A child is sliding out of control with velocity \(v_{c}\) across a frozen lake. He runs head-on into another child, initially at rest, with 3 times the mass of the first child, who holds on so that the two now slide together. The original answer is in the form \(\frac{k \cdot v_{c}}{m}\). Please find the value of \(k + m\).

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