Keys Emojis & Symbols

⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀
⠀⢀⣴⣶⠞⠭⠰⡖⠨⠝⢶⣶⣄⠀⠀⠀⠀⢰⠶⠽⣯⡀⠶⣀⣿⡧⢶⡄⠀⠀⠀⠀⠀⢀⣾⠟⢫⣽⣿⣍⠻⣷⡀
⠀⠈⠋⢸⠟⠲⢦⣤⠶⢺⢯⠉⠋⠀⠀⠀⠀⠈⠋⢁⣼⡼⣶⠇⣧⣉⠙⠀⠀⠀⠀⠀⣴⡾⠟⢰⣯⣻⣿⣽⡆⠹⢷⣦⡀
⠀⢀⢀⡾⠇⠀⠀⠀⠀⠐⠿⣆⢀⠀⠀⠀⠀⣠⢊⠉⣁⣼⣟⣦⡈⢈⠳⡄⠀⠀⠀⢼⣏⣠⣼⣆⣸⣿⣿⣇⣠⣧⣄⣹⣧
⠀⠈⣮⡄⠀⡀⠀⠀⠀⡀⠀⣼⡞⠀⠀⠀⠀⠀⡋⢰⠏⣠⣿⡀⠻⡆⢹⢀⠀⠀⠀⠸⣯⣾⣿⠋⡋⣿⣿⢙⠉⣿⣷⢿⠇
⠀⠠⢹⣵⣿⣿⣧⢰⣿⣿⣶⣽⡁⠀⠀⠀⠀⠛⠓⢾⡾⠉⡀⠉⢶⠷⠛⠛⠀⠀⠀⠀⠈⠛⠛⠻⢿⣎⣱⡿⠟⠛⠋⠁
⠀⠀⠀⠀⠟⣯⠈⠽⢿⡟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣶⢿⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠃
⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠈⢳⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣽⠇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢸
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢼⣿⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⣷⢾
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣈⣉⣅⣰⡶⡦⠀⠀⠀⠀⠀⢐⣓⣿⣧⣿⣾
⠀⠀⠀⠀⠀⠀⣼⣿⡼⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠨⢍⣩⡹⠋⠻⠶⠖⠀⠀⠀⠀⠀⠸⠇⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠸⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠁⠀⠉⠲⣄
⠀⠀⠀⠀⢀⠔⠁⠀⠀⣀⣀⠀⠀⠀⠘⡆
⠀⠀⠀⢠⠋⠀⢀⠀⠀⠀⠀⠙⡄⠀⠀⡇
⠀⠀⢠⠇⠀⢠⠆⠀⠀⠀⠀⢀⡇⠀⢀⡇
⠀⠀⣼⠀⠀⢸⠀⠀⠀⠀⢀⡼⠁⠀⣼⣁
⠀⠀⣿⠀⠀⠘⢧⣄⣠⡴⠛⠀⢀⣼⡁⠈⠙⠢⣄⡀
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠙⠲⢤⣀⠀⠉⠓⠤⣀
⠀⠀⠀⠈⠳⠦⣤⡤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠁⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⡀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠌⡜⠀⠀⠀⠀⠀⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢌⠞⠠⢂⢄⠌⡱⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠎⡠⢁⢊⠎⡰⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢼⡱⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣉⣭⣍⣁⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⡴⠛⠉⠁⠀⠈⠉⠙⠷⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠀⣀⡠⠤⢄⡀⠀⠀⠀⠀⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣏⣁⣤⣤⣀⠙⠶⣾⠛⠛⠛⠛⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣤⠿⣿⢯⡀⠀⢙⣧⡀⠈⠳⣶⣲⠖⢲⡖⢦⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠸⣧⣀⣘⣷⣝⡲⠿⣿⠇⠀⠀⣼⣁⡴⣋⡤⠞⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠉⠉⠉⠉⠉⠋⠁⠀⠀⢠⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢠⠎⠁⣰⠳⡄⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⡆⠀⢸⡀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡾⠀⢠⡇⠀⠀⠀⣰⠏⠹⣄⠀⠀⢀⡀⠀⢳⡀⠀⢻⡀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠃⠀⡾⠀⠀⣾⠛⠁⠀⠀⠈⠓⠚⠉⠉⠛⢦⢳⡀⠀⢻⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡏⠀⢰⠇⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢳⡀⠀⢻⡄⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⠁⢀⡟⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢳⡀⠀⢻⡄⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⡏⠀⣼⢁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢱⡄⠀⢳⡄⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⠀⢠⡏⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢻⡄⠀⢻⡄⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⠇⠀⡾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⢻⡼⢧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡾⠀⣸⠇⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡄⠀⢻⡄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⠃⢀⡟⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⣆⠀⢻⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟⠀⣼⢁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣆⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⣉⣉⠉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡴⠚⠋⠉⠉⠉⠉⠙⠲⢄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣉⣉⣉⣉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢧⠘⣷⣨⣍⣁⣒⠻⢭⠩⠭⢛⠲⣶⣶⣄⠻⣆⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⡴⣪⡶⠛⣧⡈⣷⡆⠀⠉⠻⡆⢻⣶⢄⠑⢌⢿⣾⠇⠙⡆⠀⠨⣳
⠀⠀⠀⠀⠀⢂⠼⠉⠀⠀⠈⠉⠡⠤⠀⠐⠀⡄⣼⡯⡀⠀⠀⠈⢫⠀⠈⠁⠀⠠⢸⡄
⠀⠀⠀⠀⢸⠸⣅⠀⠀⠀⣀⣈⣐⣈⠏⢀⠜⣰⣟⠥⡂⠄⠀⡆⣿⡴⠃⠀⠐⠀⠀⢱⣄
⠀⠀⠀⠀⠈⡆⠀⠉⢁⣲⡬⢵⣂⡠⢐⡡⣾⠞⡣⠑⠂⢀⡰⣱⠁⠐⠀⠀⠄⡠⣠⡳⢦⣍⡒⠤⡀
⠀⠀⠀⢠⡊⢀⣾⢣⢶⢌⣤⣴⣶⣾⢷⣿⣫⠵⠬⠒⠊⢩⡔⣇⠂⣁⣁⣁⣪⣜⠉⠛⠶⡬⣛⠷⣮⣑⠦⣀
⠀⠀⠀⢈⢏⢾⣮⢧⢣⣿⠃⠀⠈⠻⠿⣶⣖⣦⢶⡜⡿⡔⣱⡸⠛⠛⠋⠉⠉⠙⠳⢶⢗⡮⣙⣷⣦⣍⡛⢶⣭⣒⠤⡀
⠀⠀⠀⢃⣾⣿⢣⢣⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⡎⢷⠐⡹⡽⣷⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⣗⠉⣛⢷⣮⣙⠳⣮⣕⡢⣄
⠀⠀⢰⣿⣾⢣⢣⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⠱⢱⡹⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠛⠉⠿⡻⣦⣿⣿⣗⣷
⠀⣔⣽⢯⢣⣳⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣷⣧⢻⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⠋
⡴⣿⢯⡷⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⢿⡆⢷⡱⡀
⢻⣳⡿⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣎⣿⡘⣧⣧
⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡷⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠁
⠀⣀⡀⠀⠀⡀⢀⠀⡀⢀⠄⡀⠀⢀⠠⠀⢄⠀⡢⠐⠖⠠⠀⠔⠠⠂⢄⠠⢀⢄⠠⡀⢄⠄⡠⡀⠔⡠⢀⢄⡠⣀⠄⡀⡀⢀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠶⠚⢛⣡⠶⠞⣛⣉⣉⣛⡛⠶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⠋⠁⠀⣴⠟⣡⠾⠛⠉⢉⣩⣭⣝⣷⣯⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣼⠃⠀⠀⣼⢃⡾⠁⠀⠀⠀⣾⣁⣴⠟⣷⠈⢻⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣿⠀⠀⢸⡇⣼⠁⠀⠀⠀⠀⠻⣥⣤⣴⠏⠀⠀⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣌⢷⣄⠘⣧⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣦⣙⠳⠘⣧⡙⢧⣀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣵⠟⢁⣽⠗⠂⣼⠻⣦⣍⡛⠶⠶⠶⠶⠞⣋⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⢟⣵⠟⠁⠀⡟⠁⠀⣼⠃⢠⡟⣽⠛⢻⡖⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣴⢟⣴⠟⠁⢰⡟⠛⠃⠀⣰⠏⢠⡟⣼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⢋⣵⠟⡁⠀⣾⠛⠃⠀⠀⣰⠏⢠⡟⣸⠃⣀⣤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⠁⠀⣿⠙⠉⠀⠀⠀⣰⠏⢠⡟⣸⠇⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⢰⡟⢀⡟⣰⠏⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⣱⠏⠐⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣏⣤⡴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⣿⢿⠿⡿⠿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣛⣛⣻⣿⣛⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⣉⣀⣀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡤⠖⣋⠭⢥⢄⡀⠀⠀⠉⠉⠙⠶⣄⠀⠠⡀
⠀⠀⣠⢈⣀⡴⠉⢀⡯⠶⠒⠾⣶⢾⣆⠀⠀⠀⠠⠄⠈⠳⣄⠈⢄
⠀⣬⠇⠀⡟⠀⠀⠈⢄⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠄
⢨⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰
⠀⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠃⢀⠛
⠀⠸⣏⠦⡀⢁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠁⠀⠀⠀⢆⠂
⠀⠀⠜⢦⣙⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣄⡀⠀⢶⢄⠀⠑⠄
⠀⠀⠀⠈⠛⠛⢷⡦⣭⣩⣒⣒⣒⣒⣦⠀⠻⣷⡀⠑⢿⢢⡈⠳⡕⢄⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢋⢿⣳⣤⣿⣷⣄⠀⠑⢌⢢⡈⠣⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⠩⠻⣾⣷⣄⠂⠙⢽⣦⣌⢮⣧⣄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣶⣶⠁⠤⠤⡄⠿⠷⠦⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣄⠀⠙⢟⡷⡘⣾⡢⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣓⣴⣶⣝⢜⣮⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠑⢝⢦⡑⢟⢄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⡀⠀⠳⣽⣌⠳⣽⢦⡀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣤⣼⣿⣷⡌⠷⢽⣿

⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠳⢶⣤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⡟⠁⢀⣠⣒⡒⠢⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠳⣦⣀
⠀⢀⣿⠄⠀⣾⠛⠉⠙⣧⢸⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠷⣄
⠀⢸⣿⢠⠀⢻⣄⠀⢀⣿⠟⠀⠀⢠⡟⠉⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠙⢷⡀
⠀⠸⣿⣤⠃⠀⠉⠙⠉⠁⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢸⣷
⠀⠀⣿⡟⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡄⣿
⠀⢠⡿⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠘⣰⡟
⢀⣾⠁⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢀⣰⡿
⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠀⠀⠀⠀⢸⣿⠁
⣿⡀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⡁⣮⣷⠄⠀⠀⠀⠸⣿⣆
⢻⣧⠀⠀⠀⠀⠙⠤⢄⣀⡀⠀⠀⠐⠈⠀⠀⠀⠐⠘⣏⠢⡱⡄⠀⠀⠈⠛⠶⠶⣦⡀
⠈⢷⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢣⡑⢌⢢⠀⠀⠀⠀⠀⢾⣷
⠀⠈⠛⠷⣵⣦⣤⡤⠀⢀⣀⣈⣡⣌⣉⡲⣄⠀⠰⡉⠀⠑⡙⢄⠢⠑⠀⠀⠀⣇⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠿⣿⣧⠀⠙⢄⠀⠘⢎⠳⣄⢌⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠈⠢⡀⠀⠣⡙⢦⡡⡑⠄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣦⡑⢄⠀⠀⢄⠳⣌⢈⠢⡀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣎⣀⠤⠤⡑⣜⢦⡑⠑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣷⠈⠣⡱⣌⠨⠢⡀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠤⢀⠈⢮⠳⡄⢄⢄⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣦⡑⡄⠑⢝⢦⡣⡠⡀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠠⡱⣌⢎⠠⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣄⡀⣀⠈⢌⢣⡑⡑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⣬⣅⠀⠱⡝⢮⡢⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣬⣉⡋⢪⡳⣔⢄⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣇⣀⣱⣌⢷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣶⣾⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⣋⠬⠭⠬⣍⡛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⠕⠉⠀⠀⠀⠀⠀⠙⢷⣬⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢧⣤⣤⣤⣄⣀⡠⠤⢒⣚⣻⡶⢯⣅⡐⠊⠩⣒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⠍⠚⡇⢸⣥⠈⣿⠉⠛⢝⡻⣯⣐⠲⢦⣌⡐⠬⡓⢤⡀⠁⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠔⡁⢄⠳⡹⣮⣿⣤⣿⡤⢄⡀⠉⣪⢲⣙⢳⣬⡙⢦⡈⢪⢳⣄⠀⢣⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠤⣈⠳⠁⠀⠘⢿⣽⣿⣿⣦⠱⠀⠤⠧⢹⡧⡹⣍⢌⢳⡄⠑⢻⡄⠀⡌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⢐⠒⠠⠀⠀⠀⠠⡸⣇⣿⣿⣿⢀⢄⡒⠂⡀⡇⠠⢹⡄⢆⢿⠀⡇⣷⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠄⠀⠀⠀⠀⠀⠑⠬⠽⠟⠕⠁⡲⢌⠀⡇⡿⠘⢸⡇⠈⢸⡇⠀⣿⢀⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢬⡑⠀⢡⡇⠀⢸⡇⢠⢹⣥⢃⣟⡋⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠀⠀⢈⣢⡀⠀⠀⠀⠀⢄⠣⡱⠌⠂⢠⢃⡞⢀⡌⡾⠀⣌⡾⢣⡞⠣⡙⠸⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡞⠃⣠⢤⡖⢁⡜⠈⠢⡀⢠⢌⠢⡣⠑⢀⠔⣡⠊⡠⢊⡾⢁⢜⡞⣱⣿⡄⠀⣽⡄⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡴⠡⠊⣰⣿⠂⠀⠀⠧⢤⠤⣄⣐⣪⠔⣋⠔⣊⣴⡋⢔⣥⡞⣹⣧⡹⣧⡀⠘⢿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢊⠜⣡⠟⡉⣻⣷⣾⢛⠟⠫⣿⠋⢡⣈⣥⠲⣿⠸⠠⣾⣿⣿⠇⠸⡈⢿⣌⢳⣄⠀⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⡵⢃⡾⠃⢐⡼⢃⣿⠇⠋⠀⠸⠇⠂⣸⣿⣟⢸⡿⠸⠀⣿⣿⠿⠀⠀⢇⠀⠹⣿⡮⣧⡀⠘⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡡⢊⡴⠋⢀⡼⠋⠀⣌⡗⡘⠁⢰⣟⢘⣶⣾⢿⠀⢼⡏⢰⠚⢿⡱⠆⡆⠀⠘⡄⠀⠈⢿⣕⢅⠀⠀⠻⠶⣄
⠀⠀⠀⠀⠀⠀⣀⠼⣡⠞⣠⠏⠀⣰⠏⠀⠀⣸⡿⢡⠁⠀⡾⠋⠁⣼⡏⣽⠀⣸⡏⡃⠀⠀⣩⢀⠰⠀⢀⢱⠀⠀⠀⠹⣾⣺⣦⠀⠑⣜⣇
⠀⠀⠀⠀⠀⠸⢃⡜⣡⡞⠁⢀⠞⠁⠀⠀⢠⣟⠇⠄⠀⣸⣷⠀⠀⣿⡇⡟⠀⢿⡷⠁⠀⠀⠻⣟⠀⠆⠀⣺⡆⠀⠀⠀⠘⢿⣷⡱⡄⠈⢆⢻
⠀⠀⢀⣴⠖⡡⢀⢴⠉⢀⡴⠁⠀⠀⠀⠀⣾⠏⡐⠀⢰⡟⠁⠀⠀⣿⡇⡇⠀⣿⠁⠀⠀⠀⠀⣿⡸⠸⠀⠉⢥⠀⠀⠀⠀⠈⠻⣶⠿⣆⠀⢻⡶⢤⡀
⠀⠀⣷⢣⠊⡠⡿⠀⡠⠋⠀⠀⠀⠀⠀⢸⡯⠀⠀⠀⣿⡇⠀⠀⠀⣿⠃⡇⠀⣻⡇⠀⠀⠀⠀⠲⡆⠇⠃⠀⠪⡆⠀⠀⠀⠀⠀⠙⢷⡝⣧⡈⠿⣈⢻⡆
⠀⠀⢸⠁⣴⣗⣱⠎⠀⠀⠀⠀⠀⠀⢀⣿⢃⠁⠀⣼⣟⠀⠀⠀⠀⣿⠇⡇⠀⡿⠀⠀⠀⠀⠀⠰⣿⡀⠸⠀⠁⢷⠀⠀⠀⠀⠀⠀⠀⠻⣎⠟⢦⡙⣶⠁
⠀⠀⠘⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣮⡇⠀⠀⣰⠟⠃⠀⠀⠀⢀⣿⠢⠇⡞⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠃⢹⡘⡇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡜⠠⠀⢠⣿⡀⠀⠀⠀⠀⠘⣿⣴⠀⢱⡄⠀⠀⠀⠀⠀⠀⢸⡁⠀⠘⠘⡏⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣾⣣⠃⠀⠀⠀⠀⠀⠈⢮⢀⡼⠃⠀⠀⠀⠀⠀⠀⠈⢷⣧⣀⠀⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠋⢳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣉⣉⠳⣿⡷⠛⣉⣙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢠⠋⠁⢈⡇⢸⠀⢮⠁⠈⢳⠈⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡴⢇⠘⣄⠀⠻⠴⠋⠳⠼⠃⠀⡼⢀⡷⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡞⠀⠈⢦⡘⠦⣄⡀⠀⠀⢀⣤⠞⣡⠞⠀⠈⣇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣧⠀⠀⠀⠙⠢⢌⡉⠒⠚⣉⡤⠞⠁⠀⠀⢀⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠣⣄⣀⢀⣀⡤⠛⡹⣏⠣⣄⣀⠀⣀⣠⠞⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣭⣭⣭⡴⢋⣠⣈⠓⣦⣭⣍⡩⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡉⢩⠤⢤⡉⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠻⣀⣠⠇⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠤⠴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠛⠛⠛⠛⠛⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡭⠗⠀⢠⡖⠒⢟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢲⣆⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠀⠸⣧⣀⣽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣀⣀⣀⣀⣯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠶⠶⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠶⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠞


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠚⠛⠛⠛⠛⠛⠒⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⣀⣀⣤⣤⣴⣶⣶⣶⣦⣤⣤⣀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⢀⣤⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠦⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣋⣉⣉⣛⣶⣌⠳⣄⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠛⠉⠉⠉⠉⠉⠻⡇⠈⢳⡄⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢻⡄⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣸⠇⠀⠀⣀⡤⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⡼⠃⠀⠀⠀⣧⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢰⢃⡞⠋⠉⠳⢿⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⢸⡇⠀⠀⠀⠈⠳⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⠘⣎⣇⠀⠀⠀⠀⠀⠈⢷⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠈⣇⠀⠀⠙⢮⠣⡀⠀⠀⠀⠀⠀⠹⣞⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠳⣝⢦⡀⠀⠀⠀⠀⠸⡎⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢰⢯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣆⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠲⣝⡦⣤⣀⣀⡼⢣⡏⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⣠⢏⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿⠀⠀⠀⢀⣠⠖⣳⠀⠀⠙⢦⣀⠀⠀⠀⠙⠓⠒⠶⠚⠋⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⢀⡴⢃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⠟⠁⠀⣀⡴⠋⢀⡼⠃⠀⠀⠀⠀⠈⠛⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⠀⢀⡴⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⠟⠀⠀⠀⢀⡤⠞⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⠴⠋⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⣠⠖⠋⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⡶⠦⣄⣀⣀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠋⣁⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠟⠁⠀⢀⡴⠚⠁⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣠⡷⢦⣤⣈⣉⣉⠉⠉⠉⠉⣉⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣠⠞⠋⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⠞⠋⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⣀⡀⠀⣀⣼⠋⠀⢀⣤⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣡⡾⠟⠁
⢀⣤⠞⠋⠙⠛⠉⠀⢀⡴⠋⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣴⠞⠉
⣿⠁⠀⠀⠀⠀⣠⠴⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣡⡾⠛⠁
⣿⠀⠀⠀⣠⠞⠁⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠟⠁
⣿⠀⣠⠞⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣉⣴⠟⠁
⣿⡟⠁⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣠⡾⠋⠁
⢿⣷⡚⠧⣤⣀⡀⠀⠀⠀⢀⣠⠞⣉⣴⠟⠋
⠀⠈⠙⠻⢶⣬⣭⣙⡛⠲⣋⣠⡾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⠿⠿⠿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠙⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡾⣷⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠉⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⢠⣾⢿⣶⣤⣀⣈⣛⣿⣶⣄⠀⢀⣀⣀⣀⣀⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠘⠿⣾⣿⣿⠟⠛⠉⠉⢙⣿⡿⠟⠋⠉⠉⠛⠛⢷⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠀⣴⣿⠋⠀⠀⣠⣤⣴⣿⠏⠀⠀⢀⣠⣄⣀⣠⣾⡿⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⡀⠀⠀⠀⣼⣿⠁⠀⠀⢰⣿⣿⣿⠃⠀⠀⣰⣿⣿⣿⣿⡿⠋⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⠠⣿⡇⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠹⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢫⡟⠐⣿⡇⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣳⠏⠀⣠⣿⣷⡀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢚⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⣰⡿⠁⠹⣿⣶⣤⣤⣤⣤⣬⣿⣿⣦⣀⣀⣀⡤⠔⢊⣡⣴⡾⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠏⠀⣼⡟⠀⠀⠀⠀⠙⣿⣇⣀⣀⣠⣀⣬⣿⡿⠿⣿⣴⣾⠟⠋⢻⣇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⠆⣼⠟⠀⠀⠀⠀⠀⠀⠻⣿⡏⠉⢻⠉⢻⡏⠀⠀⠛⠻⣿⣶⣤⠀⢻⣆⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⢋⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⢠⣿⡃⠀⠀⠀⠀⢸⣿⡅⠀⠈⢿⡄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢸⠀⣿⣇⠀⠀⠀⠀⠀⠻⢿⣷⣀⠈⢷⡀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠉⢡⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠘⣷⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣁⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⣾⣏⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠘⣧⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢸⡇⢿⣏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠹⡆⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⢸⡇⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡵⠚⠋⠉⠉⠉⠓⠮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣄⣀⠀⢰⠃⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢀⡤⣤⣈⡉⢹⠀⡧⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠘⠳⠶⣤⣭⣙⡀⠱⠤⣤⣉⠓⢦⠀⠀⠀⠀⠀⠀⢀⡸⢰⡃
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣀⣠⣤⣌⣉⠛⠛⠷⢦⣬⠇⢸⡇⠀⣠⠔⢊⣩⡜⣡⡏⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⠋⠁⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⣼⣡⠞⠁⠀⢿⣥⣴⠷⠃⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⢿⠏⠀⠀⠀⠀⣠⡴⠶⠛⠳⣄⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠙⣄⠀⠀⠀⠀⣀⣼⠃⠀⠀⣰⠏⣾⠀⠀⠀⠀⠰⡏⠀⠀⣀⣤⠞⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⡀⠉⠛⠛⠛⠉⠀⠀⣠⡾⠋⠀⡏⠀⠀⠀⠀⠀⠙⠛⣋⣁⣤⡴⠶⢄⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡉⠙⠛⠶⣦⣤⣤⡶⠟⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠋⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢷⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⣄⠀⢀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣟⡞⠁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⣀⣠⠤⠖⠒⢦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣹⡆⡞⡼⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣄⠀⣾⠉⠀⠀⠀⠀⠸⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠰⡏⡼⡹⣱⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢠⡶⠞⠋⠉⠀⠀⢈⠄⠙⠷⠶⢦⣤⣤⣤⡟⠀⣷
⠀⠀⠀⠀⠀⢠⡞⡹⣱⢳⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⣇⠀⠀⠀⢀⡴⠋⣠⠖⠛⠒⠶⠤⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡾⢠⢣⢣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠹⣦⣠⡴⠋⢀⡾⠃⠀⠀⠀⠀⣀⣠⠇⠀⠀⣾⠁
⠀⠀⠀⠀⣼⢁⠃⠇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣾⣁⣀⣤⠶⠞⠋⠉⠀⠀⣠⡾⠃
⠀⠀⠀⣸⠃⠄⠌⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠈⠉⠉⠀⢀⣀⣤⣤⠶⠞⠋
⠀⠀⢠⡏⠌⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⡴⠶⠛⠛⠉⠁
⠀⠀⠀⠱⠤⠚⠉


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

⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⣦⣄
⠀⠀⠀⢀⡴⠋⣡⡴⠶⠶⢦⣄⠉⢷⣄
⠀⠀⢠⡟⢁⡾⠁⠀⠀⠀⠀⠘⢷⡀⢻⡀
⠀⢰⡟⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⢀⡿⠀⣠⡴⠿⠂⠀⠀⠀⠀⢀⣼⠃⣾⠁
⢸⡇⢰⠏⠀⠀⠀⠀⠀⠹⣟⠉⠀⣰⣿⣦⣄⡀
⢸⡇⢿⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⣿⡁⠀⠈⠙⠻⠶⣤⣀⡀
⠈⣧⠘⢧⣀⠀⠀⠀⣀⡼⠃⣰⠏⠈⠙⠳⢦⣄⡀⠀⠀⠉⠛⠳⢶⣤⣀
⠀⠈⠷⣄⡉⠛⠛⠛⣉⣤⠞⠁⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣄⡀⠀⠈⠉⠛⠶⣤⣄⡀
⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⣀⠀⠀⠉⠙⠷⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣄⡀⠀⠈⠙⠛⠶⣤⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣀⠀⠀⢉⡟⠁⣾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣾⠁⣼⠁⠈⠙⠳⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⠻⣶⣤⣀⠀⠀⠉⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢦⠉⠛⠷⢦⣄⣀⠀⠈⠙⠻⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠋⠀⠀⠀⣤⢦⣭⡷⢶⣤⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢠⡶⢤⣀⠀⠙⢶⡄⠀⠀⠈⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣤⣠⡟⠀⣰⠏⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢿⣄⣀⡿

⠀⠀⠀⠀⠀⣀⠤⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣤⣤⣤⢤⡴⡄⣬⣤⢭⣧⢤⢤⣤⣤⡤⣤
⠀⠀⠢⠖⠡⠀⠀⢠⠀⠀⠐⠈⠂⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡟⠉⠩⠉⠉⠨⠍⠛⠛⠋⠙⠉⠛⠙⠋⠙⠙⡟
⠀⡁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⣄⡀⠀⠀⠀⠀⡇⡇⠘⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠂⠀⠘⠀⠀⡧
⡟⢀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢀⡇⢀⣈⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⣿
⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⡏⠀⠀⠀⣿⢓⣶⣦⣐⣖⣀⣶⣆⣶⣶⣾⣆⣶⣶⣳⣶⣶⣷
⠰⡔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠉⠈⠉⠈⠀⠀⠁
⠀⠙⢌⠢⠈⠀⠀⠀⠀⠀⠄⡀⢌⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠉⠒⠤⢭⣠⣤⠬⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⡍⠉⢡⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠿⠿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠃⠀⢠⡟⠃⠀⠀⠀⠀⠀⠀⠀⢔⡢⠖⠊⢹⡄⠉⠛⠷⣠
⠀⠀⠀⠀⠀⠀⢀⡆⠈⠈⢰⠄⠈⠉⣏⣉⠁⠀⢰⡂⠁⠠⡀⠀⠀⠠⡞⠛⠓⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⢺⠋⠀⠀⠀⢾⡗⠀⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠘⢲⡀⠖⣠⠔⠉⠀⠠⠀⠀⢉⠂⡝⢶⢀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢈⠀⠀⣀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⡼⠁⢠⣄⠀⠀⠀⢀⡀⠑⠘⡀⠁⠀⠀⠀⠀⣷⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡈⠁⠀⠀⠀⠘⠁⠉⠋⠁⠀⠉⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠀⠉⠳⣦⡞⠋⠀⠠⠀⠁⠀⠀⠀⠀⠀⣇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠫⢄⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⣐⡅⣀⡀⣀⡠⢦⣔⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠠⠀⠀⠄⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠿⡟⠛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠒⠒⠂⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠙⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠛⠳⢶⣄⠀⠀⠀⠀⠀⢸⣻⠞⠁⠀⠀⠀⠀⢀⠀⠤⠀⠁⠼⠤⠥⢈⣀⣤⣀⡀
⠸⠧⠥⠴⠦⠤⠄⠄⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⠀⠀⠀⠀⠙⢦⣀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠰⠾⠿⠾⠿⠿⠿⠷⠶⠶⠶⠶⠶⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⡄⠀⠀⠀⠀⠀⢻⡇⢀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠘⣇⣾⡿⠁⠀⠀⠀⠀⠀⠈⠉⠏⢈⠁⠁⠀⡀⢀⠀⠀⠀⠙⢏⠁
⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣿⣶⠄⠀⠀⢠⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡅⠀⠈⠂⠀⢀⢸
⠀⠀⠈⠐⠄⠂⠂⠈⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡧⠐⠾⢿⡗⢦⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⡀⠀⠀⢀⡀⠀⣀⠀⠠⢸⠈
⠀⠀⠀⡀⠀⠀⠀⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⣰⠀⠠⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⢸⠸
⠀⠠⢁⠄⠀⠀⠀⠐⠀⠑⡄⠀⠀⠀⠀⠀⢀⡎⠀⠀⢀⣔⠁⠀⠼⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠠⠄⢀⠠⡀⠀⠀⡀
⠀⢃⠇⠀⠀⠀⠀⠀⠀⠱⢡⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⡏⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠂⠀⡞⢹⡏⠙⠂⠈⠀⠀⡀⡆
⠀⡀⠃⠀⠀⠀⠀⠀⠀⢈⢨⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠇⠉⣠⢂⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠇⡈⠡⠀⠀⠀⠀⠀⠀⠇
⢠⣗⣐⣒⣆⣓⡒⠚⠒⠒⠚⠆⠀⠀⠀⠀⠈⠠⣈⣀⣠⣤⣼⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠚⠛⠋⠙⠛⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⣀⠀⠀⠰⠆⠀⠀⢀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠈⠁⠀⢸⡇⠀⠀⠋⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠀⢀⣴⠟⠁⠀⠀⠀⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣠⠈⠁⠀⠀⠀⠀⣀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠁⠀⠀⢰⡆⠀⠀⠈⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⣤⡴⠞⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠀⠀⠀⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣷⡄⠀⢤⣤⣠⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠛⣁⣀⣸⣀⡙⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⣩⣷⡾⠟⠛⠛⠛⠛⠛⠿⢶⣿⣙⠛⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡄⠀⠀⠈⠙⢢⣦⡄⠀⠤⠤⠄⢤⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠻⠷⣶⣶⣶⣶⡶⠿⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢠⡶⣍⠲⣖⠢⣄⠀⣀⣀⣤⣤⡠⢤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠻⠧⣌⡣⠝⠻⣭⡉⠉⠉⠉⠉⠉⣟⢼⠉⠓⣯⡷⣸⡀⠀⠀⠀⣀⠉⢛⣭⠠⣝⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⣀⠤⠬⣅⡀⠀⡙⢦⡈⠓⣛⣒⣂⣀⠠⠤⠽⠶⠽⠿⠭⠶⠭⠭⠭⢉⣀⣛⣓⣒⣀⣠⠬⢤⠆⡻
⠀⠀⠀⠀⠀⠀⠀⢰⠃⢉⠞⠁⠀⠀⠀⠙⢿⠋⡠⢋⠤⠤⠒⠒⠒⠒⠒⠒⠂⠀⠉⠉⠉⠉⠉⢹⠁⠁⠀⠀⠀⠀⠀⣠⢋⡔⠁
⠀⠀⠀⠀⠀⠀⠀⡟⠀⡜⠀⠀⠀⠀⠀⣠⢋⡜⣡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⢡⠎
⠀⠀⠀⠀⠀⠀⠀⣇⣀⡇⠀⠀⠀⠀⡴⢡⢎⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠞⡰⠃
⠀⠀⠀⠀⠀⠀⠀⢻⠈⢱⠀⠀⢀⠞⡰⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⡰⢃⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢆⡠⢋⡞⣡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠜⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠈⢡⠎⡴⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠠⠤⠤⠀⠀⡀⠶⠶⠒⠒⡋⣥⣔⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣃⣘⣓⡴⢾⠛⠏⠙⠋⣳⠶⠤⠤⠔⣒⣒⣺⠿⠭⠭⠥⠾⠖⠓⠚⠛⠉⠓⠀⠈⠉⠐⠀⠶⡠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠒⠚⣳⡾⠾⠿⠛⠛⠒⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠱⣖⣉⡹⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣍⠓⢤⣉⠒⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡄⠠⠤⢦⣐⣟⠫⠭⠧⠒⠒⠋⢩⠥⠶⡆⢀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢠⡇⠙⠲⢬⡙⠲⢤⣉⠓⠦⠄⠐⠒⣒⣂⣉⠭⠥⠴⠖⢒⣚⣋⣍⠡⣴⠄⠶⡛⠉⡏⠐⠝⢠⠀⠀⡇⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⡇⠀⢔⠶⣬⠑⠢⡌⠙⡏⠉⠉⢩⠤⠶⠲⢲⠀⣴⡄⡏⢠⣾⢁⣄⠀⢰⠴⣇⠀⣿⠀⣀⠸⠀⠀⡇⢠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⢜⣿⢾⡜⣇⠀⡇⡴⢧⠀⠀⢸⠀⠀⠀⢸⠀⠉⠨⡇⢸⡟⠈⠁⣠⠀⢀⠈⢦⢿⠘⠋⢠⠀⢀⡇⢀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⡸⣇⢀⢇⡏⠀⡇⠙⡏⠀⠀⢸⠀⠀⠀⢸⠀⣶⠶⣃⣿⠁⣰⢖⣿⣖⡻⣧⠎⢸⡠⣶⢸⠀⢸⣿⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠳⣄⡨⠭⠞⠁⠀⡇⢰⢳⡀⠀⢸⡀⠀⠀⢸⡇⠀⠐⡏⠻⢶⠣⣯⣽⣋⠵⡘⡆⣸⡇⠀⠸⠦⠞⠛⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⠓⠌⡙⠶⣄⠀⠀⡇⠈⣟⠀⠀⢸⡇⠀⠀⣾⡇⠿⠃⡇⠀⠈⣆⠣⣌⣉⠞⡰⠋⢉⣀⠤⠔⠒⠂⢉⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦⡈⠢⡈⠳⣄⣳⠰⣉⡆⠀⢸⡇⠀⠀⣿⡇⠀⠀⠷⠜⠒⣊⡱⠶⠶⠟⠓⣊⣡⡤⠴⠒⠊⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠢⢌⠛⠀⢿⠀⠀⠸⠷⠶⠟⢛⣁⡠⠤⠖⠋⠉⠀⠀⠤⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡉⠂⢸⠀⠤⠤⠒⠒⢉⣁⡠⠤⠔⠒⠋⠉⠛⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢸⠀⠀⣠⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠚⠋



⣶⣶⣶⣶⣾⡷⢿⣶⣶⡾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣷⡾⣷⠿⠿⠿⡷⢶⣶⠶⠶⠷⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣉⣉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⣾⡿⠟⠛⠿⣷⣦⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⣿⠋⠀⠀⠀⠀⠈⢻⣗⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠸⣿⠀⠀⠀⠀⠀⠀⠈⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢻⣧⣶⣶⡿⠿⠿⠿⠟⠷⠾⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣷⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⣍⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣉⠀⠀⠀⠀⢀⣀⠀⣀⠤⠴⠤⢬⣬⣿⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⣽⣤⣤⡶⡿⠋⠉⣍⡠⠄⠀⠠⠂⠴⠀⢠⣿⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠐⣰⡯⠗⠋⠀⠀⣹⣧⠆⠀⠀⠂⠀⢰⣿⣿⠋⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⠁⠀⠀⠀⢀⠔⠀⢀⡀⠀⢀⣀⣀⠸⢿⢻⠀⠈⡍⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠷⣄⢀⡴⠁⠀⣼⣿⣿⡿⠿⠷⠒⠒⣾⣏⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠈⠉⡀⠀⠀⠈⢻⣟⠀⠀⢠⠀⡀⢸⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⢿⡤⠀⠀⠀⠸⡿⠀⠀⠈⢉⡥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣻⣷⣦⣄⣀⡀⣀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠿⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢠⠒⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⢠⠖⠛⠉⠙⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⢠⡟⠀⡇⠀⠀⠀⠀⠀⢸⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠆⠀⠀⠀⠀⠀⠈⣿⠀⡆
⠀⠀⠀⠀⠀⠀⣸⣇⣠⣤⣤⣤⣤⣤⣤⣀⣇⣀⣇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣿⣀⣧
⠀⠀⠀⠀⢠⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢳⡄⠀⠀⠀⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢳
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠘⠛⠋⠭⠭⠟⠿⠿⠿⠶⠶⠦⢤⣤⣤⣀⡀⠀⠀⠀⠛⠛⠉⠭⠭⠿⠿⠿⠷⠶⠶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⠔⠚⡿⠓⣦⠀⠀⣤⡄⢉⣿⠁⠀⠀⠀⠴⣦⠀⠀⠀⠖⢛⣿⣷⡆⠀⠀⣤⡄⢉⡿
⠀⠀⠀⠀⠀⠢⠤⢤⣀⣀⣀⣀⣙⣉⣁⣀⣀⣤⣥⡾⠁⠀⠀⠀⠐⠤⠤⣤⣀⣀⣀⣀⣙⣉⣀⣀⣀⣬⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠘⣿⠔⠒⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⠥⣄⣀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⣠⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠈⠉⢒⠒⠤⡥⣀⣀⠀⠀⠀⠈⡙⠋⠀⠀⠁⠐⠂⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢂⠀⠀⠀⠀⠑⠀⠀⠉⠉⠓⠒⠤⠄⣀⢀⠀⠀⠠⠄⠉⠲⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠁⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⢒⢶⠁⡄⠐⠒⠈⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⣀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢸⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⢋⣬⠝⣦⢜⡣⠄⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⡀⠄⠀⢠⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡼⢴⠚⣥⢻⡞⢰⢃⠴⡶⡟⡏⡯⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠁⠐⠀⢸⠀⠁
⠀⠀⠀⠀⢀⡠⠤⠒⠐⣻⠃⣷⣀⠡⢭⠟⢶⣈⡓⠲⠐⢨⡖⣏⣴⠂⢫⠓⢆⠀⠀⠀⠀⠀⠀⠘⡆⠀⠠⡞
⠀⠀⢸⡗⠈⠁⠀⣔⠋⣹⣻⠁⣏⣩⡊⣏⣀⡿⣀⢼⠋⡴⠙⣡⣯⢐⡂⣠⣡⣷⣢⡤⠛⠏⢰⡄⢸⡄⢀⢡
⠀⠀⠠⠀⢲⠂⠄⠉⠉⠭⠳⢴⢳⣴⢫⠛⣛⢜⠧⡤⢿⠰⢚⣛⣺⢁⣍⣩⣉⡵⢷⠛⣤⣀⣠⢗⣤⡺⣈⠄
⠀⠀⠐⠀⠸⠀⠀⠀⢘⠁⣤⣀⡁⠉⠛⠻⣭⣭⣧⣤⢯⣉⣎⠍⠳⡴⠷⠶⣏⠒⠀⠀⢻⣬⡹⠂⠉⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢨⡔⠋⢠⣶⣻⢝⡿⠹⡩⠧⢤⠟⣒⠚⢳⠳⣲⠚⡶⠛⢫⠍⠀⠀⠄⡖⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⣯⠝⠁⠈⠉⠀⠰⡈⡓⠴⣯⣉⡷⡤⠞⢋⣭⠀⠘⠃⠀⠀⠀⡇⢸
⠀⠀⠀⣄⠤⠂⠀⠀⠀⠀⠀⠀⡾⠷⠀⠀⠀⠀⠀⠀⠴⠊⠀⠐⠀⠨⠠⠀⠩⠄⠀⠀⠀⠀⠀⢀⡟⠇
⠀⠀⠀⠈⠀⠄⠁⠀⢐⠀⡀⠀⢻⣷⠀⠀⠀⠀⠀⠀⢰⢃⡆⠀⠀⠀⠀⠀⢸⠩⠀⠀⠀⠀⠀⢸⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠸⡅⠀⠀⠨⢯⣵⠀⠀⠀⠀⠀⠀⢭⠇⠀⠀⠀⠄⠀⡏⡘⠀⠀⠀⠀⠀⣨⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣤⡆⠂⡘⣿⢲⠄⠀⠀⠀⠀⠰⣊⢇⡀⠃⠀⠀⣂⡀⠀⡀⣠⠴⡺⢅⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡆⠙⢋⣩⣿⢽⠅⢠⣤⣤⡀⠀⡗⢋⡗⡒⡆⡤⠗⢢⣖⡟⠓⡿⢾⠗⠻⣟⡼⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⣉⣾⡗⠘⣶⠊⢰⣄⡩⠖⢋⣽⠀⣥⠠⣍⣠⣗⠕⠹⣖⡳⠾⢫⠹⠏⠵⠶⠾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⣏⣠⢉⣉⠍⣳⣙⡤⢼⡶⠛⡗⠐⢳⡶⢻⠛⡾⡗⠐⡿⠹⣖⠛⣢⡘⠓⠖⠦⠤⠵⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣍⣉⡴⠚⣰⣉⠖⠔⢓⡶⢧⣤⢿⠒⢤⠇⢥⡞⠛⢷⠤⢯⢹⢻⡍⡟⡷⣂⠠⠛⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠭⠓⠒⠑⢋⠚⡭⢥⠒⠢⣾⣞⣷⠈⢠⢴⠛⠖⢫⡈⣋⣈⡿⣇⢉⠗⠣⣿⣉⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠦⠤⠋⠈⠀⠙⢿⣿⣡⣨⡧⣾⡟⠧⠤⠅⡀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠃⠀⠀⠀⠀⠀⠀⠁⠛⠀⠉⠀⠀⠀⠀⠀⠓⠒⠂

⠀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢀⣀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⠔⠋⠁⠀⠉⠓⢦⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢹⠀⠀⠀⠀⠀⠀⡠⠖⠊⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢘⡆⠀⠀⠀⠀⡞⠀⠀⡠⢤⡀⠀⢣⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢦⠀⠀⣇⠀⠸⣄⣠⠇⠀⣸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠢⡀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣨⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢧⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡞⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢇⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠐⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠸⣄⡀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠉⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⣠⡾⠻⡁⢳⣤⣶⣶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⣀⣴⠟⠉⢀⣠⠿⠛⠁⠀⠹⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠈⢧⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⢀⣤⠾⠻⣄⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⣠⡶⠛⠁⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⢀⡾⣟⠳⣦⣀⡴⢻⡟⠁
⠀⠀⠀⣠⠖⠋⠉⠉⠙⠛⠶⣦⣄⠈⠳⣦⡀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣠⣾⢦⣀⡿⢮⡆⠀⠉⠛⠋
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠙⢦⡀⣠⠞⠁⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠘⣮⠻⡉⠓⠚⠁
⠀⣸⠃⠀⠀⠀⢠⡴⠶⢦⣄⠀⠀⠀⠈⠳⣤⡼⠛⠁⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣠⠞⣻⠇
⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠤⠴⠃
⠀⣿⠀⠀⠀⢸⡿⡄⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⢀⣤⣾⠋
⠀⢹⣆⠀⠀⠈⣇⠹⣄⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⣧⠘⣧
⠀⠀⢻⡄⠀⠀⠘⣦⠘⢷⣄⠀⠀⠀⠈⣷⠀⠀⠀⠀⢹⡄⣿
⠀⠀⠀⢻⡄⠀⠀⠈⠳⣄⠉⠳⣄⡀⢀⡽⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠻⣆⠀⠀⠀⠈⠙⠓⠚⠛⠋⠁⠀⠀⠀⢀⣿⡾⠃
⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⣀⣀⣀⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠋⠉⠉⠉⠉⠉⠉⠑⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⣀⡤⠴⠒⢚⣉⣉⣉⣉⣉⣉⣉⡙⠒⠲⠤⣄⡈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⡠⠖⢋⡡⠔⠚⠉⠉⠀⠀⡄⠀⠤⣄⠀⠈⠉⠙⠒⠤⣉⠓⠦⣀⠉⠲⣄
⠀⠀⠀⠀⠀⢀⡴⠋⣀⠔⢉⡴⠊⡅⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢠⡄⠉⠢⣌⠑⢤⡈⠳⣄
⠀⠀⠀⠀⣠⠋⢀⠜⢁⠔⠁⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⢄⠙⢆⠈⠣⡀
⠀⠀⠀⡼⠁⡰⠃⡴⠁⠀⠀⠀⠀⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠱⣄⠳⡀⠙⡄
⠀⠀⡜⠀⡼⢁⠞⢠⠀⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⢆⠹⡄⠘⡄
⠀⡼⠀⡼⢁⠎⠀⢸⠀⣯⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢆⠹⡄⠸⡄
⢠⠃⢠⠃⡞⠀⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⡆⢳⠀⢳
⡼⠀⡞⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢳⠘⡆⠸
⡇⠀⡇⢸⠀⢀⠄⢦⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⣀⣀⣤⠤⠤⠶⠒⠒⠻⠖⠉⠀⠐⠦⠀⢸⠀⡇
⡇⠀⡇⢸⠀⠈⢃⠟⠛⠧⠼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢄⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢘⠀⡇
⡇⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⡇
⢫⠀⢧⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇⢰
⠘⡄⠘⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠃⡼⠀⡜
⠀⢳⠀⢱⡈⢇⠀⠀⢦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠗⢀⠏⣰⠃⢰⠃
⠀⠀⢣⠀⠳⡈⢦⠀⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⠃⢠⠃
⠀⠀⠀⢳⡀⠱⣄⠳⡄⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠂⠀⠀⠀⡴⢃⡴⠁⣰⠃
⠀⠀⠀⠀⠙⣄⠈⠢⡈⠢⣀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⢀⡠⠊⡠⠊⢀⡜⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠈⠢⣌⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢤⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡤⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠑⠢⣌⡑⠲⠤⣄⣀⡀⠀⠀⠣⠘⠀⠀⣀⣀⡤⠴⠒⣉⡤⠖⠉⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡉⠑⠒⠤⠤⣍⣉⣉⣉⣉⣉⣉⡥⠤⠔⠒⠉⢁⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⢤⣀⣀⣀⣀⣀⣀⠤⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⣰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⢠⣷⣶⣤⣤⠀⠀⠀⣀⣀⣠⡀⠀⠀⢠⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⡿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⣠⠄⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⠄⣄⣀⣀⣀⡀
⠀⠀⢠⠞⡤⠚⠋⠉⠓⠀⡉⢿⣛⣛⣛⠻⠿⠯⠉⠉⠉⠉⠉⠙⠓⠒⢲⣶⡶⠤⠤⠤⢤⣄⣀⣀
⠀⠀⡏⡰⠀⠀⠀⠀⠀⠀⣙⠤⠛⣖⠒⠒⠒⠒⠶⠤⠤⠤⠤⠤⢤⣤⣻⣟⣁⣀⣀⣀⠀⠤⠤⢬⣽⣗⣦⣄
⠀⠀⠇⣧⡄⠀⠀⢀⡤⠚⠁⢀⠴⠋⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠮⡭⠭⠭⠭⢭⣿⣟⣛⣻⣧
⠀⠀⡆⢻⣿⡾⠖⠁⢀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⣀⡠⠞⣡⠴⠋
⠀⠀⢸⡜⠋⠀⣠⣴⣯⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⣉⠴⠊⠁
⠀⠀⠀⠱⣤⣊⣉⣁⣤⣤⣤⣀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠭⠿⠿⠛⠿⠿⢿⣿⣾⣷⣮⣁⡀
⠀⠀⠀⠀⡏⡛⠿⣶⣟⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣋⣛⣉⣛⣲⣶⣤⣄⣀
⠀⠀⠀⠀⡇⡏⢳⣬⡙⠻⣦⢄⣀⣀⣀⣠⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠖⢚⣛⣛⣛⣛⣭⡭⠭⠭⠭⠭⠩⠭⢵⡦⢸
⠀⠀⠀⠀⡇⡇⡶⠈⠋⠓⡀⢸⠉⣭⣥⡤⠤⠤⠶⠖⠒⠒⠂⠘⠉⠉⠉⢩⣥⣦⡄⠀⠀⠀⠀⠈⠉⠁⠐⠒⢾⡇⢸
⠀⠀⠀⢠⣇⡈⠳⠦⣿⠈⡃⣼⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⡀
⠀⠀⠀⠀⡇⣧⡀⠀⠀⠀⡇⠛⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠇⣸⡇
⠀⠀⠀⠀⠈⠪⡑⢄⠀⠀⡇⣷⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠴⠒⠒⢋⣉⣩⠤⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠲⣕⢤⡇⢻⢸⣇⠀⠀⠀⠀⣀⣀⣠⠤⠴⠆⢀⣉⣉⡥⠤⠆⠐⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⢈⠘⠟⠒⠒⢋⣉⡭⠤⠴⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠒⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⡠⠐⠁⢀⣠⣤⠶⠒⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⣤⣀⠈⠒⢄
⠀⠀⠀⠀⢀⠊⠀⣠⡴⠝⣩⠴⣚⣉⢙⡍⡑⣦⡀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠑⢄
⠀⠀⠀⡶⢡⣀⣼⠏⢀⠀⡵⠾⠛⠉⠙⠻⠽⣷⣵⠀⠀⠀⠈⠁⠒⠀⠀⠘⢷⡄⠀⢣
⠀⣘⣽⠀⠀⣼⠃⠀⠀⠂⢃⠀⠀⠀⠀⠀⠀⣡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢃
⠀⢷⡏⠀⢰⠇⠀⠀⠀⠀⠀⠉⠒⠀⠐⠒⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⡀
⠸⢿⡃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⡇
⠀⢹⣷⠀⠸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⡘⠀⢐⡇
⠈⡼⣟⣄⠀⠑⡀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠀⢀⡎⠁
⠀⢥⢿⡘⣦⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠁⢀⢀⠀⠀⡎⡸
⠀⠐⠈⢧⡅⠳⢄⠊⠈⠁⠀⠄⠀⠀⣀⣀⠀⢀⠀⠄⠒⠉⠀⠀⠀⠀⠀⠀⢠⣀⠀⠈⠣⡀
⠀⠀⠈⠁⣙⢶⣤⣉⡖⠦⠤⣄⣀⣀⠀⠀⠁⠀⠀⠀⢀⣠⡍⠂⢲⡶⣄⠀⠘⢯⡳⣄⠀⠈⢢
⠀⠀⠀⠀⠀⠈⠡⠝⠿⡗⢶⣦⣴⣭⣭⣉⣉⣏⣹⡄⠀⠻⢿⣆⠀⠙⢾⡙⣦⡀⠻⣬⠳⣔⠠⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠿⠽⠾⢛⢿⣞⢦⡤⡷⣿⣦⠀⠀⠙⢮⣛⢦⡈⠻⣜⢧⡀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠙⣛⡛⢉⣻⣮⡳⣤⡀⢠⡑⢷⣝⢦⡈⠳⣙⢦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢝⣿⣮⣻⣦⣀⣀⡙⠾⠛⠂⠈⠺⡛⠆⠀⠙⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠍⠻⠿⢿⣏⣿⠀⠘⢫⣘⣆⡈⠲⡷⣿⠗⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡿⣦⠀⠀⠙⢷⣹⢦⡈⣪⣕⢄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣙⣿⡷⢤⣤⣤⣝⢮⡳⣮⠙⢮⡵⣄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠓⠚⠿⢯⣷⡙⢇⡝⣄⠙⢮⡳⣀⢂⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢻⠀⠀⡑⢥⡳⣄⠹⣎⠲⣄⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣳⣄⠉⠀⠻⣮⡣⡈⠳⣝⢷⣄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⢿⣳⡤⡤⢬⣷⡝⢦⠬⢷⣯⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠽⡿⠻⡿⡟⢾⢯⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠉⠙⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢠⡴⠋⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡤⢟⣀⣠⣤⠤⠤⠶⠦⠤⢤⣤⣀⣘⠧⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⠷⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢯⣇
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠒⠒⠲⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⢀⣤⠞⢋⣡⡴⠶⠚⠛⠛⠛⠛⠲⠶⣤⣉⠛⢦⣄⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⣰⠋⠀⠀⠀⠀⢀⡴⠋⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡈⠳⣄⠀⠀⠀⠀⠈⢳⡀
⠀⣰⠃⠀⠀⠀⠀⢠⠞⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠘⢧⠀⠀⠀⠀⠀⢳⡀
⢠⡏⠀⠀⠀⠀⢠⠏⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣧⠀⠀⠀⠀⠈⣧
⣼⠀⠀⠀⠀⠀⡿⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠸⡇⠀⠀⠀⠀⢸
⡟⠀⠀⠀⠀⢸⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⣷⠀⠀⠀⠀⠘⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡏⠀⠀⠀⠀⢰
⢸⡄⠀⠀⠀⠀⢻⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⠃⠀⠀⠀⠀⣼
⠀⣧⠀⠀⠀⠀⠈⢷⡈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⣰⠏⣰⠏⠀⠀⠀⠀⢠⡇
⠀⠘⣧⠀⠀⠀⠀⠀⠳⣄⠹⣦⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣠⡾⢁⡴⠃⠀⠀⠀⠀⢠⡟
⠀⠀⠘⢧⠀⠀⠀⠀⠀⠈⠳⣄⠙⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⢁⡴⠋⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠙⠶⣤⣉⣙⠛⠒⠒⠒⠚⢛⣉⣡⡴⠞⠉⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢤⣄⣀⣀⣀⠀⢀⣀⣀⣀⣤⠴⠶⠛⠉
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠒⣶⠒⠦⣤⣀⠀⠀⢰⢀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⣀⡔⠾⣀⠀⠉⠀⠀⠀⠙⢷⣄⣈⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢤⣀⣬⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠹⣿⠂⠀⠀⡱⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⣸⣿⣿⣿⣿⡌⠉⠀⠀⠀⠀⠀⠀⠀⢡⢴⣿⣶⠐⠃⠱⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⣠⣠⣶⣿⢏⣿⣿⣿⡆⢀⡀⠀⢀⠠⠤⠾⣞⠁⠀⠀⢿⡅⠆⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡤⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣾⡓⠀⠀⠀⠀⠀⢫⠀⠀⠉⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⣀⣠⣿⣖⣻⣿⣾⣿⣟⣿⣿⣿⣿⣿⣶⡦⠀⠀⠀⠘⠄⠀⢢⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠸⣶⠀⢸⣦⡾⣿⣤⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠛⠛⠋⠀⠈⠉⠉⠉⣉⣉⣭⣽⣿⣬⣽⠙⠓⠒⠒⠛⠛⠁⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢒⡒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣠⠄⠠⠠⠼⠟⠁⣸⠋⣁⠀⠈⠚⠃⠀⠉⠐⠀⠀⠀⠠⠀⢀⡀⢀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠰⠟⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠬⢬⡀⠲⠄⠀⠀⠀⠀⠞⠁⠪⠀⠀⣏⠀⠀⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀⡀⠀⢠⣿⣿⣇⣡⠀⠀⠀⣈⠍⠀⡼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢠⣟⣿⣿⣿⣾⠿⠃⠈⠁⠘⣀⣾⠎⠀⠜⣷⡮⠁
⠀⠀⠀⠐⠘⠈⠘⠁⠀⢀⡀⠀⠀⠀⠀⠀⠊⠀⠾⢣⣤⣶⣆⣀⣠⣤⣤⠀⠀⠀⠀⠀⠄⠀⢀⣤⣿⡿⠏⢂⠐⠀⡀⠠⣤⣛⣻⠎⠀⡴⠀⠈⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣅⠀⠀⠀⠔⣚⣓⠠⠴⠤⡤⣬⣉⣉⣙⡛⠛⠃⠿⠿⠿⣦⣤⣄⣾⡟⡋⠂⢀⡀⢥⠉⠀⢠⣿⣿⠏⠀⠜⡟⢠⠃
⠀⠀⠀⠀⠀⠀⢠⣵⡉⠖⠋⣭⣦⢻⣾⡟⣶⠄⠀⢀⣀⠀⢀⡬⠋⣁⣀⣀⡀⠐⠀⠠⠤⢠⣬⣀⣉⣴⢈⠉⣦⡔⠠⠾⠿⠋⠀⣤⡀⣡⠃
⠀⠀⢀⠠⠀⠀⢡⣭⣹⠆⠀⠘⣯⡆⠈⣥⣠⡾⢦⢼⣣⣤⣀⠀⠀⣠⣿⣿⠿⣿⣟⡷⣶⣃⢀⡤⣭⠁⠁⠰⠗⡳⠖⠀⠤⢤⣾⣿⣷⠃
⠀⠀⢸⠐⣦⢠⣤⠀⠀⠈⠁⠚⠓⠟⠾⠿⣯⣿⣃⣶⣻⣿⢧⠷⣾⣿⠥⣻⡀⡇⢉⣷⠛⣛⡋⠀⠠⢿⡃⠵⠫⠅⣀⡤⣶⣿⠿⠛⠁
⠀⠀⢘⠀⡿⢸⣿⠀⠀⠀⠀⠉⣶⣶⠀⠀⠄⢀⠉⠁⠘⠛⠘⠪⠥⢀⣠⣍⢿⣉⣴⡨⢒⣂⣀⠰⠎⠁⢀⣤⣴⣿⡿⠛⣿⣿
⠀⠀⢸⠀⡁⢸⣿⠀⠀⠀⡅⠀⣿⣿⠀⠀⢀⡄⠀⠀⠀⠀⠂⢠⣤⣀⣀⠈⠈⠉⠓⠛⠋⠁⢀⣠⣴⣾⣿⣿⣿⣿⣧⣦⣿⣿
⠀⠄⣬⠀⠁⢸⣿⠀⠀⠀⡇⠀⣹⣿⠀⠀⣿⡅⣦⠀⠀⠀⠀⡜⠛⠷⠀⠈⢡⣷⢲⡖⠒⢲⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣤⡤⠤⢤⠤⡤⠄
⠀⠀⡿⠀⠀⢸⡿⠀⠀⠀⣿⠀⢉⣿⡄⠀⠈⠀⠸⣧⡀⠀⠀⠅⠀⠀⠀⠀⠸⣿⢻⡇⢸⡆⠻⣛⣩⣄⢠⣿⣿⣿⣿⣷⣻⣿⣿⣿⡷⣶⣗⠖⠒⠀⢀
⠀⢀⢃⠀⠀⢸⡇⠀⠀⠀⣿⠀⠈⢻⣷⠸⣤⣴⣶⣿⠀⠀⠀⢉⡃⠀⠀⠀⠀⠀⢸⡇⢸⣧⡖⣿⣿⣟⠠⢈⣭⣿⣿⣿⣿⣿⣿⣿⣷⣟⣿⣛⣋⣉⠉
⠰⠬⢽⣿⣖⣾⣦⣤⣀⣀⣿⠀⠀⠘⣿⠀⠉⣿⡇⠈⠅⠈⠀⠿⡀⢸⡇⣤⠀⠀⢸⣿⠈⣿⣿⣹⣿⠿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣻⡭⣭⣤
⠠⠠⢾⢯⣿⢽⣿⣽⣯⣿⣿⣿⣶⣶⣿⣄⣀⡈⠁⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆⠀⢸⡇⠀⣿⣿⠋⢛⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶
⠘⠀⠲⠶⠿⠿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀⠸⠇⠟⣑⠀⢸⡇⠀⣿⣿⣠⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣟⣒⡚⢛
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠯⠤⣜⣇⣰⡿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠉⠀⠀⠀⡀⠀⠠⠄⠀⢀⠀⠀⠈⠉⠙⠛⠿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⡀⠄⠒⠀⠀⠀⠀⠠⣤⠀⠠⠄⢦⡄⠀⠈⠀⠃⠠⠀⠀⠉⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⢀⠄⠈⠀⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿⠀⢀⣀⣋⣀⠀⠀⠀⠀⠀⠐⣷⠀⠀⠀⠀⢤⠀⠙⢿⣷⡀
⠀⠀⠀⠀⢀⣼⡿⠋⠀⠄⠀⠀⠀⠀⠀⢸⡀⠀⠀⣇⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠄⠀⠙⣿⣦
⠀⠀⠀⢠⣾⡟⠁⠠⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢿⣷⡀
⠀⠀⢠⣿⠏⠀⠦⠀⢠⣤⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠆⠀⢻⣷⡀
⠀⢀⣿⡏⠀⡠⠀⠀⢸⡏⠀⣸⡃⠀⢸⡇⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠤⠀⢻⣷
⠀⣼⡟⠀⢠⠀⠀⠀⢸⣇⠀⢹⡇⠀⢸⠇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⡀⠈⣿⣇
⢰⣿⠃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠚⠛⠀⠀⠀⢀⠀⢸⣿
⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⣿⡇⠀⠀⠀⣠⠂⠘⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠃⠀⢿
⣿⡇⠀⠆⠀⠙⠶⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⣬⡀⠀⠀⠶⠀⢹
⣿⡇⠀⢀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣕⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⡄⠀⣿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠐⣿⣿⣿⡟⠁⢀⠀⠀⣿
⠸⣿⡄⠀⠀⠀⠀⠀⢀⡀⢀⡐⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣦⣤⣴⣶⣾⣿⣿⣿⣦⣈⠀⢸⣿
⠀⢻⣷⠀⠙⡀⠀⠀⠸⣆⣠⠃⠙⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⡏
⠀⠈⢿⣇⠀⠐⠀⠀⣴⠉⠙⣷⣰⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣿⣦⠀⠖⠀⠈⠓⠚⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠈⢿⣧⡀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠲⠀⠀⠀⠈⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠘⣿⡧
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠃⢀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⠠⣿⣿⣿⣿⣿⣿⣿⢠⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠁⢾⣿⣿⣿⣿⣷⡀⠀⠀⠀⢰⡇⠈⢻⡆⠀⠀⠀⠀⠀⠘⠂⠈⠁⢠⠀⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠓⠀⠚⠁⠀⠀⠀⢀⠀⠤⠐⠈⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣧⠁⠉⠀⠘⠂⠀⠉⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋⠁⣼⣿⣿⣿⢿⣿⣿⡏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⣷⣶⣶⣶⣶⣾⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣏⣻⣿⣦⣈⣏⣙⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣑⣾⣿⣛⣿⣁⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⢀⠀⣀⣀⠀⠀⢀⢀⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣿⡿⢿⣿⣭⠟⠋⢿⣿⡿⠿⡿⣿⡿⣿⣿⡟⡹⡟⠿⣛⣛⣉⣙⡛⣛⣛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡋⣿⢿⣿⣿⣿⣟⡵⣾⣷⣭⣿⡶⠍⠗⠤⣜⣿⢷⡾⠰⣜⣟⠻⣿⡽⢿⡿⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣾⣿⣷⣶⣿⣿⣏⣿⣿⣯⣟⣿⣾⣿⡶⠾⣿⣇⣠⠓⣾⣿⣿⣷⣬⠙⣋⠴⣿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠈⣧⣣⣟⡛⠛⠛⠻⠻⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠟⢹⡟⠆⠙⢛⣛⣓⣛⣛⣛⣋⣉⣭⣬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⣿⠏⠉⠉⠛⢳⣼⣴⣦⣶⣤⢤⣤⡤⣤⣽⣍⣹⣿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣆⡀⢠⣴⢪⣿⡏⣿⢾⡾⠷⣠⡾⣿⣿⡟⢻⣿⣦⣿⠿⠿⣛⡩⠷⠺⢟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⠾⢾⡿⡶⣿⡇⣿⣏⠟⠂⠀⠁⣶⣾⣷⣡⣿⣿⣿⠋⣁⣤⣤⣴⣷⣾⣥⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⣿⣿⣖⣛⣏⣷⣷⢸⡇⣀⣤⣿⣿⢿⣿⡿⢙⢿⣿⣿⣾⣶⣚⡽⠻⠟⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠈⠉⠉⣛⢸⠘⠻⠿⠭⢿⣿⣭⣝⣿⣦⣾⣿⣿⣿⣧⣶⣶⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠂⠤⠄⣹⣹⣇⣪⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠳⢶⣤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⡟⠁⢀⣠⣒⡒⠢⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣦⣀
⠀⢀⣿⠄⠀⣾⠛⠉⠙⣧⢸⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠷⣄
⠀⢸⣿⢠⠀⢻⣄⠀⢀⣿⠟⠀⠀⢠⡟⠉⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠙⢷⡀
⠀⠸⣿⣤⠃⠀⠉⠙⠉⠁⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢸⣷
⠀⠀⣿⡟⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡄⣿
⠀⢠⡿⠀⠀⢠⡾⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠘⣰⡟
⢀⣾⠁⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢀⣰⡿
⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠁⠀⠀⠀⠀⢸⣿⠁
⣿⡀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⡁⣮⣷⢄⠀⠀⠀⠸⣿⣆
⢻⣧⠀⠀⠀⠀⠙⠤⢄⣀⡀⠀⠀⠐⠈⠀⠀⠀⠐⠘⣏⠢⡱⡄⠀⠀⠈⠛⠶⠶⣦⡀
⠈⢷⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢣⡑⢌⠢⠀⠀⠀⠀⠀⢾⣷
⠀⠈⠛⠶⣷⣦⣤⡤⠀⢀⣀⣈⣩⣌⣉⡲⣄⠀⠰⡉⠀⠑⡙⢌⠢⠑⠀⠀⠀⣇⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠿⣿⣧⠀⠙⢄⠀⠘⢎⠳⣄⢌⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠈⠢⡀⠀⠣⡙⢦⡡⡑⠄⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣦⡑⢄⠀⠀⢄⠳⣌⢈⠢⡀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣎⣀⠤⠤⡑⣜⢦⡑⠑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣧⠈⠢⡱⣄⠈⠢⡀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠤⢀⠈⢮⠳⡀⢄⢄⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣦⡑⡄⠑⢝⢦⡣⡠⡀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠠⡱⣜⢎⠠⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣄⡀⣀⠈⢌⢣⡑⡑⢄⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⣬⣅⠀⠱⡝⢮⡢⠣⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣬⣉⡋⢢⡳⣔⢄⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣇⣀⣱⣌⢷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣶⣾⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⢛⣉⣤⠤⠴⠒⠒⠚⠋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠛⠒⠲⠤⣌⡉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢁⡤⠚⠉⣀⣠⠴⠖⢒⣋⣉⣉⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣭⣉⣉⣛⠲⠦⣄⡉⠓⢦⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢠⡞⢁⡴⠋⣠⠔⠋⣡⡴⠖⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠷⣦⡉⠲⣄⠙⢦⠈⢷⡄
⠀⠀⠀⠀⠀⢠⡟⢀⡞⠀⡴⢁⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⢧⠈⢧⠈⣿⡀
⠀⠀⠀⠀⠀⣼⠃⣼⠀⣼⠁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⡇⠘⡇⢸⡇
⠀⠀⠀⠀⠀⣿⠀⡏⠀⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⡇⢸⡇
⠀⠀⠀⠀⠀⣿⡀⢿⠀⢿⠘⣷⠀⠀⣠⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠾⢷⣶⡿⠛⢛⣀⡏⢀⡇⢸⡇
⠀⠀⠀⠀⠀⠸⣧⠘⣆⠈⣦⣸⡷⠟⢉⣀⣀⠉⠙⠛⠿⠿⠶⢶⣶⣶⣶⣶⡶⠶⠶⠟⠛⠉⢁⣀⡤⢤⡈⠛⠿⣯⣅⡀⢺⠀⣿⡄
⠀⠀⠀⠀⠀⣀⣻⣶⠾⠟⠋⣁⡤⠞⠋⠀⠈⠙⠓⠲⠶⠤⣤⣤⣀⣀⣀⣠⣤⡤⠤⠶⠖⠛⠉⠁⠀⠀⠉⠓⠦⣌⡙⠛⠿⢷⣾⣷⣀⡀
⠀⠀⠀⠀⢿⡏⢁⣤⠴⠚⢋⡵⠶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠻⣿⡙⠓⠶⡄⢉⣿⠁
⠀⠀⠀⠀⠈⣿⠘⡇⠀⠀⠸⣷⣤⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣴⣿⠃⠀⠀⡇⢸⡟
⠀⠀⠀⠀⠀⣿⠀⣷⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⣿⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⣧
⠀⠀⠀⠀⢰⣿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣉⣭⣉⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢿⡀
⠀⠀⠀⠀⣸⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⣿⣿⣿⣿⣷⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⣇
⠀⠀⠀⠀⣿⢰⡇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⢿⣿⣿⣿⡟⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⢿⣦⢹⠀⣿
⠀⠀⠀⠀⣿⢸⠀⣰⡟⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⣿⣿⣿⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣿⡿⢸⡆⣿
⠀⠀⠀⠀⣿⢸⡄⠘⢷⣶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⣿⣿⣿⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢸⠃⣿
⠀⠀⠀⠀⢻⣆⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣼⣸⣿⣿⣿⣧⢻⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡏
⠀⠀⠀⠀⠈⢻⣆⠙⣦⡀⠀⠀⠀⠀⠀⠀⣾⠋⠙⣿⣇⠀⠀⢀⣧⣿⣿⣿⣿⣿⡜⡇⠀⠀⣼⡉⢉⣿⡇⠀⠀⠀⠀⠀⢀⡴⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠙⢷⣤⡙⠶⢤⣀⡀⠀⠀⠻⣿⣿⡿⠃⠀⠀⢸⣥⣤⣤⣤⣤⡤⢤⠷⠀⠀⠘⠿⠿⠟⠁⣀⣀⡤⠴⠚⢉⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣬⣉⣙⡓⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠛⣉⣉⣁⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣀⣈⠐⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠉⠙⠻⢷⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⡟⠀⠀⠀⠀⠀⢤⣤⣤⣄⣀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡘⠀⠀⠀⡰⢻⣆⣌⢛⣦⠀⠘⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢰⣧⣶⣠⣦⣷⣿⣼⣿⣿⣿⠀⢈⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣿⣦⠀⢸⡏⠸⡏⠀⢹⣿⣿⡇⠐⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡇⠀⡇⠀⢸⣿⣿⣷⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣅⣤⣡⣤⣬⣭⣭⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠋⠻⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠾⠷⠒⠒⠛⠛⠛⠛⠿⠦⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠀⠀⠠⠄⢀⡀⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠴⠊⠀⠀⠀⠀⣠⠔⠚⡛⠥⠐⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⢐⠵⠁⠀⠀⠀⡀⠐⢉⠤⠂⠁⠐⠂⠀⠀⠀⠀⡶⠀⡷⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣀⡀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⣠⡖⠡⠀⠐⣄⢳⡀⠀⠀⠀⠀⢠⡄⣿⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠙⢷⣝⢦⡀⠀⠀⠈⢿⣆
⠀⠀⠀⡰⠁⠀⠀⠀⣴⢃⡴⠃⠈⠀⠀⠀⠈⠳⣳⡀⠀⠀⠀⠘⠓⠛⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡙⢶⣿⣦⡀⠄⠈⢿⣆
⠀⠀⡴⠀⠀⠀⢀⡾⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠔⠈⠀⠈⢻⣿⣿⣶⣆⠀⢿⣆
⠀⢠⠃⠀⠀⠀⠞⠠⠀⠀⠒⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣡⣴⠿⠛⠀⠀⢻⣿⣿⣿⣶⠘⣿⡄
⠀⡄⠀⠀⠀⡘⢀⠁⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⣠⠖⠉⠀⠘⠋⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄⢻⣧
⡀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠸⣿
⠃⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⠀⣿⡇⢿⡇⠐⣿
⠀⠀⠀⠐⣼⡇⠰⠀⠬⠍⠉⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠿⢀⡀⣿⣟⢸⣷⡀⣿
⠀⠀⠀⠘⡎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⡿⣌⠧⠀⣿
⢠⠀⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣸⣿⡇⢟⠁⢀⣿
⠈⡤⠀⠀⢹⡿⢣⠀⠀⢀⣄⢖⣫⢱⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⣄⠀⠀⠀⠀⠀⠀⣾⣿⣶⣤⣦⡀⠀⢠⣿⡻⠼⠞⠀⣸⡏
⠀⠐⢀⠀⠘⣷⡉⣧⡀⡈⠻⣋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠉⠛⠶⠙⠀⢠⡿⡷⢡⠚⠀⣸⡟
⠀⠀⠀⢂⠀⢻⣷⡹⢿⡇⠀⠋⠀⠀⠀⠀⠀⣼⠷⠀⠀⠀⠀⢘⣧⡀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣏⡟⣡⠎⠀⣰⡟⠁
⠀⠀⠀⠀⡣⡈⣿⣿⣌⠢⣾⣦⠀⠀⠀⢀⣤⠏⠗⠀⠀⠘⠃⠈⠛⠉⠀⠀⠀⠈⠀⠻⣄⠀⠀⠠⠄⣠⣾⣯⢏⣴⠁⠀⣰⠟
⠀⠀⠀⠀⠘⢷⠈⢿⣿⣷⣝⠿⣷⣄⠀⠈⡉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⠂⢀⣤⣾⣯⠟⣡⠟⠁⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢂⠉⢿⣿⣿⣮⡝⢷⣦⢄⡀⡀⠀⠀⠀⠘⠂⠰⡆⡀⠀⠠⠀⡀⣠⡴⣞⠿⠟⢋⣴⡟⠉⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢟⢻⣿⣿⣿⣷⣮⣘⣤⠉⡀⠐⠒⠂⠤⣤⣤⣶⣶⣶⣿⣿⠿⢞⣋⣴⣶⡿⠗⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣍⣉⣁⣀⣠⣤⣤⣶⣶⣿⢿⢛⣉⢤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣥⡴⠞⠋⠁

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⣠⣾⡿⣿⡏⠉⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⡆⢻⡇
⠀⣰⡿⢿⡧⠈⢾⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢻⠀⢸⡇
⢠⡟⢀⣾⣧⠀⠈⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡄⢸⣧
⣾⠇⢸⠏⡇⠀⠀⣯⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢸⡇⢸⣿
⣿⠀⣿⠀⡇⠀⠀⢿⠀⢸⣿⣿⣿⣿⣭⣯⣭⣭⣭⣭⣭⣭⣉⣽⡇⠁⠘⠀⣿⠘⡇⠸⣿
⢻⡇⢹⣧⡇⠀⠸⡧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣼⣿⡀⠀⠀⣿⠀⣿⠀⣿
⠸⣧⠈⢿⣷⠀⢦⣷⠀⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠛⠛⢛⠻⣷⣄⠀⢿⡄⣿⠀⣿
⠀⢻⣆⢸⣿⠀⢈⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠀⢿⡆
⠀⠀⠙⢿⣿⠀⢦⣻⠀⢸⡿⠛⠛⠛⠛⠛⠛⠛⠉⠉⢉⣉⣉⣁⣀⣈⣉⣉⣁⣉⣹⡇⢸⡇
⠀⠀⠀⠀⠙⢿⣷⣿⡄⠘⠛⠛⠛⣛⣛⣋⣉⣉⣉⣉⣉⣉⣭⣭⣥⣤⣤⣤⣤⡤⠤⠤⢾⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⠛⣿⣏⣭⣭⣭⣭⣭⣭⣭⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠈⠻⣿⣿⣿⣿⣿⣿⠟⣉⣉⣹⡿⠋⠉⢉⣻⣿⣭⣭⡿⠛⢿⣆⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣄⡈⠻⣿⡏⢍⣿⠟⠩⠉⣙⣿⣿⠫⠍⠉⢻⣦⣀⣹⠶⠶⠿⢷⣄⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⠻⣿⣦⡈⠻⣷⣿⡐⠂⠐⣢⡾⠷⢔⣒⣒⣿⡿⠛⣷⣤⣀⣈⣼⣿⣷⡈⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣧⡙⢿⣿⣦⡀⠻⠿⠿⠿⠷⠶⠿⠛⠿⠟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⡻⢷⣍⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢷⣤⣿⣷⣅⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠆⢠⣶⣶⣶⣶⣶⠶⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣎⢻⣦⣽⣷⣼⣿⡿⠿⢿⣿⣿⠿⠿⠿⣿⡇⠀⣸⣧⠀⢸⣿⣚⣛⣭⣿⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡙⠳⣭⣹⣾⡟⣿⡇⠀⢸⣿⣶⠬⠭⢥⣾⡇⠀⠛⠛⠃⢸⣿⣉⣽⢽⣿⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣬⡙⢿⣧⣿⡇⢰⢸⡿⠿⠷⠶⠦⠿⠷⠶⠶⠒⠒⣺⣛⣲⣶⣻⣿⣄⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡙⣿⣿⣾⣾⣿⣿⣿⣯⣽⣯⣤⣬⣭⣭⣭⣽⠶⠶⠶⠾⠿⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⡟⡀⠀⠀⠂⠀⠀⠀⢄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠑⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⢀⢀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠙⢳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⣄⢢⠰⠹⢸⢸⢘⣸⣖⡁⡷⠀⡆⡀⡀⠀⠀⠈⢙⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢴⢦⡳⣌⡶⠷⠛⠉⠈⠉⠉⠉⠉⠙⠓⠷⠤⢡⢀⠀⠀⠈⢱⣻⣦
⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢌⠢⢕⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⡄⠀⠘⡧⠽⣆
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠌⣂⣡⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢨⣥⣾⡆
⠀⠀⠀⠀⠀⣼⠀⠀⠀⢤⡙⠭⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⡆⢒⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣏⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠤⠭⢾⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢀⣑⢒⣚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣒⣒⢾⡇
⠀⠀⠀⠀⠀⡿⠀⠀⠀⣶⣒⣒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠬⠥⠬⡇
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠒⠀⠬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠈⠉⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠤⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢘⣥⠤⢼⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡘⠛⣁⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢘⠃⠐⣺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠨⠔⠒⡇
⠀⠀⠀⠀⠀⢹⠀⠀⢐⠂⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣉⠈⢭⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⣓⣒⣺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣂⣐⣹⡇
⠀⠀⢀⣴⡜⢻⣤⣤⣈⣭⣭⣽⣿⢿⣛⡛⣉⣡⣉⣉⣉⡉⠉⠉⠉⠙⠛⠛⠛⠒⠒⠲⢶⣟⠀⣤⡭⣭⣿⣧⣤⡀
⠀⠀⡞⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠂⠠⣤⠤⢬⡽⢾⣿⡿⢿⣿⣿⣾⣿⡄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠢⣉⠒⢬⣑⠺⢭⣶⡿⣙⣯⣽⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠨⣑⠢⢍⡒⠤⣉⣶⡯⣛⣾⣭⣿⣧
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⢍⡶⢭⡔⢠⣕⠢⣟⣓⢾⣟⠲⣯⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠢⣍⠲⢬⡑⠢⣅⡂⠅⣻⢭⣿⠯⣽⡓⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⢭⡚⠽⣒⠬⣁⠂⠍⡒⠭⣹⣿⢝⣿⠯⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡛⠗⡪⣓⠤⣍⠒⢥⣲⠄⣑⣲⣽⡒⢾⣛⣿
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠛⠉⠉⠉⢿⡲⢿⡂⢬⡑⠭⠂⠍⠲⠄⣙⠸⣿⠂⣽⡟⢾⣿⡆
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⣠⣶⣾⣿⣿⣿⣿⣿⡓⠾⣷⣮⣑⠀⢀⠀⠠⠈⠁⠯⠉⠿⠅⡛⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢬⣿⣿⣑⠢⢀⠢⢀⠲⣔⡶⢃⡳⢬⣿⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡿⣿⣿⣿⣯⣤⣿⣿⣿⣿⡿⢵⣿⣿⢦⡉⠢⢌⡒⢍⣺⣿⣒⣿⣺⢽⡇
⠀⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠳⣿⣿⣿⣿⣿⣿⣿⣿⡿⣭⣾⣿⡿⢕⠪⣑⠢⣌⠒⢾⡟⠰⣟⣾⣽⡇
⠀⠀⡷⠀⠀⢐⣂⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣶⡄⣹⣿⣿⣿⣿⣟⢳⣿⣿⣿⠿⢥⣦⣕⠪⠑⠢⣝⠺⢭⣽⠧⣿⠶⣧
⠀⠀⡇⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢺⣿⣿⣿⣿⣯⣙⣿⣿⠧⣘⠢⣍⠊⢕⠢⢑⠀⣹⣿⣍⣙⡿⣽⣿
⠀⠀⡇⠀⠀⢐⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣹⣾⣿⣯⣿⡷⢌⣿⣿⡑⠮⡑⠦⣉⠲⢌⡂⢅⣚⣟⣲⣾⣻⠿⣿
⠀⠀⡏⠀⠀⠠⠌⢉⣀⡀⠀⠀⠀⠀⠀⠀⠀⢨⡗⢼⣻⣯⣿⣿⡗⠤⣿⣿⡍⠒⢌⡑⠦⣑⠢⢍⠒⢼⡟⢽⣓⣟⣹⣿
⠀⠀⡏⠀⠀⠀⣐⣂⠠⠤⠀⠀⠀⠀⠀⠀⢬⡓⣏⡘⢍⠑⠯⠛⢍⡑⢾⣿⣤⡀⠢⣌⡓⠤⣑⠢⣍⠒⢭⣿⠭⣽⣻⣿
⠀⢰⢷⡀⠀⠀⡲⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⣍⠛⢷⣿⣯⣶⣏⣲⣟⣿⣯⡷⢌⡑⠦⣉⠲⢌⡑⠢⢝⣳⣿⣻⣯⣽⣾⣗⣖⣒⡒⠚⣛⣒⣂⣀⣠⣀⣀
⠀⠀⠉⠿⠶⢦⣬⣅⣂⣀⡀⠀⠀⠀⠀⢠⢗⡦⢙⡢⢌⣛⠛⣋⠟⡛⠍⡛⢍⡓⢌⡑⠦⣙⠲⢌⣑⠲⣟⡓⢾⣉⣿⣿⣿⣭⣉⣩⣽⣿⣛⠛⠛⠻⠿⠧⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠟⠛⠿⠛⠛⠿⠟⠛⠻⠿⠓⠿⠿⠿⣿⡿⡿⠷⠾⠷⠾⠿⠶⠿⠿⠿⠭⠭⠭⠭⠭⠍⠉⠁⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣀⠀⠀⠀⠀⠀⠀⢀⣤⣴⢲⣤⣀
⠀⠀⠀⠀⠀⣼⡟⣛⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣯⣼⣧⢼⡗⢦⠀⠀⠀⢰⡿⠉⠉⠉⠈⢹⡇
⠀⠀⠀⢠⡾⢿⣿⠿⣿⡿⢿⣆⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠹⣶⣿⣾⣿⣾⣷⡞⠀⠀⠀⠈⢷⣤⣤⣤⣤⡾⠃
⠀⠀⠀⢸⣇⣸⣿⣤⣾⣏⣸⡟⠀⠀⢀⣶⡿⠟⠛⠻⣷⡄⠀⠀⠀⠉⢶⣯⣹⡷⠎⠁⠀⠀⠀⠀⠀⠈⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣼⠉⣳⡀
⠀⠀⠀⠀⠉⣿⣉⣛⢉⣿⠏⠀⠀⠀⠸⣷⣤⢴⡦⣤⣼⡇⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣴⠋⣿⠛⢿⠉⢷
⠀⠀⠀⠀⠀⠈⢻⣿⡟⠉⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠻⢦⣿⠶⣿⡴⠟
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣭⣤⠀⠀⠀⠀⣛⣛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⡖⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠀⠀⠀⠀⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣭⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠓⠚⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠀⠀⣴⠋⠉⠈⠉⢳⡄⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⣶⣿⢿⣶⣄
⠀⠀⠀⢸⡇⣀⣀⢀⣀⡀⣿⠀⠀⠀⢰⣿⠁⠈⠁⠈⢻⣧⠀⠀⠀⠀⣀⡤⣄⡀⠀⠀⠀⠀⣾⡟⠉⢀⡀⠉⢻⣷⡀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠘⣇⠈⢙⣟⠉⢀⡟⠀⠀⠀⠘⢿⣦⣤⣤⣴⡿⠃⠀⢀⣤⣞⣹⣦⡾⣝⣦⣄⠀⠀⠻⣧⡤⣿⣿⢦⣼⡿⠀⠀⢀⣾⠷⠏⠈⠷⢿⣦
⠀⠀⠀⠀⠈⠛⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠸⣇⣸⡇⣧⡇⣼⣀⡼⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⢸⡿⡤⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⣿⣛⣿⣿⣻⡟⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠻⣷⣤⢶⣴⡿⠋
⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⣤⣤⣼⣿⠀⠀⠀⠀⠀⣴⣶⣄⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⢠⣿⣧⣺⣿⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⣶⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛
⠀⠀⠀⠀⠉⠉⠹⣿⠀⠀⠀⠀⠀⠈⠉⠁⠈⣿⣿⠀⠀⠀⠀⠀⠀⠘⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉

⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⣄⢀
⠀⠀⠀⡠⠤⠐⢆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢁⣤⡡⡤⠀⠀⠀⠀⠀⠀⠀⠂⠀⣴⠙⠉⠀⠀⠉⠚⠆
⠀⠠⠴⠀⠂⣁⣀⡞⠀⢣⠀⠀⠒⠃⡘⠛⠀⠀⠀⢰⣤⡀⡟⢠⣴⡀⠀⠀⠀⠀⠀⠠⠂⠀⢻⣄⠀⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⢋⡲⠀⣀⠄⠀⠀⢷⠀⠀⠞⠀⠀⠀⠀⠳⢜⣿⠴⠃⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠉⠙⠑⠛⠋
⠀⠀⠀⠈⠀⠀⠀⠂⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⢨⡀⢀⡅⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⢠⠷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠏⠀⠀⢠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠸⠋⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠛⠀⠀⠛⠀⢀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠒⠶⣾⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⣷⣾⣿⣿⣾⣭⣍⠛⣦⠰⣧⣤⣼⠇⢾⣁⣀⣴⣾⣿⣿⣿⣿⣿⣷⣄⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⡄⢀⡞⣽⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣀⣿⣄⣀⣀⣈⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟⣧⠀⢿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡿⠀⡾⠀⣿⣿⣿⣿⣿⡟⣉⣤⠶⣛⣫⣭⣭⣿⣿⣿⣿⣿⣷⣶⣿⡟⠻⣿⣿⣿⡇⢸⠄⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⢿⣿⡿⢋⡽⠛⣩⣶⣿⠿⠛⠛⢉⣉⠉⠉⠛⠻⠿⣿⣿⣿⣷⣄⠉⠛⠧⡿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠻⣤⠞⠁⣰⠋⣠⣾⡿⢋⡅⠀⠀⠀⢸⣿⠀⠀⠀⠀⠖⠈⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⣾⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣾⡿⢡⡄⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠴⠶⠀⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢿⣸⣿⡄⠛⠃⠀⠀⠀⠀⠀⠀⢨⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⡀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡾⢸⣿⠁⢠⡏⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⣇⠀⠹⣿⣿⣧⠀⡤⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⢡⣿⠏⢀⡾⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣦⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⠞⣱⣿⠋⢠⠞⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢳⡀⠈⢻⣿⣿⣷⣤⣘⠃⠀⠀⠀⣿⠄⠀⠀⣀⣤⠞⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢻⣿⣿⣿⣿⣶⣶⣶⣶⠶⣖⣛⣭⣴⡿⢿⣯⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⠋⠛⠻⠿⢿⣿⣿⠿⠿⠟⠛⠉⠀⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋

⢀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣿⣿⡿⠻⣿⠛⡿⢿⠿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠛⠉⠙⠁⠀⠙⠛⠛⠛⠉⠉⠉⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠛⠋⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣡⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢠⡤⠤⠀⠐⠠⢖⠒⠒⠒⠒⠒⠚⠉⢉⡉⠐⠢⢤⡀⠈⠂⢄
⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⠒⢖⠚⠋⠉⣉⣉⡉⠉⠉⠡⠤⠤⠄⠀⠀⠀⠘⠲⣄⠀⠀⠀⠑⡌⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠬⠷⣄⠀⠑⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠈⠢⡀⠀⠀⠹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠒⠒⠘⢆⠀⠀⠀⠘⢆⣀⢉⡁⠀⠀⠠⠤⠄⠀⠐⠒⠊⠀⠠⠈⠄
⠀⠀⠀⡘⠀⠀⠉⡩⠓⢄⠀⠀⠀⠀⠐⡄⢀⣀⣀⣀⣀⣤⠤⠤⠤⠄⠲⠖⠒⠒⠒⠛⢠⠀⠀⠀⠨⠇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠤⠀⠠⣠⠖⠁
⠀⠀⠀⠁⠀⠀⡴⠁⢀⠜⢄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠴⠆⠐⠗⠀⠋⠀⢈⠏⠉⠀⠀⠀⠀⠀⠠⡠⠞⠁
⠀⠀⠀⠀⣀⠞⠁⢠⠎⠀⠀⠰⢀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠶⠖⠶⠒⠒⠺⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠋⠀⣠⠏⠀⡀⠀⢀⡊⣀⣀⣠⠤⠤⠤⠤⠒⠒⠒⠂⠉⠉⠉⠁⠀⠀⠀⢀⣀⣀⣀⣠⠤⠤⠤⡆⠀⠀⠠⣀⠔⠁
⠀⠀⠀⠆⢀⣰⠃⠀⠜⠀⠠⠊⠁⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠴⠖⠒⠚⠛⠉⠉⠉⠁⠀⠀⢀⣀⣀⢠⡇⢀⡠⠊⠁
⠀⠀⠀⠰⠘⡃⢀⠎⠀⠠⠦⠔⠒⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⡠⣤⠐⠛⠒⢖⡊⡉⢉⣡⡄⢀⠢⢷⣋
⠀⠀⠀⠀⠂⢣⠞⠀⡰⠁⣀⣀⣀⣤⡤⠤⠶⠖⠒⠒⠋⠉⠉⠁⢀⠄⠀⠋⠈⠀⡀⠈⠀⡤⠛⠙⠛⠦⡀⠀⢹⡆⠠⠄⠤
⠀⠀⠀⠀⠐⡈⠄⠐⠉⠉⠁⠀⠀⠀⣀⣀⣀⣤⠤⠴⢶⣖⣻⠝⠃⠀⠠⣞⢡⡍⠉⠉⢘⣗⠀⡤⠐⠒⠓⠠⢎⡀⠂⠉⠤⢻⠅⠀⠒⢢⡐⠂⠄
⠀⠀⠀⠀⠀⠠⣤⣤⡤⡶⠏⠠⡌⠟⠩⠽⠓⠚⠂⢤⡋⠀⠀⠀⠀⠤⢈⠭⠙⠋⠉⢩⣝⠁⠉⠉⣶⠤⠄⠀⡸⠂⠛⠚⢣⠛⠀⠀⠀⣈⡀⠤⠖⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠉⠲⢼⡁⠀⢠⣂⡀⠉⣦⠊⠁⠒⠉⠓⢦⠀⠒⠢⣄⠰⠤⠤⠐⠈⠀⡡⠔⠠⠤⣈⠀⠀⠙⠂⠀⠀⠂⠥⠖⠒⠉⠁⠀⠀⠀⢀⡀⠠
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠙⣏⠀⠀⠘⠀⢴⡄⠀⠀⠀⡐⠚⠀⠠⠶⡉⠀⡠⠀⠀⠀⠘⡳⠀⣀⣀⡟⢠⠔⠒⠉⠉⠀⠀⠀⠀⣀⡀⠤⠐⠒⠉⢹⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⢀⡀⠘⠁⠀⠀⢱⠗⠂⠠⠤⠄⠚⠀⢁⡒⠒⠂⠈⠘⠉⠁⠀⠀⠀⠀⢀⡀⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠸⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠀⣀⠀⠁⠀⠀⣀⠤⠆⠐⠂⠉⠁⠀⠀⠀⠀⢀⣀⡤⢤⠀⠂⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠉⠉⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠤⠒⠚⠉⠀⠀⠀⠀⠀⠀⢀⠀⠠⠰⣀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠠⠐⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠊⡁⠀⠀⠠⠀⠀⢀⡀⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠠⠀⠀⠈



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

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






⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⠿⠛⠉
⠀⠠⣤⣶⣶⣿⡿⠿⠛⠛⠟⠛⠋⠉⠙⡩⠀⠀⠀⠉⢉⡉⠁⠀⡄⢀⡀⠛⠛⠋⠁⠀⠈⠁
⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠔⢀⠀⠉⢩⠛⠀⠀⠀⠒⣄⠀⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠁⢀⠀⠀⠀⠀⠀⠄⠠⠄⠀⡀⠀⠀⠀⠐⠒⠀⢀⠀⠋⠀⠀⠀⣠⢀⣀⢀⡀⠀⣤⡀⠀⠂⠈⢀⠈⡀⠀⠀⠠⠀⠁
⠀⠀⠀⡀⠠⠀⠀⠀⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠶⠒⠒⠒⠲⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠘⡆⢠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠋⠈⠙⠲⢤⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢨⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⠀⠀⣤⡴⠶⠔⠿⠃⡀⢠⢠⡄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⡀⠀⢠⡄⠀⢦⣠⣆⠀⠀⠰⠁⠀⠀⠶⠋⢁⡀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠸⠟⡃⠄⢰⣶⠿⢈⣉⣭⣤⣤⣦⣴⣤⣿⣧⣀⡺⠻⣦⣤⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡈⠁⢀⣠⣴⣾⠛⠛⣻⣿⣿⣿⣿⠛⠛⠛⣻⣿⣿⣦⣄⡀⢠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠶⠀⠈⣁⣿⣿⣿⣿⣧⣄⡨⣽⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣠⡦⢀⣶⣿⣿⣿⣿⣿⣹⣿⡇⠋⠈⠛⠛⠋⠙⠛⠻⣿⣿⣼⣿⣿⣿⣿⣿⣷⣠⣬⠁⠀⠄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠿⢀⣾⣿⣿⣿⣿⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⣿⣿⣿⣿⣿⣿⣿⡄⢘⡁⠀⠀⢾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠘⠀⠀⠺⢦⣾⣿⣿⣿⣿⣿⣿⣿⢸⣟⠠⣠⣤⣤⣤⣤⣤⣤⣤⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣞⣁⡀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠀⠙⢿⣿⣿⣿⣿⡿⣽⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡻⠇⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⡄⠀⠙⢿⠿⠉⠀⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡎⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⡹⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣙⣿⣿⠀⠀⠈⠀⠀⢠⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⢠⠆⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣷⠀⠀⠀⠠⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠟⠁⠀⠀⠀⠀⠈⢻⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⣟⣈⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣞⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢋⡈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠛⢶⠄⠘⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⢀⣴⣦⠀⠀⠀⢸⣿⣸⣿⣿⣿⣿⣿⣿⣿⢋⡄⣿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠐⢤⡄⠈⢻⣿⣿⣿⣿⣿⢿⣿⣧⣀⣴⣿⣿⡿⢿⣦⣠⣿⣿⢿⣿⣿⣿⣿⣿⡿⠋⢘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⡄⠙⠿⣿⣿⣿⣾⠟⠿⢧⣶⣿⣿⣷⠒⠿⠿⢿⣿⣾⣿⣿⣿⡿⠋⣄⡀⢊⡙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠟⢴⣌⣻⢿⣿⣤⣈⣿⣿⣿⣿⣿⣭⣥⣤⣿⣿⣿⡿⠟⠉⠠⣶⡈⡁⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⢨⡿⠻⢂⣤⠉⡙⢻⢻⠿⠿⡿⡿⠟⠛⣛⠉⢡⡄⢀⠛⠀⠂⢀⣄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⢰⣶⡄⠻⠆⠛⢃⣈⡁⠀⠀⣿⣿⠶⣿⠁⠈⠁⣿⠁⠛⠀⠙⠋
⠀⠀⠀⠀⠀⠐⠀⢱⡜⢂⣀⣀⣠⣄⠱⣆⠀⣄⠀⣰⠃⠀⡙⠁⠾⠂⠠⠚⠛⠋⢰⣿⠀⠁⢴⡷⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠀⠀⢀⣡⣼⣿⣿⣿⡏⣩⣦⣦⣶⣿⣿⡟⠀⠢⠉⠀⠀⠀⠀⠀⡀⠀⠀⠁⡄

⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠸⠁⠀⠹⣷⡀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣶⡶⡆⠀⠀⠀⠘⢛⡷⣺⢅⠀⠀⠀⠀⢠⡞⣉⢙⠚⣛⡈⢲
⠀⠀⢶⠀⠀⢀⣯⠉⠙⠩⠔⠷⠋⠉⠉⠉⠉⠉⠉⠉⠀⢠⡄⠀⠈⠘⠉⠀⠀⡀⢠⡴⠒⠣⠒⢀⣟⠀⢦⡞⠒⢢⡀
⠀⠀⠘⠦⢤⠮⠋⠀⠀⠀⠀⠀⠀⠀⢰⢰⠀⣖⡶⢰⡶⠀⠃⡀⢀⣀⠀⠀⣰⠁⢸⡇⠀⠷⣂⠙⠶⠋⢉⠻⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣄⠌⣷⣨⣧⠀⠀⠈⢲⠼⢡⠟⠁⠀⠈⠳⠤⠤⠁⠺⣷⠆⢸⡦⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡋⠀⠉⠁⠉⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢬⣤⣻⣤⠞
⠀⠀⠀⢀⣠⠤⠄⣠⣀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⣼⡷⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⠋
⠀⠀⣴⠾⡳⢦⡀⠤⢛⠛⡀⠀⠀⠀⢸⢹⡇⠀⠀⠀⠀⠀⠀⠀⠐⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢉⣧⠀⠘⢏⠞⠁⠀⠁⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⣠⡄⠀⠀⠀⢸⣿⠃
⠀⠀⠈⠹⠶⢾⣾⡷⠶⠕⠀⠀⠀⠀⢸⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠴⣿⡃⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⠀⠀⠀⢽⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⢢⡆⠀⠀⠉⠩⠁⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣼⣾⡄⠀⠀⠀⠀⠀⠀⠐⣾⡠⡅⠀⠀⠀⣠⡄⠀⠀⠀⣸⣽⡇
⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⠀⠀⣤⡄⡽⠲⣅⣴⡀⠀⠀⠀⠂⠠⣻⠁⡇⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⣶⣶⡰⣿⡄⠀⠀⠀⡞⠁⣀⠷⣴⣋⡀⠉⢳⠀⠀⠀⠀⠛⢀⠇⠀⠀⡼⣟⡿⣄⠀⠀⣿⣿⡇
⠀⠀⠀⠛⠛⠓⠛⠃⠀⠀⠀⢧⣀⣻⡷⠳⣏⣁⣀⡼⠀⠀⠀⠀⠀⠈⠀⠀⠈⠳⡟⣷⠊⠀⢀⣿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠲⣤⠊⢿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⣧⣤⡿
⠀⠀⠀⠀⠀⣶⡿⣶⡄⠀⠀⠀⠀⠀⠴⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢳⣾⡗
⠀⠀⢀⣀⡤⢿⢠⢐⢫⠔⢲⣄⣀⣀⣀⣀⡀⠤⠤⠤⠤⠤⠤⡤⣤⠠⠤⢠⢀⡀⠀⣠⠻⢙⡺⢿⣧⡇
⠀⠀⠛⠺⣄⣀⠸⠄⣈⢀⣸⠖⠚⠒⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⠻⠶⠶⠶⠞⠁⠀⠹⣶⢃⢹⣾⣉⡇
⠀⠀⠀⠀⠀⢻⣷⣶⠏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠏⠀⠆⠀⠀⠀⠀⠀⠈⠻⣷⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⢶⡶⢶⣶⣶⡶⣶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⣿⠿⢿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⢀⣀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⣿⣇⠀⢠⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠐⣿⣿⠂⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣴⣶⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⣀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡠⢾⣉⠷⢏⣹⣆⣀⣀⣀⣀⣀⣀⣀⣴⣲
⠀⠀⠀⠀⠀⠀⢀⣾⣄⡀⠁⣀⠎⢁⣤⣴⠀⠀⠀⠀⠀⠀⠠⡤⠊⢩⠉⠀⠳⢄⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠈⠺⣩⠷⢯⡹⠋⠀⠀⠀⠀⠀⠀⠈⠻⠾
⠀⠀⠀⠀⡠⠀⢀⣀⡀⢙⣦⣿⣴⠃⠀⢀⠀⠑⢆⠀⠀⢠⠀⢐⠀⢐⡒⢂⠀⢈⠀⠀⠀⠀⢀⠀⡀⠀⠁
⠀⠀⢀⠞⢠⠖⠁⠻⠤⢾⡏⠉⢙⡦⠤⠎⠀⠀⠀⠡⡀⠀⠁⠈⠑⠎⡗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⡤⣀⠠⡄⠀⠀⠀⠀⠁
⠀⠀⠘⠀⡎⠀⠀⠀⠀⠀⠑⠦⠊⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⡞⠁⠀⢀⠀⠀⠈⢺⡄⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣷⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⢀⡀⠀⠀⠁
⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⠐⢿⣛⣾⣷⣠⠿⠃⠀⠀⠀⠀⠨⡇
⠀⠀⠀⠀⠃⢀⠀⠀⣠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⠀⠘⠛⡏⠀⠀⠀⠀⠀⠀⠈⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠓⠚⠂⢄⠀⠀⡎⠀⢰⠀⣀⡤⠐⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⠒⢉⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠀⠀⠀⠀⠀⠀⠐⠁⠲⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣀⣀⠀⠀⠀⠀⢂⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠗⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄
⠀⠀⠀⣿⣹⡇⠀⠀⢀⠜⠉⠉⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⡔⢚⣻⢢⠀⠀⠀⠀⠀⢀⡇⣼⠀⠀⠀⠀⠀⠀⠧⠀⠀⡧⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠘⢤⠤⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢀⣋⢡⣉⡼⣧⢀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⢸
⠀⠀⠀⠘⠛⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣾⡀⣴⢟⣥⠺⡮⢇⣻⡆⠀⠀⠈⢷⣾⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢨⢧⡀⢠⡀⡀⠀⢵⢺⣎⣛⠕⣳⠶⠋⠀⠀⠀⠀⠀⢹⣒⠲⠀⠀⠀⢐⣂⣀⣀⣂⡸
⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⡟⠀⠀⠨⣽⠁⠀⢹⣷⠀⠀⠘⢧⣿⡼⠃⠀⠀⠀⠀⠀⠀⢠⣼⠟⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⢀⠌⠀⢂⠀⠀⠈⢤⣄⣀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠐⣷⡀⠄⣰⡟⢤⡀⣠⠇⠀⠀⠀⠘⣿
⠀⠀⠈⠐⡆⠊⠀⠀⠀⢸⠅⠀⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠉⠈⠋⠁⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⣤⣦⠾⠾⠾⠿⢤⣀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⣇⡴⢀⡠⡆⠀⠀⠀⠀⠷⠿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡅⠀⠀⠀⠀⠸⠀⠀⡟⠀⠀⠈⠈⠁⡈⠃⠋⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡇⠀⠀⠀⠀⠀⢳⣠⠄⠤⢤⣼⠇
⠀⠀⠀⠀⠥⠀⠀⠀⠀⡠⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠀⢀⠈⠀⠀⠀⠀⠀⠀⠏⠑
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐
⠀⠀⢠⣿⣽⣧⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⠀⣤⠀⠀⠀⠀⠀⢹⠒⣰⠂⠀⠀⠀⠀⠀⢸⠘
⠀⠀⠀⠙⡟⠁⠀⠀⠀⢸⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⢀⡬⠀⠀⠀⠀⠀⢸⣸⢿⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡀⠀⣎⠀⠀⢀⡀⠀⠀⡀⠄⠀⠀⠀⠀⠠⡞⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⠈⠁⠀⠀⠀⠀⠀⠸⣤⠦
⠀⠀⠀⢀⡇⠀⠀⠀⢀⡞⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠘⠟⠁⠀⠀⠀⢣⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡷⠶⠶⠶⠚⠂⠀⠀⠀⠶⡷⠒⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⣏⣹⡷⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠔⠒⠒⣂⠙⠛⠿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠐⠒⠉⠉⢈⠉⠍⠙⠿⣿⣿⣅⠀⠀⠀⠀⠀⠙⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⠛⠛⠛⠿⢿⣿⣶⣤⡀⠀⠉⠀⠀⠀⠀⠀⠀⠙⣿⣧⡄⠀⠀⠀⠁⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⠄⠒⣄⠉⠉⠉⠒⠪⢍⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣀⠀⢀⣀⡠⠤⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢰⢠⠃⠀⠀⠈⣦⣀⠀⠀⠀⠀⠑⢜⢿⣿⡄⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⡭⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠋⠀⠊⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣹⣿⣿⡶⣾⣿⣿⣿⠟⢻⣿⣿⡏⠷⣠⠘⢹⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠠⢴⣤⣤⣶⣷⣶⣿⢿⣿⣿⣿⣿⣿⣿⣿⢟⣿⡿⣫⠀⠶⡏⢶⠷⣿⣼⠐⢿⣓⣬⣟⡥⠬⠶⢌⡆
⠀⠀⠀⠀⠀⠀⠀⣬⣿⣯⣭⣻⣛⣌⣼⡯⠍⢴⠚⠛⢣⣁⠤⠥⢼⣧⠀⣿⢆⣠⣤⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⢻⠞⢿⣿⣿⣥⢄⣐⠈⠙⠧⢤⠩⡄⣒⣿⠿⠻⣽⠉⠀⣿⣿⣿⣾⣇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⠈⠧⠎⠉⠉⢿⣧⢹⣿⣶⣦⣶⣲⡍⢠⡏⠀⠀⠟⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡯⠀⠰⣦⡄⠀⠀⠀⠉⢶⠮⢍⣻⣿⢯⣍⡹⠀⠀⠀⣿⡄⠀⠀⢹⣿⡅⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢨⡇⠀⠀⠀⢠⣤⡓⠖⢳⣿⣷⡆⠤⠐⠀⠉⢽⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠂⢼⣧⣤⣤⠀⢨⡛⠁⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⣸⣿⡇⠂⠀⠀⠀⠀⠀⣿⠤⡄
⠀⠀⠀⠀⠀⠀⣀⢹⡟⠀⠀⠀⠈⠻⠿⠻⢞⡥⠀⠀⢸⣿⣿⡇⠀⠀⢀⠀⠜⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⢀⡠⠿⠋⠁
⠀⠀⠀⠀⠀⠁⠉⠸⠷⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢂⠉⢀⡀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠳⣦⠤⣀⡀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⡀⠤⠒⠉⠁⠀⣀⣀⠲⠂⠀⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣼⣿⣿⡇⠤⠔⠒⣉⣀⠀⠀⠀⠀⡀⠿⣙⣷⣿⣆⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣒⣀⠅⢀⡀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⡟⡛⣿⡛⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⢙⡷⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣰⠋⠀⠀⠉⠛⠲⣶⠶⠒⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣇⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣀⠬⠷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠙⠛⠦⣤⣀⣸⠀⠀⠀⠀⡴⠁⠀⠀⠀⠈⠙⠛⢶⡶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠉⠛⠶⢄⣀⣀⣇⠀⠀⠀⠀⠀⠀⡴⠁⢀⡠⠤⠬⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠈⢆⠀⠈⠈⠛⠶⢦⣄⣀⡰⠁⢀⠎⠀⣠⢤⡀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣤⣼⣧⣀⡠⣄⠀⠈⡦⢀⠀⠀⠀⠀⠈⠉⠛⠢⡼⡀⠸⡀⠀⠘⢦⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠊⢦⠀⢒⠀⠄⡁⠈⡦⠊⠀⠀⠀⠂⠤⣀⠀⠀⠀⠀⠑⡕⡄⠑⣄⠀⠉⢧⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠣⡢⠑⣀⢒⠤⡒⠒⠦⢲⣉⠉⡒⠒⠲⣤⠖⠢⡉⠒⠤⣄⡀⠘⣞⣄⠈⢆⠀⠨⣇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠧⠒⢴⣉⡽⠛⠺⢭⠒⣄⠢⠄⠈⢳⠛⠲⠃⠊⠳⢱⠔⠲⡀⠈⠑⠺⢎⢆⠈⢣⠀⣿⠀⣿
⠀⠀⠀⠀⠀⣾⠻⣥⣗⣂⠤⠤⠂⠀⠀⠀⢀⠙⠴⣋⡆⣠⢞⡠⢄⡠⠬⢅⣀⣀⡃⢧⡀⢀⣀⠈⢾⢦⠀⢣⣿⢰⡇
⠀⠀⠀⠀⠀⢸⡆⢠⣉⡛⠷⣦⣄⣀⠈⠈⢩⣂⣀⣈⠉⠉⢘⠳⡾⠛⡁⠁⣀⣤⣾⠚⠳⣤⣈⡓⣌⢫⢣⠀⢁⡾
⠀⠀⠀⠀⠀⠈⣷⠀⣇⠈⠐⠢⣍⡛⠷⢦⣤⣁⡘⠛⢺⡿⠥⠄⠉⡩⣴⠾⠛⠉⡐⢁⠄⠀⡆⠉⢉⣩⣷⣷⡞⠁
⠀⠀⠀⠀⠀⠀⢹⡆⠸⡆⠀⠀⠀⠈⠑⠢⢌⣙⠻⢕⣾⣇⣀⣴⡷⢫⡃⠐⠤⠐⠃⣀⣩⣿⠷⠟⠋⢉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠾⣽⣺⢶⣷⣤⣍⣥⣖⠿⠟⠊⣉⡠⠴⠒⠉⢹⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⣟⡆⠈⢹⠉⠀⠔⠚⠉⠁⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⣧⠀⠸⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣉⠒⢴⣄⠀⠀⠀⠀⠀⠀⠈⡵⣟⡍⡀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢬⣝⡲⢄⣀⡀⠀⠀⠧⠏⠀⢃⠀⡇⠀⢱⠀⠀⠀⠀⠀⠀⠀⢸⡆⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣉⠓⠦⢄⡀⠀⢸⠀⢸⠀⢸⡀⠀⠀⠀⣀⣰⡤⠞⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣈⠑⠺⠀⢸⠀⠘⠧⠄⢐⣉⣭⡴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣬⣥⡤⠶⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠈⠛⠁⠀⠉⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣷⣦⣤⡀⠀⢀⣠⣴⣶⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⢿⣿⠀⣉⣿⣿⣷⣿⣿⣍⠀⢸⣿⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠘⣿⣶⣿⣯⣤⣄⣤⣽⣿⣷⣿⠏⠀⠀⢙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠐⢿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⠀⠀⢠⣿⠿⣿⣍⠻⣿⠟⣋⣾⡿⣿⣆⠀⠀⢨⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣾⣿⠀⣉⣿⣿⣿⣿⣿⣋⡀⢸⣿⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡿⠿⠛⠉⠀⠈⠙⠻⠿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⣠⡀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠟⠋⠉⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣦⡀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠉⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠋⠉⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠒⠒⠓⠋⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⢀⠤
⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠐⣳⣶⣿⣎⣐⣺⣿⣯⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡂⠀⣁⠂⡄⠀⠀⡀⠡
⠀⠀⠀⠀⠀⠄⠶⠐⠒⠒⠈⠓⠉⡀⠀⠤⠀⠀⠀⣰⣿⣿⣷⣿⢯⡻⣿⣿⣏⡫⣻⠏⠀⣀⡄⠀⠀⠀⠀⠀⠀⡄⢀⡘⡂⡆⠀⡁⠀⠄⢀⠐⡆
⠀⠀⠀⠀⠀⠀⣴⠀⡂⠂⠆⢐⠆⠀⡀⠀⡀⠀⠀⠠⢻⣿⣿⣿⡛⠙⠔⠛⢻⣿⠃⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠇⠘⢊⠵⠍⠀⠀⠀⠀⠈⡹⠃
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡀⠙⣺⡿⠀⠐⠀⠀⠀⢠⣿⣭⠉⢳⡄⠱⣬⣏⣟⣅⣼⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠁⠘⠠⠔⠋⠀⠣⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣶⣾⣳⣿⠙⣿⢻⣿⣿⣻⣿⣄⠀⠀⠀⠀⣀⡠⠄⢀⣤⠀⠀⣠⢀⠀⠀⠂
⠀⠀⠀⠀⡀⠤⡀⠠⠤⠄⠀⢠⣤⡄⠀⠀⣀⣄⣶⡾⡟⡩⠉⢛⣘⣿⣘⣿⣿⣿⡿⢹⠿⣟⣲⠦⡄⠈⠉⠉⠉⢻⠀⠁⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠉⠀⠙⠋⡀⢢⡄⠝⠙⠃⡥⣬⠟⠉⡡⠟⠛⢛⡉⣯⠾⠋⣀⡸⠉⣀⡿⣁⠄⠁⣀⠴⠀⠀⢀⣀⣀⡀⠈
⠀⠀⢈⠀⠀⢀⣀⠀⠀⠠⠀⣄⠀⠀⠀⠀⠐⠚⡤⡔⢋⡂⠰⡄⠃⠠⠆⠒⠐⢒⣛⡤⠚⠂⠐⠁⠀⣰⣖⣣⡔⣬⢧⡎⢠⠊⠉⠉⡄
⠀⠀⠀⡘⠉⠙⣢⠈⢦⡧⡼⢤⡟⡳⠂⠀⠈⠉⠀⠀⠡⠜⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠀⣇⣁⡻⡀⡌⠁⢱⠰⣄⡇
⠀⠀⠀⡆⣤⠨⠉⠉⠃⣈⠃⣈⡄⡀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠻⢿⠈⠄⡓⠿⢯⣀⢷⡒⠠⡀⢀⢀⠈⠐⠂
⠀⠀⠀⠅⠊⢨⡅⠀⡀⠈⢹⠷⠃⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣈⠺⢀⠭⠡⠖⠀⣄⠹⣠⠉⠵⠇⣀⠦⠉
⠀⠀⠀⠀⠀⠀⡆⠉⠁⠂⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠒⠈⠒⠉⠀⠈

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠄⠆⠠⠀⢀⠄⠀⠀⠄⣤⠀⠀⠀⠀⠀⢠⠒⠀⠒⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠒⠄⠀⠀⢄
⠀⠐⠀⠡⢄⠃⠐⠀⠀⠀⠐⠂⠬⢀⠂⠀⠀⠀⠘⠤⣀⡠⠜⠀⠀⠅⠀⠀⡀⠄⡀⠎⠀⠀⠀⠀⠀⠀⡀⢀⠁⠁⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡂⠀⠀⢸⠂⠀⠀⠀⠀⠀⡄⠀⠈⠙⠋⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠘⠁⠀⠀⠀⠀⠀⠃⠀⠀⠘⡁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠠⠟⢤⡀
⠀⠀⠀⠀⢠⠀⢄⠀⣈⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⡀⠁⠀⠀⠀⠀⠸⠰⡖⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠰⣀⣰⠂
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⣀⣀⢀⡀⠀⢀⣀
⠀⠀⠀⠈⢺⡃⠀⠀⠀⡔⠘⠁⢰⢦⠀⠀⠀⠀⡆⠒⠀⠀⠀⠁⠂⠀⠀⠀⠀⡀⠐⠀⠀⢸⡁⠀⠀⢹⠀⠼⠏
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠰⠀⠀⠀⢠⣡⠊⠀⠀⠀⠀⠀⠀⡀⣠⠈⠀⠉⠉⠀⠀⠉⠂⠊⠁⠀⠀⠀⠆⠀⠀⣠
⠀⠀⠀⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⢮⡶⠀⠀⠀⡆⠀⠀⡴⠊⠛⠦⠀⠀⠀⠀⠀⢀⡀⠅⣀⡀⠀⣀⠀⢀⠐⠈⠱⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠓⠛⠛⠆⠀⠀⠀⠀⠀⣄⠀⠀⣀⠇⠀⠄⢀⠀⠣⠀⠘⡤⠅⠀⠁⠠⠀⠀⠀⠀⠼⠀⡇⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠈⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠈⠁
⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣽⣧⣭⣯⣯⣵⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡠⣒⣶⣦⡀⢈⡽⣄⣀⠄⡴⢒⡲⠞⠣⢄
⠀⠀⠀⣧⠇⠀⣸⣷⠈⠉⢫⣇⠘⢡⣿⠁⠀⠰⠦⠹⣷⡵⡀
⠀⠀⠀⠙⠚⠯⣌⠈⣵⣄⠈⡏⠀⠀⠑⡷⡀⠀⠀⠈⢹⣿⣷⡄
⠀⠀⢠⣘⡻⠄⠈⠳⠏⣻⠩⣾⡀⠀⠀⡿⡄⢠⠔⣢⣄⡸⡇⣧
⠀⠀⠀⠉⣵⠟⡻⠤⠾⠿⣶⣿⣿⣤⣤⡠⠇⠘⠞⠁⠈⠹⡇⢸
⠀⠀⠀⢠⠖⠛⣀⠀⠀⠀⠹⣿⠁⠀⠀⠀⢀⠤⠀⠀⠀⢀⠃⡟
⠀⠀⠀⣧⡥⠊⢣⠑⣤⣶⣦⡻⠁⠀⠀⠀⢉⣴⠚⠁⠀⡀⣰⠃
⠀⠀⠀⡜⡇⢠⢄⠀⠀⢈⣭⡀⠀⢀⠴⢶⣾⢋⣶⣀⠀⡰⠃
⠀⠀⠀⠈⢉⢻⡏⣄⠀⣮⡔⠋⠀⢢⣠⠄⠛⢿⠟⣳⣾⡁
⠀⠀⠀⠀⠀⠳⢝⣿⣼⣼⣿⣦⣀⣀⣠⠤⣚⡵⢾⣿⡿⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠮⠀⠖⠚⠋⠁⠀⠀⠈⠠⡼⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡉⣻⣿⣣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣿⠿⣿⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣼⢽⣶⣶⣀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⣿⡿⡷⢄⠀⠀⢀⣴⣿⣿⣿⡃⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣙⡿⣿⣶⣴⣿⡿⠛⢉⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣼⢏⣿⣿⣄⢼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡿⠛⣿⣮⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢙⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⢼⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⡖⠛⠳⡄
⠀⢸⡇⠀⠀⡏⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⡾⠁⠀⠈⢻⡄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢰⡟⠉⠙⣧⠀⠀⠀⠀⢠⡾⠋⠉⠙⢷⡀⠀⠀⠀⣴⣷⠛⠛⠛⣾⣷
⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢰⣿⣿⣄⡀⣀⣼⣿
⣄⣹⣁⣀⣀⣏⣠⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢰⣾⣿⣶⣿⣾⡆⠀⠀⢀⠸⢇⣀⣀⣀⠸⠇⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⣇⣀⣀⣀⣘⣇⠀⠀⢨⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⣿⠿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⢹
⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡋⣿⣿⣿⠀⠀⢸⣯⣿⣿⣿⣯⣿⠀⠀⠸⣿⣿⣿⣐⣿⣿⣿⠇⠀⠈⢿⣿⣿⣿⣿⣿⣸
⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣧⣿⣿⣿⠀⠀⠈⠻⣿⣿⣿⡿⠃⠀⠀⠀⠙⢿⣿⣿⣿⠿⠋⠀⠀⠀⠀⢿⣿⣿⣿⡿

⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤⣠⣀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⢸⡇⠀⠈⣧⠀⠀⠀⠀⢰⡇⠀⠈⡇⠀⠀⠀⠀⢰⡇⠀⠈⣧⠀⠀⠀⢀⣾⠟⠉⠀⠉⠻⣧⠀⠀⠀⠀⣾⠋⠀⠈⣿⡀
⠀⢸⣃⣀⣀⡯⠀⠀⠀⠀⢸⡇⠀⠀⡷⡀⠀⠀⠀⣸⣇⣀⣀⣟⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⡀⣿⠀⠀⠀⢿⢇
⢠⣾⠟⣛⡻⣿⡆⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡿⣿⣿⡇⠀⠀⢸⣷⡀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿
⢻⣇⠸⣿⠇⣸⡿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣯⣿⣽⡇⠀⠐⣿⣿⣿⠀⠀⠀⢸⣿⣿⡂⠀⣿⣿⣿⣿⣿⣿⣿
⠈⠻⢷⣶⡾⠟⠁⠀⠀⠀⠹⣿⣿⣾⠿⠁⠀⠀⠘⠿⠿⠿⠿⠿⠇⠀⠐⠿⠿⠿⠀⠀⠀⠰⠿⠟⠀⠀⠛⠛⠻⠛⠻⠛⠛

⠀⣴⠟⠛⠻⣦⠀⠀⠀⠀⢀⡶⠛⠙⢷⡄⠀⠀⠀⠀⣰⡟⠛⠛⣧⡀⠀⠀⠀⠀⣴⠛⠙⢳⡄⠀⠀⠀⠀⣰⠟⠋⠻⣦
⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣇⠀⠀⢸⡇⠀⠀⠀⠀⣿⡀⠀⠀⣿
⣤⣿⣠⣤⣤⣿⣤⠀⠀⠠⣬⣧⣤⣤⣬⣧⡆⠀⠀⣤⣟⣀⣀⣀⣹⣧⡄⠀⠀⣮⣯⣤⣤⣬⣧⡆⠀⠀⢰⣾⣶⠶⣶⣿⣶
⣿⣿⣿⡍⣿⣿⣿⠀⠀⠠⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⡅⣽⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣧⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣶⣿⠆⣿⣿⣿⣿⡏⠉⠁⠉⠁⠈⠀⠉⠉⠉⠁⠉⠁⠉⢹⣷⣶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⢿⢧⣿⣟⣛⡿⣧⣤⡤⠤⠤⠤⠤⠤⠴⠤⣤⣄⣠⣤⣼⣿⣿⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇⢸⠷⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠿⠙⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⡈⠋⠀⠀⠀⠀⠀⢶⣾⠡⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢄⠀⠀⠀⠀⢀⡾⠁⠁⠀⢀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣾⣿⠀⠀⠀⠀⠀⢄⠀⢀⡮⠊⠀⠀⠀⠈⠷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⢿⣷⠀⠀⠀⠀⡀⠈⢸⠀⠑⠀⠀⠀⠀⠀⡾⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠈⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⡀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠀⠈⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⡃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠀⠀⠑⢾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⡇⣸⣦⢤⣤⡤⠀⠀⠀⠀⠀⠠⠧⣴⣃⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣾⣾⡞⣿⣯⣽⣷⡟⠛⠳⠒⠒⠒⠒⠲⠖⠒⠛⠋⠑⠛⢻⣷⣶⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⣿⢆⣿⣿⣿⣿⣇⡀⠀⠀⡀⢀⠀⠀⢀⡀⣀⣀⣀⣀⣸⣿⠿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠴⠶⣶⡾⠿⠿⠿⠛⠓⠀⠀⠀⠐⡄
⠀⠀⠀⠀⢀⣤⠀⢀⣀⠀⣀⣀⣀⣄⣤⣤⣴⣶⣶⣾⠿⠿⠟⠿⠿⣎⣱⣤⣶⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠂
⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠻⠿⠛⠛⠛⠉⠋⠉⠀⠀⠀⠀⠀⠻⣷⣯⣾⠛⠉⠉⠟⠁⠀⠀⣀⣀⣀⣤⣤⠴
⠀⠀⠀⣴⣶⣶⣄⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣩⣿⡖⠀⠀⡀⠻⣿⣿⣿⣿⣿⠟
⠀⠀⠾⠻⢿⣿⣿⠃⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⢿⠀⠙⢿⣿⣿⡟
⠀⢐⣶⠀⠘⠋⠋⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣻⣿⣭⣿⣿⣿⣿⣿⣦⠄⠳⣶⠈⢻⡟
⡆⢸⣿⣶⡀⣀⠂⠀⠀⣘⣛⣯⣭⣽⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣦⣠⣤⣤⡶
⠀⠀⣿⣧⣿⡏⢠⠀⢠⣿⣿⣿⣿⣿⡿⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⡿⠁
⠀⠀⠛⣿⣿⠁⠀⠀⣼⣯⣽⣷⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⣟⣛⣿⣭⣽⣶⣶⣿⡿⠿⠃
⠀⠀⠘⣿⡏⠀⠀⢰⣿⣿⣿⠿⠿⣛⣛⣯⣭⣽⣷⣶⣾⣿⡿⠿⠿⠛⠛⠉⠉⠁⠀⠀⢀
⠀⠀⠀⡿⠁⠀⠀⣭⣵⣶⣾⣿⣿⣿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⢈⡀⠠⠄⠶⠶⢛⣛⣩⣤⣤⣈
⠀⠀⠀⠁⠀⠀⠐⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⣤⠶⠶⢛⣛⣫⣤⣴⣶⡆⠀⠀⠈⠝⠀⠀⠉⠛⠛⠷⣦⣀⡀⠠
⠀⠀⠀⠀⠀⠀⢀⣉⣠⣤⡴⠶⠂⠀⣀⣨⣤⣤⣶⣾⡿⠛⠙⠃⠀⠀⠀⢈⠁⠉⠈⣄⡀⠀⢀⡂⠤⠤⠚⠛⠛⢷⣦⣈⠐⠠⢀
⠀⠀⠀⠈⠄⠉⠈⣭⣤⠀⠀⠐⠛⠻⠛⠉⠀⠈⡏⠀⠓⠀⠀⠀⠀⠃⠐⠁⣄⣄⠀⠄⢀⡀⠀⣅⠀⠀⠀⡴⠂⠀⢀⠀⠿⠶⠀⠈
⠀⠀⠀⠀⠀⠙⠂⠀⠙⠣⢄⠄⠀⠠⢁⡤⠄⠁⠀⠈⢄⡀⠀⠀⢂⡠⠀⠀⠐⠀⠀⠀⠈⡄⠀⣀⡀⠿⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢲⣜⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠉⠐⠀⠊⠉⠀⠀⢄⠉⠴⢦⢂⠀⠀⠒⠊⠁⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⣀⣨⡤
⠀⠀⠀⠀⠀⠈⠛⣦⣀⠀⡆⠀⠀⠀⠈⠉⢷⡤⠀⠠⠷⢤⠤⠄⠒⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣤⣴⣶⣿⠋⠃
⠀⠀⠀⠀⠘⠀⠀⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⠀⠀⢻⣿⣿⣿⣿⣷⣤⡀⠷⠀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣤⣵⣶⣾⣿⣿⣿⠟⣩⠉⣿⡿⢿⣿⣿⠟⠟⠉⠁⠀⠀⠀⠠
⠀⠀⠀⠀⢈⠀⠀⠸⠿⣿⢿⣿⣿⣿⣿⡷⢤⣀⣨⠤⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠻⢄⠀⢀⡀⠉⠀⣀⣤⣤⣴⡆
⠀⠀⠀⠀⠐⠀⠀⣶⣤⣌⠀⠙⠛⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠿⠿⣿⠟⠛⠁⠀⡾⣷⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠐⠀⠀⠀⢿⣿⣿⣿⣶⣤⠉⠙⡇⠀⠀⠀⠀⢸⣿⠿⠿⠿⠇⠀⠀⠀⠀⣀⣠⣤⡶⣨⠎⠋⠿⢻⣿⣿⠿⠟⠛⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⢘⠉⠀⣀⣤⣦⣤⣤⣶⣿⣿⣿⡿⠀⠀⠀⣠⡀⠈⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠲⠂⠀⡿⠄⠀⠀⣀⠠⠄⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⡇⠀⠀⠀⠀⢸⣿⣿⠿⠟⠛⠉⠡⠄⠀⠀⠀⠐⠢⡀⠤⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠿⠿⠿⠿⠿⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠿⠛⠉⠀⠀⠀⠀⣿⢸⡀⠀⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⢦⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⡴⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡞⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠤⠴⠄⠀⠀⢀⣀⣤⣤⣤⣤⣴⡛⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⡼⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⡜⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠋⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠁⣀⡴⣫⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣀⣀⠀⠀⠘⠃⠀⢀⣀⣠⣴⣾⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣦⣤⣀⢀⣤⠶⠞⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠋⠉⢻⣿⣧⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡀⠀⠀⣼⢹⣿⣧⠀⠀⠀⠀⢠⣾⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡼⣏⢿⣿⡆⠀⠀⠀⢛⡾⠁⣸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣅⣀⣤⡴⠟⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠈⠉⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣴⣿⣿⣿⣿⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⣾⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡞⣱⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣅⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⢋⣼⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠙⣦⣴⠞⠉⠘⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⡾⣡⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣷⣄⣠⡶⠋⠻⣄⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢠⣏⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⢸⣿⡿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⠀⢤⣤⣤⣤⡲⠒⠐⠒⠦⢶⣶⡒⠲⣶⣶⣶⣿⣔⣒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣊⣉⣇⠚⠤⢝⠻⢿⣿⣿⣿⣿⣿⣦⣈⠻⢿⣿⣷⣅⠀⠂⠀⠉⠿⣷⣮⡛⢿⣿⣿⣿⣷⣬⡒⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢚⣴⣿⣿⣿⣿⣿⣶⣮⡑⠦⡙⠻⠿⣿⠿⢿⠿⢷⠆⠙⠛⠻⠟⠻⠓⣶⢌⢿⠛⠛⠊⠨⠉⠹⠉⠏⠙⡃⠘⠒⢶⡄
⠀⠀⠀⠀⠀⠀⣰⢩⣾⣿⣿⣿⠿⠿⠿⠟⠛⠛⡂⠉⢙⣀⣀⡀⣀⣆⣀⣊⣠⣀⠠⣤⢤⣤⣽⠀⠀⣦⣤⡶⠶⠶⠾⠶⠓⠒⠓⠛⠛⠋⠁
⠀⠀⠀⠀⠀⣎⣩⢉⣥⡀⣦⣤⠦⣴⣷⣖⢋⣖⣳⣉⡳⢊⣅⡴⣩⢁⡆⡖⣯⠝⢺⡑⠚⢓⢶⣄⣧⢤⣬⡀⢺⣆⣦⢄⣦⢤⢄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠋⢀⣨⣐⢢⠭⢛⠛⠾⢟⠿⠯⣳⣾⣿⣷⣯⡤⠆⢶⡒⢾⣭⣴⣧⣵⢽⣿⠭⠉⠷⠤⡛⠀⠻⢈⠋⢣⠈⢛⣈⣨⣉⣃⣉⠋⡍
⠀⠀⠀⠀⠀⢸⠚⣿⣿⣿⣿⣿⣶⠈⢠⣤⣈⣁⡙⠋⠩⢿⠙⠒⠚⠛⡈⣀⣈⣉⠀⣢⣤⣥⣤⡁⠀⢀⣶⣶⣾⡇⢰⣿⣿⣿⣿⣿⣿⣿⠐⡇
⠀⠀⠀⠀⠀⠠⡆⣿⣿⣿⣿⢛⠻⢀⠸⣿⣿⣿⣿⣿⡧⢰⠠⣼⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⡇⣀⣷⣤⣿⣿⣿⠘⣻⣿⣿⣿⣿⣿⣿⣇⣹
⠀⠀⠀⠀⠀⠀⣍⠘⠛⠿⠿⠷⢸⠟⢳⣤⣸⣿⣿⣿⡇⠈⠀⣿⣿⣿⣿⣿⣿⡇⢈⣿⣿⣿⡿⠀⢛⠸⠿⣿⣿⡉⣿⣿⣿⣿⣿⣿⣿⠣⡌
⠀⠀⠀⠀⠀⠀⠣⠴⣶⣶⣤⣼⣧⣤⣞⡉⢙⠛⠻⠿⠇⠆⠀⣿⣿⣿⣿⣿⣿⡇⣹⣿⣿⣿⣷⣷⣿⣄⢋⣿⣿⠠⣿⣿⣿⣿⣿⣿⣿⢰⡇
⠀⠀⠀⠀⠀⠀⢰⠒⣿⣿⣿⣿⣿⠈⢸⣿⣿⣷⣶⣦⠀⠁⢸⣿⣿⣿⣿⣿⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡏⢙
⠀⠀⠀⠀⠀⠀⠈⣋⠛⠿⢿⣿⣿⢈⢸⣿⣿⣿⣿⣿⠈⠀⢹⣿⣿⣿⣿⣿⣿⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠺⠟⠻⠋⢉⣉⣋⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⣈⡀⠘⠻⠿⢿⣿⣿⢈⠈⣹⣿⣿⣿⣿⣿⠿⠐⠻⠟⠛⠛⠩⣉⣁⡀⠦⠤⠁⠐⠓⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠲⢶⡄⢰⣉⠐⡀⣛⠉⠭⢁⣀⡰⢤⡼⢖⡒⠂⠉⠁
⠀⠀⢰⣤⡤⠀⢠⡄⠀⠀⠀⠀⠀⣀⠐⠤⢠⣭⣐⠚⣭⡥⢤⡞⠁⠛⣭⠉⢀⢉⠁⡤
⠀⠀⠈⠛⠒⠥⠀⠀⣠⣌⡁⠁⢈⠒⠯⠹⢞⣻⣊⣉⢎⢑⡦⡥⠽⢶⣺⣉⡌⠁⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠡⡀⢐⠀⢅⡂⠉⠽⣍⠓⠊⠁⠂⢰⡇⡀⡴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⢀⡐⢀⠀⢠⣈⣉⡙⠆⠀⠸⣮⣭⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠼⠧⢀⡙⠑⠈⠉⠓⠩⠽⢦⡤⠈⠳⠶⠮⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠆⣈⣠⠀⠘⠒⠂⠀⠈⠓⣦⠂⠀⣄⣀⡤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠂⣀⠀⠀⣌⠉⠉⠀⠀⠥⢤⣱⣯⣜⢒⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠯⠬⠟⠒⠚⠀⠉⠁⠀⠶⠶⠶⠶⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⢠⣾⣿⣏⣀⣾⣷⣄
⠀⠀⠀⠀⣴⣿⣿⣿⡿⣿⣿⡿⠋⣠⣦⡀
⢀⣴⣤⡀⠙⢿⣿⣿⣦⡀⠙⠀⣾⣿⣿⡟⠀⣀
⠺⣿⣿⣿⣦⣾⣿⣿⣿⠃⣤⣄⠈⣻⣿⣦⣾⡿⠃
⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⠋⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣤⣤⣀⠀⣠⣤⣶⣶⣶⣶⣦⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄⣀⠀⣀⣀⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠐⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⠀⠀⡠⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡁⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⣶⠆⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠈⣿⡇⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠛⠁⣀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡠⢀⠐⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⠟⠂⠀⠰⠟⠶⠲⠾⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠠⡀⢄
⠀⠀⠀⠀⠀⠜⡠⠂⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠣
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠
⠀⠀⠀⠄⡐⠀⠀⢀⡀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢂
⠀⠀⠀⠐⠀⠀⠀⢸⡇⠀⣠⣟⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠀⠘⣿⠀⠀⠀⠂⠀⡀
⠀⡀⠀⠁⠀⠀⠀⢸⡇⠀⢸⡿⢿⣿⣎⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠞⠁
⢀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠈⠿⠄⠙⠿⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣤⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢦⣤⣄⡀⠀⠀⠀⠀⢸⣿
⡇⠀⠀⠀⢠⡄⠐⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣤⣀⠀⢈⣿⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀
⡇⠀⠀⠀⣿⡀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣿⣷⣦⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣠⠇
⡇⠀⠀⠀⢈⡝⠂⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠻⣿⣿⠟⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⣶
⠇⠀⠀⠀⠈⠃⠀⠚⠁⠀⠀⠀⠀⠀⠀⠠⠄⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⠤⠼⠋
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠈⠀⠀⠀⠀⠀⠀⣠⠄⠀⢤⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠃⠀⡀⠀⠀⠀⢿⣦⣀⡘⠀⠀⠁⠂⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠠⠀⠀⢰⠃⠀⠈⢻⡆⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⣿⡆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠃⠀⠈⠓⠀⠀⠛⠁⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⡈
⠀⠀⠀⠀⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠰
⠀⠀⠀⠀⠀⢢⠑⠀⠀⠀⠀⠀⠀⠀⣼⡟⣴⠶⠶⠶⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡔
⠀⠀⠀⠀⠀⠀⠑⠀⠠⡀⠀⠀⠀⢸⣿⠃⢀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⢦⣄⠀⠀⠀⠀⠀⠀⢀⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠢⡀⠀⢼⡿⠀⢈⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣰⣏⡀⢈⡉⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⢀⣽⠀⠀⠀⠀⠀⠐⢁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢄⠀⠀⢠⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⠛⠐⠋⠁⠀⠀⡠⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢂⡁⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⡄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠉⣀⣀⣀⣀⡈⠙⠻⣷⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⣠⠖⠉⠁⠀⠀⠀⠉⠙⢷⣌⢯⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡇⠃⡘⠀⠀⠀⣠⣰⣖⠄⠀⠀⣠⣼⠟⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣈⡙⠃⠀⢁⠀⠀⠀⠸⡇⠘⠁⠀⠀⠉⡛⠂⠀⢁⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣼⠉⣹⠀⠶⠑⠁⣀⣶⠶⠾⠿⠶⣶⣀⠈⠊⠶⠐⣏⠉⡧⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡈⠓⠒⠂⠤⢒⡿⣓⠒⠢⡀⢠⠔⠒⣚⢯⡒⠤⠒⠒⠛⢁⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣌⠆⠃⠔⣈⣰⠋⡞⢰⠒⠀⡅⢉⠀⠒⡎⢳⠙⣆⣁⠢⠘⠸⣡⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡆⠀⠣⠄⠀⠹⠄⠀⠀⠀⠀⡀⠈⠠⠏⠀⠠⠜⠀⢰⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⢰⡉⡆⢠⠀⢢⠀⢇⣠⣄⡸⠀⡔⠀⡄⢰⢉⡆⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡹⡏⠀⢤⠰⢀⠴⠋⢠⣿⣿⣿⣿⠀⠳⠤⡀⠆⡤⠀⢻⣟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡄⠀⠀⢀⡞⠀⠀⢈⡤⠀⢻⣿⣿⡟⠀⢄⡁⠀⠀⢳⡀⠀⠀⣤⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠀⡞⡴⠓⠆⣟⡄⠀⣾⣿⣿⣧⠀⢠⣷⠐⠓⢦⢳⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠆⢃⠁⠈⠂⠀⠉⢀⠘⠿⠿⠿⠿⠃⡈⠉⠀⠐⠁⠈⡌⠰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢠⠼⡄⠰⠎⡉⠐⢃⠔⠄⢠⡀⡠⠠⡘⠂⢉⠱⠆⢠⠣⡄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠠⠝⢦⣀⢅⡀⢈⠒⠂⢸⡇⠐⠊⡀⢀⡨⣀⡴⠣⠄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡒⠈⣓⠯⡒⢬⡀⠘⠁⢀⡥⣚⠽⣊⠁⢒⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣭⣀⠶⠁⠈⢢⡞⠁⠈⠶⣀⣵⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣍⠀⢀⣩⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠄⠄⡀⡀⠀⠀⠀⠠⠄
⠀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠐⠀⠤⠀⡀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠿⢛⡷⢔⢼⢶⡲⡄⠀⠀⠀⠀⠰⠀⠀⡃
⠀⠀⠀⠠⠄⠀⠀⠤⠀⢀⢐⣾⣒⣘⣙⣆⠡⠸⢻⢃⢉⣢⣀⠤⢀⢁⠀⡁
⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠹⠑⠛⠭⣟⣛⣺⣫⠻⣓⣔⡜⠉⢖⢺⠶⠇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⣶⠻⠛⢙⣞⡳⢙⣲⠑⢻⠋⠀⠒⠑⠀⠐⠂⠒⡂⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢸⠁⠀⠀⠐⠀⠈⠉⠀⠈⠀⠀⠀⡐⠆⠀⠴⠔⠆⠤⠅⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⣦⠀⠀⠀⠨⡀⠀⠀⠈⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⢂⣿⡧⡠⣄⠀⢓⡠⠀⢦⣴⢺⢷⠿⡖⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡃⠘⣽⣲⡭⠎⢴⡴⢾⡍⠿⡍⢎⠩⠉⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠌⢮⠁⡑⢳⣾⡐⢑⠯⡕⠛⢫⠫⠇⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠉⠻⠤⠖⠛⠈⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠒⠀⠒⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⣀⣀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⣠⡾⠛⠋⠙⠛⠳⣦⣄⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠚⠛⠛⠛⠛⠻⢷⣄⣀⣀⣠⣼⠟⠛⠉⢉⡽⣍⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⣻⠆⠀⠀⢠⣖⣲⠀⠉⠙⠋⠉⠀⠀⠀⠀⠈⠛⠛⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣟⣷⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⣐⣶⠀⠀⠀⠀⠘⡙⠛⠃⠀⠀⢀⣤⢤⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⣀⠀⠀⠀⠀⣿⣼⠀⠀⠀⠈⢓⣪⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂⠰⠾⠷⠶⠶⠖⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣦⣔⡛⠋⠉⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠛⣩⠝⠻⣿⣦⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣇⠀⣿⠀⢀⣿⣿⣧⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣰⣾⣿⣿⣿⡆⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢷⣝⢿⣿⣿⠃⠀⠀⣠⣴⣾⢧⣀⣤⣴⣶⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⡳⢾⡭⣠⣴⡿⣟⣭⣶⣿⣿⣯⣿⣷⣿⣿⣟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠈⢟⣫⣾⢟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠫⠀⢨⣝⡻⠿⣿⣿⣿⠿⠟⠋⠁⢀⣴⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠑⠀⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣠⣾⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠂⣠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⡄
⠀⠀⠀⠀⠀⣠⠞⠄⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣄
⠀⠀⠀⢀⡴⠁⠀⣠⣾⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣦
⠀⠀⣠⠎⠀⢀⣼⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⡀
⠀⠀⠀⠀⣠⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣷⡄
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣽⣿⣿⣿⣿⣿⣿⡿⣟⣯⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣮⣿⣿⣿⣿⣷⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠤⠄⡀⠀⠀⡀⣀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢦⣄⠀⠸⣿⣿⣿⣄⡀
⠀⣰⠏⠀⠀⠈⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠂⠀⠈⢻⣷⣄⠉⠿⣿⣿⣿⣆
⢠⡏⠀⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⣿⣿⡀
⣾⠀⠠⢄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⠈⠞⢑⠀⠸⡇⠙⠿⠟⣿⣿⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣲⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣿⣿⣿⡀
⣿⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⠀⠀⢀⠠⠄⠀⢠⡇⠀⠀⣠⣾⣿⣿⣿⡇
⢸⡆⠀⠓⠒⠂⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⣸⠁⣰⣿⣿⣿⠿⣿⣿⣷
⠀⢻⡄⠀⠀⠀⠠⣄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⣰⠇⠸⣿⣿⠋⠀⠀⠹⣿⣿⣆
⠀⠀⠹⣄⠀⠘⠓⠲⠀⠀⣀⠀⠀⠀⢀⡀⡀⠀⠀⠁⠈⠡⢀⡼⠃⠀⠀⢻⣿⣇⠀⠀⠀⠙⢿⣿⣷⣦⡀
⠀⠀⠀⠈⠳⣤⡀⠀⠀⠨⠬⠑⠀⠀⠐⠪⠍⠀⠀⠀⣀⡴⠛⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠈⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⣀⡀⠀⢀⣀⣀⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⢀⣹⣿⣧⠀⠀⠀⠀⠀⠀⠀⢉⣿⠿
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶