺⠃⣿⣿⣄⠀⢸⣏⠻⣿⣿⣧
⠉⠉⠉⠁⠀⠈⠀⠀⢹⣿⠟⠁⠀⠀⠀⢛⣛⣿⣿⢹⡏⣿⣿⣛⡛⠀⠀⠀⠈⠻⣿⡏⠀⠀⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⣼⣿⣿⠇⣿⢸⡇⣿⠸⣿⣿⣷⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠉⠀⣠⣾⣷⣄⠀⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⣾⣷⡙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠅⢀⠔⠁
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠠⡀⠀⢐⢐⠀⠀⡠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢠⠂
⠀⠀⠐⡖⢢⠀⠀⠀⠰⡀⠀⠈⠀⠀⠀⠀⠊⠀⠀⠃⠀⢀⡰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠠⠀⢀⡀⠉⡃⠁⣀⠀⠀⠀⡐⠀⠀⠄
⠀⠀⠀⣀⣀⡉⠤⡠⠀⠁⠀⠀⠀⠨⠀⠀⠀⠀⢘⠀⠤⠁⠀⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠬⡀⠀⢁⠐⠒⢈⠀⠀⢀⠀⢅⢋⣀
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠒⠄⠀⡀⠸⠐⠀⣄⠠⠉⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠗⢄⡐⢖⣀⠀⠳⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡐⠂⢨⢣⠣⣉⠐⠒⠀⠁
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⢰⠆⠀⢀⠐⠍⢀⠄⡀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢄⡠⠌⠂⡠⠈⠁⢈⠑⠶⡀
⠀⠀⠀⢀⣀⠀⠀⢀⡀⠘⠉⠀⠈⠀⢨⠀⠂⠀⠀⠀⢠⣦⠄⠤⠄⠀⠀⠀⠀⠀⠀⣀⠄⠘⢲⠃⠀⠀⠀⢘⡄⠄⡘⠀⠠⠂⠤⠁
⠀⠀⠀⠀⢀⣠⠄⠉⠁⢠⠀⠀⠀⠀⠰⠀⠀⠀⠀⢆⠀⠠⠈⢃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⢔⡢⡀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠔⠀⠁⢠⠃⠀⠂⢠⠔⠁⠀⢠⠀⠐⢄⠀⠘⠀⠀⠳⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⣉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠁⡀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢾⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠙⠀⠰⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣄⠀⠐⠀⠐⣦⡀⠀⢹⠀⠀⠀⣐⠂⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠠⡀⠠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⢦⣀⠀⢠⠈⠲⡟⢄⡀⢀⠃⠀⢀⣈⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠈⢆⠈⠐⢀⠴⠁⠀⠈⢀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠙⡛⠈⠁⠀⠁⠘⠀⢚⠶⡾⠋⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⢀⡆⠙⠰⠁⡈⠀⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠖⡊⠉⠄⠈⠠⠂⠄⠀⡠⠒⠀⢀⣀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⡈⠉⠀⠀⠈⠁⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⠀⠀⠀⣠⣭⡄⢂⠀⠀⠁⠀⠀⠃⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⡀⠀⠏⠀⢀⠀⠫⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠚⠉⠁⠎⢈⠇⠀⢠⠁⠂⠞⢁⣄⠀⠀⠤⠅
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢠⠔⠉⣈⠀⢤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠚⠊⠀⡀⠀⠀⡸⠂⠁⠠⣦⠀⠀⢬⠀⠀⠀⠃⠤
⠀⠀⠀⠀⠀⠉⠈⠀⠀⠆⠀⢁⡀⠀⠸⠀⡀⠀⠰⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⡟⢀⠀⠀⠁⠀⠘⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣤⠄⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠞⠀⠼⠈⢆⠒⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠓⢓⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠈⠀⠠⠀⣶⡶⠀⠀⠀⠠⠀⠀⠀⠀⠐⠖⠓⠶⠀⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣭⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠚⠓⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⣀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠑⠛⠙⠚⠓⠓⡁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠀⢠⠖⠀⠉⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠚⠐⠳⠖⠤⠂
⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠌⣠⠖⠈⠈⣧⣽⠋⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡐⠧⠢⠔⠦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡯⠀⠈⠐⠉⠈⠒⠚⠀⠙⢆⢀⡠⢀⢀⠀⢠⡄⠤⠒⢲⣴⣧⣶⡜⢚⣒⠘⢀⠀⡀
⠀⠀⠀⠀⠀⡠⠐⡐⠒⠊⠉⠁⢡⣹⣷⢰⠀⣰⣶⡆⠀⡄⣯⡴⣄⢅⠠⡌⡛⣿⣏⣴⠀⠌⠁⢉⡁⡀⣀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⣌⠄⠈⠉⢅⣐⣋⠏⡁⢹⠀⢀⡇⠀⣠⠤⠼⠂⠂⡁⠀⠈⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⢀⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠋⠉⠁⠉⠐⠄⠂⠉⠁

⢦⣼⠀⡌⣿⣿⡆⢻⡰⣴⣃⡶⠋⠹⢲⡼⠀⢠⡸⠀⠀⢸⡄⢴⠥⣰⣷⡞⢻⡈⢸⣿⣿⣧⢸⣿⣀⢼⢿
⣛⣿⣇⡿⣿⣿⣷⠋⣷⠋⢸⠀⣥⠟⡼⠓⠂⢩⡤⣰⠀⠈⢷⡘⠮⢼⡞⣷⠀⢻⣾⣿⣿⣿⣾⣿⣸⣽⣿
⠘⢧⣸⣧⣿⣿⣿⣼⠉⣧⡯⣂⡴⠚⠁⠀⢐⠘⠆⣬⡳⢦⡈⢷⡤⣉⣿⡜⣄⡄⠹⣿⣿⣿⣿⣿⢉⣿⣿
⡦⣹⡟⠈⣿⣿⣿⣇⣾⣿⡞⠉⠀⢬⠌⣠⠀⠤⠄⠰⠭⠵⢿⣿⣷⣌⣐⠿⣎⠻⣄⣿⣿⣿⣿⣿⣼⣿⡷
⣷⣼⣷⣞⣿⣿⣿⠞⢡⡿⢀⣤⣉⣤⣞⠚⠯⠀⠀⠉⠊⠀⠀⠀⠉⠻⢾⣝⠾⣿⡟⣿⣿⣿⣿⣿⣿⣯⣏
⠸⣏⣿⣿⣿⣿⣿⢸⡿⣷⣩⣠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣿⣿⣿⣿⣿⣿⣦⣾
⠀⢹⣏⣸⣿⣿⣿⣿⣋⡽⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢹
⢀⡾⠙⣿⣿⣿⣿⣏⠿⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⢸
⣼⠁⢰⣿⣿⣿⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⣿⡇⢸
⣏⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⣿⣿⣿⣿⣧⢠
⣿⡟⢰⣿⣿⣿⣿⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠤⢻⣿⣿⣿⣿⣿⣿
⣿⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⡏
⣿⣇⢸⣿⣿⣿⣿⣧⡀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⡇⠉⣿⣿⣿⣿⣿⣿⡇
⠘⣿⣾⣿⣿⣿⣿⣿⡘⣌⠂⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡇⣴⣿⣿⣿⣿⣿⣿⣇
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⡿⣝⡞⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⢸⡷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢹⣸⣿⣿⣿⣿⢿⣿⣿⡏⣿⣿
⢋⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡕⠀⠀⡀⢀⣀⣀⣠⣤⣤⡤⣶⣷⣿⣿⡟⢻⣿⣿⣆⣿⣿⣧⢻⣿
⣾⠃⣸⣾⢻⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣦⣴⡷⠛⢋⣭⢿⣉⣴⢿⣿⣿⣿⡿⣠⣾⣧⢻⣇⠟⣿⣞⣷⣻
⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣷⡟⢣⣾⣱⣷⣟⣿⣿⣿⢇⣿⣿⣿⣾⣿⣷⣻⣿⡟⣿
⣷⠀⣿⡟⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣷⣿⣿⣿⣿⡏⣽⣿⣿⠸⣿⣿⣿⢣⣿⢹⣿⣿⣿⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⣵⣦⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣷⢶⡀⠀⠰⠿⠁⠀⡠⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢇⡀⣐⣤⠐⣿⠉⠉⠧⠀⠀⢸⣞⡁⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⢇⣱⠤⢄⣖⠰⠆⢴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⢷⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢶⣶⡛⣠⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠙⣿⣿⡿
⠀⠀⠀⠀⢀⣤⣶⣾⣯⣻⣵⣾⠩⢉⡁⠊⠸⣿⣿⣴⣾⣯⣉⣻⣚⣇⠜⢶⣟⣉⣟⡛⢹⣏⣽⣿⡃⠀⠠
⠀⠀⢀⠐⣾⠃⠞⣿⣿⠀⠀⠁⠀⠀⠀⠛⣶⢞⣿⡿⠟⣛⠘⠃⠿⠿⡟⣠⣤⡿⢿⣿⠛⣹⡾⢻⣿
⠀⠀⠻⢸⣿⠠⠀⢸⣿⠀⠀⠀⠀⢀⡀⠀⢰⠘⣷⣻⡟⠈⠙⠀⠂⠀⠘⠛⠻⣧⠿⣿⣽⢿⡛⣿⣿⠄⠀⠀⣠⣤⡆
⠀⠀⠀⢟⣿⡀⡀⢸⡟⠀⠀⠀⠀⠈⠉⠀⢀⠽⣏⣿⣌⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢛⣀⠠⣛⣵⣟⣠⠐⣾⣿⡟
⠀⠀⠀⠈⢻⣇⠡⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⣩⣿⠀⠀⠀⠀⠀⠀⠀⣰⣦⡾⠩⠄⢺⣏⣻⠿⣿⡟⡹⠋
⠀⠀⠀⠀⠀⣿⡄⠉⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀⢥⡄⠀⠀⠀⠀⠀⠀⠀⠀⡽⠟⠀⠀⠀⠉⠉⠏⠀⠙⡟⠁
⠀⠀⠀⠀⠀⠸⣿⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠁⠤⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠠⠄⠀⠀⠀⠀⠒⠒⢷
⠀⠀⠀⠀⠀⠀⢠⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣀
⠀⠀⠀⠀⠀⠀⣤⡀⠸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠰⣄⣼⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠟⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠲⢰⡯⢹⢮⣴⣲⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠹⣃⡤⠶⣚⣀⢾⣧⣿⣷⣼⣿⢫⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⢿⣿⣯⣼⣻⠿⠿⣟⡿⣯⣿⣶⣝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠛⠉⣷⣯⣽⣿⣿⣿⠿⠿⣟⣧⡫⠞⣟⣀⣽⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⠀⠸⣷⠀⠈⠉⠉⠉⠉⠁⠀⠀⢰⠟⢛⣙⣿⣿⠤⣆⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⣤⣼⡃⣀⠀⠀⠀⠀⠀⣠⣰⡏⠉⠉⢁⣼⡏⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⣦⣍⣛⠻⠮⠭⣿⡟⠿⠽⠟⠉⣀⣠⡔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠶⠖⠖⠶⠾⠟⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⠂⡸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠻⡝⠻⢧⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⡏⢠⡎⠹⡆⠀⠀⠋⠒⢤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠡⡘⢷⣠⡇⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⠸⣿⡀⠀⠈⡢⣝⠛⠧⢀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠙⠷⠾⠮⣅⣉⡒⣤⣀⠁⠓⠺⢅⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣤⠞⠟⠓⣲⣦⣤⣄⡀⠈⠙⠺⣟⡲⣦⡂⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠁⠀⢀⣽⡿⠛⠉⠈⠹⣆⠀⠀⠀⠹⡄⠙⢦⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⣸⣟⠂⠀⠀⠀⢠⡿⠀⠀⠀⠀⡇⠈⠈⢷⣁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠⡋⠉⠙⢻⠒⢤⣾⡟⠁⠀⠀⣠⠞⠀⢀⠀⢸⣿⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⢄⡈⢿⡻⠶⠃⠀⠃⢈⢿⠿⢷⣿⣅⡀⠀⠈⢀⣾⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠓⢦⢹⡿⢽⣶⣄⣀⠀⠈⠁⠀⠀⠀⠈⠉⠓⠶⢮⣈⡴⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠙⣿⠁⠄⠈⠹⣿⣭⡒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⣦⡀⠀⠀⠻⣷⡙⢦⡉⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣀⠀⠀⠀⠀⠀⢈⡄⠀⠈⡆⠉⢳⠢⠄⡘⢷⡼⠟⠋⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢤⣝⢦⠈⠳⢤⣀⠀⠀⠀⢇⡀⠀⢱⣀⣨⠦⠖⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡀⠀⠀⠈⠓⢄⣀⠼⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠶⠶⠚⠋⠉
⠀⠀⠀⠀⠀⢀⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠄⠀⠀⠀⠀⠀⠀⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⡗⣊⣉⣦⢠⠤⠤⠤⠀⠛⠚⠉⠙⠛⠓⠒⠠⠤⠤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣭⣭⡵⠤⠤⠤⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠐⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⡀⣆⠀⢰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠜⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂
⠀⠀⠀⢐⠀⠀⠈⡈⠉⠈⡀
⠀⠀⠀⢸⠀⠀⠀⡁⠀⠀⡁⠀⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢈
⠀⠀⠀⢸⡥⠤⠤⡿⢶⣏⡇⢠⢴⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣸⡀⢀⡦⣤⡀⢀⣯⠒⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡠⢖⡥⠁
⠀⠀⢠⠤⡬⡬⠴⠰⡠⡇⢸⠉⠀⠀⡍⡇⠀⠀⠀⠀⠀⢀⡠⠔⠈⡅⠐⠙⠿⠿⠯⠿⠛⠀⢀⡠⠥
⠀⠀⠠⢠⡜⠀⠪⠀⠄⡜⢼⠀⠂⠀⡃⠁⠀⠀⣀⠔⠊⠁⠀⠀⠈⠀⠀⠀⠋⠀⠀⠀⠈⠀⠀⠀⢀⡀
⠀⠀⢠⣨⣟⢄⣩⣨⣠⣌⣸⣀⣀⣀⣆⣇⠔⠊⠠⠇⡄⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠒⡆⠂⠀⠀⠀⠀⠙⡆⠀⠈⢤⡀⠀⠀⠁
⠀⠀⠆⠀⠀⠊⠁⠀⠀⠀⠐⠒⠀⠓⠂⠀⠒⠁

⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⣹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡄
⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣟⠦
⠀⠀⠀⠀⣸⣯⣿⣧⠀⠀⠀⠀⠀⠀⢴⣶⣾⣶⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠐⠂⢁⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣿⣟⣆⡀
⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣶⣷⣶⣷⣶⣿⣿⡆⠀⠀⠀⠀⠀⠂⠠⢀⡀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⠉⣽⣿⣿⣿⣿⠛⠄
⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣬⣟⣛⣛⣛⣫⣭⣾⣿⡆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣤⣿⣾⣿⣿⣿⣿⣇⡀
⠀⠀⠈⠉⠉⠹⠿⠉⠛⠃⠀⠀⠀⠉⠈⠛⠋⠿⠋⠈⠉⠈⠁⠀⠀⠈⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠏⠉⠉⠀⠀⠀⠀⠈⠉⠈⠉⠹⠟⠛⠿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠄
⠀⠀⠀⠀⠀⡞⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠴⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⣰⢀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⠁⠎⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣄
⠀⠀⠀⢸⣿⣿⣿⣿⡟⢹⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠏⣨⠆⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⢰⣦⣶⣦⣶⣦⡤⡀⠀⢸⣿⣿⣿⣿⡇⠈⡇
⠀⠀⣀⣀⣻⣿⣿⣿⣋⣁⠀⠀⢀⣀⣙⣿⣿⣿⣿⣟⣉⣁⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⢀⣈⣻⣿⣿⣿⣟⣓⡃⠀⢸⣿⣿⣿⣿⡷⠚⠁
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⢰⠆⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠃⠀⠀⠀⠀⠀⢤⢰⠀⠀⠀⠀⠀⠀⠀⠂⡘
⠀⢀⡎⠠⠀⢸⡁⠀⠀⠀⠀⠀⣄⠀⠺⢢⠚⠃⢀⡄⠀⠀⠀⠲⠀⣀⣀⠐⠃⠀⠀⠀⠠⠴⠀⣀⠄⠀⠀⠤⠀⠒⠂⠀⠀⠀⠈⡟⢻⠀⢐
⠀⠀⢶⣶⠟⡇⡏⠀⠀⠀⠀⠀⠘⡗⣬⣿⡹⡄⡯⠁⠀⢀⣐⠀⠃⢴⣄⠘⡁⢂⡀⠀⣰⣦⠀⣤⠟⠀⠚⡴⠂⠈⠀⠀⠀⠑⢘⣓⣘⡣⠊
⠀⠀⠨⣐⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠗⠉⠀⠀⠀⠀⠀⠠⠧⣙⣁⠴⠑⠂⠀⠈⠘⠁⠀⠀⠉⣀⡘⠁⠀⠀⠀⢠⠀⠀⠀⠏⠹⠄⠀⠀⢀⡀
⠀⠀⠀⢿⡵⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠘⠖⠀⠀⠀⡀⠀⠀⠀⠀⢭⡥⠀⠀⠀⠀⣡⢣⠀⠀⢀⣀⠀⢠⠌⠙⠆
⠀⠀⠀⣘⢠⠀⠀⠀⢀⡷⢯⡀⠀⠀⠀⢀⢀⠀⠀⣀⠤⠄⡄⡀⠀⠸⠗⠀⠀⡀⠠⠀⡁⠠⠀⠀⠀⢰⡆⠀⢀⠀⡘⠔⢥⡏⠀⠨⠍⠀⢰⢚⠀⠈⠀⠀⡀
⠀⠀⠀⢸⡖⠀⠠⣁⠁⣷⣯⡇⠀⠤⠀⠘⢉⠀⠮⢰⡶⠳⣶⠹⡄⠀⡂⠀⠀⠀⠛⠀⡅⠘⠃⠀⠀⠈⣷⠒⠋⠀⠀⣄⣢⠀⠀⢀⣆⠀⠀⠙⠶⣀⡈⠟⠁
⠀⠀⠀⢰⢺⠀⢄⡅⢰⢾⠿⡇⠤⠄⠀⠸⠚⠀⣇⢹⣦⣤⣟⢁⠃⠈⠋⠀⠠⠀⣀⠈⠂⢀⣀⠈⠀⠀⢸⠀⠀⠀⠀⠰⠖⠀⠀⠘⡞⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢬⡷⣾⠔⠡⠀⠀⢰⣠⠀⠀⠃⠠⡅⠄⠋⠀⠀⠀⠀⠀⠐⠁⠀⡀⠸⠁⠂⠀⠀⡟⠀⠀⠀⠀⠈⡏⠀⠀⢀⢦⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣴⠄⠀⢐⣸⠀⠐⠀⠀⢠⠀⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠰⠇⠀⠀⠀⡏⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠅⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢐⠀⠀⠀⠘⠿⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠁⠀⠀⠀⠘⠦⠆⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⡄⡄⠀⠀⠀⡁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⡆⠀⠀⢠⠉⠄⠀⠀⠀⠘
⠀⠀⠀⢸⢉⠀⠀⠀⠖⠐⢠⠁⠀⠀⠀⠀⠇⠁⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠷⡆⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢦⡆⠀⠀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢐⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⡞⠰⠀⢠⢠⠄⢄⠄⠀⠀⠀⠀⠀⠀⢠⢠⠄⠀⢀⠄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠄⠰⠒⠚⠀⠀⠀⠀⠀⠞⠀⠐⠀⠰⠠⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡆⢠⣿⠃⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣸⣿⣠⡴⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣦⣀⠀⢸⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠉⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠻⣷⣼⡇⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣤⢿⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢠⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠌⢿⣶⣶⣤⣤⣀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢀⣀⣴⣶⣿⣿⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠘⠛⠻⠿⢾⣷⣄⠀⠀⠀⠀⠀⠀⢠⣿⣼⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⣆⠀⠀⠀⠀⢐⣼⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡀⢰⣞⠋⢹⣇⠀⠠⠊⠛⣿⣁⡀⠴⠶⠆⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣮⡿⠏⠁⠈⠉⠀⠈⢠⣿⡟⢡⣴⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠈⢿⡟⠁⠀⠀⠀⣄⣠⣄⠀⠘⠋⢀⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠃⠀⠀⠀⠐⠛⠛⠛⠁⠀⠀⠚⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣈⣿⣿⣿⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣭⣄⣀⣀⡀⠀⠀⣀⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠻⣿⣇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡏⠀⢻⣿⣷⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣿⣿⡋⠀⢙⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⣻⡯⠀⠀⠈⣿⣿⣿⠿⣻⣿⠃⣾⣿⡻⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡋⠀⠘⠿⠀⠀⠀⢿⣿⣥⣖⣺⣿⣵⣿⣿⣷⣮⣿⣇⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡃⠀⠈⡓⠀⠀⠀⠚⢻⣿⣿⣿⠟⢳⣿⣿⣿⡿⢻⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⣁⡀⣀⣹⣿⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⢿⣿⠇⠈⡟⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⡿⡟⣿⢻⣿⣿⡿⠀⢟⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⣿⣿⡿⠸⢿⠁⠀⠁⠀⠈⠁⣿⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣛⣿⣿⠿⠰⢷⣶⣦⣤⡀⠀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⣿⣿⣷⣶⣾⣮⣿⣿⣿⣶⣿⣿⣷⣶⣯⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢿⣿⣿⣿⣿⣾⣝⣛⡛⠿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡋⢙⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⣀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣹⣿⣿⣿⣿⣯⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣽⣿⣷⣄⠀⠀⠀⠘⠛⠛⠛⠛⢿⠟⠛⢻⡟⠉⠉⠉⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣿⣿⣿⣿⢿⣿⣦⠀⠀⠀⠀⣀⣀⡀⣼⠀⠀⢸⡇⢀⣴⣿⡿⠟⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣷⣾⣿⣿⠀⢀⣴⣿⠟⣿⣿⣿⡀⠀⢸⣧⣾⣿⣿⣿⣦⣿⣿⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⡍⠙⡏⠉⠉⠀⣴⣿⣿⣷⣦⣾⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢦⣄⠀⠈⡇⠀⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣿⣷⣼⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣾⣿⣦⠀⣷⠀⢻⠀⣼⣿⣿⣿⣿⣯⡸⣿⣿⣿⣿⣿⡇⡿⠛⠛⣻⠛⠛⠛⡟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣙⣿⣟⣡⣿⣿⣷⡰⡇⠘⠀⠘⠛⠛⠛⠻⡿⠿⠿⣿⠿⠿⠟⢡⡇⠀⠀⡏⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⠉⠉⠀⣃⣄⣀⣾⠂⠀⠀⠀⠀⡇⠀⠀⡟⠀⠀⠠⢸⠇⠀⢸⠱⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠁⢣⠀⠙⡆⠀⠀⠀⢰⡇⠀⢠⡇⠀⢀⡀⣾⠀⢀⡏⠁⢀⣴⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⣴⣶⣦⡘⡆⠀⢸⡀⠀⠀⢸⠀⠀⠘⣡⡾⣿⣿⡆⠀⡼⠀⢠⣾⣟⣿⣅⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣿⣿⣷⠀⠈⡇⠀⠀⡞⠀⢰⣾⣿⣿⡿⣿⣿⡌⠁⠀⠘⣻⠛⠿⠿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⣿⠿⡟⠋⡇⠀⢸⣶⣶⡇⠀⢸⡟⣻⠛⢻⠿⢿⠇⠀⠀⢠⠇⡌⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡀⢹⡄⢹⢀⣿⣧⣿⣿⠆⢸⠁⠇⠀⣾⡴⠃⠀⠀⢠⠏⣸⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣄⢧⠘⢷⡘⡆⠀⡏⢸⠁⠀⣿⡜⠀⣴⠋⠀⠀⡠⣠⢳⣌⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣼⣟⠈⢆⠀⢳⣇⠀⡇⢸⠀⢠⡿⠁⢰⡏⠀⣠⢞⡴⠇⡁⠈⢳⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣆⠈⢦⠘⣆⠀⠙⣄⡷⢸⠀⡼⠁⢀⡞⠀⡪⣐⢩⠴⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣄⠀⠀⠈⠣⡈⠧⠈⢧⠀⠈⠃⠘⠋⠀⠀⠚⠋⠉⠈⠀⠀⣠⠔⠋⠀⠀⢠⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠆⠀⢸⣿⣷⡄⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠰⠀⠈⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⡀⠀⣼⣿⣿⣿⣿⣷⠀⢀⠀⠸⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁