The cubic polynomial $p(x)$ satisfies $p(2) = 1,$ $p(7) = 19,$ $p(15) = 11,$ and $p(20) = 29.$  Find
\[p(1) + p(2) + p(3) + \dots + p(21).\]

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