Let \(a_1,a_2,\ldots\) be an arithmetic progression and let \(b_1,b_2,\ldots\) be a non-constant geometric progression.

We define the arithmetic-geometric progression \(c_1,c_2,\ldots\) such that \(c_k = a_k \cdot b_k.\)

If \(c_1 = 1,\) \(c_2 = 2,\) and \(c_3=3,\) what is the value of \(c_5\)?

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