{"problem":{"name":"131. The Mirrors Strike Back","description":{"content":"If you competed in the CodeRams Contest #8, you might remember a problem involving mirrors and light. This is a more difficult and updated version of that problem. You are in a hall of mirrors, and y","description_type":"Markdown"},"platform":"Codeforces","limit":{"time_limit":1000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"CF10269131"},"statements":[{"statement_type":"Markdown","content":"If you competed in the CodeRams Contest #8, you might remember a problem involving mirrors and light. This is a more difficult and updated version of that problem.\n\nYou are in a hall of mirrors, and you shine a laser beam in a certain direction. Given the position of several diagonal mirrors, figure out the path of the laser beam.\n\nYou shine your laser beam from the back of the hall of mirrors (on a computer screen, this will look like the \"bottom\" of the image).\n\n*This problem also adds \"star blocks\", which shine light in all directions when light passes through them*.\n\nThe first line of input contains two space-separated integers $n$, $m$, and $x$. $n$ and $m$ represent the number of rows and columns, respectively, and the number $x$ represents the index on the bottom row that you start shining the laser pointer at (with 0-based indices). The next $n$ lines contain $m$ characters representing the hall of mirrors. If a position does not contain a mirror, it will be pictured as a single dot. If it does have a mirror, it will be represented with either a forward slash (\"/\") or a backslash, which represents the direction that it reflects light in. See the example cases if you are confused on this. Additionally, if the position contains a star block, it will be pictured as an asterisk (\"*\"). Star blocks redirect light in all four immediate directions (up, down, left, and right).\n\nOutput the given input matrix, however, replace any positions without a mirror or star block that the laser pointer shines through horizontally with a hyphen, and replace any positions without a mirror or star block that the laser pointer shines through vertically with a pipe (\"|\"), above the ENTER key on a keyboard. If the laser pointer travels through a space in both directions, replace the dot in that position with a plus symbol (\"+\").\n\n## Input\n\nThe first line of input contains two space-separated integers $n$, $m$, and $x$. $n$ and $m$ represent the number of rows and columns, respectively, and the number $x$ represents the index on the bottom row that you start shining the laser pointer at (with 0-based indices). The next $n$ lines contain $m$ characters representing the hall of mirrors. If a position does not contain a mirror, it will be pictured as a single dot. If it does have a mirror, it will be represented with either a forward slash (\"/\") or a backslash, which represents the direction that it reflects light in. See the example cases if you are confused on this. Additionally, if the position contains a star block, it will be pictured as an asterisk (\"*\"). Star blocks redirect light in all four immediate directions (up, down, left, and right).\n\n## Output\n\nOutput the given input matrix, however, replace any positions without a mirror or star block that the laser pointer shines through horizontally with a hyphen, and replace any positions without a mirror or star block that the laser pointer shines through vertically with a pipe (\"|\"), above the ENTER key on a keyboard. If the laser pointer travels through a space in both directions, replace the dot in that position with a plus symbol (\"+\").\n\n[samples]","is_translate":false,"language":"English"},{"statement_type":"Markdown","content":"**Definitions**  \nLet $ n, m, x \\in \\mathbb{Z} $ with $ n \\geq 1 $, $ m \\geq 1 $, $ 0 \\leq x < m $.  \nLet $ G \\in \\{ \\text{`.`}, \\text{`/`}, \\text{`\\`}, \\text{`*`} \\}^{n \\times m} $ be the grid representing the hall of mirrors, where:  \n- `.` denotes empty space,  \n- `/` denotes a diagonal mirror reflecting light 90° clockwise or counterclockwise (depending on incident direction),  \n- `\\` denotes a diagonal mirror reflecting light 90° in the opposite direction,  \n- `*` denotes a star block that emits light in all four cardinal directions (up, down, left, right) upon being struck.  \n\nLet $ \\mathcal{L} \\subseteq \\{ (i, j, d) \\mid 0 \\leq i < n, 0 \\leq j < m, d \\in \\{ \\text{up}, \\text{down}, \\text{left}, \\text{right} \\} \\} $ be the set of all light rays, each represented by position and direction.  \n\nInitial ray: $ (n-1, x, \\text{up}) $.  \n\n**Constraints**  \n1. The laser beam propagates according to mirror reflection rules:  \n   - `/` reflects:  \n     - right → up,  \n     - left → down,  \n     - up → right,  \n     - down → left.  \n   - `\\` reflects:  \n     - right → down,  \n     - left → up,  \n     - up → left,  \n     - down → right.  \n2. `*` emits four new rays in directions up, down, left, right upon being hit (regardless of incoming direction).  \n3. Light propagates until it exits the grid or enters a cycle.  \n4. Each grid cell may be traversed by multiple rays.  \n\n**Objective**  \nConstruct output grid $ O \\in \\{ \\text{`.`}, \\text{`/`}, \\text{`\\`}, \\text{`*`}, \\text{`-`}, \\text{`|`}, \\text{`+`} \\}^{n \\times m} $, where:  \n- $ O[i][j] = G[i][j] $ if $ G[i][j] \\in \\{ \\text{`/`}, \\text{`\\`}, \\text{`*`} \\} $,  \n- Otherwise, for each empty cell $ (i,j) $:  \n  - If traversed **only horizontally** (left/right): $ O[i][j] = \\text{`-`} $,  \n  - If traversed **only vertically** (up/down): $ O[i][j] = \\text{`|`} $,  \n  - If traversed **both horizontally and vertically**: $ O[i][j] = \\text{`+`} $,  \n  - Otherwise: $ O[i][j] = \\text{`.`} $.  \n\nOutput $ O $.","is_translate":false,"language":"Formal"}],"meta":{"iden":"CF10269131","tags":[],"sample_group":[],"created_at":"2026-03-03 11:00:39"}}