<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 n;
    int ans2[1002];  // num of elements smaller than it right side
    int ans1[1002];  // nums of elements larger tham it right side

    void merge(int lo,int mid,int hi,vector<pair<int,int>> &v){
        int l=lo,r=mid+1;
        vector<pair<int,int>> temp;
        int right_small_cnt = 0;

        while(l<=mid && r<=hi){
            if(v[l].first <= v[r].first){ 
                ans1[v[l].second]+=hi-r+1;   // next greater
                ans2[v[l].second] += right_small_cnt;   // next smaller

                temp.push_back({v[l].first,v[l].second});
                l++;
            }
            else{
                right_small_cnt++;
                temp.push_back({v[r].first,v[r].second});
                r++;
            }
        }
        while(l<=mid){
            ans2[v[l].second] += right_small_cnt;
            temp.push_back({v[l].first,v[l].second});
                l++;
        }
        while(r<=hi){
            temp.push_back({v[r].first,v[r].second});
                r++;
        }
        for(int i=lo;i<=hi;i++){
            v[i]=temp[i-lo];
        }
    }

    void f(int lo,int hi,vector<pair<int,int>> &v){
        if(lo==hi){
            return;
        }
        int mid=(hi+lo)/2;
        f(lo,mid,v);
        f(mid+1,hi,v);
        merge(lo,mid,hi,v);
        
    }

    int numTeams(vector<int>& nums) {
        n = nums.size();
        vector<pair<int,int>> v;
        for(int i=0;i<n;i++){
            v.push_back({nums[i],i});
        }
        f(0,n-1,v);



        int res = 0;
        for(int i = 0;i<n; i++){       // for next larger
            for(int j = i+1;j<n; j++){
                if(nums[j] > nums[i]){
                    res += ans1[j];
                }
            }
        }
        for(int i = 0;i<n; i++){      // for next smaller
            for(int j = i+1;j<n; j++){
                if(nums[j] < nums[i]){
                    res += ans2[j];
                }
            }
        }

        return res;

    }
};
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