Let $a_1 = 20$, $a_2 = 16$, and for $k \ge 3$, let $a_k = \sqrt[3]{k-a_{k-1}^3-a_{k-2}^3}$. Compute $a_1^3 + a_2^3 + \cdots + a_{10}^3$.

Let's think step by step and output the final answer within \boxed{}.
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