<p>Given a string <code>num</code> that contains only digits and an integer <code>target</code>, return <em><strong>all possibilities</strong> to insert the binary operators </em><code>&#39;+&#39;</code><em>, </em><code>&#39;-&#39;</code><em>, and/or </em><code>&#39;*&#39;</code><em> between the digits of </em><code>num</code><em> so that the resultant expression evaluates to the </em><code>target</code><em> value</em>.</p>

<p>Note that operands in the returned expressions <strong>should not</strong> contain leading zeros.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> num = &quot;123&quot;, target = 6
<strong>Output:</strong> [&quot;1*2*3&quot;,&quot;1+2+3&quot;]
<strong>Explanation:</strong> Both &quot;1*2*3&quot; and &quot;1+2+3&quot; evaluate to 6.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> num = &quot;232&quot;, target = 8
<strong>Output:</strong> [&quot;2*3+2&quot;,&quot;2+3*2&quot;]
<strong>Explanation:</strong> Both &quot;2*3+2&quot; and &quot;2+3*2&quot; evaluate to 8.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> num = &quot;3456237490&quot;, target = 9191
<strong>Output:</strong> []
<strong>Explanation:</strong> There are no expressions that can be created from &quot;3456237490&quot; to evaluate to 9191.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= num.length &lt;= 10</code></li>
	<li><code>num</code> consists of only digits.</li>
	<li><code>-2<sup>31</sup> &lt;= target &lt;= 2<sup>31</sup> - 1</code></li>
</ul>

int speedUp = [] {
    std::ios::sync_with_stdio(0);
    std::cin.tie(0);
    return 0;
}();

#define opt() ios::sync_with_stdio(0);cin.tie(0);cout.tie(0);

class Solution {
    using ll = long long;
public:
    string num;
    vector<string> ret;
    int n, target;

    // Depth-first search function to generate all valid expressions
    void dfs(string &s, int index, ll res, ll mul) {
        // If reached the end of the string, check if the result is equal to the target
        // If so, add the expression to the result array
        // Regardless of correctness, return from the function
        if (index == n) {
            if (res == target) {
                ret.emplace_back(s);
            }
            return;
        }

        // Current expression length
        int m = s.size();
        if (index > 0) {
            s.push_back(0); // Placeholder to fill in the operator
        }
        ll val = 0;
        // Enumerate the length of the substring (number of digits), ensuring no leading zeros
        for (int j = index; j < n && (j == index || num[index] != '0'); ++j) {
            val = val * 10 + num[j] - '0';
            s.push_back(num[j]);
            if (index == 0) { // The expression cannot start with an operator
                dfs(s, j + 1, val, val);
            } else { // Enumerate operators
                // Addition
                s[m] = '+'; dfs(s, j + 1, res + val, val);
                // Subtraction
                s[m] = '-'; dfs(s, j + 1, res - val, -val);
                // Multiplication
                s[m] = '*'; dfs(s, j + 1, res - mul + mul * val, mul * val);
            }
        }
        s.resize(m);
    };

    vector<string> addOperators(string _num, int _target) {
        num = _num;
        n = _num.length();
        target = _target;
        string s;
        dfs(s, 0, 0, 0); // Start DFS from index 0 with initial result and multiplication value as 0
        return ret;
    }
};
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