Campfire Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡆⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠉⠀⢡⡀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⢫⠁⠀⣘⠶⣷⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠏⢀⡾⠾⠁⣠⣿⠿⣮⡓⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⣴⠻⢀⣠⠞⡽⠋⠀⡾⣿⢸⠟⣧⢻⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⢉⠻⠁⣼⡇⢸⠁⠀⠀⢹⡏⣠⡆⢸⣃⡁⣠⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢠⣌⠈⢳⡶⢼⣧⠀⠀⢰⡀⠘⠟⠉⢿⣏⠉⣧⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢠⠚⢠⢏⠄⢻⣇⠄⠹⠄⢀⣾⠛⣶⡄⠀⢀⡟⠼⢻⡇⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⣴⣧⡟⢷⣼⣮⠙⡆⣠⡟⠁⠀⣿⠇⠀⣿⢵⡆⢸⡇⣿⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⢸⡏⣼⣷⡀⠈⠻⠀⣤⣿⡄⠀⢨⣿⠻⣤⡀⠈⣿⣾⣥⠈⢹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢲⠧⣿⡈⠁⢰⡀⢸⠁⢈⠿⠀⠀⠹⡄⠙⡗⠠⠋⢰⡏⣼⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡈⢳⣌⠻⣄⢶⡇⡿⠆⠸⡄⠀⠀⠀⢁⡼⣇⠀⠀⡾⣿⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢮⡻⣄⠈⠛⢦⡙⠄⠰⡛⠀⠀⠀⠈⠀⠁⠠⠞⢛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠈⢻⣶⣥⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡻⠟⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⢢⡤⠔⠋⠈⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣤⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣊⣥⣤⡦⠞⠑⠂⠀⠀⠀⠰⠿⣶⣿⢷⠿⠿⣿⠋⠛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⢀⡤⠒⠚⠉⠁⠀⢠⣤⣶⣿⠷⢶⣿⣶⣶⡴⠶⠚⠋⠛⠒⠶⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣦⣶⣿⣟⡋⢙⣻⡁⣾⢛⣿⣿⢻⣿⣿⣴⣶⣦⣶⡷⡯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠙⠛⠻⠟⠛⠉⠳⠷⠞⠛⠈⠛⠃⠀⠀⠈⠉⠙⠛⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠚⡦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⣄⡀⠷⠀⠀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠉⠀⢺⡆⣴⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⡄⢂⢠⣦⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠠⠂⢀⡿⠀⠀⠀⡘⠃⠈⠀⢻⡆⢀⡡⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠁⣇⢺⡀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠈⠁⣀⠈⠀⠴⠁⠀⠀⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠄⠀⠠⠻⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠁⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠈⠀⢠⡟⠀⠀⠀⠀⠀⠀⣤⡴⣣⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠓⠄⠀⠀⢂⡴⠉⣴⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡷⠇⢀⣠⠏⢡⡎⢶⣄⠀⠀⢙⢦⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣴⣿⣿⣣⣷⣿⠕⣠⣻⣿⣄⢹⣄⠀⡼⢤⣙⠶⣤⣰⡟⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣿⠉⣿⣿⣿⣯⣾⣻⠥⣎⠉⠹⠥⢈⡽⠿⣦⡀⣀⣽⡷⠶⢄⣨⣯⣡⡀⠑
⠀⠀⠀⠀⠀⠀⠘⠛⢻⣋⣡⣌⣉⣙⠛⠿⠤⠿⠿⣠⠤⠾⠷⠦⠼⠿⠻⣉⡐⣢⡏⠥⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠈⠉⠉⠓⠒⠀⠶⠶⠶⠖⠒⠒⠌⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔⠀⢀⠀⠀⢰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠂⣸⠱⡄⠈⠈⢳⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠂⠀⢦⢄⡀⠂⠄⠀⣤⣀⠀⣠⠗⠊⢺⠀⠀⠌⡢⡈⠀⠀⡰⡅⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⣇⠉⢦⠀⠀⢸⠈⣶⠇⠐⠃⢸⠀⠊⣠⠥⢹⠀⠀⠁⢃⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡧⣒⠀⠀⣼⠀⠈⢇⠀⢸⠁⠆⠌⠀⠆⢸⠀⡰⠁⠀⢀⠁⣰⢀⠌⠀⢱
⠀⠀⠀⠀⠀⠀⠈⠀⠱⣌⢉⣳⣿⠂⢀⡈⠢⣜⠘⢄⠀⠀⠠⠈⢶⢃⠀⠀⣜⡴⡽⠀⢀⢦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⢧⡀⠤⢀⡈⠂⢈⠀⠀⠀⠀⠀⠡⠀⠀⠀⠐⠟⡞⢁⠔⠉⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⢱⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠈⡤⢰⡔⠁⠔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢵⣸⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠰⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢠⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠁
⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⢀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢉⣝⠁⠀⢀⡤⡤⢄
⠀⠀⠀⠀⠀⢻⣡⣀⣠⣠⡤⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠤⠶⠲⠶⠶⣿⣿
⠀⠀⠀⠀⢀⡏⠉⠉⠀⢀⠀⢀⠖⡎⠙⢦⡀⠠⣴⠅⠂⠺⡦⣷⡄⠀⠈⠉⠩⠉⠒⢦⡄⠀⠀⣸⢉⣸⡆⠀⠀⠀⡀
⠀⠀⠐⠒⠤⠆⢾⡆⠀⣉⣨⣿⡀⠁⠀⡂⡅⢀⡷⠀⠀⢵⠚⠉⢷⡐⠷⣤⣀⡂⠀⣾⡷⠛⠛⡿⢎⡼⠁⠁⠀⠁⠈
⠀⠀⠀⠀⢀⡘⠶⠓⠛⠉⠉⠉⢻⣀⣢⣴⡧⠛⢛⣉⠠⠛⣢⡒⠛⢦⣰⠀⠹⡁⠿⡿⠷⠤⣤⣶⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠹⠷⠶⠿⠁⠀⠀⠈⠙⠲⠆⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠁⠈⢻⠀⠀⠀⣤⡀⠀⠀⢸⠋⠀⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢎⣦⡴⠋⠀⠀⣰⣿⣷⡄⠀⠈⠳⣤⣎⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠁⠀⠀⢀⣼⣿⣿⣿⣿⡄⠀⠀⠀⠁⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⢻⣿⣷⣿⡄⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠈⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠍⠙⠚⠻⢦⣄⠀⢀⣠⠾⠟⢊⠭⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⣅⡀⠀⠄⠐⠩⢝⠲⠥⣀⣤⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠉⡙⠓⣶⣤⣄⡀⠀⢀⠀⠈⣙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣷⣾⡿⠟⠛⠉⠙⠻⠶⣭⣵⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠔⠁⠀⠀⠀⠘⡄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠣⡀⠀⠀⠀⠀⠀⠘⢆⠀⠣⡀⠀⠰⠀⠇⠀⠂⠀⡁⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢀⠜⢠⡀⠠⠀⠘⠀⠈⢣⠀⠈⠢⣄⠀⠀⠀⠀⠀⠣⡈⠢⡀⠀⠃⢠⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣆⠈⠢⠀⠀⠀⠀⠀⢹⠀⢡⠀⠀⠀⠈⠁⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣸⠻⣆⠀⠀⠀⠀⠀⢀⠎⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠑⠄⠀⠀⠠⣤⣀⠀⠀⠀⢀⣴⠏⡪⢚⣇⠀⠀⠀⠀⢈⠀⠃⠀⠀⠀⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠳⡄⢠⢛⠅⠋⠈⠡⢺⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⡰⢀⠎⠀⠊⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢠⠀⠀⠀⠻⡳⣄⠀⠀⢸⡗⠹⣗⠅⡠⣾⠀⡪⠼⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⠀⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢇⠘⣄⠀⠀⠀⢻⠝⠆⠀⢰⡫⠂⠁⡜⠁⠺⠈⠕⡃⠀⠀⣸⡄⠀⠀⢀⣾⡇⠀⢠⠀⠳⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢎⢢⠀⠀⡖⠐⣺⡄⢸⢗⠁⡼⠀⠀⠸⡀⠪⠅⠀⢠⠖⡹⡄⠀⣸⣻⠁⢀⢎⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⡰⠁⢰⡫⢊⡀⢷⣼⡻⢀⠃⠀⡀⠀⢣⠘⣧⠀⡾⡊⣣⢾⠀⡿⣺⣴⢟⠅⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡛⠔⡬⠱⡄⢙⡞⢸⠀⠀⢀⠀⠀⢣⠘⣶⡋⢰⡀⠹⢠⡋⠞⢑⡡⢊⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⠁⡆⠀⠈⠳⣌⢸⠀⠀⠀⡠⠀⠈⣇⠹⠃⣼⡀⣽⡿⢋⡄⡩⣪⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⡵⠀⡃⠀⠀⠀⠈⢾⡄⠀⢀⠔⠁⠀⢩⢈⠞⡸⢸⡿⣠⢺⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢸⡵⣿⡄⣇⠀⠀⠂⠀⠈⠃⠀⠀⡫⠂⠀⢸⡏⠀⣆⢀⡔⠁⢸⠨⣧⢔⡽⠛⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢦⡼⠆⠀⠐⡡⠀⠀⠀⠠⢖⠄⠀⠈⠁⡆⠈⠁⠠⠂⢸⠐⢱⠎⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⢀⣀⣀⣀⣉⣲⣌⠦⠀⠈⠔⡡⠀⠀⠀⠁⡁⠀⠀⡬⠂⠀⠀⠚⠀⡼⢸⡇⣀⣀⡀⠀⠀⢀⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣔⣍⢈⡶⢳⡲⡶⠈⢣⠀⢀⢀⡠⢄⣶⠤⡀⠀⢀⢀⠈⠀⠐⣲⢖⡺⣅⣞⣙⠀⠀⢳⠀⠠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⠒⣀⢹⢟⠕⠜⡰⣁⠄⡏⡐⠕⠇⣉⣘⣆⡤⠣⢪⢀⢀⡧⡜⣈⣸⡣⠡⠣⢢⢸⡤⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⢋⡹⠷⣤⡷⢾⡿⠾⢛⠋⠝⢛⠳⢤⣲⠈⠡⢂⡿⠐⡛⠿⣚⠾⣕⣒⣹⡟⠿⠻⠟⠛⠿⠊⠕⠂⠝⠂⠭⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠉⠀⠒⠁⠠⠛⠉⠵⠊⠁⠀⠉⠔⠊⠽⠋⠡⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⠃⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⣿⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠸⡇⠀⠀⠹⣧⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡿⠀⣿⠀⠀⠀⢹⣇⠀⣾⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⡇⠀⢹⡇⠀⠀⠀⢻⡄⢻⡀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠀⠀⠀⠀⣿⠀⠘⣷⠀⢸⡇⠀⠀⠀⠘⣧⠘⣧⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢻⣆⠀⠀⠀⢹⡀⠀⠙⠷⠾⠃⠀⠀⠀⠀⣿⠀⢸⡇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⡇⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡿⠁⠀⢸⡇⠀⣠⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣿⠀⢸⣷⣄⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣼⠋⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠈⣿⠙⣷⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣿⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢸⡏⠀⢠⣿⠀⢸⡇⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣸⡇⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⢸⡇⠀⠀⠈⣷⣤⣾⠃⠀⠸⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠟⠁⠀⠀⣿⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠿⣯⣅⡀⠈⠻⣦⠀⢸⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⣴⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⢹⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣾⠋⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⢸⡇⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⢺⣿⡇⠀⣸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢷⡀⢀⣷⡄⠀⢸⡇⠿⠾⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⣧⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠻⠾⠋⣿⠀⣼⠃⠀⠀⠀⣿⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡌⣿⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⣧⣼⠏⠀⠀⠀⠘⣷⣼⢸⡇⠀⠀⠀⠀⡀⠀⠀⢻⣿⡟⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢸⣷⣦⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣸⡇⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⠹⣧⡀⠀⣀⣤⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣀⣤⡾⠋⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡇⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⣤⣤⣤⣤⣤⣀⣀⣘⣿⣄⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣣⣴⠂⠀⠀⠀⠀⠀⣠⣿⣃⣤⣤⣤⣤⣤⣤⣤
⣼⡯⠒⢿⣆⣀⣈⣉⠉⠉⠻⣷⣄⠀⠀⠀⠈⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣽⠏⠀⠀⠀⢀⣤⡾⠛⠉⠉⠉⠀⠀⠀⣿⠀⠘
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠙⠻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠟⠁⣀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠰⡇
⢻⣆⠀⣾⠁⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣷⣦⣤⣽⣿⣷⣦⣤⣤⣀⣠⣠⣤⣤⣶⣾⣿⣿⣥⡶⣾⡿⠿⠿⠶⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⠐⣷⠢⣺
⠀⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⣀⣤⡶⠟⠋⠀⠀⠀⠈⢉⡽⢿⡿⠛⠉⠛⠙⣿⠉⠙⢧⡈⠀⡀⠀⠈⠛⠷⣤⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⣠⠔⠀⣠⡾⠋⠁⣼⡇⠀⠀⠀⠀⣿⠀⠀⠈⠻⣦⡈⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣟⣁⠄⠀⠀⠀⠔⠉⠀⣠⡾⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⡀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⣾⢏⡥⠖⠠⠈⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢸⡏⠀⠀⣠⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢷⣄⢀⠀⠀⠈⢳⠞⣉⣉⡻⢷⣄
⠀⠀⠀⠀⢸⣿⠸⠀⠀⠀⠈⢘⡇⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⣿⠃⡀⡀⠃⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣰⡏⠓⠀⠀⠙⣎⣿⡄
⠀⠀⠀⠀⠈⢿⣄⠁⢰⠀⠀⣸⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣟⣠⣧⣥⣤⣄⣀⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⡀⠀⠀⠀⠀⢉⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠷⢾⡷⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣉⠢⠀⠚⠀⠙⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣾⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⣓⠀⠀⢈⣩⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠘⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢿⣇⠀⠀⢻⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⢿⣷⣴⣿⡟⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢿⣧⠀⠀⠈⠉⠉⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⢀⣶⠀⢸⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⣿⡇⢸⣧⣄⣠⡾⣻⠀⢸⣿⣿⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢸⣿⠀⠀⠀⢀⣿⠇⢸⡇⠈⠀⢠⡏⢠⣿⢿⣿⠃⡀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⢻⣧⣀⣠⣾⡟⢀⡾⠀⠀⢠⡟⢠⣿⡏⣼⡟⢸⣿⡄⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⠙⠛⠛⠉⣠⡾⠁⠀⠀⢸⡅⠈⠻⠿⠟⢀⡾⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣟⠳⠶⠶⠞⠉⠀⠀⠀⠀⠘⣧⡀⠀⠀⣠⡾⠁⠀⣽⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠹⣆⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠉⠛⠋⠉⠀⠀⢀⡿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠉⢀⣠⣤⠶⠲⣾⠋⠉⣿⣆⠀⠀⠀⠀⠀⢀⣼⠃⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⢛⣉⣁⣤⡶⢻⢿⡀⢀⠇⣿⠻⡓⡶⢦⣤⣈⡁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣟⣛⣉⣉⠉⠑⠄⣰⢟⡱⢣⢾⣇⣎⠎⢹⡷⣷⣮⣲⣤⣈⣉⠛⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣏⡉⠉⠉⢱⢨⢂⣼⣾⣭⣕⡊⠁⣰⣿⡟⡠⠚⣇⠈⢙⣦⣍⣉⣛⣛⣛⠛⣿⠟⠛⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⡆⢀⣾⣷⣾⡿⠒⠉⠀⢹⣼⠃⢻⣷⠁⠀⢻⡀⡏⢀⡽⠿⣶⣤⣄⣸⡇⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠲⠟⠛⢡⡾⠗⠂⠉⠉⢢⡾⠁⠀⢸⣿⣤⠤⣼⡿⣧⢸⠀⣀⣬⣷⣄⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠓⢦⣤⡞⠁⠀⠀⠘⣿⠁⠀⠈⣷⠈⢿⣤⡏⣀⡼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⠴⠞⠛⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠀⠀⠙⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡇⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠀⢸⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠘⢼⠄⡸⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠁⠋⢸⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⢠⡼⠀⠀⠀⠀⢀⠀⡄⠂⠀⠀⠈⠀⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠁⠀⠀⠀⢰⡋⠁⠀⢀⡦⠀⢸⣿⡟⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⠀⠀⣴⣒⡶⣛⣷⡀⣰⣟⠷⣄⢈⠽⣯⣑⡦⣀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⣀⣀⡥⢛⡡⣠⠖⢛⣻⣿⣿⡀⠈⠻⡾⠉⠉⠙⣦⣙⠢⣄⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⢿⣧⣶⣿⣿⠃⠀⠸⠇⢠⣼⣷⣀⠀⣌⠻⣿⣿⣿⣿⣦⣶⣟⠓⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠦⠀⠀⠀⠉⠹⠿⠿⠷⢤⡴⣿⣾⣿⣿⣿⣧⣹⣴⣿⡏⠉⠁⠀⠀⢁⡀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣴⠆⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁⠀⠀⢰⣿⣿⠀⣿⣿⣿⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣇⠀⠀⠀⢾⣿⣿⠀⣿⣿⣿⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡃⣿⠀⢸⣦⢸⣿⣿⡇⢻⣿⣿⠿⣿⡟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡟⢀⣿⣿⠀⣿⢸⠃⢸⣿⡏⢠⡟⣴⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⣾⣿⠃⣸⢣⡏⠀⢸⣿⢠⠏⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⣰⠏⣼⠀⠀⣼⠏⣾⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡿⠋⠉⠛⠄⠀⣿⣀⡼⠋⠀⠙⠦⠞⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢸⣿⠃⠀⠀⠀⠀⢠⠟⠋⠀⠀⢳⡀⠀⠀⢀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⣶⡶⠶⢾⠌⠳⠼⣇⠀⠀⣴⡀⠀⠋⠀⠀⠀⣸⡎⡇⠀⢀⡾⠋⠉⠁⣀⣀⣀⡀⠀⠀⠹⡆
⠀⠀⠀⢀⡟⠀⣿⠀⠀⠀⠀⠀⠙⢦⡀⢻⡙⣦⡀⠀⠀⢠⣿⠇⣠⣴⡟⠓⠒⠂⣠⡤⠤⠤⠄⠀⠈⠙⡇
⠀⠀⠀⢸⡇⡀⣿⠀⠁⢙⣶⣶⣶⣶⣿⣶⠿⠿⣧⠀⡴⠛⡿⠻⠉⠙⠻⠟⠻⠿⣿⣿⣿⣿⣶⣦⣤⡼⠁
⠀⠀⠀⠀⣷⣴⣿⠶⠚⠉⠁⠐⠲⠿⠛⠋⠀⠀⠈⣿⡿⣦⣀⣔⡢⢄⡀⠀⠀⠀⠀⢉⣻⠶⢤⡀
⠀⠀⠀⣼⠁⠀⢈⢳⡀⠠⠤⠖⠐⠀⠀⢤⣖⣂⡤⣿⣿⡂⡙⠻⢿⡳⢤⡀⠈⢱⣶⠏⠁⡀⢲⣽⡄
⠀⠀⠀⡇⣆⠄⠇⡇⣧⠀⠀⠀⣀⡶⢿⣿⡿⠿⠚⠁⠀⠉⢻⠰⣶⣌⡑⣀⠀⢸⣧⠃⢘⡇⠀⣼
⠀⠀⠀⢻⣳⡑⠀⠃⡟⠀⣒⣫⠵⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⠈⠙⠳⢤⣼⡻⡶⠖⢄⡼⠃
⠀⠀⠀⠀⠻⢯⣤⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠚⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠠⢠⠥⢠⡀⠀⠀⠀⢀⡀⢐⠀⠀⠀⠀⠀⢇⣠⠆⠀⠀⠀⠠⣀⣸⣇⠀⠀⠂⡀⠀⢂⠎⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⡼⠻⠹⠀⠄⠀⠀⠐⠚⠛⠃⠀⠀⠀⠈⠉⠉⠐⠀⠀⠀⠀⠀⠀⠌⠒
⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⢤⣄
⠀⠀⠀⢠⢲⡀⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢉⡟⢂⣮⠖⣶⡖
⠀⠈⠸⠛⢲⠚⠑⠈⠄⠰⠀⠐⠁⠐⢄⠂⠀⠀⠐⣂⠁⠓⠥⠔⠀⣾⢉⠺⠁⠀⠀⢠⣴⣀⣀⣀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢄⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡈⠀⠀⠀⢠⠇⠀⡠⣠⠀⡠⢖⡨⢒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢰⢰⢠⣾⠀⢸⣄⠀⠀⠁⣎⣶⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⢢⠸⡜⡜⣿⡄⢆⠉⠳⣄⢂⠘⠯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⢀⠠⠌⡢⣧⡱⣼⡘⣿⣎⣇⡀⢹⡎⣇⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⢎⡦⢾⣜⢿⣿⣷⣘⣿⢸⣿⣿⢃⡟⠔⠀⣸⠀⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠙⡎⢻⣿⣿⣿⡇⣾⣟⡵⢋⣤⣢⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠇⠀⠀⠀⡀⢠⡇⢀⠻⣿⣿⣿⣿⣿⣴⣯⣍⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡛⢳⠶⡗⡤⢿⣦⣷⣿⣿⠿⠛⠛⠛⠉⠩⠑⠂⠫⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠺⠇⣸⠀⠈⠁⢂⡼⠛⠁⠁⠀⠄⠀⠀⠠⠆⠀⡀⢀⡞⡐⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠿⠧⠀⣶⠯⣥⣄⠀⠀⠀⢠⣞⣅⢠⠖⠶⠖⣾⣮⣧⣑⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⡖⠄⠀⣷⠀⠘⣮⡇⣀⠼⠛⠙⣯⣼⠀⠂⢠⡿⣿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⣈⣃⣠⣴⣿⠶⢦⣾⣟⣿⣆⠘⣀⣼⠟⠙⠏⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠚⠉⠶⠀⠀⠘⠋⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⡏⠀⠀⠀⠀⠀⠓⢻⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⡜⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡔⢹⡀⡇⠀⠀⠀⠀⡜⡇⠀⠀⠀⠀⠀⢇⣠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠁⠀⠀⠀⢸⠁⢱⠀⡀⠀⠀⠀⠘⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⣀⣊⠀⠀⠀⠀⠀⡜⠑⠃⠀⠈⠚⠹⡀⠀⠀⠀⠀⢱⢀⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠈⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠖⠇⠀⠀⠀⠀⠀⠀⠀⠸⠊⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠈⠒⠤⣀⣀⡠⠴⠊⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⣟⠒⠤⢄⣀⣀⠤⠤⢄⣀⣀⡠⠤⠒⣾⡵⢮⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡏⠏⡿⡀⠀⠀⣰⠁⡆⡞⢝⡟⣄⣀⡀⠀⡿⣝⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠯⠼⠒⠊⠉⠘⢄⠣⣉⡭⢃⠇⠀⠈⠉⠉⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⠀⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣀⡀⠈⢯⡝⣳⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣛⣳⣦⣌⠳⠃⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠹⣟⡯⢷⣚⣷⡀⠀⠀⠀⣰⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⡀⠀⣰⢿⣇⠀⠀⣿⠀⣮⡐⢊⣷⡀⠀⠰⡏⢞⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⠇⣰⡯⠥⢜⣦⢀⣿⠁⡇⢳⠐⣚⡇⠐⡶⣝⢦⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⣟⣠⡈⢗⣻⠾⠒⣸⠁⢸⡇⣲⡇⠀⣿⠽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠧⠹⡾⣟⠦⣘⠽⣰⢏⡯⢸⢠⣿⢁⡼⠗⢀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠰⣦⣄⡀⠀⢻⡍⢆⠈⢳⡇⢰⡃⡟⠬⣧⣞⡫⠁⠰⣲⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣳⠙⣞⣹⡄⢸⠿⢸⠰⡀⠁⡼⠂⡇⢨⠿⠵⠂⣐⡽⠃⣼⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⣿⣢⠷⣛⡉⡾⢘⠅⠀⠊⠀⢳⠈⢡⣶⣨⡟⢠⡀⠳⡚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣦⢻⣓⡜⢁⣵⠂⠀⠀⣰⡀⠳⡏⢻⢰⡇⢸⡽⣦⠹⠃⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⠾⠍⡜⠘⣇⠞⠀⣸⠅⠀⠀⠀⠀⢻⣀⠀⠸⡍⣻⣞⡫⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢉⡞⠀⠸⡏⠀⠀⣺⠀⠀⠀⣀⠀⠀⠡⡷⠀⢻⠉⠭⣲⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡒⡾⢀⡧⠀⠀⠀⠀⢀⠀⠀⠀⠐⡀⠀⠀⠀⠅⣸⣸⡟⠁⠘⠋
⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⡀⣯⡀⢷⠨⠯⠄⠀⠀⢀⡽⠀⠀⠀⠪⢆⠀⠀⠀⢠⣏⣾
⠀⠀⠀⠀⠀⡞⡅⡀⡀⠈⠉⠙⢷⠘⡆⠀⠁⠀⠀⣲⠍⠀⢀⡀⠘⡵⡃⠀⢀⣘⣧⢿⠑⠒⠒⠋⠉⠛⠒⡆
⠀⣀⣀⣀⡠⣗⡁⣇⣧⣷⣒⡒⠛⠉⠉⢱⣀⡀⢈⣴⣖⣋⡉⠙⠲⡌⠕⡾⠁⠈⠉⠉⠓⣶⢦⡆⣦⡆⣄⣿
⠉⠛⢛⡿⠿⣿⣻⣿⢯⠧⣖⢝⣆⢠⣄⣄⠀⡽⢁⣖⣒⡀⢍⢻⣧⣽⣎⣴⣰⡖⣤⡄⣾⣯⣧⣿⣿⣇⣏⡿⠮⠤⠤⠄
⠀⠀⠀⠀⠉⠁⢺⣏⣇⢧⣏⡼⣿⡾⣿⣿⣇⣇⢯⠯⣿⣳⢮⠇⣿⡿⠯⠽⢿⣇⣿⢸⣿⣧⣼⣿⡛⢻⠩⠭⠭⠭⠭⠭⠭
⠀⠀⠀⠀⠀⠭⢿⣛⣯⡿⠥⢾⠟⠻⣿⠟⢙⣟⠦⢍⣓⣒⣫⡾⠯⠍⠀⠀⠉⠭⢛⣾⣯⣭⣿⣷⣾⣦⣤⣀
⠀⠀⠀⠀⠀⠉⠩⠒⠲⠾⠟⠋⠉⠀⠀⠀⠀⠀⠬⠭⠭⠿⠭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠙⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠲⠀⠀⠀⠘⢀⠀⠀⠀⠆⠠
⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠐⠀⢉⠀⠈⡀⠀⠀⠀⠁⠀⠀⠀⠢⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡰⢄⠀⠌⠀⠀⡇⢀⠀⢰⡄⠀⠀⢀⡂⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠐⡀⠅⠈⢳⡀⠀⠀⡇⠈⢠⠇⢸⠀⢂⡜⠁⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⠦⣀⡅⠀⠀⠁⠀⠀⢰⠀⡌⠀⢸⡀⠀⠀⢀⠃⠀⠀⢈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠣⠀⠀⠀⠀⠀⠀⠑⡄⠀⠘⡄⠀⢀⠎⣄⣀⣬⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠠⠖⢀⠞⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠐⠀⡐⠀⢀⡀⠀⠀⠃⠂⢀⠈⠁⠀⢈⠸⠃⠔
⠀⠀⠀⠀⠀⢀⣰⡤⠤⠶⠲⢶⣀⠀⠀⠀⠀⡀⠀⠈⢁⠀⠀⡀⠈⠀⠀⠀⢀⣭⣤⣠⡀
⠀⠀⠀⠀⠀⢸⣇⣻⣿⡤⢎⡏⣁⣦⡴⢶⡊⣯⢉⡿⠻⢤⠉⢱⡄⠨⠅⡆⠀⢀⣬⢯⣳⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠛⠳⠚⠁⠘⠦⠴⠞⠙⠲⠼⠏⠉⠛⠚⠚⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢸⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡀⠀⠑⠀⠀⠀⢸⠃⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠐⠀⠀⠀⠀⢀⣴⢋⠀⡠⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠃⠀⠀⠀⢣⠸⠃⢂⠞⠐⠃⡸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠜⢆⠀⠘⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⢀⠤⠖⠒⠒⠒⢯⠀⠀⣠⣴⣕⣦⡀⠀⢠⢀⢳⠀⠀⠀⡇⡴⠋⣰⠋⠉⠑⣄⠤⣄⣀
⠀⠀⠀⢀⠤⠾⢄⠀⣠⣤⣤⣜⣁⣀⣿⣶⣶⣧⠝⠊⡀⠠⡂⠀⢀⣀⠸⠃⣤⣬⣷⡀⡀⠀⠀⢀⣌⣧⡀
⠀⠀⡴⠁⠀⠀⠸⣿⣷⣷⣿⣿⣿⡟⣫⡿⠋⣁⠀⠈⣁⡶⢀⠀⠠⡱⡅⢿⣿⣿⣶⣠⣼⣖⢤⠞⠉⠁⠉⠑⢆
⠀⠀⠹⣤⣀⣀⣠⣟⠈⠉⠙⢛⡿⢟⣧⡶⠾⢥⣾⡿⣸⣇⣾⣷⣶⣿⣝⢶⣿⣿⣿⣿⣿⣇⡰⣀⡀⣠⣂⣠⡼
⠀⠀⠀⢠⡏⠁⠀⠀⠉⢢⣶⣯⣶⠟⡥⠶⣿⣿⣿⣿⣿⡟⠉⢙⡿⠙⣿⣿⣷⣦⡿⣿⣿⡉⠣⠖⠉⠀⠈⢻⡄
⠀⡀⣤⢿⣷⡀⠀⢀⡠⠜⠛⠯⣥⢸⣿⡂⢙⣿⣿⣯⡿⠻⠉⠉⠁⠀⠙⣿⣎⡽⡠⠚⠛⠒⠒⢤⠀⠀⠀⣸⣇⡀
⠀⠀⠀⠀⠀⠉⢩⡏⠀⠀⠀⠀⠈⡷⢈⣛⡫⣜⣷⣼⣇⠀⠀⠀⢀⠤⠶⠾⠯⠮⡀⠀⠀⠀⠀⠘⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠷⠧⣤⣶⣤⡖⢉⠁⠀⠀⠀⠉⢏⡉⠡⠀⠐⠁⠀⠀⠀⠀⠀⠸⣤⣔⣤⣾⡷⠿⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡰⣼⣾⣄⠀⠀⢠⢏⢀⠀⠀⢠⡀⠀⠀⡀⣀⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷⠒⠙⠲⢶⠿⠖⠛⠛⠓⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣿⡄⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡇⢸⠟⢱⠜⣇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⠛⠁⡠⠁⠀⠈⠀⢹⡀⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠁⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣿⡷⢧⢀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣴⢪⠣⣿⡜⣿⠀⠀⠀⠀⠀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡲⠟⠁⠀⠁⡘⢧⢿⡀⢸⠀⠀⠀⡟⢟⢧⣼⡗⠀⡖⣢⡀
⠀⠀⠀⠀⠀⠀⠀⡔⣊⡀⠀⠀⢹⣷⠞⣱⣾⡷⠁⣰⡇⢸⠀⢻⣾⣄⠀⣴⡿⡀⠀⢙⢻⣼⠏⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣸⠟⠻⢮⣦⣤⡾⣣⡾⡿⣹⠁⢠⣿⠀⡇⠀⢿⣦⡌⡟⢻⣷⠸⣦⡀⣱⣭⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⣻⡿⠾⠷⢧⠤⠿⠾⢴⣧⠀⣸⣿⣷⣳⠀⠼⣷⠟⠋⠁⢉⠑⣤⠀⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⢐⣿⠀⠈⠀⠀⠀⠀⠀⢺⢻⡏⢀⣀⠀⠀⠀⠀⠘⡇⠐⠋⠁⠀⢸⣷⠊
⠀⠀⠀⠀⠀⠀⠑⢄⡀⢀⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⡴⢸⡇⠋⠀⠀⠀⠀⠀⡠⠓⠠⡤⣀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠉⠂⠦⠄⠠⠄⠀⠀⢰⡧⠹⢅⣀⣀⣀⣀⣀⡔⠀⠀⡜⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣦⠐⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⢸⢻⢰⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣼⠀⠀⠀⠀⠀⠀⠀⡾⢸⠀⢇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢻⠀⠀⠀⠀⠀⠀⣼⠃⣾⠀⠸⠄⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠁⢸⡆⠀⠀⠀⠀⣰⠋⢠⠉⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠀⠈⢷⣄⣀⣠⡾⠃⣠⠏⢸⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠀⡆⠀⢨⠉⠁⢀⡜⠁⢠⠃⠀⢀⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢤⣀⠀⠀⠀⠀⢹⣿⡀⡇⠀⠘⡆⠀⡜⠀⣰⠃⠀⠀⢸⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠳⣄⠀⠀⢸⡇⣇⠰⡄⠀⡇⠀⠃⣰⢃⣴⣿⠁⢸⠀⣼⠃⠀⠀⢰⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠆⢠⡀⢹⡆⠀⢸⡇⣿⠀⠸⠀⠃⣀⢠⣇⡞⢹⡇⠀⢸⠀⣿⠀⠀⠀⢸⡇⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⠆⠸⡃⠀⢿⣤⡾⠁⡿⠀⣠⠞⣻⠟⢸⣾⠁⠈⣧⠀⢸⠀⢿⡀⠀⢀⡾⢡⠃⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣼⠀⠀⢇⠀⠀⠀⠀⢠⢃⡾⠁⣼⠋⠀⠈⢿⠀⠀⠹⣆⠀⢃⠈⠛⠞⠛⠁⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⡇⢠⠀⠘⢆⡀⠀⠀⠸⢸⡇⢰⡏⠀⠀⠀⣼⠀⠀⠀⠹⣇⠘⡄⢀⡄⣰⡆⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⢸⠀⠀⠀⠙⢄⠀⠀⠈⣧⠈⢷⣄⠀⣠⡟⠀⠀⠀⠀⣿⠠⢃⡜⣰⢿⡇⠀⣿⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠈⡆⠀⣠⡄⠈⢣⠀⠀⠹⡆⠀⠉⠛⠉⠀⠀⠀⠀⣰⠇⡠⢊⡴⠋⢸⡇⢀⡏⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⣇⠀⢳⡀⣿⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠰⣯⣤⡶⠋⠀⠀⣿⠀⡸⠁⢿⡀⠀⣀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠹⡄⠈⣧⣻⠘⢧⣀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣰⠃⠀⠈⠙⡛⢉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠙⣄⠀⢿⣇⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠋⣀⣠⡀⠀⣠⠞⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⠳⣄⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⣛⣽⢟⡡⢚⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣌⡁⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋⢁⣠⡴⠛⠁⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⣴⠞⠛⠲⠶⣤⣀⡀⠀⠀⠙⠓⠦⣭⣛⠷⠶⠶⠶⠶⠶⠶⠶⠚⠛⣉⣁⣠⡤⠶⠛⣋⣁⣤⡤⠶⠚⠛⠉⠉⠹⣆
⠀⠀⠀⠀⢰⡏⠀⠒⠤⢀⡀⠈⠙⠻⠶⣤⣄⡀⠀⠉⠉⠛⠛⠛⠚⠚⠛⠛⠛⢋⣉⣥⡤⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢸⡇⠀⢀⡀⠀⠈⠙⠀⠀⠰⠤⣈⠉⠛⠶⢤⣤⣀⡀⣀⣠⡴⠞⠛⠉⠉⠀⢀⡀⠀⠀⠀⠉⣀⠠⠔⠚⠉⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠷⣤⣀⠀⠈⠁⠲⠦⢤⡀⠀⠀⠈⢃⣀⣤⠴⠟⠋⣉⠠⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣄⣀⣀⣀⣤⠶⠞⠋⠉⠀⠐⠚⠁⠀⠀⠀⢀⣀⣤⠄⠀⠀⠀⣀⣀⣤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠛⠉⡀⠠⠄⠀⠀⣀⠀⠀⣀⠤⠐⠊⠉⢁⣀⣤⠴⠶⠛⠉⠉⠛⠛⠶⢦⣤⣀⡀
⠀⠀⠀⢀⣠⣤⣴⠶⠛⠋⠁⠀⠠⠒⠋⣁⡤⠖⠊⠀⠀⠀⠈⣀⣠⡤⠶⠚⠋⠉⠀⢀⡀⠀⠈⠉⠓⠲⢄⡀⡀⠉⠙⠳⠶⢦⣄⣀⡀
⠀⠀⢠⡟⡁⢤⣉⠳⣤⠀⠀⠠⠐⠊⠉⣀⣀⣤⣀⣠⡴⠖⠛⠉⠙⠓⠶⢦⣤⣄⣀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⣠⠞⣉⣉⠙⠳⣦
⠀⠀⣿⢡⠛⣲⠈⡇⣽⠀⠀⡤⠔⢒⣩⡽⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢤⣄⣀⠀⠀⠤⢀⡀⠀⢰⡏⡘⣡⣬⣄⠀⢸⡇
⠀⢀⠿⣌⣳⡤⢚⣰⣯⡤⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣄⡁⣿⡀⢣⣙⢂⣨⣃⡼⠃
⠀⠠⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠦⠤⠴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⣦⢀⣷⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣶⣄⡀
⠀⠀⢀⣴⣦⣾⣿⣾⣿⡏⣾⣫⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣾⣿⡿⠟⢸⣷⣄⡀
⠀⠀⠁⠓⡾⣛⣿⢻⣿⣻⡿⣟⡿⣿⠽⠇⠀⠀⠀⠀⢀⣰⣦⣾⣿⣿⣿⣿⣋⣴⣾⣿⣹⣿⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠁⠈⠉⠀⣈⡉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣠⠢⣄⣾⠗⡙⢆⣠⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⢠⣾⠏⣰⣿⣏⡼⠁⠀⠛⢁⠈⠢⠀⠀⠀⠀⣠⣴⠆⣤⣀⣀⢀⣠⡴⣾⣁⠴⠃⠶⢤⡀
⠀⠀⠾⠛⡫⢺⠿⠙⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠛⠁⠀⠑⠍⠛⠀⠈⠀⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡀
⠀⠀⠀⢀⣠⣔⣹⣿⣮⣴⣿⡿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠟⠓⡢⣒⠆⢦⡀
⠀⠐⠚⠻⠿⢟⣿⠿⢛⡛⠉⠉⠭⠝⣛⡻⠷⠤⠄⠀⠀⡤⡖⠚⢦⡃⠁⠺⠥⠟⠠⠚⠿⡩⠒⠄
⠀⠀⠀⠀⠉⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣽⠏⣿⣿⣦⡀
⠀⠀⠀⠀⡠⠁⣠⣦⣞⣴⣤⣀⣠⠁⠀⠀⠀⠀⠀⠀⠀⢀⢠⣾⣷⡯⣿⣏⣸⣿⣿⡟⣻⣿⣦⣄⡀
⠀⢀⣴⠬⠖⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠂⠀⢠⣶⣿⣷⣾⣿⣷⣿⣿⣿⣿⣯⣾⠿⠿⠷⠿⠿⠶⠤
⠀⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡀⣰⣣⣱⣄⡴⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣅⣀⣀⣴
⠀⠀⢠⣴⣷⣼⣿⡿⠿⢿⡿⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⣴⡿⣿⡿⣬⣽⣿⣶⣶⣤⡄
⠀⠰⢋⣵⣦⣀⣠⣾⣷⣬⣤⣶⣬⣽⣮⡓⠀⠀⠀⢠⢾⡿⠿⣿⢛⣿⣷⡚⠁⠙⢾⣿⣇⣙⠿⢥⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⢀⣴⣿⣟⠃⠀⠀⠳⡄⠀⠀⠀⠀⢀⣴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠈⡳⣄⠀⠀⣠⣾⣿⣿⣿⠂⡤⠀⠀⠈⢦⡀⢀⣴⣿⡿⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⠁⠀⠙⣿⣷⣾⣿⣿⣿⣄⣱⠆⠀⠀⠀⠀⠈⠻⣿⣿⡻⣇⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⢈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⡌⠦⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⡥⠂⠀⠀⣠⣾⣿⡟⣿⣿⡇⠈⠁⣼⠀⠀⠀⠀⠀⠀⢀⠀⢤⣹⣦⣄⡀⠀⠀⠀⢀⡀⠈⠢⡀
⠀⠀⠀⠀⠚⠛⠛⠛⣛⠛⠛⣤⠐⠛⠛⠛⠐⠿⠿⠷⠾⠤⠶⠄⠀⠀⠀⠀⠀⣿⡇⠈⠙⠛⠛⠛⠃⣰⡆⠸⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⡀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⠉⠙⢦⣠⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⠀⠀⢀⣴⣿⣿⣿⠃⠀⠀⠀⠀⠙⢍⣉⣛⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣠⣤⣄⠀⣠⣾⣿⠛⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠑⢴⡛⠻⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣣⡄⣠⡾⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠙⢿⡟⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⣿⣿⡿⠁⠛⠁⠀⠀⠀⠀⠀⠀⣿⣷⣄⢀⡀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠟⣱⣿⠏⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣦⣷⣄⠀⢀⣈⠧⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠃⣾⣿⠃⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣦⣙⣻⣮⣶⣄⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⡉⢀⣤⣤⣥⣾⣿⣷⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣥⡙⠿⠦


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡇⠀⠈⢦⡀⠀⣀⣴⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⣠⣾⣿⡿⠛⠃⠀⠀⠀⠹⣾⣿⣿⣆⠙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣾⣿⠙⠷⣾⣿⣿⣿⠶⡆⠀⠀⠀⠀⠀⠈⢿⣿⢿⡀⠀⠳⣤⡞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⣿⣿⡿⠛⠀⠀⠙⣿⡿⣿⡀⢷⠀⠀⠀⠀⠀⠀⠈⠻⣷⡁⠀⠀⠙⣿⡆⠈⢷⣄
⠀⠀⠀⠀⠀⣠⡴⠛⣡⣾⠟⢛⠁⠀⠀⠀⠀⠈⠻⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠈⢻⣄⠀⠉⠳⣄
⠀⠀⠀⠀⠈⠑⢺⣿⣿⣿⣦⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠰⣄⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠸⠃⠠⡀⠀⠀⠀⢀⣴⠟⠀⠀⣀⠀⠀⠀⠀⠹⠻⡄⠀⠀⣰⡧⠀⠀⢋⡄⠀⠀⠀⠀⣠⠀⢼⠁
⠀⠀⠀⠀⣀⡄⠀⠀⠰⣿⠀⠀⠀⠀⠈⠀⠀⠠⣿⢧⡄⠀⠀⠙⠲⢀⣄⠀⠀⠀⠀⠀⠈⠙⠋⠄⠀⠈⢉⡀⠀⠀⠈⠛⠀⠀⠉
⠀⠀⠀⠛⠿⠄⠀⠀⠀⠀⠀⠀⠰⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠀⠀⠑⠀⠦


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣤⡀⠀⠀⠂⠀⠀⣰⣆⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣥⠙⠷⡄⠀⠀⠀⢼⡿⠃⠀⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡀⠀⠀⠀⠺⢿⣧⢰⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣴⣃⡀⠀⠘⠛⢿⣷⡀⠀⠀⠀⠀⠻⡦⠈⠙⠒⢀⣤⠀⠀⠀⠀⠀⣠⡆⠀⠈⠛⠻⠧⠀⠀⠀⢀⣼⡗⠆⠀⠀⠀⠀⣠⡆
⠀⠀⢞⣿⡿⠁⠀⠀⠀⠀⠙⢷⣤⣙⠓⠆⠀⣈⠁⠀⢠⣾⣯⡀⠀⠀⢠⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡉⠀⢦⠀⠀⠀⠀⠙⠷⡄
⠀⠀⠈⣿⣧⣄⠀⢠⡀⠀⠀⠀⠉⠀⠀⠀⠺⠷⠀⠀⠀⡉⠉⠈⠀⠀⠀⠀⠉⠉⣁⣴⣿⠃⠀⠀⠀⠻⠿⣷⣄⠈⠑⠄
⠀⠀⠀⠁⠙⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠘⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣼⡀⢀⣼⡿⠃⠀⣼⠃⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣿⣿⣦⣾⢿⣎⡄⠚⠁⡸⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⡀⠆⠰⡋⠙⣿⣿⣷⣀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⠟⠁⠀⠀⢁⣼⣿⣿⡧⣸⣿⣿⣿⣿⣇⢸⠃⠒⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⢀⠈⠀⠀⢸⣧⠎⠀⠀⠐⠀⣿⣿⣿⣇⣿⣿⣿⡿⠛⣿⠆⠀⠀⠨⣿⣿⣟⣻⣿⣿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⢃⣶⡟⠉⠀⠀⠀⠀⠀⠀⢺⣿⣿⡏⠀⠛⢿⣷⣾⠶⣁⠀⠀⠀⢘⣿⣿⣿⢿⣿⣿⣯⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢉⡸⠿⠃⠋⠉⠀⠀⠀⠀⠀⢰⣆⠀⢸⣿⡻⣦⣵⣤⣸⣤⡛⢿⡾⣀⢀⡀⢀⠙⢿⢹⡇⣩⠙⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠈⢰⠀⠀⡄⢀⣷⢀⣦⢸⡆⢸⣿⠀⣆⠙⠟⠀⠙⠠⠀⡈⠛⠀⠈⠁⢸⡇⠿⠷⠈⢼⠇⠿⠼⠱⠏⣰⡧⣀⣧⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⡆⣿⣦⡄⣿⣼⣿⠿⠟⠛⠋⠉⠙⠳⢿⣠⡄⠀⣄⡄⢰⣧⠸⠇⠸⢰⣸⠷⠂⠀⣤⢰⢀⡄⣠⠀⠀⠟⠐⠀⠉⠉⠃
⠀⠀⠀⠀⠀⠀⢀⡄⣾⣧⣿⣷⠿⠿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠿⢿⣾⣿⣼⡄⢉⠙⢩⡀⠀⢀⡿⠚⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢾⠇⠀⠀⣸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰

⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⡜⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣰⣿⡀⠀⠀⠀⣄⠀⠀⢀⠀⠀⢀⣾⡁⠀⠀⣄⠀⢸⣿⣇⠐⢀⠁⠀⠀⠀⠀⠀⢰⡇⣀⠀⠀⢀⣿⣿
⢀⣿⠃⠀⠀⠀⢠⠇⠀⣴⣏⠀⠀⠈⠛⢡⠀⢰⣿⠀⠈⠉⠙⢣⡀⡀⠀⣼⡄⠀⠀⣾⡧⠈⢁⠀⣴⣿⣏
⣿⣿⣖⠂⠀⠀⣼⠀⢁⣰⠉⠂⠀⠀⠄⣿⣦⣿⡗⢂⣷⢿⣿⣼⣠⣧⣶⣿⣳⣾⣤⣿⡄⣰⣼⣴⣾⣿⡿
⡟⠁⠈⢁⣀⠀⣼⣿⠂⠈⠀⠀⠀⠀⠀⡟⢿⣿⣷⣾⠉⠀⠀⠛⢻⠀⠉⢿⡿⠉⠋⠉⠛⠛⠛⢿⣿⢟⣷
⣊⡁⢦⠄⠁⠀⠈⠋⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣿⡿⢻
⠀⠁⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠙⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠓
⣠⢡⠌⠛⠀⠀⠀⠀⠀⠀⠀⣂⣠⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⢠⣠⡦
⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⢘⣿⣿⡿⠏⠉⠀⠀⠉⠉⠉⣉⠛⠻⢿⣿⣷⣾⣷⣶⣦⣤⣀⣹⣶⣶⣾⣯⣄
⣀⣀⣀⣀⣀⣠⣠⣿⣿⣷⣿⡿⠏⠁⢀⣀⡀⠀⠀⠀⠀⠈⠉⠁⣀⣦⣄⣀⣩⡉⠉⠉⠙⠙⠛⠛⠛⠻⣿
⣛⣛⣿⣿⣿⣿⣟⣛⣒⣀⣀⣀⣀⣒⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠚⣻⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⡾⣿⠃⠀⠹⢦⡀⢀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⡁⠀⠀⢷⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠔⢸⡇⣁⣹⡿⡏⠈⠀⠐⠀⠙⠃⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣧⠐⠀⢷⡄⠐⠺⠀⠀⠀⠀⠙⣄⠀⠀⠀⣀⡤⣴⠋⠀⠋⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢼⣿⡖⡠⠞⢣⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⠛⣶⠛⠀⠀⠀⠀⠀⠂⠤⠀⡀⠀⠀⠀⠀⠀⠑⠀⢰⡀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⢽⣿⣅⠀⠀⠀⠓⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣺⣇
⠀⠀⠀⠀⠀⠀⠩⣽⣿⣭⣄⣄⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢠⠀⠀⠀⣼⣇⠀⣘⢾⣿⡁⢀⢀
⠀⠀⠀⢠⣧⢄⣫⣶⣿⣟⡿⢿⣿⠿⠾⠷⠆⠄⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⣀⢀⣂⣼⣇⣼⣵⠌⠀⠧⢐⣿⣿⡀⠬⢭⣿⡖⠚⣀⣰⣀
⠀⠀⠀⠀⠀⣲⣽⣿⣛⣛⣃⢤⠖⢶⡶⠶⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⣾⣾⠿⠧⣽⣻⣿⣗⠀⠈⠉
⠀⠀⠀⠀⠀⣬⣭⣿⣿⣿⣴⠂⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠾⢧⣾⠿⣿⡂⠀⠘⠃
⠀⠀⠀⠀⠰⠶⠤⣄⠀⠈⠉⠐⠻⠆⠀⠀⣀⠀⡀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠚⠲⠆⠀⠀⠒⠂⠁⠀⠈⠠⠤⠷
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠓⠀⠛⠛⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠣⠶⠶⠠⠤⢠⣀⣀⣠⣄⡀
⠀⠀⠀⠀⠈⠉⠋⠽⠛⠉⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋

⠀⠀⠀⠀⢠⠆⠀⢀⠠⠤⢤⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠠⢀⠤⢲
⠀⢀⠀⠀⠀⠀⠀⢘⠉⠉⠉⡁⠀⠀⠀⠀⠀⢠⠕⠠⢀⡀⠀⠀⠀⠀⢠⠖⠉⠉⣐⣂⠤⠚⠚⠉⣰⠔⡇
⡠⠫⣻⠫⣀⠼⢟⣟⣀⣀⠀⠅⠀⠀⠀⠀⠀⠸⠀⠄⠀⠇⠀⠀⠀⠀⡄⠿⣴⡪⡽⢁⡠⢂⣲⡮⢔⡾⢃
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠧⠄⠀⠀⠀⠀⠀⠈⠲⢤⣉⣙⠚⠛⠕⢊⣉⣡⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⢀⣤⣄⡠⠊⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠰⢂
⠀⠀⠈⣷⠋⠀⢀⠀⠋⢂⠀⠀⠀⠀⢀⣀⠀⣦⣴⣋⣉⠖⢫⠀⢀⡀⠀⠀⠀⠀⢨⠛⠓⠚⠓⡀
⠀⠀⠐⡯⠒⠒⢒⠚⢓⢿⠇⠀⠀⠘⣄⣦⣏⣏⠁⠀⡘⡾⢞⢳⣿⡌⠀⠀⠀⠀⠸⣶⣦⣤⣾⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⣽⢧⢭⣋⠽⠶⢺⣭⣠⠵⢊⡇⠀⠀⠀⠀⠀⡆⠀⠀⢠
⠀⠀⠀⠱⣄⡀⢀⠀⣀⡜⠀⠀⠀⠈⢫⡑⠒⠦⠵⡀⢀⠾⠗⢚⡽⠁⠀⠀⠀⠀⠀⠇⠀⠀⠆
⠀⠀⠀⠀⠘⠛⠋⠛⠛⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠿⠿⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⡆
⠀⠀⣠⢴⠒⠒⢲⣤⠆⣰⡶⣢⠀⠀⠀⠀⠘⠣⣄⣀
⠀⡘⠁⡐⣰⡾⠋⡉⣞⠋⢞⠉⠀⠀⠀⠀⠀⡖⠋⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠢⢄
⢰⣷⣚⢿⣽⠀⠮⣀⠴⢀⣼⡆⠀⠀⠀⠀⠰⠛⠒⠒⠒⠚⢢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡁
⠘⣟⠻⠿⢶⣿⣶⡶⠿⠟⢻⠇⠀⠀⠀⠀⢸⠸⣄⠀⣀⠆⡀⠀⠀⠀⡤⠎⢠⡀⠁⠀⠀⠀⠀⢁⠀⠘⠄
⠀⠘⢿⣒⠤⠄⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠀⠑⢲⠛⡗⠊⠁⠀⠀⠀⠀⢇⠻⠿⠟⠀⠀⠀⠀⠰⠀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠉⠂⠀⡛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠒⢤⡀⠘⣜⣷⣦⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⡿⢶⣄⠀⠀⠀⠉⢦⡈⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⣀⣼⣿⡻⢇⢤⡀⠀⠀⠀⠹⣄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣯⣛⠇⠫⠻⠏⠁⠀⠹⢿⣶⣄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⡀⠀⠀⠠⡤⣤⣸⡟⠋⠉⠛⢯⠈⠳⣄⡄⡀⣼⣿⣿⡆⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠈⣡⣶⣺⡟⢶⣜⠦⣴⣿⡝⢯⣾⣺⣿⠇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⣴⡾⠟⣿⣿⠷⢋⣙⣷⠉⠿⠻⡾⠉⠉⠀⠀⢲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⢀⣨⡍⠀⠜⠁⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⡀⠀⣤⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡳⣄⠀⠈⢛⡞⠁⠀⠀⠀⠀⢠⠾⠋⡀⠄⣠⣾⣿⣬⣾⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⢠⡛⢦⣄⡀⠀⠏⠀⠀⠉⠀⣘⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⡀⢷⣄⠈⠙⠛⠓⠒⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠖⠢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀
⠀⠀⠀⢀⡴⠒⢋⠉⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠐⠀⠀⠠⢹⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⠄
⠀⠀⠀⡎⠤⠠⠂⠲⠅⠀⣧⠀⠀⠀⠀⠐⣿⠽⠯⠿⠽⢿⡇⠀⠀⠀⠀⠀⠐⣏⠴⣠⣄⣰⡿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸
⠀⠀⠀⢿⠀⠘⠢⢴⡁⢀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠒⢸⠀⠀⠀⠀⠀⠀⠀⠸⡆⠡⠈⣼⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣰⠁
⠀⠀⠀⠈⠓⠧⣤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠾⠃
⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢠⠖⡛⠉⠉⢉⡉⡙⢦⠀⠀⠀⠀⠀⠀⢀⣖⢒⣆⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠚⠓⠲⣄⠀⠀⠀⠀⠀⠀⣴⣞⣿⣿⣿⡶⡄
⠀⠀⠘⣎⡔⠜⠓⢼⠵⢈⠟⠀⠀⠀⠀⠀⢰⡉⠤⡠⠭⣿⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣘⡇⠀⠀⠀⠀⠀⢿⣯⢿⢹⢻⣽⠇
⠀⠀⠀⠘⣄⣸⡃⣭⣰⠏⠀⠀⠀⠀⠀⠀⢸⠄⣀⡼⡀⣼⠂⠀⠀⠀⠀⠀⣷⣦⣤⣼⣧⣤⣴⡇⠀⠀⠀⠀⠀⢻⠛⠛⠛⠛⡗
⠀⠀⠀⠀⠘⣏⡵⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡤⠜⠁⠀⠀⠀⠀⠀⠈⠲⣀⣓⠉⢀⢛⡸⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠿⠂⠀⠀⠀⠀⢠⠋⣿⢁⣾⡇⢳⠀⠀⠀⠀⠀⡶⢷⣾⡶⣄
⠀⠀⢠⢴⠞⠛⠛⠛⠻⢦⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⡜⠀⡟⠻⣴⡇⠘⡇⠀⠀⠀⠀⡏⢦⢻⣿⣿⣆
⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⢴⡖⠚⣛⣛⡒⢲⡆⠀⠀⠀⠀⠀⣧⠀⣫⡴⢛⡆⠀⡇⠀⠀⠀⠀⡇⠀⠱⡻⣿⣿⣄
⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⢸⣤⣀⣀⡀⣀⣸⠃⠀⠀⠀⠀⠀⢣⠀⢿⠙⢺⡇⢠⠃⠀⠀⠀⠀⡇⠀⠀⠙⢝⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡏⠛⠛⠻⠟⢻⠀⠀⠀⠀⠀⠀⠘⣄⣸⣞⢋⣧⡼⠀⠀⠀⠀⠀⠧⠤⣤⣄⣈⣮⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⡏⠉⢳⠏⠉⢷⠀⠀⠀⠀⠀⠀⠰⠀⠀⣀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢱
⠀⠀⠀⢠⣇⡀⣀⢆⣀⣾⠀⠀⠀⠀⢀⡤⠚⠉⠁⠆⠉⠑⢦⡀⠀⠀⠀⠀⣀⢔⠚⠿⠟⢷⢦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⡾⡛⠁⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⢸⡀⠀⣸
⠀⠀⠀⢸⡀⠀⠀⠀⠀⡞⠀⠀⠀⠸⣽⡀⠀⠀⠀⠀⠀⠀⢀⣞⡶⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠀⠸⡀⠀⡸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⠻⣖⡒⠒⣲⠾⠛⠁⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠓⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⢘⣆⢠⣾⠆⠛⠋⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⣾⣆⣀⣀⠀⠀⠀⠀⠸⠿⠃
⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⡞⣷⠀⠀⠀⢀⡤⡞⠿⠗⠀⠀⢀⡾⠿⠂⡼⠿⠆⠀⠀⣾⠁⠀⠀⠈⠙⠿⠟⠁⠀⠀⠀⠀⣀
⠀⠀⠀⢀⡴⣦⠀⠰⣞⡇⠀⠀⣵⠏⠀⠀⠀⠈⠙⠁⣤⠀⠀⠀⡜⠁⠀⡼⠁⠀⠀⠀⠠⡞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣹
⠀⠸⣿⠞⠚⠁⠀⠈⠃⠁⢀⡞⣽⠶⢶⡄⠀⢀⣤⣴⠃⠀⠀⠀⠉⠙⠚⠁⣀⠀⠀⠀⠀⠀⠀⡤⢄⣰⣩⠆⠀⠀⢠⣶⣦⠀⢰⣿
⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠸⣆⠧⢁⣸⠇⠀⠈⠙⠃⠀⡠⠤⡀⠀⠀⢀⣴⣏⠀⠀⢀⡤⣄⠀⠙⠊⠉⠀⠀⠀⣠⠏⠈⠁⠀⠈⠁
⠀⠀⠀⠯⠵⠫⠖⠃⠀⠀⠀⠈⣙⠛⠁⠀⢀⠀⠀⠀⠀⢻⠉⠀⠀⠀⠻⣴⠋⠀⠀⣸⣼⠃⠀⢀⣶⠀⠀⠀⣀⠀⢀⣴⣄⠀⢠⣖⡦
⡀⠀⠀⣀⠀⣀⡴⠛⢦⡀⠀⠀⠉⠋⠀⢠⠏⠉⢹⠃⠀⠸⠀⠀⢀⣀⠞⠁⠀⣰⢞⣽⣄⡀⠀⢘⣧⠀⠀⣼⠻⠃⡼⠉⠁⠀⠸⡇
⠃⠀⠀⠻⠟⢡⣤⣠⠜⠻⣄⠀⠀⠀⣀⡼⢀⣄⡟⠀⠀⠀⣠⣄⠻⡟⠀⠀⠰⡇⠏⠀⢨⡇⠀⠘⠀⠀⠰⢿⣗⣺⠃⠀⣼⠃⠀⠃⠠
⠃⠀⠀⠀⠀⠈⠛⡁⠀⣠⠞⠁⠀⠘⠿⠃⠈⠛⠁⠀⣴⡆⠻⣅⠀⣹⣆⠀⠀⢙⠲⠶⠛⠁⠀⡤⣤⠀⠀⢠⣄⠀⠀⣸⣧⠀⠀⠀⢰
⠀⣠⣖⠁⠀⠀⠘⠿⠋⢁⡀⢠⢞⡆⠀⠀⠀⠀⠀⠀⠹⣄⢀⡼⢾⡥⠞⠁⠀⠙⠓⠁⠀⡔⢠⡟⠁⠀⠀⢸⡸⠃⠀⠈⠉
⠈⠁⠋⣀⣀⠀⠀⠀⠀⠸⠿⠋⠉⠀⠀⠀⣠⢾⡅⠀⠀⠈⠯⠖⠋⠀⠀⠀⠀⢀⠀⠀⠀⠈⣉⠀⠀⠀⠰⣉⠇⠀⢠⠏⠙⢲⠆
⢀⡀⠈⠓⢮⣇⠀⠀⠀⠀⢀⣠⣴⣦⠀⠀⠀⠉⠀⢠⡤⠤⣤⠀⠀⠀⣀⣤⡞⠉⢳⡀⠀⠸⡯⠭⢽⠀⠀⢀⣤⣴⠋⣀⣠⠏
⣻⠏⠀⠀⠸⡿⠃⠀⠀⠐⠋⠀⠈⠉⢠⣶⡄⠀⠀⢸⠋⠉⣿⠀⠀⢺⡁⠀⣳⡀⢀⣷⡄⠀⣷⣦⢸⡇⠀⠀⠉⠁⠈⠛⠋⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⢠⣶⡆⢸⠉⠀⢀⣤⡿⠰⣿⠇⠀⠀⠀⢳⡐⠿⠃⠈⠛⠁⠀⢀⠀⠀⠿⠗⠀⠀⠀⠀⡞⣇⠀⠀⠉⠋
⠀⠀⡀⠀⠀⠀⠀⠘⡷⢄⠀⢸⡥⠤⣾⠀⠀⠀⠉⠁⠀⠀⣤⣆⠄⠀⠿⠗⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣤⡀⢈⠽⡄
⠀⣷⣿⠄⠀⠀⠀⠀⠛⠃⠀⠈⠛⠉⠉⠀⠀⠀⢀⣀⠀⠘⠱⠋⠀⠀⣤⡄⠀⠺⣝⣦⠀⢠⣶⡇⠀⢠⣶⡄⢸⠋⠁⠈⠊⠁⠀⣄
⠀⠉⠈⠀⠀⠀⢠⡒⡄⠀⠀⠀⢠⠖⣛⣛⡶⠾⡥⠾⠃⠀⠀⢠⣦⠀⡟⠁⠀⠀⠀⣿⡆⠀⠉⠀⠀⢸⣉⣤⣿⠀⠀⠀⠀⠀⣿⠋
⠀⠀⢀⡞⠀⠀⢸⡇⠀⠀⢀⣀⢿⠈⠁⢛⣷⠀⠀⠀⡄⠀⠀⢸⡁⠀⢻⠀⠀⠀⠀⠈⠀⠀⣤⣄⠀⠘⠒⠒⠋⠀⠀⠀⠀⠟⠙
⠐⣿⡟⠀⠀⠀⠘⠃⠀⠀⠈⠫⠘⠛⠲⠛⠁⠀⠀⣼⣥⡀⠀⠘⣧⠴⠞⠀⠀⣀⡀⠀⠀⠀⣿⣉⠀⠀⠀⢀⡄⠀⠀⢠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠛⠒⡆⠀⠀⠀⠀⠀⠀⠀⠸⢧⡞⠀⢠⣦⡀⠀⠀⠀⡴⠛⠛⠀⠀⠀⠸⠜⠀⠀⣰⡟⠁⠀⠀⣾⠀⠀⡇
⢠⣶⡄⠀⠀⠀⢰⠇⣀⣸⠃⠀⠀⢀⣀⠸⣿⠃⠀⠞⠀⠀⣀⡯⠇⠀⠀⠘⠁⠀⠀⠰⣖⢦⠀⠀⠀⠸⠉⠃⠀⠀⠀⢿⣶⠀⣇
⠋⠉⠁⠀⠀⠿⠟⠀⠻⠛⠀⣠⣤⠸⣟⠀⠘⣦⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⢀
⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠙⣏⠀⢈⣶⣚⡭⠇⠀⢠⠤⣄⡴⣻⠆⠀⢰⠯⣟⣷⠀⠈⠛⡁⠀⠀⢸⡭⠃⠀⠀⢀⡞⠓⢦⣤⣀⠈
⠀⢀⡤⠖⢿⠉⠈⢧⠀⠀⠀⠀⠈⢿⣭⠞⠉⠀⣀⠀⠀⠉⠁⠉⠁⢀⣀⡏⣠⣤⡏⠀⠀⣿⣿⠀⠀⠸⡇⠀⢠⣄⡾⢀⣄⡾⠀⢉⡟
⣀⣈⣧⣀⣈⣧⣀⣨⣇⣀⣀⣀⣀⣀⣀⣀⣀⣼⣻⣇⣀⣀⣀⣀⣀⣻⣟⣀⣈⣉⣀⣀⣀⣘⣀⣀⣀⣀⣁⣀⣈⣛⣁⣘⣛⣁⣀⣼⣁
⣿⣿⠋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀
⣷
⠀⢧⡀⠳⡄
⠀⠴⠇⠀⠙⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠙⠀⠀⠀⠈⠓⠢⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡀⢀⡞⠁⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠐⠂⢀⡄⠀⠈⠀⣤⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠤⠄⠀⣄⣀⣤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⢴⣶⣦⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣶⡶⢶⣶⣦⣴⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡷⡀⠙⣧⠀⠉⠈⠉⠉⠉⠉⠉⠑⠚⠛⠐⠀⠐⠂⠉⠀⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣉⠌⣻⣿⣿⡆⠈⢳⡀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠘⠛⠻⢿⣿⡀⠀⠻⡄⠀⠀⠀⠀⠉⠑⠂⠢⢤⢄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⠉⠓⠒⠤⢀⡄⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠰⢆⡦⠥⠤⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠤⢄⣀
⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠈⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠇⠀⠀⣼⣇⠀⠰⠀⠖⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢑⠚⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⣠⣠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠘⠃⢀⠀⠀⠰⣴⣶⣼⣷⣷⣿⡻⣿⣯⠀⠂⠶⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⡀⣴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠻⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠟⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠘⣆⠀⠀⣿⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡄⠀⠀⢀⠜⠀⠀⠀⣄⠀⠀⠀⠘⣆⢀⡇⠱⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠢⣄⠎⠀⠀⠀⢀⠛⡄⠀⠀⠀⠸⠞⠀⠀⢧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠘⡄⠀⠀⠀⠀⠀⠀⠘⣴⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣸⠀⠀⣄⠀⠀⠀⠀⢰⠃⠀⠀⢱⠀⠀⠀⢰⡆⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⠀⠀⡏⠱⣄⠀⠀⡎⠀⠀⠀⠈⡇⠀⡰⠃⢱⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⣄⡇⠀⠈⢣⡀⡃⠀⠀⠀⠀⣧⠊⠀⠀⢸⠀⣠⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⡏⠃⠀⠀⠀⠙⠇⠀⣷⠀⠀⠁⢀⣶⠀⠘⠋⢸⠟⡈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠹⡄⠀⠀⢶⣀⠀⣰⠁⢣⣠⠞⠁⢸⠀⠀⡰⠧⣪⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡹⡄⠀⢸⠈⠳⠇⠀⠘⠁⠀⢀⡼⢀⡴⢥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠻⠤⠬⠦⣤⠔⠒⡒⠒⠲⢾⠒⢭⡰⢄⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡴⠊⠀⠀⣠⠔⢑⣦⡈⠳⠄⠀⠀⠰⢍⠂⠉⠲⠌⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣷⡦⡜⢁⠄⣠⠔⠋⠀⣠⠵⠚⠉⠲⢄⡐⢦⡀⢄⠙⠆⡴⢊⠵⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣟⠷⡜⠀⡠⢖⡩⠖⠉⠀⠀⠀⠀⠀⠀⠉⠲⣌⠳⠁⡞⡰⣟⡼⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣒⣱⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣧⣠⠭⠒⠁


⣿⣿⢛⢿⢛⠟⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣿⠈⢷⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣴⣦⣀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣀⣀⣀⡁⢤⠀⡄⢠⠀⠀⠒⢠⣄⢠⠀⠂⠀⠀⠀⠀⣿⠀⢹⡆⠘⢧⡀⠀⠠⠛⠶⠀⠀⠀⠂⠀⠀⠀⣠⠶⣥⣛⠶⣮⠃⠀⠀⠀⠀⠀⠀⢀⣰⢶⣄
⠀⠀⠉⠙⠉⠁⠘⣄⠏⠸⠀⠀⠰⠶⠶⠶⠶⠛⠀⠀⠀⠀⠻⠇⠈⠁⠀⣰⠇⠀⠀⠤⠰⠶⠔⢠⠀⠀⠀⠰⢫⣾⢱⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⣾⡗⣬⠟
⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠦⠶⠞⠉⠀⠀⠀⠀⠀⠀⢄⠶⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠔⠢⡀
⠀⠀⠀⠸⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⣄⠾⠂⢠⠀⠀⠀⠷⣿⣷⣿⠿⠃⠀⠀⠀⠴⠀⠉⠒⢄⢀⡀⠀⠀⠀⢰⣶⣶⣶⣾⣧
⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⢠⠖⠐⣮⠙⠃⠀⠀⠀⠀⢷⡆⠉⠀⡠⠋⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⣀⣾⠃⣀⡀⠙⠷⠀⠀⠀⠈⠉⣙⠿⡇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢰⠈⠁⠈⢁⡀⠐⠀⠀⠀⢀⡀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠶⠀⠰⠏⠁⣀⣉⣁⡀⠀⠀⠀⠀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠉⠛⠉⠉⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠰⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠋⠉⠁⠂⠀⠀⠀⢠⡀⠀⢛⠿⠀⣀⠀⠀⠀⠀⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⣠⣄⢀⡀⠀⠀⢴⠁⠐⠀⠀⠀⠀⢸⡇⠀⠀⠙⠂⠀⠐⠀⠈⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⠀⢉⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠉⠁
⠀⠀⠀⠀⠀⢨⣯⡹⡭⠀⠀⠀⠁⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠟⡷
⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢐⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣦⠀⠀⠀⠀⠀⢀⡀⠲⡀⠚⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠛⠃⠀⡘⠛⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠀⠀⢁⡇⠀⠀⠀⠄⠘⠃⢀⡵⠚⠃⠀⠤⠀⠀⠈⠂⠀⠃⠀⠀⠀⡟⠻⠿⠿⠛⢋⢳⡆
⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⡄⠀⠀⣤⠀⠀⠀⠀⠀⢤⡄⠀⠠⢦⣤⠇⠀⠐⠲⠖⠲⠟⠁⠀⠀⠀⠀⠲⠐⠆⠀⢠⡤⠄⠀⠀⠀⠈⠢⠄⠀⠴⠋⠉
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⣮⠀⠀⠀⠀⠀⠈⢀⠀⠐⠀⠀⠀⠀⠀⢀⠄⠴⣿⡟⠀⠀⢀⠋⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠒⢠⡆⢠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠠⠀⢀⡤⠈⠀⠀⠀⠈⠀⣄⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⣄
⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠⠀⠀⠠⢾⣥⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠊⠀⠀⠀⠀⠀⠀⠉⠉⡆⠀⠀⠉⠀⠀⠀⠈⠀⠂⠰⣆⣶⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠀⠀⠘⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠊
⠀⠀⠀⠀⠸⠷⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢟⡻⢿⣧⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠶⠶⠀⠀⠰⠾⠦⠀⠀⠀⢰⠆⢀⡛⡄⠀⡀⠀⠀⠀⣠⣶⣶⣴⡆⠐⠓
⠀⠘⢶⣶⣄⣤⣴⣶⡞⠀⠀⠈⠛⢧⣼⣿⡶⠟⠀⠀⠀⡀⠈⠁⠈⠀⡀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠂⠀⠰⠃⣠⢻⡿⠓⢾⠉⢓⡄⠘⠿⠶⣤⡞
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠞⠃⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⢾⣤⠴⠛⠓⠋

⠀⠀⠀⢀⣤⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⢿⠆
⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠤⣴⡄⠀⠀⠀⠀⠘⠿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢷⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠖⢠⡀⠀⠀⠀⠀⠀⠂⠀⡈⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠄⢀⡤⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢲⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠓⠇⠀⠀⠄⠀⠀⠀⠙⠷⣶⣶⠶⠟⠁
⠀⠀⠀⠀⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠆⠈⠄⠀⠑⠄⠀⠀⠀⠀⠀⠀⣀⠀⠠⠔⠀⠀⠀⣤⣙⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣧⡄⠀⢀
⠀⠀⠀⠴⠆⠀⣤⣾⠃⠀⠀⠤⠤⠤⠌⠉⢫⠀⠀⠀⠀⣤⣹⣿⣧⣄⠉⠀⠀⠀⠀⠀⢠⡄⠐⢳⡄⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢠⡤⠲⡦⠴⠶⠀⠀⠂
⠀⠀⠀⠤⠄⠻⣄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⠋⣻⡑⢫⣧⠀⠀⠀⠀⠤⠸⠃⠶⠊⠀⠀⠀⠀⠿⠟⠀⣀⢠⣦⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠐⠒⠂⠐⠛⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠉⠀⠀⠀⠈⠤⢀⣀⣀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠉⠉⠉
⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡦⠄⠀⠀⠀⣀⠐⠂⠀⢠⡤⠀⠀⠀⠀⠠⠶⠶⠦⠀⢀⠀⠀⠀⠀⠛⠏⠀⠀⣤⢶⠒⣄
⠀⠀⠀⣤⣴⣶⡆⠀⠀⠀⠀⠰⠀⠀⠀⠹⠀⠀⠀⠀⣴⠿⠛⠉⠀⠀⠀⠘⠀⠀⠁⠀⣄⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡌⠄⠀⠀⠀⠁⠂⠂⠀⠐⠫⠜⠋
⠀⠀⠀⢉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠟⠃⠀⠀⠀⢈⣿⡷⠲⠶⠿⠓⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠐
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⡿⢿⣿⡿⠛⢿⡿⣿⡿⣿⠿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣥⣤⣼⣷⣠⣾⣠⣴⣷⣬⣴⣿⣥⣾⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠐⠀⠀⠀⠘⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠤⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⠂⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠁⠰⠀⠀⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡾⠛⢻⡆⠸⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠀⣴⣿⠷⠟⣀⡀⠀⠀⠀⣤⣤
⠀⠀⠀⢀⣴⣷⣄⠄⠀⣿⣶⡴⠛⠀⣠⡈⠰⣾⣥⢠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠆⣼⣿⣿⣶⠗⠀⠀⠀⠀⠀⢿⣟⡀
⠀⠘⠛⠻⠉⣀⣩⣶⣿⣀⣵⣦⡀⠔⠛⠑⠶⣦⢉⡿⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⣴⣿⢏⣤⣾⣿⣿⣿⡅⠘⢋⠀⠄⠀⠀⠀⠉⠠⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠂⠙⠋⠡⠟⠉⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠘⠛⣵⠞⢉⣤⣾⣿⠟⡿⢻⠟⣁⣼⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢻⡟⠁⢈⠀⠁⠀⠉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⡀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⣰⡟⢁⡄⠙⠛⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣥⠄⣴⣿⠟⠋⠀⠄⠀⠀⠈⢶⣾⡅⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⠟⣼⣿⣿⣿⠿⣦⠀⠀⠀⠀⠀⠛⢇⠀⠀⠻⡅⠀⠀⣸⣇⣰⣦⣀
⠀⠀⣀⠀⢀⣴⢂⡔⡿⢏⣴⣾⣿⣿⣿⠗⠀⡈⠁⠀⠀⢀⡀⠀⡈⢢⣄⡀⠀⡄⢰⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠚⠋⠀⡉⠉⠁⠠⣶⣿⡿⠋⢉⡿⠃⠀⢰⣧⠀⢰⡂⢸⣇⠀⣧⣰⡍⠉⣻⣿⣾⣋⡀⣾⣴⡷⠀⢀⡀⠀⠀⠀⢀⠔⠃⠀⠰
⠀⢠⡄⢀⣧⠀⠀⠺⠻⠋⣀⠀⠈⠀⣀⠀⣼⣿⣄⣼⣷⣾⣿⣿⣿⣿⠿⠾⠛⠋⡙⢛⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢤
⠀⣸⣿⣼⣿⣆⠀⢀⣆⣠⣿⣶⣧⣶⣿⣿⣿⣿⠟⠻⠟⡿⠛⠋⠁⠀⠀⠀⣼⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢷
⠚⠛⠛⠛⠛⠿⠦⣞⣉⡁⠀⠈⠠⠄⢛⠙⠋⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠎⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣤⣦⣀⣿⣄⠀⢰⣷⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠛⠛⠻⠿⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠢⠀⠀⠈⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠢⠄⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠐⠠⠄⣀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⣼⡇⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⢠⣿⣷⡀⠀⢀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣟⢷⡄⠘⢰⣿⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣿⠈⣿⡄⣿⣿⣿⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡟⢹⡆⢿⡇⢻⢻⡿⣷⡀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢠⣧⡿⢀⠈⠁⢸⣧⡾⢸⡇⣹⠇⡄⣸⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⢠⣾⡟⠁⠎⡄⡄⣾⠟⡁⠐⣣⣿⣾⣿⠟⢹⡇⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⣿⢀⣴⡿⠋⡤⢀⡼⠉⣰⢋⡞⢀⣾⡿⢁⠀⠠⢞⣾⠁⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣰⣿⣣⣿⡯⠂⠀⣠⡿⢁⢠⠇⡼⠀⢸⣿⣿⢋⣴⡶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⡟⠁⠀⢠⣿⠁⣼⠀⡀⠇⢰⠸⣿⣇⡿⢡⠞⢁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣿⣿⡟⠀⠀⠀⢸⣿⠀⢻⡄⡁⢀⠠⠀⠹⣿⣷⣿⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⣿⣿⠁⠈⢰⡰⡎⣿⣇⠀⡀⣧⠈⢦⠀⢠⠹⣿⣿⠈⡄⣠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣼⡿⣇⠀⠈⣇⠃⠈⢿⣆⠜⣿⡖⠘⣇⢸⣆⠻⣿⢀⣼⣿⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣇⣽⢦⡀⠸⣆⠀⢦⢹⡄⢸⣷⠀⣿⢸⣿⢸⣿⣾⣿⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡌⠻⣆⢠⡁⠀⠙⣷⣬⡁⢀⢸⡟⣸⡟⠘⣡⡿⠛⡙⢹⣿⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⠦⠐⠀⠙⣀⣦⡘⣿⢻⡌⣾⣷⡿⢁⡐⠉⢀⣊⣵⡿⠃⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣶⣵⡈⢿⡏⣿⣾⣿⣿⢋⡴⢋⣴⣾⠟⠛⠉⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣾⣿⣿⣿⣿⢱⣿⣷⣿⠋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠙⠛⠋⠙⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⣯⣓⢦⡀
⠀⠀⠀⢀⡴⢋⠡⠈⠄⡀⠍⠙⣦⠀⠀⠀⠀⠀⣰⣒⣚⣛⣛⣶⡀⠀⠀⠀⠀⠀⢸⡇⢸⡀⠀⠀⠀⠀⠀⣠⢞⣩⠴⣿⣼⣚⣍⣳⣽
⠀⠀⠀⢸⡧⠬⠤⠤⠤⠤⠤⠤⣼⠆⠀⠀⠀⠘⣧⡤⠤⠤⠤⣤⠇⠀⠀⠀⠀⠀⣞⡉⠉⣧⠀⠀⠀⠀⣼⢡⢏⠤⡅⡏⠿⢟⣫⡵⣆
⠀⠀⠀⢸⡗⠛⢒⣒⣛⣒⠒⠓⢺⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡟⠀⠀⠀⠀⠀⠸⡇⢹⡇⣸⠀⠀⠀⠀⣇⢸⠘⢒⠋⠷⢚⣭⡀⡇⣿
⠀⠀⠀⠀⢿⣙⣯⡉⠉⢉⣽⣋⡟⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡟⠉⠀⣿⠀⠀⠀⠀⠹⡜⢧⡻⠗⣡⣌⣋⡼⣡⠏
⠀⠀⠀⠀⠀⠉⠉⠙⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣸⠃⠀⠀⠀⠀⠀⢠⣇⣀⣀⣻⠀⠀⠀⠀⠀⠙⠦⣍⣛⣓⣛⣩⠖⠋
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⣠⣞⢻⡏⣿⢓⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢿⡳⣄
⠀⠀⠀⢸⠷⠾⠮⠷⠧⠾⠶⣿⠀⠀⠀⠀⠀⠀⣤⣤⣯⣯⣤⣄⠀⠀⠀⠀⠀⣠⠖⠋⡀⢀⠐⡈⠓⢦⡀⠀⠀⠀⠀⣰⢋⠀⡁⢈⠙⣿
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⢛⡧⠀⠀⠀⠙⣶⠓⡶⠚⠲⡖⢳⡞⠁⠀⠀⠀⠀⣏⠠⠀⠄⣠⡼⠁
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⣀⣿⣀⣀⣀⣀⣟⣋⣀⠀⠀⠀⠀⢸⡄⢹⠀⢸⠃⣼⠀⠀⠀⢀⣤⠞⣩⠷⠖⠛⠁
⠀⠀⠀⠀⠸⢤⡤⠤⠤⢤⠼⠀⠀⠀⠀⠈⠧⣄⣀⣀⣀⣀⣀⣀⣤⠟⠀⠀⠀⠀⠀⠳⣼⣧⡾⡴⠃⠀⠀⠀⠘⢦⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣉⣳⣆
⠀⠀⠀⠀⠀⢀⣀⡤⠞⣧⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣄⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⡄⠀⠀⠀⣸⣥⣤⣽⣧⣬⣧
⠀⠀⠀⠀⣰⠯⠤⠤⠤⠬⢷⡀⠀⠀⠀⣰⠋⠀⠀⠏⠀⠀⠟⠁⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣠⡏⠙⠓⢿⠀⠀⠀⠹⡞⠛⢚⣷⢶⡏
⠀⠀⠀⠀⡷⠶⣤⠴⢤⡴⢶⡇⠀⠀⠀⣟⣶⣒⣲⣒⣲⣒⣒⣖⣒⣶⣻⠀⠀⠀⠀⠀⢀⡞⠉⠙⠦⣄⣸⠁⠀⠀⠀⢻⣚⣩⡴⡾
⠀⠀⠀⠀⢹⡀⢸⠀⢸⠀⣸⠁⠀⠀⠀⣿⠦⠭⡽⠯⢥⡬⠵⠮⡭⠴⣻⠀⠀⢀⣠⢾⡉⠙⢦⡀⠀⢈⡗⠀⠀⠀⠀⠈⣏⣡⢴⠃
⠀⠀⠀⠀⠈⣧⣸⣄⣼⣠⡇⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠒⠚⠛⠛⠉⠁⠀⠀⠘⠲⢤⣷⣀⣠⠿⠖⠋⠀⠀⠀⠀⠀⠀⠹⣄⡟
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⢶⣤⡀⠀⠀⠀⠀⢀⣤⡀⠀⢀⣤⡀⠀⠀⠀⠀⢠⡤⠤⣤⢀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡿⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠛⣧⡾⡶⣴⢧⡀⠀⠀⣴⠋⣄⠙⣶⠋⣄⠙⣆⠀⠀⠀⠈⡏⠉⡇⠸⡷⢾⡇⠀⠀⠀⠀⠀⠀⣠⢞⡼⣳⠋⠈⢹⡄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⣿⠶⢿⣹⠋⠀⠀⠰⡇⢀⡼⠀⢸⢀⡞⠀⢸⠀⠀⠀⢀⡇⠸⣧⡞⠁⠈⢳⡆⠀⠀⢀⡴⢞⡵⣫⠟⠁⠀⠀⣸⠃
⠀⠀⠀⣴⠋⠀⠀⢀⡴⠃⣠⠞⠁⠀⠀⠀⠀⢳⡈⠃⣠⢾⡈⠇⣠⠟⠀⠀⠀⡏⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢿⢚⡿⠞⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠘⢦⣀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠙⠶⠃⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠈⠻⣄⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠲⠚⠓⠒⠶⠚⠃⠀⠀⠀⠀⠈⠉⠉

⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣻⣛⣿⣿⡿⣿⣿⣿⣿

⣞⣟⣉⣉⣉⣉⣉⠉⠉⠉⠙⣶⣶⡄
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⡏⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⣛⣙⣿⣻⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣟⠋⢹⣏⠉⢉⣿⠳⣮⣛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⣿⠀⣼⠃⣠⠞⠛⣯⡻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠤⠤⠤⠤⠼⠦⠼⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠧⠿⠶⠧⠾⣧⣤⡴⠛⣷⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣥⣤⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢦⣄⣀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢠⡞⠁⠙⣿⣦⡈⠙⢻⡿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⡴⠋⠀⠀⠀⠈⢧⣙⣶⢟⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣰⠎⠀⠀⠀⠀⠀⠀⠀⠻⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢿⣿⣿⣿⣿⡿⠋⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠛⠛⠛⠛⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣠⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⠀⠀⣿⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠉⠉⠉⠉⠀⠀⢹⡀⢀⣿⠀⠀⠉⠉⠉⠙⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣤⣄⣀⣀⡀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣀⣀⣀⣤⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉
⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠉⢹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣮⡻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠋⠀⠀⣠⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⡌⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⢀⣀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣯⣠⣶⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡁⢈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⣿⣿⣿⠟⠁
⠀⠀⠀⢀⣠⣤⣤⣤⣤⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣷⣴⣿⡿⠋
⠀⢀⣾⣿⡿⠿⠟⠿⢿⣿⣿⣿⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⣠⣾⣿⣿⣧⣀⣼⣿⡿⠋
⠀⠸⣿⣿⣄⡀⠀⠀⣀⣼⣿⡟⠀⢀⣴⣿⡿⠿⠿⠿⣿⣿⣿⡟⠁⢙⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠟⠁⢠⣿⣿⠋⠀⠀⣠⣾⣿⠟⢿⣿⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠁⣀⣠⣤⣴⣿⡿⠃⢀⣠⣾⣿⡟⠁⢀⣴⣿⣿⣿⡀⠀⠀⣴⣶⣾⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠟⠛⠉⠀⣴⣿⡿⠛⢿⣿⣶⣿⡿⠋⢸⣿⣷⠀⠀⣿⣿⡏⠉⢉⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⡇
⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⣠⣿⡿⠋⠀⠀⣸⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⣿⣿⡇
⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠟⠁⣀⣤⣾⣿⠟⠀⠀⠀⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢸⣿⡏⠀⠀⠀⠀⠰⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠛⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠘⣿⣷⠀⠀⠀⠀⠀⠈⠻⣿⣶⡄⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣿⣿⡇
⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⣀⣀⣀⡀⣿⣿⡇⠀⠀⠀⢀⣾⣿⡿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣄⣀⣀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⢺⣿⣯⡁⠀⠀⣉⣿⣿⠇⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣾⣿⡿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠈⠀⠈⠠⠀⠄⠠⠀⠀⠀⠀⢀⣤⡀⡀⠠⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⢢⡀⠈⠁⡀⠀⠈⠀⠠⡌⡙⢿⡛⢿⡀⢙⣷⠀⢰⣦⡼⢷⣀
⠀⠀⠐⡆⠀⠀⢨⠁⠀⠀⠀⣦⣦⣄⡀⠀⠀⠂⢥⠰⠌⡛⣁⠀⠼⠏⣾⡿⠛⠉⠀⠞
⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠰⠫⢻⢻⠛⢦⡣⠀⠀⠀⢡⠁⠀⠀⠀⢆⠋⠀⡀⠔⢐⣶⣤⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠡⡄⠎⠁⠀⡀⠸⠦⠀⠀⡄⠈⢃⠀⠂⢁⢣⠂⠁⠂⠀⣀
⠀⠀⠀⠂⣈⠀⠂⠀⠀⠀⠀⠀⠀⠢⠀⠘⠀⡀⠄⠋⠢⢠⠸⡦⠜⠀⠀⠀⠐⠃⠄⠀⠀⠈⣡⠆⢔⠀⠀⠂
⠀⠀⠀⡾⠁⠀⣴⡏⠐⠂⢤⣤⣀⣂⡙⠀⠴⣤⡈⠠⢖⠚⡠⠅⢂⠄⠀⠂⠨⠄⠄⣠⠔⢾⡇⢴⠂⠐
⠀⠀⠀⠀⢶⣤⣿⡇⠀⠀⠈⠛⢻⠋⠀⠄⠀⡀⠈⣂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠒⢺⠇
⠀⠀⠀⢀⣰⣿⡏⠀⡓⠊⠐⢀⣈⠀⠠⠎⠀⠀⠨⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠐⠀⠄⣁⣴⡿
⠀⠀⠀⠈⡿⠋⠀⠁⠀⠀⠀⢀⠉⢠⡄⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠙⠂⠰⢿⣽⠇
⠀⠀⠀⢷⠔⠀⢀⣤⣶⡎⣤⠜⠁⠀⢀⠄⣀⡂⣀⠀⠀⠒⠀⠀⠀⡀⠀⢀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠨⠄⠀⢸⣏⣌⣁⠙⢶⠀⠀⢸⠀⢇⠀⠀⠀⣠⠧⠀⠀⠐⢃⠀⠘⠠⣁⡀
⠀⠀⠐⠁⠀⠀⡀⠉⣱⠏⠀⠰⠀⠀⠈⡇⠈⠃⠌⠁⣀⠀⠀⠀⠐⣠⢦⡤⠚⠻⢟⡀⠖⡤⠀⠀⡠
⠀⠀⠀⠀⠐⢾⣶⣴⣿⠁⠀⠀⠀⣤⣴⣷⣄⡀⠀⠀⠅⠀⢂⣠⠶⢹⣥⣂⢐⡘⠃⠁⠾⠷⢂⣠⡀
⠀⠀⠀⠶⠀⠀⠻⣿⣃⣀⣀⣀⠈⢨⠟⠉⠉⠍⠈⠀⠸⢂⠯⠀⠩⠀⡟⠟⠅⠈⢢⠲⢟⠛⠋⠉⠀⣠⠀⡀
⠀⠀⠐⠀⢰⠀⠀⠈⡿⠏⠛⠉⠀⢀⡀⠀⡘⢆⠛⠠⠠⢹⣥⢤⠴⣎⠁⠈⢃⡔⠁⠁⡀⣘⡂⣴⡞⠛
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⣰⣾⣿⣗⠀⠄⠈⠦⠤⠀⣸⢕⠥⡊⠀⠑⡤⠋⣀⣤⠅⣬⣭⠃⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⡻⠷⠠⣤⣤⣤⣾⡃⠁⢀⠄⠠⡀⡏⠀⠀⠋⠀⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠛⠛⠉⠢⠁⡡⠠⢴⣶⣶⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠻⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⠥⠒⠀⠈⢉⡭⠙⠒⠢⠄⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⢀⠤⠰⠃⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡚⠉⠐⠉⡉⠀⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠿⠃⠀⠀⣀⠾⠁⠀⠀⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⡂⠔⠀⠠⠴⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠴⠶⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢀⣀⡀⣀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⣠⣤⣴⣶⠶⣿⣿⣿⣷⣦⣄⡀
⠀⠀⢠⡤⢴⣿⡻⠿⡿⠟⣿⠿⠙⣻⡿⠷⣤⣀⠀⠀⡀⣀⣀⣀⣀⣠⣤⣶⣶⣷⣿⣿⡏⣿⣏⣺⣗⣿⣿⣷⡿⢽⣿⡛⡛⣿⣽⡿⣦
⠀⠀⠘⠁⠈⠸⠾⠽⠿⠷⠞⠷⠶⠿⠦⠾⠙⠩⠿⠿⠛⠛⠻⠍⠙⠙⠛⠿⠋⠃⠑⠾⠿⠒⠛⠓⠊⠃⠀⠉⠑⠛⠀⠐⠋⠩⠭⠉⠋
⠀⠀⠀⣀⠀⣄⢀⣀⠸⠶⠀⠔⢠⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⣀⠠⡤⠤⠠⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣀⢀⢀⠀⠀⡀⢀⣀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠐⠀⠂⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠉⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣄⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣛⣛⠃
⠀⠀⠀⠀⠀⠀⡌⣁⣈⡈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⣟⣿⡳⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⣀⣀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⡷
⠀⠀⠀⠀⠀⠀⠈⣹⣿⡉⣀⣈⠿⠿⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢟⣇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠰⢒⣒⣒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⣍
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠘⣍⣉⣏⠌⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣿
⠀⠀⠀⠀⠀⠀⣧⡉⠐⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠆
⠀⠀⠀⠀⠀⠀⣬⣤⣤⣬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠂
⠀⠀⠀⠀⠀⠀⠙⢿⣿⢟⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠰⠾⠿⠖⠀⠻⠛⠁⠀⠀⠀⠀⠀⠀⠾⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠠⠾⠷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠈⢩⣭⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣶⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠰⣿⣿⡾⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⣰⣃⣠⡄
⠀⠀⠀⠀⠶⣿⡄⢈⣏⠰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠀⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣿⣿⡷⢄
⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠙⠋⠁

⣿⠿⠿⠿⠿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣻⣛⣿⣟⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡤⣤
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠲⡟⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡀⣤⣋⡷⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣤⠤⠞⣻⡥⣤⡷⣄⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⠦⣤⠴⠚⠉⠉⡇
⠀⠀⢀⡼⢣⡤⣼⣆⠀⠀⠙⠢⣄⣀⠀⠀⣀⠀⠀⠀⢸⣡⢇⣴⣏⣭⣷⢯⡽⠛⡆⠀⠀⠀⠀⠀⠸⣅⠀⠀⢀⡤⠶⠦⣄⠀⢳⣀
⠀⠐⠛⠲⡏⠀⠀⠉⠳⣄⡀⠀⠀⠈⠉⣻⣹⠀⠀⠀⢾⠻⠿⢾⣁⣄⣽⡞⠁⠀⢻⠀⠀⠀⠀⠀⠀⠈⡷⠀⡏⠀⠀⠀⢸⡆⠀⠈⢳⡀
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⠙⠛⠻⠿⣏⣁⡍⠀⠀⠀⠘⣆⠀⠐⠂⢠⠏⠐⠀⠀⠈⣇⠀⠀⠀⠀⣠⠞⠁⠀⠳⣄⣀⣀⡼⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠙⠦⢤⣀⣀⣀⣀⣀⣼⣁⡇⠀⠀⠀⠀⠘⢦⠀⢠⠏⠀⠀⠰⠀⠀⠸⡄⠀⠀⠀⡇⠀⠀⠀⠀⣀⣉⣁⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⢳⠇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠙⠲⠶⠒⠋⠉⠁⠉⠉⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡀⠀⠘⠀⠀⢂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠹⣄⡀⠀⣄⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡽⠛⠉⠁⠀⠀⠀⠀⠀⠀⠘⠧⣽⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⢠⡤⣼⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⡀⠀⠀⢠⡞⣠⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⢳⡼⠛⢦⡼⢾⠙⢶⣳
⠀⠀⠀⣠⠶⠶⠶⠖⠛⠛⠒⠒⠒⠒⣦⠀⠀⠀⠀⠀⣾⣛⣚⣛⣛⡇⠀⠘⣇⢳⢀⣤⣄⣴⡿⠋⠀⠀⢸⣛⡛⠳⣴⡋⡇⣾⢧⠈⢻⡀
⠀⠀⠀⠙⣟⠉⠉⠉⠉⠉⠉⠉⠉⡿⠉⠀⠀⠀⠀⣠⠟⠉⠉⠉⠉⠓⢦⠀⠟⣀⣻⡀⠈⢿⠄⠀⠀⠀⠈⣠⣸⣳⡀⠙⣧⡏⣸⣇⣠⣷⠆
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡞⠉⠙⢿⠳⠛⠀⠀⠀⠀⠀⠘⣏⠉⠉⠛⠋⠉⠉⠁⠀⡟
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡾⠶⢿⡶⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⡤⣤⣷⣠⣁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣹⠀⠀⠰⣇⠐⠀⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠚⠛⠛⠓⠛
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⢛⣿⡿⠲⠵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⡤⣤⠤⣀⣠⣀⣠⣀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⣠⠟⠛⠉⠛⠲⢤⡿⠶⣿⣶⠦⣄⠀⠀⠀⠀⠀⠀⣠⠶⡛⠉⠉⠯⠛⠲⣤⡀⠀⠀⠀⠀⠀⠀⢸⠾⠧⣤⣠⠆⠀⠀⠉⠳⣄
⠀⠀⠀⣼⠁⠀⠀⠀⠰⠀⠀⠙⢾⠓⣼⡧⣌⢧⠀⠀⠀⠀⡾⢡⠆⠙⠀⠛⠀⠘⢁⣤⠹⡄⠀⠀⠀⠀⠀⣼⠹⢦⠀⢈⡙⠲⢦⡀⢀⡀⠳⣄
⠀⠀⠀⠹⣄⠂⠀⣤⠀⠀⠈⠀⣌⢳⡘⢾⣻⣸⠄⠀⠀⠀⢹⡦⠤⠤⢤⣼⣭⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⣿⣃⡾⢀⡴⡒⣒⡶⣝⢻⣄⡤⠿
⠀⠀⠀⠀⠈⠳⢤⣀⠀⠠⠆⠀⠀⠀⢧⣸⣹⣾⠀⠀⠀⠀⢼⡳⠒⠛⣲⣶⣖⣛⣓⣶⣞⣹⠆⠀⠀⠀⠀⣟⠡⠀⣟⢾⣹⣽⠽⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠃⠀⢸⡏⡽⠃⠀⠀⠀⠀⠈⢿⡛⠉⠉⠉⠛⠯⣍⠉⢉⡟⠀⠀⠀⠀⠀⡏⣙⡦⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠚⠛⠋⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⣾⣿⣿⣿⣿⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⢠⣿⣿⣿⣿⣿⣀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠘⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⣿⣿⣿⡏⢻⣿⣿⣿⠀⠀⢹⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣿⣿⣿⣷⣼⣿⣿⡟⠀⢸⣿⣿⣿⠀⠀⠈⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⡄⠀⠀⠸⣿⣿⠀⠘⣿⣿⣿⣿⠀⣿⣿⡆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡀⢸⣿⣿⠛⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣧⠀⠀⠀⣿⡿⠀⠀⣿⣿⣿⡏⠀⣿⣿⣁⣿⣿⡇⠀⠀⢀⣴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣿⣿⡿⠀⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⡄⠀⠀⣿⡇⠀⠀⣿⣿⡟⠀⢠⣿⣿⣿⣿⣿⡇⠀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⢻⣿⡿⢿⣿⣷⡀⠀⠈⣿⣿⡇⠀⢰⡟⠀⠀⢠⣿⣿⠁⠀⣾⣿⣿⣿⣿⣿⣃⣾⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣶⣄⠀⣿⣿⡿⣿⣿⣷⠀⠘⣿⣷⠈⢿⣿⣷⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⣼⣿⠃⠀⣸⣿⣿⡟⢸⣿⣿⣿⡟⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⡇⠘⣿⣿⡀⠀⢹⣿⣧⠀⠹⣿⣇⠀⠀⣿⡿⠀⠀⠀⠀⢰⣿⠃⠀⢠⣿⣿⠏⠀⢸⣿⣿⡿⠀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠘⣿⡇⠀⠀⢹⣿⣇⠀⠸⣿⠀⠀⣿⠇⠀⠀⠀⢀⠾⠁⠀⠀⣾⣿⡏⠀⠀⣿⣿⣿⠃⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⢸⣷⠀⠀⠀⣿⣿⡀⠀⣿⠀⠀⠋⢀⣀⣠⣄⣀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⡿⠀⠀⠀⢸⣿⠁⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢹⡇⠀⠀⠀⣿⣿⡏⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠹⣿⣿⡆⠀⠀⠀⠀⠀⣸⠏⢀⣴⣿⣿⣿⣿⣿⣟⠻⣿⣿⣿⣿⣷⡄⠀⠁⠀⠀⠀⢹⣿⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠹⣿⣿⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟⢦⣍⢻⣷⣌⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⢻⣿⡇⠀⠀⠀⢀⣾⣿⣿⣿⢿⣟⠻⣦⡙⢷⡈⢿⣷⣬⣿⡟⢿⣿⣿⣄⠀⠀⢠⡟⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣦⡙⢷⡈⢿⣌⠻⣾⢷⣌⣻⣿⣦⡙⢿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⣿⠃⠀⠀⢀⣴⣿⢻⣿⣿⣏⠻⣦⣹⣶⣿⣿⣿⣦⡹⣿⣿⣦⠻⣦⣙⣿⣿⣦⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⢀⣴⡿⢫⣿⠟⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣷⡈⢻⣿⣿⣿⣷⣄⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣄⣠⣾⣏⣴⣿⣷⣾⣿⣿⠿⣛⣥⠾⢛⣡⣼⣿⣿⣿⣿⣿⣷⡜⣿⣦⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⡿⠟⠛⣉⣩⣴⡿⠟⠋⠠⠞⢋⣩⡶⠿⢛⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⡿⠟⢋⣥⡶⠞⠛⠛⠉⠁⠀⣠⣴⡾⠟⣛⣭⡴⠞⣉⣼⣿⣿⣿⡿⠿⠿⠷⣶⣤⣉⡉⠛⠿⣿⣿⣿⠿⢷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⣀⣤⣶⠿⣿⡿⠟⢉⣠⡾⠟⢋⣠⡴⢞⣁⣄⠀⠞⢋⣄⢀⣾⣏⣡⣴⡿⠟⣻⣿⣿⣿⣇⠑⠶⢦⣤⣉⣙⡛⠳⠶⢶⣤⣽⣿⣶⣤⣭⣿⣿⣿⣷⣦
⠀⠀⣀⣴⣾⠟⢋⣤⣾⣯⣴⠟⠛⢩⡴⠞⢋⣥⡾⠟⠋⣡⣶⠾⣿⣷⣞⣩⣽⢟⣫⣴⣾⣿⣿⣿⡿⠛⠛⠿⢷⣦⣬⣉⣙⣃⣀⠀⠀⢈⡉⠙⠻⣿⣿⡿⠛⠻⣿
⣠⣾⣿⣿⣿⣿⣿⠟⠋⢉⣠⡴⣀⣤⡶⠞⠋⢁⣤⣶⣿⣯⣵⡾⠟⣿⣿⠿⢿⣿⣿⣿⣿⣿⡿⠋⣀⣀⠘⠿⣶⣤⣽⣿⣿⣿⣿⣟⠓⠉⠛⠿⢶⣾⡿⢡⣆⡀⢸
⣿⡿⢋⠉⠻⣿⣧⣴⡾⠛⠉⢀⣬⣶⡴⠗⢀⣿⣿⡿⠛⣋⣡⡶⠟⣋⣥⣾⣿⣿⡿⢿⣿⣁⠀⠀⠈⠙⠿⢶⣤⣄⠉⠙⠻⣿⣿⣿⣿⣶⣶⣦⣼⣿⠇⣾⣿⢃⣿
⢻⣧⠘⣾⣷⠘⣿⣯⣤⣴⣶⠾⣿⣉⣴⣾⡿⢻⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡛⣿⡷⢦⣽⣿⣿⠶⣤⣄⣀⠶⣤⣭⣀⣻⣷⣿⣧⡈⣿⣧⣤⣉⣿⣿⣧⣼⠏⣼⡿
⠈⢿⣧⠹⣿⡃⢸⣿⣿⣿⣴⣾⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⣷⣦⣌⡉⠛⠶⣦⣭⣛⡁⠀⣉⣙⣻⣿⠋⠙⣿⣿⣭⣍⣿⣿⣄⠙⣁⣼⡿⠁
⠀⠈⠻⣷⣌⠻⡿⠋⣻⣿⣿⣷⣴⣾⣿⣿⣿⡿⠟⠋⠙⠛⠿⢿⣷⣦⣌⣙⠻⢿⣷⣤⣽⡟⠷⢶⣼⡟⠻⢿⣿⣿⣿⣿⠃⢸⣧⣿⡿⠿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠙⢿⣷⣶⣾⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣏⣛⢿⣶⣤⡿⠿⣷⣶⣤⣹⣿⡇⢶⡾⣼⡟⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠛⠛⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣶⣦⣌⡉⢻⣿⣧⣠⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿⣉⣹⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠠⠶⣷⣦⠀⠀⠀⠀⠀⠀⢀⠉⠁⠄⡀⡈⠐⠀⠀⠀⠀⠀⠀⠀⠞⢃⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢀⠁⠀⡀⠱⡐⠀⠀⠀⠀⠀⠈⢀⢁⠃⡅⠇⠀⢀⠀⠀⠀⠀⠀⠀⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠄⠀⠀⠀⠀⠈⠀⠁⠀⠀⢀⢠⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⣀⠐⠀⠀⠀⠸⢸⠀⠀⠀⡀⢆⠀⠀⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⣇⠁⠀⠀⢡
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡀⢀⠔⡀⠀⠰⠀⠰⠰⢸⠠⢀⠌⠀⠀⡁⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⡠⠶⢤⠤⡔⠀⠀⠀⡀⢃⣀
⠀⠀⠀⠈⠢⡀⠀⠃⠸⢠⢠⠰⠤⠖⠛⠁⠨⣴⣄⡆⢀⠀⠀⠀⠀⠘⡀⢞⠀⠀⠈⠁⠐⠀⠀⠀⠀⠈⠀⠀⠀⣠
⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠈⠀⠀⠀⢀⠄⠀⠀⠀⠀⠘⠀⡆⡆⠀⠀⠃⠀⡉⠐⠠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠁⢠⠄⡀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠗⠲⠶⡄⠀⠀⠀⠃⠃⠀⠀⢘⠰⠀⠀⠰⠀⠀⠀⠀⠀⡄⠈⠈⠀⡀⠀⠆⠀⢠⠃⠀⠀⠄⠤⠔⡔
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡰⢍⠀⠀⠀⠂⠀⠉⠁⠀⠀⠀⠀⠃⠀⣇⡬⠁⠈⠉⠁⠈⠁⠀⠀⠀⣠⡄⣀⣀⣀⣀⠀⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⣀⣀⡀⠔⡀⠀⠀⠀⠀⠉⠀⠈⠉⠙⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠖⠀⢂⢠⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⠈⠀⢘⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡀⣰⠋⢳⡄
⠀⠀⠀⠀⢀⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⣩⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣠⠤⣄
⠀⠀⣤⠐⠿⡞⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡞⠛⣡⢼⡛⠷⢯⣷⡀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠒⠚⠉⠀⠀⢸⡆
⢀⡞⣹⠷⠒⠿⣄⠀⠀⠈⠓⠦⢤⣄⣀⡄⠀⠀⠀⣸⣴⣣⢾⠷⠶⢿⢻⡞⠁⢧⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⡴⠛⠛⠳⣄⠀⠳⣄
⠉⠉⢻⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⣀⠿⣽⠀⠀⠀⢻⠙⠛⠻⠶⠶⢾⠏⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⡼⠀⢸⠀⠀⠀⠀⣹⠀⠀⠈⢳⡀
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠉⠛⠛⢦⣴⡆⠀⠀⠀⠈⢦⠀⠈⠁⣰⠋⠈⠀⠀⠀⢻⡀⠀⠀⠀⣰⠋⠀⠀⠈⠳⢤⣤⠴⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣤⣤⠤⢿⣸⠇⠀⠀⠀⠀⠈⢧⡀⣰⠃⠀⠀⠘⠀⠠⠈⣇⠀⠀⠀⢿⡀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⢀⠀⠀⠀⢸⡆⠀⠀⠀⠙⠓⠚⠉⠀⠀⠀⠀⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠤⠀⠈⠀⠀⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠙⢦⣀⣀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢰⢡⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⣾⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⣠⠤⡎⢉⡆
⠀⠀⠀⠀⠀⠀⠀⡟⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⣀⠀⠀⢀⡞⣠⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠙⣦⠟⠳⣦⡟⣏⠹⣞⣦
⠀⢠⠶⠶⠶⠶⠚⠓⠓⠒⠒⠒⢲⡄⠀⠀⠀⠀⠀⣞⣛⣛⣛⣛⡇⠀⠈⢇⢹⢀⣤⣄⣠⡿⠋⠀⠀⠀⣏⣛⠛⢦⣞⢹⢠⡿⣆⠘⢧
⠀⠉⢻⠉⠉⠉⠉⠉⠉⠉⠉⢉⡏⠁⠀⠀⠀⠀⡴⠛⠉⠉⠉⠉⠑⢦⠀⠞⢀⢸⣄⠈⢹⡇⠀⠀⠀⠀⢀⣈⡟⣆⠈⢻⣸⠃⣼⣄⣨⣷
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⣞⠉⠙⢿⡳⠞⠁⠀⠀⠀⠀⠀⢻⠉⠉⠙⠛⠛⠉⠉⠁⢠⡇
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡼⠶⢿⡶⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣤⣀⣷⣠⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⡏⠐⠈⠙⣏⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠒⠒⠒⠶⠟
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠻⠶⠶⣶⡿⢦⣧⡴⠋
⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠓⠲⠶⢤⣄⡀
⠀⠀⠀⣰⠿⢞⠳⠮⣍⡹⣯⣽⣇⣤⣀⠀⠀⠀⠀⠀⠁⠀⢀⣤⠶⠒⠒⣖⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⣀⡀⣠⠀⠈⠉⠳⢤⡀
⠀⢀⡞⠁⠀⠈⠀⡄⠀⠙⢦⡸⢭⣳⣈⠱⡄⠀⠀⠀⠀⣴⢋⡀⠳⠀⠶⠀⠡⠆⢈⠳⡄⠀⠀⠀⠀⠀⢸⡷⣤⡈⠉⠓⠦⣤⡄⠀⠀⠙⢦
⠀⠸⡆⠠⠀⠀⠀⠀⠀⠂⠀⠹⡄⢿⣽⣇⢹⠀⠀⠀⠀⢷⣌⣁⣀⣀⣀⣖⣀⣈⣋⣁⣹⠀⠀⠀⠀⠀⢸⡷⢈⡇⢀⣩⡤⣤⣙⠶⣞⠀⣈
⠀⠀⠘⠦⣀⠈⠁⠀⢀⠀⠈⠃⠹⡄⣹⢾⣼⠀⠀⠀⠀⣸⣭⣭⠭⣍⣉⣩⠽⢯⣍⣩⠿⡄⠀⠀⠀⠀⢸⠟⠋⢠⣯⡴⡖⣟⣮⠷⠞⠋⠉
⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠠⠄⠀⢷⡷⢺⠃⠀⠀⠀⠀⠹⣦⠤⠶⠶⠿⢶⣶⠶⠶⠖⢺⠁⠀⠀⠀⠀⢸⠶⣄⢘⣦⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣄⣸⡴⠃⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣀⣈⣹⣦⡴⠋⠀⠀⠀⠀⠀⢾⠴⠛⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡆⠀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢁⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢸⣿
⠀⠀⠀⠀⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣿⣿⣿⣾⣿⣦
⠀⠀⠀⠀⠘⢿⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⢿⣿⣿⣿⣿⣧⠀⠀⠀⠈⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡀⢰⣿⠏⠀⣤⠀⠹⣿⣿⣿⣿⡇⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⠇⠀⠀⢿⣷⣾⣿⣿⣿⣿⠇⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⡄⢻⣿⣿⣷⡀⠀⠀⡀⠀⢰⣧⠀⠀⣸⣿⡟⢀⡀⠀⠈⢿⣿⣿⣿⣿⢏⣴⣿⣿⡟⠀⠀⢤
⠀⠀⢦⣀⣴⠏⢀⠙⣿⣿⣷⠀⢀⣿⠀⣿⣿⣿⣾⣿⣿⡇⠀⣿⣦⡀⠀⠙⢿⡿⣋⢼⣿⣿⣿⣷⣤⣤⣿
⢠⠀⠈⣿⡏⠀⠸⣿⣿⣿⣏⣴⣿⠋⢀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣧⠸⣿⣿⣿⣿⣿⣿⡿
⣿⣷⣾⣿⣀⣷⡄⠘⠻⣏⣾⣿⣿⣦⣾⠆⠈⠻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠙⣦⢿⣷⣿⣿⣿⣿⣿⡟⠁
⠈⠿⣿⣿⣿⣿⡇⠀⢠⣿⣼⣿⣿⣿⠏⣾⣆⣀⣸⣿⣿⣿⣿⣿⠃⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣤⣿⣿⣿⡏⣀⣤⣾⣿⣿⣿⣿⣏⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠙⠿⠿⠿⠿⠾⠿⠿⠿⠿⠿⠿⠿⠿⠆⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗
⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⡄⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⢶⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠡⠊⡈⠀⠀⠀⢀⡆⠀⢀⡀⠀⠀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢀⣀⣼⠀⢰⡁⠉⠀⠎⠰⣋⡈⢪⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡅⢳⢸⢸⣿⠀⡌⠳⢦⡀⢄⠸⣿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠂⠣⡱⡌⢆⢣⠻⣷⡙⡄⠀⢻⡌⣆⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠀⠈⢀⢩⠑⣌⢻⣾⣷⣇⠹⣷⢹⣴⣼⡿⢸⢀⠀⠈⡇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢑⡏⣝⡳⣿⣿⣿⣶⡿⢸⣿⣟⡵⠋⡈⠀⣸⠇⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠠⠓⠂⢀⡇⠘⣿⣿⣿⣧⣿⣿⡋⣰⠾⠷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⣀⣀⡀⢐⡄⠸⣧⠰⡘⣿⣿⣿⣿⣿⣿⣷⣷⢄⡀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠀⠟⠹⢭⠏⠛⢿⣿⠿⡿⠋⠉⠁⠀⠀⠀⠃⠈⠈⢉⡶⡲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣠⠇⠀⠀⣀⠴⠊⠉⠀⠀⠀⢀⡀⠀⠀⠉⢀⣦⡄⣾⢰⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠙⠛⠂⠙⡟⠲⢖⢦⡀⠀⢠⣰⣯⣦⢀⠏⠙⠋⢹⣿⠾⢦⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⣿⡅⠀⠀⣧⠀⡌⣿⣇⣤⠞⠛⠈⢿⣾⣄⣉⣀⣿⠻⡇⠌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⣷⣀⣨⡥⢴⣿⠟⠛⢲⢿⣿⢿⣧⣈⣠⡼⠏⡉⠫⠁⠀⠀⠂⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠷⠀⠐⠈⠁⠀⠙⠉⠉

⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀⡀⠀⠀⠀⢠⠀⣥⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢰⣧⣄⣷⣶⠀⠀⠀⠀⠀⠁⠐⠂⠜⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⠀⠠⣤⣄
⠀⠀⠀⠀⠀⠈⠀⠑⠒⣿⠀⠀⠀⠀⠀⠆⠈⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠛⠀⠀⠀⠀⢰⠀⠀⠀⠀⠹⣶
⠀⠀⠀⠀⠀⢠⠀⠀⠀⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠰⠀⠓⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠐⠦⠈⠾⠃⠀⠀⠀⠀⠀⠣⠀⠠⠾⠃⠀⠀⠀⠸⠆⠀⠶⠃⠀⠀⠀⠀⠠⠄⠠⠶⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠴⠟

⠀⠀⠀⠀⣀⡀⠀⢠⣰⢄⡀⠀⠀⠀⡤⠄⠀⠠⠀⢤⠀⠀⢠⠂⠐⠒⠂⢲⡄⠀⠀⣠⡤⢀⠀⣤⣤⣄
⠀⠀⠀⠀⠹⡄⠀⠈⢹⠟⠀⠀⠀⠀⢂⠀⠀⠀⠀⠘⠀⠀⠀⢂⠀⠀⢀⠞⠀⠀⠀⠣⠀⠀⠀⠀⠀⠉⠀⠀⢠⢀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠄⠀⠀⠀⠀⠈⠈⠀⠀⠙⢻⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⡟
⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠘⠀⠀⠀⠐⠋

⠀⠀⠀⠀⣀⡀⠤⠄⠀⢀⡀⠀⠀⠀⠀⡀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⡄⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⢏⠀⡀⢀⡀⢻⡇⠀⠀⠀⠘⠓⠛⠛⢛⡗⠀⠀⠀⠂⠀⠈⠀⡶⠀⠀⠀⠀⠃⠀⠀⢹⡇⠀⠀⠀⠐⠗⢒⣆⣰⠛⣿
⠀⠀⠀⠀⠘⡀⠀⣀⢀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⠀⠀⠀⢄⡋⢿⢛⡇⣿
⠀⠀⠀⠀⠀⢟⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠁⠀⠀⠀⢀⠐⠀⢀⡀⠀⠀⠀⠀⢸⠀⠀⠐⡇⠀⠀⠀⢀⡁⠀⠒⠊⠀⡁
⠀⠀⠀⠀⠀⠘⠃⠀⠚⠃⠀⠀⠀⠀⠀⠙⠀⠒⠛⠀⠀⠀⠀⠘⠀⠐⠛⠁⠀⠀⠀⠀⠀⠂⠀⠚⠀⠀⠀⠀⠈⠓⠂⠀⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⡆⢀⠀⠲⠒⢲⡶⠶⠶⣶⠶⠖⠠⢀⣀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠓⠀⠈⠀⡀⠈⠀⠀⠀⠈⠀⠀⠀⠤⠀⠀⠰⠀⠈⢩⣟⠻⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠞⠃⠉⡀⠀⠀⠊⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠅⠀⠐⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠤⠀⠀⠀⠙⠁⣽⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢤⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣘⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠐⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣟⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣷⣤⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢛⠿⢶⣦⣤⣀⣀⣀⣀⣄⣀⣀⣠⣤⣀⡀⠀⢀⣀⣀⣀⣀⣀⣤⣀⣀⣤⣤⣤⣤⣶⣿⣿⣿⠿⣟⣛⡭⣉⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣷⣦⣈⠉⢉⣛⣛⠛⢛⣻⣟⠛⠟⠻⢿⢿⣿⣯⣿⣯⣍⣭⣭⠽⠈⢦⡉⠁⢀⣰⡟⢉⠾⠋⢗⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠰⠚⣉⡙⠿⠿⠍⣉⣿⣿⣿⡟⢻⣿⣿⣦⣤⣤⣴⠿⣷⠾⡯⠟⠛⠿⢿⡴⠋⠳⢍⣻⡿⠻⠷⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣭⡥⣄⡀⠀⠋⠙⠿⠻⠿⠛⢻⠿⢻⡿⡯⠋⠀⠋⠉⠀⠐⠂⠒⠉⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠧⣿⣿⣷⣶⣏⡋⠉⠀⠀⠒⠓⠀⠊⢀⣠⣤⣴⣶⣶⠶⣶⣶⣶⣾⣿⣿⣿⣿⡿⢛⣽⣿⣇⣼⢧⣼⡏⣨⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣌⢙⣧⣄⡙⢟⣷⣄⠀⠀⢀⣠⣾⣿⣯⣤⣶⣿⣭⣼⡿⢃⣾⠏⢀⣾⠿⣿⢹⣟⠉⣸⣿⣏⣾⡯⣴⡏⢀⡀⠘
⠀⠀⠀⠀⠀⠀⠂⠈⠉⡃⠛⢮⠛⠿⢙⣛⢻⣿⣧⣴⣿⣿⣛⠉⢻⢿⣭⣼⣿⣋⣾⡿⠿⠿⠿⢛⣴⣟⣨⡿⣿⣿⠿⡿⡗⠓⢄⣁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠶⡥⢄⡠⢀⣀⠀⠀⣈⡉⣉⣉⣤⣤⣞⡀⠒⡢⠈⠟⠀⢐⠊⣀⣀⣉⣡⣽⣄⣤⣤⣏⣀⣿⡿⠶⠾⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠙⠛⠻⠙⠿⠟⠋⠛⠻⠗⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠁⠉⠹⡟⠀⡀⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣄⠀⠀⠀⣰⣀⣼⣦⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠤⠤⠤⠤⣤⣤⣤⣄⣤⢀⣀⣠⣤⣄⣴⣶⣴⣤⠤⠴⠾⠿⠿⠒⠶⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⢦⡀⠀⠀⠀⠀⢠⠂⠈⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠰⣦⣳⡜⢡⣀⠀⠀⠀⣤⡀⢀⣀⣀⡤⣀⡠⣀⡀⠀⣸⣿⡻⠽⣟⣛⠉⠁
⠀⠀⣤⣤⡽⠂⠀⠀⣐⡛⣦⣰⢞⡀⠀⠀⠀⢀⡴⠣⢆⡼⠀⠀⠀⡀⢶⣽⠻⣾⢦⠹⡇⠀⢻⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠀⣹⣿⣿⡟⠃⠈
⢠⣖⠲⣌⣻⣆⠀⢠⡜⣿⠻⡷⢻⠟⠀⢠⢠⡿⣧⠳⣜⣷⣠⠆⠀⢸⣶⣿⣾⣷⣾⣿⣱⣇⠈⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠸⣿⣿⡿⠂⠀⠀⣄⡑⡀
⠀⠉⠛⣿⣿⠋⠀⠈⢹⣿⣿⣿⠋⠀⠀⣾⣸⣷⣿⣡⣿⡿⣿⠀⠀⠈⠙⠋⠙⠙⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⠛
⠀⣴⣿⡿⠟⠀⠴⠶⠿⠿⠿⠿⠶⠶⠆⢘⣿⢻⣿⣿⣿⣷⣿⡎⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⣄⠐⡇⢦⣠⡟⢀⡆⢀⠀⡄⠀⠀⠀⠀⠀⢀
⠀⠻⢿⣦⣤⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣾⡷⠀⠀⢠⣾⠁⠀⠀⠀⢠⢠⡷⡿⣴⡟⣼⣁⣷⢸⡄⢣⣄⣿⢴⠁⠀⠀⢀⣾⣦⣆
⠀⢀⣠⣾⣿⡷⠀⠀⠀⠀⣰⣿⣷⡆⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⢠⣎⣿⣧⣰⡀⠀⣿⢸⣧⣿⣼⣿⣿⣿⣿⣾⣷⣼⣿⣷⣾⣧⠀⢰⣿⣿⣿⣿⡄
⠀⠈⠉⠉⠉⠁⠀⠀⠀⢸⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰⠄⠀⢠⣿⣿⠃⣿⣿⣧⠀⠛⠿⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⢿⣿⠿⠏⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠻⢶⡀⠺⢗⣶⠄⠸⡌⠻⣿⠿⠃⠀⠀⢠⠀⠀⣆⢿⣰⠀⢸⣿⠁⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠛⠋⠋⠛⠛
⠀⣠⣶⡞⣡⢾⡏⠁⣶⠀⠀⠀⠋⠀⠀⣼⠁⠀⡀⠀⣸⣺⣿⡇⠀⠉⠁⠀⠘⠋⠁⠀⣤⢠⡁⠠⠀⠀⠀⠀⠀⢀⣔⣿⣿⣿⣦⠀⠀⠀⢀⣶⡄⡀
⠀⡘⢿⣦⣽⡿⢹⣿⢱⣏⠀⠀⠀⢀⢃⡈⣦⣼⠁⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠐⡘⢀⢋⣶⣿⣧⣆⡀⡀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⣠⢯⣽⣿⣿⡄
⠀⠹⢿⠶⢿⣦⣾⡿⠿⠋⠀⢰⣴⣼⣾⣷⣿⣻⣄⣆⣀⣀⢰⠀⠀⠀⠀⢠⣼⣇⣿⣾⣼⣷⣷⣿⣻⢿⣤⣄⡀⠀⢸⣿⣿⣿⡏⠀⢠⣿⠋⠸⠟⣿⣷
⠀⠀⠀⠀⠸⠛⠋⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠰⣷⡿⠿⠿⠿⡿⠾⠿⠿⠿⠿⠿⠿⡿⠀⠈⠻⠈⠟⠁⠀⠀⠻⢦⠀⢀⠿⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠑⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠢⡀⠂
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢸⣿⢮⣻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠔⡁⢨⡇⢈⡢⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣽⣿⣼⣿⣮⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠔⠁⠐⠃⠐⠃⠘⠃⠈⠢⡀
⣿⣿⣿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠣⡀⠀⣼⢇⠁⠄⣠⠈⡄⢀⠜
⣿⣿⣿⣿⣿⣿⣏⠃⢻⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣾⣾⠈⡇⠀⢀⠀⠐⡁
⣿⣿⣿⣿⣿⣷⣾⣷⣮⣯⣭⣥⣶⣶⣶⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠈⠋⠳⡓⢒⠟⠉⠉⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⢸⠂⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢠⡄⡀⣿⢷⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⢸⢻⠀⠁⣤⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣿⡆⢠⢟⡑⡇⡿⠸⡂⠀⠀⠀⠀⣆⢠⢣⡀⠀⠀⡞⢸⠀⣧⢛⡤⠀⣿⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣰⡏⣤⠏⣷⡐⠎⣱⢡⡇⡄⡇⠀⠀⠀⣸⣿⣦⠻⠃⠀⠀⡇⡌⣤⢣⡘⢞⣰⠇⣇⡘⣧⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⠻⣀⣿⠸⡼⡄⣿⠙⣯⣉⠳⣹⣇⡇⠀⠀⣼⣿⠃⡏⡷⣄⠀⠀⡇⣯⠣⢋⣹⡟⢹⢠⡸⡸⢺⣇⡸⣇
⠀⠀⠀⠀⠀⡀⠀⢸⢠⡹⣿⣼⣱⡇⣽⣶⡟⣿⢣⢧⢸⡇⢠⡇⡿⢹⠀⡗⢹⠌⣧⠀⣿⢫⣇⢻⡟⣶⣿⠘⣷⢳⣼⡟⡀⣿⠀⠀⡀
⠀⠀⠀⠀⣼⠃⠀⢸⣧⢳⣿⠣⠱⠀⣿⢿⡇⠿⢸⡃⢸⠀⣶⣼⡇⢸⠰⢻⡝⢸⣇⡆⢹⠈⣿⠸⡇⣿⢻⡄⢣⢣⢻⡇⢲⣇⠀⠀⢻⡄
⠀⠀⢀⡾⣿⠀⠀⡼⢹⣿⡇⣦⢣⠆⣿⢸⠟⣇⠘⢰⢸⣆⢻⣸⡇⡄⣇⡎⣿⣏⣹⢁⡾⣰⠙⢀⡟⢹⢸⡇⣆⢧⡤⣿⣿⠹⡄⠀⢸⡻⣆
⠀⠀⢸⡇⠹⣶⢞⣡⠇⣿⢹⡏⣿⠀⣿⠘⣇⠹⣇⢸⡇⡟⢾⡿⢀⣷⢹⠀⣿⡙⣿⠚⡇⣿⢰⡿⢠⡇⢸⡇⡿⡜⡗⣽⠁⢆⣝⣶⡞⡀⣿
⠀⢀⢸⡟⡧⢹⣬⢎⢠⣏⣤⠀⡿⡀⣿⠃⣻⡆⢦⡌⡇⣿⣾⢣⡞⡻⢸⠀⣾⢣⡿⣴⡇⡟⣴⢠⣿⡁⢼⡇⣧⡇⢰⣬⣧⣎⢬⣼⠰⡏⣿⢀
⠀⣼⢿⡇⢇⡼⠋⢋⢞⣿⡛⣦⢷⡇⡏⠈⠃⢻⢸⡔⣸⢻⣧⣻⡄⡿⢸⣿⣿⣨⣷⠈⢷⡡⣟⣾⠁⠉⢸⡇⣿⢠⠟⡿⣿⡌⠋⢻⣼⠃⣿⣿
⢰⠏⢸⣧⠞⡁⣼⣨⡞⢰⣧⡟⡏⢸⠇⣶⡀⣾⠘⣽⠃⣮⣗⡯⢿⠃⡿⢡⡇⡿⣵⢰⠈⢷⡍⢸⠀⣰⡄⣧⠁⡿⣸⣧⢹⣷⢸⡤⡹⢦⡟⠘
⡌⢄⣾⡏⣼⠁⡟⢈⢿⡿⢡⡄⣧⡿⠀⣽⡇⣯⣼⡇⡄⡏⢙⡅⡴⣸⢃⣸⢸⠣⠇⡌⢹⡜⣷⣼⡇⣿⠡⢹⣦⡇⣄⠹⣞⠏⡈⡇⣿⡌⣿⣄
⣇⣼⢻⡅⣧⡇⠷⣠⠎⣀⠋⣼⣿⠃⣶⢹⡇⠟⢹⢠⡇⣧⣾⡇⡇⣿⢸⣽⡼⡇⢸⣷⡿⠁⣿⠹⠇⣿⢡⡄⢿⣷⡉⢃⡘⢦⠰⠇⡇⣾⣿⢻
⠹⣿⠸⣇⢣⡇⡴⣣⡆⣩⡾⢋⡎⠸⡿⢸⠀⣴⢸⡆⣷⠙⠘⣇⣧⢻⢸⢣⢿⣴⡆⣸⢦⡇⣿⢠⡄⢻⢸⡇⠸⡌⠻⣎⠓⣎⢳⡀⣯⢡⡏⢸
⠀⠹⣆⠙⣎⣹⢳⢯⣾⡿⣡⠀⡇⣄⢧⡞⣸⣿⢸⣇⠛⠀⣆⢻⡘⠘⡀⢊⡌⠉⣷⢸⡔⣰⣿⢸⣷⠘⣤⢣⡄⡇⣦⠹⣿⣮⢷⢷⣩⠞⢠⡟
⠀⢰⣾⡇⢘⣿⣴⠏⣼⢡⣿⡄⣿⡿⡌⢁⡏⣿⣸⠻⡄⣾⣿⣸⣷⣄⡇⣦⠁⣤⢹⡎⢃⡟⢿⢘⡾⡇⢁⡞⣷⡇⣿⡇⢿⡈⢧⣼⡏⠀⣿⣶
⠀⢸⣿⡇⢸⡏⢟⡆⣿⠘⣏⡇⠻⠁⢻⠈⣷⢠⡟⠀⣷⢿⢹⣿⡇⣿⢃⢻⡆⢹⠀⢷⣼⢇⠸⣆⢣⡇⣼⠀⠻⢡⡏⡇⢸⢡⢸⠋⣦⢰⣿⣿
⠀⢸⡇⢹⡟⡔⣺⡌⢸⢸⣇⡏⣠⡗⢸⡞⢣⡾⢡⡷⢸⢸⡆⣹⠃⠈⡎⡼⡧⠋⢠⠈⣿⢸⡆⠹⣮⠃⣿⢸⣦⡘⣇⣷⢹⢈⣾⠰⡜⡿⢁⣿
⠀⢸⡎⢃⡐⢂⣿⠙⡿⢸⡏⡼⢁⠇⣾⣶⠟⣱⡞⡀⡿⡸⢣⣿⡆⣸⡼⣄⣿⠏⢸⡇⣿⡜⠰⣦⡙⣷⣿⡄⢆⢳⡜⣿⠸⠟⣿⡆⠃⣰⢯⣿
⠀⠘⡇⢸⠁⡾⢹⠀⣄⠺⣸⢳⡮⢠⣣⢋⠞⣿⣠⢱⠇⣰⣿⣿⢳⡿⠰⡁⡸⢠⡟⢠⡟⢷⢆⢹⡗⡌⢣⣃⢸⣇⢳⡽⢀⡄⢿⠹⡄⣹⢈⡟
⠀⠀⢻⡎⢸⡇⣾⢸⡹⡆⢱⡞⠇⣼⠃⡎⣴⢱⡇⣾⣰⡏⢹⡄⢾⡇⣼⠱⡁⡹⢠⣾⣧⣸⠸⣹⣃⡟⡄⢻⡀⡟⣾⠁⡞⣱⢸⡄⣷⢉⡾⠁
⠀⠀⠀⠙⢾⡇⢸⡎⢧⣳⡈⡷⢠⢳⠀⡟⠉⢸⡇⣿⢻⡸⣌⢷⡘⢧⠃⢣⡱⣵⢟⡀⡟⢻⠀⣿⠈⢳⠃⣄⢃⢡⠏⣰⣁⢁⡾⠀⣿⠞⠁
⠀⠀⠀⠀⠀⠉⠀⠹⣌⢿⣹⡄⡾⣸⣠⢹⣶⠸⣇⠈⡘⣇⢼⡦⠻⣌⠻⢈⣼⢫⡾⢇⡇⡈⢰⡇⣰⡿⣴⢻⢸⡄⣞⣿⢣⡞⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢿⡌⢷⣹⣦⡇⢃⣿⠈⢰⡄⡇⢸⡇⢻⡘⣷⠇⠹⣷⡾⡁⢮⡗⣾⢱⣧⢸⡇⡛⠀⡿⡞⠘⡇⣼⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠸⡝⡌⢧⢹⡇⢸⠀⣴⡘⢡⡇⢸⢸⢸⣧⢸⡆⡀⣿⡁⢻⡜⢳⣏⡸⢸⠸⣷⢈⣴⡇⢳⠀⡇⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣧⠸⡿⣇⢸⠇⣿⠧⠘⢇⢸⣸⢸⣿⢀⣰⡇⣸⣧⠻⠁⣸⡟⢷⡸⢰⠏⡼⢿⠳⢸⢰⡷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠳⣹⡸⣄⡟⢸⡇⣦⡈⠟⢸⡇⣸⣿⣄⡿⠹⣇⠿⣼⣿⠸⢃⣴⢀⣧⡘⣀⡏⣜⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡙⢷⡸⡇⢠⠛⠆⢸⣧⠀⣿⢸⡇⡄⢻⣴⡏⣿⢠⢻⢸⢨⡇⣼⠟⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣙⠓⠸⣇⢸⠸⣧⢿⡝⣾⠀⡇⣾⡿⢼⡟⣸⣨⠟⠸⢋⣵⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣬⣍⣁⢻⣧⠁⣿⠘⣴⢋⣄⣾⢁⣉⣥⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣴⠃⢛⡼⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡿⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡁⢈⣈⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⣰⡅⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⣀⠀⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢤⣿⢧⠄⠀⠀⠀⢠⣇⠀⠀⠀⣠⠀⠀⠀⣴⣯⡀⠀⠀⢠⡇⠀⠘⣿⣿⣄⠈⠀⡈⠂⠀⠀⠀⠀⠀⠀⢠⡇⢀⡤⠀⠀⠀⣼⣿⣯
⠀⣾⡟⠄⡀⠀⠀⠀⡼⠃⠀⢠⣾⡁⠀⠀⠀⠙⠟⢠⠀⠀⣼⡗⠀⠀⠉⠛⠻⢂⡄⢀⠀⠀⣼⡀⠀⠀⠀⣿⣯⠀⠉⢀⢀⣐⣿⣿⡏
⣾⣿⣇⡴⠤⠀⠀⢰⣷⡖⠂⢈⡜⠃⡄⠀⠀⢀⢠⣿⡄⢰⣿⡭⠄⣾⣸⣴⣶⣸⡇⣼⣤⣄⣿⠥⣄⣦⣨⣿⣯⠀⣆⣿⣰⣮⣿⣿⡿
⣿⠿⢉⠷⠤⠈⠀⢘⣿⣤⡀⠚⠇⠀⠀⠀⠀⠈⢠⣿⣯⣿⠿⣃⣸⠿⠏⠙⢛⣿⡿⡏⠛⢿⣿⣯⡿⢿⠿⢿⣿⣸⣿⣿⣿⣿⣿⣿⡇
⢛⣄⠀⠀⠐⠤⠀⠸⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠿⠿⣯⠷⠂⠀⠀⠀⠀⢸⡇⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠉⠉⢻⣿⣋⣸⣗
⡩⣌⢘⡶⠂⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⣿⣼⠛⢹
⠀⠃⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠘
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡀⠀⠀⠐⠀⠰⠆⠂⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⣧⠀⢰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣴⠟⠷⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣤⣴⣶⣶⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⡀⠀⠀⠛⠂⠉⠀⠀⠀⢀⠀⠀⠀⢰⣤⣿⠇
⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢹⣿⣿⣿⠿⠉⠁⠀⠀⠀⠀⠀⠉⣉⠙⠿⠿⢿⣿⣿⣷⣾⣿⣷⣿⣤⣤⡄⢈⣻⣶⣶⣾⣿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⡀⢠⣾⣿⣿⣶⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠀⣤⣈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿
⣾⣾⣿⣷⣶⣿⣿⣿⣿⠟⡋⠉⠉⠉⠁⠁⣠⡤⠾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⢷⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⣷⣶⣾⣇⣀⣤⣴⡶⠶⠶⠶⠶⠶⢶⣤⣄⣀
⠀⠀⠀⢠⣾⠟⢉⣠⣴⣶⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄
⠀⠀⠀⣿⣥⣶⣿⣿⣿⣿⣿⣿⡿⠁⣀⣤⣶⣶⡿⠿⠿⠿⢿⣶⣶⣤⣀⠀⠀⠙⢷⣦
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠿⠋⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣷⣄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠉⣹⡟⠉⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠈⢿⡄
⢀⣀⡏⠂⠀⢰⡟⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠈⣿⡄
⠿⠿⠃⠀⠀⣾⠃⠀⠀⢀⣿⡇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠸⣿⡆⠀⠀⢸⣧⣴⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣧⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣤⣿⡇⠀⠀⢸⣿⡿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣶⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄⠀⣾⠏⠀⣿⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⢻⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⢹⢹⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⡟⠀⢠⠇⠀⠀⢀⢜⣥
⠀⠀⠀⠀⠀⠘⣿⡀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣾⡾⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⢸⣇⣴⠋⠀⠀⠀⠘⢌⣡
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠙⠻⢿⣿⣿⣿⣿⠀⠀⠙⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢋⣴⠿⣿⠁⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⣰⡟⠁⠀⣹⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠀⢀⣾⡿⣿⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠋⢹⣇⠀⠘⣿⠀⢹⡀⠀⠀⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⢠⡄⠀⠀⢿⣄⠀⠈⠁⠸⣇⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠙⠿⢶⣦⣤⣿⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⢾⢳⡄⠀⠀⠀⠀⣠⣤⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠋⠀⠀⠀⠀⢰⣿⠿⠟⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣿⣿⣟⣦⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣆⡀⠀⢰⣷
⠀⠀⠀⠀⠀⣸⠀⠀⠀⣠⣾⣿⣿⣟⠉⠻⢿⣗⣦⡀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣧⠀⣤⣾⣿⣿⣿⣿⣿⣿⣦⣾⣿⣼⡇⢀
⠀⠀⠀⠀⣧⣿⣧⣦⣾⣿⣻⣛⣽⡿⢷⣄⣀⣰⣼⣿⣷⣾⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠾⣿⣭⣉⡙⠛⠿⠟⠙⢺⣿⣿⣷⣿
⠀⠀⠠⣤⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠈⠉⠛⠛⠿⣿⣽⣿⣿⡷⠦⠀⠀⠀⢀⣚⣏⣽⣿⠿⠿⠿⠿⠷⠶⠤⠤⣤⡤⢤⠴⠶⠿⠿⠿⠿⠿⠿⢦



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣄⣀⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⢀⣠⡀⢀⣴⣷⣦⣤⣤⣴⣿⣵⣾⣿⣿⣿⣦⣀⡀⣀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣄
⠀⠐⠛⢿⣿⣿⠉⠙⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠉⠋⣹⣿⣿⣿⣷⠴⠀⠀⠀⢀⣾⣷⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⣷⣄
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⡠⠔⣁⣻⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣄⣠⠏⠱⣴⢦⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣛⣋⣳⣄⠀⠀⠀⠔⠋⠹⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠹⡶⡀⠀⠀⠀⠀⠀⠘⠁⠀⠂⠀⠀⡠
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠶⠀⠠⠞⠃⠀⠰⠧⠄⠀⠀⠀⠀⠀⢀⠀⢀⣀⣴⣂⢀⣤⣄⣀⣀⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠣⣠⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠱⠋⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⠀⢀⡄⠀⢸⣜
⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⣴⡠⠚⢭⣆⣠⡏⠓⢄⡩⢳⣍⣂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣠⡤⠐⣉⣠⠐⠉⣽⠛⢻⡀⠀⠙⣀⡀⢈⣷⣌⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢟⣿⣿⡀⠀⣡⣠⣮⣿⣄⠀⠈⣻⡿⠿⠟⠛⠿⠈⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣁⠈⠉⠉⠙⠛⠛⠉⠛⢷⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⡠⠀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⢀⠠⢐⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⣀⣢⢠⣈⣀⣥⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠀⡿⠂⠀⠀⡐⢁⢲⡀⠀⠸⠀⠀⠀⠀⣠⠤⠶⠶⠿⠿⢶⡄
⠀⠘⠈⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠾⠀⠀⠀⠀⠀⠛⢾⡡⠀⢁⡆⠀⠀⠀⢹⠀⠀⠀⠄
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠁⠠⠀⠀⠀⠀⣶⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢠⠀⠀⠀⢨⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡄⠀⠀⠐⠀⠀⠀⠐⠒⠘


⠀⠀⠀⠀⢤⠉⠀⠀⠀⠀⠀⢀⣠⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢘⠆⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⠰⣆⠀⡀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⡄⠄⠀⠀⠀⠀⠸⠁
⠀⠀⠠⠂⠀⠀⠂⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠲⢎⠣⣄⠀⠀⠈⠀⡃⠈⠠⠈⠀⠠⠀⠑⠠⠀⠀⠀⢀⠀⠐
⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⢙⡧⠄⠀⠀⠀⠀⠐⠨⠤⠒⠀⠡⠀⠀⠀⠀⡁⢈⠀⠠⠋
⠀⠀⠈⠀⠈⠁⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣠⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠠⠊⠀⣀⣤⣤⣀⡈⠀⠀⠀⠀⣿⢿⡄⢸⣟⠖⠁⠀⠀⠀⠀⢀⡨⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄
⠐⠀⠂⠀⡰⢛⡅⠈⠀⠁⠀⠀⠰⠀⠀⢨⠉⠃⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡜⢋⣀⣠⡠
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠨⡀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠠⡬⠀⠀⠀⠀⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠚⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢀⠀⠀⠀⠀⠀⢀⣠⠄⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⢀⠂⠈⠂⠀⠀⠀⢠⡏⠃⠀⠀⠀⠙⡆⠀⠀⠀⡀⠔⠒⠂⠀⡀⢀⠠⠀⠀⢀⠔⠀⠀⠀⠑⠄
⠀⣀⠀⠀⠐⠚⠛⠛⢲⠇⠀⠀⢀⠈⠀⠂⠂⠀⠁⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⢈⠀⠀⠒⡄⠶⠀⠒⢄⡜⢂
⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢆⠤⠁⠔⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠈⠘

⠀⣀⠀⠀⠀⠀⡀
⢀⠀⠀⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⣰⣤⣤
⠀⠀⡀⠙⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⠃
⠀⠀⡇⠀⢣⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⢀⡜⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠⠦⡀
⠀⠀⡇⠀⢸⠀⠀⡇⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠶⠒⠃
⠀⠀⢀⠀⠸⠀⠀⠠⠀⠘⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠄⠄⠤⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠴⠛⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠁⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠠
⠀⠀⠀⢤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠛⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⡀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠴⠀⠒⡁⠀⠀⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠐⠒
⠀⡀⠀⠀⠀⠈⠤⠠⠴⠒⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡀⠉⠁⠀⠀⠀⣤
⠀⢁⠀⠀⠀⠀⠀⢀⡀⠄⠠⠀⠀⢉⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠔⠈⢀⡂⠀⠀⠀⠀⡅⠀⢀⡀⢀⡤⢠⠠⠀⠁
⠀⠈⠀⠬⠀⠈⠀⠃⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠘⡀⠀⠁⠀⠀⠀⠇⠀⢀⣤⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠛⠁⠀⠀⠄⠀⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠀⠀⢛⠉⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⢀⡀⠀⣁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢨⢀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠁⠈⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠄⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠲⠠⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠘⠆⠘⠃⠂⢂⠄⠀⠀⠀⠀⠀⠤⠄⢙⣐⣒⣉⠠⠤⡆⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⡀⠀⠀⠀⠀⣀⠤⠀⠀⠠⠤⢀
⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠸⡀⢆⢀⣄⠲⠀⢂⠀⡇⠀⠀⠀⠀⠰⢠⠂⠈⠲⠂⠀⠀⠀⠀⠀⢀⠈⠀⣀⡀⠁⡁⣀⣀⢱
⠀⠀⠀⠀⠛⠄⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⢀⠜
⠀⠀⠀⠀⠁⠀⠈⠠⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠒⠒⠒⠒⠒⠁⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠂⠀⣻⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠠⠀⠐⠄⠀⢉⡃⠀⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣏⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⣺⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⢼⣷⣀⠀⠀⠀⣸⡇⠀⠀⡀⠈⣽⣏⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢀⡾⠁⠐⠀⠀⣴⡿⠁⠀⠀⢸⡄⠁⠙⠦⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡁⠉⣰⠄⠀⡆⢸⡿⠁⠀⠀⠀⣾⡇⣾⠀⠀⢨⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠦⠚⠁⠀⢰⠇⢸⠇⠀⠀⠘⠀⣿⠁⢸⡄⠠⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⡐⠀⠀⣿⠀⠈⠇⠀⡆⠀⠀⢹⡇⠸⣿⠀⡀⣸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣿⣏⠀⠀⠟⣿⠀⠀⠀⠀⠱⣦⠀⠋⠀⢸⡿⠀⠀⣿⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢽⣿⡇⢀⠀⠀⢻⡀⠀⠄⠀⠀⢹⣇⡠⠀⢸⠁⠀⡀⢸⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣚⣿⣟⢸⡆⠀⠀⠁⠀⠀⠰⠀⢸⠋⠀⠀⠀⠀⢠⡇⣽⣓⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⠿⠾⢿⣄⠸⣄⢰⡇⠀⣴⠀⠀⣸⠀⠀⣠⣾⠷⢿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣬⢻⣷⣿⣶⣿⣶⣿⣷⣶⣿⣾⣿⠟⠁⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⣹⢟⢁⡽⠋⢛⠉⢻⡋⠙⢯⡈⣻⣿⣥⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⣿⣧⡶⠉⢂⡠⠊⢷⠀⠑⢌⡀⠈⢶⣼⣇⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢈⣿⢷⣤⠮⣤⣄⡼⢦⣠⣤⣵⣦⣾⠏⠉⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣷⣤⡿⠃⠀⠉⢳⣾⠿⢦⣶⡟⢉⠉⠙⣇⠀⠀⣸⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⠈⠛⢯⣤⣤⡔⠛⠡⠀⢀⠉⣣⣀⣁⣠⣾⣶⡶⠋⠀⢴⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠀⠀⠙⠶⠶⠶⠾⠛⠋⠉⠙⠉⠉⠁⡀⠴⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠖⠄⢠⡄⡄⡄⡀⠀⡄⠀⠤⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠀⡆⢠⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣴⠏⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡇⢃⢸⡇⠀⠀⡄⠈⣧⠀⢆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣰⡇⠘⠈⡇⠀⣼⡇⠀⡟⠀⠈⢧⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⣿⠀⡇⠀⢰⠀⣿⣧⠜⢀⠄⠀⠀⡄⢘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡇⢀⠀⢻⡄⠁⠀⠀⠀⣽⠋⢠⡞⠀⠀⢀⡇⠈⡈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⢸⡇⢸⡆⠈⠇⡀⠀⢀⣼⡇⢠⡟⠀⠀⠀⣼⠃⠀⡇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢳⠀⡇⢀⠀⡇⠀⠈⣿⠀⢸⡇⠀⠀⣸⡇⢀⣼⠃⢠⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠸⠀⠀⡜⡼⠁⠀⠀⢸⠀⠀⠁⠀⠀⣏⢣⡾⠁⢀⡞⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠈⠀⣠⠁⢰⠀⡟⠀⠀⠀⠀⠀⠛⢸⡇⠀⢸⠁⡇⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣠⠏⠀⣴⠃⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠈⡄⢃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣿⠀⠰⡇⠀⢿⡁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠸⠂⢹⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢻⡄⠀⢁⠀⠸⣇⠀⢀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⣸⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⣴⣄⣘⠳⡀⠈⣧⠀⢹⠀⣼⡇⠀⠀⣧⠀⢠⡏⠀⡄⣰⣣⣤⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣶⣯⣤⣤⣤⡤⠴⠄⢸⣷⡄⠀⣿⣇⠀⣰⠇⣴⣿⡄⢸⢰⡯⠤⢴⣶⡶⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠦⢈⠉⠉⠀⠈⠉⠀⠀⢀⡤⠜⠻⢿⡴⠋⠉⢳⣿⠿⠉⠁⠀⠙⠛⠦⣤⣤⣄⡀⠈⠃⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠊⠂⣀⣒⣦⣴⠶⠛⡉⠀⠀⠀⣰⡇⣠⠀⣀⣻⣷⣀⠄⠀⠀⠀⢱⡆⠙⠛⠻⢿⣿⣿⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣼⠿⠿⠩⢱⠓⢤⠄⠀⣀⣴⣏⠁⠁⠀⠀⠈⠉⠻⣾⣷⠃⡠⠀⢆⣈⢀⠔⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠐⠀⣶⣾⢿⡏⠉⠓⢠⣄⠀⠀⠀⠀⢿⡇⢼⠀⠀⠟⠋⠘⠊⠉⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣿⠴⠛⠁⠈⢆⢀⡅⠀⠩⠊⠀⠃⡜⠙⢧⠾⠞⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠀⠟⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠸⢿⣄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠱⡁⠀⢻⡇⠀⢡⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠘⡄⠀⠃⠀⣻⠃⠀⢈⡟⠀⠀⠀⠀⣠⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⢀⣴⡿⠗⠠⠀⠀⡀⠴⠁⠀⠀⠈⢷⡀⠢⣠⡞⢹⡇⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠂⠀⠠⣠⠟⠁⠀⢠⡖⠀⢨⠀⠈⢄⠀⠀⠀⠀⡇⠀⠻⢇⠀⠹⡆⠀⠡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⠀⠀⠀⠚⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⢨⠗⠀⢠⠎⢃⠀⠀⠀⢃⠀⠈⠢⠀⢿⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠋⠀⠀⠀⠐⠁⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⣠⡟⠀⢈⡀⠀⠀⠀⠀⠀⠀⠁⠀⠡⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠏⠁⠀⡸⠃⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⠀⠀⢰⡄⠀⠀⠒⠒⠶⠤⠤⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠒⠒⠂⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣺⣿⣷⣂⠐⣿⣗⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣿⣿⣃⠀⢀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠠⣿⣿⣿⡅⣸⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠈⠀⠂⠤⡀⠀⢀⣽⣿⣿⣧⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⣻⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⢴⣿⣿⣿⡷⠦⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⡛⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠒⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⣀⣼⣧⣷⠀⠀⣴⣿⣿⡄⠀⠀⠀⢀⢀⠀⠀⠀⠀⢲⣿⣿⣶⣿⣿⣿⣧⣿⣿⣿⣷⠀⠙⣷⢈⣣⡽⡂
⠀⠀⣤⣿⣆⢀⠀⠀⠀⢠⣿⣿⣿⣿⣶⠀⠏⢻⡿⠃⠀⠄⠆⡘⣠⣷⣆⣄⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⢸⣿⠡⢾⣿⣤⣿⡠
⠀⣰⣿⡟⢻⣿⡇⠀⠀⢾⣿⣿⣿⣿⣿⡆⠀⠀⠁⢀⣼⣸⣶⣿⣧⣯⣿⣻⢷⣥⢀⢀⢰⣦⢶⣒⣶⣶⠷⡶⠄⢲⣿⣻⣟⣡⣿⣾⡗
⠀⣿⡏⠀⠈⣿⡿⠀⠀⠈⠛⠋⠉⠛⠛⠁⠀⠀⢼⣾⡿⢿⡿⣿⠿⠿⠿⠿⠿⠿⣿⠇⢻⣿⣧⣤⡟⠶⡀⠀⠀⠀⠉⠁⡿⠿⠋⠁
⠀⠈⠉⠀⠈⠉⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⡀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣶⣿⣿⣿⡃⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣨⣮⡞⣥⡇⣆⢰⣠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠛⠉⡔⡅⢆⠀⢀⠻⣄⡴⠂⠀⠀⠀⡀
⠀⠀⠀⣧⠀⠀⣼⣻⣼⣽⣾⣿⣷⣷⣾⣿⣿⡄⠀⠀⠀⠀⢀⣴⣧⣦⣤⡀⠀⠀⠀⠀⠘⠀⠀⠺⠿⠏⠀⣜⣿⡻⢞⡿⠀⠀⣠⣹⠆
⠀⡴⢰⠏⡀⠀⠈⠙⠛⣉⠉⠉⠉⠛⠛⠛⠋⠀⡀⠀⠀⣴⣯⣿⣿⣿⣿⣷⡄⠀⢀⣠⢛⢦⡞⠀⠀⠀⠀⣨⣿⣿⣏⠀⠀⣶⣳⣹⣷
⠀⢷⡜⣧⣷⡀⠀⠀⣰⢿⣦⣆⠀⠀⠀⢶⣽⡌⢡⣀⠀⠻⢿⣿⣿⣿⣿⣿⡇⣰⣼⣹⡎⣿⣷⡎⠀⠀⠋⠉⠉⠉⠉⠉⠁⢀⣽⣿⡅
⠀⣸⡇⣿⣿⠇⠀⣼⡟⢻⡿⣾⡆⠀⡘⢶⡟⣿⢣⣹⢇⠀⢐⣿⣿⣿⣿⡏⠀⢻⣾⣿⣾⣿⣯⣧⠀⠀⠀⠀⠄⠀⠀⠀⠀⢿⣿⣁⡀
⠀⢿⣿⣿⣿⠇⠀⠻⣇⠀⢀⡿⠃⠀⢿⣿⣿⣿⣿⣿⣾⠆⠈⢻⡇⢹⡿⠂⠀⣸⣷⣿⣿⣿⣿⣽⠀⢀⣤⣶⣾⣧⣤⣤⡀⢀⣤⣿⣿

⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⡄⢀⠐⠁⠔⢸⡄⠀⠀⠀⠀⠀⠀⠁⠀⣵
⠀⠀⠀⢀⡠⠈⠋⠆⢰⠆⣝⠀⠀⠀⠐⠐⠶⠚⠙⠿⠿⣖⡂⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⢀⡼⠎⠀⠀⡀
⠀⠀⠈⠐⠂⠀⡧⠐⠒⡠⠡⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠚⣿⡷⠀⣤⢠⠌⣍⢀⣀⣈⣉⡀⢀⡀⠀⠀⠈⠙⠒⠒
⠀⠀⠀⠤⡀⢶⠸⠟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣴⣿⡇⠀⠈⠻⠴⣧⣰⣄⣤⣠⡷⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠃⠀⢰⡄⠀⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣷⣄⣤⣾⣿⣿⡿⠀⠀⢀⡀⠡⠔⠤⠀⠀⠀⡇⠀⠀⠀⠐⠒⠀⠡⡀
⠀⠀⠀⠀⠀⡠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠙⠁⠀⠀⠀⢀⣀⠀⢐⠇⠀⢠⠃⠀⣄⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠸⠀⠀⠹⡇⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠁⣀⣴⠋⠀⠀⠘⢄⣀⢀⣠⣠⣶⡞
⠀⠀⠀⠀⢀⠀⠀⢠⡇⠀⠀⠀⠀⢘⣁⠀⠀⠀⠀⠻⣷⣶⣾⣿⢃⡖⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠨⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠖⠋⠀⠀⠀⠔⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⡀⡇⠀⠀⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣸⡀⠀⠀⠀⠀⠀⠀⢂⣿
⠀⠀⠀⠀⠀⠑⠀⡾⠀⠀⠐⢦⣀⣧⣀⣴⡷⠃⠠⢂⣱⣯⢧⠷⠾⣷⣄⠀⢐⠀⣏⠀⠀⠈⠻⣤⣀⣀⣀⣰⣦⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠃⠁⠈⠉⠆⠀⠀⠈⢻⡆⠀⠀⢹⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠡⠀⠀⠀⢴⡇⠀⠈⠒⢤⣄⣠⣼⣶⠿⠋⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣤⣀⣠⡶⡀⠀⠀⠇
⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⣾⣎⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⣄⠀⠀⠏⠀⠀⠀⠀⠠⠟⠃⠙⠁⡧⠠⠄
⠀⠀⠀⠀⠰⡿⢿⣿⣿⣿⣿⣟⠓⠀⠈⠳⣦⣄⣀⣀⣀⣀⣀⣴⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠡⠙⠀⠆⠀⠀⠜⠃⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⢆⡀⠠⡄⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⡀⠀⠀⣼⣧⠄⠰⣯⡀⠀⠲
⠀⠠⡀⠀⠀⠀⠀⠀⠸⡆⠀⠰⠀⠀⠀⠀⡄⠀⠀⠠⡿⠀⢸⡁⠀⠀⠈⠁⢐⡃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⣶⣿⡤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣄⠀⢠⣿⠀⠀⡘⢿⠀⣼⣇⣘⠀⠀⢀⣼⣇⡀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠠⣤⡍⠱⣆⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⢀⡼⣯⠓⠀⣉⢉⣁⠿⢷⣽⡇⠀⡰⣭⠦⢂⠀⠀⢀⣿⠆⠀⠈⢠⡄⠀⢠⡄⢉⠤⣄⠃⠀⣤⣿⣧⡀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣽⣒⢯⣵⠓⢀⣾⣇⡉⣀⠋⢼⣿⠿⢛⣤⣈⠁⣖⣨⠄⠀⣼⣿⡷⠀⠀⠈⠀⠠⡌⣀⠼⡦⠙⠂⡀⢪⠿⠚⢁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢋⣿⣏⣹⡟⠀⠈⡿⠁⠠⢈⠒⠐⣿⣷⣙⠁⠉⣀⠈⠀⠖⠠⠿⠀⠇⠒⠀⡀⠀⣀⢹⣅⣘⠃⠀⠚⣡⡘⠃⣟⡐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠯⠀⣦⡉⠁⡤⠀⠉⠉⢉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠰⣤⡌⠀⠰⠄⠐⠋⡸⢋⠂⡈⠿⠜
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠐⢟⠑⠧⠀⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠈⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢰⡀⠀⠀⠀⡄⠀⠘⠀⠀⠰⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠹⠆⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⣡⠂⡀⠀⠀⠉⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂⠐⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠠⠄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣤⠶⠏⠓⢷⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠐⠆

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠀⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠤⣌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢰⠁⣘⠄⣀⠛⠄⡀⠂⠌⠀⢀⣀⣀⣀⠀⠤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠌⡘⢀⡏⠔⠂⠚⠂⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⡰⠤⠌⠴⠃⣘⠴⡄⠀⡤⣤⠒⣶⣒⢆⣜⣡⣌⡹⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠁⡊⠀⠀⠀⠀⠀⣯⣱⡇⠀⠁⢉⣉⣉⢉⡁⢤⢄⠀⢤⠐⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⣑⠦⠄⠀⠒⠚⢶⢯⡇⠀⣭⡿⣼⣛⠾⣽⢃⠎⣐⠢⠌⡁⠀⠉
⠀⠀⠀⠀⠀⠈⡐⠠⡎⠁⠀⣀⡤⣴⡖⣶⣸⢿⡇⠀⣿⣽⣳⣯⣿⢣⣏⡲⢠⠋⠴⠅⠀⡦⢀
⠀⠀⠀⠀⠀⠀⢀⠂⠴⠀⠀⠯⠓⣋⣩⣭⢯⣿⡇⠀⣿⣿⣟⡾⣭⣛⡾⣱⠃⡌⣑⠇⠀⡵
⠀⠀⠀⠀⠀⡀⠂⠌⡰⢇⠀⢰⢏⡟⣼⣟⡿⣽⡇⠀⣿⣿⣿⣽⢧⣟⡳⢧⣝⢰⣁⠄⠀⣇⠂
⠀⠀⠀⠀⠀⢀⠀⠠⢁⠺⡀⠀⢿⡽⠗⠛⠛⠋⠁⠀⣿⣟⡿⣯⣟⡶⠋⠁⠀⠀⠀⠀⠀⡏⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠂⡘⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡾⣹⠿⣼⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⢙⡀⠀⡆⠀⠀⠀⠀⠀⢀⡟⡳⢯⡟⠰⢦⡀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣱⠀⠸⡤⢤⣀⡠⢴⠛⠤⢁⠒⡬⢁⠆⡌⠃⠎⡐⠈⠉⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠉⠀⠀⠀⢇⠂⠬⣑⠂⡌⠑⠎⡜⠠⢌⡐⠠⠉⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢸⡌⠠⢀⡘⠠⢁⠒⠠⠁⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡜⠀⠁⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⣤⣤⣴⣾⣿⣴⣾⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠠⠒⠈⠦⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⣿⣿⠁⠈⠛⠿⠛⠻⠟⠿⠿⣿⡟⠀⠀⠀⣾⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⢸⣿⡏⠒⠓⠀⠀⠀⠀⠛⠂⠀⣿⡁⠀⠀⠀⣿⡏⠀⣀⠀⠉⠛⠛⠛⠛⢻⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡚⢿⠀⢀⠤⠄⠀⠀⠀⠠⠄⠀⡼⠘⠀⠀⢤⡿⢀⣤⡤⠀⠀⠀⢀⡉⠂⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠣⡬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢀⠀⠀⠄⣹⠘⠿⠂⠀⠀⠰⣿⠆⠀⣹⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⠀⠀⠰⣶⣶⡶⠀⠀⡰⠟⢏⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠳⢀⣀⡀⠀⢀⠀⠊⠀⠄⠎⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠁⠀⠈⠁⠐⠀⠐⡀⠠⠀⣠⣤⣄⣀⣁⣠⡄⠐⢈⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢀⠘⠂⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠢⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠴⢺⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠄⠂⠀⠀⠀⠆⠀⢲⠀⠀⠀⠀⠀⠑⠂⠠⠐⠀⠀⠐⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠃⠱⠀⠀⠀⠀⠀⠘⠁⠇⠀⠀⠀⠀⠀⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⢀⡐⠀⢀⡁⠀⠀⠀⠀⠀⣅⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠊⠁⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠂⠈⠤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠙⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠶⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡔⠂⠐⡄⠀⠀⣿⡣⠄⠙⠀⠈⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⢘⣃⡀⡀⠀⠀⢘⣖⣟⡠⠂⠀⠀⣿⡠⢄⠀⠲⢘⠀⠀⠀⠀⠀⠻⣷⠄⢀⡉⠁⠟
⠀⠀⠀⠀⢸⣥⣅⢛⣄⠀⠈⣼⠋⠀⠀⠀⠀⢸⠟⢢⠒⠂⠈⠀⠀⠀⠀⠀⠀⠘⣷⠄⡒⠊
⠀⠀⠀⠀⠀⣿⡯⠉⢩⡙⠡⠸⠀⠀⠀⠀⠀⢿⣦⠿⠰⡏⢰⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢹⣷⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠿⠓⢄⠘⢸⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠸⠙⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡦⢠⠛⢸⠀⠀⠀⠀⠀⠀⢄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⠢⠴⠰⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⢀⣷⡀⠀⡀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠖⠀⠁⣄⠁⠈
⠀⠀⠀⠀⠀⠀⢸⣿⢧⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠐⣠⠶⣌⣶
⠀⠀⠀⠀⠀⠀⢸⣿⣾⣦⠤⠦⣻⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣭⡏⠀⠩⠃
⠀⠀⠀⠀⠀⠀⠘⠛⠿⢀⡀⠀⠀⠀⠀⠀⠀⣯⣁⠌⠂⠒⢙⠀⠀⠀⠀⠀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣿⣧⢄⠀⢀⢘⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢁⢨⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⠀⠀⡀⠀⠀⠀⠀⣿⠟⠶⠄⠛⠛⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⢠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠰⢈⡘⠀⠀⢀⡿⠃⠀⠀⠀⠀⠰⣮⠇⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⡀⠀⠠⡄⠀⠀⠀⢀⠀⠀⠀⠀⣐⠀⠑⣤⠀⠉⠀⠎⠀⠀⠀⠀⠀⠙⠄⠀⠀⠂⠀⣁⣤⣤⣶⣟⣛⣓⠒⢒⠋⠀⠤⢶
⠀⠁⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢩⠐⠘⠿⣧⣦⣄⣈⠀⠀⢀⠀⠀⠈⠁⠀⠀⢀⣼⣿⠿⠁⠀⠀⠀⠀⠈⠈⠃⠘⠒⠒
⠀⠀⠀⢸⡇⠀⠀⣓⠀⠀⠀⠀⠀⠬⠀⠀⠀⠈⠁⠉⢿⣻⣳⣶⣶⡄⠦⠴⡄⡐⠋⠉⠁⠀⠀⡆⠄⠀⠀⠆⣀⡀
⠀⢀⠀⣸⡇⠀⣸⢸⣃⢀⢠⢷⡒⠾⠀⠂⠀⠒⠀⠰⠈⠓⠒⠈⢀⠆⠀⠀⠀⠈⠑⠂⢤⠀⠀⠁⠤⠈⠀⠦⠸⣽⠳
⠀⢸⠀⢸⣏⠀⢼⠍⠋⢺⠛⠉⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠐⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠪⠘⠂
⠀⠘⠋⠩⠁⠐⠉⠤⠀⢸⡁⠁⠀⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠
⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡁⠀⡤⠂⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⣄⠀⠀⠀⠀⣤⠀⡄⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠈⣠⠃⠀⠀⠀⠀⠀⠈
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠠⠃⠀⣤⣰⠦⣋⣦⠓⠀⠀⠁⠀⠀⠀⣀⡀⠀⠈⠩⠀⠀⠀⠈⢀⠀⠀⠀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠚⠓⠋⠉⠈⠘⡆⠀⠀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⠀⠈⠃⠀⠀⣦⠜⢄⠘
⠀⠀⠈⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡀⠀⢀⠀⠐⠿⠰⣜⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠏⠈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢷⠖⠁⠈⠋⣠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣾⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠰⠶⣤⡀⣤⠒⠀⢁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠳⣄⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠘⣷⡄⠀⠀⠐⠶⠀⠈⢏⠛⢦⡀⢸⣧⣀⠀⠀⠀⢰⡀⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢹⣿⡆⢀⣀⠀⠤⠤⠼⠿⠶⣇⢸⣎⠻⣆⠀⠀⢸⢷⡸⡿⢦⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⠀⡀⠤⠄⠒⠒⠘⡖⠚⠺⡿⢿⣿⠀⡀⢸⠀⢻⢃⠀⠃⢸⢿⣄⢀⡀⢀⡀
⠶⠦⣶⣶⠞⠛⠛⠓⠢⢀⡴⠊⢸⣿⣿⡉⢀⠤⠾⠂⠉⣁⣩⡅⠈⣇⠀⢻⢽⣿⣿⠀⠚⠈⣏⠉⢹⣀⢹⡆⠀⠛⢸⡆
⠀⠀⠀⠈⠉⠓⠢⠤⣠⣋⠀⡠⢺⣿⣿⠛⢧⣴⣶⣾⣥⣈⠿⠁⠒⡧⠠⠉⡐⠄⣺⡐⢨⣾⡷⠀⢺⠂⠘⣥⣐⠛⣻⠴⣆
⠀⠀⠀⠀⢶⠆⠀⢰⢣⣿⣿⣇⠸⣿⣇⢀⣿⡟⢧⡞⠙⢻⡆⠸⠿⠷⠀⠀⠀⢁⣼⠀⠉⡩⢐⣬⣽⢴⣶⣍⣿⠛⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⣤⣇⠀⢻⠿⠟⡼⠙⢿⣧⡙⡿⠥⠈⠲⣼⠃⠐⢾⡿⠀⠀⠘⠛⠋⠐⠍⡀⠿⠿⠋⠀⠉⠁⢠⡄⠠⣦⣤⠖⠁⢴
⠀⠀⠀⠀⠀⢸⣿⣿⠏⣻⠶⢤⣇⣀⢸⠉⠻⠷⠖⠒⠋⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠜⢵⣶⠎⠈⢀
⠀⠀⠀⠀⠀⣼⣿⣿⠀⣿⠀⠀⡇⠉⢙⠓⠶⣤⣀⠀⣶⣷⣀⠟⠀⠀⠀⢀⡀⣀⢀⡀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⡞⢉⡟
⠀⠀⠀⠀⠀⣿⣿⡏⠀⢸⢀⡴⣷⣶⣼⡀⠰⣿⣿⡆⠀⠉⠀⠀⠀⠀⠀⢸⡇⣿⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣧⠉⡄
⠀⠀⠀⠀⢠⣿⣿⣧⠀⠈⢿⣴⣶⡀⠙⣷⠀⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣸⣿⢿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢀⡀
⠀⠀⠀⠀⢸⣿⣿⠘⡆⠀⠘⡄⠀⢱⡀⠘⢆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣶⣶⣞⢲⡖⠀⠀⠀⠀⠀⢠⡆⢀⣀⡿⣶⡆⣾⡏
⠀⠀⠀⠀⠈⢉⡉⠙⠛⣖⠒⠻⡦⠤⠷⣤⣌⣢⠀⠸⣤⡀⣀⠀⠀⠀⠀⢸⡇⢿⡿⣿⢩⠿⣧⠀⠀⠀⠀⠀⠈⠁⡘⠻⢿⣤⠁⠉
⠀⠀⠀⠀⠀⠛⠃⢀⣀⡈⢶⣦⠘⣆⠀⢿⣿⡷⠑⣔⠿⠳⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢆⠌⠲⡄⠠⡤⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠨⠿⠶⠈⠻⣃⡠⠷⡌⢉⡱⣶⣾⡧⡀⠈⣢⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠖⠙⣦⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠢⣀⡼⠫⡀⢈⣻⣁⠈⣿⢋⣉⠫⡉⠀⠀⠀⣴⠇⠀⢰⠗⣼⠇⠜⣿⡠⠀⣨⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣿⣧⠖⠉⢢⡠⠚⠻⣿⠿⠓⢷⠾⠛⢄⠀⠑⢄⠀⠀⡄⠀⠀⢢⠜⣄⡠⠊⢣⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣼⠟⢁⣤⡊⠁⠈⠢⡀⢈⡷⣶⣾⣷⢤⡀⠑⠢⣠⣷⢆⡘⡤⢊⠔⠱⡈⢣⣾⡗⠙⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⢺⣧⠹⣷⡄⣀⠴⠊⠓⢄⠈⠙⠯⡀⠈⠢⣀⠹⠫⡀⢉⢘⣅⣠⡾⠋⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⢻⣧⡜⣿⣄⠀⠀⠀⠀⣙⡦⡖⣿⣿⣷⡌⠑⢧⣈⢔⣴⡿⠋⠀⠀⠰⣤⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡘⢿⣦⣠⠔⠊⢁⣀⠈⠛⢍⣩⠗⢆⡘⣷⡿⠋⠀⠀⠘⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠚⠉⠀⠀⣤⣈⠙⠋⢠⣾⠛⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⠼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠑⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠒⢦⡀⠈⢁⡔⢁⢍⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢨⣀⡠⠶⠖⠚⣛⣛⣉⣉⣉⢭⡽⠃
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⣇⠐⡩⢦⡘⠒⣡⡈⠢⡀⡰⣡⠔⣒⣒⢠⠃⠀⠀⠀⠀⠀⠰⠯⣥⠶⡖⠉⠙⢉⠁⠀⠈⢸⡄⠀⡇
⠀⠀⠀⠀⣶⡋⢻⡆⠙⠲⠤⢤⠔⡡⣊⡴⣛⣳⣄⠉⢶⠄⠈⠳⡇⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠠⠈⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⢎⡴⣫⣾⢟⣵⣿⣦⡀⠀⠀⡤⠜⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠠⠄⣿⠀⡇⠀⠀⢸⠃⠀⠀⠀⠃⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⢔⣹⠿⣋⣴⡷⡄⠀⠘⢆⠀⠻⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⣀⣀⣰⣦⠦⠤⠚⠛⠉⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠻⡥⣪⢟⣭⡾⢟⣦⠀⠀⢳⣀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣉⣉⣉⠥⠤⠤⠒⠒⠚⠋⠻⣧⣤⣤⣤
⠀⠀⠀⡠⠂⠀⠀⠐⠄⠄⠀⠾⠿⠀⠀⠀⠘⢷⠟⡫⣴⡿⡫⣣⡀⠀⠉⠀⢱⣀⡀⠀⠤⠠⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⢀⢀⠠⠇⠀⠀⠀⠀⠀⣀⠤⠤⠒⠒⠈⠉⡂⠀⠻⣊⢕⡪⣞⣭⣷⣂⠀⠀⠀⢷⢓⠲⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿
⡎⠀⠀⠀⠀⠀⠀⠀⠸⣴⡀⠀⠀⠀⠀⢠⠇⠤⠤⢽⣯⡪⢟⣫⣶⣷⠂⠠⣶⢸⠀⠀⠀⢔⣢⠒⣼⡂⣀⠀⠀⢀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡇
⠈⠀⠀⢒⢀⡀⠀⠀⠀⠈⠙⢶⣤⡀⠤⡼⢦⠤⣤⣃⠘⣾⣟⠟⣋⢼⡧⠤⠒⠉⠀⢰⠒⣤⠿⣋⡡⠄⠈⢷⣾⣷⣄⠈⠉⠏⠀⠙⢯⡁
⠀⠀⠀⠈⠁⠙⠦⣄⣀⠀⣰⡪⠛⢧⢠⠇⠈⠲⠃⠈⠉⢙⣶⣿⣿⡉⠀⠀⢀⡤⠒⠉⠱⡜⢧⠀⠐⠂⢀⡼⠈⢿⣿⣆⠎⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡹⠒⠒⠒⠢⣤⠀⣾⣿⣿⠋⣡⠴⠊⠁⠀⠀⠀⢀⣵⢈⡧⢴⠊⡿⣦⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠘⠂⠤⠼⠁⠀⠀⠀⠀⢹⠀⢻⡿⠗⠉⠀⠀⠀⣀⠴⠒⠉⠉⠈⢋⠀⠛⠃⠈⠁⠀⠀⡟⣠⡆⠀⠀⠀⠀⠀⡟⠫⠯⠃
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠒⠉⠀⠀⢀⡤⠖⠛⠁⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠰⣟⠄⠃⠀⠀⠀⠀⠀⠃⠀⢀⣤
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⢰⠃⠀⠀⣠⠴⠚⠉⠓⠦⡄⡀⠀⠀⢰⣿⢋⠽⢻⣾⣧⡀⠀⣀⠀⠄⠀⠈⠀⠀⠀⠀⢸⣶⣤⡀⠉
⠀⠀⠀⠀⡄⠀⣀⡀⠀⣬⠀⠀⠹⡆⠀⠀⠀⠈⠳⣍⣉⠀⠀⠀⠀⠀⢄⠈⠳⠀⠀⠸⣷⣝⠊⣠⣿⡴⠋⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷
⠀⠀⠀⠀⣷⠈⣿⣇⠀⠙⠧⣄⠼⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣶⣶⣬⣉⠀⠀⠃⠀⠀⠈⠓⠶⠿⠿⠅⠀⠰⠆⣉⠢⣄⠀⠀⠀⠀⢠⡿⠿⠟⠁
⠀⠀⠀⠀⢸⡆⢹⣜⡄⠀⠀⠀⠀⠀⠀⢀⠔⠉⢱⣶⣿⣿⣿⣿⣿⠟⠉⠛⢷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠳⢄⠀⠀⡜
⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⢀⣿⠿⠿⠻⣿⣿⣄⣀⣀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⠀⠒⠒⠾⢷⣈⠀⢀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠉⣀⡤⠔⠊⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣄⣠⣤⡶⠖⠉⠉⠀⠁⠢⣀⠀⠀⠠⡖⠒⠢⣝⢆⠀⠀⢱⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡀⠀⠀⠰⠋⠈⠀⠙⢦⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⣉⣀⣀⠤⠤⠀⢠⣤⣈⡑⢦⡔⠋⠀⠀⠀⠀⢣⠀⠀⠀⠠⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⢣⠀⠀⠀⠀⡀⡇⠀⠘⢿⣿⣿⠿⠋⠀⠀⠹⡄⠀⡰⠁⠀⢸⡿⠋⠁⠀⡍⠓⠢⠤⠤⠴⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠑⠤⠤⠤⠜⠁⠀⠀⠈⢟⠁⣀⠤⠤⠤⣄⠘⣴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠠⡄⠀⠀⠀⢀⠏⠒⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⡠⠤⡄⠀⠀⠈⢎⠃⠒⠒⠒⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠓⠤⠂⠀⣀⣠⣼⠀⠾⠭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣤⣴⣾⡿⠿⠿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣹⣛⣩⣙⣙⣟⣩⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡍⡜⠻⠄
⠀⠀⠀⠀⠠⠔⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀⣤⣶⡆⠀⠀⠀⠀⠀⠀⣴⡴⠂⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠸⢀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡆⠀⠀⡼⡿⠿⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠸⢰⣿⡿⣿⡤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠄⠒⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣾⠟⠏⠈⢣⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⣤⣯⠤⡀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣰⠃
⠀⠀⠀⠀⢀⡙⠻⠿⠷⠾⠿⣫⡄⠀⠀⠀⠀⠀⠀⠈⠲⢼⡷⠀⠀⠀⢠⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⢤⠇
⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣁⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠒⠂⠒⠚⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠘⠢⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⡇⠀⠛⣿⡿⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠉⠉⠲⠤⡌⣉⠀⡤⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⣼⡄⠀⠀⠘⠙⠓⠒⠒⣒⣲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠺⣿⣧⡠⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⢻⡏⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⣷⣂⡀⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣿⠧⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣿⣷⣦⠀⣡⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣥
⠀⠀⠀⠀⠀⢀⣿⣻⣛⡓⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠲⠤⠭⠭⠙⠂⠈⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣤⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⣴⣿⣿⣿⣿⣿⣷⣾⣿⣦⡀
⠀⠀⠀⠀⢰⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠉⠛⠃⠀⠀⠀⠀⠀⣿⠀⠈⠙⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠉⠉⠉⠁⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⠿⠼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⡴⣯⠀⠀⠀⠀⠀⠀⠀⠏⢹⣿⣿⡇
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠔⠁⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠍⠐⣆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣗⣾⣀⡤⠠⣔⣁⣀⡀⣈⠢⠀⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠐⡀⠀⠈⠑⣤⠀⠀⡀⠀⠀⡀⠀⠆⠉⠀⠐⠁⠀⠀⡠⠀⠈⠂⠀⠈⠁⠎⣀⠠⠚⡛⠛⡀⠜⠚⠷⠺⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠐⠊⠀⡆⠑⢦⠤⣚⣁⣠⠤⠞⢦⣤⡆⠀⠀⠀⠀⠀⠀⢰⠀⣤⠔⠉⠀⠀⠌⠀⠀⠀⠀⠀⠀⠡⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠖⠀⠀⠀⠀⠀⠀⠌⠀⠄⠨⠄⠀⠂⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⡎⠀⠀⠑⢠⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠠⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡀⠀⠀⠒⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠐⢦⠒⠂⠒⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠦⠦⠶⠤⠶⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢲⡀⠠⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠢⢨⡀⠀⠀⠀⠀⠀⣦⣤⣤⣤⣤⡤⠤⢴⡇⢡⡎
⠀⠀⠀⠀⠀⠀⠀⠐⣶⣮⠁⠀⠀⠐⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠨⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⡹⣄⡀⣾⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⡖⠒⠞⠳⠒⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡇⠀⠀⠀⠀⠀⠃⠀⠀⢈⡃⠈⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠁⠀⠀⠀⠀⠀⢹⠀⠀⠸⠃⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣄⠀⢨⣀⣨⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡁⠀⢈⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⢰⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠛⠛⠛⠋⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢐⣿⣿⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇
⠀⢰⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣴⣿⠉⢠⣿⡟⠋⠀⠀⠀⠄⠀⢀⠒⠀⠀⠁⢀⣾⠀⠀⠀⠀⠀⠀⠺⠟⠛
⡆⣼⢃⠀⡀⠀⢠⣶⠟⠀⠀⠀⢀⣀⠀⢰⣿⡟⠁⢀⣼⡿⠇⠀⠀⢹⠀⠀⠀⠀⠒⢂⣄⠀⣸⠟⠁⠀⣤⣾⣷⡄⠲
⠷⣿⠄⢰⡇⢀⠆⠻⣷⡀⠈⠀⠀⠁⠀⠘⣿⣥⣶⣿⣿⣧⣄⠀⠀⠈⠃⠀⠀⠀⠀⠾⠟⠀⠉⣀⠴⠊⠉⠈⢠⠀⠀⣈
⣳⣿⠏⠹⣧⢹⠅⣤⠸⠿⠧⢀⠐⠀⠀⠀⠉⠙⠛⠙⠟⠉⠻⠟⠀⣆⠀⠀⡴⠇⢀⠀⡀⣠⠀⡀⠀⡗⢠⠀⣾⢠⣆⣿⣾
⣿⣿⣧⣠⣿⣿⡄⣿⠀⡀⢰⠹⣦⢦⡀⠀⣀⠀⠀⣀⣆⣀⠀⢠⢀⠉⠀⢠⣤⣦⣸⣴⣇⣿⣾⣿⣾⡟⢺⣿⣿⣾⣿⣿⣺
⠈⠉⠉⠉⠙⠿⢷⣿⡾⠟⠛⠙⠛⠋⠁⠀⠛⠁⠀⠉⠙⠛⠙⠋⠛⢃⠀⠈⠻⠟⠿⣿⣿⠿⠿⠟⣿⣦⠾⠛⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣲⣖⠂⠁⠀⠀⠀⠀⠀⣀⣀⠐⠂⠠⠀⣄⣀⣠⣤⣤⣀⣀⣀⠀⠀⠀⢀⣁⣀⣤⣶⡶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⢤⣤⣄⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣀⠀⠉⠉⠉⠉⠉⠙⠻⠿⠟⠉⠁⠀⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣥⠶⠂
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠺⣿⣤⣀⣁⣨⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠤
⠀⠀⠈⠀⢀⣦⡄⢙⣿⣷⣶⣿⣿⣿⠿⢿⣿⣿⣿⡿⠗⠚⠃⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⣤⠤
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⣥⠀⠀⠈⠙⠻⣿⣥⠈⠉⠀⠉⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠘⠃⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠠⠀⠀⡀⠀⠔⠀⠀⠈⠀⢄⠀⠂⠥⠔⠂⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠐⠀⠀⠀⠀⠈⠀⡄⠠⠀⠀⠂⠀⢀⣀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠂⠀⠀⣁⠀⢸⠀⠀⡀⠀⠠⠀⠀⠐⢀⠈⠀⠀⠁⠢⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢸⡇⠀⣿⠀⣺⡇⠀⠀⠀⠀⠐⠀⠀⢀⡁⠀⠀⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠅⠀⣍⠀⢨⠠⢅⠀⢀⠀⠀⠀⠀⠜⢃⠀⠀⠀⠁⡄⠈⡭
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠐⠀⠍⠀⠈⠐⠀⠐⠀⠈⠀⠀⠤⠠⠄⢖⠅⢄⠀⠠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡆⠀⠀⡀⠄⠂⠈⠀⠀⠀⠀⡀⠀⠀⠀⣺⣇⣯⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡇⠀⠂⢀⠀⠀⠀⠀⠠⠀⠈⠀⠀⣿⠀⠙⠈⡛⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢄⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣿⠀⠀⢰⡇⠀⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠲⠍⠀⠀⠀⠀⢀⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠈⢹⣿⣷⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠚⢯⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠓⢠⣴⣟⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⣰⡿⠀⣴⠦⣶⣎⠃⢋⣽⣿⣿⣿⣿⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢼⣷⣳⡛⠁⣆⡜⠀⠏⠋⣠⢸⣿⣿⣿⣿⢯⡿⠂⢹⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠆⣎⣿⣿⣠⠾⠿⢁⣴⠆⠀⠏⠆⣿⣿⡿⣦⣾⠁⢠⣿⣿⠿⣿⣆
⠀⠀⠀⠀⠀⠀⢴⣿⣿⣰⣿⣿⡿⣿⣶⡀⠘⢃⡴⠀⣤⡲⢿⣿⣿⡿⠋⠀⣼⣿⣿⣿⣿⡛⣦⡀
⠀⠀⠀⠀⠀⣀⣼⣯⣹⣾⣿⡿⣿⣼⣿⡻⣦⣿⠧⠀⣹⣿⣵⠿⠋⠀⠀⢺⣿⣿⣿⣿⣿⣷⡼⣷⣄
⠀⡀⠀⣤⣾⣟⣿⢿⠿⣿⣿⢻⣾⡝⢻⣷⣾⢷⣠⣦⡿⣋⡅⠀⠀⠀⢀⣾⣻⣿⣿⣿⣿⣿⢿⣬⣿⣷⡀
⠀⣠⣾⡿⣷⣾⣿⣿⣶⣿⣿⣿⠙⣿⣾⣿⣮⣧⣹⣷⡏⢿⡶⢀⣄⣴⣿⣿⣿⣿⣿⢿⣿⣿⣜⣿⣾⣿⣿
⢸⣿⣯⣿⡿⣿⣿⣿⠸⣿⣿⡏⠀⢽⣿⣿⠃⣿⣿⡿⠀⢸⣷⡟⢹⣿⣿⣿⣿⠃⣿⣿⣿⡏⣿⣿⡿⢻⣿
⢸⡟⢹⣿⠇⢸⣿⠇⠀⢻⣿⠁⠀⠨⣿⡟⠀⢸⣿⠇⠀⠈⣿⠁⠐⣿⠇⢻⡿⠀⢸⣿⣿⠁⢻⣿⠇⠘⣿
⢸⠇⠘⡿⠀⠀⣿⠀⠀⠘⡏⠀⠀⠀⢸⠃⠀⠈⣿⠁⠀⠀⠁⠀⠀⠛⠀⠘⠃⠀⠀⣻⡟⠀⠘⡿⠀⠀⢿
⠸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠉⠀⠀⠸

⣷⣷⣿⡟⠁⣠⡶⣿⣿⣿⣿⣮⣷⡍⠻⣿⣶⠶⣖⣻⣻⣛⣰⡆⠦⣬⢈⡛⢲⢤⡙⠑⠆
⣿⣿⠏⢀⠞⡙⠂⠘⢹⣿⣿⣿⣿⣿⣿⣮⣻⣿⠁⣇⠉⠈⠹⠙⢋⠗⡶⣍⡑⠦⠛
⣿⡟⠀⡞⠰⠀⠀⠖⢙⠟⠛⢯⡿⣿⣿⣿⣿⣿⠿⠛⠿⠛⠙⠟⡶⢦⣤⣀⠛⠓⠲⢄⠀⠀⠀⠀⠀⢀⣤
⢭⠇⠀⣧⠁⠀⢻⠎⠁⠀⠀⣾⡿⢿⣾⣿⢿⣿⣯⡙⠿⢦⡘⠷⣷⣿⣭⣇⠀⠀⠀⠀⢣⠀⢀⣠⢾⣿⣽
⣧⣾⡀⢲⡀⡐⠀⠀⢰⣿⡆⢀⣴⣾⣿⣕⢌⣿⠟⣮⡢⡈⠟⢶⣽⠿⣿⣿⣿⣤⠀⠀⢠⠐⢿⣿⣿⣿⣿
⣷⣿⣇⠀⠛⠀⣠⡤⡀⠉⣰⠟⣵⣯⣙⣿⢿⣟⠻⣿⣺⡯⡢⣈⠻⣷⣜⠿⣿⣿⣷⣤⠜⠃⣀⢿⣿⣿⣿
⣿⣿⣿⣷⠀⠀⠻⠷⠇⣼⡇⣽⣿⣿⣻⣼⡻⣿⣷⣝⣻⡮⡪⡻⣅⣍⠙⢳⣞⠻⣿⣿⡄⠀⠉⣈⣿⣿⣿
⣿⣿⣿⡇⣠⡖⢦⠀⣸⠿⠗⣿⡿⠛⠛⣿⡿⠿⢿⣟⣛⣿⣛⠊⠪⡳⠕⢌⠹⣷⣮⡉⠙⢄⠀⢀⣸⣿⣿
⣿⣿⣿⠀⠻⣿⠿⠃⣿⠃⠘⣿⣁⣴⣾⡟⢂⣴⡌⣿⠿⠿⢿⣯⣦⡁⠀⠈⠳⣌⠿⡿⠀⠀⠀⠀⢼⣿⣿
⠿⣿⣿⠀⠀⡠⢄⠀⢿⡀⠀⢻⣿⣿⣿⣷⣿⣿⣷⢸⠀⠀⢀⣿⣿⣼⡢⡙⣆⠈⠀⠀⠀⠀⠀⠀⡀⣿⣿
⣷⡿⢿⡄⠸⣷⡾⠂⠸⣧⠀⠈⣿⣿⣦⢿⣿⣿⣿⣤⣷⣤⣾⣿⣿⣿⣿⣮⡺⣷⠄⠀⠀⠀⠀⠀⠑⢎⡛
⡟⣷⣾⣇⠀⠀⣠⠤⠀⢻⣇⠊⠈⢿⣿⣷⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢮⠀⠀⠀⠀⠀⠀⣇⣀⠑
⣟⡿⣿⣿⡄⠐⣿⣷⣶⠀⠁⠀⢀⡀⠀⠙⠿⣿⣾⣿⣿⣿⣄⣀⣽⣿⣿⣷⣤⣼⡇⠀⠀⠀⠀⣨⡃⡝⢴
⣿⣿⣿⣿⣿⣄⠈⠉⠁⠀⢀⣾⣯⣀⣡⠀⠀⠀⢀⣩⡿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⡪⣻⣮⡓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢳⣸⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⠁⡇⠀⠈⣷⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣾⠀⢷⠀⠀⠘⣧⠀⡞⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿⠀⢸⡆⠀⠀⢹⡄⢧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠲⣄⠀⠀⠀⢿⠀⠹⣆⣸⠇⠀⠀⠀⣧⠘⣧⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⣇⠀⠀⢸⡆⠀⠈⠁⠀⠀⠀⠀⣿⠀⣿⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣿⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⢻⣴⠏⠀⢸⠇⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡟⠀⣷⣄⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⡇⠀⣿⠈⣷⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠁⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢠⡿⠀⢸⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡾⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢴⣟⡉⠙⠳⣄⠀⣿⠀⠀⠀⠙⠋⠀⠀⠸⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⢀⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⢹⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡾⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⢻⣼⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⢦⠀⢠⣄⠀⢸⠸⣦⡾⢡⡇⠀⠀⠀⠀⠀⠀⠈⣿⢿⡀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠷⠞⢻⠀⣸⠀⠀⠀⣼⢣⢶⠀⠀⠀⠀⠀⠀⢹⡼⡇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⡴⠋⠀⠀⠀⢿⡾⢸⠀⠀⠀⢠⠀⠀⠸⣿⠃⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡿⣆⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠹⢦⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠞⠋⢸⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣼⠇
⠀⢀⣴⠶⡦⠤⠤⣀⣹⣧⡀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡾⠀⠀⠀⠀⣠⣾⡧⠤⠤⠴⢶⠶⢦⡀
⠀⢸⡇⠀⢸⠀⠀⠀⠀⣈⠻⣦⣀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠞⠁⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⣬⠀⢸⡇
⠀⢸⡇⠀⡙⠀⠀⠀⠀⠀⠀⣀⣉⣛⣶⣤⣄⣙⡳⣦⣤⣀⠀⠀⠀⠀⣀⣀⣠⣤⣖⣋⣡⣤⣶⣟⣉⣀⣀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡇
⠀⠈⠻⠾⠟⠛⠛⠛⠛⠛⠋⠉⢉⣩⠿⠛⠉⠉⠉⠙⣻⢿⡟⠛⠛⢻⡟⠻⣯⡉⠉⠉⠉⠳⢦⡈⠉⠉⠉⠉⠉⠉⠛⠒⠒⠚⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁⠀⡠⠄⢀⡴⠟⠁⣾⠁⠀⠘⠘⣇⠀⠈⠻⣄⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⣠⠶⠟⠩⡄⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⡏⠀⠀⠀⠀⣿⠀⠀⠀⠈⠳⣄⠀⠀⠀⢤⡀⣙⢳⣤⡀
⠀⠀⠀⠀⠀⣼⡁⠊⠀⠀⠜⠀⠀⢀⡴⠋⠀⠀⠀⠀⣼⠃⠀⡰⠀⠀⣿⡀⠀⠀⠀⠀⠙⢷⡀⠀⠀⣽⡉⠀⠊⡙⣦
⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⢀⣿⡾⠋⠀⠀⠀⠀⠀⠀⡟⢀⣀⣀⡀⠠⢹⡇⠀⠀⠀⠀⠀⠀⠙⢦⣼⠁⠀⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠈⠻⠶⡶⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⢸⡟⢫⠍⠉⠉⠛⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣄⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢁⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣥⣤⣤⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡜⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠱⡄⠀⠀⠀⣶⡀⠁⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡷⠀⠀⠀⢇⠙⠤⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡅⠀⠀⠀⠀⠘⢦⠀⠈⠢⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠈⢹⢀⣖⠀⠀⠀⠈⠳⡄⠀⠈⢢⠀⠀⠀⡠⢊⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡾⠈⢌⠢⠰⣤⡀⠀⠈⣆⠀⡀⣇⠀⢈⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨⢀⡀⢡⠀⠀⠈⠃⣱⢙⡆⠀⡜⢠⠀⢸⠀⢀⡖
⠀⠀⠀⠀⢠⣀⠀⠀⠀⠸⡀⠠⠑⢄⠑⢄⠀⣠⠃⠀⣧⠞⢀⢈⠀⡅⠀⣸⢱
⠀⠀⠀⠀⠈⠈⠑⢦⠀⠀⠳⡀⠣⠀⠑⡌⢷⢁⠜⠄⠉⠐⠁⡀⣰⠁⢀⠏⠨⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⣁⡀⠇⠀⠀⠙⣄⠡⠀⠈⠄⠄⠀⡠⠊⠀⠐⣠⣃⠴⠋⠀⠀⡇⠀⠀⡠⢺
⠀⠀⠀⠀⠀⠀⠈⠫⠑⠦⡀⠀⢸⡀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠋⠁⠀⡚⠁⡸⠃⠀⡊⠴⠋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡓⢤⠘⡄⢸⠁⡀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠔⢀⡼⠁⡀⢀⡔⢪⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠡⠹⠏⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡎⣠⡾⢡⠋⠠⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠄⠀⠆⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⢠⡟⡃⣇⡏⠴⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠣⠈⠙⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠂⠘⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠄⡀⠀⠈⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠐⠉⠀⠀⠀⠈