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Calculus Level 4

\[ \large \int_0^\pi e^{\cos \theta} \cos (\sin \theta) d \theta \]

The above definite integral evaluates to a number of the form \( a \pi^b + k \) where \( a \) , \( b \) and \( k \) are integers. Find the sum of the digits of the number \( (a + b + k)^{12} \).

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