\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \(a \in \mathbb{R}\). Compute the rank of the family \(\bigl((1, a, a^2), (a, a^2, a^3), (1, 1, 1)\bigr)\) in \(\mathbb{R}^3\) as a function of \(a\).
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