Black tea Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡟⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⢿⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⡘⡇⡄⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣶⠛⠉⠉⠉⠉⠉⠉⠉⠛⣶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠩⣤⣶⣶⣶⣶⣶⣶⣶⣤⠍⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠟⠻⡆⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⣠⣌⢹⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡼⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢃⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠄


⠀⠀⠀⠠⣤⣄
⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠙⠻⠷⠢⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠔⠊⠁⠂⣀⢀⣀⠔⡰⠁⠀⢀⠔⠀⡈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⢰⠁⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠯⣁⡈⠀⠀⠀⠀⠈⠁⠠⠤⠀⠀⠀⠸⣀⡂⠔⠩⠄⢐⡂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣿⣤⣤⣉⣉⣀⣂⡀⠄⠀⠂⠼⠿⠿⠟⠓⠠⠖⢀⣀⣀⣀⣀⡺⠾⣃⠿⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢽⠙⠛⠻⠏⠿⠛⠛⠛⠒⠶⠒⠲⣶⣶⣶⣶⣾⡿⣿⣿⣿⣿⠛⣿⠃⢠⠔⠊⠁⠀⣀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣒⢀⣩⣭⠿⠿⣟⣛⠀⢑⡁⢸⠃⣠⠞⠉⠉⠉⢳⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⢀⡲⢒⣚⣿⡿⠽⢖⣀⠡⡂⢻⠞⠀⠀⠀⠀⠀⠀⣧⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣤⠀⠠⠴⠶⣒⣻⡭⠄⣈⠁⣿⠀⠀⠀⠀⠀⠀⢀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠀⠒⣂⣀⣉⣭⡭⠤⠖⢀⠥⠀⡇⠀⠀⠀⠀⠀⣠⠎⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠬⠍⠭⠴⠶⣒⣛⡁⠐⠂⠀⣇⢀⡀⠤⠒⠋⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⢀⣚⣛⣭⡭⠵⠀⢋⠄⢠⡷⠂⢀⣀⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠠⠴⠶⣒⣚⣫⡅⢐⠀⣼⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠈⣉⣭⣭⣵⠖⠤⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣷⣍⠠⠀⠀⠀⠀⠀⣀⢀⣀⠐⣀⣐⣚⣿⣿⡿⠯⢈⣥⣏⣀⣀⣀⣀⣀
⠀⠀⠀⣀⣤⠴⠖⠒⠚⠹⠏⠉⠀⠀⠠⠤⠤⠿⣗⣦⣔⠀⠀⠀⠐⠀⢛⡀⣺⣿⣿⣿⣯⣧⣴⣿⠿⠿⠿⢿⣯⣍⣍⣉⠛⠛⠲⠶⣦⣤⣀
⠀⠀⢸⡁⣄⢠⡀⠀⠀⢎⠀⠀⠀⠀⠀⠰⠀⠀⠈⠣⠄⣀⣀⣀⣀⣀⣉⣍⣩⣶⣶⣶⣤⡦⠞⠁⣀⡠⠀⠀⡀⠀⠈⠉⣙⠆⠀⣠⢈⣹⣹⡇
⠀⠀⠀⠙⠳⢭⣛⡒⠶⠦⠤⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢉⠉⠉⠉⣉⣉⣡⣄⣀⣀⣀⣰⣶⣶⣶⣦⣤⣤⠤⠤⠶⠖⢛⣫⡽⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠦⣤⣤⣤⣀⣈⣉⣉⣉⡉⠉⠉⠉⠩⠉⠉⠩⠭⠉⠉⠩⠉⠉⣉⣉⣉⣉⣀⣤⣤⣤⠴⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢴⣄⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠈⣿⡆⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⢀⣴⠟⠁⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠸⣿⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠹⣷⡀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⠻⠿⠶⠶⢶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⠶⠶⠶⠿⠟⠛⣻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢸⣿⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠸⠿⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠙⠻⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣿⣶⣶⣤⣤⣤⣤⣶⣶⣿⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣍⣉⣋⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⣭⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢠⡄⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠠⣿⠁⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠹⣧⡀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣼⠇⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⣠⣴⠶⠾⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⣀⣀⣠⣤⣤⣤⣤⣴⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⡀⠀⠀⠙⣷⣀⣤⣤⣤⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣯⣿⣿⣶⢾⣿⣯⣥⢶⣤⠙⣷⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⠀⠀⠀⢻⡇⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠀⣾⡏⠁⠀⢀⣿⠃⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣷⣄⣀⠀⠀⠀⠀⠀⠀⢠⢪⡿⢀⣠⣴⠟⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⡯⠈⣿⠀⠀⠀⠀⠀⠀⢠⣿⠿⠛⣉⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠄⠀⠀⠀⠹⣷⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰⣿⠷⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⣿⣦⡀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣿⠀⠀⠀⣠⣼⣿⣄⣀⡀⠀⠀⠀⠀⠀⠐⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠋⠉⠁⠀⠈⠛⠷⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠟⠋⠀⠀⠉⠉⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⡶⠶⣄⠀⠀⠀⠀⢠⢦⣄⠀⢀⡟⠁⠀⠀⠀⢸⡇⠀⣴⢶⠀⠀⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠸⣇⠀⢀⡿⠀⠀⠀⠀⢻⡀⣽⠀⢼⣷⠆⠀⠀⠀⢸⡇⠀⣷⡞⢀⡸⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣀⣤⡶⠋⠀⢷⡾⠛⠀⢿⡶⠟⠁⠀⠀⠀⠀⠈⠛⠋⠀⠀⣧⠀⠀⠀⠀⠈⠇⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣅⠀⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢤⡈⢓⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠁⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡶⠊⠀⠘⠀⠀⠙⢶⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣿⣋⡁⠀⠀⢣⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠙⠛⠿⠭⠭⣾⣒⣒⣒⣶⣶⣶⡾⠷⠿⠛⠋⣩⠷⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠞⢻⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡼⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢙⣋⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣏⠒⠤⢀⣀⣀⡀⠀⣀⡀⠤⢒⡏⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣌⣉⣓⡒⠒⠒⠶⠶⠒⠒⢊⣉⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠛⠛⠻⠿⠿⠛⠛⠋⠉⠁

⣿⡿⠿⠿⠻⠿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠂⠀⠀⢀⣀⢀⣀⣀⣀⣁⣀⣀⡀⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠈⠀⠈⠁⡌⠁⠀⠀⠠⠾⠿⠟⢿⣷⣶⣶⣥⣉⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠚⠀⣀⠀⠀⠀⠻⠤⠴⠶⢾⠀⢀⣄⣀⠈⠉⢶⠉⢻⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠧⣄⠀⠀⠀⠀⠶⠀⠶⠄⠀⠀⠛⣳⣦⣤⣴⣿⠟⣡⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠓⢲⣬⣅⡒⠀⠀⠀⠨⠤⠿⠤⠰⠦⢒⣟⡉⠀⣺⣇⣽⡗⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠀⠀⠀⠴⠾⠃⠀⠉⠋⠀⠁⡟⣿⠇⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠻⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⢀⠾⠝⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⢠⢺⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡰⣿⣿⡯⢽⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⡷⣶⡿⢻⡤⣤⣰⡀⠀⠀⠀⠀⠀⠀⠗⠀⣠⣴⢞⢽⡟⣾⣿⣇⡄⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠻⠿⠶⢚⠁⡙⠛⠿⢿⣶⣶⣶⣶⣶⣶⣿⣿⣿⡳⡛⡕⠿⠋⠉⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠼⠋⢠⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⢀⣤⣴⠺⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣄⠀⠀⠦⠦⠤⠤⢤⣰⣤⣤⣤⠺⠏⠁⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣽⡃⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⣤⣴⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⠀⠀⠀⡼⠉⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⢀⠞⠀⡄⠀⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⢸⣠⣋⣀⡀⢿⠞⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠀⢸⡿⠿⢋⡴⠁⠀⣀⡤⠤⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢡⣄⢸⢀⡴⡫⢄⢀⣎⠀⠀⠀⠀⠀⠀⠀⢀⣉⣩⡗
⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⠛⡟⠁⢸⡄⠀⢹⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⡇⠀⠀⠀⡠⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠾⠿⠟⣇⠀⢸⠃⠀⠸⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠒⠲⠺⡦⠶⠶⠷⠴⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⢀⠴⠊⠱⢄⡈⢇⢄⠀⠀⠀⠀⠀⠠⡖⢉⣉⣉⠉⢳⡄⡇⢄⠀⠀⠀⡼⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⢡⡇⡆⠀⠀⠀⠉⠉⠻⢷⣀⣀⣀⣀⣀⡏⠉⠉⠉⠉⢹⠁⠘⢢⣴⣤⠞⠁⠀⠀⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⣌⡠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠹⡰⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡠⠄⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠈⠛⠳⢶⣦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠘⠛⠛⠃⣀⣀⣀⣠⣤⣬⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠭⠭⠿⣛⣛⣛⣛⣛⣛⣛⣛⡻⠯⠭⠽⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⠶⠦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢀⣠⣤⠴⠶⠶⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠲⠶⢦⣤⣄⡀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢳⣦⡴⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣯⠙⠻⠷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠛⢹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠛⠳⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⣷⣤⣿⣤⣤
⠀⠀⠀⢠⡶⠛⢛⡛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠗⠀⠀⠘⢷⡀
⠀⠀⠀⡿⢀⡾⠋⠉⠻⡇⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⢶⡄⠀⠀⠀⠀⣿⠁⢀⡀⣤⣔⠷⠎⠳⣤
⠀⠀⠈⣷⠘⣧⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣧⡉⠓⠈⠁⠀⣠⡶⠋
⠀⠀⠀⠘⢷⣌⡙⠛⠛⣛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣤⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⢾⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡆⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠙⠻⠶⠶⣤⣤⣤⣤⣤⠶⠶⠶⠶⠛⠋⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠉⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⠳⣌⡙⠳⠶⣤⣄⣀⣀⣀⣀⣀⡀⠀⣀⣀⣠⣤⠶⠶⠛⣯⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⣀⠀⠀⠀⠉⠙⠓⠒⠶⠯⠭⠭⠭⠭⠭⠭⠭⠭⠥⠶⠖⠒⠋⢁⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠶⠶⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠶⠶⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣱⣀⠀⠀⠀⡀⢀⠄⠴⡡⠀⠀⠀⠀⠀⠀⠀⡠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠘⢉⣠⠇⠀⠀⠀⠘⢧⡠⢆⠠⠀⠀⠀⠀⢰⡃⠁⠈⠀⢲⠀⢠⢀
⠀⠀⠀⠀⠀⣀⢠⠔⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠀⠀⢀⣿⡏⠀⠁⠈⡆
⠀⠀⠀⠘⡩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠐⠊⠘⢇⡂⠀⢀⡇
⠀⠀⠀⢾⣶⡠⠀⠀⠀⠀⠀⠀⡠⢤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠤⣀⠀⠀⠀⠀⠀⠈⠑⠐⠊
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⡎⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢕⠦⣠⠀⡰⠒⠦⠆⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⠃⠀⠀⠀⠀⠀⠀⣰⢀
⠀⠀⢐⣖⡰⠎⣀⡀⠀⠀⠀⠘⡀⣏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⡠⠤⡲⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠈⠒⢣⠖⠐⠄⠀⠀⠀⠠⢇⠑⠒⠤⠤⠤⠀⠀⠀⠀⠤⠄⠒⠊⣹⡟⠊⠀⠘⠠⠀⠀⢠⡜⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠼⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⣞⣉⣉⣉⣉⣤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠐⠒⠒⠂⢄⣀⣀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⡩⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⣹⠈⡇
⠀⠀⠀⠼⠠⠤⡴⠃⠀⠊⠀⠀⡇⠀⠀⠀⠀⠀⠀⢤⣄⣼⠀⣀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠹⠏⠀⣀⢸⢠
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠲⢾⣟⠹⣿⣁⡀⠀⠀⢀⡤⠖⠁⡄⢀⡀⠀⣵⠚⢡⠆
⠀⠀⠀⠀⠀⠼⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⣳⠾⣯⠉⠁⠀⠀⠀⠈⠒⠤⠍⢉⣉⡡⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡟⣰⣿⣇⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣩⢠⡟⣉⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⢿⣿⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⣏⣩⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠹⢿⣿⣿⣿⣿⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣧⣬⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣕⠂⠈⠳⠏⠛⢿⣧⡹⣦⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠆⠀⠈⠀⠀⠙⢷⣼⣷⡀⠀⠀⠀⡄⠾⣷⡾⠻⣿⣿⣯⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠠⢄⡀⡀⢀⣻⣿⣧⠀⠀⢠⠀⣠⣿⣧⣤⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢯⣍⣍⡉⠙⣻⣿⡇⠀⢠⣿⣶⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⣤⣽⡇⠀⡏⠁⠀⣆⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⡽⡂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠜⣻⣶⣾⣿⣻⣿⢃⣼⣯⣶⣿⣷⣾⣿⣿⠿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢿⡿⠄⠈⠙⠿⣷⣄⢹⡦⠄⠀⢀⠀⠀⠂⠀⠀⠀⣄⡀⢂⣶⣾⣿⣿⣿⣿⢾⡻⣏⠙⠛⢯⡈⠙⠷⣽⣽⡿⠎
⠀⠀⠀⠀⠀⠀⠀⢹⣄⠊⠉⢛⡲⠶⠿⢷⣇⠀⣷⣶⣅⠀⠀⢒⡊⠀⢻⣷⣾⣷⣿⣿⣛⠉⢫⣷⣴⣆⠁⠀⠀⠹⣷⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡓⠲⢶⠾⠦⠤⢼⣷⣯⠀⣽⡆⠁⣸⠃⢀⣼⣿⣟⡛⠛⢯⣉⠻⣤⡈⢊⡻⣷⣤⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⢭⣁⣀⣀⡀⠀⠉⡻⣧⣄⠿⡾⢷⡤⢻⣿⠿⠟⠻⠷⠶⠿⠶⠞⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣎⡛⠫⣍⠉⣛⠛⠓⠿⣿⣰⣿⡀⠃⡞⠁⠀⠀⠀⢀⣀⣠⣤⡶⢶⣴⢶⡶⠟⣿⣶⣶⣶⢶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣁⢀⣤⣄⡀⠈⢿⡟⡦⣼⢋⡐⢢⣴⣾⠏⣿⡋⣼⣠⡾⣳⣾⣷⣾⠻⠿⢖⡾⠋
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡟⠛⢻⣾⣿⣿⡿⠋⠀⠀⠉⢿⣷⣿⣼⡿⢿⡟⠻⠻⡭⢽⡿⣶⠟
⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡇⠀⠈⢤⠆⣨⣿⣿⣿⣿⣷⣿⣌⠙⢛⣛⡿⠁
⠀⢻⣿⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢹⣧⡀⠀⠀⠐⣼⣿⣿⣬⣟⣿⢿⣿⣾⡦⠉
⠀⠘⣯⣭⣟⣿⣷⣍⣛⣿⣷⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣾⣿⣿⣿⣿⣾⡿⢿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⢸⣿⣭⣽⣒⣭⡽⣛⣿⣿⣦⣛⠻⣦⣄⠀⠀⠀⠀⢸⢀⣿⣿⡿⣷⣾⠿⠿⠿⠟⠛⠛⠉
⠀⠀⠀⠛⣿⡻⣏⡭⢛⠯⠡⢀⣦⣿⣿⣮⣿⣷⡀⠀⠀⡏⣸⣿⠟⠈
⠀⠀⠀⠀⠈⠻⣵⣒⣯⣤⣾⠿⣙⣬⣶⡶⢿⣿⣿⡄⢠⠇⡿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣾⣟⣭⠶⣿⣿⣿⣿⣿⡇⡸⣸⠇⠀⠀⣀⣤⣶⣶⣛⣽⣿⠿⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⠿⢿⣿⠃⣿⣠⡶⣻⣟⣿⣿⣿⣟⣿⣿⣷⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⡿⢸⣿⣿⣿⣿⡻⢷⣄⡐⢻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⢁⣾⣿⣿⡻⣟⣿⣷⣭⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢣⠞⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣺⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡴⢦⠿⡾⠾⠁⠀⠀⠀⠀⠀⠀⢇⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⡄⢀⣌⢀⣤⠤⠤⢠⠶⡄⢠⠴⣤⠴⠶⠂⠈⠣⣄⠉⠉⢹⡄⠀⠀⠀⢀⠞⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠶⠃⡏⠘⡾⣧⡄⠀⣾⡀⢡⣯⠞⢹⠶⠶⠒⠂⠀⢈⣉⠉⠉⠀⠀⠀⠀⡞⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣇⢸⠁⡇⠀⠈⢻⠁⠙⢧⠀⠸⡤⣀⣀⣀⣀⣏⣸⣇⣀⣀⡀⠀⡸⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠞⠁⠙⠋⠀⢁⢀⣀⣀⣴⠀⠀⠀⠀⠀⠛⠒⠒⠒⢲⠒⠒⠒⠒⠋⠀⠧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠐⠟⠋⣙⠙⠉⠀⠀⢀⡀⢀⡴⠶⠶⠶⢾⠶⠶⠒⢲⠒⠒⣲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠅⠤⣾⠶⠀⢤⡾⠽⣳⢻⢷⣀⠙⠆⢸⡇⠀⠀⠀⢸⠀⠀⠀⢸⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⣦⡾⠘⠇⠀⠙⠃⠀⠉⠁⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢰⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⣴⠒⢛⣉⣉⣉⣉⣉⣉⣉⣩⣧⣾⣤⣤⣤⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠀⡀⠀⢿⣉⢉⣩⣁⡤⣄⡴⣤⠶⢤⣿⣦⣔⠲⠖⠲⡇⠀⠀⢾⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢧⡞⠳⠟⠟⠙⠀⢸⠉⠛⣤⠉⠀⣤⣦⠀⠐⠀⡇⠘⡏⣏⣉⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣀⣀⣀⡤⠤⠤⠼⠤⠤⢤⠀⡀⠈⣁⡀⠀⢸⠇⣠⠇⡿⠤⠤⡇⠀⠀⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣀⣏⠀⠈⠀⠀⠀⠀⠀⣸⢸⠀⣀⠀⠉⢀⠀⢸⢖⣡⣞⡓⢚⣑⣳⠀⠘⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⢻⠀⠀⠀⠀⠀⠀⢀⡟⡏⠀⠛⠁⢠⡄⢀⣟⣉⡩⢥⡬⠧⠤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⡼⢧⡀⠀⠀⠀⣀⠞⡽⢧⡀⠐⠀⠀⢀⡾⠤⠤⠤⠤⠶⠶⠆⠚⡆⠀⢀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠙⠦⣄⣨⣥⠞⠀⡴⠿⠶⠒⡚⣛⡦⠤⠤⠤⠤⠤⠤⠶⠶⠇⢀⡜⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢭⡙⢦⡀⠀⠾⠖⠛⠀⠀⠙⠛⠛⠉⠉⠉⠀⠀⠀⣀⠀⢠⣤⡤⠤⠖⣫⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢳⠀⠀⠀⠀⠀⠀⠀⡾⢦⠟⠶⠛⠶⠛⠶⠛⠂⠘⠒⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠶⠒⠒⠒⠒⠒⠛⠛⠓⠒⠒⠶⠶⣶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠶⢶⣾⣿⣿⣿⡿⠿⠿⣿⣿⣿⣶⣤⣤⠤⠞⣿⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⣁⠀⢠⡄⠀⣿⡖⢋⠭⠿⠭⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣐⠓⠀⣿⢸⡇⠀⠀⠀⢹⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡈⢀⡟⠉⠀⠀⠀⠀⡸⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⣸⠇⠀⣀⣠⠴⠚⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣿⢞⣩⣥⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣟⣡⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠚⠉⠉⢳⡀⠀⠀⢰⣆⣀⠀⠀⠀⠀⣴⣂⣿⣀⣈⣭⠽⠿⠷⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡐⠀⠀⠐⠛⠀⠈⢧⣀⣀⠈⠓⠀⠀⢒⣓⣋⣭⡽⠟⠛⠉⠉⠉⠂⠀⢀⣼⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣄⣀⣀⣀⣀⣉⣉⣉⠉⠉⠉⠉⣉⣀⣀⣤⣤⣤⣤⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠾⠭⠭⠭⠭⠭⠭⠭⠭⠵⠖⠒⠛⠉⠀⠀⣀⣀⡀




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠄⠀⠂⠀⠀⠀⠀⠐⠀⡀⠀⡤⠀⠀⢀⠀⡀⢀⠄⣤⡠⡀⠂⠀⠠
⠀⠀⠀⠀⠀⠁⠠⠀⠀⢂⣟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠄⠄⠀⠈⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡓⢌⠀⠊⠄⠀⠀⠀⠀⠀⠆⠀⠀⠀⢦⠀⠂⡀⣀⡜⠀⠁⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠉⢾⠊⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠡⡄⠀⠁⠀⠋
⠀⠀⠀⠀⡀⠀⠌⢈⠁⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠁⠄
⠀⠀⠀⠀⠈⠤⠂⢡⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠐⠃⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠋⠠⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠀⠄⠠⠀⠀⠀⠀⠀⠀⡈⠈⠀⠀⠀⠀⠀⠀⠄⠀⡀⠄⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠀⠶⠆⡛⠅⠀⠀⠀⠐⠠⠁⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡁⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠸⠆⠀⠀⠀⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⣠⡞⠁⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⢰⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠈⢿⣦⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠈⠻⣷⡀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⢹⣧⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣸⠏⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣘⣁⣀⡀⡀⠐⠋⠀⣀⣀⣀⣘⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠂⣉⣉⣁⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣈⣉⣉⠑⠒⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠁⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⢄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠘⡀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡴⠃⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⠒⠉⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⣀⣀⣀⠤⠖⠉
⠀⠀⠀⠀⠀⣠⠤⠔⠒⠂⢉⣉⣥⣤⣤⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣥⣤⣤⣬⣉⡉⠐⠒⠢⠤⣄
⠀⠀⠀⠀⠸⢄⡀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠀⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠠⠤⠌⣉⣉⣙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⣋⣉⣉⠡⠤⠔⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬⢍⣉⣉⣉⠉⠉⠈⠛⠛⠛⠛⠛⠛⠊⠀⢉⡉⣉⣉⡩⠥⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⣒⣚⣫⣭⣭⣭⣽⣷⣶⣶⣶⣾⣿⣭⣭⣭⣽⣓⣒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⣭⣶⣞⣯⠿⠛⠉⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡯⠭⣭⣿⣛⡻⢶⣮⣍⡓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣴⣿⡿⣫⣾⢋⡴⠊⣀⣤⣶⡶⠶⠶⣦⣝⣿⣿⠟⠋⣁⣀⣀⡒⠒⠦⣌⣙⠳⣮⡛⢿⣷⣮⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⣿⣿⢱⣿⠁⣿⠁⣾⣿⣟⠁⠀⠀⠀⠀⠉⢻⡷⠊⠉⠀⠀⠉⠟⣿⣶⣌⠻⣷⡌⣿⣎⣿⣿⣿⡆⢳
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠾⣆⠹⣿⣿⠘⣿⡀⠻⣧⡈⠻⢿⣷⣤⣤⣀⣀⡀⠀⠀⠀⠀⠘⠀⣀⣴⣿⣿⡿⢠⡿⢃⣿⣿⣿⣿⣿⡇⣸
⠀⠀⠀⣾⣿⡅⠀⠉⠉⠉⣛⢻⣷⣌⠻⢷⣬⣑⡀⠈⠙⠳⠶⢤⣭⣭⣉⣉⣙⠂⠀⣠⣴⠶⠿⠿⣛⠫⠥⠚⢉⣰⣿⣿⣿⣿⡿⢋⣴⡿⢶⣄⡀
⠀⠀⠀⣿⣿⣿⠿⡿⣶⣶⣼⣼⡿⣿⣿⣶⣬⣙⠻⠿⣿⣷⣶⣶⣶⣬⣭⣭⣉⡓⢸⣿⣭⣭⣥⣶⣶⣶⣾⣿⣿⣿⡿⠟⣋⣥⣾⣿⣿⡇⢅⠫⠻⣦⡀
⠀⢠⡾⣿⡿⣿⣧⠀⠪⡻⠿⢿⡃⣿⣿⣿⣿⣿⣿⣶⣶⣬⡭⣉⣛⣛⣛⡛⠻⠿⠿⠿⠿⠛⢛⣛⣛⣛⣋⣭⣭⣶⣾⣿⣿⣿⣿⠟⣿⠢⠑⠅⠀⠈⠻⣦⡀
⢠⡿⡀⠙⣷⡹⣿⣷⣤⡀⢈⣴⡿⡎⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⣰⣿⣮⡢⠀⠀⠑⡀⠘⣷
⣾⢡⠂⠀⠈⠿⣮⡻⢿⣿⣿⣷⣦⣧⠹⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⣰⢏⢻⣿⣷⡀⢀⠁⠀⠀⠸
⣿⢸⢄⡀⠀⠀⣼⣿⣶⣬⣙⡛⠿⠿⠷⣷⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠬⠐⠀⠀⠀⣴⠋⠀⠂⣿⣿⡇⠈
⢻⡌⢇⠈⠉⠒⠪⠝⣿⠻⠿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠍⠃⠀⠀⡠⢾⠃⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⢀
⠈⢿⡌⢦⡀⠀⠀⠀⠸⣄⠈⠀⠒⠉⠩⢝⡻⣿⣿⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⠁⢀⣠⣴⣿⡇⢸⠀⠀⠀⣸⡿⠃⠀⠀⠀⠀⢀⣾
⠀⠈⠻⣦⡑⢄⡀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠱⡘⣿⣿⣿⣷⣾⣭⣿⣛⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⠊⠀⢀⡼⠋⠁⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠈⠻⣦⣉⠢⣄⠀⠀⠈⠓⠦⣄⡀⠀⠙⠮⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⠄⠁⠀⠀⠉⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠙⠷⣦⣍⡒⠄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠭⣝⣛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⣋⠩⠄⠀⠀⠀⠀⠀⠀⢀⠠⢐⣨⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠀⠂⠀⠀⠉⠉⠁⠀⠀⠀⣀⡀⠠⢄⣒⣭⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⢶⣤⣤⣄⣀⣀⣀⡀⠀⠤⠤⠤⠄⠀⠀⣀⣐⣒⣂⣭⣥⠶⠶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⣽⣽⣯⣯⣗⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⣮⡾⠛⠹⠻⠟⠏⠛⢷⣵⡱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠗⡄⠀⢀⣀⣤⣤⣴⣧⣿⡋⠀⠀⣠⣽⣮⢄⡀⠀⢙⣿⣼⣦⣤⣤⣀⡀⠀⢠⠺⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣾⡟⠛⠉⠍⠉⠉⠙⢿⡶⢝⠜⢯⡽⠣⡫⢶⡿⠛⠉⠉⢩⠉⠛⢻⣷⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⡿⣯⣢⠸⡈⠀⡇⡰⢭⠃⣀⣡⣎⣀⠘⣭⣧⢘⠀⠁⠇⣔⣽⢿⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠄⣈⡚⠿⢇⣣⣤⢧⣽⣷⠞⢋⣉⣉⡉⠻⣾⡧⡼⢬⣜⡸⠿⢓⣁⠠⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣿⡇⠀⠂⠨⠍⣾⣶⣻⣟⡿⢱⣞⣭⠴⠦⣭⣳⡎⢿⣫⣟⣦⢷⠉⠁⠐⠂⢸⣿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⣵⡿⠻⣷⣄⢉⡨⣉⣷⣿⡽⡡⢎⠻⠬⢣⠎⠱⣜⠥⠟⠹⢬⢯⣾⣾⣩⣅⡉⣠⣾⠟⢿⣮⢕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⣾⠋⠀⠀⣜⢏⠧⠛⣽⠟⣱⠶⡛⡆⠠⡢⣬⡐⢂⣥⣔⠄⢰⢛⡶⣎⠻⣯⠛⠼⡹⣣⡀⠀⠙⣿⡞⣆
⠀⠀⠀⠀⣠⣀⣄⣠⣗⣿⣷⡖⣄⣞⡾⣁⠄⣸⡇⢰⢃⡟⡩⠖⢂⠻⡿⣫⣝⢿⠟⡒⠲⢍⢻⡘⡆⢸⣇⠠⣨⢷⣳⣠⢐⣾⣿⣺⣄⣠⣀⣀
⠀⠀⠀⠀⠙⠉⠋⠙⡯⣿⡿⠅⠋⢿⠷⡋⠊⢹⡇⠸⡌⣧⣑⠦⠌⣴⣷⣝⣫⣾⣦⠥⠴⣊⣼⢡⠇⢸⡏⠑⢉⡾⡽⠙⠬⢿⣿⢽⠋⠙⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣿⣄⠀⠈⢫⣎⡖⣤⣻⣦⡹⠾⣭⠇⠐⠝⢻⠡⠌⡟⠫⠂⠸⣬⠷⢏⣴⣟⣤⢶⣱⡝⠁⠀⣠⣿⢯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢕⡻⣷⣴⣿⠋⣈⣙⣋⡿⡿⣳⡑⣆⣰⢒⡜⢆⡰⢣⡒⣦⡰⢎⣞⢿⢿⣉⣚⣉⠙⣿⣦⣾⢟⡪⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣿⡇⠀⠄⠐⣀⢿⠻⣽⣫⣷⡸⢯⣛⠲⠖⣛⡽⢇⣾⣝⣫⠟⡾⣀⡀⠠⠀⢸⣿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠒⢉⢥⣶⡎⡝⠛⡖⣺⡿⣦⣈⣉⣉⣁⡴⢿⡟⢲⢚⢫⢱⣶⡬⡉⠐⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣷⣟⠝⢰⢁⠁⡇⢰⣛⡄⠉⡹⢏⠉⢠⣟⠞⢸⠀⠀⠆⠫⣻⣾⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⢿⣧⣤⣀⣂⣀⣀⣠⣾⠷⣪⢦⣞⣳⡴⣝⠾⣷⣄⣀⣁⣘⣀⣤⣼⡿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡦⠃⠀⠈⠉⠛⠛⠻⡟⣿⣅⠀⠈⠑⣺⣟⠊⠁⠀⣨⣿⢻⠟⠛⠛⠉⠁⠀⠘⢴⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⢟⢷⣤⣸⣴⣦⣇⣤⣾⡻⡱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢭⣻⣻⣿⣟⡯⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡐⠒⢦⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠰⡀⠀⠁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠸⣦⠤⠤⠤⠴⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠦⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠉⠙⢦⡀⠹⣄⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠉⣳⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⣄⣀⡤⠒⠻⣄⠈⢯⡁⠀⠀⠀⠀⠀⠀⣀⠴⠤⢄⣀⡀⣀⠀⠀⢀⣀⣀⣭⠶⠚⣿⠚⢉⣀⣀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠉⠙⠛⠛⠓⠒⠛⠒⠒⠒⠒⠒⠒⠛⠒⠒⠛⠛⠛⠉⢯⠉⠉⠁⠀⠀⠀⠀⡿⠚⠉⠁⢹⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⡼⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣼⠀⣠⠖⠉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⡇⣸⠃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡼⠛⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⢰⣷⣚⣋⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⡉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⣠⡴⠛⠉⠁⢀⣀⣠⠤⠤⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠉⠉⠉⠉⠒⠠⠀⠀⣹
⠀⠀⢠⡾⠁⢀⠠⠐⠉⠉⠀⠀⠀⠀⠀⠉⠳⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠏
⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⢻⡛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠖⠛⢉⡴⠋
⠀⠀⠀⠙⢯⡙⠒⠲⠶⠤⠤⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣱⠤⣤⣤⣤⡤⠴⠶⠖⠒⠛⠋⠉⠉⠁⣀⣀⣤⠤⠶⠚⠉
⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⡟⠀⠀⠀⠈⠳⣄⣤⡤⠴⠶⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⡀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠸⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⢀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⣀⠀⠐⠈⠉⠁⠀⠒⠒⠒⠛⠓⠲⠶⠀⠉⣉⣀⠴⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠀⠀⠐⠐⠒⠒⠛⠋⠉⠉⠉⠁⠀⠀⣸⡥⠔⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⢠⡏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣉⡉⠉⠉⠉⠀⠀⢀⡾⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⡖⠁⠀⠈⠀⢹⡆⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠳⣄⠀⠀⡠⠞⠁⠀⠀⢀⣾⠡⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣄⣀⣀⣀⣀⣀⡤⠘⠁


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

⣿⣿⣦⣀⠀⣹⠜⣶⣛⠚⣧⠆⠉⣲⠻⣯⣿⣿⣿⢿⣏⣽⣯⣯⠙⠀⠀⠿⠁⠆⠈⣿⠈⠛⠈⠛⢿⣶⡦⠟⠀⢰⠏⠀⣹
⣿⣿⣿⣿⣿⣷⣤⢼⠉⠀⠈⠛⠆⢸⠓⠿⡳⢿⣿⣿⠻⡿⣵⣿⠦⠏⠉⣸⣁⡀⡀⠆⠀⠀⠀⠀⠈⣿⠁⠀⠦⢀⢠⠤⠛
⣿⣿⡿⠿⠷⣬⣿⡦⢻⠧⡶⠄⠀⠂⢐⡷⢿⡙⠻⠏⠛⠻⠷⠎⠉⠀⣶⠻⢿⡿⡿⡀⠀⠆⡀⠀⢤⠈⡀⠊⠀⣸⣦⣴⠿
⣿⣿⣶⣶⣾⢁⣿⣧⡾⢞⣀⣤⠀⠠⠮⠗⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣨⠄⠼⠘⠀⣡⡉⠀⠀⢂⣠⣄⠀⡀⠉⡿⠁⠀⣠
⣿⣿⣿⣿⣿⣿⣿⣶⣶⣟⣛⢭⠽⠛⠛⠠⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠟⡂⢴⢈⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣹⢁⣀⡀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣠⣿⠏⣿⣾⣆⠀⠀⠰⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣾⣟⣀⡀⣠⡀⠀⠀⠀⠠⠴⠦⠄⠀⠀⠀⠀⠦⠀⢀⣻⡬⢹⡛⡿⢹⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡄⠀⠀⠀⠀⠶⠀⠀⠀⠀⠘⠀⡀⠐⠦⠀⠀⠘⣃⡤⡾⠀⠀⠂⠀⠀⠰⣦⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⡴⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⡦⠲⣀⠀⠁⠀⠀⠀⠀⠀⠰⠞⢫
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡶⣣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠒⠎⢀⠀⢀⣄⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⠅⠀⠀⠀⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢂⣶⣯⣀⣳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠸⠋⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡷⠦⢀⠀⠀⠀⠀⢀⣬⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠸⢫⣮⣤⣀⡘⣟⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⢤⣴⡇⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⡀⠨⣿⠑⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢺⣥⡾⢿⣧⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⢉⣳⣾⠯⣷⣶⠟⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡄⠀⠀⠀⠀⠀⠀⣴⠙⢻⣌⣾⡟⠓
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⣴⡶⠆⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢠⡿⠁⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠘⢷⣄⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈⢻⡆⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠟⠁⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⣠⣶⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⢀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀⠀⣸⣷⣤⡶⠶⠶⣶⣄⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⡶⠶⠿⣶⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⡄⠀⠀⠀⢻⣇⠀⠀⠈⠉⠙⠛⠛⠛⠛⠻⠿⠿⢿⡿⠿⠛⠛⠛⠛⠋⠉⠉⠀⠀⢃⣿⠃⠀⠀⢀⣾⠇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⢄⣼⡏⣀⣤⡶⠟⣁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠸⡗⠈⣿⡇⠀⠀⠀⠀⠀⠀⣴⡿⠟⠛⣉⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠄⠀⠀⠀⠙⣷⣎⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣼⣿⡷⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣦⡀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣿⡇⠀⠀⢀⣠⣾⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠉⠉⠉⠻⢷⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡶⠟⠋⠉⠉⠙⠛⠷⣦⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣄⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣤⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠿⠿⠿⠷⠶⠶⠶⠾⠿⠿⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡶⠀⠀⠀⠀⢸⡇⠀⢀⣀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⡄⠀⣴⠀⣾⠛⠛⢻⡇⠀⠀⠀⠀⣼⠛⢷⡀⠀⣿⠀⠀⠀⠀⠀⢸⡇⢀⡟⣹⠆⢀⡾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢀⣀⠈⣧⢀⣿⡀⢻⡄⢀⣼⠃⠀⠀⠀⠀⢸⣆⣼⠃⠘⣿⠛⠀⠀⠀⠀⢸⡇⠈⢿⣯⣤⠌⠷⠟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⡶⠶⠛⠁⠀⠻⠟⠉⠁⠘⣿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢹⡄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢃⣴⣿⣷⣾⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⡿⠉⡀⠛⢠⡤⠿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠻⣦⣣⣤⠒⠀⠀⠀⠀⠉⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣿⣋⣤⠖⠀⢀⠀⠀⠲⢶⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⢀⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠧⠠⠶⠾⢿⣭⣭⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡛⠓
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠻⠿⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⡻⣿⣦⡝⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡿⣯⣍⠉⠉⠀⢠
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠌⠛⠃⠠⣰⡆
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡟
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⡇⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠆⢠⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢸⣿⡇⠀⡰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡸⠋⠁⣆⠀⠈⠀⢰⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣤⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡟
⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠁⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇
⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡇⠒⣿⠇⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⣿⣿⠀⠘
⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⠀⠀⣿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠃⠀⠀⠀⠀⠀⠀⢠⡿⠁⣼⣿⠀⠀⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠂⠀⠀⠀⠀⠀⠀⠀⣾⠇⢠⣿⡇⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣾⣿⠃⠀⠀⢃⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⢠⣿⡏⠀⠀⠀⣼⣿⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⣼⡿⠁⠄⠀⢠⣾⣿
⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⣿⠁⠀⠀⢀⣾⣿⡇⠀⠀⠆
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⣼⢟⣿⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠜⢁⣾⠋⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣰⡿⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⣶⡾⠋⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⣾⡿⠋⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠀⠀⠁⠀⡀⠀⠀⠛⠁
⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠒⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠁⠀⠀⠀⠀⢰⠇
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⣀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⠐⠲⡄
⠞⠃⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡿
⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠱⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠟⢠⠂
⠀⢠⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⣇⠀⡀⠹⢦⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠈⠁⠉⠀⠀⠀⣦⣤⡄⡀
⠀⣏⠀⢱⣶⠀⣧⠀⢸⠁⣄⠀⠀⠀⠲⢿⣷⣶⣾⣿⣿⣛⣛⣛⣻⣛⣿⣿⣷⣶⡾⠟⣃⠸
⠀⠹⣦⡄⢹⣶⡻⠀⡖⣠⡘⢆⠀⢴⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣻⣿⠁⣆⣲⣦⣄⢲⡀
⣀⣀⣈⠳⣄⣹⡇⢰⠃⣿⡇⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢸⠋⠁⠀⡼⣸⠇
⠘⡍⢶⣄⣩⣭⣽⣿⣦⣿⣧⠟⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠇⣎⣠⠴⢊⣼⠟
⠀⠈⠢⣀⠜⠋⠹⠛⣿⠿⣁⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⣸⢋⣤⠾⠛⠁
⠀⠀⠀⠈⠐⠒⠒⠁⠀⠀⠙⠷⠆⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⠏⡰⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢾⣒⠁⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣫⣞⠓⢤⣤⣐⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠀⠉⠙⠋⣆⠰⢍⡛⠷⠶⠶⠶⠶⠾⡿⢿⣿⠿⠋⠻⢂⠽⠂⢉⣆⡕⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣤⡴⣞⡋⠠⡆⠀⢀⠀⢀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣻⣲⣿⣻⣿⣻⣻⣿⣷⣿⣿⣿⠿⣿⠛⠋⠉⠀⠀⠉⠙⠣⡀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣏⠙⠶⣤⣤⠤⠴⠒⠚⣷⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣿⡀⠀⠀⠀⢀⣽⣬⠿⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠮⣿⣷⣠⠤⠚⠋⠁
⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢄⠀⠀⠀⠀⠀⡄⠀⡇⢀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠈⢷⠀⠀⠀⢰⡿⣄⠐⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠜⠀⠀⠀⠈⡧⢈⠑⢄⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⣧⠀⠀⠀⠀⠀⠘⢖⠁⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠈⢸⠀⡼⠀⠀⠀⠀⠀⠑⣤⠂⢀⠙⣆⠀⠀⠀⢀⠞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⢀⠨⣸⠀⢇⠳⡀⢤⣀⠀⠀⠈⢢⠀⢐⡘⡆⠀⢠⢁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠍⡂⠑⠄⢳⠀⠀⠁⠺⠌⡮⠳⡄⠀⢸⠂⡁⠁⢵⠀⠘⣡
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⡇⠀⠘⢄⠈⠢⡀⠀⠀⢠⠇⠂⣻⢠⠏⠀⠇⠠⡍⠀⢰⢻⡀
⠀⠀⠀⠀⠀⢯⠢⣄⡀⠀⠀⢳⠅⠘⡀⠑⢄⠙⣆⣠⠋⣠⠀⠿⠉⡠⠁⠂⢠⠃⠀⡼⠐⣇
⠀⠀⠀⠀⠀⠈⠢⡀⠙⡄⠀⠀⠱⡄⠑⡀⠀⠳⡈⠃⡐⠀⡄⢀⠎⠀⡐⢠⠏⢀⣴⠇⠀⢻⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠰⣤⣅⡚⠀⠀⠀⠘⣦⠐⡀⠀⠱⡐⠀⠀⡖⠁⠀⡐⢠⣮⡴⠛⢁⡀⠀⣼⠀⠀⢀⠜⣹
⠀⠀⠀⠀⠀⠀⠀⠈⠑⡌⠓⢤⠀⠀⢸⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠁⠛⣀⠄⠒⡉⠀⣴⠏⠀⢐⡡⠞⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢠⡀⢳⡀⢸⠁⡀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠊⠀⢀⠎⢀⡤⠁⠀⠀⢈⠔⢫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⡄⢷⠟⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡘⠀⡜⢁⣴⠏⡴⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠐⠀⢠⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠘⣵⠯⢹⢰⠃⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⠃⢦⢘⠟⡌⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⠈⢦⡰⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢊⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⠀⠀⠀⣀⣀⠀⠠⠌⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠡⠐⠈⠐⠠⠀⠠⠀⠤⠂⠄⠂⠀⠠⠀⠠⠤⠆⠑⠀⠀⠁⠒⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⢀⣐⠀⠀⠀⠀⠀⠐⠂⠀⠂⠀⠀⠀⠠⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣈⣥⣴⠆⣝⠙⠀⢩⣄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⣴⣿⣿⣟⣪⣕⣶⣬⡁⠀⠀⣀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠎⣆⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⢿⣿⣿⣿⣿⠿⢿⣤⣶⣄⠶⢄⠺⠉⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣩⠇⣸⡇⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠶⢽⡻⢿⣿⣿⣶⣾⣿⣿⣶⣻⣿⣦⣤⣌⡠⠀⡀⠀⠀⠀⢠⣀⣴⣤⡾⢞⣡⠾⢹⡏⣰⣦⠾⢦⣶⣈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠑⠂⠭⢙⠛⡛⠻⠿⠷⠿⠿⠿⠿⠿⠷⠾⢛⣚⣛⣩⣭⡭⠴⠚⣋⡩⣂⣽⣿⡏⠀⠀⠀⠈⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢈⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢈⡅⠰⠶⣖⡻⣯⡷⢝⣫⣿⣿⣇⡀⠀⠀⠀⡸⢟⡇
⠀⠀⠀⠀⠀⠀⢀⢀⣤⣖⣘⣻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⠶⢬⣝⣲⣭⣶⣿⣯⣿⣿⡯⣫⣷⠞⢡⣾⠁
⠀⠀⠀⠀⡠⠂⠵⣗⡽⣌⢽⣷⣾⣧⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⠿⣟⣻⣿⣿⣷⣿⣿⣿⡿⠟⣋⣴⣾⣿⣷⣏⢂
⠀⠀⠀⡐⢰⠳⠺⡋⣾⠃⡸⡿⢻⣿⢆⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢁⣚⠻⢿⣛⣾⣿⣿⣟⣋⣥⣶⣿⣿⣥⢪⠁⠻⠀⡇⠃
⠀⠀⠀⡅⠁⠇⢇⡜⠊⠈⠈⠘⠓⠩⠻⣧⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣷⣾⣿⣿⣿⣿⣟⡿⠟⢻⡟⠾⠛⠛⡛⣁⠁⠀⡠⠁⠆
⠀⠀⠀⠱⡈⠆⠀⢀⣦⠀⠀⠀⠀⠀⠀⠈⠓⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⠚⡻⣭⣍⢺⣉⣿⣿⡿⢠⠄⠁⠀⠀⡀⠄⠀⠀⠀⠔⢁⠎
⠀⠀⠀⠀⠈⠢⡑⠤⣝⡣⡀⠀⠀⠀⠀⠀⠀⠈⠲⡅⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⣈⣃⣤⣹⢴⣿⠿⠃⠀⢀⠠⠄⠀⠀⠀⢀⠠⠊⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠠⣉⠂⠁⣀⠀⠀⠀⠀⠀⠀⠱⣈⠁⠀⠒⠒⠀⠐⠛⠂⠛⠙⠛⠋⠉⢁⣠⠞⠁⠂⠀⠀⠀⠀⠀⠠⠔⢊⡤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢤⣀⡐⠂⠀⠀⠀⠀⠀⡉⠓⠲⠶⠶⠤⠤⠤⠤⠶⠶⠒⠛⠉⠀⣀⠀⠀⠠⠀⠀⢀⡤⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠥⢀⣒⠂⠀⠤⠤⠀⠀⣉⣈⣀⣀⠀⠀⠈⠩⠤⠄⠐⣂⡀⠥⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠂⠐⠒⠂⠀⠀⠐⠒⠒⠀⠈⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡺⠉⠩⢕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⠈⣆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠾⠳⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⣀⣠⡤⠶⠖⠛⠉⢁⣀⡤⠤⣄⠀⠘⢧⡀⠘⡜⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡥⠴⠚⠋⠉⢀⡠⠤⠔⠚⠉⠉⠀⠀⠀⠈⢆⠀⠀⠻⣄⠈⠒⠡⠙⠡⢐⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⠋⠁⠀⡤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠙⣆⠀⠀⠀⠀⠀⠙⡜⡄
⠀⠀⠀⠣⡀⠀⠀⠠⠀⠀⠀⠀⠌⠀⠀⣠⠏⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠘⣧⠀⠀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⡆⠀⠀⡆⠀⠀⢀⠎⠀⠀⡴⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡤⠚⠁⣷⠀⠀⠀⠘⣧⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠘⠀⢀⡞⠀⣠⣤⡀⠀⡇⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⣨⣧⠔⠋⠁⣀⡤⠚⠁⠀⠀⠀⣠⠾⡆⠀⠀⠀⢇⢇
⠀⠀⠀⠀⠀⠀⡀⢀⠊⢀⠄⢀⡞⠀⠀⠀⠈⢻⣤⡷⠤⣄⣀⣟⠶⠶⠶⠶⢒⣫⠕⣃⡤⠖⠋⠁⠀⠀⠀⢀⡤⠚⠁⠀⣿⠀⠀⠀⠈⠢⣕⡂⠤
⠀⠀⠀⠀⠀⠔⠠⠁⠰⠁⢀⡞⡴⠾⣯⠞⠉⠙⣿⢧⣀⣀⣀⣈⣩⠭⠭⠿⠛⠛⠉⠀⠀⠀⠀⠀⣀⡤⠚⠉⠀⠀⢀⡠⠟⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢈⠀⠱⠀⠀⠀⡞⠰⡁⢿⡏⠀⣾⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠉⠉⠳⣤⣬⣿⣡⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠉⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠑⠒⠒⠲⠦⠤⠤⠤⢤⠶⢶⣶⣖⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⡠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⣠⠃⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣸⠃
⠣⣈⠢⢄⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠶⠚⠋⠓⠚⠋⣡⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠛⣉⣠⠴⠊⠁
⠀⠀⠑⠢⠬⣙⡒⠲⠦⠤⢤⣤⣤⣤⣤⣤⣤⣬⣿⣿⠥⠤⠶⠒⠚⠋⣉⣁⡤⠴⠖⠋⠉
⣀⣀⣀⣀⣀⣀⣈⣉⣙⣒⣲⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣖⣒⣚⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⣛⢟⣙⠛⣛⢏⣿⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠤⠀⢀⠀⠀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠈⠀⠀⠀⠐⠊⠀⠀⠀⠈⠂⠀⠀⠀⣐⠤⠤⠄⠠⠄⠱⣀⡀⠀⠀⢧⡋⠸⢰
⠀⠀⠠⣔⣛⣈⣡⠀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠂⠆⠶⠐⠀⠄⠀⠈⠃⢠⣶⣶⣴⣦⠀⠍⠃⠀⢀⣀⣢⣔⡃⢀
⠀⠀⠄⠙⠛⠛⠋⠀⠀⠀⠀⠓⠀⠀⠉⠀⠀⠂⠀⠀⠀⠀⠐⠀⠐⢀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠋⠀⠀⠀⠀⣆⠐⠷⠶⠂
⡀⠀⠀⠈⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢄
⠀⡀⠀⠤⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣠⡢⠀⠁
⠀⠀⢰⣶⣤⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣬⣤⣵⠆
⠀⠀⠄⠉⠛⠋⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠔
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠐⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣿⠗⢺⣯⡿⠃
⠀⠀⠀⣴⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠂⠀⠀⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⠏
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠄⠀⠄⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂
⠀⠁⠀⡤⢶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠲⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣀⠀⣀⠐⠄⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠡⠞⠶⠁⠀⠀⠀⠀⠀⠀⠴⠲⠆⠀⠀⠀⠀⠈⠀⠀⠳⠞⠉⠀⠁⠀⠀⠀⠒⠒⠐⠒⠂⠀⠀⠀⠀⠀⠐⠂⠀⠒
⠀⠰⠆⠁⠀⠀⠀⠶⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠐⡀⠀⢠⡆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠂⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⠤⠆⠀⠀⠀⠀⠴⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢶⣭⣼⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣬⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠻⢦⣄⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⣀⣼⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠶⠦⣤⣌⣉⣉⣙⣛⡛⠓⠒⠒⠛⠛⣛⣋⣉⣉⣩⣤⠴⠞⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣦⣄⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣯⣝⣓⡒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⣟⣿⡿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡟⣟⣿⢙⡖⠒⠒⠂⠉⠉⠉⡛⡛⣽⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⣿⠇⠟⠯⠘⠁⠀⠀⠀⠀⠀⠀⠅⠤⠟⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠶⠶⠶⠖⠒⠚⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢤⡀⣤⡘⢯⠻⡀⠀⡀⠒⢢⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⠀⠀⠒⠛⠺⠇⠘⣆⢻⡸⠹⠦⣠⠆⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣽⣿⣷⣶⣶⣶⠤⠨⠤⠥⠤⠤⠶⢶⡶⠚⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣿⣿⣿⣿⠢⣿⠐⠀⠀⠀⠀⠀⢸⢿⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣿⣿⣿⣿⣞⣿⠀⠀⠀⠀⠀⠀⢘⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠌⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡞⣼⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠏⣿⣿⣿⣟⣿⠀⠀⠀⠀⠀⠀⠃⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⢿⣿⣿⡇⠘⠀⠀⠀⠀⠀⠀⡇⣿⣤⢴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣶⣶⣦⠤⠤⢤⠤⠤⠴⢶⢶⠒⠛⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⣘⣯⣤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣤⣤⡄⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⠒⠛⢻⠛⠛⣿⠛⠛⠛⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣄⣀⣀⣠⣤⡤⠤⠤⠤⠤⠴⠦⠤⠤⠤⢤⣤⣄⣀⡀⣙⣿⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠳⣶⠒⠒⠛⠋⠉⠁⣿⠶⢦⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⡧⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣄⠀⠀⠀⠀⠀⢰⣇⣀⡤⠔⢁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⣴⡉⣿⠀⠀⠀⢠⠿⠤⠶⠖⠚⠉
⠀⠀⠀⠀⠀⢀⣀⣠⣤⡼⠿⣦⡀⠀⠀⠀⠀⠀⠀⣿⢾⡓⠀⢸⡄⢀⡴⠓⠶⠤⢤⣀⡀
⠀⠀⠀⠀⣾⠋⠉⠀⠀⠀⠀⢨⣿⣶⣤⣀⠀⠀⠀⠻⠾⠶⢚⣫⣴⣿⡄⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣶⣶⣶⣿⣿⠿⠟⠛⠁⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠶⠶⠶⠶⠶⠒⠚⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠳⡀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢰⠀⠀⠀⠀⠀⣧⠀⠑
⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⣀⠀⠀⠀⠀⠀⣼⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⣼⠀⠀⠀⠀⠀⡇⡜⠀⠀⠆⠅⠂⠀⠀⠀⠀⠀⠀⠸⡆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⢀⢸⠈⡄⠀⢻⡄⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⡀
⠀⠀⠀⠀⠀⠈⡜⡇⢧⠀⠀⢻⡄⠀⠌⠀⠀⠀⠀⡩⣌⣉⠩⡿⠀⠀⢰⠃⠀⠐⢈⢃⠈⠀⠀⠀⠀⣀⠀⠀⠀⣉⡄⠑⠊
⠀⠀⠀⠀⠀⠀⠘⣿⡜⣆⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡜⢦⠀⠀⠹⣆⠀⠀⠀⢀⡠⠴⠶⠖⠒⢲⡖⠲⠶⠶⠶⠒⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣮⢧⠀⠀⢹⣆⠀⠀⣏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠻⣎⢧⠀⠈⣿⡄⠀⠙⠷⠦⠀⠀⠀⠀⡇⠀⠀⢀⡀⠀⣀⣀⣷⢠⣄⠀⠠⣄⣠⢤⢠⠲⠄
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢹⡌⣇⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠇⠃⢀⠠⣿⣿⠀⢸⣀⠐⣿⢿⡰⣮⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⢷⢸⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠹⠦⠔⠛⠖⠛⠘⠙⠁⠙⠒⠉⠈⠛⠹⣧⠼⠃
⠀⠀⠀⠀⠀⠀⠀⢱⣲⡀⠀⠀⢸⡄⡇⠸⠁⠀⣀⣀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⡀⠤⠄⢀
⠀⠀⠀⠀⠀⢀⡀⢀⣻⣷⣄⠀⠘⠇⢇⡿⠟⠉⠩⢭⡭⠙⣷⢀⠄⠒⠚⢶⡀⠀⠁⣿⡁⠀⠠⠀⢰⣴⡀⠀⠈
⠀⠀⢠⠤⣁⣀⠀⠀⠈⠉⢙⣓⣂⣀⠤⠤⠄⠒⠋⠀⠀⠀⣿⠀⠀⢤⠀⣸⠀⠀⢶⡉⣌⡐⠂⠀⠘⠿⠷⠤⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠸⡄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⡟⠀⢀⠎⣠⠃⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣠⡴⠋
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⠐⣁⠔⠁⠀⠀⠀⠀⠀⡴⠛⢉⡤⠚⠁⢉⡟⢉⣉⡽⣫⠋
⠀⠀⠀⠘⣇⡹⢉⠋⠀⠀⠐⠀⠀⠂⠤⣤⣄⢀⠀⠀⢠⠟⠒⠉⠀⠀⡀⠀⠀⠀⠀⡸⢱⣆⣿⡴⠶⠚⠛⠛⠛⣻⠞⠁
⠀⠀⠀⠀⠈⢷⣎⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠂⡴⠋⠰⠀⠀⡀⠀⡄⠀⠀⢀⣠⣿⠛⢅⠀⠈⠙⠲⣶⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⢤⣤⣤⣬⠥⠴⠗⠒⠁⠀⠀⠰⢀⠀⡀⣈⠀⢀⣴⠟⠁⠈⠛⠶⠶⠶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁

⣿⣟⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⢄⠀⡀⢲⣾⣻⣍⣯⣿⣦⣾⣤⣌⡓⣒⣠⣤⡤⢤⣀⠀⠀⠀⠀⠀⠀⣰⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣺⡿⠻⠩⠿⣯⡿⠛⠀⠀⠀⠠⠤⢀⣤⣦⣲⠀⠀⠉⠙⠻⢯⠿⠷⠶⠿⢿⣶⣤⡀⡞⡵⠚⠉⠉⢝⠺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣧⣶⠐⠗⠃⠀⠀⠀⠀⣤⣄⢸⣎⡙⣣⠜⣻⠠⣴⠒⡦⣀⠀⠀⠀⠀⠀⢠⡄⢙⣿⡿⣇⢿⡀⠀⠀⠀⠀⢹⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡯⡋⡷⢻⣿⣦⠀⠀⠐⣠⣦⡽⣿⣷⡿⢿⣧⠲⣌⠶⢏⠁⠀⠀⠀⠀⠀⢀⣴⢎⡴⢣⣿⡷⡁⢸⣷⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡻⡃⠑⡘⢛⣩⢈⠀⠀⠀⠸⠾⠃⠛⣀⠨⣅⡐⠛⠭⠈⠁⠀⠀⠀⠀⠀⠀⠙⠉⢀⡠⡾⣔⣻⠿⠶⠀⠀⢀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢶⣥⣬⣉⣊⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠆⠀⠀⠀⠀⡠⢤⣲⣖⣠⣥⠶⠿⠋⠀⠖⡦⢧⠀⠧⠄⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠟⠃⢀⠀⠈⠉⠙⠛⠛⠓⠿⠿⠛⠛⠖⠲⠟⠷⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠐⡦⠔⠀⣁⡤⠐⢫⡀
⠀⠀⠀⠀⣠⣮⠀⠀⢀⠰⡇⣀⣂⠀⠄⠀⠀⠀⠄⡀⠀⢠⣄⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣄⣦⡀⠀⠀⠀⠀⠠⢋⠴⢂⣉⣉⠀⠀⠀⢻⣚⣀
⠀⠀⠀⢠⣿⡇⠀⠀⠋⡐⢛⠉⠼⣣⠀⠀⠀⠀⣼⣇⡀⢈⣁⡜⣠⠤⣶⠚⣯⢣⢄⠀⡀⠀⠛⢾⠏⡀⠀⠀⠀⠀⢀⠊⠈⣋⢬⠏⠉⠀⠀⢀⠉⢹⣷
⠀⠀⠀⢾⡏⡀⠀⠀⠒⣤⣿⡛⣾⣀⢀⠁⠀⠘⠛⠻⠛⢀⣍⣤⢧⣉⡰⢇⡿⡞⠂⡞⣉⡶⠗⠛⠛⠀⠀⠀⠀⢾⣋⠈⠁⠉⠅⠀⠀⣂⢢⢾⣴⢎⡿
⠀⠀⠀⠘⢿⣿⠿⠦⠀⠈⠓⠉⠀⠁⢴⠆⠑⡀⠀⠀⠒⠁⠠⠀⠸⣿⠊⠁⠀⠉⣡⠟⠛⠇⠀⠀⠀⠀⠀⡀⠐⠯⣭⠔⠀⠀⠀⠀⠀⣤⣴⣿⠔⠋
⠀⠀⠀⠀⠈⠙⢶⣢⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⢠⢀⡤⠀⠀⣀⣈⣤⠼⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠾⣦⢶⣤⣀⣀⡀⠀⠀⠀⢫⡑⠂⠤⢤⣤⣠⣤⣤⣔⣶⣴⡬⠴⠖⢋⣝⣀⣤⣬⣭⠬⠖⢛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⡀⠉⠉⠓⠒⠒⠛⠒⠒⠶⠦⠴⠶⠒⠒⠒⠒⠒⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⢶⣶⣤⣦⣤⣤⣤⣴⣰⣶⠤⠤⠆⣲⠖⠲⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠄⠀⠀⠀⠤⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠜⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠒⠒⣾⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣈⠲⢍⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠛⢦⣬⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢀⡟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⠴⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⠿⠛⠋⠁⠀⠀⡀⡸
⠀⠀⠀⠀⢲⠀⠀⠀⠒⠂⢄⢀⢬⠢⡐⢄⠀⠀⠀⠀⠀⠈⠛⠲⠤⠤⣤⡤⢊⡵⠛⠁⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠑⢄⡉⠉⠂⢤⠙⣦⣅⣈⡒⣬⣖⡖⣰⣀⡀⠀⠀⣔⣚⣁⠔⠋⠀⠀⣀⡀⠤⣐⠬⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢐⢢⡤⣽⣿⣦⢜⣶⠚⣅⠭⠷⣉⠯⢭⣭⣭⣭⠭⠤⠭⠭⠁⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣽⣾⢻⣛⢯⡿⡯⢽⠵⣪⣞⢴⡒⣤⡟⡳⣧⢸⡄
⠀⠀⠀⠀⠀⠈⢁⣌⠁⢯⢀⣷⠶⡪⢷⣸⣟⣛⣻⣦⣜⣸⡿⠷⠿⠿⠟⠁⠲⠼
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠢⠤⠒⠚⠁⢱⠇⠎⢤⣺⡳⢖⣙⡃⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣅⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠆⠀⠀⢀⣴⣿⣿⣿⣠⣾⣿⣿⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⣿⠟⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⠛⣥⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣧⣴⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⢀⣠⡴⠶⠶⠆⠀⠀⠀⠈⠙⣿⣿⡿⠛⡉⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢻⣩⣿⣧⠀⠉⠉⣤⢤⣤⣄⠀⠀⠀⠀⠈⠛⠋⢀⡆⠈⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡀⠀⠀⠀⠀⠀⡁⢸⡟⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⣟⠀⠀⡀⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡀⠀⠀⠀⠀⠀⠹⠟⢛⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢂⣀⠀⡄⢀⣀⣀⣀⣀⣐⣿⣥⠦⠖⠒⠀⠀⠀⠀⠀⠀⠀⢀⡬⠁⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣟⡛⠃⠀⠚⠛⠛⠋⠉⠉⠛⣦⠄⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⢀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡠⠗⣸⣇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⣠⠂⠀⠀⠀⠘⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣟⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⣶⣿⡽⠟⠒⠀⠀⠠⠎⠁⠀⠀⠀⢀⠀⠀⠈⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠛⣿⣻⣿⡿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣿⣦⣀⣠⡔⠊⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠿⠟⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠀⣸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠴⢾⣿⠟⣿⠁⠀⠀⠀⣀⣤⡤⠴⠶⠛⠁⠀⣰⣿⣿⡇⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠘⠉⣰⠋⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠘⣻⣿⡟⠁⠀⢠⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁⠀⢀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢳⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣤⡖⠂⠀⠀⣀⡀⠀⠉⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠉⠉⠀⠀⠸⡇⠀⠀⠀⠙⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡦⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠊⠑⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⢀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⢿⣿⣿⣿⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠋⠻⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠢⡀⠀⣀
⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠁⡉⠍
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⠀⢀⠀⢀⠀⢀⠀⣤⠀⢘⡇⠀⢀⠀⢀⠀⠀⠀⢠⡄⠀⣉⠀⡀⠀⢀⠀⠀⠀⡁⠖⠀⠁
⣿⡿⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢧⠀⢧⠀⠀⠃⠈⠲⢄⠀⣿⠀⢸⡇⠀⢸⡆⢾⠀⠀⠃⢸⡆⠀⣿⠸⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠑⠂⠀⠈⠀⠈⠀⠈⠀⠈⠁⠀⠈⠁⠈⠑⠀⠀⠀⠁⠀⠉⠀⠉⠒⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠐⠀⠀⠿⠀⠀⣟⠀⠈⡇⢠⠁⠀⢹⠄⢻⠀⡇⠀⠣⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠛⠀⠀⠛⠀⠀⠛⠈⠃⠀⠘⠀⢻⠀⠓⠀⠂⠀⠃⠀⠀⠰⣶⣿⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⡀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠸⣶⣝⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣷⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⡀⣀⣿⣿
⠀⢠⣀⠀⠀⠈⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡿⣿⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠌⠳⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣜⣭⣙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⣨⣭⢂
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣴⣿⣿⣿⣯⡆
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢠⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⠿⠿⠀⠘⠒⣶⡖⣶⣶⡖⣶⡶⠐⠁⠐⠿⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⢿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠛⠁⢻⣿⠇⠘⠃⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡇
⠀⠀⠀⠀⢸⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡇
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠻⠄⠀⠀⠀⠀⠀⠰⠋⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠹⡄⠒⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠒⣰⠃
⠀⠀⠀⠀⠀⠙⢶⡛⠛⠀⠀⠀⠀⠀⠀⠀⢤⣠⣤⠶⣄⣀⠄⠀⠀⠀⠀⠀⠀⠈⠛⣳⡾⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⢣⡀⠀⣠⠎⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⣶⠶⠦⠤⠽⠿⠥⠤⠶⠶⡖⠚⠋⠉⣰⠏⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡆⠀⢠⠀⠀⢠⡀⠀⡀⢱⡀⠀⣰⠏⠈⠁⠀⣿⣟⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢳⣤⢮⣴⣶⣼⣧⣤⠇⠀⡇⢠⠏⠀⠀⣠⠖⢋⣽⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⡿⠃⣀⡴⠋⠁⠀⠈⢀⣤⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠸⣿⣿⣿⣿⡿⠀⣀⡼⠓⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠒⠠⠤⠤⠤⠤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠧⠼⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢱⠤⡀⠀⠀⢠⠊⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⡸⠦⠭⠐⣂⣀⣀⣐⣒⣮⠥⠖⠉⢢⠴⠃⠀⠇⣀⣀⡤⢤⣖⣓⣶⣋⣱⣞⡙⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢸⠵⠚⠋⠉⠉⠀⠀⠀⠀⠈⠉⠛⠾⣏⢹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⡘⠀⠀⠀⠀⠀⠀⣠⣤⡖⢲⡄⠀⠀⠸⡄⠀⠀⠀⡼⠀⣠⠤⠒⠒⠋⠉⠉⠑⣦⠠⣄⠀⠈⠻⣤⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⡇⠀⠀⠀⠀⠀⠀⢧⡈⠀⡜⠁⠀⠀⠀⠈⠓⠒⣎⠁⠀⡟⠲⠤⠤⠤⠤⠖⠚⢹⠿⡨⠃⠀⠀⠈⢷⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣀⡸⣄⠀⠀⠀⠀⠀⣀⣼⠛⠒⠦⡄⠀⠀⢸⡶⢧
⠀⠀⠀⠀⠀⠀⠀⣠⠒⢺⠗⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢀⡎⠁⠀⠯⣙⣶⡶⠾⠯⠥⠞⠀⠀⣠⠏⠀⠀⢸⠷⡞
⠀⠀⠀⠀⠀⠀⢀⠽⡶⠁⠀⠀⠀⠈⠑⠲⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠋⠉⠀⠀⠀⠑⠦⠤⠤⣄⣀⣀⣀⣀⣠⠤⠖⠋⠁⠀⠀⢀⡞⠳⡇
⠀⠀⠀⠀⠀⠀⢧⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢊⣉⠩⠭⢭⣉⡭⠽⣭⣉⠑⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⠒⠁
⠀⠀⠀⠀⠀⠀⢸⡉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡰⠋⣤⠈⠰⠶⠈⢧⢈⣁⡬⢳⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⡤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⡝⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠙⠢⠤⣬⣧⡤⣴⠛⢤⣉⠤⠋⠀⢀⡼⠁⠀⠀⠀⠀⠀⢀⡴⠻⣄⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢀⣀⣀⣈⣂⣀⣽⢒⣊⣠⠤⠖⠚⠉⠀⠀⠀⠀⣀⣤⠞⠙⣦⣀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣏⢉⣳⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣀⣀⡤⢤⡖⠚⣯⡁⢸⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠳⠶⠚⣇⣈⡽⡟⢛⣶⠛⠛⣻⣟⣛⡿⣟⣋⠿⣉⡹⠧⣤⠞⠑⠶⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠑⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠻⠿⠟⠛⠉⡀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡶⠄⠀⠀⠀⠀⠀⠀⠀⣴⢿⠿⠂
⠀⠀⠀⢀⣀⣤⣤⡀⠀⠐⠾⣿⣿⣾⣿⣿⣶⣖⣒⣠⣤⣶⣾⣿⣧⡤⠶⠖⠛⠀⠀⣠⣶⣿⡆
⠀⠀⠀⠈⣿⣿⣿⣿⣶⣤⣀⣀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠉⠀⣀⣀⣀⣤⣶⣿⣿⣿⡯
⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠿⠟⠟⠛⠋⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣶⣤⡀⠀⠀⣤⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⣿⣆⣾⣿⣿⣥⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢀⣀⡀⠀⠀⢀⣾⠿⠿⢿⣿⣿⠟⠻⣿⣿⣟⠛⠛⠳⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠈⠻⣿⣧⠶⣶⣤⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠈⢿⣦⣴⣿⣿⣿⣴⣶⣶⣤⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣷⠻⠄⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠙⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⣀⣀⣀⣘⣁⣀⣀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⢿⣿⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠈⠉⠁⠙⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⣾⣀⣤⣴⣶⣾⣿⣿⡛⢿⣿⣿⣿⣿⣯⣙⣻⣛⣛⣿⣿⠿⣷⣶⣤⣀⠀⢠⡁⠀⠀⠀⢀
⠀⠀⠀⢰⡝⢿⣟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣽⣿⡷⢞⣴⣧⡶⡋⠀⠀⢍⣆
⠀⠀⠀⠘⠃⠀⠊⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⣛⣭⣴⣻⣿⣿⡿⠁⠀⠀⠀⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⢸⣿⣿⢸⡿⠟⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⠁⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣃⣀⡤⠖⣉⠔⠁
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⢟⣋⡡⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠖⠶⠾⠿⠛⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠶⠖⠒⠒⠒⠛⠛⠛⠛⠛⠛⠓⣒⣒⣶⣶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⢰⣍⠤⣄⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣢⣀⣀⣈⠁⢀⣊⣿⣆
⠀⠀⠀⠀⠀⢸⠸⠏⠙⠛⠻⠿⠭⢭⣿⣿⣿⣿⣿⣿⣭⣭⡭⠭⠿⠖⠛⠋⠉⠁⢸⣿⠀⣀⣤⡤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⢸⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠄⠀⣴⡂⠐⢸⣿⠟⢁⡴⠚⠛⠒⢦⡙⢷⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⢓⣒⠀⢸⡇⢺⡟⠀⠀⠀⠀⠀⢷⠈⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠝⠒⠂⢸⡷⠾⠃⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠭⠭⠂⠀⣿⠃⠀⠀⠀⠀⠀⢀⡴⠃⣼⠃
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠭⠤⠀⢸⡟⠀⢀⣀⣤⠴⠒⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⡪⠍⢠⣿⡵⣚⣭⣥⣶⣶⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⠂⢠⡿⠻⣫⠾⠛⠉⠁
⠀⠀⠀⠀⠀⢀⣀⣤⣬⣧⠀⠀⠀⠀⢈⣀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣒⢀⣿⠷⠾⢧⣤⣤⣄⣀⡀
⢀⡠⠴⠒⠋⠉⠀⠀⠀⢸⣆⠀⠀⠀⠠⠿⠦⠤⠤⠀⠠⠤⠌⢩⣽⣃⣸⣏⣀⣀⣾⣿⠿⠟⠻⠿⠳⢦⡄⠀⠀⡀
⢎⠀⠀⠀⠀⠠⠚⠻⠀⠀⢿⣄⠀⠀⠤⠽⠽⠭⠭⠤⠬⠿⠛⣛⣋⣭⣿⠧⠿⠋⠉⠉⠙⠂⠀⠀⠀⢠⡿⠀⠀⠀⠀⡂
⠈⠓⢦⣄⣀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⠶⠶⠶⠶⠾⠿⠛⠛⠋⠉⠀⠀⠀⡀⠀⠀⢀⣀⣠⣴⠾⠋⠀⠀⠠⠤⠤⠤
⠀⠀⠀⠈⠉⠛⠿⢿⣶⣒⡲⠶⠶⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠿⠿⠟⣛⣻⡿⠿⠛⠉⠀⠀⠀⠈⠉⠉⠉⠩⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠈⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡼⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣠⠏⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⠀⡼⠃⠐⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢹⢷⣇⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⣹⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢹⡿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀⣀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢇⣼⣦⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣬⣍⣛⡻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠀⠀⢸⡏⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠑⠒⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢋⠀⢠⡀⠀⠀⣠
⢰⡆⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⣿⣿⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠘⢣⠛⠃⢶⡄⠉
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠙⢿⣷⣌⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢳⣺⣮⠁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣮⡙⠳⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⣼⢛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣟⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣀⣠⡶⢯⣼⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠿⢿⣵⠆⢲⡎⠁⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠿⠿⠛⠛⣿⢶⣶⡠⢴⠆⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣼⣦⣤⣬⣛⣡⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣟⣻⣋⣜⣛⣟⣛⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣭⣭⣽⣿⣯⣭⣭⣭⣭⣽⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠊⠁⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⣀⣀⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠑⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠔⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠐⠊⠀⠘⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣶⣶⣦⣴⣦⣷⡘⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⢁⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣟⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢋⣾⢿⣿⣿⣿⣿⣿⠉⢿⣿⣿⣿⡷⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣇⢈⡉⠛⢿⣿⠁⠀⠀⢹⣿⣿⣟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠉⢭⢍⣉⣛⡃⣰⣿⣔⣰⠀⠛⠋⢻⣿⣿⠀⢸⡟⠈⠀⠀⢸⣿⡀⠀⠀⣸⣿⣿⣿⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⢰⢰⣶⡦⡇⡤⡤⠤⡤⠀⢻⠁⢸⣿⣿⠀⢸⣦⠀⠀⠀⠋⢸⣿⣷⣿⣿⣿⣿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠠⣿⠷⠇⣿⣿⣟⣿⠀⢸⡇⢸⣿⣿⠀⠺⠿⣧⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⢸⡄⣿⣷⡧⡀⠀⢀⢀⠀⠸⠃⠸⠃⠀⠀⢂⣁⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣋⣤⣤⠂⣭⣸⣍⡉⣀⣞⡇⠀⠀⠠⠀⢨⣭⣭⣭⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠛⠃⠙⠺⠃⠀⠀⠈⡃⠄⠀⠀⠀⢐⣓⢶⣶⠿⣿⢿⣿⠟⠁⠀⠀⠉⠉⠉⠻⢿⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠛⢳⠀⠀⠈⠁⠀⠀⠀⠀⠀⠿⠬⡧⠀⠉⠀⠻⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠖⠀⠂⠂⠀⠀⢛⣛⢠⠀⠀⠀⠀⠄⠀⠀⠀⣛⠐⡓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠒⠲⠆⠀⠆⣭⣭⠀⠀⠀⠀⠀⠑⠀⠀⠀⣭⠬⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⠀⠴⠶⠀⠀⠀⠀⣶⢄⡀⠀⠀⢛⢒⣿⠆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡐⠤⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣇
⠀⠀⣾⡏⠈⢙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠂⠌⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠹⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡦⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢠⢀⣀⠴⠊⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡌⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠚⠃⠀⠈⠙⠋⠂⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣀⣟⠶⣤⣶⣾⣿⣿⣿⣿⠶⠴⢞⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⣤⡜⢶⣛⢇⡀⠩⠘⣿⣷⣶⠆⠀⠀⠀⠀⠻⠛⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡶⠋⠐⠾⠿⠿⠋⠶⢶⣤⠀⠳⡄⠀⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠔⢉⣤⣶⣤⢦⣤⣤⣶⣶⣾⣿⠀⠀⡇⠀⣻⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠉⠜⠉⠀⢀⣤⣄⢀⣀⣤⣽⠀⠀⣷⠀⠈⣿⡟⠅⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠴⠾⠟⠛⠛⠻⠿⣿⡏⠛⠁⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⢀⣀⠴⠶⠶⢶⣶⣦⣼⡿⠄⠀⠀⠀⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⣤⡤⠶⠛⠁⢠⣤⣤⣤⣤⢀⠉⡯⠙⢦⣄⡀⠀⠀⠐⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠂⣀⣠⣁⣀⢠⣴⣦⣭⡿⠿⠿⠟⠛⠁⠀⠀⠈⠉⠉⢻⡁⠰⣿⣿⡃⠀⠀⠀⠀⠀⠹⢗⡇
⠀⠀⠀⣴⠞⠋⠁⠈⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠿⠷⠦⠤⠤⠤⠴⠖⠚⠉


⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠠⠄⠀⠀⣀⣀⠀⢁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣀⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⡠⠤⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠔⠉⠁⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢈⠀⠤⡁⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠰⠊⠉⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣌⡒⠂⠤⢄⡀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠪⠐⠠⠔⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⡶⢂⠚⡃⠈⠖⣠⣀⢲⠶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠉⣀⣠⣭⣧⣼⣄⣧⣌⣼⣾⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⡴⣼⡉⢦⣤⣙⣻⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠟⢛⣛⣯⣵⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⣟⠀⠘⣇⢸⣿⠯⠍⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡙⢄⠀⢻⡄⢻⡄⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣑⡂⠿⣄⠛⢆⢠⡻⢿⣧⡀⠀⠀⠀⠀⠀⠀⣰⣟⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠾⣷⣌⣉⢡⣼⣿⣷⣄⠀⠀⠀⢀⡴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⡶⠶⠚⢭⣿⣿⣭⣭⣛⣛⣛⣛⣺⣛⣛⣼⣓⣂⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣰⣿⣿⣿⣿⡃⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⠿⠿⠛⣛⣿⣿⣿⠖⠈⡂
⠀⠀⠲⢼⣿⣿⣶⣶⣽⠿⣿⠿⣿⣿⣦⣤⣉⣙⣛⠛⠛⠛⠛⠒⠶⠆⠀⠀⠀⠀⠛⢉⣉⣡⣴⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⢤⣄⣒⠀⠁⠉⠉⠉⠛⠛⠛⠿⠿⠷⠶⠿⠶⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠢⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢳⢦⣤⣤⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣭⣤⡽⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣟⠁⠲⣾⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣾⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢰⠇⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢠⡏⠀⢠⡟⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡾⠀⢀⡾⠀⠀⣰⠏⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡿⠀⣸⠇⠀⣼⠃⠀⣰⠏⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠛⠛⢹⡇⠀⣿⠀⢠⡏⠀⢠⡏⠀⢀⡞⠘⠛⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣩⡴⠞⣋⣁⠀⢸⡇⠀⣿⠀⢸⡇⠀⢸⡃⠀⢸⠇⠀⠀⠒⠤⣌⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢺⣿⣾⣟⣭⣤⣴⣾⣇⣶⣿⣰⣾⡇⣶⣾⣧⣦⣾⣠⣄⣈⠙⠲⣦⣝⠂⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠖⠀⣀⣴⠟⢸⣛⡷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣌⠙⠂⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⣁⣀⣠⣤⠶⠟⣫⣴⡇⣚⣛⠲⢤⡈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⢋⣉⣩⣥⣴⡾⠿⣛⣥⠀⠉⠉⠙⢷⣷⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠞⠃⢹⣟⠷⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⣉⣉⣭⣴⣶⠿⢛⡅⠀⠒⢂⣠⢿⢏⠄⣾⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠾⣛⡭⠔⣀⠀⠀⠀⢿⡛⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠛⠛⠛⣛⣋⣩⣥⣶⠾⡛⢡⡶⢞⣫⣵⡿⢋⡾⢣⣄
⠀⠀⠀⠀⢀⣴⠏⣡⠞⢫⡴⢋⣡⠔⠀⠀⠈⢿⡷⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⣛⣋⣭⣵⣶⠿⢡⣾⣿⠿⣛⣭⠾⠋⠀⠠⡉⢳⣄
⠀⠀⠀⠀⣼⠃⣸⡏⢰⡏⠠⡟⠁⣴⠉⠀⠀⠈⢻⣷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⣛⣋⣩⣭⣶⠞⣡⣿⣷⣶⣿⣿⣷⡄⢦⡘⢦⢹⡄⢻⡆
⠀⠀⠀⠀⢹⣆⠸⣧⡈⠷⣄⡻⢦⣌⠳⠤⠀⠀⠀⠙⢿⣖⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⣉⣩⣭⣭⣶⣾⣴⣿⣯⢿⣿⠹⣿⡌⣿⠈⡧⢸⠀⢁⣼⠃
⠀⠀⠀⠀⠀⠙⢦⣝⠻⣶⣌⡛⠶⢬⣝⡓⠀⠀⠀⠀⠀⠙⠿⣷⣶⠦⢤⣤⣤⣤⣬⣭⡭⢷⣿⣿⣿⣿⠿⣣⣿⡟⣸⣿⢀⣿⢣⠏⣰⠃⢈⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠰⠿⠷⢦⣍⣛⠷⠶⣤⣉⡉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠶⢖⣛⣛⣛⣭⣥⡶⠿⢛⣩⣾⠟⠡⠞⠁⢀⣠⣤⣾⣿⡿⠯⠍
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣩⣿⣷⣶⣦⣬⣅⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣉⣭⣭⣭⣭⠵⠶⠶⣛⣛⣉⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣯⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠿⠿⠿⠿⢿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠶⠶⠶⠶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⣉⣉⣉⣉⣩⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣋⣉⣉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠄⠔⣀⡠⠤⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡰⠁⣀⡨⢄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠠⠊⠀⢰⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠁⠀⡀⠸⠈⡁⡇⡠⠀⣠⣔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠃⣀⢵⣄⠀⠃⢞⡈⢉⡽⠠⠔⠁⠀⠉⠉⠉⠉⠉⠁⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⣚⠳⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠈⠉⠀⠒⠤⠄⡀⠀⣀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣼⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢊⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢑⠃⡠⠃⢠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⡀⠑⢌⡠⢁⣳⠤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠁⡀⡈⣿⣛⡠⠊⠁⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣉⠀⠐⠲⠖⠀⠉⠀⣰⢿⡵⡂⢠⠆⠀⠀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠑⣔⠀⠀⠀⠀⢀⡷⢛⡗⢞⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⢄⡀⠅⠋⠘⠤⠊⡶⢯⡶⠛⠹⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠈⠦⣙⣀⢠⠛⢄⠊⢃⠸⢷⠆⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⣿⠀⠆⡡⢚⠀⣆⠤⠢⠊⡜⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠢⠀⠠⠔⠂⣡⣂⣤⣤⣾⣽⡾⠷⠔⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠛⠒⠒⠈⠁

⠀⠀⠠⠶⠶⠦⠄⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠒⠊⠉⠀⠀⠀⠀⢀⣀⣠⣤⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿
⡤⠤⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠒⠈⠉⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⣿⣿
⠀⠀⢀⣴⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿
⢀⣘⣻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣆⠀⠠⠄⠀⠤⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠹⣿⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠙⠿⣿⣿⣶⣬⣽
⣿⣿⣿⣇⢹⣧⣤⡤⠀⠀⠀⠀⠀⠀⢠⠀⡾⠀⢀⣀⣀⣀⣄⣀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿
⣿⣿⣿⢿⣟⠛⠉⠀⠀⠀⠀⠀⠀⣀⣘⣒⣛⣤⣿⣭⣭⣿⣿⣶⡶⠿⠿⠷⠆⠁⠀⠀⠴⠶⠾⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣄⡁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣽⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⡀⠀⠀⢀⠀⢀⡀⠀⠀⡀⠀⠀⡀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⢸⣷⣤⣴⣷⠀⠇⠳⠢⠎⢡⢰⠉⠦⠴⠉⡆⣸⣷⣤⣾⣿⠀⠰⠹⠖⠴⡍⡄⠀⠀⡘⡡⠤⢼⠱
⠀⢺⣿⣿⣿⣿⠀⠀⡄⠀⣤⢸⢰⢰⡆⢠⡖⡆⣿⣽⣿⣯⢹⠆⢀⢠⠀⢠⡄⡇⠀⠀⠃⣤⠀⣠⠈
⠀⠘⣿⣿⣿⠟⠀⢦⡀⠒⢀⠌⠘⣌⠑⠊⣡⠃⠹⣿⣿⣿⡿⠀⠹⣄⠀⠀⡰⠃⠀⠀⢧⠀⠀⠈⡸
⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠙⠉

⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠾⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠛⠛⠋⠉⠉⠉⠛⠛⠻⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⢀⣴⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⢰⣿⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⣠⣤⣶⣦⣄⡀⣀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣧⣠⣼⠛⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠹⣿⠋⠁⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⢻⣦⣀⣴⣿⣯⣀⣤⡶⠾⠛⠛⠋⠉⠉⠛⠛⠷⣦⣄
⠀⠀⠀⠀⣾⡏⠉⠛⢿⣦⣀⡀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⢀⣀⣼⡿⠛⠉⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠙⠛⠷⠶⣦⣤⣤⣴⣿⣷⣤⣀⣀⣀⣀⣠⣴⣿⣧⣤⣤⡴⠶⠞⠛⠉⠁⠀⠀⠀⣿⣧⣤⣶⠶⠾⠛⠛⠛⠻⠷⣦⣄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠏⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣀⣤⣶⠾⠛⠁⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡶⠾⠛⠋⠉⠀⠀⠀⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠷⠶⠶⠾⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⡶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠠⠠⡄⢀⠀⠀⢒⠒⠒⠯⣷⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣬⣜⠋⠀⣤⣉⡻⣿⢦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡙⠋⢘⣿⣿⠄⡀
⠀⠀⠀⠀⢸⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣭⣿⡿⢠⣇⡴⠞⠉⠙⣦
⠀⠀⠀⠀⠘⣿⣮⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⣼⡟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣭⣟⣛⠻⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣻⡍⡋⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏⡙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣽⣇⠁⢤⣀⢀⠀⠀⠀⠀⠀⢀⣀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠿⠿⠦⠧⠤⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⢄⣀⠀⠀⠐⢴⡟⠆⠀⠰⣾⠧⠀⢰⢦⠀⡀⢂⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠘⠤⠄⠞⠻⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⢀⠊⡇
⠀⠀⢀⡀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠚⠉⠐⠒⠲⠤⣀⠀⠀⢀⠄⣠⠄
⠀⠀⢸⡅⢸⡁⠀⠀⠀⢸⠀⠒⢒⠢⡀⠀⠀⠀⡠⠺⠷⠶⠀⠀⠀⠀⠀⠀⠐⠛⠺⡗⢦⡀⠐⠁⢀⡀
⢾⠀⢺⠀⢮⢠⠆⠀⠀⠈⠦⠼⠷⠤⡐⠀⢀⠞⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠑⡄⠀⠹⠛⠂
⢻⠀⣻⠄⡿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣷⠶⠸⠀⣰⠀⠀⠀⠸⠿⣆⣸⠀⠀⠀⠸⠀⠠⣤⣄
⣹⠀⢻⠀⠗⣧⠀⠀⠀⠀⠂⢤⣀⠤⠀⢸⠀⠀⠀⠀⡀⠀⢎⡀⢤⠄⣤⡀⠀⠀⠉⠒⠀⠀⠀⡇⠀⠛⠁
⢹⠂⡠⠲⡄⢁⣀⠀⠀⠑⢄⣤⡕⠀⠀⢸⠀⠀⡠⠚⠀⢀⡀⠀⠀⠀⣀⣙⠦⠤⢤⠀⠀⠀⠀⡇⠀⢠⢄
⠈⠁⢈⠟⠓⠚⢤⡠⢦⠀⠀⠀⠀⠀⠀⠈⡆⠀⠁⠈⠣⠤⣀⣀⣀⣀⣀⠤⠜⠃⠄⠀⠀⠀⢠⠃⠀⠲⠄
⠀⢰⡃⠀⢰⡀⠀⢣⠀⠀⠒⠤⣀⣀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠩⢂
⠀⠈⣧⠀⠈⣧⠀⠈⡆⠀⠐⢄⣤⡶⡀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠰⠽⠇
⠀⠀⠸⡆⠀⠺⡏⠀⠸⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⣀⣀⡀⠀⠔⠊⠁⠀⠀⠀⣱⠒⠒⠄
⠀⠀⠀⢻⡀⠀⢻⡆⠀⢣⠀⠀⠀⠀⣠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⠑⠂
⠀⠀⠀⠈⢷⠀⠀⢷⠂⠈⢆⠀⠀⡠⠉⠀⡣⠀⡐⡀⡀⠀⢀⠀⠠⠂⠀⠀⢠⡀⠰⣄⣀⣛⡃⢹⠀⠈
⠀⠀⠀⠀⠘⣇⠀⠘⣧⠀⠘⡄⠂⠀⠀⠉⠀⠀⡧⠁⢣⠀⡷⠷⣧⠀⠸⡿⠂⠃⠀⠀⠀⠀⠙⠈⣄
⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⢱⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠈⠄⡀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠐⠄⠀⠑
⠀⠀⠀⠀⠀⠀⢷⠀⠸⠿⢇⣀⠇
⣀⣀⣀⣀⣀⣀⣘⣧⣴⣚⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠋⠉⠈⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣀⣤⣤⣶⠏⠐⠲⠦⢤⣀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠞⠛⠛⠛⠂⠀⠀⠀⠐⠛⠋⠛⠳⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡄⠀⠰⠞⣿⠲⡄⠀⠀⠀⡤⢲⡟⠶⠄⠀⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣇⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠈⠁⠋⠀⢀⡿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠀⠀⠀⠀⠛⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣘⡲⡄⠀⢀⣴⣲⡲⣄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢳⢳⠀⡐⣌⢁⠀⢀⠈⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡏⣇⢳⠘⣮⣉⣩⠞⠀⠀⡜⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢸⣸⡆⢳⡀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠃⠀⢳⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣷⡿⠓⠀⠉⠁⠚⢷⣶⠀⠀⠀⠀⠀⣷⣾⠗⠂⠀⠀⠒⢿⣶⡇⠀⠀⠀⠀⢺⣾⠶⠒⠀⠀⠒⠾⣷⡇
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠟⠁⢀⠀⠀⢀⡀⠀⠙⠇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠆⠀⠻⠟⠀⠘⠿⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠲
⠀⠀⠀⠀⣂⠀⠀⠀⣭⠀⠀⠀⣈⠀⠀⠀⠀⠀⢠⠀⠀⠀⣬⡄⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢈⡁⠀⠀⢀⡂
⠀⠀⠀⠀⠈⠓⠠⢀⣀⡀⠄⠚⠁⠀⠀⠀⠀⠀⠈⠓⠠⢀⣀⣀⠠⠐⠃⠀⠀⠀⠀⠀⠀⠙⠢⢀⣀⢀⡀⠔⠋


⠀⠀⠀⠀⣿⡿⠒⠀⠀⠀⠒⠿⣿⠀⠀⠀⠀⠀⣿⣿⠖⠂⠀⠀⠒⢾⣿⡇⠀⠀⠀⠀⢺⣿⠗⠂⠀⠀⠐⠶⣿⡇
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠥⠿⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠸⠁⢀⠀⠀⠀⡀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠑⠀⠀⠀⠂⠀⠆⠀⠀⠀⠀⠀⠀⠸⡾⠀⠀⢷⠇
⠀⠀⠀⠀⣀⠀⠀⠀⣭⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢬⠄⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢨⡅⠀⠀⢀⠂
⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠈⠓⠠⠀⠀⠀⠠⠐⠋⠀⠀⠀⠀⠀⠀⠙⠢⠀⠈⠁⠀⠔⠋


⠀⠀⠀⠀⣷⡷⠓⠀⠉⠀⠒⢾⣿⠂⠀⠀⠀⠀⣷⣶⠖⠂⠀⠀⠒⢶⣶⡇⠀⠀⠀⠀⢲⣶⠶⠒⠀⠀⠒⠶⣶⡆
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠟⠁⢀⠀⠀⢀⡀⠀⠙⠇⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⡞⠀⠘⡿⠃⠀⡀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠉⠁⡀
⠀⠀⠀⠀⣂⠀⠀⠀⣭⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢈⠀⠀⠀⢠⠀⠀⠀⠀⠀⢐⡀⠀⠀⢈⡁⠀⠀⢀
⠀⠀⠀⠀⠈⠓⠠⢀⣀⡀⠄⠚⠁⠀⠀⠀⠀⠀⠀⠑⠠⢀⣀⣀⠠⠐⠃⠀⠀⠀⠀⠀⠀⠙⠢⢀⣀⣀⡀⠔⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠀⢀⡴⠃⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⣠⡶⠚⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⡾⠃⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠸⣧⡀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠈⣿⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢠⡾⠋⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⠞⣀⣠⣤⣴⣶⣶⣾⣟⣛⣛⣛⣛⣛⣛⣓⡒⠶⢦⣤⣀⡀⠲⣄⠙⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⡾⠃⣿⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⣠⡟⠛⢲⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⠃⠀⢀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⠻⠿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠿⢻⠏⠀⣠⠞⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⢠⣯⡤⠞⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⢷⡄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⣠⠟⢉⡟⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠛⠛⠻⣅
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢰⡏⢀⣞⡴⠟⢛⣋⣉⣿⣶⠀⠀⠀⢀⣀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢸⣄⡞⢹⣿⣯⣭⣭⠴⢞⡟⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣝⣛⡛⠛⠦⣤⣤⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠶⠶⠦⠤⠤⠤⠴⠶⠶⠶⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⡎⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⣤⣶⣶⣺⣶⣒⣺⣖⣺⣒⣒⡲⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠖⠉⠑⠒⠈⠉⠀⠀⠀⠀⠈⡇⢠⠃⠀⡇⠀⠉⠉⠙⠒⠮⣭⣒⢤⡀
⠀⠀⠀⠀⠀⡾⢏⠀⠀⠀⠀⣀⣀⣀⣀⣀⢠⡼⠥⣯⠤⣼⣀⣀⣀⣀⣀⣀⠀⠀⠈⠛⢟⡄
⠀⠀⠀⠀⠀⡿⠦⣑⡒⠤⠭⣭⣀⣀⣀⣀⠈⠂⠀⠁⠀⠃⠀⠀⣀⣀⣀⣉⣉⣤⣤⠶⢾⠇
⠀⠀⠀⠀⠀⢻⡀⠀⠈⠉⠉⠓⠒⠒⠒⠛⠉⠭⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠁⠀⠀⣸⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⡟⠲⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⣿⣤⠤⢤⣉⠳⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠙⠛⠒⠒⠒⠶⠶⠦⠤⠴⠶⠶⠶⠒⠒⠒⠛⠉⠁⠀⢀⡏⠀⠀⠀⠘⡆⠈⡆
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⣠⠞⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣠⡤⠞⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⣀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡦⠤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠶⠶⠶⠖⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣾⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⡿⠿⠿⣿⣿⣿⣿⡿⣿⣛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣶⣭⣭⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠈⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠓⠿⠷⠀⠀⠀⠐⠛⠗⠈⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⣶⣦⠀⠀⠀⠐⣿⠆⠀⢸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠈⢙⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠐⢒⣲⠀⠀⠙⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠰⣏⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠈⠋⠁⢀⡀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠙⠛⠛⣇⣿⡇⠀⠀⠘⢷⡶⠶⠶⠖⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⣀⣿⡟⠁⠀⠀⠀⠀⠙⠶⠴⠞⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠳⣤⣀⠀⠀⠀⠀⢀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠸⡏⠛⠛⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠘⡇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢠⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⢸⣷⣤⣦⣼⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣥⣤⣹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣶⡘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠀⠀⣼⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣽⣶⡗⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣸⣿⣿⣿⣿⠏⠀⠀⠀⣸⣿⣿⣿⣧⢺⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢰⣿⣿⣿⣿⠟⠀⠈⠀⠀⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⠀⠀⠈⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡞⠿⢿⣿⣿⣿⠍⠀⠀⠐⠒⠀⠀⠀⢹⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⣠⣠⣌⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢿⣿⣼⣿⣿⣿
⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠹⣿⣿⣿⣿
⢸⣿⣿⡿⢿⣤⠆⠀⠀⠀⠀⠀⠀⢿⣿⣷⣼⡿⠿⠛⠛⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠁⠙⠿⠛⠛⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿
⠘⣿⡟⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠙⠃⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⢼⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠈⢿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⢐⡀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠈⠛⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⠿⠿⢶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠞⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠓⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⣴⡟⠉⠁⠀⢀⣀⡠⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠤⠤⠤⣄⣀⠀⠀⠀⠉⠳⣆
⠀⠀⠀⠀⠀⠀⣿⠉⠑⠲⠿⠭⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⣦⠈⣉⣑⣦⣤⣶⣿
⠀⠀⠀⠀⠀⠀⣿⢤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⢻⡟⠉⠉⠁⠀⠸⣿⢀⣤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠓⠲⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣀⣠⠴⣿⢋⣠⣤⣤⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠒⠒⠲⠶⠤⠤⠤⠤⠤⠴⠶⠒⠒⠒⣿⠋⠉⠉⠁⠀⠀⣿⠟⠉⠀⢨⡇⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢿⡀⠀⠀⠀⠀⢠⡏⠀⠀⢀⡞⢀⡾
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢉⣀⢤⣹⠀⠀⠀⠀⢸⠃⢀⡴⠋⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠦⠞⡏⠀⠀⠀⠀⣿⠴⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠶⠶⠃⠀⠀⠀⢰⣇⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⢀⡤⠖⠋⢉⣉⣈⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠙⠓⠦⣄⠀⠈⠙⢷⣄
⠀⠀⠀⣾⡏⠀⠰⣏⠉⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠛⠁⠀⠀⠀⠀⠀⣸⠇⠀⠀⢀⡿
⠀⠀⠀⠙⢷⣄⡀⠈⠉⠒⠲⠤⢤⣄⣀⣀⡈⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⢉⣁⣀⣀⡤⠤⠴⠒⠚⢉⣁⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣤⣀⣀⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠓⠒⠒⠒⠒⠒⠛⠉⠉⣉⣉⣁⣤⣤⡴⠶⠖⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⠻⠛⠛⠒⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⢀⣀⣤⠴⠶⠶⠖⠲⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⢠⡾⠛⠁⣀⣤⠶⠖⠛⠛⠛⠶⣄⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⣿⡄⠀⠀⢿⣅⣀⣠⡄⠀⠀⠀⢈⡇⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠈⠛⠷⣦⣄⣀⣉⠀⢀⣀⣀⣤⠾⠁⠀⢀⣴⡿⠃⠐⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠿⣶⣤⣄⣀⡀⠀⠉⠉⠉⠉⠉⠉⣁⣠⣤⡾⠟⠋⢀⡆⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⠀⠀⠈⡇⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⢸⡇⠒⠶⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⣀⣄⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢿⣿⣷⠆⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠐⠚⠋⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢠⣖⣂⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠻⠿⠛⠀⠀⠈⠙⠷⢤⣄⣀⣀⣀⣀⣀⣀⣠⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣤⣤⣀⣈⠙⠛⠛⠛⠛⠛⠉⠀⠀⣀⣀⣠⣤⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠾⠛⠛⠛⢛⡛⠻⠯⣶⣢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣻⢟⡭⠚⢁⠠⠤⠀⠠⣄⠈⣉⠉⠑⠲⠬⣙⠪⣝⢶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⣩⢞⣟⡵⢋⠔⡨⠔⠊⠀⠀⢈⡫⡠⠀⠀⠀⠀⠐⠪⡙⠦⡙⢿⣮⣙⡦⠄⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡴⠋⢠⢯⠞⡀⣠⠊⠀⣠⠄⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⠈⢢⠘⢎⢫⣧⠀⣀⣄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⠋⠀⢀⡏⡏⡜⠰⠃⠀⠀⠴⠄⠠⠀⠐⠀⠀⠀⠠⠆⠀⠁⠄⠀⠀⠀⠈⠈⡏⣿⠋⠀⢠⢧
⠀⠀⠀⠀⠀⠀⢠⡟⡰⠁⠀⠀⢸⢸⠁⠃⠃⠀⠀⠀⠀⠈⣄⠀⡀⠘⠇⢀⠆⠰⠀⠇⠀⢸⠀⠀⠀⠀⢸⢸⣂⡴⠛⠒⠧⠤⠀⣀
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣾⠸⡆⡆⢀⠀⡄⢂⠀⠁⠈⠓⠠⠦⣴⡅⠊⢀⠘⢀⠁⡎⠀⠀⠀⠀⢸⢸⣿⣶⣤⣄⡀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⢸⠇⠄⠀⠀⠀⠀⢻⡆⢧⠰⡈⢆⢘⣄⡢⣀⠀⠀⠀⠀⡈⢀⡼⢋⠔⡡⢊⠀⠀⠀⢰⢃⡟⣼⣿⡋⠋⠧⠙⢿⡖⠦⠤⠜
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣷⡘⣆⠀⠀⠀⠙⠻⣶⡛⠷⢶⣤⡙⠈⣴⣵⠟⠊⠀⠀⠀⡰⢃⡞⣰⣿⠙⢃⠀⠀⠀⠠⣇
⠀⠀⠀⠀⠀⣿⠰⠀⠀⠀⠀⠀⠀⡟⣷⣌⢧⡀⠐⢤⡀⠀⠈⠙⠲⢌⠻⢸⠟⠁⠀⠀⠀⡠⠞⣡⢏⣴⡿⠁⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣝⠲⣄⡈⠑⠢⠤⣄⠀⠐⠘⠀⠀⠀⢀⣠⡴⢋⣵⣾⡿⠃⠀⠀⠈⠘⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⡳⠮⣍⣓⠲⠦⠤⠤⠤⠤⠴⠒⣚⣩⡤⠞⣋⡴⠏⢀⢀⠂⠀⢀⠆⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⠙⠟⢷⣤⣈⡍⠛⠛⠛⠛⠛⠛⠋⢉⣠⡴⠞⠋⠀⠀⠡⢂⠄⢀⠌⡀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⢢⠣⡈⠲⣍⠛⠻⠿⢷⠶⠶⠷⠛⠛⠋⠉⠀⠀⠀⡀⠜⠁⡁⢠⠊⡔⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡀⢑⠢⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢐⡨⢐⡡⢊⠔⣡⠎⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⡐⠀⣀⠈⠁⠐⠂⠒⠂⠐⠀⡀⠤⠒⡡⠒⡡⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠨⢐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡠⢖⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢽⣳⡶⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⢖⣊⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠾⢭⣭⣭⣽⣭⣭⣭⠭⠶⠖⠛⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⠒⠀⠉⢀⣀⣀⣀⠀⠤⠤⠤⠄⠀⣀⣀⣈⠀⠉⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠂⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠤⠄⠄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠄⠤⠤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⡉⠳⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣇⠀⠘⡀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢲⠀⠀⠖⠚⡄⠀⠓⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡠⢒⡉⢆⠀⢯⠀⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠱⢤⡈⠢⡀
⠀⠀⠀⠀⠀⠀⢠⠊⡐⠁⠀⠁⠀⠈⠣⣄⠙⠢⠸⡄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡡⠔⠊⠉⠀⠀⠰⠗⢍⢢⠘⠄
⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⡼⠉⠒⠀⠤⠀⣀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠋⠊⠀⠀⠀⠀⠀⣠⠎⠰⠀⠀⡇⠐
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠘⠓⠉⠐⠒⠢⠤⠤⠤⠀⠀⠒⣈⡡⠴⣒⡲⠶⠶⢤⠤⠴⠚⠁⠀⠀⢀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠂⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⢒⡲⠶⠒⠂⣉⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡐⠠⢉⡐⠠⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠠⠒⠉⠀⠀⠀⣀⣀⣠⣤⣤⡤⠶⠒⢂⡁⠄⡢⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⣁⠀⠐⠀⠈⠉⠉⠁⠒⠒⠒⠒⠂⠂⠐⠒⠒⠚⠛⠛⠉⠉⠉⠁⠀⠐⠀⣀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠠⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠤⠄⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣵⣭⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠤⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆⣣⣤⡼⠃⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⠀⣱
⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⠒⠀⠀⠀⠀⠀⡐
⠀⠀⠀⡀⠀⠢⠀⠀⠐⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢁⣠⣤⣆⠏⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠈
⠀⠀⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠀⠀⠔⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠁⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⠀⠐⠄⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⣠⠤⠀⠒⠁⢀⡀⠠⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠆
⠀⠀⠀⠈⠂⠈⠉⣉⢀⣠⡤⠆⠂⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠐⠐⠀⠀⠀⠀⢀
⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠖⠀⠉⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⠴⠂⢀⠀⠀⡆⠀⠀⠙⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠂
⠀⠈⠆⠁⠆⠀⠤⠄⡔⠂⠉⠁⡤⣀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠔⠁⢠⡄⠃⠀⠀⠅⠀⠀⡇⠀⠀⠒⠂⠙⠋⠘⠃⠁
⠀⠀⠈⠐⠓⠄⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠐⠀⠔⠀⢐⠀⡇⠀⠀⠁⠀⢠⣴⠇⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⠌⠀⠀⠢⠀⠠⢍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠐⠀⠀⠤⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠠⠀⠀⠘⠀⠐⠊⠉⠀⠀⠠⠤⠄⠐⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠂⠚⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠀⠀⠀⡇⡁⢇⡡
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠁⠀⠉⠁⠘⠓
⠀⠀⠀⡄⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠓
⠀⠀⢠⠦⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⢠⠁⠀⠀⠈⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⡀⠠⠤⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⢀⠀⠀⠤⠤⠤⠀⠀⠀⠀⡀
⠀⠐⠀⠀⠁⠀⡀⠀⠀⠀⠈⠀⠀⠀⠳⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⠀⠤⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⢆⠀⠾⠡⠄⠆⡱⠀⠀⠀⠀⠀⠀⢀⡀⠘⠃⠤⠤⠛⠀⣀⣥⠀⠀⠀⠀⠀⠀⠐⠛⠐⠘⠉⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠐⠂⠀⠐⠒⠁⠀⠀⠀⠀⠀⢰⠀⢈⠈⠉⠉⣉⢉⡉⠀⢹⠀⠀⠀⠀⠀⠀⡀⢠⠀⠐⢬⡀⠀⠀⠘⠀⠀⠀⠀⢀⠚⠉⠀⠀⠆⠔⠀⠁⠓⣄
⠀⠀⠀⠀⠥⣬⠈⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠊⠜⠃⠨⠀⠄⠂⢼⠀⠀⠀⠀⠀⠀⠁⠋⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠒⠂⠀⠆⠐⠔⠂⣸
⠀⢰⠀⠀⢀⠁⠉⢀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⣉⡁⠘⠋⠁⢀⣀⠔⠁
⠀⠈⢄⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⡀⠃⠀⢂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⢰⣄⢀⡉⠉⠉⠉⠉⢁⣀⡴
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠀⠀⠉⠉⠉⠉⠀⠀⠐⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡨⠐⠂⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⡘⡠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠠⢐⠡⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡨⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠢⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠐⠁
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣐⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣂⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⣿⣿⣿⣿⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣁⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡿⣫⣵⣾⣿⣿⣷⣭⣵⣿⣿⣿⣿⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠉⣻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⢸⣿⣿⣿⣿⣿⣿⣿⡟⣡⣤⣄⠀⠠⠞⠛⠛⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⠛⣉⣀⠉⠁⠀⣠⣄⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⢻⡿⢿⣿⠃⠀⠀⠙⠛⠀⠀⣀⠙⠋⠀⠘⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣸⡇⠘⣿⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠞⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣠⢞⣿⣯⠶⠋⠉⢛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡴⠞⠁⠘⠉⠁⠀⠀⣀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢸⡇⠀⠀⣀⣤⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠋⠀⠀⠀⣠⡞⠁⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡙⠛⢳⡖⠛⠛⠋⠁⠀⠀⠀⠀⠀⣰⠋⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣧⠀⠀⠀⠀⠀⢀⡞⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠀⠀⠀⠀⢠⡞⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠰⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣿⣇⠙⢿⣿⣿⣿
⠀⠀⠀⠐⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⢀⡀⠀⠠⢠⠧⠀⠀⠒⠒⠒⠒⡋⢛⠫⠽⢻⣿⣿
⠀⠀⢀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠐⠒⠈⠉⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⡇⠀⠡⣀⢸⡿⣿
⠀⠀⠂⠀⠄⡠⠟⠋⠀⣀⠀⠤⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⠁⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠈⠀⠀⠻
⠈⠀⢀⣤⣤⡤⠤⠤⠤⡴⡶⠶⠶⠖⠒⠒⢗⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠞
⠠⠘⠉⠀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢠⢀⡄⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠉⣁⣤⠄⠤⠤⠄⠤⠲⠠⠄⠀⠀⠈⠀⠐⠒⠈⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠚⠛⠉⠉⣩⣃⡀⠀⠉⢉⣽⣧⡀⠀⠀⠀⠀⠀⠷⠿⢟⣀⣐⣀⣁⠀⠀⠀⠀⠀⠀⠉⠀⣩⣶⢦⣄⡀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠰⠾⠷⠟⠳⠖⠒⠒⠒⢒⢛⣛⣉⣙⣉⣉⣉⣉⣍⢩⣥⣤⣴⢶⣶⣶⣶⣶⡆⠀⢸⠏⠀⠀⠈⢿⠉⠀⢽⣿⣿
⠤⠒⠒⠶⡶⢶⢺⣷⡾⠟⣿⢿⣿⠻⣿⣛⠻⡿⡿⠍⠛⠛⠀⠀⠿⠻⠀⠀⠀⠁⠁⠇⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠈⠛⢿
⠀⠀⠀⠈⠛⠁⡛⠠⠀⠊⠻⠆⣙⠀⣈⢣⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠭
⠀⠀⠐⠀⠀⢠⣶⣶⠄⠈⠆⠼⠿⠇⠹⠆⠹⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠾⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣆⠋⠁⢽⠿⢆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠒⠒⠆⠀⠀⢀⠀⠠⠒⠀⠁⠐⠂⠀⠀⠀⠀⠀⣼⣋⣹⣦⣤⠞⠒⠻⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠐⠊⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⠏⠀⠀⠉⠀⠀⠀⢹⣿⠋

⠀⠀⠀⡀⠄⠂⠀⠉⠉⠉⠐⠠⢀
⠀⠠⠊⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠄⢀⡀
⠀⠁⠀⠀⢠⠐⠢⠀⠀⣀⣀⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠈⠐⢄
⡆⠀⠀⠀⠰⠀⣶⡆⠘⠋⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⢀⠄⢀⠀⠀⣀⠐⠀⠀⠑⡀
⡇⠀⠀⢠⣀⣉⣈⣀⣀⣀⣤⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠀⠀⠀⠠⠤⠄⢀⡀⠀⠀⠀⢠⠁⠀⠀⠀⡆⢀⣄⠂⣌⡀⢑
⠐⡀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⠄⣀⠤⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⣼⠀⠀⠀⠀⠑⠘⠋⠀⢿⡃⠔
⠀⠐⢄⠀⠀⠈⠙⠋⠹⠈⠡⠀⠀⣠⠆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠑⠢⠀⣠⡀⠀⠀⠀⢆⠞⠁⠀⠀⠀⠀⠀⢀⣴⠟⠛⠓⠀⠀⠀⠀⠀⠐⠛⠛⢶⡄⠀⠀⠀⠱⡀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡠⠃⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⠢⠀⠀⠀⠉⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠲⡒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢀⠎⠀⠀⢀⠫⠉⢙⠟⢻⣿⣿⣿⣿⣿⡿⠛⢏⠉⠙⢁⠀⠀⠈⢄⠀⠀⢹⣿⣿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢻⣿⠀⡠⠁⠀⠀⡠⡕⠀⠠⠃⠀⢸⣿⠁⠀⠈⢹⣷⠀⠈⢂⠀⠨⣢⠀⠀⠈⢢⠀⢸⣿⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡖⠁⠀⠀⢤⠋⠀⡐⠁⠀⠀⣾⡇⠀⠀⠀⠘⣿⡄⠀⠀⠣⡀⠐⠕⡀⠀⠀⠱⣿⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⣐⠃⢀⣜⣀⣠⣤⣾⠟⠀⠀⠀⠀⠀⠙⢷⣦⣤⣀⣑⣀⠈⢋⣀⣀⣼⡿⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠒⠒⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢠⣶⢳⡄⣠⠤⠀⠀⠐
⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢂⠉⠀⢁⠿⠀⡇
⠀⠀⠀⠁⠀⠀⣀⠤⢤⣄⠀⢶⡖⠚⡂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⠈⠀⠀⠑⠊
⠀⠀⠀⡀⠀⠀⠀⢄⣈⠝⠀⠑⠤⠔⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠴⣉⠀⠘⡄⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠢⡈⠢⡀⠀⠀⠄⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠐⠂⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⡉⠀⠀⠂⠤⣄⠀⢀⡤⢶⡀⠀⠐⡀⠀⠁⠐⠒⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠛⡇⡛⠃⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠒⠊⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠙⠀⠀⢀⡠⠚
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣶⣶⣶⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⢀⡀⠀⠒⠲⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⣀⣠⣤⡾⠛⠀⢠⡀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⣠⡶⠟⠛⠉⠀⠀⠀⠀⠀⠙⠷⣦⣤⣀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⡀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠙⢿⣆⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡄⣠⠶⠛⠛⠻⠿⠟⠀⠀⠀⠀⠺⠟⠛⠛⠛⢦⡀⢦⠹⣆⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠁⠁⣠⣴⠶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣤⣄⠁⠸⡆⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡏⠀⠘⢉⣠⣶⡆⠉⠀⠘⠀⠀⠀⠐⠋⣴⣶⣀⡙⠀⠀⣿⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⡇⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢸⡇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡈⣧⠀⠀⠀⠀⠀⠀⠰⠰⠶⠀⠠⠶⠰⠀⠀⠀⠀⠀⠀⠀⡿⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢹⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢀⣼⢛⣭⡝⢷⡀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠼⠁⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢰⡴⣄⣀⣀⢤⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠻⣦⣭⣵⠟⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⣠⡾⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⢿⣆⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠙⠻⢶⣤⣄⣤⣤⡶⠟⠁⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡶⠶⠿⠿⠿⠿⠿⠿⠿⠷⠶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣶⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣶⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠟⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣦⣤⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠁⠉⠙⠿⠋⠉⠈⠉⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣋⣠⣴⣶⣶⣶⣤⣤⣤⣶⣶⣶⣦⣤⣙⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⣿⣿⣤⣄⣀⣀⣈⣉⣁⣀⣀⣠⣤⣿⣿⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⣀⣀⣀⣀⣀⣀⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠷⢶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠟⠛⠋⠉
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⣠⡁⠀⠀⠀⠀⢲⠦⠀⠀⠀⠸⠀⠀⠀⠀⠀⠇⠀⠐⡂⠟⣓
⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⣶⣶⣶⣶⣶⡆⠀⢰⣶⣶⣶⣶⣷⡬⢹⣷⣶⣶⣴⣶⡆
⣿⢏⣈⣉⣉⣩⣟⣉⣉⣉⡉⠁⠀⠿⠛⠋⠛⠻⠇⠀⠸⠿⠿⠿⠿⠿⠀⠀⢉⣉⣉⣈⣻⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⢀⣀⠀⡀
⣿⣎⠉⠉⠉⠙⢯⡉⠉⠉⠉⠀⠀⢴⣦⣤⣤⣙⠀⠀⠀⠆⠀⠀⠀⠑⠀⠀⠉⠉⠉⢉⣽⡟⠛⢨⣿⣤⠃
⣿⣿⠀⠀⣀⡀⣾⠃⠀⢀⡀⠀⠀⠉⣿⡟⣿⠉⠀⠀⠐⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⠘⣿⡅
⣿⣿⠀⠀⢀⠀⣿⠀⠀⢀⡀⠀⣀⠀⣿⡇⣿⠀⢀⡀⠀⣿⡟⡛⢻⡇⢀⡀⢀⠀⠀⠀⣿⡇⢀⣀
⠿⠿⢀⣀⣀⡀⠿⠀⢀⣀⣀⠐⠛⠀⠛⣀⠛⠀⠘⢣⣄⠙⠁⠀⠘⢃⣤⣟⠀⠀⢀⣀⣿⡇⠈⠉⢀⣀⣀
⣤⠄⠚⢛⣿⣤⣤⠔⠛⠛⣿⣤⣤⠘⠒⠛⣲⣦⣤⡜⠛⠛⢳⣦⣤⣼⠛⠛⠲⣦⣤⣟⠛⠛⠢⣤⣬⣟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠲⠦⣤⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣠⡤⠶⠚⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣉⡷⠖⠛⠉⠉⠉⠉⠉⠙⠓⠶⣏⡁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡼⠃⢠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⠀⠻⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠁⣠⡈⣷⢀⡴⠒⠲⣄⢰⡏⣠⡀⢹⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠾⣷⣤⣤⣹⣦⣭⡴⠋⠸⣇⡀⣀⡟⠈⠳⣬⣥⣾⣡⣤⣴⡿⠶⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠉⠉⠙⠃⣀⠈⢹⠉⢀⡀⠛⠉⠉⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⠿⣶⡒⠛⠛⠛⠁⠉⠛⠛⠛⠋⠁⠉⠛⠛⠓⣲⡾⠷⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢤⣤⣤⠴⠶⠚⠋⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠤⠤⠀⣒⢒⣒⣒⣀⣀⡀⠀⠀⠀⠀⢄⢐⣒⠒⡒⡒⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣬⣤⣶⠒⠀⠅⠁⠩⠚⠉⠈⠁⠀⠀⠉⠉⠁⠀⠈⠀⠀⠀⠁⠀⠈⠘⢶⡦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡙⠿⢿⣡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠠⠄⣀⣪⡵⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡙⠟⠿⠷⠶⣿⣥⣤⣤⣠⣄⣀⣠⣤⣤⣤⣭⣥⣬⣶⢶⡶⠶⠛⠛⠋⠉⡴⢒⡸⠁⣀⣤⠄⢀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠋⣿⠀⠠⣄⣀⡠⢈⣡⡭⣫⣖⣡⠴⠒⠒⠒⢦⡌⢻⣅⢲⣆⡒⠋⡁⠘⠼⠂⠇⢹⢫⠀⠀⠐⢕⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠉⠀⠀⠀⠈⠽⠿⢯⠘⠉⠑⠃⠀⠀⠶⡄⠀⠈⠀⠻⣶⠳⠍⢢⡝⡄⠐⢤⡆⠈⠆⠀⠀⠀⠀⢩⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠄⢦⠄⣠⡌⣄⢆⣠⡄⠋⠋⡠⠛⢲⡗⠻⠗⠀⠓⢈⣆⡗⢷⡄⢴⠇⠓⡈⠈⠁⠀⠀⠀⠀⠀⢽⣄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠉⠁⠖⡨⣀⠹⢿⡉⠀⡠⢶⡄⠷⢴⡀⢛⣂⠀⠀⠈⡏⠁⠀⡇⣽⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡄⠀⣷⠀⢸⡇⠰⡦⢼⣿⡿⣸⡛⢤⣿⣷⠄⠀⠇⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇⠀⣧⠀⢸⡇⠀⠱⡏⡳⢖⡉⣡⠾⠀⠰⣶⣦⣴⣦⠀⡇⢸⠀⠀⡇⠀⠀⠀⠀⠀⠄⡸⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠇⠀⣭⣄⠸⣧⣄⣀⠀⠙⠉⠈⢀⣹⣿⡶⠟⣍⠉⣯⢄⣷⡾⠀⠀⡥⢄⠀⢀⠠⠊⠈⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠀⠀⠀⢀⣃⣀⣻⠜⠛⢿⣻⢹⢶⡐⣤⣤⣻⡿⣺⣷⣇⣏⣼⡿⣯⣸⣤⠀⠀⣦⠀⢀⠀⠀⠀⣠⠔
⠀⠀⠀⠀⠀⡀⠤⣀⣀⣐⡒⢨⣶⡀⠀⠀⢨⡻⣟⢡⣶⣄⠉⢻⣼⢢⠇⠘⠻⣿⠳⢾⡻⢶⡧⠤⢀⣴⣎⠁⠀⠀⣿⡩⣤⡴⠶⢯⣑⠢⣄⣀⡀
⠀⢄⣬⣵⡾⡛⠯⣍⡭⣺⢿⣕⡶⢁⠀⠠⡬⢟⣿⠚⢯⡝⣠⣞⢻⣻⣄⠀⣻⣿⣶⣾⡇⣈⣙⣢⣴⡯⠙⠀⠀⡠⠱⣚⣱⠾⠶⠶⡉⣏⡶⠾⠭⣝⢢
⠄⣼⢟⣇⡹⠀⠀⠀⡯⠉⠠⠤⠀⠈⢣⣀⠠⢧⣾⣿⣦⣧⡈⢁⡼⣻⣯⡀⠉⠉⢈⣧⡻⣿⡿⣿⠉⠁⠀⢀⠞⠁⠈⠁⠀⠀⠀⠈⠵⣃⠚⠻⡗⢾⡇
⠀⡻⠜⣏⠁⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠉⠳⣦⡀⢹⣯⣽⣿⣿⣿⡏⠛⠓⠀⠲⢿⣸⣍⣾⣷⠋⢀⣠⠞⢁⢄⢄⡄⠀⠀⠀⠙⠒⠠⠛⢈⣀⠴⢋⡴
⠀⠈⠒⠮⢭⣥⣀⣢⠤⡀⢀⣀⡂⠤⠨⠄⠀⠀⠈⢢⣍⢻⡿⠋⠛⠧⠀⠀⠹⣿⣻⣽⣟⣿⡿⠾⠧⠃⣀⣐⠌⣐⡤⠈⠲⢖⣀⣢⡠⠌⣃⠴⠃⠁
⠀⠀⠀⠀⠀⠢⢉⠛⠳⠾⠭⠭⠶⢶⠶⢶⠴⠭⣭⣬⣭⣭⣥⣄⣒⠰⠄⣂⣀⣙⣐⣤⢠⣤⣬⣥⣶⣶⣶⣶⣶⣶⣶⠾⠟⠋⣉⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠠⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠟⠛⢉⣉⣠⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠬⣄⣁⣐⠒⠀⠀⠠⠠⠤⠤⠤⠀⠀⠠⠤⠤⠄⠐⠒⢒⣂⣀⣩⡭⠴⠊