Approximate a percentile from buckets

You are given a histogram-style summary of query popularity. Each bucket is represented as `(left_bd, right_bd, count)`, meaning that `count` queries have `search_count` values falling in the interval `[left_bd, right_bd)` for a fixed analysis window. There are `K` such buckets, sorted by boundary, non-overlapping, and together they cover the full data range. You do **not** have the individual query-level `search_count` values. Given only these bucket summaries, estimate the `n`th percentile of `search_count`. In your answer: - Explain how to identify which bucket contains the percentile. - Discuss why simply returning the midpoint `(left_bd + right_bd) / 2` of that bucket can be a poor approximation. - Provide a better approximation method using interpolation within the bucket. - State any assumptions and edge cases, especially when bucket widths are unequal or the distribution within a bucket may be skewed.