Takeout Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠈⢁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣡⣊⣽⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⢧⣄⣴⠗⡢⡀
⠀⠀⠀⠀⠀⠀⢀⠌⡜⠙⠉⢎⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠎⠀⠀⠀⠀⠀⠑⡔⡄
⠀⠀⠀⠀⠀⢠⠂⡘⠀⠀⠀⠀⠆⠱⡀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠚⠚⠤⠤⠤⠤⠤⠤⠤⠐⠒⠤⡄
⠀⠀⠀⠀⢀⠃⠀⠃⠀⠀⠀⠀⠘⠀⢱⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡌⠀⠰⡦⣠⡄⣤⡄⡦⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠇⠿⠾⠻⠼⠧⠀⠀⠀⡇
⠀⠀⠀⢰⠀⠀⢀⡁⣉⣉⢁⡉⣉⡀⠀⠘⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣴⡔⣶⡶⣴⡔⣶⠀⠀⠀⡇
⠀⠀⠀⡆⠀⠀⠚⠓⠛⠛⠘⠓⠘⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠑⠀⠂⠀⠂⠐⠐⠂⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡜⠀⠈⠉⠁⠐⠒⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠂⠤⡀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠧⣤⣶⣦⡜⢤⡄
⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣛⣘⣔⣺⠉⠇
⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢰⢰⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠅⠀⠀⡇⠀⠈⢢
⠀⠀⡇⠀⠇⠾⣴⣶⢠⡄⠀⠈⢨⠀⢸⡤⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠃
⠀⠀⠁⡴⡆⣤⣤⢠⣈⡁⠀⡆⢸⠔⣁⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⣠⡄⣤⢠⡤
⠀⠨⣀⡀⠀⠀⠁⠉⠃⠛⠀⡇⡸⢋⡼⢃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣁⣉⢁⡉⣉⡀
⠀⠀⠀⠀⠉⠐⠒⠢⠄⣀⣀⣶⠔⠃⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠛⠛⠚⠓⠙⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠒⠒⠒⠒⠒⠒⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣘⣛⣛⣛⣛⣋⣙⣛⣛⣛⣛⣃⣙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣒⣒⣒⣒⣒⣒⣈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⣛⣛⣛⣛⣛⣛⣛⢠⡿⢛⣛⣛⣛⣛⣛⣛⠛⢻⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⣿⣏⣉⣉⣉⣍⣿⡇⣿⣿⣿⡇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢠⣿⣧⣤⣤⣤⣤⣿⢡⣿⣿⣿⢃⣏⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⠘⠿⠿⠿⠿⠿⠿⠟⢸⣿⣿⣿⢸⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠇⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⢘⣵⣾⣿⣦⡁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠃⠋⠀⠰⠶⢦⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⠤⠄⢀⣘⡋⠲⠶⣶⣾⠿⣿⣿⡳⠶⠒⠒⠂⠀⠀⣀⣠
⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⣀⣸⣿⣿
⠀⠀⠀⠀⠸⣧⠀⠄⠀⠀⡄⢠⠀⠀⠀⠐⡇⠀⠀⠠⠈⠃⠀⣠⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠋⠉⠉⠀⠀⠨⠭⠉⠉⠉⠀⠘⠛⠛⠛⠟⠙⠉⠉⠤⠄
⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠁⢸⡆⠀⠀⠀⡇⠀⠰⡆⠀⠘⣾⠇⠀⠀⢨⡇
⡀⠀⡀⠁⠀⠀⠐⠀⠀⣄⠀⠀⡀⠀⢠⠀⠀⠆⠀⢰⠀⠀⠉⢠⠦⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣙⠀⠈⠂⠂⠀⠀⡀⢬⣿⡄⠀⠀⢸⠀⠀⠰⠂⣀⢸⠀⠀⢰⣇⠀⠹⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⣷⡄⠀⠀⠀⠀⠉⠀⠘⠃⠀⠀⢸⠀⠀⠀⠀⠻⠀⠂⠀⠀⠃⠀⣀⣀⣁⣠⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠆
⠀⣄⡀⠀⠀⠘⠓⠒⠉⠛⠚⠛⠛⠛⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⣉⠁⠀⠀⠀⠀⣀⣀⡀⢀⡀⢀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡈⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢀⡤⠚⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡇
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠚⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡞⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠠⡀⢐⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡨⠄⠠⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿
⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠶⠤⠤⠶⠶⠶⠾⠿⠶⠤⠤⠄⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⢀⣀⡀⠀⢌⠐⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⢰⠠⠀⠀⠀⠀⠀⠀⡐⠀⠈⠁⢀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠐⢂⠀⠒⠂⠀⠀⠀⠖⠒⠀⠨⠃⠀⠀⠀⢠⠀⠀⠠⠀⠀⠀⡴⡏
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⢈⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠘⠄⠀⡤⠈⠀⠀⠀⡠⣜⠀⡈⠃⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⠁⣀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠜⢀⣀⢈⡉⢁⡀⠐⠓⠔⠂⠀⣀⣁⡀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⡧⠶⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠈⠀⠀⠀⠀⠀⠠⡴⠙⢆⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠤⠒⠒⠒⠐⠐⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢣⣤⣄⠀⡀⣀⣀⣄⠁⡀⢈⠁⠀⠀⢀⡀⣀⢀⣀⢀⣀⣀⠀⢤⡤⠒⠒⠂⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⠉⠚⠁⠀⣤⡀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⠐⠀⠀⠂⠀⠁⠀⠀⢀⡠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠉⠉⠂⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡆⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠀⠀⠀⡔⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⡐⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⠀⠜⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⢀⢎⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠢⠤⠄⡀⠀⠀⠀⠀⠀⡇⣃⠠⠤⢢⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠈⠡⠀⠀⡠⢚⠠⠀⡰⡡⠁⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢀⣤⣠⣤⡴⢅⣧⣽⣿⣷⣆⡀⡐⡐⢁⠠⠐⡚
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠄⢠⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠊⠁⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡞⣺⣿⣿⣿⣿⣷⣽⡿⢛⠡⢋⠡⠔⠂⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠉⠙⠛⠻⢿⠖⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⡀⠀⢸⠀⠀⡀⠄⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠡⠂⠀⠀⠀⠀⠀⠀⢀⣀⠀⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢱⠂⠀⠀⠀⠀⠀⠀⢀⣨⠀⢀⣄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⡠⠔⠢⣊⠁⠈⠆⠇⠀⠸⠠⠉⠒⡄⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠑⡀⢠⠛⠆⠀⠘⣄⠀⠀⡇⠀⢠⠵⡀⠀⠌
⠀⠀⢠⡒⢜⠛⠋⠉⠉⠡⠜⠉⠉⠉⠋⢻⠘⡆⠸⠀⠀⢻⠀⠰⠀⠀⡌⢀⠇⡜
⠀⠀⠘⠛⠋⠭⠗⠚⠓⠒⡒⠒⠓⠚⠭⠙⠘⡛⠀⠇⠀⠸⠀⠘⠀⠀⠀⠈⣰⠁
⠀⠀⠀⠇⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⢠⠉⠢⠤⣀⣄⠀⣇⣀⡠⠄⠋⠘
⠀⠀⠀⢐⠀⠀⠼⠙⠆⣀⣀⡰⠋⠳⠀⠀⠼⠄⢀⡤⢤⠀⠀⠀⣠⠤⡄⠀⡆
⠀⠀⠀⠘⠀⠀⠀⠀⠸⣿⠿⡇⠀⠀⠀⠀⡄⠆⠀⠀⠀⠀⣶⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠋⢀⢀⣀⠀⠠⣄⣸⠀⠀⠀⠈⠿⠇⠀⠀⠀⣈⠀⢀⣀⡀⠄⢀⡀
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠰⠉⠁⢙⣒⣁⣤⣍⠚⠛⠩⢉⡛⠿⠿⢛⡲⠿⢣⠭⣥⣤⣝⡌⠁⠋⠱
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠐⢄⡔⠉⠀⡆⠐⠏⣄⠩⠄⠹⢝⢓⡚⢫⠉⠉⡏⠩⠄⠤⠀⠈⠑⣄⠴
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠈⣤⡌⠀⠀⣽⡌⠀⠁⠈⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠐⠒⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⠒⠒⠐⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣰⠭⣉⢦⡀⠀⠀⠀⢠⢢⠔⣲⢦⠖⡲⣀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⢠⠞⠀⠀⠀⠘⡆⠀⠀⠀⡸⠤⠤⠼⠀⢹⡄⠀⠀⢠⡃⠰⠏⢣⡾⣹⣾⠁
⠀⠀⠀⠰⡻⠯⣙⡩⢭⣯⠽⡒⣧⠀⠀⠀⠈⢫⠉⠉⠉⢹⠁⠀⠀⠀⠈⡏⣳⠶⣉⡏⠀⠀⠀⠀⡏⠉⠛⠚⠉⠁⡎
⠀⠀⠀⠀⢗⡒⠪⣭⡍⠯⣙⡻⠁⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⣗⠶⡹⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠁⠉⠣⠟⠉⠀⠀⠀⠀⠀⠀⠘⡀⢀⢀⡇⠀⠀⠀⠀⠀⠀⠹⣠⠃⠀⠀⠀⠀⠀⠘⠂⠒⠒⠀⠴
⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⡠⢊⡭⡤⠤⢌⡱⣠⡖⢆⠀⠀⠀⢀⡠⠒⠒⡍⠙⡆⠒⢄⠀⠀⡠⠒⠉⠉⠉⠒⢤⠀⠀⠀⠀⢀⠔⠉⠉⠲⡄
⠀⠀⠀⠸⢰⢉⡮⣅⡴⢪⣊⣀⣻⣸⠀⠀⠀⡏⠀⠀⠀⠘⢄⡱⠒⠰⠂⠀⢷⠢⠲⠴⠤⡦⢶⠁⠀⠀⢀⠃⠀⠀⠀⠀⡸
⠀⠀⠀⠰⠸⡀⣊⣀⠉⠸⢋⠟⡆⠀⠀⠀⣜⠈⠢⡀⢀⡠⠊⠀⠀⠀⠀⠀⠘⡀⢇⢸⢀⠇⡇⠀⠀⠀⠀⣀⡀⠀⢀⠔⠁
⠀⠀⠀⠀⠳⣈⠂⠢⠵⠒⣁⠜⠀⠀⠀⠀⡏⠑⢦⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⠬⠔⠚⠀⠀⠀⡠⠜⡠⠛⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠦⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠀⠉⠢⠃
⠀⠀⠀⠀⠀⢀⣀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠳⢰⠇⡜⣇⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⡼⢏⡱⠮⣑⣐⣺⣆⠀⠀⠀⠀⢞⠉⠉⠉⠉⠉⡷⡖⡆⠀⠀⠀⠸⠘⣺⡅⢬⡀⠀⠀⢠⠎⢰⠑⢶⠱⠈⠲⡄
⠀⠀⠀⣴⢡⠁⠀⠀⠀⠀⠈⢢⣇⠀⠀⣀⣈⣦⡀⡀⣀⡴⣛⣛⠁⠀⠀⡰⠃⢰⠉⠀⠀⠙⡄⠀⡇⠀⢸⠀⠀⡆⢡⠀⠰
⠀⠀⠀⣜⣁⣀⣀⣀⣀⣀⣀⣠⠃⠀⠀⠑⠤⢤⠄⠀⠠⠠⠴⠊⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⡇⠀⠱⣄⠢⢀⡰⡁⠜⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠘⠒⠸⣀⠀⠀⠀⠁⠀⠀⠀⠑⠉⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡤⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢔⣵⣿⠭⣆⠀⠀⠀⠀⡙⠍⡇⠀⠀⠀⠀⠀⠀⢀⠖⢇⣜⡳⣄⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⢔⣂⡩⣶⣟⠟⠁⠀⢸⠀⠀⠀⡰⠥⠦⠗⡄⠀⠀⠀⠀⣰⣃⠤⠜⠚⠒⡎⠀⠀⡎⠉⠏⠀⠸⠋⠈⠝⠉⠑⡄
⠀⠀⠀⡎⢷⠞⠛⠋⠁⠀⠀⣠⠋⠀⠀⠀⠘⡉⢳⢀⠆⠀⠀⢀⠔⠁⠀⠀⠀⡠⠊⠀⠀⢠⠻⠲⠖⠩⠥⠖⠬⠒⠦⡞⢣
⠀⠀⠀⠈⢇⠀⠀⠀⣀⡤⠊⠀⠀⠀⠀⠀⢠⠗⠃⠘⡄⠀⠀⠧⣀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠫⡉⠭⠵⢩⡑⠩⢉⣉⡼⠁
⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⠔⠀⠀⠀⠀⠑⠤⠞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⠀⠀⠨⠀⠀⢀⠀⠈⠁⢄⠁⠠⠄⠄⠀⠀⠀⠰⠆⠐⠚⠻⠁⠸⢃⠀⠄⢀⠀⠐⠹⠀⠀⣀⠀⠉⠀⣈⠃⠠⠀⡀⠀⠀⢀⠴⠀⠐⠛⠣⠀⠫⡂
⠀⠀⠀⠀⠈⠉⡀⠀⠀⢀⠜⠀⠀⠒⠀⠀⠀⠂⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡈⠀⠀⠀⠌⠀⠠⠒⠀⠀⠐⢀⠀⠀⡀⠀⡀
⠙⠈⢃⡀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠂⠀⡄⠁⠨⢡⠀⡟⠀⠛⣰⢣⡄⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠊⢀⠈⠀⠊⢀⢠⡏
⠀⠉⢁⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⣀⡐⢀⣄⠀⠠⠀⢄⠙⡙⢚⡍⠀⠈⠉⠁⠉⠉⠀⠀⠐⠈⠀⠀⠈⠀⠀⢀⣀⡐⣀⡀⠀⠠⠀⠀⢋⡱⢫⠏⠀⠁
⠀⠀⠸⠀⠀⠸⡑⣧⠒⣗⢻⠇⠀⠖⠐⠂⡞⠀⠀⠈⠀⠀⠱⡙⠀⠀⠠⠀⠀⠀⠀⠀⠀⢻⢃⡕⢲⣇⢿⠅⠐⠂⠀⠐⠊⠀⢀⠠⠀⠀⢎⠛
⠀⠀⠠⠀⢄⠀⠀⠈⠁⠴⠉⠀⠀⢠⠀⠀⠁⠀⠀⢀⣉⠀⠀⠠⠂⠐⠠⠀⠀⢀⠄⠠⠀⠀⠀⠘⠣⠞⠁⠀⠀⠃⠀⠰⠀⠀⠀⣀⡁⠀⠈⠄⠒⠐⠠
⠀⠀⠀⠀⢀⠂⢀⠀⡀⠀⢀⠀⡀⠈⠀⠘⠠⢀⣤⡶⠋⢣⠀⠂⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠈⠀⠈⠀⣠⣾⡾⠋⠂⠘⠂⠀⠒⠐
⠀⠒⠤⠀⠀⠀⠂⠂⣊⣀⣏⠉⢄⠀⠀⠸⡛⠛⠉⠀⡠⠀⠀⠀⠄⠀⠐⠀⠀⠒⠄⠐⠀⠀⢐⢣⣁⣸⣍⠡⡀⠀⠀⢘⠛⠛⠉⠀⡠
⠁⠂⠀⠀⠀⠀⠆⠭⠀⡐⠁⠛⠀⠀⢀⠀⠁⠀⠐⠈⠀⡆⡠⠰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠅⢀⡑⠌⡁⠀⠀⠆⠀⠀⠀⠀⠁⢠⠂⡆⢰
⠈⠀⠄⠑⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠈⠁⣰⣦⠀⠀⠀⠀⢀⡏⢵⠀⠀⠠⠊⠢⡄⡕⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣲⡆⠀⠀⠀⠀⣆⠅⢅
⠀⠀⠨⠀⠀⠀⣀⠠⠄⠠⠀⠠⠄⠂⠀⠐⠉⠄⠀⠀⠀⠀⠆⠀⠀⠰⠀⡄⠀⠀⠀⠀⠠⠀⡀⠠⠀⢤⡀⠠⠐⠂⠀⠚⠁⠄⠀⠀⠈⠀⠁⠀⠀⠁⠐
⠐⠔⠉⠢⠀⠘⠁⠀⠀⠀⠀⠆⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠠⠀⠁⠔⠀⠘⠈⠁⠉⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⣀⠁⠀⠤⡧⠀⢺⠉⢉⠃⠀⠀⠉⠀⠀⠉⠀⡀⠀⠀⠀⠀⠀⢑⢀⠀⠄⠀⡀⠁⠀⣼⠦⠀⠏⠙⠹⠀⠀⠀⠁⠀⠀⠁
⠉⠀⢉⣁⠀⠀⠀⠀⠀⡠⠀⠁⠀⢀⡀⠄⠀⠄⠀⢀⠴⣀⢢⡀⢀⡀⠁⠀⠁⠈⡉⣀⠀⠀⠀⠀⢀⠤⠀⡂⠠⢰⠠⡀⠘⠀⠂⠠⠴⣐⢦⠀⢀⠈⠁
⠩⠅⠈⠳⠀⠀⣠⠂⠀⠑⢀⠅⠐⠒⠀⠈⠀⢀⠴⠃⠂⠉⠱⠀⠐⠧⠠⠄⠠⠀⠘⠐⠀⠀⣠⠊⠀⠐⢀⠃⠰⠒⡄⠀⠀⠀⠔⠀⠂⠉⠓⠀⠹⠩⠡
⣀⠀⡀⠀⠀⠈⡀⠂⡀⠀⠜⠀⠀⠂⠀⠀⠀⠣⡀⠀⡀⠊⡀⠀⠀⠀⠀⣀⡀⢀⠀⠀⢀⠈⡀⢀⠀⠠⠊⠀⠀⠂⠀⠄⠐⠄⠀⠀⠐⢀⡀⢀⠀⠀⢀
⠛⣡⣫⡖⠀⠀⠉⢀⠉⣀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠈⠁⢬⣡⠀⣟⢰⠛⣕⣣⠆⠀⠀⠁⣈⣁⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡎⠤⠎⢤⢠⡏
⠉⠉⢀⠀⠀⢐⠀⢀⠀⡀⠀⡄⠀⡤⠤⠤⢄⠀⠐⠀⠀⠈⣕⢪⡍⠀⠀⠉⠉⠁⠐⠀⠀⣐⠀⡀⢀⡀⠑⠀⢀⠤⠴⠤⠄⠀⠒⡀⠄⢩⡑⢫⠃
⠠⠀⠸⠀⠀⠘⡁⢶⠈⡯⢹⠇⠈⠇⠀⠀⡎⠀⠀⠈⣀⠀⠱⡙⠀⠀⠐⠄⠤⠀⠄⠀⠀⢻⠡⡧⢩⡯⢽⠂⠈⠀⠀⠀⠂⠀⠀⠀⡀⠀⢋⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠞⠁⠀⠀⢰⠀⠀⠀⠀⠀⣀⣈⠀⠀⠄⠀⠀⠀⢄⠀⠀⠀⠀⢄⠀⠀⠈⠣⠛⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣀⠀⢀⠄⠀⠀⠂
⠀⠀⠀⠀⢀⠃⢀⠀⡀⠀⢀⢀⠄⠈⠀⠈⢀⢠⣴⡾⠋⠡⠈⠂⠀⠒⠒⠃⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⢀⠀⡀⠈⠀⠀⢀⢠⣶⠟⠉⠂⠘⠂⠀⠒⠐
⠀⢉⠦⠀⠁⠀⠃⡓⢄⡔⣇⣉⠤⠀⠀⠘⡝⠋⠁⠀⡠⠂⠀⠀⠄⠀⠔⠀⠀⠁⠤⠀⠀⠠⢐⠳⢅⠼⣆⣁⠀⠀⠀⢹⠙⠉⠁⢀⠀⠁⠀⠠⠀⠠
⠁⠂⠀⠀⠀⠀⢂⠬⢀⣈⠁⡁⠀⠀⢠⠀⠑⠂⠀⠀⠀⠤⠔⢐⡀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠄⢀⣉⠀⡁⠀⠀⠆⠀⠑⠐⠀⠀⠠⠤⠄⣰
⠊⠐⠆⠉⠠⠀⠀⠀⠀⠀⠈⠀⠐⠀⠈⠁⣰⣲⠀⠀⠀⠀⢀⠅⠹⠀⠀⡀⠈⠐⠌⠅⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⣶⡆⠀⠀⠀⠀⣎⠅⢅
⠀⠀⠨⠀⠀⠀⡠⣀⢒⣲⡄⠐⠄⠓⠀⠚⠉⠆⠀⠀⠈⠀⠁⠀⠀⠐⠀⠄⠀⠀⡀⠀⠠⢀⡄⡰⢒⢢⡄⠰⠘⠂⠀⠚⠁⠂⠀⠀⠂⠀⠀⠀⠀⠀⠘
⠀⠂⠀⠒⠀⢈⠁⠁⠀⠀⠀⠇⠀⠀⡀⠀⢀⣀⣀⠀⠀⣀⡄⠀⠀⢀⠀⠀⠐⠂⠉⠂⠀⡊⠈⠀⠀⠀⠘⡆⠀⠀⠔⠀⢀⣀⣀⠀⢀⣀
⠀⡀⠀⠀⠀⠰⡁⠠⠀⠀⢀⠀⠀⠦⡷⠀⠻⠀⠨⠃⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⠀⠀⠀⢑⠀⠄⠐⠀⢀⠀⠀⣾⠆⠀⠏⠀⠸⠀⠀⠀⠀⠀⠀⡀
⣁⣀⣁⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣅⣀⣄⣀⣂⣀⣀⣐⣩⣲⣀⣀⣀⣁⣀⣁⣈⣁⣀⣀⣀⣀⣀⣀⣔⣀⣃⣠⣠⣠⣀⣈⣀⣀⣀⣐⣕⣢⣀⣨⣈⣁
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⠤⣄
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠉⣓⡏⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⣠⠰⠯⡞⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠒⣋⣽⡗⠾⣷⣆⠀⠀⠀⠀⠀⢠⠞⠉⠙⠒⠚⠉⠀⠈⡇
⠀⢀⣼⣩⠗⠒⠷⣄⠀⠀⠙⠒⠤⢄⣤⠀⠀⠀⣿⣎⣴⡗⠒⢿⣱⠋⠸⡄⠀⠀⠀⠀⠘⢦⡀⢠⠞⠉⠙⢲⡀⠙⢦⡀
⠀⠀⠀⠸⡄⠀⠀⠈⠓⠦⣤⣤⣤⠜⠺⠀⠀⠀⢳⠉⠉⡑⠛⡿⢁⠀⠀⢳⠀⠀⠀⠀⢀⡴⠃⠸⡄⠀⠀⣸⠇⠀⠀⢹
⠀⠀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⢉⡟⡿⠀⠀⠀⠈⢧⠀⠀⡼⠁⠀⣠⠀⠈⡇⠀⠀⠀⡏⠀⠀⠀⠉⠓⠚⠁⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠹⠇⠀⠀⠀⠀⠀⠳⡞⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠳⠤⠤⠖⠚⠉⠉⠓⠲⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡀⠀⠛⠀⠐⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⠀⠀⠀⢀⣤⡀⠘⢦⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣏⡞⠉⠀⠀⠀⠀⠀⠀⠀⠘⢲⣇⠀⠀⠈⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠲⣄⡶⢾⢉⣗⢀
⠀⠀⠀⠀⠀⠀⠀⣸⣹⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣞⣘⣦⡀⠀⢰⡇⡆⠀⠀⠀⣠⡶⠀⠀⣀⣈⣷⠉⣹⡞⢻⡈⢯⡇
⠀⠀⠠⣯⡭⠭⠭⠭⠭⠭⠭⢤⡽⠆⠀⠀⠀⢀⡧⠤⠤⠤⢧⡀⠈⡇⠃⡞⠳⢾⡏⠀⠀⠀⠿⠲⣬⠙⢧⡇⡿⢳⡀⢳⡀
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢹⠀⡵⠲⢿⣤⣼⠇⠀⠀⠀⠀⢷⠻⠷⠤⠿⠧⠟⠓⢫⠇
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⣬⣷⣄⣸⠇⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣄⣧⣀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣇⢀⢀⠀⠀⢰⠇⠀⠀⠀⠀⠀⡇⠀⠀⢾⠑⠀⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⣻⡿⠓⠷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⣴⣶⣲⡤⢤⡤⢤⡄⠀⠀⠀⠀⠀⠸⠋⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠈⠉⠙⠒⠦⣄
⠀⠀⠀⣠⠛⠉⠀⠈⠙⠺⣏⣻⡻⡟⠲⡄⠀⠀⠀⠀⡤⠚⢭⡀⣄⠓⣩⠛⢦⡀⠀⠀⠀⠀⠀⣯⡙⠒⠶⣇⣀⡀⠈⠙⢦⡀
⠀⠀⠀⡇⢀⠀⠀⠈⠀⠠⠈⢣⡙⣟⣷⢹⠀⠀⠀⠸⣅⠋⠀⠀⢡⠄⠀⠒⠃⢳⠀⠀⠀⠀⠀⣧⢙⡆⢀⣂⣉⡓⢤⡖⠀⢳⡄
⠀⠀⠀⠘⠦⣀⠈⠀⢀⡀⠀⠃⢳⢈⡿⣿⠀⠀⠀⢀⣯⣩⠭⣍⣉⣭⠿⣍⣩⠿⡄⠀⠀⠀⢰⡗⠋⢰⣫⢶⣾⣯⡷⠟⠛⠉
⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠈⡿⢻⠏⠀⠀⠀⠀⢳⡶⠖⠒⠻⠶⣖⠒⠒⣻⠁⠀⠀⠀⢸⠗⣦⣸⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠷⠋⠀⠀⠀⠀⠀⠀⠙⠲⠤⢤⡤⠬⠷⠞⠃⠀⠀⠀⠀⠸⠛⠉

⠀⠘⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠠⣠⢊⠀⠀⠀⢠⢠⣠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠
⠀⢂⡀⢀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⠐⢲⠱⠚⢂⠐⠂⠁⠂
⠀⠈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠈⠀⢀⠈⠀⢀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘⠈⠁⠀⠀⠀⠀⢀⠀⠂
⠀⡀⠀⢀⠀⠀⠀⠀⣀⣤⣶⡶⠶⡶⣶⣶⣤⣄⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀
⠀⢂⠀⠐⠀⢀⠀⢰⢟⡿⡿⡷⢶⡶⡷⠿⠿⢿⣧⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠂⠀⢀⢀⣀⢀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣨⢁⢯⢻⣿⡇⡅⠂⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠅⠀⠀⠀⠁⠈⠉⠁⠈⠁⠀⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠙⠿⢷⣿⣿⣿⣷⡾⠾⠚⠁⢀⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠃⠄⠀⠃⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠄⠀⠀⢀⠄⠀⠀⠁⠀⠁⠀⠐⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠐⠃⠀⠀⠀⠆⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠈⠳⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠠⠀⠠⡀⠀⢀⠀⠀⠠⡀⡀⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠁⡀⠀⠀⠣⢀⡸⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠐⠀⠀⠀⠀⠢⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠀⠀⠀⠈⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⡄⣶⢰⠀⠾⠿⠶⠶⠶⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠚⠉⠁⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⡀⣡⣶⠿⣛⣛⡛⠿⣶⣄⢠⣤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣫⠖⣚⡂⣼⡟⣵⡿⣛⣛⣛⢷⣌⢿⣆⢒⣓⠲⡝⢧
⠀⠀⠀⠀⠀⡴⢠⣤⣥⣤⣬⣭⣤⣭⣤⣭⣤⣭⣭⣭⣤⣭⣬⣭⣬⣭⣥⣥⣬⣤⡄⢲⡀
⠀⠀⠀⠀⡼⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢳⡀
⠀⠀⠀⡼⠁⠀⠀⢈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⠀⠀⠀⢳⡀
⠀⠀⡼⠁⠀⠀⠀⠸⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣻⣿⢃⡟⠀⠀⠀⠀⢻⡄
⠀⠀⠁⠀⠀⠀⠀⠀⢿⣜⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣿⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⣀⣀⣏⣁⡀⠀⠀⠀⠀⠀⢸⠻⡇⠀⠀⠀⠀⠀⢀⣠⠤⣤⣿⣯⣕⢦
⠀⠀⠀⡜⠁⠈⠀⠁⠈⠀⢹⡄⠀⠀⠀⣴⠓⠒⠒⠚⢳⠀⠀⠀⠀⠀⡾⠤⢧⠀⠀⠀⠀⡰⢫⠖⠋⡇⣯⣶⠾⡛⠁
⠀⠀⠀⣿⠭⢭⡩⠭⣩⠭⢽⡇⠀⠀⠀⠈⡏⠉⠉⠉⡏⠀⠀⠀⠀⢸⠓⢦⢸⡆⠀⠀⢰⡇⡟⢯⠗⣇⠭⡒⢻⢹
⠀⠀⠀⢹⡒⣖⠚⠛⢒⡶⣺⠁⠀⠀⠀⠀⣧⠀⠀⢸⠃⠀⠀⠀⠀⠘⡗⠋⢸⠁⠀⠀⠈⣇⢧⡺⡧⣀⡛⢋⠞⡼
⠀⠀⠀⠀⠉⠉⠑⠶⠋⠉⠁⠀⠀⠀⠀⠀⢸⣀⣀⣼⠀⠀⠀⠀⠀⢰⣃⣀⣸⡄⠀⠀⠀⠈⠳⢝⣚⣓⣛⡥⠞
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀
⠀⠀⠀⣤⢻⠹⡇⡟⡽⣦⡀⠀⠀⠀⠀⠀⠀⣯⣯⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠦⢤⡀⠀⠀⠀⠀⠀⢀⡴⠫⠙⣝⣧
⠀⠀⠀⣏⠉⠛⠛⠛⠉⢹⠇⠀⠀⠀⠀⢰⠒⠒⠒⠒⣦⢤⠀⠀⠀⢾⣁⣀⣀⣀⣀⣈⡳⠀⠀⠀⠀⡞⠐⠀⠀⢂⡟
⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⡧⠞⠀⠀⠀⠈⢧⠸⡇⢀⡇⣸⠁⠀⠀⢀⡴⢳⣬⡤⠖⠉
⠀⠀⠀⠘⣇⣀⣀⣀⣠⡇⠀⠀⠀⠸⣏⣉⣉⣉⣉⣉⣉⡹⠃⠀⠀⠀⠸⣄⣧⣸⣠⠇⠀⠀⠘⢯⡵⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣞⣙⣦
⠀⠀⠀⠀⣀⠤⠔⠛⢆⠀⠀⠀⠀⢀⣠⠤⣄⡤⢤⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣏⢳⡄⠀⠀⣸⢥⣤⣿⣤⣷
⠀⠀⠀⢰⠛⠒⠒⠒⠚⢣⠀⠀⠀⣞⣀⣀⣈⣀⣀⣁⣀⣀⣹⡀⠀⠀⠀⠀⢀⡴⢇⡈⠉⡧⠀⠀⠈⣏⣩⠽⢛⡏
⠀⠀⠀⢸⡉⢻⠉⣻⠉⡟⠀⠀⠘⣿⡳⢾⣷⠞⠳⣞⡷⢞⣿⡇⠀⠀⢀⡤⠾⣄⠀⠙⢲⡇⠀⠀⠀⢹⡴⢚⡽
⠀⠀⠀⠀⣧⢸⡀⡟⢰⠃⠀⠀⠀⠓⠻⠯⠭⠭⠭⠭⠽⠿⠚⠀⠀⠸⣍⣳⡀⢈⣧⡤⠚⠀⠀⠀⠀⠀⢿⢩⠇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡤⡶⣤⣄⠀⠀⠀⢀⡤⣄⠀⣠⢄⠀⠀⠀⠀⣤⣤⣤⡄⡸⢧⡀⠀⠀⠀⠀⠀⠀⢠⢾⢶⣄⡀
⠀⠀⠀⠀⠀⣠⠞⠙⡶⢳⣶⣳⠄⠀⢠⠎⢠⡈⢾⠁⣌⢳⡀⠀⠀⢸⠁⢹⢘⡗⢺⡃⠀⠀⠀⠀⢀⡴⣳⣫⠏⠀⢹⡆
⠀⠀⠀⣠⠞⠁⠀⢠⠏⢙⡷⠃⠀⠀⢸⡀⣞⠀⣸⢰⠃⢠⠇⠀⠀⡼⠀⠸⡏⠀⠀⢹⠀⠀⢠⣞⣽⣺⠖⠁⠀⢀⡞
⠀⠀⠐⢧⡀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠳⣄⡔⠙⢦⡰⠋⠀⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠘⢷⡋⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠙⢯⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠧⠤⠤⠼⠀⠀⠀⠀⠑⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢁⣼⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠀⠀⠀⠀⠀⢀⣾⡿⢡⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⣦⡴⣛⡭⠭⢭⣹⣿⢗⣛⣛⡓⢦⣾⣿⣱⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⣒⣚⣓⣯⣶⢏⣞⡵⣺⡽⢒⣫⠭⠭⠭⣓⢿⣚⣛⣓⢿⡛⣆⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠉⣱⢃⣞⣻⢶⣚⣛⣻⣸⢸⡸⡓⢬⣛⡶⠿⠿⣓⣌⣧⡿⠷⠞⠳⣜⢯⠉⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣿⡇⡞⠺⠶⢯⣖⣙⣿⢟⡷⣿⠿⢗⣛⣻⡟⠉⠉⠁⠀⠀⠀⠀⠀⠘⣎⢧⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡿⣰⠁⠀⠀⠀⠀⣼⢡⢋⡔⠓⠺⢥⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⢇⠀⠀⠀⠀⠀⡇⡏⡞⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣡⣿⠈⠉⠛⠒⠲⢼⠁⡇⣇⠀⠀⠀⠸⣿⠀⢀⣀⣠⡤⠴⠖⠒⠋⠉⠉⠀⢹⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡄⠀⠀⠀⠀⡼⢸⠖⠋⠉⠛⠂⠀⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠷⠋⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣄⣀⠀⠀⠀⠀⡇⠀⠀⣀⣠⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⢤⡧⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠙⠛⠛⠛⠛⠛⠛⠛⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣠⠿⠿⠿⠿⠿⠿⠿⣷⠀⣿⣾⠀⠀⠀⣤⣄⢠⡤⠄⣤⠀⢠⠀⡄⢀⢠⡤⢀⣤⡀⡄⢠⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣾⠃⣼⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⠀⠀⠀⣿⢹⣼⡧⠀⣿⠀⢸⠀⣧⣼⢸⡦⢸⣿⠇⢻⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣼⠇⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⠀⠀⠀⣿⠞⠸⣧⡀⢿⡄⢸⠀⢸⠇⢸⣤⠸⠟⠆⠸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠻⠷⠶⠶⠶⠶⠶⠶⠶⠟⢠⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣶⢶⣶⣶⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢠⡿⣿⡇⠀⠀⠀⠀⠀⢀⣀⠀⠛⠋⠉⠉⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⣿⠇⠀⠀⣠⣾⡿⠿⠿⠿⢿⣷⣤⡀⢸⣿⣾⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠟⠛⠿⢿⣷⣦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⣿⠀⠀⣸⣿⠋⢀⢔⣶⣦⣄⠈⢿⣷⠀⣿⢿⠀⠀⠉⠉⠉⠉⠉⢏⡿⠁⡠⢒⣴⣖⢄⠈⢿⣿⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠘⣿⣛⣛⣛⣻⣏⠀⢸⣿⠀⢸⣿⠀⢸⡿⠛⠛⠛⠿⠛⠛⠛⠿⠿⠿⢿⡇⠀⣇⣏⠀⣸⢿⠀⢸⡿⠷⠶⠶⠶⠶⢿⡆
⠀⠀⠀⠀⢿⣶⣶⣶⣾⣿⣄⠈⠛⠻⠟⠋⠀⣼⣯⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⡄⠈⠛⠿⠗⠋⢀⣾⣯⣭⣭⣭⣭⣥⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣦⣴⠶⠟⠁

⠀⠀⠀⠄⠒⠒⠀⠈⠉⠉⠉⠉⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⢀
⠀⢐⢀⠀⠀⠤⡤⠀⠀⠀⣀⠀⠐⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠈⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⠀⢨⡑⠳⠶⠄⠠⠤⠤⠤⠤⠍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠁⠀⠀⠭
⠀⠀⠙⢗⠤⠤⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠔⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠠⣄⣀⠈⠙⠂
⠀⠀⠀⠀⠁⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⡀⡀⠀⣠⡠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡈⠛⠋⠉⢉⡅⠻⠉⠉⢐⢊⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⡀⠄⣀⠀⡄⠀⢀⣉⠠⢄⣀⠀⠀⠀⠀⠘⠻⠄⠁⠤⠤⣤⣤⠬⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⢸⢯⡉⢦⣀⠛⠦⠰⠀⠡⡀⢰⠀⠀⠀⠀⠀⢂⠒⢀⠀⠀⠈⠐⠀⠀⠐⡠⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠘⢾⣴⢲⡾⠳⣜⣃⠉⡗⠘⠓⠟⠧⢁⡀⠀⠀⠑⠅⠀⠀⠀⠀⠐⢤
⠀⠀⠀⠈⢿⣉⣙⡊⢶⠉⠈⠙⠑⠒⠒⠠
⠀⠀⠀⠀⠀⠙⠿⡝⠈⠦⠃⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⡀⠛⠢⠵⢠⣄⠄⢀⡀⠂⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠦⠈⠂⠈⠁⠈⠁⠀⠀⠀⠀⠙⠤⡀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣈⣀⠀⠒⠒⠲⢶⡶⡶⡄⠐⡄⠀⠐⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⣓⠬⠤⠔⠁⠉⠛⠁⠸⠴⠤⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡤⠶⡶⠿⡴⢄⣀⣀⠀⡀
⠀⠀⠀⠀⠠⠉⠒⠂⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢮⠙⡈⢏⣰⣄⣠⢐⣄⠚⠉⠋⠁
⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢐⠢⠤⠤⠈⡧⠬⠤⠀⠫⠄
⠀⠀⠀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡤⠀⠀⠀⠈⠁⠀⠀⢀
⡄⠈⠈⡍⣉⣙⣛⡓⠒⢓⠒⠶⠶⠂⠴⠾⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⠉⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠺⠿⢿⣿⣿⣿⡿⠿⠿⠿⠿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣴⣾⠟⠋⠁⠀⠀⠀⠰⠀⠀⠀⠀⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠛⠛⠛⠛⠛⢻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⣤⣤⣤⣤⣤⣴⣶⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣤⣴⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⢷⣦⣀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠴⠶⠶⠶⠿⣿⣿⣿⣶⣶⣶⣶⣶⡶⠿⠛⠋⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢤⡤⠤⠄⢀⣀⠀⡸⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠈⡉⠉⠉⠁⠀⠀⠒⠂⠐⠈⠉⠉⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠢⠤⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠃⠘⠉⠭⠽⠟⠛⠛⠛⠋⠈⠉⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠛⠻⠛⠛⠓⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⠤⣄
⠀⠀⠀⠀⠀⢀⡴⢚⠁⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⢦⡀⠀⢀⣠⡤⠖⠛⣉⠥⠄⠠⠜⠻⣦⡀
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⠉⠀⠀⠀⠁⠀⠔⠊⠉⠴⠊⠩⡝⢳⣄⡀
⠠⢶⠤⣰⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⢣⡺⡤⣿⣌⠙⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣿⣠⢼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡲⠤⣄⣀
⡠⢀⠙⠺⠻⣷⣶⡤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣩⡾⡃⣀⣚⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣠⣤⣽⣦⣀⠤⢶⢺
⠀⠐⠣⡅⣙⢄⣋⣭⣛⣓⣶⠶⢶⣤⡤⣤⡴⠶⠶⠶⠶⠚⡟⠛⠶⠿⢱⣚⡏⠩⠈⠓⢄⡀⠀⠀⠀⠀⣀⣤⣤⣶⡶⢿⢿⠿⠿⠟⠛⠉⠉⢀⣀⣼⠿
⠀⠀⣾⣷⡔⠀⢨⠃⠂⠘⠑⠷⢶⣧⣲⡖⠃⢂⡵⠷⠒⠛⣓⣲⡒⠄⡽⣿⡀⠀⠀⠀⠀⠙⠢⣀⣴⡟⠛⢿⡟⠀⣠⣲⣷⣲⠶⢶⣒⠚⠿⢍⡳⢤⣤
⠀⠀⡟⠬⡙⠗⠦⣤⣀⠀⠀⣀⣀⠀⠀⣙⡉⠀⢀⠭⠒⣈⠡⠖⠴⠼⢂⣿⡁⢠⢠⡔⠀⠀⢠⣾⡇⠘⡳⡖⠀⡴⣱⡁⢸⡌⠉⢳⢦⣍⡒⠤⣉⣲⠖
⠀⢀⠛⣦⡘⠀⢀⡈⠉⠛⠖⠀⣀⣀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⢀⣠⠎⢱⠙⢦⡸⢃⠄⠀⣾⣿⡇⠀⣷⡉⢙⡴⣝⢧⡀⣷⠀⠈⣶⣄⣹⣲⣇⡰
⠀⠐⡀⠈⠀⠩⠴⠿⠧⠤⠤⢤⣁⣉⣥⣌⡱⣶⡴⠭⢄⣒⣂⠭⠔⠋⠀⡸⠀⠀⠙⢼⠃⢠⡟⣿⣿⠀⠸⣿⡋⠵⢾⣭⣏⣻⡖⠐⡧⣄⣈⣉⣏⣹
⠀⠀⠑⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠙⢮⡇⠈⢻⠇⠀⢳⡉⠲⣤⣄⠤⣞⠛⢟⠿⣓⠊
⠀⠀⠀⠀⠈⠉⠒⠛⠛⠉⠁⠲⠶⠶⠶⠦⠤⠤⠄⠀⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⢿⠭⡛⡘⣤⡉⢉⠇⠈⠑⢤⣙⡚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣈⣆⠱⣸⣌⣄⠉⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⡶⠶⠓⠓⠷⠶⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⢁⣤⣴⣶⣶⣿⢿⠾⢶⣶⣦⣬⣋⠛⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⣠⣾⠟⠋⠁⢉⡞⠁⠀⢀⡼⠟⠉⢻⡿⢿⣦⡐⠻⣷⣄
⠀⠀⠀⠀⠀⢠⡾⢋⣴⣿⠟⠉⠀⣀⡔⠻⠀⠀⠔⠙⠦⣀⠔⠀⠀⠠⡏⠘⢦⡘⢝⢦⡀
⠀⠀⠀⠀⢰⡟⢀⣾⠛⠻⣀⠀⠀⠉⠀⠀⠀⠃⠀⠀⠀⠉⠂⣀⠀⠀⠀⠂⡀⠑⡌⢻⣿⡆
⠀⠀⠀⢠⡟⠀⣾⢿⣆⡀⠉⠒⢄⠀⠈⠐⢦⡠⠊⠑⠤⣠⠆⠀⠑⠤⠀⠀⢀⠎⠠⠈⢿⣧
⠀⠀⠀⣼⠃⢸⣏⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠠⠖⠀⠀⡠⠛⠄⠀⡆⠈⢿
⠀⠀⠀⣿⠄⢸⡟⠀⠀⢀⠔⠣⡀⢀⡴⣣⣄⠠⣴⣛⣷⣿⣄⠁⠠⡀⠾⡁⠀⠀⠘⣿⠀⣿
⠀⠀⠀⣿⡄⢸⡇⠀⠰⢇⠀⠀⢀⣿⡀⠈⠛⠛⠻⣿⣉⢉⡻⣦⣄⠈⠀⢌⡢⣄⠀⣼⠀⣿
⠀⠀⠀⢸⡇⠘⣿⡀⠀⠀⠰⢄⡀⠉⠹⠓⢦⡀⠀⠈⠙⢿⣷⠟⠉⠓⢦⡄⠠⡨⠿⣿⢢⡿
⠀⠀⠀⠈⣿⡄⠹⣯⢢⠄⠀⠀⢹⠖⠀⠀⠀⠙⠁⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⡁⣰⢃⣾⠃
⠀⠀⠀⠀⠈⢿⣄⠙⠇⠀⣠⠞⠁⠂⠀⣠⠎⠀⢀⡴⠛⠀⠀⢀⠴⠂⠀⢀⢘⡿⢁⣼⠋
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠙⠧⠀⠀⠠⣄⠀⠀⠘⠛⡀⠀⠀⠐⠷⣄⠀⢢⡽⠋⢀⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠄⠈⢳⣤⠀⠀⠙⣦⡄⡀⠀⠠⠟⢋⣤⡼⠛⠉⠉⠉⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡈⠈⠙⠛⠋⠋⣀⣁⣤⠴⠞⢋⣥⣤⣴⣶⣶⣶⢤⣄⡀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠙⠛⠛⣿⠟⠛⠛⣩⡴⢞⣽⠛⠁⠀⠀⢠⠀⠙⠳⣬⣜⡳⢦⣀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠉⣹⠂⠀⢠⡞⠁⣤⢴⣾⣩⣷⣿⠃⠀⢀⠀⠀⠀⠀⠐⠀⢻⡟⠏⠶⣿⣷⡀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⢠⡟⠀⢠⡟⠁⠀⣾⣷⣿⡋⡇⣿⠀⠀⠀⠀⢀⢁⣄⠀⠀⢸⣿⡆⠀⠈⡟⢻⣆⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⣾⠃⠀⣾⠁⠀⣼⣿⡿⣻⠧⣅⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⡾⠙⢣⠀⢀⠇⠸⢿⡆⠀⠘⣧
⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⣿⠀⢰⡟⠀⢠⣿⣿⢹⡿⡴⢛⣣⣉⠟⠶⣤⣄⣠⣤⣾⠇⣀⣦⣰⣴⣤⠆⣳⠀⢿⠀⠀⣿⡀
⠀⠀⣴⠋⠀⣠⡤⠶⠤⣄⠀⠀⣿⠀⢸⡇⠀⠸⣿⠃⢀⡾⠛⠉⠉⠉⠛⠲⢾⡉⠀⢹⣷⠟⠉⠁⠉⠉⠙⠳⣟⠃⢸⠀⠀⢸⡇
⠀⣸⠃⢠⡞⠁⠀⠀⠀⠀⠳⠀⢸⡇⠘⣇⠀⠀⣿⢠⡟⠀⠀⣀⠀⠀⠀⠁⠀⢳⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⠀⠀⣸⡇
⠀⣏⠀⢾⠀⠀⠀⠀⠀⠀⠈⠆⢸⡇⠀⢻⡄⠀⠹⣿⠀⠀⠀⠀⣉⠀⠈⠀⠀⢸⣾⡇⠀⠀⠀⠤⢤⠬⠣⡀⢀⣿⠀⠀⢠⣿⠃
⠀⢻⡀⠘⢇⠀⠀⠀⠀⢀⠐⠀⣼⠃⠀⠘⣷⡀⠀⢻⠀⠈⠀⠉⠀⠀⠀⠀⠀⢸⡏⣷⡀⠀⠐⠀⠀⢀⠀⠁⣼⠇⠀⢠⣿⡟
⠀⠈⠻⣆⡀⠈⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠘⣷⡀⠀⠑⣤⣀⠀⠀⠀⠀⢀⣴⠃⠀⡏⠷⣤⣀⡀⠀⣀⣠⡾⠋⠀⣰⢿⡟
⠀⠀⠀⠈⠙⠻⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠈⢻⣄⡀⠈⠻⢿⡶⠖⠚⣿⠁⠀⠀⡇⠀⠈⣝⢙⡉⠉⠁⠀⣠⢾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠉⠁⠀⠛⠤⠀⠠⠴⠄⠀⠉⠀⠀⢀⣠⢾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣖⣤⣄⣀⣀⣀⠀⢀⣀⣀⢠⡴⣾⣻⠷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠶⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣠⣤⣤⣤⡄⠻⠿⠿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⢸⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⡟⠛⠻⢿⣷⣄
⠀⠀⠀⠀⠐⠶⠶⠀⠒⢨⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣤⡄⠸⠿⠻⣿⡿⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣧⣤⣤⣶⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⣵⣾⣿⣷⡜⠿⠿⠿⠿⠿⠿⠿⠿⠇⠻⠟⣵⣾⣿⣷⡝⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⣿⣿⣼⣿⠇⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠐⣿⣿⣼⣿⠇⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣀
⠀⠀⢰⡚⠛⠉⠁⡀⠈⠉⠉⠉⣩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢚⡦
⠀⠀⠀⠉⠣⣄⠀⠀⠀⠁⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠖⠋⠉
⠀⠀⠀⠀⠀⣈⣙⣓⣚⡋⠉⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⣺⠿⠗⠋⠉
⠀⠀⢰⡞⠛⠉⠀⠀⠈⠉⠉⠉⣩⠇⠀⠀⠀⠀⣀⡠⢴⣲⡯⠗⠛⠉
⠀⠀⠀⠉⢧⠀⠁⠀⠀⢀⣠⡤⠏⢀⡠⢤⣒⡭⠗⠚⠉
⠀⠀⠀⠀⠈⠓⠒⠒⢚⣉⣤⣶⠿⠗⠋⠉
⠀⠀⠀⠀⣀⣤⣴⡾⠿⠟⠻⠶⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠒⣒⣋⣽
⠀⠀⢀⣼⡿⠋⠀⠀⠄⠀⠊⣰⠋⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣴⣶⣶⣯⠭⠿⠶⠒⠛⠛⠉⠉⠁
⠀⠀⠸⢤⡀⠀⠀⠀⣀⣠⣶⣥⣤⣶⣶⠶⠿⠭⠽⠒⠚⠋⠉⠉⠉
⠀⠀⠀⠀⠾⠿⠿⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⢠⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢉⣉⣉⣉⡛⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⡟⠉⠉⠻⣶⡌⠻⣦
⠀⠀⠀⠀⠀⠐⠿⠿⠗⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠈⢿⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡄⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢼⣧⣤⣤⣤⣤⣾⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⡏⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⡾⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣶⠟⠛⠛⢷⣄⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⣿⡏⢰⡟⣿⡆⢹⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠁⣾⢻⣷⠈⣿⠛⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠻⣧⡈⠛⢛⣠⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣄⠙⠛⣁⣼⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠉⠀⠉⠀⠀⠀⣠⠤⠀⠀⠐⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠐⠒⠤⡄⠴⡴⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠞⠣⠒⠗⠒⠀⠀⠀⢄⢠⠤⠄⠤⣤⠲⠈⠀⡠⠶⠈⠑⠀⠤⣥⡔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣁⣂⣀⠁⠀⠈⠀⠀⠀⠀⠀⢂⠤⣎⣁⠠⠄
⠀⠣⠀⠐⠠⠄⠀⠀⠀⠤⠀⠀⠀⢠⠞⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠁⢀⡠⠐⠁⠀⢠⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠄⠊⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠓⠦⢀⡀⠀⡠⠖⡩⠄⠀⠀⠀⢸⡄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠐⠈⠀⠈⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠭⠔⠉⠀⠀⠀⠀⠀⠘⡇⡞⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠠⠄⢠⡄⠀⠒⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢀⡀⠀⠀⠀⠄⠠⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⠀⠈⠀⢠⠘⣠⠀⠅⠤⠀⠃
⠀⠀⠀⠒⠂⠀⠀⠀⠀⠠⢀⡌⠀⠈⠀⠸⠁⠐⠀⠀⢀⠁⠉⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠠⠴⠄⠊⠀⢀⠘⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡜⠚⠀⠁⠂⠀⠀⢄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⣴⠄⢀⡀⠙⠂⡀⢀⡀⠀⠀⠐⠧⡄⠣⡀⠀⠀⠀⠀⠀⠛⡠⠄⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⢠⣄⣀⠀⠀⠀⢀⠀⠐⠠⠤⠀⠉⠈⠒⠈⠁⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠈⠠⢌⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠀⠀⡐⠁⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠈⠀⢀⠀⠒⠂⠉
⠀⠀⠀⠀⠀⢆⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠨⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡀
⠰⢦⡀⠴⠄⠦⠔⠃⠀⠀⣀⠤⢠⡄⡶⠠⢄⠀⡆⠀⠀⠀⡀⠀⡝⠤⢀⡀⢠⡄⢤⣤⠀⣀⡠⠄⠀⡢⠄⠠⠄⠀⠀⣀⠤⠀⢨⣅⡰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⢠⠀⡥⠀⣶⠀⠀⠀⠀⠀⠀⠀⠁⢀⢈⣛⡀⠀⢠⠀⠀⠘⠡⡤⠀⠁⠀⠕⠀⢀⡀⣤⠀⠀⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠠⠄⠆⠀⠀⠐⠈⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢩⠁⠀⣠⡀⠑⠓⠐⠆⠀⠀⠀⠀⠈⠉⠀⠉⠀⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠀⣀⣀⣈⡀⠴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⠀⠀⠀⠠⠀⠀⠃
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢬⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠁⠀⠈⠁⠉⠀⠦⠤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠷⠀⠀⠐⡶⠀⡀⠀⠀⣀⡀⠀⠁⠀⠄⠀⠺⢷⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠔⠀⠀⠠⠤⠀⡀⠠
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠌⠓⠀⠀⠁⠀⠒⠀⠀⠀⠑⠠⠌⠂⠉⠐⠂⠉⠀⠀⠀⠀⠔⠀⠑⠒⠂⢀⣁⡀⠀⠈⠁⡠⠀⠀⠀⠄
⠀⣤⣄⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠆⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⡀⠀⠀⠀⢈⠄⠠⠐
⡇⠀⠀⠁⢄⠘⠦⣄⢠⡄⠔⠀⠘⣰⠀⠀⠀⣀⣦⠄⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠡⠀⠀⣤⠀⠀⠀⢠⠁⠀⢠⡏⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠄⡀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⡀⠘⢇⣀⠀⣀⠀⠔⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢋⡤⠀⠀⠀⠀⠐⢀⠁⠀⠀⠠⠒⠈⠉⠉⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠀⠀⠇⠀⠀⠀⠀⢀⠄⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠤⡀⠀⢀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⣦⠀⠀⠀⢰⠀⠀⠃⢄⣀⡀⢀⣀⡰⠞⠀⡈⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠊⠔⣈⠉⠋⣀⠲⠀⠀⡄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣊⠤⠖⠶⠤⡑⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⡐⠀⠀⣀⡀⣀⣀⠈⠁⠈⠣⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠐⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡧⠼⣀⠀⠀⠀⠀⠀⠆⠀⠴⠀⠀⢀⠤
⠀⠀⠀⠀⠓⠈⠜⢉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣰⠁⠀⠀⠀⠀⢠⠈⠁⠜⠂⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢶⠀⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⣈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⢸⣿⣯⣭⣵⣾⣿⣿⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣶⣶⣶⣶⣶⣦⡻⣿⣿⣿⣿⡿⣣⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣭⣵⣾⣿⣿⣿⣿⡇⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣯⡉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠇⡿⠿⡿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⣰⡟⣛⢻⣆⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡟⣴⢟⣛⢿⡜⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣛⣵⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣮⣛⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠓⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⢄⠀⣠⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠎⠈⠁⠃⠐⠂⡾⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠞⣤⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠤⣤⡉⢠⡈⢁⡄⠣⡀
⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⣀⠀⠀⠀⠀⠀⢹⠀⠀⠰⠊⠁⡀⢘⡏⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢼⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠉⠘⠢⢦⣴⣠⣄⣅⣸⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⣿⣽⣿⡿⣿⣿⣖⣶⣾⣟⣻⣿⣛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣸⡛⠿⠿⡭⢿⣿⣿⣿⣯⣿⡟⣛⣯⢿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⣷⣶⣻⣦⣶⠨⠶⠿⠆⠰⠓⠓⠛⠚⠉⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠙⠻⠿⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⣉⣽⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⡔⠀⢀⡄⠀⠀⠔⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢮⡀⢠⠀⠀⠋⢕⠀⣀⣈⠋⠁⠀⠀⠀⡄
⠀⠀⠀⠀⢀⡰⢧⣴⢲⡴⢬⠋⢻⡑⣶⣟⡖⠒⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡺⠷⠨⠧⣦⣇⡟⠸⠇⠈⣠⠀⠤
⠀⠀⠀⠀⠀⢹⣿⠿⣟⡙⡆⠐⠀⢉⣡⠽⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠑⣄⡁⣀⣀⢴⠋⢙⠃⡶⠑⡉⢡⡟⡃⠠
⠀⠀⠀⠀⠀⢸⡇⠈⡙⠓⢽⡋⠛⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣶⣈⣭⡈⠁⠐⣄⣫⠐⠿⣨⡤⢯⠠⠁⡼
⠀⠀⠀⠀⠀⠀⢹⢏⠅⡇⢀⡽⡛⡑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⡛⢻⣶⣧⣈⢻⣯⠙⢺⣿⢋⢙⡗⣹⡗⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠘⢦⠁⠹⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⢷⡄⠻⢿⣙⡷⣼⢇⣲⣽⣉⣛⣣⡜
⠀⠀⠀⠀⠀⠀⠀⠊⠲⣦⢎⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣥⡴⠏⡽⣩⣛⣏⣭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠈

⠶⢤⠴⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠤⠀⣀⡀
⠄⡘⢀⡇⠼⠒⠉⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣠⣤⣤⣤⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣈⠉⠙⠎⠙⡓⠲
⣒⣛⣚⣓⣗⣓⣚⡳⢤⡀⠀⠀⠀⠀⠀⣠⣾⢿⣲⡶⠤⠾⠥⢤⣄⣸⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠛⠢⢐⣤⠴⢯
⢀⠴⠒⠞⠛⠳⣦⠉⠓⣽⡄⠀⠀⠀⠀⢿⡞⣩⢂⡰⠶⢠⡢⢦⡰⠈⢗⣼⠇⠀⠀⠀⠀⠀⠀⠀⢻⣆⣠⣴⠊⠀⠀⣈⡆
⣇⢀⡁⠊⠀⢔⣽⡧⠔⡿⠀⠀⠀⠀⠀⠘⢿⣾⣧⣼⣿⣷⣶⣷⣿⣶⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⡙⠻⠶⠶⠶⠿⠷
⡈⠉⠉⠉⠉⠉⢁⡤⠊⠀⠀⠀⡄⠀⠀⠀⠀⠙⣶⠬⠭⣭⡭⠯⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠦⠤⠤⠴
⠛⠷⣶⣶⣶⡾⠟⠀⣀⠀⡴⠒⠋⠀⠀⠀⠀⠈⢻⣦⣀⣀⣀⣤⠿⠃
⠀⠀⠀⠀⠀⠀⡎⢙⡏⠳⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢶⣞⠛⠛⢶
⣠⠶⠦⣄⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣫⣥⠞⣫⣾⠋⣿⡁
⠃⢄⠀⠸⡆⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⡄⠐⠁⣪⢿⠾⠦⣄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣋⣡⠉⡀⠜⢙⠉⠃
⠀⠀⣱⢞⣁⣀⠀⠀⠀⠘⡄⠈⠁⠀⠀⠀⢀⠀⢠⠖⠛⠙⣤⣀⣽⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠞⠁⢏⠩⢚⠰
⠛⠚⡱⠋⠀⠈⠙⢦⠉⠉⠁⠀⠀⠀⠀⠠⢂⠀⢾⠐⠀⠀⡼⡇⠀⠹⡄⠐⠄⠀⠀⠀⠀⠀⠀⠀⢻⣿⢶⣤⣦⣤⣤⡴⠾
⠀⢸⠃⠈⢲⠀⠀⣺⠄⠀⠀⠀⠀⠀⠀⠀⠈⢀⠎⢠⡤⠊⠁⠛⡂⢰⡑⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⣒⣂⣰⣤
⠀⠘⢆⡀⣸⣂⠴⠋⠀⠀⠀⠀⠀⠀⠀⢸⠟⢺⢂⠘⠀⠀⠀⠀⠹⠇⠷⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢈⠝⠂⠀⠀⠀⢚⠁⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣤⣤⣀⣀⡀
⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣼⣱⢳⢳⢃⡟⣹⢋⠟⡿
⠉⠁⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⢶⠟⠀⠀⠀⠀⠀⠀⢰⣧⢏⢯⠇⡞⣴⢣⢏⠞⣰
⠖⣱⢶⡶⣤⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⣴⣟⡤⠤⢶⡷⠦⠀⠀⣿⡟⡘⡜⡘⣼⢡⢣⠊⣼⡿
⣾⠿⢷⣸⡉⣿⣿⣆⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡤⠚⠉⠹⣿⣶⣾⠟⠁⠀⢠⡞⠀⠀⠀⢰⡏⠁⠙⠁⢰⢣⠣⠃⣼⡿⣱
⣌⣱⡴⣷⣦⣙⣧⠏⠀⠀⠀⣠⣿⠃⠀⠀⣠⠋⠀⠀⣠⣾⠿⠙⢿⣦⣀⣴⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠃⠌⠀⣼⠟⣱⠏
⠈⢹⣷⠥⢿⡶⠀⠀⣀⣤⣾⠟⠁⠀⠠⡾⠿⠤⠶⠞⡿⠃⠀⠀⣸⠟⠉⠀⠀⠀⠀⠀⠀⢿⣅⣀⠀⠀⠀⠀⢸⡏⣰⠏
⣶⣶⣶⣶⣶⣶⡾⠿⠛⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⣼⣁⣀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣶⣶⣿⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⠃
⠀⠀⠀⠀⣀⠤⠒⢠⠉⠉⠉⠉⠐⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠚⣆⠒⠄⡖⢲⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡇⠀⣿⣀
⠀⢀⡴⢈⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⡙⢦⠀⠀⠀⠀⢀⣶⡦⣀⡭⣶⠀⣿⢸⢉⣇⣽⠉⣯⠉⣹⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⢸⡀⣠⠏⠉⠉⠲⣄
⢀⡮⠄⠄⠀⠀⡄⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⢡⠀⣧⠀⠀⠀⠘⢦⠑⣿⠊⢹⠀⡾⢼⡌⡍⢹⢠⡿⣄⡏⢹⠖⣆⠀⠀⠀⣧⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣸⠆
⠸⡇⠒⠀⠙⠀⢀⠀⠤⠀⠀⠠⠁⠀⠀⠤⠀⠀⠂⣾⡀⠀⠀⠀⠈⢣⠘⡇⢸⡆⡇⢀⡇⡇⢸⢸⠉⢸⡇⣯⡴⠁⠀⠀⠀⠳⡍⣓⠒⠶⠶⠶⠶⠶⠖⠒⢋⡹
⠀⠙⠦⣅⡘⠁⠀⠠⠀⠴⠀⠀⠄⠐⠂⠀⢀⣩⠞⠁⢈⡆⠀⠀⠀⢰⡷⣧⣤⣧⡥⣬⣇⡥⣼⣼⣦⣯⣧⣿⡄⠀⠀⠀⠀⠀⣇⠀⠉⠁⠒⠒⡲⠒⠞⠉⢈⠇
⢣⠔⠖⠄⠉⡉⠓⣒⡒⠒⠒⠒⠒⣒⣊⡉⢩⣤⣤⢿⡉⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠸⡀⠸⡀⠀⢸
⠈⢦⡳⡶⠮⠗⠊⠙⠙⢒⠞⠢⠴⠣⣈⢋⡩⣞⡡⢊⡼⡄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠑⠄⠙⢄⡜
⠀⡚⠿⣒⠶⠬⠭⠤⠔⠊⠀⠀⠀⠉⢉⣉⣉⠥⠔⢻⡠⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣀⠴⠃⢠⠎⡇
⠀⠱⣄⣀⠈⣉⠕⠛⠀⠀⠈⠉⠉⠉⠉⠀⠈⣓⡦⢬⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡜⠉⠀⠀⠀⣇⢰⠇
⠀⠀⠀⠙⠚⠓⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠛⠒⠒⠒⠒⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⠤⠤⠤⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠄⠐⠀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠒⠒⠒⠒⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢀⣀⣀⡀⠸
⠀⠀⠀⢀⡠⠔⢚⣉⡥⡴⠒⢺⣏⣩⠯⠤⠔⠒⠤⡉⠈⢆⠀⠀⠀⠠⠐⠁⠀⠀⠀⠄⠒⠒⠒⠤⠀⠀⠀⠈⠱⢦⡀⠀⠀⠀⠀⠀⠀⢂⡠⠢⣀⡠⠤⣀⡇
⠀⢠⣊⠥⣶⠋⣹⠤⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⠖⠃⠀⠀⡀⠀⠀⢀⠀⠀⠐⠂⠬⠭⠉⠒⠁⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⢀⣇⡤⠀⠀⠤⢨
⢠⠛⠓⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣢⠞⠀⠀⠀⠀⣇⣁⡀⡀⠀⠁⠀⠁⠀⠀⡀⠀⠄⠀⠀⣂⡈⠀⣠⡇⠀⠀⠀⠀⠀⡜⠘⠑⠀⠀⠀⠚⢣
⠘⠢⠄⢇⠀⠀⠀⠀⠀⠀⠀⢀⣒⣀⡭⠤⠖⠊⠁⠀⠀⠀⠀⠀⠈⢄⠀⠐⠤⠗⠉⠲⣄⣀⠴⠊⠢⣤⠜⠁⠈⢊⡵⠁⠀⠀⠀⠀⡼⠱⠁⠀⠀⠀⠀⠄⠠⠱⡀
⠀⠀⠀⠀⠈⠁⠐⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⠤⠄⢀⣀⣀⣀⣀⡠⠤⠔⠚⠉⠀⠀⠀⠀⠀⢸⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠐⠀⠀⠈⠁⠀⠀⢠⠃
⠀⠀⠀⠀⣀⠀⠤⠀⠐⠒⠒⠂⠠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢠⠠⢠⠃
⠀⢠⠂⡡⠔⠒⡂⠨⠍⢈⣉⠉⠀⣒⠒⠪⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⢀⠀⠇⠸⠈⢸
⠀⢸⡛⠤⠄⣀⣀⣐⠂⣀⣀⣈⣂⠠⠥⠴⣛⣯⢩⣳⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠉⠀⠀⠀⠀⠀⠀⠉⠒⠢⢀⠀⠀⠀⠀⠀⠀⢀⠏⠄⠀⠀⠀⠀⠘⠈⢆
⠀⠀⡟⠓⠶⠤⠤⠤⠤⠤⠤⠤⢰⠒⠲⠟⠋⣇⡠⢻⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⠀⠀⠀⡞⠀⠀⠴⠀⠀⠀⠂⠀⠘⣆
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⠔⠃⠀⠀⢨⣉⣀⣠⢠⣀⡄⣀⢀⡀⡄⠠⠤⠤⣤⣤⡤⣖⣲⣶⣶⣶⣾⠀⠀⢰⡇⠀⠀⠀⠀⠀⠐⠀⢠⠀⠸
⠀⠀⠄⠓⢤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠯⢐⠒⠤⡀⠀⠀⠐⡛⡯⢜⣛⣭⣴⡟⢉⣤⣢⡀⢂⣠⣌⣯⣥⣩⡭⠍⣋⣉⣩⠀⠀⠘⡇⢃
⠀⡁⠀⠀⠀⠈⠉⠛⠧⠒⠛⠫⠍⠃⠀⠀⠀⣈⠇⢈⡇⠀⠀⢉⢉⠉⠛⣦⣀⡠⣠⢚⣁⣀⣉⣀⣠⡒⢄⢀⡠⣲⣉⣁⣸⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢁⡴
⠀⠈⠒⠂⠥⠤⠄⢀⣀⣀⣒⣒⣀⣀⣀⡩⠭⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠞⠃⠀⣀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⡄⠀⡟⠈⠀⠀⠀⣼⢧⡄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⣼⠛⠦⠛⠷⣤⡀⠀⢠⡞⢀⣠⡀⢀⢸⠿⠟⠓⠛⠋⠙⠘⠓⣠⠇⠀⠀⠀⠀⣿⡀⠹⣦⠈⠷⣆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣦⣘⣧⣬⠌⠇⣤⠄⢀⡀⣤⣀⣼⠀⠀⡼⠁⠀⠀⠀⠀⠀⣼⣿⣤⣬⣷⡀⢽⣀⠀⣠⡤⣦
⠀⠀⠀⠀⠀⠀⡟⢠⣿⠀⠀⠀⠀⠈⠛⠻⣆⠀⠀⠿⢂⣙⣫⡭⠼⠿⠶⠾⠷⠆⠀⠀⠀⠠⠞⠁⠀⠀⠀⣼⣧⠀⠈⠋⢁⡴⠃
⠀⠀⠀⠀⠀⠀⣇⣼⠘⣇⠀⠀⠀⠀⠀⠀⠈⣧⠴⠛⠉⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡹⠏⠀⠀⠀⠛⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⢻⣿⣀⠼⢧⣄⠀⠀⠀⠀⠛⠁⠀⡀⠀⠀⠀⠀⠀⠀⠐⢻⣶⡦⠀⠾⢳⢞⣉⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⢤⠾⠟⣙⣷⠀⠉⠓⠀⠀⠀⠀⢀⣨⡟⠂⢀⡄⢀⣀⣿⠀⠀⠉⠀⠀⠀⢸⡏⢰⣟⣦⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠚⠉
⠀⠀⠀⠀⢾⡚⢷⡘⢿⠆⠀⠀⢀⣤⣤⣾⡶⠘⠛⠁⠀⠀⠙⢻⣽⡇⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠈⢳⣄⠉⠻⠾⣇⢰⡏⠴⠒⠊⠹⡆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡿
⠀⠀⣠⡴⠚⠁⠈⠀⠀⠀⠀⠸⣇⠰⣏⣿⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⢠⣾⣥⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠈⢉⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠷
⠾⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠄⢰⣆⢀⣴⠏
⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⠁
⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠹⣆⠀⠀⠀⢰⡄⢀⡾⠋
⠀⠀⠀⠀⠀⠉⠓⢶⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡀⣠⡈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⣉⣬⡁⠀⠀⢀⣠⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⣶⠂⠀⠀⠀⠀⠀⠀⠿⠀⠛⢁⣤⡀⢻⠀⠀⠀⠀⡀⠀⠀⠹⣆⠹⠶⠏⣼⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⢀⠀⠀⠀⠀⢀⡀⢧⣨⡇⣼⠀⠀⠈⠟⠷⠄⠀⢤⡈⠳⠶⠞⠁⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣧⡀⠀⠀⠀⠙⠶⠥⠞⠃⠰⣤⣰⣶⡖⠖⠚⠛⠁⠀⠀⠀⠀⠀⢀⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠠⣀⣀⠀⠀⠘⣧⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠟⠀⣰⡇⠀⠀⠀⠀⣸⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⢹⣇⠀⠀⠀⠈⡥⠞⠙⢦⣀⣀⠀⠰⠶⠆⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⠀⢀⣀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠞⠋⣸⠀⣠⠾⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠞⠉⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⣀⠖⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠙⠋⠁⠀⢤⣀⠄⠨⠆⠀⠀⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠡⠃
⠀⠀⠠⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠀⣀⠁⠴⠀⠤⠚⠢⠚⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢠⡀⢀⣈⡁⢁⣒⠠⣤⣄⠀⠀⠁⠀⡎⠀⠀⠘⠋⠀⠀⠀⠁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⢠⣦⠀⠀⠀⠄⠈⢫⠍⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠂⠰⠉⠀⢀⡂⠀⠀⡂⠀⠊⠄⡀
⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⠀⡄⠀⡀⠀⢀⠀⠘⢂⠀⠀⠀⢳⣶⣤⣤⣤⣴⣾⠇⠀⠁⠠⠐⠄
⠀⠀⠀⠀⠀⠀⢘⡠⠇⠀⠀⡜⠀⢠⠁⠠⠀⠀⠀⠀⠢⡀⠀⠀⠙⣿⣉⡉⠻⠋⠀⠄⠈⠁⡀⠂
⠀⠀⠀⠀⠀⠀⠜⠀⠀⠜⠠⠥⠄⠅⠄⠜⠀⠀⠀⠀⠀⠀⢱⠁⠀⠀⠲⠒⠉⠀⠀⣀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠑⠄⠀⠈⠐⠒⠚⠉⢡⣴⠂⠀⡀
⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣐⠠⠀⠀⢤⡀⠀⡀⠐⠒⠦⣀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠿⠋⠀⠀⠀⠀⠀⠒⠀⠀⡀⠈⠳⡀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⠀⠀⣁⡀
⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠊⠁⠒⠃⢘⠀⠀⢀⠐⠊⠀⠁⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⠮⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠐⠒⠀⠠⠤⠤⠤⠤⠤⠐⠙⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠤
⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠡

⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠐⠒⠲⠶⢶⣶⣶⣶⣶⣶⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⣸⣿⣿⣿⣿⡟⠛⢻⠟⠛⣿⠟⠛⠛⡟⠛⠛⣻⣿⣿⣿⣿⠁⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⢶⣶⡆⣿⣿⣿⣿⣿⠀⠘⡟⢀⠀⣿⡀⠙⢶⣿⠀⣼⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢹⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⣿⠁⣈⠀⡏⠹⠆⣸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⣾⣿⣿⣿⣿⣷⣾⣿⣶⣿⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⣀⣀⣼⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣀⣠⡿⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀⠀⣭⣿⣯⣥⠀⣠⣤⡄⠀⢀⣤⣠⢀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣙⡛⣉⣀⣀⣀⡀⠀⣀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⢻⣿⣿⠇⣼⣿⡿⠛⠋⣼⣿⡿⠁⢀⣾⣿⠏⣾⣿⡟⣰⣿⡿⢱⣿⡿⠿⠟⣰⣿⡿⢻⣿⡿⢠⣿⡟⣴⣿⠏
⠀⠀⠀⠀⠀⣰⣿⣿⢃⣿⣿⠏⣼⣿⣿⣷⠄⣸⣿⡿⠁⢀⣾⣿⠏⣸⣿⣿⣴⣿⡟⢡⣿⣿⣧⡄⣰⣿⣿⣵⣿⠟⠁⣸⣿⣿⡿⠁
⠀⠀⠀⠀⢰⣿⣿⢃⣾⣿⠏⣰⣿⡿⠉⠉⣰⣿⡿⠁⢀⣾⣿⡏⢀⣿⣿⣿⣿⠟⢠⣿⣿⠋⠉⢰⣿⣿⢩⣿⡿⠀⢀⣿⣿⠋
⠀⠀⠀⢠⣿⣿⣷⣾⣿⠏⣰⣿⣿⣷⡶⣰⣿⣿⣷⠆⣼⣿⡟⠀⣼⣿⣿⣿⠏⢠⣿⣿⣷⡶⢠⣿⣿⢣⣿⣿⠃⢀⣾⣿⠏
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠀⠉⠉⠉⠉⠈⠉⠉⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁⠈⠉⠁⠈⠉⠁⠀⠈⠉⠉

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠀⠁⠀⠀⢀⠠⠤⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠤⡀⢤⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠦⠤⠐⠀⠈⠀⠠⠀⡀⣤⠀⠤⢤⡌⠛⠂⠀⠤⠆⠉⠉⠁⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⣀⣡⡒⠦⠤
⠀⠂⠀⠀⠢⠄⠀⠀⠀⠄⠀⠀⠀⡀⠞⠀⠀⠀⠆⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠐⠀⠀⠀⢀⣀⠀⠠⠄⠊⠁⠀⠀⠀⠀⠀⠁⠒⠦⢀⡀⢀⡤⢊⠡⠀⠀⠀⠀⢠⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠔⠈⠀⠀⠀⠂⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠥⠖⠁⠀⠀⠀⠀⠀⠸⡄⠃⠀⠄⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⡤⠄⠤⠀⢤⠄⠀⢂⠀⡰
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡆⠀⠀⠄⠂⢀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠐⠄⠃⡤⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠌⠀⢀⠀⠘⠀⠐⠄⠀⠀⠁⠀⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⠀⠀⠀⡐⠁⢠⡕⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠇
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣀⠂⢀⠀⠙⠂⠀⡀⠀⠀⠀⠑⢧⠘⠀⠀⠀⠀⠀⠀⠐⢁⠤⠀⠀⠘⠀⠠⢀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⣀⠀⠈⠀⠀⠀⠀⠢⠤⠀⠈⠀⠂⠈⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠈⠠⢌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠁⠀⡄⠀⠊⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠦⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⢀⠀⣀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠘⠲⡀⠶⠄⠦⠐⠃⠀⠀⡀⠀⢠⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠁⠤⠄⠀⠀⠀⠤⠄⠀⢀⡀⠀⢀⠠⠐⠠⠀⠀⠀⠠⠄⠀⠀⢀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⣁⡀⠠⠄⠄⠀⠈⠠⠄⠈⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⠠⢤⠄⠙⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠀⠀⠠⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⠀⠀⠨⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠉⠉⠀⠦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠐⠖⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠄⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠐⠈⠁⠀⠂⠀⠀⠀⠀⠐⠒⠈⣀⣁⠀⠀⠀⡁⠀⠀⠀⠀⠃
⢀⠤⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⢀⠀⠀⢀⠀⠀⠐⠢⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠈⢂⠘⠢⠄⠀⠠⠒⠀⠈⡠⠀⠀⢀⡄⠆⠆⠀⠙⣂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠰⠀⠂⠀⠀⠀⠀⠁⠂
⠀⢀⠀⠀⠀⠱⡀⠀⠄⠒⠀⠀⢰⠁⠀⢰⠑⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⢂⠈⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠊
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⢆⣀⣀⡀⠀⠂⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠥⠄⠀⠀⠀⠀⡀⠀⠀⠀⠄⠐⠈⠉⠉⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⢀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠓⢠⠄⠀⠀⣤⠚⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠐⠢⠄⡈⠣⡙⠃⠀⠘
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠔⠒⣉⡐⠒⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣠⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⣤
⠀⠂⠀⠀⠀⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⠁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠶⠾⡄⠀⠁⠀⢂⠐⠃⠐⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠑⠀⠈⣉⠲⠀⠀⠀⠀⠀⠀⢀⡉⠀⠭⠭⣝⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣑⣤⣤⡤⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣼⣅⣇⣸⣰⣿⣿⣿⣿⣿⡇⡏⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢝⠋⡿⠿⢹⣙⡝⠹⠉⠯⢹⣿⡇⣧⣤⣤⣤⣬⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣛⣿⣿⣯⣻⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠘⢿⣿⡿⠈⠉⠉⠉⠉⠉⠁⠉⠉⠁⢿⣿⣿⠇⠉








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

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡯⠉
⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣀⢀⣀⣀⠀⢀⡀⠀⠀⠀⠀⠀⣠⡴⠖⠒⢲⡿⡏⠀⠀⠀⠀⠀⢦⣀
⡇⠀⠀⠀⠀⠀⠀⣿⣤⠏⢿⠀⣩⠏⠈⡹⠋⣤⣿⣧⠀⠀⠀⠀⠀⣿⣄⠀⠀⢀⡠
⡇⠀⠀⠀⠀⠀⠀⠙⠀⢀⣛⢃⣛⢛⡉⢛⣋⡘⢀⣙⡁⣀⡀⡀⢀⠈⠓⠒⢸⣿⠀⠀⠀⠀⢰⡷⠦⢠⣤⡤⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣹⡼⢸⣗⣸⣇⣸⡇⢻⠇⣿⡂⣷⡏⢹⡏⠀⠀⣀⣼⡿⣾⠯⠉⣴⣬⣷⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢻⡅⠲⠿⠃⢸⠟⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣾⡀⠀⠀⠠⠄⠥⠀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠲⣤⣀⠀⠀⠀⠀⠘⠛⣧⠃⠸⣄⠀⠒⠂⡨⠸⡅⢠⡄⠲⠆⡀⠼⢋⣁⡀⣴⠓⠀⠰⠶⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢌⣀⣀⣤⣤⣤⣤⣀⠈⠙⢦⡀⠈⠀⠒⠈⠀⠀⠐⠊⠙⣳⡶⣤⣦⣤⣬⣀⠀⠀⠤⠦⠭⠤⠙
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡉⠳⣄⠀⡙⠿⣍⡷⣶⠭⣤⠖⡰⠋⠉⠁⢀⡤⠤⠒
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣬⣥⣁⣠⣤⡀⣀⡀⢐⡆⠀⠀⣾⠁⡛⠀⣀⣈⡳⣌⣓⣂⣴⠏⠀⣁⣘⡀⠀⠀⠘⣟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⠄⠀⠴⠂⢈⣀⣈⣉⡟⣰⠷⠐⢆⠈⠁⢀⡄⣹⠶⠊⠖⠒⠻⣶⡏⢹⡶⠒⠒⠉
⡇⠀⠀⠀⠀⠀⠀⢸⣈⣭⣅⢠⠶⠦⠤⠤⠤⠶⠿⠭⠭⢤⣿⡶⠀⠈⢀⣀⣉⣀⣁⣀⣰⠮⣦⣤⣶⠓⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣲⠶⠶⠶⠿⣴⣖⡶⠶⠾⣿⣿⣖⣾⠽⠏⠀⠀⠀⠈⠉⣧⠀⠘⢻⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣀⣸⣿⣭⣇⣀⣀⣈⣉⣛⣳⣦⣤⢴⡖⠁⠀⠀⠹⠀⠘⢸⣿⣶⠀⢀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠛⣉⣥⠴⠒⠒⠒⠀⢀⡀⠀⡟⠉⣼⣟⠁⠀⠀⠀⠀⠄⣿⢉⡼⠶⠿⠯⢤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠒⠉⠚⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⡇⣧⠄⠀⢀⡰⢠⢀⡟⠁⠀⠀⠀⠀⠉⠺⡂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡿⠀⠂⠒⠁⠀⠀⠀⠀⠠⢴⣏⡑⠶⠶⠿⠿⠿⠿⣬⡽⢃⡾⠀⠀⢀⡴⠾⠶⣶⢶⡛
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣶⣶⣶⣶⣒⣒⣶⣞⠻⡟⠉⠉⠉⠉⡀⠀⠀⢀⣠⡾⠷⣤⠴⢋⡶⡴⣦⢀⠈⣇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣜⠯⣟⠛⠿⢋⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢻⡀⢆⣓⢛⣫⠞⢠⡏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣓⣦⣤⣤⣴⣛⣁⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣻⣦⣭⣭⣥⣞⣋⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡄
⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣴⢦⣄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿⢸⣿⡇⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣿⡏⣼⣿⡿⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⣿⡇⣿⣿⠇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠙⠋⠁⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠿

⠀⠀⠀⠀⠀⠀⢀⡤⢖⡚⠩⠭⠭⣍⡛⣒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢄⡀⢀
⠀⠀⠀⠀⣠⢞⢩⢾⡃⠠⠔⠥⠀⠀⠀⠉⠁⠨⡑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣙⠢⣑⢄⠀⢀⠀⠀⠀⣀⣀⣀⣀⠀⢀⣀⡀
⠀⠀⠀⣰⢣⡏⠀⠦⠾⠚⣆⢣⡀⠀⠀⠠⠀⠀⣸⣮⢳⡀⠀⠀⠀⠀⠀⠉⠑⠒⠚⢛⣾⣿⡿⠁⠀⠀⠀⠈⢿⠻⠣⠎⠈⠁⠈
⠀⠀⢠⠇⣿⠁⠀⠀⠀⣠⣿⡘⣿⠶⣤⣤⡤⠚⠿⠙⣇⣇⠀⠀⠀⠀⠀⠀⠀⢀⡞⣼⣿⣿⡁⣀⠀⣀⡄⢀⡸⠟⠒⢄⠀⠀⡠⢰⢦
⠀⠀⢸⢰⡇⣀⠀⢰⣯⣨⡼⡈⣷⠠⣿⣿⠀⠀⠀⠀⠨⣹⠀⠀⠀⠀⠀⠀⠀⠀⢧⣛⠿⢿⣿⣿⣾⣿⣿⣟⠁⠀⠀⢸⣷⡾⢷⣫⡞
⠀⠀⠸⡄⣿⣿⣤⣾⣻⣟⠒⠽⣯⣼⣿⠇⠀⠀⠀⠀⡖⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣿⣭⣭⣭⣭⣭⣤⣤⣶⣶⣾⣿⡿⠁
⠀⠀⠀⢳⡸⣿⣿⣿⢷⣽⣻⣟⣿⣿⣿⡤⡄⠀⢠⡔⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠙⢌⡻⣯⡺⣿⣿⣿⣿⣿⣿⣿⣧⠴⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⡿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠢⢭⣛⣛⠛⠛⣛⣛⣫⠵⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⠶⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⡀⡀⠀⠀⢠⡀⢀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣷⣿⣯⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢒⣠⡾⠏⠣⢄⣀⣀⡔⠁⠀⠈⢙
⠀⠀⠀⢀⣴⣶⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣯⠉⠀⠞⠻⣶⣧⣄⠀⠀⠀⢠⡑⠒⠂⠈⣀⠀⠀⣀⡜⠏⠋⠀⠀⠀⢀⣾⡳⡄
⠀⠀⠐⢜⡿⣾⡏⠈⢽⣷⣿⣿⠉⠉⠋⠙⠉⠉⠆⠀⠀⠀⣼⡿⣫⠆⠀⠀⠀⠀⠀⠀⠀⠈⠹⣋⡁⠀⢀⡠⠦⠴⡿⠟⣋⠴⠁
⠀⠀⠀⠀⠉⠒⠯⠤⣜⣛⣿⡿⠤⠤⠤⠤⠤⣄⣘⣢⡤⠭⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⣀⠀⠉⠉⠀⠈⠛⠛⠉⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣤⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣴⠃⠐⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠝⢱⠀⠼⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠍⠱⠀⠼⢉⡀
⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡀⠀⠀⠀⠀⡴⠋⡏⡩⠦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡴⠫⡎
⠙⢣⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠆⠅⠜⢃⡶⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄
⠀⠀⠀⠀⠀⠐⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠰⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠰⠤⠒
⡄⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠖⡼⣿⡆⢊⠀⠀⠀⠌⠄⢉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣬⣖⣾⣻⡆⠊⠀⠀⠀⠌⠆⢙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠀⠀⠀⠀⠶⢋⠘⠃⢦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡤⠀⠀⠀⠀⠶⠛⠘⠇⢦⡴⠂
⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠨⣁⠀⡤⠈⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣀⠂⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠈⠀⠀⡠⠦⡴⠐⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡁⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⢀⡠⠢⡤⠀⠀⠀⠀⠆
⠀⠀⠈⠛⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃
⠀⠀⠀⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⢀⠉⡰⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡰⠉⠆
⠀⠀⠀⠀⡶⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠃⠠⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠜⠹⠀⠼⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⠀⠍⠹⠀⠨⢡⡀
⠦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠐⠀⠀⠀⣴⠁⡏⠩⡦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡰⠉⡏
⠙⡃⡶⠀⠀⠔⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠔⠁⠜⠃⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠐⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠐⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠒
⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆
⣗⠺⣻⡆⢂⠀⠀⠀⠌⠂⢉⠿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡷⣊⣖⢾⣹⡖⠊⠀⠀⠀⠈⠂⢉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⠀⠀⠀⠀⠖⢋⠘⠃⢦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡴⠀⠀⠀⠀⠷⢛⠘⠃⢢⠴
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠈⣁⠐⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣁⠂⠤⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡰⠦⠶⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡤⠲⡴⠐⠀⠀⠀⠂
⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠈⠀⠉⠛⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃⠀⠀⢀
⠀⠀⠀⠀⢀⠀⡠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠅⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠈⠉⠰⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠉⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠃⠀⠊⠀⠀⠈⠀⠐⠀⠓⠀⠀⠀⠂⠐⠀⠀⠀⠐⠈⠒⠒⠀⠁⠁⠘⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠁⠀⠒⠉⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠚⠉⡄⠀⠀⠀⠠⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣐⠦⠤⠤⠀⠀⠀⠀⠀⠀⠀⢈⠒⠂⠁⠀⠭⠤⡄
⠀⠀⠀⠀⠀⠰⠒⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⢀⢄⠀⠎
⠀⠀⢠⠀⠀⠃⠤⢀⡀⠀⠈⠁⠀⠐⠠⠄⢀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠒⠂⡀⠀⠴⠆⡀⠔⠓⠒⠘⡗
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠐⠂⠤⣀⣀⡀⠀⠀⠀⠐⠀⢴⠀⠀⠇⠼⠓⠀⠈⠁⠀⠉⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡏⠁⠀⠈⠁⠁⠉⠉⠀⠀⠈⠉⠀⠀⠂⠠⠀⣼⠀⠀⡇⠀⣀⠀⠀⠀⠀⠀⡠⠤⡸⠷
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡈⠄⠰⣇⠘⠟⢃⠀⠀⠀⠊⠀⢀⡽⠐⠘⠀⠈⡀
⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⠀⠃⠀⠀⠀⣬⠁⠀⠀⡇⠈⠉⠑⠀⡴⠀⠀⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⠀⠀⢠⠀⠀⢀⠀⠀⡀⠀⢸⠁⠀⠀⠀⠀⠸⣷⣾⣤⣤⣤⣴⣶⣿⠇⢀⠌⠀⠠⠔⠢
⠀⠀⠀⠀⠀⡇⠀⠀⠇⣸⠂⡇⠀⢀⠏⠀⢠⠇⠀⡄⢀⠀⢸⠀⠐⡀⠀⠀⠀⠙⢿⣿⡛⠛⠛⡿⢋⠀⢂⠔⠉⠢⡘⡰
⠀⠀⠀⠀⠀⡇⠀⢀⠉⠀⠀⣧⠀⠊⠀⠀⠋⢀⢀⠄⠘⡀⢸⠀⠀⠈⠑⠠⠔⠒⠠⢈⣉⣡⡤⠒⠅⠊⠀⠀⢀⠠⠃
⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠁⠈⠀⠁⠀⠀⠀⠈⠀⠈⠀⢸⠀⠀⠀⠀⠀⢆⠀⠀⠀⢀⣂⣀⣀⣀⣤⠴⠊⢀⡀⠔⠂
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠈⠀⠑⠢⠀⠀⠀⠉⠉⠀⠀⢀⡳⡿⠁⠀⣀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠢⡄⠀⡀⠀⠠⠛⠳⢦⡉⠁⠀⠀⠐⡀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⡯⠿⠉⠀⠀⠀⠉⠒⠀⠄⠀⠀⡀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⠀⠀⠊⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠰⠀⠀⡀⠀⠀⢀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠖⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡄⣈⠀⠀⣀⠄⢠⠁⠀⠀⠁⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠛⠛⠠⠂⠀⠠⠀⡀⠈⠁⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡌⠀⠢⠤⠀⢀⣀⣀⣀⣀⣀⣐⣁⠠⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⣿⣏⣡⣞⣉⣤⠟⣁⡴⢋⣭⠟⣩⠟⢋⡽⢛⣩⠟⣫⡼⣋⣼⡇⠐⠀⠀⠀⠀⠀⠀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠉⢹⡟⠛⡟⠛⢻⠛⢻⡟⠛⣿⠛⢻⠛⠛⡟⢛⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⣿⣧⠀⠀⠀⣿⣿⣿⣗⣲⣷⣶⣾⡶⢾⣷⠶⣿⠶⣾⣖⣲⣿⣻⡇⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠀⣴⣿⣦⠘⣿⡏⠀⠀⠀⢿⣧⣭⣭⣭⣭⣉⣋⣙⣛⣛⣛⣛⣫⣭⣭⣭⣭⣽⡇⠀⠀⠀⢠⣾⣿⢃⣾⣷
⠀⠀⠀⠀⠀⠀⠀⡤⢰⣿⣿⣿⡇⠈⢃⣤⡄⠀⢸⣿⣿⡿⣿⣿⢿⣿⠿⢿⡿⠿⠿⠿⢿⡿⠿⣿⣿⠀⠀⠀⠴⠛⠉⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄⢸⣿⡇⠀⢸⣿⠇⣾⣾⢁⣷⡇⢸⣷⠆⣾⣷⢰⣿⡇⣾⣿⣿⠀⠀⠀⢤⣤⣴⣿⠿⠋⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠘⢿⣿⣿⣷⡄⢻⣇⠀⢸⣿⠀⣿⣿⢸⣏⡇⢘⣿⠀⢻⣿⠘⣻⡇⢙⣿⡿⠀⠾⢿⣷⣶⣦⣼⠿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣽⠿⢿⣿⣶⣭⠅⢸⣿⠀⣿⡇⢸⢻⡇⣿⣿⠀⣿⣿⢸⣿⡇⣹⣿⡇⠀⢶⣄⠙⢿⣿⣿⣷⡄⠙⠿⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣶⢏⣭⣶⡾⠾⠶⠀⠀⠀⠈⣿⣷⣵⣾⣶⣿⣤⣿⣿⣤⣿⣿⣬⣼⣧⣼⣿⡇⠀⢸⣿⡆⠈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⢩⣿⣿⠏⠀⠀⣠⡶⠂⠀⠀⣿⣿⠿⣧⡄⣠⡇⢸⡇⠙⠸⡏⠠⣾⡿⢿⣿⠁⠀⣿⣿⣧⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠏⠀⠀⣼⣿⡇⠀⠐⠈⣿⣿⣶⣿⡇⣸⡇⢸⢁⣤⡄⣧⠰⢾⣷⣿⣿⠀⠀⠿⠿⠃⠀⠀⢻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠿⡿⠃⠀⠀⠀⣿⡛⠛⠿⠷⠿⢿⣿⣾⣿⣷⣿⣶⠾⠿⢿⡿⠀⠀⣀⣀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣻⢿⣷⣶⡶⢶⣶⣶⣶⠶⣶⢶⣶⣾⡇⠀⠀⠿⣿⡗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢩⣟⣜⣏⢷⢞⣟⡹⡜⢶⢣⣏⣻⣽⡇⠀⣾⣷⠀⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⡏⠹⣟⣦⣾⡟⢻⡿⣤⣿⠟⣿⣿⡁⠀⠿⠏⣀⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠋⠉⣠⣶⣦⢰⣿⡇⠀⠀⠦⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡙⠟⠋⠘⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠄⠄⠀⡐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠰⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠉⠙⠓⠒⠒⠀⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠛⠉⠙⠒⠲⢤⡀⠀⠀⠀⢀⣸⠤⠔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠖⠋⠁⠀⠠⠖⠡⠟⡷⣄⡀
⠀⠀⠀⠀⠀⠐⡒⡾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠰⡄⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⡈⠹⠿⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⢟⣹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣉⣲⣄⣀⢤⠄
⠀⠀⠀⠀⠀⠀⢁⣮⠆⢽⡍⢿⣖⢶⠶⢶⠒⣒⣚⣹⠉⠞⣳⡽⠉⠈⠢⣀⠀⣀⣤⣴⡶⢛⣟⢛⡛⣉⣁⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠸⢛⠦⢄⣀⠀⠈⠉⠙⡓⠀⣉⠀⠘⠋⡥⢎⡟⢠⢀⠀⠀⣽⠉⢤⡏⠀⡎⢻⠛⢯⣄⡈⢙⣭⠗
⠀⠀⠀⠀⠀⠀⠘⠦⢀⣠⡌⠛⠁⠤⢄⢠⣈⣁⡀⠀⡀⣠⠎⠁⠣⣌⠄⢸⣿⡄⢸⣬⣖⡽⠄⣇⢀⠦⠽⢾⡜
⠀⠀⠀⠀⠀⠀⠐⢄⢀⠈⠉⠁⠈⠀⠈⠁⠉⠉⠉⠁⠈⢀⠄⠁⠀⠈⠢⣥⠙⣧⠈⣟⠬⣉⣁⡟⢴⣟⠓⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠀⠐⠒⠒⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠙⡓⡁⢶⠨⠇⠈⠂⠭⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠱⠘⠏⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠐⠂⠧⠤⠚⠉⠀⢀⠤⡄⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉
⠀⠠⠔⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠶⠚⠈⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡄⠠⠔⠀⠀⢀⣀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠤⠄
⠀⠀⠀⠀⠀⠐⠋⢁⣀⡤⠄⠂⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣀⠀⢀⠀⢂
⠀⠘⠀⠑⠀⠠⠀⢒⠉⠀⠀⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠀⠆⠃⠠⠀⠀⠀⢸⠀⠀⠓⠂⠉⠂⠉⢀
⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠁⠊⠀⢸⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡀⠀⡀⢀⠀⠀⠀⢈⡤⠤⠀⠀⠀⠀⢀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠤⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢀⢠⠡⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠐⠈⠘⠒⠘⠻
⠀⠀⠀⡀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠄
⠀⠀⡰⠤⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⣀
⠀⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠀⢀⡀⠈⠐⠀⠀⢰⠀⠀⠀⠀⠀⢀⡴⠒⢢⣄⠀⠀⠨⠀⠀⠀⠀⠀⣀⡀⡀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠃⣈⠁⠉⣈⠌⠀⠸⠀⠀⠀⡐⠢⠄⠭⠭⠬⠥⠔⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⢀⠤⠐⠒⠀⡒⠤⢄⡀
⠀⠀⠀⠠⢀⠀⠄⠀⠀⠀⠀⠀⠀⢃⠐⡠⠰⠄⠒⠐⡠⢀⡇⠀⠀⠀⠀⠘⠢⢃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⡈⢀⠀⣄⠄⢰
⠀⢀⠀⠀⠘⠑⠀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠴⠄⠀⠀⢀⠞
⠀⠀⡀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠉⠑⠒⠒⠒⠉⠁⣠
⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠀⠀⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠓⢦⡀⠀⣠⠶⠚⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣶⠃⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣄⣀⠀⢀⣿⠀⢀⣀⣠⡞⠁⠀⢀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡟⠉⠉⠉⠉⠉⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠘⢛⣛⣛⣛⣛⣛⣻⢛⣛⢛⣛⣛⣛⣛⣛⣛⡛⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣷⣶⠶⠶⣶⣶⣶⣶⣶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢹⣿⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣤⣬⣭⣭⣭⡍⣉⣭⣌⡉⢬⣭⣭⣭⣭⣭⣭⣥⣤⣤⣿⣿⠟⣋⣭⣍⡛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢰⣿⣿⣿⣿⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣼⣿⣿⣿⣿⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣶⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠟⠋⢉⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣎⠻⣷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣟⠋⢍⢢⡈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠⢶⣾⣿⣿⣿⣿⡿⠟⠣⡺⣝⢿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠈⣇⢀⣉⣁⠀⠀⠀⠀⠀⠸⣿⣦⣴⣾⣿⣿⡿⣿⣿⣿⢟⣩⣀⠀⠀⣽⣝⢷⣝⠻⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⢳⣀⣀⣤⣶⣶⢳⣟⣿⣿⣿⠙⠿⠋⣿⣿⣯⣿⣿⣠⣾⣿⣿⣦⠻⣷⣮⡻
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⠀⠈⠋⢻⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⣻⣭⣽⣿⣿⣟⢿⣿⣿⣿⣿⡀⠀⠀⣸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣜⢿⣿⣿⣿⣿⠿⣻⣿⠿⣛⣛⣛⣛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣶⣶⣶⣿⣿⡇⣾⣿⣿⣿⣿⣿⡧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣶⣭⣭⣭⣭⣵⣾⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡩⢣⠀⣤⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠓⣀⢦⠠⠤⡏⠇⠰⠰⡅⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠐⢆⡰⢌⡄⠹⡤⠿⠲⠁⣎⡹⠆⠡⠀⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣄⢀⡸⠒⠎⠘⡀⡠⠞⢢⠀⠨⠁⠈⡀⢂⠈⠎⠖⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⢹⠇⣤⠋⢄⠀⠑⢻⠲⠉⢇⠀⠁⠡⠐⠀⣃⣸⠞⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⣶⠀⠀⠀⠀⠀⣀⡇⠀⠈⢆⠈⡆⠀⠈⠓⠋⢇⢃⠘⡆⠀⠀⣣⡶⠟⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣰⠃⠀⠀⠀⠀⢸⡗⠶⠦⢤⣬⣤⣽⣄⣦⣤⣥⣸⣬⣤⡴⣶⠞⠉⠀⠀⠀⠀⠀⡇
⠀⠀⣠⡤⠔⠒⠂⠀⣈⡧⡤⠓⠒⠤⢤⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣠⣤⣤⣤⣤⣧⣤⣀⣀
⠀⣼⣏⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⣀⢟⡆⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⠉⠁⢀⠀⠀⠉⠉⠉⠛⠻⢷⣦⣤⡀
⠀⡷⠬⣍⣛⣒⠚⠛⠒⠒⠒⣒⣚⡉⠥⠚⡇⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⢉⡁⡄⠀⠀⠠⠀⠂⠀⠉⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄
⠀⡏⠀⠀⠈⠉⠉⠉⠉⠉⠁⠁⠀⠀⠈⠀⡇⠀⠸⡆⠀⠀⣀⣴⠾⠋⠁⡠⢰⠄⠉⠀⠂⢂⣠⠤⠖⠒⡈⠁⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠈⢿⣷
⢠⡇⠀⠀⠀⠀⠀⠠⠀⠀⠄⢀⡀⠀⠀⢀⡇⠀⠀⣇⣠⡾⠋⠁⠀⠀⢀⣈⣀⣤⠤⣤⠴⠿⠤⠤⣤⣉⣀⠀⢴⡴⠀⠉⠉⠠⡠⠆⠀⠀⠀⠈⣿
⢸⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⢀⣴⠟⠁⠀⢀⠀⠐⠊⠉⠁⣠⠔⠋⠀⠀⡄⠀⣀⠀⠈⠉⠛⠲⣄⠀⣆⠜⠀⠀⠀⠀⠀⠀⣸⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡀⢸⣷⠟⠁⠀⠀⠺⠈⢀⡠⠂⣤⠞⠁⠀⢀⡀⠉⠀⠠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣼⡿⠃
⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠄⡆⠀⣠⠞⠁⠐⠁⠐⠒⠁⢠⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⢻⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣰⡟⠀⠀⠀⠀⠀⠀⠋⠀⡴⠃⠀⠀⠸⠥⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⢀⣤⡞
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⣀⣤⠞⠋⣠⡇
⢸⡇⠀⠀⠁⠀⠀⠀⠀⡜⠀⠀⢨⣿⣦⣄⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠛⠉⣀⣤⠶⠛⠉⢀⣤⠞⠉⡷⢤⡀
⣿⡇⠀⠀⠀⠀⠀⢠⠎⢀⡀⠔⠉⠳⣍⡛⠿⢿⣶⣶⣴⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡿⠿⠟⠛⢉⣁⣤⠴⠞⠋⠉⠀⣀⣤⠾⠋⠁⣀⡼⠃⠀⣿
⣿⠇⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠸⣇⠙⠓⠦⢤⣈⣉⡉⠉⠉⠉⠉⠉⢉⣉⣁⣠⣤⠴⠖⠚⠋⠁⠀⢀⣠⣤⠶⠛⠉⠀⣀⣴⠞⠋⠀⠀⢠⣿
⣿⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⢦⣄⣀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⢀⣀⣤⠶⠛⠉⠀⠀⠀⢀⣴⡿⠁
⣿⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢦⣌⠉⠙⠛⠓⠒⠲⠶⠶⠶⠖⠒⠛⠛⠋⠉⠁⢀⣀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⢀⣤⡿⠋
⣿⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠓⠲⠶⠤⢤⣤⣤⣤⣤⠤⠴⠶⠶⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁
⠘⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⢛⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠛⠁
⠀⢿⣝⡛⠻⠷⠶⠿⠿⠟⢛⣋⣩⡴⠖⠛⠁⠀⠉⠛⠷⠶⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠟⠛⠋⠁
⣤⣤⣬⣭⣿⣿⣷⣾⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣭⣏⣼⣉⣯⣩⣿⣹⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠀⠀⠀⠉⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣞⢥⣼⣆⠀⠀⠀⠀⠀⢹⡆⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠈⠉⠒⠲⠦⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠉⠰⣄⡈⠓⠚⠉⠀⠀⠀⠀⢀⣠⠷⠚⠉⠀⠀⣀⠤⠒⠒⢀⣉⠭⠛⠋⠂⠀⣀⠀⠀⠀⠈⠉⠙⣒⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢥⣄⣀⠀⠀⠈⠙⠓⠶⣶⣶⠶⡷⣂⠉⠀⠀⠀⠀⠀⠸⢧⣀⡀⠈⠛⠛⠛⠛⢛⣛⣻⡿⠟⠃⠀⣠⠴⠛⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⡷⠒⠛⢻⣯⠓⢀⡇⠀⠉⠒⠢⠤⣄⣀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣠⠖⠋⠁⠀⣀⡤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⠐⠧⠀⠈⠉⠀⠀⡀⠙⠲⢤⣄⣀⠀⠈⠉⠓⠒⠦⢤⣄⣀⡀⠀⠀⡠⠒⠉⠀⠀⣠⠔⠋⠁⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡇⠀⠀⠈⣇⢰⣈⡷⠄⠀⠀⡘⠃⠤⣀⠀⠀⠉⠙⠳⠦⣤⣀⠀⠀⠀⠈⠉⠙⠁⠀⠀⡠⠔⠋⠀⢀⣠⠶⠋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠤⠾⣅⠀⠀⢀⣼⢦⣤⣤⣤⡤⠤⡇⠀⠀⠀⠉⠓⠦⣤⣀⠀⠀⠈⠙⠓⠶⢤⣄⣸⠀⠐⠈⠀⠀⣠⠴⠋⠁⣠⡴⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠞⠁⠀⠀⠀⠨⠟⢘⡉⠉⢧⠀⠀⣠⠴⠚⠓⠓⠢⣄⣀⠀⠀⠀⠈⠙⠒⠦⣄⣀⠀⠀⠀⡉⠀⠀⣠⠔⠋⠁⣀⡴⠞⢉⡀⢀⡇
⠀⠀⠀⠀⠀⠀⢀⣤⡾⡵⠁⠀⠀⠀⠀⠀⡾⠀⠈⠃⠀⠢⡾⠷⣤⣀⣀⡤⣄⡀⠀⠉⠙⠲⠤⣀⠀⠀⠀⠀⠉⠙⠲⢤⣇⠐⠉⠀⢀⡤⠞⢉⠐⡹⢩⡟⡏
⠀⠀⠀⠀⢀⡴⠋⡾⢰⡇⠀⣀⣀⡀⢀⠴⠃⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀⠀⠀⠉⠒⠦⣄⡀⠀⠀⠀⠉⠀⣠⠖⠉⠀⡞⢰⢀⡇⢸⠀⡇
⠀⠀⠀⢀⠎⠀⠀⢷⡀⠙⠚⣩⠈⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⡀⠀⠀⠉⠑⠢⠧⠐⠉⠀⠀⢀⣼⠃⡿⢸⠃⣼⣴⡏
⠀⠀⠀⡎⠀⠀⠀⠈⠛⠛⢛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠉⠑⠢⢄⡀⠀⢠⠀⢀⡤⠾⠛⠁⣠⡷⠞⠛⠉
⠀⠀⠘⣧⣀⣀⣀⣀⣀⠄⠀⢀⣀⡿⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑⢚⣑⡅⠰⠦⣤⡴⠋
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠓⠒⠋⠉⣷⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠛⢦⣀⠀⠀⠀⠀⣀⣀⡴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠈⠛⠶⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠴⠶⠛⠛⠛⠛⠳⠶⠶⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣵⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⢠⢎⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⣿⠉⠉⢹
⠀⠀⠀⠀⠀⠀⡠⣣⣿⠟⠛⠛⢛⣿⣿⣿⣿⣿⡇⠀⠀⢀⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣄⠀⠀⣿⠀⠀⢸
⠀⠀⠀⢀⡤⠪⢲⣔⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⢈⡷⣾⣍⡩⢭⢍⣉⣩⣛⢯⣥⣷⡻⠀⠀⣿⠀⠀⣼
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠙⠭⣯⣭⣚⣻⣟⣴⣛⣻⡥⠏⠀⠀⠀⣿⠀⠀⢹
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⢸⠉⢹⠀⠀⢀⠔⣊⣭⣭⣑⠢⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣊⣭⣭⣑⠢⡀⠈⢹⠉⢹
⠀⠀⢸⣶⣾⠀⠀⡎⣼⡿⢛⣛⢿⣧⢱⣖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠶⠒⡎⣾⡿⣛⡛⢿⣧⠹⠀⠘⠒⡺
⠀⠀⠈⠉⠉⠉⠉⠉⢿⣧⣑⣚⣼⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣧⡓⣊⣼⡿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁



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

⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⡠⣠⣄⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⠶
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢼⠒⠂⠚⠒⠇⠀⠀⠀⠀⠀⠀⠀⢠⢪⡀⡎⢂⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠖⠀⢔⠙⣀⡺⠵⡖⠂⡩⣀⣜
⠀⣠⢦⣀⡄⠄⢀⣤⢤⣸⠂⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠁⠾⠀⠀⠀⠀⠀⢀⠜⢬⡀⣂⡠⠶⡶⠋⠁⠀⡀⣀⠴⠚⣡⡏
⠴⢁⡫⠯⠉⢤⠜⠉⠡⠯⠤⠤⠄⠀⣃⠰⠀⠀⠀⠀⠀⠀⠈⠉⣃⢀⠉⠀⠀⠀⠀⠀⠘⣦⠘⠿⣯⣜⣽⠀⡴⣊⣩⢕⣟⣛⡩⠞⢁
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠒⠄⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣈⡉⠚⠙⠓⠒⠋⢉⣀⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣀⠀⠀⢀⠴⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⢿⢟⡷⠔⠁⡀⠙⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣠⣃⣢⣀
⠀⠀⠀⢈⣿⠁⠐⠀⠠⠀⡀⠈⣆⠀⠀⠀⠀⣀⠠⣤⡀⡸⣗⡾⠤⠤⡼⢋⠱⡄⢀⢤⠄⠀⠀⠀⠀⠀⢰⠃⠀⠄⠀⠀⣣
⠀⠀⠀⠘⡯⠭⠭⣁⣛⣋⢉⡭⢽⠃⠀⠀⠀⢡⣄⣶⣞⣭⡏⠀⠀⣀⢿⡏⣟⢓⣷⣇⡌⠀⠀⠀⠀⠀⠸⣒⣒⣒⣒⣒⡿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⣿⣖⣚⡱⢋⣻⣮⠴⣟⡉⢉⣱⡶⢉⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢻⡓⠮⣍⣉⣙⣢⠀⠀⢈⣿⣛⣥⠶⡻⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀
⠀⠀⠀⠀⠈⢳⡦⠤⠤⠤⢶⡞⠀⠀⠀⠀⠀⠀⠙⠦⣤⣈⣉⡉⢦⣤⠞⣉⣡⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣶⣤⠇
⠀⠀⠀⡠⢖⠚⠛⠛⢶⢶⡗⢀⢟⡶⢲⡄⠀⠀⠀⠀⠀⠛⠦⣄⣀⡀
⠀⢀⠎⠂⠸⢄⣤⡶⠛⠋⡰⠿⠋⡞⠋⠁⠀⠀⠀⠀⠀⢠⠖⠋⠁⠝⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠾⢤⣀
⠀⣼⢠⣼⢷⢿⠏⢀⡬⠁⠛⡁⠀⣻⡄⠀⠀⠀⠀⠀⠀⣸⣤⣄⣀⣀⣀⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠈⠣
⠀⣿⣳⢼⣵⣛⢦⣂⣙⣒⣋⣤⣴⣿⡇⠀⠀⠀⠀⠀⢸⡀⡇⠀⠁⠀⠀⡇⣹⠀⠀⠀⠀⠀⢀⠤⠤⠤⡬⠤⠤⠤⠤⠤⢬⠄⠠⢄
⠀⢻⣌⠛⠻⠿⠿⠿⠿⠿⠟⠛⠋⣸⠃⠀⠀⠀⠀⠀⠘⣆⠹⢤⣀⣠⠴⢁⡟⠀⠀⠀⠰⠖⣚⠐⣦⡀⡈⠀⠀⠀⠀⠀⠁⠄⠀⠀⡒
⠀⠀⠻⣝⡒⠦⠤⠄⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⠞⢶⠒⠉⠀⠀⠀⠀⠀⠀⢸⡘⠿⠻⠿⠃⠀⠀⠀⠀⠀⠆⠀⣠

⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⡀⠀⠀⠀⡀⠀⣐⣲⣶⡒⠠⢀⠀⠀⠀⢤⠤⠦⠴⠲⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⢄
⠀⠀⠀⠀⠀⠀⠈⠴⠀⠂⠈⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠡⠆⠐⠈⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣶⣍⣩⣥⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖
⠀⠀⠀⣠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣄⠰⠂⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢀⣴⠋
⠀⠀⡴⠟⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢰⣽⣿⣷⣶⡋⠀⠀⢠⡏⠀⠀⠀⠀⣠⠾⠁⠀⢀⣀⢀⠄
⢀⡾⠃⠀⢀⣔⡑⢂⣠⣴⠀⠀⠀⠀⠐⢤⣤⣤⣾⣧⠀⠀⠀⠀⠀⠙⠻⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⣀⡌⠐⣀⣴⡆⠀⠀⠀⠀⠦⣤
⠀⠁⠀⠀⠀⣿⣿⡏⢩⣿⠀⣀⠀⠀⠀⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⠉⣽⠇⢀⡀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠈⣷
⠀⠀⠀⠀⢶⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠶⢬⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠸⢦⣿⠀⠀⠂⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠈⢷
⠀⢤⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠛⠀⠀⠀⠀⡀⣠⠀⠀⢸⡗⢦⣬⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠘⠃
⢠⣶⡶⣃⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢉⠠⠀⠘⢧⣸⣿⣋⡔⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⣞⡀⠀⠀⠀⠘⠃
⠸⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠉⠉⣷⢤⣇⣴⣴⠀⠀⠀⠀⠻⢿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⣻⣿⣿
⠀⠀⡀⠀⠠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⢄⡃⣀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴
⠀⠒⢠⠶⠦⠶⡎⠀⠴⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠂⠂⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠂⠴⠦⠰⢶⠁⠠⠄⢠
⠐⠻⠇⢀⣛⣁⣼⣞⣲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠌⠖⠀⠀⠀⠀⢰⣄⡈⣋⡀⠿⠚⠿⠀⣈⣁⣤⣷⣓⣾⡿
⢀⣉⣤⢾⣿⣷⡟⠛⠛⠉⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣨⣿⢦⣀⣈⣡⡴⣿⣿⣾⠛⠛⠋⠁⠀⠀⠀⠀⠀⡀
⢸⣿⣿⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣶⣶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢼⣿⣿⣿⡷⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠈⠉⠀⠀⠀⠀⠀⢰⣤⠄⡀⠀⠀⠀⡀⠈⠻⢿⠿⠟⠥⣀⠀⠀⠀⢤⠤⠤⠴⠠⣤⡄⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⠠⠙
⠀⠀⠀⠀⠀⠀⠈⠰⠁⠂⠈⠀⠀⠐⣷⣿⣿⣿⣿⣿⣿⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠁⠆⠰⠈⠀⠀⠀⢾⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⣭⣉⣩⣭⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡧
⠀⠀⠀⢠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⡄⠰⠆⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣤⠏
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣴⣶⣴⣶⣴⣾⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢨⡿⠀⣰⡿⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣾⠃⢠⡟⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢇⣼⠇⢠⡿⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣼⡏⢠⡿⠃⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢰⡟⠀⣾⠗⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⢀⠠⠄⠒⠂⣈⣉⠉⠥⠀⢀⡿⢠⡿⠀⣼⠏⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠰⠁⠀⠀⠞⠁⠀⠀⠀⠀⠀⣾⠃⣾⠁⣰⡟⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢢⡀⠀⠀⠀⠈⠲⠄⣀⠙⠲⢄⠀⠀⠀⣸⠏⣼⠇⢠⡟⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣿⡙⠫⣝⢷⣦⣀⣀⣩⣭⣭⣴⣦⣤⣤⣦⣤⣶⣤⣤⣰⡟⣰⡏⢀⡿⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠉⢡⣤⡙⢿⣍⣉⣳⣈⠻⠿⣍⡠⠤⣄⠠⡤⣘⠀⠡⣄⠀⢈⣿⢡⡿⠻⣿⢁⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣀⡘⠓⢽⡿⢧⣌⠁⢞⠈⢷⠀⠈⠀⠀⠈⢄⡀⠀⠀⣸⠃⡼⠁⡼⢁⣾⠋⠙⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠈⣹⡿⠿⠷⣶⣯⣉⡑⣭⡀⢸⡶⢄⠀⠨⠀⢀⠁⢰⣶⣧⣼⡧⣾⠷⢾⣷⣶⣤⣄⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣠⣾⠋⠁⠀⢒⡂⠀⠍⠉⠻⠿⠛⣷⣶⡏⡐⠀⠈⣀⠜⡋⠀⠌⠀⡤⠀⣤⣞⣉⣍⢙⠻⣷⣌⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣠⣾⡯⠄⠀⢾⡶⣤⣥⣤⣤⣄⣘⠆⠈⠠⣼⣦⣑⣤⠞⡡⠀⡠⢀⠔⡠⠄⠀⢿⣻⡿⢻⣿⡷⠾⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣿⡟⢻⠿⣯⣽⡇⢀⠀⠈⠉⠙⣯⡀⢢⡒⢡⣾⡿⡡⢊⡤⢊⣤⡪⢶⣤⣠⣴⣼⣿⠟⠻⣿⠀⣄⣞⡷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠠⢧⣎⢘⣷⠐⠚⢇⡀⠀⠈⢻⡅⣰⡿⢋⠜⡰⣡⢜⡠⢋⠔⢉⡿⡁⠠⠁⣬⠆⢘⣿⣷⣎⣁⣠⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠉⠛⠷⢶⣾⣦⣿⣄⣐⣂⢻⣀⡐⣾⡷⣛⢖⡡⣚⢚⡵⣿⣷⣥⣶⣽⣇⣵⣾⣯⣷⣶⠿⠿⠟⠛⠋⠉⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠳⢶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠾⠋⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠟⠛⠛⠛⠋⠉⠉⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠶⠶⢶⣶⣶⣶⣶⣦⣤⣶⣶⣶⣶⠶⠶⠿⠛⠋⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣤⣤⣀
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣟⡀⠀⠀⠀⠀⠀⠀⣴⣟⣼⣃⣹⡔⣷⣸⣷⣄⠀⠀⠀⠠⠀⢄
⠀⠀⠀⢀⣠⣙⠿⣋⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣋⡿⠀⠀⠀⠀⠀⣼⣿⣿⡿⠋⠉⠛⢻⣿⣿⣿⡆⠀⠀⠇⣠⣦⣦⡤⢤⡄
⠀⠀⢠⡟⣻⣿⣿⣿⣟⢻⣄⠀⠀⠀⠀⠀⠀⢰⠋⠉⢳⡍⠁⠀⠀⠀⠀⠀⣿⣿⣿⣇⣀⣀⣀⡀⢉⣈⢹⡇⠀⠀⠀⣿⣖⣞⣰⣿⣧⣤⣤⣤⣀
⠀⢰⣿⣧⣇⠀⠛⠀⣸⣼⣿⡆⠀⠀⠀⠀⠀⠘⣷⣤⡾⠁⠀⠀⠀⠀⠀⠀⠸⣿⢝⣻⣷⣿⢻⣽⣛⣟⡿⠁⠀⠀⠐⠛⣟⡟⠛⠛⠻⠻⣛⠿⠟
⠀⠀⠈⠙⣿⣗⣀⣸⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣄
⠀⣶⣶⣶⣶⣶⣶⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢀
⠀⣿⣿⣿⣿⣿⢿⡿⣧⣤⣤⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢰⣾⣿⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣬⣀⠊⠃
⠀⣿⣿⣿⣿⣷⣴⣴⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡟⠀⠛⠉⠉⠉⠛⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠉⠉
⠀⠛⠮⠏⠛⠋⠫⠟⠛⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⣠⣴⠿⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠙⠁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣻⣿⣿⣿⣿⣶⣶⣶⣦⡤⢄⣀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⢴⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢴⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠙⢎⠛
⠀⠀⠀⠀⠀⣍⠀⢨⠄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⣶⣄⠢⠦⠰⣶⣄
⠀⠀⠀⢀⡄⣮⣷⣽⢤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢽⣿⢫⠽⣝⢿⣿⣿⣿⣿⣿⣿⢟⡽⢭⢻⣿⣿⡇⠀⠀⠹⣿⣷⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⢸⡟⡛⣿⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⠟⠀⠀⠀⠀⠀⠀⠈⠸⢦⠾⠉⠉⠉⠀⠀⠀⠀⠭⠭⠍⣽⣿⡿⠉⠉⠉⠁
⠀⠀⠀⢸⡇⡀⢰⣾⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠭⠍⠰⠿⠟
⠀⠀⠀⠘⠇⠀⠀⠁⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⢀⣂⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⣤⡤⠤⢤
⠀⠀⣤⣤⣤⡀⠀⠀⣠⡄⣿⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⣿⣯⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⡶⠀⠐⠿⠁⢀⡏
⠀⠸⠏⠀⣹⡇⠀⣴⣿⡇⠛⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣚⣿⣺⡀⠀⠀⠀⠀⠀⠀⢩⣭⠉⣩⢡⣻⣯⠇⠀⡼
⠀⠀⢀⢰⣿⣿⣿⣉⣹⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣤⣤⣤⡄⣤⣤⣤⡄⠀⣴⣿⣿⣿⣶⡦⠀⠀⠀⠀⠀⠀⢀⣀⢓⣒⣒⣒⣒⣲⣃⡀
⠀⠶⠾⠞⠻⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣿⣿⣿⣧⣿⣿⣿⣧⣾⣧⣦⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⠏⠈⠀⠸⣧⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣤⣤⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⡏⠓⠒⠯⠝⢻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠆⠀⠀⠹⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⢉⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣾⣿⣥⣤⡀⡥⠀⠀⠀⠀⠀⠀⣐⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⠞⡃⠠⡠⢹⣦⣾⣟⣽⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣻⣿⣿⣻⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣯⠙⢋⡿⠣⣮⣉⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣏⡉⠙⠁⠀⠀⠒⠦⠈⠉⠉⠉⠿⣿⣿⣿⣿⡟⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⠻⠛⠲⠶⠶⣄⣲⣀⡀⡒⠒⠺⠽⠛⠛⢋⣼⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣛⣉⣤⣶⣶⣮⠙⠉⠉⠉⠉⠉⠉⢉⣭⣶⣶⣤⣀⣾⣿⣿⠏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡿⣭⣙⡻⣿⣿⣿⣷⡄⠀⠀⢀⣴⣿⢿⡟⣛⣹⣿⣿⣿⣿⢾⣾⡏⠎⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠆⣈⡴⣋⠿⡝⢁⣙⠳⢶⡟⡇⠈⠀⢽⠋⡛⠞⢻⠁⢀⣰⡗⣵⠐⣯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⡔⢡⠗⡂⣨⣍⠳⢮⠀⣀⣚⠋⠘⠀⠀⠀⢡⣴⣿⣉⣸⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣏⡟⠳⠓⠜⠉⠉⠐⠚⠛⠿⢆⠍⣀⣎⣹⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⠤⠄⠒⠂⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡠⠖⠒⠢⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠙⠚⠀⠀⠀⠀⡷
⠀⢠⠞⠉⠉⠙⠃⠀⠀⠀⠀⠀⠠⠀⢰⠚⠁
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠉⠓⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠰⡀⠀⠠⠀⠐⠼⣀⣨⣥⣶⣧⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢡⡀⠒⣡⣶⣿⡿⠿⠿⠛⠛⠻⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢧⣾⣿⠟⠁⠀⠃⠀⠀⠀⠀⠉⠀⠃⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⣩⢤⠬⠭⠭⠭⠭⣭⢭⡉⠉⢀⡰⢲
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠮⠥⢤⣤⣉⣓⣓⣛⣓⣋⣛⣒⢋⠠⠒⢁⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⡧⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣈⣉⣇⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠡⢤⠀⠀⠀⠀⡿⣷⡖⠀⠀⢀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢀⠎⣀⣀⠴⠀⡄
⠀⠀⠀⠀⣀⣴⠋⣠⠴⢄⡀⠙⠶⠶⠶⠖⠒⠞⢯⣉⣠⠤⠤⠒⠁
⠀⠀⠀⠈⢀⣈⠉⠀⠀⠀⠈⣲⠀⠀⠀⠆⠀⠂⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢸⠟⠒⠒⠒⠒⠒⠒⡺⠛⠦⠤⣄⡀
⠀⠀⠀⠀⢠⡾⣿⠟⢣⡴⠃⢀⡤⠒⠶⠶⠒⠛⠲⠤⡄⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠘⢦⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⢀⣜⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⢟⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢠⣧⣰⣆⣾⠀⢠⣶⢲⡴⡴⣰⡆⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣟⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠒⠋⠉⠉⠉⠉⠉⠑⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⢀⡄⠠⠀⡄⡀⠀⡀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠐⡬⠀⠀⠘⠈⠂⠋⠘⠁⠊⠆⢀⠤⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⣀⡘⠖⠂⠈⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠈⠀⠀⠠⠄⠀⠀⢈⣣⣤⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣠⡀⠱⠉⠀⠀⢀⠴⠚⠉⠁⠀⠀⠀⠉⠙⠲⣄⣀⣤⣴⣶⣾⣿⣿⣿⡿⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⠀⠀⢀⡔⠁⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⣿⣿⡿⣿⡏⠹⣿⣧⠹⡇⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠰⠀⠀⣀⣠⣮⣴⣶⣾⣿⣿⣿⣿⡟⢻⣿⡍⢿⡆⢻⡇⣿⡇⣷⠹⣿⣷⡀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢀⣠⣧⣤⣴⣶⣿⣿⣿⣿⣿⢻⡏⢩⣥⣿⣿⣿⡇⣆⢻⣿⡘⡇⣄⠇⢻⣧⠛⣃⠹⣿⣷⢸⣿⣿⣿⣿⡄
⠀⠀⢰⣿⣿⡿⠿⢟⣻⣟⠉⢿⣷⢸⡇⣿⣿⠘⠟⣻⣿⣿⡇⠽⠆⢻⣷⡀⢹⡄⢸⣿⣸⣿⣷⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠸⣿⣿⣿⣷⠸⣿⣿⢸⡌⢿⡇⢀⠻⣿⡇⢻⣿⣿⣿⣇⣸⣿⣦⣿⣷⣾⣿⣿⠿⠿⠛⠛⠉⠉⠀⡜
⠀⠀⠀⢿⣿⣿⣿⡇⢿⣿⠘⣛⡘⣿⠸⣷⣌⣿⣬⣴⣾⣿⣿⡿⠿⠟⠛⠋⠉⡱⠃⠀⠀⠀⠁⠀⠀⢠⠃
⠀⠀⠀⠸⣿⣿⣿⣿⣘⣿⣬⣿⣷⣾⣿⣿⡿⡿⠛⠛⠉⠉⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠥⡀⠀⠀⢀⡎
⠀⠀⠀⠀⢻⣿⣿⣿⣿⠿⠿⠛⠋⡉⠁⠀⠀⠈⠑⠲⠤⠤⠄⠤⠴⠒⠉⠀⠀⢀⠰⡐⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠣⡀⠀⠀⠀⠀⠀⡰⠂⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣘⠢⠆⠁⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠈⠀⠚⠐⠀⠷⡀⢠⠀⠀⠀⠆⠓⠈⠁⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠉⠉

⠀⠀⠀⠀⢲⠖⠀⠀⢖⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠖⠐⢆
⠀⠀⠀⠀⣸⠀⠀⠸⣃⣼⢸⠶⣴⠓⣾⣣⠄⠀⠀⢠⢀⡆⡴⠲⣴⠀⠀⠛⠀⢑⣤⡀⣴⠃⣀⠀⠀⠀⡏⠉⠉⢹⠀⣶⣆⣄⣀⣄⣆⣤
⠀⠀⠀⠀⢀⡄⣰⣒⣄⡄⣡⠀⠀⡤⢄⠀⠀⠀⠀⣘⣺⢿⡇⡐⢿⣀⠀⠐⢠⠡⠀⠉⠀⢳⡙⠀⠀⠀⠇⠀⠀⠼⠀⠛⠉⠉⠉⠿⠉⠉
⠀⠀⠀⠀⡹⡏⠑⠘⠁⠓⠛⠀⠀⠓⠻⠀⠀⠀⠀⢈⡥⠄⠉⢡⠈⠁⢠⠊⡀⠀⠀⠨⠔⢊⡈⠂⠀⠀⡆⠀⠀⢲⠀⡄⡀⣄⢀⣀⣀
⠀⠀⠀⡇⠒⣹⡆⢹⠁⢋⢸⠁⠀⠁⠀⢈⠆⠀⠀⠸⠰⣥⣲⣴⢲⢖⠀⠢⡤⠔⠢⠤⠒⠦⡤⠂⠀⠀⡇⠀⠀⠸⠀⠛⠻⠛⠚⠛⠛⠃
⠀⠀⠀⡇⠀⡏⡇⢸⠀⢸⢸⠃⠀⠐⠶⢂⠀⠀⠀⠀⠈⠀⠀⣥⡤⢀⡴⠄⠃⢠⠀⡇⣸⠀⠇⠀⠀⠀⣆⣆⣢⣲⠀⠀⡀
⠀⠀⠀⠷⠼⠁⠇⠸⠀⠸⠸⠦⠀⠀⠀⠈⠀⠀⠀⢸⠀⢸⠀⠛⠃⣛⢃⡀⢘⢚⡚⠓⠛⠚⠀⠀⠀⠀⡏⡛⠟⣹⠀⠟⠻⠼⠿⠶⢿⠿⠳⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡤⢼⣴⢉⡧⣿⣡⠧⣾⢡⠯⢼⣼⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉
⠀⠀⠀⠀⡄⢀⡆⢀⣤⠤⣄⢠⡄⠀⣤⠀⠀⠀⠀⠘⠀⠘⠈⠁⠙⠇⠀⠀⠧⠀⠀⡴⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣧⢼⡇⣿⠁⠀⣸⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠒⠀⠐⡀⡀
⠀⠀⠀⠰⣤⡾⠁⢿⣄⡠⠃⠘⣿⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠉⠉⢹⠀⠰⠁
⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡞⢁⣤⣤⡤⣤⣤⢄⡴⠆⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠘⠒⠉⠐⠃
⠀⠀⠀⢤⡆⠀⢰⣶⣤⣾⣿⢤⡀⠀⣤⡀⠀⠀⠀⠑⠚⠛⠘⠘⠛⠛⠘⠻⠀⠀⠴⠂⠘⢀⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠁⠀⠀⠀⢀⣤
⠀⠀⠀⠤⠀⠶⠘⢿⣿⣿⠿⠚⢤⠄⢉⠀⠀⠀⠀⠀⠄⠀⠐⠽⠀⠀⠠⠀⠀⠶⠀⡀⠀⠉⠀⠀⠀⣴⢀⠄⢠⠖⣇⡯⢽⢴⡖⣶⢀⡴⡃⡀
⠀⠀⠀⢰⢓⠀⢀⡈⣙⣩⠆⢠⢆⢀⠀⠀⠀⠀⠀⡄⠁⡄⢠⠀⡄⢤⣌⠄⢀⠀⢀⠀⠒⠀⠀⠀⠀⠈⠁⠀⠈⠉⠋⠙⠉⠀⠀⢀⠉⠁⠉⠁
⠀⠀⠀⣙⣚⢙⠟⠁⢛⡟⠁⠘⠙⠛⠁⠆⠀⠀⠀⡇⢀⢃⣤⠀⡠⢄⣀⡤⣾⠓⠀⢀⡙⠀⠀⠀⠀⠇⠱⠃⠹⠂⠿⠑⠃⠃⠀⠃⠆⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢹⠙⠥⠚⠧⠻⠿⠀⠸⡤⠆⠀⠀⠀⠀⠀⠀⡄⠀⣄⡜⣶⣳⡀⠀⣄⣄⣰⣑⣤⢀⡄
⠀⠀⠀⠀⣠⠞⠋⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠓⠰⠀⠑⠊⠀⠀⡝⡋⠙⠊⠀⠉⠃
⠀⠀⠀⡼⠁⠀⢠⡇⡀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⣿⡇⠀⠀⠀⠓⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣄⠘⠀⠊⠀⠙⢀⣼⣿⣟⣧⣽⣿⣿⣿⣝⣿⣿⣇⣿⣻⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠴⠒⠛⢉⣉⣉⣉⠙⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⣒⣚⣓⡒⠦⣤⡀
⠀⠀⠀⠀⣠⠞⢁⣤⣪⡬⠶⣶⠶⡦⢯⣷⣤⠈⠳⡀⠀⠀⠀⢀⡔⣩⡴⠿⠛⢣⠹⠇⠛⠛⠲⣍⠳⣄
⠀⠀⠀⣰⠃⣽⣿⡿⠹⡇⠀⡇⠀⠇⠀⡄⠙⢷⡄⠙⡄⠀⢠⢏⣾⠧⠤⠐⢀⠀⠀⠀⠂⠘⠀⠨⠷⠙⣆
⠀⠀⠀⡇⢰⣿⢿⡵⢜⡷⠔⣤⣴⡅⣠⢡⣠⢼⢻⠀⢹⢀⡏⣾⣁⣤⠀⢂⡠⠀⠠⣄⢲⡄⢠⡠⠔⠢⢸⡀
⠀⠀⠰⡇⢸⡇⢨⠀⠸⠀⠈⠁⢸⠋⠈⠋⢹⠈⠁⠀⢸⢸⡷⣿⡄⠘⠀⢸⠀⢠⣴⡟⠈⡅⠀⡀⠠⠀⠘⡇
⠀⠀⠀⢧⠈⢧⡲⢀⣶⢀⣾⣀⡼⣀⣼⣤⣜⢰⡘⠀⣾⠸⣇⠟⠂⠀⠁⠀⠀⠻⣏⣿⠀⢀⡠⠤⠀⢷⢰⠇
⠀⠀⠀⠘⢧⠈⠿⠑⡟⠑⡗⠉⡖⠋⡙⠋⣙⠝⢁⣼⠃⠀⠹⡌⢀⠀⢀⠠⠆⠀⢟⠉⠁⡀⡀⠀⡦⢃⡞
⠀⠀⠀⠀⠈⠳⣄⡉⠓⠀⠧⠀⠧⠀⠗⠂⢀⣴⡿⠃⠀⠀⠀⠻⢦⡄⢤⣄⠀⡃⠈⠀⢈⣅⣡⢀⡵⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣤⣤⡴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⣿⣓⣐⣛⣨⠭⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣀⠤⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⣏⢀⠀⠀⠀⠀⠀⣠⣂⣒⡄⠀⠀⠀⠀⢰⣲⠒⣶⣤⢒⣆⣀
⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠜⠁⠉⠘⠒⡆⠀⠀⢸⣥⣤⣷⣸⡆⠀⠀⠀⢮⣁⡸⡌⢣⣮⠽⢾
⠀⠀⠀⠀⣯⠍⠢⣄⣢⡴⠫⢍⠇⠀⠀⠀⠀⠗⠁⠀⠈⡞⠁⠀⠀⠀⢫⠖⣮⣽⠀⠀⠀⠀⠈⢆⠉⠉⠉⠀⢀⠎
⠀⠀⠀⠀⠘⢍⡉⠙⢷⠘⣟⠏⠀⠀⠀⠀⠀⢈⠀⠀⢠⠁⠀⠀⠀⠀⠈⢟⣠⠃⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣆⠤⠼⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠰⠴⠀⠀⠚
⠀⠀⠀⠀⠀⢀⡀⠄⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⡠⠤⣤⣄⡀
⠀⠀⠀⠀⡔⣡⢒⠖⠒⣢⣼⡶⡉⢆⠀⠀⠀⡔⠁⢀⠀⠑⡀⠀⢠⡃⢛⡀⠆⢘⢙⠄⠀⢀⣀⣴⣒⠤⣀⡠⢄⣤⣀
⠀⠀⠀⢸⠸⠠⠲⠦⠮⢹⣭⣤⠷⠬⠀⠀⠀⠇⠀⠀⢀⢠⠃⠀⠀⢩⢡⠉⡏⢡⠙⠀⠀⠘⢴⠉⠙⠫⠝⠹⠛⠙⡬
⠀⠀⠀⠈⡞⢤⡀⡐⡄⢊⠔⢰⠀⠀⠀⠀⠀⠀⢆⡠⣄⠎⠀⠀⠀⠘⠄⡀⣢⢥⠃⠀⠀⠀⠸⣀⠂⠀⢀⠀⢀⡠⠃
⠀⠀⠀⠀⠈⠢⠍⢉⣋⠩⠔⠁⠀⠀⠀⠀⠀⠀⠀⣳⢬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠸⡭⢭⢊⢇
⠀⠀⠀⠀⠀⢀⡠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣆⠀⠀⠀⠀⠀⠀⢰⠸⣯⠭⡇⠀⠀⠀⠀⡤⣀⣀⠔⠢⡀
⠀⠀⠀⠀⣠⠟⡱⠶⢝⣚⣳⣄⠀⠀⠀⠀⢘⠉⠉⠉⠉⠉⣷⢲⡀⠀⢠⠜⢰⠉⠀⠈⠑⡀⠀⡜⠑⡌⠑⠄⡇⠈⢢
⠀⠀⠀⠠⣃⡎⠀⠀⠀⠀⠈⢣⡇⠀⠀⠀⡬⠄⠀⠤⠤⠾⠿⡍⠀⠀⢸⠀⠀⡀⠀⠀⠀⡇⠀⢄⡐⢄⠀⡐⠸⣀⠜
⠀⠀⠀⠀⣃⢀⣀⡀⠠⠬⠤⠤⠇⠀⠀⠀⠑⠒⠒⠒⠒⠒⠈⠁⠀⠀⠘⠤⠤⡇⣀⣀⠀⡃⠀⠀⠈⠑⠎⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁
⠀⠀⠀⠀⠀⢀⠤⠤⡒⢢⡤⣀⠀⠀⠀⠀⠀⣿⢿⠃⠀⠀⠀⠀⠀⣖⢺⣻⣗⡀
⠀⠀⠀⠀⣔⠁⠀⠀⠱⣀⣧⣈⡣⠀⠀⠀⢀⣿⣆⢇⡀⠀⠀⠀⠀⡏⠝⡿⠁⢸⠀⠀⠀⠴⠐⡶⠊⢙⠂⠙⠖⠲⠆
⠀⠀⠀⢰⡉⠢⣀⠀⡠⠜⠀⠈⠀⠀⠀⠀⠸⡤⢤⠀⡝⠀⠀⠀⠀⢎⡞⠀⠀⠸⠀⠀⢀⢶⣀⠴⠒⢂⣐⠲⠶⢦⠛⡄
⠀⠀⠀⠘⡍⠲⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠜⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀⡆⠀⠈⢛⣒⡁⠒⢉⡒⠭⠭⠯⠋
⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⣀⣀⡳⠀⠀⠀⠀⢰⣀⠤⠤⠤⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢊⣉⡡⠤⠀⠀⠀⠀⠀⠈⣉⡑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⠴⠊⠁⠀⠀⣀⡀⣀⠄⠀⠀⡀⣀⡈⠒⠤⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⠔⠁⢀⣀⡀⠀⠀⢁⠀⣩⠀⢠⠈⢀⣄⡀⠀⣤⠈⣢⡘⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶⠛⣽⠏⢷⡀⢀⡀⠀⡀⠀⠀⠀⣀⠀⢸⢏⠈⠀⠘⠂⠁
⠀⠀⠀⠀⠀⢀⠏⡰⠁⠀⠀⠀⣰⡏⠀⣼⢳⣏⣽⣼⢏⣼⢃⡞⢉⣇⣿⢋⡗⠀⠀⠀⢢⠸⡄
⠀⠀⠀⠀⠀⡞⢠⠁⠀⢾⡛⢛⣻⣦⣞⡁⠸⠞⠁⠿⠞⠷⠛⡷⣿⠸⠃⣸⠃⢲⡄⠀⠘⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣷⡂⠈⢙⣿⢤⣤⣜⢂⣃⣀⣾⠁⠀⠀⡻⣤⠼⠁⠀⠀⡁⢸
⠀⠀⠀⠀⢀⡀⡆⠐⣖⣲⡃⣿⡝⣿⣼⣿⢸⣧⣤⡍⣭⣭⣥⡎⠋⣼⣿⡌⢲⣆⣱⠂⠀⡇⢸
⠀⠀⠀⠀⠀⡇⢸⠖⣩⡉⠻⣿⣶⣿⠟⣿⢘⠋⣿⠣⠛⢫⡿⠁⢰⡿⣻⣧⠋⢥⡈⠣⠂⡁⡘
⠀⠀⠀⠀⠀⢳⠈⢄⠚⣇⢰⣿⠁⠀⠸⣿⠀⣸⡏⣰⠒⠛⢇⡇⣾⠿⠿⣿⢠⠸⡁⠀⡞⢠⠃
⠀⠀⠀⠀⠀⠀⠳⠘⣆⢽⡸⢠⡃⠀⣰⠿⠚⠉⠁⠀⠉⠢⡀⠙⠯⣀⠐⢯⠹⢘⣃⡼⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠠⡇⢠⡯⠸⢠⣎⣀⣀⣀⠀⠀⠀⠀⠀⠀⢠⡤⠈⡅⠩⠘⣯⢨⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡈⠢⠌⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠋⢀⠤⢊⡡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣉⠒⠠⠀⠀⠀⢀⡀⠀⠀⠀⠐⢊⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠋⠁⠀⠀⠀⢀⣄⡀⣀⢀⡰⢠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⣓⣖⣙⣲⣤⡀⠀⠀⠀⠔⠫⠉⠉⠁⠀⠀⠘⠚⣿⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣤⣤⣤⠤⠤⠤⠤⣉⡉⢤⣬⡍⢡⣤⣥⡤⠠⣤⣤⣤⣤⣄⡀⣾⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠛⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢃⡒⣊⡐⣐⠘⠐⠀⠀⠐⢘⡘⢂⡂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⣻⣄⣿⠀⡯⡇⢹⡼⢹⡷⣏⡁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⣉⠁⠀⠀⠀⠁⠀⠀⢈⠁⠈⢁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⡏⡅⠀⠀⣀⡄⢀⢠⡄⣏⠀⠀⠃⠀⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⡇⠿⢸⠐⠟⠇⠘⠘⠅⠋⠈⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡆⢁⡆⡀⠀⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠀⠧⠂⡇⠹⠋⠧⠙⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠈⡂⡄⠀⠀⠀⠀⠀⠀⠆⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠴⠾⠿⠟⢁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠁⠉⠉⠁

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠁
⠀⢠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠀⠀⠀⠤⠷⠀⠀⠀⣢⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢄⡀⡀⠀⠀⡀⠀⠐⣠⣀⠀⠀⠀⠂⠰⠁⠀⡀⣤⠀⠀⠀⠄⢤⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠀⠀⠈⠁⠁⠀⠀⢧⡀⠀⢊⡁⠒⠰⠂⢀⠀⠀⠀⢤⠁⠰⢀⡡⠂⠠⠍⠋⠁⣠
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠈⠛⠒⢤⣄⣉⣉⠁⠀⠑⠒⠚⠋⠑⠂⠉⣉⣀⢠⣤⠲⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠛⠟⠛⠛⣻⠛⠳⠿⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠈⠉⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠐⠒⠀⢀⡀
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⢠⠊⠀⠈⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠄⠠⠀⠉⢠⡆⠀⠁⠀⠀⠀⠀⣿⠖⠂
⠀⠀⠀⠀⠀⠋⠀⢠⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⢀⡞⢡⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⡿⢦⣄⡀⢣⣀⠀⠀⣴⠟
⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠉⠉⠀⠀⠀⢠⠆⡴⠋⠙⡀⠉⠉⠉⠻⢤⠀⡀
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠱⡈
⠀⠀⠀⠱⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⡀
⠀⡇⠀⠀⠀⠉⠓⠦⣄⣐⠒⠒⠒⠀⠀⠀⠀⠸⠧⢤⠄⣸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠆⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢂⢹⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣥⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡇
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠆⠀⠀⠇⡀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢣
⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢰⠀⠀⢀⣰⣁⣀⣀⣀⣀⣈⣆⠀⡀⠃
⢇⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠚⠀⠀⠀⠀⠀⠀⠀⢠⡶⡇⢿⠈⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀
⠀⠛⢢⣀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⣤⠀⠀⠀⠈⠱⡎⠇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠠⠀⠻⡓⢤⡀⢀⣀⠷⠔⠻⣄⠀⠀⠀⠀⠀⠀⠠⠠⡞⡼⢠⡇
⠀⠀⢆⠀⠈⠁⠀⡍⠀⠀⠉⠂⡨⢆⠀⠀⠀⠀⡀⢀⣪⠞⠁⢸⡇
⠀⠀⠀⠓⠤⠤⠬⠤⠤⠤⠤⠚⠁⠀⠘⠋⠩⠷⠞⠛⠓⠒⠒⢻⡄⠀⠀⢀⣀⣠⣤⠤⣴⠶⠶⠒⠒⠒⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠛⡛⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠁⠀⡰⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣘⣁⣀⣀⣀⡠⠞
⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣀⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠐⠒⠶⠖⠲⠆⠀⠀⠰⠒⠦⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⡍⣷⠀⠀⢀⡄⣶⣤⣠⣴⠀⣶⣄⡰⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣇⠀⠀⣽⠀⠀⠘⠃⠁⡉⡉⠀⢀⡀⠁⠀⠀⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠋⢀⡇⠸⠀⡖⠀⠏⡖⡸⠂⣾⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠛⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰⠒⠛
⠀⠀⠀⠀⠀⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡠⠒⡰⠄⢢⣎⣁⠀⠐⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠶⠶⠒⣒⡒⠲⢤⡀⠀⡤⠀⠀⠀⠀⠀⠀⢀⣠⠖⠻⡄
⡟⠓⠺⣄⠀⢀⣇⣀⣹⡄⠀⢸⣄⠀⠀⠂⠀⠀⠀⠀⢸⣁⠀⠀⠀⠀⠀⢀⠀⠈⢑⡦⢷⡀⣌⠐⢣⡄⢀⣀⡰⠟⢫⣾⠟⠁
⠇⠀⠀⠀⠉⠉⠀⠀⠀⣹⢿⣧⡎⠀⠀⠀⠀⠀⠀⠀⢸⡂⡈⠀⠀⡐⢀⡀⣈⢀⣀⣤⣾⠁⠉⠀⠛⠓⢀⣤⠶⠛⠉
⢰⠀⠀⠀⠄⠀⠀⠀⡐⠉⠀⢧⠀⠀⢐⠀⠀⠀⠀⢴⠉⠻⠽⠿⣭⠿⣯⠽⠛⣛⡻⢟⡟⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀
⠸⠀⠀⠀⠀⠀⠀⠐⢀⣠⠴⠚⠀⠀⠀⠉⠉⠁⠀⠈⠲⢍⣀⣒⣐⣒⣀⡨⠭⠄⠒⠉⠀⠀⠰⠀⠀⠀⠀⠈⠀⠈⠁⡆⠀⠀⡀⠀⠀⠨⢆
⠀⠉⠙⠛⠛⠒⠚⠞⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢀⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠀⠀⠱⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡨⠓⠀⠀⠀⡈⠓⢦⡀⠀⠀⠘⣆⠀⠀⠀⠀⣼⠃⠀⠀⠀⣤⣶⠶⣤⣤
⠀⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠰⢄⣨⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⠢⡀⠙⢆⠀⠀⠁⣆⠀⠀⠀⡇⠀⠀⠀⠀⠁⢀⠠⠐⡄
⠀⠀⠀⠀⠐⠳⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠤⠀⠀⠀⢀⠎⠀⠀⠐⠦⠀⠀⡀⢀⣨⡦⠠⠇⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⡁⢄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠄⠀⠀⢀⠀⣠⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡀⢂⣤⠾⠛⠁⠐⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⡆
⠀⠀⠀⠀⠀⠀⠄⢙⣣⡄⠀⠀⠈⠀⠉⠀⠀⠀⠀⢸⣀⣴⠟⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠘
⠀⢀⠄⠒⠢⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡤⠒⠛⠛⠉⠉⠉⠓⠦⡀⠀⠀⠀⠀⡤⢀⢄⡀
⠀⡀⢰⠀⠀⢧⠀⠀⠀⠀⣠⣄⣴⣄⠐⢃⣀⡀⠀⠀⠀⢠⠏⠀⠀⠈⠀⠀⠀⠀⢀⣠⢼⠀⠀⠤⣜⠷⠻⣇⢱⠈⡁⠀⠀⠀⠐⠀⠀⠈⠠
⠀⠄⣸⠀⠀⢸⠀⠀⢠⣾⠏⡠⠊⠀⠀⠀⠀⠉⢶⠀⠀⡟⠀⠤⠀⠀⠀⣠⣤⣶⠿⢳⣻⡀⠀⠀⣿⠄⠀⠟⣸⣄⡿⠃
⠀⡇⣇⠀⠀⣼⠀⠀⢼⡫⠜⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠳⣤⡤⠶⠊⣙⢋⠽⣷⢦⣎⡥⢴⠋⠀⡋⠘⠚⠷⠛⢛⡇⠀⠀⠀⠀⠀⡀⠠⠤
⠀⢶⢸⠀⠀⡇⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠤⠀⠠⠃⠀⠀⠈⣼⣄⣸⣭⣴⡶⠿⠋⠃⣱⠎⠀⠀⢸⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⠃⠀⠄
⠀⢸⡌⠀⢰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⢀⣀⣈⣉⣀⠤⠒⠊⠁⠀⠀⠀⠀⠃⠀⠀⠠⠓⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠳⣤⠟⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠂⠐⠚⠀⠀⠀⢀⠀⠐⡀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠟⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣠⡌⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠐⠂⠀⠀⠐⠖⠒⠲⡤⠠⢤⡶⠀⠀⢀⠤⣠⠶⠶⠚⠋⠉⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⡆⢀⣬⡦
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠂⠀⠀⠁⢨⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠘⠦⠂⣀⣒⠀⠉⠝⠃⠚⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣤⣤⣤⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠤⣽⣋⣛⣛⣛⣛⣋⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣿⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⣶⠶⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⢶⡶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠿⠶⠶⠶⠶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢠⡶⠟⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠞⠶⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣇⡀⣠⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣧⡀⣀⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠒⠒⠢⠔⠉⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠁⠀⢂⠀⠀⢀⡠⢊⣧
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠱⢶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⠁⠀⠀⠦⠀⡶
⠀⠀⠀⠀⠀⠀⡰⠒⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⣀⣀⠀⠎
⠀⠀⠀⡠⠒⠐⠧⢀⡀⠀⠀⠈⠉⠀⠒⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠧⠤⠤⠤⠤⠀⠒⠈⢁⣠⠖⠀⣠⠴⠷⠦⠱⠷
⠀⠀⠈⠃⠅⠀⠀⠀⠀⠉⡐⠒⠠⢄⣀⣀⠀⠀⠀⠉⠐⠒⠠⠄⡄⢀⡌⠀⠀⠒⣢⣶⠄⠀⠚⠋⠁⠾⠐⠊⠀⠀⠀⠠⠀⠇
⠀⠀⠀⢀⠐⠈⠶⠤⠤⠤⠤⠤⠠⠦⠴⠬⠽⠗⠶⠦⠤⢀⡀⠀⠀⠨⡇⠀⠠⠀⡁⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⣶⣄⠀⡂⢸
⠀⠀⠀⠠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡺⡇⠀⡀⠀⢀⣶⣤⠀⠀⠀⠀⡠⠞⠉⢱⣍⣁⣤⠀⠸⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠈⢆⡙⠇⠈⠉⠡⣄⠀⠀⠄⡀⢀⣀⡼⠁⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⣁⠈⠁⠀⠀⠀⢰⡆⠀⠀⣀⠧⠀⠀⠀⠒⣲⡆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⡄⠀⠀⢠⡀⠀⠀⡄⠀⠀⡄⠀⠀⡏⢡⠀⠀⠀⠀⠀⢿⣿⣷⣤⣤⣶⣶⣶⣾⣿⠃⢀⠜⠀⡀⠤⠒⢤
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⢀⡷⠀⡇⠀⠀⡞⠀⠀⣰⠃⠀⣄⢰⠀⠀⡇⠀⠣⡀⠀⠀⠀⠈⠻⣿⡿⠛⠻⠛⢿⡿⠃⡠⢊⡤⠊⠳⡀⠃⡺
⠀⠀⠀⠀⠀⠀⡇⠀⠀⡖⠋⠀⠀⡇⠀⠼⢁⠀⢰⠃⠀⢀⡁⠈⡆⠀⡇⠀⠀⠉⠢⢄⣀⠤⠤⣒⠭⠛⢒⣉⡡⢐⡩⠒⠁⠀⠀⢀⡸⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠐⠃⠀⠀⠐⠃⠐⠀⠐⠀⠒⠂⠃⠘⠀⠀⠃⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠈⠉⠙⡍⠁⠈⠁⠀⠀⣀⣤⠒⠁⢀⣀⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠋⠢⣀⠀⠀⠑⠳⠴⠶⠶⠞⠋⢉⣄⣄⡖⠉⢠⡀⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠁⠀⠀⠈⠙⠂⠀⣀⡀⠀⠀⢠⣤⠤⠟⢿⡁⢠⣤⠟⠃
⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣰⣂⠀⠀⢑⢶⡀⠤⠤⠄⠘⠉⠉⠛⢆⣀⠀⠀⠀⠈⢃⠠⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠿⠏⠁⠀⠀⠀⠀⠈⠒⠒⠒⠦⠀⢀⠀⠀⠙⢿⡄⠀⠀⠘⡀⠁⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠂⡳⡄⠀⠈⡎⡀⠀⠀⡇⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢰⠀⠀⠐⠁⠀⢰⠀⠀⠀⠀⡀⠥⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⡀⠀⠆⠀⠀⡠⠊⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠐⠀⠈⠓⠂⠀⠀⠀⠀⠀⠂⠀⠀⠸⡈⠖⠀⣤⣷⡁⣸⠀⠀⡔⠀⢀⠠⠄⠂⠐⠂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢁⠀⠀⠁⠀⠀⡠⠀⢀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⢇⠀⠀⠀⠁⠀⠀⠀⡘⠀⠈⠛⠚⠀⠤⠤⠤⠤⠌⠡⠤⠥⠤⠔⠊⠘⡄⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠚⢶⣀⣴⡾⢶⠦⣤⠀⠀⠀⡤⢀⠠⠦⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠰⠆⢲⠂⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⡦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠠⠄⣀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⢶⣾⣷⣶⠿⠿⠆
⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣧⠀⠀⠁⠀⠀⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠲⡤⠄⣶⣻⣯⣹⣿⠀⠀⠀⠀⠀⠀⠴⠌⠀⠀⠀⠀⠀⢀⡈⢀⣀⣀⡼⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠠⠆⠀⠀⠀⢻⡟⣿⣿⣿⢻⣿⡿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠒⠋⠀⠀⠐⠫⠹⠿⠿⠛⠘⠿⢷⡇
⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣤⣤⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⣿⣦⠀⠀⠀⠔⠀⠀⠀⠀⠀⠈⠁⠄⠈⠀⠀⠀⠀⠀⢓⠠⡄
⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠐⢀⢀⠄⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡄⠀⠀⠀⠐⠠
⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⣀⠀⠀⣀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠠⠤⢴⣦⠴⠆⠀⠀⠀⠀⠀⠀⣠⠂⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠔⠀⠦⣄⠀⠀⠈⠲⠤⠀⠀⠼⠃
⠀⠀⠀⠄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⣼⠛⣷⠀⢢⡄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⠤⢤⢤⣤⡄
⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁
⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠰⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢀⣥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡇
⠀⠀⣀⣀⣴⣿⣿⣿⣷⣤⣀⣀⣀⠀⠀⠀⠀⠀⠦⠴⠶⠶⠶⠦⠶⠀⠀⠀⠀⢶⡿⢿⠏⠩⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⣶⣤⣤⣄⠀⠤⢠⣩⡄

⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠟⠀⠀⢀⠁⠀⠀⠀⠙⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⡖⠀⠀⠀⠀⠁⠀⠈⠃⠐⡏⠀⢘⠆⠀⠁⢲⠀⠀⣀⠇⠀⢠⠞⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢶⠃⠀⡆⠰⢀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠇⠀⠀⠘⢧⡀
⠀⣀⠀⠀⠠⠯⢸⠢⢠⢠⡈⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡀⠀⠐⠓⡀⠀⠈⠁
⠀⡁⣀⡀⠀⠀⠙⠱⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⢀
⠐⢻⠵⠂⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁
⠄⠁⠀⠐⠀⠁⠀⠀⠀⠰⣁⠈⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⣀⣀
⠀⡠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠦⣵⢿⡷⣯⣤
⠀⣏⠀⢳⣔⡀⠀⠀⠀⡀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠅⣂⡿⢿⣟⡶⡿⣗⠈
⠀⡏⠻⠾⠏⡦⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡽⠷⣴⠷⡍⡛⠒⠀⠆
⠀⡇⡂⠘⠐⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣾⣄⢠⣾⠧⢓⡀⠀⠀⡤
⠀⠁⠑⢀⠉⢀⠂⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣜⠹⣀
⠠⡆⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣄⠀⠀⠔⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠛⠚⠋⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠁⠢⢀⣀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠑⠒⠀⢤⠀⠀⢇⠀⣠⠊⠈⠐⠀⡀⡣⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣨⣑⠨⡢⡀⠀⠪⣀⠤⠃⠙⢬⠁⠐⠤⠶⠿⠗⠀⠀⠀⠀⠀⠀⠠⣱
⠀⠀⠀⠀⢀⡠⡞⠩⠁⠈⠁⠚⠛⡖⠒⠒⠒⢢⠀⠊⠀⠀⠈⢀⣹⡀
⠀⠀⠀⠀⣅⠜⠀⠈⢲⠂⠐⠐⠐⠳⠀⠀⠀⠒⡄⠐⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠐⣒⡀⣶⠖⡖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⢂⢓⠐⠒⠀⠓⡄⡆⠀⠀⢀⠀⠀⢀⡀⠀⢀⡶⠛⠚⢩⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠒⠂⠘⠈⡆⠀⠀⠈⡇⢸⠐⠂⠀⠈⠭⢐⡒⠂⠘⠃⠀⢀⠓⢎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⡅⠸⡉⡉⠭⠓⠈⠀⠀⠀⢀⠆⠁⠀⠈⠑⢢⠀⠐⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠒⠆⣐⣓⡲⢛⠳⠄⠀⠀⠀⠀⢠⣦⣤⣀⣠⠄⠀⢰⠀⠀⢣⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠊⠀⠀⠈⢂⠀⠀⡠⠃⠀⡠⢨⠁⠀⠀⡘⠀⠀⠈⡄⢠⡶⠋⠁⠀⠀⣈⣵⠆
⠀⠠⠀⠉⠈⠑⠠⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠰⠁⠆⠀⠀⢀⡇⠀⢀⡀⣇⡟⠀⠀⢀⣄⣈⡀⠀⠱⡀⡄
⠀⢈⠀⢀⡀⢀⢀⠜⡠⢠⠀⢀⣠⣤⠤⠤⠤⡤⡄⡤⠐⢲⡶⠀⠘⢌⢲⣂⣈⣁⣈⣩⡴⠟⠉⡇⢀⠔⢡⢈⣲⠈⡀⠀⡃
⠀⠀⠈⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠘⠀⠀⠁⠤⠔⠃⠀⡎⠈⠢⠀⠀⠈⡀⠀⠠⠀⠤⠤⠄⠒⠳⢇⠀⠈⠢⠤⠚⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠑⢄⣀⣀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣄⣀⣀⣀⠔⠁

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

⠀⠀⠀⠀⠀⣀⣀⡤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡖⠭⠒⢀⡍⡁⣈⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠃⠢⠀⠒
⠀⠀⣀⠁⠀⠀⠀⠈⠀⠀⠀⢀⡰⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⡐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠂⠀⠖⠀⠀⠊
⠀⠀⠁⠀⠀⠀⠒⠒⠒⢒⣈⣩⠴⠚⡅⠀⠀⠀⠀⠀⠀⡠⠞⠍⠀⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠐⠒⠒⢀⣀⡠⠤⠖
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⡀⠠⠔⢻⠀⠀⠀⠀⠀⠀⣼⠁⠀⠐⠀⢠⠊⠁⠀⠀⠀⣠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⢀⣠⠄
⠀⠀⢸⠄⠀⡀⠀⠀⠀⠀⢀⣀⡤⢾⠀⠀⠀⠀⠀⠀⠇⠀⠀⢂⠘⠀⠀⠠⠤⠐⢊⠁⠁⡔⠊⡥⠀⠀⠀⠀⠀⠀⢩⠝⣀⢋⠼⡟⢣⠉⡄
⠀⠀⠘⠀⠄⠀⠈⠉⠉⠭⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠔⠁⠀⠞⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢼⣤⣦⣼⣤⢧⢿⠁
⠀⠀⠀⡇⠀⠀⡀⠄⠐⠀⠀⠙⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢜⣕⢕⡵⡡⡏
⠀⠀⠀⠘⠀⠀⠀⣀⣈⣀⡠⠴⡺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡯⡪⣾⢮⡇
⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢠⡆⠀⠀⠀⠀⠀⠀⣀⠄⠘⠀⠀⡚⢀⡀⠀⠃⢦⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠞⠁⠀⠀⢀⣠⠀⠀⠀⠀⣀⢄⠈⢱⠚⠇⢸⠁⠞⠂⡜⠓⢀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠐
⠀⠀⡠⠖⠂⠒⠉⠈⠁⣄⣀⣠⠞⠁⠘⠉⠀⠀⠀⠀⠁⠈⠃⣰⠸⠀⠟⠲⠀⣼⠇⠀⠈⠀⠀⠀⠀⡸⠤⠠⠄⠤⢀⡀
⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⠂⠅⠧⠼⠿⠏⠀⠀⠃⠀⠀⠀⠀⠀⣺⣁⣊⣐⣂⠐⠋⠻⠐⠠⢀⠤⠀⠈⠆
⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠌⢀⡘⡈⠀⠀⠀⠀⢄⣁⣠⠎
⠇⠀⠀⠀⠀⠀⠀⡀⠀⢈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢨⠆⠀⢀⡈⠉⠀⠒⠃⠐⠂⠐⠀⠜
⠀⠀⠀⠀⠀⣀⣴⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⡀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠀⠤⠌⠀⠉⢐⡋⣘⠁⢀⠀⢀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⣀
⠀⠀⢀⣠⠖⠊⠉⠄⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⢸⠁⠄⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢧⡀⠀⣀⡀
⠀⣠⠋⠀⡈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠀⠂⠀⠀⡄⠀⡄⠀⠠⠀⠀⢀⡀⠀⠀⠀⢸⠁⠀⠈⠃⠪⡁⢟⣠⣄
⠀⣇⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⣡⠞⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡒⢠⠉⠂⡠⠀⠀⡄⠀⠘⠀⠀⠀⠀⠈⠓⠀⠤⠘⣤⣗⣲⣄
⠈⢙⡁⠶⠄⠤⣤⠤⠤⡬⢄⠒⠚⠋⠀⡰⠁⠀⠀⠀⠀⠀⠀⠠⡔⠀⠀⡁⠂⡠⠊⠀⢀⡰⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⡢⡑
⠀⠈⡀⠀⠉⠀⠐⠈⠀⠐⣁⠤⠚⢀⡁⠴⢃⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠕⠊⠀⢀⡴⠋⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞
⠀⠘⠀⠘⠀⠤⠶⠒⠒⠒⠀⠚⠩⠉⠂⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠂⣒⣨⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⡄⠀⠀⠀⡀⢸⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣫⣭⣥⣶⣶⠶⠂⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⣿⣿⣿⡯⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⠠⡄⠠⠄⠀⡂⠀⠀⡀⣀⠀⢀⠀⠀⠈⡇⢻⣿⣿⣯⡁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣠⡀⢀⡄⡀⣀⠀⢸⣇⠋⢰⠟⣿⠁⠎⠀⠀⠀⣿⢸⣿⣿⣟⠂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢻⡇⠸⣇⢱⠸⠀⠸⣿⠀⢠⣀⡿⠀⠁⠀⠀⠀⢹⡘⣿⣿⣿⡦⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⣿⢴⣿⣼⣇⢠⡄⡟⠀⣸⣿⠃⢰⡆⣀⠀⡄⢸⣇⣻⣿⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣽⠾⠟⠛⢿⣿⣷⡇⢀⣿⣿⣤⣾⣿⣯⡄⢉⠠⣠⣌⠉⠿⠇⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠙⠿⠿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣶⣶⣿⣷⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠋⠉⠉⠀⠀⠀⠀⠈⠉⠙⠿⡟⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠠⡈⠫⡤⠴⢒⣋⣉⣹⣿⣿⣍⣉⣁⡀⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠙⠆⢶⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⢷⠀⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠑⢢⣬⣶⣾⣿⣿⣿⣷⣦⣥⣶⣿⣿⣷⣶⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⡈⠿⠯⠍⡁⠀⠠⠀⠀⠀⢀⣈⣉⠉⠁⣴⣤⡤⠏
⠀⠀⠀⠀⠲⠛⠿⠿⢿⣶⣶⣶⣶⣾⣧⡶⠶⠶⣾⣛⣛⣉⣿⣥⣭⣭⣯⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⠛⠛⠛⠛⠲⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⣠⣶⡶⣿⣿⠿⠿⠛⣿⣶⣶⣯⣉⠻⡷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⢀⣴⣿⠋⠁⢈⡴⠃⠀⣀⡜⠛⠁⠸⢟⠿⡷⢌⠻⣷⣄
⠀⠀⠀⠀⠀⠀⢠⡾⢁⣾⣿⠋⠀⠐⡴⠊⠑⢤⠐⠁⠉⠲⡎⠀⠀⠘⠃⠈⠳⡌⢮⢧⣄
⠀⠀⠀⠀⠀⢠⡟⢠⣿⡇⠈⠳⣄⠀⠠⡠⢀⠀⢁⢤⡀⠀⢈⠶⢄⠀⠀⠉⠦⡨⠆⢻⣿
⠀⠀⠀⠀⠀⣿⠀⣾⠋⢿⠂⠀⠈⠑⠀⠀⠀⠙⠁⠀⢈⠟⠁⠀⣀⠉⠀⢀⣔⠀⢈⠈⢿⡆
⠀⠀⠀⠀⢠⣯⢀⣿⠟⠁⠀⢠⡔⠀⠀⣠⢀⠀⢀⡴⢦⣤⣀⠜⠁⠀⣴⠃⠈⠁⢾⡆⢰⡇
⠀⠀⠀⠀⠸⣏⠈⡷⠀⢠⡜⠁⠈⡠⣾⠀⠻⠷⠿⣶⠾⠛⢿⣄⡀⠑⠈⡣⣀⠀⠈⣧⢸⡇
⠀⠀⠀⠀⠀⣿⠀⣿⡄⠀⠈⢢⡀⠈⠻⣷⠦⣀⠀⠈⠻⢿⣷⡾⠋⠢⣄⡜⠉⢻⢶⡯⣸⠇
⠀⠀⠀⠀⠀⢹⣇⠘⣟⢤⡄⠁⠈⢳⠖⠁⠀⠈⠛⠀⠀⣠⠋⠀⠀⠀⠈⠀⠀⠃⡼⠡⡿
⠀⠀⠀⠀⠀⠀⠻⣧⠘⢏⡀⣠⠞⠁⠀⠀⣠⠃⠀⣠⠔⠁⠀⢀⣴⡂⠀⣀⢸⡟⢡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⠧⠀⠀⠠⡄⠀⠀⠛⢃⠀⠀⠈⠻⢄⡀⣺⡿⢋⣴⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⠀⠤⠈⣳⣄⠀⠈⢳⡤⣀⠀⠠⠟⣉⣤⠟⠋⠉⠉⠉⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠦⣤⣀⣈⣉⣉⣉⣠⣤⡶⢞⡛⣭⣴⠶⠶⠶⣶⣖⠤⣄⡀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣄⠀⠉⠉⣹⠟⠋⠉⣥⣶⣋⡼⠋⠀⠀⠀⠢⠀⡉⢳⣶⣿⢷⣦⡀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⣿⠀⢀⡾⠁⠘⣶⣿⡷⡿⢿⠁⠀⠀⠀⠀⡀⠀⠁⠈⣷⡝⠘⠻⣿⣆⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣸⠇⠀⣼⠁⠀⣼⣿⣿⣇⠇⣸⠀⠀⠀⠀⠀⠘⠢⠀⢀⡟⡇⠀⠀⡇⢸⣧⠀⠀⢷
⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⣿⠀⢰⡇⠀⢸⣿⡿⣼⣏⡷⣌⠷⣤⡀⠀⠀⠀⢀⣴⠊⢁⢉⡃⡐⡃⢾⠹⣇⠀⠘⡇
⠀⠀⠀⠀⣠⠞⠉⢀⣀⣀⠀⠀⠀⣿⠀⢸⠁⠀⣾⣿⠃⢃⣮⠶⠾⠾⢥⣍⡿⠟⠛⣿⣩⠶⠟⠛⠳⠾⣥⡶⡄⢻⠀⠀⣿
⠀⠀⠀⣼⠃⣠⠞⠉⠁⠈⠉⢆⠀⢻⡀⢸⡅⠀⢸⡇⣰⠏⠀⡀⠀⠀⠀⠀⠙⣆⢠⡟⠁⠀⠀⠀⠁⠀⠀⠹⡄⠘⠀⠀⣿
⠀⠀⢰⡗⠀⡏⠀⠀⠀⠀⠀⠈⡄⢸⡇⠈⣇⠀⠘⣷⡏⠀⠀⠈⠐⡀⠀⠀⠀⣽⣾⠀⠀⠀⠀⠀⣀⠤⠀⠀⣿⠃⠀⢸⣿
⠀⠀⠈⣇⠀⠳⠀⠀⠀⠀⠀⡠⠀⣼⠃⠀⢹⣆⠀⠸⡇⠀⠈⠁⠉⠀⠀⠀⠀⣸⢻⣆⠀⠤⠤⠚⠁⠀⠁⣼⡏⠀⢠⣿⠇
⠀⠀⠀⠙⢦⡀⠈⠒⠀⠀⠈⣠⡾⠃⠀⠀⠀⢻⣆⠀⠙⢤⡀⠀⠀⠀⠀⢀⡴⠃⢸⡟⢦⣀⠀⠀⠀⣀⣼⠋⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⠛⠉⠀⠀⠀⠀⠀⠀⠙⢧⣀⠈⠻⢿⣶⡖⠒⡟⠁⠀⢸⠅⠀⢉⡛⣏⠝⠉⠀⣀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠈⠁⠀⠓⠂⠀⠜⠒⠂⠈⠁⠀⢀⣠⢾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣄⣀⣀⣀⣀⣀⣤⣴⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠛⠙⠉⠉⠉⠉⠉⠙⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢀⡠⣤⡴⠶⠒⠒⠶⠶⢤⣀⡀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⢀⣴⡾⣋⡠⣶⡶⣾⢸⠿⢷⡖⣶⣦⣝⣳⢤⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠰⣿⣶⣯⢼⡿⢿⣷⠀⠀⢸⢸⠀⢈⠃⠀⠋⣽⢸⠿⣿⣦⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⠋⠀⢠⣿⣷⣿⢸⠀⢸⡟⠀⠁⡾⠀⠀⢸⠀⠀⠀⡟⠀⠀⢹⢹⣷⡀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⢹⡆
⠀⠀⠀⠀⢠⡏⠀⢀⣿⣿⣋⡿⡲⣄⢶⢳⠄⢠⡟⣀⣠⣬⠖⠀⠀⢧⠆⠀⡞⣸⠹⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⢸⣿⡏⠿⣻⠋⠉⣦⡵⠋⠃⣈⡵⢍⣿⠟⣢⡺⡴⣗⡲⡞⢨⡂⢹⡆⠀⢸⡇⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⢠⡇⠀⠀⣿⠀⠀⢸⠁⠀⠀⡟⠏⠀⢸⠹⠁⢻⡇⠊⠉⠀⡇⠀⢸⡇⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠘⡇⠀⠘⣯⠀⢸⡀⠀⢠⡏⠀⠀⣾⠀⠀⠀⣧⠀⠀⣸⠀⠀⢀⡇⠄⠀⡰⠀⠀⢸⡇⢠⡾⠁⢀⡤⠤⠤⢤⡀⠠⢸⡇
⠀⠀⠀⠀⠀⢻⡄⠀⠹⣤⡘⡃⢄⣸⣅⢄⣨⣽⡅⢀⣤⢱⣀⣐⡟⢧⡤⣨⡁⢞⠃⠇⠀⢠⣿⢣⡏⠀⡴⠋⠀⠀⠀⠀⢙⠄⠸⡇
⠀⠀⠀⠀⠀⠘⣷⡀⠀⢻⣧⡝⢪⡽⡽⠦⣰⣿⡷⠟⠛⠛⠛⠛⠿⠿⠿⢿⢶⣯⡞⠀⢠⣿⡟⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠇⠀⡿
⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠙⣇⠀⣼⠾⠛⢋⣡⣴⣶⡾⢻⣿⡿⢿⣿⣶⣤⣉⡛⠿⡷⣿⡟⠀⠘⣇⠀⠳⡀⠀⠀⠀⠀⡠⠁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⣨⠟⠁⣴⣾⢿⢟⡟⠀⠀⡼⠛⠀⡀⠛⠂⡾⠀⠙⠶⢌⠲⣿⣷⡀⠙⢷⣀⠈⠁⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠃⣠⣾⠏⠀⠀⣾⠁⠀⡼⠉⠒⠴⢀⠀⠀⠀⠀⠓⠠⢌⠃⡌⢻⣷⡀⠀⠉⠛⠲⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣰⣿⠟⠂⢤⡞⠀⠉⠒⠄⢀⡀⠀⠀⠌⠑⠢⡄⠀⠀⣌⠀⠀⣄⠙⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⣸⡿⠿⣀⠀⠀⠈⠰⠠⢤⣀⠋⠉⠓⡞⠀⠀⣴⠀⠀⡼⠁⠉⠐⢿⡄⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⣧⡀⠈⠁⠒⠴⠀⠀⠀⠁⠀⠀⣌⣠⣤⣠⠂⠠⣘⠓⠤⣀⡀⠀⢿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢘⣿⠉⠟⠀⠀⠀⠀⠀⢀⡆⣄⡠⣾⣭⣾⠟⠻⢦⣤⣀⠀⠈⠊⠉⣻⢿⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣿⡞⠁⠀⠠⠋⠂⢀⣜⠈⠛⠛⠛⠛⢶⣶⣾⣷⡏⠈⠙⠻⠁⠀⠁⢰⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢹⡧⠀⠰⢧⡀⠀⠐⠻⠿⡶⠢⣤⣠⠀⠀⢈⡟⠀⠀⢀⠀⠀⠀⠾⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢻⣦⠀⠀⠈⠗⠠⢤⡤⠁⠀⠀⠉⠀⢀⣼⠀⠀⢀⠾⠁⢂⣾⡞⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠻⣗⡰⡖⠀⢀⡼⠁⠀⠀⡼⠀⢀⡾⠁⠀⠁⠛⠒⠤⣤⡟⣁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠻⢆⣀⡼⠁⠀⠀⣌⠀⠀⠈⠁⠲⢤⣀⢀⡀⠔⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣈⠙⠂⠀⠀⡀⠉⣲⣦⣀⣤⠾⠃⣉⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣤⣄⣀⣀⣀⣤⡶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⠈⠀⠀⠀⠀⠀⠀⠐⠒⠛⠗⠊⠀⠀⠀⡯⠥⠠⠄⠢⠴⠇⠀⢀⠂⠁⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣤⢀
⠀⠀⠀⡌⠀⠀⠀⠐⠁⠀⠀⠀⠀⡀⠀⡤⢀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠀⠁
⠀⠀⠀⠐⠄⠀⠚⠀⠀⠤⠀⠀⡰⠆⡅⠠⠃⢡⠀⢀⡞⠉⠀⢀⠀⠙⠄⠀⢠⡤⠀⠀⠄⠠⠤⡀⠀⠰⠭⠜⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠈
⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠮⠀⢰⡂⠰⢀⡀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡕⠀⠐⠀⠀⠘⠀⠂⡀⠀⠺⠀⠐⠈⠣⠐⠂⠀⠀⠀⠀⠠
⠠⠒⠂⠀⠀⠀⠀⢤⢢⣤⡀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠱⠆⣀⣀⡀⠔⠃⠀⠀⠈⠠⠄⢃⠀⠁⠲⠀⠀⠓⠀⠀⠁⠀⠀⠁⠂⠀⠀⠄⠈⠁⢀⡄⠀⠰
⠀⠊⠁⠀⠀⡪⡿⠑⣯⢹⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠠⠀⠰⠀⠀⠀⠀⠘⠒⢂⠉⠀⠀⠠⡄⠀⠀⠀⠀⠠⡮⠁⠀⠐
⠀⠀⠂⠀⠀⠀⠖⠴⠛⠁⠀⡄⠀⠀⠀⠰⡖⠀⣀⠀⠰⣃⠂⠠⠀⠐⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠊⠀⠀⣄⡐⠀⣚
⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠉⠀⠀⠀⠀⠄⠁⡤⠁⠠⠀⣤⠇⠀⠤⠀⠀⠀⡔⠁⣀⢄⡀⠀⠄⠀⢠⠀⠴⠘⠀⠀⡄⢀⡁⠦⡀⠀⠈⠚⠁⠐⠊⡀
⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣀⠠⠒⠀⠀⠠⠠⡀⣠⡀⠀⡠⠀⡄⠀⠀⠀⠐⡅⠌⠃⡐⠁⠠⠆⠀⠀⠀⠀⣀⠀⠀⠆⣠⠄⠖⢛⠀⠀⠀⠀⠀⠀⠁
⠀⣀⠀⠀⠀⠀⠀⠀⠠⠂⢉⠀⠢⠀⠀⠀⠙⢺⢃⢹⡏⡀⠇⡏⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣻⡯⠎⠃⠀⠩⣄⡠⠌⠚⠀⠀⣀⠀⠀⠀⢀
⠁⠀⠉⠀⠡⠀⠀⢠⠂⠄⠒⠀⠀⡃⠀⠀⠀⠀⠐⠀⠀⠊⠀⠙⠈⠀⠀⢀⣀⣀⡀⠀⠀⠀⣴⡿⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣰⠨⣜⡿⣃⠀⠁⢇
⠀⠒⠀⠈⠀⠀⢠⠀⢰⡀⣀⣀⣸⠀⠀⠀⠀⠀⠤⠄⠠⡀⠀⠀⠀⢀⠞⠁⢀⠀⠈⠑⠀⠀⠑⠛⠒⠂⠀⠀⠀⠀⠀⢀⡀⠀⠈⣇⢀⡹⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠉⠉⡍⠀⠀⠀⣐⢢⢤⣌⢁⠀⠐⠀⠀⢰⡀⠰⣀⠠⠀⠀⡆⠀⠀⣀⠦⣀⠀⢀⡄⡠⢄⠀⣨⢆⠀⠀⠀⢀⠠⠐⠀⠀⢶
⠀⠀⠑⠂⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠹⢮⣄⣞⣯⢋⡿⠀⠀⠀⠳⠆⣀⣀⢀⠔⠁⠀⠀⠰⠂⠂⠀⠨⣧⣤⣝⣤⢤⡜⠀⠀⠀⠀⡠⠤
⠀⠠⠄⠀⠀⡀⠈⢠⠀⠀⠀⠀⡆⢐⠀⠀⠘⣿⡎⢟⣽⠉⠀⠀⡀⠀⠀⠀⠈⠁⠀⠀⣰⡄⠀⡀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠐⡛⠂
⠀⠀⠉⠀⠀⠄⢀⠀⠀⠀⠀⠠⡁⢈⠀⠀⢰⡞⢞⡽⠀⠀⠰⠀⠀⠀⠒⠀⠀⢰⡚⠓⢀⠤⣀⣹⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⢀⠄⠀⠧⠀⠰⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠰⣟⡌⠀⠀⠀⠄⠐⠂⠤⢀⡅⠀⠈⠥⠨⠀⠀⠀⠘⠀⠀⠀⠀⠆⣉⠡⠘⢑⠀⠈⢢⠄⠀⠤⢼⠁
⠀⠀⠀⢀⠄⠀⠀⠀⡾⠀⠀⠠⠤⠤⠄⠀⠈⠊⠀⠀⠀⠒⠂⠐⠐⠀⠂⠘⠀⠀⠀⠀⣀⠀⠀⠀⢠⡀⠀⢰⡆⠀⢀⡈⢹⠀⠀⢸⣒⡚⠛⡟
⠈⠀⠀⠈⠀⣠⠀⠘⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠐⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⣀⠁⠸⠀⠈⠃⠀⠀⠀⠊⠀⠀⠀⠂⠀⠀⠁
⠀⠸⢄⠀⠀⠉⡏⡉⠉⢹⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠄⣀⢑⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠂⠑⠀⠀⢠
⡀⠂⡘⡂⠀⠀⢠⠀⢀⡞⠀⠀⢀⠈⠀⠒⣈⠀⠘⠄⠀⠄⠀⠀⠠⠀⠃⠀⠃⠙⠀⠛⠀⠀⢴⣉⡤⣉⠂⡧⠀⠀⠀⠓⠠⠓⡀⠀⠀⠀⠒⠂⠀⠀⣎
⡶⡶⢺⠃⠀⠀⠈⠋⠉⠁⠀⠀⠈⠄⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⢏⠝⣬⠀⠀⠀⢘⠀⠀⠁⡁⠀⠀⠀⠀⠘⢀⠀⢙
⠊⢶⡍⠀⠀⢀⣠⠤⢦⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⢐⣀⠀⠀⢰⡁⠀⠰⠄⠀⠀⠀⠱⣍⣫⡖⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⢘
⠓⠊⠁⠠⣞⠿⠖⠓⠋⠈⠈⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣷⡄⠀⠂⠀⢰⣀⣪⠐⠜⢓⠀⠀⠀⠀⠀⠀⢀⠤⣀⡀⠀⡀⠇⠁⠀⠀⠈⠀⠀⠁
⣀⠀⠀⠀⠈⠃⠠⠄⠀⠀⠀⠀⢀⠰⠁⠐⠄⠀⣰⣿⣿⣿⣿⡿⠃⠀⠀⠈⠡⣀⡤⠐⠋⠠⠄⠀⠀⢐⣦⠃⢶⢱⡥⠀⠀⠀⠀⢪⠰⡄⠀⠀⠀⡤
⠛⠂⠀⠀⣀⠀⠀⠀⣠⠀⠀⢰⠢⣔⠈⣉⡃⠀⠙⠿⠟⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠪⠤⠴⠚⠁⠃⠀⠀⠀⢨⡀⠼⡀⠀⠀⠃⢀
⠉⠀⠀⠀⠙⠁⠀⠤⠀⠀⠀⠸⠤⣄⠈⣸⠃⠀⣀⠠⢀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠈⠃⠀⡃⠈⢤⠀⠈⠠⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠈⡀⠀⠑⠒
⠀⠒⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠁⠀⠠⡅⠁⠈⠷⠀⠀⠀⠀⠀⠰⠗⠂⣀⡀⠀⠀⡉⢄⠰⠖⠂⠀⠀⠀⣰⠶⠂⠑⠀⠀⠑⠒⠀⠁
⠠⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡤
⠀⢀⣀⡀⡀⠙⠁⠀⠄⣀⠀⠀⠀⠀⢸⡁⣧⠘⢆⠀⠀⠮⠀⠀⠀⠀⠀⣤⣿⣮⡟⠀⠀⠀⠀⢀⣠⣄
⠀⠈⠉⠁⠘⠋⠃⠀⠐⠒⠒⠋⠀⠀⠈⣗⠈⣀⠜⠀⠀⠐⠒⢀⠀⠀⠘⠾⣽⠟⠀⠀⠀⠀⠀⠸⠧⠟
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉
⠀⠀⢚⠉⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⡰⡖⠃⡄⠀⠘⠛⠿⠟⠃⠀⠀⢂⡁⠂⢤⠀⠀⠰⢷⣾⠿⠄
⠀⠀⠀⢰⠄⠀⠀⢠⠐⠐⢃⠀⠀⠀⢀⠁⠀⠊⠀⠀⠀⠀⠀⠀⠠⠄⠰⠋⠐⣂⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠀⠀⠀⣈⠁⢲⠄⡀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠉⢳⡄
⠀⠀⠀⠠⣦⡤⠀⠰⠁⠀⠀⠀⠺⠀⠈⠁⠀⠀⠀⠀⢠⣀⣀⣤⡄⠀⠀⠙⢀⣀⠀⠀⠀⠀⠀⠀⠒⠉⠁
⠀⠀⠀⠈⠛⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠈
⠀⠀⣄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⢠⠐⣀⡤⠀⠀⠀⡀⢠⠀⠀⢀⣤⣤⣤⢀
⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⡀⠀⣄⡸⠀⠠⠤⠤⠂⠁⠀⠠⠄⠄⢀⡀⠀⠀⠐⠄⠁⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⠀⠐⠒⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⡀⣂⢠⡀
⠀⠀⠀⠀⠀⠒⠁⠀⠈⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠿⠜
⠀⠀⠀⠓⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡀⠀⠀⡀⢠⡆⠀⠀⠀⠀⣀⠀⠀⣀
⠀⢤⣤⣀⣠⡤⠀⠐⠿⣵⣦⠽⠀⠀⠀⠊⠂⠀⠀⠀⠉⠁⠀⠉⠁⠀⠰⢁⢶⠶⣔⠢⡀⠾⢏⡹
⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠉⠉
⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠰⡆
⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠒⠶⠀⠀⠀⠘⢻⡉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠻⣀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠟⠄⠀⠀⠀⠀⢦⣤⡤⠟⠁
⠀⠀⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠀⣠⡆⠀⠠⠄⠄⠒⠈⠀⠀⣀⣷⣌⡓⠀⠀⠀⢀⠀⢢⡀⠀⠀⠀⠀⠛⠃⠀⠀⢀⡤⡤⢠⠄
⠀⠀⠂⠘⠀⠐⠀⠀⢀⣀⠀⠀⠀⠠⠝⠹⠈⠗⠀⠀⢐⡘⢑⡋⠀⠀⠘⠛⣤⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⠒⠂
⠀⠀⠆⣀⡀⠀⠀⢀⠀⠉⡄⠀⠀⢀⣠⡴⠶⠂⠀⠀⠄⠂⡀⠒⠀⠀⠀⠒⠒⠂⠤⠀⠀⠘⢃⠀⢴⢂⡄
⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⢀⠀⠿⠁⠀⢀⣶⠤⠼⠦⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⢿⣿⡿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣧⣶⣷⣶⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⠋⢂
⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢱⡤⣷⣶⢢
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢿⡛⡿⠛⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⣿⡿⣻⡟
⠀⠀⠀⠀⣼⢿⡛⠍⣡⣹⣆⣿⣴⣿⣿⡿⣿⡆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢯⣿⡄
⠀⠀⠀⠀⠘⣿⣿⣬⡿⡿⣋⣫⣷⣶⣿⣿⣿⡏⣻⣯⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠆⠀⠀⠀⠀⠀⢀⣬⣿⣿⡄
⠀⠀⠀⠀⠀⢹⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣍⣿⣿⣝⢦⡀
⠀⠀⠀⠀⠀⠰⢟⣛⠛⠛⠛⢿⠛⠛⣻⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣾⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣈⣆⠀⣧⣼⣷⡿⣫⣶⣿⣿⠿⣿⣿⣷⡽⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣿⣿⣧⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣷⣟⠱⣿⣾⡼⠋⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡔⣺⡿⢿⡛⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⣿⣿⣿⠭⠿⢿⢧⡀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣥⣀⣠⣽⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣄⣈⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣹⣿⣿⣿⣿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⢄⡀⠛⠛⠛⠓⠃⠀⠀⠃⠑⠒⠒⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡇⠸⠤⣧⠀⠀⡆⠘⢵⠶⠟⠸⠿⠆⠗⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠟⢠⠀⡇⢀⠄⣤⡀⣰⢠⡄⡀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠀⠃⠃⠇⠸⠣⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠾⠁
⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⣀
⠀⠔⢂⠶⠘⣆⣁⡀⠔⠒⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⠶⠖⣒⡒⠒⠤⡀⠀⠤⠒⠀⠀⠀⠀⠀⣠⠖⢻⡀
⡏⠙⠳⣄⣀⣭⠤⣽⣄⣀⡼⡄⠀⠂⠀⠀⠀⠀⣯⠀⡀⠀⠀⠀⠀⠀⠈⢶⠼⡄⢌⢑⣦⠄⠀⠠⠟⣩⠾⠛⠁
⠇⠀⠀⠀⠀⠀⠀⠀⡜⢻⠾⠃⠀⠀⠀⢀⠀⢀⣧⣠⣀⣀⣌⣠⣄⣦⣤⣴⣾⠇⠀⠄⡉⣁⠴⠚⠉
⢰⢰⠀⠀⠀⠀⠀⠈⠀⢀⡇⠀⠀⠄⠀⠜⠀⢧⡈⢙⣛⣚⣛⡓⠋⠍⢝⡪⠟⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠋⠢⡀
⠘⠤⢤⣤⣤⣠⣦⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠚⠃⠉⠁⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢠⠆⠀⠀⠁⠰⡐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠯⠄⠉⠙⠲⣄⠀⠀⠀⢹⡀⠀⠀⠀⢠⠏⠀⠀⠁⢀⣤⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢄⣀⡒⠁⠀⠂⠀⠀⠀⠀⡠⠁⠂⠀⠀⠀⠢⡀⠑⢄⠀⠀⠃⡀⠀⠀⠘⠀⠀⠀⠈⠍⠉⡉⣝
⠀⠀⠀⠀⠲⠅⠀⠀⠀⠀⠀⠉⠁⠀⠀⢄⠀⠀⠀⣰⠁⠀⠀⠆⠀⠀⠀⣈⣦⠄⠆⠀⠀⠁⠀⠆⠀⠀⠀⠀⠀⠀⠐⠄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠡⠀⠀⢀⢀⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠃⠀⣠⡴⠚⠉⠀⠘⠀⠀⠀⠀⠀⣸⠀⠀⠀⠠⠀⡆⠀⠀⠀⠀⢰⣼
⠀⠀⠀⠀⠀⠠⠠⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠺⠶⠛⠁⢀⣠⣤⡦⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠔⢉⠉⢳⠀⠀⠀⠀⠀⢀⡀⢀⡀⡀⠀⠀⠀⠀⣠⠞⠙⠠⠁⠉⠉⠀⠉⢱⡀⠀⣀⣠⣁⣉⣡⡀⡀⠀⠀⠸⣽⣋⣦⣄
⠀⠀⢆⠀⢸⡄⠀⠀⣴⠟⢪⠟⠈⠐⠒⠣⣄⠀⢠⠏⠀⠀⠀⠀⠀⣀⣥⣾⣽⠀⠀⢉⣏⠉⠹⡟⡇⣧⠄⠀⠀⠈⠉⠀⠈
⠀⢰⡃⠀⢸⠃⠀⣺⣛⠔⠁⠀⠀⠀⠀⠀⠈⡀⠸⣄⣀⣤⠔⢲⠟⣿⣥⡴⣚⣡⡆⠀⡇⠢⢴⡽⠿⣏⠀⠀⠀⠀⠀⠀⢀
⠀⡇⡅⠀⢸⠀⠀⠈⠓⠢⠤⠤⠀⣀⠀⠀⠤⠁⠀⠈⣭⣄⣀⣋⣭⣾⡟⠋⢱⡜⠀⠀⢇⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠁⡠
⠀⢹⠁⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢅⣀⣉⣉⡠⠴⠒⠉⠀⠀⠀⠘⠀⠀⠀⡟⠀⠀⠀⠀⠀⠉⠉
⠀⠈⠣⡜⠁⠀⠀⠀⠀⠀⠁⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠂⡀
⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀⣀⣤⣴⠟⠛⠉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⢶⠈⠢⡀
⠀⠀⠀⡁⠀⠀⠈⠁⠀⠀⡙⠉⠉⠛⠉⠑⡉⠀⠠⠐⠗⠛⠉⠉⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⡿⣼⢄⣶⠟
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠆⣀⣀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⢀⡀⠀⢀⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣠⢋⣵⢯⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⡭⡇⡆⡆⣆⣯⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⣜⣻⠊⢣⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡵⠋⠀⠉⠉⠉⠁⠀⠙⢮⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⡎⠀⠀⠀⠀⢳⡈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢱⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡜⠀⠀⠀⠀⠀⠀⢣⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠚⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠚⠓⠒⢲
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠐⠥⠤⠤⠤⠤⠤⠤⠬⠂⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡸⠀⠀⣀⣀⠀⡀⠀⡀⢀⡀⣀⣀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢲⡖⢠⣶⡀⣶⣴⠂⣶⣒⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠃⠀⠀⢹⡏⣰⣿⡆⣿⢿⡁⣿⣟⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠃⠚⠉⠃⠛⠘⠃⠛⠒⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⣀⠀⡁⢀⡀⣁⣈⠉⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣶⢲⡆⣶⢰⡖⢲⡖⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣿⣹⡇⣿⣸⠇⢹⡏⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠚⠁⠙⠛⠁⠘⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⠤⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠈⠉⠓⢄
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠤⠤⠷⠶⢦⡈⢧⣤⡄
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⣹⣉⣏⣯⣹⡶⠿⡇
⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡇⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠳⡖⠐⠚⠒⡶⠃⠀⢣⡀
⠀⠀⠀⡇⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠙⢆
⠀⠀⠀⡇⠀⢿⠁⣾⣧⢰⣦⡦⢠⣄⡀⠀⡇⠀⡇⠀⠀⡇⡰⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠧⠤⠤⠤⠇⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⣀⡁⠙⠘⠋⠳⠸⣿⡀⠀⡇⠀⡇⢀⡔⠋⠁⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸
⠀⠀⢸⠀⠀⠸⣏⣹⢸⡆⣶⠰⣤⠄⠀⠀⡇⢀⡗⢋⣠⣦⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⡏⢡⣿⣇⢸⢾⡅⣿⡿⠀⠀⢸
⠀⠀⢸⡀⠀⠀⠀⠁⠈⠛⠋⠀⠗⠀⠀⢠⠃⢠⠿⠧⣤⡟⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⣈⠀⠈⠈⡀⣉⣈⠉⠀⠀⢸
⠀⠀⠀⠈⠉⠒⠲⠤⢤⣀⡀⠀⠀⠀⠀⢸⣠⢃⣴⣚⣉⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢾⣉⡷⢿⣸⡇⢹⡏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⠼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠏⠁⠀⡅⠀⠩⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⡇⠀⠀⠀⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠹⠆⠀⠀⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣆⠀⠀⠀⠀⢀⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠾⡷⠾⢟⣫⣼⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⢸⣷⣶⣿⣿⣿⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⠄⢸⣿⣿⣿⣿⣿⣿⣧⣙⣛⣛⣋⣼⣿⣿⣿⣿⣿⣿⠇⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢸⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠿⠿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢸⣿⣿⣏⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣶⣶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠧⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢋⣤⣶⣬⡻⣿⡿⣣⣴⣦⣜⢿⣿⣿⣿⣿⣿⡟⣣⣴⣦⣜⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⣏⣀⣽⡇⠛⠁⣿⣅⣨⣿⠈⠛⠛⠛⠛⠛⠁⣿⣅⣨⣿⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠰⠶⠶⠶⠾⠿⠿⠿⠶⠶⠶⠾⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠷⠶⠆⠰⠆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠶⢰⢲⡆⢶⡐⡗⠀⣿⠈⡷⣴⠦⣽⢰⢲⡰⢴⠶⢰⠒⢆⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠉⠀⠉⠀⠈⠉⠀⠈⠁⠈⠈⠀⠁⠈⠉⠈⠀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠠⢀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠈⡕⠄⠀⠀⠀⠀⠀⡀⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣀⡤⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠁⠈⠛⠛⢋⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⠄⠀⠀⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡀⠀⠀⢀⠚⢓⠀⠀⠀⠀⠀⠀⠀⠐⠒⠄⠀⠀⠀⠠⠡⡇⠔⠀⠀⠀⠦⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠠⡉⠛⠄⠇⠀⠀⠰⠀⢂⠁⠀⠀⠀⠀⢄⠀⠀⣠⠃⠀⠀⠀⠠⠄⣨⠉⠅⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡀⠀⠀⢀⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠲⠶⠛⣁⠀⠀⠀⠀⠀⠀⠁⠀⡐⢁⡤⠃⠀⠀⢠⠔⠒⢦⠀⠀⠈⠲⠦⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⣀⡀⠀⠀⠀⠆⠰⠁⠈⠀⠀⢄⡀⠸⠄⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⣉⢘⠂⠀⠀⠀⠀⠀⠀⠑⠀⡞⠁⠀⠀⠀⢀⡔⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡎⠀⠠⣬⣯⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⣧⠈⠀⠀⠀⡰⠀⠀⠀⠤⠊⠀⠀⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⣀⠆⠀⠀⠘⢠⢐⡇⠀⠈⠳⠷⠶⠛⡁⠀⢠⡄⠀⠀⠀⠀⠀⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠔⠀⠂⠀⠀⠀⠀⠀⠀⠈⣀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠐⠀⡀⠀⠀⠀⢂⡔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠰⣾⣽⣯⠇⠀⠀⢀⡀⠀⠀⠀⡴⠀⠘⢦⣠⣀⡤⠊⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢀⡀⢠⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠠⢤⡈⣷⡀⠀⣀
⠀⠀⡠⠊⠁⠀⠀⠀⠀⠈⠑⢄⠀⠀⢀⢀⣴⡞⢹⣧⣿⣱⡡⣥⢾⡼⡆⠀⠀⠀⠀⠀⣠⠀⢤⢀⣆
⠀⢰⠁⠀⠀⠠⠀⠀⠄⠀⠀⠈⡆⠀⠈⠢⣇⣧⡞⣼⡟⣸⣸⡕⡯⢁⡯⠃⠀⠀⢀⠈⠀⢱⣶⣿⣷⣀
⠀⢳⣂⡂⠀⠀⠀⠀⠀⣀⣀⣈⢗⡄⠀⠀⢹⠙⠃⠷⠴⠿⠧⠶⠜⢊⠁⠀⢠⠔⠉⠁⠈⢁⣉⣉⣀⠀⠩⠶⢤
⠀⣶⣦⣽⣭⣭⣿⣹⣷⡤⣼⢵⡞⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠖⠢⠖⣄⠖⣤⠒⠤⠲⡔⠶⣾
⠀⢫⣽⡯⠹⢖⣛⣻⣷⣾⣿⢿⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡻⠶⠶⠶⠶⠖⣖⣒⣒⣲⣟⡇
⠀⠀⠑⠂⠀⠀⠀⠀⠐⠒⠚⠁⠀⠀⠀⠀⠀⠀⢆⣀⣀⣀⣀⣀⡜⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠤⢄⣀⢀⣀⠤⠔⠒⠉⠒⢄
⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⡠⠤⡴⠁⠘⠉⠀⠀⠔⠠⠕⠀⠐⠁⢁⡈⠀⡼⢒⡶
⠀⠀⣀⣔⣊⣁⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠬⠵⠦⠀⠀⠑⢢⣘⣌⣀⣉⣀⢀⣀⣐⣒⣒⣂⣈⣡⣴⡊⣤⠋
⠀⠀⢒⣒⣶⣶⣶⣶⣶⣀⣈⣏⣉⣻⡿⠯⣿⠏⡹⡄⠀⠀⠓⣖⠪⡭⢿⡶⢲⣲⣒⣶⣿⣿⣿⡿⢔⣵⡿⠂
⠀⠀⠁⠐⠚⠒⠚⠒⠒⠂⠀⠀⠉⠀⠈⠈⠁⠀⠉⠁⠀⠀⠀⠈⠁⠒⠀⠉⠉⠓⠂⠐⠒⠓⠒⠉⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⣀⠔⣹⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡔⢣⠃⠀⠀⠀⠀⠀⢰⣍⡠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡃⢸
⠀⢸⠁⠘⠎⣓⡦⠀⠀⠀⠀⢠⠒⢶⡀⠲⢥⠇⠀⠀⣠⠤⠀⠐⢆⡹⠲⠶⢦⣀⠀⢀⢔⢒⣄⢠⠚⡟⣿⠛⠛
⡀⠮⡝⣉⣉⣩⠖⠳⡀⢠⠋⡀⢁⠉⣈⢢⠜⡄⠀⠀⢿⡓⠒⠒⠒⠒⠒⢒⣚⠏⠀⠰⠧⠄⢉⠎⣣⠿⠉⠘⠂
⠑⠶⠤⣤⢤⠾⣶⡚⠀⠘⣍⠙⠈⢉⡠⢯⠴⢸⠀⠀⠈⣦⣒⣙⣭⣋⡑⠠⡼⠀⠀⢀⠀⣰⣯⣾⡋⣦
⠀⠘⣮⡱⢪⠼⣱⠃⠀⠀⠡⢍⡉⠉⠉⣀⠔⠁⠀⠀⠀⢰⠄⣰⢀⢀⠀⢰⠃⠀⠀⡗⠀⠨⠝⠉⠀⣺⡗⠉⡄
⠀⠀⠘⡾⠿⠾⡾⠀⠀⠀⠀⠀⢀⢙⣉⣀⠀⠀⠀⠀⠀⠀⠄⢽⠈⠤⠀⡏⠀⠀⠀⠹⡉⠛⠛⠓⢈⡽⠧⠕⠁
⠀⠀⠀⠐⡄⢠⠃⠀⠀⠀⠀⢈⡀⠀⠀⢀⡹⠀⠀⠀⠀⠀⢠⣞⡘⢚⣾⠃⠀⠀⢠⡖⠚⠒⠂⠒⠋⠙⢱⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⢴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠿⠋⠉⡛⢻⡿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⢠⡟⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⣼⠃⠀⠀⠀⠸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠻⠶⢦⣤⣄⡀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣴⠶⠶⢶⣦⡈⣤⡿
⠀⠀⠀⠀⠀⠀⠘⠛⠘⠛⡛⢛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⡿⠀⠀⠀⠙⢷⣌⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣁⣉⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣧⣤⣤⣤⣤⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢩⡉⣭⣭⣭⣭⣽⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⠿⠿⠿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣼⣿⡿⣵⠟⠛⣿⠼⣿⣿⣿⣿⣿⡟⣼⡟⣵⠛⠛⣷⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠃⠙⠛⠛⠛⠛⠛⠛⠛⠛⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⣿⡿⠟⣛⡛⠛⣛⣿⡿⠛⢛⣿⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣴⠞⣉⣴⣿⠿⠿⠿⠿⠟⠻⠿⣿⡁⣠⠜⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣏⣀⣼⣟⣻⣿⣀⣀⡀⠀⠀⠀⠀⢸⡇⣀⡴⣿⡇
⠀⠀⠀⠀⠀⣠⡴⠿⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⢶⣦⣄
⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣆
⠀⠀⣴⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠙⢻⣦⠀⠀⠀⣾⠛⣷⢠⡟⣷⣴⡿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⢻⣧⠀⠀⣿⠀⣿⢸⡇⠉⢹⡇⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢿⡆⠀⣿⢀⡟⢸⡇⡄⠀⣿⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⢿⡄
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠸⣇⣸⠃⢧⡤⠃⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⢀⣀⣼⠇⠀⣿⣀⠉⠁⠀⠀⣠⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣷
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡆⢿⣿⠛⠁⠀⠀⠈⠙⢣⡀⢾⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⣿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣿⣆⠀⠀⠀⠀⠀⢸⡇⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿⡄⠀⠀⠀⠀⢸⡟⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠸⣇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡇⠀⠀⠀⠀⢸⡇⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣤⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣾⠇⠀⠀⠀⠀⠸⣧⣀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣁⣸⡆
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡾⠋⣼⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⡇⠀⢠⡿
⠀⠀⢸⣆⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣧⠶⠛⠁
⠀⠀⠘⢧⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣾⠿⠃
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⡀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢄⠀⠀⢀⡀⠢⢨⣄⠄⠠⠄⡠⠤⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⡾⠂⠂⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠⠈⠀⠆⣰⠁⠀⠀⠀⠁⠈⢀⠨⠀⠌⠁⠒⠡⠅
⢀⠀⠀⠑⠤⠐⠁⠀⢠⡆⠀⠀⠀⠐⠀⣄⠀⢀⠤⠦⠀⠀⠔⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⣡⣀⡀⠰
⠈⠆⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠰⡠⠅⠈⡀⡀⠵⣠⡠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠙⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⣞⠀⢈⠔⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⢀⣀⠀⣀⡀⠀⡀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠉⠀⠀⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⣸⡇⠸⡏⠓⠉⡏⢸⡏⠁⣼⢹⡇⣿⢹⡆⡏⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⣿⣿⢀⡙⣆⠀⡇⢸⡟⠁⣽⢸⡇⣿⢸⡇⡇⣸
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠂⢠⠀⠀⠀⠠⠀⢀⡤⠃⠀⠀⠀⠀⠈⠃⠀⠃⠘⠈⠛⠃⠀⠃⠘⠃⠀⠘⠛⠁⠙⠛⠀⠛⠋
⠀⠀⡀⢶⠀⠀⠀⢀⠀⠉⠁⠀⠀⠐
⠀⡚⠁⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈
⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣲⣀⣤⠚⠉⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⠊⠇⠉⠈⠒⠢⣠⡀⠀⠀⠀⠀⢠⣶⣶⣄⡀⣄⣦⠀⡠⢤⠴⡚⠊
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠀⠀⠀⠀⠈⡠⠈⠓⣄⠀⠀⠀⠀⠁⠀⠀⠈⠙⠩⠾⢶⣌⣭⣬⡶
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣊⠠⡢⠀⠀⠀⠀⠀⠀⠀⡀⢐⣠⣶⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠭⠿⠋
⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠁⠐⠀⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠒⠀⠀⠄⠴⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠸
⠀⠀⢸⠖⢤⠀⠀⠀⡤⠤⣤⣀⠐⣞⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⢾⡀⠙⠓⠂⠀⠲⠌⠀⡀⠔⠋⢱⠀⠀⠀⠀⠀⠀⠀⡀⣈⡅⠒⣬⢁⠒⠆⠀⢀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⡠⠚
⠀⠀⠈⠤⣄⠀⡀⠐⠂⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀⠜⠄⠀⠄⠀⠈⡈⢀⣙⡢⠊⠵⢲⣀⡰⠆⠀⠀⠈⠡⠀⠀⠠⠐⠊
⠀⠀⠀⠀⠀⠂⠬⠁⠐⠀⠉⠄⠛⠁⠀⠀⠀⠀⠀⠉⠒⠒⠾⠤⣤⠈⠁⡔⠉⣀⣒⣀⣀⣐⣂⣀⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠐⡎⢀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠐⠀⠀⠀⠤⠀⢀⢀⠐⠃
⠀⠀⠀⠀⠀⠀⢁⡠⠄⠀⠀⠀⠀⠀⢸⡦⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠸⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠄⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⢉⠀⠀⡈⠀⠁⠀⠀⠀⠀⠉⠐⠄⣀⠤⠄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡴⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⢇⠀⠀⠀⡠⠀⠀⠐⠀⠀⠀⠀⢈⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢻⠦⠐⢪⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠠⡀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠈⠖⠁⠀⣠⠊⠁⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢤⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠆⠄⠀⠀⠀⠀⠠⠂⠀⠀⠠⠀⢀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠐⠀⠜⠀⠀⡈⠀⠀⠀⢀
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠂⠃⠀⠂⠀⡘⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈
⠀⠀⠀⠉⠑⠒⠦⠤⠤⠤⠴⠚⠀⠀⠀⠀⠀⠈⠲⠐⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⡀⢀⡀
⠀⠀⠀⣄⠀⠈⠁⠀⠀⢠⡀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠊⡀⠐⡀⠑⡄
⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠐⠀⠀⠀⠀⠀⡁⠀⠀⠀⠈⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⠀⠀⠀⠀⠀⠆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⣶⢄⡀⠤⡀⡠⠄⠤⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⣿⢀⠀⡄⠐⠰
⠀⠀⠀⠀⡠⠂⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⠀⠀⠙⠒⠴⠦⠗
⠀⠀⠀⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⣠⠀⠈⠉⠀⠀⠠⣄⠀⠀⠀⠀⠀⠁⠡⢀⢀⡤
⠀⠀⠀⣀⢉⣖⠂⠤⢤⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠠⠄⠀⠀⠀⠘⠁⠀⠀⠀⠈⠁⠒⡐⠐⠂⠐⡒⠂
⠀⠀⠀⠇⠙⠨⠁⠀⠤⠤⠈⡀⠀⠀⠀⣄⠀⠀⡀⠀⠀⠀⠀⠁⠠⠀⠀⣃⢀⡂⡀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠈⠢⠄⡀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠓⠣⠄⣆⣀⠀⠊⠀⢀⣆⠀⠄⠐⠁⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⠆⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠉⣳⡤⡩⢙⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⢖⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢜⣶⣿⠀⠉⢳⡸⠀⠀⠀⠀⠀⠀⢠⣶⡤⠸⣿⡀⠮⠍⡾⢣⢟⡆⢹⣿⠀⠀⠀⠀⠀⡴⢊⠕⣺⡉⣽⣾
⠀⠀⠀⠀⣀⣀⣀⣈⣉⣉⣉⣉⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠙⣿⣶⣤⣤⣈⣬⣴⡿⠋⠀⠀⠀⠀⠀⠈⢁⡿⢯⣄⠹⣹⣧
⠀⠀⠀⠀⠐⠦⣾⡿⠿⢿⣿⠃⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠈⣿⡄⠀⠚⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠚⣸⣾⠏
⠀⠀⠀⠀⠀⢸⢃⡀⠀⣀⠘⣷⣤⣤⡞⣠⠶⣦⠀⠀⠀⠀⠀⠘⢿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠨⢿
⠀⠀⠀⠀⠀⠈⠈⠳⠶⠋⠀⠉⠉⠉⠁⠙⠶⠞⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠷⠬⠭⠭⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⢩⣭⠠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣭⣿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⡀⠀⢀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠲⣺⡟⠰⡆⠆⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠶⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢀⢀⣈⠸⠀⣀⣾⠷⠴⢹⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣍⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢼⠞⢿⠁⠼⠻⠻⢀⠮⠆⣿⠀⠀⠀⠀⠀⠀⣾⢋⣯⢩⣟⣿⢛⡟⡟⡦
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣷⣶⣾⣷⣶⣶⡄⠀⠀⠀⠀⠀⢸⣆⢸⡓⢒⢖⣠⡍⠐⠋⢶⢄⡟⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠹⣮⡁⢠⢾⠿⠟⠜⠶⣬⠟⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣙⣛⣛⣁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠷⠤⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠒⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⢀⢳⠖⠚⠉⠉⠉⠉⠓⡚⢀⡄⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠄
⠀⠀⠀⠀⠀⢎⣾⡆⠀⢀⠖⢴⠀⡘⢀⠀⣾⡇⠀⠀⠀⠀⢸⣿⡿⣟⣿⣿⣿⣽⣛⢿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠁⠀⠀⠀⠂⣼⣿⠃⠀⠀⠀⠀⣼⢱⣯⣡⣹⣹⣢⡤⢟⣼⣿⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⣐⠈⣡⠀⢸⠟⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠌⢰⡟⠁⠀⠀⠀⠀⠀⢸⣶⣽⣭⣽⣿⣿⣾⣿⣿⣿⣶⣾⣷⡀⠀⠀⠀⢴⠀⠘⠋⡰⣺⣶⣾⣿⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢳⡄⠀⠀⠀⠀⠀⢠⠟⢳⡄⠀⠀⠀⠀⠀⠀⠈⢀⢽⣿⣿⣿⣿⡳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠁⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⡤⠐⠀⠀⠀⠀⠀⣟⠟⠀⣿⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠐⠀⠀⠈⠁⠀⠀⣄⣀⠀⠂⠈⠉⠀⠈⠁⠀⠀⠀⢀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⢰⣆⣠⣀⢈⣉⣁⣤⣄⣈⢀⣺⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣛⣽⣿⡄⠀⠀⠀⠀⠘⠛⠻⢃⣬⠉⢙⡛⠛⠉⠜⠉⠉⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠉⠋⠉⠛⠁⠀⠀⠀⠀⡆⠀⠀⣤⡾⠛⡾⣿⢿⣿⠿⣷⠀⠀⠈
⠀⠀⠀⠀⣄⣰⠐⡠⠅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣴⢦⣆⣾⣶⣾⣴⣦⣴⡄⠀⠠
⠀⠀⠀⠀⠈⠉⠉⠳⠮⠥⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢈⡉⠉⠉⠛⠭⠯⣉⡙⠁⠌⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⣿⣷⡼⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠠⠀⠠⠀⠮⠽⠿⠄⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⢀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠱⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠲⣦⣖⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢉⡞⣳⣛⣻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠫⢠⣥⣼⣷⢚⣦⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⡁⠀⠀⠀⠀⠀⣰⣾⡯⣽⣿⣧⣴⡀
⠀⠀⠀⢀⣴⡀⠂⢀⠀⢦⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⣠⣴⡶⣶⣤⣤⢀⣄⡀
⠀⠀⠀⠀⠙⠛⠃⠭⠀⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠩⠿⠿⠏⠏⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠆⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣖⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠠⠀⠀⢀⣀⣠⣤⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⢿⣿⢵⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠷⠶⠒⠛⠁⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣠⢄⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠦
⠀⢀⠀⠀⠠⣰⡏⠡⢄⡅⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡶⠶⠶⠆
⠀⠀⠁⠀⠀⠀⠁⠙⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⣄⠤⣄⢠⡀⠠⠀⠀⠀⠀⠀⠀⠀⣀⠀⢐⡦⠛⢉⣠⡴⠋⣀⠆
⠀⠀⠀⠀⠀⠈⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠱⠟⠐⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⣹⠀⠴⠻⠯⠃⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠤

⠺⢉⡉⠉⠉⠒⠒⠢⠤⣤⣁⣀⡔⠉⡙⠓⡄
⠀⠀⠈⠑⠂⠤⢀⡀⡠⠒⢪⡹⠟⠒⡦⠤⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠎⠀⣠⠞⠐⡇⠀⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠲⠊⠁⠀⠀⠹⣯⡕⡒⡄
⠀⠀⢀⣀⡤⠔⠒⠉⠀⢈⠇⠀⠀⠀⠀⠀⠻⠟⠁⡇⢀⣀⣀⣀⡀⠠⠤⠄⠀⠀⠂⠀⠐⣒⡂⠀⠠⠄⠒⠒⣚⡉⣭
⢠⠊⠉⠀⠀⠀⠀⢀⡴⢏⠀⠀⠀⢀⠎⠱⡀⠀⡎⠀⠈⡤⠤⠤⠤⠀⠤⠀⠀⠈⢉⡯⡤⠤⠐⠒⠒⠈⠁⠀⠀⣐⠂
⠀⠑⠲⠤⠧⠒⠊⠁⠀⠈⢆⠀⠀⠈⡆⣤⠧⠀⠘⡄⠸⠩⡭⠤⠤⠄⠀⠀⠀⠀⢀⣒⣉⠨⠤⠤⠖⠒⠒⠚⣉⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣢⠀⠀⠸⠁⠀⠀⠀⠐⡀⠨⡉⣀⠀⠂⠀⠀⠂⠀⠀⠸⠤⠒⠐⠂⢈⣁⡀⠤⠤⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢜⡮⢧⣀⡀⠀⠀⠀⠀⠀⡸⠭⠷⠆⢀⡀⠤⢈⡀⠀⡀⠀⣳⢀⣉⠁⠤⠀⠐⢂⡀⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⣤⣈⠉⠁⡖⠦⡖⠛⠥⠀⠀⣀⠀⠤⠀⠌⣉⣙⣉⠩⠤⠀⠀⡂⣨⣍⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢉⠒⠁⠢⢄⣴⠛⠀⣴⡤⠤⣄⣉⣰⣉⠤⠀⠒⢀⣁⠠⠄⠐⠀⣉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠑⢤⡀⣠⣯⡍⣉⣢⣸⠒⠠⠤⠀⢀⣉⢀⣏⡀⠤⢐⡂⠅⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠉⠀⠀⠉⠒⡄⠈⠙⣀⣲⠤⡤⠄⡬⠤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡖⠒⠒⠒⠒⠒⠋⠙⡛⡞⢇⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⢱⠙⡎⠀⠀⠈⡄⠀⣀⢀⢀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⠀⠀⠀⠀⠀⣸⣤⠷⠖⠉⠀⣨⡿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡘⢄⠀⠀⠀⢉⠏⡇⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣄⢠⠞⢳⠟⠂⠤⠤⠒⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠶⠤⠤⠤⡆⢠⣀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⣀⣀⣀⣀⣠⡄⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠀⠀⣀⣸⠀⠀⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⣸⠀⢉⠉⠉⠙⡋⠉⠉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠆⢀⠠⠀⢠⠃⠀⠀⠀⣀⢰⡇⠀⠲⠌
⠀⠀⠀⠀⡀⠈⠀⡇⠀⠀⠃⠐⠀⡀⢀⠀⠀⢀⡡⠷⠓⠚⠉⠉⠉⠉
⠀⠀⠀⡞⢲⡺⣄⢡⠀⢀⠆⢠⡇⠁⠈⣥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⡀
⠀⠀⠀⣇⢸⡍⠳⣆⣰⠆⣠⠋⣰⡠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢫⡢
⠀⠀⠀⢻⠘⡇⠀⠈⠉⠛⠛⠛⠉⠀⣾⣦⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⣿⣿⣷⣾⣶⣤⣀⣳⣄⣦⣄⣠⡴⡊⢁⣀⣤⣖⣁⣤⠾⠁
⠀⠀⠀⠈⣧⣿⠀⠀⠀⠀⠀⠀⠀⢀⡞⢃⠠⠉⡛⡛⠛⠿⣿⠿⢿⡿⢿⣿⣿⡿⠿⢿⣿⠛⠙⠻⡤
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⡈⢀⣌⣉⡒⢒⢫⣀⡴⠶⡾⠵⠶⢥⠤⠤⠤⢤⡤⣮⣭⣵⡿⠃
⠀⠀⠀⠀⠀⢹⣇⠀⢀⣀⣀⣀⣀⣈⣍⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⢈⠀⢀⣴⣿⣿⡟⢁⠇
⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠉⠉⠉⠉⠉⠓⢤⣀⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠋⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠦⠤⠤⠀⣀⣀⡤⠴⠒⠋⠁

⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠖⠒⠶⠦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⢭⡳⢤⣀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠑⠨⣙⡲⢤⡀
⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠘⠓⠻⣧
⠀⠀⡇⠀⠀⠐⠀⠀⠰⢏⣉⣉⣉⡿⠎⠀⠀⠀⠀⠀⠀⡇⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠂⢀⣤⠼⣇
⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠄⠀⡇⠀⢻⡉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⢠⡖⠃⠶⠁⡠⠔⢺
⠀⠀⢳⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⡰⠃⠀⢸⡓⠠⢤⠖⠀⡔⠋⠐⠁⢀⠔⢠⠄⠀⠰⣃⣘⣒⣶⠥⠶⠚⠋
⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡠⠞⠁⠀⠀⠀⢻⡤⠎⠀⣀⣰⣂⣲⠮⠼⠖⠒⠒⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠓⠶⠤⣤⣤⣤⡤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠞⠆⢸⣏⠈⣇⢀⣤⠖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠲⣄⡏⠀⢸⢧⡀⡾⣜⠀⠀⠀⠏⠠⠁
⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠑⢦⡀⠀⠀⠀⢠⡂⠣⣀⠛⣇⡀⠙⣿⣿⠀⠘⢆⠀⠀⠀⡆
⠀⠀⠀⠀⡴⠃⠀⠀⠐⠀⠀⢠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢷⠀⠀⢜⡉⢿⡦⠈⠀⠈⠀⠀⠡⠻⡆⠀⠘⠄⠀⠀⠹⡀
⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡆⠀⠀⠙⠢⣀⠀⠀⠀⠅⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢰⠛⠂⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠐⣾⠁⣾⠁⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⡄
⠀⠀⠀⢹⡀⠀⠙⠲⡏⠬⠭⡍⠉⠀⠡⠅⠈⠐⠶⠚⠇⠉⠤⢧⡀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢱
⠀⠀⠀⢠⠏⠉⡐⠀⠉⠀⠀⠀⢤⡀⠀⠀⠀⠈⢠⠂⢐⡥⠀⢀⡷⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢦⠉⠀⠑⠒⠰⠶⠐⠒⠈⠁⠂⢀⠀⠉⠉⠁⢀⠔⡿⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠇
⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠉⠉⠀⠀⠤⠐⠊⠉⠀⣀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⣀⣠⡤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⠤⠀⠀⠠⠴⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣝⣫⣍⣹⣝⣽⣟⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿