<p>You have an <code>inventory</code> of different colored balls, and there is a customer that wants <code>orders</code> balls of <strong>any</strong> color.</p>

<p>The customer weirdly values the colored balls. Each colored ball&#39;s value is the number of balls <strong>of that color&nbsp;</strong>you currently have in your <code>inventory</code>. For example, if you own <code>6</code> yellow balls, the customer would pay <code>6</code> for the first yellow ball. After the transaction, there are only <code>5</code> yellow balls left, so the next yellow ball is then valued at <code>5</code> (i.e., the value of the balls decreases as you sell more to the customer).</p>

<p>You are given an integer array, <code>inventory</code>, where <code>inventory[i]</code> represents the number of balls of the <code>i<sup>th</sup></code> color that you initially own. You are also given an integer <code>orders</code>, which represents the total number of balls that the customer wants. You can sell the balls <strong>in any order</strong>.</p>

<p>Return <em>the <strong>maximum</strong> total value that you can attain after selling </em><code>orders</code><em> colored balls</em>. As the answer may be too large, return it <strong>modulo </strong><code>10<sup>9 </sup>+ 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/11/05/jj.gif" style="width: 480px; height: 270px;" />
<pre>
<strong>Input:</strong> inventory = [2,5], orders = 4
<strong>Output:</strong> 14
<strong>Explanation:</strong> Sell the 1st color 1 time (2) and the 2nd color 3 times (5 + 4 + 3).
The maximum total value is 2 + 5 + 4 + 3 = 14.
</pre>

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

<pre>
<strong>Input:</strong> inventory = [3,5], orders = 6
<strong>Output:</strong> 19
<strong>Explanation: </strong>Sell the 1st color 2 times (3 + 2) and the 2nd color 4 times (5 + 4 + 3 + 2).
The maximum total value is 3 + 2 + 5 + 4 + 3 + 2 = 19.
</pre>

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

<ul>
	<li><code>1 &lt;= inventory.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= inventory[i] &lt;= 10<sup>9</sup></code></li>
	<li><code>1 &lt;= orders &lt;= min(sum(inventory[i]), 10<sup>9</sup>)</code></li>
</ul>

class Solution:
    def maxProfit(self, inventory: list[int], orders: int) -> int:
        MOD = 10**9 + 7
        
        # Helper function to calculate how many orders we can fulfill with a given minimum value
        def count_balls_above(val):
            return sum(max(0, x - val) for x in inventory)
        
        # Binary search to find the highest value we can start selling at
        left, right = 0, max(inventory)
        while left < right:
            mid = (left + right) // 2
            if count_balls_above(mid) > orders:
                left = mid + 1
            else:
                right = mid
        
        # Calculate the profit
        ans = 0
        remaining_orders = orders
        for ball in inventory:
            if ball >= left:
                sell = min(ball - left, remaining_orders)
                remaining_orders -= sell
                ans += (ball + (ball - sell + 1)) * sell // 2
                ans %= MOD
        
        # If there are still remaining orders, sell at the found minimum value
        ans += remaining_orders * left
        ans %= MOD
        
        return ans

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