Let $c_1, c_2, \ldots, c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1, a_2, \ldots, a_{6030}$, there exist 6030 real numbers $\{b_1, b_2, \ldots, b_{6030}\}$ such that:
\[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\]
and
\[b_n = \sum_{d\mid n} c_d a_{n/d}\]
for $n=1, 2, \ldots, 6030$. Find $c_{6030}$.

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