Postage Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡤⣄⡠⢤⣠⠤⣀⡤⢤⣀⡤⣄⣠⠤⣄⣠⠤⣄⣠⠤⣄⡤⢤⣀⡤⣄⡤⢄⣠⢤⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠉⠀⠈⠀⠀⠁⠀⠉⠀⠀⠉⠀⠉⠁⠀⠈⠁⠀⠈⠉⠀⠉⠀⠀⠉⠀⠈⠀⠀⠁⠀⠉⠀⣙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡚⠀⢰⠋⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠽⠀⢸⡅⠊⠀⠀⠀⠀⠀⠀⣀⡴⢒⠫⠉⠁⠀⠀⠈⠉⠝⡲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⠀⢸⡄⠀⠂⠀⠀⠀⣠⢞⠅⠺⢄⣀⡤⣧⣴⣤⣤⢴⣀⠈⠑⢭⡳⣄⠀⠀⠀⠀⠀⠀⡇⠀⣖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⡀⢸⡇⠀⠀⠀⢀⡜⡵⠁⠀⠀⣇⣿⡏⣽⣿⣿⠑⣸⣿⣷⣤⠠⠀⢌⢦⠀⠀⠀⠀⠀⡇⠀⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⢸⡇⠀⠀⢀⡞⠜⠀⠀⠀⠀⣧⣤⣄⣈⣉⠉⠐⠿⣿⡿⢐⣴⣀⡀⠀⢳⡀⠀⠀⢀⡇⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⢸⠇⠀⠀⡼⠈⠀⠀⠀⠀⣰⡋⠒⣂⣙⠉⠙⠳⢤⣄⡁⠻⡏⠂⠀⠀⠈⣇⠀⠀⠀⡇⠀⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢸⡅⠀⢰⠃⠀⠀⠀⠀⠀⣿⠴⠶⠼⢿⣿⣷⣶⣬⣙⠻⡶⠁⠀⠀⠀⠀⢸⠀⠀⠨⡇⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⢸⡇⠀⢸⠘⠀⠀⠀⠀⠔⠁⠈⠋⠀⠀⠙⠫⠚⢿⣿⣷⣧⣀⠀⠀⠀⠀⣸⡆⠀⠠⡇⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂⢸⡇⠀⣺⢀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⢫⡇⠄⢰⡇⠀⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⠀⢸⡇⠀⢹⢸⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢸⠃⠀⢨⡇⠀⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢸⡇⠀⠸⡌⡀⠀⠀⠀⠐⢄⣀⣀⣠⠀⠀⠈⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⣸⠀⠀⠈⡇⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⠀⢸⡇⠀⠀⢧⢀⠀⠀⠀⠀⠀⢀⣠⡴⡶⠀⠟⠛⠉⠉⠙⠳⢦⠀⠀⠀⢤⡇⠀⠀⠀⡇⠀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠀⢸⡇⠀⠀⠈⢧⢂⠀⠀⠠⣞⡄⠁⠀⠀⠀⠀⠀⠀⠠⠘⡠⠎⠗⢀⣌⡞⠀⠀⠀⠠⡇⠀⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⢸⡇⠀⠀⡴⠬⠶⠷⠒⠂⠒⢌⠙⠓⠄⡀⠀⢀⡠⠚⡡⠒⠐⠒⠾⢾⠴⡆⠀⠀⠀⡇⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⢸⡅⠀⢶⡉⠀⠀⠀⠀⠀⣄⣿⠀⠀⠀⠘⠖⠁⠀⠀⢃⣠⠀⠀⠀⠀⠀⣉⠶⠀⠈⡇⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢸⡅⠀⠀⠉⠙⠒⠛⠉⣻⠀⠉⠉⣴⡒⢒⣲⣒⠒⢋⠉⠘⡏⠉⠙⠓⢋⠁⠀⠀⠐⡇⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣹⠇⣾⠃⠈⣿⡄⠀⠀⢣⠀⠀⠐⠀⠀⠀⠀⢀⡇⠀⢞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⣐⠛⠆⠙⠳⠴⠟⣀⣠⡤⠞⠁⠀⠀⠀⠰⠄⠁⠐⡇⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠀⢸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣁⣁⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⡇⠀⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡚⢀⡀⠈⡈⠁⢀⡁⢀⣁⠀⢁⡁⠀⣀⡀⠀⣀⡀⠁⢀⣀⠀⣀⠀⠁⣈⠀⢉⠈⠈⣙⠁⢀⡀⢑⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠚⠙⠒⠋⠓⠊⠈⠓⠋⠙⠚⠃⠙⠚⠁⠙⠚⠃⠙⠚⠉⠓⠚⠉⠓⠋⠑⠚⠉⠒⠃⠑⠋

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⣠⠤⠤⢤⣤⡄
⠀⠀⠀⠀⠀⡏⠉⠉⠉⠹⣇⠀⢀⡟⠉⠹⣇⡀⢀⣸⠏⠉⠙⢧⣄⣠⡾⠁⠈⢷⣄⣀⡴⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢧⣀⠀⠀⣤⣤⡤⣤⠤⠤⠤⠶⠶⠶⠖⣶⣶⣶⣶⣾⣷⣶⡶⠞⠛⠛⠛⢻⡀⠀⢀⡴⠞⠃
⠀⠀⠀⠀⠀⠀⢹⡆⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠁⠀⠀⢀⣠⣴⣾⡇⠀⢺⡀
⠀⠀⠀⠀⠀⡴⠞⠀⠀⣿⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣶⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⡇⠀⠀⠛⢾⡆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⢿⡇⠀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⡷⣤⡀⠀⣿⣶⣾⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⢸⡇⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣸⠇⠀⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⢾⠀⠀⠻⢦⣤
⠀⠀⠀⠀⠘⡟⠋⠀⠀⣿⣿⣿⠿⠋⠀⠀⠀⢀⣠⡴⡶⢶⢶⣤⡤⣤⣴⢾⠏⡇⡇⣿⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢰⣧⣄⡀⠀⣿⠀⠀⠀⠀⠀⣠⠞⣿⢸⠁⡇⢸⢸⣿⣀⣿⢸⢸⢰⣇⡇⣿⢸⠀⠀⣠⠞⠛
⠀⠀⠀⠀⠀⠁⠈⣷⠀⡧⠀⠀⠀⣠⡞⢻⠀⡇⢸⠀⡇⣸⣸⠋⢉⣳⣾⣸⠾⠛⠳⣿⣸⡀⠀⣯
⠀⠀⠀⠀⠀⣴⠾⠃⠀⡿⢲⠶⡟⢹⣇⣼⠞⠛⠿⣄⣧⣿⣁⢀⣾⣿⣿⣧⣴⣶⣷⣿⣿⠀⠀⠙⠓⢷⠄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣏⢸⠀⣧⡼⠋⢀⣤⣄⢠⣬⣽⣿⣿⣿⣿⣿⣿⣿⠟⣉⣿⣿⣿⠀⠀⠀⣠⣼⠆
⠀⠀⠀⠀⠀⠛⠛⣦⠀⣧⣼⣿⣥⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⢿⣿⣿⣿⡀⠀⣾⠁
⠀⠀⠀⠀⠀⢀⣀⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠇⠀⠻⣤⣤
⠀⠀⠀⠀⠀⣿⠉⠀⠀⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⠶⢦⡀⠀⠀⣠⠶⠶⣦⡀⠀⣠⡴⠶⣄⠀⠀⢀⡶⠒⢶⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢻⣄⣀⣠⣼⠃⠀⠀⣿⣤⣴⠏⠀⠀⢸⣧⣴⠏⠀⠀⠿⠶⠖⠾⠁⠀⠀⠿⠶⠖⣲⢿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢒⡲⢶⣆⣾⣶⡀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠰⣄⣀⡔⠣⣀⡜⠁⠘⠦⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢠⠒⠲⣀⡴⠉⠳⠤⠂⠈⠒⠃⠀⠀⠉⣀⣀⣀⣠⡤⠤⠄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠙⠢⠚⠀⠀⠉⠁⢀⣀⣀⣀⠤⠤⠤⠔⠒⠒⠒⠈⠉⡥⠤⠄⢠⢒⠀⠄⠸⠀⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠶⡉⠀⠨⠤⠶⠒⠒⠉⣉⣁⠘⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠃⢌⠸⣸⢰⠈⠀⡆⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢃⠀⠀⠁⠰⠋⠀⠀⠉⠂⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⢇⠛⢀⠄⠯⠋⠔⠀⢡⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡢⠸⠀⠀⡀⢣⡈⠖⠛⣧⣤⠚⠁⠀⠋⢂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠸⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⡁⠀⠆⠀⠀⠆⢐⢂⣄⣀⠄⠤⠥⢀⣐⡊⡀⠀⠀⢀⣠⡤⠤⢤⣀⡀⠀⠀⠀⠀⡆⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⢀⠀⠀⠀⠀⠁⣀⣤⣤⣤⣀⡀⠀⠀⠉⢁⠔⣩⠔⢂⡰⠀⠀⠉⠳⡄⠀⠀⢡⠀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⢨⠀⠀⠀⣠⠊⡠⢂⡤⠀⠀⢉⠒⣄⢀⠎⠐⢡⠊⠁⠀⠀⠀⠀⡀⠐⠀⠀⠸⠈⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⡄⠀⠀⡇⠸⢠⠋⠀⠀⠀⠀⠀⠈⠾⠀⠈⠁⠀⠀⠀⡠⢀⠌⡰⠸⠀⠀⠀⡆⢰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠃⠀⠀⡃⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠎⡠⢂⠌⡀⡐⠀⠀⠀⠃⠈⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢏⡀⢸⠀⠀⢣⠈⠒⠀⠀⠀⠀⠀⠀⠀⡠⢞⡠⢊⠔⡡⢊⡔⠡⠊⠠⠁⠀⠀⠀⠃⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⡄⠀⠀⠱⡀⠘⢔⠨⠔⡡⠔⡩⠔⣡⠔⡡⢊⡔⡡⢖⠕⢡⠃⠀⠀⠀⠀⢰⠀⣈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢢⠀⡅⠀⠀⠀⠈⠢⡀⠒⠡⠔⣉⠔⣊⠔⣊⠔⡡⢊⠔⡁⡠⠁⠀⠀⠀⠀⠀⠸⠀⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠀⢠⠀⠀⠀⠀⠀⠈⠓⢄⡈⠐⠫⠔⡊⠔⢊⠔⡡⢊⡴⣁⠀⠀⣀⡀⠀⠀⠀⡄⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠘⣀⢀⣀⣠⠋⢑⣀⠀⠈⠒⠤⡉⠐⠪⠕⡪⣲⠟⠘⠉⠁⠘⠉⠛⠀⠀⠀⢃⠘⣳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⠀⠧⠀⠈⠀⠀⢀⠈⠃⠀⠀⠀⠈⠙⠠⢌⠔⠁⢾⣉⣘⣉⠻⠉⠙⠚⠋⠀⢙⠀⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢈⠈⠉⠉⠉⠀⠒⠒⠈⠥⠄⠀⠀⠀⠀⠀⢀⣰⠛⣋⣀⠀⠤⠤⠐⠚⠉⠉⠁⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠈⡀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠀⠒⠂⠈⠉⣀⣀⠀⠀⡠⠤⡀⢡⠂⠒⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠁⠐⠒⠈⠉⠉⢀⣀⡀⢀⡄⢄⠀⡔⠒⢄⡎⠀⠈⠧⠞⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⡈⢀⡎⠙⣆⠜⠀⠱⠞⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢀⡠⠄⡠⢄⡠⢄⡀⠄⣀⠤⣂⡠⢐⡠⢔⡠⠔⣂⠤⣂⠤⣊⠤⢒⠤⢒⡠⠔⡢⠔⠢⠔⠪⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠁⠒⠁⠂⠁⠚⠁⠀⠁⠀⠉⠀⠈⠀⠀⠁⠀⠁⠀⠈

⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢘⢢⣥⡤⠄⠠⠔⠒⠒⠒⠒⠞⠥⣐⠀⠀⠀⠀⠀⠀⢠⠈⣆⠀⠀⠀⠀⠀⠀⠀⣶⠛⠉⠁⠠⠀⠀⠀⠀⠀⠉⢻⣶
⠀⠀⢸⣿⠀⠀⠠⠶⡀⠘⡄⠀⠀⠀⠒⢾⠀⠀⠀⠀⠀⢠⠾⠛⠿⢆⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠘⣹⠀⣀⣀⣿⡁⠀⢰⣷⣤⣙⣶⣾⠀⠀⠀⠀⠀⣹⣼⣀⣦⣼⠁⠀⠀⠀⠀⠈⣵⣀⡤⢀⢸⡦⠴⣀⠐⢒⡠⣀⣿
⠀⠀⠸⠩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠩⠀⠀⠀⠤⠋⠙⠛⠉⠁⠫⠜⠣⠀⠀⠀⠀⠛⠻⠿⠷⠩⠤⠤⠍⠸⠾⠿⠿⠟

⠀⠀⠀⠀⠰⠶⢠⣤⣤⣤⣤⣤⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣤⣤⣤⣤⣤⣤⣤⠄⠄
⠀⠀⠀⠀⠀⠒⠊⢉⠉⠉⠉⠉⠘⠂⠀⠀⠀⠀⣾⢒⡀⠤⠄⠀⠀⠀⠰⢲⡆⠀⠀⠀⠀⠒⠈⠉⠉⠈⠉⠉⠉⡙⠒⠆
⠀⠀⠀⠀⠈⠀⠀⠘⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠉⠉⠉⢉⣉⣀⠈⠛⠀⠀⠀⠀⠈⠒⠒⠒⠆⠀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⢳⡀⠠⠤⠀⠄⠈⠂⠀⠠⡆⠀⠀⠀⠀⢰⠀⠲⣤⣀⡐⠒⠀⠂⢑⡃
⠀⠀⠀⠀⢀⡀⠿⡇⠃⠀⠀⠁⠃⡄⠀⠀⠀⠀⢻⡧⠀⠀⡀⠀⠀⠀⠀⠒⡇⠀⠀⠀⠀⢠⣼⣴⣎⡑⠛⠀⠀⠂⡯⠇
⠀⠀⠀⠀⠘⠒⣗⣛⣋⣛⣛⣛⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⠿⠿⣉⣀⣁⣁⡀⠀⠣⡄
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠁

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

⠀⢠⣾⣿⡟⠁⠀⣴⣿⣿⠃⠀⢀⣼⣿⡿⠁⠀⢠⣾⣿⠟⠀⢀⣼⣿⡿⠃⠀⢀⣾⣿⡟⠁⠀⣠⣿⣿⠟⠀⢀⣼⣿⡿⠁⠀⢠⣾⣿
⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠤⡠⠒⠖⡙⢲
⠀⠀⠀⠀⠀⠀⢀⡴⠤⠞⣒⣋⠬⠥⠐⢒⡉⢭⢱⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⢒⣉⣒⣉⣒⣋⣒⣛⣞⣛⢹⠆
⠀⠀⠀⠀⠀⠀⢘⡆⢚⢩⠄⠂⠀⠀⠀⣠⡄⠀⠧⡆⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⣮⢸⡧⠆⡴⠒⣦⠴⣲⡢⡸⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠱⡌⣴⠀⠀⣾⣿⣶⣿⡇⠀⠸⣱⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠓⠉⢯⣼⠂⢊⠓⠙⢹⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣄⢧⡇⠀⠈⠻⢿⠟⠀⠀⢀⡃⡇⣞⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡰⢞⣙⡎⡆⢸⠀⡀⢸⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠘⡘⣀⣀⠠⠄⢐⡒⠬⣉⣂⢰⠈⡵⠀⠀⠀⠀⠀⠀⠐⡧⢸⠣⡢⠭⠤⣣⢾⣶⡸⠸⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢣⢑⠂⠭⣌⠒⠈⣍⡑⠬⠌⡆⢣⡀⠀⠀⠀⠀⠀⠘⡖⠘⠒⠈⠿⠯⠥⠤⠭⠤⠞⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡘⡀⢌⣓⣂⠬⢅⡒⣂⣍⡥⠦⠎⠁⠀⠀⠀⠀⠀⠀⠉⠚⠙⠙⠚⠉⠚⠓⠚⠚⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢥⣉⠭⡤⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⡠⢆⣀⠴⣀⣠⢦⣀⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢷⣶⠒⠒⠒⠒⠊⠑⠲⣑⡶⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢒⣞⣥⠽⠿⣟⣷⡶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢾⠉⠛⢒⢂⣀⢀⠀⠀⠘⡗⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢛⣽⢟⡥⣀⣰⣒⣒⠉⠻⣾⡓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠀⣼⠞⡩⣌⡙⢶⡀⠀⠑⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣼⠃⣩⢪⠛⣉⠉⠙⢕⢎⢪⣷⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣻⠐⡏⠌⠀⠀⢹⠨⡗⠀⠟⡧⠀⠀⠀⠀⠀⠀⠀⠀⢨⠗⡏⡆⡇⡇⠘⣌⢟⡹⠀⣿⠾⢺⣣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣻⠀⢹⣨⣒⣒⣩⡼⠉⠀⢎⡧⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣿⢠⡱⣱⣀⠘⠁⣀⢜⡜⠀⡾⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣽⠀⠰⣒⣋⣙⣒⣒⡂⠀⢧⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣘⢷⣇⡈⠿⣿⡿⠓⠉⣠⣾⣡⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡼⢄⠀⠉⠋⠋⠙⠉⠁⡠⠱⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠽⢿⣗⣦⡴⠖⣛⣉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠳⠎⠉⠶⠋⠱⠞⠉⠶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠋⠁

⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀
⠀⣴⣿⣿⠏⠀⢀⣾⣿⡟⠁⠀⣠⣿⣿⠟⠀⠀⣴⣿⡿⠃⠀⢠⣿⣿⡟⠀⠀⣰⣿⣿⠏⠀⠀⣼⣿⡿⠃⠀⢠⣿⣿⠟⠀⠀⣴⣿⣿

⣾⣿⠷⢴⣿⠿⢶⣾⡿⠶⣾⡿⠷⣶⡿⠷⣴⡿⠿⣦⡾⠿⣷⣾⠿⢷⣶⠿⢿⣦⠾⢿⣶⠾⢿⣷⡶⠿⣿⡶⠿⣿⡶⠾⣿
⢙⡇⠀⠀⠉⠀⠀⠉⠀⠀⠈⠀⠀⠈⠀⠀⠈⠀⠀⠈⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠈⠀⠀⠉⠀⠀⢸
⣿⡟⠀⢸⣿⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣿⡿⢿⡉⣻⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣼⣧⠀⢸⣿⡿⢻⣿⣿⣿⣾⣿⣿⣿⣿⠛⢥⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠻⡏⠀⢸⣿⣷⣴⣿⣿⣿⣿⢿⣿⣿⣿⡆⠀⠀⠉⣿⠟⢿⡟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣾⡿⠀⢸⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⠏⢹⣿⣿⣦⡟⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⢿
⣹⣇⠀⢸⣿⣿⣏⢙⣿⣿⣿⣿⣿⣿⣿⣿⡀⢰⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⣦⠀⣸
⢿⡏⠀⠸⣿⣿⣿⣿⣿⣿⣿⡏⠉⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⠀⢹
⣼⡷⠀⢸⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⡏⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⢾
⢻⣇⠀⠸⢿⣿⣿⣷⣾⣿⣿⣿⡿⠿⣿⣿⣿⡀⠘⣇⠈⠢⣄⣘⣉⡟⢛⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⣸
⣿⡏⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣧⠀⢻⣭⣴⢛⡩⠉⢻⡧⠤⠄⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⢹
⣨⣷⠀⢸⣿⣿⣿⣿⣧⣠⣿⣿⣿⣿⣿⣿⣿⢃⠀⠈⣸⣿⡏⠁⢉⣿⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⠀⣾
⢿⣇⠀⢸⡏⢙⣿⣿⣿⣿⣿⣿⣿⣟⡀⣿⠁⠘⢁⣴⡿⠿⣿⠐⠾⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⠀⣸
⣾⡟⠀⠘⠻⢿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣇⠀⠀⠺⠁⢸⠀⠀⠀⠐⡻⡍⠻⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻
⢙⣧⠀⠀⠀⠀⠈⠉⠛⠿⣾⣿⣿⣿⣿⣿⣿⣶⡄⢠⠇⠀⠀⣠⢊⣜⠀⣆⣽⡿⢿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⣼
⣿⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣧⠉⠀⢀⡜⣡⠎⡛⢘⢿⣿⣷⡀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣷⠀⢸
⣼⡟⠀⢸⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠇⢀⣮⠞⠁⢀⠁⢸⢀⢻⣿⣿⠀⠀⠠⠏⠀⠈⠙⠻⠿⣿⣿⣿⠀⢻
⠻⣧⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⢀⡾⠋⡀⠀⠘⠀⢸⡰⢿⣿⣿⡄⢀⣴⡇⠀⠀⠀⠀⠀⠀⠉⠛⠀⣼
⣾⡇⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⡞⠁⣰⠁⠀⠁⢀⠘⡅⢿⣿⡿⢠⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⢸
⣹⡟⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⠆⢠⠃⠀⠀⠀⢸⠀⡇⠸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⢿
⢿⣧⠀⢸⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⢠⠎⠀⠀⠀⠀⢈⠀⡇⠀⣿⡟⠁⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⠀⣼
⣼⣏⣀⣸⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠘⡇⠀⢸⣿⣀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠀⢹
⢻⣿⡉⢉⣛⡻⢿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⢀⣴⠃⠀⢐⡇⠀⠘⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢾
⣿⣯⡇⢀⡏⠈⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⣠⠞⠁⠀⡀⢸⡇⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⣸
⣨⣟⣁⣸⣅⡀⠀⠿⠀⠀⡠⣌⠻⣿⡻⢿⣿⣿⣿⡇⠀⡵⠃⠀⠀⣇⣠⣴⣾⣿⣿⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣷⠀⢹
⢿⣿⡉⢻⣿⣿⣷⣶⣤⣞⠣⣠⠇⠘⢿⣄⠈⠉⠛⠃⠊⠀⠀⠀⠀⠈⣩⡟⣻⣏⣿⣀⣀⣀⣀⣀⣀⣉⣛⣻⣿⣿⣿⠀⢾
⣾⣟⠋⠹⢿⣿⣿⣿⣿⣿⣿⣷⢶⣶⠾⢿⡶⠒⠞⠛⠛⠛⠛⣛⣿⣿⢛⣋⣻⣏⣿⣿⣿⣭⣉⣉⣉⣉⣉⣉⣉⡉⠉⠀⣸
⢹⣟⠀⠉⠛⠛⠻⠻⠿⣿⣿⡷⣿⣿⠿⣿⣿⣿⣟⡛⠛⢛⠛⣟⢻⣿⣟⣋⣻⣏⣻⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣉⡉⠀⢻
⣿⣿⠗⠀⠉⠛⠛⠛⠿⠞⢿⡿⠺⣿⣿⣿⣿⣿⣿⡟⠛⠛⢛⠟⢻⣿⢿⡏⣿⠋⢉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠉⣾
⣼⣿⣶⡙⣦⠀⠀⠀⢠⡦⢸⣷⢸⣁⡝⢹⠟⣿⠋⢀⡀⣀⣀⣰⣟⣃⣀⣃⡁⠀⠸⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⠀⢸
⠻⡟⢹⣿⣷⣄⡀⠀⠀⠀⢸⣿⠀⠉⠁⢀⡀⣿⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠟⠀⢻
⣾⣷⣾⣿⣿⣿⣿⣷⣶⣀⣾⣇⣈⣷⣶⣾⣷⣿⣀⣴⣄⣀⣴⣦⣀⣠⣦⣀⣠⣦⣀⣠⣦⣀⣠⣶⣄⣠⣶⣄⣀⣴⣄⣀⣼

⣾⠟⠒⠿⠓⠻⠛⠺⠟⠲⠿⠓⠾⠛⠲⠟⠳⠟⠛⠾⠛⠶⠛⠳⠞⠛⠶⠛⠷⠛⠳⠞⠻⠖⠛⠷⠚⠿⠒⠻⠗⠛⠟⠚⠿⠒⠻
⣼⡆⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢰
⣽⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡻⣿⣿⣿⣿⠀⢠
⣻⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡋⢕⢤⣯⢰⠟⣿⣿⠀⣠
⢻⡁⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⢋⣵⣼⠳⠿⣿⠿⠿⠿⠿⠿⢿⣿⢿⣿⣼⡷⠛⣷⡦⠳⢷⣼⣮⡟⠛⠀⢈
⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣏⠒⢶⣒⠀⢘
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢆⣀⠀⠀⠀⠈⠛⠣⣭⡉⠀⢀⡀⠶⣤⠘⡏⢻⢨⡇⠀⠘
⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢻⣧⣀⣀⣶⣤⡀⢹⣿⡍⠹⣿⣧⢨⡛⢧⡰⠦⠀⠀⠸
⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⡀⠀⠀⣀⡤⢼⠀⣀⣈⢿⡆⠆⡼⠛⠓⢾⣧⣷⠰⡈⠉⢹⣧⠈⠈⣧⠁⠀⠰
⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠛⠃⠀⣴⠜⠛⣁⣀⡴⠃⠀⢸⡆⠘⠿⢤⣷⠀⣸⠃⠀⠈⡿⣿⢠⡏⠀⠈⢿⣿⡀⢰⠀⠀⢰
⢿⡄⠀⠀⠀⠀⠀⠀⠠⣤⣿⠃⠀⢶⢰⡿⠀⠐⡏⠀⠀⠀⠀⠀⣧⡤⠀⠈⢿⣧⣿⠀⠀⠀⣾⢸⡾⡇⠀⢀⣨⣿⣿⠶⠶⠶⣶
⣾⡅⠀⠀⠖⠃⠀⠀⠒⣾⠏⠀⠀⢸⢿⠃⠀⠀⣿⠀⠀⠀⠀⠀⠉⡇⠀⠀⢨⣜⣿⠀⠀⠀⣿⣼⣷⣷⠞⡫⢡⣿⢸⠬⡁⠈⣿
⣾⡁⠀⠀⠀⠀⣠⠴⠖⢿⠀⠀⠀⠈⢺⡀⠀⠀⣿⣄⠀⠀⠀⠀⠴⣿⠀⠀⠸⣇⣿⢷⣄⠀⢸⣿⡫⢿⣷⣷⠌⣟⣈⣲⣥⣄⣉
⣽⣃⣀⣀⣠⣾⣡⣤⣤⣼⣧⣤⣴⣿⢼⠧⣴⣾⠷⠿⠶⠶⠤⠤⠤⢼⣦⡀⠀⠹⠏⢬⣏⣳⡟⠰⠈⢿⣷⣼⡞⢫⡉⠰⠀⠀⣙
⣹⣯⣭⣭⣥⣤⣤⣤⡤⣴⢶⢿⢿⠶⢶⢶⢷⢾⡷⠒⠒⠒⠲⠶⠶⠶⢾⣯⡱⡄⢠⠾⢫⡿⠷⠘⠋⢩⡟⠉⠏⠈⠁⠀⠀⣠⣿
⢻⡷⠤⠤⠴⠶⠶⠶⠶⠿⠾⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠛⠃⠙⠃⠈⠁⣾⠸⠒⠢⢀⡟⠐⠃⠀⠀⠀⣞⣴⣿⣿
⢿⣦⠤⣶⡤⣶⣤⢴⣦⠴⣦⡤⢶⣤⠴⣦⡴⣦⣤⢶⣤⠶⣤⡴⢦⣤⠶⣤⡶⣤⡴⢦⣿⠶⣶⡶⢼⣷⠿⣴⡦⣾⡷⢿⣷⠤⣿

⢾⠟⠺⠛⠺⠛⠺⠛⠺⠛⠺⠛⠲⠛⠲⠛⠳⠛⠳⠛⠳⠛⠳⠛⠳⠛⠳⠚⠳⠚⠻⠚⠻⠚⠻⠚⠻⠚⠻⠚⢿
⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣾⠂⠀⠀⢀⡠⠤⣤⡤⠤⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣾⠂⠀⣾⣻⢷⣄⡛⠁⠼⣿⢠⡿⠁⢸⣿⣶⣤⣄⡀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣾⠂⠀⠙⢷⣯⡿⣿⣷⣄⠁⠈⠁⠂⠞⠻⣿⣿⣿⠟⣰⠏⣄⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣼⠂⠀⠀⠀⠙⢿⣯⠿⣿⣷⣄⠀⠀⠀⠀⠿⠟⣿⠃⠎⣠⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣽⠂⠀⠀⠀⠀⠀⠙⢿⣹⠿⣿⣷⣀⡀⠀⣠⣾⣿⣶⣴⡏⢿⣿⠟⢻⡟⠀⣴⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⣽⠂⠀⠀⠀⠀⠀⠀⠀⢨⡓⢾⢿⠁⠁⠁⠋⣿⣿⣿⣿⣧⠈⠁⠀⠀⠈⠒⠟⣠⣿⡑⢦⡀⠀⠀⠀⠀⠀⠀⢺
⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⣙⠛⡿⢧⣤⡄⠀⠃⠀⠏⠐⠙⢀⡀⠀⣀⣀⡀⣀⣈⣘⠀⠿⠛⢄⠀⠀⠀⠀⠀⢺
⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠻⣿⡶⠋⠃⠀⣀⣀⣀⠀⢀⣼⣿⢿⣿⣿⣿⡿⣿⢿⢷⣶⣾⡟⢛⠿⠉⠹⣿
⣹⠂⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋⠙⠦⣧⠀⠀⠀⢀⣯⣼⡛⣿⣿⣛⡶⠿⠼⠶⠷⣿⣾⣿⠿⣿⢟⠀⢭⠇⠠⢿
⣻⠂⠀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣠⣤⣤⣤⣤⣴⣾⣭⣏⣻⣥⡜⠉⠁⠀⠀⠀⣴⠋⠘⢜⢀⣠⠶⡛⢫⠉⠙⢻
⣻⣏⣉⣭⣭⣭⣭⣭⡭⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠼⡾⠤⠀⠀⠀⠀⠀⣼⠁⠐⠂⢠⠟⠱⠀⠁⠀⠀⣰⣿
⣻⠧⠤⠶⠶⠶⠶⠶⠖⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠗⠒⠂⠀⠀⠀⢀⡇⠓⠒⢠⡏⠘⠀⠀⡀⢜⣶⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡤⣤⠤⣤⡤⣤⣤⣤⣤⣤⢤⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⣧⣿⣴⣿⣧⣿⣴⡿⡮⢿⢼⠇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡟⣾⠛⣾⡞⣿⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠛⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠐⣇⣿⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡠⢄⠀⡠⢤⡀⠀⠀⠀⠀⢀⡤⣀⠀⢀⠤⣀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡀⢺⣿⡅⢀⣟⠀⠀⠀⠀⣼⡀⢻⠀⣻⠀⣽⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⠀⠁⠛⠁⠀⠙⠀⠀⠀⠀⠛⠀⠉⠃⠋⠁⠘⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡖⣢⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣎⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⢆⡀⣠⠖⠲⣄⢀⡰⠒⢢⡀⢀⡖⠒⣆⠀⣠⠖⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⢉⣁⣀⣀⣈⣉⣀⣀⣀⣉⣁⣀⣀⣀⣉⠁⠀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⡏⠉⠉⢉⡉⠉⠉⣉⡉⠉⠉⠉⠉⠉⠉⡇⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⡇⢀⣼⣿⣿⣷⣯⣄⣹⠀⣠⣦⡀⠀⠀⡇⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠹⠿⠿⠿⠿⠿⠿⠃⡰⠿⠿⠿⢦⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⡇⠀⠀⠀⢀⡴⢄⠀⡼⠁⠀⠀⠀⠀⠳⡇⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⠀⡇⣀⣀⣠⣋⣀⣈⣉⠀⣠⣀⡀⠀⢀⣄⡀⠀⢀⣄⡀⠀⣠⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠃⠀⣈⡉⠉⠉⠉⣉⠉⠉⠀⢿⡀⢙⣒⣋⣀⣙⣒⣋⣀⣑⣒⠁⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⠞⠉⠙⢦⣴⠋⠉⠱⣤⡄⢀⡼⢸⣿⣿⣿⠿⠿⠿⣿⣿⣿⠀⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡠⠖⠒⢺⣿⡟⣵⢰⣿⡞⣎⢿⣿⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡑⠒⠒⣫⠔⣋⣉⣹⣿⣇⢶⢰⣶⣆⡶⣼⣿⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢍⣉⣉⠵⠊⠡⣄⢸⣿⣿⣶⣬⣭⣬⣾⣿⣿⠀⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠞⠉⠳⠤⠞⠉⠳⠤⠏⠉⠦⠤⠇




⡟⠉⣿⣛⣻⣛⣛⣟⣛⣛⣻⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣻⣿⣟⣻⣛⣛⣛

⠀⠀⠀⠀⠀⢠⡤⣄⠀⣠⡤⣤⠀⢠⡤⣤⠀⢠⡤⢤⡀⢀⡤⢤⡄⠀⣤⢤⡄⠀⣤⢤⣄⠀⣠⢤⡄
⠀⠀⠀⠀⠀⢸⣄⠈⠛⠉⠀⠈⠛⠋⠀⠈⠛⠋⠀⠀⠙⠋⠀⠀⠙⠛⠁⠀⠙⠛⠁⠀⠉⠛⠁⣠⡇
⠀⠀⠀⠀⠀⢀⣸⠇⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⠸⣇⡀
⠀⠀⠀⠀⠀⢸⣁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣨⡇
⠀⠀⠀⠀⠀⢀⣸⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⣇⡀
⠀⠀⠀⠀⠀⢸⡁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢈⡇
⠀⠀⠀⠀⠀⠀⣹⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⣏
⠀⠀⠀⠀⠀⢸⡁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢈⡇
⠀⠀⠀⠀⠀⠈⢹⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⡏⠁
⠀⠀⠀⠀⠀⢸⡋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢉⡇
⠀⠀⠀⠀⠀⠈⢹⡄⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⢰⡏⠁
⠀⠀⠀⠀⠀⢸⠋⢀⣤⣀⠀⢀⣤⣄⠀⢀⣤⣄⠀⠀⣠⣄⠀⠀⣠⣤⡀⠀⣠⣤⡀⠀⣀⣤⡀⠙⡇
⠀⠀⠀⠀⠀⠘⠓⠋⠀⠙⠓⠛⠀⠘⠓⠛⠀⠘⠓⠚⠁⠈⠓⠚⠃⠀⠛⠚⠃⠀⠛⠚⠋⠀⠙⠚⠃

⠀⢰⣿⣿⣿⣟⢛⣿⡏⠀⠀⣿⣶⣿⣿⣿⣿⠀⠀⠶⠖⠒⢂⡐⠒⠒⠒⠒⡲⠆⠀⢰⣶⣶⣶⣶⣶⣶⡆⠀⠀⠤⢄⣄⠀⠀⠀⠀⢾⡿⣿⢿⡿⣿⣿⢿⣿
⠀⠰⣿⠿⣿⣯⣸⣿⡇⠀⠀⣷⠉⣻⣽⢽⣦⠀⠀⡂⠈⠀⡽⠇⠈⠲⢀⡂⠡⡁⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣽⣷⣿⡆⡇⠀⠀⢻⣑⠄⠲⡧⢿⣿⣼⣿
⠀⠈⠏⠿⢘⣼⣿⠉⠁⠀⠀⣿⡇⢸⡿⣶⣼⠀⠀⡃⠀⢘⠃⠻⡆⠁⠚⢧⠀⠇⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣟⣿⣷⡇⠀⠀⢻⠺⣿⣾⣿⣿⣏⠉⢹
⠀⠀⠀⠀⢿⣋⣽⢀⠀⠀⠀⠙⣧⣼⣧⢤⢻⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠸⣻⢿⣿⢿⡇⠀⠀⢸⣧⣤⣭⣿⣻⣅⣴⣿
⠀⠀⠈⠁⣰⠧⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣀⡀⣀⣀⠀⠀⢠⡄⠀⠄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣤⠀⠀⡄⠀⠄⠀⢤⡄⠀⢸⣡⡟⢠⢄⡀⠀⠈⣰⢿⣆⠀⠀⢼⡇⣴⣶⡟⠀⠀⠀⢡⡶⡿⣷⡄⠀⠀⠀⣿⣿⡿⢯⣽⣭⣋⣹
⠀⠀⠀⠀⠀⢀⣠⣤⡞⢿⡇⠀⠀⡇⢀⠠⡴⢸⡃⠀⢸⣯⣼⢈⣁⣹⡖⢀⡙⠾⠋⠀⠀⣾⣧⣿⣿⣷⠄⠀⠀⢽⣿⢁⣼⡗⠀⠀⠀⣿⡿⠛⠀⠀⠀⢏⢹
⠀⠀⠀⠀⠒⠛⠛⠙⠋⠛⠃⠀⠀⠃⠈⠂⠛⠘⠃⠀⠘⠋⠛⠀⠈⠋⠀⠀⠀⠐⠘⠀⠀⠛⠙⠛⠛⠛⠂⠀⠀⠊⠁⠘⠛⠀⠀⠀⠀⠛⣣⡄⠀⠐⠂⠚⠚
⠀⣠⠀⠀⠀⡀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢉⠀⢀⢶⠄⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠃⠀⠀⣠⠄
⠀⢸⡆⡠⠺⢀⠀⡀⢰⠟⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⣸⡧⠂⢀⣿⠎⠀⠀⠀⢀⣀⢀⣾⢈⡀⠀⠀⠀⡴⠳⠀⠀⠀⠀⠀⠀⠀⣸⡗⣃⠀⠆⣴⠏⠀⡀
⠀⢸⡗⢡⣆⡾⣾⣡⡏⢀⡞⣱⠆⡴⢫⡆⢴⠟⠡⠀⢠⣏⢐⡦⣾⠃⢠⣏⣶⢂⡾⠉⣼⣷⠛⡁⠀⠀⣰⠋⡴⠀⠀⢰⠇⣴⢠⣆⡿⢸⢏⣶⣠⡏⢠⡾⠋
⠀⠟⠀⠀⠁⠀⠈⢸⡔⠁⠉⠘⠋⢙⣼⠁⠘⠒⠀⠀⠈⠃⠛⠀⣿⠐⠉⠁⠙⠊⠓⠊⠉⠈⠉⠀⠀⠀⠓⠚⠁⠀⠀⠘⠀⠿⠊⠀⠁⠘⠋⠈⢽⠔⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣱⠃⠀⠀⠀⠀⠀⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⡖⠀⣀⡾⠐⣠⡀⡐⡾⢁⣀⢰⠇⠀⣠⢠⣀⣤⣀⢠⠏⣀⡀⣠⢀⡄⠀⢠⠀⡞⡀⡀⣀⣤⢀⡀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠱⠁⠀⠓⠇⠉⢳⠋⠹⠅⠋⠓⠧⠀⢠⠓⠋⠑⠡⠿⠹⠈⠋⠛⡾⠉⠂⠀⠼⠻⠅⠋⠓⠁⠙⡟⠚⠡⠏
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⢃⣀⡀⠀⠀⠀⣀⠀⢈⣀⡀⢀⠀⠀⠀⡀⠀⠈⠁⠀⢀⣀⣀⣀⠀⠀⣀⣀⡀⣁⣀⡀
⠀⠀⣿⣿⣝⣿⣿⣿⢟⣿⠀⠀⠉⠀⠙⣿⣿⠈⠁⠀⠀⠀⣿⣁⣸⣿⣥⣾⠆⠰⠶⠀⠀⠀⠀⠤⠤⠿⠿⠿⠖⠀⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠿⠿⠿⠿⠿⠿⠿⣿⠀⠀⠀⠀⢔⣾⣇⡀⠀⠀⠀⠀⣿⣿⣿⣯⣽⣾⠀⢠⣤⣤⣤⣤⣤⣤⡄⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⡀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠇⠀⠀⠀⢠⣿⠻⣿⣿⠟⣿⠀⢸⣿⡟⡀⠀⣿⣿⡇⢨⣿⣿⠿⢿⣿⡇⠀⠀⠀⣙⣿⣿⠇⠀⣿⣿⣿⣿⣿
⠀⠀⢰⣶⣶⣶⣦⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠀⢸⣿⣧⠀⢠⣿⣿⡇⢸⣿⡿⣧⣻⣿⡇⠀⠀⠀⠀⡹⠈⠀⠀⠛⠛⠛⠛⠛
⠀⠀⢸⣹⠃⣹⣷⣼⢻⣹⠀⠀⢰⣿⣟⣿⣿⢿⣶⠀⠀⠀⢰⡔⣢⣰⣶⣆⠀⢸⣿⣿⣂⣀⣿⣿⡇⢘⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⣠⣤⣤⣤⣤
⠀⠀⢸⣿⣠⣿⠟⠀⢨⣽⠀⠀⢼⣿⢉⣻⣍⢹⣿⠀⠀⠀⢸⣧⣿⣯⣿⣿⠀⢸⣿⣟⠛⣡⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⣸⣿⣿⠿⣿⣿⣼⣎⠀⠀⢸⣏⠠⠿⠟⢈⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠸⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⠴⢤⡈⠉⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠈⢧⣤⡴⣃⣼⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣠⡴⠞⣛⣉⣤⣤⣿⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣧⣄⣤⠶⣚⣭⣴⣾⡟⠻⠛⠏⠹⣶⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⠉⠀⢁⣲⣆⠀⠼⠗⣶⠹⣆⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠲⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣛⡁⠀⠀⠀⠉⣀⡀⠀⠠⠞⡁⠀⢹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⣅⠀⠀⠀⡼⢏⠀⠀⣠⠞⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣋⣭⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠶⢶⣄⡌⠙⢻⣆⠀⠀⠀⠈⠙⠋⠹⡄⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠠⣯⠥⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣧⡀⠀⢸⣇⣹⣤⠀⢿⠀⠀⠀⠀⠀⠀⢀⡽⠀⢀⣠⣴⠻⣆⣀⣀⣀⣠⡤⠖⠛⣿⣀⡤⠿⠛⠋
⠀⠀⠀⠀⠀⠀⢠⣤⠴⣯⠁⠀⠈⣿⠚⢋⣉⡀⢹⡄⢸⣇⠀⠀⠀⢠⣞⣩⣤⣶⣿⣁⣿⡞⠹⡏⠉⠉⢀⣀⣀⣤⡿⠉
⠀⠀⠀⠀⠀⣀⣼⠷⡄⢸⣧⣤⡄⣿⠛⠉⠉⣿⣨⣷⠾⢭⣉⡛⠛⠻⢭⣀⡾⠛⢿⡈⢻⡙⡆⢻⡖⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠹⣆⠀⢹⣼⡇⠈⣇⣹⡶⢶⣿⠟⠋⠀⠀⠀⠈⠉⠛⠛⣿⡛⠳⣄⠘⣷⡄⢳⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠈⢷⣄⣿⠷⠛⠉⢀⡴⠋⠁⠀⠀⣀⣠⣴⡶⠄⠀⠀⠈⠙⠳⢾⣇⢹⣿⡌⢿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣾⠋⠀⠲⣷⡶⠛⠉⠉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⣿⣿⡘⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⢷⣄⣀⣼⣧⣀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠿⠟⣻⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣹⠉⠀⠀⡟⠛⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⠶⠛⣻⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡏⢹⡰⡜⣆⢷⣸⡇⢸⡇⠀⢀⣤⡴⠶⠂⠀⠀⣸⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠸⣧⠴⠷⠟⠛⠛⣉⣵⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⣿⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣶⠶⠟⠋⠁⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⣠⡶⠛⠛⠛⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣇⣠⣤⡶⠞⠋⠁⣀⣠⣤⣄⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣀⡛⠋⢉⡿⠻⠿⣿⠟⠉⠁⣸⡏⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠋⠛⠛⠛⠁⢀⣼⠋⠀⠀⢰⣿⣀⣼⡟⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⠿⢶⣤⣤⣤⣴⠟⠁⠀⠀⠀⠈⣿⣿⣿⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠲⡖⢦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠶⠤⢤⣀⡀
⠀⢀⣄⠐⠃⣠⠄⠁⢀⠁⡀⠀⠀⠀⠀⠀⠄⢥⠓⠃⠀⠃⠀⠞⣩⠦⠀⠀⠀⠀⠀⣠⠖⠋⠁⠐⠖⠀⢠⣂⠀⠀⣉⠢⣄
⠀⣢⠍⠃⢸⣇⣀⣠⠀⠁⡈⣄⡀⠀⠀⡀⠘⠀⢀⣀⣠⡅⠁⠀⠁⠀⠀⠀⠀⢀⡾⠁⠰⡈⠦⠀⠠⠤⠤⠈⠀⠯⡀⠀⠈⠳⡀
⠀⠀⠰⠀⠘⢿⡟⡉⠀⠀⠂⠀⠁⠀⠀⠀⠐⠛⠉⠛⠻⠻⢿⡗⠾⠧⢀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠖⠈⠀⢡
⠈⠄⠀⠲⠲⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⡇⠐⢀⠀⠀⠀⠀⡀⣀⡀⣠⣔⠀⠀⠀⠀⠄
⠀⠈⠢⢀⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⡀⢁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⠟⠛⠿⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠘⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠐⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠉⠂⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡴⠃
⠀⢀⠄⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⣠⣆⠯⡆⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠈⠀⡬⡆⠈⠇⠀⢀⡠⠊
⠀⠋⠄⠀⠀⠀⠠⠛⢇⠈⠄⠰⠀⠀⠀⠀⠙⠻⠁⠑⠀⣃⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠛⠲⠦⠤⠤⠴⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠇⠀⠈⡀⠀⠀⠇⠀⠀⠀⠀⡁⠂⠀⠛⠛⠀⠀⠀⠀⠀⠡⠀⠀⣀⡀⠀⠀⠀⠠⠄⣀⡀
⠀⡀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠬⠉⠀⠒⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⢠⠀⠤⠀⠄⠀⠀⡤
⠀⠀⢀⡠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣒⢨⣭⠙⠀⠂⢄⠀⠀⠀⠀⠋⠃⠘⠤⠀⠀⠀⠻⠼⠘⠧⠀⠈⠁⠘⠦⠆
⠀⠀⠃⠘⠀⠁⠀⠀⠁⠂⠀⡐⣄⠀⠀⠀⡔⠁⠤⠀⠀⣰⡀⠀⠀⠐⠳⡀⠀⠀⡤⠄⠠⠀⠀⠀⣢⡂⠀⠀⠀⠲⠀⠐⢀⡂⡆
⠀⢃⠀⠀⠀⠶⣾⡧⠤⠤⠴⡇⠘⠀⠀⠀⠀⡀⠀⠀⠰⣿⣣⠀⠀⢀⠄⠁⠀⠀⣀⠀⠀⡀⠀⢀⠁⠉⠀⠀⠀⠀⢀⠀⠀⠀⡁
⠀⡄⠀⠀⠐⢲⣿⡷⠶⠗⠂⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠻⡇⠀⠀⠀⠄⠀⠀⠀⢁⠀⠰⠘⠀⠀⡇⢰⠐⠀⠀⠀⠀⠀⠘⢀
⠀⠘⢄⠀⠀⡀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠠⡀⠀⡀⠐⠀⠁⠀⠀⠀⡀⠀⠀⠀⣄⢀⠀⠄⢠⠀⠀⢀⡀⠀⠀⠀⠄⡀⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠘⠏⠀⠘⠆⠞⠀⠈⠃⠀⠈⠀⠚⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠆⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢰⢀⣀⠀⢀⠀⣀⣀⡀⡀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠒⠒⡄⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠘⠒⠐⠒⠒⠒⠒⠂⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠀⠀⠀⢸⢠⣀⣀⣀⣀⣀⣀⣀⣀⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⣀⣀⣀⣀⣀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠠⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣒⣒⣒⡒⣂⣀⣀⢀⠀⠀⠀⠀⠀⠒⠒⢒⡲⢁⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢨⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠦⣤⣤⣤⣤⣤⣤⣤⣴⠴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠉⠉⠀⠀⠀⠈⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⡉⠉⡉⠉⠉⠉⠁
⠀⠀⠀⡔⢀⠀⠠⢀⠀⢀⡀⠀⣀⠀⢀⡀⠀⡀⠂⢀⠀⣀⡀⠀⣀⣀⢀⡀⡀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠀⡀⠠⠀⠀⣠⠈⢠⡤⣄⠀⠑⣄
⠀⠀⡼⠀⠦⠤⠤⠬⠤⠦⠡⠴⠌⠠⠄⠥⠴⠭⠥⠤⠥⠤⠬⠤⠬⠥⠬⠭⠴⠤⠤⠤⠅⠰⠨⠄⠀⠠⠀⠸⠄⠣⠀⠈⠓⠓⠲⠬⠦⠤⠊
⠀⠀⠳⠖⠲⠒⠒⠒⠶⠲⠖⠒⠒⠖⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠒⠒⠒⠖⠲⠒⠒⠒⠲⠒⠒⠶⠒⠒⠒⠖⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠦⣄⡴⠚⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣸⡇
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧⠀⠀⢾⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣈⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡟⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡌⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢒⡒⢒⡒⢒⡲⢒⡒⢒⡒⠒⣒⠒⣒⠒⣒⠒⣒⡒⣲⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠶⠶⠾⠿⠾⠿⣾⣿⣶⣿⣶⣿⣶⣿⡶⠿⠶⠷⠷⠶⠾⠄
⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡄




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣿⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⢯⣄⣀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡧⠶⠚⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠉⠛⠻⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠉⠉⠉⠉⠉⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


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

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⣠⣤⣤⣀⡀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢰⣿⣿⡀⠀⠀⢸⣿⡿⢿⣿⡆⠀⢀⣾⣿⠿⠟⢻⠇⠀⣿⡿⠿⣿⣿⡆⠀⢸⣿⠿⠿⠿⠋⠀⠀⣿⡿⠿⠿⠟
⠀⠀⠀⠀⢠⣿⣟⣻⣷⠀⠀⢸⣿⣿⣿⣿⣇⠀⢸⣿⡁⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡟⠀⢸⣿⣷⣾⡶⠀⠀⠀⣿⣷⣶⣶
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⢸⣿⣭⣤⣿⡿⠀⠀⢻⣷⣦⣴⣾⡆⠀⣿⣷⣶⣾⡿⠃⠀⢸⣿⣭⣥⣤⣄⠀⠀⣿⡏⠉⠁
⠀⠀⠀⠀⠛⠃⠀⠀⠈⠛⠀⠈⠙⠛⠛⠛⠁⠀⠀⠀⠙⠛⠛⠋⠁⠀⠛⠛⠛⠋⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠛⠃

⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⣠⡄⠀⠀⠀⣤⡀⠀⢠⣄⠀⠀⢠⣄⠀⢀⣤⡀⠀⠀⣤⡀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⣤
⠀⠀⠀⠀⣼⣿⠟⠛⠿⠇⠀⠀⣿⣇⠀⠀⢀⣿⡇⠀⢸⣿⠀⠀⢸⣿⣴⣿⠟⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⡄⠀⢀⣾⣿
⠀⠀⠀⠀⣿⡇⢰⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⢸⣿⣿⣯⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣄⣾⣿⣿
⠀⠀⠀⠀⢻⣿⣶⣴⣿⡇⠀⠀⣿⡏⠀⠀⠀⣿⠇⠀⢸⣿⠀⠀⢸⣿⠙⢿⣷⠀⠀⠀⣿⣷⣶⣶⣦⠀⠀⣿⡇⠻⣿⡿⢹⣿
⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠙⠁⠀⠀⠀⠋⠀⠀⠘⠁⠀⠀⠘⠃⠀⠈⠋⠀⠀⠀⠛⠛⠛⠋⠁⠀⠀⠉⠁⠀⠉⠀⠈⠋

⠀⠀⠀⠀⣶⡄⠀⠀⣴⡄⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⣶⣶⣶⣤⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⢠⣤⣤⣤⡀
⠀⠀⠀⠀⣿⣿⣦⡀⣿⡇⠀⠀⢰⣿⠟⠋⠙⢻⣿⡆⠀⠀⠀⣿⡟⠻⣿⡇⠀⠀⠀⣾⣿⠟⠛⠻⣿⣷⠀⠀⠀⢸⣿⠿⣿⣿
⠀⠀⠀⠀⣿⡟⢿⣷⣿⡇⠀⠀⢸⣿⡀⠀⠀⢀⣿⡇⠀⠀⠀⣿⣿⣿⡿⠃⠀⠀⠀⣿⣇⠀⢠⣶⣼⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⣿⠃⠀⠻⣿⠇⠀⠀⠈⢿⣿⣷⣾⣿⠏⠀⠀⠀⠀⣿⠏⠉⠀⠀⠀⠀⠀⠻⣿⣷⣾⣿⣿⡇⠀⠀⠀⢸⡿⠙⢿⣷
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠘⠃⠀⠈⠋

⠀⠀⠀⠀⣠⣶⣾⣶⣄⠀⠀⣶⣶⣶⣶⣶⡆⠀⣶⣆⠀⠀⢀⣶⠄⢠⣶⣄⠀⢠⣶⠆⠀⣤⡄⠀⠀⣠⣤⠀⢠⣤⣦⣤⣤⣤
⠀⠀⠀⠀⣿⣿⣭⣽⡟⠀⠀⠙⠛⣿⡟⠛⠁⠀⢻⣿⡄⢀⣾⡟⠀⠈⢻⣿⣶⣿⠏⠀⠀⢿⣿⣆⣴⣿⠇⠀⠘⠛⠛⣿⣿⠋
⠀⠀⠀⠀⣬⡛⠻⢿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢻⣿⣾⡟⠀⠀⠀⣠⣿⢿⣿⣄⠀⠀⠀⠻⣿⣿⠁⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠈⢿⣿⣿⣿⠟⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⢰⣿⠏⠈⢻⣿⠂⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠙⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣟⣿⣿⣷⣿⣾⣟⣿⢿⣿⣿⡷⣿⣾⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡀⠰⠀⠑⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⡛⠛⠛⠛⠛⠛⠻⠿⠟⠛⠟⠛⠻⠻⠻⠻⠻⠻⠿⡏⡉⢉⢉⡉⠙⠉⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠄⠀⠀⠁⠉⡀⠀⠂⠀⠀⠀⠁⠀⠀⢂
⠀⠀⠀⠀⢀⣀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⠀⠸⡀⢸⠀⠀⡃⠀⠀⠁⠀⠀⠀⢐
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⡄⠇⠞⡴⠃⡀⠀⠀⠂⠅⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠉⠀⠄⠄⠸⠉⠁⠀⠀⣀⠀⡀⠀⠀⡐⡒⡂⡁
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠁⠁⠀⡀⠀⠄⠀⠀⠄⠄⠄
⠀⠀⠂⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢀⠀⠄
⠀⠀⠀⠀⠄⠂⠈⠉⠰⠀⠀⠀⠀⠀⠀⠀⠀⠒⠙⠒⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠐⠐⠀⠀⠐⠀⠀⠀⠂⠂⠀⠀⠀⠀⠂⠐⠐⠀⠐⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠂
⠀⠀⠅⠀⠀⠀⡀⡀⠀⠀⠀⠠⠀⠀⠀⠊⠈⠀⠀⠀⠀⠀⠁⠁⠁⠁⠁⠁⠀⠁⠁⠁⠁⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠠⠤⠄⠨⠤⠀⠀⠤⠤⠄⠀⠀⠀⢀⢀⢀⠀⢀⠀⠀⡀⡀⡀⡀⡀⡀⠀⡀⡀⡀⡀⡀⠀⠀⠀⠀⠀⠒⠒⠒⠂
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠄⠄⠄⠄⠄⠄⠄⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠉⠁⠉⠀⠉⢃⡀
⠀⠀⠀⠀⠠⠂⠠⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠄⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠇⠀⠀⠀⠀⠐⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⠆
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⢿⣶⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠤⠤⣤⣀⣀⡉⠉⠙⠛⠿⠿⣷⣶⣶⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠀⡟⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⣤⣀⣉⠉⠛⠚⠿⠿⢷⣶⣦⣤⣄⣀
⠀⠀⠀⣀⣠⠤⠖⠛⠉⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀⣈⡉⠉⠛⢿⢷
⣴⣾⣋⣉⠀⠀⢀⣠⠴⠒⢻⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡇⠀⡟⣸
⡇⠀⣀⡉⠉⢻⠉⠀⠀⠀⡿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⢃⡇
⣇⠀⡏⢹⠀⢸⠀⠀⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡾⢸⠁
⣿⠀⣿⢸⠀⢸⠀⠀⠀⢸⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⡟
⢿⠀⣿⢸⠀⢸⡆⠀⠀⣾⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⢸⠇
⢸⠀⢸⢸⡄⢸⡇⠀⢀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡇⡾
⢸⡄⠘⠾⠇⠸⡇⠀⢸⠃⠈⠙⠓⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣼⣸⠇
⠘⡇⠰⣿⡇⠀⡇⠀⠈⠙⣒⡶⢤⣀⡀⠀⠈⠉⠙⠒⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⣏⡿
⠀⠙⠲⠦⢤⣤⣧⠴⠞⠋⠁⠀⠀⠀⢉⣽⠶⠦⣤⣀⡀⠀⠈⠉⠛⠲⠦⣤⣀⣀⠀⠀⢀⡏⠀⣼⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠯⣭⣙⡓⣶⣤⣄⣀⡀⠀⠉⠉⠛⠚⠁⢠⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢹⡟⠓⠶⠤⣤⣀⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⠳⠦⣄⣀⡀⠀⠀⠀⠀⢀⣠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣭⣛⣛⣛⣭⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⡤⠂⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣓⣺⣓⣟⣺⠓⣿⣒⣗⣺⣒⣗⣺⣗⣛⠚⣗⣻⣒⣻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇⣰⠓⣺⠂⣞⠐⡗⠺⠇⢺⠒⠿⠐⡗⢚⡆⢻⡒⢷⠒⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣾⣶⣿⣶⣷⣶⡶⠶⠶⠶⠶⠶⠶⣶⣶⣷⣾⣷⣾⣶⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇




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

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠒⠒⠀⢄
⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠑⠢⠀⢀⣀⡀⠀⠠⠂⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠏⠉⠉⠉⠩⠽⠿⠿⠿⠿⣷⠀⠀⠂⠀⡠⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠦⠂⠀⠀⢀⢂⠄⢎⣡⠀⠀⠀⠀⠀⢰⡠⠤⠔⠀⠀⠀⣿⠀⠤⠄⠊
⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠑⠢⡀⠀⠄⠀⠀⠰⠾⡀⢸⠣⠃⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⣿⠀⠀⡠⠀⠂⠈⠀⠀⠈⠐⠢⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠉⠀⠀⠀⠈⠄⠀⠀⠀⠀⠢⣙⡓⠂⠄⠀⠀⠀⠀⠠⡳⣴⠣⠄⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠂⠀⠀⡁⠀⠀⠉⠉⡡⠀⢄⠀⠀⠁⠈⠉⠉⠁⠉⠉⠍⡉⠉⠁⢸⠀⠂
⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⢀⡀⢎⠀⠀⠀⠀⢐⠒⠐⠄⠀⠀⠀⠀⠀⠀⢈⠊⠀⠓⣤⠀⠀⢸⠀⠈⠂⠀⠀⢀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠆⠈⠀⠀⠀⣠⣄⣤⡀⠀⠀⢰⠀⠀⠈⡳⢄⠴⡓⠀⠀⢸⠀⣠⠊⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⣀⣀⣀⣒⣠⣬⣦⣤⣴⣤⣤⣤⣬⣦⣶⣤⣥⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠤⠠⠄⠀⠄⢠⠈⡗⡺⣾⣽⣧⣗⣛⣗⠒⠚⠛⠛⢭⡷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡤⠬⠤⡅⢭⠸⡅⢯⠿⣿⢹⡿⣤⠀⡤⠕⣀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠤⠤⠤⠤⠤⠍⠉⠁⠨⢭⣁⣉⣉⣩⠭⠽⠬⢤⢬⠭⣬⣥⣭⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⣄⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣁⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣁⣠⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⡿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⢿⡿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢦⣄⠀⠀⠀⠀⠀⠀⣀⣴⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠛⢦⣄⣠⡴⠛⠁⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣇⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






⡿⢿⡿⠿⣿⡿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣟⣛⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⡀
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⣰⡶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⢶⣆
⠀⠀⠀⢻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡟
⠀⠀⠀⠀⢀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀
⠀⠀⠀⠀⢸⡏⣿⠀⠀⠀⠀⢠⡶⢦⡄⠀⣤⢶⡄⠀⣴⠶⠄⠠⢤⡤⠄⠀⠀⣤⠶⣄⠀⣤⠤⠤⢠⡤⠤⠄⣤⠀⣠⡶⡤⢀⡤⠤⠄⠀⠀⠀⠈⣿⢹⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⣇⣸⡇⢸⡇⠀⣿⠀⢷⣄⠀⠀⢸⡇⠀⠀⢠⡏⠀⢻⡀⣿⣀⣀⢸⣇⣀⡀⣿⠀⣿⠀⠀⢸⣧⣀⡀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⡏⠉⠀⢸⡇⠀⣿⠀⠀⠙⣷⠀⢸⡇⠀⠀⠸⡇⠀⣼⠁⣿⠉⠁⢸⡏⠉⠀⣿⠀⣿⠀⠀⢸⡏⠉⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠸⠃⠀⠀⠈⠻⠶⠋⠐⠶⠶⠋⠀⠸⠇⠀⠀⠀⠻⠶⠛⠀⠻⠀⠀⠘⠇⠀⠀⠻⠀⠙⠷⠶⠘⠿⠶⠆⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⢰⡶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠶⢶⡆⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠂⠀⣿⠀⢰⡆⠀⠀⢰⣆⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢶⡀⠀⠀⢰⡆⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⠈⣿⠀⠀⠀⣿⡀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⣇⠀⠀⠘⣧⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⠀⢹⡆⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⠀⢿⡄⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⠀⠘⣧⠀⠀⠈⣿⠀⢸⡇⠀⢸⡇⠀⢠⣤⡤⠀⢸⡇⠀⢤⣤⡄⠀⢸⡇⠀⢸⡇⠀⠀⢹⡇⠀⠀⠸⡇⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⣠⣼⣧⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣼⣧⣄
⠀⠀⠀⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⢶⣄⡀⢀⣤⠾⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡴⠟⠉⠛⠋⠉⠳⣤⡀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⢿⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣼⡷⠶⠶⠶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⡤⠠⡄⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⡷⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⣶⠀⢰⡶⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⡇⠀⠀⠀⢠⡤⠤⠤⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⡇⠀⠀⠀⠸⣧⠤⠤⠤⢤⣾⠇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⡇⠀⠀⠀⠀⣿⠠⠤⠤⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⣇⣀⣀⣀⣀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⠈⠉⠉⠉⠉⠉⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢀⣿⣒⣒⣒⣺⣇
⠀⡀⠀⣀⣀⡀⣸⣟⡛⡛⠛⡛⢛⣛⣛⣛⣛⣛⣛⣛⠛⠛⠛⣛⣀⢛⢂⠛⡛⣻⡏⣉⣉⣉⣉⣹⡇⣀
⠀⠉⠀⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢴⡶⢫⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣭⡍⢩⡍⡉⠉⡿⠉⡍⢩⣷⣿⣿⣿⠿⡟⢸⣿⠟⣥⣯⣭⣭⣿⣿⣿⣿⣿⣿⣶⠸⣧⠻⠴⠛⠚⢻⢀⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⠾⠛⠓⠈⠁⢀⠈⠉⠁⠈⠉⣉⠝⣡⡖⢁⡇⡟⣵⠋⠀⢋⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠓⠀⢸⢸⢿⡵⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⢀⣤⠠⢄⠀⠀⠀⠀⡜⣠⣾⣿⢃⡼⢱⡿⣿⠀⠀⢸⠀⠀⠀⠀⣄⣠⣤⣠⢤⠀⠈⠀⠀⠀⠀⡜⡜⣼⣏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠁⠠⠄⠀⠀⠸⣌⣉⠴⠀⠀⠀⢸⣵⣿⣟⣡⢃⡆⣼⣼⣿⡄⠀⡼⠀⠀⠀⣸⢻⣿⠿⢯⠏⠀⠀⠀⠀⠀⡰⢱⠡⣽⣾⣏⡝
⠀⠀⠀⠀⠀⠀⡠⢣⠎⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡻⣿⡿⢁⢾⡇⣿⣾⡿⠁⡴⠁⠀⠀⣆⠀⢀⠄⣀⡀⣤⣤⣤⣴⡖⣰⢣⢣⡜⣽⣿⡿⠁
⠀⠀⠀⠀⠀⡔⣱⣿⣶⣶⣶⡖⠰⡆⣴⣠⣞⣪⠉⡥⣤⣤⣤⣿⡟⢠⢋⣾⡇⣿⣧⣶⡧⢤⣤⢤⣤⢀⣤⣤⠄⠤⢤⡥⣭⠵⠒⠺⢃⣯⢻⣰⣿⡿⠁
⠀⠀⠀⠀⢾⢤⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⣤⣤⣭⣉⣉⣉⣹⣡⣏⣤⣼⣄⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣭⣭⣭⡴⠶⠒⣒⣛⡿⣱⣷⣿⡿⠁
⠀⠀⠀⠀⢘⡿⣿⡿⢗⣒⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠤⢤⣬⣿⣿⣋⠈⠀⡐⠒⠲⠶⠶⠶⢴⣶⣿⣽⣿⠃
⠀⠀⠀⠀⠸⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣽⢿⣯⠿⠿⠿⠿⠿⠿⠦⠶⠶⠿⢿⣾⣿⣿⣿⣿⣿⢿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣿⠀⠀⠀⠀⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣁⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣤⡀
⠀⠀⠀⠀⠀⢰⣿⣯⣀⢸⡇⠀⠾⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠒⠒⠖⠲⠿⠿⠿⠿⠿⠿⠿⠿⠿⠾⠀⠀⣿⢀⣨⣿⣷
⠀⠀⠀⠀⠀⢸⡇⠈⠙⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠛⠉⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠻⢶⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣴⠾⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣬⣁⠈⠉⠉⠉⢉⣉⣉⠉⠉⠉⠉⢀⣩⣼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⡾⠛⠉⠙⠻⣦⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣇⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⣿
⠀⠀⠀⠀⠀⠸⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈⠀⢸⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠓
⠀⠀⠠⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢀⠀⢨⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣄⠀⢀
⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠖⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠈⠁⠀⠁⠀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁
⣠⣀⣠⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣠⣠⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢺⢩⢣⢣⢃⠇⠟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⢇⢆⠃⡞⠆⠎⡌⡘⡜⡘⡳⣄
⠀⠀⠀⠀⠀⡤⠄⣀⣀⣀⣀⣀⣴⡞⡝⡜⡌⡎⡎⡜⡜⡸⡘⢰⡸⡰⠱⢣⢫⢦⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠓⠓⠳⠧⠷⠧⣧⣧⣧⣧⣇⢛⢦⡀
⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠤⠤⢤⡀
⠀⢀⡠⢶⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢰⠀⡄⠀⡇
⠀⢸⠳⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠇⠀⡇
⠀⡄⠀⠈⠙⢷⣂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⡄
⠀⡇⠀⠀⠀⠀⠉⠻⣥⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣝⣙⣆
⠀⡇⠀⠀⠀⠀⠀⠀⠈⠳⢖⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣿⡼⠟⢻
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⡥⣒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣲⠯⠒⠉⠀⠀⡆⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣚⠭⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⠿⠚⠉⠀⠀⠀⠀⠀⠀⡁⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣒⣤⣀⣤⣀⠄⣰⣶⠽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡇
⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠉⠑⠿⣅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇
⡿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇⢆⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⣒⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡥⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣷⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣂⡀⠀⠀⠀⠀⠀⠀⡆
⠙⠫⠤⣔⣲⣂⠤⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠬⠬⣐⣐⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡠⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⢀⠀⡀⢀⡀⣀⠀⡀⠀⠉⠉⠐⠒⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠄⠀⢸
⠀⠈⠑⠉⠚⠕⠫⠚⠝⠫⠒⠩⠖⠭⠪⠕⢪⠕⡢⢂⢔⡤⣋⢝⡱⢒⢦⡤⣤⣀⣀⡀⠀⠀⠀⠀⠻⣆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠉⠘⠁⠊⠐⠉⠋⠝⠹⠳⠖⠾⠷⡫⠪⠿⠯⠗⡵⡪⢆⢴⠖⠄⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⣰⠿⣧⡀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡟⠁⢀⡼⠋⠀⠈⢷⣄⠀⠹⣿⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠛⠳⣦⡀
⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣠⡟⠁⠀⠀⠀⠀⠹⣦⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠙⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠁⢀⡾⠃⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠈⢻⣄⠀⠹⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠃
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⠁⠀⢸⡟⠳⣤⡀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣿⠀⠀⠙⠷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⠀⠀⠀⢸⡇⠀⠈⠛⣦⣄⠀⢀⣴⡞⠋⠀⠀⣿⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣠⡾⠋⠙⠻⠛⠉⠻⣦⡀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⡞⢉⡀⣀⣀⣀⣀⣀⡀⣈⣛⣦⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣹⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⣧
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⣶⡶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢸⡇⠀⠀⠉⠻⢦⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣀⣀⣈⡀
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⡇⠀⠀⢠⡶⠛⠉⠉⠉⠉⠛⢶⣄
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡏⠀⠀⠀⠀⠀⠀⣼⠃⠹⣦⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣴⠾⠷⠶⠶⠾⢷⡄⠀⠀⠀⠀⣿⠀⠀⢸⡇⠰⣾⡷⠶⠶⠶⠶⠶⠶⠶⠶⢶⣷⠶
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⠀⢀⣤⣤⣤⣤⣤⣤⣀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣠⡿⠀⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⠀⢸⣇⣀⣀⣀⣀⣀⣿⠀⣿
⠀⠀⢸⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠁⠀⣿⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⣿
⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣇⣀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢾⣀

⠀⠀⠀⠀⠀⠀⠀⠀⢐⡋⠀⢸⠈⠳⠀⡆⠘⠋⢀⠺⠀⢰⣋⡶⢰⠀⠀⣏⠁⠀⠰⣎⠀⠈⡛⠀⣊⣑⠄⠞⠈⢢⢰⣠⠂⢸⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣅⣀⣸⣀⡔⢁⡆⢀⡀⣜⠃⣳⣘⣅⡷⣸⣀⣀⣏⣀⡀⢀⣈⡗⠀⣃⢀⣋⢙⡄⣇⣀⡜⢸⡉⢆⣸⣅⣀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣶⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠠⠤⠤⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⠔⠦⠤⢄⣀⡀⠉⠉⠉⠒⠒⠲⠦⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡼⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠢⠤⠤⣀⣀⠀⠉⠉⠙⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠢⡄⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⡠⡀⣠⠄⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡏⣺⠞⡡⢊⠜⣡⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⠁⠡⣪⠞⣡⠞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣜⠀⠚⢡⠞⠡⢚⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣀⣀⡉⠉⠁⠒⠒⠤⠬⢈⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣖⣲⡄⣭⠭⣴⣖⣠⡤⢤⣄⣀⣈⠉⠉⠁⠒⠒⠤⠤⣀⣀⡀⠀⢀⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣰⢢⡤⢤⠌⣉⡑⠒⠣⠤⠯⢁⣖⡒⣬⠭⣵⢒⣒⠤⠤⠤⠤⣈⣉⡉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣠⠞⢤⠯⣡⣟⢒⡾⢬⠏⣁⣒⢒⡆⠤⡤⢉⣉⠑⠒⠓⠨⠯⠵⢒⣖⣢⠈⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣴⠭⣼⣃⣲⠇⢤⠯⢉⣟⡓⣲⠃⢭⠭⢸⣓⢲⡧⣯⢹⣱⡀⢰⠒⢦⠤⠤⢄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣦⣤⣌⣉⣉⡐⠛⠋⠿⠅⣾⣒⣨⠯⢍⣟⣐⣺⠢⢼⠍⣿⡙⣖⣧⠈⣏⣙⣇⣻⡒⢇⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠮⠶⣭⣽⣛⢟⠗⣲⡶⡤⣬⣍⣉⣑⠒⠻⠭⠿⣐⣺⡠⢯⢽⡄⢳⡀⠴⡤⢯⣹⡄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠾⢯⣮⣎⢝⢝⢟⣱⣶⢶⢤⢤⣭⣁⣃⠈⠧⠌⠿⣙⣖⣳⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠷⠷⠥⢯⣾⣯⣎⢜⣝⣷⣶⠶⣶⣦⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⠛⠓⠶⠶⣤⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡿⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⣾⠋⢰⡾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⠾⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⣿⠀⢸⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠉⣙⣛⣛⡓⠒⠒⠒⠒⠒⠃⢀⡏⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⣿⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⠃⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡟⠀⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⠒⠒⠚⠛⠛⠀⠀⢀⡟⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢀⡟⠀⣾
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢿⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⢸⣇⡴⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⣸⠃⠀⢀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠩⠉⠉⢩⠉⠉⠉⠛⠛⠛⠛⠛⠀⠀⡿⠀⣠⡿⠉
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣀⡴⠋⠈⠉⠉⠉⠛⡟⢛⡿⠛⠓⣷⡒⠶⠶⠶⠶⢶⣦⢤⣤⣤⣤⣤⣴⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⣠⠖⠚⠁⢾⣇⡀⠀⠈⠙⠛⠓⠒⠒⠛⠁⠀⣠⡿⠀⠈⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣧⣀⣀⣀⠈⠻⠯⣿⣶⣶⣶⣶⣶⣒⣒⣾⡽⠟⠁⠀⠀⠀⠀⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡤⠤⡤⢤⡤⠤⣤⢤⣤⣄⣈⣉⣉⣉⡉⠙⠛⠛⠛⠷⠾⠯⢭⣭⣥⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠋⡿⠓⡾⠓⣲⠓⡶⠧⢤⠧⢤⡯⣤⣽⣉⡟⢀⡖⢲⠖⢦⠤⢤⡀⠀⢩⠟⣇⠀⠀⠀⠀⣠⡴⢶⢞⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⣉⡽⠉⣹⠛⢲⠟⢲⠓⢲⠿⢤⡾⠤⣼⢥⣼⠁⡜⢉⡟⢒⠟⠒⡾⠁⡰⠋⣴⠏⠀⠀⣠⣞⣁⣠⣧⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⠤⠬⣉⣉⣉⣙⣛⣒⠛⠒⠛⠲⠾⠤⠴⠧⠴⠃⢸⣉⣽⣉⡟⠓⡾⢀⡜⣡⠞⠁⠀⠀⣰⠋⠀⠀⠀⠈⠉⠲⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠓⠒⠶⠶⠶⠯⠭⣭⣭⣉⣓⣒⣒⣒⠒⠒⠦⠤⠤⢭⣭⣉⣠⢋⡼⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠲⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣴⣲⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⣾⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣧
⠀⠀⠈⢉⣽⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣯⡉⠁
⠀⠀⠀⢹⢹⠀⠀⠀⡖⢲⡄⡴⢲⡀⡖⠂⠐⣶⠂⠀⣰⠲⡄⣴⠒⢠⡖⠂⣦⢠⠖⠂⡶⠒⠀⠀⠀⡏⡟
⠀⠀⠀⣼⢸⠀⠀⠀⡗⠚⠁⡇⢸⡇⠙⢦⠀⣿⠀⠀⣿⠀⡇⣿⠒⢸⡗⠂⣿⢸⠀⠀⡿⠒⠀⠀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⠀⠀⠃⠀⠀⠓⠚⠀⠒⠚⠀⠛⠀⠀⠙⠚⠃⠛⠀⠘⠃⠀⠛⠘⠓⠂⠓⠒⠀⠀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢀⣤⣤⣤⣤⣤⣤⣤⣄⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⣦⠀⠀⣆⠀⢸⡇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⢸⡇⢰⡄⠀⢰⡀⠀⣿⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⢹⡀⠀⢹⡀⢸⡇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⢸⡇⠈⣇⠀⠘⡇⠀⣿⠀⡇⣿
⠀⠀⠀⣼⢸⠀⣿⠀⠘⡇⠀⠘⡇⢸⡇⠀⡇⠀⣤⡄⢸⡇⢠⣤⠀⢸⠀⢸⡇⠀⢻⠀⠀⢻⠀⣿⠀⡇⣏
⠀⠀⠀⣿⢸⠀⠻⠶⠶⠶⠶⠶⠶⠾⠇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠸⠷⠶⠶⠶⠶⠶⠶⠿⠀⡇⣯
⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⣤⣿⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣼⣧⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣿⣤
⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⢀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣾⡷⡾⠷⠾⠷⠿⠷⠿⢶⠿⡶⠿⠶⠿⠾⢷⠾⣷⣾⡀⠀⠀⠀⢨⣿⠿⠿⠿⠿⠿⠿⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⢴⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠴⣿⠀⠀⠀⠀⠀⠀⠠⢸⣷⠄
⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠰⣿⠀⠀⠀⠠⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⠂⠀⠀⠀⠸⣿⣤⣤⣤⣦⣤⣤⣤⣼⡷⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠈⠋⠙⠁⠙⠁⠙⠁⠙⠁⠉⠉⠋⠈⠋⠈⠋⠈⠋⠉⠂⠀⠀⠀⠚⠁⠘⠁⠉⠁⠙⠁⠉⠀⠙⠂
⠀⠀⠀⠀⠀⠀⢠⣄⠀⣠⠀⣠⡀⢀⣄⠀⣄⠀⣠⡀⢀⣄⠀⣄⠀⣠⡀⢀⡄⠀⣄⠀⣠⡀⢠⡄⢀⣄⠀⣠⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠐⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠠⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠄⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠉⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢺⣿⣤
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⣼⣿⣠
⠀⠀⠀⠀⠀⠀⠈⣹⣿⣴⣶⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣴⣤⣤⣤⣴⣦⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠛⠥⠙⠁⠈⠋⠀⠛⠀⠙⠁⠘⠋⠈⠛⠀⠙⢋⠉⠋⠈⠋⠁⠙⢁⠘⠃⠈⠋⠀⠙⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠓⠒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠤⠀⣀⡀⠃⠀⠆⠀⢀⣀⠈⠉⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡘⠢⠤⠀⠀⠉⠑⠒⠤⢄⡉⠂⠍⠐⢤⡀⠀⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢰⠁⠀⠀⠀⠀⠀⠐⠢⠤⣀⡀⠉⠒⠠⢄⣀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣀⠉⠒⠢⢤⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⣈⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠲⡇⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢠⠇⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣡⠔⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣸⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢰⠏⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⣿⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⢀⡀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠁⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⠉⣉⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡇⠀⠈⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠘⢄⡤⠃⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢳⣤⣄⣠⡔⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢠⠑⠂⠤⡀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡆⠀⠀⡹⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠹⠤⠔⢁⡀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢙⣿⠶⠒⢊⡡⠤⠚⠁
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠼⠟⠴⢾⡁
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠙⠀⠠⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢀⡀⡀⠀⠀⠀⠀⠀⣤⠀⠘⣇⡠⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣄⠀⢘⡷⢦⣤⣤⡤⠀⠀⠀⣠⠇
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡼⠁⠀⠀⣀⡤⠞⠁
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⣀⣀⡀⢀⣀⣠⠞⠁⣠⠴⠋⠁
⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠴⠚⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠒⠒⠒⠚⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⢠⠈⢶⠒⠒
⠀⠀⠀⡟⢍⠉⠉⠉⠉⢉⠝⡇⠀⠀⠀⠀⠀⢸⠣⡀⠀⠀⠀⠀⡠⢻⠀⠀⠀⠀⠀⠀⡇⠋⠙⠘⠉⠋⡏⡙⡃⠀⠀⠀⠀⡾⠍⣀⠀⠀⢀⠦⣀⡈⣇⠁
⠀⠀⠀⡇⠀⡱⢄⣀⠴⡁⠀⡇⠀⠀⠀⠀⠀⢸⠀⢈⠖⠈⠐⢎⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠒⠃⡇⠀⠀⠀⠀⠘⡄⠀⡝⠘⠉⣀⡠⠔⠛⠐
⠀⠀⠀⣧⣊⣀⣀⣀⣀⣈⣢⡇⠀⠀⠀⠀⠀⢸⣔⣁⣀⣀⣀⣀⣑⣼⠀⠀⠀⠀⠀⠀⣥⣀⣄⣄⣠⣠⣀⣄⡇⠀⠀⠀⠀⠀⠸⣴⣤⣖⣋⣉⣀⣀⣀⣀


⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⢘⡆⠟⠒⠓⠚⠇⣞⠀⠀⠀⠀⡴⠋⣀⠤⢄⡈⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠲⠖⠒⠲⣶⠶⡄⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⢨⠆⠄⠀⠀⠀⡅⢮⠀⠀⠀⢸⠀⡌⠀⠀⠀⠘⡀⢸⠆⠬⢉⣈⣀⡩⠀⠀⠀⣇⣀⣸⣃⣆⣀⣈⣣⡇⠀⠀⠀⠀⢰⠋⠉⢉⠉⠉⠉⡹⢛⡆⢸
⠀⠀⠀⢨⡃⠀⠀⠀⠀⠄⣫⠀⠀⠀⠸⡀⠡⡀⠀⠀⡰⠀⣸⣓⠒⠒⠀⠀⠒⠀⠀⠀⡇⠈⠚⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠉⠈⡇⢸
⠀⠀⠀⢰⣃⣷⣒⣖⣢⣃⣳⠀⠀⠀⠀⠙⢦⣀⣉⣁⣠⠾⠁⠤⠄⣈⣐⣂⡉⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠸⣀⣀⣈⣀⣀⣀⣀⣀⡗⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢈⠍⢹
⠀⠀⠀⡞⠥⢴⡷⠶⠶⠠⠝⡆⠀⠀⠀⠀⠀⠀⠊⢸⠘⠀⠚⢹⠈⠂⠀⠀⠀⠀⠀⠀⡏⣀⡀⢀⣀⠀⢀⡈⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠁⠀⡸
⠀⠀⠀⡇⠀⠘⠂⠤⠖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⡏⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠋⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀
⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⡇⣀⣀⣀⡀⠈⠀⠀⡇⠀⠀⠀⠀⠀⡰⠋⢉⠉⠉⠉⠉⠉⡉⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⡇⣅⣀⣀⡸⠀⠀⠀⡇⠀⠀⠀⠀⣰⢡⠁⢠⠀⢀⡄⠂⢉⠙⡆⠘⡇⠀⠀⠀⡞⠉⠹⠉⣉⣉⣉⡙⡆⠀⠀⠀⠀⢀⣤⠾⣭⣽⠉⡏⠉⠉⠉⢢
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⢣⣀⠤⠖⠉⠀⡇⠀⠀⠀⡇⠀⢀⠀⠀⠀⠈⠃⡇⠀⠀⠀⠀⢸⠉⣢⣀⣰⡄⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⡀⡀⣀⡀⢀⡇⠀⠀⠸⣄⡴⣿⢦⡤⠤⠤⠤⡤⣤⣮⢷⣄⡽⠀⠀⠀⠧⠤⠤⣤⢤⡤⠤⠤⠇⠀⠀⠀⠀⠘⠾⠤⠤⠴⣤⠤⡤⠤⠤⠼
⠀⠀⠀⠉⠈⠁⡿⠨⡇⠉⠁⠁⠀⠀⠀⠀⠙⠓⠋⠀⠀⠀⠀⠀⠈⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠇⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇

⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⠄⠢⡄⠀⠀⠠⣀⡤⠒⠂⢄⣠⠄⠀⠀⣠⠄⠐⠦⣄⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⡀
⠀⢸⠁⠈⠹⣷⡀⠀⠀⠀⠀⠀⣿⢳⣤⠇⠀⠀⣰⡏⠀⠀⠀⠈⠉⠀⠀⠀⣿⠀⠀⠀⠘⣷⠀⠀⢸⣇⡀⢀⡀⠀⠀⠈⣿⠀⠀⠉
⠀⠘⣗⡶⠀⣿⣷⠀⠀⠀⠀⠀⣿⡼⠋⣷⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡿⠀⠀⢸⡏⠀⠘⠁⠀⠀⠀⣿⠖⢦⠄
⠀⠀⠉⠀⢸⠏⣿⣇⠀⠀⠀⠀⠿⠀⠀⠛⠀⠀⠈⠻⢦⣀⣀⠤⠂⠄⠀⠀⠾⠆⠀⠴⠛⠁⠀⠀⠸⠇⠀⠠⠄⠀⠀⠠⠿
⠀⠀⠀⢀⡏⠀⠸⣿⡀
⠀⠀⠀⣼⠥⠤⠤⢽⣷⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⡏⠀⠀⠀⠏⠀⠀⠀⠀⠀⣸⠀⠀⢸⡇⠀⠀⠉⠀⠀⠘⡏
⠀⠀⢠⠇⠀⠀⠀⠀⢿⣇⠀⠀⠸⣇⠀⠀⢲⠀⠀⢀⡧⢠⠶⠼⡇⠀⠀⠀⡞⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠐⢆⠀⠀⠀⠀⡅⠀⠐
⠀⢠⣿⠀⠀⠀⠀⠀⠸⠿⣧⠀⠀⠈⣱⡆⠸⠀⠀⢸⣧⠀⠀⠀⢇⠀⠀⠀⢠⡀⠀⠀⣦⠐⠁⠀⠀⢸⡇⠀⠘⠦⠀⠀⠀⠃⠀⠬⠃
⠀⠚⠚⠀⠀⠀⠀⠀⠈⠛⠉⠳
⠀⠀⡀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠀⠀⢠⡦⠀⢠⢤⡆⠀⠀⠀⠀⣀⠄⠀⠐⠦⣄⠀⠀⠀⠰⣶⠖⠒⢦⡀⠀⠀⣠⠖⠀⠉⠑⠢⣀
⠀⠠⠗⠀⠀⠀⠀⠰⣿⡄⠀⠀⠀⠀⢈⣿⡄⠀⢸⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⣠⣀⣸⠇⠀⣼⡏⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⢸⣯⠻⣄⢸⡇⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣿⠃⠉⠁⠀⠀⢹⣇⠀⠀⠀⠀⠀⣼⠇
⠀⢀⣀⣇⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⢠⣿⠀⠈⠿⠇⠀⠀⠀⠈⠓⢄⣀⠀⠠⠘⠁⠀⠀⠰⠿⠄⠀⠸⠁⠀⠀⠈⠂⠀⠀⠐⠺⠥

⠀⠘⣿⡏⠉⠹⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⣶⠶⠒⠀⠀⠀⠀⠐⠉⡲⠂⠀⠀⠈⣿⠀⠀⠈⢿⡄⠀⠀⠀⢸⠋
⠀⠀⣿⡇⠀⢀⣷⠇⠀⠀⠀⠀⠀⠀⢸⣂⣀⠀⠀⠀⠀⠀⣴⡟⠀⢠⣦⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⡀⠀⢠⠃
⠀⠀⣿⡇⠚⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⣦⠀⠀⠀⠰⣿⡀⠀⢿⠻⠐⠄⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠹⣧⠠⠎
⠀⠀⡿⡇⠀⠀⠣⣦⣀⢰⡀⠀⠀⣦⡀⠀⢀⡟⠀⠀⠀⠀⠛⢧⣀⠀⡀⡠⠆⠀⠀⠘⢿⣦⣀⣀⡀⢀⣿⡀⠀⠀⠀⠀⢻⡇
⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⡴⣤⡤⡤⡴⠦⣄⣤⠄⠀⢶⢀⡆⠀⠀⢶⣲⡆⠀⠀⠀⣴⠶⠀⠀⠀⠀⡒⢦⣄⠀⠀⠀⠰⡆⡄⠀⠀⠦⠀⠐⠶⠤⠤⢤⣲⠆
⠀⠀⠁⠀⡶⠀⠀⠀⣰⠃⠀⠀⠸⣦⡇⠀⠀⠀⠹⣿⣄⠀⣠⠄⠀⠀⠀⠀⣰⠇⠸⣿⡆⠀⠀⠀⣿⡇⠀⠀⠀⠐⠂⠀⠀⣰⣿⠃
⠀⠀⠀⣾⡇⠀⠀⢰⡏⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⢟⣷⡃⠀⠀⠀⠀⠀⠀⠀⠀⣯⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡰⡿⠃
⠀⠀⠘⣿⡆⠀⠀⢿⣧⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⡼⠣⣿⣄⠀⠀⠀⠀⠀⠀⠀⠻⣷⣀⣀⡴⣿⡇⠀⠀⠀⠀⠀⣰⡼⠁
⠀⠀⠀⠹⢵⣄⣀⣈⡿⣦⣀⣀⣠⣟⡇⠀⠠⢴⠄⣘⡁⠀⠙⣿⢦⡀⠀⠀⣤⣄⠀⠀⠀⠉⠉⠀⡿⡇⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠁⠉⠈⠁⠈⠀⠀⠀⠚⠓⠁⠀⠀⠘⠛⠛⠀⠀⠉⠋⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠙⠚⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⢀⠀⠤⠀⠀⠀⠀⠀⠀⠁⠀⠀⡐⠢⡀⠀⠀⠀⠀⠉⠁⠈⠁⠁⠁⠈
⠀⠀⠀⢀⠀⠀⠀⠄⠀⠐⠀⠰⠁⠁⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⣀⣀⡀⠠⠤⠔⠀⠀⠰⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢰⡒⠒⠒⠒⠒⠒⣲⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡇⠀⠀⠀⠀⠀⣿⠀⠀⠐⠀⠀⠀⠛⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠀⠀⣄⠀⠀⠀⠀⢰⠆
⢺⠃⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⡀⠀⠀⠀⠀⠀⠀⠁
⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠠⡀⠀⢠⡄⠀⠀⠀⠀⠀⢠⠀⣠⡒⠒⠦⠄⠈⢂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⠉⠀⠐⠒⠒⠒⢲⠀⠀⠐⣄⠐⠦⠔⠲⠊⠀⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠈⠐⠀⠀⠀⠒⠀⠀⠀⠀⣟⣟⣿⣿⣿⣿⣿⣟⣟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠈⠢⠀⠀⠀⢀⠠⠊
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠃⣰⠏⢳⡀⠻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⡼⠁⠀⠀⠹⣆⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠏⠀⠀⠀⠀⠀⠈⢧⡈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⣀⣀⣀⣀⣀⣀⣀⣀⡞⢁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠙⣆⣀⣀⣀⣀⣀⣀⣀⣀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⠉⠻⣤⠞⠁⣾⡻⢿⣿⣿⣿⣿⡿⣛⡆⠙⢦⣼⠏⠉⠉⠉⠉⠉⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡖⠒⠒⠒⠒⠒⡆⠀⠀⣿⣿⣦⠙⠿⠟⠡⣾⣿⡇⠀⠀⢰⠖⠒⠒⠒⠒⠲⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⣿⢋⣴⣿⣷⣿⣷⣌⡻⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡟⠛⠛⠛⠛⠛⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⠛⠛⠛⠛⠛⠛⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⢀⡴⠞⠋⢹⠋⠛⠲⣤⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⣰⠏⠀⠀⠀⠘⠀⠀⠀⠈⢳⡄⠀⢸⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⢀⡞⢧⡀⠀⠀⠀⣷⠀⢸⠀⢠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢰⠟⠒⠚⢳⡆⠀⠀⣿⠀⢸⠀⣿⠛⠛⠛⠛⠛⠛⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣇⣀⣀⣀⣀⣀⡇⠀⡇⠀⠀⠘⠓⠒⠒⠚⠃⠀⠀⣿⠀⢸⠀⣿⢰⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠁⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠛⠒⠒⠂⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣤⣤⣤⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠲⡄⠀⢸⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠾⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣒⣀⣒⣀⣀⣀⣠⣏⣀⣤⣤⡤⠴⠶⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⣒⣖⣒⣶⣒⣒⣖⡒⣲⠖⠒⡶⠖⢤⠤⠭⢭⠭⢭⣍⠉⠉⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡾⢁⠾⢤⠾⢤⡾⠥⣤⠯⢬⡽⠭⢭⠿⠭⣽⢯⣉⣽⣋⡲⣾⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢠⠟⡠⠛⢲⠛⢲⠞⠒⡴⠃⢢⠞⠤⡴⠧⢤⠾⠥⣤⠮⣤⢞⡗⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⠟⠐⠿⠿⠿⠿⠿⠿⠿⠷⠾⠟⠖⠾⢷⣶⣿⣶⣺⣗⣲⡽⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⡿⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉
⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣿⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠉⠀⠀⠀⠀⠀⠠⠐⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⡀⠀⠄⠀⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡴⠡⠂⠀⡀⣤⠀⠁⠰⡄⡈⠙⠂⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠶⠁⢀⠊⠄⣷⣢⣄⡀⠘⢿⠀⣖⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⢀⡇⠀⠊⠁⣀⠤⠈⠛⢿⣶⡌⢩⢋⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠨⡄⢿⠁⠠⠘⠁⠀⠀⠠⠒⠚⠻⣆⡌⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡜⢄⣀⠀⠀⠀⠀⢠⠖⠁⠀⡾⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⡐⠂⠀⡰⠀⢠⡟⠀⢀⠳⡄⠃⠈⠁⠺⠁⠀⠀⢐⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣧⠃⠀⠀⢔⠂⠀⠸⠀⠀⠄⠀⠙⢆⡀⢀⠃⠀⠀⠀⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠂⠀⠋⠀⠀⠀⠀⠐⠀⠂⠀⠈⢿⡏⠀⠀⠀⠀⢘⢸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⣠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡼⠀⠀⡔⠀⠀⠀⠸⢂⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢨⠠⡰⠊⡌⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠀⢰⠀⠀⠀⠀⡃⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢼⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⡇⠀⡆⠀⠀⠀⡸⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡀⠀⡉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⢱⠀⡅⠀⠀⢰⠁⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁⢰⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⢄⠡⡀⠆⢁⠀⠀⣇⠎⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠠⣀⢀⢣⡎⢀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠛⠒⠳⠄⢬⣼⡀⢸⡿⠀⣠⠴⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠐⠌⠐⡄⣀⣄⡀⠀⠠⠀⠀⠀⠠⠚⠉⠁⠀⣢⣤⣈⣙⣷⣌⣴⢸⠁⠶⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⣀⡤⠖⠉⠀⢀⡀⠠⠼⠏⣿⣬⡂⠀⠀⣀⠌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⢐⡓⠂⠀⠀⠐⠒⠀⠐⠒⠚⠣⠤⠤⠀⠀⠀⠁⢀⡐⢀⠞⠇⡿⡿⢦⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠁⠀⠀⡏⠀⠀⠍⡉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠠⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢘⣶⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⡐⠆⠨⠐⠀⠀⠙⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠈⠓⢦⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⡀⠀⠀⠀⠀⠀⠈⢓⡴⠋⠙⢯⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠔⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠙⢦⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⢙⡆
⠀⠀⠀⠀⣠⠞⠁⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣠⠾⡀
⠀⠀⠀⣼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⡜⠃⠀⠙⣆
⠀⠀⠀⡇⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡴⠋⢀⣠⠔⠋⢹
⠀⠀⢠⡇⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⣠⢎⡠⠖⠉⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠈⠓⢦⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⡼⠗⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡥⠚⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡤⠖⠒⠤⣠⡴⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠠⡀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢹⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⣸
⠀⠀⠀⠸⣤⣚⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣙⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣹⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⣏⡀
⠀⠀⠀⠀⠀⢀⡼⢋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⢦⡀
⠀⠀⠀⢀⡴⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢦⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⣠⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⢦⣄
⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠁⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣶⡄⠘
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠙⠳⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠟⠋
⣿⣿⣿⣿⠿⣿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣀⡀
⠀⠀⠈⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⡀
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠢⠐⢆⠀⢀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣴⣦⣤⣤⡄⢀⣀⣀⣀⠈⠲⣶⣶⠀⣼⣷⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡿⢿⣿⣿⣿⠿⢇⢪⠴⠤⠷⢤⡀⣹⠉⠀⣹⠛⠻⡟⠋⡡⠀⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠟⠻⢄⢀⣴⣿⢀⣴⠿⠀⠀⠀⠀⠀⠁⠀⠙⠣⠀⠑⠄⠁⠼⠃⠀⠙⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣥⣤⣤⣶⡿⡛⢡⣾⠁⠀⠀⣶⣶⣿⣿⡀⠀⠀⠘⣦⢀⡀⠀⠊⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⠂⢤⣶⣷⣿⣿⣿⣿⣣⠿⠛⠛⠂⠀⠀⠛⠛⢿⣿⡷⠤⠀⠐⠁⠸⠧⡠⠴⠋⠉⠀⣄⠀⢤⠆
⠀⠀⠀⠀⠀⢰⠌⠒⣼⣿⣿⣛⣍⢳⣰⠁⠀⠐⠟⢻⡄⠀⠀⠀⠈⠋⠀⠀⢀⡖⠀⣀⠀⠦⢆⡈⠐⡿⠉⢀⢀⣀
⠀⠀⠀⠀⠀⠠⠶⠸⣿⣿⣿⣿⡟⢈⣿⡀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⣤⠟⣏⠰⣿⣿⡇⢰⣄⣥⠀⠀⠰⠚⠃⠁
⠀⠀⠀⠀⠀⠠⣤⣤⣘⢯⠟⠋⡀⢼⣟⢿⣟⢿⡋⠀⠀⣰⡄⠀⠀⠀⠈⣷⣿⠆⣈⣇⡴⡎⢶⣴⡄⠠⡦⠘⠓⢶
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣯⠕⢤⣿⣾⢿⣌⡟⣬⠁⠀⢾⣿⣿⡀⠀⠀⠀⠘⡄⠑⣾⣿⡿⣷⢠⠌⠃⠷⠞⠀⡘⠃
⠀⠀⠀⠀⠀⠀⠈⢹⣦⡘⠶⡆⠻⣯⣿⠟⢻⠆⡀⠀⠀⠟⠛⠃⠀⠀⠀⠀⠐⠀⠘⠏⡄⢻⣶⠀⣤⡸⠃⠘⣫⡄
⠀⠀⠀⠀⠀⠀⠀⣈⠻⡁⠐⣪⡿⡞⣿⣟⠟⠀⠛⠂⠒⠃⠀⠀⠀⠀⠀⠀⠀⢸⠇⣼⠁⠝⢶⠀⣀⢿⠒⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢹⠷⠿⢙⡿⢞⠚⡿⠌⢠⣿⣀⣴⡶⠖⢢⣿⣿⡄⠀⠀⠀⠈⠢⣧⠔⠂⠐⡲⣿⣿⣷⣤⣆
⠀⠀⠀⠀⢀⠀⣴⣤⣤⣴⢿⡇⢻⡷⠖⠀⣞⠛⡯⣋⡁⠰⣾⣿⣿⣷⡄⠀⠀⠀⠘⢿⡁⡄⢀⡐⣻⣷⣿⡿⠇
⠀⠀⢀⡴⠋⠛⡆⢈⣿⣥⢶⠾⣟⡙⠀⠸⣿⢶⠴⣿⠂⢀⢘⡉⣿⣿⣿⡄⠀⠀⠀⠈⠁⠦⠍⣁⠹⡿⠋⣤⣶⠒
⠀⠀⣎⠀⠀⣠⣿⣿⣿⣟⣻⠤⠿⠃⠀⠁⡭⠘⢆⠹⠃⣈⣉⠉⢳⣾⣗⣶⡄⠀⠀⠀⠰⣶⠄⡝⠰⢷⠀⠿⣇
⠀⠀⣼⠂⠀⣿⣿⣿⣿⡟⠣⣱⠶⣶⠚⣴⡆⠙⢌⣢⠀⢩⡌⠰⣰⢿⣶⠾⢛⢄⠀⠀⠀⠈⠠⣧⡁⣈⠀⢘⡿⠷⠄
⠀⠀⢹⡆⠀⠈⠿⢿⡋⢀⣠⣤⡀⠘⡇⠬⠶⠋⠀⢸⢡⡸⠇⠼⠷⠭⢿⢎⡐⠋⠁⠀⠀⣀⡀⠀⢀⡁⣤⠞⠁
⠀⠀⠀⠹⠦⠖⣀⣿⡀⠘⣿⣿⠋⢀⡾⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠓⢀⣀⣀⣱⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠒⠒⠒⠒⠂⠈⡆⠉⠐⠒⠂⣶⢂⠐⠲⢆⠀⠀⠀⠀⠰⠅⠤⢉⠉⠉⠀⠀⠒⠒⠂⠀⠀⠠⠤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠤⠄⠀⠘⠂⠀⠀⠀⠀⡝⠀⠹⢤⣀⡀⠀⠀⠈⠉⠒⠸⠧⣄⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⢺⢮
⠀⠀⠀⠀⠨⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡁⠀⠉⠉⠉⠉⠛⠒⠒⠒⠓⠒⠀⠠⠠⠤⠄⠁⢉⣀⣀⠀⠀⠀⠀⠐⠧⠤⡼⠑
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⢀⡶⠤⠀⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡤⠤⠤⠀⠒⠒⠒⠀⠉⠈⠉⠉⠉⠄⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠠⠤⠀⠐⠂⠈⣈⣀⢄⠉⢄⠀⢀⠔⠁⢸⠉⠁
⠀⠀⠀⠀⠘⠀⠀⣠⠀⠄⠰⡌⠢⠀⣀⣀⡠⠤⠤⡤⣤⡤⠤⠴⠶⠀⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠙⠻⡈⠩⡆⠸⡾⠁⠀⠀⡎⠑⠢⢴
⠀⠀⠀⠀⠀⠀⢰⠂⣤⣔⠂⠧⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣼⠿⣶⢕⠀⡇⠀⠀⠀⣇⣠⠔⢘
⠀⠀⠀⠀⠀⠄⠘⣀⡁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠉⠉⠁⠀⣧⠀⠀⢸⠁⠀⠠⠃
⠀⠀⠀⠀⠀⢠⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣞⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠁⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣤⠰⠔⠶⠒⠉⠉⠉⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡂⠀⣠⣂⢂⣀⠠⠤⠀⣠⡠⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠁⠉⠍⡂⠒⡩⠍⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠀⠀⠁⠒⢂⣃⠉⠤⠄⠐⠐⠐⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠀⠐⠂⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠄⠐⠒⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠱⠂⠐⠒⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠤⠀⠄⠀⠄⠠⠀⠀⠂⠐⠨⠀⠠⠀⠀⠀⠀⠨⠐⠉⠂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠉⠉⠁⠀⠒⠂⠤⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠶⠤⠤⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠍⡀⠆⡀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⡕⠢⢀⠐⠀⠑⠢⢝⡠⠤⢄⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢈⠐⠠⢀⡠⢄⡁⠢⡍⠃⠈⣁⡠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⠁
⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠔⠁⠄⡡⠄⣉⠐⢠⠈⠒⠀⢁⣠⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠏
⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢉⠀⢈⡑⠂⡀⠐⠀⡈⠠⠠⠐⠈⠠⠀⠈⠤⠕⠀⠀⠀⠀⠀⠀⣀⠴
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⠂⠠⠂⠂⠀⠀⠒⠀⠀⠒⠨⠀⠒⠀⠄⠂⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠤⣈⠁⠄⠀⠉⠤⠒⠁⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⣀⡄⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣠⠐⠊⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠛⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣋⣀⣀⣀⣀⣀⣈⣙⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢁⣀⣀⢈⡉⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡋⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣼⠏⠉⠉⢿⡇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⢙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⣦⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢿⣄⠀⢀⣼⣇⡀⣸⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⡿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠉⠛⠋⠁⢉⣉⣍⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⠉⠛⠛⠒⠒⠛⠛⠉⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠛⢿⣦⣀⠀⠀⣀⣴⡿⠛⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣾⢛⣁⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢈⡛⣷⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛






⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣼⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡄
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢀⣀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣆⣀
⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠙⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛
⠀⠀⣶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⡆
⠀⠀⣿⢸⡇⠀⠀⠀⠀⣿⠛⢷⡄⢠⡾⠻⣦⠀⣾⠛⠓⠀⠛⣿⠓⠀⠀⢠⡾⠻⣦⠀⣾⠛⠛⠀⣿⠛⠓⢰⡆⢀⡾⠛⠂⢰⡟⠛⠂⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⣿⣤⣼⠇⢸⡇⠀⣿⠀⠻⣦⡀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⣿⣤⣤⠀⣿⣤⡄⢸⡇⢸⡇⠀⠀⢸⣧⣤⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⠀⠈⣻⡆⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠟⠀⠀⠀⠈⠻⠾⠃⠀⠳⠾⠛⠀⠀⠛⠀⠀⠀⠈⠻⠾⠋⠀⠻⠀⠀⠀⠛⠀⠀⠘⠃⠀⠛⠿⠃⠘⠷⠶⠆⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡏
⠀⠀⣿⢸⡇⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣦⠀⠀⣶⠶⠶⠶⠶⠶⠶⣶⡶⠶⠶⠶⠶⠶⢶⡆⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡂⠀⣶⠀⠀⠀⢰⡆⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠰⣦⠀⠀⠀⣶⡄⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡇⠀⢹⡇⠀⠀⠘⣿⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⡀⠀⠀⢸⣇⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡇⠀⠘⣿⠀⠀⠀⢻⡆⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡃⠀⠀⢿⡆⠀⠀⠸⣷⠀⢸⣿⠀⠀⣿⠀⠀⣀⣀⡀⠀⣿⡇⠀⣀⣀⡀⠀⢸⡇⠀⠀⣿⠀⠀⠈⣿⠀⠀⠀⢹⡇⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⣁⣀⣀⣈⣁⣀⣀⣀⣉⣀⣸⣿⠀⠀⣿⠀⠀⠉⠉⠁⠀⢻⠃⠀⠉⠉⠁⠀⢸⡇⠀⠀⣿⠀⠀⠀⠉⣀⣀⣀⣈⣁⣀⣸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠁⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⣶⠾⠿⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠾⠿⢶
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⠉⠒⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠢⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⠀⠀⠀⠀⠀⠀⡈⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠢⢝⡒⠩⣓⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⡕⠠⢝⡰⠤⣈⡲⢁⡓⠨⢹⠒⠤⠀⠀⠀⠀⠈⠀⠀⠀⠈⠑⠂⠤⣀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠬⠘⠠⢍⡓⢼⣑⠢⠭⡒⠮⢕⢒⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠑⠢⠄⢰⠁
⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢈⠑⠢⣉⠃⢦⣁⠀⠌⡐⠠⢍⡐⠠⣀⠀⠀⠀⠀⠈⠀⠀⢠⠃
⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠍⠑⠀⠉⠃⠀⣁⠐⠌⠁⠢⢁⠐⠢⢄⡀⠀⠠⠁
⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠉⠐⠄⣀⠂⢌⠁⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠂⠢⠹⠀⠄⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⢀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉
⠀⠀⠀⠀⢠⣄⠀⠀⠀⢀⣴⠾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠶⣦⣄
⠀⠀⠀⠀⠀⠙⠁⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣴⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣦⣤⣤⣤⡄⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠃⠀⠀⠀⠀⠀⣼⡏⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠙⠗⠀⠀⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣶⣦⣤⣴⡶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣴⡶⠶⠶⠶⠶⠶⢶⣄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣶⣶⣶⣶⣶⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠾⠶⠶⠶⠶⠿⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠉⠉⣉⣉⢉⣉⡉⣉⡉⣉⣉⠉⢉⣉⠉⣉⡉⣉⡉⡉⢉⣉⢉⣉⠙⠉⢹⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⣿⠞⢹⣀⡇⠙⣦⠀⡇⠀⢾⣈⡇⡷⠂⡷⠂⡇⢯⡀⣽⡂⠀⠀⢸⣿⣿⣷⣄
⠀⠀⠀⠀⠻⠿⠿⠿⡿⠧⠤⠤⠤⠤⠤⠭⠤⠬⠥⠤⠤⠤⠤⠭⠤⠤⠤⠤⠤⠤⠤⠭⠬⠥⠤⠤⠼⢿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⢲⠀⠀⣶⠒⠒⢲⠀⠀⡖⠒⠒⣶⠀⠀⡖⠒⠒⡆⠀⢰⡖⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⡇⠀⠀⣿⠀⠀⡇⠀⠀⡇⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣸⠀⠀⣿⣀⠀⢸⠀⠀⡇⠀⠀⣿⠀⠀⣇⡀⠀⡇⠀⢸⡆⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠁⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⢹⠉⠉⠙⡏⠉⠉⣷⢸⣿⣿⣿⣿⡇⣿⠉⠉⢹⡏⠉⠉⠏⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⡇⠀⠀⣿⢸⣿⣿⣿⣿⡇⡟⠀⠀⢸⠁⠀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢠⡇⠀⠀⣿⢸⣿⣿⣿⣿⡇⡇⠀⠀⢸⡆⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⡇⠀⠀⣏⢸⣿⣿⣿⣿⡇⡇⠀⠀⢸⡇⠀⠀⡇⠀⠀⣸
⠀⠀⠀⠀⢀⡀⠀⢀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣙⣉⣁⣸⣿⣿⣿⣿⣇⣉⣉⣉⣉⣉⣙⣉⣉⣉⣉⣉⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⡂⠀⠀⠀⢠⡄⠀⠀⠀⣄⠀⠀⢠⠀⠔⠂⢀⡄⠠⣄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤⣀⠀⠦⠀⠀⠀⠀⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣶⠿⠶⣦⡀⠀⣠⡶⠿⢶⣄⠀⢀⣴⠾⠷⣦⠀⠾⢿⣿⠿⠆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣀⣀⣼⠇⢀⣿⠀⠀⠈⣿⠀⠸⢿⣤⣀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⠛⠉⠁⠀⠀⣿⡀⠀⢀⣿⠀⢀⡀⠈⠙⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣇
⠀⠀⢠⣤⣶⣾⠿⣿⣿⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠙⠿⠶⠿⠋⠀⠘⠻⠶⠾⠋⠀⠀⠸⠟⠀⠀⠀⠀⠀⣿⡿⠿⣷⣶⣤⡄
⠀⠀⢸⣿⠁⠀⠀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠈⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⣿⡇
⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⣶⣶⣶⣶⣶⣶⠀⠀⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⠀⠀⣶⣶⣶⣶⣶⣶⣦⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⣸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠻⠿⠿⢿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠻⠿⠿⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣤⣤⣤⣼⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣧⣤⣤⣤⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠛⠛⠛⠛⠛⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣾⣷⣾⣷⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡇⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢸⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠁⣿⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⢸⣿⠈⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠷⣦⣹⠇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠸⣏⣴⠞⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠛⢶⣄⣤⡾⢷⣤⣠⡶⠛⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⢀⡀⢀⣀⢀⣀⢀⣀⠀⣀⠀⣀⠀⣀⠀⣀⡀⣀⡀⢀⡀⢀⡀⢀⡀⢀⣀⢀⣀⢀⣀⠀⣀⠀⣀
⠀⠀⠀⠀⠐⢇⠈⠁⠈⠉⠈⠉⠀⠉⠀⠉⠀⠉⠀⠉⠁⠉⠁⠈⠁⠈⠁⠈⠁⠈⠁⠈⠉⠈⠉⠀⠉⠀⠉⢀⠕
⠀⠀⠀⠀⠐⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵
⠀⠀⠀⠀⠰⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵
⠀⠀⠀⠀⠰⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷
⠀⠀⠀⠀⠠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡲
⠀⠀⠀⠀⠠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡢
⠀⠀⠀⠀⠠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢
⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥
⠀⠀⠀⠀⢀⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅
⠀⠀⠀⠀⠀⠑⠊⠑⠚⠙⠚⠙⠒⠉⠒⠉⠒⠉⠒⠉⠓⠋⠓⠊⠑⠊⠑⠊⠑⠊⠑⠚⠙⠚⠙⠒⠉⠒⠉⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⣤⠤⢤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠤⠤⠤⠬⠭⠭⢉⣉⣉⣙⣛⡛⠛⠛⠛⠛⠿⠷⠶⠶⠶⠶⢶⣶⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠒⠒⠒⠒⠒⠀⠤⠤⠬⡍⢹⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⢠⠀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡠⠊⠀⠀⠀⠀⢸⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⡉⠐⠒⠂⠤⠭⢉⣉⣒⡒⠢⠤⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠡⢄⠘⢣⠔⢈⠠⢆⣂⣀⠆⢀⠬⠍⣁⠒⠒⠒⠤⠬⢍⣑⣒⡒⠒⠤⠤⢀⣀⣀⡀⠀⠀⠀⢸⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢋⠀⣐⠣⠈⠐⢂⡠⠀⠁⡒⠐⠁⠠⠆⢒⡠⠈⣁⠤⠾⣄⠩⠒⠀⢀⣄⠨⢉⠉⠑⠒⠂⠤⠭⠭⣉⣐⡚⢸⣀⡇
⠀⡀⠀⠀⠀⣀⡤⠚⠁⡠⠠⠍⡀⠐⠉⠠⠂⡐⡠⠉⣈⠀⠘⠓⠨⠜⠀⠴⠒⠉⠤⠈⠁⢀⠰⠆⠀⠰⠀⠁⡀⠀⠐⠤⠀⠰⠀⢀⡤⣼⣏⡡⠇
⠀⠀⠀⠀⠀⣏⠉⠉⠒⠒⠦⠭⠤⣚⣀⠈⠓⠀⠔⣉⡀⠀⠀⠀⠉⠁⠒⢂⠤⠀⡠⠀⠈⣃⠀⠘⠀⣤⠀⠈⡅⠐⠊⢀⡠⠔⢊⡷⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠤⠤⣤⣀⣈⠉⠉⠒⠒⠢⠤⢤⣀⣐⣂⠂⠡⠔⡂⢠⠅⢈⢀⡒⠡⠄⠖⠀⢁⡠⠔⠊⣁⠤⠚⠉
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠲⠤⠤⢄⣀⣈⠉⠉⠒⠒⠢⠤⠥⣀⣀⣁⡠⠔⠊⣁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠲⠤⠤⣄⣀⣯⡤⠖⠉

⠀⠀⡀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣷
⠀⠀⠅⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠋⠉⠉⣻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⡶⢿⡉⠻⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⣀⡀⠀⠐⢳⢄⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣁⡴⠂⠀⠀⡜⡉⠉⠢⣄
⢀⠴⡂⢶⠚⡼⠴⠶⣤⣶⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣿⡚⠃⠴⠃⠀⠀⠀⠸⣆
⠈⠀⠉⠻⡍⠁⢠⡌⢍⡍⠁⠙⢅⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠁⠋⢻⠏⠀⢠⡄⠀⠀⠀⠀⠘⢧⡀
⢸⠀⠀⠀⢹⠤⠤⠤⠤⠤⠤⠦⢾⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⢸⠠⠀⣠⠁⢠⠂⢰⣶⡦⠄⠀⣹⣆
⢸⠀⠀⠀⠸⠀⢰⠶⠒⠶⡶⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣠⡘⢀⡀⠉⠀⠘⠣⠈⢽⡷⠨⢷⢾⣿
⢸⠀⠀⠀⢸⠀⢙⣈⣩⣀⣐⠁⢨⠀⠀⠀⠀⠀⠀⠀⢠⣀⣸⡏⢳⡉⠀⡎⠑⠤⢴⣄⠬⣁⠀⢤⣾⣏⡈⢤⣀
⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠔⠊⠉⢙⡿⢌⣈⠁⣽⡠⠀⣠⢁⡀⢠⣽⠟⠁⠀⠁⠀⣩⡃
⢸⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⡤⠸⢄⣠⠤⠴⡈⠀⠱⣤⣭⣽⢇⣸⢉⠁⢉⠁⠠⢄⠀⣷⠞⠁⠀⠀⣠⡴⠞⠁⠈⢂
⠈⠉⢳⡶⣾⣷⣶⣿⠃⠨⣷⣀⠂⠀⠉⢒⣦⢱⣠⡾⠟⠉⠐⠀⣾⠿⣶⣤⡀⠀⢀⣴⠋⠉⠛⠛⠛⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠉⠲⠖⠋⠀⠀⠀⠀⠀⢠⠀⢠⡿⠀⠀⠻⣿⣿⠁⠻⣷⣄⣀⠀⢂⠀⠀⠀⠀⠀⢀
⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣼⠃⠀⠀⠀⡏⠿⣦⠀⠹⣌⡛⠀⠀⠢⡀⠀⢀⡴⠋
⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠃⠀⠀⠀⢀⡦⠀⠙⠿⢿⣿⣿⣶⣆⣠⠴⠒⠉
⠀⠀⢠⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⠀⠀⣤⡞⠀⢣⠀⢠⠀⢹⣿⡇⠉
⠀⠀⢸⠀⢀⠀⠸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡂⠀⢀⡜⣿⠀⠀⠀⡄⠈⡇⠀⠛⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠖⠒⠛⠋⠃⠉⠉⠉⠉⠉⠉⠙⠛⠛⠒⠒⠦⠴⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠒⠛⠁⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠘⠀⠀⠐⠀⠀⢀⡄⠉⠉⠐⠦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢠⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡄⠉⠓⢦⡀
⠀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⡇⢰⡏⠛⠀⢰⡀⣦⢰⠂⠀⠀⠀⠀⠰⡐⠂⠀⡞⠁⡆⠀⡟⠲⠀⢀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⡄⢳⡇⢈⡋⠉⠀⠈⣷⢹⡟⠀⠀⠀⠀⠀⠀⡅⠀⠀⣇⢀⡇⠀⡷⣆⠀⢸⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠁⠉⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⠈⠀⠈⠀⠈⣀⣀⣀⡀⢠⣄⣀⣴⣃
⠀⠀⠀⠀⠀⠀⠑⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠠⡴⠊⠁⠈⠉⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢆⣀⡀⠀⠀⠀⠀⠀⠄⠀⠠⠤⣤⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠀⢀⢀⡤⢰⡶⠾⠛⠛⠀⠒⠒⠀⠠⠄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠶⠀⢠⣄⣀⡀⣀⡀⢀⠛⠒⠒⠒⠊⠒⠒⠒⣉⣉⣀⣠⣤⡴⠾⠛⠋⠉⢁⣀⡀⢀⣄⣀⣀⣀⡀⠀⠀⠉⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠧⢭⣍⣁⣉⣉⣉⠉⣁⡭⠤⠖⠉⠀⠀⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠀⢀⣠⠄⠐⠒⠒⠛⠒⠛⠒⠂⠲⠤⡄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠒⠒⠂⠀⠈⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⢀⣀⣀⣀⣀⣀⣀⡤⠄⠖⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣫⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣭⣭⣉⠹⢤⠤⠴⠶⠖⢶⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡉⠙⣿⣿⣦⣴⣺⣿⢟⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢁⣴⡿⠿⢿⣿⣏⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠊⣠⠟⠁⠀⢠⣼⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢀⡼⠃⠀⡜⠓⠦⣄⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠤⢤⣀⠀⠀⠀⠀⡞⢀⡾⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⣴⣶⣿⣿⣿⣷⣶⣬⣉⠒⠦⠼⠃⣼⠁⢠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣷⣦⡀⢠⡏⠀⡏
⠀⠀⠀⠀⣠⠤⣤⠀⢰⡇⠀⣿⣿⣿⣿⡿⠂⢰⣿⣿⣿⣿⣿⣿⣿⣾⡇⢸⠃
⠀⠀⠀⠘⠷⠀⢸⠇⢸⡇⠀⢿⣿⣿⣿⣶⣦⣌⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠻⣦⠘⣇⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⣦⣀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⢿⣿⣿⣶⣤⣄⣀⡀⠀⠈⠉⠛⠻⢿⣿⣿⣅⡛⠶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠈⠙⢿⣿⣦⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠙⣿⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⣿⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⣿⣿⣟⣚⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣸⣿⠃⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣴⠟⠁⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣁⣼⡿⢃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠰⠿⠶⠶⠶⠶⠶⠿⢿⣿⣿⣿⣿⣿⣯⣵⠾⣟⡿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠶⠖⠛⠋⠉⠛⠶⠶⠶⠦⣤⣤⣭⣭⣭⠴⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠾⠂⢸⣿⣿⣄⡀⠀⢸⣿⣿⣿⡟⠀⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣻⣿⣿⠁⠀⠀⠸⣿⣿⡿⠋⠀⠙⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⠎⣿⣿⣿⣿⣿⣩⣭⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣯⣿⣥⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢻⣿⣿⣿⠋⠙⠉⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣽⣿⣿⣿⣧⣤⣤⣼⣿⣿⣿⣦⣀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡻⠿⣿⣿⣿⣧⢀⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠈⠛⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠋⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠁⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⠀⠀⣀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠛⢿⣶⣾⠿⠻⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⠋⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣤⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣠⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⡀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉








⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠋⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡁⠀⠀⠀⠀⠀⠀⢀⣴⣴⣶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⢦⣤⠀⢰⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⣿⣷⣜⣧⣾⣵⠀⢠⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣤⣶⣾⣿⣿⢿⠟⠛⠷⣤⣿⣶⣦⣦⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢉⣾⢻⢏⡇⠞⠈⡠⡀⠀⠆⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠁⠀⠀⠀⠀⠀⠸⣿⠛⠘⠀⣴⠀⠀⡀⣤⣾⣤⣤⣤⣤⡔
⠀⠀⠀⠀⠀⢀⣤⣲⣖⣂⡖⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⠈⠀⠀⢿⡷⠖⠋⠉⠅
⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⢠⣶⡀⢀⡀⢰⣿⣀⣤⣤⣤⡾⠁
⠀⠀⢸⣿⡿⢿⣿⣿⣿⡇⡇⠈⣄⣀⡀⠀⠀⠀⠀⣠⣼⠏⠀⠀⢀⣴⣿⣿⢿⠿⠟⡩⣥⠀⠀⠀⠀⠘⡄
⠀⠀⢘⠛⠀⠈⠈⢠⢸⡇⡇⠀⢻⣀⣀⡀⢠⣴⣿⣿⠋⠀⠀⠀⠸⣿⣅⣿⢰⣄⡀⣠⣿⣤⣤⣤⣴⣶⡇
⠀⠀⣼⣀⣧⣧⣼⣾⣿⣇⢳⠀⠸⣿⣛⣿⣭⣽⣿⠇⠀⠀⠀⠀⠀⠙⢿⣿⣼⣟⠙⠉⠉⠉⠉⠉⠉⠉⢱
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠸⡀⠀⢻⣿⠿⠿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡇⢧⠀⠘⡇⠀⠀⢸⡃⠀⠀⠀⠀⠀⢠⣶⡄⣿⠀⣿⢘⣿⡄⠀⠀⠀⠀⢤
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡸⣄⠀⠙⠀⠀⢺⣵⣄⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣾⠿⠁⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣷⣻⠀⠀⠀⠀⣄⣿⣿⣿⣶⡄⠀⠀⠀⠀⠻⠉⠿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠆
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣶⣦⣤⣤⣤⣄⡀⣀⣀⠀⢀⣀⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁


⣾⣷⠿⠿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⣭⣭⠿⠀⠀⠀⢠⠈⠡⠤⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠠⢜⠀⠀⠶⠖⣿⠍⠡⠤⠤⢵⠀⢚⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣉⣿⣉⠘⣛⣛⣽⡇⠀⢒⣚⡿⠀⠥⠵⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢤⣷⠀⢈⣁⣀⢽⡇⠈⣉⣭⣿⡀⢒⣲⡧⠀⠀⡀
⠀⠀⠀⠀⢀⠀⡀⡤⢄⡐⢒⡲⠒⠚⠓⠒⠛⠂⠀⠂⠤⢾⡃⠀⠶⣺⣯⠄⣉⡥⣿⢉⣽⡇
⠀⠀⠀⠀⠘⠙⠓⠻⢶⠀⠈⠀⠀⠀⠀⣀⣠⠴⠖⠒⠈⢉⠉⠈⢻⣽⡻⡤⠤⣶⣷⢴⣿⣧⡆⢀⣤⣤
⠀⠀⠀⠀⢀⡀⠀⢈⢹⠀⠀⠀⠀⠀⠀⡿⠿⠷⢤⣬⣦⡶⣰⣭⣿⣶⣒⣖⣒⣲⣾⣽⠿⠿⣯⣽⣿⣿⠷⠂
⠀⠀⠀⠀⠈⠙⠊⠛⠻⢦⣀⠀⠀⠀⠀⣏⠂⠀⠀⠀⠀⡏⠁⠉⠈⠉⠩⣏⠉⠉⠾⣁⣐⣐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠤⠹⠷⢶⣦⣄⣀⣇⣠⣤⣶⣾⠽⠏⠂⠐⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⡎⠁⠉⠀⠒⠀⠀⠠⠄⢀⣀
⢩⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠒⠂⠀⠠⠄⢀⣀
⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀
⠈⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢇⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⡇⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⢸⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠸⡄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢐⣨⡐⣶⣴⣤⣤⣤⢤⣤⡤
⠀⠀⠀⣇⠈⠀⠀⠈⢀⠀⠀⠤⠀⠐⢂⢀⣀⡍⣤⣤⣴⣶⣾⡿⡟⠿⠟⣻⡿⠷⠞⠛⠉⠁⠀⠀⠀⠀⠀⣀⣤⠶⠚
⠀⠀⠀⠹⢔⠂⢌⠀⢄⢔⡂⣒⠿⣛⡿⢺⣟⡿⢶⣱⣟⡻⠬⠐⠒⠊⠉⠁⣀⠀⠀⠀⠀⠀⣀⣡⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠈⠙⠂⠠⠈⠁⣥⠚⠡⠋⠁⠈⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢐⠢⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠘⢆⠀⠀⠀⠀⠄⠀⢀⣀⣀⠠⣒⠬⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢴⡇⣿⣷⣯⡷⣿⣹⣿⢺⡆⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣷
⠀⣷⣯⡻⢿⣿⣿⣿⣿⢿⣫⣷⡇⠀⠀⠀⢸⣷⣥⡻⢯⣗⣿⣾⡿⣟⣼⣾⠀⠀⠀⢸⣮⣝⢿⣿⣿⣿⣿⡿⣟⣽⣿⠿⠋
⠀⣿⣿⣿⣷⢝⣻⣋⢷⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⢗⣭⣻⣥⡻⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡮⣛⣟⡹⣾⣿⣿⣿
⠀⣿⢟⣯⣾⣿⣿⣿⣿⣮⣛⢿⡇⠀⠀⠀⢸⡿⣫⣷⣿⣿⣿⣿⣿⣮⣟⢿⠀⠀⠀⢸⡿⣻⣵⣿⣿⣿⣿⣶⣝⡿⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⢀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⣿⣿⣷⠀⠀⠀⢎⠀⣧⢈⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢹⣿⣿⣇⠀⠘⠀⢩⡾⢃⠴⢦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢿⡿⠿⢄⠄⡐⠋⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢸⣿⣷⣦⡈⠓⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠄⡈⠛⢛⣿⣷⡌⣒⣀⠥⡒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢾⣿⠀⠀⠂⢀⠔⢨⠋⣻⣀⡤⠖⠁⠈⢣⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡛⣽⣏⣀⣘⣇⢘⣲⠑⢂⣀⣸⡇⣰⠊⣁⣘⢻⣵⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡢⠀⠑⣿⣿⣻⡟⣟⢻⠛⠷⡏⣟⢹⣛⣿⣿⣿⡿⡟⣚⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠛⠉⢉⣿⣉⡆⠁⠈⠀⠀⠀⠀⠀⠛⢤⣿⣿⣿⡇⠈⠩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⠰⠉⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠁⠔⣸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠨⣐⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⣸⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉