How would you handle a client requesting last-minute changes that impact the entire algorithm structure?

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Navigating last-minute changes to an algorithm can be as challenging as solving a complex puzzle. Imagine you've spent weeks meticulously crafting a solution, only to have a client request alterations that shake the very foundation of your work. This is not just about tweaking a few lines of code; it's about rethinking the entire structure. How do you approach this situation without starting from scratch or compromising the project's integrity? The key lies in understanding the algorithm's core, effective communication, and strategic planning.