35 Emojis & Symbols

⠀⠀⠀⠀⠀⠀⣿⠓⠄⠀⢰⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⡅⢿⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⡙⠷⠄
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠆⢛⠰⢸⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠿⢶⠋⣡⠤⡀
⠀⠀⠀⠀⠀⢀⣈⠦⣄⡈⢨⡅⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⣩⡅⠀⠀⢪⣿⡖⠀⠀⢨⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣷⡄⢁⠚⠒⢤
⠀⠀⠀⠀⢰⡏⠀⢠⣞⡛⢷⣿⣶⡗⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡀⠢⢀⠿⠀⠐⢀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣋⢴⡩⠏⢀⡀
⠀⠀⠀⠀⠀⠙⢦⣄⡈⠁⠀⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⠥⠊⠀⠱⠼⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠄⣀⠨⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⢀⡤⠤⠤⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖⠠⡀⠀⠀⠀⠀⠀⡠⠐⠲⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁⢨⡇
⠀⠀⣤⠚⣻⡉⠉⢩⡖⠂⠲⠢⠤⣤⣶⠀⠀⠀⠀⠀⠀⠀⢻⠄⡘⠄⠀⠀⠀⠀⠀⠘⢃⢼⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⡧⢲⠖⢻⡆
⠀⠀⠘⢭⡿⣄⣀⠀⠀⠈⣁⡤⠒⢻⡟⠀⠀⠀⠀⠀⠀⠀⠈⢣⡤⣴⠲⢾⣶⠷⠒⣦⢤⡟⠁⠀⠀⠀⠀⠀⠀⢠⡟⢠⠟⢻⡠⢍
⠀⠀⠀⠀⠉⠻⠿⢧⠶⠋⠰⠵⢠⡟⠹⠆⠀⠀⠀⠀⠀⠀⠰⣯⡘⢹⢃⡔⠉⢦⠘⡟⢪⣿⠀⠀⠀⠀⠀⠀⠻⣿⣖⡡⢚⣛⣠⡼⣂⣄
⠀⠀⠀⠀⠀⠀⠀⣯⠟⠶⣦⡀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠗⠚⠁⠀⠈⠓⠺⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠾⠛⢀⣡⣶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠿⠿⡿⠓⠘⠉⠈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣹⢿⠟⠀⠀⠠⠶⢶⠖⠂⠀⠀⠉⠉⠿⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣝⠻⠿⠿⢿⣿⠿⠿⠛⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⡶⢶⡉⠛⠛⠛⠙⠛⠻⠿⠟⠛⠛⠛⠛⠿⠿⠿
⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣿⣿⣟⣛⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣿⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⠿⠿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠗⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⠛⢻⣿⠿⠟⠿⠟⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡁
⠀⠀⠠⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠶⠶⠶⠶⠶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⢿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠂

⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠉⢀⣨⠇⠀⣾⣀⡀⠀⠐⠀⠀⢠⢆⠀⣦⡀⢠⠠⣆⠀⡄⢠⢠⠀⠠⢰⠒⠂⢰⠒⣄⢠⠒⠀⠀⣤⠀⢰⠒⢦⠰⡀⡤
⠀⠀⠀⠀⠀⡀⠈⠉⣧⠀⠀⠀⢹⠆⢀⠀⢀⠟⠚⡄⡇⠙⣾⠰⠀⠳⡇⢸⠀⢣⠃⢸⣉⡁⢸⠙⡅⢀⣉⡆⡸⠓⢣⢸⠙⠅⠀⢹
⠀⠀⠀⠀⠀⠙⠶⠖⠃⠈⠳⠶⠋


⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣠⢞⡩⠆⢀⡀⢀⣀⠰⢝⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠲⠊⣵⠄⠠⣦⠑⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⢳⠀⢸⣉⣉⡉
⠀⠀⠠⣟⣾⠁⠀⠀⡇⠐⢦⠀⠈⣧⢻⠄⠀⠀⠀⠀⠀⢀⢡⠲⠞⠙⠀⠀⠋⠳⠆⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠘⠁⠀⠈⣆
⠀⠀⠘⡧⢿⠀⠀⠁⠁⠈⠀⠀⢀⡿⢸⠃⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣇⠸⢅⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡠⠇⢲⣀⣀⣠⣇⣀
⠀⠀⠈⢻⣭⢳⣌⡀⠀⠀⢀⣠⡾⣩⡟⠀⠀⠀⠀⠀⠀⠚⠰⠀⠀⠁⠁⠈⡀⠀⠀⠇⠓⠀⠀⠀⠀⢀⣀⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡤
⠀⠀⠀⠀⠹⢷⣭⡏⠠⠄⢹⣭⠾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢀⡿⡄⠀⠀⠀⠀⠀⠼⣿⣿⠟⠛⠉⠉⠉⠁⠀⠀⠀⠈⠛⠛⠣
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⡀
⠀⠀⠀⢠⠞⠛⢦⠀⠀⣴⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠟⠿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣮⠛⠻⣤⠀⠀⠀⣾⠷⠿⠷
⠀⠀⠀⠀⠀⢴⣟⠀⠀⠿⠶⢤⡀⢈⡁⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⣀⡀⢀⣀⠈⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⢠⡿⢤⣄
⠀⠀⠀⣄⠀⠀⢸⡇⣀⠀⠀⢈⡇⠠⠀⠀⠀⠀⠀⠀⠀⣿⣿⠅⠀⠰⣇⠸⢲⡀⠈⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀⠀⠠⠰⠔⢹⡇
⠀⠀⠀⠈⠓⠖⠋⠀⠈⠓⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡂⠀⠙⠁⠈⠋⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⠟⠀⠐⢷⣤⣴⠾⠃
⠀⠀⠀⠐⠀⠐⠀⠂⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠃⢀⢀⢠⠐⡘⠆⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⢴⢤⠤⠤⠠⠄⡤⢤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢐⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡆
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣰⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡾⣟⡿⠛⠉⠀⠈⠉⢉⠘⠳⠶⢿⠶⢻⣆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠠⣄⣾⣈⣩⣤⣾⠁⠀⠀⢀⣴⠞⣿⣋⡿⠻⠟⣦⠀⠈⣿⣧⡄⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⢠⠿⣯⣯⣹⡇⠉⣀⣤⡄⢠⣜⡔⠋⢻⣷⣆⢸⠛⡆⠀⣉⠹⢿⡀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⣾⡿⣀⢰⣿⡷⠿⣷⣄⣉⠁⢰⣿⡱⢾⡄⠀⠀⠿⢸⣶⡃⢀⣿⣇⣼⡇⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⣿⡿⢀⡬⠘⢟⣃⠅⠉⠁⠎⢠⣟⠞⠁⠀⢳⠀⠀⢀⣽⣼⣡⣼⣿⠏⣻⣿⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⣿⡇⣸⡧⡛⠛⢀⠀⠀⣼⠀⠘⣿⠁⢤⡀⢸⣧⣶⡿⠛⠉⠉⠙⠙⠻⠟⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⣽⡇⡾⠁⢿⣆⡀⠀⠀⠏⠀⣿⡏⠳⠶⡷⢾⠟⡋⠀⠀⠀⠙⣛⠂⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠈⠿⣆⠀⢿⣿⠖⢣⡄⠣⠈⣿⣇⣈⣄⠀⠈⠳⣬⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠹⠆⢺⡀⠐⣿⠷⡀⠙⣿⠛⢋⣽⡾⣆⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢹⣇⣁⣰⣼⣿⣩⣿⠀⣸⢷⣄⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⢠⣿⠏⠀⠘⣇⠉⠁⠀⠛⠂⠐⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣾⠀⠀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠤⠴⢞⡟⢛⢻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣼⠟⠁⢾⠀⢩⠇⣷⡹⣆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠋⢸⡀⢈⡴⠀⠀⠈⠓⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⡾⢃⠹⠀⣼⠀⢀⡀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠈⡑⠀⠀⢠⠘⠇⠎⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⣀⠀⠀⢼⡁⣰⡇⡏⠀⣸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠛⠛⠷⢾⠅⢹⠂⣿⠴⠿⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠟⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠒⣀⡙⢦⡀⠹⡄⢸⠀⣰⠃⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠉⠙⠲⠤⣄⡀⠙⢀⡥⠤⠤⣅⠐⢋⣀⠤⠖⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⣀⣀⣀⣉⣉⣀⣀⣀⣉⠁⡏⠀⠀⠀⠈⣧⢉⣀⣀⣒⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠚⣆⠀⠤⠤⠀⢀⣴⣄⠀⣠⠄⢧⡀⠀⠀⣠⠇⠠⣄⡀⠀⠐⠒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠷⠄⠈⠉⣷⣌⢁⡴⢉⡟⡜⣆⢀⡴⠂⡉⢉⠉⠅⠐⢤⠀⢭⠙⠲⠄⠉⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡦⠀⠴⠄⢀⡞⠉⣰⣛⢒⡭⠼⢦⡀⢸⠃⠸⠀⡰⣿⣦⠑⠄⢈⣠⠼⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣏⢀⣤⡐⠋⠀⢰⠿⠚⠁⠀⠀⢸⡽⣄⠀⡰⠋⠀⢻⢨⡳⣏⠃⠉⠀⠀⠀⢤⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣐⣺⠓⠰⣿⠆⠀⠀⠀⠀⣠⡀⠀⠀⣞⢰⣈⠻⣄⠀⠀⢯⣚⡗⡼⠓⣆⠐⠒⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠓⠪⢯⡁⠀⠀⠀⠀⢿⡧⠠⣿⠌⢳⣀⡞⠈⢣⡀⠀⠈⢯⡇⠀⠈⢴⡦⢐⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢧⠈⠻⠂⠀⠈⠓⠦⣄⣀⡽⠯⠬⠿⣥⣈⣁⡀⠀⠀⠳⡄⠀⠈⠁⠀⠀⣹⣏⢐⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡛⢷⣦⣬⣉⣀⠓⢀⡀⠙⠲⣄⠒⠂⢀⣀⡀⠉⠉⠙⠓⠛⠲⠶⠶⠶⠖⠚⠳⠦⠿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⠮⢽⣛⣛⢛⣩⠇⣈⣁⠈⠙⠦⣀⠀⢀⠀⠒⠂⠀⠓⠀⢀⠈⠁⠀⠐⠚⢂⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣁⠀⠠⠭⣉⣁⣀⣉⣁⣀⣘⣀⣈⣙⡏⠙⠯⠭⢭⡉⠭⠭⠭⠭⠉⠭⠭⠭⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡞⣽⠃⠈⠓⡀⠀⡤⠶⠶⠶⠶⠶⠿⠶⠖⠚⠡⠬⠭⠍⠉⠥⠤⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠠⠄⣈⡉⠀⠑⠮⣍⡀⢭⡍⠉⠤⠬⠭⠭⠥⠌⠉⠭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⣀⠉⢁⠀⠙⠲⢤⣙⠃⣀⣈⣉⡉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠓⠙⠊⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠁⠀⠀⠀⠀⣽⣿⣿⣿⡇⠀⠀⠀⠠⡾⢿⣿⣿⡏⠀⠀⢀⣠⣤⣴⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠀⠘⠻⠿⠿⠟⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⢟⡻⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⣫⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡵⠋⠁⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⢠⠃⠀⠀⠂⠄⠀⠐
⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠶⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠺⣿⣦⡀⠀⠀⢀⣠⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⣴⣿⣿⣿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣀⠀⠀⠀⢀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣛⡛⠛⠛⠛⠛⠻⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠙⠓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⡿⠛⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⡘⠳⢶⣤⣀⠀⠀⠀⢀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠘⠋⠙⠻⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠉⠛⠿⣿⣿⠟⠛⠉⠙⠻⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢧⣀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣠⠾⢷⣄⠀⠁⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠉⠓⠲⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⠙⠷⠦⠤⠤⢤⠶⠛⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠠⣾⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠋⣨⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣴⣶⣶⣾⣿⣿⣿⡿⠟⠁⠺⣿⣿⣿⣿⣷⣦⣤⣄⣀⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣖⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤⣒⣯⠿⠭⠷⠲⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢖⣒⣚⠓⢶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣒⣼⠮⠥⠽⣒⣒⣚⠛⠛⠛⠿⣶⣦
⣀⣀⠀⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠵⢚⣋⣉⣉⡉⠻⡌⢣⢀⡀⣀⣀⣀⣒⡺⢷⣒⣒⣒⣛⣛⣉⣉⣉⠉⠛⠻⢿⣶⣤⡀⠉
⣺⡭⠭⠭⠵⠮⡷⢄⠀⠀⠀⠀⢀⣀⡴⣚⡽⠖⠛⣋⣉⣉⠙⠛⢧⢹⢸⣆⠙⠶⣬⣭⣭⣭⣭⣭⣭⣭⣭⣉⣉⠙⠛⠿⣷⣦⣄⠈⠛⢿⣷
⣒⣚⣛⣫⣭⠭⠭⢕⣓⠴⣢⢟⢟⢕⣫⡵⠶⠟⠛⠛⠛⠛⠻⢷⣄⣿⣄⠙⠷⠶⠤⠤⠤⠴⢦⣤⣤⣄⡉⠙⠻⢿⣶⣄⡀⠙⠻⣷⣦⡀⠉
⠭⠥⠶⠖⣒⣛⣫⡭⢖⡫⢞⡵⢞⣩⣥⣶⠶⠿⠿⠿⠿⢷⣶⣤⣌⠻⣙⣛⣛⣛⣛⡛⠳⠶⢶⣤⣈⡙⠻⢿⣦⣄⡈⠛⢿⣶⣄⡈⠙⢿⣷
⣚⣋⣭⠭⠭⠵⣖⣚⠭⣚⠵⢚⣋⣩⣤⣤⣤⣤⣤⣤⣤⣤⣄⣈⣉⠛⠦⣵⣲⣶⡭⢭⣛⡳⢦⣌⡙⠻⢶⣤⡈⠛⠿⣶⣄⡈⠛⢿⣦⣄⠈
⣒⣒⣺⠯⣭⠭⡷⢖⣯⠶⠚⠛⣉⣉⣉⣀⣀⣀⣀⣀⣉⣉⠉⠛⠛⠻⠶⢦⣭⣙⠿⠷⠶⠭⠖⢮⡙⠷⣤⣉⠛⢷⣤⣈⠙⠿⣶⣄⡉⠛⢿
⢦⣝⣓⣨⣶⣭⣮⣥⣴⣚⣛⡟⠻⢛⣛⠛⠋⠛⠛⠛⠛⠛⠛⠻⠿⠷⠶⢶⣤⣭⣉⣛⡻⠭⠭⣭⣭⡭⠬⠭⠵⠤⠬⠽⠷⠦⠄⠙⠻⢷⣦
⣚⣩⡭⠥⠶⠶⠶⠶⠦⢬⣭⣉⣛⠲⠮⣭⣓⠲⢭⣛⠻⢿⣿⡶⠶⠶⢶⣦⣤⣤⣭⣍⣉⣙⣛⡒⠲⠯⠭⠭⢭⣿⣛⣛⣛⣛⣋⣉⣩⣭⣭
⣭⡴⠶⠒⠛⣛⣛⣛⣛⡒⠶⠮⣭⣛⠲⢦⣍⡛⠶⣬⡙⠲⣬⣉⠓⢶⣤⣒⠢⢤⣤⡤⣭⣭⣉⣉⣙⣛⣛⣛⣒⣒⠒⠒⠶⠶⠶⠶⠶⠶⠶
⣴⠶⠞⠛⣋⣉⣩⣭⣭⣭⣍⣛⣒⠮⠭⣓⡮⣝⡳⢤⣙⠳⣦⣉⠻⢶⣌⡙⠻⢶⣬⡙⠳⠶⣭⣉⠓⠭⢭⣍⡉⠭⣭⣭⣭⣭⣭⣭⣭⣭⣭
⣴⠶⠟⢛⣉⣩⣭⣤⣤⣤⣤⣬⣭⣭⣙⣒⠯⢕⣫⢗⡬⣓⢦⣍⠻⢦⣌⠛⢷⣤⡉⠛⠷⣦⣌⡉⠛⠷⣦⣌⡉⠛⠷⢶⣤⣍⣙⠒⠶⠦⢤
⣤⠶⠟⢛⣉⣩⣤⣤⡶⠶⠶⠶⠶⠶⠦⣬⣭⣙⣒⠯⣟⣺⢕⡮⣝⠦⣍⠻⢦⣌⠛⢷⣦⣈⠙⠻⣶⣤⣀⠉⠛⠿⣶⣤⣀⡉⠛⠻⠷⣶⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡀⠀⠀⠀⠀⠀⠀⠞⢻⡟⠀⠀⠀⠀⠎⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠈⠀⠀⣀⣠⠶⠞⠀⠠⠹⠀⠀⠀⠠⠂⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢄⡀⠀⠀⠀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢰⠁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠟⠁⠀⠎⠀⢤
⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠆⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠔⠉⠀⠀⠀⠂⠠⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠂⡀⡐⠁⠀⠀⠀⠠⠆⠀⠰⠒
⠀⠀⠀⠀⠀⠒⠲⠶⠼⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢧⡀⠀⠀⠀⠀⠀⠀⢀⠸⠀⠰⠉⢦⣗⠀⢸⡃
⠀⠀⠀⠀⠀⣄⠀⠘⠋⠷⠈⠁⡀⠀⠀⠀⠀⠀⠈⠇⠑⣼⣤⣿⠁⠀⢸⡁⠀⠀⠀⢐⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡿⢘⢒⣒⢚⡀⠀⠈⠂⡄⠀⠀⠀⠀⡀⠀⠈⠃⠀⠀⠀⠨⠀⠂⠀⠉⠀⠀⠁⠀⠀⠀⠀⠂⠻⣶⣦⡇
⠀⠀⠀⠀⠀⡀⢈⡠⠋⢙⠂⠀⠀⠀⠀⠲⣄⠀⠀⠓⠀⠀⣀⣤⣶⣴⣶⣿⠦⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠉⠳⣼⠇⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠊⠙⢿⣿⣟⣿⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣺⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣔⣒⡐⣒⡐⠈
⠀⠀⠀⠀⠀⢹⣶⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠉⠁⠀⠹⡗⠀⣀
⠀⠀⠀⠀⠀⠘⡄⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠹⣿⠁⠀⠀⠀⣸⣀⠈⠁⠀⠈⠁⠀⠐⠠⠀⡀
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⡄⢠⣟⠀⢈⠉⠛⠿⢭⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡍⠊⠙⠃⠈⣇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠁⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣒⣲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⠰⢾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡤⠀⢉⣉⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢴⡆⠀⢬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠁⠁⠀⠀⢙⢤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⠖⣤⠀⠀⠋⠓⠀⠀⠀⣠⠴⠦⢤⡀⠀⠀⠀⡼⠚⠁⠠⠄⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡸⠀⡇⠀⢀⣘⡃⠀⠀⢸⠁⢀⡆⠀⢱⠀⠀⠀⠈⠛⡙⠲⠊⡴⢋⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡘⠁⣨⣄⡞⠁⠁⠀⠀⠀⠘⣄⢈⡃⢀⡜⠀⠀⠀⠀⠀⡇⡰⢪⠊⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠃⢺⠂⠙⠇⠀⠀⠀⡴⠾⠆⠀⠉⡇⠉⠀⠀⠀⠀⠀⢀⠙⢡⠃⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢸⠀⠘⠀⠀⡇⠀⠀⡼⡆⢀⡇⠀⢰⡇⠀⠀⠀⠀⠀⢀⢸⡄⠂⠀⠀⢠⠂⡴⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠱⣄⠀⠀⢃⠀⡼⢠⠇⡏⠀⠀⠸⡇⠀⠀⠀⠀⢀⠇⠸⢭⡆⠀⢠⠃⣜⣡⠞⠀⣿
⠀⠀⠀⠀⠀⠀⡿⢆⠀⠀⠀⠀⣸⠇⠀⠀⠀⠘⣧⡀⠈⠳⡁⡞⠀⠇⠀⢤⠛⠀⢠⡄⠀⢀⠞⠀⣄⣤⡃⠠⢃⡾⠋⠁⠀⠀⢹⡀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⣿⠘⡆⠀⠀⢀⡿⠀⠀⠀⡴⠛⠉⠹⢦⡀⠘⠁⢠⢰⡳⣦⣤⡄⠚⣡⠄⡇⠰⡌⠁⠀⡰⣡⠏⠀⣄⠀⠀⠀⢸⡇⠀⠀⠀⢠⠏⣿
⠀⠀⠀⠀⠀⠀⢹⡆⠹⠀⠀⢸⣧⣤⣤⠊⠀⠀⠀⠀⣀⠙⣦⣄⣎⠸⣷⣶⣵⣄⣲⣏⠓⣹⣄⢹⣆⣀⣛⣁⣀⣀⠈⠙⠢⠀⢸⡇⠀⠀⢠⠃⣼⡇
⠀⠀⠀⠀⠀⠀⠘⣿⡀⢣⠀⣼⠀⠀⠀⢣⠀⠀⠀⢐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠐⢄⠀⠀⣧⠀⢠⠃⢰⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⣦⡇⣠⢤⣶⢒⡇⡤⠤⠞⠒⠒⠋⠉⠉⠉⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠐⡘⡆⠈⢷⡀⢹⣀⠏⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣾⠋⣴⠟⣵⠋⠀⠀⠀⠀⢀⡠⠔⣀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢃⢳⠀⠈⠁⠈⡟⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡈⠋⠸⠇⠀⠀⣀⣴⣞⡭⢴⠞⠁⣠⠀⠀⡄⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⡈⣇⠀⢀⡤⣳⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣦⣤⣴⣶⣿⣯⣭⣴⡴⣣⠂⢠⠃⠀⠀⢰⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡇⠸⣴⣋⣞⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣼⠃⢠⠇⠀⠀⠀⠘⡆⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⠋⢠⣾⠃⢀⠎⠀⠀⠀⠀⠀⢇⠀⠀⢷⡀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠸⡄⠘⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⠟⠛⠋⣩⠟⠁⠀⢰⣿⠃⢀⡎⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠏⠑⡆⠀⡀⠀⠘⠀⠀⠀⣀⣀⡇⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣏⠎⣠⠴⠋⢁⣠⣤⡀⠀⠁⠀⣾⣴⣤⡀⠀⠀⠀⠀⠀⡇⢀⣤⠴⠞⠁⠀⠈⠙⠛⠛⠛⠛⠉⣩⠏⠀⠈⣇⠀⢹⣴
⠀⠀⠀⠀⠀⠀⠀⣸⡵⠛⠁⠀⠀⡞⠁⠈⢻⡀⠀⠀⠻⡋⠀⠙⢦⡀⠀⠀⠀⠀⠻⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⠴⠶⠾⠀⠈⣿⡆
⠀⠀⠀⠀⣠⠔⣋⣁⡀⠀⠀⠀⠀⠸⡄⠀⠀⢧⠀⠀⠀⠉⢲⡀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠱⠏⠉⠉⠙⠷⣦⣄⡀⠀⣿⠀⠀⠈⠓⠶⠞⠛⠋⠀⠀⠀⠀⠙⠢⢄⣀⣀⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣭⣯⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠟⠛⠋⠠⠤⢠⣀⠀⢠⣾⣧⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⡇⠀⠀⠈⢀⣶⣷⠞⠋⠀⣠⣿⣯⣴⣋⣤⠄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣁⢀⣠⣶⡿⠁⠀⠀⠀⠀⠈⢿⣿⣭⣹⣦⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠟⠁⠀⠀⣹⣿⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣏⣀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡆⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⢀⡾⠁⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠋⠀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⡄⣀⣀⣀
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢀⣶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⠿⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡿⠿⢻⣿⣿⣿⣿⣧⣤⣸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣽⠇⠀⠀⠈⠛⣿⣿⡿⢿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣿⠿⠇⠀⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⣿⣿⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢻⠀⢨⣇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⠀⠀⢰⣿⣇⠀⠀⣿⣛⣷⠀⢰⣿⢳⡆⠀⢠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⢿⡇⠀⢸⡷⢿⡄⠀⢿⠉⠁⠀⢸⡟⠛⠁⠀⢸⣷⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⣴⣶⣶⣶⣶⠀⣠⡄⣀⣀⠀⠀⠀⠀⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠛⠃⠀⣿⣿⠀⣿⣿⣠⣤⣀⠀⠸⠃⠿⠻⠀⠀⠀⠀⠀⠠⠴⢶⣶⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣧⠀⠿⠟⠉⢻⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠚⠀⠀⠀⠀⠐⣿⣆⠀⣿⣿⠀⣤⣄⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⣒⠢⣤⠔⡒⢂⡉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢖⠢⡄⠀⠀⠀⠀⠀⠀⠀⡔⣩⡐⡄⠀⠀⠀⣀⣀⠔⡒⠢⠴⢫⡌⣣⢎⣥⢦⣶⠀⡿⣶⠀⣼⡏⣼⡇⡈⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣵⢾⡏⡔⠁⢀⣀⣀⡀⠤⠴⠒⠇⣙⠁⣩⣉⣭⢩⡴⣦⠘⡿⡿⠀⣿⣧⠁⣼⠇⣸⡏⣰⢣⣿⣠⣿⣵⣿⠖⡱⠃
⠀⠀⠀⠀⠀⠀⢀⣀⢀⠏⣼⡏⣿⠀⣩⡍⣡⣦⠉⣰⣆⣼⡿⢀⣿⢀⣿⢸⣿⣿⡷⢏⡼⢹⣇⡼⣧⣿⠟⢿⠶⠻⠟⠁⢈⣛⠁⢈⣽⢏⡜⠁
⠀⠀⠀⠀⠀⠀⡏⣠⣥⣼⣿⣼⣿⠾⣿⣷⢻⡏⣰⣿⡟⣽⣧⣾⣿⠾⣿⡼⠃⠙⠿⠋⠀⠈⠋⠀⣈⣁⠠⠤⠄⣒⣂⣩⣥⡶⠾⣛⠵⠋
⠀⠀⠀⠀⠀⡴⢃⠉⢉⣿⠏⢹⣇⠀⣿⠃⠻⠟⠁⢛⣀⣙⡉⠬⠥⠄⠲⣖⣒⣒⣊⣭⣭⣭⣽⣶⣶⣶⣿⠿⠿⢛⣛⡩⠵⠒⠋⠁
⠀⠀⠀⠀⠰⡐⣿⣧⣾⠏⢀⣀⠥⠤⢐⣒⣈⣭⣭⣤⣶⣶⣶⣾⣿⣿⣿⡿⠿⠿⠟⢛⣛⣋⡉⠭⠤⠐⠒⠈⠉⠁
⠀⠀⠀⠀⠀⠙⠢⠭⡔⠊⣥⣶⣾⣿⣿⣿⣿⣿⠿⠿⠟⣛⣋⠩⠭⠴⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⣿⡿⠟⠛⣉⡡⠄⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣈⠥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠁⠈⠀⠀⠀⠈⠁⠁⠀⠀⠂⠂⠀⠀⠀⠀⠁⠉⠀⠂⠐

⠀⠀⠀⠀⠀⢀⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠘⡃⢠⣊⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⡸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠱⡀⢰⠀⠀⠀⣀⠤⠄
⠀⠀⠀⠀⠀⡰⡥⢤⡀⠉⢣⣄⣀⠀⠀⠀⠀⠐⠒⡲⠒⠈⠉⠉⠉⠲⢞⠀⠀⠀⠀⠀⢉⠱⠦⠷⠯⠤⠤⣼⡡⠤⠤
⠀⠀⠀⠀⢸⠁⠦⠶⢣⡏⣻⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⣀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⣟⡃⣰⢀⣀⡀⡊⢟⢿⠉⠂
⠀⠀⠀⠀⠈⢇⠀⠴⠌⡩⠃⠀⠀⠀⠀⠀⠀⠀⢸⠠⣇⢆⠇⠀⣫⡱⠀⡴⠀⠀⠀⠀⠨⡉⠒⡄⠀⢤⠈⢑⠉
⠀⠀⠀⠀⣠⢴⠃⠀⠀⠁⢻⡒⢤⠀⠀⠀⠀⠀⠀⠙⢲⡄⠠⠦⠀⠀⡎⠁⠀⠀⠀⠀⠀⠉⡾⠀⠀⠸⡞⠉
⠀⠀⠀⠐⠃⢸⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡄⢀⡴⠀⡃⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢀⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⣘⣁⣀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡤⠤⡞
⠀⠀⠀⠀⠀⠀⣹⠒⠐⠂⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤
⠀⠀⠀⠀⠀⠀⠐⢄⠀⡀⠷⠼⠶⡁⠀⠀⠀⠀⠀⣀⠀⢄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣤⣏⣀⣀⢀⡀
⠀⠀⠀⠀⠤⠤⣱⠞⣡⡠⠄⣞⡷⢘⠆⠀⠀⠀⠀⢀⣽⡞⠓⢓⣠⡀⠀⠀⠀⠀⢠⢎⢥⡨⣏⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢸⡀⠃⠑⠣⠴⢠⡎⠀⠀⠀⠀⢸⠸⡧⠞⠰⣟⡼⢳⠀⠀⠀⠀⢸⡙⠓⣡⢢⢰⠶⣄⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⠇⢀⠀⠀⠀⡶⠑⡄⠀⠀⠀⠈⠲⡄⠐⠂⢀⠀⠘⣆⠀⠀⠀⢰⠋⡷⢺⠛⠉⢀⡌⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⡜⠀⠈⠦⣄⠠⠃⠀⡟⠀⠀⠀⠀⠀⣇⠙⠂⠘⠀⠀⡸⠀⠀⠀⠀⡄⢆⠈⡀⠀⡞⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⡠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢖⠒⠒⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠉⠉⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠚⡡⠔⣀⡀⠀⠀⠀⠀⠀⡤⠤⢄⣈⣵⡦⢤⡧⠴⣎⣈⣁⠀⠀⠀⠀⡤⢰⡡⢤⡀⢀⢈⢊⡃
⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⠢⡤⠄⠀⠀⠀⠀⣀⢤⠞⠱⠼⠇⠫⡸⠂⡈⢳⠈⠁⠀⠀⠀⠜⢩⠳⣂⠇⢦⣂⢶⡍⢀
⠀⠀⠀⠀⠀⡜⠀⣀⣀⡀⠀⢀⢹⠀⠀⠀⠀⠀⠁⢸⠀⠙⠋⠉⠉⢉⣉⡥⠞⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⡀⠀⠘⡧⠞
⠀⠀⠀⠀⡴⡇⠘⢥⠣⠇⠀⢷⠟⡧⡀⠀⠀⠀⢀⡟⡆⠀⠀⠀⢰⠧⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠁⠑⢤⣀⣀⣒⣒⣢⠔⠁⠃⠀⠀⠀⠀⠀⠈⢦⢀⣀⣨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣶⣶⣤⣠⡶⠛⠛⠛⠳⢦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⣠⡤⠄⠀⠀⠀⠈⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⡏⠁⠀⠀⠀⠀⣀⠀⠀⠀⣈⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⡇⠀⠀⠀⠀⠀⢿⣆⠀⠀⣿⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⢀⣼⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣒⣿⡀⠈⠻⣶⡄⠀⠀⠀⢀⣽⣧⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⢸⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣧⠙⠀⠀⠰⣿⠁⠀⠀⠀⠀⠀⠐⣆⠀⢻⡆
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⠀⠀⢀⣤⠟⠀⠀⠀⠀⠀⠀⠀⣾⠀⠘⣷⡄
⠀⠀⠀⢠⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⡴⠆⠀⠀⠈⠉⠻⡆⠈⣿
⠀⠀⠀⣠⣿⣿⣯⣿⣿⣟⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣷⣶⠦⠀⠀⠀⠙⢷⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇
⠀⠀⣰⣿⣿⣿⣏⣿⣿⣯⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⢨⡟⠛⠂⠀⠀⠀⠻⣧⡀
⠀⢀⣿⣿⣿⣿⢿⣷⣗⠿⠶⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣆⠀⠀⠀⠀⢹⡇⠀⠀⠉⠀⠀⠀⠀⠙⣧⠀⢸⣷
⠀⣾⠋⢻⣿⣿⣼⣿⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⢳⡄⠀⠀⠘⠃⠀⠀⢀⡀⠀⠀⠀⠀⠋⠀⢈⣿
⣠⡟⠁⢸⣿⡿⣿⢿⣿⣿⣿⣿⢴⣧⣾⡿⣻⣿⢿⣿⣿⣿⣿⠁⠐⠛⠓⠀⠀⢹⡄⠀⠀⠀⣀⡴⠋⠉⠉⠀⠀⠀⠀⠀⠈⢻⣆
⣿⠀⠀⠸⣿⡟⣿⣾⣿⣿⣿⣿⣿⣿⣏⣿⣿⣽⣧⣿⣿⣿⣿⡆⠀⠀⠀⠀⢰⡟⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣿
⢹⡇⠠⣶⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⢤⠀⢀⡀⠀⠀⢸⠁⠀⠀⠀⠀⠉⣿
⣾⡇⠀⢠⡠⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠄⠀⠀⢀⣾⠏⠀⠀⠀⠀⠈⢿⠆⠀⣼⠀⠀⣀⡄⠀⠀⣿
⢸⣇⠀⠠⠢⠘⣿⡽⣭⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠞⠃⠀⠀⢀⡀⠀⠀⠀⠀⣰⣯⣀⠀⠀⠹⠆⠘⣿
⠀⣿⣀⣀⣤⣤⣻⣖⣖⣒⣒⣖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⡀⠀⠀⢀⡴⠊⠉⠉⠳⣤⡴⠚⠉⠀⠉⣷⠀⠀⠀⠀⢻
⠀⢹⣿⣿⡿⠉⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣽⣿⣿⣿⣿⣿⣇⠀⠈⠁⠀⠈⠉⠀⠀⢰⠀⠀⠹⠃⠀⠀⣀⣰⠿⠶⢤⣄⠀⣿
⠀⠈⣿⣿⣧⠈⣽⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣻⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣠⡿⢦⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣿⠏
⠀⠀⠈⢿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⣀⠀⢀⣀⠀⠈⢻⡄⠀⠉⠓⠒⠀⠈⢹⡆⠀⠀⠀⢠⣿
⠀⠀⠀⠀⢻⣿⣿⡋⠛⢙⠙⠋⢿⣿⣿⣿⠟⠛⢿⣿⢻⣟⢿⡇⠈⠉⠙⠉⠁⠀⠀⢀⠇⠀⠀⠀⠀⠀⣀⣼⣇⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣽⣯⣿⣿⣿⣿⠋⠀⠊⠀⣀⣀⣀⣀⣹⡇⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠉⠉⠀⠻⣶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⣯⣉⢀⣤⣤⣄⣶⣿⣯⣯⣯⣿⡟⠉⠙⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⢿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⡞⠻⢶⠶⠿⣷⡀⢀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣟⣶⣾⣿⣿⣟⣿⣿⣿⣇⣼⠁⠀⠀⠤⠶⠿⠛⠁⠀⠀⠀⠀⣸⡷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣿⣯⣿⣿⣷⣾⢿⡄⠀⠰⣤⣤⠀⠀⠀⢀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⣿⣿⡿⠁⠀⠹⣦⣤⣀⣠⣤⣴⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⢀⢀⠀⠀⢀⠉⠉⠉⠁⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⣶⣤⡀⢠⡶⢛⣉⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣏⠀⠀⠀⣿⡇⢺⠀⣿⣿⣿⠇⠈⢛⡛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠈⠛⠀⠀⠀⣿⡇⢸⠀⣿⣿⣿⣤⣾⣿⣿⣆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣧⠀⠀⠀⣤⣤⣄⡀⣿⡇⢸⠀⢉⣀⣽⡿⠛⣿⣿⠃⠸⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⠁⠀⠀⠀⠈⠉⠙⢿⣿⡇⢸⠀⣿⣿⣿⡇⠀⣿⣿⣿⣷⣶⡌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠁⠀⠀⠀⠀⠠⣶⣄⠀⠀⠀⣿⡇⢸⠀⣿⣿⠟⠁⣀⡈⠉⠉⣹⣿⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣀⠀⠀⠀⠀⠈⣿⡆⠀⠀⣿⡇⢸⠀⣿⡟⠀⣾⣿⣿⣿⣿⣿⠿⠃⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠉⠛⠿⠃⠀⠀⢀⣿⡇⠀⠀⣿⡇⢸⠀⣿⣧⠀⢿⣿⣿⣿⣧⣤⣤⣾⣷⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⢠⣤⣤⣶⡿⠻⣿⡄⠀⣿⡇⢸⠀⣿⣿⣦⣀⠙⠛⢿⣿⣿⣿⣿⠋⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠉⠉⣀⠀⠀⠸⠿⠀⣿⡇⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣿⡇⢸⠀⣿⣇⠘⠿⠟⠋⢉⢉⣿⣿⣿⡿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣤⣴⣿⣯⣀⠀⠀⠀⠀⣿⡇⢸⠀⣿⣿⡦⠀⣴⣿⣏⠙⠛⠛⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠁⠈⠉⠻⣷⡀⠀⠀⣿⡇⢸⠀⣿⣿⡇⠀⣿⣿⣿⣿⣶⡶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣀⡀⠀⠀⠹⠇⠀⠀⣿⡇⢸⠀⣿⣿⣿⣄⣸⣿⣿⠿⠟⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣦⡀⠀⢀⣠⣾⠟⠀⠘⢧⡘⠿⣿⣿⡿⠟⣠⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠛⠉⠀⠀⠀⠈⠙⠶⠤⠤⠴⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠾⣩⠗⠒⠒⠒⠠⠤⠤⡵⠒⠉⠻⡽⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡶⠋⣀⡆⠐⢁⡤⠒⠒⠲⡤⠤⡔⠋⠉⣀⡠⠞⠒⢶⠊⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢧⡀⢀⣈⡏⠉⠑⠢⠄⢸⠀⠈⠙⢢⠋⠀⣠⠤⠐⠚⠢⣄⠑⢤⡉⠓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣡⠴⠊⣹⠋⠉⡃⠀⠉⠒⢦⠒⢠⠞⠉⠂⣀⠜⢠⠋⠀⠠⡀⠐⢤⣈⣳⣀⣈⣆⠀⣹⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣠⢾⡁⣠⠤⡞⠱⡀⠀⠑⣄⣀⠠⢄⣀⡇⠀⢀⠎⠀⠀⠮⣀⠀⢀⠗⠀⢠⠁⢠⠋⠀⠙⢏⠓⠲⢏⠱⡄
⠀⠀⠀⠀⠀⠀⢰⢃⡤⠚⢁⡠⣇⠀⢱⠋⠀⡞⠁⢢⡆⠈⡇⠀⠈⡄⠐⢄⡀⠀⢀⠏⠀⡠⠋⢀⠈⢀⡇⠀⡠⠓⢶⠲⣅⠹⡄
⠀⠀⠀⠀⠀⠀⠻⡼⠒⡲⢏⠀⠈⠆⢸⢀⡴⠁⠀⠀⢃⠀⢸⠀⠀⡏⠁⣀⡹⠖⠛⡄⢰⢇⡀⠘⡞⢁⣀⣼⠁⠀⢸⠀⠈⡇⢷
⠀⠀⠀⠀⠀⠀⠰⡇⠀⢇⡀⠙⠲⠔⠋⠉⠉⠓⠤⢄⠸⠀⣸⣀⠤⠓⠉⣁⠠⠤⠴⣁⣀⡠⠧⠒⢉⣁⣀⣀⠇⢀⠏⠉⠒⣅⣸
⠀⠀⠀⠀⠀⠀⠀⠙⠶⣎⣈⣉⡑⠢⠤⣄⣀⠀⢀⡠⠛⠋⠁⠀⠀⣠⠋⠀⡀⠀⠀⠈⢦⠀⠈⠝⠉⠀⠀⢹⠊⠉⢢⡀⠀⡼⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⢀⡏⠀⠀⢀⠔⠊⠁⠙⠄⠈⣷⡀⠀⠀⠀⠙⠢⢄⣀⣠⡇⠸⡀⠀⣀⡉⠫⡀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠈⢇⠀⠐⠃⠀⢤⡀⠀⣀⠞⠀⠈⠢⣀⠀⠀⠀⠀⡆⠠⡁⠀⠱⠊⠁⡤⠀⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣑⣤⠤⢶⡋⢀⣉⢤⡤⢤⡤⠶⠚⠛⠒⠒⢮⣀⣠⡏⠑⠒⣤⠾⠤⠚⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢭⠭⠖⢻⡀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡉⠉⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⡟⠒⠶⠦⠴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣶⡶⠶⢶⣄⠀⢀⣤⠶⠶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠙⢆⠀⢀⣿⠀⢸⣏⠀⢠⠞⠁⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢹⡄⢀⡄⠀⠀⠀⠀⣼⡏⠀⠈⣿⡄⠀⠀⠀⠐⣄⠀⣾⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⠤⠞⠁⠸⣄⣠⠎⠀⠀⢸⣇⠀⠀⣿⠀⠀⠘⢦⣀⡼⠀⠘⠦⣤⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢹⣤⡤⠄⠀⢀⡿⠁⠀⢀⠀⠈⣿⠀⢸⣿⠀⢀⠀⠀⠹⣇⠀⠀⠠⣤⣼⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡟⠁⠀⣀⣰⠞⠁⣀⡴⠁⢀⣸⡿⠀⠸⣿⣀⠀⠱⣄⡀⠙⢷⣀⡀⠀⠙⣧⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠈⣿⡀⠀⢙⣷⠀⠀⢹⣇⠀⢀⣿⠃⠀⢰⣟⠁⠀⢸⡏⠀⠀⠀⣿⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠞⢿⠁⠐⡆⠀⢹⡇⠀⣾⠃⠀⠀⣼⡏⠀⠀⢿⡆⠀⠀⢻⡆⠀⣿⠁⠀⣶⠀⢹⠟⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢻⡄⠈⠀⠸⡇⠀⣼⠇⠀⠹⣧⠀⠀⠙⣿⠀⢰⡟⠁⠀⢠⡾⠃⠀⢿⡄⠀⡿⠀⠈⠀⣼⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢳⣄⣠⡄⠀⢰⡏⠀⠀⢀⣿⣆⣀⢀⣿⠀⢸⣷⢀⣀⣼⣧⠀⠀⠈⣿⠀⠀⢀⣤⣴⡃⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠒⠛⠉⠙⠻⢦⣜⣷⡀⠀⠘⠃⠀⠉⢻⣇⠀⠀⣿⠟⠁⠀⠙⠀⠀⣠⣟⣤⠶⠟⠉⠙⠛⠂⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠶⣆⠀⠀⠚⠛⠛⢦⣀⠀⠀⠀⠀⢿⡄⣸⡏⠀⠀⠀⢀⣠⠾⠋⠛⠂⠀⠀⣴⠖⠒⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢀⣠⣿⠀⠀⠀⠀⠀⠀⣨⡿⠛⢦⡀⢸⣧⣿⠁⣠⠜⠻⣿⡀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠀⠈⠁⠹⢷⣄⣶⠀⢠⠞⠋⠀⠀⠀⠙⣾⡟⣿⡾⠁⠀⠀⠈⠛⢷⠀⠀⣆⣤⠾⠃⠈⠀⠘⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⣄⠀⠀⠉⠻⢦⣀⣠⣴⠦⣄⠀⠀⢹⣇⣾⠇⠀⢀⣤⢶⣤⣀⣠⠾⠋⠁⠀⠀⡰⠀⣀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⠓⢦⡙⠷⣦⣤⣤⣶⣿⠋⠀⠀⠈⠁⠀⢸⣿⣿⠀⠀⠈⠀⠀⠈⢻⣿⣦⣤⡤⠶⠋⢠⠟⠉⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠤⢿⡦⠀⠀⠀⠀⢹⡆⠀⠀⢦⣄⠀⣸⡿⣿⡆⠀⣠⠔⠀⠀⣸⠇⠀⠀⠀⠀⣴⠿⠄⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠹⣤⣤⠀⠀⣸⠿⣤⣀⡀⠈⠛⣿⡇⢹⡟⠋⠁⢀⣠⣴⢿⣄⠀⠀⣤⣴⠃⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣴⣿⡁⠀⠘⠁⠀⠀⠈⠉⠓⠀⢻⡇⢸⡇⠀⠚⠉⠁⠀⠀⠈⠂⠀⢈⡿⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⢷⣤⣀⣀⣀⣀⡙⠳⢦⣄⣼⡇⢻⣇⣤⡶⠞⢉⣀⣀⣀⣠⣴⠟⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⢸⡍⠉⠉⠉⠀⠀⠀⣈⣿⠇⢸⣿⡁⠀⠀⠉⠉⠉⠉⢹⡇⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣼⡧⠀⠀⠀⠉⠛⠻⠏⣿⡆⢸⡏⢹⠛⠛⠉⠀⠀⠀⣿⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠙⠛⣷⠀⢰⡿⠀⠘⣿⡀⠀⡟⠛⠉⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⣧⣤⡾⠃⠀⠀⠘⢷⣤⣽⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠇⠀⢠⠴⠋⠉⠉⠉⠉⠙⡦⡄⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠛⢸⠀⠀⠀⠀⠀⠀⡇⠛⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣄⣀⣀⣀⣀⣠⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣷⣿⣿⣿⣿⣷⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⣿⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠙⠁⠀⠀⠈⠋⠀⠀⠀⢰⡏⠉⠉⠉⠉⠉⠉⠉⣿⠉⢹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⢀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠉⠉⠉⠉⠉⢋⡿⢋⣠⠙⠛⠋⢠⡉⢻⡍⠉⠉⠉⠉⠉⠉⣻⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⣠⢾⡁⠀⠀⠀⠀⡿⣆⠙⣆⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⠀⠀⢀⡟⢰⣏⣼⣇⠀⠀⠀⢠⣧⣘⣆⢹⡄⠀⠀⠀⠀⣿⠀⠀⠀⣻⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢀⣀⣿⣀⣀⣀⣸⣇⣾⣿⠀⢀⣀⣀⣀⡈⠀⢸⣻⣄⣷⣀⣀⣀⣀⣻⣀⣀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⢸⡏⠉⠉⡇⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠙⠛⣟⠙⢻⠛⠛⠛⠛⢹⡆⢸⡟⠛⢛⡇⠀⡟⠛⠛⠛⠉⠙⡟⠋⣻⠛⠋⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠈⡇⢈⡇⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃



⣿⣿⠿⠻⡿⢛⠿⢿⠟⣿⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠖⠀⠀⡄⠈⢁⡰⠈⣀⠕⠒⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣐⠡⠤⠈⠀⠈⡹⠶⠥⡔⠁⠠⡤⠠⠂⠉⠁⡲⢭⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠁⠀⡄⠀⠰⡀⠀⠀⡠⣞⣀⡜⠁⠀⢀⠀⠀⢡⠀⡈⢑⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⡀⢸⠁⠀⠀⠈⠀⠂⠀⢀⡸⠀⠀⠀⠹⡀⠈⢺⡀⠀⢸⠀⠐⠤⠀⠁⠑⡀
⠀⠀⠀⠀⠀⢠⠀⠁⠀⠠⠋⠄⠈⠀⡀⡼⠄⠀⢳⠀⠀⢈⠦⠤⠔⠁⠀⠀⡞⠦⣀⠀⡰⡀⠘⠻
⠀⠀⠀⠀⠀⡆⡴⡡⠒⠀⢀⡠⠔⠋⠀⢣⠸⠀⠀⠃⡴⠁⠀⠀⠀⠀⢀⢞⠀⠀⠀⠉⠀⡇⠀⠒⢃
⠀⠀⠀⠀⠀⢡⠂⠁⢀⣒⠞⠀⠀⢄⣸⣰⠤⠤⣴⠿⠤⣄⡠⠔⠊⠉⢁⠼⠀⠀⠀⠆⠐⠁⠀⡧⠈
⠀⠀⠀⠀⠀⠘⡄⠀⡀⠘⠢⠤⠤⣀⡜⠁⠠⠀⡇⠀⢠⠀⠑⠒⠲⠬⠤⠤⠔⠂⠀⡸⠉⠠⠔⠳⡎
⠀⠀⠀⠀⠀⠀⠀⠄⠈⠓⠶⢤⣬⡏⠀⠀⠀⠀⡬⢇⡀⠀⣀⡠⠀⠴⠤⢤⣤⠶⠾⠴⠤⢤⡄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⢣⢀⠀⠣⡀⠀⠀⢠⠊⢀⣤⣴⡾⠛⠉⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣢⣄⣘⠀⣀⣠⠔⠛⢻⣿⣄⡀⠒⣀⣠⣤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠛⠭⡙⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡙⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣫⣾⡰⢟⡄⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠩⠙⣮⠁⣾⠶⠛⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠰⠶⣾⡇⣰⡅⠴⠋⠀⢀⣰⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡗⣶⣴⠓⣟⣁⣀⣀⣤⡴⠏⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⣿⣾⡿⠛⢹⡛⠛⠛⠓⠀⣿⣿⡆⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⠏⢀⣠⣼⣿⣁⡀⢠⣾⣿⠿⠆⢠⣶⠀⣠⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣶⡾⣷⣶⣿⠤⣼⣿⣥⡟⣿⡷⠈⠛⠁⠀⠀⢶⣌⡛⠻⣷⣿⣷⣄⢀⣀⣤⠖⠘⢧
⠀⠀⠀⠀⠀⠀⡖⠃⠀⠀⠀⠶⡀⠙⠛⠋⠉⠀⢠⡟⢻⡿⠛⠛⠁⠀⠀⠀⠀⠀⢸⡿⠛⢙⣛⠻⢿⣴⣦⠛⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⣸⣿⠛⠳⢷⣦⣀⠈⠁⢄⠀⠀⣠⡾⢷⡞⠁⠀⡀⠀⠀⡀⠀⠀⠀⠈⣿⣌⠁⠈⢳⠀⣸⠿⣟⢳⣶⠏
⠀⠀⠀⠀⢴⣿⡿⢦⣀⣠⠹⢿⣅⣶⠀⣀⠀⠉⠛⠉⠁⢠⡞⠁⠀⠀⣹⠆⠀⠀⠀⢻⣿⡆⠀⠀⢿⣙⡧⣼⣿⣽⡄⣼⠂
⠀⠀⠀⣠⠂⠀⠀⠙⠍⠁⢠⠎⠉⢀⡀⠀⢀⠀⠀⠀⠀⣸⣇⣀⣠⡼⠋⠀⠀⠀⠀⠀⠙⠻⣶⡙⢿⠉⠉⣿⡈⢻⡷⢀⡇⠀⠀⠀⢀⠖⠲⡀
⠀⠀⣰⠎⢀⡄⠀⠀⠀⠀⠇⠀⠀⣼⡇⠀⠈⡅⠀⢰⡟⠃⠘⠿⣙⣷⣄⣴⠟⠹⡀⠀⠀⠀⠀⠻⢦⡙⠚⢋⣀⠸⠿⠋⠁⠀⢀⠄⢼⡀⠀⡿⠀⣤⣄⣀⣀⣤
⠀⡖⣡⡄⠈⡇⠀⠘⡖⠊⠀⠀⢼⣿⣀⣀⣸⠇⠀⣸⡄⠀⠀⢀⡞⢯⣿⡄⠀⠊⡑⣠⠄⠀⠀⠀⠀⠈⠉⠻⣿⣷⣦⡀⠀⢤⠞⠀⠈⠳⡾⠃⢀⣏⡼⠋⣁⣸
⠀⠣⠟⠁⢠⡀⠀⠀⠀⠉⢦⣀⠟⠋⠁⠀⣠⡄⠀⣏⣻⡀⠀⠀⢳⠀⢀⣷⠶⢾⣽⠇⠀⠀⠀⠀⠀⢘⣶⡀⠀⠉⠉⠁⡶⠾⠖⠚⠁⠀⠛⠉⠙⠋⠀⠀⢸⡏
⢠⣴⡿⣿⠿⣿⣶⣦⣤⣦⣤⣌⣃⡀⠠⠚⢫⡄⠀⠸⡿⣧⠀⠀⠘⠆⠈⠉⣷⠘⠁⠀⠀⣠⣦⣴⠋⠈⢹⣇⠀⠀⠀⠀⠱⠄⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⢸⣍⡀⠀⠙⠦⠀⠀⠈⠳⠞⠉⣉⣵⣶⣤⡿⠁⠀⠀⢷⣼⣧⣄⣀⡀⠀⣴⠇⠀⠀⣀⣼⠋⠀⠿⠀⢀⡼⠛⢉⢳⡄⠀⠀⡖⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⣿⣤
⠘⣯⠷⣾⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⠟⠁⠀⠀⠀⢸⣿⣯⣿⠿⠛⠂⢹⠀⠀⠸⠛⢄⠀⠀⢲⣆⣛⠿⡀⣠⣏⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡏
⠀⠈⠓⠉⠙⠛⠿⠿⠿⠛⠁⠈⠉⠹⠟⠀⠀⠀⠀⠀⢸⠉⢿⣿⣧⠀⠀⣸⠀⠀⡀⠀⡀⠀⢀⡼⠃⣉⡀⠁⠠⠈⠁⣷⡀⠀⢚⣛⣏⠓⠒⠂⣶⣟⣉⣬⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣭⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡟⠘⠇⠀⣿⠀⢀⡇⠀⢙⡆⠀⠀⠀⣭⣤⡜⢳⡘⠟⢉⡁⢀⡈⠉⠈⠟⠛⠃⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣬⢭⡍⠙⣶⣤⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⢰⡇⢠⡿⠀⠘⡇⠘⠋⣳⣶⡶⠴⢻⡿⠥⠊⠹⠾⠧⠄⣾⣷⠄
⠀⠀⠀⠀⠀⡰⠃⠀⣷⡀⡴⠶⣾⡍⢷⢰⣄⠀⠀⠀⣸⡇⣸⡇⢸⡇⢸⣧⠀⠀⣿⢀⣸⣇⣯⡄⠀⢤⡔⢢⣤⣤⣤⣤⡴⠿⠶⣶⣦
⠀⠀⠀⠀⢀⠑⡄⠀⠀⢛⣶⡞⢹⡇⠀⢹⢿⢷⡀⠠⣿⣤⣿⣶⣾⣷⣀⣿⣄⣀⣿⣿⣿⠋⠉⠀⠀⠈⠱⣏⠀⠀⠀⣀⢀⣀⣾⠿⠋
⠀⠀⠀⣀⠀⣴⡇⠀⠀⠈⢿⡇⠈⢳⠀⢸⢸⡆⣿⡀⠉⠛⠛⠛⠿⠿⠫⠥⠹⠿⢿⣿⣿⣇⠀⠀⠀⣠⣴⣿⡿⢿⣿⣿⠛⠛⠻⣇
⡠⠊⡹⠀⢠⣭⣥⣤⣤⡀⠸⣯⠀⠀⠀⠈⠀⠁⣿⣧⠀⠀⠀⠀⠀⠀⠀⡀⣠⡀⠀⠈⠙⠻⣤⡴⢾⣿⠁⠀⠀⢸⣿⣿⠀⠀⢀⡿
⠹⣧⠈⠠⣀⣀⣀⡀⠹⠇⠀⠚⠀⠤⣀⠢⠀⢸⣿⡟⠀⠀⠀⠀⠀⣿⠇⠁⠀⣙⣆⣀⡀⠀⠀⠀⠀⣿⠁⠀⠀⣸⠇⡇⠀⠀⣸⡇
⠀⠀⠀⠀⢷⡴⠿⣭⣭⣽⣿⣿⣷⣏⣤⣴⣿⡶⠀⠐⠀⠀⠀⢀⡜⠻⡀⠑⠊⢁⡿⠀⢹⣾⠖⢢⡀⢿⡀⠀⢀⣿⢠⡇⠀⣰⡟
⠀⠀⠀⢠⡄⠀⢀⣀⠉⢹⠟⠙⠛⠛⠋⠉⡆⠀⠀⡗⠀⠀⡀⢿⠁⣀⠐⠶⣶⡞⠳⠋⠻⣧⠆⠘⡇⠈⠁⠀⠀⠁⠀⠁⠈⠋
⠀⠀⠀⠘⡏⠀⢸⣿⡿⠿⡇⠀⠀⢸⣦⣶⣇⠀⢀⠇⠀⢠⠃⣾⣶⡄⠀⠀⠈⢧⠀⣀⣠⣽⣤⣄⣼⣦⡄
⠀⠀⠀⠀⠁⠀⠀⣿⠁⠀⢰⠀⠀⠈⢻⠉⢹⡶⠋⠀⠀⣾⠀⠀⠹⠗⠈⢳⡞⠛⡿⠃⠀⢀⣩⢤⢾⣯⡀
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠘⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⣀⠼⠃⢺⠗⠲⠖⠉⠁⠀⠀⠈⢧
⠀⠀⠀⣰⣤⣤⣾⠁⠀⠀⠀⢷⡤⣴⡇⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣿⡁⠀⢀⡘⣀⠀⢳⡀⠀⠀⠀⠄⠀⢣
⠀⣠⠾⠛⠖⢰⣿⣀⣤⣄⣀⡠⢶⡴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣋⠀⠈⣷⡈⢣⠀⠀⣠⣶⣞⢀⣧⡀
⠀⠘⠦⠀⠖⢿⣿⣿⣿⣿⢿⡇⠀⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣷⣧⣬⣿⣧⣼⣿⡿⢿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠁⠙⠛⢡⡟⡟⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⢠⡟⢀⣧⡀⢀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣾⣿⣿⣿⣿⣷⣄⠀⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢧⡿⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠁⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠻⠟⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⢳⡄⠀⣠⠞⠉⠉⠉⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠁⠀⠀⠀⠀⠀⢀⣗⠀⣿⡀⠀⠀⠀⠀⠀⠘⠛⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣤⡾⠂⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣴⡶⢀⢀⠀⠀⠀⡀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢿⣷⡾⠃⠀⠀⠸⣿⠇⠀⠀⠀⠀⠘⢿⣷⡾⠁⠀⠀⠀⢧⠀⠀⠀⠚⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡠⠊⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠳⢄⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣤⠀⠀⠠⡀⡀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠲⢄⠀⠀⠠⠀⠀⢠⣤⠀⣿
⠀⠀⠀⢠⡶⢦⣠⣿⡏⠀⠀⠀⠀⠈⠒⠛⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠚⠁⠆⠀⠈⠙⣷⣿⠶⣦
⠀⠀⠀⠀⠁⠈⠻⠻⣇⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡄⠀⠀⠀⠀⠐⠀⡤⠊⠀⠈⠂⠀⠀⠀⢠⣿⡀⠀⠀⠀⠘⠁⠀⠐⢄⠀⠀⠀⠀⠀⠀⢠⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣀⣀⣴⠟⠈⠳⣤⣀⣀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣶⡀⠀⠀⠀⠀⢠⡟⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⢷⡀⠀⠀⠀⠀⢀⡶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠖⠛⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠓⠲⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣛⢉⣆⣀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⢁⠴⠋⠉⡇⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⠊⠀⠰⡃⠀⠀⡼⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣛⣵⣆⠀⠀⢸⡀⢰⠁⠀⢰⠋⠑⢆⠀⠀⠀⠀⠀⣾⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢁⢴⡠⢬⡆⠀⢣⢀⣩⣀⠀⠈⡄⠀⠀⠀⢠⣏⢀⣘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠀⠈⣿⠀⡜⠁⠀⣸⡳⠚⠀⠀⠀⢠⠞⡉⣧⠊⣱⢟⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣒⠲⡿⠘⠄⢠⠾⠁⠈⠢⢄⡀⠀⠈⡲⣉⠙⢲⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣉⢵⠀⠀⠀⠀⠀⠀⢸⢦⠀⠈⠁⠚⠶⢁⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⡇⠀⠑⣄⠀⠀⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠢⣄⠀⡜⠀⠀⠀⠀⠀⠀⡜⠀⠀⣰⡿⠓⠢⢄⣐⣀⣂⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⢻⢈⡟⠀⠀⠀⠀⠀⠀⡼⢀⢀⢜⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠻⡀⠀⠀⠀⣀⠀⣼⠗⢹⣫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡄⠀⠙⢦⡀⠀⢀⡜⠀⠀⠸⣡
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠹⡀⣀⠔⢻⡠⢺⣥⠤⠤⠤⠿
⠀⠀⠀⠀⢀⡤⠚⠁⢀⠔⠀⡔⠀⠛⠁⠠⣼⠄⡎
⠀⠀⠀⠐⣧⡀⠀⠰⢧⣤⠼⠅⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠘⡅⠈⣉⠒⢺⠹⠤⠤⢤⣤⡤⠤⠤⢤⠂
⠀⠀⠀⠀⢸⠄⠣⢍⠚⢰⡪⢝⠜⠜⠀⠀⡰⠁
⠀⣀⣀⣠⣎⠰⠭⣟⡣⠊⠐⡿⠛⠒⢢⠎
⠑⠒⠒⠛⠂⠉⠙⠛⠓⠉⠉⠓⠛⠛⠿⠯⠭⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠤⠀⠠⡤⡤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠲⢎⣩⠉⠉⠒⠠⢄⠀⠆⠘⠀⠘⠤⢤⡀⠀⠙⢇⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢏⠑⠦⠒⠀⠈⠃⠢⢄⡀⠀⢣⠟⠒⠀⢰⠀⠀⢠⡞⠉⠙⢢⡄⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⢘⠀⠒⠒⠤⡠⠊⠀⢀⡜⠃⠀⠀⢄⠀⠸⡄⠀⢀⡇⠀⠐⠘⠢⡀⠐⠢⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠔⠒⠒⡞⠁⠀⠀⠀⠀⠘⢄⠀⠀⠉⠒⠢⣄⠀⠉⠙⠣⡔⠁⠀⠀⠀⣀⡀⢸⠒⠒⠀⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⢀⠀⡇⠀⠀⠀⢦⠀⠀⠀⡏⠉⠑⠒⠀⠈⢆⠀⠀⠸⠀⠀⠀⠉⠉⠀⠈⢻⠔⠉⠑⠢⡀⢣
⠀⠀⠀⠀⠀⠀⢀⡾⠒⢢⠀⠀⡏⠀⢰⡴⠁⠀⠸⡀⠀⠀⠁⢠⠃⠀⠀⠀⢸⠀⠀⣠⠓⠒⠦⠤⣀⡀⠀⠈⠀⠴⢧⡀⠈⣎⠇
⠀⠀⠀⠀⠀⢀⠎⣆⠀⠀⠑⠚⠀⠀⠈⡇⠀⠀⠀⢙⠖⠒⠒⠒⠤⣀⠀⠀⡎⠀⠀⠃⠀⡆⠀⠀⠀⠈⠑⢤⣀⡠⠀⠙⠀⢸⢸
⠀⠀⠀⠀⠀⡜⠀⡼⠀⠠⣄⣀⣀⢤⠄⠺⡄⠀⠀⠉⠀⠀⠀⠀⠀⠈⠑⠦⣧⡀⠀⠀⠀⠳⠀⠀⡠⠂⠀⠀⠙⠖⠤⠒⠀⠈⣾
⠀⠀⠀⠀⠀⠇⢀⣧⠀⠀⡟⠀⠀⠈⡆⠀⠘⡆⠀⠀⠀⣀⣀⡀⠀⠀⣀⠠⠀⠹⡉⠉⠙⠒⢷⡋⠀⠀⠠⠤⠊⠀⢠⡀⠀⢀⡞
⠀⠀⠀⠀⠈⡞⠁⢇⠀⡜⠀⠠⢄⣀⣀⡤⠤⠳⣄⡀⠀⠀⠀⢀⣩⠟⠁⠀⠀⠀⡃⠐⢄⠀⠀⠙⢢⡀⠀⠀⢀⡠⠋⠈⣈⠟
⠀⠀⠀⠀⠀⢧⠖⠀⠰⣄⠀⠀⡼⠀⠀⠀⣀⣀⠀⡈⠀⠀⠚⠉⢸⠀⡀⠀⠀⢰⠃⠀⠀⢣⣀⣀⠀⢹⠯⠭⠥⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠈⣦⠤⢄⣀⠉⠉⠱⡀⠀⠀⠈⠉⠉⡡⠤⣀⢀⣣⣀⣀⡇⠀⠀⠋⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⡠⠔⠛⢒⣒⣶⠶⡍⣀⣀⣀⣀⣀⡼⠁⠈⠹⡓⠒⠀⣀⡴⠃⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡤⠒⡨⢥⠖⢂⡖⠓⣄⠱⣜⢤⣩⠋⢲⠤⣀⣁⣀⠀⠀⣘⣀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢯⣔⣩⣴⠫⠔⣊⠜⡹⠚⡿⣧⣒⡡⠞⠀⠈⠁⠉⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠛⠃⡿⠒⠉⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠔⠃




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

⠀⣠⣤⡀⠀⠀⣠⣶⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠂⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⣴⠂
⠀⠙⠋⡇⠀⣾⡿⠃⠀⠀⠀⠘⠆⣀⡀⠀⠀⣼⣿⠟⠁⠀⠀⠀⢀⠀⣴⠏⠀⠀⠀⠀⠀⢀⣾⢃⣒⡀⣚
⠀⠀⢰⠁⡸⠋⠀⠀⠀⠀⠀⢠⣾⣿⣿⡦⢰⠿⠃⠀⠀⠀⡀⣶⠈⡄⣀⡀⠀⠀⠀⠀⠀⠘⠁⠸⣛⡧⡟
⠀⠀⠈⠦⠄⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠇⠃⠀⠀⠀⠀⠀⠙⠲⠬⠥⠴⠒⠁⠀⠀⠠⠟⠀⠀⠸⠿⠇⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣄
⡖⠒⠒⠚⠉⠈⠓⠦⠤⡄⠀⠀⠀⡹⠋⠉⢯⠀⠀⠀⠀⠀⠀⢷⣶⠆⠀⡀⠀⠀⠀⠀⠀⣸⣿⣽⣿⣀⣠
⠀⠘⠀⠈⠀⡆⣁⠐⠀⠂⠀⠀⢰⡇⢀⡀⢈⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⢻⣿⠛⠉⠙
⠈⠢⠐⢛⠜⠠⠈⠤⠂⠀⠀⠀⠀⢙⣿⣿⡏⠁⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠀⠀⠀⠀⠀⢠⠞⠙⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠀⠰⢤⣀⡤⠂
⠀⠐⠀⠀⣶⣆⠀⠐⠀⠀⠀⠀⠙⠳⣶⣏⣲⠞⠀⠀⠀⠈⠙⠦⠍⠉⠉⠁⠀⠀⠀⠀⠀⠋⠁⣤⠉⠉
⠀⠀⢰⣾⣿⣿⣷⠄⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠐⠓⢺⣿⡗⠚
⢀⣤⣼⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⢨⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⢀⡤⠀⠤⣀⡀⣤⠠⣄⠀⠀⠀⠀⡰⠊⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠖⣒⠲⡄⠀⠀⠀⠀⠹⣿⣿⣿⠃
⢸⡇⠀⠀⢸⠀⠿⠀⣿⠀⠀⠀⠀⢣⠀⠀⡸⠀⠀⠀⠀⢀⡴⢉⡊⠷⡬⠼⠀⠀⠀⠀⠀⠀⢹⣿⡇
⢸⣇⣀⡀⢸⠀⣀⣀⣿⠀⠀⠀⠀⠀⢳⣷⠀⠀⠀⠀⠀⠠⢇⠘⠟⡈⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠈⠉⠉⠉⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠋⠑⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠟⠚⠟⠀⠀⠀⠀⢀⡠⢤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣈⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡀⠀⠀⠀⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠟⠀⠀⠒⠂⠀⠀⠀⠿⠕⠈⠀⠸⣧⡄⠀⠀⠀⠀⠀⠀⢆⡀⠂⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⠸⠄⠀⡄⠀⢴⡌⠁⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣦⠾⠃⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣤⡤⠀⠀⠀⠉⠉⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⣠⣿⣭⠴⠟⣿⣤⣤⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠶⠾⠿⠿⠿⠿⠿⠯⠥⠭⠷⠶⠄
⠀⠀⠀⢰⣴⣆⢠⣿⠿⠿⠀⠀⠀⢀⡶⠦⡀⠀⠔⠶⠞⠒⠀⠀⠸⠾⠷⠟⠀⠀⠀⡤⠤⠤⠤⢄
⠀⠀⢠⡟⠛⢻⡎⠟⠠⡊⠀⠰⠷⠿⠿⠗⠹⣣⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢰⣤⣶
⠀⠀⠀⠃⠀⠘⠁⠐⠀⠉⠀⠀⠀⠀⣀⣀⣤⡻⢻⣀⡀⠀⢀⣀⣀⣀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣾⡀⠀⠀⠀⠀⠀⠀⠀⣈⡉⠁
⠀⠀⠀⢀⣈⡁⠀⠁⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠆⠀⣤⣤⠄⠀⠀⠀⠈⠙⠿⢿⣿⡿⠟⠃⠀⠂⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣽⣯⣯⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣒⣒⡲⠤⡄⡤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠮⠁⢙⣉⢀⡀⠀⠅⠙⠖⠀⢁⢑⣦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠁⠀⠉⠀⠀⠎⠀⢸⠖⠸⠂⠈⠁⠈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠻⠀⠀⠀⠇⠀⠀⠱⠀⠀⠰⡄⠀⠀⠀⠈⠒⠘⢄⠤⠀⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡓⣁⠔⠃⠀⠂⠈⠁⠈⠁⠐⠲⢄⣠⠂⠀⠀⠰⠀⠈⠀⠀⠀⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠈⠀⠁⠀⠀⠐⠠⠢⣀⣀⠀⠀⠈⠙⠒⠂⡄⢀⡠⠄⢀⠀⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⠀⠈⠂⠒⠤⣀⠀⠀⠀⠈⠀⠤⠴⡀⠀⠀⠈⠫⡀⢀⣏⣔⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢑⣦⣤⠤⠠⡀⠀⠢⣄⢀⡠⠂⠉⢒⠁⠀⢀⣀⣀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠢⣀⢈⡵⠀⣠⣀⣙⠀⠠⢄⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢶⠒⣪⠟⠉⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⠀⠤⠤⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⡠⠤⠤⠤⠤⠤⠀⢰⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠋⣝⣇
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢹⢠⡄⠀⡄⠄
⠀⠀⠀⠀⣤⣤⠯⠤⠵⠤⠀⠀⠀⠐⠛⠛⠛⠳⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠾⠸⠧⠰⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠤⠶⠒⠒⠚⠶⠶⠄⠀⡴⠀⠀⠀⠀⢰⠀⠀⢰⣶⣿⡄⣄⡼⢴⠀⠀⠀⠀⠀⠀⠁⠐⢷⡾⠇⠂⠀⠛⠒⠚⡃⠀⢤⡤⠤⡤⠤⠠⣤⣤⣤⠍⡋⠁
⠀⠀⠀⠓⠂⠓⠒⠒⠒⠃⠀⠰⠃⠀⠀⠀⠀⢸⢂⡄⣸⡛⣿⠃⠛⠿⠬⠂⠀⣀⢀⠀⠀⠀⠀⡱⣣⡜⡇⢰⠚⡞⠞⢱⢠⣾⠿⠿⠛⢛⠃⠀⠀⠀⠓⠓⢠
⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠰⣯⢹⣿⠀⠀⠠⠐⡒⣖⢒⠢⡆⢴⠬⡇⣿⡇⠛⠀⠇⢹⡇⡟⠿⢾⠘⠋⠂⠀⣒⢐⠀⠀⠀⢀⡀⢠⡞
⠀⠀⠀⡇⢀⡿⢀⣂⣤⣤⢀⣼⣸⣤⡔⠚⢁⣘⡀⢻⡸⣿⠀⠀⠀⠘⡏⣻⠛⠟⠟⠻⠛⡃⠻⣇⡠⠀⠴⠤⠶⠗⠚⡛⠃⠀⠀⠀⣷⣼⣆⣶⢲⢲⣖⣶⠂
⠀⣤⣼⣿⣷⢾⣿⠿⠟⡄⡄⠔⠞⠛⠛⠗⠒⡝⢻⡿⠄⠨⢀⣀⣚⣻⣩⠉⠈⠁⠹⠈⠁⠇⠀⢀⠋⠓⠒⠒⠒⠘⠛⡥⢖⢠⣀⣺⡅⠈⠀⠩⠰⣤⠦⠶⠂
⠀⠧⠃⠽⠷⠶⠶⢲⡖⠶⠲⠂⠒⠒⡂⠀⠊⠛⢃⣠⣆⣘⡇⠀⠀⣛⡓⡚⣁⠀⠀⢀⠀⠠⡤⠀⠅⠀⠀⠀⠘⠒⠒⠒⠘⠀⠀⣀⣀⠠⠤⢶⠦⠤⠐⠐
⠀⠉⠈⠈⠉⠉⠁⠀⠁⠁⠬⠥⠤⢤⣍⣺⠐⠀⣻⠀⠀⠀⡄⠀⠀⡡⡗⠄⢟⠸⡏⡸⠄⣀⠐⠒⡆⠀⠐⠠⠀⠄⠄⠄⠠⡟⠛⠿⡗⠖⠒⣥⣤⣤⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠠⠭⠈⢀⠉⡀⠀⡀⠐⠂⠀⠂⠋⢲⣾⠬⠁⠀⠀⠀⠙⡋⡷⠀⠠⢠⠦⣆⣆⠀⠰⡗⠀⠂⡀⠀⠠⠶⣶⠶⠶⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠠⠇⠀⠂⠹⢶⠛⢋⠀⠴⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠠⡄⠀⠐⠊⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⡆⠀⠀⠀⠀⠀⠀⢤⣤⠤⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠑⣿⠿⣿⠿⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠄⢴⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⢺⣟⢿⡆
⠀⠀⣠⠸⠂⠛⢘⡗⠄⠀⠀⠀⢸⣿⣯⣿⡟⢂⠒⠈⠀⠀⠀⠀⠉⠀⣤⡶⣄⢹⣿⠃⠀⠀⠀⢀⣴⣾⡿⠎⠛⠊
⠀⠁⣬⠀⠀⠀⠘⠃⡀⠀⠀⠀⠀⠛⠶⠟⢅⠒⠂⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⢘⠃⠀⠀⠀⡔⠛⠋⠁
⠀⠐⠄⠛⠠⠂⠘⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⡶
⠀⠀⢀⣀⣀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋
⠀⠐⠻⡿⣿⠗⠀⠄⠀⠀⠀⢠⡶⠶⠿⠶⠶⢶⢶⡄⠀⠀⠀⠀⠄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁
⠀⠀⢠⠓⡾⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⡞⡀⠀⠀⠀⠀⠀⠀⢠⢀⣠⣿⣿⡁⠂⢀⡀
⠀⠀⢘⠁⠘⠀⠀⠂⠀⠀⠀⢸⣇⣀⣀⣀⣀⣸⣿⡇⠀⠀⠀⠀⢀⣸⠇⠙⠲⣌⠀⣄⠀⠀⠈⠂⢕⢠⠀⡀⠔⠈
⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠈⠛⠿⠆⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⣀⡀
⠀⠀⣀⢠⡠⠄⠂⠓⠀⠀⠀⠀⣉⠇⠀⢏⡅⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⡇⡤⣀⠀⠀⠀⠀⠀⠀⣹⢺⠗⣯⠤⢠
⠀⢰⣿⣴⣦⣦⣴⣴⠀⠀⠀⠀⡀⠀⢀⣀⣤⣀⣀⣀⠀⠀⠀⢶⡁⢾⡠⢇⣹⢎⡱⠆⠀⠀⠀⢠⡇⢸⠀⢸⡆
⠀⢸⣿⡿⣿⡿⣿⣿⠀⠀⠀⠀⠠⡀⠀⠙⢻⡟⠋⠁⠀⠀⠀⢸⡇⠸⠃⠘⠛⠈⣿⠀⠀⠀⠀⠘⠧⣼⣤⠼⠃
⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠂⠀⠃⠐⠓⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠚⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⡇⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣍⡉⠒⠢⠤⣀⣅⠈⡆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠎
⠀⢀⠖⡰⣳⠒⠒⠒⠤⠤⠤⢀⣀⣀⠀⠀⠀⠈⠉⠒⠤⣄⢙⡠⠾⠋⢓⣦⠔⠉⠁⠀⠉⠲⡀⢀⣀⣀⡔⢡⠊
⢠⣏⣜⣴⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠲⠤⠴⠙⡕⠁⠊⠁⡌⠀⠀⠒⠂⠀⠀⡼⣍⠀⠚⣠⠉⠈⠉⠉⢉⣙⣒⣒⣒⣤
⠀⠀⠈⠉⠒⠢⠭⣒⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⠤⠤⢤⡀⢰⠃⠀⠉⠰⠁⢸⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠬⢝⡒⠤⠄⠀⠀⠀⡠⠒⢨⠇⠀⠀⣠⠃⠀⠠⢀⢠⡁⠀⡞⠀⠀⠀⠀⠀⠈⠃⠒⠒⠲⠤⠤⠤⢄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠦⡀⢀⠇⠀⢸⠀⠀⢀⠃⠀⠛⠀⠀⠐⠸⠐⡇⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠩⢍⠉⠐⡖⢲⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⠀⠀⡆⠀⢠⠎⠀⢀⡤⢲⡤⡀⠀⠀⡇⠀⠀⠀⡅⠀⢑⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣈⠉⠩⠠⠤⢴⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠇⢠⠃⠀⣠⠊⠀⡸⠀⠐⡀⠀⡇⠀⠀⢠⠀⠀⢸⠀⣀⠤⠧⠤⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣀⣤⠃⠀⢰⠃⢀⣠⣇⠀⠀⠁⠀⠱⡀⠀⡌⠀⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡔⠋⠁⠈⡆⢠⣣⡾⠁⠀⠀⠈⡦⠰⠀⠀⡹⢄⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠒⠂⡤⡺⡃⢲⠀⠀⠀⡼⠀⣿⠀⡜⠀⠀⢱⣄⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠰⡳⠀⢵⣀⡎⣠⠞⣽⡞⠒⠐⠖⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠱⡜⠑⠒⢦⡴⠋⡼⠁⠉⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠖⠋⠁⠉⠉⠑⡄⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢤⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣿⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣛⣷⣭⣉⣏⣙⣏⣹⣟⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⣿⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠛⠛⠛⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⠿⠷⡶⠐⣾⠂⢿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡏⢀⣧⠈⠁⣾⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡦⠠⢠⡾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⢀⠤⠀⠀⠀⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣘⡁⢠⠞⠀⡀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣖⠠⣌⡀⢠⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠛⠻⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⡠
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠀⠀⠀⠀⠠⠖⠲⡄⣶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⠀⠐⠒⠒⠒⢀⡀⠛⡇⣉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢀⠀⢀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠠⠤⠤⠤⠤⠤⠤⣤⣤⣷⣶⣷⣶⣾⣷⣶⣧⣤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⠻⠿⣿⣿⣿⠿⠿⢻⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⣿⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠀⠀⠀⢰⠀⠀⠀⠸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⡀⣄⣀⢀⢀⠀⢀⢀⣤⢄⣠⡄⣠⢠⡀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⡇⠀⣿⢀⡿⡇⢀⣽⢇⢸⠖⢺⢸⠂⢸⡗⠀⣇⢀⡇⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠉⠉⠈⢩⣀⡍⣡⠀⡤⢤⠠⡬⢬⢩⡁⢭⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣿⣿⣿⣿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣧⠟⣇⡏⢻⠀⡇⢸⢸⠙⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡏⠛⠛⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠁⠀⠀⠿⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⡠⡄⠀⢠⡠⠤⡄⠀⢤⣤⡤⠄⠀⣤⡤⣄⠀⢠⠤⢤⣄⡀

⢄⡠⠄⢀⠀⠄⡀⠀⠀⠀⣀⣠⣄⣀⣀⠉⠉⠉⠀⣀⣀⠀⠀⠀⠀⠄⢀⡤⣀⠀⠀⠄⠠⠁⠈⠀⠀⠀⠀⠀⠸⢂⠀⢀
⠀⠀⠀⠛⠀⠄⢠⠀⠀⡞⠁⢘⠛⢻⠗⠷⠀⠀⢰⠉⠉⠙⡄⠀⠀⠀⡸⠉⠈⢣⠀⠀⠀⠀⠀⡀⢈⣁⣀⣀⣠⡠⠀⠐⢣
⠀⠀⠀⠁⠀⠀⠈⠀⢀⠀⠀⠀⢀⠀⠀⡘⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠁⠀⠀⠈⠀⣀⡀⠠⡇⢀⡀⢸⠒⠒⠒⡃⠀⡀⠈
⠀⠀⠀⠀⣄⣠⠀⠀⢨⡀⡀⡅⠀⢺⠀⠀⡀
⠀⢠⠒⠀⠈⠁⠀⠠⡀⢵⡃⠀⠀⠀⠒⠂⠀⠀⠀⠈⠉⠉⠀⣠⡄⢀⠀⠀⠀⠀⢠⠶⠦⡀⠰⠂⡴⢒⣤⡤⠀⡀
⠀⠈⠄⡠⠀⠀⠀⡀⣈⣈⣹⠀⠀⠀⣄⠁⠁⠀⠀⠀⠀⠀⡐⠀⠀⠙⡀⠀⠀⠀⡇⠈⠀⢧⠀⠀⠀⢨⠁⠈⠉⡇
⠀⠄⠀⠁⠀⠀⠠⠡⠠⠄⠈⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠁⠠⠦⠀⠃⠀⠀⠀⣇⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠨⠀⠁⠀⠀⠀⠀⠀⡆⠀⣆⠀⠀⠀⠀⣾⠁⠈⣧⠀⠀⠀⠀⠀⠀⠠
⠶⠶⠾⡖⠷⠀⢢⠀⠀⠀⠀⠰⠆⠀⠀⠀⠁⠀⠀⡤⡐⢄⠀⠀⠀⠀⣴⡀⠀⢢⠀⠀⠀⠀⠀⡠⠀⠀⠈⠉⠀⠠⠆
⠀⠁⠀⠖⠂⠐⠸⠀⠀⠀⡀⠀⠁⠀⠈⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡂⠀⠀⠸⠑⠅⠊⠀⠀⠀⠠⢠⠀⠀⠠⠅⠀⠀⢠
⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠰⠀⠀⠀⠃⠐⠀⠀⠀⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠭⢗⡆⠀⠀⠀⡀⣆⠈⠉⠀⠀⠀⠠⠘⠁⠀⠁⠀⠀⠀⠀⠀⠁⣲⠄⠀⠀⠀⢠⡤⠲⣤⠂⠀⠀⠀⡰⠀⠒⡄⠀⠔⠀⣴
⠀⠀⡆⠀⠀⠀⡇⠀⠀⢲⠀⠀⠀⠀⡃⠀⠀⢠⠀⠀⠀⢸⡿⠄⠼⡦⠀⠀⠀⢺⣖⣀⣻⡇⠐⠒⠂⡔⠀⠀⠘⡂⠀⠀⢋
⠐⠀⠀⠀⠀⠀⠡⡀⢂⡄⠁⠀⠀⠀⠓⠈⠠⠎⠀⠀⠀⢨⠠⠀⢠⠂⠀⠀⠀⢧⠀⠄⠀⡄⠀⠀⠀⠘⠐⠓⠈
⠀⠠⠀⠀⠀⠀⠀⠳⠾⠗⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠄⠀⠘⠁⠀⢀⠀⠪⠄⠀⠀⠂⠀⠀⠀⠀⠄⠀⠄⠀⠀⢀⣀
⡄⠀⠀⠀⠀⠁⠢⠀⠀⠀⢠⠒⠴⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠘⡄⣤⠀⠀⠠⠂⢠⡄⠀⠀⣤⠜⠍⣭⣀
⠀⠀⠀⢱⠀⠀⠀⡆⠀⠘⠧⠀⠀⠀⠠⠀⠀⠀⠀⡄⢀⠀⡞⠀⠀⠀⠁⡀⠀⢀⠃⠀⠀⠀⣿⡆⠨⣦⠀⠀⠙⣷
⠀⠀⠀⠈⢦⢂⣸⡃⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⢀⠉⢨⠀⠀⠀⠀⠀⠐⠈⠈⠀⠀⠀⢠⠁⠀⢀⡌⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠉⡁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠂⠂⠀⣀⠀⢀⠀⠁⠠⡀⠀⢀⣀⠈⠘⠀⠈⠏⠀⠀⠀⠒
⠀⢰⠁⠀⢈⣤⢌⠀⠀⠀⠤⠀⠀⠀⡄⠀⠰⠶⣄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠘⢡⠀⠑⠀⠀⢰⡀⠀⠀⠀⠄
⠀⠀⢀⠘⠀⠀⠁⢇⢠⡀⠖⠀⠀⠀⢄⠀⠀⠀⡟⠀⠀⠀⢉⠀⡀⠀⡇⠀⠀⠈⠅⠀⠀⠘⠁⠀⠀⢰⠀⠀⠀⡞⠁⠀⠰
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠈⠃⠀⠘⠀⠀⠀⠀⢀⠀⠀⡂⠠⠀⠀⠀⢆⠀⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⣤⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣶⣦⣿⣷⣶⣶⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣟⠿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣙⠿⣿⣄⡈⠛⠿⣿⣿⣿⣇⣹⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣶⣎⠛⠻⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣶⣴⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣭⣤⣤⣤⣽⣛⡻⢿⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣶⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡟⢿⣟⣷⣟⣿⣿⣿⣿⢹⡟⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⡻⢤⣈⡙⠻⣯⡻⣿⣿⣘⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⣮⡙⠲⢜⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣿⣷⣾⡅⠄⢿⣷⡘⣿⣿⣿⣽⣿⣿⣿⣧⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣤⠠⣽⣷⡈⣿⣿⣿⢹⣿⣿⣿⣿⢛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⣽⣿⣿⣿⣦⣬⣿⣷⡘⣿⣿⣸⣿⣿⡿⢁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⡛⠻⣷⡹⣿⣿⡿⢉⠁⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣛⣿⣿⣿⣿⣿⣦⣾⣷⠙⣿⣧⣾⡜⣿⣷⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⠁⢹⣿⣿⣿⣿⡟⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡛⠛⠛⠉⠄⠙⠿⣧⢹⣿⣄⡎⠹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣻⣿⣿⣿⣿⣿⣷⣶⣿⣧⢻⣿⣇⣰⡟⠈⠹⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⣍⣉⡉⠋⢛⣿⣿⣇⢻⣿⣿⠀⠀⠠⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⢻⣿⣿⣟⣻⣿⣟⣿⣿⣿⣿⣆⢻⣿⣾⠆⣰⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣿⣭⣭⣭⣿⣶⠶⣌⣻⣿⡎⣿⣯⣴⣷⠋⢩⣿⣿⣟⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠲⢆⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡥⠤⠀⢿⣿⣿⡌⢿⣿⡟⠉⣈⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠉⠙⠛⠋⠙⢶⣤⡀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠿⠿⣿⢷⣶⣶⣤⣉⣻⣿⡜⣿⣷⡞⢿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⣠⡄⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣼⠇⠈⠙⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⣥⣉⠻⣿⡜⣿⣿⠻⣿⣿⡛⠉
⠀⠀⠀⠀⣰⣿⣿⣿⣷⣁⣄⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⡉⢻⣿⣿⣴⠟⡟
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣾⣿⠧⣠⡄⣀⡀⠀⠀⠠⣾⡏⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠟⠛⠛⠶⣯⣿⠻⣿⡹⣿⡟
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣤⠄⣾⡟⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⠛⠓⠿⣷⡿⣏
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠏⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣿⣆
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣏⡶⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢠⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧⣿⢿⡇⡀⢸⡇⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠸⣿⣿⣿⣿⣟⠀⠨⠴⢨⣿⢛⡩⢉⢻⣿⣿⣿⡟⠻⣸⢃⣿⡇⠀⠀⠀⠈⠛⠷⣶⣄⠀⠀⠀⠀⣀⣛⡠⠤⢾⠋⠀⠀⠐⠲⣆⣀⠀⠀⠀⠀⠀⠀⢃
⠀⠈⠉⠙⠛⠻⣦⣴⣾⣾⣇⠀⠙⠊⢠⣿⣿⣏⣀⣠⡶⠛⠫⣥⣆⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⡙⠁⠀⠀⠈⠁⠀⠘⠳⣤⢈⡋⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠉⠙⠲⠦⣤⣌⣛⠳⠾⠷⠿⢿⡿⢟⡿⠁⠀⠉⠀⠊⠉⠑⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠀⠂⢤⣤⠀⠀⠀⠈⠻⡿⣤⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⡞⣁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠃⠤⠀⢀⠀⡀⢟⡀⠄⠀⠀⡄⠁⠀⠀⠀⠀⠀⠘⠈⠻⠦⣄⠐⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⡀⠀⠄⠀⠴⠀⠀⠀⢠⡦⠀⠀⢠⠟⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠠⠀⠐⠚⠋⠉⠀⠐⠉⠀⠀⠀⢸⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣉⣀⣉⣿⡀⣸⣉⣉⣧⣉⣸⣿⣋⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠤⢤⣖⣀⠤⠿⠶⠶⠷⢴⡒⢢⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⠞⢓⣚⠋⠉⢤⡞⢀⣦⣀⣁⡴⠊⠀⠲⠽⠚⠂⠀⣲⠬⠗⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠊⣆⡤⠊⢁⣈⠙⢦⣀⣐⣾⠒⢀⡈⢳⣌⠁⣠⠴⠚⠛⠉⠀⢀⣀⣧⠤⡑⢄
⠀⠀⠀⠀⠀⠀⣼⠖⡞⠁⡖⢠⠞⢁⡤⠖⠋⢁⡈⠹⠃⡴⠋⠈⢰⠋⠀⡷⠚⠉⠉⢻⡥⠀⡤⠿⣦⠵⣄
⠀⠀⠀⠀⠀⢠⣣⢼⠁⠘⠁⡼⠀⡎⠀⠛⠂⢸⠇⡄⢰⠇⠐⠁⡟⢀⠀⡇⢸⠦⠀⢸⡇⠈⡇⢠⠌⠣⡈⢧
⠀⠀⠀⠀⠀⠘⡇⠀⠳⣤⣀⣀⠀⠸⣄⣀⣉⠹⡄⠁⠃⠀⣇⠀⠀⣦⣀⣀⡟⠒⠒⢾⡃⠈⢉⡼⠀⢂⣸⠸⡆
⠀⠀⠀⠀⠀⠀⠙⠮⣍⣃⠀⠈⣙⣦⢤⣤⣄⣀⡼⠖⠛⠛⢻⠟⠋⢉⠉⢹⡧⠀⢢⡀⣠⠴⠋⠀⠔⢯⠸⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⣀⣀⡞⠁⣀⣀⡤⠤⢤⣄⢀⣎⣀⣈⣠⠤⠼⠿⠥⣄⡰⠴⠂⢈⠁⡷⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⡎⠉⠁⠀⡄⢠⡈⠻⣅⠀⠀⢠⡀⠀⠔⠒⠀⣻⢰⠋⣁⣰⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠈⢛⡟⢧⠀⠙⠂⠈⠁⠀⠤⠟⢲⡶⠂⠰⣇⠈⣆⣀⠀⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣉⣀⣠⣤⣴⣶⣶⡶⢶⣦⣀⣬⣷⣄⣀⡉⣷⠀⠉⠉⠈⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣟⣷⣿⣿⣽⣽⣿⣾⣿⣭⣿⣿⣶⠾⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⣿⣿⣟⡿⣿⣿⣿⡿⣿⣻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠿⠏⠈⠙⠛⠾⠗⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠉⠉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠄⡤⠚⠁⣠⠊⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⣀⣀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠞⠁⠀⢀⠜⠁⡊
⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⣤⣤⣶⣿⣾⣿⣷⣶⣶⡆⠀⣤⠀⣬⣽⣧⣝⣀⠀⢀⠶⢶⣆⢦⠠⠄⣤⠻⠶⢶⣤⣤⣏⠀⠀⠈⢀
⠀⠀⠠⡀⠈⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⠿⠿⠛⠛⠀⠀⠋⠛⠁⠈⠙⠛⠛⠂⠙⠛⠀⢻⣄⡻⠷⠾⢿⢶⡀⢀⣤⣄⠀⣀⡀⡀⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡙⠀⢲⢐⡆⢸⡷⠿⠶⡷⢼⣾⡇⠀⠀⢸⡆⠈⢣⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡄⠀⢠⠒⠒⠒⠒⠛⠚⠒⠒⠺⡀⠀⠈⠳⠼⣄⣤⣨⣄⣧⠀⠀⠀⠑⠒⠚⠂⠀⢁⡀⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠈⠁⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠂






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⢄⡁⠠⠴⢋⣁⣠⣴⣾⣿⠀⠒⠒⣒⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠠⠶⠿⠿⠿⠻⠟⣷⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠁⡄⠂⡀⢘⠃⠌⠁⠀⠀⠀⠉⢻⣿⣶⣶⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡯⠎
⠀⠀⠀⠀⠀⠀⢀⠾⠀⣧⣄⣄⣤⠀⠀⢅⢀⣀⣠⣼⣿⣿⡿⢡⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣴⣟⡧⠀⠀⣰
⠀⠀⠀⠀⠀⠀⣘⣀⣀⣛⣛⣛⣁⣀⣀⣀⣀⣉⣉⣉⣉⠁⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠓⠒⠈⠉⠉⠉⠙⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣃⡀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠁⠀⠀⠀⠀⣀⣴⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠌⡀
⠀⠀⠀⢀⢀⢀⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠈⢠⡆⢵⢫⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⣀⠀⠄⠠⠄
⠀⠀⠀⡥⢈⢷⡘⣇⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡇⠁⠀⠀⠀⠠⠈⠀⠀⢀
⠀⠀⠀⠘⢧⣰⣦⣲⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠷⠀⠀⠀⠀⠀⠚⢀⡆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠
⠀⠀⠀⠀⠀⢡⣿⠈⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣌⣨⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠂⠀⡸⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⣄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠓⠒⠓⠺⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣽⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡂⠀⠀⠀⡀⢃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⠤⠧⠤⠤⠤⠄⢷
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⢠⠿⣻⠷⠶⠶⠶⠿⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠠⣀⠸⠆⠃⠀⠀⠀⢠⠟⣲⠋⠀⠀⠀⠀⠀⠳⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠠⠒⠂⠁⠀⠀⠀⠀⠀⣠⣏⣠⡏⠀⠀⠀⠀⠀⠀⢀⣇⣜⣄⠀⠀⠀⠀⠀⠖⢦⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⢂
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣩⣤⣾⣟⠃⠂⠀⠀⠀⠀⠀⠀⠀⢠⢠⠄⠤⣶⣿⣿⠇⠀⠀⡀⣷⣿⠀⠀⠀⠀⠀⠀⠠⢰⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠟⡉⠀⠀⠀⠀⡀⢀⣀⣷⡙⠉⡸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠉⠀⠀⠀⢠⢁⡈⣁⣷⠀⠀⠀⠀⠀⠀⠓⢂⡄
⠀⠀⠀⠀⢀⢀⣄⣄⣀⡔⠐⠁⠀⠀⠀⠀⠀⡠⠋⢙⣥⣤⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢈⣂⠀⠀⠀⠀⡏⠀⠀⢋⣼⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⢠⡟⠃⠁⢠⠟⠹⣧⢱⠠⡤⠀⠀⠊⠀⣰⠟⣧⠀⠁⠈⢳⡄⠀⠀⠀⠈⠀⠀⢉⡍⢉⣈⡛⠀⠀⠀⠀⠃⠀⠆⠛⠛⡃⠀⠀⠀⠈⢀
⠀⠀⠀⡎⠀⠀⣴⣷⣄⣀⣿⠀⣯⣵⠂⠀⠀⢘⡇⠀⠘⢧⡇⠀⠀⡗⠀⠀⠀⠀⠂⠀⢘⣀⣨⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢶⣶⡟
⠀⠀⠀⢧⠀⠀⠨⠀⠀⢠⠅⠀⣙⠉⠀⠀⠀⠀⢧⡀⠀⠈⠑⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠑⠦⠤⠤⠖⠃⠀⠀⠛⠃⠀⠀⠀⠀⠀⠙⠶⠦⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠭⠭⠽⠶⠃⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀⢀⡎⠀⠀⠈⠛⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠒⠦⠔⠒⠉⠁⠀⢈⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠼⠆⠀⠀⠀⠀⡜⣡⡀⠀⠀⠘⢦⣤⣤⠤⠤⠤⠤⣀⠀⢀⣀⣀⣀⣀⠔⠁⠀⠀⠀⠀⠤⠶⢤⠤⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠢⢀⡀⡀⠀⠀⠀⢸⣎⣠⡷⠀⠀⠀⠐⢇⠹⠖⢒⠒⠒⠲⠟⠷⡄⢀⠤⣤⢤⠀⡄⠀⣀⣸⣀⡀⠀⠀⠀⠀⠀⠓⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠀⠈⠃⠺⣁⠀⠀⠀⢰⢸⢠⠘⠛⡃⠀⠃⠇⢰⠀⣿⡶⠝⠛⠊⢠⡭⠤⠄⠐⠞⢷⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⣴⠖⠈⠉⠹⠉⠉⠹⣗⠛⠘⠂⣶⠼⢴⣇⠗⠒⠒⢦⣼⡀⣏⡏⡍⡇⢸⠑⠉⠉⠑⠒⠒⠁
⠀⠀⠀⠀⠀⠀⢠⣊⣉⣉⡉⡯⠤⠤⠀⠶⠒⠒⠒⠒⠒⠒⠒⠒⢺⠋⠓⡟⢉⡑⣒⡒⢒⠒⡎⡠⢤⢸⢃⠀⣗⣇⣸⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⢀⠔⠛⠓⠒⠒⠒⢲⢸⢳⢘⠉⡇⣎⠣⠀⣇⣁⣛⣇⣸⡧⣧⣇⣯⠅⡇⢧⡸⢸⢩⡥⠁⠀⠀⠀⠀⡷⠒⢦⡀
⠀⠀⠀⠀⡐⠉⠉⢺⠉⡤⣔⢲⠶⡖⢲⣺⢨⢩⠠⠭⡅⡼⠳⡄⡇⠀⠀⠀⠀⠀⠀⠀⢻⡆⡇⠀⠀⢸⠴⢄⡀⠀⢀⣀⡀⢱⠀⡌⢳
⠀⠀⠀⠀⢱⠀⠀⠘⡀⡇⠸⢸⠀⡇⠈⣷⢨⣸⢠⠠⠇⠧⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇⣀⣤⣚⡳⠲⣌⠀⣭⠤⠤⠭⣴⠁⢸
⠀⠀⠀⡔⠁⠀⠀⠀⣇⢓⣚⡀⠓⠃⠀⣿⢰⠒⢰⣒⢰⠒⡆⢰⢒⡒⢒⠒⣒⠒⣒⠒⢒⢚⡟⠀⠀⠀⠙⢤⠀⢭⡇⣞⡲⣎⢿⠀⢸
⠀⠀⠀⠳⠤⢄⠀⠀⢸⠘⡄⣥⣏⣉⡆⣿⠸⠤⠀⠚⠈⣒⣃⢸⢩⡹⡏⡧⣇⣿⣉⠇⢘⢸⢇⠀⠀⠀⠀⠘⡄⢠⢓⡨⠝⠓⠾⢄⣸⡀
⠀⠀⠀⠀⠀⠀⠳⣤⠼⡇⠧⠼⠹⠤⠇⡇⡶⣋⡉⠥⠭⠤⠜⠘⠠⢤⡼⢢⡠⢢⡠⢦⣸⢾⠶⢷⢦⠶⠚⠯⢔⡋⢁⣀⣀⣀⡀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢃⠀⡶⢲⡄⠀⡇⠇⡖⠒⡦⡖⠲⠄⡇⠒⠊⠙⠊⠉⠈⠙⠂⠟⠚⠀⠘⣇⠀⠀⠀⠀⣿⠤⠤⠤⠤⠤⠤⠭⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠸⠀⢰⢸⡇⠀⠃⠂⠙⠒⠃⡇⢸⠀⡇⡠⣠⠊⠉⢦⡤⡤⡤⡤⡄⠀⠀⢮⣀⢀⠴⡆⢹⣒⣒⣒⣒⣒⣒⣒⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⣘⡼⠧⠤⠆⠄⠀⠀⠀⠁⠈⠀⠛⠀⠀⠀⠀⠸⠇⠇⠄⠂⠡⠀⠦⠠⠘⠀⠀⠃⠘⠃⠤⠤⠤⠤⠤⠠⠜⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⡄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢘⣿⠀⠀⣠⣿⣤⣤⣤⡄⠀⠀⠀⣼⡇⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⣸⠇⣀⠀⢹⡏⠀⠀⠉⠁⠀⠀⠀⣿⡇⠀⣠⠀⠀⣿⡶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠐⠋⢠⡟⢀⣿⠀⢰⡟⠀⠀⢻⣇⠀⣿⡇⠀⣿⠀⠀⣿⠀⢀⣴⠞⢛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢠⡶⣾⠃⣼⠇⠀⠹⣧⣤⡴⠟⣿⠀⢽⡇⠀⣿⡄⠀⣿⣄⣿⠛⠛⠛⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢠⡾⠁⠋⢰⡟⠀⠀⢀⣨⡅⠀⠀⣿⠀⢸⡇⠀⢙⣁⣠⣬⡼⣿⣄⣀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣀⣴⠟⠀⠀⠀⣾⢁⣠⡶⠟⠉⠀⣠⡾⠃⠀⢸⣷⠞⠛⠉⠉⠁⠀⠈⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠁⠀⠀⠀⠀⠛⠛⠉⠸⢷⠶⠞⠋⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢟⣩⣤⠀⠀⣸⡟⠉⠂⠒⠚⠃⠚⠉⠉⠈⠓⠐⠂⠠⠜⠀⠀⢰⠶⠖⠂⠒⠂⠀⠀⡆⠀⠐⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣶⠈⠉⠉⠉⠉⠉⠥⠄⠀⢠⠖⠀⠀⠀⠀⠀⣀⣀⣀⣠⣈⣀⣠⣤⡀⢈⠉⣉⣀⣈⣥⠾⠼⠿⠶⠶⠤⠌⠤⠄⠉⠉⠁⠉⠉⠙⠢⠀⠄
⠀⠀⠀⠉⠀⠀⢠⡀⢀⣀⡄⠤⠤⢀⣀⡈⠀⠀⠤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣨⡀⠀⠀⠉⣏⠁⠂⠐⢠⠀⠀⣀⡀⣀⠀⠀⠀⠀⢰⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠠⠄⠀⣀⡹⠀⠀⠀⠀⠐⡎⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠲⠖⠲⢦⠞⠀⠀⠀⠀⠺⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠁⠐⠈⠉⠉⠉⠀⠈⠀⠒⠀⠀⠀⠀⠀⣸⣾⢶⡄⠀⠀⠀⠀⠐⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠐⠑
⠀⠀⠀⠀⠐⢶⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠊⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣠⠄⡀⠀⡾⢹⣿⣿⢻⡇⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠊⠁⠀⠇⠘⣟⣾⣻⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠓⠒⠀⠀⠀⠀⠀⣟⣿⣻⡗⠀⠀⠀⢛⠀⠛⠛⡟⢿⣻⡗⢺⠆⣷⠓⡇
⡟⡗⠒⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢙⠀⠁⢈⡇⠉⢘⡃⢸⠀⡋⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⡏⠬⢡⠄⠀⣿⣿⣿⡇⠀⠀⠀⢨⠀⡴⢄⠀⢸⠀⠅⢸⠀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀⠀⠀⠀⠀⣤⡆⠀⢲⣶⠂⠀⠀⠀⠀⣟⣿⣻⡇⠀⠀⠀⢸⠀⣿⢋⣃⣻⡇⠆⢰⠀⠀⠀⡆
⠀⠠⠄⠤⠀⠀⠶⠀⠆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣲⣶⣾⡇⣶⠰⠶⠶⠀⠀⠀⠀⣯⣿⣯⡇⠀⠀⠀⢐⢀⢶⣿⣿⡐⠳⢂⢰⠀⠂⠀⡇
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣼⣿⣿⡧⣿⡆⠀⠀⠀⠀⠆⠀⣿⣿⣿⡇⠀⠀⠀⢸⣸⡞⠓⢹⡇⠅⢸⢻⠀⣁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣈⣽⣛⡇⢿⡗⡆⠀⠀⠀⠁⠀⣿⣿⣿⡇⠀⠀⠀⢨⣸⣇⠤⣀⠁⠂⠨⢸⠀⠄⠀⡇
⠀⠀⠀⠀⠀⢀⣠⣤⢴⣲⠿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⡛⢙⣏⡇⢼⣿⣙⡆⠀⠀⠄⠀⣯⠿⣽⡇⠀⠀⠀⠸⠸⠽⠀⠈⠂⢰⠎⠸⠀⠤⠀⠇
⢠⢠⡖⢲⡅⠘⠛⠉⠀⠂⠒⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠿⣿⡿⠿⠃⢾⡟⢻⢸⣆⠀⠀⠐⣿⣿⣿⡇⠀⠀⠀⠀⢀⠸⠀⠀⢰⣥
⡇⠐⠃⠉⠁⠐⢶⣶⣴⣤⣤⣤⠀⠀⢠⠴⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠉⠀⠀⠘⠘⣿⠄⠸⡏⣦⠀⠈⣿⣿⣿⡇
⠈⢀⠀⠀⠀⠀⠀⠉⢿⣛⠩⠇⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡄⢡⣧⠸⡇⣿⠂⠐⣿⣿⣿⡇⠀⢀⣀⠀⣀⣀⣀⣀⣀⣀⢀⣀⠀⠀⠀⣀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⣀⠀⠀⢰⠂⢀⣠⣴⢶⣿⣭⣿⠀⠀⠀⠠⠘⣼⢆⠃⢿⠀⠘⣷⣟⡾⡧⠀⢸⡇⠀⢠⣿⣿⡷⣿⣿⣾⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠠⠀⠓⠲⠤⠀⠀⠀⠈⠁⠀⠀⠀⠹⢭⢭⠽⢭⡟⣭⡍⣯⠀⠀⠀⢀⠀⢹⢹⡇⢸⠀⠐⣿⣿⣷⡇⠀⢸⡇⠸⠉⡇⣿⡇⠿⣿⣿⣿⠀⠀⠀⢸
⡆⠀⠃⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⣉⣉⣉⣿⣏⡇⠀⠀⠀⠀⡆⠈⠸⠇⣜⠀⠰⣿⣿⣿⡇⠀⢸⡇⠀⠀⣫⣿⡇⢸⣿⣿⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠚⠛⠛⠛⠟⠛⠳⡇⠀⠀⠀⠀⢁⠀⠀⠀⣿⠀⠐⣯⢿⣽⡇⠀⢸⣇⠀⠀⠑⣿⡇⠞⢻⢾⣿⠀⠀⠀⢸
⠀⠀⢀⡀⠔⠊⠀⠀⠀⠀⠰⠤⠀⠀⣀⣌⢔⡚⢂⡘⣂⣒⣐⣀⣓⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⣿⠂⠰⣿⠶⡛⡇⠀⢸⡏⠂⠀⠀⣿⡇⠀⠄⢸⣿⠀⠀⠀⢸
⠀⡠⠋⠀⠀⠀⢀⠀⠀⣀⠵⠖⠛⠁⠀⠀⠀⠀⠀⡤⢤⠤⠤⢤⡤⠤⢴⠀⠀⠀⠀⠀⠀⠇⠀⠀⠉⠀⢠⣿⣶⣻⡇⠀⢸⡇⠀⠀⠀⣿⡧⡁⠀⢸⣿⠀⠀⠀⢸
⠳⡱⣄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⢁⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⣟⣷⡛⡇⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⢸
⠀⠙⠘⠀⠀⠀⣀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠐⠀⣲⣶⣶⣶⣶⣾⣧⣶⠀⠀⠀⡂⠀⠀⢸⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣫⠒⠫⠽⠫⣢⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡟⡋⢱⢙⢶⣞⠇⠀⠀⣸⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠑⢻⡼⡜⠁⠀⠀⠀⣿⣿⣿⢿⣿⠟⠁⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠰⡁⠀⠀⢰⠙⢿⢟⣁⣨⢥⣀⣀⡤⣄⣀⣀⡙⡟⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢃⠀⠀⠀⠑⢺⠀⠀⠑⠊⠀⠈⠒⠃⠀⠀⢠⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⡀⠀⠀⠀⠀⡆⠀⠀⠀⠘⠂⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠱⠀⡠⠤⢀⣘⣄⠀⠀⠀⠂⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⡇⠀⠀⠀⠀⠑⠒⠀⠀⠐⣲⣿⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱⠀⠀⢀⣀⢀⣠⣾⣿⡋⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣾⣿⡿⢿⣿⣿⣿⣿⣶⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣻⣎⠙⢋⣾⣿⠿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠘⣿⣯⣤⣤⠤⣉⠙⠛⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠈⠀⠀⠀⠀⠀⠉⢹⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣧⢤⣤⣀⣀⣠⣤⡤⠤⢴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣤⢤⣽⣷⣀⣠⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠃⠀⠀⠀⠀⠀⢳⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⡞⠀⠀⠀⠀⠀⠀⠈⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢧⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣀⣉⣚⣺⡄⠀⠀⠀⠀⠀⠀⠿⠋⠀⠤⠽

⠀⢠⡄⠀⣧⣤⢦⡈⠲⠀⠀⠀⢠⣀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠚⠀⠀⡇⠀⡠
⠀⢸⡿⠀⠋⠈⠐⠛⠁⠀⠀⡆⠸⠛⠒⠄⠀⠀⠀⠁⠌⠀⠀⠀⠀⠀⠀⠰⠃⠀⣭⡏⠀⠀⢠⡴⠛⠀⢀
⠀⠈⠃⠀⠀⢹⠀⠀⠀⢘⡓⠂⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠱⡂⣃⡀⠀⠴⠈⠀⣠⠔⠁
⠤⠤⡃⠀⣀⣠⣀⠀⠀⢸⡇⠀⠸⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠊⠁⠛⠂⠁⠀⠀⠊⣀
⠐⡀⠀⠈⠙⠠⣼⠁⢠⢸⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⣠⣤⡎⢙⣄⣀
⠀⡃⠀⠀⠀⠀⠀⠀⠺⠄⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠄⡄⠀⠝⠓⠈⠋⠁
⠤⠧⠀⠀⠰⠄⠀⠠⣠⢀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⡀⠒⠂⠁⡠⢊⣀⣠⠤⢴
⠒⠀⠠⠤⠀⠀⠠⢤⣤⣄⣀⣀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣁⣠⠀⠀⠒⠚⢯⢥⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣏⠈⠇⠋⠉⡟⠋⠐⡈⣹⣛⡆⡃⠈⠀⠀⢀⣀⣀⡉⠀⣨⡇⠀⢀⣀⠀⢰
⠀⠀⠀⠀⠀⠀⠘⣧⠀⡄⣤⣠⡧⠀⠴⠀⣹⡽⠗⠛⠛⠛⣾⡏⣰⠉⢹⠃⠉⠉⠉⠉⠉⠉⠛⢄⡀
⠀⠀⠀⠀⠀⡀⣀⠛⢲⠉⠁⠀⡁⠀⡠⠁⠁⠀⠀⠀⠀⠀⢯⣠⠉⠃⠾⠄⠀⠀⠀⠀⠀⠀⠀⠈⠧⠉⠩
⠀⡴⢾⣿⣆⠈⠁⠀⠀⡀⠀⡠⠝⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣠⡀⣄⣀⡠⠀⠤⠖⡪⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⣠⢤⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⢠
⠄⠂⠐⠂⠀⣀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠀⠃⠀⠀⠀⠄⠆⠀⠀⠀⠀⠀⠡
⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠶⣾⢺⣅⣸⣗⡷⠖⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⢙⠉⠀⢠
⠀⠀⠀⢀⣶⣶⣦⠀⢀⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣸⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤
⠀⠀⠀⠈⠉⠉⣽⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⡀
⠀⠀⠀⣠⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠉⠉
⠀⠀⠀⢸⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠛⠛⠋⠀⠀⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣟⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⡟⠻⠿⠿⠿⠿⠿⠿⠿⢿⡿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣿⣏⣛⣟⣙⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⡰⠋⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠡⡐⢌⠳⡀⣷⣣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢶⣿⡵⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠞⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠂⢌⣿⣠⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠴⠖⠋⠑⢦⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⠘⠛⢻⢇⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⣸⣿⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢳⣀⣟⡟⠠⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⡬⢧⣿⠾⠓⠁⠀⡠⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⣲⠆⠘⠁⠈⣭⣭⡇
⠀⠀⠀⠀⠀⠀⠀⢾⡰⢧⠐⢛⣦⠀⠀⠀⡇⢰⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⢹⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣻⠀⠀⠀⣷⢸⡄⠀⠀⠀⠀⠀⠰⠰⠀⠀⠚⠸⠂⠀⠀⠀⠀⠀⠀⠀⡟⠐⡏⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⣧⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢠⠀⡀⠀⢀⡀⡄⠀⠀⠀⠀⠈⠀⢰⠇⢸⠇⠀⠀⠀⣠⢞⡵⠒⢕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⣴⡇⠀⠀⠘⡇⢷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⡟⠀⣽⠀⠀⡴⢛⡵⠋⠀⠀⠀⠱⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢧⢸⡀⠀⠀⠀⠀⠒⠒⠀⠚⠀⠰⠀⠀⠀⠀⠀⠀⢸⡇⢨⡇⠀⠈⠓⣯⠰⡋⢶⠟⣱⠀⣸⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⢄⠀⠀⠀⠀⢸⠈⣇⢀⣀⣀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠀⣼⠀⠀⠀⠀⠸⡄⠈⠠⠞⠁⢀⡏
⠀⠀⠀⠀⢀⡤⡶⠋⠀⠀⠀⢀⣤⣓⠂⠄⡀⠘⢦⢈⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⢀⣤⣤⣤⣤⣤⡤⠶⠃⠀⠀⠀⠀⠀⠙⠚⠉⠉⠉⠉⠁
⠀⠀⠀⠠⡏⠈⠂⠤⢤⣴⡶⢍⠫⠤⠴⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⢻⡴⡿⠐⣲⡄
⠀⠀⠀⠀⠙⠶⣤⠤⠤⠴⠶⠾⣐⣶⡾⠟⠁⢀⣀⣀⣀⣀⣀⣀⣈⣷⣶⣖⣒⣒⣒⣿⣁⣀⣀⡠⢤⣤⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢠⠇⣦⠗⣆⠖⢠⢄⡴⡄⣠⠀⠀⠀⠀⠀⠀⠀⠊⠇⠫⠊⢆⠀⢀⡄⡐⡋⠀⠀⠉⠐⠠⣠⣄⡀
⠀⠀⠀⠀⠀⢠⡀⠠⠂⠀⠀⠀⠀⠀⠀⢠⡏⡜⠒⢋⢛⢖⢼⢸⣷⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠘⠂⠫⠘⣉⣡⣿⣣⣤⣴⣴⡿⠀⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⢀⣃⣀⣁⣴⣋⠁⠀⠀⢀⡞⠴⡳⠰⣋⠇⠈⢢⠀⠀⠘⢖⣾⣊⣀⡀⠀⠀⠩⠆⠠⠟⢋⣉⣃⠀⠉⠉⠉⠉⠉⠀⠀⠘⢣⡀⠐⠃
⠀⠀⠀⠀⢺⡭⣤⣤⣤⡤⠤⣿⠀⠀⠈⠓⠒⠲⠶⠶⠒⠒⢺⡄⠀⣴⡟⠤⠔⠒⠒⠒⠒⠒⠚⠻⠛⠋⠉⠃⠀⣀⣤⡤⠞⠏⠃⠀⠀⠀⠈⠻⣉
⠀⠀⠀⠀⠀⣣⡀⠀⠀⠐⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⢭⠓⠈⡤⠀⣀⡜⣀⠀⠀⠀⠀⠀⠙⠫⡉
⠀⠀⠀⢰⠏⠁⠭⠖⣛⠻⠭⠆⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢃⣚⣨⢴⣺⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⣴⠶⠄⣀⢀⠀⡀⢠⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠠
⢀⡐⠀⢀⣶⡿⠆⢺⡁⢠⡾⣽⠋⠰⢆⡼⠿⢧⡿⠃⠰⢷⡆⠀⠀⠀⢈⣥⠀⢸⣿⡍⠈⣡⠀⠲⠀⢀⠋⠁⠀⠀⠛⠃⠷⣨⠿⠀⠁⠰⣷⠛⣶
⣿⡿⠀⠙⠙⠃⠠⠾⠏⠒⡦⠀⡤⡀⠈⠀⠀⠀⠀⠀⢀⠀⣿⠀⠀⢠⠛⠁⠀⠈⠉⠁⠀⠀⠀⠠⠀⢲⣷⣶⠀⠲⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⣿⠁⠀⠀⠀⠀⠀⡸⠛⢿⠃⠸⠇⠉⢠⡄⠀⠠⠀⠀⠀⣾⠀⣦⣠⡎⢀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢈⣿⣯⠀⠀⣽⣄⠠⡄⠀⠀⠀⠀⠉⠉⠿
⠋⠀⣀⣀⠀⠀⠠⡀⠀⠘⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⡃⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠶⠻⠃⠀⠀⠾⣿⡟⠀⠀⣿⡿⠀⠁
⠀⣿⣿⣿⠀⠀⢰⡇⠀⡀⠀⠀⠀⠀⠈⠀⠀⠘⠂⠀⢈⣧⠀⠈⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⢸⠇
⠀⣿⣿⣿⡀⠀⠨⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢽⣿⣿⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⣠⣆⡀⠰
⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⣄⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⡀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⠛⢿⡇⠀⢦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡸⡿⣷⠀⠘⡇⠀⢀⠂⠀⢀⣀⣿⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⡷⡏⠀⠀⠑⣠⣄⣴⣾⠟⠁⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⣿⣦⡀⠀⣿⢿⣿⣿⡷⠉⢄⣸⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣭⣿⣿⣿⠙⠐⠋⠉⠉⠉⠃⠀⢸⣿⡿⢯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣿⣹⣿⣿⣿⣿⣦⢠⣾⠟⣿⣿⣿⡿⢿⡇⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⣿⣉⡟⠉⠉⠙⢻⢸⠋⠙⠉⢻⣿⡷⢾⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠛⣙⢿⡄⠀⣠⡛⠘⣣⡀⠀⣿⣯⡁⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠘⢿⡏⣷⠀⠏⣭⣭⡌⠃⣸⣿⡷⢠⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⡌⣇⣿⣷⡀⠈⠛⠀⣰⢿⣻⠄⣘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠂⢨⣭⢩⡄⡁⡅⠰⠛⠓⠒⠻⡅⠀⢠⠀⠀⠀⠶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⢀⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⠶⠶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢋⡡⣥⠉⢻⣿⣿⡟⠄⠛⢀⡁⠛⠉⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⣬⢁⠠⣘⠀⣿⡏⠉⣐⠿⠘⠱⠟⡀⠾⢿⡛⠿⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠙⠃⣈⢉⣤⡀⠘⠛⠓⠀⠩⠴⢀⢘⠻⠇⢀⠀⢡⣦⠈⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠩⢁⣴⣿⡌⠸⣿⠁⣀⡾⠸⢋⣤⠀⠙⢈⡁⠀⣠⣴⡌⣉⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠄⢸⣿⡏⢰⣦⠙⠀⣭⡅⠀⡀⢶⡆⢀⡘⠛⡀⠀⠀⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣷⠆⠘⡭⠂⣤⣤⣴⢤⠈⠰⠇⠻⠎⠡⠈⠁⠀⠀⠀⢤⠀⢼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠀⠖⠀⠀⠘⠿⠿⠛⠀⡀⠐⠆⠰⢴⣶⣿⠀⠈⠃⠰⠂⢠⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠘⠀⠿⠆⣩⠀⢰⣷⣤⠁⢐⠈⡄⠘⣿⣿⣿⣷⣦⣶⣤⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⠀⠀⠂⣀⣘⣿⣿⣿⣶⣈⡁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⡠⠄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃⢀⡿⠃⢐⠟⠀⠚⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⣼⠁⠀⠏⠀⠀⡏⠀⣸⡀⠀⢻⡿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⡄⠸⠀⠀⠀⠀⠀⠇⠐⠁⣀⠘⣏⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⣀⣰⣇⡀⢀⣴⣦⠾⠚⠓⠾⣟⠁⠀⣟⣴⡟⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠑⣶⣶⣧⣀⣲⡿⠛⠉⠀⠈⠉⠀⠀⡠⠀⣀⣀⣀⣠⣿⠋⣠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡏⠀⢤⣤⣤⡶⠶⠶⠟⠓⠛⠛⠛⠛⠋⣡⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣁⣤⣤⣤⣴⣶⣶⠾⠿⡿⠿⣾⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠷⠷⢷⣶⣿⣿⠶⡿⠿⠿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣤⡀⢤⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡏⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠛⢋⢀⣀⠀⠀⠀⠀⠠⠀⠠⠖⠂⠂⡀⣆⣐⡀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⣀⡀⠀⠀⠀⠈⡉⠉⠉⣁⡁⠀⠀⡀⠀⠈⠈⢉⠁⠀⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⡄⠐⠛⠛⠛⠉⠠⠀⠀⠀⠀⠀⣀⠈⡀⠐⠃⠀⠂⠤⠧⣤⠄⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠤⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠒⢂⠀⠀⠀⢠⠀⠀⠐⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠆⠈⠉⠉⠉⠀⠐⠀⠀⠀⠀⠨⠭⠍⠀⠀⡀⠀⠀⠀⠥⠠⠀⠀⠐⢃⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒⠂⠀⠀⠀⠀⠂⠐⠒⠒⠂⠀⠀⠠⠀⠠⠄⠠⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⠒⠊⠉⠙⠒⠊⠉⠉⠒⠊⠉⠉⠒⠚⠉⠉⠒⠒⠉⠉

⠀⠀⠀⠀⡏⢸⢹⠋⠙⠀⡟⠉⠉⢛⣹⣿⣿⠙⡏⠀⠀⢻⢿⡇⡆⢠⠐⠀⢰⠒⠖⢐⣖⠂⢰⢶⠒⡓⡖⣾⣿⣿⡀
⠀⠀⠀⠀⡗⠴⣰⡆⠀⡆⣇⢀⣴⣿⣿⣿⣿⣾⣇⣤⣀⣠⣬⣧⣏⠸⠜⠀⠈⢁⠃⡶⡏⣾⠠⣾⣿⠉⣼⣿⣿⣿⠁
⠀⠀⠀⠀⣧⣤⣼⣧⡎⢠⣿⣿⣽⣯⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣇⠀⠀⠈⠠⢠⡇⠘⢰⣟⣯⣾⣿⣿⣿⣿
⠀⠀⠀⠀⡿⣿⣿⣿⡿⠿⣿⢿⣏⣵⢟⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⣾⣠⣶⣾⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⣿⣿⡏⣻⢃⣴⣏⣾⣿⢟⣿⣿⣿⣿⣿⣿⠟⠰⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣷⣿⢾⣿⡟⡽⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡿⢡⣿⡟⣿⠱⣻⣿⣿⣽⣿⣿⡿⣻⣿⣀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡟⢠⣏⡾⣿⣿⣿⣿⠉⣿⣿⣿⣞⣽⣿⣿⣿⡿⣽⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣷⣿⣽⡱⣿⣿⣿⣿⣇⣟⣽⣟⡿⢻⠟⣹⢋⣽⢿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠿⠯⠿⠵⠿⠿⠿⢽⣿⡿⢛⣿⣴⡿⢸⡯⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣬⡉⠁⠉⠙⠻⠧⠿⠿⠽⠿⠟⠋⠛⠛⠁⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣄⠚⠛⠛⠛⠿⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⣀⣴⣶⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⢀⣤⢖⢿⣿⣿⡿⠖⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡶⢦⣀
⢀⣼⣿⣿⣿⣿⣿⣿⡟⡟⢻⣷⡄⠀⠀⠀⣰⣿⣿⣷⣕⠁⡈⣲⣿⣿⣿⣆⠀⠀⠀⢠⣾⣿⠿⠛⠿⣿⢭⡅⢨⡭⢤
⣼⠉⠙⠛⠛⠛⠀⠀⠀⠀⠴⣿⣷⠀⠀⢰⣿⣿⣿⠿⠏⡊⢐⣏⡻⣿⣿⣿⡆⠀⠀⣾⢉⣿⣭⠀⣥⣿⣶⡆⢰⣶⣿
⣿⢸⠒⠢⠦⠀⠉⠁⠐⢢⠐⢪⣿⠀⠀⢸⣿⣿⡿⣒⢿⡇⢸⡿⣓⢿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⡇⢸⣿⣿
⠸⣮⣒⣢⣶⣦⣴⣶⣾⣯⣂⣪⠏⠀⠀⠈⢿⣿⣜⡿⠡⠤⠠⠤⢿⣣⣿⡿⠁⠀⠀⠀⠈⠙⠛⠲⠿⣿⣿⡇⢸⣿⣿
⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠻⣿⡇⡷⠛⠿⠾⢺⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠠⢤⠀⠈⠉⠛⠻⠃
⠀⠀⠀⠈⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⣴⣿⣿⣿⣛⠋⠛⠙⠦⠀⠀⠀⠀⠀⠀⣠⣾⡿⣯⠻⠟⢽⢿⣷⣄⠀⠀⠀⠀⠀⣠⣤⡍⢹⠀⣉⡋⡀⢰⣦⡀
⢠⠏⣿⣿⣿⡏⠏⢾⠏⠷⣿⢸⣆⠀⠀⠀⣾⣿⣿⣛⠀⢀⡀⠀⣛⣿⣿⣷⠀⠀⠀⠀⡿⠩⡇⢸⣶⣤⣤⣴⣾⣿⣿
⣿⣿⣿⣿⣿⣿⠸⡷⠷⠒⣿⢸⣿⠀⠀⢸⣿⣿⣿⡭⢀⢌⡡⡀⢬⣿⣿⣿⡇⠀⠀⠀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿
⢿⢠⠉⠉⠉⣅⢃⡠⠔⠀⡀⠀⠈⠀⠀⠸⣿⣿⣿⣷⠎⣼⣧⠱⣾⣿⣿⣿⠇⠀⠀⠀⣿⣩⡁⠆⠼⠹⠋⠟⣽⣹⣏
⠈⢷⡶⣶⡶⠖⠉⠀⡼⠼⡇⠀⠀⠀⠀⠀⠻⣿⣿⠏⢬⣿⣿⡅⠹⣿⣿⠟⠀⠀⠀⠀⠈⠉⠀⠢⠀⣆⠀⠀⠈⠙⠻
⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠋⢐⣿⣿⣿⣿⠀⠙⠁⠀⠀⠀⠀⠀⠀⠐⠀⠘⠀⠉

⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⣄
⠀⣰⣿⡿⠋⡏⢭⠍⡏⣿⣿⣦⠀⠀⠀⠀⢠⣾⡟⢭⠶⢦⠰⢶⣭⢻⣷⡄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣸⣿⡿⣼⠀⡿⠿⠟⡇⣿⣿⣿⣧⠀⠀⢠⣿⣿⡟⠀⠀⢜⡣⠀⠀⢻⣿⣿⡄⠀⠀⣼⡿⠿⢟⣽⣶⠖⣶⣶⡖⢾⡿
⣿⣿⡷⢟⠀⡇⠀⠀⡇⣿⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⡘⢃⠀⠀⢸⣿⣿⡇⠀⠀⣇⠤⠤⡄⠀⠀⠀⠀⡠⠤⢄
⢹⣿⣿⣿⠀⡇⠀⠀⡇⣿⣿⣿⡟⠀⠀⠘⣿⣿⣿⠂⠠⠀⠀⠄⠐⣿⣿⣿⠃⠀⠀⢿⡐⠃⣰⣶⣶⣶⣶⡄⠚⢠⣿
⠀⠹⣿⣿⣶⣥⣤⣤⣧⣿⣿⠟⠁⠀⠀⠀⠘⢿⣿⠆⣁⣀⣀⣈⠰⣿⡿⠃⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡀⢼⠀⠀⠀⠀⠀⠀⠀⠰⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⣐⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡒⡇⠚⣴⣶⠁⠀⠉⠉⠉⢘⣯⣽⠀⠉⠉⠉⠃⢀⢀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠇⠐⣾⣿⡇⠀⠀⠀⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠂⠉⠀⢠⢄⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⡂⠀⠀⢀⠀⠀⢀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⢠⠉⠀⠀⠀⠀⢀⢸⠀⠀⠀⠀⠀⠂⠀⠀⢺⠀⠀⠈⠀⠘⠀⠀⢨⠀⠀⠀⠒⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠁⠈⠐⡄⡅⣇⣴
⠀⠀⠀⠀⠀⠀⠐⠘⠂⢀⣀⠀⣆⣀⣈⠄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠇⠋⠽⠁
⠀⠀⠀⠀⠀⠀⣦⣄⣤⡄⢠⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠄⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠁⠈⢀⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸
⠀⠀⢀⡠⠀⢠⣦⢀⠉⡉⠿⠝⠀⠀⠀⠁⡠⠠⡴⣀⣠⢈⢩⡀⢳⠎⠶⠙⠀⠀⢛⣖⠐⢂⠲⠮⠏⠋⠏⠓⣸⠀⠰⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⡖⣀⣀⠀⣾⣁⠀⢀⣽⡧⢤⡞⠁⠀⠀⠁⠉⠙⠈⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠴⠀⠀⠀⡠⠂⠘⠀⠗⢆
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠋⠁⠐⠒⠒⠒⠒⠒⠒⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣌⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣷⡔⣌⢷⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣾⠉⠉⠙⢛⠿⣧⣽⣯⠉⢉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⣴⣿⣷⣤⠤⠀⠀⠀⠉⠛⠿⣿⣇⡀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠄⠒⠉⠀⣴⣿⠿⠛⠛⠿⢷⡇⠀⡀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣄⠙⠠
⠀⠀⠀⠰⠖⠚⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣴⣟⠀⠀⠀⠀⠀⠁⠨⣿⣷⣦⡄⣀⠀⠀⠀⠀⠀⠙⢦⣔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣲⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠉⢻⣿⣮⣱⠷⠡⠂⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡛⠛⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡙⠓⢶⣤⡀⠀⠀⠀⠈⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣾⣿⠃⠀⠐⠀⠀⢲⣤⣀⡀⠀⠀⠀⠀⠀⠙⠳⠶⣽⣿⣶⣄⠀⡎⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠿⣿⣿⣦⣠⣤⡀⢈⠌⢹⠛⢿⣧⣤⡀⣄⡄⡀⠀⠀⠀⠉⢿⣿⣿⣦⣤⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠁⢀⡈⡉⠙⠿⠇⢿⣆⣄⣆⠀⡄⢨⡟⠻⣿⣮⣡⣀⠀⠀⠀⡼⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣦⣜⣿⠀⢟⢻⣿⣯⠀⠀⠀⠀⠟⣿⣿⣾⣤⣲⣀⣀⢦⣿⣿⣿⣿⣿⣿⣷⣶⣤⡾⠿⣷⣄⠑
⠀⠀⠀⠠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣙⣿⠿⣿⣤⡄⣈⠁⠛⣿⣽⣿⣿⣷⣾⣿⣿⡿⠿⠛⠛⠋⠉⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣾⣥⣧⣼⣵⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠈⠡⠶⠶⠶⠶⠲⠟⠳⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠉⠉⠉⠙⠛⠛⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠤⢤⣿⣻⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⡶⠈⠚⣿⣿⣶⣿⣿⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⢠⠐⣯⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠐⠨⢴⡀⢀⡈⠛⣻⣿⡿⠿⢿⣷⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠛⢥⡀⠉⢠⠀⣄⢢⣾⣷⣿⣿⣿⣟⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠰⢄⢠⠆⠀⠀⠘⢺⣿⣟⡻⣿⣟⣽⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⠀⠁⠈⠳⣴⠋⠃⢸⣸⣿⣿⣧⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡓⢄⠈⢤⣀⠘⠁⠀⠰⣼⣿⣿⣿⡿⠟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⡿⠠⠀⠉⠀⠐⢆⣿⣿⣽⣟⣿⠞⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⠀⠉⢄⠀⠒⠐⣟⣚⣿⣿⣿⣏⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⣀⠠⠒⠉⣿⣯⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⣿⣿⣿⣿⠇⢸⣿⡿⠿⠟⣳⡀⣴⣲⡴⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⡄⢳⡞⢸⣯⣡⠟⢷⣙⣙⣬⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⠀⢈⣹⣿⣿⣿⣿⣿⢨⣅⣈⣉⣀⣀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⢿⣿⡿⠿⠿⠛⠁⠀⠘⠻⠿⠿⢿⣿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣤⡀⣤⢠⣄⢠⡄⡤⢤⡀⣠⢤⣤⠤⢤⡤⣤⣠⡤⠤⠀⣤⡄⢠⣤⢤⡤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⡄⣿⠻⣿⢸⡟⢿⡇⡿⠘⣷⡟⢸⣿⣛⢸⡷⣿⢈⣛⣷⣼⠷⢿⡸⡿⢾⡃⢸⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⣀⠔⢂⠠⠂⢀⡤⠒⠀⠀⢰⠀⠀⠀⡆⠀⠀⢠⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡠⠊⡡⠚⢁⣠⠚⢩⠀⠀⡆⠀⢸⠀⠀⠀⡇⠀⠀⢘⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣠⢎⡴⠋⣠⠖⠋⢠⠀⢸⠀⠀⡇⠀⢸⠀⠀⠀⡇⠀⠀⠀⣇⣀⡤⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⡴⣡⠋⠀⡏⠀⠀⡀⢸⠀⢸⠀⠀⡇⠀⢨⣄⣀⣀⣧⣤⣖⣚⣉⡥⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡸⠐⠉⠉⢱⡇⢐⠀⡇⠀⠀⢸⠀⠀⣇⣠⣼⠗⣋⠭⠄⠚⢩⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣀⠴⠲⡄⢠⠃⠀⠀⠀⢸⡇⢸⠀⢧⣤⡤⠴⣖⣋⢩⠒⢺⠁⠀⠀⠀⠀⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠞⠁⠀⠀⡇⢸⢰⣀⣐⣀⣸⣧⣴⣚⣻⠁⠀⠀⠀⢹⢸⠀⢸⠀⠀⠀⠀⠀⢰⣿⠀⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠀⡇⢸⢸⠀⠀⠀⡏⠉⠀⢸⣿⠀⠀⠀⠀⣼⢸⠀⢸⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⣀⣤⠔⠒⠋⡇⢸⢸⠀⠠⠀⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⢸⠀⢸⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⠀⠀⠀⠀⡇⠘⠁⠀⠀⠀⠀⡇⢸⠘⠀⠘⠃⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⢸⠀⢸⠀⠀⠀⠀⠀⠈⣏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⡃⡦⠈⣠⠀⠀⠀⡇⠀⠀⡆⡄⠀⠀⡇⢸⠈⠀⢰⠀⡇⠀⠀⣿⣿⠀⠀⠀⠀⢸⠸⠀⣸⣤⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠀⠘⠒⠛⠀⠀⠋⠁⠀⠀⡇⢐⠀⡇⡅⠀⡆⡇⢸⢘⡀⢘⡀⡇⠀⠀⣿⣿⠀⠀⠀⠀⢸⣠⣴⣿⣿⠉⣀⣲⠦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠑⡀⠅⢸⠀⠇⠇⠀⠁⡇⢸⠀⠀⢈⠀⡇⠀⠀⡏⣿⠀⠀⠀⢀⣾⣧⢨⣿⠻⢻⣿⣯⢻⣄⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⠠⠀⠄⠒⠀⠀⡅⠐⠬⡁⠀⠣⠀⣄⠁⢸⠀⠀⠈⠀⠇⠀⠀⡇⣿⠂⠀⢸⣹⣟⣿⢿⣹⣄⣾⣿⣵⣽⡿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⡍⠀⠀⢇⠀⡞⠀⠀⠀⠀⠑⠘⡄⠀⠀⠀⠉⠀⠀⠀⡁⠸⠀⠃⠐⠒⢧⠀⣀⡄⣿⢨⠈⣴⣿⣏⢛⣤⢼⣿⣿⡿⠿⣼⠃⣿⡇⡇
⠀⠀⠀⠀⠀⠀⡆⠀⠃⠀⠀⠈⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠒⠦⠤⢿⢸⡖⠻⠯⢭⣾⣹⢿⣿⣿⣤⣦⢰⠀⣿⡇⡇
⠀⠀⠀⠀⢸⣷⠁⠀⠀⠀⠀⢀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⡇⠀⠀⠀⠀⠉⢹⡏⡏⠉⠉⠀⠉⣿⡇⡇
⠀⠀⠀⠀⠀⠹⡇⠉⠁⠀⠀⠈⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⢄⡀⠀⢸⡷⠓⠀⠀⠀⠉⠉⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠸⣇⣠⡤⠤⠤⠤⠤⠤⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠠⠇⠀⢀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠴⠶⢤⡀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣄⠀⠀⠀⠉⠳⣝⣻⠿⠷⢀⡀
⠈⠙⠛⠻⠿⢭⣽⣟⣛⣛⣛⡛⠛⠛⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⠿⣿⣷⣶⣶⣶⣶⣤⣤⣄⡀⠀⠀⠀⠀⠙⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⢿⣿⣷⣶⡤⠤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠶⠦⣬⣍⣉⡙⠛⠓⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣴⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭
⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠙⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⠻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡞⠫⣾⣿⠟⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⢉⡴⢋⣾⣷⣬⠏⣀⠄⠀⠀⠈⠉⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢧⢀⡴⢋⣴⣿⣿⣿⣿⣴⣧⠞⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢠⣾⢋⣴⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⡠⠊⣠⠞⠋⡡⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⡄⣠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⡎⠀⢻⠟⢥⠊⠁⠀⠀⠑⠛⣀⣀⣀⣤⣤⣾⣵⣷⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⣿⣿⣿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⠀⢰⠁⣠⡎⠀⢀⡱⠀⣀⣠⣴⣾⣿⣿⣿⡿⢿⣿⣧⣌⠛⠛⣧⣤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠎⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡠
⠀⠀⠀⠀⣅⡼⣡⡿⣶⡶⡾⢿⣿⣿⣿⣿⣿⣿⣿⣏⠸⣿⣿⣿⣷⣶⣤⣄⣙⣳⣂⣤⢀⣠⣾⣷⡶⠀⢰⣿⣿⣿⠛⡿⣿⢿⣿⡿⡏⡇
⠀⠀⠀⣠⣿⡾⠋⠀⠻⣧⡄⢤⠙⢯⣽⣿⣿⣿⣟⡛⠲⢿⣝⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⠯⣼⣿⠃⠀⢸⣿⣿⣿⣇⣁⡇⣿⣿⡇⠇⠃
⠀⠀⠐⠿⠋⠀⠀⠀⠀⠈⠙⢦⣀⣼⠁⠈⠹⣷⠀⠈⠉⣦⠈⠻⠦⣤⣽⣿⣟⣻⣷⢤⣬⣀⣻⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⢸⠟⠒⠚⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠒⠲⠤⣤⣀⠀⠀⠀⣠⠞⠁⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠉⠉⠓⠒⠦⢤⣄⣀⡀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢦⣀⡴⠋⠀⣰⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⠀⠀⠀⠉⠀⠉⠙⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⠀⠀⠀⠠⠴⣲⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠛⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠚⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⣀⣀⣀⣀⣀⣈⡁⠒⠂⠀⠀⠀⢈⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢠⡶⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠭⢿⣿⣯⣥⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣴⠏⢀⣠⡴⠋⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠉⠛⠛⠉⠉⠁⠀⠀⠀⠀⢀⣠⢤⡀⠀⠀⢀⣀⠤⠖⠉
⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠋⠁⠀⠀⠉⠉⠉⠉
⠀⠀⠀⢀⣴⣃⡒⠉⣀⣀⣀⡤⠶⠚⠉⠁
⣠⠴⠚⠉⠁⠈⠉⠉⠉

⣷⠀⠀⠀⢀⣤⠶⣤⡃⡸⠆⠀⠁⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⠋⠀⠀⠁⠀⠻⣦⠀⠹⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠄⡀⠰⢿⣿⣷⣿⣾⣆⠀⠀⠀⠀⠀⡀⠄⠰⢷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣦⣜⠻⠂⠸⠶⠄⠺⠶⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠠
⠀⠁⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠛⠛⠀⢠⣿⣿⣿⣧⣘⠻⢆⣰⣶⣾⠃⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠇⠀⠄
⠀⠀⢤⣤⠀⠑⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡴⢦⠙⡿⡱⢲⡀⠙⠻⢿⣿⣿⣿⣿⣿⣿⠟⠀⠀⡀⠀⠉⠃⠈⠀⠀⠀⠀⠈⠋⠀⠀⠠⡀⠀⠀⠐
⠀⠀⠈⠃⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣾⣿⣿⣿⠀⠀⠀⠙⠻⠿⠟⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠁⠈
⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣤⠀⣠⡤⠀⠀⢀⣀⠀⠐⠋⠁
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠅⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⢀⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠰⣿⣿⣿⡿⠋⢠⣦⡘⠿⠆⢀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⣆⡆⠀⠀⠀⠐⠀⠃⡄⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠻⠿⠋⠀⠀⣼⣿⣿⣶⣾⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⡀⠀⠀⠀⠐⠀⠀⢺⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠡⠄⠀⠀⠀⠈⠀⠰⠂⠠⣄⠀⡉⠀⠀⠀⠀⠀⠀⢻⣿⡷⠄⠀⣠⠀⠀⠀⣸⠆⠀⠀⣀⣀⡈
⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠀⢸⡂⠀⠀⠀⠀⠀⠀⠉⠀⢀⠐⠁⠀⠀⠀⠉⠀⢰⣇⠀⠠⢉⠄
⣮⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢸⣿⣿⣶⣶⣶
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣦⠀⠀⠀⠀⠀⠀⠀⠦⢠⡤⠙⠻⢿⣿⣿
⠉⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⣿⣿⣷⣿⣿⣿⣿⣧⡀⠀⠀⠀⠇⢄⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠐⠀⠀⠂⠀⠀⠇⠀⢀⡀⠄
⡠⡀⠉⠈⠀⠀⠀⠀⠀⠀⡠⠶⠋⠁⠀⠀⠀⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠨⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⢈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠁⠀⠀⠀⠀⠈⠉⠉⣉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⠂⠀⣐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡇⠀⠀⠀⠆⠀⠦⠈⢅⣀
⠀⠀⠀⠀⠀⠀⠠⠀⢰⡶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠏⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⡉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠡⠈⣿⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠠⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⡭⣩⣭⣭⡍⣩⣭⣭⡍⣭⣭⣭⠉⣭⣭⣭⢩⣭⣭⣍⢩⣭⣭⡍⣭⣭⣭⡉⣭⣭⣭⡩⣭⣭⣭⢩⣭⣭⣥⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣱⣿⣿⡿⢰⣿⣿⡿⢰⣿⣿⣿⢸⣿⣿⣿⢸⣿⣿⣿⢸⣿⣿⣷⠸⣿⣿⣷⠸⣿⣿⣷⡹⣿⣿⣧⡙⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣱⣿⣿⣿⢣⣿⣿⣿⡇⣾⣿⣿⡏⢸⣿⣿⣿⢸⣿⣿⣿⡄⣿⣿⣿⡇⢿⣿⣿⣧⢹⣿⣿⣷⡘⣿⣿⣷⡜⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣾⣿⣿⡟⢱⣿⣿⣿⠇⣾⣿⣿⡿⢰⣿⣿⣿⠇⢿⣿⣿⣿⠸⣿⣿⣿⡇⢿⣿⣿⣿⠸⣿⣿⣿⡆⢻⣿⣿⣷⠘⣿⣿⣿⣦⠻⣿⣿⣿⣦
⠀⠀⠀⠀⢺⣿⣿⣿⡆⢿⣿⣿⣷⠰⣿⣿⣿⡇⢺⣿⣿⣿⡆⢾⣿⣿⣿⠰⣿⣿⣿⣇⢺⣿⣿⣿⡆⢾⣿⣿⣿⠐⣿⣿⣿⣧⠸⣿⣿⣿⡆⢾⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠉⠁⣾⣶⣾⣿⣶⣾⣶⣾⣷⣶⣷⣶⣿⣷⣾⣷⣶⣿⣶⣾⣶⣾⣷⣶⣷⣶⣿⣷⣾⣷⣾⣿⣶⣾⣶⣾⣷⣶⣷⣶⣿⣷⣾⣷⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠦⣄⠔⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⠤⠔⠒⠒⡤⠒⠒⠂⠀⠀⠈⠀⠀⠀⠈⡆⡀
⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⡀⣀⠀⠀⠐⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠁⡈⠃⠀⠀⠀⢀⠤⣀⡠⠤⠤⣀
⠀⠀⠀⠀⠀⠨⣀⣀⡀⠀⠀⠰⡇⣸⡄⠀⠀⠀⡏⠳⠠⠠⠤⠬⠿⢯⠉⣉⣁⢄⢠⡀⠀⣀⣇⡀⠈⠀⠀⠀⠜⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⠓⠻⡄⠀⠀⣸⣸⣀⣘⢋⡁⠛⠟⢠⢸⣷⠎⠛⠉⢴⠅⣄⣒⡚⢷⠀⣀⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠄⠐⠀⠒⠟⠉⠁⠙⢀⣈⢹⣍⣉⣡⣿⣒⣾⡜⠉⣑⠶⠾⣸⠼⢯⡇⢈⠉⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⢰⠗⠶⢠⣼⣃⣉⣉⡏⣉⡉⡥⡄⣠⡀⢈⢫⣬⠁⢠⣤⢶⢠⠌⡇⡎⡇⠨⠤⡷⠷⠒⠒⢲⡀
⠀⠀⠀⢀⠤⠾⢤⠮⢤⠤⣤⡤⢤⡅⢣⡃⣳⡂⣳⣃⢸⠒⠚⠛⠚⠙⠉⢺⣅⡇⠉⠁⣖⡋⡇⠀⠀⠀⠘⡁⢹⢇
⠀⠀⠀⠸⡄⠀⢸⢸⠉⡇⡇⡃⢸⣿⡇⣿⣁⡇⣇⣸⢸⠀⠀⠀⠀⠀⠀⠀⢫⡇⢀⣠⠳⠥⣗⠒⣖⣀⣒⣃⡆⢸⠂
⠀⠀⢀⠔⠀⠀⠈⡖⢃⡇⠓⠊⠀⣧⡜⠃⡄⠀⠲⠄⠸⠒⠒⠒⣒⢒⡒⠒⡞⡇⠉⠉⠉⢦⡈⢱⡇⣒⢶⣺⠇⢸
⠀⠀⠘⠢⠤⡀⠀⢣⢩⢨⣾⣉⡆⡇⠠⠆⠐⠈⢒⡀⠶⡌⢃⡰⡌⠹⠠⠀⠠⠇⠀⠀⠀⢀⡧⢠⣓⠩⠛⠒⠦⢼⡀
⠀⠀⠀⠀⠀⠙⢶⠚⡘⠒⠙⠒⠃⣇⠨⣄⡘⠐⠒⠂⣀⠒⣄⣑⣄⣲⣖⣜⣺⠃⢻⡒⠊⠙⠣⣍⠤⠤⠤⠤⠀⠀⣈⡇
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠸⠉⡇⠀⡏⠸⣁⣹⡏⢹⠀⠁⠉⢀⠤⡀⠀⠈⠈⠉⠀⠈⡆⠀⠀⣀⢾⣓⣒⣒⣒⡒⢒⣹⡇
⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⣇⡇⢀⣅⠃⠀⠀⠇⠘⠠⠆⠙⠃⠀⢻⣿⢹⢹⢹⣀⣀⣙⡖⠊⠉⠸⣖⣒⣒⣒⣒⣒⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠀⢀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⣿⠀⠀⣸⣯⣤⣤⡀⠀⠀⢀⣷⠀⠀⠀⠀⣿⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢠⡟⣠⠀⣿⠃⢀⡉⠁⢀⠀⢸⣿⠀⣠⠀⢰⡿⠶⠶⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠉⢠⡟⢰⡏⠀⣿⠁⠀⣻⣇⠈⣿⠀⣿⠀⢸⣇⢠⣴⣟⣭⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⡾⢿⠃⡿⠁⠀⠻⠷⠛⠉⣿⠀⣿⠀⢻⠇⠈⠿⣾⡁⠉⠀⠀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡞⠁⠀⣸⠇⢀⣠⠶⠃⠀⣰⠏⠀⣿⣠⡴⠶⠞⠛⠙⠷⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠞⠋⠀⠀⠀⠿⠶⠋⢷⣤⡴⠞⠁⠀⠀⠛⠁

⠀⣴⠲⣄⠀⢀⠀⠀⣠⢶⡄⠀⠀⣀⣀⠀⠀⠀⠉⢳⠄⠀⠀⠀⠶⠶⠶⠤⣤⣴⣤⣤⣤⣄⣀⣀⣄⣀⣀⡀
⠀⠈⠙⢼⢳⣝⣧⢞⡵⠊⠀⠀⢠⣿⣿⣿⠁⠀⣠⣼⣤⡀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣦⣤⣬⣭⣭⣭⣭⣍⣙⣛⣛
⠀⠀⠀⢀⡽⣛⡶⢏⠀⠀⠀⠀⠚⢿⣿⡟⠓⠒⠻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⠿⠿⣿⡛⠛⠻⡟⠻⠿⠿⠿
⠀⣀⣴⣫⠞⣿⡝⢮⡓⣤⡀⠀⠀⠈⠉⠀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⠁⣠⡀⠈⣷⠀⠘⠗⠀⠀⢸⣿
⠀⣙⣞⠁⠀⣿⡇⠀⠙⣶⡃⠀⠀⠀⢀⣾⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣷⡘⣷⣿⣄⣈⣀⣰⠟⠀⠀⠀⠀⠀⢸⣿
⣾⠉⠉⣷⣶⣾⣶⣶⡏⠉⠉⡆⠀⠀⢸⣯⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⣽⡇⢹⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⢸⣿
⠙⠦⢴⣋⣉⣻⣋⣉⣳⠦⠞⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡇⠀⢸⡏⠉⡏⢻⢻⠻⡇⠀⢸⣿
⠀⠀⣿⡐⣧⣼⣤⠋⢎⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢸⡇⠀⡇⢸⣸⣠⡇⠀⢸⣿
⠀⠀⠛⠒⠛⠛⠲⠓⠛⠃⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⣇⣤⣼⣷⣾⣿⣿⣿⡿⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⢸⣿
⠀⢀⣀⡙⣿⣦⣤⣤⣯⣀⡀⠀⠀⠀⢸⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⡇⢸⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⡀⢸⣿
⣴⢏⣭⣿⣿⣆⣰⢻⡿⣯⣝⢷⡀⠀⠘⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⠇⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿
⢻⣜⣛⣫⡟⠉⠀⠸⣎⣛⣋⡼⠃⠀⠀⢀⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⣀⣤⢤⣄⠀⠀⠀⢠⡤⡤⢤⢤⠤⣿⢸⣿
⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⠉⠉⠉⠀⠀⣞⣁⣤⣤⣈⣳⡄⠀⢸⣧⣤⣼⣤⣤⣿⢸⣿
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡀⠀⠀⠀⠀⠀⣇⢻⡿⠿⣿⢨⠇⠀⢸⡟⠛⠛⠛⠛⣿⢸⣿
⢀⣀⣀⢴⣾⠿⠿⢷⣦⢀⣀⡀⠀⠀⠀⠀⠠⠴⠶⠿⠿⣤⣤⡀⠀⠀⠈⠳⠤⡤⠼⠋⠀⠀⠸⠇⠀⠀⠀⠀⠿⠸⠿
⠈⢿⣿⢸⣍⠀⠀⡁⣿⢸⡿⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⢿⡿⠂⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⢀⣀⣀⢼⡿⡷⢾⢿⣿⢈⣀⡀⠀⢰⣾⡟⠛⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏⠁⠀⠀⣿⠀⠀⠀⠀⢻⡀⠈⢧⡀
⠀⢻⣿⢸⣏⠀⠀⡁⣿⢸⡿⠁⠀⢸⣿⣏⣉⣿⣿⢠⣦⠀⠀⣶⣿⣿⣿⣿⣷⣷⣶⣶⣶⣿⣶⣶⣶⣶⣾⣷⣶⣶⣿
⢀⣠⣄⢸⡿⡗⢺⢿⣿⢠⣤⡀⠀⢸⣿⣿⣿⣿⣿⠀⢹⠀⠰⡟⣩⣭⣭⡙⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣭⣍⡻⣿⣿
⠀⠻⣿⢸⣏⠀⠀⡁⣿⢸⡿⠁⠀⢸⣿⣿⣿⣿⣿⠀⢸⠀⠀⣼⣿⠉⢹⣿⠛⠛⠛⠛⠛⠛⠛⠋⣾⡟⠉⢻⣷⠉⠁
⠀⠀⠀⠻⠿⠷⠾⠿⠿⠀⠀⠀⠀⠘⠛⠿⠿⠟⠛⠑⠁⠀⠀⠘⢿⣶⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⣫⡭⠤⢤⣿⡿⣿⣿⣿⣿⡿⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣡⠚⠉⠁⡄⠒⡼⠃⢠⡿⠋⣩⠟⠉⠉⢻⡿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣥⣶⠃⠈⣦⡴⠒⠲⡇⠀⢸⠀⠀⡟⠀⡾⠤⣄⡽⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣯⣤⠀⠀⠚⠛⠀⢸⠀⠃⠀⠀⠀⣀⣀⣀⣀⠄⠸⠇⢀⠿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣠⣅⣠⡆⠈⠳⣶⣿⠷⠤⢴⠿⠛⠉⠙⠛⠟⠀⠀⣠⠟⢀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⡾⠋⠀⠀⢀⣀⣀⣠⣦⣤⣤⣤⠶⠞⠁⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⣷⣤⡿⠉⠉⠀⣀⣀⣀⣠⣤⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣿⣿⣯⣭⣭⣤⣬⣥⣬⣭⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠻⣿⣿⡋⠉⢉⣩⣿⣿⣿⣿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣇⣀⣱⡄⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠀⠈⠀⠧⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠄⠀⡞⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢈⢿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⠻⠟⠇⢰⡏⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⠀⢨⢁⢹⡘⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠮⡀
⠀⠀⠀⠀⠀⠀⠔⡐⡈⠂⠡⢘⡀⠘⡃⡠⠐⠈⠘⢯⠂⠀⠀⢠⡄⠀⠀⠀⢰⣀⠀⣀⣀⡩⢦⢀⡔
⠀⠀⠀⠀⠀⠀⠸⡷⠋⢀⡀⠀⠆⠈⢽⡄⠂⢁⠀⠎⡀⡸⣗⠠⣌⣴⡁⠑⡦⠈⠈⠍⠽⠓⡄⣩
⠀⠀⠀⠀⢀⣀⣄⠑⠐⣌⠉⡆⠌⢠⠈⠣⠄⠐⠊⠠⢘⠂⠀⠀⠒⠈⣘⣼⢦⠀⠀⠀⢀⠸⡯⠃
⠀⠀⠀⠀⠀⠈⣉⣳⠀⢧⠀⠃⠜⢇⡀⡀⢀⠀⠢⠔⡀⠸⠃⠈⠀⠀⢈⡵⢮⡅⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠈⠱⢟⠉⠀⠙⠈⠀⡠⣦⠅⠀⣍⡀⢴⡁⠡⢋⣄⠀⠚⡁⠀⡗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢠⠱⢣⠂⠀⢻⠀⢜⡠⢸⡖⠜⡟⣀⣖⡽⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⠌⠀⣰⡴⠠⢰⠆⠠⠀⢛⢴⠠⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠇⠌⢴⢯⠀⠒⠐⠠⠀⡁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡊⠁⠁⣳⣥⠆⠐⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢨⣜⠀⢾⣿⢤⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣐⢉⡶⠜⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡻⠩⠁⡛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣮⣷⣶⣼⣧⣾⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀
⠒⠒⠒⠒⠒⣿⠖⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣿⠒⠒⠒⠒⠒⠒⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣤⡀⠀⢸⣿⣦⠀⠀⠀⢸⣏⣉⣉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠻⣿⣿⣦⣄⣀⣙⠻⠿⢟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠀⢀⣼⣿⣿⣆
⣀⣤⣤⡄⠀⠀⠀⠀⠀⢀⣹⣿⣧⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣾⣿⣿⣿⣿⡄
⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⣾⣏⣿⣿⣿⣿⣿⠈⢿⣷
⣿⣿⡿⠇⠀⢴⣶⡄⢰⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣾⡦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿
⠉⠁⠀⠀⠀⢸⣿⣷⠘⣿⣿⣿⣿⣿⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡄⠉⣿⣿⣿⢿⡿⣷⠀⠀⠀⠀⠀⣀⣀⣠⣶⣾⠟⠃⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⢳⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣇⣼⣿⡇⠀⢈⡉⢈⡟⢟⠉⠉⠉⠉⠉⠉⠁⢀⣀⣀⣀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⡏⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣤⣀⣀⣴⣾⣿⡄⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⣿⣿⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⢶⠶⠄⢀⣾⣿⡿⠋⠀⠀⠀⠀⣴⣿⣿⣧⠀⠀⠀⠀⠶⣶⣶⡦⠀⠀⠀⠀⠀⣸⣿⠇⠶⣶⣾⣿⡏⠉⠉⠁⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡯⠯⠭⢁⣀⣾⣿⠟⢤⣤⣀⠀⢀⣾⡿⠋⠹⣿⡆⠀⠀⠀⠈⠽⢿⠧⠀⠀⠀⠀⠀⣿⡏⠀⠀⠹⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿⠁⠀⠀⠉⠻⣷⣿⡟⢁⣀⣠⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡟⠀⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠾⠿⠟⠛⠃⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣿⠿⠿⠿⣿⠿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣛⣿⣿⣿⣿⣿⣻⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠟⠛⠛⢿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⠀⠸⠟⣿⣿⣿⠀⠀⣀⣤⣶⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣤⣤⣀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠛⠋⠉⠛⠻⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⠘⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢟⡿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⣫⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠟⠋⠀⠀⠀⡐⠁⠐⠄⡀⠀⠠
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠻⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠉⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡁⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠘⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⢠⡄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢛⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣧⣤⠀⠀⠀⠀⣠⣿⣿⣿⣧⠀⠀⠀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣿⣿⣁⠀⠀⠀⠐⠟⢻⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢈⠀⠀⠀⠀⠶⠰⠆⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠋⠀⠀⠀⠠⠶⠿⠿⠿⠿⠆⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⠀⣶⠀⣤⠀⠰⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠀⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠸⠆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⡿⠁⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣶⡆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⢠⣦⡆⣤⣴⣴⣶⠰⡆⠀⠀⠀⠀⠀⠰⡆⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠉⠁⠘⠋⠰⠆⠦⠀⠀⠴⠳⠧⠀⠀⠰⠢⠄⠀⠀⠉⠉⢀⡁⠀⠈⠉⠀⢠⡄⠀⠀⠀⠀⠀⢀⣀⠄⠀⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⣀⣀⠀⢀⡀⢀⡀⢀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⣀⣀⣠⠄⠀⠀⠀⠀⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠈⠁⠈⠀⠈⠉⠀⠿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠋⠉⢁⡀⠀⠀⠈⠙⠉⠀⠀⠀⠀⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠆⠀⠀⠰⠆⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠈⠁⠀⠀⠛⠁⠙⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⢀⠀⠀⠘⢋⣍⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢀⡄⣤⣤⢀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠻⠿⡿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢻⣿⣿⣄⢀⣌⣤⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠛⠋⠙⠉⠋⠉⠛⠁⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡛⠀⣿⣿⣿⡼⠿⡏⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⢁⣽⣿⡇⠀⠁⠁⠈⠀⠁⠀⣤⣤⡤⢤⠄⠀⠀⠀⠀⠉⠀⠀⠄⠀⠀⠀⠉⠀⠀⠰⠟⠙⠁⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡋⠛⠀⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⠀⢰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⡀⣀⠀⠀⠠⠟⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈⠿⣿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⢀⣾⠿⠃⠀⠀⠀⢡⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠀⠛⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣴⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠃

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣹⠟⢹⡏⢻⡏⠻⣏⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣰⡟⠀⣿⡇⢸⣿⠀⢻⣆⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠿⠧⠠⠿⠇⠸⠿⠄⠼⠿⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣘⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⠿⡿⢿⡿⢿⡿⢿⠿⣿⠿⡿

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢀⠀⠀⢶
⠀⠀⠀⢠⠀⠁⡀⠀⠚⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠤⠦⠴⠴⣦⠖
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⡞⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣤⡶⣦⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠂⣤⡤⠤⠀⢀⣶⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⠁⠀⠀⢄⢸⠳⣾⣗⠇
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠘⠐⠒⠒⠖⠒⠖⠒⢛⢛⢓⠒⢺⣟⠀⠀⠀⠀⠁⠒⠚⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡨⠼⠚⠀⠀⠀⠀⠈⡉⡈⡘⡀⡘⠀⢂⡁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣷
⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⡄⠐⠐⠈⠈⡀⣐⣀⣀⣀⣠⣿⠀⠀⠀⢠⢷⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠑⣄⠢⡈⢻⠀⠀⠀⠠⣸⠀⠀⠀⠠⣤⣀⣨⡿
⠀⠀⠈⠀⠀⠀⠀⢻⣇⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠳⣈⣺⠀⠀⠀⠀⢸⣷⠀⠈⠛⠳⠧⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⣀⠤⠛⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠚⡻
⠀⠀⠀⠀⠀⢄⡠⠤⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣁⣀⣀⣉⣀⣀⣀⣀⣤⣉⣉⣉⣁⣀⣀⣀⣀⣀⣉⣉⣉⣉⣙⣉⣉⣉⣉⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠀⠀⠀⠀⢀⣈⣉⣉⣉⣉⣉⣁⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⢠⠀⠀⣉⡭⠭⣍⠉⠉⠉⠉⢍⣩⢭⠉⠉⠉⢉⠭⠥⣄⠀⠀⠈⢁⠤⠬⣉⠉⠉⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⢹⠀⠀⣇⠰⠷⣌⠀⠀⠀⡇⠸⠇⢸⠀⠀⠀⡇⢸⣷⢸⡇⠀⠀⡇⠸⡷⢨⠆⢰⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⡞⠿⡶⢸⠀⠀⠀⡇⢸⣧⢸⠀⠀⠀⡇⠸⠽⢸⡇⠀⠀⡇⢰⠒⠋⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠚⠀⠀⠙⠓⠒⠉⠀⠀⠀⠓⠚⠛⠚⠀⠀⠀⠉⠓⠒⠉⠀⠀⠀⠛⠚⠀⠀⠀⡘⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠏⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⡄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠞⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡴⠃⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣁⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⢀⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢌⣉⣉⣩⢍⡉⢉⡩⠫⣍⣍⠉⢏⡉⡉⣩⢏⡉⠉⢉⣍⠉⠉⢉⡭⣉⣉⠉⢯⡉⠉⠉⣩⡉⠉⠉⣩⣉⠉⢉⡍⡉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⢀⣀⣉⣉⣁⣀⣀⣀⣀⣀⣈⣉⡀⠀⠉⠉⢉⣀⣉⣉⣉⣀⣀⡉⠀⢀⣉⣉⣉⣀⣉⣛⣛⣁⣈⣉⣉⡈⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇⢸⡶⠒⠒⠒⠒⠒⠒⠒⠒⡆⠀⠸⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣇⢀⣀⣀⣤⠤⠤⠒⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢀⠀⠀⠀⠀⠄⠠⠤⠤⠄⠀⡇⢸⡏⡇⠰⣴⡆⡦⠰⣇⢸⡇⠄⢀⣆⠀⠀⠀⠀⠀⠠⠤⠤⢠⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠙⡇⢸⡇⢰⠘⢋⣁⣈⣀⣀⠸⡇⡇⢰⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⢸⡇⠈⠉⠉⠁⠀⠀⠀⠀⡇⣇⢸⣄⣀⣀⡀⢀⣀⠀⠀⠀⠀⢠⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⠉⠉⠉⠉⠁⠀⠉⠃⠀⠈⠛⡷⢸⢂⠤⡀⠀⠀⠀⢀⠀⠀⡇⣿⣏⢈⡉⠉⠉⠈⣉⣛⡋⠈⠛⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⠌⠐⣁⣀⣀⣀⣈⣀⠀⡇⣿⠉⢉⠉⠉⠉⠁⠉⠉⠉⠉⡉⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃⠀⠀⠀⠀⢸⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠤⢤⣤⣤⣤⣤⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⢸⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣀⣁⣀⣀⣀⣀⣈⣀⣁⣇⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇
⠀⠀⠠⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠃
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣼⠓⣔⠓⣴⢓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣈⣀⣀⣉⣘⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠀⠀⠄⠀⣿⣿⣹⣽⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠠⠐⠀⠀⢠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣁⡄⠉⠙⠢⣀⡤⠄⠁⠀⠈⠀⠀⠐⣘⢃⣴⣴⣄⠠⠄⠀⠘⠉⠛⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠜⠛⠲⠁⠈⠳⢌⠀⠀⠀⠂⠂⠐⠁⠘⠛⣿⣿⡇⠆⠀⠠⠄⠘⠁⠀⠀⠠⠁⠀⡀
⠀⠀⠀⠀⠀⠴⠈⠁⠰⠛⠠⢍⠀⠓⠀⢤⠀⠚⠁⠨⠛⠀⠖⠉⠁⣠⣦⣬⣿⡷⠀⠀⠀⠀⠀⣴⣄⠀⠀⠁⠐⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠈⠁⠚⠀⠀⠁⢈⠁⠀⠖⢀⡀⠆⠀⠋⠛⠛⠀⢉⠂⡀⡈⠻⢻⣿⣷⡦⡆⠀⠈
⠀⠀⠀⠀⠀⢣⠰⢄⠠⠒⢄⡠⠤⡄⢂⡀⠤⣶⠀⠀⠄⠚⠁⡌⣠⢄⠀⠀⠀⠇⠈⠀⠓⢾⢤⣼⣿⣿⣷⣯⣀⡈
⠀⠀⠀⠀⠀⠠⡠⠖⢲⡤⠒⢢⠶⠨⣭⠉⠆⢀⠠⠆⣊⡁⠀⣄⠀⠀⣀⠢⠀⢀⠐⠂⠀⠀⢀⡌⠙⠛⠋⢛⡔
⠀⠀⠀⠀⠠⠁⡄⠀⠠⡆⢐⣄⢠⢁⠁⡤⢄⠠⣖⠀⠉⠀⠐⠐⠀⠀⠀⠀⠀⠂⡠⢬⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠂
⠀⠀⠀⠀⠀⠀⢹⠀⠀⡉⣗⠏⢘⠿⠀⣤⡊⠄⠃⠁⢠⠄⠀⠠⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣡⠄
⠀⠀⠀⠀⠀⢰⠸⣶⣤⢿⣿⣃⣤⠀⠀⣿⣀⠀⠀⠀⠤⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠰⡀⠉⢸⣸⣾⣳⡭⡘⠁⢮⠙⠛⠒⡀⠀⠀⠀⢀⠀⠀⢠⡄⣅⠀⠀⠀⠀⠀⠈⠀⠀⠘⠁⢀⡀⠀⢦⢠⡄
⠀⠀⠀⠀⠀⠠⠄⠀⠃⠛⠛⠋⠰⠀⠀⣀⡀⢲⡄⠘⠀⠀⠊⠁⠀⠀⠀⢹⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣦⠹⣆⣋⣇⠁
⠀⠀⠀⠀⠀⠀⠀⡂⢶⠶⣤⠀⠒⡀⠀⢈⠋⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⣀⠀⠂⠀⠀⡠⡀⠘⠋⠉⠐⡎
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠈⣓⣈⡃⣀⡋⣴⢤⠤⢤⣄⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠉⢷⣢⡟⠈⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡹⢟⡛⢍⡹⠿⠿⠷⣿⡛⠻⠟⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠄⠀⠀⠀⠀⠀⠀⠙⠄⡬⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠉⠁⠀⠀⠘⠧⣄⣀⠀⠀⠀⠀⠀⡴⠋⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠙⠒⠲⠤⢤⣀⣀⠀⠈⡗⠀⠀⠀⠀⠀⠀⠀⠩⠿⡀⢠⠞⠁⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠃⠀⠀⠉⠉⠙⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠒⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⣄⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠿⠿⠿⠶⣶⣶⣶⣶⣶⣶⣆⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣠⠞⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠉⠛⠛⠋⠉⠁⠀⠀⠀⣀⡠⣄⠀⠀⣀⣀⠤⠚⠉
⠀⠀⠀⠀⣠⠊⡀⠀⠀⠀⠀⢀⣀⡤⠔⠚⠉⠀⠀⠈⠉⠉⠁
⠀⢀⣠⠴⠧⠤⠤⠤⠤⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠢⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣤⠠⣢
⠀⠀⠈⠰⠄⠀⠀⠀⢀⠀⢠⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢫⡀⠃⠀⠁
⠀⠠⢀⠀⣰⢀⠀⣀⠠⠀⠀⣥⠛⡌⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐
⡀⠀⡈⠈⠠⣮⠉⣌⡱⠀⠀⠃⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠂
⠠⡤⠤⣀⣠⡅⠈⠀⠁⠀⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠐
⠀⠙⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⡀⣆⠒⢴⠴
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⡛⣼⡎⠈⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠄
⠀⠀⠀⠀⠀⡆⠠⢀⠀⣤⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠔⠢⠀⠀⠀⠀⠒⠃⠁⢀⢀⡀⠀⡀⠀⢠⡆⠀⢀⣦⠂
⠀⡖⠀⠀⠤⣿⠏⣼⠀⠐⢶⣄⣾⣇⡳⡆⠲⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢶⢶⡶⣒⣴⡆⡤⠰⢶⡿⠀⣠⣄⠻⡦
⣴⢇⢤⣤⡻⢏⠀⠏⡜⣻⣧⣽⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠫⠟⠸⡵⢛⡭⣤⣿⣜⣏⡷⢸
⡦⣯⣾⡟⢷⣸⠋⠀⠋⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠉⠙⠙⠈⠩⣾⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢄⡀⡠⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⣄⡠⠄⢼⠀⠀⠀⠉⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠈⠁⢈⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢢
⠀⠀⠀⠀⠀⠀⠔⠒⠋⠀⠀⠀⢀⡢⠒⡄⠀⠀⠈⣲⠲⣒⠒⠒⠒⠒⣤⡄⠀⠀⠀⠐⠁⠀⠀⠀⠀⠂⠉⠑⠊⠁⠈⢙⣆⡀
⠀⠀⠀⠀⠀⠀⠑⠒⠲⣀⢀⠀⠈⣇⣺⠋⠀⠀⠀⠈⡆⠃⣤⡄⠀⡍⡩⢉⢁⣔⡙⢯⣇⠜⡠⠀⠒⠒⣢⣀⠀⠀⠀⠈⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣒⡠⠤⠠⡶⠷⠦⣄⠰⢠⡄⣁⢸⢰⡟⡍⣀⣀⡀⡏⠡⢦⠤⡄⢹⠤⠄⠦⠤⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⢰⢉⣉⣉⣩⣉⡀⠠⠤⠥⠤⠤⢽⠦⠤⡴⠻⠭⠭⠯⠤⢤⠂⣉⠓⡾⠀⡅⡇⢸
⠀⠀⠀⠀⠀⠀⡏⠉⢉⠱⠯⠤⠤⠀⠅⡤⠄⠐⢲⢠⠢⠀⠘⠠⢶⡀⢸⣆⣋⠂⣒⢸⢸⢈⠀⡗⡃⠋⠉⠉⠉⠉⡅⠀⡀
⠀⠀⠀⠀⠔⠒⠓⡞⢃⣰⠶⣤⡶⢶⠾⣙⣂⢸⢚⢈⠿⡄⡌⠉⠉⠉⠉⠀⠉⠉⢧⢸⠀⠁⠀⣏⡅⡆⠀⠀⠀⠀⠇⠀⠛⡄
⠀⠀⠀⠀⠆⠀⠀⠇⢰⠈⠀⡀⡇⠘⣀⢇⡨⣀⢸⢸⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠀⣀⣔⡳⠤⣍⠀⣉⣉⣉⣁⡔⠀⡏
⠀⠀⠀⠔⠁⠀⠀⢸⠘⣒⡂⠓⠃⠀⣿⠔⠒⢰⠂⠔⢲⠀⡖⡐⢒⠒⣒⢒⣒⠒⠲⢻⠂⠀⠀⠘⢤⠀⢡⡇⣒⡲⡲⠆⠀⡇
⠀⠀⠀⠣⠤⣄⠀⠈⡄⣧⢩⣎⣉⡆⡿⠣⠄⠘⠂⢐⣊⡀⢿⣩⢇⠡⣇⣽⡨⠀⡀⡾⡄⠀⠀⠀⠀⡆⢠⢓⡨⠝⠒⠦⢄⣇
⠀⠀⠀⠀⠀⠈⠢⡄⢣⠹⠴⠸⠤⠇⡄⡾⡉⠩⠃⠧⠔⠀⢠⠢⡜⢲⡄⢢⡔⢆⡷⡓⠚⡖⠶⠛⠻⢤⡉⡀⣀⣀⡀⣀⣀⣀⡙
⠀⠀⠀⠀⠀⠀⠀⠇⠘⡄⠸⠙⡆⠀⡇⠀⡏⢉⣶⠒⡒⢸⠈⠊⠈⣉⠈⠁⠈⠹⠋⠃⠀⢻⡀⠀⠀⠀⣿⠂⠲⠶⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠸⠀⡇⠀⠁⠇⠉⠉⢸⠀⡇⢸⡀⠠⠊⠀⢳⡒⠒⠖⠖⠄⠀⠐⠢⡄⠖⡏⢸⠭⠭⠭⠭⠭⠨⠥⠅⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠤⠗⠑⠒⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠃⠀⠀⠀⠋⠉⠀⠉⠀⠀⠁⠀⠉⠉⠀⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢷⡄⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⡶⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣸⠇⠀⠰⣿⠛⠛⠿⠆⠀⠀⠀⣿⠀⠀⠀⠀⢸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢰⠟⢰⡇⢰⡏⠀⣴⠀⠀⣤⡄⠀⣿⠀⢰⡇⠀⢸⡏⠉⢉⣤⣴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⡄⣿⢀⣾⠀⢸⣧⡀⣠⡼⣷⠀⣿⠀⢸⣇⠀⢸⡇⣰⠿⠷⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⡾⠹⠇⣸⠇⠀⠀⠙⡛⠉⠀⣿⠀⣿⠀⠈⠟⠀⣘⣃⣿⡀⠀⠀⣀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠟⠁⠀⢠⡟⢀⣠⡴⠛⠁⢀⣴⠏⠀⣿⣤⠶⠛⠛⠋⠁⠘⠻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠛⠁⠀⠀⠀⠘⠿⠛⠁⢷⣤⠶⠛⠁⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⣴⠐⠒⠶⡶⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⣀⡀⠃⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠸⠀⣀⣀⢿⡀⢀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢰⡟⡉⡀⠀⠀⠀⠀⠀⠀⡐⢃
⠀⠀⠀⠀⠀⢀⡐⠒⠖⠀⠋⠀⠯⠐⠚⠋⠉⠈⠃⢿⡿⣿⣶⠄⠀⠀⠀⠀⣴⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠘⠀⠀⠀⠀⠀⠈⡧⣬⠁
⠀⠀⠀⢀⡴⢛⣗⣾⣶⡆⡆⠀⠀⠀⠀⠀⠀⠀⢠⣫⣶⣿⠒⣦⠀⠀⠀⠀⠈⠂⣹⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⣠⠍⡉⠀⠀⠀⠀⠀⢀⠐⠉⠈⠂⢠
⠀⠀⠀⠀⠀⠈⠸⣷⡾⠇⠉⠉⠉⠉⠉⠀⠈⠉⠉⠿⣦⡿⠀⠁⠀⠀⠀⠀⠀⠀⠀⢮⡂⠀⠀⠀⢀⣴⣷⢼⣦⠀⠀⠛⡠⠴⠇⠀⠀⠀⠀⢸⣟⣛⣛⣻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠱⡀⠀⠀⠉⠀⠘⠁⠀⠀⠀⢀⡀⣰⡄⠀⠀⠀⡆⣿⣿⠿⣟⣦⠂
⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠙⡧⠀⠀⠀⣿⣟⣷⣜⣋⣟⡀
⠀⠀⠀⠀⠠⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠀⠀⠀⠀⢐⢳⡞⠂⠀⠀⠀⠙⢄⠀⠀⠀⠐⠲⣤⣰⣶⣤⡾⢷⣶⠄⠀⢙⢈⣽⢭⡭⣍
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⣧⠀⠀⠀⠀⠀⠀⣐⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡓⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⣿⣷⣿⣷⣶⠉⠀⠀⢸⣾⣿⢸⣿⣿
⠀⠀⠀⢠⣤⣯⣄⣠⣤⣤⣴⣤⡄⠀⠀⠀⡞⠉⠀⠙⡆⠀⠀⠀⠀⠀⠂⠒⠊⣿⠾⠉⠑⠒⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢨⡿⢣⠼⠭⢥
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠓⢄⣄⠰⠃⠀⠀⠀⠘⣆⠀⠀⢀⡚⣷⠀⢀⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣣⣬⢩⣭⡭
⠀⠀⠀⠀⠀⢏⣹⠶⢶⣛⡷⠠⡄⠀⠀⠀⢆⡉⠙⠛⡆⠀⠀⠀⠀⢙⣿⢧⠴⣇⢰⠦⠾⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢼⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⠿⡿⢓⣬⣷⣙⠿⡟⡀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⢠⡇⠑⢶⣀⣶⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣦⢀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠐⠂⠀⠀⠀⠀⠀⢰⢴⣷⢰⣾⠆
⠀⠀⠀⠀⠲⠾⠟⠟⠻⠿⠳⠞⠀⠀⠀⠀⢶⡾⡿⠻⠻⡛⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣤⣷⠸⠧⠙⠉⣳⠾⠂⠀⠀⠀⠀⣀⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢐⡲⢤⡲⢶⣖⣾⣿⣶⠷⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣴⠾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠸⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣷⣟⣳⣭⣿⣭⡷⣻⣿⣿⡇
⠀⠀⠀⢴⡶⢦⣀⠤⢲⠾⠛⠒⣦⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣩⣍⣹⠿⣿⡿⣏⡻⣭⡛⠇
⠀⠀⢰⣿⡷⡌⢳⢶⡶⠾⢟⣉⠻⠄⠀⠀⠠⠤⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⠤⣶⠀⠠⠀⠀⠀⠒⠉⣍⡤⠀⠀⠀⢀⠉⠴⣫⣿⣇⣿⡗⣿⣿⣮⣙⡇
⠀⠀⢘⡭⣛⢴⠒⠒⠈⣁⣀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⠓⠛⠀⠁⠀⠀⠀⠀⠀⠠⠤⠿⠀⠀⠈⠁⠈⠉⠀⠁⠉⠉⠉⠉⠉⠉
⠀⠀⠈⣇⡈⢸⠀⠀⠘⡁⣯⣠⢰⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣄⣀⠀⡀⠀⠀⠈⠃⠀⠐⠀⠀⢀⣀⣀⣀⣀⣀⠀⣀⠀⡀⠀⣀⢀⣀⡀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⣇⡙⣿⠀⠈⠙⠇⢹⣀⡈⠀⠀⠀⠀⠀⢠⣴⠛⡉⠉⡅⢉⡿⢷⣄⡀⠀⠀⠀⠀⠀⢸⣋⣁⡻⣋⣀⣈⣟⢉⡈⢹⡉⡉⡹⡍⠉⢉⠟⠉⠉⠈⡏⠁
⠀⠀⠀⣏⡙⣿⠀⠈⠈⠈⠸⠀⡆⠀⠀⠀⠀⣀⡎⠀⠲⡜⢼⢃⣞⡤⠞⠙⡇⠀⠀⠀⠀⠀⠹⠀⠀⠀⠐⠖⠒⠀⠈⠐⠂⠉⠀⠃⠘⠛⠛⠓⠛⠋⠉⠘
⠀⠀⠀⡏⠙⣷⠉⠈⠉⠀⠹⠀⡇⠀⠀⠀⠀⣸⠁⣀⡅⢘⣿⣿⡥⢀⠀⠀⢻⠃⠀⠀⠀⠀⠂⠀⣦⡶⠦⣤⣤⣄⡀⣤⠄⣤⣤⠴⣀⣴⣦⣠⡤⣤⣤⣤
⠀⠀⠀⣏⡛⣿⠀⠐⠠⠋⢱⠀⡇⠀⠀⠀⠀⠘⣇⠠⠖⢡⣿⣿⡙⠦⡈⢉⡟⠂⠀⠀⠀⠀⠀⢀⣥⡴⢃⣤⠤⣬⡇⢀⠀⣏⢀⣀⠙⠁⠀⢸⡏⠁⠀⠃
⠀⠀⠀⡏⠙⣿⠀⠰⠀⠋⢹⢉⠇⠀⠀⠀⠀⠈⠙⢧⣄⢰⠇⢿⡇⣀⡴⠛⠂⠀⠀⠀⠀⠀⠶⠿⠿⢟⡛⠠⠤⠤⡷⠴⠤⡷⠶⠦⢶⣤⣦⣼⣇⡀
⠀⠀⠸⣦⣕⣻⠀⠘⠊⡉⣉⣀⡠⠦⠀⠀⠀⠀⠀⠈⠉⡙⠒⠾⡏⠉⠀⠀⠀⠀⠀⠀⠀⠰⢰⡶⣶⣦⡿⠤⠴⠶⡧⢤⡤⣧⠠⣶⠾⣯⡽⣿⣿⣇⡠⢀⡀
⠀⠀⠀⠀⠉⠋⠀⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠙⠉⠩⠋⠩⠋⠙⠉⢙⠉⠉⠋⠉⢙⠉⠉⠛⠉⠙⡛⠒⠒⠒⢒⠒⢒⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠂⠀⠴⠁⠐⠃⠀⡆⠤⠼⠤⠄⡦⠤⠼⠤⠄⠤⠀⠀⡇⠀⢰⠀⠨⡆⠈⢦⠈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠼⢄⡠⠣⣀⠜⠤⠠⠓⠄⠜⠆⡠⠳⣀⠼⠣⡠⠟⢄⡰⢣⣀⠞⣄⡠⢧⣀⠼⢄⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⠴⠒⡊⣉⣉⡓⠲⣄⠀⡖⡆⡒⠒⣒⡒⣒⣒⣒⣒⣒⡒⠒⠐⣦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⡔⢋⣥⣼⡆⠋⠒⡌⡆⠇⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡧⠾⢿⢥⣽⣶⣆⡇⡆⡆⠀⠀⠀⠀⠊⠀⠄⢠⡌⠈⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡀⠀⣠⣶⣤⡀⠀⡀⡇⡀⠀⠀⠀⠀⠂⠀⠀⠀⡄⠑⣤⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡀⢀⠉⠀⠉⣁⠀⡁⡇⠁⠀⠀⠠⠁⠀⠀⠘⢆⡧⠀⣡⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⡒⠐⢒⠁⠀⡁⡇⠀⡀⠀⠀⠀⠠⡄⠠⠼⠇⠘⠧⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠉⠙⠛⠋⠀⠁⡇⠀⢐⠐⢘⠀⠠⠇⠀⠀⠓⡓⡗⠓⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠓⠒⠐⠛⠛⠛⠋⠀⠒⠓⠒⠞⠒⠒⠲⠒⠒⠶⠶⠷⠗⠓⠓⠓⠒⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣷⣶⣦⣤⣤⣄⠀⠀⢠⣿⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⡿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⠿⠟⠋⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠁⡖⠋⠉⠀⠀⣄⠀⠀⠀⠀⠀⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⠖⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣸⠀⠀⡃⠁⠀⢨⣴⢀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠆⠀⠠⠂⠀⠀⠀⣀⠀⠠⡶⠆⠸⠯⢀⠠⡇⠀⠁⠈⢁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⡂⠀⠀⠀⠀⠸⠞⠁⠀⠄⠠⢈⠀⠲⠛⠁⠀⠂⠀⠐⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠈⠀⠀⠡⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠂⠸⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠀⠁⣲⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⢐⠨⠀⠐⡀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠈⠀⠀⢱⠀⠀⠀⠄⢡
⠀⠀⠀⢀⠄⠊⠀⠃⣸⠀⠀⠀⡀⠀⠀⠁⠀⠠⠀⠈⢀
⠀⠀⠀⠀⠒⠒⠂⠁⠀⢃⠠⠰⠀⠀⠀⠀⢨⡆⠀⠀⠰⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢀⠒⠈⠉⠁⠈⠒⠒⡀⡀⠉⠉⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡐⠚⠄⣢⣠⣈⣁⣀⡠⢆⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡇⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠁⡏⢀⠔⠂⠐⠐⠈⠉⠈⠈⢄⠔⢔
⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠂⠠⢱⣐⡐⠀⠀⠂⢀⠀⠀⡀⠁⠀⠐⢀
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢀⡔⠁⠀⠀⠜⠌⠉⠉⠉⠁⠹⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠉⢒
⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠂⠉⠁
⠀⠀⠀⠀⠉⠁⠉⠈⠉⠁⠈⠀⠁⠐⠁⠀⠊⠀⠁⠒⠀⠂⠐⠀⠀⠂⠂