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Let the numbers $2, b, c$ be in an A.P. and $A = \begin{bmatrix} 1 & 1 & 1 \ 2 & b & c \ 4 & b^2 & c^2 \end{bmatrix}$. If $\det(A) \in [2, 16]$, then $c$ lies in the interval:
(1) $[2, 3)$
(2) $(2 + 2^{3/4}, 4)$
(3) $[4, 6]$
(4) $[3, 2 + 2^{3/4}]$
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