Let \( p(x) = c_1 + c_2 \cdot 2^x + c_3 \cdot 3^x + c_4 \cdot 5^x + c_5 \cdot 8^x \). Given that \( p(k) = k \) for \( k = 1, 2, 3, 4, 5 \), compute \( p(6) \).

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