Ramen Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠐⢄
⠀⠠⠈⠀⠁⠀⢀⠈⠀⣶⠃⠀⠀⡄⠀⣠⠊⡠⠒⠚⠂⠠⠀⠀⠑⠂⠀⠃⠀⠀⠞⢁⣀⡄⠀⠐⠉⠈⠀⠀⠀⢳
⠀⠀⣰⡀⢀⠀⢈⠁⠀⠈⠀⠀⠀⡆⠀⡇⢈⠀⠈⠀⠀⠒⠆⠀⠀⢀⡴⠀⠀⠈⠀⠘⠋⠃⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠇⠑⠂⠀⠀⠀⠀⡀⠀⠀⠀⡠⠁⠀⠁⡘⠧⢥⣄⠚⠀⣠⡀⠀⠈⠁⠀⠀⠘⣄⠀⠀⠀⠠⠀⠢⣄⠀⠀⢀⡜
⠀⠀⠐⢀⠁⠙⠁⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⢠⣀⡈⠁⠀⠉⠀⢀⠈⠀⠀⠀⠀⠈⠳⠤⢤⣄⠈⣀⠠⠤⠴⠏
⠀⠀⠀⢀⠁⠀⠀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⡉⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡓⠀⠀⡀⠀⡀⠁
⠀⠀⠀⠚⠀⠚⠆⠋⠂⠝⠓⠂⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠒⠂⠂⠐⠋
⠀⠀⢀⠀⠤⣐⣒⣒⠒⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⢀⠔⢁⣼⣿⣿⣿⡌⠱⠖⢦⠹⣷⠀⠀⠀⠀⠰⣶⠊⣱⠀⣀⣀⡄⢀⠀⠀⢠⠘⢧⠀⢀⡠⠤⠶⢒⠒⡒⠠⢄⢀⡠
⡾⢠⢶⡾⣻⠾⢷⣣⣴⡛⢧⣤⢹⡆⠀⠀⠘⠼⠟⢪⡃⢁⠦⣀⡀⡼⠀⠀⠈⠓⠌⣙⠻⢆⣖⣘⣁⠔⠁⠰⡄⠁⠂
⢧⠸⠛⠛⢥⣤⠺⢄⠉⠙⣿⡇⣾⠀⠀⠀⠀⠀⠀⢳⠂⠘⠦⣄⣠⠞⠀⠀⠀⠀⠀⠻⣾⣻⣭⡿⠗⠏⠀⣀⡀
⠘⢦⠀⠀⠀⡁⠀⠀⢄⡀⢈⣴⠃⠀⠀⠀⠀⠀⣀⠘⠦⠤⠖⠋⢀⠠⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠋
⢀⣈⣳⣦⣠⣭⣤⣤⡴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠛⠛⠛⣋⣀
⠘⠛⠛⠛⢛⠛⠛⠛⠓⠒⠃⠀⠀⠀⠀⠀⣾⢄⡀⠤⣢⢀⠠⢀⠆⣠⠄⠀⠀⠀⠀⠓⠒⠀⠀⠀⠀⠐⠛⠈⠉⠁
⠀⠀⠀⠀⠘⠀⠌⠇⠛⠗

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠉⠀⠀⢰⣶⡾⠉⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣯⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠐⠳⣶⣿⣿⣶⡄⠨⠙⠻⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠛⢛⣉⣉⣉⣥⣤⣤⣤⣤⣤⣤⣤⣍⣉⣉⣉⠛⠛⠲⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠚⣉⣥⠴⠖⠛⠋⠉⠉⠉⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⣈⠉⠉⠉⠛⠓⠶⢦⣄⣉⠛⠲⣤⣀
⠀⠀⠀⢀⣴⢟⣥⠖⠋⣉⣀⣀⢀⣀⣤⣤⡴⠞⣋⣩⣥⠤⠤⠶⠦⠤⢤⣭⣍⣙⡓⠶⣤⣀⡀⣀⣠⠿⠿⠻⣤⣽⠳⣄
⠀⠀⢠⡞⣵⠋⠁⣰⣿⣿⣿⣿⣿⣧⡀⠠⠶⣛⣭⡤⠶⣶⣶⠶⠒⣶⣶⣶⢤⣭⣙⠓⢦⢍⣹⡏⣩⠶⡛⢳⡄⢻⣦⡘⣷
⠀⠀⣸⣹⠃⣤⠆⣿⣿⣿⠟⠋⠙⠻⣿⣆⠈⢉⣠⠶⢛⣡⡴⠞⣋⣩⣤⣶⣶⠒⠛⠛⠳⢈⡧⣼⢣⡞⢁⡼⢃⣹⡎⢷⢸⡇
⠀⠀⣿⢸⡟⠁⠀⢿⣿⠃⠀⠀⠀⠀⢸⣿⣴⠟⣡⠾⢋⣥⠶⠛⣩⠶⠛⠉⠉⠉⠛⠳⢦⡙⣷⠙⠈⠓⠋⣠⣼⠉⠻⣾⢻⡇
⠀⠀⢸⡄⠻⣆⡀⠈⢿⣧⡀⠀⠀⢀⣼⡿⠁⣌⣡⠶⠋⠁⢀⡾⢡⣶⠀⠀⠀⠀⢀⣠⠾⠛⠛⠿⠷⣿⣟⠃⠀⠀⣰⠏⢸⡇
⠀⠀⠘⣇⠀⠈⠛⢦⣄⡙⠻⠿⣿⠿⠟⣱⣮⣟⠻⣷⣄⡀⢸⡅⢻⣧⡀⠀⠀⡴⢋⣁⠀⠀⠀⠀⠀⠀⢙⣧⡴⠟⠁⠀⣾
⠀⠀⠀⠹⣶⣄⡀⠀⠈⠙⠛⠶⢤⣤⣀⣀⠉⠻⣷⣬⣟⠿⠇⠻⢦⣙⣃⡤⢾⠁⣿⣏⣀⣀⣠⣤⠴⠞⠋⠁⠀⠀⣀⣼⠃
⠀⠀⠀⠀⠹⣿⣿⣷⣦⣤⣀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠶⠶⠶⠶⠾⠷⠖⠚⠛⠛⠛⠉⠉⠁⠀⠀⢀⣀⣤⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠘⢿⡿⢿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⣩⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⣀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣟⣒⠶⠶⠶⠶⠶⠶⠒⣛⣫⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣭⣽⣿⣭⣭⣿⣭⣭⣿⣯⣭⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡾⠁⠀⣼⠋⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠇⠀⢰⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⡆⠀⠘⣇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢿⡀⠀⢻⡄⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⡇⠀⠈⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠇⠀⢰⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⡟⠀⢀⡟⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠖⢿⠟⠛⣿⠋⠉⢻⠟⠛⢻⠟⠛⠒⠶⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⣋⣭⣤⣤⣀⣤⡤⢾⡷⠶⢿⠟⠛⢻⣟⠛⢿⡷⠶⠶⣤⣤⣀⡀⠈⠉⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣼⡋⠀⢀⣀⠉⠻⣄⠈⢷⠀⣸⣷⣾⡿⠿⠿⢿⣷⠀⠀⠀⠀⠉⠙⢻⡞⢻⣦⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⠞⠋⠉⠙⠛⠿⣿⣿⣆⠹⡇⢠⡞⠉⠋⠁⠀⠀⠀⠀⢿⠀⠀⠀⣀⣠⡴⠟⣰⡏⣽⠛⢦⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠞⣿⠀⠀⣰⣾⣶⣦⡈⢻⡏⢠⣿⣿⣴⣾⣿⡿⢿⣷⣦⠀⢸⣧⣤⠶⢿⣅⣠⠞⠉⣽⠃⣰⠿⣭⢷⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠘⣧⡀⠘⢿⣿⣿⡇⠈⣿⣿⠟⠉⠛⠉⠀⠀⠀⠀⢹⣷⣿⢡⡴⠶⣄⢉⣿⣠⠞⠁⣴⢋⡴⢋⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣬⣻⢦⣤⣉⣁⣠⡾⢻⡇⢀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣟⢾⣡⢟⣠⣿⣁⡴⠾⣷⣫⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣯⣭⣅⣀⣈⠳⣬⣙⣛⣛⣿⣛⣃⣠⡼⠛⠧⣿⣤⣶⣞⣁⣬⣵⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣯⢿⣿⣿⣟⢻⣿⣿⣿⡛⣿⣯⣭⣍⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣰⣾⣣⣿⣧⣲⣟⣾⣿⣲⣟⣼⣿⣗⣶⣣⣿⣧⣓⣏⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠉⠀⠀⠀⠀⠀⠀⣀⣴⠷⠞⠋
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⢀⣀⠤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠸⠒⠃⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⢀⣀⣾⡶⠋⠀⢀⣀⡤⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⡟⢹⠄⠂⢺⣝⡴⡶⢶⠋⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡇⠀⢣⣀⡟⠁⠱⠀⠨
⠀⠀⠀⠀⠀⠀⠛⠃⠀⠁⠀⠠⠀⠀⠱⡀⢈⠈⡧⡄⠈⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡵⣽⢶⣬⡌⢃⣻⡀⠀⠐⠀⠀⡑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⢺⣁⣀⡤⠽⠿⣶⡅⠑⣦⠤⠆⡺⣻⣿⣶⣦⣴⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄⢀⣤⡀⠀⡀⢀⠘⣿⡻⠁⣴⢫⣟⣯⣿⢿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠰⣻⣿⣇⣵⠶⢋⣌⡉⣶⣶⣮⠅⣠⣯⡟⠁⠀⢈⣀⣤⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⣷⣾⣟⣈⢄⢜⡁⠤⠠⠻⢛⣝⡶⡄⢀⠠⢄⣛⣣⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠆⠀⠀⠉⠉⠙⠚⠛⠿⠿⠿⠾⠟⠓⠚⠛⠋⢉⢁⠀⢼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣄⣀⡀⠀⣀⠀⣀⣀⢀⣈⣤⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠟⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⣛⣛⣉⣭⣭⣭⣭⣭⣭⣭⣭⣉⣙⣛⡛⠳⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢞⣫⣥⠶⠖⠛⠋⠉⠉⢉⣁⣠⣤⣤⣤⣤⣤⣤⣉⣉⠉⠙⠛⠳⠶⢦⣭⣙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠶⣻⣭⠶⠦⣤⣤⡴⡶⢞⣛⣭⠴⠶⠶⠶⠶⠶⠶⠬⣭⣟⡳⢦⣄⣀⣀⣼⠿⡟⠷⣾⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⣼⠏⠀⠀⣀⣀⠙⢦⡞⣻⡽⠶⣺⣿⠿⢛⣻⣿⠿⠶⢶⣮⣝⡻⢦⣽⢏⣴⣿⡙⣷⢘⣿⣎⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣵⠟⣿⠀⣰⠟⠉⠉⠻⣎⣷⣀⣴⢞⣫⡴⢞⣫⣵⣶⣾⣿⣿⣏⣉⡙⣳⡟⣾⣽⣋⣴⠏⣼⣷⢹⣾⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠹⣧⠀⠹⣇⢿⡀⠀⠀⢀⣿⣹⠟⠴⢟⣥⠾⠛⣡⡾⣋⠁⠀⠀⠈⠉⠻⣮⣿⣭⣈⣭⣵⠾⠏⠈⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣤⡙⢿⣿⡶⠶⠟⣡⣟⡰⣶⣿⡁⠀⢰⡏⣼⡏⠀⠀⠀⠀⠀⢠⡟⠀⠀⠛⠉⠉⠀⣠⡾⠋⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠈⠙⠳⢿⣿⣛⡛⠉⠛⠻⣶⣝⠻⣶⡼⢷⣘⠿⢄⣠⣤⡴⠶⠟⠁⠀⢀⣠⣤⠶⠛⠉⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡛⠷⣦⣄⡀⠀⠉⠙⠛⠓⠶⠾⠿⢿⣦⣤⣤⣽⣿⣿⣭⣤⠶⠶⠶⠛⠛⠋⠉⠀⢀⣠⡴⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣤⣍⡙⠛⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠶⠛⢋⣩⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣍⠙⠛⠶⠶⣤⣬⣉⣉⣛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣉⣉⣥⣤⠶⠶⠛⣫⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣟⣛⣛⣛⣛⣛⣻⣭⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡤⣤⠒⠒⠶⠶⠃⠉⠁⠀⠀⠀⣀
⠀⠀⠀⣴⣿⣯⡿⣷⣾⣧⣤⠼⢿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢶
⠀⠀⠘⢧⡏⠉⠉⠉⠁⢙⡇⠀⢀⡾⠃⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⡦⠀⠀⠀⠀⠀⠘⣿⢰⣶⣾⣄⢀⠰⡆
⠀⠀⠀⠀⠉⠓⠒⠲⠶⠎⠓⠒⠈⠀⠀⠀⠀⠀⠀⣶⣠⣿⣿⠿⠿⡿⢝⣶⡆⠀⠀⠀⠰⢿⣿⣿⣿⣷⢿⡓⠅
⠀⠀⠀⠀⠀⠰⠤⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢿⡟⠿⠁⠀⠐⠱⠀⠛⡗⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⣤⣤⣤⠄⠀⠀⠀⠉⠢⡄⠀⢠⡤⠆
⠀⠀⠀⠀⢠⢜⣧⡟⣼⣻⣶⢶⡠⣢⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠁⠀⢛⣿⡇⣡⣶⠟⢉⢨⡼⠀⠀⠀⠠⣸⣯⠾⡛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠃⠜⠒⠁⠀⠀⠀⣰⢲⣿⡿⡴⠷⡧⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣤⠀⠀⠅⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢺⡿⣩⠖⠜⠁⠀⠀⠂⡀⠀⠀⠘⣿⣿⣿⡿⠀⠀⠀⣀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣶⠄⠀⠂⠄⠀⠀⠀⠀⣠⡿⢁⡤⠀⠀⠀⢀⠌⠀⣠⣶⣾⢿⣿⣟⣷⣶⣦⣄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣤⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡸⠀⣿⣿⣽⡞⠛⠛⠛⣿⠛⢻⣿⣧⢀⡆⡀
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠠⠟⠿⠷⠊⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣻⣏⣀⠀⠐⠂⣸⣄⢉⣿⣯⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢄⣈⣀⣀⠄⠀⠀⠀⠀⠉⢩⣭⣍⡛⠛⠛⣛⣻⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠓⠰
⠀⠀⠀⠀⠠⣐⣢⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢈⣾⠋⠉⠙⢻⣿⣿⣷⠀⠀⠀⢴⣾⡿⠙⢃⡤⠚⢻⣿⣿⣿⢳⣆⠀⠀⢀⠔⠀⢀
⠀⠀⠐⣾⡃⡻⠛⠈⠙⢇⣀⣿⡀⠀⠀⠘⢿⣄⣠⠎⠀⠂⠰⣙⣿⣿⣿⠟⠀⠀⢸⡀⢤⠃⠀⡖⠁
⠀⠀⠀⢺⣿⡇⠀⠀⠀⢸⣿⡿⠄⠀⠀⠀⠀⠈⠙⠛⠒⠞⠛⠛⠛⠉⠁⠀⠀⠀⠀⠙⠺⢧⣘⣇
⠀⠀⠀⠈⠙⠻⠶⡶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠨⠙⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠟⢛⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⣿⣿⠿⢛⡝⠀⣹⠿⠘⠣⡚⢛⣋⠙⠙⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⡿⠋⠀⠀⡤⠿⠶⠟⠁⠐⢋⡴⠛⡚⢛⣡⣷⢈⠻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣥⢤⣤⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈⢻⡽⣞⢻⣷⢘⡗⠌⡩⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣿⣿⣿⡄⢃⠀⣾⣿⣿⠖⣲⠖⠡⣤⣤⡄⣶⢘⣧⡻⣟⡉⢚⣽⢪⡻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⡟⠉⠉⡇⠈⣤⠴⠼⢫⠤⠬⣧⣼⡶⢢⣾⡿⣿⠎⢷⡙⢠⢠⣁⢿⣿⣿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠀⠀⠀⡇⠀⠁⠤⠊⠀⢠⡀⠀⠙⣄⡊⣉⣷⣿⣿⣧⡻⣎⢣⡭⣿⣿⠍⣿⣿⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⣀⠀⡏⠀⢀⠀⠆⣴⢻⣦⡀⠀⣟⢣⡝⢿⢇⡹⣏⣳⡹⣧⡹⡿⠿⢰⣿⣿⣟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⡷⠗⠚⡇⠀⠀⠈⣿⢋⠁⡸⠜⢹⣷⣯⢻⣿⣿⢻⡿⣘⢷⡜⢶⣶⣿⣿⡏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⠋⠀⠀⠀⣡⡀⠀⠈⠁⢀⣴⠇⣻⣿⢿⣻⣟⠋⣭⢿⣷⣿⣎⢿⣎⢻⣿⣿⡇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⡸⠀⠀⡴⠛⠋⠳⠐⣶⣶⣿⠟⠻⣷⠿⢦⣭⣿⣷⡟⠋⢉⠄⠀⢨⣿⡲⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⡆⠀⠈⠓⠔⡀⠒⢐⣮⡙⢀⣾⠃⣼⠦⠈⢉⢻⠋⠀⠈⢀⣿⣿⣿⡿⢷⣟⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣶⣶⠠⠰⠀⣮⠍⠉⡄⠁⠰⠒⡖⢠⠊⣿⡄⠀⠘⠉⠈⠉⠁⢃⣸⣿⡏⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⠦⡀⠓⢀⣤⣧⠁⡴⠿⠞⡦⣠⣌⡊⣷⣶⣤⣴⠶⣲⡿⠟⠉⢿⣆⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⣴⣶⣿⣿⣿⡀⠀⣀⣶⣊⣵⣤⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⢻⣦⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣟⡡⠔⠋⠀⠀⠀⠀⠀⠀⠻⣧⡀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⣄⢠⣞⣵⣤⡿⡿⣿⢰⡿⣠⡾⣿⠀⠀⠀⢾⡀⡴⢻⣧⣴⣢⣇⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣎⠉⠈⠈⠀⠀⣿⠈⠉⠉⠀⣏⠀⢠⡴⠟⠁⠉⠉⠀⠉⠉⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⢧⣯⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣿⡟⣉⡴⠿⠿⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠟⠀⢲⣼⠇⢾⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⡁⡟⣰⠋⠀⠤⣀⠀⠈⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⢿⠄⢾⣌⣫⣬⣞⣾⣃⣤⠤⠀⠀⠀⠒⠀⢨⡭⠿⠿⠇⣱⠃⠜⠨⣦⠜⣷⡶⢉⡀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠐⠶⢄⣵⣮⣻⣿⣿⣿⣿⡿⣍⣸⣞⣀⣄⡤⢔⡤⢦⡬⠻⣷⢉⣠⣶⣦⣈⣴⡟⢠⣿⠀⢰⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢿⣝⠻⢻⣿⠫⣷⣿⣽⣉⢁⣐⡚⣛⣳⣾⣿⢿⣷⣿⡦⣝⣯⢳⣮⢁⢋⡙⣀⣯⡈⢀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠙⠷⣦⣝⣷⣽⣿⣿⣆⢶⡲⠬⣽⣛⣛⣛⣿⣛⣩⣴⣷⠶⠭⣶⣯⠟⢿⣉⠉⠠⢟⣫⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠉⠉⢋⣵⢿⠧⣝⣾⡿⣟⣓⣋⣭⣶⣤⣴⣬⠿⠋⠂⠼⠿⠓⠀⠩⠴⠾⢟⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡑⠄⠀⠙⣶⣄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠀⠐⢰⢢⠴⢊⣵⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠮⠸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⠊⢫⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⡫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣴⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣨⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢦⣄⣀⣈⡀⠀⣀⣀⠀⢀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠙⠛⠈⠁⠘⠉⠁
⠀⠀⠀⢠⡄⡀⠀⠀⠀⢀⠀⠀⠀⢀⡀⣀
⠀⠀⠐⠲⠾⠷⠾⠿⠷⠿⠾⠷⠿⠿⠿⠿⠶⠶⠖⠒⠒⠒⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠐⠒⠲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠔⠃
⠀⠀⢬⣬⣹⣧⣴⣦⣤⣷⣴⣤⣦⣶⣖⣧⣄⣀⣀⣀⣀⣀⣀⣐⠒⢒⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⡟⠻⣿⡿⢡⡔⣻⣿⣿⢿⡿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣻⣿⠿⠿⠋⢁⡼⣀⣤⡟⢋⣀⠘⣢⢠⢴⣆⢠⡙⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠁⠀⠀⠵⠛⠚⠛⠁⢀⠈⠡⠞⣉⣬⢩⣤⣼⡧⢠⡻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣼⣿⡖⢤⣶⠂⠀⣀⣀⠀⠀⠀⠀⠙⢀⠀⠹⠮⢷⡙⣿⡇⣸⡙⢎⡁⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣿⣿⣿⣿⠀⣃⠀⣼⣿⣿⡶⠒⡾⠛⠥⣦⣤⡤⢲⠂⣻⣎⢻⣫⡁⢊⣭⢢⣝⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢄⣿⣿⠋⠉⢸⠀⠁⣤⡼⣴⠟⣁⢠⣽⣧⢰⡶⢦⣴⣿⣿⡷⠹⣦⠻⣡⢨⡋⢾⣿⣿⣿⣿⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⠇⠀⠀⢸⡄⠀⠁⣀⡐⠁⢀⣀⠀⠈⢻⡠⠞⠛⢵⣬⣿⣿⡜⢷⡘⢧⣛⣷⣿⣟⢻⣿⣿⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠛⠀⠀⠀⠸⡇⠀⠀⠀⡄⢠⡴⣿⡄⢁⠀⡿⢇⡙⣿⣿⠙⣿⠻⣎⢿⣌⢳⣿⣿⠇⣼⣿⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⢐⣷⣐⣧⣤⡌⠀⠃⠈⢇⣿⠛⠆⣸⠳⠼⣧⣜⣣⣿⣞⣿⣿⣦⠻⣧⡹⣆⣠⣽⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⣼⠟⠉⠀⠀⠃⠀⠀⠀⠼⠁⠂⣰⠏⣤⣼⣿⢿⣤⣿⠿⠷⣼⣯⣷⡹⣷⡜⢷⣿⣿⣿⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⡇⠀⠀⢀⣠⡾⣆⠀⣀⣀⣠⣾⣿⣡⣞⣿⠙⣴⣿⣦⢰⣿⡾⠿⠻⠷⠙⣿⣌⢿⣿⡿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠃⠀⢰⡏⠉⠀⠈⠂⠹⢿⡿⠏⣁⠙⣿⠛⠹⣿⣿⣿⠟⠁⡀⠐⣀⣀⣐⣾⣮⢻⣿⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠡⠊⠄⠉⢈⣽⣦⡁⢸⡟⠀⠿⠓⠀⠂⣌⡇⠀⠀⣅⣠⢿⣿⣿⣿⢛⣿⣹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⠀⠘⠀⡷⠦⠉⠠⢀⠀⠻⠛⡟⠠⠋⢼⡷⡀⠀⠈⠀⠀⠀⠀⢃⣠⣿⣿⡄⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⠦⣀⠸⠂⣠⣼⣎⠀⠜⡿⠟⣷⢤⣤⣚⡹⣷⣶⣶⣶⠶⢻⣿⠟⠉⠙⣿⡄⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⣰⣶⣾⣿⣿⣇⠈⢀⢀⣴⣚⣥⣦⣽⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⠟⣃⡤⠚⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠋⠀⣠⠄⠀⣠⣤⣀⡀⢀⣄⠀⣀⣀⠀⠀⠀⢀⣀⠀⢀⣠⡀⢀⡀⣀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⢻⣦⢸⡷⢿⢾⡟⠿⢹⡇⢿⣫⡾⠋⣿⠀⠀⠀⢙⣷⠰⣏⡽⠓⣿⣵⣯⣼⡷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠿⠀⠐⠾⠋⠁⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣴⣶⡿⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢦⣴⣸⣾⣇⠾⠿⣋⠉⢁⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢟⠽⢩⣶⢩⣶⢡⢺⡿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡈⡜⡑⡆⡈⣾⡈⡜⣀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠹⠀⢡⡱⠘⡷⡄⢋⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡇⠇⠈⡌⣰⢡⢣⢻⢨⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠃⡐⠀⠁⢹⠃⠀⡘⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣟⣿⣶⡾⢀⡇⢸⢸⣿⠘⣱⢷⠘⣷⠶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢯⣿⣿⣿⢟⣋⣍⡅⠈⠱⠸⠆⢿⢀⢻⣠⢀⠿⠿⠮⢿⣿⣫⣤⣮⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⢃⣿⣿⣿⣟⣻⣯⣿⣷⡐⠃⣰⢸⡄⠈⠎⣏⣸⣿⣿⣿⣿⣎⣿⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⡏⣸⣿⣿⣿⣹⢿⢊⣭⣩⣷⠾⠫⢿⣿⣾⢿⣿⣄⣉⣩⣿⣿⠋⣁⣤⣿⣿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⡇⣿⣿⣽⡏⡟⣾⡤⠍⢻⣷⢾⣾⡷⠟⢋⣙⣉⣛⡿⠏⠿⠿⢿⣿⠟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⣇⠙⣿⡇⠇⡇⣏⡰⠷⣾⣿⡿⠋⠠⣦⣿⣿⠟⠁⣴⣿⣿⣷⣆⢹⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⣧⡈⠳⠀⠃⠈⢃⠿⢀⣆⣨⡟⠀⠀⠀⠹⣿⠏⠀⠘⢿⣿⣿⣿⠟⣸⣗⠭⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣦⣀⣀⢼⡖⢋⣤⠤⠷⣤⠤⠤⠤⣎⠀⠀⠀⠀⣉⣉⡁⣤⣿⣡⡄⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣿⠉⣉⣠⣍⣡⣶⠀⠘⡛⠛⣇⠿⠶⠶⢲⣭⢋⣈⠑⡄⣿⣿⡃⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣾⣯⣀⣿⣿⣿⣿⣿⣻⣻⣿⣿⣿⣻⣿⣿⣯⣿⣿⣿⣾⡅⠹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠶⠒⠒⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠐⠒⠒⠂⣉⣉⣉⣉⣀⣠⣀⠠⠤⠤⠤⠤⠤⠤⠤⠄⠐⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⡤⢾⡀⠀⠠⡀⢘⣄⢂⣸⢋⠔⣦⢀⠋⠁⠀⣄
⠀⠀⠀⠀⠀⠸⣿⣍⣈⣁⢂⡀⠆⡀⣈⠂⣹⣶⣼⣀⣘⠚⣡⣴⡿⠆
⠀⠀⠀⠀⠀⠀⠹⡀⠈⠉⠉⠉⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠙⢛⣻⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⣭⠿⣭⡿⢯⣽⠿⣽⠿⠯⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⡏⠀⡼⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⡄⠀⣧⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢷⠀⠸⡆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠀⢀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⡏⠀⣼⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠒⠛⣿⠉⣿⣉⣹⣏⣉⣟⠙⠛⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⡵⠖⠛⠲⢶⡛⢻⡉⢹⣉⣽⣿⣭⣿⠉⠛⠓⠲⠦⣤⣤⣉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⣰⠏⢀⡴⠚⠓⠲⢾⣷⣦⢹⡄⡵⠚⠛⠋⠀⠀⢹⡇⠀⠀⣀⣠⢞⡴⣻⠦⣄⠙⣦
⠀⠀⠀⠀⠀⠀⢻⡴⢻⡄⠀⣶⣷⣦⠙⣏⣸⣿⣷⣾⠿⠿⢷⣦⢀⡧⠴⠚⣧⣴⠊⣹⠃⡼⢫⠷⣾
⠀⠀⠀⠀⠀⠀⠈⢷⣄⡳⣄⡙⠿⠿⢁⣿⡟⠁⠈⠀⠀⠀⠀⢿⡿⢳⣯⢽⠄⣿⠞⣡⣞⡵⣫⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣛⣋⠘⢧⣘⣶⣶⣶⣆⣀⡼⢻⣹⣈⣭⢟⣙⣫⣥⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⣿⣿⣿⣽⣿⣿⣽⣿⣿⣯⣿⣿⣿⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⠿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⣤⠪⣿⡏⣼⣿⡛⠛⠉⠉⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠖⡷⣋⣮⢛⣭⢋⢴⣶⣿⣷⡿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⢰⢣⡉⠘⣽⡘⡈⡄⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢣⠃⢃⢧⠹⢧⢱⠾⣼⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⡄⡆⠘⡒⢠⢃⠇⢾⠙⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠇⢧⢠⠃⢺⡞⠈⠸⣄⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣤⣠⡘⠀⡄⣰⢠⣼⠠⢎⣶⠙⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣿⣿⣷⣾⠿⠿⠃⠘⣇⢸⡌⣿⠀⢿⡈⠄⣿⣿⣝⣿⣿⡷⠂⣠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢧⣿⣿⣿⣯⣶⢾⣿⡇⢀⠇⡎⢰⠈⠸⡘⡿⢠⣯⣿⣷⣭⡙⢾⣿⣿⣿⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⣾⣿⣿⡿⢷⣟⠿⠿⠿⣞⣤⡖⣼⣧⣠⣵⣼⠞⠿⠿⢿⣿⡿⠾⠋⢹⣿⣿⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣿⢰⣿⡿⣿⡧⣿⣍⠘⠿⣾⣏⣩⣩⣝⣿⣿⢽⣿⣷⣶⡶⢿⣿⣤⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⣿⠸⣿⣿⢻⡇⣿⡟⢉⣅⣽⢿⣽⡿⠋⡁⣴⣶⣶⠶⠚⣩⡭⣍⡙⠻⣏⣼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣧⠘⣆⠘⢻⡎⢸⣿⣷⠊⠛⢿⣿⠋⠀⠘⢿⣿⡿⠁⠀⣾⣿⣿⣿⣿⡄⣿⣻⡝⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣮⣷⣌⢁⠀⠀⠈⢴⣉⡼⠷⠾⡇⠀⠀⠀⠈⢿⠃⠀⠀⠹⢿⣿⠿⠋⢰⡿⠒⠒⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⡤⡖⣿⠉⣴⠖⣋⠙⢻⣙⣚⡛⣿⡀⠀⠀⣀⣤⠶⣖⠒⢿⣧⣾⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⡏⢠⣦⣵⣦⣾⣿⣀⣨⣥⢈⣧⣙⣋⣉⣹⣶⣧⣴⡄⢱⢸⣿⣧⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⣿⡇⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡸⢸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⣼⣿⣿⣿⣿⣯⣽⣽⣿⣯⣿⣽⣿⣿⣿⣾⡟⣿⣿⣷⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣶⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⣫⠗⣿⣿⣿⡏⣿⢨⠏⡼⠹⢛⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣯⢶⣫⢷⡟⠛⠻⢿⣅⣿⣾⢜⡁⣀⠈⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠼⠿⣻⣿⣿⣏⠿⠇⠼⣭⣦⢹⣿⣿⣿⣧⠉⠰⡾⡿⢫⡿⣿
⠀⠀⠀⠀⠀⢀⡠⠖⢉⡠⠂⡻⣿⡇⣞⠱⣧⣼⠂⢮⣡⣾⣿⣿⣿⣿⠇⢀⢱⠃⢼⣷⠏
⠀⠀⠀⠀⠀⠉⣠⠖⠉⠀⠀⠙⣅⡳⣹⣷⡂⢀⡒⢋⠉⡋⠙⠯⠥⣄⠰⢁⠚⠀⢟⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣭⣝⣓⡀⢄⣠⢤⢀⣠⠄⢒⣛⣹⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡆⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢻⡀⠀⠀⣼⡥⠦⡄⠀⣠⢤⠀⢠⢤⡟⠀⣼⠇⣼⡥⠋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠈⣧⠀⢀⡿⣾⣡⣽⢾⠻⣼⣤⡏⡼⡇⣰⡟⠀⠻⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠸⣇⢸⠃⢷⣴⠏⠘⠞⠁⠘⠛⠀⠓⠹⣇⣤⠤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠹⡿⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣀⡤⠶⠒⠋⠩⠍⠂⠀⠀⠀⢀⣀⣀⣏⡁⢎⠉⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⢠⠖⠒⠦⠤⣤⣒⣓⡶⢏⢧⡢⡑⢕⢌⢳⡀⠀⠀⠙⣗⠤⣀
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⣀⣠⣇⠀⠁⢀⡠⠿⠤⣞⡄⢏⠣⡑⢌⢮⢣⢣⡣⠞⠑⡔⣉⠢⠈⠳⣄
⠀⠀⠀⠀⠀⢰⠏⠈⠀⢸⠉⢀⣨⠟⢓⠭⠒⣒⣊⠡⠽⣞⢄⢣⣈⣣⠣⡉⢱⡱⣄⠘⠤⡀⠀⠱⡀⠈⠳⡄
⠀⠀⠀⠀⠀⣾⣠⠴⣒⣚⣚⣕⣋⠭⣓⣶⣭⠴⣚⣩⠈⢹⡞⡄⠾⣉⠑⢽⣦⡫⡙⢕⢄⠱⠤⠴⡷⢠⠀⠹⡄
⠀⠀⠀⠀⠀⢹⡗⡩⠄⢒⡯⠴⠾⡉⠀⡴⣊⠅⡢⠤⢒⠊⢳⡸⣄⠈⠦⠤⢳⡳⠽⣬⣵⠷⠶⠯⠤⣼⠀⠀⣷
⠀⠀⠀⠀⠀⠸⣷⡩⡼⠁⢀⡀⠀⠈⢾⠀⠧⣈⡂⠤⠬⣢⣹⡙⢬⣍⣂⠬⡵⠊⡝⡹⡇⠀⠠⠀⠀⠘⡆⠀⡟
⠀⠀⠀⠀⠀⠀⣏⠻⣅⡔⠉⠙⡆⠀⢸⡗⢲⡤⣌⠱⣄⣘⣌⣏⠉⡠⡡⢒⠦⢭⠞⠉⠉⠉⢳⡄⢀⡼⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠸⡄⠈⠳⣤⡀⡇⠀⢸⡇⠰⠶⠶⢛⡶⢶⠦⣝⠛⠚⣴⣥⠤⢶⡀⠀⠐⠂⠀⠻⠋⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⠳⢦⣼⣐⠛⠛⠛⣓⠾⡫⠓⠉⠉⡹⠁⠀⠄⠀⢧⠀⠀⣀⣠⠾⠋⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⢮⣤⣤⣤⣤⣧⣤⣤⡤⠶⠾⠒⠋⠉⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠀⠀⠀⠁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⢤⣤⣤⣤⣤⠤⠶⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣷⣶⣤⡄⠈⠁⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⢄⣿⠿⢩⡼⢟⣩⡭⣤⣤⡯⢉⠉⠉⠀⠀⣤⣤⣥⣤⣍⣙⢻⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠔⣸⣿⣿⣛⣉⣹⡿⣾⣯⣶⣿⣴⣟⣳⠾⣯⢛⣿⣥⣦⣴⣿⣭⢻⠉⠉⠙⠿⢻⠿⣻⣿⣿⢿⣶⣄
⠀⠀⠀⠀⠀⡀⢠⣾⡿⠟⢉⣩⣁⡀⠀⠀⠉⠉⠻⢟⣉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣴⣀⣤⣾⠿⠛⢁⣤⣾⣮⡛⢿⣦⣤⡄
⠀⠀⠀⠀⠐⢠⣿⡟⢁⠀⠀⣾⣿⣿⣦⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠀⠀⠀⠈⠁⠛⠿⠿⣿⡿⣿⣿⣿⣆
⠀⠀⠀⠀⡄⣿⡟⠀⢸⣷⠿⢟⣛⡛⠉⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠈⠋⢀⡀⢿⣿⣿
⠀⠀⠀⠀⡇⢿⡇⢠⠊⢠⡶⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠉⡟⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠇⠙⠿⣷⣆⣐⣖⡀⣀⡼⣦⣼⣿⡟
⠀⠀⠀⠀⠱⠸⣧⠀⠀⣏⠀⠹⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⡇⠘⢟⣼⠿⠉⠘⠉⠛⠚⢿⡏⠀⠀⠀⠀⠈⠙⡻⢿⣿⠿⣤⣾⣿⡿⢠⠁
⠀⠀⠀⠀⠀⠱⠙⢦⣄⠘⣦⡀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠘⠃⠀⠭⢿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷⠟⣉⣿⠟⡰⠁
⠀⠀⠀⠀⠀⠀⠘⣦⠻⣦⡬⢝⣛⣛⡉⠀⡀⢀⣀⣰⡠⠤⣹⠞⢲⠞⠻⢦⣄⡀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢀⣿⠃⣨⠟⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣮⡙⠦⣬⣉⠹⣻⣟⡟⡓⠇⢀⡀⠀⢠⠞⠶⠀⠈⣋⠹⠶⢶⣾⣿⣦⣀⡤⠀⢀⣠⣿⡷⢛⣥⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣹⣿⣷⣮⣝⡻⠶⣶⣴⣂⡐⢀⣤⠄⠀⠉⢠⣤⣤⣐⣁⠀⠛⣻⠟⢛⣻⣿⡿⠟⣋⣥⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣶⣬⣭⣉⣙⣓⡒⠒⠲⠾⠷⠾⠿⠿⠿⣿⣟⣻⣯⣭⣶⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡾⣻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠿⢉⣿⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⣔⡆⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢹⠏⡼⠋⢀⡀⠈⢠⡀
⠀⠀⠀⠀⠀⠀⠐⢿⠁⣶⡒⣏⣥⣶⣇⣀⡤⠀⠀⠰⠆⢉⣻⠿⠿⢌⡼⠡⠂⣦⡸⣦⡜⠋⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠲⢄⣴⣿⣿⣿⣿⣿⣿⣉⣰⣖⣀⡠⠔⡨⢦⡈⠿⡎⣃⣤⣦⣌⣴⡏⣸⡏⠀⣾⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⣝⠛⣿⡻⢿⣿⡯⠉⢠⢚⣛⣻⣾⠿⣿⣿⣻⢮⡿⢟⡦⣀⣉⣃⣼⡥⢀⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠳⢮⣽⣾⡿⣿⠟⡫⣑⣻⣭⣭⣭⠿⠧⠷⣿⣞⠓⢾⣷⡛⢯⡄⢀⣫⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠛⢠⣀⠑⠯⠿⠓⠚⠛⠛⠉⠷⠶⠿⠶⠛⠛⠁⠀⠉⠉⠉⢀⠚⣛⡭⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠀⠀⢹⢗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⢊⠥⢮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢃⠴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣦⣤⡈⣀⣀⣀⠄⣀⣤⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁
⠀⠀⣀⣒⣶⣀⣤⣀⣶⣠⣠⣆⣶⣶⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠭⣭⠉⣉⠉⣍⣉⢉⣉⣍⣭⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠉⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠉⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢄⡀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔⠀⢀⠈⠇⠐⠀⠀⠀⠀⠀⠈⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠂⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠡⠀⠒⠃⠀⠀⡀⠐⢀⠃⣁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠐⠀⠀⠄⠀⠈⠀⠀⠲⠀⠀⠀⠄⠈⠠⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠐⢤⠀⠀⠄⠀⡀⠀⠀⠐⠉⠂
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡁⠀⠀⠀⠀⠀⡀⡀⠁⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡈⠁
⠀⠀⠀⠀⢀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢙⠈⠐⠉⠀⠐⠒⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⡀⠐⠀⠁⠈⠉⠀⠱⠂⠀⡠⠀⠉⠑⠂⠀⠤⠄⠀⠄⠀⠀⠀⠒⠀⠀⢊⠀⠀⠀⠀⡄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⢀⡄⢀⠂⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠠⠒⠠⢤⡂⠀⠤⠀⠀⠂
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠐⠚⠈⠀⠀⠀⠀⠦⠀⠈⠁⠀⠈⠈⠀⠀⠀⠀⠀⣀⠸⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠆⠀⡆⠀⠐
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠂⠀⠀⠀⢰⡃⠀⠀⢀⡀⠒⠀⠀⠀⠆⠀⠀⠄⠀⠀⠀⠀⠀⣀⡀⢈⠀⠈⠀⠀⠀⠀⢨⡀⠈⠀⠀⠀⠀⠀⠀⠰⠃⣰⠇
⠀⠀⠀⠀⠀⠈⠣⡀⣄⠀⠀⠀⠀⠀⠈⠲⠀⠐⢤⠀⢠⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢀⠄⠀⢠⠁⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠅⡀⠀⠀⠀⠁⠀⠁⢀⠘⠀⠀⠄⠀⢀⡠⠄⢀⠀⠀⠣⠁⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡤⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠨⠅⣑⠀⠀⠥⠀⠠⠙⡄⣁⠃⠈⠀⣀⠀⠀⠴⠁⠀⠀⠀⠀⠀⣀⢁⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠁⠀⠅⠀⠉⠀⠈⠈⠀⠉⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣽⣭⣿⣵⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣶⣶⣿⣷⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⣀⣀⣀⣈⣀⣀⣀⣀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢖⣛⣫⣭⣭⠭⠥⠤⠤⠤⠭⢭⣭⣭⣙⣛⡲⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⣻⡽⣖⣋⠉⠁⣀⣀⡤⠶⣒⣒⣒⣒⣒⡶⠦⣤⣀⡉⠉⠛⢶⣬⣽⣳⣤⡀
⠀⠀⠀⠀⠀⠀⣴⣿⠟⣱⠟⠉⠉⠙⢯⣩⣵⣞⣯⣭⣵⡶⢦⣭⣭⣽⣓⡾⣿⣳⣖⣫⡿⢮⠹⣯⡻⡄
⠀⠀⠀⠀⠀⢀⣿⣧⠖⣿⢀⡴⠛⠛⢦⣻⣌⣥⢞⣯⠷⣛⣭⣵⣶⣾⣛⣛⢳⣿⣾⣿⣫⡞⣸⣏⣷⣿
⠀⠀⠀⠀⠀⠀⡏⢿⡀⠹⣼⣇⠀⠀⣸⢿⡟⠷⣫⠶⠋⣡⢞⡍⠀⠀⠈⠉⢳⡽⢯⣬⣥⡖⠋⢘⡿⣿
⠀⠀⠀⠀⠀⠀⢿⡀⠙⠶⣽⣿⣿⣿⣥⢾⣼⡻⢧⣄⢰⡇⢿⡅⠀⠀⠀⣀⡾⠀⠀⠁⢀⣠⡴⠋⢠⡏
⠀⠀⠀⠀⠀⠀⠈⣿⠦⣄⡀⠉⠛⠲⠦⢤⣬⣻⣶⣿⣛⣙⣶⣭⣶⣛⣉⣩⣤⡤⠶⠚⠋⠁⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣬⣉⡛⠶⢤⣤⣀⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⣀⣀⣤⠴⠖⣛⣩⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣙⠛⠲⠦⣬⣭⣉⣛⣛⣛⣛⣛⣛⣛⣛⣋⣉⣭⡤⠶⠖⣛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣶⣦⣤⣤⣤⣤⣤⡤⢴⣶⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠒⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡠⠀⠛⡟⢛⡒⠦⢀⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣤⣬⡁⢈⠁⢚⡀⠖⢋⠻⠆⠈⢿⣭⠀⠄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⢤⡀⠀⡀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠸⠖⠃⠐⠦⠀⣕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣹⡤⠽⠃⢀⣄⣁⠀⠀⠖⠁⠀⠀⠀⠀⢀⣀⣬⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠜⠉⠁⠀⠔⠀⠙⠉⠉⠠⢀⡆⠀⢠⠔⠀⢀⣀⣠⣄⣄⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⠀⣀⣀⠉⠀⠀⠀⠀⠀⠀⠀⡁⢀⠁⠀⡴⠉⠀⠀⠀⠀⠲⠹⢆⠻
⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠸⠀⡎⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡜⠃⠙⠿⡀⠀⡰⠞⠛⠲⡄⠂⠀⠈⠀⠀⠀⠀⠑⠂⠀⠀⠲⡀⠀⠀⠠⣄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢿⡒⠀⠀⠀⣷⡀⢁⠀⠀⢠⢀⡴⣤⡶⢤⢤⠄⠀⠀⠀⠀⠀⠀⠳⡤⠀⠂⡁⡤⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠀⠀⢘⣛⠢⠈⠬⢅⡛⣐⠷⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⠇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠁⠉⠐⠤⢴⠟⣀⡄⠀⡀⠀⠀⠀⠀⠀⠄⢠⡾⠃⢾⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠈⠀⢨⣩⣭⡄⠰⠓⡄⠀⠀⠒⠋⠠⠟⢠⡆⠘⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⡗⠀⢃⠈⠻⠚⠀⡀⠘⠀⢀⢀⡐⠃⡤⢤⡟⠁⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠑⠀⠀⠉⠀⠀⠀⠀⣠⠀⡃⠜⠳⣤⣤⠈⣃⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢀⣶⣶⣒⣠⣇⠙⠂⢡⡔⢄⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠘⠃⠀⠉


⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠤⠤⠤⠶⠒⠒⠂
⠀⠀⠀⠀⢁⣤⡶⢟⣿⣟⣻⣿⠛⠛⣷⣶⣬⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣶⡆⠒⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠠⢀⠒⠂
⠀⠀⠀⢰⠿⣿⣿⣿⣻⡿⠿⣿⣿⠗⠑⢻⢟⣽⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⣨⢉⠒⠀⢉⡇⠁
⠀⠀⠀⠙⢶⣏⠀⠀⠀⠀⠀⠀⣟⠀⠀⢀⡿⠃⠀⠀⠀⠈⠀⠀⠀⢶⣿⣿⣿⣿⡇⣿⡗⠀⠀⠀⠀⠀⠀⡀⢻⣧⢸⣿⣾⣧⡄⣀⠰⡇
⠀⠀⠀⠀⠀⠈⠙⠒⠶⠴⠶⠮⠙⠒⠒⠉⠀⠀⠀⠀⠀⠀⠂⣶⡆⣴⣿⣿⡿⠿⣽⠿⣜⢲⣷⠀⠀⠀⠀⠑⠿⣿⣿⣯⣿⣿⣮⣿⣚⡁⠂
⠀⠀⠀⠀⠀⠁⠠⣄⣀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⣽⠃⠀⡐⢎⣧⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠠⠍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠀⢀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣄⠴⣟⣂⣤⣀⣀⡀⠐⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠓⠀⠁⠛⠛⠁
⠀⠀⠀⠀⠀⢰⡆⣿⢿⢱⣾⣹⣿⠏⢳⣔⣽⠆⡀⡀⠀⠀⠀⠀⠀⡀⣤⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣳⣿⣗⢈⣶⡾⠟⢀⠠⣤⠏⠀⠀⠀⠀⢎⣼⣧⠾⡟⢡⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠹⢁⡲⠖⠊⠁⠀⠀⠀⢀⡆⣼⣿⡿⣠⣿⢾⡀⠀⠰⠀⠀⠀⠸⠀⢀⣀⣠⣤⣤⣄⠀⠀⠀⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⣿⣇⣷⡿⠗⣡⢀⡿⠀⠀⠀⡆⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠄⠀⡀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⢾⣧⠟⠁⢊⠀⠀⠀⠀⠀⠔⠂⠀⠀⣀⣽⣿⣿⣿⣿⣀⡀⠀⠀⠐⢆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡿⠒⠉⠀⠁⠁⠀⠀⠀⠀⠘⠢⠿⡇⠤⠖⠀⠀⠀⠀⢠⠇⢀⣠⣾⣿⣿⣯⣿⣿⣷⣟⣿⣿⣷⣦⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⡀⢸⣿⢷⣶⡟⠉⠉⢍⠙⢿⠛⠙⣻⣿⣷⠀⣖⢠
⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⡼⠿⠟⠷⠎⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣟⣷⣏⠀⠀⢀⠤⠂⢘⡇⠈⢙⣿⣏⡼⠉⠋
⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠄⡒⠀⠉⢡⠀⠀⠀⠀⠀⠙⠛⢋⠙⠛⠿⠷⠶⠾⠿⠿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠠⠿⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⠉⠉⠉⠀⠀⠐⠋⠉⠙⠛⠋⢀⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠁
⠀⠀⠀⠀⢀⠔⣨⣬⢾⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠾⠟⠛⣿⣷⣶⣦⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠠⢨⣿⡛⠉⠈⠛⢻⣿⣷⣿⣎⠀⠀⠀⠠⣶⣿⡿⠀⠋⢰⡔⠚⢹⣿⣿⣿⣟⢹⣦⠀⠀⠀⢀⡔⠂⠀⣀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠆⣿⠃⢖⡾⠃⠓⠻⣇⠀⢨⣿⠀⠀⠀⠀⢻⣿⡀⠀⡴⠋⠀⠖⢨⡟⢯⣶⣿⢻⡿⠁⠀⠁⢸⡃⢀⡼⠁⠀⡴⠂⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠁⣿⣦⡯⠀⠀⠀⠀⠈⣿⣿⣿⡈⠀⠀⠈⠀⠈⠛⠿⢷⣤⣤⣤⣴⠿⣾⠧⠟⠋⠀⠁⠀⠀⠘⢷⣤⣿⠀⢸⡇⠀⠀⠐⠀⠀⠀⡠
⠀⠀⠀⠈⠸⢿⣷⣀⢀⢀⣀⣴⡿⢿⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠶⠷⢤⣤⠌⠄⠐⠉
⠀⠀⠀⠀⠀⠈⠑⠛⠻⠏⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⢰⣷⣼⣿⣿⣿⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢢⡤⠔⠤⣄⣀⡀⠀⡠⠚⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡴⠋⠀⠀⠀⠈⢣⠈⢹⢃⠖⡭⠛⠦⣌⣷
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢇⣀⠀⡼⠁⣠⠖⠒⠢⡀⠈⣧⢃⡎⡞⢠⠤⢄⠈⢻⡀
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠈⢻⣿⣿⣿⣿⣿⡿⠋⣀⠤⠤⢤⣙⡇⢰⠁⢰⡂⠀⢸⠀⢱⢊⠜⣠⠃⠀⠀⡣⢸⢷⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠞⠉⣉⣉⡉⢉⡠⠎⣡⠴⠒⠦⢌⣇⠈⠣⣀⣀⣠⠞⢀⡟⠁⡞⢡⠶⢄⡠⣇⡜⡌⢳
⠀⠀⠀⠀⠀⠀⣷⡀⡰⠁⡔⠉⣀⠤⠬⣁⡠⠞⢡⢶⡒⠒⠢⣌⠢⣀⣠⢀⣀⣠⠞⢇⢰⣉⡵⣶⣶⣄⠻⢼⠀⢸
⠀⠀⠀⠀⠀⠀⢱⠑⢇⡸⢀⠞⢁⡤⠤⠤⠤⢴⡑⢦⡉⠓⠒⠚⠓⠒⠦⣀⡟⠢⣱⣸⡯⢭⣻⢜⣠⣯⠣⡊⡠⡟
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠙⠾⣴⠃⡴⢚⣉⡓⢦⡘⠦⣈⣉⣉⣉⡉⠑⢦⡈⠳⣀⣿⣟⣲⡠⡯⢽⣋⣕⡶⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢻⠢⣀⠀⠀⠉⠓⠣⠤⣌⣢⡙⠢⠤⠤⠤⢤⡉⠢⡀⠹⡄⢹⢸⣃⣸⣇⠤⠒⠋⠁⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⠙⠲⢄⣀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠒⠛⠒⠛⠒⠛⠉⠉⠁⠀⠀⠀⢀⣀⠤⠚⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠉⠑⠒⠦⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠴⠒⠊⠉⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⣉⣉⡉⠉⠉⠉⣉⣉⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠶⡇⠁⣴⢍⣠⠀⢓⣡⢶⣈⠁⢲⣼⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠡⠛⠓⢧⠐⠦⣈⡡⠀⡉⠡⠌⠍⠁⣀⢻⡖⠀⠂⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠲⣄⡀⣐⡀⠀⠀⢘⡇⠀⠀⠀⢀⠀⠀⠚⠋⠁⢉⡛⢄⡲⠝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⣀⣹⡦⠬⠫⠀⣤⣘⡀⠀⠀⠞⠁⠀⠀⠀⠀⠀⢀⣀⣤⠷⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢩⠞⠉⠁⠀⡰⠂⠈⠛⠉⠉⠤⡀⡆⠀⠀⡤⠀⠈⠀⣀⣀⣀⣀⣀⣀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡇⠀⠀⣀⠟⠀⠀⠀⠀⠐⠀⠀⠈⠑⠀⡘⠀⢀⡤⠋⠉⠀⠠⠀⠠⠭⣣⠄⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠈⣇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡇⢀⡎⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡦⠤⣤⡏⠀⠀⢀⣤⡤⢤⡀⢠⠀⠀⠐⠃⠀⠀⠁⠘⣆⠠⠀⠀⣀⡀⠀⠁⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡀⠄⢈⢹⡀⠀⠞⠁⠀⠀⠙⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠈⡸⠇⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⡄⠀⠀⠸⢷⣄⠢⣀⢆⡠⢃⣴⢻⡾⠋⠙⢛⠋⠀⠀⠀⠀⠀⠀⠀⠘⢻⠁⢄⡡⠰⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠩⢬⠑⣀⠒⠒⠭⢁⣜⠃⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡾⢀⠐⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠀⠋⠈⠁⠉⡐⠢⠤⣞⢋⣠⡆⠀⡄⠀⠀⠀⠀⠀⢀⠆⠀⡾⠛⠘⡇⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡄⠈⠁⠀⢌⣭⣍⡀⠰⠟⢀⠀⠀⠐⠒⠋⢀⡺⠂⡶⠁⠻⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⠈⣥⠀⢰⠀⠻⠧⠛⠀⠀⠠⠃⠀⠀⢂⠰⠃⣠⣤⣼⠃⢠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠐⠣⠀⠈⡛⠢⠀⠀⠀⠁⠀⠀⠴⢁⠿⣧⡀⢁⠀⠿⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡀⢀⣀⣀⣠⠠⡤⠺⡇⠐⠀⡄⠈⠉⠙⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⠿⠷⠶⣮⢻⣄⠀⠠⠟⠈⠓⠉




⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⢰⣾⣆⣿⣿⣿⣿⣿⣇⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⠙⢷⠒⠢⠤⡼⠁⣠⠤⣤⡀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⢧⠀⣰⠃⡜⣡⠊⠉⠉⠓⢼⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢧⣄⣀⠀⡞⠀⢀⠔⠚⠙⠲⢄⠀⠈⡶⠃⡜⢰⠃⣠⠖⠦⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣀⣀⣠⣀⣈⢹⠇⠀⡏⠀⣴⠂⠀⠈⡇⠀⣧⠞⣠⠏⣰⠃⠀⠀⠘⡄⢸⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣀⠽⠟⠛⠻⠿⠿⠛⠉⣠⠞⢉⣀⣀⣀⡉⠹⡄⠀⢧⡀⠈⠁⠀⣠⠇⠀⣧⠞⢡⠞⢁⣀⡀⠀⢰⠃⡼⣸⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⡜⠁⣠⠴⠖⠒⠲⠶⠖⠊⢁⡴⠋⢀⣀⣀⣉⠛⢷⡀⠀⠙⠒⠖⠚⠁⢀⣼⠃⢀⡷⠤⢏⡀⠙⠊⢙⡶⢣⠇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⣼⠀⡴⠁⢀⡤⠖⠒⠒⠶⠖⠋⢀⡞⠻⣄⡀⠉⠙⢦⠙⠢⠴⠒⠲⢤⡴⣏⠈⢧⣈⣧⣴⣩⢺⠭⣝⣆⠙⣷⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢧⣠⠃⣰⠋⢀⡤⠖⠒⠶⠤⠶⣏⠙⢦⡀⠉⠛⠛⠛⠉⠙⠓⠦⣤⡟⠒⢦⢳⣰⡧⠲⣌⡿⢌⣠⢞⡜⢸⡁⣠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠙⠲⣏⣠⠋⢀⠴⠒⠙⠒⢦⡀⠱⣄⡈⠙⢒⣋⡉⠉⠑⠢⣄⠈⠳⡀⢸⣿⠟⠢⡄⣼⣫⣗⠒⣩⢶⡾⠟⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣄⠀⠀⠀⠉⠓⠧⢴⣋⡉⠑⢦⠙⢦⣀⣉⣉⣉⣉⠉⠓⢦⡀⠈⢧⠀⠹⡏⣿⠋⠹⡧⠈⣀⣨⠿⠛⠉⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠑⠦⣄⠀⠀⠀⠀⠈⠉⠛⠚⠳⠦⠤⢤⣄⣀⣈⣙⣦⣀⣹⣄⣈⣧⣠⠧⠽⠶⠖⠛⠉⠁⠀⠀⠀⢀⡠⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠉⠒⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠤⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠴⠒⠚⠉⠉⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣓⠲⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠖⣚⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⡈⠉⠉⢀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠒⠛⠛⢋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⠛⠛⠓⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⣉⣡⣤⠶⠶⠛⠛⠛⠉⠉⠉⠉⠉⣉⣉⣉⠉⠉⠉⠉⠛⠛⠓⠶⠶⣤⣤⣉⠙⠛⠶⣤⣀
⠀⠀⠀⠀⢀⣴⠞⣩⡴⠞⠛⠉⠀⠀⠀⠀⣀⣀⣤⠶⠞⠛⣛⣉⣉⣉⣉⣉⣙⡛⠓⠶⢦⣄⡀⠀⠀⠀⠉⢹⣷⣶⣦⣉⣳⣦⡀
⠀⠀⠀⣠⠟⣵⠞⠉⣠⣶⣾⣿⣶⣶⣟⠛⠙⣩⣤⠶⢛⣋⣉⣉⣭⣭⣭⣉⣉⣉⡛⠓⠶⣬⣝⡛⠶⢶⠶⢛⣨⣥⣌⠉⣿⣄⠹⣦
⠀⠀⢰⣏⡾⠁⢀⢰⣿⣿⣿⣿⠿⢿⣿⣷⡈⠡⠶⠛⢋⣩⡿⠛⣋⣥⡴⠾⠛⣋⣉⣛⣷⣦⣌⡁⣶⠛⣴⢏⡵⠂⣸⠇⢼⡟⣧⠸⡇
⠀⠀⢸⢻⣧⡾⠋⢸⣿⣿⠋⠀⠀⠀⠈⢻⣷⠀⣠⡶⢋⣡⠶⠛⣉⣤⠶⣞⣿⠿⠯⢤⣤⣀⡈⢡⣾⡱⣇⢾⣁⡴⠋⢠⡾⣧⢸⡇⣿
⠀⠀⢸⡎⢿⡀⠀⠈⢿⣿⠀⠀⠀⠀⠀⢸⣿⡿⠋⡴⠛⣡⡴⠛⠉⣠⠞⢋⠀⠀⠀⠀⠀⠈⠙⣷⣸⣧⣤⢀⣉⣠⡶⠛⠀⠈⣿⠃⣿
⠀⠀⠸⡇⠈⠻⣦⡀⠈⠻⣷⣄⣀⣀⣤⣿⡿⠀⣴⣤⣞⠁⠀⠀⣼⠋⣼⡟⠀⠀⠀⠀⣠⡴⠛⠉⠉⠉⠙⠛⠿⣭⠀⠀⣠⡾⠃⢠⡇
⠀⠀⠀⢿⡀⠀⠈⠙⠷⣤⣈⡛⠻⠿⠛⠋⠰⣷⣮⣝⠻⣷⣤⣀⢿⡀⠻⣷⡀⠀⢀⡾⢋⣤⠀⠀⠀⠀⠀⠀⢀⣸⡷⠞⠋⠀⠀⣼⠁
⠀⠀⠀⠈⣿⣶⣄⡀⠀⠀⠉⠙⠓⠶⢦⣤⣄⣀⣉⢻⣷⣮⡟⠿⠁⠙⠶⣬⣥⠶⢿⡁⣿⣿⣄⣠⣤⡴⠶⠚⠋⠁⠀⠀⢀⣠⣾⠇
⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠓⠒⠒⠚⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⠿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⣩⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣶⢦⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⣺⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠦⠤⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⡤⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⣀⡉⠑⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⢿⡿⣿⣶⣦⡤⠀⠀⠁⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇⣄⣿⠟⢩⣼⢟⣏⣭⢤⣴⣴⡇⢠⠊⠉⠀⠀⣠⣤⣤⣤⣦⣬⣙⣻⣿⣷⣦⣀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢀⠔⣹⣿⣿⡛⠛⣉⣿⣿⣾⣯⣶⣟⣵⡿⣻⣶⢿⡽⢻⣿⣁⣂⣴⣿⣯⡿⡟⠉⠉⠛⠿⢿⠿⢿⣿⣿⡿⣿⣦⡀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠛⢋⣉⠁⠀⠀⠉⠙⠛⠿⣿⠛⣉⣴⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⡿⣀⡴⢀⣠⣼⣿⠿⠛⢉⣤⣦⣝⠿⣶⣤⣄⡀⠁
⠀⠀⠀⠀⠀⡌⢰⣿⡿⠁⠄⠉⢱⣿⣿⣷⡀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⠛⠋⠉⠉⠀⠐⠊⠹⢿⣿⣿⣷⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠘⠀⣿⡿⠀⢸⣤⣴⣿⣿⠿⠿⠷⠠⠄⠠⠀⠀⠀⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠄⠀⠀⠀⠁⠸⠝⠉⠙⣿⣿⣿⡆
⠀⠀⠀⠀⢸⢠⣿⡁⠀⡼⠛⣉⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⡟⢷⣶⣦⣀⠄⢀⣀⠀⠀⣰⣂⢘⣿⣿⡇
⠀⠀⠀⠀⠸⡀⣿⡇⠘⠀⣸⠋⠹⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⡀⣾⡟⣿⣿⡿⢛⡿⣿⣟⣿⣿⠅⠉⠁⠀⠈⠙⢿⣶⣾⣥⣤⣶⠋⢻⣿⣿⡟⢠
⠀⠀⠀⠀⠀⢡⡹⣇⠀⠀⢷⡀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⠀⠉⡵⣿⠙⠀⠀⠀⠈⠉⠉⣿⡄⠀⠀⠀⠀⠀⠀⠸⢛⣻⣟⣻⣾⣿⣿⡟⢠⠃
⠀⠀⠀⠀⠀⠀⠑⡘⢿⡢⠈⠷⣄⣘⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣉⡀⢐⠚⣿⣆⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢁⢤⣿⢏⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡙⢦⣖⠤⣭⣉⣁⡀⣀⣀⢠⡀⢼⠠⠴⢬⡟⠋⣙⠋⠹⢷⣤⣀⠀⠀⠀⣸⣿⠄⠀⠀⠀⠀⠀⠀⣼⠟⢀⡴⠋⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣙⠷⣤⣄⡉⠟⣻⣟⡟⣓⠣⢀⣀⠀⠀⡴⠛⠖⠠⠀⢉⠩⠻⠶⣶⣿⣿⣦⣀⣤⢀⠀⣀⣼⣿⠞⣋⣴⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣻⣿⣿⣦⣍⡛⠿⣶⣶⣤⣄⠠⠂⣠⣄⠀⠈⠁⢠⣤⣄⣐⠃⠀⠘⢛⣻⠿⢻⣻⣿⣿⠿⢛⣩⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣶⣬⣍⣉⣛⠛⠷⠶⠶⠦⠶⠾⡷⣿⣿⡿⠿⠶⠿⠿⣞⣛⣩⣥⣶⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣧⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇⢠⡞⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠹⡆⠀⠀⢠⣿⢛⡙⢶⠀⣠⡞⣧⠀⣰⢛⣿⠀⢠⣿⠀⣿⠖⠋⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢻⡄⠀⣸⢿⣿⠤⢾⢛⡟⢳⡿⢶⡇⡜⣿⢀⣾⡇⠀⠙⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣷⠀⣿⠘⢧⣴⠟⠈⠳⠋⠀⠈⠛⠁⠙⠋⢾⣤⣤⠴⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⣠⡬⠴⠖⣒⣛⣉⠉⠉⠉⠉⠉⠉⠉⣱⠒⢫⠝⠛⠓⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⢀⣀⣀⡀⠀⠀⣠⡤⢤⣀⠒⡏⢹⠻⡑⢄⠑⣄⠀⠀⠀⠑⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⢰⠃⠀⠄⠉⠉⡾⡖⢒⡞⠁⡷⡙⢮⠲⣌⢮⠳⡘⡆⢀⣤⠀⢈⢦⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⡀⠀⢀⣠⠤⠴⡷⡄⢀⣠⠤⠞⠓⣓⣻⡝⣄⠣⡈⠢⡑⢌⢦⣳⡑⣌⠋⠀⡩⠍⠔⠑⡄⠈⠹⣦⡀
⠀⠀⠀⠀⠀⠀⢰⠃⠈⠀⠀⡏⠀⣀⣠⡟⢉⡩⠖⢛⣛⣩⠤⠒⠻⣜⢆⠹⣄⣈⣎⢧⡀⠱⣜⢦⡀⠑⠢⡄⠀⠈⣆⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⣿⢀⡤⠖⠒⠛⣊⣕⣋⡩⢕⣒⣚⣁⠴⣚⣉⡉⠭⢹⡜⡄⠰⣉⠈⠑⢝⢦⡻⡝⢮⡲⡄⠣⣄⠤⣻⠖⡄⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢿⡯⢚⡭⠭⠽⣋⣥⣔⡒⠉⢀⡤⢒⡩⢅⣲⠬⢔⡊⢧⡸⡄⠀⢣⣀⣤⢧⠳⣝⣒⣛⣮⣦⣴⣮⣥⣀⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢸⣷⠑⣒⡶⠋⠁⠀⠀⠙⢦⡞⢰⡋⠐⢏⣀⡀⠣⣀⢎⠳⣍⠲⠤⠤⢐⣪⡽⢒⠞⣿⠁⠀⢀⠀⠀⠈⢧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡟⢧⡜⠀⣠⠖⢦⡀⠀⠈⣷⢄⣉⠑⠒⢦⡉⠙⢮⢻⡦⠬⠟⣹⢋⣉⡪⠴⣋⡴⠽⠦⠤⣄⠀⠀⠀⣼⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡈⠻⣶⡃⠀⠀⣿⠀⠀⣿⠀⣚⠩⠓⣦⣙⠲⠞⠢⢷⣄⠴⡱⢩⠶⠋⠛⠋⠀⠀⢀⠀⠘⣷⣠⠞⠁⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠙⢶⣄⣿⠀⠀⣿⠀⣘⣛⣛⣓⠭⣛⣽⡛⢦⣉⣉⣰⠷⠒⠚⠛⣄⠀⠀⠈⠀⠀⢘⣁⣠⢞⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠈⠙⠳⢶⣏⣉⡑⠛⢛⣓⡯⢛⡵⠋⠀⠈⢩⠇⠀⠀⠄⠀⢸⡀⠀⢀⣀⣤⠾⠋⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⢾⣥⣤⣤⣤⣄⣾⣤⣤⣤⣤⡤⠶⠷⠛⠋⠁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠄⠀⠀⠐⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠒⠒⠚⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠁⢺⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠄⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠂⠀⠒⠀⠀⠀⠀⠀⠂⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢐⠀⠐⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐⠀⠀⡘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠀⠐⠀⠠⢀⠄⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠄⠠⡔⠂⠁⡄⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠐⠂⠀⠄⠀⢀⠀⠒⠠⠀⢄⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠒⠀⠀⠀⠀⠀⢀⢁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⡔⠁⢠⠀⠐
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⡀⠐⠃⠁⠀⠀⠀⠆⠀⠉⠀⠀⠈⠀⠀⢀⠀⢀⠸⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠈⠐⠀⡆⠀⡇
⠀⠀⠀⠀⢻⡄⠀⠀⠈⠀⠀⠀⢸⠁⠀⠀⠄⢈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⡘⠁⡜
⠀⠀⠀⠀⠀⠈⠠⠢⠀⠀⠀⠀⠀⠩⠀⠑⡆⠀⠀⠀⠀⠀⠆⠀⠀⡀⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠰⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄⡀⠀⢄⠀⠰⠀⠄⠠⠀⢒⡍⠀⠀⠀⠈⢠⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠩⠈⠀⠃⠂⠀⠀⠘⠀⠐⠁⠀⠀⠐⠀⠒⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣽⣽⣭⣬⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣯⣯⣽⣿⣮⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣴⠆⣼⣿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡦⠙⠀⢿⠛⣋⠀⣏⣀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣼⠇⣰⡶⢠⡟⢹⠿⡿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⠀⡇⠸⠀⢹⡀⢃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢃⣴⠀⠰⡀⢿⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠰⠈⡤⠃⢸⡇⢸⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⢠⠀⠀⡼⢡⡎⠠⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⡀⠸⠀⡀⢋⠃⠠⠁⠌⣁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⢠⣾⣿⣿⣿⡇⠀⠀⠰⠀⠀⡃⠠⠐⠏⠀⠛⢻⣇⣸⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡁⠀⠼⠛⢻⣿⡇⠀⠀⡀⢇⠀⠀⠄⠂⠘⠶⠀⡰⠛⢧⢈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⣿⣿⣿⣧⠀⣨⠀⠘⠿⢁⠀⢟⡀⢸⠀⠀⢀⣈⣶⡿⠿⣿⠀⣼⣿⣥⣄⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⠍⣿⡋⢙⡷⠐⣆⢀⢻⡆⠛⠣⠈⡀⢘⣀⣼⣾⣿⣿⡇⠀⢰⣶⡿⣿⣷⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣤⠛⠄⢘⠋⠘⡇⠐⢠⠆⠈⠃⠀⢀⣀⣤⣼⣿⣿⣿⡏⠉⠓⣴⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⣶⣾⣿⣿⣷⣦⠈⢰⣄⠠⣾⣧⣿⣦⣾⣭⣿⣿⣦⣀⣿⣿⣹⣿⣿⣿⣻⣿⡊⢿⡍⣿⣿⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⡿⢙⣛⣋⣉⡡⠤⠤⢈⠁⢠⣼⠿⡿⢯⠀⠀⠉⡍⠈⠟⠃⡀⢛⣿⠿⠿⠤⠟⠀⠀⢿⣳⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⣿⢣⣾⣿⡿⢋⡀⠠⢤⠞⣋⠺⢿⡶⠊⠁⠀⠀⠀⠸⠆⢠⡀⠀⠀⠱⠂⠈⠀⡿⣦⠔⠺⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡻⣿⣿⣿⣤⢿⣷⣤⣴⠛⢉⣤⣂⠀⠀⠀⠀⠀⠀⠀⠢⠀⢯⠀⠰⡀⢣⠀⠀⣤⠞⠶⢟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣿⠿⢻⣿⣿⣬⣣⡀⠈⠛⠛⠛⠶⣤⡀⠀⠀⠀⠀⠃⠈⢄⠀⢡⠔⠑⢢⣤⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠀⡏⢠⣼⣛⡛⠙⠿⠟⠁⢠⣶⠿⠛⠲⣄⡀⠀⢈⡀⢼⡅⢻⣶⡆⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣄⣅⠈⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣷⣶⣾⣿⡿⠃⢸⡯⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡇⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⢀⠠⠄⠀⣀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠒⠂⠀⠀⠀⠀⠀⠠⣄
⠀⠀⢀⠔⠈⠠⠀⠐⠈⠘⠄⠀⢀⢢⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢁⣀⣤⣌⠀⠂⠀⠀⠠⣄⡀⠈⢢⠀⠀⠀⠀⡴⠃⢀⠀⠀⠀⠀⣠⠔⠓⠀⠠⠀⠈⠳⡄
⠀⠐⠂⠀⠀⠀⠥⠀⢰⣄⡠⠀⠽⡇⠀⠀⠀⠸⠀⠀⢠⠏⢀⡞⠑⡀⠒⢲⠀⠉⠀⠁⠀⠀⠉⠀⠀⢃⠀⠀⡼⠁⠠⣥⣼⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⣼⣄⠀⢀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⣹⠀⢁⠀⠁⠀⠀⠀⠈⠛⠏⠀⠀⠀⢀⣼⠇⠘⠀⠀⠃⠀⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸
⠀⢷⠀⠣⢦⡀⠀⠀⠀⢀⠀⠐⠂⠀⠀⠀⠀⡼⠀⠀⠉⠀⠹⢦⣉⣀⡩⠴⠃⠀⢐⠆⠉⠀⠙⠋⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⡀⠠⣆⡀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⢠⠐⠒⠠⢤⠖⠐⠂⠂⠀⠂⠁⢀⡎⠀⠀⠀⠀⠀⠂⠀⠈⢙⠳⠄⠀⠰⠿⠃⠀⠀⠠⠔⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠠⠀⠆⠈⠙⠒⠀⢀⣤⠟
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⡔⠊⠀⠀⠀⠀⠀⠀⠀⠉⢶⣰⡀⠀⢀⣀⣀⣀⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⡓⠶⠆⠰⠤⠒⠒⢛⡩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢉⠉⠀⠀⢀⡀⠀⣀⠈⠁
⠀⠀⠀⠀⢸⡁⠀⠟⡤⠰⠃⠇⢸⠷⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⠷⠀⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⡀⠦⠤⠰⠄⠁⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⢀⡠⠐⠈⣉⣭⣤⣬⣤⡉⠉⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡴⠋⣀⣠⣿⣿⣿⣿⣧⠀⠸⣏⠳⢦⠈⣿⡆⠀⠀⠀⠀⠀⠐⣞⡝⠂⣠⢆⢄⣀⣀⢠⡅⢀⡀⠀⠀⢠⠀⠻⣄⠀⠀⣀⣤⠤⠶⠖⠒⠒⠒⠒⠤⣄⡀⢀⡰⠃
⣼⠃⢰⠥⣼⡿⢫⣕⡿⡽⠄⣠⡶⠶⣍⡀⢸⣿⠀⠀⠀⠀⢸⣑⣶⠯⡘⠏⠙⠠⡍⠉⠉⠠⣸⠃⠀⠀⠈⠳⢤⠘⠃⠞⣭⡖⡦⠎⣼⠫⢩⠍⡁⠠⡀⠙⢉⡤⠜
⣯⠠⣯⣽⣟⡶⠋⠉⢹⡟⠛⠞⣷⢦⣽⡟⢸⡿⠀⠀⠀⠀⠀⠉⠉⠀⠝⡓⠀⣇⠈⠒⠒⠊⣣⠀⠀⠀⠀⠀⠈⠁⣾⡀⣍⡶⠷⠲⣦⡚⣥⠖⠀⢈⠃
⢹⡀⢻⠉⠉⠱⠶⣞⠋⠑⢤⡀⠀⡱⣿⠇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⠀⠉⠓⢶⣤⠞⠁⠀⡀⠀⠀⠀⠀⠀⠘⢿⣮⣽⣆⡾⠛⠃⠇⢀⣀⣄⡄
⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠁⠀⢀⣴⠟⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠶⠤⠴⠶⠋⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠹⣦⣄⡀⢘⣛⣃⣀⣀⣀⣭⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠳⢄⣀⣀⣀⡴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡿⣿⣿⠿⠿⠇
⠀⣶⣶⣶⣿⣷⣾⣿⣿⣿⣿⡉⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⣀⡀⠀⠀⠀⠀⢀⣰⡶⠶⠆⣤
⠀⠉⠉⠉⠉⠉⡍⠉⠉⣉⠉⠉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠠⢿⠂⡤⠰⠖⠸⠂⠤⠐⠂⠜⠃⣹⠖⠁⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠏⢀⡄⣹⠇⠛⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⡄⣰⡿⠿⠛⠛⠉⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢰⠈⣀⠟⣻⣀⣥⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠋⠐⡟⢸⠃⡏⠋⡙⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⣡⠈⡆⢹⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡥⠃⢼⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠃⠀⢀⣄⠠⠇⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⡆⠀⠀⡀⠀⠄⠀⡼⠀⣴⣦⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡗⠈⠹⠿⣿⣧⠀⢀⢰⠀⠁⠀⠀⠦⠀⡸⢾⢙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠠⠀⠛⢃⡀⡯⠀⠂⠀⣀⣴⡾⠻⠇⣸⣿⣥⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⡆⢻⡅⣽⠁⢇⡈⢷⠈⠁⠀⠀⣤⣾⣿⢿⣇⠀⣼⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⣩⣴⣦⣼⣀⠈⣁⠀⣠⣔⣞⣻⢿⣿⣏⡀⣴⣿⣿⣿⣿⢿⡿⢿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⢛⣛⣛⣉⠄⠍⠃⢈⣽⢿⣿⡉⠉⢛⠉⠿⠋⠙⣿⠿⠯⠹⠊⠁⣿⣯⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣽⣿⣳⣾⡿⢛⡅⠠⠄⣋⠙⠷⠖⠁⠀⠀⠀⠇⠠⡀⠀⠘⠂⠀⠸⢦⠐⢻⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣾⣿⢲⡏⠁⣶⣧⣀⡀⠀⠀⠀⠘⡀⠱⠀⠃⠐⢀⠐⠋⣛⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢡⠋⣹⣿⠷⢦⡴⠀⢨⣽⡷⣄⠀⠀⠀⠀⡗⢰⣅⡀⣹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣈⡀⢻⣿⣷⣶⣶⣤⣬⣤⣤⣤⣿⣤⣼⣷⡟⢀⡟⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣇⢘⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣢⡴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡤⣀⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡴⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣦⣠⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⡄⠀⠀⠀⠀⢀⣄⣀⣦⠞⠋⠁⠀⠀⠀⡀⢀⣀⡤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣾⠁⡄⠀⣀⣴⣺⠟⠋⢀⢀⣀⣤⡤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠃⢹⠆⠘⠜⢻⣴⡣⡻⡿⠿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣻⠀⠀⡧⡄⢠⣿⠘⠁⢇⠀⠂⡉
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⡄⠀⠤⢄⠀⠀⠀⠀⠀⢸⡄⡀⠀⠏⢹⡆⡐⠀⡎⠐⠂⠊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⢂⡀⢀⠀⠱⡔⠂⣆⠰⣟⢷⠀⠐⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢡⡴⣌⣿⣖⣬⡊⠩⣂⢻⡌⠀⠀⠀⡂⠀⠀⠈⠢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡿⠁⠉⠀⠈⢿⡿⣦⣍⡳⠙⣄⡀⣀⣀⡠⡼⠿⣷⣶⢠⣄⡀⣠⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡛⡲⠒⠟⠈⠉⠉⠹⣿⣧⣣⣸⠞⠉⣩⣤⢎⡵⣿⣿⢼⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣆⠀⣼⢶⡄⠀⣀⡄⠀⠤⠌⡷⣧⠛⠀⣾⣏⣴⣇⠿⡾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⣿⡇⢉⣾⢶⣋⢁⠁⠛⣳⣤⣤⣾⣹⠀⡠⢅⣹⠋⠁⠀⠁⠋⠀⢀⣤⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣽⣿⣿⣯⣅⠤⣼⡍⠊⡁⢉⡿⠏⠉⡡⢰⣿⣿⡽⣦⣀⠀⠁⠚⠛⠋⢵⣿⡯⢻⣿⣟⣵⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠙⠻⠻⠿⣷⣤⣞⣵⢊⣤⣀⣛⣐⡋⣐⠢⣾⣖⠛⠻⣀⣴⣤⣛⣵⡤⢶⠿⢾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡻⠆⠀⠀⠀⠀⠈⠉⠉⠛⠓⠛⠟⠛⠻⠿⠿⠿⠛⠓⠛⠛⠋⠉⠈⡀⢀⠊⠠⣾⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠣⣼⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣔⣮⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣾⣾⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣦⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣬⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣶⣿⣷⣾⣿⣿⣿⣿⢿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⠀⠀⠀⠺⡀⢀⣷⠄⢠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠉⠙⠻⢦⡄⠀⢿⢋⢿⠂⠀⡇⠀⠰⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠉⠻⣤⡀⠀⢠⢹⣤⠇⣼⡋⠀⠠⠀⢈⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠈⠻⡀⠀⠀⢻⡄⠹⡧⠂⠀⣶⠉⣠⠿⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⡀⠀⠀⢹⡆⠀⢸⡇⠘⡇⠀⠀⡇⣼⢃⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⣿⡀⠸⣇⢠⡏⠀⠘⢰⡗⠀⠀⠀⠀⠀⠀⠀⠠⠒⡄⠀⣀⠼⠀⠀⢰⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠸⣇⠀⠻⠸⡏⢰⠃⡼⠇⠀⠀⠀⠀⠀⠀⠊⠠⣼⠏⣾⠁⠀⠀⠀⡴⠀⠀⠷⠈⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠘⣇⠀⣦⢡⢏⡼⠃⠀⠀⠀⠀⠀⠀⢘⠀⠐⣿⡽⠃⠀⠀⠀⡠⠀⠀⢀⡞⠀⢘
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢹⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣿⡟⠀⠀⠀⣰⠕⠀⠀⢠⡇⠀⣈⡀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣾⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣯⠀⠀⢀⡟⡀⠀⠀⣸⠀⢠⡟⠁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠹⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣳⣦⡌⣥⡄⣠⠾⢁⣠⠟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢙⠋⠉⠡⠔⢳⡖⠛⡁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠐⠁⠐⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣙⡼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣷⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢻⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⣀⡀⠈⠁⠀⢨⣳⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠳⡄⠀⠀⡌⠀⢀⠀⣾⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠶⠖⣒⣸⠶⢄⠀⠀⠈⢢⣰⡟⠀⢠⣾⣯⠴⠖⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡀⠀⠀⠙⠻⣦⡰⠙⢶⣄⡘⠘⠛⠓⠞⠋⠁⠀⢀⣠⠄⣒⡭⠟⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠀⠀⠙⢻⣄⡂⠈⠻⣯⡑⠯⠠⠤⠛⠷⠧⠚⠈⠁⡠⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⣤⣉⣀⣙⣓⣤⣜⡨⣿⢲⠦⣀⣠⣤⣤⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢀⡀⠤⠟⠋⠀⠉⠻⢊⣁⡀⡀⠀⠀⢀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⡄⢐⣦⣤⡀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟
⠀⠀⢠⣾⣤⠀⠸⠿⢿⣿⣆⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠈⣿⠀⠀⠀⠀⢸⣿⡇⠀⣾⠁
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣆⠰⠀⠀⠀⠀⣠⠊⣡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠁⠀⠀⠀⣿⡄⠀⠀⠀⢸⣿⡇⣶⡏⠀⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⡄⠀⠀⠀⢰⢁⡞⠁⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣿⡇⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⣿⡇⣿⡇⢸⡿
⠀⠀⠀⠀⠙⠛⠓⠒⠈⠁⠀⠀⠀⡸⠙⣿⡀⠀⠀⠘⢿⡝⢤⣀⣀⣇⡀⠠⠄⠒⠒⠀⠀⠀⠀⠀⠈⢛⡿⠦⣀⠀⠀⠘⣿⡆⠀⠀⠛⣿⣥⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠀⢹⣿⠀⠀⠀⠀⠙⠿⣍⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡄⠉⣧⠀⠀⢻⣎⠀⠀⠀⣿⣸⣿⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡔⠀⠀⢼⡿⡀⠈⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣾⡆⠀⠈⢿⣆⠀⠀⢹⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣾⣆⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠀⢠⣴⣇⣿⣿⡿⠃⠀⠀⠈⢿⣧⠀⢸⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠻⣶⣇⡀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⣌⠀⠀⠞⠓⢻⡿⠋⠀⠀⠀⠀⠀⠈⡿⣦⠀⣿⣿⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⢻⣧⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⢿⣶⣶⣿⣏⠀⠐⠀⠀⠀⠀⠀⠀⢰⡈⢲⣿⡟⢠⠆
⠀⠀⢀⠢⢈⡠⣤⣄⣀⡤⠀⠀⠀⠀⠸⣿⠟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡏⠀⠀⠀⠀⠉⠉⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⡇⡟
⠀⠀⠀⡐⣶⣿⣿⣧⣹⣶⣶⣀⣀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣠⠂⢳⣀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟
⠀⠀⠈⠈⠁⠈⠻⠋⠿⠿⠟⠻⠊⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⢀⣴⣮⡀⠀⠀⠏⠁⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠛⠇
⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⡟⠀⠀⠶⡞⠠⠀⠀⢸⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠘⡿⣍⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠀⠀⠀⠀⠀⠠⢠⡇
⠀⠀⠀⠀⠀⠁⠈⠂⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠃⠀⠄⠀⡀⠀⠀⢀⠎⠀⠀⠈⠀⠀⠀⢀⡄⣰⠀⠗⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠠⠄⣤⡴⠟⠁⠀⠀⢀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠴⠤⠓⠃⠤⠂⢾⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡝⣰⠁⢿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡀⠐⠋⠈⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢠⠀⡿⠀⠀⠀⠀⠀⠀⠁⠀⠈⢰⣀⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣹⠿⠓⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡀⠀⠉⠛⠂⠀⠀⠘⠉⢋⠀⠀⠀⠀⢠⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠃⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⣀⠤⠄⠠⠄⣢⡀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⣰⡧⠀⡆⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠈⢠⠖⠂⠾⣤⣷⣀⣄⠀⣀⣤⣆⣰⣰⣦⢀⣤⡎⠁⠀⠀⠀⠀⠀⠹⣶⠤⠄⠀⠀⠀⣀⣠⠀⠀⣠⣾⠟⠁⢀⠟⠂
⠀⠀⠀⠀⠄⣀⠀⠀⠈⠀⠀⠈⣫⣽⣿⣿⣿⣿⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⣶⣾⡷⠞⠛⠋⠉⠉⠁⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣨⣿⠋⠉⠉⠛⢿⣿⣿⡇⠀⠀⢄⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⡄
⠀⠀⢀⣀⡀⠀⠀⢀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠘⠟⠛⠁⠀⠐⠂⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣇⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠘⠿⠃⠀⠀⢿⣷⡄⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣬⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⠄⢤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠤⠠⠄⠀⠀⠀⠄⠄

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

⠀⠀⢀⠠⠒⠒⠢⢄⠀⠖⡙
⠀⠀⠙⠖⠢⣄⠀⠀⢻⠸⡠⠀⢀⣤⣀
⠀⠀⠀⢨⠀⠈⠳⡀⠀⣯⣁⠻⣄⣀⠀⠀⢀⣀⣠⣤⣤⡶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣀
⠀⠇⠀⠈⠉⠐⠲⠽⠦⡏⠙⡿⣛⠭⣛⡛⠋⠉⠀⠀⠤⠄⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⢘⢀⣀⡠⠒⠒⠢⠔⢻⠹⡈⠢⡉⠙⠛⠫⠍⡑⠒⠢⢄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠈⠁⠀⠑⠉⢉⡇⠀⠈⡄⠁⠀⠈⠒⢄⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⣠⡞⠀⢰⠀⡇⢀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⣷⡀
⠀⠀⠀⢠⡞⠁⠀⠀⠃⠀⡅⠈⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⢷⡀
⠀⠀⣠⠏⠀⠀⠀⡀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⣧
⠀⢰⠏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠰⡀⢻
⢀⡟⠀⠀⠀⠀⠠⠈⠀⠀⡇⠀⠀⠐⡀⠀⠀⠀⡀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡆⠀⢇⢸
⢸⠃⠠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠱⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⢰⠀⢸⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢧⠀⠀⠀⠱⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠸⡄⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⣆⠀⠀⠀⠱⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡇⠀⡿
⣿⠀⢀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠘⣆⠀⠀⠀⠱⡀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣇⠀⡇⣸⠇
⢻⡀⠀⡀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠘⣆⠀⠀⠀⠈⢆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⣷⡟
⠘⣇⠀⠱⠀⠀⠄⠀⠀⠀⠀⠘⢄⠀⠈⢦⠀⠀⠀⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣷⡟
⠀⢹⡄⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⣇⣼⠏
⠀⠀⢻⣄⠀⠱⡀⠀⠀⡀⠀⠑⡀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠘⣄⡴⢿⣟⡢⠄
⠀⠀⠀⠙⢦⡀⠙⣄⠀⠈⠢⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣆⣠⠾⠋⠅⠂⠑⠀⢄⢀
⠀⠀⠀⠀⠀⠙⢦⣌⠳⢄⠀⠙⢤⡀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢳⣠⡴⠟⠁⡄⡄⠲⢤⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣭⣒⣀⡉⠢⢄⡈⠑⠲⠤⣄⣉⣐⣒⣤⣤⡴⠾⠛⠉⠉⠃⠠⠘⢀⡆⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⠈⢄⠠⠀⡀⠠⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠚⠀⠀⠀⠈⠀⠅⠀⠀⠀⠈⠀⡄
⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⢀⠄⠔⢁⠀⠀⠀⠐⠠⠀⠒⠉⠢⣀⡑⡀⡀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⣑⠀⠄⠐⡀⠀⠀⠠⠀⠀⢀⠤⠤⣄⡀⠀⠀⠑⠂⠠⡀⠰⡠⠀⠙⠢⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠋⠂⠀⡔
⠀⠀⣀⠀⡀⠀⣦⠀⢀⠀⠀⢈⠀⠀⠀⠈⠣⣀⠀⠄⠀⢈⠼⠁⠀⠀⠀⢲⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⢄⠀⠀⠥
⠀⠘⠁⠘⠀⠀⣸⡰⢠⠀⡈⢂⡀⠀⠀⠀⠀⠀⠑⣦⠊⠁⠀⠰⠊⠀⢀⠂⢂⡉⠃⠀⠀⣠⡬⠐⠀⠁⠀⡈
⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠹⣄⠈⢂⠀⠀⠀⠙⢄⡀⠰⠂⣀⠔⠁⠀⠀⠄⠠⠒⠋⠉⠀⠂⠀⠀⠁⠀⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠹⠂⠈⠉⠐⢂⠀⠉⠢⠴⢄⡀⠠⠀⠝⠚⠉⠁⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄
⠀⡀⠂⠂⠀⠔⠁⠀⠀⠀⠀⠀⠀⠱⢀⠐⡀⠆⠀⡀⠀⠀⠡⠀⡀⠈⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠈
⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⢀⠀⠀⡉⠄⠂⠀⠀⠀⠀⠀⣀⠀⢀⠰⢀⢀⠠⠄⠀⠊⠄⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠑⠂⠊⠀⠱⡀⠀⠀⠀⠀⠀⠠⠁⢸⡆⢸⠀⠈⠀⠀⠀⠤⢴⡇⠀⠀⡇⠀⡏⠀⠀⠀⠀⠀⠁⠀⡀⠁⠀⠘⠠⠀⠀⠆⡀
⠀⠀⠀⠀⢀⡀⠈⠑⠒⠒⠢⢀⠀⠀⠘⡇⣼⣴⡶⢠⣶⣦⣾⢾⢠⣻⡆⡗⡦⡷⣾⠇⢿⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⢰⠀⠀⠀⠻⠇⠙⠒⠊⢻⡇⠉⣈⣂⣉⣣⣯⣅⣀⣈⡃⠚⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⡀⠀⠐⠂⠀⢄⣂⣀⣀⠈⠃⠠⠄⡀⠒⠒⠃⠐⠚⠀⠉⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠰⠣⢀⠘
⠀⠀⠀⠈⠀⠉⠀⠀⢠⢦⣗⢦⠁⠀⠀⠀⠀⠂⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡅⠀⠂
⠀⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠱⣄⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀⠀⡀⠀⠈⠉⠐⠒⠒⠆⠀⠀⠀⣀⣀⣀⣀⣐⡀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠂⠐⠒⠦⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⠂⠠⡀⠄⠀⠀⠔⠐⠀⠀⢔⠒⠁
⠀⠒⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢆⠀⠡⠀⠀⠀⠠⠀⠐⠠⠀⠠⠠⠈⠀⢠⠀⠀⠆⠀⠀⠀⠀⠀⠒⢠⠄
⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠄⠄⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⡀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠀⡠⠄⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠁⠢⠀⠴⠁⢒⠠⠈⢤⠀⠊
⠀⠀⠀⠀⠀⢀⡈⣀⠀⠀⢄⠀⠀⠀⠀⣀⢀⡀⠠⢈⠐⠀⠀⠀⠂⠀⠁⠈⠐⠀⠀⠀⠀⣀⣀⠀⠃⠠⠁⠀⠠⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠅⠀⠀⠘⠑⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣹⣿⡀⢜⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⡸⠁⢸⡇⠈⠢⣹⡷⢤⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⣧⠀⠀⠙⡿⣦⠙⠢⡈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡾⠁⠀⠀⢸⣿⠀⠀⠀⠈⢾⣇⠀⠈⢷⡄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⢀⣾⠃⠀⠀⠀⢈⣿⠀⠀⠀⠀⠘⣿⡄⠀⠈⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⢠⢰⡿⠀⠀⢺⣿⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⣿⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢸⣧⠀⠀⠻⣿⠀⠀⠀⠀⠀⢾⣇⠀⠀⠀⠀⢿⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⡄⠀⢰⣿⡆⠀⡀⣄⣦⣿⣿⠀⠀⠀⢀⣾⢇⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⣦⣺⣿⣿⣇⣿⣼⣿⣿⣿⡠⠐⡆⣼⣿⣦⣾⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⣾⣿⣿⣿⣿⣿⡟⢻⡟⣽⣼⡡⣼⣰⣏⡐⠶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢙⢻⢿⣿⡿⣿⣿⣿⡷⣞⡞⣇⢻⠿⠇⢡⡝⣉⣡⣴⠛⠊⠉⠉⠉⠉⠉⠙⠒⠂⠶⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠿⠿⣻⣼⣢⣿⠧⠤⠄⠐⢿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠀⡀⠈⠁⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⣄⣀⡀⣀⣀⣠⣴⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠿⠻⠿⠟⣿⣷⣾⡿⢟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⣀⣤⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⣿⠟⠉⠀⠀⠀⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠾⠛⠛⢿⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⣾⣷⣶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⣧⠀⠀⠀⠈⣇⠀⠀⢀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠘⠀⣠⣿⣷⡀⠀⠀⠘⡄⠀⢸⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠋⠉⠛⠲⢄⠀⠙⠀⢸⡇⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⡀⠀⠘⠀⣠⠏⠀⠀⠀⠀⠀⣀⡤⠶⠻⢇
⠀⠀⠀⠀⣴⠋⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⢻⢿⣦⣤⡴⠶⠛⠋⠀⠀⠀⠀⠘⡆
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢈⡿⠁⠀⢀⡀⣠⠀⠀⠀⠀⢀⠇
⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣹⢋⣀⡀⠠⠬⠝⣹⣶⣶⣶⠖⠁
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⡀⠄⠤⠒⠛⠉⠁⢀⣀⡀⠤⠐⠀⣰⠃⠘⣿⡇
⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣴⠶⠶⣄⠀⠐⠂⠀⠀⠀⠀⣀⠀⢂⣴⠇⠀⠀⣸⠁
⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣠⡟⠀⠀⠀⠈⠉⠉⠉⠀⣢⣿⢿⣤⣤⠶⠋
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢌⠡⡉⠳⠶⣤⣤⣤⣤⠶⣺⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠄⠀⠄⠀⡀⠀⠀⠀⠽⢮⣕⣀⣀⣀⣀⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣧⠸⡇⠀⣀⠀⠀⠀⠀⣰⠃⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⡹⡋⠉⠒⠉⠙⢦⠀⠀⣸⡆⢻⡏⣨⠇⠀⢀⡴⠃⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⣀⠾⣤⡀⠀⡰⠁⡠⢀⣠⡤⠤⣿⡿⢹⡇⠘⡿⢧⣠⠚⣡⠂⠀⠀⠀⠘⡄⠀⢄⠱⣄
⠀⠀⠀⠀⠀⠀⠰⡁⠄⠀⡹⠒⡳⠎⠈⠁⠂⢉⡽⠋⠀⠈⡇⠀⣯⡀⠈⡛⢧⡀⣼⠀⠀⠀⢱⡀⠈⢆⠈⣦
⠀⠀⠀⠀⠀⠀⠸⡔⠨⠐⢣⣒⣃⣤⡀⠀⣠⡟⠀⠀⠀⡇⣷⠀⢸⡴⠀⠀⠀⠘⢧⠀⠀⠀⡎⣧⠀⠘⡎⣾⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠸⡆⠀⢠⠪⠉⢇⢊⠋⠳⡏⡅⠀⠀⠀⠇⢹⡧⠀⡷⠀⠀⡇⠀⢸⠂⠀⢰⣰⣿⠴⠞⠛⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠦⢿⠀⠀⡘⢦⡑⠢⣇⣵⣄⠀⠀⠘⠈⣿⠀⣧⢃⢀⠇⢀⣾⣄⢀⣹⢧⣀⡀⠀⠀⣀⡴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣬⠶⠟⠓⠚⢮⣻⣷⣤⣄⣀⣼⡄⠸⣆⣈⡤⢞⣈⣉⡋⠻⣆⠄⣈⣉⡉⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠓⠒⠈⠛⡽⢚⡩⠭⠲⠭⣷⣮⡑⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣋⣀⠠⠤⠀⠀⢉⣿⠶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠥⠶⠤⠤⠔⠚⠋

⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⠀⡌⢳⠪⡿⣥⢯⠧⣘⠉⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠤⢄⠀⠀⠀⢀⡩⢵⣴⣦⠀⡉⢺⢉⢺⠙⡯⠭⠨⠼⠐⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠁⠸⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠰⣤⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢆⠀⠀⠀⠀⠀⠄
⠀⢀⣀⡠⠖⠋⠀⠀⠀⠐⢈⣿⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠿⣾⡁⠀⠀⠀⠀⠠⡀
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⠉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣝⣥⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠀⠀⠀⠀⠀⣀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠁⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⡀⠀⠀⠀⠀⠀⠀⣤⢀⡔⣹⣆⡴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⢀⡤⠀⠐⠒⠒⠀⠤⡀
⣵⣶⡄⠀⠀⠲⠶⣷⡈⣷⣶⠇⠢⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⢠⡀⣀⠀⠀⠈⠁⠀⠀⠒⠒⠈⠁⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠉⣹⡗⠀⠀⠀⠀⢈⡇⠛⠃⠀⠀⢀⢤⣖⡢⣀⣠⡀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢈⠙⠚⠃⠀⠀⡔⢁⡀⡀⠉⠀⢠⡿⣿⠾⡯⣕⣳⠵⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠠
⠈⠀⠀⠀⠀⠰⢀⣻⣯⢧⡏⣛⡷⣜⡧⡧⠝⢒⠓⢡⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡄
⡀⠀⠀⠀⠀⠀⢨⣾⢻⣅⣓⢹⣜⡯⣷⡕⡝⣶⣻⠍⢰⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠈⡄⠣⡟⡬⣗⢝⡝⠔⣳⢟⣏⢟⡺⢁⠃⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠈⠢⡙⢷⣡⣟⣊⢆⣿⡃⠵⣉⠔⠁⠀⠀⠀⠀⠑⢴⡀⡀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠰⡄⢠⠄⠀⠀⠁⠒⠀⠤⠤⠴⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠬⠈⠡⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⡳⣷⡞⠐⠀⠙⠈⠁⠀⠀⢀⣀⡀⠀⢰⡟⢁⡴⠀⣰⠂
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠁⠀⠀⠀⠀⡠⠖⣉⣠⣤⠄⠉⠉⢾⠿⡷⢞⣩⣄⠀⠀⠀⠀⠀⠀⠀⢈
⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡰⠣⣓⣿⡷⢺⠀⠐⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⠂⠄⠳
⠟⠩⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠙⠂⠈⠼⠓⡀⠀⠀⠀⠀⠀⠀⠸⢠⣒⣊⣝⣱⣭⣅⣢⢤⣠⢤⢠⢀⢂⠀⠀⠀⠀⠀⠀⢠⠹⣒
⢶⣾⠧⡰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠁⠀⠀⠀⠀⠀⠀⣸⡓⢻⢭⠷⣟⢼⢾⠾⢺⢿⣯⠏⠀⠄⠀⠀⠀⠀⠠⡀⢭⡿
⣾⡖⣃⠂⢈⣶⣤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⠘⢿⣓⡟⡞⣤⣳⠧⣝⣫⣧⡗⢮⣣⠀⠀⠀⠀⠀⠀⠐⢌⠁
⠂⠐⠅⠐⠘⣿⡿⠃⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡉⠵⡾⢞⣫⠾⣍⡻⡼⣼⡻⢩⠎⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡷⠉⠃⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠳⠿⠹⠣⠋⡋⠔⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⣀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⢒⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠
⠒⠈⠉⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢱⢤⢀⡻⢧⣺⣕⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠫⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠖⠒⠒⠒⠒⠒⠒⠒⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⡠⠄⠒⠲⢤⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠆⠀⠀⠀⢎⠀⠀⠀⠀⠀⣺⡇⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠑⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⠤⠶⠒⠛⠛⠛⠛⠛⠒⠲⠒⠚⠋⠁
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⡾⠒⠋⠉⠉⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣸⣿⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠳⢶⣤⣤⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⣿⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣔⢜⣾⠏⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣶⣤⣴⣤⣤⣼⣶⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⠂⠀⠠⠤⠤⠤⠤⠴⠷⢿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣳⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣀⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡟⢻⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢤⠛⠳⠴⠴⢦⣠⢖⡥⢽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⠚⠛⠲⡞⠋⠉⢋⠀⣐⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠸⡄⠰⣼⡇⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠒⠛⠋⢉⠇⠈⢿⡉⠙⠛⠒⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠃⠀⠀⢈⣈⣿⣿⠃⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣀⣩⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠾⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⠶⠾⠾⠷⠷⠾⠿⠛⠋⠁


⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠢⠄⠀⠀⠀⣀⠂⡀⠀⡄⠀⠀⠀⠀⢀⠎⠁
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠆⠀⠀⢀⡀⠤⠤⠠⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠄⠸⡿⠁⡀⠀⠠⠀⡈⠐⣄⣼
⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠚⠄⣀⡊⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠀⠀⠀⠀⠐⠦⣼⣧
⠀⠀⠀⢐⡀⠒⠄⠀⡔⠠⢶⡀⠀⠀⠙⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠸⠿⠆⡀⠀⠃⣄⡀⠀⠀⠀⠐⡢⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠀⠀⠁⢠⠶⠃⠉⠀⠀⠀⡀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠠⠀⠀⠘⢄⣠⣠⠆⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⢠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⠀⠉⠉⠒⠤⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠢⠢⡤
⠀⠀⠀⠀⠈⠛⠂⠉⠙⠓⠂⢤⣌⠳⡀⠀⠀⢀⠀⠀⡀⠠⠀⠀⠀⠄⠀⡀⠀⠀⠱⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣗⣄⠀⠸⠀⠀⠃⠀⠀⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⢠⠠⠀⠀⡀⠀⠁⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠍⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⡀⡀
⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣀⠀⠀⠀⠀⠀⠀⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠥⠤⠀⠀⠀⠀⠀⡀⠀⡀⠀⡠⠀⠐⠀⠀⠘⠛⠐⠀⣀⠔
⠀⠀⠀⡐⠠⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⡤⠀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⠀⢢⠞⠁⠀⡐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⡀⠀⠘⠀⠀⡁⠀⢱⠖⡘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠁
⠀⠀⡔⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠇⠀⠀⠀⠀⡁⠀⠃⠀⠀⠀⠀⠀⠀⠐⠈⠈
⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⣆⠀⣀⠂⠀⢀
⠀⠀⠀⠣⢠⡀⠀⠀⠀⠀⢁⠠⠄⠀⡀⠠⡄⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⠂⠈⡆⠫⠀⠐⠁⢰⠶
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣴⡶⠟⠁⠀⠀⠀⠘⢄⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⠀⠀⠀⠀⣠⡀⠀⠀⠀⢠⡾⠀⠀⠀⣀⢀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣄⣠⣽⣿⣄⠀⠀⠀⢿⣦⣠⢈⣽⠻⣿⣿⣿⣿⡄⠀⠀⠀⣐⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡟⠛⠛⠻⠿⢿⣿⡿⠿⠿⠀⠀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠾⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡇⣡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢛⣃⠉⣉⡁⣠⡄⡄⡄⠀⢠⣤⣀⣤⡈⣉⡙⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⡭⠘⣿⡃⡷⠀⣑⡆⡗⣇⠀⢸⡞⢻⣃⡇⣿⡟⣸⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⡆⠀⠀⠀⠃⠀⠁⠁⠉⠁⠈⠀⠁⠀⠀⠈⠀⠀⠉⠀⠁⠁⠃⠛⠀⠀⠀⡆⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⣤⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡂⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠑⠂⢄⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠔⠂⠐⠢⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢢⠀⠀⠀⠘⠀⡌⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠀⡐⠀⠡⠀⠀⠈⢆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂
⠣⣀⠄⠀⠀⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀⠀⠀⠀⡀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠱⡆⠀⠀⠉⠀⠀⣀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡟⢦⠶⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡠⠳⣄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢃⠞⢁⡴⠃⣰⣦⢾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠖⠋⡠⠔⠁⡰⠋⠀⡴⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠀⢠⠊⠀⢠⠞⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋⠀⠀⠀⠀⡰⠃⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⠃⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⠎⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢸⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⠎⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠋⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⠞⠁⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢀⣀⡴⢋⠔⠉⢀⠔⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠤⠤⠴⢿⣿⠛⣁⡠⠖⠁⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢥⣄⣴⠀⣀⡠⠴⠊⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠉⡉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣼⠁⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⠀⣸⠃⠠⡏⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣧⠀⠀⠀⠀⠀⢰⡏⠀⢸⡇⠀⠀⢰⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡄⠀⠀⠀⠀⢸⠇⠀⢸⡇⠀⢰⡟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⢇⠀⠀⠀⠀⣾⠀⠀⢸⡇⠀⣾⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⡟⠀⠀⢸⡇⣰⠏⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⣤⠀⠈⣧⠀⠀⠀⡇⠀⠀⢸⣇⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⠀⢹⠀⠀⠻⡄⠀⠀⡇⠀⠀⢸⣿⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢳⣄⡦⠀⠀⣿⠀⠀⡇⠀⠀⢸⣿⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠹⣿⡀⠀⠸⣇⠀⣷⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠘⢷⡄⠀⢸⣆⣿⠀⠀⣸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠈⢷⡀⠀⢿⣿⠀⠀⣿⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢿⡀⠈⣿⠀⠀⣿⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠘⢧⡀⢹⡄⠀⣿⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⣧⠈⢃⠀⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣄⠘⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢻⡄⠸⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢃⠀⢺⡅⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣱⡄⠀⢸⡇⠸⣇⣀⡼⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣧⣤⡜⠳⣤⠟⠉⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠋⠁⣠⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠶⠛⠋⠁⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⣠⠤⠚⠋⠁⠀⠀⠀⢠⠀⠀⠀⠘⡀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⠖⠈⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⢀⡾⠃⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠘⠃⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⢦
⠀⠀⣠⠟⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠃⠀⠀⠈⢧
⠀⢰⡏⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠈⣵⠀⠀⠀⠀⠀⠈⣷
⢠⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⢸⡇
⡜⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⢹
⣷⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀
⠈⢷⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⣸
⠀⠘⢷⡀⠀⠀⠀⠀⠈⢄⡀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠶⠶⠶⠶⢾⣦⡴⢶⣤⡶⣤⡾⠛⠛⠛⠛⠛⠛⠛⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠁⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⣉⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣤⡶⠶⠿⠿⠛⠛⠋⠛⠛⠀⠀⢤⣤⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠈⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠋⢦⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢾⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠄⠀⢼⣿⡆⠀⠀⠀⠀⠀⣤⠔⠋⠁⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣻⠟⠀⠀⠴⢠⣷⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢾⣯⢀⡄⠀⢀⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠿⣿⣿⣿⣿⣿⡿⣋⣼⠷⣼⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠛⡁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠿⣿⠛⠏⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣁⣾⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢻⡧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠉⠛⢀⠀⣹⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣦⢠⣄⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠈⠛⢷⣶⣶⣶⣶⣶⡶⠦⢴⣶⣶⣶⣶⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

⠀⠀⠀⠀⢀⠄⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠔⠁⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⠀⠀⠁⠐⡆⠀⠀⠀⠀⠘⠄⠀⠀⠀⠕⠀⡀⠀⠀⠀⠥⢥⡐⠂
⠀⢀⢂⠂⠀⠐⠄⠀⢀⠴⠈⠢⠀⠀⠀⢸⠀⠀⠀⡔⠀⠉⠐⠢⡀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⡐⠀⠀⠀⠚
⠀⠈⠂⠀⠀⠀⡜⢹⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠄⠄⠋⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⠤⠄
⠀⠀⠀⠀⠀⠀⣀⡀⠄⣀⡀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠑⠲⢄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⢤⣦⠀⠀⢀⠄⠃⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⣈⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡪⠂⢀⠎⠀⠀⠘⠍⠁⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⢄
⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⢆⠄⠄⡕⠁⡰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⡁⠡⠀⢊⣔⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⠐⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⡀⠈⠀⠀⠀⢈⡀⠀⠀⠀⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠊⢑⡦⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠊⠀⠐⠂⠀⠀⡀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠁
⠀⠑⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡡⠄
⠀⠀⠀⠈⠁⠀⠐⠁⠀⠀⠈⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠁⠂⠀⠐⠀⢱
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢁⠐⠀⣀⠠
⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠄⠀⠠⡜
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠖⠄⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⢶⣶⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠹⣷⣶⣭⡻⣿⣶⡤⠖⠚⠋⠉⠛⠻⣿⡷⡶⠾⠟⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠠⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠒⠒⠒⠒⠒⠢⠤⠤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣧⠀⣀⣶⣿⣿⣿⡿⠋⠀⠀⠈⠙⠲⣤⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣀⢔⣪⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⢛⣶⣦⣼⣷⣽⠉⠛⢿⣿⠟⠲⣾⣏⣿⣷⣄⠈⠃⠀⠀⠀⣠⡘
⠀⠀⠀⠀⠀⢀⢞⣵⣿⣿⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠏⣿⣿⣿⣿⣦⣤⣼⣿⣤⠀⠠⠛⠛⢿⣿⣷⣄⠀⠀⠀⠉⠁⡀
⠀⠀⠀⠀⢠⣿⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣸⣿⣿⢃⢂⢀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣶⡛⣿⣿⣦⡇⢾⠀⠀⠀⠘
⠀⠀⠀⢀⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣾⣰⡀⡆⢀⠀⠀⠀⠀⠀⠀⠀⣤⣷⢸⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣷⣝⣿⣿⣷⣤⣀⣤⡦⠊
⠀⠀⢀⡾⠃⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣼⣆⣆⣇⣧⢀⢀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣛⣹⠟⠋
⠀⢀⠞⠁⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡛⢿⣿⣿⣿⣿⣿⣷⡮⠭⠍⠉⠉⠉
⠀⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡌⠈⠀⠀⢸⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⣼⣿⣿⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢿⣿⡃⠀⠀⢠⣿⣿⣭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣼⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠸⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠁⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠈⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢹⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠘⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣧⣀⡆⠀⠀⠘⠿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡀⠀⠀⠙⢻⣿⣿⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣤⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⣠⠖⠓⠒⢒⡦⠒⠒⠒⠒⠚⠉⠉⠳⠤⣀
⠀⠀⠀⠀⠀⣀⣠⠊⠁⢠⠞⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣆
⠀⠀⠀⠀⡼⠉⠁⠀⠀⢧⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⡔⠒⠚⠉⠉⠛⢤⡀
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⢱⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠓⢢⠀⠀⢰⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠈⡆⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣳⡀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣄⣀⠀⠀⠘⠦⣄⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠲⡄⠀⠉⣶⠚⡄⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⠤⠟⠀⡇⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡿⠟⠋⠁⡀⠈⠛⠿⣿⠿⠛⠉⢀⠈⠙⠻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢀⣠⣴⣿⣿⣷⣤⣀⠀⣀⣤⣾⣿⣷⣦⣄⡀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⢀⣤⣶⣦⣄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⢰⢿⣷⣷⣿⣷⣝⢿⣿⣿⡿⣣⢞⣿⣿⣝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣏⡿⣿⣿⣿⣿⣿⣿⣳⣾⣷⣻⣽⣿⣿⣿⣿⣼⡆⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣲⡶⣭⣿⣻⣭⣞⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣝⠿⠿⢿⣂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⣿⣿⣿⡿⣱⣿⣿⢹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣽⣿⣯⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣧⢹⣿⣿⢰⡟⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠧⠌⣸⣿⣿⢸⡏⣱⠃⣼⣿⠿⠛⣛⣛⣛⣛⣛⢿⣿⣿⣿⣿⡿⣿⣹⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣛⣯⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣦⣈⣡⡾⢋⣴⣾⠟⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣷⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡴⣻⣿⣿⣾⣿⣿⠟⣡⣤⣄⠈⢿⣿⣿⣿⡿⠋⠀⣨⣿⣿⣿⡟⣰⠿⠛⠹⠞⣿⣿⠟⢋⣴⣿⣿⣿⣿⣿⣿⣯⣿⠮⣽⣅⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⡄⠘⣿⣿⡿⠁⣠⣾⣿⣿⣿⣿⡇⠀⠛⠓⠛⠛⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣦
⠀⠀⠀⠀⠀⠀⣙⢯⡻⣿⣿⣿⠀⢿⣿⡿⣿⡇⠀⣿⡿⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟
⠀⠀⠀⠀⢀⣾⣿⣷⣻⣽⣿⣿⣧⣀⣉⣀⣿⡇⢠⡿⠁⣼⡿⠟⠛⠻⣿⣿⠿⠛⠛⠿⢿⣿⣿⠟⠛⠻⠟⣿⣿⠟⣻⠿⠟⢿⣿⣿⣿⣿⡿⣋⡀
⠀⠀⠀⠀⠘⣿⣿⡿⣻⢿⣿⣿⣿⣿⣿⣿⣿⠁⢸⠃⣼⠋⢠⣿⡇⢀⡿⠁⣴⣿⡇⠀⣼⠏⢀⣾⣿⠀⣰⣿⠃⠀⣡⠆⢠⣿⣿⣿⣷⡟⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠈⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣸⣿⠀⠙⣋⣤⡾⠃⠰⣿⡟⠀⣸⠟⠀⢾⣿⠃⢠⣿⠏⢀⣾⡏⠀⣾⢋⣿⣿⣟⣧⣻⣿⡿⠁
⠀⠀⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⣿⣿⣦⣤⣭⣥⣾⣷⣤⣤⠆⢀⣴⣾⣦⣬⣵⣤⣬⣴⣤⣾⣿⣧⣤⣴⣿⣿⣿⣿⣯⡳⣌
⠀⠀⠀⠀⠀⠹⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡿⠋⡡⠀⣼⣿⠟⢛⣛⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢉⣟⡳⣿⣻⣿⣿⣿⣿⣿⠟⣩⢶⡟⢫⠄⣽⣿⡟⠁⣾⡇⢠⣿⣿⡄⢦⡉⣟⣻⣦⡻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣯⠾⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⢷⣿⣿⣿⣿⢯⣾⣿⣾⠵⢋⣼⣿⣿⣇⠀⠿⢁⣾⣿⣿⣿⣤⡙⠻⠿⣿⣷⣽⣿⣿⣿⣿⣿⢿⡿⣻⣭⣧
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡟⣾⣿⣿⣿⣿⣿⣛⣭⣥⣶⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡜⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣻⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⣶⣶⣶⣝⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⢟⣯⣿⣛⠾⠿⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠾⡿⣿⣿⣿⣿⢿⣯⠿⡿⢯⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⣿⣿⡵⢫⣵⣿⣿⣷⣝⢿⣿⢿⣿⣻⠏⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠙⠻⠿⠛⠁⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡰⢻⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢸⠳⡀
⠀⠀⠀⠀⢰⠁⠘⡄⠀⠀⠈⠙⠒⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠒⠉⠀⠀⠀⡜⠀⢳
⠀⠀⠀⠀⣸⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢸⡄
⠀⠀⠀⠀⣿⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⢀⣀⠤⠄⠒⠒⠒⠒⠤⢄⣀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠇⢀⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢹⠀⠀⠀⠀⠐⠆⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠆⠀⠀⠀⠀⢸⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡸⠀⠀⠀⢰⣿⣆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣾⣷⠀⠀⠀⠸⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⠁⠀⠀⠀⠌⠙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠋⠄⠀⠀⠀⢱⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⡇⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠁⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⢉⣉⣉⡁⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢁⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⢸⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢸⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡂⠈⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡈⠀⡺⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⠀⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠇⢠⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢦⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠴⠶⠄⠀⠀⠴⡶⠄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⠎⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠣⣳⡀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢠⡡⠋⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⠠⠤⠤⢀⣀⡠⠔⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠤⢄⡀⠀⠀⢀⡀⠤⠐⠈⠉⠁⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡽⠛⠒⢄⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⢷⡤⠀⣤⠀⠉⠺⠋⠒⢐
⠀⠀⠀⠀⠀⠀⠢⠤⠥⢈⠁⠀⠛⠞⠒⣶⣆⠀⠀⢻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠏⠀⠓⠲⣿
⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠙⠀⠀⢒⠀⠀⠀⠀⠀⢠⢠⠀⣀⡀
⠀⠀⠀⠀⣤⠔⢄⠀⠀⠄⠀⠀⠀⠀⠀⠐⠢⠔⠁⠀⠉⠙⢞⢀⠎⢀⠀⠂⣎⡀⢶⢉⡒⠀⠀⣀⡄
⠀⠀⠀⠀⡇⠀⠘⠳⢤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡳⠛⠀⣐⢋⡉⠀⠛⠠⢁⡀⠈⠉⣀⠤
⠀⠀⠀⠡⡇⠐⠀⠀⠀⠑⠂⠉⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠘⢀⣈⣩⣥⡉⠠⠀⠀⡀⠀⠀⠀⠠⠄⠶⣴⣤⡄
⠀⠀⠀⠀⠈⠀⢀⡀⡀⠀⠀⠀⠀⠀⢶⣿⠀⠂⠀⠀⠀⠀⢀⡤⢹⠓⠘⠃⠀⣠⣬⣍⠉⠄⠀⠁⠀⠬⢿⣿⣧⣀
⠀⠐⠀⠀⠀⠠⣼⢣⡷⠚⠀⠀⠀⡠⠆⠀⠀⠂⠀⠀⠀⠀⠁⠀⣠⣄⠀⢀⣬⡀⠀⠀⢹⠍⢉⠀⠲⠶⠼⠉⠁
⠀⠀⠀⠀⠀⠀⠙⢉⣁⠹⠉⠀⠞⠁⠀⠀⠀⠀⠀⠄⠀⣣⠀⠀⠉⠁⠀⠈⠀⠀⡀⠀⠀⣴⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠉⠁⠀⠂⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠘⠒⠚⠁⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣆⣠⣤⣄⣀⣀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⡤⠖⠚⠉⠉⠀⠀⠀⠈⢳⠀⠀⠀⠀⠉⠙⠓⠦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠇⢀⣠⠤⠴⠖⠒⠒⠛⠛⠛⠛⠛⠛⠒⠲⠦⣤⣀⡀⢻⠲⣄
⠀⠀⠀⠀⠀⣠⠞⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣉⣡⣤⣤⣀⠀⠀⢠⣈⡉⠻⢤⣈⠳⣄
⠀⠀⠀⠀⢰⢃⡴⠋⠀⢀⡴⢋⣠⠤⠶⠚⠉⠉⠛⠋⠁⠀⠀⠀⠈⠳⠦⠤⣤⡉⠂⠀⠈⢳⡹⡄
⠀⠀⠀⠀⣿⡞⠀⠀⠐⢁⡴⠋⠀⠀⠀⠀⢀⠴⠊⠉⠉⠉⠲⡄⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢹⣇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢯⣀⡀⠀⠀⣀⣠⠇⠀⠀⠀⠀⣀⡼⢁⡇⠀⠀⢰⡿
⠀⠀⠀⠀⢹⠙⢦⡀⠀⠘⢤⣖⣂⡀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠟⠩⠖⠉⠀⢀⡴⢫⡇
⠀⠀⠀⠀⠈⣧⠀⠙⠳⠦⣤⣀⣂⠙⠶⠤⠴⠿⠿⠧⢤⣤⣤⡤⠴⠚⠁⠀⢀⣠⠴⠚⠉⢀⡞
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠉⠉⠛⠲⠶⠦⠤⠤⠤⠤⠤⠤⠶⠖⠚⠋⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⢿⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣄⣀⡀⠈⣿⠀⠀⢸⡟⠀⠀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣀⣄⣰⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠍⠒⠒⠨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠒⢲⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠛⠚⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠋⠉⠉⠓⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⡤⠴⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⡎⣿⡱⣿⣎⣿⢸⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆
⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣷⣿⣾⣿⣿⣷⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠉⠛⢿⣉⣿⡿⣿⣿⠛⠉⠉⠁⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⣿⣿⣿⣏⣹⠟⠋⠁
⠀⠀⠀⠀⠀⢰⣿⣿⡿⠟⠛⠁⠀⠀⣀⣤⣴⣶⣾⣿⣃⣤⡀⢀⣀⣀⣠⣤⣄⡀⠈⠛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢀⣄⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⡀⢀⣀⡻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣬⣤⣶⠶⠆⠸⢟⣻⣛⣽⣿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠁⠲⠂⢁⡤⠞⠉⠁⠀⠀⢀⣤⠾⢿⣿⣿⣿⣷⣿⣼⣿⠟⠛⣿⡿⠯⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⣠⢾⢿⣻⣶⣿⣿⣿⣿⣿⠿⠉⠉⠀⠀⠃⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⣶⣾⡟⠉⣡⣿⣿⢾⣿⣿⡋⠸⣿⣿⣿⣶⣿⡶
⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠈⠏⠀⠀⠻⠞⠿⣻⣿⣟⣇⠀⠈⠻⢿⣿⡁⠀⠀⠀⠀⠀⢀⣀⣴
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠁⠀⠀⠀⠀⠙⠻⠆⠀⢠⣶⣾⣿⣿⣿
⠀⠀⠀⢠⣶⣿⣧⣿⣿⣿⡿⣶⣦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣧⣼⣷⣤
⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡇⣿⣿⣼⡿⣿⣿⣿⣿⡿⣿⣿⣿⣟⣿⣿⣿⣯⢿⣮⡟⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣷⣿⣧⣿⣿⣿⣹⢿⣿⡇⣿⣿⡿⣿⣿⣿⣹⣿⣯⣿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠁

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

⠀⠀⡀
⠀⠀⣸⡍⠢⠴⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡳⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡈
⠀⠀⢧⡁⢕⣶⣟⣋⠀⠀⠀⠀⠀⠀⢀⡴⠛⠋⠁⠈⠉⢢⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣽⣷⣿⣻⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠉⠉⠉⢳⡼⠁
⠀⠀⠀⠉⠙⠛⠛⠻⠶⠀⠀⠀⠀⠀⡈⠃⠀⠀⠀⠀⣠⠏⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠷⠀⠀⠀⠀⠀⢸⣻⠁⠴⠶⠖⣢⣞⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠋⠓⣂⡡⠋⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣿⣛⣛⣻⠟⠁⡾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠍⣱⠏⠉⠀⢀⣾⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣻⡟⠉⡕⣀⣼⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣮⣷⣶⣿⡛⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣿⣶⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⢀⢦⠠⣀⣀⣈⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡡⣤⣄⣉⣛⣷⣤⡀
⠀⠀⠀⠀⠰⠁⠀⠳⢀⠀⠈⠉⠁⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢻⣦⡈⠉⠉⠉⣉⢿⣄
⠀⠀⠀⠀⠸⡄⠀⢰⠀⠁⢤⡀⠀⠈⢈⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⣸⡇⠉⢦⣔⡚⠋⢣⣻⣷⡀
⠀⠀⠀⠀⠀⠙⠷⣾⢄⣀⡀⠈⢱⣤⣤⢤⣭⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣠⣶⣉⢻⣶⣿⣿⣯⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠫⠏⠤⢿⣽⣿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⠿⣿⣿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⡠⠖⠒⠒⠤⡀⢠⢊⠝
⠀⠀⠚⢲⠤⣄⠀⠀⠘⡇⣆⠤⠀⢀⣀⡀
⠀⠀⠀⢸⠀⠈⠳⡄⠀⢱⣞⠒⢮⡁⠀⠁⠀⠀⢀⣀⣀⣤⣤⡴⠶⠶⠶⠶⠶⠶⢦⣤⣤⣀⡀
⠀⡎⠉⠉⠑⠒⠤⠬⢦⣸⠙⠛⣿⠭⣛⠛⠛⠛⠉⠉⠁⠌⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⣄
⠀⢰⠂⠀⢀⠤⠠⠤⡀⠼⡖⢎⠢⣍⠒⠛⠶⠖⡒⠒⠠⢄⡀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠳⣄⡀
⠀⠀⠂⠉⠻⠔⠒⢲⠀⠀⢡⠘⡀⠀⠑⢄⠀⠀⠀⠈⠂⠄⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⣠⠏⠀⡆⢸⠀⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⣧⡀
⠀⠀⠀⠀⣠⠞⠉⠀⢀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⢷⡀
⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠘⣷
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸
⠀⣸⠁⠀⠀⠀⠀⡀⡄⠀⢸⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠈⡀⠀⢣
⢠⡇⠀⠀⠀⠀⠀⠁⡇⠀⠈⡄⠀⠀⠈⡄⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢰⠀⠸⡄
⣸⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⢣⠀⠀⠀⠘⡄⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡆⠀⣧
⣿⠀⠈⠀⠀⠀⠀⠀⠘⡀⠀⠈⢧⠀⠀⠀⠘⡄⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢇⠀⠀⡇⠀⣿
⣿⠀⢀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⣆⠀⠀⠀⠘⡄⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡆⠀⣿⠀⣿
⣿⠀⠘⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⣆⠀⠀⠀⠈⢆⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⢹⠀⢹⢰⡏
⢹⡄⠀⠄⠀⠀⠀⠀⠁⠀⠀⠑⡀⠀⠘⢆⠀⠀⠀⠀⠡⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡇⣼⡿
⠈⣷⠀⠈⠄⠀⠐⡀⠀⠁⠀⠀⠘⢄⠀⠈⢧⡀⠀⠀⠀⠈⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⡆⠀⣷⡿⠁
⠀⠸⣇⠀⠘⡄⠀⠐⠀⠀⠀⢄⠀⠀⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣇⣴⠏
⠀⠀⠹⣦⠀⠘⢆⠀⠈⠂⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⢀⠀⠀⠘⣦⡼⠻⡧⣑⠤
⠀⠀⠀⠘⢷⣄⠈⢣⡀⠀⠑⢄⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣆⣤⠞⠋⠨⠂⠙⠂⠡⣀
⠀⠀⠀⠀⠀⠙⠳⣤⡙⠦⡀⠀⠑⢄⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⣧⣤⡶⠛⠡⡀⡀⠒⢦⡀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⣕⣂⠠⠑⠦⣄⠈⠓⠦⢤⣄⣈⠐⠲⢦⣤⣤⠶⠿⠛⠋⠉⠆⠀⡖⢂⢆⠀⠀⢀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣛⣚⣿⣿⣶⣶⣾⣛⣛⣛⣋⣉⣀⣀⣀⣀⣊⣈⣂⣙⣋⣕⣀⣁⣀⣁⣀⣀⣀⣀
⣿⣿⠋⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠻⠲⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠖⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠈⡆⠀⢄⠠⡀⠀⠀⠀⠀⠀⠀⣠⠖⡩⠐⢀⠠⠐⠂⡉⠄⢂⠀⠀⠀⠀⠀⠰⠘⡄
⠀⠀⠀⠀⠀⢠⠇⡀⠠⡑⠨⢑⡒⠤⢤⣀⡔⠁⠀⠔⠈⣀⠄⠂⠁⡀⠔⠁⠀⠀⠀⠀⠘⡄⠀⠘⡄
⠀⠀⠀⠀⢀⠎⠃⠃⠀⠈⠒⠠⢀⣉⠐⠂⠨⢙⠲⢤⡈⠀⢀⠔⠈⠀⠀⠀⠀⠀⠁⠀⠀⠰⡀⠀⠸⡄
⠀⠀⠀⢀⠎⠌⡘⠠⠀⠀⠀⠀⠐⠠⢈⠁⠒⠄⡉⠢⡙⢎⢀⠔⠀⡠⠀⠀⠀⠀⠀⡀⠀⠀⢃⠀⠀⢣
⠀⠀⠀⡎⡜⠠⠁⠂⠀⡄⠀⠀⠀⠐⢄⠉⠂⠀⠘⡄⠱⡈⣇⠀⠊⢀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠇⢸
⠀⠀⡸⠰⢀⠃⡈⢀⠀⠁⠀⠀⠀⠑⠄⠑⠀⠀⠀⠰⠀⠃⢹⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡸
⠀⠀⡇⠇⡄⠀⠁⢸⠀⡀⢀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢲⠃
⠀⠀⡇⠀⠃⠀⠀⠀⠆⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⢃⠟⡇⠀⠒⠀⠀⠀⠊⠀⢀⠔⢁⠀⡰⠃
⠀⠀⢇⢇⠘⠄⠀⠀⠈⠢⡑⠀⠀⠀⠀⠀⠀⠂⡶⡡⢃⠞⣰⠃⠀⠤⠀⠀⠀⠀⠀⠀⢐⡡⠚
⠀⠀⠘⢎⠢⡈⠐⠤⣀⠀⠠⠄⠀⠀⠀⠀⠀⢀⠔⡡⠋⣰⣏⠀⠀⠀⠀⠀⣀⣀⠤⠒⠉
⠀⠀⠀⠈⠳⢌⡐⠢⠄⢉⣀⠐⠂⠀⠀⠀⣀⡤⢺⣿⡴⠋⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠶⢶⣖⣒⣲⣒⣚⣻⣿⡿⢿⣏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓


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

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⠶⠶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠋⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣾⣏⣀⡀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⡿⠀⠀⠀⠀⢻⡄⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠶⢻⣏⣹⠃⠀⠀⠀⢀⡾⠁⠀⠀⠀⣷⠀⠀⠀⣀⣺⡄⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢀⡼⠛⠛⠛⠉⠛⠀⠀⠀⠀⠸⣧⡀⠀⠀⢀⠘⠷⠶⠟⠉⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢠⡟⣀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⢾⡄⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣠⠟⢉⡽⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⡠⠀⠀⠷⣤⡄⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣾⣧⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⣠⣷⡀⢀⣀⡘⣷⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠙⠉⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣤⣤⣤⠴⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⢤⣤⣄⡀⠀⠀⠈⠻⠶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⢴⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠘⠞⢁⣤⡾⠃⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⢀⡠⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⣤⣤⣤⣤⣤⣤⠶⠞⠛⠁⠀⠀⠀⠀⠀⢀⣀⣭⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣷⣄⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠾⢛⣛⣿⣿⡶⠶⠶⠶⢶⣝⡲⠶⣤⣂⣤⣤⣤⣤⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⡼⠇⠀⣠⢴⡆⠉⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠤⠶⠀⠀⠀⠀⠂⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠊⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡊⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⣤⣧⣴⣶⣶⡶⠷⠶⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢟⡿⢏⣠⣶⣿⠟⠁⠀⠀⠀⠀⠀⢀⣠⣾⣇⠜⠉⢂⡇
⠀⠀⠀⢠⠀⠀⠀⠀⠈⠐⠋⢽⢶⢿⠿⠛⠁⡀⠀⣀⡁⠀⠄⠀⠉⠛⠋⠹⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠭⢅⠀⠀⠀⣠⣀⢲⠢⣄⣄⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⢀⠤⠊⠰⠆⠀⠀⠀⠀⠈⠄⢀⡀⠳⠷⠀⠉⠉⠡⠤⠋⠋⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠀⠦⠀⠢⢀
⠀⠈⠒⠄⠀⠀⠀⠀⠀⡇⠀⢠⠀⢀⠀⠀⡉⠉⠀⠀⠀⠈⢉⢁⠀⡀⠠⡀⠰⢄⠀⠀⠀⠀⢀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⢅⠀⠀⠀⠗⠀⡜⠀⡏⢀⠀⡇⠀⢨⠀⠀⢇⠈⡈⡀⡇⠀⠱⡄⠀⠓⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡈⠀⠁⠀⠘⠀⠀⠘⠀⠃⠁⡀⠀⠀⠄⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠜⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠠⠂⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⣄⠠⠀⣀⣨⢶⠤⣄⡀
⠀⠀⠀⢀⠥⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠋⠄⠄⠘⠊⠀⠀⠈⠙⠀⡒⠒
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠐⢊⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠊⠀⢀⢼
⠀⠀⠈⣀⠑⠄⠐⠀⠂⠉⠀⠀⠀⠜⠀⠀⣠⠎⠜⠀⠀⠀⢀⠀⢈⡆
⠀⠀⠀⠉⠐⠠⠙⠀⠀⠀⠀⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠻⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠉⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⠁⠀⠀⠀⠀⠄
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠆
⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣰⣜⠃⠀⠀⠀⢀⠃
⠀⠀⠠⠌⠙⠣⣄⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡕⠁⠉⠀⠀⠒⠂⠈
⠀⠀⠀⠠⠀⠀⡎⠁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠆⠀⠀⠀⠀⠀⣠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣴⣞⣠⢶⡆⠶⡶⣰⢋⡴⢋⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠟⠁⠈⠉⠉⠈⠉⠈⠁⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⡿⢟⣟⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠈⣿⣿⣿⣿⣿⣿⣿⢋⡾⣿⣿⣿⠀⣻⣿⣿⣿⣿⣿⣿⡏⡄⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⣿⣿⣿⣿⣿⣿⡇⣾⡇⣼⣿⠟⣠⡿⡿⢿⡿⠿⠿⣿⠀⢡⠿⣿⣿⣿⣿⣿⣿⣿⠿⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠑⣿⣿⣿⣿⣿⣿⠀⣿⡅⣨⣷⣾⡟⢠⢰⡄⢿⢰⠆⣿⠀⣴⠗⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⢻⣿⣿⣿⣿⣿⣆⢿⠇⣿⣿⣿⣷⡘⠷⢀⣴⣿⠐⣫⠀⣤⡙⢟⣼⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢟⣭⠙⠛⠛⠻⠟⠛⠛⠛⠛⠛⠋⠉⠉⣶⣍⢿⣿⣿⡟⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠛⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠒⠛⠉⢋⠛⠛⠲⠶⠶⠟⠛⠋⠉⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢢⡖⠉⠀⠀⠀⠀⠠⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡜⠀⠀⠀⠀⠀⢠⠃⢸⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢇⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠁⢠⠃⠉⡆⠘⠉⠄⢸⠇⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠸⣆⠀⠀⠀⠀⠘⠀⣸⠀⠀⠀⠘⢄⠐⠏⠀⢸⡀⢺⠛⢦⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢀⣤⣿⠿⠿⠷⢦⣤⡤⠤⠶⠶⠾⠛⠛⢻⣿⣦⠀⠀⠐⠟⠗⠒⠒⠒⠂⠐⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠠⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢳⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠙⣄⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠶⠀⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠈⠻⠿

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

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

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡔⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⡏⡆⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⢀⠎⡜⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⢶⡶⣄⡀⠀⠀⠀⠹⣦⠀⠀⠀⡇⠸⡆⠀⠀⠀⠀⢠⡞⢸⠀⣰⠃⡜⠀⠀⣴⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠲⢄⡀⠀⠹⣷⡀⠀⢠⠀⢻⡀⠀⠀⢠⡏⠀⡇⢰⠃⡾⠁⢀⡞⠁⣺
⣀⣀⣀⣀⠀⠀⠀⠀⠀⠘⢦⡀⠙⢦⣄⠹⣷⡄⢸⠀⠀⢳⡀⢀⡞⠀⢸⢠⠃⣼⠃⢠⠎⠀⠀⡇
⠉⠉⠉⠉⠛⠛⠷⢦⣤⣀⠀⠙⠢⡀⠈⠳⣭⣿⡄⡆⠀⠀⠱⡜⠀⠀⣸⠆⢰⡏⢠⠏⠀⠀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣤⣈⠠⠀⠈⢻⣟⣷⡄⠀⢀⠃⠀⢠⡿⠀⡏⢠⠏⠀⠀⢠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣓⢷⣄⠀⠙⡌⢳⡀⠘⠀⠀⢸⡇⢸⢠⡟⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡝⢷⡄⠘⠀⠹⡆⠀⠀⠈⠑⣼⡏⠀⠀⠀⡸⠀⠀⠀⣀⠤⠴⠒⠉⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⢦⠹⠄⠁⠀⠀⡀⠀⠸⠁⠀⠀⡰⠁⣠⠶⠋⠁⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢷⡘⡄⠀⠀⠉⢠⡄⠀⠀⣸⡵⠋⢀⣀⡤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠹⡡⠀⠀⠀⢨⠀⢀⡾⠋⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢱⢃⠀⣠⡌⢀⡟⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⡏⠻⠃⠹⠇⢹⡀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠑⢄⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠳⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣴⢇⣤⣼⣶⣄⢶⣀⡴⠖⠚⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣛⣉⣉⣀⣿⣁⣽⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿