Usa flag mushroom Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⢆⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠉⠁⣀⣀⣠⣤⣤⣤⣤⣀⡀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⡴⢞⣟⣍⢽⡇⠀⣻⣎⢌⣿⣿⣷⡆⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠸⣿⣿⣫⣞⡸⡇⠀⢸⣇⣏⣽⡿⢟⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⡀⠀⠀⠀⠈⠓⠦⣭⣭⣉⣹⡇⠀⢸⡯⠭⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠃⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠟⠋⠉⠀⠀⠀⠀⢝⣧⡀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡾⠀⢀⣠⡴⡾⠋⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢠⣴⢶⢶⡶⢤⣦⣤⣄⡈⠙⡆⠀⠀⠀⠀⡼⠁⠀⢰⠇⢀⡾⠋⠂⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣙⣛⢿⠇⢸⠷⠷⠿⢋⡰⠃⠀⠀⠀⣸⠃⠀⠀⡾⠀⠀⢷⡁⠀⠀⠀⠀⠀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠀⢸⡟⠛⠉⠉⠀⠀⠀⠀⢰⠇⠀⠀⣸⠃⠀⠀⠀⠉⠓⣦⣤⣀⣀⣀⣀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢰⠇⠀⠀⠀⠀⠀⣰⠋⢐⡟⠉⠉⠉⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢻⡀⠀⠀⠀⠀⠀⣼⠁⠀⠀⡾⠀⠀⠀⠀⠀⡴⠃⠀⡼⠁⣀⣤⣴⠾⠋⠁⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢳⡀⠀⠀⠀⢠⡇⠀⠀⢰⡇⠀⠀⠀⢀⡾⠁⠀⡼⠁⢰⡃⣤⣤⣀⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠚⠓⢦⡀⠀⢳⡀⠀⠻⣄⠀⠀⢸⡁⠀⠀⢸⠀⠀⠀⠀⡾⠁⠀⡼⠁⠀⠈⠳⣝⣳⡿⢉⣿⡦⣄⠙⡆
⠀⠀⠀⠀⠀⠀⠀⣴⢿⠃⠀⠀⠀⠙⠳⣆⢳⡀⠀⠘⢦⠀⣿⠇⠀⠀⣾⠀⠀⠀⣼⠁⠀⢸⠃⠀⠀⠀⠀⣠⠟⠀⡼⠯⢭⡭⠼⠁
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢀⣨⠟⠀⢻⡀⠀⠈⢷⣿⠀⠀⡇⣿⠀⠀⢠⣏⢀⢠⡿⠀⠀⠀⣠⠞⠁⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⣶⠒⢿⡉⠀⠀⠀⠀⢳⣀⠀⠨⣿⡄⣰⢹⣽⡇⠀⣸⡿⣼⢸⡇⢀⡴⠛⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⠳⣄⠀⠀⠀⠈⣧⢂⡐⣻⣇⡟⡜⡿⣿⡀⣿⡷⣿⠘⣿⡋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⣛⠿⠿⢶⣼⡏⢠⠀⣿⡽⡰⡱⣸⡇⣿⡏⣻⠸⣿⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣴⣾⣽⡿⠃⣿⡅⡆⢸⣿⢮⣬⠞⠁⠸⣗⠅⢆⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣧⠾⠃⠀⠀⠀⠀⠀⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣉⣿⣿⠉⣿⡟⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢻⡶⠋⢿⠏⠸⡿⠁⣰⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣁⢿⡇⣦⣄⠀⢀⣤⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢇⣼⣇⠿⣿⣧⠸⠟⠀⣹⣡⣿⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⢻⡏⢸⣿⠉⢀⡈⠁⠀⣶⡄⢿⡟⣯⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣸⣿⣿⢸⣿⣿⣷⡀⠘⣿⣧⠀⢿⣿⠀⢰⣿⢛⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣿⡿⠚⢿⡏⠉⣷⡀⠈⣿⡇⠘⠛⠀⠛⠁⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠇⢠⡜⢻⣆⠻⠇⠀⢹⣿⡀⢀⣶⡖⢀⣀⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⠀⢸⡇⠘⠟⢀⠀⣲⢸⣿⠀⠈⠛⠁⣿⣿⠁⢰⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⣿⠀⠐⠿⢀⡟⠀⠀⢠⣿⣦⠀⣙⠋⠀⣾⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢻⣷⣿⡇⠀⣴⣿⠃⣿⡇⠸⣿⣿⡄⢸⣷⠀⠉⢰⣿⠛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⢸⣿⣿⢧⣼⣿⠟⠁⠛⠀⠀⣉⠹⠇⠀⠛⢀⣀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠈⣻⡟⠈⠉⠛⠀⣴⣦⠀⠀⣿⣷⣴⣿⠆⢾⡏⣾⡇⢠⣿⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣽⣿⠀⣿⡁⢰⣿⣦⠀⣿⣿⠀⠀⢻⣿⣿⡿⠀⠀⠀⣿⡇⢈⡁⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⣿⡏⢹⡀⢿⣷⠀⠉⠁⠀⠘⠋⠀⠀⠀⠹⣿⠃⠀⣶⡄⢿⡇⠸⣧⠚⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣿⣿⣿⠃⣸⣷⣦⠉⣼⣿⣿⡆⣶⢰⣶⣦⡀⠀⢠⣦⡀⢻⣧⠀⠀⣤⡀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣸⣿⡟⣿⣿⣯⠘⠋⠀⠈⠙⠛⠁⣿⠘⣿⣿⣷⡀⢹⣿⠇⣄⠛⢀⡀⢻⣧⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣽⣿⣧⣿⢉⣟⢰⣶⣆⠀⣿⣷⡾⣿⠀⣿⣿⣿⣷⡀⠀⣸⣿⠀⢿⣿⢸⣿⡀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⣿⣿⡿⢻⣿⡟⠸⣿⡟⠀⠹⠟⠃⠹⣆⠹⣿⣿⣿⠇⢀⣿⣿⠀⠸⣿⠘⠿⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣷⣿⡋⢷⣾⣿⡄⠀⠀⣠⣤⡀⠻⣧⠀⠉⠁⠰⠿⠋⠻⠆⠀⢀⡀⠀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⢿⣿⣿⡏⢷⣬⣿⠛⢃⡀⠘⣿⣿⣷⠀⣀⣤⣦⡄⢠⣾⣷⣰⣿⠄⣾⣇⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⣏⢱⡈⣿⣿⣷⡼⣿⡆⠹⠿⠃⠘⠿⠿⠟⠁⠀⠛⠛⠈⠋⠘⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣽⣿⣿⣦⢹⣷⣜⢿⡇⢿⣅⡀⠀⣶⡀⠀⣴⡶⢀⣼⣿⣤⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⢿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣷⣾⣷⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡏⠙⠀⠁⠀⠀⠁⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣿⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⢸⣿⠀⢰⣆⡄⠀⡀⠀⠀⣴⡿⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⢸⣿⡀⠘⣿⣧⠸⣷⣆⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡍⣿⣷⣘⣷⢱⣤⣿⣜⣶⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣌⣿⣿⣿⣧⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⣿⣿⣿⣿⣿⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢿⣭⠞⣋⡴⢛⡽⢟⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣷⠞⣩⠶⣫⠶⣋⣴⠞⣹⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢦⣤⣤⣴⡷⢛⡵⠞⣡⠞⠉⠉⠙⣧⠞⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣫⡴⢛⣥⠞⣩⠶⢋⣠⠞⣻⣄⠀⠀⢀⣿⠶⢋⡴⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠞⣡⠶⢋⡴⠞⣁⡴⠋⣡⠞⢉⡽⢛⣾⠟⣡⡴⢫⡞⠋⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡴⢛⡵⠞⠛⠛⠾⣏⡴⠟⣡⡶⢋⣴⠿⣥⡞⣡⡶⣿⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⣿⠁⠀⠀⠀⠀⢸⣧⠞⣡⡶⣿⠀⠀⠈⣿⣡⡶⣫⡷⣤⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣿⣀⠀⠀⠀⠀⣸⣧⠞⢁⡴⢛⣶⠶⣾⠟⣡⠾⣫⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⣾⣿⣷⣶⣯⣴⣿⣤⡾⠷⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⠻⠿⠿⠿⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠆⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠖⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡴⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣠⠞⢀⡀⠀⠀⠀⠀⣸⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢰⡟⢻⡆⢀⣤⣄⠀⠀⢀⡴⢶⣿⣛⠳⢶⣤⡀⠀⠀⠀⠀⣠⣤⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⣖⣶⣤⠤⢤⣀⠀⠐⠯⠿⠀⢀⣴
⠀⠀⠙⠋⠀⠈⠛⠛⢀⣴⡯⠩⠄⠀⠀⠄⠀⠈⠙⡲⢤⣀⠀⠀⠀⠀⠘⠛⠁⠀⣴⣶⡄⠀⡠⢊⡽⠛⢓⡾⢳⠤⠬⣇⠀⢻⡛⢦⡀⠀⠀⠘⠛
⠀⠀⣴⢶⡄⠀⠀⣠⡟⠁⠠⠀⠀⠀⠀⠀⠀⠀⠊⠔⡀⠩⡳⢦⣤⣀⠀⠀⠀⠀⠛⠛⣠⠾⠒⣻⢠⠋⠉⣳⢸⣤⣀⡼⠓⠺⡅⣞⠙⢦
⠀⠀⠛⠞⠃⢀⡼⠛⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡆⠀⠀⠀⠀⡴⣧⠤⣞⣽⠃⢙⠮⣵⠃⠀⠈⣷⣤⡤⡿⠾⣗⢮⠳⡄
⠀⠀⠀⠀⢀⣞⣡⣬⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠟⠀⠀⠀⢀⣾⠗⠋⢹⣆⠎⠀⣇⡀⣸⠶⠤⠖⠡⡞⠙⢳⡀⠈⣿⠓⢾⣦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣶⣶⣶⡿⠿⠛⠛⠉⠀⠀⠀⢀⣴⡧⠻⠤⠖⢋⣾⣀⣀⣠⣭⣥⠤⠤⠤⠤⠽⠶⣶⣿⣶⣛⣦⡼⢇
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⢻⡋⠋⠛⠉⠉⠛⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣮⠶⡒⠛⡋⠉⠉⣁⣠⠤⠐⠒⣒⣂⣒⣒⠂⠤⣄⡀⠀⠉⢙⠉⠛
⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡇⠀⠀⢀⡀⠀⠐⠂⠀⢰⡏⠀⠀⡀⠋⣄⠊⠰⠸⡇⠀⠰⠋⣉⣀⣀⡉⡱⠀⢨⠎⡆⠈⡄⡁⡄
⠀⠀⠛⠛⠀⣤⣦⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡇⢶⣶⣾⠇⠀⠀⠀⠀⠈⠳⣄⡀⠐⠀⠈⠒⡠⠈⠘⢓⣒⣿⠛⠛⠛⣿⠳⠛⠓⠩⠐⠊⠐
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡇⠀⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⡉⠛⠒⠲⠶⠿⠿⠶⠶⠦⣿⠀⠀⠀⣿⠤⠤⠤⠤⠦⠴⠶⠖⠚
⠀⢀⣼⣷⠀⠀⠀⠀⠀⠀⢴⡷⢹⡀⠀⠀⠀⣷⠀⠀⠀⢀⣴⣿⠏⠉⠁⠈⠉⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⠀⠀⠀⢀⣠⣄
⠀⠈⠉⠁⠀⠀⠀⢀⣀⡤⣄⣀⣀⢳⡀⠀⠀⠘⡆⢀⡴⢫⣿⠃⠀⠀⠀⠀⢀⠤⠐⠚⠙⢷⡄⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⣅⣹
⠀⠀⠀⠀⠀⣠⢶⣉⡿⠦⠾⣄⣀⣿⢧⠀⠀⠀⣹⠟⡣⡟⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠉⠁
⠀⢀⣀⢀⡴⢛⣇⣠⠞⠂⠘⢧⡀⠀⢹⡆⠀⣴⠋⠀⠁⠀⠀⠀⠀⠀⡀⣀⣀⣀⣤⣤⣶⣾⠿⠋⠀⠀⢰⠇⠀⠀⠀⣿⠀⠀⠀⣀⡤⠖⣾⣩⠿
⠀⠈⣹⣿⣩⣏⣀⣹⡀⢀⡀⠮⣨⠛⣍⢹⠘⢧⣄⣀⣃⣠⣤⣶⠶⠿⢿⡿⠿⠟⣻⠟⠋⢳⡀⠀⠀⠀⡼⠀⠀⠀⠀⡇⢠⣴⣾⣵⡒⠏⢤⢞⣒
⠀⠀⠻⣾⣾⣧⣦⣬⣭⡉⢠⢤⣷⣾⣿⣾⡆⠀⠹⣹⡏⠉⠉⢻⠀⠀⢸⡇⣤⡚⢁⠢⠀⠀⢻⡄⠀⢰⠃⠀⠀⠀⢠⡇⠈⠉⠛⠻⠿⣿⣿⣶⣤
⠀⢀⡀⠀⠉⠙⠛⠛⠛⡆⠀⠀⠟⠛⠉⠁⡇⠀⠀⢿⡇⠀⠀⠘⡇⠀⢈⡟⠻⠿⢿⠿⡿⢷⣦⠗⠀⡞⠀⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⣴⡍⠉⢻
⠀⠈⠘⠀⠀⠀⠀⠀⠀⡉⠀⠀⡄⠀⠀⠀⡇⠀⠀⢸⣇⠀⠀⠀⣧⠀⠸⡇⠀⠀⠸⠀⠇⠀⠀⠀⢠⡇⠀⠁⠀⠀⢸⡇⠀⢀⣀⣀⡀⠀⠀⢀⠇
⠀⠀⠀⠀⣀⣤⢤⣀⠀⡇⠀⠀⡇⠀⠀⠀⡇⠀⠀⠘⣿⠀⠀⠀⡏⠀⢠⡇⠀⠀⡇⢰⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠸⣇⣴⠟⠉⠉⠽⣳⡄⢸
⠀⠀⠀⣼⠃⠀⠈⠪⢳⣄⠀⢀⡇⠀⠀⠀⡇⠀⠄⠀⣿⠀⠀⢰⠃⠀⢸⡇⠀⢀⠇⢸⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣰⣟⣁⠂⠁⠀⠀⠀⠙⢾
⠀⠀⢸⣏⠃⠀⠀⠀⣠⡼⠆⢸⡇⠀⠀⠀⡇⠀⠀⠀⢹⡇⠀⡞⠀⠀⢸⡇⠀⢸⠀⢸⠀⠀⠀⠀⡏⠀⢠⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢿⣵⠧⣶⡾⢻⡸⠀⠀⠀⢇⠀⠀⢸⡇⠀⠀⡇⠸⡷⣰⠃⠀⠀⠸⡅⠀⡾⠀⠈⡄⠀⠀⣸⠁⠀⠸⠀⠐⣏⠂⢤⡀⢀⣒⡒⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⢸⠉⠀⢇⠀⡀⣄⡼⢀⠀⠘⡇⠀⠀⠈⠀⢳⡟⢰⠀⠀⠀⢣⡐⠧⠂⣤⣿⠀⣵⠃⠀⠀⠀⠀⠀⠈⠛⢲⣾⣦⠤⠭⢴⡖⠚⣩
⠀⠀⠀⠀⠐⢾⠀⠀⠘⡷⢿⣿⣦⠒⠻⢶⡹⡐⡄⢇⢸⡆⠗⢁⡄⠠⠀⠀⠙⢿⡿⡿⢿⡟⠁⠀⠀⠀⠀⢠⡀⠶⠄⠘⣻⡇⠀⠀⠸⣸⠎⠠⣊
⠀⠀⠀⠀⠙⢾⣤⣀⢀⡼⠇⠀⠀⠁⠀⠒⠿⢿⣷⣌⣼⣷⠖⠨⢶⣷⠋⠚⠁⠊⠁⠁⣿⠀⡀⠀⢀⣠⠀⢛⠕⠀⣰⣶⡟⠀⢀⣀⠀⠹⣾⡉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠈⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠈⠛⠉⠀⠛⠁⠀⠀⠁⠀⠛⠛⠶⠛⠀⠁⠘⠛⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠶⠿⠿⠃⠀⢾⣿⣿⡇⠀⠀⣩⣽⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣵⣶⣄⠀⣴⣶⣶⡄⠉⠉⠀⠀⠘⢿⣿⡿⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣏⠀⠛⠿⠟⠁⠀⣾⣿⣿⡆⠀⢀⣀⠀⢰⣶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣼⡟⠻⢟⣛⠻⣿⣿⣿⣦⣀⣴⣿⣷⡌⠛⠛⠁⠰⣿⣿⡇⠈⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⢀⣠⡄⠀⠀⠀⢸⡁⠙⠒⠦⢤⣉⣘⣛⣿⣿⣿⣿⣿⡿⠁⣰⣾⣶⡄⠈⠉⠁⣰⣶⣶⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣌⣯⣼⣿⣿⠁⣀⠀⠀⠸⡏⠓⠒⠢⠴⠤⠤⠤⠶⣻⣿⣿⣿⣿⣶⣿⣿⣿⣇⣴⣶⣦⠘⠛⠛⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠿⣿⠟⠛⣵⣾⣿⡆⠀⠀⠹⣖⠒⠒⠒⠒⠚⠋⠉⢁⣾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣷⣆⣹⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡸⣆⠸⣿⣿⣿⣅⣀⣀⠀⠈⠓⢶⡄⠶⠒⠂⣉⣴⠇⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⣆⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠉⠓⢾⣍⣡⠟⠀⠀⠀⠀⠀⠉⢛⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢯⣀⣈⣭⣭⣥⣤⣤⣀⡀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⢸⣷⡈⢦⠙⢦⡙⣮⣱⡤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢿⣿⡟⠁⠉⣿⣿⣿⣦⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠚⠛⠋⢉⣿⣅⠀⠀⣠⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣄⣀⣰⣿⣿⣶⣶⠟⠛⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠘⢿⣿⠇⣸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠻⣿⣿⠋⠉⢻⣿⣿⣀⣀⣼⠟⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⣤⣤⣤⣤⣀⣀⣽⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣀⣀⣾⣿⣄⣠⡼⠿⣿⣿⣿⣿⣦⣠⣾⣷⡀⠀⠀⠀⠀⢀⣴⣾⠛⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿⣿⣿⣿⣿⡀⠀⢀⣿⣿⡁⠀⣹⣿⣿⣿⣦⠀⠀⢰⣿⣯⣨⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⡈⠠⠐⢊⢝⣿⢻⠿⠿⣿⣿⣿⣿⡿⢿⠛⠛⢛⡞⠀⠀⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠐⢊⡵⢋⠼⣿⠀⠀⠀⠀⠀⢧⣁⠂⣬⡽⠋⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢧⣼⡿⠀⠀⠀⠀⠀⠈⣿⠋⠁⠀⢀⣼⠃⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢸⡇⠀⠀⠀⠀⠀⠀⠘⣧⣰⢠⣿⡏⠀⣀⢀⣞⡴⠋⠙⠻⠿⠿⢿⡿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣸⣿⣾⠃⢀⠀⠀⠀⣄⠀⠰⣿⣿⣾⣿⣇⣾⣿⡾⠋⠀⠀⠀⠀⠀⠀⣸⣥⣴⡶⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣿⣿⣿⣿⣿⣿⣿⡀⢸⣷⣀⠘⣿⣄⢀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡆⣀⡤⠀⣀⣴⣿⣿⣿⣿⣿⣿⡏⠉⠛⠛⠳
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⣰⣿⠿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠟⣿⣿⣿⡿⠿⠷⠾⣿⣿⣿⡟⠻⢿⣇⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠸⠛⠉⠁⠀⠈⠙⠿⠙⠷⠀⠙⢿⠃⢸⡿⠋⠁⠀⠀⠀⠀⠀⠉⠻⠇⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠟⠉⠉⠀⠈⠃⠉⠻⡟⠟⢷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡿⠋⣹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠻⣿⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢜⡵⠋⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠔⡵⠃⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠸⠁⠈⢳⡀
⠀⠀⠀⠀⢀⠎⠎⠀⠀⠀⠀⠀⣇⠀⠀⣀⣠⣄⣠⣄⣠⣠⣴⡦⣤⣶⠴⢦⠴⠦⠴⣧⠤⢴⣼⡟
⠀⠀⠀⠀⢸⣴⣾⣶⢶⣶⣾⣛⣛⣋⣛⣿⣿⣿⡟⠛⠻⢿⣿⣿⣷⣦⣤⣠⣤⣄⣤⣶⠿⠛⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠙⠛⠛⠛⠿⠿⠿⠿⠿⢿⠁⠀⠀⢸⣿⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡤⣀⣀⣿⠟⣩⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣟⠛⢛⣷⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠉⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠿⢿⡻⣶⣄⠀⡇⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⢻⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣸⠿⣾⡇⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⢏⠀⠘⣷⡀⡞⡇⢠⠀⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠽⠟⠻⠻⠮⠼⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⢀⡀⠀⠀⠉⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠉⠀⡰⠁⠀⠐⠀⠀⠈⠀⠀⠙⠒⠚⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠉⠛⠋⠀⠀⠀⠈⠲⠴⠊⠀⠀⣀⡀⠀⠀⠀⡸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣎⠁⠉⡃⠀⠀⢄⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⡹⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣠⠀⢠⠖⠒⢦⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠐⠆⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠚⠁⠀⠰⣄⣠⠞⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⠀⠘⠦⣤⡤⠂⠀⣠⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠠⠒⠙⡆⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⡷⠶⣶⣶⣤⣧⣀⣀⠀⠀⠀⠀⠁⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⡀⠈⠒⢊⣤⣴⠾⠟⠟⠙⠉⠃⠉⠀⠀⠈⠻⣧⠀⠀⠀⠀⠉⠋⠻⣿⠷⣦⣄⡓⠶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣙⣠⣾⠛⠫⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢹⣇⣤⣼⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠁⢠⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢀⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣇⣘⣀⣀⣀⣴⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢿⡟⠉⠉⠉⠉⢹⡏⠉⠙⠛⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⣴⣿⡏⠁⠀⠀⣿⠀⠀⠀⢀⣈⠛⠶⠶⠞⠃⢀⣽⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠈⠳⣦⣤⡾⠋⠀⢠⡾⠋⠉⠻⣦⠀⠀⠀⢾⡁⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢶⡄⠀⠀⠀⠀⠀⣀⣀⣸⣇⣀⣀⣀⣿⡀⠀⠀⠈⠳⢦⣴⠾⠋⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣀⣠⣿⣶⣶⠿⢻⡟⠛⢻⠋⠉⢻⠋⠉⠛⡟⠛⠻⡷⠶⢦⣤⣀⡀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠉⢷⡀⠘⠆⠈⠃⠀⠈⠀⠀⠘⠀⠀⠘⠁⠀⠸⠃⠀⡼⠋⢉⣿⠷⣮⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⠄⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠋⡁⢀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣴⣏⣀⠀⣴⠀⠀⣿⡇⠀⠀⠀⠘⣿⠀⠀⢄⢀⣰⣤⣤⣿⡿⠟⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣿⠁⠀⠀⠀⡄⣿⡿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⣧⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⡟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢀⢀⠇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡜⡘⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣸⠀⠇⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⢠⠇⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠉⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣤⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⡟⣿⡷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠈⠛⠛⣁⡀⣼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⡿⠓⠂⠀⠀⠀⣠⡾⠃⣀⡀⠀⠃⠀⠛⠃⠈⠛⡋⠠⣿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣴⠟⠀⠀⠛⠁⢀⣤⡀⠀⠆⠀⠘⠟⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠖⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠆⠀⠀⠀⠀⠉⠀⠀⠀⢀⡄⠀⠀⠀⠀⠉⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣏⣤⣴⣶⠾⢛⣫⠟⣻⢿⠟⢹⣿⣿⣿⢹⡙⢯⡛⢭⡙⠢⢭⡉⠓⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀⠀⠀⠀⠀⣼⡟⣩⠖⠉⣠⠞⠉⡴⠋⢰⣋⣀⣼⣿⣿⣿⠾⣷⣤⣷⡀⠹⣄⠀⠙⢦⡀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠀⠀⠀⠻⢿⣧⣤⣾⣶⣶⡾⠿⠟⠛⠋⠉⣽⣿⣿⡿⠀⠀⠈⠉⠛⠷⢾⣦⣄⣀⣳⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣀⣤⣤⣀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡟⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡾⢻⣿⣿⠏⡻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢩⠀⢿⣿⠟⢈⡉⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⢶⡆⠉⣵⡄⠁⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣭⡴⠶⠶⣤⣄⡀⠀⠀⢀⢸⡆⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⡀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠽⠒⠚⢋⣩⠿⣿⣿⣦⡀⠈⠘⣇⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⣁⣠⣤⣶⣶⣾⣿⣿⣟⡛⠓⠶⢬⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⣠⠖⠋⣠⠎⣿⡟⡟⣿⣦⡀⣿⠀⠀⠀⠀⣰⣿⣿⣿⣿⣶⠟⣫⠟⢹⣿⡟⣆⠙⢎⠙⠫⣝⠢⣄⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣞⣁⢀⣰⣃⣼⣿⢃⠇⢹⢻⠻⣾⠀⠀⠀⣸⣿⣿⣿⣿⣷⣃⣼⡷⠶⢿⣿⡟⠛⠶⣤⣷⣀⠈⠳⣌⠳⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢻⣿⣿⠛⠶⠾⠾⠶⠛⠀⠀⣰⣿⣿⣿⣿⡟⠛⠉⠁⠀⠀⢸⣿⡇⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⢺⣷⡄⠀⠻⣇⠀⢺⡆⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⠀⢠⣤⠀⢰⣶⠀⠀⣴⣦
⠀⠀⠀⠀⢴⡍⢧⡀⠀⠸⡄⢸⠃⠀⣼⣿⡿⣿⣿⣿⣧⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣧⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⡏⢀⡞⠁⠀⣸⠟⣃
⠀⠀⠀⠀⠈⠙⣄⠳⣄⠀⢷⡏⢠⡄⠻⢿⡇⢟⢻⢻⢿⣄⠀⠀⣿⣿⡇⣿⣿⣿⡟⣿⣿⡆⠀⢻⡿⡟⢻⣿⡟⠿⠂⠀⢀⡀⢳⣼⠁⠀⡴⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠈⠳⢽⣷⣌⢧⡈⢳⡀⠈⣿⡸⡌⣦⠳⣝⢧⡀⠛⣿⢃⣹⢻⡟⢣⣸⢸⣧⠀⣿⢁⢇⡜⢻⣧⠀⠀⠀⣸⠇⣸⠏⣠⣞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣌⡙⠻⣿⣶⣽⣷⣾⣷⣷⢨⠳⡈⠳⡿⣆⢹⣾⡇⠸⣻⡈⣿⡀⢿⡄⣿⡏⡌⡇⣸⢻⣆⣤⣞⣡⣴⣿⡾⠛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠉⠀⠈⠻⣿⣦⣽⣄⣿⡟⠘⣿⢻⠀⢿⡇⠙⣷⡍⣷⢻⣄⣇⣣⣏⣼⠟⠛⠛⠛⠛⠒⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠘⠿⣧⣼⣇⣰⣿⣰⡿⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠛⠋⠉⣿⣿⣿⡇⡀⢀⢍⢉⡻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠸⣿⣟⠕⡔⡰⡰⡀⢻⣿⣿⡟⠌⢎⢢⠱⣹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣧⡀⠙⢿⣮⣤⣁⣃⣃⣼⣿⣿⣿⣼⣼⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠉⠉⠙⠙⣿⣿⣿⡏⠉⢁⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠿⠛⠛⢻⣿⣿⠛⠻⠿⣿⣿⣦⠀⠀⠀⢠⣿⣿⣿⣇⣴⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡠⢊⢀⢀⢻⣿⣿⡆⠄⠄⡔⢸⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢶⣤⣼⣿⣿⣮⣬⣼⡾⠟⠁⠀⠀⣸⣿⣿⣿⡿⡛⡛⠛⣻⣿⣿⠛⠻⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⡀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣷⣬⣼⣰⣼⣿⣿⣐⣂⣢⣼⣿⠏
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⣄⠀⠈⣿⣿⣿⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃⠀⠉⠉⢛⣿⣿⡿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⢠⣿⡿⠟⢛⣿⣿⡏⠛⠛⢿⡆⠀⢹⣿⣿⣷⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠈⠻⢤⣤⣤⣿⣿⣤⣤⠷⠟⠁⠀⠀⢿⣿⣿⣧⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⣰⣿⣿⣿⠀⣀⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⣿⣿⣿⣿⡇⠀⠀⢀⢼⢿⣿⡿⢣⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⣠⣿⣶⣶⡶⠇⠺⢿⡛⠻⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣀⣾⣿⣿⣿⡿⠃⠀⠀⠈⢉⣿⣿⡿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣰⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⣿⣿⣿⣿⢿⣿⣿⠏⠀⠀⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠁⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠄⠤⡀
⠀⠀⠀⠐⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠠⠀⠠⢀
⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⡠⠤⠒⠂⠁⠀⢀⠔⠁⠀⠀⠀⡀⣀⠑
⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠒⠒⠒⠒⢚⠉⠀⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡦⠄⠀⠀⠀⠀⠀⠆⢀⠂⠀⠀⠀⠑⠘⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢂⡀⠠⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⡀⠤⠀⠒⠀⢊⡀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢉
⠀⢀⠐⠈⣀⡠⠤⠀⠀⠚⡤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠆
⠀⠁⢀⠀⡉⠃⠀⣀⠠⠄⠁⠀⠀⢀⠐⠀⢠⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣀⠀⠀⢀⠀⠠⠈⠀⠀⠐⠂⠀⠀⠀⠀⢰⠌⠠⢖⣁⠤⠄⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠰⠀⠄
⠀⠘⠠⠀⠈⠂⠀⠰⠕⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠰⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⡁
⠀⠀⠀⠀⠠⠐⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠂⢰⠀⠀⡀
⠀⠀⠀⠀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠⠯⠐⣠⣂⡰⢠⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠶⠒
⠀⠀⠀⠠⡘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠉⠀⠉⠁⠋⣰⣥⠅
⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⢚⡯⠟⠛⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣖⣾⣻⣷⣿⣿⣿⠿⢿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣠⠤⠤⠀⡠⠄⠊⢟⡰⠊⠁⠀⠀⠤⡤⣀⠀⠘⠄
⠀⠀⢀⣴⣾⣿⣭⣿⣱⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣾⣻⡇⠀⠀⠀⢠⠞⠉⠀⠘⠃⠼⠓⢁⠔⠉⠠⠀⠀⠀⠀⠀⠀⠀⡳
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠔⠁⠐⠄⠀⢤⠄⢀⡀⠀⠈⠀⢠⠂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣍⣻⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠘⡃⠀⣀⠀⠀⠀⠀⡠⢤⠀⠘⠿⠂⢈⠉⠀⠠
⠀⠀⠀⠙⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠊⠁⠀⠀⠀⠀⠈⠆⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣉⣹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢀⠠⠘⠒⠲⣦⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⣿⣿⢿⣿⣿⣇⣶⣿⣷⣤⣤⣠⠴⣦⣀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠄⠤⢠⣀⠀⠙⠋⠁⠈⠙⢷⣻⣿⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣾⣯⣽⡿⣿⣿⣿⣿⣷⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠘⠀⠛⠢⣄⠀⠀⠀⠉⠉⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣿⣿⣟⣻⢿⣿⣻⠿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⠤⢰⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠟⣻⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⠂⠀⢀⠰⠀⠠⠐⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⠀⠀⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⣿⣿⣿⣷⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⣆⠀⠀⠀⠀⠀⠠⠁⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣾⣿⣿⣿⣿⣿⣿⡁⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⡆⠀⠀⠀⠆⠀⢀⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏⢿⢿⣿⣿⣿⣿⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠘⠳⣷⡀⢧⣠⡆⡀⠀⢱⣧⠷⠖⠢⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠉⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠲⠽⠿⠟⠿⣷⠤⠏⠁⠀⠨⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠂⠒⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⡽⢒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠙⠀⠀⠀⠀⠀⠈⠙⡻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣭⣤⣤⣭⣽⣻⡆
⠀⠀⠀⠀⠀⢀⡠⣚⡍⠁⢀⢀⣀⡀⠀⠀⠀⠀⠀⢺⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣋⣼⠾⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣿
⠀⠀⠀⠀⠰⡿⣾⣭⣽⣿⣷⣿⡿⠛⠛⣶⣶⣤⣄⣀⠈⠱⣄⠀⠀⠀⠀⠀⠀⢠⣶⣿⣫⣷⣿⣿⣿⡿⠁⠋⠻⣿⣿⣿⣿⣿⣽⣿⠟⠃
⠀⠀⠀⠀⠀⠙⠿⠿⣽⣿⣿⣿⡟⠋⠋⠓⣿⣛⣿⣿⡿⢎⣿⠀⠀⠀⠀⠀⢀⡾⣻⣿⣻⣿⣿⣿⣿⣿⣦⡀⠤⢸⣿⣿⣿⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢄⣤⡀⠀⠀⢉⠟⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠸⣵⡏⠘⢛⣻⣿⣿⣯⣿⡟⠀⠀⠈⣿⠉⣁⣠⡆⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠠⡄⠄⣼⠋⠉⠙⢿⠀⠙⠛⠄⢀⡀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠋⠉⠉⠉⠉⠹⡄⡀⠀⠈⠻⢉⣿⢓⣀⣀⣀⣀⣀⣐⠯⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢂⠀⠀⠀⠀⠀⡄⡆⢠⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣡⣷⣈⣀⣤⣾⣿⣿⣿⠿⠻⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⣇⣼⣰⣿⣇⡄⠀⠀⣄⡄⢛⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⢻⡏⠉⠙⢿⠉⠙⠛⢛⠀⠀⡟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⣼⣯⣤⣿⣿⣽⣧⣶⣿⣾⣿⣿⣯⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⣗⠀⠀⣋⠀⠀⠀⢸⠀⢰⡇
⠀⠀⠀⠀⠀⠰⠻⢯⣽⣭⡟⣋⣻⠟⢻⣿⣉⣉⡹⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⢈⣺⡷⠚⢻⠇⠀⣤⣿⣥⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣻⣯⣤⣴⣶⣶⣶⣶⣶⣦⣭⣛⡣⠆⢺⠀⠀⠘⠋⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠛⠿⠿⣿⣿⣿⠛⠙⡿⠾⢾⣿⠟⣃⠺⡀⢰⠁⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⠀⢸⡿⡐⠙⡷⠀⣾⡀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠐⠐⡪⢂⠀⠀⠤⠀⠺⠶⠦⠴⣾⠋⠸⠄⠃⠀⢠⣼⠀⠡⡞⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⠀⠙⣳⣟⠀⠀⠀⠁⣾⡄⠀⠀⣹⠊⡿⢦⠀⠀⢸⡟⠀⢰⠃⠠⠀⢠⡉⠁
⠀⠀⠀⠀⢧⣶⠛⢿⣿⣿⣿⣿⡟⣻⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⢈⣿⣠⠀⡀⠀⠹⡿⠀⠀⠘⣄⡋⠀⠁⠛⢸⡇⠀⣾⡈⡃⢀⠦⢉⠤⠄
⠀⠀⠀⠀⠀⡉⠑⠒⠚⠛⠛⠏⠀⠈⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠳⣾⣆⣴⠚⣇⠀⠀⣶⣾⣧⠀⠀⠀⠘⣿⢄⡼⡇⣷⠏⢌⣔⡁
⠀⠀⠀⠀⠀⠳⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠉⢻⣏⣸⠦⢹⡄⠈⠀⠸⣿⡷⣳⡀⠀⣹⣾⡧⢥⣏⣲⢮⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⣻⣿⠃⣿⣤⣿⣾⢇⠀⠘⠀⣱⠟⡁⡙⢦⣽⡿⣯⡇⢡⡗
⠀⠀⠀⠀⠀⠀⠀⢦⠂⣀⣸⡀⠀⢠⢤⣠⣿⣿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⡽⣿⣽⣿⣿⣷⠦⢠⢸⢿⣶⣿⡟⣷⣽⣷⣿⣵⠏⣴⠁
⠀⠀⠀⠀⠀⠀⠰⣬⣆⣹⣿⣷⣶⣾⣾⣤⣿⣿⣷⢿⣄⡀⠀⠀⠀⠀⠀⠈⠠⣀⡷⣏⣿⣿⣿⡛⢰⠣⡤⣾⣷⣶⣽⣾⣿⣿⣽⣿⠧⣿⣾⣾
⠀⠀⠀⠀⠀⠲⢿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⢻⠻⡏⢛⣷⣾⣆⠁⢿⣿⠱⡿⢿⣿⢿⢿⠿⠀⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠤⠤⠀⣀⡹⠛⠳⢟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠟⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠄
⠀⠀⠀⠀⢀⣴⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⡀
⠀⠀⢠⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡔⠉⠉⢹⣿⣿⣿⣷⣾⠂
⠀⣰⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣾⣿⣿⣧⠀⠀⠈⢿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣤⡀
⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡆⠀⠃⠘⣿⣿⡟⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⠀⠀⠀⠛⠟⠃⢰⣶⣾⣇⠀⠀⠀⠀⠀⠀⣠⣤⣌⣿⣿⣿⡏⢾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡀⠀⠈⠉⠙⠋⢀⣠⣤⣰⣶⣶⣾⡿⠟⠃⠀⠀⠀⢀⣤⣼⠁⠈⣿⣶⣿⣟⣁⣸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡎⠛⠁⣽⣿⣿⡇⠀⠀⠀⠀⠀⣴⣌⣻⡟⠛⠂⠀⠀⠘⠛⢻⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠈⣹⣿⣇⠀⠀⠀⠀⢈⣉⡛⠉⠷⠀⠀⢠⡀⠀⠀⢈⢙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣤⣿⣿⡄⠀⠀⠀⠈⠛⢃⣠⡤⠀⠀⠘⠻⣆⢰⣿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⡿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⣁⣀⡉⠠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡎⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠶⠤⣤⡀⠀⠀⠀⢱⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠘⠛⠻⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡟⣿⣿⡇⠀⠀⠐⠂⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠋⣽⣿⣷⠀⠘⠀⠀⢀⡤⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢩⣿⣿⡇⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣿⣿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣸⣿⣿⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣰⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⣀⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠋⠐⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⢃⢺⣷⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⢷⣸⣶⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣌⠛⣦⢘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠆⣡⠾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⡰⠀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⠀⠀⠤⡔⠚⠆⢤⠄⠚⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡠⠀⢎⠀⠄⡢⡄⡈⠦⠄⣀⣑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠧⣴⡮⣴⣴⢥⣤⣬⣅⣂⣝⣈⡂⠀⠱⣄⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠉⣩⣀⣥⣀⠤⠀⣘⣣⣤⣃⣀⠭⠀⢸⠉⠙⡿⢯⣥⣐⣖⠒⠊⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⢿⡅⠈⢄⠉⠻⡛⢇⠉⠈⣡⢩⠙⡛⠳⠓⠶⢧⡤⣯⣹⣟⣟⣶⢦⣝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣀⠀⠀⠐⢌⢢⡘⢆⠀⡀⠀⠈⠁⠒⠦⡤⠀⢈⡠⢊⠹⠛⢫⣽⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠈⠓⠤⣑⣨⣑⠫⢾⡐⠀⠀⠍⠀⠀⡞⠇⠨⠐⠀⡁⠀⠀⠠⢿⠖⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢭⡛⠢⣜⣉⣿⡶⣤⣤⣤⣤⡭⢭⣛⢇⠁⠈⠀⠂⣿⠤⠤⠤⠤⠤⢄⣀⣀⡤⠾⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣢⡶⡞⢛⣬⠶⠛⠉⡯⢹⠉⠛⣮⣽⢲⡀⠈⠉⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⡞⠉⡤⢞⠉⡇⡇⠀⢰⠃⡈⠀⢀⣿⠏⢸⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⣤⡼⠛⡄⠈⠀⢸⣅⣄⣌⠐⠁⢠⠟⠁⠀⢸⠀⠀⠀⢸⠀⣠⣠⣤⣤⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⠞⠃⢌⠑⠬⢦⡴⢊⠥⠤⢜⡡⠖⠁⠀⠀⠀⡟⠀⠀⠀⢸⢸⣿⢉⣙⢻⣟⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢲⡏⠑⠂⠄⣀⣀⡻⠮⢀⣴⠂⡻⠀⠀⠀⠀⠀⢸⠃⠘⠀⠀⢘⡄⠙⢻⠁⢑⡝⣯⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡭⠻⠖⢶⠶⣶⢲⣿⠟⠻⡞⠉⠀⠣⠀⠀⠀⠀⠆⠀⠀⣴⠀⠀⢇⢠⠏⠀⡌⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠶⠾⠶⠛⠋⠁⠀⠀⠙⡌⣀⡈⡆⠀⠀⡜⠀⠀⡄⢸⡄⠀⠘⢇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡔⠂⢹⠀⡐⠀⠀⢀⡀⡇⢣⠄⠀⠘⢂⠀⣦⠤⣐⣢⣒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢦⣄⠀⡀⠀⠓⢶⡂⠤⣀⠀⠡⡴⠆⢳⡄⠀⠀⠈⠀⣋⠘⡄⠀⠀⠈⡞⣡⢞⠞⠉⠉⠻⠼⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡉⡓⠦⣀⠩⡀⡈⠓⢤⢳⠄⠀⠐⢆⡀⠀⠣⢘⠀⡇⠀⢠⠃⡰⠁⣈⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⠳⡈⠙⣧⠱⠀⠀⠻⣆⢄⠩⠚⢧⠀⢸⡐⢀⠇⡔⢁⡴⢁⡾⠉⣠⠬⢝⣷⠶⡲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⠢⣻⠀⠃⠀⠀⣯⠈⠕⣠⠞⡆⠸⠏⣼⠔⡡⠎⢡⠞⡀⠋⣠⠞⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⢦⡈⠡⡄⣯⡴⡏⡃⢀⣷⣄⣼⡋⠚⣡⡴⠯⢊⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠓⠚⠓⠶⠾⠛⠋⠉⠉⠉⠉⠉⠛⣍⣁⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡥⠊⠉⠁⠤⡡⣄⡂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠰⠦⠁⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠼⠃⠀⠀⡞⠐⣐⠂⠀⢄⠀⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡘⢡
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣛⡹⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣬⠑⠀⠀⣠⣿⣼⣇
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⢠⠿⠿⠁⠀⠀⡀⠀⠀⠀⢀⡡⠤⠀⠀⠀⠀⠐⣤⠇⠸⢿⠀⠂⠔⣿⣟⡠⠞⣷
⠀⠀⠀⠀⠀⠀⢸⡰⡆⠀⠀⠀⠀⢀⣶⡶⠀⠀⠀⠀⣳⡜⠀⠀⢘⡁⢠⡦⠀⡀⠀⠸⡆⡾⠀⠣⠈⠠⣼⣿⠟⠁⠀⣠⡇
⠀⠀⠀⠀⠀⣴⡾⠀⠀⠀⡀⠀⠀⠚⡉⠀⠀⠤⡄⠀⡶⢰⡖⣠⡜⠃⠀⠁⠘⢇⣠⡴⠇⠽⢤⣀⠀⡇⠿⢃⣤⠶⠋⠁⣿⠀⢸⡀
⠀⠀⠠⠁⣿⣿⠟⢉⡁⢈⣯⡠⠰⡿⠇⠀⢀⣶⡇⢠⣧⣼⡅⠘⠀⣤⣼⡟⣴⣿⣁⣀⣀⣀⣤⡬⠷⠁⢴⠛⣵⠀⣀⠴⠀⢀⣎⢸⡄
⠀⠀⣇⢰⣏⡩⣶⢁⠀⡂⠀⡱⣤⠁⠸⠿⠤⠙⠁⡞⠁⠈⣧⠀⠀⠏⠙⠃⣾⣅⡀⣀⡀⣀⡄⡠⣾⣷⠈⢆⣿⠊⢉⠄⠀⢘⡁⢀⣙
⠀⠀⠸⡢⣿⣧⡜⢂⣉⠔⣤⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡀⠀⠀⠀⢻⣯⣽⣯⣼⣧⠯⠻⢿⠇⠀⠀⠡⣔⠉⠀⠀⣫⡀⢀⣈⣇
⠀⠀⠀⠘⠻⢿⡿⠧⠁⠈⠔⠁⠀⠀⢀⣼⣤⣀⣀⠀⠀⠀⠀⡟⢃⠀⠀⢀⠀⠙⠶⣶⠶⠖⢒⡾⠋⠀⠀⠀⠀⠀⠀⣠⣾⣯⣥⣤⣤⣭⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣏⠉⢄⠀⠀⢰⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠂⠘⣧⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠌⡌⠐⠀⠆⠱⠈⠆⡆⠀⢘⠀⠠⠋⠰⠈⠀⠘⢠⠁⠚⠛⢠⢰⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠘⠛⠛⠛⣧⠀⢰⡆
⠀⠀⢀⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⢠⠄⣀⠀⡀⠀⡄⢠⠀⠲⠈⠁⢀⣿⡇⠀⢠⣤⣄⠈
⠀⠀⠈⣡⡤⢠⠄⠀⣴⡇⠀⠀⠀⠀⠀⢠⡄⠀⡤⡀⢄⠀⠁⠀⠁⠀⠀⠀⠀⠀⣸⣿⣿⠀⠹⠿⠟⠀⠀⢀⣄
⠀⠀⠀⠀⠰⠈⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠰⡿⣄⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠉⠁⠈⠉⠀⠁⠡⠶⢿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠐⡅⠀⢠⠃⠀⠀⠃⠈⢀⣴⣷⡀⠛⠛⢁⣀⣀⡀⠀⢀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠑⢸⠘⠀⠀⠐⠰⠾⣿⡿⠆⠀⠀⠀⠀⠈⠀⠈⠁⠀⠐⢛⡉⣉⢳⡀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣀⠀⠀⢠⣈⣠⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠖⢳⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠀⠀⠀⠙⠀⠀⠘⠖⠃⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⢀⣠⣶⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⢄⡒⠀⠀⠸⠀⠀⠀⠚⠁⠀⠛
⠀⠀⠀⠛⠛⠿⠇⠀⠛⠋⣤⡶⠀⡀⠀⠀⠈⠥⠀⠘⠂⠀⠀⢄⡐⠀⠀⠀⠸⡀⠀⠀⠀⡀⠀⠀⠛⠀⢀⢀⣀
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⡀⠀⠀⠁⢀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⠻⣿⣿⣿⣄⠀⠀⣶⡄⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠸⠇⡌
⠀⠀⠀⠻⠿⢿⣿⡷⠾⠛⠀⠀⢨⣯⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⡎⠀⠀⠀⣌⠀⠤⠀⠀⢠⡄⣬⠀⠀⢠⡆⠅
⠀⠀⠀⠈⠁⠈⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠙⠲⠤⣀⡀⣀
⠀⠀⠀⠀⢠⡠⠀⠈⠀⠀⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⠀⠀⠈⠉⠀⠐⠀⠈⠀⡁⠝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠰⠈⣁⠖⠁⠀⠀⢀⠄⠐⠀⠀⠀⠀⠐⢄
⠠⢈⠀⢀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠢⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⠠⠄⢀⠄⡤⠒⠒⠉⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡠⣄⡀⢁
⠡⢈⣀⠀⠠⠔⠒⠉⠡⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⠴⠖⠛⠃⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡇⠀⠄⠒⠀⠀⠀⠈⠀⠀⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⡀⠠⠀⠀⢱⣒⡒⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡆⠀⠀⠀⠀⠀⡆⢠⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠸⠔⠓⠠⠀⠀⠀⠀⠀⠀⠸⠀⠠⠃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢡⠈⢀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⡠⠂⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠠⠀⠂⠀⠄⠀⠂⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠠⠀⠀⠀⠀⠐⠈⡀⠂⠂⡈⢠⠀⠄⢧
⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠢⢌⠐⢀⠨⠀⡀⡈⡀⠐⢂⠀⣀⠟
⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⡀⣀⡤⠠⠤⠤⠤⣀⠀⠁⡀⠀⠀⠀⠁⠀⡀⠀⠀⣀⠀⠀⣀⣀⡀⠄⠁⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⠐⠲⠂⠂⠈⠁
⠀⢀⠀⠀⢀⡠⠤⠒⠋⠁⠀⠀⣀⣠⠤⠄⠄⠈⡃⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠐⡀
⢀⠂⡤⠚⠁⢀⣀⡠⠒⠀⠓⠀⠈⠳⠖⠂⣀⠠⢂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢱
⠈⢘⡀⠀⠈⠐⠛⠛⡀⠀⠀⣀⣀⠊⠉⠡⠐⠂⠀⡠⠔⡀⢄⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢠
⠀⠐⠌⠈⡁⠈⠥⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠉⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠥⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠐⠄⠀⠁⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢀⢈⠀⠀⠒⠁⠀⠠⡀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠒⠈⠉⠉⠉⠐⠂⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⡄⠀⠑⠊⢀⠜⠀⠠⢀⡤⠔⠀⠀⠀⠐⠒⠂⡢⣀⠉⢀⡥⠖⠁⠀⠀⠀⠀⠀⠀⣀⣠⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠄⠀⠒⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠐⠒⡇⠀⠈⠐⠋⠉⠉⠉⠁⠀⠀⠀⠉⠁
⡈⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠈⠀⠒⠀⠀⠂
⠀⠈⠙⢤⠀⠀⠀⠂⠄⠀⢹⠌⠀⠀⠀⠀⠀⠀⠘⠄⣀⣀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠰
⠀⠀⠀⠀⠑⡌⢀⠀⠀⠀⡄⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠀⣀⣀⣀⣀⣈⠒⢄⡀
⠀⠀⠀⠀⠀⠘⡄⠡⠀⠁⢡⠀⡤⠄⠀⠀⠀⠀⠈⠀⠀⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠠⠂⡁⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠉⠀⡀⣀⠀⠀⠠⠓⠀⠘
⠀⠀⠀⠀⠀⠀⡔⠘⡀⢁⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠠⠠⠥⣀⠀⠀⠀⠠⢀⠄⠀⠀⠈⡀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠈⡄⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠰⢟⠀⠡⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡀⢴⡮⠰⠂⢔⡢⢰⠄⡄⡀⠡⡀⠀⠀⠀⠐⠀⠐⠂⠅⠀⠀⠱⡴⠟⠐⠀⠁
⠀⠀⠀⠀⠀⡀⣾⠀⠐⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠈⠁⠒⠷⠀⠻⠿⠋⡷⠂⡵⠈⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠱⢈⠀⣀⠐⠁⠀⠀⠀⠀⠐⠂⠄⣀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠆⠀⠀⠀⠸⠛⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠒⠀⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄

⣀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠿⠿⣿⠿⠗⠀⠀⠀⠀⠉⠀⠀⢀⣴⣄⠀⠹⠇⢀⡀⠀⢰⣿⣿⣷
⣿⡀⠀⠀⣿⣿⣿⠄⠀⠆⠀⣴⡁⠀⠀⠀⣾⣿⣷⠄⠦⠀⣾⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⠀⠉
⣿⣷⡀⢴⣿⣿⣿⣧⠀⠀⣼⣿⣷⡀⠀⣠⣿⣿⣿⡄⠀⠰⣿⣿⣿⡷⠀⢀⣄⣀⠺⠿⢿⣿⠿⠟⠂
⠻⠋⠁⣼⡿⣿⣿⣷⠀⢀⣼⣿⣷⡀⠀⣹⣿⣿⣿⡏⠀⠼⠿⣿⠿⠿⠄⣹⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⢈
⠀⠠⠄⠀⠀⠙⢠⡄⠀⢉⣿⣿⣿⡉⠀⠛⠛⣿⠛⠛⠂⣄⢠⠘⠀⠀⠰⣿⣿⣿⠧⠀⠉⣀⠀⠐⢻⣤⡞
⣤⣤⣤⠄⠀⠀⣾⣿⡀⠚⠛⣿⠛⠛⠂⠀⠀⠉⠀⣸⡄⣏⢹⠀⣼⡆⠼⠿⣿⠟⠷⢀⣾⣿⣷⡀⠿⠛⣿
⠿⢿⢿⠀⠀⣼⣿⣿⣧⠀⠀⠉⠀⢦⡼⣆⠀⠃⢠⣿⣧⠈⠉⣸⣿⣿⡀⠀⡉⠀⠀⣸⣿⣿⣿⡇⠀⠀⣿
⠀⠸⢸⠀⣰⣿⣿⣿⣿⣇⠀⠋⠀⣾⡿⠉⣀⠀⠛⢿⠿⠓⢰⣿⣿⣿⣷⠀⠉⠀⠰⢿⡿⣷⣿⡿⠆⠀⠏
⠀⠀⣰⡀⠛⠛⢻⡿⠛⠋⠀⠀⠀⠙⠶⠿⢫⣾⣿⣄⠀⢀⣿⣿⣿⣿⣿⣇⢰⡆⢰⣿⣿⣿⣿⣿⡆
⠀⣰⣿⣷⡀⠀⡀⡀⠀⠀⣴⣶⣶⡀⠀⠀⢿⣿⣿⣿⠆⠀⠀⠉⠻⠉⠁⠀⣾⣷⡀⠀⠀⠟⠀⠀⠀⠀⠈
⠀⣹⣿⣿⡉⢰⡍⣵⠀⢈⣿⣿⣿⡅⠀⣰⣿⣿⣿⣿⢧⠠⢶⣀⡴⠃⠀⠾⢿⡿⠷⠀⢀⡄⠀⣼⣶⣶⡀
⠴⣿⣿⣿⡿⠆⠉⠁⠀⢾⣿⣿⣿⡿⠄⣹⣿⣽⣿⣽⣯⠀⣾⢿⣷⣀⣁⠀⢀⡀⠀⢠⠀⠀⢠⣿⣿⣿⡄
⣰⣿⣿⣿⣷⣄⠀⠂⢀⣾⣿⣿⣿⣷⡄⠛⠛⢻⡟⠛⠛⠃⡄⠀⣿⣿⣿⣿⣿⠀⢠⣿⣧⡀⢻⣿⣿⣿⡟
⠀⠀⢻⠁⠀⠀⠀⠀⠈⠁⠀⢿⠀⠀⣄⠀⢄⠈⠁⢀⠀⢀⣿⡀⡟⡇⠀⢸⢸⠀⣠⣿⣿⡀⠟⠛⣿⠛⠛
⠀⣧⠀⠀⠂⣰⣶⣶⡀⠀⠃⠀⠀⠀⣸⣶⣏⠁⠀⣀⢀⣾⣿⣧⠀⠁⠀⠈⠈⠘⣻⣿⣿⡛⠀⠀⣉⠀⠤
⣼⣿⣆⠀⠀⣽⣿⣿⡏⠀⣶⣶⣶⣶⣿⠋⠙⠁⢠⣿⣎⠙⠟⠉⠁⢠⣾⣦⡀⠘⠛⢻⠛⠛⠂⣴⣿⣷⡀
⠻⡿⠟⠂⢾⣿⣿⣿⡿⠄⡟⡟⠛⢻⣿⠀⠀⠐⢻⣿⣟⠂⠀⠀⢰⣯⣿⣿⣷⡀⠀⡈⠀⠀⢠⣿⣿⣿⡅
⢠⡄⣀⠀⣼⣿⣿⣿⣿⣄⠧⠃⣤⠘⠟⠀⢀⣠⣿⣿⣿⣦⡀⠀⣴⢿⣷⢿⣿⣄⢰⡟⣳⠀⣹⣿⣿⣿⣏
⢾⣽⡅⠈⠁⠀⢹⠁⠀⠀⠀⣸⣿⡆⠀⠤⠀⣰⣿⣿⣿⣆⠀⠸⣿⡿⣿⣿⢿⡿⠀⠓⠃⡄⠉⠉⢿⠉⠉
⣤⣙⡟⠀⠐⠂⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠘⠋⠉⣿⠉⠉⠃⣰⣿⣿⣿⣿⣿⣿⡆⠀⣸⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣠⣿⡇⠀⠀⠀⠀⠀⡇⠰⠖⠘⡸⢙⡤⢤⢐
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠙⠋⠀⠀⠀⠀⠘⠐⠻⠈⠒⠀⠱⢎⣀⢒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⠀⠀⠀⠀⣰⣀⡀⢠⡴⡴⢤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠖
⠀⠀⠀⠀⠀⠀⠀⣿⣾⢿⣽⠀⠀⠀⠀⠩⠸⠿⣿⡖⣲⡀⠿⡗⠳⠗
⠀⠀⠀⠀⠀⠀⣴⣯⣿⠿⣷⠆⠀⠀⠀⢴⣶⢶⡂⣟⣸⣶⣶⡶⠲⠶⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠐⣿⣯⣿⢿⣿⣾⠄⠀⠀⢸⡿⣢⣴⣷⣶⣶⣶⡔⠤⠖⠀⠀⠀⠀⡼⠀⠀⠀⠔⠀⡰
⠀⠀⠀⠀⠀⣼⣹⣹⣿⣿⢻⣿⣦⠀⠀⠐⣪⣼⣿⣿⠟⢻⣿⡇⠄⡀⠀⠤⠀⠈⠑⠀⢀⣴⠃⢀
⠀⠀⠀⠀⢠⣟⣻⣿⣿⢿⣿⢿⡾⢀⠀⠈⠁⠐⣿⠁⢀⡈⢻⣇⠈⡇⢀⡰⢄⣠⣦⣾⢟⠋⢀⠄
⠀⠀⠀⠀⡷⣿⣿⣿⡏⠈⣹⣗⣿⠟⠢⣤⡀⠈⡿⡬⠿⠧⣿⡇⠀⠃⠀⠀⠈⠀⠁⢸⠁⣆⠊⣀⡀
⠀⠀⠀⠀⢛⠻⠁⢘⠟⢱⡏⠛⠀⡟⢰⡄⢀⠠⡴⣦⣾⡧⠾⠭⣦⠀⠀⠀⠀⠀⠀⠈⡇⡃⠀⠈⣤⠰⡃
⠀⠀⠀⠀⢷⡤⠀⣰⠀⠈⠀⢀⣿⡟⣾⠁⣸⡇⠠⠾⠿⠷⠶⠶⠶⠶⠺⠁⠀⠂⡤⠈⠁⠀⣿⠀⢻⠞⣧
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠉⠀⠉⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⢁⠠⠐⣶⡄⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣟⡉⠫⣽⡚⣿⣴⡂⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⠝⠛⣿⡿⢛⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⠿⢿⣿⣤⣤⣠⣴⠾⡷⠁⠙⢿⣾⣏⠖⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠈⠁⠈⠀⠙⠃⠀⠀⠀⠈⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣶⠙⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⡅⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣿⣷⣤⣄⣤⡄⣀⣀⣢⡶⠂⠀⠀⣀⢻⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⣀⣴⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠉⠙⠛⢻⣶⣷⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠹⣿⣿⣶⣶⠦⣻⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡶⠒⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣄⣀⠜⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠃⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣧⣬⣵⣄⡀⠀⠀⣀⣀⣀⢈⣿⣿⣿⣞⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢿⣿⣿⣢⡞⠉⠀⠁⠀⠙⢿⣿⡟⢠⠀⠀⠀⠡
⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⣿⠿⠋⠀⢸⠿⠛⣿⣧⣀⡀⠀⡀⠀⠀⢻⣇⠈⠀⠀⠀⠈⡄
⠀⣠⣿⡟⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⢸⠀⠀⠈⣹⣿⡿⢿⣧⣀⣠⣾⣏⣀⣠⡀⠀⠀⠙⢆
⢰⣿⡟⠀⠀⢠⣤⣴⣿⣿⡇⠀⠀⠀⠈⣆⠀⣰⣿⡏⠁⠀⠉⣿⣿⣿⡇⠈⠙⠻⠀⠀⠀⠀⠣⠀⠀⠀⠀⠆⢴
⠀⢿⣯⣀⣿⣿⠇⠈⠛⢿⣷⣤⣤⣄⣀⣿⣰⣿⡁⠀⠀⠀⣀⣹⣿⣿⡀⠖⠛⠳⡄⠀⢀⠀⠀⢁⠀⠀⡘⢠⡇⠀⡠⠄⠀⠁
⠀⠀⠉⠁⠈⣿⣇⢀⣠⣴⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣦⠀⠀⠁⣤⣤⣶⣾⣾⣴⣄⣁⣴⡇⣠⣶⣦⣦⡄⢀⡠
⠀⠀⠀⠀⠀⠸⠿⠿⢟⢏⡉⠁⠀⢿⠛⠟⠟⠋⠉⠿⠛⠛⠗⠞⠋⠉⠛⢿⡿⣷⠿⣿⢾⠿⢿⡿⣿⢾⡿⠏⠻⠋⠹⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⢀⡀⠀⢐⠀⠀⢀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠒⢂⢤⣤⡀⠀⠀⠀⠀⢀⣾⠋⡇⠆
⠀⠀⠆⠘⠅⢢⠐⠀⡆⠏⢐⠀⠀⠀⠠⢃⡤⠤⠄⣀⠀⡄⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠈⠡⢷⡄⠀⠀⠀⡎⣬⠊⡄⣷
⠀⠀⠙⠀⠀⢸⠠⠀⠃⠀⡄⠀⠀⠀⠀⠋⡆⠀⣴⡭⠿⠃⠀⠀⠀⠸⣿⣆⣲⡂⣀⣀⣠⣾⣿⠇⠀⠀⢰⢧⠉⠀⠇⡿
⠀⠀⠀⢂⣬⡥⠘⡄⡐⢀⠀⠀⠀⠀⠀⠀⠉⠀⣈⣀⢠⠄⠀⢀⠲⠀⠀⠉⠛⠛⠛⡛⠛⠉⠀⠀⠀⠀⠈⠛⠦⠴⡴⠃
⠀⠀⠈⠒⣎⣇⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⢠⠎⠉⠙⠈⠃⠈⡳⠟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠸⢻⣴⢠⡇⡞⠈⠀⠀⠀⠀⢀⣽⣤⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⢸⣿⢸⣧⢧⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠈⠿⠸⠿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣶⣒⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⣛⡋⠉⢉⣭⣙⣋⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣷⣿⡟
⠀⠀⠀⠀⠀⡐⠚⣛⠛⠁⠙⠢⠀⠀⠀⢠⣶⠾⠟⠛⣛⣭⡦⣯⡈⠉⠉⠉⠉⠙⠛⠆⠀⠀⢀⣀⣀⡀⠘⠻⣿⠁
⠀⠀⠀⠋⠉⠀⠀⠈⠰⣳⠆⢀⣄⠀⠀⠈⠁⠀⠀⠹⠟⠚⠛⠛⣛⣠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠀⠀⠉
⠀⠀⠀⠠⠶⠻⠂⢀⡀⠙⠶⢯⠆⠀⠀⠀⠀⠀⠁⠀⠀⡀⢰⣟⢻⣿⠀⠀⣿⣦⣼⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⢷⠄⠀⠀⠀⠀⠀⠀⢆⣰⡀⠀⣦⡬⢿⣿⡀⠀⢿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⡇⠀⠈⠃⢸⣧⡆⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠤⠤⠖⠲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠧⠀⠰⣄⣾⠛⠁⠠⠆⠀⠀⠀⠀⣀⡀⠀⢠⣿⣤⣄⣠⣤⣤⣾⡾
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⣡⣬⡿⠂⠀⠁⠀⡏⠉⠉⠋⠁
⠀⠀⠀⠀⠉⠓⠷⣤⢀⠀⠀⠡⠖⠚⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠢⠐⠒⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢈⠀⢐⣿⡿⠆⢰⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⡶⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⡅⠀⠀⠐⠿⠀⠰⠄⠀⠀⠀⠀⠀⣠⠀⠀⢀⠂⠐⡓⠻⠁⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢐⠇⠀⠁⠀⠆⡉⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠀⠀⢀⡐⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⡀
⠀⠀⠀⠋⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠐⠄⠃⠔
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠭⢹⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠄⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⠒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⢰⠄⠀⢀⠀⠀⠠⢠⠀⠀⠀⢀⠀⢸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠐⠀⠀⠀⠀⠠⠉⠀⡀⠀⡄⠀⡀⠘⠀⠀⠘⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡈⡴⢶⡎⣀⠀⠠⠄⠀⠔⠁⠑⠄⠀⠁⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡙⡚⠣⡀⠀⠀⠀⠀⠢⠀⠠⠃⠀⠀⠀⠀⠁⠀⠀⠲⠀⣨⠡⡠⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢴⣟⣳⠡⠠⠤⠀⠀⠀⠀⠀⠁
⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⢀⠀⠀⣠⠐⠫⠁⠃⡄⠀⢠⠀⢀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠠⢔⡀⠄⠀⠀⠤⠀⠌⢢⣀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠐⢨⢇⠜⡕⠂⠀⠀⣀⠀⠀⠈⠑⠀⠀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⠄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠀⡱⠒⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⣀⣤⠶⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠤⠤⢀⢀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡔
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠄⡀⠂⠀⠄⠀⢀⠀⢘⡀⠀⠀⠀⡠⠀⠀⠨⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠊⢁⡀⡋⠁⠀⠈⠂⠀⠀⢠⠃⠀⠀⠉⠀⢤⠈⠀⠀⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢄⠀⠀⠀⢐⢯⠀⠀⠀⠀⠖⠀⠀⣀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣶⣾⣧⣜⢦⠀⠀⣞⠀⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠀⠀⠀⠈⠙⢦⣠⠻⣧⣾⠇⠀⠀⠀⠀⡀⠀⠠⠂⢀⠀⠀⠀⠺
⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠇⠀⠀⠀⢰⣧⣼⡦⢿⣦⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⠀⠀⠀⠀⠁
⠀⠹⢄⠀⠀⠀⡄⠀⢳⠀⠀⠀⠀⣠⡆⠀⣨⣍⣭⠖⣩⡿⠉⠀⠀⠀⠀⢀⠀⠀⠀⣀⠶⠁⠀⠀⠐⠁
⠀⠀⠀⠛⠢⡈⢧⠀⠘⠀⠀⠀⠀⠉⢃⠀⠙⠛⢿⣶⣿⡇⠀⠀⠀⢀⠔⣙⣀⡚⣀⢠⠀⠀⢀⣠⡦⣉⣠⠊
⠠⡄⠀⢀⠻⣤⠿⠗⠀⠘⠄⠀⠀⠀⠀⣠⠋⠀⠀⠀⢸⡷⠀⣠⡴⠟⠛⠛⠙⡁⠉⣠⣦⣴⣿⢿⠔⣠⠃
⠀⠀⠀⠀⠄⠻⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⡄⣦⡀⠸⢿⠟⠢⢀⠀⠐⠗⠀⠻⠛⠋⣹⣿⡿⠀⢀⡿⠤⠄
⠀⠀⠀⠄⠈⠂⢒⣇⠀⠀⠀⠠⠀⣼⣷⠄⠘⠿⠟⢹⣶⣿⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠸⡟⢐⠀⠈⠁
⠀⠀⠀⠀⠐⠌⣻⣿⣦⡀⣠⢤⡾⠋⠀⠀⠀⣶⣤⣷⡟⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠹⣍⡻⠦⠈⠁⠐⠒⠀⠀⠈⠉⠉⣶⣤⣿⠀⠀⠀⣀⠀⠀⠠⠐⢀⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⣶⡿⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠛⢠⣤⣀⠀⠀⠙⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠁⠀⢸⡟⡤⠄⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⠣⣀⣀⡈⢒⠀⠐⠀⠀⠀⡀⠀⠊
⠠⠒⠂⠀⣰⠏⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂⡾⠀⢀⣄⣶⣤⣀⣦⢀⡾⠁⡌⠸⠛⠉⠠⠘⠋⠀⠀⠄⠘
⠀⠀⠀⠐⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢈⣻⡿⠍⢻⣿⡊⠀⠀⠀⠊⠁⠠⢄⠀⡠⠒⠀⠉⠛
⠀⠀⠀⢇⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡁⣨⡴⠚⠍⠉⢐⠦⣄⠈⠑⢦⡀⠀⠤⠒⠉⠉⠑⠒
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠍⠀⢾⣦⡷⠻⣾⡿⠀⠀⠀⠁⠀⠢⣤⣞⡁
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠷⠲⢤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢠⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢬⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠄

⠀⠀⠀⠀⠀⠄⠀⠐⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠻⢤⣤⣤⣴⠟⠁⠀⠀⣠⡄⠀⣄⠀⠀⠀⠀⠀⠀⡔⠃⠀⡠⠀⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢪⡀⠀⠀⠈⣵⠀⠀⠀⠀⢰⡷⠟⠟⣛⣺⡿⢷⣃⡇⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠰⠀⠀⠀⡇⠀⠀⠀⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠳⠀⠉⠁⠀⠘⣧⣈⡓⢄⠀⠂
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⢹⡏⡇⢀⡀
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠺⣧⡀⠀⠐⠈⠻⣄⣾⡿
⠀⠀⠀⠀⠈⠳⠦⣤⡤⠔⠁⠑⠤⠀⢠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⣠⢍⠙⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣙⣌⣷⠂⠘⠰⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⡀⡽⠒⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡉⠉⠁⠀⠀⠁⠠⡔⠷⣄⣀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⣼⢥⣤⠽⠃⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢠⠊⢉⡷⠐⠀⠐⡀
⠀⠀⠀⠀⠀⠠⠖⠀⠸⠃⣯⢱⢠⡇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢸⡯⣘⠇⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⢤⣄⣀⣀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⣱⢿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠁⠀⠀⠈⠁⠀⠀⢀⢖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢹⠘⡇⡆⠀⠀⠀⠀⠀⡀⠀⠀⢀⣔⡊⠀⢲⠀⠀⠀⠰⢤⣨⡇⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠂⢸⠀⣿⠁⠀⠀⠀⠀⠀⠉⠙⠉⢁⠉⡉⠙⠋⠀⢐⢂⠀⡔⢠⠇⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡾⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢴⡀⠑⢄⣠⠋⢀⠟⠁⠀⡈⢄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⢀⣃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠃⠀⡘⢀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃⠀⠈⠉⠁⠈

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀
⠀⠀⠿⠯⢠⣾⣧⡀⠐⠀⠀⣾⠁⢨⡷⠀⣴⢒⣦⠀⢉⡀⢰⢿⠂⣰⡿⣛⠻⣇⠀⡞⠀⠈⢳⠋⠀⠙⡆
⠀⠀⠀⣰⣿⡟⣿⣷⡀⠀⠀⠹⢤⣀⡀⠀⢻⣌⣉⢰⠫⣽⠂⠀⠀⠹⣷⣭⣾⡏⠀⠳⡀⠀⢺⠀⠀⡸⠃
⠀⠀⣴⣿⣷⡅⣥⣿⣿⣄⠀⣶⠆⢨⠟⣉⠉⠉⡉⠙⢳⡀⠀⠃⠀⠐⢶⠭⠶⠆⠀⠀⠙⢦⣸⣠⠞⠁
⠀⠈⢉⣽⡿⠇⡿⣿⣏⠉⠀⡀⠀⠘⠦⠿⠀⠀⣉⣀⣰⣧⣄⣀⣀⡘⠻⠖⠿⠃⢀⣤⡀⠀⠙⣡⡶⣤⡀
⠀⠠⠾⢿⣿⡆⣾⣿⡿⠧⠀⡁⢀⠀⠀⣀⣶⣀⠙⣿⣿⣿⣿⣿⡿⠟⠀⠄⠀⠀⠈⠛⠃⠀⠸⣕⡿⣫⡇
⠀⠀⣠⣿⣯⡃⢋⣿⣷⡀⢸⣎⣹⡁⠐⣺⣿⣗⠂⢸⣿⣿⡟⠁⠀⢠⡄⢶⢦⣤⠀⠸⣿⠟⠀⢉⣉⠉
⠀⠐⢛⠛⠛⣷⡟⠛⠛⠛⠈⠑⣋⡁⠀⠀⠛⠀⠀⢸⣿⣿⣧⠀⢸⡗⢠⣶⣦⠐⣷⠀⠀⠀⣰⢿⣿⡷
⠀⠀⣉⠠⡄⢉⠁⠀⣠⡀⠠⣾⣿⣿⣆⠀⠀⠙⠀⢸⡿⡟⡿⣧⠘⢫⡄⣩⣡⣬⠋⠀⢼⣿⣇⣠⡿⠃
⠀⢠⣭⡷⢳⣯⣄⠘⠿⠏⠀⠘⢿⣄⣸⣿⣯⠄⢀⣿⡁⠀⠀⣽⣧⠀⠀⠉⠁⠀⠀⠄⠀⠀⠉⣉⠀⢠
⠀⠀⠴⢋⡏⠳⠀⠀⢀⠀⠀⢀⣀⣉⣉⣉⣤⣶⣿⡿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣦⠀⠀⢀⣼⣿⣦
⠀⠀⠀⢠⣤⡄⠘⠁⠘⢷⣾⣿⣿⣿⠿⠻⣻⣿⡃⠀⢐⣻⣿⣿⡿⠛⠉⢀⣿⠏⣿⡆⢠⣾⣯⣩⣿⣷⡀
⠀⢰⣿⡟⣙⢻⣷⠄⠀⢸⣿⣿⣿⣧⣀⢀⢸⣿⣷⣦⣷⣿⡿⠋⠀⠘⠃⢸⡿⠀⣿⠇⢀⣼⣿⢙⣿⣧⡀
⠀⠀⢽⣧⣭⣾⡏⠀⠶⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣁⡤⢤⣀⠀⣰⡞⠁⣴⠟⠀⠀⣼⣟⠛⣿⣇
⠀⢰⣶⣭⢀⣭⣶⡆⠀⢰⣿⣿⣿⣿⠋⠉⠀⣠⣀⠀⡼⢃⢮⣷⡂⢧⠈⡀⠀⢡⡀⠀⠈⠉⠉⡿⠉⠉⠁
⠀⠈⠻⠿⠿⠿⠛⠀⢠⣿⢣⣿⡟⠁⠖⠀⠀⠛⠋⠀⠸⣆⡛⢟⣂⠏⠈⠁⣰⣿⣿⡄⢀⡄⠀⠠⡞⢳⠄
⠀⠀⠠⠀⠀⠒⣠⣴⡿⠃⣼⡟⠀⣰⠚⣶⣴⣾⣶⡀⠀⠀⣍⢉⠀⠀⠀⠐⣿⣧⣾⡟⠂⠀⢀⡀⢉⠉
⠀⢠⡶⣲⢶⣰⡿⠋⣠⣾⠟⢁⠀⠹⣾⣿⣿⣿⡿⠀⠐⢶⡿⢻⡷⠂⠀⠰⣿⣧⣾⡿⠂⠐⠶⢟⡻⠶⠂
⠀⠸⣞⣿⣳⠇⠀⠀⡿⠋⠐⣏⣱⠀⠘⢿⡿⠋⠀⠆⠐⠛⡿⢿⠛⠃⠀⠰⠛⢧⡾⠿⠂⠐⢋⡶⢾⠙
⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢹⣿⠠⢝⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⡴⠁⠉⠠⠀⠆⠺⠟⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡼⠁⠀⠋⢠⡄⠐⠀⠃⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⡼⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⠀⣠⠖⡋⡽⡫⢻⣿⢻⡻⡿⡓⢖⡲⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⠀⢸⣧⢞⡕⢁⣞⡼⠥⢼⣿⠉⠓⠷⣙⡆⠙⣎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⣾⡿⣀⣤⣀⠈⠙⠒⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢰⣿⡟⣆⠀⠀⠀⠀⠀⢠⣿⠟⡹⠟⠨⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠵⠶⣤⡉⠀⢸⠀⠀⠀⠀⢀⣾⡏⠀⠁⠘⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠄⠒⡠⣸⣿⣄⠘⡇⠀⠀⢀⣾⣿⢀⣠⣤⣶⣶⣿⣭⣓⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⠊⡴⢡⡟⠛⣷⡇⠀⢀⣾⣿⡿⢹⣃⢸⣷⠷⣝⣟⢮⡳⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⡟⠒⠷⠛⠀⣼⣿⡟⠓⠋⠁⢸⣿⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⢰⣿⣿⣇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⢸⣿⣿⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢘⢇⠈⢣⠘⠁⣿⣿⣿⣇⠀⠀⣿⢸⣿⣿⣧⠀⢠⣿⣿⣧⠀⠀⠶⠀⠋⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡳⡈⡇⢤⠻⡇⡟⢿⣄⢰⣿⡸⣿⡟⣿⡆⢻⢿⣿⡿⠂⢀⠀⡀⢀⢈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢴⣌⣢⣻⡴⣆⠻⢦⢹⣦⠹⢃⡌⢧⣏⡟⢻⡏⠀⢌⣠⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠙⢧⣁⡿⠛⣿⠀⠇⠳⢸⠻⣄⣨⠟⠛⠛⠋⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠑⢺⣇⣀⣀⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣘⣿⣿⣿⣿⣿⣷⣦⣍⢁⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡖⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⢀⠀⢀⣱⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣤⣶⣾⣿⣿⣿⣽⣭⣯⣧⢽⣿⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⡿⠉
⠀⠀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠁⠀⠻⠻⢿⣿⣽⣿⣿⣛⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣽⣾⣿⠿⠋
⠀⢺⣿⣿⣿⣿⣿⢛⣻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣯⣭⣿⣟⣿⣿⣿⣿⣿⠙⠛⠟⠛⠉⢀⠀⠀⠁
⠀⠀⠁⠻⣿⣿⣿⣶⣾⣷⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣹⣿⣿⣿⣿⣾⣿⣿⣿⠿⠟⠙⠉
⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣛⣉⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣠⡀⠀⠀⠘⠻⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠘⠛⠶⣦⣤⣀⣠⣝⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⡂⠀⠀⠀⢀⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠉⠀⠉⠁⠈⠉⠉⠉⠉⠉⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡢⡄⢀⣴⡆⣶⣄⠀⡤⣲⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣌⢳⡡⢼⣟⡓⢻⣿⠤⣱⢋⡜⠀⢀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠹⣷⣦⡀⠈⢿⢿⡛⡧⠹⣼⢦⣙⡮⢨⡾⠡⡟⣻⢿⠏⢀⣴⠿⠋⣀⣤⡤
⠀⠀⠀⠀⠀⠀⣤⣙⣿⣷⡤⠜⠛⣲⢼⣶⣽⢰⢂⣶⡝⢨⠐⣴⣆⠲⢸⣵⣾⠻⣍⣁⠒⢾⠿⣷⣶⠆
⠀⠀⠀⠀⠀⣴⣾⠋⣱⣞⣀⣼⣻⣷⣬⣿⡮⢨⣟⢞⡓⠄⠔⣻⢞⣯⠨⣼⣯⣵⣌⡱⠦⣌⢿⣆⠙⢿⣆
⠀⢀⡄⢲⣜⠋⢀⣼⠋⠉⠅⠀⠉⠉⠉⠁⠈⠁⠁⠉⠁⠀⠀⠉⠁⠁⠉⠈⠉⠙⠋⠡⠠⢀⠈⠻⢄⠀⢉⣶⠂⣄
⣠⣬⣌⣿⣿⠊⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠈⢺⣿⣏⣬⣤
⠈⠙⢿⠿⣻⠠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⢸⡿⢾⠟⠉
⠀⠐⠈⢶⢟⣀⡌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠨⣄⡘⢷⠎
⠀⠀⠀⠀⣉⣿⡁⠀⠀⠀⠀⠃⠈⠀⢣⠢⢂⠔⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠈⠐⠂⢀⠀⠀⠀⠀⠀⣹⣏⡁
⠀⠀⢐⠾⣦⠉⢓⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢐⡋⢥⡾⢖⠂
⢀⣤⣾⢖⣿⠒⠟⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠗⢺⣗⢾⣢⣄
⠉⢛⢉⢿⣿⠄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢼⣿⢏⢙⠋
⠈⠈⠃⠘⢏⡀⠈⢲⣄⠈⠀⠐⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠠⠀⠂⡀⢰⠈⠀⠀⡙⠄⠋
⠀⠀⠀⠀⠈⢿⣶⡈⢻⣎⠛⢶⣻⣿⠾⣷⠮⠱⣗⣽⣃⠤⢄⣻⣕⡷⠩⢿⣿⣾⣗⣢⡴⠎⣴⡟⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠾⢛⣷⣾⠦⠌⢉⡳⢿⣿⢾⢨⡙⣾⢇⠰⢀⢿⡞⣩⢸⢾⣿⣋⠟⠛⣀⣼⣭⡶⢛⡍
⠀⠀⠀⠀⠀⠀⠐⠛⠉⢡⣶⡾⠋⠀⣽⣴⣋⣦⢠⢶⡽⢞⡦⠴⢦⠠⣎⣳⣼⡅⠙⢶⣤⡌⠛⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠉⢉⠟⣱⠫⢼⣗⣋⣸⣾⠬⢳⡙⢏⠉⠙⠀⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡣⠇⠘⢿⡇⣿⠟⠀⠧⣻⣼
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠀⣀⣠⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⢾⢙⡈⠋
⠀⠀⠀⠀⠀⠀⠠⣠⠎⣀⠀⠀⡈⢆⡀⠀⠀⠀⠀⠀⠀⣀⠙⣆⣒⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡶⠁⠀⠀⠑⡾⢁⣄⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠒⠒⠋⠂⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣤⡀⠀⢹⠩⡯⡯⢹⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⢚⣼⣯⡧⣄⠀⠀⠀⠀⢀⣿⡿⠿⢸⠿⠿⣦⠙⠶⠚⠳⠾⠃⠀⠛⣠⠾⢟
⠀⠀⠚⠟⠂⠸⢍⠻⡟⣛⠇⠀⠀⠀⠀⢸⡅⢉⣻⠏⡀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣸⢀⠀⣀⢣⢠⡄
⠀⠀⠀⠀⠀⠰⣶⢿⣯⡁⠶⠆⠀⠀⠀⠘⢷⣬⣙⣊⣡⣶⠏⠀⠀⠀⠀⠀⠀⠀⠘⠺⠦⠷⠊⠈⠁
⠀⠀⠀⠠⣶⣶⡌⠙⡿⠭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⡤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⣒⡮⠭⠭⠛⡒⢚⡉⠉⣉⠈⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣤⠀⠀⠀⢸⡸⡅⢲⣖⡶⠾⠿⠹⠿⠏⠉⠉⠀⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⣄⠀⣵⡂⠀⠀⡇⡇⠈⡁⣄⣠⣰⡔⠖⠀⠀⠀⠀⡘⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠄⠰⠀⠐⠀⡚⠀⠀⢱⢡⢀⡉⢁⣁⣠⠀⢰⡶⠶⠾⠈⢁⣧⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⣐⣀⡀⠓⠁⠀⠀⠘⡘⡜⠑⣉⣈⠥⠤⠤⣒⣒⣛⣉⠭⠥⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢧⡨⠥⠤⠖⠒⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠓⠚⠋⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠼⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⣤⣄⣤⣴⣤⣶⣤⣶⣿⣶⣷⣶⣾⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⠿⠿⢿⢿⠿⢿⠟⡿⡛⡻⠟⠛⠙⠛⡙⢋⠛⠛⢟⠟⠻⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠉⠙⠋⠋⠘⢓⡁⠈⠈⠃⠀⠀⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⢜⣅⣹⣿⣿⣿⣿⡟⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠁⠁⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠐⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠠⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠠⢳⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⡄⠀⠀⠀⠀⠀⠀⠀⣂⠸⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣱⣼⣇⠀⡀⣀⣠⡴⣴⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠛⠻⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠻⢀⠘⡀⠃⠀⠶⠧⠊⠃⢡⡴⠀⠶
⠀⠡⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣼⠁⠀⠠⣤⢠⣆⣃⠦⢯⢡⠑⣄⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⠉⠓⠁⠈⠀⠀⠘⠑⠀⡄⠉⠉⢠⠀⠊⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠩⠁⠀⠒⠐⠊
⠀⠀⢠⡀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡀⠡⠀⠀⠀⠂⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣦⢤⡶⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠤⡨⠂⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢰⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡤⠔⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠖⠛⠲⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠙⠦⡀⠀⠘⢦⠁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⢢⡀⢈⡟⠁⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢷⡼⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠗⠋⠛⣟⠋⢿⡟⣶⠚⣿⢻⣿⡇⠀⠀⠀⢀⣠⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⢧⣺⠃⣿⢀⣿⠈⡇⠉⠛⢻⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⣿⠀⢸⠂⡏⢀⡇⠀⢀⡞⢸⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡙⣿⢳⡞⠻⣽⠀⢸⢀⡇⢸⠁⠀⣜⣠⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇⢸⡆⠀⢸⠀⢸⣸⠁⡾⢀⡾⢫⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣇⢸⡷⣶⢺⠁⣻⣟⣢⣷⣏⣘⣁⣰⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⡇⡿⠚⠩⠵⠂⢸⣿⣿⡏⠁⠖⠂⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣯⢣⣀⠀⢠⡴⡫⡀⠙⢿⣆⢀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣼⣿⠉⢿⢧⠀⠀⠀⣸⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢿⣿⠀⢸⡌⣿⠛⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⢧⠀⠘⡇⣿⠀⣿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡀⠈⢷⠀⣿⣿⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣈⣇⡿⣿⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⠄⠀⠀⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣶⣾⣿⡁⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣶⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⢿⣿⣿⣿⣿⣿⣧⡥⢠⣆⣴⡆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠙⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠦
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀
⠀⠀⠀⠀⠀⣤⢾⡀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⢸⡇⠸⣤⠀⠀⠀⣠⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⡼⠀⠀⠃⠀⢀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡘⢱⠻⠿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣧⡄⠄⠀⢉⠓⣿⣿⣿⢻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠁⠀⠘⡇
⠀⠀⠀⠀⠘⠆⠀⢀⡏⠀⠶⠀⢸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⡁
⠀⠀⠀⠀⠀⠄⠀⢸⣿⣆⢰⡀⢿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣜⣧
⠀⠀⠀⠀⠀⠆⠀⢈⣿⡿⠿⠁⠀⠀⠓⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀
⠀⠀⠀⠀⠀⣠⠐⠁⠀⢀⠁⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠀⠉⠁⠒⠒⠂⣄⣀⡀
⠀⠀⠀⡀⠎⠁⠀⢀⠀⢀⠀⠀⠀⢙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠂⢄⠀⠀⣀⣀⣀⣀⡀⠀⣀⣉⣀⡀⠀⠱⡄
⠀⠀⡔⠁⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⠔⠀⠁⠀⡴⠊⢁⡠⡔⠢⣌⠉⠀⢀⡤⢤⡀⢀
⠀⠀⡆⢎⡀⠙⢭⣍⠉⠩⠉⣘⣉⡩⢓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠊⠀⢀⡨⡇⡇⢸⠀⡵⠴⠋⠀⠀⠉⠁
⠀⠀⠀⠐⠫⠝⠛⠃⠀⠀⢠⠴⠔⠢⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠑⠤⠤⠐⠚⡁⢇⢹⠀⢸⠸⣠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠌⠞⠘⠺⡜⠁
⠀⠀⠀⠀⠀⠀⠰⣤⠠⡀⠤⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡜⠈⠡⠀⢈⠀⠢⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⠠⠎⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⢷⠀⠀⠀⠀⣗⣼⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠐⠀⠀⠂⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⣉⠉⣉⢛⡛⣩⣽⡶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠬⠄⠰⣯⢠⣶⡌⠁⡾⠆⣴⡆⢉⣲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠰⠆⠀⠀⠀⡄⠂⠠⠀⠒⠒⡃⠦⢤⣬⣍
⠀⠀⠀⠀⡠⠐⡨⠀⠈⠈⠀⠈⠉⠒⠠⣀⠀⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡐⠁⠀⠊⠀⢀⣀⣤⣴⣤⣤⣄⣀⣈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠁⠐
⠀⠐⣶⣶⣶⣾⣿⣿⡿⠿⠻⣿⡿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⣀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠁⠀⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⠉⢀⣀⡀⠠⠄⠒⠁


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

⠀⠀⠀⠠⢀⠀⠀⠀⠀⡀⠀⢀⠀⠐⢠⡄⠀⠀⠀⠠⣸⣀⠤⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢄⣸⣠⠄⠀⠀⠀⢀⡄⢂⠀⡀⠀⢀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠱⠘⠀⠀⠀⠠⠄⣻⠄⠑⠆⣲⢒⡐⠂⠀⠐⠛⡝⠒⠀⠈⠒⣀⢀⠰⢀⠄⡠⠎⠀⠐⠊⡍⠓⠀⠀⠀⢂⡚⢖⠀⠎⠡⣞⠠⠄⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠂⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠲⢈⡁⠆⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠐⠀⠀⠈⠁⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⠀⠀⠀⠘⢘⡠⢣⢜⡳⠜⠀⠁⠁⠈⠀⠀⠄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⠀⠀⠄⡹⠃⡭⠀⠀⠀⢀⠐⠑⠠⠍⠂⠑⢦⡀⠀⠀⠨⢐⠟⡠⠀⠀⠛⠀⢀⠀⠀⢠⠀⠀⠀⠀⡀⠠⠄
⠀⢀⠀⢿⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠂⠀⠀⠐⠄⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠀⠀⠀⠀⠀⠂⣼⠆
⠀⠀⠂⠠⡄⠀⠀⡀⠀⠀⠠⡇⠀⠆⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠆⢨⠅⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⠤⠢⠀⢠⠀⠀⠐⠀⢰⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢠⡆⠀⠀⠀⠈⡠⠀⠔⠤⠤⠄⠀⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠖⠂⠒⠀⣷⠀⠀⠀⠑⣄⠀⣀⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⣀⠌⠀⠀⠀⣴⠀⠔⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠒⣀⠋⠰⠀⠀⠙⠛⠀⢣⠆⠀⠀⠀⡄⠀⠀⠐⡞⠂⠛⠊⠀⠀⠒⠘⣀⠐⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⡄
⠀⠀⠀⠀⢤⠘⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠌⠀⠀⠀⠀⠀⠘⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⣠⠁
⠀⠀⠀⠀⠀⡀⠊⢠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠤⠑⢂⠄
⠀⠀⠀⠀⠀⢀⠐⠆⢀⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠘⠢⢜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⡰⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠔⠛
⠀⠀⢠⡄⠀⠀⠀⠉⡈⠂⡑⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠐⢃⡈⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠤⠄
⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠄⠄⠡⠄⠤⠀⠀⠠⠀⠀⠂⠀⠀⠀⡈⢀⣀⠀⠄⢂⠀⠀⠀⠀⣀⡰⠤⡀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠃⠀⠀⠀⠀⠂⠀⠄⠀⠀⠤⠠⠤⠀⠠
⠀⠀⠀⠐⠈⢀⠂⠀⠀⠀⠀⠘⢣⣽⡀⣁⢒⢡⡤⠭⠒⢠⠊⠉⠀⠀⠀⢉⣇⣀⠟⠀⠀⠀⠘⡙⡄⠐⠭⠤⡌⠐⣀⢀⣼⣔⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠐⠋⣕⠉⠉⠀⡃⠠⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⠘⡉⢢⠙⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠔⢲⢺⠀⡉⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡗⢒⠢⠔⠂⠀⠀⠀⠠⡄
⠀⠀⠀⢈⣰⠀⠀⠀⠀⠂⠒⢋⠁⡈⠐⠀⠀⠀⠀⠀⠀⠀⠗⠔⠃⠠⠄⠀⡀⠀⠀⠃⠘⠀⠃⠌⠀⠀⠀⠀⠀⠀⠈⠐⢉⠀⡉⠒⠀⠂⠀⠀⠁⣦⠀⠆
⠀⠀⠀⠈⡃⡄⠀⠀⠀⠈⠄⢨⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⡁⠠⠁⠀⠀⠀⢁⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠃⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠠⠀⠀⠐⠀⠀⠀⠄⢰⡈⠣⢀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠓⡄⠀⠀⠀⠀⠐⠂⢲⠀⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠁⠀⠀⢀⠞⠁⠃⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡯⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣙⣳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⡙⠿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⣤⣴⣿⣷⣆⠈⠁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠻⠟⢻⣿⠿⣿⣿⣿⣿⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣠⠴⠛⠃⣤⣾⣿⣿⣿⣬⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⣧⠄⢀⣴⣿⣅⢀⣿⣿⣍⡻⡟⣳⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⢆⢀⡷⣼⣿⣿⡿⠟⢁⡰⠎⣩⣿⣿⠏⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⢿⣿⢿⣿⣿⡟⠉⣃⢲⡛⠛⠛⠻⢆⣈⣤⣤⠴⠋⣹⠇⠀⠀⢀⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⢀⢉⡼⠋⠁⢠⣾⣿⣯⠝⢶⡤⡴⠾⠿⠟⢽⢳⣾⣥⣤⣤⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢶⠞⠛⣿⡀⢀⡀⣀⣾⣿⣿⣿⣶⣤⣤⣤⣷⣶⣶⡘⣟⣿⢿⣿⠿⢿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣁⢀⣤⠂⠀⢀⣨⣷⣨⣿⣿⣿⣽⣿⣿⣿⡇⠀⢹⣿⡟⠿⡿⢧⣿⣶⣤⣴⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣒⠉⠙⠓⠲⠧⢬⣁⠛⠉⣫⣼⣄⡠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡿⠛⢹⢿⣻⣽⠟⠋⢱⣾⠿⢿⣦⡽⠟⠛⠿⣿⡷⣶⡄⢠⡼⢋⢀⡀⠀⣠⣶⢶⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡂⣀⠠⣦⠰⠟⠉⠀⣀⣠⣾⣿⣄⣼⣿⣖⠶⢤⣤⡬⢿⣶⠊⠁⠙⠿⣿⣿⣶⣟⣁⣾⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⡆⣉⢩⣅⠀⠀⠘⣧⡀⣠⣾⠟⠛⠋⣠⣦⠀⠛⢿⣦⣆⡀⠀⠀⠙⠿⣷⣠⣤⣀⠈⠉⣉⠉⣭⣁⣴⣶⣾⠇
⠀⠀⠀⠀⠀⡸⢸⣿⣿⣦⡆⠀⠙⠣⣾⠟⣿⡷⢀⣷⣷⡟⠾⠁⣀⣴⣿⠧⣿⣷⡀⠀⠙⣿⣿⠂⠀⠀⡖⠉⢻⡟⣿⣿⣶⣿⣶⡏⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠐⣷⣾⡿⢣⡟⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠁⠀⠀⢻⣿⣥⣿⠒⢿⣿⡇⠀⠀⠈⢿⡿⣿⣇⠰⢲⠞⠀⠀⠙⢿⣇⡁⠀⠀⠀⠀⠀⣠⡄⢰⣞
⠀⠀⠀⠀⢀⡿⠙⢱⡟⠀⣴⣶⣶⣶⠀⣠⣴⠿⠋⢀⡰⣷⣾⣿⣿⠿⢿⠦⢾⣿⣷⣶⠀⢀⠈⠙⠻⢿⣿⣛⢿⡷⢦⣄⣠⣈⢉⠓⢦⣤⣤⣾⢣⢿⣼⣿⡇
⠀⠀⠀⠀⢸⠃⠀⡾⢸⣿⡘⣿⣿⣿⡟⢉⠀⠀⣰⣿⣷⡸⣿⣷⣿⣤⣴⣄⣴⣿⣟⣻⣾⣿⡆⢰⣶⣾⣯⠉⠳⣄⠸⠟⠛⠛⢾⣿⣤⣦⣤⣿⣿⡜⠛⡿⠤⠴⠂
⠀⠀⠀⠀⢸⡆⢸⡇⠀⣿⣧⠹⣿⣿⣿⡘⢷⣿⠟⢛⣉⢷⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠙⣿⠇⣾⣿⣿⡟⢀⡼⠋⣀⣀⠀⠀⠀⣽⢻⡽⠿⣿⡍⠀⢸⠷⠿⠂
⠀⠀⠀⠀⢸⣇⠈⢷⡖⠛⡿⠃⠉⣀⣀⣰⣞⢳⣼⣿⣿⠿⠿⢿⣿⣿⡇⠀⢀⣤⣤⣄⢸⠟⠀⠿⢿⣿⣧⣾⣵⣿⠟⠻⣿⡆⠀⣷⢰⣇⣴⡟⣠⢆⡇
⠀⠀⢀⡴⠾⣷⡀⠈⢿⡾⣧⣴⣿⣿⠋⠉⣿⣿⣿⣿⡇⠀⠀⠸⡏⢿⡇⣰⠿⠋⠉⢻⡟⠶⠦⠶⣦⣿⣿⣿⣿⠀⠀⠀⢹⣿⢠⡷⠖⢛⣫⣾⣡⣾⣀⡀
⠀⠀⠈⢰⣿⣿⠧⡀⢨⢷⣿⣿⠷⠛⠀⠀⣸⣿⣿⡏⠤⠀⠀⠀⣿⡘⣹⣿⠀⠀⠀⣼⠀⠁⠀⠀⣜⢹⣿⠟⣿⠄⠰⠆⣼⣾⣯⠶⢒⣻⣿⣟⣥⣤⢀⡉⠁⣀
⠀⠀⠀⠀⠉⠃⠀⢻⣿⣼⣟⢣⣀⠐⣠⢤⣿⣿⣿⣿⠔⣄⠶⢰⣿⠇⢻⣿⠣⡜⠃⠻⡇⠶⠀⠀⠉⣻⣟⢦⡀⠶⣤⠴⠂⣠⢿⣾⡟⠋⢠⡅⠈⡿⠏⠛⠃
⠀⠀⠀⠀⠀⣼⢻⡏⠙⢯⠙⠿⢿⣯⣤⡞⠁⣉⣿⡀⠀⠀⠑⠉⠉⠀⣸⡈⣳⣶⣶⡾⠿⢆⠰⡶⠚⢿⣿⣿⣿⣶⣦⣀⠶⠋⣀⠬⠛⢁⡾⠶
⠀⠀⠀⠀⢰⡇⡀⠃⣀⢈⣷⣀⣀⠈⢩⡙⠚⠻⠿⣷⣤⣀⣀⣀⣄⣰⣿⣿⣯⢿⣿⣇⢀⡜⠉⢱⣄⣼⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⣸⣧⣄⠀⠈⡄
⠀⠀⠀⠀⣸⢟⣡⠋⣹⣿⣿⣿⡿⢣⢸⡏⡷⣶⠀⠀⠉⠉⠻⠿⠿⢿⣿⣿⡿⠿⠿⠿⠛⠛⠀⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡟⠀⠀⡧
⠀⠀⠀⢀⡾⢹⡇⠀⢿⢻⠝⡏⠀⠀⢸⡇⡇⣿⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⣤⢸⠆⢶⠀⠀⠀⠀⠀⠀⠀⣤⢀⣀⣠⡄⣲⠴⢷⠀⣸⡿⢥⣠⠔⠋
⠀⠀⠀⢘⣷⠟⢳⡀⢀⡟⣰⠀⠀⠀⠘⠃⠗⠿⢸⡏⢸⡇⢸⡇⢰⡆⡆⠀⣿⠀⠀⢼⠐⡆⣷⠁⣷⠛⣤⣿⢸⣿⣿⠷⠟⠃⠈⢠⡏⠀⢠⡗
⠀⠀⠀⢸⡏⠀⢸⣿⣼⠁⠈⠉⠛⠲⠦⠤⣤⣀⣀⡀⠈⠁⠈⠃⠈⠃⡇⠀⠙⠀⠀⠚⠚⠃⠛⠀⠉⠀⠉⠀⠀⠀⣀⣀⠤⠤⠒⠉⢷⣠⠟
⠀⠀⠀⠈⢿⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣄⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⢴⡦⠶⠶⠖⠙⠃⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⣀⣤⣶⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⢀
⠀⠀⢀⣾⣿⡛⢿⣗⡀⠠⠉⠢⡀⠀⠀⠀⠀⠀⢠⡉⢬⣯⣒⢀⡀⠀⠀⠔⠀⡤⠈⠒⠀⠀⠀⠀⣼⣽⣋⣿⡄⢈⣨⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦
⠀⠀⠸⠛⠛⠛⡶⠾⣥⣀⡀⠀⠹⣄⠀⠀⠀⠀⡈⠿⠩⠽⠦⠬⢥⡀⠀⠀⠁⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⢻⣷⣾⣿⡇⠀⢀⣜⣐⡆⠀⡾⠃⠀⢀⣈⡟⢏⣄⡀
⠀⠀⠀⠀⠀⠄⠁⠁⠈⡍⠛⠶⣦⣽⡆⠀⠀⣒⠃⢲⣒⣲⣒⣉⣊⡀⠀⠀⠘⠀⢠⡔⠲⢦⡄⢸⣷⣿⢵⣿⣿⣿⣿⡇⠀⠈⠉⠙⠁⠀⠀⠀⠀⠈⠋⠁⠀⠛⠃
⠀⠀⠀⢀⠊⠀⠀⠀⠸⠁⠀⠀⠀⠙⠁⠲⢄⣉⣩⣥⢥⣞⣃⣐⣎⣥⡀⠀⠀⠀⠀⠁⠀⠶⠋⠈⠙⠻⠛⠟⠀⠋⠉⠀⠀⠀⠀⠂⠒⠓⡀
⠀⠀⠠⠁⠀⠀⠀⠀⢠⠀⣴⠞⠉⢿⣦⠀⠀⠙⢮⡭⠴⢏⣑⠫⠘⠉⠁⠈⠛⠛⠋⠀⠀⣷⣤⣤⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢰⣤⣶⣿⠂⠈⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠿⠃⠀⠈⢉⡙⣲⣒⡛⠓⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠋⠀⠀⠀⣀⣠⣤⣭⣁⠉⠀⠀⠈⠉⠉⣡⡆⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣻⣆⠀⣴⣿⠁⢹⣭⣪⡿⠁⣢⠘⠃
⠀⠀⠀⠑⢶⣤⣤⡤⠄⠀⠈⠐⠶⠂⠁⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⠇⣿⠀⠫⣁⣠⠞
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠘⢿⣙⡝⠿⢛⣣⡴⠃⠀⠀⠀⠀⠀⠀⣀⣁⠉⠁⠉
⠀⠀⠀⠀⢀⢠⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣈⠛⠟⠀⠀⠀⠐⣮⣣⣒⠄⠀⡔⠁⠀⠀⠈⢂⣀⠀⠉⠉⠉⢉⣥⣤⣦⣀⠀⠀⠾⣿⣿⣿⣟⠁⠰⣿⣦⣤
⠀⠀⠀⠀⠿⠆⠙⠁⠀⠀⠁⠀⠀⡀⠀⢸⣿⣿⣿⣿⣷⣴⣤⡀⠀⢸⣿⣿⣿⠋⠀⢧⠤⠤⠦⢶⣾⡀⢑⠀⠀⠀⣿⡁⠧⣡⠽⠧⠀⠀⠙⣛⣏⣀⡀⢀⡈⠉⠉
⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⠐⠆⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡧⠀⠈⣿⣿⡿⠀⠀⠀⢸⡇⠈⠈⠁⠰⠏⠰⠒⡶⠈⠻⠶⠿⣆⡓⠀⡃⠰⣿⡽⢿⣿⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⣠⠄⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠃⠀⠈⠡⠌⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠁⠀⠉⠋⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠂⠉⠀⠄⠐⡌⢻⣿⣿⣯⣽⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠐⠀⠔⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⢹⣿⣿⣽⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠄⢹⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⠈⢆⣿⣿⣿⣿⡋⣹⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⣀⣀⠤⠤⠤⢤⣤⣤⣾⣿⣿⣿⣿⣿⡿⣿⠟
⠀⠀⠀⠀⠀⡀⠀⠀⠠⠐⢊⣁⣭⠭⢿⠀⠀⠀⡏⠉⠙⠒⠓⠒⠶⠾⠟⠛⠁
⠀⠀⠀⠀⠀⠈⠐⠓⠊⠉⠉⠀⠀⠀⠈⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⡠⢬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⣈⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⠚⢹⣿⢻⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠐⠂⠤⠤⢤⠥⠀⠀⠀⠀⠀⠈⠀⢾⡇⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢺⠃⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣧⢀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠐⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣙⠇⢀⢀⣠⣀⣠⠼⢷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢠⡐⠂⢰⠐⠄⢠⠀⡆⠀⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣐⠀⡀⠀⢱⣎⣈⠐⠡⣀⡨⡅⠀⠉⠠⡀⢀⠴⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠃⠠⡆⠀⡂⠙⣏⡥⢀⠛⢿⢏⠀⠃⣤⡀⣿⡏⡄⠧⢀⢀⠁⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢠⠀⣜⣠⢾⠛⡏⢫⠿⣦⣼⠒⡿⣠⡁⠀⠷⣷⠂⠜⡠⠁⠖⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣠⢉⣽⡈⢛⣶⣬⣏⠘⢁⣍⠿⢻⣿⡴⠛⢻⣿⡅⡲⢎⣽⣇⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢘⡟⢃⣞⣱⣧⣿⣵⣾⣷⣞⣿⣿⣽⡤⣿⣬⣤⣤⠜⢃⡈⡛⠂⠊⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠚⢆⠈⣋⣀⡍⢫⣯⣽⡍⠥⣿⣿⣿⣿⣿⣷⣿⣞⣷⣾⣿⢂⣻⣶⣎⡹⡇⢀⢦⡆⠁⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠹⠀⢠⣿⠈⡙⠘⣬⣿⣷⣶⣿⣿⡿⠿⣿⣿⢿⣿⣿⣿⣿⣿⢻⣏⣻⡓⢻⠮⣨⣁⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡾⢗⣁⣈⢓⣻⣿⣿⣿⣯⠿⣾⣷⣾⣯⣈⣿⣿⡶⣟⣻⡿⣿⠛⢫⣶⡄⢞⠣⢙⠁⠰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡈⢠⡏⠉⣴⣡⣽⣯⣷⣮⣿⣴⣯⣿⣿⣿⣿⣿⣟⣛⣿⣿⠟⡿⣋⣿⣵⣦⢎⠟⣈⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠈⡴⢣⡍⢟⣲⠼⣿⠿⣿⣯⣾⣿⣿⣿⣟⣺⣿⣿⣇⡟⠸⣧⡟⢬⠉⢁⡀⠤⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠴⢊⠋⠉⠷⣬⣿⢶⣸⣻⡿⣿⣿⠻⣿⣿⢛⣽⣿⣛⣿⡅⠿⣞⢃⠁⣸⣻⠃⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢂⢃⡀⢽⣯⣀⢾⣽⢮⣤⡤⣿⣿⡟⣿⢹⣿⡷⣼⣿⣟⠻⠟⢟⡰⠅⠱⠒⠊⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠌⡙⠳⠆⡸⠚⠫⠙⢻⣾⢗⣾⠟⣟⠱⢦⡽⢻⢷⣬⡸⢋⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⡀⠠⠓⠃⠒⠉⣼⣠⣥⡴⢦⠐⣀⣼⢿⡐⢻⣯⡴⣤⠐⠴⠻⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢀⠠⠒⡀⠉⢻⠳⠛⠊⢁⡱⠥⠚⠙⠂⡁⠀⠀⢀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠋⠁⡀⠁⡀⠈⠴⢆⢉⡆⠹⠄⠋⠒⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠤⠄⠂⠐⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⢛⣛⣛⡉⣉⠙⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢩⣖⣦⣶⣆⢸⣿⠃⢹⡇⠸⢷⣿⠀⣿⠉⠉⠉⠛⣲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⣿⢸⡟⠁⠟⢻⠈⠋⠳⠈⠁⠒⢚⣁⣀⣯⣀⣀⣀⣘⠛⢭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠷⡶⠀⠀⣻⣄⣡⣤⡤⠴⠶⠚⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠙⠋⠀⣀⡴⠃⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠠⠀⡇⢨⡄⠀⣼⣡⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⡆⠀⠀⠀⣹⣶⢶⡖⢲⢶⡶⡾⢯⣹⡇⠀⠀⣸⡎
⠀⠀⠀⠀⠀⠀⢸⡇⣇⢸⡀⠀⠉⠁⠀⢠⣤⡤⢤⠤⣤⢲⣶⢿⡁⢹⣌⣿⠀⠀⠀⠿⠋⠈⢷⠇⠈⠻⠃⠀⠙⠃⢸⡇⣿⡇⢠⣤⡄
⠀⠀⣴⢒⠊⠉⠙⠋⠛⠙⢓⠒⢦⡀⠀⢸⣿⡇⢸⣧⣽⠸⣿⠾⡇⠀⠋⠈⠀⠀⠀⠀⢸⡇⠀⢰⣢⠀⠀⣴⣦⠀⢸⣧⠻⠇⠈⠛⠉
⠀⠀⡧⢀⣀⢉⣭⡉⣩⣅⢠⣤⡀⢻⠀⠈⠈⠀⠈⠀⡁⠀⠁⠀⠐⣦⣀⢿⣠⢿⠀⢰⢟⣧⠀⣰⠿⡀⠀⢸⣻⣦⠈⠿⠀⢠⣦⣀⡀⠻⠃
⠀⠀⢷⡟⢻⣿⣸⣇⣟⣻⡿⢹⡇⡿⠀⠀⠀⠀⣶⣾⡿⣰⠀⠀⠀⠹⡉⣂⡡⠼⠀⣯⣵⡿⠀⣿⠃⢷⡀⣾⣭⣼⠇⠀⠺⣼⢿⣻⠅
⠀⠀⣸⣗⠀⣤⠙⠿⣃⠉⢠⣌⣻⠧⣄⢠⡖⠒⢻⣿⡏⢉⣳⡄⠀⡶⣄⢠⢺⠂⠀⢀⡀⣀⠀⢙⡓⠛⠁⠀⠀⠀⠀⢀⣀⡙⠋⠁⣀⢀
⠀⢸⠁⠈⠉⠓⠒⠚⣉⠉⠉⠉⠀⠀⠘⡇⢳⠚⠹⣿⡏⠉⣹⠁⣤⣷⣿⣿⣾⣧⠀⡸⠖⢶⣉⡳⣆⡵⠀⡶⠉⢲⣤⠟⠁⢸⡆⠀⠻⠟
⠀⢸⣴⢶⣄⡞⣷⣼⢻⡆⣾⢻⢠⡟⣦⣿⢸⡀⠀⣿⡇⠀⡟⠀⢸⣏⠋⠋⣭⡏⠀⠁⠈⠁⢀⣉⡉⠀⠀⣷⡀⠀⠉⠀⠀⣼⠃⣶⣤⡤
⠀⠀⣏⠈⠿⠃⠻⡿⠈⠷⣏⠈⢿⡃⣸⡅⠀⠣⢀⠛⠓⠚⠁⠀⠈⣿⣀⣴⣉⡇⠀⠀⠀⣰⠋⠀⠙⣦⠀⠈⠙⠲⢤⣠⡼⠃⠀⠻⠟⠁
⠀⠀⢿⠺⡧⢮⡥⣬⢤⡴⠶⢋⣛⣬⠾⠃⠀⣼⢁⢹⣠⡖⢒⡄⠀⠈⠉⠉⠉⠀⢀⡴⠒⣿⠀⠀⠀⣿⠞⠉⣶⠀⠀⣽⠁⠀⢀⡤⠖⠂⢤
⠀⣠⡀⠉⠓⠒⠒⠒⠒⠛⠉⠉⠀⢠⣤⡀⢠⡬⠽⡿⠻⣿⠏⣤⢰⣿⣤⣶⠀⣤⠞⠳⣄⢿⠀⠀⢀⣟⣠⠤⣿⡁⠀⠙⢦⠀⡞
⠲⡋⠹⡃⢤⢤⣶⣶⡾⣹⠀⡤⣄⢨⣿⡁⠸⡇⠀⣇⢠⡧⠀⢼⢻⣿⣿⡅⣸⠃⠀⠀⢹⣮⠧⣴⠻⡿⠁⠀⠈⣿⠀⠀⣼⠀⡇⠀⠀⠀⢠
⠘⠛⠉⠁⠘⠚⠉⢁⡀⠀⣠⢿⡅⠺⠾⠇⠀⣇⠀⣿⢸⠇⣀⣿⣸⣿⣸⠇⢻⡀⠀⢀⣾⡏⠀⢸⣿⡇⠀⠀⢀⡿⠀⠴⠃⠀⣸⠶⠴⠒⠋
⢸⠖⢦⠀⠀⣠⠚⣵⠿⠀⢑⣛⣁⡔⢰⠀⠀⠈⠉⠉⠉⠉⠁⣀⣀⠀⠀⠀⠀⠙⢶⠋⣿⡄⠀⣸⡿⠙⣦⡴⢛⣀⣀⡀⠀⢠⡇⠀⣸⣷⣶
⠉⠻⠁⠐⢊⣲⠗⠁⣴⠰⣿⣌⠿⠃⠘⢡⣾⠀⠀⠀⢀⡴⠺⣅⡽⢦⡀⠀⠀⠀⠀⠀⠈⢳⣤⠟⠠⠀⡄⡴⠋⠀⠀⠈⢢⠈⢇⠈⠛⡽⠋
⠘⠃⣠⡤⣤⣁⣀⣀⡈⠙⠋⠀⠀⢀⠴⢭⠭⠵⢦⠀⢯⢌⣓⣉⣫⣭⡽⠀⣔⠀⠀⠡⣠⠀⡅⢀⠃⠜⠐⡇⠀⠀⠀⠀⣸⠇⢸⡆⢠⣤
⠀⠀⣿⠀⠖⠀⢀⣠⡾⣷⣀⠀⠀⡾⢉⢚⣩⢉⡹⣄⢸⢸⣻⢹⢸⣿⠃⠀⣾⡆⠀⠀⠈⢳⡁⢀⠎⠀⠀⢻⣄⣀⣀⡴⠋⢀⡼⠀⠙⠛⠁
⠀⠀⢻⣆⣤⡾⠛⢋⣠⠾⣹⠃⠈⣶⣦⡭⡶⣤⡖⡟⠀⠉⠛⠛⠛⠁⡔⠀⠉⠙⠒⢄⡀⠀⣿⠀⠀⠀⢠⠞⠋⠁⠀⠀⣶⡄⠀⣶⣤⠄
⠀⠀⠀⠉⣷⣤⠴⢛⡵⠚⢡⣄⡄⠹⢾⣷⣷⣿⠼⠁⡠⣠⣶⣄⠀⢺⡄⠀⠀⠀⡀⣠⡗⠀⣿⡀⠀⠀⠸⡀⠠⣿⣖⠀⣀⣠⡆⠘⠁
⠀⠀⠀⠘⠧⠔⠚⠉⠀⠀⢮⢿⠗⠀⠀⠀⠀⠀⠀⠀⣱⢺⡟⡞⠁⠀⠘⡏⡏⣿⣏⣿⠃⠐⠁⠀⠀⠀⢀⡇⠀⠁⠀⣠⣯⡀⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣈⣁⣀⣀⣀⣷⣿⣟⣛⣛⣀⣀⣀⣀⣀⣠⣏⣀⣀⣀⣀⣈⣀⣈⣃⣀⣀⣀⣀
⣿⣿⠛⡻⢛⡛⠛⣟⢛⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠒⠒⢒⣒⣒⣒⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢋⠉⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠈⢙⡄⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⣎⣉⣩⠗⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡫⠴⠚⠛⠒⢦⡀⠀⠉⠀⠀⣀⡤⠤⠤⢤⣤⣀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣄⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡎⠀⠀⠀⠀⠀⣠⠇⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡞⠁⠀⠀⠀⠀⠈⢦⠀⠈⢧⡀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠑⠦⠶⠒⠒⠋⠁⠀⠀⠀⣏⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠹⣄
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⣀⠀⠀⠀⠈⠒⠒⠉⠉⠉⠀⠀⠀⠀⣠⣄⣀⠈⠲⠶⠶⠖⠚⠁⠀⠀⠀⡠⠤⣌⢧
⠀⠀⠀⠀⡾⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢸⡾⡆
⠀⠀⠀⠀⡇⠀⠀⠸⡅⠀⠀⠀⠀⠀⢀⣠⡤⠚⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⠀⠀⠀⢀⡴⠲⢤⡀⠀⠙⠉⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠉⠂⠀⠰⠒⠊⢉⣠⠴⢖⡋⠉⠙⢯⠀⢹⠀⣯⢸⠀⡜⢀⠎⠀⡠⠋⣩⠛⢒⣦⣈⠳⣄⣀⡷⠀⠀⠀⠀⡧
⠀⠀⠀⠀⢹⡀⠀⡤⠴⢤⠀⠀⣠⡖⠋⠒⠤⣀⡉⠲⢄⡈⢣⠸⡆⢸⢸⢰⢣⠏⣠⢞⡴⢋⡤⠚⣉⡠⠜⠳⣄⠀⠀⠀⡠⢄⢠⠇
⠀⠀⠀⠀⠀⠙⢆⠃⠀⠜⠁⡼⠡⠬⢍⣛⣒⠦⣍⣙⠒⠽⣦⣱⡧⠴⠶⠾⠧⢼⣟⣵⠾⠵⢒⣋⡥⡤⠖⣒⣙⣇⠀⡎⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⢷⠒⠒⠲⠒⠒⠛⢛⣛⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣛⣛⣋⠭⠭⠥⠤⢠⡟⢀⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣻⣮⣉⣉⠩⠭⠶⢚⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣦⣤⠶⠶⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠓⠋⠉

⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢰⠖⠟⠳⢲⡆⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣰⣿⣿⣿⣿⣿⣦⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⢰⣶⣶⡆⢱⣶⣶⡆⠀⠀⠀⠀⠀⠇⠀⠀⡀⠀⠾⢽⢯⠷⢀⢀⠆
⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⡷⠀⠀⢸⣿⣿⡇⢸⣿⣿⠁⠀⠀⢾⣦⡀⠀⠀⠀⠈⠀⢰⣿⢾⣿⣄
⠀⠹⣿⣿⣿⣿⣿⠟⢀⣴⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⡇⢸⣿⣿⠀⠀⠀⣸⣿⣧⣤⣄⠀⠀⠐⢿⠿⠀⠿⠿⠁
⠀⠀⠀⠉⠉⠉⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠸⠿⠿⠇⠸⠿⠿⠀⠀⠿⠟⠛⠉⠛⠛⠃⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣉⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⡁
⠀⠀⠀⠀⣸⣇⠀⠀⠀⠈⣚⣷⡴⠀⠀⢦⣴⣗⠚⠀⠋⣤⠀⠀⠀⠀⠀⢈⡉⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⢀⡻⢟⡀⠀⠀⠈⠁⢖⣿⣶⣶⣾⣳⠋⠁⠀⠈⢃⡄⠀⠀⠀⠀⣈⡠⠄⠀⠸⠿⠆⠀⣀⠂⠀⡀⠀⠺⠿⠂
⠀⢻⣿⣸⣿⣿⣇⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣤⠄⠀⢀⣈⣁⡤⠖⠀⠀⠀⠀⠀⠈⠉⠹⣷
⠀⠈⠿⠋⠀⠀⠙⠻⠋⠀⠀⠈⣿⠛⠛⢻⠏⠀⢠⣦⠀⠀⢡⡆⠀⠈⣡⣀⡠⠵⠚⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣰⣾⣶⣶⣧⣄⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠀⠎⠸⠂⠀⠈⠈⠁⠀⠂⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⣠⣶⣿⠓⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⣾⣿⣿⠟⠃⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠐⠻⠿⠿⠀⠀⣤⣄⣻⣿⣿⣟⣠⣠⠀⠀⠀⢀⣴⣄⡘⠇
⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠂⠀⠀⠙⢿⣿⣷⣆⣹⡿⠋⠀⠀⣴⣿⣤⣬⣿⡆
⠀⠰⠿⠿⠿⠿⢿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣄⠀⠀⠀⣿⣿⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠋⠈⠿⢿⠷⠀⠀⠿⠿⠀⠸⠿⠇
⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⢀⣬⠁⠱⠆⠀⠀⠀⠀⠀⠀⠀⣔⣰⣶⣠⣾⣧⢀⣤⠀⠀⣴⡟⠛⠉⠉⠙⢳⣆⠀⠀⢀⣴⣾⣦⡀
⠀⠀⠘⢆⡛⣡⡿⡿⠳⠇⢀⠀⠠⠀⠀⢀⣹⣷⣽⣿⣿⣿⣿⠃⠀⢘⣿⠀⠀⠀⠀⠀⠀⣯⠀⢠⣿⡿⠟⢻⣿⡆
⠀⠘⠛⣤⠢⣤⠷⢆⠈⣘⠋⠀⢇⠀⠀⠸⢿⣿⣿⠿⠿⠟⠃⠀⠀⠀⢿⣤⣀⡀⢀⡀⣰⡟⠀⢺⣿⣭⢀⠰⣿⣿
⠀⠀⠀⠉⠀⠀⠀⠘⠁⠙⠁⠀⠀⠑⠂⠤⠤⠤⠤⠤⠄⠂⠀⠀⠀⠀⠀⠉⠙⡿⡿⠛⠉⠀⠀⠈⠻⢿⣿⣿⠟⠃

⡏⠉⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠛⠛⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠈⠁⠐⠒⢄
⡇⠀⠀⠀⢠⢤⣦⣄⠂⠀⣴⠾⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠂⣀⢀⠀⠤⡴⣶⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠛⢹⠂⠀⠁⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⣼⡷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡀⡄⠀⢀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢠⣯
⡇⠀⠀⠀⠀⠀⠀⢰⡆⠀⢹⠀⠁⠀⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡄⠀⠱⠀⠀⠤⣿⡗⠀⢾⠀⡼
⡇⠀⠀⠀⠀⠀⣧⠸⠀⠀⢸⡆⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⡄⢷⠀⠀⣧⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠘⠗
⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠁⠀⢸⡆⣀⠀⠀⠀⠀⠀⠀⠢⡐⢎⣧⢠⣿⡆⣠⠆⠀⠀⠀⢀⣀⡀
⡇⠀⠀⠀⠀⠀⠳⢿⣀⣠⣤⡀⣀⣼⣿⠟⠀⠀⡀⠀⠀⠀⠀⠘⠌⢿⡌⢯⣹⠟⠀⠀⣀⣤⣿⣿⣿⣷⢶⣤⣀⠀⠀⠀⠈⣶⡷⢶⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⢁⣈⡁⠀⠀⠘⠃⠀⠀⠀⡀⠀⠘⠿⠟⠛⠓⠀⣠⣾⣿⡟⢿⣿⡿⠟⢃⣉⡙⠆⠀⠀⠀⠈⠀⡜⠋
⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠶⠋⣭⠉⠀⠀⠀⠀⢀⢆⣀⣀⣊⣠⡀⠀⠀⠀⢰⠿⣿⣿⣀⡈⣥⣴⣿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠙⢻⣶⡄⠀⠀⠘⢹⡿⣿⠆⠂⠀⠀⢀⣠⣥⠈⠉⠉⠉⠀⠀⠀⠘⠛⠿⠟⢿⣿⠟⢃⡞⠁
⡇⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⢠⠀⣷⡄⠀⠀⠀⠀⠉⠁⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⠁⠀⠀⠀⢠⣀⣶⠖⢶⣶⣶
⡟⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠷⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⡆⠈⠛⠙⠉⠙⠛⠀⢸⡏
⡇⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⣙⠟⢰⣄⠀⠁⠀⠀⠀⢀⠀⢸⠀⢈⣷⠶⣦⣤⣀
⡇⠀⠐⠷⢾⡎⠁⠀⠠⣄⣀⣀⣀⣀⣀⣠⡤⠤⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠷⠦⣿⡁⠈⠀⠀⠀⠛⠀⠀⠀⠈⡯⠀⣭⠉⠉
⡇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠉⠉⡏⠁⠀⠀⠀⠀⠀⠀⣀⠈⣽⠏⣿⢿⠷⠆⠀⠀⠀⠈⡗⢉⣉⡥⠀⣤⠀⢠⡀⠀⠀⠀⣇⠀⡇⢀⢠⡤
⡇⠀⠀⠀⢰⡇⠀⠀⠐⠀⠸⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⡸⠀⠉⢲⡥⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠸⣿⣿⠀⠀⡀⢻⣤⡷⠘⣤⠁
⡇⠀⠀⠀⠘⠁⠀⢄⠀⠻⡀⠀⠀⡀⠀⠀⢰⡀⠀⠀⠲⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⢞⣿⠚⠛⡀⠃⠋
⡇⠀⠀⠀⠀⠀⠀⢀⣤⣴⡓⣶⣶⣿⣿⠿⡿⠋⠀⠀⢸⠀⠀⠀⢸⠁⡄⠀⠀⠀⠀⠐⢶⡄⠀⢀⣤⡤⠀⠀⠀⠀⠀⠈⠁⠀⠉
⡇⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠈⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢾⠀⠀⠀⣠⣄⠀⠤⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⣿⠺⡖
⡇⠀⠀⢰⣐⡒⠶⢶⣶⠶⠶⣶⣤⣤⠀⡀⠀⠀⠀⠀⢸⠆⠀⠀⢸⡧⡄⠀⠀⢽⡀⠀⢸⠀⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⠀⣦⠄⠁⠐
⡇⠀⠀⠀⠉⠙⠛⠻⢿⠆⢠⡾⣿⣻⣿⣿⡿⠇⠀⣧⠀⡌⣾⠀⢸⣷⡇⠀⠀⢸⡇⠀⠚⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣾
⡇⠀⠀⠀⢀⣠⣤⣤⣿⠀⢸⣧⠀⠀⠀⠀⠀⠀⢀⡛⠀⠛⠃⠀⠘⠻⠃⠀⠀⣼⠃⠀⠀⠀⠀⠠⢃⣠⠄⠢⠌⠀⠀⠀⠀⣿
⡇⠀⢶⠶⠿⣿⠆⠘⣿⠀⠸⠟⠀⠀⠀⣀⣤⣶⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠖⠰⠞⠒⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠈⠃⠈⡿⠀⠠⡄⠀⠀⢐⣩⣄⣙⠛⣿⠛⣿⣷⡄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⡀⠀⠀⢀⡇⠀⠀⡀
⡇⠀⠀⠀⠀⡀⢀⢀⡇⠀⠀⣰⢀⠀⠀⠉⠛⠻⠿⠧⢶⣯⣿⣵⣄⠀⠀⠀⠀⠙⠛⠿⠁⠀⠶⠾⠓⠚⠁⠛⠉⠀⠀⣬⣿⠇⢠⠀⠃
⡇⠀⠀⣷⡶⣿⡌⡼⠁⣤⠀⢸⠿⠆⠀⠰⢦⡄⠐⠀⠋⠉⠈⠁⠀⠀⠀⢀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠀⠘⢱⠀⣴
⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣤⠀⠀⠁⠀⡶⠶⡆⠀⠈⠛⠚⠿⠏⠁⠀⠀⠀⠀⠐⠛⠛⠁⠀⠀⠀⠀⠄
⡇⠀⠀⠀⢀⡀⠀⣀⣷⣠⡀⠤⠖⢯⢦⡀⠈⠙⠟⢠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣤⣠⣄⣠⣴⡆
⡇⠀⠀⡘⢿⣟⠻⢿⡙⠇⠾⠗⣾⢻⡇⢃⣠⣤⡀⠈⠛⠃⠈⠋⠈⠻⢶⡀⠀⣠⣾⠏⠙⠀⠀⠀⠀⡄⠀⠀⠀⣤⣤⣤⠀⠉⠈⠉
⡇⠀⠀⠦⣼⣿⣿⡉⠁⠀⠀⠀⡸⠈⠀⡉⢀⣿⠈⣦⡀⠀⠀⠀⢠⣤⠀⠀⠀⣽⡇⠀⠀⢀⠀⠀⢈⡳⠖⠄⠀⠈⢉⠁⠀⠀⠀⠀⠂
⡇⠀⠀⠀⣼⣿⢀⡶⢿⣴⡶⠿⠟⠚⠿⠷⢻⠟⡿⠢⠃⠀⠀⠀⠈⠀⢠⡄⠀⢿⡇⢰⣿⡭⠀⠀⣸⠁⠀⠀⠀⠀⠘⣶⣷⣾⡣⣴⠆
⡃⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠒⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡄⠟⠤⠺⠟⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠿⠻⠿⠋⠁
⡇
⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡟⠉⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣾⣿⣿⣿⣿⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⢻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⣀⣴⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣇⣀⣈⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢋⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠛⠻⠯⢍⡛⢍⠙⢏⢹⠉⠛⡙⣋⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⡋⠩⠭⠍⠑⠒⠲⢤⣬⣶⣷⣬⣴⣬⣬⡶⡡⡒⢕⡊⠁⣐⡋⠉⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣤⣄⣒⣈⣉⠅⠀⡻⣿⡏⡢⣄⣢⠁⢹⣿⣽⣶⣿⣿⣶⣶⣒⣲⡶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⢻⡇⠧⠀⠀⠠⢹⡟⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠃⢀⣈⢀⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠷⠛⠛⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠰⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡈⠀⠀⠈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡞⠀⠰⠀⠀⠀⠰⠈⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⡾⠀⠠⠀⢠⡟⠀⠀⠀⣧⠀⣰⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⢸⣾⠃⣰⡆⢰⢯⡇⠀⢠⡶⣿⡼⠁⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡇⠀⣿⡇⡟⢸⢠⣆⣞⣤⣿⡴⢚⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⡟⢿⢶⡇⣿⠇⣿⡿⣿⠟⣿⠋⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠓⠈⠇⠁⠈⠀⠛⠀⠉⠀⠀⠀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⢀⣄⠀⠀⠀⠰⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠻⠑⠀⠀⠈⠉⠀⠀⠀⠀⠀⡴⠎⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠾⠛⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⣀⠀⠈⠙⠒⠒⠀⠤⠄⠀⢀
⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠟⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠤⢴⡟
⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠒⣛⣋⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢮⣯⣛⠛⣿⡶⠶⠶⢶⡶⠶⠶⠶⠶⢶⢶⠶⠒⢒⠛⢛⠛⣋⣉⣭⡀⠨⠔⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⣦⣿⣦⣤⣤⣤⣾⡆⢂⡶⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀⠀⢸⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⢷⢦⣄⡄⡤⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢿⣦⣴⣠⡤⠼⠳⡏⠀⠀⠀⣰⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠐⡄⢠⠀⢸⣿⠋⠁⠀⠀⠀⢹⡀⠀⢠⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⢄⠱⣞⠀⣸⡗⠀⠀⠀⠀⠀⠀⢣⠀⣸⠀⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⣎⢶⡟⢶⣿⠁⠀⠀⠀⠀⡀⠀⠈⡄⡿⢰⠃⡼⠀⠀⣠⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢿⣼⢷⣠⣿⣦⠀⠀⠀⢀⠁⠀⠀⠠⡇⡎⣠⡗⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣾⠋⣿⣿⡏⣿⠀⣀⠀⡈⠀⠀⠀⠀⣷⠏⣼⠀⡾⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢰⡇⣿⡇⣽⠂⡇⢀⠆⠀⠀⠁⠀⣿⢠⠏⣸⣃⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⣷⣿⡇⣿⣴⠀⢸⠀⡆⠀⢴⢰⣿⣯⣴⢟⣉⠥⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣆⢸⡀⣇⢰⣼⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⣿⣿⣿⣿⡿⠻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⢶⠶⢶⡶⠒⠄⠄⣠⠀⢀⠀⠀⠀⠀⡶⠟⠛⠿⠶⠤⢾⣞⠿
⠀⠀⢀⣀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣭⣦⡶⠶⠛⠒⠺⠿⠿⣿⢾⣿⠖⢟⠥⢀⣀⣜⣀⣤⣆⣜⣹⡷⠖⠂
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣦⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⠭⠯⠛⠉
⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣆⣀⠀⠙⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠿⣶⣾⠇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢐⣆⠀⡀⢠⣿⣿⣿⣿⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉

⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢡⢠
⠀⠀⠘⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠆⠰⠈
⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠛⡘⠛⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠰⡇⠀⠀⠠⠤⠖⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠠⠄⠀⠀⠠⠠⠄⡄⠀⠀⠀⠀⡂⠆⠅⠀⠀⠀⠰⠰⠀⠀⠀⠀⡀⠀⠀⣠⢭⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠰⠠⠀⣃⣀⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡀⠀⠀⡀⢀⢀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⢸⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⡀⢇⣃⠸⡤⡀⣧⣅⠈⣆⠀⠀⠀⠁⠀⠀⠸⡿⠇⠀⠀⠀⠀⠀⡀
⠀⠀⣩⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⢀⡄⡄⠀⣄⠀⢠⠀⡀⡀⢠⡀⠀⡄⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⢀⣿⡇⠀⠈⠫⣶⡽⠃
⠀⠀⡡⢖⠀⠀⠀⠀⠀⠀⢀⠀⠀⠣⡀⠐⠇⠧⠀⠷⠄⠸⠀⠬⠄⠀⠧⠀⠧⠉⠆⠮⠐⠦⠰⠥⠀⠀⠠⠻⠛⠃⠀⠀⠀⣿
⠀⠀⠖⠉⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠁⠈⠁⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠁⠀⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⢴⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢄⠐⠀⣠⠆⠀⠀⠀⡆⠀⡆⠀⢀⢀⠀⠀⠀⢀⣀⡀⠀⠀⣄⠀⠀⠀⢰⡿⠁⠀⢈⠀⠄⠀⠀⠁⢈⠄
⠀⠀⠀⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⢀⠀⢤⢿⣿⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣿⠿⠿⠿⠟⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢀⡀⠀⠠⣤⣮⣤⣔⡒⣶⠖⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀⢀⡀⠈⢿⣶⣤⣄⣀⣀⣈⣉⠐⠀⠀⠀⢀⣩⣭⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⢿⣾⣿⣿⣿⣶⡦⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣬⠇⠀⣸⣿⣿⣿⣿⡿⢿⣷⡆⠀⠀⠀⠀⢿⣿⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣡⣤⣶⣾⣿⣿⣿⣿⣯⣽⢶⡇⠀⢠⡿⠟⢉⣻⣯⣤⣾⡿⠓⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠘⣿⣿⠿⠁⠘⠇⠀⠘⢁⣴⣏⣉⣹⣿⡍⠒⡄⠀⠀⣔⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠉⠛⠛⠩⣤⡈⠀⠀⠀⠀⠠⠀⢼⣿⡿⣿⢿⡟⠓⠀⠀⠀⢠⣾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢏⠑⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢦⠀⢸⡀⠀⠀⠀⣞⢀⠀⢠⠀⠀⠀⠀⠀⠀⠴⠋⣄⡀
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠒⢶⣿⣿⣶⣄⣁⠀⠀⢠⣟⣿⡀⠈⣶⡄⠀⣄⣰⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠉⠀⠉⣤⠀⣾⡀⠙⠉⣻⣿⣧⢤⢶⣾⣿⣿⣿⣶⣿⣶⡏⣾⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠸⡿⠙⠙⣻⣷⠿⣿⣿⣿⣿⣆⢹⣿⣿⠛⠙⠋⢸⠛⠏
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠂⠹⢿⡿⢯⣾⣿⡿⠿⣿⢿⠗⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠀⠀⢽⣶⡄⠀⠀⠃⠾⠏⠻⠄⠱⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠰⠇⡛⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣰⣿⠀⠀⣿⣄⡀⢸⡀⠀⠀⠱⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀
⠀⠀⠀⣀⣀⣀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⣀⣄⠀⠑
⠀⠀⠊⠉⠉⠉⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣀⣀⣐⣲⠷⣶⠈⠀⠀⠩⡍⠁⠀⠀⡀⠀⡄⠀⠀⠀⠰⢶⠋⠀⠀⢰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⡀⠀⠀⡇⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠘⡆⠉⠀⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⣤⠀⠐⡀⠀⡀⣿⣦⠀⡤⢀⡇
⠀⠀⠀⠀⢳⡀⡅⠀⠘⣇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⠀⢸⡀⠇⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⢄⡁⠀⠀⠙⡧⠀⠁⠘
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠉⠀⢸⣇⢠⠀⠀⠀⠀⠀⠀⢌⠠⢸⡀⣼⣷⢀⡆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠰⠾⣤⣀⣄⡀⢀⡠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠲⠹⣧⠙⣋⠟⠁⠀⢀⣴⣿⣿⣿⡶⣤⣀⠀⠀⠀⠈⣶⡾⣦⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠈⣈⡁⠀⠀⠘⠃⠀⠀⠀⡀⠀⠒⠟⠚⠛⠀⢠⣾⣿⡏⢻⡿⠋⠓⢈⡉⠳⠀⠀⠀⠈⠀⠟⠃
⠀⠀⢀⡀⠀⠀⠀⠀⢠⠒⠈⣰⠉⠄⠀⠀⠀⢀⣠⣀⣄⣠⣄⠀⠀⠀⢀⠾⣏⣻⣠⠈⠥⣼⡿⠿⠛⠉⠀⠀⠀⠐⠀⠛
⠀⠀⠀⠙⡻⠶⠀⠀⠈⠀⠿⢻⠇⠂⠀⠀⠠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠿⠿⠈⢠⠉
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠄⠸⠆⠀⠀⠀⠀⠈⡀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣠⣤⣶⠂⢹⣿⠖
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠞⠃⠀⠀⠀⠀⠠⣤⣀⠀⠀⢨⠀⠈⠈⠁⠉⠈⠁⢸⣅
⠀⢀⠀⢀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⣁⠸⣴⢈⡀⠀⠀⢠⡄⠘⠄⠰⡿⠉⢿⠶⠄
⠀⠈⠉⢻⠀⠀⠀⠓⠶⠤⠤⢴⠶⠖⠒⠂⠁⠀⠀⠀⠠⣤⢴⣤⣤⡀⠀⠀⠀⢩⡍⠿⠤⠀⠀⠀⠈⠀⠀⠀⠀⡇⢰⡆⠀⢀
⠀⠀⠀⢸⡇⠀⠀⡀⠀⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡆⠐⠏⡺⠉⠋⠉⠀⠀⠀⠀⠃⠒⠚⠁⠘⠁⣴⣧⡀⠀⠀⣧⢸⡄⢆⠛
⠀⠀⠀⠸⠀⠀⠀⢠⠌⠀⠀⡁⠀⠀⢀⠀⠀⠀⠄⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠧⠀⣦⠸⠟⠀⠺⠇
⠀⠀⠀⠀⠀⠈⠁⣀⣱⢀⣠⣴⣶⣦⣼⠖⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠤⣤⡀⠀⢀⣀⣀⠀⠀⠀⠀⠘⠁⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠻⠟⠻⠚⠛⠉⠉⠀⠉⠀⠰⠀⠁⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠘⡇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⢀⣀⠀⠤⢀⡁⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⡟⣶⠂⠀⠄
⠀⠀⢠⡀⠠⠄⣤⡤⠤⢤⣄⣀⡀⠀⠀⠀⠀⠁⢸⠀⠀⠀⣷⢄⠀⠀⢹⡀⠀⢸⠀⠀⠇⢀⠀⢀⠀⠀⠀⠀⠀⡇⠀⠰⠇
⠀⠀⠀⠀⠐⠒⠸⡧⠀⡴⠿⠏⠳⠦⠿⠆⠀⡆⠨⠀⡇⠀⣿⣾⠀⠀⢸⡇⠀⠚⠀⠀⢀⡏⠀⠀⢄⠀⠀⠀⠀⡇⠀⠀⠐
⠀⠀⠀⢀⣠⠤⢼⡖⠀⣷⡀⠀⠀⠀⠀⠀⢀⣃⠀⠁⠀⠀⠙⠃⠀⠀⣼⣅⠀⠀⢀⠀⢨⣤⡖⠩⠂⠀⠀⠀⢀⠇⠀⠀⠆
⠀⠲⠚⠛⢟⠀⢘⡇⠀⡏⠀⠀⠀⡠⠔⠺⡛⣿⢷⣄⡀⠀⠄⠀⠀⠀⠈⠉⠁⠘⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠠⠀⠀⠀⠴⣶⣦⣀⠞⡈⠷⡨⠆⠀⠀⠀⠀⢤⢶⣄⠀⠀⣀⣤⣠⡤⣶⠞⠃⠀⣀⣾⠀⠀⡄
⠀⠀⣀⣀⡄⠀⢰⠀⡀⠀⣦⡄⠀⠀⠀⠈⠉⠉⡿⠟⠛⠛⠀⠀⠀⠀⠀⠠⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣇⡀⣸
⠀⠀⠙⠋⠈⠀⠉⠀⠃⠀⠉⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣤⣤⠀⠘⠶⠶⣶⠞⠁⠀⠀⠀⠠⡶⠶⠂⠀⠀⠁⢀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠄⠤⡀⠀⠹⠗⢀⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣄⣤⣠⣤⡆
⠀⠀⠠⠒⡒⠠⢜⠻⠈⠶⠠⡔⡆⠁⢀⣀⡀⠈⠛⠀⠘⠁⠐⢶⡄⠀⣰⡾⠋⠁⠀⠀⠀⣄⠀⠀⠀⣤⣤⠄
⠀⠀⠀⢰⡷⢶⠀⠐⠀⠀⠀⠀⠁⠀⢀⣶⠈⠢⠀⠀⠀⠀⣶⠀⠀⠀⣿⡇⠀⢀⡄⠀⢀⡝⠓⠂⠀⠀⢂⠀⢀⡀
⠀⠀⠀⠀⠯⠀⠔⠃⠔⠚⠛⠁⠛⠒⠸⠛⠏⠀⠀⠀⠀⠀⠀⠀⣤⠀⣿⠁⣸⡷⠀⠀⠸⠃⠀⠀⠀⠀⠈⣿⣾⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉

⡀⠙⠶⣄⡀⢠⡤⠈⢳⠀⠀⠸⣇⠘⢦⠴⣄⡤⣿⠀⢀⡄⠀⡀⢀⠺⣠⠖⠚⠉⠈⠙⢦⡀⠦⠀⣀⡤⢶⡎⠀⡆⠀⠒
⡃⠀⠀⣀⡽⠓⣦⠀⢸⡆⣤⡴⠏⠶⢄⠀⠀⣀⡼⠞⠉⠉⠛⠺⢧⡀⠇⠀⠀⠀⠉⠀⡀⠑⡄⠚⠁⠀⣉⠀⠀⠎⠀⠠⠄
⠛⢀⡾⠉⢀⣀⡸⠘⢫⠄⣀⡖⠀⣠⠼⠶⢦⡟⠁⠀⠚⠀⠠⣆⡤⠖⠉⠉⠙⠢⣄⠈⢁⡄⠥⠤⠠⣄⡀⢩⠶⡀⠀⠀⢀
⣀⡼⣣⡖⠛⠉⠙⠲⠠⣄⠴⢄⡾⠁⠀⡆⠈⢧⡀⠹⠤⠦⠀⠚⠀⢀⣀⣉⡀⣀⠈⢱⡉⠀⠀⠠⠄⠈⢣⡐⠲⠕⡶⠊⢍
⣯⣵⠋⠀⠰⠄⠀⡀⠀⢸⠂⢸⠀⠀⢄⡀⡀⢈⣱⠿⠿⠿⠿⠿⠿⠛⠛⠛⢻⣶⣶⡄⡇⠠⠄⠀⠀⠄⠀⢑⠀⢀⠈⠳⡄
⠼⡇⠰⢦⠀⠀⡄⡷⠀⢺⡀⢀⡱⠦⣈⠉⢨⣼⠛⠀⠀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣿⡇⠀⠀⠂⠀⠀⢀⡼⠀⠀⠀⢀⡞
⣾⢿⡄⠀⠠⠴⠗⠀⢀⠀⢳⡌⠀⡦⠀⢸⣿⠁⠀⠀⢸⣧⠀⠀⠀⢻⡇⠀⠀⠀⠀⣿⡇⠀⠉⠙⠢⣄⡀⠀⠀⠑⢤⡨
⣿⡀⠻⢦⣤⡴⠖⠛⣹⠋⣯⠁⢠⡀⢤⣾⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣶⠈⠀⠠⠀⠁⠄⠂⠒⠀⠉⡀
⠘⠷⠶⣤⢦⡦⣤⡀⢾⡁⣙⢦⣤⣈⢸⣿⠀⠀⠀⢠⣦⠀⠀⣶⣆⣀⣰⠿⠀⠀⠀⠀⢸⣿⢠⠀⠀⠀⠠⠃⠀⢀⣀⠀⠱
⠷⢦⣤⣤⣦⣄⡈⢯⢙⡜⠃⠀⠴⠉⠸⣿⣀⡀⠀⠀⠘⠿⠟⠃⠘⠛⠛⠀⢀⣀⣰⠶⠿⠩⠌⢀⡀⡀⠔⡀⠀⢀⠀⠉⠑
⠷⣤⡶⠾⣋⣙⣻⣯⠸⡀⠀⢆⡀⡀⠀⠀⠛⢳⣶⣶⣀⣀⡀⠀⠀⠀⠀⣤⡟⠛⠃⠀⠆⠀⠀⠠⠀⠀⢆⠈⣂
⠀⠀⠙⢶⡍⡉⠀⢠⡀⣙⢦⡀⠀⠈⠉⣀⣀⣰⠀⣀⠛⠛⢳⣆⠀⠀⣀⡿⠇⡈⠑⠜⠀⠡⠀⠀⠀⠄⠀⠃⠐⠀⠐
⠀⠋⠀⠀⣻⠀⠀⡀⠴⣃⡀⠗⢀⡴⠋⣭⠀⠰⠄⢸⡈⠁⡘⠛⣤⠀⣿⡇⣤⣭⣂⣀⡀⠜⠁⠀⠀⠀⠀⠈⠠⠀⠀⠄⢒
⠒⠀⣀⣼⣯⡒⣺⠗⠉⡉⠉⠛⠲⢤⠀⠏⣠⠤⠒⠓⠶⢤⣈⡠⠄⠻⠃⠀⢠⠀⠀⡀⠈⠒⡄⠢⢠⢀⡄
⣦⠞⠉⡀⢀⡟⠁⠀⠀⠃⠀⠴⠀⠀⢹⠄⠉⠀⠰⠂⠀⡄⠈⠳⣄⠾⠀⠀⠈⠀⠀⠁⠀⠀⠇⠀⢀⡄⡀⠈⢣⠀⠀⠀⠂
⠉⠀⠀⠁⠈⢧⡀⠘⠦⢶⣀⡀⠀⠀⢞⠀⠀⢲⣀⡄⠀⠁⠀⢀⡏⠰⣄⡀⠑⠊⠲⠈⠀⢀⡸⠀⠀⠀⢀⠀⠰⠀⠐⠀⢀
⣄⠈⠳⠖⠣⠬⠳⣆⠀⠀⠁⣀⣠⣤⠟⠘⢧⡀⠁⠘⠋⠁⠀⢈⡇⠀⠢⠝⢂⡀⠀⣠⠛⡉⠀⠀⢠⠀⠈⠀⠀⢀⣀⡤⠞

⠀⠀⠀⠀⠀⣀⣀⣤⣄⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠤⣤⢤⣤⣤⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀
⠀⠀⠀⠐⠒⠾⢿⡁⠀⠀⢀⡿⠟⠁⠉⠀⠀⠀⠀⠀⠈⠐⢶⣁⠀⠀⠀⠀⢀⠽⠟⠀⠀⠀⠀⠀⠀⠀⠹⠶⣧⡀⠀⠀⢈⡹⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠚⠁⠈⠁






⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠁
⠀⠀⠀⠀⠀⠰⣦⠀⠉⠙⠷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⣁⠀⠀⠀⣠⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠱⠂⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢍⠁⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠴⠃⠀⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡶⠦⠾⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠙⣄⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⢠⡤⠠⣄⡜⠋⠹⡀⠀⠀⢠⡆⢀⠦⣄⣀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⡀⠀⠀⠀⣰⢿
⠀⠀⠀⠀⠀⠉⠀⠀⠀⢇⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⢸⠷⡄⠀⠀⢀⡇⠀⢳⡀⠀⣰⠃⢸⡀⠀⠀⠀⣀⠀⠀⠀⢀⡖⠲⡆
⢀⡠⠊⠁⠀⠀⠀⠀⠁⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠙⢦⡀⡼⠀⠀⠀⠻⡴⠃⠀⠈⡇⠀⣠⠞⡻⠰⡖⠂⠋⠀⠀⠻⢠⣤
⠉⠒⠠⣤⠀⠀⢀⢀⣀⡀⠠⠖⠋⠀⠀⠀⠀⠀⣟⠒⠦⣄⡀⢸⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⡇⠀⢈⠆⠀⠀⠀⠂⣾
⠀⠀⠀⢸⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⣄⣀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⣸⣤⣴⡷⠤⡤⠟
⠀⠀⠀⠀⣧⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠃⠀⢈⠻⡀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠈⣹⠛⢦⡀⠀⠀⠀⠘⠛⠉⠀⢠⠏
⠀⠀⠀⠀⠀⠀⡴⠤⣄⠀⠀⠀⢰⡖⠒⠛⠳⡆⢠⡤⠐⠋⠀⠀⡠⢶⣉⠁⠀⠀⠀⢀⠔⠁⠀⠀⠈⠲⢄⠀⠀⠀⠀⣴⣃
⠀⠀⠀⠀⠀⡜⠀⠀⠽⠦⢤⣤⡀⠙⠢⣄⠀⠛⠚⠁⠀⠀⠀⡜⠁⠀⠈⠉⠒⠤⣤⠋⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠉⠉⠒⠒⢤⣤
⠀⠀⢶⡐⠈⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣈⣳⡄⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣦⡀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠈⡦⠀⠀⢀⡀⠀⠀⠸⣞⠉⠉⠉⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⡸⠁⠰⣎⠁⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⠓⣻
⠀⠀⠀⠀⣇⣀⡠⠞⠉⠉⠘⠚⠉⠑⢤⡀⠀⠀⠀⢀⠇⠒⠀⠀⠤⠤⠀⣀⠀⠀⠀⣃⠀⠠⠔⠊⠁⠀⠀⠀⠀⠰⠁⠀⠀⠈⠉⠲⢄⠀⠀⠀⠀⠛⠢⡄⢀⣀⣨
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣠⠶⠒⠒⠋⠁⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⣀⣉⡶⠀⠀⠀⠀⣧⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠚⢛⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢠⡴⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠎⠋⠀⣠⠔⠉⢇⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠙⢆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢀⡰⢻⣄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣀⡞⠉⠳⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠙⠉⠓⢄⠑⠄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⡶⠦⠿⢦⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⣀⣀⠀⠉⠂⢵⣼⡆⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⣄⠀⢀⡞⠀⠀⠀⠀⠀⠀⠙⡄⠀⣸⣠⡟⠲⠛⠁⠀⠀⠀⠀⡴⠋⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠁⢈⠉⠉⠉⠁⠀⢠⣦⠀⠀⠀⠀⠀⠀⠑⡄⠀⣷⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠲⢍⡁⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⠉⣇⠀⠀⢀⡴⠋⠸⡆⠀⠀⢀⣶⡀⠀⠈⢦⣿⣮⣀⠀⠀⠀⢸⡟⠒⠦⢤⣀⠹⡄⢀⢀⠀⠀⠀⠀⠀⠀⢸⣀⡶⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⣰⠏⠀⠀⠀⢳⡀⢀⡞⠀⢳⡀⠀⣼⠋⠀⠈⠛⠲⢤⣘⡆⠀⠀⠀⠈⠙⢋⠏⠈⢳⡀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⠳⠟⠀⠀⠀⢳⡀⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠠⠤⠄⠚⠀⠀⠀⠛⠒⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠙⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⢠⣀⠀⢁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠊⠀⠀⠀⠀⠀⢉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⣀⡔⠁⠉⠳⠾⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⠀⠀⠀⠀⠀⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠟⠒⠤⠦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⢿⣾⠿⡇⠀⢠⣄⠀⢠⢀⣴⣤⡶⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠿⠛⠉⠁⠘⡄⠀⠁⠁⠈⠙⠛⠈⡎⠉⢘⠋⣤⠀⢰⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⡄⠁⠀⠀⡈⠈⠀⠘⣿⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⡄⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⢿⣇⢠⣷⣏⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣿⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⠻⠿⣿⡳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⣹⡄⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠤⠆⠶⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠰⠀⠀⠨⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⠀⠏⠀⡀⠀⠀⠀⠀⠀⠀⢀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⢰⢧⢰⡄
⠀⠀⠀⠀⠀⠀⢰⣆⡆⠘⠃⢠⠀⠀⠿⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠘⣾⠉
⠀⠀⠀⠀⠀⠀⢠⣿⣅⠠⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠈⠣⠄⠀⠀⠀⠀⠀⡆⡆⠀⣿⣶⡄
⠀⠀⠀⠀⠀⢰⣾⣿⠈⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠐⠆⠀⠀⣀⣤⣾⣿⣿⣿⠟⠛⠃⡀⣲⠀⢳⠰⢀⣿⣿⣿
⠀⠀⠀⠀⠀⣼⡿⡏⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⣶⣦⠶⠆⠀⠀⣀⡀⣀⣀⣤⣤⡶⣶⣾⣿⣟⡻⢯⣖⠯⠙⠀⠀⠀⠀⡠⠁⠀⢸⡆⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣷⡇⠀⡁⠀⠀⠀⠀⠀⠀⢀⣴⡃⠛⢀⣠⠶⣿⣿⣭⡙⢄⣸⣼⣷⣿⣿⣿⡥⠚⣉⠅⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⣄⠀⢻⣿⣿⠇
⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣴⣖⠛⣉⣠⣽⣿⣿⣿⣿⣿⣿⣿⠿⠛⠰⣶⣿⣿⣿⡿⣶⣾⠶⠁⠀⠀⠀⠀⠀⠸⢰⣼⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⡀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋⠛⠋⠉⠵⠒⢫⡿⢛⣁⣤⣀⣀⣠⣾⣿⣿⣿⠿⠛⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠘⣿⣿⡄⠛⠀⠀⠀⠸⢿⣿⣿⣿⣾⣦⠄⠀⢀⢀⢔⣾⣷⣿⣿⣿⠛⠹⠏⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠒⠒⡟⠋⠉⠉⠉⠉⢻⣡⠀⠀⠀⠀⠠⢀⠂⠀⠄⠀⠀⢀⠀⠀⡀⠀⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣀⢀⠀⠞⠄⠀⠀⠀⠀⠀⠀⠆⠀⠈⠄⠀⠀⡀⠀⠘⡟⠀⠀⠀⠀⠀⣈⠣⠠⠇⠀⠀⠈⠀⠂⠀⠀⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡉⠳⣄⠈⠐⢢⡀⠠⡀⠀⠀⠀⠀⠒⠂⠀⠀⡘⠀⡷⠀⠀⠄⣰⣿⠉⠀⠀⠀⠀⠀⠀⢄⣡⠞⣵⡏⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣴⠄⠀⢢⡀⠈⠀⠹⣰⣂⠀⠀⠀⠀⠀⠘⡇⡀⣿⡆⠘⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⢠⣛⡉⠰⣤⠀⣿⣿⣖⣀⣤⣤⠀⠂⡁⠀⠘⣿⠆⠀⠹⣿⠿⣷⣶⣦⣀⣀⣀⡀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⣹⣴⣦⣿⡞⢿⣿⣿⠟⢻⣿⣟⣴⠁⠀⢀⡿⠂⠠⡅⠠⣰⡟⠻⣧⠋⢉⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢾⣿⣽⣷⣯⠛⠿⣾⡁⠀⢤⣿⣦⣄⢈⡤⠔⡑⠀⢠⣿⠀⢲⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢇⣃⡀⢠⣾⣿⣿⣿⣿⣏⣷⡆⣰⣶⣿⡿⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠈⠁⠀⠉⠙⠛⠃⠀⠀⠑⠠⠀⡐⠉⠁⠀⠀⠡⡀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠈⠋⠐⠀⠐⠆⠘⠂⠒⠒⠀⠀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⡄
⠀⠀⠈⢢⠀⡇⠈⠚⠉⠁⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠀⠀⠀⠀⢀⢠⡀⠀⠀⡇⠰⠷⠒⠀⠲⠲⡇⠐⡀⡇
⠀⠀⠀⢦⣃⠈⠁⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⢆⠀⠀⠀⡀⠄⡵⠈⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢉⡥⣍⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠢⣈⡡⠚⡃⡅⠀⠀⠇⠀⠄
⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁⠈⢆⢀⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⡀⢸⠀⠀⠀⠀⠀⠀⡠⢢⣉⡀⣸⠁⠀⠐⢀⣤⡀⠐⣤
⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⡀⠀⢀⡠⠸⠈⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸⠀⠀⠀⠀⡠⠊⠰⢿⣿⣿⣿⡿⠀⠄⠀⡸⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢘⠀⠁⠀⣀⣤⢎⡶⣀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⢄⣂⠘⡛⣛⣟⡅⠀⠀⢈⠃⠀⠒⠒⠂⠠
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠤⣀⡠⢴⡁⠀⣻⣧⡢⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠊⢙⠀⠀⠀⠱⢼⡄⠤⠊⠀⠁⠐⠲⠂⠂
⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡀⠀⠀⢸⣷⣾⠿⢿⠤⠴⠖⠡⠀⠀⠀⠀⠀⠀⡤⠤⠖⠃⠀⠀⠀⠀⠀⠀⠘⠀⡀⣀⡀⣀⠃⠀⠀⠰⠆⠰⠶
⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠃⠀⡇⡄⠈⠙⢇⡞⠓⠀⠠⡄⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢠⠀⠀⠀⠠⡤⠤
⢇⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠃⠈⢀⢡⡥⢤⠎⠀⣥⢀⣰⡒⣬⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣄⠀⠀⠀⠀⠀⠀⠀⢰⣄⡃⢈⠀⠀⠀⠠⡤⠔⠤⠠⢤⡄
⠀⠈⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⠺⠤⠤⠄⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⢌⠁⠀⠀⣀⡠
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢠⣄⣂⣉⡃⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠈⠉⠀⠀⠀⠖⠀⠀⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠠⠤⡈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠑⠢⠀⠠⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠤⠴⠶⣤⡄⠀⠤⠞⠛⠂⠀⠀⡇⠀⠠⠐⠈⠉⠀⠀⠉⠉⠀⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆⠀⠠⠒⠁⠂⠀⠀⠀⠀⠐⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡤⠀⢠⠀⠀⠀⠠⠀⠀⠀⠀⠰⠀⠀⠑⠀⠀⠀⠀⠤⠒⠒⠒⠐⠒⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⢀⡀⠲⠶⠦⠦⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⡀⠀⠀⠀⠀⠉⢩⡉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠈⢳⠠⢄⣀⣀⡠⠓⡀⠉⢓⠲⠤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠀⠀⠐⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⡩⠄⠀⠀⢀⡴⢏⡨⠁⠀⠀⠀⠊⢄⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣃⠀⠀⠀⠀⠀⠀⢠⡮⣕⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠋⠀⠀⠀⠀⠀⢰⠗⠥⠀⠀⠀⢈⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⣀⠀⠀⠀⠀⠀⢀⠢⢀⠄⠀⠀⠀⠀⠀⠐⢠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠑⠀⡀⢀⠀⠀⠀⠀⠀⢠⠀⢃⠀⠂⠀⠀⠈⢀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⠀⠀⠄⠀⡀⠀⠀⣔⢂⣨⠜⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠦⡀⠀⠈⠲⠐⠤⠊⠁⠀⡀⢀⠀⠀⠀⠀⠀⠴⢆⣄⠉⠸⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠆⠌⠀⠀⠔⠁⠀⢈⠦⡀⠚⠋⢄⠀⠀⠀⠄⠠⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠄⠄⠀⡠⠀⠀⡀⠐⠁⠀⢀⠰⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠈⠀⠀⠐⠈⠀⠄⠀⠈⠁
⠀⠀⠀⢠⠂⠔⡡⠀⡀⠄⡂⠠
⠀⠀⢠⣏⣀⡨⠔
⠀⠀⠚⠋⠁

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⣦⢄⡀
⣤⣦⣤⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⣿⣿⡄⠀⠀⠀⠀⠀⠀⣤⣷⣮⣶⣤⣄
⣿⣿⣿⣿⣿⣿⣿⣷⣱⠮⠀⠀⠀⣠⣀⢠⣄⣨⣵⣤⣶⣶⣴⡏⠸⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠉⠻⢄⡂
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠒⠛⠀⠐⣤⣾⣷⣿⣿⣿⣿⣿⣿⣧⣤⣴⡿⡿⣿⡟⡛⠀⠀⠀⠀⣄⡀⠀⠹⡕⣄⠀⠀⠈
⠛⠛⢛⠻⣽⣿⣿⣿⡥⠤⢤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣐⣿⣾⡄⠀⢀⣴⣾⡇⠀⠀⠀⠘⣎⢦⣀
⠀⠀⠀⠙⣿⣿⣿⣿⠀⠀⣀⡀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⡟⠻⢧⣄⣾⣿⣿⣧⠀⠀⠀⠀⠈⢧⡙⣄
⠀⠀⠀⠾⣿⣿⣿⣿⣇⠀⢹⣿⣿⣿⣿⠿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡅⣴⠃⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢳⡜
⠀⠀⠀⠀⠀⠘⠛⠛⢻⡀⢸⡟⠿⠯⣿⣶⣤⡈⢀⣠⣴⣿⣿⣿⣿⡿⣿⡿⠿⠟⢻⠛⣿⠒⠀⢠⣾⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢷⠀⣿⠀⠀⠣⣄⣈⣷⣎⣁⣀⣹⡿⠁⢂⣴⣿⡇⠀⠀⠀⡠⠁⠀⣠⠇
⠀⢀⡀⠀⠀⠀⠀⠈⠈⠈⢧⠘⣇⠀⠀⠀⠤⣄⣀⣭⡽⣱⠟⠙⣿⣿⣿⡇⠀⢀⠔⠁⣠⠞⠁
⣿⣯⣤⣤⣤⣶⣶⣇⡡⣸⠃⢀⣿⣦⢀⣀⢀⣙⣤⣤⣾⢋⠀⠀⣿⣿⣿⣇⡠⠂⣠⠞⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢶⣼⢀⣜⠿⣡⣿⣾⣌⣹⠏⣸⣿⣿⣿⣿⠿⡏⠀⠈⢻⣴⠄⠀⣀⣷⢰⠐⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣯⢹⣿⣇⣴⣿⣵⡿⣏⣩⣥⣔⠒⢧⣠⣤⠾⣿⣠⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣭⣿⣟⠛⢡⣿⣿⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⠟⣴⣾⣿⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⠏⠿⢳⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣅⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣤⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⠀⠁⠀⢻

⠀⠀⠀⠀⠀⠀⢀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡴⠚⠓⣶⡼⢿⣾⣷⣾⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣶⠏⠀⠀⠀⡰⣿⠃⢋⣴⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⣠⠋⠀⠀⢺⡂⠁⠀⠐⠛⠛⠛⠿⠛⠶⠇⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢽⠀⠀⡁⠀⠠⢔⣾⡅⠀⢀⣻⣛⣻⡟⡟⢛⡟⠉⠉⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢀⡾⠏⠀⡇⠀⢀⡼⣻⢸⣷⣦⣽⣏⢙⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⣇⠐⠡⣰⡇⠻⠛⣧⡈⣻⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠀⢀⠀⠀⠀⢀⣠⣤⣆⣀⣨⣿⣿⣧⠶⢻⣿⣿⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣗⠲⢶⡿⠟⠛⢉⣽⠟⠉⠁⢈⣿⡿⠛⣧⣠⣼⡿⠿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠏⠋⠌⣿⠂⢤⣾⡟⠊⠁⢉⣿⢻⡗⢂⡾⣻⡟⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢀⢀⣁⣈⣆⠔⠋⠀⠀⢠⣿⣿⣿⣴⠪⣭⣬⡙⢦⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢚⣿⣿⣏⣀⠀⣠⣟⣯⡿⠋⠁⢚⣃⣨⣤⣤⣤⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣥⣽⣿⣿⣿⡿⠿⠿⠿⠿⠿⠻⠛⠛⠋⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣶⣾⣷⣷⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣴⣶⣤⣤⣦⣤⣀⢀⡀⢀⣄⠿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣭⡟⠻⠟⠻⠿⡿⠻⠿⠿⡿⣿⣿⣿⠿⣿⣿⣿⡏⠉⠙⠙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠾⠟⢶⣴⡶⣮⣥⣽⠏⠹⠓⠒⢻⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠭⠉⠉⠿⠿⠀⠀⠀⠀⢸⣿⣯⣻⣿⣿⣿⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠘⠒⢺⣿⣿⣿⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢻⣥⣀⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠘⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⣀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣥⣄⣀⡀⠀⣶⢖⣦⣶⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢪⡙⣿⣿⣿⡆⠀⣼⡟⠀⠀⠔⢾⣿⣷⡆⣤⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠱⣾⣿⠄⠀⠀⠀⠀⠀⢀⣠⣿⣼⣿⡀⣄⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠄⠐⢻⣿⠃⠸⠟⠂⠀⠀⠙⠿⠿⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣤⣤⣄⣴⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣴⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣔⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠻⠿⠿⣷⢛⣿⣿⣿⣿⡟⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣟⠟⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠿⠉⠈⠉⠁⠼⠛⠛⣿⣿⣿⣿⣿⠿⣿⡿⢿⣟⡿⠿⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠳⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠚⠉⠉⠉⠉⠀⠈⠉⠉⠙⠛⠛⠛⠃⠈⠉⠓⠈

⠀⠀⣤⣤⣀⠀⣀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⢀⣀⣀⡠⠀⢤⣤⣤
⠀⠀⢹⣿⠋⠀⠹⣿⡏⠀⠹⣿⡯⠉⢽⣿⡷⠀⠲⣿⡶⠂⠰⣾⡶⠂⠰⣟⡶⠃⠈⢻⡿⠏⠈⢻⡿⠇⠀⠛⣿⠇⠀⠐⢿⠏
⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⠛⠀⠀⠈⠛⠀⠀⠀⠻⠃⠀⠀⠸⠃⠀⠀⠸⠇⠀⠀⠈⠏⠀⠀⠀⠛⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⡀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣀⡴⠄
⠀⠀⠀⠀⢛⣼⡺⠀⠀⠀⠀⠀⣻⣿⣿⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣿⣿⡟⠁⠀⠀⠀⠀⢠⣆⣀⠀⢿⣿⡘
⠀⠀⠀⠀⠀⠉⠁⠐⣶⣾⣇⠀⠈⠀⠉⠀⣾⣿⣿⠀⠀⠀⠀⠀⢴⣿⣿⡃⠈⠁⠈⠁⠀⠀⣀⠀⠘⢻⣿⣿
⠀⠀⠰⠖⠀⠀⠀⠰⠿⢿⡏⠁⠀⠀⠀⠀⠈⠉⠉⠀⢠⣾⣷⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⡞⡹⠇⠀⠈⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠘⠿⠾⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠒⠀⠀⠀⠀⠀⢀⠀⠁⠁
⠀⠀⠀⢀⡾⠿⠞⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣦⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠖⠄⠉⠃
⠀⠀⠀⣿⣧⠀⠀⢈⣿⡇⠀⠀⣀⣀⡀⠀⠀⡀⠀⡀⠀⣀⡀⢘⢋⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⡀⢀⡀⣀
⠀⠀⠀⠻⣿⠀⠔⢺⡿⠃⠀⠨⢛⣧⣧⠀⠀⡟⠶⢻⠀⡧⠄⢸⢴⠃⣯⢼⠁⢱⠎⠀⠀⠈⠁⠁⠀⠀⢛⠋⣉⣡⣟⡃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠈⢀⠃⠃⠈⠀⡁⠁⠈⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠊⠀⠀⠀⠀⠙⠁⠘⠃⠑⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠈⣠⣤⡀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣛⣛⣂⠀⠈
⠀⠀⠀⠰⣾⣷⣾⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡄⠀⠀⠐⠽⠿⠯⠛⠄
⠀⠀⠀⢀⣝⣛⡿⠿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⡖⠀⠀⠠⣄⣀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠉⠹⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠠⡾⠃⠀⠀⠀⡞⠛⠆⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣠⠤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⣀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠤⠄⠀⠀⠠⠤⠀⠄⠀⠀⠀⠀⠿⠿⠶⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣀⡀⢮⣷⡄⠀⠀⠀⠀⢀⡴⣆⣠⣄⣤⣀⡄⠀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢮⠅⠘⠛⠀⠀⠀⠈⠀⠠⢤⣀⠀⢠⣀⣈⡀⠀⠁⠠⠄⢀⣠⡼⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⣿⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠈⠀⠀⠀⠁
⠀⠐⠀⠀⠄⠤⠀⠀⠀⢀⣠⡝⠁⡈⣿⠇⠀⠀⠀⠀⠀⠀⢠⣦⣴⣦⣴⣦⣤⠀⠀⠀⠀⢿⡏⠋⠁⠀⠀⠀⠀⠒⠒⠲⠆⠐
⠀⠀⠀⠀⡀⠀⠙⠿⠛⠛⣿⣇⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠒⠒⠒⠘⠋⠙⠀⠀⠀⠀⠀⠐⠋⠋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠙⠉⠉⠉⠉⠀⡩⡶⠒⠒⠒⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣭⣍⣩⣤⣤⣤⣄⠀⢠⡼⠀⠁⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣛⣛⣹⣿⣯⣭⣭⣾⣏⣉⣁⣤⣈⣐⠒⠂⢤⣀⣀⠀⠈⠑⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠟⢿⣿⣿⣿⣿⣦⣴⡛⠶⣀⡀⡀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣮⡁⣀⡀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣧⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡴⠃
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣮⡏⠉⠓⠉⣿⣆⢤⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠉⠉
⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠫⠏⣉⡟⠂⠀⠑⠀⠂⠀⠚⠉⢳⣆⠈⢳⠀⠀⠀⠀⠐⠀⠀⠀⣀⣤⣤⣴⣶⣦⡤⣤⣀
⠀⠀⢀⣴⣿⣿⣿⡿⠤⠀⠀⠀⠋⠀⠀⠀⠠⠠⢶⣄⡀⡈⢰⣿⢇⢹⡇⠀⠀⠀⠀⠀⣴⣿⣻⣟⠽⠅⠉⠉⠙⠑⠈⠙⠲⣄
⠀⢠⣿⣿⣿⣿⡏⠁⢀⡀⠀⠀⠀⠀⣀⣂⣴⣿⣿⣿⣷⣍⠻⢟⣾⣽⢿⠀⠀⠀⠠⣹⣯⣽⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⣿⣿⣿⣿⣿⣶⣴⣿⣇⠀⣥⣠⢶⠋⠉⠻⣿⣿⣿⣿⣿⣿⠛⠖⠀⠈⠀⠀⢠⣺⣿⣟⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⠅⠂⠀⠀⡀⠀⠹⣿⣿⣿⣟⠑⠃⠀⠀⠀⠀⠀⣷⣯⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣹⣶⠿⠿⠿⠿⣷⢶⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠙⠙⠳⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚
⠀⠀⠻⣿⣿⡷⠿⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡿⠶⣶⣶⣼⣧⣤⣤⡀⠀⢠⠀⠀⠀⠀⢤⡙
⠀⠀⠀⠀⠁⠀⠀⠘⣷⣿⣿⡟⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠐⣰⠀⠀⠀⠀⠀⠀⢀⣷⣿⠟⠻⣇⠀⢈⠀⠉⠘⠢⢆⢠⣤⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣶⣆⢰⣆⡄⡇⠀⢠⣜⣘⣿⣿⣇⠀⠀⠀⠀⡀⢸⣿⣿⠊⠀⠀⠉⠈⠃⢀⠲⢄⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣧⠂⠀⠀⠀⠈⡉⢹⠛⠀⣠⡶⣸⣧⣾⣿⣏⡓⠀⠀⠀⠀⠀⠠⢨⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠸⣿⣟⠀⠀⠀⢰⣿⣹⣆⣴⣟⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠲⠶⠶⠼⣿⣟⣡⣿⣿⣧⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠒⠋⠒⠲⠶⠦⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⢄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣅⣄⣤⣾⡟⢧⠀⠀⠀⣤⣄⠀⠀⡀⠀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⣿⣷⣦⣶⠀⠀⠙⠛⡇⠀⠘⠋⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⠘⠻⢻⣿⠟⠀⢀⡀⠀⣹⣶⡀⠀⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⡀⠋⠀⠀⠀⣸⣿⣤⢀⡀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠖⠀⠀⠀⠀⠀⣀⣀⠀⠋⠀⢀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠙⠣⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⢧⡄⠀⠀⠀⣷⣀⡶⠋⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣤⡈⢦⡀⢠⡀⠀⠠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡚⣷⠀⠀⢰⠃⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⣿⣿⣶⡁⢨⡟⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠓⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠘⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⣧⠇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣷
⠀⠀⠀⠀⠀⠀⠀⣶⠿⠿⠗⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠙⣿⣟⣿⣿⣿⣿⣿⡟⠛⠻⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⡀⠒⠛⣿⣇⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠂⠉⠀⠀⠀⢹⣿⡀⠀⠀⣈⠀⢀⣶⣄⣠⡖⠉⠠⠀⠀⡈⠋⢠⠄⠀⠀⠈⠘⠃⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⡆
⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠂⠀⣹⠀⠀⠹⠿⣿⡄⠀⣆⠠⣾⠀⣄⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢸⣿⣟⣻⣿⢿⣿⣁
⠀⠀⠀⠸⠀⠀⠀⠂⠀⣽⡿⠃⢀⡿⠀⠀⠀⠤⠈⠉⠛⣡⣴⡇⣼⠏⠰⠀⠀⠈⠀⠀⠀⠉⣁⠀⣿⣿⣿⣿⣿⣾⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⣿⠉⠹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢼⠁⠀⠀⠀⠀⠀⣿⣿⡇⢀⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡿⠿⣿⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⡀⠀⢸⣿⣘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣷⡆⢸⣿⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡟⠋⠁⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣤⢴⡰⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣿⣿⣥⣿⣿⣿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢠⣴⣿⣿⣷⡶⠖⢀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢁⠀⠀⣿⣿⡿⠟⠧⠄⠀⠉⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⢿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠛⠉⠉⠁⠀⠀⠈⠓⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⠋⠀⢀⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣠⡤⠚⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠈⠉⠉⠀⠀⠀⠀⢰⡆⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢰⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⣼⣿⣷
⠀⠀⠀⠀⢀⣀⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣤⣀⣤⠄⠀⠈⠻⢿⣦⣀⣀
⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣠⡤⢶⢶⠚⠛⣿⢟⢶⢶⣤⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⡟⡕⡱⢡⢹⡆⠀⢻⠈⢎⢣⢣⡟⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⠦⣍⣓⠳⠧⢾⡇⠀⢸⣚⣚⣯⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠊⠉⠙⢦⠀⠀⠀⠀⠀⠈⠉⠉⢹⠀⠀⣼⠁⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⡟⠀⠀⡏⠀⠀⣀⡤⠖⠋⠉⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⡤⠤⢤⡤⢤⣤⣄⣀⠈⢦⠀⠀⠀⠀⣸⠁⠀⢸⠃⢠⠏⠁⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⣛⡺⢿⠀⢸⠧⠶⠿⣃⠞⠀⠀⠀⢰⠃⠀⢀⡏⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠀⢸⠛⠋⠉⠁⠀⠀⠀⢠⠏⠀⠀⡼⠁⠀⠀⠀⠙⢲⡤⣄⣀⣀⣀⣠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡄⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⠃⠀⠀⠀⠀⢀⡞⠀⡼⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢧⠀⠀⠀⠀⠀⢸⠃⠀⠀⡟⠀⠀⠀⠀⢠⠏⠀⣸⠃⣠⠤⠒⠋⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢹⡀⠈⢧⡀⠀⠀⠀⡟⠀⠀⢸⠃⠀⠀⠀⣰⠋⠀⣰⠃⠐⡇⢶⡶⢤⢄⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠉⠳⢤⡀⢳⡀⠀⠳⡄⠀⢰⡇⠀⠀⢸⠀⠀⠀⢰⠇⠀⣰⠃⠀⠀⠈⠲⢽⠏⢠⣿⣳⡄⢸
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠉⣦⢳⡄⠀⠙⣆⢸⠀⠀⠀⢸⠀⠀⢀⡏⠀⢀⡏⠀⠀⠀⢀⡴⠋⢠⠞⠛⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⣀⣤⠴⠚⠁⠀⢻⡀⠀⠈⢿⠀⠀⠀⢸⡄⠀⢸⠁⠀⢸⠁⠀⣀⠴⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⣄⠀⠀⠀⠀⢳⠀⠀⠸⡆⠀⠀⠈⣧⠀⡿⠀⠀⢸⣤⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠳⠤⠤⣤⡈⣇⠀⠀⢳⠀⠀⠀⠸⡆⡇⠀⠀⢸⡁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⢀⡽⢹⡀⠀⠈⣧⣀⣀⣰⠃⢻⠀⠀⢈⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⢧⣀⣠⠟⠈⠉⠀⠀⠈⠳⠤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⠿⣿⠻⡇⠀⢰⣄⡀⠠⣠⣴⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠟⠋⠁⠀⠈⠆⠀⠁⡀⠀⠙⠃⠀⡆⠀⠈⡁⡆⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠄⠀⠀⢀⣀⠀⠀⢹⣇⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢰⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠿⠄⣾⢿⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠹⡇⠸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠋⠛⠻⣯⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡿⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠛⠀⠀⠐⠒⠄⢀⠀⠀⠀⠀⠀⢠⡍⠀⠀⠁⠀⠀⢠⠀⣠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⠇⠘⠀⢠⡄⠀⠀⠀⠀⠀⢀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⢀⠛⣄⡧
⠀⠀⠀⠀⠀⠀⢿⣤⠇⠈⢀⡆⠀⠘⠃⠀⠀⠀⠀⠀⠛⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⢀⣸⣿⡦⠁⠘⢣⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠁⣤⣠⣬⣎⣁⣄⡀⠀⡀⢸⠐⠀⢸⣿⣷
⠀⠀⠀⠀⠀⣾⣿⡇⠀⠰⠀⢸⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣈⣁⣤⣤⣶⣿⣻⡿⣟⠋⠀⠄⠐⠸⠃⠈⡇⠃⣾⣿⣿⡀
⠀⠀⠀⠀⠀⣿⣛⠃⠐⠆⠀⠀⠀⠀⠀⠀⠀⡿⢻⡏⠉⢀⣤⣦⡤⢶⠖⢺⢻⣟⣿⣿⣿⠗⣩⠮⠗⠉⠀⠀⠀⠀⠐⠁⠀⡀⠁⠀⢾⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⣀⣤⠾⠍⢾⣿⣿⣦⣵⣾⣿⣿⣿⣿⣛⣡⣤⣾⣥⣤⣀⣄⡠⠈⠀⠀⠀⠀⢸⣆⣸⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣧⠀⠆⠀⠀⠀⢠⣾⣿⣿⣿⡿⣢⡶⠾⢟⣛⠿⣻⡿⠟⠛⠋⠉⢀⣼⣿⣿⣿⣿⡽⠿⠋⠁⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⠠⡆⠀⠀⠀⣿⣿⣿⣿⣯⣅⠀⠀⡀⠈⣡⣼⣯⣶⣾⣿⡾⢶⢿⣿⣿⠿⠋⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠷⠤⠌⣶⠽⠟⠻⠿⠿⣿⢀⠀⠀⠉⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⢸⡮⠀⠀⠀⠀⠀⠻⡀⢸⡀⠀⠀⠀⠀⠈⠀⢠⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡕⠲⣄⠈⠂⢀⡀⠀⡀⠀⠀⠀⠀⠤⠄⠀⠁⠰⠀⣇⠀⠀⠀⣠⡔⠛⠀⠀⠀⠀⠀⢀⠐⣠⢦⣽⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⣬⠁⠤⡀⠙⠀⢑⠂⡆⠀⠀⠀⠀⠀⠱⡇⣀⣧⠄⠐⢰⣿⣿⣀⠀⠀⠀⠀⠀⠀⠛⠁⢸⡏⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⡸⠟⠠⣄⠆⣼⣿⡧⠀⣀⣀⠀⠐⠃⠁⠹⣷⡄⠈⢿⡿⣷⣤⣤⡀⠀⠀⠀⠀⢠⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠙⣏⣤⣽⣷⢿⣿⣿⡿⠿⣿⣗⣤⠃⠀⠀⣿⠀⢀⡁⠁⣸⡿⢿⡍⠋⠉⣩⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢤⣿⣟⣿⣾⡛⢿⣶⡂⠉⢍⣿⣆⡀⢘⣡⠔⠂⠀⢠⡟⠀⣾⣷⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣰⡀⠀⣿⣿⣿⣿⣿⡏⣷⡄⢡⣶⣿⡧⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⣼⡇⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠰⣶⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣿⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣷⣀⣀⣻⣀⣀⣀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠟⠻⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⣻⣿⡇⢸⣿⣷⠈⣿⣿⡏⣿⣿⡿⠿⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠻⣿⠃⢸⣿⣿⠀⠻⠟⠁⣿⣿⡧⠀⢸⣿⣿⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠛⠟⠁⠀⠸⣿⡿⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⢭⣍⣉⣉⡀⠀⠀⠀⠀⠀⠀⡄⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⢸⠃⠀⣧⠀⣿
⠀⠀⣀⣠⡤⠴⠶⣶⣶⣿⣿⣿⡿⠿⢿⣿⡿⠿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⠀⠀⣿⣧⠀⣾⣿⣿⣿⣷⡀⢹⣿⣿⣿⡏⠙⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢰⣿⣿⣇⠈⠛⢸⣿⣿⣷⠀⠙⢻⣿⣷⠀⢸⣿⣿⠀⠉⣹⣿⣷⣹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢿⣿⠏⠀⠀⠘⢿⡿⠃⠀⠀⢸⣿⣿⠃⢸⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠋⠀⠀⠻⠿⠟⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠲⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⣿⠏
⠀⠀⠀⠀⠀⢀⠠⠤⠰⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠲⠶⠖⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⣀⣀⣤⣤⣤⣴⡶⠶⠶⠶⠶⠾⠿⠿⠿⠟⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠶⠶⠶⠶⢶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⣿⣷⣶⣶

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣭⣿⣿⣯⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠑⣄
⠀⠀⠀⣠⠴⣶⣦⡂⠀⠀⣴⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⡀⠠⢀⠀⠀⣴⢶⡷⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣭⡽⠛⢿⠀⠈⠁⠀⠘⠀⠀⠀⠀⠁⠄⠀⠢⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⢠⣷⣆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢳⠀⠙⠀⢰⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⣆⠀⠀⢆⠀⠀⡀⣿⣦⡀⢠⡄⠀⡾
⠀⠀⠀⠀⠀⣮⠀⠿⠀⠀⢸⡇⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⠐⡄⠀⠀⣤⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⠈⠂⠀⠓⠻⢿⡁⠘⠃⣘⠃
⠀⠀⠀⠀⠀⢸⡇⠃⠀⠀⠸⡗⠀⠀⣿⠀⠀⠀⠀⡄⠀⠀⠀⢄⠐⠀⢹⡀⠀⢿⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠂⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⡻⡆⠀⠀⠀⠀⠀⠀⣿⣄⣆⠀⠀⡂⠀⠀⠀⠀⠳⠄⠀⣷⠀⣼⣿⣠⡾⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠘⠛⢦⣄⣤⣤⡀⣀⣤⣻⠿⠋⠀⢀⠃⠀⠀⠀⠀⠀⠘⠎⠼⡇⢹⢹⡟⠁⠀⠀⣀⣴⣿⣿⣿⣿⠳⢶⣤⡀⠀⠀⠀⢀⣷⡾⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡉⠀⠀⠀⠘⠛⠀⠀⠀⢀⠀⠀⠒⠾⠟⠛⠛⠂⢀⣴⣿⣿⠏⢹⣿⠹⠞⠓⢂⣉⠛⠆⠀⠀⠀⠀⠀⢸⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢩⡏⠓⡀⠀⠀⠀⠀⡠⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣴⣿⠳⡿⢀⠈⢋⣄⣶⣿⣿⠿⠛⠀⠀⠀⠀⠐⠂⠘⠇
⠀⠀⠀⠻⢶⢶⣤⡀⠀⠀⠘⠠⣶⣿⣎⠀⠆⠀⠀⠀⠀⣉⠉⠙⠛⠋⠙⠁⠀⠀⠀⠶⢧⣼⢷⣾⣿⠭⠇⢙⠋⠁
⠀⠀⠀⠀⢠⠋⠉⠀⠀⠀⠀⠀⠉⢸⡇⠀⠀⠀⠀⠀⠛⠛⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠋⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⠃⠈⠋⠀⠀⠀⠀⠀⠀⠀⣤⡆⣴⢄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣤⣤⢿⣿⣯⠀⢸⡿⠋⠀⠀⠈⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠞⠋⠀⠀⠀⠀⠀⠀⢴⣦⣄⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⣀⠀⠸⣧⠀⡀
⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⠁⠺⣤⡀⢀⠀⠀⠀⢀⣄⠀⠣⠀⢠⡿⠉⢹⣿⣶⡄
⠀⠈⠛⠚⣷⠈⠀⠀⠰⢦⣤⣀⣄⣀⣤⣤⡤⠶⠆⠋⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⠀⠀⠀⠀⠈⢛⡛⢿⣅⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡟⠀⣶
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠁⢸⠇⠀⠀⠀⠀⠀⠀⠈⢠⡀⢠⡾⠃⣿⠿⠿⠗⠀⠤⠀⠀⢈⡁⣀⣤⡴⠂⠰⠀⢀⡤⢀⠀⠀⠀⣇⠀⣿⢀⡄⣶⠄
⠀⠀⠀⠀⣸⠁⠀⢀⠐⠀⢠⠂⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢢⡧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⡀⢻⣄⡿⠈⢣⡄
⠀⠀⠀⠀⠏⠀⠀⣂⠀⢶⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠐⠂⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⢀⣿⠘⠛⣁⠈⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⡧⣤⣤⣴⣷⣶⣶⣾⠗⠀⠀⢠⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠠⢦⣤⡀⠀⢄⣤⣤⡀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠺⡿⠿⠷⠜⠛⠛⠈⠉⠀⠉⠀⠀⡄⠘⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡶⣤⡤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠀⠀⠘⢿⡁⠈⠁⠞⠁⠀⢠⠀⢀⠀⠀⠀⠀⠀⠈⠉⠉⣇⠸⠃⢠⠆
⠀⠀⠀⠣⠄⠈⠐⢼⣿⠛⠋⢛⣿⣶⢶⣀⡠⣄⡀⠀⢀⠀⠘⠄⠠⠀⣿⣷⡀⠀⠀⠈⣧⠀⠀⣽⠀⠀⠈⢀⡞⠀⢰⠇⠀⠀⠀⠀⠀⡏⠀⠀⠘⢀⡀
⠀⠀⠀⠀⠀⠈⠉⠁⢻⠃⠀⡾⠛⠒⠓⠛⠑⠚⠃⠀⢺⡀⢸⢰⠇⠀⢻⣿⡇⠀⠀⠀⣿⠀⠀⠁⠀⠀⠀⡾⠁⢀⡀⢂⡄⠀⠀⠀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⣀⣠⡤⢤⣿⠀⠀⣗⡄⠀⠀⠀⠀⠀⠀⢀⣘⡁⠈⠈⠀⠀⠈⠛⠃⠀⠀⣼⣇⠀⠀⠀⣀⠀⢀⣱⣶⡍⠩⡧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇
⠀⢰⡔⠶⠿⠿⠀⠘⢻⠀⠀⡏⠁⠀⠀⠀⣠⠴⠾⢻⣿⣷⣦⡀⠀⠀⠄⠀⠀⠀⠀⠈⠓⠈⠃⠛⠋⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠘⠃⠀⣿⠀⠀⠇⠀⠀⠀⢈⣀⣄⣀⠀⢹⡏⢹⣿⠳⡀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⢀⡀⢀⣀⣄⣠⡴⠀⠀⢀⣼⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠸⡀⠀⠀⠀⠉⠛⠻⠷⠦⠴⣮⣥⣦⣥⡄⠐⠀⠀⠀⠘⠋⠻⠇⠀⠰⠾⠟⠑⠛⠉⠉⠁⠀⠀⢰⣾⡟⠀⢠⠘⠀⣠
⠀⠀⢰⣦⡔⣆⠘⡐⠇⢠⡀⠀⡽⠇⠀⠀⠀⠤⠀⠀⠀⠐⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠋⠶⠀⡏
⠀⠀⠈⠉⠀⠈⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠘⠀⢀⣶⣶⡆⠀⠈⠛⠚⠿⠿⠋⠀⠀⠀⠀⠀⠲⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠤⠠⡄⠀⠀⠹⠿⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⣠⣀⣤⣀⣤⣶
⠀⠀⠀⡤⠶⠄⢀⣤⠻⡧⢨⣤⣀⡴⢧⠀⠆⠀⢀⠀⠀⠻⠟⠀⠐⠃⠲⢴⣤⠀⠀⢀⣴⡾⠛⠃⠀⠀⠀⠀⡄⠀⠀⠀⣀⣀⣠⠀⠈⠁⠉⠉⠀⢀
⠀⠀⠀⠀⢸⣅⣤⠀⠃⠈⠀⠀⠀⠉⠘⠀⠀⠉⢭⡓⢄⡀⠀⠀⠀⠀⢀⡀⠈⠁⠀⢸⣿⠀⠀⠀⡀⠀⠀⠸⣷⢤⠄⠀⠉⠹⠉⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠘⣯⠙⠀⣠⠀⣀⣠⣶⣄⣠⣄⣀⣰⡿⣤⠂⡵⠀⠀⠀⠀⠙⠃⠠⠀⠀⢸⡏⠀⣤⡖⠙⠀⠀⣻⠀⠀⠀⠀⠀⠀⢆⣰⢴⡖⢠⡦
⠀⠀⠀⠀⠂⠛⠀⠋⠉⠐⠉⠉⠉⠀⠀⠉⠁⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣻⣇⢰⣿⠏⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣸⠿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠒⠒⠂⣁⣀⣀⣀⣉⡀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⣠⣤⣶⡾⢿⢿⠿⣿⣿⣿⣿⣷⣦⡀⢀⣇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣠⠞⠋⠉⠀⠆⠰⡀⠀⠜⡜⢽⣿⣿⣿⣿⣾⣥⣽⣷⣄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡛⠀⣠⠏⣄⠂⠂⠀⠈⠀⠁⢦⢠⠏⡸⡿⡇⠼⣿⡻⣛⠩⣻⣿⡆⢰
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠀⢰⡷⣤⢬⡲⠄⣈⡃⣹⡀⠞⢧⡼⡧⣿⠁⣜⠹⢻⣷⣮⣮⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⣠⠋⠌⠛⠦⠏⠒⠉⠀⠀⠈⠀⠀⠈⠀⠃⣤⡿⢡⣭⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⣧⣄⡁⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⣴⣶⣿⣏⡁⠁⡍⠑⣆⣀
⠀⠀⠀⠀⠀⠀⢸⡏⢀⣟⣻⣿⣿⣦⣶⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣤⣶⣶⠛⠉⠉⠉⠉⠃⠀⠈⠃⠙⠙⣿⡄
⠀⠀⠀⠀⠀⠀⣼⠇⣸⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠏⠀⠀⣴⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢳⡀
⠀⠀⠀⠀⠀⠀⢻⡆⢻⣿⡛⢿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⣠⣾⣿⣿⣧⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢿⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠾⣍⠻⢦⣟⢿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡻⢯⡾⠅⠒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣝⣿⣿⣿⣿⣿⠟⢿⡿⠛⠓⠀⠀⠀⠤⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡗⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⢶⠞⠋⠀⠈⠛⠟⡛⠋⠁⢀⣾⡦⠠⠄⠀⠀⠀⠀⠈⣀⠀⠀⠀⠈⠻⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⡟
⠀⠀⠀⠀⠀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⢙⣦⢀⣿⢏⣴⣿⣿⣿⣿⣿⣿⣷⣎⢁⠀⠀⠀⠀⠈⠛⠿⢦⣤⣴⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⣶⡿
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣹⣼⣿⢸⣿⣿⣿⣿⡟⠛⢻⣿⣿⣬⣷⡀⠀⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣴⣦⣤⣴⣦⣦⣀⣿⡿⠟⠁
⠀⠀⠀⠀⢾⡇⠐⠄⠀⠀⠀⣀⣠⢞⣉⣻⣎⡟⣿⣸⣿⣿⣿⡝⠀⠀⠀⢹⣿⣿⡏⢘⢦⢈⢃⣰⣿⣿⡿⠟⠹⢿⣿⣿⣿⠇⠀⠈⠉⠉
⠀⠀⠀⠀⢸⣯⣀⣨⠄⣀⣸⠿⡟⠟⠋⠉⢏⠀⠈⠙⠛⢿⣿⡆⠀⠀⠀⠨⡟⢫⠻⣿⣷⣿⣿⣿⣟⡛⠁⠀⠈⠁⠈⠙⠹
⠀⠀⠀⠀⢀⣿⣯⣁⣀⣼⡿⣦⠈⠀⠀⠀⠀⠑⢦⣴⣷⡾⣿⡁⠀⠀⠀⠀⡁⠾⣷⣼⣉⣿⣿⣿⣷⣁⠀⠀⠄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡌⣦⠀⠀⠀⢀⠀⠀⢻⣿⣷⣿⠣⠀⠀⠆⠤⠻⠷⢦⣿⠿⣿⣿⣿⣷⢿⣤
⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⣿⡷⢸⠗⠰⠆⣻⠄⢀⡄⢿⢻⡿⠟⣀⣌⠂⠑⡄⣮⣌⡉⠓⢿⣿⣿⣿⣏⠀⠀⠀⠀⠐
⠀⠀⢠⣴⣴⣀⡱⣟⣷⣶⣾⢿⡞⣧⠳⣾⣇⠠⣋⠃⠮⠉⡞⢋⣴⣇⣯⣑⢸⣷⢳⣧⣴⠀⠀⠙⠓⣥⠰⠰⡖⡰⢠⠆⡀⡀⢄⠀⢀⠀⠀⡀⠀⠀⣠⡤
⠀⢼⣯⣯⣬⠟⣿⡿⢙⣋⢳⣾⠟⣟⢆⠛⡺⣿⣿⣿⠀⡑⠈⣿⠟⣿⣽⣿⣶⣟⣾⣧⢿⢠⣾⢷⣴⣀⣆⡚⠞⢇⣁⡞⠝⠻⢺⡆⠀⡄⣐⠼⣤⣤⠻⠁
⠀⠀⠩⣙⣫⠭⠘⢀⡛⣾⠅⠙⠉⢙⣈⢣⡛⠢⡛⣞⣷⠐⠂⠅⡚⠻⠟⣹⣧⣛⣱⣀⣸⣁⣿⣾⣩⡜⢹⣵⣻⠜⠛⢿⢿⡶⠘⠣⢜⠴⣉⠬⠀⠉
⠀⠀⠀⠀⠉⠁⠐⠂⠀⠁⠈⠀⠀⠙⠀⠀⠁⠂⠄⠹⡏⠛⠛⠛⠙⠼⠿⠛⠿⠲⠷⠎⠉⢩⠉⠉⠌⠉⢤⠘⠦⠌⠀⠀⠾⠀⠀⠀⠊⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠐⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠈⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡠⣄⠘⠀⠠⠊⠀⠈⢐⠄⠀⢀⣤⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢲⣂⡀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⢧⡄⠀⠀⢀⡤⣄⣠⡀⠀⡇⠀⠀⠀⣦⡀⠀⠀⠀⢀⠖⠉⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⢺⡀⠡⣀⠵⣤⣇⡀⣀⣰⠋⡱⠂⠀⢠⠁⢀⠴⠒⠀⠀⠈⠀⡤⠄
⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠑⡄⠃⠙⡄⠀⠀⠙⠟⠀⢀⣁⠠⠐⠑⡀⢸⣶⠔⠈⠁⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⢰⠙⣆⠈⠙⠀⠄⠢⠂⠉⠀⠀⠀⠀⠀⢇⣸⠃⠀⠀⠀⠀⠀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⡄⠀⠘⣆⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⠀⠠⠄⠒⠉⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⢾⠁⠀⠢⠀⡇⠀⠀⠈⠑⡿⠸⠀⠀⠀⠀⠀⠀⣐⠞⠋⡏⠉⠄⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠁⠀⠀⠈⠳⣦⠀⠀⠀⠐⡄⢁⠀⢀⢴⠾⠊⣁⠄⠰⢦⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢀⠀⠀⠀⠀⠈⠓⣤⠄⠀⠈⡌⠋⣡⠆⠒⠲⡕⠢⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡵⠶⢀⡰⣦⣴⠁⠀⠀⢀⠃⡾⠁⠀⠀⠀⠘⠢⡤⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⠀⠀⠹⡁⠀⠀⡠⢂⡜⠁⠀⠀⣀⡀⠀⡰⠁⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢊⣁⣤⣤⣼⠖⠊⢠⡋⠀⠀⠀⠘⣏⠱⣯⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠅⢀⡠⢛⡁⠐⡀⣈⡫⣍⡉⠉⠀⠈⠓⢽⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⠈⣉⠜⡡⠃⠀⠀⢣⡈⠁⠛⢷⡀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠄⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠀⠐⠦
⠀⠀⠀⣶⡤⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⣾⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⣽⢷⡀
⠀⠀⢠⢽⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⡟⠷⡄⠂⠀⠀⠀⠀⠀⠀⠀⣿⠄⣿⠀⠰⠀⠀⠀⠀⠀⠀⠘⠀⣸⡇⣸⡇
⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠁⠉⣠
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠁
⠀⠀⠀⠀⠀⣽⣿⢿⣿⡇⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⠈⣿⣷⠀⠀⠀⠀⠀⠸⢿⣿⣷⣶⡀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⣦
⠀⠀⠀⠀⠀⢿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠉
⠀⠀⠀⠀⠀⠾⠁⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⣤⣀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠛⠋⠀⠀⠀⠠⡄⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠠⠄⢀⠤⠒⠒⠓⠒⠦⣄⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡠⠞⠁⡀⠢⣸⢠⣰⢀⠀⣹⡄⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⣾⣥⣦⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⣆⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣠⣴⣭⣽⣿⣿⢿⣿⣿⣿⣿⡏⢙⣿⣖⣭⡽⠶⠖⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣀⣤⠾⠋⠁⢿⣆⡙⠛⠋⠛⣠⣼⠇⠙⢿⣿⣭⣭⠥⠤⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠉⠉⢛⣿⣷⡀⠈⠛⠧⠖⠖⠺⠕⠋⠀⣠⣶⣟⠉⠉⠀⠀⢀⡄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡦⢠⡾⠟⠋⢙⣿⠞⢦⣄⣀⣠⣴⣤⣶⡂⢋⡹⠛⢷⣄⠀⠀⠘⠇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠉⠀⠀⡠⢿⢯⡀⠀⣼⣿⠋⠀⠉⣉⠿⢯⡀⠀⠀⠉⠂⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⢹⣿⠗⠒⠒⠊⠁⠀⠀⠁⠀⠀⢤⠀⠀⠀⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠀⠐⠲⠀⠀⠀⠀⠸⠙⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠈⠓⠶⠊⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⠤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠉⠉⠙⡻⢻⣿⠛⠛⠚⠛⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠊⢀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⢸⡇⠀⠀⠀⣉⡾⣿⠀⠀⢀⡖⡀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠡⡇⠀⠀⣼⠃⠀⠀⠐⠋⠁⢻⣇⠀⡾⠉⢠⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠁⠀⢠⡟⠀⠀⠀⠀⠀⠀⠊⠻⣆⠃⡏⠈⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣙⣷⡅⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⣦⣤⣶⠶⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⣭⣽⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠉⢱⣿⣿⣿⡿⣄⣴⣶⡖⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣴⠾⣿⢻⣟⢛⣿⣾⠟⢻⣷⣟⣩⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⡀⣀⠑⠾⣍⠈⣶⣾⣿⣾⣿⡟⣵⣢⣶⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⣢⡅⠐⣿⣣⣤⠞⣟⠷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣥⣾⣖⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣄⠀⠀⠘⣿⠁⢸⣿⡤⡴⡆⠈⡇⣿⣿⣿⣿⡿⣿⣶⢮⠙⠡⠷⠈
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⠀⠀⣸⣿⣧⣿⡟⣀⣠⣯⣾⠿⣿⣿⣿⣿⣧⡜⠋⡀⣰⠆⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠐⢻⡿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠟⠁⠸⡦⠚⣋⢤⣿⢿⣿⣿⣿⣿⣟⡿⢛⣿⣿⣿⡛⣟⠒
⠀⠀⠀⢀⠀⠀⠀⠠⠇⢙⢿⣿⣿⣿⣿⡿⠋⢀⣰⠓⣲⢶⣶⣇⣈⢀⣾⣿⣿⣿⣯⡾⢛⡋⢒⡀⠀⠈⠛⠁
⠀⣤⣶⣿⣿⣆⡀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⠴⠛⡷⣤⣌⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠿⠛⠁
⠀⢿⣿⣿⣿⣿⣷⠻⣧⠀⣤⡜⣿⣿⣿⣿⡤⢬⡛⠻⠟⣿⡟⣿⣿⣿⣿⣿⣿⣿⠉⡜⠉⢩⠓⠂⣠⣤⣤⣀⠀⢀⣀
⠀⠈⠻⢿⣿⣿⣷⣄⣿⣾⡏⣰⣿⣿⣿⣿⣗⠛⢨⡿⠷⣬⣥⣤⣿⣿⣿⣿⣿⣿⣶⣿⡶⣾⡂⣼⣿⣿⣿⣿⣶⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⡇⡋⢀⣻⣿⣟⣻⣿⣿⡿⠛⣻⠿⠻⣿⣿⣯⡥⡀⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⠉⠀⢿⣿⣿⣿⣿⣿⡾⠋⣟⣼⣟⣻⣿⣿⡇⠀⣿⠀⢀⣿⣿⣿⣫⠥⢈⠀⠀⡉⣉⣀⣾⣿⡿⠿⢿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⣾⣿⣿⣿⣿⣿⣷⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⡶⠢⠁⢠⠚⠙⠛⢛⣿⣿⠁⠀⢨⣿⣿⣿⣿⣏
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⢠⣿⣿⣿⣿⣿⣿⡟⠛⣤⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⣘⡇⠀⠠⠐⠴⠿⠾⠛⠛⠁⠀⠀⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠈⢻⣿⣿⣷⠀⢸⣿⣿⣿⣿⣿⡿⣷⢾⣏⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣋⣷⣤⡀⣷⣶⣂⣠⣾⣧⠀⠀⢠⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⢘⣿⡿⣿⣿⠁⠠⢼⣧⣿⣿⢟⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⡽⠿⠻⡅⣟⢀⠀⢸⣿⣿⣿⣿⡅
⠀⠀⠀⠀⢰⣿⡏⣿⡟⠀⢸⣿⠁⠸⣿⣰⡶⠛⠙⠁⠀⣮⠀⠀⢸⠟⠇⠸⣀⠟⠛⢿⣛⣿⠀⠀⢰⡇⠁⠀⡇⢸⠀⠀⢸⣿⣇⢸⣿⣷
⠀⠀⠀⠀⠈⣿⣹⣿⠇⠀⢾⣿⡀⢴⣿⣏⠁⠀⠀⠀⠀⣿⠰⠄⣻⠶⠀⠄⣿⠀⠀⠀⠿⠀⠀⠀⢸⡇⠀⠀⡇⠈⠀⠀⠸⣿⡟⠸⠿⣿⠇
⠀⠀⠀⠀⠈⠛⠻⣿⣷⣄⠈⢿⠄⠀⠸⠿⠀⠀⠀⠀⠀⠿⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢾⠿⠃⢤⣾⡿⠇
⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⠤⣤⠤⣤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠀⠀⠆⠀⠀⠀⠀⠹⠇⠀⠠⠀⠀⠖⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠾⢧⣶⣦⣦⣴⣦⣶⣴⣶⡾⠷⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠉⠁⡀⠉⠁⢀⠈⠁⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠛⠿⠟⠛⠿⠿⠛⠶⠶⠛⠿⠷⠛⠻⠿⠋⠉⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⠶⢧⣤⣤⣶⣤⣤⣦⣠⣄⣴⣠⣄⣶⣤⣄⡴⠶⠶⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣀⣀⡀⠉⠁⠀⠉⠉⠀⠉⠉⢀⠉⠉⠀⠉⠉⢀⡀⣀⣦⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡁⣈⠛⠋⠻⠶⠾⠻⠶⠶⠻⠶⠶⠟⠶⠶⠟⠶⠶⠛⠙⠛⣁⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡙⠛⠛⠶⠾⢧⣴⣦⣶⣰⣠⣦⣔⣢⣴⣔⣂⣶⣴⣦⡼⠷⠾⠛⠛⢋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⠻⠿⢾⣄⣠⣄⠉⢄⣀⢉⢁⣄⠉⡉⣠⠉⡉⣤⠈⡁⣠⣄⣠⡷⠿⠟⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣤⣉⣁⡉⠛⠛⠉⠛⠛⠉⠛⠛⠉⠛⠛⠉⠛⠛⢉⡉⣁⣤⣴⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠛⢻⣿⣮⠷⠻⠾⠷⠻⣬⡥⠟⠾⠵⠟⠾⣷⣿⡟⠛⠉⠁⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⣀⠺⡉⡀⠠⠶⠦⠀⢰⡂⠐⣆⠀⠴⠢⠀⣀⢩⠇⣀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⢀⣤⣷⣴⣠⡀⠠⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣀⠀⠀⠀⠀⢰⡄⠀⠀⠀⢠⡿⠟⠛⠛⠻⢿⠀⠀⠀⠀⣀⡆⠀⠀⠀⠀⣀⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⢤⠡⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⡇⠃⢀⠀⠀⠄⡤⠀⠀⠀⡉⠐⠂
⠀⠀⠀⠀⠀⠀⠁⠀⢀⠋⠘⠎⡠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠈⢀⡀⠘⠀⠀⠀⠀⠀⠀⢀⠳⠁⠋⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠉⠠⠤⠵⠴⣤⠉⠣⠠⠆⢀⡀⠤⠀⢸⡿⢶⣶⣶⡾⢿⡇⠀⠤⢀⡀⠠⠀⠜⠈⠤⠶⠮⠤⠄⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢦⡴⣟⠢⣶⣤⣌⣤⠞⣠⣄⡰⢦⢀⠀⡞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⣶⣿⠦⢤⡄⣀⣀⢅⣀⣰⣸⡀⡄
⠐⣻⡅⠋⠑⠿⡟⢫⠈⢨⢈⣟⢫⠉⣿⠓⣹⣩⢇⡀⡀⠀⣤⠀⠀⣀⣀⡼⢀⠮⠂⢪⡇⣿⣣⡉⣿⡿⢁⠺⡘⣯⠼⡦⢿
⠈⢇⢓⡦⡟⡾⢹⡼⢗⣙⢈⡷⣈⡻⠖⢢⠒⣶⡐⣿⡟⣿⡙⢿⣿⢽⡿⣺⣍⣌⣇⣰⣆⠺⠧⡑⠾⣷⠈⠠⡌⢓⢂⡟⡶
⠀⠀⠼⡆⢉⠗⢜⠓⠎⠴⢖⣄⡹⠟⢵⣡⠟⡀⢌⡙⢶⠉⣐⠈⠷⠠⠃⠄⡂⣹⣧⢧⢸⠌⣣⡘⢲⢦⠈⠇⢉⠘⠾⠑⢹
⠀⠀⢠⠀⡌⠁⠄⡐⠀⠀⠈⠀⠁⠚⠂⠛⠀⠸⠾⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠄⢄⠉⠠⡌⠳⠶⠌⠀⠁⠈⠈⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠁⠀⠑⠀⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀⣹⣿⣿⣿⣿⣿⡷⢾⣿⣿⡿⠟⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣹⣿⣿⢿⠟⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⣈⣿⣿⠟⠁⢀⣬⣿⣿⣿⣿⣿⣿⡿⠟⠛⠙⠱⠈⠀⢀⠐⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣷⠘⣿⡿⠵⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⠿⠟⠋⠉⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⢀⣤⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣤⣞⡄⣬⣿⣿⠿⠟⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠘⠤⣀⠀⠀⠀⠀⠀⣶⣦⠖⠀⢠⣤⣾⣶⣿⣿⠿⠗⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣆⠸⡿⠿⠛⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢠⠀⠀⠀⠀⠀⠀⠠⠾⠟⢰⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢄⠄⠀⠀⡀⡠⣠⣤⣤⣶⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⡴⢖⣿⣿⣿⣿⣧⣿⣿⣿⠋⠙⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⢰⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢈⠋⠀⠀⠀⠀⠀⠀⠀⣤⣀⣿⡾⠁⠀⠊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⢀⡀⣠⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡋⠹⠻⡀⢨⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣏⡈⣁⣠⣼⡇⢀⡀⠸⣇⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣇⠈⢣⠀⢻⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣸⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠄⡀⢀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠠
⠈⠆⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠰
⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠉⠉⠉⠀⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣷⣤⣶⣟⡀⠲⠤⣀
⠀⠐⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣵⡀
⠀⠀⠀⠀⠀⠻⣷⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⢀⡀⠀⠀⠀⠈⠛⠛⠉⣿⣿⠀⠈⠀⠙⠿⣿⣿⣿⣛⣿⣤⣤⣘⣇⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⣾⣿⢿⢿⣟⣛⠛⡻⣿⡿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠀⠀⠀⠛⠛⠻⠷⠛⠻⠾⠟⠋⠀⠀⠀⠀⠀⠀⠠⠀⣄⠀⡀
⠀⠀⠀⠀⠀⢶⣶⡀⣀⣠⠂⠀⠀⠀⠈⢄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠞⢠⣀⢀⡀
⠀⠀⠀⠀⠀⣿⡿⢑⣥⣷⣢⣤⣾⣀⣋⣿⣕⣗⣰⡲⡶⣶⣒⣒⣶⡖⠒⠲⣶⠀⠀⣛⣹⣷⡀⢞⡃
⠀⢀⠀⠀⠀⠃⣹⣿⣿⣿⣿⡿⠿⡿⠿⠟⠻⠵⠛⣼⠇⠽⠿⠿⠚⠓⠦⠤⠻⠗⠛⠿⢿⣿⣿⠿⠃⠀⠀⠀⡀
⢀⠆⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠰
⠀⠀⠀⠂⠁⠈⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠁⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡀⠀⠀⠀⣀
⠀⠀⢀⡀⠠⠓⡀⠀⠀⠄⠀⠦⠊⠛⢲⠤⠀⢀⣀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⡴⠛⠉⠋⠀⠄⠀⣐⠐⢀
⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠟⠁⠓⠘⠇⠘⠋⢲⠞⠒⠁⠾⠁⠠⠈⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡻⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⣸⠟⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠅⠀⠉⠒⠂⠀⠰⠀⠀⢠⠤⠀⠦⠀⠀⠀⠴⠂⢤⠄⠀⠚⠁⠘⠉⠉⠁⠀⢀⠀⠀⠁⠀⢸⢥
⠀⠀⠀⠀⠀⠐⠀⠀⠚⠠⡬⠑⠀⠸⠗⠀⢈⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⢠⡀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠥
⠀⢀⣀⣤⣄⡤⠀⠂⠀⠈⠀⠀⠀⢀⠀⠀⠀⠉⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠫⡄⠀⠐⠂⠀⠲⠌⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⣤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠌⠀⠀⢄⠘⢁⠀⠀⠘⠁
⠀⠀⠀⡐⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⡐⣒⠀⠀⠂⢀⡀⠀⡄⢸⡆⡀⠀⢄⠈⠉⠀⠀⢀⡴⠀⠨⠀⠀⠀⠀⠇⠀⠀⠀⣠⡄
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠁⡀⠄⠀⡔⡷⠀⠯⠼⣧⢼⢦⠼⠷⠄⠀⠀⡖⠀⠀⠀⠀⠐⠀⠀⠚⠂⢠⠀⠈⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠎⠀⠃⠞⠚⠀⠀⠂⡇⠀⠀⠀⠀⠀⡄⠀⠀⢠⡀⢠⠀⠀⠀⢀⠆
⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢖⡦⢀⢻⠀⢾⡘⠺⠈⢿⡚⠊⠓⡆⠀⠋⠀⠈⠀⠀⠀⠀⠀⠠⠈⠘
⠀⠀⡀⠀⠀⠀⠈⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠃⡀⠛⢀⡓⠀⣉⠀⠋⠁⢂⡑⠷⠀⠀⠆⠀⠂⠀⠀⠀⢀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢨⠅⠀⣤⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠈⠁⡀⣠⠀⠀⠀⢠⡄⠀⠙⣓⠃⣼⠃⠠⣤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⡾⠀⠦⡀⠀⠠⠄⠀⠀⢀⣀⣀⡀⠀⠀⠐⠊⠛⠀⠀⠀⠈⠀⠠⢀⠉⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠂⠀⠀⠂⠀⠚⠋⠀⠀⠀⠀⠈⠀⠝⠲⢤⣉⣀⡀⠀⠉⠀⡀⠀⠄⠐⢊⠠⠸⢀⡀⠀⠀⠀⣀
⠀⠀⠀⣮⢧⢆⣄⠀⠀⠀⠀⠀⠀⠈⠫⠉⠁⠀⠐⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠈⡀⠀⠀⠀⢀⠠⠀⠄
⠀⠀⠀⠁⠁⠀⠁⢐⢄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣼⡃⠠⠠⠀⠀⠀⠀⠀⠘⢞⡛⣭⡴⣄⣶⠀⠀⠄⠀⠨⠀⠀⡆⢸⠀⡆⡆
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠳⣧⣾⣗⣻⡇⡮⠀⠄⠠⠸⠀⠀⠛⠘⠀⠃⠃
⠀⠀⠀⠠⠄⠐⠆⠠⠤⠀⠀⠢⣀⠀⠆⠘⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠷⠍⠗⠀⢦⡤⢲⡀⠠⣆⠀⠀⠄⠲
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠙⡇⠀⠀⢤⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⡤⠀⡀⠀⠀⠀⠛⠀⠀⠲⠄⠀⠀⢀⡤⠘⣷⠀⠀⠀⢀⠀⠀⠠⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⣀⠀⠄⠀⠰⡤⠀⠀⠀⠀⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⠀⠈