<p>There are <code>n</code> soldiers standing in a line. Each soldier is assigned a <strong>unique</strong> <code>rating</code> value.</p>

<p>You have to form a team of 3 soldiers amongst them under the following rules:</p>

<ul>
	<li>Choose 3 soldiers with index (<code>i</code>, <code>j</code>, <code>k</code>) with rating (<code>rating[i]</code>, <code>rating[j]</code>, <code>rating[k]</code>).</li>
	<li>A team is valid if: (<code>rating[i] &lt; rating[j] &lt; rating[k]</code>) or (<code>rating[i] &gt; rating[j] &gt; rating[k]</code>) where (<code>0 &lt;= i &lt; j &lt; k &lt; n</code>).</li>
</ul>

<p>Return the number of teams you can form given the conditions. (soldiers can be part of multiple teams).</p>

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

<pre>
<strong>Input:</strong> rating = [2,5,3,4,1]
<strong>Output:</strong> 3
<strong>Explanation:</strong> We can form three teams given the conditions. (2,3,4), (5,4,1), (5,3,1). 
</pre>

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

<pre>
<strong>Input:</strong> rating = [2,1,3]
<strong>Output:</strong> 0
<strong>Explanation:</strong> We can&#39;t form any team given the conditions.
</pre>

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

<pre>
<strong>Input:</strong> rating = [1,2,3,4]
<strong>Output:</strong> 4
</pre>

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

<ul>
	<li><code>n == rating.length</code></li>
	<li><code>3 &lt;= n &lt;= 1000</code></li>
	<li><code>1 &lt;= rating[i] &lt;= 10<sup>5</sup></code></li>
	<li>All the integers in <code>rating</code> are <strong>unique</strong>.</li>
</ul>

class Solution {
public:
    unordered_map<string, int> dp;
    int ans = 0;

    int numTeams(vector<int>& rating) {
        int n = rating.size();
        for (int i = 0; i < n - 2; ++i) {
            ans += dfs(rating, i, 0, n);
        }
        return ans;
    }

private:
    int dfs(vector<int>& rating, int ind, int continueNum, int n) {
        if (continueNum == 2 || continueNum == -2) return 1;
        if (ind == n) return 0;

        // Construct the key using ostringstream
        ostringstream oss;
        oss << ind << "#" << continueNum;
        string key = oss.str();

        if (dp.find(key) != dp.end()) {
            return dp[key];
        }

        int harvest = 0;
        for (int i = ind + 1; i < n; ++i) {
            if (rating[ind] > rating[i] && continueNum >= 0) {
                harvest += dfs(rating, i, continueNum + 1, n);
            } else if (rating[ind] < rating[i] && continueNum <= 0) {
                harvest += dfs(rating, i, continueNum - 1, n);
            }
        }

        dp[key] = harvest;
        return harvest;
    }
};
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