C# Backtracking with Validation Solution
Intuition\n Describe your first thoughts on how to solve this problem. To solve the problem of generating all possible original coordinates from a given string 
# Intuition\n<!-- Describe your first thoughts on how to solve this problem. -->To solve the problem of generating all possible original coordinates from a given string s, we need to explore all ways to split the string into two parts and then generate valid coordinate representations for each part.\n___\n\n# Approach\n<!-- Describe your approach to solving the problem. -->//  Extract the Core String: Remove the outer parentheses from the input string s using Substring(1, s.Length - 2).\n\n//  Split the String into Two Parts: Iterate through the string to create all possible splits. For each split position i,\n//  generate two substrings: the left part and the right part.\n\n//  Generate Valid Decima Placements:\n//  *For each part, use the GenerateValidNumbers method to generate all valid representations (both integer and decimal) \n//  of the substring.\n//  *\'IsValidNumber\' checks if the substring is a valid integer (it should not start with \'0\' unless it is exactly "0").\n//  *\'IsValidFraction\' checks if the substring can be a valid fraction (it should not end with \'0\').\n\n//  Combine the Results: For each valid representation of the left part of each valid represeentation of the right part,\n//  combine them to form coordinate pairs and add them to the result list.\n___\n\n# Complexity\n- Time complexity:\n<!-- Add your time complexity here, e.g. $$O(n)$$ -->O(n^3) where n is the length of the string (excluding parentheses). This is because we generate all possible splits, and for each split, we generate valid decimal representations, which involves checking each substring.\n___\n\n- Space complexity:\n<!-- Add your space complexity here, e.g. $$O(n)$$ -->O(n^3), due to the storage required for all possible combinations of valid coordinates.\n___\n\n# Code\n```\nusing System;\nusing System.Collections.Generic;\n\npublic class Solution {\n    public IList<string> AmbiguousCoordinates(string s) {\n        var result = new List<string>();\n        \n        // Remove the parentheses\n        s = s.Substring(1, s.Length - 2);\n        \n        for (int i = 1; i < s.Length; i++) {\n            var leftParts = GenerateValidNumbers(s.Substring(0, i));\n            var rightParts = GenerateValidNumbers(s.Substring(i));\n            \n            foreach (var left in leftParts) {\n                foreach (var right in rightParts) {\n                    result.Add($"({left}, {right})");\n                }\n            }\n        }\n        \n        return result;\n    }\n    \n    private IList<string> GenerateValidNumbers(string s) {\n        var validNumbers = new List<string>();\n        \n        if (IsValidNumber(s)) {\n            validNumbers.Add(s);\n        }\n        \n        for (int i = 1; i < s.Length; i++) {\n            var left = s.Substring(0, i);\n            var right = s.Substring(i);\n            \n            if (IsValidNumber(left) && IsValidFraction(right)) {\n                validNumbers.Add($"{left}.{right}");\n            }\n        }\n        \n        return validNumbers;\n    }\n    \n    private bool IsValidNumber(string s) {\n        // A valid number should not start with \'0\' unless it is exactly "0"\n        return !(s.Length > 1 && s[0] == \'0\');\n    }\n    \n    private bool IsValidFraction(string s) {\n        // A valid fraction should not end with \'0\'\n        return s[s.Length - 1] != \'0\';\n    }\n}\n```
['String', 'Backtracking', 'C#']