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⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣒⣲⠦⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⡴⠁⠀⠀⠀⠀⠉⠀⠱⣤⣴⣶⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣿⣿⣿⣿⣷⣿⣦⠀⠀⠤⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣼⠁⠀⠀⠀⠀⠀⠂⣀⣠⡽⣿⡯⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⣽⡿⠿⣃⡤⠔⠋
⠀⠀⠀⡴⢺⣽⣛⣹⣤⣤⣤⣶⣶⠿⠿⠛⠛⡹⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠉⠀⣛⣛⡛⠫⢭⣑⣒⣬⣵⠆⠀⡀
⠀⠀⠀⠉⠛⠛⠛⢋⠻⣍⠁⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢁⠀⠲⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⢿⡿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⢫⣾⣿⣷⣶⣵⣿⠛⠛⢛⢿⣧⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠁⠀⠈⠉⠃⠙⣿⣿⣿⢿⣧
⠀⠀⠀⢰⣿⣿⡿⠟⠃⣿⣿⣿⣿⡿⠿⠟⠃⣓⣬⣾⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⣐⣀⣤⣼⣿⣽⣾⣿
⠀⠀⠀⠈⣿⣕⣦⣘⣧⡉⣋⣤⣤⣶⣶⡿⣹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣿⣿⣶⠾⠿⠛⠛⠋⠉⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣭⣛⣻⣛⣯⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣦⣤⣀⣠⣴⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠷⠿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠉⠙⠻⢿⣿⡿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣄⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣟⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⣀⣀⣤⡤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⣿⠛⠛⠛⠛⠻⣷⣶⠚⠉⢉⣠⡶⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣨⣿⣶⣿⣿⠗⢁⣨⣽⣶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠺⠿⠟⢉⣁⣤⣴⣾⣿⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⣤⡶⠾⢛⠿⠟⣻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠷⠒⠂⠀⠀⠀⠀⠀⠀⠁⠀⠹⣅⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣷⡶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⡉⠙⠛⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠙⠁⠀⠀⠀⠈⠉⠙⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣴⣶⣶⡦⣤⣴⣄⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠷⠄⠈⠈⠀⠉⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠐⠀⠀⠀⠂⠂⠂⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⡶⠤⣄⢀⡤⠖⣺⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⢿⡷⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⠋⢁⣠⡾⠋⢙⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠃⡉⠩⠅⠀⠀⠁⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⡴⠾⠋⢁⡤⢞⡽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣥⣶⠶⠶⠶⠓⠛⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢀⣀⡴⣚⡭⠞⡩⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠠⠀⠄⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⢠⣾⡵⠖⠋⠛⢏⣥⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣶⣡⠍⣭⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⢀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⣼⢽⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠈⠀⠁⠈⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣆⠠⣾⢏⣼⣟⠀⢣⠀⠀⠈⡱⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⠠⠂⠀⠠⠀⠐⠠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢏⠁⠀⠹⣿⢿⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠄⠀⢀⡐⠐⡠⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⢀⣷⣄⡰⢃⣥⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠄⠈⠉⠀⢀⠖⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⠛⠻⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢁⡐⠴⠒⠢⣂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⣠⠊⠁⠀⠊⠀⢈⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠎⢑⢦
⠀⠀⠀⠀⠄⢂⠀⠀⠉⠁⢠⠃⠐⠁⠈⢀⢀⠀⢌⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠈⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠄⠠⡠⠤⠀⠮⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠅⣦⠀⠀⠀⠀⡠⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠄⠢⡀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣠⢊⠀⠀⢀⡠⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠀⠮⠔⠚⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣠⠖⠋⢩⢭⠉⠙⠲⣄⠀⣀⣀⢇⣀⡀⢀⡴⠚⡯⢹⠉⠉⠓⢦⡀⠀⠲⣎⡶⠂⠀⣠⡶⠛⠋⠍⡩⢛⠶⣄
⠀⠀⡞⠡⠈⠁⢈⡋⠀⠐⠃⠈⣇⠀⡵⢼⡍⢠⠯⠤⠬⠤⡈⠒⠒⠒⣒⣻⡄⠜⠁⠙⠀⡼⢃⡠⠀⣂⠉⠤⠎⡠⠊⣧
⠀⢸⡇⢎⣉⠆⠀⣀⣁⠺⠗⠀⢸⠀⠀⠀⠀⣾⠤⠤⠄⠤⠊⠘⠑⠘⠐⠒⣷⠀⠀⠀⠀⡗⣁⡠⠊⣀⠠⠐⣉⠤⢚⢽⡇
⠀⠘⣇⢠⢤⠀⠘⠦⠞⠀⣤⠀⣼⢀⡶⡲⠆⠸⣖⠲⠀⠶⠀⠦⠉⠉⠭⢭⡇⢰⢩⢵⠀⢯⡴⢜⠡⠐⠋⠤⠐⠊⠜⣺⠁
⠀⠀⠘⢮⡉⠀⠃⠀⢠⠀⣠⡼⠁⠈⠧⠥⠆⠀⠹⣖⣒⣒⣒⣒⡦⢠⣲⠞⠀⠀⠁⢁⠀⠈⠳⣕⢪⠔⢒⡀⠉⢀⡴⠃
⠀⠀⠀⡄⠉⠓⠒⠲⠒⠛⠁⠂⠁⠈⠀⢀⣠⢤⡀⠀⠉⠓⠒⠒⠚⠋⠁⢆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⢛⠉
⠀⠁⢊⢈⡀⠀⠀⠀⠀⣴⣦⠀⠂⠀⠀⣾⢼⡻⠯⠀⠀⠀⠀⠀⠐⠂⡁⠈⠀⠔⠂⠠⢀⡂⡀⠀⢾⡿⠆⠀⠻⠃⠰⠄
⠀⠀⠀⠀⠃⣀⣠⢤⣀⣉⠉⠀⠀⠀⠀⠈⠛⠛⠁⠀⢀⣀⣤⣄⣀⡐⠒⠂⢴⠀⠐⠂⠻⠟⡀⠀⣈⣉⣤⣄⣀
⠀⠀⢀⣴⡛⠀⠀⠀⠀⠨⠷⢦⠀⢠⠋⠔⡈⢛⣿⣟⠉⠁⠒⠒⠉⠙⠳⣄⠀⠀⢀⡈⠀⠁⡴⠊⣅⠀⠄⣠⢌⠙⢦⡀
⠀⢀⡏⠀⠉⠐⠤⣀⣀⣐⣩⣂⣷⠀⢆⠡⠐⢄⠙⢬⣉⣉⣛⣛⣋⣉⣩⠹⣆⠀⠙⠁⠀⡾⡰⠄⠀⠀⠨⠩⠄⠘⠃⢳
⠀⢸⡉⢑⡒⠠⠤⠀⠀⠀⠠⠤⢼⠀⠀⠑⢄⠙⠷⠢⣈⠒⠶⢶⣒⠒⠖⠒⣿⣶⣄⠀⠀⡇⠈⡡⠤⡈⠀⠐⠂⠘⠃⢸⡇
⠀⠀⢧⠘⢋⠠⠀⠀⠐⢒⡒⢂⡞⠀⠀⠀⠀⠙⡦⢌⡑⠺⢖⡀⠤⢉⡑⠂⠛⢈⠟⠳⡀⢳⡀⡑⠥⠂⠀⡔⡒⢄⠉⡼⠁
⠀⠀⠈⠻⣤⡒⠀⠌⠀⢢⡥⠞⠀⠒⠠⢐⠃⣀⠈⠲⣌⡳⠦⢤⣀⡀⠀⠛⠉⠁⠊⣀⠇⠀⠳⢤⣀⠐⠀⠑⢒⣨⠞⠁
⠀⠘⠀⠠⠀⠉⠙⠛⠉⠉⠆⠀⠀⡼⠐⠪⡄⠀⠠⠀⠀⠉⡙⠛⠉⠩⠉⠈⠁⠁⢉⣢⠀⢾⠆⠀⠉⠉⠛⠋⠉⠀⢠⣤
⠀⠐⠈⠀⠁⠂⠀⠀⠈⡀⡀⡈⠀⠀⡜⢢⠀⢁⠀⠀⠀⠂⡐⠀⠀⢄⠘⠀⠄⡤⠒⠉⠃⠄⠄⠀⠾⠆⠀⠀⠀⠄⠈⠁
⠀⠀⠀⠀⣀⡤⠴⠶⠤⢤⡀⠀⠴⡉⠀⠀⣁⠦⠀⠀⣠⡤⡴⢦⠤⣄⡸⠇⠀⠠⡀⠀⠀⠈⠀⣀⡤⠴⠶⢦⢤⣀⠈⠃
⠀⠀⢠⠞⠁⠈⠉⢸⠀⢀⡞⢳⡀⠀⢂⡜⠀⠀⡰⠛⠁⢁⢊⣘⡦⡼⣻⢦⠀⠓⠀⠈⠑⢢⠾⠭⠤⠤⠬⠤⠒⣩⠷⡄
⠀⢠⡟⡀⢴⡒⠉⠀⢀⡴⠛⠛⢷⠀⢀⡀⠀⢸⠳⡄⠐⣇⠖⣀⣀⢣⠳⠬⡇⠀⠠⠀⠀⡿⠬⣉⣓⢒⣒⣒⣋⣀⣤⣻⡄
⠀⢸⡄⠈⠐⡈⠉⠒⠄⠓⢢⠀⢸⠀⠣⠜⠀⢸⠳⣌⢉⡢⠈⠀⢇⠘⠒⢒⡟⠀⠀⠀⠈⡯⢕⣒⣂⣤⣬⠵⠖⢚⠽⢺⡇
⠀⠀⢳⡀⠀⢰⠀⡠⠃⠀⠐⣇⠏⢠⡶⡲⣄⠘⣦⣓⣈⣀⠇⢀⠀⢬⣅⡼⠃⢀⣰⣄⠀⢻⡔⠒⠀⠩⣈⣉⣉⠅⢒⡟
⠀⠀⠀⠙⠦⣄⣒⣁⣀⣠⠔⠋⠀⠘⢦⣥⠟⠀⠈⠳⢤⣀⣒⣄⣳⡽⠚⠁⠀⠈⠹⠋⠀⠀⠙⠶⣤⣐⣒⣂⣤⠼⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢀⡀
⠀⠀⠀⠀⠀⢀⠚⠀⠀⡉⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠀⠀⠀⠉⣂
⠀⠀⠀⠀⠀⢘⠀⠀⠙⠇⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠁⢨
⠀⠀⠀⠀⠀⠀⠢⣀⠀⢀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠈⣀⡀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣤⣀⣤⡤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢
⠀⠀⠀⠀⠀⠀⡔⠊⠁⠈⠑⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠃⠈⠁⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠭⠀⠉⠲⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡤⠀⣀⣀⢸
⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣈⡂⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠁⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠗⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡠⢀⡂⢀⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⡀⢠⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠐⠽⠞⠀⠀⠀⠀⠱⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠸⢨⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢉⣇⡀⠀⡀⠄⠀⢨⡉⠆
⠀⠀⠀⠀⠀⠣⡳⡀⠀⠀⣰⣭⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠚⠉⠒⠈⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠂⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⠄⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠢⢄⠘⠚
⠀⠀⠀⠀⠀⢀⠃⠀⢀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡋⠀⠀⠀⠀⡸⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠼⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⠒⢄⣠⠒⠉⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠘⠢⠀⠄⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠓⠎⣀⣀⡱⠚⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡰⠊⠁⠀⠈⠓⢄⡄
⠀⠀⠀⠀⠠⣲⣇⡀⠀⠀⠀⣀⣸⣖⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣰⣀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠱⡉⠈⠉⠉⢁⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⠑⠦⢤⢜⣢
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⢻⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⠀⠀⣀⡀⣠⡤⠤⠀⢸⣧⣤⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣾⠿⠀⠘⠋⠀⢀⡀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⢀⣀⠀⣶⣶⣾⣿⣧⣀⣰⣬⠻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⢀⣴⣿⡟⢀⠈⢿⢿⠏⢽⣿⣿⡗⠆⢻⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⣾⣿⣿⣷⡆⠀⠰⠖⠀⠀⠙⠋⠁⣀⣀⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠘⣿⣾⠟⠯⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣿⣯⠂⠀⡀⢨⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣯⣤⣤⣄⠙⣷⣳⣢⠀⠀⠀⠀⠀⠀⠙⢀⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⠋⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣹⣿⣿⣿⣿⣿⣏⢹⣦⡔⡂⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣶⣿⣿⡈⠁⢀⠈⢀⡀⠀⠀⠀⣾⢿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣽⣶⡟⠀⠀⠈⠁⠈⠁⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣿⢋⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣤⡀
⠀⠀⢹⣿⣿⣿⣿⣿⣼⣷⣦⣤⣀⣀⡀⠀⠀⠉⠛⠷⣶⣦⡀
⠀⠀⠀⢻⣿⣿⣿⣿⡟⠻⠉⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢹⡏⠙⠈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣆⠀⠀⢠⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣌⠀⠀⠀⣿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠻⡭⢿⡒⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣿⣿⣶⣿⣜⠇⣠⢿⢷⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⢀⣷⠀⠀⠀⠀⠀⠀⣴⣿⣿⣤⣿⣏⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⣟⢿⡃⠀⢸⣿⡆⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠷⠾⠀⣤⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣾⣿⣿⡿⠽⣷⣄⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣀⣀⣀⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡇⠀⠘⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⠈⠀⠀⠀⠀⠙⢿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⠠⠚⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢲⠋⡁⠀⠀⠌⠉⢘⠯⣥⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⢀⡠⠔⠆⡉⠹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⣠⠀⢠⡷⠺⠇⠀⠰⣰⡟⢸⠀⠂⠄⠀⠀⠀⠀⠀⠀⣾⠧⢤⠶⠋⣠⠂⠁⠀⠀⢀⣻⣤⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠂⣠⢰⡇⡦⠜⠁⢀⠄⣫⢞⡱⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠠⠀⠀⠁⠐⡡⠊⣁⡉⢸⣏⡈⠉⠁⠘
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠛⠊⠁⠀⠀⣀⣤⣮⣵⡟⢁⣼⡟⠀⠀⠀⠀⠀⠀⠴⠊⡉⣿⠴⠟⣋⡠⠖⢃⠄⠡⠈⠵⢀⠋⠁⢤⡀
⠀⠀⠀⠀⠀⣀⡀⢸⡌⠨⣥⢤⡤⡶⢶⠛⠛⣭⡴⠋⠀⣾⡟⢁⠀⠀⠀⠀⠀⠄⣀⣀⣐⡘⢓⣀⣀⣀⣈⠬⡤⠐⠒⣨⠟⠘⠀⠈
⠀⠀⠀⠀⠀⠂⠖⡜⠳⣶⡠⣆⠵⣖⣯⣿⡿⠋⠀⢀⡞⣣⣤⡛⠃⠀⠀⠀⠀⠀⠀⢀⡈⡄⠀⠙⠳⢮⣤⣴⣦⠶⢚⡁
⠀⠀⠀⠀⠀⠀⣜⠀⠀⠈⠙⠓⠲⢾⠿⠋⠀⠀⣠⠋⢨⠁⢿⢻⠀⠀⠀⠀⠀⠀⠀⠈⡡⠐⠂⢘⡃⠆⠘⠀⠐⠀⠀⢠⠀⠆
⠀⠀⠀⠀⠀⠀⠿⡀⠫⣇⡀⠤⠂⠁⠀⠀⢀⠔⠁⠀⠀⠒⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⢀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠰⠀⠐
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢠⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠀⠀⠀⠐⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⣿⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢢⢐⣅⠒⡀⠂⢠⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⠀⡀⠁⢈⠛⠟⣹⣄⠁⢲⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠠⣿⠀⠀⠀⠀⢸⣿⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠛⡉⠀⠠⠀⠀⠉⠀⠀⠈⠁⠀⢉
⠀⠀⠀⠀⠀⠀⠐⠈⢰⣿⣾⣿⠒⢀⠀⢠⠝⠋⠋⠀⣆⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠏⢲⣄⣴⡟⡐⠞⡈⠀⣄⠶⣿
⠀⠀⠀⠀⠀⠀⡛⠢⢉⣯⣹⣍⢁⣀⣠⡂⠧⠤⠔⢊⠓⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠌⠀⣈⠈⢽⡇⡀⠠⠖⠯⣼⡾⠃
⠀⠀⠀⠀⠀⠀⢠⠔⡈⠉⣠⠁⠉⠋⣼⠀⢠⢄⡌⠀⡂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣁⠀⠈⠁⠘⠓⣀⡰⠏⠁
⠀⠀⠀⠀⠀⠀⠈⢢⡕⠠⡀⠀⠠⡴⢜⣄⡠⠀⢄⡉⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣦⣰⢄⠁⠨⠀⢢⣤⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠛⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⠀⢀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⠁⠀⠀⡈⢉⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡠⠆⠒⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠒⢠⠡⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣷⠀⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠀⢰⠁⠈⠀⠀⠐⢠⠀⡀⠈⢃⠃⠀⠀⡨⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡇⠐⠒⠉
⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⡘⢀⡈⠓⠀⠄⠸⠀⠐⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⣠⠀⡇⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⠀⠈⡤⠀⠈⣿⠠⣄⣈⠆⢄⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠰⣯⡄⣱⡀⠀⣀⠶⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠆⠉⡍⠂⠄⢺⠀⠈⠆⠀⠈⠈⠛⢃⠌⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡠⢀⡝⠀⡀⣰⠀⠀⣀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠙⠻⠿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠣⢤⣤⣄⡄⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠃⠀⠐⢠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠀⠀⠁⠀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⡿⣻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣭⣽⣭⣯⣯⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⡠⠒⠊⠉⠉⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠉⠉⠉⠒⠢⣀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠉⠉⠁⠀⠠⣀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠉⠉⠁⠀⠢⢀
⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⢀⠞⠀⠀⠀⠄⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢉⠀⠐⠃⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠢⡀⠀⠀⠂⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠁⠂⠤⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠈⠒⠄⢀⠠⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠈⠒⠤⠀⠀⠀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠤⠒⠁


⠀⠀⠀⠀⣀⠀⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠤⣀⠄⣂⡤⠴⣮⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢔⡨⠞⠹⠅⠀⢠⢏⠃⠀⡠⠂⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⠑⢄
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠐⠀⢀⠠⣪⠔⠋⠀⠀⠀⠈⢶⡋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢘⠀⠀⠀⢀⢔⡥⠚⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈
⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡠⣻⡊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠢⣀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⢈⠞⠁⠑⣄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⡀⠌⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠉⠁⠀⠐⠈⠁⠀⠀⠀⠀⠺⣤⠤⢖⠏⠙⠐⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠐⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣠⠒⠋⠙⠹⠷⣦⡀⠀⠀⠀⠀⠀⠀⠈⠹⡿⣳⣦⡀⠀⠀⠀⠀⢀⣰⣎⡹⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣦⡀
⠀⠀⡠⠄⠀⠀⠀⠀⣍⢻⣿⡄⠀⠀⠀⠀⠀⢈⠀⣩⢉⢻⣿⡄⠀⠀⢀⠏⠀⠍⠀⠢⣤⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠸⢻⣿⡆
⠀⠀⡛⠀⠰⠄⠀⠦⡀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠇⢭⣰⢾⣿⡇⠀⠀⠘⣷⣶⡄⠀⢠⣿⣿⣿⡇⠀⠀⠀⠐⠒⠶⠿⢾⡶⢿⣿⡇
⠀⠀⠐⢠⠀⠀⠀⠀⣉⣾⡿⠀⠀⠀⠀⠀⠀⢀⠑⣾⣿⣿⡿⠁⠀⠀⠀⠈⣿⣇⢀⣿⣿⣿⡿⠁⠀⠀⠀⠀⢐⠉⠓⡿⠟⣿⡿⠁
⠀⠀⠀⠀⠐⠠⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠶⠷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠚⠓⢶⣤⡀⠀⠀⠀⠀⠀⠰⠞⢙⢿⣶⣦⡶⢿⣷⠀⠀⠀⠠⠈⠂⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣦⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣉⣽⣷⡀⠀⠀⠀⠀⠀⠐⠁⠋⣴⣿⣷⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡄
⠀⠀⠀⢀⣈⣉⣉⣁⣁⣅⣿⡇⠀⠀⠀⠀⠀⠀⠐⢋⣽⠟⣡⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⡇
⠀⠀⠀⠀⠈⠙⠛⠃⠙⢫⡿⠁⠀⠀⠃⠀⠀⠀⠀⣉⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠠⠶⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠠⠬⠿⠟⠋⠀⠀⠈⠂⠀⠒⠐⠾⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠠⡴⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠉⢉⡽⠛⢹⡏⠻⣯⡙⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⢀⣴⠋⠀⠀⢸⡇⠀⠈⠻⣄⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣇⡀⠀⠀⢀⡾⠁⠀⠀⠀⢸⡇⠀⠀⠀⠙⣧⠀⠀⠀⢀⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠈⠙⠳⢦⣾⣅⣀⣀⠀⠀⢸⡇⠀⠀⣀⣀⣸⣧⡴⠞⠋⠁⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⣸⠇⠉⠉⠉⠛⠛⢻⠛⠛⠛⠉⠉⠉⢻⡄⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠻⡟⠛⠛⠛⠛⠛⠛⢻⠛⠛⠛⠛⠛⠛⠛⣿⠒⠒⠒⠒⠒⠛⢺⡇
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⣸⣧⡴⠶⠶⠖⠛⢻⠛⠛⠳⠶⠶⢦⣿⣄⡀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣴⠞⠋⠉⢻⣆⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣼⠃⠉⠙⠳⣦⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠻⣆⠀⠀⠀⢸⡇⠀⠀⢀⣼⠃⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⣀⠀⠙⢷⣄⠀⢸⡇⢀⣠⠟⠁⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣿⣷⣼⣷⣿⣥⠶⠶⠛⠋⠁

⠀⠀⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀
⠀⠀⠀⣴⣿⣿⣿⡿⠥⣄⡀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠒⠲⠀⢀⡀⠀⠀⠀⠀⠀⢀⣞⠁⣀⣠⣾⣿⣿⣿⣦
⠀⠀⡸⣿⣿⣿⣏⠀⠀⠀⠀⠀⣦⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠈⠢⣾⣧⡀⠀⠀⠀⣿⣿⣶⣭⡿⠿⣿⣿⣿⣿⣇
⠀⠀⠃⠈⢻⣿⣿⣿⣷⠀⠀⣰⣿⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⣼⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠉⢻⣿⣿
⠀⠀⠘⡄⣼⣿⣿⣿⣿⢀⣴⣿⠏⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⣰⣿⣿⡿⠁⠀⠀⠀⠻⣿⣿⣿⣿⡆⢀⣠⣿⣿⠃
⠀⠀⠀⠀⠙⠿⢿⠿⠿⠻⠛⠁⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣾⡿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣴⣿⠿⠛⠁

⠀⠀⠀⢀⠔⠈⠁⠀⣹⣿⣶⣄⠀⠀⠀⠀⠀⢀⠤⢢⣛⠾⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠈⠑⠢⡀
⠀⠀⠰⠁⠀⠀⢠⣾⣷⣮⣽⣿⣧⠀⠀⠀⢀⠊⣼⣿⣿⣿⣿⣿⢿⠟⢷⡀⠀⠀⠀⠌⠈⢿⣉⠁⠀⠀⠀⠀⠈⠄
⠀⠀⣷⣦⡀⠀⠈⠛⠻⣿⣿⣿⠁⡆⠀⠀⠈⠀⠛⠛⣿⣿⣿⣿⡿⠀⠈⠃⠀⠀⢸⠀⠀⠀⠀⣼⣿⣶⣄⡀⠀⢸
⠀⠀⢻⣿⡏⠀⠀⠀⠠⣿⡿⠃⢠⠁⠀⠀⠰⠀⠀⠀⢸⣿⣿⡿⣀⠀⠀⠆⠀⠀⠈⡄⠀⠀⠀⠘⢿⣿⣿⠃⠀⡘
⠀⠀⠀⠻⡅⠀⠀⠀⠀⠋⢁⡠⠁⠀⠀⠀⠀⠑⢄⠀⠀⠻⠟⠁⠁⡠⠊⠀⠀⠀⠀⠈⢄⡀⠀⠀⡿⠋⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠁⠒⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠐⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⡀⠀⠤⠖⠒⠢⡀⠀⠀⠀⠀⠀⣀⠠⠐⠒⠈⣁⠀⢸⡀
⠀⠀⠀⠀⢠⣀⣱⡉⠙⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠬⠂⢀⣄⠈⢁⢄⣦⠐⠛⠈⠆⠀⠸⣇
⠀⠀⠀⣠⡏⢧⠀⠡⠀⠰⣶⣶⡄⠀⠀⢀⠀⠀⠀⠀⢸⣿⣷⣽⠛⠂⠀⠀⠀⠂⠀⠀⠀⣟⡀
⠀⠀⢸⡝⡗⣎⠇⠘⣇⠀⢻⢿⣷⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⡿⡄⠐⠀⠀⠂⠀⠀⠀⠀⣿⡔⡀
⠀⠀⠈⣇⢻⣿⣾⡄⠸⣆⠀⠄⠊⠀⠀⠂⠀⠁⠉⠀⠈⣿⣇⠚⣇⠐⠀⠀⠀⠀⠀⠈⠃⠹⣧⢱
⠀⠀⠀⢹⡄⣽⣿⣹⠀⢿⡆⠐⠃⠀⠐⠀⠀⠀⠀⠀⠀⢸⣿⣅⣹⡄⠀⡀⠀⠀⠤⠤⡀⠀⢹⡇⡆
⠀⠀⠀⠀⣷⠹⣯⢿⣇⠈⢿⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃⣈⣤⣶⠶⠶⠦⣤⣤⣼⣷⢽⡄
⠀⠀⠀⠀⠸⡆⢹⣿⣽⠀⢸⡟⠤⡀⢐⣀⠀⠀⣀⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣶⣮⣽⣿⣟⣎⢆
⠀⠀⠀⠀⠀⢻⠀⣿⡇⡇⠀⠀⠀⠘⣾⣿⣿⣿⣷⣿⣿⣟⣿⣿⣿⣿⡿⠿⣿⣿⣿⠿⢷⣾⣽⣾⣯⣽⢥⠓
⠀⠀⠀⠀⠀⠘⠆⢸⣿⠰⠤⣴⣶⣿⣿⣿⣿⣿⠿⠋⠉⠻⣯⣿⣻⣿⠃⠀⠀⠀⠉⠙⠛⠺⠿⣭⣟⣿⠿⣟⠁
⠀⠀⠀⠀⠀⠀⢰⠀⠫⠻⠿⣿⣻⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠗
⠀⠀⠀⠀⠀⠀⠀⠑⠐⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣾⣳⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣣⡾⠻⢿⡿⠟⢷⣜⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠔⡀⠀⢀⣀⣠⣤⣯⣾⠃⠀⣀⢵⡮⣀⠀⠈⣷⣽⣤⣤⣀⡀⠀⢀⠢⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣛⠉⠡⠈⠉⠙⢷⢪⠞⢫⡝⠳⡕⡾⠋⠉⠀⡍⠉⣛⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠘⣿⣿⡆⡄⠀⡠⣳⢏⣤⣴⣦⣄⡹⣾⢄⠀⢠⢠⣿⣿⠃⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠂⠀⠭⢻⣶⣻⣗⡿⢋⡤⢶⡶⢤⣙⢿⣺⣟⣶⡟⠭⠀⠐⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣯⣿⣇⠀⠀⡠⢿⢾⣺⢟⡧⠏⢾⡩⢍⠷⠹⢼⡻⣷⡷⡿⢄⠀⠀⣸⣿⣽⡶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣩⡾⠁⠀⡹⣓⡽⢞⡿⢋⣭⠏⡀⡠⣈⡁⢈⣁⢄⠀⠹⢭⡙⢿⡳⢯⣚⢏⠀⠈⢷⣍⣆
⠀⠀⠀⢀⣀⣀⣠⣷⣿⣆⢄⣜⡾⣁⢀⣾⢠⣏⡞⡳⠧⠸⢿⣳⣞⡿⠇⠼⢞⢳⣹⡄⣿⡀⣈⢷⣣⡠⣰⣿⣾⣆⣀⣀⡀
⠀⠀⠀⠈⠉⠉⠹⡾⣿⠏⠊⢫⢷⡉⠈⢿⠘⣏⢧⡵⡦⢰⣾⡽⢯⣷⡆⢰⢮⡼⣹⠃⣿⠁⢉⡾⡝⠑⠹⣿⢿⠏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣝⢷⡀⠀⣱⡭⣳⢮⣷⣌⣓⡆⠀⠑⣉⡅⢨⡉⠊⠈⢰⣚⣡⣾⡵⣞⢭⣎⠀⢀⣾⣏⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣟⣿⡏⠀⠀⠑⣾⢾⢿⣮⡖⣆⢶⣑⣊⡷⣰⢲⡵⡿⡷⣷⠊⠀⠀⢹⣿⣻⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠤⠀⣒⣼⠿⣽⡯⣷⣌⠓⠾⠷⠚⣡⣾⢽⣯⠿⣧⣒⠀⠠⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢠⣿⣿⠇⠃⠀⠑⡽⣎⠛⠻⠟⠛⣱⢯⠊⠀⠘⠘⣿⣿⡄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣭⣀⡐⠀⣀⣠⡾⢜⢦⣜⣣⡴⡣⢷⣄⣀⠀⢂⣀⣭⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠢⠁⠀⠈⠉⠛⠛⣟⢿⡀⠀⠉⡺⢗⠉⠀⢀⣿⣻⠛⠛⠉⠁⠀⠈⠔⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡝⢷⣴⣾⣷⣦⡾⣿⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢿⡽⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⡿⠿⢿⠛⠿⣿⣿⣷⡶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⢿⣃⡾⠀⡐⠌⠐⠋⠈⠀⠙⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡿⢛⡛⣉⣉⣁⣄⣀⣤⣀⣀⣀⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⢻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣬⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⢿⠿⠻⠛⠛⠛⠃⠒⠟⠛⠹⠟⠟⠻⠿⠟⠻⠟⠻⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠹⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⣤⣤⣤⣄⠀⠀⠀⠀⢀⢛⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠝⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⣿⣿⣿⣿⣷⣾⣿⣿⣧⡄⠀⠀⡤⢈⢩⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣉⠛⠂⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⢎⣉⣉⡿⠿⠷⠾⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣘⣛⣿⣷⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡠⡀⣀⢀⣤⢀⣄⡄⣠⠾⠆⠤⠐⠐⠒⠊⠁⠆⠀⠀⠀⠉⠛⡿⠉⠋⢭⣿⣿⣧⣍⢋⡉⠹⣿⣿⣷⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⠟⠙⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠈⠙⠧⡄⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣋⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣏⣀⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣨⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣾⣿⣿⣭⣤⣴⣤⣶⣶⣶⣶⣶⣾⣷⣶⣪⣴⣶⣶⣆⠴⣦⡶⣦⣤⣄⣀⣐⣠⣤⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠋⣨⣴⣿⣿⣿⣿⣿⣿⠿⡛⠛⠙⠉⠋⠉⠉⠉⠉⠛⠉⠀⠁⠀⠘⠻⠻⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣡⢀⣂⣤⣠⣤⣴⣶⣶⣶⣤⣤⣴⣤⣀⣀⠀⠀⠀⠀⣹⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣹⢻⣿⣿⣷⣾⣷⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡛⠏⢛⣿⣛⡛⠻⠙⢟⢿⣿⣿⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣽⣀⡆⠠⠦⡌⠠⡝⢛⡚⢢⣮⣾⣿⣿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣥⣶⣷⣿⣿⣷⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⣀⡀⢀⣀⠀⢀⣀⠀⣀⡀⠀⣀⡀⢀⣀⠀⣀⣀⠀⣀⡀⢀⣀⡀
⠀⠘⠧⠇⠀⠀⠀⠻⠟⠀⠙⠃⠀⠛⠃⠀⠀⠀⠘⠂⠀⠀⠀⠐⠁⠀⠻⠃⠀⠧⠟⠘⠿⠇⠸⠿⠃⠣⠼⠀⠧⠟⠘⠷⠇⠸⠿⠃⠧⠟⠘⠧⠟⠸⠾⠃
⠀⠀⠶⠀⠀⠖⠀⠸⠗⠀⠰⠂⠀⠾⠆⠀⠂⠀⠰⠂⠀⠐⠀⠀⠀⠀⠀⠀⠠⣞⣷⢰⣟⡦⢰⣹⡆⣞⣳⠀⣮⡑⢰⢛⡆⢰⣻⡆⡞⣷⠰⣿⡦⢰⣿⡆
⠀⠠⣤⠄⠀⠤⠀⢠⡄⠀⠀⠀⠀⢠⠀⠀⢤⠀⠰⠆⠀⢴⠄⢰⣆⠀⠀⠀⢀⡬⢥⢀⠍⠄⢠⣥⡄⣠⣥⠀⡬⢄⢀⠭⡄⢠⢥⡀⡤⣤⢀⡬⣄⢠⢤⡄
⠀⠀⣀⡀⠀⡀⠀⢀⡀⠀⣠⡄⠀⣀⡀⢀⣤⡀⠀⡀⠀⢀⠀⠀⣁⠀⢀⡀⠀⢓⡋⠈⣊⠂⠘⣛⠁⢙⡛⠀⢓⡋⠈⣒⡃⠘⣚⠁⢛⡋⠈⣛⡃⠘⣚⠁
⠀⠀⠙⢃⠀⠉⠀⠈⡁⠀⠀⠀⢀⠉⠁⠘⠚⡁⠈⠁⠀⢉⠀⠈⠋⠀⡘⠋⠐⢧⠿⠸⢿⠗⠸⣽⠇⠿⡿⠀⢷⠿⠘⠿⠇⠸⡿⠃⢯⠿⠸⠥⠇⠸⡿⠇
⠀⠀⡔⠉⢹⠀⢠⣾⣿⡆⠀⠌⢉⢡⡄⠀⢿⣿⣦⠀⠈⠉⢡⡄⠀⠉⠉⣤⠠⡞⣲⢰⣞⡆⢰⣷⡆⣶⣷⠠⣾⣶⢰⣟⡆⢴⣓⡄⣞⣶⢰⣞⡆⢴⣷⡆
⠀⠀⠙⠒⠋⠀⠈⠛⠛⠀⠀⠀⠒⠛⠁⠀⠘⠛⠋⠀⠀⠐⠚⠁⠀⠐⠚⠋⢀⡬⣅⢀⣭⡄⢠⢭⡀⣨⢥⢀⣬⣅⢀⣭⡄⣨⢭⡀⡬⣅⢀⣭⡄⢠⣭⡀
⠀⠀⡜⠛⢡⡀⠀⣿⣿⣄⠀⠔⠉⢡⡄⠀⣾⣿⣤⠀⠰⣻⢧⡄⠀⣺⣿⣄⠈⢓⠋⠈⣛⠃⠘⣛⠁⢙⡚⠈⢋⠋⠘⣓⠃⠙⡚⠁⢓⠋⠈⣓⠃⠙⣛⠁
⠀⠀⠘⠲⠛⠁⠀⠛⠿⠋⠀⠀⠲⠞⠁⠀⠛⠿⠋⠀⠀⠺⠟⠁⠀⠙⠿⠋⠐⢏⡽⠸⣍⠅⠺⣽⠇⢯⡽⠐⢏⡽⠸⣭⠇⠺⣹⠃⢏⡽⠸⣏⠇⠺⣿⠇
⠀⠀⠐⠀⠀⣒⠀⠰⠆⠀⠠⠆⠀⠶⠀⢠⡖⠀⠰⠆⠀⠰⠆⠀⠀⠀⢰⡳⠐⣦⣠⡶⠀⣠⡄⠀⠄⢀⠀⣦⣄⡀⠀⠀⢠⣤⠀⢆⡄⢰⡂⢰⠄⣶⣴⡄
⠀⠀⠄⡀⢀⣤⠀⢀⡄⠀⣀⡄⢠⣤⡄⢠⣄⠀⣀⡀⠀⣤⡄⢀⣤⡀⢀⣄⠀⡉⣭⠁⠀⣩⠀⢀⠎⠁⠆⠀⢠⡆⠠⠤⣌⣈⠀⠃⠀⣌⡃⣦⠒⠠⣤⠄
⠀⠀⠁⠀⠀⣉⠀⠀⠁⠀⠉⠀⠀⠉⠀⠈⡁⠀⠈⠁⠀⠉⠁⠀⢉⠀⠈⡁⠀⣑⠒⠿⠟⠰⣿⠈⡂⢙⠃⠀⢾⡷⠠⢈⠛⠂⠀⡁⠈⠓⠟⢨⣤⣤⣤⡄
⠀⠀⠛⠀⠈⠒⠁⠘⠛⠀⠂⠐⠀⠀⠀⠀⠛⠀⠹⠗⠀⠋⠁⠀⠹⠀⠀⠟⠀⣨⣀⢀⣀⡄⠉⠀⠁⢸⡽⠇⠄⠈⠂⠈⠲⠌⢀⣬⠭⣤⡀⠸⡿⠿⣿⡇
⠀⠀⢔⣠⡀⠀⠀⡶⡖⠀⠀⠐⣮⠀⠀⠀⠀⡀⡆⠀⢐⢀⢁⠀⠀⣅⣀⡂⠀⠉⠠⠟⢣⡤⠀⠀⢐⠀⠀⠀⠈⣶⡄⡁⣤⡀⢾⣟⣿⣿⣿⠀⣦⡓⢤⠄
⠀⠀⠽⢟⠓⠀⠀⠻⠃⠅⠀⠀⠏⠘⠀⠀⠙⠀⠃⠀⠈⠛⠸⠀⠀⠼⠀⠄⠀⠶⠺⠶⠶⠄⣠⢃⣋⡀⠄⣀⣘⢛⡃⡦⢀⣄⡘⠷⣿⠾⢋⠒⠙⠃⠀⣄
⠀⠀⠀⡐⠀⠄⠀⠀⡀⠠⠀⠀⣀⢀⠀⠀⠀⣀⠀⠀⠀⢀⣀⡀⢠⠀⠀⡠⠀⠛⠂⠘⠛⣇⢀⠈⠛⠁⡘⠿⠿⣨⣭⣾⡿⡋⢴⣳⣄⠒⠋⠤⠘⠃⠘⠁
⠀⠀⠀⠻⠉⠇⠀⠘⡿⢶⠀⠀⠀⠈⡀⠀⡀⢟⢇⠆⠀⠛⠉⠑⠈⡲⠟⢀⠀⡚⢃⡐⠟⠈⢸⠘⣃⠋⡇⣴⣲⠄⣌⣿⣧⡑⡦⠄⣉⣸⣻⣿⡎⡀⢟⠂
⠀⠀⠄⠠⠀⠄⠄⠀⠠⠀⠄⠄⠠⠠⠀⠄⡤⠠⠀⠠⠀⠄⠠⠠⠀⠄⠄⠠⠈⣋⡀⠄⠄⣚⢈⢿⣭⡿⠣⠶⠀⣠⢄⡉⠍⠲⡇⢃⠻⠿⠿⢋⣬⣥⢨⢅
⠀⠀⠒⠘⠀⠂⠀⠐⠐⠀⠀⠂⠒⠀⠀⠂⠒⠐⠐⠀⠀⠂⠂⠘⠀⠂⠐⠘⠈⠛⠃⠐⠚⠛⠒⠒⠒⠒⠂⠒⠒⠓⠒⠒⠛⠛⠂⠂⠘⠂⠒⠚⠻⠒⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⣀⣀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⢶⡏⢿⣿⣿⢿⠛⠋⣥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣶⡦⢯⣿⣷⣄⡀⠚⠟⠀⣰⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡌⠹⣿⣿⣿⣶⣶⣾⣿⣿⣿⠿⠯⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣆⠈⠙⢿⣿⣿⣿⣿⣿⣿⡲⣄⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣿⠏⠙⢻⣿⣶⣀⣀⣉⠙⢛⣻⣷⣹⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠟⠀⠠⠘⢋⣿⢿⣿⣿⣷⣿⣿⣳⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⡿⢻⣿⣿⣽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠸⠛⢛⣿⠟⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣧⣀⣐⣾⣂⠉⠀⠀⠀⠀⠀⠶⣤⣤
⠀⠀⠀⠀⠀⠀⡰⠂⣀⠀⠀⠀⠀⠀⠀⠰⠁⣼⣿⣯⣙⣛⣿⣬⣵⣦⣤⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⣼⣿⣿⣿⣿⣿⠿⠋⠁⠹⠡⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣨⣌⡽⠏⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⣿⠋
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⢀⡛⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣟⣀⣤⠄⣀⣀⠀⠀⠀⣀⣀⠽⣧
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣅⣀⠀⠀⢩⡏⣁⠀⢠⣿⣿⣿⣧⡈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀⠀⠀⢀⡘⠛⠛⠛⠛⠛⠛⠒⠀⠀⠸⠷⠾⠿⠿⠿⠿⣷⡘⣆⠀⠀⠀⠀⣡⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠖⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠄⠀⠼⠿⠟⠋

⠀⠀⠀⢀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⡴⣔⡁⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⢃⠀⠀⠄⠀⢀⠠⠀⠀⠀⠀⠀⠀⠔⠖⠚⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⡀⠂⠃⠀⠀⠁⢢⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢐
⢀⠑⠚⠰⠦⠖⠁⠲⡤⢮⡆⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⢀⠰⣠⠃
⠘⠈⠰⠐⠠⢀⠠⢶⠄⠄⠋⠀⠀⠀⠀⠨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡉⠰⠂⠂⠈⠒⠀⡈
⠸⡷⠄⠄⠀⠀⢲⣦⡄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠀⡄⠊⠠⠔⠁
⠀⠘⢤⡠⠔⠂⠀⢤⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⡄⠒⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠖⠀⠀⠀⠂⡀⢀⠱⡄⠒⠄⠤⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠁⠀⡠⠏⠀⠒⠀⠈⡠⣜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡤⠔⠀⠡
⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠁⣐⢠⠀⠠⣅⣉⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀
⢀⠀⠀⠄⠀⠀⠀⠉⠉⠉⠀⠃⠀⠀⠀⠀⠉⠔⠆⠀⠘⠄⠀⠈⣁⠋⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠔⠃
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣖⣯⣼⣟⢲⣦⣽⣤⡀⢠⠀⠀⠈⠀⣠⣦⡙⣅⠀⠙⠷⣦⡀⠀⠀⠀⠀⣠⡴⠖⣲⢤⡀
⠀⠀⠀⠀⣰⣶⣯⡿⣿⣿⣿⣿⣿⣿⣦⣼⣿⡌⠄⣄⠀⠐⢅⠀⠑⢤⡐⢄⡀⠈⢑⡀⠀⠀⠐⠋⡀⠈⢁⣼⣿⡀
⠀⠀⠀⠀⣿⢄⣽⣷⡌⠻⠿⢿⣿⠭⣭⣁⣿⣷⡠⠏⠀⠀⠀⠑⢄⠀⠉⠲⢌⡑⠺⡇⠀⠀⠘⣁⠀⣠⠔⡻⣿⠁
⠀⠀⢀⣈⡟⢻⡿⣛⣿⣿⣷⣾⣧⡴⣿⣿⡟⡿⠃⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⣨⡿⠀⠀⠀⠀⠈⠛⠿⠷⠚⠁
⠀⢠⡼⢃⣀⣘⣿⡏⠙⣋⠈⠛⢋⡄⠈⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠵⠈
⠀⠀⠈⠑⠻⢏⠙⣻⣿⣧⣦⣿⣼⡳⢾⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠉⠉⢉⡿⠿⣿⠟⠋⠑⠁⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠂⣐⣦⣤⡀⠀⠀⣤⣄⡰⣧⢠⡇
⠀⠀⣤⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣚⣃⣀⡀⠐⣒⣀⡀⣘⣻⣦⠀⠀⠲⠥⠤⠚⠁
⠀⠸⣏⣉⠉⢉⠀⠀⠀⢀⣴⣯⣤⠀⡟⣳⣦⡀⠀⠀⠉⠉⠙⠇⠈⠈⠉⠉⠛⢨⠙⠩⡇⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠛⠛⠋⠀⠀⠀⢶⣶⠿⣏⠛⣉⡿⢿⣷⠀⢰⣶⣦⣤⡤⣤⣶⣄⠠⢤⣤⣤⣼⡧⠀⠀⣤⡾⠿⠿⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⣷⣾⣿⡀⣠⡟⠀⠈⠋⠛⡛⠿⠿⠓⠛⠛⠙⠉⠉⢿⠇⠀⢸⣏⠤⠴⠛⠿⠶⣿⡆
⠀⠀⠀⠀⣴⣞⢦⠀⠀⠀⠉⠉⣉⣿⣿⣿⠟⠀⠀⠀⠈⠺⣉⣛⡛⡛⢛⣛⣛⣿⡿⠋⠀⠀⠘⣷⣶⣶⣶⣖⣲⡾⠁
⠀⠀⠀⠀⠘⠻⠛⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠴⢋⠁⣤⣤⠈⠉⠳⣄⠀⠀⠀⠀⠀⠀⣠⡖⢋⡧⢸⠁⠈⠙⠲⡄⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⢈⡠⢋⠝⢦⡀
⠀⠀⠀⠀⠀⢰⠏⢂⡈⠀⢨⠁⡀⠘⠃⡈⣧⠀⠀⠀⠀⢰⠇⠀⠂⠠⣄⠉⡉⢉⣉⡹⡆⠀⠀⠀⠀⡼⢃⠄⠂⡀⠤⠤⠊⡠⠊⢹⡄
⠀⠀⠀⠀⠀⢾⠀⢅⡨⠀⢀⠤⡄⠻⠃⠀⢸⠀⠀⠀⠀⣿⠤⠤⠄⠒⠃⠘⠋⠘⠒⠂⣻⠀⠀⠀⠀⣟⣁⠄⢊⡀⠄⢂⡡⠔⢊⢵⡇
⠀⠀⠀⠀⠀⠸⡆⡀⡄⠀⠘⠶⠋⠀⠤⢀⡞⠀⠀⠀⠀⠸⡦⠴⠀⠚⠀⠫⠭⠥⠤⢠⡏⠀⠀⠀⠀⢳⡦⢊⠅⠂⠉⠐⠒⡈⠐⣹⠇
⠀⠀⠀⠀⠀⠀⠙⢮⡁⠀⠁⠀⠤⢀⣠⠞⠀⠀⠀⠀⠀⠀⠙⢮⣭⠭⠥⠤⠶⢐⡶⠋⠀⠀⠀⠀⠀⠀⠻⢵⣨⠔⡂⠉⠍⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⢞⡉⠒⠀⠤⠤⠬⠓⢦⠀⠰⠁⠄⡐⢉⣻⣟⣉⠁⠀⠂⠀⠉⢳⣄⠀⠀⠀⠀⠀⠀⡴⠋⣡⠀⠄⣒⡢⠉⠳⣄
⠀⠀⠀⠀⠀⢰⡇⠀⠉⠒⠤⢀⣀⣐⠍⣈⣧⠈⢄⠡⠑⢄⠑⢬⣉⣉⣉⣉⣉⣉⠉⠼⣇⠀⠀⠀⠀⡼⢱⠄⠀⢀⠂⠁⠑⠈⠛⠹⡆
⠀⠀⠀⠀⠀⢸⠉⣑⡒⠀⠤⠤⠤⠤⠤⠀⢾⠀⠀⠑⢄⠙⠟⠢⣄⠀⠲⠄⡀⠰⠐⠒⣿⣷⣄⠀⠀⡇⠀⠠⡠⠌⠀⠀⠂⠘⢃⠀⡇
⠀⠀⠀⠀⠀⠘⣆⠓⡁⠄⠀⠀⠀⢀⡀⢰⡏⠀⠀⠀⠀⠹⡶⢌⡁⠸⠃⡀⠠⠉⡉⠀⠛⢀⠙⠱⡄⢱⡀⡑⠉⠂⠀⠰⣂⢠⠈⣰⠃
⠀⠀⠀⠀⠀⠀⠈⠣⣄⠐⠀⠀⠐⣢⡵⠋⠀⠀⠀⠀⠀⠀⠈⠶⣌⡳⠶⣤⣁⣀⠀⠛⠁⠀⠈⢀⠄⠀⠙⢦⣀⠐⠀⠈⢀⣨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠈⠉⠁⠈⠉⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢦⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠴⢤⣤⣀
⠀⠀⠀⠀⠀⠀⣰⠊⠀⠈⠉⡝⠀⢀⡟⢦⡀⠀⠀⠀⠀⠀⡰⠛⠡⠡⠈⢈⡤⠸⢛⢦⠀⠀⠀⠀⠀⢀⡴⠯⠥⠬⠥⠤⠔⢋⠷⣄
⠀⠀⠀⠀⠀⢰⢣⠀⢤⠐⠁⠀⢀⡬⠛⠼⣷⠀⠀⠀⠀⢸⠃⡀⠰⡅⠴⢀⣀⢣⠱⠬⣇⠀⠀⠀⠀⡾⠬⣉⣑⠒⠀⠒⢊⣁⣠⢼⡆
⠀⠀⠀⠀⠀⢸⠀⠁⢰⠉⠉⠲⡌⠒⢦⠀⢸⠀⠀⠀⠀⢟⠢⡍⠀⡊⠉⠁⢎⠘⠲⢒⡿⠀⠀⠀⠀⡯⢑⣒⠂⢀⣩⡽⠴⢒⡪⠭⡟
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢡⠀⡰⠁⠀⠘⣠⠏⠀⠀⠀⠀⠸⣆⠃⠈⢀⠆⢀⠀⠉⠉⣴⠃⠀⠀⠀⠀⠹⣆⢀⠀⠉⠁⣒⡒⠉⠀⣺⠃
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣘⡘⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠣⣤⣀⡠⣈⣳⡽⠞⠁⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⣠⡬⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠠⠘⠇⠀⠀⠂⠀⠀⠀⠠⠌⡃⠰⠤⠇⢀⠀⡈⢋⠀⡉⢀⠉⡀⠨⠨⠀⠀⠨⠀⠀⢉⠶
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣄⠒⢀⡀⡤⠀⠀⠀⣠⣀⠀⠀⠀⡀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠈⠀⠀⠐⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠚⠂⠀⠉⠀⠐⠀⡀⢀⠀⠂⠀⠀⠂⠀⠂⠠⠀⠀⡀⣘⠢⡀⠀⠠⠀⢐⠀⠂⣉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠧⠀⠂⠀⠀⠀⠀⠀⠀⡆⠸⠀⠀⠀⠀⠀⣲⢒⣖⠀⠀⠴⠀⢀⠀⠨⡀⢁⡁⡠⠀⠀⠀⢐⠀⠀⠐⠙⡀
⠀⠀⢈⢀⡀⠐⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣄⢀⠀⡀⣀⢀⠀⡀⢀⠀⠀⠀⠠⡆⠀⠐⠖⠒⠀⠀⡆⠀⠀⠀⠀⠈⠂⠊⠀⠀⠠⠀⠰⠀⠀⢉⠶
⠀⠀⢐⣒⡂⠀⠀⢀⠀⠀⠛⠀⠃⠀⠀⠀⠀⠀⠘⢎⠀⡇⠈⠾⠀⣿⠈⠶⡁⠀⠀⠆⢠⠀⢠⠀⠘⠂⢖⠀⠀⡅⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⢈⠀⠀⠀⢠⡄
⠀⠠⠄⠠⠄⠈⠀⠀⠀⠈⡹⣈⣁⣉⣉⣉⡇⠀⠉⠈⠀⠁⠈⠀⠁⠉⠈⠀⠁⠀⠂⠂⢸⠀⢸⠀⣳⡆⡀⡄⠠⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢴⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢶⠶⠶⢶⢶⣶⡆⢸⣻⡅⠀⠀⠤⠄⠠⠠⠤⠀⠀⠀⠀⠆⢸⠂⢸⠀⢹⠃⠈⠁⠀⠀⠀⠀⢀⡀⡀⡀⣀⢀⡀⣀⠀⡀⡀
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠉⠀⢈⠉⠉⠉⠁⠘⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢸⠂⢀⣀⡀⠀⢀⠔⠒⠢⡄⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉⠉⠁⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠁⠘⠈⠀⠘⠐⠀⠩⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠡⠂⠘⠀⠠⠒⠀⠘⡸⡀⠛⢀⡕⠄⠀⠈⠑⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣀⠘⠁⠋⠁⠈⠘⠀⠃⠀⠀⠐⢤⡀⢀⣄⠀⣤⠀⢀⣀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠰⠂⠀⠀⠨⢯⡉
⠀⢀⡄⡀⣠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⢸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣴⡄⠃⠀⠀⠀⠀⠀⠀⠀⢈⣉
⠀⠀⣀⣁⣀⣈⠀⠀⠀⠀⠀⢠⡄⠀⡀⠠⡄⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠔⠁⠀⢰⣦⠀⠂⠀⠀⠐⡾⠂
⠀⠀⠉⠉⠙⠉⠀⠁⠀⠀⠀⠈⠀⠀⠁⠀⠀⠈⠁⠈⠀⠀⠁⠀⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⠀⠀⠀⠆⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠋⢉⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⡂⠀⡤⠠⠀⠀⢨⡿⣤⠀⠀⠀⠀⠀⠀⠀⣴⢃⡠⠔⠋⠐⠈⠀⠈⢷⣀⡀
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢸⠡⠄⠁⠀⢐⠔⡩⢡⡏⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⡈⢹⣇⠉⠁
⠀⠀⠀⠀⠀⠀⡇⠀⡀⠁⠈⠀⣀⣤⡶⢧⠟⢁⣾⠇⠀⠀⠀⠀⠀⠊⠀⢷⡒⠉⠴⠞⠁⠀⠀⠐⣀⠁⠐⠄
⠀⠀⠀⠀⠀⡀⣻⣀⠚⠒⢚⣉⢭⣤⡶⠋⢀⡾⠋⣤⠀⠀⠀⠀⠐⠂⠀⠈⠠⣤⣒⠐⣐⢂⣈⠴⠃
⠀⠀⠀⠀⠀⡘⠀⠉⠣⠯⣵⣾⡿⠋⠀⡠⠋⠚⣿⢦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠉⠉⠉⠀⠃
⠀⠀⠀⠀⠀⢇⠐⢰⡀⠄⠂⠀⠀⢀⠔⠁⠀⠂⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠢⣀⣀⡀⠤⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡀⠁⠑⠙⣂⠀⠐⡄⠘⡀
⠀⠀⠀⠀⠀⠀⡀⢠⠐⡏⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⣁⠀⠀⢀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⣤⠀⠿⠯⠽⠀⠀⠀⠀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⠁⠶⠎⡋⠈⢁⣠⢰⣻⠇
⠀⠀⠀⠀⠀⢀⠀⠚⢚⠓⠲⢛⠃⢀⡉⠄⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⠐⠂⠀⠇⠠⠍⠘⡽⠃
⠀⠀⠀⠀⠀⠈⣀⠀⠊⠀⠀⠌⣂⢈⠈⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠀⠂⠂⠉
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣠⣄⠀⠄⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⠒⠀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀
⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠠⢀⠈⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣆⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡀⠀⠀⠃⠀⠀⠠⢡⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠀⠐⠊
⠀⠀⠀⠀⠀⡂⠀⠀⠀⢀⡈⠀⢀⠐⠀⠈⠀⠈⠀⠀⠀⠀⠀⢀⡄⢘⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⡃⠀⡄⠁⠢⢀⣤⠺⠀⠀⠀⠀⠀⠐⢢⢬⢦⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠁⡇⠀⢀⠀⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⠿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⠮⣀⣀⣈⠦⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⣿⠓
⠀⠀⠀⠀⠀⠀⢀⣀⡴⢛⡓⠈⠙⢷⡀⢠⣤⢄⣀⣤⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⣠⣤⣶⣿⣿⣿⣿⠏⠀⠀⠀⠁⠀⠙⠶⡉⠉⠙⢦⢤⣾⠿⠷⢤⣄⡠⠛⠻⠆⡀
⠀⠀⠈⠉⣃⣙⠃⣴⣿⡿⠂⠀⠀⣠⣶⣾⣧⢦⣀⠀⠀⠶⢤⡄⠠⠔⠛⣿⠁⠀⠀⢀⠉⠒⠛⠘⠢
⠀⠀⠀⠀⠉⠉⠃⠈⠉⠦⣤⣴⣾⣿⡿⠿⢿⣤⣤⡀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⢀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠉⠁⠀⠘⡽⠿⠻⠀⠀⠀⠀⢴⣿⣏⠀⠀⠀⣸⣿⡇⢀⡀⡆
⠀⠀⣿⣷⣦⣤⣀⠀⠀⠀⠀⣠⠀⠀⠁⠄⠊⣀⣶⣦⣶⣄⠀⠀⠀⠈⠿⢶⠂⢀⣿⣿⣿⡃
⠀⠀⠈⠁⠀⣿⣿⣷⣿⣴⣆⣿⣇⣴⡄⠀⠀⢘⠛⠿⣍⡁⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣯
⠀⠀⠰⢴⣷⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢀⣿⣶⡆⠀⠉⠂⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣷⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⠛⠛⠿⠻⠿⠿⠛⠿⠿⢿⣄⣼⡆⠀⠀⣠⣠⣿⠿⠿⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⡀⠈⠈⠙⠂⠀⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢰⠀⠁⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠘⠳⡄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠰⠀⠉⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⣠⠀⠈⠏⠀⠀⠀⠀⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡀⠀⠄⠀⠄⠒⢁⠀⠀⠀⠐⠊⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡄
⠀⠀⠀⠀⠀⠐⠔⠀⠤⠋⠀⠀⠊⠙⡇⠀⠐⠰⣦⠸⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡠⡤⣄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢎⠀⠀⠀⠠⡄⣀⠀⠈⠃⠀⠜⠒⠃⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠱⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣈⣁⠙⠃⠀⠀⠀⡠⠀⠀⠀⢤⡆⠀⠀⠀⡀⠒⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠘
⠀⠀⠀⠀⠀⠘⣦⠀⠐⠁⠈⢧⠀⠀⠀⢰⠇⠀⠀⠈⠁⢿⡄⠀⠀⠀⣀⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⢸⡇⠀⢻⣀⡀⠀⠀⢀⡴⢚⠁⠀⠔⠋⠀⠈⠉⠻⣻⣄⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠈⡑⠂⡊⠀⠢⠟⠫⠉⠁⠊⠐⠻⠁⠀⠀⠀⠀⠀⠁⠀⠀⠻⣿⠇⠀⠀⠀⠀⠀⢀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⡀⠆⠀⠀⠉⠛⠃⠠⣤⡐⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⣁⠀⠈⣀⠀⠠⣀⠈⠂⠀⠀⠀⢀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⢸⡉⠙⢿⢥⡀⠀⢀⣀⣀⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡉⠀⠀⢙⠟⠀⠐⠠⠉⠝⠀⠀⠀⠀⠀⠀⠀⠀⠉⡹⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⠈⠉

⠈⠓⠂⠀⠀⠛⡯⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⢠⠈⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⣀⣀⡀⢠⡤⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠄⢀⡀⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠊⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⠀⠀⠈⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢢⠀⠀⠐⠂⠄
⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⡀
⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠦⠤⠄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠒⠂⠀⠒⠀⠀⡀
⠀⡀⠀⣀⡀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠉⡡⣤⡀⠈⠉⠉⢠⠉⠳⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠫⢟
⣥⣰⣀⣉⣁⣀⣞⣋⣲⣚⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣀⣀⣐⣁⣀⣀⣀⣀⣀⣀⣀⣿⣲⣐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣄⣤⣶⣶⣤⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢀⣀⢴⠴⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠓⠾⠃⠀⠀⠀⠀⠀⠀⠀⣤⠀⠙⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠲⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣗⣾⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡠⠒⠊⠉⠉⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠉⠉⠉⠒⠢⣀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠉⠉⠁⠀⠠⣀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠉⠉⠁⠀⠢⢀
⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⢀⠞⠀⠀⠀⠄⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢉⠀⠐⠃⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠢⡀⠀⠀⠂⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠁⠂⠤⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠈⠒⠄⢀⠠⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠈⠒⠤⠀⠀⠀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠤⠒⠁


⠀⠀⠀⠀⣀⠀⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠤⣀⠄⣂⡤⠴⣮⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢔⡨⠞⠹⠅⠀⢠⢏⠃⠀⡠⠂⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⠑⢄
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠐⠀⢀⠠⣪⠔⠋⠀⠀⠀⠈⢶⡋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢘⠀⠀⠀⢀⢔⡥⠚⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈
⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡠⣻⡊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠢⣀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⢈⠞⠁⠑⣄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⡀⠌⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠉⠁⠀⠐⠈⠁⠀⠀⠀⠀⠺⣤⠤⢖⠏⠙⠐⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠐⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠐⠚⢻⣯⠀⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠀⠀⠀⠀⣤⣄⣹⡾⠛⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡁⣀⠀⠀⢼⣿⠖⠀⠀⣐⠻⣮⣿⢹⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣿⠴⠁⠰⣿⣶⠀⠿⠗⠉⠀⠻⠁⠛⠯⠹⢿⣿⡿⣷⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣴⣝⣯⣟⣏⣔⣠⢦⣼⡿⡄⠀⠀⠀⠀⠀⠆⠀⢀⠀⠉⣉⠂⣽⣷⣤⣤⣆
⠀⠀⠀⠀⠀⠀⢰⡿⣼⡿⢾⣿⣿⣿⣿⣿⣿⡯⠎⠈⢠⣶⢊⠜⠀⠀⠀⢈⠅⠀⢸⣧⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠒⠁⡯⣀⣼⣿⣿⣹⣿⣿⡿⠀⡀⠀⠈⠁⠈⠀⠀⠠⠈⠁⠀⠀⣴⠎⠙⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⢢⡈⣹⣿⣿⣿⣿⣧⡀⠀⠀⢀⣶⣄⣠⡴⡆⠀⣤⠒⠂⠀⠘⠀⠁⣤⣿⠁
⠀⠀⠀⠀⠀⠀⢷⡞⠻⣿⢟⡺⣿⠿⣿⣷⣧⣴⢲⣬⣽⣿⣿⠁⢁⡚⠀⠀⠀⠀⠁⠀⢀⣽⡏
⠀⠀⠀⠀⠀⠀⠸⣟⠆⠺⣮⢱⣊⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⢂⣶⣿⣆⡀⠀⢐⣧⣄⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠁⠛⣿⣿⣿⣿⣿⡿⠟⠻⢿⣿⣿⡿⣭⠟⠉⠀⠀⠀⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣁⠀⠀⠁⣜⡛⢿⡿⠐⠀⠀⠀⠘⣿⡷⠁⠆⠰⠀⠤⠀⣀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡃⠈⠙⠑⠻⠿⣄⡀⠀⢐⣶⣿⣤⣐⣤⢀⣂⣤⣿⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣒⣦⠠⣰⣦⣥⢉⣼⣿⣟⢻⣿⣿⣯⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠺⠯⠍⢟⣶⣿⡫⠴⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⢤⡀⠀⠀⠀⠄⠐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣽⣿⠂⠀⠀⠀⣰⡄⠀⠀⢰⠁⠀⠀⠀⠀⢹⠀⠀⠀⡤⠀⢀⢠⡜
⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠉⠁⠀⠀⠘⡀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠃⠓⠋⠁
⠀⠀⢀⢀⡀⠀⠁⠀⠀⠀⠀⣀⢘⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠒⠲⠖⠊
⠀⠀⠼⣿⠷⠀⠀⠀⠀⠀⢨⡀⢰⠹⡄⠀⠀⠀⠃⢐⡄⠀⠀⢤⣴⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢻⣖⠚⠖⠓⠉⠀⠀⠀⠸⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠈⠁⠄⠀⠀⢀⠀⠀⠀⠀⠈⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢽⡿⠄
⠀⠨⠄⠀⠀⠀⠀⠆⠀⠈⠛⠃⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠐⢤⠀⠁⠀⠀⠰⠀⠀⠀⠀⠁
⠀⠀⠐⠀⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣄⡐⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⢹⠀⠀⠀⠀⠘⣄⠀⠀⠀⡀⠀⢀⣄⠀⠀⣿⡟⠡⣾⣿⣷
⠀⠀⠀⣼⣿⣅⠀⢀⠄⠀⠀⠀⠀⣀⡖⠁⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠈⠙⠁⠀⠱⣦⣄⠀⡉⠁
⠀⠀⡜⠘⠁⠀⢰⠁⠐⣄⣀⡤⠚⡡⠀⠀⣤⡤⠀⠀⠀⠀⠀⡄⠀⢸⠉⠐⢢⣤⡀⠀⠀⠈⠀⠀⡀
⠀⠀⡂⠀⢂⠀⠈⠓⠒⠋⠉⠈⠁⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⢸⣠⠖⡋⢀⠙⢄⠀⠀⠀⠿⠃
⠀⠀⠠⠀⡂⠀⠀⠀⠀⠀⡔⠉⠈⡁⣐⡀⠀⠀⠀⠈⠀⠀⠀⣠⣀⠈⠣⡀⠀⠊⠀⡈⠢⣀⣀⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠐⠀⡀⡀⠀⠀⠀⠰⠆⠀⠀⠛⠇⠀⠀⠙⢆⠈⠢⢃⣔⠁⠈⢢⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣈⡨⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠚⠹⡀⠙⣄⡠⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢶⣾⣄⠀⠀⢰⠉⠢⡀⢈⠦⠜⡆
⠀⠀⠀⠐⠦⣠⡠⠔⠁⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠀⠀⠀⠑⠢⢬⣣

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⣤⣀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠉

⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⠿⠛⣛⠫⠭⠉⡀⢶⠀⠀⠀⠀⣼⣏⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⠈⠀⠀⠀⠀⠀⠀⠀⠰⠿⠶⠶⠶⡿⢀
⠀⠀⠀⠀⢠⡖⣶⣿⣿⠀⠀⠀⠀⠗⠂⠀⠀⠀⠀⢿⣟⢻⠀⡀⠀⠆⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡞⠄⠀⠀⠀⠀⠀⠀⠹⢞⡖⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠷⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠰⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠄⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡟⠉⠐⠠
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⢡⠀⠀⠀⠀⠀⢠⣾⣿⣿⠇⠀⠀⠐⡄⠀⠀⠀⠀⠀⢸⣿⣭⡴⠶⠒⠛⢻⡆⠀⠀⠀⠀⢸⣿⣿⠀⣀⡤
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣀⣠⡾⠀⠀⠀⠀⠀⠸⣿⡛⢉⡀⠀⠀⠈⡇⠀⠀⠀⠀⣴⢿⣿⣟⡀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠘⢿⣿⣦⡀⢰⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠙⠷⣿⣷⣦⡤⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⡤⠀⠀⠀⠀⠀⠀⢀⣄⣠⠄⢠⣤⣤⣴⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠛⣻⢿⣿⠀⠀⠀⠀⢀⣤⣶⣖⡒⠄
⠀⠀⠀⠀⠀⠀⠸⠿⢾⣶⣋⠐⠚⠀⠀⠀⠀⠀⠂⣠⡝⠀⠘⠛⣃⠁⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⣴⠎⢻⡶⠃⠀⠀⠀⠀⣾⣿⡿⠖⠈⠂⢠
⠀⠀⠀⠀⠀⠀⠻⣿⣟⣳⣦⠶⠒⠀⠀⠀⠀⠀⠀⣿⡀⠀⣰⠀⠛⢿⣦⠀⠀⠀⠀⠀⠀⠁⢠⢻⣧⠶⢛⡕⠀⠀⠀⠀⠀⢻⣿⣷⣬⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠶⠖⠀⠀⠀⠀⠀⠀⠸⠏⠁⠀⠁⠰⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⢎⠓⡚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠦⠀⠀⠀⠀⢾⣯⣿⠖⣂⠀⠀⢠⡾⣉⣀⣉⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⠀⠀⢿⣭⢤⡤⠀⠀⠀⢩⠽⠿⣤⠷⡤⠀⠀⠀⡄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠙⠻⠟⠛⠀⠀⢠⢾⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠂⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢀⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠒⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡽⣞⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⢛⣉⣤⠆⠀⠀⠀⠔⠀⠀⠀⠤⢶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⡄⠀⢀⣠⡤⠞⠋⠁
⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠶⠒⠫⢉
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣗⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⣻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⢿⡄⠀⠀⠁⠀⠀⢴⠄⠀⠸⠿⠟


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠲⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⡋⠹⣗⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠲⣄
⠀⠀⢀⡤⢶⡏⠀⠀⠀⠀⠀⠈⢳⠶⢤⡀⠀⠀⠀⢠⣿⠇⠀⠙⠶⢬⡥⢿⡀⠀⠀⠀⠀⠀⢠⣏⣠⣤⠤⠶⣚⣩⣧
⠀⠀⠻⣤⣸⡇⠀⠀⠀⠀⠀⠀⣸⣠⡴⠃⠀⠀⠀⠸⣟⣉⡓⢦⡰⣏⣿⢸⠇⠀⠀⠀⠀⠀⢸⡶⠶⠖⠚⠋⠉⣀⣾
⠀⠀⠀⠀⠈⠻⣏⡙⠛⠛⣉⡽⠋⠀⠀⠀⠀⠀⠀⠀⠹⣶⣏⢻⣝⢾⡶⠏⠀⠀⠀⠀⠀⠀⠀⠳⣤⡶⠖⠚⣋⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⣴⣟⣻⠆⣀⣉⣳⣄⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠈⠳⣞⠛⢳⡆⠀⠀⠀⢀⡴⠋⠘⣧⠀⢨⡷⣄
⠀⠀⠀⠀⢸⣇⣀⣀⣘⡳⠒⠒⢿⠀⠀⠀⠀⠀⠀⣠⡇⡀⠀⠀⠀⠀⢀⣹⡶⠋⠀⠀⠀⠀⢸⠁⣀⡾⠛⠁⢨⠇⢹
⠀⠀⠀⠀⠘⣷⣒⣶⠞⠃⣴⢶⡾⠀⠀⠀⠀⢠⡞⣹⠋⠉⢳⣤⠴⠚⢋⡿⠀⠀⠀⠀⠀⠀⠸⣏⣉⣡⣄⣀⣈⢛⡟
⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⠼⠋⠀⠀⠀⠀⠀⠈⠙⠛⢦⣤⠾⠤⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢥⣬⣭⠽⠋



⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⣴⣋⣠⡟⠀⣏⡽⣆⠀⠀⠀⠀⠀⠀⠀⡼⠯⠴⠶⠚⢩⣿⣦⠀⠀⠀⠀⣾⠁⢿⡟⠁⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢸⣇⣉⡁⠀⠀⣠⠴⢿⠀⠀⠀⠀⠀⠀⢸⣯⣭⣭⣙⠛⠲⠿⢿⡇⠀⠀⠀⠈⠳⣌⠛⢦⣄⡀⠀⠀⠀⢹⣦⡀
⠀⠀⠀⠀⠈⢿⡀⣹⠀⠀⣇⢀⡞⠀⠀⠀⠀⠀⠀⠈⢷⡶⢀⣈⣉⣛⣛⡿⠀⠀⠀⠀⠀⠀⠘⣟⠲⢬⣍⡛⠒⠶⠿⠋⢹⡆
⠀⠀⠀⠀⠀⠀⠙⠷⠤⠴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠭⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠽⠟⠛⠓⠛⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣦⣀⣈⡳⠿⣜⡓⠺⡴⠈⢱⡟⢿⣼⢿⣿⢿⣯⣽⣯⣯⠙⠀⠘⠟⠠⠆⠹⡇⠙⠃⠙⠹⣷⣦⠼⠃⢑⠟⠀⣹
⣿⣿⣿⣿⣿⣶⡤⢭⣁⠀⠉⠳⠀⡟⠾⢋⣾⣻⣿⠟⡟⣷⣿⢶⠍⠁⣿⣠⣀⡰⠀⠀⠀⠀⠀⢹⠃⠀⠓⢠⡡⠴⢛
⣿⣿⠿⠿⢷⡼⣿⠎⠿⠲⠖⠀⠐⠀⣾⠾⡍⠙⠛⠙⠙⠛⠇⠈⠀⣗⠛⢷⢟⢇⠀⠰⢀⠀⠰⠄⡀⠈⢁⣠⣧⠴⠟
⣿⣿⣷⣶⣿⣰⣿⠾⠟⣣⣤⣆⠀⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡴⠂⠛⠚⠀⣤⠁⡀⠈⠤⣤⠀⠀⠈⡿⠀⠀⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣕⠖⠋⠉⠁⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣓⢒⡆⡽
⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⡇⣤⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢾⣟⣼⣿⣷⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣰⡆⠀⠀⠀⠐⠚⠙⠀⠀⠀⡀⠀⠃⠀⠨⠷⠊⣮⠹⢓⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠗⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢀⣄⠈⠃⠀⢀⡩⠔⢸⠁⠀⠁⠀⠀⠘⢿⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⣫⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠱⣆⡀⢀⠄⠀⠀⠀⠈⠉⣵
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢞⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠉⠀⡄⠐⠲
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡏⠀⠀⠀⢸⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠆⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠿⠻⡶⠚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠺
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠂⠀⠀⠀⠰⠖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠀⠡⡶⣂⣂⠀⠐⡟⢦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣡⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢋⣀⣏⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣿⠃⠼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠦⣎⣴⣷⣗⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣿⣻⣼⡿⠗⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠀⠀⠀⠀⠀⠀⢠⡕⠛⣯⣰⡿⠳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢡⡦⠦⠙⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡈⢻⠗⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⠚⢬⠤⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⠄⠀⠀⠐⢲⠒⣰⢂⡏⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣨⠀⠀⣢⠀⠹⢒⣽⢺⢋⡀⠸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡀⢤⣠⣆⣵⣿⣿⣿⡏⠛⣟⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠁⣘⣾⠿⣿⣿⣻⣯⣿⣿⣅⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢡⠄⠄⢸⣷⣼⡇⠀⠙⣿⣦⣿⣿⣿⡯⡵⣯⠸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡤⢀⠀⠀⠀⢐⣦⡭⠶⢠⡻⣿⣿⢇⠀⠈⢿⣾⣿⣾⣿⣽⣂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡇⠀⠀⠠⠀⣴⣬⣹⣖⣿⣿⣿⣿⠁⠀⢀⠀⠈⢹⣿⣿⣿⣿⣛⢀⡀⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡅⢀⠴⢶⡀⠀⠋⠈⠙⠛⠛⠉⠀⠀⠀⠊⠀⠀⡁⣿⣿⣿⣿⢿⡟⠡⢆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⢟⣯⢹⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢴⢀⠁⣿⣿⣿⣿⣟⣞
⠀⠀⠀⠀⠀⠀⢤⡤⢉⣿⣿⣟⣾⣽⡿⠁⠀⠀⢀⡀⠆⠄⠀⠂⠀⡘⠀⠀⠠⣟⢤⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⡲⣷⣾⣿⠿⣿⣻⣿⠃⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⢠⠄⢨⣿⣺⣛⣿⡛⠻⣾⡇
⠀⠀⠀⠀⠀⠆⢸⣿⣿⣿⣾⣿⠟⠁⢀⣴⣿⣦⠐⠄⡂⠈⠀⠐⠠⠀⠀⡧⣦⡟⣻⣟⣋⣽⡶⣟⣗
⠀⠀⠀⠀⠀⣰⢿⠿⣿⡛⣟⣋⣀⣀⣿⣿⣿⣿⣄⣀⠁⢂⠄⠀⢁⡀⣬⣼⣿⣤⣿⣿⣿⣿⡿⣿⢷⠄⡀
⠀⠀⠀⠀⠀⠉⠈⠒⠛⠛⠿⢿⡟⠻⡟⢻⣿⣿⡷⢿⣧⣤⣤⣶⣴⣶⣿⣿⣿⣿⣿⢿⣧⠙⡗⡟⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠺⡈⢉⡿⢿⣯⡻⣿⣿⠏⠙⠛⠛⠉⠀⠈⠀⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠘⣟⣠⡘⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣰⣿⠀⠀⣨⠔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⣴⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤⡧⣶⠛⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⣿⠀⢀⣴⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣷⠠⠗⠀⠀⠀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡏⡄⣋⡀⠰⠊⠁⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠜⠋⠉⢹⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠘⠛⠛⠛⣉⠉⠀⠀⠀⠀⢀⣀⣤⣴⣾⣾⡏⣄⠀⠀⠀⢰⡄⣠⠐⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣄⠀⠀⠀⠀⢀⡀⠀⣂⣀⣠⣿⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⠟⢠⣿⡈⠒⠂⣾⡇⠀⡼⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⠀⢀⢤⣠⣷⠀⠀⠁⠀⣦⡀⣠⠤⠰⢦⣈⡙⣿⣿⣿⡿⣟⠿⡿⠿⣿⡟⠛⠀⠀⢼⣿⡆⠁⠋⣿⣷⠠⣤⠇⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⣧⣿⠶⠃⠀⣩⠗⠮⣂⠄⠘⡥⠈⠁⡄⢰⠈⣉⠭⠛⠩⠑⢊⠀⠊⢠⠈⠂⠒⠥⣢⡠⣀⠀⡌⠛⠜⠛⠿⡂⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠧⠊⠁⠠⠀⡠⢞⡄⢠⠀⠀⠁⡀⠀⠉⠀⠟⠃⠋⠀⠀⣀⡠⠂⠨⠂⠀⣾⡆⠐⢄⡀⠈⠉⠈⢸⠷⣦⠀⢂⠂⢘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠤⢊⠁⠀⢀⠰⠀⡀⢸⡀⣆⠀⠙⠒⠁⠀⠀⠀⢠⠀⠂⠀⠀⠄⠀⣿⢠⣿⣇⠀⠂⠀⡀⠀⣁⠤⠊⠀⣦⢸⠀⢸
⠀⠀⢰⠐⡀⠄⠀⠀⡀⠀⠀⠀⠀⠁⠈⣁⣀⣀⡛⠗⠉⠀⣠⡠⠄⠀⡀⠀⣾⡆⠤⣌⣀⣀⠼⠿⠾⠟⠛⠋⠉⡉⠔⠈⠀⠀⠐⠀⡏⢸⠀⠘
⠀⠀⠸⠀⠤⡌⠒⠠⠄⠀⠉⠑⠂⠀⠿⢿⣿⡭⠿⠇⠀⠀⠂⠀⠀⠈⠝⠐⠿⠧⡈⢁⡠⠂⠅⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠀⠈⠤⢆⡀⠰⢠⣦⣤⣄⡀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠛⠿⠃⠒⠀⠀⣤⣀⢀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠭⢃⣆⠶⠀⠀⠀⠐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⠶⠾⠿⠿⠿⠿⠿⠿⠷⢶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⠉⠉⠛⠻⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠿⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⢀⣠⣤⣶⡾⠿⠿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⡀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⢀⣴⡿⠋⢁⣠⣶⠿⠛⠋⠁⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⣴⡿⠋⣴⠆⣼⡇⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⣰⡿⠋⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⢀⣾⠟⠀⢸⡿⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠋⠀⠀⣠⣾⠋⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⢿⣦
⠀⠀⠀⢠⣿⠋⠀⠀⠘⠁⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡶⠾⠿⠛⠛⠉⠀⠀⣠⣾⠟⠁⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠻⣧
⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠹⣧
⠀⢀⣿⠃⠀⠀⠀⣀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠁⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⣠⣤⣶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⣼⠇⠀⢀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠿⠟⢉⣀⣤⣴⠾⠛⠁⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢿⡄
⢠⡿⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠉⢁⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⠛⣿⡆⠀⠀⠀⠀⣼⡟⠁⠘⣧
⢸⣧⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⢠⣿⠃⠀⠀⢀⣼⠟⠀⠀⠀⢿
⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⢀⣴⡿⠃⠀⠀⢠⣾⠋⠀⠀⠀⣠⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣰⡿⠃⠀⠀⠀⣰⡿⢹
⣿⠀⠀⢀⣤⣴⡶⠶⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢸⣏⣀⣀⣤⣾⠟⠋⠀⠀⠀⢀⣼⠟⠁⠀⠀⢀⣾⠟⠀⢸
⣿⢀⣴⡿⠋⠁⢀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⣰⡿⠋⠀⢠⣿
⢹⡿⠋⠀⢀⣴⡿⠋⠉⠀⠀⠉⠉⠛⠻⢷⣦⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣠⣾⠟⠀⠀⣠⡿⣿
⠸⣧⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⣠⣼⠟⠁⠀⠀⣴⡿⢁⡿
⠀⢿⡄⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣠⣾⠟⠁⠀⠀⣠⣾⠟⠀⣼⠇
⠀⠘⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⢀⣤⡾⠟⠁⠀⠀⣠⣾⠟⠁⠀⢰⡟
⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⠋⠁⠀⢀⣠⣶⠟⠋⠀⠀⢀⣤⣾⠟⠁⠀⠀⢠⡿⠁
⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠟⠋⠁⢀⣠⣴⣾⠟⠋⠀⠀⢀⣤⣶⠟⠋⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠛⠋⠉⠀⢀⣠⣴⠾⠟⠋⠉⠀⢀⣠⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠉⠀⠀⠀⠀⣠⣴⡾⠛⠉⠀⠀⣀⣤⣶⠾⠟⠋⠁⠀⠀⣰⡆⣰⡿⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⣸⡟⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣠⣴⡿⠟⠉⠀⠀⠀⠀⣰⡿⠰⡿⠁⠛⣡⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣅⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⢀⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⣀⣼⡏⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣤⣄⡀⠀⠀⢀⣾⠟⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⢿⣿⣤⣤⣤⣴⣿⣥⣤⣤⣤⣤⣶⡶⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠈⢦⡀⠀⠀⠀⠀⣠⠤⣖⠒⠢⠤⣀⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡺⠃⠀⣠⠤⢤⡀⠀⠈⠉⢉⡽⠋⠀⠀⠀⠱⣀⡖⠋⠀⠀⠀⠀⠙⣗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⣰⠃⠀⠠⡇⠀⣀⠝⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⢹⠀⡠⠖⠚⠑⡆⠀⠘⣆⠉⠢⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢠⠇⠀⠀⠀⣇⡼⠁⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣰⠃⠀⠀⢸⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⢸⠀⠀⠀⠀⣀⠤⠖⠒⠒⠦⣄⠘⡆⠀⠀⠀⠀⠀⠀⠓⠦⣄⠀⡇⠀⠀⣠⠎⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠓⠤⢖⣋⣁⣄⡤⠤⢤⡀⠈⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡓⠋⠁⠀⠀⢀⡠⠖⠒⠻⡄
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠏⠀⠀⠀⠙⠚⣋⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢄⠀⡎⠀⡴⠛⢢⠹⡀
⠀⠀⠀⡼⠀⠀⡠⠲⡄⠀⢀⠞⠀⢀⡞⣼⠀⠀⠀⠀⠀⡼⠁⠀⠘⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢧⡀⠹⠤⠼⠀⢣
⠀⠀⢠⡇⠀⣸⠁⢠⡇⢠⠏⠀⠀⡜⠀⣿⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠙⠲⢄⠀⠀⢸⡄
⠀⠀⢸⠀⠀⣿⠀⢸⠀⠀⠓⠒⠊⢀⡴⢻⠀⠀⢰⠃⠀⡠⠖⠒⠒⠦⣄⠀⣠⠏⠀⠀⠀⠀⠀⢀⠴⠚⡅⠀⠀⠀⠀⣸⠀⠀⠀⡇
⠀⠀⢸⠀⠀⢸⡴⠋⠙⡄⠀⠀⢠⠏⠀⢸⠀⠀⠘⠦⠞⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣇⠀⣰⠃⠀⠀⣀⠴⠃⠀⠀⠀⡇
⠀⠀⢸⠀⢠⠏⠀⠀⠀⠹⣄⣀⡸⢠⢤⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠦⢾⡁⠀⠀⠀⠀⠀⡇
⠀⠀⠘⡇⠘⣆⠀⠀⠀⠀⢳⡈⠀⡇⡞⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡀⠀⠀⢸⠃
⠀⠀⠀⢳⠀⠸⡀⠀⠀⠀⠀⠳⣄⠛⣠⠞⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⡴⠁⠀⠀⡞
⠀⠀⠀⠈⣇⡴⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠙⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣎⠉⠁⠀⠸⣅⠀⠀⣸⠁
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣀⠀⠀⠀⠉⣱⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠉⣹⠃
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⣠⠤⠖⠊⠉⠉⢢⠀⠀⠀⠀⠀⠀⡤⠋⠉⠑⠒⠊⢧⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⡰⠛⠤⠤⠤⠤⠤⠒⡏⠀⠀⠀⠀⢀⣞⣠⠤⠶⠤⢤⣀⠈⠣⡀⠀⠀⠀⣰⠃⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣇⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⢒⡿⠋⠀⢀⡤⢤⣀⠈⠙⠒⠚⠓⠒⠊⠁⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠸⣀⣀⣈⡱⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠴⠒⠒⠒⠒⣒⣦⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠤⠤⠤⠤⠤⠤⠶⠒⠚⠉⠁

⠀⠀⠀⠀⠀⡀⠄⡀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡖⠤⡀
⠀⠀⠀⠀⣘⡓⠠⢑⡂⢒⢰⡀⠀⠀⠀⠀⠀⣸⣉⣀⣠⣤⡤⣴⡀⠀⠀⠀⠀⠀⠀⠻⣿⣏⠁⠀⢻⡄
⠀⠀⠀⠀⢃⢈⠘⡠⠄⣤⣸⠁⠀⠀⠀⠀⠀⢉⠉⡍⠋⠍⢡⢴⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣼⠇
⠀⠀⠀⠀⠈⠲⢶⣿⡶⠟⠁⠀⠀⠀⠀⠀⠀⠈⠳⢶⣶⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠲⢤⣾⣿⠽⠋
⠀⠀⠀⠀⠐⠶⠶⠠⠶⠂⢨⠇⠀⠀⠀⠀⠀⠀⠲⠴⠦⠆⠠⡄⠀⠀⠀⠀⠀⠀⠀⠶⠶⠒⠆⠰⠆

⠀⠀⠀⠀⢀⣀⡯⢾⡷⣶⠄⠀⠀⠀⠀⠀⠀⢀⡶⠯⠽⠽⠳⣄⠀⠀⠀⠀⠀⠀⢀⣔⣿⣽⣭⣷⣦⣤⣤⡄
⠀⠀⠀⠀⣿⡯⢓⣾⣶⡿⣻⠄⠀⠀⠀⠀⠀⡞⠀⠠⠥⣄⠤⢸⡄⠀⠀⠀⠀⠀⣼⣟⣛⣛⣛⣿⣿⡟⠋
⠀⠀⠀⠀⠹⣿⣬⣿⣭⣿⡟⠀⠀⠀⠀⠀⠀⠹⣥⣴⣾⣿⣿⡿⠀⠀⠀⠀⣴⣿⣿⣷⣿⣟⣛⣿⡿⠃
⠀⠀⠀⠀⠀⡈⠛⠛⠛⢉⠀⠀⠀⠀⠀⠀⠀⢀⠉⣛⣛⢛⠉⣀⠀⠀⠀⠀⠀⠁⢀⣈⣛⠛⡛⡋⣄
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠁⠛⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠁⠙
⠀⠀⠀⠀⠀⢀⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢄⡀
⠀⠀⠀⠀⣴⣯⣭⣭⠭⠿⢧⣤⣄⠀⠀⠀⠀⣰⣓⠒⣒⣒⣚⣳⡀⠀⠀⠀⠀⠀⣰⡏⠘⠋⢰⣿⣷⡄
⠀⠀⢀⡤⣧⣴⣶⣶⣶⣶⣾⠛⠁⠀⠀⠀⠀⠧⠿⠿⠿⠧⢤⣤⠇⠀⠀⠀⠀⠀⢿⡁⠘⠃⣀⢨⣿⡇
⠀⠀⠈⠉⠙⢿⣭⣭⣭⡽⠋⠀⠀⠀⠀⠀⠀⠘⢮⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠣⣷⣤⣿⣿⠟
⠀⠀⠀⠀⢠⣤⣤⣬⣤⡀⣦⠀⠀⠀⠀⠀⠀⣤⣤⠤⣭⣤⠀⣶⠀⠀⠀⠀⠀⠀⠀⣤⣤⡭⣥⢰⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⢲⢶⠀⢰⣱⠐⣣⣴⡾⣖⠛⣦⠒⣾⡒⠰⠻⠿⠷⠶⠶⠷⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠐⠿⠶⠞⠓⠷⠏⠪⠿⠷⠆⠿⠐⠾⠞⠐⠫⠿⠳⠗⠸⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣤⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠓⠀⠀⠀⠀⠀⠀⠀⢻⡍⠉⠉⠙⡧⠀⠈⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠃⣰⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⠀⡞⠹⡇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠶⣦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⠀⣐⠰⣟⠀⠀⠀⢰⠉⢻⠇⢠⠀⢀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣰⠋⠙⣿⠀⠀⠀⠀⠐⠪⠃⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠠⡀⠳⡴⠛⠀⢀⡈⢁⣀⡪⢊⣠⠾⠛⠟⣻⣿⡷⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⣖⣫⣿⣶⣮⣝⢿⢜⣋⣠⢦⣿⠁⠀⠀⠀⢱⣿⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢿⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⡆⢿⡘⣿⣧⠻⢆⡀⣀⣠⣬⠁⠃⠂⠀⠀⠀⢀⣴⢿⣿⣦⠀⠀⠀⠀⣀⠁⠈⡇
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣇⣛⣽⣿⣿⣿⣿⣿⣿⣏⡚⠋⡁⡘⠉⠁⠁⢀⡀⠀⠀⣼⠀⠀⠘⡿⢃⠀⡄⠀⠘⢆⠀⡄⠰⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣹⣿⣿⣿⣷⡔⠜⠆⣤⠝⠳⠟⠀⠀⠈⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣀⢄⣀⣠⣤⣴⣄⣤⡤⡄⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠙⢿⣿⣿⣿⣿⣿⣿⣅⠹⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠰⠆⠀⢠⠉⢸⠟⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⡄⠀⠛⠁⠀⣀⣴⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣒⡔⣄⣶⣶⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣿⣿⣿⣿⣿⣿⣿⣆⠹⠛⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣛⣉⣩⣭⣽⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⡂
⠀⠀⠀⠀⠀⠐⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒⠈
⠀⠀⠈⠀⠀⢠⡀⣠⠇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠁⠠⠖⠊⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠄⠀⠒⠀⠀⠀⠀⠀⡀⠐⠐⢲⢤⡀⠀⠀⠀⠀⠀⠔⠀⠀⠤⠒⠖⠉⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠚⠶⠆⠚⠛⠷⠛⡀⠀⠀⠀⢁⠄⠈⠀⠀⣠⢼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢿⣶⣾⠷⠹⠞⢻⠁⠀⠀⡠⠁⣀⡤⡲⠋⠁⡘
⠀⠈⠀⠈⢀⠄⠁⠈⠀⠀⠈⠫⠭⠭⠭⠟⠃⠀⠠⠊⠀⡢⠕⠂⠒⠐⠈⠀⠀⠀⠈⡌⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⡀⢴⣒⣶⣤⣄⠀⠀⣀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⢀⡤⠖⠒⠂⢄
⠀⠀⠀⠀⢀⠉⠉⠉⢐⣓⣏⣉⡤⠂⠄⠀⠀⠀⠀⠀⠀⡀⠰⠓⠰⠂⠀⠀⠀⢀
⠀⣀⠤⢦⢂⣴⣾⣾⠭⣟⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡀⡞⠉⠈⠈⠂⡀⠂
⠨⠗⠒⠛⠻⡏⠁⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠈⠲⠀⢀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠠⠄⠀⠀⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠈⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⢁⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣴⣿⣿⣿⡿⣿⣿⣿⢏⠩⠟⢿⣿⣿⣿⣆⠱⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⣸⣿⣿⣿⣿⣷⣼⠟⠻⣦⣿⡞⣾⣿⣿⣿⣿⡇⢿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⣿⣿⣿⣿⣿⣝⢿⡷⢀⣸⣿⡟⣿⣿⣿⣿⣿⡏⠁⣅⡀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⢀⢹⣿⣿⣿⣿⣿⣿⣥⣼⣯⣵⣾⣿⣿⣿⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⢶⣦⣎⢿⣿⣿⣿⡿⠿⠿⠻⣻⠿⠿⣿⣿⣿⣿⡟⣰⢠⡍⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢆⡿⡛⠻⠮⠛⠩⣄⠒⠂⠀⣠⣌⡁⢀⣠⡬⠻⢋⡴⣫⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⢿⣶⣦⣐⠀⢀⠀⡀⢀⡈⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⣀⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣷⣀⠙⢿⣯⠙⠻⠦⣥⣄⣀⣀⠀⠀⢀⣀⣬⣤⠼⠶⣛⣿⡿⠋⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡁⠀⠀⣼⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣏⠀⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢽⣿⣿⣿⣿⣿⡁⣤⣙⠻⠟⣫⠀⢺⣿⣿⣿⣿⣿⡅⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠀⠀⣿⠏⠹⠙⠁⠀⠀⠀⠈⠀⠀⠀⠀⠟⢻⠟⠿⣧⠁⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⣴⣤⣦⣴⣀⣦⠀⡇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠉⠉⠈⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠠⢤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠠⠄⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢳⣠⣷⠠⠘⠀⢀⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢛⢻⣿⣶⠟⣷⠀⢻⡀⠀⠀⠀⠀⠀⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢾⣿⣭⣵⣿⣷⡖⡈⠇⠀⣀⣄⣴⡖⠀⢰⣿⣷⠾⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠻⠟⣻⣿⣿⣷⠿⢿⣿⣿⣿⣷⣿⣿⣿⣿⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⠙⣿⣿⣿⣛⠿⠒⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡿⠻⣱⣿⣧⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡉⠁⠈⠁⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⣤⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢛⢤⡰⣀⠀⠠⠺⠙⣄⠀⠀⡀⣠⣬⣿⣏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⣽⣿⣿⣾⣶⣄⣀⣶⣼⣷⢾⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣟⣛⣷⠿⣯⣽⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣿⣿⠿⡿⠋⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣽⣽⣯⣬⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⠀⢀⣀⣤⣤⣤⣀⣀⣤⣤⣤⣀⡀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢿⣤⣤⣶⡎⠀⡴⠛⣩⣿⠧⠤⢀⣠⣄⡀⠭⠽⣿⣍⠛⢦⠀⢱⣶⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢟⡔⠀⢀⠍⠁⠶⠶⡿⣟⣿⣿⠶⠶⠈⠩⡁⠀⠳⡛⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣴⢟⡡⣯⠎⢀⣠⡄⠀⢀⠈⢹⡟⢻⡏⠁⡀⠀⢰⣄⡀⠐⣟⢌⡻⢦⡀
⠀⠀⠀⠀⠀⢀⡽⣣⢯⡾⠃⢸⠁⢨⢝⣙⣦⢣⠈⡀⢀⠁⣜⣶⣋⡫⡅⠈⡇⠈⢷⡽⣌⢯⡁
⠀⠀⠀⠀⢠⣾⡄⠁⠈⢀⣇⠀⣀⡈⢚⣷⣫⣤⡕⠹⠏⢪⣤⣭⣷⡓⢁⣀⠀⣸⡀⠁⠀⢠⣷⡄
⠀⠀⠀⠀⢞⣽⠏⣠⣷⡿⣏⡌⢁⣾⣟⡛⠉⠀⠀⢀⡀⠀⠀⠈⢛⣹⣷⡌⢡⣽⢿⣾⣄⠹⣿⡳
⠀⠀⠀⠐⣻⠅⢈⣼⡟⠁⣼⢁⣿⣙⠎⠙⠲⣚⠛⠓⠚⠛⣓⠖⠋⠱⣋⣿⡈⣷⡈⢿⣧⡁⠈⣟⠂
⠀⠀⠀⠀⣿⢠⡎⢸⣿⠘⢿⡝⣿⣟⠃⠀⠀⠻⢧⣄⣠⡴⠟⠀⠀⠉⣻⣿⢫⡿⠃⣿⣇⢱⣄⣿
⠀⠀⠀⠀⣿⣫⡿⠉⡹⣦⠘⡫⣽⣟⠷⢤⣤⣤⠴⢿⡿⠧⣤⣤⡤⠶⢻⣷⢝⡃⣴⢏⡈⢿⡟⣿
⠀⠀⠀⠀⠘⡿⢷⣇⢃⠹⠊⢳⣿⢱⢠⡆⣰⣿⣾⣿⣿⣿⣿⣆⢰⡄⡎⣯⠞⠈⡃⡐⣸⣿⢿⠇
⠀⠀⠀⠀⠀⠹⣾⣿⣤⡆⠃⠀⢿⠘⣾⣇⣿⡟⢿⣿⣿⡿⢻⣿⣸⣷⠃⡿⠀⠘⢱⣤⣿⣷⠎
⠀⠀⠀⠀⠀⠀⠉⠘⣿⣧⣠⣤⡀⠀⣽⣿⣿⣿⣄⠉⠉⣠⣿⣿⣿⣯⠆⢀⣤⣴⣼⣿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢿⢹⣿⣆⣿⣿⣷⣤⣽⣿⣿⣿⣤⣾⣿⣿⣰⣿⡏⣿⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠏⣧⠙⢟⣛⣫⡿⠀⠘⢮⣝⡛⡻⠛⣽⠹⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣦⣍⠁⠀⡄⣀⡄⠁⣨⣴⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡑⢥⣽⣯⡤⢊⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠚⠀⢲⢿⣿⡇⠀⠃⠉⠃⠀⠀⠂⠀⠀⠐⠃
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣾⣿⠇⢀⡀
⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠏⠀⠀⢇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠚⠂⠀⠀⠀⠀⡀⠀⠀⠃⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⢠⣿
⠴⠁⠀⠀⠀⠘⠂⠀⠀⠀⠘⠓⣀⣀⣀⠀⣀⠤⠄⠤⣀⠀⠀⠑⠇⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣵⣿⠟
⠂⠒⠒⠢⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢀⡀⣀⡀⢐⡤⢤⠀⠀⠀⠀⠀⠀⠀⠂⠠⢷⣶⣾⣿⣶⣾⣿⣿⡷⠊
⡀⠀⠀⠐⠀⡀⠀⠂⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠉⢻⣿⣿⢿⣿⣿
⠛⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⡄⠠⠖⠁⠀⠈⢩⣿⠛⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢘⠋
⠀⠀⠀⣀⠐⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⡷⠤⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠉⠻⠦⠀⠀⠀⠀⠠⠤⢄⣿⣿⣇⠀⣿⣀⣠⣌⠀⠈⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠤⠿⢿⣿⣼⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⣅⠀⠀⡰⠁⠀⡀
⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⡉⠁⠁⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠐⠀⠋
⠀⠁⣀⣧⠀⠀⠀⢀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠰⢂⣨⡽⠋⠉⠛⢧⠈⠀⠀⠀⠒⠤⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡠⠄
⠀⠀⠀⠈⠋⠀⡄⠀⠀⠀⠀⠀⠀⠁⠒⠒⠂⢩⠊⠀⠐⠁⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠤⠴⣦⣤⣤⠀⠀⠀⠀⠀⠀⢀
⠐⠂⠀⠀⠀⠀⠀⠀⠐⢄⣀⡄⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠀⠀⠀⠉⠉⠈⠓⠲⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⢈⡀⠀⡀⠀⠀⠀⠈⠙⠈⠈⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⢳⡀⠸⣇⣀⣠⣴⠿
⣀⠀⠀⠬⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢷⣶⡀⠀⠀⣸⠂⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠸⣷⣾⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⢿⣷⣄⠰⠁⠀⠀⣠⣿⡷⠂⠀⠀⠀⠀⠀⠀⢰⠟⡻⢧⡀⠀⠀⠁
⣀⠀⠐⠀⠐⠀⠀⢀⠀⠀⠀⢀⠐⠀⠈⠀⠀⠈⠓⠦⠀⠀⠘⢿⣿⣶⢾⠀⣾⣿⢟⠀⠒⠀⠀⠀⠀⠈⢀⠈⠀⠀⠀⠙⠒⠤⠄
⠀⠀⠂⠀⠀⠀⠀⠀⠰⠀⠀⡂⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣤⣌⢋⡰⢤⢻⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠁⠀⠀⠠⠒⠂⠈⠉⠝⠛⠛⢻⡿⢿⠁⣡⡐⣽⣿⣅⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠠⠄⢀⠀⠀⠀⠸⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡣⠉⠉⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⡁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⣀⣠⡤⢤⣄⡀⠐⠠⠔⢂⣠⣐⡈⠠⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣶⣖⣶⠂⠠⡀
⠀⠀⠀⠀⠀⢴⡝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⣿⡿⠟⠉⠀⠀⠈⢦⡀
⠀⠀⠀⠀⢘⡀⠀⠀⠀⡀⠀⢤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠈⠁⢀⡀⠀⠱⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣿⡿⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡟⠁⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛⠋⠉⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⣄⠀⠀⣀⣀⣤⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠈⣩⣠⣽⡛⠋⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠸⠶⠀⠀⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡏⠀⠀⠀⠀⠀⣰⠇⠀⠀⠸⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⢉⠀⠀⠀⢈⣛⣛⣰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⢶⣿⣄⣀⣠⣿⣿⣏⣀⣨⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⣦⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⣽⣿⡟⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠐⠛⠚⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡴⠋


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣀⣠⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠿⠟⠛⠉⣩⣶⣶⡄⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠏⣄⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⡗⠀⠘⣷⣄⠀⠀⢩⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇⠀⠀⢹⣿⣷⡀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠁⠀⠀⣼⣿⣿⣿⣦⣈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣀⣀⣸⣿⣿⣿⣿⠀⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⣿⣿⣿⣿⣿⢱⢶⣼⣿⣿⣿⢃⣠⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⡘⣛⣿⣿⣿⣿⡚⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣿⣿⣿⣿⣿⡏⣿⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠠⠈⠉⢉⣿⣛⣿⣵⠂⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣶⠀⣶⠀⠀⣼⡷⢟⡐⢇⣸⣿⣿⣿⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠋⠡⠟⢀⣼⣿⣿⣻⣧⡌⠀⢻⣿⣷⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⣿⣿⡿⠁⠀⠀⢿⣿⣿⠟⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡻⢾⣿⡿⠁⠀⠀⠀⠈⢿⣿⣿⣽⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣿⡽⠁⠀⠀⠀⠀⠀⠈⢿⣿⣥⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢿⣷⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⢿⢟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣾⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠀⠱⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠉⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠖⠚⠏
⠀⠀⠀⠀⠀⠀⠀⢀⡝⠟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⢀⣾⣫⣾⣿⣿⣷⣮⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠀⠀⣺⣿⣿⣿⣯⣯⣿⣯⡷⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠐⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠚⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢻⠀⠀⠰⣆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⣀⡛⣓⣚⣋⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠉⠉⠉⠀⠉⠉⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⠓⣄⠀⢢⠀⠇
⠀⠀⠠⠀⠀⠀⠑⢤⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠈⠁⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢠⣤⡤⢀⡀⠀⠆⠀⠀⠱⣼⣿⡆
⠀⠀⠀⠀⢰⣾⡦⢀⠂⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠌⠀⠸⠋⠀⠀⠀⠀⠀⠉⠛⢅
⠀⠴⠶⣄⠀⠀⠇⠈⠀⠃⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣷⣆⠖⠋⠉⠀⢨
⠀⠀⠀⠈⠑⠄⠈⠀⠀⠤⠀⠹⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠈⠀⠀⠀⠂⠀⠉⠃⠄⢀⠀⣠⠸
⠀⠀⠀⠀⠀⠀⠁⡘⠆⣀⢠⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠠⣾⡦⠠⠀⠀⢦⡄⢀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠝⠈⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠁⢀⡐⠁⠁⠀⠄⠀⠀⠨⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠘⢠⣀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣇⠘⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣿⠀⠀⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⠀⠀⢄⢰⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠃⠰⠃⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢮⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣄⠀⠀⠈⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⡸⣦⢸⣇⠄⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡠⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣜⣿⣿⣞⠄⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢧⡀⠘⡄⢸⣷⠆⠀⠐⠤⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉
⢀⣀⡀⠲⢦⣤⣔⣤⣄⣀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡹⣆⢌⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⣷⡄⢧⣼⣿⠀⠀⠀⠀⠙⣿⡿⠛⣹⡶⠟⢅⣀
⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣟⡷⣤⡀⠀⠀⠀⠹⣿⣿⡽⣾⣏⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢹⣧⢸⣿⣿⠀⠀⠀⣀⣴⢞⣡⣾⣏⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⣄⡀⠀⠙⣿⣿⣿⣆⢧⠀⢻⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠘⣿⣼⣿⣿⠀⡴⣵⣟⣵⠞⢋⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⣿⣶⣄⠘⢿⣿⣿⣞⣧⠘⡆⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣧⣸⣿⣿⣿⣾⣿⢟⡿⢃⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣷⣮⣿⣿⣿⣿⣷⣷⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⣡⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢏⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣄⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣇⠀⠀⠀⠀⠀⠀⣨⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⢀⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠙⠢⠀⠘⠢⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠊⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠴⣄
⠀⠀⠐⠁⠀⠀⠀⠃⠀⠀⠀⡈⠙⠗⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠌⠀⠀⠗
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⡀⠀⠀⠀⠀⢀⡘⢿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣠⠀⠀⠳⠀⠀⠂⠴⢤⠈⠁⡈⣀⠄⢴⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃
⠀⠁⠀⡈⠀⠀⠀⠀⠀⢼⠀⠀⣄⣩⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠻⣦
⡁⠁⠀⠀⠀⡗⢸⡄⠀⠀⠀⠈⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠
⠀⡄⠀⠀⡴⠲⣄⠀⠀⠀⠀⢠⠂⠀⠀⢀⠱⣄⠀⠀⠀⠀⠈
⠀⢷⠀⠀⠀⠀⠙⡄⠀⠀⡀⠚⠀⠀⠀⠈⠀⣻⠆⠀⠀⣀⠤⠶⡂⢀⣠⡀⠀⠀⠀⠀⠀⠀⢲⡄
⠀⠈⠀⠀⡀⠀⠈⡟⠀⣈⣧⠠⣀⠀⣀⠴⢎⡥⠠⠀⠈⠀⠀⡄⠀⠈⢫⣷⣄⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠈⠁⠉⠀⠈⠊⠋⠂⣀⣈⠐⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⡀⢸⠇
⠀⠀⠀⠠⠀⠀⡀⠄⠘⠀⠀⢀⠈⠙⠀⠰⣆⣄⠠⢄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⡀⠀⠀⠘⠀⠊
⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⢀⡁⠀⠠⠀⠀⠠⢄⠀⠁⢀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠑⠆⠀⣇⠉⠳⡮⣄⠀⠀⢀⢀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠁⠀⠈⠩⠗⠀⠀⠄⠉⠝⠀⠁⠀⠀⠀⠀⠀⠀⠌⢉⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣥⣤⣾⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⡄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢠⢦⠀⡄⠀⠶⠀⠀⢠⠃⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⢤⠄⢼⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢴⢸⠀⡇⢠⡆⠀⠀⠀⠀⣀⠀⢠⠗⠺⢖⣋⠏⠐⡄⢀⠀⠠⠃⢀⠔⠀⠀⢀⠀⠀⠐⠂⠀⢰⠃⠸⠀⢠⠒⢄
⠀⠀⠀⣆⣋⠀⣱⡣⠓⠠⠤⠤⣼⠈⣇⢸⢀⢀⡏⡠⡔⢄⡸⠚⠉⠉⠙⠢⡔⣂⡀⢠⠤⠀⠐⠒⠀⠘⡄⠄⡆⡏⠀⢸⠀⠀⣹⠀⠚⣀⡀
⠀⠀⠀⠀⠈⠀⣇⠀⠀⠀⠀⠀⠘⡄⠻⠭⠀⠀⣧⠇⣗⠸⠀⠀⠀⣀⣀⠀⠷⠀⠒⠂⠐⠰⣆⢰⠶⠂⠳⣄⢈⠉⢀⠸⣤⡴⣹⠀⠇⠰⠁
⠀⠀⠀⠀⠀⠇⠉⠀⠀⡶⡄⠀⠀⠘⠢⢤⡀⠀⡔⠊⠁⠙⠓⠮⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⡀⠀⠀⠈⡇⠀⠀⠖⠊⠁⠉⠓⠀⢄⡀
⠀⡁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢣⡄⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠒⠂⠄⠠⠄⢀⠀⢀⢰⠈⠀⡀⠀⠀⠀⠀⠀⠀⠉⠃
⠐⠗⠚⠒⠒⠒⠒⠚⠛⠓⠒⠒⠒⠒⠒⠘⠂⠐⠛⠓⠓⠒⠛⠒⠐⠂⠒⠒⠒⠒⠓⠒⠒⠒⠒⠐⠒⠒⠚⠒⠒⠛⠐⠒⠒⠒⠒⠒⠒⠒⠛⠓⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣶⣶⣶⣶⣶⣶⣶⣿⣤⣐⣢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠟⢛⣋⣭⣤⣶⣾⣿⣿⣿⡿⠛⠻⠛⠛⠽⠻⠉⠉⠓⠀⢀⡄
⠀⠀⠀⠀⠠⣔⣒⡬⠄⢀⣠⣴⡾⠛⠁⣀⣴⣾⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠃
⠀⠀⠀⢦⡻⢛⠁⠀⠚⠛⠛⠁⠀⠴⠾⣿⣿⣿⣟⠽⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠠⠀⠀⢀⠎⠐⣺⣿
⠀⠀⢇⡀⠫⠘⠀⠀⣠⣠⢀⢀⢀⠀⠀⠀⢰⠟⢁⡀⠀⠴⠶⢦⡀⠀⠀⠀⣼⠛⠁⢰⠀⠀⢀⣀⠀⢀⠀⣼⡿
⠀⠀⠀⠀⠀⢤⣤⠀⣄⣉⣉⣈⣐⡂⠀⠀⠀⢄⡐⢃⡀⠷⠞⡸⠁⠄⠀⠀⠀⠀⢠⡇⣠⣴⣿⣿⣿⣿⡷⠋
⣠⣾⣶⣶⣽⣧⣀⠀⠀⠉⢉⣛⠛⠛⠛⠷⠶⠖⠲⠄⠳⠒⠋⠂⠀⠀⠠⠀⠀⣐⣺⣾⣿⣿⣿⣿⣿⠏
⠻⢿⣿⣿⣿⣿⣿⣿⣷⣖⣀⣿⣿⣿⣦⡄⢀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡜⢸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣾⣶⡆⡏⢀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⢿⣒⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠁
⠀⠀⠀⠀⠀⡤⢤
⠀⠀⠀⠀⠀⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠀⠙⢧
⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣀⣶⠇
⠀⢁⠠⣁⢀⡖⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣁⣊⣔⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣁⠆⠀⠀⢀⠴⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⠻⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢀⣄⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⣀⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠋⠀⠀⠀⠀⠈⠉⠛⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⡠⡳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⡨⠀⠐⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⢸⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠁⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⣤⡄⠀⠀⠀⠀⠀⠻⠦⠄⡠⡄⠀⠐⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⣿⡟⢿⣛⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠂
⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⠙⠁⡉⠀⠄
⠀⢀⠴⠾⠗⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⣿⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠙⠈⡉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⠀⠄
⠀⢈⠘⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⢀⠐⢢⠄⠛⠀⠀⠀⠀⠀⠀⠄⠁⠠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⠀⠀⡄⡀
⠀⠃⣄⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢘⢉⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⣿⡇
⠀⠤⠜⢅⠘⣩⡷⡤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠖⠀⠀⡼⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣿⣿⠇
⠀⡆⠀⠓⠀⠀⠐⢉⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢹⣿
⠀⠫⠰⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⣿
⠀⠀⢤⣄⠀⠀⠀⣿⡉⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡏
⠀⠀⠈⠛⠿⢷⣾⣿⣿⣿⣿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠁⠑⢤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠭⠀⠀⠀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡀⠉⢁⣒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⣀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠕⠁⠀⠀⠀⠀⠉⠢⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⢃
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⡀⣀⠀⠀⠀⠈⠢⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢨⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢨⠀⠀⠡⠀⠀⠀⠀⠱⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⢠⠀⣀⣀⠉⠁⠒⠠⠄⣀⠀⠀⠀⠀⠸⡆⠀⠀⠑⠄⠴⠀⠀⠀⠀⠀⢡⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠒⠈⠉⠅
⠀⠀⠀⢈⠱⡀⠀⠈⠁⠐⠒⠠⠄⣈⠑⠒⠤⢀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⠀⠤⠀⠂⠒⠈⢉⣀⠀⠤⠤⠤⠄⢠
⠀⠀⠀⠘⡀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⣀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠃⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠃⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠳⣤⡀⠀⠀⠀⠀⡠⠂⡜⢀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠤⠭⠴⠊⠂⠁⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠤⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⣉⠠⠤⠀⠀⠄⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⢀⣀⣀⣀⣀⣠⠤⠀⠀⢀⡠⠒⠁⡠⠐⠉⠀⠀⠀⢀⠔⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⢌⠀⢀⣀⠠⠄⠊⠁⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢍⡁⠒⠀⠤⠤⠄⠄⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠁⠈⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠰⡷⢦⣄⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣀⡴⠖⠛⠉⠛⠲⣄
⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠹⣆⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠛⠁⠈⢻⡄⠀⣿⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣰⣏⡀⠀⠀⠀⠈⢳⣄⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠁⠉⠉⠳⣄⣠⣤⡤⢽⣦⣀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⢸⠃⢀⣀⠀⠀⠀⠀⣼⣿⡇⠀⢸⣿⣿⡏⠙⣿⣿⡟⠛⣿⣟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠸⣆⠁⠀⠀⠀⠀⢀⣿⣿⠃⠀⣿⣿⣿⠃⢰⣿⣿⡇⠀⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣤⣾⣿⠋⠀⣼⣿⣿⠏⠀⣼⣿⣿⠁⢰⣿⣿⠁⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣾⣿⣿⠏⠀⣰⣿⣿⠏⢀⣿⣿⡏⢀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡁⢀⣼⣿⣿⠏⢀⣾⣿⡟⢀⣾⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣧⣤⣿⡿⠿⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠄⠀⠀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠒⠀⠈⠉⠉⠉⠉⠉⠁⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠾⠏⠀⠀⠀⠀⠀⡾⠄⠀⠀⠀⠀⠀⠀⠘⡋⠀⠐⡆⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡧⠔⠊⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⢁⡠⡇⢄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠉⠀⣀⠠⠔⠂⡏
⠀⠀⠀⠀⡠⠔⠉⠀⢀⠔⠁⠀⠰⡀⠁⠀⢀⣀⣀⠤⠤⠒⠊⠉⠁⣀⡠⠤⠒⠊⠉⠀⠀⠀⡰
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠉⠀⢀⣀⠠⠤⠐⠒⢋⡁⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠑⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⠒⠫⡉⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣐⠐⠁⠀⠀⠀⠀⠀⢐⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡉⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣠⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⠀⠀⢰⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⢠⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⣿⠟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠈⠙⠛⠛⠃⠀⠀⠀⠈⠋⠉⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠛⢿⡿⢿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣿⣀⠠⠤⣔⣫⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠲⠦⠶⠂⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣭⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣠⣹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁



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