Alice Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢾⠟⠉⠀⠐⠉⠁⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠛⠛⠉⠉⠉⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠂⠀⠉⠙⢻⣷⡢⣀⠀⠀⠀⡰⠋⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⣀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠈⠻⣿⣆⠑⢄⡀⠀⡀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣾⣿⡿⠟⠻⣛⣛⣉⣉⡛⠷⣦⣕⣾⣿⣷⡄⠡⡀⠀⠀⠀⠀⠠⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⣿⣿⠿⢏⡙⢦⡙⢿⣿⣿⣿⣿⣆⠘⡀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⠆⠀⣿⣿⣀⣈⣿⣄⠁⠈⣿⣿⣿⣦⣤⡤⡄⢁⠀⠀⠃⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠁⢀⣀⠐⠿⠛⠉⠉⠋⠀⠀⢺⣿⣿⣿⣉⣁⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣠⣤⣤⡀⢆⠀⠀⠀⠀⠀⠀⣰⠟⡙⣷⣿⣿⡬⠄⣀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⣀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠠⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣤⡌⠉⠉⠃⠀⠀⠀⠀⠀⠉⠀⠉⠻⣿⣷⣜⢿⡉⠀⠀⠀⠀⠀⠐⠁⠺⠂⠠⠄⠐⠏⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢐⠀⡁⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⣀⣿⣿⣿⡄⠇⠀⠀⠀⠀⢸⡁⣰⠂⠐⠒⠚⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠠⢀⠀⠀⠀⠠⠻⣿⣿⣿⣟⣻⣶⡄⠀⠀⠀⠀⠀⢀⡼⠀⠘⠛⠻⣿⣿⣷⠀⢠⠀⠀⠀⣼⠁⠀⣤⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠤⠀⠀⠀⠰⢺⣦⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡾⡟⠀⠀⠀⢀⣿⣟⣿⠀⡎⢠⣾⣖⠛⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠈⠀⠤⢂⣴⣾⣶⡌⣽⢧⡈⢿⣿⠿⢋⠀⠀⠀⠀⢠⢿⠇⠀⠀⠀⣼⣿⣿⣿⡜⠀⣼⡿⣦⡉⠓⠧⠤
⠀⠀⢠⠀⠀⠀⠀⠃⢀⣠⣴⣶⣶⣶⣶⣴⣿⣿⣿⣿⣿⡚⠓⣷⡈⢻⣯⣉⣁⣤⣴⣤⣏⠉⠀⠀⠀⣰⣿⡿⠟⠋⠀⢀⣷⣤⣀⠀⠉⠀⠂
⠀⠀⠀⢆⢪⣦⣶⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠟⢾⣭⡆⡙⣿⣿⣿⣯⣽⠋⠀⢀⣠⠒⠿⡶⠒⠒⡶⢶⣾⣟⠻⠋⠙⠻⠞⠁
⠀⠀⠀⠈⢻⣿⣿⠋⣿⠟⣯⡿⠀⢛⠾⣿⣿⣿⣿⣿⣯⣿⡃⢈⣙⠛⠛⡿⣛⣋⠙⡧⠐⠊⢻⡟⢿⠆⢀⣄⠘⠠⣿⣿⠙⠁⠈⢡⠊
⠀⠀⠀⢀⣿⣿⣿⣻⡁⠀⠀⠑⠆⣱⣶⣟⣻⣿⣿⣿⣿⣿⠇⢾⣭⣿⠿⡇⣿⠉⢂⣙⠛⠓⠄⢀⣀⠀⢃⣌⠕⣠⣾⣣⣤⣀⠀⡌
⠀⠀⢀⣾⣿⣿⣿⣭⣟⠀⠀⢀⣴⣿⣿⡽⠋⠀⡰⠻⡿⢿⣃⡠⠍⣥⣴⣿⣿⣶⣼⡉⠛⠄⠀⡏⣲⣦⠀⣠⣾⡿⢻⣿⣿⣯⣭⡦
⠀⣼⣿⢿⣿⣿⣿⣿⣿⡀⢀⠁⣿⣿⣿⠇⠀⠀⠀⡈⠀⠔⠉⠉⠉⠙⠛⠿⣭⣛⢻⣿⢷⣶⣀⣈⣭⣴⣾⣿⢯⣽⣭⣿⣿⣿⣿⠃
⢠⣿⣼⡿⣿⣿⣿⣿⣿⠃⢀⠀⠘⢿⣿⣇⣠⣾⣦⣾⣬⠶⣄⠀⠀⠀⢀⡐⠲⠬⢙⣙⢢⠉⠛⢿⣿⣿⣿⣿⣆⣤⣬⣿⣿⣿⣿⠇⠀⠠
⢈⠑⠀⢿⣿⣿⣿⣿⣿⡦⠀⠀⢺⣥⡤⠈⠛⢛⣿⣿⣿⣷⡈⢷⣦⡰⠶⢍⠳⣌⠂⡙⠆⠈⠂⣀⠹⣿⣿⣿⣿⣟⢻⣿⡟⠋⠀⡀
⠀⢀⡐⠪⡿⠿⢿⣿⣿⣶⠀⣴⢤⣤⡄⠔⢀⣾⣿⣿⣿⣿⣧⢸⣿⣿⡲⠄⠀⠈⠛⠙⠈⠲⣄⠫⡻⣿⣿⣿⣿⡿⢿⣿⡁⢀⠀⠀⠀⠀⡀
⠀⠀⠄⠀⢀⡀⡼⠶⠤⢉⠝⢻⠛⠁⠀⠀⠀⠈⠁⢸⡿⣿⣿⣼⣿⣿⣇⢠⠀⠀⠀⠀⠀⠀⠈⠆⠈⠘⠛⠁⠻⣏⠛⣿⠀⠙⢶⣤⡴⠖⠂
⠀⠀⠀⠀⡋⠀⠀⠂⠀⢸⢿⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠶⠿⠚⠃⡴⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⣠⣾⡀⣿⡟⣰
⠀⠀⠀⠀⠀⠀⠈⠳⠶⣶⣬⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣍⣍⣻⣷⣽⡀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⢠⣤⣾⣤⣬⣹⣷⣥⣤⣤⣤⣴⠦⠶⠂⠛⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠄⠈⠛⠿⢷⣾⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠚⠛⠀⣤⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠑⢋⣠⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⠿⣿⡿⣿⠿⠿⠿⠿⠿⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣽⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⣏⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣦⠀⠀⠀⠀⣤⠴⠂⠉⠀⠈⢿⣿⣿⣗⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡻⣷⣜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⢁⣤⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠾⠋⠀⠀⢰⣿⣿⣿⣿⣿⣿⡏⠉⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠇⠀⠀⠀⠀⠻⠿⠿⠿⣿⠿⠟⢰⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⠙⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠻⠫⠶⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠂⠀⠀⠰

⣠⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠠⣰⡄⠀⡀⠀⠀⡈⠁⠀⠀⠙⢞⡳⠈⣁⠀⡲⣦⠴⠋⠹⡄⣠⠀⠨⠁⠁⢀⡀⢠⡾⣧⠀⠀⠀⠸⠟⠛⠀⠀⢚⢴⠭⢒⣏
⣿⣤⣯⣿⡻⠓⠀⠀⠀⢰⢦⠊⣸⣀⢁⣥⠀⠀⣀⣀⡈⣿⡦⠄⠙⠲⠿⠖⠋⢁⡀⠀⠠⢸⠛⢦⣰⣷⣠⠾⢿⡄⢀⠈⠻⣿⣇⣈⣤⣤⣶⣶⠞⣽⠷⠾⠂
⠙⠲⠔⠉⣁⣀⣀⣀⢀⡶⠿⢞⡋⢁⣼⣯⢰⣿⢿⣿⡿⣿⡁⠀⣀⠀⠀⠀⠀⠛⣿⣦⢾⡎⢤⡄⠈⠉⠁⠀⠻⠇⠿⠇⠀⠀⣿⠋⢻⣏⠋⢠⣿⠏⠀⠀⣀
⣰⠀⠀⢠⣿⣿⣿⣿⠀⠓⠒⣿⣓⣿⠿⠃⡀⣼⣷⠟⠁⢿⣃⡠⠄⡗⠒⠀⠐⠀⢟⠃⢐⢑⠂⢀⠀⠀⣄⢀⣀⠀⣀⡀⠀⠀⠹⡄⠀⡿⣐⢺⡹⡄⠀⠸⣿
⣱⠀⡈⣶⣿⣿⠿⣿⠆⠀⠀⠟⠉⠣⠄⢀⠀⣿⣷⠀⠀⠀⡀⠀⠀⠐⠂⠐⠀⠀⠀⠄⠀⣩⠀⠀⠀⢀⣸⡄⠉⠀⡛⠉⠀⢶⡂⢧⣤⣿⠇⠈⣟⠁⡄⢀⡙
⠋⠀⣱⠀⣱⡇⣠⣶⠟⠀⠀⠀⠀⠀⠠⡾⣢⣿⣿⠀⠀⠐⣯⣶⣤⣠⡇⠀⠘⠂⠀⠀⠀⠁⢀⣳⣟⣨⣿⣁⠞⣸⣧⡀⠤⠀⠛⠂⠙⠃⠆⠠⢿⣶⣤⢸⡆
⣀⠀⢿⣄⠙⣶⣿⢟⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⣟⠛⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠘⠉⠃⢀⡏⠀⠀⠹⣿⡄⠀⡀⠀⠀⠀⠐⡀⢀⣀⡵⠋⠷
⠋⣀⣤⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⢀⣰⡞⠀⣀⠂⡀⡄⠠⠀⠀⠐⠀⠀⠀⠀⣀⠀⠀⡄⢀⠀⢸⣶⣶⡶⢶⣿⠟⠀⠟⠀⠀⠀⠿⣷⢆
⠊⠉⠉⠛⠉⠉⠉⢉⣍⣩⣽⠒⠀⠘⠐⠖⠛⠟⢿⡄⠙⢀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠠⣄⣀⣼⣷⠟
⠀⠳⢀⠀⠀⠀⠘⠞⠃⠀⠚⠶⠂⠀⠀⠛⠲⠉⠀⠀⣀⡛⠀⠀⠀⠀⠀⠀⠀⠰⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠂⠀⠛⠧
⣀⡀⢀⠸⣶⣶⠤⡀⡄⠀⠀⠀⠠⠀⠐⣤⡄⠀⠜⠛⠿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣶⠀⠈⠛⠀⠀⠀⡄⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠄
⢀⡀⠈⠠⠿⣉⣶⣿⣇⡳⢠⣶⡆⠀⠀⠀⠀⠘⠤⣀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⡄⠀⠛⠛⠛⠟⡇⠀⠀⠀⣿⣿⣿⣿⡏⢁
⠉⡉⠂⠀⣶⣿⣿⣿⡇⠀⠈⠁⠂⠀⠀⠀⢠⣤⣤⣤⡄⠀⠀⠀⢠⣤⣤⣤⣤⡄⡄⡄⣤⠀⠀⠀⠈⠐⠁⠀⠀⠛⠟⠟⠟⡇⠚⡆⠀⢻⣿⣿⣿⡇⠚⠃⠶
⠐⠁⢀⠸⣿⣿⣿⣿⡧⣖⢀⣌⠃⠀⠀⠀⢼⣯⣿⣽⡇⠀⠀⠀⢸⣿⣿⣿⣿⡇⡇⡇⣿⡤⠰⠚⠣⠀⠀⠀⠀⠚⠟⠟⠟⡗⠲⠃⠀⣼⣿⣿⣿⣇⡀⠀⡄
⢀⡀⠈⢀⣿⣿⣿⣿⡧⠈⠉⠘⠣⠀⠀⠀⣺⣿⠿⣿⡇⠀⠀⠀⢸⣿⣿⣯⣿⡇⡇⡇⣿⠄⢠⡱⠶⣿⣿⢂⠀⠺⠟⠟⠗⡇⠀⠀⢀⣿⣿⡟⡉⠉⢳⣆
⠋⡁⠌⠨⣿⣿⣿⣿⡇⣲⢠⣶⡆⠀⠀⠀⢺⣿⣚⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⡇⡇⡇⣿⢩⣤⠀⠀⢌⡗⠃⠀⠺⠷⠷⠗⡇⠶⡆⠀⢻⣿⡇⠀⠀⣼⣿⣷
⠛⢷⣦⣴⣿⣿⣿⣿⡇⠀⠉⠉⠂⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣾⣿⡇⡇⡇⣿⠀⠠⠤⣤⣘⡃⠀⠀⠲⠷⠷⠷⡇⠶⠇⠀⣾⣿⣿⣶⣦⣿⣛⣿
⢈⠁⠘⢹⣿⣿⣿⣿⣧⡔⠀⡐⠂⠀⠀⠀⢸⣯⣿⣽⡇⠀⠀⠀⢸⣿⣿⣯⣿⡇⡇⡇⣿⣀⠀⢺⣿⡋⠉⠀⠀⠶⠷⠷⠷⡇⠀⠀⢀⣿⢿⣿⡟⢸⡿⣿⣿
⠊⠑⠄⠨⣿⣿⣿⣿⣇⠈⠛⠙⢋⠀⠀⠀⢾⣿⠿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⡇⡇⡇⣿⢁⣼⣸⣿⠜⠀⠀⠀⠶⠶⠷⠷⡇⠶⡄⠀⢿⣶⣭⡁⠾⠀⣾⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⢀⠀⠈⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠺⠁⠀⠀⠐⠀⠂⠞⠁⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣩⡥⠶⠀⠀⢀⠀⠀⠀⢲⡞⠋⠁⠐⠀⡄⠀⠠⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠤⢄⡈⢹⡏⠀⡜⢀⡇⣸⠋⠀⠰⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⠄⣀⠱⣤⠀⡾⣡⣟⣰⣯⠀⠈⠀⠀⠀⢠⡇⠀⠆⠃⠀⠀⢰⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⣷⡿⠘⠁⠿⣿⢯⣷⣄⠀⠀⠀⠀⢰⠇⠀⡒⢀⠆⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚⠈⠻⠋⠀⠀⣸⠏⠙⢿⣿⣷⡆⠀⠀⢸⠀⠀⠀⡄⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠓⢤⣀⠀⠀⠠⠶⠛⠁⠀⠀⠀⠓⠮⢷⣆⡀⠘⠀⢈⡜⣠⣾⠟⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⣠⠒⡀⠐⠁⢠⡼⠻⢿⣶⣶⡄⣄⡴⠃⠃⠀⠀⠀⠀⠀⢹⡖⡄⠀⢹⣿⣿⣧⣤⠀⠀⠀⠀⣠
⠀⠀⠀⡠⠀⠴⢋⡼⠁⠔⣠⠂⠀⡈⠠⠀⠀⣿⠻⢿⡆⠀⠀⠀⠀⢸⢀⡂⠀⡼⠐⣡⣿⣿⣿⣿⣿⣿⣿⣄⡄⠀⣿
⠀⠠⢊⠄⠀⢀⡟⠁⠊⣰⡄⠀⢰⠇⠀⠀⠐⠃⠀⢺⣷⠀⠀⠀⠀⠸⢸⡀⠀⠇⢠⣿⠺⣿⣿⣿⣿⣿⣿⣟⡇⠀⣹
⠀⠖⠁⠀⣰⡞⠀⠀⢠⣿⡇⠀⠘⠀⠀⠀⠀⠀⠀⠈⣻⡿⣄⠀⠀⣰⡷⢄⢰⠇⢸⡇⠘⣿⣿⣿⣿⣿⣿⣿⡗⠀⢿
⠀⠈⡀⠰⠋⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⣽⣧⢠⣿⣥⣿⣿⡀⣿⡇⠀⢽⣿⣿⣿⣿⠷⢶⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠉⠢⠤⣄⣀⠀⠀⠀⣀⣄⣄⡹⠟⣋⢿⣿⢾⣯⠀⣿⣯⠀⢸⣿⣿⣯⡙⠀⢻⡗⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡟⠀⠀⠀⠉⠉⠉⠀⢼⣿⣮⣀⣹⣿⣿⣿⣿⠀⢨⣿⠀⡌⣿⣿⡇⠀⠀⢈⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠛⢠⣤⣤⣤⣤⣤⣠⣤⣿⣿⣿⣿⣿⣿⠛⣿⠀⣸⣿⡀⢀⣽⣿⡃⠀⠀⠀⠀⢠⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⣿⣿⡟⠿⢿⡟⢿⣿⣿⣿⣿⣿⡿⣿⣵⣿⠀⠸⣿⡇⢈⢻⣿⡇⠀⠀⠀⠀⢸⡇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠃⠀⠀⡇⠈⢿⣿⣿⣿⣿⣷⣟⢿⢹⠀⠠⣿⠁⣀⢸⣿⠇⠀⠀⠀⠀⣿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⡇⢀⣈⣿⣿⣯⣽⣿⣿⣶⣿⣄⢸⣿⣤⣤⣿⣽⡇⠀⠀⠀⠀⠸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⣰⣤⣤⣽⡦⠤⢀⠀⠰⢈⡩⣹⣿⣿⡭⠿⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠰⠶⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⠓⠒⠚⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⠋⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢲⡜⣾⢁⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠲⡖⠶⢦⡄⠀⠦⠀⠀⠸⡙⠃⢺⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣿⣇⢡⡀⠡⠴⡶⠧⠶⠟⢵⠟⣠⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢫⢿⠏⣾⣿⣾⡷⣷⣬⣤⣼⣽⣶⣤⢾⣧⣀⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠗⠈⠛⣽⡷⠀⢈⠟⣛⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠋⠐⠸⣿⡾⠁⣶⡴⣾⡧⣌⡷⠲⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⠟⠉⣧⣬⠀⠀⣟⡁⠘⣛⡂⢻⡷⠙⠀⢉⣦⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⠗⠈⠁⠂⠘⢿⣿⣾⣿⠛⠀⠀⢠⣶⢿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠁⠀⠰⣶⢂⠀⠀⠈⠻⣿⡿⠂⠶⠶⣿⡷⣹⣧⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⣤⣤⣤⣤⣭⣤⣬⣴⣤⠠⣤⣽⣯⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⡀⠀⠀⠀⢀⣿⠟⣿⠃⢠⣿⣿⣿⣯⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈⠀⠻⣆⠟⢁⠜⠁⠀⠉⢻⣿⣿⣿⣿⣿⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⣽⡀⢿⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡸⠗⠓⠌⢑⣶⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⣤⠤⠤⠤⠠⠴⠤⠤⠤⠦⠤⠰⠦⠀⠐⠲⠶⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡄
⡀⠀⣀⣀⡁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠈⠉⠐⠐⠀⠀⠀⠀⠀⠠⠤⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⡾⠀⠀⠀⠀⢻⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡴⠒⠢⡀⠀⠀⣀⠀⠀⠀⣠⡤⣄⠘⠃⠀⠀⠀⠀⠀⠓⢀⠰
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠀⠒⠚⢄⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢔⠕⠸⠣⡉⣀⠐⠀⠀⠀⢀⠀⠀⢤⡀⠀⠶⠀⠀⠶⠀⢀⡔⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠑⠂⠄⠀⠀⠀⠀⠀⠇⡇⢺⡃⠀⠀⢀⡏⠈⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢠⠃⠀⡔⢉⡐⡄⠀⣠⠀⢀⠀⠀⠀⠀⠈⢀⠀⠛⠀⠀⠘⠃⠀⠉⡧⠀⠀⠀⠤⢀⠈⠉⠉⣠⡤⢤⣄⠈⠀⠀⠀⠀⠀⠀⠉⣠⠌
⠀⠀⠀⠀⠀⢸⡀⠀⠣⣈⡡⠳⡶⠁⠄⠈⠀⡀⠀⠀⠀⡀⠠⣄⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢀⠀⡘⠂⠓⠀⠀⠄⡀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⢳⠀⠠⠈⠀⠈⠁⠀⠀⠀⠀⡋⠀⠀⡌⠀⢀⢈⠛⠁⢡⠀⠀⡀⠄⢀⠂⡀⢀⠀⠁⠀⠀⠀⠠⠦⡄⠀⠀⠀⣠
⠀⠀⠀⡄⣀⠀⠀⠁⠀⠀⡀⠘⠛⠀⠀⠀⠀⠀⡄⡀⢈⢀⠀⡘⠀⠀⠘⣇⠀⡡⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⠬⠕⢀⠀⠀⠋⠀⢀⡀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠤⠀⠆⡀⠀⠈⠀⣖⠊⡁⠀⠀⠀⠘⠛⠛⠛⠁⠀⠅⣬⠀⠀⠀⠀⠀⢉⠀⠈⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠒⠀⠀⠀⠀⠱⠖⠴⠶⠚⠀⠀⠠⠀⠀⠀⠀⠀⠄⠈⠡⠀⡀⠀⢈⣀⠀⠀⠀⠐⠂⠀⠲⠖⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠑⠒⠶⣤⠶⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⡄⢠⠀⠄⣀
⠀⠀⠀⡰⠛⢃⡐⠒⡀⠁⢀⠀⡀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠠⠤⠀⠈⠂⠀⠀⠠⠇⢰
⠠⠄⠔⣷⡏⠌⣠⡚⢠⠀⠉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠠⡀⠉⠀⠢⣤⠞⠢⡄⠂
⠀⠀⡀⠀⠈⠂⡎⠁⠃⣤⢀⠀⠀⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡀⠈⠀⡉⠀⠄⢁⣦⣀⡀⠼⢄⡤⠇⡀
⠀⠀⠝⠀⠀⠀⠁⠀⠀⠘⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠄⠀⣿⡟⠀⠀⠠⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠄⠐⠂⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣀⠰⡂⠀⢲⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠛⡇⠀⢀⢸⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡶⠨⠆⠦⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⢸⣿⣾⣥⣶⠋⢰⠀⠀⠀⠀⠀⠀⠉⠐⢂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠸⠀⢠⣄⠀⣿⠰⡆⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠙⣿⣿⣿⣿⣿⣷⣤⣆⡀⢀⠀⠀⠀⠀⠟⢷⡀⠀⠀⠀⠀⠀⢀⠈⡀⡛⢢⣇⠘⠮⠽⠷⢄⡀⠀⣠⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢄⣿⡟⢻⣿⡿⢹⣿⣿⣾⣧⣮⣵⡆⠠⣀⠀⠁⠀⢀⠀⠀⠀⠀⡆⠘⠃⠘⢛⠷⣶⣢⠞⠀⢿⠾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢠⣿⣧⣿⡉⣁⣼⣿⣿⣿⣿⣿⣿⣥⣤⡽⠃⠀⠀⠀⠀⠐⡄⠀⡷⢀⡀⡠⠈⣤⡿⠻⣆⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢻⣿⣹⣿⣼⣿⠏⣡⠿⠛⡉⠛⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢦⡷⡮⢿⡷⠶⢾⣏⣤⣽⣤⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣷⣬⣿⣼⣿⣿⢧⣾⣿⣾⡷⡌⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠌⢱⠀⠀⢸⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢻⡉⢹⡴⣫⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠿⣿⣿⣿⣿⡿⣿⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⣶⣿⣿⡿⠯⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣿⡷⣽⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣶⡄⣤⣄⠀⠀⠀⡀⢀⡀⠀⢀⣾⣿⣿⣿⠛⠁⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠋⢿⣿⣿⣿⠃⡈⠀⢀⡿⠀⠻⣦⣷⠉⠋⠀⠀⠋⠹⠋⡁⠀⠈⣽⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢢⡄⢻⣿⣿⣾⣷⢀⣞⢁⠀⢀⣿⠉⡆⠀⠀⢤⠀⠀⠀⠁⠀⠀⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢷⣿⣿⣦⣙⣿⣿⣯⣾⡃⢎⠑⠸⡃⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⠲⣹⡛⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣧⣾⣿⠉⠩⣍⠛⠿⣷⠚⢷⣾⠋⡺⠁⢀⣀⠀⠀⠀⢆⠁⠙⠓⢻⡟⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⡄⠀⠙⠀⠀⣨⡅⢻⡗⠂⠀⠈⠀⠀⠀⠀⠀⠻⠙⢺⡖⣾⠇⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⣿⡏⠘⣶⣄⠀⠐⠏⢰⣿⣿⣿⣷⠃⠀⠀⠀⠀⡀⢠⢀⣼⣿⡏⠲⣦⣐⢹⣿
⠀⠀⠀⠀⠀⣀⡖⠀⠀⢀⠀⠀⣸⢠⣿⣿⣿⣄⣀⣀⠀⣀⢸⠟⠁⠈⠹⠄⠀⠀⠤⠚⠁⣀⣠⠋⢀⣿⣧⡠⠊⠉⠚⢿⡄
⠀⠀⠀⢀⣴⡿⠀⠀⢀⡄⠀⢠⡟⣼⣿⣟⡹⢿⣿⠛⣷⣾⣿⢶⣠⣿⣗⣀⣤⣤⠖⢰⣿⣿⣿⣷⣾⢻⣿⣿⣆⡀⠀⠸⡇
⠀⡄⢠⣾⡿⠁⣴⣀⣿⡀⢠⡧⢳⣿⣯⡸⣧⣼⡧⢾⣿⣿⣷⡞⣿⣿⣽⣏⠋⠀⠀⠐⠀⠊⠈⠀⣿⠖⠾⠛⠿⣿⢄⠐⡃
⠀⣳⣾⡿⢃⣴⣸⣿⡏⢴⡟⢠⣿⣿⠉⣷⣿⣻⣷⣿⣿⣿⠿⣿⣿⣿⡟⠛⠣⡤⠀⢄⠀⠀⠀⠁⠀⠙⠂⣠⠄⡿⢬⡀⢹⡀
⣼⣿⣿⣧⣾⣿⣿⣿⣷⣿⠃⢾⣿⣿⣾⣿⣼⣻⣿⣿⢿⡟⠀⣿⣟⣿⡄⠶⢀⠇⢱⡦⠀⢠⣄⠀⠀⠶⣿⣥⠀⢰⢸⣧⣸⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⣿⣿⣿⡿⣿⣽⣿⣿⣿⣿⣅⣼⣿⡿⠻⣷⣤⢾⡴⠃⣀⠀⣈⣹⣷⠖⠻⠇⢣⠀⢰⣴⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⣧⣾⠀⡄⠈⠙⢪⠟⠛⠋⡞⣿⠤⣄⠀⠀⢰⡾⣿⣿⣿⣿⡅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛⣛⡛⠛⠛⠛⡟⠙⠛⠻⠞⠰⠻⢄⣼⣿⣿⣻⣿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣛⣛⠛⠻⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣷⣦⣀⣠⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⣈⣻⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⠿⠿⣶⣾⠷⠶⠦⠄⠀⠀⠀⠀⠉⠉⠓⠛⠙⠛⠛⠋⠛⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣯⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣠⣄⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣴⠂⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠀⠀⠀⠐⠂⠠⣦⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⢤⣤⠀⠀⣰⡟⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⠈⠉⠀⠀⣿⡆⣬⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⡄⠀⠀⣼⢻⣿⠃⣀⣀⠀⢻⢘⠇⣸⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣿⣟⠀⠀⣸⡟⣿⣿⠟⠿⡇⠀⢸⢀⣴⡿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⠏⠀⢻⣿⣿⣿⠟⠀⠀⠀⠀⣼⣿⣇⠀⣄⡙⣿⣄
⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠻⠟⠆⠀⠀⢻⣿⣏⠁⠀⠀⠀⠰⠿⣱⣷⣾⣯⣤⣿⡿⢦
⠀⠀⠀⣀⢀⠀⠀⠀⠉⠀⠀⠀⠀⠘⡄⠀⢺⣿⣿⡆⠀⠀⠀⠀⢀⣿⣯⠀⠀⠹⡟⠃⠈
⠀⠀⠠⠼⠺⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⢄⣿⣿⣿⣇⣀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡄⠉⢁⣿⡿⠁⢀⣴⣿⣶⡶⢦⠀⡀
⠀⢺⣿⣿⡿⠶⠷⠿⠿⠿⠿⠿⠛⣿⠏⢀⣴⣿⡏⠁⢠⣿⣿⡿⡏⠐
⠀⢾⠛⠉⣿⣿⣿⣿⣿⣿⡿⠭⢬⠃⣼⣿⣿⡿⠀⠀⢠⡟⠋⠀⠁⠀⠀⠳⡀⠀⢄
⠀⣾⢘⢠⣿⣿⣿⣿⣿⡏⣿⠀⡄⢠⣿⣿⢻⡆⠀⠀⠈⠳⠄⠀⠃⠀⠀⠀⠘⣄⠀⠳⡄
⠀⠁⢸⠨⢹⣿⣷⣿⣿⠃⠸⡏⣳⣿⣿⠋⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠹⣆⠀⠘⣆
⠀⠀⢸⣷⣿⣿⣿⣿⣿⠀⡧⡇⣿⣿⡟⠀⠀⠐⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⡀⠀⠹⣤⠀⠘⡆
⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⡇⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢻⣆⠀⠘⢦⡀⢀
⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⣇⡃⣿⣷⠖⣶⣶⡶⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⠘⠀⠀⠈⣿⣇⠀⠀⢳
⠀⠀⢼⣿⣿⣿⣿⣿⡇⠀⡇⠀⣿⡏⢠⣿⠅⠀⠲⠆⠉⠀⣿⡉⠀⠀⠀⠀⠛⠁⠀⠀⠨⠿⠂
⠀⣀⣸⣿⣿⣿⣿⣿⡇⡆⣇⡄⠉⠁⠺⣇⣀⣀⠀⠀⠀⠀⣿⡇
⠀⠉⢸⡿⢿⣿⣿⣿⡇⣇⡟⠁⠀⠐⠀⠀⠀⠀⠉⠉⢳⣶⣿⣷⣶⣶⣿⡄⠀⡄
⠀⠀⠀⢨⣿⣿⣿⣿⠁⣿⣷⠄⠀⢿⣄⣿⣿⣦⠀⠀⠘⣿⣿⣿⣿⠟⠋⢳⠈
⠀⠀⢀⡾⠿⠿⠿⣿⣦⣿⣿⠀⣈⡛⠉⠋⠉⠈⠁⠀⠀⢹⣿⡇⠘⡆
⠀⡀⠀⠀⠒⠲⢶⠋⠁⠙⣿⠀⠛⠛⠿⠿⢶⣦⣤⣀⣀⣼⣿⣿⠀⢻
⠀⠣⣄⠀⣀⡀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⢈⣽⣿⣦⠴⠿⢧⡾⠃⢠⡄
⠀⠀⠶⢿⣿⣿⣿⣶⣤⣴⣿⠟⢀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣼⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠈⠉⠑⠙⠿⠏⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠖⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⢎⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡎⢿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣾⡿⠿⠃⣤⠶⢤⡀⠰⠹⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠂⠀⠀⠐⢰⣿⢃⢠⡙⢀⣁⠀⢈⠀⠀⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡞⢋⠈⠁⠁⠀⡄⠀⢸⡁⠎⠀⣀⡻⣬⠇⠀⡀⢐⣐⣷⣄⠀⠀⠀⠀⠀⢰⡆⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⡇⠨⣻⡶⠤⠀⣇⠐⢻⣧⠀⢀⠉⠐⠀⠀⠀⠀⠈⣬⠏⠉⠀⠀⠀⠔⠀⣾⠁⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⡀⢿⣄⣼⣿⡀⠢⠑⠒⠂⣀⠈⢀⣼⡿⠁⠀⠄⠀⡄⠀⠸⠋⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⢿⡆⠻⡅⠀⣿⣿⣤⣀⢢⣄⣰⡟⢛⠃⠀⠀⢀⠉⠀⣴⡇⢀⠶⡀⣋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠰⠚⢻⣿⠟⢻⢿⣿⣄⠘⢧⠀⣛⣿⡁⠉⠿⢯⠉⠀⠂⠀⠄⠀⠀⠀⠐⣿⣇⠸⢷⡛⠞⠧
⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⡤⠃⠀⣿⠃⠼⢻⣿⣿⡿⣷⡘⠄⠉⠸⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢼⣿⣿⣶⡾⠛⡀⠦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⢻⡆⢠⣨⣿⣿⣿⣿⣆⠀⠀⠀⠃⠀⡆⠀⠀⠀⠀⠀⠀⠔⠂⠀⠠⢤⣤⣤⣌⣀⠠⠀⠤⠶⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠘⠛⢿⠿⣿⣷⣌⣻⣿⡆⣤⠂⠀⠘⢠⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠹⢾⣿⣿⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣶⣦⣀⣀⠀⠀⠀⡿⠉⢻⣿⣿⡧⠂⢀⡰⡛⠀⣉⣦⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⣿⣏⣀⣀⣤⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡋⠁⠀⠀⠠⠇⠀⠀⣿⣿⠁⠀⢸⠛⡇⠈⢉⠀⣀⣀⢠⠀⠐⠀⠀⠀⠀⠀⢸⡿⠛⣿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡍⠁⠀⠀⢰⡶⠀⠀⣰⣿⠁⠀⠀⠿⡛⢟⣠⣟⠄⠻⠉⠀⠀⠘⠃⠀⠀⠀⠀⠘⣇⡜⢿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠎⠁⠀⠀⣀⣼⠁⠀⠈⢻⣇⡀⠀⠀⢛⣃⡛⢛⣙⣦⣀⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢮⠀⣰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠹⠟⠀⠀⠀⠘⣿⣿⣟⣿⣿⣿⣿⠿⢿⣿⡾⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠙⠛⠻⣿⣿⣿⣿⣿⠿⢑⣿⡄⠀⠀⢀⣠⠄⠄⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⡿⠿⢿⡿⡟⢻⡟⢹⠌⢠⣴⡟⡁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣁⣽⠁⠀⠈⢷⡝⠻⠀⠏⠀⠀⠀⢷⣶⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠉⢀⠀⠰⠦⠀⠀⠀⠀⠠⠦⠀⠀⠀⠛⠉⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠒⣴⡆⠀⠀⡀⠀⠀⠀⠰⠲⠄⠀⠀⠀⣀⣀⠙⡄⠊⢰⠄⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢤⡘⠿⠼⢦⡈⠷⠠⢄⣤⣄⣠⡤⠞⡥⢠⡼⠖⢧⡐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⢝⡴⣷⢤⣤⢞⡁⣔⢂⠀⠀⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⣙⠉⠁⣩⡕⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⡙⢆⡈⠼⠏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣆⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⢿⣿⡏⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣒⠀⠻⣷⡆⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣽⣥⢼⣴⣿⣧⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢿⢛⣀⡀⠈⠀⠋⢻⣿⣦⣧⠠⡀⠀⢇⢰
⠀⠀⠀⠀⢠⣔⠒⠛⠉⢿⣅⠀⠀⠈⠛⠀⠀⠀⠀⠀⠉⡻⠟⠿⢟⢬⠻⠗
⠀⠀⠀⠀⠀⢿⠆⠀⠀⠘⢯⠀⠄⠀⠈⠙⠣⠄⢺⣧⡐⠁⠀⠀⣰⣟⡁⠰⡄⡐⡀
⠀⠀⠀⠀⠀⠀⢻⣄⡀⢀⣈⣷⣶⣶⣶⠦⠀⠤⠸⢿⣯⠀⠀⢷⣿⣿⣿⡾⠋⠑⣴⡆
⠀⠀⠀⣀⣠⣤⣤⣼⠀⠈⠉⠉⠝⠁⠀⠰⠶⠶⠶⠞⣀⠉⢷⠄⠹⠿⠛⠁⠀⠀⠈⠁⠀⠙⢶⣄
⠀⠀⠀⠀⠈⠉⠉⡉⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⢤⣀⣀⣤⣤⣤⡾⢇⡀⠀⠰⡘⣷⣄⣀
⠀⠀⠀⠀⢀⢤⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣿⠀⠀⠀⢀⣬⣿⡿⠋⠉⠀⠀⠈⠐⠄⠘⢿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⣿⡿⣿⣿⠿⠏⢀⣻⠂⠀⢠⣿⡬⠙⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠓⠊⠀⢀⠐⡟⠈⠀⠀⢸⠿⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⢱⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠀⡇⡇⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠙⢧⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⡆⡇⣇⠠⠀⠀⣀⣾⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⡇⡗⣿⢨⠀⠘⣿⠇⠀⠀⢸⡇⠀⠀⢀⣀⣀⡀⡠⠄⢂⣀⢶⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⡇⡇⣿⢸⠀⠀⣿⣄⣁⣈⣼⣅⣈⣡⣤⣤⣤⣤⣤⣾⣿⣿⡄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⡇⢱⡟⠰⠀⣾⣿⣿⣿⢻⣿⣿⢍⣩⣿⡇⢨⣿⠿⢻⡿⠟⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⡇⢸⡇⠀⠀⠉⠙⠿⠧⣿⣽⣿⣿⣿⣿⣿⣤⠿⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⣿⠇⣿⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠇⢸⣵⣦⠀⠀⠀⠀⠀⣇⣀⣿⠶⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⠄⠈⠩⢿⠀⠀⠤⢀⣴⣿⢿⣇⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⡆⢸⣼⣯⣤⣶⣿⣿⡿⢯⣽⣿⣿⡟⠓⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢿⢿⢲⡿⠶⢴⣤⣤⣤⣤⣴⢿⣿⣛⣑⡇⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹⠹⢀⣷⣤⣀⣀⡀⠉⠉⠁⠉⠁⠀⠁⠀⢀⡀⠠⠐⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠬⠈⡗⣾⣬⣿⣿⣿⣿⣿⣿⡉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠿⢿⡏⢷⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣆⣠⣤⠶⠶⢿⡿⣽⡏⢶⣦⣄⡐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣹⣿⣧⣼⡆⢸⣿⣿⣯⣿⣿⣿⣿⣿⣷⣄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣷⢿⣿⣹⡻⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣾⡿⣯⣽⡟⠛⢻⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣿⠉⠻⣷⣟⡿⣿⣿⡇⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⠿⣈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣇⠐⣿⣿⡖⡺⣿⣧⠈⡷⠾⣿⣿⣿⠿⠿⢻⣿⣖⢻⣹⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣓⢛⢷⣅⣿⣷⣷⡰⠳⢦⢖⣳⣿⣿⡶⠾⠿⢻⡛⢿⣿⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⢯⣭⣛⠾⣍⣿⣿⡟⢿⣿⣧⣿⣿⣿⡇⠒⠀⠀⢈⣿⣾⣿⡏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢻⡿⣷⣌⠻⣿⣇⠈⣟⠉⠃⣝⣿⣧⣔⡀⢀⣼⣿⣿⡟⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠈⠲⣝⡃⠙⠻⠴⣿⣴⣠⣾⣯⠭⠿⢿⣿⣿⡿⠋⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢊⢻⡿⠀⠠⣤⡈⢻⣿⣯⣻⣦⣠⠾⣿⡯⢤⣀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢤⣸⣿⣇⡀⠀⠙⠃⠈⣟⠉⠉⠙⢷⣾⢻⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣼⣿⣿⣍⣳⣈⣦⣼⣿⣿⣿⣿⣤⣹⣏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣋⣿⣿⣻⡟⠈⣿⣿⣿⣯⣼⣿⣿⠟⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⢦⣾⣷⣿⣶⣾⣿⣿⣿⣿⣿⠃⠒⠀⢀⡣⡙⣷⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢿⣿⣿⣿⣿⣿⣿⣉⠙⣷⢀⣤⣧⣷⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⠈⠉⠉⣿⡷⡿⡆⣸⣧⡯⢾⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣵⣄⣀⣀⣠⠞⢹⣿⣟⣾⣟⡏⠀⠀⢻⡸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⣿⣿⠄⢀⣠⣿⣿⣿⣟⣷⣖⡸⣿⢇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⡤⠿⠿⠿⠿⢏⡀⠶⢯⡿⠾⠃⠀⠿⠿⠠⠿⠾⠿⠿⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠠⠈⠵⡀⣬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠾⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡠⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠣⠶⣿⣿⣿⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠾⢥⡤⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢡⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⣀⡄⢐⣿⣿⣿⣯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠦⠀⠀⣨⣽⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠙⠀⢸⣧⣼⣿⣯⣿⡾⣿⡀
⠀⠀⢰⣆⣄⣀⠀⠀⠀⣼⣿⡈⢿⣿⣿⡟⣿⣦⣴⣶⣶⣶⣤⣀⠀⠀⠀⣸⠏⣿⠀⠀⠈⢹⠛⠛⠋⠻⠍⠁
⠀⠀⠘⣷⣋⢽⣿⣶⣾⣿⣿⠈⠈⠿⣟⣿⣾⣿⣿⣿⣿⣭⣉⠉⠀⢀⡶⠣⣿⣿⣼⠀⠐⠊⠁⣄⣐⡶
⠀⠀⠀⠺⢭⠿⢟⠛⠟⠋⢸⡏⡄⣷⣽⣿⣿⣿⣿⡿⠛⠛⠛⠋⢀⣸⣆⡀⢉⣻⡭⣤⠀⠠⣾⣿⣿⡇⠀⠀⠀⠀⢀⡠⠔⠆⡀
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠂⠚⡧⠱⣄⠉⣱⣋⠀⠀⢉⡁⢡⠀⣠⣾⠿⢿⣷⠈⠀⠀⠈⠀⢸⣿⣿⣿⣧⣀⠀⠀⠀⡀⠈⡖⠀⠖⠀⡀⢀
⡀⢻⣍⠋⠀⢀⣴⣒⠒⠲⠬⣇⠀⢀⠽⠇⠙⠓⠖⣄⢈⢫⣷⣿⣶⣯⣿⡟⠀⠀⠀⠀⠀⢸⡟⡿⠿⢯⠻⢿⣍⠉⣿⢢⡧⢶⠀⠂⠓⢙
⠀⠀⠘⠂⠀⢠⡉⠉⠛⠻⠶⣦⣭⡼⢀⣆⣀⣼⣀⣯⣼⣿⣿⣿⣿⣿⣿⡀⣧⡙⠃⠀⠀⠋⠀⢡⠀⠀⠁⠀⣈⣠⣿⠞⠁⠞⠀⠀⠉⠈⠅
⠀⠠⠶⠶⢶⣶⣵⣤⣀⡰⡄⢀⢨⣤⣼⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⠈⠀⠀⡔⣃⡧⠀⠈⠘⠐⢛⠋⠉⡠⣋⣤⠦⣤⡊
⠀⠀⠀⠴⣔⡀⣈⣉⠻⢿⣾⣾⣾⠟⠿⣿⠋⣀⣤⣀⠙⢿⣿⣿⣿⣿⣿⣿⠀⣀⠀⡹⠫⠄⡼⢀⡴⣿⠁⣠⡶⠛⣉⠤⡠⠀⠉⠒⠠
⠀⠀⠀⡠⠛⢁⣠⠤⢭⣶⣿⢹⠏⡟⠀⠈⢣⡿⢩⠍⢟⢡⠙⠿⣿⣿⣿⣷⡦⣿⣶⣦⣤⣴⡟⣻⠸⠿⠰⠿⡣⡊⢁⠜⢀⣤⢔⡡⠔
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠹⢿⣿⣾⠛⢁⡇⠒⡭⣴⣢⡼⣼⠆⢰⣈⣏⣏⢁⣠⡿⣿⡛⡟⠉⣠⠇⠀⠀⠀⠘⠐⠁⠀⠐⠁⠐⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣥⣬⣥⣤⣤⣬⣥⣥⣥⣥⣤⣽⣯⣭⣬⣥⣤⣽⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣸⣯⣙⣿⣭⣙⣽⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀⠀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣯⣭⡿⡋⢄⣙⠉⠉⠃⣀⣴⡲⣑⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⣿⣄⠀⠀⠀⣰⠊⢀⣾⠋⠀⠈⡀⢉⣀⣿⣕⡞⢻⡿⣟⢳⣿⢿⡿⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡈⢈⣻⠏⣯⠓⠀⡾⠋⣷⣿⠃⠀⣰⣽⣀⡨⢿⣿⡿⢿⣾⠋⠲⣌⠉⠀⠀⣾⣹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠙⠃⠀⠀⠀⠀⢴⣿⣿⡟⠛⠃⠸⠀⠉⠙⡓⠛⡿⠓⠠⣯⡅⣠⣿⣄⠀⠀⢭⣳⣿⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⣿⡗⠀⠈⠀⠀⠀⠉⠀⠀⠁⠀⣭⣽⣿⣧⠻⠿⠿⣓⡛⠛⢋⣯⠗⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣿⠁⠀⠀⢀⡤⠔⢶⣶⡰⣖⠒⠉⢛⠻⠿⣛⣋⠠⣉⡤⣿⡷⠋⢀⣐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠈⠀⠀⣀⠈⡂⠀⠀⣽⠧⠀⣄⠀⠀⠀⠀⢈⣼⠿⠿⢯⠀⠠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡤⡞⢻⠿⣿⡷⣦⣀⠀⠀⠈⠃⠀⠀⠀⠸⠅⠀⠀⠀⠡⠂⡠⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⡤⢲⣤⢠⣶⣅⠂⠀⠀⣴⡿⠀⠠⠀⠀⠀⢁⡈⠉⠛⡈⠒⡆⡀⠀⣀⣀⠀⠀⠀⠀⠀⠐⠐⣂⣠⠀⠂
⠀⠀⠀⠀⠀⠀⠈⣱⣿⣿⢯⡿⣿⡤⠀⣴⡧⡄⠈⠀⠀⠘⠀⠉⠁⠀⠀⠀⠀⠀⠉⠙⠥⢀⣀⡀⣀⠐⠀⠀⠀⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠚⡅⣿⠷⠞⠁⠙⠃⡐⣻⣿⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠙⡿⠇⠀⠀⠀⠀⠀⢸⣿⣟⣇⣀
⠀⠀⠀⠀⡀⣄⠈⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⣹⠂
⠀⠀⠀⠀⡁⣿⠀⡏⠀⠀⠀⠀⠀⠀⠠⢟⣍⠆⠉
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠈⠁⣶⡀⠀⣿
⠀⠀⠀⠀⠀⢻⠃⠀⢰⣿⣇⠼⣷⠗⠀⣾⠡⢘⢠⡀⡀
⠀⠀⠀⠀⠀⣾⡇⢰⡟⠋⣹⣔⣿⣆⠀⢸⣦⠃⠰⣤⡔
⠀⠀⠀⠀⠀⢸⠁⣾⡇⠘⣏⢻⣟⠉⢦⡌⠿⠀⠘⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⣽⣿⣿⠒⠞⠋⠀⠀⠀⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠉⠉⠃⢀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡴⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠎⠄⠀⣣⠁
⠀⠀⠀⠀⠀⠀⠀⠁⡔⠀⠁⢠⠀⠀⠸⡇⠇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⣠⠁⠀⠀⢸⠀⠀⠀⡻⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠾⢾⡇⠀⠀⠀⠤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⣘⠿⠦⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⠰⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⢀⠑⢆⣼⠋⠀⢀⡾⠋⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣾⢁⣼⠀⠈⣯⠀⢠⡾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠛⠻⣖⠃⢴⣶⠶⠠⡄⠀⠄⠤⠄⠀⠀⠀⠀⢷⡌⠉⠀⠀⢉⣾⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣍⣤⣄⣀⣸⡇⠀⠀⠀⠀⠶⠆⠀⠀⠀⣼⣷⠄⠀⣠⣿⠟⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⣽⣿⣿⡿⣟⣷⠀⢰⡆⡀⠀⠄⢴⣶⣿⣿⡇⠴⠾⢋⠀⠀⠘⠇⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡾⢣⢿⣿⡟⢡⣿⣿⡇⠀⣷⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢀⡟⠸⠏⠀⣾⡟⢙⣵⣿⣄⢛⡻⣶⢶⣦⣴⣦⣤⣶⣿⣿⢠⠖⠀⢻⡗⢀⣠⠾⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⣿⣇⣾⢿⣿⣿⣄⠻⣿⡇⠀⠙⠟⠁⣨⣿⠁⠀⠀⠀⠸⣿⠻⠟⣨⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠆⣸⠟⠀⠀⠀⠀⠸⣿⣿⠁⠀⠀⠠⠀⠀⢀⠴⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⠁⠀⠂⠀⠀⣴⣄⡿⠁⢰⣤⣀⢤⣴⣿⠁⡠⠀⢨⡁⣤⣄⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣷⣶⠀⠿⠉⠀⠀⠀⠀⠀⠀⠈⢹⡟⠀⠀⠘⣿⠏⠀⣿⣿⠀⠱⣼⠟⠁⠐⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢸⣿⡿⠁⠀⠠⠤⠤⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠉⠀⠀⠀⠐⣀⣤⣶⣿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠁⣴⣄⠀⠀⠀⠂⠀⠀⠀⠀⣜⠿⢗⠂⣠⣶⣿⣶⠀⠈⠉⠀⠀⠀⠀⣀⣴⣾⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣰⣦⣤⠀⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣛⡖⢺⣿⣾⠋⠀⠀⠀⠀⠀⣀⣾⠟⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⠀⠈⠀⢀⠂⠄⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠏⠀⠀⠀⠀⣠⢾⣿⡁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠀⡀⠀⣀⠀⠘⠡⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢹⠟⠃⠀⠐⠛⠃⠰⣿⣿⠟⢀⣼⣷⣦⡀⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣤⣄⣀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋⠉⠉⠉⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⠀⡁⢀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⢡⡿⠁⠀⠀⠀⡃⢠⡟⢠⣁⡄⣽⣿⣾⣿⣯⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⣴⡿⠋⠀⣰⠏⠀⠀⠴⠀⢘⡁⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣦⠀⠋⠀⢠⠔⠁⠀⠀⠀⠀⠀⠈⢭⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡀⠘⣷⠀⠀⠀⠀⠀⠀⣀⣀⣲⣿⣿⣿⣿⣿⣿⡿⠟⠦⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡤⢦⠀⠈⠃⠠⣶⣶⠿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠐⠛⠉⠀⠀⠀⠀⠀⠈⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣀⠐⠂⡀⠀⣰⣶⣶⣶⣶⣶⣶⠶⠶⠶⣶⢶⣶⣶⣶⣶⣶⣶⡶⠶⢶⠶⠖⠒⠒⠖⠲⠶⠖⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠈⠋⠲⢿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣶⡅⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⡠⠀⠀⠀⠀⠀⢠⣄⣁⣿⡆⢴⣶⣿⠇⠀⠀⠀⠀⠀⢀⡾⣉⠠⢾⣧⡰⣦⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⡾⢡⠋⠁
⠀⠀⠀⠀⠀⠀⢀⠐⠉⠻⣶⠀⠠⠀⠀⠀⢀⠀⠉⣤⣿⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⢾⠶⢿⠀⢠⣿⠀⠉⠀⠀⠀⠀⠀⠈⠹⢶⣴⡏⣴⠁
⠀⠀⠀⠀⠀⠀⢸⣇⣖⠀⠉⠿⠀⠀⠀⢸⡏⠁⠀⢩⣟⡟⠁⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡘⢯⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢿⣻⣤
⠀⠀⠀⠀⠀⠀⢠⣿⢛⡟⠀⠀⠀⠀⠀⠘⠟⠙⠇⠈⠀⠗⢻⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⢳⡀⠀⠀⠀⠀⠀⠀⠼⡿⣿⢀⣼⠛⠁
⠀⠀⠀⠀⠀⠀⢸⣧⠞⠁⠀⠀⠀⠀⠀⢴⠆⠀⠴⠀⠀⠀⡀⠰⠀⠀⡄⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡈⣷⡄⠀⠀⠀⠀⠀⠀⢰⣄⣄⡙⠳
⠀⠀⠀⠀⠀⠀⣼⡀⠀⢣⡀⠀⠀⠀⠀⠨⠆⠀⠦⠀⡘⢧⠀⠠⠀⠒⠈⠁⠀⠀⠀⠃⠀⠀⢰⡂⠉⠻⠟⠛⠀⠀⠀⠀⠀⠀⢸⡿⣨⡋
⠀⠀⠀⠀⠀⠀⠀⢻⣆⡈⠁⠀⠀⠀⠀⣀⣠⡁⠀⠠⠇⠠⣄⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢷⠏⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠶⠀⠀⠀⠈⠉⠀⠙⠀⠀⠀⠀⠀⠀⠀⢀⡰⠧⠄⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠚⣛⣣⣤⣤⣤⣤⠶⠀⠀⢀⢹⣦⠀⠀⢀⣀⣀⠀⠀⣤⠂⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠻⣿⠛⣉⣽⠿⠉⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣷⠾⣿⡄⠀⠀⠀⠀⣿⡥⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠟⠀⠛⠀⡟⠉⠁⢀⠀⠀⠀⠀⣸⣿⣥⠀⢈⡛⣿⢋⡠⠀⢨⣿⠀⠀⠀⢰⣿⠉⣵⠘⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣧⠀⣴⠀⠀⠸⣷⠀⠀⣦⠀⠀⣰⣿⣿⡧⣠⣿⡄⠈⢩⣥⡄⢰⣟⠁⠀⠀⠘⠛⠐⠿⠏⠀⠀⢀⡃⣀
⠀⠀⠀⠀⠀⠀⣴⡾⢾⡿⠟⣿⣿⠾⠿⠄⠀⠀⠀⠀⢀⣿⠀⠀⠋⣿⣿⣶⣬⠟⠗⠀⠚⠯⢤⣤⠀⠀⠀⠀⠘⠛⢛⣿⣶⣶⠶⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠤⠞⠃⠀⠀⠀⠛⠿⣛⠗⠀⠈⠯⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡾⣿⣿⡇⠀⠈⠻⠇
⠀⠀⠀⠀⠀⢀⡀⠃⢀⠘⠀⠀⠁⢐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀
⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠺⣧⠴
⠀⠀⠀⠀⠀⠰⠆⠀⠠⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢠⡔⠁⠈⣿⣇⠀⠀⠀⠀⠀⠀⣀⢀⠉⠊⣷⠋⣀⣠⡄
⠀⠀⠀⠀⠀⢠⡄⠁⢠⠈⠁⠂⠀⠀⠈⠁⠀⠀⠀⠐⣧⣶⣄⠀⠻⠁⠀⠀⠀⠀⠀⠈⢀⡀⠤⠈⢉⠛⠛⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⡛⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡿⠎⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⡉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠰⣆⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠋⠉⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢀⠛⢿⣧⠀⠀⠀⠘⣧⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢼⣷⠼⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣼⣆⠀⢀⣮⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣀⣀⣠⣥⠴⣦⠀⠀⠀⢀⣤⣀⣀⣀⡐⠶⠶⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠘⠋⢹⣿⣾⡷⢶⡶⠀⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠈⠻⣿⣿⡄⠀⠀⠀⠀⠀⠈⠙⢻⣿⠛⠛⢷⣶⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠾⠡⢴⣆⠹⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠖⠀⣿⣿⣿⣆⢀⡀⠀⠀⠀⠀⡠⠛⠛⡇⠺⡇⠰⣿⣶⡾⠃⠀⠀⠀⠀⠀⢀⠹⠆⠁⠀⣶⣴⠆⠁⡠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠋⠙⠀⠙⠀⠀⢀⡜⠀⢰⣾⠁⠀⢡⡄⢈⣿⣅⠀⠁⠀⠀⠀⠀⠀⠀⣄⠁⠀⢈⣋⡀⠉⠣⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⠀⠀⠸⢷⣼⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⡀⠜⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣻⡤⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⢠⠀⠀⠀⠈⠉⠁⠀⠋⠀⠙⠁⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠟⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠃⠀⢀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡃⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠦⠀⠀⢤⣀⡴⢋⠘⣡⡟⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣾⢁⣿⠀⢻⡁⠀⣴⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠈⠃⡘⠲⣶⣖⡄⠀⠀⠠⢤⠀⠀⠀⠀⣧⡘⠛⠀⠈⣶⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⣦⣤⣼⣿⡀⠠⠀⠐⠶⠀⠀⠀⣸⡧⠀⠀⣸⡿⠟⣽⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⣿⣿⢡⣽⡇⠸⡆⠀⠀⠰⠾⠿⠿⠷⠶⠚⠋⠀⠰⡿⢀⣦⢄⣱⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡟⢱⢹⣿⠃⣼⣿⣿⢆⡘⣤⣀⣀⡐⠐⠠⠂⠐⣀⣀⢱⣚⣿⣾⣻⠻⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢙⡀⠁⠀⣿⡗⣾⣿⣷⡌⢫⣿⡟⠻⣿⣿⡿⡟⣻⣿⡿⣿⣿⣯⣭⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣤⢾⣽⠇⠀⠈⠉⢠⣾⣷⠋⠈⢛⡙⠿⠙⢃⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠋⠄⠀⠀⣤⡌⣿⠃⣀⠀⠈⢁⣠⡞⠙⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣭⣿⠛⠉⠀⠀⠀⠀⠁⠺⡻⠁⠀⢻⣿⠊⣿⣿⡀⠂⢠⡷⠛⠐⠀⠀⠀⠀⠀⢀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⡿⠃⠀⠀⡄⢀⠀⠀⠀⠀⠺⠁⠀⠐⠈⠁⠀⢸⣿⣇⠈⠯⠀⠀⠠⡀⢀⣀⣤⣴⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣦⣀⠘⠃⠊⢀⠀⠀⢄⣶⣶⢀⣀⣴⣶⡄⠈⠛⠉⠀⠀⠀⢀⣠⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⡆⢸⡿⠋⠀⠀⠀⠀⠀⠣⠸⣿⣿⣸⣿⣿⡟⠁⠀⠀⠀⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠙⠋⠀⠀⠀⢰⣷⣶⠄⠀⠀⠀⠀⠹⣿⣿⣿⣿⠃⡀⢰⣄⣴⢿⣿⡅⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠛⠃⠈⠀⠀⠀⠀⠀⠈⠻⠿⠛⠀⠀⡙⠋⠘⠿⠟⠃⣠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⢻⣿⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⡿⠋⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠁⠺⠁⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡀⠐⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠦⠶⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⢀⡀
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⠿⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⢠⠃⠈⠀⠀⠀⣽⠃⠀⠀⠀⠀⠀⠀⠲⣤⣿⣇⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣅
⠀⠀⠀⢀⡄⣠⠁⠀⢠⡾⢁⡆⠀⠀⠀⠀⠀⠀⢀⣺⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡜
⠀⠀⠀⡻⠸⠁⠀⠀⡻⠄⢊⣴⣧⠀⠀⠀⠀⠀⣸⣿⠛⠀⠠⣄⠀⠀⠀⠠⢤⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠐⡆⠐⠂⡀⣰⣿⣿⠟⠀⠀⠀⠀⠀⠈⠱⠀⠀⠀⠈⠀⠀⠀⠀⣿⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆
⠀⠀⠀⠀⠀⣷⡆⠨⡐⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠁⡠⠐⣆⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠉⠹⡇
⠀⠀⡆⠄⢠⣿⡼⣷⣿⠏⠀⠀⠀⠀⢀⣀⣷⡦⠀⠄⠀⠰⢸⣆⠁⠀⠀⢢⡈⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⠀⠇
⠀⠀⠀⠀⠀⣩⣾⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⠁⠀⣾⣤⣤⠀⠀⠀⠷⣱⡄⠀⠁
⠀⠀⠀⠀⢠⢹⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣭⡥⡀⠀⠀⢀⡏⣿⠋⠀⠀⠀⠈⠻⣷⠀⠀⡀⠀⠀⠠⠀⠀⠀⠢⠄⣀⣸⣆⠂
⠀⠀⠀⠀⠀⠃⠠⣹⣧⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⠀⠀⢸⠁⡇⠀⠀⠉⠀⠀⡈⠛⠳⠾⣷⣴⡆⠀⠓⠀⠀⠀⠀⠀⢻⠛
⠀⠀⠀⠀⠀⠀⠀⠄⠋⠀⠀⠀⠀⠉⠉⠾⡟⣿⡟⣧⣀⡀⢀⣿⠾⡐⠆⠀⢠⢧⠀⠀⠀⠀⠈⠉⠐⠂⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⣤⣀⣄⢀⢚⣴⣿⣷⣿⣿⣿⣿⡿⠶⠁⠀⠠⠀⢀⠁
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠴⠒⠋
⠀⠀⠀⠀⠀⠀⣦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠛
⠀⠀⠀⢀⠀⠀⡘⠗⠀⠋⠓⠀⣀⠀⠐⠀⣀⣸⠂⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣧⣤⣤⣀
⠀⠀⠀⠈⠀⠀⠀⠴⡺⣀⠀⠀⠀⡄⠀⠚⠿⡋⠛⡛⢹⣿⣿⣿⣿⣿⣀⠀⡀⠀⠰⠟⠻⣿⣿⣿⣿⣿⣻⡗⡉⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠑⣹⣴⡆⠘⠁⠀⠉⠢⣍⠠⣀⢸⣿⣿⣿⣿⣯⣽⢴⠏⠀⠀⠀⠀⠀⠉⢹⣉⡈⠉⠈⠱⡀⠙⢨⡄
⠀⠀⡰⠀⢰⣆⠀⠈⢷⠘⣿⡿⠀⠀⠀⠀⠀⠙⣦⠻⢾⣿⣿⣿⣿⡿⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠘⣦⡀
⠀⠀⠃⠀⣦⡉⠀⢠⣀⠀⢹⣯⢠⠀⠀⠀⠀⢰⡇⠀⠘⠻⣿⣧⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣤⢹⣆⣹⠿
⠀⠀⠀⢰⢻⠇⠀⠀⠙⢷⡌⢻⡆⢾⣀⠀⠀⡼⠃⡀⠀⠀⠈⣯⠻⡟⡣⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠈⢁⠀⠉⠁⣒⠛⢀
⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⢹⡏⡷⢸⣿⠏⠀⠀⢰⠁⠀⠀⠀⢸⣷⠆⠀⠂⠀⠀⠀⠀⢢⣴⣶⢶⣶⣶⠂⠀⡐⣧⣿⣿⣷⡄
⠀⠀⠀⠀⢰⣀⠀⠀⢰⣶⠉⢀⢳⠈⡽⠆⠀⢠⡟⠀⠀⠉⠀⠸⡿⠂⠀⠀⠀⠀⠀⠀⠀⠙⠇⠘⠛⡿⠀⠀⣾⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠾⣙⢆⠀⡆⠀⠁⢨⣿⣇⠃⠀⢠⠈⢁⠄⠀⠀⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣴⣼⣿⠟
⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠛⠋⠿⠀⠀⠀⣴⣿⠀⠀⠀⢠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠚⢋⣴⠿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣿⠃⠀⠀⢀⠶⣿⠀⠀⠀⡀⢀⣴⠂⠀⠾⠛⠁⠀⠲⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠠⠀⠀⢀⠀⢀⡸⠉⠀⠀⠀⠈⠀⠟⠂⠀⠰⢿⠿⠡⠴⠶⣤⠶⠅

⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⣼⡿⡇⢤⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢤⣿⣷⣦⢀⢠⣿⡤⠁⣦⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⠞⠁⠈⠷⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠌⠀⢼⣿⣿⣾⣶⣶⡄⠀⠘⣷⣄⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣄⠀⠄
⠀⠀⠀⠀⢀⡄⠡⠀⠀⢂⠔⠀⠀⠹⣿⣿⣿⡀⠀⠀⠀⣿⣿⡞⢧⡀⠀⠀⠀⢀⠠⠂⠉⠀⢻⣂
⠀⠀⠀⠀⢸⠄⡄⢰⣿⣷⠄⢀⢀⠀⣹⣿⣿⠁⠀⠀⠀⢻⣍⣡⡀⡗⠀⠀⢀⡸⣷⢀⡀⡆⠈⣿⡄
⠀⠀⠀⠀⢸⡀⢠⡎⣁⠱⡧⣤⣦⢀⠘⣿⣿⡞⠀⠀⢀⣼⣻⣾⣛⠀⣱⠏⠙⠓⣿⢧⣩⡥⣀⣿⠁⠀⠀⢶
⠀⠀⠀⠀⢘⣇⢸⠁⠷⣏⣁⠀⢻⡷⡤⠈⠍⠹⢇⣲⡍⠋⠀⠉⠡⣿⣿⣾⢯⣇⣿⡿⢿⠟⡿⢻⡄⠰⣌⣘⡷
⠀⠀⠀⠀⢸⣿⡦⠇⠀⠀⢀⣤⠀⠀⠀⠀⣀⠀⠐⠿⠿⢾⠌⠃⠰⠇⠈⠈⠟⠛⠧⠄⢀⠓⠂⠈⠗⠂⠈⢿⡁
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠶⠀⠀⢀⣀⣀⣄⡀⠀⠀⠈⢁⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⣿⣯⠉⠙⢻⡿⢿⣷⣶⡾⠿⠶⠄
⠀⠀⠀⠀⠀⠀⣰⣠⣀⣀⣰⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⠷⠐⠀⠀⠈⣩⣭
⠀⠀⠀⠀⠀⠐⠿⡿⠿⢿⣿⣿⢿⣿⠿⠶⢶⣶⣶⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢿⣤⣀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠈⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⠀⠐⠛⢶⣄⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠀⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠒⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠁⠷⠄⠀⣴⠀⣴⡿⢿⣆
⠀⠀⠀⠀⠀⠮⠄⣀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣶⠋⠀⢸⠀⠀⠀⢰⠋⣿⠟⣿⢿⣿⣷⣄
⠀⠀⠀⠀⡰⢃⣠⡄⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠆⢀⣄⣤⣤⣇⢸⣏⡒⢪⣿⣿⣿⣿⣗⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠬⠀⠀⠀⣀⠈⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠩⡇⠸⢦⣿⠇⠙⠏⠀⢼⢯⡃⣾⣻⣿⣿⣧⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠈⢠⣿⠀⠀⠀⡀⠀⢬⣱⣾⣿⡠⠀⡄⠀⠀⢻⠀⠈⠉⡄⠀⣷⠀⠹⢿⣷⠈⠻⣽⣿⣿⣟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠚⠃⠀⠀⠀⢽⠸⢿⣿⣿⣿⣶⣦⣽⡄⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⣠⣿⡆⠀⠈⢻⡿⣦
⠀⠀⠀⠀⠀⡀⠀⠁⠀⢠⡄⠀⠀⢤⠀⢰⡏⠄⢉⣿⣿⣿⣿⣿⣿⢁⢀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢿⢿⡿⡄⠀⠀⠡⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⣾⠁⠘⠀⠈⣿⣇⣸⡜⠁⠀⢠⠤⠀⠁⠀⠁⢀⣠⡾⠁⠈⠈⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢠⡌⠀⣀⠀⠀⠀⠈⠁⠀⠀⠄⢀⡀⠀⣿⡿⠋⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⢻⡦⠀⠀⠀⠀⠠⡄⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠒⠐⣀⠀⠀⠈⠁⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋⠀⠀⡞⠀⢲⣧⡸⣀⠀⢀⡀⣿⣇⠀⡀
⠀⠀⠀⠀⢰⡇⠀⢠⢀⣶⡄⠀⠀⠀⢠⣶⠀⠁⢀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⡸⢻⡄⠘⣿⣷⡄⠀⠀⠠⢸⣿⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠙⣃⡀⠀⢸⣷⡷⣾⡏⠀⣰⠋⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⡾⠀⠇⠀⠀⠀⠈⣿⣿⣦⣴⣤⠳⠚⠃
⠀⠀⠀⠀⡇⠀⢰⡄⠚⠃⠀⣾⣿⣿⡿⢀⣴⠃⠀⡜⠀⠀⠃⠀⠀⠀⢠⠋⠀⢰⠇⠀⠃⠀⠀⠀⠀⠸⣿⣿⠿⠀⢀⠁⢸⡆
⠀⠀⠀⠀⠁⠀⠛⠀⠀⢤⠤⢿⣿⠏⠀⣾⠃⠀⠀⠃⢀⣼⠂⡠⣤⣀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠘⣇⠀⠀⠙⣷⠊
⠀⠀⠀⠀⠀⢀⡀⢰⠀⠘⣴⠞⢿⣦⣼⠃⢀⠎⠀⠀⣺⠇⡰⠁⠘⠻⠛⠒⠒⠠⠴⠶⠖⠲⡦⢴⣶⣶⣶⣿⡄⠀⠀⠟⠛
⠀⠀⠀⠀⡆⡈⠠⠛⠀⣠⣿⡀⢀⣹⣿⡆⠈⠀⠀⣐⠋⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢷⠀⠀⠈⢻⣿⣇⣤⢀⣶⣄⡀
⠀⠀⠀⠀⠒⠀⠀⠀⠀⣅⡿⣀⡈⠉⠙⠛⢠⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠞⠀⣠⡌⠓⠀⠀⢈⣹⣿⡿⣿⡿⠝⡉
⠀⠀⠀⠀⢀⡀⠀⡀⠚⠋⠃⠀⠀⠀⠈⠐⠺⠿⣿⣷⣽⣿⣟⣿⣯⡿⣿⣿⣶⡶⣾⣿⣿⣿⣇⣰⣶⣿⣿⣿⣿⣞⣻⣿⣿⣧
⠀⠀⠀⠐⠛⣩⡟⣽⡿⢋⣽⣦⣤⣤⣶⣤⣀⣀⡈⠉⣿⡿⠛⣿⡟⠀⠙⢻⣿⡃⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠻⢿⣽⣿⡇
⠀⠀⠀⠐⠋⠉⡁⠉⠀⢸⡿⠛⡩⠟⠛⠉⠉⠉⠻⣟⠉⡀⠀⠈⠁⠀⠠⠏⢺⠇⠹⢿⣿⣿⢻⣷⣶⣴⡿⠟⠛⢛⢡⡌⢋⠙⠁
⠀⠀⠀⢠⡀⢀⠀⠀⠀⠀⡀⢀⣠⣶⣿⣿⣿⣷⣦⣀⠀⠈⠳⠀⠀⠠⠤⠠⠄⣠⣤⠶⠋⢰⢿⡷⣃⠀⠀⠠⢶⡇⢀⣄⢈⠛
⠀⠀⠀⠀⠀⠈⠓⠀⣠⡾⣛⣿⣿⣟⠉⠠⠌⠛⠛⠉⠉⠁⠐⠋⠉⣧⣶⣒⣨⣍⣤⣶⣿⣶⣯⡀⠀⢠⣤⡇⢼⣏⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⠋⢥⣶⣾⠿⠟⠁⠉⠀⠀⠻⣶⣶⠛⠈⠑⠂⠘⠉⠀⠈⠈⠉⠁⠀⡿⠸⠋⠀⠻⠄⢀⠖

⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠉⠉⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣠⣾⣿⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠿⠿⠶⢦⣤⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⡄⠀⠀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⢾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⣀⠀⠀⢠⡟⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠆⠀⣿⠀⠀⠀⠀⠀⠀⢲⠀⢀⡔⠴⠆⠀⠀⣿⠀⢠⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠋⠀⠈⠁⠀⠀⠀⣿⢠⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠻⠆⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠛⢿⣿⣻⣿⣿⣿⣿⡿⠋
⠀⠀⢠⣄⢀⣤⠀⣿⣿⣿⣿⡏⠹⠆⣠⣤⡻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⡁⣠⣄⢉⣉⣯⣯⠉⠀⠀⣠⠤⢤⣤⡄
⠀⠀⠸⢿⣾⠿⠺⢿⣿⣿⣿⣃⡀⠀⣿⣿⠇⠈⠻⢦⣄⣀⣀⣀⣀⣠⡴⠞⠁⠀⢷⣿⣿⡦⠀⣿⡇⢠⡴⠶⠧⣦⡀⡿⠁⣀⡀
⠀⠀⢰⣾⣿⣶⣶⣾⣿⣿⣿⣿⣿⣶⣾⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣷⣾⣿⣿⣶⣶⣿⣿⣶⣾⣿⣿⣾⣶⣿⣷⠀⢙⡷⣾⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣭⡤⡶⠋⣧⡈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡴⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠉⠀⢲⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠐⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⣈⠉⣠⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠘⠀⣤⠀⠀⠠⠀⠀⠉⡿⢛⣿
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⠻⣿⡄
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡇⣰⠞⣁⣿⣟
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠐⢶⣶⠂⠀⠠⡟⣽⣎⣼⣻⣿⣿⣄
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂⠀⡷⠐⠀⣿⡿⣦⠀⣸⣶⠟⠃⠺⣿⡿⣿⣿⣇
⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣗⢀⡎⠞⠀⠀⢠⣿⣧⠀⠀⠀⠀⢙⣿⣿⡿
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠁⠈⠈⠉⠉⠛⠷⣤⣀⠀⢻⣿⣯⡁⠠⠞⢙⣿⡿⣿⡇
⠀⠀⠀⠀⠈⠑⠀⢀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡏⠙⠳⣏⠉⠐⠬⠇
⠀⠀⠀⠀⠀⠀⠀⢤⣀⡈⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠤⠄⠒⠼⠿⠟⠓⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⡀⢰⣦⣤⠀⠀⢀⣀⠁⠀⠀⢀⣀⣀⡀⣀⣀⠀⣀⣀⣸⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣄⡀⠐⡠⢄⣴⠛⣉⣙⣿⣿⡿⠿⠛⠻⠿⣽⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠛⠛⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢘⠀⢸⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣷⣄⠀⠀⣘⠀⠘⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣿⣿⣿⣿⣆⠀⡿⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⠋⠠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠁⠀⠠⣤⣤⣠⣴⣶⠾⠛⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢻⠿⠟⠆⠸⠀⠀⠚⠀⠀⠀⣾⣿⣿⡿⣿⣿⣻⣿⣿⢿⣦⡀⠀⠻⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⡟⢟⣿⡧⠉⠟⠻⣿⣿⡁⠀⠙⢿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠖⠀⠀⡰⠆⠀⠀⠐⠒⠐⠲⣶⣾⣿⣿⣿⠭⠿⠓⠂⠉⠀⠈⠉⠓⠂⠀⠀⠹⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠪⠜⠒⢀⠐⡀⢀⠄⠀⠀⠀⠀⣼⢿⣿⣿⠁⠀⠀⠀⠀⠀⠂⠶⣄⣀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠄⠠⠐⠀⠀⠀⠀⠀⡏⠸⣿⠇⠀⠀⠀⠀⠀⠀⠂⢰⣮⣟⣁⣈⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⡌⠀⠀⠀⠀⠀⠶⠖⠀⠘⢿⠘⠅⠤⠛⠛⠒⠈⠄⡄
⠀⠀⠀⠀⠀⠠⠄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⡆⢸⠁⠀⠀⠀⢀⡄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠶⠽⣿⣾⣇⠸⠦⣶⠶⠟⣣⣴⣿⠻⢿⡿⢿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠼⣿⡇⠠⠾⢷⣾⣿⣿⡄⠀⠠⣳⣿⣿⣿⡿⣿⠛⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠈⡦⠀⢠⣤⣀⠀⠀⠀⠀⠘⢿⣆⡀⢰⣺⣿⠿⠖⠀⣦⣿⣟⢻⠟⠓⠘⢛⣛⣛⣿⡇⢘⣧⣙⡿⠛
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠘⡁⠀⠠⢤⡄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠛⠾⠉⠉⠉⠉⠉⠈⠉⠉⠉⠙⠛⠋⠉
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⡤⠀⣀⠀⠐⠒⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠉⠀⠀⠀⠀⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⢀⡴⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣠⠟⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣠⡾⠃⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢠⣾⢃⢀⣼⡏⠁⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢤⡋⢠⠟⡁⢀⣾⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢒⡄⠹⡟⠈⢠⣾⡵⠚⠋
⠀⢠⣄⣀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠓⢿⠇⠀⢁⣼⣿⠋
⠀⠺⠋⠉⢩⠍⠹⠟⢽⣽⠋⠉⠙⠓⡀⠀⠀⠀⣷⠀⠀⠀⢰⣿⣿⣇⠀⢠⡀
⠀⠀⠀⣰⡏⣶⣦⣶⡼⣿⠀⣄⠀⠨⣁⢀⣠⣤⣴⣁⣠⣾⠟⣉⣸⡟⠀⠀⠑⣄
⠀⠀⠀⢹⣷⣿⣿⡿⢹⣿⣇⠸⣇⠀⢈⢛⠛⠋⠙⡉⠉⠡⠂⠙⠛⠀⣰⣿⠾⠿⣁
⠀⢠⣁⡾⢩⠹⣿⠃⣿⣿⣿⣤⡙⣦⣤⣄⣀⣀⣀⣰⣀⣴⣦⣤⣴⢾⣷⢄⢀⣶⣧⣄
⠀⠀⢙⠁⠘⠀⠉⠀⣿⡗⣿⣿⣿⡎⢿⣿⡟⠻⣿⣿⠿⣿⡟⠛⠁⠘⣿⣾⣿⢿⣿⣿⡟
⠀⠀⠈⠀⠉⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠁⢼⣿⡿⠁⠀⢀⡍⠛⣠⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃⠀⢠⣶⣶⡼⣿⠃⣤⡄⢈⣠⣶⡏⠁⢛⣧⣄⣄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⢿⣻⣿⣿⡿⠋⠁⠀⠀⠀⢫⠿⣿⠃⠈⢿⣿⠋⣿⣿⡝⢷⣼⡟⣩⠍⠡⠀⠀⠀⣠⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠌⣴⣿⡏⠉⢁⣀⣀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠁⠀⢸⣿⣧⣈⠷⠈⠉⠰⣄⣀⣴⣾⡿⠟⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⡀⠀⠘⣿⡿⠀⠀⠀⠀⢠⣶⠄⢀⣴⣶⣄⠈⠟⠋⠀⠀⠀⣀⣴⣿⡿⠛⠋⠒⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣿⣿⠂⠉⠀⠀⠀⠀⠐⣯⣿⡄⣿⣿⣿⠟⠀⠀⠀⠀⣠⣾⡟⢻⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠛⠁⣠⣀⡀⠀⠀⠀⠀⢻⡿⣷⣿⣿⡇⠀⠄⣠⢀⡾⣿⡏⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⣿⠇⡀⠀⠀⠀⠀⠁⢿⣿⣿⠇⠀⠠⠿⠋⢾⡿⠿⣀⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠔⣿⣀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⢶⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠻⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⣀⠀⠀⣰⡾⠟⠁⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠏⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢘⣷⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡂⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣿⣤⣷⡀⠙⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠤⠶⠞⠉⠀⠀⠐⠂⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⡑⢮⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠘⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⢻⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣼⣿⣿⣿⣿⣶⣒⡛⠘⠠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠉⠁⢿⡿⠛⠛⠃⠀⠀⠈⢿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣦⣄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢠⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣦⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠷⣤⡤⠄⠀⢀⡀⠛⠛⠢⠀⠀⠈⠙⠻⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⠋⠀⠀⢠⣄⢀⡀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⢙⡋⠀⠀⠀⠀⣶⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠶⣶⣴⣾⣷⣤⣶⠄⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⡿⠟⠻⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠋⠀⠋⠁⡈⣣⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣦⣠⣤⡇⠘⠀⠀⠀⠀⠰⢤⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⢠⡄⠀⠀⢀⣀⠈⠀⠂⠀⠀⠀⠀⠀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣷⣜⠂⣼⠋⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⢀⠀⠀⣾⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣟⣿⣭⣿⣿⣧⠀⠀⠀⠈⢿⣿⣿⣿⡏⠀⠀⠀⠘⢷⣾⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⢻⣿⣿⣿⡄⠀⠀⠀⠀⡹⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠿⠤⠤⠄⠐⠚⠉⠉⠉⠉⠁⠉⠀⠈

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠘⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡄⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠇⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⣿⣿⣿⣆⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⢹⣿⣿⣿⡀⠀⠀⠀⢸⣿⣷⣷⣶⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢠⣿⣿⣿⣇⣠⣀⣠⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠃⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠉⢼⣿⣿⣷⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⣿⡿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠿⠿⠟⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣁⣤⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠈⠁⠀⠠⠄⠈⣤⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⣀⡀⠀⠀⢀⣭⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠹⠗⠀⠀⢸⡅⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⣷⣦⠀⠀⠀⠀⣾⠃⡞⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢰⣿⠇⠀⠀⡐⡅⣼⣧⣤⣴⡖⠀⢸⠀⠃⢰⣿⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣰⣿⣿⠀⢀⣴⣿⢀⣿⡟⠁⠙⠂⠀⢸⠀⡼⡝⠙⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⣿⣿⠃⠀⠈⣿⣿⣿⡟⠁⠀⠀⠀⠀⢊⡶⢥⠀⢤⡠⢉⣿⣆
⠀⠀⠀⠀⠀⠀⠈⠉⢐⡀⠀⠈⠛⠋⢁⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠐⠻⣀⣶⡿⡷⣴⣶⣿⡏⠂
⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⣿⣿⣟⡂⠀⠀⠀⠀⠀⣽⣿⡀⠀⠈⢻⠏⠃⠈
⠀⠀⠀⠀⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢸⣿⣿⣿⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠙⠉⣠⣿⡏⠁⢀⣠⣶⣶⣶⢆
⠀⢰⣶⣶⣦⡤⠴⠴⠶⠤⠴⠦⠶⠶⢶⡾⠉⢀⣾⣿⠋⠀⣠⣾⣿⣿⡟⠁
⠀⣰⠾⠛⠻⢶⣶⣶⣶⣤⣤⣤⣀⣀⡞⢡⣾⣿⣿⡿⠀⠀⢉⡿⠟⠁⠁⠀⠀⠠⡀
⠀⢠⠀⠀⣐⣭⣿⣽⣿⣿⠻⣏⠀⠀⠀⣸⣿⡿⡇⠀⠀⠀⠹⣆⠀⠀⠄⠀⠀⠀⠘⢄⠀⠀⣄⠀⢀
⠀⠛⠀⠀⠙⣼⡿⢃⣿⣿⠀⡹⡤⢊⡼⣿⡿⠁⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢦⠀⠁
⠀⠀⠀⠀⢠⣿⣧⢘⣾⣿⠀⡀⠇⣼⠟⠏⠇⠀⠘⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠘⡆⠀⠀⢣⡀
⠀⠀⠀⣾⣿⣟⣪⣿⣿⡏⠀⡇⠀⢸⣷⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣧⠀⠀⢃
⠀⠀⠸⣿⣴⣺⣿⣿⣾⠁⠀⠃⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣦⠀⠀⠱⠄
⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⡧⡁⣿⣿⠏⣹⣿⠗⠲⠶⣦⣤⣄⣀⣤⡄⠠⠤⠦⠄⣀⠀⠀⠈⢿⡦⠀⠀⡹⡄
⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠈⠃⠀⢿⡏⠀⣿⠇⠀⠀⠒⠀⠀⠀⣿⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠿⡂
⠀⢘⠰⣷⣾⣿⣿⣿⣿⠀⠀⡄⠀⠈⠁⠘⢟⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠸⢸⣿⢻⣿⣿⣿⣟⠀⠃⡇⠀⠀⠀⠂⠀⠀⠀⠈⠈⠁⢶⣶⣿⣧⣤⣶⣶⡆
⠀⠀⠀⠈⣸⣿⣿⣿⣟⠀⣿⡇⠀⠀⢸⡇⣰⢜⣦⡀⠀⠀⠸⣿⣿⣿⣿⠿⠿⠁⡀⠃
⠀⠀⠀⢠⡟⣿⣿⣿⣯⣰⣿⣿⠀⠐⠲⠓⠛⠛⠂⠁⠀⠀⠀⢻⣿⡏⠘⡆
⠀⠀⠀⠁⠠⢤⣤⣬⠟⠛⠻⣿⠀⠾⢶⣶⣦⣤⣀⡀⠀⠀⠀⢸⡿⣧⠀⢱⡀
⠀⢤⠀⠀⠀⠀⠀⠑⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠉⠛⠛⣷⣖⡉⣙⣿⣇⣨⡧⠀⡀
⠀⠀⠑⢠⠤⣤⣤⣀⡀⠀⢠⣿⢀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠟⠋⠉⠉⢙⣿⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠉⠓⠺⠯⣹⠿⢷⣺⡿⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡂⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢠⡀⢀⣦⣀⠀⠀⣸⡟⠫⡛⠃⠀⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡛⠙⣿⣿⣷⣦⣟⡋⠁⢻⣄⠀⠁⡟⠀⠀⠀⠀⠀⠀⠐⢶⣶⡋⠄⠀⠈⠑
⠀⠀⠀⠀⢐⠁⢀⢄⡀⠉⠛⣿⣿⣿⢫⠀⠀⠙⣷⣶⣧⡀⠀⠀⠀⠀⢀⡀⠟⠉⣷⡈⠀⠀⠀⠠
⠀⠀⠀⡇⡀⢠⣶⣎⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⣿⡛⡃⢹⠄⠀⠀⢸⣯⠀⠀⡄⢻⣷
⠀⠀⢰⡄⢁⡜⠋⢻⣆⣠⡑⡌⣿⣿⡽⠆⠀⢀⡼⣿⣿⣎⠐⡶⠚⠇⣾⢏⣓⡅⢘⡿⠂⠀⢠⡄
⠀⠀⠀⣇⢸⠁⢷⣞⡃⠸⣿⣤⡉⠝⠛⣇⣤⡞⠋⠉⠙⢸⣾⣿⣟⣆⣽⣿⣿⠿⡟⢷⠀⢶⡙⣷⠄
⠀⠀⠰⣿⡮⠇⠀⠀⢩⡄⠀⠀⠀⢀⡀⠸⠿⠿⣶⠘⠀⠿⠉⠋⠹⠛⠣⡄⢈⠳⠤⠸⠆⠀⠹⡏
⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠐⠷⠀⣀⣀⣀⣴⣀⣀⠀⡈⣡⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠟⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⣿⡉⠉⠻⠿⢿⣿⣷⠿⠿⠖
⠀⠀⠀⠀⣸⣴⣤⣤⣇⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠟⠙⠀⢀⠀⢬⠽
⠀⠀⠀⠈⠛⠻⠛⣻⣿⡿⠿⠟⠛⠛⢿⣿⢶⣶⣶⣶⣤⣤⣤⣀⣀⡀⢀⡀⠻⣶⡤⠂⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⣆⠃⠀⠀⠀⠰⣟⣇⠀⠀⠀⡀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⡆⠄⣆⠀⠀⠽⢿⡿⠟⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠚⠛⠊⠀⠀⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠠⠃⠀⠀⢀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠀⣰⢄⣀⠀⠀⠀⠐⡶⠾⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⣿⡿⡁⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢀⠂⠀⠀⠀⢂⡾⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡄⠀⠀⢴⣆⢀⣀⣀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠛⠻⠟⠛⢋⠀⠀⠀⠀⠠⡄⠀⠀⡀⠀⢀⠆⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⠀⠉⠀⠀⠀⠀⠘⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢀⠀⢄⠀⠀⠀⠀⠀⠞⠁⣀⠀⠘⠁⠃⠐⠃⢠⠎⠠⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⣷⣦⣄⣀⡄⣠⣤⣶⢇⣿⣇⣀⣠⣤⣀⣤⣧⠀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⠋⡩⠁⣹⠏⠀⠀⠀⠉⠿⢫⠉⠉⠉⠉⢁⡼⣻⠟⠋⠉⠀⠀⢀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠁⠉⠁⢀⡞⢠⠆⢀⢠⠁⠀⠀⠀⠀⠘⠛⠧⢼⣥⠀⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠓⠛⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠐⢰⠀⠀⠀⠠⢄⡀⠀⠀⢸⣿⣷⡞⣐⡀⡠⠴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢼⣂⣆⢶⣿⣿⣟⠀⡀⣻⠟⢻⣏⡛⠃⠀⢂⠇⣶⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢨⣴⣄
⠀⠀⠀⠀⠀⢊⣽⣿⣿⣿⣿⣷⣤⠲⢿⠟⠙⠈⣿⣌⠀⠘⠀⣟⠀⠀⠀⠃⠀⠀⠀⠀⠀⠻⣶⣞⢩⠄⠀⢆⠈⠉⠂
⠀⠀⠀⣠⠄⠠⠒⠉⠹⠿⣿⣿⣿⣿⣯⠾⡀⠀⠘⢿⣷⣄⣼⣿⡀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡟⠹⣧⡈⠉⠀⠀⠀⢈⡆
⠀⠀⠈⢵⠀⠀⢞⡴⠖⠑⠂⣿⣿⣿⣿⣶⠇⠀⠀⠀⢹⣿⣿⡟⠷⡄⠀⠀⠀⠀⣴⣼⠃⠀⠀⠀⣿⣯⣥⡤⠒⠉⠁
⠀⣆⢀⡄⢰⣿⣿⣿⡎⣉⠁⠃⢹⣿⣿⣿⣸⠀⠀⠀⠸⣿⣿⣷⠘⡷⢀⠀⢀⣀⡿⣿⣇⣁⣸⠆⢈⣿⣿⠀⠀⠀⣾⣄
⠀⣇⠈⠁⣼⠟⠻⣿⣁⢩⡌⢦⠙⣿⣿⣿⣿⡧⠀⠀⣠⡟⣿⣿⣿⣅⠘⣾⠛⠛⠻⣿⡯⣟⣿⠼⣐⣿⡏⣦⣻⣾⣿⣿
⠀⣿⠀⣺⠗⢿⣦⣿⡏⢹⣿⣖⠃⠛⠿⠿⣾⣆⣰⣾⣿⠏⠉⠛⠋⣶⣿⣧⣿⢿⣆⣿⣿⣿⣿⢿⣿⢻⣧⠹⣿⣟⢿⣿⡆
⠀⣿⣧⢻⡄⠛⣿⠿⠶⠀⠛⠿⠟⠛⠈⠑⠘⣛⣾⣿⣇⣆⢴⡆⢀⣿⠻⢿⠻⣿⣿⣟⢋⠿⢿⣾⣭⣑⣿⣤⠇⠙⣿⡿⡇
⠀⣿⣿⡿⠃⠀⠀⠀⣴⣷⡍⠄⠀⠀⢠⣶⡄⠋⠻⠙⠛⢿⣟⠃⠈⠉⠀⠈⠈⠉⢀⠈⠉⠁⠺⠤⠤⠹⠛⠓⠚⠃⠘⠗
⠀⣿⣿⠇⣶⡇⠀⠀⡆⠀⠀⠀⠉⠀⠈⠉⠋⠀⠀⠀⠀⠘⠟⠓⠀⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣼⣷⣎⣁⣤⣤⣤⣄
⠀⢻⡿⠀⡏⡇⠀⢠⣷⠆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣈⣉⣿⣿⣿⣄⠀⠙⡿⠿⣿⣿⣿⣿⡟⠛⠛⠓
⠀⠀⡏⢸⡇⡇⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠉⠛⣿⣿⢿⣿⣿⣿⣓⣶⣉⣹⣿⣿⣧⡊⠂
⠀⠀⢡⣾⣿⣿⣿⣿⣿⣶⣾⣷⣶⣶⣦⣤⣤⣤⣀⣀⣸⣀⣀⡀⠀⠀⠀⠉⠁⢸⡏⠀⠈⠻⣿⣤⣤⣽⣿⣯⣠⣦⡀
⠀⠀⡌⠉⠛⠿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣧⣴⣶⣶⣾⣿⣿⣿⣿⣽⣿⡝⠁
⠀⠈⠀⠀⠀⠀⠀⠉⠛⠛⠛⠙⠂⠚⠛⠛⠻⠟⠋⠛⠛⠻⠹⠛⠛⠛⠛⠛⠻⠿⠿⠿⠟⠻⠿⠿⠛⠛⠛⠛⠛⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣾⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠓⠀⠓⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⡄⠀⠀⠀⢦⢤⣤⣤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡕⠨⠀⠄⠀⣠⣿⡌⠀⢉⡧⢿⣦⠀⠄⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢀⠀⠀⠀⠀⢠⡋⠁⠀⠀⠠⣘⣧⠘⢫⣒⡻⢷⡈⣰⡆⠀⢠⣄⣴⣽⣿⡄⠡⢄⣤⣤⣲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣴⣶⣿⣶⡋⠀⠀⠀⠀⢐⣿⣿⣧⣀⣤⡄⢡⡍⠻⣎⣀⠉⢋⡟⠻⣿⣧⣾⣿⣾⠛⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢼⣿⣿⣿⣿⣿⣿⣿⠯⠁⠀⠀⡀⢀⣽⣿⣿⣿⣿⣿⠦⠉⠑⠮⣼⣿⣦⠾⠀⢀⡾⠛⠛⢛⣛⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢿⣿⣛⡛⠛⠇⠀⠒⠀⠉⠀⠀⡼⠟⠉⠉⠉⣀⣀⣠⣤⡴⢶⣿⣿⣧⠸⠋⠀⢺⡟⠉⣿⡖⢿⡆⢘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣈⠙⠲⣶⣄⠻⠦⠀⠸⣣⠂⡐⠀⠀⠈⡏⠀⠙⢿⠻⣿⣿⣿⠃⣶⣷⡿⠀⣼⡿⠁⣾⠃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡉⠙⠻⢶⣤⣽⣷⡀⠀⠴⣿⣶⡷⣤⣤⣀⣿⣀⣤⣼⣶⣿⣿⣿⣿⣿⡿⠁⣼⣿⣣⣾⣿⣷⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠙⠟⠿⣶⣦⣼⣿⣿⣿⠀⠂⠴⣿⢡⣿⡇⢸⡏⢹⣿⣿⣿⢻⣿⣿⢿⡟⣡⣾⣿⣿⣿⠟⣻⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⣻⡿⠒⣤⣌⣉⢻⣿⣿⣿⣷⣥⣶⠿⢸⣫⡁⢀⣁⠈⠙⡟⢻⣗⣿⣅⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣺⠁⠀⣠⠾⠟⠟⠛⠟⠛⠛⠛⠉⠁⠀⠉⠀⠀⠊⠁⠀⠈⠀⠀⠈⡉⠙⢉⣽⣻⠷⠾⡦⣼⣭⣽⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠉⠀⠀⠀⠀⠐⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⢀⣐⠀⠀⠀⠀⠀⠐⠂⠀⠂⠀⠀⠀⠠⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣈⣥⣴⠆⣝⠙⠀⢩⣄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⣴⣿⣿⣟⣪⣕⣶⣬⡁⠀⠀⣀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠎⣆⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⢿⣿⣿⣿⣿⠿⢿⣤⣶⣄⠶⢄⠺⠉⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣩⠇⣸⡇⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠶⢽⡻⢿⣿⣿⣶⣾⣿⣿⣶⣻⣿⣦⣤⣌⡠⠀⡀⠀⠀⠀⢠⣀⣴⣤⡾⢞⣡⠾⢹⡏⣰⣦⠾⢦⣶⣈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠑⠂⠭⢙⠛⡛⠻⠿⠷⠿⠿⠿⠿⠿⠷⠾⢛⣚⣛⣩⣭⡭⠴⠚⣋⡩⣂⣽⣿⡏⠀⠀⠀⠈⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢈⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢈⡅⠰⠶⣖⡻⣯⡷⢝⣫⣿⣿⣇⡀⠀⠀⠀⡸⢟⡇
⠀⠀⠀⠀⠀⠀⢀⢀⣤⣖⣘⣻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⠶⢬⣝⣲⣭⣶⣿⣯⣿⣿⡯⣫⣷⠞⢡⣾⠁
⠀⠀⠀⠀⡠⠂⠵⣗⡽⣌⢽⣷⣾⣧⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⠿⣟⣻⣿⣿⣷⣿⣿⣿⡿⠟⣋⣴⣾⣿⣷⣏⢂
⠀⠀⠀⡐⢰⠳⠺⡋⣾⠃⡸⡿⢻⣿⢆⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢁⣚⠻⢿⣛⣾⣿⣿⣟⣋⣥⣶⣿⣿⣥⢪⠁⠻⠀⡇⠃
⠀⠀⠀⡅⠁⠇⢇⡜⠊⠈⠈⠘⠓⠩⠻⣧⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣷⣾⣿⣿⣿⣿⣟⡿⠟⢻⡟⠾⠛⠛⡛⣁⠁⠀⡠⠁⠆
⠀⠀⠀⠱⡈⠆⠀⢀⣦⠀⠀⠀⠀⠀⠀⠈⠓⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⠚⡻⣭⣍⢺⣉⣿⣿⡿⢠⠄⠁⠀⠀⡀⠄⠀⠀⠀⠔⢁⠎
⠀⠀⠀⠀⠈⠢⡑⠤⣝⡣⡀⠀⠀⠀⠀⠀⠀⠈⠲⡅⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⣈⣃⣤⣹⢴⣿⠿⠃⠀⢀⠠⠄⠀⠀⠀⢀⠠⠊⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠠⣉⠂⠁⣀⠀⠀⠀⠀⠀⠀⠱⣈⠁⠀⠒⠒⠀⠐⠛⠂⠛⠙⠛⠋⠉⢁⣠⠞⠁⠂⠀⠀⠀⠀⠀⠠⠔⢊⡤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢤⣀⡐⠂⠀⠀⠀⠀⠀⡉⠓⠲⠶⠶⠤⠤⠤⠤⠶⠶⠒⠛⠉⠀⣀⠀⠀⠠⠀⠀⢀⡤⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠥⢀⣒⠂⠀⠤⠤⠀⠀⣉⣈⣀⣀⠀⠀⠈⠩⠤⠄⠐⣂⡀⠥⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠂⠐⠒⠂⠀⠀⠐⠒⠒⠀⠈⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⢀⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⣾⠋⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⡀⠀⣴⠟⠁⢀⣼⠋⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠂⡄⠈⢾⠃⠈⢠⣾⠥⠄
⠀⠀⠀⠀⠀⠀⠀⢿⡦⣤⡤⡤⣀⣀⣄⢀⠀⠀⠀⠀⠂⠀⠀⠀⠰⡇⠺⠧⠀⠸⢇⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡱⢀⠉⠛⢰⣦⠁⠀⠀⠈⣀⡀⠀⠀⠀⣿⣄⠀⠀⢠⣾⡿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⣷⣿⢿⣿⠀⠘⠀⠀⠉⢀⣠⣤⣶⣿⣁⣠⣴⠟⠁⠀⣾⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣼⣿⣿⠋⣾⣿⡇⠘⣇⠀⢠⠉⡉⡉⠉⢀⠉⠁⠀⠂⠀⠀⠀⠀⠀⠶⠖⠃
⠀⠀⠀⠀⠀⠀⠨⣴⠋⣼⠨⡿⠁⣼⡿⢿⣧⣤⡙⠷⣶⣤⣄⣄⣀⣨⣤⣴⣶⡆⣀⠀⠸⣷⠀⠀⣴⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣿⣇⣿⣿⣿⣷⣸⣾⣿⡉⠙⢿⠿⠛⣻⡟⠁⠉⠀⠀⣿⢾⣿⢑⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣯⢷⣿⠟⠀⠀⠀⠀⢾⣿⡟⠀⠀⠀⡀⠉⠀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠉⠀⠂⠐⠒⣶⣸⡟⠀⣤⡀⣀⣤⣴⡏⠀⠀⢙⡇⣀⣀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣿⣿⡿⣋⠁⠀⠀⠀⠀⠘⠘⢿⠏⠀⠀⢿⡿⠃⢿⣿⡇⠙⣄⡾⠋⠀⠈⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⡿⠁⠀⢠⠄⢀⡀⠀⠀⠀⡰⠃⠀⠀⠐⠀⠀⠀⠸⣿⣧⠀⠙⠀⠀⠀⠐⠀⣠⣴⣾⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⣤⣄⡀⠰⠆⠟⠀⠀⠀⠀⡰⡷⠄⠀⣠⣾⣷⡆⠀⠉⠁⠀⠀⠀⢀⣠⣾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢰⣶⣤⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣷⠐⣏⣿⣯⡟⠁⠀⠀⠀⠀⢀⣶⠟⠋⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠷⠃⠀⠈⠀⢰⡢⡄⠀⠀⠀⠀⠀⠸⣟⢶⣿⣿⡖⠀⠀⠀⠀⣀⢾⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠀⢀⣀⠀⠨⠷⠂⠀⠀⠀⠀⠀⠀⠈⠺⢿⡛⠛⠀⠀⠚⠋⠑⠿⠿⢛⠀⣴⣿⣦⡀⠀⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣶⣤⣤⣀⣀⣀⣀⡀⢀⠀⣀⣀⣀⣀⣤⣀⠀⠀⠀⠠⡀⠀⠉⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠋⠉⠉⠉⠉⠉⠛⠛⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠄⠈⢿⡿⠟⠠⢁⡀⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢃⣾⠃⠀⠀⢀⡰⠆⣾⣲⣶⣆⣶⣿⣿⣿⣿⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠠⠀⠠⠌⠀⠀⠲⣦⠀⢠⡾⠛⠁⢀⠞⠁⠀⠀⠂⢨⣌⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⣦⡈⠁⢀⡖⠉⠀⠀⠀⠀⠀⠀⠌⣥⣽⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⢰⡏⠰⠀⠘⣧⠀⠀⠀⢀⣀⣤⣤⣬⣿⣿⣿⣿⢿⡿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⢀⡀⠾⠟⠛⠣⠀⠀⣀⠻⣿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⡀⢀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠤⣤⢨⡼⠻⡯⠟⠛⠻⠙⠙⠋⠝⠛⠙⠻⠿⠿⠿⠟⠛⠛⠙⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⠤⠆⠀⠀⠀⠀⠴⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢶⣭⣼⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣬⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠻⢦⣄⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⣀⣼⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠶⠦⣤⣌⣉⣉⣙⣛⡛⠓⠒⠒⠛⠛⣛⣋⣉⣉⣩⣤⠴⠞⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣦⣄⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣯⣝⣓⡒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⣟⣿⡿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡟⣟⣿⢙⡖⠒⠒⠂⠉⠉⠉⡛⡛⣽⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⣿⠇⠟⠯⠘⠁⠀⠀⠀⠀⠀⠀⠅⠤⠟⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠶⠶⠶⠖⠒⠚⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢤⡀⣤⡘⢯⠻⡀⠀⡀⠒⢢⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⠀⠀⠒⠛⠺⠇⠘⣆⢻⡸⠹⠦⣠⠆⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣽⣿⣷⣶⣶⣶⠤⠨⠤⠥⠤⠤⠶⢶⡶⠚⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣿⣿⣿⣿⠢⣿⠐⠀⠀⠀⠀⠀⢸⢿⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣿⣿⣿⣿⣞⣿⠀⠀⠀⠀⠀⠀⢘⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠌⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡞⣼⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠏⣿⣿⣿⣟⣿⠀⠀⠀⠀⠀⠀⠃⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⢿⣿⣿⡇⠘⠀⠀⠀⠀⠀⠀⡇⣿⣤⢴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣶⣶⣦⠤⠤⢤⠤⠤⠴⢶⢶⠒⠛⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⣘⣯⣤⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣤⣤⡄⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⡀⠀⠀⣰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⠻⣿⣄⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⢻⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣄⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣯⣿⣿⣶⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣷⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⢻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠢⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠙⢿⡿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠲⣴⣦⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢹⠏⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⢹⣿⣿⡿⢿⡇⠀⠀⢠⡆⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠃⠐⠋⠀⠀⠀⠈⠓⠒⠓⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⢤⡟⣼⡔⣹⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠐⡟⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠽⣽
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣚⣿
⠃⠀⢀⣠⣤⣤⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣭⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⣹⣻
⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠶⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⣤⡀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣶⣍⣿⣿⣧⣩⣏⣆⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠐⢿⣿⣟⠈⠉⠈⣿⣿⡿⢠⠀⠀⢀⣶⣶⣞⣿⣼⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⢤⡀⠀⢾⣹⣻⡏⢛⣿⣻⣿⣉⣿⣿⣿⢻⣿⢽⣟⡿⣿⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⡇⠀⣥⣾⡿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣯⣿⣯⣟⣻⣿⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣟⣿⡿⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣿⣭⣿⣟⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⡇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣛⣙⣿⣿⣿⡻⠿⠿⠏⠹⠿⠿⠿⢿⡿⠿⠛⢻⣿⡿
⣿⣿⣿⣿⣿⣯⣼⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣖⣲⣷⣶⣶⣶⣶⣘⣖⡆⠄⠀⠉⢉⣇
⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠀⠒⠘⠋⣽⣿⣿⣿⡇⠀⠀⠀⢠⣄⢀⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢩⡿⣿⣥⣤⣤⣤⣌⣉⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⢡⠦⣄⠀⣠⣤⠀⠈⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣀⡠⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⡶⢾⠾⠿⠿⠿⠿⠟⠛⠛⠛⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠟⠛⠿⠿⠻⠿
⠀⠀⠀⠀⠀⣇⢸
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠄⠀⠀⠀⠤⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠜⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠒⠒⣾⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣈⠲⢍⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠛⢦⣬⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢀⡟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⠴⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⠿⠛⠋⠁⠀⠀⡀⡸
⠀⠀⠀⠀⢲⠀⠀⠀⠒⠂⢄⢀⢬⠢⡐⢄⠀⠀⠀⠀⠀⠈⠛⠲⠤⠤⣤⡤⢊⡵⠛⠁⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠑⢄⡉⠉⠂⢤⠙⣦⣅⣈⡒⣬⣖⡖⣰⣀⡀⠀⠀⣔⣚⣁⠔⠋⠀⠀⣀⡀⠤⣐⠬⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢐⢢⡤⣽⣿⣦⢜⣶⠚⣅⠭⠷⣉⠯⢭⣭⣭⣭⠭⠤⠭⠭⠁⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣽⣾⢻⣛⢯⡿⡯⢽⠵⣪⣞⢴⡒⣤⡟⡳⣧⢸⡄
⠀⠀⠀⠀⠀⠈⢁⣌⠁⢯⢀⣷⠶⡪⢷⣸⣟⣛⣻⣦⣜⣸⡿⠷⠿⠿⠟⠁⠲⠼
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠢⠤⠒⠚⠁⢱⠇⠎⢤⣺⡳⢖⣙⡃⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⡄⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡈⠂⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⡄⣸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠧⠸⣿⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⣯⣭⡲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⣐⣾⣛⡀
⠀⠀⠀⠠⠀⢀⠀⠀⠀⠄⠀⠁⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⡋⠁⠐⡆
⠀⠀⠀⠀⠑⢈⠀⢀⠀⠠⠀⠀⠘⠈⠉⣠⣶⣶⣶⣶⣾⣿⣿⠿⠿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣜⣤⡤⣦⣾⣿⣿⣿⠟⣻⣓⠀⠙⠖⢲⣶⣮⠙⣿⠇⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠂⠀⠀⠀⡄⠈⠀⡎⠀⠀⣰⣿⣿⣿⣿⣿⡛⠉⣿⠅⠀⠨⡟⠿⣿⡇⢸⠀⠀⠀⠞⡠⠀⠀⠈⠀⢀⢁⣐⣋⠉
⠀⠀⠀⠀⠀⠀⠘⡄⠀⡄⠀⢀⣿⣿⣿⣿⣿⣿⣿⣴⣧⢀⣤⢢⣶⣼⢻⢁⠇⣀⠌⠀⠛⣀⣠⠈⠉⠀⠃⠀⡀⢁⡤
⠀⠀⠀⠀⠀⠄⠀⠈⡄⠐⡀⢸⣿⣿⡿⠟⢉⣡⣈⠙⢿⣿⠏⠐⠙⠻⠁⣼⠖⠁⠔⠀⠈⡡⠀⢀⠔⠁⠀⠔⠀⠁
⠀⠀⠀⠀⢀⠀⢁⡤⣇⣀⠀⣿⡿⢋⣴⣶⣿⣿⣿⡆⢸⣷⣿⡞⠀⠀⢸⡇⠀⠘⡀⠂⠩⠖⠋⠁⠂⠀⠀⠀⠲⠶⠄
⠀⠀⠀⠀⠀⠐⠀⠀⢣⣿⣶⣿⣷⣿⣿⠿⢿⣿⣿⡇⣼⣿⣿⡿⠀⠀⣿⡇⠀⠈⠀⠒⢀⠀⠀⣀⠀⣀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠉⠀⠀⠈⡾⠟⢰⣿⣿⡟⢹⠀⠀⣿⠃⠀⠠⠀⠲⣅⢤⡄⠀⠀⠄⠉⠠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⢀⢠⠀⣋⣠⠀⠀⢀⣿⣿⣋⣀⣈⣇⠀⠩⠄⠀⠀⠀⠀⠈⠂⠈⠈⠀⠈⠑⠤
⠀⠀⠀⠀⠀⠀⠀⢀⣰⣬⣅⣬⣿⣿⣿⣿⠀⠀⣾⣿⣿⣛⣉⣩⠽⡀⠀⠢⠄⡀⣂⣠⣀⢀⣤⡀
⠀⠀⠀⠀⢠⣤⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠈⠉⠉⠉⠉⠀⠀⠀⢙⣤⣀⡠⢸⣿⣿⠿⣿⠻⠇⠂⢶⠤⠤
⠀⠀⠀⠀⠘⠛⠛⠛⠙⠉⠉⠛⠻⠿⠷⠶⠒⠛⠛⠛⠉⠙⠋⠉⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⢇⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸⠈⡆⠀⠀⠀⠀⠀⢠⠃⡆⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠸⡆⢣⠀⠀⠀⠀⠀⡎⢰⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢇⠘⡆⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠸⡄⡇⠀⠀⠀⡏⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡀⠀⠀⠀⠀⡇⣧⠀⠀⠀⡇⠀⢸⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢡⠀⠀⠀⠀⣇⢻⠀⠀⢰⡇⠀⢸⡄⠀⢀⠇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⣯⢸⠀⠀⢸⣿⣢⣸⡇⣄⡾⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣤⣟⣿⣛⡛⠲⠷⣿⣛⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⡋⠩⠋⠋⠄⠀⠉⠈⠢⡙⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠎⠀⠀⡀⠀⠀⠀⠀⠀⠐⣴⣦⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⣿⠨⠂⠀⠀⣤⠀⠀⠀⠘⢼⣿⡾⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠓⠬⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⠀⠀⣃⡀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠒⡀⡀⡖⠠⠀⠙⠮⣍⡴⠟⠀⠀⢠⣄⡀⠠⠤⢴⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢿⣿⣶⣶⣿⠐⠒⠄⠀⣈⡀⠀⠀⠀⠘⠂⠁⠍⢭⡯⠉⡉⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠙⠻⢿⣿⣷⣖⠒⠊⠀⠈⠙⠒⣋⣾⣯⡾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠚⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣀⡠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢐⣶⠀⣾⣿⡂⠀⢀⣿⠻⣗⠶⠂⠀⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⠐⠄
⠀⠀⠀⠀⠀⠈⢿⢧⣿⣿⣯⣠⢸⡷⠳⠙⣷⡀⠀⠃⣿⠀⠀⠀⠀⠀⠀⠀⠰⣶⣖⠁
⠀⠀⠀⣠⠄⠀⠊⠉⠿⢿⣿⣿⣿⡿⠂⠀⠘⢿⣢⣴⣷⠁⠀⠀⠀⠀⠀⢀⠀⠿⠛⣷⡀
⠀⠀⠂⠀⠀⣀⣠⡞⠁⠁⠸⣿⣿⣤⡇⠀⠀⠀⣿⢿⡯⠹⡆⠀⠀⠀⠀⢪⠀⠀⡐⢻⣿⡀
⠀⢀⡇⠀⠘⣿⣿⣟⠠⠄⠀⢾⣿⣿⣣⠀⠀⠀⢹⣷⣶⣬⠃⣄⡤⢶⣦⣧⠐⣦⡇⠘⢿⠃
⠀⠈⣇⠠⣾⣥⣜⣿⠲⢷⡄⠄⢿⣿⣯⡅⢀⡴⣿⠞⠿⠞⣄⢾⣣⠤⠈⣿⣾⣿⣷⣶⢷⡔⣦⡆⢠⣦
⠀⠀⣿⡘⣇⢚⣿⢷⡄⠘⠿⠶⠂⠀⠀⢷⣿⣿⣡⢀⡄⢠⣟⠹⡿⣿⡷⡟⡩⢙⣴⣉⡚⣧⡤⠙⢿⠇
⠀⠀⢸⣿⠞⠀⠀⠀⠾⠀⠀⠀⠀⢴⣆⠈⠛⠙⠻⣦⡃⠈⠉⠀⠀⡁⠀⠈⠁⠐⠦⠆⠟⠋⠐⠂⠘⠂
⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠤⣴⣶⣿⠶⠦⣤⣤⣾⣿⣤⣤⣤⣤⣄
⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟⣥⡀⠀⠀⠀⠉⠛⡽⠋⠀⠉
⠀⠀⠀⣠⣷⣾⣤⣤⣦⣀⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠹⠁
⠀⠀⠀⠈⠉⠻⢹⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣽⣏⣿⣎⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣭⣭⣭⣽⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠉⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣠⡤⠤⢤⣄⣀⣀⡏⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣇⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣿⠈⠳⣄
⠘⢦⣀⣀⣀⡠⠴⠚⠁⢠⠃⠀⠀⠀⠀⣴⣏⣉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣿⡆⠀⠀⠀⠀⢻⠀⠀⠈⠓⠦⣄⣀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠛⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⣤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠠⣤⠤⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢆⠀⠀⠀⠀⠀⠘⡆⠀⢠⠁⠀⠀⠀⠀⠀⠠⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠘⢄⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⣠⠃⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⠴⠒⠒⢬⡑⠦⢄⣀⠼⠁⠀⠈⢦⣀⣠⠤⢚⡥⠒⠒⠦⡀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠰⠁⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠈⣆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⣀⣀⠤⠶⠷⠶⠶⠶⠶⠶⠞⠦⠄⣀⣀⣀⡀⠤⠚⠁

⣤⣤⣄⣀
⣿⣿⣿⡿⣿⣶⣦⣤⣀⡀⠀⠀⠀⣀⣠⣤⢶⣶⢿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⣶⣿⣭⣿⠩⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣷⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⡏⢽⠮⢩⣴⣶⣶⣶⣶⣶⣿⡿⣿⣿⣶⣤⣀
⣍⣛⡛⣿⣿⠿⣻⢻⣿⣿⣿⣿⣿⠽⢿⣏⣫⣗⣤⢤⡾⠻⠾⣿⣿⣿⣿⣼⣿⣿⣷⠘⡿⣿⣿⣿⣷⣤
⣿⣿⣿⣿⣿⣿⡾⣿⢻⣿⣿⣻⠧⢉⡥⢞⣝⣔⡿⣿⣶⣶⡜⢿⣯⣟⠿⠿⠛⠍⠛⠤⣼⣿⣿⣿⣿⣿⣷⣤⡀
⣩⡭⠶⠿⠒⢺⣵⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣻⣿⣿⣿⣿⣯⣐⣍⣉⣋⣇⣀⣴⣷⣿⣿⣻⣻⣻⡟⣻⡿⢟⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣾⡻⡿⢿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡗⢠⠙⠻⣿⡿
⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣽⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣏⠞⡠⠰⠀⠀⡸⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⣿⣿⣿⣻⡟⠉⢉⡁⣠⣴⡟⣿⣷⣉⡯⣴⣠⢃⣂⡌⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣋⣞⠀⠈⠁⠉⠛⠷⠒⣟⢰⠟⠚⠹⣻⣿⣿⣿⣿⣾⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣶⠺⢶⠄⡐⠁⠀⠀⠀⠀⠀⣠⡷⠏⠁⠁⡈⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣶⣳⢺⣶⢦⣴⣾⡷⠟⠰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠌⠿⣿⣿⣿⡿⡿⣣⠟⠛⠿⡏⣿
⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⢸⣅⣙⣮⠑⠒⠺⡁⠁⢀⡙⣏
⣭⣿⣿⣶⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⠁⠈⠤⠄⠀⣀⣵⡇
⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠐⣳⠀⠀⠀⠀⠀⣌⡻⠁
⣿⣷⣿⢙⡟⣿⠿⠛⠻⠟⣿⢯⡻⣿⣿⣿⣟⡷⣮⢆⡀⠀⠀⣾⡿⠃
⣿⣿⣏⣞⣩⠿⢦⢭⡄⣡⠀⢀⣿⠑⢿⣿⣿⣧⠀⢦⠄⠆⣾⣿
⣿⣿⣿⢻⢜⢷⡄⠆⠂⢈⡘⣆⠉⠃⠸⣿⣿⡟⡛⠃⢀⣴⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⣌⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢹⣷⡘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠀⠀⠛⠀⠸⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡤⠐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⣄⠀⠀⠀⢠⡇⢹⡄⠀⠀⠀⠀⠀⠀⣠⡶⣩⡾⠋⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠙⠂⠀⠀⢸⡇⢸⣧⠀⠀⠀⠀⣀⠞⠙⢠⡟⢰⡶⢃⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠠⣄⢀⢸⠀⣼⣿⠀⠀⢠⡾⠛⠀⠀⣾⣇⣼⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢻⣿⠟⢀⣿⣿⠀⢀⣾⣷⠀⡄⢸⣿⡿⠃⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⡏⢠⢾⣷⣿⣀⣾⣯⣬⣶⠀⣿⠏⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠀⠠⠖⠀⣀⢠⣄⠀⠈⠉⠛⣤⠊⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡔⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠞⢠⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢿⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠁⠀⠀⠐⡄⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢿⢿⡇⢠⠀⠀⣀⣠⠀⠀⠂⢌⠀⢀⠀⣆⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠪⣿⠞⠁⣌⠁⢼⣿⠋⢀⡰⢿⠈⣠⣤⡀⠈⠂⠈⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠅⠀⠐⣄⣠⠄⢸⣷⣄⢙⠀⠈⠁⠀⠀⢸⠋⠀⠀⠀⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠘⠀⠈⠀⠘⠛⠟⠻⠷⢦⣄⣤⡄⠞⢂⣠⣤⠶⣛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡖⡀⠀⠀⠀⠀⣨⣦⡄⠀⠀⠀⠀⠀⠈⡽⢿⠟⠻⠛⡉⠀⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡀⠁⠀⢀⡄⠀⠈⠉⠁⠀⠀⠘⠀⠀⠀⠐⢠⠀⠈⠀⠀⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⡖⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠢⣿⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣸⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣢⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠡⠀⡠⠁⣼⡿⣿⡆⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠀⠀⣤⢟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡉⢀⠈⠁⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⢀⣿⡏⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣌⠀⠈⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠀⢸⣿⠀⠠⢿⠁⠄⠠⡇⠀⠀⢀⠀⠀⠀⠀⡀⠙⠀⢠⢀⣾⠃⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣼⣿⠀⠀⠀⠀⠈⢠⣠⣤⢿⣿⡀⠀⠀⠈⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠹⣜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⢂⠀⠀⠀⢀⣾⣿⠛⣤⣿⣿⣆⠀⠀⠠⠀⠀⣼⣿⠇⠀⠀⡀⠀⢤⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⡛⠶⣿⣿⠈⠀⠀⠀⢸⣿⠇⠚⠿⠟⢿⣷⡄⠀⠀⡀⢀⣿⣿⡆⠀⠛⠿⠒⠾⠿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⣞⡍⣠⣾⣇⣀⠀⠘⠂⢸⣿⣿⣷⣶⣶⣾⣹⡇⠀⠀⠁⢻⣿⣿⣦⣠⣴⠟⢣⠀⠀⠈⣷
⡐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠰⠚⢿⣟⣱⡿⠛⠁⠋⠀⢀⣾⠛⠋⠙⠋⠛⠛⢿⣿⠐⠺⠉⢳⡿⣟⠛⠿⣯⣄⣨⣇⣠⣦⣙⣴⡄
⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠷⠶⠶⠶⠿⠏⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠓⠀⠙⠛⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠘⠀⠀⠀⠐⢀
⠀⢹⠂⡄⠀⠈⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠒⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠠⠄⠀⠀⣀⣀⠀⢁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣀⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⡠⠤⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠔⠉⠁⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢈⠀⠤⡁⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠰⠊⠉⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣌⡒⠂⠤⢄⡀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠪⠐⠠⠔⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⡶⢂⠚⡃⠈⠖⣠⣀⢲⠶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠉⣀⣠⣭⣧⣼⣄⣧⣌⣼⣾⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⡴⣼⡉⢦⣤⣙⣻⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠟⢛⣛⣯⣵⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⣟⠀⠘⣇⢸⣿⠯⠍⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡙⢄⠀⢻⡄⢻⡄⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣑⡂⠿⣄⠛⢆⢠⡻⢿⣧⡀⠀⠀⠀⠀⠀⠀⣰⣟⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠾⣷⣌⣉⢡⣼⣿⣷⣄⠀⠀⠀⢀⡴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⡶⠶⠚⢭⣿⣿⣭⣭⣛⣛⣛⣛⣺⣛⣛⣼⣓⣂⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣰⣿⣿⣿⣿⡃⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⠿⠿⠛⣛⣿⣿⣿⠖⠈⡂
⠀⠀⠲⢼⣿⣿⣶⣶⣽⠿⣿⠿⣿⣿⣦⣤⣉⣙⣛⠛⠛⠛⠛⠒⠶⠆⠀⠀⠀⠀⠛⢉⣉⣡⣴⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⢤⣄⣒⠀⠁⠉⠉⠉⠛⠛⠛⠿⠿⠷⠶⠿⠶⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠢⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢳⢦⣤⣤⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣭⣤⡽⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣟⠁⠲⣾⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀
⠀⠀⠀⠀⠀⢠⡔⠂⠀⢀⣀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⢀⢺⣶⣶⣾⣿⣿⠿⣿⣿⣿⣿⣭⣭⣽⣿⣿⣿⣿⣿⣷⣶⣶⣿⣥⣴⣾⣯⣭⣯⡙⠿⠿⣿⣷⣦⣄⠀⡸⢀⡆⠀⢀⡠⠤⠄⠠⣄
⠀⠀⠀⠀⢸⣦⠙⠿⣟⣩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣯⣿⣿⠟⣡⣾⣧⣶⢛⡀⠀⠀⠠⢤⣷
⠀⠀⠀⠀⠘⠿⠀⠂⠚⠙⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣫⣴⢞⣿⣿⣿⣷⡏⠁⠀⠀⠀⠀⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢛⣛⣛⡛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠍⠡⠖⠀⡰⠊⣿⣿⣿⠍⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢿⣿⣿⢁⣿⠛⠋⠀⠀⠀⠀⠀⡄⢀⠆
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⢿⣿⢻⡏⠀⠀⠀⠀⢀⡠⠊⡠
⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⡟⠀⠀⣀⠠⠖⢋⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⠋⣁⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⣾⣼⠿⠟⡫⡶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣶⣤⣠⣤⣤⣄⣤⣤⣤⣤⣴⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠁⠈⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠈⣿⣿⣿⢇⠀⠀⠀⠀⣠⣾⡟⢡⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠃⢾⣿⡏⢸⠀⠀⢀⣼⣿⡿⠁⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢸⣿⡇⣸⡆⠀⣼⣿⣿⡇⡞⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⣿⢡⣿⣇⢀⣿⣿⣿⠓⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢛⣛⣫⣭⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⣍⠉⡘⢿⣿⣿⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢇⣿⠀⡱⣟⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣏⠀⣥⣤⣴⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⣛⣯⠙⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣩⣿⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣂⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⠾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⠛⣿⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿⣆⣹⣿⡿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠰⡶⣾⣿⣿⠁⠀⠈⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡛⠙⠏⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠆
⠀⠀⠀⠀⠀⠰⣿⣿⡍⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣷⣻⣿⣷⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⢀⣿⣿⡧⣀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡞⣩⠆⣥⡤⢠⣤⣄⣄⣀⣀⠀⡘⠃
⠀⠀⠀⠀⠘⠂⣿⣿⣿⣛⣃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣟⣿⣿⣿⣿⣿⣿⡟⠙⣧⡀⢀⣠⠂
⠀⠀⠀⠐⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠙⣿⣿⣿⣿⡟⠋⠈⢸⣿⣿⣿⣿⣧⣶⣏⠆
⠀⠀⠀⠀⠈⠿⠿⢿⣿⣿⡟⠙⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠘⣿⣿⣿⣿⣿⣀⡸⣿⣿⣿⣿⣿⢟⣬⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⡿⠷⣿⣶⣿⣿⣿⣿⡿⣿⣿⣿⡿⠇⠀⠶⢩⣿⡟⣿⣿⣿⣷⣿⡟⢿⡿⠿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠉⠁⠀⠀⠉⠁⠀⠀⠐⠃⠈⠑⠁⠈⠉⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠀⢀⡴⠃⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⣠⡶⠚⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⡾⠃⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠸⣧⡀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠈⣿⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢠⡾⠋⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⠞⣀⣠⣤⣴⣶⣶⣾⣟⣛⣛⣛⣛⣛⣛⣓⡒⠶⢦⣤⣀⡀⠲⣄⠙⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⡾⠃⣿⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⣠⡟⠛⢲⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⠃⠀⢀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⠻⠿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠿⢻⠏⠀⣠⠞⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⢠⣯⡤⠞⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⢷⡄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⣠⠟⢉⡟⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠛⠛⠻⣅
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢰⡏⢀⣞⡴⠟⢛⣋⣉⣿⣶⠀⠀⠀⢀⣀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢸⣄⡞⢹⣿⣯⣭⣭⠴⢞⡟⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣝⣛⡛⠛⠦⣤⣤⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠶⠶⠦⠤⠤⠤⠴⠶⠶⠶⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣷⣄⡀⠀⠀⠀⠀⠀⢶⣤⣦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⢻⣿⣿⣶⣶⣶⣿⣿⣿⣿⣷⣶⡖⠀⣀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣴⣦⠀⠀⠀⠀⢴⣠⣽⣿⣿⡿⣿⡿⣯⣾⣿⣍⣩⣦⣤⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⢾⣿⣿⣿⣀⣀⣠⣿⠇⣿⣿⣿⠟⣫⣷⠸⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⡿⣿⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠦⣤⣤⣤⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⠃⣼⣿⣿⣆⣺⣿⣿⣿⣿⣿⠉⢹⣿⠋⠀⢸⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠁⠄⠉⠡⢸⣿⣿⠿⣿⡇⢸⡛⣿⣿⣿⣅⢸⡟⢁⣿⣿⣦⣾⣿⣇⡀⢼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⢸⣿⡟⠀⠿⠀⠘⠃⠈⠙⠛⠈⠙⢿⣷⣾⣿⣿⣛⡛⠛⣛⣿⣛⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢄⣀⡀⠸⣿⣿⡷⠀⢀⣴⣶⣿⣿⣇⢻⠿⣿⣏⡛⠛⢿⣿⣝⣻⣯⣿⣿⣿⡿⡿⠶
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⠀⠀⠉⠁⠀⣰⣿⣿⣿⠛⢿⡿⠇⢷⡄⠈⠁⠐⠀⢠⡿⠿⢿⣿⣿⣶⣶⣴⣶⡄
⠀⠘⠒⠀⠀⠀⠀⢠⣤⣾⣿⡷⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣷⣄⡀⠀⠀⠙⠆⡠⢄⠀⢻⡀⠀⠀⠀⣀⠿⣿⠷⠎⠙⠇
⠀⠀⠀⠀⢲⣴⣦⣤⣿⣿⣌⠃⠀⠀⣾⣿⣿⣿⣟⡛⢿⣿⣯⠙⠻⠇⠒⠦⣀⠀⠀⣀⣀⣁⡀⠤⠀⠀⠠⠁⠒⢀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⢀⣾⣿⣇⠼⠿⣿⣿⡛⠛⠛⠣⢦⣄⠀⠀⠈⠑⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢠
⠀⠀⠀⠘⡟⢹⣿⡟⠉⠀⠀⠀⢸⣿⣿⠿⡀⠀⠀⠉⠁⠀⠀⠀⣬⠉⢣⢄⡀
⠀⠀⠀⠾⣿⡏⠀⠁⠀⠀⠀⠀⢸⣿⣿⡗⠈
⠀⠀⢀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣘⠀⢀
⠀⠸⢸⡄⠘⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡉⠀⠈⠀⠀⠀⠀⠀⠰⡆⠀⢰⡆⠀⠀⣶⠀⡔⠀⠐⠀⡆⠀⠀⠖⢴⡀⠂
⠀⠐⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⢠⠃⠻⡄⢸⡇⠀⠀⣿⠘⣇⠀⠀⠀⣏⠈⠀⠀⢀⠙⡷
⠀⠀⣿⡇⠀⠀⠀⠀⠀⢤⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠈⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠘⣿⣟⠀⠀⢀⣤⠀⣸⣿⡆⠀⣿⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡀⠄⠄⠀⢀⠠⢀⠀⡄
⠀⠘⣿⣿⠀⢀⠘⣿⣾⣿⣿⡆⠀⢿⣤⡇⢘⡇⢠⡀
⠀⠀⣿⡿⠀⣌⠀⣶⠃⣿⡿⣧⡀⠸⣿⠇⢈⣿⣿
⠀⠀⢹⠃⢸⣿⡀⡼⢱⣿⡇⠚⣻⡇⠹⠀⡈⣿⠷⠦
⠀⠀⠂⠀⠀⠉⢻⣿⣿⣿⠏⠉⠉⠀⠀⠐⣃⣤⣄⠀⠀⠘⠞⠄⠀⠀⠂⠑⠄⠀⠁⠂⠀⠐⠆⢸⠀⠀⠳⠀⠳⠄⠀⠘
⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⢀⣤⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⣠⠚⢻⢷⡄
⠀⠀⠀⠀⠀⠀⠠⣼⡏⠁⠀⠈⣾⠿⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⡍⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠄⠐⠂⠀⠐⠀⠠⠀⠠⠂⠁⠁
⠀⠀⠀⠰⢠⡃⠀⢠⠠⡇⠀⠀⠈⡷⠀⠋
⠀⠀⠀⠀⠈⠁⢰⣾⢿⡿⠀⠀⠀⣠⣤⣴⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣴⣿⣿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡂⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠏⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠿⣿⣇⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠁⠀⠀⠀⠀⠀⠄⠸⠧⣤⡄⠀⠀⠀⠀⠀⣠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠀⣀⢀⣀⣀⣀⣀⡀⠀⠂⠝⣿⣆⠀⠀⢠⡞⠁⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣾⡶⠋⠉⠙⠋⠉⠙⠳⢦⣀⠘⣿⣧⡈⢋⠀⠂⠚⡀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⢹⣿⠟⠋⠀⠀⠙⣿⣿⣿⣿⣄⠱⠀⠀⠈⢀⠻⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢿⣿⣿⣟⣹⠂⠀⠸⣿⣀⣨⣧⡀⠀⢸⣿⣿⣍⣉⣓⠁⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡠⠂⡠⠀⡁⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⢠⡀⠻⠋⠁⠀⠀⠀⢨⣿⢿⣿⣟⣋⠈⠐⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠡⠑⡦⠄⠁⠀⠀⢀⣩⣿⣿⣧⠐⠲⢦⢷⡀⠀⠀⠀⠀⢠⡞⢸⣷⣟⣿⠍⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⠃⠀⠀⠘⠃⠀⢰⡄⠀⠈⠀⢀⠘⣿⣯⡙⣏⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⢀⠀⠽⢿⣿⣾⣤⡀⠀⠀⠀⠀⠁⠀⠀⠀⠿⣤⣿⣿⣧⢨⠀⠀⠀⠀⢸⠀⠄⠀⠀⠶⢂⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠄⠀⣤⡀⠈⢛⣶⣾⣿⡄⠀⠀⠀⠀⢰⡆⠀⠀⠈⢙⣏⣿⡇⠀⠀⠀⠀⣾⠁⢠⡄
⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡆⠃⠄⠀⢻⣿⣿⠃⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⣏⣿⠁⠀⢠⣿⠍⠃⠀⠼⠁
⠀⠀⠀⠘⡟⡐⠀⠉⣤⣤⣶⣷⣷⣶⣦⣾⣿⡟⣿⣿⡔⠞⣂⠘⢿⣖⠖⢀⣀⢀⣴⠇⠀⠀⠀⢰⣿⣿⡿⠂⠀⣾⢍⠒⠀⠀⠚
⠀⠀⠀⠀⢽⣂⣼⣿⠿⠃⠈⠉⠉⠏⢷⣿⣽⢿⣴⣾⣷⠚⣦⡀⠈⣹⣿⣿⠛⡛⠉⠀⠀⠀⢠⣌⣁⠀⣀⣀⣰⣿⡷⢦⣤⣀
⠀⠀⠀⠀⠀⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⡟⣡⡿⠀⣈⠉⠙⢻⢙⣀⠀⡁⠀⠀⢾⣷⡆⠀⢀⠐⠁⣽⡿⠃⠀⠀⠨⠁
⠀⠀⠀⢀⣼⣿⡗⠀⠀⠀⠀⠀⣠⡋⠀⠀⠉⠩⣽⣿⡅⠀⢿⣶⠟⣿⣿⠻⢠⣉⠙⠲⠈⠉⠁⢾⣩⡿⣠⣿⣉⡀⠀⠀⠁
⠀⠀⣴⣯⣿⡇⣷⣆⠀⠀⠀⢘⣿⣿⡎⠀⠀⠀⠀⠀⠁⠀⠀⢀⣦⣿⣿⣾⣾⡉⠳⠀⢸⣿⡧⠀⢀⣴⠿⠿⢿⣿⣕⣂⠀⠀⡀
⠀⢀⣏⣴⢾⣷⣸⣧⠀⠀⠀⠈⢿⣿⡇⢀⢀⠀⣀⢠⡀⠀⠀⠀⠀⢙⠢⠍⠹⡛⢿⣶⣤⣭⣤⣶⣿⠁⠀⣹⣿⣿⣻⡏⠀⠀⠀⠀⠀⡀
⠀⠀⠛⢸⣎⡛⣿⡕⡢⠀⠀⠀⡆⠍⠙⠾⠟⢛⣿⣾⡙⣦⡀⢀⣐⠦⡙⠓⠤⡜⠆⠉⠻⢿⣿⣿⣿⣆⣀⣼⣿⣿⣿⡇⠐⣂⠄
⠀⠀⠀⠚⠋⠑⠛⠢⣤⡄⣀⣄⡠⡄⠠⢀⡰⣏⣿⣿⣷⣾⣿⡦⠉⠁⠈⠢⠄⡀⠀⠀⠀⡼⣿⣿⣿⠁⡯⢿⡏⣿⢽⡡⡶⠃⡆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠒⠃⠀⠀⠚⠁⠈⢻⣿⣿⣿⡿⣿⣦⣁⠆⠀⠀⠀⠈⣷⡀⠐⠿⠛⢻⣏⢩⣼⡇⣿⠴⡮⠥⢨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⢠⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠿⣯⠟⣿⣴⢄⠆⠀⠀⢇⠹⡆⠀⠀⠀⠹⠉⣾⠀⠛⠶⣿⣛⡉⠁⠘⠳⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⣿⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠶⣿⣿⡥⠀⠀⠀⢠⡧⠠
⠀⠀⠀⠀⠀⠀⠐⠿⠷⠶⡮⢷⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠰⢾⣿⣓⣛⣃⡓⣤⡤⠄⠘⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠷⣤⣈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠉⠛⠛⢀⡀⠔⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⠋⠙⠛⠿⠻⠉⠛⠂⠛⠛⠈⠁
⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠖⠀⢀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣠⠎⢠⡤⢦⣤⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣧⠀⣴⠈⡿⢦⠀⡈⠑
⠀⠀⠀⠀⠀⣾⡿⠘⠀⠀⠀⢀⠗⢿⣦⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⣵⠀⠀⠀⠀⣀⠚⠉⠡⡧⣴⣿⠀⢻⣮⣷
⠀⠀⠀⠀⢈⣿⡇⡴⡞⠒⠒⠘⣶⡌⣿⣇⠀⠀⠀⠀⡈⠁⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⡎⠁⠀⠁⠀⠈⠁⠉⢱
⠀⠀⠀⠀⢸⣿⣀⣙⣦⣘⣋⣱⣿⢣⡿⣿⠀⠀⠀⠰⠇⠀⠀⡀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠑⢦⠀⠀⠀⠀⢀⣶⠎
⠀⠀⠀⠀⢠⡻⠋⠋⠛⡟⠛⠻⢉⠛⠇⢻⡇⠀⠀⠀⠀⠠⡒⠓⠀⠀⠈⠀⠀⠀⠀⠀⠰⠃⠀⠁⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠘⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⠻⠛⠛⠒⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⢀⣀⣤⠴⠶⠶⠖⠲⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⢠⡾⠛⠁⣀⣤⠶⠖⠛⠛⠛⠶⣄⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⣿⡄⠀⠀⢿⣅⣀⣠⡄⠀⠀⠀⢈⡇⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠈⠛⠷⣦⣄⣀⣉⠀⢀⣀⣀⣤⠾⠁⠀⢀⣴⡿⠃⠐⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠿⣶⣤⣄⣀⡀⠀⠉⠉⠉⠉⠉⠉⣁⣠⣤⡾⠟⠋⢀⡆⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⠀⠀⠈⡇⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⢸⡇⠒⠶⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⣀⣄⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢿⣿⣷⠆⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠐⠚⠋⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢠⣖⣂⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠻⠿⠛⠀⠀⠈⠙⠷⢤⣄⣀⣀⣀⣀⣀⣀⣠⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣤⣤⣀⣈⠙⠛⠛⠛⠛⠛⠉⠀⠀⣀⣀⣠⣤⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠈⠢⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠠⠊⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡇⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡐⠁⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠐⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢂⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⢀⠂⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠢⡀⢀⠤⠘⠈⠁⠀⠀⠀⠀⠀⠀⠉⠘⠠⢀⠀⡠⠂⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠹⠿⠃⠀⠀⠀⠀⠀⠀⠻⠿⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠉⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠁⡉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⠀⠀⠤⠤⠄⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠠⠄⠄⠀⢀⡀⣠
⠀⠀⠀⠀⠀⣀⠄⠀⠂⠉⠀⠈⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⢩⠁⠀⠀⠄⠀⠀⠀⠀⠀⢀⠔⠁⠈⠁⠂⠠⢀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠈⠀⠈⠁⠉⠀⠈⠀⠀⠀⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠂⠀⠁
⣟⣿⣿⣿⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣻⣿⣟⣻⣛⣟⣻⣿⣛⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡱⠋⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡜⠁⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠜⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡾⠁⠈⠀⠀⠄⢚⡉⠄
⠀⠀⠀⠀⠀⠀⠀⠴⠊⠀⠀⠀⠞⠁⠀⠀⡴⠒⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⡀
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠂⠀⢂
⠀⠀⠀⠀⠀⠉⠉⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠀⠀⠀⠀⢀⣀⢀⡀⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡄⠀⠀⠀⡩⠶⠦⠤⠤⠄⠀⠐⠀⠧⣄⣀⠀⠀⠀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠒⢀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠠⠀⠀⠀⣀⢀⠏⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⠉⠒⠊⠁⠉⢀⠄⠀⠁⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠀⠐⠂⠀⠠⠀⠠⠄⠀⠉⠭⠀⠀⢸⠀⢀⢀⡴⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡟⠛⠩⠵⠖⢂⡀⢀⠀⠀⠄⠠⠬⠭⠤⠤⠀⣈⠐⡂⠈⠀⠩⢠⣖⣮⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⡿⣿⣿⣷⠶⣶⣶⣬⣤⣀⣈⣉⣉⣉⣁⣀⣨⣥⣴⣶⣶⣶⣶⣶⣭⡵⠋⠁⣠⠴⠒⠛⠙⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡃⠀⠀⠉⠀⠁⠀⠀⠀⠀⠉⠀⠈⡛⠛⣛⠛⠻⠯⣽⣿⣿⣿⡇⢸⡣⠀⣯⠄⢀⣠⠴⠶⢤⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠒⣐⣒⣛⣻⣭⡭⣖⡂⠠⢒⢠⣇⡴⠋⠀⠀⠀⠀⢹⡄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠠⠍⠉⠒⢚⣛⣋⡭⠴⠀⡵⢨⡏⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠐⢒⡂⠈⣉⣭⠭⠶⢒⡋⠠⠆⢸⠇⠀⠀⠀⠀⠀⠀⡼⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠬⠄⠠⠤⠴⠶⠖⢚⣛⣩⠀⢒⠁⢸⠀⠀⠀⠀⢀⡠⠞⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠂⣐⣛⣩⣭⠥⠖⢀⡉⠀⢸⠀⠄⠐⠊⠉⠀⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠅⠠⠬⠵⠶⢒⣚⣁⠀⢒⠀⡾⠉⣀⡤⠴⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠐⣚⣛⣭⡭⠵⠖⢈⠅⣸⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⢠⣴⣶⣾⠏⣑⢃⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣤⡤⠤⠬⢷⣄⠒⠀⠀⠀⠀⠀⡄⢨⣤⠈⣥⣤⣭⣽⣿⡟⠳⢀⣴⣿⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⣠⠴⠖⠚⠉⠉⠙⠒⠀⠀⠠⠤⠤⠒⠒⣛⣻⠶⠥⠀⠀⠀⠈⠄⢩⣀⣼⣿⣿⡿⠧⠽⢶⣿⣛⠲⠿⠿⠿⠤⣤⣭⣉⡉⠉⠛⠳⣶⣦⣄
⠀⠀⠸⣅⢤⣠⣆⠀⠀⠪⠅⠀⠀⠀⠀⠀⠣⠀⠀⡈⠒⠠⠤⣤⣀⣀⣀⣀⣂⣀⣶⣿⣿⣧⡷⠞⠁⠠⠤⠂⠀⠠⠄⠀⣀⣨⠽⠂⣀⡄⣈⠵⣸⠇
⠀⠀⠀⠈⠛⠾⣭⣛⡓⠲⠶⠦⠤⢤⣭⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣨⣥⣤⣀⣀⣀⣰⣶⣶⣶⣾⣥⡤⠴⠶⠶⠒⢛⣉⣭⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠶⢤⣤⣤⣄⣀⣀⣀⣀⣉⡉⠉⠉⠩⠉⠉⠉⠭⠉⠉⢉⣉⣉⣉⣉⣀⣀⣀⣤⣤⡤⠶⠶⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠈⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠢⡀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠⠘⢀⠄⠂⠉⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠉⠒⢄⠀⠀⠀⠀⢸⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠈⠀⠀⠈⡀⠀⠀⠀⠀⠎
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⢸⠀⢸⠀⠈⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⡏⠉⠢⡀⢡⡀⠀⠸⠀⢸⠀⠀⡇⠰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠳⡀⠀⠙⡄⢣⠀⠀⡆⢸⠀⠀⡇⡘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠑⣄⠀⠘⡄⡆⠀⢱⣸⣄⣠⣤⡃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠢⣠⠜⠊⠉⠀⠀⠀⠀⠀⠀⠉⠑⠢⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣀⡈⠑⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣩⡛⢄⠀⠘⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⢠⣀⣀⡀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠀⠀⢰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠨⣿⣵⣦⣩⢢⡀⠀⠀⠀⠀⠹⢝⣋⡼⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠘⣿⣿⣿⢃⡇⢀⠀⣂⡤⡀⠲⢆⠀⠀⠀⡜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠉⠒⠉⢠⠁⠺⣄⣀⠇⠀⠀⠃⠀⡸⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠬⢦⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠝⠀⠀⢀⣀⠜
⡇⠀⠀⠀⠀⠀⢀⡠⣄⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⡇⠀⠀⢠⠊⠉⠛⠀⠀⠓⣹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠐⡂⠀⠉⠁⢼⢤⣤⣤⠤⠤⢀⡤⢤⣀⠤⢄⣀⢀⣀⣞⠀⠈⢳⠤⢄
⡇⢰⠁⠈⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡎⠀⠀⠀⠀⠀⡸⠀⠄⠀⠄⢠⡁⣏⠀⢀⠊⠀⠑⠀⠀⡷⠤
⡇⡽⠃⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡼⠁⠀⠀⠀⠀⢾⠀⠠⣀⡀⠂⠀⡱⠈⢳⠀⠢⠀⠀⠀⢸
⡎⢧⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢀⠀⠀⠀⢰⠒⠁⡇⠀⠀⠀⠀⠀⠈⠁⠲⡀⣀⠿⣍⣵⠆⠀⢳⠤⠖⡖⠐⠊
⡇⢯⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣹⠀⠀⠀⠸⡀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⠁⠀⠀⠙⠿⢶⠇
⡇⠀⠈⢋⠀⢀⡀⠀⢀⡀⠱⡄⠀⠀⠀⠀⠀⠀⣰⡇⠐⢶⡊⠁⠈⡄⠀⠀⠀⠱⢄⠀⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡼
⡇⠀⠀⠀⠉⠁⠑⠂⠊⠈⠉⠈⠢⢄⡀⠀⠀⠀⠉⢉⣹⠷⠃⠀⠀⠘⣄⠀⠄⢀⣸⠖⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡴⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠁⠊⠉⠁
⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡑⠦⠀⠀⢸⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡷⠀⠈⠃⣿⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠁⠀⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄
⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⢠⣾⣷⣾⡟⠁⠀⠀⣐⣂⣨⣥⣤⣀⠐⠶⣻⣿⡇⡄
⠀⠀⠀⠀⠀⠻⣄⠳⡀⠀⠀⠀⠉⠙⠛⠛⠻⠧⠦⠬⠉⠿⠧⠌⠻⠓⢊⣻⣥⣾⣷⡟⠞⠐⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠍⣻⣿⣿⡿⢻⠋⠀⠐⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠛⠹⡇⠀⢀⠄⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢴⡾⢱⢿⠟⠁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⢂⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⠷⠖⠊⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠻⠧⣕⢽⣦⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⣿⣿⣿⣷⡒⠢⣀⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⢔⡀⠙⢿⣷⡀⠠⣤⣤⣀⣠⣿⣷⣾⣿⣿⣿⡿⠟⠊⠀⠈⠀⢹⣿
⠀⠀⠀⠀⠀⠈⠀⠈⠠⢄⠀⠓⠞⣐⡔⠀⠹⡌⠀⠀⠀⠌⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⣤⣄⣀⡀⠷⠦⠄⠀⠀⠀⠀⠀⡠⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠙⠛⠛⠿⢟⠗⠛⠛⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⣴⡿⠋⠀⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⣷⠀⠀⢿⣧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠘⡆⠀⠘⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠃⣤⠀⡇⠀⠀⠀⠀⡄⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⡄⠀⣿⠀⠀⠀⠀⠀⡜⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠹⠀⢸⡃⢆⠀⠀⠀⣁⡼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣠⡆⠘⠃⠈⢶⣶⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡟⠙⠡⠰⢇⠈⠈⠟⣤⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠋⢀⣠⠤⠀⠘⢿⣄⢠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢿⣷⣴⣦⣡⣌⠻⡏⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢀⣾⠀⣿⣿⣿⡏⢹⣿⢄⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢺⣭⣿⣿⣈⣉⣯⡥⠖⠁⣰⡿⣯⣴⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠚⠣⠟⠉⠡⠉⠉⠻⣯⡀⠀⠀⠁⠀⠈⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⢃⠤⢀⣼⡄⠂⡀⠐⣦⣤⣿⣷⣤⣤⣀⣠⣬⠍⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣰⣾⣾⡿⣩⣤⣴⡌⠀⡍⠉⣤⡀⠀⠀⠉⠁⣠⡤⠌⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣯⡄⢠⢤⣾⡿⣷⣤⠠⠒⠉⠁⠀⠰⠦⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢥⣤⣄⣐⠒⠛⠋⠁⠀⠀⠀⣤⠀⠀⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠹⠟⠁⢖⡚⠁⢀⡆⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡌⠀⠰⠺⣮⡁⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠸⠃⠀⠀⠀⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⢰⡀⠀⠆⠀⣠⠆⠈⠛⠀⠻⠀⠀⠀⢀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⡇⠀⠙⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢠⠄⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣾⠠⣿⣿⠐⡆⠀⠀⢠⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⢿⣿⣆⠀⠀⠀⠀⠀⠰⢿⣇⣀⡀⠐⠄⠀⠀⢠⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡿⠻⣧⣧⠈⣿⡇⠀⠀⢀⠀⠀⢀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠷⠀⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢿⠁⡿⠀⠻⣆⠘⣷⡄⣠⠀⠀⠀⠀⣻⣿⣿⣿⠷⠂⠀⠀⠀⡀⠀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠘⣷⠀⠀⠀⠛⢄⠈⡏⠁⠀⠀⠀⠀⣻⣿⣿⡅⠀⠀⢠⡀⠀⠁⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⢀⣘⣿⣇⠀⠀⠀⠀⠹⣶⠖⠋⠀⠀⣿⣿⠟⠻⡀⣶⠆⠀⠀⠘⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠿⣧⠀⠀⠀⠀⢻⡃⠀⠀⠀⡿⠀⠀⠀⢻⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠙⠇⠀⠀⠀⠸⠃⠀⠀⠀⠓⠀⠀⠀⠸⠀⠀⠀⣤⠇⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢘⡀⠀⠀⠀⢤⣷⠾⠂⢀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠣⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣜⢿⣾⠃⡆⢤⡼⠟⠁⢣⢸⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣰⣿⣤⠾⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠛⠛⠻⠆⠑⠂⠀⠀⠀⠈⠈⠀⠈⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⢀⣀⡀⢀⣀⠀⣀⡀⢀⣀⠀⣀⡀⢀⣀⠀⣀⣀⢀⣀⡀⣀⣀⠀⣀⡀⢀⣀⠀⣀⣀⢀⣀⡀⣀⣀⢀⣀⡀⣀⣄⢀⣠⡀⢀⣄⠀⣠⣀⢀⣤⡀
⠀⠰⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⣭⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠄
⠀⢠⣿⠀⠀⠀⢀⠀⣤⠀⡀⠀⠀⢀⡤⠤⣄⠀⣀⣀⡀⠀⠀⠀⢠⣴⣶⣄⣼⣿⡇⢻⠐⢂⠀⢀⣀⣀⠀⣠⠤⢤⡀⠀⠀⢀⠀⣤⠀⡀⠀⠀⠀⣷⡀
⠀⢈⣿⠀⠀⢀⣈⡳⠉⢚⣁⡀⢠⠏⡴⠶⠀⡿⠁⠰⡿⠂⠀⠀⠀⢻⣿⡻⠿⢿⣶⣏⡘⠿⠖⠿⠆⠈⢿⠀⠶⢦⠹⡄⢀⣈⡓⠋⢞⣁⡀⠀⠀⣟
⠀⠘⣿⠀⠀⠀⢩⠖⣤⠲⡍⠀⡞⠀⠙⠒⠚⠀⠀⠀⠀⠀⣀⠀⠀⠀⢉⠛⠛⢻⡄⠘⢴⣷⠀⠀⠀⠀⠈⠓⠒⠋⠀⢷⠀⢩⠖⣤⠲⡍⠀⠀⠀⡿⠁
⠀⠰⣿⠀⠀⠀⠀⠀⠉⠀⠀⢠⡟⢉⣱⡄⠀⠀⠀⠀⠀⠀⢻⣷⡄⣴⡿⠀⠀⠀⢀⣀⠀⠀⢠⡄⠀⠀⠀⠀⢠⣎⡉⢺⡄⠀⠀⠉⠀⠀⠀⠀⠀⣿
⠀⢠⣿⠀⠀⢠⠞⢉⠑⢦⠀⡼⠀⠈⠉⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣷⣶⣶⠆⣾⣿⣷⣴⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⢧⠀⡴⠛⡉⠳⡄⠀⠀⣷
⠀⢀⣽⠀⠀⠸⣄⣻⠇⠈⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⠿⣿⣦⡀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠁⠸⣟⣠⠇⠀⠀⣯
⠀⠈⣻⠀⠀⢀⡀⣶⡄⣰⣷⣾⡷⠆⠀⠀⢸⣦⢀⣴⡟⠸⠟⠀⣿⠁⢎⢉⡉⠀⠈⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠰⠾⢶⣾⣆⢠⣶⢀⡀⠀⠀⣟
⠀⠘⢿⠀⠀⠈⢿⠬⠛⠉⠉⠻⡦⠀⢤⣤⣤⣿⣿⣿⣴⣤⡄⠀⢨⣄⠸⠞⠁⠀⢠⣿⣿⣿⣿⣿⣿⣻⣶⡄⠀⠀⠀⢴⠟⠉⠉⠛⠥⡿⠁⠀⠀⡿
⠀⠰⣿⠀⠀⠀⡠⣶⢾⡷⣶⣆⡀⠀⠀⣨⣿⡿⣿⣿⣮⠁⠀⠀⠀⠈⢙⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣰⣶⢾⡧⣶⣄⠀⠀⠀⣷
⠀⢀⣾⠀⠀⣸⡿⠎⢁⡀⡁⢿⣟⠀⠀⣍⢡⠃⣿⠃⠙⠃⠀⢀⣠⣤⣶⣶⣶⣶⣾⣿⣿⣿⢿⣿⣿⣿⣾⣿⠗⠀⠀⣻⡿⢈⢀⡀⠱⢿⣇⠀⠀⣧
⠀⠀⣿⠀⠀⢻⣯⡄⠈⠃⠀⣭⣟⠁⠀⠈⢛⣠⠆⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠋⠀⠀⠀⠈⣻⣭⠀⠘⠁⢠⣽⡟⠀⠀⣏
⠀⠈⣿⠀⠀⠈⠹⣾⢵⣦⢿⡎⠉⠀⠀⠀⠘⠄⠀⣰⣿⣿⡿⢿⣾⣿⣿⢿⣿⣿⣿⣿⣿⣧⣿⣿⡿⢿⡄⠀⠀⠀⠀⠉⢱⡿⣴⡮⣷⠍⠁⠀⠀⡿
⠀⠐⣿⠀⠀⠀⢀⣀⠈⠁⢣⡴⠉⣽⠀⠀⠀⠀⣐⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣷⣾⣿⣟⣿⣿⣷⠀⠀⠀⣯⠉⢦⡜⠈⠁⣠⡀⠀⠀⠀⣿
⠀⢠⣾⠀⠀⠀⢸⣿⣿⣦⢸⣇⡠⠴⠶⠦⡀⢠⣟⣿⣿⣷⣿⣿⣿⣹⣿⣿⣿⢿⣿⣟⣿⣿⣿⣿⣿⣿⡏⢀⠴⠶⠦⢄⣸⡇⣴⣿⣿⡇⠀⠀⢰⣷
⠀⢀⣽⠀⠀⠀⠀⠻⢿⣿⣷⠋⠀⠀⢀⣄⡹⢨⣿⣿⣿⣶⣿⣷⣿⣿⣿⣟⣿⣿⣿⣿⣻⣿⣿⣿⣿⡟⠀⣏⣠⡀⠀⠀⠙⣾⣿⡿⠛⠀⠀⠀⢸⣯
⠀⠈⣻⠀⠀⠀⠀⠀⠀⢨⣧⣦⡄⠀⠀⠉⠀⢸⣩⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠉⠀⠀⢠⣶⣼⡅⠀⠀⠀⠀⠀⢸⡟
⠀⠐⢿⠀⠀⣤⣴⣴⣤⣸⡉⣛⣁⠀⠀⠀⠀⣴⣟⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⢻⣿⡋⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣈⣉⢉⣇⣤⣶⣦⣤⠀⢸⡿
⠀⠠⣿⠀⠀⠙⠿⣿⣿⣿⣷⡿⠟⣀⣀⠀⠀⠻⣯⣿⣿⠿⣿⠾⣿⣿⣼⣯⣿⣿⣿⣟⣷⠈⠛⠻⠷⠿⠆⠀⠀⣀⣀⠻⢿⡾⢿⣿⣿⠿⠋⠀⢸⣷
⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣙⣴⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣋⡀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⣹⠀⠀⠀⠀⠀⢠⣴⣿⣿⡿⠛⠲⠤⢤⣀⣀⣀⣀⡀⢰⡇⠘⣁⡠⠘⠠⢀⡈⠃⢸⡆⢀⣀⣀⣀⣀⡤⠤⠖⠛⢿⣿⣿⣶⠄⠀⠀⠀⠀⢸⣏
⠀⠀⢻⠀⠀⠀⢤⡀⡇⣨⠍⢁⡀⠀⠀⠀⣴⣾⣿⠋⣠⠞⠛⠛⠲⣾⣇⣾⣆⣿⣦⠖⠛⠛⠳⣄⠙⣿⣿⣦⠀⠀⠀⢀⡈⠩⣅⢸⢀⡤⠀⠀⢸⡟
⠀⠀⣿⠀⠀⠰⠶⡁⠄⡱⠶⠉⢉⣳⡄⠼⠿⠿⠃⣸⠁⣠⠦⡄⠘⣿⣿⣿⣿⣿⡟⠀⢀⠴⣄⠈⣇⠘⠻⠿⠧⢠⣞⡉⠉⠶⢎⠠⢈⠶⠆⠀⢸⣿
⠀⠀⣾⠀⠀⠀⠚⠁⡷⠙⠆⢰⣩⠉⠙⣄⠀⠀⢀⠹⡀⠘⠁⡽⠀⠈⠙⠉⠙⠉⠀⠀⢯⠈⠃⢠⠇⡀⠀⠀⣠⠋⢉⣍⡆⠰⠋⢼⠘⠓⠀⠀⢸⣧
⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠙⠛⠉⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠉⠛⠋⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⠻⠷⠖⠶⠶⠲⠶⠖⠶⠶⠲⢶⠖⠲⡶⠖⢶⠶⠲⡶⠖⢶⡶⠲⢶⠖⢶⡶⠲⢶⠶⠶⡶⠖⢶⡶⠲⣶⠖⢶⡶⠲⣶⠶⢶⣶⠶⣶⡶⢶⣾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠡⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⠀⢰⠧⣤⡔⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⣠⠏⠁⠀⠀⡠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣼⣿⡿⠗⠀⠀⠀⠀⠀⢱⠛⢄⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⣰⠃⢼⠀⠸⠙⢠⣿⠆⠀⠀⠀⠀⠀⠀⠉⢦⣤⣾⣿⠀⠀⠀⠀⠀⢀⠔⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡃
⠀⠀⠀⠀⣰⠇⢠⠎⠀⢀⣠⡾⢁⡄⢀⠀⠀⠀⠀⠀⠀⡀⣿⣿⡟⠇⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣬⡿⠁
⠀⠀⠀⠀⡟⢰⠏⠀⠀⢸⡛⢠⠞⣠⣏⠀⢀⠀⠀⠀⠀⢹⣿⡿⠆⠀⣀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢸⠇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠘⠇⣹⠀⠄⠘⠛⠉⣠⣾⣿⣿⠀⠀⠀⠀⠀⠐⠻⣿⠁⠀⠀⠹⣦⠀⠀⠀⠀⣩⡏⠀⠀⠚⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠁⣧⡀
⠀⠀⠀⣾⠤⢸⡶⢈⠁⡜⣶⣿⣿⡿⠋⠀⠀⠀⠀⠒⠀⠀⠈⢆⠀⠀⠀⡀⠀⠀⠀⠀⠸⣦⠐⠾⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⢿⣀
⠀⠀⠀⣟⠀⢸⣷⣦⡸⣿⣽⣿⠋⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⢶⢄⡆⢀⡴⠘⢷⡀⠒⡿⣁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠛⡿⠋⢡⠸⣯
⠀⠀⠷⢸⡆⣼⣿⣧⣿⣿⡿⠁⠀⠀⠀⠀⢄⣀⣻⣿⡇⠀⢄⠀⢸⠆⣻⣌⡄⠠⠀⠳⡄⣹⠃⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠘⠁⠐⠀⢀⡿⠀⡏
⠀⠀⠀⠈⢧⡜⣭⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣺⣿⣏⡀⠀⠈⠋⠀⢰⣿⣧⣤⠀⠀⠀⢹⣮⣧⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤
⠀⠀⠀⠀⠹⡜⢿⣿⣿⣿⠁⠀⠀⠀⠀⠐⣶⣿⡟⣛⣃⠀⠀⠀⠀⣸⢿⣿⣏⠀⠀⠀⠀⢻⣾⣧⠀⠀⠐⠀⠀⠀⡀⠐⠀⠀⢠⣀⠀⠐⢺⡇⠶
⠀⠀⠀⠀⠀⢻⣍⢿⢿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⣇⠀⠀⠀⡿⢸⡿⠀⠀⠶⠓⠂⠘⣸⣧⣤⣠⣶⠀⣤⠀⠰⣬⠃⠀⠀⠈⠙⠻⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠙⣟⠊⣿⣧⠀⠀⠀⣀⢤⢾⣿⣿⣿⣿⣿⡆⠀⢠⠃⢾⣄⣀⣀⠀⠀⠀⢸⡀⠀⠉⠙⠛⠻⢿⣤⣠⡇⠀⠄⠀⠀⠀⠈⡟⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⢤⠀⠀⠀⠀⠀⢉⣽⣛⣿⣿⣿⣦⣤⣤⣾⣿⣿⣿⡿⠀⢠⠞⢧⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢈⠳⣤⣠⣖⢶⣚⣻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⡰⠃⣀⡩⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡝⢋⠛⠛⠿⠻⠛⠛⠋⠉⠀⢀⣀⡤⠶⠛⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⣫⡄
⠀⠀⠀⠀⠂⠈⢡⣦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠛⠋⠡⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⠀⠹⣷⠤⢿⡷⣶⠒⠤⠤⢠⣄⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣤⣶⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠘⢧⡀⠀⣄⢑⣶⡜⠁⠀⡔⠓⠂⠡⢠⣶⣿⣷⣾⣿⡓⣶⣶⣶⣤⣶⣶⠀⠉⢀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠦⣄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠙⠀⠀⠈⢿⢳⣞⣥⡈⢤⡿⠀⢡⣉⠻⣦⠐⣮⢷⣿⣿⣿⣿⣿⣿⣴⠖⣼⡆⠰⠉⠉⠙⠻⢿⣿⣿⣿⡏⣿⣿⣷⡘⢫⣷⣤⡀
⠀⠀⠀⠀⠀⢡⠄⠈⢦⡌⠾⣯⣿⡇⠎⡀⠀⠀⠈⢳⣄⠱⣶⣚⣿⣿⣿⣿⣿⣿⡾⣿⡿⢁⠃⠀⠀⠊⠀⠀⠘⠿⣴⣦⡔⢋⡤⣷⡀⠙⣶⡧
⠀⠀⢠⠃⠀⣼⣷⠠⠀⠻⡞⢿⣷⣇⠀⠈⠣⡀⠀⠘⣿⡆⠻⢹⣿⣿⣿⣿⣿⣿⢡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⡈⣤⣟⡑⢿⡀⠸⣷⣄
⠀⠀⠘⠀⢰⣿⡏⠀⣤⣄⠈⠈⣿⣿⢦⡀⠀⠑⠀⠀⢿⣧⠈⣨⠛⢿⣿⣿⣿⢿⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣀⣋⣵⡜⣷⣆⣹⡿
⠀⠀⠀⠀⣼⣿⡇⠀⠀⠙⢿⣦⠹⣿⣿⣷⡄⠀⠀⠀⣹⠇⢰⢻⠀⠈⢻⡟⢿⣿⣿⡄⠀⠀⢰⡟⠓⠃⠀⠀⠀⠀⠚⠋⣉⠉⠹⠟⠰⣬⠟⢁
⠀⠀⠀⠈⠏⠙⠁⠀⠀⠀⠀⠹⣷⣽⡧⢨⣿⣶⢀⡊⢀⣠⡟⡈⠀⠀⠀⣿⣾⡇⠹⠆⠀⠀⠈⠃⠠⣤⢠⣤⣤⣤⣤⣤⠌⠀⠐⣶⣮⣿⣶⣶⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡰⠟⠹⣷⠸⢿⣏⡘⠀⢸⣿⡔⠁⠴⠂⠠⣿⣿⡁⠀⠀⠀⡀⠀⠀⠀⠘⢿⣿⠻⢿⣯⣿⠀⠀⣼⣾⣽⣿⣿⣿⡗
⠀⠀⠀⠀⠀⣻⣧⡐⣀⢘⠛⣧⢈⠲⣿⡄⣿⠛⠀⠠⣿⡏⠁⠌⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡏⠀⠀⡿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠘⠛⣎⢷⠈⣽⠆⠐⢄⣿⢿⣧⠟⠀⠀⡖⠈⣰⠃⠀⠀⠀⣴⢹⠆⠀⠀⠰⠀⠀⠀⠀⠀⢀⢠⡇⠀⠀⠐⢀⣠⡾⢿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠀⠀⠈⠛⠋⢻⡀⠀⠐⢡⣾⣿⠀⠔⠀⠠⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣠⠄⠐⠋⢁⣴⠿⠋
⠀⠀⠀⠀⣀⣀⠀⠀⡄⠈⡀⠒⠀⠀⠐⠄⠁⠀⡰⢻⡿⡇⠀⠀⠀⣼⢿⣯⠀⠀⠀⢀⠀⣠⡶⠋⠀⠿⠟⠋⠀⠐⣶⡶⠋
⠀⠀⠀⠐⠊⠁⠀⠀⢀⣀⣀⣀⣠⡈⣁⢀⠀⠰⢁⡿⠙⠀⠀⠀⠀⠻⢸⡿⠄⠀⢠⣼⣾⣟⣡⣤⣶⣶⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣆⠀⢀⣤⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣤⣿⠉⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣾⡇⠀⠀⠀⠠⢀⣸⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣧⢴⣦⣠⡧⣹⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣿⢁⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠌⣿⣿⣿⡿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⢿⡿⠿⠆⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⡀⠀⢾⡿⠛⠁⡀⠘⠂⢾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣂⡀⠉⠉⠹⠿⡿⠿⠇⠀⠈⠻⠀⠈⢶⣾⠁⣿⣧⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡍⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢹⣿⣷⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇⠀⠉⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣇⠀⠀⣰⣶⡄⢸⣿⣿⡿⡿⠃
⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠛⠉⠀⣀⡀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⣼⣿⣿⣷⠄⠉⡟⠃
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠋⠀⠀⣸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢙⣿⣿⠋
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣦⣶⣿⣿⣿⣿⣿⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁
⠀⠀⢠⣿⠋⠁⡾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠿⢛⣧⣾⣿⣶⡄⠀⠀⠀⠈⠉
⠀⠀⠸⠿⢀⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣤⣼⣿⣿⣿⡿⠃⡀
⠀⠀⠀⠀⢼⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦⣼⣧⣴⣶⣶
⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⠿⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣧⣈⣻⣿⣿⣿⣿⡌⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⠿⠿⠁
⠀⠀⣾⣿⣷⣆⠀⠀⠒⠛⠻⣿⣭⣥⣤⣤⡄⠘⣀⣀⠀⠀⠀⢀⡔⣷⡀⠀⠀⠰⣶⡄
⠀⢸⣿⣿⣿⣿⣧⡀⠀⢀⣀⣨⣿⡿⠿⠟⠁⠺⠿⠟⠀⢀⣶⣿⣿⣿⡿⠂
⠀⠈⠻⣿⣿⣿⣿⣿⣶⣌⡙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢶⣦⣤⡴
⠀⠀⠀⠀⠉⠙⠛⠋⠉⠉⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠶⠶⠿⠶⠟⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠚⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⣶⡄⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⣿⣿⡿⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠉⠉⠉⠛⢶⣄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣼⣦⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⢹⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣦⡀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢸⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⣀⢀⢦⢑⠓⠀⠀⠈⣿⠀⣰⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠛⠶⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠂⠀⣤⠀⣠⡀⢰⣿⣿⣿⣿⣿⠹⠇⢠⣤⡘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⠀⣠⣀⠉⡉⣿⣿⠉⠀⠀⠀⣤⠶⢤⣠⡄
⠀⠀⠀⢿⣶⣿⠇⢾⣿⣿⣿⣿⠇⠀⠀⢼⣿⣿⠈⠛⢶⣤⣀⠀⠀⠀⠀⣀⣠⡴⠟⠁⢸⣦⣿⣿⣆⠀⢻⡿⠀⣠⡤⢼⣥⣄⠀⣿⠃⢀⡀
⠀⠀⠀⣴⣿⣧⣤⣤⣿⣿⣾⣿⣿⣿⣦⣼⣿⣧⣴⣶⣶⣮⣽⣿⣿⣿⣿⣯⣽⣶⣶⣦⣤⣽⣿⣿⣥⣶⣿⣿⣶⣧⣴⣶⣄⠉⠻⣥⣾⣿⡿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⣏⡙⣛⣁⣴⠺⣍⠙⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣍⠁⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁