Prostrate Emojis & Symbols

⠀⠀⠀⣀⢤⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄
⠀⢠⡶⠀⠒⠂⢶⡆⠀⠀⠀⣴⡖⠐⠒⠐⣶⡀⠀⠀⢠⡶⠀⠒⠀⢶⡆
⢀⡾⠇⡀⣠⢀⡸⠿⡄⠀⣰⠿⣀⢀⡀⣀⠻⢧⠀⢀⡾⢇⡀⣀⢀⡸⠿⡄⠀⠀⢀⡀⠀⠀⠤⠤⠤⠤⠤⠤⣄⠀⣤
⠘⠒⠒⠒⠐⠒⠒⠂⠁⠀⠛⠒⠒⠒⠒⠒⠒⠊⠂⠘⠒⠒⠒⠂⠒⠒⠒⠓⠀⠀⠚⠛⠒⠒⠒⠒⠀⠒⠒⠓⠚⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠘⣆⢀
⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠰⠿⠷⠦⠤⠤⠤⠤⠤⠭⠶⠯⠸⠷
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣥⣤⣤⣤⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⣠⡄⠀⠀⠀⠤⠀⠤⠄⠤⡀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡆⠀⠀⢀⠀⠀⢀⠄
⠀⠁⠁⠒⠐⠒⠒⠒⠒⠒⠒⠃⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄
⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡄
⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⢶⡆
⠀⣠⣄⠀⠀⠀⠡⠤⠤⠤⠤⡀⢤⣄⠀⠀⠀⠀⢀⡿⠇⠀⣀⠀⠸⠿⡄⠀⠀⠀⣤⡀⠀⠀⠀⠤⠤⠤⠤⢤⡀⣤⡀
⠐⠛⠓⠒⠀⠒⠒⠒⠒⠒⠒⠃⠘⠛⠂⠀⠀⠀⠘⠐⠒⠒⠀⠒⠒⠒⠁⠀⠀⠘⠛⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠘⠛
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠒⠒⠒⠶⠤⠤⠴⠶⠶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠚⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⡱⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡎⡞⠀⢀⡔⠃⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⢠⠙⢄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠋⠀⠲⡀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡜⢄⠀⡇⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⢰⢃⠸⡀⢷⠀⠈⣆⠠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠈⡏⢧⠹⠼⡄⠀⢻⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢰⠀⡑⠦⠳⣀⡟⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢠⢎⠙⢾⡆⣈⠓⢀⠟⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⢠⡀⠀⠀⠆⠀⠀⠀⡇
⠀⠀⠀⠀⠈⡏⡇⢠⠇⠈⢰⠏⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣀⣠⣀⣙⡂⠀⢸⠀⠠⠚⡇
⠀⠀⠀⠀⠀⣟⡃⠹⡄⠀⡇⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⡞⢁⡤⣤⡤⢝⢦⣸⡀⢀⣰⠃
⠀⠀⠀⠀⠀⡇⠀⠀⢳⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⣰⠂⠀⡼⢠⠋⠀⠀⣁⢬⡽⢧⣸⡉
⠀⠀⠀⠀⠀⠙⠮⠤⠖⡟⠳⠠⠀⠀⠀⠀⢀⠠⢤⣒⡥⣞⠁⠀⠀⣇⢸⠀⣠⡎⠁⣸⣀⣴⡚⢉⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠢⠤⠤⢴⣶⡮⣉⠀⠀⠈⠉⠋⠉⠉⢻⣸⠁⢧⢀⡏⠁⠀⠉⠻⣄⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢌⡠⠤⠤⠔⠊⢁⡴⠃⠀⢑⣤⣄⠀⠀⠀⠀⠈⡿⡆⠈⡿⠀⠀⠀⠀⠰⢹⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢗⣦⡐⠉⠀⢀⡴⡻⡪⠞⢧⣀⡀⠀⠀⣇⠹⣄⡇⠀⠀⠀⠀⢈⡏⣇⠗⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠭⢤⣨⠤⠿⣠⣄⠀⠈⠉⢙⠷⣎⠲⣄⣱⣄⠀⠀⠴⣪⣾⢳⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣕⠀⣜⠺⣝⠾⡗⠈⠙⠛⠛⠚⠛⠁⠈⠋⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠓⠛⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠓⠀⠘⠛⠛⠛⠛⠛⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣤⠶⠖⠛⠒⠶⢦⡄⠀⠀⠀⣼⠆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢠⣄⠛⠛⠻⣿⣦⣤
⠀⠀⠀⠀⠀⠀⣠⠞⢁⡞⢉⣤⠶⣦⠀⣀⣀⣀⣀⡰⢦⣼⠏⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⡔⠀⠀⠀⠀⠀⣰⠟⠀⢸⣿⣧⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⡐⠉⢰⡟⣿⣠⢿⡾⠛⠁⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⠎⠀⣠⠂⠀⠀⣴⠏⠀⢀⣾⡿⠛⣼⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⣧⠀⣼⡿⠁⠀⠀⠈⠲⢤⡀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣰⠃⢀⣴⠇⠀⠀⠾⠃⢀⣴⠟⠁⢀⣾⡇⢀⣸⡟⠛⠿
⠀⠀⠀⠀⣀⢸⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣾⠃⣤⣾⠃⢀⣀⣠⣴⣾⡿⠃⠀⣠⣾⠟⠛⠛⠋⠁
⠀⠀⢀⡴⠋⠀⠀⠘⣦⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠙⠛⠛⠛⠉⠉⠉⠉⠙⠛⠻⠿⠋
⠀⠀⠘⢧⠀⠀⠁⠀⡘⠗⢰⡀⠀⠈⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠶⢾⡿⠛⠉⠁
⠀⠀⠀⠈⣧⣠⡴⢋⡀⠀⠀⠉⠉⠠⠀⣠⡾⠋⢀⡀⠀⠀⢠⡾⢁⣠⠟
⠀⠀⠀⠛⠛⠋⠀⢰⡆⠀⠀⠀⠶⣤⣾⣿⣶⡟⠋⠁⢀⣴⠛⠉⠉⠁
⡤⠂⣴⢂⣤⠄⠀⠻⠆⠀⠠⣄⣠⡙⠉⠉⠉⡉⠛⠛⠿⠃
⠀⠚⠡⠎⠈⠀⠀⠀⠀⠀⢀⡬⢝⣵⣿⣿⠜⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣼⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⣾⠣⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆
⠀⠀⠀⣾⣏⠃⡈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣧⣄⠀⡄⠀⡀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣼⣧⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⣹⣟⠉⠛⢿⣿⣿⣦⠤⠋⣉⣿⣷⣌⣉⣀⣠⡶⠖⠛⣲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⢸⣿⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠒⠒⠉⢀⣴⣿⣿⣉⡙⠓⠀⢹⡇⠀⠈⠉⠉⠉⠉⠉⠉⢀⣴⡾⠉⠉⠉⠉⠑⠦
⠀⣼⡟⠇⠀⠀⠀⢀⣀⡶⠶⠾⠛⠋⠙⠛⠉⠉⡉⠀⠀⣠⣾⡿⠛⣽⣿⣿⡿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠑⢶⠒⠶⠖⠃
⠀⣿⡇⠀⠀⠀⣰⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⣾⡿⠛⣠⠞⢩⣿⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⣿⡇⠀⠀⠦⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⣁⠴⠛⠁⠀⠈⣿⣿⡁⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⢹⡇⢀⠘⠀⣿⡗⠈⢤⠀⠀⠀⠀⠀⢠⠾⢿⢿⡎⠁⠀⠀⠀⠀⠀⣿⡟⠃⠀⠀⢸⣦⠈⠻⣦⣄⡀⠀⠀⢸⣿⡇
⠀⠀⠁⠀⠀⢸⣿⡇⠀⠘⠀⠀⠀⠀⠈⠀⠀⠈⠸⣯⡄⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⢸⠉⢷⣄⡈⠛⣿⣷⣤⣀⣿⣷
⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢡⠀⠀⠀⠀⠀⣿⣿⡗⠀⠀⢸⠀⠀⠈⠛⢠⣤⣉⠙⠿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠘⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⡇⡆⠀⣐⠨⠁⠙⠛⢳⣶⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣷⣟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢸⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⢸⡇⠀⠀⣸⠂⠀⠀⢠⣿⡿⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⡄⠀⠀⢀⣀⡄⠀⡀⠀⠀⣸⠂⠀⠀⠀⢠⣯⣿⠀⠀⠀⠈⢧⠀⠀⣿⠆⠀⠀⣼⣿⠁⣤⠀⠈⠃⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⠉⣁⡼⠋⠀⠀⠀⣹⡆⠀⠀⠀⣶⣿⣿⣦⠀⠀⠃⠘⠀⠀⠈⣄⠀⢰⣯⣿⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠸⣿⣿⣿⣅⣠⠃⡀⢠⡇⠀⠀⣿⠀⢸⣿⣿⡴⠈⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡸⣷⠉⣿⣿⣿⣯⣾⡇⠻⣤⡀⠀⢠⣦⠀⢉⣿⣷⣆⡀⢀⣀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⡻⠇⠀⣿⣿⣿⣿⣿⠁⠆⠉⣿⡀⠈⢳⡄⢸⡇⠈⠛⠷⠦⠾⠷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠃⢠⣜⠃⠀⢀⡿⣿⣿⣥⣭⠀⢄⠀⠰⣗⡀⠈⣣⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢾⠋⠀⢀⣾⣿⣿⣿⣿⣿⠃⢶⠀⠀⠛⡽⡀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡿⠆⠀⢰⣏⡿⣿⣿⣿⣿⣿⠆⣸⠀⠀⠀⠱⣯⡆⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠃⢀⠴⡱⠏⠀⣿⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠈⢾⡳⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⣿⡀⢊⠖⠁⠀⠀⣿⣿⣯⣾⣿⠀⢻⠀⠀⠀⠀⠀⠀⢹⣽⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⣿⣟⡴⠋⠀⠀⠀⠀⣿⣿⣿⣿⠏⣦⠈⠀⠀⠀⠀⠀⠀⠀⣿⣮⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣿⢳⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣼⣿⣆⣗⠀⠀⠀⠀⠀⠀⣿⡉⠑⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⡾⢣⡇⢀⠀⠀⠀⠀⠀⣰⠿⣿⡿⠟⠋⠀⠈⠙⢶⣄⠀⠀⠀⠀⢿⠀⠀⠉⡾⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⣎⠱⠈⠃⢠⠀⠀⠀⢀⡾⣁⣾⣿⠛⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣾⠘⠀⠀⠰⢟⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢧⠈⠄⠀⠁⣼⠀⠀⠀⠘⢧⣿⣟⡿⠆⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⣿⠀⠀⠀⠀⣼⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⡆⠀⠀⢰⠇⣀⢀⣀⣇⢸⣟⣛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⢀⣘⡄⠀⠀⠀⠨⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠸⡆⠁⠀⠀⢀⢶⣫⣿⠻⣿⣿⣿⣯⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣛⠙⣿⣯⣷⣦⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⣤⢱⠎⠙⢮⣴⣿⣼⣿⣿⣿⣅⣠⡄⢀⣀⠀⡀⢀⣼⣏⣵⠋⠀⠈⠳⣡⠀⠀⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣌⠇⠀⠀⢠⠙⢻⣿⣿⣿⣿⣶⣬⣤⣿⣿⣷⣼⠿⣿⠟⠱⣆⠀⠀⠀⠛⣠⣌⠁⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⠘⣿⠀⠀⣠⠏⠀⢎⡙⠁⠙⣿⣿⣿⣿⣿⣿⡟⠁⠀⠱⢛⣄⠈⠦⠀⠀⠀⡟⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢻⣄⣿⠀⣠⠃⢀⣘⡜⠁⢀⣼⣯⡍⠹⠏⠉⣋⠙⣦⠀⠀⠈⠻⣷⣤⠖⠀⠀⣿⠉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠙⣿⡏⡀⠈⠀⠉⠉⠀⠀⣿⣿⡋⠑⣠⣶⣶⡍⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⣦⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠛⠾⣇⡀⠀⠀⠀⠀⠀⠙⠿⣿⣾⣿⣿⣿⣷⣶⠶⠀⠀⠀⠀⠀⠀⠀⣰⡸⢧⡀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠰⠄⣻⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠇⠀⠀⢉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡠⠜⢋⡛⢿⡦⢀⠀⠀⠀⠀⣾⣿⠛⢀⠜⠁⠙⠢⡀⠀⠀⠀⠀⢀⢇⠙⢀⡛⠃⠈⠉⠉⠙⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣷⡶⠾⣷⣤⡻⠟⢧⠀⠀⠀⣿⡟⠀⠋⢠⣶⣷⠀⠘⠇⠀⠀⢀⠋⣸⠁⢸⡃⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠿⠋⠀⠀⠀⢿⣷⠿⡜⢣⠀⠀⣿⣇⣠⡀⣸⣿⣿⠀⠀⢸⠀⠀⡾⠀⡷⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠹⡆⠀⠀⠀⠀⢸⣏⠀⢹⡶⣦⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⠀⢸⣷⢸⣇⠰⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠇⠐⠂⠀⠀⠀⠀⢸⣿⡀⠘⣧⣀⠀⣿⣿⣿⣿⣿⣿⡧⠀⠐⢾⠀⢸⣇⢸⡏⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣃⠀⠀⠀⠀⠀⢿⡷⠆⢻⠁⠀⢹⣿⣿⡿⣿⣿⠇⠀⠀⢴⠀⢼⡇⢈⣿⠆⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣬⣯⣽⣷⡖⠀⠀⠀⣸⡄⠀⠙⣤⠀⠀⠻⢿⣷⡿⠛⣀⠀⠀⣀⡔⠸⣿⠘⠇⠀⢠⣿⣿⠋⣶⡄⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡋⠉⠃⢀⠀⠙⣿⣶⠶⠁⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢹⠇⠀⠘⠓⠒⠛⠛⠻⢿⣿⣿⣷⡤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢷⣾⣿⣾⠗⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣧⣠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣔⣶⣦⣤⣶⣿⣿⣷⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢻⣿⣿⢿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡤⠞⣋⡴⢡⣿⣿⣾⣿⣿⣿⣿⣿⡯⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠞⠊⠉⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⢹⣇⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣧⣤⣤⣴⣯⣿⣽⣿⣿⣿⣿⣿⣿⢿⣿⣶⠀⠐⡟⠀⠀⠀⢀⢧⠠⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣯⠀⢽⣿⣶⣾⣿⣿⣿⣿⣿⡏⢩⣿⣿⠀⢰⡇⠀⠀⠀⠈⠚⠀⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣦⠜⣿⣿⣿⣿⣿⣿⣿⡿⠁⣼⡿⣿⡀⠈⠇⡄⠀⠀⢀⣠⣦⡟⣿⣴⣿⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⣿⣿⣗⠸⣿⣿⢿⡇⠀⣸⣧⣴⣾⣿⢭⠼⠃⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣥⣬⣥⣶⡟⠋⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠝⠻⣿⣿⡿⠿⠟⠉⠀⠀⠈⠉⠁⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣤⠴⠶⠷⣦⣠⣤⡴⠖⠒⠙⣲⢦⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣉⡤⠖⠒⠚⠛⠛⡋⣀⡶⢶⡀⠙⠳⠿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠿⠛⠓⠀⠀⠀⠀⠸⢧⣈⣙⠟⠧⣄⡀⠀⠈⠑⢦⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀⠀⠉⢦⡄
⠀⠀⠀⠀⠀⠀⠘⣟⣿⣿⠛⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢛⡻⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠈⠻⠆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣶⡾⠟⠋⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠛⠯⠆⠀⣲⣀⣨⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢸⣇⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣰⠆⢀⣿⠃⢄⠀⠀⣀⣤⣔⠛⠉⠀⠀⡈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⢛⠁⠀⠀⠈⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⢰⣇⣠⡾⣾⣠⠴⠒⠋⠁⠀⠙⢧⡀⠀⠀⠓⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡞⠀⠀⠀⠀⠀⠀⣀⢈⣑⣀⡀⢀⣀⡉⠓⠦⠷⡀⠀⠀⢀⣟⣛⡿⠷⣍⢁⡄⠀⠀⠀⠘⠀⠀⢹⠀⣠⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠾⢧⡤⠖⠲⣟⠻⣿⡿⠋⠁⣼⠃⠈⠓⠒⠾⢿⣶⣾⣷⢶⡿⣁⣠⣶⣤⣿⠀⠀⠀⠠⡄⠀⠀⠀⣿⠀⠀⠀⢀⡄⠘
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⡾⠔⠊⠀⠙⢦⡈⠛⢦⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠋⠉⠉⠛⣿⠀⠀⠀⠀⠀⠀⠀⣤⣿⠀⠀⠀⠈⠁
⠀⠀⠀⣠⠖⠀⠀⠀⠀⣀⠴⠚⠱⠀⠀⠀⠠⣄⢀⠙⢦⡘⠿⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠈⣿⠀⣠⡄
⠀⣠⡾⠃⠀⠅⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃⠀⣷⡀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠁⠀⡀
⣾⠋⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠈⠰⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠛⠃⠀⢰⡇⠀⠀⠀⠀⠒⡟
⠘⠳⣤⡀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠉⠙⠶⢤⣄⡀⣄⡀⠀⠻⡌⣧⢤⡰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠠⡄⠀⣠⡶
⠀⠀⠀⣽⣷⣦⣄⡀⠀⠀⠀⠀⠦⠀⠀⢠⣄⡺⠾⠛⠻⠦⣤⡀⢷⡘⣦⠀⠐⠿⣶⢄⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⣠⠟⠋⠉
⠀⠀⠀⣽⣿⠏⠂⠙⠷⠶⣄⣀⣤⠀⠀⠀⠀⡉⠙⠛⠲⠶⠾⣯⣿⣛⠺⣭⣤⠀⢻⠀⠉⠒⠀⠀⠁⠀⠀⠀⣀⣤⠤⣀⡠⠞⠛⠛⠋
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀⢀⣻⣀⣀⣤⠴⠶⠞⠛⠉⠉⠙⠒⠀⠍⠛⠛⡛⣛⣿⠛⢛⠿⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⡿⡏⣹⣷⠞⣿⡝⢋⣵⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⠛⡻⢷⣿⣿⣿⣞⢶⣷⢿⣽⣱⡛⢮⣿⣿⣿⣒⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿⣏⢽⣣⣣⠘⣽⣿⣿⣙⢋⡿⢚⣿⣿⣿⣿⣫⢡⣿⣟⠀⣼⣾⣟⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣽⠞⠾⣟⡛⣿⢧⠹⣿⢿⣿⣿⣷⢼⣻⡅⣿⣷⣧⡏⢸⣿⡁⠀⣽⣯⡋⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣹⣿⣄⣾⣹⢋⢿⡾⡆⣿⣿⣧⠻⣿⣶⣿⠡⣿⡦⡿⡷⠌⣿⣿⣦⠀⢿⣗⡿⣜⠽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⠀⢿⣯⠼⠻⢿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⢰⣿⣷⢾⣿⡅⣷⡿⠙⠃⢨⡟⠿⢼⣧⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⠀⢸⣧⠾⢿⣾⣿⣜⡇⢸⣭⢁⣿⡁⣽⣿⢰⣿⣟⣿⣿⣼⣿⣷⣤⡄⠹⣻⠇⣴⣉⠘⣯⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⢴⠀⣿⡝⣿⣯⠶⣾⠃⣼⠁⢸⠉⣥⣇⣿⠨⣿⣿⣯⡿⣻⣷⣿⡁⠠⣄⣿⡅⢿⣇⣴⢁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣏⠙⠀⠀⢻⣿⡿⣿⢺⠿⡇⢹⡁⣾⡾⢿⡟⢿⡁⢿⣿⣟⡸⢛⡭⠿⣿⣆⠈⢿⣷⣾⣤⣿⣫⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢈⡄⡤⢀⣟⣿⣇⠙⡏⠘⡇⠀⣿⠝⡏⠂⠿⢿⡃⣾⣦⣄⠛⢃⣠⣤⡘⠛⠠⣞⣻⢿⣿⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⡀⢰⣾⣿⡗⠀⠷⣽⡼⣧⠀⣷⣆⠘⠀⣄⣸⠃⢸⣿⢿⣿⣿⢿⣿⣿⠄⢻⡽⠗⣰⣿⠛⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡇⣸⣿⣿⣷⠀⠰⣿⣾⣿⠀⢸⣾⣷⣼⣏⣟⠀⣰⠐⣿⡿⠞⣠⡟⢉⠶⠾⠕⣾⠞⢳⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢿⡇⣿⣿⣿⠝⠃⢀⣿⣿⣿⡂⢈⣿⣿⣿⣿⠇⢰⣷⡀⢷⢛⣾⠏⢠⣶⢶⡟⠛⢀⢒⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣇⢼⣿⣿⣦⣄⠈⠹⣿⣿⣿⠀⣿⣿⡿⣿⠆⣹⣿⣿⣦⣹⢏⣤⣼⡷⣍⣷⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣿⣿⡆⠀⠻⢿⡅⠀⠈⡁⡆⡇⢀⣬⣠⣿⠏⠀⠀⢨⠬⣺⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣿⣦⠓⣾⠁⠿⠛⠗⣾⠛⠿⠏⠶⣶⢾⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣧⣺⣦⣶⣶⣖⣴⣇⣴⣶⣶⡷⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣛⣛⡷⢮⣙⣿⡇⢠⠾⠟⢻⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⢦⡹⢿⣷⣥⠐⠲⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⣿⣿⡇⠀⠀⣸⠟⠁⠀⠀⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⢦⡴⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⢧⡄⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠡⢀⡤⠀⠀⠀⠀⠰⠄⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡄⠀⠉⠀⣶⣄⠀⢀⡀⠰⠿⠟
⠀⠀⠀⠀⠀⢠⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠈⠿⣷⡌⢉
⠀⠀⠀⠀⠀⢸⠈⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠋⣀⠀⠀⠀⡀⠀⠈⠁⢸⡄
⠀⠀⣀⡀⠠⡌⠠⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠘⢿⣦⡰⠆⠀⠲⠾⠛⠁⠀⠀⠀⠀⢧⠀⠀⡀
⠀⠀⠀⠙⠓⠀⢀⢀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠤⠆⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⢀⣴⣿⠀⠀⠀⢀⡀⠂⢠⣦⡴⠃⠀⠀⠀⠀⠀⠀⠇⢀⣤⣴
⠀⠀⠈⠻⢶⣶⡶⠤⠀⠠⡾⢠⣿⠿⠃⠀⠲⣄⠈⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠈⠁⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠘⠷⠦⠀⠄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢠⢀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⣤⠀⠈⠆⢠⡄⢡⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⢀⣙⠁⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⡠⢀⣤⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠊⠐⠉⠛⠻⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⠀⢀⡀⢀⡀⠀⡀⠀⣀⠀⣀⠀⢀⠀⣀⠀⣀⡘⢆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⣀⠀⣀⠀⣀
⡏⡇⣏⣑⢸⢹⢸⡉⠀⡏⢸⢉⡇⣏⡇⢹⠀⣏⠀⠈⠛⠀⢳⠂⢸⣄⠇⡏⡇⢹⠁⣇⣬⠏⡃⠀⣋⠀⡏
⡏⠁⣿⡇⢸⢸⢠⢹⠀⡇⢸⣿⠀⡏⡇⢸⠀⡏⠁⠀⠀⠀⡼⢇⢸⠸⡇⡁⡇⢸⠀⣿⢻⡘⡉⠀⡏⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀⣀⠀⠀⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠃⣿⣿⣿⣿⣿⠛⠉⠁⣸⣿
⠀⠀⢠⣾⣿⣷⣦⡀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⣾⣿⣿⣿⣿⣿⣶⣦⣄⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣄⣀
⣾⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠻⠛⠋⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠉⠉⠉⠉⠙⠉⠁

⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢢
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣠⠟⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⡾⠃⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⣴⠿⠷⠛⠙⣿⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣀⡀⣠⣤⣤⠶⠖⠚⠋⠉⠁⠀⢀⣀⣠⣤⡴⠿⠟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠖⠛⠛⠋⠀⠀⢀⢀⣠⣤⣤⡶⠿⠛⠛⠋⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠴⠶⠚⢛⡿⠁⣠⠞⠁⠀⠀⠀⢀⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⣀⡴⠶⠟⠋⠉⠀⠀⠀⣴⠋⢀⣼⠃⠀⠀⠀⠀⠀⠘⠁⠈
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣴⠟⠉⠁⠀⠀⠀⠀⠀⣤⡞⠁⣠⠟⢁⠀⠀⠀⠀⠀⠀⡆
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠅⠀⠀⠀⠰⢆⠀⢀⣴⠏⢠⡜⠃⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠈⣠⡟⠃⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢣⡾⠉⠰⡀⠀⢣⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠁
⠀⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠛⢁⣴⠏⠀⠀⠀⠈⠀⠀⠡⠀⠀⣀⠀⠀⠀⡇⠀⠀⢀⠐⠊
⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢀⠀⡀⢀⣼⠋⢛⡟⠉⠉⠀⠀⠀⠀⢠⣀⠀⠘⢂⠀⡀⠀⢰⡧⠄⠉⠁⠀⠀⠀⠀⠀⠂
⠀⢸⡄⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠛⣧⠛⡁⢠⠋⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠈⠁
⠀⠈⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⣿⠋⣰⡿⠋⠀⠀⠀⠀⠀⣄⠠⠀⠀⠀⠀⠀⠀⠀⠘⠣⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠚⢂⡽⠋⢠⡿⠛⢿⣣⡄⠀⠒⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄
⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸⡇⠀⣼⢧⣸⠏⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⣤⠀⠀⠀⠠⠀⠀⢀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⡇⣠⠻⣾⠏⠀⠀⠀⠀⠘⣷⠒⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⣠⠀⠀⠱⠄⠈⠀⠀⢠⠞
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣷⡞⢀⡻⠀⠀⠀⠀⠀⠀⢿⡀⣧⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧⣸⣇⣀⠀⢀⡀⣀⠀⠘⢀⡀⠀⠀⠀⠀⠀⢤⠀⠀⠉⢀⠀⠀⠀⢀⠀⠀⠀⠛⡀
⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠘⣿⡏⠁⠀⠈⠉⠙⠋⠋⢠⡿⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⣀⠃⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⢸⡗⢠⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⠀⠀⠀⠀⠃⡄⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠉⠁⠀⠐⢢⠀⠐⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⠀⠂⣀⠀⢀⠀⠠⠤⠀⠒⠒⠛⢨⣧⠀⠀⠸⢂⣤⡟⠂⠘⡀⢦⡀
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢽⡆⠀⠀⠀⠀⠀⠁⣿⠃⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⢀⣠⣧⠤⣧⠴⠏⠀⣵⣤⡀⠡⡀⠙
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣟⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢀⡤⢲⡦⠐⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠠⠆
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠙⣦⠀⠀⠀⠘⡟⠁⠀⣀⠀⠀⠁⠀⣛⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠛⠀⠀⠀⣴⡟⠀⠀⠀⠀⣠⠞⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣾⣉⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠐⢉⠽⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣹⠇⠀⢠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⠉⠛⠛⠃⢹⣿⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⣏⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣿⠀⡀⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢻⡆⠁⠘⠒⢀⠀⠀⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣘⣃⣀⣈⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⢀⡀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⢀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠃⠀⠀⠀⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣼⣿⣿⡿⠋⠀⠀⠀⠀⢀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠀⠀⠀⣀⣴⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠉⠀⠀⠀⢀⣤⣾⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⣈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣍⠉⠋⠉⠉
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠎⠿⣦⡀
⠀⢀⡀⠐⠛⠃⢠⡄⢶⣦
⠀⠸⡷⠰⣦⡄⠀⠿⣤⡉⠈⢳⣄
⠀⠀⠁⠐⠘⢧⡀⢀⠀⠹⣶⣄⠈⠳⣦⡀
⠀⠐⠀⣄⠀⠈⠓⡀⠁⠀⠘⣿⣿⣄⠀⠻⢶⣄⠀⠀⠀⠀⠀⠀⣀
⠀⡇⣼⡏⢠⣶⣄⠙⢦⡢⡀⠈⠻⣿⣷⣦⡀⠹⣷⣄⠀⠀⠠⠊⢹⡇
⠀⠀⣿⡇⠀⢩⣽⣷⠀⠹⣟⢦⡀⠈⢻⣿⡷⡄⠸⣿⣆⢸⡇⠀⠘⣄
⠀⢸⠁⢀⡀⠘⢿⡽⣷⣄⢈⣷⣝⢶⡀⠹⣿⣆⠀⠉⢿⣿⡗⡗⢄⣻⡄
⠀⠇⠀⣾⣿⡄⠈⢿⣜⢿⡁⠀⢙⣿⣧⠀⠘⢿⣦⡀⠀⠿⠇⠹⡈⠹⣷⡀
⢸⠀⢰⣿⢿⣷⡀⠈⠹⣿⣿⣆⠀⠈⢻⣧⠀⠈⣿⣿⣦⠀⠀⠠⣵⣄⠙⢷⡀
⡇⠀⠈⣿⠘⠿⣷⡀⠀⠘⢻⣞⣿⣄⠈⢿⡆⠀⠘⠉⢦⣄⠀⠀⣿⣿⣕⢾⣿⣄
⡇⠀⠀⣿⠃⠈⣿⣷⡀⠀⠈⢻⡞⢿⡄⠀⠁⠀⠀⠀⠀⠙⠗⢀⠘⣧⠙⣷⠈⠛⢷⡄
⠀⠀⢀⣿⣇⠀⠸⣿⣧⠀⠀⠈⣿⣎⠟⠀⠀⠀⠀⠀⠀⢀⡀⢸⣇⠈⢳⣟⠀⠀⢰⣍
⠐⡄⠀⠀⠻⣧⠀⢻⣿⡆⠀⠀⠸⣿⡆⠀⠀⢀⠀⠀⠀⢸⣷⡀⢿⠀⠀⢹⡜⣆⠈⣿⣆
⠀⠘⢇⠀⠀⠘⣷⣼⣿⣿⡀⠀⠀⣿⡃⠀⠀⢸⣦⣤⡀⠈⢿⣿⡄⠀⢀⡀⠠⣿⣆⢹⣿⡄
⠀⠀⠈⢧⠀⠀⠘⣱⣿⣿⣇⠀⠀⢸⣿⣦⣆⠈⢻⣿⣿⣆⠸⣿⣿⡀⢸⣧⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠈⣷⠀⠀⠀⠹⣿⣿⠀⠀⠈⠉⠻⣿⠃⠀⢻⣿⣿⣦⠹⣿⣷⡌⢿⣷⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠉⠀⠀⠈⢿⣿⣿⡇⠙⣿⡛⠀⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠙⣄⠀⠀⠈⠙⣿⣆⠀⠀⠀⠀⢾⣷⣦⣀⠻⣿⡇⠀⣾⣇⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠀⠉⠙⢻⣧⡈⠀⠀⠿⠛⢃⡀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⣿⣿⣆⢠⣄⠀⠀⠈⠛⠃⠀⠀⢀⣴⣿⡇⠀⠈⠙⠻⠿⣿⣿⣷⠆⣴⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠐⣿⣿⣿⣷⡛⢿⣖⠀⠀⠀⠀⠀⠉⠙⠛⣧⣄⡀⡀⢸⣶⣤⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⠟⣋⣙⡻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣯⣄⣀⠀⠀⠹⢿⣿⣿⣿⣦⣝⢿⣦⣀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⠋⠁⠀⠀⢾⣿⣿⣿⣿⣿⡿⢡⣾⣿⣿⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣸⡟⣿⣿⣿⣷⣀⠀⠉⠹⣿⣿⣿⣿⣷⣌⡛⢿⣶⣶⣿⣛⠀⠀⠀⠀⠁⢀⠀⠀⠀⢨⣿⣿⣿⣿⠟⣱⣿⠟⠉⠀⢀⣴⠶⠃
⠀⠀⠀⠀⠀⠀⡟⠈⢿⣿⣿⠻⣷⠂⠀⣤⣿⠿⠋⢋⣵⣿⡿⣶⣮⠻⣿⣿⣤⡀⠀⠀⠀⠘⠓⠶⠶⣄⢙⣿⣿⢃⣴⣿⠃⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⡇⢀⣿⣿⣿⠀⠈⠓⠀⠀⠠⠴⣾⣿⡿⠏⣰⣿⣿⣷⣮⡛⡿⠃⠀⠀⠀⠀⠀⠀⠀⢻⡌⠀⢡⣿⠟⠁⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠆⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⣰⣿⣿⣿⣿⣿⠃⡀⠀⠀⢠⣆⠀⠀⠀⠀⠈⢷⠀⣾⠇⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢊⣠⡖⠀⢩⣿⣿⣿⣿⠟⢸⡇⠀⠀⣿⣿⣦⡔⢀⡀⠀⠈⢠⣿⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⡘⠻⠿⠿⣿⣟⣛⣤⣴⣾⣿⠏⢀⣴⣿⣿⣿⣿⠏⠀⢸⡇⠀⠀⠻⣿⣿⣿⣿⢖⣠⡀⣸⡇⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣴⣿⣿⣿⣿⣿⣿⡿⠃⠀⠈⢡⣿⡿⠟⠁⠀⠀⠈⢷⣤⣤⣼⠶⠍⠉⠁⠉⠀⢀⣿⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣭⣀⣀⣠⠄⠀⠀⡴⠟⠋⠀⠀⠀⠀⠀⣠⣸⣯⣵⡖⠂⠀⠀⠀⠀⢀⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⣿⣿⣥⠤⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠒⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣤⣤⠤⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⣰⡾⠿⢿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠐⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⢀⠀⠀⠀⠀⠀⣠⠞⠉⠹⣦⡀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣁⣴⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠋⠈⠛⠿⢿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣆⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⠻⠿⢿⡿⠿⠿⠟⠋⠁
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⢀⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⢀⣠⣴⣶⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠋⠀⠀⣠⣶⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠿⠛⠋⠉⠛⠛⠋⠁⠀⣀⣤⣤⣤⣤⣄⡀⠀⠈⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠉⣠⣤⣤⡄⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⢟⣵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣏⣤⣾⣿⣿⡿⠃⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⡀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢀⣴⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⡿⠃⠀⣰⣿⡆
⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⠙⠛⠉⠁⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣀⣀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢠⣾⣿⣿⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⣴⣿⣿⡿⠋⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣶⣿⣿⣿⣟⣠⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⡀⠙⢆⠱⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠑⠀⠀⠑⠄⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠄⠀⢦⡀⠀⡆⢠⠄⢦⡎⡏⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣦⠊⠢⠀⢵⠀⠀⠈⢆⠀⠑⣇⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⡆⠀⡁⠀⠀⢄⠀⠀⠣⠀⡈⠦⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠈⢆⠈⠰⠀⠀⢣⠀⠄⠁⠀⢄⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠱⡄⢺⡀⠀⠁⠀⠇⠀⠀⠀⠀⠈⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠈⡞⠃⠀⠔⠀⠀⠀⠀⠀⠀⡀⠀⢣⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣧⠀⠀⠀⠀⢆⠀⠂⠀⠘⡀⠀⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠀⠀⠈⠢⠀⠀⠀⠐⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠂⠤⠀⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠈⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠚⠛⠑⣄⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠈⠳⣄⡰⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⢻⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⠏⠀⠀⠀⠀⠀⢀⣾⠣⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡄⠀⠀⠀⢀⠔⠃⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡚⣿⣿⣿⣿⠿⣟⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣎⢻⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢻⣆⠹⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠹⣶⣝⢿⣿⣿⣿⡿⠟⢛
⠀⠀⠀⠀⣰⣻⣖⣲⣾⣿⣷⣶⣷⣾⢒⣒⡒⠶⢝⣆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡟⢋⣡⣤⠴⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠓⠉⠁


⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢪⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⡶⢿⡿⣷⣦⣀⠀⠀⡀⠔⠊⠡⠖⠷⠀⠀⠀⠀⠀⢰⣠⡆⢤⣄⣀⠀⠀⢀⣠⠤⠒⠚⢁
⠀⠀⠀⠼⠷⠶⠮⠭⠽⠭⢥⡄⠤⠴⠾⠿⠿⠿⢧⠀⠀⠀⠀⠶⠧⠤⠾⠿⠷⠤⡶⠿⠭⠭⠭⠤⠬⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡵⠚⡛⠲⣯⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠘⢀⣠⠷⠶⢾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠠⠄⠀⠤⠄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⣄⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣨⠁⠠⠞⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠋⢠⣀⡀⠙⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢻⡀⠀⡏⣿⣧⠀⠀⣼⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡄⠀⠓⢰⠇⣿⠸⠀⠚⠁⢀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡗⠀⣠⠉⢀⡏⠀⠀⣀⠀⢸⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣀⡼⠟⠋⠁⣠⠞⠀⠱⣄⠈⠳⢾⣧⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⣠⡴⠛⠁⠀⠀⠀⠈⠳⣦⣀⠈⠙⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠶⠛⠉⠰⡀⠀⠀⠀⠀⠀⠀⣴⠙⠳⠶⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⡀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠛⠛⠒⠒⠀⠀⠨⣽⣷⣶⣶⠶⠶⠾⠿⠛⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⠈⠉⠙⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⠛⠻⠶⢶⣤⣤⣄⣀⣀⣀⣤⠤⢴⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⢤⣤⣤⣴⡶⠾⠟⠛⠉⠉⠀⠀⠀⠈⠙⠛⠳⠶⠶⠶⠶⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡽⠆⢀⡤⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⢧⣤⣄⠳⣿⠟⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢠⡁⣄⡹⣶⠀⠈⢛⠒⠊⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠧⣉⠁⠈⢷⡀⠀⠙⢆⠀⠈⢦⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠙⡄⡄⡘⡧⠀⠀⢻⡆⠀⠀⠹⣄⠀⠳⣤⠯⣠⠟⠁⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⣷⡈⠁⠘⣦⠀⠀⠻⣦⠀⠁⠘⣄⣀⠈⢷⣿⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢺⣿⣷⡀⣀⠘⣆⠀⠀⠘⣷⠀⠀⠘⢯⠀⠈⠻⡖⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⢾⣿⣷⡹⠀⠙⢦⠀⠀⠘⣧⡄⠀⠈⢳⣄⠀⠹⣆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⢀⢸⡿⣿⣧⠃⢀⠙⣾⡄⠀⠈⢿⡆⠀⠀⠁⠀⠀⠹⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢻⣷⣴⣿⣯⢻⡁⠘⢷⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⢸⣿⣿⣿⣿⣆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠽⠿⠟⠉⡉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⡘⠟⢿⣿⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣜⡖⠁⢁⠀⢺⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣦⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠨⠅⣙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣀⠈⠩⢭⣿⣷⣔⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡾⢁⣿⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢫⠛⣿⣿⣅⢦⡀⠀⠀⠀⠢⠀⣤⢾⠔⠟⠛⠋⢠⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣲⣿⠀⠀⠀⣀⠿⣿⣿⣯⢻⣞⣿⣦⠴⠋⠀⠀⠀⠀⢀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡺⠿⢿⣤⣴⠞⠁⣸⣿⡟⢛⣿⠿⠋⠀⠀⠀⣀⠄⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣄⢠⣴⠟⢁⡴⡋⣾⢃⠀⠀⢀⣀⣀⣤⠶⡛⠁⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣦⡼⠟⠋⠀⠀⠀⠺⣿⣿⠀⣓⠚⠛⠍⢀⣠⠖⠚⠉⠩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢁⡀⠀⠠⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠸⣿⣿⠇⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⡷⠀⠘⣿⡄⠀⠀⢹⡏⠀⠀⢠⣿⠃⠀⣶⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⡿⠟⠋⠀⠀⠀⠘⡇⠀⠀⠘⠃⠀⠀⢸⠃⠀⠀⠈⠙⠿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⣴⣾⣷⣾⣷⣦⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⢀⣀⣀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀⣀⡀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠹⠿⠛⠛⠉⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠙⠛⠿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠹⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡀⠀⠀⠀⣴⢃⡼⣄⢳⡄⠀⠀⠀⣠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣙⠦⡀⣸⢃⠎⠀⠈⢆⢻⡀⡠⠞⣡⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⡇⡏⠈⠁⢸⣇⠀⢠⣄⣤⠀⢀⣿⠀⠉⠈⡇⣿⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣛⡲⢦⣸⡀⠀⠀⠀⠸⠀⠀⣿⣧⣿⡇⠀⠘⠀⠀⠀⠀⣽⣤⠶⣚⡋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⠉⠀⠈⣷⢄⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⢀⢴⡏⠀⠈⢹⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⠀⠀⠀⠘⢆⠁⠀⣠⣤⣴⣾⣧⣿⣶⣤⣤⡀⠀⢁⠞⠀⠀⠀⠸⢸⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⢧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⢏⣀⣀⣀⡀
⠀⠀⠐⢾⡛⢉⣩⡭⠭⠍⠉⠙⢦⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⠞⠉⠉⠭⠭⣭⡉⠙⣻⠖
⠀⠀⠀⠀⠙⢦⡙⢆⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠏⢸⣿⣿⣿⣿⣿⣿⠈⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢀⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣉⡀⢸⣿⣿⣧⣿⣿⣿⠀⣈⣙⣿⣿⣿⣿⣿⣷⡀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣳⣤⣉⡁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣉⣡⣴⣋
⠀⠀⠀⠀⢀⣴⣞⡉⠴⣄⠀⠁⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠁⢀⡴⠌⣙⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀⡀⠀⠀⠀⠀⠈⠉⠻⠿⠟⠛⠿⠿⠿⠿⠉⠁⠀⠀⠀⠀⣀⣠⠴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠚⠋⠉⠁

⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠈⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠶⣄⠀⠀⢀⣴⠿⠁⠀⠀⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢻⡀⠀⠀⠀⣀⣀⣀⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠄⠀⠀⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣾⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢡⠇⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⣄⠀⠀⠀⢷⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⢀⡴
⠀⠀⠀⠸⡄⠀⠀⠸⡆⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⢷⠀⠀⠀⢻⡀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⡼⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠈⣧⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢩⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⠃⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⡄⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣰⠇⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢷⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡰⠋⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠛⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠉⠀⠀⠀⠀⣎
⠀⠀⠀⠀⠀⣠⠖⠁⠹⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⡇⠐⣆
⠀⠀⠀⠀⡼⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣰⢃⠄⠀⠀⠀⠀⠀⠀⢠⠃⠀⠘⢧
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⡘⣆⠀⠀⠰⡷⢋⠆⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⣧
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢷⣜⣦⠀⠀⠀⡼⢀⡶⠀⡄⢀⡀⣸⠁⠀⠀⠀⠀⢸⡆
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣄⠸⡄⠉⠀⠀⢠⡇⣸⠇⢼⠃⣾⢁⠇⠀⠀⠀⠀⠀⠀⣷
⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⢻⠰⡆⢹⡆⢻⡄⢡⠀⠀⠀⠸⠠⢹⠠⢿⢰⢹⠸⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⣼⠀⠀⠀⠀⠀⡴⠹⡆⠀⠈⣇⣿⡈⡿⡌⣿⣼⠀⠀⠀⠀⠃⠨⠜⠀⠌⠈⠀⠀⢰⠣⡄⠀⠀⠀⠀⣧
⠀⣿⠀⠀⠀⠀⠀⡇⠀⢿⠀⠀⢩⡇⢣⢳⢳⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢻⡀⠀⠀⠀⢹⡄
⠀⡏⠀⠀⠀⠀⢸⠇⠀⠸⡆⠀⠀⠀⠘⠞⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠘⡇⠀⠀⠀⠈⣇
⢰⡇⠀⠀⠀⠀⣼⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢻⠀⠀⠀⠀⢻⡀
⢸⠇⠀⠀⠀⠀⡟⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⡆⠀⠀⠀⠸⡇
⢸⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⢿
⡏⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠸
⠃⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸
⠀⠀⠀⠤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠾⠟⠛⠛⠛⠉⠉⠉⠉⢉⣛⠛⠛⠻⠷⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠉⠉⢛⡛⠛⠂
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠓⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠀⠐⠂⠀⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡠⠤⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡔⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡜⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⣸⠁⠎⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣦⣄⢸⠀⣤⣾⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢀⡏⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠸⡀
⠀⠀⠀⠀⡞⠀⢀⣼⣤⣤⣄⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣷⡄⠀⠀⢧
⠀⠀⠀⣸⢁⣰⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⣛⣛⡛⠙⢿⣿⣿⣿⣿⠋⢉⣉⣙⣛⣛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠶⠀⠸⡄
⠀⠀⠰⠧⠴⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠦⣄⣉⣉⣉⣡⠴⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠓

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⢉⡙⢩⡍⢿⣿⣿⣟⠏⠈⠀⠀⣠⣿⣝⠻⢍⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣄⣲⠄⠀⠀⠀⠀⣀⣶⣶⣦⣤⣤⣄⡴⠼⣯⠿⠄⠰⠓⠀⠛⠛⠛⠑⠐⣿⠟⠻⢿⣿⣿⣿⣟⠛⣀⡈⢙⠻⠟⠛⣿⣿
⣉⣉⣙⣁⡀⠀⠀⠀⠀⠉⠻⠿⣿⣿⠛⠿⣿⣶⡂⠀⠀⢠⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⣀⢀⣀⣀⣐⣻⣿
⠈⠉⠉⠉⠉⠋⠉⠉⠀⠀⠈⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⢿⣿⣿⣿⠆⠀⠀⠁⠀⠀⠈⠀⠐⠂⠚⠛⠛⠋⠉⠉⢉⣉⣉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⡀⠀⢸⣿⠿⣿⣿⣿⣿⣿⢻⣿⣀⣀⣀⣴⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠹⡿⠋⠀⠀⢀⠀⠚⠉⡛⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⠀⠀⠠⠱⡱⣠⣠⠃⠀⠐⠄⠃⠀⠀⠈⢛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠦⣄⣣⣿⣿⣿⠃⣠⡀⠀⠀⠀⠲⢄⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠉⠉⢻⣿⡀⢹⣷⡀⠀⠀⠃⠈⠴⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡤⠶⠟⠛⠾⠟⠋⠉⣩⡽⠟⠋
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠙⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠈⠻⠿⠿⠿⠷⠦⠄⣀⣤⠔⠉⠒⠦⡤⠠⠄⠀⠀⠛⠿⠷⠟⠛⠁⠀⠀⠀⠀⠀⣴⣾⣿⣷⣆⡀
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠐⠶⣖⠀⢡⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⡠
⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣈⠲⣄⠀⠀⠀⠀⠀⠀⣸⣷⣦⣄⣀⣀⠤⠀⠀⠀⠀⠀⠀⠀⡀⠿⠛⣇⠈⢻⡇⠰⡀
⠀⠀⠀⠀⠀⢹⣶⣄⡀⠠⡀⠀⠁⠀⣱⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢰⣿⠀⠘⠀⠀⡇
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣶⣬⣲⣤⣰⡾⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢿⡄⠀⠀⣸⠇
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣃⣡⣼⡿⠿⠿⠿⠿⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⢠⠃⢡⠀⠀⢻
⠐⠿⠿⠟⠫⣿⣿⣿⣿⣿⣿⣿⡛⢻⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⣸⠁⠀⠘
⣀⣀⣀⣀⣀⣉⣛⣉⣀⣙⣛⣛⣃⣻⣿⣿⣿⣿⣛⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣋⣀⣀⣀⣃⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣯⣿⣯⣽⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠋⠲⣂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢀⠝⠁⣰⠉⡙⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠊⠀⢠⠂⠀⡔⠀⢉⣴⠇⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡅⡸⠁⠀⡔⠁⢀⠄⠀⠠⢆⡀⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠟⠁⠀⡸⠀⠠⠋⠀⠰⠁⠀⢠⠃⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠔⠀⢠⠁⠀⡴⠁⠀⠀⡿⣄⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠖⠁⠀⠀⠘⠀⣠⠂⠀⠘⠀⠀⠀⠜⠀⡇⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⢀⠷⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠴⡃⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⡠⠊⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢠⠞⠓⠆⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⣧⣀⠄⠈⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠠⣖⠉⣉⣉⡳⣽⣗⠈⠂⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠈⠙⢏⠀⠈⠻⣦⣄⠀⠀⠀⠀⣰⣿⣤⣤⣴⣢⣤⣶⡋⠀⠀⠀⠀⢀⡔⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⠜⣧⠤⠂⠀⠉⠉⣿⡯⠊⢻⣮⡛⠿⣿⠛⠷⠒⠉⡷⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣅⣌⡄⣿⡌⠄⠈⠳⣤⣬⡿⣡⣿⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⢻⣛⡓⠢⣀⣤⡇⢹⣿⣧⠹⣧⠀⠀⠀⠈⠻⠿⠿⠏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⡻⠋⠁⠀⠙⠓⢾⠉⡛⢦⡙⠂⠈⠢⠵⠖⡶⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⢀⣠⠇⠀⠀⠉⠉⠐⠒⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠒⠀⡖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⠘⢃⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⣼⠀⠀⢈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⣀⡈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣰⡇⢔⠇⠀⠀⠀⣦⣴⠀⠀⢰⠖⢒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣄⡀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡾⠟⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣇⠴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⠀⠀⢈⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠃⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠿⡆⠀⠠⢄⣀⣹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢱⡆⠀⠀⠠⣿⡟⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠚⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠰⠽⣀⣠⡶⠉⠀⠀⠀⠀⠀⠀⡐⢠⠖⠢⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⠂⠻⡘⠐⣀⢂⠐⣶⠀⠀⢐⠀⠸⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠧⠄⠃⠈⠻⠇⠠⠀⠈⡁⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⣒⣴⣦⠬⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠽⢉⡀⠀⠙⠙⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠂⠘⠀⠀⠀⠐⠂⠐⠀⠒⠀⠰⠀⠀⠀⠀⠐⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠉⠉⠉⠉⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣀⣤⠤⠒⠊⠁⠀⠱⡄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣰⡋⢈⣉⡁⢀⠀⢈⣉⡁⢨⣷⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢻⡇⠀⠀⠀⣼⠀⠀⠀⠀⢸⠜⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠱⡀⠀⠀⢀⡀⠀⠀⢠⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣠⠔⠛⢢⣄⣀⣀⣠⡖⠛⠢⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⡁⠀⠈⠙⢦⠀⠀⡴⠋⠀⠀⢈⣂⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠒⠤⣀⢣⡜⣀⠄⠒⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢠⠏⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠹⡄⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠠⢄⡀⠀⠀⠀⣰⣋⣠⠧⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⣄⣙⣆⠀⠀⠀⢀⡠⠄⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡯⣄⡀⠀⠙⡆⣠⠞⠉⠁⠀⠀⠀⠀⠉⠒⠢⠤⠤⠤⠤⠔⠒⠉⠀⠀⠀⠀⠈⠉⠳⣄⢰⠋⠀⢀⣠⢽⡄
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⢣⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡜⠁⠀⠀⠙⣆
⠀⠀⠀⠀⢠⡇⡎⡜⢠⡆⢸⠀⠒⠒⠒⠒⠒⠒⠒⠚⠛⠓⠒⠒⠤⣀⠤⠒⠒⠚⠛⠛⠓⠒⠒⠒⠒⠒⠒⠂⠀⡇⢰⡄⢣⢱⢸⡄
⠀⠀⠀⠀⠘⢷⠀⠃⡏⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠒⠲⣄⡤⠔⠤⢄⡀⠀⠀⠀⠀⠀⠀⠘⡼⢱⠘⢀⡼⠃
⠀⠀⠀⠀⠀⠈⠛⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠙⠲⢄⣀⠀⠀⠀⠀⠀⠈⠒⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣆⣀⣤⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢀⢿⣿⠀⠀⠀⠀⢠⡶⠄⠀⠙⠇⠀⠀⠈⢀⣴⠦⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⣿⠀⢰⣿⢳⢸⣧⡄⠀⠀⡀⠀⢀⡀⠸⣧⡐⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣸⣿⠖⢺⣿⠀⣠⠻⣷⣄⣠⣿⡀⢸⣇⠀⡹⣷⡄⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⡴⠋⢸⣿⠀⠈⠻⠴⠁⠀⢈⢿⡆⠻⠷⠛⣿⡰⠁⢘⣿⡏⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⠁⠀⡀⡜⠀⠀⢸⣿⢰⡿⢹⠀⠀⠀⢤⠼⠁⠀⣤⠀⠈⠁⠀⡊⣸⠃⠀⢠⠀⠀⡠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠢⣈⡇⠀⠀⢧⣀⣤⣿⠃⣸⣇⠊⠀⠙⠇⠀⠀⠀⠀⠁⣀⠀⣠⡀⠉⢠⣦⠄⠀⠁⣸⡠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠈⠙⠉⠁⠀⢹⡇⠀⢀⣴⡆⠀⣤⢾⡁⣸⡏⠃⣿⡄⠀⠘⠋⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣃⠀⠆⠀⢀⢀⡀⢸⣇⠀⣸⡇⢙⠋⣿⢀⠉⠹⣇⡰⠉⢻⣦⠒⠀⠐⠀⣬⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠽⠦⣆⠀⠘⣿⠇⠀⠻⠖⠀⠉⠁⠀⠈⠋⠀⠀⠈⠀⠀⣊⠟⠀⢀⣤⠞⠃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢧⡄⡈⠀⣠⠀⢀⣄⠀⢀⣤⡀⠀⣠⡄⠀⣤⠀⠀⠀⠠⡞⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⢸⣿⢀⣿⠛⠶⠖⣿⡇⢠⣿⠀⣠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⡀⠙⠓⢻⣿⠉⠻⠦⠆⠀⠙⠓⠃⠻⡶⠃⢀⣴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡰⠉⣿⠀⡀⠀⡔⠐⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣰⠇⠰⣅⠀⠉⣠⠃⠀⢰⡦⣰⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⣄⣀⣀⣀⠈⠕⠚⠁⠤⠐⠋⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⣿⣿⡿⠃⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣟⣀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⡀⠀⢹⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⣤⠲⠶⢄⠀⢤⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠁⠄⣈⠀⢸⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⢀⠀⠁⠀⠐⠀⠀⠀⠀⢀⣬⠃⢀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⣿⣿⠀⠘⣶⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⢀⠀⠶
⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠉⠋⡠⠐⡀⠤⢀⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⡤⣤⠬⠁⠲⠃⢠⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡩⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡧⠻⠋⠀⢀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⣀⣤⣤⣿⣿⣧⠸⡇⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⣿⣿⡏⢙⣿⣿⣿⣿⣿⣿⣿⠣⠞⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠿⠿⢿⣿⣇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣴⣶⣾⣿⣿⠇⠁⠉⠀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠙⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⡿⠂⡀⠀⠀⠀⠀⠘⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠩⠉⠥⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠂⠀⠔⠈⠀⠀⠀⠀⢠⠀⠀⠄⠠⠀⠄⠠⠄⢠⠄⠀⡄⠀⠰⠀⠀⠠⠄⠰⠀⠀⠔⠀⠄⠤⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠇⠀⠀⠀⡄⠀⢀⠠⠀⠀⠀⠀⠀⠀⠄⠀⢀⠈⠀⡀⢠⢠⠄⠠⠠⠄⢐⠀⣠⣀⣤⠀⣤⡀⢤⠀⡄⣤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡶⠀⠀⠀⠀⢀⢀⠀⠀⢀⠀⠀⢀⡈⠀⡁⠄⠈⠀⠀⡀⢀⠈⡀⠉⡀⠀⠀⡀⠀⣀⢀⣀⡀⡄⠂⡀⣀⣄⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠁⠀⠁⠀⠁⠐⠀⠉⠈⠈⠁⠈⠉⠁⠉⡈⠀⠋⠉⠉⠉⠁⠀⠁⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⠀⠂⠀⠂⠀⠛⠋⠀⠁⠘⠀⠂⠀⠘⠄⠁⠀⠚⠐⠋⠀⠛⠒⠓⢓⠃⠘⠃⡄⠑⠛⡛⠀⠛⠍⠃⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠁⠋⠀⠃⠂⠀⠀⠊⠒⠒⠂⠀⠖⠀⠐⠰⠀⠀⠐⠇⠲⠔⠐⠾⠀⠐⠰⠇⡘⠂⠐⡰⠁⠠⠂⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠀⠐⠆⠀⡀⠀⠐⠁⠲⠢⠂⠆⠀⠀⠢⠄⠀⠶⠀⠠⠀⠠⠄⠴⠠⠀⠶⠀⠆⠐⠶⠢⠾⠥⠐⠀⠶⠰⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⡀⡀⠐⠀⠀⡠⠄⠄⠄⠀⢀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣴⣦⣶⣶⣦⣤⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣴⣴⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣁⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣽⣿⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣾⣿⡇⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠸⣿⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠹⣿⠃⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣿⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣿⣿⣿⣿⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⣿⣿⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣿⣦⣿⣿⣿⣿⣧⣿⣷⣶⡆⠀⠀⠀⠀⠀⢀⣀⡀⠀⣿⣿⣿⣿⣿⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣿⣿⣿⡟⣷⡄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣿⣿⣿⠁⠙⢧⡄⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣿⣿⣿⣿⡆⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢧⣼⣿⣿⣿⣧⡼⠿
⠀⠀⠀⠀⠀⢰⣯⣄⣀⣰⣿⣿⣿⣧⢀⣠⣼⡆⠀⠀⣀⣤⠴⣿⣍⣀⢰⣿⣿⣿⣿⣇⣀⣩⣿⠦⣤⣄⠀⠀⠀⢠⣤⣄⡀⣾⣿⣿⣿⣇⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠈⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠁⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⡟⢿⣇
⠀⠀⠀⠀⠀⠀⢀⣰⠿⠀⣿⣿⣿⡁⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡹⣿⣿⣿⣿⢏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣥⣿⣿⣿⣿⣧⣼⡿⠄
⠀⠀⠀⠀⡴⢶⣯⣅⣀⣠⣿⣿⣿⣇⣀⣨⣽⣶⢦⠄⠀⠀⠀⣤⣤⣌⣷⣿⣿⣿⣿⣾⣡⣤⣤⠀⠀⠀⠀⠀⠀⣶⣶⣤⣀⣿⣿⣿⣿⣯⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⣶⠶⢶⠶⡶⣶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⡴⠒⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠷⢦⣄⣀⣾⡏⣀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣼⡿⠀⢠⢶⣦⠀⣴⡄⣠⣶⡄⣠⡶⢦⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢼⢸⢻⢶⠁⠀⢀⣿⡇⣴⡿⣰⣿⣰⣿⡵⠉⠟⣰⡟⠀⣼⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⣄⡿⠋⣿⡟⠀⠀⠀⣿⣇⡴⣿⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡖⢱⡿⠣⢄⣀⣀⣀⣀⣈⣀⣀⣀⣀⣈⣩⣤⣍⣅⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡄⠀⣤⣤⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⠀⢠⡄⠀⠀⠀⠁⠉⠀⠋⠋⠃⠙⠀⠀⠀⠀⠀⠀⣤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢀⣿⢤⣿⡧⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⣼⡿⢿⡿⠓⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢿⡄⠀⢸⡟⢠⣷⡰⣾⢢⡞⡣⢰⣧⢴⡷⢀⣴⠟⣓⢀⣿⢃⣿⢡⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢸⣿⠀⣿⢁⣾⡟⠁⠁⣿⠉⣠⣿⠃⣿⢁⣼⡏⣰⡿⣸⡏⢸⡿⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠿⠃⠀⠛⠋⠛⠀⠀⠀⠙⠋⠉⠃⠀⠙⠁⠘⠛⣽⠇⣿⠁⣸⠁⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠈⠒⠋⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⡲⢖⡷⠀⣷⣷⡒⣆⠀⠀⠀⣆⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⣷⠀⣶⡄⠀⠀⢀⡀⠀⠀⠀⠀⣶⡆⠀⠀⢀⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠰⠟⣸⡿⢀⡀⠀⡸⠋⠀⣀⠀⣸⡿⠀⢀⡀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠘⣿⡄⢠⣿⡱⣿⡏⢰⡿⢀⣾⣩⢃⣿⠃⣰⡟⢩⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⡇⢸⡿⠁⣿⡠⣿⣃⢾⣏⣀⢼⣏⡴⣿⣠⣾⣇⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠋⠀⠘⠁⠀⠉⠁⠈⠁⠀⠉⠁⠈⠉⠀⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠲⣄⣰⣼⣷⢶⢶⠀⢰⣆⢿⡕⠐⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣷⣶⣦⣤⣄⣀⠀⠀⢶⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠉⠙⠓⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠉⠻⠷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⡿⠛⠻⢹⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠞⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠤⠴⠖⠒⠲⡒⠉⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⢼⠉⠑⢌⠻⢆⡀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣣⣀⣨⠶⣀⡸⠀⠀⢉⣲⠖⠂⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⣟⢰⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠤⠤⠤⠴⠆⠀⠀⣠⣿⠘⡀⠀⢠⣗⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⠤⠒⠊⠈⣿⠀⣇⢤⠖⠥⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⢾⣻⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢭⢈⣀⣉⠶⢢⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⡀⠀⢨⠀⢈⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⣰⠚⠧⡀⠸⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡏⠬
⠀⠀⠀⠀⠀⠀⠀⠳⣚⣉⣽⡙⠒⣳⠸⡿⢣⡌⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠿⠓⢤⠇
⡰⠊⠉⠉⠉⠉⠉⠉⠉⠀⠈⠛⠉⠑⡖⠀⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⢂⠔⠀⠰⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⣯⣄⣠⠀⢧
⠈⠢⣄⣀⣀⣠⡤⠔⢪⠋⠀⠀⠀⡸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⢽⠗⡸⢁⣠⠔⠊⠉⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⡠⠞⠁⠀⠀⢠⠀⠀⠀⠀⠀⢠⡄⠤⢴⣒⠒⢡⡪⢊⠜⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⣄⣀⣀⣀⣀⡀⢸⢳⠊⠢⣀⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠱⣤⣄⡀⠀⠀⠑⠒⠐⠒⠒⠊⠉⠉⠙⠦⠬⠥⡀⢹⠳⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢫⣉⣉⣉⣙⣉⣍⠁⠀⠀⠀⠈⠋⠰⠢
⠀⠀⠀⠀⠀⡅⠀⠑⠂⠄⠀⠀⠀⠀⠀⢀⣉⠁⠂⠆⢠⣀⣀⣀⣀⠀⠀⣀⣠⠇⠆⣸⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠒⠢⠤⢤⠀⣀⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠘⡀⠀⠀⠀⠀⠀⠀⠈⠓⠓⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢍⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣄⡀⡰⠲⡄⠀⠀⠀⠀⠀⡆⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠎⠀⠀⠈⢡⠎⣀⠀⠀⠀⠀⠀⡇⠒⠒⠒⠂⠠⠬⢁⣀⣀⣀⡀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠑⡄⠰⢦⠀⠜⠀⠀⠀⠀⠀⠓⠊⠉⠉⠉⠁⠒⠒⠂⠤⠤⠀⠀⢠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⢁⠐⣿⣿⣿⣤⡄⠀⠀⠀⠁⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠑⡀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⠀⠀⠄⠀⠠⠂⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠄⠀⠀⠀⠐⡄⠀⠀⢻⣿⣿⣿⣿⠇⠘⠀⠠⠀⠠⠂⠀⠐⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠀⠀⠁⢀⠀⠢⠀⠈⠀⠀⠀⠀⠈⢿⣿⣿⠏⠀⠀⠠⠁⡐⢀⠔⠁⠀⠔⠁⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠈⠂⢀⠑⢄⠀⠄⠈⠂⠁⠀⠀⣸⣿⣿⣄⠀⠀⠡⠂⠀⢁⠀⠊⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⡀⠀⡑⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠊⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠩⠀⠀⠈⠀⠀⢀⠀⠠⠄
⠀⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⠁⠀⠀⠀⠤⠈⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣐⠀⠈⠁⠄⠤⠄
⠀⠀⠀⠀⠠⠤⠄⠠⠤⠤⠤⠤⠀⠄⠠⠤⠀⠀⠀⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠒⠂⢀⣼⣿⡟⠀⢸⣿⣿⣿⣿⣿⣿⠁⢸⣿⣧⡀⠀⠂⠠⠀⡉⡁⠀⠀⠀⠀⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠁⠀⢀⣴⣿⣿⠟⠁⠀⢠⣿⣿⣿⣿⣿⣯⠀⠈⢿⣿⣷⡄⠁⠂⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣤⣿⡿⠋⠁⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠙⠻⣿⣦⣤⣴⡀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠀⠀⠂⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣶⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⢤⡯⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⠟⠛⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢿⣿⣿⣷⢟⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢋⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⣼⠛⢻⡏⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠈⠳⢤⣿⡹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣀⣀⣀⣠⣿⣿⣆⣀⣀⣀⣰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⡿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢐⣟⣿⣻⣛⣛⣿⣟⣿⣻⣿⣻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠟⢿⠟⠁⠀⠀⠀⠀⠀⠙⢿⠇⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⡞⡟⠳⣄⠀⣀⡀
⠀⢰⢻⣌⣖⠋⠻⣄⣹⠶⣄
⠀⠘⣟⢸⡉⣳⡀⠈⠳⣄⠈⠳⣄
⠀⢰⠻⡄⠉⠉⢳⡀⠀⠀⠙⢆⠈⠙⢦⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣿⠛⢦⠀⠈⠲⣄⠀⠀⣱⣦⡀⣙⢷⣄⢀⣠⠞⡭⣷
⠀⢀⣴⣇⠒⠚⢇⠀⠀⠈⢧⡀⠈⡠⢷⡄⠋⠙⣯⡟⠀⠳⢿⡀
⠀⢸⠁⠙⡄⠀⠈⢆⠀⡠⣢⡙⣦⠀⠀⢻⡄⠀⠘⣧⡀⠀⠸⡇
⠀⡆⠀⢀⠿⣄⠀⠈⠣⡀⠀⠘⠉⣧⠀⠀⠙⣆⠀⠈⠳⣤⡀⠙⣆
⢰⠇⠀⢹⠀⢹⡆⠀⡴⡟⢦⠀⠀⠘⣧⠀⠀⠘⢦⠀⠀⠈⠻⣄⠙⣆
⣼⠀⠀⢸⡀⠉⢻⡄⠀⠀⠞⢦⡀⠀⠘⢷⡀⠀⠀⠑⠀⠀⠀⠈⠓⠮⠳⣄
⡇⠀⠀⠜⢧⡀⠈⣧⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠹⡄⠀⠀⠀⠱⣄⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠙⣆⠀⠀⠀⣹⠁⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠈⢦⠀⠀⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⡆⠀⠀⠀⠀⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣧⠈⣧
⠀⠀⠀⠀⢳⡀⠀⠀⠔⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠸⡄⢸⡄
⠀⠀⠀⠀⠀⢹⡗⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣳⡈⢧⣀⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⣨⠷⠋⠁⠀⠀⠈⠛⢳⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⡸⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠙⠳⠦
⠀⠀⠀⠀⠀⠀⡈⠀⠙⣆⠀⠀⠀⠁⠀⣸⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⣇⠀⠀⠀⢀⠼⠁⣠⣿⣿⠳⣄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢸⠀⠀⢀⡏⠈⠓⠦⠀⠀⠒⠾⢡⠞⠁⢀⠈⠻⢦⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⢸⠀⠀⠸⣧⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⣸⠀⠀⡞⡽⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠨⢸⣄⣇⣿⣆⣀⣤⠤⠞⠁⠀⠀⠁⠀⡏⢀⡾⠁⡇⠀⣄⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠳⡉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣧⠀⠙⣦⣀⣀⣠⠤⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢲⠒⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⣀⣼⠷⠒⠊⠉⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣀⡴⠋

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⢠⢁⣠⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠆⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⣎⣂⣁⡠⣭⣿⣥⡂⢄⣴⣛⣛⠒⠂⠠⠈⠷⠀⠀⠀⢬⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠟⢃
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣜⡶⠯⠟⣿⣿⣦⡀⠀⠀⠠⠒⠉⠀⠀⣀⣛⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣮⡭⠤⠥⠭⠭⣡⣀⠤⠤⠶⢿⣿⣿⣿⣷⣦
⠀⠀⠀⣠⣤⣶⣾⣿⣿⢿⣿⣿⣿⣧⡀
⠀⠀⢰⣤⢹⣿⣿⣿⣟⣥⡵⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠐⣀⡀
⠀⠀⠈⢻⣼⡎⠻⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠙⢿⣆⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣦⣬⢻⣿⣿⣿⣿⣿⠿⠛⢉⡀⠀⠀⣼⣅⠀⠀⣀⣀⣀⡀⠀⣀⡀⠤⠤⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⣼⣿⠟⠉⣀⣠⣤⠾⠋⠀⠀⡿⢛⣛⣙⣛⣻⣟⣿⣿⣿⣿⡯⣋⣉⠒⠓⠖⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡶⠟⠋⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠘⠈⢹⡋⠉⠁⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⢀⣴⣦⡀⠀⠀⠀⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡃⠰⠂⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡤⠖⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⠀⠀⠰⠆⠘⠛⠛⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⠀⠀⠿⠥⠦⠶⠤⠴⠧⠴⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠶⠶⠶⢶⣤⣄⡀⠘⠛⠃⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠐⠆⠀⣰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠉⠻⣦⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⢠⡏⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠇⠀⠀⠀⠀⢸⣿⠀⢻⡄
⠀⠀⠀⠀⠠⡆⣋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣛⢰⠄
⠀⠀⠀⠀⠐⠆⣡⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣌⠰⠂
⠀⠀⠀⠀⠀⣓⣠⠄⢸⡇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠠⣄⣚
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⡀⠀⠄⠀⠴⠂⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣤⣤⣀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠀⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⣿⠄⠀⢀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⡹⠀⠀⢸⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠈⠀⡠⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡴⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠛⠛⠛⢁⣠⣤⣤⡄⠀⠀⠀⡀⠈⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣡⣴⡿⠃⠀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠈⠚⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⣿⣿⣿⡿⠋⠀⢀⣤⢻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠀⢀⣴⣿⣿⣷⣽⡻⠿⠿⠋⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣠⡄⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⡿⢃⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠁⠀⣠⣾⣿⣿⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⢸⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠎⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆
⠀⠀⠀⣠⣶⢛⣤⣄⠀⠥⠀⠀⠀⠀⠀⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⣤⡖⢠⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⢂⣤⡄⠀⠥
⠀⠀⠛⢛⣵⣾⢟⣱⠆⠀⠀⠀⠀⠀⠉⢁⡴⠊⠁⠀⠀⠀⠀⠀⠀⠛⢋⣴⡿⢋⡀⠀⠀⠀⠀⠀⠛⢛⣡⡾⢛⣡⠄
⠀⠀⠀⠈⠛⢱⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠘⠋⢴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⢰⡿⠋
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⣴⡾⠿⣷⣄
⠀⠀⢰⡟⠀⠀⠀⠻⣷⣤⣾⠿⢷⣦⣀
⠀⠀⢸⣧⣀⡀⠀⠀⠈⢿⣇⠀⠀⠈⠻⣷⣄
⠀⠀⣿⡟⠛⠿⣷⣄⠀⠀⠻⣧⡀⠀⠀⠈⠻⣷⣄
⠀⠀⣿⡇⠀⠀⠙⢿⣦⠀⠀⠙⢿⣦⡀⠀⠀⠈⠛⢿⣦⡀
⠀⢠⣿⣿⠀⠀⠀⠈⢻⣷⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠙⢿⣦⣄
⢠⡿⠀⢻⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠉⢻⣷⡄
⢸⡇⠀⠀⣴⣶⣦⡀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠙⢿⣆⣠⡶⠟⢿⣇
⢸⡇⠀⢸⣿⡟⠙⣿⣆⠀⠀⠀⠻⣿⣆⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠈⢿⣏⠀⠀⠸⣿
⢸⣧⢀⣼⣿⡇⠀⠘⢿⣧⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠹⣷⡄⠀⠀⠀⠘⣿⡄⠀⠀⣿⡆
⠈⣿⣿⠃⢿⣿⠀⠀⠈⢿⣧⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠹⣿⡄⠀⠀⠀⢹⣷⠀⠀⢸⣷
⠀⢸⣿⠀⠈⣿⡇⠀⠀⠈⢻⣧⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠹⣿⡀⠀⠀⠀⢻⣧⠀⠀⢿⣧
⠀⣿⡏⠀⠀⠸⣿⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠹⣷⡀⠀⠀⠀⢻⣧⠀⠘⣿⣆
⢰⣿⠁⠀⠀⠀⢻⣇⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠻⣧⠀⠘⣿⣇
⣸⡿⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠙⢷⡀⠘⢿⣆
⣿⡇⠀⠀⠀⢠⡀⠘⣷⡀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠁⠀⠈⢿⣧
⣿⡇⠀⠀⠀⠸⣧⠀⠹⣧⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⢻⡇⠀⠀⠀⠀⢻⣇⠀⢻⣇⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠈⢻⣤⠀⠀⠀⠀⢻⣧⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠈⢻⣧⡀⠀⠀⠀⠻⣧⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠹⣷⡀⠀⠀⠀⢹⣷⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⢹⣇⠀⠀⠀⣼⡿⠛⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⣿⡀⠀⠈⠁⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠉⢻⣷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡏⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣶⣾⣷⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⢀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠸⠿⠂⠀⠸⣡⠞⠁⠠⠊⣠⠆⣀⣠⠆⢀⣀
⠀⠀⠀⠀⠀⠃⠉⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⠃⠁⠀⠀⡴⢻⢵⢿⢡⢶⠟
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⣶⣧⢷⠷⠇⠀⠸⠚⠀⠀⠈⠓⠀⠀⠀⢀⣀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⢭⣵⣥⣶⢌⠁⠀⠀⠀⢰⣶⡆
⠀⠀⠀⠀⠀⠈⠁⠀⠑⠜⠉⠉⠉⠙⠲⡅⣀⠀⠀⠀⠀⠀⠀⢁⡄⠲⢤⣵⡄⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠂⠀⠃⠀⠀⠒⡄⠀⠀⠀⠘⢎⠁⠐⠀⠂⠀⡄⢄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠑⠂⡀⠠⣀⣀⡠⠀⠀⠈⠀⠀⠐⠉⠀⠈⣳⢤⡤⢤⠮⣀⢄⠉⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠦⢤⡤⡤⠖⠣⡀⠀⠀⠈⠑⠊⡠⠀⠀⠉⠁⠀⠈⠑⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⣐⡀⠀⠀⠈⢂⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠶⠦⡄⠀⠀⣠⠤⢤⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⣤⣴⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴⠋⠚⠀⠀⠀⡟⡴⠀⢙⠖⠁⠀⠔⠃⢀⠀⢀⣴⠃⠀⠀⠟⢿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠘⠂⠆⠀⡠⢊⣾⠀⠀⢸⡀⠀⢀⡠⡆⠿⠲⠹⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠞⠀⠀⠀⠀⢨⡛⢔⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⡠⠐⠈⠉⠉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⢀⠎⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠘⠛⠛⠛⠛⠁⠀⠀⠶⢶⣤⣀
⡾⠀⠀⠀⠀⣀⠤⠀⠉⠀⠀⠀⠀⠀⡀⠀⠤⠄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⢿⣶⣦⡄
⣇⠀⣤⡾⠋⠁⠀⢰⠁⠀⠀⠀⠀⠀⠈
⠹⣄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡠⠒⠊⠉
⠀⠙⠿⣦⡀⠰⠊⡀⠀⠀⠀⢀⣤⡆⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠘⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⣠⣤⠤⣤⠀⠀⠀⠈⠑⠲⠶⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⠛⠛⠉⠁
⣼⣿⢱⡟⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠈⠳⢤⣀⣀⡀⠀⢀⣀⣀⣀⣠⣤⡤⠤⠤⠴⠶⠒⠒⠒⠛⠛⠛⠛
⠹⣿⣌⠛⢁⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠉⠉⠁⠀⠱⣄⠀⠀⠀⠀⣀⠤⠖⠋

⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣦⢤⡴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠰⢦⣾⠿⣾⠗⠿⠖⠓⠦⠀⠀⠀⠀⠀⠀⡿⠛⡿⠟⠾⣷⢿⣦⡴⠀⠀⠀⠀⠀⠀⣴⢾⣿⣧⣯⣿⣦⠀⠀⠀⠀⣀⣤⣿⣿⣺⡷⢿⣤⠆
⠀⠀⠀⠀⠀⠛⠡⠀⢀⠆⠔⡠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡀⠿⠀⡑⠠⠘⠛⠀⠀⠀⠀⢀⡿⠏⠛⠁⠘⠀⠉⠁⢀⠀⠀⠀⠉⠉⠟⠛⢻⡟⠀⠙⡥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣴⣮⣷⡾⠷⠦⠀⠀⠀⠐⠏⡃⠀⠀⢧⠀⠀⡀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⡌⠀⢀⡧⡂⠀⠀⠀⠀⠁⠐⡠⠀⠀⢄⢢⡥
⠀⠀⠀⠀⠀⣀⠧⣀⠀⠱⠈⢀⠀⠠⠀⠀⠀⠀⠀⠈⣋⠴⠁⠀⡆⠀⢠⡘⣾⠃⠀⠀⠀⠀⠀⢀⡦⡀⠘⠀⢀⠻⡴⠁⠀⠀⠀⠀⠀⠀⡇⢠⠐⠌⠴
⠀⠀⠀⠀⠀⣌⣠⠇⡀⠀⢡⡈⢀⠀⠀⠀⠀⠀⠀⠀⠌⣒⠁⡖⡇⢀⡈⣧⣫⠆⠀⠀⠀⠀⠀⣌⣠⠁⠀⠀⠀⣷⣋⠁⠀⠀⠀⠀⠀⠠⢀⢈⣂⣛⠉
⠀⠀⠀⠀⠀⢨⣅⠚⡡⠀⠈⡰⣮⠜⠀⠀⠀⠀⠀⠀⠠⠤⠼⣀⠠⠰⣙⢾⡀⠀⠀⠀⠀⠀⠀⢬⡰⢃⠀⢀⣸⠿⠃⠀⠀⠀⠀⠀⠀⠰⠈⠙⠀⠁
⠀⠀⠀⠀⠀⠶⢬⣎⠁⡶⢈⢀⡿⣸⠇⠀⠀⠀⠀⠀⢄⡀⠈⠉⠸⠀⣽⣾⡄⠀⠀⠀⠀⠀⠐⢮⣒⠃⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠲⠤⠤⢽⣀⢠⠀⡙⣫⠖⠀⠀⠀⠀⠀⠈⠷⣆⡆⠐⢀⡜⠁⠀⠀⠀⠀⠀⠀⠐⢤⣭⡞
⠀⠀⠀⠀⠀⠀⠀⠐⠊⢉⠳⠀⢻⣧⠆⠀⠀⠀⠀⠀⠀⢀⣋⣔⡀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⢆⠀⠀⠀⠀⡀⢀⣦⣸⣆⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⠀⠇⣸⠟⠋⣴⠂⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹⠟⠋⠟⠉⠉⠁⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠨⣌⣡⠄⡸⠁⢿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⣄⣰⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠠⣏⠵⠀⠡⣈⣬⡭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠛⣿⣍⣿⣽⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠓⢂⡄⢠⠸⠋⠀⠀⠀⠀⠀⣰⢤⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠉⡁⢂⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠁⠀⠀⠀⠀⠀⠀⠺⠿⠛⠟⢿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠠⡿⠾⠿⠟⠛⠃⠀⠀⠀⠀⠀⢁⠔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠂⠀⠀⠀⠀⠀⢀⣓⠂⠒⠃⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⣝⡸⢋⡡⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣕⡲⠄⣠⠖⢃
⠀⠀⠀⠀⠀⠀⢠⣿⠀⡀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠐⡆⢀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⠖⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣉⡛⢃⡯⠥
⠀⠀⠀⠀⠀⠀⠠⣿⠋⢁⠀⠀⠀⠀⠢⣽⣄⠀⠀⠀⠀⠀⢻⡥⢃⡀⠀⠀⠀⢻⡸⠇⠀⠀⠀⢀⠻⣊⠄⠀⠀⠀⠀⢠⡄⠀⠀⠀⡘⠒⠦⢅⡩⠖⡂
⠀⠀⠀⠀⠀⣄⢷⣿⡷⠞⠁⠀⠀⠈⡓⠶⣛⠀⠀⠀⠀⢀⡚⠇⣹⠂⠀⠀⢘⡎⠙⢠⠆⠀⠀⠀⡣⠤⠀⠀⠀⠀⣀⢃⡜⠀⠀⠀⠉⠛⠲⢤⠴⠊⡁
⠀⠀⠀⠀⠀⢸⣎⣇⣀⣤⡼⠀⠀⠀⠉⣻⠅⣀⠀⠀⠀⠀⠙⢺⢋⡴⠀⠀⢨⠘⠟⢃⡀⠀⠀⣆⠡⢼⡅⠀⠀⠀⣝⠩⠞⠀⠀⠀⠀⣙⠓⠂⢒⡉
⠀⠀⠀⠀⠀⠠⣥⣙⠋⣀⣴⠀⠀⠀⠶⣈⡶⠃⠀⠀⠀⠀⢻⣜⢋⣤⠀⠀⡀⠛⠶⠋⠀⠀⠀⣄⡓⢋⡄⠀⠀⠰⣌⠔⠊⡄⠀⠀⠀⠤⣉⠋⠔
⠀⠀⠀⠀⠀⠀⢠⣉⣷⠟⠁⠀⠀⠀⠲⣿⣷⠀⠀⠀⠀⠀⠀⣼⣯⣡⠀⠀⠘⠳⢄⣰⠀⠀⠠⣌⡛⠉⠀⠀⠀⠤⣌⣱⠚⠀⠀⠀⠀⢉⣀⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⣿⠖⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⣠⡟⠉⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠈⠑⡞⠀⠀⠀⠀⠠⠼⢴⠀⠀⠀⠀⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣞⣷⠦⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢦⡈⠉⠳⢬⡙⢾⣅⢦⣀⠀⠀⠁⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⢄⠈⠛⠯⡙⠣⣝⢳⣄⠀⠀⠀⠔⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠹⢦⣍⠻⣌⢷⡀⠀⢀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠉⠙⠛⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠺⣦⣄⠈⠙⢦⣀⠙⠳⠈⠀⢷⡈⢩⢟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⢠⡆⠀⠀⠀⠹⣷⡀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡄⠘⠈⠷⣄⠀⠁⠀⠀⠀⠀⠘⢷⡉⢀⣿
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣸⣿⠃⠀⢠⠿⣦⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⠀⠀⠀⠀⠈⢂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⣧⠉⠯
⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⣿⡇⠀⢰⣃⠜⠁⠀⠀⠀⠀⢸⣇⠀⠀⠀⠉⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣶⡧
⠀⠀⠀⠀⠀⠀⠌⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣴⣿⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡇⢀⡀⠈⢁⡟⠹⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⡀⠀⠀⠀⠀⠀⠀⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣴⠋⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⡷⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠰⠄⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣵⣿⣿⠀⠀⠀⠀⢀⠏⢠⠓⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠙⠏⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢿⡇⠀⠀⢠⡾⠋⣽⠎⠀⠴⢣⣾⣿⣿⠇⡀⠀⠐⠁⠀⠠⣂⣾⣿⢿⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠐⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⢿⣥⠾⣇⠀⢐⣵⣮⣿⣿⣁⠀⠀⠀⠀⢀⠔⠀⠹⡎⠁⢹⣷⡀
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠹⡪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⣠⢒⣀⣭⣶⠿⠚⣧⠙⠂⠀⠀⠂⠀⠀⠀⠀⠀⢀⣼⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⣻⠖⠥⠶⣣⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣐⠈⠒⠉⠙⣇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃


⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣄⡀⣀⢠⣀⣀⡄⠀⠀⠀⠀⠀⢄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⠠⠤⠤⠤⠀⠀⠀⡀⠀⢠⣠⣀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⣀⡄


⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡄
⠀⠀⠀⠀⠃⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⢙⠃⠀⠁⠂⠀⠀⠀⠀⠀⠀⠉⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣤⠆
⠀⠀⠀⠀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡄⠀⠀⠀⠀⠀⢰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠀⠀⠀⠀⢀⡄
⠀⠀⡶⠚⣿⣿⣿⠳⡆⠀⠀⠀⠀⢰⠒⣿⣿⣿⠒⡆⠀⠀⠀⠀⢰⠖⢻⣿⣿⡗⢶⠀⠀⠀⢰⡞⣿⣧⣾⡜⢲⠀⠀⠀⠀⠈⣼⣷⣄⠀⡀⠀⣿⠁
⠀⠀⡇⢀⡿⠿⠿⠀⡇⠀⠀⠀⠀⢸⠀⠻⠿⣿⠀⡇⠀⠀⠀⠀⢸⠀⢸⠿⠿⠇⢸⠀⠀⠀⢸⡅⠹⠟⠻⠗⢸⠀⠀⠀⠀⢠⠛⠿⠟⠀⠁⠀⠈⢀⣤
⠀⠀⠛⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⠘⠓⠛⠒⠒⠒⠃⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠛⠀⠀⠀⠘⠓⠒⠒⠒⠒⠛⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠂⠐⠀⠄⠀⠤⠴⠦⠄⠀⠀⠀⠀⠠⠤⠐⠴⠤⠦⠴⠀⠀⠰⠀⠶⠦⠆⠰⠀⠶⠶⠀⠀⠀⠂⠰⠦⠤⠀⠂⠠⠴⠂⠀⠀⠂⠀⠐⠀⠤⠲⠴


⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀
⠀⢸⢡⣤⡀⢲⣿⣷⣿⠁⠀⠀⢸⠋⢡⣤⡀⢻⣿⣷⡇⠀⠀⠀⡏⢡⣄⡀⢺⣿⣷⡏⠀⠀⠀⣿⠁⡀⠀⣤⣶⣦⣿⠀⠀⠀⢸⠉⠀⠀⢠⣴⣿⣿⡁
⠀⢸⣈⠙⣿⣿⠛⣋⣹⡆⠀⠀⢸⣇⠈⠙⠻⡿⠋⢁⡇⠀⠀⠀⣇⡈⡙⣻⣿⡛⣁⣿⠀⠀⠀⣏⣘⣻⣷⣼⣿⢛⣹⠀⠀⠀⢸⣄⣛⣻⣿⣿⡟⣹⡆
⣤⣬⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣤
⣿⣿⣋⣝⣛⣩⣛⣻⣻⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⢸⣿⣿⡇⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠈⣿⣿⠁⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢿⡿⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⡇⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⣼⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⠀⢀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣄⠀⠀⣿⣿⣿⣿⣿⠟⢁⣠⣾⣿⣿⣿⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣿⣿⣿⣿⣇⢰⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⢀⠞⠙⢦
⠀⠀⠀⠘⡄⠀⠤⠳⣤⠤⢤⡀
⠀⠀⠀⢠⠫⠿⡄⠀⠙⣆⠒⠉⠲⣄
⠀⠀⢠⣷⠘⠤⠽⣦⠀⠈⢳⣄⠀⠈⠳⣄
⠀⠀⢻⠙⣧⠀⠀⠈⢷⡀⠀⠙⢦⡀⠀⠈⠻⣦⡀
⠀⠀⢸⠀⣆⠘⣆⠀⠈⢳⡄⠀⠀⠙⢦⡀⠀⠈⣙⣦⡀⠀⠀⠀⠀⡴⢋⡙
⠀⠀⣼⢶⡏⠛⠹⣆⠀⠀⠹⣄⠀⠀⣤⠻⣄⠈⠵⠊⢳⣄⣠⠶⠻⠁⠈⠳⡆
⠀⢸⠃⠀⢇⠀⠀⢻⡄⠀⠀⠘⢧⡀⠘⠁⠹⣆⠀⠀⠀⠹⣯⠀⣴⠀⠀⠀⡇
⠀⣼⣄⠀⡸⣆⠀⠀⢻⡄⠠⢶⡨⣷⡀⠀⠀⠘⣧⡀⠀⠀⠸⣷⡇⠀⠸⣿⡇
⠘⠇⠘⣇⡇⠹⣆⠀⠀⠙⣦⠀⠁⠀⢱⡀⠀⠀⠹⣷⡄⠀⠀⢻⣷⠀⠀⠸⣇
⢸⠀⠀⣿⠀⠀⠹⣧⠀⢻⠊⣧⠀⠀⠀⢷⣀⠀⠀⠙⠛⠂⠀⠈⢿⣦⠀⠀⢿⡄
⡆⠀⢀⣿⠀⢧⠀⢹⡄⠀⠀⠘⣦⠀⠀⠀⠙⠗⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠈⢷⡀
⢧⠀⢸⠟⢳⡈⣆⠀⣷⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠘⣧
⠈⢧⠀⠀⠈⢧⠸⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠈⣧⠀⠀⠈⢣⡹⠖⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⣇⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠠⢋⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠉⠉⠉⣙⣿⣿⣿⣶⣦⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠰⠹⣿⣿⣿⣶⣶⣤⣤⣶⠖⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⠟⠋⠹⣷⣄⡀
⠀⠀⠀⠀⠀⣴⠟⢿⣦⣀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⢿⣿⡉⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⢴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⢠⡾⠁⠀⠈⣿⠿⠿⠂⠀⠀⠀⠀⠉⠁⠀⠀⢀⣿⣿⣄⠀⠀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢈⢵⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⣰⠟⠀⠀⠀⣼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⣿⣿⣿⣦⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⢸⡏⠀⠀⠀⠀⣿⣿⣄⣀⠀⠀⢀⣠⣤⣶⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⣿⠃⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠃⠀⠀⠀⠀⠀⢀⣴⣿⠟
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢋⣁⣴⡾⠁⠿⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⢀⣴⣿⠟⠁
⢸⡟⠀⠀⠀⣠⣤⣤⣤⣤⣶⣶⣿⣿⡿⠋⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⢀⣴⣿⡿⠋
⣸⡇⢰⣶⣤⣽⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⣠⣿⣿⠟
⣿⣿⡈⠛⢿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠇⢀⣠⣾⣿⡿⠃
⠘⢿⣿⣦⣄⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣶⣿⡿⠟⠋
⠀⠀⠙⠻⣿⣿⠶⠄⠀⠀⢠⣾⣿⣿⣶⣿⣷⣴⠟⠋⠁⠀⠻⣿⣅⡠⠴⣾⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠈⠻⣶⣶⣶⣶⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠶⠒⢒⣒⠂⠝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡖⠒⠚⠉⠉⢭⡁⠐⠒⠒⠈⠉⠁⠀⠉⠲⢦⣄⠊⢧
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠐⡈⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⢣⡀
⠀⠀⠀⠀⠀⠀⠰⣠⠀⠀⠀⢈⣄⢀⠇⠀⠀⠀⠀⠀⠀⠀⣀⡠⢄⠀⠀⠀⠀⠀⢠⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣠⢊⠕⠚⢋⠁⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠢⠤⡄⠀⠀⠀⠡⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠁⢀⢔⡩⢧⠀⠀⠀⠀⠀⠘⡉⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠁⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡇⠉⠀⠀⠁⠁⠀⠀⠀⠀⠐⠀⠈⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠐⡐⠀⠀⠀⠀⠀⠀⠈⢆⣀⡸⠀⠀⠀⠀⠀⠀⠐⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣀⠀⡠⠺⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢎⢖⡵⣻⠎⢢⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⡎⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣋⠎⠁⠀⢡⡑⠀⠀⢀⣃⣐⣒⣒⣒⣂⡀⠀⠠⠵⠦⠭⠥⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⣶⠠⢻⣏⣁⣨⠉⠉⠠⠾⠯⠭⠭⠭⠟⠒⠒⠒⠒⢂⣀⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣡⣟⡲⢶⣶⡬⠍⠉⠐⣖⣚⢋⣠⣦⣶⣦⣬⡯⠬⠵⠬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠖⠒⠒⠛⠛⠛⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⣠⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⢶⣄⠀⣾⣷⡄⣠⡶⠃⠀⠀⠀⠀⠀⠀⠀⣰⠆⠿⡿⠃⠀⠀⠀⠀⠀⠿⡿⢋⡷⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⢛⣿⠋
⠀⠀⠀⣿⡇⠀⠀⠀⠙⢷⣭⣭⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢾⣿⣿⡷⣦⠀⠀⠀⠀⢰⣿⡏⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠛⠁
⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⡩⡿⠋⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣯⢻⣿
⠀⠀⠀⣿⣇⣶⣶⣆⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⠟⠘⠟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⢿⡿

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⣠⣶⣳⣶⣤⠀⠀⠀⠀⠀⣴⠶⠶⠾⠻⠷⣶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠸⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃
⠀⠘⠿⣿⠉⣿⢸⡇⠀⠀⠀⣸⡏⠀⠀⠀⠀⠸⣿⠇⢹⣆⠀⠀⠀⠀⠀⣴⣿⡏⣿⣦⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣶⣾⣿⡄⣠⣶
⠀⠀⠀⣿⣿⣿⢘⣇⠀⠀⢰⡿⣶⣿⣿⣿⣿⣿⣿⣿⣶⢿⡆⠀⠀⢀⣤⣬⣿⣧⣿⣤⣽⣿⣿⣄⠀⠀⠐⠒⢶⣶⡿⠛⠛⢉⣹⣿⡇⣿⡏
⠀⠀⠀⣿⣿⣿⠸⠿⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⢹⢹⣿⢿⢿⣿⠉⠉⢹⡇⠀⠀⠀⠀⠀⣿⠀⢠⣾⠟⣛⣿⣥⣿⡇
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢘⣿⡟⠉⠉⠉⠉⠉⠉⠉⢻⣿⠃⠀⠀⠀⣾⠈⣿⣾⣼⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣼⡇⣾⣿⣿⣿⣿
⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⠈⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣐⠯
⠀⢰⣾⣿⡄⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣀⠠⠊⠃
⠀⠀⢻⣿⣿⡄⣾⣶⡶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡿⢶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢰⣶⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣶⡄⢀⠀⠀⠀⢸⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⠿⠿⠛⠛⠻⠿⠟⠸⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠿⠃⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠓⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠔⠒⠒⠒⢶⡇⢠⠖⠉⠁⢠⠋⠉⠑⢖⠲⡄
⠀⠀⠀⠀⠀⠀⣠⠖⢉⡽⠛⠛⢦⡤⠒⠒⠊⠓⠚⢷⡤⠒⠛⠲⣄⡠⠞⠒⠛⠲⣔⠒⠦⣄
⠀⠀⠀⠀⠀⢰⠋⠀⠘⢄⠀⠀⠀⢱⣀⡠⠖⠊⠑⡞⢡⠖⠉⠉⠹⣇⣀⡀⣀⠀⠀⣳⠀⠙⢣
⠀⠀⠀⠀⠀⠈⢦⢄⣀⠀⠉⠒⠴⠤⡿⣁⣀⠀⠀⢸⡇⠀⠀⡔⢉⣀⣀⡀⠤⠓⠉⠁⡀⢀⠞
⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⣲⢤⣀⡀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠁⠀⢀⣀⡤⠴⠒⠛⠛⢹
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠁⠀⠉⠉⠒⠶⠶⠶⠦⠶⠒⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢻
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣄⣀⣀⣸⠆
⠀⠀⠀⠀⠉⠒⠢⠤⠤⠔⠲⠆⠠⠖⠒⠲⠤⢄⣀⣀⣀⣀⡠⠴⠞⠒⢛⣉⠭⠤⢴⣉⠉⠉⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠙⡆⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣀⣀⡤⠞⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⣦⠀⠀⣴⠋⠀⢠⡏⢀⡴⠶⢶⡄⠀⠀⠀⠀⠀⠀⣴⠶⢦⣄⠸⣇⠀⠈⢷⡄⠀⢸⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣿⢀⣾⡇⢀⣠⣾⣷⡏⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⢳⣄⠀⠘⣷⣿⣦⣀⠀⣿⣆⢸⡄⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢴⣯⣿⠏⠁⢀⣿⠋⠁⠀⣸⣯⠞⢻⡆⠀⠀⠀⠀⣾⠳⣮⣧⡀⠀⠈⢻⣅⠀⠈⢿⣿⣷⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⣿⡉⠀⠐⢻⡏⠀⠀⢾⡿⠃⠀⣼⠃⠀⠀⠀⠀⢻⡄⠈⠻⣿⡆⠀⠈⢻⠁⠂⠀⣽⣿⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠹⣿⠁⢀⣀⡾⠁⣀⣠⣿⠁⠀⢶⠇⠀⠀⠀⠀⠀⠀⢳⡄⠀⢸⣧⣀⣀⠸⠧⢀⡀⢸⣟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⠻⠚⠁⠀⠀⠀⠉⠉⠉⠀⢤⣼⠀⠀⠀⠀⠀⠀⠀⢸⣇⠤⠈⠁⠈⠁⠀⠀⠀⠈⠛⠋⠂⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⣀⣀⣀⣀⣸⣧⣠⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠤⠤⠴⣿⠀⠀⠀⠀⠀⢀⣿⠤⠤⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣿⣠⣤⡤⢤⣤⡀
⠀⠀⢠⡞⠉⠉⠉⠉⠙⢿⡟⠛⠛⣽⣿⣏⣉⠒⠂⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠒⢉⠉⠁⠀⠀⠀⠈⣽⠟⠁⠀⠀⢀⣀⣿
⠀⠀⠀⠙⠛⠶⣦⡀⠀⠀⠙⠶⣤⣬⣭⣶⣿⡇⡢⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠶⠟⠁⠀⠀⣠⡞⠋⠉
⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⡆⣄⡀⢀⡀⠀⢿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢱⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠾⣿⠷⣹⣿⡆⣾⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠋⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⣇⣤⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⣭⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡟⠻⢿⠃⠀⢹⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣩⣤⡙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⣿⠇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢟⣃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⡧⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⡠⢆⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⣠⣔⡒⠂⠉⣀⣈⢦⡀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⢋⠀⠤⠄⡀⠀⠠⠤⢠⣹⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠈⢻⠀⠀⠀⠃⠀⠀⠀⡸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠵⢄⠀⠙⠁⣀⠴⢅⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⠁⠀⠺⣍⠀⢈⠝⠂⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠄⡈⣦⣎⡀⠄⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⢹⠉⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢀⠟⡏⡠⠞⠉⠉⠉⢹⠉⠉⠉⠙⢤⡈⡏⢇⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⣀⠀⠀⠀⣠⣏⣨⣧⣄⡀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣇⣧⣈⣧⡀⠀⠀⢀⡠⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠤⡀⠈⢣⣀⠔⠉⠉⠀⠀⠀⠉⠑⠒⠠⠬⠤⠐⠒⠉⠁⠀⠀⠈⠉⠑⢄⣀⠁⠀⡀⠼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠀⠀⠈⠲⠋⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠲⠊⠀⠀⢈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡌⢰⠃⠆⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⢤⡊⠉⠉⠉⠀⠉⠉⠉⠉⠉⠁⠀⠀⠆⢳⠈⡄⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣇⠎⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⡗⠚⠉⠉⠲⢄⠀⠀⠀⠀⠀⠘⠉⢆⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠓⠦⡄⠀⠀⢀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


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

⠀⠀⢀⡀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⢀⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠐⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⢢⣤⠀⠀⠀⠬⡄⠀⠀⠀⠀⠀⠘⠛⢩⡁⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⠀⠀⠀⠉⠀⠀⠀⠀⠉⠐⡀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠴⠿⠟⠫⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠀⠁⡀
⠀⠀⡎⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢤⡄⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠅⠒⠂
⠀⠀⠘⢄⠀⠠⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠃⠄⠅⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠈⠒⠀⠄⠀⠀⠠⡀
⠀⠀⠀⠀⠂⡺⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠳⠀⠀⠀⡀⠆⠀⠀⠄⠃⠀⠌⠀⠀⠐⢂⡀⡔⢀⡀⡄⡠⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠌⠄⠀⠁
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⢈⠆⠀⠀⠀⠀⠀⠀⠛⠱⢃⡙⡾⠜⢗⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠆⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡰⠀⠀⠀⠀⠀⠀⠄
⠀⡀⠀⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣤⡄⣠⠡⣶⣦⠀⡀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠠⢸⢇⡴⢁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠄⠀⢺⣿⣷⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢨⠁⢹⠄⠈⠉⣴⠁⠀⠈⠁⠠⠀⠀⠀⠀⠀⠀⡐⣮⠀⠁⠀⠠⠀⠀⢀⡀
⠀⠲⢦⡀⠛⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⣈⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣹⠇⡜⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢐⢳⡌⠀⠀⠁⠀⠀⠚⡇⠐
⠀⠂⠀⣈⣆⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠬⢽⠈⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠁⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠋⠠⣀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠉⠀⠀⠀⢀⠁⢹⣆
⠀⠀⠀⠀⠀⣰⠿⠁⠘⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⠤⣀
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⣀⣀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣏⣈⣷⡶⠶⠶⣭⣓⣲⣤⣄⡀
⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠀⠉⠙⣿⣦⡤⠞⠉⠁⠀⠙⢿⣍⣷⣦⡀
⠀⠀⠀⠀⠈⠛⠲⢦⣴⣴⠶⠞⠛⠛⠛⠋⠉⢀⣠⠴⠶⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⡆⠞⠀⠀⠀⠀⠀⣻⠟⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠁⠀⠀⠀⠿⠉⠀⠀⠙⠉⡙⣦⡄⠀⠀⢰⣷⢡⣀⡀⠀⠀⢠⡟⣦⣶⡀⠀⠀⠀⢀⠀⢹⢇⡽⢃⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠟⣿⡄⠀⢾⠈⠈⢻⡷⠄⢠⣾⣥⡿⠟⠛⠳⠶⣶⣿⣣⣯⡞⢠⠀⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡇⠀⠘⡆⠀⠀⣿⡙⠛⢿⠋⠀⠀⠀⠀⢀⡈⣿⣿⠏⣰⡟⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢶⣶⣤⣄⣀⠈⠻⠶⣶⣶⠀⠀⠀⠀⣀⠀⢀⣤⣴⠟⠁⠀⣼⢷⠀⠀⡘⠿⣶⣿⣶⡾⠴⠶⠶⠟⠛⣛⡡⠚⠃⢠⡿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠈⠁⠛⡻⠿⢿⣄⠀⠈⠛⠷⠶⠶⠷⠿⠾⠛⠛⠁⠀⠀⢠⠃⠀⢳⡀⠙⠷⡦⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⢠⡾⠛⠉⠽⠇⠀⠀⠰⠀⠃⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠙⢦⣄⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⠋⠉⢀⣿⠇
⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⢛⣛⣶⣾⣿⣿⣿⣯⣀⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⠛⢶⣶⣶⡶⠟⠛⠉⠁⠀⠀⠀⠀⠻⠿⢛⡷
⠀⠀⠀⢿⣄⠀⠀⠂⠀⢀⣼⡇⠠⠀⣴⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠚⠋⠁
⠀⠀⠀⠀⠙⠳⠶⠖⠒⠛⠋⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠤⠶⠶⠾⠿⣷⡾⠶⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠊⠉⠉⠉⠒⠦⣄⣀⣀⣀⡀⠀⠀⢀⡤⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠋⠀⠀⠀⠀⠀⠙⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠑⠲⢄⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⢢⣀⠀⠀⢃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⣀⣤⣥⣬⠷⠚⠉⣉⡙⠋⠉⠙⠲⠶⠶⠒⠳⢤⡀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⡂⠤⣠⠞⠁⠀⠀⣠⠴⠚⠉⠉⠉⣉⠿⠿⢷⣀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠘⡟⢦⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⢀⣠⠜⠋⠀⢀⡤⠞⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠨⣙⠲⣄⠀⠀⠘⡇⠀⠀⠀⠀⠧⠀⠹⣦⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢸⠄⠀⠀⠀⢀⡴⠋⢀⣀⢀⡴⠋⠀⠀⠀⠀⠀⡴⠋⠁⠠⡀⠀⠀⠀⠀⠠⡈⣧⠀⠀⢻⠀⠀⠀⠀⠘⣄⠀⠈⢧⠀⠀⠈⠛⠳⢦⡀
⠀⠀⢀⡴⠋⠀⠀⠀⢠⡏⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢄⠀⠀⠀⠀⠐⠘⣆⠀⠈⠳⣄⠀⠀⠀⠀⠈⠳⡌⣧⠀⠀⠀⠀⠀⢳
⠀⡴⠁⢠⡞⠉⠳⢤⣸⡀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⣠⠜⠁⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢄⠈⢧⡀⠀⠈⠓⠦⣄⠀⠀⠀⡇⢸⠀⠀⠀⠀⢠⠏
⠀⡇⠀⢸⡇⠀⠀⠀⠻⠷⠖⠓⢾⡀⠀⠀⠀⠀⡔⠉⠀⠀⠐⠀⠢⢀⠀⠀⠀⠀⠡⠀⠀⠈⠄⠀⢳⡀⠀⠀⠀⠈⠻⣆⠀⠙⣼⠀⠀⠀⠀⢸
⠀⢹⠀⢸⡇⠀⠀⠀⠀⠀⠀⠄⠀⢳⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⠀⢃⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⢸⡆⠀⢸⠀⠀⠀⠀⠸⡆
⠀⠀⣧⡼⠃⠀⠀⢀⠀⠀⠊⠀⠀⣸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠄⠀⠀⠀⠀⢸⡆⠀⠀⠀⣠⠞⠀⠀⢸⠀⠀⠀⠀⠀⠙⣆
⢠⡞⠁⠀⠀⠀⡔⠁⠀⠀⠀⠀⣠⠇⠀⠀⣀⠴⠁⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠀⠑⢜⠀⢀⠀⠀⣸⠀⠀⢠⡞⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢷
⣸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢐⡏⠀⠀⢸⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⣌⠂⢀⡟⡄⠀⠸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈
⡟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⣇⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠌⠣⣼⠀⠈⠓⠄⢹⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⡾
⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⢦⡀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠑⠄⣠⡴⠋⠀⠀⠀⢀⡞⠀⡂⠔⠁⠀⡴⠋
⢻⠀⠀⢠⠀⠀⠀⠁⡀⠀⠀⠀⠠⠊⠙⠲⢤⣌⣣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣰⠃⠀⠀⠀⠀⢀⣼⡃⠀⠀⠀⠀⢸⡇
⠸⡇⠀⠠⠀⢀⠀⠀⠀⠀⠂⠄⣀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠶⠤⠤⠤⠤⠤⣤⣀⣀⡀⠀⣿⠀⡞⠁⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⢸⡇
⠀⢣⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠠⡀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡩⠓⠿⣾⠁⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠈⢧⡀⠈⠢⠘⢄⠀⡈⠀⠀⠐⠁⠀⠀⠈⠁⠉⠁⠀⣠⠊⠀⠈⠑⠂⠄⠀⠀⠀⠔⠁⠀⠀⠀⢸⠀⠀⠀⠀⡴⠏⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠳⣄⠀⠑⠄⠑⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣮⣤⡤⠴⠊⠑⠈⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⡀⠀⠐⠀⠀⠄⣀⣀⣀⠀⠀⣂⠁⠀⠀⠀⠀⠀⠀⣠⠞⣁⣤⠤⠦⠤⣤⡤⠶⠚⣩⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠈⢄⠀⠀⠀⠀⠀⠀⣠⠞⢉⣭⣙⢦⣀⠀⠀⣠⠞⣡⠞⠁⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠾⢶⣶⡖⠒⠋⠁⢀⡏⠀⠈⠓⢌⠛⠋⣡⠞⠁⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⢾⣀⠀⠀⠀⠈⠙⠚⠁⠀⢀⣀⡤⠶⠚⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣷⣶⣶⣶⣶⣾⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠂⠈⠉⠉⠉⠉⠉⠉⠉⠓⠒⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⠀⠉⠓⣄⠀⠀⠀⣿⠀⠀⠀⡇⠉⠙⠲⡄⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣆⣀⡠⠏⠀⠀⠀⣿⠀⠀⠀⣇⣀⡤⠔⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠿⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠈⠳⣤⠀⡄⠀⡏⠠⠄⠰⠇⠀⠤⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢆⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠁⠈⠄⣀⣀⣀⣁⣀⣀⣀⣀⣒⣀⠀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠤⣄⠀⠀⠀⠊
⠀⠀⠐⠲⡄⠀⢀⠟⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠒⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠂⢀⡤
⠀⠈⢷⡄⢠⡆⡸⠀⡀⠀⢠⡀⠀⠀⢀⣀⠢⢀⠐⠀⢇⢈⡀⠀⠀⠀⣄⠀⢀⠄⠀⠀⡠⠒⡀⢠⠀⠀⠀⢇⢔⠜⠁
⠁⠀⠘⡘⠎⠂⠀⠨⠴⠀⠠⠱⠰⠄⡈⠀⠙⠀⠹⠇⠂⠀⠛⠒⠀⠀⠈⠀⠑⠈⠠⠊⠄⠀⠣⠀⠃⠀⢀⠀⠀⠒⠃⠀⢀
⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠻⣦⠀⠀⣙⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⡇
⠀⠀⠀⠚⠛⠛⠀⣇⠀⠀⠀⢸⡇⠘⠛⠛⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⢻⡆⠀⠀⠀⠀⠀⠀⢸⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣠⠘⠳⠶⠶⠋⣤⡀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠙⠳⢦⣤⣤⠄
⠀⠀⠀⠀⠀⠞⠁⠀⠀⣾⠀⠀⠈⠻⠆⠀⠀⠀⠀⢰⡏⢷⡀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⠻⣄⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠷⣄⡀⠈⠛⢶⡋⠁⠀⠀⠀⢀⡴⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠉⠛⠶⢤⣽⣆⠀⠀⢀⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣤⡼⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠶⠶⠶⠶⠖⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠰⠶⠶⠶⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⢩⡟⠀⠀⠀⠀⣠⡴⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⣏⠉⠉⠉
⠀⠀⠀⠀⠀⠚⠛⠛⠛⡿⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠛⠛
⠀⠀⠀⠰⠶⠶⠶⠶⣾⠃⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃

⡆⢑⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡊
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⡞⠑⢆⣶⠉⡏⢀⠶⡀⣾⣱⠀⡆⠀⣎⡁⠀⢰⣉⠈⢻⠁⣞⣱⢠⠊⠑⡄⣆⠖⢰⣉
⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣧⣤⣃⣟⣀⣇⣼⣉⣃⣻⣼⣃⣧⣄⣧⣄⣀⣠⣼⣂⣘⣀⣏⣹⣘⣦⣴⣃⣇⣱⣘⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣾⠋⠁
⠀⠀⢿⡈⠁⠀⠚⣍⠙⠦⡀
⠀⢠⡾⠁⠘⣶⠋⠉⠳⢄⠘⠷⣄
⠀⠀⢷⡈⠀⠈⢳⡀⠀⠀⠙⠲⣄⠑⠢
⠀⠀⢨⣷⠀⠈⠵⠻⢄⠀⠀⠀⠘⢷⣄⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⡄⢹⣿⠀⠘⡆⠀⠀⠀⣄⠀⠠⡶⣌⢳⡀⢀⣧⣀⠀⠀⠀⠈
⠀⢻⠘⣿⠀⣀⡀⠀⠀⠀⠈⠳⣄⡀⠘⡇⢿⡉⠈⢿⣴⠃
⠀⠈⠀⣹⡜⠃⠀⠰⣄⠀⠘⠉⠣⣿⠄⠀⠈⠻⡄⠀⠙⡆⠐⣦⡴
⠀⠀⢀⣿⣿⣆⠀⠀⠈⠣⡀⠀⠀⠆⢠⠀⠀⠀⠹⣆⠀⠀⠀⡙⠦⣼⡄
⠸⠀⢸⣿⣿⣿⣄⠀⠠⢶⡌⣆⠀⠀⠀⢷⡀⠀⣠⣼⠇⠀⠀⠘⣦⡀
⠄⠀⢸⣿⣿⣿⣿⣆⠀⠀⠁⠘⣧⠀⠀⠈⠃⠀⠋⠙⢷⣦⡀⠀⠈⠻⣖⢤⣀
⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠃⠙⠀⠀⠀⢸⠀⠙⠧⡄
⢇⠀⠈⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡀⠀⠻⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠹⣇⠀⠙⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠹⠀⠀⠈⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠹⣄⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⡀⠇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣷⡾⡀
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⡄⠀⠢⣀⣤⣶⣶⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠸⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠔⣹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢨⣷⡀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣴⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⢠⠃⢸⣿⣄⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⢰⠋⠘⠷⠤⠤⢄⡀⠈⢿⣿⡿⠁⣠⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢀⡔⠁⠀⠀⠀⠀⠀⠀⢿⣆⠀⣿⠃⣴⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣰⣇⣼⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⢛⣛⣛⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣦⡀⠀⠀⠀⢀⣀⡼⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣧⣠⣿⣿⣿⠶⠖⠋⠉⠀⢀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⣠⣾⣿⣿⣿⡇⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠸⣿⣿⣿⣿⠟⠁⠀⠀⢀⣾⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⣀⣴⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠋⠉⠹⡿⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣷⣿⣿⣵⣾⣷⣯⣿⣛⣭⣾⣆⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣟⡿⣿⣿⣧⣄⣿
⣿⣿⣻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⡿⣵⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⣿
⣿⣿⢿⣿⠀⠰⡶⣤⣄⣠⣤⣄⣄⣤⠀⠀⠀⣿⣿⡇⠀⠀⠊⠀⠀⡀⠉⠀⠃⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽
⣿⣿⣿⣿⠀⢀⠁⠈⠘⠋⠙⠈⠈⠉⠀⠀⠀⣿⣿⡇⠀⠆⠀⠂⠐⠃⠀⡼⠱⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⣿⣫⣾
⣿⣿⣿⣿⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⣿⣿⡇⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⠀⠀⢠⠀⠀⠀⠀⣉⣉⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣯⣿⠀⠀⡼⠉⡀⡞⠑⣔⠐⠄⠀⠄⠀⣿⣿⡇⠀⠀⡍⠉⠁⠈⠀⠐⠀⠂⠀⠀⢸⣿⣿⠀⠠⠨⠀⠀⠀⣀⠛⠋⠁⠀⠀⠀⣿⣿⣟
⣿⣷⣿⣿⠀⠈⠙⠶⠃⠳⠴⠻⠴⠃⠈⠀⠀⣿⣿⡇⠀⠀⠀⠀⠰⠬⠀⠀⠸⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣽
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢀⠀⠁⠀⠀⡀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡇⠀⠀⠃⠈⠀⠀⠁⠁⠀⠈⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣿⣾⣧⣄⣤⣠⣠⣀⣈⣄⣡⣀⣠⣠⣄⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣿⣿⣿
⣿⢿⣯⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⢿⣟⣿⣿⣿⡿⣟⣿⣻⣿⣿⣿⣿⣻⣛⢿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿
⣷⣾⣾⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣷⣿⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠛⠛⠛⠙⠛⠛⠙⠛⠛⠛⠉⠉⠋⣿⣿⡟⠙⠋⠉⠉⠋⠋⠙⠛⠋⠙⠋⠋⢻⣿⣿⠋⠛⠛⠋⠋⠛⠙⠙⠋⠙⠛⠋⠛⣿⣿⡟
⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠆⢀⠄⠀⠣⢂⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⣿⢻⣟⣿⠀⠀⠀⠀⠀⣀⠀⠐⢄⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢈⡆⠄⣌⢠⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢠⠤⣄⠞⢣⣰⢤⡀⠀⠀⣿⣿⣿
⣿⢺⣿⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⡄⠀⠀⣿⣿⡇⠀⠀⢀⡈⢀⠀⠁⡌⣀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢯⣰⡾⡀⣸⢻⢀⡟⠀⠀⣿⣿⣿
⣿⣧⡹⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢨⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠉⠀⠈⢄⠈⠉⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠠⠌⠄⠀⠈⠉⠈⠇⢀⠔⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣀⢁⢲⣁⠃⠈⡁⡀⠐⠄⠀⣿⣿⣿
⣟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⣿⣿⡇⠀⠀⣀⢆⠀⠀⠀⠀⡄⠀⠀⠀⢸⣿⣿⠀⠀⠄⠸⠏⠷⢴⠺⠗⠏⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠲⢠⠄⠤⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠲⠄⢠⠀⠤⠐⠄⠀⠀⠀⣿⡿⣻
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡟
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣟⡧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢿⣽⣿⣯⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽
⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣶⣠⡄⡄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣠⠀⣤⠀⡀⠀⠀⠀⣥⠏
⠀⠀⠀⠀⠀⠀⠀⢀⡰⢡⣏⣻⠸⠄⠀⡻⡇⠃⠀⢀⣀⠀⠀⠀⠀⠠⠄⠐⠃⠄⠈⢁⠄⠘⠁⠛⠀⠀⠀⠘⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢉⠋⠁⠀⡂⠀⡇⠀⠀⠀⡎⠀⠀⠀⠠⠄⠄⠂⢀⠦⠚⠰⠂⠀⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠨⠇⣀⠀⠀⠀⠀⢧⡀⠣⢀⢦⠈⠠⠁⡇⠐⠘⠀⠀⠀⠐⢀⢄⠭⠃⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢳⡄⠀⠁⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣠⠁
⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠒⠂⠤⠤⠠⠐⠀⠀⠀⠀⠂⡄⠠⡠⠀⠆⠀⣠⠀⠀⣠⠀⠀⠀⠀⠀⠀⢲⠦
⠀⠀⠀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠥⠈⢒⠋⢰⡌⣇⡌⠃⡀⠀⠠⣾⠀⠀⢈⠀⠐⢀
⠀⠀⠀⠻⠂⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠔⠚⠥⠀⠀⠀⠙⠁⠘⠁⠀⠀⠀⠁⠠⠠⠀⠀⠁
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠰⠅⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠈⣂⠰⡤⣀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⢰⣋⠀⠀⠠⠀⠀⠘⠀⠀⠀⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠁⠚⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠘⠀⢀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠲⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠀⠄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠐⠁⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣠⣶⣦⡄⠁⡀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠄⣀⠠⣾⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠁⠀⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣿⣿⣻⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠉⠀⠠⠄⠀⠀⠠⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠛⠀⠀⠀⢴⣾⣿⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣷⣤⣀⡀⣀⣠⣾⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠚⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⡷⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠉⠻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⣀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⠚⠛⠛⠛⠓⠒⠲⠶⠤⢤⣄⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠈⢧
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈
⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠴⢄⣀⠀⢀⣠⡾
⢸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⡏
⡻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣢⠀⠀⠀⠀⠀⠀⢀⣴⠏
⢹⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁
⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⣀⣠⣴⠟⠁
⠀⠈⠳⣄⠀⠀⠀⠀⠠⠤⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠁
⠀⠀⠀⠀⠙⠲⠤⣄⣈⡛⠊⠀⠀⠀⠀⠀⠙⠒⠒⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠀⠀⠸⣿⣿⡆
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣄⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆⠀⠀⠀⠻⣿⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠛⢿⣿⣆⠀⠀⠀⠹⣇
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠙
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠹⣦⡀⠀⠀⠀⠙⠂
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⠻⢦⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⣿⣿⣿⣷⣦⣄⡀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢂
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠟⣡⣴⣿⣷⡄
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⢋⣴⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⠀⠀⠀⢀⣤⡿⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⣠⣶⠟⣋⣤⣤⣼⡗⠀⠀⠀⠀⠀⠀⣠⡾⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣴⠿⠋⠁⣰⡟⠁⠈⢻⡇⠀⠀⠀⠀⠀⢰⣿⠁⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢁⣤⣄⠀⣿⠇⠀⠀⠸⣿⡀⠀⠀⠀⠀⢸⣇⠀⠈⢿⣆⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣾⡏⠙⠿⠟⠀⠀⠀⢀⣽⡷⠀⠀⠀⠀⠘⣿⡄⠀⠀⠙⠿⢶⣶⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⣀⣠⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠙⠿⣦⣄⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠻⣦⣀⠀⠀⠉⠙⢻⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠈⠛⠿⢶⣦⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣼⠟⠁⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⢀⣀⣤⣤⣶⣄⣼⠏⠿⠋⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⠾⠿⠛⠋⠉⠁⠘⣿⡏⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⡀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⢿⣿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠹⣿⣤⣤⣀⣤⣤⣤⣤⣤⣴⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢀⣿⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣿⣷⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⢦⡉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠓⢄⠀⠉⠳⠦⡀⠉⠀⣄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢢⠀⠑⠦⣀⠀⠈⢢⡀⠀⠙⠦⡅⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⣄⠀⠱⣄⠀⠈⣧⡀⠀⠙⢦⡀⠀⠘⣦⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣦⠀⠈⠣⡀⠈⠻⣆⠀⠀⠑⠀⠀⠈⠳⣄⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⣧⡀⠀⠐⣄⠀⠈⠳⡄⠀⠀⠁⠀⠀⠀⠑⠺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⠉⢿⡀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡄⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢚⢦⡈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠹⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢱⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠈⠀⠻⣆⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⣆⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⡿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣄⣻⣤⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣬⣉⠛⠟⡡⠖⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢠⠛⣿⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⢠⡝⢁⡾⠁⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠘⠁⠜⠉⢳⢦⣀⡀⠀⢀⠄⠀⣠⠟⠁⣾⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡟⠳⣄⡀⠀⠀⠀⣸⣿⠞⠁⣠⣾⡏⠀⢸⠏⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠐⠒⠈⠙⠣⢠⣿⡟⣡⣴⣿⠛⠉⢀⣠⠞⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡸⣆⠀⠀⠀⠀⢸⠏⣼⢫⣿⣃⣤⣶⠟⢉⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠳⠄⡁⠀⠈⢩⣏⣼⢣⣿⣿⣿⣿⣥⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣾⣿⣿⣿⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠾⠛⠋⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣻⣫⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣽⣯⣽⣿⣿⣿⣿⣯⣭⣿⣿

⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠙⠿⢿⣿⣧⠌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⠒⠀⠀⠄⠊⠹⣷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⠲⠤⠀⠀⣠⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠰⢻⠉⠛⠒⢞⠋⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⡠⠾⣿⡿⠋⠸⣟⣧⡀⣆⠀⠀⠈⠀⠀⣸⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢣⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⡘⡠⠛⠦⣄⣀⣀⡙⠿⣿⠛⠗⠒⠃⠀⢠⣇⡤⠜⠋⠉⠉⠉⠉⠉⠒⡴⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸
⡿⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠄⠂⠈⢢⡿⢴⡀⠀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣤⠃⠀⠀⠀⠀⠤⢤⣀⠀⠀⠀⠀⢸
⣷⠀⢀⠄⠀⢼⠘⢧⡀⠀⣥⠄⠀⣠⠞⠁⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⢳⡄⡀⠀⢸
⣿⠀⣾⠀⠀⠈⢿⡄⠙⠓⠲⠖⠛⠁⠀⠀⠀⠀⣸⠀⣠⡴⠖⠲⠦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣡⣀⣠⠤⠴⠚⠀⣷⠀⢸
⣿⠀⠘⢷⣄⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣣⠞⠁⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⢸
⣿⠀⠀⠀⠙⠻⢦⣤⣈⠑⠒⠢⠴⠖⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿
⣿⣿⣿⣿⣛⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣿⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠐⠂⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢂⠁⢄
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⢃
⠀⠀⠀⠀⠠⠀⠀⢀⣀⣀⡀⠀⠀⠀⣈⢀⡀⠀⢀⡠⣄⣀⠠⢁⢀
⠀⠀⠀⠀⠁⠀⠀⠀⡇⠁⡉⠄⠠⠀⢸⠀⠀⠀⢨⠂⠀⡀⠢⠀⠠⠁
⠀⠀⠀⠀⠃⠆⠀⠀⡇⠟⢕⠀⠀⠀⢸⠀⠀⠀⢸⠠⠛⠓⠀⠀⠰⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠆⠸⠠⠠⠄⠼⠰⠄⠄⠼⠰⠄⠤⠈⠀⡠⠪
⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡂
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⢀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢇
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡩
⠀⠀⠀⠀⠇⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠠⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⡄⠚⠚⠐⠐⠉⠉⠉⠀⣏⠓⠒⣐⣤⡤⣠⣀⠀⠀⢀⡇
⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠓⠀⠀⠀⠂⠀⠀⠧⢀⣋⡀⠲⢙⠽⣆⣀⡆
⠐⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠒⡠⣰⠀⠴⡈⠉⢏⣿⡟⢶⣤⡀
⠀⠀⠹⡀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⡀⠈⠂⠀⠁⠈⠀⠿⢓⣘⠂⠟⠲⠖⠙⠛⣦⡀⠄⠊⠡⠐⠀⢀
⠀⠀⠀⠈⠓⠤⢄⢀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠋⠲⣖⢐⣠⠔⢺⠞⠁⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠖⠲⠖⠲⠒⠲⠶⠦⠦⠖⠛⠒⠖⠓⠉⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⣈⡻⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⢈⣉⠁⡄⢈⣉⠉⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠃⠀⠀⢀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣄⠈⠉⢁⣠⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⢿⣁⣀⣘⠿⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢷⠀⠀⠀⠀⠀⠀⢠⢇⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠘⡀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣀⠇⠀⠀⠀⠀⠀⢸⡸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⡄⠀⠙⢤⣀⣀⣀⡀⣀⠜⠁⠀⣼⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⣀⣀⡈⠢⢄⣀⡬⠭⠆⠈⠀⣀⡤⠊⠀⣀⣉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠑⠲⠤⣉⡓⠒⣉⠭⠕⠒⠋⠉⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣄⣘⣷⣤⣄⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠀⠀⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅⠀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠄⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⣤⡍
⠀⠀⠀⠀⠀⠀⢀⡄⡸⠋⠀⣠⣗⢠⡦⠀⠀⠀⠀⠀⠀⠀⠰⢦⣄⡀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠀⣀⠘⠷⣀⣤
⠀⠀⠀⠀⢈⡏⠞⠓⠁⢀⣤⣼⡏⢈⢡⣴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡟⡋
⠀⠀⠀⠠⠈⠁⠀⠀⢀⣾⣇⠈⠁⢨⣶⣉⠛⠙⠃⠀⠀⠀⠀⠀⡀⢠⠂⣠⡾⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠇
⠀⠀⠀⠀⠀⣀⠞⠃⢌⡼⢿⣷⣶⡎⠉⠁⠀⠀⠀⠀⠀⣀⣀⣠⣤⣽⠾⠛⠁⣀⣁⣤⣭⣱⡦⠀⠀⠀⠀⠀⠰⠟⠛⠓⠚⠒
⠀⠀⠀⠀⠀⠀⠀⠐⡞⠉⢻⢿⠟⢧⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠂⠙⠛⠲⠛⠋⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⡌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣷⣎⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⣐⣶⣄⡄⠀⠀⣼⣿⣿⣿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠆⠰⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⣋⠛⠁⠀⠈⢙⢉⣹⣿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣟⣿⠓⠀⢘⡻⡿⡟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇⠀⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⢹⡿⠀⠀⠀⠀⠀⠀⠀⠲⠬⣿⣾⡆⣸⣿⡿⣛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢺⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠾⣿⠛⣿⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠞⠉⠁⠐⡎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⣀⣤⣴⡶⠾⠿⠿⠿⠿⠿⢷⣶⣤⣶⣦⡄⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣠⡴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⠉⠛⠋⡉⢉⡀⢠⡀⠀⢦⣰⠆⠀⠀⣴⡟⠙⢿⣾⣇⡴⠟⠋⣉⠉⠉⠉⢛⡛⠛⠛⠛⢉⡥⠀⠘⣧⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣄⣤⣾⣿⣻⠃⠀⢹⣀⣇⣿⠹⣾⡇⠀⠀⢿⠀⠀⢰⣿⠑⠲⣬⣿⣋⠄⠀⢸⡽⣇⢹⡀⣯⣿⡈⡇⠀⢸⡖⠃⠀⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⠀⠀⠸⠉⠟⠙⠞⠁⠛⠛⠀⠘⠀⠀⢸⡏⠠⢤⣈⣿⡵⠂⠀⠸⠷⠟⠘⠃⠘⠚⠃⠛⠚⠁⠙⠁⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣾⣿⣧⣿⣿⡟⠀⠀⠠⣤⣤⡤⠤⠀⠀⠤⠶⠶⠶⠆⠀⣿⠀⣄⣈⣿⠖⠋⠀⠀⣠⠤⠤⠤⠀⠀⢀⣠⣤⠴⠆⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠐⠒⢒⣚⡃⠀⠀⢘⣉⣉⣉⣉⠀⣿⡆⣀⣈⣿⡖⠋⠀⠀⠖⢒⣚⣒⠀⠀⠠⠤⠴⠚⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⠸⢿⣿⣿⠀⠀⠀⠩⣭⣭⠭⠤⠀⠀⠡⠤⠴⠦⠤⠀⣿⣇⣀⣙⣿⡴⠂⠀⠀⢛⣉⣉⣁⠀⠀⠘⡒⣚⣋⡥⠀⠀⢸⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⣼⡿⠀⣸⣽⡿⠀⠀⠀⠲⠶⠖⠒⡃⠀⠀⠐⠒⠛⠛⠓⠂⣿⣇⣀⣉⣿⡴⠂⠀⠀⣩⠤⠶⠦⠀⠀⠀⣍⣩⡤⠴⠂⠀⠘⣿⣿⡇⣿⢻⣿
⠀⠀⠀⠀⢀⣿⡇⡿⣄⣾⡇⠀⠀⠀⣉⣉⣉⣉⠁⠀⠀⢈⣉⣉⣉⣉⠀⣿⣇⣀⣈⣿⡴⠚⠀⠀⠤⠶⠒⠒⠂⠀⠀⠤⠤⠴⠚⢁⠀⠀⣿⣿⡇⣿⢸⣿
⠀⠀⠀⠀⢸⣿⢰⠇⠁⣾⠁⠀⠀⠀⣍⣩⣭⡤⠄⠀⠀⢀⣤⣤⣤⣤⠀⣿⣇⣀⣩⣿⡖⠉⠀⠀⠀⠛⠉⠉⠉⠀⠀⠐⠒⠒⠋⣉⠀⠀⣿⣿⡇⠻⠈⣿⡂
⠀⠀⠀⠀⢸⣿⡆⣾⣄⣿⠀⠀⠀⠀⠤⠤⠤⠶⠂⠀⠀⢀⡤⠤⠤⢤⠀⣿⠇⢄⣉⣿⡶⠚⠁⠀⠐⠛⠉⠉⠉⠀⠀⠐⠶⠖⠛⠁⠀⢀⣿⣿⣇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠰⠶⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⣸⡿⠲⠤⣬⣿⣡⠶⠂⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣤⡄⣿⡇
⠀⠀⠀⢠⣿⢻⣿⣿⣿⣷⣤⣄⣀⣀⣀⣀⣀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣷⣤⣄⡈⣿⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣟⣿⣿⣿⡏⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣀⣀⣀⣀⣀⣈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠇⠀⠒⠀⠀⠀⠐⠂⢨⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡦⠀⠉⠀⠀⠀⠈⠁⢠⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⣲⠤⣀⡠⢴⡋⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠿⠀⠀⠈⠳⡀⡰⠋⠀⠀⠸⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠉⠂⢵⠥⠊⠁⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡋⠀⠀⠀⠀⠀⡰⢫⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⡀⠀⠀⠀⠀⠈⣣
⠀⠀⠀⠀⠀⠀⢠⡎⠉⠉⢢⠀⢀⠾⠓⠋⠉⠑⠂⠤⣀⣀⣀⡀⣀⡠⠄⠒⠉⠉⠓⠻⢆⠀⢠⠊⠉⠉⢦
⠀⠀⠀⠀⠀⡴⠋⠀⠙⢦⡰⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣠⠞⠉⠈⠳⡄
⠀⠀⠀⠀⢸⢱⢠⠂⡤⢸⠃⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠤⣀⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⢸⠠⡄⢢⢰⢹
⠀⠀⠀⠀⠘⢦⠈⢸⠱⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠒⡤⠤⠒⠒⢄⡀⠀⠀⠀⠀⠈⠶⢹⠘⣠⠞
⠀⠀⠀⠀⠀⠈⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡇⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠤⠤⠧⠤⠤⠤⠤⠤⠤⠤⠤⠷⠴⠒⠒⠉

⠀⠤⢤⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀⣀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠈⠁⠀⠁⠐⠀⠈⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⡄⠀⠀⢘
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠌⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⡀⠀⠀⢀⠐⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸
⢨⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⡄⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠀⠀⠈⠀⠀⠈⠀⠀⠀⠠⠈⠄⢸
⢸⠀⠀⠀⠄⠀⢠⢀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠄⡀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠂⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠸⠈⠆⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠈⠃⠉⠐⠂⠁⠀⠆⠌⠀⠀⠀⠀⡀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠
⢸⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠠⢤⣄⠀⠈⠁⠀⠄⠀⠀⠠⠀⡀⠀⢤⠀⠀⠀⠀⢠⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣉⣤⣼
⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⡤⠤⠤⢤⣤⡴⠄⢰⢼
⢸⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⡀⠀⠀⢀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢃⠈
⢸⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⢠⠐⠄⠡⠀⠀⠀⠂⠀⠐⠚
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⠀⠀⠀⠀⠀⠀⠂⠲⠀⠀⡐⠀⠀⠐⠀⠀⠀⠐⠀⠈⠀⢀⠁⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠐⠀⠐⠂⠀⠀⠀⠀⠀⠉⠁⡝
⢈⠀⢠⠒⠀⠗⢀⠁⠀⡞⣀⠁⡶⠀⠂⠀⠁⠀⠀⢀⠀⠠⡀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠋⠀⠠⠀⢠⠀⢀⠀⠀⢀⠉
⠀⠀⠀⠀⠀⢠⠋⠀⠀⢣⢉⠀⢦⠀⠀⠀⠀⠀⠀⠀⢀⠀⡿⠀⠀⠆⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⢂⡙⡄⠠⠟⠀⠀⢠
⠀⠀⠀⠀⢠⡈⠲⠀⠀⡃⢈⠀⢁⠀⠀⠀⠀⡀⠀⠠⣀⠈⠀⠀⡰⠀⠣⠄⠀⠠⠀⢀⢀⡀⠀⠐⠀⠀⠀⠂⠀⠀⢡⠌⠀⣀⠀⠀⠇
⢀⠀⣀⠀⠿⣀⣸⠄⠂⠇⠸⠦⢈⠀⠀⠀⠀⠁⠀⠠⡀⠓⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⢠⠀⢰⠐⠃⠀⠷⠄⠰
⠠⠀⠛⠀⠀⠘⢿⠤⠁⠀⢠⠐⣁⢀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠂⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠑⠚⠀⠘⠰⡀⠀⠀⡆
⠐⠀⠀⠒⠰⣖⢠⠀⡄⢂⠸⠓⣛⠉⣠⠀⠀⠀⠀⠀⠒⡀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠴⠂⠐⠠⠲⠖⠀⡷⠲⡆⢳⠈⢶⢶
⢠⠀⠀⠀⠀⠈⠇⠙⠀⠹⠀⠀⠃⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⢰⠠
⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠂⢘⠀⠂⠐⠛⣀⠉⡇⢸
⢸⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢀⠀⠄⠀⠈⠁⠀⠀⠀⠀⠐⡇⢸
⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠠⠀⠀⠀⠀⠀⠀⢤⡤⢀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⣀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠠⠀⠀⠁⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⢽
⢸⠀⠀⠹⠒⠀⠆⠀⠐⠂⠀⠰⠆⠀⠴⠴⠆⠀⠀⠀⡄⠀⠀⠣⡀⠀⠂⠀⠉⠁⠈⠓⠀⠀⠀⠀⠀⠐⠀⠐⠐⠆⠀⠒⠒⠐⠛⠂⠂⠒⢸
⢸⠀⠀⢠⠃⠀⠀⠂⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠠⠙⠀⠀⠐⠈⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠂⠀⠐⠈⠉⠃⠘⠃⠂⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠴⠀⠉⠀⣤⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾
⢸⡁⢰⠀⠀⠀⠀⠀⣠⠄⢉⠡⠐⠒⠋⠀⠀⠀⠀⡀⠗⠀⠑⠀⠃⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼
⢰⠉⠀⠀⡈⠓⠀⠀⠶⠁⠰⢀⣀⠀⠀⡀⣀⠀⢀⡀⣈⢀⠀⢀⠀⡀⢀⠀⡀⠀⠄⠀⢀⣀⠀⣠⠀⠀⠀⡆⠈⠃⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠘⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⠀⠠⠀⠶⠂⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠘⠂⠐⠲⠶⠶⠖⠷⠠⠖⠂⠀⠒⠲⠒⠂⠀⠀⠘
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣀⡤⠤⠴⠶⠒⠲⠶⠦⠾⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠒⠛⠷⢦⣄⡀
⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣾
⠀⢧⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⠁⠀⠉⠻⡓⠦⣄⣀⣀⣀⣠⡤⠊
⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠴⠒⠛⠐⠢⢤⡀
⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⡨⠷⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠀⠀⠀⠀⠀⠀⠈⢧⡀⡀⠸⡄⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣄⣀⣀⣠⣾⠟⢮⣦⣝⣦⡀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠛⠿⠿⠦⣄⣸⣷⡤⡀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣤⣄⣀⣶⡮⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⡴⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢡⠉⠑⠒⠢⠤⣀⠘⢏⠻⣄⠀⢀⢷⣤⠞⣀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢶⡀⠀⠀⠈⢢⣘⡆⠹⣧⣼⡼⠁⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢆⠀⠑⣄⡀⠀⠀⠛⢧⠀⢷⡟⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣆⠀⠀⠀⠘⡀⢠⠏⡇⠀⠀⠀⠈⡇⢸⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⣿⠃⠀⣧⡀⠀⠀⠀⢡⢸⠀⡠⠊⡠⠃
⠀⠀⠀⠀⣠⡤⢤⣤⣤⣄⣀⠀⠀⠀⠀⢸⠀⣸⠀⠀⠀⢀⠇⠀⢀⡏⢳⡀⠀⠀⡸⣸⡰⣡⠞⣠⡴⠖⠛⠋⠙⠒⠦⠤⡀
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢉⠢⣀⠀⠘⡄⢿⡴⣷⠀⡸⠰⠀⣸⠷⣄⠱⡀⢀⣧⡏⣔⣡⣾⠏⠤⠤⠤⠄⠀⢀⠀⠀⡈⠢⡀
⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠐⠤⡑⢄⠐⣼⡇⡇⢀⢇⡀⢰⠃⠀⠈⢦⣇⢸⡟⣼⠟⢉⣀⡀⠀⠤⠠⣀⡀⠀⠀⠐⠀⠀⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠈⠢⣧⠘⣿⡁⢸⡼⢠⠃⠀⠀⠀⠀⢿⡼⣱⠵⠊⠁⠀⢀⡄⠀⠀⠀⠈⠑⠢⣀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢌⣣⠘⣧⣸⢧⠃⠀⠀⠀⠀⠀⡾⢻⠁⠀⠀⠀⣠⠣⢩⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣇⠇⠀⠀⠀⠀⠀⢸⢁⠏⠀⠀⠀⡴⠁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣎⢻⠀⠀⠀⠀⠀⢀⠇⡞⠀⠀⢠⠎⠀⠀⢸⢁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⡄⠀⠀⠀⠀⡞⡸⠁⠀⢠⠃⢀⠀⡠⢣⠏⡜⠀⢀⡠⠄⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠚⠛⢲⣤⡀⠀⠀⢧⡇⠀⠀⠀⡸⢱⠃⠀⢀⡎⣰⣋⠔⡱⢃⣼⡴⣊⡡⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⡠⠒⠋⠀⠀⠀⠀⠐⠉⠁⠀⠈⣉⣰⣿⠸⠀⠀⢰⢣⠇⠀⠀⣼⣴⠟⢁⣞⡠⠾⠟⠛⠛⠒⠒⠦⣄
⠀⠀⠀⠀⣠⠊⡀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⠳⡄⢻⡇⢠⢏⡎⠀⣠⢼⠝⣁⣴⡯⠵⠶⠶⠶⢿⣂⣀⢀⠀⠀⠑⣤
⠀⠀⢀⡜⠡⠈⠀⠀⠀⣀⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠘⡆⣷⢏⣮⣴⣿⠿⠷⠾⠿⣷⣖⣒⠒⠦⠤⣀⡀⠀⠉⠉⠉⠉⠉
⠀⠀⢮⡤⠤⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⣸⠟⣁⣀⡀⠤⠤⠤⠤⠤⣈⣙⠲⢄⡀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣴⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢶⣄⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡡⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢒⡡⠖⠉⣤⠴⠆⢠⣦⣄⡀⢀⡴⠶⠄⠀⢴⠖⢒⡂
⠀⠀⠀⠀⠀⠐⢏⡠⠔⠊⠁⠀⠀⠀⠙⠲⡄⢸⣇⣠⢷⡘⣇⠠⠞⡷⢸⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠃⢸⠏⠁⠀⠇⠈⠶⡼⠃⠸⡦⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣭⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣀⠺⣿⣿⣿⣿⣿⣿⣿⢇⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣬⡻⠿⣿⠿⣋⣵⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣌⣙⠺⠙⠿⠿⠟⠋⣁⣡⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⣌⣡⣶⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣮⡻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠈⠻⣿⣿⣿⣿⢟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠻⢟⣵⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠐⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢦⣄⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡄⠀⠻⡆⠀⠈⠙⠂⠀⢠⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⡧⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠈⠛⠛⠁⠀⠀⢀⡀⠀⠀⠈⠛⠛⠁⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣶⣿⠿⠛⠁⠈⠛⠿⣿⣶⣶⣿⠿⠛⠁

⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤
⠀⠀⢰⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⣦⡀
⠀⠀⠀⠀⠈⠉⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀
⠀⠀⠀⢀⡴⠒⠁⠈⠐⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⣠⠒⠀⠀⠀⠒⢦
⠀⠀⠀⠸⣆⠙⣀⣄⡘⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠃⣀⡀⠃⣠⠃⠀⠀⠀⠀⠀⢣⡈⠃⣠⡀⠃⣸⠃
⠀⠀⠀⠀⠈⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉
⣶⡶⢶⣦⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣹⣷⣮⣿⣷⣮⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⠉⠉⠀⠀⠀⠉⠉⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣠⣶⣶⣤⡀⠀⠀⣤⡀⠀⢠⣶⣶⣦⡀⠀⠀⠀⣧⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⣿⣬⣿⠃⠀⠀⣿⡇⠀⢸⣿⣀⣽⡿⠀⠀⠀⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠻⣷⡄⠀⠀⣿⡇⠀⢸⣿⠛⠛⠁⠀⠀⠀⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠸⠟⠀⠻⠇⠀⠀⠻⠃⠀⠘⠿⠀⠀⠀⠀⠀⠀⣿⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠀⠀⣟⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣞⡿⠞⠃⠀⢰⣯⣙⡏
⠀⠀⠀⠀⠀⠀⣄⠀⣿⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⢸⣧⣹⡇
⠀⠀⠀⠀⠀⢤⣿⢷⣿⢀⣾⣇⣠⡶⢀⡤⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⢠⣄⠀⢸⠷⣽⡇⣀
⠀⠀⠀⠀⠀⠈⢿⡄⠙⠚⠙⠛⢻⣷⣯⣤⣤⣤⣤⣿⣦⣸⣧⣤⣤⣿⡿⢿⠳⣾⠞⠿⣿⡿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣭⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣧⣦⣷⣾⣼⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣯⣭⣬⣿⣬⣭⣧⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⠿⠿⠿⠿⠿⠿⠷⠦