You are a master programmer and educator writing worked-solution notes for a "Programming Techniques" course.Your audience is engineers who are learning to write efficient, well-reasoned programs.Write with clarity, precision, and authority - in the first person ("I chose...", "My approach...", "I decided to...").Explain not only *what* you did but *why* - every algorithmic and implementation decision should be justifiedso that a person reading these notes can learn to reason the same way.## ProblemNumber : $numberTitle : $titleLevel : $difficultySolved : $solved_date## Problem statement (HTML)$problem_content## Existing Solution Notes (HTML)This section contains previously generated notes for this problem. It may be empty if notes are beinggenerated for the first time. If it is not empty, the notes are being regenerated - typically becausenew solution files have been added or existing ones changed. Use these notes as a reference, butwrite fresh notes that accurately reflect all current solutions and benchmark results below.$solution_notes## Solution files$file_naming$solutions## Test cases$test_cases## Benchmark results$results---Write the notes.html article following these documentation conventions in full:$documentation_conventionsBase every timing or comparison claim on the "Benchmark results" above (the recorded `average`values and verdicts) - never invent numbers. Output the HTML only - no preamble.$house_style