<p>You are given an array <code>trees</code> where <code>trees[i] = [x<sub>i</sub>, y<sub>i</sub>]</code> represents the location of a tree in the garden.</p>

<p>Fence the entire garden using the minimum length of rope, as it is expensive. The garden is well-fenced only if <strong>all the trees are enclosed</strong>.</p>

<p>Return <em>the coordinates of trees that are exactly located on the fence perimeter</em>. You may return the answer in <strong>any order</strong>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/24/erect2-plane.jpg" style="width: 400px; height: 393px;" />
<pre>
<strong>Input:</strong> trees = [[1,1],[2,2],[2,0],[2,4],[3,3],[4,2]]
<strong>Output:</strong> [[1,1],[2,0],[4,2],[3,3],[2,4]]
<strong>Explanation:</strong> All the trees will be on the perimeter of the fence except the tree at [2, 2], which will be inside the fence.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/04/24/erect1-plane.jpg" style="width: 400px; height: 393px;" />
<pre>
<strong>Input:</strong> trees = [[1,2],[2,2],[4,2]]
<strong>Output:</strong> [[4,2],[2,2],[1,2]]
<strong>Explanation:</strong> The fence forms a line that passes through all the trees.
</pre>

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

<ul>
	<li><code>1 &lt;= trees.length &lt;= 3000</code></li>
	<li><code>trees[i].length == 2</code></li>
	<li><code>0 &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 100</code></li>
	<li>All the given positions are <strong>unique</strong>.</li>
</ul>

class Solution {
public:
    /*int gcd(int a, int b)
    {
        if(a == 0 or b == 0) return a + b;
        return gcd(max(a, b)%min(a, b), min(a, b));
    }*/
    vector<vector<int>> helper1(vector<vector<int>>& trees, int &ind, int mod)
    {
        int x = trees[ind][0], y = trees[ind][1];
        int a2 = 100, b2 = 1;
        int a1 = -100, b1 = 1;
        vector<vector<int>> ans1, ans2;
        bool f2 = false, f1 = false;
        for(int i = ind + 1; i < trees.size(); i++)
        {
            int k = trees[i][0], l = trees[i][1];
            if(l <= y && mod == 1)
            {
                
                if(f2)
                {
                    if(k == x) {ans1.push_back({k, l}); ind = i;}
                }
                else
                {
                    if(k == x)
                    {
                        f2 = true;
                        ans1.resize(0);
                        ans1.push_back({k, l});
                        ind = i;
                    }
                    else
                    {
                        int a3 = l - y, b3 = k - x;
                        if(b2*a3 == a2*b3) {ans1.push_back({k, l}); ind = i;}
                        else if(b2*a3 < a2*b3)
                        {
                            ans1.resize(0);
                            ans1.push_back({k, l});
                            a2 = a3;
                            b2 = b3;
                            ind = i;
                        }
                    }
                }
            }
            if(l >= y && mod == 0)
            {
                if(f1)
                {
                    if(k == x) {ans2.push_back({k, l}); ind = i;}
                }
                else
                {
                    if(k == x)
                    {
                        f1 = true;
                        ans2.resize(0);
                        ans2.push_back({k, l});
                        ind = i;
                    }
                    else
                    {
                        int a3 = l - y, b3 = k - x;
                        if(b1*a3 == a1*b3) {ans2.push_back({k, l}); ind = i;}
                        else if(b1*a3 > a1*b3)
                        {
                            ans2.resize(0);
                            ans2.push_back({k, l});
                            a1 = a3;
                            b1 = b3;
                            ind = i;
                        }
                    }
                }
            }
        }
        vector<vector<int>> ans;
        for(auto &k: ans1) ans.push_back(k);
        for(auto &k: ans2) ans.push_back(k);
        ans.push_back(trees[0]);
        return ans;
    }
    bool solve(vector<int> &a, vector<int> &b)
    {
        return a[0] >= -b[0];
    }
    vector<vector<int>> outerTrees(vector<vector<int>>& trees) {
        set<vector<int>> last;
        vector<vector<int>> trees1 = trees;
        vector<vector<int>> trees2;
        for(auto &k: trees) trees2.push_back({-k[0], k[1]});
        sort(trees1.begin(), trees1.end());
        sort(trees2.begin(), trees2.end());
        int ind1 = 0;
        int ind2 = 0;
        while(true)
        {
            auto ans1 = helper1(trees1, ind1, 0);
            auto ans2 = helper1(trees2, ind2, 0);
            for(auto &k: ans1) last.insert(k);
            for(auto &k: ans2) last.insert({-k[0], k[1]});
            
            
            if(solve(trees1[ind1], trees2[ind2])) break; 
        }
        ind1 = 0;
        ind2 = 0;
        while(true)
        {
            
            auto ans1 = helper1(trees1, ind1, 1);
            auto ans2 = helper1(trees2, ind2, 1);
            for(auto &k: ans1) last.insert(k);
            for(auto &k: ans2) last.insert({-k[0], k[1]});
            if(solve(trees1[ind1], trees2[ind2])) break; 
        }
        vector<vector<int>> result(last.begin(), last.end());
        return result;
    }
};
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