<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)

    # If a garden is already complete, clamp it to the target.
    flowers = [min(flower, target) for flower in flowers]
    flowers.sort()

    # All gardens are complete, so nothing we can do.
    if flowers[0] == target:
      return n * full

    # Having many new flowers maximizes the beauty value.
    if newFlowers >= n * target - sum(flowers):
      return max(n * full, (n - 1) * full + (target - 1) * partial)

    ans = 0
    leftFlowers = newFlowers
    # cost[i] := the cost to make flowers[0..i] the same
    cost = [0] * n

    for i in range(1, n):
      # Plant (flowers[i] - flowers[i - 1]) flowers for flowers[0..i - 1].
      cost[i] = cost[i - 1] + i * (flowers[i] - flowers[i - 1])

    i = n - 1  # flowers' index (flowers[i + 1..n) are complete)
    while flowers[i] == target:
      i -= 1

    while leftFlowers >= 0:
      # To maximize the minimum number of incomplete flowers, we find the first
      # index j that we can't make flowers[0..j] equal to flowers[j], then we
      # know we can make flowers[0..j - 1] equal to flowers[j - 1]. In the
      # meantime, evenly increase each of them to seek a bigger minimum value.
      j = min(i + 1, bisect_right(cost, leftFlowers))
      minIncomplete = flowers[j - 1] + (leftFlowers - cost[j - 1]) // j
      ans = max(ans, (n - 1 - i) * full + minIncomplete * partial)
      leftFlowers -= max(0, target - flowers[i])
      i -= 1

    return ans
784