<p>You are given an integer array <code>cards</code> of length <code>4</code>. You have four cards, each containing a number in the range <code>[1, 9]</code>. You should arrange the numbers on these cards in a mathematical expression using the operators <code>[&#39;+&#39;, &#39;-&#39;, &#39;*&#39;, &#39;/&#39;]</code> and the parentheses <code>&#39;(&#39;</code> and <code>&#39;)&#39;</code> to get the value 24.</p>

<p>You are restricted with the following rules:</p>

<ul>
	<li>The division operator <code>&#39;/&#39;</code> represents real division, not integer division.

	<ul>
		<li>For example, <code>4 / (1 - 2 / 3) = 4 / (1 / 3) = 12</code>.</li>
	</ul>
	</li>
	<li>Every operation done is between two numbers. In particular, we cannot use <code>&#39;-&#39;</code> as a unary operator.
	<ul>
		<li>For example, if <code>cards = [1, 1, 1, 1]</code>, the expression <code>&quot;-1 - 1 - 1 - 1&quot;</code> is <strong>not allowed</strong>.</li>
	</ul>
	</li>
	<li>You cannot concatenate numbers together
	<ul>
		<li>For example, if <code>cards = [1, 2, 1, 2]</code>, the expression <code>&quot;12 + 12&quot;</code> is not valid.</li>
	</ul>
	</li>
</ul>

<p>Return <code>true</code> if you can get such expression that evaluates to <code>24</code>, and <code>false</code> otherwise.</p>

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

<pre>
<strong>Input:</strong> cards = [4,1,8,7]
<strong>Output:</strong> true
<strong>Explanation:</strong> (8-4) * (7-1) = 24
</pre>

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

<pre>
<strong>Input:</strong> cards = [1,2,1,2]
<strong>Output:</strong> false
</pre>

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

<ul>
	<li><code>cards.length == 4</code></li>
	<li><code>1 &lt;= cards[i] &lt;= 9</code></li>
</ul>

class Solution {
    double eps = 1e-6;
    vector<pair<double, string>> dp[4][4];
public:
    bool judgePoint24(vector<int>& cards) {
        vector<double> nums;
        for (auto& c : cards) {
            nums.push_back((double) c);
        }
        sort(nums.begin(), nums.end());
        while (true) {
            dfs(nums, 0, 3);
            for (auto [x, str] : dp[0][3]) {
                if (abs(x - 24) <= eps) {
                    cout << str << endl;
                    return true; 
                }
            }
            for (int i = 0; i < 4; i++) {
                for (int j = 0; j < 4; j++) {
                    dp[i][j].clear();
                }
            }
            if (next_permutation(nums.begin(), nums.end()) == false) return false;
        }
        return false;
    }
    
    void dfs(vector<double>& nums, int left, int right) {
        if (!dp[left][right].empty()) return;
        if (left == right) {
            string str = "";
            str += nums[left] + '0';
            dp[left][right].push_back({nums[left], str});
            return;
        }
        for (int i = left; i < right; i++) {
            dfs(nums, left, i);
            dfs(nums, i + 1, right);
            for (auto [l, strx] : dp[left][i]) {
                for (auto [r, stry] : dp[i + 1][right]) {
                    dp[left][right].push_back({l + r, "(" + strx + "+" + stry + ")" });
                    dp[left][right].push_back({l - r, "(" + strx + "-" + stry + ")" });
                    dp[left][right].push_back({l * r, "(" + strx + "*" + stry + ")" });
                    if (r != 0) {
                        dp[left][right].push_back({l / r, "(" + strx + "/" + stry + ")" }); 
                    }
                }
            }
        }
    }
};
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