Noodles Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣸⠃⠀⣤⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣧⠀⠀⠀⠀⣿⣤⠤⣄⠀⠀⢀⣤⡀⠀⢀⣤⣴⡇⠀⢠⣿⠀⣾⣡⡼⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢹⡆⠀⠀⢸⣿⣡⠆⠘⣧⣴⣿⡀⡿⢀⡾⢡⣿⠁⢀⣾⠇⠀⣿⠁⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡀⠀⣾⠉⣟⠛⢻⡏⢾⣀⡿⠛⢹⣇⡞⢸⣀⡞⣿⠀⠀⠈⠛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⣷⢀⡿⠀⠻⠶⠛⠀⠈⠋⠁⠀⠀⠉⠀⠈⠉⠘⠷⠶⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⠇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣀⣤⡤⠶⠒⢛⣋⣩⡍⠉⠉⠉⠍⠉⠉⠉⢩⠗⠋⡭⠛⠛⠛⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠉⠀⢀⣀⡀⠀⠀⠀⢀⣀⣤⣀⣠⠒⡏⢹⠛⣟⢦⡈⢢⡀⠀⠀⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣰⠋⢉⡉⠉⠓⢲⣯⠔⢶⡞⠉⣦⡑⢌⠲⣌⠳⡙⣄⢹⡀⠀⣀⠀⠀⣧⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣀⣤⣅⠀⠀⠀⢀⣠⠾⠾⠤⣾⢣⡘⢇⠙⢌⠳⣌⢣⡹⣌⢦⠳⠞⠉⠓⡚⢁⠛⢄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠐⠃⠀⣰⠋⠉⠀⢘⡾⠳⢖⡫⠭⠤⠴⠊⣉⣉⣧⢳⡈⢣⠀⠙⢎⢧⠉⠙⣆⠳⡀⠀⠯⣀⡁⠀⠈⢣⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣀⣀⣸⣠⠴⣻⠿⠶⣒⣋⣠⠴⠋⣉⡭⠤⠤⠜⣧⢹⠀⢙⡷⠛⠦⡓⢄⠸⣗⢭⡢⢄⠀⢹⠀⠀⢀⢧⢀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠐⣇⡔⣋⡥⠤⠤⣴⡾⠕⢒⣊⡵⠒⠒⢒⣫⠥⢖⡒⠂⢁⣹⡆⢧⠈⠓⢦⠀⠈⢢⡱⡌⢆⠙⣦⡱⡌⠑⢒⡡⠊⠘⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠊⠥⠒⠒⣫⡥⠖⠛⠦⣄⠀⣠⠞⣡⠖⣫⠥⠒⢊⡭⠂⣹⡌⢧⡀⠘⠢⠒⠚⣳⣝⡢⣥⣤⡾⠞⠛⠛⠒⠒⢤⡇⠠⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡩⢤⡟⠁⠀⠀⠀⠀⠈⢳⡏⠐⣅⡀⠓⠤⢤⣈⡳⣄⢣⠘⢦⣉⣓⣒⣈⡭⢶⠎⣩⠋⣼⠀⠀⠀⠄⠀⠀⠀⢳⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠸⡏⢷⣜⠀⣠⠞⠛⢦⡀⠀⠈⣿⠦⣄⣉⡉⠓⢦⡀⠈⢧⠹⡗⠤⠏⢉⢏⣠⠬⠥⢖⣡⠾⠖⠓⠒⢦⡀⠀⠀⢀⡾⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢦⣇⠀⠀⠘⡇⠀⠀⣿⠈⣀⠒⠙⢳⣄⡑⠒⠚⠒⢿⣄⡰⣫⠏⡴⠚⠉⢹⠁⠀⠀⠀⠀⠀⠙⣆⡠⠚⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠷⣤⣀⡇⠀⠀⣿⠀⣒⠛⠛⠋⣊⠽⢛⢝⡛⢦⣉⠙⣁⣾⠴⠒⠒⠺⡄⠀⠀⠀⠁⠀⠀⢻⢀⣠⠞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠈⠙⠳⢦⣴⣏⠒⠚⠿⠭⠭⢒⡪⢟⡭⠒⠉⠉⠉⡿⠁⠀⠠⠀⠀⢹⡀⠂⠀⠀⣀⣤⠾⠛⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠶⣶⣭⣥⣖⣉⣀⣀⠀⠀⢰⡃⢀⣀⣀⣀⣀⣨⣧⡴⠞⠛⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠠⠄⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠉⠁⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⡿⡿⠿⡟⢛⣋⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣤⣤⣴⣶⣶⣤⣤⣴⣶⣾⠿⠿⠛⣛⣋⣩⣭⣤⣴⣶⣿⡿⠿⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⠿⣿⢿⢛⡏⡟⡇⣿⣿⣿⣿⣶⣶⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⡏⢹⢸⠀⡇⢸⢰⡇⡏⡇⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⡿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⢾⢸⠀⡇⢸⠸⡆⡇⡇⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣾⢸⠀⡇⢸⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡏⢸⠀⡇⢸⠸⣤⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢸⠀⡇⢸⢸⣿⡅⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢸⠀⡇⣼⢸⢻⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⣾⢸⡇⡟⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⡇⣿⢸⡇⡇⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣨⣿⡇⣇⣿⣸⣇⣧⣸⣀⣇⡇⣿⣧⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣋⠭⢭⣉⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣯⣤⠤⠬⣝⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣾⣷⣾⣿⣿⣿⣥⣴⠚⣉⣉⣭⣉⣍⣉⣙⠒⠢⢬⡙⠿⣷⣶⣬⣷⣼⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⣋⡭⠥⠒⠒⠒⠒⠢⠤⢍⣉⠛⢿⣷⣤⠤⠤⠤⠤⢬⣉⡓⢦⣬⣷⡿⢛⣉⣉⣉⣛⠻⢿⣦⣄
⠀⠀⠀⠀⠀⢠⣿⠟⣵⠚⣡⠖⣊⣉⠭⠭⠭⣉⡓⠒⠤⣉⠲⣌⢻⣷⡖⠒⠒⠒⠦⣍⣾⡟⣡⠞⣩⠤⠴⠤⢬⡙⢦⡙⣿⣆
⠀⠀⠀⠀⣠⣼⣿⣴⣧⣾⣵⣾⣷⣶⣶⣶⣶⣶⣿⣷⣶⣿⣷⣾⣷⣿⣿⣿⣿⣿⣶⣾⣿⣶⣿⣼⣧⣶⣿⣿⣦⣽⣤⣷⣼⣿⣄⡀
⠀⠀⠀⠀⣿⣟⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠈⣿⡆⠛⠃⠀⠛⠛⠀⠘⠛⠀⠛⠛⠀⠘⠛⠀⠛⠃⠀⠘⠛⠃⠘⠛⠃⠀⠛⠛⠀⠘⠛⠀⠘⠛⠃⠘⠛⠃⠛⢃⣿⡇
⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⡿⠋
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠛⠛⠛⠿⠿⠿⠿⠿⢿⣶⣾⡿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⣿⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠃
⠀⢰
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣆⡀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠁⢾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢂⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠻⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣍⢻⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣤⣿⢹⣟⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⡏⡾⢸⢿⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⡾⣡⠏⣼⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡴⣫⡴⣫⣾⣛⣛⠷⣤⡲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡲⢾⣉⡞⣵⢫⡞⠋⢩⠾⠿⣿⢮⣻⣯⠍⣙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣈⠳⣬⣙⠒⢸⡇⣧⡻⢮⣽⢳⡎⠃⣼⣛⠿⢧⣛⠩⠶⢋⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣶⣿⣿⣤⣿⣮⣉⣻⣿⣟⣥⣞⣩⣿⣻⣶⣼⣷⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣤⣀⣉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣀⣠⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣺⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠕⠊⠁⣀⠤⠒⠉⣀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⠭⠒⢉⡠⠤⠒⣉⠤⠔⣂⠥⠄⠺⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⡪⠕⢊⡠⠄⣂⠥⢔⣂⠥⠐⣊⣁⣤⣴⠦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠝⠊⣁⣴⠮⢙⡂⠥⠒⠈⣁⡀⠤⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠮⢛⣩⡴⣞⠫⠕⠂⢉⡡⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⣺⣿⣽⢿⠅⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣏⣿⢧⣿⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⡽⣟⢯⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣳⢯⣟⡾⡽⢤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠛⠉⠉⣉⣉⣤⣤⣤⣤⣤⣤⣧⣏⣿⣾⣽⠁⢀⣀⣀⣀⣀⣀⣀⠉⠉⠉⠉⠛⠓⢲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣉⣠⡴⣞⣿⢽⣷⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡦⢤⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠂⠭⣙⡛⠛⠾⠿⠿⠶⠶⣿⣿⣷⣶⣿⣭⣟⣶⣝⣳⣽⣿⣿⣿⣿⣿⠭⠿⠷⠾⠟⠛⣛⡋⠥⢐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠬⠅⠀⠀⠉⠉⠐⠒⠒⠂⠠⠶⠤⠤⠤⠤⠀⠀⠠⠆⠀⠀⠠⠆⠀⠀⠐⠒⠒⠈⠉⠁⠀⠀⠨⢽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡒⠒⣂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠔⢉⠄⠒⡒⡀⠉⠪⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡬⠐⠈⠀⡜⠁⠈⢣⠁⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠹⡀⣼⣀⠈⠁⠀⡤⢀⠌⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⠢⢈⣉⣀⡉⠄⡪⠊⠀⠀⠀⠀⠀⠀⢀⡀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡒⠤⠤⣀⣀⣀⠀⠀⠀⠉⠒⠂⠐⠒⠉⠀⠀⣀⣀⠀⠤⠄⢀⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⠤⠤⢭⣭⣭⣭⣭⣭⣭⣭⡥⠤⠤⠶⠖⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠄⠀⣐⣂⣀⣀⣀⣠⣤⠤⠤⠤⠤⠶⠶⠒⠒⠒⠚⠚⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣛⣚⠀⡄⣇⣩⣉⣩⠩⠅⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠶⠧⠦⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⡡⢭⣍⠂⠘⠒⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢰⡇⠛⠀⠠⠥⠀⢄⣍⡐⡠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⣄⠄⠘⡇⠀⢤⡤⡄⢴⡄⠈⡆⠄⡈⡴⡂⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣍⢀⡘⡙⠀⠄⠓⠟⠁⠓⢄⠻⢋⠤⠣⡀⠀⣀⣱⠈⢑⣤⡉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢀⠭⢄⠁⠂⡀⠉⣐⡂⣮⠤⠶⠞⣠⣀⡀⠃⠉⢉⣀⡄⢍⡸⢼⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡐⠏⡣⡁⠬⢧⡀⠀⣀⡀⠋⢐⠁⣈⡢⠊⠬⠂⠛⠶⡄⡘⢂⠂⣇⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡛⢷⣾⣤⣃⣀⣚⠂⠊⠀⠀⡀⠴⡄⢾⢗⣀⣀⣄⣠⣬⣥⡶⠾⢛⣭⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠳⢦⣭⣉⣙⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⣉⣥⡤⠶⠛⠉⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⢉⠉⠉⠛⣿⠟⣿⡿⠟⠿⣿⠿⠿⢿⣬⠻⣷⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠿⣛⣛⣛⠿⣾⣯⣼⢯⣼⢟⡟⠖⣾⠿⣶⣽⣧⣽⡇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⢛⣉⣙⠻⢾⡷⡿⡿⣻⢿⢻⢣⠇⠹⣟⣉⣉⡙⢷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣉⣙⢿⡬⡇⣼⡳⣛⣾⣺⣿⡿⣧⡞⣉⣉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⢸⣧⣹⣟⢿⣮⣻⡷⠿⢃⣿⢰⡟⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡿⣮⣝⣧⣿⢏⣴⠞⣛⣡⡾⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⠿⠿⠾⠿⠾⠟⠛⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⠶⣤⡀
⠀⠀⠀⠀⡤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⡠⠬⢍⣔⠵⢀⠊⢴⡈⠅⠠⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠂⠀⠘⠄⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⣀⢠⣀⠜⠫⠛⠁⠀⣨⡗⠁⡙⡉⠒⠤⣶⡍⠓⣀⡀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢀⠒⢀⡤⠧⠦⣮⠁⢉⡏⢀⣀⣋⢄⢈⠢⣑⠶⡼⢀⠋⢩⠀⣀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠆⠀⠰⡀⠀⠀⠀⢸⠃⠀⠀⡠⣮⠀⡞⢋⠓⣄⡌⠀⢸⣌⢲⡂⢒⡡⠂⡎⠀⠀⣅⡫⠂⠈⠠⠊⠑⣀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⣿⠀⠀⡜⠀⠀⣸⠁⢰⠂⡉⠀⠀⡁⠀⠉⡋⣱⠄⠀⣩⡀⠀⣀⣀⢠⣶⠊⠔⠃⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢢⠀⠀⢹⣄⠨⡇⣰⠛⠉⢆⢂⠠⠐⠢⠰⢆⠖⡉⢹⢹⡜⡲⢧⣀⡀⢉⠔⠊⢐⠈⣿⡶⠿⠩⠢⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢣⠀⠘⣿⣨⣵⣏⠀⢠⡟⢹⠛⠲⡤⠀⡇⢀⠞⢣⢸⢄⠀⢸⣭⣉⣊⣀⡀⠴⠀⠈⢸⠑⡂⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏⣙⣲⣬⢥⣄⠑⠀⠰⢄⣼⣾⢂⠨⠘⣃⣀⠠⠨⢈⣛⡞⣐⡨⠀⠀⠖⠄⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠆⠀⠈⢷⣆⡎⠈⣄⠁⠬⣡⣄⣠⠉⠈⠳⣄⣦⡦⠛⠉⠍⢉⢘⠀⠉⡔⠣⣀⢈⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠆⠀⠈⢋⠻⣶⣄⢙⠖⠋⣀⣈⡄⠔⠀⣥⠦⢤⣄⡐⢆⠲⠆⢂⠜⡀⢀⣴⠟⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠱⡄⠙⠻⢶⣬⣁⣩⢀⠁⢊⠲⠇⠀⢒⡑⣠⣋⣩⣥⡶⠟⡋⢐⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⠈⢦⡀⠀⠀⠉⠙⠛⠻⠿⠿⠷⠿⠿⠿⠛⠛⠉⡑⠂⠅⣪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠉⠀⠈⠂⠀⠀⠀⠀⣂⡤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠀⠀⠀⣟⡳⠦⣤⣄⣀⣀⣤⣀⣀⣤⡴⢞⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢀⢳⠀⡄⠀⠀⠀⠻⣿⣷⣶⣬⣭⣭⣭⣭⣴⣶⣾⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡜⡆⢻⠹⡄⠀⠀⠀⠈⠑⠪⠭⣭⣭⣭⣭⡭⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⡄⢧⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣋⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠶⠶⠶⣶⡶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⢶⣶⣶⡾⠿⠿⣦⣶⠿⠻⢿⣶⡿⠻⢿⣭⣭⣉⣉⣩⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡅⠀⠀⣿⠃⠀⠀⢰⣿⠀⠀⣾⣏⣉⣉⣉⣉⠁
⠀⠀⠀⠀⠀⠠⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣻⣇⠀⠀⢿⡇⠀⠀⣸⡇⠀⢠⣿⣋⣉⣉⣉⣹⣷
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡄⠀⢸⣷⠀⠀⣿⡇⠀⢸⣿⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⡿⠀⠀⣿⡇⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣾⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣿⡇⠀⢰⣿⡀⠀⠀⣼⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠉⠁⠀⠀⣿⡇⠀⢸⣿⠻⣷⣴⡿⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣿⡿⠁⠀⠀⣠⣴⡾⠟⠻⣷⡀⠀⢿⡆⠈⢻⣧⠀⠀⣰⡿⠁⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢰⡿⠁⠀⢠⣾⠟⠁⠀⣠⣴⣿⡇⠀⠸⣷⠀⠀⢻⣇⣴⣿⡿⠿⠛⠛⠛⠛⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⣠⣾⡇⠀⠀⣿⠃⠀⢀⣾⠏⠁⣼⡇⠀⢀⣿⠀⠀⢸⣿⠋⠁⢀⣀⣤⣤⣤⣀⡀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢰⡿⢿⡇⠀⠀⣿⠀⠀⠸⣿⠀⣰⡿⠀⠀⣼⡟⠀⠀⣼⣧⣴⠿⠛⠋⠉⠉⠉⠙⠻⢷⣄⠀⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠸⣿⡀⠀⠻⣷⣄⣠⣿⡿⠋⠀⢀⣼⠟⠀⠀⣰⡿⠋⠁⣀⣤⣶⠾⠿⣷⡄⠀⢸⣿⠀⠀⠀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⡿⠶⠶⠿⠷⠶⠿⠿⠶⠶⠾⠿⠿⠿⠶⠶⠶⠿⠿⠶⠶⠾⠿⠷⠶⠾⠿⠿⠶⠶⠶⠿⠷⠶⠿⠿⠶⠶⠶⠶⠾⠿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠆⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣴⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣻⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣲⣾⠿⢿⣛⣿⣿⣯⣿⣯⡭⡭⣍⣽⡿⣿⣯⣽⢿⣿⡿⣟⠿⠿⣷⣶⣦⣤⣀
⠀⠀⢀⡤⢖⣯⣷⣛⣩⣥⣶⣿⢿⣿⣿⣿⣯⣿⣿⣹⣾⡿⣗⣶⢖⢾⣻⣷⣛⣿⣵⡵⣶⣶⣮⣭⡛⠾⣽⡲⢄⡀
⠀⡴⣫⣾⢛⣽⣿⣿⣵⣿⡯⣺⣿⣿⣿⣿⡿⣿⣷⢿⡟⡟⣿⢿⡞⣟⣽⣵⢞⣯⣽⣿⣺⣿⣿⣿⣽⢢⣌⠛⣷⣝⢦
⡼⣱⣿⡱⡿⢿⡶⣿⣿⣿⣟⣽⣿⣿⣿⣯⣿⣿⣿⡜⣺⣿⣿⡞⣿⣻⡿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⢽⣣⣈⢿⣎
⡇⢿⣇⠓⠻⣿⣿⣶⣽⣿⢿⣿⣿⣿⣮⣛⣿⣿⣆⣿⣿⣟⣿⣞⣯⣯⢻⡜⣿⡿⢿⣿⣟⣿⡿⣿⣿⣯⣾⣿⣿⣸⡿
⠹⡜⢿⣄⠀⠻⠿⣿⢿⠿⣯⣿⡿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢗⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠿⣫⡿⢣
⠀⠈⠢⣝⠷⣤⣀⠻⣿⣭⣷⣾⣿⣿⣾⣿⣿⣿⣿⣟⢭⣝⣿⣿⣿⣿⣽⣿⣷⣿⣿⣿⣻⣯⣿⡿⠟⣋⣤⠾⣫⠔⠁
⠀⠀⠀⠈⠙⠲⢽⣛⠶⣬⣍⣉⡉⠛⢿⣷⣾⣽⣿⣿⣛⣥⣿⢿⣿⣿⣿⣿⣽⣿⠿⢛⣛⣡⣤⠶⣛⡯⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠭⠭⣛⣓⣺⡿⠿⠶⠦⠤⠤⠤⠭⠥⠶⠶⢶⣖⣛⣛⠭⠭⠓⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠒⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢸⣃⠠⠤⡀⣬⣠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠐⠼⠃⡀⠊⢰⣷⡌⡰⠀⢀⡀
⢀⣀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠶⠟⠉⠁⠁⢰⢈⠆⡴⠓⡆⢜⠩⠀⠁⠀⣆⠠⡀
⠀⠀⠀⡈⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⠀⠐⠁⠄⠎⠀⠚⢣⣨⠀⠄⠀⢰⣀⡰⢄
⠶⠞⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⢤⣀⠀⣀⣦⠀⠃⠃⠞⡀⠄⠺⢃⠔⢤⡓⠄⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⡮⢊⠀⠔⠂⠄⢀⡌⡀⠀⠦⠀⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣀⢤⡞⠁⢀⠀⠁⠀⠁⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠇⠸⠀⡴⠁⠀⠀⠀⣴⣦⢚⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢃⠀⠀⠀⠀⢡⠀⠀⠀⠀⠘⠋⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢁⠃⠀⠀⠀⠂⠳⠀⠰⠃⢈⣆⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡆⠘⢯⠀⡁⠀⠀⠀⡇⠠⢀⠀⢠⠳⡷⣄⠈⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠉⠁⠠⡇⢀⠆⡼⠀⠀⠀⢈⠆⠔⠉⠂⠈⡟⣀⠀⠆⠰⠈⡛⠻⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⢀⠐⠚⢩⠂⠀⠐⢱⠛⠚⡁⠊⡐⠁⠀⡂⠰⡇⠂⢠⢄⠤⢯⡠⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⡸⠃⠀⢀⠼⡄⡄⠀⡃⠀⠁⠀⡖⠡⢦⠘⠀⠀⠏⠎⡇⠉⠐⠄⣘⡁⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⡀⠀⣁⠆⠐⠉⠀⠍⢀⡐⢧⠰⡀⠀⠀⠘⠺⠀⠀⡜⠠⡄⠈⢶⡄⠸⠑⠐⠡⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢳⢰⣏⡼⣨⢂⠀⠀⠀⣱⡈⡇⠡⠀⠀⡠⠘⡐⠞⡁⢠⠹⡀⢰⢥⠘⢠⠐⠀⡋⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣾⠤⠊⠐⡉⠄⠀⠀⠈⠽⡇⡃⠅⠀⠈⠀⣆⠉⡞⠀⣠⣜⣀⣎⡤⠗⡄⢚⢸⠄⠈⠂⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠐⡄⢠⣡⡤⣤⠂⠀⢢⠀⠀⠀⡴⠡⠁⠐⠠⠠⢀⡿⠃⣷⢀⣐⡀⠀⡀⡠⠊⠀⠲⣄⣯⡰⠜⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⠘⢄⢲⠰⡇⠀⢀⢰⠆⠐⠀⠤⠀⢀⡋⠉⢁⡈⠀⠠⠟⢧⡞⠁⠁⣡⣛⠃⠀⠄⠨⣟⠏⣆⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⢈⠓⡹⠀⠚⠦⣀⡴⡲⠶⡒⢇⠚⠛⠂⢶⣈⠣⠀⠀⣀⡴⠊⠁⠩⠀⠀⠀⠓⢤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢭⡳⢬⣀⡠⠶⢦⡈⠀⢀⣘⠃⠄⣀⡤⢈⠀⠈⠁⠀⠀⠀⠇⠀⠀⢈⣃⣠⣴⠾⢛⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠉⠛⠿⠶⣶⣤⣌⣁⣀⡀⡀⣇⢀⢀⢁⣐⣦⣤⣌⣤⣴⡶⠿⠛⠋⢑⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⢐⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠁⠐⢂⠨⠀⠈⣂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣈⡑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠴⠊⠁
⣿⢿⢿⠿⣿⢟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣻⣿⣟⣟⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣠⣤⣠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⢃⠁⠄⡀⠀⠀⠀⠈⠙⠯⣟⡶⣥⣒⠠⣀
⠀⠀⠀⠀⠑⢄⠈⠒⠄⣀⣀⣀⣠⣤⡽⢿⣭⣻⣷⣮⣔⡠⢤⣀⣀⡀
⠀⠀⠀⠀⣠⡤⠧⡀⠐⠚⠻⣄⠀⠀⠀⠀⠀⠈⠙⠳⢭⣋⠗⡲⢭⣓⠿⢳⠶⣤⣀
⠀⠀⠀⡎⠋⠀⠀⠀⠀⠀⠀⠘⣧⣤⣴⣶⠖⣶⣶⣲⡶⠮⣿⣶⣥⣊⠝⡲⠦⣄⡛⠷⢀⡀
⠀⠀⠀⡗⢦⣠⣴⣧⣸⣄⣀⣠⡞⣎⠙⣷⢿⢿⣿⣽⣾⢾⣯⣽⣷⣿⣿⣶⣌⢐⠠⢦⡔⠠⠅⡐⠠⠀⡀
⠀⠀⠀⢡⠀⠉⠙⠛⠿⠶⠿⢦⣏⣧⣮⣮⣷⣷⣿⣷⣯⣩⣿⡿⠿⢿⡿⠻⠍⠓⠫⣒⠨⣐⠠⠀⠈⠀⠆⠇
⠀⠀⠀⠈⡍⠐⠠⠄⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⠠⣴⣮⡏⠈⠐⠨⣐⣬⡇
⠀⠀⠀⠀⠘⣏⠩⡳⡦⠀⠩⡴⠶⣆⠐⠒⠒⣒⣶⣒⠐⠒⠒⣰⡞⠿⢦⠤⡺⡡⡞
⠀⠀⠀⠀⠀⠈⢮⣾⣮⣶⡾⠏⣂⢴⣦⣭⣟⠫⠉⣳⣬⠀⢙⢕⣪⣲⣿⣷⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠑⢿⠐⠚⠂⠕⡪⠩⠩⢵⠫⠿⠿⣿⡛⣿⡛⠝⠂⢈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣖⡂⠀⠤⠤⠤⠤⠤⠤⠤⠀⠐⡒⣲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠛⠋⠁⠀⠐⠛⠚⠒⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠒⠒⢠⡀⢁⠁⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⣄⠔⠋⠀⢁⠕⠢⠤⡀⠀⠀⠀⠀⣁⠐⠐⢦⡄⠤⠠⠀⣀
⠀⠀⠀⢀⠤⠖⠚⠉⠁⠀⠀⠀⠐⠁⠈⠀⢀⠂⠀⠀⠀⠐⠈⠁⠂⠂⠀⠈⠠⢄⠀⠀⠀⢄⠀⠈⠙⠂⠤⡀
⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠐⢄⠀⠨⠒⠂⠀⡄⠈⠐⢀⢀⠠⠄⠒⠂⠀⠀⠈⠛⠩⡀⠀⠀⠀⠀⠈⡆
⠀⠀⠈⠓⢄⡀⠀⠀⠔⠉⠑⠁⠁⠀⠀⠔⡁⠠⠀⢘⡀⠀⠠⠐⠨⠀⠀⠀⡀⠈⠀⠀⠢⠀⠀⠀⠀⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠈⠑⠓⠦⣤⣤⣄⣀⠀⠀⠀⠀⠀⠚⠀⠀⠐⠀⠀⠀⠁⠀⠪⠀⣀⣀⣀⣠⣤⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣍⡉⠉⠙⠛⠛⠛⠛⠒⠓⠒⠖⠛⠒⠛⠉⠉⣉⡽⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⢤⣤⣤⡬⠯⠤⠤⠤⠴⠒⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣭⣽⣿⣿⣯⣿⣽⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣽⣿⢸⢸⢸⢸⣟⠛⣷⣄⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣴⣶⠶⠶⠶⠶⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣻⣿⢸⢸⢸⢸⣿⠀⠈⠛⣛⣛⣛⣛⣛⣉⣉⣉⣁⣀⣀⡀⠀⠀⢀⣀⣀⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⢸⢸⢸⣿⣀⣠⡿⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⢸⢸⢸⢸⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⢸⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⢸⠀⣿⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣴⣿⣾⣾⣾⣤⣿⣴⠾⣛⣛⡭⢍⣛⣛⠻⣶⣄⣀⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣠⡾⢋⣥⣶⡾⢛⡩⠤⠒⣒⣒⣒⣒⠬⢝⡻⣾⣥⠖⢒⣒⠒⠬⣹⣾⠟⣛⡩⠭⠭⠭⠭⠭⣙⡻⢶⣤⡀
⠀⠀⠀⠀⢰⡿⢰⢣⣿⢃⠞⣡⢒⣩⠭⢒⣒⣒⣨⣍⣓⣌⣢⣻⣯⣭⣭⣍⣿⣟⣰⣊⡵⢚⣩⠭⠭⠭⢍⡒⠬⡳⣝⢿⣆
⠀⠀⢰⣶⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠷⠿⠿⠿⠿⢶⣶
⠀⠀⠀⠹⣷⣜⠛⠀⠛⠛⠀⠚⠃⠀⠒⠓⠀⠐⠛⠂⠘⠛⠃⠀⠛⠋⠀⠘⠛⠁⠘⠛⠀⠘⠛⠀⠚⠓⠀⠐⠒⠀⠶⠶⣠⣾⠏
⠀⠀⠀⠀⠈⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣋⣉⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣉⣉⣉⣉⣉⣉⣉⣹⡏
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢈⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠶⠒⠒⠒⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢲⠀⠸⡇⠈⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠸⠁⠀⠈⠄⠀⠃⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠃⠀⠀⣳⠀⢀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠠⠢⡉⠀⠈⠁⠉⠀⠀⠀⠂⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⡇⠀⢀⡀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⣠⡐⡄⠀⠙⣁⠈⠅⠼⠟⠒⠢⣀⢬⠔⢂⡀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⡘⠀⠀⠀⠀⢀⠄⠄⠸⠄⠤⠲⠀⣵⠁⡬⢀⠀⡌⡶⢦⣄⠐⠀⠐⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠁⠀⠀⢰⢀⢁⠀⠚⡛⠒⡠⠞⠹⠁⠔⠉⢠⣆⠀⡀⡀⠜⡏⠓⠊⢀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠈⣬⢈⠄⢀⠀⠘⡺⠦⠠⣤⠤⠊⠁⠀⢰⠆⠀⣐⠀⣠⠴⠀⢀⠀⠀⠀⡘
⠀⠀⠀⠀⠸⡄⠀⠐⠈⠛⠈⠁⡰⠠⠄⠔⠂⠈⣛⠲⠧⢴⠾⣂⠀⠀⠀⠆⠀⠀⠁⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠙⣄⠀⠀⠂⢤⠂⡀⠀⠀⠤⠂⠈⠁⠀⠰⠔⠓⠂⠔⢶⠂⣠⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠁⠠⢄⠐⠀⡐⠂⡀⢶⠂⠀⠀⠀⠀⠁⠈⠁⠀⡀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠂⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣠⣤⣤⣶⡆⣴⡄
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⠤⠤⢤⢄⣀⣀⣀⡀⠀⠀⠀⢀⠤⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡠⠊⠀⠀⠈⢫⠑⢲⢃⢔⡲⠦⣈⣧
⠀⠀⠀⠀⠀⠀⢀⠜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⠰⠁⡠⠄⠠⢄⠀⣧⢃⢎⠎⡠⢤⡈⢻⡀
⠀⠀⠀⠀⠀⢰⠃⠀⠈⢿⣿⣿⣿⣿⡿⢋⡤⠤⠤⢍⡇⢸⠁⢴⠀⠈⡆⠰⣪⢎⡜⠀⠀⢱⢸⢷⡀
⠀⠀⠀⠀⠀⢂⠀⢀⠔⢉⡡⠬⢍⡡⠔⢁⠔⠂⠒⠢⢧⠀⠳⠤⠤⠞⢀⡞⢁⣇⠔⠲⠤⢧⣌⠎⠃
⠀⠀⠀⠀⠀⢸⢄⡌⢰⠃⡠⠒⠒⠒⠒⢁⡞⠣⣈⣉⣢⣑⡂⠒⢢⡴⢏⠣⡸⠴⡻⡿⢷⡌⣻⠀⡆
⠀⠀⠀⠀⠀⠸⡈⠳⢇⡜⡠⠊⡩⠭⢍⡓⢌⠒⠤⠤⠤⠤⢌⠑⢾⡉⢫⣾⣋⢱⡿⡶⢿⣁⡧⢚⠇
⠀⠀⠀⠀⠀⠀⢷⣄⠀⠉⠓⠮⢴⣒⠲⡝⠦⣉⣒⣉⡑⠢⣄⠑⡄⠱⢻⡗⢺⡌⣉⠵⠞⠉⠀⡸
⠀⠀⠀⠀⠀⠀⠈⡄⠑⠢⢄⡀⠀⠀⠉⠉⠐⠒⠒⠦⠬⠷⠼⠦⠞⠒⠛⠃⠉⠁⠀⠀⣀⠴⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠈⠉⠒⠂⠤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠒⠚⠉⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⠠⠤⠤⠀⣀⣀⠠⠤⠤⠄⣒⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠤⠤⠀⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢳⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢇⠀⠀⡏⠉⠉⠀⠀⠀⠈⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠐⠀⠉⢹⠀⠀⣱⣖⣒⣚⣛⣛⣒⣂⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣸⡀⢀⣀⣀⡀⢸⣀⣀⢸⡏⠉⠉⠉⠉⠉⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣭⣭⣭⣅⣀⣀⣀⣀⣀⣀⣀⣈⣉⣭⣭⣭⠷⣾⠓⠀⠉⠿⠷⠿⠿⠛⠒⠛⣿⡇⠀⠀⠀⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣄⣀⣀⣀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣀⣤⣴⢾⣿⡶⠶⠶⣶⣶⣶⡶⠶⠶⢶⣸⡇⠐⠒⠛⠒⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠥⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣈⣭⣤⣤⠤⢴⣿⣍⣉⣉⣉⣉⣉⣉⣉⣉⣭⣿⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣵⣭⣤⣤⣤⣤⣤⣤⣤⣤⡤⢭⣭⣤⣒⣺⠟⠁⠀⠒⠒⣿⣿⡒⠂⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠈⠉⠭⠭⢭⣭⣭⣉⣭⣭⠩⠭⠭⠉⠉⢁⡏⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠉⠀⠀⠀⠀⣸⠀⠀⠀⡄⠀⢀⡀⠀⡀⢀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠂⢃⠈⠷⠣⠃⠀⢀⡀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠐⢶⣶⡶⢶⠆⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠆⠀⢀⡀⠀⠀⠀⠀⢀⡼⣧⢀⠀⠠⠀⠈⣇⠀⡇
⠀⠀⣀⣀⣠⣤⣀⣀⣀⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢘⠀⠀⠀⢈⣉⣻⣇⣭⣀⣀⠀⠀⡀⢸⡅⣿⣉⣉⡀⠀⠀⣿⠀⡇
⣶⣋⠡⠤⢤⣤⣤⣄⣉⠙⠳⣤⢀⡠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⡏⠀⠀⠀⠋⠀⢠⠛⠁⠀⠀⠀⠀⠀⠀⡷⢻⠁⠐⠒⠒⠀⢻⠀⡇
⠈⠙⠲⢤⡀⠀⠈⠐⠪⣑⢦⡈⠀⡀⠀⠠⠤⠀⣄⣀⢀⠑⣴⠀⡖⠂⠉⠉⠉⠀⠁⠀⢠⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡇⠀⠀⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠑⠯⣳⣌⡢⣀⠀⠀⠀⠀⠈⠓⢿⡾⠗⠒⠒⠒⠈⠉⠀⣰⡁⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⢹⣌⣧⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠛⢿⣛⠷⢦⣤⣄⣖⠁⠠⠛⢦⣀⣀⣀⠀⠀⣂⣼⣿⠛⠓⢶⣶⡖⠮⠐⠋⠀⠀⢨⡟⢿⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠐⠨⢙⡓⠲⠬⠭⠭⠭⠭⠭⠿⠖⠚⠋⠉⢀⠋⠁⠙⣅⠀⡇⠀⠀⠀⠀⠀⠀⠘⡀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⣠⠃⢠⠇⠀⠀⠀⠀⠀⠀⠀⠋⣼⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⢄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⣠⣅⠤⠔⠊⠁⡴⠧⠀⠈⠑⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠤⠶⠀⠀⠀⢄⠀⠀⢁⠀⠀⡇⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢍⣂⠤⢀⣀⣀⣀⢀⡜⡅⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⡇⠀⠀⢀⡾⠡⠤⠤⠤⢀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠉⠩⠽⠿⠿⠹⠀⢡⠀⠸⠀⠈⣠⠀⠀⠀⠄⠀⠀⠀⠠⡐⠀⣸⠇⠤⣠⠞⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢁⠀⠣⡈⠀⠰⠀⠀⠄⠈⠀⢀⣤⠁⢠⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠶⠶⣶⣦⣶⡶⠾⢦⡐⠥⡀⢈⠲⠤⠄⠀⠢⢤⡰⠞⢁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢽⣶⣌⡐⠠⠤⠤⣄⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⣪⠔⠂⠀⢱⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⡴⠂⠀⣡⠀⠀⠀⠀⠀⠀⠀⢠⠋⠈⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡾⠀⣩⠃⡠⠂⣴⢀⠔⢠⡆⢀⡆⢰⠂⣰⠂⣠⠂⠁⠀⠀⠀⡼⠁⠀⠈⢉⠗⣠⠂⡀⣠⠖⣰⢋⠆⣠⠆⣠⠔⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠛⠈⠁⣠⠃⠉⠈⠈⠃⠉⠀⠙⠀⠙⠀⠑⠀⠀⠀⠀⠐⠃⠀⠀⠀⠙⠀⠉⠈⠁⠉⠀⠾⠈⠀⠙⠀⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡱⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢷⠀⠀⠀⢠⣇⣀⣀⠀⠀⢀⡀⠀⠀⣀⣼⠀⢀⡿⢀⡞⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⣇⠀⠀⣸⣏⡤⠙⣇⣴⣏⣹⢀⡾⢹⡏⠀⣼⠇⢸⡏⢁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢹⡆⠀⡟⢹⠛⣹⠇⣿⣸⠟⠹⣧⠇⣧⡼⡿⠀⠀⢉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢿⣸⡇⠈⠛⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⣛⣀⣤⠤⠤⠤⠶⠶⠶⠶⠤⠤⢤⡤⢤⢤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠀⠐⠂⠉⠁⣀⣀⠈⡠⡄⠢⢷⢄⠙⢄⠀⠀⠈⠣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⢠⠋⢉⠙⠒⢢⣮⠭⣼⠋⢣⡳⡕⢌⠳⡕⡌⢧⠀⢀⠀⠈⣍⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣀⣠⢤⣧⡀⢀⣀⡠⠾⢶⢶⡟⡄⢧⡑⢌⠢⡱⣜⣜⢎⠒⠁⢑⣚⡠⠳⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡾⠀⠁⠀⢾⠁⢀⣠⠟⢙⡭⠒⣒⣚⡡⠤⠼⣜⢆⠳⣀⣱⡝⣄⠈⢆⠣⡀⠙⠦⡄⠀⠘⡄⠀⠙⢦
⠀⠀⠀⠀⠀⢸⡇⡠⠴⣒⣚⣚⣕⣋⡭⢗⣲⣊⡥⢖⣋⡉⠭⢹⡜⡄⠸⣉⠉⠪⡳⣜⢏⠺⡑⢄⠳⢄⠴⡻⢰⡀⠈⢧
⠀⠀⠀⠀⠀⠘⣿⢖⠭⠤⢜⣫⠥⢴⡊⠁⣠⠔⡪⢕⣲⠬⢖⡪⢳⡸⡀⠀⠧⠠⠼⣜⠮⣓⣊⣿⠴⠶⠮⠤⣈⠇⡀⢸⡆
⠀⠀⠀⠀⠀⠀⣿⣖⢉⡽⠋⠀⠀⠀⠘⢦⡃⢮⡀⠻⠤⣄⣑⢄⣻⠱⣌⣓⢒⣒⡨⡽⢋⡟⣹⠁⠀⠀⠀⠀⠈⣇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡹⣾⠀⡰⠛⠳⡄⠀⠘⡗⠤⣌⣉⠓⢦⠀⠱⡹⡗⠚⢉⣍⡤⢬⣐⣪⠴⠓⠓⠒⢦⡀⠀⣠⡏⢀⡾
⠀⠀⠀⠀⠀⠀⠈⣇⠈⠻⣧⡀⠀⣿⠀⠀⣟⠀⣭⣌⡽⣦⣍⣉⣉⡻⢦⡪⢊⠜⣉⣀⡍⠀⠀⢠⡄⠈⢷⠞⠋⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠙⠶⣯⣀⢰⠇⣰⣮⣭⣥⣲⣥⠾⡮⠽⠦⢴⡞⠉⠁⠀⢲⡄⠀⠀⠀⠀⣰⡴⠞⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠉⠛⠳⠶⣤⣤⣐⣊⡕⠊⠀⠀⠀⡼⠀⠀⠈⠀⣀⣧⣠⡤⠶⠛⠉⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⠶⠶⠶⠶⠖⠚⠛⠉⠁

⠀⠠⠐⠂⠉⠉⠉⠉⠉⠉⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠀⢨⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠎⠉⠂⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠉⠒⠢⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠄
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⠎⠠⠀⢢⣤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⡼⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠄⣀⠐⠠⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⠠⠄⠐⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⣀
⠀⠀⠶⠤⠤⣄⣀⡀⠈⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⣤⡴⠴⠶⠛⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢁⣄⠼⠁⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠂⠀⠀⠀⠒⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠠⢄⣀⡀⠞⠁⡁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡚⢭⠽⠒⠖⠀⠈⠰⠭⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⢀⠀⠀⠀⠀⠠
⠀⠀⣀⠤⠀⠒⠀⠀⠀⠀⠂⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠦⢄⡀⠀⣀⣀⣀⣀⡀⠀⣀⠤⠊⠀⠀⠀⢰⡇⠼⠀⠃⠀⣀⠀⠠
⠀⠈⠝⢄⣀⣀⣀⠀⢀⣀⣀⣠⡬⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠠⠘⠉⠉⠉⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡀
⠀⠈⠂⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⣈⡡
⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢄⣀⣀⡀⠤⠀⠀⠀⠀⠆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠀⠀⠀⠀⠀⠐⢄⠀⠂
⠀⠀⠀⠀⠈⣠⣄⠰⠦⠦⠄⡤⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠄⣀⣀⣀⣀⣀⡀⠠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄
⠀⠀⠂⠠⠉⢁⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠠⠀⠈⠙⠚⠙⠛⠈
⠀⠈⠁⠀⠀⠒⡔⢢⡤⣔⠊⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⡄⠄⠄⠒
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀⢀⠀⠀⡀⣀⣀⣠⠴
⡔⣚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⢑⣠⠀⠀⠀⠀⠀⣈⡉⣁⡀⣈⣁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠇⠀⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉⠘⠀⠀⠀⠀⠀⢀⣀⣉⢉⠉⡀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠂⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⡀⠈
⠁⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠤⠄⢈⡀⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠘⠡⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡄⠁⡀⡀⢀
⠀⠀⠀⠈⠾⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠙⠋⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠙⠚⠚⠃⠈⢀⣟⡀⠁⠉⠋
⠀⠀⢸⢀⠈⠀⠀⠀⠀⠀⠀⠠⢤⠖⠀⠀⠀⠀⠀⠐⠠⠤⣀⣀⣠⣄⠤⠤⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢠⠂⠀⠀⠀⠐⠀⠁⢀⡄
⠀⠀⢸⠈⠀⠛⠊⠚⠁⠈⣴⠑⠀⠄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠜⠁⣀
⠀⠀⢸⠄⠀⠀⠀⠀⠀⣘⠋⠀⢈⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⡀⠈⠠⠀⠄⡔⠈⢀⢀⠔⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⡞⠄⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⣀⣀⣉⠁⠴⠀⠀⠀⠀⠀⠀⠀⠉⠉⠠⠙⠳⠏⠐⠂⠁
⠇⠀⠘⠂⠀⠀⠀⠀⠀⠆⠐⠤⠀⢂⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠚⢀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠒⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠋⠉⠁⠀⠉⠉⠉⠛⠛⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡴⠶⠊⠛⠉⠉⠉⠉⠉⠉⠉⠑⠲⠶⠶⢦⣀⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⣴⠖⠛⠉⠉⠙⠛⠢⣤⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⡠⠾⣦⡀⠀⠀⠀⠀⠀⣠⡾⢧⠀⣀⣀⣀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠉⠀⠀⠀⠀⠉⠛⠓⠒⠒⠂⠁⠀⠀⠀⠈⠉⠉⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠃⣠⠔⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠤⠀⠈⠢⠀⠀⢻⡆
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠘⣧⡀
⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠁⠀⢹⡇
⠀⠀⠀⣾⠁⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⠀⣰⣿⣋⣡⣤⣄⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⢻⣄⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣟⡉⠉⠉⠛⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣴⡶⠀⣸⣿⣿⣿⣟⠛⠛⠻⢷⣦⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⣸⠇⠈⠀⠀⢿⠀⠀⠀⢠⡾⠋⠁⣠⣾⣿⠛⠉⠙⠻⣷⡀⠀⠈⣿⣧⠀⠀⠀⢸⡇⠀⠀⠀⠇⠀⢻⡄
⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⡿⠀⠀⢠⣿⠃⠀⠘⣿⡿⠁⠀⠀⠀⠀⠉⠀⠀⠀⣸⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⡀⠀⠠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠂⠀⠹⣇
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣠⡦⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⢷⣦⣤⣴⡶⢿⣿⣤⣤⣤⡾⠟⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠄⠀⠀⣿⠁
⠀⠀⠀⠀⠀⢻⡆⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠃⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠆⠀⠘⣧
⠀⠀⠀⠀⣾⠉⠀⠀⠀⠀⠡⠄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⣿⠀⠐⠂⢀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠈⠻⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣤⢠⣶⣶⣶⣶⡆⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⠀⣀⣀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⡴⠋⠁⠈⠙⢶⠒⠤⢤⠋⣀⠤⣤⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢣⢀⡎⡰⣡⠋⠀⠉⠳⢟⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠒⠤⣄⡎⠀⡔⠋⠁⠉⠳⡀⠸⠏⡰⢱⠃⡴⠋⠳⡄⠈⣧
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠋⣠⠔⠒⠒⠢⢬⡇⠸⡁⠐⠧⠀⠀⡇⠀⣟⡴⢃⠜⠁⠀⠀⢸⢀⠏⣇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡔⠉⣀⡠⠤⣈⣀⠤⠚⢠⠖⠂⠑⠒⠦⣇⠀⠑⠤⣀⡤⠞⠁⣸⠋⢠⣃⡰⠓⠦⠴⢇⡼⡸⠈⡆
⠀⠀⠀⠀⠀⠀⢸⣆⠀⡎⢀⠞⠁⣠⠤⠤⠤⠤⠔⢁⡴⢏⡉⠙⠒⢌⠳⢄⠤⠤⣄⣠⡖⠹⡄⢺⣀⢞⢕⣖⣦⡘⢲⡁⢀⡇
⠀⠀⠀⠀⠀⠀⠈⡎⠻⣅⡞⢠⠎⢀⡤⠤⠤⠤⠴⣎⠳⢄⠉⠓⠒⠚⠓⠒⠢⢄⡼⠓⢌⢂⣼⠥⢬⣻⢌⣠⢇⠇⣎⢀⢾⠁
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠑⠮⣰⠋⡠⠒⣊⡓⠢⡄⠱⣄⣉⣁⣉⣉⠉⠑⠢⡀⠑⣄⣸⡿⣛⢦⢸⡯⢕⠋⡵⣲⠟⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠘⡷⢄⠀⠀⠈⠉⠓⠮⢭⣉⣳⡌⠲⠤⠤⠤⠤⣌⠙⠢⡀⠘⡄⠘⡇⣿⠁⢹⣀⡠⠤⠛⠉⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠉⠲⢤⣀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠶⠶⠷⠤⠾⠶⠞⠒⠛⠛⠉⠉⠀⠀⠀⠀⣀⠴⢊⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠈⠉⠒⠢⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠔⠚⠉⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣒⠢⠤⠤⠤⠄⣀⣀⠀⠤⠤⠤⠔⢒⡭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠢⠤⠤⠤⠤⠤⠤⠴⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣀⣤⣤⣴⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⠶⠶⣤⣤⣤⣀⡀
⠀⠀⢰⠿⠿⠯⠤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣿⣿⣿⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⢻⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠋⢿⣿⣿⣿⠟⠉⠀⢸⡇
⠀⠀⠘⢶⣤⣄⣠⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠟⠿⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣋⣉⣀⣤⣤⣾⠇
⠀⠀⠀⢸⣿⣿⡏⢻⠏⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢡⡟
⠀⠀⠀⣸⣿⣿⣧⣸⠆⢀⣰⣾⣿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢉⣭⣽⣥⣤⣤⣤⣤⡀⢸⡇
⠀⠀⣼⣿⣿⠟⠻⣿⣾⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠁⠀⠀⠸⣿⢿⣿⣿⣿⣿⣿⣿⣮⢻⣄
⠀⢼⣿⣿⠉⢠⣾⢟⢻⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉⠉⠉⠹⣿⣿⣿⡆
⠀⢸⣿⣿⢰⣿⠁⡀⣿⣿⢿⣿⣯⡶⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⣴⣆⣀⣠⣤⣤⣠⣤⡀⠀⠀⢸⣿⣿⡇
⠀⢸⣿⡇⢸⡇⠠⡇⣿⣟⢸⣿⣷⣶⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣼⣿⣿⡇
⠀⢸⣿⡇⢸⡇⠠⡇⣿⡟⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣿⣿⣿⡇
⠀⢸⣿⡇⢿⡇⠀⠃⣿⡇⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣾⣇⣀⣀⣿⣇⣨⣿⣿⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉⠁⠀⣿⡇
⠀⢸⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⢸⣿⡍⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣴⣿⠻⣶⡀⢶⣷⠶⣷⣶⢶⣶⡄⠰⣿⣻⣷⡄⠻⣿⡿⢁⣾⡟⢹⣷⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⢻⣿⣀⣿⡇⢸⣿⠀⣿⣿⢸⣿⡇⣼⣿⣻⣿⣇⠈⣿⣧⡀⢿⣧⣼⠏⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠛⠉⠀⠛⠛⠉⠛⠛⠙⠛⠛⠉⠛⠛⠛⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣹⠃⣠⣤⣤⡀⠀⣀⣀⣀⣀⣀⠀⢀⣤⣤⣤⢀⣤⡶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡙⠻⣿⣇⣨⣿⢿⣿⡆⢹⣿⡇⣿⣿⠄⣿⣯⠙⠋⣸⣿⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠟⠉⠻⠷⠾⠿⠧⠘⠿⠷⠿⠿⠆⠻⠿⠿⠟⠈⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣴⣾⣉⢠⣿⠉⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⠉⠁⠀⠁⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠉⣽⣿⡿⠛⠟⢻⡆⠀⢠⡄⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⣾⣿⠏⠀⠀⠈⣻⣿⡆⣈⣽⢢⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢷⣼⣿⣿⡇⠀⠀⠀⣿⣿⣿⡏⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠉⠻⠋⠁⠀⠀⠀⣿⡿⠋⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⢿⠀⠘⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⣿⣦⣀⠀⠀⠤⠤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠤⠦⠴⠶⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠦⠀⣹⣿
⠀⠀⣿⣿⣿⡿⠿⠷⣶⣦⣤⣤⣤⣈⣉⡙⠛⠛⠒⠒⠶⠴⠶⠦⠤⠤⠤⠤⠤⠄⠠⠄⠀⠀⠀⠀⢀⣉⣭⣭⣤⣶⣿⣿⣿⣿⣿
⠀⠀⢹⣿⡿⠃⠀⠀⠙⢿⣏⠉⠉⠛⠛⠛⠛⠛⠒⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⠶⠶⠶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣉⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣶⣤⣤⣀⣀⣀⣀⡀⠀⠀⢀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣶⣶⡿⠿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢶⣦⣬⣭⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣥⣤⣤⠶⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠖⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠓⠢⠤⣄⣀⡀
⠀⠀⠀⠀⠀⣠⠒⠋⠭⠭⠅⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠈⠁⠂⣀⣀⠀⠉⠙⠒⠢⣤⣀⡀
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡆
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⡐⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠒⠒⠠⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠡⡀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠆⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⡀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚
⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠂⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄
⠀⣀⠤⠔⠒⠊⢳⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠒⠒⢤
⡎⠁⠀⠀⠀⠄⠀⢀⣉⠉⠉⠉⠉⠉⠓⠋⠀⠀⠘⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠂⠀⠀⣤⣤⡤⢄⣀⠗⠂⠀⢀⡠
⠈⠓⠢⠤⢄⣀⠀⠀⠀⠈⠉⠉⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠐⠒⠂⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣒⡤⠦⠤⠭⠖⠒⠚⠉⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠀⠀⢀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠤⠤⠰⠐⠒⠒⠒⠚⠈⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⣛⣘⠠⡄⣅⣉⣉⣉⠉⠅⠠⠀⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠠⠤⠦⠦⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠁⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⡡⢭⣝⠀⠘⠒⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡏⠛⠀⠠⠭⠀⢔⣍⡐⠠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⣌⠄⡘⡇⠐⢤⡤⡀⢴⡤⠈⡦⠄⡈⡴⡂⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣝⣀⡘⡉⠀⠄⠓⠞⠁⠗⢄⠻⢋⠤⠫⡄⠀⣀⣱⠘⢓⣤⡉⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢠⠭⢄⠁⠂⡀⠉⣀⡆⣮⠤⠶⠟⢀⣀⡀⠇⠉⢉⣈⠄⢍⡺⢿⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡐⠏⡫⡁⠬⣧⡀⢀⣀⡈⠋⢀⠁⣀⡢⠈⠬⠒⠛⠴⡄⡈⢀⠂⣇⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢍⠛⠶⣾⣤⣃⣀⣚⠂⠊⠀⠀⡐⠴⡄⢾⢗⣀⣀⣄⣠⣬⣥⠶⠞⢛⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣬⣉⣉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣉⣠⡤⠴⠚⠉⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢈⠉⢉⠙⠛⠛⠛⠟⠛⠛⠛⢉⠩⠉⠉⠈⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣔⠡⡀⠀⠈⠅⢒⣤⣀⢤⠀⢐⡖⠖⣦⢀⠀⠠⠀⠄⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣊⢐⠀⢀⠀⢸⡷⡿⡿⣻⢯⢫⢣⠍⠹⡅⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⢸⡇⣼⡳⣻⣾⣺⣿⠿⠧⠆⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⢀⠠⠩⠄⠍⠈⠡⠓⠁⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀⠀⠀⢀⡃⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠶⠶⠶⠶⠛⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡄⠲⡶⠶⠦⠄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠴⠭⡀⢍⣘⡁⠐⣋⡓⠂⠹⡭⠡⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠓⢆⡀⢄⡀⠀⢹⠀⠀⢀⠀⠀⠛⠁⠈⠂⡰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⠼⠓⢊⠠⣴⠶⠄⢀⠁⠀⠀⡀⠀⠀⠐⠛⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠐⠁⠀⠀⠀⠈⠐⠆⠀⠁⢀⠔⠒⠋⠙⢐⢾⡀⢆
⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⢣⠀⠉⠁⠀⠀⠀⠀⠀⡐⠀⠚⢠⠃⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡧⠓⠒⢯⠀⢠⠶⠒⠢⠐⠀⠀⠀⠀⠀⠘⠂⠀⠀⢒⠀⠀⠠⣄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠸⡇⡀⠀⠸⣦⠀⠀⢀⠐⣰⢧⠶⠲⡶⠀⠀⠀⠀⠀⠀⢷⠂⡐⠡⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢀⠀⡼⠇⢀⠒⠺⢡⡝⠀⠀⠁⠀⠀⠀⠀⠀⢀⣞⢰⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠨⠉⡫⠴⠗⢠⡄⠀⠀⣀⡤⢀⣜⣉⠨⡄⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⠀⡆⠺⡿⠇⠈⠰⠁⠀⠀⠔⣈⣡⠿⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠂⠀⠀⠁⣀⢰⠸⠻⣄⡁⢘⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢀⣶⣖⣂⢦⠙⠀⡤⠢⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠤⠖⠒⠒⠒⠉⠉⠉⠁⣀⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠒⠒⢒⣂⣉⣉⣭⣤⣤⡤⠤⠶⠶⠖⠒⠒⠛⠒⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣭⡝⠉⡧⠽⠤⠤⠤⠤⠤⠤⠤⠤⠴⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⢀⠁⢀⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢤⡤⡤⠡⢡⠦⠄⠈⢸⠤⠤⠤⠤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠉⡉⠄⠤⠤⣄⠀⢀⡀⢸⣰⢠⣘⡘⣄⠠⠤⠀⢤⡠⣀⠀⠈⠉⠓⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⡠⠒⠀⠈⠀⠀⠀⠒⠒⠠⢄⣀⠶⠶⢈⣈⠗⢍⠐⣂⣓⣂⡓⢄⡂⠁⠂⣁⠢⡀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⡠⠐⠒⢒⣒⠒⠒⢤⣉⠢⡑⢵⡄⢀⢒⣄⠂⣉⠨⢝⡢⢴⣊⠅⢀⠲⠶⠒⡪⢕⠄⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠙⠒⠦⢄⣄⡀⠈⠬⢁⡶⢋⠡⠤⢌⡙⠲⣶⠞⡫⠍⠁⠬⢍⣒⣲⡛⠓⢦⢠⣐⣦⠤⢒⠋⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠉⠙⠛⠛⠒⠒⠶⠲⠮⠷⠼⠧⠶⠭⠽⠒⠒⠒⠒⢚⡋⠭⠁⠀⠀⠈⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠓⠒⠤⠤⢀⣀⣀⣀⣀⠡⠤⠤⠖⠋⠁⢘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠈⠑⠂⠂⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠖⠢⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⣤⣄⢀⣀⡀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠻⠿⠿⣿⠿⠃⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠶⣶⣶⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣄⠀⠀⠉⣩⠟⠛⠻⢅⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⢡⣶⡿⠀⠹⣾⣿⣂⣀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⢀⣀⣄⡀⠀⢸⣿⡿⣶⡄⠙⣿⣿⡋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⡶⠷⠒⠀⠀⠀⠀⢽⣿⣿⣿⣿⡇⠀⠙⠛⠛⠁⠀⠈⠻⠿⠷⠟⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⡇⠀⠀⠀⠈⠿⠿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢴⠂⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣮⠀⠀⠀⣸⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠠⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⡀⠀⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣀⠀⠀⠙⢟⠛⢋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⡳⠀⠀⠀⢤⣄⠀⠈⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⢀⣀⣿⣿⣧⠀⢿⣿⠏⠙⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡰⠁⢻⣿⣿⣿⠏⠀⠈⠙⠀⠀⠖⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣀⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣠⠴⠋⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠶⡤⠀⠀⣀⣠⠤⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⡀⠤⠐⠂⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⢤⠄⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣞⠛⣳⠄⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⢁⣠⣎⣉⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠍⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢯⣁⠡⠝⠃⠀⠀⠀⠀⣐⠲⣤⠤⠀⠒⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣩⣤⡤⣀⡀⠀⠀⣀⣀⡀⠤⠤⠐⠒⠂⠉⠉⠈⢁⣀⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣡⡼⣇⠈⢊⡾⠋⠉⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⢠⠞⠻⠮⡺⣝⢷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⣯⠀⠀⢘⡩⣲⡟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢠⣞⣳⠄⠀⡀⠀⠀⠀⠀⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣴⡟⢫⡄⠐⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠉⠁⠀⠀⠀⠀⠀⢀⡞⠛⠥⣀⡝⢵⡄⠀⠀⠘⠀⠀⠀⠈⠈⠑⠈⠁⠀⠀⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢆⠦⠀⠘⣧⡀⠀⠀⢊⡼⠁⠀⠀⠀⢰⠒⣦⠀⠀⠀⠀⠀⠀⠀⠓⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠠⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠋⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠴⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠴⠒⠒⠂⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠒⠒⠃⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠓⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢄⠀⠀⠀⠀⢀⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡀⠉⠲⢄⠑⠀⠐⠚⠛⠛⠛⡃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⣏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣀⣤⣼⣧⠤⠤
⠀⠀⠀⠀⠁⠒⠢⢤⣀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠔⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⢉⣿⣿⡂⡇⣀⠀⠀⢀⢀⣠⡋⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢀⣭⣦⢕⣛⣤⡄⠀⠀⠀⢈⣡⣶⡀⡀⠟⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢃⣽⣾⣿⣿⣷⡄⠀⢠⣞⣈⠶⡘⣧⣼⡉⠉⡉⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⡠⠚⠋⠁⢉⣷⣾⢻⡷⡤⣽⣷⣴⠉⡟⠁⠈⠉⠃⠀⠘⢿⣽⡆
⠀⠀⠀⠀⠂⠀⠀⠀⠸⣷⡀⠀⢀⡞⠁⠀⠀⠈⠀⠀⠀⠀⠈⠉⠛⢿⡿⣯⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⢿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣿⣆⠀⠀⠀⠀⡀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡆⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠖⠒⠚⠛⠛⠒⠀⠒⠲⠼⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢹⡀⠀⠀⠀⢸⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⢜⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠂⠠⠀⡄⠉⠒⠂⠤⠤⠠⠤⠄⠀⠤⠄⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣤⣀⣀⣀⣀⡴⣤⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣛⠕⣿⣭⠟⣀⢨⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠉⠙⠛⠳⠶⠛⣁⠤⢋⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠶⠶⠷⠿⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⡤⠶⢤⡀⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠍⠀⠀⠀⠈⠳⠤⠔⠁⠀⠀⠀⠀⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠈⢁⡴⠒⢲⣤⠤⠤⠴⠖⠂⠀⠃⠀⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⠷⠖⠒⠒⠋⠉⠉⠳⠤⠼⠁⠀⠀⠀⢀⠤⢤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡼⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⢠⠖⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⡼⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠐⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡋⠁⠉⢻⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠿⠿⠟⢛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠿⠿⠟⠁⠀⠀⢩⣭⣭⣴⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⠋⠙⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢠⠆⢠⡇⣠⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⣧⠀⠀⣿⡩⢹⣄⣾⢱⢠⠎⣿⠀⣾⠁⣗⢉⡴⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠸⡆⢸⠹⣏⡹⠹⡤⠋⠹⠞⠹⠞⡏⣀⣈⡭
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠹⡾⠀⠈⠀⢀⣀⣀⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⡤⠔⠒⠋⠭⠉⠀⠀⠀⣀⠈⣽⡉⢌⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠠⠒⠒⠒⢤⡎⢩⠞⣄⢜⠬⡪⡢⡱⠀⢀⠈⢻⠲⢄⡀
⠀⠀⠀⠀⠀⡰⢋⠀⣀⠤⠰⣧⣀⡠⠴⢛⠒⢿⢂⠣⠑⢕⢕⠴⡕⡝⠁⡥⠥⠁⠄⠙⢦⡀
⠀⠀⠀⠀⢰⠃⢀⣀⣇⠤⡶⢥⣔⡉⠔⣒⠦⠨⢷⢣⢑⡴⠎⡢⡸⡮⣦⠈⢲⠀⢈⡄⡀⠱⡄
⠀⠀⠀⠀⢸⡞⢕⠂⢶⣛⣀⠝⠊⡩⢝⠐⣈⠡⠘⣏⢆⠁⢆⣈⣟⢜⠌⢺⣦⣭⣕⣁⡇⠀⢳
⠀⠀⠀⠀⠈⣷⢁⡴⠉⠀⠀⠙⣼⠱⣁⠪⠄⣘⣤⣊⢪⣓⣂⣀⣬⢓⠟⣿⠀⠀⠀⠈⢳⠀⢸
⠀⠀⠀⠀⠀⢯⠳⣄⠞⠉⢦⠀⢸⡒⠤⢬⡑⢄⣹⣽⡑⢁⠬⢔⡒⢭⠚⠉⠉⠲⡄⢀⠞⢠⡏
⠀⠀⠀⠀⠀⠘⡄⠈⠳⣄⣸⠀⢸⠃⡰⠖⢛⠶⣲⠶⣝⢓⣱⠥⠔⢦⠀⠀⠀⠀⢹⣁⠴⡟
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠉⠓⠿⣤⣍⣉⠩⠛⠕⠁⠀⢠⠁⠀⠀⠀⣇⣀⣤⠴⠚⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠒⠚⠒⠛⢛⠋⠉⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⣀⣀⣀⣀⣠⠤⠶⠚⠉


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠤⠤⠶⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⣀⣀⣰⢥⣤⡚⢻⠿⣿⣷⡶⡖⠂⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡆⠀⠍⠁⠀⠀⠀⠀⠓⡶⠛⠿⣏⠀⠀⠙⠣⣭⣥⠅⠀⠒⠲⠒⠒⠓⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠂⠄⠀⠀⠁⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠠⠀⠀⡄⠀⠀⠠⣀⣠⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⢾⣄⠠⠀⢴⡄⠀⣱⣄⣉⣄⡟⢡⠚⢳⠄⡘⠉⠁⠀⢰⣄⠀⠀⠀⠀⠀⠀⠈⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠘⣜⣿⠑⢊⠩⠰⠆⠀⡰⠐⠀⠑⠄⢙⣵⣬⣝⠁⠈⠲⠷⢃⡀⣾⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠸⡏⠙⠛⠒⠿⠾⠯⠤⣾⣾⣷⣶⣿⣮⣿⣿⣿⣿⣿⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣠⣤⣴⣿⣿⣿⣿⣿⣿⣟⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⢀⣐⣒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣄⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡶⠻⠿⢭⣽⣿⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠚⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠶⠒⠒⠒⠒⠒⠒⠒⢶⣶⣒⣒⡲⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠉⠀⠀⠀⠀⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠂⢤⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡏⠀⠀⠀⠀⠈⠉⠒⡤⣄⡀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠁⠂⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡂⠀⠀⠀⠁⠒⢤⡜⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⢀⣀⠀⠀⠀⠘⠓⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⡇⣿⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠰⣇⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⡇⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⡇⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣶⣷⠀⠀⠀⠀⡀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⢷⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡁⠰⠀⠀⠀⡀⠀⠀⠉⠀⡤⠀⠀⠀⠀⠀⠀⢧⣾⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⢈⠂⢄⠀⠀⠾⠃⠀⠀⠀⠀⠐⠀⡄⠀⠠⡿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣌⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠘⠇⡇⠀⠀⢠⠃
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣤⣤⣀⣀⠀⠈⠙⠓⠤⣐⡠⠐⢶⡄⠀⠀⠀⣣⠀⡠⠃
⠀⠠⣾⣿⣿⡟⣿⢻⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠉⠑⠒⠤⢤⣠⣼⣿⠔⠁
⠀⠀⣿⣿⣿⣏⣿⣣⣃⣕⣩⣼⣿⣿⣿⠇
⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⢚⣛⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣤⡤⠶⠚⣛⣉⣭⡴⠶⠚⠛⠉⠉⣀⣀⣤⣴⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⢛⣋⣩⣤⠴⠖⠛⠉⣁⣠⣤⣤⣶⡾⠿⣟⣛⣭⡭⠷⠖⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡞⣱⢞⣫⣭⢹⡶⣦⣤⡤⠶⢚⣛⣩⣥⠶⠖⠛⣉⣉⣤⣴⣶⠾⢿⣛⣛⣭⡭⠷⠖⠛⠋⠉⠁⠀⠀⠀⢀⣀⣀⡀
⠀⠀⡀⠀⣀⣤⣾⠿⢿⣷⣿⣼⣿⣿⣾⠿⣯⣶⠞⠛⣉⣉⣤⣤⣶⠾⢟⣛⣭⡭⠶⠚⠛⠉⠉⠁⠀⠀⣀⣀⣠⣤⡤⠶⠖⠛⠛⠉⠉⠉
⠀⠈⣠⢞⣥⣾⣿⣿⣟⣿⣿⡿⣯⡻⣎⢳⡙⣏⢿⣿⣻⣿⠿⠿⠛⠛⠉⠉⠀⣀⣀⣤⣤⠴⠶⠒⠛⠋⠉⠉
⠀⣼⣿⣿⡿⠟⢛⣋⣩⣿⣝⣿⣎⣷⢹⡄⣿⣿⢈⣏⣁⣀⣠⣤⡤⠶⠞⠛⠛⢉⠁
⢀⣿⣿⣴⣶⣿⣿⣿⡭⠷⣿⣿⡟⣿⢸⣷⣇⡟⣼⣿⡉⠉
⣸⢿⣿⣿⣋⣉⣤⣴⣶⣾⣿⡿⣱⢇⣾⣽⣿⣷⡟⢿⣻⡄
⣿⣾⠈⠉⠉⠹⣟⣿⣿⣿⣿⣳⢫⣿⡿⣿⢿⣟⣧⠈⠳⠛
⠈⠁⠀⠀⠀⠀⣿⣿⢸⣹⡇⡏⣾⢸⣿⣼⣻⣿⣾⡷⢦⡀
⠀⠀⠀⠀⠀⢠⢟⡿⣯⣿⣧⢷⣿⡿⣿⡏⣿⠉⣿⣿⣷⢻
⠀⠀⠀⠀⠀⢻⣸⣷⣿⣿⢹⣧⡻⣷⣧⢻⡘⣿⠵⠋⡟⣿
⠀⠀⠀⠀⢠⣞⣽⣾⣿⡿⣿⣏⣷⣜⣿⣿⣳⣼⢷⡄⢿⣻⣄⡀
⠀⠀⠀⠀⢸⣽⡇⠈⢻⣿⣿⣿⣿⣽⣞⣿⣿⢿⣷⣻⠀⠙⢾⣝⢶
⠀⠀⠀⠀⣸⣿⠃⠀⠸⣷⣿⣿⡻⣯⣹⣿⡿⣿⣿⣿⠀⠀⠀⠿⠾
⠀⠀⢀⠀⣿⢿⡀⠀⠀⣿⣿⡏⣿⣶⢿⣿⡟⠋⢿⣿⡀
⠀⠀⠘⠆⠘⢯⣷⡀⠀⡟⣿⡇⣿⣷⠞⣼⠃⠀⠈⢻⣝⣦
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢷⡻⣷⣿⢣⡟⠁⠀⠀⠀⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠀⢠⣿⣿⢫⣷⣿⣶⣤
⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⢣⢿⣿⣿⣿⣯⣴⣞⢲⡄
⠀⠀⠀⠀⠀⣾⣫⣷⣶⠖⠋⠀⢻⣷⡌⠉⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠛⠁⣿⣿⠀⠀⠀⠀⢻⡿⣦⡀⠠
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠙⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠘⡄⠀⣠⣶⣀⠤⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⢖⠒⠠⡀⠀⠀⠀⠁⠀⠉⠳⢿⣟⠶⣭⣒⡢⢄⡀
⠀⠑⣄⠈⠒⢄⡀⠀⠀⠀⠀⠀⠉⠛⠶⣝⡻⣷⣮⣕⡢⠤⣀
⠀⠀⠈⠢⡀⠀⠈⠒⢶⣲⠶⠿⠿⠝⠛⠛⠛⠳⢮⣽⡿⢗⡶⣭⣓⠲⢖⣢⡤⢄⣀⡀
⠀⡠⡶⠟⠛⡆⢰⠉⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⣕⡪⢝⡲⢮⣍⡛⠥⣛⡻⢶⣄
⡜⡜⠀⠀⠀⡇⠘⡀⠀⠀⠈⢟⣤⣴⣶⠶⡶⢶⣲⣲⣲⡶⡶⡾⣿⣷⣮⣕⡢⢍⡓⠶⣬⣉⡛⠷⢄⡀
⡿⣧⣀⢀⡠⢧⡀⣇⠀⠀⢀⣨⠷⡭⡝⣯⣾⣿⢿⣟⣫⣾⣷⣿⡏⣹⣿⣭⣿⣷⣦⣍⠒⠭⣲⣤⣒⠤⣉⠒⠠⠄⣀
⢱⠈⠙⠫⠿⣛⣽⠽⣹⣽⣹⣲⢸⠣⡓⠝⣖⣱⡩⢾⣝⠜⡴⡫⢲⡻⣯⣿⣟⣿⣾⠿⣺⣕⡢⢍⡓⠨⢑⡊⠕⣂⠤⢉⠰
⠘⡦⢄⡀⠀⠀⠈⠉⠉⠒⠛⠛⠋⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠭⠭⠭⠍⠒⠒⠈⠉⠀⢀⡨⣷⠨⣑⠢⢌⡑⠢⣌⠉⠉
⠀⠱⣀⣈⣑⡒⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠐⣾⡿⢿⠇⠀⠀⠉⠓⢮⣿⡽⠃
⠀⠀⠹⣏⡨⣝⢗⠤⣀⠤⡸⠿⢿⣍⡉⠉⠑⣒⣲⣶⣖⡒⠒⠒⢂⡩⡻⠿⠶⡖⠴⢲⢢⢚⠇
⠀⠀⠀⠘⢮⡪⢷⣵⣲⣾⡮⠭⢭⡿⣮⠭⢽⠟⣋⣐⣽⡮⣑⣒⠝⡫⣒⢭⣽⣾⣷⣵⣷⠋
⠀⠀⠀⠀⠈⠫⣵⣿⣺⣿⢟⡚⢅⡛⠾⠿⢷⣽⠦⠴⢿⣿⡷⡶⣿⣾⢿⡫⠭⠽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣀⠈⠑⠂⠭⠭⠝⠊⠉⠉⠉⠛⠛⠍⠓⢂⣁⣠⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣭⡍⣉⣁⣀⣀⡀⠀⠀⠂⣈⣉⡉⠭⢔⡲⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠶⠒⠚⠋⠉⠉⣡⠞⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠴⠶⠶⠶⠒⠚⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢸⡿⠥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠉⠛⠲⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⢦⣤⣄⣀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡴⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇⠀⠀⣾⡷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣞⣱⠆⠀⠀⠀⠀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣅⣠⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠺⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠶⢤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠦⣤⣤⣀⣀⣀⣿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢡⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⡀⢢⢣⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡤⢤⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠧⢣⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⡄
⠀⠀⠈⠙⠻⠏⡐⠦⢄⣀⠄⡀⠂⣠⡇⠀⠁⢡⠈⠢⠀⠀⠀⡀⠀⠀⢐⠀⠿⠋⠁⠈⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠨⠁⡒⠠⢀⣂⣤⣀⣐⡠⣌⢄⣀⡀⠈⢤⣠⣄⡉⣠⡁⠰⠄⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡤⠂⠉⢐⢾⣧⣻⡟⢉⣵⣾⣿⣿⣥⣶⣺⢶⡤⡤⣤⡙⢋⡱⠢⢤⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠛⠾⡦⣽⣵⣄⣰⣿⣿⣿⣧⡿⣿⣯⣿⣻⢦⣽⢻⣻⣯⣬⣷⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠛⠛⠛⠛⠛⠛⠛⠟⠛⠒⠉⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠉⠛⢻⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⠿⠿⢛⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣅⣀⠐⠀⠁⢀⠈⠉⠉⠉⠉⠉⠍⠉⠉⠀⡀⠀⢠⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠿⢷⣶⣶⣤⣤⣤⣤⣤⣥⣬⣴⣶⠶⠿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠈⠉⠉⠉⠁⠀⠠⠔⠊⠁
⠀⠀⠀⢠⣤⣤⢤⣤⣤⣤⡤⣤⣤⡄⠀⠀⠀⠀⠁⠒⠒⠀⠀⠐⠒⠂⠁
⠀⠀⠀⢸⣿⡏⡸⠙⠋⠋⠀⠛⣿⣧
⠀⠀⠀⠙⠿⠿⠷⠶⠷⠷⠶⠾⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⡴⢠⣾⡿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶⠈⠀⢟⣛⡁⣸⣀⣀⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣷⠋⠴⡟⢰⠏⣾⢻⠛⠟⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠃⠘⡄⢱⡌⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠐⠚⡄⢹⡄⣷⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⠀⠋⠀⣸⢀⠟⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⡀⠈⠀⠀⠘⠆⠀⠃⠚⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀⡆⠀⠰⠃⠰⠿⣇⢰⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡃⠁⠼⠛⢿⣿⠀⠀⡀⢷⠀⠁⠀⠀⠳⠄⢀⠟⢿⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⢠⠀⠹⠟⣀⠸⡁⠈⠀⠀⢀⣸⣶⠟⢿⠀⣼⣿⣥⣄⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡰⠸⣏⢉⡟⠀⠇⠠⢻⠀⠉⠀⠀⠀⣀⣾⣿⣿⣿⠀⢀⣷⣟⣿⣿⣿⣿⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡖⠑⠂⣍⣀⡿⠈⠐⠃⠈⠁⣀⢦⡶⢾⣿⣿⣿⠃⢈⣱⣷⣿⣾⣿⣿⣿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣾⡿⣿⣿⠿⠷⠀⠺⢦⠚⠿⣿⣿⣿⠖⠻⠿⠷⢼⣿⠷⢿⣿⣿⣽⠻⣄⠷⢸⣿⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⣿⢁⣶⣶⡶⠆⠉⠉⣦⠄⣾⣏⡻⠝⠂⠀⠀⣂⠈⠀⠐⠀⢟⡉⠉⠈⣄⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣿⣿⡋⣴⣦⣈⣩⠴⠆⡩⠋⠁⠀⠀⠀⠀⠈⡀⢱⠀⢠⠈⠄⠀⠀⣉⣧⣤⣍⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣽⣿⣿⣿⣿⣹⣏⠀⠸⠿⠶⢤⣀⠀⠀⠀⠀⠰⠀⢃⠀⢃⡨⠢⣀⣉⣠⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠔⢁⣽⣿⠟⠳⡶⠖⠀⣤⡿⠷⢦⣀⠀⠀⢀⠠⣃⢺⣤⡄⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣅⣠⠈⣻⣿⣿⣷⣾⣿⣤⣦⣶⣶⣶⣭⣶⣶⣿⣾⠇⢸⡏⢀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡇⠐⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠰⡠⢄⡀
⠀⠀⠀⠀⢾⠉⠁⢠⠴⠀⠩⣀⠀⠤⠊⠁⣤⣀⡄⠈⠃⢀⢄⠤⠄
⠀⠀⠀⠀⠀⠙⡖⠀⠀⢈⣐⠂⠐⡀⠘⣢⣠⣶⣲⡄⢩⣵⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⣾
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⠈⠛⠁⡨⢀⣼⠛⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡾⠃⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⠀⠀⠀⢀⣠⡾⠿⣗⡤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⣀⣈⣉⣹⡑⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠠⡤⠠⣤⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣈⠉⣉⣩⡥⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠁⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣤⡄⠐⢲⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⠀⣀⣀⠀⠀⠀⣀⠀⠈⣻⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⢀⡀⠉⠀⠘⣓⣤⣢⡍⣳⣄⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠁⠀⠀⠀⠔⠠⠀⠉⠉⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠒⠀⠙⠉⠉⢉⡖⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠈⠉⠁⢱⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣈⣵⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠔⠈⣉⡡⢿⣶⣄⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⠁⠀⠀⠀⠀⣿⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠘⡄⠀⠀⡄⠀⣠⡽⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⣀⡠⠽⢂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠐⠠⠤⠐⠃⠀⠀⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⣤⣶⡛⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠈⢁⠀⠠⠀⠀⢀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠒⠒⠒⠒⠒⠲⡊⢸⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠁⠒⠲⢻⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⠄⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠘⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢋⠵⢺⣿⡟⢡⠴⣲⣶⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠴⠶⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⡔⠿⠋⠀⠀⣾⡿⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣧⠖⣻⠁⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⣉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠃⣰⣣⠞⠁⢀⣠⠴⢂⣡⣔⠚⠙⠺⠿⠃⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡜⡱⢃⡴⠚⡛⣡⢞⣡⠂⠀⠀⢦⡘⢾⠅⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⡇⡞⠀⣀⣸⢟⡠⠂⠀⠄⢀⣴⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⣿⣇⠀⣩⡿⠋⠀⠀⣴⠀⠈⠙⢛⣉⠂⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡎⢀⣈⣾⣿⣿⣶⣶⡾⠃⣀⠘⢀⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣹⡿⠾⣿⣿⣿⠟⠾⠛⣅⣾⣿⣿⢟⠄⠠⡀⠀⠀⠀⠀⠀⠀⣙⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣙⣷⠅⡼⣿⣦⣝⣷⠾⢿⣁⣈⢕⣱⣂⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢩⣿⣷⡂⣸⣯⣷⣈⣙⣛⣿⣵⠻⢿⡄⣃⡐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⡾⠿⣿⣷⡇⠘⠷⣿⢯⣹⣖⣷⣄⣀⢀⣹⣼⠆⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⣤⣤⠙⣡⣶⡀⡺⢓⣽⣢⣿⡿⠛⣻⢻⠆⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠿⠾⠿⠋⠙⢿⣄⣉⠹⡿⢿⣾⣶⣿⡆⠰⠀⠀⠀⠀⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠞⠃⣠⠀⣤⣬⣿⣷⣿⣿⣶⡶⠌⢯⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣴⣧⠞⠈⢀⠶⠿⣋⠄⡼⠁⠐⠚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⣀⣀⣀⣲⣽⣾⠥⠊⡠⢀⠄⠸⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠒⠋⠀⠀⠀⠀⠀⠐⠒⠛⠐⠓⠚⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠶⠶⠖⠒⠒⠒⠲⠶⠶⠶⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢖⣛⣭⠥⠶⠒⠛⠛⠋⣉⣉⣉⣉⣉⣙⡛⠛⠒⠶⠦⣭⣝⡛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⢞⡽⢞⣯⠥⢤⣤⣤⣤⠶⣛⣭⠭⠶⠶⠶⠿⠭⢭⣟⣲⢦⣄⣀⣈⡿⢿⠷⣽⡷⣄
⠀⠀⠀⠀⠀⠀⣸⣿⠏⣠⡏⠀⢀⣀⣈⠳⣖⢻⠿⢖⣿⠿⣛⣻⡿⠿⣒⣶⣭⣟⣢⡽⣧⣾⡏⣳⢘⡷⣌⣧
⠀⠀⠀⠀⠀⠀⣿⣿⠞⢹⡇⣰⠋⠁⠈⢳⣽⣦⠞⣫⠶⣛⣭⢶⣾⠿⠿⠿⢭⣌⡹⣟⠿⢯⡴⢋⣾⢷⣽⣿
⠀⠀⠀⠀⠀⠀⢿⠘⢧⡀⢳⣽⣆⣀⣀⡼⣹⢇⣛⡵⠟⠉⣴⢏⡵⠀⠀⠀⠀⠈⣷⠛⢲⡶⠶⠋⠁⣠⡟⣼
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠙⠳⣽⣿⣯⠭⠶⠻⣾⣝⢷⣦⣀⣧⠸⣧⠀⠀⣀⣀⣴⠃⠀⠀⠀⣀⣤⠞⠉⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠹⣟⢦⣄⡀⠈⠙⠛⠲⠶⢤⣽⣷⣾⣉⣈⣛⣶⣶⣋⣭⣥⡤⠴⠶⠚⠋⠉⢀⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣤⣉⡛⠲⠦⣤⣄⣀⣀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣀⣀⣀⣤⡤⠶⠚⣋⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡙⠛⠲⠶⣤⣭⣉⣉⣛⣛⣛⣛⣛⣛⣛⣋⣉⣭⣥⠴⠶⠚⣫⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣦⣤⣤⣤⣄⣀⣤⣤⣤⣴⣶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠶⠶⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠤⠄⠐⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠀⣒⣒⣒⣀⡩⠥⠤⠤⠤⠄⠒⠒⠒⠒⠚⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢭⠋⠻⠉⠀⠀⠀⠀⠀⠀⠒⠒⠐⠒⠒⠒⠒⠂⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠄⠒⠒⠀⠒⠂⠀⡖⠂⠀⠂⠠⠤⠄⣀⡀
⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⠀⠀⠀⠂⠀⢀⣭⠘⢢⠐⠨⠡⢄⢄⠂⠀⠀⠈⠉⠐⢤
⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠐⡒⠐⢤⡀⠈⣧⡀⠨⣕⣈⡈⣓⠤⣅⠁⠄⠈⠒⠂⢀⣠⠃
⠀⠀⠀⠀⠀⠀⠆⠉⠐⠒⠤⠄⣠⣏⣀⣒⣒⣍⢻⣉⡀⣀⣀⣀⡺⣍⠷⠤⠖⠒⠉⡘
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠐⠒⠂⠠⠀⠄⠐⠒⠊⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠒⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⣄⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣄⣀⣤⡴⠶⠶⠶⠟⠁⠟⠹⠛⣻⣆⣼⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⠏⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⠹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠈⠻⣿⣷⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁
⠀⢰⣕⡄⢀⣤⠀⠀⠸⠵⠁⠀⢰⡏⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠸⡄⡶⣏⢳⣀⡤⠤⠾⠷⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢿
⠀⠀⣀⣀⠀⠙⣧⢿⣆⢳⣾⣿⣿⡿⣟⢦⡙⣆⠀⠀⢀⣴⣶⣦⡄⠀⠀⠀⠀⣼⣿⠿⣷⣼
⠀⠀⠳⢬⣓⣚⣷⡏⣿⣛⡿⣫⣶⣽⣾⣤⡇⣿⠀⣰⣿⣻⢭⢻⡷⠀⠀⠀⠀⢿⡯⣏⣿⡿
⠀⠀⠀⢀⣿⠍⠛⡿⢽⠷⣾⣯⣤⣍⡛⡆⣷⣿⢠⣿⡇⣷⡾⣼⠃⢀⢠⣖⣦⠈⢷⣿⣿⣃⣤⠤⣤
⠀⠀⠀⠿⡶⣤⠀⠴⢋⣩⣴⣚⣿⣿⣿⣷⣾⣏⠀⢿⣿⣾⠾⠃⢀⣼⢷⣟⣛⣿⣿⣿⣻⣶⣒⣛⡿
⠀⠀⠀⠀⠰⣯⣾⣷⣿⣯⣭⣭⠖⢋⣥⣾⣯⢟⡿⣷⣟⠒⠒⢾⢻⡿⣿⢹⡏⣿⣿⡿⢿⣿⠉
⠀⠀⠀⢀⢀⡀⠀⠀⠉⠙⠛⠛⣿⣟⣿⣟⣯⠞⠻⢿⣿⣿⣷⣚⣿⣿⣿⠸⣧⠻⣿⠤⢾⣋⣤⣠⡴⡶⣦⣾⣽⣇
⠀⠀⠀⠙⠓⠃⠀⠀⠀⠀⠀⠀⠈⠉⢹⣟⠷⣤⣀⣀⠀⠀⢀⡿⢰⣷⠹⡯⣿⡟⣉⣼⠋⢻⣿⣿⢳⣾⣣⣩⢻⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡟⣏⢻⡶⣬⣭⣉⣛⣛⣱⣿⣿⠷⠚⣩⡾⣿⠇⣿⠈⣿⣽⣼⢡⡟⣷⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⡿⣾⣵⡿⢟⣭⣿⣿⣿⢹⣿⣷⡾⣏⣳⣿⣮⣿⣤⣿⣾⡻⣿⣾⣿⡿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣾⢟⣽⠿⣿⣿⣰⣾⣿⣛⣾⣿⡟⣿⣹⡿⠿⢶⣽⣎⢷⡹⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⢇⣿⢃⣴⠟⣻⡿⣿⣿⣽⣿⣿⡇⣿⣿⣼⣻⣷⡽⡿⣏⢷⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢻⢹⣿⡇⣼⡏⣼⡏⣰⡟⣹⣿⣿⣇⢿⡘⣿⣿⡋⢠⣇⡇⣿⡾⣇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣼⣧⣿⣧⣿⠀⣿⢀⣿⠀⣿⢿⣿⡹⣮⣻⣌⡻⠿⣿⣿⣡⣿⣿⣛⣻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇⣿⠸⣿⠀⣿⠸⣿⠿⣬⣛⢿⣿⣶⣾⣿⣿⣿⢟⣯⡥⢶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣿⣇⣿⣧⣽⣷⣮⣭⣭⣿⣿⣿⡍⠛⠿⠿⠷⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣩⠶⣛⣿⡷⠶⠾⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⢏⣩⣿⠀⠀⠀⠈⠻⠿⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣩⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠟⠛⠛⠛⠛⠛⠻⢿⣿⠿⢿⡶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣆⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣠⣤⠴⣶⣶⣶⡶⢿⣶⣾⣿⣦⣀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⣣⣤⣤⣶⡿⠓⠒⠶⠿⡿⠃⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣈⣉⣀⣀⣴⣶⣶⣦⣀⣀⣀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣉⣉⠉⠉⢉⣉⠉⠉⠉⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠉⠉⠉⠉⠀⠈⠉⠀⠿⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣀⣀⣀⠀⠲⠦⣤⣤⣤⠰⠞⢀⣠⣤⣤⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⢛⡻⢷⣤⣄⣀⣤⣤⣴⣶⡿⠟⠉⠙⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠨⠙⣿⡿⠛⠛⠛⡇⠀⣴⣼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠃⢠⡄⢉⡄⠀⠀⠀⢇⠀⠈⡏⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢠⣀⣿⠀⠀⢹⡿⠃⠀⠀⠀⠸⠀⠀⠁⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠻⣿⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢣⡄⠀⢹⡅⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢃⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢡⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⡼⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⢡⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⡴⠀⣸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠠⣆⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⢠⡞⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣿⣦⠀⠀⣠⠈⢀⣀⣴⠃⠀⠀⠛⠀⣤⡀⠀⠀⠀⠉⠀⣠⠀⠘⠁⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠃⠈⠀⠾⠿⠇⠸⠿⠿⠄⠀⠀⠰⡦⢹⠧⠦⠀⠶⠶⠎⣿⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠤⠤⠤⠶⠶⠶⣶⣦⣤⣤⣶⣶⣶⣶⣾⣶⣤⣤⣤⣴⣶⡿⣷⡶⠴⠶⠶⢤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣠⠄⠤⢤⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⠆⠀⣀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠀⠀⠀⢿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠈⠀⠀⠀⢳⣦⣄⠀⠀⠀⠀⠀⠀⡄⣼⡇⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢰⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⢀⣄⡀⠀⠉⠛⢿⣄⠀⠀⠀⠀⢣⣿⡇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠙⠂⠀⡀⠘⣿⠀⠀⠀⠀⢸⣿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿⡀⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢿⠀⠀⠀⡇⠘⣿⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⢿⡇⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⢠⡄⠀⠀⠀⢡⡀⢹⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⢀⠀⢠⡞⠀⠀⠀⠀⢸⡇⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠸⣿⡀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⢀⣀⡤⠎⠀⠀⠀⠀⠀⢸⡇⢸⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⠀⣿⡇⠀⡀⠀⠀⠀⠉⠀⠲⠦⣄⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⢀⣼⣇⣼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⢹⣿⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣿⢨⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣧⠸⣿⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢿⣶⢿⣇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣼⣿⠀⠹⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠏⠈⠁⠘⠿⠛⠓⠶⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠿⡿⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣤⠀⢠⡄⠈⣯⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⡶⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠘⠁⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡈⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣶⣿⣁⣀⣀⣀⣀⣤⣤⣄⠀⠀⢀⣤⣤⣤⣤⣤⣤⣼⣿⣶⡶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡿⠟⠛⠿⠿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⢀⣀⢀⣀⣀⣀⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁
⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⢀⣴⣦⠶⠛
⠀⠀⠀⠀⠀⠀⠊⠓⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⣀⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⣦⢴⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⣴⣤⠖⠋⠁⠀⢀⡀⡀⠴⠒⠉
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢯⢠⠀⣼⡻⣫⣥⣤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⢸⡀⠨⡄⠀⡞⠃⢏⠉⢸
⠀⠀⠀⠀⠀⠀⡦⣴⠀⠠⠄⠀⠀⠀⠨⠃⢀⠸⢱⡇⠀⠪⠐⠊⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣀⢤⣈⢮⠰⡌⣯⠃⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡾⠉⠋⠙⣿⢦⡙⠞⢦⣀⣈⡀⣠⢶⣤⣄⡀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢹⠖⠻⠈⠉⠙⢿⣧⣬⠗⢁⡥⣮⡷⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠤⣚⣳⠄⠤⣦⡒⠐⡿⠎⠀⠿⣽⠇⠛⢻⠟⠛⡿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣋⣡⣒⠖⠚⢻⡟⠛⡣⣲⣷⢧⣀⠀⠠⠴⠞⣳⣿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠙⠻⠿⢷⣞⣥⣥⣄⣀⣜⣙⣠⣷⡛⢓⣤⣦⣼⡶⢷⠟⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⡠⢰⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣝⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣰⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣤⣄⣀⣤⣀⣀⣤⣠⣠⣷⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁
⢀⣀⡀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠈⠉⠓⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣊⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠃⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠘⠤⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠤⠚⡆
⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⣀⣀⣀⣀⠤⠤⠤⠖⠒⠒⠒⠉⠉⡩⠓⠦⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠚⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⡴⠊⠲⡄⠀⠀⠸⡄⠀⢀⡗⠀⠀⠀⠀⢠⠊⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠳⢄⡤⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠈⠉⠢⡀⢰⠁⠀⠀⢳⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡇⠈⠢⠤⠤⠃⠀⠀⡔⠉⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡴⠒⠀⠒⢦⡀⠀⣉⣒⣋⣀⠤⠤⠤⠃
⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⢀⣀⣀⣀⠠⠤⠤⠤⠒⠊⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠘⠒⠒⠂⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⣠⣤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⣄⣀⠀⠉⠛⠶⢤⣀
⠀⠀⡇⠈⠉⠓⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠙⠲⠦⢤⣽⠗⠀⠀⠀⠀⠉⠛⠶⣤⣀
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠉⠙⣳⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠉⠛⠶⣄⡀
⠀⠀⡿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠴⠛⠒⠶⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⡀
⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⠖⠒⠛⠛⠒⠦⡀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣽⠒⠒⠒⠲⠶⠦⠤⠤⢽⡇
⠀⠀⡇⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⠁
⠀⠀⡇⠀⠀⠀⢀⡴⠛⠉⠳⡄⠀⠀⠀⠀⠸⣄⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡏
⠀⠀⡇⠀⠀⢰⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⣿⠀⠀⠈⠳⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡏
⠀⠀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠓⠶⠤⣤⣄⣀⡀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠐⠃⠀⠀⠀⠀⣰⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠿⣅⣀⡀⠈⠉⠳⡄⠀⠀⠀⡀⠀⠀⠙⢦⣄⣀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠦⣿⣀⡀⠈⠛⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀⠀⠀⠀⠉⠙⠛⠲⠦⣤⣄⣀⢸⠀⠀⠀⠯⠭⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⠟⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠛⠿⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠉⠀⣀⠀⠀⠄⣄⣤⠶⠶⠶⣦⣔⡠⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠁⠀⣠⣤⡶⢛⣨⣤⣶⠿⠋⠀⢐⣤⠠⣝⣻⡿⣿⢻⡿⢶⣜⡐⢰⢀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⣰⡾⠟⣡⡾⠟⠛⢽⠓⠀⠀⡥⠀⠀⠀⠲⠿⢿⣾⠦⠀⠀⣙⣛⡻⣾⠻⣦⡄⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⣶⢿⠉⣼⠋⡄⠀⠀⠈⠀⠀⢐⣻⡅⠀⠘⠆⠀⠀⠲⣦⠀⠀⠘⠋⠁⠼⠳⢮⡁⣧⠀⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⢸⣟⢃⡾⠓⢃⠠⠄⠀⢤⡖⠀⠀⠛⢻⣤⣠⡤⠀⠶⠀⠀⠁⠀⠀⠀⠀⠡⠆⡺⣿⡅⢻⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠸⣾⣏⢸⠇⣩⡄⠀⠉⠀⠿⠷⠂⠀⠀⠀⠈⠙⠋⠴⠂⠄⠀⠀⠠⠄⠀⠀⣂⠘⣷⢠⢿⣿⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠹⣏⢸⡀⠉⣭⡤⠀⠀⠘⠫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⠩⣾⡵⣯⢸⡏⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠈⢿⣿⡡⠀⠶⣃⡀⠀⠀⠀⠁⠐⠒⠶⠂⠖⠠⠤⠀⠀⢠⣼⣆⣤⢀⣠⡴⣀⣴⠟⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⠻⣦⣄⡉⠐⠣⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠴⣚⡭⠵⢚⣫⣵⠾⠋⠁⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠉⠛⠻⠶⢦⣤⣤⣀⣐⣒⣒⣲⣶⣂⣤⣤⣴⠶⠟⠛⠉⠀⠠⠀⢡⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢠⡧⠘⠰⠔⢀⠠⠀⢉⠉⡉⠉⠙⠉⠉⠉⠉⣀⠀⠀⠀⣰⠀⣰⢀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢳⡀⠀⠀⠈⠁⡀⠀⠀⠁⠘⠀⠐⠀⠟⢣⡅⠀⣠⠀⢣⣾⠃⢨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡝⠀⠀⠀⢠⡁⠀⠀⠀⠀⣧⡀⠀⣶⣸⠇⣸⣿⠀⡼⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⣀⠀⠀⠰⡼⡇⠻⠻⡇⢸⣿⠛⠀⠟⢁⣠⣵⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⢶⣤⣤⣤⣤⣤⣤⣴⠶⠾⠟⠋⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⣶⣶⣶⣶⠶⢶⣶⠶⠶⣶⡶⠶⣶⡶⠶⠶⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⡾⠿⠶⣶⠿⠷⢶⡾⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣼⠃⠀⢰⡟⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠘⣧⠀⠀⢻⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢹⡆⠀⠈⣷⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣷⠀⠀⢹⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡿⠀⠀⣸⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣼⠃⠀⢠⡟⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣼⡧⠴⣾⡷⠶⠶⣿⠶⠶⣾⡧⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠉⠉⠁⢀⣸⣇⣀⣼⣧⣤⣤⣿⣤⣤⣿⣀⣀⣀⠀⠉⠉⠙⠛⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⣴⠞⠛⠛⠛⠾⣟⠋⠹⣏⠉⠹⣇⣀⣤⣿⣦⣤⣽⡍⠉⠉⠛⠛⠶⢶⣤⣀⣤⣀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⢀⣼⠋⠀⢀⣤⣾⣧⣤⣠⣴⣦⣄⠘⢷⡀⠙⣠⣴⣿⡿⠟⠋⠉⠙⠛⡿⠀⠀⠀⠀⠀⠀⣠⡟⢠⡿⣧⣄⠀⠈⢳⣄
⠀⠀⠀⠀⠀⣾⠃⢀⣴⠏⠁⠀⢀⣀⡉⠛⢿⣿⡷⠈⣧⣰⠏⢀⣠⣤⣤⣀⡀⠀⠀⣿⠀⠀⠀⣤⡶⠞⢉⣴⢿⣴⠏⢉⡿⣦⡀⢻⡆
⠀⠀⠀⠀⠀⢻⣶⠋⢻⡄⠀⢰⣿⣿⣿⣦⡀⢻⣅⣰⣿⣿⣾⣿⡿⠿⠛⠿⢿⣦⠀⣼⡷⠾⠛⠻⣷⣴⠟⠁⣰⠏⢠⡟⢛⡮⠻⣿⠃
⠀⠀⠀⠀⠀⠈⢿⣦⡈⢻⣄⠀⠻⣿⣿⡿⠃⢸⣯⡿⠋⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⠿⣰⢞⡛⣦⠀⣾⣡⠾⠁⣠⢟⣴⠛⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣬⣻⢦⣤⣤⣤⣴⠟⢹⡇⢀⣀⣀⠀⠀⠀⠀⠀⠀⢿⢿⡄⣇⠿⠵⢏⣤⢿⣥⡴⠞⢳⣿⣵⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣶⣤⣄⣀⣈⠻⣦⣍⣙⣛⣻⣟⣋⣠⣴⠟⠘⠷⣿⣴⣶⣛⣡⣤⣤⣶⣾⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣟⣛⣟⢻⣿⣿⣛⣟⢻⣿⣟⣛⡛⣿⣿⣛⣛⡻⣿⣿⣿⣟⢻⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠵⠽⢷⣿⣏⠦⠿⣸⣿⡅⡭⢇⣿⣿⡱⣯⢓⣿⣏⣲⣝⣸⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠶⠶⠶⠾⠿⠷⣦
⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠛⠛⠛⠛⠋⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⠶⠶⠶⣶⣶⣶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡛⠻⠶⣦⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⣤⡙⠻⣦⡀
⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡆⠘⣿⡀
⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢀⣿⡇
⠀⠀⠀⠀⠀⢻⡿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡾⠟⢁⣤⡾⣿⠃
⠀⠀⠀⠀⠀⠘⣷⠈⠙⠻⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣉⣤⣴⠾⠛⠉⢰⡟
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠈⠉⠛⠛⠿⠷⠶⠶⢶⣶⣶⣶⣶⠶⠶⠶⠿⠟⠛⠋⠉⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠉⠛⠷⣦⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⡶⠟⠋⢁⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣤⣀⡀⠉⠉⠉⠉⠉⠉⠉⠁⣀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢄⣀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢘⡆⠀⠀⢈⠿⣤⣤⣬⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢧⡶⠿⠟⠋⢷⣤⣤⡀⠋⠉⠉⠓⢬⣿⣶⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣍⠓⠒⠲⠶⠦⠼⡿⡿⠶⠾⠶⠾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠀⠀⠀⠀⠐⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢴⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⠀⠠⢶⣦⣤⠓⠋⠀⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢣⡀⢂⢼⣄⡀⠀⠐⠖⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡁⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡠⠴⠶⠆⠀⡱⣅⡀⠀⠈⠒⠈⠀⠀⠀⠐⠚⢿⠿⡱⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠠⡷⠀⠈⠐⠂⢀⡀⠀⠀⠀⠀⠈⠃⢾⡗⠉⠉⠆⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⠀⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡅⣀⣀⠶
⠀⠀⠀⠀⠀⠀⠠⠂⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣧⣀⡀⠬⠵⠟⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠰⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠟⢛⠉⠀⠀⠀⠈⠀⠀⠀⢀⠞⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣸⡅⠀⠀⠀⠀⠀⠉⢃⠻⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣮⡽⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠲⠛⠋⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢉⡩⠍⠿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡈⠌⢑⠒⢄⠀⠀⠀⠀⠀⣷⠁⡠⠒⠉⠉⠙⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡐⢁⡤⠤⢤⣈⠢⡑⣄⠀⠀⢀⢿⣴⠁⠀⠀⠀⠀⠀⠘⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠃⠀⠀⠀⠈⠓⢮⣙⠂⢤⣎⣼⠃⠀⠀⠀⠀⠀⡘⢀⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⡰⣡⣾⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⢀⣴⣷⣧⣄⠀⠀⢀⣀⣀⣄⢀⣠⣤⠶⠳⠄⠀⢸⠻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣀⠈⠉⠁⢀⣈⣉⠉⢻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣾⡿⠷⠶⠤⣼⣦⡙⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣆⠀⠙⠋⠁⢀⣾⣿⣿⣿⣿⣿⣿⡟⠁⠈⠹⣶⣿⣾⣆⠀⢀⣼⣿⣿⣿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢼⠏⣿⣿⣿⣿⣿⡋⢼⠇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣴⣿⣿⣿⣿⣿⣷⣄⣀⣴⡿⠿⠿⠿⠿⢿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⠶⣶⢾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠟⠛⠉⠉⠉⠉⠁⠉⠉⠉⠙⠛⠛⠷⠶⣤⣀
⠀⠀⠀⠀⢀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⠶⠚⡡⣚⠒⠤⣒⠲⠄⣀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⢃⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⣀⡴⢾⡄⠉⠚⢪⡡⠺⢔⠿⣋⠀⠈⠤⡄⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⡀⠖⠐⡾⠠⢂⠅⠆⢤⡔⠚⢀⡐⠒⢬⡀⣍⡛⠓⡀⢰⠤⠄⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⡄⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⡴⠁⢴⣯⣙⠕⠻⠆⢠⠛⡔⠂⠀⢒⠆⡐⢹⠖⡙⡖⣡⡆⠀⠀⠄⠠⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠸⡄⠀⠀⠀⢸⡇⠀⠀⡀⠚⢃⢘⠎⠀⠈⣢⠐⠀⠀⠒⢌⣯⢇⠠⠊⠠⣃⠀⠀⠦⠔⠀⠀⠀⡋⢂⠉⠄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⡸⠁⠀⣠⣏⠀⠸⠀⠄⢀⠀⠴⠀⠀⣨⡀⣳⢀⠠⡴⠄⠀⢶⡌⠔⡦⠏⣐⠃⠩⣀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠱⡀⠀⢸⣇⡬⢂⢀⡞⠁⠘⢦⣀⠐⠌⠑⠘⢒⡔⡨⠀⠀⣹⠈⠙⢷⣄⣀⠈⡒⠁⠀⠇⠨⡻⡟⠋⠪⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢣⠀⠀⢿⣇⢴⣾⡆⠀⣠⠏⠹⠛⠷⠖⠒⡎⠀⡰⠉⡀⢻⠀⠀⠨⣤⣍⠘⡀⣠⠴⢂⠂⢀⠯⠉⡓⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢿⡜⢉⠚⣢⡌⢠⣄⠑⠀⠐⠄⢈⣶⡑⠐⠄⠚⢄⠀⠂⠠⢈⣛⡒⢁⢊⠅⠀⠸⠸⡌⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢆⠀⠀⢻⣦⡰⠇⢌⡌⠀⠌⣡⣄⣂⠍⠁⠙⢮⣐⣐⡢⠛⠫⠋⠉⠑⡃⠁⢁⠚⢄⡀⢁⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢆⠀⠀⠹⡻⢶⣔⡀⠐⡶⢊⠀⣀⣥⠀⠀⢠⣠⣠⢠⡀⠭⡄⢢⡭⠔⠠⠃⠀⢩⣵⠟⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⠻⢾⣤⣔⠋⠙⠃⠒⠡⠪⣤⠀⠉⠭⡣⠈⠒⠒⣈⣤⣽⠾⡋⢕⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠐⡄⠀⠀⠀⠳⡄⠀⡈⠉⠛⠿⠶⢷⣾⣶⣶⣴⣶⣾⣶⡷⠿⠟⠛⠉⢐⠠⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⠀⠀⠀⠀⠈⠳⣀⠀⠁⠀⠀⠀⠀⢀⣂⣀⡀⠀⠄⠀⠀⠀⠀⠌⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠱⡀⠀⠀⠀⠀⠀⣿⣶⣭⣙⡛⠶⠶⠾⠿⠶⠞⣛⣫⣥⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢆⢳⢀⣆⠀⠀⠀⠈⠻⣟⡻⠿⠿⢿⣿⡿⠿⠿⠿⣛⡯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡝⣆⢣⠸⣆⠀⠀⠀⠀⠀⠉⠛⠒⠒⠶⠶⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⡆⢇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠿⠿⡿⢿⣟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣻⣛⣻⣿⣻⣻⣟⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡭⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⢤⣴⣾⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣛⣿⡽⣷⣿⢿⣻⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⣶⣾⣿⣧⣿⣿⣯⣿⠽⠿⠽⠯⠭⣭⣭⣽⣿⣿⣖⣲⣶⣶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⡤⣖⣺⣭⡷⠶⢛⣛⣭⣿⣿⣿⡶⣿⣷⣞⠳⡳⣭⠽⣟⣛⡿⢿⣾⠿⢗⣭⣻⢭⡛⠛⠷⢾⣭⣗⣲⢤⣀⡀
⠀⠀⠀⢀⡤⢞⣭⣷⣟⣋⣭⣴⢶⣿⣿⠿⣿⣛⣛⣿⢮⣾⣿⣇⣷⣧⡿⣟⣾⡿⡚⢯⡻⣯⣝⣓⢿⡵⠼⣶⣿⣷⣶⣭⡙⠻⣾⣭⡳⢤⡀
⠀⢀⡖⣩⣾⠟⣫⣿⣿⣿⣩⣿⢟⡭⣲⣿⣿⣿⣿⣿⣿⡽⡏⡆⣿⢿⡹⣻⢿⣪⢺⡿⣻⢳⣮⢭⣝⡻⣽⢺⣯⣯⣿⣿⣜⡦⣄⠙⠻⣷⣝⢲⡀
⢠⠏⣼⡿⢃⣾⣿⣯⣭⢟⡿⣗⣽⣞⣿⣿⣿⣛⣿⠿⣏⣿⢽⣇⣏⣾⢷⣿⣇⣟⢬⣚⣾⠿⠵⠿⢾⣿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣄⠘⣿⣷⣻
⡿⢸⣿⠳⡋⣯⣽⣿⣕⡻⣿⢟⣏⣾⡿⣿⣿⡸⣿⣿⢿⣯⢺⢸⣴⣿⣯⣷⡏⣿⠵⢿⣯⠲⠷⠷⡿⠷⠳⡛⠿⠟⠟⠟⢿⣿⢿⣿⣿⣿⣼⣿⡇
⣷⢸⣿⡄⠉⠙⣷⣿⣿⣿⣮⢻⣟⣿⢿⣿⣿⣿⡶⣶⢯⣿⣻⣸⣿⣟⢵⣿⣻⣽⣿⣷⣹⢻⢻⠿⣛⣛⣛⣻⣽⣿⣛⣿⣿⣿⣽⣿⣿⢿⢿⣿⡇
⠘⢦⠻⣿⣄⠀⠙⠫⢿⡶⢿⡿⢿⣾⣸⢿⣿⡾⣿⢟⣿⣿⣿⣕⣝⣿⣿⣿⢳⣾⣿⣿⡇⣿⡧⣫⣿⡾⣹⣿⣿⣿⣿⣿⣛⣽⣸⣻⠕⣣⣿⠟⡴
⠀⠈⠳⢌⡛⢷⣤⡀⠐⣿⡿⣟⣛⣦⣿⣻⡿⣽⣿⣾⣿⣿⣿⣿⢛⣟⣛⣻⣿⣿⣿⣟⣽⣿⢷⣾⣯⣶⣿⡿⣿⣛⣻⣿⡿⠟⢁⣤⡾⢛⡡⠞⠁
⠀⠀⠀⠀⠙⠲⢭⣛⠷⣦⣍⣙⡒⠒⠓⠻⢿⣮⣾⣭⣟⣻⣿⣿⡷⡯⢋⣿⣛⣿⣿⡿⡿⣻⢟⣿⣿⢿⣿⣿⠟⢉⣉⣥⣴⠾⣛⡭⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠬⣝⣛⡿⠶⢶⣤⣬⣛⣻⣿⡿⠛⠋⠑⠚⠛⠚⠯⠽⠛⠓⢛⣛⣛⣛⣯⣥⡶⠶⠿⣛⣫⠥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠮⠭⠭⣭⣭⣽⣛⣛⣛⣛⣛⣛⣛⣛⣿⣭⣭⠭⠭⠵⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⣥⣴⡶⣟⢋⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡿⣿⣿⣀⡀⣩⣯⡅⢈⣉⣉⣛⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⢿⣿⣶⢣⢀⢉⡾⠋⠙⠳⣾⡿⢩⣦⣌⢲⠍⠻⢻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠉⠁⠐⠩⠊⠠⠉⣚⣤⡖⡠⣄⠿⠀⠤⣀⠙⣆⢏⠖⠁⠀⣙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢛⠟⠀⠳⠀⠗⣤⡴⡲⠚⠣⣄⠀⠾⣲⣶⠀⢠⣌⢳⠘⣌⠢⢞⣴⢋⠄⠜⢁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢱⡆⠈⠈⢄⣔⠋⠻⠐⠄⠀⠀⡈⢳⣦⠛⠙⢧⡙⢿⣮⣄⢸⣷⢦⡬⢭⣑⣢⣴⡤⠍⣠⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠎⠂⠄⡴⠋⢉⡀⠀⠒⠆⠀⠀⢸⢸⡋⢀⠀⢈⡛⢌⠿⡛⣖⡿⢿⡒⠾⣽⡿⣯⠄⣾⣷⣆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⡀⡁⠐⠓⣤⠘⡷⢶⢦⠞⣀⡤⠾⢿⡟⠁⢁⡼⢱⣤⢉⣷⣾⢬⣿⣶⡐⣠⣾⠟⠛⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠉⠀⢡⠰⣬⣿⡼⢀⣷⢠⣪⠇⠀⠀⣄⢹⡿⠹⢠⣿⣿⣎⣿⡿⠋⠈⢀⣈⠛⣱⠆⡤⠀⠸⡶⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⣮⣀⣝⡤⣌⣽⣫⣿⠄⠀⣄⣠⣄⡰⢿⡎⡀⢨⣿⠞⡻⣿⠃⢀⣾⡟⠙⣿⠀⡄⠀⢂⢘⠦⢻
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠶⠏⡴⢈⢹⠉⢤⡉⠙⠰⡐⠈⣟⡉⠀⣸⢇⢠⡟⢿⣿⣿⣿⡄⠈⡿⣏⣡⣽⢃⡤⠤⠀⠲⠈⣻
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠁⠔⠑⢂⡌⠀⣼⣿⣞⣳⣽⣄⠈⠬⢋⣙⢛⡛⠛⠌⡁⠙⣿⣷⣄⠈⠛⢉⡡⠈⠄⢉⠍⠀⢠⠙
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⡶⠸⠾⢴⣧⣀⠻⢿⠵⢿⣿⠥⠿⣶⡏⠀⣈⠓⢲⢶⣿⣶⣦⡙⢻⣿⡻⣮⣦⣂⢑⠂⠀⠁⢬⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣴⡂⢞⠻⣧⡧⠀⠀⠀⠀⡟⠀⠀⠀⢀⠀⠉⣷⢩⣿⡙⣸⢿⣿⣆⢻⣿⣾⣿⣷⣧⣦⡤⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠤⠠⠄⠾⠁⠀⠀⠀⠀⣟⣀⡀⣤⢾⠟⣶⢶⡾⣟⠫⠀⠺⣿⢿⡿⠿⣹⣦⡽⠛⠛⠗⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣆⡒⠀⠀⠀⠀⣀⡰⠶⢏⡀⢙⠷⢀⠀⢘⠞⢰⣾⠶⣦⠤⡿⢿⠃⠘⡈⠢⠐⠊⠀⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣦⡀⠀⠤⠺⡁⠀⠀⠙⢦⡤⢾⢆⣤⣿⡾⣷⠃⢐⢆⠁⢉⣬⠲⠣⠐⣠⣩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡀⡈⢄⡓⢀⣠⠾⠿⠶⣞⣙⢿⠁⡅⠱⡌⠀⢀⠀⠛⠖⢦⣾⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣦⣾⣧⣮⠩⢗⣤⣴⣖⡄⢠⡎⠁⣠⠮⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣋⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠷⠤⣄⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢴⣎⣀⣤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣀⣀⣠⠤⠴⠚⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡎⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠚⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠚⠉⠉⠀⠀⠙⢤⣀⡤⠋⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⣀⣀⡤⠤⠶⠒⠊⡟⠉⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡶⠓⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠳⠤⠖⠃⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⡴⠋⠙⢦⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠁⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠹⢤⣠⣞⣠⠇
⠀⠀⠀⠀⠀⠉⢲⡀⠀⠀⠣⣀⣀⡰⠃⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠒⠋⠉⠁
⠀⠀⠀⠀⠀⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢹⠀⣀⣀⡤⠤⠒⠚⠉⠉
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠿⠔⠒⠋⠉⠁
⠀⠀⠀⠀⠀⡇⠀⣀⣀⠤⠴⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠤⠤⣤⢤⣄⣀⣀⡀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠻⠯⠵⣾⣻⣿⣟⣛⣛⣛⣿⣟⣶⣦⡬⠁⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⠦⠤⣤⣀⣀⣈⣉⣉⣉⣉⣉⣀⣤⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢠⣿⠁⠀⣠⣴⠶⠿⠿⠿⠷⣶⣄⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⠶⣦⣤⠀⢸⡇⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣴⠟⣿⡇⠀⠀⠀⢀⡀⠀⠈⠀⣼⣇⣴⣿⣳⡚⡶⢤⡀⠀⠀⠀⠘⣿⣦⡘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⡄⠋⢀⠀⠀⠀⢫⣔⣆⠀⠀⣿⣿⠃⠙⢦⠀⠁⠲⣅⡀⠀⣰⡟⠋⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣿⡱⣸⡟⠀⠀⣼⣿⣿⣿⠀⠀⣿⣿⡀⠀⠸⡾⣻⠹⢉⣗⠾⠙⣿⣟⢫⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠷⣮⣄⣀⣼⣿⣿⡛⣿⣧⠀⣿⡿⣷⣄⢸⠈⠓⠲⡞⠺⠴⠴⠋⣠⡿⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉⠀⢸⡇⠈⠛⠿⣶⣤⣤⣧⣤⣴⣶⠿⠋⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠒⠒⠒⠒⠛⠋⠁⠀⠙⠷⣶⣤⣤⣀⣀⣀⣀⣀⣤⣤⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⢀⡐⠦⣄⠀⠀⠀⣀⠀⠀⠠⠤⠀⢤⡤⠀⠀⡀
⠀⠀⠀⠀⠀⢰⠀⠀⠀⣹⠄⠒⠂⠠⡄⠀⢀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠀⠐⠀⠴⠾⠼⣓⠀⠀⠡⡄⠁⢀⠗⠀⢠⣈⣑⣀⢐⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⢠⠃⠐⠉⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣥⡞⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠊⠀⢂⠀⡴⠒⢤⡐⣤⣤⢰⠟⠀⢠⠄⠀⠲⡄⠀⠀⠀⠀⠀⠀⠄⣠⠞⠁⠀⠀⠀⠀⠈⠲⡄⠀⠀⣀⡼⣯⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠏⠀⠒⠘⢻⡈⠉⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠽⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⣀⡼⢄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠈⠁⠈⠑⡄⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠑⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠘⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣆⢀⠠⠈⠀⠈⠳⡄⠀⠀⠀⠀⠘⠀⠤⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠄⢀⣴⠇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡄⠀⠀⢠⠆⠀⠀⠀⡀⢐⠠⠤⠀⠀⠀⠠⠀⠀⠀⠀⢄⣿
⠀⠀⠀⠀⠀⠀⠘⠷⠾⠛⠁⠀⢤⣄⡀⠈⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡉⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⡇
⠀⠀⠀⠀⠠⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠠⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢳⢀⡇
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠄⠀⠀⢁⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠘⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠐⠛⠀⠀⡀⠀⠠⠈⠒⠊⠀⠤⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⢀⠀⡼⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡀⣇
⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠞⠋⠁⠀⠀⠀⠂⠄⠤⠔⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠑⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢈⣁⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣾⣵⣿⣽⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⡴⢂⣀⣀⣀⡠⠤⠴⠶⠶⠶⠶⠶⠤⢤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣀⣀⣀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⡿⢿⣥⡽⣋⣭⣶⠾⠿⠟⠛⠻⠿⠷⢶⣶⣤⣄⣈⠉⠩⢿⣷⣶⡶⢺⡟⢲⣶⠄
⠀⠀⠠⡿⠯⠭⠭⠭⠽⡾⣣⡾⠟⠁⠀⢀⡠⢤⣤⣤⣀⣠⣤⣤⣬⣉⡛⠿⣶⣄⡉⠻⣇⣢⣧⢀⣿⠃
⠀⠀⠈⣭⣭⣭⣭⣽⢎⣼⠟⠀⢀⡠⠤⣞⣀⣸⡥⠴⢯⡳⡍⠉⠉⣩⠽⢷⣂⠙⢿⣦⡀⠑⢔⢻⣭⠅
⠀⠀⠰⣾⣿⡷⣠⢃⡾⠁⢠⡎⣯⣴⣋⡵⣲⣶⣛⣓⣓⠻⣧⡴⠿⣁⡄⠀⠈⠱⢦⡘⢿⣦⠀⠁⣹
⠀⠀⠈⣿⣯⢭⠇⣾⠃⠀⡎⡹⢿⢿⣿⣛⡥⠤⠶⠤⣭⣍⡙⡟⣾⢀⣟⡖⢶⣌⡀⣽⠀⠻⣧⠀⢹⡀
⠀⠀⠀⣿⣯⡞⢸⡏⠀⢠⠟⢧⣀⣾⣭⠴⠒⠒⠚⠉⣉⣙⣿⣷⠿⣼⣼⣧⣿⠀⠉⡏⠀⠀⣾⡇⠘⠇
⠀⠀⠀⣿⡿⡇⡯⡇⠀⡾⢠⣯⣞⣡⠤⣶⡶⠖⠋⠉⢁⣰⠟⢻⡄⠀⢈⣩⠟⣋⣍⡓⢦⢄⣿⣿
⠀⠀⠠⠿⠶⡇⡇⡇⠀⡇⡜⠙⣿⡤⣾⣥⠤⠖⠒⠋⠉⣀⡤⠞⣠⡖⣩⡴⣿⢏⡿⣯⣢⡱⢹⡟
⠀⠀⢐⣟⣻⡇⢧⡇⠀⢠⠑⠼⣅⣸⣁⣀⣤⣴⣶⣌⣉⠁⠐⢾⡁⠳⣙⢋⣾⣷⣿⢿⣿⢧⢸⡇
⠀⠀⠠⣷⣶⣿⠸⣾⡀⠈⣆⠀⠀⠀⠀⣿⣻⣟⠟⢻⣿⣤⢞⡭⠽⣧⣬⠽⠿⣟⡛⣡⢞⡼⢸⡇⢀⡄
⠀⠀⢈⣉⣭⣭⣇⠹⣷⡀⠈⢦⡀⢀⣶⡿⣿⣻⣿⣿⣿⣯⢸⡀⢞⣢⠙⣍⣳⣌⠻⡵⠋⢠⣿⠁⣸⠃
⠀⠀⠠⠭⠶⠶⠈⣆⠹⣷⡄⠀⠑⢼⣯⣾⣿⣿⣷⣿⣾⣿⣷⣷⣜⣿⣷⣮⣷⣽⠀⣷⣰⠟⠁⣰⢿
⠀⠀⠐⠿⠛⣛⣛⣛⣦⠙⢿⣦⣀⠀⢻⣵⢿⣮⣿⣿⣿⢧⣭⡿⣬⣭⣭⡭⠷⢞⣿⣽⡇⢀⡼⠯⢭
⠀⠀⠘⣛⣉⣭⣭⠭⠽⠷⣄⠙⠿⣷⣦⣝⡻⣏⣿⣎⣹⠟⠉⠀⣀⠉⠭⠛⠛⠉⠁⣉⣴⣿⣿⣟⣛
⠀⠀⠰⠷⠶⠒⠚⣛⣛⣋⣙⣓⣦⣀⠙⠛⠿⠿⠛⠛⠒⠊⠉⠁⠀⠀⠀⢀⣠⣴⣚⣛⣛⣛⣛⣛⣛⠂
⠀⠀⠐⠛⠛⣛⣉⣉⣭⣥⣤⣤⣤⣴⠿⠷⠶⠤⢤⣤⣤⣤⣶⣶⣶⣾⣿⣭⣭⣭⣍⣉⣉⣉⣙⣛⣛⡁
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠴⠶⠶⠶⠶⠶⠖⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⣛⣻⡽⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⠤⢤⣤⣤⣀⣀⣀⣤⠴⠶⠛⣋⣩⡴⠶⠞⣛⣉⣥⣶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠛⠋⠉⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⢋⣉⣥⡴⠶⢛⣛⣭⣵⡖⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⢀⣀⣠⡤⠖⠛⣉⣡⣤⠶⠶⢛⣋⣭⠴⠞⠛⠉⠉⠀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣁⣤⡴⠶⠛⢉⣉⣩⣤⡴⠶⠛⢋⣡⣤⠶⢿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠟⠛⢉⣉⣤⡴⠶⠛⠋⠉⠁⣀⣠⣶⡞⡏⢩⡴⣻⡿⠛⠛⠛⢿⡍⠙⠳⣶⣄⡀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⣴⣶⠉⢁⣠⣼⣿⣷⡞⠛⠉⠀⠀⣀⣤⣴⣞⣻⣿⣥⣿⣿⣷⡏⢰⡏⠀⠀⠀⠀⢀⡿⠀⠀⣻⣽⣿⣷⣦⡀⠀⠀⠀⣧
⠀⠀⠀⠀⠘⢷⣿⣾⠋⠉⠀⠈⣿⣿⣤⠴⣶⡟⣿⣿⣿⠿⢿⡽⠉⣿⣿⣟⢷⣘⢷⣤⣤⡤⠶⠛⣁⣠⣴⣿⣿⣿⠟⠛⡿⣦⠀⢠⡿
⠀⠀⠀⠀⠀⠸⣧⣽⠷⠶⠛⢻⣿⡄⠀⣼⣻⣿⣿⣿⣟⣶⣾⠇⠀⣬⠻⣿⣿⡟⣿⣿⣷⣶⣶⣿⣿⣾⡿⠛⣛⣻⣿⡿⢷⢿⣀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⠻⣧⣙⣿⣽⣿⣼⣿⣟⣰⡶⢩⣤⠘⣿⣿⣻⣿⠛⣿⣵⡟⣉⣤⣶⣿⠿⣿⣿⣵⣿⡾⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠉⠻⢿⣿⣿⡏⢸⡿⢾⣧⣽⣿⣿⡿⠿⣿⣾⣏⣾⣿⣿⡟⠉⣠⣾⣯⠿⠛⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣄⠀⠀⠀⠈⠙⠛⠶⠶⣶⣿⣥⣿⣾⣟⣿⣈⣹⣿⣙⣿⣿⡿⠾⠟⠋⠁⠀⠀⢀⡴⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⢿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣴⣞⣿⣫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠿⣯⣷⣿⣭⠻⣟⢲⡶⠶⣤⠤⣤⣤⡤⠤⣶⣶⣖⢲⡶⢶⡖⣾⣛⠻⣧⣬⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠉⠙⠛⠲⠿⠾⠿⣧⣸⡇⢧⡮⢻⡼⣧⡶⠙⣿⣼⣧⡽⠗⠛⠉⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⡦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⢖⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⠿⠿⠿⠿⠿⠿⠿⠿⣯⡭⠭⠵⠖⠚⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡴⠶⠶⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⠶⠶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠛⢉⣉⣀⣤⣤⣤⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣤⣤⣤⣀⣈⡉⠙⠛⠻⠶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⣉⣤⡴⠶⠛⠛⠉⠉⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠉⠉⠙⠛⠷⠶⣤⣄⡈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⣠⡾⢋⣤⡶⠛⠉⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠟⢛⣉⣩⣤⣤⣤⣤⣤⣤⣤⣬⣉⣉⡛⠛⠶⣦⣄⡀⠀⠀⠀⢀⣿⣿⡿⢷⣤⣬⡿⢦⡀
⠀⠀⠀⢀⡾⢋⣴⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⡉⠉⢉⣡⡴⠞⢛⣉⣉⣤⣤⣤⣤⣤⣤⣤⣬⣉⣉⡙⠛⠶⣦⣍⡛⠛⢾⡟⠛⣡⣴⠶⣤⡉⢿⣧⡀⠻⣦
⠀⠀⠀⣾⢡⡟⠁⢀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⠡⠶⠟⠛⣉⣭⠾⠛⢋⣡⣴⠶⠟⠛⢋⣉⣛⣻⣶⣦⣍⡁⣶⠛⢡⡾⣫⡶⠀⢸⡇⢠⡟⢻⣆⢹⡇
⠀⠀⢀⡿⣾⣁⣴⠟⢹⣿⣿⣿⠋⠁⠀⠀⠈⠻⣿⣇⠀⢀⣴⠾⢋⣡⡴⠞⠋⣉⣤⡴⢾⣿⣿⣭⣭⣉⣉⠁⠉⢣⣾⠃⣿⢰⣏⢀⣴⠟⢀⣴⣿⡀⢻⡄⣿
⠀⠀⠘⣧⢹⣟⠁⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⣻⣿⣴⠟⣡⡴⠟⢉⣤⡶⠛⠉⣡⡾⠋⠁⠀⠀⠈⠉⠉⠛⢷⣌⠙⣷⢉⠀⠙⠛⢁⣤⣼⠇⠈⢻⣾⠃⣿
⠀⠀⠀⣿⠀⠻⣦⡀⠀⠘⢿⣷⡀⠀⠀⠀⠀⣠⣿⣿⠁⢀⡉⢠⡾⠛⠁⠀⢠⡾⢋⣴⡆⠀⠀⠀⠀⠀⠀⣠⡴⠿⠿⠿⠿⢷⣾⣿⠿⠁⠀⠀⢀⣼⠏⢀⡿
⠀⠀⠀⢻⡆⠀⠈⠻⢦⣄⠀⠙⢿⣷⣶⣶⣾⣿⡿⢣⣀⠿⢿⣷⣄⡀⠀⠀⣿⠁⢺⣿⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⣀⣴⠟⠁⠀⣸⠇
⠀⠀⠀⠈⣿⣀⠀⠀⠀⠉⠛⠶⣤⣌⣉⠉⠉⠁⠀⠛⠿⣿⣦⣉⠻⣿⣶⣤⡹⣦⡈⠻⣷⠄⠀⣠⡿⢡⣶⡖⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠋⠁⠀⠀⢠⡟
⠀⠀⠀⠀⠘⣿⣷⣦⣄⡀⠀⠀⠀⠉⠙⠛⠷⠶⣦⣤⣄⣀⣙⣿⣿⣶⡍⠛⠀⠈⠛⠶⣦⠶⠟⢻⣇⣸⣿⣷⣤⣤⣴⠶⠟⠛⠉⠁⠀⠀⠀⣀⣤⣶⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢛⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⣛⣩⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⠶⠶⠶⠶⠶⠶⠞⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠛⠛⣉⣉⣉⣀⣄⣨⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⢉⣠⠶⠞⠛⣉⣉⣡⣤⣄⣉⣿⠙⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⣠⠞⠋⣠⣴⠾⠟⠛⢋⣉⣉⣉⣩⣿⠙⠶⣄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢠⡞⢁⣴⠟⠋⣠⣴⠶⠛⠋⣉⣉⣉⣽⡟⠷⣦⡈⠳⣄⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⠛⠛⠛⠛⠛⠛⠿⢯⣤⣴⡾⠿⠿⠿⣍⡙⠳⣦⠘⢿⣆⠙⢧⡈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠁⠀⠀⢠⡖⠀⠉⠀⠲⡄⢹⣦⡴⠶⠶⣶⣿⣿⣦⣽⣷⣄⣻⣧⠈⢷⡈⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢻⡛⢤⡀⠀⠀⠁⠂⠀⠀⢈⣠⠞⣻⡴⢚⣋⣭⠥⠤⡶⠶⢶⡄⠈⢻⣧⠘⣧⠘⣏⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⢷⣄⠉⠙⠓⠒⠒⠒⠛⠉⠀⣰⣿⠀⠻⠿⠶⠦⠾⠶⠚⠋⢁⣠⣾⣿⡇⣻⣄⣿⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠉⠛⠶⢽⣷⣦⣤⣤⣤⣤⣤⡴⠾⠿⣿⣓⠒⠲⠶⠒⠒⠒⠛⢋⣉⣡⣿⢋⣻⣿⠿⠟⠋⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠉⠉⠛⠛⠳⠶⠶⠶⠤⠬⢽⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣤⠀⠀⠀⠀⠀⠀⠀⢠⠴⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠐⠾⠷⠆⠀⣤⣤⣤⠤⠴⠤⢤⣤⣤⡄⠀⠶⠿⠖⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⣀⠀⠀⠀⠀⠹⣟⡁⠀⠀⠀⠀⣹⠟⠁⠀⠀⠀⢀⣀⣤⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠛⠳⠶⠤⣤⣤⣍⣉⣛⣋⣉⣥⣤⡤⠴⠶⠛⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣦⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⣀⡀⡀⡀⢀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠉⠁⠀⠀⠉⠀⠀⢉⡋⠁⠐⠒⠒⠀⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠀⠀⠉⠿⠿⠷⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣼⣿⠿⠿⣶⣶⣶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⠿⠟⠛⠋⣹⣿⣿⡷⣶⣶⣮⣭⣭⣙⣛⣛⠻⠿⠿⠿⠿⣶⣶⢶⣶⣲⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⠁⠀⣰⣶⣾⡿⠉⠀⠀⢰⣿⣿⣮⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠇⣀⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣼⣿⣾⣿⢿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡑⢦⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⠿⣿⣿⣿⠀⠀⠂⢠⣽⣿⣛⣻⣿⡦⣬⣽⣿⣿⣿⣿⡛⣻⣿⣿⣿⣿⡇⠀⢻⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣿⣿⣿⡿⣿⣿⣤⣠⣄⡤⣫⣉⢿⣣⣥⡵⣽⡻⢗⣶⣶⣿⣿⣯⣭⣭⣿⣿⣧⣶⣟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠉⣵⣿⣿⣿⣽⢧⣷⢺⣿⣉⣰⣿⣿⣷⣯⢥⣻⣵⣲⣶⢏⡟⡿⣿⣿⣿⢻⣹⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢰⣿⣿⡿⣹⡇⣦⠟⠁⠘⡻⠾⠟⠋⠻⣮⣻⣮⣯⣽⢋⣽⡜⣿⣶⣝⢿⣿⣿⣵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣸⣿⣿⣣⢿⣿⠏⠀⢀⡬⡍⠙⠀⠀⠶⠾⠝⠛⠀⢳⣮⢫⣾⣿⣿⣿⡇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣿⣿⣯⡯⣾⣿⠀⠀⢸⣷⡜⠀⠀⢠⣎⣻⠀⠀⠀⠘⣷⣷⣿⢻⣿⣿⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⣿⣿⡟⢶⡽⣿⡀⠀⠀⠀⣬⣄⣂⠈⠛⠋⠀⠀⠀⢠⣿⡟⣻⣼⣿⡟⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠸⢿⣿⣟⣛⠯⣌⠳⣄⠀⠀⠻⣤⡴⠟⠁⠀⠀⠀⢠⣾⣿⡿⣮⣽⣿⠷⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⢿⣿⣷⣽⣦⣷⣽⡗⠤⣀⣀⣀⣀⣀⣠⣤⡾⠛⢬⣿⣜⣾⣿⡿⠄⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣀⠀⠀⠈⠊⠙⠛⠿⠿⠿⣿⣿⣧⡿⣯⣻⣽⣷⣿⣴⣭⣭⣽⣿⣿⡿⠝⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣧⡶⣆⢤⡀⢀⡀⣠⠀⠀⠀⠉⠙⠳⠟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠘⢻⡷⠿⣾⡟⣯⠟⣿⢻⣦⣶⣦⣴⢦⣤⣴⠀⢠⡶⠆⣀⡀⡀⠀⢀⡀⣀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡟⣼⣶⣤⣌⣀⡀⠀⠸⠿⠟⠃⠚⠙⠧⠄⠿⠼⣧⣿⠀⣈⣻⣦⣼⣿⣏⣿⡏⢸⠛⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⣷⣿⣿⠋⠓⠒⠲⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠉⠁⠉⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⣸⢱⣿⣿⣿⡟⢻⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⡏⡼
⠀⠀⠀⠀⠀⠀⡿⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣽⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⢰⡇⣿⣯⣼⣿⣿⣿⠃⠀⠀⢀⣿⣭⣽⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⢸⠁
⠀⠀⠀⠀⠀⣸⣵⡶⣭⣟⡻⢿⣿⣀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡏⡘
⠀⠀⠀⠀⠀⢹⣿⣿⣟⠻⣿⣿⡇⠈⠉⠙⣛⣛⣻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣿⡿⠃⡇
⠀⠀⠀⠀⠀⠀⠻⢷⣿⣛⡾⣿⣧⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⠶⣻⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣶⣤⣤⣌⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠻⠿⠿⠷⠶⠶⠿⢿⡾⠷⠿⠿⠛⠋

⠀⠀⠀⠀⣀⣀⠤⠤⠐⠒⠒⠊⠉⠉⠉⡉⣉⢉⠉⡉⠉⢒⠒⠒⠠⠤⢄⣀⡀
⢀⢶⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠕⠘⠚⠻⠲⠀⠀⡹⡆
⢸⠏⠑⠫⢔⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡧
⣿⠀⠀⠀⠀⠈⣑⠊⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⢟⡸⣧
⢻⡄⢀⡀⠀⢸⡹⡷⡀⠀⠉⠑⠫⠖⣢⣄⣀⠀⠀⠀⠀⠀⠀⠀⡠⣻⡿⡆⣡⣿
⠀⡇⠘⠗⠀⠈⠧⣀⡅⠀⠀⠀⠀⣠⡀⠉⠑⠛⠵⣖⡤⣀⡀⡰⣵⣝⣷⡛⣿⣿
⢀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠙⠁⠀⣠⢲⡶⣄⠈⠙⠺⣻⡇⢹⡿⢁⡿⣿
⢸⡆⠀⣠⣤⡀⠀⠀⣾⣹⠼⠟⡆⠀⠀⢸⣷⠅⠀⢈⠀⠀⠀⣿⣼⡎⠟⢛⢳⣿⡶⢶⣖⠒⠂⠠⠤⠤
⢸⡇⢸⣿⡟⠚⡆⠀⠙⢇⣀⠰⠁⠀⠀⠀⠙⠤⠴⠊⠀⣠⠶⡿⠙⡕⡀⣸⠛⣿⡛⢉⣉⠉⣉⣉⣈⡁⠉
⢸⡧⡀⠙⠢⠤⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠠⣷⣵⠂⢸⣷⡟⣏⣿⡿⠔⠀⠤⠦⠤⠔⠂
⠀⠙⠪⣕⡤⣀⠀⠀⠀⣴⢦⠀⠀⠀⣾⣶⡵⠛⠆⠀⠀⠉⠓⣔⣳⣿⣷⣿⣿⣃⣉⣉⣉⣭⡭
⠀⠀⠀⠀⠙⠳⢽⣦⣀⠈⠊⠀⠀⠀⠙⢏⡀⣀⠇⠀⠀⠀⠀⣿⠄⠦⢶⡿⠷⠺⠿⠖⢳⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣄⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⣿⢿⡏⣿⠍⠉⠉⠉⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⣒⣄⡀⠀⠀⢠⡶⡄⠰⣷⢁⣼⣽⡯⠍⠉⠉⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣗⣤⡀⠉⠀⠀⠈⡿⣿⣟⣓⣒⣐⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⣦⢄⡀⠀⣧⡯⠌⠥⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠾⣶⣿⣿⠛⠛⡏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠪⠍⡄⡐⠈⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠇⡥⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⡆⢀⠔⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡀⠴⠄⢬⢠⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠄⠂⢀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠢⠤⢆⡀
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠈⠁⠠⠧⠀⠁⠀⠀⠈⢆⠐⠀⡀⠑⠉⢠⠊⠀⠀⡀⠀⡁⠈⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠀⠀⠘⡆⠠⡐⠂⢠⠃⠀⠀⠀⢀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠠⠕⠂⢲⡀⠀⠀⠀⠀⠀⠈⠀⢃⠐⠀⢀⡒⠠⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⠙⣀⠤⠀⠐⢦⢌⣖⡙⠀⣾⣿⠧⡥⢲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣘⡁⠕⠀⠁⣳⠈⠁⠀⢹⢞⡔⠄⢆⠛⢼⣻⠆⡄⢀⠖⠉⣠
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⡤⢉⡙⠒⠐⠓⠦⠤⠦⠾⠞⠒⠛⠊⢉⡉⠤⢒⡇⠀⠁⠊⠁⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢀⡜⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠠⠈⢍⣉⠁⡀⢀⠀⣉⣉⠁⠠⠃⠒⢈⠜⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⢇⡀
⠀⠀⠀⠀⠈⢄⣱⣄⡠⠀⠀⢠⢗⣟⠄⠀⠓⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣱
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠤⠖⣏⢀⡀⠀⠤⠔⠃⢐⠤⣄⢀⠀⠀⠀⢠⠔⠊⠣⢄⣀⡠⡀⢀⣳⡣⣄⠤⠒⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠁⠀⠑⠉




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

⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤
⠀⠀⠀⠀⠴⢃⠄⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠀⠀⠀⢀⠾⠄⠀⠀⠀⠀⢠⣬⡄
⠀⠀⠀⢰⡀⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠁⣉⡠⠄⠀⠀⠀⠀⠀⢀⣤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⡰⠆⠀⠂
⠀⠀⠀⢿⣏⡂⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠁⢀⠀⠀⠸⡗⠦⠤⠄⠤⠤⠤⢶⠏⠀⠀⣀⠠⡣⣉⠴⢙⢲⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠈⠛⢿⣶⣄⠀⠀⢀⠀⠀⠀⣀⠀⡀⠀⢄⠀⠐⢤⣀⢀⡀⢀⡤⠆⠀⠐⠊⠩⠻⠌⠑⢯⠁⡱⠋⣘⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡀⠀⠀⠀⠀⠀⠙⠳⢶⣿⣄⠀⣸⣯⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠺⡅
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠸⡿⢿⣷⣼⣶⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⠀⢀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣀⢤⠀⠀⠁⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣦⣘⣦⣤⡌⣐⣤⠀⠰⣄⠀⠀⣀⡒⠉⠀⠀⠀⠘⠉⢙⣯⡇⠀⠀⠀⠀⡄⠂⠀⡀⣠⡄
⠀⠀⠀⠓⠢⠤⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠃⢁⠈⠟⠀⠀⠀⠉⠉⠉⠁⣠⠂⠐⠂⢸⣤⢒⣬⡅⠀⠀⢀⡀⠐⢄⡚⠟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⢠⣽⣦⡄⠀⢠⠈⠉⠛⠁⠀⠈⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠐⠯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠴⠀⡀⠀⠀⠀⠀⡿⢉⣿⠀⠀⠀⠀⠀⠀⢠⠀⠘⠂⠀⠀⡰⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠒⠅⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢂⠄⠒⠒⠂⠈⠀⣰⡦⠀⢰⢡⢸⡇⠀⠀⠂⠀⠀⠀⠜⡀⠠⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠃⠄⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⡆⢸⡏⠀⠀⡌⠀⠀⠀⠀⠲⠁⠁⠒⠀⠀⠀⠀⠂⠀⠒
⠀⠀⠀⠀⠄⠀⠀⠀⣄⡀⠀⣄⣤⠀⠀⠀⠀⠀⠀⠠⢀⠀⠘⠁⠀⠀⠀⠀⡄⠀⠀⠀⡀⠰⠟⠀⠀⢀⠀⠀⠀⠀⠀⠢⢀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠁⠀⠀⠀⠦⡙⡀⢗⣄⢀⠀⠀⢄⠀⠀⠀⠀⠃⠀⠀⠀⠁⡀⠠⠀⠉⠀⡚⠩⠖⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠠⠀⡐⡡⠆⠀⡀
⠀⠀⠀⠀⠀⠀⢘⣆⣽⠵⣿⠞⠁⠀⠀⠀⠀⠀⠀⠂⠁⢠⣤⡤⠞⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠈⠀⠇⠀⠀⢀⠐⠲⠤⠄⠀⠰⠒⠁⢀
⠀⠀⠀⠢⡄⠀⠘⠛⠩⣶⠂⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠄⠊⠀⠀⠀⢄⠁⡧⠤⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⠀⠾⠀⠀⠀⠀⠈⠒⠒⠒⠂⠀⠐⠋⠁
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠰⡏⠀⠑⠾⠴⠂⠀⠑⠀⠀⠀⠒⠤⠄⠄⠤⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠑⠂⡤⠀⠈⠀⠠⠠⢀⠀⢀⡠⠐⠘⠀⠀⠈⡅⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡠⠀⠠⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢁⣂⣉⠀⠅⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢤⠀⠀⡉⢀⡤⠖⠗⠀⠀⠑⠂⠒⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣻⣛⣻⣻⣙⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣭⣿⣽⣽⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣡⣤⣤⣀⠀⠈⠉⠑⢲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⢸⣵⣿⣿⣿⢇⠀⢰⢰⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⣤⣤⡀⠀⠉⠹⠿⠗⠃⠀⠘⣞⢷⣝⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⢴⣶⣤⡀⠈⠉⠁⢀⣤⠶⠶⠶⢦⣄⠀⠈⠙⠚⠭⠉⠉⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠘⠻⣿⡿⠏⠀⠀⢰⣿⣶⣿⣿⣿⣿⣿⣷⠒⠒⢒⣛⡉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⣠⠔⣋⣀⣀⣀⣤⠤⠤⠴⠖⠒⠒⠋⠉⠉⠻⢿⣿⣿⣿⣿⠿⣫⠂⠀⢿⣿⣷⡄⠀⢸
⠀⠀⠀⠀⠀⣾⠉⠉⣽⣂⡀⠀⠀⣠⢖⣒⣒⡂⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠋⠀⠀⠀⠈⠙⠋⣀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠸⠿⠇⠀⠘⣱⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⢔⣂⡀⠀⠀⢀⡴⣫⢷⣶⣿
⠀⠀⠀⠀⠀⣿⣀⣀⡠⢄⡀⠀⠀⢿⣿⣿⣿⢿⡟⠀⢀⡤⢤⠤⠀⠀⠀⣿⣿⣿⡆⠀⡞⡾⣱⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⢶⣝⢦⠀⠀⠙⠛⠚⠋⠀⠀⠋⣾⣿⣿⣷⡀⠀⠙⠒⠋⠀⠀⡜⣇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⢻⣆⢧⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣛⡿⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⠿⠿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⢀⣀⣀⣀⡤⠬⠥⠤⠶⠒⠒⠚⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢿⣿⡿⠿⠿⠿⠓⠒⠒⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⣊⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢖⣚⣩⠥⠴⠒⠋⢉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⡜⠀⠀⠀⠀⠀⠀⣀⣠⠤⢖⣒⣯⣭⣒⣚⡭⠭⠤⠖⢒⣚⣋⣩⡭⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇⠀⠀⠀⣠⣾⣽⣷⣚⡟⢻⢹⡏⡿⢲⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⠒⠀⠀⠉⠉⠉⠁⠀⡇⢸⠀⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡥⠤⠶⠶⠶⠦⠤⢤⣇⣸⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠖⣉⡤⠖⣚⣉⣩⣉⣛⠒⠦⢌⡙⠲⢇⡇⣾⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⡩⠋⢠⠞⢁⡴⠋⣁⠤⠤⢤⣬⠽⠓⠒⠛⠓⠒⠿⢼⡀⠀⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⡷⢤⣀⠀⡼⠁⣰⠃⣰⠋⢀⡞⢁⡠⠞⠉⣠⠖⠋⠉⠉⠉⠓⠲⢤⡈⠓⢄⣀⣀⡴⡏
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠉⠙⠒⠓⠲⠷⠤⠼⠴⠯⠤⣴⣮⣤⡴⠮⠭⠭⠭⠽⠷⠖⠛⠛⠋⠉⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣈⣉⣉⣉⣉⣉⣉⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⠃⠀⠶⠶⠂⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠧⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠁⠀⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠸⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⢿⣿⣿⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣽⠟⠁⠈⠻⣧⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡃⠀⠀⠀⠀⢈⣿⣿⣿⡏⠀⠀⠀⢹⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠋⠀⠀⠙⢿⣿⣿⣿⣇⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠀⢀⣠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠃⣀⣀⣀⣠⣤⣤⣬⣿⣿⣿⣿⣷⣾⣿⣿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠋⠉⢁⡀⠈⣉⠉⠉⡉⠙⠛⠻⢿⣟⠋⠁⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣿⣿⣿⣿⠻⣾⣤⣤⣐⣛⡀⠐⣒⣀⣀⣠⣤⠾⠃⠀⠀⠀⠸⠷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣾⣯⣿⣿⣿⣯⣿⣿⣭⣷⣶⣿⣤⣤⣴⣶⣶⣶⣶⣶⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⣈⡉⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣖⠮⠭⢖⣒⡪⠭⠥⠖⢒⣂⡩⠭⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢉⣿⠯⢐⣒⣂⠩⠥⠤⠒⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢰⠏⢹⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠸⢸⢸⠀⡜⠀⠇⢠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠸⢸⠀⣇⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⢐⢤⣀⠤⠤⠤⢾⠀⣿⠀⣿⠸⠀⠀⠀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠒⢤⠀⠀⢣⠱⡀⠀⠸⢸⢠⢻⠀⣟⠀⡇⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠣⡳⡀⠀⣀⣡⠤⠬⢾⡸⢸⠀⡇⡇⢸⠉⠉⠁⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣆⠀⠀⠀⣽⠮⠤⡀⠀⠀⢠⢿⠈⡭⠵⣓⠼⣀⣂⣀⡒⣂⡀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢣⡀⠸⡻⣀⣀⣜⡠⠔⢁⠞⠒⠦⢀⣀⣑⠢⢄⡀⠀⠀⢈⡦⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣌⣉⣉⠁⠙⠛⠛⠓⠒⠖⠃⠤⠠⠹⠦⠀⠶⠒⠒⠒⠈⠉⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣶⡳⡨⢭⡍⣩⣮⠁⣶⣖⠒⡶⠲⣖⣶⡲⣀⣝⢫⠉⡍⢧⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⢎⣑⣚⣁⡓⠛⠐⠤⠋⠳⠷⡥⠃⠷⠽⠛⠛⠝⠐⢛⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡤⠤⠄⠀⠀⠀⠀⠠⠤⠤⠤⢤⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠀⠀⠀⠀⠀⠈⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢂⡉⠠⠤⠀⠒⠒⠒⠂⠀⠀⠀⠤⠤⠄⢈⡁⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⣉⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠠⠊⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⣶⡋⠉⢉⣀⡴⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠣⡀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⢀⠤⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⠠⣈⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠊⡀⠄⠊⡠⠂⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⡁⠒⠠⠈⠁⠀⠒⠀⠀⠀⠀⠒⠂⠀⠉⠠⠀⠒⢈⠁⠄⠂⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠀⠒⠀⠀⠠⠤⠤⠤⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⠀⠠⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉⠀⠀⠀⠀⠀⢀⡀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⣂⠔⠁⠀⠀⠀⠀⠀⠀⢀⠰⢟⡀⣃⢩⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⠀⠂⢄⡠⠂⢁⡠⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⡡⠂⠀⠀⠀⠔⠈⠉⠉⠐⡤⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠆⠀⠠⠠⠀⠀⣠⠀⠀⠄⠉⠲⠄⠀⣀⠀⣀⠀⠀⢀⣀⠀⠠⠄⣀⣀⡀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠑⠤⢦⠀⢀⠄⡢⠙⠉⠁⠀⡆⢀⠬⢉⠲⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢎⠀⠀⠀⠀⠀⢁⠃⠀⠀⠈⠪⡇⠀⠈⠁⠀⠐⠒⠒⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⢉⠀⣤⡈⢹⣄⡦⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡾⠒⠚⣀⣼⣦⠤⣄⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⡡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠼⢛⢆⢐⣧⠲⠙⠀⠀⠀⠆⢀⡀⠀⠀⠀⣀⡀⠠⠀⠂⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠉⢀⠤⠊⠀⠉⠨⠛⡃⡀⠀⠀⡼⢁⡌⠀⠀⠀⢀⣀⠠⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠜⣄⠀⠀⠀⠀⠀⡁⠀⢀⣸⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠒⠀⠀⠈⠀⠀⠀⠀⠉⠙⠛⠛⠛⠀⠈⠉



⣶⣶⣶⣶⣶⡾⢶⣶⣶⡾⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣾⠶⣷⠶⠶⠶⢶⣶⡶⠶⠶⠾⠶⠾⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⢱⣵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⣿⠋⣦⠞⠋⠛⠢
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠞⡁⣈⡷⠒⡘⡆⠀⠀⠀⠀⠀⠀⠀⠤⣤⡄⣿⡀⢃⡞⢁⠀⠂⢄⠀⢱⡦⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⢈⣿⣤⣴⣶⣽⣧⡴⢖⣀⡀⠀⠈⠁⡲⠖⠋⣉⡃⣾⠀⠁⠰⡷⢹⡿⢂⡤⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠈⠙⢲⣿⡾⣿⣿⣿⣿⣾⣱⣤⣯⠴⠶⣒⣡⣚⠭⣤⣍⣟⣠⠶⣿⣿⣲⠿⢁⠾⡇⢀⡿⠇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣍⠿⣬⡉⢽⣎⠻⢿⣿⣂⢄⣈⢳⣤⣾⣿⣛⣻⣿⠽⢃⡿⢛⡼⣣⠌⣤⡤⠾⠔⢠⡠⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣌⠛⠷⠮⣿⢟⣻⣟⢨⣓⣀⣴⣖⣦⠵⠿⠓⢒⢫⣽⠍⠙⣿⣎⡉⠳⠄⣀⣭⣖⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠋⠠⣄⡀⠻⠎⠙⠃⠚⠉⠋⠒⠒⠚⠛⠛⠛⠋⠁⠀⠀⠀⠁⠀⢀⠉⣋⠁⢨⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢘⠷⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠃⠠⢎⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢊⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡈⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠈⠛⠚⠙⠉⠔⠚⠉
⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠢⠦⠴⠦⠾⠷⠶⠷⠾⠾⠿⠿⠽⠦⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠠⠤⠤⠤⠤⠤⠴⠖
⠀⠀⢀⣐⣠⣀⣠⣤⣠⣤⣤⣤⣤⣦⣰⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠁⠉⠉⠉⠁⠈⠉⠁⠉⠉⠁


⣿⣟⢹⣟⠻⣿⣛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿