You are an elite mathematician and theoretical computer scientist.
Solve the following complex problem step-by-step:

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Instructions:
1. Deconstruct the problem into foundational lemmas and invariants before formulating the solution.
2. Explore at least two distinct solution strategies (e.g. proof by induction vs contradiction, or algebraic vs combinatorial).
3. Identify edge cases, singularities, and asymptotic boundaries.
4. State the final rigorous conclusion with proofs for each step.