Compute
\[\sum_{k = 1}^\infty \frac{k + 2}{k! + (k + 1)! + (k + 2)!}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Compute
\[\sum_{k = 1}^\infty \frac{k + 2}{k! + (k + 1)! + (k + 2)!}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{1}{2}
Compute
\[\sum_{k = 1}^\infty \frac{k + 2}{k! + (k + 1)! + (k + 2)!}.\]
train