<p>You are given two jugs with capacities <code>x</code> liters and <code>y</code> liters. You have an infinite water supply. Return whether the total amount of water in both jugs may reach <code>target</code> using the following operations:</p>

<ul>
	<li>Fill either jug completely with water.</li>
	<li>Completely empty either jug.</li>
	<li>Pour water from one jug into another until the receiving jug is full, or the transferring jug is empty.</li>
</ul>

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

<div class="example-block" style="border-color: var(--border-tertiary); border-left-width: 2px; color: var(--text-secondary); font-size: .875rem; margin-bottom: 1rem; margin-top: 1rem; overflow: visible; padding-left: 1rem;">
<p><strong>Input: </strong> <span class="example-io" style="font-family: Menlo,sans-serif; font-size: 0.85rem;"> x = 3, y = 5, target = 4 </span></p>

<p><strong>Output: </strong> <span class="example-io" style="font-family: Menlo,sans-serif; font-size: 0.85rem;"> true </span></p>

<p><strong>Explanation:</strong></p>

<p>Follow these steps to reach a total of 4 liters:</p>

<ol>
	<li>Fill the 5-liter jug (0, 5).</li>
	<li>Pour from the 5-liter jug into the 3-liter jug, leaving 2 liters (3, 2).</li>
	<li>Empty the 3-liter jug (0, 2).</li>
	<li>Transfer the 2 liters from the 5-liter jug to the 3-liter jug (2, 0).</li>
	<li>Fill the 5-liter jug again (2, 5).</li>
	<li>Pour from the 5-liter jug into the 3-liter jug until the 3-liter jug is full. This leaves 4 liters in the 5-liter jug (3, 4).</li>
	<li>Empty the 3-liter jug. Now, you have exactly 4 liters in the 5-liter jug (0, 4).</li>
</ol>

<p>Reference: The <a href="https://www.youtube.com/watch?v=BVtQNK_ZUJg&amp;ab_channel=notnek01" target="_blank">Die Hard</a> example.</p>
</div>

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

<div class="example-block" style="border-color: var(--border-tertiary); border-left-width: 2px; color: var(--text-secondary); font-size: .875rem; margin-bottom: 1rem; margin-top: 1rem; overflow: visible; padding-left: 1rem;">
<p><strong>Input: </strong> <span class="example-io" style="font-family: Menlo,sans-serif; font-size: 0.85rem;"> x = 2, y = 6, target = 5 </span></p>

<p><strong>Output: </strong> <span class="example-io" style="font-family: Menlo,sans-serif; font-size: 0.85rem;"> false </span></p>
</div>

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

<div class="example-block" style="border-color: var(--border-tertiary); border-left-width: 2px; color: var(--text-secondary); font-size: .875rem; margin-bottom: 1rem; margin-top: 1rem; overflow: visible; padding-left: 1rem;">
<p><strong>Input: </strong> <span class="example-io" style="font-family: Menlo,sans-serif; font-size: 0.85rem;"> x = 1, y = 2, target = 3 </span></p>

<p><strong>Output: </strong> <span class="example-io" style="font-family: Menlo,sans-serif; font-size: 0.85rem;"> true </span></p>

<p><strong>Explanation:</strong> Fill both jugs. The total amount of water in both jugs is equal to 3 now.</p>
</div>

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

<ul>
	<li><code>1 &lt;= x, y, target&nbsp;&lt;= 10<sup>3</sup></code></li>
</ul>

class jugs {
public:
    int x;
    int y;

    jugs(int x_curr_, int y_curr_) 
        : x(x_curr_), y(y_curr_) {}

    int sumCurr() { return x + y; }

    string toStr() const { return to_string(x) + to_string(y); }

    bool operator<(const jugs& rhs) const {
        return toStr() < rhs.toStr();
    }
};

class Solution {
public:
    bool canMeasureWater(int x, int y, int target) {
        set<jugs> visitedState;
        queue<jugs> toVisit;

        jugs startJugs(0, 0);
        toVisit.push(startJugs);
        while (!toVisit.empty()) {
            jugs curr = toVisit.front();

            toVisit.pop();

            if (curr.sumCurr() == target) return true;

            int tToX = min(x - curr.x, curr.y);
            int tToY = min(y - curr.y, curr.x);

            bool inserted;

            jugs next(x, curr.y);
            std::tie(std::ignore, inserted) = visitedState.insert(next);
            if (inserted) toVisit.push(next); 

            next.x = 0;
            std::tie(std::ignore, inserted) = visitedState.insert(next);
            if (inserted) toVisit.push(next); 

            next.x = curr.x; next.y = y;
            std::tie(std::ignore, inserted) = visitedState.insert(next);
            if (inserted) toVisit.push(next); 

            next.y = 0;
            std::tie(std::ignore, inserted) = visitedState.insert(next);
            if (inserted) toVisit.push(next); 

            next.x = curr.x + tToX; next.y = curr.y - tToX;
            std::tie(std::ignore, inserted) = visitedState.insert(next);
            if (inserted) toVisit.push(next); 

            next.x = curr.x - tToY; next.y = curr.y + tToY;
            std::tie(std::ignore, inserted) = visitedState.insert(next);
            if (inserted) toVisit.push(next); 
        }
        return false;
    }
};
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