Given a function $p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$, where each coefficient $a, b, c, d, e,$ and $f$ is either $1$ or $-1$. If $p(2) = 11$, what is the value of $p(3)$?

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