Consider a sequence of integers \(a_1, a_2, a_3, \ldots\) such that \(a_1, a_2, a_3\) form an arithmetic progression and \(a_1, a_3, a_5\) form a geometric progression. If \(a_1 = 1\), find the smallest positive integer \(a_5\) which is also a perfect square. Let's think step by step and output the final answer within \boxed{}.
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