Maximize Score of Numbers in Ranges
<p>You are given an array of integers <code>start</code> and an integer <code>d</code>, representing <code>n</code> intervals <code>[start[i], start[i] + d]</code>.</p>

<p>You are asked to choose <code>n</code> integers where the <code>i<sup>th</sup></code> integer must belong to the <code>i<sup>th</sup></code> interval. The <strong>score</strong> of the chosen integers is defined as the <strong>minimum</strong> absolute difference between any two integers that have been chosen.</p>

<p>Return the <strong>maximum</strong> <em>possible score</em> of the chosen integers.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">start = [6,0,3], d = 2</span></p>

<p><strong>Output:</strong> <span class="example-io">4</span></p>

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

<p>The maximum possible score can be obtained by choosing integers: 8, 0, and 4. The score of these chosen integers is <code>min(|8 - 0|, |8 - 4|, |0 - 4|)</code> which equals 4.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">start = [2,6,13,13], d = 5</span></p>

<p><strong>Output:</strong> <span class="example-io">5</span></p>

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

<p>The maximum possible score can be obtained by choosing integers: 2, 7, 13, and 18. The score of these chosen integers is <code>min(|2 - 7|, |2 - 13|, |2 - 18|, |7 - 13|, |7 - 18|, |13 - 18|)</code> which equals 5.</p>
</div>

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

<ul>
	<li><code>2 &lt;= start.length &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= start[i] &lt;= 10<sup>9</sup></code></li>
	<li><code>0 &lt;= d &lt;= 10<sup>9</sup></code></li>
</ul>
Greedy; Array; Binary Search; Sorting
class Solution:
    def maxPossibleScore(self, start: List[int], d: int) -> int:
        def check(mi: int) -> bool:
            last = -inf
            for st in start:
                if last + mi > st + d:
                    return False
                last = max(st, last + mi)
            return True

        start.sort()
        l, r = 0, start[-1] + d - start[0]
        while l < r:
            mid = (l + r + 1) >> 1
            if check(mid):
                l = mid
            else:
                r = mid - 1
        return l
