Math homework Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠠⠤⠖⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠋⣀⠀⠀⠀⠀⠤⠄⢒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢀⡸⣿⠛⠛⠳⠖⠚⠛⠛⣯⠀⠀⣀⣴⣶⣶⣶⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣦⣤⠟⠀⠿⠀⠀⠿⠀⠀⣿⢀⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣉⡀⠀⠰⣆⠀⠃⡰⠂⢠⡏⣾⣿⣿⠿⠟⠹⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⣄⣀⠉⢉⣀⣴⣋⠀⣿⡿⠀⣴⡄⠀⣠⡝⠻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠻⢭⣭⡭⠿⠋⠉⣳⣿⣧⠀⡈⠠⠄⠉⠁⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⢀⢠⡶⠒⠒⠲⠦⣤⠞⠉⠰⣿⣿⣦⡙⠢⠄⠚⢀⣤⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠻⣦⣹⣿⣷⠲⡄⠀⠀⢹⡆⢀⣠⡴⠶⡏⠙⢿⠶⠛⠋⣷⣭⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣸⠃⠈⠁⠀⣟⠀⠀⠈⣷⠛⠁⠀⠰⣧⡴⠾⣄⡀⣰⠇⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣠⠶⠋⠀⢀⣀⣼⠻⠧⠀⣠⠷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠋⠁⢻⠀⠀⡴⠃⢀⢈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣀⣠⡴⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⠶⠿⠦⢤⣀⣨⣭⠿⣇⠀⠈⠻⣿⡀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣿⣯⣀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⣼⣯⣄⡜⠷⡇⠀⣈⣙⡦⣤⡈⠉⠉⠳⡤⢤⡄⣼⡿⠶⢾⣷⠟⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣻⣾⡃⠘⠇⡸⠀⠀⣗⠚⣿⠍⣰⢫⡴⣤⡀⣿⢃⢾⡟⠛⠁⠀⠀⠀⣀⣤⡴⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⠤⠭⠬⠥⠄⣴⣁⠤⠤⠽⠿⢭⡼⢡⠣⡜⡼⢩⢃⢎⠞⣿⠶⢤⣤⠴⣏⠁⠀⢀⣀⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⠴⠶⠶⠶⢤⣼⣤⣤⠶⠶⠶⡾⠭⠿⠿⠿⠿⢧⣎⣿⣀⣥⠴⠶⠶⠶⠿⠾⠛⠛⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠚⠋⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠙⠉⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀⠔⡁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⠋⡀⢀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠢⠉⠔⢊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢀⠀⢀⠄⠀⠀⠀⠐⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡫⠄⠀⠀⠾⢀⠀⠀⠁⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠘⠈⠐
⠀⠀⠀⠀⠀⠀⠀⣠⣠⡤⠀⠀⠀⠀⠀⣤⠀⠀⢸⠄⠒⢄⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⡠⠀⠀⠀⡄
⠀⠀⠀⠀⠰⠂⠟⠃⠀⠀⠀⠀⠠⠂⠀⠑⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠢⡀⠀⠀⠀⠀⠀⠀⣀⠀⠊⠀⠘⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠴⠀⠇⠤⠠⠆⠀⠇⠠⠤⠤⠤⠤⠀⠀⠾⠀⠠⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠈⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠿⣱⣄⠀⣤⣤⡀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢾⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠅⠐⠟⠋


⠀⠀⠀⠀⠀⠀⠀⣤⠠⠀⠀⠀⠀⠀⣄⠠⠀⢤⠄⠠⢠⡄⠠⣄⠀⠀⠀⢤⠄⢤⠀⠤⠄⠀⠀⠀⠀⢄⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠀⠀⠒⠂⠀⠂⠂⠒⠂⠀⠀⠒⠂⠀⠀⠂⠀⠂⠐⠀⠀⠀⠐⠒⠒⠒⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠈⠉⠁⠀⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡖⡒⠒⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣴⣦⣿⣤⣄⡁⠐⢢⠀⠀⠀⠀⣬⡎⠈⣁⠉⠀⡟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⢠⠊⠙⠉⡉⠏⠳⠉⢛⣾⣀⣼⠁⠀⠀⠀⠈⠣⢄⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⣰⢫⣟⠛⠛⠒⠉⠓⣄⣸⠀⠀⠀⠄⡫⠀⠀⣸⠏⡽⠃⠀⠀⠀⠀⠀⡴⣋⠀⢉⡲⣄
⠀⠀⠀⠀⠀⠀⠸⣍⠀⠀⠐⠀⠄⢀⡾⠉⠢⠀⢀⡀⢀⠤⠚⠁⡰⠁⠀⠀⠀⠀⠀⡎⠊⠘⡀⢸⠀⡋⠳
⠀⠀⠀⠀⠀⠀⠀⣈⡵⢦⢤⡔⠂⢹⣁⡄⠀⠀⠴⣕⠝⠁⠀⢠⡃⠀⠀⠀⠀⠀⠀⠙⠢⠤⡇⣼⡮⠂
⠀⠀⠀⠀⢀⡴⠊⡀⠀⢉⣄⡼⠖⡉⠄⣇⠀⠀⠀⣿⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡆
⠀⠀⠀⠀⣸⠦⣒⣦⡄⠀⠉⠒⠉⠀⠘⡟⠀⠰⡀⣿⡀⠀⠀⠀⡧⠤⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⡏⠑⢲⢟⠁⠀⢠⡀⣀⢤⣐⠃⢀⡤⠟⠜⠷⠖⢂⣖⡇⠀⠀⠈⣱⠀⠀⠀⢀⡇⠀⠀⢻
⠀⠀⠀⠰⡇⠀⢸⠀⠀⠹⢍⣁⠀⢸⡀⠽⠁⠐⢀⢤⡁⠖⣁⠴⠁⠀⡠⠚⠁⠀⠀⠀⡸⠀⠀⠀⢸
⠀⠀⠀⠀⡇⢀⢸⠀⠀⠀⢸⣁⡱⡌⠓⠒⠒⠚⢹⢸⢧⢸⠁⢀⠀⢰⠃⠀⠀⠀⠀⠀⠉⠀⢺⠏⡏
⠀⠀⢸⠻⢧⢸⠚⠀⠀⠀⣿⣣⠏⣏⢸⠹⠉⠁⢸⢸⢸⢸⠘⡿⣸⢸⡆⠀⠀⠀⠀⠀⠀⢀⣸⡀⡇
⠀⠀⠈⠂⠸⠖⢂⠄⠀⠸⠷⢧⡂⣇⣀⣃⣆⡃⢸⢸⠺⠬⢐⣣⠁⣉⣓⠀⠀⠀⣀⠤⠤⢉⣁⠍⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠉⠀⠀⢀⣔⡩⠋⠀⠀⠀⣤⠞⠚⡚⠃⠂⠉⣀⣔⠔⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡜⢠⣄⡀⠀⡀⣀⣤⢀⠇⠊⢱⠴⣑⡇⠳⢆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠁⠈⠀⠑⠊⠀⠀⠈⠉⠀⠈⠒⠒⠁

⠀⠀⢀⣴⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣦⡀
⠀⠀⢸⡏⠀⢠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⢹⣇
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠏⠁⢹⡆⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠉⣿⠀⠀⠀⠀⠀⠀⣀⣹⣀⣠⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠀⠀⠀⠰⠿⠒⠓⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⢸⡇⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡶⠾⠿⠿⠿⠿⠷⠿⢶⣦⣿⡇⠀⢸⣿
⠀⠀⢸⣇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣷⣶⣶⣶⣶⣶⣶⣶⣾⠟⠉⠁⠀⣸⡏
⠀⠀⠘⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣛⡃⠀⡀⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠦⠀⠀⠀⠀⠀⣤⣀⡕⠀⡽⣧⠀⡿⢀⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣦⢀⡆⣺⡀⣖⡒⠒
⠀⠀⠀⠀⠀⠀⣶⠚⣋⠉⠁⡄⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡿⠀⠉⠻⠁⣌⡽⠃
⠀⠀⠀⢰⠀⡆⢸⡆⡏⢴⣄⠛⢻⡋⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⣀⠀⡠⠀⢀⠴⣄
⠀⠀⠀⠻⠚⣏⠀⣷⠛⠾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣀⣀⠀⠀⠀⠉⣿⠁⠠⣶⣯⢼⣗⣻⠎⣻⠇
⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⣛⣉⠉⠉⠉⠙⠛⠿⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⢷⣶⣄⠀⠀⢀⣀⡓⠆⠹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣶⠄⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⠁⠀⠉⢉⡇⢰⡟⠿⠛⣦⠀⣷
⠀⠀⠀⠀⠀⢀⡤⠀⠀⣨⣽⠆⠀⠀⠀⠀⠀⣀⢰⠇⠀⠙⢦⣤⡴⠋⠠⠼⢧⡀⢀⣼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⡞⢀⣤⣄⠀⠀⠀⠀⠀⠀⢰⡏⠉⠟⠀⠤⣴⣄⣠⡀⣄⠀⠀⠀⢹⠟⠁⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣙⢲⠀⠀⠀⠀⠀⠈⣉⡷⠂
⠀⠀⠀⠀⢸⡁⠻⢀⡟⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠹⣇⣠⡿⠘⠿⣶⡶⠟⠀⠀⠠⠷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⢮⡿⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⠀⠀⠀⠈⠻⢷⣄⣀⣀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢰⣟⣓⡾⠁⢠⣄⠀⠤⡄⠀⢸⠂
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⡤⠤⠤⠴⠶⢶⣿⣷⣀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣠⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣹⠃⠐⠛⠛⣠⡴⠋⠀⣠⠴⠊
⠀⠀⠀⣶⠛⠉⠉⠉⠉⠀⠀⠀⠀⣀⣀⣤⠤⠤⠽⣿⣿⣿⣦⡀⠀⠀⠀⢀⣀⣤⡾⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠇⠀⠀⠲⣌⡉⠉⠁⠈⠁⣈⡽⠆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⠀⠀⢀⡼⠛⠁⠀⠀⢸⡿⠿⣯⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣰⠏⠀⠀⠀⠀⠀⠉⠓⠦⣄⡈⠛⠲
⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⡀⠀⠀⠙⠶⠤⣤⡤⠞⠁⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⣿⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⣤⠶⠲⣾⢃⣾⡴⠒⣆
⠀⠀⠀⣿⠀⢀⣴⡿⠋⠤⠤⠤⠤⢤⣤⣤⢤⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣦⣀⣀⡇⠀⠀⣼⠟⠁⠀⣠⠏
⠀⠀⠀⣿⣴⣿⠟⠁⠀⠚⢛⣻⣿⠾⠛⣦⡀⢹⡧⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⣠⠶⠛⠉⣿⣄⣼⠃⢀⣠⠾⢯
⠀⠀⠀⣿⣿⡏⠀⠀⢀⡶⠛⠉⢀⣀⡀⠈⣷⣴⣷⣼⣇⠀⣠⠴⠖⠛⠓⠶⢤⣄⡀⠀⠘⠃⠀⠀⢰⣷⣯⠞⠛⠉⠁⣠⠏
⠀⠀⠀⢿⣿⣧⡀⠀⣿⠀⣤⣤⣤⣭⠟⠋⠉⠀⠀⠀⠉⠻⣇⠀⠀⠀⣠⠴⠖⠛⠛⠲⢤⣄⡀⠀⠀⣿⡿⣦⠶⠖⠛⣹⠷⠦⠤⣤⣤⣤⡤⠤⠤⠤⣶⡶
⣠⣤⣤⡼⣿⣿⣿⣿⠟⠛⠛⠋⠁⠀⠀⠀⢀⣠⡴⠖⠚⠓⠻⣷⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣾⡿⠻⢶⣶⣛⡿⠀⠀⠀⠀⠀⣀⣀⣤⣶⠿⠛⠒⠒⠛⢻
⢻⠀⠀⢠⣴⡞⠋⠁⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⣈⣡⣤⣤⣴⣶⡾⠟⠛⠉⢉⣉⣭⣿⣿⡗⠀⢸
⢸⡇⢶⣾⣏⡉⠓⢶⣤⣤⡤⠖⠋⠁⠀⠀⠀⠀⠀⢀⣀⣀⣴⣾⣟⣳⣦⠤⠶⠶⠶⠶⠶⠶⠤⠤⠤⢤⣤⣤⣿⠿⣟⠛⠛⠛⠛⠾⠿⠟⠛⠛⠛⠛⠛⠛⠋
⢸⡇⠀⠙⠦⣝⠳⢦⣈⣽⣀⣀⣀⣤⡤⠶⠖⣛⣋⣉⣭⣥⣴⣟⣛⣛⣹⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉
⢸⡇⠀⠀⠀⠈⠙⠦⣬⣭⡭⠽⠶⠒⠛⠋⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁
⠘⠷⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⢤⡤⠤⠤⠤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⡿⠟⠛⠛⢋⣤⠀⠀⠉⠻⣿⠁⠀⠀⠀⠀⠀⢠⡴⠯⠽⠋⠉⢈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣒⡿⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣄⠀⢀⣠⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢶⡀⠀⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣀⣀⣸⣇⣀⣠⣾⣧⣤⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠈⣻⡄⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠻⣦⡾⠁⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠉⠉⣽⠉⠁⠈⢿⣀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⠞⠋⣷⠀⠀⠀⠚⠛⣿⠉⠉⠀⠀⣠⠿⢧⡀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠾⠋⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⠟⠙⠻⢦⣄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠁⠀⠀⠀⠀⠉⣷
⠀⠀⠀⢸⡇⠀⠀⠀⣠⠾⠛⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠶⣤⣀⠀⠀⠀⢠⡟
⠀⠠⣤⣾⡦⠶⠶⢶⡿⠶⠆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠉⠛⠶⣦⡟
⠀⠀⠀⣿⠀⠀⠀⠈⢷⣤⣠⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢀⣀⣶⡟
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢀⡀⣹⠟
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢠⣞⣹⠏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⠃
⠀⠀⠀⣿⠀⠀⠀⢠⡾⠋⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⡴⢫⡾⠁
⠀⠘⠻⣿⠛⠛⠛⢿⡛⠛⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⣠⠄⣸⠏
⠀⠀⠀⣿⠀⠀⠀⠈⠳⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠾⣁⣿⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⢀⡴⢛⣶⣶⠏⣸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠶⣤⣄⡀⡴⢎⠙⣧⡞⠁⢘⣿
⠀⠀⠀⣿⠀⠀⠀⠀⣠⡴⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠙⠳⣦⡿⠋⠀⠘⢯⡿
⠀⣀⣀⣿⣀⣀⣀⣼⣋⣀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣠⡴⠟⠉⠀⠀⠀⡰⣾⡇
⠀⠉⠉⢻⡉⠉⠉⢿⡉⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡶⠛⠁⠀⠀⠀⠀⠀⢀⣤⣾⠁
⠀⠀⠀⢸⡇⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⢴⣋⢼⡟
⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠂⢀⡜⢡⡔⠁⢠⣴⠛⢤⡔⣩⡾⠁
⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣀⣴⢛⣴⣋⣳⣜⣣⣜⣃⣈⣶⣋⣨⣷⡼⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢒⡶⠀⢚⣁⡀⠖⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠐⠻⡍⠀⠀⠾⠱⠀⢿⡗⡈⠀⢬⣽⡟⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⢀⣤⠀⢡⡤⠐⠥⠔⠀⠀⠉⠄⠠⠀⠠⠀⠀⠉⣶⡖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⡀⠀⠍⣀⠚⢈⡬⣭⠇⣸⠐⠋⡑⠃⠀⠐⢲⡄⠀⠈⠳⠛⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢩⠟⠀⣠⠤⠔⢲⠁⠢⡂⡏⠒⠀⠄⣀⠐⠮⠟⢠⡄⠱⠂⠊⠀⠚⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠤⣠⠄⢀⡀⢿⣉⠆⠀⣮⡵⡼⣷⡀⣀⢰⠘⡣⠊⠀⢾⠷⠄⠀⠞⣷⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠼⠙⣦⠀⢠⡀⣄⡚⠁⢹⡄⠛⣠⠋⠀⠙⠀⠀⠀⠘⠀⣀⠿⣀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⢼⣗⢀⠀⠀⠒⢰⣲⡄⢀⣌⡩⠉⢳⣰⠇⠀⠐⠂⠀⣶⠲⡴⠏⢀⡀⠓⠱⠋⠜⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠇⠀⠐⡄⣾⣭⠇⠀⡽⠷⢀⡌⢿⡈⣀⠀⠘⣇⠛⠚⢀⠀⠃⠃⡀⠀⢰⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠴⣆⠀⠀⠀⠈⠓⠒⠚⡻⠟⠛⡛⠷⢮⣷⡄⢰⠀⢢⠈⠉⢻⡆⢠⣾⡇⠀⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠤⠤⠀⢊⡿⢳⠄⠁⠀⠀⠙⣿⣎⠀⣀⣄⠀⢨⡇⠀⠘⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⣤⠤⠈⠓⠋⠐⠏⡷⢀⣤⠘⣿⡔⣷⢲⠄⣼⣤⡤⠞⢤⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣙⣛⡆⡤⠀⢠⣚⣓⣂⣀⠠⢹⣷⢈⣥⠎⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠒⠉⠉⠙⠈⠉⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⠀⠈⢻⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠼⠿⠿⠿⠤⠤⠤⠀⠜⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⢀⣤⣶⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠟⢿⣿⣿⣿⠟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣦⣼⣿⣿⠟⠋⠁⠀⠀⠘⢿⣿⠃⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠛⠻⣿⣿⣿⣿⣶⣄⡀⣠⣴⣶⣶⣶⣦⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⢸⣿⡀⣿⡟⠀⠈⠙⠿⣿⡿⢋⣥⣤⡈⠹⣿⡆⠀⠘⣿⣿⡿⠛⠉⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠠⣾⡿⠇⠈⠁⠀⠀⠀⢸⣿⡇⣾⣿⣿⡇⢀⣿⣿⣶⣶⣬⣭⣤⣄⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣤⣤⣶⣶⠀⠀⠀⠀⠈⢿⣧⣙⠛⣉⣠⣾⣿⣥⣴⣿⠛⢉⡉⠙⢿⣦⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣿⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⡿⠿⢿⣿⠃⢰⣿⣿⣆⠈⣿⣇⡀⠀⠀⢻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⣴⣶⡀⠘⢿⣿⣧⡀⢸⣿⡀⠹⢿⠿⠃⣰⣿⣿⣿⣷⣶⣤⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⣿⣿⡀⠀⠀⠀⠹⣿⣿⣄⠀⠛⠋⠁⠀⠻⣿⣶⣤⣴⣾⡿⠋⠀⠈⠙⠛⢿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⡿⠿⢿⣷⡄⠹⣿⣷⡀⠀⠀⠀⠙⢿⣿⣿⣶⣤⣤⣶⣶⣦⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠿⣿⣿⣿
⠀⠀⠀⠀⢀⣠⣴⣶⣿⠿⠛⢉⣠⣤⣤⡈⢻⣿⡄⠙⢿⣿⣄⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⣰⣶⣶⠆⣻⣿⣿
⠀⣠⣴⣾⡿⠿⠛⢉⣠⣴⣾⣿⣿⣿⣿⡇⠀⢿⣿⡄⠈⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡇⣿⣿⠀⣠⣿⣿⠏
⣾⣿⡟⠁⣠⣶⣾⣿⣿⣿⣿⣿⠿⠛⠉⢀⣤⡀⢿⣿⡀⠀⢀⣹⣿⣿⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⡿⠟⢻⣿⣷⣬⣥⣶⣿⡿⠃
⢻⣿⣧⠘⣿⣿⣿⡿⠿⠛⠉⣀⡀⣴⣷⡜⠷⠟⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⢿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣥⣀⠀⠀⠙⠻⠿⠿⢛⣭⣶⣤⣄
⠈⢿⣿⣧⠈⠉⠁⠀⢰⠶⡜⢯⡿⠈⢉⣥⡀⣾⣿⠌⢿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣾⣿⣀⠀⠀⢈⣿⣿⣿⣿⣽⣷⣤⡀⠀⠀⣰⣿⣿⣧⣛⣻
⠀⠈⢿⣿⣷⣶⣿⣷⠈⠛⣃⡀⣴⢳⡜⠿⠟⠈⢉⣤⡌⢿⣿⣿⡿⠿⢿⣿⣧⣴⣿⣿⣿⣿⣧⣤⣽⣿⣭⣿⣽⣽⣽⣯⣯⣴⣾⡿⣵⢿⡿⣿⣿
⠀⢀⣿⢋⡾⣹⣿⣥⢦⡔⢯⡿⠈⢛⣡⡀⣾⣷⡌⠻⠋⠈⢿⣇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⠞⣱⢋⣾⣿⠃
⠀⠸⣿⠈⠰⢫⣾⡟⠛⣃⡀⣴⢶⡜⢧⠿⠈⢉⣤⡄⠀⠀⠈⣿⡆⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠻⢿⣿⣿⢟⡽⢻⡞⣵⣿⡛⠻⣷
⠀⠀⠹⣷⣄⣘⣿⣷⡐⣯⣿⠈⠛⣁⡀⣾⢳⡜⠿⠟⢀⣠⣶⠿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢰⣿⣧⡞⡵⢫⣾⣭⣭⠁⢠⣿
⠀⠀⠀⠈⠛⠛⠻⣿⣷⡀⠀⠀⠘⢯⡿⠈⢉⣠⣴⡾⠟⠋⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣇⠀⣼⡏⠙⢿⣷⣿⡷⠶⠒⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⣀⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠃⠀⠈⣿⣿⣿⣿⣿⣿⠀⣿⣿⠾⠛⠉⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡟⠛⢻⡿⠃⠀⠀⠀⠸⣿⡿⠛⢿⣟⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⣿⣷⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠯⡄⢤⡀⢀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⣀⡤⠤⠒⡚⡉⠉⠀⠀⠀⠘⠒⠃⠈⠓⠸⠰⠮⠄⠀⠀⠀⠀⠀⠀⠀⠩⣷⠸⢼⢊⣳⡅
⠀⠀⢠⢠⠸⣤⡧⣄⠳⡟⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠩⣁⢀⠄⢠⢭⡀
⠀⠀⠙⠹⠄⠯⠙⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠹⠀⠲⠟⠺⠿⠊⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⣐⣶⡀⠀⠀⠨⠭⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⢀⡏⢸⠁⠉⢹⠂⡞⠟⢳⡀⡏⠁
⠀⠀⠀⢀⡔⠀⠐⠚⠁⠀⠀⢠⠤⣼⠀⣈⠓⠚⠁⠘⠓⢤⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠠⠤⣀
⠀⠀⠀⣼⠰⢋⡆⠀⠀⠀⠀⢻⠀⠀⠈⠻⡋⣱⠹⣦⣤⠞⠀⢀⣅⡀⠀⠀⠀⠀⠀⠀⡾⢷⡾⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠈⠛⣦⠀⠀⠙⠻⢤⣄⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⣸⢯⡾⠁⣤⠀⢤⡀⠸⠀⡀
⠀⠀⣠⠤⠤⠶⠖⠒⠒⠛⠉⢉⣙⣿⣷⣤⡀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⢀⡏⣰⠃⠨⡁⠶⠯⠀⠖⠭⡄
⠀⠀⢻⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠉⠉⣽⠇⠀⠀⣴⠾⣏⢸⡇⠀⠀⠀⠀⠀⠀⡼⢰⠃⠀⠀⠈⠑⠢⣄⡐⠻
⠀⠀⣿⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⡄⠀⠈⠓⠲⠖⠋⠀⠀⠹⣇⠀⠀⠀⠀⣀⣰⢃⡏⡀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⣿⠀⢀⣼⢟⣁⣀⣀⣀⣀⣀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⢰⡏⠉⣻⠞⠋⢹
⠀⠀⣿⣴⡿⠁⠐⢒⣶⡶⠻⣅⠹⡧⣄⠀⠀⠀⠀⠀⠀⠀⡴⢋⡩⠟⢋⣧⣴⣃⣠⠴⡇
⠀⠀⣿⣿⠀⢀⡴⠋⠀⠠⣤⣼⠗⠓⠿⣤⠔⠋⠉⠙⣓⣦⣄⡈⠀⠀⢸⣿⡏⣉⣠⢴⣁⠀⠀⠀⠀⠀⠀⣀⡀
⣀⣀⣸⣿⣷⣼⠷⠿⠟⠛⠁⠀⢀⣠⠤⠬⢷⣄⠔⠋⠁⠀⠀⠉⠓⠦⣤⣿⢿⣭⣤⣾⠉⠉⠉⠉⢉⣩⣭⠿⠥⠤⢶
⣿⠀⢠⣴⣛⠁⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⣭⣤⣴⣶⣶⣛⣋⣉⣭⣿⣶⠂⢸
⣿⠘⢿⡧⣌⡛⢶⡞⠉⠀⢀⣀⣠⠤⠤⠶⣻⣿⡿⣿⣯⣭⣭⣭⣿⣿⣶⣶⣶⠿⠽⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸
⢹⠀⠀⠈⠳⢭⣛⣳⠿⠿⠿⠒⠒⠛⠋⠉⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠉⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠉⠀⠀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠋⠁⠀⠀⠀⣀⠠⠄⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⠀⠀⠀⣀⠠⠔⠂⠉⠀⠀⠀⠀⠀⠠⠄⠒⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡀⠀⠐⠉⠀⠀⢀⣀⠠⠀⡀⠉⢀⡀⠄⠐⢛⣈⡁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠇⠀⠀⠄⠒⠈⠄⡤⠔⠘⢂⡈⠄⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⢢⠀⠀⠀⠀⠘⢈⣀⠥⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⢰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠤⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠶⣞⠋⠈⣓⠶⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠐⠀⠁⠀⠀⠀⢠⡯⠶⣄⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⡇⠀⠀⠒⠈⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⣀⢀⠤⠄⠈⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡅⢠⢤⡀⠀⡇⠀⠀⠀⠀⠀⢀⡀⠄⠀⠐⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠛⠁⠀⠀⠀⠀⠈⠁⠁⠀⠀⢀⣸⣋⡡⠬⠀⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠐⢆⡭⠁⠁⠀⠀⢀⡀⣸⡄⠀⠈⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢧⠀⠐⠂⠈⠏⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⢀⡀⡄⠄⠀⠂⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠄⠤⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠤⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠐⠒⠀⠁⠀⠀⠀⢀⣀⡤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⠞⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⡠⠤⠖⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠤⣤⡤⠤⠤⠤⠔⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠘⠲⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠩⠀⠐⠒⠛⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⢁⣠⣈⠄⠄⠠⠔⢒⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠈⠜⣿⠉⣙⠛⠛⣛⠋⠉⣿⠀⠀⣠⣶⣾⣿⣿⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⡶⠶⠋⠀⠻⠀⠀⠛⠀⠀⣾⢀⣾⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣬⣄⠀⠐⢦⣀⣁⠔⠂⢠⡏⣾⣿⡿⠿⠛⠁⠻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣀⣀⣀⣠⣶⣏⡀⣿⡿⠁⢴⡆⠀⢰⡎⠙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠉⠓⠦⠭⠥⠞⠁⢈⣹⣿⣷⠀⣀⠀⠄⢈⠁⠀⢘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⣀⣰⡖⠚⠛⠓⠒⣴⡊⠉⠰⢿⣿⣷⣌⠐⠒⠈⣁⣤⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠙⢦⣽⢿⣷⠒⣆⠀⠀⢸⡇⢀⣤⠶⠒⡿⠈⠻⠿⠛⠉⣽⠦⣭⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠁⠀⠀⠀⣏⠀⠀⠀⡿⠋⠀⠀⠀⢧⡴⠛⢦⣀⣠⡏⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡴⠞⠁⠀⣀⣠⣼⠛⠳⠀⣠⠟⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣠⡞⠉⠀⢹⣀⠀⡴⠁⢀⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⣤⡴⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⡴⠞⠛⡶⠤⣄⣬⠽⠿⢧⡀⠀⠙⢿⣀⣀⠀⠀⠀⠀⠀⣼⡟⠀⣾⢿⣀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣬⣧⡄⡞⠺⡇⢀⣠⣽⡶⣤⡀⠉⠉⠹⡶⢶⡆⣸⣿⠓⠶⣿⠞⠋⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣻⣿⡃⠘⠃⡜⠀⢀⣗⠚⣷⠆⣴⠣⠷⢤⡄⣾⢍⢾⣟⠛⠉⠀⠀⠀⣀⣤⠴⠞⢧⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠥⠤⠭⠬⠥⠤⣼⡡⠤⠤⠽⠿⢥⡼⠡⠭⠌⠬⠩⢍⢎⠞⣿⠓⠦⡤⠴⢿⠁⠀⣀⣀⣀⣻⣆
⠀⠀⠀⠀⠀⠀⠈⢿⣤⠶⠶⠶⠶⢦⣼⣤⡴⠶⠶⠶⡾⠭⠿⠻⠛⠛⢫⣎⣾⣀⣥⠶⠶⠒⠲⠾⠷⠛⠋⠉⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠋⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⣅⠠⣄⠄⢠⠀⣀⡀
⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⡒⠛⠀⠀⠀⠀⠰⠶⠏⠐⠛⠆⠿⢤⣞⡁⠀⠀⠀⠀⠀⠀⠀⠀⢩⣽⡀⣧⣿⠸⢯⡍
⠀⠀⠀⡀⡀⢻⢰⢂⣀⣧⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡚⠁⢀⠉⢘⡛⠁
⠀⠀⠸⠷⣗⢸⡿⠮⠟⠀⠉⠀⠀⠀⠀⣴⣾⣿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣀⠀⠀⠀⠐⢿⠂⢨⣽⣢⣿⣶⠙⡾⠁
⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠛⣛⡋⠉⠉⠙⠻⠿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠀⠉⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡴⠲⣶⣄⠀⣠⣬⣑⠘⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠩⢿⡃⠀⠀⠀⠀⠀⢰⠇⢸⡁⠀⢀⡞⢸⡏⠋⠘⡆⢹
⠀⠀⠀⠀⣴⠋⣀⠘⠛⠁⠀⠀⢀⡴⠶⡟⢀⣀⡉⠋⠁⠀⠉⠙⢦⠾⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⡄⠀⠀⠀⠐⠲⢤⡄
⠀⠀⠀⢰⡇⠾⣹⠇⠀⠀⠀⠀⠈⣇⠀⠀⠀⠹⣏⢩⡇⠳⣦⣤⠟⠀⠀⣿⣤⠀⠀⠀⠀⠀⠀⠀⢰⠯⣷⡏⠀⠀⠀⠀⠞⠉
⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠙⠲⣆⠀⠀⠈⠻⢦⣄⣀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣿⣳⡞⠀⣤⡀⠤⡄⠀⠇
⠀⠀⠀⣀⣠⣤⣤⠤⠤⠶⠖⠒⠒⠒⠲⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠁⠀⠀⠀⠀⠀⠀⡼⠀⡼⠁⣘⠉⣤⣞⡁⠠⢖⣋⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣤⣶⠖⠚⠛⠛⠛⢻⣿⠄⠀⠀⣤⣶⡾⠋⡇⠀⠀⠀⠀⠀⠀⠀⢰⠇⣼⠁⠀⠈⠓⠦⣄⡀⠀⣴⣞⠁
⠀⠀⢸⡇⠀⠀⠀⠀⣰⣿⠏⠁⠀⠀⠀⠀⠀⠻⣤⣀⣀⣸⠏⠉⠙⢶⣿⠀⠀⠀⠀⠀⠀⢀⡟⣰⠃⠀⠀⠀⠀⠀⠀⠉⠒⠤⠄
⠀⠀⢸⡇⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣇⡆⠀⠀⠀⠉⠀⠀⠀⠀⠈⠻⣄⠀⠀⢀⡤⠶⣾⢡⣯⠤⣄
⠀⠀⢸⠀⣠⣾⠟⠡⠤⠶⠤⣤⣤⠤⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢦⣤⣸⡀⠀⡼⠋⠀⣠⠏
⠀⠀⢸⣾⣿⠋⠀⠈⣩⡽⠿⠋⠳⣄⢸⡗⣦⠀⠀⠀⣀⣀⡀⠀⠀⠾⢁⡴⠚⠉⢸⣷⢾⣁⡤⠞⢻
⠀⠀⢸⣿⣇⠀⢠⡞⠁⠀⠐⢦⡤⠿⠛⠛⠻⢤⡖⠋⠉⠉⢉⣹⡶⣤⣈⠁⠀⠀⢸⣿⣋⣀⣠⠴⣏⣀⡀⠀⠀⠀⠀⣀⣀⣀
⣀⣀⣸⣿⣿⣶⣼⠷⠿⠟⠛⠉⠀⠀⢀⣠⠤⠤⢿⣦⡠⠞⠉⠀⠀⠀⠈⠙⠲⢤⣨⣿⢿⣥⡤⢾⠏⠉⠉⠉⠉⠉⣉⣭⡿⠯⠤⠤⣶
⣿⠉⠁⣤⡽⠛⠁⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣌⣉⣉⣤⣤⣴⡶⠟⠛⢉⣉⣲⣶⡶⠀⢸
⣻⠰⢶⣿⣉⠓⠶⣤⡴⠚⠉⠀⠀⠀⢀⣀⣀⣤⣴⣾⣿⣷⡶⠒⣒⣒⣒⣒⣶⣦⣤⣤⣴⣿⣻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⢸
⢹⠀⠀⠙⠮⣝⠲⢼⢧⣤⣴⣖⣚⣯⡭⠥⠶⠶⠺⢯⣽⡿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣆⣀⣀⣀⣈⣙⡒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠓⠀⠀⠘⠢⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠀⠸⡷
⠹⠀⠀⠀⠀⠔⠠⣶⡀⠀⠠⠀⠀⡴⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠰⣇⣀⡔⠄
⠀⡀⠀⢠⠊⠉⠀⠈⠀⠀⠀⠀⠀⠉⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠁⠀⠀⠀⠀⢨⠉⠁⡀⠐⠂
⠀⠀⢀⠈⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣤⠀⡴⠆⠻⣿⠔⠋
⡟⠶⡆⠀⢰⣶⣾⠉⠀⠀⠀⠀⠆⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠀⠀⢸⡀⢀
⠀⠈⠁⠀⠰⡄⠁⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣤⣤⠀⠁⠈⠀⢀
⠁⠂⠀⣤⡄⠁⠀⠈⠁⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠈⠀⠀⠀⠘⠃⠀⣶⡶
⠂⡠⠀⠈⠰⠶⠆⢀⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠄⡀⠈⢀⠀⠊
⣤⡄⠀⠁⠀⠀⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢃⡀⠀⢀⠀⢿⠄⢉
⢞⠹⣞⠀⠈⠁⢩⠾⠶⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠓⠃⠀⣴⢛
⠋⠁⠙⢛⡐⠂⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⣀⡀⠀⠁⠐⠈⠉
⢶⠇⢀⠀⠀⢢⠀⠈⠀⠀⠀⠜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠢⠃⠐⠁⠀⠀⢀
⣥⠇⢀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣳⢀
⢿⡀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠆⠀⠀⠀⠀⠀⠃⠀⠀⣠
⡌⠃⠀⠠⠇⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣠⣦⠀⢀⣀
⠈⠠⠤⣆⠂⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠄⠀⠀⠀⠀⠉⠘⠂⠈⠉
⠀⠄⠠⠮⠂⠀⠀⠀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠇⠀⠹⣤⣖
⠀⠀⠀⠀⠔⡄⠀⠀⠀⢁⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢩
⠀⠈⠁⠀⠄⡬⠤⠀⠀⠀⠉⠆⠀⠀⠑⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⢸⣿⠟⠀⢺⣭⠧
⠣⠛⠀⠀⠯⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠠⣒⠀⠀⠀⠀⢄⠀⠀⠀⢀
⢄⠰⠁⠀⢀⡆⠀⢠⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⣤⠀⣀⢀⠀⠙⡆⠘
⢚⣰⣳⠄⠈⠀⠀⠀⢙⠀⠀⠘⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡁⠀⠂⠀⢰⣛⡅⠸
⢤⠏⡁⠘⠀⠀⠀⠈⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⢄⠀⠐⠊⠁⠀⡻
⡞⠀⢛⡒⠀⠀⡦⠀⢀⠈⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⠀⠀⠀⢢⠀⠈⠤
⠁⠀⠀⣷⡆⠀⠀⠀⠈⢩⠁⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣲⠀⠈⠠⣞⢳⠀⠀⠈⠁⢀
⠳⠀⡠⣍⠠⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⡀⠀⠁⠘
⠇⠀⠁⣰⠋⣸⡋⠀⠀⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠰⠟⠃⠀⢀
⠻⢃⠘⠉⠹⢥⡿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⠀⢀⠀⠀⠀⠠⠄⢀⡈
⠐⠿⡆⠀⠀⠀⣶⠆⠦⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠘⠲⠀⠀⢦⡁
⡆⢠⡀⢹⣻⣇⠀⢠⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢖⣙⢤⠲⡅
⣀⣧⠙⠈⢫⡟⠈⠛⠀⠀⠈⠓⠂⠀⠀⠀⡤⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠘
⢋⡄⠦⣴⠄⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠦⠀⠰⡞⠓⢬⠴⢎⡁⠐
⣋⣠⠀⡀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⡁⢶
⠈⠉⡀⠀⡞⣳⠀⠰⣿⣇⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⡠⠂⠭⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠲⢄⠀⠀⠀⠀⠀⢠⠋⢰⠢⢄⣀⡀⠈⣝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⢀⠄⡠⡀⠈⢧⠀⠀⠀⠀⢸⢠⠏⡀⠀⠄⠈⠙⢩⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣠⠾⠊⠘⠁⠙⣆⠸⠀⠀⠀⠀⠀⡇⣇⠈⣤⠀⢀⠎⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠹⡄⠉⢀⡀⢡⠇⢸⠀⠀⠀⠀⠀⢣⡈⢱⡦⠒⡇⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⣘⣷⠒⣾⣉⠐⠁⠀⠀⠀⠀⢀⠔⠉⢹⠓⠔⢻⠀⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⢅⢰⠉⡻⣨⠏⠉⢦⠀⠀⢠⠋⠀⠀⠈⡆⠀⡄⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠪⡈⢧⠃⣱⠀⠀⠀⡆⠀⡎⠀⠀⠀⠀⢹⣀⡃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣀⠀⠈⢀⠈⠁⠀⠀⠀⡇⠀⡇⠐⣦⡀⠀⣏⣈⣹⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡤⠿⢆⠀⠈⠀⠀⠀⡄⠀⡇⢸⢃⠞⢟⡳⡆⠀⠀⠀⣀⡞⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢰⠁⡨⣿⡇⠲⠀⠀⡠⠧⣀⢸⠈⠻⣨⢟⠕⠗⠐⢾⢿⠀⠙⢆⡜
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠛⠚⠓⡺⠿⢽⡖⣲⢾⣖⡀⠈⡟⠒⠒⢺⠎⠀⠀⣴⠟⢻⣄⡠⢿⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢰⠘⠿⠛⠛⡄⠀⣰⠃⠀⠀⠀⠀⢠⡎⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡀⠷⣒⣢⢤⣀⡰⠃⢣⠀⠀⠀⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠚⠃⠀⠀⢸⣉⣉⠥⠤⠼⠶⠖⠛⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⢩⣟⣃⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧⣠⠖⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠙⢓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠙⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⣀⣤⣤⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣸⠆⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠀⣠⠞⠁⠀⠙⢮⡁⠀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢱⠐⣆⡑⠒⠤⠄⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⢧⠀⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⣚⣷⠿⣋⣽⡶⠶⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠈⣏⣰⣾⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣶⢸⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⠛⢉⡴⠶⠶⠦⣄⠀⠀⠀⠀⢀⡴⠶⠶⣍⠛⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣾⣄⢸⢠⠏⠀⠀⠀⢀⠘⣧⠀⠀⠀⡟⠀⠀⣤⠈⢳⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣼⠁⣀⠀⠙⠾⡟⠀⠀⠀⠀⠙⠁⢸⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⢁⡼⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢿⠀⢨⢷⡀⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣇⠀⠀⠀⠀⣸⠛⠏⣠⠴⠄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣇⢯⣈⠁⠀⠘⢦⣀⠀⠀⢀⣠⠏⣠⠶⣤⡈⠓⠦⠴⠞⠁⠀⡸⢹⠆⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣭⣤⡇⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣀⠁⠜⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⣿⠴⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⣤⣤⡤⠤⠶⠚⠋

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆
⠀⠐⠷⠶⠶⠶⠿⣽⠂⠀⠀⠐⠳⠶⠶⠶⠆⠀⠀⠈⠐⠂⢵⣤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠐⠀⠀⠀⠂⠀⢀⣄⠀⠀⠀⠀⣄⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈⠩⠀⠀⠀⡀⠙⠀⠈⠁⠁⡌⠀⡁⠈⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠃⠁⠋⠀⠀⠀⠀⠀⠂⠘⠐⠀⠀⠄⠈⠀⠀⠀⠘⠐⠂⠀⠀⠀⠀⠀⢀⠀⠀⠘⠒⢚⠒⠒⠀⠀⠀⠀⣰⠉⢀
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠂⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠘⠂⠀⠄⠤⠤⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⠝⠁⡀⠀⢀⣜⣄⠀⠀⠀⠀⠄⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⠀⣆⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣁⡤⣲⡄
⠀⢀⠂⡀⠀⠀⠀⠀⠓⠃⠲⠃⠀⠀⠈⠈⠀⠀⠀⠠⠈⠀⡀⠀⠀⠀⠀⠃⠂⠀⠀⠚⠁⠀⠀⠀⠀⠀⠈⠉⠈⡉⡀⠘⠁⠈⠈⠀⠀⠀⠙⠘⠀⠻⠻⠿⠚⠉
⠀⠀⠀⢹⡢⠄⣀⡀⠀⠀⠀⠀⢀⢂⠀⠀⠀⢀⠆⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠄⠀⠀⠀⠃⠃⠃⠽⠐⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⡀
⠀⠀⠀⠈⣅⠜⠉⠀⠀⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠆⠀⡀⡀⠀⠁⠀⠀⠘⠀⠀⠀⣆⠂⠂⠀⠠⠰⠂⠐⠒⠘⠀⠂⠑⠀⠁⠀⠁⠁
⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡉⢂⠀⠓⠀⠉⢀⠃⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠆⠀⠀⠀⠛⠀⠀⠨⠀⠀⠚⠀⠀⠃⠁⠀⠀⠀⢀⡀⠀⠀⠀⠠⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢀⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠖⠀⠀⠀⠐⡀⣀⠐⠂⡄⠀⠀⠀⠀⠠⠠⠀⠐⠂⠀⠀⢸⣁⡦⠤⠸⠇
⠀⠀⣀⣀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠁⠠⠃⣀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⢠
⠀⠀⠛⠋⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣄⡀⠀⠀⠀⠈⢀⣀⡀⠀⠀⠀⠐⠐⠂⠙⠂⠁⠠⣿⣛⣿⣷⠰⠈⠐⠀⠡⢚⠇⠜⠈⠀⠘⠀⠀⡀⣠⣄
⠀⠀⣀⠠⠮⠅⠀⠀⠈⠉⠉⡉⠃⠀⠘⠛⠋⠋⠙⠐⠀⠀⢸⣿⢛⣿⠀⠀⡀⢀⣠⠀⡀⠀⠀⠀⠛⠙⠛⠉⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠳⢝⡫⠼⡇
⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⣀⣀⣀⠀⠀⠀⡀⡀⠀⠀⠀⠀⡀⠀⢈⣁⣈⠁⠀⠀⠀⠀⢀⣯⢠⠤⣤⣄⢦⢰
⠈⠁⠒⢲⡠⠤⢄⡄⠄⠀⢠⠂⠀⢀⡎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⣻⣾⡿⣧⠀⠀⠘⠁⠉⠈⠃⠁⠀⠀⢠⢉⡅⠀⠉⠼
⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠊⢀⠤⠄⠀⠋⠈⠀⠀⠀⠀⠀⠀⠘⠯⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣄⠀⠀⠈⢠⢀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠤⣄
⠀⠀⡌⠀⠀⠀⡐⠀⡄⢀⡀⠂⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⣤⣀⠀⣤⢠⢠⡄⡆⠀⠀⠄⠀⠀⠄⢀⡀⠘⢧⡼⠇⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠛
⠀⠀⠀⠈⠀⠑⠉⡀⠀⠄⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⣀⣸⣿⣧⣤⡄⠈⠐⠤⡀⠀⠀⠀⠃⠀⠈⠀⣠⣤⢤⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠾⠰⠂⠇⠔⠀⠌⠀⠀⠀⠀⢼⣯⣭⡽⣷⠀⢸⣿⣿⣾⣿⣿⣷⣶⣆⣶⣦⠀⠙⠀⠀⠀⠀⢸⡷⣾⣻⣽⡇⠀⠈
⠀⠀⠨⠴⢅⣇⣐⠣⠀⠀⠀⠀⠉⠈⠁⠈⢁⠌⣬⠉⠉⠀⠀⠀⠀⠀⠙⢦⣽⣧⠟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⠀⠀⠀⠀⠘⠿⣭⡯⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⢤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⡤⢤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣵⣷⢸⡿⠟⢿⣯⣾⡆⣿⡻⠻⣯⣾⡇⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠵⢿⣿⣦⣿⡿⠫⠾⣿⣭⣭⣿⡯⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⠿⠿⠿⠋⠀⠀⠀⠀⣴⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⣆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢿⣿⣿⠷⡙⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣉⢴⣿⣿⣶⣤⣤⣄⣀⢴⣿⣿⡿⣃⡋⣒⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣘⣛⣛⣛⣓⣈⣛⣛⡻⠿⠿⠟⣘⣛⣛⣃⣛⣃⣛⣛⣛⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⢂⠄⠀⠀⢀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠂⠀⡁⠀⠀⠀⠀⠀⠄⢀⠄⠀⠀⠀⠡⡀⠀⠀⠀⠰⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠃⠀⡀⠀⠈⠀⠀⠂⠀⠀⠀⠄⡀⠂⠀⠀⠀⠀⠈⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠰⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠁⠀⣠⠀⠀⠀⠁⠀⠃⠀⡀⠀⠈⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢐⠦⠀⣀
⠀⠀⠘⠀⠂⠀⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠈⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⣎⠁⠀⠁⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠈⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡶⠌⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠈
⠀⠀⡆⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠈⠠⠃⠆⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⢌⠂⠀⠀⠀⠀⠀⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢉⠈⠀⠈⡀⠀⠠⠀⠀⣀⠸⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡅⠀⠀⠠⠀⠂⠀⠀⠀⠀⢀⠄⠀⠀⢀
⠀⠀⠈⠄⠀⠀⢁⠀⠁⠘⠀⠀⢀⠀⠀⠐⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠂⠈⠀⠀⢀⠀⠈⠀⠀⠀⠀⠄⠑⡀⠠⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⡀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡦⠁⠀⢁⠀⠀⡄⢀⠄⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⡀⠀⠈⠇⠀⠃⠈⠂⠀⠈⠆⠀⠀⠀⠈⠂
⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠙⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠴⣀⠀⠠⡀
⠀⠀⠔⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠠⠂⡠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⡰⠂⠁
⠀⠘⠀⠀⠀⠆⠀⡥⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠄⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠂⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠁⠂⠠⠀⠄⠀⠀⠀⠊⠀⠀⡡⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠁⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠐⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⢀⡁⢀⠢⠀⡄⢁⠘⢀⠀⡠⠊⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⠀⠄⠀⠀⠂
⠀⠀⠠⡂⠀⡐⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⡠⠀⠀⡠⠁⠁⠈⠀⠁⠐⢁⠜⠀⠀⡀⠀⠐⠠⠀⢸⠀⣒⡀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠓⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠂⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠣⠄⠈⠀⠀⠀⡀⠰
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠒⠀⠀⠀⢀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⢄⠀⠀⡄⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⡴⠖⠛⣋⣽⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠯⢭⡉⠙⠳⠤⠞⠁⠀⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⢀⡞⣁⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⠧⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠁⢠⠞⢀⠀⠀⠐⣦⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠉⢷⠀⠀⣷⠈⠛⠦⣽⣟⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢀⣀⠌⠉⠛⠲⠦⣤⣀⡀⠹⣏⠙⠛⠲⣶⠤⣤⣈⣛⣦⣄⠉⠉⠉⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⣩⠖⠚⠓⠶⣤⡀⠀⠀⠈⠉⠉⠁⣀⠀⠹⡄⠀⠀⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠇⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠙⣦⠀⢀⣤⠖⠒⠛⢧⠀⠹⡄
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⡾⣠⠏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⣧⠞⠁⠀⠀⠀⠘⡇⠀⣹⠦⠤⡄
⠀⠀⠀⠀⢠⡟⢁⣀⣀⣀⠙⠻⣵⠋⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⢉⠟⠒⠳⡄
⠀⠀⠀⠀⠈⣇⠀⠀⠙⠮⠿⡄⠙⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢀⡀⢠⡇⢠⣤⣄⠀⠀⢸⡇⠀⠈⢷⣉⠛⣧
⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣯⣿⡿⠣⣷⣾⡿⢀⣠⠟⢀⣠⠤⣄⠸⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠰⣄⣈⢙⣓⣶⠞⠁⠀⠀⠉⠉⠉⠀⠠⠋⠀⠀⣸⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠞⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡄⠀⠀⠉⠓⠦⢤⣀⡀⠀⠀⠀⠀⢀⡄⢀⡴⠒⠛⣚⣧⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡼⠀⣼⠀⠀⠀⠀⢠⡏⠉⠙⢲⣒⣒⣾⣤⣬⠵⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⣇⣠⡇⠀⠀⠀⠀⢸⡗⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⠚⢧⡞⠀⢸⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⡑⠀⠁⠀⡞⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠤⠤⠤⠞⠁⠀⠀⠀⠀⢀⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣦⠤⠶⠶⠶⠒⣾⢻⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⣿⣸⡤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠈⠀⠉⣹⣉⣁⣼⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠢⠤⠤⢴⢶⢖⠟⣻⢀⣈⣷⡤⣾⠇⠀⠀⠀⠀⠀⠀⠲⠶⠶⠤⢤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢝⣒⣒⣒⣛⣛⣚⣉⡩⠥⠖⠋⠉⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⡶⠒⠒⠀⠠⠤⠬⠭⠉⠉⠻⠿⠿⠷⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠶⠶⠯⠭⠥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣉⠀⣠⣶⣿⣷⣦⡀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠏⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠋⢹⣿⣿⣿⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠸⠄
⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣿⣿⣿⣿⣷⣶⣤⡀⣴⠄
⠀⠀⠀⠰⠟⠋⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠻⣿⣄⠀⣀⣠⣤⣶⣶⣿⠿⠿⠿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢃⣠⣴⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠛⠉⠉⠉⠉⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡝⠻⣿⣷⣄
⠀⢠⣀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⣀⠠⠤⠐⠒⣘⣫⠬⢇⠀⠈⠛⢿⣷⣄
⠀⢀⠭⠉⠙⠛⠛⠛⠓⠛⠛⠛⠋⠉⠉⣉⣉⣉⠘⠒⠒⠤⡀⠀⠀⠀⠀⠀⣀⠤⣒⠉⠁⠀⠀⣀⡀⠀⠀⠀⣘⣉⣥⣤⣴⣶⣾⠿⠇⠀⠀⠙⠿
⠀⠀⣷⠤⠤⠤⠄⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠑⠒⠤⢄⣑⢄⠀⣠⣮⣖⠯⠵⠒⣀⣀⣥⣤⣶⣶⡿⠿⠟⠛⠋⠉⠁
⢾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣄⣈⣙⣷⣯⣥⣶⣶⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠘⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⢸⢰⠳⠕⢤⣠⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀⡠⢠⢤⢱⠀⠀⠀⠀⠀⠀⠸⣹⠀⠀⠀⢠⠜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢱⡋⠁⠀⠈⢸⠈⡇⠀⠀⠀⠀⠀⠀⡇⠳⣤⠤⣎⢠⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣽⣄⠀⢀⡼⢠⠇⠀⠀⣀⠀⠀⠘⣜⣾⠽⠶⠯⡿⢣⠗⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣹⠂⠹⡽⢟⡟⢿⣎⠀⣠⣼⠋⠉⢦⣀⡏⡀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣫⠦⡀⠘⠃⠀⢸⠀⣇⠻⣿⠤⢲⠞⠙⠷⡅⠀⠀⠀⠀⢸⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⣤⠜⠁⢀⣷⡀⠀⠀⣸⡀⢸⡀⠀⠀⠀⠱⣄⢀⡇⠀⢀⣀⡤⠼⠄⠀⢸
⠶⠶⠶⠶⠶⠶⠶⠶⠾⢆⣘⣶⣶⣿⣷⡿⣖⣾⣉⣁⣠⡷⠶⠶⠶⠶⣶⠛⠋⠉⣹⣄⣀⣀⣀⣀⡼⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⠷⢶⡶⣶⣬⣥⣖⣶⣶⣶⡗⠀⢀⡴⣾⣛⣲⡤⠾⠔⣒⣒⣒⢤⠀⠹⡜⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣇⠘⡌⠁⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠉⠛⠿⠛⠉⣩⣙⣛⣁⣀⣣⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⡤⠤⠤⠼⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠤⠤⠤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⣤⠀⠂⠀⠀⠀⡴⢦⡈⣇⠀⠀⠀⠀⠒⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠑⠠⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠀⣤
⠀⠀⠀⠔⠢⡀⠀⠙⠛⠁⣿⢀⡀⠀⠀⠀⡄⠀⠀⠉⠀⠀⠀⠀⠐⠃⠀⠀⠀⠰⠶⠆⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢸⠃⢠⡏
⠀⠀⠘⠀⠀⢱⠀⠀⠀⠀⣿⠸⣟⡷⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠤⣾⠃
⠀⢸⠀⠀⠀⠀⢃⠔⠁⠀⠛⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⠀⠀⠀⡁⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⡄⠀⠀⠀⡄⠠⡤⠀⠀⡟⢀⡟
⠀⠃⠀⠀⠀⢀⠛⠀⠀⠠⠀⢠⠀⠀⠁⠀⠀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠄⠀⢂⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠁⡀⢀⣈⡉
⠀⠠⠀⠀⠀⠊⠀⠁⠙⠀⠀⠀⠀⢀⢀⠀⠃⠀⠀⠀⠀⠀⠇⠀⠄⠀⠀⠆⠼⠀⠀⢎⠧⠀⠁⠈⠃⠉⠀⠀⠀⠠⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⢀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⡀⢀⡞⣧⠀⠀⠦⠍⠉⢀⣀⠁⠀⠀⢠⣼⣶⣦⣤⣤⣄⠄
⠀⠀⠀⠀⠀⠀⠹⡀⠂⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡤⠀⠀⡖⠀⠀⣿⠃⡄⢸⣎⠀⢸⠃⠹⠀⢀⠀⢒⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠁
⠃⠀⠀⠀⠄⠈⠀⠁⠐⢖⠒⠈⠁⠀⠀⠀⠀⠀⡜⠃⠀⡏⠀⢸⡇⢠⠀⠏⠀⠀⠛⠈⠀⠈⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠐⠀⠀⡀⠀⠀⣼
⠀⠀⠀⠁⠀⠀⢀⡄⠀⠀⡱⠀⠀⠀⠈⠄⠀⢰⠃⠉⠐⠃⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠃⠄⠁
⠀⠀⡀⡄⠀⡀⠈⠀⠀⠴⠥⠤⠤⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⢀⡀⠀⠐⠀⠀⠀⢠⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠅⠀⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡀⠐⠈⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠁⠀⠀⠀⠈
⡁⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠘⠁⡀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠨
⠀⠀⠀⠁⠀⠀⠘⠍⢁⠀⠀⠀⠀⠀⡠⠋⢀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠠⠀⠀⣀⠀⠀⠤⠀⠌⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⢀⠄⠰⠔⠚⠉⠁⠀⠀⠀⠄⠀⠀⠐⠀⠁⠈⠀⠀⠀⠋⡽⢩⠖⠃⢰⠀⠀⠀⠀⠀⠀⣀
⠀⣀⣀⣠⢤⡀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠃⠀⠀⠀⢠⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⣣⠄⠀⠀⡄⠀⠀⠀⡆⠀⠂⠬
⠸⠃⠈⡏⠐⠃⠈⠒⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢀⠀⠀⢠⠤⢤⣄⡀⠀⠁⠀⠀⠴⠀⠀⡰⠃⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⢰⠀⠀⢠⠀⠀⢛⠒
⠀⠀⢠⡇⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⡄⠀⠰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣀⠄⠀⠀⠈⠒
⠘⠧⠞⠛⠴⠃⠀⠀⠡⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⢠⠾⢧⣀⢰⡄⠀⠐⠃⠁⠀⠀⠃⢸⠀⢹⠀⠇⠸⠀⣦⠀⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠉⠉⠉⠉⠉⠉⠒⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠲⢦⣤⣤⣀⡉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡉⠉⠒⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠎⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡒⠦⢄⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢠⠏⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡙⠒⡶⠦⠤⢄⣀⣀⣀⠀⠀⠈⠳⢄⠈⠙⠢⢄⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣐⠿⠀⠀⠀⠀⠀⠤⠤⠒⠒⠛⠉⠡⠒⣸⠁⠀⢄⠀⠀⠀⠉⠉⠉⠛⠒⠚⠃⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠵⠃⠀⠀⠀⠀⠀⠻⡃⠀⠀⠀⠀⡴⠉⠉⠳⣄⡴⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⡇⠀⠀⠀⠸⡀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⠀⠀⠀⠀⢧⠀⠀⠀⢻⣟⣀⣀⣠⠇⣀⣀
⠀⠀⠀⠀⠀⠠⣁⠀⠀⠀⠀⠀⠀⢸⠉⠉⣉⡓⠊⠀⠀⠀⠀⠘⠦⢄⡠⠞⠀⠉⠹⠖⠉⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⢤⣤⡼⣄⠘⠐⠋⠓⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠖⠉⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠿⣗⢲⡶⠤⠤⠔⠒⠉⠀⠊⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡌⠉⠁⣉⡷⣀⡀⠀⠀⠀⡞⠋⠸⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⠀⠀⠀⣸⣷⢄⡴⢁⡄⠀⡴⢮⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⡟⠚⠋⠉⠀⢈⣧⣤⠎⠈⢦⡃⣸⠑⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣇⣱⠤⠒⠒⠋⠉⠀⠙⢢⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⢽⠘⠦⠴⠶⠶⡖⡖⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣧⡼⠋⠀⠀⡇⡇⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠁⠀⠀⠀⠀⠀⣀⣼⣴⠧⠄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⢛⣻⡗⢦⣀⠀⠀⠀⠀⠀⠀⡭⠉⠁⣀⣭⡾⠿⢍⡓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣁⡔⢻⡀⠘⠞⠀⠉⠒⠤⢤⣄⣴⡓⠚⠙⠞⠁⢹⠦⣀⠙⣆⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠸⠿⠽⠭⠭⠭⠭⠭⠍⠉⠛⠓⠬⠭⠭⢍⣉⣿⣿⣏⣩⡯⠥⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡴⢠⡄⢠⠠⣦⠄⡤⢦⡄⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠧⠞⠀⠟⠀⠳⠼⠃⠏⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢾⡽⠛⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡶⣅⣈⡾⠤⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣭⢸⣾⠧⣤⡟⠲⣤⣃⡼⢸⡓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠻⣤⣏⣙⢶⣁⠜⠁⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠃⡇⡇⣰⣹⡆⠉⠉⠉⢀⣞⣳⠀⡇⡏⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⡇⣇⣃⣀⣘⣀⣀⣀⣘⣀⣀⣃⡇⡇⠀⠀⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠰⣛⣒⣒⣓⣒⣒⣒⣒⣚⣒⣓⣓⣚⡷⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠤⢤⣄⣈⣉⣩⣭⠉⠉⠉⠉⠉⠉⠉⢉⣩⠤⢤⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢡⢶⡆⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡴⠷⠚⠉⠉⠒⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣔⣿⡿⣪⣵⡖⢶⠚⠣⠤⠀⠀⠀⢠⠖⢺⠓⠒⡲⣄⣀⣀⣀⣸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢻⡭⣯⠭⢽⢭⡽⢸⡞⢆⠀⠀⠀⠀⢸⠀⡞⣾⡖⠒⠒⠒⠒⢾⢻⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠙⢾⣄⡧⢊⠀⢸⠇⠀⠳⡄⠀⠀⡏⠉⢧⣿⠏⠉⠉⠉⠉⢹⡼⠟⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠠⣞⡆⠉⡀⠻⢃⡜⠀⠀⠀⠈⢦⡀⡇⠀⠈⠦⣭⣭⣭⡍⠉⠉⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣈⣀⠘⠟⠀⡞⠀⠀⠀⠀⠀⠀⠉⢣⠀⠀⣇⣠⣤⣤⣤⣤⣀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⢶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⡒⠒⠶⢤⣀⣠⠴⠋⢁⡤⠴⠾⠯⠤⠤⠤⠤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠽⠷⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣝⠂
⠀⠀⠀⠀⠀⠀⠀⢰⡯⠴⠚⢛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠠⢄⠀⠀⢸⠒⢤⡀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡆⠀⢸⡄⠀⠈⠑⠘⠂⠉⠒⠦⣤⣀⡀⠀⠀⠀⠀⢤⡄⢀⣀⠀⠀⠀⠀⠀⢤⡤⠤⠤⠤⠤⣀⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⠀⠀⠀⠀⠄⠐⣀⣀⠈⠉⠛⠂⠠⠀⠈⢳⣄⠀⠈⢹⡛⠒⠶⠦⣬⣷⣦⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠁⠀⠀⠀⠀⢀⡾⠋⠉⠉⠙⠢⣄⠀⠀⠀⠀⠀⠀⣤⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡟⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠈⢣⡀⢀⡤⠚⠉⠉⠙⡄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⣀⠀⡼⢡⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⠀⠀⠀⠀⠀⢸⠀⢀⡟⠒⢲⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⣤⣤⡈⠙⢷⡟⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⢸⠀⠈⡽⠛⠚⢳⡄
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠙⠢⠽⢦⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢀⡀⠀⡇⠀⢤⣤⡀⠀⠀⡿⠀⠀⠹⢯⣉⠙⡆
⠀⠀⠀⠀⠀⠀⠈⠂⠄⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⢤⣿⡞⠓⢴⣾⡿⠃⣀⡼⠃⣠⡤⢤⡀⢹⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⢒⡶⠞⠁⠀⠀⠉⠈⠉⠁⢀⡜⠁⠀⠀⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠞⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠉⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⠄⠀⠀⢂⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠄⠀⠀⠀⠀⠀⠀⢰⡟⠓⠲⢦⠤⠤⣼⠃⢀⣓⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⣸⠁⠀⠀⠀⠀⢸⣥⠤⠶⠚⠓⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣾⢀⠋⠉⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⠢⠀⠛⠀⢀⡇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⠀⠂⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣉⡉⠁⣀⣀⣀⣀⣤⣤⠔⢲⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡍⠉⠉⠉⠀⠀⡆⣰⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠤⠷⣿⠆⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⣀⣠⡷⠦⠴⡞⠋⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣉⠙⠛⢉⣟⣍⣝⣀⣧⠤⢴⣿⡞⠿⠃⠀⠀⠀⠀⠀⢀⠉⠉⠙⠛⠛⠓⠲⠶⠶⠶⢤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠭⠭⠭⠭⠭⠭⠥⠔⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⢿⡿⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠄⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠐⠀⠰⠰⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡶⠛⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢏⣥⡏⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣤⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⠒⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⢀⣴⡖⠶⠶⠶⠦⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⠛⢩⣭⣭⣭⣝⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⢿⣆⢸⣿⠓⠒⠒⠒⢺⢡⣶⣦⢤⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠘⣿⠀⢨⣭⣭⣭⣍⢾⠉⠉⡟⢿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⡏⠻⢺⣿⡆⠀⠀⠰⣼⣂⣩⣧⠀⠁⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠸⣦⣼⠀⢸⣿⣿⣿⠻⢸⠖⠚⢿⣾⣿⣿⣿
⠀⠀⠀⠐⢿⣿⢿⣿⠿⠀⢸⣿⡇⠀⠀⠀⠈⡟⢹⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⠀⠠⠀⢤⣿⡿⠀⢸⣿⣿⣿⣷⣦⡦⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠘⣿⠷⠶⠶⢶⡖⠛⢿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣲⠀⢺⣿⣷⡄⣀⣛⣛⡛⢻⣧⡅⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠸⡿⠈⣿⠀⠀⠀⠿⠄⠀⠀⠼⠇⠀⠿⠹⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣀⣿⣀⣹⣿⣿⣃⣹⣿⣿⣏⣘⣿⣇⣼⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡠⠏⣀⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⡤⢄⣀⣀⣀⣀⣤⠤⠖⠒⠳⠬⠙⠃⠉⠀⠛⠒⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⠓⠺⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠿⢝⡂
⠀⠀⠀⠀⠀⠀⠻⣖⣒⣒⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡴⠃⣀⣤⠴⠶⠶⠦⣄⡀⠙⢆⡀
⠀⠀⠀⣀⡴⠚⠉⢉⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠤⣄⣹⣞⡴⠚⠉⠀⠀⠀⠀⠀⠈⠹⣄⠀⠙⢦⡀
⠀⠀⠀⠈⠉⢉⠟⠁⠀⠀⠀⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠹⠏⠀⠠⠶⢤⣄⡀⠀⠀⠀⠀⢹⡄⠀⠀⣳
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⢬⣙⠳⡄⠀⠀⠘⡇⠀⠀⣧
⠀⠀⠀⣼⠖⢹⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⡇⠀⠀⢻
⠀⠀⠀⠁⠀⣿⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⠶⠒⢸
⠀⠀⠀⠀⠀⢸⠠⠤⢼⡄⠀⣀⣀⣀⣀⣤⣤⣴⠾⠛⠋⣴⡄⠀⢠⣆⠀⠀⠀⠀⠀⢀⣿⠀⢠⡏⠑⠂⢸
⠀⠀⠀⠀⠀⢸⠀⠈⠉⡇⠀⠈⢻⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⣼⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⢧⠀⠀⢳⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⣀⣤⡿⠁⠀⢠⡇⠀⠀⣰⣧⣄⡀
⠀⠀⠀⣠⠶⠒⠚⠢⠤⠼⠂⠀⠀⠀⠈⠉⠛⠛⣟⡿⠋⠀⠀⠠⠖⣦⠀⠀⠉⠉⠉⠀⠀⠀⠳⠶⠊⣡⡴⠒⠙⣆
⠀⠀⠀⡏⢀⡴⢲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠫⣿⠀⠀⢹
⠀⠀⠀⢷⡀⠀⠼⠈⡧⢸⡀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠒⠒⠂⠁⠐⠒⠒⠤⣄⡀⠀⠀⠀⠀⢠⣆⠙⠀⠋⠀⢀⡿
⠀⠀⠀⠀⠛⢦⣄⣀⣠⠼⢧⠀⠀⠀⢠⣿⣿⣿⣶⣶⣤⣤⣤⣀⣀⣠⣤⣾⣿⣿⡆⠀⠀⠀⣼⠉⠛⠶⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢧⡀⠀⠘⢿⢿⣿⢿⣟⣋⣛⣛⣛⣿⣿⠿⠿⣿⣿⡇⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠉⠲⠤⢄⣀⣀⣉⣁⣀⣀⣠⠴⠞⠉⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠚⠛⠋⠉⠁
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣟⣟

⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣄
⠀⠀⠀⠀⡐⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠋⠉⠁⠀⠀⠀⠀⠀⠐⠖⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠅⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠬⠉⢩⠉⠉
⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣸⣄⣸⣇⣸⣇⡀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠉⠉⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠊⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠠⣔⠨⠸⠀⠀⠁
⠀⠀⠐⠀⠀⠄⠀⠠⠀⠠⠀⠀⠀⠀⠀⡏⢹⡯⢹⡿⢹⠄⠀⠀⠀⠀⠀⠀⠑⠀⠰⠊⠠⠀⠀⠀⠀⠀⠸⡄⠀⠍⠅⠀⠤
⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠈⠊⠁⠀⠀⠀⠀⠀⠀⠑⠐⠀⠈⠜⠈⠁


⠀⠀⠀⠀⠀⣀⠀⠀⠤⠁⠀⠀⠀⠀⠈⣶⣶⣶⡶⢲⣶⡅⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷
⠀⠀⠀⣁⡀⠤⠄⠐⠖⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠚⢻⡆⠀⠀⠀⠀⠀⠸⣿⠧⠼⣿⠇⠀⠀⠀⠀⠀⠀⠐⠈⠃⢁⠙
⠀⠀⢀⠀⢀⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⢀⣶⣆
⠀⠀⠀⢀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠠⠀⠀⠀⠀⠀⠀⠀⢠⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠐⡃⢠⠔
⠀⠀⠀⠀⠀⠀⡀⢇⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠉⠀⠈⠀⠀⠀⠀⠀⠀⠤⠄⠰⡷⠀⠦⠄
⠀⠀⠀⠀⠀⠂⠀⡇⠀⠀⠀⠀⠀⠀⢈⠀⠀⣆⡄⣼⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠄⠀⠀⠓⠂⠀⠀⠀⠀⠀⠘⠛⠚⠛⠃⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠟⠛⠛⠛⠛⠻⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⡴⠞⠿⠆⠀⠀⠀⠀⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢀⣔⣩⣄⡀⠀⠀⣀⡀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢀⡾⠉⠀⠀⠹⣎⣻⠟⠛⠻⣿⠶⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠸⡇⠀⠐⠂⠀⣿⣇⠐⠶⠀⣹⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠙⢦⣤⣤⠾⠃⠼⢷⣶⣾⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣫⠅⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣙⣀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢀⣘⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢿⣦⣤⣀⣀⣀⣠⣤⣴⣾⡿⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠐⠋⢻⣤⣤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⢦⡀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠳⠶⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡴⠶⠛⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⢻⣿⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⡇⢸⡇⠀⠀⠀⠀⠉⠉⢹⡏⠉⣿
⢀⣤⣤⣤⣤⡤⠶⠖⠛⠛⠉⠉⠉⠉⠀⠈⣷⡀⠀⠀⠀⠀⠀⢸⡇⠀⠾⠷⢦⣤⣤⣤⣀⣀
⠛⠶⢤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⢶⣤⣀

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠋⢸⡇⢸⡇⠙⣿⠛⢠⡞⠙⢷⡀⣿⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠸⢧⡼⠇⠀⠿⠀⠈⠷⣤⠾⠁⠿⠙⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣯⡽⠛⢉⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣤⡞⠁⠙⢲⣾⣀⢸⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣟⢀⡞⠉⠓⢦⣟⠀⠉⣻⠓⣷⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡴⠄⢿⣹⡗⠶⣤⡏⠈⠙⣶⣇⣀⡏⠰⢦⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠈⠻⣇⣰⠋⠉⣳⣞⠁⣠⠟⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢸⡇⡇⠀⣠⢦⡌⠙⠺⠿⠷⠞⠋⢡⠴⣄⠀⢸⢸⡧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⢸⡇⡇⣠⠷⠴⢧⡀⠠⠠⠄⠄⢠⠾⠦⠿⡄⢸⢸⡇⠀⠙⠳⢤⣀
⠀⠀⠀⠀⢠⡴⠋⠁⠀⠀⠀⠀⢸⡇⣇⣛⣀⣀⣀⡃⣀⣀⣀⣀⣘⣀⣀⣀⣓⣸⢸⡇⠀⠀⠀⠀⠉⠓⢦⡄
⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣼⠧⠭⠭⠭⠭⠭⢭⣥⣭⣭⣭⡭⠭⠭⠭⠭⠥⠼⣧⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠙⠦⠴⠶⠦⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠶⠶⠶⠋⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠤⢤⣤⣤⣀⣠⢤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠦⠤⣔⡋
⠀⠀⠀⠀⠀⢀⣴⢃⣴⢶⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⠴⠚⠋⠙⠓⠶⣽⣦⡀
⠀⠀⢀⡴⢛⣩⣇⣼⣉⡽⢃⣀⣀⣀⣠⣿⠰⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣁⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⢰⠋⣴⣯⣥⣯⣞⣡⣶⣿⣧⡙⣧⠀⠈⠉⠉⠁⠀⠀⠀⠀⣼⠁⠈⡇⠆⠆⠠⠘⠦⠤⠤⠤⠤⠤⢼⡧⠤⣤
⠀⡿⠀⢿⡤⢼⡧⠤⢤⣿⠤⣼⠇⢸⡎⢳⡀⠀⠀⠀⠀⠀⠀⣿⠀⣼⢻⢻⡖⠶⠶⠶⠶⠶⠶⢲⡏⡟⣷⠀⢸
⢀⡟⠀⠈⠳⣄⢷⠀⣰⢃⡾⠁⠀⢸⡇⠀⠙⢦⠀⠀⠀⠀⣶⠛⠒⣿⢸⢸⡗⠒⠒⠒⠒⠒⠒⢺⡇⡇⣿⠒⠛
⣼⠁⠀⠀⢀⡈⠻⣷⡿⢋⡴⣄⠀⣸⠁⠀⠀⠈⠳⣄⠀⠀⢻⠀⠀⠈⣟⠛⣥⣤⣤⣤⣤⣤⣤⣬⣿⡍⠁
⠉⢹⠀⠰⣏⣹⠆⠈⢀⠀⠙⠋⣠⠏⠀⠀⠀⠀⠀⠘⢦⡀⢸⡀⠀⠀⠈⠳⠤⠤⣴⣒⡆⠀⠀⠀⢰⡇⠀⠀⢀
⠀⢸⣄⠀⠈⠁⠀⠰⣯⠷⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⡇⠀⠀⢸⡄⣤⣤⣬⣭⣤⣤⣤⡄⢸⠃⠀⠀⢸
⠀⠀⠉⠛⠛⣿⠛⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⢸
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⠀⠈⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠉⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠉⠉
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠰⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢷⣦⡀
⠀⠀⠀⠀⣰⡿⠛⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠹⣷⡄⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⢲⡦⣀⠀⠀⠀⠀⠻⡶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠾⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠙⠢⣄⣀⠀⣹⠀⠙⢦⡀⠀⢲⣄⡀⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠳⡖⠶⠚⠃⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠒⠛⠀⠙⢦⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡼⠆⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⣀⠀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⣠⠴⠒⠲⢤⡀⠀⠀⠀⠀⠁⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢰⠃⠀⠀⠀⠀⠹⡄⢀⣞⡉⠑⢦⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡼⠀⠀⠀⢸⠀⠀⠀⠀⠀⣄⡷⠸⡏⠁⠀⢸⠆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣥⣬⡙⠞⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⢩⠇⠀⠙⠦⠤⠞⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣄⣤⠬⢦⠀⠀⠀⠀⠀⠘⢦⡀⢀⣠⡎⠀⠀⠀⠄⠠⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠢⠤⠴⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠉⠓⠦⣄⣀⠀⠀⠀⠒⠀⢀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⣠⠃⠀⠀⠀⠀⡟⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⡏⠈⡹⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠻⠬⠉⠀⢰⠃⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣠⣤⠞⠦⢤⣄⣀⣀⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣏⠀⠀⠐⠀⡴⠖⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⡀⠈⣏⣙⣿⣷⡾⢛⣛⣿⠋⢉⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠴⠶⠒⠋⠁⠈⠑⠚⠿⠭⠽⠛⠁

⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⢀⡠⠖⠀⠀⠀⢠
⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠃⠀⠀⠀⠋⠁⠀⠈⠀⠀⠀⡰⠃⢀⣈⠀⢀
⠀⠁⢀⢀⡀⢠⡶⣤⢹⡆⠰⠒⢨⢀⢁⢡⡂⠦⣤⣤⢀⡂⠢⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⡼⠀⣾
⠀⠀⠎⠀⠱⡀⠙⠋⢸⡇⣀⡁⠀⠀⠀⠀⠀⠀⠉⠈⠈⠁⠀⠀⠄⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡣⢴⠇
⠀⡆⠀⠀⠀⢇⡠⠂⢸⡇⢿⣿⠀⡀⠈⠀⠀⠀⢀⠀⢰⡁⠀⢠⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠸⣁⡜
⠐⡃⠀⠀⠀⠞⠄⠀⠈⠃⠀⠠⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⡈⠤⠀⠀⠠
⠀⠆⠀⠀⠘⠀⠑⠘⠀⠰⠂⠰⠀⠰⠀⠀⠀⠀⠀⢀⣄⣀⣀⠀⠠⠀⠀⠀⠀⠈⠀⠐⢠⠐⠃⠰⢍⣁⠀⠒⠈
⠀⠄⠀⠂⡀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠐⠀⠀⠀⠀⠙⠛⠟⠛⠛⢁⠀⠀⠠⡄⠆⠀⠀⣀⠒⠀⢌⢀⠒⠠⠀⠁
⠀⠀⠀⠀⠣⠄⠀⠀⠄⠉⠀⠀⣀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⣠⠀⠠⠄⡁⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠄⠀⠀⠀⠠⠀⠀⢀⠀⢰⠋⡆⡿⠀⣼⠉⠧⡏⠙⢲⣶⡄⢸⢻⠀⣸⢻⡆⣀⠄⠠⠂⠀⠐⠃
⢀⡈⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⣼⡇⡇⠀⡇⢲⡆⡧⠄⢨⡿⣅⢸⣏⠀⡿⠶⡇⠠⠆⠀⠴⠀⠐⡁⠄
⠘⠁⠀⠐⠇⠐⠀⠀⢀⠁⢸⡇⢸⠇⠧⠤⢻⡼⢹⣇⣀⢸⡇⣸⢸⠈⣧⡇⠀⠷⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠
⠀⣐⣀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⢉⣀⣉⣅⠈⠀⠀⠀⠀⠀⠀⠂⠀⠃⠂⠀⠀⠁⠀⣤⠤
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡏⠘⠁⠀⠀⠀⡀⠒⠀⠀⠀⢠⠀⠇⠈⠀⠀⣈⡉
⠀⠀⠀⠀⠀⠞⠆⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠒⠂⣄⣰⣇⡠⠀⡀⠐⣀⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣈⡉
⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⡀⣈⡁
⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠃⠀⣠⠖⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠖⡳⢶⣤⡤⠁⠀⠀⠀⠀⣄⣀
⠀⠑⠄⠀⠀⠀⡄⣀⠄⡈⠁⠀⠈⠀⠀⢠⠀⠁⠈⠀⠀⢀⠀⠀⢀⡀⠀⠀⢀⡼⠁⡎⡀⠀⢡⠀⠀⠸⠀⠤⣅
⠀⡄⠀⠀⡀⠀⠘⠄⠚⠀⢀⠀⠀⠀⠀⠁⠈⠀⢀⠤⠤⣬⠀⠀⠎⠀⠔⠈⠉⠀⠘⠛⠁⠀⠀⢆⢀⠂⠀⠢⢄
⠀⠀⠀⠰⠀⠀⠆⠈⢄⠀⡈⠀⠈⠀⠀⠀⠀⠄⠀⣠⠞⠁⠀⠀⠀⠀⠐⠃⠐⢀⠔⠂⠀⠀⠀⠀⠁⠀⠀⠠
⣀⣀⣈⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣂⣀⣀⣈⣀⣊⣙⣲⣛⣀⣀⣃⣀⣐⣃⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡯⠈⠀⣾⣧⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠇⣨⡝⠂⣸⣿⡏⠀⠀⠀⢋⠁⢤⠤⠒⢂⣷
⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⠥⠄⢹⣿⣿⣿⣿⣿⡿⠀⠀⠒⣞⠀⠈⡀⢠⣿⣿⠁⠤⢤⣄⡐⢠⢯⢐⣦⣍⡿
⠀⠀⠘⡄⠀⠀⠀⢀⣂⣀⣭⣵⣀⣿⡿⠟⠋⠁⣀⣀⣀⣀⠈⠉⠒⠋⠚⠛⠁⠀⠀⠀⠀⠫⠉⠭⠜⠛⠉
⠀⠀⠀⠙⠛⡛⢛⡻⠛⠉⠀⣹⠛⠛⢋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠀⠀⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⢠⠉⣐⢎⠐⠁⠀⠀⠀⠀⡇⢠⣿⣿⣷⣶⣶⣶⣶⣦⣤⣬⣤⣤⡀⠀⢿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢸⢰
⠀⡼⠣⢍⡁⢱⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠟⠻⣧⣐⣲⣆⡼⣸
⡘⢠⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⠛⣙⣤⣶⣿⣿⣿⢟⡠⠋⠔⠁
⠇⢸⡀⠀⠀⠀⠀⢀⣀⡀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢾⣟⠁⣿⣿⣿⣿⡟⠉
⠈⠀⢱⢤⡒⣈⣭⣤⠄⠀⢀⠗⠘⠛⢛⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣶⣿⣿⣿⢤⣤
⠀⡔⠁⠀⠚⠛⠋⣁⡤⠖⠉⠛⢶⣤⣤⣀⣈⣉⠉⠙⢌⠻⣿⣿⣿⣥⠖⣺⣿⣿⣿⢿⣫⣷⣿⣿⠀⣀⡀
⠀⠈⠒⠒⠒⠛⠛⠳⠤⠄⠀⣶⠻⠿⣒⣚⠛⠛⠛⠓⣚⣓⣍⠻⣿⣿⣷⠞⠛⣫⣵⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣐⡒⠫⠉⠩⠉⣙⣓⠒⠓⠾⠛⠿⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠠⠬⠥⢉⣑⣒⠒⠂⠨⠥⠄⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⡠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠠⢍⣉⠑⠒⠒⠤⠬⣭⣉⣉⣉⠀⢻⣿⣿⣿⠟⠁⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣈⣉⠛⠓⠶⠤⣤⣄⣀⣀⠀⠀⠉⠁⠀⢸⣿⡿⢃⡴⠊⠑⣢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⡂⠭⢉⡓⠒⠦⠤⢤⣉⣉⡙⠛⠻⠶⠶⠾⠋⠠⡏⠑⢢⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣚⠁⠀⠀⠈⢉⠚⠂⠀⠨⠉⠐⠂⠀⠀⠉⣍⣩⢿⠒⠒⢂⡄⠈⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣀⣀⡉⠒⠒⠀⠀⠀⣠⢮⢟⣰⢃⢞⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣦⣤⣽⣷⣶⡖⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣀⣈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⣠⣤⣤⣴⣶⣶⣦⣤⣤⣄⠀⠀⣀⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⢿⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⢤⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⣄⣀⠀⠀⠀⠀⠈⠛⠙⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣧⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠻⣿⣄⣤⣤⢠⣿⣿⠟⢋⠀⣉⣧⣤⣄⣴⣿⠋⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠁⠞⠉⢀⣠⣴⣿⣿⣿⠻⣯⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉⠀⠀⠀⠀⠉⠓⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣟⣻⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢻⡏⠉⣻⡿⣀⣿⠁⠀⠀⠀⠀⠀⠀⢠⣿⠿⢿⡉⠉⠙⢿⡿⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⢃⣼⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⡾⠃⠀⠀⢸⡟⣶⣈⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣯⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⢿⣿⡄⠀⠀⠀⠀⢰⣿⣛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠙⠻⢿⣽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠉⠙⣿⠙⣼⣿⠇⠀⠀⠀⠀⠈⠻⠿⠋⣀⣀⣠⣤⣤⣴⣶⣶⣦⣤⣽⣿⠛⠀⠀⠀⠀⠀⠀⠈⠿⣾⣿⠇⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⣰⣿⣿⡿⠃⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⢀⣴⣿⡿⠋⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣠⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⡄⠀⠀⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⢠⣶⣿⣿⣿⠿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠙⣿⣷⠀⢀⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⣦⡀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣦⣬⣉⣙⠛⠛⠉⠉⠉⠉⠀⢠⣿⣿⡇⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⠇⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣯⠍⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⢰⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣽⡗⠀⠀⣠⣶⣶⡾⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣧⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠆⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⡿⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⠀⠀⠀⢠⣴⡶⠶⠲⠶⣄⠈⢷⡄⠀⢠⣿⠀⢰⡿⢋⣉⠉⠻⠃⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠹⣿⣿⣿⡇⠀⠀⠿⠟⠁⢾⣿⠂⣈⣀⣈⠁⠀⠈⠁⠀⢀⣀⣼⣿⣃⣀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢻⣶⣌⠻⣿⡇⠀⠀⠀⠀⠴⠞⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠁⢩⠀⠀⠀⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⣰⡿⠁⣨⣷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢠⣴⣾⠟⢡⡄⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⠀⠀⣀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣨⡿⠷⢶⣄⢻⡂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠺⠂⠀⠀⠛⠰⠃⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠁⣠⠀⠈⢿⣾⣿⣆⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠷⠴⠶⠿⠿⠶⣄⣸⣿⣿⣿⡿⠉⣿⡀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⢠⣶⣰⠆⠀⠀⠀⠀⠀⡤⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠤⠤⠤⢤⠄⢹⣿⣿⣿⡿⠁⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢠⣿⠀⠀⢀⡀⣀⡀⠀⣾⣿⣿⣿⠃⠀⠀⠈⣿⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⢸⣿⠀⠠⣤⣿⣏⠛⣠⣿⣿⣿⣿⠀⠀⠀⠀⢸⣧⠀⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⠻⣿⣿⣿⠟⠉⠀⣿⠀⠀⠀⣠⠾⣿⠀⠈⢻⣿⣿⣿⣷⣤⣀⣀⠀⠀⢀⣀⣀⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⣿⡟⠀⠀⠀⢹⠀⠀⢠⠏⠀⣿⡇⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠇⠀⠀⠀⣸⠀⠀⡼⠁⠀⠘⢷⣄⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⠿⠛⠉⣿⡿⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⣼⣿⠀⢀⣀⣴⡏⠀⢠⡇⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⢈⣻⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⢤⣼⣿⣿⢰⣿⣿⡟⠠⠖⠻⠁⢸⠀⠀⠀⠀⠀⠈⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣠⣾⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢘⡇⠐⠿⣿⠇⠐⣿⡟⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⢠⡌⠻⣆⠀⠀⠀⠀⠀⠀⠀⠾⠃⢀⣼⣿⣿⡇⣸⣧⣤⣄⣀
⠀⠀⠀⣠⣴⠖⠛⠉⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣷⠀⠹⡄⠀⠀⠀⢠⣿⡟⠀⢹⣆⠀⠀⣀⠀⠀⠀⠀⣠⣿⠟⢿⣿⡇⣿⠀⠉⠙⠛⠛⠿⠷⣶⣶⣄
⣠⠀⠈⠁⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⣾⣿⡇⠀⠈⣿⡄⠀⢿⡄⠀⢀⣼⡟⠁⠀⠀⣹⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⣼⣿⡿⠀⠀⠀⠘⣷⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠻⣅⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠙⣷⣄⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⣠⣦⣠⣄⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡏⠤⢸⡇⣹⡆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣄
⠀⠀⢠⣿⠇⣠⣾⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⣠⣤⡀⠀⠀⠀⠀⠀⠠⣤⣥⣭⣤⣤⣥⣭⣥⡤⠀⠀⠀⠀⠀⠀⢹⠉⠛⠉⡟
⠀⢠⣿⣿⣾⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⠀⢀⣤⣶⣾⣶⣦⡄⠻⡿⠃⠀⠀⠀⠀⠀⠀⣿⠉⢹⡏⠉⣿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠸⡇⠉⢐⠏
⠀⣾⣿⡯⠀⠀⠈⠻⣿⡿⠋⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣏⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⣿⣶⣾⣷⣶⣿⣶⣾⡇⠀⠀⠀⠀⢀⣀⣤⡈⣂⠁⣤⣀⣀
⠀⠹⣿⣿⠦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢾⣭⣭⣥⣤⣭⣿⣥⣤⣬⣭⣭⡇⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⣇⣿⢸⡿⠿⣿⣧
⠀⠀⠈⠻⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀⠠⠿⠿⠿⠿⠿⠷⠀⠸⠇⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠾⠂
⠀⠀⢀⣺⡿⢂⡤⠤⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣸⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠈⡟⠙⠉⠉⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⢙⣟⣡⣴⠶⠛
⠠⣶⢿⣿⣷⣌⡖⠒⠒⢻⣧⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⠀⢠⣄⠀⠀⡀⠀⠀⠀⠀⣠⣾⣿⣿⢷⣷⠶⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠙⢿⣿⡏⢹⠛⠀⠀⣼⣿⠀⠀⠀⠀⠀⡿⢸⣿⡏⠛⣳⣾⣏⠀⢠⡇⠀⠀⠀⠀⣿⢸⣿⣿⠀⠙⠚⠓⡖⠒⠚⠁⠀⠀⠰⣿⠀⢠⡟⣿⣿⡇
⠀⠀⣾⣿⡇⠸⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⢳⣷⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠉⠁⣿⢿⡇
⠀⠀⣿⠘⣿⠉⢱⡖⠚⢿⠉⠀⠀⠀⠀⠀⠀⣿⠋⣿⠀⠙⢿⢻⡿⢸⣿⠀⠀⠀⠀⠀⢸⡇⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⢿⡄⠀⢠⣿⢸⣿
⠀⠀⠋⠀⠛⠀⠘⠃⠀⠛⠃⠀⠀⠀⠀⠀⠀⠛⠀⠙⠀⠐⠉⠙⠃⠘⠋⠀⠀⠀⠀⠀⠙⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠁⠀⠘⠃⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣤⣄⡀⢀⠀⠀⠀⢠⣄
⣾⠋⠉⠛⠛⠲⡶⠒⢸⣿⠈⢻⡆⠀⠀⠀⠀⣿⠉⣭⡍⣉⠉⠉⠉⣿⠀⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⣿⣿⣴⠀⠀⠀⠀⣸⠟⣛⢻⣿⠟⣛⢿⡀⣰⣷⢸⡿⢶⡟
⣿⠀⠀⠀⠀⠀⡇⠀⣸⣿⠀⢸⡇⠀⠀⠀⠀⣿⢀⠀⡀⠙⠿⣄⠀⣿⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣿⣿⣿⠀⠀⠀⠀⢻⣈⠛⣰⢿⣜⢛⣸⣇⣉⣬⣥⠀⠈⠉
⣿⠀⠀⠀⠀⠀⡇⠀⠙⠛⠀⢸⡇⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⣟⣿⣿⣿⣿⣿⣿⠿⠿⠟⠃
⣿⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⡇⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠛⠿⠿⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠟
⠛⠛⠛⠛⠻⠿⠿⠿⠛⠛⠛⠛⠃⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⠘⠻⠿⠿⠿⠿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠸⠿⠹⠿⠆


⠀⠀⢠⣄⡀⠀⠀⢰⣶⣆⠀⠀⠀⠀⠀⠀⢠⣶⡿⣷⣆⠀⣴⣾⣿⣶⡄⠀⠀⠀⠀⢢⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⣀⣻⣷⣉⣀⠈⣿⣿⣧⠀⠀⠀⠀⠀⢿⣭⡀⣨⣿⠀⣿⣭⣭⣽⡿⠀⠀⠀⠀⢸⣿⣄⠸⡿⣃⡄⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣶⣦⣤⣄
⠀⣾⣿⣿⣿⣿⣿⣷⠸⣿⣷⠀⠀⠀⠀⠀⠈⠛⣻⡛⠁⠀⠘⢛⣛⠛⠁⠀⠀⠀⠀⢸⣿⣿⣧⡈⢿⣿⣦⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿⣿⣿⣿⣿⠋⠉⢻⡀
⠀⣿⣿⣿⣿⣿⣿⣿⠀⠙⣛⣅⠀⠀⠀⠀⣰⣿⡏⡿⣷⠀⣾⡛⠟⣻⣆⠀⠀⠀⠀⢸⣿⠙⢿⣷⣄⠹⣿⣷⡄⠀⠀⠀⠀⡤⡄⠉⠉⢩⣭⡍⠉⠁⠤⡀⣹⡇
⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠙⠿⠀⠀⠀⠀⠻⣶⣖⣶⡿⠀⢿⣥⣤⣽⠏⠀⠀⠀⠀⢰⣿⣶⣶⣿⣿⣦⡘⠟⠋⡀⠀⠀⠀⢧⡇⣀⢀⣠⣿⣇⡀⢀⣠⠇⠈⠁
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⣿⢿⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣦⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣀⠀⠉⣻⡏
⠀⠀⠀⠀⠀⠀⣠⣶⣤⡀⠀⠀⠀⠀⠀⢠⣴⣤⡀⠀⠀⠀⠀⠀⢀⣴⣦⣄⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢨⣿⠻⣶⡟
⠀⠀⠀⣀⣀⣤⣿⠀⢸⣧⣤⣤⣤⣤⣤⣿⠁⢸⣧⣤⣤⣤⣤⣤⣼⡇⠀⣿⣤⣤⣤⣤⣤⣼⡏⠀⣿⣆⣀⣀⠀⠀⠀⠀⠀⠀⣼⠏⢠⣾⠃⣰⡟
⠀⣴⡾⠋⠉⠉⣿⠀⢸⡏⠉⠉⠉⠉⠉⣿⠀⢸⡯⠉⠉⠉⠉⠉⢹⡇⠀⣿⠉⠉⠉⠉⠉⢹⡇⠀⣿⡋⠉⠙⠿⣦⠀⠀⠀⣼⠏⢠⣾⠃⣰⡟
⣼⡏⠀⠀⠀⠀⣿⡀⣸⡇⠀⠀⠀⠀⠀⣿⣀⣸⡏⠀⠀⠀⠀⠀⢸⣇⢀⣿⠀⠀⠀⠀⠀⢸⣇⢀⣿⠇⠀⠀⠀⠹⣷⠀⣼⠏⢠⣾⠃⣰⡟
⣿⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⣿⣼⠟⢀⣾⠃⣰⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⢠⣾⠃⣰⡟
⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠏⢀⣾⠃⣰⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢀⣾⠃⣰⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢀⣾⠃⣰⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢀⣾⠃⣰⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣾⠃⣰⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⣾⠃⣰⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣾⠃⣰⡿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠈⠛⣿⡟⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣹⠇⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢋⣿⠆⠀⠀⠀⠀⠀⠀⢠⣥⣀⣠⡾⠋⠀⠀⣿
⣿⠀⠀⠀⠀⠀⣠⣤⣄⠀⣿⣧⡄⠀⠀⢠⡄⠀⠀⢸⣿⣤⣤⡀⠰⣿⣧⠄⠀⣀⣀⣀⡀⠀⣾⣣⣿⠋⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⢀⣥⣤⣿⠆⠀⠀⠀⠠⠶⣾⡷⠦⠀⢸⣿⠁⠈⣿⠀⠀⠀⠀⠈⠋⠉⠉⠉⢠⣿⠟⠁⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠸⣧⣤⣿⠇⠀⠀⠀⠀⠀⠸⠃⠀⠀⠸⣿⣤⣴⠟⠀⠀⠀⠀⠐⠶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠏
⣀⣈⣙⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢻⣦⣤⣀⠀⠀⠀⠀⢠⡴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠁⠀⠈⠳⣄⠀⠀⢻⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⢹⡷⢦⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠉⠉⠉⢷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⣄⠀⠈⠙⠋⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠸⣧⣤⡟⠀⠀⠀⠀⢀⣸⣀⣀⣀⣸⣀⣀⣀⣸⣇⣀⣀⣸⣃⡀⠀⠀⠀⠀⠘⢷⣤⡿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⣻⠀⠀⠀⡇⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦⠀⠀⠀⠀⠀⡇⠀⠀⢼⡋⢙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠀⠀⠀⠓⠒⠒⠒⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠃⠀⠀⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⣨⡷⠒⠒⠒⠒⠒⠒⠒⠲⣿⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡘⠃⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⣿⣀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣻⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣀⣀⠀⠀⠀⠀⢿⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠸⠧⠤⢴⡏⢉⣷⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣦⣀⣀⣠⠞⠁⠀⠀⣾⠋⢹⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠀⠀⠁⠀⠀⠀⠀⠘⠛⠛⠁⠀⠈⠳⣄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠺⣇⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⢀⠠⠀⠀⢀⣀⢰⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠀⢀⠄⡆⡆⡄⡀⣀⢀⢀⠀⠀⠀⠰⢲⠀⡖
⠀⠀⢀⠤⡀⠘⠾⠛⡇⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⢀⡀⠀⢋⠁⠀⠀⠺⠀⠈⠀⣠⠁⠀⠁⠁⠘⠈⠈⠁⠀⠀⢇⣼⠃
⠀⢀⠀⠀⢸⠀⢀⠀⡇⠻⡷⠀⠀⠗⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠀⠁⠀⠠⠄⠀⠒⠈⠀⠀⠀⡈⠀⠂⠀⠀⠀⠰⣇⠎
⠀⠞⠀⠀⢀⡗⠁⠀⠉⠠⠂⠀⠀⡀⠄⠊⠀⠀⠀⠄⢀⠄⡎⠁⠔⠈⢰⠀⠀⠀⠀⠁⠀⠐⠄⠀⠀⠀⠂⠀⠀⡀⠀⠀⢠
⠀⠂⠐⠀⠚⠐⠰⠂⠀⠀⠀⠀⠃⠀⠀⡘⠀⠁⠁⠀⠀⠀⠱⡄⠀⡠⠊⠀⠀⠐⠈⠀⠔⠀⠀⠂⠐⠒⠀⠀⠀⠩⠄⠐
⠀⠄⠀⠀⠀⠀⠀⢇⠐⠀⠀⠀⠄⠈⠀⢀⣤⢠⠀⢀⣤⢀⣀⡈⡊⠀⣀⠀⢀⣤⠀⠘⢁⠀⠀⠀⠁⠀⠀⠀⠄⠂⠈
⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⡼⢸⢸⠀⣾⣀⢹⡀⢹⣿⠀⣯⠇⣾⣸⠀⠁⠀⢴⣷⣶⣶⡶⠂⠀⠀⠀⠀⡁
⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⡀⢀⡏⢹⢸⣀⢧⡸⢻⠉⢸⡏⣷⡟⣆⡏⢸⡆⠀⠀⠈⠁⠁⠀⠀⠀⢠⡘⠀⠀⠁
⠀⢐⠀⠀⠀⠀⠊⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠈⠉⠉⠛⠁⠁⠈⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠐⠈⠀⠀⠠⠀⠲⠆
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠰⠥⠤⠄⠀⠠⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢀⠀⠈⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⡠⠂⠀⠀⠈⠂⠉⠉⠀⠀⠂⠀⠐⠀⠀⠀⠈⡼⡩⠓⠀⠀⠀⠀⠀⠀⣈⡁
⠀⠋⢹⠙⠀⡀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠂⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠗⠀⠀⢠⠀⢀⠀⠀⠦⠄
⠀⣄⡾⡠⠀⠐⠅⢪⡂⡠⠀⠀⠀⠀⠀⠁⠀⠀⠐⢒⠖⠀⠀⠀⠀⠘⠀⠀⢊⠐⠀⠂⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠂⠈⢄⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⠧⣰⠆⠀⠀⠎⠀⢰⠄⠀⢆⡆⣂⢴⢰⢠⡀⡆⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠦⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⠿⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⣀⠀⠀⣀⣀⠀⠀⠉⠓⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⢀⠔⠂⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢘⣶⠾⠷⠒⠒⠲⠶⠤⣄⡀⠀⠀⠀⢱⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢸⡀⢀⠀⣄⣸⠙⣄⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⣠⠟⣇⡾⣦⠏⠉⢳⠘⡄⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠀⠀⠈⢭⣍⣀⠈⠁⠀⠀⠀⠀⠀⡆⢳⠀⠀⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠀⠀⠀⠘⢛⣛⣋⣉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠓⠀⠀⠀⣷⢸⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⡼⣧⠚⠁⠀⠀⢀⡔⠉⢀⣤⡄⠙⣆⠀⠀⠀⢀⡔⢋⣉⠉⠑⢦⠀⠀⠀⠿⣼
⠀⠀⠀⠀⠀⠀⢾⠀⡇⠀⠀⠀⠀⢸⠀⠀⣿⣿⣷⡶⢸⠀⠀⠀⢸⢸⣿⣿⣤⡄⠀⡇⠀⠀⠀⢸⠳⡄
⠀⠀⠀⠀⠀⠀⠈⠦⣧⠀⠀⠀⠀⠀⠦⠤⣈⣛⠋⢀⡔⠀⠀⠀⠸⣌⠛⠛⢋⣀⣠⠃⠀⠀⠀⣜⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢠⠏⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠘⢧⡈⠳⢤⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠦⠤⣄⠀⠀⠀⠀⠀⡰⠋⠀⠀⣀⡠⠴⠛⡦
⢀⣤⣐⣒⡒⠐⠉⢂⣀⠈⢢⠈⠱⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⡔⠊⠁⠠⢤⣍⣁⡀
⣾⠭⠴⠶⠀⠀⠀⠀⠈⢳⢸⠁⠀⠀⠉⠳⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡞⠉⠀⠀⠀⢸⢇⡞⠉⠃⠀⠀⠸⠯⢽⣶⡀
⢳⣤⡤⠤⠤⠤⠤⢴⡖⠋⠸⡆⠀⠀⠀⢀⡞⢸⠉⠉⠻⢯⣋⣉⡽⠋⠈⣏⢧⡀⠀⠀⢀⡞⠈⠧⣄⣀⣀⡀⠈⠉⠓⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠹⢤⣤⠴⠋⠀⣼⠀⠀⠀⢠⣤⣤⡀⠀⠀⢹⠀⠙⠦⠤⠞⠀⠀⣰⠃⠀⠉⠉⠉⠓⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⣀⡴⡇⠀⠀⠀⣤⣤⣿⣿⠀⠀⠘⣧⣀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠛⠉⠁⠀⡇⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⢻⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣀⣀⣤⣤⣤⠤⠤⠤⠤⠤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢰⡶⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠐⠂⠀⠈⡗⠒⠒⠢⠴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣤⣄⣀⣠⣧⣀⣠⣤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠚⠳⣄⠀⠀⠀⡹⠞⠳⢤
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣤⣤⣤⣼⣀⣀⣸⣧⣤⣤⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠋⠉⠲⣄⠀⠀⠀⠀⠀⢠⠋⢸⠢⣄⣀⡀⠊⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⢀⠄⡤⡀⠈⢧⠀⠀⠀⠀⢸⢠⢏⡀⠀⡄⠈⠛⣹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣠⠾⠋⠘⠁⠙⣆⢸⡄⠀⠀⠀⠀⡏⣇⠈⣤⠀⢀⠟⠁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠹⣄⠉⢀⡂⣡⠇⣸⠁⠀⠀⠀⠀⢣⡈⢳⡶⠲⣏⣀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⣺⣷⣲⣾⣝⠒⠁⠀⠀⠀⠀⢀⠖⠉⢹⠓⠔⢻⠈⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⢇⢸⠉⣻⣨⠏⠉⢦⠀⠀⢠⠏⠀⠀⠀⡇⠀⡌⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠪⣈⢧⠃⣹⠀⠀⠐⡆⠀⡏⠀⠀⠀⠀⢹⣠⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣄⠀⠈⣀⠈⠁⠀⠀⠀⣧⠐⡇⠐⣷⡀⠀⣏⣉⣹⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡤⠿⢆⠀⠉⠀⠀⠀⡄⠀⡧⢸⢃⠞⣟⡷⡆⠀⠀⠀⣀⡞⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢸⠃⡨⣿⡇⠺⠀⠀⣠⠧⣀⢱⠈⠻⣸⢞⡕⠟⠰⢿⢿⠀⠙⢦⡞
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠛⠛⠓⡺⠿⢿⡖⣶⣾⣷⡀⠈⡟⠒⠒⣻⠎⠀⠀⣴⠟⢻⣄⣠⢿⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢺⠘⠿⠟⠛⡇⠀⣰⠃⠀⠀⠀⠀⢠⡏⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢹⠀⠀⠀⠀⠸⡀⠿⣖⡢⢤⣀⣰⠃⢳⠀⠀⢀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠛⠒⠓⠚⠓⠚⠃⠀⠀⢹⣉⣉⣥⠤⠼⠶⠞⠛⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⣲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⣠⠊⣁⠤⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠲⣄⠀⡴⠋⠀⠀⠀⣼⠵⠋⢀⣞⣡⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠼⠀⠈⠛⠀⣀⠀⠀⠘⠁⠀⠀⠈⠁⢠⣎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠚⠀⠀⠠⠀⠑⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠓⠀⠀⡀⠀⣰⠇⡰⠋⠘⣤⠟⢆⢀⠞⠢⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠀⠀⠀⠀⡜⠹⠜⠙⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⠱⠆⠈⢆⢠⢆⠀⠀⣀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠆⠀⣰⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⠈⣆⡀⠁⠀⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣎⡄⠀⡀⡇⠀⠀⣀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢀⡀⠸⢣⠀⠀⢹⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢛⠃⢰⠛⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠘⡄⠃⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠀⠀⠀⢀⠄⢒⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣠⣤⣄⠀⠀⣧⣦⠷⡵
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠓⠼⠀⠀⠀⠂⠀⢺⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣿⣿⠃⠀⠉⠙⠓⢢
⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⡀⠀⠀⠀⢆⠀⠈⠙⠛⢁⠄⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠙⠛⠁⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠑⠀⠀⠒⠁⠀⠀⠀⠀⠐⠢⡀⠀⠀⠁⠂⠀⠒⠁⠀⠀⠀⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⠤⠤⠶⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⠀⠀⡣⢿⡄⣼⣿⡄⢸⡇⡄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⡀⢠⣾⣿⣿⠀⢸⣧⢸⣿⣿⢧⡄⣿⣷⣶⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠈⠀⣬⠉⣀⣻⣿⣿⣿⡿⣤⣷⣿⣯⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⣣⡄⠀⠈⢀⣸⣿⣿⣿⣿⠙⢻⣧⣿⣧⣿⢯
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠓⠶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠒⣀⣮⣿⣿⣿⣻⣿⣿⡘⣿⣯⡿⢿⣿⣿
⠀⠀⠀⠀⠈⣿⣤⡀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠤⡐⠻⠟⠉⠻⠆⢾⣿⣿⣿⣽⣿⡏⣻⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣌⠉⠲⣶⣤⣤⣀⣀⡀⠹⣷⣤⠀⠀⠀⣀⠀⠀⠀⠘⠀⠀⠀⠀⣠⠁⠀⠀⠘⠃⠀⢠⣤⡼⠃⠀⣨⣽⣿⣿⣿⣿⣿⣿⢻⣿⣿
⠀⠀⠀⠀⠀⢿⡟⠳⣦⣤⣬⣤⣭⣬⣭⣽⣿⣿⠿⣶⣶⡦⠼⢦⣄⣀⣀⣀⣘⡃⠈⠁⠀⣠⣤⣀⡀⠈⣡⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠘⢿⣶⣾⣾⣿⣿⣾⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⢱⣿⣿⣿⡿⠿⣿⣿⣿⣀⣤⣤⣸⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡼⣿⣿⡁⡀⢠⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠆⠀⠞⠀⣤⣤⠀⢰⣿⣿⣧⡀⣁⠀⠙⠿⢀⣤⣴⣶⣾⣿⣿⣷⣿⣿⣷⣿⣷⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠂⠀⠀⢠⡦⡀⣀⡀⠁⠀⠸⣿⣯⣷⣿⣯⡈⣹⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠀⠀⠀⠺⠗⠀⠁⠇⠀⡰⠂⣶⠢⣿⣿⣿⣿⣿⡏⠹⣿⣿⣿⣿⠿⣿⢷⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠛⠋⠙⠉⠿⠷⠈⢿⡿⠿⠿⠆⣿⣵⣿⣿⣿⣟⣿⡟⣿⣙⣿
⣿⠿⠿⠿⡿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠆⠘⠀⠀⠀⡖⣦⢹⡀⠀⠀⠀⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡐⠒⠀⠀⠀⠀⡴⢱⢇⡎⣠⠂⡴⢠⠄⣀⠀⡴⡄⢀⡄
⠀⠀⡀⠈⢆⠀⠈⠉⢸⡇⣤⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠁⠈⠈⠁⠉⠐⠃⠀⢰⠁⢸⠇
⠀⡠⠀⠀⠘⡆⢀⠀⢸⡇⠻⠋⠀⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠆⠀⠀⠀⠀⠀⡉⠈⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣨⡉⡿
⢀⠇⠀⠀⠀⢼⠁⠀⠀⢀⠀⠀⠆⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠄⠀⠀⠀⠁⠀⠁⡄⠐⠃⠐⠀⠀⠘⠁⠀⠙⠴⠃
⠀⠀⠀⠀⠘⠀⠂⠒⠀⠀⠁⠀⡀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⠀⠂⠀⠀⠼⠀⠀⠘⠀⠀⣠⠂⠀⠀⠀⠀⡀⠰⠀⠀⠈⠉⠉⠁⠀⠂
⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠮⠀⠀⣀⠂⠀⠀⢠⣶⣶⣤⣤⣄
⠀⠀⠀⠀⠀⠸⡀⠂⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⢠⠀⠀⠀⠁⠀⠀⢄⠀⠉⠀⠠⠁⠂⢀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠁⠀⢂⠀⠀⡠⠄⠈⠫⡁⠈⠀⠀⠀⠀⣀⡀⠀⠀⠐⠈⣀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⣋⡀⢠⠁⠀⠀⠀⠀⠀⠂⠂⠁⠀⢠⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠆⠀⣠⠃⠀⣀⠀⠠⠀⠜⠣⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⠀⠈⠀⠐⠀⠀⠀⠘⠀⠁⠀⢀
⡀⠈⠨⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⠡
⠂⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⢸⢀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠇⠈⠀⠀⠀⠀⠀⢐
⠁⠀⠀⠀⠀⠐⠛⠀⠀⠠⠀⠀⠐⠃⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠌⠈⠀⠀⠀⠄⠁⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠀⠀⡴⠈⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢡⢋⠟⠋⠰⠀⠀⠀⠰⠀⠠⢁
⢠⡴⢤⠖⡆⠠⢀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠂⠀⡞⠀⠀⡀⠀⠀⡠⠀⠀⠀⠒⠁⠸⠒⠀⠀⡀⠀⠀⡄⠀⣁⡀
⠈⠀⢸⠀⠁⠀⠀⠢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠋⢉⡶⠂⠀⠀⠀⠈⠁⠀⠘⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⠀⢠⠀⠀⢠⡀
⢠⣆⡿⣄⠔⠀⠀⣀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣴⡉⢠⡀⠀⠬⠰⠀⠀⡄⡤⠠⡄⡄⢠⢀⡄⢠⠀⠈⠀⠀⠑⠁⠀⠀⡀

⠀⡔⠂⠀⠀⠂⠄⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠒⠤⣀⢀⡎
⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠿⠀⠀⠀⠀⠀⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣶⡶⠂⠀⠀⠀⠰⣿⢆⡀⠀⠀⠀⢰⠁⠀⠀⠀⠑⣿⠗⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣶⡇⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⠀⠂⢤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⢿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋⠀⠀⠀⠀⡼⠉⠉⠐⠠⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠞⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠀⠀⠀⢀⡰⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣠⡾⠁⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⢸⣿⣦⣤⣶⡆⠀⠀⠉⣿⡶⢾⣧⣀⡀⠀⠀⠀⠉
⠀⠀⠰⣿⣾⡿⠋⠉⠙⢻⣧⣤⡜⢿⡏⠀⠀⢘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⣿⡟⠙⠿⢷⣤⣤⣾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢉⠶⠎⠁⠈⢳⡀
⠀⠀⠙⠛⠻⠂⠀⠀⠀⠐⠛⠳⠀⠀⠘⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠔⠚⠛⠛⠛⠓⠦⠘⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠰⠀⠰⠀⠂⠂⠀⠀⠰⠂⠀⠀⠀⠖⠀⠰⠐⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠄⠠⠀⠀⠀⠂⠰⢀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠄⠀⣄⠀⠀⠰⠀⡀⠀⠚⠠⠄⠀⡀⡁⠀⠠⠁⠀⠀⠀⠀⠀⠀⠠⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠀⠀⠀⠀⡜⠀⠀⢀⠤⠊⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠠⠀⠊⠀⠀⠀⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠈⠀⢀⠀⢀⠀⠀⠄⠀⠀⠀⢀⠃⠀⠀⢀⠀⡰⠀⠀⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠉⠉⠉⠛⠓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⣀⣴⡶⠿⠿⣶⡄⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠤⢀⡀
⠀⠀⠀⠀⡀⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣼⡟⠉⠀⠀⠀⢀⠁⢀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠶⣶⣦⣄⠈⠱⢦⡀
⠀⠀⡠⠂⢀⣤⣤⣄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⠔⠃⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣄⠈⢃
⠀⣤⠀⢠⡿⠋⠉⠻⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⠀⠀⢠⡼⠋⢀⡼
⠀⠇⢀⣿⠃⠀⠀⠀⠟⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢀⣀⣀⣀⣀⣀⣤⡞⠃
⠀⣤⣀⣿⡀⠀⠀⣴⡀⠀⠿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢀⣴⠟⠛⠛⠛⠛⠋
⠀⠻⠟⠉⠀⠀⠀⠙⠿⣦⣄⠈⢱⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠛⢋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⢷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠋⠀⢲⣦⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣶⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠴⠚⠛⠛⠛⠛⠛⠛⠛⠓⠒⢠⣤⣤⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠛⠁⠀⠀⠀⠀⣀⣤⡶⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠛⢶⣄⣻⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⢀⣴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡆⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡶⠀⠀⠀⠀⢀⣼⠏⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠀⠘⠆⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⢈⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠉⠁⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⠁⠀⣈⠃⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠞⠁⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⡀⠀⢈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠻⠷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⣠⣄⣠⣴⢧⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡴⣦⣄⣠⣄
⣀⠁⠀⠀⠄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠻⣷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣾⡟⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠈
⣿⣀⣀⡀⠀⠀⠀⠈⣦⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠹⣿⡀⠈⠻⣦⡀⠀⠀⠀⠀⣴⠁⠀⠀⠀⢀⣀⣀
⠀⠉⠉⠉⠀⠺⣆⠀⠈⠿⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⢸⡇⠀⠀⣠⠿⠁⠀⣰⠗⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠙⢦⡴⠋⠙⢧⣄⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡼⠋⠙⢦⡴⠋⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣀⠀⠀⠀⠀⣠⡞⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⣄⠀⠀⠀⠀⣀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⣀⣠⠞⠃⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠘⠷⣄⣀⣴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣦⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠛⠛⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠛⢻⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢀⣀⣀⣀⣀⡀⢰⣿⡀⣀⣀⠀⣀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠉⠉⠉⣿⠉⠉⠉⢹⣿⠉⠉⣹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡿⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢀⣼⡇⠀⠀⠀⢸⣿⣄⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⠀⣻⡇⠀⠀⠀⢸⣿⣁⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣶⣷⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣿⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⣿⣿⣿⣿⣿⠛⠋⠙⠟⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡿⠟⠋⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⠀⠀⢀⠤⠾⠿⠿⢿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛
⠀⠀⠀⠀⣀⣤⣶⠟⠋⠁⠀⠀⢀⣠⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠰⣶⣶⣤⣄⣀
⠀⣠⣴⣾⠿⠋⠀⠀⠀⢀⣠⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⣽⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣷
⣿⠟⠋⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⠁⠀⡿⠟⠋⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀
⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣋⣽⣿⣿⣿⡿⠟⢿⣷⣶⣤⣀
⣿⣿⣿⣿⣿⣾⠉⠉⠛⠻⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢈⠍⢺⣿⡇⠀⢸⣿⠟⠉⠙⠛⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⣠⣶
⣿⣿⣿⣿⣿⣿⣦⣤⣶⡾⠛⠃⣀⠖⢶⣄⠀⠀⠀⣤⠿⠟⠃⣰⣄⣀⣴⡾⠛⣁⠀⠛⠃⣀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣦⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣼⣿⡿⠄⠀⢿⢿⡾⢛⣽⠟⢶⣴⠿⠀⠐⢋⣠⣍⣿⠏⢀⣼⠋⠑⠂⠀⢀⣬⣿⠶⢦⣤⣼⣿⠏⠁⢸⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡘⠁⡿⢣⣾⠿⠁⠰⠿⣶⠛⠋⢀⣄⣉⣿⠿⣿⣷⣾⣿⣿⡧⢠⣾⣿⣦⣄⣠⣿⣦⡾⣿⣿⣿⣷⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⣃⠀⠀⢠⣤⣉⣷⣶⣾⡿⣿⠷⣶⣤⣾⠋⠀⣉⡳⢾⠛⢍⣙⣿⠛⠻⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣩⣿⣷⣶⠎⣛⢉⣩⡿⠀⠀⠤⢱⡌⠛⠁⣧⣀⢨⣶⡿⣷⣶⣀⣻⣿⣷⣦⣤⣩⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣭⠤⣿⣦⣩⣶⣿⢤⠄⠀⣻⡀⠈⣥⣼⣿⣭⠟⠻⠟⢋⣿⣿⣿⣿⣿⣿⡿⢻⣷⣽
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⣷⣿⣽⠶⣿⣉⢛⣿⣿⡟⣯⣾⠿⣿⣿⣿⣽⣯⣀⣀⠀⠀⠀⢉⣿⣿⣦⣭⣿⣿⡟⠻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣷⣿⣿⡳⡝⠲⣶⣬⣽⠟⠁⣠⣵⣏⡃⠺⣦⠟⢈⣿⠉⣿⣿⣿⣴⣾⣿⣿⣿⣿⡿⢷⣼⣿⣷⣾
⣋⠿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⡳⠏⣧⣉⣙⣷⣶⠿⠷⣶⣾⣿⣿⣤⣙⢛⣛⣶⣿⠟⣷⣿⣽⣿⣿⣿⠻⠖⣿⣿⣷⣿⡋⠘⣿⣿