<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:
    int numTeams(vector<int>& ratings) {
        // Find the maximum rating
        int maxRating = 0;
        for (int rating : ratings) {
            maxRating = max(maxRating, rating);
        }

        // Initialize Binary Indexed Trees for left and right sides
        vector<int> leftBIT(maxRating + 1, 0);
        vector<int> rightBIT(maxRating + 1, 0);

        // Populate the right BIT with all ratings initially
        for (int rating : ratings) {
            updateBIT(rightBIT, rating, 1);
        }

        int teams = 0;
        for (int currentRating : ratings) {
            // Remove current rating from right BIT
            updateBIT(rightBIT, currentRating, -1);

            // Count soldiers with smaller and larger ratings on both sides
            int smallerRatingsLeft = getPrefixSum(leftBIT, currentRating - 1);
            int smallerRatingsRight = getPrefixSum(rightBIT, currentRating - 1);
            int largerRatingsLeft = getPrefixSum(leftBIT, maxRating) -
                                    getPrefixSum(leftBIT, currentRating);
            int largerRatingsRight = getPrefixSum(rightBIT, maxRating) -
                                     getPrefixSum(rightBIT, currentRating);

            // Count increasing and decreasing sequences
            teams += (smallerRatingsLeft * largerRatingsRight);
            teams += (largerRatingsLeft * smallerRatingsRight);

            // Add current rating to left BIT
            updateBIT(leftBIT, currentRating, 1);
        }

        return teams;
    }

private:
    // Update the Binary Indexed Tree
    void updateBIT(vector<int>& BIT, int index, int value) {
        while (index < BIT.size()) {
            BIT[index] += value;
            index +=
                index & (-index);  // Move to the next relevant index in BIT
        }
    }

    // Get the sum of all elements up to the given index in the BIT
    int getPrefixSum(vector<int>& BIT, int index) {
        int sum = 0;
        while (index > 0) {
            sum += BIT[index];
            index -= index & (-index);  // Move to the parent node in BIT
        }
        return sum;
    }
};
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