Robo Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⢰⣉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠜⠓⠒⠚⡙⣾⠻⡛⡛⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢗⠄⠪⠊⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⠀⢸⠑⢕⠕⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢘⡢⡠⡢⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠑⠂⠀⠒⠁⠀⢨⠆⠔⣀⢨⡄
⠀⠀⢀⡀⢎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠤⣤⢤⣤⣼⠗⠒⠛⠋
⠀⠀⠈⠻⢬⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠈⠉⠀⠉⠉⠉⠉⠉⠩⢹⡛⠉⡭⠭⡍⢽
⠀⠀⠀⠀⠀⠑⢌⠢⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⢬⡇⠀⠠⡶⢈⡶⠮⠶⠀⠀⠀⢸⡦⢞⣣⣶⣘⢼
⠀⠀⠀⠀⠀⠀⠀⠑⢜⢄⠀⠀⠀⢀⠠⠴⢸⣦⠵⢊⡉⡃⠀⠀⣤⠶⠶⠶⠖⡺⠗⣢⢥⡖⠋⠝⡡⡉⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡕⣲⡬⠕⠋⠈⠁⠈⠉⠓⠢⢭⣤⣆⣿⢤⠔⣎⡥⠖⣿⠁⠀⡣⡢⡪⡢⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠠⠛⢁⡴⡁⣿⠀⠠⢆⢔⡺⡢⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠐⠀⠉⠀⠂⢻⠁⢸⢗⢕⢰⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠉⠉⠀⠀⠁⠀⢸⢗⢕⠋⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣂⣀⡀⠀⠀⠀⠀⠀⠠⣸⣗⣇⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠏⣽⠉⠋⠉⡿⠉⠉⠉⢈⢪⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠊⡛⠓⡞⢣⠀⠀⠑⠕⢑⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⡀⣢⣨⡇⢸⡀⠀⠀⣢⣢⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠈⠉⠉⡉⡁⡽⠁⠉⠉⠉⠫⡉⠁⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⣀⣀⡀⣀⣬⣮⣇⠄⢀⠀⠀⠈⡴⣋⣰⡇
⠀⠀⠀⠀⠐⠂⠐⠀⠐⠀⠂⠀⠒⠒⠂⠒⠀⠒⠒⠂⠄⠞⠲⠞⠒⠛⠖⠚⠈⠛⠫⠟⠫⠛⠭⠿⠥⠐⠄⠠⠀⠄⠠⠤⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡆⠊⠱⡆⡇⣸⠀⡀⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡁⣟⡬⣻⢍⡭⣝⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠋⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣯⣧⣽⣬⣧⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣉⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠁⠀⢀⡠⠖⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⢤⡀⠀⠀⢻⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⢸⡇⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢸⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⡇⠀⠀⢠⠖⠲⡀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⢸⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⡇⠀⠀⠘⠢⠞⠁⠀⠀⠀⠀⠀⠀⠳⠴⠃⠀⠀⢸⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣧⠀⠘⣄⠀⠀⠀⠀⠀⠉⠓⠒⠒⠉⠁⠀⠀⠀⠀⢠⠏⠀⣼⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠳⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠖⠁⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⣉⣉⣉⡉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠉⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠤⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣄⠀⠀⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⢀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⢶⡏⠉⠉⠲⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⣾⢿⣇⣒⢐⢀⣻⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠘⡏⠉⠉⠛⠉⢉⣍⣈⡉⠉⠉⢉⣉⡉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⣏⡴⠃⠀⠀⠋⡈⠉⣾⣧⡞⢰⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠁⠈⠘⣠⠆⠀⣶⣙⣂⡝⢋⠻⣬⣙⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⡀⢄⣠⡏⠀⠀⠈⠉⠉⠀⠈⠀⠈⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⢿⡃⠀⠀⢰⣟⣛⣚⣿⣘⣋⣹⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⡾⠀⠀⠀⠸⡯⠽⠥⠯⠼⠷⠻⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠒⢳⣶⣶⣷⣶⣶⣶⣶⠶⢶⣒⣒⡒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣬⣿⣿⣿⠛⠛⢫⡉⣿⠿⠿⠿⠿⠿⠶⠶⠣⠤⠤⠀⠘⠛⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⡀⠁⡀⠀⠡⠀⠀⡇⠀⠀⡐⠲⡄⠀⠀⠀⠀⠀⠀⠀⠰⣄⡿⠀⢸⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⡁⢹⠀⠠⠀⠀⡀⠀⡇⠀⠀⠙⠀⢀⣀⣀⣀⣄⣀⣀⣤⣤⡤⣄⠀⢸⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⡄⠙⠉⢀⣤⣶⣾⢀⡄⡇⠀⠀⣎⠶⠿⠛⣯⣽⠟⡟⣫⣉⡁⢹⡇⠀⢸⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠻⢶⣶⣿⣿⣿⡟⣼⡇⡇⠀⠀⣿⠀⡞⠙⠁⣆⠀⡀⠀⠀⡟⢠⡇⠀⢸⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⠀⠀⠀⣼⣿⣿⣿⣿⢃⣿⢰⡇⠀⠀⣿⠀⠘⠆⠊⠁⠀⠀⠈⣉⣀⢸⡇⠀⢸⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠳⣄⠀⣼⣿⣳⣿⣿⣿⢸⣿⡆⡇⠀⢀⡏⡴⠂⠀⠈⠀⢠⣀⡀⣈⠦⢸⡇⠀⢸⣷⣈⡉⠳⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠐⠛⠆⢸⣿⣿⣵⣿⣿⣿⡇⣿⣧⠀⡇⠀⠜⣧⠉⠂⢉⡀⣀⠀⡅⠠⠄⠶⠸⡇⠀⢸⣿⡏⠀⠀⠀⠀⡉⠓⠢⣤⣀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠀⠂⢸⡀⢠⡼⣿⣿⣿⢳⣿⠃⠀⡇⠀⢀⡏⠀⢉⣈⣁⣀⣤⣤⣤⡴⠟⠖⠁⠀⢸⡿⣷⡶⡶⠖⠀⢠⠛⠛⢣⡀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⣴⣯⠀⠀⣷⣿⣿⣿⣼⡿⢠⡇⡁⠀⠀⠙⠛⠋⠹⠛⠉⠋⠁⠈⠀⠀⠀⠀⠀⢸⠛⠋⠡⠄⣀⡀⠃⠠⣈⠂⣿⣷⡖⠴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠁⠀⠀⠀⢹⣿⣿⣿⣿⡇⣾⠇⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⠳⡶⠚⠛⠉⠉⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣧⠀⠀⠀⠈⣿⣿⣿⣿⢠⣿⠀⡇⠀⠀⠀⠀⠀⡅⠀⠀⠀⢀⣀⠄⠀⠀⢀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠄⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡸⣧⠀⠀⠀⠿⠿⢿⣿⣾⡏⠀⣷⠀⠀⠀⠀⠀⣉⣉⣉⣤⣴⣶⣶⣟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠩⡙⢶⣖⠓⢢⡀⠈⣿⣿⣿⣶⣿⣶⡾⠿⠿⣿⣿⣿⡟⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠣⠇⣨⠈⠆⠀⠱⠀⣿⣿⣿⡏⠁⠀⠀⠀⠀⢿⡟⠁⡅⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢱⠊⢻⡄⠀⠀⣈⠀⠿⢿⣿⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⣀⣸⡧⠴⠾⠋⠀⢀⢸⡟⠀⠀⠀⠀⠀⠀⢸⣤⢀⡄⢀⣀⣠⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠧⠦⣄⡀⠀⠀⠀⢈⣄⠘⡏⠉⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⢄⠈⢿⠒⠊⠉⠁⢹⠀⣷⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠈⡇⠈⣦⣤⣶⠶⢧⡴⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡈⠘⠁⣠⠋⠀⠀⠀⢼⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⡶⣾⠟⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡂⣸⠇⣿⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡃
⢀⣀⣤⣶⣾⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⡇⣿⢰⡇⠀⠀⠀⠀⠀⠋⢀⣀⣠⣼⠇⠒⠂⠤⠤⣀
⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⡈⠀⠀⠀⠀⠀⠀⢹⣿⣭⣁⠀⠀⠀⢀⣀⣤⣼⡂
⠀⠀⠈⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⡛⠠⠭⢲⣤⣀⣀⣤⡴⠟⠛⠻⠿⣿⣶⢛⡫⠭⠒⣚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣉⡉⠀⠀⠀⠀⢀⣀⣤⣽⡩⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⢿⣿⣾⣭⣟⣻⠿⣿⣿⣿⣿⣿⣦⠶⠛⠉⠁⠐⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣬⣭⣛⡻⡟⣒⡨⠤⠒⠂⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⣀⣀⣀⣡⠡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠠⠐⡖⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⢂⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢧⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⠀⢄⠺⠿⢃⠀⠀⠀⠠⡾⢃⠀⠀⠀⢸⣼⣠⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠀⠉⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⢸⠹⣾⠟
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠐⠄⡀⠀⠀⠐⠂⠀⠀⠀⢀⠔⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⢘⡀⢆⡀⠀⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠁⠒⠀⠀⠀⠀⠂⠁⣀⡀⣼⠤⠏
⠀⠀⠀⠀⠀⢈⠀⢸⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣛⣟⣿⠅
⠀⠀⠀⠀⠀⠐⡒⠒⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠒⠃⠀⠛⠛⢳⠶⡖⠠⢀
⠀⠀⠀⠀⠀⠀⠈⢻⠛⢻⡀⠀⠀⠀⠀⢀⠠⠐⢨⡁⣦⡥⠊⠀⠐⣛⡛⠛⣿⡋⡗⡶⠉⠉⠁⠐⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠠⢣⡀⠀⡠⠐⠁⢰⣠⡇⠀⡇⠀⠀⠀⡠⢊⠈⠀⣿⠉⢹⠙⢦⡤⠤⣄⣀⡀⠀⠠⣨⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠁⣷⠊⡀⣀⠔⠋⠀⠀⡀⣇⠀⠀⠘⠄⠁⠀⠁⣿⡀⢸⠁⢸⠆⠀⠀⠀⠉⠁⢳⠋⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢤⡱⠚⠁⠀⠀⠀⡆⠀⠂⠀⡘⠛⠿⠿⠿⠿⠿⠉⣸⠈⣘⠀⠀⠀⠀⠀⢀⠆⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⢀⡟⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⡀⠀⠀⢀⣀⣀⣤⣸⣿⣿⡿⠁⠀⠀⠀⠀⡼⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣠⣤⡄⠀⣶⣤⣤⣍⠁⠀⠀⠀⠀⠀⣀⢀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⣻⠀⠀⠘⠙⠛⡏⠀⠀⠀⠀⠖⢉⠙⠟⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠀⠀⠀⠀⢄⡇⠀⠀⠀⠀⠑⠀⠁⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢸⠀⠀⠀⠀⢊⡅⠀⠀⠀⠀⠀⡐⣡⠂⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⣷⠆⠀⣸⠀⣥⣇⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠳⣄⣀⣀⡄⡰⢀⡀⢀⣁⣡
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⠀⠀⠀⠐⠂⠄⠰⡅⣤⣴⣠⣤⡿⣇⣀⣄⣉⣉⣳⠖⢀⠴⠀⠀⠠⠐⠀⠔⠰⠀⠀⠄⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣽⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣀⣀⣿⢸⣿⣿⡿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⡏⢀⣆⣸⣦⡀⢹⣿⣿⣿⣿⠁⣰⣈⣷⣆⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣇⠈⢿⣿⡿⠃⣸⣿⣿⣿⣿⡀⠹⢿⣿⠟⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠿⣿⢸⣿⣿⣧⣄⣀⣠⣴⣿⣿⣿⣿⣿⣷⣤⣄⣀⣤⣾⣿⣿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⣀⣀⡀⢸⡇⣿⡟⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠙⣿⣿⣿⡇⢀⣀⣀
⠀⠀⠀⢠⣿⡿⢿⣿⣿⣇⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣷⣿⡿⢯⣻⡄
⠀⠀⠀⠘⣿⣿⣾⣿⣿⣿⣿⡇⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⣿⣿⣿⡿⣿⣿⣿⣿⠁
⠀⠀⠀⢠⣿⣿⣿⠃⢸⣿⣿⡇⠀⢀⣤⡀⢀⣤⡀⢀⠤⣄⢠⢤⡄⣠⢤⡄⣿⣿⣿⡇⠈⣿⣿⣿⡄
⠀⠀⠀⣾⣿⣿⠇⠀⢸⣿⣿⣇⠀⠘⠛⠃⠈⠛⠃⠈⠛⠋⠘⠛⠃⠙⠛⠁⣿⣿⣿⡇⠀⠘⣿⣿⣷
⠀⠀⢰⣿⣿⡏⠀⠀⢸⡏⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡇⠀⠀⢹⣿⣿⡆
⠀⣠⣿⣭⣿⡀⠀⠀⠘⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣿⣭⣿⣄
⣾⣿⣧⣨⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⠁⠀⠀⠀⠀⢰⣿⣿⣅⣽⣿
⠻⡿⠀⠀⠘⡿⠃⠀⠀⠀⠀⢰⡶⠶⠶⢾⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⣾⠀⠀⠀⠀⠀⠘⢿⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣶⣄⡀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⣾⣿⣿⣿⣿⣿⣿⣷⡄⢠⡾⣡⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠤⢐⣒⣒⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣩⣥⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣮⣗⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢹⣷⣮⡳⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢀⠉⠀⠉⠉⠉⠉⡛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠿⠿⠛⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⣤⣤⣀⠀⠐⠺⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠞⢟⠝⠒⢄⠨⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠋⠁⠈⣝⢷⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣿⠀⣆⠀⠀⠀⢠⣤⣼⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡿⢟⠂⢄⠀⠀⠀⣸⡇⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢚⡷⠃⣑⣥⣤⡶⢻⣀⠘⠗⠷⠶⠶⠶⠶⠶⠲⠶⠶⠶⠶⠒⣶⠶⠶⢖⣶⠖⠲⠞⢓⡾⢂⡐⠂⠐⣈⠟⠓⠒⠋⠁
⠀⠀⠀⠀⠀⠀⡤⡖⠒⠯⣠⠊⠁⠀⢠⠏⠃⠀⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⠁⠀⠈⣧⢔⠯⠟⣣⡴⠋⠈⠉⠉⠁
⠀⠀⠀⠀⠀⢸⡏⡀⠀⡀⣙⠀⠀⠀⠸⣷⢀⣀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⠰⢙⣿⣃⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⢳⣬⠸⣃⡕⠀⠀⠀⠀⠙⢮⡌⠱⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠂⠉⢮⣳⣻⠛⣣⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠹⣄⠀⠀⠀⠀⢸⡄⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠃⢸⡇⢹⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢇⣈⢀⡇⠀⠀⠀⠀⠈⢷⠤⠰⣇⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠘⣦⠼⠇⠀⢨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⢙⣒⣒⣒⡒⢦⣿⣀⣀⣮⣤⣦⣤⣌⡙⢦⠀⠀⠀⠀⠀⠀⢠⢷⣤⣴⣿⣿⣟⠀⠀⣙⣛⣒⣒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣯⣷⣶⣿⣿⣿⣿⣿⣦⣸⡿⠿⠿⠿⠿⠿⠿⠿⠾⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢹⣫⣶⣶⣿⣿⣿⣿⣿⣧⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⠀⠀⢠⡴⠏⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠁⠀⠀⣿⣿⣿⡄⠈⣹⠾⠃⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠋⠉⠉⠉⠠⠔⠀⠀⠀⠀⣀⢤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢳⡀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣍⡳⠂⠀⠀⠠⣄⠠⠀⣀⣊⠁⠀⠀⠀⠀⠀⠁⢣⢈⣄⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⢠⠀⠁⠀⠀⠀⢀⠺⢦⡴⠀⠉⠸⠠⠒⠂⠀⠀⠀⠈⠩⡁⠈⠂⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⠀⢸⠀⠀⠀⣠⣀⠋⠀⢻⠐⠄⠆⠀⠀⢐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⡀⠂⠸⠀⠀⠀⢣⣤⣄⡀⠈⠀⠀⣣⣴⡴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠚⡛⠉⢙⠗⠾⠶⠛⣿⠈⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢤⣠⠶⠋⠀⠀⠀⠀⠘⠛⢤⣇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠁⠀⣠⡾⡁
⠀⠀⠀⠀⠀⠐⠃⠀⢀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠉⠂⡁
⠀⠀⠀⠀⠀⢷⡀⠀⠈⣁⡐⠁⠀⠀⠀⢠⢀⡘⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢍⡜⠈⢋⠀⠀⠀⠀⠘⠻⠞⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡰⣤⣤⠤⡀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠀⠀⠠⠃⠀⠤⡀⠀⠀⠀⠀⠀⠀⠛⠛⠻⣧⣄⣠⡄⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠄⠀⠀⢻⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠰⠚⠁⠛⠲⠗⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⡀⣈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠛⠂⠀⠀⠀⠀⠘⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠐⣄⠘⠗⣀⠖⢀⠐⠗⢀⡼⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠋⠉⣁⠀⠀⢉⡉⠉⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⠯⣥⢤⣭⠟⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⣤⣤⣤⣤⡴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠒⠚⠒⠚⠚⠛⠒⠛⠛⠚⠒⠒⠒⠓⠚⣶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢉⠠⡆⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⡾⠀⢨⠀⣷⠀⢠⡞⠁⢀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢀⣽⣄⠈⠋⣿⠀⠀⢧⡀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣟⣁⣴⡋⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡴⠋⠀⠉⠛⠶⠾⡆⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠙⢦⡀⠙⢦⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠶⠉⠉⠃⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⠀⢸⡇⠀⠀⠀⠀⠙⢷⡎⠁⡀⠙⠆
⠀⠀⠀⠀⠀⠀⠘⣀⠐⠗⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣾⠀⠀⠀⠀⠀⠀⠐⣆⡈⢉⣠⡆
⠀⠀⠀⠀⠀⠀⢀⡟⢳⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠁⡄⠀⠀⠀⠀⠀⠀⡰⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⠙⣏⠹⣇
⠀⠀⠀⠀⠀⢀⡞⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠘⠢⠄⠀⠀⠤⠚⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸
⠀⠀⠀⠀⠀⣾⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣧
⠀⠀⠀⠀⡸⠃⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢠⡿⠶⠶⠶⠶⢶⡟⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡄
⠀⢀⡴⠞⢳⠔⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡾⠁⠀⠀⠀⠀⠀⢻⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⣟⠙⠓⢄
⢰⠏⢀⡴⠛⠶⢄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢧⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡎⠉⠙⢦⠀⢇
⣇⠀⡞⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⠀⠀⣘⠀⡈
⡟⠀⢻⡄⠀⢀⡼⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠿⢤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⣹⣤⣀⠀⠀⠀⠀⠀⠈⢳⡀⠙⡇⠀⠙⠶⠁
⠹⣆⢀⡿⠀⠀⠳⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠀⠙⣧⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠛⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⢾⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⣤⣄⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣉⣉⠁⡄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⡀⣼⡀⣤⣠⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢠⡄⣸⣠⡇⢀⠀⣀⣀⣀⡀
⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠉⠳⣄
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠿⠀⠤⢤⠤⡤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣦⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠖⠚⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣈⠉⠉⠉⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠰⠞⠋⠁⠀⠸⠿⢦⡀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢰⣾⠀⠀⠀⢰⣹⡆⠀⠰⣽⡷⠀⠀⣾⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣦⣀⠀⠀⠉⠀⠀⠀⠀⠀⣠⣼⣿⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢻⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⡿⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠦⣤⣉⣙⡋⠀⠀⠀⠀⠀⠙⠛⠉⢁⡰⠏⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠠⠄⠀⣀⠉⠉⠉⠛⠛⠓⠂⠉⠉⠉⡀⠀⠤⠐⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠈⡂⠈⠀⠃⡄⠀⠀⠀⠀⠀⡇⠀⠈⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣏⠀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡊⠀⣠⡄⠠⠀⠀⠀⠀⣀⠀⡇⠀⢀⣇⠀⣹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠻⣄⡼⠋⠙⢦⡀⡸⠀⠀⠀⠀⠀⠀⠀⠐⢇⠘⠋⠀⠐⠀⠀⠀⠀⠁⠀⣇⣠⠞⠉⠳⣇⢨⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡿⣀⡟⠀⠀⠀⠀⠉⡛⠀⠀⠒⡀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠘⣇⢈⣇
⠀⠀⠀⠀⢀⠴⠚⠓⠾⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠙⠀⠀⠂⠀⠐⠀⠈⠁⠈⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⣹⠤⠿⢤⡀
⠀⠀⣀⣤⡏⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⡄
⣴⠋⢁⡴⠛⣄⣀⣀⣠⠛⢦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡤⠻⣄⠀⠀⢀⡼⣍⠓⢦
⠈⢳⣏⠀⡼⠀⢸⠇⠘⢦⣠⠷⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⢾⣀⡴⠋⢹⡏⠙⣆⠈⢳⣠
⠀⠀⠈⠙⠧⠤⠼⠆⠀⠀⠀⠀⠀⠀⠀⠉⢲⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⢶⠀⠀⠀⠀⠀⠈⠀⠀⢸⣇⣀⣽⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠉⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣈⣀⡀⡸⣄⠀⠀⠀⠀⠀⠀⡧⣐⠀⢀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⠙⢳⠾⠿⠗⠂⠉⠑⢤⢀⡴⠊⣙⣳⠶⠤⠬⠒⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠎⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⢹⡔⠉⠀⠀⠑⢄⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⡀⠀⠈⢢⣀⣀⣀⠠⢤⠂⠀⠀⠀⠀⠀⢣⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⠁⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⢠⠃⠈⠀⠀⠀⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡀⡀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠸⡀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⢀⡌⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⢀⠜⠁⠀⡀⠄⠊⠘⢄⠀⠀⠀⢀⡴⠁⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡄⢀⣤⣖⣒⣒⣒⡒⣒⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡘⠠⠤⡀⠀⣈⡉⡇⠀⢸⠀⠀⠀⠀⠀⠀⢠⢒⡆⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠂⠔⠁⠸⠄⠝⡇⠀⠘⡇⠀⠀⠀⠀⠀⣎⠐⣅⡠⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠤⣄⣀⡠⡤⡀⢿⠀⠀⡇⠀⠀⠀⠀⠀⣼⠲⣄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠙⠒⠧⠦⠓⠃⢸⢸⣠⠟⠀⠀⠀⠀⡼⠣⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⡟⢻⠛⠓⠛⠋⠀⠀⠀⠀⢀⡞⢤⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠔⠒⠛⠧⠾⠛⠓⠒⠒⢲⣤⣄⡴⠫⣹⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡅⠈⠉⠉⠉⠉⠓⠒⠒⡖⠉⣯⢺⠹⣬⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢤⡇⠀⠒⡂⡒⠒⠒⢲⠀⡇⠀⣙⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⢤⣷⠀⠟⠲⠃⠀⠀⢸⡗⣷⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣸⢹⠀⢰⣞⡳⠀⠀⢸⣷⣿⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣀⡼⢸⡄⠘⠒⠒⠒⠒⠚⠁⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣤⡇⠈⡇⠀⠀⣤⣤⣤⣤⣤⣿⠔⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⣤⠗⡄⠓⣒⢲⡖⠒⠒⠒⠲⡿⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎⠀⢇⣸⠀⣍⣹⠀⠀⠀⠀⠀⡷⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠘⠉⠀⠆⢾⠀⠀⠀⠀⠀⣏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⡟⠀⠀⠀⠀⠀⢹⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⣇⠀⠀⠀⠀⢠⢾⠚⡧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠘⠒⠃⡇⠀⠀⠀⢻⣈⡉⠀⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣓⠒⣒⣒⡭⠃⠀⠀⠀⠈⠒⠬⠭⡩⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⠤⣄⠀⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠀⢸⡇⠀⢸⡇⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠛⣶⠛⢀⡀⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢻⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣼⣤⡤⠤⠤⠤⢤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠃⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⡶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣿⠟⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⢸⣀⣠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⣄⣿⡀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉⠀⠀⠀⢀⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⡶⠒⢳⡟⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡞⠛⠛⠋⠉⠉⣉⠛⠛⠓⠒⠲⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢯⣡⠖⣿⠀⠀⠀⠀⠀⢿⣬⠇⠀⠀⠀⠀⢹⣍⣙⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢧⣠⠞⠁⠀⡇⠀⠀⠀⠀⠀⣖⣛⡆⠀⠀⠀⠀⢸⠀⠙⢦⣀⣽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠛⠦⣴⠋⠀⠀⠀⣷⠀⠀⠀⠀⠀⣨⠭⡀⠀⠀⠀⠀⣼⠀⠀⠈⢻⣄⡬⠿⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⠤⢀⠀⢸⠀⠀⠀⠀⠸⣦⣄⣀⣀⣀⣙⣲⣃⣀⣀⣀⣰⠏⠀⠀⠀⢸⠃⢀⡤⣄⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⢸⣇⣸⠃⠀⠀⠀⠀⢀⣏⣀⣯⠈⠉⠉⠉⠉⣽⣁⣹⡀⠀⠀⠀⢸⡀⣸⡀⠈⠳⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣸⣀⠁⣿⠀⠀⠀⠀⠀⣿⠈⣈⣧⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣏⡉⠉⢹⣆⠀⠀⢀⣰⡏⠉⢉⣹⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⠉⠛⠉⠉⢳⣰⠋⠈⠉⠛⠋⠉⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠶⠾⠆⡈⠚⠀⠁⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠰⣄⣠⢂⡁⠀⠀⠀⠀⠈⠀⠿⢢⠀⠀⡀⠀⠀⠀⢀⡀⠒⠀⠀⡈⢙⠒⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⣿⣻⣇⠀⠀⠀⢠⠄⡀⢰⠀⢀⠟⡹⠚⠓⠛⡟⠂⣰⡶⠒⡷⠾⠓⠖⠾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡾⠁⢿⡏⠀⠀⠀⠞⠠⠿⢻⠀⠈⠁⣄⣴⠟⠀⠛⢿⡇⢰⡀⠐⡀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠳⠀⠘⠃⢂⣴⡶⠀⡐⠚⣜⠀⣸⠀⠾⢯⣛⣶⣶⣿⡇⢸⣇⠀⢃⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠄⢀⡾⠾⠿⣧⣤⡾⠋⣹⡋⣠⠿⣄⡰⢿⡄⠀⣰⣿⠃⠀⢿⡆⢸⡄⠀⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣱⣷⡾⡿⠲⢦⣈⠉⣷⣾⣟⡷⠃⠀⠉⠀⡘⣿⣿⣿⣿⠀⠀⠈⠻⣿⣿⣦⢙⣷⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣷⠄⣿⠒⢤⡈⢛⣾⡿⠛⠀⠀⠀⠀⠀⠱⠘⠛⢻⠃⠀⠀⠀⠀⢈⣿⣁⠀⢈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠉⢹⡄⢿⠠⢤⣉⣻⣿⣧⠀⠀⠀⣈⠀⠀⠀⠐⢠⠂⠀⠀⠀⠀⠀⣼⡯⢀⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⣹⠁⠘⠆⢤⣽⣿⣿⣿⡀⠀⠘⢿⠃⠀⠀⠀⠸⠀⠠⠀⠀⠀⣰⢻⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⠾⣿⣇⡿⢦⣄⠀⢼⡟⠸⠿⣿⣿⡤⢴⠿⠿⠿⢿⡷⠿⣶⣶⣦⣤⣴⣿⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⢀⣿⣿⠓⠀⠸⠁⠘⠀⠀⣀⣼⣧⠈⢉⠓⠒⠂⠴⠲⠶⣿⣿⣿⣿⣿⣝⠣⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⡏⠈⢻⠁⠀⠀⠀⠀⣧⡞⢉⠜⠉⡛⠋⠀⠐⠒⠂⠚⠛⠿⡿⠿⣿⣧⠈⠳⣄⡁⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣸⠀⠈⠒⠉⠠⠀⠀⣿⠉⠙⠲⢶⠁⠀⡜⠀⠀⠀⢠⡆⠀⢸⠗⢾⣟⠃⢄⠈⠙⢾⣇⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠁⠀⡇⠠⠀⠀⠀⠀⣹⠀⠀⠀⠀⡀⢀⣧⠀⠀⠀⠘⣷⡆⡆⠀⠀⢿⡏⠀⠑⠀⢠⣟⣴⡇⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⠀⠀⢹⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⢡⣤⣿⣧⠀⠀⣾⣿⣷⠀⠀⠀⠘⣿⡀⠀⠀⣾⢪⠟⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⠀⠀⣠⣶⣤⣾⠀⠀⠀⠀⢸⣿⣿⠟⠐⣴⣿⣿⣇⠄⠀⠀⠀⢿⣿⣔⣾⣿⡉⡷⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡴⢋⣴⣾⣿⣿⣿⡏⠀⢈⠀⠀⢸⣿⠇⠀⠀⠘⢿⣿⣏⠀⡄⠀⠀⠘⣿⣿⣿⣟⠁⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣰⡿⡡⣿⣽⣿⣻⡟⢻⡇⠀⠘⠀⠀⣹⣿⠀⠀⠀⠀⠈⢻⡿⣦⠐⠤⠀⠀⠈⢿⠛⠁⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣾⣿⠉⠙⡿⠋⠀⠘⣗⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⣤⣧⣽⣇⠂⡀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢈⡇⠀⠀⠀⡿⠻⠶⣶⣶⡿⣿⡇⠀⠀⢠⢸⣿⡟⠛⢿⡟⠛⠛⢋⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⠇⠀⠀⠀⣿⡶⢦⣽⣿⢘⣿⠇⢀⣀⣼⡠⠏⣠⠇⢈⢿⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢹⠁⣼⡿⣿⢻⣿⣀⠘⣿⢽⣧⣶⠇⠀⣚⠋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣤⣤⣤⣶⣿⣧⠈⠁⣸⣿⣿⣿⣿⣷⡦⠥⢾⣤⡶⣿⣿⣉⠉⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣤⣴⣾⣿⣿⣿⣿⣿⣆⣴⣏⣼⣧⣿⣿⣿⡆⠀⠀⠈⠁⠘⣿⣿⣿⣿⣿⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠿⠋⠀⠀⠘⣿⢿⣿⣷⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⣼⣿⣿⣶⣄⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠈⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣠⣶⣿⣿⣿⣿⣿⣋⠉⠁⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⢀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠻⠿⠿⠿⣿⡿⠿⠟⠛⠉⠉⠁
⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠅⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠰⣶⣄⣀⢴⣶⢦⡀⢱⠀⠈⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⣿⠒⠀⠉⠙⠥⠤⠽⠿⠿⠘⢀⡤⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢭⡁⠀⡇⢀⣀⣀⣀⣀⣀⣀⣀⠀⠐⢸⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡇⠸⣆⠀⠀⠀⠀⠉⣹⠃⡆⢸⠀⠀⣀⡾
⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣈⣓⠢⠤⠦⠖⠋⠀⡇⠈⠉⢹⡏
⠀⢴⠛⢎⢧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠉⠉⠉⠙⠛⠛⢷⡚⠓⠒⢟⠋
⡔⢦⠳⡌⣮⠗⠋⢳⠤⢦⢠⢦⠤⠤⠤⣴⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⣄⠀⢀⣳⣀⣀⣀
⠈⠢⣱⠎⠀⠀⠀⢸⠚⠉⢸⢸⠀⠀⢰⡇⠀⠀⠀⠅⠀⠀⠀⡀⠀⠀⠀⢸⠰⡇⠀⠀⠀⠈⡟⡇
⠀⠀⠳⢄⠀⠀⠀⢸⠀⠀⡃⢸⠉⠛⣿⠀⠀⠀⠀⡀⠀⢠⠁⢹⠙⡄⠀⠘⡇⠙⠛⢻⠛⢺⠇⡇
⠀⠀⠀⠀⠉⠒⠲⡛⢆⠀⡇⢸⠀⠀⡏⠀⠀⠀⠀⠄⠀⠘⢤⣎⡰⠁⠀⢀⡇⠀⠀⢸⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡱⣆⡛⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢐⠠⠀⢠⠀⢸⠇⠀⠀⢸⠀⢸⠀⡇⠀⠀⠀⠀⠴⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠗⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠐⠚⠒⠊⠀⢸⠀⠀⠀⢸⠀⢸⢠⡇⠀⠀⠀⢀⣸⡤⠗⢻⣗⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠠⢸⠀⠀⠀⢸⠀⢯⣹⣏⣉⣉⣹⣏⠀⠀⠀⠈⣗⣒⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣸⣇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠬⢽⠓⠒⠲⠶⠤⠤⢤⣄⣀⣀⣀⣀⣿⣀⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢇⣀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡩⠛⠻⠳⠛⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠉⠉⠙⢄⠀⠀⠀⠀⠀⠀⠀⣰⠁⢧⣀⣀⣀⣠⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⢿⠀⠀⠀⠀⠙⠒⠦⠤⠤⠤⠴⠥⠔⠒⠉⠀⠀⠈⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡽
⠀⠀⠀⠀⠀⠀⠀⢠⢤⣂⠉⠳⠞⠈⠉⣲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠒⠲⠖⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢴⠀⡧⣀
⠀⠀⠀⠀⠀⠠⢞⡉⠙⠒⠤⣀⢸⣄⡠⠖⠉⢈⡵⠂⠀⠀⠀⠀⠀⠀⡟⠓⠤⣄⣈⠒⢃⣀⠨⠕⢦
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⣀⠉⣇⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠈⢹⠉⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣈⠉⠒⠦⢄⣸⣀⡤⠔⠊⢉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣄⣈⣇⣠⡤⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣦⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⣦⢄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⣿⠀⠋⢰⡟⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠏⠈⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⡟⠀⠀⢸⠁⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠘⠳⠦⠤⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⠖⠛⠛⢶⣀⡇⠀⠀⣼⠀⠀⣀⣀⣀⠈⢳⡄⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢿⡀⠀⠀⣸⢿⡇⠀⠀⡿⢠⡞⠁⠀⠈⢷⠀⢿⠀⠀⠀⠀⠀⠀⠘⢷⠦⣤⣄⣀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣈⠛⠲⠚⠉⢸⠀⠀⠀⡇⠈⢧⣄⣀⣠⠟⠀⢸⡇⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢩⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠷⠶⠦⢬⣭⣥⣤⣄⣸⡇⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠀⡶⠴⠤⠤⠄⠀⣠⣤⣀⣀⣀⠀⠀⣀⠀⠀⠀⠈⡅⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⡀⢀⣀⣀⣀⣧⣠⣠⣤⣄⣤⣤⣤⣤⣄⣀⣀⣉⣉⣉⣹⣇⣈⣁⣀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⢸⣧⡁⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⢸⠤⢴⠇⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⡏⠀⡆⢸⣿⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠓⠒⠶⣤⡀⠀⠀⠀⠀⠀⣸⡇⢸⠀⡾⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣸⠓⠒⡂⠈⣿⡄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣿⠃⢸⣺⡃⠀⣀⡴⠋
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⡏⠀⠀⠁⠀⢹⣷⡀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣼⡏⠀⢸⠈⠉⠉⠁
⠀⠀⠀⠀⣾⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢀⣼⠟⠀⠀⢸
⠀⠀⠀⢰⡇⠀⠀⠀⢰⠇⠀⠀⠀⠃⠀⠀⠀⠙⢿⣦⡀⠈⠳⣄⣀⠀⠀⠀⣀⣤⠞⢁⣠⡿⠋⠀⠀⠀⢸
⠀⢀⣤⣾⣁⡀⠀⠀⡿⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⣉⣛⣛⣋⣉⣤⡶⠟⠉⠀⠀⠀⠀⠀⢸
⡴⠋⠀⠀⠈⠉⠙⠛⢳⡄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⡇⠀⣿⠙⠒⣶⠄⢰⠏⠀⠀⠀⠓⠒⡔⠐⢶⡶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⢲⡶⠢⠞
⠀⢻⣀⣻⡄⣰⠋⢠⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⢦⡀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠋
⠀⠀⠉⠉⠉⠛⠲⠟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣦⣀⣀⣿⢦⣄⣠⣤⣿⣀⣀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠤⣤⣤⣬⡉⠉⡏⠀⠀⠀⠀⡏⠉⣽⠧⠶⠶⢦⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠾⠃⠀⠀⠀⠈⠷⢶⡇⠀⠀⠀⠀⣷⠞⠃⠀⠀⠀⠈⠓⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠖⠛⠋⠉⠙⠛⠶⣄⣿⠀⠀⠀⢸⣠⠶⠛⠋⠉⠉⠛⠲⣤⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⠒⠚⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠀⠉⠁⠀⠁⠀⠀⠉⠉⠁⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠁⠞⠁⠀⡐⠂⢂⡐⠀⠄⠀⠀⠀⠀⠀⠀⠰⠂⢂⣀⠐⠄⠀⠈⠣⡈⢲⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠁⠎⠀⠀⠀⡇⣎⠙⠴⡇⡇⠀⠀⠀⠀⠀⠀⠐⠈⠉⠦⢷⠀⠀⠀⠀⠱⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣸⠀⠀⠀⠀⠣⠌⠵⠊⠠⠁⠀⠀⠀⠀⠀⠀⠰⠌⠱⠎⠡⠆⠀⠀⠀⠀⣇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⣀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠉⢠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡔⠉⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠠⠄⠺⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⠗⠠⢄
⠀⠀⠀⠀⢀⡰⢄⠀⠀⠙⠢⠘⡇⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡵⠀⠀⠀⠀⠀⠸⠃⠖⠋⠀⠀⡠⢆⡀
⠀⠀⢀⠔⠁⠀⠀⠑⠂⡄⠂⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠸⠀⠐⢠⠐⠊⠀⠀⠈⠢⡀
⠀⠠⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⠁⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈
⢆⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⡀
⢨⠓⠠⢀⣀⠀⡠⠊⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣬⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⢄⠀⣀⡀⠄⠚
⢨⠒⠐⠒⠣⡀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⠜⠒⠐⠒
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠂
⢰⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠑⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠊⠀⠀⠀⠀⠀⠀⢠⡀
⠀⢆⠀⠀⣠⠰⡎⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⡏⢙⠁⠀⠀⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢱⠆⣄⠀⠀⡰
⠀⠈⠁⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠈⠁
⠀⠐⢀⠀⠈⢄⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⡠⠁⠀⡀⠁
⠀⠀⠀⠐⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⡄⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⠀⠀⣀⡴⠂⠀⠀⠐⣄⡀⠀⠀⢀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠁⡁⠀⠀⠀⠰⡌⠀⠀⠀⠀⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⠍⠈⠉⠉⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⡈⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠠⠤⠀⠀⠠⢀⡀⠸⠀⠀⠀⢀⡠⠄⠀⠀⠤⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠑⣮⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⠑⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠄⠀⣀⣽⣤⣭⣤⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣭⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣄⣽⣧⣈⠃⣿⣗⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢹⣿⣿⣿⣿⣿⣛⡿⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣶⣾⣿⣿⣿⣿⣿⣷⡶⣦⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢸⣿⣷⣿⠀⣨⣟⠛⠛⠛⠋⡀⢸⣼⡟⢹⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⣿⠛⣿⣄⣧⣿⡀⢰⣿⡆⠀⣼⢻⣇⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⢿⣿⣿⣿⣃⣀⣉⣀⣠⡟⠀⢻⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⠆⠀⠸⣿⣯⣴⣶⡶⣾⣭⣽⠇⠀⢺⣿⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⣀⠛⣛⡷⣷⡛⢛⡀⠀⠀⠀⠹⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⢸⣿⡿⠿⠷⠾⠿⣿⣽⡄⠀⠀⢀⣜⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠛⠆⠀⣿⡿⣿⠀⠀⠀⢸⣏⠉⠁⠀⠠⠟⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠁⠀⢿⣿⣿⡀⠀⠀⢸⣿⣷⠏⠀⠈⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢿⡿⠟⠀⠀⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⡏⣸⠖⠀⠀⠻⣿⣩⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠶⠆⠿⢷⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣼⣶⣶⡴⠤⣴⣶⣧⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⠿⠿⠿⢿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⣴⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⣿⠃⠛⠤⢌⣁⣬⣥⣌⡡⢶⠛⠈⣿⡇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⣤⣿⣿⠀⠀⣦⣯⡾⠋⠙⢳⡟⣤⠀⠀⣿⣿⣤⣤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣽⣻⣿⣷⡆⣿⣼⣠⣤⣤⣀⣷⣿⢰⣾⣿⣟⣯⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⢿⠉⣿⣿⡇⣿⣷⣿⣾⣷⣿⣿⣿⢸⣿⣿⠉⡿⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⣶⣿⠀⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢿⣶⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠇⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠸⣟⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠟⠀⠀⢠⣾⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣷⡄⠀⠀⠻⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⣤⠀⠀⣿⣾⣇⠀⠉⠛⠋⠉⠀⣸⣷⣿⠀⠀⣤⣾⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡽⠟⠁⠀⢟⢿⣟⠀⠀⠀⠀⠀⠀⣻⡿⡻⠀⠈⠻⢯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⡇⠀⠀⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⣭⣿⠷⠂⠀⠀⠐⠾⣿⣭⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀⣀⣀⡀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⠿⠿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠻⣿⣿⠟⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢠⣄⠀⠀⣠⡄⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣾⣿⣿⣷⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠂⢈⠁⠀⠀⠈⡁⠀⠀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⡀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿




⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⠼⠤⠤⠤⠤⠤⠤⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠤⢀⠀⠀⠀⢀⠠⠤⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⡂⠀⠄⡂⠁⠀⠀⠁⡊⠄⠀⡆⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠐⠠⠩⠔⠁⠀⠀⠑⠬⠡⠐⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂⠂⠘⠈⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠈⠈⠀⠂⠃⠀⠘


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠐⠆⠀⠀⠖⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠶⠶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢾⣯⣾⣿⣧⡤⣤⣿⣿⣿⣿⣿⣿⣿⡤⢤⣼⣿⣿⣞⡿⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⣿⣿⢯⣶⣻⣻⣮⣻⣿⣿⣿⡿⣵⣞⣟⣶⣝⣿⣿⡇⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⢹⣿⣿⣿⣺⢿⣿⣿⣿⡇⣿⡿⣿⣗⡿⣼⣿⡇⠀⠀⠀⣿⣿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢸⡿⣿⣷⣝⣛⣛⣯⣿⣿⣿⣿⣿⣮⣛⣛⣫⣽⣿⣿⡇⠀⢀⣼⠏⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠻⣿⣿⣿⣿⣏⡍⡉⡍⠉⡏⠙⣿⣿⣿⣿⣿⣿⡿⡿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣭⣭⣯⣯⣯⣿⣿⣿⣿⣿⣿⣟⡿⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣺⣵⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣅⣨⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣽⣿⢾⠇⠍⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣳⣤⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣤⢾⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣟⣿⣻⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣿⣿⣇
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠯⠵⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴⡁⡖⡗⡄⡔⡟⡆⡱⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠧⢧⠭⠯⠤⠬⠾⢥⠥⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⣉⣉⣉⣉⣉⣉⣉⡉⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠫⡃⠀⠥⠧⠭⠼⠬⠥⠧⠄⠀⣫⢳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠒⠚⢯⡣⡱⠇⠀⠀⠀⠀⢎⠆⠀⠀⠀⠀⠷⣡⢪⠟⠒⡐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢱⠈⠋⠀⠀⠠⠔⢦⢤⠔⢤⢤⠖⠤⠀⠀⠈⢛⠰⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠃⠛⠀⠃⠃⠀⠃⠃⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄⠀⣤⠀⢠⡄⠀⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣒⣿⠀⢸⣗⡒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣒⣿⠀⢸⣗⣒⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡒⣿⠀⢸⡗⣒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠒⠛⡆⡞⠓⠒⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠃⠓⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠐⠑⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠖⠠⠠⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢱⠀⢀⡀⠤⠐⠒⠒⠒⠒⠠⠄⣀⣠⠞⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣀⡤⣤⡀⠀⢰⠛⠁⠀⣀⠀⠉⠓⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⡇⡴⠌⣹⠀⡏⡀⠀⢾⣿⠇⠀⠀⡂⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠛⠛⠉⠀⠈⠢⣤⣤⣤⣰⡶⠞⠃⠐⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠐⠂⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠂⠠⠤⢀⣀⣈⣉⣀⣉⣉⡈⣁⣀⣀⣀⣠⡠⠶⠚⢡⠤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣮⠂⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⠀⠐⣐⣨⠇⡀⡞⠉⠑⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠈⢀⣄⣽⣵⡄⠀⠈⠋⠁⠀⠐⠀⠀⠀⠄⠖⠖⠒⠒⠊⠉⠁⠔⢠⣺⡇⠀⠀⠀⣈⢧
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⣏⣮⣁⣩⠗⢶⣎⣑⣀⡤⠀⠄⠒⠃⣉⣫⡿⢯⣉⣙⣶⡁⣐⣙⣷⡂⠀⠀⠚⢏⡂
⠀⠀⠀⠀⠀⠈⠀⠀⣐⡠⣏⣽⣭⣽⠀⠔⠛⠉⠀⠀⣀⣤⣔⠋⠉⠀⠀⠀⠀⠀⠈⠉⠙⠄⠠⢀⠀⠤⠰⠀⠃
⠀⠀⠀⠀⠀⡄⠈⢉⠈⠉⢩⠙⡋⠀⣀⣠⢤⠶⠛⠉⠉⠆⠩⢉⢿⠿⠷⠿⠿⠾⣾⣧⣷⠷⢚⡶⣶⣾⣾⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠬⠀⠤⠄⠃⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠴⠖⠠⠐⠴⠴⠨⠷⠾⠿⠟⢿⠛⠁
⠀⠀⠀⠀⠀⠑⠄⣐⢀⢀⣢⠎⠠⠈⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⢞⠖⢒⢲⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠨⠉⠉⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠁⠀⠹⠋⠉⠙⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣀⣴⣠⣤⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠤⠬⢬⣅⡉⣀⡀⠂⠈⠀⠶⠾⡆⡶⣿⠿⠿⠾⠛⠛⠛⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡼⠁⠀⢸⡄⠀⠀⠀⠀⠀⡫⠈⠸⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢪⣤⣬⣁⣁⣁⣴⠆⠀⠀⠀⠀⠀⢪⣰⣀⣂⣊⣂⡱⠀⠀⠀⢀⢀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠙⠉⠀⠁⠈⠀⠀⠀⠁⠉⠋⠁⠉⠉⠉⠀⠀⠁⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢷⡀
⠀⣠⡶⠶⣦⠀⠀⠀⠀⠀⠀⠘⣷⣤⡾⠁
⠀⠻⣤⣶⣿⡀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠙⢿⣎⢷⣄⠀⢸⡟⣿
⠀⠀⠀⠀⠀⠈⠻⣧⡙⢷⣾⠃⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢨⣿⠚⠛⠀⢉⣉⣿⣤⠤⠴⠶⠶⢦⣄
⠀⣶⠛⠛⠛⠛⠛⠛⡉⠉⠉⠉⢁⣀⠀⠀⠀⠀⠀⠀⠼⠋⠛⢷⡄
⠀⣿⠀⠀⠀⠀⠀⠀⠃⠀⢀⢾⢋⣉⡹⡄⠀⠀⠀⠀⠀⠀⣉⡀⣿
⠀⣿⡀⠀⠀⠀⠀⠀⠐⠀⢸⠘⠸⠀⡁⢰⠀⠀⠀⠀⠢⣤⣈⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢢⣡⣌⣡⠎⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⠶⢦⣴⠟⣹⠟
⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠘⣛⠀⣿
⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⠀⠉⠛⢳⡄⠀⢠⡾⠁⠀⠀⠀⠀⠀⢀⣿⣄⣈⣿
⠀⢀⣨⡿⠒⠠⠤⠀⢀⣠⡀⣀⡀⠤⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⠟⠀⠀⠀⠀⢠⡶⠟⠋⠉⠉⠉⠁
⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠁⠀⠀⠀⠀⣰⠟
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢠⠔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠤⠀⠘⠦⣠⠔⠒⠒⢶⡟
⠸⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣴⠟⠛⠛
⠀⢻⡇⠀⠀⠀⠀⡀⠉⠀⠀⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠨⢿⣀⣴⡆
⠀⠘⣿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣦⣤⣤⡿⠁
⠀⠀⢿⡇⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢸⡏
⠀⠀⠸⣿⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠸⣧
⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡭⠂⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣆⣁⠀⠀⠈⠛⠷⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠉⠒⣦⣤⣶⡶⠶⠟⠛⠋⠉⠉⠉⠙⠻⣦⡄⠀⠀⠈⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠉⠘⠻⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁
⠀⠀⠀⠀⡠⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠻⠬⢅⡀⠀⠀⠀⠀⠉⠁⠒⠤⢀
⠀⠀⠀⠀⢆⣁⠌⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⠤⠄⣒⣒⡂⠀⢀⠹⡄⠀⠀⠀⠀⣤⣒⡬⣥⣍⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⠙⢢⡀⠀⠞⡡⠞⢉⣉⡉⠉⠳⣄⠙⣿⡀⠀⠀⠀⣯⣶⣶⣤⡈⢷⡌⢆
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠘⡄⠀⢟⠎⡜⢡⢾⣿⠟⠛⢷⡀⠘⣧⠘⡇⠀⠀⠀⣿⣿⠋⠉⣷⠈⣿⠸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⡇⠀⢸⣴⡇⣿⣿⣿⣀⣀⣼⣷⠀⣻⠆⣷⠀⠀⠀⣿⣿⣷⣶⡿⠀⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⣧⠀⢸⣿⣇⢻⣿⣿⣿⣿⣿⡏⠀⣿⢀⡇⠀⠀⠀⣟⠻⠿⠛⠁⣴⠃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠯⣴⠟⠡⡹⣦⣙⠿⠛⠛⠊⣀⣼⢇⠜⠀⠀⠀⠘⠻⣟⡲⠶⢛⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣌⡪⢝⣻⣿⣿⣛⡋⠕⠋⠀⠀⠀⠀⠀⢀⡠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣲⣦⣤⣤⣤⣤⠤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣴⣿⠛⠋⠉⠻⠃⠈⢲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠔⣿⣯⡀⠀⠀⠀⢀⠀⢸⠓⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⣿⡛⠍⠀⠐⠐⣛⠀⢸⠀⠀⠙⣖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠇⠀⢰⠋⠙⣇⠀⠈⡡⣁⢴⠋⢰⠀⠀⠸⡄⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡖⠘⠀⠀⠈⠷⠊⠈⠓⠖⠋⠀⠀⣻⠁⠀⠀⠀⠻⢮⠿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⡯⠿⠏⠀⢀⣔⡣⠄⡀⠀⠀⠀⢀⣶⠵⠒⠒⢢⠀⠉⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠅⠆⠁⣴⣿⡯⠀⠀⠹⡀⠀⢀⣿⣻⠄⠀⢀⣀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⣣⣴⡶⠶⢧⣀⣼⣗⡛⣠⣞⣿⠉⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣷⣿⣿⣇⣀⡈⠟⠛⠿⠿⠿⠿⠿⠤⠄⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣍⠉⣉⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠈⠉⠁⣈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠠⠖⠂⠀⠤⠀⣀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣈⠉⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠈⢣⣀⠐⠂⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⠀⢈⠁⠀⠀⠌⣀⡑⠀⠀⠀⠀⠀⡘⣁⡈⡆⠀⠀⠀⠇⠸⢙⡏⢢⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠆⠀⢨⠀⠈⠀⠀⠀⡄⠻⢁⠀⠀⠀⠀⠀⢇⠻⠃⡇⠀⠀⠀⠀⠀⠸⠁⡃⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠚⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⢸⡠⢅⣀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢤⠀⠀⠠⠼⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠤⠀⠀⠀⠀⢀⡁⢀⣈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢄⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠉⠁⠀⠀⡎⠡⠀⡠⠔⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤⠘⢌⠂⠀⠠⡄⠀⢀⠾⠊⠐⠤⢀⡀⠀⠀⠀⠀⠀⢀⠠⢔⡌⣪⡀⢀⠀⣀⠀⠀⠀⠁⢐⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠓⠚⠂⢌⠁⡄⠐⢀⠈⠀⠐⡑⠂⠀⠤⠄⠀⠀⡀⠀⠀⠤⠀⢬⠀⠉⠉⠃⠐⠁⠀⠀⠀⢀⢠⠄⠬⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⡆⠀⠐⠉⠀⢰⣬⠇⠀⠀⡠⠒⠂⢄⠀⠎⡱⠀⠀⠘⢠⣤⡤⠧⡘⣤⠖⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠂⡤⠤⠠⡤⠀⠃⠀⠀⠀⠀⠸⠓⠀⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⢀⡀⠦⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠂⠐⠣⠀⠀⠀⠀⠀⠀⠀⠍⠐⠀⠀⠱⠦⠤⠥⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢘⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢫⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⢄⠀⡀⠀⠀⠀⠀⠀⠈⠲⡤⣤⣀⣀⡀⠀⠀⠀⣀⣀⢠⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡀⢘⠉⠀⠀⠀⠠⠀⠀⠴⠧⠤⠤⠵⠄⠀⠀⡨⣀⣀⣀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠐⡄⠈⠄⢘⠁⠀⠀⠸⡀⠀⠀⠐⠂⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃⠀⠀⠀⠀⠀⠛⠒⣤⠀⠀⠀⠀⠈⠓⠦⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠐⠚⠛⠒⢋⠒⠲⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠁⠀⠀⠉⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⡀⠀⠀⠀⠀⠀⠀⢐⠈⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢇⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠒⠒⠃⠐⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣦⡀⠀⠀⣉⠐⠄⠀⠀⡀⠀⠌⣠⠀⠀⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠠⠉⠀⠂⠀⠀⢄⠙⠋⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣄⣠
⠀⠀⠀⣀⡴⠶⣤⡀⣀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠵⠋⣠⠇
⠀⠀⠀⠈⢮⠂⠀⠈⠙⠻⢶⣠⣀⢀⠀⠀⠉⢁⣀⣲⣶⢶⣖⣒⣐⣒⠊⠀⠀⢀⣀⠔⠉⠀⠀⡘
⠀⠀⠀⠀⠀⠑⢕⠄⠀⠀⠀⠈⢹⠋⢤⡄⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠙⢺⡋⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠸⠀⣖⠄⢸⠀⠀⠀⠀⠠⠀⠀⢨⣍⣩⡅⠀⠁⢀⡠⡢⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⢐⡀⢸⠀⠠⠤⠦⡠⠅⡀⠐⢒⡔⠂⠀⡏⢣⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⢻⠀⢔⡀⠀⢉⣉⣉⣈⡀⠈⠁⠨⡵⠮⡅⠀⡇⣠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡸⠀⠀⠀⠀⠈⠁⠛⢸⠁⠀⠀⠐⠪⠭⠂⠀⣌⠛⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢡⡀⠀⠀⠀⠀⠐⠒⠚⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⠛⠛⠯⠭⠭⠭⠿⣩⡟⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠸⢄⡀⠀⠀⠀⠀⠀⡀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⠀⠄⡄⢹⠄⠀⠀⢀⡈⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠈⠒⠲⠶⠖⠒⠉⠀⠀⠀⠀⠁⠲⠆⠤⡤⡤⠪⠀⠀⠀⠀⠂⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣦⣤⣾⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢛⣻⣿⣿⣿⠟⢛⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢾⣿⣿⣿⣿⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣾⣿⣿⣿⣷⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣯⣽⣯⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠛⠛⠛⢻⠛⠛⠛⢿⠛⠛⠛⢻⣿⣷⣤⣀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣾⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣄⣰⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣤⣤⣤⣼⣤⣤⣤⣾⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⠉⠈⠙⠋⠉⠉
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣼⣿⣿⡿⠿⠟⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢀⣾⠿⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠾⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠛⠛⠛⢻⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⠉⠀⠈⢹⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡟⠋⠀⠈⢻⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⡀⣰⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣿⣿⣿⡿
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣿⣾⣷⣶⣶⣿⣾⣷⣿⣾⣷⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠋⣶⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⣒⣒⣒⣒⡶⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⢞⣫⣶⣿⣿⣿⣿⣿⣿⣿⣷⣮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠱⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣠⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢶⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣦⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢠⣿⣿⣿⡇⡀⣠⠾⣟⡳⣆⠀⠀⠀⢀⡴⢿⡛⢶⡀⠀⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠇⢿⣶⠿⢇⣿⠀⠀⠀⠸⣷⡾⠿⣸⡇⡆⣿⢸⣿⠟⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠉⠛⠛⠋⠀⡇⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠠⡀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⢀⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢶⡶⠶⠦⠶⠶⢶⠶⠶⠟⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣤⡶⠶⠿⠿⠶⢦⣤⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⡀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢬⠀⠀⠒⢸⠐⠂⢈⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠑⠢⠒⠁⠉⠊⠁⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡠⢄⡀⠀⠀⠀⠀⠀⠀⣀⠤⡀⠀⠈⢦
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠘⢄⡀⠈⠙⠓⠒⠒⠛⠉⠀⣀⠜⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢀⡀
⡔⠉⠈⠑⡄⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠉⠓⠓⠂⠰⠄⠒⠓⠊⠁⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⡔⠉⠈⠑
⠙⢄⣈⣤⡃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣣⣌⣀⠔
⠀⠀⠈⠣⡑⢄⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⢀⠔⡡⠋
⠀⠀⠀⠀⠉⠢⢝⣒⢒⡇⢸⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⠀⠀⢸⠀⣗⢒⣚⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠈⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣶⣶⣶⡖⠒⠒⠒⠒⠒⠒⣶⣶⣶⣶⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠿⣻⠃⠀⠀⠀⠀⠀⠀⢻⡛⠟⡿⠋
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⢧⠀⠀⠀⣰⢫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠾⠾⠷⠶⠾⠷⠿⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠀⢀⣤⣄⡀⠀⠀⣀⣤⣀⠀⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢰⣯⣾⡮⣷⠀⢸⢱⣷⣞⡇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠀⠓⣿⣷⣃⣀⣈⣳⣿⡞⠁⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠷⠿⠶⠿⠶⠿⠿⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣿⣛⣛⣛⣛⣛⣛⣛⣻⣛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖⢾⠋⠉⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⡷⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠋⠉⣿⠀⠀⠀⠀⠀⠀⠀⣰⠖⠲⡄⣿⠉⠉⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠃⣿⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⣿⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢺⡀⢀⡇⣿⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⡾⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣿⠀⢰⢿⣶⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡿⢷⡿⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡼⠿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠶⢶⠶⠶⠶⠶⠶⠶⣶⠶⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣓⣒⣺⡇⠀⠀⠀⠀⠀⠘⣟⣩⠞⣩⣧⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡯⠯⢽⠇⠀⠀⠀⠀⠀⠀⠈⠳⣏⣁⡼⠟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠾⠷⠶⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣿⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣼⠇⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠒⣛⠒⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠀⠀⢀⡀⠀⠘⠀⢀⡄⠀⠀⠱⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠐⠀⠀⠈⠁⠀⠜⠀⠈⠁⠀⢀⠆⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠲⠂⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠑⢼⡀⠀⢹⠀⠀⢸⠀⠀⢸⠅⠀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠠⠼⠄⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡯⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡎⢹⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢣⠀⣀⡄⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⠀⡇⠀⠀⠀⠀⢀⡇⢸⣀⠀⡝⠒⢄
⠀⠀⠀⠀⠀⠀⢀⠚⢄⣠⠞⠉⠀⠘⠺⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠀⡇⠀⠀⠀⠀⢸⠗⠃⠀⠉⠢⣄⡠⠓⡀
⠀⠀⠀⠀⠀⢠⣁⠀⡜⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⢧⠀⣀⡄
⠀⠀⠀⠀⢀⠃⠀⡝⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢏⠁⠐⡀
⠀⠀⢰⠊⡉⢉⡘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠂⣉⠉⠑⡄
⠀⠠⠊⠈⠁⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣤⠀⢠⠄⠀⠀⠀⠀⠀⢠⡄⠀⠤⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠁⠀⠈⠉⠓⢄
⠀⠀⠀⠂⠁⠉⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⡇⠀⠀⠀⠀⠀⢸⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠉⠈
⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠚⡇⠀⠀⠀⠀⠀⢸⠓⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠲⠲⡇⠀⠀⠀⠀⠀⢸⠖⠒⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣗⢒⠲⡇⠀⠀⠀⠀⠀⢸⠖⢒⣺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠀⠀⠀⠈⠙⢆⠀⠀⠀⡰⠋⠉⠀⠀⠀⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⡠⠤⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣈⢂⡎⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠸⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠚⠛⠛⠛⠛⠛⠛⠋⠀⠀⠛⠛⠓⠒⠒⠒⠶⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠒⠂⠛⠀⠐⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠀⠴⡚⠒⠀⠛⠴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠇⠲⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡠⠀⢤⣤⡄⠀⠀⠀⠀⠀⠀⣀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠊⠀⡤⣄⠈⠌⠀⠀⠀⠀⠀⡞⢀⣀⠈⠹⠀⠀⠀⠀⢀⠀⠀⠈⠀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠄⠀⠓⠊⢀⢃⠆⠀⠀⠀⠀⡀⠈⠊⢀⡄⡸⠀⠀⠀⢸⠀⠀⠀⠀⢃⠀⣸⠃⠀⠀⠀⠀⡞⠋⠙⢦⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠒⠀⠒⠋⠁⠀⠀⠀⠀⠀⠈⠐⠒⠛⠈⠁⠀⠀⠀⢸⠀⠀⠀⠂⠐⢺⠃⠀⠀⠀⠀⠀⠘⢤⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡇⠀⢲⠂⠀⢰⡆⠀⢰⠀⠀⢹⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡶⠛⠛⠒⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⣃⠀⢸⠀⠀⢸⡇⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠉⢳⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠈⠁⠉⠀⠉⠀⠒⠚⠓⠂⠛⠒⠒⠛⠒⠒⠂⠀⠀⠀⠚⠃⠁⠀⠀⠀⠀⢸⠀⠀⢀⡠⠎⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠖⠋⠣
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠚⡇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⡇⠀⠀⣯⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠓⡄⠀⠀⠀⡄⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢤⡤⠔⠒⠚⠉⠁
⠀⠀⡾⠀⠀⠀⠀⠈⢀⣤⠶⠓⠒⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡃⠀⠀⠀⠀⡶⠋⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⣈⠹⢆⣰⣶⠾⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡜⠁⢀⣤⡀⠈⢣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠳⠶⠏⠀⠷⠤⠜⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠼⣃⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠉⠉⠉⠁⠉⢁⡞⠛⠒⠒⠒⠲⠦⠤⢤⠤⠤⠤⠔⠛⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠶⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⢉⠉⣁⡽⢦⡄⠀⠀⠀⠀⣠⠿⣁⡀⠐⢶⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⣾⡁⠀⠀⠀⠉⠁⠘⣆

⠀⠀⠀⠀⡰⠛⠉⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀
⠀⠀⠀⢰⢓⠊⠛⠛⠀⠌⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠏⠀⢈⡇
⠀⠀⠀⡟⢻⠒⠒⠚⠃⠂⠡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⢰⣿⣿⣷⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⠃⠐⠁⣼
⠀⠀⢰⡒⠰⠶⠶⠶⠶⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠒⠦⠼⢿⡿⣿⢿⡯⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡂⠀⠀⠀⢾⡀
⠀⠀⢸⣍⣄⣈⣈⣁⣍⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢙⣿⣿⣟⣉⣀⠀⠀⣄⣀⡀⠀⠀⠀⠀⠀⢺⡧⠄⠀⠁⠀⢙⡇
⠀⠀⢸⡎⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠁⠀⢺⠉⣿⣥⣤⣼⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡾⡄⠰⡔⠊⣷
⠀⠀⠀⢿⡣⠉⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠹⣏⢉⡿⠁⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⡴⠛⣬⣿
⠀⠀⠀⠘⣿⣋⣍⡉⢥⡴⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠙⠟⠁⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁
⠀⠀⠀⠀⠘⢦⣤⣶⠎
⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣤⣴⡟⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⣴⣿⣿⣿⣷⣄⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣾⣿⣿⣿⣿⣽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⣿⣿⣿⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢷⣿⣿⣿⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⣀⠴⠖⠒⠶⢄⡀⣀⡀⠴⠋
⠀⠀⠀⠀⢀⠔⠈⠀⠀⠉⠐⢤⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡞⠁
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠋⠈⢻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⢂⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣶⡄⠀⣴⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠏⣸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣥⣶⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠠⠾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⡿⠛⠁⠀⠀⠀⠀⣀⡏⠀⠷⡀⢰⣶⡄⠈⠛⢿⣿⣿⣿⣿⠟⠋⢀⣴⣖⠀⠸⠆⠸⡄⠈⠙⢿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⣶⣿⣿⠃⠰⠆⠀⢿⡼⠻⠀⠀⠀⠙⣯⡏⠁⠀⠀⠘⠻⡾⠂⠠⠦⠀⣧⡀⠀⠀⠈⠻⣿⣿
⠀⣠⣶⣿⣿⣿⡿⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣦⡀⠀⠀⠈⠻
⣾⣿⣿⣿⠟⠁⠀⠀⠀⣸⠟⠃⠀⠀⠀⠀⣀⠀⠀⠛⠃⠀⣀⡀⠀⠀⠀⠀⠛⢧⠈⠻⢿⣿⣿⣦⡀
⣿⣿⠟⠁⠀⠀⢀⣴⣾⡟⠀⠀⠀⠀⢀⠄⠉⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠄⠺⣄⠀⠀⠙⢿⣿⣿⣦
⠟⠁⠀⠀⢀⣴⣿⣿⡟⠁⠀⠐⠶⣤⣬⡀⠃⠀⠀⢤⣀⠀⠀⢛⣭⣤⠴⠆⠀⠀⠹⣷⡄⠀⠈⠻⣿⣿⣷
⠀⠀⠀⣴⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠡⣮⡿⠇⠀⢸⡿⠁⠀⠀⠀⠀⠀⠉⠛⢦⠀⠀⠹⣿⣿
⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠁⢠⡀⠀⣠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠹⣿
⢀⣾⣿⡿⠉⢧⣀⢀⣀⣀⣀⣀⣤⣤⡤⢀⢘⠛⡀⠀⠀⢚⠻⠀⠦⣤⣤⣄⣀⣀⣀⡀⢀⣴⣿⡆⠀⠀⢻
⠾⠿⠿⠥⠤⣤⠾⠿⠿⠿⠿⠥⠤⠤⠶⠤⠤⠤⠤⠤⠤⠬⠥⠥⠤⠦⠤⠬⠿⠿⠿⠿⠿⠿⠿⠿⠤⠤⠬

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡏⣴⡌⣿⣿⡿⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⠻⢿⣿⠀⣿⣏⢻⣿⣿⣿⣿⣮⣵⣿⣿⣧⣟⣽⡇⠀⠀⣠⣤⠀⢷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣦⣼⡿⠰⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠐⢿⣧⡠⣠⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣤⣬⣉⣁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣽⣿⡇⠀⠈⢲⣷⣾⣷⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⣿⡏⢻⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠛⠃⠸⣿⣴⣟⠿⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⠀⢶⣿⣿⣿⣇⣁⣿⣿⣿⣿⣿⣯⣿⣾⣿⡏⣀⢙⣿⣮⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣰⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⠀⠸⣿⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⠿⢿⣿⣿⣿⣻⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣾⢿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣌⣻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⢀⣸⣿⢿⣿⣿⢛⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡙⠻⠿⠟⠃⠘⢿⢣⣿⣿⡟⢻⣿⡇⠉⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⢿⣷⣶⣦⠀⠀⢮⡿⢭⣛⢻⣿⡟⠆⢸⣭⣽⠻⣿⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣾⣧⣼⣿⠀⡛⠀⣿⣟⣽⣼⣿⣧⠂⢸⣿⣻⣴⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣅⣈⠙⠁⠀⣙⠀⢿⣿⣿⣿⣿⣇⠀⢸⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠻⣷⡄⠀⢬⠀⣸⣿⣿⣿⣿⣿⡖⣾⣿⣿⣿⣿⣿⣖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⢸⣿⠀⢶⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠋⠀⠛⠂⠞⢿⣿⣿⣿⣿⣿⣿⣶⢿⣿⣿⣿⣿⣿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⠛⢻⣄⣀⣀⣠⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢤⣤⡾⠛⠀⠀⠈⠉⠉⠉⠉⠀⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠁⢀⡤⠶⠚⠛⠛⠛⠲⢦⣄⠀⠙⢷⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⢀⡴⠋⠀⠀⠀⡀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠟⠁⢠⡟⢠⣀⠀⣰⠉⣿⠀⠀⢀⠶⣶⡄⠘⣧⠀⠸⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⠀⠀⣾⠀⣸⡅⠀⠘⠿⠋⢀⣀⠘⠶⠟⠀⢠⣽⡆⠀⠈⢙⡇⠀⠀⠀⣀⣀⡀⠀⠀⠀⣴⠞⠋⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⣿⠀⠉⠀⢀⡀⠀⠀⠻⣤⣿⠆⠀⠐⠃⠙⣷⠶⠿⠿⠃⠀⠀⠀⢿⣿⣿⣟⠳⣾⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢴⣋⣽⡦⢄⣀⣀⡠⣴⣷⠀⠀⣸⠇⠀⠀⠀⠀⣀⣠⣤⠾⠟⠂⠈⠀⠸⡄⠀⠀⠀⠘⠓⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠙⠉⠛⠻⠿⠿⠟⠛⠁⠀⣰⠟⣀⣤⣴⠾⠟⠋⠉⠀⠀⠀⠀⢀⣀⣤⣿⣦⣀⣀⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⢀⣤⠾⢿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢿⣏⣿⠿⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⠀⠀⠀⠀⣿⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⢀⣸⠷⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠟⠉⢿⣦⠒⢲⠖⢉⣱⣄⠀⠀⠸⡀⠀⣀⣠⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣯⡉⠀⠀⠀⠈⠙⢷⡾⠿⠛⠛⠁⠀⠀⠀⢹⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡛⠟⠿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠃⠁⠀⢀⣾⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⣾⠏⠀⢿⡦⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⢙⡆⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡏⠀⠀⠘⣿⢀⠀⠀⣴⠛⠉⠁⠀⣠⡶⠛⠁⠀⣘⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣧⠀⠀⠀⢹⣆⣄⣀⣼⠇⠀⠀⠀⠸⣧⠀⣤⣿⣿⠿⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠉⠛⠒⠒⠛⠋⠉⠹⣷⠀⠀⠀⠀⠀⢻⡄⠂⠉⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢠⣶⡀⠀⣶⡆⠀⢹⣧⣀⣠⡴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⠀⠉⠀⢀⣀⣠⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠉⠀⢻⣇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⢰⣶⠄⠀⠀⠀⠀⠀⢀⣠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢿⣷⣄⡀⠀⠁⢀⣀⣠⣴⣶⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢋⡁⠀⠀⠉⠙⠿⠶⠛⠛⠿⠛⢾⣿⣿⣿⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠍⠲⣌⠋⢁⠀⣀⠀⢀⣀⣤⣀⣤⡁⠀⢨⠿⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠙⠠⠐⠈⠛⣡⣄⡚⠉⠉⢉⡉⠀⠀⠀⠀⠉⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⡀⠀⠀⠀⠀⣠⢶⡿⠛⠋⠀⠀⢰⣿⠃⠀⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠇⣰⠠⡄⠐⣷⠁⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⡟⢀⠃⠀⡿⠀⠀⠀⠀⠀⣼⠏⢀⡔⢻⣆⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠘⠀⠀⡇⡄⠀⠀⠀⣸⡟⢀⠏⠀⣼⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠁⡇⠀⠀⢠⡿⠁⡎⠀⢰⡟⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢰⡇⠀⢀⣿⠃⢰⡁⣰⡟⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⠃⠀⣼⠇⠀⠀⠛⠋⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠼⠣⠤⢀⡀⠀⢸⢳⣾⣏⣀⡀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⢶⣴⡶⠀⠈⣉⣹⣾⠛⣷⣄⠀⠀⣈⣈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣤⣤⣶⡶⠿⠛⠉⠁⠀⠈⠛⢷⣄⡉⠛⠿⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢶⣶⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⣴⣿⠉⣿⠉⡟⠉⣉⣻⠙⣿⡟⢹⣿⡟⢉⡙⢿⠉⣿⣧⠀⠈⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢿⣿⠀⣤⠀⡇⠀⠶⢿⠀⠿⠇⠘⠿⣇⠸⠃⣸⠆⣿⣿⠃⠀⣿⠀⢀⣴⢾⡏⣿⢹⡏⣿⢻⡷⣤⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⣿⣴⠟⠁⠈⠁⠉⠈⠁⠉⠈⠁⠈⢻⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⠿⠛⠛⠉⠁⠀⠀⠀⢰⡟⢹⡏⣴⢛⣿⡟⠛⠛⠛⠛⣻⣿⠻⣆⣿⡉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢸⡇⢷⣜⣿⣋⣀⣀⣀⣀⣹⣟⣡⡟⢸⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⢿⣟⡛⢛⣻⠟⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣭⣍⣭⣥⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣿⣀⣀⣀⣈⣿⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⣠⣴⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯
⢀⣾⣿⣿⣦⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧
⠀⠻⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄
⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠏⢀
⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀
⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣠⣤⣄⡀
⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⣶⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢿⣿⡿⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢾⠿⠿⠿⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢀⣀⣀⣀⣀⣀⣀⠺⠏⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⠙⣿⠟⢁
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣦⡀⣴⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃⣰⡿⠋⣉⣉⡉⠛⣿⣿⣿⠟⠋⣉⣈⡉⠻⣿⣿⣿⠛⢉⣁⣉⠙⢿⣿⣿⡿⠋⣉⣈⡉⠻⣷
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠰⣿⠁⣼⣿⣿⣿⡆⢸⣿⡟⠀⣿⣿⣿⣿⡀⢹⣿⡇⢰⣿⣿⣿⣧⠈⣿⣿⠀⣾⣿⣿⣿⡆⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢿⣆⠘⠿⡿⠿⠃⣸⣿⣿⡀⠻⢿⡿⠟⢀⣾⣿⣧⠈⠿⢿⠿⠋⣰⣿⣿⡄⠙⠿⡿⠟⢁⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⣛⣷⣦⣤⣴⣾⣿⣿⣿⣿⣶⣤⣤⣶⣿⣿⣿⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣷⣦⣤⣶⣿⣋
⣿⣿⠉⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⠊⠄⢈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⡑⢢⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡡⡄⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⢱⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠆⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡱⡀⠀⠀⠀⠀⠀⠀⣜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡵⡀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⠀⠀⠀⠀⠉⠉⠀⠐⠒⠚⠛⠀⠒⠒⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⣄⡄⠀⠠⠐⠒⠒⠤⡀⠀⠀⠀⠔⠀⢉⡒⢄⠀⠁⠀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡿⡇⠀⢂⠀⠿⠆⠀⡁⠀⠀⢘⠒⠢⣴⡖⠊⠇⠀⠠⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠐⠊⠀⠀⠀⠈⠢⢄⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⠒⡄
⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣧⣐⣠⠂
⠀⠀⠀⡄⠬⢊⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠑⠠⢀⡀⠀⠀⠀⡀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣫⠞⠁⣜⡇
⠀⠀⠀⠀⠙⣟⢅⠉⠙⠒⠭⢖⡠⢄⡀⠀⠀⠀⠀⠈⠑⠠⠤⠤⢀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⠜⠀⠀⠀⠀⢀⠔⣢⠞⠁⠀⡐⡻⠁
⠀⠀⠀⠀⠀⠈⢦⡳⡀⠀⠀⠀⠈⠓⠮⢵⣦⣄⣠⠠⠀⠤⠀⢠⠤⡤⠿⢿⠟⠿⢯⡤⠤⠤⣤⠤⢀⡀⠀⡠⣊⠴⠊⠀⠀⠀⠔⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣌⠢⡀⠀⠀⠀⠀⠀⠈⡿⠁⡖⣤⠀⢰⠒⠒⠂⠀⠀⠀⠤⠤⡀⠀⠀⠀⠀⠀⠹⢙⡟⠁⠀⠀⠀⢀⠊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⢢⡀⠀⠀⠀⠀⡇⠀⣨⡁⠀⢸⠈⠉⠀⠀⠀⠀⠀⠀⠂⠀⠲⠚⣚⠚⠃⠀⠙⠀⠀⠀⡠⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢆⠀⠀⠀⠇⠀⠓⠋⠀⢸⠀⠀⠀⠀⣠⠀⠢⡀⠁⠀⣙⠓⠂⠛⡁⠀⢠⣤⡴⢮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⣤⢄⡀⠀⢨⠤⠂⢸⠀⠀⣚⠒⠃⡶⠔⠂⡆⠀⢉⡩⠤⢍⡁⠀⣸⠀⢱⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⢞⡗⠀⠐⢢⠀⠀⢉⢉⣉⣉⣉⠀⠀⠈⠁⠀⢨⣑⡚⣋⡀⠀⣿⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⢗⠀⠈⠉⠀⠀⠈⠻⠹⣿⠉⡟⠁⠀⠀⠀⢠⡤⠒⢪⡄⠀⠉⠼⣫⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⣷⢹⠀⠀⠀⠀⠀⠀⠐⣀⠃⠀⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠛⠢⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⢾⣘⡂⢀⠀⠀⠀⠀⣉⢛⡉⠃⢀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣶⠿⠻⠇⠩⠉⣀⣈⠉⣉⣉⣙⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠉⠀⢈⠉⠳⡀⠀⠀⠀⠀⠀⡠⠂⠉⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⢀⠀⢤⢠⡎⠀⠀⠤⠑⠸⠀⣃⡗⠠⠐⡀⠀⢰⡄⠢⢠⠠⠀⡀⡀⠉⢶⡠⠀⡀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠁⠐⠈⠐⠂⠙⠒⠺⠓⠷⠚⠍⠂⠅⠂⠥⠂⠥⢂⠍⠋⠖⢲⠤⣤⢤⠴⠪⠐⠍⢂⠔⡭⠢⠑⠡⠢
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠄⡤⠤⢄⠀⠀⠀⡠⠤⠤⠠⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡌⡎⠉⠱⠃⠀⠘⠔⠉⠑⠑⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠱⢑⣀⠪⠆⠀⠐⢔⣠⢚⠌⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⠉⠉⠉⠿⠉⠉⠉⠉⠉⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⣀⠀⠀⠀⢸⡇⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢃⠀⠀⢀⣿⠀⠀⠀⠈⠳⠶⠶⠶⢶⡶⠶⢶⡶⠶⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣒⡴⣯⠚⠀⠀⠀⠀⣀⣀⣀⣀⣸⣇⣀⣸⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡈⢻⡁⠀⠀⠀⠀⢰⡏⠉⣩⣭⣍⡉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢠⠒⢛⠦⢤⠤⣤⢼⣇⡊⠂⣀⠊⢙⡄⠀⠀⠀⠀⠄⢸⡧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⣇⠀⡁⠸⡀⢸⡇⠁⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⢸⡇⢈⠆⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⢹⡇⠁⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⢸⡏⠑⠒⢬⡀⠜⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢳⠤⠔⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡼⠦⢤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⢀⢞⠝⠉⠙⡞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠇⠀⠀⣿⠀⠀⠀⣸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣍⠂⢲⠃⠀⠀⠀⠀⢸⡂⠐⣹⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠤⠥⡏⠀⠀⠀⠀⠀⢸⡢⠭⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢍⣒⣸⠁⠀⠀⠀⠀⠀⢸⠧⠉⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠦⠤⢾⡀⠀⠀⠀⠀⢰⠟⠛⠒⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣸⠀⠀⠀⠀⠀⠘⠒⠤⠤⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣾⣿⣿⡿⠟⠙⡡
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⡎⢀⣿⣿⣿⣿⣧⣤⣦⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠿⣿⠯⠚⣤⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢆⡿⣧⣽⣿⣿⣿⣿⣗⠙⠻⢟⠂
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠆⢸⣠⣾⣿⣿⠋⡾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⠄⠀⠀⠈⣻⣿⣿⣿⣿⣿⠖⠀⠀⠐⣾⣦
⠀⠀⠀⠀⠀⠀⣿⣿⡌⠠⣦⣼⣿⡿⠛⢛⠀⠁⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠫⠀⠀⠀⠈⠙⠿⠋⠉⠿⠋⠀⠀⠀⠀⠿⠏
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⣿⣿⢿⣶⣾⣻⡶⣶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠑⡀⠀⢀⣀⡀⠤⠤⠤⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⣤⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣈⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⣀⣤⣄⣀⢿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠤⠄⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⡍⢀⠀⠀⢀⠀⠀⠀⣀⠀⢀
⠀⢀⣴⣟⣋⢰⣿⠛⢿⡆⠹⡛⣿⡿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣺⢩⣠⣤⣤⠒⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⠃⢸⠀⣿⠀⢹⠀⡏
⠀⢸⡟⣰⣿⣾⣯⡶⣿⣷⣤⣹⣭⡥⢙⠟⠳⠖⠋⠴⢶⠖⣠⠀⢬⣥⢸⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⣸⡀⠛⢀⢸⡀⡇
⠀⢸⣿⣿⣿⣿⡏⡈⠀⠉⠃⡿⠁⠀⠉⠀⢁⣀⣀⣠⣤⣤⣿⣤⣛⣿⣾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠉⠉⠁⠉⠉⠉⠈⠉⠁
⠀⠀⠙⣿⣿⣿⣿⣿⣷⣦⣄⣱⡶⠿⢛⣛⣿⡿⠟⠛⠋⠉⠉⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⢻⠿⠿⠟⣼⡛⣧⣭⣭⣝⣛⣋⢉⣁⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡖⠀⠀⠀⠀⠀⠐⡈⢂
⠀⠀⠀⠀⣶⣦⠀⢰⣿⣿⣿⣿⣿⡿⣿⣽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠌⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁
⠀⠀⠀⠀⢻⣿⣆⠀⠉⣽⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠡
⠀⠀⠀⠀⠀⢿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⡤⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢘⣉⠉⠉⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣙⡀⢀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⣏⣿⡋⣋⡇⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⢟⠂⠭⠽⠿⠯⠇⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡀⠄⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠀⠀⣿⠋
⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⠛⣷⣀⡀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣤⡿⠇⠀⠀⠈⠛⠛⠛⠋⠀⠈⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠈⠁⢀⣤⡴⠶⠶⠶⠶⢦⣤⡀⠀⠙⢧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠁⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠴⠟⠀⣰⠏⡀⠀⠀⡐⢻⣧⠀⠀⠀⣴⣤⠀⠹⣆⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢠⡏⠀⢟⠁⠀⠳⡾⠏⠀⠀⠸⣤⡿⠀⠀⣻⡄⠀⠙⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⣾⡇⠐⠋⠀⠀⠀⠀⠀⣾⣁⣶⡀⠀⠠⠌⢻⣧⣤⣤⣶⠟⠀⠀⠀⢸⣿⣯⣶⠦⣤⣾⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢀⣴⠟⣦⣀⠀⠈⠙⢋⣠⣤⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⣀⣸⣿⠯⠙⠂⠉⡷⠀⠀⠀⠀⠿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠈⠻⠾⠻⢿⣵⣶⣷⣿⠿⠛⠀⢀⡿⠁⠀⣀⣤⣴⡶⠞⠋⠁⠀⠀⠀⠀⢀⣸⣆⡀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣶⡾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⢺⡿⢯⣿⡿⠷⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣄⠀⠀⠀⠀⣶⠋⠈⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣻⣯⠟⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⣿⢀⡀⢀⡤⠿⢀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣀⣤⠤⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠿⠟⠉⠀⠀⠻⣿⣅⣾⣤⣴⣾⠗⠀⠀⠈⢧⣠⠴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠣⠈⠉⢙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠈⠀⠀⢀⣾⡏⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡐⠉⣫
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⡏⠀⠘⣿⣃⠀⠀⠀⢀⣀⠴⠶⠛⠋⢁⣨⡗⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⠀⠀⠹⣏⡄⠀⠀⣿⡁⠀⠀⠀⢾⡋⠁⠀⣀⣤⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⢻⣦⣀⠀⢀⣻⣤⣤⣴⡿⠃⠀⠀⠀⠘⣷⠀⠲⠿⠟⠋⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠘⣷⡀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⢾⣷⠀⠸⠿⠆⠀⢹⣧⣤⠴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⡀⢀⣠⣤⢴⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢿⡔⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣠⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠻⣷⣄⡀⠀⠀⠀⣀⣤⢶⢾⣿⣿⡿⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢉⠀⠀⠀⠀⠙⠿⠖⠛⠛⢻⡇⠈⣴⢿⡌⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢫⡑⠢⣉⠂⠀⠀⠀⠀⠀⠀⢠⡰⣬⡟⠁⠀⠛⢠⠑⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⠓⠄⠲⠍⢊⡁⣄⡲⠝⠋⢩⡟⠀⠀⠀⠀⠈⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⠀⠀⢀⣤⣞⠷⠙⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠻⠀⡆⢀⠀⢸⣷⠁⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣸⠇⠸⠀⢸⡏⠀⠀⠀⠀⠀⣼⡏⠀⡰⢶⣆⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠇⠀⢸⡇⠀⠀⠀⠀⣸⡟⢀⠞⠀⣸⡟⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⢡⡇⠀⠀⢠⣿⠁⡜⠀⢠⣿⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢸⠇⠀⢀⣾⠃⢠⠁⢠⣾⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣸⠀⠀⣼⠇⠀⠈⠶⠟⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⣀⡀⠀⠀⠀⠹⢠⣸⡏⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⢉⡄⣤⡍⠙⠒⠄⣤⣼⢿⣇⠐⠂⠀⠀⠰⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣈⣙⣉⣡⣤⣶⡾⠟⠋⠀⠹⢷⣄⡀⠳⣶⣤⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣤⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⣉⣉⣉⠉⠉⢉⣉⣉⣉⣉⣉⣉⡉⠉⣿⠋⢻⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠛⠛⠛⠁⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⣿⠀⢸⡟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⣿⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣾⣿⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⣿⣶⣾⣷⣶⣿⣤
⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣼⣿⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣿⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⢹⣿⠀⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁⠀⣿⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⠋⣠⣴⡾⠿⠿⠿⠿⢷⣦⡀⢠⣶⣶⣶⣶⣶⣶⣶⡆⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣀⣄⣰⡿⠁⢠⡄⠀⠀⣠⡀⠙⣿⣄⣠⣀⢀⣤⣤⣤⣤⡄⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠘⢿⠿⢿⣇⠀⠈⠁⠀⠀⠉⠁⠀⣿⡿⢿⠿⠈⠉⠉⠉⠉⠁⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⣀⣀⣀⣀⣈⣿⣦⣀⣀⣀⣀⣀⣠⣾⣿⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⣿⡇
⠀⣼⡟⠛⠛⢻⣿⠛⣿⡟⢿⣟⠛⠛⠛⢿⣏⠀⣰⡿⢿⣯⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⡀
⠀⣿⡇⠀⠀⣸⡿⢀⣿⠇⢸⣿⠀⠿⠇⢸⣿⣾⠏⣠⣾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⠀⣿⡇⠀⢠⣿⠃⣼⣿⣶⣾⣿⣶⣶⣶⣾⣿⣥⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠘⠿⣶⣿⣿⣾⣿⣿⣀⣀⣀⣀⣀⣀⣿⣏⡉⠁
⠀⠀⠀⢀⣾⠟⡛⢛⡛⢛⡻⣿⣟⣛⣻⣿⣛⣻⣷⡀
⠀⠀⠀⠘⣿⣌⣋⣘⣃⣘⣣⣿⣟⣛⣻⣿⣛⣻⣿⠃
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣊⣰⠀⠀⠀⢾⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢻⡀⠀⠀⠀⢀⡿⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠋⠉⠉⠉⠻⠉⠉⠉⢉⣩⠽⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠭⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⡴⠞⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡖⢺⡇⣴⣶⣦⢤⠀⠀⢀⣤⣴⣶⣄⢸⡇⠀⢀⡞⠉⡟⡱⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⢸⠇⠸⣿⡿⢿⠀⠀⢸⢿⣿⣶⡏⢸⡇⠀⠸⡄⠘⡇⢇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⠀⠐⢲⠄⠀⠀⠀⠀⠠⣤⡈⠁⠀⡇⠀⠀⠑⠒⢛⡏⠁
⠀⠯⢭⡙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣀⣈⣙⣒⠲⠶⠶⠶⠾⠆⠀⠀⡇⠀⣀⡤⢾⡉
⡜⠳⠴⢧⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠉⠉⠉⠛⠛⠛⠒⠲⠶⠶⠷⠺⡅⠀⠀⠙⢦⡀
⠈⠒⠒⠙⠲⡏⣸⠣⣄⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⣤⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠙⠳⣄⠈⠙⠛⠛⠛⠉⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⡏⠀⢀⠀⠀⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠛⢻⡇⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⠛⠋⠉⢻⠻⣄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠄⢠⣄⣀⣀⣀⣐⡂⠁⠤⠀⠠⠄⠠⡄⠇⢿⠀⣀⡤⢒⣹⡄⢹⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠤⠤⠤⠄⣀⣀⣉⣉⣉⣉⣉⠛⠓⠒⠒⢻⣋⠤⠚⠁⢸⣷⡞⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢻⠀⠀⠀⠀⢸⣧⡟⣲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣠⡞⡵⠛⢫⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⢲⡶⠶⢶⡤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣿⣠⡴⠛⠁⣇⣧⠀⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣸⡁⠀⠀⠀⠀⠀⣼⠉⠉⣿⠉⠉⠁⠀⠀⠀⠈⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠶⢾⣷⡀⠀⠀⠀⠀⣹⣀⠠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡛⠁⠀⢀⡴⠋⠀⣇⠀⣀⠴⠚⠁⠉⠛⡿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⠭⣭⣭⣽⠴⠖⠋⢱⡻⠷⠤⢤⣄⣀⡞⢁⡼⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣒⣶⣶⣷⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⢠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢀⠔⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠛⠛⠛⠛⠓⠿⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⢸⠃⠘⠶⠴⠂⠀⠠⣄⣠⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣾⠀⣶⣶⣤⣄⣀⣀⣀⣀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⠀⠉⠛⠿⢿⣾⣿⣾⠿⠃⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⣿⣷⣾⣶⣦⣤⣤⣄⣀⣀⣴⡿⠀⠀⠀⢀⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣽⠀⣀⣀⣀⣀⡈⠉⠉⠛⢿⣏⠀⠀⠀⣠⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⡏⢰⡏⠹⠉⠭⠩⠍⢹⠆⢸⣿⣤⣤⠾⠋⢸⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⡄⢸⠃⢸⡇⠀⠀⠀⠀⠀⣾⠀⣾⠀⣀⣀⣠⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⢶⣴⣾⠛⠋⠉⣰⠇⢾⠀⢸⠁⠀⠀⠀⠀⠀⡟⠀⢼⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⣹⣶⢶⡿⠉⠀⣽⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠟⠉⠀⢸⡇⠀⠀⢹⠀⢸⡇⠀⠀⠀⠀⠀⢸⡄⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠈⣷⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡆⠀⢷⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⡇⠀⢸⡀⣀⠀⣄⢤⠰⣞⡇⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢿⠀⠀⢷⠴⠶⢦⡀⣶⣟⠁⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⡆⠀⠀⠀⠀⠈⣛⣷⠹⡆⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢀⡒⠀⣷⠀⠠⠄⠀⢺⣭⠥⠖⠋⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⡀⢹⣄⣀⣤⣤⡶⠶⠛⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿⡇⠀⠀⢰⡞⠋⠙⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠟⠛⠛⢿⡇⠀⠀⢿⣿⣿⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣿⣿⠇⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠛⠛⠛⠛⠛⠿⢶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠐⣿⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢰⣿⣤⣤⣤⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⡇⠀⠿⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⠹⣷⡀⠀⠘⠟⠀⠀⠀⠿⠀⠘⠇⠐⠿⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃⠀⠀⠀⠈⠻⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⢀⣿⠀⠀⢀⣤⣶⣶⣶⣶⠾⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣟⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣾⣿⣿⣿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡟⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠋⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⡿⠻⠟⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣾⣭⣛⡋⠂⠐⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⠉⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⡿⠿⣿⣿⠀⣴⡾⣶⣄⣸⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⣠⣿⣿⠈⣿⣤⣼⡿⣿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⠆⠈⠉⠉⢠⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣻⢿⠿⣀⡀⠠⠤⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣾⡧⠠⣤⣖⣲⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⢰⣾⣿⣷⣿⣿⣿⣷⣦⣿⣿⣿⣾⡟⠑⠆⠀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣿⣤⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠊⠀⢰⣠⣤⣿⡎⠇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⠇⠉⠉⠙⠛⠏⠉⠀⠀⠀⠀⢠⣼⣿⣿⣿⣷⢸⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣧⡶⣖⣶⣶⣶⣲⣖⢦⠀⠀⠀⠙⣿⣿⣿⣿⢸⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡿⡘⢰⣲⠀⠀⣿⣿⣿⡟⢸⡇⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠉⠀⢡⣿⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡿⣿⣿⠏⣰⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠐⠁⠀⠀⢂⣾⣿⣄⠹⣿⣿⢏⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠂⣹⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢷⡀⢀⣾⣿⣿⣏⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠉⠉⠻⠟⠉⠀⠀⠀⠀⠀⠀⠈⠛⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣶⣾⣦
⠀⠀⠀⠀⠀⣴⣶⣿⣶⣦⠀⠀⢰⣔⠀⠀⠀⠀⠀⠈⠉⣿⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠛
⠀⠀⠀⠀⠀⠀⣀⣿⣅⡀⠀⠀⢸⡇⠀⠀⢾⠶⢶⡏⠿⣟⠿⢻⠶⢾⡗⠀⠀⠀⢀⣤⣤⡄⣴⣿⣧⡠⣤⣤⣄
⠀⠸⡗⠛⠒⢿⣿⠿⢻⣿⠓⠚⠺⠇⠀⠀⣿⡀⠀⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⠶⣿⠁⠀⣼⣿⠙⡻⣿⡀⠈⢹⡷⠄
⠀⢸⣇⠀⠀⠸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠻⠁⠐⣏⣿⣹⣿⡹⠀⠸⠏⠀⠀⠀⠀⣀⣦⠿⣿⡷⣷⠿⣧⣆⡀
⠀⠈⠑⠀⠀⠰⣷⠉⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⢨⡅
⠀⠀⠀⠀⠀⠐⠛⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠂⢴⣶⣿⣷⣶⠔⠃
⠀⠀⠀⠀⠀⠀⢉⣿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⠛⠛
⠀⠀⠀⢺⡗⠛⡿⣿⢿⠛⠺⡗⠀⠀⠀⠀⠀⢰⣶⣿⡟⠛⣿⣶⡦⠀⠀⠀⠀⠀⢰⡦⠦⡾⣿⣿⢿⣿⡶⠤⣶
⠀⠀⠀⣸⣇⠀⣯⣿⣸⠀⢸⣇⠀⠀⠀⠀⠀⢰⡇⢿⡧⢾⣿⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢁⣿⣺⣸⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠸⣷⣾⡏⡺⡗⠀⠀⠀⠀⠀⠻⠟⠀⣟⣻⣟⣿⣻⠇⠘⠿
⠀⠀⠀⠀⢀⡤⢤⡿⣤⢤⡄⠀⠀⠀⠀⠀⠀⠀⣀⢀⣸⢇⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠉⢹⡏
⠀⠀⠀⠀⠘⠋⠙⠃⠛⠘⠋⠀⠀⠀⠀⠀⠀⠀⠛⠙⠛⠘⠃⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷
⠀⣀⣀⡀⠀⣿⣾⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠉
⠀⢹⡿⠁⠀⠈⢻⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡉⠉⠀⠀⠀⠀⠀⢠⣶⠤⣾⣿⣿⣻⣻⣿⣿⡤⢴⡦
⠀⢰⣧⢤⣾⣿⠉⣭⡉⣿⣿⣤⣴⡆⠀⠀⠀⣰⣾⣿⣛⠻⢛⣿⣿⣆⠀⠀⠀⢠⣷⠀⠈⢿⣿⣉⣸⣿⠉⠀⢸⡇
⠀⠀⠀⠀⠉⣿⠠⠉⠄⣿⠉⠀⢸⡇⠀⢐⣾⡟⢈⣿⠒⣠⠢⣿⣇⠙⣷⡆⠀⠘⠙⠀⠀⢰⣽⠛⢫⣵⠀⠀⠙⠙
⠀⠀⠀⠀⠀⢨⣽⠛⣻⡅⠀⠀⠿⠿⠀⠀⠀⢴⠛⠛⠻⠿⠟⠛⠛⢶⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠠⣬⡿⠀⢹⣧⠄⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠛⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⣄⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠘⠛⣐⠛
⠀⠀⠀⠀⠀⠛⠁⠀⠈⢻⣿⣿⠳⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⢈⣌⠉⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠛⠃⣤⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⠿⠿⠿⠿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠿⢿⣿⣿⡿⠿⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣟⠀⠀⣸⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⠶⠶⠶⠶⠶⠶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣶⣄⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣠⣶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆⣀⣀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣀⣀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡿⠁⠉⠉⢠⣿⣿⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿⣿⣿⠀⠉⠉⠈⢿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣄⣠⣿⣿⣄⣠⣿⣿⣄⣀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢹⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠠⠀⠀⠈⠀⠀⠀⢀⠄⡈⠄⠀⠀⠀⠀⠹⣿⣿⣿⡇⠀⠀⠀⢠⠀⡆⠀⠀⢰⠀⠀⡆⢸⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠐⣠⠖⠃⠀⠀⡀⠀⠈⠹⡆⠀⠐⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠘⠐⠓⡀⠀⢸⠀⠀⠁⢸⣿
⠀⣼⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠈⠃⢰⡏⠀⠀⠀⠘⢦⡔⠈⠀⣿⠀⠀⠀⠂⠀⢸⣿⣿⡇⠀⠀⢸⠀⠀⠀⠃⠀⢸⠀⢸⠀⢸⣿
⠀⢹⣿⣿⣿⣿⣿⣿⡇⠁⠀⠠⢀⠀⠈⠁⠢⠀⠀⣶⠀⠓⠠⢈⠁⠀⠈⠀⠄⠁⢸⣿⣿⡇⠀⠀⡈⠀⠈⠀⢠⠀⢸⠀⢸⠀⢸⣿
⠀⢻⣿⣿⣿⣿⣿⣿⡇⢂⢠⡀⠀⠈⠀⠈⠀⠈⢰⢸⠀⠀⡀⡀⠈⠀⠁⢼⣈⠀⢸⣿⣿⡇⠀⠀⡇⠀⠀⡀⢸⠀⢸⡄⠸⠀⢸⣿
⠀⣾⣿⣿⣿⣿⣿⣿⡇⠈⣞⡀⢀⠦⠂⠀⠠⣧⣤⣼⣴⣄⡟⠀⠀⠠⡇⢠⣟⠀⢸⣿⣿⡇⠀⠀⡇⡌⠀⢱⢸⠀⢰⡇⡇⠀⢸⣿
⠀⣼⣿⣿⣿⣿⣿⣿⣷⣾⡿⣿⣿⣀⣀⣴⣿⣿⣿⣽⡿⣿⣿⣷⣈⣠⣿⣿⣿⡿⢿⣿⣿⡇⠀⢠⠃⠃⠀⢸⢸⠇⢸⡇⡇⠀⢸⣿
⠀⢹⣿⣿⣿⣿⣿⣿⡇⠉⠙⠛⠿⢯⣽⣿⣾⣿⣿⣿⣿⣿⣯⣿⠿⠿⠛⠋⠁⠀⢸⣿⣿⡇⠀⢸⠀⡇⠀⡸⢸⡆⠸⡇⡇⠀⢸⣿
⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣩⠭⢍⡉⠠⡄⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⠀⠘⠤⠃⠘⡇⠀⣿⠀⠀⢸⣿
⠀⣾⣿⣿⣿⣿⣿⣿⡇⢰⠀⠀⠀⠀⠀⡴⡫⠖⠉⠨⠅⠈⠉⠳⢍⢢⡀⠀⠀⠀⢸⣿⣿⡇⠀⢸⠀⠀⠰⠀⠀⡇⠀⣽⠀⠀⢸⣿
⠀⣼⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠀⡜⡜⠁⠀⠠⠀⠠⢠⠀⠀⠈⢳⣡⠀⠀⠀⢸⣿⣿⡇⠀⠸⠀⠀⠀⠀⠀⠣⠀⣴⡄⠀⢸⣿
⠀⣹⣿⣿⣿⣿⣿⣿⡇⢇⠀⠀⠀⠀⢱⠁⣀⡀⠀⠀⢀⠀⠂⠄⢦⠈⡆⡆⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⢻⣿⣿⣿⣿⣿⣿⡇⡆⠀⠀⠀⠀⡌⣶⣼⠄⢠⠀⠉⠠⢠⠟⣾⣿⢡⠁⠀⠀⢸⣿⣿⣿⣷⣤⣤⣤⣤⣤⢠⣤⣤⣤⣶⣾⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠄⠜⢍⠀⠀⡁⠀⠀⠀⢀⣠⢏⠎⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿
⠀⣼⣿⣿⢿⣿⣿⣿⡇⠊⠀⠀⠀⠀⠀⠀⠠⡁⡂⠠⠤⠤⠖⡫⢓⡁⠀⠀⠀⠀⢸⣿⣿⣯⣋⣼⣿⣿⣿⣿⣿⣮⡻⣿⣿⣿⣿⣿
⠀⣹⣿⣿⣿⣿⣿⠛⢇⠒⡄⠀⠀⢠⢖⣫⣤⣌⠫⡉⠉⠉⣰⢊⣵⣶⣦⡕⡄⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⣿⣿
⠀⢻⣿⡿⣿⣿⣿⣤⣿⠀⢸⠀⠀⡇⣾⣿⠉⢻⡇⠡⠀⠀⣏⢾⣿⡀⣹⣏⠀⠀⢸⡇⣿⣿⣭⣴⣮⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⠉⣽⠀⡸⠀⠀⠳⣜⠿⣿⠟⡃⠂⠀⠀⠙⢮⡛⠿⠛⠡⠃⠀⠸⣿⣿⣿⠿⣛⢿⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿
⠀⣼⣿⣿⢿⣿⣿⣦⡟⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⣐⣲⢰⠀⠀⠀⢸⣿⣿⠟⠛⣿⣶⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿
⠀⣹⣿⣿⣿⣿⣿⣿⡇⡄⠀⠀⠀⠀⠀⢰⣾⣿⡿⠿⠿⠟⠛⡛⠿⠿⣼⠀⠀⠀⢸⣿⣿⣃⠐⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⡿⡿⠛⣇⠁⠀⠀⠀⠀⠀⠘⠀⠒⠚⠲⠦⣀⠀⠀⠀⠂⠍⠒⢤⡴⠊⠀⠉⠛⠉⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⠛⢩⣁⠤⠛⠊⠉⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠂⡤⢀⠂⠬⠑⠦⣄⠀⢀⠈⠀⠀⠙⠻⢿⢿⡿⢿⡿⢿⣿⣿
⠀⡼⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉⠑⠦⣁⠀⠀⢁⣠⡤⠒⠀⠈⡄⠀⠐⢘⣂⡀⣠⠀⠀⠈⠉⠿⡀
⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠋⠉⠀⢠⠀⠤⠴⠴⠆⠁⠀⠘⢻⠀⠀⠀⠀⠀⠁
⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠄⠤⠀⠄⠀⠠⣐⠀⠀⠀⠀⠀⠀⠀⢘⠀⠆⠀⠀⠀⠁
⠀⠣⠀⠤⠄⠠⠄⠠⠜⠒⠦⠀⠤⠀⠤⠄⠠⠄⠠⠤⠀⠤⠀⠤⠄⠤⠄⠠⠄⠠⠶⠒⠤⠀⠤⠄⠤⠄⠠⠄⠠⠤⠙⠤⠀⠤⠄⠴⠃

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣾⣽⣿⣯⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣯⣿⣽⣵⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠴⣼⡀⠀⠀⢆⣠
⠀⠀⠀⠀⠀⠀⠙⠻⡦⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣶⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣾⣿⠀⠀⡾⠁
⠀⠀⠀⣰⠜⣿⣿⣿⣿⣿⣿⡟⢴⠀⠀⠀⠀⣠⠝⣿⣿⣛⣛⢻⣿⡟⢵⡄⠀⠀⠀⢰⡾⢷⣶⣶⣿⣶⣶⡾⢷⡆⠀⠀⠀⢀⡞⢹⣿⣿⣿⣽⣿⠗⠚⠁
⠀⠀⠀⣇⠀⣿⣿⣿⣿⣿⣿⡇⢸⡄⠀⠀⠀⣯⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⢸⡁⢸⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⢸⡇⠸⣿⣿⣿⣿⣿⠅
⠀⠀⠀⢩⠧⠙⢛⡟⠛⣿⠛⠷⢯⠀⠀⠀⠀⢫⢥⡙⢻⡟⠛⣿⠛⠧⢽⠃⠀⠀⠀⠞⠃⢸⣿⣿⣿⣿⣿⠁⠛⠇⠀⠀⠀⠃⠘⢀⠬⣿⣿⣿⢧
⠀⠀⠀⠀⠀⠀⢸⡕⠀⣽⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⡗⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠉⠉⠉⠙⢷⡀⠀⠀⠀⠀⢀⣠⣞⠉⠀⠀⠀⠉⣷⣄
⠀⠀⠀⠀⠀⠀⠴⠷⠰⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠰⠿⠇⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⠀⠘⠉⠉⠀⠀⠀⠀⠘⠉⠉


⠀⠀⠀⠀⡀⠀⠀⣀⣀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡶⣶⢶⣆
⠀⠀⠀⠀⠑⢢⣶⣿⣀⣿⣦⠒⠁⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢟⣛⠿⠟⠁
⠀⠀⠀⠀⣠⢼⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⢀⡿⠻⣿⣿⣿⣿⣿⠟⢻⡀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣮⡀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⠿⣶⣶⣤⣤
⠀⠀⢀⡼⡅⢸⣿⣿⣿⣿⣿⠀⡽⣄⠀⠀⠀⣾⣇⠀⢸⣿⣿⣿⡏⠀⢸⣷⡀⠀⠀⠀⠀⠻⠟⠛⡟⠛⠿⠃⠀⠀⠀⠀⢠⠋⠀⣿⣿⠆⠀⠈⣿⡇⠈⢣
⠀⠀⠈⣷⠋⢸⣿⣿⣿⣿⣿⠀⢳⡏⠀⢠⠶⢿⡟⠀⢀⣯⠀⢸⡄⠀⠸⣿⠷⣄⠀⠀⡄⣆⣿⣿⣿⣿⣷⡦⡀⠀⠀⠀⢸⣆⠀⠿⢿⡿⠶⢾⡿⠇⢀⣾
⠀⠀⠀⠟⢃⡴⠟⠿⠿⠿⠿⣄⠚⠇⠀⠈⠀⠈⠀⠀⣾⣿⠀⣾⣿⡄⠀⠁⠀⠉⢀⣼⡻⠁⢻⠿⠿⠿⠃⠻⣳⡀⠀⠀⠘⠉⠀⠀⣿⠀⠀⠈⣷⠀⠋⠘
⠀⠀⠀⣴⣾⠇⠀⠀⠀⠀⠀⢼⣶⡄⠀⠀⠀⠀⠠⠶⠿⠿⠀⠻⠿⠶⠄⠀⠀⠀⠈⠉⠀⠀⠼⠅⠀⠿⠀⠀⠉⠉⠀⠀⠀⠀⠀⣠⣿⣄⠀⣴⣧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⣶⣶⣶⠶⣦⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⣄⣤⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠭⣿⣿⡷⠥⠿⠀⠀⠀⠀⠀⠀⠀⠈⢠⣠⣤⣄⡌⠁
⠀⠀⠀⠀⠀⢾⣥⣽⣿⣤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠉⠿⠿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣜⢫⣿⡜⣼⡆
⠀⠀⠀⣾⣷⣿⠉⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⣿⣾⣿⡿⠛⠻⣿⣷⣿⡇⠀⠀⠀⣾⡇⣿⣿⡟⠛⠛⣿⢺⣿⠀⠀⠀⠀⣰⣶⣾⣛⡟⠛⠛⢻⣷⣶⡄
⠀⠀⠀⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣇⣀⣠⣿⣿⣿⡇⠀⠀⠀⣿⡇⣿⣀⣀⣀⣀⣿⣺⣿⠀⠀⠀⠀⣷⡇⢸⣿⣿⣿⣿⣿⡇⢻⣻
⠀⠀⢠⡿⠇⢿⣿⣿⣿⣿⣿⠸⢿⡄⠀⠀⠀⢠⡿⠴⣿⣿⣿⣿⣿⡷⢿⡇⠀⠀⠀⣿⠇⢹⡏⠉⠉⢽⡏⠼⣿⠀⠀⠀⠈⣷⠆⢸⣿⣿⠿⣿⣿⡇⠼⣿
⠀⠀⠀⠁⣠⠞⠛⠿⠿⠛⠻⣤⠈⠀⠀⠀⠀⠀⠉⠀⣿⠿⠿⠿⣿⠁⠉⠀⠀⠀⠀⠈⠁⢻⡟⠀⠀⣿⡇⠈⠁⠀⠀⠀⠀⠈⠀⠀⠙⢰⣶⡆⠋⠀⠀⠁
⠀⠀⢠⣶⣿⠀⠀⠀⠀⠀⠀⢼⣷⡆⠀⠀⠀⠀⠀⢰⣿⡦⠀⢴⣿⡆⠀⠀⠀⠀⠀⠀⠀⣾⣷⠄⠠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇


⣿⣿⡿⠿⣿⠿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣴⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣤⣀⣤⣄⣀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⡀⠀⣸⠀⠀⣄⠒⣠⠇⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⠉⠀⠀⠐⠀⠈⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⠴⠤⠤⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠶⠶⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢠⣴⣶⣶⣶⣦⡄⠀⠀⠀⠀⠠⠀⠀⡷⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣎⣻⣿⡇⠐⣄⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡟⠛⣻⣿⣭⣿⣿⡛⠛⢿⡇⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠓⡆⠈⠀⠀⠀⢸⡇⠀⣽⣿⠁⠀⠈⠳⠤⢾⡇⠀⣯⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣾⣿⡄⠀⠀⢸⣿⣿⣿⡟⢳⠲⠖⣶⢶⣿⡇⠀⣿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣿⡿⣿⠛⠃⠀⣿⠛⠛⠛⡗⢻⠛⡷⠻⠄⢻⡇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠼⠁⠀⠀⠀⠀⠙⠷⠓⠚⠛⠛⠶⠟⠺⠷⠿⠁⠀⣿⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡶⢶⠶⣶⣶⣶⣶⣶⡶⣶⣶⣶⡾⠛⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢀⡇⠀⠀⠀⠀⠀⠀⢠⣶⣾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠛⠂⠘⠁⠀⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧⣤⣷⣶⠀⠀⠙⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣆⣀⠀⠀⣿⣿⣿⣯
⣤⣠⣀⣄⣀⣀⣀⣀⣠⣀⣤⣤⣠⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⢙⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣤⣿⣿⣶⣿⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⣛⣛⡉⠭⠽⠦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡤⠒⠦⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠘⣄⢀⣠⠃⠀⠀⠀⠀⠀⠀⡉⠉⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠉⠀⡔⠐⠂⠈⠉⠉⠀⠀⠱⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⡄⠀⠀⢀⣀⣀⣀⣤⣤⣧⣤⡼⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⣠⡤⠷⢲⣋⣉⣁⣈⣧⣀⡀⠀⠀⠀⠀⠀⣠⠴⠚⡏⠉⠉⢹⢶⣋⣠
⢀⡀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠙⠉⣁⡤⠖⠋⠉⠉⠀⠀⠀⠀⠉⠳⣄⠀⡠⠺⡁⠀⠀⢱⣀⣠⡼⢯⠻⡄
⣏⣙⣶⡏⠉⠀⠀⠁⠉⠙⢲⣄⡀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠁⠀⢹⡴⠚⠉⠁⠀⠀⠈⠛⠁
⢀⡜⡻⠷⠶⠶⠴⣤⣀⠀⢨⠀⠙⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠞⠁
⠈⠚⠃⠀⠀⠀⠀⠀⠈⠙⠻⢤⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠶⠒⠒⠒⠋⠉⠉⠉⠉⠉⢧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣧⣤⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣉⣉⣹⠛⠛⠛⠋⠉⠉⠉⠉⠹⣄⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡒⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠤⠿⣄⠀⠀⠀⠀⠀⠀⠀⢰⣏⣉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣤⣤⣤⡼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢸⣿⢈⡁⡇⢐⣾⠸⣿⠸⣿⠰⠆⡧⣼⣿⠀⣿⠀⢀⣤⢶⢶⣶⡶⣤⡀⠀⣿
⠀⠀⠀⠀⠈⠛⢾⣷⣷⣶⣾⣶⣾⣶⣾⣷⣾⡷⠟⠁⢀⣿⡴⢋⣘⣘⣈⣉⣁⣃⠙⢦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⣿⣿⠉⠉⠁⠀⠀⠀⡞⢹⣷⡫⣿⡍⠉⠉⢩⣿⢽⣾⡏⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠙⢾⡇⠛⠛⠒⠒⠒⠒⠛⠛⢸⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡐⢿⣛⣛⡿⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡶⠶⠶⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠞





⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠷⣤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⠶⣿⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠓⢲⠆⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⣽⠀⠁⢸⠋⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣰⠋⠀⢿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣀⣀⡀⠀⡏⠀⠀⢸⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠉⠛⠒⠺⠆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢰⠏⠁⠉⠻⣦⡇⠀⠀⣿⠀⣠⠴⠦⢤⡀⢳⡄⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠘⣦⣀⣀⡴⢻⠇⠀⠀⡏⢸⡇⠀⠀⠈⡧⠀⢧⠀⠀⠀⠀⠀⠀⢹⠟⠓⠲⠤⣤⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠤⠤⠤⢤⣭⣥⣀⣸⣤⣄⣠⡇⠀⠛⠦⠴⠞⠁⠀⢸⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠾⠂⠀⠀⠀⠀⢸⠃⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠀⠤⠤⢤⠤⠴⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⣶⠰⠒⣶⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠛⠛⠛⠛⠛⠒⠲⣶⡖⡆⠀⣸⠁⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢠⡏⢱⠀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏⢩⡇⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⢀⡬⠀⢿⡇⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⢠⣿⠀⡧⡼⠀⠀⢀⡴⠃
⠀⠀⠀⠀⢠⠇⠀⠀⠀⢰⠇⠀⠐⠀⠘⣿⡀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣾⠇⠀⡇⠓⠖⠚⠋
⠀⠀⠀⠀⡾⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⣼⠏⠀⠀⡇
⠀⠀⠀⢠⠇⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠈⢧⣀⠀⠀⠀⠀⠀⣠⡞⠁⣠⡾⠋⠀⠀⠀⡇
⠀⢀⣀⣾⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⡉⠓⠒⠒⠒⢛⣁⣴⠞⠋⠀⠀⠀⠀⠀⡇
⣴⠋⠁⠉⠉⠛⠒⠾⢤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠘⡇⠘⡟⠒⠲⣦⠀⣰⠃⠀⠀⠰⠤⢤⠤⠶⣶⠦⠦⠤⠤⠶⣤⠤⠤⠤⠤⣤⠤⠤⠤⠶⠶⣤⠦⢤⠇
⠀⢷⣀⣳⠀⡼⠃⢰⠇⠀⠀⠀⠀⠀⢸⠀⠀⠘⢦⡀⠀⠀⠀⣾⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠃⠀⢰
⠀⠀⠉⠉⠘⠓⠲⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⡀⠀⣿⢦⣄⣀⡤⣧⠀⣀⡴⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠤⠤⠤⢤⡍⠉⡏⠀⠀⠀⠀⡏⢉⡿⠶⠶⠶⣦⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡖⠋⠀⠀⠀⠀⠛⢶⣇⠀⠀⠀⢀⡷⠛⠀⠀⠀⠀⠘⠓⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣀⣤⣤⣤⣀⡀⠀⢸⠀⠀⠀⢸⠀⢀⣀⣤⣤⣤⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠋⠁⠀⠀⠀⠀⠉⠳⣾⠀⠀⠀⣸⠞⠋⠀⠀⠀⠀⠈⠙⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠗⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠒⠒⠛⠉⢀⢠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⡇⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠠⠚⠟⠀⠀⠀⠰⠟⠀⠀⠀⣿⣥⣿
⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⡇⢻⠋
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⣷⡸
⠀⠀⠀⠀⢀⠈⣆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢲⣷⣷
⠀⠀⠀⠀⠐⣂⣻⢀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⣉⢉⣖⣒⠀⢄⡀
⠀⠀⠀⠀⠀⠈⠏⠙⡄⠀⠀⢀⠀⠀⠀⠄⡷⠋⠁⠀⠭⠍⣽⡧⢿⣇⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠐⡀⠸⡄⠀⠀⡠⠐⠁⠀⡇⠀⠀⠈⠀⠀⢺⠃⢰⠈⡏⠉⠑⠒⠂⢤⠾⢤
⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⠶⠊⠀⠀⠀⠀⠁⢀⠺⠷⠶⠶⠿⠃⣸⢀⡇⠀⠀⠀⠀⡌⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣿⣿⠃⠀⠀⠀⡼⢡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠐⡖⠚⠿⠿⠋⠀⠀⠀⠰⠁⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⢿⠇⠀⠟⠿⠿⠀⠀⠀⠠⠔⠦⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⠀⠐⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡀⠀⡇⠀⠀⠀⠀⡄⠔⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠛⠃⢀⠂⠰⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣈⣘⣓⣿⡎⠐⠒⠒⢧⠄⢀⠀⠀⠀⠀⡀⡀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣉⡉⣉⡉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠁⠑⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣈⣉⣉⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⣀⣿⣇⣀⣀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡤⣼⠀⡀⢀⠤⢄⠀⡀⢸⡤⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠈⠹⣀⣇⣎⣀⣈⣆⣇⡸⠉⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡚⠀⠀⠀⠀⢨⣿⣯⠀⠀⠀⠀⠘⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⡇⠀⡞⠉⡞⠁⠀⠀⢙⡎⠙⡆⠀⣿⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢟⣿⠀⢇⢀⡯⠝⠽⠿⠯⣇⢀⠇⢰⣟⢇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣩⠁⠀⠀⠀⠀⢩⣹⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠵⠀⠀⠀⠰⠥⠬⠇




⣏⣈⣏⣯⣟⣯⣿⣹⣽⣿⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⢀⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠖⠚⠛⠛⠛⠓⠛⠚⠛⠛⠻⠷⠶⠾⠿⠿⠿⠶⠶⠶⠶⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⠟⣩⣤⣌⠻⣆⠀⠀⠀⠀⠀⣴⠟⢩⣭⣍⠙⣦⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣸⠃⠰⣿⣿⣿⠀⣿⠀⠀⠀⠀⢰⡏⠀⣿⣿⣿⠀⠈⣷⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⡀⠀⠘⠛⠁⠀⣿⠀⠀⠀⠀⢸⡇⠀⠈⠛⠋⠀⣠⠏⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⣄⣀⣀⣠⡼⠃⠀⠀⠀⠀⠀⢿⣄⣀⣀⣠⡾⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢛⣛⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣄⣀⡀⠀⢀⣶⠖⢿⡀⠈⠛⠉⠉⠉⠉⠁⠀⠀⠀⣠⡶⠶⣦⣤⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠶⠶⠶⠶⠿⠿⠟⢻⣿⠀⢸⣿⣛⣻⣿⣿⣿⣿⣛⣛⠛⠛⣿⡇⠀⣿⠤⠤⠴⠶⠶⠶⠶⠶⣶⡆
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠦⠾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⡿⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢰⡏⠉⠁⠀⠀⢀⣠⣿⣤⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣧⣿⣍⠉⠉⠉⠉⠙⣷
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⢀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⣀⡀⣹⡆⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⢻⡏⢉⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡟⣿⠋⠉⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠘⣧
⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡇⠀⠀⠀⠀⠀⠀⢿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠈⣛⡿⢿⡿⣷⣶⠶⣦⣼⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠃⠀⠀⣀⣀⠀⠀⢈
⣠⡾⠋⢰⣿⣅⠈⠙⢶⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢳⣶⠾⣿⠟⠻⣿⡉
⢿⡀⢼⠛⠁⠈⢻⡦⠀⣿⠀⠀⢿⡤⠤⠶⠶⠶⠶⠶⠒⠂⣿⡇⢸⣟⠛⠛⠛⠛⠛⠛⠛⠋⣉⣉⣹⡇⢐⣿⣛⣛⠛⠒⠲⠶⠶⢄⣸⡇⢰⡟⢁⣸⡿⢶⣄⡈⠛
⢸⡇⠸⡇⠀⠀⢸⡇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠛⢿⣆⣸⡟⠛⠛⢻⣦⠀⠀⠐⣿⠉⠉⢹⣧⡾⠋⠉⠉⠛⢻⡆⠀⠀⠀⠉⠃⠘⣇⠘⣏⠀⠀⢩⡇⢰
⠈⣷⢀⣿⠀⠀⣸⡃⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠁⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣼⠁⢸
⠀⠸⠟⠉⠀⠀⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣆⡿⠀⠀⢿⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⣶⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠁⠳⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣇⠀⠀⠀⢀⣀⣤⣤⣤⣤⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠖⣾⣿⣿⣿⣿⣿⣿⣿⡛⠿⠟⠀⠀⠈⠛⢻⡿⠿⠿⠿⠿⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠉⠉⠉⠉⠉⠉⢷⡀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣀⣠⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣄⠀⣾⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠋⠉⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⠖⠒⠒⠒⠲⠦⠤⢤⣀⣀⡀⠀⠉⠉⠙⠓⠲⣤⡀
⠀⣀⣠⢶⡖⢲⠀⣠⡄⠀⠀⠀⠀⠀⠀⡟⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⢤⡀⠈⢻⠶⣄
⢿⡉⢣⠀⣇⣸⠏⣡⡟⠀⠀⠀⠀⠀⢰⠃⢀⡇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡸⠀⠈⠳⣄
⠀⠙⣆⣳⣯⡧⣴⠋⠀⠀⠀⠀⠀⠀⡞⠀⣸⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⡾⢀⠃⠀⠀⠀⠈⡇
⠀⠀⣹⠥⠶⠒⢻⠇⠀⠀⠀⠀⠀⢰⡇⠀⡏⠀⠀⠀⠀⠛⠛⠒⠲⠦⠤⣤⠀⠀⠀⠀⠀⢐⠇⡌⠀⠀⠀⠀⢠⡇
⠀⠀⣟⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣾⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠁⠀⠀⠀⠀⣸⠁
⠀⠀⣿⠤⠤⠀⠈⣧⠀⠀⠀⠀⢠⡏⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⠀⠀⠀⠀⠀⡟
⠀⠀⢸⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸⡇⠀⠷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⡇⠀⠀⠀⠀⢸⠃
⠀⠀⠘⣷⠒⠉⠀⠀⠹⡄⠀⠀⠈⠳⣤⣀⡀⠀⠉⠉⠛⠒⠲⠶⢤⣤⣀⡀⠀⠀⠀⢠⠇⢸⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠹⡄⢀⡤⠂⠀⠙⢦⡀⠀⠀⠀⡽⠋⠙⠓⠲⠤⢤⣄⣀⣀⡀⠈⠉⠉⠛⠛⠁⢀⣎⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠹⣏⠀⠀⢀⠀⠀⠙⠳⠶⡟⢃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠶⠶⠖⠋⠉⠹⡆
⠀⠀⠀⠀⠀⠘⢦⣴⠋⠀⠀⡰⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⣀⣤⠿⠶⠶⠶⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣴⣁⣀⡼⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠋⠨⡇⠀⠀⣶⠀⠀⣰⠁⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠟⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⡀⢀⣀⣀⣀⣀⣀⠀⠁⠀⠀⢠⠞⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠽⠉⠁⢸⠁⠉⠉⠛⠶⣄⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⠤⠤⠤⠤⠤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⢷⡀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⣤⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⢦⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣸⠧⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠈⠀⠀⠀⡟⠀⠀⢀⣠⡤⠖⢛⡗⢲⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠘⠧⠶⠛⡟⠙⢻⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣼⣀⣤⣼⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣧⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⢴⠶⣤⡀⢀⣀⣠⠖⠺⡁⠀⠘⡀⠉⠛⠒⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⠶⠶⠶⣶⠒⠋
⠈⢧⠀⠙⢫⡈⠱⡀⠀⠸⠄⠀⠑⠀⠀⢀⣠⡞⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠘⠛⠳⠖⢦⡀
⠀⠈⢧⠀⠀⢣⠀⠳⠀⢀⣀⣀⡴⠓⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡠⠂⠁⠀⢠⠃⠀⣤⠀⡸⠛⡆
⠀⠀⠈⢧⡀⡼⠦⣤⠴⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⢠⠃⠀⢀⠃⡜⠀⢀⡇
⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠻⠤⠴⠿⢶⡇⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣁⣀⣀⣀⣲
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠟⠛⠛⠘⠛⠛⠃⠚⢻
⠀⠈⠐⠐⠒⢅⠔⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⢠⠑⠀⠀⠀⠀⠠⢸
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⠀⠀⠰⠐⠁⢸
⠀⠀⠀⠀⢠⠓⡀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠧⠤⠤⠤⠤⠤⠄⠤⠞⠲⠲⡆
⠀⠀⠀⠀⠈⠉⠉⠑⠒⠠⠤⣀⣠⢺⡇⠀⠖⠙⠆⠀⠀⠀⠆⠀⠡⠀⠀⡷⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⡇⠀⠓⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⡷⠘⠉⠑⠒⠤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠖⠀⠀⠀⠀⠈⠉⠉⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠂⣀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡆⠉⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⡄⠠⡞⡢⠤⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠧⠀⠀⠀⠤⠠⠤⠤⠇⠀⡇⠀⠀⠀⢀⠈⠀⠀⠀⠡⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⢤⠄⠀⠀⠀⠀⠄⠤⠤⠐⠒⡇⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠤⠀⠀⠀⠀⠤⠀⠀⠴⠐⠚⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠤⠤⠤⠤⠤⠤⠤⠀⠐⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⢤⡤⠤⠤⠤⢤⠄⠒⠒⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠄⠀⢰⡆⠀⠀⠀⢸⠓⠂⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⡠⡇⠀⠀⠀⢸⠂⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡇⠀⠀⠀⡞⠤⠤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠉⠉⠉⡇⠀⠀⠀⡧⣀⡀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠓⠒⠒⠐⡇⠀⠀⠀⡇⠀⠈⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢷⠶⠤⠤⡖⠃⠀⠀⠀⢹⡶⠶⠶⢞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠳⠤⢆⡔⠀⠀⠀⠀⢻⠳⢤⠴⢠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣯⣭⣤⣤⣤⣤⣤⣤⣽⣶⣶⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⠶⠶⠾⠷⠶⢶⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⠀⢠⡶⢶⣄⢠⣿⡟⠁⣴⠶⢦⡀⢻⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⠀⠀⠹⣯⣼⠏⢸⣿⣇⠀⢿⣬⡾⠃⣸⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣤⣤⣶⣿⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⡁⠀⠀⣀⡀⠀⠀⠀⠈⣹⣿⢿⣿⣟⢸⡟⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡇⠀⠀⠈⠻⠶⣶⡶⠟⠉⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣀⣠⣟⠀⠀⠀⣠⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡏⠉⣹⣦⠀⢴⣿⣿⡆⢹⣷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣸⣷⣾⠁⠘⣷⡀⠉⢹⣇⠸⣿⣠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⢉⣹⡟⠁⠘⢷⣄⠈⣷⣤⣤⣿⡶⠟⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⠀⠀⣠⡟⠉⢷⣄⠀⠀⣹⣿⣯⣤⣤⣤⣄⠀⠀⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⠏⠀⣿⣠⡾⠛⢷⡀⠀⠙⣷⡾⠋⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⠈⠻⣦⡀
⠀⠀⠀⠀⠀⣴⠟⣷⣴⠟⠹⣧⡀⠈⠻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⣀⣨⣿⣄
⠀⠀⠀⠀⢸⣿⠾⢿⡄⠀⠀⠈⢳⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣟⠉⠁⠀⠹⣆
⠀⠀⠀⢀⣿⠃⠀⠈⢻⣄⢀⣤⠾⣯⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠀⠀⣿⢻⣆⠀⠀⠀⠹⣦
⠀⠀⠀⠈⣿⡀⠀⣀⣠⣿⠟⠁⠀⠈⠙⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⠀⣿⡶⢶⣦⣶⣿⡆
⠀⠀⠀⠀⠈⠛⠛⠋⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣤⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣧⣤⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⠾⠁⠀⠘⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡿⠀⠀⠉⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⡿⣧⡀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢠⣞⠋⠉⠉⠉⢉⣉⣉⣉⣉⣉⣉⣹⡇⠀⣿⡀⠈⠻⣶⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⣿⢿⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⣶⣆⠀⠀⣤⣤⣤⣤⣤⣤⡴⠄⠀⠀⣿⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣤⣬⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⢶⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠁⣿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡏⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣼⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⠲⠞⠓⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⠶⠾⠓⣿⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⣛⣿⣍⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠞⠛⠛⠛⠻⣦⣄⠀⠀⠀⠀⠀⢀⡿⠋⠉⠁⠈⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢀⣿⣦⣶⣶⣶⢶⡶⣶⣶⢶⣶⣾⣇⠀⠀⢸⣟⣿⣿⣿⡟⣿⣿⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⣰⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠻⢯⡽⠏⠀⠀⠀⠀⠀⠀⠀⠘⠿⣭⠿⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣽⣦⠀⢸⠏
⠀⠀⢀⡔⢾⣿⣿⣿⣿⡖⣆⠀⠀⠀⣰⠶⣿⣟⣛⣻⣷⢶⣄⠀⠀⠀⡴⢶⣶⣾⣷⣶⡶⢦⠀⠀⠀⢠⡖⢻⣿⣿⣿⣿⠒⠋
⠀⠀⢸⡀⢽⣿⣿⣿⣿⡇⣸⠀⠀⠀⣧⠘⣿⣿⣿⣿⣿⢀⡿⠀⠀⠀⣇⢸⣿⣿⣿⣿⡇⣸⠄⠀⠀⡸⢇⠸⣿⣿⣿⡿
⠀⠀⠀⠟⠀⢸⡅⢸⡀⠋⠃⠀⠀⠀⠙⠃⠉⣿⠉⣿⠉⠛⠁⠀⠀⠘⠈⣸⠿⠿⠿⠿⣇⠁⠀⠀⠀⠁⠀⡴⠻⠿⠻⢦⡀
⠀⠀⠀⠀⠀⣸⣅⣸⣅⠀⠀⠀⠀⠀⠀⠀⢠⣯⣠⣯⡄⠀⠀⠀⠀⠀⠼⠇⠀⠀⠀⠀⠼⠆⠀⠀⠀⠰⠚⠃⠀⠀⠀⠴⠛⠆


⠀⠀⠀⢀⠀⢠⡤⢤⡄⢀⠄⠀⠀⠀⠀⠀⠀⢰⣭⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣦
⠀⠀⠀⠀⢸⣿⣷⣾⣿⡁⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠐⣀⣀⣀⣀⠂⠀⠀⠀⠀⠀⡀⣀⣛⣶⣛⣁⢀
⠀⠀⢀⡞⢹⣿⣿⣿⣿⠋⣧⠀⠀⠀⣼⡇⠘⣿⣿⣿⡏⠀⣷⡀⠀⠀⠀⢿⡿⢿⡿⢿⡇⠀⠀⠀⢀⡞⠋⣿⡟⠉⠙⣿⠛⢣
⠀⠀⢳⡞⢸⣿⣿⣿⣿⠘⣧⠃⢀⡴⢿⠇⠀⣹⠉⢽⡀⠈⣿⠧⡄⠀⢀⣔⣶⣾⣷⣖⣦⡀⠀⠀⢸⣆⠀⠿⣷⠦⢾⡿⠀⣴⡅
⠀⠀⠸⢁⡴⠛⠻⠟⠛⢤⠛⠀⠀⠁⠈⠀⢸⣿⠀⣿⡷⠀⠈⠀⠁⣠⡟⠁⢻⠛⢻⡃⠙⣶⡄⠀⠘⠈⠀⢸⡇⠀⠈⡇⠈⠈
⠀⠀⠐⠛⠃⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠘⠛⠛⠀⠙⠛⠓⠀⠀⠀⠀⠀⠀⠚⠂⠘⠂⠀⠀⠀⠀⠀⠀⠀⠶⠷⠀⠾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣫⣿⣿⣝⣿⠀⠀⠀⠀⠀⠀⠒⣀⣀⣀⡐
⠀⠀⠀⠀⢼⣭⣿⣯⣽⠀⠀⠀⠀⠀⠀⠀⢼⣋⣻⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢶⣺⣿⣕⣷
⠀⠀⢸⣿⣿⣩⣿⣿⣿⣿⣷⠀⠀⠀⢘⣿⣿⡟⠉⠻⣿⣿⣷⠀⠀⢸⣿⣿⠿⠉⠉⣿⣿⣿⠀⠀⠀⣰⡿⣿⣿⣉⣉⣽⢷⣦
⠀⠀⢸⣿⢿⣿⣿⣿⣿⢿⡿⠀⠀⠀⢸⣿⢿⣷⣤⣾⣿⣿⡿⠀⠀⠸⣿⢿⡦⠤⠤⡿⢿⡿⠀⠀⠀⣿⠅⣿⣿⣿⣿⣿⠈⣿⠄
⠀⠀⠛⢀⣹⢿⣿⣿⢿⡀⠘⠀⠀⠀⠘⠊⢹⣿⣿⣿⡟⠙⠋⠀⠀⠘⠊⢸⡧⠀⢸⣯⠘⠋⠀⠀⠀⠛⠁⠹⠏⣭⡝⠟⠀⠛
⠀⠀⣤⣾⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⠀⣼⣧⠀⣼⣧⠀⠀⠀⠀⠀⠀⣼⣧⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣴⠶⠶⠘⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣼⡷⠀⠀⠀⢀⣶⠄⠀⠀⢸⣰⢛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡆⠀⠁⠊⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⢻⣼⡟
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠃⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢸⠀⣼
⠀⠀⠀⠀⠀⢈⡀⠆⡀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠈⠐⣀⣀⣀⣀⠀⠀⠀⠸⠧⠃
⠀⠀⠀⠀⠨⠀⠸⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠹⣿⣿
⠀⠀⠀⠀⠀⠉⣈⣇⡐⠀⠀⠀⠀⠀⠀⠀⢠⠐⠠⣈⣉⣀⣉⣉⣽⣳⣀⠀⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠹⠉⠹⡀⠀⠀⢀⠀⠀⢀⡐⢸⠟⠁⠀⢀⡧⠄⢹⡷⢼⣿⡀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠱⣀⠔⠁⣀⠔⠚⠀⢸⠁⠀⠀⠊⠀⠀⢸⡇⠀⡆⢹⡏⠉⠒⠒⠦⢤⡷⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣁⡴⠎⠁⠀⠀⠈⠸⠀⠀⠶⢶⡶⣶⣾⡗⢰⡃⠸⠁⠀⠀⠀⠀⠐⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠒⠀⠀⢸⡿⡿⠀⠀⠀⠀⣰⢃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠀⠀⠀⣀⣠⣤⣤⣤⡤⠀⠀⠀⠀⢠⠇⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣬⣥⡀⠀⣦⣤⣭⡅⠀⠀⠀⠀⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢹⠀⠀⠉⠉⢹⠀⠀⠀⠀⡋⠡⠙⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⢀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣜⡄⠀⠄⠠⢺⠀⠀⠀⠀⠐⠄⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠉⡁⢠⠉⠈⠛⠢
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠄⠄⢼⣀⣈⣉⣉⣷⣆⢀⠀⠛⢻⡦⠀⡠⠀⠀⠀⠀⠠⢤⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠁⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣧⡀⡘⣼⣤⣅⣤⣃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡦⢷⠣⡖⡞⡞⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⢀⡀⠀⠀⡀⠀⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣤⠶⣾⠁⠀⡠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⢹⡶⢦⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣟⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⡇⠀⠈⠯⠭⠏⠀⠀⠀⠀⠈⠯⠭⠏⠀⠀⡇⠘⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⠀⡇⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠙⠒⢿⡀⠸⡄⠀⠀⠀⠊⠁⠀⠀⠀⠉⠂⠀⠀⠀⡼⠁⣸⠗⠚⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠒⠤⣄⣀⢀⣀⡀⠀⣀⣀⡤⠔⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠈⠁⠀⠉⡀⠈⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠁⠠⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡄⠀⠀⠀⠀⡘⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠄⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⢀⡀⠀⣀⠀⢀⡀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠈⠆⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡐⢂⠄⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠶⠶⣶⡦⢄⠀⠀⠀⢠⠘⠂⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡐⠀⢀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⢸⠀⠀⠀⠈⢄⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⠀⠀⠀⠀⠀⠂⠀⠀⠁⠈⠀⠀⠀⠀⠀⠁⠀⢸⢼⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣽⠁⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠠⠤⠤⠤⠤⢤⠀⠸⡏⠀⠀⠀⠀⠀⢸⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡋⠍⡀⠀⠀⠀⠀⠀⡄⠁⠀⠈⠂⠄⢀⣀⣀⠠⠂⠀⣯⠆⠀⠀⠀⠀⠠⣛⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠐⢄⠳⠠⢀⠀⢀⠰⣀⠀⣐⠶⠦⠤⠤⠤⢤⠤⣴⠊⠀⠀⡠⢀⠞⡣⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢴⡢⣀⠁⠀⠀⠉⠉⠟⠛⢻⣻⢋⠉⠁⣀⡀⡡⣂⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⣾⠍⠐⠀⠀⠠⠀⢀⢀⣀⣀⣛⠳⣬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣽⣿⡁⠀⢸⠰⠶⡠⠠⠈⡆⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢹⣿⡂⠀⠆⡀⡠⡡⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⡂⠀⠀⢇⠀⠰⠁⢀⠃⠀⠀⠀⠀⠀⢀⡀⠤⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠿⠖⠀⠊⠤⠄⠀⠁⠠⠑⠀⠀⢀⠄⠊⢁⠤⠰⢄⡈⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡏⢡⣰⣮⠖⠈⠀⢀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⣥⠄⠈⠀⠀⡀⠀⠀⠀⢠⠀⡀⣢⠟⠼⠋⠀⠀⠀⢠⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⣠⣿⠿⠷⠒⠀⠉⠉⠁⠀⢉⠩⠉⠥⠂⠀⠀⠀⠀⠀⡶⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠠⢪⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡁⢲⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⢄⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⣁⡬⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠈⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⠁⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢀⣀⣀⣀⣠⠶⠀⠄⠀⠀⠄⠠⠐⠀⠀⠄⠀⠀⠀⠐⠀⠠⠐⠀⠂⠄⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⡀⠂⠀⠐⠀⠐⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠠⠀⠀⠀⠒⠒⠐⠀⠂⠈⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠒⠄⠀⠀⣠⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣈⣓⠄⠀⠀⠈⠩⠊⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⠀⠠⠀⠀⠐⠀⠀⠂⠀⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⣢⢄⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠬⠤⡤⠤⠤⠤⠤⠤⠦⠤⠶⠤⠒⠐⠒⠃
⠀⠀⠀⠀⠀⠀⠀⡰⣅⡠⠇⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⡠⢊⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⡠⠊⡠⠊⠀⢀⠀⡀⠀⠀⠀⠀⡠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⢀⠔⠈⠑⣎⠀⠀⠠⠀⠐⠈⠒⠒⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗⠴⠧⡀⠀⠀⠀⠐⠀⠈⣄⠀⠀⡊⠀⠑⠄⠀⠀⡎⠀⠀⡃
⠘⣄⠠⢂⡴⠀⠒⠚⠣⠄⠔⠋⠀⠀⠈⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⡀⠈⠀⠀⠀⠀⡄⣠⠗⠒⠒⠳⠤⠞⠁⠉⠉⠙⠖⠒⠁
⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠚⠓⠂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⡀⡀⠀⠈⠈⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣾⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢹⠌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠤⢼⡀⠀⠀⠀⠀⠀⡧⠄⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡼⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡀⣳⠀⠀⠀⠀⢞⠀⠐⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠊⠧⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢶⠃⠀⠀⠀⠀⠈⡗⠒⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢼⡀⠀⠀⠀⠀⠀⡧⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠺⣀⣠⡇⠀⠀⠀⠀⠀⢇⢀⠜⠐⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠀⠒⠒⠒⠒⠒⠚⠓⠳⠀⠀⠀⠀⠋⠀⠀⠐⠒⠒⠒⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠀⠀⠈⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⡻⢾⣿⣿⣿⣿⡿⡏⣿⣿⡟⢹⡇⡖⣷⣾⢉⡇⢸⡏⣿⢹⣿⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡙⠛⠛⠛⠃⠈⠉⠻⠿⠻⠿⠳⠜⠻⠛⢸⡇⡇⣿⣿⠨⡇⢸⡇⣿⢸⣿⣿⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⣻⡟⣿⡿⣿⣿⡟⢻⣿⣸⣿⡆⣤⣤⣤⢸⠀⢠⣀⣀⣄⣠⣧⣀⣀⣸⡇⡇⣿⣿⠀⠀⠀⡇⣿⢸⣿⣿⣿⣿⣿⢻⣽⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣨⡛⢬⣶⣘⠒⣿⣿⡇⠀⠀⠈⣿⡇⠂⠀⠀⣸⠛⠛⠉⠉⠁⠀⠀⠀⠉⠉⠙⠛⠻⣿⠀⠀⠀⡇⣿⠚⠛⠛⠛⠛⢿⣿⣿⣯⣿⣿⣿⣿
⣿⣿⣿⢿⣟⣙⣩⣕⣮⢻⠁⠀⠈⠅⠀⠀⢰⣿⠃⡀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⡟⣿⠀⠀⠀⠀⠀⢸⡟⠻⢿⣿⣻⣿⣿
⣿⣿⣿⣿⣿⣏⠙⢯⠀⠀⠀⠀⢰⣷⣿⣷⠀⠀⠅⠀⠀⠀⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣿⡀⡀⢐⡇⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣾⣝⣭⣭⠼⢦⣤⠀⠀⢘⣿⣿⣿⠀⠐⠃⠀⢀⣤⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⣄⠀⢸⡇⢸⠀⠀⠀⠀⠀⢸⡇⠐⠞⣿⠛⣿⣿
⣿⣿⣯⣿⣙⣫⣞⢳⡇⢹⡀⠀⠈⠛⠛⠛⠀⠀⡆⣴⢻⡇⡇⢀⣤⣦⡀⠀⠀⠀⠀⠀⢠⣴⣤⠀⣿⢹⢳⠸⣇⣼⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠐⣿⣿
⣿⣿⡿⢋⠿⢃⣔⢺⣇⣠⣾⣿⣿⣿⣿⣿⣿⠠⠃⣇⢸⡇⣇⠈⠻⠿⠃⠀⠀⠀⠀⠀⠘⠿⠟⠀⡿⢸⢸⢀⣯⣽⣤⣤⣤⣤⣤⣼⡇⠀⠀⣗⢀⣿⣿
⣿⣿⡿⣾⣟⣩⡁⢥⢽⣿⠁⠀⣿⢉⡉⠉⠉⣸⠀⠈⠘⠧⣿⠓⠶⠶⠒⠒⠒⠒⠒⠒⠒⠶⠶⠚⣿⠞⠈⠀⡧⢹⡏⢩⣭⣭⣿⣿⣿⣿⣷⣿⣤⣯⣿
⣿⣿⣿⣶⢍⣍⡁⡆⢸⣿⠀⠀⠿⠹⠀⠀⢸⠐⢶⣶⣶⣆⣿⠒⠒⢒⣒⣒⣒⣒⣒⣒⣒⠒⠒⢚⣿⠀⠀⡄⠇⢹⣿⣏⠛⠋⠛⢻⣿⣽⢿⣿⣿⣿⣿
⣿⣛⣿⣗⡊⠀⢰⡧⢸⣿⠀⡆⠀⠀⠀⠀⠸⣀⣄⣈⠉⠉⢹⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⡄⠀⢸⣿⣿⠀⠀⢀⣘⣿⠿⠿⢿⣛⣛⠛
⣿⣯⡿⣭⠁⠀⢸⡇⢸⣿⠀⠇⠀⠀⢀⣠⣤⣤⣦⣤⣤⣤⣟⠓⠀⠘⠛⠛⠛⠛⠛⠛⡛⠃⠐⠚⣿⣄⣀⣤⢤⣼⣿⣿⣀⠀⠀⠀⢸⡇⢏⣿⣿
⣿⣷⣿⣙⠿⠿⣻⡧⢼⣿⠀⣡⡴⢞⣩⠅⠀⠀⠀⣧⣤⢸⣷⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣿⣿⢠⣤⡇⠀⠀⠀⠈⢭⣛⢶⣤⣼⢳⡿⣿⣿
⣿⣿⡿⠭⠯⢤⢽⡇⠀⣀⣼⢏⣴⠏⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠈⢷⣝⢿⣼⡟⢻⣿
⣿⣯⣿⣿⡷⢞⣨⢏⠹⣯⡟⣼⡏⠀⠀⢀⣤⣄⠀⣿⣿⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢸⣿⡇⠀⢀⣤⣄⠀⠀⠀⢿⡎⣿⠛⠛⣿⠈
⣿⣿⣿⣿⣿⢼⠁⣠⣤⣤⡇⣿⡇⠀⠀⠻⣿⡿⠀⣿⣿⣼⣿⣿⣿⣿⣷⣤⣄⣀⣤⣶⣿⣿⣿⣿⣿⣼⣿⡇⠀⠸⣿⣿⠃⠀⠀⢸⣧⣿⠶⠦⠾⢠⣤
⣿⣿⣛⣾⠾⢴⠀⣹⡇⠀⣇⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢸⣿⣿⠀⠀⠋⠉⣿
⣿⣶⣿⡃⣁⣺⠀⡿⠃⠀⣿⣿⡇⠀⠀⠀⢻⣇⠀⠀⠀⢸⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⠀⠀⠀⠀⢀⣿⣷⠀⠐⠀⢸⣿⣿⠂⠂⢦⢰⢻
⣿⣿⣿⣭⣾⣿⠀⡇⠀⠀⢹⣿⡇⠀⠀⠀⢸⣿⡆⠀⠀⢸⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⠀⠀⠀⢠⡿⢹⣿⠀⠀⠀⢸⣿⡟⠀⠀⢻⢿⣶
⣿⣿⢿⡥⠴⢿⡦⣇⢀⠀⢸⣿⡇⠀⠀⠀⢸⡟⣷⠀⠀⢸⣿⣿⣿⣎⢻⣿⣿⣿⣿⣿⢋⣼⣿⣿⣿⠀⠀⠀⣾⠁⢸⣿⠀⠀⠀⢸⣿⣇⢀⡀⣸⣿⣿
⣿⣿⡯⢶⣼⠞⡁⣿⠀⠀⠈⢹⡇⠀⠀⠀⢸⡿⢸⡆⠀⢸⣿⣿⣿⣿⣷⣝⣛⣟⣛⣵⣿⣿⣿⣿⣿⠀⠀⣸⡷⢶⢸⣿⠀⠀⠀⢸⡏⠁⠀⠀⣿⡟⣿
⣿⣿⣟⣛⡷⣟⠃⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⢸⢿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⢻⣾⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣿
⣿⣿⣿⠛⠃⣨⢋⣿⣀⢀⣀⣼⡇⠀⠀⠀⢸⣇⠀⣿⣷⠈⠙⠛⠛⠛⠛⣛⣛⣛⣛⣛⠛⠛⠛⠛⠉⠀⣼⡧⠚⣾⢻⣿⠀⠀⠀⢸⣇⣀⣀⢀⣿⣿⣿
⣿⣿⣶⣷⢣⣯⠀⣹⣿⡿⠛⣿⡇⠀⠀⠀⢸⡟⠀⣨⣹⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⡿⠀⣀⡟⢸⣿⠀⠀⠀⢸⣿⢹⣿⡿⢿⣿⣿
⣿⣟⣿⣟⣿⣽⠟⢻⡇⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⣧⡀⢿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣼⡇⢐⣿⡇⢸⣿⠀⠀⠀⢸⣷⢈⣙⢩⣾⣿⣿
⣿⣟⣿⣛⣟⢋⣶⣿⡇⠀⠀⢸⡷⠀⠀⠀⢾⡇⠸⣯⡄⢸⡆⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⢠⡟⢰⣾⣿⡇⢸⣿⠀⠀⠀⢼⣿⣀⡸⢀⣿⣿⣿
⣿⣿⣿⣿⡿⣻⣷⣾⡇⠀⠀⢸⣷⠀⠀⠀⢸⣷⠀⠙⡀⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣿⣻⣿⡇⢸⣿⠀⠀⠀⣾⣿⠀⢀⢐⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠛⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣛⣛⣛⡛⠿⠿⢿⣿⣿⣿⣿⣿⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⣿⣿⣿⣶⣮⣝⡻⠟⣩⡅⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢋⣴⣿⣿⢖⣴⣭⡻⣿⣿⡏⣼⣿⢇⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢷⣿⣿⢟⣵⣿⣿⣿⢇⣿⣿⡃⣿⡇⣾⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠿⣃⢻⣿⣌⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣭⣭⣭⣭⣭⣭⣭⣶⣶⣿⣿⣦⣝⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣆⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣛⣛⢿⣿⣿⣿⣿⣆⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⠰⣿⣿⡇⣿⣿⣿⣿⣿⣆⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⢏⣵⣶⣍⢻⣿⣿⣷⣦⣄⠀⠀⠀⠹⣿⣿⣿⣿⣷⣭⣭⠼⢛⣛⣿⣿⣿⣿⣆⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠘⣿⣿⡿⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠹⣿⣿⣿⣿⡟⣱⣾⣿⣿⣿⣿⡿⠟⣣⣾⡍
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⣛⡛⠳⢦⣤⣶⣿⣿⣿⣿⣿⡏⣴⠀⠀⠀⠀⠘⣿⣿⣿⣷⣿⣿⣿⣿⣿⢃⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣷⣦⡙⣿⣿⣿⣿⣿⡟⣼⡿⠀⠀⠀⠀⠀⠘⣿⣿⠿⢛⣩⡛⠻⠏⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣍⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⢿⣿⣿⠿⠿⢿⣿⣿⣿⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣶⣬⣉⡋⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⠀⠉⠛⠿⠘⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣤⣤⡤⠄⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡐⣤⣀⣀⣀⣀⣐⣀⣀⣦⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠋⠉⠉⠉⠉⠉⠉⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡀⠀⠀⣄⠀⠀⢀⡄⠀⠀⢠⡀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠀⠴⠀⢸⡆⠀⢸⡄⠰⠆⢨⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠰⠤⠴⠋⡀⣀⣀⠙⠲⠖⠋⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢀⢀⣀⣀⣉⣉⣉⣀⣤⣤⣤⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣤⡤⠤⠤⠶⠶⠶⠶⠾⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣋⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡴⠋⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠁⠀⠀⢈⠷⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⠘⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣁⠀⠈⢀⣸⣷⡆⠀⠀⣿⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⠑⡇⠲⣶⣽⣿⣿⣿⣿⣿⣿⡆⢠⡞⠛⠉⠙⢿⣄⢰⣿⣿⡏⠉⠙⣿⠿⠆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⣤⣴⣦⣤⣿⣿⣿⣿⣧⣤⣤⣿⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠘⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠚⠒⠦⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠀⠀⠑⢄⡠⠂⠁⠀⠀⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢰⠀⠀⣀⡀⠀⠈⡅⢀⡀⠀⠀⠐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠘⡀⠀⠛⠃⠀⢀⠁⠹⠟⠀⠀⠠⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢘⠢⢄⠈⠂⠤⠄⠒⠁⠐⠠⠀⠠⠔⠁⢀⢰⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢈⠀⠨⠁⠢⠤⣀⡀⠀⠀⠀⣀⠠⠄⡖⠁⢀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢠⠀⢘⠀⠀⠀⠈⠁⠀⠈⠀⠉⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠢⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⡿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡶⠶⢤⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠑⠢⠄⣠⣀⡀⢀⣀⣠⠠⠄⠛⠉⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠘⣇⠀⢧⠀⣠⠃⠀⡸⠓⢤⡀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⣦⠚⠉⠉⠲⡞⠁⣠⠜⢧⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⣲⣤⣤⣤⣤⣤⣤⣖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣍⣀⠤⠜⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠂⢹⠋⠀⠋⠉⠁⠀⠉⠧⠈⢳⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣄⣀⣀⡤⠋⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⡻⠀⠀⠀⠘⣄⡀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠉⠳⣤⣤⣀⠀⠀⠀⠀⠀⣠⠞⠑⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢾⠃⠀⢰⠃⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠚⠊⠁⠀⠀⠀⠀⠀⢻⡀⠈⢣⡄⠤⠖⡋⠁⢠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠇⠀⣿⠀⠀⡟⠀⢀⣾⣿⣷⣶⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡇⠀⣀⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠣⣀⡼⠋⠈⠧⣸⠃⠀⠘⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⢇⣠⠋⠀⠀⠀⠀⡿⠀⠀⠀⠈⠛⢿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋
⠀⠀⠀⠀⠀⣠⠖⠛⠶⣏⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢀⣀⣼⠁⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣠⠞⠉⣠⠿⣄⠀⠀⠀⡰⠳⡀⠀⠀⠙⠦⣄⡐⠉⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⢁⠄⠀⠀⣸
⠘⢦⡞⠁⢰⠏⠉⡟⠉⢣⡀⢙⡦⠀⠀⠀⠘⠈⠉⠒⠢⠤⠤⣀⣀⣉⣀⣀⣀⣀⣀⣀⡠⠤⠤⠲⠊⠉⠁
⠀⠈⠙⠲⣿⣀⣀⡇⠀⠀⠙⠉⠀⠀⠀⠀⠀⡷⠋⠉⠓⢦⡀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠲⣄⠀⡘
⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠟⠁⠢⡀⢹⡄⠀⠀⠀⣰⠃⣠⠖⠀⢦⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢠⡿⠾⠧⠀⠀⠀⡸⢦⣧⡀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠙⠒⢾⡀⠀⠀⠀⠀⠀⠀⠀⢨⠶⠒⠋⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣀⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣀⣠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠛⠉⢵⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠙⠛⠋⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠚⠙⢛⠒⠲⠦⢬⣧⠀⠀⠀⠀⠀⠀⢀⣞⡤⠴⠖⢶⠛⠲⠖⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠤⠀⠀⣀⠀⠀⠈⠳⣄⠀⠀⠀⡴⠛⠁⠀⢀⡀⠀⣁⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⡤⠤⢤⡤⠤⢤⡤⠤⢤⡤⠤⡤⢼⡆⠀⢸⣅⣄⣠⣠⣄⡠⣤⣤⣤⣤⣤⢤⣤⢤⣵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠆⢨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡶⠒⣏⣀⣀⡤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⣒⣩⣭⡕⢷⡤⠷⠬⠛⠓⠊⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⢄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠄⠀⣀⠀⠙⠀⢰⡆⠀⠗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠐⠠⢀⡉⠀⠀⠧⣀⣀⠎⡐⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠁⠀⠈⠐⣁⣀⣠⠀⠀⠁⠁⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣄⣀⣀⣀⣀⢠⠤⢤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠆⠀⠉⠀⠀⠘⡖⠒⠒⡶⣤⠀⢰⢶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠡⣦⢀⠀⠒⡒⠆⢻⠉⠋⠉⠉⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣃⣯⣭⠬⣼⣑⠗⡺⠘⡆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠐⠤⠀⠒⢁⠟⠙⠙⠋⠊⢱⣺⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠢⢄⡤⠴⠧⠴⠦⠖⠖⠒⠋⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠖⠐⠒⠒⠒⠲⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠁⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠶⠀⠀⠀⢀⠀⠀⠃⢀⠁⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠓⠠⢀⡀⠀⠀⠀⠀⣀⢠⠒⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠜⠀⠀⠀⡇⠀⠀⠀⠀⣠⠴⡄⠀⢰⠓⠲⡄⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠐⠒⠂⠒⠒⠈⠀⠀⠀⢀⣠⠔⠁⠀⠀⠀⠀⠱⡄⣱⡤⢮⣀⠴⠉⣳⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣀⣀⣀⣀⣀⣀⡀⠤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠹⡖⠊⢁⡷⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⡖⠉⠟⠛⠉⠙⣏⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣄⠀⢀⡴⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣻⠁⠀⠈⢦⣀⠀⠀⠈⠀⠀⣠⠎⠀⠈⠱⡤⢤⡀⠀⠀⠀⢀⡴⠉⡼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡟⠀⢠⠇⠀⢀⣀⠀⣈⠉⠒⠛⠃⠉⠀⠀⠀⠀⠀⠱⡀⠙⡦⠒⠛⠁⢰⠖⠁
⠀⠀⠀⠀⠀⠀⠀⣴⠋⢡⡤⢧⠀⡜⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣺⠤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⣼⠀⡴⠋⠀⠀⢹⠃⠀⠈⠻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠃
⠀⠀⠀⠀⢀⠴⠧⢼⠁⠀⠀⠀⠸⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⣀⣠⠃⠀⠀⠀⠇⠀⠀⠀⡇⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣔⠋⣡⠞⣦⣀⣀⣠⠛⣄⠀⠀⠓⢤⣄⣁⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠁⠄⣀⣠⠇
⠈⠻⣁⣰⠃⢨⡍⠈⢦⡬⠃⠀⠀⠀⣀⡈⢉⡛⠒⠒⠒⠠⠤⠤⢀⣲⣶⡒⠚⠻⠉
⠀⠀⠀⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⣏⡤⠄⡈⠳⠀⠀⠀⠀⡴⠉⡀⢤⣹⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣸⡤⢇⣀⠀⠀⣧⣮⡀⠀⠹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠑⠓⠲⣅⠀⠀⠀⠀⠀⠀⢸⠷⠒⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣤⣤⠞⠀⠀⠀⠀⠀⠀⠻⣤⣤⣤⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣇⣈⡉⠐⢆⠀⠀⠀⠀⠀⠀⢸⠃⠉⣉⣸⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠁⠠⠁⠈⠉⠚⢦⡀⠀⠀⠀⡠⠟⠊⠉⠈⡤⠈⠁⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀⣀⣈⣀⣀⣁⣀⣀⣀⣱⠀⠀⡞⣀⡀⣀⢀⣐⣀⣁⣀⣀⣀⡀⣷
⣀⣀⣀⣀⣀⣀⣀⣳⣤⣦⣤⣼⣤⣤⣧⣤⣤⣤⣼⣼⣀⣀⣧⣭⣤⣭⣤⣬⣥⣤⣥⣤⣬⣤⣽⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠑⠄⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠄⡄⠀⠀⠀⢀⠀⠴⠓⠂⠀⠀⠀⠀⣴⣤⣤⣤⣀⠀⠀⠀⠀⠋⠱⠀⡃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠉⠁⠀⠁⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡌⡇⠀⠀⠀⢸⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⠊⠀⠀⠀⠀⠀⠀⠀⠠⠐⠢⠀⠀⠀⠔⠒⠂⠀⠀⡃⡇⠀⠀⠀⠀⠑⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠃⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢁⠀⠁⠀⠀⠁⠉⠉⠀⢨⢸⠀⠀⠀⠀⠀⠀⣾⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣔⠂⢆⣀⠀⠀⠀⠀⠰⠀⢀⣀⠀⠉⠒⠐⠀⠐⠀⠀⠀⡰⡍⠀⠀⠀⠀⣀⡀⠆⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢝⠀⡂⡀⠀⢥⠁⠤⠤⢭⣉⣙⣓⣒⣒⣒⣲⣲⠅⠀⢀⠄⠂⠜⡩⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⢽⣢⣄⣀⡀⠀⠀⠘⠭⠉⠽⠽⠃⠄⢀⢤⠠⣠⡪⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠇⢀⣁⣀⠀⠀⣂⡀⠀⠁⠤⢍⡍⠲⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠨⠱⠲⣠⠒⠄⢻⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣛⣛⠀⢸⢂⡀⡠⠠⠀⡈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢽⣿⡀⠸⠈⣇⠈⠐⠁⢀⡜⠀⠀⠀⠀⠀⠀⠀⡀⡠⠈⠈⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠨⡯⠄⠀⠊⠌⠡⠩⠅⠭⠍⠂⠀⠀⢀⠄⠂⢁⡠⢰⣄⣈⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⡀⠀⠀⠀⠀⠄⠀⠀⢀⢀⠀⢸⠒⣁⢄⣬⡥⠂⠁⢀⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢘⢀⠈⠋⠀⠀⠂⠊⠂⠈⢀⠈⠉⠤⣸⠇⡿⠛⠉⠀⠀⢀⡾⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣷⣴⣤⠀⠀⠀⠀⠐⠀⠂⢖⣒⣈⡀⠀⠀⠀⠀⠀⠀⣾⠃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠠⣲⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈⠁⠀⠀⠀⠀⠀⡀⡠⢏⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠪⢄⢔⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠣⢄⠄⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠆⠘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⣎⠊⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠈⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣌⣀⣀⣀⣀⣀⣤⡼⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⣠⡴⠶⢾⡷⠶⣦⣄⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡯⠀⠀⠸⠃⠀⠹⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣶⣤⡀⠀⠀⠀⠈⣩⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡉⠛⠛⠛⡏⢻⠛⠛⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣿⢿⣴⣊⠀⠐⠊⠈⣷⣮⣿⣿⣷⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠿⢯⣍⠙⣿⣦⣉⠳⣶⣬⣭⣭⣭⣭⣵⣾⠟⢋⣵⣿⠋⣹⡿⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⢀⣿⣧⣿⣿⡟⠓⠚⠛⠛⠛⠛⠛⠛⠓⠚⢻⣿⣿⣸⡏⠀⠀⢈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣤⣤⣾⢧⣿⣿⣿⠷⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠘⣿⣿⣿⣦⣤⣤⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣰⣿⣿⢻⣯⣭⣷⠛⠿⣿⣿⠖⢲⠒⠒⠒⠒⠒⠒⡖⠲⣿⡆⢹⣿⠿⠛⢿⣯⣿⡟⢻⣿⣆
⠀⠀⠀⠀⠀⢠⣿⣿⡭⠛⠻⢿⡁⠀⠀⣿⣿⣛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣇⣾⣿⠀⠀⣈⡿⠛⣛⢯⣹⣿⠄
⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠈⢳⠶⠿⣿⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⣵⣿⠿⠶⡞⠁⢀⣠⣿⣷⢹⣿
⠀⠀⠀⠀⠀⠀⠉⠻⣾⣿⣶⣶⣾⣷⣄⠀⣸⣯⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣽⡇⠀⣠⣿⣶⣿⣿⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠿⣷⣿⣿⡃⠀⠀⠀⢰⡆⠀⠀⠀⢹⣿⣿⣿⠿⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠿⠿⠶⠶⠿⠿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣤⣤⣤⡴⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⢰⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠗⠒⠉⠉⠉⠑⠒⢄⡉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠓⠒⠒⠒⠢⠤⢀⠀⣀⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠛⠉⠑⠒⢄⠀⡸⠁⠶⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠂⠤⠮⠤⠀⠊⠀⣇⠘⢄⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡖⠒⠒⠒⠒⠈⠁⠈⠒⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠀⠤⠤⠖⠋⠀⠀⣀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠘⣍⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⢰⠊⠁⠀⡎⠉⡆⠀⠀⠘⣎⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢰⠉⠉⡆⠀⠈⠑⡆⠈⢆⠀⠀⠈⠫⠔⠒⠄⠀⣸⣆⣧
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠤⢬⣓⠚⠁⠀⠀⣰⢻⠀⠈⢆⠀⠀⠀⠀⠀⠰⠋⡱⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡦⢄⡀⠀⠀⠈⠃⠀⠀⢠⢃⡎⠀⠀⠈⠢⠴⠊⠭⠓⠫⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢧⣀⡠⠤⣀⠀⣠⠃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡠⠴⠤⣉⣃⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠦⠀⠀⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡔⠓⢦⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠘⢧⣤⡞⠀⠐⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣧⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⣀⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣎⡆⢾⡆⠀⠀⠀⠀⠀⠀⠀⢸⡁⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⣧⣀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⢿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢿⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢸⡇⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣸⡆⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⣼⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣛⣛⣛⣻⣏⣉⣻⣉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢒⡾⠁⠀⠀⠀⡤⢤⡀⠀⠉⠉⣷⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡹⠊⢩⡇⠀⠀⠀⠀⡹⢯⠀⠀⠀⠀⢸⠓⢞⠙⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠯⣈⡿⠁⠀⠘⡇⠀⠀⠀⠀⢙⣚⠀⠀⠀⠀⣸⠀⠀⢳⣏⣈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⠄⢀⠈⡇⠀⠀⠀⢻⣄⣀⣀⣀⣙⣴⣁⣀⣀⣠⠇⠀⠀⠀⡟⠀⡠⣄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢾⣠⡇⠀⠀⠀⠀⣼⣅⣽⠈⠉⠉⠉⢹⣅⣽⡄⠀⠀⠀⣄⣸⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣄⣼⡀⠀⠀⠀⣸⣄⡤⣇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠺⠧⣤⡼⠧⡄⢀⠴⠿⣤⡤⠿⠒⠲⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⠟⠛⠿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢠⡄⠀⣠⣤⣄⠀⣀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⣇⠀⠹⣴⠟⠀⠼⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠈⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣤⣤⣠⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣶⣶⣶⣶⠶⣶⠶⢶⣶⣶⣶⣦⣤⣤⣼⡇⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠻⠿⢧⣤⣤⣀⡀⠀⠀⠀⠀⢤⣤⣤⣤⣀⣤⣤⣤⣽⡇⠀⠀⠀⣸⡟⠿⠷⢶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠉⠛⠛⠛⠶⠶⢦⣤⣬⣉⣋⡉⠠⠶⠾⠷⢤⣄⣠⣽⣛⣳⡶⣶⣶⣤⣭⣉⡛⠛⠿⠿⢶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠷⠶⠶⠶⢤⣤⣤⣤⣄⣀⣀⣉⣉⣁⣀⣀⣀⣀⣀⣈⣉⣻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠖⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⢀⡴⠟⠋⠉⠉⠉⠙⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢦⣼⠃⠀⠀⠀⠀⠀⣿⣿⣶⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠛⠿⠟⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⣿⣤⠖⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠞⠷⣦⡀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠉⠋⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⣄⡀⣸⠇⠀⡀⠀⠙⠳⣄⠀⠀⠀⢀⣠⠞⠉⢷⣄⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⢀⣀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠁⠀⠘⠿⠀⠀⠀⠈⠳⣦⠚⠋⠁⠀⠀⠀⠙⠳⣦⣀⣀⣴⠟⠁⠀⠀⢸⡏⢙⣧⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⣴⠛⡦⠀⡙⠉⠁⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⣾⠛⣶⣶⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠈⠉⠁⢮⡿⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢦⣼⣿⣤⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠰⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠈⠛⠿⢶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⣴⡿⠛⠿⢿⣿⠀⠹⣿⣄⡀⠀⠀⢸⣏⠉⠉⢉⣟⣿⡷⠶⢶⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⡟
⢸⣿⠀⠀⠀⠀⠉⡀⠀⠙⠛⠿⠿⣷⣾⣿⣀⡀⢀⡉⠀⠀⠀⠀⢻⡏⢹⣏⠉⠛⠛⠛⠛⠛⢻⣿⠿⣿⣿⣿⣿⣏⣉⣁
⣿⣇⣀⠀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣿⠀⠀⠀⠀⠀⠘⣿⠀⣿⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠉⠉⠛⠛⠿⢷⣶⣶⣤⣤⣀⣀
⠈⠙⠛⠻⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢹⣇⠹⣧⠀⠀⠀⠀⠀⠀⠻⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣦⣤⣀⡀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⢿⡄⢿⡄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠺⠿⠟⠉⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⣧⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠉⠙⠛⠻⠿⠿⣷⣶⣦⣤⣄⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠘⣷⠶⣶⠀⠀⠀⠀⢀⣀⣨⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠹⡷⠶⠛⠛⠛⢋⣉⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⣀⣸⠇⠀⠀⠀⠀⣀⣀⣤⣴⠶⠟⠛⠉⣉⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⣠⣤⣶⠿⠟⠋⣁⣠⣤⣴⠾⠛⠛⠉⠁⠀⠀⠀⢠⡾⠋⠉⠙⠛⠿⣶⣤⣀⡀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⣠⣤⡴⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣾⠟⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⠿⠟⠛⠿⢷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠻⠷⣦⣄⣤⣴⣶⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢶⣶⣤⣀⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⢉⣁⣀⣀⣀⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠟⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣀⣠⣤⣤⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿