Consider the set of all ordered $6$-tuples of nonnegative integers $(a, b, c, d, e, f)$ such that:
\[ a + 2b + 6c + 30d + 210e + 2310f = 2^{15}. \]
In the tuple with the property that $a + b + c + d + e + f$ is minimized, what is the value of $c$?

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