Vieux norrois Emojis & Symbols

⢰⠀⢠⠄⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠄⠒⠀⠀⠂⠀⠀⠀⠐⠋⠐⠂⠀⠀⠀⠀⠐⠐⠐⠂⠂⠀⡀
⠈⠃⠀⠀⠀⠰⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠨⣗⣢⣴⡰⡫⢐⣠
⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠈⠀⡀⣀⠀⠀⠀⠀⢐⠀⡈⣿⣟⢟⣃⣻⡍⠂⠀⠁⠐
⠀⠸⠁⡀⠆⠁⠀⠈⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠤⠈⠁⡄⠁⡑⠂⠀⠀⠁⢀⠾⡼⣄⡙⣿⠁⠀⠀⠐
⠀⠀⠀⠀⠃⠀⠀⠀⢀⣤⡐⠚⢁⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠙⣄⠀⢀⠆⠠⣻⣹⠃⢼⢹⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠚⣿⡃⠀⠘⣠⣆⠀⠀⢠⠸⣀⠈⠉⠁⠀⢻⣷⣘⣅⠀⠙⠿⠦⠈⠟⠀⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⢛⣧⠀⠀⠀⠙⠀⠀⠀⠃⠈⠀⠠⠆⡬⠀⣙⢧⢝⡃⡀⣰⣃⠀⠀⠀⡋⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⡠⢠⣂⡈⠂⠀⠇⠀⠸⣇⡏⠠⠠⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠁⠀⠀⠀⠀⠀⠀⣾⣿⢠⠃⡀⠄⠀⠀⠀⠘⢓⠀⣡
⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠇⠀⡶⠀⠀⠀⠀⠀⠀⢻⡿⣠⢿⡗⢍⠂⡀⠄
⠍⠀⠀⠀⠀⠀⠀⢢⣺⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⢨⡃⠀⢪⠃⠀⢀⢨⡄⠠⠀⠀⠀⡄⠀⣠⣦⠤⢶⠒
⢱⣶⣄⠀⠀⠀⠐⠀⢻⣿⣼⣿⠿⠀⣰⠁⠾⠀⡀⠈⢠⠀⠠⠀⠘⢌⠛⡈⢀⠄⣤⢶⣶⣺⠯⠙⠀⣬⠯
⠀⠁⠈⣣⠀⠀⠀⠀⠀⢫⠩⠍⠁⠈⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠺⠟⠀⠀⠀⠀⠐⠉⠛⠂⠀⠃⠀⠴
⠀⠀⠀⠈⡂⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠊⠐⠄⠀⠀⠀⠀⠀⢀⢄⠀⠀⢀
⡄⠀⠀⠀⠀⠀⠀⠰⠂⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠸⠂⠀⢠⢈
⢡⠀⠀⡀⠂⠀⠀⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠒⠀⠠⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠐⡀⠀⢀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡈⠃⠀⠀⠀⠀⠀⠔

⠀⠀⠀⣀⣀⠐⡟⠁⡍⠀⡄⠀⠀⡄⠀⢠⠉⠉⠩⠉⢰⠀⠀⠴⠀⠀⠄⠀⠀⠈⢩⠉⠉⠉⠉⠉⠀⢤⡄⠉⢛⡇⣠⢄
⠀⠀⠀⢷⣾⠁⣇⠀⡄⠀⡤⡀⠀⡄⠀⢠⡀⠀⣀⠀⢀⢀⠀⡀⡀⣀⢀⠀⢠⠀⢠⣀⠀⠀⣄⡀⠀⡄⣀⠄⣩⢽⣿⡟
⠀⠀⠀⠘⢸⢰⡧⠀⠇⠀⠭⠆⠠⠧⠀⠰⠘⠈⠍⠀⠸⠀⠀⠏⠀⠚⠃⠀⠰⠀⠠⠁⠀⠀⠇⠀⠀⠈⠇⠀⠿⠶⣟⢁
⠀⠀⠀⢁⣼⢁⣷⡤⠄⠀⠀⠀⠀⡨⠋⠡⣎⠉⣄⠀⠀⠀⠀⢰⢀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⣿⣄⡀
⠀⠀⢴⢛⣾⢸⡟⢓⡂⠀⠀⠀⠀⣃⠀⠐⢃⡈⠃⠀⠀⠀⠤⠼⢥⠤⠄⠠⠘⠀⠤⠄⠀⠀⠄⡀⠀⠀⠀⠀⠀⡨⡿⣟⠻
⣠⣀⠬⠿⣷⠸⡇⠠⡥⠤⠤⡦⠄⡧⠀⠀⢀⠄⠄⠁⢤⠌⠀⠀⠈⠠⠤⠣⠀⠡⠤⠄⡡⠤⠰⡧⠤⠤⣨⠤⠌⠃⣿⠿⠇⠀⣤
⠉⠨⣼⢿⣟⠒⡇⠀⡗⠄⢰⡅⠀⡇⠁⠰⢹⠀⢰⢆⠀⢛⠀⠊⠁⠀⠈⠃⠀⣭⠀⢰⠱⡀⡀⡧⠀⢈⡱⠓⠀⡅⣻⡿⢫⠂⠈
⠀⠀⠻⣾⣽⢸⡟⠒⡑⠒⠀⠁⠒⡓⡒⠒⠀⠀⢈⠀⠀⠀⡠⠒⢒⠒⠒⠒⠀⢀⠐⢠⢀⢁⠓⠡⠀⠠⠄⠠⢀⡄⣟⣷⠞
⠀⠀⠀⣈⢻⡆⣇⠀⡗⠀⠀⠀⠀⡲⣄⠀⠀⠀⣈⡅⠀⡎⠁⠀⢡⠀⠤⠤⠀⢺⠀⢠⢟⢠⠀⠀⠀⠤⠤⠔⠓⡃⡿⠳
⠀⠀⠀⢠⢼⡇⣯⠀⠁⠒⠒⠀⠀⠀⠀⠀⠐⠒⠂⠐⠀⠒⠒⠒⠀⠐⠒⣀⠐⢂⡐⢀⡀⠀⠀⠐⠒⠒⠂⠤⠤⡆⣿⡆
⠀⠀⠀⠘⢿⡇⣏⠀⠀⠀⠀⠀⠀⠐⠀⠘⠑⢀⣈⣈⣈⣈⡀⣀⡀⠀⠁⠉⠉⠀⠀⠈⠈⠀⠀⠀⠀⠐⠂⠠⢶⣇⢿⣻⠄
⣤⣤⣤⣤⣤⣤⣿⣶⣦⣤⣤⣴⣶⣶⣶⣤⣤⣬⣭⣽⣿⣥⣤⣼⣥⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣴⣶⣶⣶⣿⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣠⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢥⣼⡿⠋⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣹⣿⣿⠿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠞⣷⣿⣾⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢫⣿⣭⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣿⢿⣏⣿⣿⢹⣿⣵⣷⣮⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣾⣿⣿⣿⣞⣿⣹⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣽⡻⢿⣿⣿⣿⣿⣿⣿⣿⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⣤⠄⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣷⣮⣭⣃⣙⡿⡿⣿⣷⣄⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⢀⣴⠀⠀⠀⠀⢸⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣦⣜⢯⣜⡃⠹⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⡤⠀⠀⠀⠘⢿⣿⡷⠀⠙⣿⣿⡆⠋⠛⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠛⠃⠈⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣾⠏⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣯⡕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢡⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⣙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡴⠀⠀⠀⠠⠀⠀⠀⢠⣾⡇⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⢀⣤⡾⠿⢷⣤⡴⠛⠛⠛⠶⠦⡄⠈⠙⢳⣄⠀⠀⠀⠀⠀⠈⠙⠻⠿⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣶⡴⠃⠉⠤⠀⠀⣠⡬⠁⠀⠀⢠⡀⠀⠀⠀⠀⠙⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⣀⣀⠀⣉⣉⣉⣉⣀⠀⠀⠀⠀⠀⣤⠤⠖⠚⠓⠲⠶⢤⣤⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠶⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣀⡀⣀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⡠⢀⡀⡯⠀⠆⠀⠆⠀⠀⠄⠀⠀⠆⠀⠐⠀⠐⠀⠀⠐⠀⠀⠒⠀⠐⠀⠐⠆⠀⠀⠃⠀⠀⠐⠆⠀⢈⣿⣠⣴⡄
⠀⠀⠀⠀⠀⢻⣿⠀⡧⠀⡆⠀⣖⠌⠉⡄⠀⢠⢄⠀⣤⠉⢡⢠⠄⢠⠀⢤⡂⠄⢰⠀⠀⣆⠆⠀⢰⠄⠀⢠⣀⡔⢡⡖⢻⣿⡃
⠀⠀⠀⠀⠀⡈⢸⠀⡗⠀⠃⠀⠒⠃⠀⠓⣀⣘⢈⡀⠀⠀⠘⠀⠀⠘⠀⠈⠁⠂⠘⠀⠀⠃⠀⠀⠘⠀⠀⠀⠛⠀⠐⠛⢸⡅⡄
⠀⠀⠀⠀⠀⣡⢾⠈⣿⡿⠿⠀⠀⠀⠀⡆⠂⠠⣲⠀⣲⠄⠀⠀⠀⢐⡇⠀⠀⠀⠰⠆⠐⡀⠀⠀⠀⠀⠀⠀⠈⠿⠷⠟⢸⣷⣤⡀
⠀⠀⠀⠀⣮⣙⡿⠈⡏⢉⡅⠀⠁⠠⠤⣥⠤⠈⠁⠄⠙⡀⠀⠀⠄⠀⢣⠂⠀⡀⠀⢀⠀⠂⢀⠀⠀⢢⠀⠀⢀⡀⠈⡡⢸⢿⣛⠃
⠀⠀⣤⢀⢲⣛⣷⠀⡇⠀⡄⠐⠒⡒⠒⠦⡀⠀⢠⠀⠀⡐⠒⠒⡖⠒⣲⡒⠒⢒⠒⢒⠒⠂⢀⠒⠒⢻⠒⠒⣂⡀⠂⠈⢸⣟⣛⡢⠠⡆
⠀⠀⠀⠐⣼⢿⣛⠀⣇⠀⡗⠂⠰⡇⠀⡀⠁⠘⢩⠀⢀⡃⡀⠈⡇⠈⢠⠈⢸⢸⠀⢠⠀⠀⠏⢆⢠⢸⢂⠀⢠⡊⠁⢰⢨⣿⠋⡐
⠀⠀⠀⠀⠛⣿⣼⠀⡇⠐⡆⠀⠀⠀⠒⠀⡀⠀⠀⠀⣀⡀⠀⢰⣠⠀⣀⡆⠀⢀⡀⠀⣄⠀⣄⡄⠄⠀⠒⠐⠂⠀⢄⠠⢸⣽⠟⠁
⠀⠀⠀⠀⠀⡄⢹⡇⣧⠀⡏⠀⠀⠀⠀⠜⢢⠀⠀⠀⠨⠅⠀⢸⠀⠀⠁⡇⠀⠰⠂⠈⡇⠀⠞⠗⡄⠀⠀⠀⠀⠒⠛⢫⢸⡏⠆
⠀⠀⠀⠀⠀⢰⣺⡇⣧⠀⠉⢉⠉⠀⠀⠀⠀⠀⡀⠀⡀⠀⢀⢀⢀⠀⡀⠀⠀⣀⢄⠀⡀⡀⢀⡀⠀⠄⠀⠀⠀⠀⠀⢺⢸⢧⡀
⠀⠀⠀⠀⠀⠘⠳⠇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣀⣀⣀⣈⡀⣀⣀⠁⠀⠈⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠶⢿⡘⢯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠛⠛⠒⠒⠂⠂⠀⠀⠛⠒⠒⠒⠓⠒⠒⠒⠂⠀⠀⠀⠒⠂⠀⠐⠒⠒⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⣿⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⠀⠀⠀⠀⠀⠀⠀⢀⣴⡗⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⢾⣷⣿⣷⣦⣀⣀⣠⣶⣶⣶⣲⣿⣿⠗⢦⣄⡀⠀⠀⠀⢀⣿⡿⢧⠄⠀⠀⠠⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠤⠴⢿⢿⣿⣿⣿⣿⣦⣙⢻⣽⣏⣸⣿⣿⠿⠶⡆⠘⣿⣷⣤⡀⠈⠉⠀⠀⠀⠂⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⣸⠇⠀⠀⠀⠀⢸⣿⡍⣿⡛⣷⣧⡺⠿⠛⢋⣁⠀⢀⡀⠊⠁⢹⣿⣿⣿⣿⠂⠀⠀⠀⠀⢐⣿
⠀⠀⠀⠀⠀⠀⠀⣽⠀⡏⠀⠀⠀⠀⠀⠀⣾⣿⣿⠌⣭⣻⣯⣍⣶⠀⠻⢿⣿⡿⠃⠠⢂⢸⣿⣿⣿⣿⡷⣤⡀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣭⠄⠻⠄⢀⣧⣐⣂⣠⣿⣿⣿⣿⢛⣿⣿⣇⢄⠀⠀⢸⣿⠉⠀⣀⡄⠙⠾⢸⣿⡀⠈⡸⢿⡄⠀⠐⣻
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⠡⠀⢠⡵⢾⣿⡏⠉⠛⣶⣠⣂⣸⣿⡇⠄⢀⠀⠙⣦⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣷⣶⣿⣿⣿⣿⣯⣿⣛⡻⢏⢀⣴⣾⣿⣿⣿⣷⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣗⣒⠋⠠⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠺⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣝⢿⡶⠋⠉⢹⣿⣿⣿⣿⣿⠛⠻⠏⣿⣿⣿⣿⣿⠿⣴⣷⠦⢈⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣃⠀⣀⡲⢸⣿⣿⣿⣿⣿⠀⠠⡀⣿⣿⣿⣿⣿⠛⠚⠋⠀⢐⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⠺⠭⢽⣿⣿⣿⣿⣿⣆⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡋⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡃⠀⠀⠀⠠⡿
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢐⣿
⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢼⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⣭⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠐⣻
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣻⡇⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⢿⡧⠀⠄⢸⣿⣿⢿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⡼⠛⠿⣷⡭⣥⢼⢣⣷⢀⠞⢧⠆⡟⢿⠿⡷⡷⢸⢉⠀⣬⢭⣾⠿⠓⢧⡀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠴⠀⠀⠘⠿⠻⣏⢸⠀⡯⡺⡄⢸⠇⡷⡛⠸⡄⡆⢸⠓⠀⣽⠛⠿⠀⠀⠀⠦⡀⠀⢐⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠻⠀⠛⠀⠱⠃⠑⠇⠹⠀⠙⠇⠸⠮⠄⠉⠀⠀⠀⠀⠀⠀⠃⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠾⣿⣿⡷⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠱⣽⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣶⣶⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣤⡖⠫⠗⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣻⡿⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⢠⣟⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣶⣉⢈⢿⡇⢠⣦⣄⣀⠀⠀⠀⠀⠀⠂⠀⢸⣿⣿⣿⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣯⡟⠃⠘⣯⡼⣾⢁⣀⣽⣇⣿⣧⡀⠀⡀⠤⣶⠄⢾⣿⣿⣿⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣶⣶⣤⣭⣿⣽⣿⣿⣿⣿⣿⠀⠀⠀⠟⠀⢸⣿⣿⣿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⠋⣿⣿⡟⠁⠉⠈⠀⠘⢿⣿⣿⣿⣿⣷⣤⡀⠂⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀⠀⠘⢻⣿⣦⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⡟⣠⣤⣤⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠉⣿⣷⣂⠀⣉⣻⠟⠲⣶⡀⠀⣜⣤⣿⣿⡟⠙⠛⠿⣿⣿⣏⢰⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠟⣿⣿⣿⣾⣷⠰⡿⢿⡿⠾⢦⣿⣿⣿⡇⢀⣄⠰⡇⠀⢹⣿⣿⣿⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⡾⠁⢰⡇⠀⣼⣿⣿⣿⠘⢧⣸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⣧⣤⣈⡇⠘⣿⣿⣿⣿⠎⠁⢈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹⡄⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⢹⡏⢰⣿⡇⢼⣿⣿⣿⣿⠃⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣠⣾⡈⠙⠣⡞⢻⣿⣿⣿⠆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢈⣿⡟⢷⡾⢿⣧⣤⣤⣴⣶⣶⣴⣿⡿⠿⢿⣷⣄⣴⣷⢸⣿⡿⢳⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣗⠀⣉⠀⠋⠉⠉⠹⠟⣿⠙⢿⣥⢤⡶⣿⣁⡿⢇⢸⣟⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢤⠶⣿⣿⣶⣿⡿⢷⠶⣶⣿⣿⣿⣳⣞⣻⣷⣼⠟⣿⠁⠘⠃⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣧⠀⠠⣼⣿⣿⣟⣿⣾⣿⣿⠏⢰⠇⠀⣀⣒⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦⡀⠀⠠⠼⣿⣿⣿⠛⠟⢻⡿⠋⢩⣘⢻⣿⣿⠃⢀⡏⠀⠀⠉⠉⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠐⡢⠼⢷⡾⣿⡗⣢⣤⠙⠛⠋⠀⠈⢾⣧⠀⣸⢛⣻⣿⠁⠀⣼⠁⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠐⢀⠂⠀⠀⠀⠑⠃⡚⣿⣏⣀⠉⡠⠞⠁⣀⣸⠚⢿⡏⠠⠀⠮⢿⣷⣀⣠⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠴⠖⣾⣭⡄
⠀⠀⠀⠀⠀⣀⣀⣀⣤⢂⡀⠀⣀⣉⣓⣉⣥⣤⠴⠶⠿⠛⠉⠋⠟⡃⠀⠹⠤⠴⠾⡟⠛⠿⠛⢦⠀⠰⣶⢦⠀⣴⣶⣶⣿⣿⠿⠯
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⢿⣿⣿⣿⠉⠉⠀⠄⠀⡀⠀⠀⠘⠾⣧⡀⠀⠀⣀⣀⣴⣶⣾⣿⣇⣀⡀⠀⠈⠀⠀⠠⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠛⠁⠀⠀⠀⠀⠀⣤⡼⠒⠾⠿⠦⣤⣀⣈⣉⣰⡬⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⠛⠒⠈⠛⠉⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣶⣾⣿⣟⣁⣀⢫⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢻⡿⠋⠈⣻⡁⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣆⠈⠀⣱⣦⣿⣀⠖⠂⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠈⣱⣤⣀⠉⠁⣀⣀⣤⣄⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣯⣿⣿⣿⣶⣶⣿⣝⡏⠛⠓⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣠⣤⣾⣿⡿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠟⢻⣷⡶⠞⢻⢯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣴⣿⣿⣯⣿⣿⣿⣿⣷⣦⡿⠿⠛⠛⠋⠈⠉⠉⠙⠒⣡⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣯⣼⣿⣽⣿⣿⢉⡁⠀⢀⡀⠀⠀⢀⠀⣤⡁⢻⣳⣿
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⣠⣿⣿⣿⣿⣧⡇⠓⢀⠐⢞⣿⣶⣿⣿⣨⣥⣴⣿⣿⣿
⠀⠀⠀⠀⠁⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢞⣿⣿⣿⣿⣿⣿⣿⣿⢀⠈⠀⠉⠙⠻⣿⣿⣿⡟⠿⠛⠛⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣦⣑⣄⠐⠤⢂⣽⣿⡛⠋⠃⠈⠁
⠑⠀⠀⠀⠀⠂⠒⠂⠆⠠⠤⠄⠀⠀⠀⠀⢀⢀⢬⣿⣿⣿⣽⡏⢤⠀⣈⣄⢀⣹⣟⣻⠞
⠄⠀⠀⠀⠀⠀⠍⠀⠈⠄⠠⠀⠄⠀⡠⣤⣠⣴⣟⣿⣿⢻⡿⠁⣺⣾⡟⠉⠋⢣⠉⠡⠀⠀⠀⠀⠀⢀⢠
⣂⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⢾⣾⣿⣿⠿⠋⠤⠇⡐⠛⠛⠋⣁⠽⡖⠚⠉⠀⠀⠀⠀⠠⠄⠘⠻

⠀⠀⠀⠀⠀⠀⢶⡆⠀⣀⡴⠶⠀⠀⣶⡀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠐⣿⠶⣤⣀⠀⠀⠀⢺⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢸⡷⠞⠉⣠⣤⠀⠀⣿⠻⡄⠀⠀⠀⢸⣿⠷⢦⣄⡀⠀⠀⣿⡀⠀⠉⠛⠃⠀⢸⡇⠀⣉⡿⠗
⠀⠀⠀⠀⠀⠀⢸⣧⠴⠋⠁⠀⠀⠀⣿⠀⠹⣄⠀⠀⢸⣿⢀⣠⠜⠋⠁⠀⣿⠙⠳⢦⣄⡀⠀⢸⣿⣍⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠹⣆⠀⢸⣿⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠘⠋⠀⠛⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠘⠛⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⠖⠀⠀⠻⣧⠀⠀⢀⡟⠁⠀⣿⡟⠳⢦⣤⣀⠀⢹⡇⠀⠀⠈⣿⠀⠀⠀⠘⣿⠀⠀⠀⠈⣿⠀⠀⠀⣠⠖
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠙⣷⣴⠋⠀⠀⠀⣿⣇⡤⠖⠋⠁⠀⢸⣿⣦⣄⡀⣿⠀⠀⠰⣶⣿⡀⠀⠀⠀⣿⠀⠀⣾⡁⠰⣦⡀
⠀⠀⠈⠻⣦⡀⠀⠀⠀⢠⠞⢿⣆⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠉⠛⣿⠀⠀⠀⠀⣿⠙⠁⠀⠀⣿⠀⠀⠈⠛⢀⡼⠃
⠀⠀⠀⠀⠙⠟⠀⠀⣴⡋⠀⠀⠻⣧⠀⠀⢿⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣿⡄⠀⠀⠀⣿⡀⠀⠀⠀⣿⡀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣤⡀⠀⠀⢀⡀⠀⠀⣀⠀⣀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢸⣿⠙⠻⢶⣤⠀⠀⠀⠀⣿⡟⢦⣄⠀⠈⣿⠛⢦⡴⠛⠁⠀⠀⠹⣦⣹⡇⣠⠋⠀⠀⢀⠞⠉⠁⠀⠀⠀⢠⠔⢋⣿⠻⢦⣄
⠀⢸⣿⣤⠖⠋⠁⠀⠀⠀⠀⣿⡇⠀⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠁⠀⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠈⣿
⠀⢸⣿⠙⠛⢶⣤⠀⢀⣄⠀⣿⡇⠀⠀⠀⠀⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⢷⠄⠀⠀⠀⠀⣿
⠀⠘⣿⡤⠖⠋⠀⠀⠀⠉⠻⣿⡇⠀⠀⠀⠀⣿⡴⠋⠙⠷⠄⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣤⡴⠃⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⢶⣄⠀⠀⠀⢰⣦⠀⠀⢰⣦⡀⠀⠀⢠⣶⠀⠀⢰⣶⣤⣀⠀⠀⠀⢀⡴⠋⣀⠀⠀⠀⠀⣶⡄⠀⠀⠀⣠⣆⠀⠀⠀⢀⣴⣦⡀
⠀⢸⡟⢷⣄⣴⢻⣿⠀⠀⢸⣿⠻⣦⣴⠋⣿⠀⠀⢸⡇⠀⠉⠙⠒⠀⠈⡻⢿⣍⡀⠀⠀⠀⣿⡿⣦⠀⡴⢻⣿⠀⠀⣴⠋⠀⠙⢿⣦
⠀⢸⡇⠀⠙⠁⢸⣿⠀⠀⢸⣿⡴⠋⠛⢷⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠿⣦⣙⠻⡶⠀⠀⣿⡇⢘⣿⣅⢸⣿⠀⠀⠙⢷⣄⣠⠞⠁
⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢨⡽⢿⣦⠀⠀⣿⣧⠏⠀⠻⣾⣿⠀⠀⠀⢀⡿⢷⣄
⠀⠘⠗⠀⠀⠀⠀⠿⠂⠀⠈⠿⠀⠀⠀⠀⠻⠃⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⠞⠁⠀⠀⠙⠃⠀⠀⠀⠘⠟⠀⠀⠛⠋⠀⠈⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⢀⡀⠀⠀⡀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢸⣇⣴⠟⠃⢸⡿⠷⣦⣄⠀⣿⠀⠀⠀⠀⣿⣶⣄⡀⠀⣿⡷⢦⣄⡀⠀⠀⣠⣾⠇⠀⢿⡄⣰⡟⢸⣿⠶⣤⣀
⠀⢸⣏⣥⡶⠂⢸⡇⠀⠀⣿⠀⣿⠷⣤⣀⠀⣿⣦⣉⠛⠀⣿⣇⣴⠟⠁⢀⣴⠟⠁⠀⠀⠘⣿⡿⠀⢸⣿⣠⡾⠋
⠀⢸⡟⠁⠀⠀⢸⡇⠀⠀⣿⠀⣿⡶⠟⠉⠀⣿⠈⠙⠷⠄⣿⣿⣅⠀⠀⠈⠻⣦⡀⠀⠀⢠⣿⣷⠀⢸⣿⠋
⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠙⢷⡄⠀⠀⠘⢿⡆⢀⣾⠃⢹⣧⢸⣿
⠀⠈⠁⢀⠀⠀⠈⠁⠀⠀⠉⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠉

⠀⢸⡇⠀⠀⣷⠀⣀⢰⡆⠀⠀⠀⢸⡇⠀⠀⠐⢶⡀⠀⠀⠀⠀⣶⣤⡀⢰⣦⡀⣰⡷⠀⢶⣼⣧⡶⠀⠀⣰⡇
⠀⢸⣷⡄⠀⣿⠀⢻⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⢈⣿⣦⡀⠀⠀⣿⠈⠁⢸⡏⠻⠏⠀⠀⠈⢿⡿⠁⠀⢰⡿
⠀⢸⡏⠻⣦⣿⠀⠀⢻⣧⠀⠀⠀⢸⡇⠀⠀⢴⣟⣵⠛⠁⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠛⠛⢻⡿⠁
⠀⢸⡇⠀⠙⣿⠀⠀⢸⡿⣧⠀⠀⢸⡇⠀⠀⠀⠻⣦⡀⠀⢰⣤⣿⠀⠀⢸⣷⠾⣷⡀⠀⠀⢸⡇⠀⠀⠀⢀⣾⠃
⠀⠘⠃⠀⠀⠛⠀⠀⠘⠃⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠃⠀⠀⠈⠙⠀⠀⠘⠁⠀⠈⠋⠀⠀⠘⠃⠀⠀⠀⠘⠃

⠀⠀⠀⣠⡀⠀⢠⣄⡀⠀⠀⠀⣄⠀⠀⠀⡄⢀⠀⠀⠀⣠⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⢀⣄⡀
⠀⢠⡾⣿⡿⡆⢸⣿⢙⣷⡦⠀⣿⣷⣤⣾⡇⢸⣷⣤⣾⣿⢸⣿⠷⣦⣀⠀⠀⢀⣄⠀⠀⣿⣆⢠⣾⡇⢴⣟⠉⣷⡆
⠀⠀⠀⣿⡇⠀⢸⣿⡛⠁⠀⠀⣿⠀⠉⢸⡇⢸⣿⠟⢿⣿⢸⣿⠀⠈⠉⠀⣴⡋⠙⣷⠀⣿⢹⣿⢻⡇⠈⢻⣾⠏
⠀⠀⠀⣿⡇⠀⢸⣿⠛⣶⣤⠀⣿⠀⠀⢸⡇⢸⡇⠀⠀⣿⢸⣿⠀⠀⠀⠀⠈⠻⠾⠁⠀⣿⡾⠻⣿⡇⠀⣼⢿⣆
⠀⠀⠀⣿⠇⠀⠸⠿⠛⠉⠀⠀⢿⠀⠀⠸⡇⢸⡇⠀⠀⣿⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠙⠇⠼⠏⠀⢻⠆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠿⠿⠿⣿⣿⣿⣿⢸⣿⣿⠿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣹⠎⢹⣆⢸⠉⠉⣀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠈⢢⡀⠀⠙⠶⠏⠠⣴⡏⢸⠀⡮⣶⣼⡌⢻
⠀⠀⠀⠀⠀⠈⢷⣄⣠⠤⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠁⠈⢿⡇⠀⠀⠀⠃⢀⠀⠀⠀⠀⠉⠁⢸⡴⠟⢻⠟⠋⠙
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠑⢀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣿⠐⠐⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⠀⠆⠀⠀⠀⠀⠂⠡⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⡏⠐⠀⠈⠻⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠙⠀⠀⠀⠃⡰⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠤⠀⠂⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣰⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢸⡇⣠⣤⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀⣀⢀⠃⠈⠀⠘⢿⣿
⠀⠀⠀⠀⠀⢀⠴⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⣤⣼⣿⣿⡇⣿⠁⠀⢀⡀⢠⡞⠁
⠀⠀⠀⠀⢠⢈⣤⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⣿⣦⣰⡾⠟⣿
⠀⠀⡄⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣸⣿⣿⣯⣿⣟⣹⠀⣿⣿⣿⣿⡗⠈⠳⣤
⠀⠀⠤⠤⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣶⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⡿⠀⢀⠀⣤
⠀⠀⠀⣈⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⢀⢐⣾⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⢸⣿⣿⣿⠃⠀⠀⢠⣿
⠀⣠⣄⣀⡠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⡟⢸⣿⣿⣿⠀⠀⠀⣼⣿
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣶⣶⣶⣿⣿

⠀⢠⣀⣴⠄⠀⠀⠀⠀⠀⢰⡶⣤⣀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢸⠟⣡⠄⠀⠀⠀⠀⠀⢸⡇⠈⠙⣷⠀⠀⠀⠀⠀⢸⡟⢷⣤⡀⠀⠀⠀⠀⠀⠀⣿⠙⠷⠀⠀⠀⠀⠀⠀⠀⣇⣸⡿⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉
⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⣴⠿⠋⠀⠀⠀⠀⠀⠀⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⠀⣾⣏⠁
⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠀⠀⠀⠀⠀⠀⠀⣿⠙⠷⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄
⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠛⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠏⠐⠀⠲⠶⠀⠀⠀⠀⠀⠹⠂⠶⠐⠂⠀⠀⠀⠁⠀⠒⠐⠂⠀⠰⠐⠀⠀⠀⠘⠇⠶⠀⠶⠀⠂⠀⠀⠀⠻⠠⠐⠶⠛⠒⠆⠀⠀⠀⠸⠆⠖⠐⠂⠶


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠘⢷⡀⠀⣠⡟⠀⠀⠀⠀⠀⡟⠻⢶⣤⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠈⢻⣶⠋⠀⠀⠀⠀⠀⠈⣷⡶⠛⠁⠀⠀⠀⠀⠀⢸⡷⣤⡀⣿⠀⠀⠀⠀⠈⠻⣶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠞⣡⠟
⠀⢠⡾⠛⣧⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣧⠀⠀⠀⠀⠀⠀⠈⡿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⡁⠞⠁
⠀⠛⠀⠀⠈⠻⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠄⡀⣄⣀⠀⠀⠀⠀⠀⣤⡀⠀⠀⣀⣀⠀⠀⢀⣠⢀⣀⠀⡀⠀⡀⠀⠀⠀⠀⢠⡄⠀⣀⠀⣀⢀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠈⠉⠀⠉⠉⠀⠀⠀⠀⠀⠁⠈⠁⠀⠈⠉⠀⠀⠀⠈⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠈⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠈⠈⠁⠈


⠀⠀⣤⡀⠀⣠⡆⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⣤⡀⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⣶⣦⣀
⠀⠀⡏⠹⠾⠋⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⠀⠀⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠞⠉⣿⠙⠳⠀⠀⠀⠀⠀⣿⣀⣽⠇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠛⠛⣷⠖⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⢯⣄
⠀⠐⣧⡾⠻⣦⡀⠀⠀⠀⠀⠘⢷⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣀⡽⠗
⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⡄⠀⢀⢀⠀⠀⠀⠀⠀⠀⣠⢀⡀⢀⡀⠀⠀⠀⠀⣤⢠⣀⡀⠀⠀⠀⠀⠀⢄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠄⢀⣀⡀⡀⠀⠀⠀⠀⣄⡀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠈

⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣿⢷⣤⡾⣿⠀⠀⠀⠀⠀⠿⣦⣤⢾⡇⠀⠀⠀⠀⠀⣼⢷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⣿⣦⡀⢀⣼⣿⠀⠀⠀⠀⢀⣴⠾⣦⡀
⠀⠀⣿⠀⠉⠀⣿⠀⠀⠀⠀⠀⣶⠟⠻⣾⡇⠀⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⠀⠀⠀⣴⠏⠈⢻⡆⠀⠀⠀⠀⣷⠈⣷⡞⠁⣿⠀⠀⠀⠀⠻⣧⣀⣼⠟
⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣴⠟⠁⠀⠀⠀⠀⣯⣴⠏⠻⣦⣿⠀⠀⠀⠀⢀⣼⠿⣧⡀
⠀⠀⠻⠀⠀⠀⠻⠀⠀⠀⠀⠀⠇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠘⠟⠀⠀⠀⠀⠙⠁⠀⠈⠛

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠠⣄⢀⣀⡀⡀⠀⠀⠀⠀⡀
⣤⣴⣶⣶⣶⣴⣶⣶⣤⣤⣴⣿⣾⣷⣶⣶⣶⣶⣤⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣤⣴⣦⣾⣦⣥⣶⣦⣤⣤⣤⣴⣦⣴⣷⣶⣶⣤⣤⣤⣤⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⠀⠀⠀⠀⠀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠠⡀⠀⠀⠀⠀⣽⣧⠐⡀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⢀⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⣶⣼⣦⡀⢀⣼⠟⣏⠀⠙⢦⡀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣛⣿⣿⣿⣷⡾⠃⣠⣼⠆⠀⠀⣧⣸⡇⠀⠀⠀⠀⠀⢈⡀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠻⣿⣿⡿⣠⡾⣿⣧⣈⠘⢻⣿⣿⣿⡆⠀⠀⠠⣀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣬⣿⣾⡟⠀⠀⠀⠉⠙⠁⣿⣿⣿⠇⠀⠀⣤⣌⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣋⠀⠀⢀⡀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠁⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠿⠦⠀⠸⠷⠌⠀⢰⣆⡀⠀⢀⡀⠐⢤⡻⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠓⠂⣛⣹⣋⣙⣛⣓⣻⠿⣥⣀⡸⠧⠤⣐⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡓⠦⣟⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⡜⠟⠻⣿⣿⡹⠛⠛⠉⣿⠇⠀⠈⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢠⡀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣴⣶⣶⢷⡄⠀⠀⠀⠀⠸⢠⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡁⠈⠁⣻⠀⠀⠀⠀⠀⠀⢿⣄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣶⣶⣆⢹⡀⠀⠀⠀⠀⠨⢿⣿⣦⣀⣼⡦⢀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡯⠻⢿⣿⣿⣦⡀⢻⣿⣶⡣⢿⣿⣿⣿⣽⡀⠀⢠⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⢰⢀⡔⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⠁⠉⣁⣾⣿⡿⠶⢺⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⣄⡀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡿⠏⡙⠃⠀⠀⠀⠀⣼⣿⣿⣿⡏⡄⢸⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⣠⣾⣿⠻⣿⣿⡟⠻⢿⣿⣿⢟⡛⠉⠠⠃⠀⠀⠀⠀⢀⣿⣿⣿⣯⠚⠁⢸⣿⣿⣛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠸⣿⣿⣀⠀⠉⠀⠀⠀⠻⣿⡁⠁⠰⢀⡀⠀⠀⠀⠀⢻⣿⣿⣿⠓⠀⠀⢸⣿⣿⣿⣦⣈⠉⢙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⢸⣿⡇⣠⣔⣺⣧⣤⡄⢀⣰⣿⣿⣿⠿⠉⠀⠀⠘⢿⡟⢿⣿⣿⡆⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡋⠝⠻⢿⣿⣿⠏⣼⡿⣿⢹⣿⣦⡀⠀⠀⠀⠠⢁⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠚⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⢻⣬⢈⠀⢿⡏⢹⣆⡿⣿⣉⠙⠋⠙⠙⣦⣴⣆⠄⣛⡽⢿⢾⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡗⣸⠛⢮⣟⣢⣿⠇⠀⠁⠀⠀⠀⠈⢉⣒⡾⠷⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡀
⠀⠀⢀⣤⣴⣦⣄⣀⣀⠀⠀⠀⣴⢟⣿⣿⣯⠀⢀⣭⣿⢟⣇⠀⠀⠀⠀⢀⣴⣯⣅⡀⠀⠀⠀⠉⠳⡆⠀⠀⣄⡄⠀⠀⠂
⠀⠀⠈⠋⠉⠉⠉⠉⢽⣿⣶⣦⣿⣼⣿⣿⡋⠀⠴⣾⣙⣟⣤⠀⠀⢹⠠⡿⠉⠉⠙⢿⣦⣀⡄⠀⠀⠄⠰⣿⣿⠃
⠀⠀⠀⠀⠠⡀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣛⣇⠀⢀⠹⣿⣿⣿⣦⡀⠀⣿⡁⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠹⡤⢤⠦
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣶⠀⠀⠓⠣⠝⣿⣷⣷⣤⡈⣳⣶⣴⣶⡿⠀⣿⠗⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡀⢢⣤⠀⢤⣈⠙⢻⣿⣿⣿⣽⣿⣿⡅⠀⢹⣞⣶⣄⢳⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠀⠀⠀⠉⠉⠋⠁⠙⠛⠛⠋⠈⠁⠀⠈⠙⠙⠛⠉⠛⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣁⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⠀⡼⢁⠀⠀⠀⠀⡶⡀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢰⣃⡀⡰⢂⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣯⠞⡰⠋⠀⠀⠀⢰⠇⠸⡄⠀⠀⠀⡏⣳⡄⠀⠀⢸⣋⣉⡴⠋⠀⠀⠀⠀⡾⢈⡷⠀⠀⠀⡿⠀⠀⠀⠀⠀⢆⠀⣰⠀⠀⠀⣰⡷⣤⡀
⠀⣯⠞⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡟⠁⠀⠀⠀⣸⠁⠈⠀⠀⠀⠀⠀⠀⡇⠘⢆⠀⠀⠀⡿⣄⠀⠀⠀⠀⢈⣞⠁⠀⠀⠘⠓⡟⠁
⠀⠇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠇⠀⠀⢰⠇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠈⠂⠀⠀⡇⠈⠂⠀⠀⠴⠋⠈⢦⠀⠀⠀⠀⡇



⠀⠀⡦⣄⠀⠀⢠⣄⢀⡄⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⢀⡇⠀⠀⠀⢸⠈⠉⠀⠀⢠⣇⣤⡄⠀⠀⠀⣯⠵⠶⠮⣿⠀⠀⠖⢉⡇⠙⠢⠀⠀⠀⢸⢈⡳⠀⠀⠀⠀⢸⠑⠤⠞⡏⠀⠐⠦⣼⣀
⠀⢸⠇⠀⠀⠀⡿⠀⠀⠀⠀⠈⠉⢰⠁⠀⠀⠀⡇⠀⠀⠀⡿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡼⠛⢤⡀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⢸⠈
⠒⢼⠀⠀⠀⠀⣧⠞⠲⠀⠀⠀⠀⠸⠀⠀⠀⠀⠇⠀⠀⠀⠗⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢧⠴⠊⠁⠀⠀⠀⠏⠀⠀⠀⠇⠀⠀⠀⠸


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢠⣀⠀⣠⠀⠀⡤⡀⠀⠀⠀⢠⢄⠀⠀⠀⠀⢠⠀⠀⠰⣤⢶⢄⠞⠀⠀⠀⠀⡶⣤⠖⠀⠀⠀⢰⠀⠀⠀⠀⡖⠢⡀⣀⠴⡖⠀⠀⠢⣼⡴
⢰⢌⠛⠁⠀⠀⣇⠘⠆⠀⠀⡼⠈⢆⠀⠀⢐⣾⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⡇⠀⠉⠁⠀⡇⠀⠀⠀⢸
⣶⠀⠀⠀⠀⢠⠉⠃⠀⠀⠀⡟⡖⢺⠀⠀⠀⡇⠀⠀⠀⠀⡾⠀⠀⠀⠀⣀⡀⡇⠀⠀⠀⣠⠞⡏⠢⡀⠀⢠⣃⡴⠒⠤⣠⡁⠀⠀⠀⣸
⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠉


⠀⢦⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⡄
⠀⡸⢆⡰⢧⡀⠀⠀⢰⠦⠀⠀⠀⠈⣳⢾⡁⠀⠀⠀⢀⡴⠛⢲⡄⠀⠀⡷⣄⠀⣠⠴⡆⠀⠀⠀⣇⣹⠆⠀⠀⠀⣧⣀⠀⡇⠀⠀⠀⡇
⠀⢱⠞⢧⠊⠀⠀⠀⣾⠀⠀⠀⠀⠀⢙⣶⠋⠀⠀⠀⠘⣦⡔⠁⠀⠀⢸⢀⡨⠻⣅⢰⠇⠀⠀⢰⡏⠀⠀⠀⠀⠀⡇⠀⠙⡇⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣂⡛⠊⠻⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠚⠉⠉⠀⠀⠻⢸⡶⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠈⠓⠀⠀⠀⠀⠉⠻⣆⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⡅⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠈⠧⠈⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⢀⡀⠀⠀⢀⣀⣀⣀⠶⠁⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢾⣯⣡⠏⠉⠉⢹⠍⡫⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠶⠦⢄
⠀⠀⠀⠀⠀⠀⣠⠤⠤⢄⡀⠀⠀⢀⣴⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠠⠴⠀⠀⢉⠙⠲⣄
⠀⠀⠀⠀⢀⡴⠁⢒⡚⠂⠉⠒⠦⣼⠀⢠⠊⠔⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠒⠀⠀⢸⡄
⠀⠀⢀⡤⠋⠀⡼⠉⠉⠳⡀⠀⠀⣨⠀⡆⠐⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠓⠀⠀⠈⠉⠐⠒⠒⠒⠒⠒⠲⠦⣄⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⡴⠋⢠⣄⡼⢁⠔⠠⢀⡘⣦⣻⡇⢠⠃⠀⠀⠀⠀⠀⠀⢀⢜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠼⠁⢠⠶⡐⠀⠀⠀⠀⠀⢱
⠈⡇⠀⠳⣄⢰⠃⠰⢀⡀⠀⠉⠉⠁⠺⠆⢀⠀⠀⠀⠀⠀⠑⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠤⠔⠃⢀⠆⠀⠀⠀⠀⡄⢀⠁
⠀⡇⠀⠀⣸⠘⠀⢰⠋⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⡆⠀⢀⡎⠀⠀⠀⠀⢰⢁⡎
⠀⠙⠤⣠⡏⠠⠀⠘⡆⠀⠀⠈⠳⢤⡀⠀⠀⠀⢀⡀⣠⡖⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠀⣿⠀⠈⣇⠀⠀⡀⠀⠸⣎⣣⣠⠄
⠀⠀⠀⠀⠀⢰⠀⠀⣹⠀⠀⠀⠀⠀⠉⠙⠓⠦⣭⠞⠻⢭⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢻⠀⠀⠈⠢⡱⣄⢌⣆⠹⡍⠉
⠀⠀⠀⠀⠀⢸⡄⠸⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠦⠤⠤⠤⠴⠶⠶⢶⣶⣀⣀⡀⢀⡀⠀⢀⠀⠂⠐⠄⢸⠂⠀⠀⣀⣷⢟⡼⢿⢂⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠭⢷⣄⣈⠉⠃⠂⠀⠈⠀⡙⢄⠀⠀⠀⠀⣉⣁⣼⠞
⠀⠀⠀⠀⠀⠀⠈⢄⣁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢦⡀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⢶⣤⣄⡀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢿⠙⢆⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⣆⠈⢧⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠀⣀⣸⡄⠘⡅⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢠⠏⠀⠀⢀⡇⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⣷⠀⠀⠀⣸⠓⠂⣠⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣀⣀⣤⠴⠋⠀⢠⠞⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠴⠒⠂⠀⠉⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠉⣠⠔⠊⣀⣀⣈⢉⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠞⣁⡴⢧⣄⠠⠶⠶⠤⠰⠶⣄⡀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡾⢁⠀⢀⣿⠷⠆⠀⠤⣭⠏⢴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠸⠋⠀⣀⣉⣫⡥⠩⣟⣓⡉⠙⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⠈⢸⠘⠀⡄⠀⢀⠐⢒⡂⣄⣓⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠸⠰⣇⠋⠰⠈⡀⠡⠍⠰⠸⢀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠁⡀⡀⠃⠘⠧⠙⢳⣤⣤⣤⣤⡾⢋⡹⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⡀⢃⡅⢰⡘⠦⠠⠀⡿⠁⠈⢻⠧⠠⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⣸⣿⣄⠉⠂⡴⠀⡇⠀⠀⠨⠗⠄⠞⢃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠻⣷⣾⠁⠀⠐⢦⡴⠂⠀⣸⣾⠛⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣾⣿⡉⢦⡀⠈⠳⢦⠀⠚⠃⢠⡴⣋⣠⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡟⢳⡄⠁⠀⠀⠈⠛⠾⠟⢩⡾⠋⢉⡡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠘⠻⣦⡀⠠⣀⣀⠀⠠⠏⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣄⡀⠀⠙⢦⡈⠙⠛⠃⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠢⡄⠁⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢰⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣈⣠⡿⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⣀⠀⠀⠁⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡾⣄⠀⠀⠏⠙⠄⠉⠳⣄⡙⢳⣤⡀
⠀⠀⠀⠀⠀⢠⡞⠁⠙⡆⠀⡀⠀⠀⠀⢤⣀⣄⠉⠓⢦⣻
⠀⠀⠀⠀⠀⣿⠁⠀⠀⣠⠴⠁⠀⢠⡄⠀⢻⣿⣷⡄⠀⠹⠑⡄
⠀⠀⠀⠀⠀⣸⠀⠘⡞⠁⠀⠀⠀⠘⠁⠀⢸⠿⣿⣷⠀⠀⠀⣇
⠀⠀⠀⠀⢰⠃⠀⠹⡁⠀⠐⠀⠀⠀⠄⣠⣾⠀⢿⡿⠀⠀⠀⠈⣧
⠀⠀⠀⠀⢧⢰⡇⠀⢱⠀⢠⠀⠀⠀⢠⡟⢸⡆⠸⠃⠀⠀⠀⢀⣟
⠀⠀⠀⠀⠈⠺⣧⢣⠘⡀⠰⣀⡻⠂⣸⡇⠈⡇⠀⠀⠀⣠⡴⠋⠀⠙⠒⠠⠄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠘⠣⣶⣿⡷⠠⠃⢀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⢦⣄
⠀⠀⠀⢠⠖⠛⣛⠉⠀⠀⠀⠀⢸⣿⣿⠃⢠⠖⠁⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⣸⢎⡽⠋⠁⠀⠀⠀⡀⢼⣿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠒⣿⣫⡴⣐⠔⠂⠉⢁⣠⠞⢻⡀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣧⠀⠀⠀⠀⠀⠘⣿
⠀⠀⢸⣿⣹⠁⠃⠖⠉⠉⠀⠀⠠⢾⣷⣿⣶⣤⣼⣶⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠉⣿⠀⢀⣷⠀⠀⢀⡠⠀⣼⣿⣿⡿⠛⣿⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣋⣄⠀⠀⠀⠀⠀⢀⣿⠉⠛⠲⢦⣄
⠀⠀⠀⠀⠘⣧⢻⠁⡠⠊⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡀⠠⣴⣾⣿⣷⣶⣿⡿⠿⣿⣷⣒⣻⣽⣿⣿⡿⢳⢄⠀⠀⠀⠈⣿⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢤⣀⣤⠟⠈⠀⡀⠀⠀⠀⠀⠀⠀⡞⠁⣤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠟⠁⢸⠀⠳⣄⢀⡀⠈⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠈⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠸⠟⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⠀⠀⢸⡀⠀⠙⣿⠀⠘⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠘⢦⣴⢀⣄⠀⠀⠀⡘⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⢿⡄⠀⠀⢳⡀⠀⢹⡄⠀⠀⠀⠀⠀⠰⣧⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⠀⠀⠠⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢷⡄⠀⠀⠳⡄⠸⡇⠀⠠⠀⠀⠀⠀⢿⡙⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⣰⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣼⡇⠀⠀⠀⠹⣄⢻⡶⠀⠀⠀⢠⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣠⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡟⠀⠀⠀⠀⠀⢹⡈⡇⠀⠀⠀⢸⢧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⢸⠈⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣦⣸⠃⠀⠀⠀⠀⠀⢀⡼⠀⣧⣠⡆⢀⡟⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⢀⡋⠉⠀⣀⣀⣤⠤⠖⠋⠀⠀⣿⣋⣴⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢁⡴⠯⠭⠶⠶⠤⠤⠤⠴⠺⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⣿⣆⠀⢀⣤⣤⣤⣴⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡶⢶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣌⠠⠔⠛⠢
⠀⠀⠀⠀⠀⠸⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⡿⠙⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢶⣤⡈
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡬⠁⠢⣙⡒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣶⣤⡁
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⠀⠈⢉⡻⠿⠖⠂
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠄⠠
⠀⠀⠀⠀⢀⣾⠏⢁⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⢀⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣼⣿⣿⣿⣿⡟⣽⣿⣿⡿⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠈⠿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⢡⢾⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠋⠀⠁⠈⣠⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⣹⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠿⢶⣶⣶⣶⣿⣿⣿⠿⣋⣾⣾⡿⠟⣡⠟⢀⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⢛⣉⣤⣴⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⢉⣠⡴⠚⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⠿⠛⣋⣉⣥⡤⠤⠤⣬⣭⣭⠿⠿⠟⠛⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢾⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⡌⠂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⣸⠀⠈⠙⠃⠀⠐⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⣍⡆⠀⠀⠀⠠⠀⠀⠈⠀⡏⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠉⢸⠆⠀⠀⠀⢁⡦⠀⠀⢇⡏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡤⢿⡷⠐⣏⣠⣖⡒⠉⡁⠀⣠⠊⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⢠⡾⢋⠅⡼⢟⣡⣄⠄⣰⠃⡀⠀⡀⠉⠳⣄⠐⠁⠀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣮⡰⣇⠈⢧⠸⢊⣴⢋⡜⠁⣼⠃⢸⠀⠀⠈⠛⢆⠈⠢⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡲⣆⠹⣿⠇⢸⢠⢺⡎⢸⢀⢸⡎⢀⡟⠀⠈⠉⠒⠄⠀⠀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡉⠟⣸⠇⠠⠟⠈⢯⠃⣟⢸⠸⡆⠸⠁⠀⠀⢀⠀⠀⠀⠴⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠓⣀⠀⠈⣦⠀⠀⠀⠈⢧⠙⠮⠧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣤⣦⠀⠶⠆⠛⠛⠋⠀⣤⣤⡤⠀⠙⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢀⣄⠤⠠⠖⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂⠀⠒⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⢄⣀⣀⠉⠁⠀⠀⠀⠀⠀⣈⣉⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⠦⣈⠛⢦⠀⠀⢀⠴⣶⠶⢤⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠁⡎⣸⣿⣿⣆⠙⣿⣞⡾⣥⣿⣍⣉⣙⣒⠲⢤⣤⣄⢂⡤⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⡆⢫⠙⣿⣟⠛⢋⢿⣝⢿⣿⣶⣌⠙⢷⣶⣖⠚⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⣹⢿⣿⣷⠘⢧⡀⠛⠃⣸⡼⣯⡼⡟⢻⣿⣿⣦⣮⡻⣿⣃⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣱⣾⣃⣏⣻⣿⣇⢀⣹⣧⠀⢡⠄⡏⠃⣿⠇⠟⢉⣿⣿⣿⣮⡻⣤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠴⠶⠶⠦⢬⡻⠿⢿⣶⡶⠚⡈⠀⣼⣿⡿⠄⠀⠉⢹⣿⣷⣽⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣤⣒⠢⡀⢀⡀⢄⠒⠤⠄⡴⢺⣿⠟⠋⠀⢀⣴⢻⡿⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣌⡑⠛⠦⠕⠒⠊⠀⠀⠈⠓⠂⠒⢫⡍⣨⢁⣾⢻⣿⣯⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣷⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢈⡏⣼⢿⣿⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡞⢆⠰⣤⣄⣀⠀⠀⠀⠀⠀⣉⣾⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢧⡀⠳⣜⠿⣮⠙⢶⣄⠀⠀⢈⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠋⣻⣷⣦⡈⣆⣼⣿⣷⣬⡛⠿⣄⢼⣿⣷⡄⠈⣺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠿⢿⣿⣿⢿⣿⣿⣿⣾⡿⡟⠿⣹⣷⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⡀⢀⡀⠀⠀⠛⣦⣹⣿⣷⠻⣿⣿⡈⠁⠉⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢶⣷⠀⠀⣿⣿⣿⣿⣯⠈⢉⣟⣿⡶⠒⢹⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⢀⣮⣿⠀⣷⣿⣿⣿⢿⡟⢰⠿⣫⣟⣡⣤⣼⣶⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⠏⣼⣿⢇⣿⡟⣼⡠⠴⣺⢿⣻⣿⣿⣿⠻⣿⣟⠀⠙⠆
⠀⠀⠀⠐⢶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⡶⠶⠀⠀⠀⣠⣴⣶⣿⣿⡿⠋⣰⡿⢋⣾⠟⡼⠉⠀⢘⣵⢿⣿⣿⠿⠏⠙⠈⠛⣿⡄
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⢂⣼⡿⣛⣽⣾⣿⠁⡈⠉⣴⡿⠃⠈⠀⠀⣰⠟⣵⠿⠟⠉⠀⠀⠀⠀⠘⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠫⣾⣿⠿⠋⠛⠛⠁⠸⠋⠀⠀⠀⠀⠈⠁⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣛⡛⠛⠓⠒⣀⣈⣭⡷⠤⠤⠤⠶⠶⠤⠤⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⣉⡭⠶⠛⠩⡭⠀⠰⠖⠀⣤⠀⠛⢁⠀⠀⡀⠀⠀⢀⠈⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠚⣁⠀⠘⠀⢤⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣞⠷⢽⠆⠀⠀⣀⣈⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢒⡀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⣁⣤⡶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠴⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠈⢧⠀⠀⠀⠀⠛⠛⠛⠊⣩⠇
⠀⠀⠀⠀⠀⢀⡴⢻⡏⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠤⠒⠂⠀⠈⠁⠀⠀⠈⠃⠀⠀⠀⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⣾⠄⣸⠀⢀⡴⡒⠂⠀⠀⣀⡤⠴⠖⠒⠛⠳⣉⣄⣀⡴⠶⠶⠶⠶⠖⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠹⣮⣿⠀⠸⣥⣢⠔⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢰⠋⡷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢠⠇⠀⣌⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣀⡠⠔⠃⠀⢦⣈⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢄⠀⢦⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⣆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⠛⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠙⠁⢸⡇⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⣿⠀⠸⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡀⠀⠀⠀⠀⣿⠀⠀⠠⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢴⡿⢴⠀⠀⠀⣿⡇⠀⠀⠀⠀⠣⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⡀⠀⠀⠚⠁⠀⠀⠐⠀⠀⠑⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣴⣠⣶⢆⣠⠅⢀⣴⡀⠀⢂⣀⣠⣽⣾⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡿⢿⣯⣿⢾⣛⣭⣶⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠠⠀⠉⠀⢀⣥⣶⣿⣩⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠐⣶⡆⣤⠶⠚⢉⣩⣻⢟⣿⣿⡟⢿⣿⣿⠿⠛⠋⠉⠉⣹⢿⣿⣫⣔⡲⢶⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⣷⣒⣩⣤⣾⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⢠⠀⡴⠻⣺⣿⣿⣟⢏
⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣶⣶⣿⣿⡿⠿⠟⠛⠉⠀⢀⠀⠀⠀⣀⣥⠄⠀⠈⣀⢰⣟⠀⠀⠈⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠤⣄⣶⣤⣀⣀⠀⠀⠀⠀⢀⣠⣦⢾⣿⣿⣿⣿⣟⡛⠁⠀⣰⣿⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣿⡿⠉⣠⣬⣿⣿⠟⡡⠋⢡⠾⠿⠛⠻⠷⠢⠀⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣝⠿⡷⢽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣶⣶⠒⠒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣠⣿⣶⣷⣾⣿⣶⣿⣶⣿⣾⣷⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⠀⣤⣤⠄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠮⡥⠉⢡⡽⠷⡜⠁⣤⢾⣷⡈⠟⡃⣸⡷⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⢈⣧⣶⢶⡰⣟⣽⢳⣧⢻⣿⢯⣶⣷⣾⡽⣿⡿⠈⠽⠛⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⣏⡬⢿⣿⠛⠓⠈⠀⠀⢀⣀⣠⣤⣀⣉⡉⠁⠻⣿⣃⣴⣿⣼⠏⢤⡀
⠀⠀⠀⠀⠀⠀⣼⣃⡏⢰⣿⣶⠊⠁⠀⢀⣤⢞⡟⢋⣹⠯⣌⣿⣷⣿⣂⣦⠀⠐⡍⢻⣟⠁⢃⠷⡀
⠀⠀⠀⠀⠀⢸⡩⢮⡕⠆⣻⠁⠀⣠⡾⡹⢿⣭⡽⣶⣿⣿⡶⠟⡛⠈⠀⠁⠀⠀⠐⠀⠘⣿⣯⠚⡧
⠀⠀⠀⠀⠀⢺⣶⢄⡟⢈⣿⠀⠀⣿⣟⣷⢭⣍⡁⠻⣿⣿⣷⣋⢽⣀⡀⠀⢀⡔⠀⠈⠀⢸⡷⢾⡶
⠀⠀⠀⠀⠀⢨⡹⠾⣷⡈⠿⣦⡴⣿⡋⠀⠠⣄⠴⠷⣌⠳⣤⡘⢦⣈⠙⠓⠋⠀⠀⠃⠀⠐⡷⢶⡁
⠀⠀⠀⠀⠀⠀⢻⡄⠾⣛⠛⣫⣽⡏⠂⢀⣲⡃⠀⠀⠀⠙⠚⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⣼⡯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠨⢙⠘⠛⠛⠊⠁⠀⣤⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣸⡷⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠆⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡞⣩⣹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣻⢯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⠆⠳⣿⣵⡸⠏⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣤⣴⣴⢶⣾⣾⠉⢿⣿⣣⣾⣾⣿⣿⣿⣿⣿⡶⢦⢤⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢠⡤⣦⢦⣼⠿⢛⡻⠟⣷⣀⣪⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣟⡷⢼⠯⠇⢀⣦⣄
⠀⠀⠈⢀⣤⣶⠿⠿⠳⣷⣿⠝⠓⠉⠉⢁⡴⢻⡟⣿⣿⡿⠟⠋⠉⡁⠀⠠⠠⠠⠠⠡⠉⠛⠷⣤⣶⢿⣮⣛⣉⣡⡄
⠀⢀⠶⣿⠯⣍⢿⣿⠟⠉⠀⠀⠀⠀⢠⣯⣷⣮⣛⡿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⡃⠛⠋⢩⣿⡆
⠀⣼⠲⠞⠧⠙⣼⡏⠀⠀⠀⠀⠀⠀⣾⡌⢻⣿⣿⣇⡀⠀⠀⢀⣀⣠⣤⡔⢲⣖⣶⣶⣶⡄⣀⠀⠀⠈⡿⣿⣿⡷⣿⢛
⠀⣿⠛⡿⢷⡞⢿⠃⠀⠀⠀⠀⠀⠀⢺⣵⡆⢥⡄⣿⣷⠶⢋⡽⠛⢛⣯⠒⣩⡭⣬⠉⣷⠀⣾⣗⣄⠀⣷⣮⣁⠸⢿⡞
⠀⢹⣺⢳⣿⢏⡐⣜⡦⠀⠀⠀⠀⠀⠈⢷⣶⣗⣲⢿⡶⢃⣴⣲⣮⣅⣀⡼⠿⠿⠯⢿⣯⠈⢀⣡⣥⣿⣭⣻⣭⣧⡍⣠
⠀⠈⢿⣿⣥⢾⣿⢩⢮⢀⣷⣄⡀⠀⠀⠀⠉⠻⠿⢮⢡⣼⠽⠿⠛⠉⠁⠀⠀⠀⠀⢀⣨⣿⢮⡽⢿⣿⣆⣫⣭⠟⢠⠇
⠀⠀⠀⠻⣿⢛⣏⡘⢫⡠⣽⡶⢶⣝⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣯⠿⣉⡧⠔⠘⣇⢻⡟⠋⣰⠃
⠀⠀⠀⠀⠈⠙⡯⣿⣌⣱⣮⠿⣑⡛⢈⡳⢠⣾⢻⣿⠱⡴⢇⣾⣉⣻⣻⣿⣿⣿⣿⣿⣷⣯⣼⢀⡜⣿⣶⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠍⡙⠓⠾⢿⣿⣌⣡⣿⡴⢟⣙⡃⢀⣾⣟⣻⣿⣓⡿⣿⣿⣿⣿⠿⠞⢹⠌⢻⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⢀⡉⠉⠁⠙⠛⠻⠿⠛⠿⠟⠟⠇⠃⠛⠉⢉⡀⠄⠒⡧⢀⣼⡃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⢜⡑⢿⢐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠃⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⢦⣤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢀⠶⣆⣰⣯⡿⠟⢻⣽⣿⣦⣅⠀⢀⡠⢲⣭⣵⣟⣻⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢽⣿⢿⣁⣀⣀⣀⣀⣩⣥⡿⣸⡆⢣⣵⣟⣻⣾⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣻⣖⣎⣝⢣⠮⣝⣿⣿⡆⡾⣷⡸⣃⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⣉⡛⣩⣟⡿⠛⠉⣿⣿⢚⡷⠟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⣀⣼⣫⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣄⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣻⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣻⣾⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣻⣾⣿⣿⡿⠿⠿⢃⣀⣀⣀⣀⣤⠴⠶⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⢟⣩⣴⣶⣶⣾⣿⡿⣻⣿⡟⠡⡜⢿⣼⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣤⣀⠸⣿⣿⠟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⢷⢹⢸⡿⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣮⡉⣸⣿⡿⣿⣿⣿⣿⣿⣿⡟⠛⠀⡟⠀⡟⡾⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣇⢹⣿⡇⣿⣿⣧⣈⣉⡙⢿⣷⣶⣤⣀⡀⠀⢙⣦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣛⣛⣻⣽⡔⢮⣅⣈⣛⣛⣫⣥⣾⣷⣮⣝⠻⢿⣿⡿⠿⠛⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠿⡿⣦⣝⡛⠿⠟⠛⢋⣛⣿⣿⡿⠿⠿⠟⠁⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠈⠉⠉⢉⣴⠟⠉⠉⠀⠙⠃⠀⠀⠀⠐⠳⣶⣶⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣼⡿⠿⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣿⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⣤⣤⢀⣤⣤⣤⣤⢠⣤⡄⣤⣤⡤⣤⣤⢠⣤⣄⢠⣤⡄⢀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡌⠁⢸⣿⠘⢿⣷⣬⣉⠀⣿⣦⣼⣿⠀⣿⡇⢸⣿⣿⣦⣿⠀⣿⡏⢈⣛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⣾⣿⠼⡿⢶⣿⠿⢰⣿⡆⢼⣿⠤⣿⣧⢼⣿⠌⠿⣿⠀⠻⣿⡶⢿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠠⣠⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀⠀⢀⡀⠀⠘⣧⡄⢠⢰⠒⠒⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢣⠀⣧⠀⠈⠁⠀⠀⠸⡿⠜⠚⠓⢤⡄⠀⠀⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⡀⠘⢧⡀⠠⠀⠀⡇⠀⠀⠀⠀⠀⢀⡀⣸⡿⠆
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡠⠄⢀⣀⠈⠙⠓⠘⠂⠛⠐⠚⠛⠛⠋⠈⠁⣀⠠⣀⠀⠀⠀⠈⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⣸⠤⠀⡰⢧⠀⣏⡁⣰⠋⠳⣤⡏⠉⢣⢹⠀⢘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠓⠂⠰⠉⠹⠀⠇⠀⠹⠤⠴⠃⠳⠤⠊⠈⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠛⠛⠛⠒⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡆
⠀⠀⠀⠀⠀⠀⠀⣦⣧⡀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⡞⠛⠛⠛⠒⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣻⣷⣄⣀⣠⣴⡿⠋⠉⣿⣿⣿⡇⠀⠀⢂⣤⣴⣤⣰⣬⣙⠳⣦⣄
⠀⠀⠀⠀⢀⡴⣾⠿⢿⣛⣿⣿⣿⡏⠀⡆⠀⠻⣼⣿⡇⠀⠀⠀⠙⠻⣿⣿⢿⣿⡷⣮⡻⣷⣄
⠀⠀⠀⠀⠋⠞⢁⣤⠾⠿⠻⣻⣿⡇⠀⠑⣧⠀⡉⠻⣷⣀⠀⠀⠀⠀⠘⢿⣧⠹⣷⡈⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣾⣁⣀⡴⠞⣹⣯⢣⢰⠀⠙⢷⣴⣰⠀⡙⠳⣄⢢⡰⡄⠈⢿⣇⠘⣧⠀⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢻⣿⠋⠀⠀⣿⡹⣆⠳⢷⣄⠀⠙⠻⣷⡘⣷⡈⢻⣷⢹⡄⠘⣿⡄⠈⠀⠀⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⡍⠀⠀⠀⠹⠙⠮⡷⣤⣉⣗⡦⣄⣀⠙⠺⣿⣦⢻⡇⣿⠀⢻⣇⢰⡄⠸⡆⢻⣿⣿⣧
⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⡿⠮⣕⠦⡀⠙⣿⣿⣿⡇⢸⣿⢸⣷⠀⣿⢸⡿⣿⡜⡆
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⠻⣆⠈⠱⣝⣄⠘⣿⣿⡇⢸⣿⢸⣿⡆⣿⣿⡇⣿⣷⡐
⠀⠀⠀⠀⠀⠀⠈⣿⠀⡄⠀⠀⠀⠄⠀⠀⠈⣛⡍⠀⢹⡄⠀⠸⣿⡀⣿⣿⡇⢸⣿⣾⣿⣇⡿⣿⣇⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣯⠀⢸⠀⠀⢠⠀⠀⠀⠀⠛⠀⠀⢸⡇⠀⠀⣿⠇⣿⣿⠀⣼⢏⡿⢻⣿⠃⣿⢹⣿⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⣼⣿⠇⢠⡟⠘⠃⢸⠏⡀⢁⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⡇⠀⢸⠀⠀⢠⠀⢀⢀⣴⡿⢀⠀⢸⣿⣿⡟⢠⣿⠁⠀⠀⠈⣸⢃⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⠀⢸⠀⠀⡎⠀⣼⠏⣸⠃⣾⠀⢈⣾⡿⢠⣿⠃⡔⢀⡗⣰⣿⡞⣽⣿⣿⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⡄⠀⣇⢰⠃⢠⡟⢠⡇⠀⣾⡿⢡⣿⢃⡼⢁⣾⣷⡿⢋⣾⢻⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠇⠀⠸⠃⠀⢹⡇⠀⣼⡇⢸⡇⣸⣿⢡⣿⢃⡾⣡⣿⠿⢋⣴⠟⣱⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⣀⠸⡇⠀⣿⠀⠀⣷⣿⣧⣿⣷⣿⡿⠟⢁⡴⠋⢁⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⡄⠀⠀⡼⠛⢣⡇⠀⣿⡀⢰⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡁⠀⠀⢈⣀⡼⠁⠀⢸⡇⢘⣿⣿⠁⢀⡀⠀⢀⡠⢊⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠒⠚⠋⠀⠀⠀⢰⣿⡨⣿⣧⣴⣫⢆⡴⢋⣴⢟⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠘⣷⣿⣿⣿⡷⢋⣴⠟⣡⠟⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠞⢿⣿⣿⡴⠛⣡⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⣾⣴⣄⡿⣿⣧⡾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣿⣿⣿⢠⣿⡟⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠁⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠛⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⢺⠒⠒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⢻⠀⣀⡼⠀⠀⠀⠠⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢸⡀⣿⣤⣬⣯⣦⣥⣧⣄⣢⣽⣶⡶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣟⣠⣼⣿⣿⡏⣠⣤⣶⠿⣟⣿⣿⣿⣿⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣟⣿⣟⣿⠟⠉⠀⢰⣿⣿⣿⣿⡏⢻⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣤⡀⠀⠀⢉⢿⣿⣿⣿⣹⠇⢀⣤⡾⠟⠉⣼⣿⣿⢀⣼⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣹⣿⣿⣿⡿⠀⢀⣰⡿⢛⠁⠈⣩⡶⠋⠁⣀⣸⣿⣿⣿⠇⣾⣿⡯⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣼⣿⣷⡾⢛⡅⢸⣿⣿⠛⠁⣴⣾⣿⣿⣿⠟⠛⣿⣷⣿⣿⠟
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠍⣽⣿⣎⣾⣿⣿⣾⣿⣭⣅⣠⣼⡟⠛⠁⣀⣅⣴⣿⣿⣿⣿⣿⣦⣀⠀⠖
⠀⠀⠀⠈⢿⣦⣵⣦⣀⣤⣀⣶⣶⣦⣌⠙⠿⣿⣿⣤⣿⠏⠙⠻⢿⠿⠟⠛⠉⠀⠀⠀⠈⠉⠃⠀⠙⠛⠋⠻⣿⣯⣅⣀
⠀⠀⠀⠀⠀⠉⠋⠉⠿⣿⠿⣿⢿⣿⣿⣿⡿⠿⢿⣿⡿⠟⠁⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣋⣉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠱⠔⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠉⠁⢹⡿⠛⡟⢻⣛⡟⣿⡿⡻⣿⢻⡟⣿⣿⢛⢿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢒⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠉⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡿⢤⡀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢼⣆⣀⠀⣀⣤⣶⣿⣟⣛⣿⣿⣿⣿⣿⡟⢿⣷⣶⡅⠀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⣼⣷⡘⢷⣽⡿⠿⠿⣛⡋⡉⠛⠛⠻⣾⣿⠏⣴⣿⣶⣄⠀⠀⠓⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣞⠉⠀⣴⣿⣿⣿⡿⠈⠁⠀⠀⠀⠀⠀⠈⣉⣉⣉⣡⣄⣜⣷⣿⣿⣿⣷⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡁⢈⣧⣾⣷⣾⣿⡯⣄⡀⠀⠀⣠⠴⡒⡋⠉⠉⠀⠈⠠⠀⠀⣉⣻⣿⠿⢿⡿⢿⡿⣿⡿⢟⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢍⣉⣿⣿⡿⠋⠀⠀⠉⣽⠟⠉⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠀⠀⠞⠐⣛⠛⠈⢀⡼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⢠⠀⣱⠟⠇⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣅⠀⠠⣄⡈⣿⣦⣾⣥⣇⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢫⠀⣒⠿⡈⠠⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⣛⠶⣄⠈⠁⠉⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢦⣄⡠⠀⠀⠀⣤⠀⠀⣰⡦⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡷⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⢖⣛⣿⡦⠖⢻⣿⣿⣿⠦⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡿⠉⣹⠃⠀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣄⠀⠀⠀⠀⠀⡀⢰⢖⡓⢲⠶⠚⠁⠀⢰⠇⣰⠏⢇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⣿⣿⣿⣷⣀⠀⡀⡠⠚⠙⡟⡇⡿⠀⠀⠀⠀⢸⢞⣅⣴⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢻⣿⣿⣿⣷⣾⣄⡜⡇⢀⣿⠇⠠⣀⠀⣀⡄⠀⢻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⡱⢄⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣆⠀⢹⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡐⠀⡄⡌⠙⢻⣶⣯⣿⠿⢿⣿⡿⠿⠿⣿⣿⣿⠿⠛⡿⠲⠁⠀⡠⡀⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢳⠮⠥⡘⠃⠀⠀⠀⠙⢿⣷⣆⠈⠚⢿⣿⣿⣿⡿⠁⠀⡰⠃⠀⠀⠀⠈⠛⣀⠭⢶⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⣤⠀⠀⠀⠀⠙⢻⡄⠀⠈⠹⣿⣿⠀⠐⡏⠀⠀⠀⠀⢀⡔⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣥⠀⠀⠀⠀⠀⠀⠿⠒⠦⣤⣿⢟⠁⠀⣧⠀⠀⠀⠀⢤⡁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠻⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⢶⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣀⣀⣼⠃⠀⣄⠠⠤⢶⢀⡉⠀⠀⢀⣦⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⣿⠀⢰⣿⠀⠀⠀⠋⠈⠁⠈⡤⠟⢻⣿⡇⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡀⡙⣄⢼⣏⡀⠀⠀⠠⠑⢲⡁⠠⠾⠾⢿⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠸⡇⠀⠘⣿⣦⣤⡀⠀⠨⠦⠀⠄⠀⠀⠉⠙⠁⠉⡏⣶⠂⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⣭⣇⠀⢷⡀⠀⠘⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣤⣄⠀⠀⠀⢀⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⢸⣿⣤⣀⠹⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢀⣠⡾⠟⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠄⠀⠈⠛⠷⣝⣟⠻⣷⣟⣷⣤⡀⠀⠂⠁⠀⠀⢀⣠⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠈⠛⠿⣿⣿⣏⣉⠀⠀⠀⢀⣠⣴⣿⣿⠋⠀⢀⣤⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠑⢦⣀⡀⠈⠛⢿⣿⣷⣤⣀⣀⣀⣴⡿⢁⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠁⣽⣦⣤⣤⠛⣩⣿⡿⠿⠿⠛⠂⠉⡉⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠙⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠚⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⢟⣽⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢸⣧⣤⣤⣤⣤⣤⡖⠁⠀⣼⢏⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⡟⠸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢻⣿⣿⣿⠞⠁⢿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠈⢋⣿⣁⠀⠀⠈⠻⣬⣛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⣠⡾⢡⡾⠋⠁⠀⠀⠀⠀⠈⠙⠻⠷⣶⣦⣭⣭⣭⣭⣻⣿⣿⣿⣿⡟⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣣⣾⡿⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⣿⠛⠛⠻⠿⢿⣶⣤⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠟⠁⣠⠂⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣇⠀⠀⠀⠀⠀⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠏⠀⠀⢸⡏⠈⢳⡄⠀⠀⠀⢀⣀⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⠀⢸⠁⠀⠀⠹⣦⣈⣩⣭⡙⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⢰⡆⠈⠀⠀⠀⠀⢸⡀⣾⣿⣷⡀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠘⣿⣈⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠀⠀⠀⠀⢷⡀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠈⣿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢷⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠸⡇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢹⡀
⠀⠀⠀⢠⣿⡟⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢷
⠀⠀⠀⣸⣿⠁⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠸⡇
⠀⠀⠀⣿⡏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣶⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠐⣄⣻⡀
⠀⠀⢰⣿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢹⣿⣿⣄
⠀⠀⢸⡏⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⠃⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣿⣿⣿⡇
⠀⠀⣾⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢸⣿⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⣴⣶⣷⠲⢬⢻⣿⣿⣿⣿
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⣿⣿⠀⠀⠀⠀⠀⠀⠳⡈⢶⣄⠘⢿⣿⣿⣷⡄⠏⣿⣿⣿⡉
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢡⡀⢻⣷⡀⠙⠛⠋⣱⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠹⣿⣷⣶⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠛⠻⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⢸⣿⣿⣿⡄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⢸⣿⣿⣿⣧
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢸⣿⡏⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢸⣿⠁⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⣿⡇
⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢠⡞⠁⠐⠂⠬⢍⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⠭⠄⠂⠈⠙⣦
⠀⠀⠀⢸⡖⠚⢋⣉⣓⡒⠦⢝⡮⠷⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠴⠭⣞⠥⠖⣚⣋⣙⠛⠒⣿
⠀⠀⠀⠈⢷⡉⣉⠀⠨⢭⣍⣻⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣼⣋⣭⠍⠀⣈⡉⣸⠇
⠀⠀⠀⠀⠘⢧⡠⠤⠖⠒⠒⠒⠛⠦⣄⠀⠀⠙⢦⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠞⠒⠒⠒⠒⠦⠄⣰⠏
⠀⠀⠀⠀⠀⠈⢷⣐⠊⠉⢉⣉⣉⡙⠛⠛⠒⠦⣄⡳⣐⣢⡴⣁⡤⠖⠛⠛⠛⣉⣉⣉⠉⠉⢒⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠧⣤⣤⠬⣭⣭⣭⠯⠭⢽⣏⢘⣿⠭⠭⣯⣭⣭⡭⢤⣤⡬⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣁⠤⠄⠐⣶⠤⠒⠊⡝⣿⢾⡿⡑⠒⠤⢴⡒⠀⠤⢄⣉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⣉⠤⠒⢈⡏⠀⠀⣠⢾⢀⡏⡘⡇⠹⢆⡀⠀⠙⣏⠐⠢⢌⠓⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠤⠂⡩⠳⡴⢺⠁⠀⣸⢀⢃⢹⠸⠀⢻⠲⡴⠫⡁⠢⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⢠⠊⠀⡜⠀⠇⠀⢰⠇⠈⠃⠈⣿⠀⠈⡄⠑⡀⠈⢦⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠀⡘⠀⠀⡆⢀⡞⠀⠀⠀⠀⠸⣆⠀⠇⠀⢸⠀⢈⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠶⠚⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡰⠃⡤⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡼⢡⠞⡤⢊⡴⣉⠖⡢⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⢮⡀⣇⢼⠤⢋⠜⢡⢊⢔⡥⣢⢭⡢⡄⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⠻⣿⣿⣿⡫⠻⣟⣟⣿⡿⡾⢾⣷⣦⣔⡃⠀⣠⣦⣢⢀⠌⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⣀⣤⣾⡿⣻⢫⠂⢌⢯⣿⣿⠑⡁⢞⣿⣿⡥⡁⣡⣿⣿⣿⣿⡿⡻⣶⣿⣵⣫⣪⣶⡏⡂⠀⠀⠀⠀⠀⠀⢀⣤⢞⠤
⠀⠀⣀⡔⣺⢋⣠⡀⠀⠉⠑⠪⠦⠲⡻⢿⠙⢈⢜⣿⣿⣽⢉⠱⢻⣿⢿⣷⣜⡜⡹⣿⣿⡯⣹⣻⣦⣤⡤⣄⣤⣤⡤⣴⢻⡷⢓⡀
⣴⠞⢅⠐⠧⣩⠌⢀⠀⠆⠰⣤⡁⡀⡁⡀⠎⢎⢌⢾⣿⣧⢆⢄⠑⣿⣿⡿⡈⢬⢹⣿⣿⢟⠿⠿⣿⡿⠴⢿⣾⢿⢷⢽⢩⣿⠳⠔⠺
⠉⠲⡃⠙⣶⠘⠇⡻⠀⠀⠱⠒⡀⠈⢲⣴⡟⠂⠀⠂⡛⠔⡥⢣⢎⣼⣿⡱⠵⡱⡸⡿⡯⠪⢣⡣⣋⣥⡮⠵⠒⠘⠒⠦⣽⣿⣟⡑⠂
⠰⠤⢽⣟⣏⡫⠘⠱⢖⡋⠀⠀⠼⡐⣿⣍⡉⠀⠀⡰⣝⡬⡺⣔⢵⢟⣃⡌⡬⡪⠨⠈⣎⣶⡵⣾⠾⠻⢦⠔⠀⠀⠀⠀⠈⠳⠦⣳⣔
⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⣂⣄⣨⢚⡥⢂⢙⠣⢑⠡⢰⢡⢊⡊⣗⣬⣮⣬⠧⠿⠒⠋⠉⠋⢉⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠛⣛⣩⠍⣉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣻⣞⢿⠣⡉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠷⡶⣤⣄⣀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⡇
⠀⠄⢤⠶⢤⡀⢠⣷⡤⢴⡆⣸⣇⣠⣤⡄⣐⣀⣠⣀⣠⣀⡀⡏⡀⠀⠀⣠⣰⡞⢁⠀⠀⠀⡄⠀⢀⠀⢸⣖⠀⠀⠀⠀⠀⠀⠀⢀⡁⣀⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠠⠹⣀⣼⣤⣬⣿⣦⣌⣤⣹⣯⣬⣭⠉⠿⣿⠙⣩⣹⣿⡗⢭⣶⣶⣖⣾⠻⣿⣿⣤⡼⠯⠷⣶⢸⠂⠛⠛⠠⠀⠤⠀⠀⡀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇
⠀⠀⠀⣭⡇⠀⠀⠉⢙⠋⠉⠋⠁⠿⡟⠛⠛⣿⠒⠾⢿⣿⠟⠛⠛⢛⣛⣿⣿⣿⣿⣭⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠂⢈⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡇
⠀⠂⠠⠴⠄⠠⠄⠀⠈⡈⠈⠀⠀⠀⠀⠀⠀⣄⡀⡤⠌⠙⡟⠛⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠇⠀⡇
⠀⠀⣼⢻⠂⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⡟⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡤⠀⠐⠇
⠀⠤⠾⢿⣯⣭⡁⠈⠁⠀⠀⠀⠂⠀⠀⠀⣶⡶⣄⢰⣶⠦⠀⠀⠀⢠⣤⣧⣀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⣻⡎⠀⣀⣀⣀⣀⣀⣀⣤⡤⣤⣤⣶⣲
⠀⠀⠀⠀⢼⣿⡟⡉⣠⣀⣄⡀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣴⣶⣦⣀⡀⠀⠀⡈⠉⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠿⠿⢭⣭⣶⣶⠿⠿⠿⠿⠿⠛⡟
⠀⠀⠀⠀⠈⠛⠻⣤⣾⠿⢿⡾⠚⠛⠉⠉⠀⢴⠟⠛⠛⠉⠉⠁⠈⠉⠃⡤⠀⠀⠀⢀⣠⣿⣿⠿⠟⠓⠀⠀⠀⣀⡀⠀⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣶⣶⣶⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣠⣴⣾⠯⠉⠛⠛⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⠫⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⡟⠛⠿⠟⠋⠀⠀⠀⠀⠀⢻⡏⠉⠁⠀⠈⠉⠀⠐⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠉⠛⠩⢤⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣘⡻⢝⠻⡙⣌⠦⠰⠒⠋⠋⡇⠉⠉⢉⣀⣂⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠒⠦⢈⠓⢓⡙⠁⢠⣇⠀⠀⠀⠁⠀⠀⡟⢟⢉⡇⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠼⠷⠭⢍⣢⣄⣀⠴⠋⣢⠔⠃⢹⡄⠀⢀⣄⠀⠀⠙⠒⠛⠁⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠠⣞⣭⣍⣑⡶⠋⢀⣤⣶⠛⠁⠀⠀⢰⡇⢠⠞⠉⠳⣄⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡯⠖⠒⣲⠏⠀⠀⠐⠋⠛⢷⡀⠀⢀⡾⡿⡧⡀⠀⠀⠈⣇⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠁⢠⠟⠒⠒⠶⣤⣄⡀⠀⠿⡤⠋⠈⠀⠁⢳⣠⠖⠀⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠙⣏⠢⡄⢻⠀⠀⣀⡤⢺⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡧⢴⠶⣾⠲⠒⠢⢜⡄⠘⣼⣠⠚⠁⠀⢸⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠱⡀⠀⠀⢡⢀⡟⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⠦⣧⣄⡀⣼⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢀⣾⣿⣦⣑⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⢻⢿⢿⡺⣷⣤⣤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠟⡇⠃⠈⢫⡳⡑⠢⡉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡇⠘⠀⣀⣳⡈⠢⣈⠂⠌⣢⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡜⠀⠀⡸⠀⠈⠑⠤⠤⠌⠭⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻⠜⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⣏⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡴⠶⠶⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⡰⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣤⠒⠋⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠬⠳⡀
⠀⠀⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⣠⡇⠀⠲⣄⠀⠀⣠⡟⣡⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣇⢠⣀⠀⣀⣹⢦⡀⠻⣄⣴⠟⡇⠀⠀⠹⡳⡜⡝⡴⢹⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣿⢀⣙⠻⡦⣹⣾⣌⡦⠾⠴⠴⠣⠶⣦⣄⣸⢠⡷⢁⠟⣿⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢹⡀⠉⢳⣝⣜⢿⠏⠀⠀⠀⠠⠊⠀⠈⢷⡇⢸⡇⡜⢸⣿⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢱⠛⠍⠉⢻⣾⡀⠀⠀⠀⠀⠀⠀⣀⣼⣇⣸⣷⠋⠉⠉⣹⠉⢹⡄⠀⡇⠀⢀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁⠀⠉⠑⠚⠉⠉⠉⠉⠀⠀⠈⠁⠈⠓⠒⠚⠉⠀⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⡀⡀⡀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠔⢡⠔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠎⡐⢁⠞⡠⣊⢔⡁⢒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣰⣘⣃⣃⣊⣐⣡⣊⣊⠕⡰⠲⠕⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⣴⣲⠟⢹⣿⡿⡪⠋⠛⣽⣽⡝⡟⠹⣷⣷⣯⣀⣠⡾⠪⣂⢐⢌⢂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⢀⣤⠔⠛⠿⠱⠑⡀⡢⠙⣿⡇⡄⠜⣾⣿⣟⣄⣜⣿⣿⣻⡟⠊⠻⣿⣷⡶⢵⣯⡇⠀⠀⠀⠀⠀⠀⣠⣶⣫⡂
⢀⣠⠖⣁⠣⠖⠇⠀⠀⠀⠀⠍⠉⠎⢅⢄⠌⢜⣿⣞⢀⠈⠜⣿⣽⡟⠔⠴⣽⣽⣟⣔⢜⣿⣿⢁⣦⣿⣷⣘⡮⣿⣥⠂⢀
⠫⣁⠑⢬⡁⣧⢠⠆⠀⠂⡙⠂⠃⣀⢂⡎⠊⠊⠻⠏⠔⠐⡠⣻⣿⠏⡊⡀⣾⡿⣫⠎⠨⠛⠋⠉⣉⢉⢩⣉⡡⣿⣆⠡
⢀⠈⣢⢤⠳⡈⠜⣠⠤⠈⠀⡠⢠⡾⠛⠇⠀⠀⡐⠄⡸⡈⢀⡿⠋⠈⢁⠁⠙⠈⢄⢠⣢⣥⢾⣉⠀⠀⠀⠀⠙⠯⣍⠒⡠
⠀⠀⠈⠁⠉⠐⠲⠤⢤⡄⠀⠐⠰⡋⠅⠲⡀⠄⡠⢑⠀⠀⠎⡈⣖⣱⣕⡤⠴⠚⠻⠛⠈⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢺⣷⠖⣚⣒⠛⠒⠓⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣲⡽⡟⢌⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠤⠤⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠆⡀⠁⠀⠀⠀⢄⠀⠰⣄⠀⠀⠀⢀⣀⡀⠔⠊⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠁⠀⠃⠀⠁⠀⠀⠀⠐⢄⡑⠂⢈⣂⣠⠍⠊⠁⠀⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠒⠉⠉⠂⢐⡛⣀⠐⢄⠀⢀⡠⠴⣶⡾⠋⠉⠛⠉⠀⠀⠀⣀⠠⠔⢊⠕⠁
⠀⠀⠀⠀⠀⠸⣅⡀⠀⡠⠤⠂⠁⢂⣈⣤⢔⠊⠡⠖⠂⠉⠁⠀⠀⠀⣀⡠⢴⡲⠭⣒⠤⠚⠁
⠀⠀⠀⠴⠊⠁⠀⣀⠄⠁⣠⠖⠉⠉⠑⠒⠉⠀⠀⢀⣀⠠⠄⢒⠈⠍⣂⠬⠐⠊⠁
⠀⠀⠀⠀⠀⠠⠚⠁⢠⠞⠁⠀⠀⠀⠂⠀⠤⠒⢈⠁⠀⣠⠔⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠄⠀⢀⡠⠄⠀⡠⠊⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠊⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣵⡀⠀⠀⠀⠀⣄
⠀⠀⢀⣴⣶⣶⣶⣴⣶⣶⣶⣶⣶⣤⣄⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⠀⠈⠿⠟⠀⠀⠀⠀⠀⠈⠙⢪
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣀⠀⠠⠒⠒⠐⠒⠤⣤⠠⢤
⢀⣿⡿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢤
⠸⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⡍⠻
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠏⠙⠛⠛⡟⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⠀⠀⢀⣌⡸⠀⠀⣰⡇
⠀⠀⠀⣠⣾⣿⣿⡿⠋⠀⠀⠐⢒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣥⡤⡀⠀⠀⣿
⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⣀⠀⡍⠀⠂⢀⡤⢖⠰⠂⠀⠀⠀⠀⢀⡶⠶⠶⠒⡟⢻⠀⠑⢤⠄⢣
⠀⠀⢸⡏⣿⡟⠀⠀⠀⠀⠘⡍⡁⠀⢰⠎⠀⠈⠇⠀⢲⣿⢓⠀⡄⠀⠀⠀⠀⢱⠈⡇⠀⠀⢢⣈⢄
⠀⠀⣿⣇⣿⠀⠀⠀⠀⠀⢰⠱⠈⣐⠈⠄⠀⠀⠸⡀⣿⣿⠀⠄⢸⠀⠀⠀⠀⠘⣤⠇⠀⠀⠀⠏⣸
⠀⠀⣹⠹⣿⡀⠀⠀⠀⠀⠈⡀⠀⠀⣀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠸⠀⠀⠀⠀⢰⠸⠀⠀⠀⠀⠸⢹
⠀⠀⠈⠙⠯⠿⠀⠀⠀⠀⠀⡃⠀⠀⠃⠀⠀⠀⠀⡃⠇⠚⠩⠞⡦⠂⠀⠀⠀⡎⡃⠀⠀⠀⠀⢀⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣒⠼⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠂⠀⠀⡔⢐⠄⠀⠀⠀⠀⢸⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢸⠀⠀⠀⠀⢠⢸⠀⠀⠉⠁⠀⠀⠀⠀⠀⣇⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢠⡤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣀⣀⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠠⡀⠀⠑⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡟⠉⣶⣄⣻⣯⣿⠿⢟⣻⡦⠀⠀⠰⡆⠀⠄⠀⢨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡇⠀⢿⣿⣿⣏⠀⢸⣽⣽⠇⠀⠀⠀⠀⠀⠸⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡰⣿⣿⣦⣀⣬⣙⣛⣷⠿⠋⠁⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠙⣻⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠻⢿⣿⣿⠟⠛⠋⣀⣤⠞⠋⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣀⣤⠤⠴⢶⣯⣵⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⣿⣿⣿⣷⡕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⣿⣿⣿⣿⣿⡟⣻⠷⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡹⣿⣿⣿⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢁⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠖⠙⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⡌⠀⠹⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠐⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣆⣶⣶⣶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣧⠺⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠤⠤⠤⠤⢤⣤⣄⣀⣀⣀⣠⣦⠀⠀⠀⢸⠃⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣄⠉⠉⢹⡆⠀⠀⢀⣠⣤⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣤⣿⣿⣽⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠉⠁⢀⣀⣀⡀⠄⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⣉⣭⣝⣻⡷⠀⠀⢀⠀⠀⠀⠒⠶⠤⠐⠒⠂
⠀⠀⠈⣀⣤⣤⣴⣶⣿⡿⠿⠿⢷⣶⣿⣷⣾⣿⡟⣻⣿⣿⢻⣿⣭⣭⣵⣶⣶⣶⣿⣿⣿⣿⣿⣿⣽⣦⠄⠀⠐⠛⠛⠓⠒
⢀⣀⣴⣶⣦⣤⣤⣍⣿⣷⣿⣿⡿⢿⣽⣿⣿⣛⣚⣛⣿⣧⣴⣶⣿⢿⠿⠿⢿⣿⣿⣟⣻⣿⣿⣿⣷⡶⠶⠶⠆⠈⠉⠉
⠀⠁⢭⣉⣉⣋⣉⠉⠛⠳⢤⣄⣀⣭⡅⠉⠳⣿⣿⣉⡀⣾⣿⣿⡿⠉⠛⠉⠀⠀⠐⠺⣶⣚⣛⣿⣿⡆
⠀⠀⠀⠀⠉⠉⠁⠀⠚⠓⠶⠦⣤⣤⡤⠀⠀⠈⠉⠁⠶⠶⢿⣿⠇⠀⠈⠉⠉⠉⠉⣉⣀⣤⠴⠆⠀⠀⠀⠋⠉⢉⠉⠓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠘⠛⠃⠀⠀⠉⠉⠉⠒⠒⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⡀⠀⠠⠀⠄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣽⠇⠀⠀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⡀⠀⠀⠀⠦⠹⠴⠋
⠀⠀⠀⠀⢷⡶⠃⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠻⡆⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠖⠀⠀⠀⠀⣴⡀⠀⠀⠓⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠉⠉⠤⠄⠈⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠟⡆⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⡰⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠐⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⢄⣶⣿⣿⣤⣰⣿⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣟⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣿⣿⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⢠⡀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⣷⢾⣿⣻⣿⣦⣤⣤⣶⣄⣄⡤
⠀⠀⠀⠀⠀⣀⣼⢿⡀⠈⠛⠛⠙⠻⢿⣿⣿⣿⡶⠒⠂
⠀⠀⠀⠀⢀⠞⠁⠁⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣋⡁
⠀⠀⢀⡴⠁⠀⠀⠰⡀⠀⢠⡀⠀⠀⠀⠀⠀⠙⣿⣿⣟⠁
⠀⠀⢸⣜⠃⢠⠄⠐⠋⠀⢹⣧⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⣿⣅⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠋⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡷⣶⣾⣿⣿⣿⣷⣶⣿⣅⣀⣀⣠⣤⣤⣴⣶⡿⠿⠿⠛⠛⠉⠉⠙⠻⣿⣷⣤⡀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠉⠛⠿⠿⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠉⠉⠉⠉⠁⠈⠉⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⢄⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠷⠀⢤⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⣅⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏
⣿⣷⣄⣑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣿⣿⣿⣿⣿⡿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⢶⠶⢶⠶⣶⣶⡶⢶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⣀⡤⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠠⠖⠁⠀⠀⠈⠀⡀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⡀⠈⠀⠀⠀⠑⠦⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⡥⠤⠠⠤⢠⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⠠⠤⠠⠤⡅
⠀⡰⠀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠰⠃⠠⣀⡠⠐⠀⠀⠂⠀⠀⠀⠀⠒⠀⠐⠀⣀⡀⠀⠳⠀⠀⣠⠄
⠀⢀⠀⠀⢀⠀⠀⠀⠀⢰⠂⠁⠀⠠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠀⠀⠀⢲⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠩⡉⠀⠀⠀⠀⠀⠈⢀⡀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⣀⠈⠀⠀⠀⠀⠈⠈⡩⠁
⠀⠀⠀⠈⠀⠀⠠⠀⠄⠀⠀⠈⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⠀⠀⠀⠄⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠢⣀⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⣀⡠⠒
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⣀⠢⡀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡠⢂⡈⠉⠉⠀⠀⢀⡀
⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣀⠀⠐⠂⠳⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡴⠓⠒⠀⢀⡀⠀⠀⠀⠀⣠
⠰⠂⠈⠋⠀⠀⠀⠈⠉⠉⠀⠀⢀⡉⠆⠛⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠃⠎⢁⠀⠀⠈⠉⠉⠀⠀⠀⠈⠋⠀⠲
⠀⢀⠀⠀⠀⠈⠣⠐⠀⠀⠀⢀⠁⠀⠤⠊⠁⡈⠒⠀⠤⠤⠤⠔⠐⠊⡄⠈⠢⠄⠀⠁⠀⠀⠀⠐⠠⠋⠀⠀⠀⢀
⠓⠉⠀⠀⡀⠔⠃⠀⠀⠀⠤⢀⣀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠀⠀⠀⠀⠀⠓⠄⡀⠀⠈⠑
⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⡀⠀⢀⠸⠀⠀⠀⠀⠀⣄⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢒⡄⠀⠀⠀⠀⠸⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢼⡇⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⢸⡇⣷⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⣿⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣀⣠⡤⠶⠚⠛⠉⠉⠉⠀⠀⠀⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣥⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠖⠚⠛⠋⠉⠉⠉⠉⢹⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⡿⣆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣀⣤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡇⣿⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⠋⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡾⠁⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡇⣿⠀⠀⢀⣴⠏⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⢇⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⣿⠀⣰⠟⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣷⠞⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⡇⣿⠓⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⣀⣬⣿⠦⠴⠶⠤⠤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠦⣤⣤⣀⣀⣀⣤⣤⣤⣤⠴⠶⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠲⠶⠶⠶⠶⠿⠓⠒⠛⠛⢳⡆
⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠻⠶⠴⠶⠶⠒⠒⠛⠓⠒⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⠁

⠀⠀⠀⠀⠀⠛⠛⠓⠶⣦⡄⢀⡤⠀⠀⠀⠀⠀⢠⣂⣴⣶⣶⣲⣀⢠⡆⠀⠀⠀⠀⠀⢀⣀⣤⣔⣆⣄⠀⣠⡀
⠀⠀⢀⠀⠀⠀⣀⣀⠰⠄⡀⢸⡇⠀⠀⠀⠀⣶⣿⣿⢿⣛⡽⠏⠁⠻⠀⠀⠀⠀⣠⣠⠀⢠⣤⠤⠒⣚⠄⢵⠇
⠀⠀⠈⠀⠀⢀⣌⠉⠐⠛⠁⠀⠙⠀⠀⠀⠂⠈⠉⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡁⠈⠉⠁
⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠈⠐⠂⠈⠻⠀⣠⠀⠀⠀⠀⠀⠀⠔⠎⣿⣿⣿⣿⡿⠷⣺⡇⠀⠀⠀⠀⠀⢀⠈⠙⣿⣿⣿⡆⣀⡄
⠀⠀⠀⠀⠀⠐⠓⠉⣠⠈⣿⠀⠀⠀⠀⠀⠘⠒⠈⠀⢽⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠤⠌⠀⠀⢾⣿⣿⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠇
⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⡄
⠀⠀⠀⠀⢾⡛⡟⣿⢹⠀⢀⡄⠀⠀⠀⠀⠀⣠⡞⢩⣍⢹⢇⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠠⣄⡀
⠀⠀⠀⢳⠸⡧⣿⣸⢸⠆⢽⡇⠀⠀⠀⠀⠀⣽⣧⣼⣼⣾⣼⠉⠳⠀⠀⠀⠀⠀⠀⠈⠃⠰⠻⠉⡿⣛⠓⢲⡿
⠀⠀⠉⠚⠪⣿⡃⠚⠻⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠻⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠋⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣶⣶⣦⣤⣄⠀⣠⠄⠀⠀⠀⠀⠀⢀⡀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣶⣴⣶⡄⢀⡀
⠀⠀⠀⢀⠨⢿⣿⡟⣛⠉⠉⠿⠀⠀⠀⠀⠀⠀⠘⢀⠀⠐⠈⠑⠀⣤⡀⠀⠀⠀⠀⡴⠿⠈⠏⣯⣿⣿⠷⢿⡗
⠀⠀⠢⠀⠀⣬⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠠⣀⡟⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡑⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠈⠀⠀⣠⡧⢱⠍⠲⣾⠄⠀⠀⠀⠀⠀⠀⠀⡭⢽⣿⣿⠗⠲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⠙⠻⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠠⠠⠤⠀⢀
⠀⢀⣰⣶⣟⠲⢶⢖⠳⣘⣫⣶⠿⠿⠞⠛⢓⢹⠄⠂⡈⣡⢰⣺⣦⣷⣠⡄⠒⠚⠚⠇⡴⡂⢄⣃⣀⡀⢀
⠀⣠⡼⢖⣀⣤⣯⣥⠴⠂⣿⠃⠀⣀⠀⣄⣾⢻⠀⣀⠝⠨⠧⢸⡽⣏⠉⠳⢄⣄⣤⢤⡀⣤⣴⣛⣃⢼⣌⡙⣤
⢀⠏⡀⠈⠁⢈⢇⣘⣆⢀⡼⡆⠒⠉⢣⣀⢯⡘⡆⠀⠠⠤⠀⠈⢃⡷⠲⠆⠀⠀⠐⠊⠏⠁⠀⠉⠹⡀⠀⢈⣛⠁
⠊⢿⡾⠠⣿⣾⣹⡽⣝⢧⣀⣈⣀⠀⣉⡠⠀⠘⣇⠀⠐⢠⣯⠴⣟⢲⣄⢀⣐⣆⡀⢀⠶⠂⠈⠛⠂⠀⠃⣀⠉
⠀⢀⠇⠀⠀⡞⡩⠙⣶⡾⠟⠛⠻⡛⠻⣟⡣⢾⣿⣭⣶⣿⠋⠛⠉⣠⣟⠦⢌⠿⣻⣿⣾⠛⡯⡀⠠⠂⠀⠀⢀
⠀⢨⠔⠊⡰⣠⠇⢡⣿⠷⣶⡶⠀⠹⢷⡟⠙⠷⢼⣮⢻⣻⣧⠄⠞⢻⣯⣿⣃⣼⠇⠀⠈⠛⣧⣼⡁⠀⠀⠀⢨⢰⡀
⠰⠖⠶⠂⠤⠀⠀⠈⣩⡗⠈⠀⢴⡄⠀⡓⢊⠒⠶⠶⣾⡽⡈⠋⣄⣀⡏⣩⣷⡋⠀⢀⣀⣀⢸⣿⣖⡆⢸⣄⠱⣣⠂
⠀⠰⠀⠀⢘⠄⠀⠀⠀⠱⢤⣀⠤⡄⣰⡀⠙⢂⠀⠀⢪⡄⠇⡠⡤⠀⣤⢸⡂⠛⠃⢀⠁⢉⠾⣟⣿⠇⠀⠈⣄⠉
⠀⠃⣈⠉⠩⠓⠒⢲⡞⢁⠘⢿⣟⣻⠷⠖⢶⣿⣦⣐⣿⣧⣤⣀⢀⡼⠗⠛⠗⠢⡄⠛⠋⠜⠃⣁⣀⡀⠀⣀⡄⢀
⠀⢶⠉⡚⡶⠞⠶⡖⡽⠎⠃⠼⢸⣵⠴⠚⠛⠛⠿⢏⡙⠛⡛⠛⣻⠆⠀⠀⠀⠀⠙⢶⣵⣧⣼⡿⣭⣙⢷⣄⣀⠆⣃
⠀⠈⢻⠉⠀⠀⠃⠞⠀⠀⠀⠀⢈⢻⣾⣷⠄⠺⣿⡆⣿⡇⠉⣉⣿⠆⠂⠰⠶⠈⠐⢀⣏⣽⠻⠋⣹⡛⠿⠻⡆⠀⣹
⠀⠀⢩⠁⠀⠀⠀⠀⣀⡀⡵⠖⣶⠞⣇⠀⢀⣀⠀⢀⡋⠀⠘⠛⠣⡄⠷⠀⠀⠀⠀⠊⠉⠈⡆⣀⣭⣩⠀⠜⠀⠙⠃
⠀⠀⢨⡇⠤⢖⠨⠱⢠⠀⠀⠀⠀⠀⣨⣁⠠⣠⠀⡈⠀⠀⠀⠀⣰⣬⠀⠂⢀⠀⠀⠀⠀⠀⣙⢏⣁⠤⠀⠰
⠀⠀⣚⠦⣿⡴⣴⠂⠟⠀⠸⠁⠀⠀⣸⣭⠓⠻⢺⣾⡃⢤⢠⢤⣼⣿⣲⢶⠃⠘⠿⠗⢀⣶⠳⠾⠲⣄⠀⠒⠿⢻
⠀⠀⠀⡞⢿⣠⣤⠠⠒⠓⡆⠈⠀⣜⢟⣁⡈⠚⠓⢋⣷⠻⠇⡆⠨⡿⢿⠀⠀⠀⠀⠀⠀⣹⠈⠁⠰⠟⠛⡖⣺⣿⢧
⠀⠀⠀⢀⠈⠎⠘⢲⣥⠀⢈⢶⣺⢶⠊⠛⡛⡁⠸⡈⢻⠀⠀⠓⠼⠀⢸⡏⣡⠶⠄⠤⣸⠛⠟⢀⡄⠀⠀⢰⠿⠛⠈
⠀⠀⠀⠉⢠⡔⠓⠂⣶⠒⡡⡶⠀⠛⠀⠀⢀⡁⠀⢉⢠⠀⠀⠚⠻⢂⡞⠼⢿⡷⢶⣟⡹⢠⡀⠀⠀⠊⡠⠀⠆⠄
⠀⠀⠀⠀⠠⠱⡤⠀⠄⠀⡄⠀⠉⣥⠀⡠⢬⣁⣀⡠⠈⠀⠀⠤⠀⠈⠊⠉⠸⠉⠓⠛⠀⠀⠘⠃⠄⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⣀⡀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣌⣽⣿⡊⣩⣿⣷⠾⠷⢾⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣨⣿⣮⣽⡿⠋⠁⢀⣀⣤⠞⢅⣨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣾⣿⣶⣿⡽⠋⠀⢠⡶⢫⣿⣇⠀⢘⠋⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡽⠛⠉⠀⠀⣴⠏⣴⣻⠏⠈⠳⢾⡓⠀⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣤⣄⡤⠒⣫⡍⠁⠉⠉⠉⠉⠁⠀⠀⠳⠞⠃⣸⢇⣿⠀⠀⠀⠀⠑⣤⣶⠆
⠀⠀⠀⠠⢽⣦⣽⣶⢿⣼⣿⠏⠀⣜⣿⠀⠀⠀⠀⠀⠀⠀⠇⡀⠀⢀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠤⣄⣉⢝⣣⣼⣿⡏⠀⢸⢻⡸⣄⠀⠀⠀⠀⢰⣤⡀⠈⠓⠄⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠚⠿⢶⣾⣟⣿⣿⣄⠈⠊⢷⣿⣷⣶⣤⣤⣾⣿⣿⡇⢀⡞⠳⢿⣿⣿⣿⣗⢄⣀
⠀⠀⠀⠀⠘⢛⣯⣶⠞⠛⠁⠈⠻⢷⡖⠈⠻⣿⡿⠛⠛⠛⠛⠛⠪⡦⣀⣒⣌⣻⣿⣟⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢿⣷⣮⣮⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⡇⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠿⢗⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣹⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⢳⣄⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣆⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣟⠿⣿⡿⠿⠿⢯⣭⣿⣿⣾⣿⣮⣜⡿⣆⠀⢉⣦⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣒⠚⠓⠈⢀⣀⣀⡼⢥⣿⣿⠼⠋⠛⠻⢿⣼⡄⠀⢀⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢠⣾⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠒⠒⠢⣤⣤⠰⢿⣿⡦⠌⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⣤⣤⣤⡾⠙⠏⢀⣼⠅⠀⠀⠒⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⣀⡔⢿⣿⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠸⠀⠁⠈⠀⠀⠀⠀⠡⠀⠀⠀⠀⢠⠃⠀⠄
⠀⠀⠀⢠⣄⣀⠀⢪⣩⣆⢠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⡇⠰⠀⠀⠀⠐⠀⡔⠀⠁⠀⡁⠀⠆
⠀⣀⠐⢦⡀⠙⣷⣘⢳⠏⢰⢀⠏⠰⠀⠀⠀⠀⠀⡔⠡⡐⢤⠤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⠀⠀⠈
⠀⠸⣆⠀⠙⠄⠈⣿⢼⠇⠰⣎⡴⠓⠄⢀⢔⣤⡞⠀⢀⠇⠈⡆⢠⠔⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠈⠑⢄⢸⣆⢻⣶⡆⠐⣽⠁⠀⠀⢈⠘⣿⡇⠀⢨⡆⠀⢿⢀⡟⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⢦⠻⡈⣻⢃⢣⡏⠀⠀⠀⠀⠄⠻⣷⡄⠸⣆⣀⡞⣼⢁⠇⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⡄⠀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠣⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠣⠼⡿⠿⠋⠜⡿⠑⠁⠀⠀⠀⠀⠀⠀⠈⠯⣤⠀⢳⠆⠀⢰⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠘⠀⠀⠀⠀⠀⠀⡴⠒⠂⣴⠶⢄⡀⢳⣼⣇⣼⣿⠷⠚⠃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠑⣄⡀⠈⠷⣇⡙⢶⣍⢯⠉⣤⣤⡶⠿⢛⡋⡳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⠀⠹⢦⣦⡭⣷⣶⠽⠷⠦⣤⠤⡤⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣉⣀⣀⣑⣀⣐⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⣟⡛⢻⡟⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⣿⣿⡆⠀⠀⣟⠓⠂⠀⠀⢰⣿⣿⣿⣷
⠀⠀⠀⣼⣿⣿⣿⡿⣥⡀⠀⠈⠫⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠋⠀⠀⠀⠈⠀⠀⠀⠀⠈⠿⣿⣿⡿
⠀⠀⣼⣿⣿⣿⣿⣇⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⢀⣿⣿⣿⣿⡟⣽⣿⣿⣇⠀⠀⠀⠀⢠⡴⠟⢷⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⡄
⢰⣀⣀⣤⣼⣤⣼⣭⣭⣿⣿⣦⡀⠀⠀⠀⢀⣤⣾⣿⡟⢁⠀⠀⠀⠀⠀⠘⢷⣶⣶⣶⣶⣷⣿⣷⣤⣴⣶
⠈⢙⣋⣉⣿⣉⣛⣉⣝⣋⣙⠉⠀⠀⠀⣓⣫⣝⣛⣍⣛⣫⣑⣊⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠉⠉⠀⠉⠉⠁⠈⠉⠁⠀⠀⠀⠀⠈⠀⠈⠁⠀⠉⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤
⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⠀⠀⠛⠛⠋⢩⠟
⢠⣄⡀⣿⡟⠁⠀⠀⣽⠀⠀⠀⠀⣀⣤⡄⠀⠀⣸⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣿⣄
⠀⠙⠛⢿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠀⠀⠀⡟⣿⣾⠀⠀⠲⣶⣶⣤⣴⣶⣶⣴⣿⣿⣶⣶⣶⣶⣶⡶
⠀⠀⠰⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣬⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⡁⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣇⣀
⠀⠀⣾⣿⣷⡀⠀⠀⣤⣭⣥⠀⠀⠀⠀⠀⠬⠴⠮⠭⠴⠮⠤⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⠙⠛⠟⠀⠀⠀⢻⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡦
⠀⠀⠀⠀⢘⡃⠀⠀⣿⣿⣿⠀⠀⠀⠀⢀⣐⣚⣍⣓⣚⣉⣚⣀⠀⠀⠀⠀⠉⠉⠙⠃⠀⠀⠀⠀⣠⡆
⠀⠀⢀⣴⣿⣿⣦⡀⣿⣿⣿⠀⠀⠀⠀⠀⠈⢁⣀⣉⡉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣾⣿⡤⠴
⠀⠀⠀⣿⡏⢹⣿⠀⣿⣿⣿⠀⠀⠀⠀⣀⣀⣰⣿⣭⣿⣿⣿⣿⡆⠀⠀⢲⣶⣶⣶⣶⣯⣥⣤⣤⣭⣥⣤
⣀⣤⣤⣿⣷⣾⣿⣀⣿⣿⣿⣧⣄⠀⠀⠈⠿⠿⣿⣿⣽⣤⣧⣤⡇⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠿⠛⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⣼⣿⡄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣾⣿⣿⣿⡵⣶⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣍⣙⠳⣾⡀⢬⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣿⣿⠁⡈⠋⡽⢿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠃⣾⣷⠔⠁⠘⠷⡀⠀⠀⢸⣿⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⢀⡛⠛⣉⡌⠀⠀⠀⣿⡀⠀⠈⠁⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠋⠀⣡⠞⠀⠀⠀⢰⠟⣧⡀⠀⠀⠀⠀⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⣼⡿⠃⠐⣳⠞⠀⣷⣄⠀⣀⣴⡟⠀⠈⠻⣄⠀⠀⢀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣾⢏⣄⣶⠀⣠⣠⣴⡾⠛⠛⠛⣿⣿⡆⠀⠀⠈⠀⠀⠙⠋⣿⣿⣿⣿⠆
⠀⠀⠀⠀⣿⡌⠉⡀⢶⠞⠋⠁⠀⠀⠀⢀⣿⣿⠇⠀⢰⠀⠀⠀⠀⠀⠻⣿⣿⣷⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠙⠓⠾⠷⠟⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⢰⡆⢉⡀⠀⠀⠀⠀⠉⠛⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⢸⡇⠀⠀⠀⣠⣿⠃⠈⠻⡆⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠛⣿⠀⠀⢸⡇⠀⠀⠀⠁⣿⠀⠀⠀⠁⠀⠀⠀⢰⣄⠀⠀⠈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠿⠀⣿⠀⠀⣼⠃⠀⠀⠀⡼⠃⠀⠀⢰⡀⠀⠀⠀⠀⢻⣦⡀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⢻⡇⠀⣿⠀⠀⠀⠌⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀⣿⡇⠀⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠚⠀⡌⠷⠐⠃⣠⠀⠀⠳⣶⠆⠀⠀⠀⢹⡇⠀⠀⠀⢸⡏⠁⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣼⠃⡴⠦⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣸⡇⠀⠀⢸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠈⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢐⣿⠀⠀⢀⣿⡇⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⣼⡏⠀⣠⣾⣿⠁⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠘⣦⣄⣠⣦⣀⡀⢀⣰⡟⠁⠀⡀⠀⢰⣿⣷⣿⣿⣿⠃⠀⢀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⢶⣿⡿⢿⣿⣿⣿⣿⣿⠀⡠⠞⠁⢠⡿⠛⠛⣿⣿⡇⠀⢀⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⢁⡾⠋⠀⣼⣿⣿⣿⣿⠏⠀⢠⣶⠾⠛⠁⠀⠀⣿⡿⠁⢀⣿⣿⣿⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⢰⡷⠋⠀⠀⢠⣿⣿⣿⡿⠉⠀⢶⡿⠁⠀⠀⠀⠀⢠⣿⢃⣴⢿⣿⣿⣿⣿⣿⣿⡟⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⢻⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣾⡇⠀⠀⠀⠀⢀⣾⣿⣾⠏⠀⠙⠉⢻⣿⣿⣿⣿⣆⠉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣼⠃⠀⠀⠀⠈⢿⣿⣿⣿⣄⢰⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠻⣿⠳⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢸⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⢹⡀⣿⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣼⡇⠻⣆⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢾⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣧⡀⣽⣇⣀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠈⢷⡀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠘⣿⣿⣿⡏⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠸⣇⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠸⣿⣿⣄⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠈⣿⣦⠀⣸⣿⡿⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢫⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠿⠿⣿⣿⣿⣿⠿⣿⣿
⣿⣶⣷⣶⣷⣶⣶⣶⣶⣾⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡎⢠⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⠀⠀⠀⠀⢀⠀⠀⠐⠀⣀⠤⠀⠀⢀⠀⠀⠀⠘⠄⠀⡀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠾⠀⠉⠠⠀⠘⠓⠁⠀⠀⡐⠂⠾⡐⢄⡀⠀⢠⡘⠀⠄⢼⣷⡤⠅⢷
⠀⠀⠀⠀⠀⠠⠄⢰⡀⠀⠂⠠⠀⠀⠀⠀⣰⡿⣣⣾⣧⣁⣈⠀⣇⠉⠀⠀⠀⡀⠱⡄⠈
⠀⠀⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢀⠀⠈⠆⠠⠄
⠀⠀⢳⡀⠀⠈⠛⠿⠿⠷⠾⠲⠶⠶⠶⢶⣶⣒⣛⣛⣛⣛⣛⣉⠉⣿⠀⡸⠃⠀⠀⠀⠀⠂
⠙⠛⣉⣉⣐⣛⣛⣛⣋⣁⠀⠀⠈⠛⠿⠯⠭⢥⣤⠀⠀⠀⠀⠀⠀⢹⡗⠰⠀⠀⠀⢱⠀⠀⠁⠀⠀⣠⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣦⣬⣭⠉⠁⠀⡀⢠⠀⠀⠀⠬⣤⣴⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠘⡉⠀⡀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣧⠀⠀⠀⣿⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠃⢰⡀⠀⠀⠛⠁⠀⠀⢰⡀⠐⠠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣄⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡼⠁⠠⠄⠀⠀⠀⠀⠀⠀⠁⠐⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡓⠀⠀⠀⠄⠈⠛⠉⠓⠄⠀⠠⠀⠀⠐⠊⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠽⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⣀⣀⣤⣤⣶⣦⣀⣀⣀⣓⣠⣤⣤⣤⡀⠀⠀⠶⠿⠿⠛⠛⠉⣉⣀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠉⠉⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣷⡿⠛⠋⣿⠽⡿⢷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠋⠀⠀⠊⡅⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⡟⠀⠀⢀⡄⠈⣀⣠⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣀⣿⣿⣿⠿⠋⠀⠀⠀⠀⢠⡆⠀⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣄⡀⡀⠀⠰
⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠙⠣⣱⣦⣀⡆
⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⣧⣿⣿⣿⡀⠀⠀⠳⠼⠋
⠀⠀⣰⣿⣿⣿⠃⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⢰⣿⣿⣿⣿⣷⡀
⠀⠈⣿⣿⣿⣿⡆⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠿⣿⣷⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠻⣿⣿⣿⣷⡀⠀⠻⣆⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠙⣿⣿⣿⣷⠀⠀⠈⢿⣶⣄⠘⢷⣦⣄⡀⢠⣤⣽⡄⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣠⣿⣿⡿⠃⠀⠀⠀⠘⢿⣿⣇⠀⠙⠻⣿⣿⣿⣿⣿⡀⠀⠐⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠈⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⢿⣿⠉⠙⠛⠃⢾⠀⢈⣻⣿⡛⠉⠙⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣸⢀⣾⡟⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣄⠀⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡝⢿⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢟⢮⣷⣦⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠇⠀⠱⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠙⣦⡀⠀⠀⠀⠀⠀⠀⢰⡆⢸⣿⡀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠟⢷⣀⠀⠀⠀⠀⠘⠾⠿⠞⠉⠀⠀⠀⠀⠀⠸⡿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣄⢴⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠻⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠟⢂⢀⣉⣀⣀⠀⠀⢐⡤⢀⣀⣀⣉⣑⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣥⠞⣻⣿⣿⣿⠟⢀⣾⣿⣿⣿⣯⠤⠤⠤⠈⠀⠀⣈⣒⡒⠀⠒⢒⡒⠂⠫⠉⢩⠉⠉⡯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠊⢐⣛⣛⣛⣉⠀⠉⠉⢡⠀⣠⣄⣶⣲⣆⣴⣶⡧⠤⠤⠤⠤⠤⠤⠤⠄⣙⣛⣓⣒⣦⠷⠖⠂
⠀⠀⣀⣄⠀⣀⡤⣖⠯⠤⠤⠛⠓⠒⠒⠛⠛⠉⠉⠉⠉⠉⢉⣀⣒⣒⣒⣒⡂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠤⠋⠁⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠈⠉⠉⠙⠻⣦⣶⣤⣤⣠⣤⣤⣤⣴⠷⢶⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠤⠀⠠⠤⠀⠀⠤⣄⣀⠠⣄⣀⡀⣀⣀⣤⣄⣀⣠⡀⢀⣀⣄⣀⣉⡷⠘⠁⠀⠀⠀⠈⠋⠉⠉⠻⣦⠤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠳⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣄⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠘⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⢱⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⢻⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⡀
⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠟⠁
⠀⠀⠀⠀⣿⠃⠀⢠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⢀⣄⠀⢁⣤⣄⡀⠀⣤⡀⢠⣄⠀⣤⡀⢠⣄⠈⢩⡍⠁⢀⣤⣀⡀
⠀⠀⠀⢰⡟⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠋⠀⣿⠀⣿⠀⠘⠇⠀⣿⠀⢸⡇⠀⣿⠁⠸⣿⡄⢸⡅⢠⡟⠀⠘⠇
⠀⠀⠀⠸⡇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣴⠀⠀⣿⠀⢿⣦⣀⠀⠀⣿⣀⣸⡇⠀⣿⠀⢀⡻⣇⢸⠇⣾⡷⣆⣀⣀
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠛⠀⠀⣿⠀⠈⠙⢻⣷⠘⣿⠉⢹⡇⠀⣿⠀⠰⡆⢻⣼⡅⢿⡇⠀⢸⡏
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣆⠀⠀⣿⠀⣿⠀⢸⡇⠀⣿⠀⢰⡇⠘⣿⠀⠸⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣃⡀⠀⢐⣟⣀⠛⣳⣖⠁⢀⣿⣆⢸⣷⡀⣿⣇⣸⣧⡀⢹⡀⠀⠙⣳⣞⢿
⣿⣿⡿⠿⣿⡿⠿⠿⡿⣿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣾⣶⣾⣶⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣔⣂⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣣⣾⣀⣴⡅
⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⢀⠆
⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⡻⣿⣿⣿⣿⣿⡶⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢟⣽⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⡎⡻⣿⣿⣿⣶⣶⠶⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢿⣿⣿⣿⠿⣿⣿⣷⢸⣿⣿⣿⣷⡝⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⠏⠀⣸⣿⡿⣼⣿⣿⣿⣛⣋⣈⢻⡿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡷⢠⣿⣿⠿⣛⣿⣿⣿⣿⣿⣿⣧⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⢀⣤⣤⢄⣀⣈⠉⠁⣰⣾⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣄⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣄
⠀⠀⢀⣠⣴⣾⣿⡟⢫⣍⣙⣿⢿⣿⣿⣷⡼⠛⣩⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣟⣿⣍⠛⢻⣿⢿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⢠⣾⣿⡟⠉⠀⣼⣿⣿⣿⠿⢿⣷⣾⣭⣭⣥⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣮⢻⡆⠊⠙⠀⠀⡿⠸⢹⣿⣿⡿⣿⣿⢻⣿⣿⣿
⢻⣿⣯⠁⣠⣾⣿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⠀⠀⠀⠜⠁⠤⠟⣿⣿⡇⣿⡏⣼⣿⣿⣿
⡞⢡⡟⢠⡛⢻⣿⠆⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⢹⡿⣿⢠⡿⢹⡏⠁
⠻⠞⠀⠀⢹⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠩⣿⣿⣿⣎⠻⡼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠃⠑⠝⠦⠓⠈⠷⠄
⠀⠀⠀⠀⢸⡋⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣶⡶⢦
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣷⣯⣿⣿⠿⢿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣧⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣀⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⡏⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⡗⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⡏⢸⠹⠿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⣴⣦⣷⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣧⣀⠦⣶⣿⣿⣿⣯⣿⣿⣿⣦⣴⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣿⣿⣿⣿⡿⠟⣋⣭⡍⠙⠁⡯⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⣶⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⣀⡜⠀⣄⣀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠚⣿⣿⣿⣿⠟⣳⡿⠁⠀⠀⠀⠀⣸⣿⠀⠀⢈⡉⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⢠⢏⣿⣄⠀⢀⡁⢀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡤⠤⠤⠤⣤⣀⣠⣤⣾⠟⠁⠀⢰⡀⠀⠀⠀⠀⠀⢀⣿⠇⠈⠓⢿⣡⡴⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢰⡌⣧⠀⠀⣠⠂⠀⣼⣿⠀⠀⠀⠀⢹⡶⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠁⢠⡖⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⢷⣨⡁⠀⠁⠀⠀⣿⣿⡀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⢀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⢿⡇⠀⢹⣰⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠙⠝⠡⣶⡇⢰⡟⢹⡇
⠀⠀⢠⡶⣫⣽⣿⣟⣻⣿⣿⣿⠏⠊⠁⠸⣷⡀⠈⠛⡆⠀⠀⠀⠀⠀⠀⠀⢸⠂⠁⠀⠀⢀⡤⠀⢻⡃⡞⠀⢸⡿
⠀⢀⡿⠿⠋⠞⠉⠐⠒⠚⠋⠁⠀⠀⠀⠀⢹⣷⣄⠀⠹⣦⡀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠉⣠⢀⠀⠀⠀⣄⣼⡇
⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⢨⣿⣶⣦⣤⣤⣼⣿⡄⠀⢠⢶⡉⠻⠈⠀⠸⠰⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣿⣿⣿⣿⣿⣿⠿⠷⢴⣀⡡⠈⠑⡝⠶⢻⡿⠾⠎⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠘⣿⣿⡿⠀⠀⠀⠈⠉⠒⠒⢦⣢⢸⡟⢧⠀⠾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⢳⡀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣾⡿⠀⠀⠈⢷⡘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡘⢼⣻⣿⡄⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣸⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⠙⢿⣷⡀⢀⣤⣶⣤⢂⣿⠁⠀⠀⠀⠐⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⣿⣾⣿⠿⠿⠛⢩⣶⣿⣦⠐⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡂⠙⣿⣿⣆⠀⠀⠈⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠀⠼⣇⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠼⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠃⠉⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠴⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⠔⡂⠄⠈⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡡⠀⠠⠊⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⡐⢁⠄⠀⠀⠐⠈⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⣮⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠿⣾⣶⠶⠶⣵⣾⣥⣴⣎⠀⠀⠀⠀⠀⠛⠂⡀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⢞⣩⠃⠂⠀⢈⠹⢁⡬⢬⢻⠋⣍⢙⣛⢶⡞⣀⠀⠀⡀⠀⠀⠀⢀⡾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⣀⡀⠀⠀⢰⣿⣾⠶⢟⣱⠀⠀⠀⢃⠀⠐⠀⠓⠘⢄⠖⠀⢘⠏⠫⣻⢋⠀⠀⠀⢠⣟⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠈⠉⠀⣠⣴⣿⣛⣬⣤⡟⠁⣸⣿⡡⠀⠇⠀⠀⠀⠀⢊⡀⢀⢀⡈⠂⠠⠀⠈⣲⣾⣿⣿⣏⠕⠀⠀⠀⠀⠀⠀⢀⠜
⠀⢸⡶⠁⣀⠾⡿⠿⣻⡭⠉⠙⠛⠒⠋⢈⠛⠀⠀⢰⣠⡠⡃⠀⢷⢱⡋⠀⠀⠀⠀⠀⠛⠄⣿⣿⣿⡄⠀⠀⠀⢀⡴⠊⠁
⢠⠋⠀⠈⡔⣩⡍⠑⡌⠃⠀⣨⠟⠀⠀⠄⠀⠀⠲⠈⢐⢃⠒⠀⠑⡚⠃⠄⠀⠀⠀⠂⠢⣴⣿⣿⠅⠀⠀⠀⠀⠸⡅
⡏⢳⡀⠀⢇⡛⠓⣠⠇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠬⠂⠀⢕⣐⠡⣠⢀⡴⠞⠛⠳⣾⣿⡿⣧⡀⠀⠀⠀⠀⠀⠙⡄
⠐⠀⠙⠀⠀⣉⣉⠥⠀⠀⠀⠀⠀⢀⠐⢁⠆⠀⠀⣠⠄⡞⣠⣜⣧⣠⣴⡏⠙⡀⠀⠀⢿⣿⡿⠌⠓⠄⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠑⢆⠀⠀⡄⢀⠀⠀⠀⠠⠘⠁⣴⡟⢄⠀⠄⢀⠆⢢⣾⣿⣿⣿⡟⠀⠀⠈⠐⠒⢺⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠉⠓⠦⠀⠀⣀⣤⣒⣀⢴⣿⣧⡀⠃⠀⠀⣰⢾⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⢿⣽⣿⣿⣿⣿⡽⣿⣾⣿⣿⡷⠀⠀⠀⠀⠄⠀⣸⠁⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠳⣽⠿⠿⠿⠿⢿⡿⢯⣁⠀⠀⠀⠐⠊⠀⠀⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠙⠲⣤⡀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡀⠀⠀⠀⠟⠦⣀⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⣰⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠐⠄⠀⠀⠅⠀⠀⠉⠑⠀⠀⠀⠀⠀⢘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⣴⣆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⢠⣶⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣴⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣫⡖⣦⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⠵⣯⣼⣿⣿⢸⣧⡴⢞⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡔⢻⣶⣿⣿⡿⣟⠈⢷⠄⠙⠃⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣼⢋⣴⣉⣿⣿⠿⠋⣵⠄⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣴⠟⣵⣧⣼⡿⢋⡅⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣿⠀⠀⠀⠀⢀⡾⢫⣾⣿⠟⠉⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡇⠀⠀⠀⣠⡟⣱⣿⡿⣁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⣰⡟⣼⡿⢫⡄⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⢸⣟⢱⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣧⠀⢰⡟⣼⠟⣥⡌⠀⠀⠀⠀⠀⣠⣾⡿⠿⣿⣿⣿⠿⣿⣿⣶⣄⠀⠀⣿⣿⣋⣣⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡻⣤⣿⢱⠏⣦⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⢸⣿⣿⠀⠈⢻⣿⣿⣦⣼⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡴⠛⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⣦⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠛⡟⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢈⣿⣶⡄⠈⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣟⢛⡿⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⣷⣦⣤⣟⢷⣭⡻⢶⣦⣤⣤⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡋⣻⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⢻⣿⣿⣷⣬⡛⢶⣭⣛⠿⣍⣽⣿⠛⣿⠟⣿⠟⣿⣏⣹⣧⠼⣛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆⠈⠙⠻⠶⣮⣭⣽⣛⣓⣛⣛⣫⣭⣭⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣶⢦⣀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣄⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣿⣟⣷⣼⣿⣿⣿⡿⢟⣯⣽⡟⠟⡴⢋⣴⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣛⢷⣶⣿⣿⣿⣿⡿⠋⠑⢋⡽⢛⡅⣠⢿⠟⡉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠻⠿⣿⡘⣿⡇⡄⠀⠀⠀⢉⡞⠟⢸⣴⣷⣦⣿⡤⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⠀⣘⣶⣶⣝⠻⡧⢳⣄⣀⢀⣼⣤⡄⣨⡭⠭⣂⣀⡨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠽⣿⣿⠀⢷⣆⠛⠛⠛⢛⡭⠀⢰⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣾⣷⣤⠙⠿⠶⣟⣫⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠩⢭⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⣠⠴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣣⣸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢹⣿⣿⡿⠃⠀⠀⠀⠀⢀⣾⣿⢿⣿⣿⡏⠉⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⢿⣿⣧⠀⠀⠀⠀⠀⠈⣿⡇⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠈⠻⣿⣦⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠈⢻⣷⣄⠀⠀⠰⠿⣧⣀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡇⠀⠀⠀⢿⣿⣷⣦⡀⠀⠀⠉⢿⣿⡟⠋⣉⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⡇⠀⣦⡄⠈⠿⠟⠙⠻⣷⣤⣤⣾⣟⠳⣌⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣧⠀⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⢻⡝⢳⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⢦⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠒⠲⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⣀⡘⢻⣿⣿⣷⣦⢤⣠⣶⣟⣿⣿⣷⡶⢰⣶⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣉⣛⣿⣿⣿⡿⢋⣥⣾⣿⣿⣿⢸⣿⣿⡇⢸⡱⣆⠹⠛⠋⣻⠿⠃
⠀⠀⠀⠀⠀⠀⣉⣛⣻⣿⡿⢉⣴⣿⣿⣿⣿⣿⠟⢼⠿⠉⠃⡈⠛⠚⠁⠀⠘⠧⢤⡄
⠀⠀⠀⠀⠀⢈⣉⣵⡿⠋⣴⣿⣿⣿⣿⣿⣿⣷⡄⠑⠆⠀⠀⠀⠰⠀⠀⠠⠀⢀⠔
⠀⠀⠀⠀⠈⢁⣺⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠂⠀⡀⣄⠢⠄⣠⠖⠉
⠀⠀⠀⠀⠚⢙⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠷⠤⣀⠀⠂⠈⠛⠋⠁
⠀⠀⠀⠀⠴⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⠁⠀⠁
⠀⠀⠀⢀⣴⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣀⣀⡠⠤⠄
⠀⠀⠀⠀⣰⡇⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠀⣴⣤⣶⣿⠟
⠀⠀⠀⠈⢨⡇⢸⣿⣿⣿⣿⣿⣿⣻⠯⢿⡯⢚⢤⣮⡿⠛⠋⠁
⠀⠀⠀⠀⠉⣵⢸⣿⣿⣿⣿⣿⣿⠆⢙⠈⠉⠙⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡄⠙⡌⣿⣿⣿⣿⣿⣿⠅⡀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⢿⣿⣻⣗⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠐⡖⠰⡘⣿⣿⣿⣿⣿⡟⡡⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠉⢯⡻⠿⣗⣄⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠹⣇⠛⡸⣿⣿⣿⣿⣷⡿⡁⠀⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⢡⠀⠘⣫⡇⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠹⠆⣿⢈⡻⣿⣿⣿⣿⣽⡂⠰⢯⠯⠥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡯⠀⢀⣀⣴⡏⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠿⡈⢷⣝⠿⣿⣿⣿⣿⣆⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡀⠀⢠⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠋⢌⣱⡮⠙⠿⢿⣿⣶⣷⣤⣀⠀⠀⠀⠀⠀⣀⠐⠀⢀⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⣤⣬⣙⢛⡛⣿⢿⣿⣿⣿⣿⡿⠿⠷⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⣶⣶⣾⣷⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⢋⡄⠀⠀⠀⠀⢲⡴⠋⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⣠⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠘⢃⠀⠿⢿⣿⣿⣦⡾⠁⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠻⣿⣿⡥⢾⠀⢠⣶⡇
⠀⠀⠀⠀⠠⠀⠀⠸⠀⠀⠀⠀⠀⠈⠳⠀⠘⠟⠁⣿⠇⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠙⠻⢿⣧⠀⠀⠀⠀⠀⢠⠀⠀⠉⠉⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣷
⠀⠀⠀⠀⣰⠿⠁⠀⠈⠉⠀⠀⠀⠀⠈⠈⠓⢀⡀⠤⠤⠄⠻⠀⠀⠀⠸⠹⡤⠀⠀⡐⠀⠔⠈⡇⠀⠀⠀⠀⠀⠲⠈⠀⢀⣀⡀⢻⡆
⠀⠀⠀⢠⠏⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢰⡆⢛⠶⠄⠀⠀⠀⠀⢀⠑⠀⠀⣶⣄⠀⠁⡠⠂⠀⠀⠀⠀⠀⠀⠠⣍⠡⣦⡁⣀
⠀⠀⠂⠈⢢⡈⠛⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠰⠋⣠⡎⠀⠀⠀⠘⢿⡀⡠⠖⣁⡠⣴⠊⠀⠀⠀⠀⠀⠀⠙⢶⣾⣿⣶⣶⠟
⠀⠀⠐⠗⠤⣅⣀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠀⠀⠀⠀⠀⠀⠀⠻⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠉⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣽⣯⣯⣭⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⠶⠶⣟⠛⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⢴⡶⠖⠋⢝⡃⠠⠠⠀⢸⢠⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⣾⢟⣫⠛⠋⠄⢀⠀⣀⢱⣔⢸⡇⣼⠈⠙⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣀⣀⢀⣠⣄⣀⢼⣵⠒⠻⠿⣿⡁⠀⠀⠠⢀⣠⣶⣾⣧⠃⣴⣧⡴⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠾⢋⡎⠀⠀⢀⣨⣿⣿⠟⠁⠀⣠⡴⣛⣽⣄⣬⣷⣿⠿⣉⣥⣾⠿⠿⡿⣤⣬⡇
⠀⠀⠀⠀⢀⣠⡖⠋⠈⢸⣿⡄⣼⣴⣾⠧⠞⠛⣿⡟⠀⠀⠞⢧⣦⣿⣿⣿⢿⡍⣧⣷⣿⣿⠉⠉⠉⡛⣯⣿
⠀⠀⢀⣴⡿⠋⠀⠀⣰⣿⠟⢃⠋⠉⠀⠀⠀⣬⠟⠁⣰⠂⣼⣿⣯⠿⣿⣿⣾⡿⠛⠛⠿⡋⣉⡙⢻⠁⣼⠋
⠀⣰⣛⠿⢅⣀⣴⠛⢻⡿⠀⠈⠀⠀⠀⠀⢠⠟⢂⠰⠷⠀⣸⣥⠴⣶⣿⣿⡫⠛⣷⡄⠀⠀⣸⡧⠚⢲⠏
⣰⣿⡟⢠⣾⡿⢿⢀⣼⡇⠀⠈⠇⡀⠀⣀⡍⣱⢦⣤⣾⠏⠁⠀⠀⠸⣿⣦⠀⠀⣨⣿⣠⣦⠾⣥⠤⠏
⣿⣿⣧⣿⠃⠀⠸⣼⠟⠀⠀⠀⠀⢉⡽⣿⢾⡟⠁⠈⠀⠀⠀⠀⠀⠀⠈⢻⣧⣀⣺⣾⠉⠁
⢻⣿⣿⡟⠀⠠⠚⠁⠀⠀⠀⠀⡰⠋⡼⠁⠻⡇⠀⠀⠀⠀⢀⡀⠠⣴⣦⣴⣿⠟⣿⣿⡴⠒⠒⠒
⠈⠻⣽⠇⠀⠀⠀⠀⠀⠀⠀⡼⠁⠐⡇⠀⠀⣈⣒⣶⣾⣿⣋⣭⠟⠋⠉⠁⠁⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣫⣭⣥⣬⣉⣽⠏⠁⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠋⠉⠉⢉⠉⠭⠟⢱⡀⠀⢢⣾⣾⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣄⣄⠀⠀⠠⠠⠒⠉⠀⠉⠂⣬⣽⡟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠵⣤⣶⣶⣦⡴⣶⠾⠟⠋⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⣿⣿⣻⠿⣿⣻⢿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣾⣾⣿⣾⣾⣾⣾⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⣿⣼⣿⣻⣟⣩⣿⣾⣀⢴⡿⣿⠯⢀⣩⡐⢺⠦⣿⠙⣽⣾⢻⣟⣹⡯⣿⢻⠤⠦⢀⣶⣂⠤⣤⣤⣄⠀⠃⣶⣴⡄
⠀⠀⠀⠀⠉⠉⠁⠉⠉⠁⠉⠉⠀⠀⣩⣿⣿⡿⠻⠙⢷⣄⡙⠀⣇⣼⡞⣯⣿⣷⣿⣨⣧⡏⠈⠁⠝⡘⣯⡿⢹⠋⢰⢞⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡟⢋⣠⣤⣦⣿⣿⣿⣀⠈⠭⠵⠯⠌⠸⠟⠟⠇⢤⡤⠙⣿⣿⠇⠀⣴⣗⠼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣧⠀⢸⡧⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣯⣿⣻⣿⣿⢟⣻⣿⣿⣷⣯⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣼⡇
⠀⠀⠀⠀⢀⡀⣴⢿⣿⣿⣿⣿⣿⣿⡛⠩⡽⠎⠈⠓⠊⠁⠉⠙⢼⡽⠃⢘⣿⡯⣵⠉⢻⣿⣷⣄⡀⠀⠀⠀⠀⠉⠛⣻⡷⠇
⠀⠀⢰⡾⣏⡿⢋⣿⣿⡟⣿⣿⣿⠁⠂⠀⠀⢀⣀⠀⠀⠀⠀⢠⡶⠁⠀⠈⠉⠁⠀⠀⠈⠴⠬⠻⣧⡀⠀⠀⠘⠳⠾⠙⠇
⠀⢠⢤⣷⡽⣤⣿⣿⣿⣴⣿⡿⠉⠀⠀⠀⣀⠜⢿⣦⠙⢷⣤⣸⣿⠳⠤⠀⠀⠀⣀⡔⠀⣀⣤⠴⠒⣬⣶
⠀⢹⡄⢿⣷⣿⣿⣿⣿⡿⣿⡇⠀⠀⠘⠋⣁⡈⠀⠙⠃⠘⠻⠿⠌⠳⣄⠈⠶⠬⠄⡁⠈⠙⢁⣴⢋⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠈⣿⣷⠙⣿⣻⣿⣿⣿⣿⣷⣤⠶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣤⣴⣽⣿⣦⣤⣤⣬⣷⣤⣥⣈⣙⣳⢦⡆⠀⠀⠀⣿⣿⠋
⠀⢀⣿⣿⠀⠛⢿⣿⠻⣿⣿⣾⣧⢀⣿⣿⣿⣿⣿⡿⣻⣾⣿⣿⣿⣿⠿⣻⣯⣥⣼⡿⣿⣦⢈⠉⢭⡿⠎⠀⠀⠀⠀⠈⠉
⠀⠈⣽⣿⣧⢀⣠⠙⣷⡌⠛⠚⢿⣿⣿⡇⣾⣿⣿⠷⠾⣫⣾⣿⣿⣿⣿⠿⠋⢉⣿⣿⠇⠀⠀⠀⠄
⠀⠀⠸⣻⣿⣦⡈⢁⣙⠻⣦⣀⠈⠉⢻⡇⢸⣿⣯⡿⠟⣵⣿⡿⢃⣴⣶⣶⠏⢉⠛⠉
⠀⠀⠀⠹⣿⠿⠿⢦⣝⣄⠀⠉⠳⣀⣠⣇⢸⣿⠏⢀⣠⣷⣻⠿⠋⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠸⠟⠙⠋⢂⠰⡶⠾⣿⡧⢿⡇⠀⣰⣶⣦⣤⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠵⣮⠓⠉⠁⢾⡇⠀⠀⠀⠤⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⣿⠿⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣾⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣠⠊⣠⠖⢉⡤⠝⣷⣤⣀⠀⠀⢀⣴⠟⡹⠋⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣱⣱⣃⡞⣡⠞⣁⠔⢋⣡⣬⡿⠃⢀⡾⢃⡞⠀⣰⠋⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠋⠉⠉⠩⠭⢝⣛⡻⢾⣥⣖⣉⢤⣿⠀⢀⣾⢁⡞⠀⣰⠃⣠⠎⡴⣿⠏
⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠉⠑⠮⡙⢷⣾⠃⠀⣼⢇⡞⢀⣼⠃⣰⠇⢀⡾⠃
⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠈⠣⡙⢷⣤⣿⣟⣤⠟⣡⡾⢃⢠⡿⠁
⠀⠀⠀⣤⣿⠀⠀⠀⠀⢼⣟⣹⡄⠘⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⣛⠭⠖⢁⡿⠁
⠀⠀⠀⣿⡙⠶⠂⠀⠀⠈⠛⠋⠀⠀⡇⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣒⣒⡂⢸⡇
⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⣸⠃⠀⢀⣿⣄⠀⠀⠀⠀⠀⠀⢠⣴⠶⣦⣤⡑⠢⣄⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣤⠄⠐⠁⠀⣠⡾⣿⠮⣛⢶⣄⠀⣀⣴⣿⢧⣄⠈⢿⣿⡳⣦⡙⠢⡹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢶⣶⣿⣋⣀⣈⣿⣮⣿⣿⣾⣏⠑⢬⡓⣭⣿⠌⢿⡻⣆⠙⢦⡀⠈⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢏⣿⣯⡙⢮⡑⢝⢷⣹⡶⠟⠋⠁⠀⠈⣷⡘⢷⡈⠻⡄⢈⡲⢝⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⣿⣷⡙⢦⡙⢮⣻⣿⠀⠀⠀⠀⠀⠀⠘⣷⡈⢳⡀⠙⣆⢹⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⢮⣿⣦⡙⢦⡙⣿⡄⠀⠀⠀⠀⠀⠀⠘⢷⡄⠳⡄⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⡿⣯⡢⣱⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣽⣦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣼⡷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡃⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣻⡏⢻⡾⠿⣒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⡀⣿⣹⣿⣰⡿⢆⢰⡆⢀⣀⠀⢀⣄⣷⢟⣼⣡⣿⠀⠀⣤⣀⢻⣇⣙⣗⣶⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡟⢏⣿⡇⣿⣿⣿⣵⣇⢹⠃⠸⡿⢿⡿⢿⠃⣡⣞⣿⢿⣰⣷⡝⣿⣾⣿⣿⢦⣸⠎⣿⢿⣶⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⢿⡇⡟⣿⣇⣼⣿⡿⠙⠇⣿⣿⢻⡞⣧⣄⠀⠸⡿⢸⡧⠱⢟⡉⣸⣿⡿⣿⣿⣿⣟⠻⣦⡙⠿⣽⣼⣿⢻⠿⠟⠓
⠀⠀⠀⠰⠶⠶⠛⠡⠾⠦⠽⠿⣿⣿⠿⠷⠦⠀⠋⠿⠾⠿⠿⠿⠾⠷⠿⠦⠶⠿⠍⠁⠀⠉⠛⠉⠉⠁⠘⠓⠶⠿⠧⠬⠷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢳⣄⠀⠀⠀⠀⠀⠀⢀⡤⢚⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⣾⣾⣿⣿⣿⣶⣤⣤⣾⣍⣴⠋⣀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡱
⠀⠀⢀⣠⣤⣶⡾⢿⣿⢿⣿⠿⠿⣿⡿⡿⢿⢿⡟⣻⣿⡿⣿⢿⢿⡿⣿⣿⣷⣤⣄⣀⣀⠀⠀⠀⣰⠋⣰⠁
⠰⢾⠋⠩⣽⣿⠃⠀⠉⢷⣶⣶⠶⠾⠿⠷⠚⠳⢶⣷⡌⠀⠁⠉⠚⠁⠉⠁⠋⠙⠛⠛⢛⣿⣾⣾⣿⣿⠃
⠲⢦⣧⡶⠈⠁⠀⢀⡄⣾⣸⣄⣀⠄⠒⠒⠒⣈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠟⠋⠋⠉⠉⠙⣿⠿⢇
⠀⠀⠈⠉⠛⠲⢦⣼⣿⣟⣻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣶⣶⣶⣾⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣟⠿⡿⠛⠛⠛⠛⠋⠉⠉⠉⠙⠯⣉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢬⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⣀⠀⠀⢠⣞⡁⣷⢀⣠⣤⣴⡶⢶⣂⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⠿⢷⣤⣜⣾⡇⣿⣿⡿⠟⢛⣋⣉⣭⠤⠬⠭⢽⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣖⣩⣭⣤⣤⣤⣤⠾⢻⡿⠠⢿⣿⣝⣛⣛⣛⣛⣛⣛⣉⣀⣀⠀⠉⠻⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⢏⣴⣾⡿⠛⠁⠀⠀⠸⠄⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣯⣍⠉⠻⢷⣦⣹⣆⠙⠳⣦⣀⠀⠀⠀⠀⠀⢠⣿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠸⢋⡿⠋⠀⠀⠀⢀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠉⠻⢿⣿⡳⢦⣌⠀⠻⣿⠀⡀⠈⠛⢷⣶⣤⣤⣶⣿⠏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣿⠁⠀⢪⣶⣿⠋⢰⡆⠀⠀⠀⠈⢿⣦⣄⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠟⠀⠈⠒⠤⣤⣤⣭⠴⠟⣁⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠉⠡⠖⠋⠀⠀⠀⠀⠀⠈⣿⣿⣧⡀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠠⠤⠤⠖⠛⣩⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣅⠀⠀⠠⠤⣤⣴⣶⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⣶⣆⠤⠀⠀⠀⣈⠉⠻⣿⣆
⠀⠀⠀⠀⠀⣀⣤⡖⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⢍⣩⣿⠷⣀⣿⠇⢀⣾⡿
⠀⠀⠀⢀⡾⢋⣭⣻⡖⠀⠀⠀⠀⠉⠉⠉⠻⢶⣦⣤⣀⣀⣤⣶⣿⣿⣿⡇⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣩⣴⢾⣟⣭⠶⠾⠻⢯⣀
⠀⠀⠀⠈⣧⠈⡿⠹⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡿⠿⠿⠿⠿⠛⠋⢠⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠞⣿⣿⠸⠏⠀⢲⣤⡀⠀⠙⣷⡀
⠀⠀⠀⠀⢸⡄⠀⢀⣠⠔⠉⢠⠞⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⣿⠀⠀⠀⣸⢣⣿⡿⠀⣿⠇
⠀⠀⠀⠀⠈⠛⠻⠻⢿⡶⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⣾⠿⠋⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡾⠟⣋⣤⠶⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢋⡡⠖⢋⣉⣤⣴⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⢖⣫⣶⢿⣿⠟⢋⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣡⣾⡿⠛⠉⢀⡞⢁⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⠟⠉⠀⠀⠀⢸⣵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁

⠀⠀⠀⠀⠀⠀⠄⠀⠂⡀⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⢀⠀⣤⣀⠴⡀
⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠃⡀⠀⠈
⠊⠋⠄⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠑⠠⠄
⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢖⣼⠷⣄⠀⠖⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠂⠰⠶⠀⠀⠋⠁⠙⠊⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡤⠖⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠄⠉⠀⢠⡿⠃
⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠟⠁⠀⢰⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠌⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣬⣿⣿⣷⣶⣤⣶⣿⡟⠀⠀⠀⠀⠀⢈⣰⣶⣀
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⢤⣄⡤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⡤⢂⣌⡫⠥⠄⠠⠶⠖⠲⠶⠄⠀⠒⠰⠚⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠶⡀⠀⢁⢐⣒⣠⣶⣠⣴⡾⠶⠟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⡄⠀⡄
⠀⠀⠀⠀⠀⣰⣧⣾⣄⣴⣵⣿⣤⣼⣿⡿⠿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠦⣀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠛⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⢀⡂⠀⠀⠀⢀⠄⠀⠀⢀⣀⣼⣷⣶⣶⣶⣿⣶⣶⣶⢤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡠⠀⠈⠐⠀⠀⣠⣾⣋⣽⠟⠛⢛⣩⠋⣩⣉⡙⢻⣿⣻⣿⡾⣿⣶⣆⡀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠒⡠⣀⣄⣴⣿⢿⣛⣏⣵⣿⣼⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⢻⣿⣷⣞⢪⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠂⢀⠤⢠⣀⣜⣻⣯⣟⣋⣵⣾⣿⣿⣿⣿⡿⠟⢟⣿⢟⣹⣿⣻⣛⠛⢿⣿⣿⣯⣾⣿⣯⣛⣿⣿⣯⣝⡠⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⣨⢫⡜⠛⣼⣿⢿⢿⣾⣿⢿⠇⡐⢷⣼⣷⣿⣿⣿⡛⠉⠉⠒⡿⠉⠻⣿⣿⣿⣿⣻⡯⣩⣿⣿⣍⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⣹⡼⢈⣼⠟⠋⠁⢸⣿⣿⣾⣤⣽⣿⣿⣿⣿⣿⣿⡅⠀⠀⢠⣷⡶⠀⠹⡟⠿⣿⣿⣿⣿⣧⣭⣹⣷⠉⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠸⠃⢒⣴⡟⣠⣾⡃⠀⠀⠀⣼⣿⣿⢙⣽⣿⣿⣿⣿⣿⣿⣿⣿⡗⠶⣿⣿⣗⠂⢤⡇⠀⠙⠻⣿⣿⣿⣿⣿⠞⠁⢀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⣀⠀⠔⢛⣯⣼⣿⣿⡇⠀⢀⠀⢸⣿⣿⣿⣠⣽⣿⡿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠗⠀⣸⠇⠀⢉⣫⣿⣿⣿⣿⣟⠛⢋⠁
⠀⠀⠀⠀⠀⠀⡀⠀⣤⣼⣿⣿⣿⣼⣯⣆⡨⠆⠈⣿⣿⣟⡋⢽⣿⣷⣭⣙⣻⣿⣿⣿⣿⣿⡿⠂⣬⡟⠀⠀⣠⣿⡿⣫⠿⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠻⠿⣿⣿⣿⣿⣓⠅⢀⠈⠿⣿⣿⣾⣿⣹⣿⣿⣿⣿⣿⣿⢿⢫⣴⣾⢟⣁⣴⣾⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠙⠻⣿⣾⣧⣄⣀⡈⠻⢿⣿⣿⣷⣄⣯⣿⣾⣾⣶⣿⣿⣷⣞⡯⢿⠛⠛⠀⠉⠀⠀⣤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⢶⠶⠶⢷⣿⣿⣿⣿⠿⣿⡿⣿⣿⡛⢯⠄⠒⠉⠁⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠠



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡈⠙⠻⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠉⠻⢿⣶⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⢿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡿⠿⢿⣶⣦⣄
⠀⠀⠀⠀⠈⢿⣧⠉⠛⠻⢿⣶⣄⡀⠀⠀⠀⢀⣤⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠈⠙⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠉⠻⢿⣦⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⢀⣠⣴⡿⠟⠛⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⢀⣿⣟⣠⣴⣾⡿⠟⠋⠀⠀⠀⠀⠈⠙⠿⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⢾⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⡿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⣀⣤⣾⡿⠛⠛⠛⠿⠿⠿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣞⣛⣻⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢾⣟⣛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡯⠤⠼⣧⣿⣿⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣿⣿⣼⠧⠤⢼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡾⠛⠛⠯⣵⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⣿⠵⠛⠛⢷⣶⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠦⠤⠤⠬⢦⡀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢀⡴⠥⠤⠤⠶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⣀⣀⣀⣀⣙⣦⣀⠀⠙⢷⣄⠀⠄⢠⠀⣠⡾⠋⠀⣀⡴⣋⣀⣀⣀⣀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠙⢧⣾⣿⣼⠋⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣈⣼⡟⢹⣷⣥⣄⣤⣤⣶⣶⣶⣶⣶⡶⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠃⠉⢋⠭⠉⠉⠉⡈⢿⣿⣼⣿⢁⠈⠉⠉⠤⣚⠉⠉⠛⠳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠒⠀⠀⡠⠎⠀⢀⡤⠊⡀⠋⣿⣿⠉⢀⠁⠀⡀⠀⠘⢤⡀⠀⠐⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠔⠂⢀⣄⣠⠤⠲⠋⠀⠐⢀⠂⣿⣿⡀⠀⠣⠀⠈⢓⠠⢄⡠⠀⠀⠂⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⠊⠀⡌⠀⣴⠃⢀⠎⢀⢂⠎⢸⣿⠐⠔⠀⠱⡀⠈⢦⠀⠱⡀⠀⢦⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣞⠀⣠⠁⢀⠎⠀⡠⠂⠀⠘⠏⠀⠈⢦⠀⠐⡄⠀⣃⠀⣱⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣾⣤⡾⠁⠀⠀⠀⠀⠀⠀⠈⠳⣤⣼⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⡄⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⢀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣿⣶⣩⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⣤⣶⣦⣠⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣯⡾⣫⡿⣽⣾⣿⡂⣴⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣹⣿⣿⠟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⢹⡿⣤⣤⢄
⠀⠀⠀⠀⠀⣠⡾⢣⣿⣿⣿⣿⣿⣿⣿⣿⣎⠀⠀⠀⠀⠀⣰⢓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣧⣽⣄⣬⡶⣀⡀
⠀⠀⢀⣴⣿⡟⠀⣾⣸⣿⣿⣿⣿⣿⠏⠙⣞⣦⠀⠀⠀⢸⣷⣾⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣦⢾⢃
⢀⣼⣿⡟⠋⢀⣰⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⣦⠀⠀⢠⣿⣿⣿⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣶
⣿⠟⢛⣅⣴⡹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣾⡿⢿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⣫⣴⣿⣿⡿⠀⢺⣿⣿⣟⣿⣿⣿⡿⣿⣿⢻⡷⠀⢀⣾⡟⢀⣿⡆⣰⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⢿⡿⣿⠆⠀⢻⣻⣿⣿⡿⣼⠇⢓⠌⣻⠀⡃⠀⣂⣹⣶⣿⣿⣐⣿⣿⣿⣿⣿⣿⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣠⣾⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⡠⠷⠌⠁⢷⣾⣿⣿⣳⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠣⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵
⢀⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⠋⠻⡓⢀⣤⡀⣨⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢟⣿⣿⣾⣿⣿⣿⣿⢟⣼
⠘⣿⣿⣇⠀⢼⣿⣿⣿⡿⠃⠀⠀⠁⣜⣿⣿⣿⣿⠻⣿⠟⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⣿⣿⡟⠻⠁⠈⠉
⠀⠈⢻⣿⣠⣼⣿⣿⣿⠁⠀⠀⠀⠀⠘⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⢇⡶⠄⠦⠞
⠀⠀⠀⠈⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⢿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⢀⡀⠀⠀⣀⠘⠻⢛⠛⠋⠉⠉⠉⠉⠁⠐⣒⡄⠀⢄⣀⡀
⠀⠀⠀⠀⠸⣷⡴⠾⠳⠦⠾⠛⠉⠉⠀⠀⠀⠀⠈⠀⠀⠄⡀⠀⠀⠑⠿⢿⡯⠝⣚⣩⢠⠄
⠀⠀⠀⢀⡀⢽⡇⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠀⠃⣀⣉⡴⢀⣤
⠀⠀⠀⠸⠟⠛⠋⠀⠀⣠⣌⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠀⠠⢐⠁⠀⠈
⠀⠀⠀⠀⠀⠐⢠⠤⢿⠀⢦⠀⠀⠀⠠⣲⡦⠤⠄⠀⠀⠀⠀⣰⡠⠟⠓⠁⣰⠄⡿⠤⡄⠂
⠀⠀⠀⠀⡠⠒⠁⠀⠸⠖⠉⠁⠀⠀⠲⠀⢀⡀⠀⠀⠀⢀⡀⠀⠔⠂⠀⠈⠉⠲⠇⠀⠈⠐⠄⡀
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣷⣄⣿⣵⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢦⡟⠈⢻⣿⣿⣿⡀⢚⡠⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢿⣿⣷⣾⡿⣶⡜⣿⣿⣿⣃⢛⣷⣿⣯⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⣿⣿⣿⣿⣿⣿⣿⣾⢻⣿⣿⣿⣫⣿⣯⣥⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠿⣿⣻⣿⣾⣻⣿⣿⣆⣾⠉⣧
⠀⠀⠀⠀⢠⣦⣀⠀⠀⣀⡉⣿⣷⢻⣿⣿⣿⣿⣿⣫⢻⢫⣾⣿⣿⣿⣿⣷⣯⣾⣽⣿⣿⣧⣿⡆⢀⣠⣦
⠀⠀⠀⣰⣿⣿⣿⣿⣶⣽⣇⣘⣿⡇⠻⣿⣿⣬⣿⣿⡃⣿⣿⣿⣿⣿⣿⣿⡇⢿⣷⣟⣟⣿⣷⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣷⢶⣿⣯⣿⣵⣭⣭⣭⣭⣽⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⠯⣻⣹⣹⢿⢹⣟⣿⣯⣿⡯⣿⣋⣷⣿⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⡠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠆⢶⣤⣦⠘⠿⠿⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠠⠀⠀⠀⠀⡼⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠃⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢁⣼⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣴⡿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣵⣿⣶⣴⣥⣤⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣧⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⡿⢘⣫⠁⠀⠀⡀
⠀⠀⠀⠀⠤⠤⠄⠀⠐⠂⠀⠈⢪⣽⠿⡿⣿⣿⣿⣿⣷⣯⣤⣤⣁⡀⠀⠀⢠⣾⣿⣿⡏⠀⣀⠤⠠⠖⠓⠈⠉⠀⢸⣿⣿⠇⢸⡿⠁⠀⠈⠁⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠒⠒⠀⠩⠭⠀⠠⢶⣯⣓⣒⠬⠭⢟⣻⣧⣿⢿⣿⣿⣿⣿⣿⣿⣏⣉⡀⠤⣤⠠⢖⠲⣟⠙⢋⣀⡄⣹⡏⠀⠈⡄⠀⡀⠀⠐⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠠⠥⠤⠥⠤⠤⣤⣤⣝⣿⣿⣿⣿⣳⣲⡦⠤⠤⣤⣤⣤⣤⣠⣀⣀⣀⠀⠠⠤⠑⣚⣢⣊⣽⣽⣥⣴⡿⢆⡁⠀⢣⠀⣀⠛⠛⢃⣀⣀⣀
⠀⠐⠒⠀⢒⣓⣒⣒⣒⣀⣀⣀⣀⠐⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⡄⠂⠀⣾⠤⠤⠄⠀⠀⣀
⠀⠀⠀⠀⣉⣉⣭⢭⣁⠈⠉⢉⣿⣿⣧⣤⣤⣀⡀⢀⣈⡉⣙⠛⣋⠉⠛⠛⠿⠛⠛⠿⡿⠿⠟⣿⡿⠛⠉⡀⢀⣀⣀⡤⣆⡉⠀⠙⠋⠛⠒⠂⠀⠀⠁⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠂⠉⠁⠀⠁⠈⠁⠀⠸⣄⣤⣭⣓⠢⠘⡛⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⡿⣶⣦⣄⢀⡀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢟⣫⣽⠖⠂⠀⠀⠀⠉⣿⣷⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣫⣶⣿⣿⡇⠀⠀⠀⠀⠀⠠⠿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⣿⡿⢫⣾⣿⡿⠿⠿⠏⠀⠀⣠⠔⠀⠀⢴⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⠛⣰⠟⠋⠀⠀⠀⠀⠀⣠⡾⢋⡞⠀⠀⢲⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⣾⡿⢁⣾⣧⠀⠀⠸⡇⠹⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⣾⣿⣿⡇⠸⣄⠃⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⡟⠀⠀⠠⠴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⢢⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠘⠻⢿⣿⣿⣿⣿⠖⢀⡀⠀⠃⢰⣿
⠀⠀⠀⠀⠀⠀⣠⣶⣿⢡⠀⠀⠀⠀⠀⠀⠀⢿⣧⠰⡆⠀⠀⠀⣀⡀⣴⣿⡿⣿⣿⡿⢀⣿⡿⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣴⣿⢻⣿⢸⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⣷⢹⣿⠧⢟⣻⣿⠟⠉
⠀⠀⠀⠀⢰⣿⡏⢸⣿⢸⡇⠀⣠⡀⢀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠋⠀⣿⣾⣿⡿⠁
⠀⠀⠀⠀⢰⣿⡇⢸⠇⠀⣿⣼⣿⣿⣿⣿⣷⣶⣿⣿⡟⠛⠉⣁⡀⢀⣰⡿⠋⠉⠁
⠀⠀⠀⠀⣿⣿⡇⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣧⡼⠛⠁
⠀⠀⠀⣼⣿⣿⣷⠘⠀⢠⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉
⠀⠀⣰⣿⣿⣿⠏⠀⢀⣾⡿⠋⠀⠈⢻⣿⣿⣿⣿⠏
⠀⢜⣿⣿⠟⣡⠀⢀⡾⠋⠀⠀⠀⢀⡄⢿⣿⡿⠃
⢠⡿⠋⠀⠀⠁⢠⣿⠁⠀⠀⠀⠀⠟⢀⣼⠟⠁
⡟⠀⠀⠀⠀⠏⣾⠃⠀⠀⠀⠀⢠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⡄⠀⣿⠃⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠘⣷⡾⠏⠀⠀⠀⢰⣇⢸⡟⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⢛⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢻⡭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠇⠀⠀⠉⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠿⢻⠟⠀⠉⠋⠀⠀⠀⠀⠸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⢻⣿⣿⣿
⣿⣿⣿⢿⡿⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠛⠁⠀⠀⠀⠀⠀⠀⠹⠿⣿
⣿⣿⢥⠟⠓⠠⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣏⠛⠓⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢳⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠹⠟⠛⠻⠛⠉
⠘⠂
⡌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠂⠀⠰⠆⢠⣩⣅⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠄⣀⣀⠀⠁⠀⠀⠀⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⣿⠋⢀⣴⡄⠠⣀⡀⣀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣿⠏⠒⠛⠉⠀⠀⠀⠀⣉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣠⣾⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢿⡄⠀⠀⠐⣦⣿⣧⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠈⠻⣦⡀⠀⠀⠈⠛⠋⠓⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠹⣿⣿⠀⠀⠀⢀⣠⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣦⡀⠀⠉⠁⠀⠀⠀⠈⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠁⠙⠷⣦⣄⣀⡀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⣤⡀⠀⣿⢻⡀⢀
⠀⠀⠀⣨⢿⣷⡿⠿⠿⣿⣿⣿⣦⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⣄⣀⡀⢰⣦
⠀⠀⠰⠁⠀⠀⠻⢦⡄⠀⠉⠻⣿⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⢀⣴⣤⡍⢑⣿⣿
⠀⠀⢀⠀⠀⢠⣶⡆⠀⠀⡀⠀⠈⠻⣷⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣣⡾⣟⢻⣇⠰⠛⠁⣷⣶⠖⠁
⠀⠀⢸⠀⠀⠀⠉⠂⠀⠀⢿⣦⡀⠀⠈⢿⣿⣗⡂⠤⠤⠀⠀⠀⠀⠀⠀⠈⠉⠀⠹⢦⣉⣋⡒⠢⣹⣿⣿⣶⡶⠞⠁
⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⣼⣿⣿⣦⡀⠀⠻⡿⡿⠗⠒⠂⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠈⣭⣍⢓⣾⡹⣿⣿⣷⣄⡀
⠀⠀⣀⠀⠀⠀⣠⣿⠷⣿⣿⣿⣿⣿⣿⠀⠀⠹⣎⡳⣦⠤⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠛⠁⢸⣿⡇⠈⣿⣿⣿⣿⣗
⣷⣾⣿⣷⡿⠛⠉⠀⢸⣿⣿⣿⣿⣿⠟⠀⠀⠀⠙⢿⣎⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠸⣿⣿⣿⣝⡂⠠
⠈⠻⠿⠿⠃⠀⠀⠀⠸⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⢻⣷⣷⣦⣄⡀⠀⢸⠀⠀⠀⠀⡀⠀⣠⣾⣿⣿⡇⠀⠻⣿⣿⣿⣛⡲⠶⠂⠀⠀⣠⣶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠭⠀⠠⠀⠀⢸⡀⢀⡀⢠⣷⣿⣿⣿⣿⣿⠀⠀⠀⠙⣯⠭⠽⢭⣉⠁⢀⣾⣫⣶⣶⣶⣦⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠰⠀⠀⠀⠀⠙⢾⣿⣿⣿⣿⣯⡻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣿⣿⢱⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠹⣿⡿⠿⣿⣿⣿⢿⡏⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃⢸⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠈⢻⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⠀⠀⠈⣿⢰⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡄⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⣿⡇⠀⠻⠿⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣴⢰⣿⡟⠀⠀⠀⣿⢸⡇⡿
⠀⠀⠀⠀⢀⠀⠀⣀⣤⣾⣿⡿⠀⠀⠀⠀⠀⢀⣼⣿⣷⣄⡀⠀⠀⠀⢸⡅⠀⠀⠀⣠⣿⣷⠀⠀⠀⠀⠀⠀⠀⡄⢀⣇⠀⠀⢀⣼⠏⣼⣿⡇⠀⠀⠀⣿⠸⠁⡇
⠀⠀⠀⠀⠁⣸⣿⣿⣿⡿⠋⠀⠀⠀⣀⣠⣴⣿⡿⣿⣿⣿⣿⣿⣶⡶⠊⠀⣀⣴⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣸⠇⣼⣿⣿⣿⣿⢋⣼⣿⣿⠁⠀⠀⠀⡏⠀⠃
⠀⠀⠀⠀⢾⣿⠋⡹⠋⠀⣀⣤⣶⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠐⠁⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⡿⣰⣿⣿⣿⢟⣵⣿⣿⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⣿⠀⠁⣀⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠛⠛⠛⠋⠉⢸⣿⡗⠀⠀⢀⣴⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⣿⣰⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣠⣿⣿⢻⣿⠁⠀⠸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢠⡈⡹⣿⣿⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⣿⠟⣿⢰⣿⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢹⠃⣹⠇⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⣸⣿⠀⣿⣿⡏⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣴⡟⠑⢤⣟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠙⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⣿⣿⠃⠀⠀⢸⣿⣿⠏
⠀⠀⠀⠀⢸⣿⣧⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⡿⣿⡏⠀⢠⣿⣿⡇⠀⣰⠟⣿⡟
⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⡸⢀⠀⣹⡇⢀⡿⣿⣿⣧⠀⣿⣠⣿⡇
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡈⣠⣿⡇⠸⣇⣽⠿⠁⠀⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡇⢘⣿⠁⠀⠀⠀⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠃⢸⣿⡀⠀⠀⠀⠀⠀⡸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⣶⡞⣼⣿⣿⡇⠀⠀⠀⠀⢠⣥⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⢿⣿⣿⣿⡇⠀⠀⠀⠠⡁⠀⣿⡟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢄⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣴⠟⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢉⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡵⢟⣵⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⢸⠀⠀⠀⠀⠀⣀⡴⣻⡷⣫⣴⣿⣿⠟⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢸⡆⠀⢀⣤⠞⣫⠞⣫⡾⣋⡵⢋⡴⣊⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣧⠾⢋⡵⢊⡵⢛⡥⢞⡡⢚⡡⢚⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢹⡞⣩⠖⣫⢔⡭⢒⣡⠖⣩⡶⢋⡵⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⢻⡵⡫⢞⣡⠞⣋⣴⠞⣩⢞⣡⠞⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢞⣻⠀⠀⠀⠀⠀⠀⠀⠀⢋⠔⢋⣠⠾⢋⠔⣫⠴⣫⡵⣛⣵
⣛⡻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣟⠿⣒⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢋⡥⢚⡵⢊⡵⢞⣥⠾⠛⠛
⢯⣝⡓⠦⣝⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⢟⣋⡥⠶⣛⣩⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⢞⣫⢞⣯⢞⣋⣥⣶⠾⠟
⠶⢬⣙⡳⢬⡛⠦⣍⠻⢶⣞⡛⠛⠻⠭⠝⢛⣋⡩⠤⠒⣊⣩⣤⣶⡿⠿⢛⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠯⠞⠛⢉⣡⣴⣶⣿
⢝⡲⢬⣙⠲⣭⡳⢮⡙⢦⣉⠻⣶⣒⣉⣩⣭⡤⢶⣾⠿⠟⢛⣩⠥⢒⣋⡥⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡴⠶⠛⢋⣩⣴⣿⢟
⢶⣬⣑⠮⠙⠦⣝⠲⣝⠳⣬⡳⣌⠛⢧⡀⠐⠉⠥⢒⣚⣩⠥⠖⠋⢉⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⠟⠛⣹⡿⣫⠞
⠒⠌⡙⠷⣦⡀⠀⢅⡈⡛⢿⣯⡲⣕⠈⠻⣄⠚⠋⠉⣁⡠⠴⠖⢛⣩⣵⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⣤⣴⣿⢟⡴⢋⡴
⣛⠳⢦⣄⡈⠙⠳⣦⣌⠲⣄⢝⢿⣾⣟⢮⡈⠣⠈⠭⢀⣒⣤⡵⠞⠫⢥⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣵⢟⡵⢋⡴⠋⣠
⣿⣽⣶⣮⣝⡳⢦⣄⠉⠻⣮⡑⢿⣫⣻⣧⣽⠶⠞⠛⢯⣭⣄⡈⠙⠳⠦⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⢋⡴⢟⣵⠟⢁
⠶⠮⠭⣽⣭⣭⣦⠾⠷⠶⠛⠛⢋⠛⠷⠦⠬⣛⣛⠷⢶⣤⣉⠙⠻⢦⣄⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣸⡴⢋⣴⠟⣡⡴⢟
⠿⣷⢶⣬⡑⠲⠦⣤⣄⣀⣈⠉⠙⠛⠛⠛⠓⠶⠮⣭⣛⡚⠿⠿⣶⣦⣬⣙⣻⣤⣤⡐⢒⣒⠂⠀⠰⢯⣴⢟⣁⣴⣚⣉⣼⠀⡇⢚⣩⠞⣋
⠷⣌⠙⠶⢭⣟⡒⠶⠦⣭⣭⣙⣛⣛⠛⠛⠛⠛⠒⠲⠿⠿⠿⠷⣶⣿⣿⣿⣿⡏⣿⠉⠁⠀⠀⠒⠐⢒⡒⠀⠀⠠⠶⠾⠯⠿⠿⡯⠖⠛⣉
⡳⢬⣝⡶⣤⣉⡙⠳⠶⢮⣬⣀⣛⡛⠿⠿⣿⣿⣗⣛⣛⣲⣶⣶⣿⣭⣽⡟⠀⣧⢸⡀⠀⠀⠀⠀⠀⠨⠄⠀⣐⣂⣤⣶⣶⠶⠿⠿⢛⣛⣉
⣍⣓⠮⣭⣑⣻⡿⢿⣷⣶⣦⣬⣭⣭⣭⣭⣥⡴⠶⠶⢿⣿⡿⣟⣫⣭⠿⢷⣾⣾⡔⣧⣶⣢⣄⣀⣤⣶⡶⠿⠛⠋⣉⣠⣤⣶⡾⠿⢛⣋⣉
⡚⢿⣿⣿⣿⣿⡿⠿⠟⢛⣻⣿⣿⣷⣶⣶⣶⠶⠿⢛⡩⠤⠴⠶⠖⣒⣚⣋⣭⠿⢳⣯⣿⣿⠿⠛⠉⣀⣤⣴⣾⣿⣿⡿⠿⠟⠛⠛⠉⣉⣉
⢿⣷⣶⣮⣭⣭⣭⣍⣛⣿⣿⠿⠟⣛⣋⣡⣤⣴⣒⣂⣀⣀⣉⣭⣭⣭⠵⣊⣶⡿⠟⠋⣉⣠⣴⣶⡿⠿⠛⣋⣉⣡⡤⠴⠶⠚⠛⠛⠉⠉⠉
⢽⠶⠿⣿⣿⣿⡿⣩⣭⠖⠘⣉⡡⠭⠤⠶⠖⠒⠒⠲⠶⠶⠶⣶⣿⠿⠛⢉⣠⣴⠶⠟⢛⣉⣥⠴⠖⠛⠋⢉⣁⣀⣀⣤⣤⣤⣶⠶⠶⠶⠒
⣉⣩⣭⡭⠭⢒⣫⡥⠖⢚⣩⠥⠖⢒⣛⣋⣉⣉⣉⣉⣩⠿⠋⣉⣤⡶⠟⢋⣩⡤⠖⣚⣩⣥⣴⠶⠶⠟⢛⣛⣉⣩⣭⣤⡴⠶⠶⠶⠶⠖⠒
⠶⣴⠶⠟⣛⡩⢤⣒⣭⣥⣶⣾⣿⣭⣭⣭⣭⠿⣛⣭⡴⠾⣛⣩⣴⣶⣿⡿⠶⠛⣛⣉⣭⣤⣶⠶⠾⠛⣛⣉⣭⡭⠴⠶⠒⠚⢛⣛⣋⣉⣉
⠿⠷⠾⠫⠴⠾⠿⠿⠿⠿⣶⣶⣦⣴⣾⡿⠖⣛⣭⠵⣾⡿⠿⠛⣋⣭⡤⠶⠞⣛⣛⣭⡭⠴⠖⢒⣛⣭⣭⠤⠶⠶⠚⢛⣛⣛⣉⣉⣉⣉⣉

⠀⠠
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠄
⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣲⠶⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠂⠃⠀⠑⠢⠂⣠⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠄⠀⠀⠈⠔⠁⠀⠣⣄⠀⠀⠀⠀⢀⠤⠒⣩⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠤⣖⡩⢀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠑⠂⠠⠀⡀⠊⠀⠀⠀⠀⠀⠁⠈⠆⠀⠀⠀⠁⠀⠀⠈⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⢃⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⠀⠀⠐⡀⠀⠀⠈⠀⢀⠀⢠⠁⠀⠁⢀⡀⡀⠘⠀⡤⠀⠀⠈⠩⠍⠑⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⡀⠈⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠃⠀⠀⡕⠀⠀⠘⠢⠈⠁⠀⠀⠂⠂⠀⣀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⠤⠀⠀⡶⣶⣶⣶⡀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠠⠀⢋⠀⠐⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⢀⣠⠒⠁⠄⠒⠀⠒⠒⠀⠈⠈⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠉⠉⠁⠂⠀⢀⡀
⠀⠠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠉⢀⡠⠒⠉⠀⡀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⢀⡀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⣀⡸⠒⠂⠈⡁⢀⣤⡀⣀⣀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠘⠦⢤⣀⠤⠤⠀⣩⡴⠐
⠀⠀⠀⣀⠱⠤⠀⠁⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢂⡜⠉⠀⠁⠀⢀⣀⣁⡠⠀⠀⠀⠙⠶⠶⣒⠐⠒⠂⠤⠤⠴⠚⠉
⠀⠀⠘⠁⢀⡠⠄⠀⠛⠀⡱⡀⠧⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡁⢙⣒⢒⣉⠁⠂⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠤⢶⡤⣀⣀⣀⠠⠴⠒⣧⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⠘⣧⠀⠉⠙⠈⠶⠀⠄⠛⢪⣀⣠⠤⢶⣛⡋⠉⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠈⠁⠀⠸⡄⠈⢋⠀⣤⣇⠀⠀⠀⠉⠂⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣹⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⢈⠃⠘⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⠤⠤⣤⣶⡶⠶⠾⠛⠋⠀⠀⠀⠴⠶⠷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠤⠒⠊⠉⠉⠉⠒⠤⠔⠷⢂⡀⠐⠸⠱⣎⡀⢤⠤⢤⣤⠀⣀⠀⠀⠤⢤⡴⠟⠀⠀⢠
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠖⣄⡐⠂⣡⠤⡏⠂⠞⠓⣛⠛⣟⣩⣭⣤⣤⡶⠖⠊⠁
⠀⠀⢠⣄⡠⠔⠒⠋⠉⠀⠀⠀⠉⠀⠈⠑⠳⢤⡭⢄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⣀⠠⠄⣀⠀⢀⣐⢁⠽⠛⠳⠰⠶⠤⠖⣞⣳⠻⣛⢉⡉⣉⠁⠉⠂
⢠⠔⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡎⢀⡀⠀⠀⠈⠑⠂⠠⠤⠤⠀⠀⠊⠉⠈⠉⠙⢧⠦⣊⢀⠴⠶⠘⠛⠉⠀⠀⠉⠙⠛⠛⠀⠀⠢⣄
⡘⠈⣡⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠖⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢅⣤⣼⣙⢦⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⠳⠽⠧⠼⠐⠶⠆⣠
⠘⡏⠁⠀⠀⠈⠛⢎⡙⡛⠓⠀⠛⠛⣒⢀⡉⠉⠀⠀⢈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠇
⠀⠈⠂⢀⡀⠀⢀⣸⠟⠉⣀⠀⠀⠬⠵⢶⣭⣆⡉⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉