When \(p = \sum_{k=1}^{6} k \ln{k}\), the number \(e^p\) is an integer. What is the largest power of \(2\) that is a factor of \(e^p\)? If the answer is \(2^x\), what is the value of \(x\)?

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