Tip over Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⡏⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠂⠀⠀⠀⠀⠓⠀⠀⠒⠀⠄⡀
⠀⠀⢠⡶⠛⠛⠳⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣠⠤⠤⠤⠤⠤⠤⣤⣀⡀
⠀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠉⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡀
⢰⡏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⡒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠛⡶⠀⠀⠀⠂⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠒⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠉⠉⠛⠒⠒⠲⠖⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠙⠦⣤⣀⣀⡀⠀⠀⠂⠉⠉⠉⠀⣀⣀⣀⣀⠀⠀⠀⠀⠐⠒⠲⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡚⠛⠋⠁
⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⠓⢄⠀⠀⠀⠤⣀⠀⠀⠈⠉⠙⠒⠲⠦⠤⠤⠀⠀⠀⠀⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠀⠀⠀⠀⣰⢋⠤⠈⠱⣄⠑⡀⠀⠀⠀⠉⠲⢤⡀⠀⠠⢀⣀⠀⠀⠀⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢦⣀⡴⠋⠀⠁⠀⠀⠀⠀⣀⣀⣈⠂⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡛⠋⠀⠀⠙⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡞⠙⠲⢤⣀⠀⠘⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⠟⠀⠀⠀⠀⠉⠛⠦⣌⡛⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⣠⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣴⠃⣠⣄⡀⠀⠀⠀⠀⢠⣤⣠⠟⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡼⠃⠐⠿⣿⣿⣶⣾⡀⠀⠈⣿⣿⡄⠀⠀⠙⣩⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⡾⠁⠀⠀⠀⠀⠉⠿⠟⠃⠀⣰⠃⠹⢿⣦⣤⣀⢿⣿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠿⣿⢾⣿⣿⣿⣿⡗⢰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⣴⣿⣿⡿⠁⠈⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣀⣴⣾⣿⣿⡿⠁⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⠏⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢰⣾⠿⢋⣵⣿⠟⠁⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⣠⣇⠸⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢠⣿⡏⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄
⠀⠀⠀⠀⣰⣿⣿⠇⠀⠈⠙⠶⣤⣀⠀⠀⠀⣼⠃⠀⠀⠀⣾⡿⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⣴⣿⣿⠃⠀⠀⠀⣀⣀⣀⣉⡛⣦⣾⣁⣀⣀⣀⣀⣈⣀⣀⣀⣀⣿⣧⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠘⣿⣿⣦
⠀⠀⢸⣿⣿⡃⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⢘⣿⣿⡇
⠀⠀⠀⢿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿
⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⡤⠀⢀⣀⣠⠖⠄⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠧⡲⢤⡀⠰⢎⡉⠛⣄⠎⠈⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠚⠶⠵⡄⠑⡄⠑⣇⠀⡇⢸⠅⢀⡴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡎⠉⠉⠁⢪⠶⠚⠢⢹⡏⠉⠉⠺⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⢒⠴⠶⡟⠓⠘⠀⡔⠀⢀⣠⢧⣤⣒⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠻⣦⡀⠐⡀⠀⣶⢁⠴⠓⢁⠔⢁⡨⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠊⡠⠓⠒⠛⡦⣽⣧⣦⠐⠛⢍⢫⠣⢄⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⡴⡁⠀⠀⠛⢡⣉⡁⠉⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⢠⢋⠍⠁⢀⠄⡀⠆⡇⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠀⡀⠮⠉⠡⣂⡡⠛⠫⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⢊⠀⡀⡢⢪⠈⠀⢠⠀⠀⣀⣀⢤
⠀⠀⠀⠀⠀⡤⠀⠐⠒⡰⠊⠚⣻⣃⣼⢀⠀⠆⠁⠁⢈⠼⣀⢰⢠⡄⣀⠄⠚⠀⠒⠠⢄
⠀⠀⠀⠀⢐⡧⢤⡔⠂⠘⠋⠁⠀⠀⡙⢀⠁⠒⣂⠍⢀⠄⠡⡘⠉⠉⠁⠀⢈⣒⠐⠒⠀⣃⣀⡀
⠀⠀⠰⣍⣉⣹⠛⣠⣤⣤⣄⠒⣯⣀⠸⠄⠂⠀⠠⠴⣁⠀⢘⣸⣐⣲⢉⠉⠀⠉⠒⣠⠸⠤⠤⠼
⠀⠀⠀⠀⡄⠚⢡⣀⣧⣤⡬⠛⡉⠀⠀⠀⣀⣀⡀⣀⣀⢻⡓⢮⣤⣤⣷⠿⠓⠁⠈⡩⠂
⠀⠀⠀⠀⡙⢛⠷⢤⡬⠆⡲⠭⠽⡒⣒⠿⠤⢶⣃⡖⠂⠛⠛⠛⠉⠉⠉⠉⡖⠂⣟⣁⣐⡄
⠀⠀⠀⠀⠀⠐⣀⡀⠤⠔⠓⠒⠚⠁⠀⢂⠈⠁⠘⠂⣠⣤⣤⠴⣒⣒⡒⠒⠋⢀⠝⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠓⠒⢢⠉⠧⠤⠤⠇⠀⠀⠈⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⢛⣉⣉⣹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⠿⠋
⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠈⠻⣿⠋
⣿⣿⣿⣿⢡⣶⣶⡝⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣷⣄⣠⠞⠁
⣿⣿⣿⣿⡘⠿⠿⣣⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠙⠁
⣿⣿⣿⣿⠟⣿⣟⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃
⣿⣿⣿⣿⢸⣿⣿⠇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⣿⣿⣿⣿⣶⣭⣥⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⣿⣿⣧⣿⣤⣤⣤⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠠⡢⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠙⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡠⡪⡑⡐⣌⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣔⠔⠌⣮⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠽⠋⠛⢦⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⠪⠢⢙⡾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠄⠉⢶⣄⠀⠀⠈⠳⡄⠀⡠⡐⠑⢕⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣪⣰⣻⡷⣄⠀⠀⢈⢊⢀⠈⣌⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡿⡷⣻⠛⢿⣧⠕⠕⢤⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠔⠀⠀⠀⠀⣰⣿⣽⡇⠀⠀⠀⢨⢞⡼⠁⠀⠀⠈⠙⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⣠⢋⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⢸⡃⣾⠁
⠀⠀⣠⣶⣶⣶⢤⡀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠔⠁⠀⠀⠀⠄⠔⡀⣀⣀⣸⣽⠃
⠀⠀⢻⡨⢦⡈⢳⡹⡄⠀⠀⠀⠀⢀⠀⡠⠃⠀⢀⣀⣠⢵⣾⡿⠟⠛⠋⠉⠁
⠀⠀⠀⠳⢭⡝⣦⠗⡇⠀⠀⠀⡠⢊⠔⣀⡤⣚⣉⡁⠈⣉⣀⣀⣒⠶⠶⠒⠀⠚⠛⠋
⢴⡦⣀⡀⢀⣹⢶⣷⣳⡶⢐⣻⣎⣡⠾⠥⠤⠤⣴⣶⡞⣛⠀⠁⣀⣠⠬⠤⠀⠉⣉⣉⣉⡛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢁⡈⣩⢵⣾⢿⡿⠿⠿⠛⠿⣿⣶⣶⣤⣄⠈⠙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣣⠤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣷⣦⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣀⣠⣤⣴⣾⣿⣶⡶⢿⣿⣶⣤⣴⣶⣄⣀⠀⠀⠙⢧⡀⢻⡏⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠛⠉⠉⠉⠉⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠢⣌⠳⡜⡇⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⣀⣤⣤⣶⣦⣤⣤⣾⣿⣶⣶⣶⣶⣶⣦⣤⣴⣶⠶⢦⡄⠰⣿⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠛⠉⠉⠉⣩⣽⣭⣉⣉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣠⡖⢸⡏⢷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣿⡿⣋⣞⣿⢹⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣙⡛⠳⣦⣤⣤⣤⣤⣤⣶⣶⣦⣶⣾⣿⡿⠛⣛⠍⣠⣾⣿⣿⣟⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣟⣋⣈⣉⣉⡭⠍⣉⣡⡭⣖⣠⢴⣪⣵⣿⣿⡿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠿⢿⣿⢿⣿⣿⣿⣿⢟⣩⣽⣿⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠾⠿⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣽⣯⣿⣿⣿⣯⣯⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠠⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⢠⣄⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠙⠏⠁⠀⠉⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⠟⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢴⡂⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⡍⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣤⡖⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣩⣥⣴⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠛⣛⣭⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⣠⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠠⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠰⠃⠂⠈⠀⠂⠀⠠⠀⢀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⡀⠀⢀⣀⡃⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠉⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣠⠀⠸⠠⠄⠀⠀⠀⠂⠀⠀⠀⠒⠀⠀⠖
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠐⠀⠂⠀⠀⠀⠂⠀⡠⠀⠁⠀⠀⢀⠀⠀⠀⢂⣀⠀⠀⠁⠀⠈
⣿⣻⣿⣻⣟⣻⣻⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠁⠀⢀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠤⠄⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡌⢿⣿⣿⢛⡛⡟⣿⡟⡟⡿⣛⣿⢻⠟⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠟⠁⠀⠈⠹⣿⣿⣿⣿⣿⡈⣿⣿⢸⠇⡇⣿⡇⡇⡇⣿⣿⢈⢺⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⡄⠀⠀⠀⢀⣿⣶⣿⣿⣿⡇⣿⣿⠶⠶⡿⡶⠾⢷⡷⠶⣾⣾⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣫⣿⣆⠀⢠⣿⣯⣿⣿⣿⣿⢁⣿⣿⡇⣾⡇⡇⣛⣸⣜⡛⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⢃⣾⣿⣿⣧⣿⣧⣧⣿⣿⣭⣥⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⡆⠀⠀⢠⡄⠀⠀⣰⠀⠀⠀⣤⠀⠀⠀⣄⠀⠀⢠⡄⠀⠀⢠⡄⠀⠀⢰⡀⠀⠀⣤⠀⠀⠀⣦⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣷⠀⠀⣾⡇⠀⠀⣿⡇⠀⢠⣿⡀⠀⢸⣿⠀⠀⢸⣿⠀⠀⣼⣧⠀⠀⣿⡇⠀⠀⣿⡆⠀⢰⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⣿⣿⠀⠀⣿⣷⠀⢰⣿⡇⠀⢸⣿⡇⠀⢸⣿⡄⠀⣾⣿⠀⠀⣿⣿⠀⠀⣿⣷⠀⢸⣿⡇⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡆⢠⣿⣿⡄⢰⣿⣿⡀⣸⣿⣷⠀⣼⣿⣧⠀⣾⣿⡇⢀⣿⣿⡄⢠⣿⣿⡄⢰⣿⣿⠀⣸⣿⣧⠀⣼⣿⣇⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡇⣿⣿⣿⠀⣿⣿⣿⠀⣿⣿⣇⢸⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⡇⣿⣿⣿⠀⣿⣿⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢰⣿⣿⣧⣼⣿⣿⣧⣼⣿⣿⣧⣿⣿⣿⣤⣿⣿⣿⣴⣿⣿⣷⣼⣿⣿⣧⣼⣿⣿⣧⣾⣿⣿⣧⣿⣿⣿⣤⣿⣿⣿⣴⣿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠛⠛⠙⠙⠛⠛⠉⠛⠙⠋⠉⠛⠛⠋⠉⠛⠛⠉⠙⠛⠛⠙⠙⠛⠛⠉⠛⠛⠛⠉⠛⠛⠋⠉⠛⠛⠋⠉⠋⠛⠉⠙⠛⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⠇⢸⡹⠠⣩⠄⠳⠠⢧⢸⠳⡇⠗⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢀⣤⣀⠀⢤⢠⠄⣤⣤⢄⡤⣄⡄⢠⣤⢠⣄⡄⠀⠀⠀⠀⠀⠀⣮⣹⣿⠟⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⡀⠂
⠀⠀⠀⠀⠀⢠⣤⣤⡤⢠⣤⣤⡄⣤⣦⣤⡄⣤⣤⣤⡄⣤⣤⣤⢠⣤⣤⣤⢠⣤⣤⡄⢠⣤⣤⡄⣤⣤⣤⡄⣬⣤⣴⣀⣤⣤⣤⢠⣤⣤⣤
⠀⠀⠀⠀⠀⢸⣿⣿⡗⢸⣿⣿⡇⣻⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿⣿⢸⣿⣿⡇⢸⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⢠⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⡇⠘⣿⣿⡇⢹⣿⣿⠃⢻⣿⣿⠁⣿⣿⣿⠈⣿⣿⡏⠘⣿⣿⡇⢸⣿⣿⠃⢹⣿⣿⠁⢻⣿⣿⠀⣿⣿⡟⠈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⢸⣿⣿⠀⢸⣿⣿⠀⣿⣿⣟⠀⣿⣿⡇⠀⣿⣿⡇⢸⣿⣿⠀⢸⣿⣿⠀⢸⣿⣿⠀⣻⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⢻⣟⠁⠈⣿⡟⠀⠘⣿⡏⠀⠙⣿⠃⠀⢹⣿⠃⠀⢻⣿⠁⠀⢿⡟⠀⠈⣿⡏⠀⠘⣿⡇⠀⠹⣿⠃⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⢀⡀
⠀⣰⠏⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠰⠋⠀⠙⠖⠋⠁⠀⠈⠉⠉⠋⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠁

⠀⠏⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡳⣽⣿⣿⣷⠀⠸⣿⡟⠹⡁⢣⠀⠀⠀⠀⠀⠆⠈
⠘⡄⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠅⢀⣀⣈⣡⣀⠡⠈⢆⢲⣄⡀⠀⢘⡄
⠀⠹⣼⣟⣛⠀⠒⠲⠄⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣧⡈⠤⣿⣿⣧⡀⠻
⡆⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠤⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣷⣤⡈⠹⣿⣿⣦
⢿⡀⠀⠀⣉⠁⠀⠐⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠋⠁⠀⠈⠻⣿
⠸⣧⠳⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢻⣧⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣿⣷⣤⡀
⠀⠀⠙⠀⣀⢻⣆⠀⠀⠀⠀⠀⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⡀
⠀⠀⠀⢸⣿⣿⠟⠂⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣄⣶⡆⡀⠀⠀⡀
⠀⠀⠀⠀⠛⠓⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡱⣄⠀⠃⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠘⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉⠉⠉⠉⠉⠙⢻⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⣷⣄⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⣁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣦⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠆⠀⠀⣠⣤⣤⣤⣤⡄⠀⠀⠸⣿⠄⠀⠀⢀⣿⡇⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⠇⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠉⠉⠉⠁⠀⠀⠀⠉⠀⢀⣠⣾⡿⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⠿⢋⣤⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠁⠀⢀⣤⣶⣶⣶⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⢸⡿⣡⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⣰⣿⢋⣿⡟⠁⠀⠀⠀⠀⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⠀⠀⠀⠁⢿⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡿⠋⣰⣿⢃⣾⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⣿⡀⠀⠀⠀⠈⠙⠛⠛⠛⣹⣿⠟⠙⠛⣋⣉⣉⣉⠀⠀⣰⣿⣧⣬⣯⣹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⣤⣤⣤⣤⣤⣶⡿⠟⠁⠀⠀⣰⣿⢛⣿⡟⠀⠀⠈⠉⢉⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣰⣿⢃⣾⣿⣀⣀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣬⣟⢻⣿⡟⠁⠀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢙⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣴⣧⠀⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



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

⠀⠀⢠⣴⣶⣦⣤⢤⣤⣤⣤⡄⠐⣖⢻⠓⠒⠐⠒⠒⠂⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠐⠒⠂⠐⠒⢶⡶⠶
⠀⠀⢐⣒⣒⡇⣀⢀⡀⠤⠀⡇⠉⠉⣘⣤⢤⣠⣤⣤⣤⣤⣤⣶⣶⣤⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⣰⣖⣶⡿⠿⠿⠿⠛⠋⠇⡹⢟⣿⣿⣾⠿⠿⠿⠿⠭⠭⠭⠯⠿⠯⠭⠽⠥⠤⠄⠀⠀⠀⠀⠀⢀⡴⠊
⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⡟⠟⠋⠩⠭⠍⠉⠁⠀⠀⠠⠀⠤⠤⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠍⠉⠉⢀⠀⠒⠒⠂⣐⠂⢐⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣄⣀⣀⣀⣀⡀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣒⣒⣖⢒⢒⣒⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⢠⠒⣄⡠⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡠⠄⢘⠃⠀⠀⡖
⠀⠀⠈⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠘⠃⡼⠀⠃⣠⠁
⠠⡞⣒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠔⣦⡉⠁⠀⡀⣾⠀⠀⠠⡠⢾⡨⢈⡃⢖⢀⠇⣬⠀⠤⠒
⠀⠉⠉⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⢀⡈⠁⠀⠘⠇⠀⠀⠀⠀⠀⠘⣍⣜⢂⡸⢈⠱
⠀⠀⢰⢊⡔⠤⣀⠀⠀⠀⠀⠀⠀⠠⣴⡇⢈⡙⠀⢰⠂⠀⢀⣀⠰⠞⠀⠀⢀⡯⠑⠨⢧⠌⠂⠈⠒⠆
⠀⠀⠀⠉⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠹⡚⠁⠀⡄⡀⠀⠙⠉⠀⠀⠀⠀⠀⠀⣀⡀⠈⠗⡀⠢
⠀⠀⠀⠀⣀⠀⠀⠀⡇⠀⠀⢀⡀⠀⠀⠀⡿⠢⠄⢽⣃⠀⠀⠚⠗⠀⠀⠀⠀⠘⢤⠳⢰⠶⠅
⠀⠀⠀⠀⢀⡷⡒⢂⠅⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⢠⢤⣠⠏⠀⠀⠖⠆
⠀⠀⠒⢦⣮⠀⡁⢰⠁⣙⢠⠀⠠⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⢦⠀⠀⠀⠀⠀⠀⡰⠀⡀
⠀⠀⠀⢸⣤⢀⡅⠸⣠⡝⢰⠀⠀⠀⠀⠀⠀⣔⠢⠔⠒⠈⠀⠁⠀⠀⠀⠀⢀⣸⠄⢠⠈⠒⠀⠀⠐⠚⠁
⠀⠀⠊⠉⢣⣰⠁⡜⢂⡄⠃⠀⠉⠓⠂⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠋⠓⢭
⠀⠀⠀⠀⡰⠋⠒⢡⡄⠁⠘⢆⠀⠀⠀⡠⠴⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⡈⠇⠀⢠⠀⡀⠀⠀⠙⠹⡄
⠀⠀⠀⠀⠀⠀⢯⡘⠤⣤⣜⠀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⡀⣲⠀⠇⢰⠂⢀
⠀⠀⠀⠀⣀⡀⡰⠁⠀⢡⠤⠀⠀⠀⠀⠀⠀⠣⠎⠑⠤⢤⠀⠀⠀⠀⠀⠀⠀⢒⡔⢩⡂⠰⠄⢤⠀⠁⠴
⠀⠀⠀⠐⢧⠭⡁⠀⠀⠀⠀⠀⣀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⡿⠶⢠⡄⢛⠀⢤⠐⠇⠐⠚
⠀⠀⠀⠀⠀⠀⡇⠀⡀⠣⠀⠀⠈⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣷⠆⢠⡂⢙⠠⠎⡘⠅⢀⢉⠁
⠀⠀⠀⠀⠐⢍⡵⠒⢍⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢦⡮⠀⡌⢁⡄⠓⠠⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡖⢢⡀⢶⠈⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢿⣿⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣶⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢟⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢶⣄⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣤⣽⠆⠀⣼⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠛⣳⠆⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠋⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⣿⣿⡿⠁⢠⡴⠖⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠠⠶⣄⠙⢿⣿⠟⠀⠀⠀⠀⠀⠀⢰⣿⣁⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⢠⣿⣿⠃⠀⠈⠙⠛⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⣀⠀⣸⣿⠏⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣬⠋⠀⠿⣯⡀⠀⠘⣧⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠙⠛⠁⠀⠀⠒⢸⣷⠀⠀⠈⠻⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠻⠿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠊⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠷⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣠⡴⠟⠋⠁⠀⠀⠀⠀⣰⡟⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠈⣱⠟⠛⢷⡄
⠀⠀⠀⠀⣠⡶⠟⠉⠉⠉⠉⠀⠀⠀⢀⣠⣶⣦⣴⣟⡀⠀⠀⠙⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠟⠋⣿⠀⠘⢳⣾⠋⠀⣠⡞⠻⣦
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠠⣾⠛⠉⠀⠀⠈⣹⡿⠛⠂⠀⠀⠙⠻⢶⣭⣙⣛⣻⣿⡿⠛⠛⠛⢿⡍⠀⠀⢀⣿⠆⠀⠀⠙⠳⢾⡋⠀⣰⡟
⠀⠀⠀⠈⢻⡟⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⡆⠀⠀⣻⣦⣴⠟⠁⠀⠀⢀⣤⣀⣈⣿⠿⠋
⠀⠀⠀⠀⠈⠻⣦⡖⠀⠀⣴⠂⣴⠟⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡾⠃⠉⠁⠀⠀⠀⣠⡾⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡴⠟⠛⠉⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢶⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⣴⠾⢶⣄⠀⠀⠀⠀⣿⡷⢶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⣠⡾⢃⡴⠋⠹⣧⡀⠀⠀⢹⣇⠀⠀⠈⠉⠙⠛⠳⢶⣦⣤⣄⣀⡀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⣠⡾⠋⡠⠋⠀⠀⣰⡟⠻⢦⣄⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⢻⣶⣦⣤⣾⠃
⠀⢀⣴⠏⣠⠞⣡⣜⠳⣾⠋⠀⠀⠀⠙⠷⣿⣍⡛⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠈⠻⣧
⣰⠟⣡⠞⠉⡞⣡⢘⡶⠻⣦⡀⠀⠀⠀⠀⠀⠈⠛⢦⣬⡿⠛⠻⠶⢶⣤⣤⣤⣿⠁⠀⠀⠀⢹⡇
⠛⢶⣧⣤⣴⠿⠋⠉⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠈⣽⠏⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⣦⣀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢯⡀⠀⠀⠉⡿⠛⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⠓⢦⣀⠉⠿⣤⡾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣌⡑⠲⣎⡁⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡙⠢⣼⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣷⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠞⠛⠏⠉⠍⠉⣉⠋⠉⠫⠽⡶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠏⠀⣁⣠⠀⠀⠘⠳⢤⠄⠀⠀⠈⢈⡒⠟⣻⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡟⣿⣿⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠹⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⢏⣿⣏⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⣦⣷⣹⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠇⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⣸⣿⣾⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⠁⠀⠐⠚⢀⠀⠀⠀⠀⠀⠼⠀⠀⢰⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⣱⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣴⣿⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡏⢱⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡞⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡷⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠿⠛⠛⣹⣛⣉⣴⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣰⢏⢴⡿⢿⠉⠀⢠⡆⢺⣿⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣦⡀⠀⣾⣧⣤⣾⠃⠏⠀⠀⠸⠑⣿⣿⢇⣿⣿⣿⣿⣿⣿⡧⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⡟⢻⣿⡟⠈⠀⠀⠀⠀⢰⣿⣿⣻⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣾⣿⣟⢹⢿⣿⡧⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠋⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣮⡻⣿⣿⣿⣿⡿⠿⣿⣛⡻⣿⣿⣤⡤⡄⠀⠀⠀⠀⠠⣿⣟⣫⠽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣶⡟⣿⣿⣿⣿⣶⣾⣿⡿⠶⠿⠉⡁⣀⣀⠀⠀⠀⣴⣦⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠟⠰⢶⠆⢁⣛⠁⢟⠁⢴⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⠦⠀⠤⠥⠄⠸⢶⣀⡀⢘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣶⣶⣶⣦⣀⣉⠁⢠⢩⡽⣿⣿⣿⣿⣿⢻⣿⣷⣿⢹⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⠻⠿⠿⠿⠻⠇⠤⠄⠚⢳⢿⣿⣿⣿⡭⢿⡿⣟⣶⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠈⠉⠙⠛⠛⠛⠛⠉⠀⢠⣤⣤⣴⣶⣶⣶⣮⣿⣻⣿⠛⠿⠻⠗⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⠿⢿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠆⠃⠀⠄⠈⠦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⡄⠀⠰⠀⢂⠀⡆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣀⣩⡴
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣠⠤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠈⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠘⠁⠠⠊⠡⡊⠐⢡⡊⠚⠉⣀⣤⡿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣐⡢⠔⣱⠀⠐⠐⢀⣐⡡⢄⠠⢲⡩⠄⡴⠶⣁⡀⠔⠒⠀⠶⢤⣦⡀⠀⠀⠀⠠⠐⠀⣁⡤⡂⢀⡈⠁
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⣿⠀⡤⠀⠯⢀⡂⠴⠒⠳⢤⣄⠰⡗⠑⣶⡿⠒⣠⠖⠖⠃⠰⢶⣮⢴⣂⣰⠝⠛⠉⠁⠄⠀⢙⡾⢆⠐⢪⣚⠄⠈⢁
⠀⢀⠀⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠍⠀⢞⠫⠥⢤⣅⡐⠛⣲⠄⠀⠀⠀⠀⠈⠁⠐⣀⣴⠊⠱⡷⢶⠛⠁⠀⠀⠀⢂⢐⠂⠊⠉⠀⠁⠰⠫⢶⡀⠰⠅
⠆⠀⠔⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⢀⡠⠀⠀⠀⢂⡱⠄⠂⠀⠀⠄⢀⡠⡞⢁⣏⡴⠉⠀⢸⡄⠸⠰⠚⠀⠀⠀⠴⠀⣔⣦⠀⠒⢤⣌⣱⠀⡄
⠀⠀⠀⠀⡀⠀⠠⡄⢶⣶⣶⡾⣿⣿⣿⣏⢉⠁⠒⠐⠾⠐⡒⣀⠀⠒⢐⡂⠢⠒⡫⢸⣯⠞⢠⠞⣁⣠⡄⠱⠿⣷⣴⣄⠈⠀⠀⣐⣀⣠⣄⡉⠌⢷⣌⠭⠉⠁⠁
⢠⢠⠀⠀⠁⣬⢼⡇⢶⣽⣿⣿⣿⣿⣿⡏⠀⡀⠀⠀⠨⢟⣄⣸⣊⠉⠉⢀⠕⠋⠀⠊⢉⡰⢃⡇⠈⠁⠠⠔⠶⢴⣶⣭⣍⠙⠀⠌⣉⣽⠀⠉⠚⣂⣳⣶⡌
⠀⠇⠀⠀⡰⢄⣼⡿⣿⡯⣿⣿⣿⣿⣿⡧⠀⢊⠁⣙⣋⡰⠶⠀⠀⠻⢁⣀⠤⢠⡾⠀⣁⠗⠚⠻⢤⣉⠈⠋⠑⠨⠙⢷⣞⡳⣡⣠⠄⡀⣷⡑⢠⠢⢀⠄
⠀⠀⢠⠾⠓⠉⢸⡓⠿⣿⣿⣷⣿⣿⣿⣇⠈⠉⣠⡄⢀⣀⣀⣠⡶⠓⠉⣀⡖⡫⠶⠾⣅⠀⠀⠣⠀⠠⣀⡠⢄⠠⠖⠀⠉⠉⠉⠢⠔⠂⠈⠉⠐⠂⠁
⠀⠀⠀⢀⠀⢋⢼⣧⣿⣿⣿⣿⣿⣿⣿⣇⠀⠉⢐⣀⣀⣨⣟⣁⠀⠀⠚⠉⠀⠀⠞⠉⡉⣤⡄⠰⡲⠤⣤⡀
⠀⠈⠀⠋⢀⠞⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠚⠒⠈
⠀⠀⢠⢂⠞⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠁⠘⠋⠉⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⣀⠊⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢰⠲⢄⡖⢦⣰⣄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠒⠊⠓⠋⠘⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢸⡇⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠘⠇⠀⠸⠇⠀⠀⠀⠀⠰⠆⠸⡇⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠖⠛⠒⠶⢦⣤⡀⢠⡴⠶⠶⠶⠶⠶⠶⠶⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣄⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⢰⡄⠀⠀⠀⠀⠀⢸⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣿⣹⡇⠀⠀⠀⠀⣤⠀⠀⢸⣇⣈⣀⣀⣀⣀⣀⣀⣸⡷⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣈⣉⣀⣀⣀⣤⡴⠿⣦⣀⡀⢹⡏⢉⡉⢉⡉⢉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⠉⢉⣉⠉⠉⠀⣀⣀⠀⢹⣧⣼⡇⠘⠁⠘⠇⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣿⠀⠺⠻⠀⠀⠀⠻⠟⠀⢸⡇⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣿⠀⠀⠀⣀⣄⣀⠀⠀⠀⣸⡷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠿⣧⡀⠐⠋⠉⠙⠃⢀⣴⠿⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⢸⡟⠷⣶⣶⣶⠶⢻⡿⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⢸⣧⡾⣏⠀⣹⡷⣾⡇⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠛⢿⡄⠈⢁⣀⣀⣀⣀⣀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡇⠀⢸⣏⣉⣉⣉⣽⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣈⣷⣀⣈⣉⣉⣉⣉⣉⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣿⣿⢿⠟⠿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣏⣉⣽⣽⣇⡄⢠⣇⡤⣧⣠⣽⣿⣿⣿⣯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⠛⣿⠍⠙⠁⠻⡌⠛⣞⣉⣡⠿⡽⢿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠲⡿⢿⡿⠛⣻⠛⣿⠿⣶⡾⠷⣿⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣿⢿⣿⣿⠿⢿⡿⣿⢻⡟⡟⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡏⠁
⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⣿⣿⢿⢶⣿⣿⢿⣿⡿⠛⣿⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⠿⠿⣟⠿⠳⢿⣷⣶⣾⡾⠟⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣟⣻⣿⡿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⡿⠟⣣⣿⣿⣿⡄⠀⣿⣿⣬⠻⠋⣿⡙⡿⣷⠘⠎⢶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣾⣟⣨⣉⣋⣙⣷⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⣷⣦⣭⣽⣇⣩⣤⣷⣾⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⠿⣻⣿⠿⣿⣿⣟⣿⣿⣿⣿⣿⠻⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⡐⢿⡇⢻⣋⣴⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⣾⣇⣻⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡗⢼⡇⢉⣶⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣼⣿⣿⣼⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣾⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢒⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠚⠀⠉⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⣍⡑⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣠⠀⢄⡀⣧⡖⠼⠳⠒⠉⠉⠁⠀⠈⠩⠙⢖⡀⠀⡴⠏⡌⢼⢈⠀⠀⠀⠀⠀⠀⠀⠀⢲⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢄⢈⡀⠀⢈⡷⣌⣿⡿⠛⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢀⣔⠠⢊⠔⣠⠗⡜⠘⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠙⢶⡕⢍⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣼⠀⠓⡣⠺⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣸⠏⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⠈⠠⡃⣰⡱⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⢀⡤⠚⣕⡈⣀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠒⣕⣢⣀⠈⢡⠋⡘⢧⠀⠀⠀⠀⠀⢰⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠇⠀⠀⠀⢘⣩⠒⢩⣿⢱⣨⠃⠀⠀⠀⠀⠀⠀⢸⣤⣪⣇⡝⢯⠓⠀⢢⢁⣤⠃⠀⠀⠀⠀⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡜⠀⠀⠀⠀⡇⠐⢍⣘⡿⢾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⣟⣃⢊⡇⠀⠈⡽⢎⠙⠆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⡄⢱⡇⠈⠃⠇⠀⠀⠀⠀⠈⠒⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢉⣙⠐⠀⠀⠈⠉⣰
⠀⠀⠀⠀⠀⡟⠛⠇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⢔⡉⡇
⠀⠀⠀⠀⠀⠓⠶⡄⢎⢸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠑⢰⡅⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡘
⠀⠀⠀⠤⠀⠓⠆⠰⠖⠀⠹⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢫⠎
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢢⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠄⠀⠠⢄⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⡀⠀⣀⠐⣥⠛⠊⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⡞⠁⢀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⡄⠁⠀⠈⢆⢀⡀⢠⣿⠈⠀⠀⠀⠀⠀⢤⠤⠤⠄⠙⠛⢰⡊⢀⠌⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠆⠀⣀⠦⡟⠀⠀⠀⣀⡀⠀⠈⠑⠒⠲⠶⢦⡀⣤⡖⠃⣧⠤⠖⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠦⠤⠄⠀⠚⡁⠪⣇⠀⠀⣰⠀⠀⢉⣨⠾⢟⠋⡸⢻⣍⡉⠀⡼⠈⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠁⠂⡉⠹⠁⣴⠬⠴⠀⠚⠋⠛⠂⠈⠯⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡄⠀⠀⠀⠀⠀⠀⠳⡆⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣀⠰⣏⠀⠀⠀⣀⣀⣤⣼⠃⢿⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⠈⠳⠚⠋⣡⠤⠶⢤⠀⠈⠙⠓⢦⣀⠀⠀⢸⡍⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠸⣅⣀⣤⡞⠁⠀⠀⠀⠀⠈⠳⠞⢋⡴⢦⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢹⠀⠳⡄⠀⠀⠀⠀⠀⠀⠐⣏
⠀⠀⠀⠀⠀⠀⢰⡏⠙⢧⣤⡴⠏⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠸⣇⣴⠖⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⣤⣈⣹⠞⠙⡆⠛⠶⠴⠏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣀⡾⢀⣤⠆⠀⠀⠀⠀⠀⠈⣇⣀⡴⢳⠞⠛⢳⣄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠄⠀⠀⣏⠉⠙⢿⡁⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠏⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠳⠞⣡⠴⢦⡀⠈⠓⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣘⡇⠀⠀⠉⠳⢤⣀⡀⠀⠀⠀⠀⣀⣠⡤⢦⡈⢷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⣷⢠⡟⠋⠉⠉⠀⠀⠀⣹⠆⠉⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠸⢦⠀⠀⠀⠀⠀⠀⠙⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠞

⠀⢶⣶⣄⠀⠀⠀⠀⣴⡄⢀⠀⣤⡀⡀
⠀⠀⠫⠻⣶⣦⡀⠒⢟⣿⣤⣆⣼⣿⣿⣆⡑⢀⣰⣤⡴⣤⣁⡐⣤⣴⡖⠰⣤
⠀⠀⠘⣷⠞⣿⢗⡛⡌⣹⣿⣿⠿⣯⠈⢹⠞⠿⢻⣴⣟⣿⣿⠐⢿⡿⣻⠃⠘⣣⣰⡆⠶
⠀⢤⡠⢄⡀⢺⠾⣷⣿⢿⣿⣏⢀⡩⡙⢷⠄⠀⠀⠈⠙⠉⠛⠀⢸⡇⢷⣤⡼⢫⡽⢃⠀⣴⠀⣄
⠀⠀⢙⢻⣯⡾⢷⣿⣾⣿⣿⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢿⡀⠱
⠀⠀⣄⠱⣾⠃⢺⣿⣿⠋⠁⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⢢
⠀⠀⣼⣄⠀⠉⣾⣿⡇⢼⡦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃
⠀⠀⠘⣿⣶⣷⣿⡟⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⢰⣾⡿⣿⣿⣿⡟⡟⢡⣄⣤⣀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠘⣿⣿⣿⣿⢋⡤⠸⣧⠄⠀⠀⠉⠙⠛⣷⣶⠶⠚⠂⠐⠂⠤⢀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣼⣄
⠀⠀⡈⢛⣿⢿⡇⣾⠗⠀⡿⠂⠀⠀⠀⠀⠀⠈⢷⠶⢦⡂⠀⠀⠀⠀⠠⢠⣄⣤⠾⠋⠀⠀⣠⡾⠈⣠
⠀⠀⠀⢈⣱⣾⡇⢿⢀⣾⠓⠀⠀⠀⠀⠀⠘⣀⣈⢠⣿⣦⣈⣿⠟⢣⡰⠿⢟⢻⣶⣦⣀⠈⢤⠛⠀⡿
⠀⠀⢀⠺⡟⢿⣇⡀⢸⣿⡆⢠⣴⠀⠀⠈⠛⢻⣮⠻⢯⠚⣿⢹⣴⣯⠿⠀⣾⢈⡿⠷⠹⣿⢣⡄⡘
⠀⠀⢺⠒⢿⣾⣿⢡⠈⠻⣇⠀⠀⠀⠀⠀⠠⠜⢻⠀⠀⢹⡏⠙⠁⡇⠀⠀⢩⡆⠰⠑⡆⠚⢇
⠀⠀⠙⠀⢨⣽⣿⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⠁⠀⠀⣸⡏⢼⡂⠙⢦⠈⢥⠀⠆
⠀⠀⣤⠀⣾⣿⠳⣷⣄⣀⣼⣏⠀⡤⠀⠀⠀⢀⠢⡄⢰⠃⡐⠀⠀⢀⠀⢠⡟⠙⠾⠹⡆⠀⢣⢈⣆
⠀⠀⠈⣻⡉⢿⠀⢹⣿⡏⢙⡿⠀⢼⠀⠀⠀⡻⡄⠀⢸⠀⠀⠀⠀⠀⠀⣼⡇⠀⢠⡆⠘⠀⢀⡤⠡⡄
⠀⠀⠸⠿⣷⣿⠐⢈⣿⣿⣀⠀⠀⣼⡀⠀⡸⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠃⠀⠘⣧⠀⢠⠘⢃⡀
⠀⠀⠶⢀⣿⣟⢠⣼⣿⣿⡏⠀⠀⠸⣀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠐⡏⠀⠂⣸⡇⠀⠑⡀⠘⡇⠀⠇
⠀⠠⢰⡇⣿⣧⠋⢹⣿⣿⡇⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⡄⠀⠳
⠀⢀⣾⣿⣿⣿⠆⠈⢿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⠟⠀⠀⢈⡤⠀⠀⠰
⠀⠸⣿⣿⣿⣿⠀⢐⢾⡟⢿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠈
⠀⠀⣿⣿⣿⡿⠀⠠⠛⡁⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢀⣶⡏⠀⠀⠀⠀⠀⢀
⠀⠀⡿⢻⣿⠗⠰⢷⡄⠐⣀⣤⡅⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠐⠄⣸⠇⠀⠀⠀⢠⣶⠟⠇
⠀⠀⠛⣼⣏⣤⣤⣤⣼⣶⣿⣿⣟⠦⣀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⣻⣿⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠈⠀⢨⡄
⠀⠀⠚⠛⣿⡿⢻⡿⣷⠶⠿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣿⡤⠄⠀⠀⠀⠀⠀⠀⠂⠻⣤⣄⡀
⠀⠀⠀⠠⠘⣷⢿⣿⡗⠀⠈⠚⢀⠐⣧⢴⣄⣄⠀⠈⠉⠁⠀⠁⡈⢠⣡⣗⠌⠀⠀⠀⠀⠀⠀⠀⣹⣿⠆
⠀⠀⠀⠀⠂⢸⡾⠟⣏⣀⠠⠀⣈⡀⠀⠀⠈⢿⡻⣎⠊⣡⣅⡘⢰⠈⡳⣿⣄⠀⠀⠀⢀⠀⢀⣾⡟⢳⡶⣤
⠀⠀⠀⠀⠀⠋⠀⠀⠸⢿⣿⠲⠁⠙⠚⣿⠻⢦⡗⠸⡇⠀⣿⡇⢠⣤⢠⡿⢏⠂⣤⣄⠎⠈⡳⡈⠢⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣠⣸⢷⣬⣄⢸⡇⠀⠀⢠⠈⠃⢈⢹⠈⠀⠘⠿⣿⠏⢀⡀⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⢀⠍⠞⠘⢃⣀⣄⣀⣀⡶⢀⣈⣠⢰⣤⢰⡄⠚⠿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⢠⡖⠳⠃⠀⣿⠋⠠⠁⠘⠏⠈⠙⢻⠀⠀⠀⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠐⢠⡄⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⣄
⠀⠀⠀⠈⠂⣩⣤⠤⠀⠀⠀⠘⠂⠳⢆⡀⠀⠀⠀⠀⠀⡆
⠀⠀⢀⠀⣀⠀⠙⣭⣶⣦⣴⣄⣀⡀⢈⠱⣀⠀⠀⠀⠀⢻⠀⠀⠀⢀⡤⠖⠒⠒⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⢿⣿⡏⠀⠱⢀⡀⣈⡡⢈⣙⢳⣷⣎⠑⠱⠶⣿⣼⠂⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣠⣤⣶⡆⠢
⠀⠀⠸⠸⡿⠁⠄⡀⣸⣄⢈⠑⠀⠒⠚⠈⠛⣾⡈⣧⣿⣿⡣⣼⡆⠀⠀⠀⠀⠀⢈⠢⡀⢄⡀⠀⠤⢀⠠⣈⣦⣿⣿⣿⠇⡀⣽
⠀⠀⠀⣇⠇⠞⡴⢣⡋⡀⠀⠃⣀⣤⣴⡾⣿⣟⣻⣾⣿⣱⡇⢁⣿⢀⣀⣀⣀⢀⣀⢁⣪⣀⣨⣙⣓⡂⢍⣺⠃⠀⠀⢙⠋⠩⡞
⠀⠀⠀⣿⠀⣝⣴⣿⣏⢤⣶⡾⣽⡿⢟⣩⡵⠋⢹⣿⡽⡟⠿⣿⣿⣽⠆⠂⢬⠪⠀⠻⢭⡂⠠⡉⠛⣿⣿⣿⣶⠮⢑⣪⣿⢎⣤
⠀⢀⡀⠈⠤⣈⣿⣿⣷⣾⣿⡿⣫⣿⡝⠶⠗⢶⣿⡟⣣⣽⡤⠛⡿⠉⠫⠉⠛⣧⡑⣔⢿⣿⣷⣶⣴⣶⣿⣿⣿⣿⣶⣿⣶⣿⡿
⠀⠀⠀⠀⠡⠁⠹⣿⠛⢿⣻⣯⠃⡞⠀⠀⣤⣿⣛⣣⢷⣿⣄⣾⣷⣶⡞⣇⠀⠇⣽⢿⣷⡋⠀⠈⠉⠛⠛⢛⠉⠙⢹⣿⣿⣘⡃
⠀⢀⣱⡄⣟⠀⠀⠓⠀⢸⣿⡧⠈⠱⠀⣮⣎⡹⢏⡿⢿⣿⣿⣿⣿⣿⠛⣣⠈⣿⠿⠯⣿⣿⠷⣦⣴⣠⠌⠃⠀⠀⠈⣻⣿⡏
⠀⠠⠙⠡⠘⠁⠀⠀⠀⢸⣿⠁⠀⡐⢸⣿⡌⠉⡤⣿⣾⣿⣿⣿⡿⠁⢠⠇⢠⣿⡄⠀⢻⡟⠀⠐⠌⢹⣷⣄⠀⠀⣐⣾⡝
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡏⠀⢰⣿⣿⣿⡇⢸⠀⣿⣾⣿⣿⡤⠀⠀⠖⠀⣾⣿⡅⠀⠈⡇⢠⢀⢂⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣷⣿⣿⣿⣿⡄⡾⣄⣿⣿⢻⣿⣗⠀⢸⡇⠀⠏⣿⡷⡀⠀⠀⣧⣆⣾⣿⡿⡿⣛⡿⠟⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠾⠧⢴⠏⣿⣿⣿⣿⣆⠀⠈⢿⡟⢻⣿⣿⣦⣿⡆⠀⡰⣿⣧⡀⠀⠐⣿⣿⣿⣿⢁⡔⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣆⠙⣿⣿⣿⣿⣦⣄⢘⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⠉⢻⣿⣿⣿⣟⣿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⠀⠻⣿⣿⣿⡬⠻⢿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⠀⠀⠁⠂⢿⣿⣿⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠨⠻⠀⠀⠹⣿⣿⡃⠄⠿⠀⢼⣿⣿⣟⣿⢿⣿⣿⣿⡏⢱⣐⣶⣰⣶⣾⣿⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡈⠀⡆⠀⢸⣿⣿⣿⡛⠹⣿⣯⣿⡶⠀⠙⠿⣿⣿⠟⠋⡔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠸⠀⠇⠀⠸⣿⣿⣩⠙⠀⠲⢽⡇⠹⠄⠀⠀⠘⣇⠰⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣿⡇⠈⠀⠀⠀⠀⠻⡆⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠘⠄

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

⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣿⣿⣿⣿⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢷⣿⣽⣿
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣶⣯⣿⣿⣯⣻⣿⣿⡿⣿⠿⢿⣿⣿⡦⡀⠀⢠⣶⣔⠲⢂⣤⣀⠀⠀⠀⠀⢀⣀⣀⣤⣾⣿⣿⣿⡿⢿⡿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⣤⡿⣿⣷⣿⣧⣼⣿⣤⣉⠟⢿⣭⣭⣭⣭⠭⠽⠿⠛⠻⠃⠐⠈⠙⠿⠻⣦⣾⣿
⠀⣾⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣳⠙⠿⣿⣷⣿⣿⣿⣿⣿⣿⠛⣙⣿⠿⢿⣿⣥⣴⠶⠃⠀⠒⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿
⠈⠙⠛⠋⠁⠙⠻⠿⢿⣟⣿⡟⢿⡿⠟⡽⡿⣿⣴⣴⣿⣿⣿⣿⡿⠿⠿⢿⡿⠿⣿⣿⣦⣤⡀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠠⢦⣛⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⡀⠀⠙⠻⢿⣿⣿⣿⣿⣯⣄⣁⣠⣤⣴⣷⣾⣿⢿⣶⣤⣤⣤⣷⣶⣶⠟⠋⠁⠀⠀⠀⢦⣤⣭⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣵⡀⠀⠀⣴⣿⣿⣿⡯⣿⣷⡾⠟⣿⣿⣥⣴⣶⣶⣶⣤⣄⣉⣉⠛⠛⠋⠁⠀⠀⠀⠀⠈⠉⠉⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡻⣿⠙⣦⣼⣿⡟⠃⢸⣇⣿⣿⣏⣹⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠘⣧⣿⣾⣯⣝⡿⣐⠛⠂⠙⠻⣾⡿⣦⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⢏⡀⠉⢈⣁⠀⣀⠀⠀⠐⡚⢷⠟⠃⠀⠈⡅⠉⠄⢐⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠙⠳⢖⡀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⣠⠞⠛⠈⠍⠉⠉⢹⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠗⠀⠀⠀⠈⠂⣄⢠⣄⢼⡄⠊⠀⠀⠘⢂⣠⣤⣴⣿⣿⣿⣿⣽⡿⣶⣶⣶⣦⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠗⠀⠔⠒⠛⠛⠉⠀⠉⠛⠛⠻⠟⠛⠙⢿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⡀⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⢰⢂⢄⠀⠤⣤⣄⣀⣴⣉⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠣⢙⡆⠻⢦⡀⠘⡴⠊⠉⢳⡄⠀⠀⠈⠺⢺⡗⡀⠀⢨⠻⠋⠙⡟⢢⡈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠁⢀⣀⣅⣸⡀⠀⠰⣅⡴⠄⠀⠀⠀⠀⠀⠀⢠⠃⡷⠀⠢⠀⠸⡇⣜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣦⡀⠀⢀⣸⣿⡧⠝⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⢧⠤⡤⣤⣼⣤⣤⢤⣤⠀⠀⠀⣠⠤⠴⠶⢾⠀⠉⢀⡰⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠉⠉⠉⠉⠉⠉⠉⢉⡏⠉⠉⠁⠀⠀⢀⣎⣡⡤⢲⣶⠋⠀⣠⣏⠀⠀⠀⠀⠈⠉⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⣀⣤⣶⣿⡿⠋⢴⣿⣿⣾⡿⢋⣩⣷⡄⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣼⠋⠀⣠⣔⣊⣀⡿⠿⡏⠀⠀⠀⠈⠙⠁⣰⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠤⡾⠓⠉⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠅⣼⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠁⠀⠀⠀⢠⣾⣿⣿⡿⠋⢹⠉⠙⡄⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣾⣿⣟⣫⠤⠤⠔⠒⠚⠉⠉⠀⠀⠀⠀⠀⢀⡠⠾⠟⠛⠛⠒⠈⠉⢇⠀⢩⠑⡄⡜
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣠⣤⣄⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡾⠋⢛⠿⣿⣿⣿⣿⣿⡉⣉⣙⡛⠻⠿⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠎⠀⠂⠉⠻⣿⣿⡇⠀⠀⣀⣉⡉⡚⠈⠑⡄⠀⠀⠀⠀⠀⠀⠸⠠⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡴⠁⠀⠀⠦⠤⡔⠁⢹⣗⣒⣂⣁⣭⣉⡇⠀⢀⠯⠽⠀⣒⣒⣒⣐⠁⢰⠤⠄⠬⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢃⠎⠀⠀⠀⠀⢢⠎⠀⣠⠎⠑⠒⠩⠭⣉⡉⡓⠈⠁⠀⢀⣀⡠⠤⠀⠘⠒⠊⠹⢹⢹⠉⠉⠐
⠀⠀⠀⠘⠮⠶⢌⣁⠎⣤⡀⠀⣀⠔⢁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠈⡏⠒⠎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣳⣤⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣟⣿⣿⢩⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠂⠀⢀⣼⣿⣏⣯⡷⠛⠀⠀⠀⠀⠙⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠠⣶⣤⣄⣴⣷⣦⠀⢠⣤⣄⠀⠀⠀⢰⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⢸⣿⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠿⢭⡿⢧⡄⠀⠀⢨⡏⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⡄⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠳⣦⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠉⠋⠉⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠙⣧⠐⣦⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠄⠀⠀⠀⠀⠀⠀⢱⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⡀⢂⣀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣍⠉⠉⠀⣖⠈⢉⡀⠀⠀⠀⠀⢀⠀⠀⠋⠑⢶⢤⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠉⢹⠀⡌⢀⠀⠁⠀⢰⠀⠀⠀⢠⠀⠀⠉⠒
⠀⠀⠀⠈⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡂⢢⡈⡋⠀⠀⢀⠃⠀⠀⠀⠈⡆⠀⠀⠈
⠀⠀⠀⠀⠺⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠉⠂⠀⣀⠀⠀⢉⢦⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠈⢹⣿⣦⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⣷⠼⠋⠜⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠠⢹⣯⡉⠠⠠⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣄⣀⠀⠀⠀⠀⠂⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠢⠤⠶⠦⠄⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠰⠀⡁
⠀⠀⠀⠀⠀⠀⠁⠛⠁⠐⠒⠒⠒⠂⠳⠐⠂⠐⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠤⡀⠀⢀⡤⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣃⢀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠃⠀⠠⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⠤⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠲⡾⠉⡀⠙⡇⠀⠀⢈⡁⠀⠠⠿⣶⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠚⠀⢦⡀⠀⢠⡇⠀⠳⣤⣤⠾⠁⠀⠀⠾⠃⠘⣆⠘⠒⠲⣄⠀⠀⠀⠀⠀⢀⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡤⠄⠀⢀⣠⣍⡛⢉⣠⠖⠚⠋⠉⠛⠲⢦⡀⢤⡀⠠⠟⠛⠒⠀⠸⣆⡀⠀⠀⣠⠍⡿⠳
⠀⠀⠀⠀⢠⣄⣠⠀⠀⠀⢀⡞⠁⠏⠀⠀⠀⢉⣤⠄⣹⠋⠀⠀⠀⢠⡆⠀⠀⠀⠈⢾⣃⡼⠶⠶⣞⠁⠀⠀⠙⡆
⠀⠀⠀⠀⠛⣿⠿⠉⠁⠀⠘⡇⠀⢠⠄⠀⠹⣏⣀⢰⠃⠀⠀⢰⡄⢸⣧⠀⣶⠀⠀⠀⢿⠀⠀⠀⠘⠇⠀⠐⢦⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⣀⣼⣄⠐⢦⣄⣁⣼⠀⠠⣀⠸⣿⣼⣿⣼⡟⢀⡀⠀⢸⢠⡞⠉⠙⡆⠀⠀⢋⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⠁⠀⢀⣀⠀⠀⣷⠸⡄⠀⠈⠳⣿⣿⣿⣿⠿⠋⠀⠀⡾⠋⣀⡀⠀⠁⢤⡴⢋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠾⠉⠉⢳⠈⣡⠶⠿⣄⠀⠈⠉⠉⠏⠉⠉⠀⣠⡞⠱⠋⠉⠉⢳⡀⠀⡷⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⣆⡀⠸⣄⣁⡼⠀⣿⠙⠲⣤⣤⣤⡤⠴⠚⠉⠳⣤⣤⠞⠀⢠⠇⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡁⠀⠀⡀⠀⠀⠈⠓⠚⠋⠰⡃⠀⠀⠲⣤⠤⠀⠀⣀⠀⢀⣰⠃⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠦⠤⠖⠛⠶⠤⠖⠛⠶⠤⠖⠛⠶⠴⠞⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠂⠀⢰⣦⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠋⢠⣤⡈⣘⣿⣿⣿⣿⣷⠀⠀⠈⢠⠐⢻⣿⣿⣿⣷⣾⣶⣄⢳⢲⣄⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢹⠋⠉⠉⠃⢸⣿⡇⣻⣿⣿⣿⣿⣿⣧⣤⣤⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠶⠶⡿⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢻⠛⠛⠛⠛⠛⠛⡿⣿⡿⢸⣿
⢠⣄⣀⣀⣀⠀⢸⠀⠀⠀⠀⢸⡿⠟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢉⣷⢸⣿
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢘⠋⢹⣿
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾⡇⢻⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⣿⠁⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⣷⠈⣧⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠀⠀⠀⠄⠀⢘⣴⠿⠀⢠⠈⡟⣷⣄⢀⣀⣀
⠀⠀⠀⠀⠀⠰⠶⢿⣿⣷⠂⠿⠇⠸⣦⣤⣶⣶⡶⢸⡗⠂⠀⠀⣿⠀⠀⠀⠠⢀⠉⠁⠠⠀⢸⣇⠨⣿⣿⣷⣉⠁⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⢠⣭⣽⣿⣿⣷⢺⡇⠀⠀⠀⡟⠀⠀⠀⠀⢀⣼⠃⠀⠀⠈⠁⠀⣉⣿⣿⣿⣷⣾⣶⣶⡄
⠿⠢⠀⠘⠛⢷⣶⣶⣶⣶⣤⣠⣤⣤⣟⠉⢹⣿⣿⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠉⠛⠙⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣽⣿⣿⡯⠐⢸⣿⣿⢹⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣷⣾⣶⣿
⠀⠀⠀⠀⠀⠠⠀⠈⢏⣿⣿⣿⣿⣿⡇⠉⢸⣿⣧⢸⡧⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢙⠛⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⣿⣿⣿⡗⠠⢸⣿⡿⢺⡷⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⣿⣿⣿⠁⠄⢸⣿⣏⢸⡧⠀⠀⢐⡁⠀⠀⠀⠂⠀⢀⣿⣿⣿⣿⣿⣻⢛⡏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡎⠍⠀⠀⣼⣿⣏⢸⣇⣬⢰⣨⠃⠀⠀⢨⠀⠀⠀⣿⣿⣿⣿⡿⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣠⠀⢀⣿⠿⠉⢸⣿⣿⣼⣿⡄⠀⠀⠨⠁⠀⠀⡿⠿⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠘⣿⣷⢿⣿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣜
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠑⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⣀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠲⠶⢤⣌⣳⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⢸⠛⣿⠷⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠺⠥⠤⠤⠤⠴⠶⠶⠾⠿⣷⠢⣄⢻⠀⠈⠳⠤⣄⠉⠓⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠙⢆⠣⣄⡀⠀⠀⠉⠒⢤⣀⠈⠉⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠒⢬⣳⠤⣉⣓⠦⣄⡀⠀⠈⠙⠓⠦⢤⣀⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠙⠢⢄⣈⠙⠒⠦⢭⣙⡛⠳⠶⣤⣄⣀⣀⣀⣩⡍⠉
⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠋⣉⣤⣴⣒⣶⣾⡇⠀⠀⠀⠉⠙⠓⠲⠾⠭⢿⣶⣦⣤⣀⣉⠉⣉⣹⣿⠆
⠀⠀⠀⠀⠘⢴⣶⣶⣶⡾⠷⠞⠛⠻⢤⣽⣦⣁⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡇⠀⠈⠉⠉⠙⠓⠒⠒⠦⠤⠤⠤⠤⠤⠄⠀⠀⠒⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⣽⣦⣈⠉⠛⠛⠶⠶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠵⣚⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠶⠦⠤⠬⠭⠽⠿⠷⠶⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠲⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠠⠐⠉⠀⠌⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠡⢀⣀⠴⡵⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢐⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⢄⣶⠠⡄⣀⣀⠀⠉⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣅⣀⠀⡌⠩⢕⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠦⣌⣉⣩⡀⠀⠀⠀⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢱⡀⠀⠀⠀⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢶⠀⢃⠀⢠⠟⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠈⣄⠀⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠀⠙⢓⡒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⠈⠁⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣽⣧⠀⢢⠀⠀⠀⠀⠈⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢂⢀⡾⠘⡷⢤⠗⠀⠀⡄⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠈⠀⠁⣑⠿⠴⠚⠤⣀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠈⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢺⠋⠈⢠⣤⠐⠒⠀⠙⠃⡁⠀⠀⠀⠀⢀⠄⠒⠉⠁⠁⠀⡀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠉⠀⠂⠇⠀⠀⠀⠀⠀⠈⣦⠀⣠⡶⣥⣦⢄⣸⠇⠐⢾⠻⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣷⢾⣖⠉⡛⡀⠀⠀⠢⢄⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠙⢄⠙⣧⠀⠀⠀⠀⠀⠣⣙⠢⣀⡀⠙⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⠀⠑⢌⠺⡄⠀⠀⠀⠀⠈⠓⢌⠃⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠁⠀⠀⠀⠀⠀⠀⠀⢀⣨⠤⠇⠀⠀⠀⠶⣀⠳⡈⢢⡀⠀⠀⠀⠀⠈⠂⣸⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢞⠉⠉⠉⠉⠉⠉⠉⠐⣷⠛⠉⢛⡻⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠛⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠆⠰⠂⠀⠐⠀⠐⠐⠀⠀⠐⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⡿⠛⠟⣋⣭⣧⣤⣦⣶⣶⣷⣾⣭⣽⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢛⣯⣴⣦⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠻⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⠀⠀⠀⠀⠀⢀⡀⢄⡀
⠀⠀⠀⠀⠀⢪⣀⣀⣈⣲⡀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⣔⣉⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣾⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⢠⠋⠉⣩⠷⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣋⣡⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣷⠀⠸⡇⠀⠀⠀⠺⣍⡉⠙
⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⢠⡞⠉⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⣿⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⢀⡟⠀⠀⠀⠀⠀⢀⣘⡧⠀⠀⠀⠀⠀⠀⣾⣁⣀⡈⣷⠀⢸⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠘⣷⠶⠶⠚⠛⠻⠏⢹⠇⠀⠀⠀⠀⠀⠀⠸⣄⣉⣩⠟⠀⢸⣧⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠙⢦⣀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣤⠀⠀⠀⠉⠁⠀⠀⢸⣷⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠻⢿⡿⠀⠀⠀⠀⠉⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠞⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠞⠋⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣬⣿⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣸⣃⠹⢿⣿⣿⡏⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⢀⡟⠉⣻⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠉⠙⠙⠿⡿⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡶⠛⠓⢲⡇⠀⡸⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⣛⡿⠿⢿⣶⣤⢤⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣇⠀⠤⣾⢧⣴⠁⢰⠏
⠀⠀⠀⣀⡴⠶⠛⠛⠛⠉⣠⡶⠋⠁⠀⠀⠀⠈⣧⣄⣀⣙⣷⣀⠀⠀⠀⠀⠀⠀⣠⣴⣟⠁⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⢁⣀⠨⠛⠁
⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡧⠀⠙⢛⠿⠶⠶⠾⠟⢹⣄⣽⣧⡄⠀⠀⠠⠄⣠⡴⢋⣤⡾⠃⠠
⠀⠀⠙⣶⠃⠀⡀⠀⢀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡿⠻⣷⣶⣶⣶⣿⣯⠾⠛⠁
⠀⠀⠀⠘⠷⣾⠁⢘⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠖⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣦⠀⠀⠀⠀⠀⠀⢰⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⡾⢋⡔⠙⣧⠀⠀⠀⠀⠀⣿⡯⠛⠛⠻⠶⠶⣤⣤⣀⣀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⢀⣴⠟⣱⠋⢀⣴⢻⣆⠀⠀⢀⣠⡿⣶⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢻⡟
⠀⠀⠀⣠⡾⢃⠜⠁⣼⣿⢷⡎⢻⡄⢀⣿⣏⠀⠙⢷⣄⣀⠀⠀⠀⠀⠐⠶⢦⣴⣟
⠀⢀⣼⠏⡰⣏⠀⢀⡿⠃⠈⢷⡀⠙⢋⣿⠟⢦⡴⠛⠉⠙⠻⢶⣤⣀⣀⣠⣴⣿⣷⣤⣀⡀
⣠⡿⢡⠞⠁⣸⣷⠟⠁⠀⠀⠈⠻⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠙⠻⢦⡌⠙⣷
⠛⠷⠿⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⢲⣾⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠙⠛⢿⠿⠟⠉⠛⢛⣷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣉⣓⡒⠶⠒⠦⠤⣤⣤⣏⣀⣀⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠻⠿⠶⢶⣶⣦⣬⣭⣍⣹⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠲⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⢁⠀⣀⣀⣀⣠⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠲⣌⣃⠈⠉⢻⡟⠋⠉⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⢻⠃⠀⣀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⢀⣤⣤⠘⠠⣰⣟⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡋⣿⣧⠀⣴⡙⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢧⡇⠈⡃⠀⠈⢣⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⣼⠇⢀⠇⢀⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠁⣾⠉⠀⢸⡃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠀⢸⠁⠀⠀⠀⠁⢸⠟⢆⠀⠀⠀⠀⢀⣠⠔⠚⣛⣿⡙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⣝⠀⠀⠀⠀⠀⠀⢀⣘⣀⣀⣰⢼⡋⢠⣶⠿⠟⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠃⡇⠀⠀⠀⠀⠀⠀⠘⠯⠩⣉⠈⣡⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⣀⣤⣤⣴⠶⢁⠠⠁
⠤⠤⠤⠄⠀⠀⣠⣤⣠⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣾⣄⢷⣤⣶⣿⠋⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢉⣹⣿⣿⠿⡏⠉⠉⠉⠈⠉⠉⠉⠁⠉⠉⠈⠓⠀⠀⠀⠘⠓⠂⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠍⠟⠛⠋⠹⡷⣾⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠉⠉⠻⠛⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣦⠀⠀⠀⢀⡗⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡀⠀⠀⠈⣑⠂⢀⡼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⠀⠀⠈⠃⠀⠀⢰⣿⠀⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⠀⠀⠀⠀⢀⣿⠀⠀⠁⢀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠈⡟⠀⠀⠀⠘⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠛⣆⠐⢣⡀⢸⠁⠀⡄⠀⠀⠀⠀⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡀⣿⠰⡄⠌⠂⠀⠀⢸⠀⠀⠀⠆⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣜⢧⡍⠓⠀⠀⠀⠀⠀⠸⡇⠈⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⢻⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢠⡽⣥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⡷⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡿⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠟⠓⠿⠃⡗⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⢦⡐⣢⣬⣱⣁⡜⠁⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣽⣶⣿⣾⡳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠉⠉⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠛⣿⣻⡟⠁⣀⡻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠖⠛⠉⠙⠓⢦⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⣀⣀⡠⠞⠁⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠙⠶⣤⣤⣀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⣾⣿⣭⣿⣿⡄⠀⢀⡟⠁⠀⠀⠀⠀⢀⠀⢀⣄⠀⢀⣄⠀⢀⠀⠀⠀⠀⠀⠈⣇⡀⠀⣼⣿⣯⣽⣿⡆
⠀⠀⠀⠀⣿⣯⢺⣯⣿⣿⣏⠉⠀⠀⣀⣠⣴⠷⠛⠋⠉⠉⠀⠈⠉⠉⠛⠳⢷⣦⣀⡀⠀⠈⢉⣿⣿⣯⣿⢲⣿⡇
⠀⠀⠀⠀⣇⡌⠀⢿⣿⣿⡻⣿⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣾⡿⣫⣿⣿⠇⠀⣄⡗
⠀⠀⣠⣴⣿⣿⡀⠈⠻⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⡿⠋⠀⣘⣿⣷⣤⡀
⠀⢠⠟⠀⢿⣯⣻⣤⣠⣿⣿⠟⣩⣠⠖⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⣦⢤⡙⢿⣿⣧⣄⣼⣫⣿⠇⠘⢧
⠀⢿⡄⠀⠀⠛⢿⣿⡛⡿⠁⠀⠁⠀⠀⢠⡿⠿⣿⣦⠀⠀⠀⠀⠀⢠⣾⡿⠿⣧⠀⠀⠀⠉⠀⠹⡟⣻⣿⠟⠃⠀⠀⣼⠇
⣠⠟⠁⠀⠀⠀⢠⣿⣏⠀⢂⡄⠀⠀⢀⣿⣿⡹⣻⣿⣦⠀⠀⠀⢠⣾⣿⡻⣻⣿⣇⠀⠀⠀⣄⠂⢈⣿⣧⡀⠀⠀⠀⠙⢧
⠻⣄⠀⠀⠀⠀⠘⢿⠁⠀⣾⠀⣠⣶⣿⠳⣼⣽⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣽⡴⢻⣿⣦⡀⠸⡆⠀⢹⠟⠄⠀⠀⠀⢀⡼
⣠⠾⠆⠀⠀⠀⠀⢠⣞⠞⣽⠂⠸⣿⣿⣶⣾⣿⢿⣿⡗⣧⠀⢀⡗⣿⣿⢿⣿⣶⣾⣿⡿⠀⢺⡝⢎⣦⠀⠀⠀⠀⠀⠾⢦
⣏⠀⠀⠀⠀⠀⠐⠉⠓⠂⢄⠁⠀⠈⠻⠿⠛⢡⢞⣿⣧⣿⣦⣾⣧⣿⣟⢦⠙⠻⠿⠋⠀⠀⠂⠄⠒⠋⠑⠀⠀⠀⠀⠀⢈
⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠐⢾⡏⠉⠉⠉⠉⣿⠆⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠹⣤⣀⣀⠀⠀⠀⠀⠉⠐⠆⠀⠀⠀⢠⠀⢆⠀⠀⠁⠿⢦⠾⠇⠁⠀⢀⡆⢠⠀⠀⠀⠀⡔⠈⠁⠀⠀⠀⢀⣀⣠⡼⠃
⠀⠀⢴⡛⢿⣄⡀⠀⠀⠀⠀⣼⠤⡀⠀⠈⠠⣸⡁⠀⠀⡀⠘⠂⡀⠀⠀⣹⡠⠈⠀⠀⡠⠼⡄⠀⠀⠀⠀⣀⣼⠟⣳⠄
⠀⠀⠀⠈⠉⠀⢳⡆⠀⠀⠀⠀⠀⠘⠞⠉⠳⠪⣿⡿⣩⣤⣠⣠⣬⡹⣿⡯⠳⠈⠉⠞⠀⠀⠀⠀⠀⠀⣶⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣈⣁⡀⣉⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢜⣧⢤⣶⣶⣯⣗⠲⢤⡤⠛⠈⠀⠀⠀⠀⠀⠈⠘⠣⣤⠴⠒⣫⣷⣶⣴⠒⠾⣅
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⣈⡀⠹⡟⠀⣽⣦⣶⠖⠢⣶⠶⡿⢻⢶⣦⣤⣰⡴⠛⣿⠁⠻⡏⣤⣦⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠐⢾⡇⠠⣿⡇⠀⠆⠐⠟⢋⠃⣤⡄⠀⠀⠽⠟⠀⠀⠸⣿⠇⠀⢸⠆⠀⡇⠘⠿⢓⡛⠓
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢻⠇⠀⢀⣈⠚⣏⠀⣿⣁⣤⡄⢰⣷⠠⠀⠈⠟⠀⡄⠠⡄⠀⠁⠀⠛⠏⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣤⣀⠘⠋⣠⣟⠀⠛⠛⢻⣧⠈⠁⣴⠀⠀⠀⣼⣗⢀⠣⠀⣺⣷⣤⣤⠴⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠙⠿⠶⣤⠤⠜⠻⣦⠴⠿⣧⣄⡸⠙⠟⠻⠟⠉⠁
⣿⣷⠿⠿⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣶⣿⣷⣶⣶⣾⣿⣷⣿⣾⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⡴⢚⡩⣙⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡭⠭⣝⢦
⠀⢀⣀⣀⣀⣸⣇⣇⡇⢨⣇⣸⣀⣰⢳⠴⣲⣀⣀⣀⡀⣠⣒⣒⢄⣀⡀⠀⣰⣺⣲⣠⣚⣄⣀⡀⢻⡘⣇⠀⡟⣸⠃⢀⠀⠀⣀⡀
⢙⡬⠉⠉⠀⠉⠉⠀⡇⢸⠁⠁⠁⡹⠜⠎⠉⠉⠉⠉⠉⠉⠉⠙⡜⡍⠉⠉⠉⠉⣧⢳⡉⠉⠉⠉⠉⠉⢹⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿
⠹⣇⣩⣍⠩⠍⠩⠁⡇⢸⡗⠛⢸⠱⣳⠇⠀⠐⠀⠒⠂⠃⡄⠀⢿⠐⡀⠀⢀⣺⣘⣓⣓⣶⣒⡒⣒⠒⣺⢰⡗⢒⠒⠒⢒⣒⡒⣐⣲⡿⡃
⠻⠶⠿⠷⠤⠮⠶⠦⣇⢸⠷⣞⢿⣸⡿⠮⣶⡶⣰⠿⢿⣿⣿⢴⢤⣇⢷⡴⣾⠤⣿⣿⢸⡿⣹⣾⣷⡶⣾⢸⡧⣤⣤⣶⣶⣶⣶⣿⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠈⠓⠋⠀⠀⡇⡏⠺⢵⣲⢽⣀⣀⣀⣈⣓⣚⡡⣔⡵⠉⠳⠬⠵⠃⠀⠀⢸⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⣧⠁⠀⠀⢸⣿⠛⠚⠻⠛⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣽⠀⠀⠀⢸⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⢸⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠉⠉⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣧⡀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠈⠙⢦⣀⡾⠁⠹⣆⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⡿⢧⡀⠀⠘⣦⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡾⠀⠀⠙⢦⡀⠸⣇⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⣇⠀⠀⠀⠀⠙⢶⣟⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠘⢦⣄⣀⣀⣤⠞⠙⢧⣄⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠉⠁⠀⠀⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠖⠛⠋
⠀⠀⠀⠀⠀⢀⣤⠶⠲⠦⣤⣤⡤⠶⠲⠦⣤⣤⡤⠶⠲⠦⣤⣠⣤⠶⠶⠶⣤⣄⣤⠴⠖⠶⢤⣄⣠⡤⠶⠲⢦⣄
⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠴⠶⠶⠶⠶⠶⠦⠀⠀⠀⠤⠦⠶⠶⠤⠦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠒⠒⠶⠶⠶⠶⠶⠶⠶⠂⠀⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠻⣆⠀⢀⡴⠋⠳⣄⠀⢠⡾⠛⢷⡀⠀⢠⡞⠙⢧⡀⠀⢠⡞⠙⢷⡀⠀⢀⡼⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠓⠛⠁⠀⠀⠙⠛⠋⠀⠀⠀⠙⠶⠋⠀⠀⠀⠛⠖⠋⠀⠀⠀⠙⠶⠛⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠂⠐⠐⠂⠒⠐⠂⠂⠐⠀⠂⠒⠐⠂⠒⠐⠀⠂⠐⠀⠂⠒⠐⠂⠂⠐⠀⠂⠒⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠂⠐⠐⠂⠒⠐⠂⠂⠐⠀⠂⠒⠐⠂⠒⠐⠀⠂⠐⠀⠂⠒⠐⠂⠂⠐⠀⠂⠒⠐⠂
⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣶⡾⣿⣿⣿⣿⣷⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⣷⣿⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣯⡿⣶⠋⠋⠈⠁⠻⠣⡟⣿⣿⣿⣿⣿⣿⣿⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡆⠐⣿⠼⣿⣿⣿⣿⡟⠋⡟⠁⠟⠀⠀⠀⠀⣊⣸⡇⡟⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢰⣿⡀⠀⢁⡇⠉⠘⢾⣿⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣁⢀⣨⣸⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡟⠙⠀⠈⠀⠀⠀⠀⠙⠘⣿⡇⠀⠀⠀⠀⠎⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⢠⡄⠠⠐⠉⢁⡤⠤⠀⠈⠙⠻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢸⣿⠀⢠⡀⢀⣠⣄⣀⣤⣄⠀⠀⠄⠀⢠⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠈⢫⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠸⠈⣿⣿⣿⣿⡿⠇⠀⢠⡤⡤⡸⠃⠀⠀⠀⠀⢀⣄⣤⣤⣦⣿⣅⣿⢠⠀⢻⣿⣷
⠀⠀⠀⠀⠀⢸⣿⠦⠰⢠⣿⣿⣿⡋⠀⠀⠀⠘⢻⣷⠇⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢻⣿⣿⡄⠈⣿⣟
⠀⠀⠀⠀⠀⢸⣿⠀⣰⣿⣿⣿⣿⣧⠀⠀⠀⠀⣸⣿⣄⠀⠀⣠⣶⠿⣷⣄⣠⡤⠄⠀⣸⣿⣿⣿⡀⢿⣿
⠀⠀⠀⠀⠀⢸⢇⣴⣿⣿⣿⣿⣿⠉⠀⡀⠀⠀⢻⠃⣁⡀⠀⠾⠋⠈⣽⡿⠟⠀⠀⠀⣿⣿⣿⣿⣷⣿⡿
⠀⠀⠀⢴⣤⣼⣿⣛⣿⣿⣿⣿⣿⣶⣶⡷⠶⠶⠞⠛⢓⣦⣤⣤⣴⣦⣤⣧⣄⣀⣀⠀⠾⣿⣿⣿⣿⢱⡇
⠀⠀⠀⠙⠛⠋⠭⣭⣿⣿⣷⣤⣤⣤⣼⣿⣶⣴⣿⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣷⣽⣶⣾⣿⣿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠘⣿⢫⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣼⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠁⠹⠛⢻⡿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠛⠿⠻⠿⠛⠉⠹⠋⠉⠉⠋⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟⣿⣿⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⡾⠿⠿⠿⠿⠿⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠿⠛⢋⣩⣥⣶⣶⣶⣦⣄⡀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⢀⣠⣴⠾⠟⠋⠁⠀⠀⠀⠈⠙⠿⣶⡄⠀⠀⢻⣷
⠀⠀⠀⠀⣀⣴⡿⠿⣶⣶⡾⠋⠁⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⢻⣷⡀
⠀⠀⠀⣴⡿⠁⠀⠀⣠⡼⠃⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⣷⡄
⠀⢀⣾⠟⠁⠀⣠⡾⠋⠀⢺⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⡄⠀⠀⢸⣿
⢠⣾⠃⠀⠀⣰⠋⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢀⣾⠇
⣼⡇⠀⠀⠸⠇⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⢀⣤⡿⠋
⣿⡇⠀⠀⠀⠀⠀⣴⡾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⢿⣇⠀⠀⠀⢀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠀⠀⠀⣿⣧⣤⣶⠶⠶⠾⠿⠿⠿⠿⠛⠛⠿⢷⣦⡀
⠈⢿⣆⣀⣴⠟⠁⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⣀⠀⣠⡶⠟⠛⠉⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠻⣶⣀
⠀⠀⠙⠛⠻⠿⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣄⠉⠀⠉⣀⣤⣤⣶⠿⠛⠋⠉⠉⠉⠉⠛⠻⠿⠿⣦⣄⡀⠈⠻⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⡀⠀⠉⠀⠀⢸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⠀⢀⡀⠀⠉⠙⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣿⣷⣄⠀⠀⠀⠙⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠁⠘⣷⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠟⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠘⣧⠀⠀⠀⠀⠘
⠀⠀⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⢀⣠⣶⡶⠿⠿⠻⠿⣷⣦⣄⣀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢿
⠀⠀⢿⣤⡿⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⣤⡾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⢻⣦⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣤⡀⠀⠀⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠈⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠁⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄⠀⠄⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⣠⣴⠿⠛⠁
⠀⠀⠀⠀⠀⢀⣠⡾⠟⠉⠁⠀⣴⡜⠿⠷⠶⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣆⡀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⣴⡿⠁
⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣀⠀⢰⣿⠁
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣾⡟
⠀⠀⠀⠀⠀⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣷⠀⠀⠀⠀⢀⣤⣾⠟⠉⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⠋⠁⠀⣀⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠀⠀⠀⠀⣰⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⡀⠀⠀⠀⠉⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠐⢿⣦⣤⣄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣄⠀⠀⠀⠈⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡄⠀⠀⠀⢈⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣰⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⣰⡿⠃⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣠⣿⠋⠀⠈⢻⣷⣾⣟⠁⠀⠀⢸⣿⣶⡿⠟⠿⣷⣤⣴⣶⠾⠿⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡾⠿⠟⢿⡏⠀⠀⠉⠻⠏⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠙⣿⡀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠹⣷⣶⣄⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠐⠿⠿⢷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣄⣀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠙⣿⣾⡿⠿⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⣶⣶⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢠⡄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⡾⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⢷⡀⠀⠀⠀⠀⢀⣀⣸⣇⣀⡀⠀⠀⠀⠀⢀⡾⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⠶⠛⠋⠉⢉⣉⡉⠙⠛⠶⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢠⡶⢶⡄⠀⣿⣉⣿⠀⣠⣴⣤⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣂⠀⠀⠀⠀⣴⠏⠀⠀⠘⠷⠾⠃⢀⣨⣍⠁⠀⠻⣦⡿⠁⢀⡹⣦⠀⠀⠀⠀⣐⣧
⠀⠀⠀⠀⠀⠀⠰⠏⠉⠛⠳⠶⣾⠇⣴⠶⣦⠀⢀⣀⡀⢸⣇⣽⠇⠀⢀⣀⡀⢸⣏⣹⡿⣷⠶⠞⠛⠉⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠻⠶⠟⠀⣿⣉⣿⠀⣈⣁⠀⠀⢿⣉⣿⠆⠙⠋⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⣄⠀⠀⠈⠉⠁⢸⣏⣹⡇⢀⣤⣉⠁⣴⠟⢷⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⣧⣼⠇⣼⡛⣻⡄⠀⢉⣉⡀⢿⣥⣿⠀⠙⠻⠛⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠰⣦⣤⡴⠶⠛⠻⣆⠈⠁⢀⣈⡛⠋⠀⠀⣿⣉⣿⠀⠉⢠⡾⠻⣦⠀⣰⠟⠛⠶⢦⣤⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠹⣦⡀⢾⣉⣿⠆⢀⣤⣌⠉⠁⢀⣤⣄⠛⠛⢃⣴⠏⠀⠀⠀⠀⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣉⠁⠀⢺⣧⣽⠇⠀⢿⣤⣿⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠙⠻⠶⣤⣬⣤⣤⣤⣤⠾⠟⠋⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣠⣦⡾⢻⡿⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣅⡠⠾⣿⣿⡵⠟⠁⠀⠀⠀⠹⣶⠀⠀⠀⣴⡎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠊⠉⠁⣠⣶⣴⣤⠀⠀⠀⠀⠀⢀⡀⡀⣠⠶⠋⠑⠀⢦⠀⠤⠄⣠⡀⠀⠀⠀⠀⠈⠉⢿⠾⠁⢀⣶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠄⠛⠁⠀⠀⠀⢠⡾⠛⠉⠀⠙⠷⣤⡀⢠⡀⠛⢩⠟⢁⠀⠠⠃⠄⢀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢀⣶⡥⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠙⠳⣄⡈⠁⠘⠊⠉⠀⠀⠀⠀⣿⣷⠀⣸⣿⣿⠀⢀⡴⢊⠿⠋⠠⠀⢠⠆⡴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣠⠀⡀⠀⠀⠀⠀⠀⢀⡈⢻⣦⡼⠋⠀⡠⠞⠀⢀⣿⣿⣤⣿⣿⣿⣗⠁⠀⠁⠀⠀⣠⠐⢋⣠⣾⡿⠟
⠀⠀⠀⠀⠀⣼⣆⡀⠀⠀⠚⢁⠔⠁⠈⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⠀⠀⠈⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⢊⣠⣴⠟⠋⠉
⠀⠀⠀⠀⠀⠙⠛⠿⣿⣧⣰⠋⣶⣄⢀⠀⠀⠀⠀⠀⠀⢠⣟⣁⠀⠀⠀⠸⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣇⣠⣦⠀⢡⣉⠁⢰⣷⣾⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⠿⣿⠟⠛⠛⢳⣦⡀⢀⢠⣼⣿⠟⠁⠀⣠⣦⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡄⣿⣿⣷⣾⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣄⠀⠀⠀⠀⠘⠿⢾⣿⣿⢁⡀⠀⣄⣿⣿⣷⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⠉⢀⡀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠛⠳⣤⡄⠀⠀⠀⠉⠉⠉⠙⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⢀⠠⠀⠊⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⣴⣾⡟⣀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣠⣶⣶⠗⠁⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠛⠓⠴⢄⡄⠀⠀⠈⠿⣆⣀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣰⣎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠙⠿⣧⣶⣦⣨⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠿⠛⠋⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢀⣴⠟⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠈⠙⠿⠛⣿⣿⣿⣿⣿⣿⣯⠉⠀⠁⠀⠀⢀⠀⠀⢀⣤⣾⣿⡶⠛⠁⠀⠀⠔⠋⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠿⠿⠟⠋⠉⠈⠀⠀⠀⠀⠀⠀⠁⠀⠐⠛⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⢺⣿⡏⣩⣭⣭⣭⣭⣭⣭⣭⣭⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣋⣴⣿⢘⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠋⣼⣶⣮⣭⣉⣛⣛⢩⣭⣴⣾⣿⣿⡿⢸⣿⡇⠛⢛⣛⠻⢿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣛⡛⢛⡉⣼⣧⠹⣿⣿⣿⣿⣿⣿⣷⣬⣙⠛⣛⡛⢡⣿⣿⣷⢸⣿⣿⣿⣶⣬⣙⣛⡁⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢸⣇⠻⠿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣧⢹⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⢃⣾⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣇⠛⠛⠛⠛⠛⠛⠛⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⡿⠿⠛⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣶⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⡰⢿⣿⣿⣿⣿⣿⣿⣿⡷⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠉⠉⠉⠉⠛⠓⣀⣛⡛⠿⢿⣿⣿⡿⠿⢛⣃⣤⣚⣋⠉⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢀⣤⣶⣿⣿⡿⠿⣒⣤⣤⣐⠻⢿⣿⣿⣿⣿⣿⣿⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠃⢠⣿⣿⠟⣋⡥⢶⣿⣿⣿⣿⣿⣿⣶⣮⣝⡻⢿⣿⡁⠘⠉⣠⣤⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢘⣩⣶⣿⣿⣿⣮⠙⠿⣿⣿⣿⣿⣿⣿⡿⠟⣲⣬⡃⠀⠀⠈⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡴⠋⠸⣿⣿⣿⣿⡿⢋⣾⣿⣶⣬⣭⣭⡉⢴⣶⣿⣿⣿⠇⠈⠳⣄⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣨⣭⡍⣴⡞⡻⠿⢿⣿⣿⡿⣛⡛⢷⣦⢉⣭⣥⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠆⠸⣿⣿⡇⣿⡔⢧⣻⢆⣿⣿⠸⣥⢝⣆⣿⢸⣿⣿⡇⠐⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣿⣿⣶⣶⣿⣿⣿⣷⣶⣿⣿⣿⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣧⣤⡄⠀⠘⢿⣷⣿⣶⣵⣾⣷⣤⣾⣿⠃⠀⠀⠀⣾⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏⠀⠀⠀⠀⠙⠿⢿⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠙⠃⠀⠀⠀⠀⠀⣰⣷⣶⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣐⢾⣿⣿⣿⣿⣿⣿⡷⢂⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣝⠿⢿⡿⠿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃
⠀⠀⠀⠀⠸⣿⣿⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⠇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⢹⣿⣿⣿⣿⣏⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⣰⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣷⡀⠀⠈⣿⣿⣿⣿⠁⠀⢀⣾⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣷⡀⠀⢻⣿⣿⡟⠀⢀⣾⣿⣿⣿⣿⠏⠁
⠈⠉⠛⠿⣿⣶⣶⣶⣦⣤⣀⡀⠀⠛⢿⣿⣿⣿⡀⠘⣿⣿⠃⢀⣿⣿⣿⡿⠋⠀⢀⣀⣤⣤⣶⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣦⣤⣈⠹⢿⣿⡀⢹⡏⢀⣿⡿⠏⣁⣤⣴⣿⣿⣿⣿⣿⣿⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣷⣦⣉⠳⣼⣧⠞⣉⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣛⣛⣭⣭⣭⢿⡿⣭⣭⣭⣛⣛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣿⡿⠛⠁⢸⡇⠈⠛⢿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⠟⠟⠋⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠙⠻⠻⠿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣶⣦⣤⣀⠀⢀⣾⡿⠃⠀⠀⠈⠻⣿⣶⣤⡀⣠⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⠋⠉⠉⠉⠉⠉⠛⢿⣷⣿⡟⠁⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣴⡶⢶⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠁⠀⢈⡗⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⣧⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⢻⣄⡀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⢸⡇⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣷⣤⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠆⡆⢴⠠⠀⠀⠤⠤⢠⢆⠃⠠⠀⠠⠲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⡿⣿⣿⣿⣶⣶⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣿⠟⡛⢋⣥⣬⣵⣷⣦⣴⣦⡙⣋⣛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣡⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣍⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢈⠀⠈⠲⣄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁⠉⠉⠈⠻⣿⣿⣿⣿⣿⡄⠀⠀⢀⡔⠉⠀⢈⡆
⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣴⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡇⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⢰⠋⠀⣰⠖⠀⠀⠘⣿⣿⣿⣿⣿⠏⠀⠀⢠⡀⠀⠈⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣶⠀⢿⣿⣿⡇⠀⠀⠀⠐⠳⣈⠉⢳
⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⢀⣀⣉⡶⠋⠉⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠀⠀⣼⣋⡀⠙⣯⡄⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠸⣧⠴⠶⠒⠛⠿⠉⡿⠀⠀⠀⠀⠀⠀⢫⡉⠙⣛⠃⠀⢿⡁⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⣧⠀⠀⠀⠀⠹⣄⡀⠀⠀⣠⠞⠁⠀⠀⠀⢀⠀⠀⠀⠈⠉⠀⠀⠀⢸⣦⣾⣃⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠟⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠛⠛⠋⠹⡆⠀⠀⠀⠀⠀⠀⣠⣼⡶⠾⣇⡑⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣰⢿⣿⣿⣷⣾⠃⠀⠀⠀⠀⢠⡾⠃⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠧⠤⣝⣿⣿⡏⠀⠀⠀⢀⡴⠋⠀⠀⠀⣀⣼⡁⣸⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⡄⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢀⣴⠟⠀⠀⠀⠀⢸⠁⡀⢈⣿⠀⡸⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣹⠿⢷⡟⠙⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠘⣇⠀⠙⠋⠛⣧⠔⠃
⠀⠀⠀⠀⢀⡄⠀⢤⣀⣀⣀⣰⠃⢰⠃⠀⢠⡟⠛⣾⡟⣿⠶⢦⣤⣀⣠⡤⡶⣿⢷⣄⠀⠀⠀⠀⠀⠀⣰⠋⢀⡤⠀⠊⠁
⠀⠀⠀⠀⡌⠀⡽⠋⡀⠀⠈⠁⠀⠘⠒⠦⣼⡗⢀⣿⠇⡗⠀⠲⣟⣨⡷⠂⠀⢻⣾⣿⣦⣄⣀⣠⣴⣋⡥⠞⠉
⠀⠀⠀⢰⠁⠀⡿⠃⢁⡀⢀⡴⠞⠳⣤⣀⣼⣧⣼⡟⣸⠃⠀⣼⠋⢉⡗⠀⠀⢸⣿⡇⠈⠙⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⢳⡔⠉⣰⠏⠀⠀⠁⠘⠭⣝⣿⠿⠛⠁⠀⣸⠃⠀⠀⣳⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣞⠀⢠⡏⠀⠀⠀⠀⠀⢰⣩⡿⡀⠀⠀⠀⢻⡄⠀⠀⡼⠀⠀⢸⡿⠁
⠀⢠⡄⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⣮⣼⣏⠛⠳⠶⢦⣤⣼⣟⣋⡀⠀⠀⢸⡇
⠀⠐⠀⠈⢡⣦⣀⡀⠀⠀⠀⠀⠀⠀⢰⣵⣿⣤⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢳⡿
⠀⠈⠀⣴⠟⡡⠋⢱⣶⣄⡀⠀⠀⠐⡞⣾⠇⠙⢷⣄⣄⠀⠀⠀⠀⠒⠦⢤⣿⠁
⠀⣰⡾⢋⠼⠅⠀⣸⠟⠀⠹⣇⠂⣶⡿⠟⠶⠖⠋⠈⠙⠻⠶⣤⣄⣠⣴⣾⣿⠦⣤⣀
⣴⡟⡴⠋⢀⣼⠞⠁⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⢳⡆⠉⣿
⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⡾⠃
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣡⣿⣥⣤⣤⣤⣀⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣀⣀⣉⠙⠛⠃⠀⠈⡽⠛⠳⣤⡀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⣾⣥⣭⣍⣍⣛⣓⣚⠳⠶⠤⠬⣷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡾⢫⢏⣼⡿⡵⠁⠀⠀⠀⠀⣼⣿⢛⠶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⢋⡜⡱⢣⢎⣼⠑⠁⠀⠀⠀⠀⢸⡟⣡⢋⠞⡴⣩⢛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡴⣡⢋⠜⡴⢣⢎⡜⣦⣄⠀⠀⠀⠀⠘⢿⣥⢋⠞⡴⣡⢋⠞⡽⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⠞⡴⣡⢋⠞⡴⣡⢋⠜⡵⢻⡇⠀⠀⠀⠀⠀⡿⢫⢞⡼⣡⢋⠞⡴⣡⠟⣭⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡡⢫⠞⡴⣡⢋⠞⡴⣡⢋⠜⡵⢡⠇⠀⠀⠀⠀⡼⡱⢣⢎⡜⡡⢫⠞⡴⣻⢸⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡜⡱⢫⠞⡼⣡⢋⠞⡴⣡⢋⠞⣴⡏⠀⠀⠀⠀⡍⢿⣱⣧⣎⣜⡱⢋⣮⡴⢿⢸⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢏⡜⡱⢣⢎⡼⡡⢫⠞⡴⣡⢋⠞⣽⠀⡀⠀⠀⠀⠉⠒⠈⠀⠈⠙⠻⠟⠋⠀⢸⢸⣯⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡜⡱⢣⢎⡜⡱⢣⢎⡴⡡⢋⠞⡼⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡵⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡜⡱⢣⢎⡜⡱⢣⢎⡜⡱⢋⠎⣴⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢞⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⡞⡱⣡⢎⡜⡱⢣⢎⡜⡱⢣⢎⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣫⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣞⡰⣡⢎⣾⣵⠷⠾⣾⡱⢣⢎⡜⡹⢻⢟⡶⡶⢦⡀⠀⠀⠀⠀⢀⣀⣀⠀⢸⢸⡵⡇⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡈⠙⠛⠛⠉⠀⠀⢠⡘⢿⣧⣎⡜⡱⢣⢎⡼⡱⢋⠟⣧⣄⣀⣴⡿⣩⢋⠗⣿⢸⢟⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠉⠲⠮⠍⠛⠻⣧⢎⡜⡱⢫⢞⡼⣩⢋⠞⡴⣡⢫⣾⠇⣼⣣⣿⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⡱⢣⢎⡜⣱⢋⢞⡽⠳⠟⢋⠾⣿⣿⣿⡿⣜⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡷⢣⢎⡔⡱⢣⢾⡄⣶⠰⠿⣾⠁⠀⠈⢾⢾⡆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡀⠀⠀⠀⠀⠐⣧⢚⡟⡱⢣⢎⡜⡱⣣⢾⣾⣿⡰⣷⡿⣦⣠⣤⡶⣶⠆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⣀⣀⡈⠛⠿⣽⣂⡀⠀⠀⢹⢏⠞⡱⣣⢊⣜⣵⡗⠋⠁⠀⠳⣜⠿⣽⣽⣵⠵⢋⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⠭⢍⠉⠚⠛⠛⠛⠛⠛⣛⡋⠉⠉⠉⢉⡥⠭⢍⣉⠙⠓⠚⠓⠲⠶⠖⠚⠛⠚⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠲⠤⠄⠀⢉⣉⣉⣉⡤⠀⠀⣀⠀⢀⣤⣤⣀⡀⠉⠉⠓⠒⠊⠁⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⠂⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡖⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⣤⣤⣴⣿⡀⠀⠀⠀⣷⣶⣿⣶⣶⣿⣶⣾⠀⡀⠀⠀⡆⣿⣤⣤⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠩⠉⠉⠉⣭⣿⣷⣠⣤⣼⡿⠉⠉⠉⠉⠉⠉⠉⠀⠑⠐⠒⠓⠛⠛⠛⠛⠛⠓⠀⠐⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⣤⡴⣶⡄⣄⠂⣀
⠀⠀⠀⠀⠰⠶⠶⠶⠆⠀⠀⣶⣶⣶⣶⣶⣾⣷⡛⠛⠈⠉⠀⠀⢐⣶⣒⡂⠀⠀⣀⡀⠀⠀⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⣄⠛⠯⣿⡾⣿⣷⠀⠀⠛
⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⢿⣿⢿⠿⠿⢿⣿⡟⠒⢲⣶⣶⣶⣾⣿⣿⣧⣶⣶⣿⡇⠘⠚⣿⡿⠿⠿⠿⠿⠿⢀⠀⡆⠀⠐⠀⡃⠃⠉⠄⠒⠁⠀⢏⣣
⠀⠀⠀⠤⣘⠻⡟⢿⣇⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠈⠿⠛⠉⠉⠉⠉⠉⠉⠉⠿⠇⠀⠀⠛⡇⠀⠀⠀⠀⠀⠘⡄⠍⠀⠛⠀⠀⠀⢴⠾⣈⠀⠀⡈⠞
⠀⢐⠁⠤⢼⡂⠀⡠⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡄⠈⠁⡴⢨⠉⢆⠈⠁⠀⠁
⠀⠀⠀⠀⠀⡠⢠⠨⡌⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⠐⠂


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

⣿⣿⣧⣀⠀⢉⣣⢳⣜⡓⠚⣧⠆⠍⣳⠽⢿⣽⣿⣯⣿⣿⣎⣻⣯⢭⡝⠁⠀⢹⠋⢰⠀⠹⡇⠈⠻⠈⠛⢿⣵⣆⡼⠃⠈⡾⠃⠀⣛
⣿⣿⣿⣿⣿⣿⣤⣄⣛⠃⠀⠘⠲⡄⠘⡷⣼⠟⠾⠷⣿⣿⢿⣻⣭⣟⣧⡖⠙⢐⡗⡀⠐⠠⡄⠀⠀⠀⠀⠀⣿⠃⠀⠰⠄⡑⣀⡤⠻
⣿⣿⣿⣿⣿⣍⣻⣷⡄⣿⣆⣤⡄⠀⠌⠀⣴⣼⡛⠻⢿⠟⠾⢳⢯⣎⠐⠁⣰⠼⣿⣿⣿⡇⠀⢠⡄⠀⠀⣄⠘⠀⠠⠂⠀⢻⣁⣠⣷
⣿⣿⣥⣄⣤⡟⢹⣿⣡⣯⡴⠁⣀⠄⠀⣀⡻⠄⠉⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⣂⢀⡟⡜⠘⠁⠖⠐⠃⠄⠀⡁⠁⠀⠐⢺⡟⠉⠈
⣿⣿⣿⣿⣿⣷⣿⣟⣭⣥⣾⡿⣛⢢⡬⠽⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠃⠀⠀⠀⠀⠛⠂⣶⣀⢀⠁⣿⠂⠐⠀⠘⠀⠀⠈⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢲⠛⠈⠀⠀⠀⠉⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣤⡦⢽⣔⡏⠀⠀⠀⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣀⣾⣰⣶⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢺⢻⣥⡿⡽⡽⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⣴⡆⠀⠀⠀⠀⢚⠉⠉⠀⠀⠀⢀⠀⠈⠃⢀⠤⠚⠋⢸⡅⠊⢱⠛⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⢠⡀⠈⠃⠀⠀⡈⠥⠖⡿⠀⠀⠁⠀⠀⠀⠛⣷⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⠾⢛⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠗⠙⢤⠀⠀⠀⡀⠀⠀⠀⠀⠘⢉⣱
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡾⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠙⠁⠛⠀⣀⠀⢤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⣾⡅⠀⠀⠀⠀⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡆⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⡨⣾⣯⣀⣣
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⢁⣹⠋⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣶⣄⠀⠀⠁⠀⠀⠀⢀⢘⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠳⠀⢱⡶⣂⣂⠀⠀⢺⠹⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢎⢡⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠺⠟⣒⢰⡇⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢁⣽
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡜⢁⡦⣤⠀⠘⠇⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢂⣈⣉⣁⣧⣿⢙⢾⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢥⣠⣭⢿⠛⣚⡿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡄⠀⠀⠀⠀⠀⠀⠠⢾⠈⠹⠧⣽⡟⠉⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠠⢞⡒⠂⠈⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠶⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡒⠒⠒⢢⡄⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠛⠒⢤⡀⠀⠀⠀⢇⠉⠓⠒⠋⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠰⣛⣲⡄⠀⢀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠙⠒⠶⠖⠒⠚⠉
⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⢀⣠⣀⠛⠬⠗⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠈⠓⠚⠃⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⡢⢄⠀⠀⠀⠀⠀⡼
⠀⠀⢮⣉⣲⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⣫⣭⣉⠉⠒⠻⠄⠙⢦⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣋⣉⣉⣉⠉⠀⠀⠀⠀⠀⠀⠹⡅
⠀⠀⢀⡤⠒⣤⠀⠀⠑⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⣤⠀⢶⠘⡄
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⡴⣄⠀⠀⠀⢀⠶⢤⣧⠀⠉⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠒⠋⠀⠀⠙⠒⠒⠉⠀⠀⢸⠀⠀⠰⡏⡆⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠓⠃⠀⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⢀⣴⣾⡦
⠀⠀⠀⠀⠈⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣤⣶⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣴⣶⣶⣶⣶⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣶⣶⣶⣶⣦
⠉⠉⠉⠉⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣤⣶⡆⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⢀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢀⣾⠋⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⣧⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢠⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣧⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢠⡿⠃⠀⠀⣿⡇⢠⡠⠆⢤⡠⠠⣄⠔⠢⡀⢸⡇⠀⠀⠹⣷⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡀⠀⣠⡿⠁⠀⠀⠀⣿⡇⢄⣠⠄⢄⡠⠦⣀⠔⠤⡀⢸⡇⠀⠀⠀⠹⣷⣀⠀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⣿⡇⢄⣀⠤⢄⣠⠠⣀⠐⢤⠀⣼⡇⠀⠀⠀⠀⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠁⠀⢈⡴⣦⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⠀⠃⢸⡇⣻⠀⡉⢰⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠠⠀⡔⢸⡇⣽⠈⢇⠰⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢡⠀⠇⢸⡇⣻⠀⡔⢠⠃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⠀⡌⢸⡇⢾⠀⢇⠸⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢃⠀⡇⢸⡇⢻⠀⡬⢀⠃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠘⢸⡇⣿⠐⡆⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠎⠀⡁⢸⡇⣹⠀⢙⠀⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⠀⠀⣸⡇⢿⡄⠀⠈⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣶⠟⠁⠘⢷⣦⣴⡾⠋
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⡴⡺⢿⠽⡖⠢⠤⢔⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⣱⡷⣲⣮⣀⡀⠀⠀⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣏⡞⠃⠀⠣⠀⡙⠖⢢⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢇⠀⠀⠖⠂⠁⣰⡿⠥⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡤⠤⣤⣤⣟⣏⣱⠀⠀⣀⣀⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠬⣿⠺⣛⠛⠓⠉⠁⢈⡏⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡣⢝⣿⣿⢷⠒⠹⣇⣀⣼⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢹⣊⣷⣾⠍⣉⡧⣔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠒⠤⠤⠽⢦⡽⠛⠋⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡈⠀⢀⡤⠚⠉⠉⠲⣄⠀⠁⢀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⡄⠀⠳⡀⠀⠀⠀⠈⠑⢄⡀⠡⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣜⣄⣀⣸⠆⠀⠀⠀⠀⠀⠙⢦⣈⣦⡤⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⣀⣀⣀⡸⠂⠀⠀⠀⠀⠀⢯⣉⣀⣀⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣿⡀⠀⢀⣤⠶⠛⠋⠉⠉⢀⣀⡉⠉⠙⠛⠶⣤⡀⠀⢀⣿⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡶⠋⠁⣠⠶⣤⠀⠀⣿⠉⣹⠆⠀⣠⣤⡈⠙⢶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢿⣤⡾⠀⠀⠈⠙⠉⠀⠸⣧⣀⡿⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⣤⣄⠀⠀⠀⠀⠀⢴⠋⢹⡆⠀⠀⠀⠉⠀⢠⡖⠲⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢿⣀⣽⠀⢰⠞⢲⡄⠈⠛⠛⠁⠀⢠⡶⠳⣦⠘⠷⠾⠃⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣦⣀⣀⣸⠁⠀⠀⠉⠀⠀⠘⠶⠾⠃⢠⡶⢦⡀⠀⠈⠳⠶⠋⢀⣀⡀⠀⠈⣇⣀⣀⣴⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠟⠉⠉⢹⡄⢠⡶⢶⡄⠀⣀⣀⡀⠀⠹⢦⡼⠃⢠⡶⢶⡄⠀⣾⡁⣹⠆⢠⡏⠉⠉⠻⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⢦⡼⠃⠸⣏⣈⡿⠀⠀⣠⣤⣄⠘⢧⡴⠇⠀⡈⠉⠉⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣀⣉⠉⠀⠀⠀⢿⣀⣼⠃⠀⠀⢰⡏⠙⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⢸⣅⣸⡇⠀⢀⣠⣀⠈⠀⠀⣀⣄⠈⠛⠛⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⣄⡉⠉⠀⠀⢿⣁⣹⠆⠀⢸⣇⣨⡇⢀⣠⠾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⣿⠁⠀⠈⠛⠶⣤⣄⣀⣉⠁⠀⣀⣀⣩⣭⠶⠋⠁⠀⠈⣿⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⠿⠿⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠻⠿⢶⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠉⠙⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⡾⠛⠉⠁⠀⠈⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⣠⣿⣷⣶⣦⣤⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣤⣴⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⢀⣴⡿⠋⠉⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠉⠛⢿⣦⣠⣴⡾⠿⠿⠿⠷⣶⣄⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠟⠛⠛⠿⣶⣄⣹⣧
⠀⠀⢘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿
⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣦⡀
⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠙⢿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⣠⣴⡿⠟⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠿⠛
⠀⣼⡟⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠉⠀⠉⢁⣤⡁
⠀⣿⡇⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣀⣴⣾⢿⣷
⠀⠘⠿⣷⣶⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⣿⡆⠀⠀⠀⣠⣾⠟⠋⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⢻⣧⡀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠋⠁⠀⣿⡇⠀⠀⢸⣿⠁⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⣿⡇⠀⠀⠛⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⢀⣿⠏⠀⠀⠀⢠⣿⠃⠀⠀⠸⣿⣄⣀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⢻⣷⠀⠀⢿⣧⣀⣀⣠⣿⠏⠀⠀⠀⠀⠈⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠀⠀⢸⣿⠀⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢻⣧⣀⣀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣾⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠈⠑⣤⠀⠄⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠉⠂⢀⡤⠒⠂⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡀⡀⣀⣀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠤⣄⠀⠀⡀⢠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⢀⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠁⠀⠀⡠⠐⠒⠂⠀⠀⣀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢴⡾⠃
⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡖⠋
⠀⢸⠁⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠸⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣾⠗⠦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢣⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣿⡃⠀⠀⠀⠈⠀⠒⠀⠤⢀⡀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠉⠒⠤⠀⠀⠒⠤⣀
⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠠⡀⠈⡐⠠⢈⠐⠂⠄⠀⠀⠁⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠑⠠⡈⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠊⠉⠁⠙⢄⡀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠙⠶⣅⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠶⠶⠒⠚⠋⠁⠀⠀⢀⠎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠠⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠄⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠚⣉⠉⠀⠀⠀⠈⠉⠻⢿⠲⢤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡅⠀⢀⠜⢹⡀⡞⠦⣄⡀⠀⠀⣀⠀⠀⠉⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⠗⠛⠛⠂⠻⠇⠀⠐⠛⢿⡓⠻⡆⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠀⡾⢶⡄⠀⠀⢀⣤⣤⡀⠀⠀⢹⠀⠀⠀⢸⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠗⡻⠷⡖⡶⣆⠀⠀⠀⠀⠙⠀⠀⢸⢀⣠⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠖⣫⠔⠶⡙⣦⣳⣹⡿⡇⠀⠀⠀⠀⠀⠀⠘⣫⠄⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠘⡇⢰⡧⠚⠛⠋⢻⠙⠛⢉⡇⠀⠀⠀⠀⠀⠀⣠⣁⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⡇⠀⠀⢻⣴⣳⣤⣤⣤⠾⢶⣛⠉⠀⠀⠀⠀⣀⣤⡞⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⣇⠀⠀⠈⢻⡇⠀⠀⠘⠀⢴⡛⠙⡟⠛⢃⡬⠟⠉⣻⡉⢉⡟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⢦⣀⡴⠋⠀⠀⠀⡀⠀⢠⡇⢰⣧⠔⠋⠀⢰⡏⠉⠀⠈⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠋⢀⣀⣠⠤⢴⠇⠀⠀⢻⡼⠁⣀⡤⢾⠯⣅⣀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠛⠋⠉⠁⠀⠀⣼⠀⠀⠀⣼⠗⠋⠁⢀⡟⠲⠦⣌⡓⡆⣀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢰⡇⠶⠀⠀⢸⡇⠀⠀⠀⢹⠇⣼⠳⣄⣤⠦⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠈⠛⠶⠤⠴⠏⢰⣧⠴⢯⡉⠁⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇⠶⠀⠀⠀⠀⠀⠀⢀⡤⢿⡁⠀⠀⠙⣆⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⣫⠴⠒⠛⣆⠀⠀⠘⡆⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⢠⡄⠀⠀⠀⠀⠻⣷⣒⣛⣩⡌⣇⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⢶⠛⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⣿⠓
⠀⠀⠀⠀⠀⠀⢀⣀⡴⢛⡓⠈⠙⢷⡀⢠⣤⢄⣀⣤⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⣠⣤⣶⣿⣿⣿⣿⠏⠀⠀⠀⠁⠀⠙⠶⡉⠉⠙⢦⢤⣾⠿⠷⢤⣄⡠⠛⠻⠆⡀
⠀⠀⠈⠉⣃⣙⠃⣴⣿⡿⠂⠀⠀⣠⣶⣾⣧⢦⣀⠀⠀⠶⢤⡄⠠⠔⠛⣿⠁⠀⠀⢀⠉⠒⠛⠘⠢
⠀⠀⠀⠀⠉⠉⠃⠈⠉⠦⣤⣴⣾⣿⡿⠿⢿⣤⣤⡀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⢀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠉⠁⠀⠘⡽⠿⠻⠀⠀⠀⠀⢴⣿⣏⠀⠀⠀⣸⣿⡇⢀⡀⡆
⠀⠀⣿⣷⣦⣤⣀⠀⠀⠀⠀⣠⠀⠀⠁⠄⠊⣀⣶⣦⣶⣄⠀⠀⠀⠈⠿⢶⠂⢀⣿⣿⣿⡃
⠀⠀⠈⠁⠀⣿⣿⣷⣿⣴⣆⣿⣇⣴⡄⠀⠀⢘⠛⠿⣍⡁⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣯
⠀⠀⠰⢴⣷⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢀⣿⣶⡆⠀⠉⠂⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣷⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⠛⠛⠿⠻⠿⠿⠛⠿⠿⢿⣄⣼⡆⠀⠀⣠⣠⣿⠿⠿⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀
⠀⠀⠁⢲⣦⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠀⠀⠙⠆⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠄⠰⠀⣠⡇⠗⡄⠻⣦⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠁⠀⠀⢀⣦⠄⠀⠀⠀⠀⠙⠦⡀⣀⡀⠀⠐
⠀⠀⡀⠀⠀⢳⢰⠙⠀⢻⣆⠀⡀⠀⠀⠂⢈⡹⠓⠀⠀⠀⢀⡶⠂⠀⠀⢠⡄⠀⠀⠀⢁⣤⡿⠁⠀⠀⡼
⠀⠀⢫⣶⠀⠈⣧⣱⠰⡌⠛⢣⡜⢒⡀⠁⠀⠀⠀⠈⡀⠀⡀⢀⣀⡴⢢⢞⠷⠂⠉⠉⣡⠝⢃⠄⡰⢀⠀⠀⠉
⠀⠀⠀⠘⢧⣰⠘⠻⠄⢳⣀⠠⢱⡶⡛⣄⠀⠘⣦⡠⠛⡁⣤⠖⠁⣰⠎⠀⣀⠔⢠⠾⠃⠖⠁⠊⠀⠠⢀⡇⢠⡰
⠀⠀⢄⠀⠀⠈⠡⠄⠀⠀⠉⠋⠓⠣⡙⠼⣾⣄⢀⠰⡘⠋⡤⠐⠺⠁⠠⠖⠁⢤⠏⠐⠀⠀⠐⠀⠊⠁⠊⠰⠋
⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⠈⠁⠐⣣⣺⢃⡤⠔⠊⣀⠼⠂⠡⠀⠄⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠂
⠀⠀⠀⠀⡠⠄⠐⠵⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣁⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠂
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣀⣤⠤⠲⠶⠞⠛⠋⠛⣋⠟⡉⠉⠙⠛⠁⠒⠒⢂⢤⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠶⠤⢄
⠀⠀⠀⠀⠀⠀⣠⡾⣠⠋⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⢈⠓⣤⡀
⠀⠀⠀⠀⣠⣾⠟⣰⠀⣠⣿⣷⡄⠀⠀⣠⣴⣿⣷⡀⠀⠀⠀⠀⢠⣾⣧⠀⠀⠀⢠⣼⣿⣷⣌⠀⠀⢀⡀⠘⡂
⠀⠀⠀⢀⣻⣿⣴⣿⣶⣿⣿⣿⣿⣄⣴⣿⣿⣿⣿⣿⣄⡀⣀⣴⣿⣿⣿⣷⣦⣠⣿⣿⣿⣿⣿⣿⣤⡐⠒⢺⠇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠠⠌⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣽⣦⠴⠖
⠀⠀⠀⠀⠀⠀⠒⠒⠻⠛⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁⠀⠤⣾⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⣉⠂⠒⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠚⠩⠟⠋⠉⠉⣿⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⡠⡄⠀⢀⣹⢾⣁⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣀⠤⠧⢄⡀⠀⣤⡀
⠀⠀⠀⠓⡯⠚⠁⠀⠀⠀⠉⠊⡗⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⠋⣶⠐⡟⠠⡮⠻⡉
⠀⢀⡀⣰⠁⠀⠀⠀⠀⠀⠀⠀⢱⡀⣀⠀⠀⢀⣀⠀⣀⡠⠿⠤⣀⠀⢀⡀⠀⠀⢀⡀⣸⠸⠇⣴⢜⡋⠠⡦⠻⢳⡀⣀
⠀⠣⠟⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠧⠃⠀⠈⡿⠋⠀⠿⠄⠀⢀⣝⢻⠃⠀⠀⠙⠉⢹⠾⠦⡶⢈⣏⠾⢆⡛⢸⠉⠉
⠀⠀⠀⠈⡦⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⡸⠁⢴⡆⠄⠀⠀⠁⠋⠀⢣⠀⠀⠀⠀⠈⣣⣺⠃⣤⡉⣤⣝⣱⠃
⠀⠀⠀⢏⡟⠉⠒⠤⡠⠔⠊⠙⣏⠗⠀⠐⠛⢧⣠⣆⠈⠀⢮⡵⠀⢦⣀⢸⠛⠃⠀⠀⠻⠃⠈⠑⠒⡖⠊⠉⠀⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠺⠭⠂⠀⠀⠀⠀⠀⠀⠀⠘⣏⠁⠀⢠⢀⠄⠈⠙⢻⠏⠀⠀⠀⠀⠀⠀⣀⡀⠘⠛⠰⣒⡆
⠀⠀⠀⠀⢀⡀⠀⠀⠈⡏⠀⠀⠀⣀⠀⠀⠀⠰⠿⠓⠤⣾⣗⣀⡤⠖⠻⠆⠀⠀⠀⠀⠀⠘⠴⡧⠤⣲⣞⠙⠤⣀⠀⡴⢄
⠀⠀⠀⠀⠀⠑⣄⠤⠒⣓⠢⢄⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡜⠁⠀⢳⠺⡀⠀⠈⠩⡛⠋
⠀⠀⣤⣀⣀⢞⡀⠻⢀⣍⠐⠗⣙⣆⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢯⠽⢌⠀⠀⠀⠀⠀⠀⠀⣄⣀⣇⠀⡀
⠀⠀⠀⠀⢽⢘⠃⠾⢊⣅⢐⠗⣉⣹⠁⠀⠀⠀⠀⠀⣀⠀⢀⡠⠼⠃⠤⢄⡀⠀⣀⠀⠀⠸⡀⠀⣰⣠⠄⠀⠘⢺⣹⡟⢯⡼
⠀⠀⣄⡠⠼⡛⢓⡿⠀⣧⠛⢣⡍⡸⠤⣀⡀⠀⠀⠈⣪⠉⠁⠀⠀⠀⠀⠀⠈⠛⣕⠁⢴⢶⠥⢌⠢⠗⠀⠀⠀⢠⠎
⠀⠀⠈⠀⠀⠑⣜⡃⢶⠆⢶⣎⡔⠁⠀⠈⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⠋⠀⠀⢉⣦⡖⠚⠉⢱⣛⠆
⠀⠀⠀⠀⠠⡞⠀⠈⠉⡏⠉⠀⠘⡦⠀⠀⠀⢰⣦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⡄⠀⠀⠘⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠓⠀⠠⡖⢢⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠯⡸⣀⠤⣤⠛⢧⡀⠀⢠⢄⠀⠈⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⡄⢲⣀⣹⣀⡶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠺⡴⡇⠀⠈⠑⡵⠚⠀⠸⠕⠙⠢⢄⣀⠀⣀⡠⠔⠛⠾⠂⠀⠴⣠⠟⡭⡀⢶⠆⡩⡛⢅⡰
⠀⠀⠀⠀⢰⣙⣖⠃⠀⠀⠀⠁⠀⠀⢀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠶⠴⣡⡄⢈⡀⠶⠆⣉⡡⣾⢧⠴
⠀⠀⠀⠀⠀⠁⠘⡇⠀⢀⡄⡀⠀⠐⢺⣋⣯⢷⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡇⣨⡀⣙⠁⣶⢆⣝⢡⣢⢸⠴⠆
⠀⠀⠀⠀⠀⢠⠤⣇⣀⠣⠽⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠻⡉⣁⡛⠁⠶⠎⠉⣶⢄⠞⠶
⠀⠀⠀⠀⠀⠈⠓⠃⠀⠙⢢⡠⠄⠒⠒⡕⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢙⡮⣁⠸⠗⠐⣵⣤⠊⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠽⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠾⠉⣭⠉⠷⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠄⡀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠀⠸⡞⠁⢀⡂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠
⢀⠆⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠃⠀⠉⢀⡀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠆⠁⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢂⠊⠀⠠⡆⠀⠀⠁⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⢁⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⡤⠀⢀⠀⠀⠀⠠⠂⢀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐
⢀⡆⠀⠀⠀⠀⠁⠀⡔⠀⡀⠀⠀⠀⠀⠒⡂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠠⠀⠀⠀⠁⠀⠤⠀⡀⢀⠀⠐⠀⡀⠈⠀⠆⠀⠀⠀⠀⢀⠀⡐⠤⠀⠀⠀⠢⠀⠀⠀⠀⡀⠀⠀⠊⠄⡀
⠴⠂⠀⠀⠃⠀⠄⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠁⡀⠀⣀⠄⠀⢀⠂⠀⠀⡀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠐
⠀⠀⠀⢨⡇⠀⠀⡀⠐⠤⡁⠀⠀⠀⠀⠀⠆⠒⠊⢀⠀⢀⠘⠂⠀⠐⠆⠀⢁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⣀
⠀⢠⡔⠈⠁⠀⠀⠁⠀⠲⣉⠀⠈⠘⠀⠀⠀⠐⠒⢄⠀⠀⠀⡠⠀⠂⠀⠔⠀⠀⠉⠀⠂⠀⠀⠀⠒⠒⠀⠒⠘⠂⠤

⠀⠀⠀⠀⠀⠀⢸⠋⠉⣇⣀⠀⣀⢀⣀⠀⣀⠀⣀⡀⢀⣀⠀⣀⡀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⣿⠀⡇⠈⡇⠀⡏⠀⣿⠀⢸⠇⠀⣿⠀⠀⠀⠀⡤⠀⢤⠀⡤⢠⡤⣤⢤⡤⣠⠄⠀⠀⢀⠀⡇
⠀⠀⠀⠀⢶⣤⢾⠀⠀⡇⣿⢸⡇⢸⡇⢨⡇⠀⡏⠀⢸⠀⠀⡿⠀⠀⠀⠀⠀⠀⢸⠀⣇⢸⡇⣿⢸⡇⣸⣀⣦⠀⢸⠧⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⣸⣇⣸⣇⣸⣇⢰⣧⠀⣸⣄⢀⣧⣀⣀⣀⣀⡉⠉⢹⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠄⠀⠀⠀⠠⡀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠺⡗⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⢈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⡇⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠚⠚⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⢰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠉⠉⠉⠛⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠋⠉⠉⠙⠒⢤⣀⡤⠤⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⢰⠃⠀⡤⠀⠀⠀⣷⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⢣⡀
⠀⢀⣠⣤⣀⠀⢿⠀⠘⢇⠀⠀⣠⠏⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⡴⠉⡠⠶⠌⢳⠘⣆⠀⠀⠉⠉⠁⠀⠀⣀⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡀
⣇⠀⠧⣀⣠⠞⠀⠀⠑⠢⢤⣤⣤⣴⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⣉⣹⡉⢉⣉⣉⣉⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠻⣍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠰⠚⠓⢦⠀⠈⠉⠓⠚⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⣀⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠳⣄⣀⠀⠀⠀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠉⠁⠀⠈⠑⠦⢤⣀⣀⣠⠤⠖⠉⠀⠀⠀⠈⠉⠉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⢰⡆⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⣤⣤⣤⣤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠺⠻⣿⠉⠀⠀⢸⣯⠀⠀⠈⣹⡿⠷⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⢿⣏⠀⢀⣠⣿⡷⠶⠶⠿⠿⠶⠶⣶⣿⣅⡀⠀⢹⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⣹⡷⠟⠋⠁⠀⠀⠀⣴⡶⣦⡀⠀⠈⠙⠻⢶⣿⠁⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⣦⣤⡿⠋⠀⢰⡟⠻⣷⠀⠀⢿⣤⣼⠇⠀⣴⠿⢷⡄⠙⢷⣄⣠⡾⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⢀⣼⠏⠀⠀⠀⠘⠷⠾⠟⠀⢀⣠⣍⡀⠀⠀⢿⣦⡾⠃⠀⣀⡹⣯⡀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢷⣦⣾⠏⣠⡶⣶⡄⠀⠀⠀⠀⠀⣿⡋⣹⡷⠀⠀⠀⠀⠀⠀⣾⠛⢻⣿⣷⣤⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡟⠀⢿⣤⣼⡟⢠⣾⠛⢿⡄⠈⠛⠛⠁⠀⢀⣾⠛⢿⡆⠛⠷⠟⠁⢻⡏⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣄⣼⡇⠀⠀⠈⠁⠀⠈⠻⠶⠿⠁⣴⡶⢶⣄⠀⠀⠻⠶⠿⠃⣀⣄⡀⠀⢸⣧⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⢹⡇⢠⣶⢶⣦⠀⢀⣠⣀⠀⠀⢻⣦⣼⠏⢠⣶⠶⣦⠀⢸⣟⢉⣿⠆⢸⡟⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣸⣧⠸⣧⣤⡿⠀⣿⡋⣹⡧⠀⠀⣀⣤⣄⠸⣷⣤⡿⠁⠀⠙⠛⠋⠀⣼⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠛⢿⣆⠀⠀⠀⢀⣈⡛⠛⠁⠀⠐⣿⣉⣹⡇⠀⠀⢀⣾⠛⢷⡄⠀⣰⡿⠻⢷⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈⣻⣆⠀⢠⣿⠛⣿⡆⠀⠀⣀⠉⠛⠋⠀⠀⣀⠀⠻⠶⠿⠃⣰⡟⠁⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠋⠙⢷⣄⠛⠿⠟⠁⢠⣿⠛⢿⡆⠀⢰⡟⠛⣷⠀⠀⣠⡾⠛⠻⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⢀⣿⠷⣦⣄⡀⠀⠻⠶⠟⠁⠀⠈⠻⢾⣿⣴⢾⣏⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣇⠀⠈⢙⣿⠿⠶⠶⣶⡶⠶⠶⢿⣿⠋⠁⠀⣹⣷⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⢶⣾⣏⡀⠀⠀⣹⡇⠀⠀⣀⣿⣦⡦⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠛⠛⠛⠛⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣲⣆⠀⣀⡀⢘⣀⣀⢠⡄⡄⢩⣀⠀⠃⠀⠠⡤⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⠀⠙⠓⠘⡖⠀⠀⠘⡆⢰⠖⠂⢀⠀⠠⣧⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⢀⡄⡄⠀⠀⠀⠀⡄⠀⠀⠐⠞⠇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠈⠁⠀⠳⠆⠠⠞⠁⠓⠀⠀⠲⠀⠀⠛⠋⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡄⢡⣄⠀⣰⣆⣶⡘⢧⣄⡀⣤⠦⠠⣄⣀⡆⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠈⠀⠀⢿⣶⠋⠉⠈⢉⣈⠉⠀⠙⠳⣤⣼⡿⣽⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⡷⠀⠀⢀⣤⠾⠛⠛⠙⠛⠻⢶⣄⠀⠈⠀⣿⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣼⠃⠀⡔⠩⠄⠀⡀⠀⠀⠂⠄⠀⠩⠑⡄⠈⣿⣤⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⡞⣰⡿⠦⣾⢄⡀⠀⣠⢸⣧⠶⣦⡀⠀⠉⢉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⡅⣯⡾⠛⣿⠘⢿⣝⠛⢻⠿⠿⣮⡃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠋⠀⠀⠘⢢⣨⣍⣤⠚⠈⠀⠈⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢀⠀⠀⢉⡁⠁⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠁⣀⠀⡀⠈⠃⠒⢤⠀⡀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡎⠀⠀⠹⣧⡘⠒⠋⣀⣸⠆⠀⢺⡇⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⡇⢤⡠⠇⣿⡇⠀⢸⡿⢡⠀⠀⢸⠓⠚⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣻⣟⣀⣉⣀⠻⣿⣄⣼⣇⢸⣦⠀⣸⡏⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣯⣿⣿⣿⣼⡆⢸⡏⠈⠋⢨⣤⣾⣯⡤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⣿⣿⣿⠋⠘⠁⣄⠘⠁⠠⠗⢸⡾⣿⣟⣿⡁⢧⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢩⠝⠀⠺⢁⣀⣤⣾⣿⡀⠀⣴⣀⡀⠁⠸⣏⠀⣡⠀⢛⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⣈⣹⣿⣧⡻⣿⢷⣰⣿⠻⣿⡖⢦⣤⣄⣸⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣁⡏⣻⠯⠭⢄⣿⣆⠹⣧⡱⢼⣿⣧⡙⣿⣿⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⣺⣽⣿⣿⢿⣻⡆⢽⣇⠈⡉⣻⣀⣹⡁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠁⠈⠿⢟⡁⣹⣻⡇⢈⣹⣦⣥⣿⣭⠍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠀⠉⣿⣺⡇⠈⠉⣻⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠰⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠈⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⠈⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⡠⠒⠙⡄⠀⢀⣀⢀⠀⠄⠀⠀⣀⠀⢠⠠⠖⡱⡦⢀⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠀⢀⡴⢀⣰⣄⣿⡾⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠂⠉⡄⢀⡴⠛⠴⡘⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣡⡀⠀⡨⡗⢮⣿⠿⠋⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠠⣞⠄⢋⡴⢡⡞⢃⠎⠘⡄⠀⠀⠀⠀⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠛⢷⠝⢤⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠨⡇⢈⢸⠰⢻⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣹⠇⠀⠂⠀⠈⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠁⠀⡼⡁⢻⠱⡄⠀⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⣠⠔⠩⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⣨⣢⣄⡀⠙⡰⠁⡝⣶⠀⠀⠀⠀⠀⠙⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠀⠀⠀⠐⣛⡄⢚⣯⡝⡄⡽⠀⠀⠀⠀⠀⠀⠀⠸⣇⢔⣭⣉⠿⡑⠦⠑⡅⠣⣇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⢅⡎⠀⠀⠀⠀⡞⠙⣌⠚⢻⣥⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡛⠿⠠⢙⠀⠀⠘⣽⠋⡞⡄
⠀⠀⠀⠀⠀⢀⠀⣴⠀⠳⠙⠃⠀⠀⠀⠀⠑⠤⠞⠛⠅⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⠽⠓⠏⠄⠀⡀⠱⠛⡇⡆
⠀⠀⠀⠀⠀⣙⢶⡿⠠⠀⣔⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡠⠇⠃
⠀⠀⠀⠀⠀⢿⣀⡀⢀⠄⣴⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⢢⡀
⠀⠀⠀⠀⠀⠀⠈⠡⠈⢳⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠢⠙
⠀⠀⠀⠀⠀⣀⣀⠀⡈⡟⠸⡆⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣃⠌
⠀⠀⠀⠙⠀⠈⠈⠀⠉⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣠⡴⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣴⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠛⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠶⠚⠛⠛⠓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠀⢨⢏⣴⡆⠀⠸⠿⣷⡀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠇⠀⠀⠀⢩⠴⡄⠀⡴⠒⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⣠⢤⣾⡄⡟⢸⣁⠀⢸⡦⠔⠒⠒⠒⠠⠀⠀⠀⢀⠴⠒⠛⠛⠓⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡴⠚⠉⡏⠀⠀⠈⢓⣁⣘⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠳⣄⡀⠀⢄⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠐⠂⠀⠤⠄⠀⠄⠀⢠⣇⣀⠀⠀⣰⠛⠒⠛⠛⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡤⠇⣀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⠁
⠀⠀⠀⠀⠀⠂⡀⠀⠄⠒⠒⢧⡀⠀⠀⠙⠛⠙⠒⠦⣤⡀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⡠⠈⠙⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠒⠀⠀⠀⠀⠈⢦⣀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⢈⣧⣀⠀⠀⠀⠀⠀⣀⡼⣄⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠖⠋⠀⠈⠓⠶⠶⠖⠛⠁⠀⠈⠳⠦⠤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⣀⡀⠀⠀⠘⣦⡞⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣹⣁⠀⠀⠀⢀⡀
⠀⠀⠀⠺⢤⡧⠤⠒⠋⠁⠉⠑⠦⡤⣧⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⠴⣊⡉⢡⣮⢉⣓⢤⡞⠋
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣫⣄⠙⠁⣴⡅⠘⠋⣤⣷⡀
⠀⣰⢦⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⢦⠀⠀⢠⣦⣀⡠⢴⡞⠛⠓⠲⢤⣀⣴⡄⠀⠀⢠⣦⣤⡇⢉⠃⠻⠇⣴⡄⢘⠟⣨⡁⣧⣤⣦
⠀⠘⠚⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⠚⠀⠀⠀⡹⠋⠀⠸⠛⠂⠀⢀⣴⡌⢻⡀⠀⠀⠀⠀⠀⢳⠻⠃⡿⠆⢨⣤⠻⣣⣝⢡⠇
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣰⠁⢰⣷⠲⠀⠀⠀⠀⠚⠀⠀⢧⠀⠀⠀⠀⠀⠈⣳⣜⠟⢠⣦⠁⣴⡌⣡⡎
⠀⠀⠀⢺⣙⡞⠙⠲⠤⡄⡤⠴⠚⠙⣟⣹⠀⠀⠸⠿⢿⣀⡰⠀⠁⠀⢞⣹⠀⠠⣀⠀⢸⠿⠗⠀⠀⠀⠿⠃⠈⠙⠒⢳⠖⠚⠉⠀⠹⠗
⠀⠀⠀⠀⠈⠀⠀⠀⢰⡛⣣⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡟⠋⠁⠀⢄⠀⡀⠐⠒⠾⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⢠⠒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⣀⠀⣼⣾⠁⠀⢀⣤⣿⡀⠀⠀⠀⠀⠀⠀⢰⣋⣹⠀⣀⣀⣨⢳⡉⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠰⢆⠀⠀⠀⣸⡀⠀⠀⠀⡶⠄⠀⠀⠀⠈⠉⠀⠈⠙⠛⣷⡖⠊⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢈⡧⠉⢱⣊⣱⠀⠈⠑⢦⢼⣇⡷
⠀⠀⠀⠀⠀⠀⢈⡷⣊⡉⢩⣭⢉⡒⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⣠⠏⠀⠀⠀⠹⠙⠂⠀⠀⠀⢺⡀
⠀⠀⢰⠦⣄⣠⢋⡀⠙⠃⣴⡅⠘⠚⣠⣵⣄⡤⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠘⠧⠟⢧⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⣀⣇⢀⣀
⠀⠀⠀⠀⠀⡏⠙⠋⠺⠗⣨⡅⠸⠽⣈⡁⣧⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡠⠤⠟⠹⠤⣄⡀⠀⢀⣀⠀⠀⠸⡄⠀⣀⣜⡠⠀⠀⠈⠹⣇⢾⠞⢯⡸
⠀⠀⢀⢀⣀⣷⠻⠓⢾⠆⢨⢅⠿⢃⣙⢃⡧⣀⣀⡀⠀⠀⠀⠻⡝⠚⠉⠀⠀⠀⠀⠀⠀⠉⠓⢫⠞⠀⡤⢴⣣⣄⠳⢬⠇⠀⠀⠀⠀⣠⠋
⠀⠀⠈⠃⠀⠈⢣⡸⠷⢠⣬⠁⣤⡜⣫⠞⠀⠀⠙⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠑⠼⠁⠈⠙⢢⣄⡤⠤⠒⠒⡾⢦⡄
⠀⠀⠀⠀⠀⠀⣠⠋⠓⠢⢭⡤⠟⠋⠳⡄⠀⠀⠀⠀⠀⣀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⡀⠀⠀⠀⣞⣙⡆⠀⠀⠀⠓⠋
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢸⡀⠀⠀⠀⠚⠁⠀⠀⠀⠸⠿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠻⠇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡈⠁⠀⠀⢏⣩⠆⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⠣⠤⢶⣶⡋⠱⢤⡀⠀⡠⢦⡀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⢠⡄⠘⣯⣠⡽⣄⣽⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠻⡴⣏⠀⠀⠈⠛⡵⠒⠃⠀⠯⠞⠙⠢⢤⣀⡀⢀⣀⡤⠔⠋⠳⠽⠀⠀⠶⣠⠞⢩⢤⠀⣖⡆⣠⢝⠣⣀⣦⡀
⠀⠀⠀⠀⠀⢰⠛⡦⠞⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡮⠀⠀⠀⠀⠀⠀⢀⠦⣴⢃⢄⠈⡉⢠⣶⠄⠈⡋⣴⣿⣄⣤
⠀⠀⠀⠀⠀⠈⠋⠉⡇⠀⠀⠀⡀⠀⠀⠀⢤⣳⠶⡜⠦⠖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡟⠈⠋⠘⠽⢣⣤⡀⠺⠝⣨⡁⢹⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⡶⢷⠋⠀⠀⠀⠑⢚⠏⠉⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⣷⠫⠗⢴⡳⠈⡩⡔⠿⢂⡑⠃⣼⣀⠁
⠀⠀⠀⠀⠀⠀⠰⡏⣱⠒⢦⣙⠊⠀⠀⠀⣀⡀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣈⢦⡰⣶⠄⣠⡙⢁⣤⠛⢋⣼⣁⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢈⡵⣔⠉⠉⠀⢹⣈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡟⠢⣤⣑⣃⣈⣷⠗⢏⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠇⠠⠧⠀⠟⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠄⠠⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠄⠀⠀⢰⠂⠀⠀⠂⠀⢀⠀⠀⣀⠠⡈
⠀⠀⠀⠀⠀⠐⠠⠊⣄⡀⠇⠐⠀⠀⠀⢀⡤⠀⠀⡆⠀⠀⢀⠀⣦⡀⠀⡎⠀⠀⢈⠀⠀⢁⣀⠎⠀⡀⢀⡘⡀
⠀⠀⠀⠀⠀⠰⡠⡞⠀⠀⠢⡤⠒⣞⠉⠈⡃⠆⠀⢦⡀⠀⣠⡷⠦⢴⠞⠀⠀⠀⢸⡀⠀⠀⠀⣴⠀⠀⠀⣧⢰⡀
⠀⠀⠀⠀⠀⠰⠎⠢⠤⣵⠞⠀⣤⣀⣀⣰⣧⣊⣀⣪⡴⠛⠁⠀⢒⡉⠀⠀⢀⡠⠃⠈⠓⡲⠋⠐⢀⡠⢷⢒⣲⡇
⠀⠀⠀⠀⠀⠀⠑⢔⣆⡸⠦⠤⣠⠀⣩⢟⡫⣉⡿⣻⡴⢴⣖⡤⠤⠖⠒⠒⢉⣇⣀⣀⣉⡵⣢⡼⠋⢠⢮⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣶⣶⡾⣳⣾⠞⣥⡔⢫⠀⣺⡟⡦⣤⡐⢢⣚⣉⣀⣠⣤⠀⢱⣥⣤⣴⣯⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⣷⠿⣾⣆⣿⢶⠷⠸⣞⣀⣋⣿⣀⣭⣍⣛⣛⣛⣯⣵⣿⣛⣉⣽⣚⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢿⣿⡻⣥⣟⣒⠒⡿⡿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠛⣿⠿⣿⢿⣿⢾⢿⠋⡇⢸⢿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⡟⢠⡧⣼⡇⠘⢸⠀⡇⠈⠘⣿⠉⠿⡿⠿⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⢸⡇⠀⠈⠀⠇⠀⠀⣇⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⡤⠛⠒⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢚⣓⣚⠧⡄
⠀⠀⠀⠀⠀⠀⡏⡞⢻⠉⣷⠘⡆⠀⡤⣄⡠⣄⠀⠀⠀⠀⢀⠤⢤⡀⠀⠀⠀⢀⡤⢤⠀⡤⡄⠀⠀⠸⡁⣿⠀⢸⡇⣽
⠰⣒⠚⠛⠛⠛⠛⠛⢻⠀⡟⠛⠛⠚⡇⢯⠖⠚⠒⠒⠒⠒⠓⠚⢳⡹⡒⠒⠒⠚⠚⢧⢻⠖⠛⠒⠒⠒⠛⠺⡆⢸⠚⠓⠒⠒⠒⠒⠒⠒⠒⢒⣶
⢶⡭⠤⠤⠤⠤⠤⠤⢸⠀⡧⠤⠤⡜⣴⣰⠤⠤⠤⠤⠤⠤⠄⠀⠄⢧⢡⠤⠄⠤⢠⠪⢦⢧⣄⡤⠤⠤⡤⠤⡇⢸⡤⣤⣤⣤⣤⡤⠀⢠⣀⣶⠟
⣬⣛⣻⣛⣊⣭⣠⣀⣸⠀⣏⣨⢽⡇⡷⣋⠈⣁⡀⢁⣡⣤⡭⠭⠉⠙⡌⣦⢤⡰⠎⠷⢷⢺⣾⡳⡤⢤⣄⣮⡇⢸⠠⡶⢀⣴⣶⣶⣴⣶⣿⣿⣶
⠀⠉⠉⠉⠉⠉⠉⠉⢹⠀⡏⠙⢮⣻⠟⠉⢹⢻⢇⡫⢍⣹⣻⠉⠉⠉⠳⣜⣩⠟⢉⢝⡽⢆⡻⢴⡿⠉⠉⠉⡇⣾⠉⠉⠉⠙⠉⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡇⠀⠀⠀⠀⠀⢸⢸⠀⠈⠑⢺⣽⣖⣲⣶⣶⣖⣒⡮⠕⠋⠀⠀⠉⠉⠀⠀⠀⠀⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡇⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⢸⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠶⠚⠛⠛⠻⠿⠷⠶⠶⢶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⢀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿
⣿
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⣶⣾⠅⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⣸⣻⢏⢟⡟⠿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠳⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⣿⢻⣀⣀⣠⣤⣤⡄⠀⠀⠀⠀⠀⡿⣯⡞⣼⢩⣛⢹⣩⢥⡿⡿⣿⢱⣶⢿⣷⣿⣶⡄⠀⠀⠀⠀⣤⡤
⣾⠀⠀⠀⠀⣤⣀⠀⠀⣰⣿⡇⠸⣿⡿⡍⠉⠙⠛⠛⠛⠛⠛⠿⠶⠶⡧⣤⣹⣿⣼⣿⣼⢼⣌⣿⣇⣿⠸⠇⢸⣿⣿⣿⡇⠀⠀⠀⣠⡟
⠸⢀⣀⠀⠀⠉⠙⠛⣴⠟⠁⡇⢰⣿⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⢿⠆⢸⣿⣿⣿⡇⠀⠀⢀⣷
⢾⣿⠏⢷⡖⠒⠤⣀⠀⠀⠀⡃⢘⣿⡃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡄⢸⣿⣿⣿⡧⣄⠀⡾⠇⣤
⢼⣏⢊⣴⠴⢦⢀⢈⡟⢋⢶⣧⣰⣿⣿⡦⡿⣦⠤⢠⠀⠀⣰⠂⠉⢸⠇⠤⢄⡀⠀⠀⠀⠀⠀⢠⠒⠛⢼⡇⢸⣿⣿⣿⡎⠈⠉⠀⠈
⣼⣣⠊⠋⠀⠂⠈⠻⢇⠈⢨⣇⣯⣿⣿⡞⢉⠉⡀⠀⡥⡀⠋⢀⡴⠺⠛⠲⡄⠘⠖⠆⠀⣠⠶⡷⣾⠃⢸⡄⢸⣿⣿⣿⡇⠀⣀⡎⢀
⣾⠇⡁⢀⠀⠀⠠⢀⠀⢀⡞⠁⣸⣿⣷⡛⡌⠓⠆⠐⠘⠋⠀⣾⢁⣀⡀⠀⠸⡆⠀⠈⠀⢹⣀⡛⠁⡄⢸⡇⢸⣿⣷⣿⡟⠃⠉⠁⠈
⢸⠊⠁⠙⠋⠓⠙⠛⢋⡏⠈⣴⣟⣿⣿⣿⣟⢧⣈⡁⠁⢁⠻⣿⣿⣍⣹⣻⣿⣿⠋⣥⠀⠀⠉⠐⠒⠂⠉⠃⠘⣿⣿⣿⣗⡒⠒⠒⠀⢀
⣾⠀⠈⠀⠀⠠⠄⠀⡾⠀⣰⣿⣿⣿⣯⣿⢷⣛⣿⡧⠀⠀⠈⣯⠈⠉⠉⠉⢸⡏⠀⠀⢀⢁⣀⣀⠀⠄⠀⠉⣰⣿⣿⣿⣯⡡⡄
⢻⠀⠀⠀⠀⢈⣀⣘⠁⣰⣧⣿⢿⣿⢿⡶⢋⣨⣥⡴⡶⣶⢶⣿⠀⠀⠀⠀⠸⣷⠦⢤⣤⣄⠤⠀⠀⣰⠁⢠⡿⣿⣿⣿⣿⣿⣎⣹⠶⠲
⢹⠤⣀⠀⠘⠛⠓⣤⢞⣩⣷⣾⠟⣋⠈⣴⣿⣿⣧⠀⢒⣿⡆⣷⠤⣤⣤⣴⣾⣿⠆⣾⣻⣿⣷⡄⢠⠁⢀⡾⣽⣿⣿⣿⡿⠟⠉⢀⡠⠒⠒
⣧⣾⣮⡏⣤⣳⣤⡿⠚⠛⠁⠀⠀⠀⠤⠿⣿⣿⡿⠆⣹⣿⣯⣿⢦⣿⣿⣿⣿⣿⢤⣿⣿⣿⠟⣡⠋⢀⣾⣿⣿⡿⢛⢑⣢⣶⡟⠉
⣿⣌⠂⣰⢉⡾⣳⡟⠆⢀⠄⠛⠒⠒⠂⠀⠀⠙⠛⠛⠿⠿⠿⣿⣿⣿⣿⡿⠿⡿⠶⢟⣫⣵⡚⣿⣯⣿⠟⠫⢀⣰⣷⣟⣵⠻⠦⠦
⡿⣰⣶⡤⣮⡵⢾⠅⠋⠨⢁⠏⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⠉⠉⡟⠁⣀⠄⠛⠁⢻⣿⠛⠉⠛⣉⣉⣉
⣿⣽⣿⢿⣿⣿⣻⣷⣦⣥⣾⢰⠇⣀⣀⡀⠀⣀⡀⠀⠀⠈⢉⡯⠁⠤⠤⠤⠤⠄⠀⠀⠛⠛⠲⢶⠋⢀⣄⡀⠉⣉⡉⣉⠉⠈⠀⡤⠉
⣿⣯⣏⣟⣿⣻⢟⢻⣿⣓⣲⣿⣻⣷⡇⣋⣉⣛⣉⣉⣉⣀⡃⠖⠒⠒⠒⠒⠒⢒⡖⡀⣒⣾⡗⣳⣛⣛⣛⣛⠋⣉⣶⣷⠖⠋⣰⠐⢠⣁⣀
⣿⣷⣾⢿⣷⣾⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⣻⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣴⠶⣄⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠻⢤⣈⣱⡄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⡞⠉⢉⣹⠆⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠛⠿⣿⣿⣿⠁⠀⠀⠀⠀⢰⣶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⢹⣿⣿⣿⣿⡟⠀⣠⡴⠶⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠛⠆⠘⠟⠁⠀⠀⠀⠀⠀⠸⣿⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⢸⣿⣿⣿⡿⠁⠀⠈⠛⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⡷⠀⣾⣿⣿⡿⠁⠀⢴⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠻⠟⠁⠀⠭⢹⡟⠁⠀⠀⠈⢧⡀⢹⡆
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡼⠃⠀⠀⠀⠀⠀⠀⢰⣤⡾⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣦⣄
⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠋⢁⣀⣈⣻⣄⠀⠀⠀⠀⠀⠀⠀⠘⣥⣈⣉⣹⡿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠉⢙⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠙⢯⡿⣿⣿⡆
⠀⠀⠀⠀⣠⡶⠟⠛⠛⠋⠁⠀⠀⣀⣤⣀⣾⠁⠈⠙⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢿⡟⠀⢦⣴⠟⠀⢠⠿⣧
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣴⠞⠋⠀⠉⢛⡷⠲⠀⠀⠙⠿⣭⣛⣛⣳⣶⡶⠶⠞⢿⣏⠁⠀⢈⣷⠀⠀⠛⠦⣴⠋⢀⣾⠁
⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠰⡄⠀⣹⣧⣴⠟⠁⠀⢀⣠⣀⣨⡷⠛⠁
⠀⠀⠀⠀⠙⢷⡆⠀⢠⣎⣴⠏⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠋⠀⠁⠀⠀⣠⡾⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣄⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⣷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⢀⡼⢋⡴⠛⢷⡀⠀⠀⢿⡀⠉⠉⠛⠛⠶⢦⣤⣄⣀⡀⠀⠀⠀⠀⢠⡟
⠀⠀⢀⣴⠏⡠⠋⠀⢠⡾⠛⢦⣄⠈⣷⣄⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠶⣶⣤⡾⠁
⠀⣠⠟⣡⡎⡠⣎⢱⡟⠀⠀⠀⠙⠻⠃⠙⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⣾⣁⠞⠁⣉⣧⠼⠋⠙⢦⡀⠀⠀⠀⠀⠀⢀⣴⠏⠉⠙⠛⠲⠶⣦⡄⠀⠀⠀⢸⡇
⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠙⠷⣤⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⣯⣀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡃⠈⠉⢻⠞⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣍⠒⢤⣈⠺⣅⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣌⡓⠮⣁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣑⣿

⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⡖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⠛⠛⠓⣆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢋⠹⣿⡟⣕⠒⣪⢻⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣴⢦⡀⢠⡞⣭⡝⣦⠀⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡟⣡⣿⣷⡘⢧⣙⠶⣋⣼⣿⡇⠀⢸⡇⠀⠀⠀⢀⡾⠁⠈⢳⡌⠷⣭⡵⠏⠀⣿
⠀⠀⠀⢸⣿⣿⣿⠏⣴⢋⣼⣿⣿⣌⢻⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⣠⠟⣰⠇⠀⠀⠹⣆⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣿⣿⡏⢸⠃⣾⣿⣿⣿⣿⡆⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⡟⢸⠃⠀⠀⠀⠀⢹⡆⠀⠀⠀⣿
⠀⠀⠀⢸⣿⣿⣷⠸⣧⣿⣿⣿⣿⡿⢃⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⢷⡘⠁⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿
⠀⠀⠀⢸⣿⣿⣿⣷⣌⡛⠻⠛⣋⣥⣿⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⠈⠳⢦⣤⣤⡤⠞⠁⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡟⣻⢻⠿⣿⢿⢛⠟⢻⢻⣛⣿⢟⡟⢻⡇⠀⢸⣧⣄⣤⣤⡤⣤⣤⣠⣄⣠⢠⢤⣠⣤⣤⣿
⠀⠀⠀⠸⣧⣵⣼⣶⣿⣾⣬⣶⣼⣶⣿⣿⣮⣷⣾⠇⠀⠘⣟⣛⣛⣛⣃⣃⣛⣙⣓⣛⣚⣘⣛⣛⣛⡟
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠉⠻⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⡀⠀⠀⠀⠀⣸⣿⣾⡷⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠟⠉⠛⠾⣿⣿⢿⣰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣽⣿⣦⣝⣿⣄⠑⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⣷⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⣿⣿⣿⣽⣟⣿⡿⢿⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣿⡏⡹⣼⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⣿⣤⣿⣾⣿⣿⣿⣿⣧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣀⢠⢤⠀⠀⣀⠲⡇⣀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣳⠖⠀⣠⣴⣠⣤
⠀⠀⠀⢠⣭⣶⣦⣴⠀⣩⣅⡿⢯⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠠⣵⣤⢤⣾⡍⣁⣤⣤⡤
⠀⠀⠀⠀⡀⠿⣿⡯⠁⣼⣿⢆⢀⠰⠾⠲⠄⠀⠀⠀⠀⠀⠀⠀⢰⠆⢀⣦⣿⣿⡙⣟⠿⣯⢦⡀
⠀⠀⠀⠀⣹⣃⠀⣀⡀⠈⠀⠈⠉⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠘⠛⠁⠈⠁⣩⡭⠀⢸⡌⠁⠀⣀⡀
⠀⠸⣿⣦⣽⠿⢿⠟⡏⠁⠸⠀⠒⠀⡀⠈⢹⡇⠉⠁⠀⢀⠀⠀⠀⠤⠤⠤⠤⠶⠶⠾⣧⠰⠒⠂⠀⠐⠀⠀⣺⣷⣶
⠀⠈⠉⠉⢛⠛⠛⠛⡟⠁⠀⠛⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⡀⠀⠀⢻⠀⠐⣦⠀⠄⣀⣸⣿⢿⣿⡶
⠀⠀⠠⠠⠼⠄⠀⠂⠂⠀⠘⠀⠀⠀⠀⠓⠂⣐⠛⢂⣨⡟⠳⠖⠀⣶⠛⠀⣼⣧⠐⠀⢸⣧⢶⡿⠒⣏⢛⠙⠿⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢑⣆⣐⣧⣀⣀⡀⠴⣿⣖⣻⣙⣉⢗⠉⢹⣿⣎⣹⣿⣿⣇⣀⣸⣿⠋⠁⠀⢠⡄⠴⠶⠄⠐⠂
⠀⠀⢲⣾⣗⣺⣓⢚⣛⡛⢀⡘⠊⠉⠀⠁⠈⣿⠀⠀⠀⠋⣿⠀⠀⢻⡇⠀⠘⣦⣀⠀⠀⠀⠀⣤⣀⣸⣄⣠⣴⣀⡠
⠀⠀⠀⠈⠉⠉⠉⢩⣥⠤⠄⢀⣀⣮⡥⠀⠐⠻⠤⠀⠤⣤⣿⠓⠀⠘⡃⣀⠀⡌⢿⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠠⠒⠚⣛⣲⣶⣿⣡⣿⠧⠞⠒⠓⠒⠒⠀⠀⠀⠀⠀⢀⢀⣠⣤⣤⣬⣡⣤⣤⠴⢿⠿⡟⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠉⠁
⣀⠀⢀⠀⠀⢀⡀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢶⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⢶⠃⠀⠸⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⡇⠀⠀⣷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣄⠀⢶⣄⣀⡸⠛⢢⣤⣠⣴⣛⡦⣤⣤⣼⢳⣦⣤⠟⠲⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⢀⠀⠀⠀⠀⢸⠙⠋⢠⡤⠔⠊⢁⣀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠉⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢸⡉⠭⢦⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠂⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠉⠉⢱⠖⠚⠁⠀⠐⠮⡠⢄⡀⢀⡀⠔⠀⠀⠢⠤⠀⠀⠀⣠⠖⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡤⣄⡀⠀⠀⠈⠁⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⠄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⣥⣈⣵⡤⠇⠀⠀⠀⠀⠀⣀⡼⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠁⣄⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⣀⡴⠛⣇⠀⠀⠀⣰⣧⠞⠀⠙⠒⠦⢬⣭⣉⣉⣻⠉⠉⠉⠉⣉⣀⣀⣀⣠⡼⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠈⠛⠛⠛⢡⡟⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠐⢰⡖⠋⠉⠉⠇⢚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢧⠀⠀⠀⢸⡼⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⣇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⠀⠸⣤⣀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣨⣦⡀⠘⠓⢦⠤⠼⠷⢤⡾⢻⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡤⠚⠉⠀⠀⠀⠙⠛⠛⠋⠙⠛⠒⠚⠁⡞⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠓⠒⠶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⣒⣲⠶⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⡠⠚⠉⠁⠀⠀⠀⠀⠀⠈⢉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣋⣀⣀⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠳⣄⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠤⣄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢦⠀⢀⣞⣥⣄⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⣶⣶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⡸⠀⠀⠀⠀⠈⢇⠈⠁⣠⡴⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠟⠉⣿⣿⡿⠿⢻⣿⣶⣦⣤⣀⡀⠀⠈⠑⢦⡀⠀⠀⠀⠀⢸⠀⠈⠉⠙⠳⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⣀⠀⠀⠀⠀⠀⣿⠋⠀⠠⠖⠒⠦⠀⠀⠀⠛⠛⠋⠉⠹⣿⣷⣦⣄⡀⠙⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣿⠀⢀⣿⠀⠀⠀⠀⠀⡎⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢦⡀⠀⢹⣦⣼⡄⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠾⠉⢹⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠁⠀⠀⢿⣿⣿⡄⢰⠃
⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣐⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠏⠈⣷⠀⠀⠀⠀⢸⣿⣿⡟⠁
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⡿⢿⣿⣿⣿⣿⡷⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⡄
⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠑⠀⠁⠀⠻⡽⢻⣿⡧⠀⠑⠒⠚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⢀⣋⣃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠤⠤⠚⠁
⠈⠳⠤⠤⣄⣀⣀⣀⣀⣀⣂⣼⣿⣄⠀⠀⠀⠀⠀⢸⣿⡧⠚⠉⠙⠯⠤⣀⣀⡀⠀⠀⣀⣀⡠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠸⡅⣀⣠⣤⠶⢶⣶⣏⠀⠈⠉⠉⢀⣠⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢣⠀⠀⠀⠀⢹⡍⠴⠒⣅⢈⠻⣿⣿⣿⣿⡿⡿⠿⠿⢷⣔⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠘⡄⠀⠀⠀⠀⠻⣔⣜⣆⣾⡟⣋⠽⠖⠚⠑⠀⠻⢂⠀⢻⡆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠱⡀⠀⠀⠀⠀⠈⠉⠉⠉⣇⢩⡔⠂⠀⠀⢀⡀⠀⠁⣸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠷⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢷⠀⠀⠀⡑⠏⢒⣵⠟⢵⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣠⣤⠼⣷⠸⡔⣓⣩⣼⠾⠻⡕⠊⢱⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠒⠈⢦⠻⠾⠛⠛⠭⠔⠣⠆⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠤⠤⠤⠤⠤⠀⠀⠤⠤⠤⠤⠤⠤⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠉⠁⢈⡩⣽⣿⣿⣿⣿⣿⣯⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⣀⣀⠀⠀⠈⠘⢟⣿⣿⡿⣿⣿⣿⣷⡝⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠁⣴⣿⣷⣀⣀⣠⠀⠀⠀⠉⠈⠋⣹⣿⣿⡿⠘⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⣹⣇⣼⣿⣿⣿⣿⣿⠿⠆⠀⠀⢀⣠⠰⠋⠛⠹⣧⠀⢻⡌⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢁⢴⣿⣿⣷⣶⣶⣷⣶⣾⣿⣿⣿⡿⢟⡿⠀⠀⠀⠀⣿⠀⠈⣿⡄⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢡⢎⣾⣿⣿⣿⣿⠿⠿⢿⣿⣭⡉⠀⠀⠉⣀⣤⡴⠶⢼⣿⣀⣿⣿⣿⡄⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣾⡾⠿⣿⣵⣿⣧⣤⣶⢶⣶⣿⣅⠀⠀⠀⢉⣅⣤⣦⣄⢿⠚⠻⡿⠛⠛⢷⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⣋⣠⣴⣿⣿⣿⡏⠉⠓⠲⠟⠉⠈⠀⠀⠀⠚⠀⠽⠟⠁⢸⣧⡀⠀⢶⡀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⠃⢸⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠺⠳⠶⠢⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⣠⣾⠃⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⢸⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣿⣿⣿⣤⣾⠟⠁⠇⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡈⣇⠻⣿⣿⣿⣿⣿⣿⣷⠀⠀⠉⠻⣯⣭⡽⠋⠀⠀⢠⣿⣿⡿⠟⠉⠀⠀⡸⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡘⢄⠀⠀⠀⠀⠉⠙⢻⣗⡀⠀⠈⠀⠀⠀⠀⡠⢴⠟⠛⠋⠀⠀⠀⠀⣰⢃⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡈⢦⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⠖⠋⠀⠘⡀⠀⠀⠀⠀⢀⡴⢁⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⡑⢄⠀⠀⠀⠀⢸⣿⣿⡿⠋⠁⠀⠀⠀⢀⡿⠋⢠⣀⡴⢋⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣙⠢⢍⣠⣼⣿⡇⠀⠀⠀⠀⢀⡔⠁⠀⢀⡼⠉⠑⠻⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠓⢺⡟⢸⣿⡇⠀⠀⢃⡴⠋⠀⠀⣠⠞⡁⡀⡀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣾⣿⠁⢸⡟⠀⣠⠞⠉⠀⠀⢀⣴⣷⣾⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠻⠋⠁⠀⠀⠀⠶⠟⠛⠋⠉

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠰⠂⠀⠠⢀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⢠⢂⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠐⠀⠀⠀⠠⠤⣄⠀⠐⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⡀⠉⠐⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠙⠂⠀⠀⢠⡀⠀⠐⣄⣠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠁⠀⠉⠀⠀⠀⠀⠀⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⡀⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠂⠀⠀⣀⠀⠀⣠⠔⠂⠁⠉⢁⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠
⠀⠀⠀⠀⠀⠸⢁⠐⢬⠞⣠⡇⠀⠀⠀⠀⠂⠊⢀⡠⠬⠁⣂⡀⠀⠀⠚⠀⢀⠀⠀⠈⠁⠞⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠑⡄⠀⠧⢈⠵⠘⠀⠀⠀⠘⠀⠒⠋⠀⢀⡀⠄⠀⡰⠀⠰⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠁⠁
⠀⠀⠀⠀⠀⡐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⡀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⢰⠖⢀⠥⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⡀⠀⠀⡈⡃⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡂⠄⠩⠰
⠀⠀⠰⠆⠀⠘⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⢀⣤⠞⠋⠙⡆
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣠⠟⠀⠀⠀⢰⠃
⠀⠀⠀⠀⡼⠁⠀⠀⢀⡼⠃⠀⠀⠀⢠⠏
⠀⠀⠀⡼⠁⠀⠀⢀⡾⠁⠀⠀⠀⣰⠋⠀⢠⠄
⠀⠀⢰⠃⠀⠀⢀⣾⠁⠀⠀⢀⡴⠃⠀⠀⡾
⠀⠀⡟⠀⠀⠀⣾⠃⠀⠀⣠⠞⠀⠤⠴⣾⠗⠒⠂
⠀⠀⡇⠀⠀⣸⡏⠀⣠⠞⠁⠀⠀⠀⢀⡿
⠀⠀⢳⣀⣀⣿⡷⠚⠁⠀⠀⠀⠀⠀⣼⡇⠀⠰⣏⣳⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⣠⢲⡄
⠀⠀⠀⠈⠉⢿⡇⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣰⠋⣷⠀⠀⣴⡟⠁⣨⡀⠀⢀⣤⡄⠀⠀⣠⣶⢠⠃⣾⠇
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⣠⣾⡏⠀⠀⣰⠃⢠⡿⠀⣼⡟⠀⣼⡿⠀⢠⡞⣿⠇⠀⣰⠃⣿⠏⢠⣿⠀⡼
⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⣠⠞⠁⣿⡇⠀⣰⠃⢠⣿⠃⡼⠹⣧⠞⣿⠁⣰⠋⢰⡟⠀⡼⠃⠸⠏⠀⢸⣇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣿⡞⠁⠀⠀⣿⠀⣰⠃⠀⣾⢇⡞⠁⠀⠀⠀⠙⠊⠁⠀⠸⣧⠞⠁⠀⠀⠀⠀⠸⠞
⠀⠀⠀⠀⠀⣠⠴⠋⢸⡧⠀⠀⠀⠙⠚⠁⠀⠀⠘⠋
⠀⠀⠀⣠⠞⠁⠀⠀⣸⠇
⠀⣠⠞⠁⠀⠀⠀⣠⠟
⣰⠋⠀⠀⠀⣀⡴⠋
⢿⣀⣤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠻⢏⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠑⢤⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠒⠒⠒⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⡤⣤⡽⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⢳⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⡟⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠹⣦⠴⠦⠶⠶⢶⡶⠶⠶⠶⢶⡶⠦⠶⠶⠷⠞⠋⠀⠙⢦⡀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢠⠶⠋⣿
⠀⠀⠀⠀⠀⠠⡶⠒⠚⠛⠲⣆⣠⡤⠤⠶⠶⣤⣤⠶⠶⠤⠿⠴⣯⡀⠀⣠⣤⠤⠤⠴⢴⡄⢠⡶⠶⠿⠶⠛⠒⠚⠛⠛⠓⢶
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠉⠻⠀⠀⠀⠀⠾⢱⠀⠀⠀⠀⠀⠈⠹⣦⡏⠀⠀⠀⠀⠀⢱⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡄⢸⣿⠃⠀⠀⢀⠀⠀⠈⠇⠲⠶⠖⢶⠀⠀⠀⢀⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠳⡄⢯⡿⠀⠀⠀⠫⠇⠀⠀⠘⣦⠤⢤⠈⠁⠀⠀⢈⡿⣆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠾⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⢸⢰⠀⠀⠀⠈⡇⠘⢧⡀
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠋⢹⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢀⡶⠒⣆⠀⠀⠀⢦⡀⢸⡀⣀⣀⣀⡇⠀⠈⢷⠀⢀
⠀⠀⠀⠀⠀⣴⠛⣦⡤⠤⠘⢃⣼⠆⠠⠤⠤⠄⢃⡶⠤⠤⠤⠤⠴⠷⠤⠤⠤⠞⢀⣀⠘⠓⠒⠛⠛⢹⣾⡉⠉⠉⣹⠁⠀⠀⠈⣷⠛
⠀⠀⠀⣠⠞⠁⠀⠈⠙⠛⠉⠉⢁⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛⠉⢻⡟⠛⠛⠛⡏⢹⡟⠛⠛⠉
⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡠⠞⠀⠀⢁
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠋⠀⠀⠀⠀⠸⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠
⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠐⡶⠀⠀⠀⠀⠀⠀⣸
⠸⣆⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⡟
⠀⠹⣄⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠈⠲⣀⣀⣀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣤⣤⣀⡞⠀⠀⢀⡴⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣒⣒⣒⣉⣉⣉⣛⣚⣉⣀⣀⣀⣉⣳⣦⣤⣶⣛⣉⣉⣉⣛⣶⣤⣖⣉⣀⣀⣉⣓⣶⣚⣉⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠶⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣏⠁⠀⠀⣤⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠈⠳⣤⡼⠁⢳⡀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⡞⠛⢦⡀⢻⡄⠀⠀⠈⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⡁⣆⠀⠙⢦⡷⠀⠀⢰⠇⠀⠀⠲⣄⠘⠃⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⠳⣬⣉⣠⠾⠛⢦⣠⠟⠀⠀⠀⠀⠈⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⠀⢀⣀⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣭⣉⣉⣉⣭⣉⢉⣉⣭⣉⠉⣉⣭⣉⡉⣉⣩⣍⡉⢉⣩⣍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⠬⠭⠥⠤⠬⠭⠥⠤⠬⠭⠥⠤⠬⠭⠥⠤⠤⠭⠭⠤⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡒⠒⠒⠒⡒⠒⠒⠒⡒⠒⠒⠒⢒⠒⠒⠒⢒⠒⠒⠒⠒⡒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠳⡤⠞⠉⠳⡤⠞⠉⠳⣤⠞⠉⠳⣤⠴⠋⠙⢦⡴⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢀⡠⠞⠉⡠⠊⢀⣮⠴⠚⠋⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⠋⠀⢀⠞⠀⣠⠊⢁⡤⠴⠶⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⣠⠷⠖⠚⠉⠉⠉⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢠⠒⠐⠠⡀⠀⠀⣤⣼⠃⢰⡥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⡠⠒⠐⡆
⠀⠀⠀⠀⠀⠈⠑⠒⠒⠛⠀⢰⠏⠀⡵⠋⠀⠀⢀⣶⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⠓⠒⠒⠁
⠀⠀⠀⠀⠀⡔⠋⣹⠖⠀⠀⡿⣄⣴⠀⠀⠀⠀⢈⣋⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡆⠈⣇⠀⠀⠐⢎⡋⢳
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⣏⡠⣿⠀⠀⠀⣰⠋⠁⠀⠀⠙⣦⠀⠀⠀⡀⠀⢀⡤⠶⢤⠀⢹⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣹⡀⠀⠀⣇⢀⣀⣀⣤⡤⢾⠀⠀⠀⡁⠀⢸⠧⠤⢤⠇⢸⡗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⢧⠀⠀⠻⡉⠀⠀⠈⣠⠏⠀⠀⠀⢧⠀⠀⠑⠒⠊⠀⠀⣇⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⢦⣈⡄⠀⠀⠉⠓⠒⠛⠁⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⠀⠀⣠⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⣠⣾⣿⣟⣲⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⣍⡹⢿⣿⡟⠀⠀⠀⢀⡴⠋⠀⠀⠀⡞⢙⡧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠸⡆⠀⠀⠀⠀⠉⠙⠻⠁⠀⢀⡴⠋⠀⠀⠀⢸⠉⠑⣺⠀⣘⠓⡄
⠀⠀⠀⠀⣀⣤⣄⡤⠚⣋⡭⠖⠛⠛⢻⠖⠒⢷⡀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⠀⠀⠀⠸⡈⠘⠟⠚⣧⠜⠁
⠀⠀⡴⠋⠁⠀⠀⠀⢾⠁⠀⠀⠀⠀⠈⠛⢻⡟⠝⡶⢤⣤⣤⡤⠶⡟⢳⡄⠀⠀⠀⠀⢀⡴⢁⡤⠂⠈
⠀⠈⢣⡔⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⠁⠀⠀⠀⠀⢘⢿⡶⠿⣦⣤⣤⣶⡭⠖⠋
⠀⠀⠈⠓⣾⡀⡸⠛⠁⠀⠀⠀⠀⠀⠀⣸⠃⠀⡀⠀⣸⠆⢰⡄⠀⢸⡆⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠉⠁⢀⡀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣁⡈⠋⠐⠾⠁⡂⢸⡇
⠀⠀⠀⠀⠀⢀⡼⢋⢿⡄⠀⠀⠀⠀⣿⠶⢤⣤⣈⣉⡙⠛⠛⠛⠀⢸⡇
⠀⠀⠀⠀⣠⠟⡵⠃⣠⢷⡀⠀⠀⣀⣿⡄⠀⠀⠈⠉⠙⠛⠓⠶⠶⡾
⠀⠀⢀⡾⢫⠞⢰⣿⢷⡊⢧⡀⣺⡏⠈⠳⣄⣠⠀⠀⠀⠐⠢⢤⣾⠁
⠀⣰⠟⡴⢧⠀⡼⠋⠀⢳⡈⣋⡽⠛⠶⠚⠁⠉⠛⠶⢤⣤⡴⢾⣿⠶⢤⣀
⣾⣣⣎⣠⡼⠛⠁⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠆⣸⠇
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢅⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠓⠞⢿⢿⠛⠓⢲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣝⣒⣒⡒⠠⠤⠤⠏⣀⣈⣳⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣙⣛⣛⣛⣷⣶⣾⣶⣿⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⠛⠀⠙⠛⠛⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⣀⣠⣼⠏⠀⠘⠖⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠛⡋⠉⣀⠀⢻⡆⠀⠀⠈⠉⠀⠸⠆⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⣿⠀⢙⡀⠀⠀⢶⡄⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⣷⠀⠀⠀⠸⣦⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣀⠀⠀⠀⠈⠁⣀⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢋⡀⠀⠀⠀⠀⠈⠁⠀⣀⡘⠿⣿⣿⣿⣿⣿⡿⠛⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣋⠀⠀⠀⠘⣁⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠉⠉⠉⢁⡀⠈⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣋⠀⠀⠀⠀⢻⡄⠀⠀⣼⠏⠀⠀⠀⠀⠀⢀⡀⠀⢀⣿⣷⡀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⣀⠀⢰⡏⠀⠀⠀⠀⠀⢀⣾⣷⡀⢾⣿⣿⡷⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⠀⠙⠁⢸⡇⠀⠀⠀⠀⠀⠸⣿⣿⠇⠈⠙⠋⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣦⣄⣠⣴⣦⣄⣠⣴⣤⣄⣤⣶⣤⣀⣤⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠿⠿⠛⠛⠿⠿⠛⠻⠿⠟⠛⠻⠿⠟⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣶⣶⣿⣷⣶⣶⣿⣷⣶⣶⣿⣷⣶⣾⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠙⠛⠛⠋⠙⠛⠛⠉⠙⠛⠛⠉⠛⠛⠛⠉⠀⠛




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠁⠀⢈⡇⠀⠀⠀⠀⢀⡤⠖⠒⠐⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠱⠖⠀⠀⠀⠉⡅⠀⢀⠴⠂⠛⠀⠀⠀⠀⠀⠸⠃⠐⢶⠊⠉⠉⠙⣦⣀
⠀⠠⠴⠬⠭⠉⢉⡁⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠍⣉⣉⣉⣀⣀⣀⡠⠀⠀⠀⡞⠁⠀⡏⠀⠀⣞⡉⠉⠱⡄⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⡉⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠒⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠉⠁⠀⡤⣄⠀⠀⠈⣉⡉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡠⢀⠈⠙⠒⠠⠤⠀⣀⣀⣀⣀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠵⠤⠀⠒⠀⠁⠀⠀⠀⠀⢠⣀⣀⡠⠞⠀⠀⠈⠁⢸⠇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢌⡑⠂⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠤⣀⠀⠀⠀⠀⣉⠀⢂⣤⣤⠴⠖⠛⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⠒⠦⠤⣤⣈⣁⡀⠒⠒⠒⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠐⠒⣒⠒⠤⢭⡒⢄⡀⠀⠀⠄⢤⠀⠀⠀⠀⠁⢀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡲⣄⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠲⠶⠤⠤⣤⢤⡤⣀⡀⠀⠉⠉⣳⡀⡀⠈⡆⠈⠙⠃⠚⠁⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⠑⢄⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠸⣀⠀⠀⡠⠉⢳⠖⠋⠹⣄⣀⣠⠃⠀⠀⠀⠀⠀⠐⢆⢀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣉⠓⠴⢯⡀⠀⠈⣓⣢⠀⠀⢀⠋⠁⠈⠤⠤⠞⠀⠀⠀⠀⠉⠀⠀⠀⠉⢢⡀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣙⣃⠀⠀⠀⠀⠀⣩⠆⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⠴⠋⠉⠁⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠴⠞⠳⠤⠤⠴⠋







⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⡿⠙⠃⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣵⣿⣿⣿⠁⢠⠄⠁⡸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣿⣿⣿⣆⠔⠁⡀⣼⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡽⣿⣿⣿⠟⢁⣴⡊⠠⢑⠔⠂⢰⡤⠒⠂⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠂⠉⠢⣞⢿⣇⠀⠖⠶⠈⣹⣘⠋⠉⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠿⡿⠀⠑⠢⠖⠁⠀⠈⠉⠐⠀⠀⠀⠀⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠒⠂⠀⠀⡀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⢺⠀⠀⠁⠀⡀⠀⢄⡀⠀⠀⠀⠐⠃⠀⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⠀⠁⠀⢀⠉⠒⢤⣀⠀⠀⠀⠁⠀⠀⠀⠘⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠃⢀⡀⠀⠆⢘⠀⠀⠀⠈⠙⠒⠢⠤⠤⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡤⠒⠒⠐⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣱⠃⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠬⠀⠉⠲⣄⠑⠒⠊⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠑⠢⢀⣀⣀⣀⣠⠞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⢦⢄⣀⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢀⠇⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡌⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⡰⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠈⠀⠀⠈⠒⠪⡍⠀⠒⠀⠀⠀⠠⠤⠤⠠⠄⠄⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⣰⢆⣠⠀⠀⣼⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠡⠤⠄⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠏⢩⢴⠚⠳⠎⢰⠀⠟⠀⠀⠙⠋⠃⠀⡠⠔⡡⠋⠢⠤⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣿⢀⡑⣄⠀⢸⠀⠀⢢⠀⣠⠀⠀⢰⠀⠀⢧⢂⠔⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⡇⠑⠽⢬⠢⣼⢀⠀⢸⡉⣀⣀⣀⣸⠤⣀⡀⠉⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣒⡒⠁⠀⢀⣈⣱⣏⣴⡖⢒⣏⣙⣦⣶⣞⡩⠍⣱⢀⣀⡱⣦⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠈⠛⠛⠻⠿⠿⡧⢺⠷⠾⠿⣿⣿⣿⣿⣿⣷⠶⠿⠿⠿⠿⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠌

⠀⠀⠰⠆⣰⡇
⠀⠀⢰⠄⣿⡄
⠀⠀⢸⠆⢾⠂
⠀⠀⠸⠀⢹
⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡀⣀
⠀⠀⠀⠀⣿⠀⢠⠤⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⢻⣿⣿⣿⣿⢿⣿⣿⣿⢻⣿⠗⠒⠂⠈⡁⢀⡀
⠀⢸⡀⢸⢻⡔⡂⢀⠀⡀⣠⠄⠀⠀⠀⠀⣿⣿⣴⣝⣯⣯⣿⣾⣿⣿⣌⣻⣻⠀⠀⠀⠀⡅⠬⠀⠀⠀⣇⠠⠀⡇
⠀⠸⣷⠸⢸⠛⡇⠀⠁⠀⠛⠛⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⡿⢯⠀⠀⠀⠿⠶⠇⠃
⠀⠀⠉⢺⣿⠀⠆⠀⠀⠀⠆⢀⣅⣀⣀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⡇⢘⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢸⢿⠀⢆⡀⠄⣠⠀⠀⠋⠉⠉⠀⣿⠉⠋⠉⠉⠉⠉⠙⠋⠛⠉⠉⠙⡿⠀⠉⠀⠁⠀⠀⢀
⠀⠀⠀⠙⢛⠀⠀⡇⠀⢹⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣇⠀⠀⠀⠀⠈⡏⠀⣉
⠀⠀⠀⣾⣶⠀⠀⡇⠀⢸⣄⣀⣀⣀⣀⣀⣿⡄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢠⣷⠀⠀⠀⠀⢀⡇⠀⡇⠀⢠⡄⣀⣠⣤⡤⣤⣤
⠀⠀⢈⣿⣿⠀⠀⡇⠀⢸⠀⠀⢀⠼⠃⠀⣿⠀⠀⠀⠀⠀⠐⠂⠒⠒⠒⠒⠘⡿⠐⠂⣀⠀⠈⡇⠀⡇⠀⠈⠧⠉⠉⠉⠉⠉⠉
⠀⠀⢀⣿⣿⠀⠀⡇⠀⢸⢀⠄⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠈⠲⣀⡇⠀⡇
⠀⠀⢰⣿⣿⠀⠀⡁⠀⢸⠍⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠙⡇⠀⡇
⠀⠀⠘⠻⠛⠀⠀⡅⠀⣼⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡋⠆⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡁⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠆⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠋⠶⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡃⠀⠘⠆⡀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⢠⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠀⠀⠙⠁⢀⠀⠇⠀⠁
⠀⠀⠀⠀⠀⣿⠈⠉⠉⠉⢹⡂⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢠⠀⢸⠖⠒⠐⠒⠦
⠈⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠙⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠈⢠⠚⠢⠀⠀⠠⠚⠀⠤⠄⠤
⠀⠀⠀⠂⠀⠠⠐⠀⣀⡀⠀⢸⡧⠤⠤⠄⣿⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⢸⠄⡠⡄⡀⡀⣀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⢿⠂⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡿⠂⠀⠶⠀⠠⠒⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠯⠀⣬⠇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠈⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠁⢀⠄⠀⢰⠂
⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⢀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠐⠈⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢀⡀
⣿⣿⡿⠛⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿