Number symbols Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⢤⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠤⡆⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠉⠉⠀⡀⠀⠀⠀⠀⠀⢀⣤⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⠠⢀⠌⠀⠀⠀⠀⠀⠐⠊⠑⢢⠀⠰⡀⠀⠰⠟⠂⢀⡷⠀⠀⠀⠀⢣⣤⢠⠄⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠀⠀⠀⠀⠀⠠⠤⠄⠻⠹⠀⠂⠀⠀⢦⡤⠄⣐⡡⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⠀⠀⠀⠀⡀⢊⡀⣀⡄⠀⠀⠀⠀⡀⢀⢤⡀⠀⢠⠈⢦⡄⣀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⢸⠀⠂⠀⠀⠀⠀⠀⢸⠁⠛⢁⠴⠃⠀⠀⠀⠀⠫⠝⠋⠁⡠⢸
⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠈⠓⠈⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠉⡔⠊⠀⠀⢀⠠⠴⠆
⠀⠀⠀⠀⠠⠁⠐⡶⠈⡄⠀⠀⠀⠼⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡗⢠⠂⠁⣀⡀⠀⠆
⠀⠀⠀⠀⢰⡈⠃⠂⠀⠢⡀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠁⢠⢢⠀⠀⠀⠀⠀⠈⠐⠃⠄⠐⠋⠀⠚⠃⠈⠙⠃⢐
⠀⠀⠀⠀⠈⢩⠁⠊⠿⠂⠀⠀⠀⠀⠀⠀⡀⠘⠉⡀⢤⡀⢀⠽⡀⠀⠀⠀⠀⠀⠠⠀⢂⡈⢣⠀⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠀⠀⠠⠈⠡⠄⠀⡼⠀⠀⠀⠀⠀⠈⠲⣈⠤⠊⠁⠲⠊⠁⠀⠀⠀⠀⠀⠂⠀⡸⠓⡀⠀⠀⠀⠀⢈⡤⠞
⠀⠀⠀⠀⠀⠈⠓⠐⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢰⡄⠀⠝⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⠁⠐

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⣾⠋⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠈⠛⠛⠻⠋⢀⡴⡾⠃⠀⠀⠀⠀⠀⢤⣀⣼⡋⠁⠀⣰⡇
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⢫⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠚⢋⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠱⠆⠀⠀⢀⡟⠀⠀⣠⣼⠄⠀⠀⣼⠀⠀⠀⠀⠀⢀⠃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠃⠀⠀⣵⠀⠘⠀⠀⢀
⠀⠀⠀⠘⠢⢤⣤⣠⡴⠎⠁⠀⠀⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡘⡈⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⢤⣀⠀⠀⢀⣴⠇⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠐⢲⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠶⠿⠃⠀⠀⠀⡠⡦⡄⠀⠹⡆⠀⠀⠈⠉⠛⠋⠁
⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⣀⣀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠶⠶⠿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠾⠁⠀⢸⡇⡀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠖⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠾⠉⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠘⠙⠓⠚⠀⠀⢸⠞⠛⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠀⠀⢤⣤⣤⣤⣀⣤⣼⡇⠀⠀⠀⠒⠀⠀⠑⠚⠋⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠉⢂⠀⠀⠀⠀⢿⡉⠉⠂⠀⣷⠀⠀⠀⠀⠀⠰⡟⠙⠂⠀⢸⠆⠀⠀⠀⠈⠓⠦⠤⠤⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠛⠀⠀⠀⣻⠀⠀⠰⣄⡀⠀⠀⡀⢔⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠶⠆⠀⡿⠀⠀⠀⠀⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⡪⠒⠆⠀⠰⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠘⠦⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⢀⣀⢤⡺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⢤⠤⠦⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠶⠀⠟⠿⢾⣹⠀⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⡀⢠⣤⡏⠀⠀⢀⡄⢪⡄⠀⠀⠀⠀⠛⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢶⣶⣶⡶⣷⠀⣃⠘⠀⠩⠀⠙⠂⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⣭
⠀⠀⠀⠀⠀⠀⠀⠠⠦⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠩⠀⢀⡆⠀⠄⠀⢺⡿
⠀⠀⠀⠀⠀⠀⠀⠰⠖⠐⠊⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⡀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⡄⠠⠀⢴⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⣤⣤⠤⣄⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⡟⢻⣛⣋⢛⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠤⠀⠉⠉⠁⠁⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠾⢛⡟⠻⡙⠃
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠤⡄⠀⠀⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠖⠶⠂⢠⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢚⡒⠀⠀⠖⠖⠂⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠻⠋⠄⠉⠀⠀⠀⠀⠠⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣶⣰⠔⡆⡄
⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡸⠻⠷⠗⠷⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⣥⣬⢤⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠟⠚⠎⠃⠈⠁⠁⡘⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣌⣡⢤⣌⠥⣬⣡⣤
⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠧⣿⢸⠄⠐⠛⠒⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣴⠋⣿⢀⣿⣰⠏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠛⠏⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠛⠋⠉⣉⣁⣛⠛⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢁⣀⣄⡂⣄⢀⡀⠉⠁⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠋⠛⠋⠛⠂⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠟⠿⠿⠿⠹⠇⣤⠈⣀⣀⠀⣀⢀⠁⡀⡀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⢠⡖⣾⣦⣾⢲⡆⢀⣀⣻⡻⠓⠃⠓⠀⠀⠀⢀⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡬⡭⠉⠁⠀⠀⠀⠒⠂⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠛⠻⣷⣄⠀⠀⠀⢀⣴⠿⠛⠋⠙⠛⠿⣦⡀⠀⠀⠀⣠⣶⠟⠛⠉⠛⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⢀⣤⣾⠀⠀⠈⢻⣦⠀⢠⣿⠁⠀⣴⠛⠛⣦⠀⠘⢿⡄⠀⣼⡟⠁⠀⠾⠋⢳⡄⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠀⣿⠀⠀⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⣠⠟⠀⠀⢸⡇⠀⣿⠁⠀⠀⠀⠰⢾⡅⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⣿⠀⠀⠀⣸⡟⠀⠸⣷⡀⠀⣴⣟⣁⣀⠀⢀⣾⠇⠀⢻⣇⠀⠀⢦⣀⣠⠟⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣈⣀⣤⡾⠟⠀⠀⠀⠙⢿⣦⣍⣉⣉⣩⣴⡿⠋⠀⠀⠀⠻⢷⣤⣀⣉⣁⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁


⠀⠀⠀⣠⣴⡶⠶⠶⢶⣦⣄⠀⠀⠀⠀⠀⣠⣶⠶⠶⠶⢶⣤⡀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠶⣶⣄⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⢶⣦⣄
⠀⠀⣾⠟⠁⠀⢀⣤⠀⠈⠻⣷⡀⠀⢠⣾⠋⠀⣴⠶⠶⠄⠙⢿⣦⠀⠀⣴⡿⠉⢀⣤⠶⣄⠀⠙⣷⡄⠀⢀⣾⠟⠁⠤⠤⢤⣤⠈⠻⣷⡀
⠀⢸⡏⠀⠀⣠⠞⣿⠀⠀⠀⢹⡇⠀⣾⡇⠀⠀⣿⠴⣤⡀⠀⠀⣿⡆⢰⡿⠀⠀⣼⣡⢤⣌⠀⠀⢸⣷⠀⢸⡏⠀⠀⠀⢀⡾⠁⠀⠀⢹⡇
⠀⢸⣇⠀⠰⠧⠤⣿⠤⠀⠀⣸⡏⠀⢿⡇⠀⢠⡄⠀⢈⡇⠀⠀⣿⠇⠸⣿⡀⠀⢹⡁⠀⢸⠆⠀⢸⡿⠀⢹⣇⠀⠀⠀⣾⠁⠀⠀⠀⣸⡇
⠀⠀⠻⣦⡀⠀⠀⠛⠀⢀⣴⡟⠀⠀⠘⢿⣆⡀⠙⠒⠋⠀⣠⣾⠏⠀⠀⠹⣷⣄⠈⠛⠒⠋⢀⣠⡿⠃⠀⠀⢻⣦⡀⠀⠋⠀⠀⢀⣴⠟
⠀⠀⠀⠈⠛⠷⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⠿⠛⠁⠀⠀⠀⠀⠈⠛⠿⠶⠶⠾⠟⠋⠀⠀⠀⠀⠀⠈⠛⠿⠶⠶⠿⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⢉⣉⠉⠛⢿⣦⠀⠀⠀⣠⡾⠟⠉⣉⡉⠉⠻⣷⣄⠀⠀⠀⣴⡿⠋⠉⣉⡉⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢰⡏⠉⣳⠀⠀⢹⣧⠀⢰⡿⠁⠀⣾⠉⠉⣧⠀⠈⢿⡆⠀⣼⡏⠀⢠⡞⠉⠹⡆⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⠟⠛⢷⡀⠀⢀⣿⠀⢸⡇⠀⠀⠻⢦⠴⣿⠀⠀⢸⡇⠀⣿⡀⠀⢸⡇⠀⠀⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠹⢦⣤⡼⠃⢀⣼⠟⠀⠘⣷⡄⠀⠲⣤⣤⠏⠀⢠⣾⠃⠀⠻⣧⡀⠈⠷⣤⡼⠃⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣤⣠⣤⣴⠿⠋⠀⠀⠀⠈⠻⣶⣤⣄⣠⣤⣶⠟⠁⠀⠀⠀⠙⠿⣦⣤⣄⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣶⣶⣶⣯⣓⢄⠀⠀⠀⢀⢔⣫⣷⣶⣶⣯⣓⢄⠀⠀⠀⢀⢔⣫⣷⣶⣶⣮⣕⢄
⠀⠀⠀⠀⠀⠀⠀⢠⢣⣿⣿⢟⡁⣾⣿⣿⣮⢇⠀⢠⢯⣾⣿⣡⣶⣦⢹⣿⣮⢧⠀⢀⢯⣾⣿⣡⣶⡦⢹⣿⣧⢣
⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⡇⣿⣿⣿⣿⢸⠀⢸⢸⣿⣿⣿⠟⣡⣾⣿⣿⢸⠀⢸⢸⣿⣿⣿⣯⣤⠻⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢎⢿⣿⣿⣇⣿⣿⣿⢫⠇⠀⠈⢏⢿⣿⣁⣚⣛⣻⣿⢏⠏⠀⠀⢯⢻⣿⣌⣛⣋⣴⣿⢏⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⣟⣛⣛⡯⠖⠁⠀⠀⠀⠀⠑⠯⣛⣛⣛⡫⠕⠁⠀⠀⠀⠀⠑⠯⣟⣛⣛⡯⠕⠁


⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢀⢔⣭⣷⣶⣶⣯⣓⢄⠀⠀⠀⢀⢔⣯⣷⣶⣶⣯⣓⢄⠀⠀⠀⢀⢔⣫⣷⣶⣶⣯⣗⢄⠀⠀⠀⢀⢔⣫⣷⣶⣶⣯⣕⢆
⠀⢠⢣⣿⣿⡿⢃⢸⣿⣿⣮⢇⠀⢠⢫⣿⣿⠁⣶⣶⣾⣿⣮⢇⠀⢀⢯⣾⣿⢣⣶⣦⣽⣿⣧⢧⠀⢀⢏⣾⣿⣶⣶⠆⣼⣿⣧⢣
⠀⢸⢸⣿⡟⠱⠿⠸⣿⣿⣿⢸⠀⢸⢸⣿⣿⣴⣶⣦⢹⣿⣿⢸⠀⢸⢸⣿⣿⢠⣶⣦⢹⣿⣿⢸⠀⢸⢸⣿⣿⣿⠇⣾⣿⣿⣿⢸
⠀⠈⢎⢿⣷⣶⣶⣰⣿⣿⢣⠇⠀⠈⢏⢿⣿⣌⣛⣋⣼⣿⢯⠇⠀⠀⢯⢿⣿⣌⡛⣋⣼⣿⢏⠎⠀⠀⢧⢻⣿⣿⣸⣿⣿⣿⢏⠎
⠀⠀⠈⠓⠭⣛⣛⣛⡫⠖⠁⠀⠀⠀⠀⠑⠭⣛⣛⣛⡫⠕⠁⠀⠀⠀⠀⠑⠯⣛⣛⣛⡫⠕⠁⠀⠀⠀⠀⠑⠮⣛⣛⣛⡫⠕⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣫⣽⣶⣾⣭⡲⣄⠀⠀⠀⢀⠴⣫⣽⣶⣾⣭⡳⢄⠀⠀⠀⢀⡴⣪⣽⣶⣾⣭⡓⢄
⠀⠀⠀⠀⠀⠀⠀⢠⢫⣾⣿⠡⣶⡎⢻⣿⣎⢇⠀⢠⢏⣾⣿⢡⣶⣮⢻⣿⣎⢧⠀⢀⢎⣾⣿⢋⣶⣦⢻⣿⣮⢣
⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⢃⣬⣄⢻⣿⣿⢸⠀⢸⢸⣿⣿⣌⠛⣋⢸⣿⣿⢸⠀⢸⢸⣿⣿⢸⣿⣿⢸⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢎⢿⣿⣜⠻⢋⣼⣿⢏⠇⠀⠈⢏⢿⣿⣝⠿⣋⣼⣿⢏⠏⠀⠀⢯⢿⣿⣌⠻⢋⣼⣿⢏⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢽⣛⣛⣛⡫⠕⠋⠀⠀⠀⠈⠓⠭⣛⣛⣛⡫⠕⠃⠀⠀⠀⠈⠓⠭⣛⣛⣛⣫⠕⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⣈⡁⠢⠀⠀⠀⠀⢀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠛⠂⢀⡼⠃⢀⠂⠀⢰⣄⠀⠑⠞⠁⠀⡀
⠀⠀⠀⠀⣠⠤⠼⠃⠀⠈⠘⠒⠢⢲⠀⠀⠀⣇⠀⠀⢀⡀⠀⠀⠸⠀⠀⠀⢀⣤⠊⢀⠔⠁⠀⠀⠀⣙⠗⡀⠀⠀⢎
⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⣀⣀⠼⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢈⡧⠐⠁⠀⠀⠀⠀⠺⣅⡀⠀⠲⣄⠀⡣
⠀⠀⠀⠀⠙⠒⠂⢻⡆⠄⢸⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠊

⠀⠀⠀⠀⢀⡤⣀⠀⢀⣴⠞⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⢀⡀⠀⠀⠀⢀⡖⠈⢡
⠀⠀⠀⠀⠘⢅⡡⢣⡿⠃⡀⡠⠂⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠤⡀⠀⣶⠒⠚⠁⠘⠋⡩⠀⠀⠀⠀⣿⠀⠸
⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⡔⠁⠀⠀⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠖⠀⡀⠸⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢀⣾⠎⠀⢁⠊⠀⡏⠍⡂⠀⠀⠀⠉⠂⠒⠒⠀⠀⠉⠈⠉⠁⠀⠀⠈⠒⠋⠙⠒⠁⠀⠀⠀⠀⠈⣓⣒
⠀⠀⠀⠀⠈⠣⢄⠒⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠦⣄⠀⠀⢠⡤⠄⣀
⠀⠀⠴⠟⠉⣀⠑⡄⠀⠀⢰⡆⠀⠀⠈⠛⣦⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⡠⠊⠀⠀⠈⠓⠠⢀⡀⠀⠂⢀
⠀⢐⡘⠀⢀⣾⠇⢸⠀⠀⣼⠃⠌⠀⠀⠀⣭⡀⢆⠀⠀⠀⠰⣿⠋⠀⠰⣋⡀⠀⠀⠀⠀⢀⣠⡶⠟⢁⠔⠁
⠀⠘⣧⠈⠚⠋⢀⠇⠀⠀⢰⡄⢸⠀⠀⠀⠉⡁⡆⠀⠀⠀⠀⠀⠉⠒⠤⣀⡈⠙⡲⠀⠰⡟⠁⢀⠔⠁
⠀⠀⠀⠑⠒⠊⠁⠀⠀⠀⠘⠧⢈⠀⠀⠻⢋⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠑⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢨⣛⣋⣀⠀⠀⠀⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠈⠉⠈⠁⠀⠀⠀⠀⡀⢁⠀⠀⠀⢀⣒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣅⢀⡀⢠⣤⡠⡄⠤⠀⠀⠐⠂⠀⠉⠃⣥⣤⣤⣤⡄⢠⣀⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡤⠀⣷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡯⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠤⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠛⣈⣒⣠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠙⠛⠉⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠘⠃⠀⠀⠀⠐⠀⠂⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠻⠛⠛⢲⠄⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⠂⠀⠀⡘⠒⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡶⠶⠦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣄⡄⠀⠀⠀⠀⠀⠩⠭⠀⠒⢲⠂⢀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⢠⡀⠀⣶⡂⠀⠀⠀⠈⠉⠀⣟⡻⠟⠏⠛⢀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣤⡁⣤⣤⡄⣿⠇⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡆⠚⠒
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠁⢀⣀⡉⠁⣤⣈⣐⠀⢠⣤⡤
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠁⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠈⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠃
⠀⠀⠀⠀⠀⠀⣤⠀⢠⠀⠀⠀⢀⠚⠃⠸⠀⠀⠀⠀⡶⠈⢆⠀⠀⠀⠀⣤⠤⠇⠀⠀⠀⠒⢀⡇⢀⣛⣀⠇
⠀⠀⠀⠀⠀⢄⣁⣠⠇⠀⠀⠀⠈⠉⠃⢸⠀⠀⠀⠐⠠⠔⠃⠀⠀⠀⠤⠤⠀⢳⠀⠀⠀⣤⠌⢀⠀⠉⠁
⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠲⠖⠃⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠠⢤⠴⠃⠀⠀⠠⣀⡠⠌⠀⠀⠐
⠀⠀⣀⣉⠀⣉⡄⠀⠀⣤⠈⡷⠀⠀⠀⠈⣡⠃⠀⠀⠐⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠸
⠀⠀⠀⠈⠲⠯⠀⠀⠈⢁⠆⠀⡦⢄⠉⢁⠄⠀⠀⠉⠙⣀⣶⠛⠀⠀⠀⣀⣀⠀⢰⠀⠀⠀⠀⠀⢀⠃⠈
⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠒⠚⠁⠀⠉⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠁⠀⠀⠀⠀⠐⠞
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠦⠁⠀⠀⠀⠐⠾⠀⡇⠀⠀⠀⠀⣠⡵⠀⠀⠠⠰⡤⡷⠀⠄⣴⣄⢰⠀⡄⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠒⠗⡠⠇⣄⠀⣀⠀⢀⠄⢤⡇⠀⠀⠀⠰⠆⢀⠀⡄⠀⠐⢦⠀⠀⠀⠃⠰⠞⠀⢳⡶⢌⣷
⠀⠀⠀⠀⠠⠀⠀⠠⠤⠖⠋⠀⠈⢠⠋⠀⠀⠀⠀⠀⠈⠐⠒⠋⠀⠑⠟⣁⠟⠀⠀⠉⠙⣉⠀⡀⠙⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠤⠛⠀⠀⠢⢤⠀⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠰⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠾⣿⡇⠀⠀⠀⠀⠀⠸⠋⠉⠙⣿⡆⠀⠀⠀⠋⠀⠈⣿⠆⠀⠀⠀⠀⠀⣼⢻⣿⡅⠀⠀⠀⠀⣸⠉⠉⠉⠋
⠀⠀⢀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣁⡼⠟⢡⡀⠀⠀⠉⠳⠺⣧⣄⠀⠀⠀⠀⡼⢋⢸⣿⠀⣤⠀⠀⠀⠛⠛⠒⢲⢦⡠⠄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣶⣷⣟⠁⠀⢂⡄⠃⠀⢀⡄⠀⠄⢸⡍⡄⠀⠀⢘⣷⡿⢺⣿⡟⠃⠀⠀⢀⣤⠀⠀⢀⣯⠏
⠀⠀⠀⠐⠛⠓⠀⠀⠀⠀⠘⠙⠛⠛⠛⠛⠃⠀⠀⠈⠉⠒⠒⠉⠐⠇⠀⠀⠈⠉⠉⠚⠛⠛⠁⠀⠀⠈⠉⠓⠒⠛⠁

⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⣠⣠⣀⣄⡤⠀⠀⠀⠀⢀⡀⢤⣀⠀⠀⠀⠀⠀⣀⡠⢄⡀⠀⠀⠀⠀⠀⠀⣠⠤⣤⡀
⠀⢠⣿⣟⣀⣈⠧⠄⠀⠀⠀⠁⠀⢀⡼⠁⠀⠀⠀⣀⣿⣤⣀⣿⠆⠀⠀⠀⡼⡏⠀⠂⢻⢆⠀⠀⠀⢄⣾⡇⠀⠘⣿⡄
⠀⢸⣿⣯⣄⠙⣷⡄⠀⠀⠀⣀⣀⣾⣡⠀⠀⠀⠀⢹⣾⢟⢿⣿⣄⠀⠈⢋⠻⣧⣀⣀⣼⣽⠀⠀⠀⢸⣿⡠⣤⣠⣿⡇
⠠⣼⣿⡅⠔⢀⡿⠇⠀⠀⠀⠦⣿⣿⣁⠀⠀⠀⢀⣿⣇⠉⠄⣿⣿⡁⠀⠈⠀⣄⠉⢹⣿⣟⠿⠁⠀⠈⢿⣇⠨⢹⡿⢛⠓
⠀⠣⠈⠉⠉⠉⠑⠂⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠘⠄⠉⠙⠉⠉⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣴⠄⠙⡆⠀⠀⠘⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⠀⣤⣄⠙⣦⠀⠀⠀⠠
⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠈⡆⠀⠀⡈⠓⠀⡰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⢄⣈⣁⠀⢸⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠣⡄⠀⠉⠛⠛⠊⠀⠀⠀⠀⠀⠠⣤⣤⠄⢨⡇⠀⠀⠀⠀⠀⠠⣠⣶⠆⢈⡇⠀⠀⠀⠀⠈⠉⠁⠀⣿⠀⠀⠀⣀⡀⠲⡄
⠀⠀⠀⠀⠀⠀⢦⣤⣤⠀⠠⣤⡷⠀⢀⡁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠁⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⢥⡀⠙⣅⠀⠀⠀⠙⢶⢀⡄⡢⠃⠀⠀⠓⠿⠀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⠇⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣄⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⡠⡎⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠨⡄
⠀⠀⠀⠀⢠⣄⠀⢱⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⠀⠈⠉⠁⠀⢰⡄⠀⠀⠀⠀⠀⠀⠉⠉⠁⡀⢰⡦⠈⣳⠀⠀⠀⠀⠀⢰⡄⠲⠶⠲⠖⠿
⠀⠀⠀⠈⢻⣉⠐⢾⠃⠀⠀⠀⠀⠀⠀⢀⡔⡆⠀⠇⡄⠀⠀⠃⠀⠀⠸⠟⠫⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⡀⠰⡏⠀⠀⠠⠟⠻⢛⠃⠀⠀⠄
⠀⠀⠀⢀⠀⠙⠀⡸⠇⠀⠀⠀⠆⠀⠀⠀⠉⠁⠀⢹⠇⠀⠀⠀⠀⠐⣶⠷⠀⠽⠀⠀⠀⠀⣠⡀⠱⡄⠀⠀⠸⠟⠀⢸⠀⠹⢝⢻⠆⢸⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠙⠛⠓⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⣤⣿⡇⠀⠀⠀⠀⠐⢤⣦⡺⠂⠀⠀⠑⠾⠟⠁⢰⠇⠀⠐⠶⠔⠄⠊⠀⠐⣤⢤⢤⡺⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠼⠋⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢹⡆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⣉⠉⠠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠘⢦⠀⠀⠀⠰⢤⣤⣴⡴⠻⠀⠈⠛⠛⢀⠒⠒⠲⠀⠀⢠⠎⠴⡿⠇⠸⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠉⢠⢔⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠀⢴⡕⠀⠀⢀⠀⠀⠀⢈⠥⠴⠶⠆⠀⠠⠆⠴⠄⠰⠇⠀⠸⠿⢶⣶⡄⢰⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊⠁⠀⢤⣀⡄⢀⣽⠀⠀⠀⠀⠀⢀⡤⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠖⠀⠀⠀⠒⠆⢴⡆⠀⠤⠤⠀⠉⠻⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⢠⠻⠋⠀⠀⠀⠀⢄⠀⠁⠀⣀⠟⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⢸⡆⠀⠀⠀⠳⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠁⠀⠀⠀⠀⠀⠀⠙⠪⠯⠚⠀⠀⠀⠀⠀⠀⠀⠑⠦⣲⡊⠝⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⢄⡀⠀⠀⠀⠀⢀⡠⣤⣤⢄⡀⠀⠀⠀⠀⢀⡠⢤⣤⢤⣀
⠀⠀⠀⠀⠀⠀⢀⢞⣷⣿⠟⣿⣷⣿⢆⠀⠀⣔⣵⡿⢛⣛⢿⣮⣢⠀⠀⡰⣿⣾⢟⣛⢿⣾⡷⡀
⠀⠀⠀⠀⠀⠀⢸⣾⣿⣷⡇⣿⣿⣿⣿⡆⢸⣹⣿⣷⣿⠟⣼⣿⡗⡇⢰⣿⣿⣿⣿⣋⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⡽⣿⣿⣇⣿⣿⡿⡻⠀⠘⣟⣿⡟⠱⠿⢿⣿⣻⠁⠀⢟⢿⣿⡙⠟⣡⣿⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠪⠽⠛⠻⠭⠊⠀⠀⠀⠈⠛⠽⠟⠻⠯⠛⠁⠀⠀⠀⠑⠽⠟⠛⠯⠕⠁


⠀⠀⢀⢤⣲⣶⣖⡢⣄⠀⠀⠀⢀⡠⣒⣶⣶⣢⢄⠀⠀⠀⠀⡠⣔⣶⣶⣒⢄⡀⠀⠀⠀⣠⢔⣲⣶⣖⡤⡀
⠀⢰⣳⣿⣿⢏⢹⣿⣯⢣⠀⢠⢮⣾⡏⣭⣭⣿⣷⣱⠀⠀⣎⣾⡿⣩⣭⣻⣷⡵⡄⠀⡼⣽⣿⣭⡭⢩⣿⣞⡆
⠀⢀⣿⡿⠡⠿⠸⣿⣿⣿⠀⢸⢿⣿⣧⣶⣦⢻⣿⡇⠇⠸⢸⣿⡇⣵⣦⢹⣿⣿⡇⠀⡿⣿⣿⡿⢱⣿⣿⣿⡠
⠀⠘⢜⢿⣿⣿⣼⣿⢟⠎⠀⠈⢾⢿⣧⣝⣥⣾⢟⠝⠀⠀⠫⡻⣷⣜⣫⣼⡿⡷⠁⠀⠱⡻⣿⣧⣿⣿⡿⡣⠃
⠀⠀⠀⠙⠚⠋⠛⠚⠁⠀⠀⠀⠀⠙⠚⠋⠛⠓⠁⠀⠀⠀⠀⠈⠒⠛⠙⠓⠉⠀⠀⠀⠀⠈⠓⠛⠙⠓⠋


⠀⠀⠀⠀⠀⠀⠀⣠⣖⣭⣿⣿⣕⣦⡀⠀⠀⢀⢔⣮⣿⣿⣵⡢⡀⠀⠀⢀⣴⣪⣽⣿⣥⣢⣄
⠀⠀⠀⠀⠀⠀⢰⣵⣿⡏⢶⡎⣿⣷⣵⡀⢠⢳⣿⡏⣴⣦⢹⣿⡞⡄⢀⣮⣿⡿⢱⣶⠹⣿⣮⡆
⠀⠀⠀⠀⠀⠀⢸⢿⣿⠫⣶⡦⢻⣿⢿⠃⢸⢸⣿⡷⣭⣥⢸⣿⣇⡇⠘⡿⣿⣇⢻⣿⢁⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠈⠫⡻⢷⣭⣵⡿⡫⠋⠀⠀⠳⣿⢷⣮⣵⡿⣿⠞⠀⠀⠙⢝⢿⣮⣥⡾⢟⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠂⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀⠀⠀⠀⣠⣤⣄⠀⣀⣤⣤⡀⠀⠀⠀⢀⠤⢤
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇⠸⣇⣀⡀⠀⠘⢆⠀⠳⠃⢠⠞⠀⠀⣀⣀⣘⣤⣴⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡀⠀⣀⣀⡇⠀⠀⢈⠇⠀⠐⢇⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⣸⡇⠀⠀⠀⣰⠋⢀⠶⣄⠈⢳⡀⠀⠀⠀⢰⠉⢙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣉⣉⣉⣉⣉⣹⠀⢰⠒⠒⠒⠒⠒⠒⡆⢰⠋⣉⣉⣉⣉⣉⣉⣉⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⢘⡶⠶⠶⠶⠶⠶⡇⢸⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢞⠉⡆⢸⠉⠙⡇⠸⠤⠤⠤⠤⠤⠤⠇⠸⣅⡇⡠⢒⡒⡄⢠⢒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢨⣟⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢇⠸⠇⣼⢃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢭⠀⢠⠤⣀⡀⠀⠀⠀⠀⢠⠏⢹⠀⠈⠉⢩⠃⣼⠋⣍⢱
⠀⠀⠀⠀⠀⠀⠀⡤⠖⠋⣁⡠⠼⠀⠘⠦⣄⡉⠓⠢⣄⠀⡼⠀⡏⠀⠀⢠⣃⡞⠸⣄⣋⡼
⠀⠀⠀⠀⠀⠀⠘⠧⣀⡉⠓⠦⢄⠀⢀⡤⠖⠋⢁⡠⠞⢀⠇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠼⠀⠸⠤⠖⠋⠁⠀⠀⣼⠀⡟⠀⠀⠀⠀⡏⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠁⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣤⠲⡄⠀⡰⡶⠀⠀⠀⢠⡒⡄⠀⡠⡶⠀⠀⠀⠀⣔⣒⢄⠀⢠⡲
⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⡰⡑⡡⢄⠀⠀⠈⡇⡇⡰⣱⠡⢄⠀⠀⠀⠘⢥⢪⡰⡱⡡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⡗⣝⠜⣵⢸⡀⠀⠀⠓⡗⡝⠉⢉⢂⠇⠀⠀⠋⠑⡺⣵⢏⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣜⠎⠈⠀⠲⡼⠁⠀⠀⢼⠜⠀⠴⠁⠓⠆⠀⠀⠀⢼⠜⠉⠀⠢⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠁⠂⠀⠉⠊⠇⠃⠈⠐⠃⠒⠐⠋⠀⠃⠒⠁⠁⠂⠕⠉⠐

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⢀⡠⠰⢶⣀⡀⠀⠀⠀⠀⠀⠀⣠⡀⣰⡆⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠠⠂⣼⠀⠀⠀⠀⢠⣿⡅⠚⠚⠛⣷⢥⠀⠀⠀⠀⠀⣾⣷⡖⠁⠛⢈⢗⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿
⠀⠀⠀⠀⠀⠀⢷⠚⣁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⢿⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣿⠏⠓⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣉⡵⠇
⠀⠀⠀⠀⠀⠀⢸⡏⢔⠀⠀⠀⠀⠀⠀⢀⣤⡶⣦⣨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣏⢿⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡟⢉⣶⡇
⠀⠀⠀⠀⠀⠀⢈⡗⢱⠀⠀⠀⠀⠀⣀⠸⣌⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣅⠀⠀⠀⠀⠀⡀⡸⠶⣏⡀⣷⣿⣀⡀
⠀⠀⠀⠀⠀⠀⢈⣠⣼⠀⠀⠀⠀⠀⣤⡙⣄⣀⡀⡀⢀⠀⠀⠀⠀⠀⠳⣺⠆⢀⠀⡘⢿⡏⠀⠀⠀⢀⣼⣿⣭⣧⣼⣯⣿⡽⣿⡁
⠀⠀⠀⠀⠀⠀⢨⢷⡿⠀⠀⠀⠀⠠⠍⠾⠏⢻⡶⠞⣿⠀⠀⠀⠀⠀⠀⠋⠴⠶⠤⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇


⠀⠀⠀⠀⠀⠀⢠⣄⡤⣤⡄⣄⠀⠀⠀⠀⠀⠀⢀⣀⢠⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡤⣠⣀⡀⡄⠀⠀⠀⠀⠀⠀⢤⡤⡀
⠀⠀⠀⠀⠀⣠⠚⡻⠙⠿⠤⠃⠀⠀⠀⠀⠀⣀⠞⠧⠘⠿⠤⢃⡄⠀⠀⠀⠀⠘⠀⠞⠓⠘⣯⡀⠀⠀⠀⠀⣀⣿⣹⠌⠷⠛⣧⡀
⠀⠀⠀⠀⠀⢸⣧⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠄⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡷⠀⠀⠀⠀⠀⢣⠙⡃⠀⠀⢘⠫⡁
⠀⠀⠀⠀⠀⠚⢯⣡⣔⣤⣄⡀⠀⠀⠀⠀⠀⠾⢽⣃⡴⣠⡄⡀⠀⠀⠀⠀⠀⠀⠀⢰⣛⡽⠁⠀⠀⠀⠀⠀⠸⣾⡖⢶⣠⡇⣤⠂
⠀⠀⠀⠀⠀⠐⠁⠈⠛⠚⣛⣻⡄⠀⠀⠀⠀⣴⠏⡉⠛⠸⠿⢿⣦⠀⠀⠀⠀⠀⢠⢯⠙⠃⠀⠀⠀⠀⠀⠀⣨⣟⡱⠈⠛⠀⢉⡀
⠀⠀⠀⠀⠀⣶⣖⡀⠀⢀⣿⣹⠇⠀⠀⠀⠀⣷⣛⣃⠀⢀⣸⣻⣇⠀⠀⠀⠀⢀⣾⣛⠃⠀⠀⠀⠀⠀⠀⠀⡍⣿⣀⡀⢀⣼⣫⠏
⠀⠀⠀⠀⠀⠁⢼⣢⠗⠺⠋⠝⠀⠀⠀⠀⠀⠀⠸⡁⠞⠼⠛⠽⠁⠀⠀⠀⠀⡼⢁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⣤⡴⠆⣀⠀⠀⠀⠀⠀⠀⢠⡴⣦⣲⢆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢪⡴⠉⠓⠻⣛⡃⠀⠀⠀⠀⢠⣷⣷⠟⠉⠊⡈⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣨⡄⠀⢀⡠⣷⡖⠀⠀⠀⠀⢰⢢⡈⠀⠀⠀⢳⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣶⣞⠁⣇⣨⢓⠀⠀⠀⠀⠸⣟⡀⠀⠀⠀⡷⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠉⠀⠀⠀⠀⢰⣥⡶⠀⠀⠀⣄⣽⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡝⣶⣀⠠⣼⣟⠿⠀⠀⠀⠀⠘⣅⡸⣦⠀⠠⠟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢯⡭⠌⠃⠀⠀⠀⠀⠀⠀⠙⠷⢷⣤⠞⠓⠁

⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠃
⠀⠀⠀⠀⠟⢻⢛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠻⠿⠻⠛⣾⣿⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠀⠀⠀⣻⣿⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⢟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⠀⠀⡀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣵⣿⣷⡀⠀⠀⠀⠀⠀⠀⡾⡿⡿⡏⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣚⣟⣟⣛⣻⣋⣁⣀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣀⣠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠙⠙⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠛


⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣀⡀
⠀⠀⠸⣿⣿⣿⣽⣿⣿⣿⣿⢽⠅⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⡧⣿⡇⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆
⠀⠀⢨⣭⣭⡏⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢁⣷⣧⣿⡇⠀⠀⠀⠀⠀⢸⣿⢿⢻⢻⣿⣿⣿
⠀⠀⢸⣿⣿⡇⣺⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢣⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⢽⣿⣿⢿⡀
⠀⠀⢸⣿⣿⡇⠿⠛⠻⣿⣿⣿⣆⠀⠀⠀⠀⢠⣿⣿⣿⠃⠛⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠛⠻⣵⣿⣿⡀
⠀⠀⢀⣀⣀⡀⠀⠀⠀⢘⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡁⠀⠀⠀⣹⣿⣿⡇
⠀⠀⠈⣿⣿⣿⣦⣤⣴⣿⣿⣿⠏⠀⠀⠀⠀⠸⣿⣿⣿⣦⣤⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⣤⣶⣿⣿⣿⠃
⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠋⠉⠙⢿⣿⡿⡇⠀⠀⠀⠀⢰⣿⣿⣿⠏⠉⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⡀⠀⠀⣼⣿⣧⡇⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⣽⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣳⣿⣿⣿⠃⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⢀⠈⠙⠻⣦⠀⠀⢠⡾⠛⢉⣀⣈⠙⠻⣦⠀⠀⢠⡾⠛⢉⣀⣀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠐⢻⠀⠀⠀⢹⡇⠀⣿⠁⠀⠚⠀⢸⡇⠀⢹⡇⠀⣿⠁⠀⠘⢀⣸⠇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⣷⠀⠀⢀⡴⠋⠀⠀⢰⡇⠀⣿⠀⠀⣀⠀⠉⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣧⣄⠀⠘⠀⣀⣴⠟⠀⠀⠘⢷⣄⡛⠛⠛⣃⣴⠟⠁⠀⠘⢷⣄⡈⠓⠋⢁⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁


⠀⠀⢀⣴⠟⠛⠛⠻⢶⣄⠀⠀⠀⢀⣴⠞⠛⠛⠻⢶⣄⠀⠀⠀⢀⣴⠞⠛⠛⠛⢶⣄⠀⠀⠀⢀⣴⠾⠛⠛⠛⢶⣄
⠀⢰⡟⠁⠀⢠⣾⠀⠀⠹⣧⠀⢰⡟⠁⢀⡟⠛⠃⠀⠹⣧⠀⢠⡟⠁⢠⡖⠛⠦⠀⠙⣷⠀⢠⡿⠁⠐⠛⢛⡾⠀⠙⣷
⠀⣾⠀⠀⣴⣃⣿⡀⠀⠀⣿⠀⢸⡁⠀⠘⠋⠙⣦⠀⠀⣿⠀⢸⡇⠀⢸⡞⠛⢦⠀⠀⣹⠀⢸⡇⠀⠀⢀⡞⠀⠀⠀⢸⠆
⠀⠸⣧⡀⠉⠉⠿⠁⠀⣰⡟⠀⠸⣧⡀⠘⠦⡤⠏⠀⣰⡟⠀⠘⣷⡀⠘⠧⡤⠞⠀⣠⡿⠀⠘⣷⡀⠀⠸⠁⠀⠀⣠⡿
⠀⠀⠈⠻⢶⣤⣤⣴⠾⠋⠀⠀⠀⠈⠻⢶⣤⣤⣴⠾⠋⠀⠀⠀⠈⠻⢶⣤⣤⣴⠾⠋⠀⠀⠀⠈⠛⢶⣤⣤⣴⠾⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⢀⣤⣄⠈⠻⣦⠀⠀⣠⡾⠋⢀⣤⣄⠈⠻⣦⡀⠀⢠⡾⠋⢀⣤⣄⠈⠛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢻⣄⣼⠀⠀⢸⡇⠀⡿⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⠀⣾⠀⠈⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣏⠀⢸⡇⠀⣸⡇⠀⣿⡀⠀⣈⠙⢹⠇⠀⣸⡇⠀⣿⡀⠀⢿⠀⢠⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡈⠉⠉⣠⣴⠟⠀⠀⠘⢷⣄⡈⠉⠉⣀⣴⠟⠀⠀⠘⢷⣤⣈⠉⠉⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⠉⢉⠉⠛⢷⣄⠀⠀⢀⣴⠟⠋⢉⣉⠉⠻⣶⡀⠀⠀⣠⡾⠛⠉⣉⡉⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠠⠶⣿⠀⠀⠀⢿⡆⠀⣿⠁⠀⠸⠉⠉⣷⠀⠈⣿⠀⢰⡿⠀⠀⠚⠁⣹⠆⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⣿⠀⠀⠀⣸⡇⠀⣿⠀⠀⠀⣠⠞⠁⠀⠀⣿⠃⢸⣇⠀⠀⡀⠈⠙⡆⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠿⠀⠀⣰⡿⠁⠀⠻⣧⡀⠾⠷⠶⠶⢀⣼⠟⠀⠈⢿⣆⡀⠙⠶⠞⠃⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⡶⠶⠟⠋⠀⠀⠀⠀⠈⠛⠷⢶⠶⠾⠟⠁⠀⠀⠀⠀⠙⠻⠶⠶⠶⠾⠛⠁


⠀⠀⠀⢀⣤⣴⣶⣦⣤⡀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣄⡀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⣀
⠀⢀⣼⠟⠁⠀⣠⡄⠈⠻⣧⠀⠀⢠⡾⠋⢁⣤⣤⡄⠉⢻⣦⠀⠀⣰⡿⠋⢀⡤⣤⡈⠙⢷⡄⠀⠀⣴⠟⠁⣤⣤⣤⡌⠻⣷⡀
⠀⣸⡇⠀⠀⡴⢻⡇⠀⠀⢸⣇⠀⣿⠁⠀⢸⡧⢤⣄⠀⠀⢻⡆⢰⡿⠀⠀⣿⡤⣬⡁⠀⠈⣿⠀⢸⡏⠀⠀⠀⣰⠏⠀⠀⢸⣧
⠀⢹⣇⠀⠾⠷⢾⡷⠀⠀⣸⡇⠀⣿⡀⠀⢠⡀⠀⣿⠀⠀⣼⠇⠸⣷⠀⠀⣿⠀⢈⡗⠀⢀⣿⠀⢸⣇⠀⠀⢠⡏⠀⠀⠀⢸⡟
⠀⠈⠻⣦⣀⠀⠈⠁⣀⣴⠟⠀⠀⠘⢷⣄⡈⠙⠋⢁⣠⣾⠏⠀⠀⠙⢷⣄⡈⠛⠋⢁⣠⡾⠃⠀⠀⠻⣦⣀⠈⠁⠀⣀⣴⠿⠁
⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠿⠶⣦⣄⠀⠀⠀⠀⢀⣤⡶⠾⠷⠶⣦⣀⠀⠀⠀⠀⣠⣴⠶⠾⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢀⡴⠶⣄⠀⠙⣷⡀⠀⣴⡟⠁⣠⠶⠶⡄⠈⠻⣦⠀⢀⣾⠏⠀⣠⠶⢦⡀⠉⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢈⣷⠤⣟⠀⠀⢹⡇⠀⣿⠀⠀⢿⣀⣀⣿⠀⠀⣿⡄⢸⡏⠀⠀⡏⠀⠘⡇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠸⣇⣀⣼⠀⢀⣾⠇⠀⣿⡀⠀⢠⣉⣁⡟⠀⢀⣿⠀⠸⣷⡀⠀⢷⣀⣸⠇⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣈⣉⣀⣤⡾⠋⠀⠀⠈⠻⣦⣄⣉⣁⣠⣴⠟⠁⠀⠀⠙⢷⣦⣀⣉⣁⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⠀⢜⠇⠀⠀⠀⠀⠀⠀⠀⠀⢡⡜⡏⠐
⠀⠀⠔⡛⠛⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⢤⡢⠂⠀⠀⣀⣤⣀⣒⠀⠀⠀⠉⠀⢜⠛⠃⣠
⠀⠀⠀⠭⠾⠋⠀⠀⠀⠠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⣤⠀⠈⡃⠀⠀⠀⠀⠀⠠⠭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⠀⡀⢔⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠛⠀⢠⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠠⠂⢠⡄
⠀⠀⠀⠀⠀⠰⠟⠂⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠶⠿⠹⠂⠐⠀⠀⠀⢢⣶⣤⣥⢰⡆
⠀⠀⠀⠀⠘⠀⠀⢤⢐⠅⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⣀⣉⣀⡀⠒⠀⠀⢘⡂⠀⢚⡀⠀⡾⣯⣛⣡⡾
⠀⠀⠀⠀⢰⠙⠂⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡊⠃⠀⠀⠥⠁⠙⠁⠈⠀⠀⠈⠈⢐⠟⠁⠀⠁⠸⠟⠋⠁
⠀⠀⠀⠀⡀⣀⡀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⠈⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠃

⠁⠐⠓⡆⠀⠀⠀⠀⠘⠋⠀⠀⠀⠠⠠⡄⠀⠀⢠⠖⢋⠒⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡷⠀⠀⠘⠦⠞⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢼⠁⣹⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⡇⠀⠀⢀⡰
⠀⠀⠀⠀⠀⠀⠩⠟⠀⠀⠀⠚⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢖⠢⠀⠀⠀⠀⠀⠈⡰⠀⠀⠀⠀⠃⠀⠀⠈
⠀⠊⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠀⠀⠀⡐⢦⠀⠀⠘⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀
⠀⠀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠂⠀⠀⠀⠁⠀⠀⢞⢈⡆
⠀⠀⠀⡀⠀⠀⠀⡇⠑⠒⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠚
⠀⠀⠲⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠨⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⡴⠲⢦⣄⠀⠀⠀⢸⡙⢦⣀⣸⠀⠀⠀⠀⠀⠱⠄⠀⠀⣀⣀⣀⣀⠀⠀⠀⠐⠋⠀⠀⠀⣰⡍⣰
⠀⠀⢧⣤⠜⢸⠆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⢄⠀⠀⠀⠈⠣⠟
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠁
⢀⣠⠀⠀⠀⠀⠀⠯⡄⠀⠀⠀⡀⠀⠀⢇⣘⠉⠳⠀⠀⢀⠔⣀⠀⠀⠀⠀⠐⣀⠀⠀⣀⡀⠀⠀⡖⠀⡆⡆
⠈⠐⠄⠀⠀⠀⠀⠁⠁⠀⠀⠸⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠊⣉⠇⠀⠀⠳⠉⠁⠀⠀⠒⠉⠀⠀⠙⠚⠁
⠀⠀⢀⡀⠀⠀⠀⢀⠀⠤⠀⠀⢀⢤⡀
⠀⠀⠈⢣⠀⠀⠀⠀⠓⠀⠀⠀⢳⡀⢹⠄⠀⠀⠀⠀⠐⡉⠀⠀⠀⠄⣞⡀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠀⠀⠀⠀⠀⠀⠋⡡
⢀⡴⡀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠀⠀⠀⠀⣰⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠃⠃⠀⠀⠤⡀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠼⠊⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡽⠀⠀⠀⡔⠒⠂⠀⠀⠀⡀⢤
⠀⠀⠀⠀⠀⠠⠉⠀⠀⢀⠀⠀⠀⢀⢀⠄⠀⠀⠀⠀⠀⠀⣀⡖⠀⠀⠘⠒⠚⠁⠀⠀⠀⠑⣖⢀⡄⠀⠀⠉⠈
⢠⢤⡀⠀⠀⠀⠀⠀⠀⡟⡠⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠐⠒⠛⠛⠛⠓⠛⠛⠛⠃
⠀⠀⠀⢀⠆⡆⠀⠀⠀⢀⠤⠶⢦⣅⠀⠀⠀⠀⡤⠀⠦⠾⡤⠀⠀⠀⠀⡠⠹⢡⠀⠀⠀⠀⠀⠄⠴⠤
⠀⠀⣂⣁⠀⠆⠀⠀⠀⠃⠔⠒⠀⢸⠀⠀⠀⠀⠁⠐⠂⣔⠀⠀⠀⠀⡰⠁⠖⢠⠀⠀⠀⠀⠀⠀⠎⢁⠈
⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⢀⠌⢀⠎⠀⠀⠀⠀⠈⠒⠆⠀⢡⠀⠀⣀⡀⠈⠀⠈⣰⠀⠀⠀⠀⠁⢢⠀⠱
⠀⠀⠀⢨⣀⡄⠀⠀⠀⠰⣁⣀⡀⠀⡱⠀⠀⠀⣏⣉⣁⡠⠎⠀⠀⠀⠀⠀⢁⢸⠁⠀⠀⠀⢸⣉⣁⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⣤⡉⣄⠀⠀⠀⡀⠀⣠⡌⢢⡀⠀⠀⠀⣀⣀⣃⠀⠀⠀⠀⢀⡀⣠⢀⡀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠞⠀⣔⡒⠂⠀⠀⠠⠤⠤⡄⠀⣠⠇⠀⠀⡌⢠⣀⠀⡃⠀⠀⢠⠃⢠⠒⡆⢸⠀⠀⢀⠜⠡⠐⠢⡈⢆
⠀⢘⠀⠠⠀⡀⢳⠀⠀⠀⢀⠎⢀⠞⠁⠀⠀⢀⠜⢀⣀⠐⢅⠀⠀⠈⠦⣈⡉⠁⢀⠀⠀⢸⠀⡇⠀⠀⡇⢸
⠀⠘⡄⠐⠔⠃⡸⠀⠀⠀⡌⠀⡎⠀⠀⠀⠀⢸⡀⠡⠤⠃⣰⠀⠀⠀⡤⠐⠉⣀⠌⠀⠀⠘⢆⠈⠒⠊⢀⡜
⠀⠀⠈⠁⠒⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠐⠒⠊⠁⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠉⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⣤⣤⣤⣤⣄⣠⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠏⠀⠀⠘⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠼⠷⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢾⠖⠇⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠆⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛
⠀⠀⠀⠀⠀⣠⣶⡦⠀⠀⠀⠀⣠⣶⡶
⠀⠀⠀⣀⠀⠙⠉⠀⠀⠀⠀⠀⠉⠉⣠⡆⠀⠀⠀⠀⣼⣷
⠀⠀⠀⠁⠀⣤⣤⣀⠀⠀⠀⠀⢠⣤⣄⠁⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠈⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⠀⠙⠻⠗⠀⢠⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠳⡄⢤⡀⢀⠈⣆⣠⠖⠊⠀⠀⢠⣴⡆⠐⢶⡄⣤⡀⢀⠘⣆⡤⠶⠃⠀⠀⢠⢸⠀⠔⢷⡌⣤⡀⣀⠘⣂⡤⠖⠃⠀⠀⢠⢼⠀⠔⢶⡀⣄⡀⣀⠰⣀⡤⠖⠃
⠀⣃⡈⠉⠉⠤⢥⡄⠀⠀⢀⠄⠊⠀⢰⢀⠀⣇⡈⠉⠁⠤⢥⡄⠀⠀⣀⠀⠈⠀⢠⣀⠀⣇⠈⠉⠁⠤⢥⡀⠀⠰⡀⠀⠉⠀⠀⡀⠈⣇⠈⠉⠁⠤⣥⡀⠀⠐⡀
⠐⠉⢇⠄⢀⣀⡀⢀⠀⠀⡿⠀⠘⢳⡜⠁⠐⠉⠇⠀⢀⣀⠀⢰⢀⠀⡟⠀⠘⢷⠘⠁⠈⠉⢃⠀⢀⣀⠀⢠⠀⢀⠇⠀⠛⢷⠘⠁⠈⠙⢃⠀⣀⣀⠀⣀⠀⢨⠇
⡶⠖⢸⣦⠀⠈⠷⢸⠸⠣⣆⠀⠀⡸⠁⢶⡶⠖⢸⣦⠀⠈⠷⢸⠸⢠⡆⠀⠀⡜⠀⢶⠶⠖⢹⡄⠀⠉⠇⣾⠺⢠⡆⠀⠀⠜⠀⣶⠶⠆⢾⡄⠀⠙⡇⣷⠿⢠⡆
⠃⢠⡤⠛⢒⣦⣤⡈⠠⠴⣿⠛⠁⣀⣃⡈⠁⢠⡬⠛⢒⣤⣤⠈⠠⠶⣿⠛⠁⣀⣃⡘⠀⢠⠜⠛⣒⣤⣤⠈⠠⠶⣟⠋⠀⣀⣃⡘⠀⢠⠜⠛⣒⣤⣄⠉⠠⠶⣟
⠆⠀⣤⡴⠉⠀⠹⢷⠀⠀⢸⡤⢨⡁⠈⠹⠆⠀⣤⡖⠁⠀⠹⣶⠀⠀⢸⡄⢨⠁⠈⢻⠀⠀⣤⡖⠁⠀⠹⣶⠀⠀⢸⡄⢨⠁⠈⢻⠀⠠⣤⠆⠁⠀⢹⣶⠀⠀⣻
⣠⠴⠚⣇⠀⣀⠀⠘⠷⢶⡄⢰⠈⠳⠶⣂⣤⠴⠚⣇⠀⣀⠀⠘⠷⢶⡀⢸⠈⠳⠶⣂⡤⠖⢺⣆⠀⣀⠀⠘⠷⢶⡀⣶⠘⠳⠶⣂⡤⠖⢺⡄⠀⡀⠀⠹⠶⣶
⣀⡀⠀⠙⣻⣥⣰⣄⠀⠈⢳⡜⢧⡐⠿⣭⣀⡀⠀⠙⣫⣥⣸⣄⠀⠈⢱⠘⣦⡐⠿⣭⣀⠀⠀⠙⣫⣅⣸⣀⠀⠘⢳⠸⣦⡰⢿⣥⣀⠀⠈⠛⣫⣄⣾⡀⠀⠈⢲
⠉⢛⠀⠰⣄⠈⢳⠉⠉⢀⠀⢀⣿⡛⠃⠀⠉⣛⠀⢰⣄⠈⣷⢉⠉⣀⠀⢀⣛⡛⠁⠀⠉⣋⠀⢰⣄⠈⡇⠉⠉⡀⠀⣀⣟⡛⠀⠀⠉⣣⠀⢸⡀⠈⡇⠉⠉⡀
⠚⠁⠶⠀⠽⠁⠈⠼⠓⢸⡄⠀⠀⢻⠀⠰⠚⠁⠶⠀⠝⠁⠀⠼⠓⢹⡄⠀⠈⢳⠀⠴⠊⠁⠶⠀⠝⠁⠁⠼⠓⢹⠀⠀⠈⢳⠀⠴⠊⠑⠶⠀⠝⠀⠡⠾⠓⢻
⠖⠋⠀⠀⢠⣴⡇⠐⠻⡆⢴⣀⣀⠀⣇⡤⠒⠃⠀⠀⢠⢼⡄⠐⢻⡄⣶⣀⣀⠈⣇⠤⠒⠃⠀⠀⢠⢸⠀⠘⢳⡄⣦⣀⣀⠘⣆⠤⠚⠃⠀⠀⣠⢼⠀⠚⢳⡀⣦
⠀⠀⣀⠀⠈⠀⢀⣀⠀⣇⡀⠉⠁⠴⢶⡄⠀⠠⣠⠀⠈⠀⢀⣀⠀⣁⡀⠉⠡⠴⢦⡄⠀⠀⣀⠀⠈⠀⢈⡀⠀⣇⡈⠉⠁⠴⢦⡀⠄⠀⡀⠀⠁⠀⢀⡀⢀⣁⠈
⠁⠀⠿⠀⠈⢻⡎⠁⠀⠉⢃⠀⢀⣄⡀⢠⠀⠄⠏⠀⠉⢳⡎⠁⠈⠉⢣⠀⢀⣄⡀⠀⠀⡄⠏⠀⠉⢳⠈⠀⠈⠉⢇⠀⢀⣄⠀⡘⠀⠠⠇⠀⠉⢷⠈⠀⠊⠙⣇
⠸⠃⣆⠀⠀⠘⠁⢻⡶⠒⢸⣆⠀⠈⠗⢸⠸⠠⣆⠀⠀⠘⠀⢿⠶⠊⢸⡆⠀⠈⠇⢿⠸⢰⣆⠀⠀⠞⡀⢿⠶⠂⢸⡆⠀⠈⠇⣧⠻⢰⣆⠀⠀⠎⡀⣷⠶⠂⢿
⠠⠒⢿⠉⠀⣠⣇⣈⠁⠠⠖⠙⣛⣧⣤⠈⠠⠲⢯⠉⠀⣠⣇⡈⠀⠠⠞⠙⣻⣧⣤⠈⠠⠲⣯⠉⢀⣠⣅⡈⠀⠠⠞⠙⣻⣦⣤⠈⠐⠲⣏⠉⢀⣸⣅⡈⠀⠠⠎
⡀⠀⢸⡤⢨⡁⠀⠹⠆⠀⡴⡖⠁⠀⠸⢷⡀⠀⢸⡄⢨⡁⠀⢹⠀⠀⠶⡆⠁⠀⢸⢷⠀⠀⢸⡄⢨⠁⠀⠹⠀⠠⠶⠎⠀⠀⠹⣦⠀⠀⢨⡄⣭⠀⠈⠻⠀⠠⢶
⠳⢶⡆⢸⠈⠛⠖⣂⡤⠖⠛⣇⣀⣀⠀⠘⠳⢶⡄⢰⠈⠛⢖⣂⡤⠖⠳⣇⣀⡠⠀⠘⠷⢶⡀⣷⠈⠳⢖⣀⡤⠒⢻⣇⢀⡀⠀⠘⠷⢶⠀⣇⠈⠳⢒⣠⡴⠚⢹
⣀⡀⠱⡌⢷⣐⠻⢧⣄⡀⠀⠙⠩⢥⣸⣄⣀⡀⠱⠜⢧⣐⠿⢥⣄⡀⠀⠙⠯⣤⣸⣄⣀⠀⢱⠘⣦⡐⠿⣤⣄⡀⠀⠙⠭⣤⣺⣄⣀⠈⢳⠘⣦⡘⠿⣤⣀⡀⠈
⡉⢀⠂⢀⣭⡙⠁⠀⠈⣉⠀⠸⣄⠀⠷⢈⡉⢠⠀⣠⣭⡉⠁⠀⠈⣩⠀⢸⣄⠈⠇⢉⠁⣀⠀⣠⣭⡙⠁⠀⠈⣩⠀⢰⣄⠈⡇⢉⠁⡀⠀⣠⣭⡉⠀⠀⢉⣡
⠛⠸⠆⠀⠀⢹⠀⠴⠊⠁⠲⠀⠜⠁⠀⠜⠛⢹⠄⠀⠀⢹⠀⠴⠋⠁⠳⠀⠜⠁⠀⠜⠛⢹⠄⠀⠀⢿⠀⠔⠋⠀⠷⠀⠝⠀⠠⠞⠋⢻⠀⠀⠈⠇⠀⠖⠋⠐⠷
⠻⡆⢰⣄⣀⠈⣧⠴⠚⠋⠀⠀⢠⢼⠇⠘⠻⡆⢶⣀⣠⠈⣦⠴⠚⠁⠀⠀⢠⢼⠀⠘⠻⡄⢶⣀⡠⠘⣧⠴⠚⠁⠂⠀⢰⢾⠀⠚⢻⡄⣦⣀⡠⢐⡦⠔⠛⠁
⠀⣡⡀⠈⠀⠐⠶⡄⠀⠀⣠⠀⠈⠀⢰⣠⠀⣧⡀⠈⠀⠔⢶⡄⠀⠀⣤⠀⠀⠀⢰⡠⠀⣧⡀⠉⠀⠒⢶⠄⠀⠐⡠⠀⠁⠀⢀⡀⢈⣥⡀⠁⠀⠒⢶⠄⠀⠀⡤
⠈⠀⢃⠄⢀⣤⡀⢰⠀⡀⠟⠀⠈⢻⡎⠀⠀⠀⢃⠄⣀⣤⡀⢰⢀⠀⠇⠀⠈⢻⠈⠀⠈⠈⢣⠀⣠⣤⡀⠰⢀⠠⠇⠀⠈⢻⠈⠀⠊⠈⢣⠀⣠⣄⡀⡤⢀⠰⠇
⡷⠊⢸⣶⠀⠀⠛⢸⡘⠡⣆⡀⠀⠜⡀⢻⡖⠋⢸⣶⠀⠀⠟⢸⠘⠰⣇⠀⠀⠜⡀⢿⠖⠃⢸⣆⠀⠈⠇⢷⠛⠰⣇⠀⠀⠞⡀⢿⠖⠃⢻⡆⠀⠈⠇⣷⠛⢰⣇
⠁⠠⠖⠉⣫⡵⣦⡀⠀⠒⢿⠉⢁⣠⣥⣌⠁⠠⠖⠉⣫⡵⣦⠈⠐⠒⢯⠉⢀⣤⣥⡈⠀⠰⠞⠉⣫⣵⣦⠈⠐⠒⣯⠉⢀⣬⣥⡈⠀⠰⠎⠙⣫⣷⣦⠀⠐⠲⣯
⠇⠀⠶⡞⠀⠀⠘⢿⡀⠀⢸⠤⢨⡀⠀⠙⠆⠀⠶⡆⠀⠀⠘⢿⡀⠀⠸⢄⢨⡀⠀⠹⠀⠀⠶⠎⠀⠀⠸⢾⠀⠀⢸⠄⢨⡀⠀⠹⠀⠠⠶⠆⠀⠀⠹⣷⠀⠀⠹
⠴⠞⠋⣧⣀⡠⠀⠘⠻⢾⡆⢸⠀⠛⢚⣣⠴⠚⠉⣧⣀⡠⠀⠘⠳⢾⡄⢺⠈⠛⢚⣡⠴⠚⠹⣆⣀⡠⢀⠘⠳⢾⡀⣷⠈⠛⢚⣡⠴⠚⢹⣆⣀⡄⢀⠘⠳⢿
⣤⣀⠀⠉⠭⢦⡼⣆⣀⠀⠹⠎⣷⣜⡻⢦⣤⣀⠀⠉⠽⢦⡸⣄⣀⠈⢹⠈⣷⣜⡻⢦⣄⣀⠀⠉⠽⢦⣸⣄⣀⠈⠹⠘⣧⣘⠻⢦⣄⡀⠀⢉⠽⣦⣺⣄⣀⠈⠳
⠀⣩⠄⠸⣦⡀⢻⢈⡁⢠⠂⡠⢭⣍⠁⠀⠀⣩⠀⠸⣄⡀⠷⢈⡁⣀⠀⡠⢭⡉⠁⠀⢈⣩⠀⠸⣄⠀⡇⢈⡁⣄⠀⡨⢭⡉⠁⠀⢈⡩⠀⢺⣄⠈⡇⢈⠁⣐⠁
⠋⢀⠻⠀⠜⠁⠀⠚⠛⠸⠆⠀⠀⠹⠀⠔⠋⠀⠳⠀⠞⠁⠀⠚⡋⠸⠆⠀⠀⢻⠀⠖⠉⠀⠓⠀⠎⠀⠀⠚⡋⢹⠀⠀⠀⢻⠀⠖⠉⡈⠗⠀⠋⢀⠠⠚⠋⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⢸⣿⠀⣾⢋⣤⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⢹⡇⠀⠰⠂⠀⠀⡿⠀⠃⠈⠉⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⢀⠀⠟⠀⠀⠀⠀⠀⣀⠀⠀⢀⣤⡀⠀⣠⡶⢶⣿⠁⠀⠠⠄⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⣠⡾⢉⠇⣰⡿⢠⠇⣴⡟⢠⣿⠇⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣧⡼⣾⡏⢻⡾⢾⡟⣠⢿⡿⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣤⣴⣶⡄⣼⡏⢀⡞⢠⣿⣠⠏⠀⠸⡷⠋⣾⡇⠀⢀⡾⠁⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠉⢁⡼⠃⠹⠷⢋⣠⣬⡿⠷⠦⣄⠀⠀⠀⣿⡇⢀⡾⠁⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⡶⠛⠉⠀⣴⠂⠀⡼⠀⠀⠀⠘⠳⠛⡁⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠸⣿⣄⠀⢀⣠⡾⠋⡀⠀⣼⠏⠀⠀⠀⢰⡏⣠⠞⣱⠆⢀⣾⠀⣰⠟⣹⠇⠀⠀⠛⣥⣴⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠈⠛⠛⠛⠁⠀⠀⠁⢸⡏⠀⠀⠀⢀⣿⢿⡅⢠⣿⡴⢻⡏⣸⣿⠞⠁⣠⠄⣀⠀⠰⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠀⠀⠀⠀⣠⣴⡾⠿⠿⠿⣷⣄⠀⠀⣼⡇⠀⡟⠀⠉⢀⣾⠷⣿⠁⠀⣰⠏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠗⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠈⠛⠀⠀⣿⠀⡼⠁⢀⡴⢻⡟⠀⠻⣶⠾⠋⠀⠀⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠻⠟⠁⣠⠏⠀⣾⠁⢠⣶⡆⠀⠀⠀⢰⡿⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⠀⢠⡏⢀⣼⠃⢠⡞⣯⡀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠛⢁⡀⠀⠉⠀⢁⣀⣠⣿⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⣠⡞⢹⠄⠀⣴⠋⡷⠀⢀⣾⠟⢩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⣿⢃⣾⣿⠀⣜⣠⣾⢿⣴⣃⣠⣿⠁⣠⢿⡟⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⢀⣼⠇⣼⡇⠘⡿⠉⢸⡇⣴⠋⠉⢸⣇⡴⠃⢸⡇⠀⣠⡟⠀⠀⠛⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣀⣀⣀⣤⣿⣿⠀⣿⣠⡞⠁⠀⠸⠟⠁⠀⠀⠀⠉⠀⠀⣼⠛⠞⠋⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⢁⣾⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡆⣸⠇⢀⣤⠄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢠⣦⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⣼⠀⣰⡇⢀⣾⠃⠀⣴⣿⣾⠿⠋⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⣼⢇⣼⣿⠀⣼⠇⣠⢞⣿⣿⡁⠀⠀⠀⢀⣾⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣾⡿⠀⠀⠀⢀⣀⡀⣿⠟⢸⣧⡾⠻⠶⠃⣸⠇⠘⢷⣤⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⣿⠷⠶⠿⠿⠟⠛⠁⠀⠀⠀⢉⡀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠀⠀⢀⡾⢹⡇⠙⠀⠀⠀⠰⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠋


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠆⢾⣋⣉⣛⣋⣩⣉⣉⣋⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢘⠄⠀⠀⠀⠀⢀⣄⠤⠀⠤⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆
⠀⠀⡀⠛⠀⠀⠀⢤⢈⠇⠀⠀⠀⠀⠀⠀⡀⢾⡆⣴⠀⣷⡆⠀⠀⠀⠰⠰⣶⠀⣀⠙⠃⠰⠐⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢉⡡⢒⠂
⠀⠀⠯⠀⠂⢆⢠⡎⠀⠀⢀⠀⠀⠀⠀⠀⢹⡶⡦⢴⡶⣿⠀⠀⠀⠀⠀⠀⢀⣈⠛⢠⣤⠀⠀⠀⡐⢻⡆⠀⠀⠀⠀⢰⣧⡄⠀⣐⢰⣯⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠿⠀⠈⠀⠀⠀⠀⠈⠋⠛⠋⠛⠙⠁⠀⠀⠀⠀⠀⠈⠋⢰⡦⠀⡀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠇⠀⠆⠈⠀⣇⡀⠐⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠙⠛⠆⠀⠀⠀⠀⠒⠀⠀⠀⠀⠅⠓⠀⠰⠆⣻⣣⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠿⡿⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠠⣀⣶⠀⠀⣀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠄
⠀⠀⠐⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢆⠀⠉⠉⠀⠜⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠒⡦⠐⠀⣂⠀⢀⠀⠀⠶⢤⡀⠠⡀
⠀⠀⠀⠀⠀⣴⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠰⣿⡗⠀⢰⣾⡄⠀⠀⠀⠀⡠⢀⣴⠞⠈⣠⣄⢼⠇⢷⣛⣃⠛⠱⠿⠛⣀⣤⠀⡹⠀⡇
⠀⠀⠀⠀⠀⠙⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠀⠀⠀⡠⢊⡼⠋⠠⣶⢷⠌⠉⠀⠑⠈⠁⠈⠀⠉⢈⠛⡻⣻⡵⢁⠜⠁⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠂⡛⠀⠀⠀⠀⠁⠸⠀⢰⣴⣶⠦⠀⠀⠀⠀⠀⣀⠠⣤⣀⣛⠿⠛⡁⠔⠁⠀⣀⣔⡛⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠠⠛⠛⠳⣄⠀⠀⠀⠀⢀⠀⠈⠡⣠⣶⡿⢡⣿⡯⠸⠿⠖⢛⡉⠤⠐⠀⠀⠀⠀⠰⣯⣯⣭⣼⣿⡆
⠀⠀⠡⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡰⠁⠀⢶⡆⠙⢦⡀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢾⣣⠀⠈⠳⠀⠀⢀⡀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃
⠀⠀⠀⠀⠀⢀⠀⢠⡤⠄⡀⠀⠀⠀⠀⠸⣁⠀⠀⠀⠀⢀⠀⠊⠀⠀⠟⠇⠀⠑⣰⣾⣾⠀⠀⡀⠀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⢀⠀⠀⢀⣀⡀
⠀⠀⠀⠆⠢⣍⠀⠐⣂⠀⠶⢑⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠕⢺⣿⠟⠁⠀⡇⡄⠀⡒⠂⠒⣤⡔⣀⣂⣒⠒⡄⠒⣒⣒⡐⠂⠆
⠀⠀⠀⠀⢠⡄⠉⠠⣭⣶⠻⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⡷⡇⠛⡻⠊⢸⣻⡇⠋⠛⠈⢸⠘⡀⢋⠃⠁⢠⡇
⠀⠀⢄⠀⠈⠁⠀⢰⣿⠇⢒⡇⠅⠀⠀⠀⢰⠀⢀⡀⠀⠀⠈⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇⠀⠓⠀⠈⡟⡇⠘⠓⠀⠀⢠⡁⠘⠂⠀⢘⡆
⠀⠀⠀⠑⠢⣀⠀⡆⠶⠀⠈⠑⠀⠀⠀⠀⠀⠉⠒⠀⢼⡂⢰⣾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠙⠛⠓⠉⠛⠛⠉⠋⠀⠛⠙⠋⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⣀⣀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⣀⣤⣤⠄⠠⣤⣀⠀⢀⣤⣄⠀⣀⣀⣀⣀⣀⣀⣤⣤⣀⡀⠀⣀⣤⣤⣤⣤⣀
⠀⣿⣿⠀⠰⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣷⠀⠸⣿⣿⣿⠀⠘⠿⠿⠃⠘⡧⣿⠀⠿⢿⣿⣿⣟⣛⣿⡟⠛⠃⣾⣿⠁⠀⠀⠈⢻⣷⡀
⠀⣿⣿⣶⣶⣿⣿⣷⡆⠀⣶⣶⣶⣾⣿⡟⠀⢀⣤⣿⣿⠛⢀⣠⣀⠀⠈⣙⡉⠀⠀⠀⠀⣀⠉⠉⠉⠁⠀⠀⣿⣧⠀⠀⠀⠀⢸⣿⠇
⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠙⠉⠉⠉⠁⠀⠀⠈⠻⠿⠿⠄⠹⠿⠛⠀⢸⣏⣛⠀⠀⡴⠋⠉⡉⠳⡄⠀⢀⡀⠙⠿⣦⣤⣤⣤⡿⠏
⠀⡖⠛⠀⠾⣿⣦⣾⢻⣦⣴⡟⣷⣄⡖⠲⣆⠀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠶⠟⠀⠘⣇⠀⠀⢹⠆⣿⠀⢸⡇⠀⣾⠀⠀⠀⠀⣠
⠀⠓⠲⡄⢠⣿⢡⣿⢿⣧⢿⣧⣿⣿⡇⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⠀⠰⣿⣦⠀⠀⠈⠛⠛⠁⢠⠏⠀⢸⣇⣀⣿⡀⠀⠐⠛⣿
⠀⣤⠴⠃⠸⠟⠘⢿⣼⠟⠲⣦⡿⠋⠧⡤⠏⠀⠛⠛⠛⢻⣿⣿⣿⠀⠀⠿⠇⠀⠀⢀⣀⣠⠴⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⢀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠠⡦⠴⠶⠀⠀⣸⣿⣿⣿⠀⣀⡀⣴⢦⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⣿
⠀⠀⢠⠈⠀⠀⣾⣿⠁⠀⠀⠀⠉⣿⣧⢀⡷⣔⣓⠀⠀⣿⣿⣿⡟⠀⠈⠙⠷⠾⠁⠀⢸⡟⠲⣄⡀⠀⠀⠀⠈⠙⣶⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣴⡾⠛⠁⠀⠷⠨⠧⠀⢰⣿⣿⣿⡇⠀⢠⣤⣤⣤⡄⠀⢸⣧⡀⠀⠉⠓⠶⠦⠤⠶⠛⠁⠀⠀⣿
⠀⠀⠀⠜⠀⠀⢠⣾⡟⠁⠀⠈⢻⣷⡄⠈⠻⣙⡋⠀⢸⣿⣿⣿⠃⠀⢸⣿⣿⣿⠇⠀⣠⣠⣀⣤⡄⣠⢀⡀⣠⠤⣀⠀⠀⠀⣿
⠀⠀⠀⠊⠀⠀⠀⠛⠿⠶⠶⠶⠟⠋⠀⠐⠿⠒⠙⠀⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⠀⠈⣿⢾⣡⡾⡁⠻⠬⢸⡁⠀⢈⠆⠀⢀⣿⡀
⠀⠐⢲⡶⠂⠀⢀⣠⣀⠀⣀⣀⢀⣤⡀⣀⣀⠀⠀⢀⣿⣿⣿⡇⠀⠀⣿⣿⣿⡟⠀⠀⠉⠉⠈⠛⠁⠘⠛⠀⠙⠒⠋⠀⠀⠉⠉⠉
⠀⠀⢸⡇⠀⠀⡏⠀⠈⡇⡊⢥⡈⣽⠏⣿⢼⡁⠀⢸⣿⣿⣿⠇⠀⢸⣿⣿⣿⡇⠀⣤⡶⠶⠆⠀⢀⣤⡴⠶⠶⢶⣤⣀⠀⠀⢆
⠀⠀⢸⡇⠀⠀⠙⠒⠚⠀⠋⠚⠘⠛⠂⡛⠘⠁⠀⣼⣿⣿⣿⠀⠀⣸⣿⣿⣿⠁⠀⡤⣭⠦⡀⠀⠺⣿⣄⠀⠀⢀⣿⡿
⠀⠀⢸⡇⠀⠀⢠⣶⠾⠛⠲⠶⣤⡀⠀⠈⣿⠀⠀⠛⠛⠛⠛⠀⠀⣿⣿⣿⣿⠀⠀⣳⡀⢶⠀⠀⣀⣬⡿⠟⠻⠿⣧⣄⠀⠀⠀⡀
⠀⠀⢸⡇⠀⠀⠘⢧⣀⠀⠀⠀⠀⠉⠓⢤⣿⠀⠀⣰⠟⣶⣄⡀⢀⣿⣿⣿⡇⠀⠀⠥⠍⢾⠀⠀⣿⣯⠀⠀⠀⠀⣨⣿⠇
⠀⠀⢸⡇⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⣃⠈⠁⢸⣿⣿⣿⠃⠀⠀⠻⠟⠛⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁
⠀⠀⢸⣇⠀⠀⠀⣸⣧⣤⣤⠀⠀⢀⡴⠚⠉⠉⠀⠀⢰⢿⡇⠀⢸⣿⣿⣿⣤⣤⣤⣄⢠⠖⢲⣄⣶⠶⣦⢰⡶⣶⡆⢰⡖⢀⡔⠚⠂
⠀⠀⢸⡿⠋⠀⠀⢸⡇⠀⣿⠀⢀⡏⢀⣠⣤⣀⠀⠀⠈⠻⠃⠀⢸⣿⣿⣿⣿⣿⣿⣟⢸⠀⢸⣿⣿⢻⣷⢸⣷⣿⡇⣿⠇⠘⠧⢤
⠀⠀⠀⠀⢀⣀⠀⠘⠃⠀⣿⠀⢸⡇⢾⠀⠀⠈⡇⠀⢸⣿⡁⠀⠘⠛⠛⠛⠛⠛⠛⠋⠸⢦⡼⠋⢿⣼⡿⠸⣧⣿⠿⣿⣤⠄⣤⣼
⠀⣴⣾⠛⠉⠉⠛⣷⣆⠀⠀⠀⠀⠳⣌⣁⣀⡴⠃⠀⢰⣼⠇⠀⢠⣶⣾⡀⠙⣿⣶⣦⠀⠀⣀⣠⣤⣤⣤⠀⢀⣤⣤⣤⣤⣤⣤⣤⡄
⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢠⣤⣤⣀⣉⡀⠀⠀⠀⢠⡶⡄⠀⠀⠉⠉⠠⢴⣿⣟⠋⠀⣼⣿⡿⠿⠿⠿⠀⠘⠿⣿⣿⠿⠿⣿⣿⡇
⠀⢿⣷⡀⠀⠀⢀⣾⡿⠀⣤⣼⣿⣯⣭⣿⣿⣿⣿⡆⢸⣿⡇⠀⢴⣶⣶⠄⠀⣿⣿⣿⠀⢿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⠇⠀⣿⣿⡇
⣀⣀⣈⣉⣛⣛⣉⣉⣀⣀⣉⣉⣛⣉⣉⣉⣉⣉⣉⣀⣀⣉⣀⣀⣀⣉⣙⣀⣛⣋⣉⣁⣀⣀⣈⣉⣛⣛⣛⣛⣛⣋⣉⣁⣀⣀⣉⣉⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠇⡀⡀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠲⠠⠑⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠗
⠀⠀⠀⠀⠀⠀⠄⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠢⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠑⠀⠒⠀⠈⠘⠐⠀⠀⠀⠀⠦⠀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠃⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡲⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⡀⢀⠀⠢⠀⠀⠀⠀⢀⡀
⠀⠀⠁⠙⠀⠄⢀⠀⡀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢑⣀⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡤⠄⡃⠀⠀⠀⡴⢀⠂⠀⠀⠀⣀⡐⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⢠⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⡴⣀⡀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠰
⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠓⢠⠀⠀⠀⠀⠐⡂⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠦⠀⠈⠓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⡷⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠠⣤
⠀⠀⠀⣸⡟⢻⣟⣿⡏⢻⡆⠀⠀⠀⢂⡀⠀⠂⠀⠰⠀⠁⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⢀⣴⣶⣦⣀⡿⠁⠀⠀⠀⠀⠀⠓⠙⠀⠀⠀⠀⠀⠀⠄⠘⠀⠡⢀⠁
⠀⠀⠀⠘⣻⠛⣿⣿⠛⡋⠀⠀⢀⡀⡀⡰⣨⡄⢩⠀⠠⠀⡄⠀⠀⠀⠙⠛⠋⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⢀⣻⣻⣆⣾⣹⣟⣖⠀⠀⠀⢀⡄⠁⢀⡈
⠀⠀⠀⠰⠿⣦⡉⣁⣼⠿⠄⠀⠀⠀⠠⠐⠠⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⣿⣿⣧⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣍⣷⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢠⣤⣼⠟⠿⣤⣤⡀⠀⠀⠉⠀⢀⡂⠀⠀⠙⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⣿⣏⣹⡿⠛⣿⣿⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠛⠁⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠙⣷⢸⡿⠁⠀⢻⢸⡇⠀⠀⠀⠼⠛⢻⠟⠛⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⠀⠀⠀⠀⠀⡟⣸⠃⠀⠀⢸⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⣀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣠⣴⣿⣿⣿⣷⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣷⠖⠀⠀⠀⢻⣼⠇
⠀⠀⠀⢰⢻⠆⠀⠀⣿⣿⣧⣽⣿⣿⣥⣿⣿⠇⠀⣠⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣀⡀⠀⢰⣦⣿⡆⢰⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠛⠻⠛⢉⣰⠏⠀⠀⣿⣿⣟⠛⠀⠀⠀⠈⢛⣿⣿⡿⠁⠀⠀⣠⣾⣿⣿⣿⣦⣨⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣤⡄
⠀⠀⠀⢠⣴⡄⠀⠀⠀⠀⠉⠛⠓⠚⠋⠁⠀⠀⠀⠘⠻⣿⣿⣦⠄⠀⢴⣿⣿⡟⠁⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢀⣴⣾⣿⣦⡄
⠀⠀⠀⢸⣿⡇⠀⠀⣠⠄⠒⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣴⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠋⠉⠉⠙⠛⠃⠀⠀⣀⡀⠀⠀⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠈⠉⠁⠀⠀⣿⣁⡀⣘⣹⣇⣶⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⡀⣀⣠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢸⣿⣅⣽⣿⣉⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣷⣾⣿⡿⣿⣻⣿⣿⠟⠃⠀⣴⣷⣤⣄⠀⠀⠀⢀⣠⢀⣦⢰⣷⢾⡿⢙⡏⠀⠀⠀⢹⣦⣀⡧⠀⣾⠀⠀⡀⠀⠀⠀⢸⣧⠙⠻⠟⠟⣸⣿
⠀⠀⠀⠀⠀⠀⠘⠻⣿⣽⣿⣿⣿⡟⠉⠀⠀⠀⠀⣿⣿⣿⡧⠀⠀⢻⣿⣼⠿⠾⠁⠘⠀⣀⡁⠀⠀⠦⣤⣽⣿⣷⡀⢹⣷⠾⣡⠖⠀⠀⠀⠻⠿⠶⠶⠾⠿⠃
⠀⠀⠀⣠⣄⡀⠀⠀⠈⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⠛⠛⣿⠆⠀⢈⣉⢀⣤⢰⡆⣿⢸⣿⠇⠀⠀⢲⣦⣄⣸⣿⣷⣾⡇⢀⣿
⠀⠀⠈⠹⠛⠆⠀⠀⠀⠀⠀⠀⠉⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⢿⠸⠤⠻⠊⡋⠀⠈⠋⠉⠉⠙⣿⣿⣿⣿⡿⠛⠁⠀⠰⣒⣦⣤⣞⠁⠂
⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣷⣦⠀⠀⠀⠀⠘⠋⣈⣤⣤⣶⢶⣿⡿⠀⠀⠀⠀⠀⠀⣹⣿⡿⢾⡇⠀⠀⠀⠀⠋⠉⠉⠉⠛⢷⣄⣀⡀
⠀⠀⠀⠀⠀⢯⣽⣿⡽⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢿⣿⡏⣿⡀⣿⠄⠃⠃⠀⠀⠀⠀⣀⣠⣿⣦⣤⣼⣧⣀⠀⠀⢰⡀⠀⠀⠀⠀⠀⣿⣿⠗
⠀⠀⠀⢀⣤⠤⣽⡿⠤⢤⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠋⠁⠉⠀⠀⠀⠀⠀⢄⣤⡀⠀⠉⠛⠻⠻⠟⠙⠏⠉⠁⠀⠀⡿⣦⣤⠀⣤⡾⣿⠻⢶⡄
⠀⠀⠀⢸⡏⣿⣿⣿⢾⢽⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⡏⢛⣹⣏⡛⢹⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠠⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⢸⣧⡿⣿⣿⢯⣼⡇⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣿⡿⢿⣿⣧⠀⠀⠈⠛⢿⣷⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⢿⡿⡇
⠀⠀⠀⢸⡿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣠⣶⣾⣾⣿⣧⡀⠀⠀⠀⠘⠷⣿⣾⢧⣼⣿⡟⠀⠀⠀⠀⠀⣾⣿⣿⣛⣡⣤⣶⣾⡁⠀⠀⠀⢸⣷⣾⣿⣶⣿
⠀⠀⠀⠀⢸⣧⠀⠀⣾⡇⠀⠀⠀⠀⠀⢀⣾⣿⠟⣈⢱⣿⡿⣱⠀⠀⠀⠀⠀⠛⠛⠀⠉⠁⠀⠀⠀⠀⣀⣴⡿⣿⣿⣿⡿⣿⣿⠿⠇⣠⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣟⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⣿⣴⣿⣿⡿⠿⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⢦⡜⣛⣥⣶⣿⣿⣿⣗⣠⡤⣶⣿⣿⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣷⡙⣟⣛⣭⣵⣶⣾⡿⠋⠀⠀⠀⠀⣀⠀⢢⠀⡦⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠃⠈⠀⠀⠀⠁⠀⠀⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠸⠟⠀⠀⠻⠇⠀⠀⠀⠈⢻⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠁⠐⠀⠥⠅⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠙⣛⣻⣉⣁⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⣾⠋⠀⠀⠀⠉⠛⢿⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠺⢟⣿⠥⠤⠄⠹⢿⢾⡿⡟⠇⣿⡀⠀⠀⠀⠀⠀⠀⠙⢦⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢾⣩⣷⣶⠰⣿⠀⠀⠀⠀⠀⠀⠀⠇⠙⡄⠘⢳⣄⠀⢀⠀⠀⠀⠀⠈⣯⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠉⣵⠋⠹⣿⣿⡓⠙⠷⣆⣀⠀⠀⠀⠀⠈⢆⠈⣢⣶⣝⠷⣤⣀⠀⠀⠀⢀⣾⠈⠳
⠀⠀⠀⠀⠀⠀⣀⣤⡔⣂⡠⠽⣿⣿⡌⢿⣄⠀⠈⠢⡉⠂⠄⣨⣭⣛⣓⡲⠶⠦⠴⠷⢿⣿⣿⣿⣮⠙⠛⠛⠛⣋⣡⣄⡰
⠀⠀⠀⠀⠠⠊⠉⣵⠋⠁⠀⠀⢿⡿⣟⠂⠙⠿⢦⣤⡄⣁⣂⣿⣿⣿⣿⣿⣿⠤⠤⠤⠤⠙⣿⣿⡟⠒⠠⠤⠌⣿⣿⡿⠁
⠀⠀⠀⠀⣷⣶⡌⢿⣀⡀⠀⠀⠈⠳⢌⠑⠒⠤⣴⣶⣷⣿⣭⣏⣀⢀⡉⠍⢛⠰⢀⡠⠚⠛⡋⠢⢬⣅⣐⠂⠐⠛⣉⣀⡀⠠⠠⢤⢠⡀
⠀⠀⠀⠀⢻⡿⢟⡀⠙⠛⠻⣶⣶⡶⠶⠿⠒⠢⠿⣿⣿⣿⣿⡇⠀⠤⠀⠂⠁⣼⣛⠻⠄⣈⢉⣀⠄⡫⠜⡆⠀⠸⡉⢫⣀⣀⠀⠀⠤⠮⠦⡀
⠀⠀⠀⠀⠀⠙⠢⢀⠉⠐⢾⣿⣿⣿⣿⡇⠀⠂⠉⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠁⠒⢨⡗⣩⠀⠚⡇⠀⠈⠈⠾⠁⠀⠀⠀⢒⣀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡆⠀⠀⡇⠁⠀⢰⡇⠀⢀⠆⠈⠀⠀⠀⣀⡈⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⡇⠚⡠⠘⠀⠀⠈⠄⠢⠀⣀⡀⠉⠁⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠓⠈⠀⠀⠀⠀⠀⠀⠻⠀⠉⠁⠤⠀⠒⠊



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⢭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⢁⣤⣤⣶⣶⠦⡄⣼⣿⠋⢱⡆⣠⣾⡿⣀⠀⣰⣿⠏⢠⣿⡟
⠀⠀⠀⠀⠀⠀⢠⣿⡟⢼⣿⠟⣻⣿⠃⠀⢸⣿⠃⢀⣿⢰⣿⠏⢠⣿⢰⣿⠏⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣴⣿⠃⠀⠀⢿⡟⢀⣼⠃⣿⡿⠀⣾⠇⢸⣿⣠⡟⢹⣿⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠘⣿⣇⣠⣾⠟⠁⠀⠀⠀⠈⠛⠚⠁⠀⢿⣧⡼⠋⠀⠀⠉⢡⣾⡆⠙⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⢀⣾⣿⠀⢀⣀⠀⠀⣤⣾⣿⣧⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⣾⣿⡏⣠⣿⡟⢀⣾⡿⠫⣿⡟⣠⣾⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⢸⣿⣟⣴⣿⣿⠁⣼⣿⠁⠀⠈⠀⣼⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⢈⠻⠛⠁⣿⡇⠀⢿⣇⣠⡴⠂⢠⡿⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣠⠏⠀⠀⠀⠘⠓⠚⠁⠈⠉⠀⠀⠘⠃⠀⠹⣿⣧⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⢀⠄⢀⠄⠀⠀⠀⠀⣀⠠⠌⠒⠈⠁⠀⡀⠀⠀⠀⠀⡉⢠⠉⠈⠀⡄⠀⠀⠀⡇⢄
⢀⠔⣲⠦⣄⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠈⠠⡄⠀⠀⠉⠀⠀⠀⠀⠀⢁⡀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠈⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠕⣄⠀⠈⠁⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠄
⠒⡘⣢⣤⣔⣓⣂⣂⣄⡀⠀⠀⠀⡠⠜⡩⠃⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠐⠀⠀⠀⠀⡠⢒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢆⡠⠊⠀⢀⠴⢳⡀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡀⠌⠗⡠⡔⠁⠬⠒⠚⠃⠀⠀⠀⠐⠀⠤⠀⠘⠀⠀⠀⠀⢀⡌
⠀⠄⡙⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠄⠁⠀⠀⠙⠣⠔⠁⠀⠀⠀⠀⠈⠰⠀⠁⠀⠀⣀⠀⠤⠀⠀⠁
⠀⠪⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⠛⠛⠒⡄⠀⠀⠀⠂⡤⠤⠤⠩⠒⠑⠙⠒⠒⠓⠂
⠀⠀⠈⠉⠉⠀⠀⠀⡀⠀⢀⠠⣀⡀⠀⠀⠀⠀⢀⠋⠧⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⡀⠠⠂⠀⠀⠀⠠⣄⣀⣀⠀⠀⠀⠀⠠⠄
⠀⠠⢄⠀⠢⠴⠔⠐⠊⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⡀⠀⠀⠉⠉⠒⠐⠌⠤⣄
⠀⠀⠀⡀⠀⠀⢀⣀⠠⢤⢐⣒⣭⠥⢄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⠤⠐
⠀⠀⠠⢒⠈⠬⠃⠀⠈⠉⠁⠀⠀⠀⠠⠀⠀⡐⡩⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠄⠀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⠔⠘⠲⣈⡀⠀⢰⡰⠂⠙⢢
⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡂⠒⠀⠩⠲⢂⣋⣀⡐⠤⡄⠀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠘⣿⠀⠀⠀⡝⡂⠀⠘⢧⠀⠀⢠
⠀⠁⣀⠀⠒⡠⠜⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣀⣀⠀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠐⡀⠀⠀⢀⣀⡀⠀⠀⠁⠒⠈⡇⡂⠀⠠⠀⠉⠃⠁⠀⠀⣐
⢄⠀⠈⠂⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡽⠙⠓⠢⠀⠀⠀⠀⢀⡀⠀⠐⠀⢄⠀⠀⠈⠂⠊⠉⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠹⠂⠀⠀⢔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠠⠁⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⢀⠤
⠀⢀⣤⡀⠀⠡⡀⠀⠀⠀⠈⠩⢤⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⠖
⠂⠀⠀⠻⣄⡀⠐⠀⠀⠀⠰⣀⠀⠀⠈⠑⠂⠤⡄⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⡲⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⢀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⠈⢧⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⡄⠄
⠀⠀⠀⠀⠀⠂⡀⠀⣐⣄⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⢀⠌⢠⠎⠀⡀⠀⠀⠀⡀⠀⠈⠉⠈⠉⠐⠒⠂⠤⠤⠀⠀⠀⠱⠀⠀⠸⡀
⠀⠀⠀⠀⠠⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠁⠂⢤⠀⡰⠁⠀⡔⠀⠀⠀⠂⣀⣐⢢⠤⠐⠀⡤⠄⢀⣀⣀⠀⠀⠀⠀⢣⠀⠀⢡⠡
⠀⠒⠀⠀⠈⠀⡁⠀⠀⠀⣖⣠⣄⠀⠀⠀⠀⠁⠺⢦⢄⠀⠀⠈⠀⢀⠎⠀⠀⠀⠀⠠⡀⠀⠈⠉⠁⠑⠛⠑⠒⠒⠤⠀⠀⠀⠀⠀⡂⠀⠘⡄
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢩⠆⠀⠀⠉⠂⢄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⠀⠀⡸⠂
⠀⠀⠰⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠊⡴⠁⠀⢠⠆⣦⡀⠀⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⢄⡀⠐⠁⠀⠈⠙⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠀⠀⠠⡀
⣀⣀⣀⣀⣀⣀⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣒⣂⣀⣀⣀⣀⣀⣀⣁⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡠⠤⡀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⡞⡠⢄⠈⡆⠀⠀⢀⠞⠁⠀⡇⠀⠀⢀⣀⡐⢂⢱⠀⠀⠀⠾⣤⠒⡀⡤⠀⠀⠀⢠⠎⠀⢸
⠀⠀⠀⢸⠅⡀⢸⠇⡇⠀⠀⠘⠋⡇⠀⡇⠀⠀⠀⠉⢀⡼⡸⠀⠀⠀⠀⠰⢞⡻⡅⠀⠀⢠⢃⡞⢻⣸⡀
⠀⠀⠀⠸⣆⠳⠞⢠⠃⠀⠀⠀⠀⡇⢸⡇⠀⠀⢀⡴⠛⠒⠠⡄⠀⠀⡔⠲⣀⠇⡐⠀⠀⠿⠬⡭⡁⣴⠇
⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠚⠒⠒⠒⠒⠃⠀⠀⠈⠒⠢⠒⠁⠀⠀⠀⠀⠀⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠐⠀⡀⢀⠄⠀⠀⢀⠞⡁⣉⡂⠀⠀⢚⣋⡛⠋⠙⡇⠀⠀⡜⢁⠤⡍⢢⠀⠀⢠⡞⣋⣉⠲⡄
⠀⠀⠀⠀⣇⣰⡆⠄⠀⠀⠀⢾⠸⠄⠥⡄⠀⠀⠀⠀⢨⠃⡄⠀⠀⠀⣻⠌⠒⢃⡜⠀⠀⡟⢸⡃⣽⠀⢱
⠀⠀⠀⠀⣀⡁⢘⡆⡆⠀⠀⠘⠀⠀⠂⢹⠀⠀⠀⢀⠇⡀⠀⠀⠀⠀⡇⠰⡁⡇⢹⠀⠀⠙⣮⠿⢷⠀⡜
⠀⠀⠀⠈⢧⡁⢋⠐⠁⠀⠀⠐⢀⣁⣠⠎⠀⠀⠀⢸⣠⡇⠀⠀⠀⠀⠷⣄⠈⣀⠞⠀⠀⠘⢍⡑⢋⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠠⡒⠀⠀⠀⠀⠀⠀⠀⢀⠬⡀⠀⠀⠀⢀⢊⡠⠀⢳⠀⠀⠀⠀⠀⠀⠠⠀⡀⣰⠀⠀⠀⠀⠀⠀⠀⠿⡶⠏⡆
⠀⠀⠀⢀⡴⠢⠟⠫⣯⠶⣄⠀⠀⠀⠀⢀⡤⠧⣘⡆⠀⣠⠞⠉⠙⠛⠾⠻⣄⠀⠀⢠⠆⠰⢕⣺⣰⡾⢦⡄⠀⠀⠀⢠⠟⠚⠑⢲⡇
⠀⠀⢠⠋⠀⡴⠚⠛⢢⡀⠈⣧⠀⣠⠴⠋⠀⠀⡀⡇⠘⢧⣀⡴⠒⢦⠀⠀⢹⠀⠀⠸⠦⠤⢤⠄⠀⣠⠏⠀⠀⠀⣰⠃⢀⣼⠀⢸⡇
⠀⠀⡏⠀⣸⠀⠀⠀⠀⢷⠀⢸⡄⢻⣠⠴⡆⠈⠀⡇⠀⠀⠈⠀⢀⡞⠀⢀⡟⠀⠀⠀⢠⠖⠁⠀⠰⣇⠀⠀⠀⡰⠃⢠⣎⣸⠀⢸⡇
⠀⠸⡇⠀⢻⡀⠀⠀⠀⡞⠀⢸⡇⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⣠⠎⠀⣠⠞⠀⠀⠀⠀⠸⠤⠤⣄⠀⠀⢱⡀⡴⠁⠀⠀⠀⠀⠀⠀⠉⢹
⠀⠀⢷⡀⠀⠓⠶⠶⠚⠀⠀⡼⠁⠀⠀⠀⡇⠀⠀⡇⠀⠀⡎⠁⠀⠈⠉⠉⠉⢹⠀⢀⣀⣀⣠⠟⠀⠀⢀⡇⠉⠉⠉⠙⠛⣻⠀⠀⡏⠉
⠀⠀⠀⠳⢤⣀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⣇⣀⣀⡇⠀⠀⠧⠤⠤⠤⠤⠤⠤⠼⠀⣼⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⢿⣀⣠⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢀⡀⠀⡰⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⡄⠀⠀⠀⠀⠀⠀⢀⠊⣠⠚⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⢴
⠀⠀⠀⠀⡤⠆⣤⡣⣌⠤⢤⠀⠀⠀⠀⢀⣤⣬⣤⡸⢤⠀⠀⣾⠛⠛⠛⠚⠳⣍⠴⠛⡇⠀⠀⠰⠃⡔⠛⡄⠀⠀⠀⢀⡤⢆⣰⣷⣀⢧⡀
⠀⠀⠀⠀⡇⠀⣠⠭⠭⠤⠼⠀⠀⢀⡔⠉⠀⣀⡬⢤⡜⠀⠀⣯⣀⣀⣠⣀⠀⠀⠈⣠⡇⢀⠔⠛⠛⠛⠟⠳⡄⠀⢠⡏⠀⣠⠔⠲⣄⠀⣹⡄
⠀⠀⠀⠀⡇⠀⠿⢤⣀⠀⠀⠀⠀⡞⠀⠀⠺⠓⠶⢤⡀⠀⠀⠀⠀⢀⡴⠁⠀⣠⠞⠁⠀⣾⠀⠰⣍⣈⠆⠀⣿⠀⢸⠀⠀⢧⣀⣀⡟⠠⠟⡇
⠀⠀⠀⠀⠷⠶⢤⡀⠈⠱⣄⠀⢰⠇⠀⢀⡤⠦⣄⠀⠹⡄⠀⠀⢠⠞⠀⢀⡼⠁⠀⠀⠀⢘⡦⠀⠈⠁⠠⣔⡁⠀⠈⢧⣀⠀⠈⠁⠀⠀⢀⡇
⠀⠀⠀⢀⣀⠀⣀⡇⠀⠀⢹⠀⠸⡆⠀⢸⡀⠀⣸⠀⠀⡇⠀⢀⡞⠀⠀⣼⠁⠀⠀⠀⣰⠋⠀⣠⠖⠲⣄⠀⠙⡆⠀⠀⠈⢉⣹⠇⠀⠀⡼⠁
⠀⠀⠀⡜⠉⠉⠁⠀⠀⣠⠏⠀⠀⠳⣄⠀⠉⠉⠁⢀⡼⠁⠀⢸⣁⣀⣰⡇⠀⠀⠀⠀⢧⠀⠀⠳⣤⡤⠏⠀⢀⡇⠀⡏⠉⠉⠀⠀⣠⠞⠁
⠀⠀⠀⠉⠓⠒⠒⠒⠋⠁⠀⠀⠀⠀⠈⠓⠒⠒⠚⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣀⡤⠞⠀⠀⠙⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠖⠃⠀⠀⠇⠀⠀⠀⠀⠔⢐⠐⡀⡂⠀⢐⠀⠂⡂⠂⢐⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⡄⢼⢿⡧⡿⣿⢼⢾⣧⡧⣿⢿⢼⡇⢠⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁
⠀⠀⢧⠀⠁⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠂⠐⠀⠐⠀⢺⡇⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠂⠓⢀⠀⠀⢄⣀⠀⠁⠀⠀⠀⡀⢸⢽⡇⣿⣿⢸⢸⡇⡇⣿⢺⢸⡇⠰⠀⠀⣿⡇⠈⠃⠘⠀⠙⠀⢻⡇⠀⠀⠘⢿⣶⣿⠟⠁
⠀⠀⠀⠀⠼⠑⠦⠔⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⡇⠈⠁⠈⠀⠈⠀⢹⡇⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⢰⡖⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠉⠀⠂⠠⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⢰⡟⡩⠉⠉⠘⠩⠛⠉⠙⣷⠀⠀⠀⠙⢦⣄⠀⢀⣠⡴⠖⠊⠁
⠀⠀⠒⠚⠛⠛⠃⠀⠀⠁⠒⠐⢲⡆⠀⠀⠀⠘⠶⠔⡥⠤⠒⠁⣀⡼⠀⠀⠀⠀⠈⠃⠀⠀⠀⠤⣤⣤⣤⣼⠿⠀⠀⠀⠀⣠⣽⣿⣿⠉
⠀⠀⠀⠀⢠⣄⠀⠀⢀⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠸⣇⣀⡾⠛⠉⠀⠀⠀⠀⠀⢠⡶⠞⠒⠒⠒⠂⠒⢒⣶⣦⠀⠀⣠⣾⣿⠟⠉⢻⣷⣄
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠛⣦⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⢀⣀⣐⣿⠀⠈⢻⠟⠁⠀⠀⠀⢻⣿⠆
⠀⠀⠀⠀⠀⢸⣧⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠙⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢦⣠⡾⠋⠈⠳⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡀
⠀⠀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⠀⠀⢀⣤⠤⠤⠤⠙⠻⠠⠤⢤⣄⠀⠀⠀⠀⠰⣄⠀⠀⠉⠀⠀⣠⠌⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣼⣿⠂
⠀⢺⡃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣠⡞⠁⠀⠀⠀⠈⢦⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠁⠀⠀⠀⠁⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⢀⡾⠿⠛⠛⠋⠉⠀⠀⠀⣾⡋⠀⠀⣠⠾⣄⠀⠀⢙⡦⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⡴⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠞⠁⠀⠈⠳⣴⠋⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⢉⠁⢀⡀⠈⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⣀⣶⣦⣿⠁⢸⡇⠀⣿⣷⣰⡦⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⡌⠻⣿⠏⠛⠀⠘⠃⠘⠋⠸⢿⠁⠴⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣶⢶⡶⢶⡶⣶⣶⢶⣶⣶⡶⣶⢶⢶⣶⣶⢶⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠿⡿⠿⠿⢿⡿⡿⣿⣿⣿⣿⣿⣷⡄⠀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠘⣇⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠃⠀⠒⠀⠀⠀⠀⠒⠆⠀⠀⠀⠂⠦⠈⡇⠀⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠠⡶⠰⠶⠀⠀⣤⡆⣴⡆⠀⢠⡦⢰⡦⡇⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡇⠀⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⡟⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⠦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠇⠀⣶⢶⡆⢲⡆⠀⠀⢰⣶⡆⠐⢶⣤⡶⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⣿⠟⠃⣿⣧⣤⢀⣾⠿⣿⡀⢸⣿⠀⠁⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠨⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠂⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠂⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠻⠿⠃⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⣂⡄⡀⡔⠒⡀⠀⢀⠀⡤⠔⠒⠠⡀⠀⠀⢀⠀⡤⠔⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡈⠀⡨⣜⢰⢅⠃⡰⢁⠊⠅⡄⠂⠄⠑⠀⡰⢡⠊⠁⡄⠢⡀⠱⠀⠀⠀⠀⠀⠀⠀⢸⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠡⡀⡁⡘⢹⠁⢀⠃⡌⠀⠃⡈⣒⠽⠀⢠⠁⡀⠀⢆⡈⣒⢽⠀⠇⠀⠀⢀⠠⠤⢤⣃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠊⠀⠀⠆⢘⠀⡇⠀⠀⠀⠀⠀⠀⣸⠀⠇⠀⠀⠀⠀⢸⠀⡀⢀⢊⠔⠁⡑⣦⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠇⠀⠀⠀⠸⠀⠠⠀⠀⠀⠀⡘⠀⠿⠀⢠⠀⠀⠀⠀⡌⠀⠃⠁⢌⠀⠀⠇⠁⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⢀⠄⠄⠀⢣⠀⠑⠤⠤⠊⠀⡔⠀⢣⠀⠑⠤⠄⠊⠀⡌⠀⡀⠐⡀⠀⠈⢻⠧⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠤⠊⠀⠀⠀⠑⠠⠀⠠⠄⠊⠀⠀⠀⠑⠠⠀⠠⠔⠊⠀⠀⠑⠢⠄⠁⠩⠊⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡄⠀⠀⠀⠀⠀⢈⠀⢡⠀⠠⠪⢪⢢⠀⠀⠀⡠⡊⠥⢈⠑⢄⠀⠈⡀⡅⢀⠆⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢤⠀⠀⠈⡄⢸⠀⢀⢠⠌⡘⠀⠀⠌⠔⠀⠄⡂⠂⢘⠀⠀⢠⢠⡌⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠈⠀⠀⠀⠀⡈⠀⠈⠎⢀⠁⠀⡘⠘⠀⠀⠈⠢⠤⠊⠀⠀⢠⠌⡞⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠆⠁⠀⠌⠀⡌⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡐⡡⡆⠃⠈⢢⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠧⠊⢀⠌⠈⠀⠆⠁⠰⠀⠀⠀⠀⠀⠠⠀⢰⠀⡇⠀⠀⠀⢃⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡇⣗⠂⠠⠤⠤⢲⡆⠀⠀⢀⠌⠀⠀⠀⢰⠐⠄⠑⠄⣀⠀⢠⠖⠀⠠⣀⡣⠁⠀⠈⢌⠒⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠜⠀⠈⠁⠒⠂⠁⠑⠤⠐⠁⠀⠀⠀⠐⠊⠀⠀⠑⠒⠒⠂⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⠀⢀⡀⠀⣀⠀⣀⣀⡀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣾⣟⡋⠘⢸⢶⡏⢻⡎⣓⡄⠉⣹⠁⠀⣇⣘⠉⣿⡁⠄⣶⠄⢱⣜⡧⠤⣿
⠀⠀⠀⠀⠀⠀⠝⠛⣿⢠⢸⡆⣴⣏⣠⡿⣦⢰⡯⠀⠰⡇⣲⢐⠶⡃⡅⠛⡈⣸⡏⣓⣶⡋
⠀⠀⠀⠀⠀⠀⣾⣇⣿⢠⢌⠨⠉⠉⢹⡧⣬⢰⠛⡆⢀⣅⣹⠹⢼⣧⣭⣟⠻⡟⠁⢉⣠⡿
⠀⠀⠀⠀⠀⠀⠨⢭⣡⠚⡏⡆⠀⠐⠸⣧⣿⢨⠛⡅⢀⣻⠙⢁⠈⠁⣾⢘⠋⠑⡄⣿⣥⡄
⠀⠀⠀⠀⠀⠀⠒⡴⢁⠮⢡⡶⣦⠛⣼⠠⣤⠘⠤⠃⠀⣇⡕⠘⠰⢐⡃⣿⣔⡊⡀⢿⣼⢧
⠀⠀⠀⠀⠀⠀⠈⠉⢸⣹⢸⡷⣿⢰⣶⠈⣳⠰⢯⡁⠀⣮⡇⡘⣴⢸⡇⣿⢸⡅⠗⠊⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠛⠛⠘⠋⠓⠋⠙⠚⠁⠈⠓⠁⠀⠛⠁⠈⠙⠛⠘
⠀⠀⠀⠀⠀⠀⡖⡂⢖⡶⣶⡖⣢⡞⣶⣶⢐⢴⣲⠀⢠⠂⢲⢰⡆⠀⠀⠆⠴⠶⣦⢰⡂⢴
⠀⠀⠀⠀⠀⠀⣇⡃⠿⢹⡆⣴⣃⡴⠇⡈⣐⠒⣿⠀⠀⢱⠎⢀⣷⣖⠲⣷⣾⣛⣋⠂⣉⠘
⠀⠀⠀⠀⠀⠀⡄⣿⣀⢻⡆⢿⠟⣹⢋⢅⢡⡆⣻⠃⠰⣿⡦⣷⡄⣍⡇⣯⣩⡏⠉⠴⢣⢦
⠀⠀⠀⠀⠀⠀⠓⠋⠙⠟⠱⢐⠘⣿⣮⣜⠈⡙⣿⠀⠀⠩⣲⡮⢶⡟⣶⣼⣌⠀⡂⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠺⣿⣼⠞⡇⠚⢸⢁⠙⢿⡇⢙⠡⠀⠠⣠⠿⠸⢠⡞⣷⡛⠳⣷⠀⢣⡶⣧
⠀⠀⠀⠀⠀⠀⢲⡶⠾⠗⢿⣽⠈⠁⡀⢭⡄⢀⢈⠁⠰⢳⡆⡇⣾⣷⢳⡅⣰⣹⢔⠘⠧⣿
⠀⠀⠀⠀⠀⠀⠚⡇⢮⡹⠸⡽⠀⠀⠀⢤⡇⠄⠠⠀⠰⣯⠄⡟⢯⡼⠿⠰⠈⠁⠸⠟⠷⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠚⠙⣷⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠖⠛⣷⠂⡀⠛⠻⠁⠀⠘⠉⠉⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠰⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⡿⡃⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⣿⡟⠐⠃⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⢿⣿⣶⣿⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠚⢛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣖⡒⠶⠶⣤⣀⣾⢁⠀⡀⠀⠀⢉⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡒⠂⠨⣀⡢⢍⡻⣿⣆⠞⡴⢀⠀⠀⠔⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⠀⣀⡒⠭⣓⠯⣺⢎⡞⠐⢁⠄⡀⠐⡿⠀⣀⣤⣤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⠀⠀⠬⣕⡪⢝⠊⢳⠀⠀⠀⠈⣠⡾⣡⠞⡙⠀⠁⠐⠀⠋⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣤⣤⣙⢦⣤⣙⡒⠍⠀⠀⢠⡆⡔⠀⣠⠏⡴⠏⠰⢁⠄⡠⢀⠀⠀⠀⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⠁⠀⠂⠈⢢⡙⢷⣮⡙⠺⣄⡀⢀⢸⡏⡴⡴⢃⠞⠁⠀⠀⢈⠜⡡⢃⢎⡄⡀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⠢⣈⠢⠁⠀⠀⠉⡳⢤⡙⠦⣙⣏⣞⡵⠳⢂⣞⡰⠤⠤⣞⢼⣱⢿⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠐⠬⣑⠮⣑⠀⠀⠠⣑⣊⣑⣉⣓⣻⣿⣿⣖⠻⡛⠉⠀⠀⢤⡲⢝⠮⡓⢍⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣦⣽⣶⣹⢦⢟⢀⠀⠁⠉⠱⢫⠞⢩⠿⡽⢍⡓⢦⣀⢀⠢⡑⢌⠢⡑⢌⠢⠁⢈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⡵⡱⢣⢣⢎⠄⡄⡀⣴⠋⣰⢏⠜⣷⠒⠝⢦⣌⠳⣵⡘⠂⠁⠉⠀⠀⡐⣌⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⠰⢡⢣⠏⡎⡜⣰⡼⢡⢞⠁⠀⠀⠺⣏⠣⠀⠈⠳⣮⡙⠦⣷⣼⣤⣦⠽⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⢀⠁⠀⠈⠀⠀⢠⣟⣵⠟⣩⢖⡄⡠⠀⣐⢀⠀⣀⠒⠤⢉⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣮⣴⣠⣥⡷⠞⠉⣾⠅⢈⠔⣫⢞⠔⡱⣯⡢⣕⠢⣉⠀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠰⡷⠂⠀⢊⠵⣫⢎⢜⣵⢝⠮⡑⠀⠀⠀⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⣿⡥⢀⠁⢊⠔⠁⣾⠿⣧⠁⠀⠀⠀⣈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡟⠀⠀⠀⠀⠀⠘⢶⣥⣔⣡⡴⠞⠁⠀⠈⠛⠶⠾⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣖⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⣊⣁⠀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠒⠒⠒⠉⠀⣿⠀⠯⠿⠀⢠⣤⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⣉⡁⠀⠀⠹⠶⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀⡀⠀⠀⠀⠀⠀⠀⡿⢀⣴⣆⠰⣾⣿⡦⠀⠀⠰⣾⣿⡦⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣉⡁⠀⡄⣀⠀⠀⢶⡗
⠀⠀⠀⠀⠀⠀⢀⡀⣷⠀⣧⣇⠀⢀⡀⠀⠀⠀⡇⠈⠻⠋⠀⠈⠁⠀⢠⣤⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠶⠂⠀⢋⣁⡤⣅⠙⠃
⠀⠀⠀⠀⡴⠊⣁⠀⠸⡆⠙⣉⠀⣼⣿⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⢿⣿⠶⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠒⠉⠁⠀⠀⠀⠑⢄⠘⠷⢰
⠀⠀⠀⠀⢣⠀⡼⢣⠀⣧⠀⢿⡗⠈⠛⠃⢠⣷⡇⠀⠀⠀⠀⢰⣶⠀⠈⠉⠀⢰⣶⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠐⠿⠆⠀⠀⣠⣤⠀⠀⠀⣘
⠀⢀⡴⠊⠉⢣⠀⣤⡀⢹⡀⠀⠀⠀⠀⠀⠘⢻⡇⠀⠀⠀⠘⢿⡿⠗⠀⠀⠘⢿⡿⠗⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠈⣉⠤⠖⠊⠁
⠀⠘⣇⠸⡧⢸⡆⠻⠏⠸⡇⠀⠀⠀⠀⣶⣄⢠⡇⠀⠀⠀⠀⠀⠀⢀⣰⣶⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡈⠂⠒⠛⠉
⠀⠀⠘⡄⢁⣄⣳⠀⠀⠀⢿⠀⠀⠀⠘⠿⣿⢹⡇⠀⠀⠀⠀⠀⠀⠈⢹⣿⠉⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⠔⠡⠾⡀⠀⢀⡤⠉⠑⢶
⠀⠀⠀⠘⡜⠿⠛⣇⠀⠀⠸⡇⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⢀⣴⣧⣄⠀⠀⢠⣴⣶⣄⢀⣴⣦⢨⡇⠀⠀⢀⠴⠊⣵⡆⠀⢚⣡⠀⠈⡠⠀⢠⡜⣇
⠀⠀⠀⠀⠘⡄⠀⠹⡄⠀⠀⢻⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢹⡿⠁⠀⠀⠈⠹⡿⠁⠀⠹⠏⢸⡇⠀⠀⢦⠀⠀⠀⠠⠤⢌⣉⡠⠊⠀⠀⠛⠃⢹⡄
⠀⠀⠀⠀⠀⠰⡄⠀⢳⠀⠀⠘⡇⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡶⢸⡇⠀⠀⠈⡆⠀⠀⣀⠀⠀⠀⢧⠀⠀⢰⡷⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢳⠀⠈⡇⠀⠀⢷⠀⠀⠀⠈⠻⣅⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠹⠗⣸⠇⠀⠀⠀⢱⠀⠀⢿⡷⠀⠀⠘⠀⢀⣤⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢇⠀⢹⡄⠀⢸⡀⠀⠀⠀⠀⠈⠉⠩⠉⢩⠉⠭⠉⣟⣻⡟⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠈⠋⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢳⡀⠀⣇⠀⠀⠀⠀⣀⣀⣤⢤⣴⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⢄⣀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⣇⠀⠈⠳⠲⣚⣩⣭⣷⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠘⠳⣶⣞⡿⠟⠛⠉⠁⠀⢀⣠⣴⠆⠉⠹⠛⠻⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠎⠁⠀⠀⠀⠀⠀⠀⠀⠰⠛⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⣰⠂⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣠⠀⣀⠀⠀⠀⠀⣀⣠⡿⠋⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⣧⣴⡷⢤⣤⡶⠾⠋⠁⢀⣠⣼⣿⠏⠀⠀⣀⣀⣀⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡿⠯⢶⣶⣾⣶⣿⠿⠛⠉⠁⠀⠀⠈⠁⠀⠉⠀⠀⠘⠉⠛⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⡾⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⡉⠰⠂⠀⠀⢀⠀⠀⠱⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠧⠶⢤⣰⣴⠶⠶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠶⠀⠘⠛⠛⠀⠠⠶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢛⣡⣄⠀⠀⠀⠀⠀⢀⣠⣜⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠿⢿⠁⠀⢸⣿⣿⡆⠀⢹⠿⢧⠀⠀⣀⣄⠀⠀⠶⠶⠄⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡟⠂⠀⣶⣶⡎⠀⠘⣿⣶⣀⣲⠶⠎⠁⠀⠀⠀⠀⠀⠀⠈⠱⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣥⣦⠄⠛⣛⣃⡀⠀⢭⣯⣭⠉⢰⣿⡟⠀⢠⣶⣶⡄⠀⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠶⠒⣘⠛⠇⠀⠀⠿⣿⣇⣀⠀⠻⣟⣤⣦⠌⠁⣀⣈⡛⠛⢧⣶⣤⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠁⠀⣴⣶⣶⠀⠈⣻⣿⠄⠶⢾⢇⣀⣀⡿⣿⣏⣀⠀⠿⣿⣌⣉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡖⠉⠉⣿⣶⡆⠈⠙⣿⣷⣠⡾⢿⣇⡀⠰⣿⣿⡇⠀⢈⣿⠗⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣷⠖⠀⢻⣿⣧⣠⣴⠏⠁⠾⠟⣩⡽⡅⠀⣾⣿⣧⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⢀⣼⡯⠀⠉⠀⠀⠀⠀⣴⣶⣦⠀⠐⠻⢟⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠆⣀⣀⠀⢠⣴⣦⡀⢀⣙⣋⠀⠠⣤⣬⡍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢺⣿⡇⠀⠘⠿⠋⠁⢸⣿⣿⠇⠀⠸⠟⠃





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣍⠄⢰⠀⠀⠀⠀⢠⠒⣠⡆⣅⣀⠀⣄⡩⣛⡸
⠀⠀⠀⢀⡤⠶⠴⢾⢎⡎⠀⢀⡀⣠⠺⣁⣀⣀⠉⠛⢯⠀⠀⠀⠀⡎⠀⡴⠮⠭⢠⡟⠀⡤⠀⡇
⠀⠀⡰⠋⢠⠔⠒⠪⡉⢹⡤⠊⠑⢺⠀⠈⠀⣸⠃⠀⡼⠀⠀⠀⠰⢥⣀⠑⠢⣀⠞⢀⡎⡑⠀⡇
⠀⢰⠃⢠⠇⠀⠀⠀⡷⡏⢀⡄⠐⢸⣠⠤⠊⠁⣠⠞⠡⢀⠴⣀⣀⠀⢈⠆⠀⢹⡀⠀⠀⠁⠀⠉⢹
⠀⢸⠀⠘⢆⡀⣀⡴⠁⢹⠃⣧⠀⢸⡀⠀⠀⠛⢳⠔⠳⠚⠿⠻⡀⠉⠁⠀⢀⡜⠉⠉⠉⡇⠀⡏
⠀⠈⠣⣀⠀⠉⠁⠀⣠⠏⠀⢸⠀⠘⡏⠓⠒⠤⣾⠀⠸⣁⠕⠀⡟⠛⠒⡋⠁⣀⠀⠀⠀⠓⠒⠃
⠀⠀⠀⠈⠉⠓⠚⠉⠁⠀⢀⠸⢤⠤⠇⠀⠀⠀⡨⠓⠀⠀⠀⠺⢄⢀⣠⣤⣃⣹⣀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⡖⠤⠤⣡⡖⡍⢃⠀⠀⠀⢸⠀⠠⡏⠉⣱⠀⡼⠉⢀⡠⠬⣥⣏⣄⡿⠦⠓⠢⡀
⠀⠀⠀⠀⠀⠀⠸⣅⣀⣀⡀⠈⠒⠁⡧⢄⡌⡼⣒⡇⠈⠉⠁⡜⠀⠀⠈⠉⠲⢤⠇⠀⡖⠉⢢⢀⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡤⠚⠧⠄⣈⠉⠚⡭⠋⠉⠉⡇⠀⡜⠉⢱⠀⢸⣇⠀⠓⠔⠋⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⡴⠁⠀⠀⡔⠊⠁⢠⡊⠀⠀⠀⠀⢧⡀⠙⠒⠊⠀⡸⠉⠲⠤⠴⢶⠀⠀⡞
⠀⠀⠀⠀⠀⢀⡎⠀⢠⠏⠀⠀⠀⠀⠓⠒⡄⠀⠘⡆⠀⠀⠀⠀⠙⠒⠒⠒⠋⠀⠀⠀⡤⠚⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠈⠓⠒⠚⠀⠀⠀⢰⠓⠒⠒⠁⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⠤⠒⠉
⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⢟⣻⣙⣯⣍⠛⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⠋⠃⠀⠘⠛⣛⠛⠋⠀⠀⠀⣠⣄⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣿⣷⠄⣀⣴⣾⠿⠿⠿⠿⠿⣷⣦⣈⠻⣿⣆⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣾⣿⣯⣿⣿⢁⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣊⠁⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠚⣿⣿⣿⣻⣿⢯⡍⢉⣉⡉⠛⠒⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⢻⣿⡦⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡄⠀⠀⣿⣿⣷⣺⣇⣻⡆⣏⠀⣹⢰⠟⠳⡄⣠⣌⡉⠓⠶⢤⣀⡀⠀⣿⣯⣺⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣸⡿⠛⣿⣿⣷⣤⡤⣌⣉⣁⡘⠦⠾⢡⡏⣀⡗⠀⢠⠆⠈⠙⠳⢿⣟⣺⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣋⡀⠙⣿⠁⢠⣿⣟⡺⠿⢠⣿⡏⠉⠉⠛⠓⠦⣌⣉⡀⠠⢯⡀⢠⡟⢀⡆⢈⡙⠧⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⡴⠋⠀⢸⣿⡒⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⢤⣘⠲⠟⣴⠋⠙⠆⣈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⣾⣿⢭⣀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⣌⡓⠃⡾⣷⠒⠂⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣹⣿⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠧⣿⡟⢳⣤⣙⠀⠀⣰⠏⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⢾⣿⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠋⠁⠀⢠⣿⠃⠙⢳⡞⠁⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⢣⣅⠀⣠⣿⠃⠀⢀⣾⣤⣤⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣤⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⣲⣿⠏⣴⣿⠁⠀⠀⢸⡟⣫⡃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⠉⠀⠀⠀⠀⠀⢀⣴⣿⡿⠍⠈⠻⢃⣾⡿⠁⠀⠀⠀⠀⣿⠋⠉⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣯⡀⠀⠰⣿⣿⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠖⠒⠚⠉⠉⠉⠉⠁⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠔⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡀⠀⠀⠀⢧
⠀⠀⠀⣠⢴⠊⠉⠀⠀⠀⠀⠀⠀⠀⡴⢲⠀⠀⡴⡿⠛⣍⢳⠀⠛⠉⠑⡜⡆⠀⠀⠘⡆
⠀⠀⣰⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⢞⣴⡾⡄⠀⠓⠃⠀⣸⡞⠀⠰⣦⡔⢡⠇⠀⠀⠀⢱
⠀⢀⡇⡜⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⣇⠀⢀⡤⠔⠵⢧⡄⢠⠤⠜⢨⠇⠀⠀⠀⠀⢇
⠀⢸⣰⠀⡈⠈⡆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠓⠛⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⢏⠈⢠⠣⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⢸⡀⠇⠀⡼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢇⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠘⡄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢳⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠈⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣵⡆⡏⡇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠸⡄⠀⠀⢸⡀⠀⠀⡖⢇⢀⣀⠀⠀⠀⠀⠀⠓⠽⢻⢳⠀⣗⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢇⠀⣤⠛⡈⣥⠼⠀⡞⠒⠢⠤⡄⠀⠘⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠘⡄⠘⠋⢣⣸⠄⠀⠉⠓⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠒⠒⠊⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠱⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⠤⠤⠔⠒⠒⠊⠉⠉⠀⠀⣀⣀⠠⠤⠤⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⡧⠒⠚⠉⠉⠉⠀⣀⣀⣀⡀⠤⠤⠒⠒⠒⣈⣉⠁⡉⠤⠤⢄⣒⣒⡶⠭⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡜⠀⠒⠀⠈⠩⠭⠤⠤⠤⠄⠒⣒⣀⣨⠭⠤⠔⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠧⠤⠤⠤⠶⠖⠒⠒⠊⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠓⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⡶⣠⣶⣿⣿⣟⠙⠇⠿⠼⢭⣿⣿⣷⣦⡰⣶⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠷⠝⠿⣿⣿⣏⣛⣇⣶⣰⣤⣼⣿⡿⠛⠴⠿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠶⣦⢤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣤⠤⠶⣾⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⡀⣿⢸⣿⡇⢹⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⣿⣸⣿⡿⢟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⢻⠿⣷⣾⢿⣿⣿⣿⣿⣶⡶⠿⣿⠟⠛⣩⣵⣾⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣸⣀⣿⣿⢸⣿⣿⣿⣿⣿⠀⠀⣿⡀⢸⣿⣿⣿⠘⡿⢛⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⣿⢻⣿⣿⢻⣿⣿⣿⣿⣿⠟⠛⣿⠋⠉⣇⢸⣉⣡⣿⠀⢡⣿⣿⣷⡄⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣧⣿⣸⣿⣿⣸⣿⣿⣿⣿⣿⣀⣀⣿⣀⣤⣿⢸⣭⣴⣾⣇⣸⣿⣿⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⡏⣿⣿⢹⣿⣿⣿⣿⣿⡏⠉⢹⠁⠀⣿⢛⣋⣿⣿⣿⣼⡇⠐⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣷⣾⣧⣿⣿⣼⣿⣿⣿⣿⣿⣥⣤⡼⠦⣶⠸⣿⣿⣿⣿⣿⣷⣚⡀⣵⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⣿⢸⣿⡏⢹⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⣿⣠⢹⣿⣿⣿⣿⣿⣿⣄⣼⣾⣿⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠛⡾⢿⣷⣾⣿⣿⣿⣿⣿⣴⠶⢿⡾⠛⢻⡏⣧⢻⡛⠋⠙⢿⠃⠻⣿⣿⡿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣀⡇⣸⣿⡇⣿⣿⣿⣿⠿⢇⣀⣨⣥⣤⣤⣥⡴⠀⠓⠀⠲⠟⣷⡀⠪⣍⣡⣧⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⢛⠉⠁⠀⢙⣿⣿⣿⣿⡟⣟⢻⣿⣶⣶⣥⠲⣿⣿⣿⣦⣹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣤⡴⣠⣾⣿⣿⣇⣈⡁⣁⣉⠉⣻⣿⡿⠋⠜⠛⣻⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠧⠙⠻⠿⣿⣧⣬⣷⣿⣞⣻⣿⣍⣀⡤⠖⢿⠙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⡶⢤⡤⣤⣼⡿⣿⣿⣿⣿⣿⡏⠉⢹⡇⠀⣸⡧⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠘⣇⢹⣿⡇⣿⣿⣿⣿⣿⣧⡤⠼⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄⠀⠀⠀⠀⠀⠀⡠⠤⠤⠠⢄⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⡤⣤
⠀⠀⠀⠀⠀⢈⡭⢾⣋⣻⠀⠀⠀⠀⠀⠘⠴⠚⠒⢦⡀⢐⠀⠀⠀⠀⠀⠣⠒⠓⢲⡀⠱⠀⠀⠀⠀⠀⠀⢀⣞⣿⡍⢹⠀⠀⠀⠀⠀⠀⡷⢾⠗⠛⠓⠛
⠀⠀⠀⠀⠀⠀⠀⢸⠶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣒⣺⠀⠀⠀⠀⠀⠀⢤⢤⠺⣁⠜⠀⠀⠀⠀⠀⢠⠯⡼⠁⠥⢼⠀⠀⠀⠀⠀⢠⣟⣟⢒⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋⠀⠀⠀⠀⠀⠀⠀⣠⣞⣒⡶⠁⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⠑⡄⠀⠀⠀⢰⠯⣿⢤⣴⢯⢽⠦⠀⠀⠀⠀⠈⠁⠉⠙⢳⡀⢘
⠀⠀⠀⠀⠀⠀⠀⢰⠠⡆⠀⠀⠀⠀⠀⢠⢮⢵⠶⢧⠤⢤⠄⠀⠀⠀⢀⠦⣤⡤⡴⠏⢀⠇⠀⠀⠀⠘⠛⠛⠛⠛⡟⢺⠚⠀⠀⠀⠀⢠⢤⣀⣠⡾⠁⡔
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠚⠚⠒⠒⠒⠒⠚⠀⠀⠀⠀⠈⠓⠃⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠀⠀⠀⠀⠀⠘⠚⠒⠓⠛⠉


⠀⠀⠀⠀⠀⠀⣀⡤⡤⡖⠀⠀⠀⠀⠀⢰⢶⡖⡖⢲⢲⠲⠀⠀⠀⠀⠀⢀⠤⡒⣒⠦⣄⠀⠀⠀⠀⠀⢀⢄⠒⠒⠂⢄⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⢤⡀
⠀⠀⠀⠀⣰⠫⣤⠞⠛⠉⠀⠀⠀⠀⠀⠘⠚⠓⠓⣻⣾⡾⠀⠀⠀⠀⢰⠅⣼⠉⠉⡯⠨⡅⠀⠀⠀⢠⠍⢀⡟⠙⢣⠀⢓⠀⠀⠀⠀⢰⠁⡼⠋⢻⡁⠱
⠀⠀⠀⢠⣇⣷⢟⣏⡉⠐⢄⠀⠀⠀⠀⠀⠀⠀⣰⠯⡴⠁⠀⠀⠀⠀⠀⢣⡨⢷⣞⣭⣞⠀⠀⠀⠀⠸⠀⠸⣆⣀⢼⡭⢸⠀⠀⠀⠀⡇⢀⡇⠀⠀⡧⢩⡇
⠀⠀⠀⢸⡻⢿⠁⠀⢹⣂⢈⠀⠀⠀⠀⠀⠀⣰⠿⣻⠁⠀⠀⠀⠀⠀⠰⠫⣶⠋⠙⣦⠈⢢⠀⠀⠀⠀⠉⠂⠤⠐⡾⠋⡹⠀⠀⠀⠀⡧⢐⡇⠀⢠⡇⢸⠇
⠀⠀⠀⠀⢃⡘⠣⡤⠾⠁⡜⠀⠀⠀⠀⠀⢰⣋⣹⠁⠀⠀⠀⠀⠀⠀⠨⣀⠹⢤⢴⠟⢁⡆⠀⠀⠀⠀⢠⢠⠶⠋⢁⠘⠁⠀⠀⠀⠀⠹⡀⠹⠦⠞⢁⠜
⠀⠀⠀⠀⠀⠈⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠒⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠊⠁⠀⠀⠀⠀⠀⠘⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠊⠈



⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⢀⠀⢤⠀⠆⠀⠐⠇⠓⠙⠈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠺⠇⠈⠧⠘⠀⠘⠂⠁⠀⠛⠈
⠀⠀⠀⠀⠀⠃⠈⠀⠍⠁⠀⠀⠁⠉⠈⠰⠋⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⠈⣠⠂⢀⠀⠀⠀⠰⠁⠶
⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢑⣄⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⣄⡀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠑⢀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠁⠀⠀⡀⠡⡑⠬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢡⣀⠰⢸⣷⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⢠⠈⠂⢄⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠄⠀⠀⢡⠀⠌⢰⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⢸⣿⢸⢸⣿⠠⠀⠀⠀⣶
⠀⢀⠀⠀⢠⠀⠙⠓⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠦⠒⢁⡀⠀⠀⠐⠘⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⢸⣿⢸⢸⣿⠀⣿⡏⠠⣿
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⢠⡤⠠⠀⠑⠄⡰⠃⠠⡀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠀⠀⢠⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⡉⠉⢉⠉⠉⣭⠉⠉⡏⠉⠛⠀⠃
⠀⢀⡔⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⣠⠊⢀⡴⢂⠐⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠼⠎⠁⠔⠀⡰⢆⡰⠆⠀⠀⠀⠀⠡⢄
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠰⠋⢒⣈⠀⠨⠆⠀⠀⠄⠒⢀⣉⣠⡤⠴⠴⠂⠀⠈⠁⠀⠀⠀⠀⡥⡀⠨⡤⠈⠚⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠾⡰⠁⠀⠠⠀⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠘⣆⡄⠀⢸⠖⠀⠀⠁⠀⠀⢠⠂⡀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⡀⠀⠐⡄⠀⠀⠁⠁⠀⠀⠀⠉⠐⣦⣠⣄⡀
⠀⠀⢰⠉⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⣨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢢⠤⢤⡀⠀⠀⠘⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⡟
⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⢀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡆⠀⠢⠙⣻⠷
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠛⠲⢞⡷⠾⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡸⠂⠓⠐⣠⣤⣼⣤
⠀⠀⠀⠀⠀⠀⡀⠢⣀⠀⠐⠒⠒⠒⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠐⢮⢁⡥⠀⠀⠀⠀⠀⠀⠘⢅⣀⣩⠼⠁⠛⠛⠛
⠀⠀⠀⡌⠀⠤⠴⠀⠉⠀⠀⢸⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠁⡀⠀⠉⠶⠀⠀⠀⠀⠀⠀⢠⢃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢣⠀⠀⠀⢀⠠⣰⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠘⠣⠀⠀⠧⠐⢾⡄⠦⠡⢹⠠⠄⢟⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡆⠈⠀⡇⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡅⠒⣨⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠠⣃⠀⠀⠀⠀⠀⠀⠇⡄⠀⠀⠁⠀⣠⡨⢦⡀⠀⢀⡔⠔⠦⡀⠀⠀⠀⡠⠊⡴⠋⠡⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠒⠢⡀⠀⠀⠀⠀⢀⠛⠃⠀⠀⠄⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣆⣇⢀⣿⠀⠀⣿⠀⠀⠑⠀⣮⣤⣤⣴⡟⠀⢀⠈⡠⠊⠀⠰⠀⡆⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⠂⠀⡀⠴⠆⠀⠂⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠔⢡⠞⠀⠀⠀⠀⡆⠇⠀⠀⠀⠀⠀⠈⠢⡀⠀⢠⠚⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠀⠀⠀⠀⢃⢸⠀⠀⠀⠀⠀⠀⢰⣯⣭⣽⠃⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢀⡀⠀⢀⣀⡀⠀⡀⠀⣤⠀⠀⠀⠀⠀⠀⠛⡼⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠃⠈⠁⠈⠛⠂⠀⠀⠘⠂⠈⠀⠁⠀⠉⣠⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠉⠀⡤⠤⠃⠹⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⢢⣱⡀⠀⠀⠀⠐⠈⠀⠀⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⣠⠄⠞⠉⠀⠀⠀⠂⠀⠐⠀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠔⠊⠉⠁⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⠈⠱⠀⠀⠀⠀⠄⠒⡀⠀⠀⠀⠀⠀⠀⡂⠀⡀⠀⠀⢀⡔⠊⠀⠃
⠀⠀⠀⠀⠰⠄⠀⠀⠤⠄⠀⠄⠀⠀⠀⠀⠐⠏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠁⠀⠀⠀⣀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠁⢀⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠂⠤⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⠀⠀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠈⡇⠀⠀⢸⠀⠀⢸⠀⠀⣰⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠘⡆⠀⡄⢡⠀⢀⠈⢰⠀⡆⠰⠀⠇⠀⠀⢠⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠘⢆⠀⢦⠐⠀⠱⠈⠀⠸⠀⠸⠀⠁⠇⣠⣤⠚⠠⠃⢀⠀⣠⠎
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠢⡀⠑⣄⠂⠀⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⡟⢠⡇⠁⡰⢃⠜⠁⡀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠑⠦⣀⠈⠢⡈⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⣀⣿⡷⠛⠀⠈⠀⠀⡀⠊⡠⢚⡁⠀⠀⠀⡀
⠀⠀⠀⠀⣀⠀⠀⠀⢌⡑⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡇⠀⠀⠀⠀⢰⠏⠀⡠⢸⡟⠠⠖⠋
⠀⠀⠀⠀⠀⠉⠓⠢⠄⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣇⠀⠀⠀⣠⠟⠀⠉⠀⢸⡇⡠⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡞⢹⣿⢻⠟⢦⣠⠼⠋⠀⠀⠀⠂⢸⡇⠀⠂⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣤⠀⠀⢠⣶⣤⠞⣹⣿⠀⣿⠘⠻⠟⢰⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠒⠒⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⣿⡄⠀⣾⣿⠈⡇⠸⢿⠴⠋⠀⠀⠀⢸⣧⠀⣀⠀⠀⠀⠀⠀⢸⠇⣀⣈⠁
⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⣿⡇⢰⣿⣷⣼⢿⣿⣰⡇⠀⠀⠀⠀⢀⣠⡀⢸⣿⣼⣿⠃⠀⠀⠀⡆⢠⡀⠠⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠈⢿⣧⡞⠘⣿⣇⠀⢬⡉⠀⠀⠀⠀⠀⣾⡟⠷⣸⣿⣿⡃⠀⠀⠀⢸⡇⠘⠃⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣿⣿⠀⢸⣷⠀⣶⡄⢰⡀⣿⡇⢀⡟⣿⡏⢿⡄⠀⠀⡾⠠⢄⣀⠁⠒⠲⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⢠⡏⢸⣿⡆⢸⣿⠀⢿⣇⢸⡇⣼⠛⠊⠀⠈⠋⠀⠻⣆⣼⢃⠀⠀⠌⣉
⠀⠀⠀⠀⠀⠀⣀⡤⠒⡡⠖⠁⡾⢀⠈⣿⡇⠸⣿⡇⣼⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠈⠙⠲⠄
⠀⠀⠀⠀⠀⠊⠁⠀⠈⢀⡤⢰⡇⠀⢠⣿⠁⠀⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡑⢦⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⢸⡇⠆⣼⠃⠀⢠⠈⠛⠁⠀⠀⠀⠀⢀⠀⠀⢠⠀⢀⠱⡄⠘⢦⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠛⠚⠁⡰⢀⠇⠀⠀⡆⠀⠀⡆⠀⠀⡆⠀⡄⢧⠀⢣⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣰⠁⠀⠀⡜⠀⠁⢸⠀⠃⠀⡇⠁⠀⢱⡀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢰⠇⠀⠀⢸⠀⠀⠀⢷⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠐⠂⠀⠀⠀⠒⠀⠄⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⡀⠂⣰⠌⢡⣆⠐⢄⠀⠀⠀⠀⠀⠀⡀⠀⢀⡠
⠀⢠⠓⠀⠉⠀⠀⠈⠉⠀⠒⠀⠀⠚⠂⠀⠀⠒⠀⠀⠀⠀⠓⠀⠀⠀⠀⡀⠰⠐⠄⠁⠈⠰⠂⠆⢀⠀⠀⡔⠀⠉⠀⠀⠠⢀⢀⠀⠀⠁
⠀⣹⠀⠐⠐⠊⠃⠑⠂⠂⠀⡃⠀⠀⣛⡋⠁⣙⣛⠃⢍⣛⡛⠀⠀⠀⠀⠀⠀⠐⠀⠂⠐⠀⠂⠐⠀⠀⠀⠁⠀⢸⠀⠀⠁⠀⠀⢁⠅
⠀⢼⠀⠀⠀⡀⠂⢄⠀⠀⠀⡅⠀⠀⠴⠃⠂⠠⠇⠀⠂⠞⠀⠀⠀⠀⠀⠀⠾⠷⠀⠾⠷⠀⠾⠗⠀⠀⠀⠈⡀⠀⠀⢘⠀⠀⠁⠁⠠⠁
⠀⢺⠀⠀⠀⡇⠀⠘⠀⠀⠀⠆⠀⠀⠚⠓⠂⠚⠛⠃⠒⠛⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠂⠐⢀⠀⠀⠀⠰⠤⠁⠚⠋⠑⠂⠀⠂
⠀⠘⠀⠀⠀⠃⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣠⡴⣮⣥⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠄⢒⢂⠠⡅⢄⠀⠀⠀⠀⠀⢔⢥⡄⠉⠛⠉⢠⡅⢀
⠀⠀⠀⠀⠀⠀⠀⢛⠀⠈⣱⠀⠀⣀⣊⣅⣥⣋⣉⣣⣤⣋⣢⡀⠀⠀⠀⡐⠴⠀⠁⠀⠀⠀⠀⠆⠂⠀⠀⠀⠌⣌⠀⠀⠀⣦⠀⠀⢀⣂⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡼⠃⠀⠀⠉⢆⡈⢇⠀⡇⢀⠏⠀⠌⠁⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⢈⢰⠀⠀⠀⠀⡀⠀⠀⠀⠛⣴⠛⠀⠀⠡
⠀⠀⠀⠀⢀⠑⠔⠁⢸⠃⠀⠀⠀⠀⠀⠙⢮⣂⣁⡊⠊⠁⠀⠀⠀⠀⠀⠃⢢⠡⠀⠀⠀⡀⠌⡴⡀⠀⠀⠀⠐⡸⡁⠀⡀⠀⢀⠀⡙⡢⠁
⠀⠀⠀⢀⣼⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠨⠀⠐⠄⠀⡐⠀⠀⠀⠀⠀⠈⠊⠈⠁⠂⠙⠈⠊
⠀⠀⠀⠀⠁⠉⠉⠉⠁
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠑⡀⠀⠀⠀⠀⠀⠀⡀⠄⠠⢄⠀⠀⠤⠠⠄
⠀⠀⠀⠀⡐⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠱⠀⠀⠀⠀⠐
⠀⢀⠔⠈⠀⠀⠈⠐⠄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⢠⠀⠀⢀⣄⠀⠀⢀⠂⠀⠀⠀⠠⡀⠀⠀⡴⢦⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠁⠀⠝⠈⢅⠀⠁⠀⠀⠀⠀⠀⠀⠉⡡⠁⠈⢌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡠⠊
⠀⠀⠐⠒⠒⠒⠂⠓⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣀⠄⠐⠀⠂⠠⢀⠀⠀⠀⡀⢂⣀⠀⠀⠄⠠⠀⠀⢀⠀⠀⠀⠀⠈⣥⠉⠩⡍⠂
⠀⢀⠐⠂⠀⠀⠀⠠⠀⠈⠒⡀⠀⠙⢣⣇⠤⢄⠠⠤⠠⠤⢤⠂⠀⠀⠀⠀⣀⠘⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠
⠀⠠⠀⠀⠀⠐⠈⠂⠀⠈⠀⠴⠀⠀⢸⣧⠸⢼⠈⠧⠀⠹⢼⡀⠀⠀⠀⠀⠉⠀⠈⠁⢂⠔⡐⠄⠀⠀⠀⠀⠠⠰⠂⡔⠒⠒⠓⡶⠀⡄
⠀⢊⠀⠄⠀⠂⠤⠀⢀⠀⠀⡑⠀⢀⢘⡏⠁⠉⠉⠉⠈⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⢢⡅⠀⠁⢈⡢⠀⠀⠀⠀⠠⠤⠓⠐⠀⡌⠣⠠
⠀⠀⠒⣄⠀⠄⠒⠠⠑⢡⠒⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢠⡀⡨⠑⠁
⠀⠀⠀⠈⠐⠐⠀⠂⠂⠁⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⡀⠀⢀⠀⠀⠀⠠⡤⠤⠤⠤⠤⠤⠤
⠀⠈⡆⠀⠀⠀⠐⠄⠀⢸⠤⠭⠀⠀⠀⠠⠀⠠⡀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣙⢦⡀⠐⠀⠰
⠀⠀⠄⡄⣤⢠⠄⠀⠀⠀⠀⢨⠀⠀⢂⠠⠉⠁⠂⠀⠀⠀⠥⠭⠀⠀⠀⠀⠀⠀⠀⠀⡜⢨⠇⠙⢦⡀⠙
⠀⠀⣭⣯⣿⣹⠋⡄⠀⠠⡤⢤⠀⠀⠀⠀⣷⡿⠀⠀⢠⡀⠈⠄⢀⡄⠀⠌⠀⣠⠀⣜⣱⠋⠀⠀⠀⠙⢦⡀
⠀⠨⠐⠓⠛⠊⠒⠡⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢌⠂⠁⠀⠀⠈⠁⣀⠀⠀⠁⠀⠛⢉⠔⢉⠈⢈⠓⢄⠘
⠀⣤⣒⣒⣒⣒⣲⠄⠀⡠⠂⠙⣿⣹⣦⡀⠀⠀⠀⢮⠴⡀⠀⠀⠀⡐⠉⠀⠀⠀⠀⠀⠌⠠⠁⠀⠀⠡⠈⣆
⠀⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠿⠾⠷⠇⠀⠀⠜⠒⠳⠃⠀⠀⡐⠀⠀⠈⠄⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠤
⠀⡇⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠔⠀⠀⠰⠊⠀⠀⠐⠆⠐⠷⣶⣶⣶⠾⠀⠀⠀⢀⡄⠠⢠⠄⠀⡀
⠀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠈⢀⠀⠁⡀⠀⢀⠀⡀⢠⠀⣀⠀⠀⠀⠀⠠⣤⡤⠄⠤⠤⡀⠀⠀⠀⠀⣀⡀
⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣄⡇⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡐⠈⠉⣿⣽⣳⡀
⠀⢼⠉⢍⡩⣉⢍⢉⡆⠀⠀⠀⢀⢐⠀⠀⣠⠂⡡⠀⠄⠁⢆⠀⠀⠀⢲⣵⣤⣆⣶⡄⠀⢰⠀⠀⠀⠟⠛⠺⠃
⠀⢸⢈⢈⠡⡉⣈⠈⡇⠀⠀⡇⢰⢐⠀⡠⠆⠨⠤⠤⠤⠅⠈⠤⠀⠀⣱⢺⠲⡗⣯⠀⠀⠀⠢⡀⠀⠀⢀⠌
⠀⡸⠑⠜⠣⠛⠜⠘⡇⠀⠀⡀⠀⣲⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠁⠈
⠀⠠⢍⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⡀⠀⠀⡀⠀⡀⠀⢀⠀⠀⠀⢠⡄⠀⠀⠀⠞⠭⠭⠭⠝⠆⠀⠈⠉⠁
⠀⠀⠀⠳⣄⠀⠀⠀⠀⡔⣈⡐⡀⠀⠈⠀⠈⠁⠀⠁⠀⠉⠀⡀⠀⠀⡉⢎
⠀⠀⠀⡦⡈⠳⣄⠀⠠⢡⠀⡐⠁⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⢠⣹⢣⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄
⠀⠀⠀⠃⠈⠆⠈⠳⢄⠈⠿⠃⠀⠀⢀⡀⠉⠁⢀⡀⠀⠉⠁⡀⢀⡁⠁⠀⢄⡄⠀⠃⠄⠀⠠⠀⠃⠀⡈⠛⠁
⠀⠀⠉⡉⢭⣥⣍⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⡀⠤⠐⠀⠀⠀⡌⡓⡀
⠀⠀⠊⠀⢸⣿⣿⡧⠀⠀⠚⠀⠁⠛⠀⠐⠀⠀⠀⢀⠠⠤⢤⢤⢤⢤⠠⠠⡄⠀⠀⡀⠙⢀⠀⠀⡐⠀⡇⠐⡀
⠀⠀⢄⠀⠀⠀⢀⠁⠀⠀⢴⣙⠀⠀⠀⠈⢸⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⢶⡆⠀⠰⣶⣶⣳⣶⠶
⠀⠀⠀⠁⠒⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣨⠴⠒⠛⠓⠲⢬⡀⠄⠀⠀⠀⠀⠀⠀⠀⢀⢄⣨⠶⠚⠛⠓⠶⣽⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠄⠒⠒⠒⠦⣅⠂⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢸⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⠀⢰⠖⣦⠀⠀⠹⣌⠄⠀⠀⠀⠀⢀⠂⠌⠀⠀⢴⠛⣆⠀⠈⠱⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣠⠟⠀⠀⠀⢹⠘⠀⠀⠀⠀⠄⠘⠀⠀⠀⠀⣤⡗
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠰⢿⠀⠀⠀⢸⠁⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⣀⠀⣇⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠁⡱⣄⠀⠀⠘⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠪⣷⣄⠀⠘⠒⠖⠀⣀⡼⡱⠁⠀⠀⠀⠀⠀⠂⠡⣀⠀⠘⠛⠁⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠈⠀⠉⠓⠲⠒⠛⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠿⢶⣶⣶⠿⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠲⠶⠖⠊⠁⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡵⠊⠉⠀⠀⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠛⠉⠉⠉⠉⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠠⣨⠞⠉⠁⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⡇⢸⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣘⡟⠀⠀⠀⣿⠉⠁⠀⠀⠹⣆⠀⠀⠀⠀⠀⠐⣼⠁⠀⠀⣾⠙⠇⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⣇⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠿⠲⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠄⣇⠀⠀⠀⣿⠶⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠜⣧⠀⠀⠀⠀⢹⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢙⣧⠀⠀⠀⣤⢘⡇⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⣿⢀⡇⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠪⡳⣄⣀⠀⠘⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠨⡳⣄⣀⠈⠉⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠐⢝⢦⣄⣈⠉⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠠⢐⡬⠶⠶⠶⠮⣔⡠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⠤⠴⠶⠬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠭⠴⠯⣤⣢⢄
⠀⠀⠀⠀⠀⢄⡴⠋⠀⠠⠤⣤⠀⠈⠹⣌⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⡤⣄⠀⠈⠑⢄⢀⠀⠀⠀⠀⠀⠀⣼⡾⠉⠀⢀⡤⣄⠀⠉⢷⡁
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢰⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢱⠇⠀⠀⢸⡇⣸⠀⠀⠀⠘⡆⠄⠀⠀⠀⠀⣘⣿⠀⠀⠀⢸⠀⣿⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣼⠁⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠸⡄⠀⠀⢰⡏⢻⠀⠀⠀⢠⡇⠂⠀⠀⠀⠀⢉⣧⠀⠀⠀⠈⠓⣿⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠊⢣⡀⠀⠠⠇⠀⠀⠀⣠⢏⠀⠀⠀⠀⠀⠀⠡⠱⣄⠀⠘⢧⠾⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠈⢜⢦⡀⠀⠸⠦⠟⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠑⠻⣷⡦⣤⡤⢶⡚⠕⠁⠀⠀⠀⠀⠀⠀⠀⠐⠈⢓⠦⢤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣤⣤⢶⡫⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⡤⠶⠶⠶⢬⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠞⠁⠀⢠⠶⣄⠀⠈⠳⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡏⠀⠀⠀⢸⠀⣿⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⢸⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡹⣄⠀⠀⠸⠦⠟⠀⠀⣠⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢷⡦⣤⣤⣤⡴⣚⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠒⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠉⠉⠉⠙⠻⣷⣄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⢠⡾⠛⣷⠀⠁⠀⠀⢠⣿⠇⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⡀⠀⠀⠀⣾⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⣿⠃⠀⡟⠀⣀⠀⢀⣾⣿⣴⣾⠿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢠⣿⠁⢀⣠⡀⠀⠀⣿⣇⣼⠁⠀⠀⢀⣼⣿⠟⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠤⠀⠀⠀⠀⢀⣿⡏⠀⣠⡿⠃⢠⡟⢁⡟⠀⠀⢻⣿⡁⠀⢀⡴⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣥⡾⠛⠁⢀⣿⣡⠞⠁⠀⢠⡾⣿⣧⣴⠏⠀⣿⣿⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⢠⣿⡟⠙⢳⣄⣀⣼⡿⠁⠀⢀⣴⠋⢀⣿⣿⠁⠀⢸⣿⡟⠀⠰⠂⠀⠀⢰⡆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠙⠿⣷⣦⣤⠆⣸⣿⠇⠀⠀⣿⠁⢸⣧⣀⣴⠟⠁⣠⡿⢻⣿⠀⠀⣼⣿⡇⠀⠀⠀⠀⢠⡿⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⣿⠀⠀⠙⢋⣁⠀⣴⠏⠀⣼⣿⠀⠀⢿⣿⡇⠀⠀⠀⣴⡿⠁⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣿⣿⠀⠀⣼⠇⠀⠀⢀⣾⠏⣼⠃⠀⢀⣿⡏⠀⠀⠘⣿⣧⣤⣴⣾⠟⠀⠀⠉⠴⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⠀⢻⣿⣀⣼⠟⠀⡤⣄⡼⠁⢰⡏⠀⢀⣾⠟⠀⠀⡀⠀⠈⠙⢛⣿⠃⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠛⠛⠉⠀⣼⣳⢯⣧⠀⠈⠷⠶⠟⠉⠀⠀⠀⠁⠀⠀⠀⣼⡏⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠗⠀⠀⠀⢰⡏⠀⠛⠁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⢰⣿⠁⣠⠏⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣠⡄⠀⠀⢀⣿⠇⠀⠀⠀⢿⠃⠀⠀⠀⠉⠀⠀⠀⢠⣾⠟⠙⢃⣿⡏⣰⢃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣿⠀⢠⣿⠋⠀⠀⣹⡛⢳⡻⠟⠁⠀⢰⡆⠀⠀⢠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠾⠇⠀⢀⡾⠀⢠⣿⡇⢀⣾⡏⠀⣠⣾⣻⣻⡛⠁⠀⠀⠀⠀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⣾⠇⢠⢿⣿⢀⣾⣿⠁⣴⠏⢸⣿⡟⢿⣦⣀⠀⠀⢀⣠⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⣰⣿⣠⡟⢸⣿⡾⠁⠻⠟⠁⠀⢸⣿⠁⠀⠙⠻⠿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⢀⣤⣶⠿⠛⣿⠟⠀⠈⠛⠁⠀⠐⠟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣠⣾⠿⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠸⠂⠀⠈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⣀⡀
⠀⠀⢀⣾⣿⣿⣿⠀⠀⢀⣾⣿⢿⣷⣄⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⣤⣶⣶⠶⠆
⠀⠀⠈⠛⢻⣿⣿⠀⠀⠈⠛⠛⣼⣿⡏⠀⠀⠀⢰⣿⣿⣯⡀⠀⠀⢠⣿⣟⣴⣶⡄⠀⠐⠿⠛⠻⣷⣤⡀
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠈⠉⢻⣿⡷⠀⢠⣿⠿⠿⣿⣿⠇⠀⠀⠀⠀⠀⣹⣿⡇
⠀⠀⠀⠀⣸⣿⣿⡄⠀⢴⣿⣿⣶⣿⠃⠀⢠⣄⣀⣴⣿⡿⠃⠀⠀⠀⠀⢠⣿⣿⡂⠀⢴⣷⣦⣶⡿⠟
⠀⠀⠀⠈⠛⠛⠛⠁⠀⠈⠉⠉⠉⠉⠀⠀⠘⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠈⠉⠁
⠀⠀⠀⣀⣤⣀⠀⠀⢀⣀⣀⣀⣀⣠⣄⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣠⣀⣀
⠀⢀⣾⣿⠿⠛⠁⠀⠈⠛⠛⢻⣿⣿⠏⠀⠀⣾⣿⡋⣹⣿⠂⠀⠀⣰⣿⠿⣿⣿⣦⠀⠀⣰⣿⡿⠻⣿⣷⡀
⠀⢸⣿⣿⣶⣶⣤⡀⠀⠀⢀⣾⣿⡏⠀⠀⠀⢨⣿⣿⣿⣿⣷⡄⠀⢿⣿⣄⣼⣿⣿⠀⢰⣿⣿⠁⠀⢹⣿⡇
⠀⢸⣿⣿⡟⢈⣿⡇⠀⢀⣾⣿⣿⠁⠀⠀⢠⣿⣿⡇⠀⣹⣿⠇⠀⠀⡉⠋⣹⣿⠟⠀⠘⣿⣿⣆⣀⣿⣿⠃
⠀⠀⠛⠿⠿⠾⠛⠀⠀⠈⠛⠿⡟⠀⠀⠀⠀⠻⠿⣷⠶⠛⠉⠀⠀⠘⠻⠾⠛⠉⠀⠀⠀⠘⠻⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢻⣿⡄
⠀⢀⣠⣿⣧⣤⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⣶⣶⡦⠀⠀⢀⣠⣽⣥⣄⠀⠀⢴⣶⣶⡶⠀⠀⠹⠇⢀⣼⣿⠏
⠀⠈⠉⢿⠏⠉⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠙⠛⠋⠀⠀⠈⠉⣽⡍⠁⠀⠀⠛⠛⠛⠋⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡍

⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⠀⠠⣾⠟⠛⣿⡆⠀⠀⠀⠀⠀⠾⠟⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠻⠗
⠀⠀⠀⠀⠈⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠰⠿⣿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣏⣀⡀⠀⠀⠀⠀⠀⣤⣄⢀⣼⡷⠀⠀⠀⠀⠀⠀⠀⢶⡦
⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠈⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀
⠀⠀⠀⠀⢀⣼⣿⣿⠀⠀⠀⠀⠀⢀⣿⣟⣛⠛⠁⠀⠀⠀⠀⢀⣾⡏⣉⠻⠃⠀⠀⠀⠀⠀⠻⣿⣶⣿⡿⠃
⠀⠀⠀⢠⣾⠋⣿⣿⠀⠀⠀⠀⠀⠸⠿⠛⠻⣷⡄⠀⠀⠀⠀⢸⣿⡿⠛⣿⡆⠀⠀⠀⠀⠀⣴⣿⣿⣿⣧⡀
⠀⠀⠀⠻⠿⠿⣿⣿⠗⠀⠀⠀⠀⢰⣶⣤⣴⣿⠇⠀⠀⠀⠀⠈⢿⣧⣠⣾⡟⠀⠀⠀⠀⠀⠙⠋⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⠀⠀⠀⠸⠿⠿⢿⣿⡇⠀⠀⠀⠀⢠⣾⠿⢿⣦⠀⠀⠀⠀⠀⠀⣴⡿⠿⣶⡄⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠘⣿⣷⣾⣟⠀⠀⠀⠀⠀⠘⣿⣇⣠⣿⣷⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣿⡆⠀⠀⠀⠀⠀⣈⡛⠛⣻⡿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁

⠀⠀⠀⠀⢠⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⢰⣿⡏⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣇⣀⣀
⠀⠀⠀⢸⣿⡄⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠿⠿⢿⣿⡿⠿⠟
⠀⠀⠀⠘⢿⣷⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇
⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⢠⣯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡠⠒⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡤⠀⡠⢀⠤⠀⣀⠄⣢⠄⢀⡠⠂⡾⠁⠀⠀⣼⠁⣠⠄⠀⣠⠄⢀⡤⡰⠄⣰⣏⠤⣶⠄
⠀⠀⠀⠀⠀⠻⢷⣤⣀⠠⢴⡞⠁⢾⡤⠜⠓⠸⠧⠤⠛⠂⠻⠖⠈⠃⠀⠀⣰⠇⠐⠿⠖⠈⠻⠄⠺⠧⠄⠸⠋⠐⠮⠅
⠀⠀⠀⠀⠤⠤⠤⠤⠤⣤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣴⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠄
⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢀⠤⠒⠚⠛⠓⠒⠶⢤⣄⡀⠀⢀⣾⠋
⠀⠀⠀⠐⠒⠚⠉⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⠿⢧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⢀⣀⣀⣠⡤⠶⠛⠉⠀⠀⠀⠈⠉⠛⠓⠲⠶⠒⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⣤⣤⣤⣿⣿⣤⣤⣤⡀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠙⢿⣷⣾⣿⠟⠁⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠿⠿⠿⠿⠿⠿⠿
⠛⠛⠛⣿⣿⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⣠⣾⣿⢿⣷⣄⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣴⣶⣶⣶⣶⣶⣶
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠁⠀⠙⢿⠗⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⣴⡟⢻⣆⠀⢀⣴⣦⠀⠀⢰⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣧⠀⠀⣀⣀⣀⣀⣿⣷⣀
⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠙⠷⠾⢃⣴⣿⠟⠁⠀⠀⠀⠉⠛⠻⠿⣿⣷⣦⠀⠀⣤⣶⣿⡿⠿⠛⠉⠁⠀⠀⠿⠿⠿⣿⣿⠿⠿
⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⠟⣵⣶⣦⡀⠀⠀⢀⣠⣤⣶⣿⡿⠿⠀⠀⠻⢿⣿⣶⣦⣄⣀⠀⠀⠀⣶⣶⣾⣿⣿⣶⣶
⠘⢿⠟⠁⠀⠀⠀⠀⠀⠀⠐⠿⠟⠁⠈⢿⣤⡿⠃⠀⠸⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⡟⠀⠀⠉⠹⣿⡟⠉⠉⠉

⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⣠⣶⡆⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⣠⣤⣤⣿⣿⣤⣤⣤⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠙⢿⣿⣾⣿⠟⠁⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⠀⠀⠻⠿⠿⠿⠿⠿⠿
⠙⠛⠛⣿⣿⠛⠛⠛⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⣠⣾⣿⢿⣿⣦⡀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⣠⣤⣤⣤⣤⣤⣤
⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠁⠀⠙⠿⠃⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣴⣿⠆⠀⠀⣾⡏⣹⡧⢀⣴⣾⡇⠀⠀⠸⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡿⠀⠀⣠⣤⣤⣴⣿⣿⣤
⠀⠀⢀⣴⣿⡿⠃⠀⠀⠀⠈⠛⢛⣡⣿⡿⠋⠀⠀⠀⠀⠀⠉⠙⠛⣿⣿⣷⠀⠀⢰⣿⣿⡟⠛⠉⠀⠀⠀⠀⠙⠛⢛⣿⣿⠛⠛
⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⣴⣿⡿⢃⣾⠟⣷⡄⠀⢀⣤⣶⣾⣿⠿⠟⠋⠀⠀⠈⠛⠿⣿⣷⣶⣤⣄⠀⠀⢾⣿⣿⣿⣿⣿⣿
⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠻⠶⠟⠁⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠀⠀⠀⠈⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⣠⠐⠒⠲⣶⣄⠀⠀⠀⣠⠐⠒⢶⣦⡄⠀⠀⢠⣶⣶⡀⠀⣀⡄⠀⠀⢠⣶⡶⢶⣶⣤⣤⠆
⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⣿⣄⣤⠀⣹⣿⡇⠀⠀⣿⣤⠄⢈⣿⡿⠀⠀⠈⠀⢹⠇⢻⣿⡇⠀⠀⢸⣿⡇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠙⠛⠋⣰⣿⡿⠁⠀⠀⠈⠁⠀⠾⣟⡁⠀⠀⠀⠀⠌⠀⢸⣿⡇⠀⠀⠸⠛⠓⠒⢶⣤⣄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⣀⣀⠀⠀⢹⣿⡆⠀⠀⡈⠀⠀⢸⣿⡇⠀⠀⠀⣀⣀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣜⣋⣀⣀⣀⣀⠀⠀⢼⡿⠛⠃⠀⢸⣿⡟⠀⠀⠁⠀⠀⢹⣿⡏⠃⠀⢼⡿⠻⠃⠀⣸⣿⡇
⠀⠀⠀⠀⠀⢰⣿⣿⠄⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠈⠣⠄⠀⠤⠿⠋⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠈⠣⠄⢀⠴⠿⠋

⠀⠀⠀⠀⠀⣠⡶⠊⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⣴⣾⠉⢿⣦⡀⠀⠀⢠⣴⠏⠉⢳⣦⡀⠀⠀⠀⠀⣤⡾⠉⠹⣶⡄
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣠⠏⠀⠀⠸⣿⣿⠀⣸⣿⠇⠀⢀⣿⣿⠀⠀⠘⣿⣷⠀⠀⠀⣾⣿⠃⠀⠀⢻⣿⡆
⠀⠀⠀⣸⣿⣿⠋⠈⣷⣦⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⢙⣻⠦⣿⣋⠀⠀⠈⣿⣿⠀⠀⢰⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣷
⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⢰⣿⡟⠀⠀⠀⠀⣴⣿⠃⠀⠈⣿⣧⠀⠀⠈⠻⠦⠠⠾⣿⡿⠀⠀⠸⣿⣿⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠸⣿⣇⠀⠀⣸⣿⠇⠀⠀⣿⣿⡇⠀⠀⠀⠀⢿⣿⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢻⣿⡄⠀⠀⣾⡿⠁
⠀⠀⠀⠀⠙⠻⠦⠤⠟⠋⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠈⠛⠷⠄⠼⠛⠁⠀⠀⠀⠀⠤⠞⠋⠀⠀⠀⠀⠀⠀⠙⠷⠄⠼⠛⠁

⠀⠀⠀⠀⢀⣴⣿⢰⠆⠀⠀⠀⣠⡾⣫⠝⢷⡄⠀⠀⠀⠽⠿⠾⠷⢍⡿⣫⠖⠀⠀⠀⠀⢀⣼⡇⡶⠀⠀⠀⢸⡏⡽⠿⠿⠯⠶⠂
⠀⠀⠀⣠⡾⡁⣿⣸⠀⠀⠀⢀⣿⣼⠁⠀⢸⡇⡇⠀⠀⠀⠀⢀⣴⠋⠼⠋⠀⠀⠀⠀⣠⡟⢹⡇⡇⠀⠀⠀⢸⣇⣥⢴⣤⣀
⠀⠀⠈⢫⡾⢹⡏⣿⠀⠀⠀⠘⣿⢹⠀⠀⣼⢣⡇⠀⠀⠀⠀⢻⡵⠎⢷⡄⠀⠀⠀⣴⢏⡶⢸⣿⡇⠀⠀⠀⢼⡟⣵⠋⠁⢹⡆⡀
⠀⠀⠀⠀⠀⢸⡇⡏⠀⠀⠀⠀⠐⠛⢀⡼⣣⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢀⡾⢣⠟⠀⣼⡿⠃⠀⠀⠀⠐⠚⠁⠀⠀⢸⡇⣷
⠀⠀⠀⠀⠀⣼⢷⡇⠀⠀⠀⠀⢀⣴⣋⡾⠋⠀⠀⠀⢠⣄⠀⠀⠀⢀⡾⣳⠇⠨⠭⠭⠭⠏⣿⢯⡿⠷⠂⢶⡀⠀⠀⠀⢀⣾⢳⠇
⠀⠀⠰⣶⠶⢿⣎⣀⣀⡀⠀⢰⣟⣰⣯⣦⣤⣄⣤⡄⠀⠙⠶⠴⣶⣫⡵⠋⠀⠀⠀⠀⠀⠀⠻⢸⡇⠀⠀⠀⠙⠶⣠⡶⣟⡵⠋
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⢀⣤⢖⡔⢶⣄⠀⠀⠀⢶⣶⣶⣲⣶⢲⡶⣠⠀⠀⢀⣴⣲⡖⢶⡀⠀⠀⠀⢀⡴⣒⠶⣄⠀⠀⠀⠀⢀⣴⢖⡒⢶⡄
⠀⠀⠀⣾⣳⠏⠀⠀⠹⢇⡄⠀⠀⠀⠀⠀⢀⣾⣱⠏⠀⠀⣼⣷⡏⠀⢸⡧⡆⠀⠀⣿⣼⠁⠀⢹⡇⡀⠀⢀⣾⢳⠏⠀⠀⢿⣤
⠀⠀⢸⡏⣩⢖⣲⡒⢦⡉⠁⠀⠀⠀⠀⢀⣾⢳⠏⠀⠀⠀⠹⣧⣁⣤⡞⣵⠃⠀⠀⣿⡹⠀⠀⢸⣷⣇⠀⢸⡇⡏⠀⠀⠀⢸⡇⡇
⠀⠀⢸⣿⢯⡟⠁⠀⠘⣷⢡⠀⠀⠀⢀⣾⢳⠏⠀⠀⠀⢀⣾⢫⠞⠃⢹⡆⡀⠀⠀⠈⠻⣦⣴⢿⡿⣿⠀⢸⡇⡇⠀⠀⠀⢸⢧⡇
⠀⠀⠘⣿⠸⠀⠀⠀⢀⡿⣸⠀⠀⢀⣾⢣⡟⠀⠀⠀⠀⢸⣧⡟⠀⠀⣸⡇⡧⠀⠀⣄⠀⠈⠉⣸⣇⡟⠀⢸⣇⡇⠀⠀⢀⡟⣸
⠀⠀⠀⠙⠷⣄⣀⡤⢞⣵⠃⠀⠀⣼⢣⡟⠀⠀⠀⠀⠀⠈⠷⣅⣠⡴⢟⡼⠃⠀⠀⠙⢧⣀⡴⣋⡾⠁⠀⠀⠻⣤⣀⡤⢞⣵⠋
⠀⠀⠀⠀⠀⠐⠒⠛⠉⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠉⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠁⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⢀⠀⠀⠀⢀⡀⠀⠀⠀⠦⣄⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣼⣇⣣⠤⡄⢀⠀⣤⣤⣤⣤⣤⣤⠀⡀⠙⢷⣤⢟⣵⠏⠀⠀⠀⠻⠟⢲⠀⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠘⠟⠃⢹⣇⡟⠛⠛⠋⠀⠙⠛⠛⠛⠛⠛⠛⠁⣠⡾⣛⢷⣄⠀⠀⠻⠏⣡⣬⢰⠷⠾⠂⠀⠀⠀⢀⡽⢟⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠞⠁⠀⠩⠵⠆⠀⠀⠉⠭⠟⠀⠀⠀⢀⣠⢖⣫⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣫⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢾⣿⢰⠀⣠⠞⣵⠟⢀⣼⢫⡵⠒⠉⣷⡄⡀⠀⣶⢤⠀⠀⠀⠀⠀⣠⡴⣞⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⣫⠞⣡⠞⠁⠀⠈⢻⠮⠀⣀⣴⠿⣹⠃⠀⣿⢸⠀⢀⣤⢶⣫⡵⠞⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⢟⣥⡄⡀⠀⠀⠀⢰⡟⡽⠶⠚⠁⠀⠀⣿⣾⠀⠙⢦⣋⠁⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⢒⠠⠆
⠀⠀⠀⠀⠀⢠⠾⣫⠞⠁⠈⢛⣵⠇⠀⠀⠀⣬⣅⠇⠀⠀⠀⠀⠐⡛⡿⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠨⠭⠿⠿⠶⠖⠖⠃
⠀⠀⠀⠀⠀⠀⠉⠁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠙⣥⠏⠀⠀⠀⠀⠘⢟⣴⠀⠀⠀⠀⠀⠀⠉⢣⡤
⠀⠀⠀⠀⠀⢠⡴⠋⣩⣟⣃⣽⢷⣆⠀⠀⠀⠀⠀⣴⢦⣄⣾⣲⠆⠈⠀⠀⣤⢶⣄⢀⠀⠀⠀⠀⢠⡄⡀
⠀⠀⠀⠀⢰⡿⢠⡿⣱⠆⣿⢣⡞⣿⢡⡀⣀⣀⣰⣏⢚⣼⡇⢏⠀⠀⠀⢀⣿⡟⡵⠋⠀⠀⠰⢦⣸⣧⣴⢗⡴⠀⠠⣶⢆⡄
⠀⠀⠀⠀⢿⡇⢻⡇⡇⣸⡏⡞⢰⡿⣼⢁⣘⣢⣿⡐⣸⡏⠞⠛⠛⠁⣴⢟⡍⢻⡄⣠⠀⡐⡖⢲⣿⣿⡖⢲⣆⣤⠀⣠⡛⠁
⠀⠀⠀⠀⠸⣷⡈⠻⠶⡿⠷⠶⣛⡵⠃⠐⠃⣿⢣⢸⡟⡽⠯⠶⠂⠸⣏⠟⠀⣀⡿⣯⡚⠁⠸⣫⢸⡇⡝⠲⣠⠀⠀⢹⡟⡷
⠀⠀⠀⠀⠀⠈⠛⠶⣭⢭⣖⣬⠍⠀⠀⠀⠘⣧⡇⢟⣼⠃⠀⠀⠀⠀⠈⠗⣾⡭⠖⠈⢣⡴⠀⠉⠈⠡⠇⠀⠁⠀⠀⠭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⢤⣄⠀⢀⣤⠖⠚⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠒⠚⠋⠉⣷⡀⣾⠁⠀⠀⠀⠸⣤⠎⢹⡆⠀⠀⢀⠀⠀⠀⠀⢹⣤⢶⠲⠦⢤⣀⡀
⠀⣀⡤⠶⢶⣄⡀⠀⠀⠘⡆⠀⠀⠀⣠⠠⠂⠹⡟⠀⠀⠀⠀⠀⢱⡀⢸⡇⠀⠀⠫⠇⠀⠀⠀⢸⠃⡏⠀⠀⠀⠀⢹⢀⡶⠶⢤
⢻⠁⠀⠀⠀⠄⠙⠦⣄⠀⢧⠀⠀⠀⢠⡀⠀⠀⢻⠀⠀⠿⠂⠀⠀⣧⠈⠀⠀⠀⠀⠀⢀⡤⠖⠁⢀⠇⠀⠀⠀⠀⢸⡞⠀⠀⢀
⠈⣧⠀⠀⠀⢠⠀⠀⠈⠳⢼⡄⠀⠀⠀⢉⠇⠀⡇⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⢳⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠸⡆⠀⠀⠀⢢⡀⠀⠤⠐⣧⠀⠀⠀⠘⠂⠰⡇⠀⠀⠀⣀⠀⠀⠀⢹⡇⠀⠀⠰⡄⠀⠀⢣⠀⡦⠀⠀⢀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣧⠀⠀⢠⠍⣴⠊⣉⠳⣧⢀⣀⡇⠈⠦⣀⣈⣶⡇⠀⠀⡎⢣⠀⠀⠀⢀⡏
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢀⡀⠷⠄⠤⠤⠔⠒⢛⠁⠉⠁⣠⣥⡘⠀⢂⡷⣤⣀⣀⣴⣄⣀⣀⡄⠉⠑⠚⢁⠀⢆⡀⠀⣼
⠀⠀⠀⠘⣆⢀⡀⠄⠒⠉⠁⠀⣀⣶⠤⠴⠖⠒⠋⠛⠛⠉⠁⠀⡯⠀⢸⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠛⠳⣄⣉⣹⠏
⠀⠀⠀⠀⠈⠳⣤⡤⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠉⠻⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠛⢠⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⡅⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡟⢳⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠇⠀⢇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠠⠀⠀⠘⡄⠀⠈⠆⠀⠀⠀⠀⢀⡠⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣉⡒⠠⠄⣀⣀⠃⠀⠀⡇⠀⠀⠀⢱⠀⠀⠘⣄⠤⠔⢊⣡⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⡌⠀⠀⢘⡛⡛⢛⢛⣙⡁⠀⠀⢤⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡎⠈⠉⠉⠀⠀⢱⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠄⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠋⠙⣛⠲⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠹⠿⠂⡀⠞⢃⡴⠋⠉⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠋⢸⠄⠀⠻⠷⠀⠀⠀⣨⣉⠛⢲⡄⠀⠀⠀⣠⠴⣋⡀⠛⣁⣤⠛⣡⠞⠉⠀⢠⡄⠘⠇
⠀⠀⠀⠀⠀⠀⣠⠏⠀⢸⡆⠀⣀⠀⠰⣷⡆⠈⠛⠀⠸⡇⠀⢠⠾⣥⠤⠭⠤⠤⠭⢦⡟⠁⢠⣄⠀⠘⠛⠀⢸⡄
⠀⠀⠀⠀⠀⣰⠏⣰⠀⢸⣧⡀⠻⠿⠀⠀⠀⣠⣄⠀⠀⡇⠀⠘⡇⠀⠀⢀⠀⠀⠀⠈⣧⠀⠘⠟⠀⠀⠀⣠⡀⢳⡀
⠀⠀⠀⠀⠀⣿⠀⠉⢠⡟⠉⠉⠙⠒⠶⠤⣤⣈⣛⠁⠀⡇⠀⠀⠸⡄⠀⢻⣷⠀⠀⠀⠘⣧⠀⠀⢀⡀⠀⠿⠇⢈⡷
⠀⠀⠀⠀⠀⣿⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⣀⣀⠉⠙⣿⠇⠀⠀⠀⢱⡀⠀⠀⠀⢀⣄⡀⠹⡄⠀⢿⡇⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⣿⣠⠉⢀⣀⡀⠀⣴⣶⡆⠈⠛⠛⠁⣰⠏⠀⠀⠀⠀⠀⢷⡀⠀⠀⠈⠻⠃⠀⢻⡀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⢻⣏⡀⠘⠻⠛⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠈⢧⣤⣤⣤⣤⡤⣤⣬⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⢤⣄⣀⣀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁

⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠁⠂⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠁⡀⠀⠀⡀⠄⠐⠈⠀⠀⠀⠀⠀⢂⠀⠀⢀⠃⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠐
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⢤⠀⠀⠀⠀⢠⠂⣄⠀⠀⠀⠁⠢
⠀⠀⠀⠀⡀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⡁⠀⠀⠀⠀⡐⠁⠀⠁⠄⠐⡄⠀⠀⠀⠂⠉⠀⠀⠂⠀⠀⡐⠁⠀⠁⠀⠀⠀⠀⠀⠀⠁⠂⢀
⠀⠀⠀⠈⡀⢀⠀⠀⠀⠀⡍⠐⠀⠄⡀⠈⠆⠀⠠⠀⠀⠀⠁⠀⠀⠀⠘⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⢡⠔⠀⠀⠀⢀⠄⠀⠠⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠄⠉⠀⠀⠀⠁⠀⠀⢀⣈⡀⣀⣄⡈⠄⠀⣠⠀⢤⢀⣰⠇⠠⡄⡡⠀⠀⠂⠃⠢⠀⢠⣮⠤⡴⣎⣉⣟⣒⡰⢤⡅⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣠⠔⢫⡤⠓⠒⠚⠢⣌⢹⡋⡀⣗⠉⠀⠀⠉⢱⢄⠘⡇⠀⠉⢰⠤⣊⠽⠖⠒⢶⠀⣵⡀⣈⡷⠈⣱⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠄⣞⢹⡆⣈⠷⢤⡤⣤⠴⢎⣻⢟⣲⠮⢉⣖⡲⣌⠭⢶⣹⣇⡧⣴⠣⣞⠁⠀⣀⡴⣋⠖⡴⢯⡄⣸⡮⢉⡆⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣵⡟⠧⣭⣽⢴⠼⢯⠽⣻⡿⣏⠩⢞⢓⡲⣮⢼⠗⠩⡴⠚⢣⣔⣩⠏⢭⣗⣊⡳⢮⠭⠯⣮⠉⡶⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣺⠿⢥⣸⣚⣋⠻⣂⣽⣧⣏⠉⢿⠓⠒⣯⣽⠷⣬⡑⠒⡤⢤⠞⢓⣌⢿⣿⠯⣯⠭⠽⣭⠾⠋⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡷⢬⣶⣒⣛⡾⢭⣧⣷⠿⠭⣿⠒⠲⣯⢽⣋⡷⣦⣽⣍⣉⣹⣄⣘⡷⠾⢭⢯⡤⠴⡏⡗⡼⠛⢦⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢣⣟⣓⣳⣒⣚⣖⠿⣩⠴⢢⡀⣛⡗⠲⢋⢽⡷⢷⠀⣳⡀⢲⣏⢉⠛⢅⣠⢬⡙⣯⢽⡗⢳⢶⠀⠀⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢭⣆⣀⣯⠭⢽⢏⡜⠁⢀⣜⣀⡝⢉⠛⣿⣽⣿⣟⣓⣾⡶⡁⠨⡌⠄⡰⠈⢂⠹⡌⣯⠿⡬⡎⢦⣀⡼⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⢆⠽⣭⢭⡇⠸⣀⣠⣉⣤⠌⡀⠉⠀⣀⠑⣮⣡⣌⣉⣇⢡⡀⢁⡐⠈⢐⠀⡄⢫⢹⠏⠙⢼⢆⡈⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠣⣠⣞⠁⠀⠁⠄⠁⠀⡄⢀⠶⢙⡁⠀⠀⠀⢆⠠⠌⠁⠤⠌⣠⠓⠃⣨⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠀⡘⠀⠈⢂⡤⠋⠗⢨⡅⠀⠘⠃⠀⠘⠠⠘⠀⠜⠁⠿⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠘⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠶⣄⡔⠋⠙⢦⡴⠊⠉⠙⠶⠁⠀⠀⣾⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠈⠀⢀⣤⣌⠀⢠⡶⣄⠀⢠⡶⡄⠀⣴⠿⣶⡿⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⠆⠁⠀⢠⣿⠋⢹⡄⠹⣿⣸⣠⡾⣧⣇⣼⡏⠀⣿⡇⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣾⡇⢀⢸⣧⢾⣏⡝⠸⣇⡜⠉⠹⠔⠁⢿⡇⠀⣼⠃⠀⣠⠟⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣷⠀⠘⠛⢁⣼⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠾⠋⠀⢊⡥⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⡟⠀⠀⠀⡞⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣼⣿⠞⠀⠀⠀⠀⠳⠋⠀⠀⠀⠀⣠⣤⡀⠀⢰⡿⣠⠖⣳⠄⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣀⣤⣿⡇⠀⠀⠀⢀⡄⠀⠀⣼⠀⠀⣼⡏⠘⠃⢀⣿⣿⣡⠾⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢤⠀⠈⠉⠉⠁⣿⡇⠀⠀⠀⢸⡇⠀⣼⣿⠀⢠⣿⠀⠀⢠⠎⣿⢿⣧⡀⠀⠀⠀⠀⠀⢀⣾⠏⢠⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⡆⠀⢻⡇⠀⠀⡰⢻⣇⡼⠸⣷⣠⠏⠻⣦⠴⠋⠀⠿⠀⠙⠿⣶⣦⣤⣤⣶⠿⠋⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⠻⠶⠞⠁⠈⠉⠀⠀⢈⡁⠀⠀⠀⣤⠀⠀⠀⣠⣷⡀⠀⠈⠉⠉⠀⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣤⠞⠷⣤⣠⣤⠞⠙⠶⠶⠚⠉⠓⠒⠚⠉⠈⠙⠶⢦⣤⣤⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣴⣿⣷⡄⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠘⠿⠿⠿⠿⠿⠿⠿⠟⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠛⢻⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣦⣿⣡⣥⣀⣀⣀⣡⣄⣹⣯⣈⣁⣀⣀⣀⣀⣸⣷⣾⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⣫⠀⢍⠉⠉⠉⠉⠉⠉⠉⠉⢉⢉⠉⢹⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠿⠀⠀⠙⠌⠱⠶⠃⠀⠼⠤⠦⠼⠄⠸⠿⠿⠿⠿⠿⠦


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣰⣋⡌⣶⡀
⠀⠀⠀⠀⢬⡤⠉⠀⠀⠀⠀⠉⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡋⠀⢰⣶⣶⠀⠈⡑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠴⠛⢹⣿⣧⣼⣿⠁
⠀⠀⠀⢨⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⡜⠡⣾⡿⣠⣴⣿⣷⣤⡸⣿⡦⠙⡄⠀⠀⠀⠀⠀⠀⠤⢖⡿⠛⣿⣥⣾⡿⠛⠁
⠀⠀⠀⠠⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⡸⢁⡀⢀⣾⣿⣿⠟⢿⣿⣿⣆⠀⣀⠸⡀⠀⠀⠀⠀⠘⣶⣾⣷⣿⠿⢛⡉
⠀⠀⠀⠀⠀⠀⣿⠓⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡇⢾⣿⢺⣯⠙⠋⠀⠈⠛⢉⣿⢸⣿⠇⡇⠀⠀⠀⠀⠀⡴⠭⣩⣟⡲⠛⣷⣶⢋⡀⡀
⠀⠀⠀⢰⠀⠀⠛⠀⠛⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⣻⣿⣿⣿⣿⣿⡿⣃⠀⠀⣸⠀⠀⠀⠀⠀⠀⣷⣾⣏⣹⣿⣂⣹⣷⣶⠾⢃⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣤⡄⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠳⡜⢿⡿⠉⢛⣟⣛⠋⠹⣿⠟⡰⠃⠀⠀⠀⠀⠀⠀⣬⢻⣿⣿⣿⣿⣿⣿⣿⣾⡟⠁
⠀⠀⠀⠘⢤⣤⣤⡄⠀⠐⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠘⠿⠟⣀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣷⣾⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠈⠙⠳⠥⡀⠠⠠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⠞⣶⣿⣩⣿⢿⡶⣣⠖
⠀⠀⡠⠤⠤⠤⢤⣤⠤⠤⠤⠤⣤⣤⠤⠤⠤⢤⡀⠀⠀⠀⠀⠙⢦⡀⠐⢿⣿⡿⠖⠀⣠⠞⠁⠀⠀⠀⠀⡬⢤⣭⣭⣭⠭⣌⠸⣿⠗
⠀⣾⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢹⠀⠀⢀⡀⠀⡀⠘⢦⡀⠙⠁⣠⠞⠁⣀⡀⠀⠀⠀⠀⣷⣾⣿⣿⣿⣶⣧⡟⣿⠗⣲⢦⣄⡀
⠀⣿⠘⢋⣩⣟⠁⡇⠛⣉⣹⡏⢸⠛⢋⣉⣿⠃⢸⠀⠰⣿⣿⣿⣿⠇⠀⠙⣦⡞⠁⢠⣾⣿⣿⣷⡄⠀⠀⣿⣿⣿⣿⣿⣿⡧⠛⡿⢃⡻⢶⠛⣿⡽⠆
⠀⣿⠀⠈⠛⠁⠀⡇⠀⠘⠛⠀⢸⠀⠀⠛⠃⠀⢸⠀⠀⠈⠙⠛⠁⠀⣠⠞⣁⡙⢦⡀⠙⠻⠟⠋⠀⠀⠀⠧⠼⠿⠿⠿⠤⠗⢙⣴⣿⡟⢻⠎⢥⣶⡍⠁
⠀⠹⢄⣀⣀⣀⣠⣷⣄⣀⣀⣀⣼⣧⣀⣀⢀⣠⠞⠀⠀⠀⠀⠀⣠⠞⢁⣼⣿⣿⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢻⣿⣿⣿⣿⣿⠏⠚⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠈⠛⣻⡛⠋⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⡟⢻⠏⣘⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡿⠿⣿⠻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠂⠄⡀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⢤⣀⠈⠐⠤⣀⣼⠁
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠠⠃⢀⣀⠀⠀⡞⠀⠀⠹⡆⠀⠀⠀⣿⠈⠁
⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠢⠤⠀⠀⠀⢰⡁⠈⣧⠀⢷⡄⠀⠀⡇⢀⡔⠛⣿⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣠⡄⠀⠀⠀⢸⣷⢠⡟⠀⣾⡛⠶⣴⠗⠺⣧⡼⠻⣆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢘⣿⣀⣀⠔⠁⣉⣉⡀⠀⠘⠳⠒⠁⠀⠀⠀⣠⠀⠉⠓⠦⠄⠊
⠀⠀⡠⠒⠉⠉⠉⠉⠉⠉⢹⡏⠁⠀⢠⠏⠀⠀⢹⠀⠀⠀⠀⣠⠤⡀⢰⡇⠰⣦
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢄⣿⣀⢀⠀⠊⠀⣄⠀⣼⠃⠀⠀⣸⣇⡴⠃⠀⠀⡠⠔⠒⢦⡀
⠀⠸⣆⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢸⣏⠀⠀⢸⠇⢰⣏⠀⣿⣆⣀⠔⠹⠏⠉⠓⡶⣮⣀⣀⣀⡼⠁
⠀⠀⠙⠳⠦⠤⠤⠖⠋⠀⠀⠀⠀⠀⢿⣄⣀⠾⠧⠜⠛⠊⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⡀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢠⣄⠀⠀⠀⠀⢀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠋⢀⣀⡙⢛⣗⡒⢒⣺⡏⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣰⣗⣿⣙⡿⣿⣓⡷⣏⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⣏⣀⣍⠀⠈⠁⠀⣼⢦⠁⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣿⠁⢋⡙⣷⠶⣶⢾⡟⣁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣟⠀⢸⣇⣏⣾⡉⢹⡷⣏⠀⢛
⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⢮⣍⣉⠈⠉⠈⠁⠹⣧⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣤⡤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣛⣻⣛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣼⡇⠀⠀⠀⠸⣼⣧⡤⠤⢤⣄
⣤⠤⠶⠶⠶⠖⠒⠛⠛⠛⠛⠛⠋⠉⢿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢈⡇⠀⠀⠀⠀⠀⠀⠀⣷⣤⠶⠶⠶⢶⡄⠀⠀⠀⢰⠛⢛⣛⣻⣯⣤
⠸⣷⠶⠶⠚⠛⠛⠛⠛⠛⠛⣛⣉⣉⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢻⡇⠀⠀⠀⢸⠀⣿⠉⠉⠉⠁
⠀⠈⠻⠶⠶⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣭⣤⣴⠶⠏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠻⢦⣀⠀⣠⡞⠁⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⢰⠏⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠉⠻⠋⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠘⣷⣤⣞⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣼⠇
⠀⠀⠀⠀⣀⣀⣀⣈⣉⣭⣥⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⠀⠀⠀⠀⠀⠈⠳⢿⣁
⠀⢰⡟⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡀⠀⠀⠀⠀⠉⠛⣦
⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣼⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⣠⠎⣙⠶⣄⡀⠀⠀⣰⣻
⠀⠘⢷⣖⣛⣻⣭⣭⣭⣭⣭⣤⣤⣤⡴⠶⠟⠀⠀⠀⠀⠀⠀⠀⢰⣯⡀⠀⠀⢀⡴⢣⡾⠛⠻⣦⡙⠲⠾⣱⠟
⠀⠀⠈⠉⠉⠁⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡳⠴⠟⣰⠟⠀⠀⠀⠈⠙⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣅⣀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠹⢯⣭⡿

⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡶⠶⠶⠶⣦⣄⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠉⠙⠛⠿⣦⣄
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⡀⠀⠈⢻⣦
⠀⠀⠀⠀⢀⣿⠁⠠⠂⣁⠀⠐⠄⠀⠀⠈⣿⡀⢀⣿⠁⠀⠀⢠⠞⢁⠤⡤⠌⢢⡀⠀⢻⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠐⢀⡈⠁⡄⠀⠀⠀⣸⡇⢸⣿⠀⠀⠀⢄⠀⢃⠀⠘⡃⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠈⣿⡄⠀⠀⠘⢤⠀⠉⠁⢀⠜⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⣿⣦⣄⡀⠀⠀⢀⣠⣴⠟⠁⠀⠀⠘⢿⣄⠀⠀⠀⠈⠁⠈⠁⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⣿⡇⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣤⣠⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠉⠉⠉
⢿⡿⠛⠛⠻⣿⠟⠛⠛⠛⠛⢿⡿⠛⠛⠛⠛⣿⡿⠋⠀⠀⣸⡟
⠘⣿⡄⠀⠀⠿⠀⣀⡀⠀⠀⠸⠇⠀⠀⣠⡾⠏⠀⠀⠀⢠⣿⠁
⠀⠈⢿⣄⠀⠀⠀⣿⡇⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠻⢷⣤⣀⣿⣇⣠⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⠶⢶⣶⠂
⠀⠀⠀⠀⠀⣽⡟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠈⣿⡍⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⡶⠾⠿⠿⣿⡇
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣼⣿⠶⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣶⣶⠶⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠶⠾⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠉⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢠⡴⠶⣄⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢀⡿⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠞⠛⠁⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⣾⡿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⣀⣀⣤⣤⠤⣤⣤⣀⣀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠉⠉⠁⠀⠀⢌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠔⣡⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⡞⠁⠀⠀⠀⠀⠀⠀⣰⢲⡔⡄⠑⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⢀⡠⢂⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇⠀⠀⠱⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠃⣀⡀⠀⠀⠎⣰⡟⠁⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⢀⡄⢸⣇⡠⠤⢄⣀⣀⡀⠀⢐⢠⡀
⠀⠀⠀⠀⠀⠀⣼⠇⠀⢈⣉⠓⠀⢠⡿⠀⠀⠀⠿⠀⠀⣀⣀⢤⠀⢀⣐⡒⡄⠀⠀⠀⢀⣇⡾⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡑⢝⠂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⣉⡉⠁⣼⡇⠀⠀⠀⠀⠀⣸⠧⣌⣧⣀⣿⢧⣹⡇⣀⡴⠚⠉⣿⠃⢀⣀⠐⠚⠋⣁⡀⠀⠀⠀⠀⠤⠀⠀⠈⢻⡜⠂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⡠⠀⣿⡀⠈⠉⠉⢳⡆⢿⠀⢠⡏⠈⣏⠀⣸⠏⢹⡁⠀⣰⢿⣠⣟⡽⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠚⠀⠀⠘⣧⡀⠀⠀⣸⡇⠈⠑⠩⠄⠀⢙⠚⡩⠀⠈⢑⣊⠅⠀⠉⠻⠤⠄⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠪⣙⠒⣚⠏⠀⣠⠖⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⢀⠄⠀⠀⣀⣠⣄⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣾⢸⠇⠀⠀⠀⠀⠀⠀⠻⠟⠃⠀⠀⢀⡿
⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⣿⠀⣠⠟⠁⠀⠀⣸⠃⠀⣾⣷⢠⡟⠉⠓⠀⣿⡿⢖⡦⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⢀⠄⠀⠉⠓⠒⢿⡛⠁⠀⠀⠀⣴⢿⠀⠀⢷⡿⢸⣇⠀⠀⢀⣿⣏⠁⠀⢀⣄⡀⠀⠀⠀⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠘⣧⠈⢆⠀⠀⠀⠀⠀⠀⠃⠆⠀⠘⣎⠳⢤⣀⣠⠼⠋⠈⠓⠤⠚⠓⠋⢉⡓⠚⠍⠃⠈⠙⠢⢿⡞⠁⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡙⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⣀⠤⠖⠋⠀⠈⠳⣄⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⣌⣉⠉⠛⠛⠋⢉⣩⠴⠂⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡢⢊⣴⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣄⣀⣀⣀⣀⣀⣀⠤⠖⠋⡡⢞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠐⠁