<p>Alice is a caretaker of <code>n</code> gardens and she wants to plant flowers to maximize the total beauty of all her gardens.</p>

<p>You are given a <strong>0-indexed</strong> integer array <code>flowers</code> of size <code>n</code>, where <code>flowers[i]</code> is the number of flowers already planted in the <code>i<sup>th</sup></code> garden. Flowers that are already planted <strong>cannot</strong> be removed. You are then given another integer <code>newFlowers</code>, which is the <strong>maximum</strong> number of flowers that Alice can additionally plant. You are also given the integers <code>target</code>, <code>full</code>, and <code>partial</code>.</p>

<p>A garden is considered <strong>complete</strong> if it has <strong>at least</strong> <code>target</code> flowers. The <strong>total beauty</strong> of the gardens is then determined as the <strong>sum</strong> of the following:</p>

<ul>
	<li>The number of <strong>complete</strong> gardens multiplied by <code>full</code>.</li>
	<li>The <strong>minimum</strong> number of flowers in any of the <strong>incomplete</strong> gardens multiplied by <code>partial</code>. If there are no incomplete gardens, then this value will be <code>0</code>.</li>
</ul>

<p>Return <em>the <strong>maximum</strong> total beauty that Alice can obtain after planting at most </em><code>newFlowers</code><em> flowers.</em></p>

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

<pre>
<strong>Input:</strong> flowers = [1,3,1,1], newFlowers = 7, target = 6, full = 12, partial = 1
<strong>Output:</strong> 14
<strong>Explanation:</strong> Alice can plant
- 2 flowers in the 0<sup>th</sup> garden
- 3 flowers in the 1<sup>st</sup> garden
- 1 flower in the 2<sup>nd</sup> garden
- 1 flower in the 3<sup>rd</sup> garden
The gardens will then be [3,6,2,2]. She planted a total of 2 + 3 + 1 + 1 = 7 flowers.
There is 1 garden that is complete.
The minimum number of flowers in the incomplete gardens is 2.
Thus, the total beauty is 1 * 12 + 2 * 1 = 12 + 2 = 14.
No other way of planting flowers can obtain a total beauty higher than 14.
</pre>

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

<pre>
<strong>Input:</strong> flowers = [2,4,5,3], newFlowers = 10, target = 5, full = 2, partial = 6
<strong>Output:</strong> 30
<strong>Explanation:</strong> Alice can plant
- 3 flowers in the 0<sup>th</sup> garden
- 0 flowers in the 1<sup>st</sup> garden
- 0 flowers in the 2<sup>nd</sup> garden
- 2 flowers in the 3<sup>rd</sup> garden
The gardens will then be [5,4,5,5]. She planted a total of 3 + 0 + 0 + 2 = 5 flowers.
There are 3 gardens that are complete.
The minimum number of flowers in the incomplete gardens is 4.
Thus, the total beauty is 3 * 2 + 4 * 6 = 6 + 24 = 30.
No other way of planting flowers can obtain a total beauty higher than 30.
Note that Alice could make all the gardens complete but in this case, she would obtain a lower total beauty.
</pre>

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

<ul>
	<li><code>1 &lt;= flowers.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= flowers[i], target &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= newFlowers &lt;= 10<sup>10</sup></code></li>
	<li><code>1 &lt;= full, partial &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution:
    def maximumBeauty(self, flowers: List[int], newFlowers: int, target: int, full: int, partial: int) -> int:
        n = len(flowers)
        flowers.sort()
        initialComplete = sum(flowers[i] >= target for i in range(n))
        incompleteGardens = n - initialComplete

        minIncompleteIndex = -1
        minFlowerLevel = 0

        def maximizeMinFlowers() -> None:
            nonlocal minIncompleteIndex, minFlowerLevel, newFlowers
            if j == 0:
                return
            if minIncompleteIndex == -1:
                minIncompleteIndex = 0
                minFlowerLevel = flowers[0]
            while minIncompleteIndex < j:
                nextFlowerLevel = flowers[minIncompleteIndex + 1] if minIncompleteIndex + 1 < j else target - 1
                maxIncrease = min(nextFlowerLevel - minFlowerLevel, newFlowers // (minIncompleteIndex + 1))
                minFlowerLevel += maxIncrease
                newFlowers -= maxIncrease * (minIncompleteIndex + 1)
                if minFlowerLevel != nextFlowerLevel:
                    break
                minIncompleteIndex += 1

        def addCompleteGarden() -> bool:
            nonlocal j, newFlowers
            if j - 1 < 0 or newFlowers < target - flowers[j - 1]:
                return False
            newFlowers -= target - flowers[j - 1]
            j -= 1
            return True

        def removeCompleteGarden() -> None:
            nonlocal j, newFlowers
            assert j < incompleteGardens
            newFlowers += target - flowers[j]
            flowers[j] = flowers[j]
            j += 1

        j = incompleteGardens

        while addCompleteGarden():
            pass

        maximizeMinFlowers()

        maxBeauty = full * (n - j) + partial * minFlowerLevel

        while j < incompleteGardens:
            removeCompleteGarden()
            maximizeMinFlowers()
            maxBeauty = max(maxBeauty, full * (n - j) + partial * minFlowerLevel)

        return maxBeauty
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