In an acute triangle \( \Delta ABC \), let \( \overline{AD} \), \( \overline{BE} \), and \( \overline{CF} \) be the altitudes from vertices \( A \), \( B \), and \( C \) respectively. Given that
\[ 9 \overrightarrow{AD} + 4 \overrightarrow{BE} + 7 \overrightarrow{CF} = \mathbf{0}, \]
compute the measure of angle \( \angle ACB \) in degrees. Let's think step by step and output the final answer within \boxed{}.
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