<p>Given an integer array <code>nums</code> with possible <strong>duplicates</strong>, randomly output the index of a given <code>target</code> number. You can assume that the given target number must exist in the array.</p>

<p>Implement the <code>Solution</code> class:</p>

<ul>
	<li><code>Solution(int[] nums)</code> Initializes the object with the array <code>nums</code>.</li>
	<li><code>int pick(int target)</code> Picks a random index <code>i</code> from <code>nums</code> where <code>nums[i] == target</code>. If there are multiple valid i&#39;s, then each index should have an equal probability of returning.</li>
</ul>

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

<pre>
<strong>Input</strong>
[&quot;Solution&quot;, &quot;pick&quot;, &quot;pick&quot;, &quot;pick&quot;]
[[[1, 2, 3, 3, 3]], [3], [1], [3]]
<strong>Output</strong>
[null, 4, 0, 2]

<strong>Explanation</strong>
Solution solution = new Solution([1, 2, 3, 3, 3]);
solution.pick(3); // It should return either index 2, 3, or 4 randomly. Each index should have equal probability of returning.
solution.pick(1); // It should return 0. Since in the array only nums[0] is equal to 1.
solution.pick(3); // It should return either index 2, 3, or 4 randomly. Each index should have equal probability of returning.
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 2 * 10<sup>4</sup></code></li>
	<li><code>-2<sup>31</sup> &lt;= nums[i] &lt;= 2<sup>31</sup> - 1</code></li>
	<li><code>target</code> is an integer from <code>nums</code>.</li>
	<li>At most <code>10<sup>4</sup></code> calls will be made to <code>pick</code>.</li>
</ul>

from random import randint
class Solution:

    def __init__(self, nums: List[int]):
        self.nums_i = []
        for i, n in enumerate(nums):
            self.nums_i.append((n, i))
        self.nums_i = sorted(self.nums_i, key=lambda x: x[0])
        self.memo = {}

    def pick(self, target: int) -> int:
        if target in self.memo:
            l, r = self.memo[target]
        else:
            l, r = self.left(target), self.right(target)
            self.memo[target] = (l, r)
        return self.nums_i[randint(l, r)][1]

    def right(self, target) -> int:
        l = 0
        r = len(self.nums_i) - 1
        while l <= r:
            m = (l + r) // 2
            if self.nums_i[m][0] <= target:
                l = m + 1
            else:
                r = m - 1
        return r
    
    def left(self, target) -> int:
        l = 0
        r = len(self.nums_i) - 1
        while l <= r:
            m = (l + r) // 2
            if self.nums_i[m][0] < target:
                l = m + 1
            else:
                r = m - 1
        return l
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